Properties

Label 378.2
Level 378
Weight 2
Dimension 980
Nonzero newspaces 16
Newforms 50
Sturm bound 15552
Trace bound 11

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Defining parameters

Level: \( N \) = \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 16 \)
Newforms: \( 50 \)
Sturm bound: \(15552\)
Trace bound: \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(378))\).

Total New Old
Modular forms 4248 980 3268
Cusp forms 3529 980 2549
Eisenstein series 719 0 719

Trace form

\(980q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 10q^{8} \) \(\mathstrut +\mathstrut 24q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(980q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 2q^{4} \) \(\mathstrut +\mathstrut 12q^{5} \) \(\mathstrut +\mathstrut 12q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 10q^{8} \) \(\mathstrut +\mathstrut 24q^{9} \) \(\mathstrut +\mathstrut 12q^{10} \) \(\mathstrut +\mathstrut 36q^{11} \) \(\mathstrut +\mathstrut 6q^{12} \) \(\mathstrut +\mathstrut 24q^{13} \) \(\mathstrut +\mathstrut 16q^{14} \) \(\mathstrut +\mathstrut 36q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 48q^{17} \) \(\mathstrut -\mathstrut 12q^{18} \) \(\mathstrut +\mathstrut 24q^{19} \) \(\mathstrut -\mathstrut 12q^{20} \) \(\mathstrut -\mathstrut 24q^{21} \) \(\mathstrut +\mathstrut 24q^{22} \) \(\mathstrut +\mathstrut 38q^{25} \) \(\mathstrut -\mathstrut 40q^{26} \) \(\mathstrut -\mathstrut 54q^{27} \) \(\mathstrut +\mathstrut 2q^{28} \) \(\mathstrut -\mathstrut 48q^{29} \) \(\mathstrut -\mathstrut 72q^{30} \) \(\mathstrut +\mathstrut 24q^{31} \) \(\mathstrut -\mathstrut 2q^{32} \) \(\mathstrut -\mathstrut 54q^{33} \) \(\mathstrut +\mathstrut 12q^{34} \) \(\mathstrut +\mathstrut 30q^{35} \) \(\mathstrut -\mathstrut 12q^{36} \) \(\mathstrut +\mathstrut 32q^{37} \) \(\mathstrut +\mathstrut 26q^{38} \) \(\mathstrut +\mathstrut 84q^{39} \) \(\mathstrut +\mathstrut 12q^{40} \) \(\mathstrut +\mathstrut 60q^{41} \) \(\mathstrut +\mathstrut 24q^{42} \) \(\mathstrut +\mathstrut 80q^{43} \) \(\mathstrut +\mathstrut 48q^{47} \) \(\mathstrut +\mathstrut 12q^{48} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut -\mathstrut 38q^{50} \) \(\mathstrut -\mathstrut 108q^{51} \) \(\mathstrut -\mathstrut 24q^{52} \) \(\mathstrut -\mathstrut 180q^{53} \) \(\mathstrut -\mathstrut 36q^{54} \) \(\mathstrut -\mathstrut 144q^{55} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 126q^{57} \) \(\mathstrut -\mathstrut 96q^{58} \) \(\mathstrut -\mathstrut 318q^{59} \) \(\mathstrut -\mathstrut 72q^{60} \) \(\mathstrut -\mathstrut 96q^{61} \) \(\mathstrut -\mathstrut 208q^{62} \) \(\mathstrut -\mathstrut 246q^{63} \) \(\mathstrut +\mathstrut 10q^{64} \) \(\mathstrut -\mathstrut 432q^{65} \) \(\mathstrut -\mathstrut 144q^{66} \) \(\mathstrut -\mathstrut 212q^{67} \) \(\mathstrut -\mathstrut 150q^{68} \) \(\mathstrut -\mathstrut 216q^{69} \) \(\mathstrut -\mathstrut 174q^{70} \) \(\mathstrut -\mathstrut 312q^{71} \) \(\mathstrut -\mathstrut 48q^{72} \) \(\mathstrut -\mathstrut 168q^{73} \) \(\mathstrut -\mathstrut 232q^{74} \) \(\mathstrut -\mathstrut 204q^{75} \) \(\mathstrut -\mathstrut 90q^{76} \) \(\mathstrut -\mathstrut 192q^{77} \) \(\mathstrut -\mathstrut 144q^{78} \) \(\mathstrut -\mathstrut 200q^{79} \) \(\mathstrut -\mathstrut 24q^{80} \) \(\mathstrut -\mathstrut 60q^{82} \) \(\mathstrut +\mathstrut 12q^{83} \) \(\mathstrut -\mathstrut 120q^{85} \) \(\mathstrut -\mathstrut 40q^{86} \) \(\mathstrut -\mathstrut 72q^{87} \) \(\mathstrut -\mathstrut 42q^{88} \) \(\mathstrut +\mathstrut 102q^{89} \) \(\mathstrut -\mathstrut 36q^{90} \) \(\mathstrut +\mathstrut 78q^{91} \) \(\mathstrut +\mathstrut 24q^{92} \) \(\mathstrut +\mathstrut 12q^{93} \) \(\mathstrut -\mathstrut 12q^{94} \) \(\mathstrut +\mathstrut 120q^{95} \) \(\mathstrut -\mathstrut 12q^{96} \) \(\mathstrut +\mathstrut 36q^{97} \) \(\mathstrut +\mathstrut 46q^{98} \) \(\mathstrut +\mathstrut 72q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(378))\)

