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Properties

Label 85.4
Level 85
Weight 4
Dimension 742
Nonzero newspaces 10
Newform subspaces 17
Sturm bound 2304
Trace bound 8

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Defining parameters

Level: \( N \) = \( 85 = 5 \cdot 17 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 10 \)
Newform subspaces: \( 17 \)
Sturm bound: \(2304\)
Trace bound: \(8\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(85))\).

Total New Old
Modular forms 928 834 94
Cusp forms 800 742 58
Eisenstein series 128 92 36

Trace form

\( 742 q - 8 q^{2} - 20 q^{3} - 32 q^{4} - 14 q^{5} - 32 q^{6} - 28 q^{7} - 16 q^{8} + 30 q^{9} + O(q^{10}) \) \( 742 q - 8 q^{2} - 20 q^{3} - 32 q^{4} - 14 q^{5} - 32 q^{6} - 28 q^{7} - 16 q^{8} + 30 q^{9} + 40 q^{10} + 112 q^{11} + 144 q^{12} - 4 q^{13} - 288 q^{14} - 340 q^{15} - 800 q^{16} - 298 q^{17} - 1112 q^{18} - 376 q^{19} - 56 q^{20} + 120 q^{21} + 688 q^{22} + 556 q^{23} + 3728 q^{24} + 1154 q^{25} + 1712 q^{26} + 616 q^{27} - 400 q^{28} - 436 q^{29} - 1448 q^{30} - 1560 q^{31} - 2880 q^{32} - 2240 q^{33} - 4136 q^{34} - 980 q^{35} - 3008 q^{36} - 1892 q^{37} + 1440 q^{38} + 2568 q^{39} + 2472 q^{40} + 2748 q^{41} + 5936 q^{42} + 2028 q^{43} + 3376 q^{44} + 1086 q^{45} + 1344 q^{46} - 2028 q^{47} - 6368 q^{48} - 2474 q^{49} - 616 q^{50} - 3300 q^{51} - 5568 q^{52} + 596 q^{53} + 4560 q^{54} + 2184 q^{55} + 1952 q^{56} + 3760 q^{57} + 2544 q^{58} + 712 q^{59} + 1552 q^{60} + 508 q^{61} + 2848 q^{62} - 3676 q^{63} + 3152 q^{64} - 1504 q^{65} - 9888 q^{66} - 4540 q^{67} - 16528 q^{68} - 17192 q^{69} - 17664 q^{70} - 7528 q^{71} - 17408 q^{72} - 6540 q^{73} - 2160 q^{74} - 1756 q^{75} + 448 q^{76} + 3600 q^{77} + 13696 q^{78} + 6400 q^{79} + 20680 q^{80} + 18902 q^{81} + 22208 q^{82} + 18284 q^{83} + 48704 q^{84} + 20542 q^{85} + 30928 q^{86} + 18136 q^{87} + 24176 q^{88} + 8636 q^{89} + 16632 q^{90} + 5592 q^{91} + 3248 q^{92} - 2528 q^{93} - 13536 q^{94} - 6240 q^{95} - 25344 q^{96} - 10580 q^{97} - 33736 q^{98} - 13488 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(85))\)

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
85.4.a \(\chi_{85}(1, \cdot)\) 85.4.a.a 1 1
85.4.a.b 1
85.4.a.c 1
85.4.a.d 2
85.4.a.e 3
85.4.a.f 3
85.4.a.g 5
85.4.b \(\chi_{85}(69, \cdot)\) 85.4.b.a 24 1
85.4.c \(\chi_{85}(84, \cdot)\) 85.4.c.a 24 1
85.4.d \(\chi_{85}(16, \cdot)\) 85.4.d.a 2 1
85.4.d.b 16
85.4.e \(\chi_{85}(21, \cdot)\) 85.4.e.a 36 2
85.4.j \(\chi_{85}(4, \cdot)\) 85.4.j.a 48 2
85.4.l \(\chi_{85}(26, \cdot)\) 85.4.l.a 72 4
85.4.m \(\chi_{85}(9, \cdot)\) 85.4.m.a 104 4
85.4.o \(\chi_{85}(3, \cdot)\) 85.4.o.a 200 8
85.4.r \(\chi_{85}(12, \cdot)\) 85.4.r.a 200 8

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(85))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(85)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(17))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(85))\)\(^{\oplus 1}\)