Properties

Label 156.2
Level 156
Weight 2
Dimension 270
Nonzero newspaces 12
Newform subspaces 32
Sturm bound 2688
Trace bound 10

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Defining parameters

Level: \( N \) = \( 156 = 2^{2} \cdot 3 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 32 \)
Sturm bound: \(2688\)
Trace bound: \(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(156))\).

Total New Old
Modular forms 792 310 482
Cusp forms 553 270 283
Eisenstein series 239 40 199

Trace form

\( 270 q - 12 q^{4} - 6 q^{6} + 4 q^{7} - 10 q^{9} + O(q^{10}) \) \( 270 q - 12 q^{4} - 6 q^{6} + 4 q^{7} - 10 q^{9} - 12 q^{10} + 12 q^{11} - 12 q^{12} + 12 q^{15} - 12 q^{16} + 6 q^{17} - 24 q^{18} - 8 q^{19} - 48 q^{20} - 34 q^{21} - 72 q^{22} - 24 q^{23} - 54 q^{24} - 78 q^{25} - 60 q^{26} - 36 q^{27} - 72 q^{28} - 30 q^{29} - 54 q^{30} - 24 q^{31} - 60 q^{32} - 54 q^{33} - 60 q^{34} - 24 q^{35} - 24 q^{36} - 30 q^{37} - 28 q^{39} + 24 q^{40} - 66 q^{41} + 12 q^{42} - 32 q^{43} + 60 q^{44} - 60 q^{45} + 108 q^{46} - 24 q^{47} + 42 q^{48} - 176 q^{49} + 108 q^{50} - 24 q^{51} + 96 q^{52} - 48 q^{53} - 18 q^{54} - 12 q^{55} + 108 q^{56} - 58 q^{57} + 84 q^{58} + 12 q^{59} + 84 q^{60} - 90 q^{61} + 60 q^{62} + 62 q^{63} + 48 q^{64} - 18 q^{65} + 72 q^{66} + 40 q^{67} + 66 q^{69} + 48 q^{70} + 24 q^{71} + 72 q^{72} + 16 q^{73} + 104 q^{75} - 12 q^{76} + 72 q^{77} + 108 q^{78} + 72 q^{79} + 50 q^{81} - 12 q^{82} + 36 q^{83} + 120 q^{84} + 54 q^{85} + 48 q^{86} + 84 q^{87} + 48 q^{88} + 72 q^{89} + 144 q^{90} + 52 q^{91} + 72 q^{92} + 74 q^{93} + 84 q^{94} + 48 q^{95} + 120 q^{96} + 60 q^{97} + 108 q^{98} - 18 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(156))\)

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
156.2.a \(\chi_{156}(1, \cdot)\) 156.2.a.a 1 1
156.2.a.b 1
156.2.b \(\chi_{156}(25, \cdot)\) 156.2.b.a 2 1
156.2.b.b 2
156.2.c \(\chi_{156}(131, \cdot)\) 156.2.c.a 2 1
156.2.c.b 2
156.2.c.c 8
156.2.c.d 12
156.2.h \(\chi_{156}(155, \cdot)\) 156.2.h.a 8 1
156.2.h.b 16
156.2.i \(\chi_{156}(61, \cdot)\) 156.2.i.a 2 2
156.2.k \(\chi_{156}(31, \cdot)\) 156.2.k.a 2 2
156.2.k.b 2
156.2.k.c 2
156.2.k.d 2
156.2.k.e 10
156.2.k.f 10
156.2.m \(\chi_{156}(5, \cdot)\) 156.2.m.a 4 2
156.2.m.b 4
156.2.p \(\chi_{156}(35, \cdot)\) 156.2.p.a 8 2
156.2.p.b 40
156.2.q \(\chi_{156}(49, \cdot)\) 156.2.q.a 2 2
156.2.q.b 4
156.2.r \(\chi_{156}(23, \cdot)\) 156.2.r.a 4 2
156.2.r.b 4
156.2.r.c 40
156.2.u \(\chi_{156}(41, \cdot)\) 156.2.u.a 4 4
156.2.u.b 16
156.2.w \(\chi_{156}(7, \cdot)\) 156.2.w.a 4 4
156.2.w.b 4
156.2.w.c 24
156.2.w.d 24

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(156))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(156)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(26))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(39))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(52))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(156))\)\(^{\oplus 1}\)