Properties

Label 8008.2.a.r
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 9
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: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{3} + \beta_{6} q^{5} - q^{7} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{3} + \beta_{6} q^{5} - q^{7} + ( 1 + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} - q^{11} - q^{13} + ( -1 + \beta_{3} + \beta_{6} ) q^{15} + ( -\beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{8} ) q^{17} + ( 2 + \beta_{4} + \beta_{7} ) q^{19} + ( -1 + \beta_{1} ) q^{21} + ( \beta_{1} + \beta_{5} + \beta_{6} ) q^{23} + ( -\beta_{1} - \beta_{3} + \beta_{5} + \beta_{6} ) q^{25} + ( 2 - 2 \beta_{1} - \beta_{2} - \beta_{5} + \beta_{7} - \beta_{8} ) q^{27} + ( -\beta_{1} + \beta_{3} - \beta_{4} ) q^{29} + ( 2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{31} + ( -1 + \beta_{1} ) q^{33} -\beta_{6} q^{35} + ( 2 - \beta_{1} - \beta_{2} + \beta_{6} ) q^{37} + ( -1 + \beta_{1} ) q^{39} + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{41} + ( -1 + \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{43} + ( -2 - \beta_{1} + 2 \beta_{3} - \beta_{5} - \beta_{8} ) q^{45} + ( 2 - \beta_{1} + \beta_{5} + \beta_{6} + 2 \beta_{8} ) q^{47} + q^{49} + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{8} ) q^{51} + ( 2 - \beta_{1} + 2 \beta_{3} + \beta_{5} - \beta_{6} - \beta_{8} ) q^{53} -\beta_{6} q^{55} + ( 3 - 3 \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{57} + ( 4 + 2 \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{59} + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{61} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{63} -\beta_{6} q^{65} + ( 3 + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{67} + ( -2 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + 3 \beta_{6} - 2 \beta_{7} ) q^{69} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{8} ) q^{71} + ( -2 + 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} ) q^{73} + ( 4 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} ) q^{75} + q^{77} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} ) q^{79} + ( 3 + 3 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} - \beta_{8} ) q^{81} + ( 2 - \beta_{1} - \beta_{3} + 2 \beta_{4} - \beta_{6} ) q^{83} + ( 1 + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{85} + ( 3 + 3 \beta_{3} + 2 \beta_{7} - \beta_{8} ) q^{87} + ( 2 - 2 \beta_{1} - 2 \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{89} + q^{91} + ( 1 - \beta_{3} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{93} + ( \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} ) q^{95} + ( 2 - 2 \beta_{1} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} ) q^{97} + ( -1 - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9q + 5q^{3} + 3q^{5} - 9q^{7} + 8q^{9} + O(q^{10}) \) \( 9q + 5q^{3} + 3q^{5} - 9q^{7} + 8q^{9} - 9q^{11} - 9q^{13} - 3q^{15} - 7q^{17} + 13q^{19} - 5q^{21} + 9q^{23} - 2q^{25} + 5q^{27} + q^{29} + 10q^{31} - 5q^{33} - 3q^{35} + 14q^{37} - 5q^{39} - 2q^{41} + 5q^{43} - 15q^{45} + 13q^{47} + 9q^{49} - 3q^{51} + 22q^{53} - 3q^{55} + 