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Properties

Label 2016.2
Level 2016
Weight 2
Dimension 46746
Nonzero newspaces 60
Sturm bound 442368
Trace bound 40

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Defining parameters

Level: \( N \) = \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 60 \)
Sturm bound: \(442368\)
Trace bound: \(40\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(2016))\).

Total New Old
Modular forms 113664 47646 66018
Cusp forms 107521 46746 60775
Eisenstein series 6143 900 5243

Trace form

\( 46746 q - 48 q^{2} - 48 q^{3} - 48 q^{4} - 44 q^{5} - 64 q^{6} - 44 q^{7} - 120 q^{8} - 96 q^{9} + O(q^{10}) \) \( 46746 q - 48 q^{2} - 48 q^{3} - 48 q^{4} - 44 q^{5} - 64 q^{6} - 44 q^{7} - 120 q^{8} - 96 q^{9} - 160 q^{10} - 42 q^{11} - 64 q^{12} - 60 q^{13} - 76 q^{14} - 132 q^{15} - 88 q^{16} - 64 q^{17} - 64 q^{18} - 138 q^{19} - 80 q^{20} - 96 q^{21} - 144 q^{22} - 90 q^{23} - 64 q^{24} - 138 q^{25} - 8 q^{26} - 72 q^{27} - 160 q^{28} - 164 q^{29} - 32 q^{30} - 106 q^{31} - 8 q^{32} - 152 q^{33} - 24 q^{34} - 66 q^{35} - 112 q^{36} - 76 q^{37} + 160 q^{38} - 36 q^{39} + 184 q^{40} + 32 q^{41} - 52 q^{43} + 224 q^{44} + 32 q^{45} + 48 q^{46} + 42 q^{47} + 144 q^{48} - 2 q^{49} + 216 q^{50} + 16 q^{51} + 192 q^{52} + 36 q^{53} + 112 q^{54} - 60 q^{55} + 88 q^{56} - 144 q^{57} + 216 q^{58} + 46 q^{59} + 48 q^{60} - 108 q^{61} + 24 q^{62} - 6 q^{63} - 288 q^{64} - 4 q^{65} - 64 q^{66} - 22 q^{67} + 40 q^{68} + 64 q^{69} - 48 q^{70} + 16 q^{71} - 64 q^{72} - 304 q^{73} - 80 q^{74} + 176 q^{75} - 48 q^{76} - 44 q^{77} - 208 q^{78} + 86 q^{79} - 296 q^{80} + 128 q^{81} - 464 q^{82} + 336 q^{83} - 192 q^{84} - 184 q^{85} - 424 q^{86} + 268 q^{87} - 376 q^{88} + 48 q^{89} - 352 q^{90} + 72 q^{91} - 632 q^{92} - 64 q^{93} - 248 q^{94} + 478 q^{95} - 336 q^{96} + 64 q^{97} - 112 q^{98} + 164 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(2016))\)

