Properties

Label 8004.2.a.h
Level 8004
Weight 2
Character orbit 8004.a
Self dual Yes
Analytic conductor 63.912
Analytic rank 0
Dimension 13
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8004.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9122617778\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(5\) \(x^{12}\mathstrut -\mathstrut \) \(27\) \(x^{11}\mathstrut +\mathstrut \) \(158\) \(x^{10}\mathstrut +\mathstrut \) \(180\) \(x^{9}\mathstrut -\mathstrut \) \(1652\) \(x^{8}\mathstrut +\mathstrut \) \(65\) \(x^{7}\mathstrut +\mathstrut \) \(7388\) \(x^{6}\mathstrut -\mathstrut \) \(3259\) \(x^{5}\mathstrut -\mathstrut \) \(15445\) \(x^{4}\mathstrut +\mathstrut \) \(7832\) \(x^{3}\mathstrut +\mathstrut \) \(15102\) \(x^{2}\mathstrut -\mathstrut \) \(5184\) \(x\mathstrut -\mathstrut \) \(5832\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(1690006925\) \(\nu^{12}\mathstrut +\mathstrut \) \(14190017491\) \(\nu^{11}\mathstrut +\mathstrut \) \(29382303177\) \(\nu^{10}\mathstrut -\mathstrut \) \(452564017120\) \(\nu^{9}\mathstrut +\mathstrut \) \(232155583236\) \(\nu^{8}\mathstrut +\mathstrut \) \(4668264761068\) \(\nu^{7}\mathstrut -\mathstrut \) \(6261574688221\) \(\nu^{6}\mathstrut -\mathstrut \) \(19340319330322\) \(\nu^{5}\mathstrut +\mathstrut \) \(35348665957787\) \(\nu^{4}\mathstrut +\mathstrut \) \(31079563327595\) \(\nu^{3}\mathstrut -\mathstrut \) \(68386010563462\) \(\nu^{2}\mathstrut -\mathstrut \) \(13353849263538\) \(\nu\mathstrut +\mathstrut \) \(30728510682372\)\()/\)\(1564335784584\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(13787660437\) \(\nu^{12}\mathstrut +\mathstrut \) \(39779102519\) \(\nu^{11}\mathstrut +\mathstrut \) \(446589014949\) \(\nu^{10}\mathstrut -\mathstrut \) \(1195536481460\) \(\nu^{9}\mathstrut -\mathstrut \) \(4749857805276\) \(\nu^{8}\mathstrut +\mathstrut \) \(11607589617668\) \(\nu^{7}\mathstrut +\mathstrut \) \(22030722743971\) \(\nu^{6}\mathstrut -\mathstrut \) \(45096431232206\) \(\nu^{5}\mathstrut -\mathstrut \) \(51594415603229\) \(\nu^{4}\mathstrut +\mathstrut \) \(70597924476775\) \(\nu^{3}\mathstrut +\mathstrut \) \(64700494527334\) \(\nu^{2}\mathstrut -\mathstrut \) \(37940862111138\) \(\nu\mathstrut -\mathstrut \) \(34093521863244\)\()/\)\(1564335784584\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(15536082095\) \(\nu^{12}\mathstrut +\mathstrut \) \(45388236577\) \(\nu^{11}\mathstrut +\mathstrut \) \(513140254587\) \(\nu^{10}\mathstrut -\mathstrut \) \(1374176976448\) \(\nu^{9}\mathstrut -\mathstrut \) \(5688540158004\) \(\nu^{8}\mathstrut +\mathstrut \) \(13514485061380\) \(\nu^{7}\mathstrut +\mathstrut \) \(28390816616465\) \(\nu^{6}\mathstrut -\mathstrut \) \(54320468285350\) \(\nu^{5}\mathstrut -\mathstrut \) \(72779616014599\) \(\nu^{4}\mathstrut +\mathstrut \) \(92492719714457\) \(\nu^{3}\mathstrut +\mathstrut \) \(95990682979838\) \(\nu^{2}\mathstrut -\mathstrut \) \(54048711679398\) \(\nu\mathstrut -\mathstrut \) \(52850941100892\)\()/\)\(1564335784584\)
