Properties

Label 8008.2.a.s
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{9} q^{5} + q^{7} + ( 1 + \beta_{5} - \beta_{6} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{9} q^{5} + q^{7} + ( 1 + \beta_{5} - \beta_{6} ) q^{9} + q^{11} - q^{13} + ( -1 + \beta_{1} + \beta_{6} - \beta_{9} ) q^{15} + ( -1 - \beta_{2} - \beta_{4} - \beta_{9} ) q^{17} + ( 1 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{19} -\beta_{1} q^{21} + ( -2 + 2 \beta_{1} - \beta_{5} + \beta_{7} + \beta_{8} ) q^{23} + ( -1 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} ) q^{25} + ( -2 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} ) q^{27} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} ) q^{29} + ( -2 + \beta_{2} + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} ) q^{31} -\beta_{1} q^{33} + \beta_{9} q^{35} + ( 1 + \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{37} + \beta_{1} q^{39} + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{8} - \beta_{9} ) q^{41} + ( 1 + \beta_{1} + \beta_{2} + \beta_{7} - \beta_{8} + \beta_{9} ) q^{43} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} ) q^{45} + ( -4 - \beta_{5} - \beta_{7} + \beta_{8} ) q^{47} + q^{49} + ( -1 + 4 \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{51} + ( -3 - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - 3 \beta_{9} ) q^{53} + \beta_{9} q^{55} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} ) q^{57} + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} ) q^{59} + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{61} + ( 1 + \beta_{5} - \beta_{6} ) q^{63} -\beta_{9} q^{65} + ( -2 + \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{67} + ( -4 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} ) q^{69} + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{7} ) q^{71} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{73} + ( -1 - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 3 \beta_{6} - \beta_{8} ) q^{75} + q^{77} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{79} + ( 2 + 4 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 4 \beta_{7} + 2 \beta_{9} ) q^{81} + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{3} - 3 \beta_{4} + \beta_{6} + \beta_{7} - \beta_{8} - \beta_{9} ) q^{83} + ( -3 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} + 2 \beta_{7} - 2 \beta_{9} ) q^{85} + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} ) q^{87} + ( -4 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - 3 \beta_{7} + \beta_{8} - 2 \beta_{9} ) q^{89} - q^{91} + ( 1 - \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} - 3 \beta_{7} - \beta_{8} - 2 \beta_{9} ) q^{93} + ( -3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{95} + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{9} ) q^{97} + ( 1 + \beta_{5} - \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 3q^{3} - 4q^{5} + 10q^{7} + 9q^{9} + O(q^{10}) \) \( 10q - 3q^{3} - 4q^{5} + 10q^{7} + 9q^{9} + 10q^{11} - 10q^{13} - 5q^{15} - 11q^{17} + 2q^{19} - 3q^{21} - 8q^{23} + 2q^{25} - 15q^{27} - 8q^{29} - 23q^{31} - 3q^{33} - 4q^{35} + 10q^{37} + 3q^{39} - 18q^{41} + 12q^{43} - 10q^{45} - 36q^{47} + 10q^{49} + 9q^{51} - 21q^{53} - 4q^{55} - 30q^{57} - 13q^{59} - 2q^{61} + 9q^{63} + 4q^{65} - 4q^{67} - 26q^{69} - 24q^{71} - 23q^{73} - 28q^{75} + 10q^{77} + 14q^{79} + 30q^{81} - 9q^{83} - 17q^{85} + 7q^{87} - 18q^{89} - 10q^{91} + q^{93} - 4q^{95} - 9q^{97} + 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} - 15 x^{8} + 43 x^{7} + 66 x^{6} - 173 x^{5} - 127 x^{4} + 246 x^{3} + 99 x^{2} - 82 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( -2 \nu^{9} + 41 \nu^{7} + 7 \nu^{6} - 255 \nu^{5} - 64 \nu^{4} + 501 \nu^{3} + 157 \nu^{2} - 173 \nu + 16 \)
