Properties

Label 54.9
Level 54
Weight 9
Dimension 170
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 1458
Trace bound 1

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Defining parameters

Level: \( N \) = \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) = \( 9 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(1458\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(\Gamma_1(54))\).

Total New Old
Modular forms 678 170 508
Cusp forms 618 170 448
Eisenstein series 60 0 60

Trace form

\( 170 q - 256 q^{4} + 1764 q^{5} + 768 q^{6} - 5504 q^{7} + 25908 q^{9} + O(q^{10}) \) \( 170 q - 256 q^{4} + 1764 q^{5} + 768 q^{6} - 5504 q^{7} + 25908 q^{9} + 8448 q^{10} - 91512 q^{11} - 21504 q^{12} + 67720 q^{13} + 188928 q^{14} - 70002 q^{15} + 32768 q^{16} - 182784 q^{18} + 364486 q^{19} - 112896 q^{20} - 2071668 q^{21} - 260352 q^{22} - 758142 q^{23} + 178394 q^{25} - 4137588 q^{27} - 4352 q^{28} - 4842 q^{29} - 861696 q^{30} - 1426196 q^{31} + 10547334 q^{33} - 4325376 q^{34} - 12450834 q^{35} - 5104128 q^{36} + 10218826 q^{37} + 15378048 q^{38} + 4239102 q^{39} - 1081344 q^{40} - 24892128 q^{41} - 11028480 q^{42} - 27217772 q^{43} - 12558942 q^{45} + 13718016 q^{46} + 50154444 q^{47} + 3440640 q^{48} + 744210 q^{49} - 55489536 q^{50} - 3281850 q^{51} - 8668160 q^{52} + 16379136 q^{54} + 58852728 q^{55} + 24182784 q^{56} - 13553658 q^{57} - 28512000 q^{58} - 180471384 q^{59} - 26125056 q^{60} - 23245784 q^{61} + 108329070 q^{63} + 96468992 q^{64} + 148758480 q^{65} + 152091648 q^{66} + 181987522 q^{67} + 18888192 q^{68} - 40190130 q^{69} - 222851328 q^{70} - 251437608 q^{71} - 60948480 q^{72} - 3128600 q^{73} + 6778368 q^{74} + 138868662 q^{75} + 90880000 q^{76} + 555602292 q^{77} + 201203712 q^{78} + 198320380 q^{79} - 23838948 q^{81} - 110350848 q^{82} - 597057804 q^{83} - 128300544 q^{84} - 345701700 q^{85} + 125141760 q^{86} + 34353810 q^{87} + 104546304 q^{88} + 375969762 q^{89} + 452712960 q^{90} - 47381116 q^{91} - 70783488 q^{92} + 285095622 q^{93} - 372903168 q^{94} - 823047192 q^{95} - 88080384 q^{96} - 516486314 q^{97} - 234233856 q^{98} + 26756370 q^{99} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(\Gamma_1(54))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
54.9.b \(\chi_{54}(53, \cdot)\) 54.9.b.a 2 1
54.9.b.b 4
54.9.b.c 4
54.9.d \(\chi_{54}(17, \cdot)\) 54.9.d.a 16 2
54.9.f \(\chi_{54}(5, \cdot)\) 54.9.f.a 144 6

Decomposition of \(S_{9}^{\mathrm{old}}(\Gamma_1(54))\) into lower level spaces

\( S_{9}^{\mathrm{old}}(\Gamma_1(54)) \cong \) \(S_{9}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(\Gamma_1(27))\)\(^{\oplus 2}\)