Properties

Label 8004.2.a.d
Level 8004
Weight 2
Character orbit 8004.a
Self dual yes
Analytic conductor 63.912
Analytic rank 1
Dimension 8
CM no
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 8 x^{6} + 19 x^{5} + 19 x^{4} - 35 x^{3} - 10 x^{2} + 18 x - 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} + ( -1 - \beta_{7} ) q^{5} + ( \beta_{6} + \beta_{7} ) q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + ( -1 - \beta_{7} ) q^{5} + ( \beta_{6} + \beta_{7} ) q^{7} + q^{9} + ( -\beta_{3} + \beta_{7} ) q^{11} + ( -1 + \beta_{4} ) q^{13} + ( -1 - \beta_{7} ) q^{15} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{17} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{19} + ( \beta_{6} + \beta_{7} ) q^{21} + q^{23} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{25} + q^{27} + q^{29} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + ( -\beta_{3} + \beta_{7} ) q^{33} + ( -3 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{35} + ( -2 - \beta_{2} - \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{37} + ( -1 + \beta_{4} ) q^{39} + ( -1 - 4 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{7} ) q^{41} + ( -1 - \beta_{1} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{43} + ( -1 - \beta_{7} ) q^{45} + ( -2 + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{7} ) q^{47} + ( -3 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{49} + ( -1 + 2 \beta_{1} + \beta_{3} - \beta_{4} - \beta_{6} ) q^{51} + ( -1 - 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{53} + ( -3 + \beta_{2} + 2 \beta_{3} - \beta_{6} - 2 \beta_{7} ) q^{55} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{57} + ( 2 - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 4 \beta_{6} + 2 \beta_{7} ) q^{59} + ( -\beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{61} + ( \beta_{6} + \beta_{7} ) q^{63} + ( 2 + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{65} + ( -1 - 2 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} ) q^{67} + q^{69} + ( 3 \beta_{1} + \beta_{2} + 3 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{71} + ( -5 + 5 \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{6} + 3 \beta_{7} ) q^{73} + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{7} ) q^{75} + ( \beta_{1} - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{77} + ( -2 + 4 \beta_{1} - \beta_{2} - 3 \beta_{4} + \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{79} + q^{81} + ( 2 - 2 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + 7 \beta_{6} + 4 \beta_{7} ) q^{83} + ( -\beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} ) q^{85} + q^{87} + ( -2 + \beta_{1} + \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{89} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{91} + ( -\beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( 1 + 2 \beta_{1} - 2 \beta_{5} + 3 \beta_{7} ) q^{95} + ( 3 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - 4 \beta_{6} - 3 \beta_{7} ) q^{97} + ( -\beta_{3} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} - 5q^{5} - 4q^{7} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} - 5q^{5} - 4q^{7} + 8q^{9} - 5q^{11} - 4q^{13} - 5q^{15} - 3q^{17} - 5q^{19} - 4q^{21} + 8q^{23} - 5q^{25} + 8q^{27} + 8q^{29} - 2q^{31} - 5q^{33} - 15q^{35} - 10q^{37} - 4q^{39} - 11q^{41} - 7q^{43} - 5q^{45} - 14q^{47} - 18q^{49} - 3q^{51} - 15q^{53} - 17q^{55} - 5q^{57} + 4q^{59} + q^{61} - 4q^{63} - 5q^{67} + 8q^{69} - q^{71} - 21q^{73} - 5q^{75} - 8q^{79} + 8q^{81} + 3q^{83} + 8q^{87} - 20q^{89} - 7q^{91} - 2q^{93} - 3q^{95} - 7q^{97} - 5q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 8 x^{6} + 19 x^{5} + 19 x^{4} - 35 x^{3} - 10 x^{2} + 18 x - 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{7} - 13 \nu^{6} - 46 \nu^{5} + 79 \nu^{4} + 131 \nu^{3} - 131 \nu^{2} - 108 \nu + 50 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -5 \nu^{7} + 13 \nu^{6} + 46 \nu^{5} - 79 \nu^{4} - 131 \nu^{3} + 133 \nu^{2} + 106 \nu - 56 \)\()/2\)
