Properties

Label 6032.2.a.bd
Level 6032
Weight 2
Character orbit 6032.a
Self dual Yes
Analytic conductor 48.166
Analytic rank 1
Dimension 12
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6032.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1657624992\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
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.

\(f(q)\) \(=\) \( q\) \( + ( -1 + \beta_{1} ) q^{3} \) \( + \beta_{6} q^{5} \) \( + ( -1 - \beta_{5} ) q^{7} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -1 + \beta_{1} ) q^{3} \) \( + \beta_{6} q^{5} \) \( + ( -1 - \beta_{5} ) q^{7} \) \( + ( 2 - \beta_{1} + \beta_{2} ) q^{9} \) \( + ( -1 + \beta_{8} ) q^{11} \) \(+ q^{13}\) \( + ( \beta_{4} - \beta_{6} ) q^{15} \) \( + ( 1 - \beta_{1} + \beta_{5} - \beta_{8} + \beta_{11} ) q^{17} \) \( + ( -1 - \beta_{2} - \beta_{5} - \beta_{9} - \beta_{10} ) q^{19} \) \( + ( \beta_{5} + \beta_{7} + \beta_{8} - \beta_{11} ) q^{21} \) \( + ( -1 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{23} \) \( + ( 1 + \beta_{4} - \beta_{11} ) q^{25} \) \( + ( -3 + 2 \beta_{1} + \beta_{5} + \beta_{7} + \beta_{10} - \beta_{11} ) q^{27} \) \(+ q^{29}\) \( + ( -1 + \beta_{1} + \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{31} \) \( + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} + \beta_{11} ) q^{33} \) \( + ( -1 - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{11} ) q^{35} \) \( + ( -2 - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{10} ) q^{37} \) \( + ( -1 + \beta_{1} ) q^{39} \) \( + ( 1 - \beta_{1} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{41} \) \( + ( -1 - \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{9} ) q^{43} \) \( + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{8} - \beta_{10} + \beta_{11} ) q^{45} \) \( + ( -1 - \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{47} \) \( + ( 3 - 2 \beta_{1} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - 2 \beta_{10} + \beta_{11} ) q^{49} \) \( + ( -1 + \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{51} \) \( + ( 3 + \beta_{1} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - 2 \beta_{9} - \beta_{10} - 3 \beta_{11} ) q^{53} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{55} \) \( + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{6} - \beta_{7} + \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{57} \) \( + ( -2 + \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} ) q^{59} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{61} \) \( + ( 2 - \beta_{1} - 2 \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - \beta_{9} + \beta_{11} ) q^{63} \) \( + \beta_{6} q^{65} \) \( + ( -5 + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} + \beta_{9} + 2 \beta_{10} + \beta_{11} ) q^{67} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} ) q^{69} \) \( + ( -3 - \beta_{1} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{9} + 2 \beta_{11} ) q^{71} \) \( + ( -4 + 2 \beta_{1} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{73} \) \( + ( -3 - \beta_{1} - \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{75} \) \( + ( 