Properties

Label 4016.2.a.j
Level 4016
Weight 2
Character orbit 4016.a
Self dual Yes
Analytic conductor 32.068
Analytic rank 1
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\)  \(=\)  \( q\) \( - \beta_{1} q^{3} \) \( + \beta_{4} q^{5} \) \( + ( -1 + \beta_{9} ) q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} \) \(+O(q^{10})\) \( q\) \( - \beta_{1} q^{3} \) \( + \beta_{4} q^{5} \) \( + ( -1 + \beta_{9} ) q^{7} \) \( + ( 1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{9} \) \( + ( -1 + \beta_{8} ) q^{11} \) \( + ( - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{13} \) \( + ( - \beta_{4} - \beta_{8} - \beta_{9} - \beta_{13} ) q^{15} \) \( + ( \beta_{1} + \beta_{6} ) q^{17} \) \( + ( -2 - \beta_{5} + \beta_{13} ) q^{19} \) \( + ( - \beta_{2} - \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{11} ) q^{21} \) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{23} \) \( + ( 3 + \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{13} ) q^{25} \) \( + ( -1 - \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} ) q^{27} \) \( + ( -1 - \beta_{2} + \beta_{9} - \beta_{10} + \beta_{11} + \beta_{13} ) q^{29} \) \( + ( -2 + 2 \beta_{1} + \beta_{2} - \beta_{10} - \beta_{12} ) q^{31} \) \( + ( 3 \beta_{1} + \beta_{2} + 2 \beta_{5} + \beta_{6} + 2 \beta_{7} - \beta_{12} + \beta_{13} ) q^{33} \) \( + ( -1 + \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - 2 \beta_{7} + \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{35} \) \( + ( - \beta_{1} - \beta_{2} - \beta_{5} - \beta_{6} - 2 \beta_{10} + \beta_{12} ) q^{37} \) \( + ( -1 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} ) q^{39} \) \( + ( 2 - 2 \beta_{1} - 2 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{12} - \beta_{13} ) q^{41} \) \( + ( -2 + \beta_{1} + 2 \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{12} ) q^{43} \) \( + ( -1 - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{45} \) \( + ( -1 + \beta_{1} - \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} ) q^{47} \) \( + ( 2 + 2 \beta_{1} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{10} - \beta_{11} - 3 \beta_{12} + \beta_{13} ) q^{49} \) \( + ( -4 - \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} + \beta_{12} ) q^{51} \) \( + ( \beta_{1} + \beta_{2} - \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{12} ) q^{53} \) \( + ( -4 - \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - 2 \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} + \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{55} \) \( + ( 1 - 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} + 2 \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{57} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{59} \) \( + ( 1 - \beta_{1} + 2 \beta_{3} - \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{61} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{7} + 2 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{63} \) \( + ( 1 + \beta_{2} + \beta_{3} + \beta_{10} - \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{65} \) \( + ( -1 + \beta_{2} + \beta_{4} + \beta_{5} - \beta_{9} + 2 \beta_{10} + \beta_{12} - \beta_{13} ) q^{67} \) \( + ( - \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{9} + \beta_{10} + 2 \beta_{11} - 2 \beta_{13} ) q^{69} \) \( + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - 2 \beta_{13} ) q^{71} \) \( + ( -1 + 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{73} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} - 3 \beta_{6} + 6 \beta_{7} + 3 \beta_{9} + \beta_{10} - 2 \beta_{11} - 3 \beta_{12} + 3 \beta_{13} ) q^{75} \) \( + ( 1 - 2 \beta_{3} + 3 \beta_{5} - \beta_{8} - 2 \beta_{9} + 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} ) q^{77} \) \( + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{4} + 2 \beta_{7} + \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{79} \) \( + ( \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - 4 \beta_{10} + \beta_{12} ) q^{81} \) \( + ( 2 + \beta_{1} + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + \beta_{6} - 2 \beta_{7} - \beta_{9} - \beta_{10} + 2 \beta_{11} - \beta_{13} ) q^{83} \) \( + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{4} - \beta_{5} + \beta_{8} + 2 \beta_{9} - 2 \beta_{10} + \beta_{12} + 2 \beta_{13} ) q^{85} \) \( + ( -1 - \beta_{1} + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 4 \beta_{5} - \beta_{6} - 4 \beta_{7} + \beta_{8} - 2 \beta_{10} + 3 \beta_{12} - \beta_{13} ) q^{87} \) \( + ( -3 - \beta_{2} + 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + \beta_{12} + \beta_{13} ) q^{89} \) \( + ( -3 + \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 3 \beta_{12} - \beta_{13} ) q^{91} \) \( + ( -4 + 2 \beta_{1} - \beta_{3} + 3 \beta_{5} + 2 \beta_{6} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{93} \) \( + ( 2 \beta_{1} - \beta_{2} - 2 \beta_{4} + 3 \beta_{5} - 3 \beta_{8} + \beta_{9} - \beta_{10} - 3 \beta_{11} - 2 \beta_{12} ) q^{95} \) \( + ( 2 - \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} - 3 \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{97} \) \( + ( -3 \beta_{1} - \beta_{5} - \beta_{6} - 2 \beta_{7} + \beta_{8} - \beta_{9} - 3 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)  \(=\)  \(14q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut -\mathstrut 9q^{11} \) \(\mathstrut -\mathstrut q^{13} \) \(\mathstrut -\mathstrut 14q^{15} \) \(\mathstrut -\mathstrut 27q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut 26q^{25} \) \(\mathstrut -\mathstrut 15q^{27} \) \(\mathstrut -\mathstrut 25q^{31} \) \(\mathstrut +\mathstrut 16q^{33} \) \(\mathstrut -\mathstrut 21q^{35} \) \(\mathstrut -\mathstrut q^{37} \) \(\mathstrut -\mathstrut 33q^{39} \) \(\mathstrut +\mathstrut 10q^{41} \) \(\mathstrut -\mathstrut 35q^{43} \) \(\mathstrut -\mathstrut 4q^{45} \) \(\mathstrut -\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 36q^{49} \) \(\mathstrut -\mathstrut 48q^{51} \) \(\mathstrut -\mathstrut q^{53} \) \(\mathstrut -\mathstrut 41q^{55} \) \(\mathstrut +\mathstrut 14q^{57} \) \(\mathstrut -\mathstrut 30q^{59} \) \(\mathstrut +\mathstrut 3q^{61} \) \(\mathstrut -\mathstrut 31q^{63} \) \(\mathstrut +\mathstrut 7q^{65} \) \(\mathstrut -\mathstrut 22q^{67} \) \(\mathstrut -\mathstrut 17q^{69} \) \(\mathstrut -\mathstrut 6q^{71} \) \(\mathstrut +\mathstrut 5q^{73} \) \(\mathstrut -\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut -\mathstrut 56q^{79} \) \(\mathstrut +\mathstrut 26q^{81} \) \(\mathstrut +\mathstrut 28q^{83} \) \(\mathstrut -\mathstrut 23q^{85} \) \(\mathstrut -\mathstrut 11q^{87} \) \(\mathstrut -\mathstrut 24q^{89} \) \(\mathstrut -\mathstrut 38q^{91} \) \(\mathstrut -\mathstrut 55q^{93} \) \(\mathstrut +\mathstrut 4q^{95} \) \(\mathstrut +\mathstrut 6q^{97} \) \(\mathstrut -\mathstrut 12q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(3\) \(x^{13}\mathstrut -\mathstrut \) \(27\) \(x^{12}\mathstrut +\mathstrut \) \(79\) \(x^{11}\mathstrut +\mathstrut \) \(274\) \(x^{10}\mathstrut -\mathstrut \) \(747\) \(x^{9}\mathstrut -\mathstrut \) \(1422\) \(x^{8}\mathstrut +\mathstrut \) \(3287\) \(x^{7}\mathstrut +\mathstrut \) \(4161\) \(x^{6}\mathstrut -\mathstrut \) \(6861\) \(x^{5}\mathstrut -\mathstrut \) \(6676\) \(x^{4}\mathstrut +\mathstrut \) \(5599\) \(x^{3}\mathstrut +\mathstrut \) \(4627\) \(x^{2}\mathstrut -\mathstrut \) \(359\) \(x\mathstrut -\mathstrut \) \(196\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(458652\) \(\nu^{13}\mathstrut +\mathstrut \) \(10207449\) \(\nu^{12}\mathstrut +\mathstrut \) \(8865462\) \(\nu^{11}\mathstrut -\mathstrut \) \(298701457\) \(\nu^{10}\mathstrut -\mathstrut \) \(45787181\) \(\nu^{9}\mathstrut +\mathstrut \) \(3246408968\) \(\nu^{8}\mathstrut +\mathstrut \) \(146603840\) \(\nu^{7}\mathstrut -\mathstrut \) \(16541854354\) \(\nu^{6}\mathstrut -\mathstrut \) \(1455748504\) \(\nu^{5}\mathstrut +\mathstrut \) \(39709795344\) \(\nu^{4}\mathstrut +\mathstrut \) \(7703220112\) \(\nu^{3}\mathstrut -\mathstrut \) \(36804146400\) \(\nu^{2}\mathstrut -\mathstrut \) \(13258305511\) \(\nu\mathstrut +\mathstrut \) \(1861877352\)\()/\)\(762886946\)