\(\beta_{4}\)\(=\)\( 5 \nu^{7} - 13 \nu^{6} - 45 \nu^{5} + 76 \nu^{4} + 126 \nu^{3} - 122 \nu^{2} - 102 \nu + 50 \)
\(\beta_{5}\)\(=\)\((\)\( -11 \nu^{7} + 29 \nu^{6} + 98 \nu^{5} - 171 \nu^{4} - 269 \nu^{3} + 273 \nu^{2} + 210 \nu - 104 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -13 \nu^{7} + 33 \nu^{6} + 120 \nu^{5} - 195 \nu^{4} - 339 \nu^{3} + 317 \nu^{2} + 272 \nu - 132 \)\()/2\)
\(\beta_{7}\)\(=\)\( 7 \nu^{7} - 18 \nu^{6} - 64 \nu^{5} + 107 \nu^{4} + 179 \nu^{3} - 175 \nu^{2} - 143 \nu + 71 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(5 \beta_{7} + 4 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + 10 \beta_{3} + 9 \beta_{2} + 16 \beta_{1} + 20\)
\(\nu^{5}\)\(=\)\(20 \beta_{7} + 17 \beta_{6} + 14 \beta_{5} + 12 \beta_{4} + 31 \beta_{3} + 26 \beta_{2} + 63 \beta_{1} + 48\)
\(\nu^{6}\)\(=\)\(77 \beta_{7} + 60 \beta_{6} + 48 \beta_{5} + 34 \beta_{4} + 120 \beta_{3} + 98 \beta_{2} + 211 \beta_{1} + 199\)
\(\nu^{7}\)\(=\)\(279 \beta_{7} + 223 \beta_{6} + 180 \beta_{5} + 141 \beta_{4} + 413 \beta_{3} + 326 \beta_{2} + 766 \beta_{1} + 633\)

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
−1.85152
0.386133
−1.03502
3.55127
−1.58311
1.24887
0.411238
1.87214
0 1.00000 0 −3.41642 0 2.05098 0 1.00000 0
1.2 0 1.00000 0 −2.77471 0 0.556399 0 1.00000 0
1.3 0 1.00000 0 −1.84132 0 −2.79101 0 1.00000 0
1.4 0 1.00000 0 −1.29538 0 2.07218 0 1.00000 0
1.5 0 1.00000 0 −0.270840 0 0.409136 0 1.00000 0
1.6 0 1.00000 0 0.138622 0 −3.33662 0 1.00000 0
1.7 0 1.00000 0 1.71910 0 0.210007 0 1.00000 0
1.8 0 1.00000 0 2.74094 0 −3.17108 0 1.00000 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
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 8004.2.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{8} \)
$5$ \( 1 + 5 T + 35 T^{2} + 123 T^{3} + 515 T^{4} + 1432 T^{5} + 4547 T^{6} + 10490 T^{7} + 27216 T^{8} + 52450 T^{9} + 113675 T^{10} + 179000 T^{11} + 321875 T^{12} + 384375 T^{13} + 546875 T^{14} + 390625 T^{15} + 390625 T^{16} \)
$7$ \( 1 + 4 T + 45 T^{2} + 155 T^{3} + 966 T^{4} + 2785 T^{5} + 12546 T^{6} + 30168 T^{7} + 107038 T^{8} + 211176 T^{9} + 614754 T^{10} + 955255 T^{11} + 2319366 T^{12} + 2605085 T^{13} + 5294205 T^{14} + 3294172 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 5 T + 76 T^{2} + 294 T^{3} + 2545 T^{4} + 7973 T^{5} + 50715 T^{6} + 131560 T^{7} + 672774 T^{8} + 1447160 T^{9} + 6136515 T^{10} + 10612063 T^{11} + 37261345 T^{12} + 47348994 T^{13} + 134638636 T^{14} + 97435855 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 4 T + 90 T^{2} + 319 T^{3} + 3697 T^{4} + 11374 T^{5} + 90515 T^{6} + 235529 T^{7} + 1442141 T^{8} + 3061877 T^{9} + 15297035 T^{10} + 24988678 T^{11} + 105590017 T^{12} + 118442467 T^{13} + 434412810 T^{14} + 250994068 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 3 T + 102 T^{2} + 230 T^{3} + 4831 T^{4} + 8730 T^{5} + 143335 T^{6} + 216129 T^{7} + 2915821 T^{8} + 3674193 T^{9} + 41423815 T^{10} + 42890490 T^{11} + 403489951 T^{12} + 326567110 T^{13} + 2462032038 T^{14} + 1231016019 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 5 T + 107 T^{2} + 478 T^{3} + 5359 T^{4} + 21939 T^{5} + 169074 T^{6} + 622661 T^{7} + 3759380 T^{8} + 11830559 T^{9} + 61035714 T^{10} + 150479601 T^{11} + 698390239 T^{12} + 1183575322 T^{13} + 5033909267 T^{14} + 4469358695 T^{15} + 16983563041 T^{16} \)
$23$ \( ( 1 - T )^{8} \)
$29$ \( ( 1 - T )^{8} \)
$31$ \( 1 + 2 T + 104 T^{2} + 429 T^{3} + 5928 T^{4} + 32756 T^{5} + 261266 T^{6} + 1458383 T^{7} + 9221706 T^{8} + 45209873 T^{9} + 251076626 T^{10} + 975833996 T^{11} + 5474632488 T^{12} + 12281905779 T^{13} + 92300382824 T^{14} + 55025228222 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 10 T + 251 T^{2} + 1913 T^{3} + 27402 T^{4} + 167849 T^{5} + 1781631 T^{6} + 9069749 T^{7} + 78490558 T^{8} + 335580713 T^{9} + 2439052839 T^{10} + 8502055397 T^{11} + 51355759722 T^{12} + 132654989741 T^{13} + 643997328659 T^{14} + 949318771330 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 11 T + 217 T^{2} + 2015 