Properties

Label 4012.2.a.g
Level 4012
Weight 2
Character orbit 4012.a
Self dual Yes
Analytic conductor 32.036
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0359812909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( - \beta_{1} q^{3} \) \( + \beta_{9} q^{5} \) \( + ( - \beta_{2} + \beta_{8} - \beta_{9} ) q^{7} \) \( + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{9} \) \(+O(q^{10})\) \( q\) \( - \beta_{1} q^{3} \) \( + \beta_{9} q^{5} \) \( + ( - \beta_{2} + \beta_{8} - \beta_{9} ) q^{7} \) \( + ( \beta_{1} + \beta_{4} + \beta_{5} ) q^{9} \) \( + ( -1 + \beta_{2} - \beta_{4} - \beta_{5} + \beta_{9} + \beta_{11} ) q^{11} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{8} - \beta_{9} ) q^{13} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} ) q^{15} \) \(- q^{17}\) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{7} ) q^{19} \) \( + ( 1 + \beta_{3} - \beta_{8} + \beta_{9} ) q^{21} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{11} ) q^{23} \) \( + ( 2 \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{9} ) q^{25} \) \( + ( -2 - \beta_{1} + 2 \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{27} \) \( + ( -1 + \beta_{3} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{29} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{31} \) \( + ( -1 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{11} ) q^{33} \) \( + ( - \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{35} \) \( + ( 1 + 3 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{37} \) \( + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{39} \) \( + ( -2 - 2 \beta_{1} + \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{11} ) q^{41} \) \( + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{7} - 3 \beta_{9} ) q^{43} \) \( + ( 2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{4} + 3 \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} ) q^{45} \) \( + ( -5 - 2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} - \beta_{7} + \beta_{8} + \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{47} \) \( + ( 2 \beta_{1} - \beta_{4} - \beta_{6} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} - \beta_{11} ) q^{49} \) \( + \beta_{1} q^{51} \) \( + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{7} + 2 \beta_{9} - \beta_{10} ) q^{53} \) \( + ( -2 - \beta_{1} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{55} \) \( + ( 3 + 5 \beta_{1} - \beta_{2} + \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} - \beta_{11} ) q^{57} \) \(+ q^{59}\) \( + ( -3 - \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} - \beta_{10} ) q^{61} \) \( + ( -2 + \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} ) q^{63} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{8} + 3 \beta_{9} - \beta_{10} ) q^{65} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{10} ) q^{67} \) \( + ( -2 + 3 \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{6} + \beta_{8} + 2 \beta_{9} + 3 \beta_{11} ) q^{69} \) \( + ( -1 - \beta_{3} + 2 \beta_{4} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{71} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{11} ) q^{73} \) \( + ( -5 - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - \beta_{7} + 3 \beta_{8} - 4 \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{75} \) \( + ( -1 - \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} - \beta_{11} ) q^{77} \) \( + ( 5 + 4 \beta_{1} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{11} ) q^{79} \) \( + ( -1 - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{81} \) \( + ( -1 - 3 \beta_{2} + \beta_{3} + 2 \beta_{4} + \beta_{8} - 2 \beta_{9} + \beta_{10} - \beta_{11} ) q^{83} \) \( - \beta_{9} q^{85} \) \( + ( -2 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{8} - 2 \beta_{9} + 2 \beta_{11} ) q^{87} \) \( + ( -2 - 3 \beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} - \beta_{5} + 3 \beta_{7} - 5 \beta_{8} + 6 \beta_{9} + 3 \beta_{10} + 4 \beta_{11} ) q^{89} \) \( + ( -3 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - 5 \beta_{9} - \beta_{10} + \beta_{11} ) q^{91} \) \( + ( -2 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{6} - \beta_{7} + 2 \beta_{8} - 4 \beta_{9} - 2 \beta_{11} ) q^{93} \) \( + ( -5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{4} - 3 \beta_{5} - \beta_{6} + 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{95} \) \( + ( -2 - 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 4 \beta_{9} + \beta_{10} + \beta_{11} ) q^{97} \) \( + ( -1 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{10} + 3 \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 6q^{9} \) \(\mathstrut -\mathstrut 5q^{11} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 16q^{15} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 15q^{19} \) \(\mathstrut +\mathstrut 5q^{21} \) \(\mathstrut -\mathstrut 22q^{23} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 16q^{27} \) \(\mathstrut -\mathstrut 17q^{29} \) \(\mathstrut +\mathstrut 5q^{31} \) \(\mathstrut -\mathstrut 33q^{33} \) \(\mathstrut -\mathstrut 9q^{35} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 24q^{39} \) \(\mathstrut -\mathstrut 18q^{41} \) \(\mathstrut -\mathstrut 16q^{43} \) \(\mathstrut +\mathstrut 30q^{45} \) \(\mathstrut -\mathstrut 60q^{47} \) \(\mathstrut +\mathstrut 10q^{49} \) \(\mathstrut +\mathstrut 4q^{51} \) \(\mathstrut -\mathstrut 24q^{53} \) \(\mathstrut -\mathstrut q^{55} \) \(\mathstrut +\mathstrut 41q^{57} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut -\mathstrut 17q^{61} \) \(\mathstrut -\mathstrut 23q^{63} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut -\mathstrut 22q^{67} \) \(\mathstrut +\mathstrut 6q^{69} \) \(\mathstrut -\mathstrut 10q^{71} \) \(\mathstrut +\mathstrut 4q^{73} \) \(\mathstrut -\mathstrut 62q^{75} \) \(\mathstrut -\mathstrut 6q^{77} \) \(\mathstrut +\mathstrut 48q^{79} \) \(\mathstrut +\mathstrut 4q^{81} \) \(\mathstrut -\mathstrut 20q^{83} \) \(\mathstrut -\mathstrut 3q^{85} \) \(\mathstrut +\mathstrut 13q^{87} \) \(\mathstrut -\mathstrut 29q^{89} \) \(\mathstrut -\mathstrut 28q^{91} \) \(\mathstrut -\mathstrut 31q^{93} \) \(\mathstrut -\mathstrut 48q^{95} \) \(\mathstrut -\mathstrut 26q^{97} \) \(\mathstrut -\mathstrut 3q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(4\) \(x^{11}\mathstrut -\mathstrut \) \(13\) \(x^{10}\mathstrut +\mathstrut \) \(60\) \(x^{9}\mathstrut +\mathstrut \) \(48\) \(x^{8}\mathstrut -\mathstrut \) \(289\) \(x^{7}\mathstrut -\mathstrut \) \(89\) \(x^{6}\mathstrut +\mathstrut \) \(602\) \(x^{5}\mathstrut +\mathstrut \) \(161\) \(x^{4}\mathstrut -\mathstrut \) \(555\) \(x^{3}\mathstrut -\mathstrut \) \(197\) \(x^{2}\mathstrut +\mathstrut \) \(181\) \(x\mathstrut +\mathstrut \) \(82\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 2338 \nu^{11} - 11665 \nu^{10} - 20131 \nu^{9} + 163345 \nu^{8} - 30162 \nu^{7} - 687740 \nu^{6} + 380492 \nu^{5} + 1189576 \nu^{4} - 606967 \nu^{3} - 905001 \nu^{2} + 253934 \nu + 253800 \)\()/2599\)
