Properties

Label 8008.2.a.w
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{6} q^{5} \) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( + \beta_{6} q^{5} \) \(- q^{7}\) \( + ( 1 + \beta_{2} ) q^{9} \) \(+ q^{11}\) \(+ q^{13}\) \( + ( 1 + \beta_{1} + \beta_{3} - \beta_{7} + \beta_{9} ) q^{15} \) \( + ( 1 + \beta_{2} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{17} \) \( + ( 2 - \beta_{7} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( 1 - \beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{10} ) q^{23} \) \( + ( 1 + \beta_{2} + \beta_{3} - \beta_{6} + \beta_{8} ) q^{25} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{27} \) \( + ( 1 + \beta_{1} + \beta_{9} - \beta_{10} ) q^{29} \) \( + ( 1 + \beta_{6} - \beta_{10} ) q^{31} \) \( + \beta_{1} q^{33} \) \( -\beta_{6} q^{35} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{9} + \beta_{10} ) q^{41} \) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{7} + \beta_{8} ) q^{43} \) \( + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{45} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{10} ) q^{47} \) \(+ q^{49}\) \( + ( 1 + 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{51} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{10} ) q^{53} \) \( + \beta_{6} q^{55} \) \( + ( -2 + 4 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} ) q^{57} \) \( + ( \beta_{1} + 2 \beta_{4} - \beta_{6} + \beta_{9} ) q^{59} \) \( + ( -2 - 2 \beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{61} \) \( + ( -1 - \beta_{2} ) q^{63} \) \( + \beta_{6} q^{65} \) \( + ( 1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{10} ) q^{67} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} - 2 \beta_{5} - 2 \beta_{6} + \beta_{10} ) q^{69} \) \( + ( 1 - \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{8} + \beta_{9} ) q^{71} \) \( + ( \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{5} - \beta_{7} - \beta_{8} ) q^{73} \) \( + ( \beta_{1} + \beta_{2} - \beta_{4} + 2 \beta_{5} + 5 \beta_{6} + 2 \beta_{7} - 2 \beta_{8} - \beta_{10} ) q^{75} \) \(- q^{77}\) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} + \beta_{10} ) q^{79} \) \( + ( -2 + \beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{81} \) \( + ( 3 + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{8} - \beta_{10} ) q^{83} \) \( + ( -1 + 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{10} ) q^{85} \) \( + ( 2 - \beta_{1} + \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} - \beta_{10} ) q^{87} \) \( + ( 1 + \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{5} - 3 \beta_{6} - \beta_{7} + 2 \beta_{8} - 3 \beta_{9} ) q^{89} \) \(- q^{91}\) \( + ( 2 \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{8} - \beta_{10} ) q^{93} \) \( + ( 1 + \beta_{1} - \beta_{4} + 2 \beta_{6} - 2 \beta_{9} ) q^{95} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} ) q^{97} \) \( + ( 1 + \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 11q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut 11q^{11} \) \(\mathstrut +\mathstrut 11q^{13} \) \(\mathstrut +\mathstrut 7q^{15} \) \(\mathstrut +\mathstrut 9q^{17} \) \(\mathstrut +\mathstrut 20q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut +\mathstrut 13q^{25} \) \(\mathstrut +\mathstrut 15q^{27} \) \(\mathstrut +\mathstrut 8q^{29} \) \(\mathstrut +\mathstrut 7q^{31} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut +\mathstrut 2q^{35} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut +\mathstrut 24q^{43} \) \(\mathstrut -\mathstrut 6q^{45} \) \(\mathstrut +\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 11q^{49} \) \(\mathstrut +\mathstrut 17q^{51} \) \(\mathstrut +\mathstrut 3q^{53} \) \(\mathstrut -\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut q^{59} \) \(\mathstrut -\mathstrut 22q^{61} \) \(\mathstrut -\mathstrut 14q^{63} \) \(\mathstrut -\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 14q^{67} \) \(\mathstrut -\mathstrut 22q^{69} \) \(\mathstrut +\mathstrut 6q^{71} \) \(\mathstrut +\mathstrut 3q^{73} \) \(\mathstrut -\mathstrut 11q^{77} \) \(\mathstrut +\mathstrut 8q^{79} \) \(\mathstrut -\mathstrut 9q^{81} \) \(\mathstrut +\mathstrut 29q^{83} \) \(\mathstrut -\mathstrut 9q^{85} \) \(\mathstrut +\mathstrut 19q^{87} \) \(\mathstrut +\mathstrut 20q^{89} \) \(\mathstrut -\mathstrut 11q^{91} \) \(\mathstrut -\mathstrut q^{93} \) \(\mathstrut +\mathstrut 18q^{95} \) \(\mathstrut -\mathstrut 25q^{97} \) \(\mathstrut +\mathstrut 14q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(3\) \(x^{10}\mathstrut -\mathstrut \) \(19\) \(x^{9}\mathstrut +\mathstrut \) \(55\) \(x^{8}\mathstrut +\mathstrut \) \(128\) \(x^{7}\mathstrut -\mathstrut \) \(361\) \(x^{6}\mathstrut -\mathstrut \) \(343\) \(x^{5}\mathstrut +\mathstrut \) \(1012\) \(x^{4}\mathstrut +\mathstrut \) \(215\) \(x^{3}\mathstrut -\mathstrut \) \(1090\) \(x^{2}\mathstrut +\mathstrut \) \(240\) \(x\mathstrut +\mathstrut \) \(160\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( 225 \nu^{10} - 279 \nu^{9} - 6263 \nu^{8} + 5843 \nu^{7} + 64532 \nu^{6} - 49233 \nu^{5} - 279091 \nu^{4} + 198312 \nu^{3} + 420607 \nu^{2} - 316334 \nu - 66384 \)\()/18712\)
\(\beta_{4}\)\(=\)\((\)\( 195 \nu^{10} + 226 \nu^{9} - 5272 \nu^{8} - 4448 \nu^{7} + 48287 \nu^{6} + 28437 \nu^{5} - 178414 \nu^{4} - 58755 \nu^{3} + 226681 \nu^{2} - 805 \nu - 36014 \)\()/4678\)
\(\beta_{5}\)\(=\)\((\)\( -1275 \nu^{10} + 1581 \nu^{9} + 29253 \nu^{8} - 26873 \nu^{7} - 247172 \nu^{6} + 166715 \nu^{5} + 914121 \nu^{4} - 468848 \nu^{3} - 1323093 \nu^{2} + 551330 \nu + 394888 \)\()/18712\)
\(\beta_{6}\)\(=\)\((\)\( 687 \nu^{10} - 1039 \nu^{9} - 15443 \nu^{8} + 17903 \nu^{7} + 125246 \nu^{6} - 103919 \nu^{5} - 430327 \nu^{4} + 241190 \nu^{3} + 540389 \nu^{2} - 227560 \nu - 127096 \)\()/9356\)
\(\beta_{7}\)\(=\)\((\)\( 393 \nu^{10} - 768 \nu^{9} - 8538 \nu^{8} + 13418 \nu^{7} + 68961 \nu^{6} - 81877 \nu^{5} - 244566 \nu^{4} + 206887 \nu^{3} + 326801 \nu^{2} - 196659 \nu - 68048 \)\()/4678\)
\(\beta_{8}\)\(=\)\((\)\( 1671 \nu^{10} - 3569 \nu^{9} - 35785 \nu^{8} + 62605 \nu^{7} + 279164 \nu^{6} - 368631 \nu^{5} - 924797 \nu^{4} + 822368 \nu^{3} + 1102313 \nu^{2} - 596866 \nu - 262480 \)\()/18712\)
\(\beta_{9}\)\(=\)\((\)\( 3391 \nu^{10} - 7573 \nu^{9} - 67653 \nu^{8} + 122449 \nu^{7} + 499488 \nu^{6} - 667647 \nu^{5} - 1607281 \nu^{4} + 1414604 \nu^{3} + 1929137 \nu^{2} - 1072966 \nu - 444360 \)\()/18712\)
