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Properties

Label 2016.4
Level 2016
Weight 4
Dimension 141138
Nonzero newspaces 60
Sturm bound 884736
Trace bound 40

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Defining parameters

Level: \( N \) = \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 4 \)
Nonzero newspaces: \( 60 \)
Sturm bound: \(884736\)
Trace bound: \(40\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_1(2016))\).

Total New Old
Modular forms 334848 142038 192810
Cusp forms 328704 141138 187566
Eisenstein series 6144 900 5244

Trace form

\( 141138 q - 48 q^{2} - 48 q^{3} - 48 q^{4} - 52 q^{5} - 64 q^{6} - 60 q^{7} - 120 q^{8} - 96 q^{9} + O(q^{10}) \) \( 141138 q - 48 q^{2} - 48 q^{3} - 48 q^{4} - 52 q^{5} - 64 q^{6} - 60 q^{7} - 120 q^{8} - 96 q^{9} - 384 q^{10} - 42 q^{11} - 64 q^{12} + 188 q^{13} + 148 q^{14} - 228 q^{15} + 552 q^{16} - 272 q^{17} - 64 q^{18} - 10 q^{19} - 208 q^{20} + 192 q^{21} - 512 q^{22} - 186 q^{23} - 64 q^{24} + 598 q^{25} - 88 q^{26} + 216 q^{27} - 560 q^{28} - 172 q^{29} + 64 q^{30} + 1302 q^{31} - 1288 q^{32} - 1016 q^{33} - 1112 q^{34} + 414 q^{35} + 3920 q^{36} + 2220 q^{37} + 7616 q^{38} + 444 q^{39} + 3320 q^{40} - 1616 q^{41} - 1440 q^{42} - 4644 q^{43} - 8288 q^{44} - 4000 q^{45} - 8784 q^{46} - 3030 q^{47} - 9840 q^{48} + 502 q^{49} - 10056 q^{50} - 1232 q^{51} - 1920 q^{52} - 1860 q^{53} - 944 q^{54} + 2020 q^{55} + 2744 q^{56} + 6960 q^{57} + 9672 q^{58} + 5390 q^{59} + 12144 q^{60} + 10412 q^{61} + 8568 q^{62} + 666 q^{63} - 10272 q^{64} - 5108 q^{65} - 64 q^{66} - 5254 q^{67} - 3544 q^{68} - 8384 q^{69} - 1464 q^{70} - 7024 q^{71} - 64 q^{72} + 7904 q^{73} + 6896 q^{74} - 4048 q^{75} + 10192 q^{76} - 1996 q^{77} - 10288 q^{78} + 26902 q^{79} - 16552 q^{80} + 6080 q^{81} - 15184 q^{82} + 21936 q^{83} - 4224 q^{84} - 1736 q^{85} + 3880 q^{86} + 1516 q^{87} + 9928 q^{88} + 480 q^{89} + 18656 q^{90} - 13848 q^{91} + 36488 q^{92} - 64 q^{93} + 38312 q^{94} - 44962 q^{95} + 25776 q^{96} - 7248 q^{97} + 24976 q^{98} - 29596 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_1(2016))\)

