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.

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

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

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

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