Properties

Label 8016.2.a.w
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $7$
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( -\nu^{5} + \nu^{4} + 7 \nu^{3} - 2 \nu^{2} - 8 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{5} - \nu^{4} - 7 \nu^{3} + 3 \nu^{2} + 8 \nu - 1 \)
\(\beta_{3}\)\(=\)\( -\nu^{6} + \nu^{5} + 7 \nu^{4} - 2 \nu^{3} - 9 \nu^{2} - \nu + 1 \)
\(\beta_{4}\)\(=\)\( \nu^{6} - \nu^{5} - 7 \nu^{4} + 2 \nu^{3} + 9 \nu^{2} + 3 \nu - 1 \)
\(\beta_{5}\)\(=\)\( -\nu^{6} + \nu^{5} + 8 \nu^{4} - 3 \nu^{3} - 15 \nu^{2} + 6 \)
\(\beta_{6}\)\(=\)\( 2 \nu^{6} - 2 \nu^{5} - 15 \nu^{4} + 6 \nu^{3} + 23 \nu^{2} - 2 \nu - 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{4} + \beta_{3}\)\()/2\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 2\)
\(\nu^{3}\)\(=\)\((\)\(2 \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 5 \beta_{3} + 2 \beta_{2} + 2 \beta_{1}\)\()/2\)
\(\nu^{4}\)\(=\)\(\beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 7 \beta_{2} + 7 \beta_{1} + 7\)
\(\nu^{5}\)\(=\)\((\)\(16 \beta_{6} + 18 \beta_{5} + 15 \beta_{4} + 29 \beta_{3} + 24 \beta_{2} + 22 \beta_{1} + 4\)\()/2\)
\(\nu^{6}\)\(=\)\(13 \beta_{6} + 21 \beta_{5} + 11 \beta_{4} + 15 \beta_{3} + 50 \beta_{2} + 49 \beta_{1} + 34\)

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
2.80982
1.47270
−0.674271
−2.05123
1.31154
−1.20126
0.332704
0 1.00000 0 −3.20729 0 −3.68779 0 1.00000 0
1.2 0 1.00000 0 −3.01562 0 1.84677 0 1.00000 0
1.3 0 1.00000 0 −1.68509 0 2.23045 0 1.00000 0
1.4 0 1.00000 0 −0.597616 0 −2.60994 0 1.00000 0
1.5 0 1.00000 0 0.0621653 0 −0.782308 0 1.00000 0
1.6 0 1.00000 0 1.82640 0 −2.26944 0 1.00000 0
1.7 0 1.00000 0 3.61705 0 −2.72774 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.w 7
4.b odd 2 1 4008.2.a.g 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4008.2.a.g 7 4.b odd 2 1
8016.2.a.w 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{7} + 3 T_{5}^{6} - 15 T_{5}^{5} - 50 T_{5}^{4} + 23 T_{5}^{3} + 133 T_{5}^{2} + 56 T_{5} - 4 \)
\( T_{7}^{7} + 8 T_{7}^{6} + 11 T_{7}^{5} - 55 T_{7}^{4} - 147 T_{7}^{3} + 37 T_{7}^{2} + 336 T_{7} + 192 \)
\( T_{11}^{7} + T_{11}^{6} - 31 T_{11}^{5} - 38 T_{11}^{4} + 225 T_{11}^{3} + 271 T_{11}^{2} - 248 T_{11} + 16 \)