\(\beta_{3}\)\(=\)\((\)\(458652\) \(\nu^{13}\mathstrut -\mathstrut \) \(10207449\) \(\nu^{12}\mathstrut -\mathstrut \) \(8865462\) \(\nu^{11}\mathstrut +\mathstrut \) \(298701457\) \(\nu^{10}\mathstrut +\mathstrut \) \(45787181\) \(\nu^{9}\mathstrut -\mathstrut \) \(3246408968\) \(\nu^{8}\mathstrut -\mathstrut \) \(146603840\) \(\nu^{7}\mathstrut +\mathstrut \) \(16541854354\) \(\nu^{6}\mathstrut +\mathstrut \) \(1455748504\) \(\nu^{5}\mathstrut -\mathstrut \) \(39709795344\) \(\nu^{4}\mathstrut -\mathstrut \) \(7703220112\) \(\nu^{3}\mathstrut +\mathstrut \) \(37567033346\) \(\nu^{2}\mathstrut +\mathstrut \) \(12495418565\) \(\nu\mathstrut -\mathstrut \) \(4913425136\)\()/\)\(762886946\)
\(\beta_{4}\)\(=\)\((\)\(8519764\) \(\nu^{13}\mathstrut +\mathstrut \) \(100995904\) \(\nu^{12}\mathstrut -\mathstrut \) \(475048121\) \(\nu^{11}\mathstrut -\mathstrut \) \(2501409580\) \(\nu^{10}\mathstrut +\mathstrut \) \(7651912803\) \(\nu^{9}\mathstrut +\mathstrut \) \(22192164735\) \(\nu^{8}\mathstrut -\mathstrut \) \(47188811121\) \(\nu^{7}\mathstrut -\mathstrut \) \(94922312539\) \(\nu^{6}\mathstrut +\mathstrut \) \(114053323154\) \(\nu^{5}\mathstrut +\mathstrut \) \(203636821844\) \(\nu^{4}\mathstrut -\mathstrut \) \(66751900430\) \(\nu^{3}\mathstrut -\mathstrut \) \(178059623419\) \(\nu^{2}\mathstrut -\mathstrut \) \(50327477697\) \(\nu\mathstrut +\mathstrut \) \(9359448304\)\()/\)\(4577321676\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(4195687\) \(\nu^{13}\mathstrut +\mathstrut \) \(59312328\) \(\nu^{12}\mathstrut -\mathstrut \) \(28844005\) \(\nu^{11}\mathstrut -\mathstrut \) \(1424333823\) \(\nu^{10}\mathstrut +\mathstrut \) \(2228743711\) \(\nu^{9}\mathstrut +\mathstrut \) \(11739065967\) \(\nu^{8}\mathstrut -\mathstrut \) \(21011735390\) \(\nu^{7}\mathstrut -\mathstrut \) \(43345751602\) \(\nu^{6}\mathstrut +\mathstrut \) \(74926731662\) \(\nu^{5}\mathstrut +\mathstrut \) \(72932658519\) \(\nu^{4}\mathstrut -\mathstrut \) \(103565031713\) \(\nu^{3}\mathstrut -\mathstrut \) \(45673704976\) \(\nu^{2}\mathstrut +\mathstrut \) \(36469265332\) \(\nu\mathstrut +\mathstrut \) \(2862099808\)\()/\)\(1525773892\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(5336880\) \(\nu^{13}\mathstrut -\mathstrut \) \(865584\) \(\nu^{12}\mathstrut +\mathstrut \) \(175902437\) \(\nu^{11}\mathstrut +\mathstrut \) \(7885974\) \(\nu^{10}\mathstrut -\mathstrut \) \(2142375619\) \(\nu^{9}\mathstrut -\mathstrut \) \(43040057\) \(\nu^{8}\mathstrut +\mathstrut \) \(11998211113\) \(\nu^{7}\mathstrut +\mathstrut \) \(1393620799\) \(\nu^{6}\mathstrut -\mathstrut \) \(32452371932\) \(\nu^{5}\mathstrut -\mathstrut \) \(8827404356\) \(\nu^{4}\mathstrut +\mathstrut \) \(41134202654\) \(\nu^{3}\mathstrut +\mathstrut \) \(12822573733\) \(\nu^{2}\mathstrut -\mathstrut \) \(22034766939\) \(\nu\mathstrut +\mathstrut \) \(1054324184\)\()/\)\(1525773892\)
