Properties

Label 8016.2.a.y
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 8
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 23 x^{6} - 3 x^{5} + 163 x^{4} + 13 x^{3} - 418 x^{2} + 4 x + 269\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4008)
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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{3} q^{7} + q^{9} +O(q^{10})\) \( q + q^{3} + \beta_{1} q^{5} -\beta_{3} q^{7} + q^{9} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{11} + ( -1 - \beta_{2} - \beta_{5} ) q^{13} + \beta_{1} q^{15} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{17} + ( 2 \beta_{1} + \beta_{4} - \beta_{7} ) q^{19} -\beta_{3} q^{21} + ( -1 + \beta_{3} ) q^{23} + ( -\beta_{6} + \beta_{7} ) q^{25} + q^{27} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{29} + ( 3 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{31} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{33} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{35} + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( -1 - \beta_{2} - \beta_{5} ) q^{39} + ( -1 + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{41} + ( 2 \beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{43} + \beta_{1} q^{45} + ( 5 + 2 \beta_{2} + \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{47} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + 2 \beta_{6} ) q^{49} + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} ) q^{51} + ( 2 - \beta_{1} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{53} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} ) q^{55} + ( 2 \beta_{1} + \beta_{4} - \beta_{7} ) q^{57} + ( 2 - \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 2 \beta_{5} + \beta_{6} - 3 \beta_{7} ) q^{59} + ( -1 - \beta_{1} + 2 \beta_{4} - \beta_{5} ) q^{61} -\beta_{3} q^{63} + ( -4 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} + 2 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{65} + ( -3 + \beta_{1} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{67} + ( -1 + \beta_{3} ) q^{69} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{6} + \beta_{7} ) q^{71} + ( -5 + \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{73} + ( -\beta_{6} + \beta_{7} ) q^{75} + ( 4 + \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{6} ) q^{77} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} + \beta_{7} ) q^{79} + q^{81} + ( 4 - \beta_{2} + 2 \beta_{4} + 2 \beta_{6} ) q^{83} + ( 2 - \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{85} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} ) q^{87} + ( 2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - \beta_{7} ) q^{89} + ( 3 + \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + 3 \beta_{6} - \beta_{7} ) q^{91} + ( 3 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( 8 + \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{7} ) q^{95} + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} ) q^{97} + ( -\beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{3} + q^{7} + 8q^{9} + O(q^{10}) \) \( 8q + 8q^{3} + q^{7} + 8q^{9} + 3q^{11} - 8q^{13} - 7q^{17} + q^{21} - 9q^{23} + 6q^{25} + 8q^{27} + 17q^{29} + 23q^{31} + 3q^{33} + 15q^{35} + 8q^{37} - 8q^{39} - 8q^{41} + 2q^{43} + 34q^{47} + 