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Properties

Label 56.7
Level 56
Weight 7
Dimension 302
Nonzero newspaces 6
Newform subspaces 9
Sturm bound 1344
Trace bound 2

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Defining parameters

Level: \( N \) = \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) = \( 7 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 9 \)
Sturm bound: \(1344\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(\Gamma_1(56))\).

Total New Old
Modular forms 612 322 290
Cusp forms 540 302 238
Eisenstein series 72 20 52

Trace form

\( 302 q + 6 q^{2} - 2 q^{3} - 46 q^{4} - 62 q^{6} - 6 q^{7} + 516 q^{8} - 3314 q^{9} + O(q^{10}) \) \( 302 q + 6 q^{2} - 2 q^{3} - 46 q^{4} - 62 q^{6} - 6 q^{7} + 516 q^{8} - 3314 q^{9} + 3834 q^{10} + 5154 q^{11} - 8630 q^{12} - 5766 q^{14} - 10836 q^{15} + 20954 q^{16} - 5736 q^{17} + 38632 q^{18} + 5602 q^{19} - 23172 q^{20} + 12516 q^{21} - 67826 q^{22} - 26718 q^{23} + 21550 q^{24} - 37138 q^{25} + 61284 q^{26} - 65732 q^{27} + 48042 q^{28} + 64458 q^{30} + 139854 q^{31} + 237936 q^{32} - 77584 q^{33} - 64130 q^{34} - 128754 q^{35} - 479798 q^{36} + 54180 q^{37} + 22176 q^{38} + 342120 q^{39} + 120624 q^{40} - 33408 q^{41} + 241530 q^{42} - 580256 q^{43} + 1362 q^{44} + 15204 q^{45} - 112416 q^{46} + 252246 q^{47} - 692642 q^{48} - 744346 q^{49} + 190938 q^{50} + 247478 q^{51} + 1532796 q^{52} - 76860 q^{53} + 1707118 q^{54} - 735216 q^{56} + 273992 q^{57} - 2051292 q^{58} - 743610 q^{59} - 850560 q^{60} + 345492 q^{61} + 140856 q^{62} + 534858 q^{63} - 738886 q^{64} + 443880 q^{65} + 1079942 q^{66} + 475390 q^{67} + 543618 q^{68} + 2056770 q^{70} + 3367260 q^{71} + 1615348 q^{72} - 619604 q^{73} + 811446 q^{74} - 6700652 q^{75} - 2259374 q^{76} - 2039544 q^{77} - 5301312 q^{78} - 1648590 q^{79} - 3543228 q^{80} + 1486402 q^{81} - 83990 q^{82} + 5572152 q^{83} + 1647078 q^{84} - 455112 q^{85} + 3034326 q^{86} + 1646820 q^{87} + 6927298 q^{88} + 1001508 q^{89} + 10102608 q^{90} + 142056 q^{91} - 1259142 q^{92} + 1342404 q^{93} - 4550490 q^{94} - 5938974 q^{95} - 13280126 q^{96} + 4528192 q^{97} - 9482778 q^{98} + 67088 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{7}^{\mathrm{new}}(\Gamma_1(56))\)

We only show spaces with odd 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
56.7.c \(\chi_{56}(41, \cdot)\) 56.7.c.a 12 1
56.7.d \(\chi_{56}(15, \cdot)\) None 0 1
56.7.g \(\chi_{56}(43, \cdot)\) 56.7.g.a 36 1
56.7.h \(\chi_{56}(13, \cdot)\) 56.7.h.a 2 1
56.7.h.b 2
56.7.h.c 2
56.7.h.d 40
56.7.j \(\chi_{56}(5, \cdot)\) 56.7.j.a 92 2
56.7.k \(\chi_{56}(11, \cdot)\) 56.7.k.a 92 2
56.7.n \(\chi_{56}(23, \cdot)\) None 0 2
56.7.o \(\chi_{56}(17, \cdot)\) 56.7.o.a 24 2

Decomposition of \(S_{7}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces

\( S_{7}^{\mathrm{old}}(\Gamma_1(56)) \cong \) \(S_{7}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)