# Properties

 Label 6004.2 Level 6004 Weight 2 Dimension 648734 Nonzero newspaces 64 Sturm bound 4.4928e+06

## Defining parameters

 Level: $$N$$ = $$6004 = 2^{2} \cdot 19 \cdot 79$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$64$$ Sturm bound: $$4492800$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(6004))$$.

Total New Old
Modular forms 1130220 653966 476254
Cusp forms 1116181 648734 467447
Eisenstein series 14039 5232 8807

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(6004))$$

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
6004.2.a $$\chi_{6004}(1, \cdot)$$ 6004.2.a.a 1 1
6004.2.a.b 1
6004.2.a.c 1
6004.2.a.d 8
6004.2.a.e 24
6004.2.a.f 25
6004.2.a.g 27
6004.2.a.h 31
6004.2.f $$\chi_{6004}(3951, \cdot)$$ n/a 780 1
6004.2.g $$\chi_{6004}(3001, \cdot)$$ n/a 132 1
6004.2.h $$\chi_{6004}(5055, \cdot)$$ n/a 720 1
6004.2.i $$\chi_{6004}(1445, \cdot)$$ n/a 240 2
6004.2.j $$\chi_{6004}(1265, \cdot)$$ n/a 260 2
6004.2.k $$\chi_{6004}(4605, \cdot)$$ n/a 268 2
6004.2.l $$\chi_{6004}(1793, \cdot)$$ n/a 268 2
6004.2.q $$\chi_{6004}(791, \cdot)$$ n/a 1560 2
6004.2.r $$\chi_{6004}(1737, \cdot)$$ n/a 264 2
6004.2.s $$\chi_{6004}(419, \cdot)$$ n/a 1440 2
6004.2.t $$\chi_{6004}(1557, \cdot)$$ n/a 268 2
6004.2.u $$\chi_{6004}(2583, \cdot)$$ n/a 1592 2
6004.2.v $$\chi_{6004}(315, \cdot)$$ n/a 1592 2
6004.2.w $$\chi_{6004}(767, \cdot)$$ n/a 1592 2
6004.2.x $$\chi_{6004}(3105, \cdot)$$ n/a 268 2
6004.2.y $$\chi_{6004}(1319, \cdot)$$ n/a 1592 2
6004.2.bl $$\chi_{6004}(3579, \cdot)$$ n/a 1592 2
6004.2.bm $$\chi_{6004}(293, \cdot)$$ n/a 268 2
6004.2.bn $$\chi_{6004}(4131, \cdot)$$ n/a 1592 2
6004.2.bo $$\chi_{6004}(633, \cdot)$$ n/a 780 6
6004.2.bp $$\chi_{6004}(529, \cdot)$$ n/a 798 6
6004.2.bq $$\chi_{6004}(213, \cdot)$$ n/a 798 6
6004.2.br $$\chi_{6004}(381, \cdot)$$ n/a 1440 12
6004.2.bs $$\chi_{6004}(2157, \cdot)$$ n/a 798 6
6004.2.bu $$\chi_{6004}(371, \cdot)$$ n/a 4776 6
6004.2.bw $$\chi_{6004}(2315, \cdot)$$ n/a 4776 6
6004.2.bz $$\chi_{6004}(631, \cdot)$$ n/a 4776 6
6004.2.cc $$\chi_{6004}(687, \cdot)$$ n/a 4776 6
6004.2.ce $$\chi_{6004}(789, \cdot)$$ n/a 804 6
6004.2.cg $$\chi_{6004}(1739, \cdot)$$ n/a 4680 6
6004.2.ci $$\chi_{6004}(261, \cdot)$$ n/a 798 6
6004.2.cl $$\chi_{6004}(1051, \cdot)$$ n/a 4776 6
6004.2.cn $$\chi_{6004}(191, \cdot)$$ n/a 8640 12
6004.2.co $$\chi_{6004}(1177, \cdot)$$ n/a 1584 12
6004.2.cp $$\chi_{6004}(1443, \cdot)$$ n/a 9552 12