5q^{49} - 7q^{51} + 12q^{53} + 7q^{55} + 16q^{59} - 2q^{61} + q^{63} - 14q^{65} - 21q^{67} - 9q^{69} + 29q^{71} - 38q^{73} + 6q^{75} + 20q^{77} + 12q^{79} + 8q^{81} + 32q^{83} + 23q^{85} + 17q^{87} + 11q^{89} + 5q^{91} + 23q^{93} + 67q^{95} + 8q^{97} + 3q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 23 x^{6} - 3 x^{5} + 163 x^{4} + 13 x^{3} - 418 x^{2} + 4 x + 269\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 185 \nu^{7} - 1337 \nu^{6} - 3411 \nu^{5} + 24599 \nu^{4} + 18970 \nu^{3} - 110475 \nu^{2} - 18219 \nu + 87357 \)\()/8453\)
\(\beta_{3}\)\(=\)\((\)\( -211 \nu^{7} - 714 \nu^{6} + 3799 \nu^{5} + 13889 \nu^{4} - 10670 \nu^{3} - 61564 \nu^{2} - 16368 \nu + 64354 \)\()/8453\)
\(\beta_{4}\)\(=\)\((\)\( -253 \nu^{7} - 776 \nu^{6} + 5076 \nu^{5} + 14290 \nu^{4} - 24572 \nu^{3} - 48820 \nu^{2} + 35699 \nu + 9019 \)\()/8453\)
\(\beta_{5}\)\(=\)\((\)\( -312 \nu^{7} + 747 \nu^{6} + 4656 \nu^{5} - 11512 \nu^{4} - 10289 \nu^{3} + 48782 \nu^{2} - 26206 \nu - 39128 \)\()/8453\)
\(\beta_{6}\)\(=\)\((\)\( 545 \nu^{7} + 402 \nu^{6} - 10734 \nu^{5} - 10235 \nu^{4} + 53600 \nu^{3} + 47164 \nu^{2} - 66009 \nu - 38506 \)\()/8453\)
\(\beta_{7}\)\(=\)\((\)\( 545 \nu^{7} + 402 \nu^{6} - 10734 \nu^{5} - 10235 \nu^{4} + 53600 \nu^{3} + 55617 \nu^{2} - 66009 \nu - 80771 \)\()/8453\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} + 5\)
\(\nu^{3}\)\(=\)\(3 \beta_{7} - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} - \beta_{2} + 9 \beta_{1}\)
\(\nu^{4}\)\(=\)\(14 \beta_{7} - 17 \beta_{6} - 5 \beta_{4} + 2 \beta_{2} + 2 \beta_{1} + 41\)
\(\nu^{5}\)\(=\)\(54 \beta_{7} - 59 \beta_{6} - 22 \beta_{5} - 36 \beta_{4} + 47 \beta_{3} - 18 \beta_{2} + 97 \beta_{1} + 12\)
\(\nu^{6}\)\(=\)\(191 \beta_{7} - 247 \beta_{6} - 4 \beta_{5} - 102 \beta_{4} + 9 \beta_{3} + 29 \beta_{2} + 61 \beta_{1} + 422\)
\(\nu^{7}\)\(=\)\(804 \beta_{7} - 902 \beta_{6} - 332 \beta_{5} - 531 \beta_{4} + 624 \beta_{3} - 240 \beta_{2} + 1139 \beta_{1} + 333\)

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
−3.24252
−2.57145
−2.12535
−0.938106
1.02139
1.94540
2.18690
3.72373
0 1.00000 0 −3.24252 0 3.21024 0 1.00000 0
1.2 0 1.00000 0 −2.57145 0 −1.34479 0 1.00000 0
1.3 0 1.00000 0 −2.12535 0 −2.08900 0 1.00000 0
1.4 0 1.00000 0 −0.938106 0 −4.96672 0 1.00000 0
1.5 0 1.00000 0 1.02139 0 1.14465 0 1.00000 0
1.6 0 1.00000 0 1.94540 0 4.16489 0 1.00000 0
1.7 0 1.00000 0 2.18690 0 −0.195760 0 1.00000 0
1.8 0 1.00000 0 3.72373 0 1.07650 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

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 8016.2.a.y 8
4.b odd 2 1 4008.2.a.h 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.h 8 4.b odd 2 1
8016.2.a.y 8 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(167\) \(1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{8} - 23 T_{5}^{6} - 3 T_{5}^{5} + 163 T_{5}^{4} + 13 T_{5}^{3} - 418 T_{5}^{2} + 4 T_{5} + 269 \)
\(T_{7}^{8} - \cdots\)
\(T_{11}^{8} - \cdots\)
\(T_{13}^{8} + \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 - T )^{8} \)
$5$ \( 1 + 17 T^{2} - 3 T^{3} + 173 T^{4} - 62 T^{5} + 1217 T^{6} - 551 T^{7} + 6789 T^{8} - 2755 T^{9} + 30425 T^{10} - 7750 T^{11} + 108125 T^{12} - 9375 T^{13} + 265625 T^{14} + 390625 T^{16} \)