T^{3} + 24755 T^{4} + 190244 T^{5} + 1764337 T^{6} + 11438832 T^{7} + 86678840 T^{8} + 468992112 T^{9} + 2965850497 T^{10} + 13111806724 T^{11} + 69951713555 T^{12} + 233450245015 T^{13} + 1030772620297 T^{14} + 2142297012691 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 7 T + 258 T^{2} + 1386 T^{3} + 30118 T^{4} + 127901 T^{5} + 2161035 T^{6} + 7532761 T^{7} + 108671872 T^{8} + 323908723 T^{9} + 3995753715 T^{10} + 10169024807 T^{11} + 102967448518 T^{12} + 203753701998 T^{13} + 1630911666642 T^{14} + 1902730277749 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 14 T + 330 T^{2} + 3314 T^{3} + 42963 T^{4} + 335929 T^{5} + 3171577 T^{6} + 20936253 T^{7} + 167081468 T^{8} + 984003891 T^{9} + 7006013593 T^{10} + 34877156567 T^{11} + 209645734803 T^{12} + 760049353198 T^{13} + 3557141058570 T^{14} + 7092723686482 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 + 15 T + 297 T^{2} + 2791 T^{3} + 29831 T^{4} + 178666 T^{5} + 1370849 T^{6} + 5070700 T^{7} + 50414860 T^{8} + 268747100 T^{9} + 3850714841 T^{10} + 26599258082 T^{11} + 235380938711 T^{12} + 1167183620963 T^{13} + 6582815255313 T^{14} + 17620667097555 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 4 T + 245 T^{2} - 1104 T^{3} + 32870 T^{4} - 142016 T^{5} + 3010387 T^{6} - 11639812 T^{7} + 205110130 T^{8} - 686748908 T^{9} + 10479157147 T^{10} - 29167104064 T^{11} + 398297656070 T^{12} - 789276426096 T^{13} + 10334230742045 T^{14} - 9954605939276 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 - T + 282 T^{2} - 746 T^{3} + 40915 T^{4} - 129123 T^{5} + 4080190 T^{6} - 11958870 T^{7} + 292906840 T^{8} - 729491070 T^{9} + 15182386990 T^{10} - 29308467663 T^{11} + 566502584515 T^{12} - 630068840546 T^{13} + 14528745569802 T^{14} - 3142742836021 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 5 T + 367 T^{2} + 2067 T^{3} + 65226 T^{4} + 359097 T^{5} + 7477626 T^{6} + 36309399 T^{7} + 596860648 T^{8} + 2432729733 T^{9} + 33567063114 T^{10} + 108003091011 T^{11} + 1314377018346 T^{12} + 2790708596169 T^{13} + 33198226256023 T^{14} + 30303558026615 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 + T + 309 T^{2} - 403 T^{3} + 50633 T^{4} - 107515 T^{5} + 5709802 T^{6} - 13531994 T^{7} + 465330730 T^{8} - 960771574 T^{9} + 28783111882 T^{10} - 38480801165 T^{11} + 1286669644073 T^{12} - 727104428453 T^{13} + 39582987731589 T^{14} + 9095120158391 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 21 T + 479 T^{2} + 6828 T^{3} + 103348 T^{4} + 1162895 T^{5} + 13471003 T^{6} + 125077854 T^{7} + 1189125482 T^{8} + 9130683342 T^{9} + 71786974987 T^{10} + 452385924215 T^{11} + 2934901410868 T^{12} + 14154932837004 T^{13} + 72489094392431 T^{14} + 231995368901037 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 8 T + 441 T^{2} + 3855 T^{3} + 87828 T^{4} + 828813 T^{5} + 10853967 T^{6} + 103733333 T^{7} + 975633828 T^{8} + 8194933307 T^{9} + 67739608047 T^{10} + 408637132707 T^{11} + 3420907714068 T^{12} + 11862052418145 T^{13} + 107201567884761 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 3 T + 339 T^{2} - 265 T^{3} + 59257 T^{4} + 2766 T^{5} + 7528439 T^{6} + 900678 T^{7} + 722055524 T^{8} + 74756274 T^{9} + 51863416271 T^{10} + 1581562842 T^{11} + 2812237727497 T^{12} - 1043845770395 T^{13} + 110832786572091 T^{14} - 81408152968881 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 + 20 T + 264 T^{2} + 3299 T^{3} + 52221 T^{4} + 576266 T^{5} + 5870883 T^{6} + 60574359 T^{7} + 659088331 T^{8} + 5391117951 T^{9} + 46503264243 T^{10} + 406249665754 T^{11} + 3276462567261 T^{12} + 18421812122251 T^{13} + 131203060813704 T^{14} + 884626697910580 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 7 T + 592 T^{2} + 3692 T^{3} + 165275 T^{4} + 908555 T^{5} + 28552838 T^{6} + 135175014 T^{7} + 3332566836 T^{8} + 13111976358 T^{9} + 268653652742 T^{10} + 829213617515 T^{11} + 14631676917275 T^{12} + 31704460228844 T^{13} + 493119426917968 T^{14} + 565587991346791 T^{15} + 7837433594376961 T^{16} \)
show more
show less