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Properties

Label 56.10
Level 56
Weight 10
Dimension 454
Nonzero newspaces 6
Newform subspaces 12
Sturm bound 1920
Trace bound 2

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Defining parameters

Level: \( N \) = \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) = \( 10 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 12 \)
Sturm bound: \(1920\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_1(56))\).

Total New Old
Modular forms 900 474 426
Cusp forms 828 454 374
Eisenstein series 72 20 52

Trace form

\( 454 q + 30 q^{2} - 22 q^{3} + 850 q^{4} + 1128 q^{5} - 9374 q^{6} - 10710 q^{7} + 6756 q^{8} + 90314 q^{9} + O(q^{10}) \) \( 454 q + 30 q^{2} - 22 q^{3} + 850 q^{4} + 1128 q^{5} - 9374 q^{6} - 10710 q^{7} + 6756 q^{8} + 90314 q^{9} - 52790 q^{10} - 109740 q^{11} - 109526 q^{12} + 206306 q^{13} + 72330 q^{14} + 314088 q^{15} - 371238 q^{16} + 634566 q^{17} + 165320 q^{18} - 1817510 q^{19} - 3571596 q^{20} + 2943792 q^{21} + 7570166 q^{22} - 5748564 q^{23} - 3901474 q^{24} - 2219638 q^{25} - 367236 q^{26} + 13187576 q^{27} + 12011722 q^{28} - 2309880 q^{29} - 1736662 q^{30} - 4646404 q^{31} + 8642520 q^{32} + 567050 q^{33} - 2412514 q^{34} + 6980778 q^{35} - 97721838 q^{36} + 35316786 q^{37} + 54201432 q^{38} + 81417072 q^{39} + 197846040 q^{40} - 33322860 q^{41} - 188007526 q^{42} - 155812828 q^{43} - 97663182 q^{44} + 104101042 q^{45} + 89557232 q^{46} + 370400796 q^{47} + 415936830 q^{48} + 201468054 q^{49} - 184663566 q^{50} - 541357004 q^{51} - 313448692 q^{52} + 94238238 q^{53} + 140242190 q^{54} + 764533464 q^{55} + 17253984 q^{56} - 178155744 q^{57} - 648126364 q^{58} - 969745074 q^{59} + 97009608 q^{60} - 259563688 q^{61} + 1166893608 q^{62} + 736710350 q^{63} + 108228682 q^{64} + 1381042164 q^{65} - 1837673082 q^{66} + 389466076 q^{67} - 847931862 q^{68} - 1766068372 q^{69} + 863891970 q^{70} - 821249448 q^{71} + 1294672412 q^{72} + 1600163338 q^{73} + 840167766 q^{74} + 323013094 q^{75} - 2023662638 q^{76} - 669158382 q^{77} - 3385225744 q^{78} - 894155956 q^{79} - 1523889972 q^{80} - 663551064 q^{81} + 4969474714 q^{82} + 3203650926 q^{83} + 6023866454 q^{84} - 1100866064 q^{85} - 2336582010 q^{86} - 3359587596 q^{87} - 6475900318 q^{88} - 2056741182 q^{89} - 1401819448 q^{90} + 36010254 q^{91} + 838716138 q^{92} + 279310198 q^{93} + 4076770758 q^{94} + 6970432788 q^{95} + 593112338 q^{96} + 3634176604 q^{97} - 1641533370 q^{98} - 9262065476 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_1(56))\)

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
56.10.a \(\chi_{56}(1, \cdot)\) 56.10.a.a 3 1
56.10.a.b 3
56.10.a.c 4
56.10.a.d 4
56.10.b \(\chi_{56}(29, \cdot)\) 56.10.b.a 26 1
56.10.b.b 28
56.10.e \(\chi_{56}(27, \cdot)\) 56.10.e.a 2 1
56.10.e.b 68
56.10.f \(\chi_{56}(55, \cdot)\) None 0 1
56.10.i \(\chi_{56}(9, \cdot)\) 56.10.i.a 18 2
56.10.i.b 18
56.10.l \(\chi_{56}(31, \cdot)\) None 0 2
56.10.m \(\chi_{56}(3, \cdot)\) 56.10.m.a 140 2
56.10.p \(\chi_{56}(37, \cdot)\) 56.10.p.a 140 2

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_1(56))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_1(56)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)