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} + \beta_{9} + \beta_{10} ) q^{77} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} ) q^{79} \) \( + ( 5 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - 4 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} - 2 \beta_{10} - \beta_{11} ) q^{81} \) \( + ( -2 + \beta_{1} + \beta_{2} + 3 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} - \beta_{9} - \beta_{10} - \beta_{11} ) q^{83} \) \( + ( -1 + \beta_{1} + 2 \beta_{3} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} + 3 \beta_{11} ) q^{85} \) \( + ( -1 + \beta_{1} ) q^{87} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{89} \) \( + ( -1 - \beta_{5} ) q^{91} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{8} - 3 \beta_{9} - 3 \beta_{10} + \beta_{11} ) q^{93} \) \( + ( 2 - 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} ) q^{95} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{8} + 3 \beta_{9} + 2 \beta_{10} + 2 \beta_{11} ) q^{97} \) \( + ( -\beta_{1} - \beta_{2} + 2 \beta_{4} + 3 \beta_{5} - \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} - \beta_{11} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut -\mathstrut 6q^{3} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut -\mathstrut 14q^{11} \) \(\mathstrut +\mathstrut 12q^{13} \) \(\mathstrut -\mathstrut 8q^{15} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut -\mathstrut 11q^{19} \) \(\mathstrut -\mathstrut 5q^{21} \) \(\mathstrut -\mathstrut 15q^{23} \) \(\mathstrut +\mathstrut 5q^{25} \) \(\mathstrut -\mathstrut 24q^{27} \) \(\mathstrut +\mathstrut 12q^{29} \) \(\mathstrut -\mathstrut 13q^{31} \) \(\mathstrut -\mathstrut q^{33} \) \(\mathstrut -\mathstrut 18q^{35} \) \(\mathstrut -\mathstrut 23q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 2q^{41} \) \(\mathstrut -\mathstrut 26q^{43} \) \(\mathstrut -\mathstrut 9q^{45} \) \(\mathstrut -\mathstrut 15q^{47} \) \(\mathstrut +\mathstrut 16q^{49} \) \(\mathstrut -\mathstrut 21q^{51} \) \(\mathstrut +\mathstrut 31q^{53} \) \(\mathstrut -\mathstrut 10q^{55} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut -\mathstrut 7q^{59} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut 25q^{63} \) \(\mathstrut +\mathstrut 3q^{65} \) \(\mathstrut -\mathstrut 47q^{67} \) \(\mathstrut -\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 32q^{71} \) \(\mathstrut -\mathstrut 25q^{73} \) \(\mathstrut -\mathstrut 31q^{75} \) \(\mathstrut -\mathstrut 4q^{77} \) \(\mathstrut -\mathstrut 7q^{79} \) \(\mathstrut +\mathstrut 64q^{81} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 7q^{85} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut 6q^{89} \) \(\mathstrut -\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 17q^{93} \) \(\mathstrut -\mathstrut q^{95} \) \(\mathstrut -\mathstrut 7q^{97} \) \(\mathstrut -\mathstrut 44q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(6\) \(x^{11}\mathstrut -\mathstrut \) \(10\) \(x^{10}\mathstrut +\mathstrut \) \(98\) \(x^{9}\mathstrut +\mathstrut \) \(10\) \(x^{8}\mathstrut -\mathstrut \) \(585\) \(x^{7}\mathstrut +\mathstrut \) \(151\) \(x^{6}\mathstrut +\mathstrut \) \(1524\) \(x^{5}\mathstrut -\mathstrut \) \(445\) \(x^{4}\mathstrut -\mathstrut \) \(1567\) \(x^{3}\mathstrut +\mathstrut \) \(273\) \(x^{2}\mathstrut +\mathstrut \) \(424\) \(x\mathstrut +\mathstrut \) \(68\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 4 \)
