Computation regarding linear dependence
This computation is supplement to the paper:
MODULAR FORMS WITH NON-VANISHING CENTRAL VALUES AND LINEAR INDEPENDENCE OF FOURIER COEFFICIENTS
by Debargha Banerjee and Priyanka Majumder,
This is a computation regarding linear independence. In this computation, we are interested
to find computationally how big`` d" is needed to ensure that
T_1 e, T_2 e ...T_d e is linearly independent.
Level N=23 and weight 4
> M := ModularSymbols(23,4);
> T2 := HeckeOperator(M, 2);
> E:= WindingElement(M, 2);
> E; (242*X^2 + 22*X*Y + 1/2*Y^2)*{-1/22, 0} + -1/2*Y^2{oo,0}
> E*T2;
(768/13*X^2 + 96/13*X*Y + 3/13*Y^2)*{-1/16, 0} + (-100/13*X^2 - 40/13*X*Y - 4/13*Y^2)*{-1/5, 0} + (-100/13*X^2 - 10/13*X*Y - 1/52*Y^2)*{-1/20, 0} + (-169/4*X^2 - 13/2*X*Y - 1/4*Y^2)*{-1/13, 0} + (2527/26*X^2 + 133/13*X*Y + 7/26*Y^2)*{-1/19, 0} + (1701/52*X^2 + 189/26*X*Y + 21/52*Y^2)*{-1/9, 0} + (-867/52*X^2 - 51/26*X*Y - 3/52*Y^2)*{-1/17, 0} + (-3375/52*X^2 - 225/26*X*Y - 15/52*Y^2)*{-1/15, 0} + (121/26*X^2 + 11/13*X*Y + 1/26*Y^2)*{-1/11, 0} + (3267/13*X^2 + 297/13*X*Y + 27/52*Y^2)*{-1/22, 0} + -7/13*Y^2*{oo, 0}
> T3 := HeckeOperator(M, 3);
> E*T3;
(256/13*X^2 + 32/13*X*Y + 1/13*Y^2)*{-1/16, 0} + (75/13*X^2 + 30/13*X*Y + 3/13*Y^2)*{-1/5, 0} + (-900/13*X^2 - 90/13*X*Y - 9/52*Y^2)*{-1/20, 0} + (845/4*X^2 + 65/2*X*Y + 5/4*Y^2)*{-1/13, 0} + (4332/13*X^2 + 456/13*X*Y + 12/13*Y^2)*{-1/19, 0} + (-1539/52*X^2 - 171/26*X*Y - 19/52*Y^2)*{-1/9, 0} + (-15317/52*X^2 - 901/26*X*Y - 53/52*Y^2)*{-1/17, 0} + (-1125/52*X^2 - 75/26*X*Y - 5/52*Y^2)*{-1/15, 0} + (-2057/26*X^2 - 187/13*X*Y - 17/26*Y^2)*{-1/11, 0} + (1089/13*X^2 + 99/13*X*Y + 9/52*Y^2)*{-1/22, 0} + -9/26*Y^2*{oo, 0}
> T4 := HeckeOperator(M, 4);
> E*T4;
(256/13*X^2 + 32/13*X*Y + 1/13*Y^2)*{-1/16, 0} + (75/13*X^2 + 30/13*X*Y + 3/13*Y^2)*{-1/5, 0} + (-2200/13*X^2 - 220/13*X*Y - 11/26*Y^2)*{-1/20, 0} + (-507*X^2 - 78*X*Y - 3*Y^2)*{-1/13, 0} + (-14801/26*X^2 - 779/13*X*Y - 41/26*Y^2)*{-1/19, 0} + (-243/26*X^2 - 27/13*X*Y - 3/26*Y^2)*{-1/9, 0} + (4624/13*X^2 + 544/13*X*Y + 16/13*Y^2)*{-1/17, 0} + (3375/13*X^2 + 450/13*X*Y + 15/13*Y^2)*{-1/15, 0} + (2904/13*X^2 + 528/13*X*Y + 24/13*Y^2)*{-1/11, 0} + (27830/13*X^2 + 2530/13*X*Y + 115/26*Y^2)*{-1/22, 0} + -50/13*Y^2*{oo, 0}
>
Looking at the coefficients of {oo,0}, we get an equation c1* (-1/2)+c2 (-7/13)+c3 (-9/26)+c4 (-50/13)=0;