16q^{57} + 43q^{59} - 10q^{61} - 8q^{63} - 3q^{65} + 26q^{67} - 30q^{69} + 18q^{71} - 8q^{73} + 28q^{75} + 9q^{77} - 9q^{79} + 33q^{81} + 4q^{83} + 5q^{85} + 33q^{87} + 7q^{89} + 9q^{91} + 13q^{93} + 7q^{95} + 2q^{97} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9} - 4 x^{8} - 9 x^{7} + 36 x^{6} + 23 x^{5} - 89 x^{4} - 20 x^{3} + 51 x^{2} + 18 x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -2 \nu^{8} + 59 \nu^{7} - 166 \nu^{6} - 565 \nu^{5} + 1504 \nu^{4} + 1441 \nu^{3} - 3276 \nu^{2} - 520 \nu + 992 \)\()/139\)
\(\beta_{3}\)\(=\)\((\)\( 15 \nu^{8} - 95 \nu^{7} - 6 \nu^{6} + 832 \nu^{5} - 577 \nu^{4} - 1981 \nu^{3} + 1079 \nu^{2} + 981 \nu + 205 \)\()/139\)
\(\beta_{4}\)\(=\)\((\)\( -41 \nu^{8} + 167 \nu^{7} + 350 \nu^{6} - 1505 \nu^{5} - 721 \nu^{4} + 3895 \nu^{3} + 118 \nu^{2} - 2876 \nu - 375 \)\()/139\)
\(\beta_{5}\)\(=\)\((\)\( -44 \nu^{8} + 186 \nu^{7} + 379 \nu^{6} - 1727 \nu^{5} - 967 \nu^{4} + 4597 \nu^{3} + 1042 \nu^{2} - 3378 \nu - 833 \)\()/139\)
\(\beta_{6}\)\(=\)\((\)\( 66 \nu^{8} - 279 \nu^{7} - 499 \nu^{6} + 2382 \nu^{5} + 686 \nu^{4} - 5297 \nu^{3} + 661 \nu^{2} + 2287 \nu + 207 \)\()/139\)
\(\beta_{7}\)\(=\)\((\)\( 95 \nu^{8} - 370 \nu^{7} - 872 \nu^{6} + 3277 \nu^{5} + 2230 \nu^{4} - 7913 \nu^{3} - 1321 \nu^{2} + 4406 \nu + 418 \)\()/139\)
\(\beta_{8}\)\(=\)\((\)\( 141 \nu^{8} - 615 \nu^{7} - 1085 \nu^{6} + 5569 \nu^{5} + 1693 \nu^{4} - 13812 \nu^{3} + 496 \nu^{2} + 7609 \nu + 1232 \)\()/139\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 2 \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} + 2 \beta_{7} - 3 \beta_{6} + 4 \beta_{5} + 3 \beta_{3} + \beta_{2} + 11 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(3 \beta_{8} + 13 \beta_{7} - 17 \beta_{6} + 20 \beta_{5} - 2 \beta_{4} + 18 \beta_{3} + 4 \beta_{2} + 36 \beta_{1} + 19\)
\(\nu^{5}\)\(=\)\(18 \beta_{8} + 41 \beta_{7} - 63 \beta_{6} + 81 \beta_{5} - 9 \beta_{4} + 64 \beta_{3} + 20 \beta_{2} + 157 \beta_{1} + 35\)
\(\nu^{6}\)\(=\)\(64 \beta_{8} + 188 \beta_{7} - 273 \beta_{6} + 342 \beta_{5} - 49 \beta_{4} + 289 \beta_{3} + 81 \beta_{2} + 590 \beta_{1} + 187\)
\(\nu^{7}\)\(=\)\(289 \beta_{8} + 695 \beta_{7} - 1068 \beta_{6} + 1375 \beta_{5} - 203 \beta_{4} + 1105 \beta_{3} + 342 \beta_{2} + 2418 \beta_{1} + 561\)
\(\nu^{8}\)\(=\)\(1105 \beta_{8} + 2895 \beta_{7} - 4357 \beta_{6} + 5578 \beta_{5} - 883 \beta_{4} + 4590 \beta_{3} + 1375 \beta_{2} + 9470 \beta_{1} + 2452\)

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
3.98888
2.28175
1.66280
1.14890
−0.0694951
−0.302339
−0.688949
−1.74591
−2.27563
0 −2.98888 0 −0.559279 0 −1.00000 0 5.93339 0
1.2 0 −1.28175 0 0.660545 0 −1.00000 0 −1.35712 0
1.3 0 −0.662800 0 3.68243 0 −1.00000 0 −2.56070 0
1.4 0 −0.148900 0 −0.337922 0 −1.00000 0 −2.97783 0
1.5 0 1.06950 0 0.381633 0 −1.00000 0 −1.85618 0
1.6 0 1.30234 0 −2.00169 0 −1.00000 0 −1.30391 0
1.7 0 1.68895 0 3.11266 0 −1.00000 0 −0.147453 0
1.8 0 2.74591 0 −3.50460 0 −1.00000 0 4.54004 0
1.9 0 3.27563 0 1.56622 0 −1.00000 0 7.72976 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{9} - \cdots\)
\(T_{5}^{9} - \cdots\)
\(T_{17}^{9} + \cdots\)