\(\beta_{7}\)\(=\)\((\)\(34205344\) \(\nu^{13}\mathstrut -\mathstrut \) \(272353484\) \(\nu^{12}\mathstrut -\mathstrut \) \(487766789\) \(\nu^{11}\mathstrut +\mathstrut \) \(6852436190\) \(\nu^{10}\mathstrut -\mathstrut \) \(985278789\) \(\nu^{9}\mathstrut -\mathstrut \) \(60820846611\) \(\nu^{8}\mathstrut +\mathstrut \) \(33603498195\) \(\nu^{7}\mathstrut +\mathstrut \) \(248314486025\) \(\nu^{6}\mathstrut -\mathstrut \) \(134060247376\) \(\nu^{5}\mathstrut -\mathstrut \) \(476618765200\) \(\nu^{4}\mathstrut +\mathstrut \) \(150113282698\) \(\nu^{3}\mathstrut +\mathstrut \) \(352160580251\) \(\nu^{2}\mathstrut +\mathstrut \) \(5264197347\) \(\nu\mathstrut -\mathstrut \) \(18482722832\)\()/\)\(4577321676\)
\(\beta_{8}\)\(=\)\((\)\(5967155\) \(\nu^{13}\mathstrut -\mathstrut \) \(22669783\) \(\nu^{12}\mathstrut -\mathstrut \) \(134567812\) \(\nu^{11}\mathstrut +\mathstrut \) \(542735784\) \(\nu^{10}\mathstrut +\mathstrut \) \(1026279357\) \(\nu^{9}\mathstrut -\mathstrut \) \(4439965553\) \(\nu^{8}\mathstrut -\mathstrut \) \(3819469690\) \(\nu^{7}\mathstrut +\mathstrut \) \(16339663095\) \(\nu^{6}\mathstrut +\mathstrut \) \(8758574070\) \(\nu^{5}\mathstrut -\mathstrut \) \(28676881891\) \(\nu^{4}\mathstrut -\mathstrut \) \(12289960304\) \(\nu^{3}\mathstrut +\mathstrut \) \(22212436924\) \(\nu^{2}\mathstrut +\mathstrut \) \(6353079607\) \(\nu\mathstrut -\mathstrut \) \(2832505686\)\()/\)\(762886946\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(13587249\) \(\nu^{13}\mathstrut +\mathstrut \) \(82033945\) \(\nu^{12}\mathstrut +\mathstrut \) \(220632596\) \(\nu^{11}\mathstrut -\mathstrut \) \(2025748304\) \(\nu^{10}\mathstrut -\mathstrut \) \(153782879\) \(\nu^{9}\mathstrut +\mathstrut \) \(17354603633\) \(\nu^{8}\mathstrut -\mathstrut \) \(10380694436\) \(\nu^{7}\mathstrut -\mathstrut \) \(66840155539\) \(\nu^{6}\mathstrut +\mathstrut \) \(50524361632\) \(\nu^{5}\mathstrut +\mathstrut \) \(117771424507\) \(\nu^{4}\mathstrut -\mathstrut \) \(72326314222\) \(\nu^{3}\mathstrut -\mathstrut \) \(79088806890\) \(\nu^{2}\mathstrut +\mathstrut \) \(14179372355\) \(\nu\mathstrut +\mathstrut \) \(6261708580\)\()/\)\(1525773892\)
\(\beta_{10}\)\(=\)\((\)\(42935729\) \(\nu^{13}\mathstrut -\mathstrut \) \(93792808\) \(\nu^{12}\mathstrut -\mathstrut \) \(1159294432\) \(\nu^{11}\mathstrut +\mathstrut \) \(2336791369\) \(\nu^{10}\mathstrut +\mathstrut \) \(11502377406\) \(\nu^{9}\mathstrut -\mathstrut \) \(19984751130\) \(\nu^{8}\mathstrut -\mathstrut \) \(54780817125\) \(\nu^{7}\mathstrut +\mathstrut \) \(74365149889\) \(\nu^{6}\mathstrut +\mathstrut \) \(130314127684\) \(\nu^{5}\mathstrut -\mathstrut \) \(119198922599\) \(\nu^{4}\mathstrut -\mathstrut \) \(140109115921\) \(\nu^{3}\mathstrut +\mathstrut \) \(74681823445\) \(\nu^{2}\mathstrut +\mathstrut \) \(47947897389\) \(\nu\mathstrut -\mathstrut \) \(22751872012\)\()/\)\(4577321676\)
