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:

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.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

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

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T )^{8} \)
$5$ \( -18 - 124 T + 196 T^{2} + 9 T^{3} - 125 T^{4} + 50 T^{5} + 7 T^{6} - 7 T^{7} + T^{8} \)
$7$ \( -16 + 48 T + 84 T^{2} - 255 T^{3} + 63 T^{4} + 65 T^{5} - 19 T^{6} - 4 T^{7} + T^{8} \)
$11$ \( -1596 + 1316 T + 1804 T^{2} - 435 T^{3} - 691 T^{4} - 104 T^{5} + 41 T^{6} + 13 T^{7} + T^{8} \)
$13$ \( 56 + 412 T + 994 T^{2} + 973 T^{3} + 320 T^{4} - 52 T^{5} - 37 T^{6} + T^{8} \)
$17$ \( 7822 - 41000 T + 30254 T^{2} - 4863 T^{3} - 1801 T^{4} + 626 T^{5} - 20 T^{6} - 11 T^{7} + T^{8} \)
$19$ \( 4384 - 8448 T + 1000 T^{2} + 2645 T^{3} - 288 T^{4} - 298 T^{5} + 5 T^{6} + 12 T^{7} + T^{8} \)
$23$ \( 384 - 928 T - 312 T^{2} + 585 T^{3} + 119 T^{4} - 114 T^{5} - 19 T^{6} + 7 T^{7} + T^{8} \)
$29$ \( 88 + 652 T - 1858 T^{2} - 231 T^{3} + 1007 T^{4} + 8 T^{5} - 103 T^{6} - T^{7} + T^{8} \)
$31$ \( -1017696 + 1322576 T - 275064 T^{2} - 50249 T^{3} + 13680 T^{4} + 581 T^{5} - 206 T^{6} - 2 T^{7} + T^{8} \)
$37$ \( -42872 - 83724 T - 26006 T^{2} + 18797 T^{3} + 2913 T^{4} - 894 T^{5} - 107 T^{6} + 9 T^{7} + T^{8} \)
$41$ \( -859446 + 962494 T - 154490 T^{2} - 50327 T^{3} + 10530 T^{4} + 793 T^{5} - 188 T^{6} - 4 T^{7} + T^{8} \)
$43$ \( 114 - 622 T + 1258 T^{2} - 1137 T^{3} + 400 T^{4} + 23 T^{5} - 38 T^{6} + 2 T^{7} + T^{8} \)
$47$ \( -68732 - 89328 T + 111504 T^{2} + 26255 T^{3} - 6384 T^{4} - 1811 T^{5} - 33 T^{6} + 17 T^{7} + T^{8} \)
$53$ \( 20326 + 8186 T - 10334 T^{2} - 4463 T^{3} + 1042 T^{4} + 407 T^{5} - 53 T^{6} - 9 T^{7} + T^{8} \)
$59$ \( -21056 + 122192 T + 23524 T^{2} - 20601 T^{3} - 5490 T^{4} + 459 T^{5} + 269 T^{6} + 29 T^{7} + T^{8} \)
$61$ \( 51244 + 24192 T - 106134 T^{2} + 23067 T^{3} + 4848 T^{4} - 1028 T^{5} - 99 T^{6} + 12 T^{7} + T^{8} \)
$67$ \( -1226366 + 1010722 T - 178430 T^{2} - 36139 T^{3} + 10402 T^{4} + 320 T^{5} - 177 T^{6} + T^{8} \)
$71$ \( 32816 + 24360 T - 14648 T^{2} - 21115 T^{3} - 8371 T^{4} - 1345 T^{5} - 34 T^{6} + 13 T^{7} + T^{8} \)
$73$ \( -9386184 - 4503628 T + 816678 T^{2} + 283203 T^{3} - 8468 T^{4} - 5200 T^{5} - 169 T^{6} + 20 T^{7} + T^{8} \)
$79$ \( 39988958 - 23441222 T - 3540828 T^{2} + 541873 T^{3} + 67944 T^{4} - 3851 T^{5} - 464 T^{6} + 8 T^{7} + T^{8} \)
$83$ \( -2562224 + 656928 T + 485892 T^{2} - 21827 T^{3} - 25110 T^{4} - 1629 T^{5} + 257 T^{6} + 33 T^{7} + T^{8} \)
$89$ \( -216312 - 393620 T - 188850 T^{2} + 561 T^{3} + 12156 T^{4} + 398 T^{5} - 199 T^{6} - 4 T^{7} + T^{8} \)
$97$ \( -24584 - 1293780 T + 547194 T^{2} + 124853 T^{3} - 26723 T^{4} - 5065 T^{5} + 50 T^{6} + 31 T^{7} + T^{8} \)
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