Looking at the coefficients of {-1/22, 0}, looking at the coefficient of X^2, we get an equation
242 c1+ 3267/13 c2 +1089/13c3+27830/13 c4=0
Looking at the coefficient of X*Y, we get an equation
22 c1+ 297/13 c2 +99/13 c3+2530/13 c4=0
Looking at the coefficient of Y^2, we get an equation
1/2 c1+ 27/52 c2 +9/52c3+115/26c4=0;
We get a system of equation that we solve by SAGE:
var('c1 c2 c3 c4')
eq1 = c1 *(-1/2)+c2*(-7/13)+c3 *(-9/26)+c4* (-50/13)==0
eq2 = 242* c1+ (3267/13)* c2 +(1089/13)*c3+27830/13* c4==0
eq3 = 22* c1+ 297/13* c2 +(99/13)* c3+(2530/13)* c4==0
eq4 = (1/2)* c1+ (27/52)* c2 +(9/52)*c3+(115/26)*c4==0
solve([eq1, eq2,eq3],c1,c2,c3,c4)
with [[c1 == -40*r1 + 9*r2, c2 == 30*r1 - 9*r2, c3 == r2, c4 == r1]]
Looking at the coefficients of {-1/11, 0}, looking at the coefficient of X^2, we get an equation
121/26 c2 +(-2057/26)*c3+2904/13* c4=0
Looking at the coefficient of X*Y, we get an equation
11/13 c2 +- 187/13 c_3+528/13 c4=0
Looking at the coefficient of Y^2, we get an equation
1/26 c2 +17/26c3+24/13c4=0;
We get a system of equation that we solve by SAGE:
var('c1 c2 c3 c4')
eq1 = c1 *(-1/2)+c2*(-7/13)+c3 *(-9/26)+c4* (-50/13)==0
eq2 = 242* c1+ (3267/13)* c2 +(1089/13)*c3+27830/13* c4==0
eq3 = 22* c1+ 297/13* c2 +(99/13)* c3+(2530/13)* c4==0
eq4 = (1/2)* c1+ (27/52)* c2 +(9/52)*c3+(115/26)*c4==0
eq5 = (121/26)* c2 +(-2057/26)*c3+(2904/13)* c4==0
eq6 = (11/13)* c2 +(- 187/13)* c3+(528/13)* c4==0
solve([eq1, eq2,eq3,eq4,eq5,eq6],c1,c2,c3,c4)
[[c1 == -13*r1, c2 == 3*r1, c3 == 3*r1, c4 == r1]]
Level N=23 and weight greater than 4:
We can do the computation but no surprise.
M := ModularSymbols(23,4);
> T2 := HeckeOperator(M, 2);
> E:= WindingElement(M, 2);
> E*T2;
(768/13*X^2 + 96/13*X*Y + 3/13*Y^2)*{-1/16, 0} + (-100/13*X^2 - 40/13*X*Y - 4/13*Y^2)*{-1/5, 0} + (-100/13*X^2 - 10/13*X*Y - 1/52*Y^2)*{-1/20, 0} + (-169/4*X^2 - 13/2*X*Y - 1/4*Y^2)*{-1/13, 0} + (2527/26*X^2 + 133/13*X*Y + 7/26*Y^2)*{-1/19, 0} + (1701/52*X^2 + 189/26*X*Y + 21/52*Y^2)*{-1/9, 0} + (-867/52*X^2 - 51/26*X*Y - 3/52*Y^2)*{-1/17, 0} + (-3375/52*X^2 - 225/26*X*Y - 15/52*Y^2)*{-1/15, 0} + (121/26*X^2 + 11/13*X*Y + 1/26*Y^2)*{-1/11, 0} + (3267/13*X^2 + 297/13*X*Y + 27/52*Y^2)*{-1/22, 0} + -7/13*Y^2*{oo, 0}
> T3 := HeckeOperator(M, 3);
> E*T3;