\(\beta_{11}\)\(=\)\((\)\(20741705\) \(\nu^{13}\mathstrut -\mathstrut \) \(78046665\) \(\nu^{12}\mathstrut -\mathstrut \) \(457715319\) \(\nu^{11}\mathstrut +\mathstrut \) \(1856372484\) \(\nu^{10}\mathstrut +\mathstrut \) \(3173974206\) \(\nu^{9}\mathstrut -\mathstrut \) \(14783803588\) \(\nu^{8}\mathstrut -\mathstrut \) \(8133068789\) \(\nu^{7}\mathstrut +\mathstrut \) \(49588145194\) \(\nu^{6}\mathstrut +\mathstrut \) \(3889798330\) \(\nu^{5}\mathstrut -\mathstrut \) \(65520134901\) \(\nu^{4}\mathstrut +\mathstrut \) \(7319903428\) \(\nu^{3}\mathstrut +\mathstrut \) \(20909058893\) \(\nu^{2}\mathstrut +\mathstrut \) \(3171818988\) \(\nu\mathstrut -\mathstrut \) \(46956196\)\()/\)\(1525773892\)
\(\beta_{12}\)\(=\)\((\)\(118858355\) \(\nu^{13}\mathstrut -\mathstrut \) \(427006354\) \(\nu^{12}\mathstrut -\mathstrut \) \(2693628022\) \(\nu^{11}\mathstrut +\mathstrut \) \(10377729199\) \(\nu^{10}\mathstrut +\mathstrut \) \(20013238272\) \(\nu^{9}\mathstrut -\mathstrut \) \(86552904492\) \(\nu^{8}\mathstrut -\mathstrut \) \(63314476149\) \(\nu^{7}\mathstrut +\mathstrut \) \(321067037137\) \(\nu^{6}\mathstrut +\mathstrut \) \(92843969962\) \(\nu^{5}\mathstrut -\mathstrut \) \(533348181419\) \(\nu^{4}\mathstrut -\mathstrut \) \(87018856909\) \(\nu^{3}\mathstrut +\mathstrut \) \(312683090797\) \(\nu^{2}\mathstrut +\mathstrut \) \(70791640479\) \(\nu\mathstrut -\mathstrut \) \(7920902056\)\()/\)\(4577321676\)
\(\beta_{13}\)\(=\)\((\)\(40998083\) \(\nu^{13}\mathstrut -\mathstrut \) \(152031178\) \(\nu^{12}\mathstrut -\mathstrut \) \(851304577\) \(\nu^{11}\mathstrut +\mathstrut \) \(3546579085\) \(\nu^{10}\mathstrut +\mathstrut \) \(5069396045\) \(\nu^{9}\mathstrut -\mathstrut \) \(27563296343\) \(\nu^{8}\mathstrut -\mathstrut \) \(7226354746\) \(\nu^{7}\mathstrut +\mathstrut \) \(92002461912\) \(\nu^{6}\mathstrut -\mathstrut \) \(18695643274\) \(\nu^{5}\mathstrut -\mathstrut \) \(131587919995\) \(\nu^{4}\mathstrut +\mathstrut \) \(43902940955\) \(\nu^{3}\mathstrut +\mathstrut \) \(64100998940\) \(\nu^{2}\mathstrut -\mathstrut \) \(5970357810\) \(\nu\mathstrut -\mathstrut \) \(3159888728\)\()/\)\(1525773892\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{12}\mathstrut -\mathstrut \) \(4\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\)
\(\nu^{5}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(15\) \(\beta_{11}\mathstrut +\mathstrut \) \(13\) \(\beta_{10}\mathstrut +\mathstrut \) \(17\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{7}\mathstrut -\mathstrut \) \(19\) \(\beta_{6}\mathstrut -\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(57\) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)