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
2016.4.a \(\chi_{2016}(1, \cdot)\) 2016.4.a.a 1 1
2016.4.a.b 1
2016.4.a.c 1
2016.4.a.d 1
2016.4.a.e 1
2016.4.a.f 1
2016.4.a.g 2
2016.4.a.h 2
2016.4.a.i 2
2016.4.a.j 2
2016.4.a.k 2
2016.4.a.l 2
2016.4.a.m 2
2016.4.a.n 2
2016.4.a.o 2
2016.4.a.p 2
2016.4.a.q 2
2016.4.a.r 2
2016.4.a.s 3
2016.4.a.t 3
2016.4.a.u 3
2016.4.a.v 3
2016.4.a.w 3
2016.4.a.x 3
2016.4.a.y 3
2016.4.a.z 3
2016.4.a.ba 4
2016.4.a.bb 4
2016.4.a.bc 4
2016.4.a.bd 4
2016.4.a.be 5
2016.4.a.bf 5
2016.4.a.bg 5
2016.4.a.bh 5
2016.4.b \(\chi_{2016}(1567, \cdot)\) n/a 120 1
2016.4.c \(\chi_{2016}(1009, \cdot)\) 2016.4.c.a 8 1
2016.4.c.b 10
2016.4.c.c 16
2016.4.c.d 16
2016.4.c.e 20
2016.4.c.f 20
2016.4.h \(\chi_{2016}(575, \cdot)\) 2016.4.h.a 16 1
2016.4.h.b 16
2016.4.h.c 20
2016.4.h.d 20
2016.4.i \(\chi_{2016}(881, \cdot)\) 2016.4.i.a 8 1
2016.4.i.b 88
2016.4.j \(\chi_{2016}(1583, \cdot)\) 2016.4.j.a 72 1
2016.4.k \(\chi_{2016}(1889, \cdot)\) 2016.4.k.a 48 1
2016.4.k.b 48
2016.4.p \(\chi_{2016}(559, \cdot)\) n/a 118 1
2016.4.q \(\chi_{2016}(1537, \cdot)\) n/a 576 2
2016.4.r \(\chi_{2016}(673, \cdot)\) n/a 432 2
2016.4.s \(\chi_{2016}(289, \cdot)\) n/a 240 2
2016.4.t \(\chi_{2016}(193, \cdot)\) n/a 576 2
2016.4.v \(\chi_{2016}(71, \cdot)\) None 0 2
2016.4.x \(\chi_{2016}(55, \cdot)\) None 0 2
2016.4.z \(\chi_{2016}(505, \cdot)\) None 0 2
2016.4.bb \(\chi_{2016}(377, \cdot)\) None 0 2
2016.4.be \(\chi_{2016}(1201, \cdot)\) n/a 568 2
2016.4.bf \(\chi_{2016}(31, \cdot)\) n/a 576 2
2016.4.bg \(\chi_{2016}(689, \cdot)\) n/a 568 2
2016.4.bh \(\chi_{2016}(95, \cdot)\) n/a 576 2
2016.4.bm \(\chi_{2016}(1231, \cdot)\) n/a 568 2
2016.4.bn \(\chi_{2016}(367, \cdot)\) n/a 568 2
2016.4.bs \(\chi_{2016}(271, \cdot)\) n/a 236 2
2016.4.bt \(\chi_{2016}(1025, \cdot)\) n/a 192 2
2016.4.bu \(\chi_{2016}(431, \cdot)\) n/a 192 2
2016.4.bz \(\chi_{2016}(239, \cdot)\) n/a 432 2
2016.4.ca \(\chi_{2016}(257, \cdot)\) n/a 576 2
2016.4.cb \(\chi_{2016}(527, \cdot)\) n/a 568 2
2016.4.cc \(\chi_{2016}(545, \cdot)\) n/a 576 2
2016.4.ch \(\chi_{2016}(1247, \cdot)\) n/a 432 2
2016.4.ci \(\chi_{2016}(1265, \cdot)\) n/a 568 2
2016.4.cj \(\chi_{2016}(767, \cdot)\) n/a 576 2
2016.4.ck \(\chi_{2016}(209, \cdot)\) n/a 568 2
2016.4.cp \(\chi_{2016}(17, \cdot)\) n/a 192 2
2016.4.cq \(\chi_{2016}(863, \cdot)\) n/a 192 2
2016.4.cr \(\chi_{2016}(1297, \cdot)\) n/a 236 2
2016.4.cs \(\chi_{2016}(703, \cdot)\) n/a 240 2
2016.4.cx \(\chi_{2016}(223, \cdot)\) n/a 576 2
2016.4.cy \(\chi_{2016}(529, \cdot)\) n/a 568 2
2016.4.cz \(\chi_{2016}(607, \cdot)\) n/a 576 2
2016.4.da \(\chi_{2016}(337, \cdot)\) n/a 432 2
2016.4.df \(\chi_{2016}(929, \cdot)\) n/a 576 2
2016.4.dg \(\chi_{2016}(1103, \cdot)\) n/a 568 2
2016.4.dh \(\chi_{2016}(943, \cdot)\) n/a 568 2
2016.4.dk \(\chi_{2016}(125, \cdot)\) n/a 1536 4
2016.4.dm \(\chi_{2016}(253, \cdot)\) n/a 1440 4
2016.4.do \(\chi_{2016}(323, \cdot)\) n/a 1152 4
2016.4.dq \(\chi_{2016}(307, \cdot)\) n/a 1912 4
2016.4.ds \(\chi_{2016}(391, \cdot)\) None 0 4
2016.4.du \(\chi_{2016}(407, \cdot)\) None 0 4
2016.4.dw \(\chi_{2016}(457, \cdot)\) None 0 4
2016.4.dz \(\chi_{2016}(761, \cdot)\) None 0 4
2016.4.ea \(\chi_{2016}(89, \cdot)\) None 0 4
