Properties

Label 8016.2.a.x
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 1
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: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 8 x^{6} + 28 x^{5} + 9 x^{4} - 64 x^{3} + 17 x^{2} + 23 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 + \beta_{2} ) q^{5} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 + \beta_{2} ) q^{5} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{7} + q^{9} + ( -2 + \beta_{1} + \beta_{4} ) q^{11} + ( -\beta_{5} - \beta_{7} ) q^{13} + ( -1 - \beta_{2} ) q^{15} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{19} + ( -1 + \beta_{1} - \beta_{3} - \beta_{5} ) q^{21} + ( -1 + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{23} + ( \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{25} - q^{27} + ( 1 - \beta_{1} - 2 \beta_{3} - \beta_{7} ) q^{29} + ( -1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} ) q^{31} + ( 2 - \beta_{1} - \beta_{4} ) q^{33} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{35} + ( -1 + \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{37} + ( \beta_{5} + \beta_{7} ) q^{39} + ( 1 + \beta_{1} - 2 \beta_{3} + 3 \beta_{6} - \beta_{7} ) q^{41} + ( \beta_{2} - \beta_{3} - \beta_{4} ) q^{43} + ( 1 + \beta_{2} ) q^{45} + ( -2 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{47} + ( -3 \beta_{1} + 3 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{49} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} ) q^{51} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{53} + ( -1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{7} ) q^{55} + ( 2 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} ) q^{57} + ( -4 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{59} + ( -1 + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{61} + ( 1 - \beta_{1} + \beta_{3} + \beta_{5} ) q^{63} + ( 1 - \beta_{4} - 2 \beta_{5} - 2 \beta_{7} ) q^{65} + ( -1 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{67} + ( 1 - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} ) q^{69} + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{71} + ( -3 - 4 \beta_{2} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{73} + ( -\beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{75} + ( -5 + 5 \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} ) q^{77} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} - 4 \beta_{6} + 2 \beta_{7} ) q^{79} + q^{81} + ( -5 + \beta_{1} + 2 \beta_{2} - 2 \beta_{4} - \beta_{5} + 2 \beta_{7} ) q^{83} + ( -4 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{85} + ( -1 + \beta_{1} + 2 \beta_{3} + \beta_{7} ) q^{87} + ( \beta_{1} - \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{89} + ( -1 + 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{91} + ( 1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( -1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{95} + ( -6 + 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{97} + ( -2 + \beta_{1} + \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{3} + 7q^{5} + 4q^{7} + 8q^{9} + O(q^{10}) \) \( 8q - 8q^{3} + 7q^{5} + 4q^{7} + 8q^{9} - 13q^{11} - 7q^{15} + 11q^{17} - 12q^{19} - 