Properties

Label 8016.2.a.v
Level $8016$
Weight $2$
Character orbit 8016.a
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
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: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 20 x^{5} + 2 x^{4} + 87 x^{3} + 46 x^{2} - 48 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1002)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + q^{9} +O(q^{10})\) \( q - q^{3} + ( 1 - \beta_{1} ) q^{5} + ( -1 + \beta_{3} ) q^{7} + q^{9} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{11} + ( \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{17} + ( 1 + \beta_{2} - \beta_{5} - \beta_{6} ) q^{19} + ( 1 - \beta_{3} ) q^{21} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{23} + ( 1 + \beta_{1} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{25} - q^{27} + ( -1 - \beta_{2} - 2 \beta_{4} + \beta_{5} + \beta_{6} ) q^{29} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} ) q^{31} + ( 1 + \beta_{2} + \beta_{5} - \beta_{6} ) q^{33} + ( 1 + 2 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} ) q^{35} + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{37} + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} ) q^{39} + ( -2 - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{41} + ( -\beta_{1} + \beta_{3} + \beta_{4} + \beta_{6} ) q^{43} + ( 1 - \beta_{1} ) q^{45} + ( 1 + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{47} + ( 2 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{4} ) q^{49} + ( -\beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{51} + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} ) q^{53} + ( -2 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} ) q^{55} + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{57} + ( 3 - \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{59} + ( 3 + \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{61} + ( -1 + \beta_{3} ) q^{63} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{65} + ( 3 + 3 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{6} ) q^{67} + ( \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} ) q^{69} + ( 4 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{71} + ( 4 - \beta_{1} + 3 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{73} + ( -1 - \beta_{1} + \beta_{4} + \beta_{5} - \beta_{6} ) q^{75} + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} ) q^{77} + ( -\beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{79} + q^{81} + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{5} + \beta_{6} ) q^{83} + ( -3 - 4 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{85} + ( 1 + \beta_{2} + 2 \beta_{4} - \beta_{5} - \beta_{6} ) q^{87} + ( -1 - 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{89} + ( 2 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{91} + ( 1 + \beta_{1} + \beta_{2} + \beta_{4} ) q^{93} + ( 6 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{6} ) q^{95} + ( -2 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} ) q^{97} + ( -1 - \beta_{2} - \beta_{5} + \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 7q^{3} + 5q^{5} - 7q^{7} + 7q^{9} + O(q^{10}) \) \( 7q - 7q^{3} + 5q^{5} - 7q^{7} + 7q^{9} + 6q^{13} - 5q^{15} + 6q^{17} + 2q^{19} + 7q^{21} + 12q^{25} - 7q^{27} - 4q^{29} - 7q^{31} + 13q^{35} - 3q^{37} - 6q^{39} - 12q^{41} + 2q^{43} + 5q^{45} + 11q^{47} + 10q^{49} - 6q^{51} + q^{53} + 2q^{55} - 2q^{57} + 19q^{59} + 12q^{61} - 7q^{63} - 10q^{65} + 17q^{67} + 20q^{71} + 10q^{73} - 12q^{75} - 24q^{77} - 2q^{79} + 7q^{81} + 7q^{83} - 18q^{85} + 4q^{87} - 3q^{89} - 4q^{91} + 7q^{93} + 24q^{95} - 3q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 2 x^{6} - 20 x^{5} + 2 x^{4} + 87 x^{3} + 46 x^{2} - 48 x - 2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 5 \nu^{6} - 21 \nu^{5} - 33 \nu^{4} + 41 \nu^{3} + 12 \nu^{2} + 162 \nu - 14 \)\()/104\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{6} + 23 \nu^{5} - \nu^{4} - 243 \nu^{3} + 128 \nu^{2} + 610 \nu - 158 \)\()/52\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{6} + 105 \nu^{5} + 269 \nu^{4} - 621 \nu^{3} - 892 \nu^{2} + 542 \nu + 486 \)\()/104\)
\(\beta_{5}\)\(=\)\((\)\( -17 \nu^{6} + 61 \nu^{5} + 237 \nu^{4} - 389 \nu^{3} - 800 \nu^{2} + 406 \nu + 162 \)\()/52\)
\(\beta_{6}\)\(=\)\((\)\( -59 \nu^{6} + 227 \nu^{5} + 743 \nu^{4} - 1399 \nu^{3} - 2388 \nu^{2} + 1042 \nu + 290 \)\()/104\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} - \beta_{4} + 3 \beta_{1} + 5\)
\(\nu^{3}\)\(=\)\(4 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + 18 \beta_{1} + 11\)
\(\nu^{4}\)\(=\)\(24 \beta_{6} - 28 \beta_{5} - 15 \beta_{4} - 4 \beta_{3} + 13 \beta_{2} + 83 \beta_{1} + 80\)
\(\nu^{5}\)\(=\)\(119 \beta_{6} - 148 \beta_{5} - 59 \beta_{4} - 24 \beta_{3} + 74 \beta_{2} + 437 \beta_{1} + 342\)
\(\nu^{6}\)\(=\)\(623 \beta_{6} - 763 \beta_{5} - 328 \beta_{4} - 119 \beta_{3} + 401 \beta_{2} + 2196 \beta_{1} + 1865\)