6004.2.cu $$\chi_{6004}(45, \cdot)$$ n/a 3216 24
6004.2.cv $$\chi_{6004}(49, \cdot)$$ n/a 3216 24
6004.2.cw $$\chi_{6004}(125, \cdot)$$ n/a 3168 24
6004.2.cx $$\chi_{6004}(761, \cdot)$$ n/a 2880 24
6004.2.cy $$\chi_{6004}(183, \cdot)$$ n/a 19104 24
6004.2.cz $$\chi_{6004}(145, \cdot)$$ n/a 3216 24
6004.2.da $$\chi_{6004}(235, \cdot)$$ n/a 19104 24
6004.2.dn $$\chi_{6004}(31, \cdot)$$ n/a 19104 24
6004.2.do $$\chi_{6004}(217, \cdot)$$ n/a 3216 24
6004.2.dp $$\chi_{6004}(7, \cdot)$$ n/a 19104 24
6004.2.dq $$\chi_{6004}(387, \cdot)$$ n/a 19104 24
6004.2.dr $$\chi_{6004}(151, \cdot)$$ n/a 19104 24
6004.2.ds $$\chi_{6004}(37, \cdot)$$ n/a 3216 24
6004.2.dt $$\chi_{6004}(39, \cdot)$$ n/a 17280 24
6004.2.du $$\chi_{6004}(69, \cdot)$$ n/a 3168 24
6004.2.dv $$\chi_{6004}(179, \cdot)$$ n/a 19104 24
6004.2.ea $$\chi_{6004}(5, \cdot)$$ n/a 9576 72
6004.2.eb $$\chi_{6004}(9, \cdot)$$ n/a 9576 72
6004.2.ec $$\chi_{6004}(101, \cdot)$$ n/a 9648 72
6004.2.ee $$\chi_{6004}(63, \cdot)$$ n/a 57312 72
6004.2.eh $$\chi_{6004}(29, \cdot)$$ n/a 9576 72
6004.2.ej $$\chi_{6004}(67, \cdot)$$ n/a 57312 72
6004.2.el $$\chi_{6004}(33, \cdot)$$ n/a 9648 72
6004.2.en $$\chi_{6004}(155, \cdot)$$ n/a 57312 72
6004.2.eq $$\chi_{6004}(175, \cdot)$$ n/a 57312 72
6004.2.et $$\chi_{6004}(35, \cdot)$$ n/a 57312 72
6004.2.ev $$\chi_{6004}(51, \cdot)$$ n/a 57312 72
6004.2.ex $$\chi_{6004}(345, \cdot)$$ n/a 9576 72

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

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(6004))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(6004)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(79))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(158))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(316))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(1501))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(3002))$$$$^{\oplus 2}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ ($$1 + 2 T + 3 T^{2}$$)($$1 + 3 T^{2}$$)($$1 - T + 3 T^{2}$$)($$( 1 + 3 T^{2} )^{8}$$)
$5$ ($$1 + T + 5 T^{2}$$)($$1 - T + 5 T^{2}$$)($$1 + 3 T + 5 T^{2}$$)($$1 - 4 T + 25 T^{2} - 84 T^{3} + 337 T^{4} - 948 T^{5} + 2874 T^{6} - 6880 T^{7} + 17138 T^{8} - 34400 T^{9} + 71850 T^{10} - 118500 T^{11} + 210625 T^{12} - 262500 T^{13} + 390625 T^{14} - 312500 T^{15} + 390625 T^{16}$$)