\(\beta_{5}\)\(=\)\((\)\(905712243\) \(\nu^{12}\mathstrut -\mathstrut \) \(4656709097\) \(\nu^{11}\mathstrut -\mathstrut \) \(23117172527\) \(\nu^{10}\mathstrut +\mathstrut \) \(141573490824\) \(\nu^{9}\mathstrut +\mathstrut \) \(129313852652\) \(\nu^{8}\mathstrut -\mathstrut \) \(1367979642768\) \(\nu^{7}\mathstrut +\mathstrut \) \(247986187723\) \(\nu^{6}\mathstrut +\mathstrut \) \(5236246246106\) \(\nu^{5}\mathstrut -\mathstrut \) \(2654434165681\) \(\nu^{4}\mathstrut -\mathstrut \) \(8069925791881\) \(\nu^{3}\mathstrut +\mathstrut \) \(3766957425458\) \(\nu^{2}\mathstrut +\mathstrut \) \(3910054440014\) \(\nu\mathstrut -\mathstrut \) \(557666645400\)\()/\)\(86907543588\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(1019705775\) \(\nu^{12}\mathstrut +\mathstrut \) \(3494592671\) \(\nu^{11}\mathstrut +\mathstrut \) \(31563088001\) \(\nu^{10}\mathstrut -\mathstrut \) \(106568815746\) \(\nu^{9}\mathstrut -\mathstrut \) \(307208588408\) \(\nu^{8}\mathstrut +\mathstrut \) \(1055875622376\) \(\nu^{7}\mathstrut +\mathstrut \) \(1201415865197\) \(\nu^{6}\mathstrut -\mathstrut \) \(4279468555196\) \(\nu^{5}\mathstrut -\mathstrut \) \(2214476607911\) \(\nu^{4}\mathstrut +\mathstrut \) \(7364886658123\) \(\nu^{3}\mathstrut +\mathstrut \) \(2710161806296\) \(\nu^{2}\mathstrut -\mathstrut \) \(4524879364430\) \(\nu\mathstrut -\mathstrut \) \(2052309240216\)\()/\)\(86907543588\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(2437175963\) \(\nu^{12}\mathstrut +\mathstrut \) \(8549688593\) \(\nu^{11}\mathstrut +\mathstrut \) \(79092677443\) \(\nu^{10}\mathstrut -\mathstrut \) \(269326838404\) \(\nu^{9}\mathstrut -\mathstrut \) \(855454394380\) \(\nu^{8}\mathstrut +\mathstrut \) \(2816864826292\) \(\nu^{7}\mathstrut +\mathstrut \) \(4165302849813\) \(\nu^{6}\mathstrut -\mathstrut \) \(12416428807394\) \(\nu^{5}\mathstrut -\mathstrut \) \(10949657689331\) \(\nu^{4}\mathstrut +\mathstrut \) \(23603889525985\) \(\nu^{3}\mathstrut +\mathstrut \) \(16842794267506\) \(\nu^{2}\mathstrut -\mathstrut \) \(15333631200502\) \(\nu\mathstrut -\mathstrut \) \(11343802257396\)\()/\)\(173815087176\)
\(\beta_{8}\)\(=\)\((\)\(9734759813\) \(\nu^{12}\mathstrut -\mathstrut \) \(29112317203\) \(\nu^{11}\mathstrut -\mathstrut \) \(328022585121\) \(\nu^{10}\mathstrut +\mathstrut \) \(908769962656\) \(\nu^{9}\mathstrut +\mathstrut \) \(3757635764364\) \(\nu^{8}\mathstrut -\mathstrut \) \(9404404673716\) \(\nu^{7}\mathstrut -\mathstrut \) \(19504667463587\) \(\nu^{6}\mathstrut +\mathstrut \) \(40596989990194\) \(\nu^{5}\mathstrut +\mathstrut \) \(51248521715941\) \(\nu^{4}\mathstrut -\mathstrut \) \(73851918539075\) \(\nu^{3}\mathstrut -\mathstrut \) \(67107507868658\) \(\nu^{2}\mathstrut +\mathstrut \) \(45137834103882\) \(\nu\mathstrut +\mathstrut \) \(34162657809876\)\()/\)\(521445261528\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(5847934169\) \(\nu^{12}\mathstrut +\mathstrut \) \(19644634477\) \(\nu^{11}\mathstrut +\mathstrut \) \(194857600527\) \(\nu^{10}\mathstrut -\mathstrut \) \(623292257566\) \(\nu^{9}\mathstrut -\mathstrut \) \(2206830144924\) \(\nu^{8}\mathstrut +\mathstrut \) \(6604463634076\) \(\nu^{7}\mathstrut +\mathstrut \) \(11434971721055\) \(\nu^{6}\mathstrut -\mathstrut \) \(29695433860456\) \(\nu^{5}\mathstrut -\mathstrut \) \(31080010349149\) \(\nu^{4}\mathstrut +\mathstrut \) \(58070744996285\) \(\nu^{3}\mathstrut +\mathstrut \) \(45149229350024\) \(\nu^{2}\mathstrut -\mathstrut \) \(40098763498014\) \(\nu\mathstrut -\mathstrut \) \(28467417835332\)\()/\)\(260722630764\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(41709748121\) \(\nu^{12}\mathstrut +\mathstrut \) \(127653995371\) \(\nu^{11}\mathstrut +\mathstrut \) \(1373816531673\) \(\nu^{10}\mathstrut -\mathstrut \) \(3931489120324\) \(\nu^{9}\mathstrut -\mathstrut \) \(15115618592820\) \(\nu^{8}\mathstrut +\mathstrut \) \(39743603527756\) \(\nu^{7}\mathstrut +\mathstrut \) \(73867437754463\) \(\nu^{6}\mathstrut -\mathstrut \) \(165869322951934\) \(\nu^{5}\mathstrut -\mathstrut \) \(181800066250657\) \(\nu^{4}\mathstrut +\mathstrut \) \(290823676955747\) \(\nu^{3}\mathstrut +\mathstrut \) \(228971110289870\) \(\nu^{2}\mathstrut -\mathstrut \) \(170502012658314\) \(\nu\mathstrut -\mathstrut \) \(119261517833700\)\()/\)\(1564335784584\)
\(\beta_{11}\)\(=\)\((\)\(30303637499\) \(\nu^{12}\mathstrut -\mathstrut \) \(106595365123\) \(\nu^{11}\mathstrut -\mathstrut \) \(972434668869\) \(\nu^{10}\mathstrut +\mathstrut \) \(3320160658246\) \(\nu^{9}\mathstrut +\mathstrut \) \(10288162455228\) \(\nu^{8}\mathstrut -\mathstrut \) \(33996477904156\) \(\nu^{7}\mathstrut -\mathstrut \) \(48102446520869\) \(\nu^{6}\mathstrut +\mathstrut \) \(144611937263500\) \(\nu^{5}\mathstrut +\mathstrut \) \(118462643729539\) \(\nu^{4}\mathstrut -\mathstrut \) \(262240045314851\) \(\nu^{3}\mathstrut -\mathstrut \) \(167464172157692\) \(\nu^{2}\mathstrut +\mathstrut \) \(166083004109142\) \(\nu\mathstrut +\mathstrut \) \(106105578701832\)\()/\)\(782167892292\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(20460244711\) \(\nu^{12}\mathstrut +\mathstrut \) \(65987984381\) \(\nu^{11}\mathstrut +\mathstrut \) \(668494464807\) \(\nu^{10}\mathstrut -\mathstrut \) \(2046823086404\) \(\nu^{9}\mathstrut -\mathstrut \) \(7277849991516\) \(\nu^{8}\mathstrut +\mathstrut \) \(20921586976244\) \(\nu^{7}\mathstrut +\mathstrut \) \(35331329478001\) \(\nu^{6}\mathstrut -\mathstrut \) \(89115795792098\) \(\nu^{5}\mathstrut -\mathstrut \) \(88709659600319\) \(\nu^{4}\mathstrut +\mathstrut \) \(162763438597669\) \(\nu^{3}\mathstrut +\mathstrut \) \(120560040508930\) \(\nu^{2}\mathstrut -\mathstrut \) \(104540651148630\) \(\nu\mathstrut -\mathstrut \) \(70761852857268\)\()/\)\(521445261528\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(18\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(24\) \(\beta_{10}\mathstrut +\mathstrut \) \(19\) \(\beta_{9}\mathstrut +\mathstrut \) \(22\) \(\beta_{8}\mathstrut -\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(66\)
