Properties

Label 315.8
Level 315
Weight 8
Dimension 16692
Nonzero newspaces 30
Sturm bound 55296
Trace bound 9

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Defining parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 30 \)
Sturm bound: \(55296\)
Trace bound: \(9\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(315))\).

Total New Old
Modular forms 24576 16964 7612
Cusp forms 23808 16692 7116
Eisenstein series 768 272 496

Trace form

\( 16692 q - 82 q^{2} + 88 q^{3} + 330 q^{4} - 1206 q^{5} - 4972 q^{6} + 1810 q^{7} + 25986 q^{8} + 12584 q^{9} + O(q^{10}) \) \( 16692 q - 82 q^{2} + 88 q^{3} + 330 q^{4} - 1206 q^{5} - 4972 q^{6} + 1810 q^{7} + 25986 q^{8} + 12584 q^{9} - 8590 q^{10} - 61972 q^{11} + 4040 q^{12} - 10224 q^{13} + 138138 q^{14} + 58444 q^{15} - 116082 q^{16} - 312880 q^{17} - 59288 q^{18} + 337444 q^{19} + 222910 q^{20} + 42708 q^{21} + 601708 q^{22} - 357504 q^{23} - 1077516 q^{24} - 332898 q^{25} + 336836 q^{26} + 1921012 q^{27} + 2256498 q^{28} + 1546264 q^{29} - 1109074 q^{30} - 1274452 q^{31} - 692954 q^{32} + 1221200 q^{33} + 475668 q^{34} - 1719316 q^{35} + 316996 q^{36} - 5684632 q^{37} - 2434132 q^{38} + 1734928 q^{39} - 1251798 q^{40} + 2582212 q^{41} + 6116268 q^{42} + 13042736 q^{43} + 11584720 q^{44} - 3974788 q^{45} - 22182500 q^{46} - 16182412 q^{47} - 17412292 q^{48} - 934532 q^{49} + 2865026 q^{50} + 7475576 q^{51} + 22242432 q^{52} + 10820828 q^{53} + 21903920 q^{54} - 2581316 q^{55} + 12595074 q^{56} + 2062504 q^{57} + 3973908 q^{58} + 15538796 q^{59} + 8415242 q^{60} + 27844328 q^{61} + 3927240 q^{62} - 40972164 q^{63} - 8416002 q^{64} - 62362384 q^{65} - 34764692 q^{66} - 72215348 q^{67} - 4956836 q^{68} + 47267232 q^{69} + 76165026 q^{70} + 141202168 q^{71} + 116540496 q^{72} - 602148 q^{73} - 37447820 q^{74} - 83460104 q^{75} - 45528092 q^{76} - 149538384 q^{77} - 82626064 q^{78} - 60529804 q^{79} - 25695326 q^{80} - 17988280 q^{81} + 90487120 q^{82} + 175927032 q^{83} + 193421892 q^{84} + 37124180 q^{85} + 224876840 q^{86} - 4121516 q^{87} - 18634980 q^{88} - 214056492 q^{89} - 341732282 q^{90} - 101603892 q^{91} - 202700712 q^{92} - 36361452 q^{93} - 34456500 q^{94} + 131328758 q^{95} + 333545776 q^{96} - 107410224 q^{97} + 196512970 q^{98} + 76039336 q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(315))\)

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
315.8.a \(\chi_{315}(1, \cdot)\) 315.8.a.a 1 1
315.8.a.b 1
315.8.a.c 2
315.8.a.d 2
315.8.a.e 3
315.8.a.f 4
315.8.a.g 4
315.8.a.h 4
315.8.a.i 4
315.8.a.j 4
315.8.a.k 4
315.8.a.l 4
315.8.a.m 5
315.8.a.n 6
315.8.a.o 6
315.8.a.p 8
315.8.a.q 8
315.8.b \(\chi_{315}(251, \cdot)\) 315.8.b.a 36 1
315.8.b.b 36
315.8.d \(\chi_{315}(64, \cdot)\) n/a 106 1
315.8.g \(\chi_{315}(314, \cdot)\) n/a 112 1
315.8.i \(\chi_{315}(106, \cdot)\) n/a 336 2
315.8.j \(\chi_{315}(46, \cdot)\) n/a 188 2
315.8.k \(\chi_{315}(16, \cdot)\) n/a 448 2
315.8.l \(\chi_{315}(121, \cdot)\) n/a 448 2
315.8.m \(\chi_{315}(8, \cdot)\) n/a 168 2
315.8.p \(\chi_{315}(118, \cdot)\) n/a 276 2
315.8.r \(\chi_{315}(184, \cdot)\) n/a 664 2
315.8.t \(\chi_{315}(101, \cdot)\) n/a 448 2
315.8.u \(\chi_{315}(59, \cdot)\) n/a 664 2
315.8.z \(\chi_{315}(104, \cdot)\) n/a 664 2
315.8.bb \(\chi_{315}(89, \cdot)\) n/a 224 2
315.8.be \(\chi_{315}(236, \cdot)\) n/a 448 2
315.8.bf \(\chi_{315}(109, \cdot)\) n/a 276 2
315.8.bh \(\chi_{315}(169, \cdot)\) n/a 504 2
315.8.bj \(\chi_{315}(26, \cdot)\) n/a 152 2
315.8.bl \(\chi_{315}(41, \cdot)\) n/a 448 2
315.8.bo \(\chi_{315}(4, \cdot)\) n/a 664 2
315.8.bq \(\chi_{315}(164, \cdot)\) n/a 664 2
315.8.bs \(\chi_{315}(52, \cdot)\) n/a 1328 4
315.8.bv \(\chi_{315}(23, \cdot)\) n/a 1328 4
315.8.bx \(\chi_{315}(2, \cdot)\) n/a 1328 4
315.8.bz \(\chi_{315}(73, \cdot)\) n/a 552 4
315.8.cb \(\chi_{315}(13, \cdot)\) n/a 1328 4
315.8.cc \(\chi_{315}(92, \cdot)\) n/a 1008 4
315.8.ce \(\chi_{315}(53, \cdot)\) n/a 448 4
315.8.cg \(\chi_{315}(157, \cdot)\) n/a 1328 4

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

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(315))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(315)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(35))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(45))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(105))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(315))\)\(^{\oplus 1}\)