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Properties

Label 2013.2
Level 2013
Weight 2
Dimension 108623
Nonzero newspaces 84
Sturm bound 595200
Trace bound 16

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Defining parameters

Level: \( N \) = \( 2013 = 3 \cdot 11 \cdot 61 \)
Weight: \( k \) = \( 2 \)
Nonzero newspaces: \( 84 \)
Sturm bound: \(595200\)
Trace bound: \(16\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_1(2013))\).

Total New Old
Modular forms 151200 110743 40457
Cusp forms 146401 108623 37778
Eisenstein series 4799 2120 2679

Trace form

\( 108623 q + 9 q^{2} - 227 q^{3} - 439 q^{4} + 18 q^{5} - 231 q^{6} - 456 q^{7} + 5 q^{8} - 247 q^{9} + O(q^{10}) \) \( 108623 q + 9 q^{2} - 227 q^{3} - 439 q^{4} + 18 q^{5} - 231 q^{6} - 456 q^{7} + 5 q^{8} - 247 q^{9} - 466 q^{10} + 3 q^{11} - 539 q^{12} - 438 q^{13} + 12 q^{14} - 252 q^{15} - 487 q^{16} - 6 q^{17} - 251 q^{18} - 460 q^{19} + 26 q^{20} - 256 q^{21} - 571 q^{22} + 32 q^{23} - 235 q^{24} - 447 q^{25} + 26 q^{26} - 197 q^{27} - 392 q^{28} + 50 q^{29} - 186 q^{30} - 404 q^{31} + 149 q^{32} - 217 q^{33} - 958 q^{34} + 64 q^{35} - 219 q^{36} - 426 q^{37} + 80 q^{38} - 238 q^{39} - 410 q^{40} + 6 q^{41} - 228 q^{42} - 468 q^{43} - 59 q^{44} - 592 q^{45} - 504 q^{46} - 116 q^{47} - 487 q^{48} - 789 q^{49} - 241 q^{50} - 406 q^{51} - 1326 q^{52} - 98 q^{53} - 251 q^{54} - 782 q^{55} - 600 q^{56} - 400 q^{57} - 850 q^{58} - 120 q^{59} - 694 q^{60} - 917 q^{61} - 552 q^{62} - 196 q^{63} - 1239 q^{64} - 108 q^{65} - 301 q^{66} - 1416 q^{67} - 542 q^{68} - 418 q^{69} - 1048 q^{70} - 24 q^{71} - 355 q^{72} - 658 q^{73} - 138 q^{74} - 367 q^{75} - 820 q^{76} - 56 q^{77} - 494 q^{78} - 480 q^{79} + 138 q^{80} - 247 q^{81} - 302 q^{82} - 8 q^{83} - 192 q^{84} - 396 q^{85} + 116 q^{86} - 270 q^{87} - 495 q^{88} + 110 q^{89} - 306 q^{90} - 344 q^{91} + 164 q^{92} - 294 q^{93} - 328 q^{94} + 60 q^{95} - 171 q^{96} - 406 q^{97} + 73 q^{98} - 297 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_1(2013))\)

