Properties

Label 756.2
Level 756
Weight 2
Dimension 6710
Nonzero newspaces 32
Newform subspaces 71
Sturm bound 62208
Trace bound 21

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Defining parameters

Level: \( N \) = \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 32 \)
Newform subspaces: \( 71 \)
Sturm bound: \(62208\)
Trace bound: \(21\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(756))\).

Total New Old
Modular forms 16452 7030 9422
Cusp forms 14653 6710 7943
Eisenstein series 1799 320 1479

Trace form

\( 6710 q - 18 q^{2} - 32 q^{4} - 48 q^{5} - 24 q^{6} - 6 q^{7} - 30 q^{8} - 60 q^{9} + O(q^{10}) \) \( 6710 q - 18 q^{2} - 32 q^{4} - 48 q^{5} - 24 q^{6} - 6 q^{7} - 30 q^{8} - 60 q^{9} - 16 q^{10} - 12 q^{11} - 6 q^{12} - 54 q^{13} + 9 q^{14} + 18 q^{15} + 16 q^{16} + 18 q^{17} + 18 q^{18} + 4 q^{19} + 72 q^{20} - 30 q^{21} - 42 q^{22} + 48 q^{23} - 12 q^{24} - 38 q^{25} - 12 q^{26} + 54 q^{27} - 90 q^{28} - 6 q^{29} - 54 q^{30} + 10 q^{31} - 108 q^{32} + 48 q^{33} - 52 q^{34} + 51 q^{35} - 120 q^{36} + 2 q^{37} - 54 q^{38} - 6 q^{39} - 4 q^{40} + 96 q^{41} - 90 q^{42} + 24 q^{43} - 84 q^{44} - 114 q^{45} - 36 q^{47} - 174 q^{48} - 34 q^{49} - 120 q^{50} - 72 q^{51} + 44 q^{52} - 30 q^{53} - 192 q^{54} + 72 q^{55} - 63 q^{56} - 132 q^{57} + 8 q^{58} + 36 q^{59} - 240 q^{60} - 18 q^{61} - 234 q^{62} + 69 q^{63} - 110 q^{64} + 78 q^{65} - 210 q^{66} + 80 q^{67} - 186 q^{68} + 12 q^{69} - 117 q^{70} + 120 q^{71} - 60 q^{72} + 48 q^{73} - 210 q^{74} + 156 q^{75} - 144 q^{76} + 150 q^{77} - 24 q^{78} + 170 q^{79} - 138 q^{80} + 12 q^{81} - 76 q^{82} + 144 q^{83} - 24 q^{84} + 64 q^{85} - 18 q^{86} + 126 q^{87} - 72 q^{88} + 174 q^{89} + 120 q^{90} + 28 q^{91} + 162 q^{92} - 126 q^{93} - 48 q^{94} + 240 q^{96} + 90 q^{97} + 90 q^{98} - 54 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(756))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
756.2.a \(\chi_{756}(1, \cdot)\) 756.2.a.a 1 1
756.2.a.b 1
756.2.a.c 1
756.2.a.d 1
756.2.a.e 1
756.2.a.f 1
756.2.a.g 2
756.2.b \(\chi_{756}(55, \cdot)\) 756.2.b.a 4 1
756.2.b.b 4
756.2.b.c 12
756.2.b.d 12
756.2.b.e 16
756.2.b.f 16
756.2.e \(\chi_{756}(323, \cdot)\) 756.2.e.a 24 1
756.2.e.b 24
756.2.f \(\chi_{756}(377, \cdot)\) 756.2.f.a 2 1
756.2.f.b 2
756.2.f.c 2
756.2.f.d 4
756.2.i \(\chi_{756}(37, \cdot)\) 756.2.i.a 2 2
756.2.i.b 14
756.2.j \(\chi_{756}(253, \cdot)\) 756.2.j.a 6 2
756.2.j.b 6
756.2.k \(\chi_{756}(109, \cdot)\) 756.2.k.a 2 2
756.2.k.b 2
756.2.k.c 2
756.2.k.d 4
756.2.k.e 6
756.2.k.f 6
756.2.l \(\chi_{756}(289, \cdot)\) 756.2.l.a 2 2
756.2.l.b 14
756.2.n \(\chi_{756}(19, \cdot)\) 756.2.n.a 4 2
756.2.n.b 84
756.2.o \(\chi_{756}(179, \cdot)\) 756.2.o.a 88 2
756.2.t \(\chi_{756}(269, \cdot)\) 756.2.t.a 2 2
756.2.t.b 2
756.2.t.c 2
756.2.t.d 4
756.2.t.e 12
756.2.w \(\chi_{756}(341, \cdot)\) 756.2.w.a 16 2
756.2.x \(\chi_{756}(125, \cdot)\) 756.2.x.a 16 2
756.2.ba \(\chi_{756}(71, \cdot)\) 756.2.ba.a 72 2
756.2.bb \(\chi_{756}(611, \cdot)\) 756.2.bb.a 88 2
756.2.be \(\chi_{756}(107, \cdot)\) 756.2.be.a 4 2
756.2.be.b 4
756.2.be.c 28
756.2.be.d 28
756.2.be.e 64
756.2.bf \(\chi_{756}(271, \cdot)\) 756.2.bf.a 32 2
756.2.bf.b 32
756.2.bf.c 32
756.2.bf.d 32
756.2.bi \(\chi_{756}(307, \cdot)\) 756.2.bi.a 4 2
756.2.bi.b 4
756.2.bi.c 80
756.2.bj \(\chi_{756}(451, \cdot)\) 756.2.bj.a 4 2
756.2.bj.b 84
756.2.bm \(\chi_{756}(17, \cdot)\) 756.2.bm.a 16 2
756.2.bo \(\chi_{756}(85, \cdot)\) 756.2.bo.a 54 6
756.2.bo.b 54
756.2.bp \(\chi_{756}(193, \cdot)\) 756.2.bp.a 144 6
756.2.bq \(\chi_{756}(25, \cdot)\) 756.2.bq.a 144 6
756.2.bs \(\chi_{756}(11, \cdot)\) 756.2.bs.a 840 6
756.2.bt \(\chi_{756}(103, \cdot)\) 756.2.bt.a 840 6
756.2.bx \(\chi_{756}(41, \cdot)\) 756.2.bx.a 144 6
756.2.ca \(\chi_{756}(173, \cdot)\) 756.2.ca.a 144 6
756.2.cc \(\chi_{756}(139, \cdot)\) 756.2.cc.a 840 6
756.2.cd \(\chi_{756}(31, \cdot)\) 756.2.cd.a 840 6
756.2.cf \(\chi_{756}(155, \cdot)\) 756.2.cf.a 648 6
756.2.ci \(\chi_{756}(95, \cdot)\) 756.2.ci.a 840 6
756.2.ck \(\chi_{756}(5, \cdot)\) 756.2.ck.a 144 6

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(756))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(756)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(108))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(126))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(189))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(252))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(378))\)\(^{\oplus 2}\)