Properties

Label 4008.2.a.k
Level 4008
Weight 2
Character orbit 4008.a
Self dual Yes
Analytic conductor 32.004
Analytic rank 0
Dimension 11
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4008 = 2^{3} \cdot 3 \cdot 167 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0040411301\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( + ( 1 - \beta_{1} ) q^{5} \) \( + \beta_{3} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( + ( 1 - \beta_{1} ) q^{5} \) \( + \beta_{3} q^{7} \) \(+ q^{9}\) \( -\beta_{10} q^{11} \) \( + ( 1 - \beta_{4} ) q^{13} \) \( + ( -1 + \beta_{1} ) q^{15} \) \( + ( 1 - \beta_{4} + \beta_{5} + \beta_{7} ) q^{17} \) \( + ( -1 + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{19} \) \( -\beta_{3} q^{21} \) \( + ( -1 + \beta_{6} + \beta_{8} - \beta_{9} ) q^{23} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{25} \) \(- q^{27}\) \( + ( 2 - \beta_{7} ) q^{29} \) \( + ( -\beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{31} \) \( + \beta_{10} q^{33} \) \( + ( 2 + 3 \beta_{3} - 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{35} \) \( + ( 1 - 2 \beta_{2} - \beta_{6} + \beta_{9} - \beta_{10} ) q^{37} \) \( + ( -1 + \beta_{4} ) q^{39} \) \( + ( 2 + \beta_{3} - \beta_{5} + \beta_{9} ) q^{41} \) \( + ( 1 - \beta_{1} + 2 \beta_{2} + \beta_{4} + \beta_{6} - \beta_{8} ) q^{43} \) \( + ( 1 - \beta_{1} ) q^{45} \) \( + ( -1 + \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{8} ) q^{47} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + 2 \beta_{4} - \beta_{8} - \beta_{9} ) q^{49} \) \( + ( -1 + \beta_{4} - \beta_{5} - \beta_{7} ) q^{51} \) \( + ( 4 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{10} ) q^{53} \) \( + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - 3 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{55} \) \( + ( 1 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} ) q^{57} \) \( + ( \beta_{3} - \beta_{4} - \beta_{6} + \beta_{10} ) q^{59} \) \( + ( 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{10} ) q^{61} \) \( + \beta_{3} q^{63} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{65} \) \( + ( -1 - \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{67} \) \( + ( 1 - \beta_{6} - \beta_{8} + \beta_{9} ) q^{69} \) \( + ( -2 - \beta_{2} - \beta_{4} + 2 \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} + \beta_{10} ) q^{71} \) \( + ( 2 - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} ) q^{73} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{75} \) \( + ( 4 + 2 \beta_{1} - \beta_{2} + 3 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} ) q^{77} \) \( + ( -1 - 2 \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} ) q^{79} \) \(+ q^{81}\) \( + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{7} + \beta_{8} ) q^{83} \) \( + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} + 2 \beta_{8} - 2 \beta_{9} + \beta_{10} ) q^{85} \) \( + ( -2 + \beta_{7} ) q^{87} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 3 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{89} \) \( + ( -2 \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{91} \) \( + ( \beta_{2} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} ) q^{93} \) \( + ( -1 + \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{9} + \beta_{10} ) q^{95} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - 3 \beta_{6} - \beta_{10} ) q^{97} \) \( -\beta_{10} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(11q \) \(\mathstrut -\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(11q \) \(\mathstrut -\mathstrut 11q^{3} \) \(\mathstrut +\mathstrut 10q^{5} \) \(\mathstrut -\mathstrut q^{7} \) \(\mathstrut +\mathstrut 11q^{9} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut +\mathstrut 10q^{13} \) \(\mathstrut -\mathstrut 10q^{15} \) \(\mathstrut +\mathstrut 17q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut -\mathstrut 3q^{23} \) \(\mathstrut +\mathstrut 21q^{25} \) \(\mathstrut -\mathstrut 11q^{27} \) \(\mathstrut +\mathstrut 17q^{29} \) \(\mathstrut -\mathstrut 15q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut +\mathstrut 11q^{35} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 10q^{39} \) \(\mathstrut +\mathstrut 16q^{41} \) \(\mathstrut +\mathstrut 10q^{43} \) \(\mathstrut +\mathstrut 10q^{45} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut +\mathstrut 22q^{49} \) \(\mathstrut -\mathstrut 17q^{51} \) \(\mathstrut +\mathstrut 42q^{53} \) \(\mathstrut -\mathstrut 5q^{55} \) \(\mathstrut -\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 2q^{59} \) \(\mathstrut +\mathstrut 12q^{61} \) \(\mathstrut -\mathstrut q^{63} \) \(\mathstrut +\mathstrut 10q^{65} \) \(\mathstrut -\mathstrut q^{67} \) \(\mathstrut +\mathstrut 3q^{69} \) \(\mathstrut -\mathstrut 9q^{71} \) \(\mathstrut +\mathstrut 24q^{73} \) \(\mathstrut -\mathstrut 21q^{75} \) \(\mathstrut +\mathstrut 22q^{77} \) \(\mathstrut -\mathstrut 30q^{79} \) \(\mathstrut +\mathstrut 11q^{81} \) \(\mathstrut +\mathstrut 16q^{83} \) \(\mathstrut +\mathstrut 25q^{85} \) \(\mathstrut -\mathstrut 17q^{87} \) \(\mathstrut +\mathstrut 37q^{89} \) \(\mathstrut +\mathstrut q^{91} \) \(\mathstrut +\mathstrut 15q^{93} \) \(\mathstrut +\mathstrut 5q^{95} \) \(\mathstrut +\mathstrut 4q^{97} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{11}\mathstrut -\mathstrut \) \(x^{10}\mathstrut -\mathstrut \) \(33\) \(x^{9}\mathstrut +\mathstrut \) \(22\) \(x^{8}\mathstrut +\mathstrut \) \(417\) \(x^{7}\mathstrut -\mathstrut \) \(151\) \(x^{6}\mathstrut -\mathstrut \) \(2470\) \(x^{5}\mathstrut +\mathstrut \) \(272\) \(x^{4}\mathstrut +\mathstrut \) \(6584\) \(x^{3}\mathstrut +\mathstrut \) \(292\) \(x^{2}\mathstrut -\mathstrut \) \(5687\) \(x\mathstrut +\mathstrut \) \(242\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 6 \)
\(\beta_{3}\)\(=\)\((\)\(2623\) \(\nu^{10}\mathstrut -\mathstrut \) \(70944\) \(\nu^{9}\mathstrut -\mathstrut \) \(193391\) \(\nu^{8}\mathstrut +\mathstrut \) \(2378035\) \(\nu^{7}\mathstrut +\mathstrut \) \(2748114\) \(\nu^{6}\mathstrut -\mathstrut \) \(23458607\) \(\nu^{5}\mathstrut -\mathstrut \) \(15128561\) \(\nu^{4}\mathstrut +\mathstrut \) \(74577279\) \(\nu^{3}\mathstrut +\mathstrut \) \(34805559\) \(\nu^{2}\mathstrut -\mathstrut \) \(38238885\) \(\nu\mathstrut -\mathstrut \) \(7710406\)\()/10323192\)
\(\beta_{4}\)\(=\)\((\)\( -1379 \nu^{10} + 55664 \nu^{9} - 65533 \nu^{8} - 1226719 \nu^{7} + 1417390 \nu^{6} + 9091923 \nu^{5} - 7692731 \nu^{4} - 25269067 \nu^{3} + 11595133 \nu^{2} + 19499297 \nu - 2478234 \)\()/1876944\)