\(\nu^{6}\)\(=\)\(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(65\) \(\beta_{10}\mathstrut +\mathstrut \) \(31\) \(\beta_{9}\mathstrut +\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\) \(\beta_{7}\mathstrut -\mathstrut \) \(35\) \(\beta_{6}\mathstrut -\mathstrut \) \(17\) \(\beta_{5}\mathstrut +\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(119\) \(\beta_{3}\mathstrut +\mathstrut \) \(101\) \(\beta_{2}\mathstrut +\mathstrut \) \(77\) \(\beta_{1}\mathstrut +\mathstrut \) \(227\)
\(\nu^{7}\)\(=\)\(13\) \(\beta_{13}\mathstrut +\mathstrut \) \(6\) \(\beta_{12}\mathstrut -\mathstrut \) \(186\) \(\beta_{11}\mathstrut +\mathstrut \) \(146\) \(\beta_{10}\mathstrut +\mathstrut \) \(220\) \(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{8}\mathstrut +\mathstrut \) \(135\) \(\beta_{7}\mathstrut -\mathstrut \) \(259\) \(\beta_{6}\mathstrut -\mathstrut \) \(253\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(210\) \(\beta_{3}\mathstrut -\mathstrut \) \(104\) \(\beta_{2}\mathstrut +\mathstrut \) \(510\) \(\beta_{1}\mathstrut -\mathstrut \) \(20\)
\(\nu^{8}\)\(=\)\(7\) \(\beta_{13}\mathstrut +\mathstrut \) \(158\) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{11}\mathstrut -\mathstrut \) \(849\) \(\beta_{10}\mathstrut +\mathstrut \) \(365\) \(\beta_{9}\mathstrut +\mathstrut \) \(326\) \(\beta_{8}\mathstrut +\mathstrut \) \(170\) \(\beta_{7}\mathstrut -\mathstrut \) \(462\) \(\beta_{6}\mathstrut -\mathstrut \) \(224\) \(\beta_{5}\mathstrut +\mathstrut \) \(438\) \(\beta_{4}\mathstrut +\mathstrut \) \(1316\) \(\beta_{3}\mathstrut +\mathstrut \) \(1065\) \(\beta_{2}\mathstrut +\mathstrut \) \(641\) \(\beta_{1}\mathstrut +\mathstrut \) \(2187\)
\(\nu^{9}\)\(=\)\(119\) \(\beta_{13}\mathstrut +\mathstrut \) \(143\) \(\beta_{12}\mathstrut -\mathstrut \) \(2190\) \(\beta_{11}\mathstrut +\mathstrut \) \(1644\) \(\beta_{10}\mathstrut +\mathstrut \) \(2590\) \(\beta_{9}\mathstrut +\mathstrut \) \(171\) \(\beta_{8}\mathstrut +\mathstrut \) \(1326\) \(\beta_{7}\mathstrut -\mathstrut \) \(3171\) \(\beta_{6}\mathstrut -\mathstrut \) \(3198\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2623\) \(\beta_{3}\mathstrut -\mathstrut \) \(1572\) \(\beta_{2}\mathstrut +\mathstrut \) \(4876\) \(\beta_{1}\mathstrut -\mathstrut \) \(913\)
\(\nu^{10}\)\(=\)\(-\)\(50\) \(\beta_{13}\mathstrut +\mathstrut \) \(1940\) \(\beta_{12}\mathstrut +\mathstrut \) \(64\) \(\beta_{11}\mathstrut -\mathstrut \) \(10374\) \(\beta_{10}\mathstrut +\mathstrut \) \(3908\) \(\beta_{9}\mathstrut +\mathstrut \) \(4210\) \(\beta_{8}\mathstrut +\mathstrut \) \(1497\) \(\beta_{7}\mathstrut -\mathstrut \) \(5487\) \(\beta_{6}\mathstrut -\mathstrut \) \(2660\) \(\beta_{5}\mathstrut +\mathstrut \) \(5180\) \(\beta_{4}\mathstrut +\mathstrut \) \(14845\) \(\beta_{3}\mathstrut +\mathstrut \) \(11613\) \(\beta_{2}\mathstrut +\mathstrut \) \(5205\) \(\beta_{1}\mathstrut +\mathstrut \) \(22786\)
