Properties

Label 8016.2.a.p
Level 8016
Weight 2
Character orbit 8016.a
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 5
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: \(5\)
Coefficient field: 5.5.36497.1
Defining polynomial: \(x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1\)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{5} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{9} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{11} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + 2 \beta_{4} ) q^{13} + ( 2 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{15} + ( -3 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} ) q^{17} + ( 3 + 3 \beta_{1} - 2 \beta_{3} + 3 \beta_{4} ) q^{19} + ( -1 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 2 - \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{23} + ( 2 + \beta_{1} + 3 \beta_{2} + \beta_{3} + \beta_{4} ) q^{25} - q^{27} + ( -1 + \beta_{1} - 3 \beta_{2} - 3 \beta_{3} + \beta_{4} ) q^{29} + ( 4 - 3 \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{31} + ( -2 - \beta_{1} + \beta_{2} + \beta_{4} ) q^{33} + ( \beta_{1} + \beta_{2} - 4 \beta_{4} ) q^{35} + ( 1 - 3 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} ) q^{37} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} ) q^{39} + ( -4 + 6 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{41} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + 5 \beta_{4} ) q^{43} + ( -2 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{45} + ( 2 - 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{47} + ( 1 - \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{49} + ( 3 - 2 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} ) q^{51} + ( 1 - 4 \beta_{1} + 3 \beta_{2} ) q^{53} + ( -1 - 7 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{55} + ( -3 - 3 \beta_{1} + 2 \beta_{3} - 3 \beta_{4} ) q^{57} + ( 7 - 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} ) q^{59} + ( -1 - 6 \beta_{1} - 4 \beta_{2} + 5 \beta_{3} - 2 \beta_{4} ) q^{61} + ( 1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{63} + ( -6 + 5 \beta_{1} - \beta_{2} - \beta_{3} - 3 \beta_{4} ) q^{65} + ( 3 - 5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{67} + ( -2 + \beta_{2} + \beta_{3} - 2 \beta_{4} ) q^{69} + ( -5 + 2 \beta_{1} - 6 \beta_{2} - \beta_{3} + 3 \beta_{4} ) q^{71} + ( -1 - 2 \beta_{3} - \beta_{4} ) q^{73} + ( -2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - \beta_{4} ) q^{75} + ( -9 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} - 2 \beta_{4} ) q^{77} + ( -3 - 2 \beta_{1} - \beta_{2} - 3 \beta_{4} ) q^{79} + q^{81} + ( 2 - 4 \beta_{1} + 2 \beta_{2} + 6 \beta_{3} - \beta_{4} ) q^{83} + ( 8 + 3 \beta_{1} + 9 \beta_{2} + 2 \beta_{3} + \beta_{4} ) q^{85} + ( 1 - \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - \beta_{4} ) q^{87} + ( -7 - 5 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} - 5 \beta_{4} ) q^{89} + ( 6 + 6 \beta_{1} + \beta_{2} + 7 \beta_{4} ) q^{91} + ( -4 + 3 \beta_{1} - \beta_{3} + 3 \beta_{4} ) q^{93} + ( -9 - \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{95} + ( -2 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - 2 \beta_{4} ) q^{97} + ( 2 + \beta_{1} - \beta_{2} - \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{3} - 9q^{5} + 4q^{7} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{3} - 9q^{5} + 4q^{7} + 5q^{9} + 15q^{11} + 9q^{15} - 11q^{17} + 16q^{19} - 4q^{21} + 9q^{23} + 6q^{25} - 5q^{27} - q^{29} + 18q^{31} - 15q^{33} + 4q^{35} + 7q^{37} - 10q^{41} - 6q^{43} - 9q^{45} + 7q^{47} + 11q^{49} + 11q^{51} - 9q^{53} - 17q^{55} - 16q^{57} + 37q^{59} - 2q^{61} + 4q^{63} - 16q^{65} - 9q^{69} - 13q^{71} - 6q^{73} - 6q^{75} - 2q^{77} - 14q^{79} + 5q^{81} + 5q^{83} + 29q^{85} + q^{87} - 30q^{89} + 33q^{91} - 18q^{93} - 43q^{95} - 9q^{97} + 15q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{5} - 2 x^{4} - 3 x^{3} + 5 x^{2} + x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 2 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 3 \nu^{2} + 4 \nu + 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{4} + 2 \beta_{3} + 5 \beta_{2} + 5 \beta_{1} + 7\)

