Properties

Label 350.4
Level 350
Weight 4
Dimension 3180
Nonzero newspaces 12
Sturm bound 28800
Trace bound 4

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(28800\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(350))\).

Total New Old
Modular forms 11136 3180 7956
Cusp forms 10464 3180 7284
Eisenstein series 672 0 672

Trace form

\( 3180 q + 8 q^{2} - 20 q^{3} - 16 q^{4} - 10 q^{5} - 68 q^{6} - 56 q^{7} + 32 q^{8} + 374 q^{9} + O(q^{10}) \) \( 3180 q + 8 q^{2} - 20 q^{3} - 16 q^{4} - 10 q^{5} - 68 q^{6} - 56 q^{7} + 32 q^{8} + 374 q^{9} + 100 q^{10} - 134 q^{11} - 80 q^{12} - 166 q^{13} - 92 q^{14} - 160 q^{15} + 192 q^{16} - 1166 q^{17} - 772 q^{18} - 460 q^{19} + 160 q^{20} + 680 q^{21} + 1320 q^{22} + 1994 q^{23} + 528 q^{24} + 1286 q^{25} - 868 q^{26} - 752 q^{27} - 56 q^{28} + 200 q^{29} + 288 q^{30} + 466 q^{31} - 192 q^{32} + 2886 q^{33} + 1108 q^{34} + 1228 q^{35} + 2760 q^{36} + 2868 q^{37} + 1420 q^{38} - 108 q^{39} - 240 q^{40} - 2296 q^{41} - 2716 q^{42} - 3476 q^{43} - 1432 q^{44} - 922 q^{45} - 4744 q^{46} - 3250 q^{47} - 608 q^{48} - 900 q^{49} - 700 q^{50} + 3638 q^{51} - 616 q^{52} + 5520 q^{53} + 816 q^{54} - 1184 q^{55} + 816 q^{56} - 5960 q^{57} - 1224 q^{58} - 12280 q^{59} - 5440 q^{60} - 3608 q^{61} - 8504 q^{62} - 6280 q^{63} + 128 q^{64} - 2642 q^{65} - 1808 q^{66} + 8822 q^{67} + 4296 q^{68} + 20940 q^{69} + 4800 q^{70} + 8264 q^{71} + 512 q^{72} + 23318 q^{73} + 12264 q^{74} + 27824 q^{75} + 3704 q^{76} + 14882 q^{77} + 17264 q^{78} + 10778 q^{79} - 160 q^{80} - 1936 q^{81} - 1608 q^{82} - 16926 q^{83} - 2448 q^{84} - 21418 q^{85} - 11496 q^{86} - 56240 q^{87} - 6336 q^{88} - 50480 q^{89} - 27516 q^{90} - 19198 q^{91} - 12592 q^{92} - 20138 q^{93} - 7216 q^{94} + 6808 q^{95} + 832 q^{96} + 17744 q^{97} + 12536 q^{98} + 33188 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(350))\)

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
350.4.a \(\chi_{350}(1, \cdot)\) 350.4.a.a 1 1
350.4.a.b 1
350.4.a.c 1
350.4.a.d 1
350.4.a.e 1
350.4.a.f 1
350.4.a.g 1
350.4.a.h 1
350.4.a.i 1
350.4.a.j 1
350.4.a.k 1
350.4.a.l 1
350.4.a.m 1
350.4.a.n 1
350.4.a.o 1
350.4.a.p 1
350.4.a.q 1
350.4.a.r 1
350.4.a.s 1
350.4.a.t 1
350.4.a.u 1
350.4.a.v 1
350.4.a.w 3
350.4.a.x 3
350.4.c \(\chi_{350}(99, \cdot)\) 350.4.c.a 2 1
350.4.c.b 2
350.4.c.c 2
350.4.c.d 2
350.4.c.e 2
350.4.c.f 2
350.4.c.g 2
350.4.c.h 2
350.4.c.i 2
350.4.c.j 2
350.4.c.k 2
350.4.c.l 2
350.4.c.m 2
350.4.c.n 2
350.4.e \(\chi_{350}(51, \cdot)\) 350.4.e.a 2 2
350.4.e.b 2
350.4.e.c 2
350.4.e.d 2
350.4.e.e 2
350.4.e.f 2
350.4.e.g 2
350.4.e.h 4
350.4.e.i 6
350.4.e.j 6
350.4.e.k 6
350.4.e.l 8
350.4.e.m 8
350.4.e.n 12
350.4.e.o 12
350.4.g \(\chi_{350}(293, \cdot)\) 350.4.g.a 16 2
350.4.g.b 24
350.4.g.c 32
350.4.h \(\chi_{350}(71, \cdot)\) n/a 184 4
350.4.j \(\chi_{350}(149, \cdot)\) 350.4.j.a 4 2
350.4.j.b 4
350.4.j.c 4
350.4.j.d 4
350.4.j.e 4
350.4.j.f 4
350.4.j.g 8
350.4.j.h 12
350.4.j.i 12
350.4.j.j 16
350.4.m \(\chi_{350}(29, \cdot)\) n/a 176 4
350.4.o \(\chi_{350}(143, \cdot)\) n/a 144 4
350.4.q \(\chi_{350}(11, \cdot)\) n/a 480 8
350.4.r \(\chi_{350}(13, \cdot)\) n/a 480 8
350.4.u \(\chi_{350}(9, \cdot)\) n/a 480 8
350.4.x \(\chi_{350}(3, \cdot)\) n/a 960 16

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(350))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(350)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(175))\)\(^{\oplus 2}\)