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Properties

Label 350.8
Level 350
Weight 8
Dimension 7418
Nonzero newspaces 12
Sturm bound 57600
Trace bound 4

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Defining parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 12 \)
Sturm bound: \(57600\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(350))\).

Total New Old
Modular forms 25536 7418 18118
Cusp forms 24864 7418 17446
Eisenstein series 672 0 672

Trace form

\( 7418 q - 16 q^{2} + 130 q^{3} - 256 q^{4} - 50 q^{5} + 2608 q^{6} + 2592 q^{7} - 1024 q^{8} - 41452 q^{9} + O(q^{10}) \) \( 7418 q - 16 q^{2} + 130 q^{3} - 256 q^{4} - 50 q^{5} + 2608 q^{6} + 2592 q^{7} - 1024 q^{8} - 41452 q^{9} + 4080 q^{10} + 50148 q^{11} + 8320 q^{12} - 3006 q^{13} - 62608 q^{14} - 67960 q^{15} + 114688 q^{16} + 537320 q^{17} + 351584 q^{18} - 176042 q^{19} - 107520 q^{20} - 531502 q^{21} - 167648 q^{22} + 1072816 q^{23} + 433152 q^{24} + 1432446 q^{25} + 838032 q^{26} - 722300 q^{27} - 816896 q^{28} - 3826748 q^{29} - 2462592 q^{30} - 717772 q^{31} + 262144 q^{32} + 10150812 q^{33} + 2736144 q^{34} + 976028 q^{35} + 2790144 q^{36} - 2232926 q^{37} - 451184 q^{38} - 2638752 q^{39} + 424960 q^{40} - 1934356 q^{41} + 66896 q^{42} - 7199908 q^{43} + 23808 q^{44} - 2872882 q^{45} + 279616 q^{46} - 658532 q^{47} + 532480 q^{48} + 7911090 q^{49} + 6767600 q^{50} + 7483220 q^{51} - 2504064 q^{52} - 9638226 q^{53} - 16067424 q^{54} - 25528944 q^{55} + 244736 q^{56} + 5982940 q^{57} + 12795648 q^{58} - 5778978 q^{59} + 3706880 q^{60} - 2662406 q^{61} + 22741600 q^{62} - 32958088 q^{63} + 2097152 q^{64} - 53440882 q^{65} - 29281472 q^{66} - 19899304 q^{67} - 16796160 q^{68} + 42333744 q^{69} + 1138880 q^{70} + 5074328 q^{71} + 3838976 q^{72} + 42468476 q^{73} + 57550016 q^{74} + 73744904 q^{75} - 9534080 q^{76} - 58841248 q^{77} - 22054720 q^{78} - 45648640 q^{79} - 204800 q^{80} + 56915216 q^{81} - 60233920 q^{82} - 39486530 q^{83} - 1578624 q^{84} + 28203862 q^{85} - 57522656 q^{86} + 20751820 q^{87} + 20834304 q^{88} + 253203846 q^{89} + 171598704 q^{90} + 55894918 q^{91} - 53459968 q^{92} - 347940572 q^{93} - 73271456 q^{94} - 80561352 q^{95} - 23920640 q^{96} - 130122508 q^{97} - 136714000 q^{98} + 3952412 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(350))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
350.8.a \(\chi_{350}(1, \cdot)\) 350.8.a.a 1 1
350.8.a.b 1
350.8.a.c 1
350.8.a.d 1
350.8.a.e 1
350.8.a.f 1
350.8.a.g 1
350.8.a.h 1
350.8.a.i 2
350.8.a.j 2
350.8.a.k 2
350.8.a.l 2
350.8.a.m 2
350.8.a.n 2
350.8.a.o 2
350.8.a.p 2
350.8.a.q 2
350.8.a.r 3
350.8.a.s 3
350.8.a.t 3
350.8.a.u 3
350.8.a.v 4
350.8.a.w 4
350.8.a.x 4
350.8.a.y 4
350.8.a.z 6
350.8.a.ba 6
350.8.c \(\chi_{350}(99, \cdot)\) 350.8.c.a 2 1
350.8.c.b 2
350.8.c.c 2
350.8.c.d 2
350.8.c.e 4
350.8.c.f 4
350.8.c.g 4
350.8.c.h 4
350.8.c.i 4
350.8.c.j 4
350.8.c.k 4
350.8.c.l 6
350.8.c.m 6
350.8.c.n 8
350.8.c.o 8
350.8.e \(\chi_{350}(51, \cdot)\) n/a 176 2
350.8.g \(\chi_{350}(293, \cdot)\) n/a 168 2
350.8.h \(\chi_{350}(71, \cdot)\) n/a 424 4
350.8.j \(\chi_{350}(149, \cdot)\) n/a 168 2
350.8.m \(\chi_{350}(29, \cdot)\) n/a 416 4
350.8.o \(\chi_{350}(143, \cdot)\) n/a 336 4
350.8.q \(\chi_{350}(11, \cdot)\) n/a 1120 8
350.8.r \(\chi_{350}(13, \cdot)\) n/a 1120 8
350.8.u \(\chi_{350}(9, \cdot)\) n/a 1120 8
350.8.x \(\chi_{350}(3, \cdot)\) n/a 2240 16

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(350))\) into lower level spaces

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