\(\beta_{10}\)\(=\)\((\)\( 1773 \nu^{10} - 2947 \nu^{9} - 37751 \nu^{8} + 49411 \nu^{7} + 289956 \nu^{6} - 286537 \nu^{5} - 939171 \nu^{4} + 685480 \nu^{3} + 1084287 \nu^{2} - 637230 \nu - 173940 \)\()/9356\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(25\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{10}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(27\) \(\beta_{2}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\mathstrut +\mathstrut \) \(20\)
\(\nu^{6}\)\(=\)\(-\)\(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(38\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(31\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(114\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(187\)
\(\nu^{7}\)\(=\)\(-\)\(14\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(87\) \(\beta_{8}\mathstrut +\mathstrut \) \(186\) \(\beta_{7}\mathstrut +\mathstrut \) \(58\) \(\beta_{6}\mathstrut +\mathstrut \) \(144\) \(\beta_{5}\mathstrut +\mathstrut \) \(37\) \(\beta_{4}\mathstrut -\mathstrut \) \(54\) \(\beta_{3}\mathstrut +\mathstrut \) \(316\) \(\beta_{2}\mathstrut +\mathstrut \) \(180\) \(\beta_{1}\mathstrut +\mathstrut \) \(260\)
\(\nu^{8}\)\(=\)\(-\)\(34\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(69\) \(\beta_{8}\mathstrut +\mathstrut \) \(529\) \(\beta_{7}\mathstrut -\mathstrut \) \(158\) \(\beta_{6}\mathstrut +\mathstrut \) \(387\) \(\beta_{5}\mathstrut +\mathstrut \) \(198\) \(\beta_{4}\mathstrut -\mathstrut \) \(131\) \(\beta_{3}\mathstrut +\mathstrut \) \(1199\) \(\beta_{2}\mathstrut +\mathstrut \) \(124\) \(\beta_{1}\mathstrut +\mathstrut \) \(1566\)
\(\nu^{9}\)\(=\)\(-\)\(154\) \(\beta_{10}\mathstrut -\mathstrut \) \(60\) \(\beta_{9}\mathstrut -\mathstrut \) \(768\) \(\beta_{8}\mathstrut +\mathstrut \) \(2168\) \(\beta_{7}\mathstrut +\mathstrut \) \(264\) \(\beta_{6}\mathstrut +\mathstrut \) \(1551\) \(\beta_{5}\mathstrut +\mathstrut \) \(518\) \(\beta_{4}\mathstrut -\mathstrut \) \(731\) \(\beta_{3}\mathstrut +\mathstrut \) \(3572\) \(\beta_{2}\mathstrut +\mathstrut \) \(1168\) \(\beta_{1}\mathstrut +\mathstrut \) \(3013\)
\(\nu^{10}\)\(=\)\(-\)\(419\) \(\beta_{10}\mathstrut -\mathstrut \) \(80\) \(\beta_{9}\mathstrut -\mathstrut \) \(1060\) \(\beta_{8}\mathstrut +\mathstrut \) \(6566\) \(\beta_{7}\mathstrut -\mathstrut \) \(1714\) \(\beta_{6}\mathstrut +\mathstrut \) \(4509\) \(\beta_{5}\mathstrut +\mathstrut \) \(2282\) \(\beta_{4}\mathstrut -\mathstrut \) \(2003\) \(\beta_{3}\mathstrut +\mathstrut \) \(12822\) \(\beta_{2}\mathstrut +\mathstrut \) \(1082\) \(\beta_{1}\mathstrut +\mathstrut \) \(14264\)

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
−2.60864
−2.37601
−1.93952
−1.59424
−0.291357
0.846033
1.12948
1.36523
2.48144
2.65994
3.32763
0 −2.60864 0 −4.23324 0 −1.00000 0 3.80501 0
1.2 0 −2.37601 0 2.73467 0 −1.00000 0 2.64540 0
1.3 0 −1.93952 0 −0.283518 0 −1.00000 0 0.761736 0
1.4 0 −1.59424 0 0.804472 0 −1.00000 0 −0.458395 0
1.5 0 −0.291357 0 −2.53284 0 −1.00000 0 −2.91511 0
1.6 0 0.846033 0 −0.529248 0 −1.00000 0 −2.28423 0
1.7 0 1.12948 0 2.32569 0 −1.00000 0 −1.72428 0
1.8 0 1.36523 0 −2.50606 0 −1.00000 0 −1.13614 0
1.9 0 2.48144 0 1.56116 0 −1.00000 0 3.15756 0
1.10 0 2.65994 0 3.55706 0 −1.00000 0 4.07531 0
1.11 0 3.32763 0 −2.89815 0 −1.00000 0 8.07314 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

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