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Properties

Label 238.2
Level 238
Weight 2
Dimension 551
Nonzero newspaces 10
Newform subspaces 32
Sturm bound 6912
Trace bound 5

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Defining parameters

Level: \( N \) = \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 10 \)
Newform subspaces: \( 32 \)
Sturm bound: \(6912\)
Trace bound: \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(238))\).

Total New Old
Modular forms 1920 551 1369
Cusp forms 1537 551 986
Eisenstein series 383 0 383

Trace form

\( 551 q + 3 q^{2} + 8 q^{3} - q^{4} + 6 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} + O(q^{10}) \) \( 551 q + 3 q^{2} + 8 q^{3} - q^{4} + 6 q^{5} - q^{7} + 3 q^{8} + 11 q^{9} - 2 q^{10} - 20 q^{11} - 8 q^{12} - 10 q^{13} - 13 q^{14} - 72 q^{15} - 9 q^{16} - 21 q^{17} - 49 q^{18} - 16 q^{19} - 2 q^{20} - 40 q^{21} - 20 q^{22} - 8 q^{23} - 16 q^{24} - 31 q^{25} - 2 q^{26} - 16 q^{27} - q^{28} + 2 q^{29} + 24 q^{30} - 24 q^{31} + 3 q^{32} - 16 q^{33} + 23 q^{34} - 26 q^{35} + 11 q^{36} - 30 q^{37} + 8 q^{38} - 88 q^{39} + 6 q^{40} - 106 q^{41} - 48 q^{42} - 84 q^{43} - 52 q^{44} - 90 q^{45} - 40 q^{46} - 88 q^{47} + 8 q^{48} - 65 q^{49} - 91 q^{50} - 48 q^{51} - 42 q^{52} - 94 q^{53} - 64 q^{54} - 88 q^{55} + 3 q^{56} - 88 q^{57} - 46 q^{58} - 88 q^{59} - 40 q^{60} - 50 q^{61} - 72 q^{62} - 85 q^{63} - q^{64} + 12 q^{65} + 32 q^{66} + 44 q^{67} + 3 q^{68} - 32 q^{69} - 26 q^{70} - 24 q^{71} - 25 q^{72} - 98 q^{73} - 30 q^{74} - 120 q^{75} + 16 q^{76} - 68 q^{77} - 56 q^{78} - 96 q^{79} - 26 q^{80} - 65 q^{81} - 82 q^{82} - 48 q^{83} - 24 q^{84} - 162 q^{85} - 4 q^{86} - 192 q^{87} - 52 q^{88} - 90 q^{89} - 58 q^{90} - 138 q^{91} - 8 q^{92} - 144 q^{93} - 104 q^{94} - 168 q^{95} - 74 q^{97} - 5 q^{98} - 20 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(238))\)

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
238.2.a \(\chi_{238}(1, \cdot)\) 238.2.a.a 1 1
238.2.a.b 1
238.2.a.c 1
238.2.a.d 1
238.2.a.e 1
238.2.a.f 2
238.2.b \(\chi_{238}(169, \cdot)\) 238.2.b.a 2 1
238.2.b.b 6
238.2.e \(\chi_{238}(137, \cdot)\) 238.2.e.a 2 2
238.2.e.b 2
238.2.e.c 2
238.2.e.d 2
238.2.e.e 6
238.2.e.f 10
238.2.g \(\chi_{238}(183, \cdot)\) 238.2.g.a 8 2
238.2.g.b 8
238.2.j \(\chi_{238}(67, \cdot)\) 238.2.j.a 4 2
238.2.j.b 8
238.2.j.c 12
238.2.k \(\chi_{238}(15, \cdot)\) 238.2.k.a 16 4
238.2.k.b 24
238.2.n \(\chi_{238}(81, \cdot)\) 238.2.n.a 4 4
238.2.n.b 4
238.2.n.c 16
238.2.n.d 24
238.2.p \(\chi_{238}(27, \cdot)\) 238.2.p.a 48 8
238.2.p.b 48
238.2.q \(\chi_{238}(9, \cdot)\) 238.2.q.a 16 8
238.2.q.b 32
238.2.q.c 48
238.2.s \(\chi_{238}(3, \cdot)\) 238.2.s.a 96 16
238.2.s.b 96

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(238))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(238)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(34))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(119))\)\(^{\oplus 2}\)