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Properties

Label 98.8
Level 98
Weight 8
Dimension 659
Nonzero newspaces 4
Newform subspaces 30
Sturm bound 4704
Trace bound 1

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Defining parameters

Level: \( N \) = \( 98 = 2 \cdot 7^{2} \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 4 \)
Newform subspaces: \( 30 \)
Sturm bound: \(4704\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(98))\).

Total New Old
Modular forms 2118 659 1459
Cusp forms 1998 659 1339
Eisenstein series 120 0 120

Trace form

\( 659 q - 8 q^{2} + 120 q^{3} - 192 q^{4} - 222 q^{5} + 2496 q^{6} + 664 q^{7} - 512 q^{8} - 9183 q^{9} + O(q^{10}) \) \( 659 q - 8 q^{2} + 120 q^{3} - 192 q^{4} - 222 q^{5} + 2496 q^{6} + 664 q^{7} - 512 q^{8} - 9183 q^{9} + 2928 q^{10} + 15180 q^{11} + 7680 q^{12} - 44838 q^{13} + 26832 q^{14} - 2856 q^{15} - 12288 q^{16} - 210630 q^{17} + 2712 q^{18} + 139104 q^{19} + 82560 q^{20} + 174918 q^{21} + 8736 q^{22} - 371832 q^{23} - 184320 q^{24} - 434085 q^{25} + 170096 q^{26} + 1116864 q^{27} - 11264 q^{28} + 108774 q^{29} - 1039296 q^{30} - 1091304 q^{31} - 32768 q^{32} + 2156856 q^{33} + 825648 q^{34} + 56910 q^{35} - 996288 q^{36} - 498562 q^{37} - 3275632 q^{38} - 8360766 q^{39} - 92160 q^{40} + 5145582 q^{41} + 5768208 q^{42} + 3949632 q^{43} + 3006336 q^{44} + 4833876 q^{45} - 4435920 q^{46} - 12143460 q^{47} - 1056768 q^{48} - 17608886 q^{49} + 1702888 q^{50} + 10539432 q^{51} + 2544896 q^{52} + 11880594 q^{53} + 11361456 q^{54} + 7454226 q^{55} + 5179392 q^{56} + 4292844 q^{57} - 6551184 q^{58} - 6864804 q^{59} - 5537664 q^{60} - 31612132 q^{61} + 7869584 q^{62} + 37224732 q^{63} - 7077888 q^{64} + 10757292 q^{65} - 3772032 q^{66} - 10719540 q^{67} - 13480320 q^{68} - 15785136 q^{69} - 14363664 q^{70} - 9868824 q^{71} + 173568 q^{72} - 1026294 q^{73} + 11980496 q^{74} + 51106392 q^{75} + 25589760 q^{76} + 15014832 q^{77} + 21462720 q^{78} - 36425424 q^{79} - 909312 q^{80} - 43093911 q^{81} - 39522960 q^{82} - 46932684 q^{83} - 13975680 q^{84} - 31830624 q^{85} - 43612768 q^{86} + 31600968 q^{87} + 11126784 q^{88} + 142772694 q^{89} + 132569904 q^{90} + 76722398 q^{91} + 18826752 q^{92} - 34877316 q^{93} - 47118144 q^{94} - 219103152 q^{95} - 11796480 q^{96} - 154649058 q^{97} - 31638144 q^{98} - 182230524 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(98))\)

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
98.8.a \(\chi_{98}(1, \cdot)\) 98.8.a.a 1 1
98.8.a.b 1
98.8.a.c 1
98.8.a.d 2
98.8.a.e 2
98.8.a.f 2
98.8.a.g 2
98.8.a.h 2
98.8.a.i 2
98.8.a.j 2
98.8.a.k 2
98.8.a.l 4
98.8.c \(\chi_{98}(67, \cdot)\) 98.8.c.a 2 2
98.8.c.b 2
98.8.c.c 2
98.8.c.d 2
98.8.c.e 2
98.8.c.f 2
98.8.c.g 4
98.8.c.h 4
98.8.c.i 4
98.8.c.j 4
98.8.c.k 4
98.8.c.l 4
98.8.c.m 4
98.8.c.n 8
98.8.e \(\chi_{98}(15, \cdot)\) 98.8.e.a 102 6
98.8.e.b 102
98.8.g \(\chi_{98}(9, \cdot)\) 98.8.g.a 192 12
98.8.g.b 192

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(98))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(98)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(98))\)\(^{\oplus 1}\)