Properties

Label 6032.2.a.be
Level 6032
Weight 2
Character orbit 6032.a
Self dual Yes
Analytic conductor 48.166
Analytic rank 0
Dimension 13
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{12}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{7} q^{5} \) \( -\beta_{5} q^{7} \) \( + ( 2 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{3} \) \( -\beta_{7} q^{5} \) \( -\beta_{5} q^{7} \) \( + ( 2 + \beta_{2} ) q^{9} \) \( + \beta_{8} q^{11} \) \(- q^{13}\) \( + ( -1 + \beta_{1} - \beta_{3} - \beta_{7} ) q^{15} \) \( -\beta_{10} q^{17} \) \( -\beta_{11} q^{19} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} + \beta_{9} + \beta_{12} ) q^{21} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{10} + \beta_{12} ) q^{23} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{11} - \beta_{12} ) q^{25} \) \( + ( 2 + \beta_{3} - \beta_{4} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} ) q^{27} \) \(+ q^{29}\) \( + ( -1 + \beta_{1} + \beta_{4} - \beta_{12} ) q^{31} \) \( + ( 2 - \beta_{7} + \beta_{8} + \beta_{10} ) q^{33} \) \( + ( 1 + \beta_{2} - 2 \beta_{5} + \beta_{7} - \beta_{10} + \beta_{11} ) q^{35} \) \( + ( 2 + \beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{12} ) q^{37} \) \( -\beta_{1} q^{39} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} + \beta_{12} ) q^{41} \) \( + ( 3 \beta_{1} + 2 \beta_{2} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} - \beta_{12} ) q^{43} \) \( + ( -2 \beta_{1} - 2 \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{12} ) q^{45} \) \( + ( -\beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{47} \) \( + ( 1 + \beta_{1} + \beta_{2} - \beta_{6} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{49} \) \( + ( -2 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{51} \) \( + ( 2 - \beta_{3} + \beta_{6} + \beta_{9} ) q^{53} \) \( + ( \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{8} - \beta_{11} ) q^{55} \) \( + ( 2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{12} ) q^{57} \) \( + ( -1 + \beta_{1} - \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} ) q^{59} \) \( + ( 1 + 2 \beta_{1} + \beta_{2} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - \beta_{11} - 2 \beta_{12} ) q^{61} \) \( + ( -1 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} + 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{63} \) \( + \beta_{7} q^{65} \) \( + ( 2 - \beta_{1} - \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} + 2 \beta_{12} ) q^{67} \) \( + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} ) q^{69} \) \( + ( -3 - 3 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + 2 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{11} + 2 \beta_{12} ) q^{71} \) \( + ( 2 - 3 \beta_{1} - 3 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + 4 \beta_{8} + \beta_{9} + \beta_{10} + \beta_{12} ) q^{73} \) \( + ( 3 + 3 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{6} - \beta_{7} - 2 \beta_{8} - 2 \beta_{11} - 