$7$ \( 1 - T + 26 T^{2} - 10 T^{3} + 293 T^{4} + 170 T^{5} + 1973 T^{6} + 3807 T^{7} + 11987 T^{8} + 26649 T^{9} + 96677 T^{10} + 58310 T^{11} + 703493 T^{12} - 168070 T^{13} + 3058874 T^{14} - 823543 T^{15} + 5764801 T^{16} \)
$11$ \( 1 - 3 T + 41 T^{2} - 53 T^{3} + 839 T^{4} - 654 T^{5} + 13251 T^{6} - 4788 T^{7} + 155452 T^{8} - 52668 T^{9} + 1603371 T^{10} - 870474 T^{11} + 12283799 T^{12} - 8535703 T^{13} + 72634001 T^{14} - 58461513 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 8 T + 57 T^{2} + 370 T^{3} + 2162 T^{4} + 10395 T^{5} + 46959 T^{6} + 194063 T^{7} + 739966 T^{8} + 2522819 T^{9} + 7936071 T^{10} + 22837815 T^{11} + 61748882 T^{12} + 137378410 T^{13} + 275128113 T^{14} + 501988136 T^{15} + 815730721 T^{16} \)
$17$ \( 1 + 7 T + 98 T^{2} + 559 T^{3} + 4645 T^{4} + 22152 T^{5} + 138662 T^{6} + 556330 T^{7} + 2823464 T^{8} + 9457610 T^{9} + 40073318 T^{10} + 108832776 T^{11} + 387955045 T^{12} + 793700063 T^{13} + 2365481762 T^{14} + 2872370711 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 59 T^{2} - 162 T^{3} + 1930 T^{4} - 7581 T^{5} + 59167 T^{6} - 175147 T^{7} + 1382418 T^{8} - 3327793 T^{9} + 21359287 T^{10} - 51998079 T^{11} + 251519530 T^{12} - 401128038 T^{13} + 2775706979 T^{14} + 16983563041 T^{16} \)
$23$ \( 1 + 9 T + 189 T^{2} + 1307 T^{3} + 15143 T^{4} + 83642 T^{5} + 688539 T^{6} + 3080972 T^{7} + 19690828 T^{8} + 70862356 T^{9} + 364237131 T^{10} + 1017672214 T^{11} + 4237632263 T^{12} + 8412300301 T^{13} + 27978783021 T^{14} + 30643429023 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - 17 T + 223 T^{2} - 1799 T^{3} + 11399 T^{4} - 45172 T^{5} + 82549 T^{6} + 525004 T^{7} - 4747660 T^{8} + 15225116 T^{9} + 69423709 T^{10} - 1101699908 T^{11} + 8062296119 T^{12} - 36899557051 T^{13} + 132645600583 T^{14} - 293247897253 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 23 T + 291 T^{2} - 2559 T^{3} + 19167 T^{4} - 133797 T^{5} + 902061 T^{6} - 5635784 T^{7} + 32738735 T^{8} - 174709304 T^{9} + 866880621 T^{10} - 3985946427 T^{11} + 17701127007 T^{12} - 73261997409 T^{13} + 258263571171 T^{14} - 632790124553 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 - 8 T + 223 T^{2} - 1097 T^{3} + 18733 T^{4} - 47772 T^{5} + 848091 T^{6} - 655265 T^{7} + 30323167 T^{8} - 24244805 T^{9} + 1161036579 T^{10} - 2419795116 T^{11} + 35108658013 T^{12} - 76070320829 T^{13} + 572156989207 T^{14} - 759455017064 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 + 8 T + 176 T^{2} + 1141 T^{3} + 16364 T^{4} + 93186 T^{5} + 1035222 T^{6} + 5288225 T^{7} + 49290458 T^{8} + 216817225 T^{9} + 1740208182 T^{10} + 6422472306 T^{11} + 46240753004 T^{12} + 132191925341 T^{13} + 836018346416 T^{14} + 1558034191048 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 - 2 T + 162 T^{2} - 289 T^{3} + 15170 T^{4} - 24474 T^{5} + 990912 T^{6} - 1421947 T^{7} + 48358106 T^{8} - 61143721 T^{9} + 1832196288 T^{10} - 1945854318 T^{11} + 51863211170 T^{12} - 42485440027 T^{13} + 1024060813938 T^{14} - 543637222214 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 - 34 T + 671 T^{2} - 9008 T^{3} + 92085 T^{4} - 741241 T^{5} + 5008223 T^{6} - 30523154 T^{7} + 196537247 T^{8} - 1434588238 T^{9} + 11063164607 T^{10} - 76957864343 T^{11} + 449345424885 T^{12} - 2065939823056 T^{13} + 7232853485759 T^{14} - 17225186095742 