Properties

Label 6024.2.a.o
Level 6024
Weight 2
Character orbit 6024.a
Self dual Yes
Analytic conductor 48.102
Analytic rank 1
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1018821776\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{3}\) \( -\beta_{1} q^{5} \) \( + ( 1 - \beta_{7} ) q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{3}\) \( -\beta_{1} q^{5} \) \( + ( 1 - \beta_{7} ) q^{7} \) \(+ q^{9}\) \( + ( -2 - \beta_{9} ) q^{11} \) \( + ( 1 + \beta_{3} ) q^{13} \) \( + \beta_{1} q^{15} \) \( + ( \beta_{2} - \beta_{8} + \beta_{9} ) q^{17} \) \( + \beta_{12} q^{19} \) \( + ( -1 + \beta_{7} ) q^{21} \) \( + ( -2 - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{11} - \beta_{12} ) q^{23} \) \( + ( 2 + 2 \beta_{1} - \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} - \beta_{11} ) q^{25} \) \(- q^{27}\) \( + ( -2 - \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} + \beta_{9} ) q^{29} \) \( + ( \beta_{1} - \beta_{5} + \beta_{8} - \beta_{11} ) q^{31} \) \( + ( 2 + \beta_{9} ) q^{33} \) \( + ( -4 - \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{35} \) \( + ( 1 + \beta_{5} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{13} ) q^{37} \) \( + ( -1 - \beta_{3} ) q^{39} \) \( + ( -\beta_{2} - \beta_{4} + \beta_{5} + \beta_{9} + \beta_{13} ) q^{41} \) \( + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} ) q^{43} \) \( -\beta_{1} q^{45} \) \( + ( \beta_{1} + \beta_{2} + \beta_{6} - \beta_{8} + 2 \beta_{9} - 2 \beta_{11} + \beta_{12} ) q^{47} \) \( + ( 2 - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{12} ) q^{49} \) \( + ( -\beta_{2} + \beta_{8} - \beta_{9} ) q^{51} \) \( + ( -3 + \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} - 2 \beta_{9} + \beta_{10} + \beta_{11} - \beta_{12} ) q^{53} \) \( + ( 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{10} + \beta_{13} ) q^{55} \) \( -\beta_{12} q^{57} \) \( + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} + \beta_{13} ) q^{59} \) \( + ( -1 + \beta_{2} - \beta_{5} + \beta_{7} + \beta_{8} - \beta_{9} + \beta_{10} - 2 \beta_{13} ) q^{61} \) \( + ( 1 - \beta_{7} ) q^{63} \) \( + ( 1 - \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{9} - \beta_{10} - \beta_{13} ) q^{65} \) \( + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{6} - 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{67} \) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} ) q^{69} \) \( + ( -2 - \beta_{3} + \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 2 \beta_{11} + \beta_{12} ) q^{71} \) \( + ( -\beta_{5} + 3 \beta_{8} - \beta_{9} ) q^{73} \) \( + ( -2 - 2 \beta_{1} + \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{75} \) \( + ( -2 - \beta_{3} - \beta_{6} + 3 \beta_{7} - 2 \beta_{9} - \beta_{13} ) q^{77} \) \( + ( 1 + 3 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} ) q^{79} \) \(+ q^{81}\) \( + ( -1 + \beta_{1} + \beta_{2} + 3 \beta_{4} - \beta_{5} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} + 2 \beta_{12} + \beta_{13} ) q^{83} \) \( + ( -1 - 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{11} + \beta_{12} ) q^{85} \) \( + ( 2 + \beta_{3} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{9} ) q^{87} \) \( + ( -2 + \beta_{1} + \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} + 3 \beta_{8} - 4 \beta_{9} + \beta_{10} - 3 \beta_{12} ) q^{89} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} ) q^{91} \) \( + ( -\beta_{1} + \beta_{5} - \beta_{8} + \beta_{11} ) q^{93} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{7} - 2 \beta_{8} + \beta_{9} - \beta_{11} + \beta_{12} + \beta_{13} ) q^{95} \) \( + ( 2 - 2 \beta_{1} + \beta_{5} - 2 \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{10} ) q^{97} \) \( + ( -2 - \beta_{9} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 7q^{7} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut -\mathstrut 22q^{11} \) \(\mathstrut +\mathstrut 9q^{13} \) \(\mathstrut +\mathstrut 3q^{15} \) \(\mathstrut -\mathstrut 7q^{21} \) \(\mathstrut -\mathstrut 17q^{23} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut -\mathstrut 18q^{29} \) \(\mathstrut -\mathstrut 7q^{31} \) \(\mathstrut +\mathstrut 22q^{33} \) \(\mathstrut -\mathstrut 27q^{35} \) \(\mathstrut +\mathstrut 9q^{37} \) \(\mathstrut -\mathstrut 9q^{39} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 14q^{43} \) \(\mathstrut -\mathstrut 3q^{45} \) \(\mathstrut -\mathstrut 15q^{47} \) \(\mathstrut +\mathstrut 15q^{49} \) \(\mathstrut -\mathstrut 11q^{53} \) \(\mathstrut +\mathstrut 2q^{55} \) \(\mathstrut -\mathstrut 36q^{59} \) \(\mathstrut +\mathstrut 2q^{61} \) \(\mathstrut +\mathstrut 7q^{63} \) \(\mathstrut +\mathstrut 8q^{65} \) \(\mathstrut +\mathstrut 3q^{67} \) \(\mathstrut +\mathstrut 17q^{69} \) \(\mathstrut -\mathstrut 29q^{71} \) \(\mathstrut +\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 25q^{75} \) \(\mathstrut +\mathstrut 8q^{77} \) \(\mathstrut +\mathstrut 23q^{79} \) \(\mathstrut +\mathstrut 14q^{81} \) \(\mathstrut -\mathstrut 55q^{83} \) \(\mathstrut +\mathstrut 7q^{85} \) \(\mathstrut +\mathstrut 18q^{87} \) \(\mathstrut +\mathstrut 9q^{89} \) \(\mathstrut -\mathstrut 22q^{91} \) \(\mathstrut +\mathstrut 7q^{93} \) \(\mathstrut -\mathstrut 27q^{95} \) \(\mathstrut +\mathstrut 17q^{97} \) \(\mathstrut -\mathstrut 22q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(3\) \(x^{13}\mathstrut -\mathstrut \) \(43\) \(x^{12}\mathstrut +\mathstrut \) \(119\) \(x^{11}\mathstrut +\mathstrut \) \(679\) \(x^{10}\mathstrut -\mathstrut \) \(1667\) \(x^{9}\mathstrut -\mathstrut \) \(4890\) \(x^{8}\mathstrut +\mathstrut \) \(9662\) \(x^{7}\mathstrut +\mathstrut \) \(16575\) \(x^{6}\mathstrut -\mathstrut \) \(20277\) \(x^{5}\mathstrut -\mathstrut \) \(25196\) \(x^{4}\mathstrut +\mathstrut \) \(8040\) \(x^{3}\mathstrut +\mathstrut \) \(10776\) \(x^{2}\mathstrut +\mathstrut \) \(912\) \(x\mathstrut -\mathstrut \) \(416\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(102160228984125\) \(\nu^{13}\mathstrut -\mathstrut \) \(223956217420031\) \(\nu^{12}\mathstrut -\mathstrut \) \(4551967250809079\) \(\nu^{11}\mathstrut +\mathstrut \) \(8918302875541603\) \(\nu^{10}\mathstrut +\mathstrut \) \(75149691440341379\) \(\nu^{9}\mathstrut -\mathstrut \) \(127027480494896559\) \(\nu^{8}\mathstrut -\mathstrut \) \(572354612125554522\) \(\nu^{7}\mathstrut +\mathstrut \) \(760936910664972734\) \(\nu^{6}\mathstrut +\mathstrut \) \(2047465404860353907\) \(\nu^{5}\mathstrut -\mathstrut \) \(1688156493700203161\) \(\nu^{4}\mathstrut -\mathstrut \) \(3059424933048010188\) \(\nu^{3}\mathstrut +\mathstrut \) \(655007521450873280\) \(\nu^{2}\mathstrut +\mathstrut \) \(951009769878186960\) \(\nu\mathstrut +\mathstrut \) \(268101597699285680\)\()/\)\(113480274736800592\)