4q^{21} - 7q^{23} - 5q^{25} - 8q^{27} + q^{29} + 2q^{31} + 13q^{33} + 4q^{35} - 9q^{37} + 4q^{41} - 2q^{43} + 7q^{45} - 17q^{47} - 2q^{49} - 11q^{51} + 9q^{53} - 7q^{55} + 12q^{57} - 29q^{59} - 12q^{61} + 4q^{63} + 8q^{65} + 7q^{69} - 13q^{71} - 20q^{73} + 5q^{75} - 22q^{77} - 8q^{79} + 8q^{81} - 33q^{83} - 31q^{85} - q^{87} + 4q^{89} - q^{91} - 2q^{93} - 3q^{95} - 31q^{97} - 13q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 3 x^{7} - 8 x^{6} + 28 x^{5} + 9 x^{4} - 64 x^{3} + 17 x^{2} + 23 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{7} + 38 \nu^{5} + 2 \nu^{4} - 147 \nu^{3} - 11 \nu^{2} + 168 \nu + 15 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{7} + 15 \nu^{5} + 3 \nu^{4} - 70 \nu^{3} - 20 \nu^{2} + 98 \nu + 19 \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{7} - 7 \nu^{6} - 16 \nu^{5} + 64 \nu^{4} + 21 \nu^{3} - 135 \nu^{2} + 21 \nu + 25 \)\()/7\)
\(\beta_{5}\)\(=\)\((\)\( -3 \nu^{7} + 7 \nu^{6} + 24 \nu^{5} - 61 \nu^{4} - 28 \nu^{3} + 122 \nu^{2} - 42 \nu - 27 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 2 \nu^{6} + 10 \nu^{5} - 18 \nu^{4} - 25 \nu^{3} + 37 \nu^{2} + 8 \nu - 5 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( 4 \nu^{7} - 7 \nu^{6} - 39 \nu^{5} + 58 \nu^{4} + 98 \nu^{3} - 95 \nu^{2} - 56 \nu - 13 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} + \beta_{5} + \beta_{3} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{6} + \beta_{4} - \beta_{2} + 5 \beta_{1}\)
\(\nu^{4}\)\(=\)\(6 \beta_{7} + 7 \beta_{5} + \beta_{4} + 8 \beta_{3} - 2 \beta_{2} - \beta_{1} + 15\)
\(\nu^{5}\)\(=\)\(\beta_{7} + 9 \beta_{6} + 8 \beta_{4} + 2 \beta_{3} - 9 \beta_{2} + 28 \beta_{1}\)
\(\nu^{6}\)\(=\)\(37 \beta_{7} + \beta_{6} + 45 \beta_{5} + 8 \beta_{4} + 57 \beta_{3} - 21 \beta_{2} - 8 \beta_{1} + 84\)
\(\nu^{7}\)\(=\)\(13 \beta_{7} + 65 \beta_{6} + \beta_{5} + 53 \beta_{4} + 27 \beta_{3} - 71 \beta_{2} + 165 \beta_{1} + 4\)

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
−0.459587
2.63734
1.60389
−2.45154
−0.0452510
−1.71120
2.18779
1.23856
0 −1.00000 0 −2.63865 0 0.604857 0 1.00000 0
1.2 0 −1.00000 0 −1.29727 0 2.24503 0 1.00000 0
1.3 0 −1.00000 0 −0.122001 0 −3.47013 0 1.00000 0
1.4 0 −1.00000 0 1.48532 0 5.10210 0 1.00000 0
1.5 0 −1.00000 0 1.52778 0 −0.428515 0 1.00000 0
1.6 0 −1.00000 0 2.45529 0 −2.43610 0 1.00000 0
1.7 0 −1.00000 0 2.52133 0 0.307143 0 1.00000 0
1.8 0 −1.00000 0 3.06821 0 2.07562 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.x 8
4.b odd 2 1 501.2.a.e 8
12.b even 2 1 1503.2.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.e 8 4.b odd 2 1
1503.2.a.e 8 12.b even 2 1
8016.2.a.x 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\(T_{5}^{8} - \cdots\)
\(T_{7}^{8} - \cdots\)
\(T_{11}^{8} + \cdots\)
\( T_{13}^{8} - 37 T_{13}^{6} - 52 T_{13}^{5} + 320 T_{13}^{4} + 973 T_{13}^{3} + 994 T_{13}^{2} + 412 T_{13} + 56 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{8} \)
$5$ \( 1 - 7 T + 47 T^{2} - 195 T^{3} + 785 T^{4} - 2416 T^{5} + 7321 T^{6} - 18114 T^{7} + 44442 T^{8} - 90570 T^{9} + 183025 T^{10} - 302000 T^{11} + 490625 T^{12} - 609375 T^{13} + 734375 T^{14} - 546875 T^{15} + 390625 T^{16} \)