$7$ ($$1 + 5 T + 7 T^{2}$$)($$1 + 4 T + 7 T^{2}$$)($$1 + T + 7 T^{2}$$)($$1 - T + 13 T^{2} - 16 T^{3} + 96 T^{4} - 40 T^{5} + 411 T^{6} + 679 T^{7} + 1814 T^{8} + 4753 T^{9} + 20139 T^{10} - 13720 T^{11} + 230496 T^{12} - 268912 T^{13} + 1529437 T^{14} - 823543 T^{15} + 5764801 T^{16}$$)
$11$ ($$1 - T + 11 T^{2}$$)($$1 - 2 T + 11 T^{2}$$)($$1 + 11 T^{2}$$)($$1 + 7 T + 45 T^{2} + 240 T^{3} + 1092 T^{4} + 4810 T^{5} + 19147 T^{6} + 69981 T^{7} + 249270 T^{8} + 769791 T^{9} + 2316787 T^{10} + 6402110 T^{11} + 15987972 T^{12} + 38652240 T^{13} + 79720245 T^{14} + 136410197 T^{15} + 214358881 T^{16}$$)
$13$ ($$1 + 2 T + 13 T^{2}$$)($$1 + 4 T + 13 T^{2}$$)($$1 - 5 T + 13 T^{2}$$)($$1 + 12 T + 104 T^{2} + 670 T^{3} + 3932 T^{4} + 20094 T^{5} + 93464 T^{6} + 382864 T^{7} + 1450726 T^{8} + 4977232 T^{9} + 15795416 T^{10} + 44146518 T^{11} + 112301852 T^{12} + 248766310 T^{13} + 501988136 T^{14} + 752982204 T^{15} + 815730721 T^{16}$$)
$17$ ($$1 - 3 T + 17 T^{2}$$)($$1 - 4 T + 17 T^{2}$$)($$1 + 17 T^{2}$$)($$1 + 7 T + 103 T^{2} + 548 T^{3} + 4706 T^{4} + 20574 T^{5} + 133017 T^{6} + 495463 T^{7} + 2645882 T^{8} + 8422871 T^{9} + 38441913 T^{10} + 101080062 T^{11} + 393049826 T^{12} + 778081636 T^{13} + 2486169607 T^{14} + 2872370711 T^{15} + 6975757441 T^{16}$$)
$19$ ($$1 + T$$)($$1 + T$$)($$1 - T$$)($$( 1 - T )^{8}$$)
$23$ ($$1 - 8 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 + 6 T + 23 T^{2}$$)($$1 - 10 T + 148 T^{2} - 1150 T^{3} + 10460 T^{4} - 65118 T^{5} + 443372 T^{6} - 2270570 T^{7} + 12421126 T^{8} - 52223110 T^{9} + 234543788 T^{10} - 792290706 T^{11} + 2927136860 T^{12} - 7401794450 T^{13} + 21909311572 T^{14} - 34048254470 T^{15} + 78310985281 T^{16}$$)
$29$ ($$1 + 6 T + 29 T^{2}$$)($$1 + T + 29 T^{2}$$)($$1 - 6 T + 29 T^{2}$$)($$1 - 5 T + 123 T^{2} - 632 T^{3} + 8076 T^{4} - 40332 T^{5} + 367881 T^{6} - 1687507 T^{7} + 12353870 T^{8} - 48937703 T^{9} + 309387921 T^{10} - 983657148 T^{11} + 5712001356 T^{12} - 12963046168 T^{13} + 73163268483 T^{14} - 86249381545 T^{15} + 500246412961 T^{16}$$)
$31$ ($$1 + 31 T^{2}$$)($$1 - 5 T + 31 T^{2}$$)($$1 + 10 T + 31 T^{2}$$)($$1 - 3 T + 85 T^{2} - 104 T^{3} + 4332 T^{4} - 2254 T^{5} + 173379 T^{6} + 53815 T^{7} + 5357270 T^{8} + 1668265 T^{9} + 166617219 T^{10} - 67148914 T^{11} + 4000692972 T^{12} - 2977431704 T^{13} + 75437812885 T^{14} - 82537842333 T^{15} + 852891037441 T^{16}$$)