(256/13*X^2 + 32/13*X*Y + 1/13*Y^2)*{-1/16, 0} + (75/13*X^2 + 30/13*X*Y + 3/13*Y^2)*{-1/5, 0} + (-900/13*X^2 - 90/13*X*Y - 9/52*Y^2)*{-1/20, 0} + (845/4*X^2 + 65/2*X*Y + 5/4*Y^2)*{-1/13, 0} + (4332/13*X^2 + 456/13*X*Y + 12/13*Y^2)*{-1/19, 0} + (-1539/52*X^2 - 171/26*X*Y - 19/52*Y^2)*{-1/9, 0} + (-15317/52*X^2 - 901/26*X*Y - 53/52*Y^2)*{-1/17, 0} + (-1125/52*X^2 - 75/26*X*Y - 5/52*Y^2)*{-1/15, 0} + (-2057/26*X^2 - 187/13*X*Y - 17/26*Y^2)*{-1/11, 0} + (1089/13*X^2 + 99/13*X*Y + 9/52*Y^2)*{-1/22, 0} + -9/26*Y^2*{oo, 0}
> T4 := HeckeOperator(M, 4);
> E*T4;
(256/13*X^2 + 32/13*X*Y + 1/13*Y^2)*{-1/16, 0} + (75/13*X^2 + 30/13*X*Y + 3/13*Y^2)*{-1/5, 0} + (-2200/13*X^2 - 220/13*X*Y - 11/26*Y^2)*{-1/20, 0} + (-507*X^2 - 78*X*Y - 3*Y^2)*{-1/13, 0} + (-14801/26*X^2 - 779/13*X*Y - 41/26*Y^2)*{-1/19, 0} + (-243/26*X^2 - 27/13*X*Y - 3/26*Y^2)*{-1/9, 0} + (4624/13*X^2 + 544/13*X*Y + 16/13*Y^2)*{-1/17, 0} + (3375/13*X^2 + 450/13*X*Y + 15/13*Y^2)*{-1/15, 0} + (2904/13*X^2 + 528/13*X*Y + 24/13*Y^2)*{-1/11, 0} + (27830/13*X^2 + 2530/13*X*Y + 115/26*Y^2)*{-1/22, 0} + -50/13*Y^2*{oo, 0}
>
> E; (242*X^2 + 22*X*Y + 1/2*Y^2)*{-1/22, 0} + -1/2*Y^2*{oo, 0}
For weight greater than 6; we can do the computation but it is tedious.
Level N=257 and weight 2:
Level N=257 (Note that 257 is a prime).
M := ModularSymbols(257,2);
> E:= WindingElement(M, 1);
> E;
-1*{oo, 0}
> T2 := HeckeOperator(M, 2);
> E*T2;
-1*{-1/128, 0} + -3*{oo, 0}
> T3 := HeckeOperator(M, 3);
> E*T3;
-1*{-1/84, 0} + {-1/131, 0} + -1*{-1/171, 0} + -1*{-1/128, 0} + -4*{oo, 0}
> T4 := HeckeOperator(M, 4);
> E*T4;
-1*{-1/191, 0} + {-1/88, 0} + -1*{-1/84, 0} + {-1/131, 0} + -1*{-1/64, 0} + -3*{-1/128, 0} + -7*{oo, 0}
We get a linear independence!
> M := ModularSymbols(263,2);
> E:= WindingElement(M, 1);
> E;
-1*{oo, 0}
> T2 := HeckeOperator(M, 2);
> E*T2;
-1*{-1/131, 0} + -3*{oo, 0}
> T3 := HeckeOperator(M, 3);
> E*T3;
-1*{-1/96, 0} + {-1/156, 0} + -1*{-1/68, 0} + -1*{-1/159, 0} + {-1/186, 0} + -1*{-1/46, 0} + {-1/255, 0} + -1*{-1/189, 0} + {-1/246, 0} + -1*{-1/149, 0} + {-1/224, 0} + {-1/80, 0} + -1*{-1/251, 0} + {-1/25, 0} + -1*{-1/83, 0} + {-1/146, 0} + -1*{-1/230, 0} + -1*{-1/175, 0} + -4*{oo, 0}
> T4 := HeckeOperator(M, 4);
> E*T4;
{-1/156, 0} + -1*{-1/68, 0} + -1*{-1/159, 0} + {-1/186, 0} + -1*{-1/46, 0} + {-1/255, 0} + -1*{-1/189, 0} + -1*{-1/197, 0} + -2*{-1/131, 0} + -7*{oo, 0}
>
We get a linear independence!
