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Properties

Label 690.4
Level 690
Weight 4
Dimension 8680
Nonzero newspaces 12
Sturm bound 101376
Trace bound 4

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Defining parameters

Level: \( N \) = \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(101376\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(690))\).

Total New Old
Modular forms 38720 8680 30040
Cusp forms 37312 8680 28632
Eisenstein series 1408 0 1408

Trace form

\( 8680 q - 28 q^{3} - 8 q^{5} + 8 q^{6} - 80 q^{7} + O(q^{10}) \) \( 8680 q - 28 q^{3} - 8 q^{5} + 8 q^{6} - 80 q^{7} + 80 q^{10} - 64 q^{11} + 16 q^{12} + 304 q^{13} + 128 q^{14} + 1224 q^{15} + 256 q^{16} + 848 q^{17} - 144 q^{18} - 912 q^{19} - 672 q^{20} - 3760 q^{21} - 2608 q^{22} - 3944 q^{23} - 96 q^{24} - 2048 q^{25} - 1456 q^{26} + 1004 q^{27} + 384 q^{28} + 2192 q^{29} + 1328 q^{30} + 4496 q^{31} + 2944 q^{33} + 448 q^{34} + 2412 q^{35} - 1024 q^{36} + 3616 q^{37} - 864 q^{38} - 720 q^{39} + 192 q^{40} - 1448 q^{41} + 2176 q^{42} - 1568 q^{43} + 1984 q^{44} + 2728 q^{45} + 1408 q^{46} - 7280 q^{47} + 64 q^{48} - 10776 q^{49} + 352 q^{50} - 1664 q^{51} - 704 q^{52} + 680 q^{53} - 6464 q^{54} + 376 q^{55} + 640 q^{56} - 3112 q^{57} - 5376 q^{58} + 8776 q^{59} - 2408 q^{60} - 1120 q^{61} - 2016 q^{62} + 5320 q^{63} - 2016 q^{65} + 10560 q^{66} + 688 q^{67} + 576 q^{68} + 15344 q^{69} + 5760 q^{70} + 3456 q^{71} + 4800 q^{72} - 896 q^{73} - 160 q^{74} + 10438 q^{75} + 1408 q^{76} + 3456 q^{77} + 4824 q^{78} + 9864 q^{79} - 128 q^{80} - 5832 q^{81} - 2688 q^{82} + 632 q^{83} - 8144 q^{84} - 940 q^{85} - 2912 q^{86} - 17908 q^{87} - 2688 q^{88} - 10296 q^{89} - 10768 q^{90} - 21552 q^{91} - 288 q^{92} - 1408 q^{93} - 4864 q^{94} - 39712 q^{95} + 640 q^{96} - 38184 q^{97} - 23776 q^{98} - 11244 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(690))\)

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
690.4.a \(\chi_{690}(1, \cdot)\) 690.4.a.a 1 1
690.4.a.b 1
690.4.a.c 1
690.4.a.d 1
690.4.a.e 1
690.4.a.f 1
690.4.a.g 1
690.4.a.h 1
690.4.a.i 2
690.4.a.j 2
690.4.a.k 3
690.4.a.l 3
690.4.a.m 3
690.4.a.n 3
690.4.a.o 3
690.4.a.p 3
690.4.a.q 3
690.4.a.r 3
690.4.a.s 4
690.4.a.t 4
690.4.d \(\chi_{690}(139, \cdot)\) 690.4.d.a 14 1
690.4.d.b 16
690.4.d.c 16
690.4.d.d 22
690.4.e \(\chi_{690}(551, \cdot)\) 690.4.e.a 48 1
690.4.e.b 48
690.4.h \(\chi_{690}(689, \cdot)\) n/a 144 1
690.4.i \(\chi_{690}(47, \cdot)\) n/a 264 2
690.4.j \(\chi_{690}(367, \cdot)\) n/a 144 2
690.4.m \(\chi_{690}(31, \cdot)\) n/a 480 10
690.4.n \(\chi_{690}(89, \cdot)\) n/a 1440 10
690.4.q \(\chi_{690}(11, \cdot)\) n/a 960 10
690.4.r \(\chi_{690}(49, \cdot)\) n/a 720 10
690.4.w \(\chi_{690}(7, \cdot)\) n/a 1440 20
690.4.x \(\chi_{690}(77, \cdot)\) n/a 2880 20

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(690))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(690)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(115))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(230))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(345))\)\(^{\oplus 2}\)