Properties

Label 8008.2.a.v
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 11
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: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} - q^{7} + ( 1 + \beta_{3} + \beta_{9} + \beta_{10} ) q^{9} +O(q^{10})\) \( q -\beta_{1} q^{3} + \beta_{3} q^{5} - q^{7} + ( 1 + \beta_{3} + \beta_{9} + \beta_{10} ) q^{9} + q^{11} - q^{13} + ( -\beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{15} + ( -1 + \beta_{7} - \beta_{8} ) q^{17} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{19} + \beta_{1} q^{21} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{23} + ( \beta_{1} + 2 \beta_{2} - \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{8} + \beta_{10} ) q^{25} + ( -1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{27} + ( -\beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{29} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} + 2 \beta_{10} ) q^{31} -\beta_{1} q^{33} -\beta_{3} q^{35} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{37} + \beta_{1} q^{39} + ( 1 + \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{10} ) q^{41} + ( -2 - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{43} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{45} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} + \beta_{9} ) q^{47} + q^{49} + ( -1 + 2 \beta_{1} - \beta_{3} + 2 \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} ) q^{51} + ( -1 - 2 \beta_{2} + \beta_{4} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{53} + \beta_{3} q^{55} + ( -4 + 2 \beta_{1} + 3 \beta_{2} + 2 \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{57} + ( 1 - \beta_{5} + \beta_{9} ) q^{59} + ( 1 + \beta_{1} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} ) q^{61} + ( -1 - \beta_{3} - \beta_{9} - \beta_{10} ) q^{63} -\beta_{3} q^{65} + ( -3 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} - 2 \beta_{9} ) q^{67} + ( 1 - 2 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{69} + ( 1 + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} + \beta_{8} ) q^{71} + ( -2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{73} + ( -2 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 2 \beta_{5} - 3 \beta_{6} + \beta_{7} - \beta_{9} - 2 \beta_{10} ) q^{75} - q^{77} + ( -3 - 3 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} ) q^{79} + ( \beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{81} + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{83} + ( -2 - \beta_{1} - 4 \beta_{2} - \beta_{3} + \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{85} + ( -2 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} ) q^{87} + ( 1 + \beta_{1} - 3 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} + \beta_{8} - \beta_{9} ) q^{89} + q^{91} + ( -3 - \beta_{1} - 3 \beta_{3} + \beta_{4} - 3 \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} ) q^{93} + ( -1 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{95} + ( 3 \beta_{1} + \beta_{2} + \beta_{3} + 4 \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{97} + ( 1 + \beta_{3} + \beta_{9} + \beta_{10} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 2q^{3} + 2q^{5} - 11q^{7} + 9q^{9} + O(q^{10}) \) \( 11q - 2q^{3} + 2q^{5} - 11q^{7} + 9q^{9} + 11q^{11} - 11q^{13} - 7q^{15} - 4q^{17} - 16q^{19} + 2q^{21} - 3q^{23} + 11q^{25} - 11q^{27} + q^{29} + 14q^{31} - 2q^{33} - 2q^{35} - 8q^{37} + 2q^{39} + 4q^{41} - 30q^{43} + 13q^{45} - 3q^{47} + 11q^{49} - 14q^{51} - 5q^{53} + 2q^{55} - 22q^{57} + 11q^{59} + 15q^{61} - 9q^{63} - 2q^{65} - 41q^{67} + 12q^{69} + q^{71} - 8q^{73} - 24q^{75} - 11q^{77} - 26q^{79} + 19q^{81} - 31q^{83} - 27q^{85} - 25q^{87} + 2q^{89} + 11q^{91} - 37q^{93} - 6q^{97} + 9q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11} - 2 x^{10} - 19 x^{9} + 33 x^{8} + 120 x^{7} - 178 x^{6} - 296 x^{5} + 380 x^{4} + 280 x^{3} - 295 x^{2} - 87 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3471 \nu^{10} - 1645 \nu^{9} - 68837 \nu^{8} + 9923 \nu^{7} + 439510 \nu^{6} + 44393 \nu^{5} - 1009563 \nu^{4} - 166370 \nu^{3} + 809815 \nu^{2} + 91282 \nu - 165726 \)\()/5542\)
