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Properties

Label 180.4
Level 180
Weight 4
Dimension 1071
Nonzero newspaces 12
Newform subspaces 28
Sturm bound 6912
Trace bound 4

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Defining parameters

Level: \( N \) = \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 12 \)
Newform subspaces: \( 28 \)
Sturm bound: \(6912\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(180))\).

Total New Old
Modular forms 2752 1127 1625
Cusp forms 2432 1071 1361
Eisenstein series 320 56 264

Trace form

\( 1071 q + 6 q^{3} - 26 q^{4} - 11 q^{5} - 58 q^{6} - 20 q^{7} - 48 q^{8} - 42 q^{9} + O(q^{10}) \) \( 1071 q + 6 q^{3} - 26 q^{4} - 11 q^{5} - 58 q^{6} - 20 q^{7} - 48 q^{8} - 42 q^{9} + 62 q^{10} - 82 q^{11} + 4 q^{12} - 216 q^{13} + 144 q^{14} + 12 q^{15} - 350 q^{16} + 164 q^{17} + 200 q^{18} + 112 q^{19} + 34 q^{20} + 340 q^{21} + 422 q^{22} - 180 q^{23} - 42 q^{24} + 1281 q^{25} + 620 q^{26} - 744 q^{27} + 136 q^{28} - 342 q^{29} + 714 q^{30} + 528 q^{31} - 170 q^{32} + 1686 q^{33} + 594 q^{34} + 128 q^{35} - 2478 q^{36} - 640 q^{37} - 2530 q^{38} - 344 q^{39} - 1010 q^{40} - 1448 q^{41} - 2228 q^{42} - 1934 q^{43} - 444 q^{45} - 552 q^{46} - 804 q^{47} + 4670 q^{48} + 1791 q^{49} + 1864 q^{50} + 3150 q^{51} + 3408 q^{52} + 5384 q^{53} + 4526 q^{54} + 3676 q^{55} + 2788 q^{56} + 326 q^{57} - 2656 q^{58} + 914 q^{59} - 2246 q^{60} + 3054 q^{61} - 3988 q^{62} - 1504 q^{63} - 2756 q^{64} - 464 q^{65} - 7196 q^{66} - 1754 q^{67} - 10514 q^{68} - 3008 q^{69} - 3408 q^{70} - 7272 q^{71} - 2982 q^{72} - 3872 q^{73} + 924 q^{74} + 1446 q^{75} + 1470 q^{76} - 7260 q^{77} + 392 q^{78} - 4328 q^{79} + 3152 q^{80} - 6066 q^{81} + 1388 q^{82} - 2196 q^{83} + 2980 q^{84} + 5264 q^{85} + 8558 q^{86} + 196 q^{87} + 9254 q^{88} + 3130 q^{89} - 2334 q^{90} + 10016 q^{91} + 9808 q^{92} - 2216 q^{93} + 11900 q^{94} - 228 q^{95} + 1016 q^{96} + 6858 q^{97} + 1710 q^{98} + 1420 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(180))\)

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
180.4.a \(\chi_{180}(1, \cdot)\) 180.4.a.a 1 1
180.4.a.b 1
180.4.a.c 1
180.4.a.d 1
180.4.a.e 1
180.4.d \(\chi_{180}(109, \cdot)\) 180.4.d.a 2 1
180.4.d.b 2
180.4.d.c 4
180.4.e \(\chi_{180}(71, \cdot)\) 180.4.e.a 24 1
180.4.h \(\chi_{180}(179, \cdot)\) 180.4.h.a 4 1
180.4.h.b 32
180.4.i \(\chi_{180}(61, \cdot)\) 180.4.i.a 2 2
180.4.i.b 8
180.4.i.c 14
180.4.j \(\chi_{180}(17, \cdot)\) 180.4.j.a 12 2
180.4.k \(\chi_{180}(127, \cdot)\) 180.4.k.a 2 2
180.4.k.b 2
180.4.k.c 2
180.4.k.d 8
180.4.k.e 12
180.4.k.f 28
180.4.k.g 32
180.4.n \(\chi_{180}(59, \cdot)\) 180.4.n.a 8 2
180.4.n.b 200
180.4.q \(\chi_{180}(11, \cdot)\) 180.4.q.a 144 2
180.4.r \(\chi_{180}(49, \cdot)\) 180.4.r.a 36 2
180.4.w \(\chi_{180}(77, \cdot)\) 180.4.w.a 72 4
180.4.x \(\chi_{180}(7, \cdot)\) 180.4.x.a 416 4

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(180))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(180)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 2}\)