\(\nu^{5}\)\(=\)\(66\) \(\beta_{12}\mathstrut -\mathstrut \) \(18\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(45\) \(\beta_{9}\mathstrut +\mathstrut \) \(73\) \(\beta_{8}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(76\) \(\beta_{6}\mathstrut +\mathstrut \) \(20\) \(\beta_{5}\mathstrut -\mathstrut \) \(47\) \(\beta_{4}\mathstrut +\mathstrut \) \(25\) \(\beta_{3}\mathstrut +\mathstrut \) \(23\) \(\beta_{2}\mathstrut +\mathstrut \) \(108\) \(\beta_{1}\mathstrut -\mathstrut \) \(44\)
\(\nu^{6}\)\(=\)\(-\)\(296\) \(\beta_{12}\mathstrut -\mathstrut \) \(39\) \(\beta_{11}\mathstrut +\mathstrut \) \(465\) \(\beta_{10}\mathstrut +\mathstrut \) \(312\) \(\beta_{9}\mathstrut +\mathstrut \) \(440\) \(\beta_{8}\mathstrut -\mathstrut \) \(82\) \(\beta_{7}\mathstrut -\mathstrut \) \(21\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(185\) \(\beta_{4}\mathstrut +\mathstrut \) \(269\) \(\beta_{3}\mathstrut -\mathstrut \) \(65\) \(\beta_{2}\mathstrut +\mathstrut \) \(138\) \(\beta_{1}\mathstrut +\mathstrut \) \(944\)
\(\nu^{7}\)\(=\)\(1219\) \(\beta_{12}\mathstrut -\mathstrut \) \(317\) \(\beta_{11}\mathstrut +\mathstrut \) \(374\) \(\beta_{10}\mathstrut -\mathstrut \) \(812\) \(\beta_{9}\mathstrut +\mathstrut \) \(1556\) \(\beta_{8}\mathstrut +\mathstrut \) \(251\) \(\beta_{7}\mathstrut -\mathstrut \) \(1532\) \(\beta_{6}\mathstrut +\mathstrut \) \(372\) \(\beta_{5}\mathstrut -\mathstrut \) \(963\) \(\beta_{4}\mathstrut +\mathstrut \) \(523\) \(\beta_{3}\mathstrut +\mathstrut \) \(430\) \(\beta_{2}\mathstrut +\mathstrut \) \(1611\) \(\beta_{1}\mathstrut -\mathstrut \) \(656\)
\(\nu^{8}\)\(=\)\(-\)\(4765\) \(\beta_{12}\mathstrut -\mathstrut \) \(982\) \(\beta_{11}\mathstrut +\mathstrut \) \(8657\) \(\beta_{10}\mathstrut +\mathstrut \) \(5067\) \(\beta_{9}\mathstrut +\mathstrut \) \(8645\) \(\beta_{8}\mathstrut -\mathstrut \) \(1396\) \(\beta_{7}\mathstrut -\mathstrut \) \(933\) \(\beta_{6}\mathstrut +\mathstrut \) \(502\) \(\beta_{5}\mathstrut -\mathstrut \) \(3965\) \(\beta_{4}\mathstrut +\mathstrut \) \(4782\) \(\beta_{3}\mathstrut -\mathstrut \) \(1088\) \(\beta_{2}\mathstrut +\mathstrut \) \(3116\) \(\beta_{1}\mathstrut +\mathstrut \) \(15162\)
\(\nu^{9}\)\(=\)\(21176\) \(\beta_{12}\mathstrut -\mathstrut \) \(5934\) \(\beta_{11}\mathstrut +\mathstrut \) \(10269\) \(\beta_{10}\mathstrut -\mathstrut \) \(13604\) \(\beta_{9}\mathstrut +\mathstrut \) \(31979\) \(\beta_{8}\mathstrut +\mathstrut \) \(4777\) \(\beta_{7}\mathstrut -\mathstrut \) \(29062\) \(\beta_{6}\mathstrut +\mathstrut \) \(6965\) \(\beta_{5}\mathstrut -\mathstrut \) \(19105\) \(\beta_{4}\mathstrut +\mathstrut \) \(10852\) \(\beta_{3}\mathstrut +\mathstrut \) \(7622\) \(\beta_{2}\mathstrut +\mathstrut \) \(27310\) \(\beta_{1}\mathstrut -\mathstrut \) \(7476\)
