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Properties

Label 360.6
Level 360
Weight 6
Dimension 6999
Nonzero newspaces 18
Sturm bound 41472
Trace bound 10

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Defining parameters

Level: \( N \) = \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 18 \)
Sturm bound: \(41472\)
Trace bound: \(10\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(360))\).

Total New Old
Modular forms 17664 7107 10557
Cusp forms 16896 6999 9897
Eisenstein series 768 108 660

Trace form

\( 6999 q - 4 q^{2} + 14 q^{3} - 48 q^{4} - 67 q^{5} + 104 q^{6} - 172 q^{7} - 1216 q^{8} + 298 q^{9} + O(q^{10}) \) \( 6999 q - 4 q^{2} + 14 q^{3} - 48 q^{4} - 67 q^{5} + 104 q^{6} - 172 q^{7} - 1216 q^{8} + 298 q^{9} + 476 q^{10} + 2166 q^{11} + 2172 q^{12} + 1250 q^{13} - 2284 q^{14} - 472 q^{15} - 10004 q^{16} - 2550 q^{17} + 8960 q^{18} - 6528 q^{19} + 7934 q^{20} - 496 q^{21} - 9172 q^{22} + 5964 q^{23} - 18112 q^{24} - 41493 q^{25} - 37448 q^{26} + 16712 q^{27} + 10344 q^{28} + 17562 q^{29} + 38650 q^{30} - 19536 q^{31} + 12756 q^{32} + 6894 q^{33} + 21064 q^{34} - 36372 q^{35} - 133668 q^{36} - 9910 q^{37} - 20500 q^{38} + 36168 q^{39} - 30654 q^{40} - 9012 q^{41} + 190180 q^{42} - 45062 q^{43} + 72100 q^{44} - 4490 q^{45} - 171424 q^{46} - 237700 q^{47} - 255664 q^{48} + 221125 q^{49} + 256898 q^{50} + 122422 q^{51} + 140992 q^{52} - 4366 q^{53} + 141912 q^{54} + 14948 q^{55} + 203072 q^{56} - 168430 q^{57} - 5912 q^{58} + 22266 q^{59} - 473350 q^{60} + 34334 q^{61} - 712256 q^{62} - 728 q^{63} + 158952 q^{64} + 133384 q^{65} + 199720 q^{66} + 353670 q^{67} + 923620 q^{68} + 76828 q^{69} + 560626 q^{70} - 553864 q^{71} + 389196 q^{72} - 67126 q^{73} - 698072 q^{74} - 488090 q^{75} - 821884 q^{76} + 329736 q^{77} - 558596 q^{78} - 168216 q^{79} - 198704 q^{80} + 495858 q^{81} + 1530340 q^{82} + 1167532 q^{83} + 364928 q^{84} + 6938 q^{85} + 1224340 q^{86} + 47604 q^{87} + 209596 q^{88} - 599162 q^{89} - 319166 q^{90} - 236640 q^{91} - 1279780 q^{92} - 203692 q^{93} - 1010504 q^{94} - 535500 q^{95} - 73280 q^{96} + 189428 q^{97} - 480056 q^{98} + 156156 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(360))\)

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
360.6.a \(\chi_{360}(1, \cdot)\) 360.6.a.a 1 1
360.6.a.b 1
360.6.a.c 1
360.6.a.d 1
360.6.a.e 1
360.6.a.f 1
360.6.a.g 1
360.6.a.h 1
360.6.a.i 1
360.6.a.j 2
360.6.a.k 2
360.6.a.l 2
360.6.a.m 2
360.6.a.n 2
360.6.a.o 3
360.6.a.p 3
360.6.b \(\chi_{360}(251, \cdot)\) 360.6.b.a 40 1
360.6.b.b 40
360.6.d \(\chi_{360}(109, \cdot)\) n/a 148 1
360.6.f \(\chi_{360}(289, \cdot)\) 360.6.f.a 6 1
360.6.f.b 8
360.6.f.c 8
360.6.f.d 16
360.6.h \(\chi_{360}(71, \cdot)\) None 0 1
360.6.k \(\chi_{360}(181, \cdot)\) 360.6.k.a 18 1
360.6.k.b 20
360.6.k.c 22
360.6.k.d 40
360.6.m \(\chi_{360}(179, \cdot)\) n/a 120 1
360.6.o \(\chi_{360}(359, \cdot)\) None 0 1
360.6.q \(\chi_{360}(121, \cdot)\) n/a 120 2
360.6.s \(\chi_{360}(17, \cdot)\) 360.6.s.a 28 2
360.6.s.b 32
360.6.t \(\chi_{360}(127, \cdot)\) None 0 2
360.6.w \(\chi_{360}(163, \cdot)\) n/a 296 2
360.6.x \(\chi_{360}(53, \cdot)\) n/a 240 2
360.6.bb \(\chi_{360}(119, \cdot)\) None 0 2
360.6.bd \(\chi_{360}(59, \cdot)\) n/a 712 2
360.6.bf \(\chi_{360}(61, \cdot)\) n/a 480 2
360.6.bg \(\chi_{360}(191, \cdot)\) None 0 2
360.6.bi \(\chi_{360}(49, \cdot)\) n/a 180 2
360.6.bk \(\chi_{360}(229, \cdot)\) n/a 712 2
360.6.bm \(\chi_{360}(11, \cdot)\) n/a 480 2
360.6.bo \(\chi_{360}(43, \cdot)\) n/a 1424 4
360.6.br \(\chi_{360}(77, \cdot)\) n/a 1424 4
360.6.bs \(\chi_{360}(113, \cdot)\) n/a 360 4
360.6.bv \(\chi_{360}(7, \cdot)\) None 0 4

"n/a" means that newforms for that character have not been added to the database yet

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(360))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(360)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 16}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(10))\)\(^{\oplus 9}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(30))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(60))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(90))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(120))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_1(180))\)\(^{\oplus 2}\)