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
2016.2.a \(\chi_{2016}(1, \cdot)\) 2016.2.a.a 1 1
2016.2.a.b 1
2016.2.a.c 1
2016.2.a.d 1
2016.2.a.e 1
2016.2.a.f 1
2016.2.a.g 1
2016.2.a.h 1
2016.2.a.i 1
2016.2.a.j 1
2016.2.a.k 1
2016.2.a.l 1
2016.2.a.m 1
2016.2.a.n 1
2016.2.a.o 2
2016.2.a.p 2
2016.2.a.q 2
2016.2.a.r 2
2016.2.a.s 2
2016.2.a.t 2
2016.2.a.u 2
2016.2.a.v 2
2016.2.b \(\chi_{2016}(1567, \cdot)\) 2016.2.b.a 8 1
2016.2.b.b 8
2016.2.b.c 8
2016.2.b.d 16
2016.2.c \(\chi_{2016}(1009, \cdot)\) 2016.2.c.a 2 1
2016.2.c.b 4
2016.2.c.c 4
2016.2.c.d 4
2016.2.c.e 8
2016.2.c.f 8
2016.2.h \(\chi_{2016}(575, \cdot)\) 2016.2.h.a 4 1
2016.2.h.b 4
2016.2.h.c 4
2016.2.h.d 4
2016.2.h.e 8
2016.2.i \(\chi_{2016}(881, \cdot)\) 2016.2.i.a 8 1
2016.2.i.b 24
2016.2.j \(\chi_{2016}(1583, \cdot)\) 2016.2.j.a 24 1
2016.2.k \(\chi_{2016}(1889, \cdot)\) 2016.2.k.a 16 1
2016.2.k.b 16
2016.2.p \(\chi_{2016}(559, \cdot)\) 2016.2.p.a 2 1
2016.2.p.b 4
2016.2.p.c 4
2016.2.p.d 4
2016.2.p.e 4
2016.2.p.f 4
2016.2.p.g 16
2016.2.q \(\chi_{2016}(1537, \cdot)\) n/a 192 2
2016.2.r \(\chi_{2016}(673, \cdot)\) n/a 144 2
2016.2.s \(\chi_{2016}(289, \cdot)\) 2016.2.s.a 2 2
2016.2.s.b 2
2016.2.s.c 2
2016.2.s.d 2
2016.2.s.e 2
2016.2.s.f 2
2016.2.s.g 2
2016.2.s.h 2
2016.2.s.i 2
2016.2.s.j 2
2016.2.s.k 2
2016.2.s.l 2
2016.2.s.m 2
2016.2.s.n 2
2016.2.s.o 4
2016.2.s.p 4
2016.2.s.q 4
2016.2.s.r 4
2016.2.s.s 4
2016.2.s.t 4
2016.2.s.u 6
2016.2.s.v 6
2016.2.s.w 8
2016.2.s.x 8
2016.2.t \(\chi_{2016}(193, \cdot)\) n/a 192 2
2016.2.v \(\chi_{2016}(71, \cdot)\) None 0 2
2016.2.x \(\chi_{2016}(55, \cdot)\) None 0 2
2016.2.z \(\chi_{2016}(505, \cdot)\) None 0 2
2016.2.bb \(\chi_{2016}(377, \cdot)\) None 0 2
2016.2.be \(\chi_{2016}(1201, \cdot)\) n/a 184 2
2016.2.bf \(\chi_{2016}(31, \cdot)\) n/a 192 2
2016.2.bg \(\chi_{2016}(689, \cdot)\) n/a 184 2
2016.2.bh \(\chi_{2016}(95, \cdot)\) n/a 192 2
2016.2.bm \(\chi_{2016}(1231, \cdot)\) n/a 184 2
2016.2.bn \(\chi_{2016}(367, \cdot)\) n/a 184 2
2016.2.bs \(\chi_{2016}(271, \cdot)\) 2016.2.bs.a 12 2
2016.2.bs.b 32
2016.2.bs.c 32
2016.2.bt \(\chi_{2016}(1025, \cdot)\) 2016.2.bt.a 32 2
2016.2.bt.b 32
2016.2.bu \(\chi_{2016}(431, \cdot)\) 2016.2.bu.a 8 2
2016.2.bu.b 8
2016.2.bu.c 48
2016.2.bz \(\chi_{2016}(239, \cdot)\) n/a 144 2
2016.2.ca \(\chi_{2016}(257, \cdot)\) n/a 192 2
2016.2.cb \(\chi_{2016}(527, \cdot)\) n/a 184 2
2016.2.cc \(\chi_{2016}(545, \cdot)\) n/a 192 2
2016.2.ch \(\chi_{2016}(1247, \cdot)\) n/a 144 2
2016.2.ci \(\chi_{2016}(1265, \cdot)\) n/a 184 2
2016.2.cj \(\chi_{2016}(767, \cdot)\) n/a 192 2
2016.2.ck \(\chi_{2016}(209, \cdot)\) n/a 184 2