$37$ ($$1 + 4 T + 37 T^{2}$$)($$1 - 3 T + 37 T^{2}$$)($$1 - 2 T + 37 T^{2}$$)($$1 + 15 T + 237 T^{2} + 2302 T^{3} + 23122 T^{4} + 175446 T^{5} + 1383461 T^{6} + 8836885 T^{7} + 59177890 T^{8} + 326964745 T^{9} + 1893958109 T^{10} + 8886866238 T^{11} + 43334350642 T^{12} + 159629789014 T^{13} + 608077158933 T^{14} + 1423978156995 T^{15} + 3512479453921 T^{16}$$)
$41$ ($$1 + 4 T + 41 T^{2}$$)($$1 - 7 T + 41 T^{2}$$)($$1 - 6 T + 41 T^{2}$$)($$1 + 5 T + 191 T^{2} + 866 T^{3} + 17988 T^{4} + 72014 T^{5} + 1116991 T^{6} + 3913103 T^{7} + 51879078 T^{8} + 160437223 T^{9} + 1877661871 T^{10} + 4963276894 T^{11} + 50829788868 T^{12} + 100331470066 T^{13} + 907269910031 T^{14} + 973771369405 T^{15} + 7984925229121 T^{16}$$)
$43$ ($$1 + T + 43 T^{2}$$)($$1 + 3 T + 43 T^{2}$$)($$1 + 4 T + 43 T^{2}$$)($$1 - 10 T + 295 T^{2} - 2540 T^{3} + 39711 T^{4} - 293262 T^{5} + 3191662 T^{6} - 19896490 T^{7} + 167715516 T^{8} - 855549070 T^{9} + 5901383038 T^{10} - 23316381834 T^{11} + 135764006511 T^{12} - 373401445220 T^{13} + 1864802099455 T^{14} - 2718186111070 T^{15} + 11688200277601 T^{16}$$)
$47$ ($$1 + 3 T + 47 T^{2}$$)($$1 - 3 T + 47 T^{2}$$)($$1 + 9 T + 47 T^{2}$$)($$1 - 18 T + 263 T^{2} - 2560 T^{3} + 25259 T^{4} - 188914 T^{5} + 1472786 T^{6} - 9491518 T^{7} + 71064236 T^{8} - 446101346 T^{9} + 3253384274 T^{10} - 19613618222 T^{11} + 123255862379 T^{12} - 587123217920 T^{13} + 2834933631527 T^{14} - 9119216168334 T^{15} + 23811286661761 T^{16}$$)
$53$ ($$1 - 14 T + 53 T^{2}$$)($$1 + 9 T + 53 T^{2}$$)($$1 - 6 T + 53 T^{2}$$)($$1 - 9 T + 181 T^{2} - 464 T^{3} + 10156 T^{4} + 7868 T^{5} + 802565 T^{6} - 1836179 T^{7} + 63746742 T^{8} - 97317487 T^{9} + 2254405085 T^{10} + 1171364236 T^{11} + 80135725036 T^{12} - 194042708752 T^{13} + 4011749364349 T^{14} - 10572400258533 T^{15} + 62259690411361 T^{16}$$)
$59$ ($$1 - 6 T + 59 T^{2}$$)($$1 + 6 T + 59 T^{2}$$)($$1 - 3 T + 59 T^{2}$$)($$1 - 8 T + 252 T^{2} - 1240 T^{3} + 26964 T^{4} - 107256 T^{5} + 2224100 T^{6} - 9475496 T^{7} + 154140598 T^{8} - 559054264 T^{9} + 7742092100 T^{10} - 22028130024 T^{11} + 326732522004 T^{12} - 886506130760 T^{13} + 10629494477532 T^{14} - 19909211878552 T^{15} + 146830437604321 T^{16}$$)
$61$ ($$1 - 9 T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)($$1 - 8 T + 61 T^{2}$$)($$1 - 13 T + 413 T^{2} - 4122 T^{3} + 76238 T^{4} - 621696 T^{5} + 8486267 T^{6} - 57502613 T^{7} + 627276258 T^{8} - 3507659393 T^{9} + 31577399507 T^{10} - 141113179776 T^{11} + 1055579226158 T^{12} - 3481425952722 T^{13} + 21277914611093 T^{14} - 40855656868273 T^{15} + 191707312997281 T^{16}$$)