2016.4.ec \(\chi_{2016}(361, \cdot)\) None 0 4
2016.4.ef \(\chi_{2016}(25, \cdot)\) None 0 4
2016.4.eg \(\chi_{2016}(185, \cdot)\) None 0 4
2016.4.ei \(\chi_{2016}(599, \cdot)\) None 0 4
2016.4.ek \(\chi_{2016}(199, \cdot)\) None 0 4
2016.4.en \(\chi_{2016}(103, \cdot)\) None 0 4
2016.4.ep \(\chi_{2016}(23, \cdot)\) None 0 4
2016.4.eq \(\chi_{2016}(359, \cdot)\) None 0 4
2016.4.es \(\chi_{2016}(439, \cdot)\) None 0 4
2016.4.eu \(\chi_{2016}(41, \cdot)\) None 0 4
2016.4.ew \(\chi_{2016}(169, \cdot)\) None 0 4
2016.4.ez \(\chi_{2016}(85, \cdot)\) n/a 6912 8
2016.4.fb \(\chi_{2016}(293, \cdot)\) n/a 9184 8
2016.4.fc \(\chi_{2016}(11, \cdot)\) n/a 9184 8
2016.4.fg \(\chi_{2016}(19, \cdot)\) n/a 3824 8
2016.4.fh \(\chi_{2016}(187, \cdot)\) n/a 9184 8
2016.4.fk \(\chi_{2016}(347, \cdot)\) n/a 9184 8
2016.4.fl \(\chi_{2016}(107, \cdot)\) n/a 3072 8
2016.4.fm \(\chi_{2016}(115, \cdot)\) n/a 9184 8
2016.4.fo \(\chi_{2016}(5, \cdot)\) n/a 9184 8
2016.4.fs \(\chi_{2016}(205, \cdot)\) n/a 9184 8
2016.4.ft \(\chi_{2016}(37, \cdot)\) n/a 3824 8
2016.4.fw \(\chi_{2016}(269, \cdot)\) n/a 3072 8
2016.4.fx \(\chi_{2016}(173, \cdot)\) n/a 9184 8
2016.4.fy \(\chi_{2016}(277, \cdot)\) n/a 9184 8
2016.4.gb \(\chi_{2016}(139, \cdot)\) n/a 9184 8
2016.4.gd \(\chi_{2016}(155, \cdot)\) n/a 6912 8

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

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_1(2016))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_1(2016)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_1(6))\)\(^{\oplus 20}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(8))\)\(^{\oplus 18}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(9))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(12))\)\(^{\oplus 16}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 15}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(16))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(18))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(21))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(24))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(32))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(36))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(42))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(48))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(56))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(63))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(84))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(96))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(112))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(126))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(144))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(168))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(224))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(252))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(288))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(336))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(504))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(672))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_1(1008))\)\(^{\oplus 2}\)