3 \beta_{12} ) q^{75} \) \( + ( 1 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} + 3 \beta_{11} + 2 \beta_{12} ) q^{77} \) \( + ( \beta_{1} + 2 \beta_{2} + \beta_{4} - \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} - \beta_{12} ) q^{79} \) \( + ( 6 - \beta_{1} + \beta_{3} - \beta_{4} + \beta_{6} + 3 \beta_{7} + 2 \beta_{8} + 2 \beta_{9} + 2 \beta_{10} ) q^{81} \) \( + ( 1 - \beta_{1} + \beta_{3} - \beta_{5} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{83} \) \( + ( 1 + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{7} - 2 \beta_{8} - \beta_{10} - 2 \beta_{11} - \beta_{12} ) q^{85} \) \( + \beta_{1} q^{87} \) \( + ( 3 + \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} ) q^{89} \) \( + \beta_{5} q^{91} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 2 \beta_{8} - \beta_{10} - 2 \beta_{11} - 3 \beta_{12} ) q^{93} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{95} \) \( + ( 2 - \beta_{1} + \beta_{4} - 2 \beta_{5} - \beta_{7} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{97} \) \( + ( 3 - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{12} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(13q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(13q \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 5q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut -\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 13q^{13} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 4q^{17} \) \(\mathstrut +\mathstrut 3q^{19} \) \(\mathstrut +\mathstrut 5q^{21} \) \(\mathstrut -\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut 24q^{25} \) \(\mathstrut +\mathstrut 16q^{27} \) \(\mathstrut +\mathstrut 13q^{29} \) \(\mathstrut -\mathstrut 21q^{31} \) \(\mathstrut +\mathstrut 21q^{33} \) \(\mathstrut -\mathstrut 8q^{35} \) \(\mathstrut +\mathstrut 23q^{37} \) \(\mathstrut -\mathstrut 4q^{39} \) \(\mathstrut +\mathstrut 4q^{41} \) \(\mathstrut +\mathstrut 24q^{43} \) \(\mathstrut +\mathstrut 9q^{45} \) \(\mathstrut -\mathstrut 7q^{47} \) \(\mathstrut +\mathstrut 41q^{49} \) \(\mathstrut +\mathstrut q^{51} \) \(\mathstrut +\mathstrut 17q^{53} \) \(\mathstrut +\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 24q^{57} \) \(\mathstrut -\mathstrut 11q^{59} \) \(\mathstrut +\mathstrut 26q^{61} \) \(\mathstrut -\mathstrut 17q^{63} \) \(\mathstrut -\mathstrut 5q^{65} \) \(\mathstrut +\mathstrut 35q^{67} \) \(\mathstrut +\mathstrut 30q^{69} \) \(\mathstrut -\mathstrut 30q^{71} \) \(\mathstrut +\mathstrut 17q^{73} \) \(\mathstrut +\mathstrut 43q^{75} \) \(\mathstrut +\mathstrut 34q^{77} \) \(\mathstrut -\mathstrut 3q^{79} \) \(\mathstrut +\mathstrut 37q^{81} \) \(\mathstrut +\mathstrut 18q^{83} \) \(\mathstrut +\mathstrut 9q^{85} \) \(\mathstrut +\mathstrut 4q^{87} \) \(\mathstrut +\mathstrut 38q^{89} \) \(\mathstrut +\mathstrut 6q^{91} \) \(\mathstrut +\mathstrut 25q^{93} \) \(\mathstrut +\mathstrut 9q^{95} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 