\( T_{13}^{7} + 2 T_{13}^{6} - 43 T_{13}^{5} - 136 T_{13}^{4} + 100 T_{13}^{3} + 335 T_{13}^{2} + 100 T_{13} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( ( -1 + T )^{7} \)
$5$ \( -4 + 56 T + 133 T^{2} + 23 T^{3} - 50 T^{4} - 15 T^{5} + 3 T^{6} + T^{7} \)
$7$ \( 192 + 336 T + 37 T^{2} - 147 T^{3} - 55 T^{4} + 11 T^{5} + 8 T^{6} + T^{7} \)
$11$ \( 16 - 248 T + 271 T^{2} + 225 T^{3} - 38 T^{4} - 31 T^{5} + T^{6} + T^{7} \)
$13$ \( 4 + 100 T + 335 T^{2} + 100 T^{3} - 136 T^{4} - 43 T^{5} + 2 T^{6} + T^{7} \)
$17$ \( 4036 + 1368 T - 1523 T^{2} - 287 T^{3} + 218 T^{4} + 6 T^{5} - 11 T^{6} + T^{7} \)
$19$ \( 1712 + 1544 T - 3171 T^{2} + 1116 T^{3} + 52 T^{4} - 65 T^{5} + 2 T^{6} + T^{7} \)
$23$ \( -928 - 1920 T - 1251 T^{2} - 85 T^{3} + 234 T^{4} + 103 T^{5} + 17 T^{6} + T^{7} \)
$29$ \( -15036 - 1320 T + 5413 T^{2} + 573 T^{3} - 428 T^{4} - 57 T^{5} + 7 T^{6} + T^{7} \)
$31$ \( 2656 + 4624 T + 1667 T^{2} - 468 T^{3} - 285 T^{4} - 6 T^{5} + 10 T^{6} + T^{7} \)
$37$ \( 252 - 696 T - 5099 T^{2} - 3447 T^{3} - 374 T^{4} + 101 T^{5} + 21 T^{6} + T^{7} \)
$41$ \( 9516 - 10884 T - 6689 T^{2} + 1848 T^{3} + 551 T^{4} - 82 T^{5} - 8 T^{6} + T^{7} \)
$43$ \( 399928 + 28888 T - 60865 T^{2} + 2018 T^{3} + 2107 T^{4} - 156 T^{5} - 12 T^{6} + T^{7} \)
$47$ \( -54336 + 134160 T + 3809 T^{2} - 12158 T^{3} - 1503 T^{4} + 111 T^{5} + 25 T^{6} + T^{7} \)
$53$ \( 740588 - 389092 T + 17793 T^{2} + 14642 T^{3} - 931 T^{4} - 205 T^{5} + 7 T^{6} + T^{7} \)
$59$ \( 1928 - 3520 T - 161 T^{2} + 862 T^{3} - 65 T^{4} - 51 T^{5} + 3 T^{6} + T^{7} \)
$61$ \( 144508 - 157812 T + 40951 T^{2} + 4744 T^{3} - 1734 T^{4} - 123 T^{5} + 14 T^{6} + T^{7} \)
$67$ \( -3736 - 5752 T + 5269 T^{2} + 1858 T^{3} - 284 T^{4} - 87 T^{5} + 4 T^{6} + T^{7} \)
$71$ \( -3808 - 8048 T - 5111 T^{2} - 463 T^{3} + 647 T^{4} + 234 T^{5} + 27 T^{6} + T^{7} \)
$73$ \( 1354996 + 1011580 T + 212917 T^{2} + 2400 T^{3} - 3442 T^{4} - 225 T^{5} + 12 T^{6} + T^{7} \)
$79$ \( -13328 + 11352 T + 4649 T^{2} - 2348 T^{3} - 1071 T^{4} - 90 T^{5} + 8 T^{6} + T^{7} \)
$83$ \( -17907304 - 954432 T + 538813 T^{2} + 32070 T^{3} - 5135 T^{4} - 329 T^{5} + 15 T^{6} + T^{7} \)
$89$ \( -8244 - 3252 T + 6347 T^{2} + 4326 T^{3} + 598 T^{4} - 91 T^{5} - 14 T^{6} + T^{7} \)
$97$ \( 19084 - 6816 T - 13157 T^{2} + 5859 T^{3} + 539 T^{4} - 204 T^{5} - 3 T^{6} + T^{7} \)
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