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 12 T + 287 T^{2} - 2704 T^{3} + 40155 T^{4} - 320963 T^{5} + 3624181 T^{6} - 24616764 T^{7} + 227551679 T^{8} - 1304688492 T^{9} + 10180324429 T^{10} - 47784008551 T^{11} + 316842264555 T^{12} - 1130800613072 T^{13} + 6361171644023 T^{14} - 14096533678044 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 - 16 T + 35 T^{2} + 1428 T^{3} - 7709 T^{4} - 96093 T^{5} + 1088279 T^{6} + 2962600 T^{7} - 81956161 T^{8} + 174793400 T^{9} + 3788299199 T^{10} - 19735484247 T^{11} - 93412735949 T^{12} + 1020911898972 T^{13} + 1476318677435 T^{14} - 39818423757104 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 2 T + 217 T^{2} - 120 T^{3} + 20030 T^{4} - 78881 T^{5} + 1125723 T^{6} - 9676201 T^{7} + 58648922 T^{8} - 590248261 T^{9} + 4188815283 T^{10} - 17904488261 T^{11} + 277332195230 T^{12} - 101351556120 T^{13} + 11179921236337 T^{14} + 6285485672042 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 21 T + 574 T^{2} + 8515 T^{3} + 136593 T^{4} + 1558001 T^{5} + 18283819 T^{6} + 166136202 T^{7} + 1528683795 T^{8} + 11131125534 T^{9} + 82076063491 T^{10} + 468589054763 T^{11} + 2752502070753 T^{12} + 11496315286105 T^{13} + 51923111365006 T^{14} + 127274943711783 T^{15} + 406067677556641 T^{16} \)
$71$ \( 1 - 29 T + 698 T^{2} - 11502 T^{3} + 167613 T^{4} - 2002193 T^{5} + 21936902 T^{6} - 209526928 T^{7} + 1874048096 T^{8} - 14876411888 T^{9} + 110583922982 T^{10} - 716606898823 T^{11} + 4259328087453 T^{12} - 20752245995202 T^{13} + 89413998176858 T^{14} - 263758484593339 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 38 T + 1109 T^{2} + 22304 T^{3} + 378276 T^{4} + 5224831 T^{5} + 63151333 T^{6} + 652952139 T^{7} + 5994430214 T^{8} + 47665506147 T^{9} + 336533453557 T^{10} + 2032548081127 T^{11} + 10742373012516 T^{12} + 46237788810272 T^{13} + 167829656954501 T^{14} + 419801143725686 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 - 12 T + 320 T^{2} - 1893 T^{3} + 34010 T^{4} - 101016 T^{5} + 2948272 T^{6} - 11033963 T^{7} + 281803786 T^{8} - 871683077 T^{9} + 18400165552 T^{10} - 49804827624 T^{11} + 1324692254810 T^{12} - 5824867763307 T^{13} + 77787985766720 T^{14} - 230446907833908 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 - 32 T + 889 T^{2} - 16658 T^{3} + 281459 T^{4} - 3833129 T^{5} + 47889213 T^{6} - 508519056 T^{7} + 4992494871 T^{8} - 42207081648 T^{9} + 329908788357 T^{10} - 2191733331523 T^{11} + 13357571570339 T^{12} - 65616539031094 T^{13} + 290649991925041 T^{14} - 868353631668064 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 11 T + 522 T^{2} - 5609 T^{3} + 136333 T^{4} - 1294131 T^{5} + 22088115 T^{6} - 179114938 T^{7} + 2381362433 T^{8} - 15941229482 T^{9} + 174959958915 T^{10} - 912322236939 T^{11} + 8553837942253 T^{12} - 31320989449441 T^{13} + 259424233881642 T^{14} - 486544683850819 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 - 8 T + 554 T^{2} - 4269 T^{3} + 144129 T^{4} - 1076092 T^{5} + 23543035 T^{6} - 162571999 T^{7} + 2688085377 T^{8} - 15769483903 T^{9} + 221516416315 T^{10} - 982120113916 T^{11} + 12759636741249 T^{12} - 36659355557133 T^{13} + 461466490730666 T^{14} - 646386275824904 T^{15} + 7837433594376961 T^{16} \)
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