40q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{13}\mathstrut -\mathstrut \) \(4\) \(x^{12}\mathstrut -\mathstrut \) \(22\) \(x^{11}\mathstrut +\mathstrut \) \(96\) \(x^{10}\mathstrut +\mathstrut \) \(159\) \(x^{9}\mathstrut -\mathstrut \) \(827\) \(x^{8}\mathstrut -\mathstrut \) \(362\) \(x^{7}\mathstrut +\mathstrut \) \(3029\) \(x^{6}\mathstrut -\mathstrut \) \(142\) \(x^{5}\mathstrut -\mathstrut \) \(4316\) \(x^{4}\mathstrut +\mathstrut \) \(208\) \(x^{3}\mathstrut +\mathstrut \) \(2228\) \(x^{2}\mathstrut +\mathstrut \) \(376\) \(x\mathstrut +\mathstrut \) \(16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(2367\) \(\nu^{12}\mathstrut -\mathstrut \) \(10978\) \(\nu^{11}\mathstrut -\mathstrut \) \(51546\) \(\nu^{10}\mathstrut +\mathstrut \) \(275072\) \(\nu^{9}\mathstrut +\mathstrut \) \(363465\) \(\nu^{8}\mathstrut -\mathstrut \) \(2514391\) \(\nu^{7}\mathstrut -\mathstrut \) \(735380\) \(\nu^{6}\mathstrut +\mathstrut \) \(10085083\) \(\nu^{5}\mathstrut -\mathstrut \) \(941380\) \(\nu^{4}\mathstrut -\mathstrut \) \(16858056\) \(\nu^{3}\mathstrut +\mathstrut \) \(1820292\) \(\nu^{2}\mathstrut +\mathstrut \) \(9950724\) \(\nu\mathstrut +\mathstrut \) \(744768\)\()/56872\)
\(\beta_{4}\)\(=\)\((\)\(2470\) \(\nu^{12}\mathstrut -\mathstrut \) \(8161\) \(\nu^{11}\mathstrut -\mathstrut \) \(59988\) \(\nu^{10}\mathstrut +\mathstrut \) \(200172\) \(\nu^{9}\mathstrut +\mathstrut \) \(515082\) \(\nu^{8}\mathstrut -\mathstrut \) \(1786081\) \(\nu^{7}\mathstrut -\mathstrut \) \(1787469\) \(\nu^{6}\mathstrut +\mathstrut \) \(6968222\) \(\nu^{5}\mathstrut +\mathstrut \) \(2010677\) \(\nu^{4}\mathstrut -\mathstrut \) \(11205734\) \(\nu^{3}\mathstrut -\mathstrut \) \(565546\) \(\nu^{2}\mathstrut +\mathstrut \) \(6365604\) \(\nu\mathstrut +\mathstrut \) \(527848\)\()/56872\)
\(\beta_{5}\)\(=\)\((\)\(6995\) \(\nu^{12}\mathstrut -\mathstrut \) \(32106\) \(\nu^{11}\mathstrut -\mathstrut \) \(145536\) \(\nu^{10}\mathstrut +\mathstrut \) \(783032\) \(\nu^{9}\mathstrut +\mathstrut \) \(916693\) \(\nu^{8}\mathstrut -\mathstrut \) \(6888979\) \(\nu^{7}\mathstrut -\mathstrut \) \(918558\) \(\nu^{6}\mathstrut +\mathstrut \) \(25979393\) \(\nu^{5}\mathstrut -\mathstrut \) \(6440700\) \(\nu^{4}\mathstrut -\mathstrut \) \(38815050\) \(\nu^{3}\mathstrut +\mathstrut \) \(7530800\) \(\nu^{2}\mathstrut +\mathstrut \) \(20983768\) \(\nu\mathstrut +\mathstrut \) \(1582928\)\()/113744\)
\(\beta_{6}\)\(=\)\((\)\(4738\) \(\nu^{12}\mathstrut -\mathstrut \) \(19707\) \(\nu^{11}\mathstrut -\mathstrut \) \(103972\) \(\nu^{10}\mathstrut +\mathstrut \) \(478768\) \(\nu^{9}\mathstrut +\mathstrut \) \(746898\) \(\nu^{8}\mathstrut -\mathstrut \) \(4196767\) \(\nu^{7}\mathstrut -\mathstrut \) \(1654419\) \(\nu^{6}\mathstrut +\mathstrut \) \(15809122\) \(\nu^{5}\mathstrut -\mathstrut \) \(975429\) \(\nu^{4}\mathstrut -\mathstrut \) \(23798686\) \(\nu^{3}\mathstrut +\mathstrut \) \(1706354\) \(\nu^{2}\mathstrut +\mathstrut \) \(13122276\) \(\nu\mathstrut +\mathstrut \) \(1339208\)\()/56872\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(4841\) \(\nu^{12}\mathstrut +\mathstrut \) \(16890\) \(\nu^{11}\mathstrut +\mathstrut \) \(112414\) \(\nu^{10}\mathstrut -\mathstrut \) \(403868\) \(\nu^{9}\mathstrut -\mathstrut \) \(898515\) \(\nu^{8}\mathstrut +\mathstrut \) \(3468457\) \(\nu^{7}\mathstrut +\mathstrut \) \(2706508\) \(\nu^{6}\mathstrut -\mathstrut \) \(12692261\) \(\nu^{5}\mathstrut -\mathstrut \) \(1919756\) \(\nu^{4}\mathstrut +\mathstrut \) \(18032620\) \(\nu^{3}\mathstrut +\mathstrut \) \(167636\) \(\nu^{2}\mathstrut -\mathstrut \) \(8797820\) \(\nu\mathstrut -\mathstrut \) \(724184\)\()/56872\)
\(\beta_{8}\)\(=\)\((\)\(3551\) \(\nu^{12}\mathstrut -\mathstrut \) \(13520\) \(\nu^{11}\mathstrut -\mathstrut \) \(77429\) \(\nu^{10}\mathstrut +\mathstrut \) \(318378\) \(\nu^{9}\mathstrut +\mathstrut \) \(544139\) \(\nu^{8}\mathstrut -\mathstrut \) \(2664327\) \(\nu^{7}\mathstrut -\mathstrut \) \(1077153\) \(\nu^{6}\mathstrut +\mathstrut \) \(9279180\) \(\nu^{5}\mathstrut -\mathstrut \) \(1347528\) \(\nu^{4}\mathstrut -\mathstrut \) \(11878575\) \(\nu^{3}\mathstrut +\mathstrut \) \(2031906\) \(\nu^{2}\mathstrut +\mathstrut \) \(5565662\) \(\nu\mathstrut +\mathstrut \) \(351916\)\()/28436\)
\(\beta_{9}\)\(=\)\((\)\(8417\) \(\nu^{12}\mathstrut -\mathstrut \) \(32349\) \(\nu^{11}\mathstrut -\mathstrut \) \(188234\) \(\nu^{10}\mathstrut +\mathstrut \) \(773776\) \(\nu^{9}\mathstrut +\mathstrut \) \(1403947\) \(\nu^{8}\mathstrut -\mathstrut \) \(6625222\) \(\nu^{7}\mathstrut -\mathstrut \) \(3511507\) \(\nu^{6}\mathstrut +\mathstrut \) \(23954233\) \(\nu^{5}\mathstrut -\mathstrut \) \(103513\) \(\nu^{4}\mathstrut -\mathstrut \) \(32942642\) \(\nu^{3}\mathstrut +\mathstrut \) \(1970822\) \(\nu^{2}\mathstrut +\mathstrut \) \(15635808\) \(\nu\mathstrut +\mathstrut \) \(1345080\)\()/56872\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(10575\) \(\nu^{12}\mathstrut +\mathstrut \) \(45316\) \(\nu^{11}\mathstrut +\mathstrut \) \(222236\) \(\nu^{10}\mathstrut -\mathstrut \) \(1081564\) \(\nu^{9}\mathstrut -\mathstrut \) \(1442093\) \(\nu^{8}\mathstrut +\mathstrut \) \(9213729\) \(\nu^{7}\mathstrut +\mathstrut \) \(1907216\) \(\nu^{6}\mathstrut -\mathstrut \) \(32937193\) \(\nu^{5}\mathstrut +\mathstrut \) \(7670382\) \(\nu^{4}\mathstrut +\mathstrut \) \(44376366\) \(\nu^{3}\mathstrut -\mathstrut \) \(8588108\) \(\nu^{2}\mathstrut -\mathstrut \) \(21895664\) \(\nu\mathstrut -\mathstrut \) \(1655392\)\()/56872\)
\(\beta_{11}\)\(=\)\((\)\(6406\) \(\nu^{12}\mathstrut -\mathstrut \) \(27371\) \(\nu^{11}\mathstrut -\mathstrut \) \(136809\) \(\nu^{10}\mathstrut +\mathstrut \) \(658084\) \(\nu^{9}\mathstrut +\mathstrut \) \(926640\) \(\nu^{8}\mathstrut -\mathstrut \) \(5679929\) \(\nu^{7}\mathstrut -\mathstrut \) \(1590984\) \(\nu^{6}\mathstrut +\mathstrut \) \(20833127\) \(\nu^{5}\mathstrut -\mathstrut \) \(3340897\) \(\nu^{4}\mathstrut -\mathstrut \) \(29700847\) \(\nu^{3}\mathstrut +\mathstrut \) \(4349914\) \(\nu^{2}\mathstrut +\mathstrut \) \(15262538\) \(\nu\mathstrut +\mathstrut \) \(1268504\)\()/28436\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(15926\) \(\nu^{12}\mathstrut +\mathstrut \) \(63373\) \(\nu^{11}\mathstrut +\mathstrut \) \(347324\) \(\nu^{10}\mathstrut -\mathstrut \) \(1512948\) \(\nu^{9}\mathstrut -\mathstrut \) \(2455250\) \(\nu^{8}\mathstrut +\mathstrut \) \(12933141\) \(\nu^{7}\mathstrut +\mathstrut \) \(5057513\) \(\nu^{6}\mathstrut -\mathstrut \) \(46757110\) \(\nu^{5}\mathstrut +\mathstrut \) \(5100207\) \(\nu^{4}\mathstrut +\mathstrut \) \(64867606\) \(\nu^{3}\mathstrut -\mathstrut \) \(8111670\) \(\nu^{2}\mathstrut -\mathstrut \) \(32552268\) \(\nu\mathstrut -\mathstrut \) \(2862032\)\()/56872\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(42\)
\(\nu^{5}\)\(=\)\(13\) \(\beta_{10}\mathstrut +\mathstrut \) \(13\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(15\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(43\) \(\beta_{1}\mathstrut +\mathstrut \) \(36\)
\(\nu^{6}\)\(=\)\(-\)\(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(30\) \(\beta_{10}\mathstrut +\mathstrut \) \(31\) \(\beta_{9}\mathstrut +\mathstrut \) \(29\) \(\beta_{8}\mathstrut +\mathstrut \) \(48\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(13\) \(\beta_{4}\mathstrut +\mathstrut \) \(18\) \(\beta_{3}\mathstrut +\mathstrut \) \(84\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(389\)
\(\nu^{7}\)\(=\)\(-\)\(5\) \(\beta_{12}\mathstrut -\mathstrut \) \(4\) \(\beta_{11}\mathstrut +\mathstrut \) \(141\) \(\beta_{10}\mathstrut +\mathstrut \) \(147\) \(\beta_{9}\mathstrut +\mathstrut \) \(155\) \(\beta_{8}\mathstrut +\mathstrut \) \(184\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(36\) \(\beta_{5}\mathstrut -\mathstrut \) \(123\) \(\beta_{4}\mathstrut +\mathstrut \) \(162\) \(\beta_{3}\mathstrut +\mathstrut \) \(64\) \(\beta_{2}\mathstrut +\mathstrut \) \(346\) \(\beta_{1}\mathstrut +\mathstrut \) \(514\)
\(\nu^{8}\)\(=\)\(-\)\(55\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(359\) \(\beta_{10}\mathstrut +\mathstrut \) \(384\) \(\beta_{9}\mathstrut +\mathstrut \) \(327\) \(\beta_{8}\mathstrut +\mathstrut \) \(606\) \(\beta_{7}\mathstrut +\mathstrut \) \(80\) \(\beta_{6}\mathstrut -\mathstrut \) \(48\) \(\beta_{5}\mathstrut -\mathstrut \) \(149\) \(\beta_{4}\mathstrut +\mathstrut \) \(247\) \(\beta_{3}\mathstrut +\mathstrut \) \(828\) \(\beta_{2}\mathstrut +\mathstrut \) \(128\) \(\beta_{1}\mathstrut +\mathstrut \) \(3788\)
\(\nu^{9}\)\(=\)\(-\)\(148\) \(\beta_{12}\mathstrut -\mathstrut \) \(110\) \(\beta_{11}\mathstrut +\mathstrut \) \(1469\) \(\beta_{10}\mathstrut +\mathstrut \) \(1609\) \(\beta_{9}\mathstrut +\mathstrut \) \(1586\) \(\beta_{8}\mathstrut +\mathstrut \) \(2128\) \(\beta_{7}\mathstrut +\mathstrut \) \(116\) \(\beta_{6}\mathstrut -\mathstrut \) \(470\) \(\beta_{5}\mathstrut -\mathstrut \) \(1221\) \(\beta_{4}\mathstrut +\mathstrut \) \(1771\) \(\beta_{3}\mathstrut +\mathstrut \) \(1001\) \(\beta_{2}\mathstrut +\mathstrut \) \(3040\) \(\beta_{1}\mathstrut +\mathstrut \) \(6587\)