\(\beta_{3}\)\(=\)\((\)\(139579303918975\) \(\nu^{13}\mathstrut -\mathstrut \) \(348803377277369\) \(\nu^{12}\mathstrut -\mathstrut \) \(6077411015267013\) \(\nu^{11}\mathstrut +\mathstrut \) \(13717313487730393\) \(\nu^{10}\mathstrut +\mathstrut \) \(97599481407614489\) \(\nu^{9}\mathstrut -\mathstrut \) \(191901021333777173\) \(\nu^{8}\mathstrut -\mathstrut \) \(721976110664782158\) \(\nu^{7}\mathstrut +\mathstrut \) \(1132282727011900110\) \(\nu^{6}\mathstrut +\mathstrut \) \(2559020949405016801\) \(\nu^{5}\mathstrut -\mathstrut \) \(2613787912685535495\) \(\nu^{4}\mathstrut -\mathstrut \) \(4174940629309736220\) \(\nu^{3}\mathstrut +\mathstrut \) \(1836550728177771676\) \(\nu^{2}\mathstrut +\mathstrut \) \(2113188179508749096\) \(\nu\mathstrut -\mathstrut \) \(100571358867112368\)\()/\)\(113480274736800592\)
\(\beta_{4}\)\(=\)\((\)\(360141139074321\) \(\nu^{13}\mathstrut -\mathstrut \) \(1239104231363285\) \(\nu^{12}\mathstrut -\mathstrut \) \(15089905841992425\) \(\nu^{11}\mathstrut +\mathstrut \) \(49579547453945521\) \(\nu^{10}\mathstrut +\mathstrut \) \(229348796620102189\) \(\nu^{9}\mathstrut -\mathstrut \) \(704509837983309381\) \(\nu^{8}\mathstrut -\mathstrut \) \(1556108755747599016\) \(\nu^{7}\mathstrut +\mathstrut \) \(4219524522207086246\) \(\nu^{6}\mathstrut +\mathstrut \) \(4821180861657057315\) \(\nu^{5}\mathstrut -\mathstrut \) \(9858656163670110843\) \(\nu^{4}\mathstrut -\mathstrut \) \(6564333951753128446\) \(\nu^{3}\mathstrut +\mathstrut \) \(7039789131713671532\) \(\nu^{2}\mathstrut +\mathstrut \) \(1815396785928819632\) \(\nu\mathstrut -\mathstrut \) \(956081207119446224\)\()/\)\(226960549473601184\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(210653563625063\) \(\nu^{13}\mathstrut +\mathstrut \) \(555783816244845\) \(\nu^{12}\mathstrut +\mathstrut \) \(9239758965660829\) \(\nu^{11}\mathstrut -\mathstrut \) \(21678952567628985\) \(\nu^{10}\mathstrut -\mathstrut \) \(150007569171007185\) \(\nu^{9}\mathstrut +\mathstrut \) \(295109531697037877\) \(\nu^{8}\mathstrut +\mathstrut \) \(1121725643694139398\) \(\nu^{7}\mathstrut -\mathstrut \) \(1604781962689563666\) \(\nu^{6}\mathstrut -\mathstrut \) \(3943837242891010673\) \(\nu^{5}\mathstrut +\mathstrut \) \(2697306301042614707\) \(\nu^{4}\mathstrut +\mathstrut \) \(5785539595387658028\) \(\nu^{3}\mathstrut +\mathstrut \) \(663099775296066744\) \(\nu^{2}\mathstrut -\mathstrut \) \(1481303134680859336\) \(\nu\mathstrut -\mathstrut \) \(363486457391226928\)\()/\)\(113480274736800592\)
