Properties

Label 6004.2.a.d
Level 6004
Weight 2
Character orbit 6004.a
Self dual yes
Analytic conductor 47.942
Analytic rank 0
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(47.9421813736\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 4 x^{7} - 15 x^{6} + 56 x^{5} + 87 x^{4} - 248 x^{3} - 241 x^{2} + 340 x + 248\)
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{5} -\beta_{4} q^{7} -3 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{5} -\beta_{4} q^{7} -3 q^{9} + ( -1 + \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{11} + ( -2 + \beta_{1} - \beta_{2} - \beta_{4} - \beta_{6} ) q^{13} + ( -1 - \beta_{5} + \beta_{6} ) q^{17} + q^{19} + ( 2 + \beta_{2} - \beta_{3} + \beta_{7} ) q^{23} + ( \beta_{1} + \beta_{2} - \beta_{5} + \beta_{6} ) q^{25} + ( 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( -1 + 2 \beta_{1} - 3 \beta_{2} - \beta_{4} + \beta_{5} - 2 \beta_{7} ) q^{31} + ( -2 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + 2 \beta_{7} ) q^{37} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{6} ) q^{41} + ( 1 - \beta_{2} + \beta_{3} + \beta_{4} ) q^{43} -3 \beta_{1} q^{45} + ( 2 - 2 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{47} + ( 4 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{49} + ( -\beta_{1} + 2 \beta_{2} + 3 \beta_{3} + \beta_{5} ) q^{53} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{55} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{7} ) q^{59} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + 3 \beta_{4} q^{63} + ( 2 - 3 \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{65} + ( 3 - \beta_{1} - \beta_{5} + \beta_{6} ) q^{67} + ( 6 - \beta_{1} + \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{71} + ( -3 - \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} + ( 1 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{77} + q^{79} + 9 q^{81} + ( \beta_{2} - \beta_{3} - 2 \beta_{5} + \beta_{7} ) q^{83} + ( -3 + 2 \beta_{3} - \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{85} + ( -2 - \beta_{1} + 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} ) q^{89} + ( 6 + 5 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{91} + \beta_{1} q^{95} + ( -4 - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} - 2 \beta_{7} ) q^{97} + ( 3 - 3 \beta_{1} + 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} + 3 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{5} + q^{7} - 24q^{9} + O(q^{10}) \) \( 8q + 4q^{5} + q^{7} - 24q^{9} - 7q^{11} - 12q^{13} - 7q^{17} + 8q^{19} + 10q^{23} + 6q^{25} + 5q^{29} + 3q^{31} - 11q^{35} - 15q^{37} - 5q^{41} + 10q^{43} - 12q^{45} + 18q^{47} + 31q^{49} + 9q^{53} - 17q^{55} + 8q^{59} + 13q^{61} - 3q^{63} + 4q^{65} + 21q^{67} + 44q^{71} - 20q^{73} + 15q^{77} + 8q^{79} + 72q^{81} - 4q^{83} - 9q^{85} - 10q^{89} + 48q^{91} + 4q^{95} - 22q^{97} + 21q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{7} - 15 x^{6} + 56 x^{5} + 87 x^{4} - 248 x^{3} - 241 x^{2} + 340 x + 248\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} - 4 \nu^{5} - 10 \nu^{4} + 36 \nu^{3} + 37 \nu^{2} - 68 \nu - 56 \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 3 \nu^{6} + 14 \nu^{5} - 26 \nu^{4} - 73 \nu^{3} + 31 \nu^{2} + 140 \nu + 72 \)\()/16\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} - 4 \nu^{6} - 10 \nu^{5} + 40 \nu^{4} + 25 \nu^{3} - 104 \nu^{2} + 4 \nu + 64 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{7} - 5 \nu^{6} - 8 \nu^{5} + 56 \nu^{4} + 7 \nu^{3} - 171 \nu^{2} + 32 \nu + 112 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( \nu^{7} - 6 \nu^{6} - 4 \nu^{5} + 66 \nu^{4} - 29 \nu^{3} - 200 \nu^{2} + 92 \nu + 128 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( \nu^{7} - 7 \nu^{6} + 76 \nu^{4} - 69 \nu^{3} - 229 \nu^{2} + 188 \nu + 160 \)\()/8\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{2} + \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 9 \beta_{1} + 4\)
\(\nu^{4}\)\(=\)\(-6 \beta_{7} + 21 \beta_{6} - 15 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 11 \beta_{2} + 17 \beta_{1} + 37\)
\(\nu^{5}\)\(=\)\(-36 \beta_{7} + 84 \beta_{6} - 52 \beta_{5} + 10 \beta_{4} + 12 \beta_{3} + 14 \beta_{2} + 97 \beta_{1} + 68\)
\(\nu^{6}\)\(=\)\(-132 \beta_{7} + 365 \beta_{6} - 249 \beta_{5} + 60 \beta_{4} + 88 \beta_{3} + 137 \beta_{2} + 265 \beta_{1} + 369\)
\(\nu^{7}\)\(=\)\(-598 \beta_{7} + 1464 \beta_{6} - 970 \beta_{5} + 268 \beta_{4} + 312 \beta_{3} + 352 \beta_{2} + 1225 \beta_{1} + 1032\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.49694
−1.94482
−1.76571
−0.654712
1.60188
2.45095
2.82169
3.98766
0 0 0 −2.49694 0 4.08498 0 −3.00000 0
1.2 0 0 0 −1.94482 0 −0.969069 0 −3.00000 0
1.3 0 0 0 −1.76571 0 2.40269 0 −3.00000 0
1.4 0 0 0 −0.654712 0 −2.24646 0 −3.00000 0
1.5 0 0 0 1.60188 0 −2.96099 0 −3.00000 0
1.6 0 0 0 2.45095 0 −5.04197 0 −3.00000 0
1.7 0 0 0 2.82169 0 4.85844 0 −3.00000 0
1.8 0 0 0 3.98766 0 0.872374 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(19\) \(-1\)
\(79\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6004.2.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6004.2.a.d 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6004))\):

\( T_{3} \)
\(T_{5}^{8} - \cdots\)

Hecke characteristic polynomials

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