\(\beta_{3}\)\(=\)\((\)\( -2282 \nu^{11} + 16131 \nu^{10} + 9692 \nu^{9} - 248259 \nu^{8} + 163630 \nu^{7} + 1388641 \nu^{6} - 1280151 \nu^{5} - 3383822 \nu^{4} + 3048679 \nu^{3} + 3293805 \nu^{2} - 2419836 \nu - 852468 \)\()/65776\)
\(\beta_{4}\)\(=\)\((\)\( -6401 \nu^{11} + 33417 \nu^{10} + 92699 \nu^{9} - 597451 \nu^{8} - 381637 \nu^{7} + 3756664 \nu^{6} + 14221 \nu^{5} - 9634183 \nu^{4} + 2385714 \nu^{3} + 8494061 \nu^{2} - 2534708 \nu - 1171164 \)\()/131552\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(16367\) \(\nu^{11}\mathstrut +\mathstrut \) \(101865\) \(\nu^{10}\mathstrut +\mathstrut \) \(132713\) \(\nu^{9}\mathstrut -\mathstrut \) \(1588671\) \(\nu^{8}\mathstrut +\mathstrut \) \(265149\) \(\nu^{7}\mathstrut +\mathstrut \) \(8922602\) \(\nu^{6}\mathstrut -\mathstrut \) \(4612487\) \(\nu^{5}\mathstrut -\mathstrut \) \(21371169\) \(\nu^{4}\mathstrut +\mathstrut \) \(12153396\) \(\nu^{3}\mathstrut +\mathstrut \) \(19074001\) \(\nu^{2}\mathstrut -\mathstrut \) \(9323964\) \(\nu\mathstrut -\mathstrut \) \(3423820\)\()/131552\)
\(\beta_{6}\)\(=\)\((\)\(17223\) \(\nu^{11}\mathstrut -\mathstrut \) \(109739\) \(\nu^{10}\mathstrut -\mathstrut \) \(138813\) \(\nu^{9}\mathstrut +\mathstrut \) \(1780553\) \(\nu^{8}\mathstrut -\mathstrut \) \(425221\) \(\nu^{7}\mathstrut -\mathstrut \) \(10457092\) \(\nu^{6}\mathstrut +\mathstrut \) \(6357337\) \(\nu^{5}\mathstrut +\mathstrut \) \(26262073\) \(\nu^{4}\mathstrut -\mathstrut \) \(17298418\) \(\nu^{3}\mathstrut -\mathstrut \) \(24602727\) \(\nu^{2}\mathstrut +\mathstrut \) \(13195940\) \(\nu\mathstrut +\mathstrut \) \(4767844\)\()/131552\)
\(\beta_{7}\)\(=\)\((\)\(10119\) \(\nu^{11}\mathstrut -\mathstrut \) \(60913\) \(\nu^{10}\mathstrut -\mathstrut \) \(97717\) \(\nu^{9}\mathstrut +\mathstrut \) \(985031\) \(\nu^{8}\mathstrut +\mathstrut \) \(31695\) \(\nu^{7}\mathstrut -\mathstrut \) \(5750850\) \(\nu^{6}\mathstrut +\mathstrut \) \(2040411\) \(\nu^{5}\mathstrut +\mathstrut \) \(14264261\) \(\nu^{4}\mathstrut -\mathstrut \) \(6191772\) \(\nu^{3}\mathstrut -\mathstrut \) \(13011125\) \(\nu^{2}\mathstrut +\mathstrut \) \(5035512\) \(\nu\mathstrut +\mathstrut \) \(2366212\)\()/65776\)
\(\beta_{8}\)\(=\)\((\)\(26939\) \(\nu^{11}\mathstrut -\mathstrut \) \(166263\) \(\nu^{10}\mathstrut -\mathstrut \) \(237481\) \(\nu^{9}\mathstrut +\mathstrut \) \(2671453\) \(\nu^{8}\mathstrut -\mathstrut \) \(268833\) \(\nu^{7}\mathstrut -\mathstrut \) \(15551452\) \(\nu^{6}\mathstrut +\mathstrut \) \(7449581\) \(\nu^{5}\mathstrut +\mathstrut \) \(38773061\) \(\nu^{4}\mathstrut -\mathstrut \) \(21268834\) \(\nu^{3}\mathstrut -\mathstrut \) \(36029363\) \(\nu^{2}\mathstrut +\mathstrut \) \(16798324\) \(\nu\mathstrut +\mathstrut \) \(6458996\)\()/131552\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(38513\) \(\nu^{11}\mathstrut +\mathstrut \) \(239513\) \(\nu^{10}\mathstrut +\mathstrut \) \(332907\) \(\nu^{9}\mathstrut -\mathstrut \) \(3856379\) \(\nu^{8}\mathstrut +\mathstrut \) \(492779\) \(\nu^{7}\mathstrut +\mathstrut \) \(22531384\) \(\nu^{6}\mathstrut -\mathstrut \) \(11171843\) \(\nu^{5}\mathstrut -\mathstrut \) \(56665175\) \(\nu^{4}\mathstrut +\mathstrut \) \(31063282\) \(\nu^{3}\mathstrut +\mathstrut \) \(53890813\) \(\nu^{2}\mathstrut -\mathstrut \) \(23786804\) \(\nu\mathstrut -\mathstrut \) \(10216572\)\()/131552\)