\(\beta_{3}\)\(=\)\((\)\( -2324 \nu^{10} + 1582 \nu^{9} + 45542 \nu^{8} - 15875 \nu^{7} - 286486 \nu^{6} + 25969 \nu^{5} + 641822 \nu^{4} + 2582 \nu^{3} - 483424 \nu^{2} - 19195 \nu + 90446 \)\()/2771\)
\(\beta_{4}\)\(=\)\((\)\( -6303 \nu^{10} + 2204 \nu^{9} + 125636 \nu^{8} - 3690 \nu^{7} - 803561 \nu^{6} - 162019 \nu^{5} + 1830536 \nu^{4} + 457295 \nu^{3} - 1424035 \nu^{2} - 271189 \nu + 275704 \)\()/5542\)
\(\beta_{5}\)\(=\)\((\)\( 7983 \nu^{10} - 3381 \nu^{9} - 158391 \nu^{8} + 15867 \nu^{7} + 1007688 \nu^{6} + 137771 \nu^{5} - 2279981 \nu^{4} - 438162 \nu^{3} + 1760411 \nu^{2} + 246972 \nu - 350902 \)\()/5542\)
\(\beta_{6}\)\(=\)\((\)\( 8768 \nu^{10} - 4533 \nu^{9} - 173457 \nu^{8} + 32913 \nu^{7} + 1101741 \nu^{6} + 62246 \nu^{5} - 2494913 \nu^{4} - 328435 \nu^{3} + 1925042 \nu^{2} + 233075 \nu - 382328 \)\()/5542\)
\(\beta_{7}\)\(=\)\((\)\( 16188 \nu^{10} - 8115 \nu^{9} - 320665 \nu^{8} + 55521 \nu^{7} + 2041865 \nu^{6} + 150968 \nu^{5} - 4651071 \nu^{4} - 700915 \nu^{3} + 3628688 \nu^{2} + 504589 \nu - 715838 \)\()/5542\)
\(\beta_{8}\)\(=\)\((\)\( 16691 \nu^{10} - 7773 \nu^{9} - 331173 \nu^{8} + 45871 \nu^{7} + 2112530 \nu^{6} + 223051 \nu^{5} - 4826435 \nu^{4} - 848530 \nu^{3} + 3813573 \nu^{2} + 572058 \nu - 792496 \)\()/5542\)
\(\beta_{9}\)\(=\)\((\)\( -1088 \nu^{10} + 503 \nu^{9} + 21531 \nu^{8} - 2861 \nu^{7} - 136641 \nu^{6} - 15636 \nu^{5} + 308417 \nu^{4} + 57859 \nu^{3} - 236944 \nu^{2} - 38459 \nu + 45924 \)\()/326\)
\(\beta_{10}\)\(=\)\((\)\( 23144 \nu^{10} - 11715 \nu^{9} - 457111 \nu^{8} + 80387 \nu^{7} + 2895869 \nu^{6} + 213874 \nu^{5} - 6526733 \nu^{4} - 988767 \nu^{3} + 5000438 \nu^{2} + 692193 \nu - 983768 \)\()/5542\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10} + \beta_{9} + \beta_{3} + 4\)
\(\nu^{3}\)\(=\)\(\beta_{10} + \beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(11 \beta_{10} + 11 \beta_{9} - 2 \beta_{8} + 2 \beta_{7} - \beta_{6} - \beta_{4} + 11 \beta_{3} + \beta_{2} + \beta_{1} + 27\)
\(\nu^{5}\)\(=\)\(13 \beta_{10} + 13 \beta_{9} - 11 \beta_{8} + 14 \beta_{7} - 4 \beta_{6} + 14 \beta_{5} + 14 \beta_{4} + 25 \beta_{3} + 7 \beta_{2} + 58 \beta_{1} + 19\)
\(\nu^{6}\)\(=\)\(107 \beta_{10} + 108 \beta_{9} - 25 \beta_{8} + 31 \beta_{7} - 24 \beta_{6} + 9 \beta_{5} - 6 \beta_{4} + 107 \beta_{3} + 10 \beta_{2} + 26 \beta_{1} + 228\)
\(\nu^{7}\)\(=\)\(149 \beta_{10} + 153 \beta_{9} - 107 \beta_{8} + 162 \beta_{7} - 80 \beta_{6} + 170 \beta_{5} + 158 \beta_{4} + 266 \beta_{3} + 35 \beta_{2} + 521 \beta_{1} + 269\)
\(\nu^{8}\)\(=\)\(1039 \beta_{10} + 1056 \beta_{9} - 266 \beta_{8} + 391 \beta_{7} - 371 \beta_{6} + 204 \beta_{5} + 21 \beta_{4} + 1047 \beta_{3} + 63 \beta_{2} + 420 \beta_{1} + 2128\)
\(\nu^{9}\)\(=\)\(1691 \beta_{10} + 1771 \beta_{9} - 1047 \beta_{8} + 1801 \beta_{7} - 1178 \beta_{6} + 1956 \beta_{5} + 1689 \beta_{4} + 2789 \beta_{3} + 76 \beta_{2} + 4912 \beta_{1} + 3393\)
\(\nu^{10}\)\(=\)\(10280 \beta_{10} + 10517 \beta_{9} - 2789 \beta_{8} + 4670 \beta_{7} - 4888 \beta_{6} + 3216 \beta_{5} + 1103 \beta_{4} + 10519 \beta_{3} + 170 \beta_{2} + 5734 \beta_{1} + 20944\)

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
3.33407
2.73758
1.49423
1.48564
0.983398
0.447939
−0.674923
−0.979305
−1.62645
−2.29224
−2.90994
0 −3.33407 0 −0.131360 0 −1.00000 0 8.11603 0
1.2 0 −2.73758 0 4.10495 0 −1.00000 0 4.49435 0
1.3 0 −1.49423 0 −2.57953 0 −1.00000 0 −0.767273 0
1.4 0 −1.48564 0 −0.395242 0 −1.00000 0 −0.792869 0
1.5 0 −0.983398 0 −1.20732 0 −1.00000 0 −2.03293 0
1.6 0 −0.447939 0 3.28103 0 −1.00000 0 −2.79935 0
1.7 0 0.674923 0 −4.42380 0 −1.00000 0 −2.54448 0
1.8 0 0.979305 0 2.33286 0 −1.00000 0 −2.04096 0
1.9 0 1.62645 0 1.90839 0 −1.00000 0 −0.354663 0
1.10 0 2.29224 0 0.279184 0 −1.00000 0 2.25438 0
1.11 0 2.90994 0 −1.16917 0 −1.00000 0 5.46776 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{11} + \cdots\)
\(T_{5}^{11} - \cdots\)
\(T_{17}^{11} + \cdots\)