Level N=257 and weight 4:
M := ModularSymbols(257,4);
E:= WindingElement(M, 2);
E; (32768*X^2 + 256*X*Y + 1/2*Y^2)*{-1/256, 0} + -1/2*Y^2*{oo, 0}
> T2 := HeckeOperator(M, 2);
> E*T2;
(59168/5*X^2 + 688/5*X*Y + 2/5*Y^2)*{-1/172, 0} + (-84872/5*X^2 - 824/5*X*Y - 2/5*Y^2)*{-1/206, 0} + (-2738/5*X^2 - 148/5*X*Y - 2/5*Y^2)*{-1/37, 0} + (-61347/5*X^2 - 858/5*X*Y - 3/5*Y^2)*{-1/143, 0} + (-15129/5*X^2 - 246/5*X*Y - 1/5*Y^2)*{-1/123, 0} + (20736/5*X^2 + 288/5*X*Y + 1/5*Y^2)*{-1/144, 0} + (-19881/5*X^2 - 282/5*X*Y - 1/5*Y^2)*{-1/141, 0} + (1521/5*X^2 + 78/5*X*Y + 1/5*Y^2)*{-1/39, 0} + (13872/5*X^2 + 136/5*X*Y + 1/15*Y^2)*{-1/204, 0} + (-10092/5*X^2 - 116/5*X*Y - 1/15*Y^2)*{-1/174, 0} + (96774/5*X^2 + 762/5*X*Y + 3/10*Y^2)*{-1/254, 0} + (-5070*X^2 - 78*X*Y - 3/10*Y^2)*{-1/130, 0} + (5780*X^2 + 68*X*Y + 1/5*Y^2)*{-1/170, 0} + (-16/5*X^2 - 8/5*X*Y - 1/5*Y^2)*{-1/4, 0} + (60492/5*X^2 + 852/5*X*Y + 3/5*Y^2)*{-1/142, 0} + (21875/3*X^2 + 350/3*X*Y + 7/15*Y^2)*{-1/125, 0} + (114264/5*X^2 + 1104/5*X*Y + 8/15*Y^2)*{-1/207, 0} + (245*X^2 + 14*X*Y + 1/5*Y^2)*{-1/35, 0} + (-43264/5*X^2 - 416/5*X*Y - 1/5*Y^2)*{-1/208, 0} + (-77976/5*X^2 - 912/5*X*Y - 8/15*Y^2)*{-1/171, 0} + (8192/15*X^2 + 128/15*X*Y + 1/30*Y^2)*{-1/128, 0} + (98304/5*X^2 + 768/5*X*Y + 3/10*Y^2)*{-1/256, 0} + -2/5*Y^2*{oo, 0}
> T3 := HeckeOperator(M, 3);
> E*T3;
(-21675/2*X^2 - 255*X*Y - 3/2*Y^2)*{-1/85, 0} + (-81356/5*X^2 - 946/5*X*Y - 11/20*Y^2)*{-1/172, 0} + (-148526/5*X^2 - 1442/5*X*Y - 7/10*Y^2)*{-1/206, 0} + (-9583/10*X^2 - 259/5*X*Y - 7/10*Y^2)*{-1/37, 0} + (-429429/20*X^2 - 3003/10*X*Y - 21/20*Y^2)*{-1/143, 0} + (-105903/20*X^2 - 861/10*X*Y - 7/20*Y^2)*{-1/123, 0} + (36288/5*X^2 + 504/5*X*Y + 7/20*Y^2)*{-1/144, 0} + (-139167/20*X^2 - 987/10*X*Y - 7/20*Y^2)*{-1/141, 0} + (10647/20*X^2 + 273/10*X*Y + 7/20*Y^2)*{-1/39, 0} + (24276/5*X^2 + 238/5*X*Y + 7/60*Y^2)*{-1/204, 0} + (-17661/5*X^2 - 203/5*X*Y - 7/60*Y^2)*{-1/174, 0} + (290322/5*X^2 + 2286/5*X*Y + 9/10*Y^2)*{-1/254, 0} + (43940*X^2 + 676*X*Y + 13/5*Y^2)*{-1/130, 0} + (17340*X^2 + 204*X*Y + 3/5*Y^2)*{-1/170, 0} + (-48/5*X^2 - 