Properties

Label 39.8
Level 39
Weight 8
Dimension 280
Nonzero newspaces 6
Newform subspaces 13
Sturm bound 896
Trace bound 1

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Defining parameters

Level: \( N \) = \( 39 = 3 \cdot 13 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 13 \)
Sturm bound: \(896\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(39))\).

Total New Old
Modular forms 416 304 112
Cusp forms 368 280 88
Eisenstein series 48 24 24

Trace form

\( 280 q - 12 q^{2} + 48 q^{3} + 172 q^{4} - 780 q^{5} + 318 q^{6} + 3144 q^{7} - 6576 q^{8} - 6 q^{9} + O(q^{10}) \) \( 280 q - 12 q^{2} + 48 q^{3} + 172 q^{4} - 780 q^{5} + 318 q^{6} + 3144 q^{7} - 6576 q^{8} - 6 q^{9} + 15228 q^{10} - 6012 q^{11} - 39528 q^{12} - 24842 q^{13} + 22464 q^{14} + 62202 q^{15} + 156116 q^{16} + 74754 q^{17} - 206052 q^{18} - 289836 q^{19} + 284352 q^{20} + 338724 q^{21} + 449784 q^{22} + 648 q^{23} - 575430 q^{24} - 616700 q^{25} - 788352 q^{26} - 114834 q^{27} + 576128 q^{28} + 626670 q^{29} + 2368098 q^{30} + 1119068 q^{31} - 277572 q^{32} - 1524600 q^{33} - 3288660 q^{34} - 1339080 q^{35} + 1563372 q^{36} + 1154858 q^{37} + 103440 q^{38} + 2649948 q^{39} - 1804296 q^{40} - 3374046 q^{41} - 3922440 q^{42} - 602724 q^{43} + 4231308 q^{44} - 1194138 q^{45} + 11216124 q^{46} + 1337904 q^{47} + 4686858 q^{48} - 3982646 q^{49} - 8351376 q^{50} - 4834404 q^{51} - 22739048 q^{52} - 10538076 q^{53} - 3646146 q^{54} + 7394520 q^{55} + 24813996 q^{56} + 14166852 q^{57} + 31152828 q^{58} + 11603148 q^{59} + 7494696 q^{60} - 5796178 q^{61} - 20431980 q^{62} - 8658996 q^{63} - 55259768 q^{64} - 20983470 q^{65} + 8420808 q^{66} + 19617804 q^{67} + 9410544 q^{68} - 5583240 q^{69} + 23445840 q^{70} - 1774776 q^{71} + 15875772 q^{72} + 19429224 q^{73} + 19353384 q^{74} + 13962006 q^{75} - 71983020 q^{76} - 44307960 q^{77} + 13690176 q^{78} - 1085056 q^{79} + 23491392 q^{80} + 13938870 q^{81} + 87398988 q^{82} + 29446764 q^{83} - 18415488 q^{84} + 44588490 q^{85} + 10016640 q^{86} - 61326378 q^{87} - 141562800 q^{88} - 83314860 q^{89} - 20907720 q^{90} - 41529480 q^{91} - 29759256 q^{92} + 72039648 q^{93} + 68676564 q^{94} + 128021880 q^{95} + 57058596 q^{96} + 74768900 q^{97} + 80470632 q^{98} + 8040024 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(39))\)

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
39.8.a \(\chi_{39}(1, \cdot)\) 39.8.a.a 2 1
39.8.a.b 3
39.8.a.c 4
39.8.a.d 5
39.8.b \(\chi_{39}(25, \cdot)\) 39.8.b.a 8 1
39.8.b.b 10
39.8.e \(\chi_{39}(16, \cdot)\) 39.8.e.a 14 2
39.8.e.b 16
39.8.f \(\chi_{39}(5, \cdot)\) 39.8.f.a 60 2
39.8.j \(\chi_{39}(4, \cdot)\) 39.8.j.a 16 2
39.8.j.b 18
39.8.k \(\chi_{39}(2, \cdot)\) 39.8.k.a 4 4
39.8.k.b 120

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(39))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(39)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(13))\)\(^{\oplus 2}\)