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Properties

Label 522.4
Level 522
Weight 4
Dimension 5971
Nonzero newspaces 12
Sturm bound 60480
Trace bound 10

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Defining parameters

Level: \( N \) = \( 522 = 2 \cdot 3^{2} \cdot 29 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(60480\)
Trace bound: \(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(522))\).

Total New Old
Modular forms 23128 5971 17157
Cusp forms 22232 5971 16261
Eisenstein series 896 0 896

Trace form

\( 5971 q - 8 q^{2} - 6 q^{3} + 16 q^{4} - 24 q^{5} + 36 q^{6} + 8 q^{7} + 16 q^{8} + 210 q^{9} + O(q^{10}) \) \( 5971 q - 8 q^{2} - 6 q^{3} + 16 q^{4} - 24 q^{5} + 36 q^{6} + 8 q^{7} + 16 q^{8} + 210 q^{9} + 24 q^{10} - 54 q^{11} - 48 q^{12} - 28 q^{13} - 136 q^{14} - 180 q^{15} + 64 q^{16} - 96 q^{17} - 312 q^{18} - 460 q^{19} + 212 q^{20} + 72 q^{21} + 476 q^{22} + 920 q^{23} + 48 q^{24} - 280 q^{25} - 138 q^{26} + 320 q^{28} - 1190 q^{29} + 576 q^{30} - 1044 q^{31} - 128 q^{32} - 1530 q^{33} - 614 q^{34} - 2072 q^{35} - 1032 q^{36} + 340 q^{37} + 1012 q^{38} + 1164 q^{39} + 824 q^{40} + 498 q^{41} + 1632 q^{42} - 154 q^{43} + 720 q^{44} + 1404 q^{45} - 336 q^{46} + 2612 q^{47} + 288 q^{48} + 2184 q^{49} - 872 q^{50} + 306 q^{51} - 112 q^{52} - 2991 q^{53} - 1908 q^{54} - 3048 q^{55} - 544 q^{56} - 1974 q^{57} - 192 q^{58} - 2206 q^{59} + 1872 q^{60} - 3160 q^{61} + 3680 q^{62} + 3588 q^{63} - 896 q^{64} - 435 q^{65} - 576 q^{66} - 3094 q^{67} - 1320 q^{68} - 2988 q^{69} - 16440 q^{70} - 33100 q^{71} + 48 q^{72} - 9643 q^{73} - 20368 q^{74} - 19342 q^{75} - 3352 q^{76} - 3480 q^{77} + 2872 q^{78} + 9044 q^{79} + 1344 q^{80} + 24458 q^{81} + 17088 q^{82} + 24156 q^{83} + 11952 q^{84} + 43632 q^{85} + 36500 q^{86} + 27284 q^{87} + 336 q^{88} + 41388 q^{89} + 21584 q^{90} + 29464 q^{91} + 17568 q^{92} + 15124 q^{93} + 3432 q^{94} + 13680 q^{95} - 768 q^{96} - 6809 q^{97} - 20316 q^{98} - 21904 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(522))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
522.4.a \(\chi_{522}(1, \cdot)\) 522.4.a.a 1 1
522.4.a.b 1
522.4.a.c 1
522.4.a.d 1
522.4.a.e 1
522.4.a.f 1
522.4.a.g 1
522.4.a.h 1
522.4.a.i 2
522.4.a.j 2
522.4.a.k 3
522.4.a.l 3
522.4.a.m 3
522.4.a.n 3
522.4.a.o 3
522.4.a.p 4
522.4.a.q 4
522.4.d \(\chi_{522}(289, \cdot)\) 522.4.d.a 6 1
522.4.d.b 8
522.4.d.c 8
522.4.d.d 16
522.4.e \(\chi_{522}(175, \cdot)\) n/a 168 2
522.4.g \(\chi_{522}(17, \cdot)\) 522.4.g.a 8 2
522.4.g.b 20
522.4.g.c 32
522.4.h \(\chi_{522}(115, \cdot)\) n/a 180 2
522.4.k \(\chi_{522}(181, \cdot)\) n/a 222 6
522.4.l \(\chi_{522}(41, \cdot)\) n/a 360 4
522.4.n \(\chi_{522}(91, \cdot)\) n/a 228 6
522.4.q \(\chi_{522}(7, \cdot)\) n/a 1080 12
522.4.r \(\chi_{522}(89, \cdot)\) n/a 360 12
522.4.v \(\chi_{522}(13, \cdot)\) n/a 1080 12
522.4.x \(\chi_{522}(11, \cdot)\) n/a 2160 24

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(522))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(522)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(29))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(58))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(174))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(261))\)\(^{\oplus 2}\)