Properties

Label 1216.4
Level 1216
Weight 4
Dimension 76954
Nonzero newspaces 24
Sturm bound 368640
Trace bound 49

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Defining parameters

Level: \( N \) = \( 1216 = 2^{6} \cdot 19 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 24 \)
Sturm bound: \(368640\)
Trace bound: \(49\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(1216))\).

Total New Old
Modular forms 139536 77702 61834
Cusp forms 136944 76954 59990
Eisenstein series 2592 748 1844

Trace form

\( 76954q - 128q^{2} - 96q^{3} - 128q^{4} - 128q^{5} - 128q^{6} - 92q^{7} - 128q^{8} - 106q^{9} + O(q^{10}) \) \( 76954q - 128q^{2} - 96q^{3} - 128q^{4} - 128q^{5} - 128q^{6} - 92q^{7} - 128q^{8} - 106q^{9} - 128q^{10} - 56q^{11} - 128q^{12} - 272q^{13} - 128q^{14} - 340q^{15} - 128q^{16} - 432q^{17} - 128q^{18} - 126q^{19} - 272q^{20} - 104q^{21} - 1072q^{22} - 92q^{23} - 2128q^{24} - 262q^{25} - 48q^{26} + 276q^{27} + 1392q^{28} + 672q^{29} + 4512q^{30} + 628q^{31} + 2352q^{32} + 1868q^{33} + 1872q^{34} + 860q^{35} + 1632q^{36} + 912q^{37} - 576q^{38} - 200q^{39} - 3408q^{40} - 2208q^{41} - 6448q^{42} - 1768q^{43} - 2128q^{44} - 3096q^{45} - 128q^{46} - 1980q^{47} - 128q^{48} - 2350q^{49} + 5584q^{50} - 9044q^{51} + 6496q^{52} - 944q^{53} + 3328q^{54} - 1244q^{55} - 912q^{56} + 908q^{57} - 5024q^{58} + 8824q^{59} - 9920q^{60} + 2032q^{61} - 6112q^{62} + 15260q^{63} - 12224q^{64} + 4196q^{65} - 11200q^{66} + 11952q^{67} - 4256q^{68} + 1144q^{69} - 4160q^{70} + 804q^{71} + 1168q^{72} - 2208q^{73} + 5136q^{74} - 14832q^{75} + 5816q^{76} - 6520q^{77} + 3856q^{78} - 20252q^{79} - 8656q^{80} - 13862q^{81} - 14048q^{82} - 5216q^{83} - 8416q^{84} - 3440q^{85} + 912q^{86} - 92q^{87} + 6112q^{88} + 6976q^{89} + 18592q^{90} + 6572q^{91} + 25104q^{92} + 16624q^{93} + 17728q^{94} + 6792q^{95} + 25568q^{96} + 18608q^{97} + 24080q^{98} + 9392q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(1216))\)

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 the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1216.4.a \(\chi_{1216}(1, \cdot)\) 1216.4.a.a 1 1
1216.4.a.b 1
1216.4.a.c 1
1216.4.a.d 1
1216.4.a.e 1
1216.4.a.f 1
1216.4.a.g 2
1216.4.a.h 2
1216.4.a.i 2
1216.4.a.j 2
1216.4.a.k 2
1216.4.a.l 2
1216.4.a.m 2
1216.4.a.n 2
1216.4.a.o 2
1216.4.a.p 2
1216.4.a.q 3
1216.4.a.r 3
1216.4.a.s 3
1216.4.a.t 3
1216.4.a.u 3
1216.4.a.v 3
1216.4.a.w 3
1216.4.a.x 3
1216.4.a.y 5
1216.4.a.z 5
1216.4.a.ba 5
1216.4.a.bb 5
1216.4.a.bc 5
1216.4.a.bd 5
1216.4.a.be 7
1216.4.a.bf 7
1216.4.a.bg 7
1216.4.a.bh 7
1216.4.b \(\chi_{1216}(607, \cdot)\) n/a 120 1
1216.4.c \(\chi_{1216}(609, \cdot)\) n/a 108 1
1216.4.h \(\chi_{1216}(1215, \cdot)\) n/a 118 1
1216.4.i \(\chi_{1216}(577, \cdot)\) n/a 236 2
1216.4.k \(\chi_{1216}(305, \cdot)\) n/a 216 2
1216.4.m \(\chi_{1216}(303, \cdot)\) n/a 236 2
1216.4.n \(\chi_{1216}(255, \cdot)\) n/a 236 2
1216.4.s \(\chi_{1216}(31, \cdot)\) n/a 240 2
1216.4.t \(\chi_{1216}(353, \cdot)\) n/a 240 2
1216.4.u \(\chi_{1216}(151, \cdot)\) None 0 4
1216.4.v \(\chi_{1216}(153, \cdot)\) None 0 4
1216.4.y \(\chi_{1216}(321, \cdot)\) n/a 708 6
1216.4.z \(\chi_{1216}(49, \cdot)\) n/a 472 4
1216.4.bb \(\chi_{1216}(335, \cdot)\) n/a 472 4
1216.4.bd \(\chi_{1216}(77, \cdot)\) n/a 3456 8
1216.4.be \(\chi_{1216}(75, \cdot)\) n/a 3824 8
1216.4.bj \(\chi_{1216}(161, \cdot)\) n/a 720 6
1216.4.bl \(\chi_{1216}(223, \cdot)\) n/a 720 6
1216.4.bm \(\chi_{1216}(127, \cdot)\) n/a 708 6
1216.4.bq \(\chi_{1216}(121, \cdot)\) None 0 8
1216.4.br \(\chi_{1216}(103, \cdot)\) None 0 8
1216.4.bs \(\chi_{1216}(15, \cdot)\) n/a 1416 12
1216.4.bu \(\chi_{1216}(17, \cdot)\) n/a 1416 12
1216.4.bw \(\chi_{1216}(27, \cdot)\) n/a 7648 16
1216.4.bx \(\chi_{1216}(45, \cdot)\) n/a 7648 16
1216.4.ca \(\chi_{1216}(9, \cdot)\) None 0 24
1216.4.cb \(\chi_{1216}(71, \cdot)\) None 0 24
1216.4.cg \(\chi_{1216}(5, \cdot)\) n/a 22944 48
1216.4.ch \(\chi_{1216}(3, \cdot)\) n/a 22944 48

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(1216))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(1216)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(19))\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(64))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(76))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(304))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(608))\)\(^{\oplus 2}\)