\(\beta_{5}\)\(=\)\((\)\(64985\) \(\nu^{10}\mathstrut -\mathstrut \) \(195192\) \(\nu^{9}\mathstrut -\mathstrut \) \(1989097\) \(\nu^{8}\mathstrut +\mathstrut \) \(5477813\) \(\nu^{7}\mathstrut +\mathstrut \) \(21384366\) \(\nu^{6}\mathstrut -\mathstrut \) \(52505737\) \(\nu^{5}\mathstrut -\mathstrut \) \(92700271\) \(\nu^{4}\mathstrut +\mathstrut \) \(195304833\) \(\nu^{3}\mathstrut +\mathstrut \) \(124569801\) \(\nu^{2}\mathstrut -\mathstrut \) \(220910235\) \(\nu\mathstrut +\mathstrut \) \(34175614\)\()/10323192\)
\(\beta_{6}\)\(=\)\((\)\(101765\) \(\nu^{10}\mathstrut -\mathstrut \) \(492056\) \(\nu^{9}\mathstrut -\mathstrut \) \(2608397\) \(\nu^{8}\mathstrut +\mathstrut \) \(12499993\) \(\nu^{7}\mathstrut +\mathstrut \) \(25094846\) \(\nu^{6}\mathstrut -\mathstrut \) \(110447061\) \(\nu^{5}\mathstrut -\mathstrut \) \(109201699\) \(\nu^{4}\mathstrut +\mathstrut \) \(387400045\) \(\nu^{3}\mathstrut +\mathstrut \) \(176633261\) \(\nu^{2}\mathstrut -\mathstrut \) \(427193567\) \(\nu\mathstrut +\mathstrut \) \(39160638\)\()/10323192\)
\(\beta_{7}\)\(=\)\((\)\(116141\) \(\nu^{10}\mathstrut -\mathstrut \) \(242652\) \(\nu^{9}\mathstrut -\mathstrut \) \(2909401\) \(\nu^{8}\mathstrut +\mathstrut \) \(5468837\) \(\nu^{7}\mathstrut +\mathstrut \) \(25357890\) \(\nu^{6}\mathstrut -\mathstrut \) \(43356601\) \(\nu^{5}\mathstrut -\mathstrut \) \(87108655\) \(\nu^{4}\mathstrut +\mathstrut \) \(142378425\) \(\nu^{3}\mathstrut +\mathstrut \) \(86939145\) \(\nu^{2}\mathstrut -\mathstrut \) \(163661007\) \(\nu\mathstrut +\mathstrut \) \(20257270\)\()/5161596\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(522481\) \(\nu^{10}\mathstrut +\mathstrut \) \(1421984\) \(\nu^{9}\mathstrut +\mathstrut \) \(13457169\) \(\nu^{8}\mathstrut -\mathstrut \) \(32908293\) \(\nu^{7}\mathstrut -\mathstrut \) \(124595270\) \(\nu^{6}\mathstrut +\mathstrut \) \(263695489\) \(\nu^{5}\mathstrut +\mathstrut \) \(481600775\) \(\nu^{4}\mathstrut -\mathstrut \) \(850380841\) \(\nu^{3}\mathstrut -\mathstrut \) \(618583841\) \(\nu^{2}\mathstrut +\mathstrut \) \(928868747\) \(\nu\mathstrut -\mathstrut \) \(65756350\)\()/20646384\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(596041\) \(\nu^{10}\mathstrut +\mathstrut \) \(2015712\) \(\nu^{9}\mathstrut +\mathstrut \) \(14695769\) \(\nu^{8}\mathstrut -\mathstrut \) \(46952653\) \(\nu^{7}\mathstrut -\mathstrut \) \(132016230\) \(\nu^{6}\mathstrut +\mathstrut \) \(379578137\) \(\nu^{5}\mathstrut +\mathstrut \) \(514603631\) \(\nu^{4}\mathstrut -\mathstrut \) \(1213924881\) \(\nu^{3}\mathstrut -\mathstrut \) \(743357145\) \(\nu^{2}\mathstrut +\mathstrut \) \(1176264339\) \(\nu\mathstrut -\mathstrut \) \(13787246\)\()/20646384\)
\(\beta_{10}\)\(=\)\((\)\(303385\) \(\nu^{10}\mathstrut -\mathstrut \) \(1032256\) \(\nu^{9}\mathstrut -\mathstrut \) \(7769641\) \(\nu^{8}\mathstrut +\mathstrut \) \(25611773\) \(\nu^{7}\mathstrut +\mathstrut \) \(71466478\) \(\nu^{6}\mathstrut -\mathstrut \) \(223268481\) \(\nu^{5}\mathstrut -\mathstrut \) \(278511527\) \(\nu^{4}\mathstrut +\mathstrut \) \(789881825\) \(\nu^{3}\mathstrut +\mathstrut \) \(385046857\) \(\nu^{2}\mathstrut -\mathstrut \) \(908813755\) \(\nu\mathstrut +\mathstrut \) \(47745918\)\()/10323192\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{5}\)\(=\)\(-\)\(4\) \(\beta_{10}\mathstrut +\mathstrut \) \(14\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(19\) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(73\) \(\beta_{1}\mathstrut +\mathstrut \) \(50\)