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
2013.2.a \(\chi_{2013}(1, \cdot)\) 2013.2.a.a 11 1
2013.2.a.b 11
2013.2.a.c 12
2013.2.a.d 12
2013.2.a.e 13
2013.2.a.f 13
2013.2.a.g 13
2013.2.a.h 14
2013.2.b \(\chi_{2013}(2012, \cdot)\) n/a 244 1
2013.2.e \(\chi_{2013}(1585, \cdot)\) n/a 104 1
2013.2.f \(\chi_{2013}(428, \cdot)\) n/a 240 1
2013.2.i \(\chi_{2013}(562, \cdot)\) n/a 204 2
2013.2.j \(\chi_{2013}(538, \cdot)\) n/a 248 2
2013.2.m \(\chi_{2013}(782, \cdot)\) n/a 416 2
2013.2.n \(\chi_{2013}(70, \cdot)\) n/a 496 4
2013.2.o \(\chi_{2013}(34, \cdot)\) n/a 400 4
2013.2.p \(\chi_{2013}(367, \cdot)\) n/a 480 4
2013.2.q \(\chi_{2013}(58, \cdot)\) n/a 496 4
2013.2.r \(\chi_{2013}(1522, \cdot)\) n/a 496 4
2013.2.s \(\chi_{2013}(400, \cdot)\) n/a 496 4
2013.2.v \(\chi_{2013}(230, \cdot)\) n/a 488 2
2013.2.w \(\chi_{2013}(1024, \cdot)\) n/a 212 2
2013.2.z \(\chi_{2013}(197, \cdot)\) n/a 488 2
2013.2.ba \(\chi_{2013}(163, \cdot)\) n/a 496 4
2013.2.bd \(\chi_{2013}(1040, \cdot)\) n/a 976 4
2013.2.bf \(\chi_{2013}(314, \cdot)\) n/a 976 4
2013.2.bl \(\chi_{2013}(497, \cdot)\) n/a 976 4
2013.2.bn \(\chi_{2013}(62, \cdot)\) n/a 960 4
2013.2.bo \(\chi_{2013}(131, \cdot)\) n/a 976 4
2013.2.bq \(\chi_{2013}(1559, \cdot)\) n/a 976 4
2013.2.bu \(\chi_{2013}(41, \cdot)\) n/a 976 4
2013.2.bv \(\chi_{2013}(895, \cdot)\) n/a 496 4
2013.2.bx \(\chi_{2013}(487, \cdot)\) n/a 496 4
2013.2.ca \(\chi_{2013}(64, \cdot)\) n/a 496 4
2013.2.cc \(\chi_{2013}(430, \cdot)\) n/a 416 4
2013.2.cd \(\chi_{2013}(296, \cdot)\) n/a 976 4
2013.2.cf \(\chi_{2013}(1064, \cdot)\) n/a 976 4
2013.2.ci \(\chi_{2013}(182, \cdot)\) n/a 976 4
2013.2.ck \(\chi_{2013}(149, \cdot)\) n/a 976 4
2013.2.cl \(\chi_{2013}(796, \cdot)\) n/a 496 4
2013.2.cp \(\chi_{2013}(95, \cdot)\) n/a 976 4
2013.2.cq \(\chi_{2013}(650, \cdot)\) n/a 824 4
2013.2.ct \(\chi_{2013}(406, \cdot)\) n/a 496 4
2013.2.cu \(\chi_{2013}(757, \cdot)\) n/a 992 8
2013.2.cv \(\chi_{2013}(25, \cdot)\) n/a 992 8
2013.2.cw \(\chi_{2013}(361, \cdot)\) n/a 992 8
2013.2.cx \(\chi_{2013}(169, \cdot)\) n/a 992 8
2013.2.cy \(\chi_{2013}(199, \cdot)\) n/a 816 8
2013.2.cz \(\chi_{2013}(16, \cdot)\) n/a 992 8
2013.2.da \(\chi_{2013}(191, \cdot)\) n/a 1952 8
2013.2.dd \(\chi_{2013}(28, \cdot)\) n/a 992 8
2013.2.df \(\chi_{2013}(85, \cdot)\) n/a 992 8
2013.2.dh \(\chi_{2013}(236, \cdot)\) n/a 1952 8
2013.2.di \(\chi_{2013}(389, \cdot)\) n/a 1952 8
2013.2.dj \(\chi_{2013}(377, \cdot)\) n/a 1952 8
2013.2.dn \(\chi_{2013}(23, \cdot)\) n/a 1664 8
2013.2.do \(\chi_{2013}(175, \cdot)\) n/a 992 8
2013.2.ds \(\chi_{2013}(172, \cdot)\) n/a 992 8
2013.2.dt \(\chi_{2013}(211, \cdot)\) n/a 992 8
2013.2.du \(\chi_{2013}(145, \cdot)\) n/a 992 8
2013.2.dw \(\chi_{2013}(38, \cdot)\) n/a 1952 8
2013.2.dy \(\chi_{2013}(83, \cdot)\) n/a 1952 8
2013.2.ec \(\chi_{2013}(202, \cdot)\) n/a 992 8
2013.2.ed \(\chi_{2013}(431, \cdot)\) n/a 1952 8
2013.2.ef \(\chi_{2013}(380, \cdot)\) n/a 1952 8
2013.2.ei \(\chi_{2013}(107, \cdot)\) n/a 1952 8
2013.2.ek \(\chi_{2013}(65, \cdot)\) n/a 1952 8
2013.2.el \(\chi_{2013}(100, \cdot)\) n/a 848 8
2013.2.en \(\chi_{2013}(4, \cdot)\) n/a 992 8
2013.2.eq \(\chi_{2013}(136, \cdot)\) n/a 992 8
2013.2.es \(\chi_{2013}(49, \cdot)\) n/a 992 8
2013.2.et \(\chi_{2013}(161, \cdot)\) n/a 1952 8
2013.2.ex \(\chi_{2013}(266, \cdot)\) n/a 1952 8
2013.2.ez \(\chi_{2013}(164, \cdot)\) n/a 1952 8
2013.2.fa \(\chi_{2013}(74, \cdot)\) n/a 1952 8
2013.2.fc \(\chi_{2013}(260, \cdot)\) n/a 1952 8
2013.2.fi \(\chi_{2013}(134, \cdot)\) n/a 1952 8
2013.2.fk \(\chi_{2013}(167, \cdot)\) n/a 1952 8
2013.2.fn \(\chi_{2013}(97, \cdot)\) n/a 992 8
2013.2.fp \(\chi_{2013}(26, \cdot)\) n/a 3904 16
2013.2.fr \(\chi_{2013}(7, \cdot)\) n/a 1984 16
2013.2.fs \(\chi_{2013}(40, \cdot)\) n/a 1984 16
2013.2.ft \(\chi_{2013}(139, \cdot)\) n/a 1984 16
2013.2.fx \(\chi_{2013}(10, \cdot)\) n/a 1984 16
2013.2.fy \(\chi_{2013}(254, \cdot)\) n/a 3296 16
2013.2.gc \(\chi_{2013}(185, \cdot)\) n/a 3904 16
2013.2.gd \(\chi_{2013}(212, \cdot)\) n/a 3904 16
2013.2.ge \(\chi_{2013}(71, \cdot)\) n/a 3904 16
2013.2.gg \(\chi_{2013}(79, \cdot)\) n/a 1984 16
2013.2.gi \(\chi_{2013}(250, \cdot)\) n/a 1984 16
2013.2.gl \(\chi_{2013}(152, \cdot)\) n/a 3904 16

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

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_1(2013))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_1(2013)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_1(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(61))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(183))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_1(671))\)\(^{\oplus 2}\)