\(\beta_{3}\)\(=\)\( -5 \nu^{9} + 15 \nu^{8} + 62 \nu^{7} - 180 \nu^{6} - 159 \nu^{5} + 452 \nu^{4} + 108 \nu^{3} - 245 \nu^{2} + 42 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( -11 \nu^{9} + 29 \nu^{8} + 147 \nu^{7} - 343 \nu^{6} - 474 \nu^{5} + 827 \nu^{4} + 525 \nu^{3} - 356 \nu^{2} - \nu + 10 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -19 \nu^{9} + 41 \nu^{8} + 277 \nu^{7} - 469 \nu^{6} - 1090 \nu^{5} + 1015 \nu^{4} + 1547 \nu^{3} - 140 \nu^{2} - 241 \nu + 18 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -19 \nu^{9} + 41 \nu^{8} + 277 \nu^{7} - 469 \nu^{6} - 1090 \nu^{5} + 1015 \nu^{4} + 1547 \nu^{3} - 142 \nu^{2} - 241 \nu + 26 \)\()/2\)
\(\beta_{7}\)\(=\)\((\)\( -19 \nu^{9} + 33 \nu^{8} + 299 \nu^{7} - 365 \nu^{6} - 1350 \nu^{5} + 705 \nu^{4} + 2169 \nu^{3} + 156 \nu^{2} - 505 \nu + 62 \)\()/2\)
\(\beta_{8}\)\(=\)\((\)\( 13 \nu^{9} - 9 \nu^{8} - 241 \nu^{7} + 71 \nu^{6} + 1352 \nu^{5} + 71 \nu^{4} - 2493 \nu^{3} - 684 \nu^{2} + 735 \nu - 76 \)\()/2\)
\(\beta_{9}\)\(=\)\((\)\( 15 \nu^{9} - 11 \nu^{8} - 277 \nu^{7} + 91 \nu^{6} + 1550 \nu^{5} + 45 \nu^{4} - 2875 \nu^{3} - 732 \nu^{2} + 895 \nu - 96 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{6} + \beta_{5} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} + 8 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(2 \beta_{9} + 4 \beta_{7} - 12 \beta_{6} + 10 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 4 \beta_{1} + 29\)
\(\nu^{5}\)\(=\)\(18 \beta_{9} - 14 \beta_{8} + 6 \beta_{7} - 17 \beta_{6} + 14 \beta_{5} - 27 \beta_{4} + 26 \beta_{3} + 17 \beta_{2} + 79 \beta_{1} + 26\)
\(\nu^{6}\)\(=\)\(42 \beta_{9} - 6 \beta_{8} + 63 \beta_{7} - 132 \beta_{6} + 102 \beta_{5} - 39 \beta_{4} + 23 \beta_{3} + 31 \beta_{2} + 80 \beta_{1} + 263\)
\(\nu^{7}\)\(=\)\(245 \beta_{9} - 159 \beta_{8} + 123 \beta_{7} - 233 \beta_{6} + 174 \beta_{5} - 314 \beta_{4} + 291 \beta_{3} + 234 \beta_{2} + 846 \beta_{1} + 321\)
\(\nu^{8}\)\(=\)\(635 \beta_{9} - 138 \beta_{8} + 807 \beta_{7} - 1454 \beta_{6} + 1077 \beta_{5} - 571 \beta_{4} + 371 \beta_{3} + 533 \beta_{2} + 1233 \beta_{1} + 2642\)
\(\nu^{9}\)\(=\)\(3061 \beta_{9} - 1746 \beta_{8} + 1849 \beta_{7} - 3016 \beta_{6} + 2148 \beta_{5} - 3568 \beta_{4} + 3200 \beta_{3} + 2956 \beta_{2} + 9340 \beta_{1} + 4081\)

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
3.44977
2.46329
1.79551
1.79314
0.364324
0.119595
−1.01146
−1.48379
−1.49066
−2.99972
0 −3.44977 0 0.464152 0 1.00000 0 8.90091 0
1.2 0 −2.46329 0 −1.35093 0 1.00000 0 3.06778 0
1.3 0 −1.79551 0 3.68086 0 1.00000 0 0.223871 0
1.4 0 −1.79314 0 −4.09989 0 1.00000 0 0.215350 0
1.5 0 −0.364324 0 2.29944 0 1.00000 0 −2.86727 0
1.6 0 −0.119595 0 −2.15143 0 1.00000 0 −2.98570 0
1.7 0 1.01146 0 −0.0780254 0 1.00000 0 −1.97695 0
1.8 0 1.48379 0 1.24336 0 1.00000 0 −0.798370 0
1.9 0 1.49066 0 −2.23108 0 1.00000 0 −0.777924 0
1.10 0 2.99972 0 −1.77646 0 1.00000 0 5.99829 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)
\(13\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{10} + \cdots\)
\(T_{5}^{10} + \cdots\)
\(T_{17}^{10} + \cdots\)