\(\nu^{6}\)\(=\)\(-\)\(39\) \(\beta_{10}\mathstrut +\mathstrut \) \(17\) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(35\) \(\beta_{7}\mathstrut +\mathstrut \) \(68\) \(\beta_{6}\mathstrut +\mathstrut \) \(20\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(122\) \(\beta_{2}\mathstrut +\mathstrut \) \(74\) \(\beta_{1}\mathstrut +\mathstrut \) \(432\)
\(\nu^{7}\)\(=\)\(-\)\(89\) \(\beta_{10}\mathstrut +\mathstrut \) \(167\) \(\beta_{9}\mathstrut -\mathstrut \) \(151\) \(\beta_{8}\mathstrut +\mathstrut \) \(77\) \(\beta_{7}\mathstrut +\mathstrut \) \(294\) \(\beta_{6}\mathstrut -\mathstrut \) \(163\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(54\) \(\beta_{3}\mathstrut +\mathstrut \) \(241\) \(\beta_{2}\mathstrut +\mathstrut \) \(722\) \(\beta_{1}\mathstrut +\mathstrut \) \(656\)
\(\nu^{8}\)\(=\)\(-\)\(576\) \(\beta_{10}\mathstrut +\mathstrut \) \(244\) \(\beta_{9}\mathstrut -\mathstrut \) \(31\) \(\beta_{8}\mathstrut +\mathstrut \) \(516\) \(\beta_{7}\mathstrut +\mathstrut \) \(1101\) \(\beta_{6}\mathstrut +\mathstrut \) \(116\) \(\beta_{5}\mathstrut -\mathstrut \) \(55\) \(\beta_{4}\mathstrut -\mathstrut \) \(186\) \(\beta_{3}\mathstrut +\mathstrut \) \(1378\) \(\beta_{2}\mathstrut +\mathstrut \) \(1072\) \(\beta_{1}\mathstrut +\mathstrut \) \(4199\)
\(\nu^{9}\)\(=\)\(-\)\(1462\) \(\beta_{10}\mathstrut +\mathstrut \) \(1922\) \(\beta_{9}\mathstrut -\mathstrut \) \(1489\) \(\beta_{8}\mathstrut +\mathstrut \) \(1378\) \(\beta_{7}\mathstrut +\mathstrut \) \(4200\) \(\beta_{6}\mathstrut -\mathstrut \) \(1860\) \(\beta_{5}\mathstrut +\mathstrut \) \(147\) \(\beta_{4}\mathstrut +\mathstrut \) \(704\) \(\beta_{3}\mathstrut +\mathstrut \) \(3209\) \(\beta_{2}\mathstrut +\mathstrut \) \(7555\) \(\beta_{1}\mathstrut +\mathstrut \) \(8094\)
\(\nu^{10}\)\(=\)\(-\)\(7771\) \(\beta_{10}\mathstrut +\mathstrut \) \(3303\) \(\beta_{9}\mathstrut -\mathstrut \) \(341\) \(\beta_{8}\mathstrut +\mathstrut \) \(7202\) \(\beta_{7}\mathstrut +\mathstrut \) \(15782\) \(\beta_{6}\mathstrut -\mathstrut \) \(172\) \(\beta_{5}\mathstrut -\mathstrut \) \(44\) \(\beta_{4}\mathstrut -\mathstrut \) \(1868\) \(\beta_{3}\mathstrut +\mathstrut \) \(15861\) \(\beta_{2}\mathstrut +\mathstrut \) \(14338\) \(\beta_{1}\mathstrut +\mathstrut \) \(43210\)

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
3.54432
3.11942
2.80306
2.38119
1.09328
0.0427374
−1.60189
−2.17907
−2.34947
−2.83671
−3.01685
0 −1.00000 0 −2.54432 0 0.525270 0 1.00000 0
1.2 0 −1.00000 0 −2.11942 0 −0.802640 0 1.00000 0
1.3 0 −1.00000 0 −1.80306 0 −4.10861 0 1.00000 0
1.4 0 −1.00000 0 −1.38119 0 −0.260099 0 1.00000 0
1.5 0 −1.00000 0 −0.0932775 0 3.86231 0 1.00000 0
1.6 0 −1.00000 0 0.957263 0 −0.898491 0 1.00000 0
1.7 0 −1.00000 0 2.60189 0 −3.58131 0 1.00000 0
1.8 0 −1.00000 0 3.17907 0 0.651548 0 1.00000 0
1.9 0 −1.00000 0 3.34947 0 3.54581 0 1.00000 0
1.10 0 −1.00000 0 3.83671 0 4.48179 0 1.00000 0
1.11 0 −1.00000 0 4.01685 0 −4.41557 0 1.00000 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
\(2\) \(1\)
\(3\) \(1\)
\(167\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4008))\):

\(T_{5}^{11} - \cdots\)
\(T_{7}^{11} + \cdots\)