\(\beta_{10}\)\(=\)\((\)\(46969\) \(\nu^{11}\mathstrut -\mathstrut \) \(299085\) \(\nu^{10}\mathstrut -\mathstrut \) \(354899\) \(\nu^{9}\mathstrut +\mathstrut \) \(4688943\) \(\nu^{8}\mathstrut -\mathstrut \) \(1186075\) \(\nu^{7}\mathstrut -\mathstrut \) \(26615124\) \(\nu^{6}\mathstrut +\mathstrut \) \(15634207\) \(\nu^{5}\mathstrut +\mathstrut \) \(65014311\) \(\nu^{4}\mathstrut -\mathstrut \) \(39863766\) \(\nu^{3}\mathstrut -\mathstrut \) \(60353793\) \(\nu^{2}\mathstrut +\mathstrut \) \(29111868\) \(\nu\mathstrut +\mathstrut \) \(11916636\)\()/131552\)
\(\beta_{11}\)\(=\)\((\)\(25420\) \(\nu^{11}\mathstrut -\mathstrut \) \(159523\) \(\nu^{10}\mathstrut -\mathstrut \) \(208810\) \(\nu^{9}\mathstrut +\mathstrut \) \(2535167\) \(\nu^{8}\mathstrut -\mathstrut \) \(428768\) \(\nu^{7}\mathstrut -\mathstrut \) \(14597111\) \(\nu^{6}\mathstrut +\mathstrut \) \(7551271\) \(\nu^{5}\mathstrut +\mathstrut \) \(36085832\) \(\nu^{4}\mathstrut -\mathstrut \) \(20112733\) \(\nu^{3}\mathstrut -\mathstrut \) \(33453693\) \(\nu^{2}\mathstrut +\mathstrut \) \(15258344\) \(\nu\mathstrut +\mathstrut \) \(6086412\)\()/65776\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{4}\)\(=\)\(-\)\(5\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(5\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{5}\)\(=\)\(-\)\(29\) \(\beta_{11}\mathstrut +\mathstrut \) \(15\) \(\beta_{10}\mathstrut -\mathstrut \) \(6\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(31\) \(\beta_{7}\mathstrut +\mathstrut \) \(7\) \(\beta_{6}\mathstrut +\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(68\) \(\beta_{2}\mathstrut +\mathstrut \) \(97\) \(\beta_{1}\mathstrut +\mathstrut \) \(74\)
\(\nu^{6}\)\(=\)\(-\)\(129\) \(\beta_{11}\mathstrut +\mathstrut \) \(47\) \(\beta_{10}\mathstrut -\mathstrut \) \(42\) \(\beta_{9}\mathstrut -\mathstrut \) \(18\) \(\beta_{8}\mathstrut +\mathstrut \) \(140\) \(\beta_{7}\mathstrut +\mathstrut \) \(40\) \(\beta_{6}\mathstrut -\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(67\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(308\) \(\beta_{2}\mathstrut +\mathstrut \) \(335\) \(\beta_{1}\mathstrut +\mathstrut \) \(371\)
\(\nu^{7}\)\(=\)\(-\)\(608\) \(\beta_{11}\mathstrut +\mathstrut \) \(236\) \(\beta_{10}\mathstrut -\mathstrut \) \(163\) \(\beta_{9}\mathstrut -\mathstrut \) \(45\) \(\beta_{8}\mathstrut +\mathstrut \) \(675\) \(\beta_{7}\mathstrut +\mathstrut \) \(199\) \(\beta_{6}\mathstrut +\mathstrut \) \(21\) \(\beta_{5}\mathstrut +\mathstrut \) \(332\) \(\beta_{4}\mathstrut -\mathstrut \) \(31\) \(\beta_{3}\mathstrut +\mathstrut \) \(1302\) \(\beta_{2}\mathstrut +\mathstrut \) \(1498\) \(\beta_{1}\mathstrut +\mathstrut \) \(1246\)