\(\beta_{3}\)\(=\)\((\)\( -2547 \nu^{11} + 12262 \nu^{10} + 23568 \nu^{9} - 172850 \nu^{8} + 11984 \nu^{7} + 736444 \nu^{6} - 344410 \nu^{5} - 1285735 \nu^{4} + 587872 \nu^{3} + 974824 \nu^{2} - 256909 \nu - 271839 \)\()/2599\)
\(\beta_{4}\)\(=\)\((\)\( -4271 \nu^{11} + 20370 \nu^{10} + 40491 \nu^{9} - 289158 \nu^{8} + 7917 \nu^{7} + 1252500 \nu^{6} - 537826 \nu^{5} - 2258499 \nu^{4} + 946617 \nu^{3} + 1803505 \nu^{2} - 421447 \nu - 531563 \)\()/2599\)
\(\beta_{5}\)\(=\)\((\)\( 4271 \nu^{11} - 20370 \nu^{10} - 40491 \nu^{9} + 289158 \nu^{8} - 7917 \nu^{7} - 1252500 \nu^{6} + 537826 \nu^{5} + 2258499 \nu^{4} - 946617 \nu^{3} - 1800906 \nu^{2} + 418848 \nu + 523766 \)\()/2599\)
\(\beta_{6}\)\(=\)\((\)\( 5174 \nu^{11} - 24852 \nu^{10} - 48663 \nu^{9} + 352472 \nu^{8} - 13528 \nu^{7} - 1523042 \nu^{6} + 656567 \nu^{5} + 2730194 \nu^{4} - 1130395 \nu^{3} - 2141268 \nu^{2} + 488631 \nu + 604080 \)\()/2599\)
\(\beta_{7}\)\(=\)\((\)\( 7177 \nu^{11} - 33844 \nu^{10} - 70684 \nu^{9} + 485510 \nu^{8} + 20451 \nu^{7} - 2146631 \nu^{6} + 789903 \nu^{5} + 3978285 \nu^{4} - 1463177 \nu^{3} - 3242526 \nu^{2} + 666268 \nu + 949639 \)\()/2599\)
\(\beta_{8}\)\(=\)\((\)\( 7808 \nu^{11} - 36691 \nu^{10} - 76783 \nu^{9} + 523720 \nu^{8} + 23527 \nu^{7} - 2291872 \nu^{6} + 831696 \nu^{5} + 4179055 \nu^{4} - 1509014 \nu^{3} - 3334225 \nu^{2} + 673024 \nu + 955056 \)\()/2599\)
\(\beta_{9}\)\(=\)\((\)\( 359 \nu^{11} - 1664 \nu^{10} - 3574 \nu^{9} + 23728 \nu^{8} + 1593 \nu^{7} - 103673 \nu^{6} + 36826 \nu^{5} + 188615 \nu^{4} - 68695 \nu^{3} - 150396 \nu^{2} + 31304 \nu + 43294 \)\()/113\)
\(\beta_{10}\)\(=\)\((\)\( 8258 \nu^{11} - 38536 \nu^{10} - 81149 \nu^{9} + 547518 \nu^{8} + 25074 \nu^{7} - 2374515 \nu^{6} + 868264 \nu^{5} + 4272685 \nu^{4} - 1552486 \nu^{3} - 3360792 \nu^{2} + 674617 \nu + 952399 \)\()/2599\)
\(\beta_{11}\)\(=\)\((\)\( -9490 \nu^{11} + 44107 \nu^{10} + 93749 \nu^{9} - 627896 \nu^{8} - 32509 \nu^{7} + 2735205 \nu^{6} - 1007803 \nu^{5} - 4961670 \nu^{4} + 1855766 \nu^{3} + 3959645 \nu^{2} - 838245 \nu - 1146553 \)\()/2599\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{5}\)\(=\)\(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(7\) \(\beta_{9}\mathstrut -\mathstrut \) \(10\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(8\) \(\beta_{6}\mathstrut +\mathstrut \) \(22\) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(54\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{6}\)\(=\)\(39\) \(\beta_{11}\mathstrut +\mathstrut \) \(17\) \(\beta_{10}\mathstrut +\mathstrut \) \(26\) \(\beta_{9}\mathstrut -\mathstrut \) \(5\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(28\) \(\beta_{6}\mathstrut +\mathstrut \) \(107\) \(\beta_{5}\mathstrut +\mathstrut \) \(62\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(78\) \(\beta_{1}\mathstrut +\mathstrut \) \(125\)
