Properties

Label 414.4
Level 414
Weight 4
Dimension 3748
Nonzero newspaces 8
Sturm bound 38016
Trace bound 3

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Defining parameters

Level: \( N \) = \( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 8 \)
Sturm bound: \(38016\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(414))\).

Total New Old
Modular forms 14608 3748 10860
Cusp forms 13904 3748 10156
Eisenstein series 704 0 704

Trace form

\( 3748 q - 8 q^{2} - 6 q^{3} + 16 q^{4} - 24 q^{5} + 36 q^{6} + 8 q^{7} + 16 q^{8} + 210 q^{9} + O(q^{10}) \) \( 3748 q - 8 q^{2} - 6 q^{3} + 16 q^{4} - 24 q^{5} + 36 q^{6} + 8 q^{7} + 16 q^{8} + 210 q^{9} + 24 q^{10} - 54 q^{11} - 48 q^{12} - 28 q^{13} - 136 q^{14} - 180 q^{15} + 64 q^{16} + 80 q^{17} - 312 q^{18} - 680 q^{19} - 448 q^{20} + 72 q^{21} - 400 q^{22} - 536 q^{23} + 48 q^{24} - 88 q^{25} + 412 q^{26} + 496 q^{28} + 1196 q^{29} + 576 q^{30} + 1288 q^{31} - 128 q^{32} - 1530 q^{33} - 684 q^{34} - 1426 q^{35} - 1032 q^{36} + 296 q^{37} - 724 q^{38} + 1164 q^{39} + 96 q^{40} - 52 q^{41} + 1632 q^{42} - 1870 q^{43} + 720 q^{44} + 1404 q^{45} - 168 q^{46} - 764 q^{47} + 288 q^{48} - 834 q^{49} - 872 q^{50} + 306 q^{51} - 112 q^{52} - 418 q^{53} - 8156 q^{54} - 10404 q^{55} - 6352 q^{56} - 10422 q^{57} - 5928 q^{58} - 3460 q^{59} + 1696 q^{60} + 2528 q^{61} + 8828 q^{62} + 11948 q^{63} - 896 q^{64} + 25500 q^{65} + 10336 q^{66} + 3770 q^{67} + 7656 q^{68} + 13466 q^{69} + 16608 q^{70} + 22668 q^{71} + 4976 q^{72} + 6416 q^{73} + 7304 q^{74} + 1546 q^{75} + 680 q^{76} - 3120 q^{77} - 3168 q^{78} - 5654 q^{79} - 2880 q^{80} - 16118 q^{81} - 9048 q^{82} - 29242 q^{83} - 9056 q^{84} - 26710 q^{85} - 15328 q^{86} + 6408 q^{87} + 336 q^{88} + 5918 q^{89} + 864 q^{90} - 7804 q^{91} + 1728 q^{92} + 5268 q^{93} - 1608 q^{94} + 6648 q^{95} - 768 q^{96} + 6964 q^{97} + 2092 q^{98} - 2304 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(414))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
414.4.a \(\chi_{414}(1, \cdot)\) 414.4.a.a 1 1
414.4.a.b 1
414.4.a.c 1
414.4.a.d 1
414.4.a.e 1
414.4.a.f 2
414.4.a.g 2
414.4.a.h 2
414.4.a.i 2
414.4.a.j 2
414.4.a.k 2
414.4.a.l 3
414.4.a.m 4
414.4.a.n 4
414.4.d \(\chi_{414}(413, \cdot)\) 414.4.d.a 24 1
414.4.e \(\chi_{414}(139, \cdot)\) n/a 132 2
414.4.f \(\chi_{414}(137, \cdot)\) n/a 144 2
414.4.i \(\chi_{414}(55, \cdot)\) n/a 300 10
414.4.j \(\chi_{414}(17, \cdot)\) n/a 240 10
414.4.m \(\chi_{414}(13, \cdot)\) n/a 1440 20
414.4.p \(\chi_{414}(5, \cdot)\) n/a 1440 20

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(414))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(414)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(46))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(207))\)\(^{\oplus 2}\)