$7$ \( 1 - 4 T + 37 T^{2} - 131 T^{3} + 637 T^{4} - 2096 T^{5} + 7091 T^{6} - 21477 T^{7} + 57412 T^{8} - 150339 T^{9} + 347459 T^{10} - 718928 T^{11} + 1529437 T^{12} - 2201717 T^{13} + 4353013 T^{14} - 3294172 T^{15} + 5764801 T^{16} \)
$11$ \( 1 + 13 T + 129 T^{2} + 897 T^{3} + 5403 T^{4} + 26878 T^{5} + 120351 T^{6} + 466726 T^{7} + 1652716 T^{8} + 5133986 T^{9} + 14562471 T^{10} + 35774618 T^{11} + 79105323 T^{12} + 144462747 T^{13} + 228531369 T^{14} + 253333223 T^{15} + 214358881 T^{16} \)
$13$ \( 1 + 67 T^{2} - 52 T^{3} + 2166 T^{4} - 2407 T^{5} + 46871 T^{6} - 49521 T^{7} + 723870 T^{8} - 643773 T^{9} + 7921199 T^{10} - 5288179 T^{11} + 61863126 T^{12} - 19307236 T^{13} + 323396203 T^{14} + 815730721 T^{16} \)
$17$ \( 1 - 11 T + 116 T^{2} - 683 T^{3} + 4251 T^{4} - 18412 T^{5} + 96214 T^{6} - 371378 T^{7} + 1794794 T^{8} - 6313426 T^{9} + 27805846 T^{10} - 90458156 T^{11} + 355047771 T^{12} - 969762331 T^{13} + 2799958004 T^{14} - 4513725403 T^{15} + 6975757441 T^{16} \)
$19$ \( 1 + 12 T + 157 T^{2} + 1298 T^{3} + 10390 T^{4} + 65307 T^{5} + 390291 T^{6} + 1947317 T^{7} + 9226946 T^{8} + 36999023 T^{9} + 140895051 T^{10} + 447940713 T^{11} + 1354035190 T^{12} + 3213976502 T^{13} + 7386203317 T^{14} + 10726460868 T^{15} + 16983563041 T^{16} \)
$23$ \( 1 + 7 T + 165 T^{2} + 1013 T^{3} + 12309 T^{4} + 65238 T^{5} + 541223 T^{6} + 2417292 T^{7} + 15329148 T^{8} + 55597716 T^{9} + 286306967 T^{10} + 793750746 T^{11} + 3444562869 T^{12} + 6520015459 T^{13} + 24425921685 T^{14} + 23833778129 T^{15} + 78310985281 T^{16} \)
$29$ \( 1 - T + 129 T^{2} - 195 T^{3} + 6633 T^{4} - 16732 T^{5} + 181393 T^{6} - 805780 T^{7} + 4241976 T^{8} - 23367620 T^{9} + 152551513 T^{10} - 408076748 T^{11} + 4691394873 T^{12} - 3999674055 T^{13} + 76732208409 T^{14} - 17249876309 T^{15} + 500246412961 T^{16} \)
$31$ \( 1 - 2 T + 42 T^{2} + 147 T^{3} + 2272 T^{4} - 556 T^{5} + 120062 T^{6} + 147459 T^{7} + 2714766 T^{8} + 4571229 T^{9} + 115379582 T^{10} - 16563796 T^{11} + 2098239712 T^{12} + 4208485197 T^{13} + 37275154602 T^{14} - 55025228222 T^{15} + 852891037441 T^{16} \)
$37$ \( 1 + 9 T + 189 T^{2} + 1437 T^{3} + 17491 T^{4} + 112148 T^{5} + 1044441 T^{6} + 5719578 T^{7} + 44753916 T^{8} + 211624386 T^{9} + 1429839729 T^{10} + 5680632644 T^{11} + 32780950051 T^{12} + 99647266209 T^{13} + 484922291301 T^{14} + 854386894197 T^{15} + 3512479453921 T^{16} \)
$41$ \( 1 - 4 T + 140 T^{2} - 355 T^{3} + 11350 T^{4} - 28966 T^{5} + 691586 T^{6} - 1546337 T^{7} + 31338264 T^{8} - 63399817 T^{9} + 1162556066 T^{10} - 1996365686 T^{11} + 32072387350 T^{12} - 41128951355 T^{13} + 665014593740 T^{14} - 779017095524 T^{15} + 7984925229121 T^{16} \)
$43$ \( 1 + 2 T + 306 T^{2} + 625 T^{3} + 42368 T^{4} + 81466 T^{5} + 3468520 T^{6} + 5843465 T^{7} + 183436652 T^{8} + 251268995 T^{9} + 6413293480 T^{10} + 6477117262 T^{11} + 144847760768 T^{12} + 91880276875 T^{13} + 1934337092994 T^{14} + 543637222214 T^{15} + 11688200277601 T^{16} \)
$47$ \( 1 + 17 T + 343 T^{2} + 3782 T^{3} + 46162 T^{4} + 389283 T^{5} + 3631945 T^{6} + 25382322 T^{7} + 198853598 T^{8} + 1192969134 T^{9} + 8022966505 T^{10} + 40416528909 T^{11} + 225255834322 T^{12} + 867382816474 T^{13} + 3697270857847 T^{14} + 8612593047871 T^{15} + 23811286661761 T^{16} \)