2016.2.cp \(\chi_{2016}(17, \cdot)\) 2016.2.cp.a 8 2
2016.2.cp.b 56
2016.2.cq \(\chi_{2016}(863, \cdot)\) 2016.2.cq.a 8 2
2016.2.cq.b 8
2016.2.cq.c 8
2016.2.cq.d 8
2016.2.cq.e 32
2016.2.cr \(\chi_{2016}(1297, \cdot)\) 2016.2.cr.a 8 2
2016.2.cr.b 8
2016.2.cr.c 12
2016.2.cr.d 16
2016.2.cr.e 32
2016.2.cs \(\chi_{2016}(703, \cdot)\) 2016.2.cs.a 16 2
2016.2.cs.b 16
2016.2.cs.c 16
2016.2.cs.d 32
2016.2.cx \(\chi_{2016}(223, \cdot)\) n/a 192 2
2016.2.cy \(\chi_{2016}(529, \cdot)\) n/a 184 2
2016.2.cz \(\chi_{2016}(607, \cdot)\) n/a 192 2
2016.2.da \(\chi_{2016}(337, \cdot)\) n/a 144 2
2016.2.df \(\chi_{2016}(929, \cdot)\) n/a 192 2
2016.2.dg \(\chi_{2016}(1103, \cdot)\) n/a 184 2
2016.2.dh \(\chi_{2016}(943, \cdot)\) n/a 184 2
2016.2.dk \(\chi_{2016}(125, \cdot)\) n/a 512 4
2016.2.dm \(\chi_{2016}(253, \cdot)\) n/a 480 4
2016.2.do \(\chi_{2016}(323, \cdot)\) n/a 384 4
2016.2.dq \(\chi_{2016}(307, \cdot)\) n/a 632 4
2016.2.ds \(\chi_{2016}(391, \cdot)\) None 0 4
2016.2.du \(\chi_{2016}(407, \cdot)\) None 0 4
2016.2.dw \(\chi_{2016}(457, \cdot)\) None 0 4
2016.2.dz \(\chi_{2016}(761, \cdot)\) None 0 4
2016.2.ea \(\chi_{2016}(89, \cdot)\) None 0 4
2016.2.ec \(\chi_{2016}(361, \cdot)\) None 0 4
2016.2.ef \(\chi_{2016}(25, \cdot)\) None 0 4
2016.2.eg \(\chi_{2016}(185, \cdot)\) None 0 4
2016.2.ei \(\chi_{2016}(599, \cdot)\) None 0 4
2016.2.ek \(\chi_{2016}(199, \cdot)\) None 0 4
2016.2.en \(\chi_{2016}(103, \cdot)\) None 0 4
2016.2.ep \(\chi_{2016}(23, \cdot)\) None 0 4
2016.2.eq \(\chi_{2016}(359, \cdot)\) None 0 4
2016.2.es \(\chi_{2016}(439, \cdot)\) None 0 4
2016.2.eu \(\chi_{2016}(41, \cdot)\) None 0 4
2016.2.ew \(\chi_{2016}(169, \cdot)\) None 0 4
2016.2.ez \(\chi_{2016}(85, \cdot)\) n/a 2304 8
2016.2.fb \(\chi_{2016}(293, \cdot)\) n/a 3040 8
2016.2.fc \(\chi_{2016}(11, \cdot)\) n/a 3040 8
2016.2.fg \(\chi_{2016}(19, \cdot)\) n/a 1264 8
2016.2.fh \(\chi_{2016}(187, \cdot)\) n/a 3040 8
2016.2.fk \(\chi_{2016}(347, \cdot)\) n/a 3040 8
2016.2.fl \(\chi_{2016}(107, \cdot)\) n/a 1024 8
2016.2.fm \(\chi_{2016}(115, \cdot)\) n/a 3040 8
2016.2.fo \(\chi_{2016}(5, \cdot)\) n/a 3040 8
2016.2.fs \(\chi_{2016}(205, \cdot)\) n/a 3040 8
2016.2.ft \(\chi_{2016}(37, \cdot)\) n/a 1264 8
2016.2.fw \(\chi_{2016}(269, \cdot)\) n/a 1024 8
2016.2.fx \(\chi_{2016}(173, \cdot)\) n/a 3040 8
2016.2.fy \(\chi_{2016}(277, \cdot)\) n/a 3040 8
2016.2.gb \(\chi_{2016}(139, \cdot)\) n/a 3040 8
2016.2.gd \(\chi_{2016}(155, \cdot)\) n/a 2304 8

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(2016))\) into lower level spaces

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