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
378.2.a \(\chi_{378}(1, \cdot)\) 378.2.a.a 1 1
378.2.a.b 1
378.2.a.c 1
378.2.a.d 1
378.2.a.e 1
378.2.a.f 1
378.2.a.g 1
378.2.a.h 1
378.2.d \(\chi_{378}(377, \cdot)\) 378.2.d.a 4 1
378.2.d.b 4
378.2.d.c 4
378.2.e \(\chi_{378}(37, \cdot)\) 378.2.e.a 2 2
378.2.e.b 2
378.2.e.c 6
378.2.e.d 6
378.2.f \(\chi_{378}(127, \cdot)\) 378.2.f.a 2 2
378.2.f.b 2
378.2.f.c 4
378.2.f.d 4
378.2.g \(\chi_{378}(109, \cdot)\) 378.2.g.a 2 2
378.2.g.b 2
378.2.g.c 2
378.2.g.d 2
378.2.g.e 2
378.2.g.f 2
378.2.g.g 4
378.2.g.h 4
378.2.h \(\chi_{378}(289, \cdot)\) 378.2.h.a 2 2
378.2.h.b 2
378.2.h.c 6
378.2.h.d 6
378.2.k \(\chi_{378}(215, \cdot)\) 378.2.k.a 4 2
378.2.k.b 4
378.2.k.c 4
378.2.k.d 8
378.2.l \(\chi_{378}(143, \cdot)\) 378.2.l.a 16 2
378.2.m \(\chi_{378}(125, \cdot)\) 378.2.m.a 16 2
378.2.t \(\chi_{378}(17, \cdot)\) 378.2.t.a 16 2
378.2.u \(\chi_{378}(43, \cdot)\) 378.2.u.a 6 6
378.2.u.b 12
378.2.u.c 24
378.2.u.d 30
378.2.u.e 36
378.2.v \(\chi_{378}(67, \cdot)\) 378.2.v.a 72 6
378.2.v.b 72
378.2.w \(\chi_{378}(25, \cdot)\) 378.2.w.a 72 6
378.2.w.b 72
378.2.z \(\chi_{378}(41, \cdot)\) 378.2.z.a 144 6
378.2.ba \(\chi_{378}(47, \cdot)\) 378.2.ba.a 144 6
378.2.bf \(\chi_{378}(5, \cdot)\) 378.2.bf.a 144 6

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(378))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(378)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(126))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(189))\)\(^{\oplus 2}\)