$67$ ($$1 - 2 T + 67 T^{2}$$)($$1 + 7 T + 67 T^{2}$$)($$1 - 2 T + 67 T^{2}$$)($$1 - 21 T + 647 T^{2} - 9526 T^{3} + 166524 T^{4} - 1867294 T^{5} + 23315863 T^{6} - 205726833 T^{7} + 1978152066 T^{8} - 13783697811 T^{9} + 104664909007 T^{10} - 561612945322 T^{11} + 3355645273404 T^{12} - 12861291769282 T^{13} + 58526573263343 T^{14} - 127274943711783 T^{15} + 406067677556641 T^{16}$$)
$71$ ($$1 - 16 T + 71 T^{2}$$)($$1 - 6 T + 71 T^{2}$$)($$1 - 9 T + 71 T^{2}$$)($$1 - 44 T + 1282 T^{2} - 26938 T^{3} + 456972 T^{4} - 6426094 T^{5} + 77190686 T^{6} - 800022320 T^{7} + 7221053478 T^{8} - 56801584720 T^{9} + 389118248126 T^{10} - 2299969729634 T^{11} + 11612426689932 T^{12} - 48602330257238 T^{13} + 164224563986722 T^{14} - 400185286969204 T^{15} + 645753531245761 T^{16}$$)
$73$ ($$1 - T + 73 T^{2}$$)($$1 + 13 T + 73 T^{2}$$)($$1 - 2 T + 73 T^{2}$$)($$1 + 20 T + 577 T^{2} + 9036 T^{3} + 146493 T^{4} + 1823236 T^{5} + 21290890 T^{6} + 213010104 T^{7} + 1939271878 T^{8} + 15549737592 T^{9} + 113459152810 T^{10} + 709269799012 T^{11} + 4160143518813 T^{12} + 18732274914348 T^{13} + 87319848568753 T^{14} + 220947970381940 T^{15} + 806460091894081 T^{16}$$)
$79$ ($$1 - T$$)($$1 + T$$)($$1 - T$$)($$( 1 - T )^{8}$$)
$83$ ($$1 - 4 T + 83 T^{2}$$)($$1 - 8 T + 83 T^{2}$$)($$1 + 83 T^{2}$$)($$1 + 4 T + 576 T^{2} + 2392 T^{3} + 149900 T^{4} + 606992 T^{5} + 23200192 T^{6} + 84755532 T^{7} + 2349690630 T^{8} + 7034709156 T^{9} + 159826122688 T^{10} + 347070134704 T^{11} + 7114002317900 T^{12} + 9422185218056 T^{13} + 188317655060544 T^{14} + 108544203958508 T^{15} + 2252292232139041 T^{16}$$)
$89$ ($$1 - 6 T + 89 T^{2}$$)($$1 - 12 T + 89 T^{2}$$)($$1 - 15 T + 89 T^{2}$$)($$1 + 10 T + 432 T^{2} + 4376 T^{3} + 97892 T^{4} + 937012 T^{5} + 14548976 T^{6} + 124904378 T^{7} + 1524844726 T^{8} + 11116489642 T^{9} + 115242438896 T^{10} + 660564412628 T^{11} + 6141963455972 T^{12} + 24435844148824 T^{13} + 214695917695152 T^{14} + 442313348955290 T^{15} + 3936588805702081 T^{16}$$)
$97$ ($$1 + 8 T + 97 T^{2}$$)($$1 - 16 T + 97 T^{2}$$)($$1 + 13 T + 97 T^{2}$$)($$1 + 22 T + 760 T^{2} + 11450 T^{3} + 229228 T^{4} + 2655822 T^{5} + 39640072 T^{6} + 375560034 T^{7} + 4585904678 T^{8} + 36429323298 T^{9} + 372973437448 T^{10} + 2423897032206 T^{11} + 20293390025068 T^{12} + 98325045942650 T^{13} + 633058723746040 T^{14} + 1777562258518486 T^{15} + 7837433594376961 T^{16}$$)