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
5.12652
2.30288
0.561628
−0.0402333
−1.55552
−2.09849
−2.29679
0 −1.00000 0 −4.12652 0 −3.19234 0 1.00000 0
1.2 0 −1.00000 0 −1.30288 0 −1.53952 0 1.00000 0
1.3 0 −1.00000 0 0.438372 0 2.51945 0 1.00000 0
1.4 0 −1.00000 0 1.04023 0 −4.50614 0 1.00000 0
1.5 0 −1.00000 0 2.55552 0 −3.69934 0 1.00000 0
1.6 0 −1.00000 0 3.09849 0 2.06938 0 1.00000 0
1.7 0 −1.00000 0 3.29679 0 1.34851 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.v 7
4.b odd 2 1 1002.2.a.k 7
12.b even 2 1 3006.2.a.u 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1002.2.a.k 7 4.b odd 2 1
3006.2.a.u 7 12.b even 2 1
8016.2.a.v 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8016))\):

\( T_{5}^{7} - 5 T_{5}^{6} - 11 T_{5}^{5} + 93 T_{5}^{4} - 110 T_{5}^{3} - 110 T_{5}^{2} + 208 T_{5} - 64 \)
\( T_{7}^{7} + 7 T_{7}^{6} - 5 T_{7}^{5} - 99 T_{7}^{4} - 28 T_{7}^{3} + 448 T_{7}^{2} + 96 T_{7} - 576 \)
\( T_{11}^{7} - 48 T_{11}^{5} - 12 T_{11}^{4} + 624 T_{11}^{3} + 368 T_{11}^{2} - 1344 T_{11} - 1152 \)
\( T_{13}^{7} - 6 T_{13}^{6} - 30 T_{13}^{5} + 244 T_{13}^{4} - 288 T_{13}^{3} - 552 T_{13}^{2} + 1088 T_{13} - 384 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \)
$3$ \( ( 1 + T )^{7} \)
$5$ \( -64 + 208 T - 110 T^{2} - 110 T^{3} + 93 T^{4} - 11 T^{5} - 5 T^{6} + T^{7} \)
$7$ \( -576 + 96 T + 448 T^{2} - 28 T^{3} - 99 T^{4} - 5 T^{5} + 7 T^{6} + T^{7} \)
$11$ \( -1152 - 1344 T + 368 T^{2} + 624 T^{3} - 12 T^{4} - 48 T^{5} + T^{7} \)
$13$ \( -384 + 1088 T - 552 T^{2} - 288 T^{3} + 244 T^{4} - 30 T^{5} - 6 T^{6} + T^{7} \)
$17$ \( -1808 - 2960 T - 160 T^{2} + 1108 T^{3} + 184 T^{4} - 56 T^{5} - 6 T^{6} + T^{7} \)
$19$ \( 7808 - 22912 T + 3312 T^{2} + 3936 T^{3} + 44 T^{4} - 116 T^{5} - 2 T^{6} + T^{7} \)
$23$ \( -1024 - 4608 T - 3296 T^{2} + 1808 T^{3} + 140 T^{4} - 92 T^{5} + T^{7} \)
$29$ \( -67328 - 107904 T + 9168 T^{2} + 7936 T^{3} - 396 T^{4} - 160 T^{5} + 4 T^{6} + T^{7} \)
$31$ \( 17536 - 19072 T + 3608 T^{2} + 1712 T^{3} - 371 T^{4} - 69 T^{5} + 7 T^{6} + T^{7} \)
$37$ \( -333456 - 93904 T + 21854 T^{2} + 6552 T^{3} - 455 T^{4} - 145 T^{5} + 3 T^{6} + T^{7} \)
$41$ \( -171936 - 8208 T + 34200 T^{2} + 3124 T^{3} - 1252 T^{4} - 104 T^{5} + 12 T^{6} + T^{7} \)
$43$ \( -125792 - 38944 T + 13032 T^{2} + 4912 T^{3} - 144 T^{4} - 134 T^{5} - 2 T^{6} + T^{7} \)
$47$ \( -11184 + 28400 T - 21648 T^{2} + 3440 T^{3} + 1255 T^{4} - 137 T^{5} - 11 T^{6} + T^{7} \)
$53$ \( 5072 - 4184 T - 2330 T^{2} + 2018 T^{3} + 87 T^{4} - 101 T^{5} - T^{6} + T^{7} \)
$59$ \( 189800 - 47560 T - 55722 T^{2} - 712 T^{3} + 2059 T^{4} - 49 T^{5} - 19 T^{6} + T^{7} \)
$61$ \( -2048 - 10752 T - 9856 T^{2} + 5984 T^{3} + 1104 T^{4} - 132 T^{5} - 12 T^{6} + T^{7} \)
$67$ \( 17569912 - 150208 T - 731998 T^{2} + 34026 T^{3} + 6697 T^{4} - 371 T^{5} - 17 T^{6} + T^{7} \)
$71$ \( 2048 - 12800 T + 20256 T^{2} - 9536 T^{3} + 1420 T^{4} + 32 T^{5} - 20 T^{6} + T^{7} \)
$73$ \( -395712 - 283392 T - 31008 T^{2} + 11680 T^{3} + 1292 T^{4} - 180 T^{5} - 10 T^{6} + T^{7} \)
$79$ \( 502400 - 392960 T + 1024 T^{2} + 20132 T^{3} - 276 T^{4} - 260 T^{5} + 2 T^{6} + T^{7} \)
$83$ \( 327912 - 48344 T - 47694 T^{2} + 6184 T^{3} + 1321 T^{4} - 175 T^{5} - 7 T^{6} + T^{7} \)
$89$ \( 5628816 - 1370480 T - 211096 T^{2} + 62912 T^{3} - 39 T^{4} - 481 T^{5} + 3 T^{6} + T^{7} \)
$97$ \( -416 - 1168 T - 20 T^{2} + 1372 T^{3} + 177 T^{4} - 137 T^{5} + 3 T^{6} + T^{7} \)
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