\(\nu^{11}\)\(=\)\(920\) \(\beta_{13}\mathstrut +\mathstrut \) \(2294\) \(\beta_{12}\mathstrut -\mathstrut \) \(25342\) \(\beta_{11}\mathstrut +\mathstrut \) \(18962\) \(\beta_{10}\mathstrut +\mathstrut \) \(29328\) \(\beta_{9}\mathstrut +\mathstrut \) \(2457\) \(\beta_{8}\mathstrut +\mathstrut \) \(12925\) \(\beta_{7}\mathstrut -\mathstrut \) \(37107\) \(\beta_{6}\mathstrut -\mathstrut \) \(38396\) \(\beta_{5}\mathstrut -\mathstrut \) \(564\) \(\beta_{4}\mathstrut -\mathstrut \) \(32149\) \(\beta_{3}\mathstrut -\mathstrut \) \(21153\) \(\beta_{2}\mathstrut +\mathstrut \) \(48906\) \(\beta_{1}\mathstrut -\mathstrut \) \(16315\)
\(\nu^{12}\)\(=\)\(-\)\(2101\) \(\beta_{13}\mathstrut +\mathstrut \) \(23661\) \(\beta_{12}\mathstrut +\mathstrut \) \(2860\) \(\beta_{11}\mathstrut -\mathstrut \) \(123400\) \(\beta_{10}\mathstrut +\mathstrut \) \(40080\) \(\beta_{9}\mathstrut +\mathstrut \) \(51131\) \(\beta_{8}\mathstrut +\mathstrut \) \(11385\) \(\beta_{7}\mathstrut -\mathstrut \) \(61898\) \(\beta_{6}\mathstrut -\mathstrut \) \(29604\) \(\beta_{5}\mathstrut +\mathstrut \) \(59402\) \(\beta_{4}\mathstrut +\mathstrut \) \(169642\) \(\beta_{3}\mathstrut +\mathstrut \) \(129551\) \(\beta_{2}\mathstrut +\mathstrut \) \(40862\) \(\beta_{1}\mathstrut +\mathstrut \) \(247897\)
\(\nu^{13}\)\(=\)\(6170\) \(\beta_{13}\mathstrut +\mathstrut \) \(31190\) \(\beta_{12}\mathstrut -\mathstrut \) \(291368\) \(\beta_{11}\mathstrut +\mathstrut \) \(223100\) \(\beta_{10}\mathstrut +\mathstrut \) \(326994\) \(\beta_{9}\mathstrut +\mathstrut \) \(29713\) \(\beta_{8}\mathstrut +\mathstrut \) \(127390\) \(\beta_{7}\mathstrut -\mathstrut \) \(425018\) \(\beta_{6}\mathstrut -\mathstrut \) \(448904\) \(\beta_{5}\mathstrut -\mathstrut \) \(14838\) \(\beta_{4}\mathstrut -\mathstrut \) \(390117\) \(\beta_{3}\mathstrut -\mathstrut \) \(269763\) \(\beta_{2}\mathstrut +\mathstrut \) \(507870\) \(\beta_{1}\mathstrut -\mathstrut \) \(238643\)

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
3.30568
2.87232
2.53364
2.37559
1.90340
1.46572
0.224635
−0.194346
−0.788858
−1.48367
−1.64615
−1.93078
−2.20631
−3.43087
0 −3.30568 0 0.595872 0 −0.878141 0 7.92752 0
1.2 0 −2.87232 0 −0.801100 0 4.32848 0 5.25020 0
1.3 0 −2.53364 0 2.70770 0 −1.66309 0 3.41934 0
1.4 0 −2.37559 0 −4.08952 0 −3.67423 0 2.64345 0
1.5 0 −1.90340 0 3.78264 0 0.441155 0 0.622940 0
1.6 0 −1.46572 0 0.817924 0 −4.79612 0 −0.851657 0
1.7 0 −0.224635 0 −2.42875 0 2.24830 0 −2.94954 0
1.8 0 0.194346 0 2.87489 0 −0.213356 0 −2.96223 0
1.9 0 0.788858 0 0.735420 0 −1.11520 0 −2.37770 0
1.10 0 1.48367 0 −3.74411 0 4.87233 0 −0.798715 0
1.11 0 1.64615 0 −4.00715 0 −2.94760 0 −0.290201 0
1.12 0 1.93078 0 3.14832 0 −3.83969 0 0.727926 0
1.13 0 2.20631 0 −0.868402 0 2.87616 0 1.86782 0
1.14 0 3.43087 0 −0.723735 0 −3.63900 0 8.77086 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3}^{14} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4016))\).