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
2.31801
−1.55629
0.410375
1.33419
−0.506287
0 −1.00000 0 −3.48658 0 −3.13190 0 1.00000 0
1.2 0 −1.00000 0 −3.33576 0 0.969164 0 1.00000 0
1.3 0 −1.00000 0 −2.19483 0 4.52759 0 1.00000 0
1.4 0 −1.00000 0 −1.19539 0 −1.77240 0 1.00000 0
1.5 0 −1.00000 0 1.21255 0 3.40756 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
Significant digits:
Format:

Atkin-Lehner signs

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

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 8016.2.a.p 5
4.b odd 2 1 501.2.a.b 5
12.b even 2 1 1503.2.a.d 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
501.2.a.b 5 4.b odd 2 1
1503.2.a.d 5 12.b even 2 1
8016.2.a.p 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{5} + 9 T_{5}^{4} + 25 T_{5}^{3} + 12 T_{5}^{2} - 39 T_{5} - 37 \)
\( T_{7}^{5} - 4 T_{7}^{4} - 15 T_{7}^{3} + 49 T_{7}^{2} + 55 T_{7} - 83 \)
\( T_{11}^{5} - 15 T_{11}^{4} + 69 T_{11}^{3} - 36 T_{11}^{2} - 447 T_{11} + 761 \)
\( T_{13}^{5} - 41 T_{13}^{3} - 32 T_{13}^{2} + 412 T_{13} + 687 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( ( 1 + T )^{5} \)
$5$ \( 1 + 9 T + 50 T^{2} + 192 T^{3} + 586 T^{4} + 1433 T^{5} + 2930 T^{6} + 4800 T^{7} + 6250 T^{8} + 5625 T^{9} + 3125 T^{10} \)
$7$ \( 1 - 4 T + 20 T^{2} - 63 T^{3} + 230 T^{4} - 573 T^{5} + 1610 T^{6} - 3087 T^{7} + 6860 T^{8} - 9604 T^{9} + 16807 T^{10} \)
$11$ \( 1 - 15 T + 124 T^{2} - 696 T^{3} + 3040 T^{4} - 10921 T^{5} + 33440 T^{6} - 84216 T^{7} + 165044 T^{8} - 219615 T^{9} + 161051 T^{10} \)
$13$ \( 1 + 24 T^{2} - 32 T^{3} + 503 T^{4} - 145 T^{5} + 6539 T^{6} - 5408 T^{7} + 52728 T^{8} + 371293 T^{10} \)
$17$ \( 1 + 11 T + 75 T^{2} + 390 T^{3} + 2037 T^{4} + 9105 T^{5} + 34629 T^{6} + 112710 T^{7} + 368475 T^{8} + 918731 T^{9} + 1419857 T^{10} \)
$19$ \( 1 - 16 T + 156 T^{2} - 1114 T^{3} + 6331 T^{4} - 30103 T^{5} + 120289 T^{6} - 402154 T^{7} + 1070004 T^{8} - 2085136 T^{9} + 2476099 T^{10} \)
$23$ \( 1 - 9 T + 114 T^{2} - 658 T^{3} + 4866 T^{4} - 20699 T^{5} + 111918 T^{6} - 348082 T^{7} + 1387038 T^{8} - 2518569 T^{9} + 6436343 T^{10} \)
$29$ \( 1 + T + 56 T^{2} - 30 T^{3} + 1306 T^{4} - 3579 T^{5} + 37874 T^{6} - 25230 T^{7} + 1365784 T^{8} + 707281 T^{9} + 20511149 T^{10} \)