\(\nu^{10}\)\(=\)\(-\)\(1014\) \(\beta_{12}\mathstrut -\mathstrut \) \(296\) \(\beta_{11}\mathstrut +\mathstrut \) \(4018\) \(\beta_{10}\mathstrut +\mathstrut \) \(4439\) \(\beta_{9}\mathstrut +\mathstrut \) \(3400\) \(\beta_{8}\mathstrut +\mathstrut \) \(7110\) \(\beta_{7}\mathstrut +\mathstrut \) \(545\) \(\beta_{6}\mathstrut -\mathstrut \) \(776\) \(\beta_{5}\mathstrut -\mathstrut \) \(1690\) \(\beta_{4}\mathstrut +\mathstrut \) \(3105\) \(\beta_{3}\mathstrut +\mathstrut \) \(8507\) \(\beta_{2}\mathstrut +\mathstrut \) \(2704\) \(\beta_{1}\mathstrut +\mathstrut \) \(38125\)
\(\nu^{11}\)\(=\)\(-\)\(2891\) \(\beta_{12}\mathstrut -\mathstrut \) \(2028\) \(\beta_{11}\mathstrut +\mathstrut \) \(15207\) \(\beta_{10}\mathstrut +\mathstrut \) \(17444\) \(\beta_{9}\mathstrut +\mathstrut \) \(15658\) \(\beta_{8}\mathstrut +\mathstrut \) \(24109\) \(\beta_{7}\mathstrut +\mathstrut \) \(748\) \(\beta_{6}\mathstrut -\mathstrut \) \(5498\) \(\beta_{5}\mathstrut -\mathstrut \) \(12069\) \(\beta_{4}\mathstrut +\mathstrut \) \(18983\) \(\beta_{3}\mathstrut +\mathstrut \) \(13815\) \(\beta_{2}\mathstrut +\mathstrut \) \(28586\) \(\beta_{1}\mathstrut +\mathstrut \) \(79661\)
\(\nu^{12}\)\(=\)\(-\)\(15778\) \(\beta_{12}\mathstrut -\mathstrut \) \(5782\) \(\beta_{11}\mathstrut +\mathstrut \) \(43817\) \(\beta_{10}\mathstrut +\mathstrut \) \(49936\) \(\beta_{9}\mathstrut +\mathstrut \) \(34068\) \(\beta_{8}\mathstrut +\mathstrut \) \(81073\) \(\beta_{7}\mathstrut +\mathstrut \) \(2626\) \(\beta_{6}\mathstrut -\mathstrut \) \(10750\) \(\beta_{5}\mathstrut -\mathstrut \) \(19094\) \(\beta_{4}\mathstrut +\mathstrut \) \(37495\) \(\beta_{3}\mathstrut +\mathstrut \) \(89969\) \(\beta_{2}\mathstrut +\mathstrut \) \(40683\) \(\beta_{1}\mathstrut +\mathstrut \) \(392822\)

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
−2.90994
−2.69571
−2.47527
−0.993990
−0.863805
−0.0989896
−0.0772773
1.32867
1.82663
2.07047
2.30310
3.23329
3.35280
0 −2.90994 0 −1.66472 0 −2.42557 0 5.46773 0
1.2 0 −2.69571 0 2.65571 0 −3.33796 0 4.26683 0
1.3 0 −2.47527 0 2.58551 0 4.52168 0 3.12695 0
1.4 0 −0.993990 0 3.24551 0 −4.74513 0 −2.01198 0
1.5 0 −0.863805 0 −2.53485 0 −1.34341 0 −2.25384 0
1.6 0 −0.0989896 0 −2.29988 0 3.37309 0 −2.99020 0
1.7 0 −0.0772773 0 0.908464 0 −0.210501 0 −2.99403 0
1.8 0 1.32867 0 −1.97424 0 1.77375 0 −1.23465 0
1.9 0 1.82663 0 2.84424 0 3.25048 0 0.336589 0
1.10 0 2.07047 0 −1.28770 0 −4.21640 0 1.28687 0
1.11 0 2.30310 0 4.32944 0 −4.01657 0 2.30428 0
1.12 0 3.23329 0 1.87064 0 2.94834 0 7.45419 0
1.13 0 3.35280 0 −3.67813 0 −1.57180 0 8.24127 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
\(2\) \(1\)
\(13\) \(1\)
\(29\) \(-1\)

Hecke kernels

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

\(T_{3}^{13} - \cdots\)
\(T_{5}^{13} - \cdots\)