Properties

Label 6004.2.a
Level 6004
Weight 2
Character orbit a
Rep. character \(\chi_{6004}(1,\cdot)\)
Character field \(\Q\)
Dimension 118
Newform subspaces 8
Sturm bound 1600
Trace bound 3

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Defining parameters

Level: \( N \) \(=\) \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6004.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(1600\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(6004))\).

Total New Old
Modular forms 806 118 688
Cusp forms 795 118 677
Eisenstein series 11 0 11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(19\)\(79\)FrickeDim.
\(-\)\(+\)\(+\)\(-\)\(32\)
\(-\)\(+\)\(-\)\(+\)\(26\)
\(-\)\(-\)\(+\)\(+\)\(27\)
\(-\)\(-\)\(-\)\(-\)\(33\)
Plus space\(+\)\(53\)
Minus space\(-\)\(65\)

Trace form

\( 118q - 4q^{3} + 2q^{5} - 2q^{7} + 126q^{9} + O(q^{10}) \) \( 118q - 4q^{3} + 2q^{5} - 2q^{7} + 126q^{9} + 2q^{11} + 12q^{13} + 4q^{15} + 2q^{17} + 2q^{19} + 4q^{21} - 12q^{23} + 120q^{25} + 8q^{27} + 12q^{29} - 28q^{31} - 44q^{33} - 2q^{35} - 8q^{37} - 8q^{39} - 4q^{41} - 6q^{43} + 2q^{45} + 14q^{47} + 132q^{49} + 20q^{51} + 4q^{53} - 18q^{55} - 12q^{59} + 30q^{61} + 2q^{63} + 24q^{65} + 4q^{67} + 4q^{69} - 12q^{71} - 10q^{73} - 12q^{75} - 6q^{77} + 134q^{81} + 26q^{85} + 16q^{87} - 24q^{89} - 20q^{91} - 28q^{93} - 2q^{95} - 4q^{97} - 14q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 2 19 79
6004.2.a.a \(1\) \(47.942\) \(\Q\) None \(0\) \(-2\) \(-1\) \(-5\) \(-\) \(+\) \(-\) \(q-2q^{3}-q^{5}-5q^{7}+q^{9}+q^{11}-2q^{13}+\cdots\)
6004.2.a.b \(1\) \(47.942\) \(\Q\) None \(0\) \(0\) \(1\) \(-4\) \(-\) \(+\) \(+\) \(q+q^{5}-4q^{7}-3q^{9}+2q^{11}-4q^{13}+\cdots\)
6004.2.a.c \(1\) \(47.942\) \(\Q\) None \(0\) \(1\) \(-3\) \(-1\) \(-\) \(-\) \(-\) \(q+q^{3}-3q^{5}-q^{7}-2q^{9}+5q^{13}+\cdots\)
6004.2.a.d \(8\) \(47.942\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(4\) \(1\) \(-\) \(-\) \(-\) \(q+\beta _{1}q^{5}-\beta _{4}q^{7}-3q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots\)
6004.2.a.e \(24\) \(47.942\) None \(0\) \(1\) \(9\) \(2\) \(-\) \(-\) \(-\)
6004.2.a.f \(25\) \(47.942\) None \(0\) \(4\) \(-8\) \(2\) \(-\) \(+\) \(-\)
6004.2.a.g \(27\) \(47.942\) None \(0\) \(-4\) \(-10\) \(-8\) \(-\) \(-\) \(+\)
6004.2.a.h \(31\) \(47.942\) None \(0\) \(-4\) \(10\) \(11\) \(-\) \(+\) \(+\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(6004))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(6004)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(79))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(158))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(316))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1501))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(3002))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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