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Properties

Label 112.3
Level 112
Weight 3
Dimension 391
Nonzero newspaces 8
Newform subspaces 16
Sturm bound 2304
Trace bound 3

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Defining parameters

Level: \( N \) = \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) = \( 3 \)
Nonzero newspaces: \( 8 \)
Newform subspaces: \( 16 \)
Sturm bound: \(2304\)
Trace bound: \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(\Gamma_1(112))\).

Total New Old
Modular forms 852 437 415
Cusp forms 684 391 293
Eisenstein series 168 46 122

Trace form

\( 391 q - 8 q^{2} - 5 q^{3} + 4 q^{4} + q^{5} + 4 q^{6} - 5 q^{7} - 32 q^{8} - 21 q^{9} + O(q^{10}) \) \( 391 q - 8 q^{2} - 5 q^{3} + 4 q^{4} + q^{5} + 4 q^{6} - 5 q^{7} - 32 q^{8} - 21 q^{9} - 84 q^{10} + 27 q^{11} - 116 q^{12} - 28 q^{13} - 24 q^{14} - 18 q^{15} + 68 q^{16} + 41 q^{17} + 136 q^{18} - 21 q^{19} + 156 q^{20} + 77 q^{21} + 80 q^{22} - 57 q^{23} - 108 q^{24} + 23 q^{25} - 204 q^{26} - 140 q^{27} - 68 q^{28} - 102 q^{29} - 116 q^{30} - 105 q^{31} - 28 q^{32} - 163 q^{33} + 140 q^{34} - 53 q^{35} + 80 q^{36} - 87 q^{37} - 92 q^{38} + 284 q^{39} - 92 q^{40} - 36 q^{41} - 420 q^{42} + 138 q^{43} - 580 q^{44} - 546 q^{45} - 608 q^{46} - 153 q^{47} - 804 q^{48} - 73 q^{49} - 536 q^{50} - 441 q^{51} - 536 q^{52} - 447 q^{53} - 244 q^{54} - 528 q^{55} + 240 q^{56} + 186 q^{57} + 592 q^{58} - 325 q^{59} + 1148 q^{60} + 257 q^{61} + 1032 q^{62} + 231 q^{63} + 604 q^{64} + 624 q^{65} + 916 q^{66} + 755 q^{67} + 544 q^{68} + 1228 q^{69} + 724 q^{70} + 1030 q^{71} + 652 q^{72} + 457 q^{73} + 172 q^{74} + 974 q^{75} + 364 q^{76} + 485 q^{77} + 144 q^{78} + 159 q^{79} - 476 q^{80} - 238 q^{81} - 620 q^{82} - 648 q^{83} - 244 q^{84} - 62 q^{85} - 548 q^{86} - 1086 q^{87} + 4 q^{88} - 135 q^{89} + 740 q^{90} - 1268 q^{91} + 540 q^{92} - 647 q^{93} + 1116 q^{94} - 2097 q^{95} + 1948 q^{96} - 612 q^{97} + 972 q^{98} - 1942 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(\Gamma_1(112))\)

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
112.3.c \(\chi_{112}(97, \cdot)\) 112.3.c.a 1 1
112.3.c.b 2
112.3.c.c 4
112.3.d \(\chi_{112}(15, \cdot)\) 112.3.d.a 2 1
112.3.d.b 4
112.3.g \(\chi_{112}(71, \cdot)\) None 0 1
112.3.h \(\chi_{112}(41, \cdot)\) None 0 1
112.3.k \(\chi_{112}(43, \cdot)\) 112.3.k.a 48 2
112.3.l \(\chi_{112}(13, \cdot)\) 112.3.l.a 4 2
112.3.l.b 56
112.3.n \(\chi_{112}(73, \cdot)\) None 0 2
112.3.o \(\chi_{112}(23, \cdot)\) None 0 2
112.3.r \(\chi_{112}(79, \cdot)\) 112.3.r.a 4 2
112.3.r.b 6
112.3.r.c 6
112.3.s \(\chi_{112}(17, \cdot)\) 112.3.s.a 2 2
112.3.s.b 4
112.3.s.c 8
112.3.u \(\chi_{112}(11, \cdot)\) 112.3.u.a 120 4
112.3.x \(\chi_{112}(5, \cdot)\) 112.3.x.a 120 4

Decomposition of \(S_{3}^{\mathrm{old}}(\Gamma_1(112))\) into lower level spaces

\( S_{3}^{\mathrm{old}}(\Gamma_1(112)) \cong \) \(S_{3}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 2}\)