24/5*X*Y - 3/5*Y^2)*{-1/4, 0} + (10584*X^2 + 252*X*Y + 3/2*Y^2)*{-1/84, 0} + (-51483/2*X^2 - 393*X*Y - 3/2*Y^2)*{-1/131, 0} + (105861/5*X^2 + 1491/5*X*Y + 21/20*Y^2)*{-1/142, 0} + (153125/12*X^2 + 1225/6*X*Y + 49/60*Y^2)*{-1/125, 0} + (199962/5*X^2 + 1932/5*X*Y + 14/15*Y^2)*{-1/207, 0} + (1715/4*X^2 + 49/2*X*Y + 7/20*Y^2)*{-1/35, 0} + (-75712/5*X^2 - 728/5*X*Y - 7/20*Y^2)*{-1/208, 0} + (9747/5*X^2 + 114/5*X*Y + 1/15*Y^2)*{-1/171, 0} + (-385024/15*X^2 - 6016/15*X*Y - 47/30*Y^2)*{-1/128, 0} + (-688128/5*X^2 - 5376/5*X*Y - 21/10*Y^2)*{-1/256, 0} + 9/5*Y^2*{oo, 0}
> T4 := HeckeOperator(M, 4);
> E*T4;
(-92450*X^2 - 860*X*Y - 2*Y^2)*{-1/215, 0} + (28900/3*X^2 + 680/3*X*Y + 4/3*Y^2)*{-1/85, 0} + (414176/15*X^2 + 4816/15*X*Y + 14/15*Y^2)*{-1/172, 0} + (-594104/15*X^2 - 5768/15*X*Y - 14/15*Y^2)*{-1/206, 0} + (-19166/15*X^2 - 1036/15*X*Y - 14/15*Y^2)*{-1/37, 0} + (-143143/5*X^2 - 2002/5*X*Y - 7/5*Y^2)*{-1/143, 0} + (-35301/5*X^2 - 574/5*X*Y - 7/15*Y^2)*{-1/123, 0} + (48384/5*X^2 + 672/5*X*Y + 7/15*Y^2)*{-1/144, 0} + (-46389/5*X^2 - 658/5*X*Y - 7/15*Y^2)*{-1/141, 0} + (3549/5*X^2 + 182/5*X*Y + 7/15*Y^2)*{-1/39, 0} + (-272214*X^2 - 2556*X*Y - 6*Y^2)*{-1/213, 0} + (52920*X^2 + 420*X*Y + 5/6*Y^2)*{-1/252, 0} + (-94090/3*X^2 - 970/3*X*Y - 5/6*Y^2)*{-1/194, 0} + (32368/5*X^2 + 952/15*X*Y + 7/45*Y^2)*{-1/204, 0} + (-23548/5*X^2 - 812/15*X*Y - 7/45*Y^2)*{-1/174, 0} + (18432*X^2 + 192*X*Y + 1/2*Y^2)*{-1/192, 0} + (-1419352/15*X^2 - 11176/15*X*Y - 22/15*Y^2)*{-1/254, 0} + (-153790/3*X^2 - 2366/3*X*Y - 91/30*Y^2)*{-1/130, 0} + (36481/3*X^2 + 382/3*X*Y + 1/3*Y^2)*{-1/191, 0} + (-7744/3*X^2 - 176/3*X*Y - 1/3*Y^2)*{-1/88, 0} + (-6936*X^2 - 408*X*Y - 6*Y^2)*{-1/34, 0} + (40460/3*X^2 + 476/3*X*Y + 7/15*Y^2)*{-1/170, 0} + (608/15*X^2 + 304/15*X*Y + 38/15*Y^2)*{-1/4, 0} + (48050*X^2 + 620*X*Y + 2*Y^2)*{-1/155, 0} + (-5766*X^2 - 372*X*Y - 6*Y^2)*{-1/31, 0} + (486*X^2 + 108*X*Y + 6*Y^2)*{-1/9, 0} + (-7056*X^2 - 168*X*Y - Y^2)*{-1/84, 0} + (17161*X^2 + 262*X*Y + Y^2)*{-1/131, 0} + (89888*X^2 + 848*X*Y + 2*Y^2)*{-1/212, 0} + (-25088*X^2 - 448*X*Y - 2*Y^2)*{-1/112, 