\(\nu^{7}\)\(=\)\(200\) \(\beta_{11}\mathstrut +\mathstrut \) \(102\) \(\beta_{10}\mathstrut +\mathstrut \) \(119\) \(\beta_{9}\mathstrut -\mathstrut \) \(98\) \(\beta_{8}\mathstrut -\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(49\) \(\beta_{6}\mathstrut +\mathstrut \) \(231\) \(\beta_{5}\mathstrut -\mathstrut \) \(24\) \(\beta_{4}\mathstrut +\mathstrut \) \(33\) \(\beta_{3}\mathstrut -\mathstrut \) \(146\) \(\beta_{2}\mathstrut +\mathstrut \) \(456\) \(\beta_{1}\mathstrut +\mathstrut \) \(150\)
\(\nu^{8}\)\(=\)\(533\) \(\beta_{11}\mathstrut +\mathstrut \) \(220\) \(\beta_{10}\mathstrut +\mathstrut \) \(390\) \(\beta_{9}\mathstrut -\mathstrut \) \(102\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(306\) \(\beta_{6}\mathstrut +\mathstrut \) \(1050\) \(\beta_{5}\mathstrut +\mathstrut \) \(502\) \(\beta_{4}\mathstrut +\mathstrut \) \(49\) \(\beta_{3}\mathstrut +\mathstrut \) \(27\) \(\beta_{2}\mathstrut +\mathstrut \) \(740\) \(\beta_{1}\mathstrut +\mathstrut \) \(1045\)
\(\nu^{9}\)\(=\)\(2293\) \(\beta_{11}\mathstrut +\mathstrut \) \(1067\) \(\beta_{10}\mathstrut +\mathstrut \) \(1531\) \(\beta_{9}\mathstrut -\mathstrut \) \(991\) \(\beta_{8}\mathstrut -\mathstrut \) \(87\) \(\beta_{7}\mathstrut +\mathstrut \) \(226\) \(\beta_{6}\mathstrut +\mathstrut \) \(2466\) \(\beta_{5}\mathstrut -\mathstrut \) \(186\) \(\beta_{4}\mathstrut +\mathstrut \) \(408\) \(\beta_{3}\mathstrut -\mathstrut \) \(1192\) \(\beta_{2}\mathstrut +\mathstrut \) \(4122\) \(\beta_{1}\mathstrut +\mathstrut \) \(1479\)
\(\nu^{10}\)\(=\)\(6431\) \(\beta_{11}\mathstrut +\mathstrut \) \(2602\) \(\beta_{10}\mathstrut +\mathstrut \) \(4889\) \(\beta_{9}\mathstrut -\mathstrut \) \(1482\) \(\beta_{8}\mathstrut +\mathstrut \) \(204\) \(\beta_{7}\mathstrut -\mathstrut \) \(3113\) \(\beta_{6}\mathstrut +\mathstrut \) \(10537\) \(\beta_{5}\mathstrut +\mathstrut \) \(4247\) \(\beta_{4}\mathstrut +\mathstrut \) \(765\) \(\beta_{3}\mathstrut +\mathstrut \) \(202\) \(\beta_{2}\mathstrut +\mathstrut \) \(7578\) \(\beta_{1}\mathstrut +\mathstrut \) \(9443\)
\(\nu^{11}\)\(=\)\(25282\) \(\beta_{11}\mathstrut +\mathstrut \) \(11239\) \(\beta_{10}\mathstrut +\mathstrut \) \(17863\) \(\beta_{9}\mathstrut -\mathstrut \) \(10168\) \(\beta_{8}\mathstrut -\mathstrut \) \(604\) \(\beta_{7}\mathstrut +\mathstrut \) \(164\) \(\beta_{6}\mathstrut +\mathstrut \) \(26631\) \(\beta_{5}\mathstrut -\mathstrut \) \(914\) \(\beta_{4}\mathstrut +\mathstrut \) \(4595\) \(\beta_{3}\mathstrut -\mathstrut \) \(10026\) \(\beta_{2}\mathstrut +\mathstrut \) \(39156\) \(\beta_{1}\mathstrut +\mathstrut \) \(15700\)

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
3.27119
2.43162
2.36915
1.85534
1.18762
0.859886
−0.660935
−0.692763
−0.864716
−1.18842
−1.70453
−2.86344
0 −3.27119 0 3.82141 0 −1.10063 0 7.70067 0
1.2 0 −2.43162 0 −0.659543 0 4.52406 0 2.91279 0
1.3 0 −2.36915 0 2.61284 0 −2.25519 0 2.61289 0
1.4 0 −1.85534 0 −3.75753 0 −3.33364 0 0.442279 0
1.5 0 −1.18762 0 1.03337 0 2.52681 0 −1.58957 0
1.6 0 −0.859886 0 2.49803 0 −3.53399 0 −2.26060 0
1.7 0 0.660935 0 2.45345 0 3.10518 0 −2.56317 0
1.8 0 0.692763 0 −3.58411 0 3.81064 0 −2.52008 0
1.9 0 0.864716 0 −1.40497 0 −2.45790 0 −2.25227 0
1.10 0 1.18842 0 0.677556 0 1.46203 0 −1.58766 0
1.11 0 1.70453 0 −0.637713 0 1.14740 0 −0.0945840 0
1.12 0 2.86344 0 −0.0527883 0 −1.89476 0 5.19930 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(17\) \(1\)
\(59\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4012))\):

\(T_{3}^{12} + \cdots\)
\(T_{5}^{12} - \cdots\)