$31$ \( 1 - 18 T + 245 T^{2} - 2311 T^{3} + 17740 T^{4} - 108663 T^{5} + 549940 T^{6} - 2220871 T^{7} + 7298795 T^{8} - 16623378 T^{9} + 28629151 T^{10} \)
$37$ \( 1 - 7 T + 108 T^{2} - 806 T^{3} + 6788 T^{4} - 38669 T^{5} + 251156 T^{6} - 1103414 T^{7} + 5470524 T^{8} - 13119127 T^{9} + 69343957 T^{10} \)
$41$ \( 1 + 10 T + 85 T^{2} + 521 T^{3} + 2706 T^{4} + 9015 T^{5} + 110946 T^{6} + 875801 T^{7} + 5858285 T^{8} + 28257610 T^{9} + 115856201 T^{10} \)
$43$ \( 1 + 6 T + 89 T^{2} + 79 T^{3} + 1118 T^{4} - 15267 T^{5} + 48074 T^{6} + 146071 T^{7} + 7076123 T^{8} + 20512806 T^{9} + 147008443 T^{10} \)
$47$ \( 1 - 7 T + 178 T^{2} - 799 T^{3} + 13385 T^{4} - 45389 T^{5} + 629095 T^{6} - 1764991 T^{7} + 18480494 T^{8} - 34157767 T^{9} + 229345007 T^{10} \)
$53$ \( 1 + 9 T + 138 T^{2} + 183 T^{3} + 2319 T^{4} - 34225 T^{5} + 122907 T^{6} + 514047 T^{7} + 20545026 T^{8} + 71014329 T^{9} + 418195493 T^{10} \)
$59$ \( 1 - 37 T + 764 T^{2} - 11069 T^{3} + 121669 T^{4} - 1047931 T^{5} + 7178471 T^{6} - 38531189 T^{7} + 156909556 T^{8} - 448342357 T^{9} + 714924299 T^{10} \)
$61$ \( 1 + 2 T + 74 T^{2} - 366 T^{3} + 8477 T^{4} + 9987 T^{5} + 517097 T^{6} - 1361886 T^{7} + 16796594 T^{8} + 27691682 T^{9} + 844596301 T^{10} \)
$67$ \( 1 + 116 T^{2} - 558 T^{3} + 2479 T^{4} - 75523 T^{5} + 166093 T^{6} - 2504862 T^{7} + 34888508 T^{8} + 1350125107 T^{10} \)
$71$ \( 1 + 13 T + 97 T^{2} + 1015 T^{3} + 10325 T^{4} + 77487 T^{5} + 733075 T^{6} + 5116615 T^{7} + 34717367 T^{8} + 330351853 T^{9} + 1804229351 T^{10} \)
$73$ \( 1 + 6 T + 350 T^{2} + 1712 T^{3} + 50031 T^{4} + 186001 T^{5} + 3652263 T^{6} + 9123248 T^{7} + 136155950 T^{8} + 170389446 T^{9} + 2073071593 T^{10} \)
$79$ \( 1 + 14 T + 439 T^{2} + 4305 T^{3} + 72074 T^{4} + 504519 T^{5} + 5693846 T^{6} + 26867505 T^{7} + 216444121 T^{8} + 545301134 T^{9} + 3077056399 T^{10} \)
$83$ \( 1 - 5 T + 246 T^{2} + 331 T^{3} + 18969 T^{4} + 134411 T^{5} + 1574427 T^{6} + 2280259 T^{7} + 140659602 T^{8} - 237291605 T^{9} + 3939040643 T^{10} \)
$89$ \( 1 + 30 T + 662 T^{2} + 9524 T^{3} + 118385 T^{4} + 1164809 T^{5} + 10536265 T^{6} + 75439604 T^{7} + 466689478 T^{8} + 1882267230 T^{9} + 5584059449 T^{10} \)
$97$ \( 1 + 9 T + 437 T^{2} + 3063 T^{3} + 80677 T^{4} + 428251 T^{5} + 7825669 T^{6} + 28819767 T^{7} + 398838101 T^{8} + 796763529 T^{9} + 8587340257 T^{10} \)
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