\(\nu^{8}\)\(=\)\(-\)\(2645\) \(\beta_{11}\mathstrut +\mathstrut \) \(897\) \(\beta_{10}\mathstrut -\mathstrut \) \(813\) \(\beta_{9}\mathstrut -\mathstrut \) \(308\) \(\beta_{8}\mathstrut +\mathstrut \) \(2957\) \(\beta_{7}\mathstrut +\mathstrut \) \(945\) \(\beta_{6}\mathstrut -\mathstrut \) \(150\) \(\beta_{5}\mathstrut +\mathstrut \) \(1541\) \(\beta_{4}\mathstrut -\mathstrut \) \(135\) \(\beta_{3}\mathstrut +\mathstrut \) \(5640\) \(\beta_{2}\mathstrut +\mathstrut \) \(5962\) \(\beta_{1}\mathstrut +\mathstrut \) \(5478\)
\(\nu^{9}\)\(=\)\(-\)\(11734\) \(\beta_{11}\mathstrut +\mathstrut \) \(4015\) \(\beta_{10}\mathstrut -\mathstrut \) \(3407\) \(\beta_{9}\mathstrut -\mathstrut \) \(1083\) \(\beta_{8}\mathstrut +\mathstrut \) \(13217\) \(\beta_{7}\mathstrut +\mathstrut \) \(4287\) \(\beta_{6}\mathstrut -\mathstrut \) \(488\) \(\beta_{5}\mathstrut +\mathstrut \) \(7028\) \(\beta_{4}\mathstrut -\mathstrut \) \(354\) \(\beta_{3}\mathstrut +\mathstrut \) \(24076\) \(\beta_{2}\mathstrut +\mathstrut \) \(25752\) \(\beta_{1}\mathstrut +\mathstrut \) \(21400\)
\(\nu^{10}\)\(=\)\(-\)\(50638\) \(\beta_{11}\mathstrut +\mathstrut \) \(16440\) \(\beta_{10}\mathstrut -\mathstrut \) \(15395\) \(\beta_{9}\mathstrut -\mathstrut \) \(5457\) \(\beta_{8}\mathstrut +\mathstrut \) \(57179\) \(\beta_{7}\mathstrut +\mathstrut \) \(19154\) \(\beta_{6}\mathstrut -\mathstrut \) \(3790\) \(\beta_{5}\mathstrut +\mathstrut \) \(31050\) \(\beta_{4}\mathstrut -\mathstrut \) \(1316\) \(\beta_{3}\mathstrut +\mathstrut \) \(103252\) \(\beta_{2}\mathstrut +\mathstrut \) \(107549\) \(\beta_{1}\mathstrut +\mathstrut \) \(91361\)
\(\nu^{11}\)\(=\)\(-\)\(219785\) \(\beta_{11}\mathstrut +\mathstrut \) \(71157\) \(\beta_{10}\mathstrut -\mathstrut \) \(65762\) \(\beta_{9}\mathstrut -\mathstrut \) \(21803\) \(\beta_{8}\mathstrut +\mathstrut \) \(248754\) \(\beta_{7}\mathstrut +\mathstrut \) \(83997\) \(\beta_{6}\mathstrut -\mathstrut \) \(16064\) \(\beta_{5}\mathstrut +\mathstrut \) \(136569\) \(\beta_{4}\mathstrut -\mathstrut \) \(3677\) \(\beta_{3}\mathstrut +\mathstrut \) \(441426\) \(\beta_{2}\mathstrut +\mathstrut \) \(460758\) \(\beta_{1}\mathstrut +\mathstrut \) \(377635\)

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
−2.41502
−2.05683
−1.89571
−1.22268
−0.285855
−0.260195
0.848556
1.32893
2.19118
2.59227
2.89888
4.27647
0 −3.41502 0 1.12461 0 3.99026 0 8.66236 0
1.2 0 −3.05683 0 −3.01704 0 1.81391 0 6.34423 0
1.3 0 −2.89571 0 2.36446 0 −4.24816 0 5.38516 0
1.4 0 −2.22268 0 2.71513 0 −1.08088 0 1.94031 0
1.5 0 −1.28586 0 −3.44298 0 −3.94227 0 −1.34658 0
1.6 0 −1.26019 0 0.631127 0 −0.921981 0 −1.41191 0
1.7 0 −0.151444 0 3.67772 0 0.901102 0 −2.97706 0
1.8 0 0.328928 0 −1.23337 0 3.96198 0 −2.89181 0
1.9 0 1.19118 0 3.27457 0 −4.85960 0 −1.58109 0
1.10 0 1.59227 0 −1.41565 0 −0.109839 0 −0.464663 0
1.11 0 1.89888 0 −0.496327 0 −2.15081 0 0.605764 0
1.12 0 3.27647 0 −1.18224 0 0.646306 0 7.73529 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
\(2\) \(1\)
\(13\) \(-1\)
\(29\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6032))\):

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