\(\nu^{10}\)\(=\)\(-\)\(76136\) \(\beta_{12}\mathstrut -\mathstrut \) \(21554\) \(\beta_{11}\mathstrut +\mathstrut \) \(160366\) \(\beta_{10}\mathstrut +\mathstrut \) \(83086\) \(\beta_{9}\mathstrut +\mathstrut \) \(169068\) \(\beta_{8}\mathstrut -\mathstrut \) \(22934\) \(\beta_{7}\mathstrut -\mathstrut \) \(28208\) \(\beta_{6}\mathstrut +\mathstrut \) \(12948\) \(\beta_{5}\mathstrut -\mathstrut \) \(80531\) \(\beta_{4}\mathstrut +\mathstrut \) \(87884\) \(\beta_{3}\mathstrut -\mathstrut \) \(16771\) \(\beta_{2}\mathstrut +\mathstrut \) \(66616\) \(\beta_{1}\mathstrut +\mathstrut \) \(257569\)
\(\nu^{11}\)\(=\)\(358520\) \(\beta_{12}\mathstrut -\mathstrut \) \(114667\) \(\beta_{11}\mathstrut +\mathstrut \) \(249828\) \(\beta_{10}\mathstrut -\mathstrut \) \(219382\) \(\beta_{9}\mathstrut +\mathstrut \) \(647100\) \(\beta_{8}\mathstrut +\mathstrut \) \(82937\) \(\beta_{7}\mathstrut -\mathstrut \) \(541943\) \(\beta_{6}\mathstrut +\mathstrut \) \(131300\) \(\beta_{5}\mathstrut -\mathstrut \) \(376299\) \(\beta_{4}\mathstrut +\mathstrut \) \(225555\) \(\beta_{3}\mathstrut +\mathstrut \) \(132811\) \(\beta_{2}\mathstrut +\mathstrut \) \(494701\) \(\beta_{1}\mathstrut -\mathstrut \) \(43483\)
\(\nu^{12}\)\(=\)\(-\)\(1210903\) \(\beta_{12}\mathstrut -\mathstrut \) \(447830\) \(\beta_{11}\mathstrut +\mathstrut \) \(2982387\) \(\beta_{10}\mathstrut +\mathstrut \) \(1377739\) \(\beta_{9}\mathstrut +\mathstrut \) \(3304019\) \(\beta_{8}\mathstrut -\mathstrut \) \(374305\) \(\beta_{7}\mathstrut -\mathstrut \) \(725941\) \(\beta_{6}\mathstrut +\mathstrut \) \(296585\) \(\beta_{5}\mathstrut -\mathstrut \) \(1608212\) \(\beta_{4}\mathstrut +\mathstrut \) \(1647159\) \(\beta_{3}\mathstrut -\mathstrut \) \(246869\) \(\beta_{2}\mathstrut +\mathstrut \) \(1391621\) \(\beta_{1}\mathstrut +\mathstrut \) \(4510744\)

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
−4.03097
−2.79682
−1.91772
−1.42689
−1.03716
−0.672529
1.23293
1.59293
1.79014
2.19606
2.69157
2.93518
4.44328
0 −1.00000 0 −4.03097 0 −3.90486 0 1.00000 0
1.2 0 −1.00000 0 −2.79682 0 0.517096 0 1.00000 0
1.3 0 −1.00000 0 −1.91772 0 −2.53447 0 1.00000 0
1.4 0 −1.00000 0 −1.42689 0 −0.237900 0 1.00000 0
1.5 0 −1.00000 0 −1.03716 0 0.840656 0 1.00000 0
1.6 0 −1.00000 0 −0.672529 0 −0.215981 0 1.00000 0
1.7 0 −1.00000 0 1.23293 0 −4.84147 0 1.00000 0
1.8 0 −1.00000 0 1.59293 0 3.02509 0 1.00000 0
1.9 0 −1.00000 0 1.79014 0 −5.03804 0 1.00000 0
1.10 0 −1.00000 0 2.19606 0 4.05334 0 1.00000 0
1.11 0 −1.00000 0 2.69157 0 −2.34023 0 1.00000 0
1.12 0 −1.00000 0 2.93518 0 2.71487 0 1.00000 0
1.13 0 −1.00000 0 4.44328 0 −0.0380915 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.13
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\(T_{5}^{13} - \cdots\)
\(T_{7}^{13} + \cdots\)