$53$ \( 1 - 9 T + 371 T^{2} - 2932 T^{3} + 62840 T^{4} - 427509 T^{5} + 6314527 T^{6} - 36165056 T^{7} + 411010840 T^{8} - 1916747968 T^{9} + 17737506343 T^{10} - 63646257393 T^{11} + 495837826040 T^{12} - 1226149185476 T^{13} + 8222977978859 T^{14} - 10572400258533 T^{15} + 62259690411361 T^{16} \)
$59$ \( 1 + 29 T + 741 T^{2} + 12436 T^{3} + 187204 T^{4} + 2234733 T^{5} + 24274943 T^{6} + 220913290 T^{7} + 1841244926 T^{8} + 13033884110 T^{9} + 84501076583 T^{10} + 458967228807 T^{11} + 2268418448644 T^{12} + 8890798582364 T^{13} + 31255775427981 T^{14} + 72170893059751 T^{15} + 146830437604321 T^{16} \)
$61$ \( 1 + 12 T + 389 T^{2} + 4096 T^{3} + 72802 T^{4} + 647219 T^{5} + 8262029 T^{6} + 61325593 T^{7} + 615125834 T^{8} + 3740861173 T^{9} + 30743009909 T^{10} + 146906415839 T^{11} + 1008004916482 T^{12} + 3459466448896 T^{13} + 20041425626429 T^{14} + 37712914032252 T^{15} + 191707312997281 T^{16} \)
$67$ \( 1 + 359 T^{2} + 320 T^{3} + 64940 T^{4} + 71061 T^{5} + 7533739 T^{6} + 8111583 T^{7} + 600908932 T^{8} + 543476061 T^{9} + 33818954371 T^{10} + 21372519543 T^{11} + 1308613797740 T^{12} + 432040034240 T^{13} + 32474559198671 T^{14} + 406067677556641 T^{16} \)
$71$ \( 1 + 13 T + 534 T^{2} + 5116 T^{3} + 118293 T^{4} + 877603 T^{5} + 15080094 T^{6} + 90574920 T^{7} + 1280201724 T^{8} + 6430819320 T^{9} + 76018753854 T^{10} + 314103767333 T^{11} + 3006023980533 T^{12} + 9230437359716 T^{13} + 68405551613814 T^{14} + 118236562059083 T^{15} + 645753531245761 T^{16} \)
$73$ \( 1 + 20 T + 415 T^{2} + 5020 T^{3} + 66722 T^{4} + 623383 T^{5} + 6619959 T^{6} + 52721729 T^{7} + 512092382 T^{8} + 3848686217 T^{9} + 35277761511 T^{10} + 242506584511 T^{11} + 1894787436002 T^{12} + 10406819396860 T^{13} + 62803703909935 T^{14} + 220947970381940 T^{15} + 806460091894081 T^{16} \)
$79$ \( 1 + 8 T + 168 T^{2} + 573 T^{3} + 22756 T^{4} + 69216 T^{5} + 2102300 T^{6} + 2692689 T^{7} + 175872908 T^{8} + 212722431 T^{9} + 13120454300 T^{10} + 34126187424 T^{11} + 886348043236 T^{12} + 1763153316627 T^{13} + 40838692527528 T^{14} + 153631271889272 T^{15} + 1517108809906561 T^{16} \)
$83$ \( 1 + 33 T + 921 T^{2} + 17544 T^{3} + 295768 T^{4} + 4076215 T^{5} + 50726539 T^{6} + 543414180 T^{7} + 5301266758 T^{8} + 45103376940 T^{9} + 349455127171 T^{10} + 2330726746205 T^{11} + 14036652685528 T^{12} + 69106529040792 T^{13} + 301112083872849 T^{14} + 895489682657691 T^{15} + 2252292232139041 T^{16} \)
$89$ \( 1 - 4 T + 513 T^{2} - 2094 T^{3} + 127678 T^{4} - 487693 T^{5} + 19972765 T^{6} - 67413913 T^{7} + 2130074694 T^{8} - 5999838257 T^{9} + 158204271565 T^{10} - 343808446517 T^{11} + 8010803846398 T^{12} - 11693020486206 T^{13} + 254951402262993 T^{14} - 176925339582116 T^{15} + 3936588805702081 T^{16} \)
$97$ \( 1 + 31 T + 826 T^{2} + 15984 T^{3} + 265829 T^{4} + 3793587 T^{5} + 48345108 T^{6} + 548722798 T^{7} + 5707233480 T^{8} + 53226111406 T^{9} + 454879121172 T^{10} + 3462304428051 T^{11} + 23533650238949 T^{12} + 137260046667888 T^{13} + 688034876071354 T^{14} + 2504746818821503 T^{15} + 7837433594376961 T^{16} \)
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