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Properties

Label 350.7
Level 350
Weight 7
Dimension 6360
Nonzero newspaces 12
Sturm bound 50400
Trace bound 4

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(50400\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(350))\).

Total New Old
Modular forms 21936 6360 15576
Cusp forms 21264 6360 14904
Eisenstein series 672 0 672

Trace form

\( 6360 q + 32 q^{2} - 256 q^{3} + 360 q^{5} + 896 q^{6} - 432 q^{7} - 1024 q^{8} + 1848 q^{9} + O(q^{10}) \) \( 6360 q + 32 q^{2} - 256 q^{3} + 360 q^{5} + 896 q^{6} - 432 q^{7} - 1024 q^{8} + 1848 q^{9} + 4560 q^{10} - 3908 q^{11} - 8192 q^{12} - 18456 q^{13} + 4752 q^{14} + 6640 q^{15} - 24576 q^{16} + 34992 q^{17} - 65392 q^{18} - 108804 q^{19} - 12800 q^{20} + 14188 q^{21} + 123168 q^{22} + 314600 q^{23} + 10752 q^{24} - 161244 q^{25} - 140896 q^{26} - 1003552 q^{27} - 129024 q^{28} - 28736 q^{29} + 323328 q^{30} + 594204 q^{31} + 49152 q^{32} + 826604 q^{33} + 66000 q^{34} - 340072 q^{35} - 95104 q^{36} - 813732 q^{37} - 441296 q^{38} - 2065828 q^{39} - 99840 q^{40} + 120992 q^{41} + 601184 q^{42} + 685608 q^{43} + 674688 q^{44} + 3037908 q^{45} + 495216 q^{46} + 668180 q^{47} + 262144 q^{48} - 277836 q^{49} - 173200 q^{50} - 1481876 q^{51} - 335232 q^{52} - 1577476 q^{53} + 386064 q^{54} - 345504 q^{55} - 189440 q^{56} - 1542888 q^{57} + 883104 q^{58} + 3440216 q^{59} + 844800 q^{60} - 1732212 q^{61} - 4051456 q^{62} + 1568512 q^{63} + 393216 q^{64} - 2691572 q^{65} + 1646912 q^{66} + 5436432 q^{67} + 2177536 q^{68} + 11211200 q^{69} + 3852960 q^{70} - 1386088 q^{71} + 1017856 q^{72} - 5334948 q^{73} - 2139648 q^{74} - 7032816 q^{75} - 1612800 q^{76} - 7905416 q^{77} - 12275072 q^{78} - 1471560 q^{79} - 368640 q^{80} + 11494572 q^{81} - 1229088 q^{82} + 2109920 q^{83} + 949248 q^{84} + 2133192 q^{85} + 8284704 q^{86} + 1530348 q^{87} + 1242624 q^{88} - 2871784 q^{89} - 3888176 q^{90} - 14574408 q^{91} - 9291520 q^{92} - 9125252 q^{93} + 1698480 q^{94} + 10336688 q^{95} + 1490944 q^{96} + 22065672 q^{97} + 24246912 q^{98} + 32687904 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(350))\)

We only show spaces with odd 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
350.7.b \(\chi_{350}(251, \cdot)\) 350.7.b.a 4 1
350.7.b.b 16
350.7.b.c 16
350.7.b.d 16
350.7.b.e 24
350.7.d \(\chi_{350}(349, \cdot)\) 350.7.d.a 8 1
350.7.d.b 32
350.7.d.c 32
350.7.f \(\chi_{350}(43, \cdot)\) n/a 108 2
350.7.i \(\chi_{350}(199, \cdot)\) n/a 144 2
350.7.k \(\chi_{350}(101, \cdot)\) n/a 152 2
350.7.l \(\chi_{350}(69, \cdot)\) n/a 480 4
350.7.n \(\chi_{350}(41, \cdot)\) n/a 480 4
350.7.p \(\chi_{350}(93, \cdot)\) n/a 288 4
350.7.s \(\chi_{350}(113, \cdot)\) n/a 720 8
350.7.t \(\chi_{350}(31, \cdot)\) n/a 960 8
350.7.v \(\chi_{350}(19, \cdot)\) n/a 960 8
350.7.w \(\chi_{350}(23, \cdot)\) n/a 1920 16

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

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(350))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(350)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(70))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(175))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(350))\)\(^{\oplus 1}\)