0} + (141148/5*X^2 + 1988/5*X*Y + 7/5*Y^2)*{-1/142, 0} + (153125/9*X^2 + 2450/9*X*Y + 49/45*Y^2)*{-1/125, 0} + (266616/5*X^2 + 2576/5*X*Y + 56/45*Y^2)*{-1/207, 0} + (13872*X^2 + 408*X*Y + 3*Y^2)*{-1/68, 0} + (-83167*X^2 - 1526*X*Y - 7*Y^2)*{-1/109, 0} + (-512*X^2 - 128*X*Y - 8*Y^2)*{-1/8, 0} + (3844*X^2 + 124*X*Y + Y^2)*{-1/62, 0} + (-24336*X^2 - 312*X*Y - Y^2)*{-1/156, 0} + (-8978*X^2 - 268*X*Y - 2*Y^2)*{-1/67, 0} + (9065/3*X^2 + 518/3*X*Y + 37/15*Y^2)*{-1/35, 0} + (-302848/15*X^2 - 2912/15*X*Y - 7/15*Y^2)*{-1/208, 0} + (11449*X^2 + 214*X*Y + Y^2)*{-1/107, 0} + (-4900*X^2 - 140*X*Y - Y^2)*{-1/70, 0} + (10240*X^2 + 640*X*Y + 10*Y^2)*{-1/32, 0} + (96800*X^2 + 1760*X*Y + 8*Y^2)*{-1/110, 0} + (274776*X^2 + 2568*X*Y + 6*Y^2)*{-1/214, 0} + (-23716*X^2 - 308*X*Y - Y^2)*{-1/154, 0} + (-4096*X^2 - 128*X*Y - Y^2)*{-1/64, 0} + (-181944/5*X^2 - 2128/5*X*Y - 56/45*Y^2)*{-1/171, 0} + (1040384/45*X^2 + 16256/45*X*Y + 127/90*Y^2)*{-1/128, 0} + (6258688/15*X^2 + 48896/15*X*Y + 191/30*Y^2)*{-1/256, 0} + -94/15*Y^2*{oo, 0}
>
Suppose we have
c_1 T_1 e+c_2 T_2e+c_3 T_3 e+c_4 T_4 e =0.
Looking at the coefficients of {-1/31,0}, we get c_4=0.
Looking at the coefficient of {-1/204, 0}, we get c_3=0.
Looking at the coefficients of {oo,0}, we get an equation c_1* (-1/2)+c_2 (-2/5)+c_3 (9/5)+c_4 (-94/15)=0;
Looking at the coefficients of {-1/256, 0}, looking at the coefficient of X^2, we get an equation
32768 c_1+ 98304/5c_2 +-688128/5c_3+6258688/15 c_4=0
Looking at the coefficient of X*Y, we get an equation
256 c_1+ 768/5c_2 + - 5376/5 c_3+48896/15c_4=0
Looking at the coefficient of Y^2, we get an equation
1/2 c_1+ 3/10 c_2 +- 21/10c_3+191/30c_4=0;
We get a system of equation that we solve by SAGE:
var('c1 c2 c3 c4')
eq1 = c1 *(-1/2)+c2*(-7/13)+c3 *(-9/26)+c4* (-50/13)==0
eq2 = 242* c1+ (3267/13)* c2 +(1089/13)*c3+27830/13* c4==0
eq3 = 22* c1+ 297/13* c2 +(99/13)* c3+(2530/13)* c4==0
eq4 = (1/2)* c1+ (27/52)* c2 +(9/52)*c3+(115/26)*c4==0
solve([eq1, eq2,eq3],c1,c2,c3,c4)
[[c1 == -40/3*r1 + 6*r2, c2 == r1 - 3*r2, c3 == r2, c4 == r1]]
We get even c_1=c_2-0.