\(\beta_{6}\)\(=\)\((\)\(497673575650939\) \(\nu^{13}\mathstrut -\mathstrut \) \(2031706876941115\) \(\nu^{12}\mathstrut -\mathstrut \) \(19414247019505775\) \(\nu^{11}\mathstrut +\mathstrut \) \(81253766710476831\) \(\nu^{10}\mathstrut +\mathstrut \) \(259114577888169163\) \(\nu^{9}\mathstrut -\mathstrut \) \(1150321700462232203\) \(\nu^{8}\mathstrut -\mathstrut \) \(1334175436317358548\) \(\nu^{7}\mathstrut +\mathstrut \) \(6807804040762609634\) \(\nu^{6}\mathstrut +\mathstrut \) \(1944619223607752809\) \(\nu^{5}\mathstrut -\mathstrut \) \(15320575164739001557\) \(\nu^{4}\mathstrut +\mathstrut \) \(405023675285817474\) \(\nu^{3}\mathstrut +\mathstrut \) \(9690844697273457388\) \(\nu^{2}\mathstrut -\mathstrut \) \(3471198890654064\) \(\nu\mathstrut -\mathstrut \) \(1245424436536462640\)\()/\)\(226960549473601184\)
\(\beta_{7}\)\(=\)\((\)\(282322126790709\) \(\nu^{13}\mathstrut -\mathstrut \) \(865164088681703\) \(\nu^{12}\mathstrut -\mathstrut \) \(12267181842569951\) \(\nu^{11}\mathstrut +\mathstrut \) \(34559663959115219\) \(\nu^{10}\mathstrut +\mathstrut \) \(196913026046634483\) \(\nu^{9}\mathstrut -\mathstrut \) \(489273620291239679\) \(\nu^{8}\mathstrut -\mathstrut \) \(1453335453568403898\) \(\nu^{7}\mathstrut +\mathstrut \) \(2894637963175190190\) \(\nu^{6}\mathstrut +\mathstrut \) \(5079989292169782787\) \(\nu^{5}\mathstrut -\mathstrut \) \(6398240083437233001\) \(\nu^{4}\mathstrut -\mathstrut \) \(7800292963246667004\) \(\nu^{3}\mathstrut +\mathstrut \) \(3187077622129062824\) \(\nu^{2}\mathstrut +\mathstrut \) \(2924304227425531376\) \(\nu\mathstrut +\mathstrut \) \(59117336444021248\)\()/\)\(113480274736800592\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(145424724115102\) \(\nu^{13}\mathstrut +\mathstrut \) \(555371801647469\) \(\nu^{12}\mathstrut +\mathstrut \) \(5856658684055137\) \(\nu^{11}\mathstrut -\mathstrut \) \(22136489252665683\) \(\nu^{10}\mathstrut -\mathstrut \) \(83136857423130809\) \(\nu^{9}\mathstrut +\mathstrut \) \(311384618679919627\) \(\nu^{8}\mathstrut +\mathstrut \) \(495156550711999939\) \(\nu^{7}\mathstrut -\mathstrut \) \(1815287127345479202\) \(\nu^{6}\mathstrut -\mathstrut \) \(1180362333535082848\) \(\nu^{5}\mathstrut +\mathstrut \) \(3885151382733937299\) \(\nu^{4}\mathstrut +\mathstrut \) \(1106400353597761585\) \(\nu^{3}\mathstrut -\mathstrut \) \(1854398370549607904\) \(\nu^{2}\mathstrut -\mathstrut \) \(271973509242329460\) \(\nu\mathstrut +\mathstrut \) \(86686275530816976\)\()/\)\(56740137368400296\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(75770107003455\) \(\nu^{13}\mathstrut +\mathstrut \) \(304955783475835\) \(\nu^{12}\mathstrut +\mathstrut \) \(3006391066330991\) \(\nu^{11}\mathstrut -\mathstrut \) \(12239603188220543\) \(\nu^{10}\mathstrut -\mathstrut \) \(41500706658397755\) \(\nu^{9}\mathstrut +\mathstrut \) \(174210218522018513\) \(\nu^{8}\mathstrut +\mathstrut \) \(231910116232681998\) \(\nu^{7}\mathstrut -\mathstrut \) \(1040079786752910940\) \(\nu^{6}\mathstrut -\mathstrut \) \(457244144571890015\) \(\nu^{5}\mathstrut +\mathstrut \) \(2369546825549565973\) \(\nu^{4}\mathstrut +\mathstrut \) \(227523215290734920\) \(\nu^{3}\mathstrut -\mathstrut \) \(1445973730133987604\) \(\nu^{2}\mathstrut -\mathstrut \) \(42915193060201410\) \(\nu\mathstrut +\mathstrut \) \(67395399980501772\)\()/\)\(28370068684200148\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(333456052243539\) \(\nu^{13}\mathstrut +\mathstrut \) \(1108710482211023\) \(\nu^{12}\mathstrut +\mathstrut \) \(14065127560125175\) \(\nu^{11}\mathstrut -\mathstrut \) \(44381240552077439\) \(\nu^{10}\mathstrut -\mathstrut \) \(215809398490302811\) \(\nu^{9}\mathstrut +\mathstrut \) \(630455172738299419\) \(\nu^{8}\mathstrut +\mathstrut \) \(1486041406241881372\) \(\nu^{7}\mathstrut -\mathstrut \) \(3754133104704608606\) \(\nu^{6}\mathstrut -\mathstrut \) \(4721360587001236177\) \(\nu^{5}\mathstrut +\mathstrut \) \(8472745227809098009\) \(\nu^{4}\mathstrut +\mathstrut \) \(6852710827847343566\) \(\nu^{3}\mathstrut -\mathstrut \) \(4912514216771641832\) \(\nu^{2}\mathstrut -\mathstrut \) \(3267059514933151040\) \(\nu\mathstrut +\mathstrut \) \(360858743825480272\)\()/\)\(113480274736800592\)
\(\beta_{11}\)\(=\)\((\)\(687568838052891\) \(\nu^{13}\mathstrut -\mathstrut \) \(2232831647964323\) \(\nu^{12}\mathstrut -\mathstrut \) \(28895623562412591\) \(\nu^{11}\mathstrut +\mathstrut \) \(88533970484338863\) \(\nu^{10}\mathstrut +\mathstrut \) \(440032346472219811\) \(\nu^{9}\mathstrut -\mathstrut \) \(1237863978362435707\) \(\nu^{8}\mathstrut -\mathstrut \) \(2981596068649173852\) \(\nu^{7}\mathstrut +\mathstrut \) \(7146904819615719730\) \(\nu^{6}\mathstrut +\mathstrut \) \(9139469180380986209\) \(\nu^{5}\mathstrut -\mathstrut \) \(14937934247020585373\) \(\nu^{4}\mathstrut -\mathstrut \) \(12270716022185567878\) \(\nu^{3}\mathstrut +\mathstrut \) \(6317447348833702460\) \(\nu^{2}\mathstrut +\mathstrut \) \(4457435940448532192\) \(\nu\mathstrut +\mathstrut \) \(241122880378396768\)\()/\)\(113480274736800592\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(1437608246229121\) \(\nu^{13}\mathstrut +\mathstrut \) \(5076120118488849\) \(\nu^{12}\mathstrut +\mathstrut \) \(58914509485450421\) \(\nu^{11}\mathstrut -\mathstrut \) \(202301647050933461\) \(\nu^{10}\mathstrut -\mathstrut \) \(860414979803144985\) \(\nu^{9}\mathstrut +\mathstrut \) \(2852931504646788297\) \(\nu^{8}\mathstrut +\mathstrut \) \(5402468584922669604\) \(\nu^{7}\mathstrut -\mathstrut \) \(16775008908771238990\) \(\nu^{6}\mathstrut -\mathstrut \) \(14332622337994375115\) \(\nu^{5}\mathstrut +\mathstrut \) \(37017961598353240159\) \(\nu^{4}\mathstrut +\mathstrut \) \(15665919370430684002\) \(\nu^{3}\mathstrut -\mathstrut \) \(21133578636271541724\) \(\nu^{2}\mathstrut -\mathstrut \) \(4289950497625583680\) \(\nu\mathstrut +\mathstrut \) \(2437368238033035472\)\()/\)\(226960549473601184\)
\(\beta_{13}\)\(=\)\((\)\(1573378793415459\) \(\nu^{13}\mathstrut -\mathstrut \) \(5220619154611247\) \(\nu^{12}\mathstrut -\mathstrut \) \(65952957071802091\) \(\nu^{11}\mathstrut +\mathstrut \) \(208321575309308339\) \(\nu^{10}\mathstrut +\mathstrut \) \(999279614844233815\) \(\nu^{9}\mathstrut -\mathstrut \) \(2944518757895850607\) \(\nu^{8}\mathstrut -\mathstrut \) \(6692860674237136328\) \(\nu^{7}\mathstrut +\mathstrut \) \(17374262131005461586\) \(\nu^{6}\mathstrut +\mathstrut \) \(19906041873294275385\) \(\nu^{5}\mathstrut -\mathstrut \) \(38405526987464180273\) \(\nu^{4}\mathstrut -\mathstrut \) \(24732980773534038746\) \(\nu^{3}\mathstrut +\mathstrut \) \(20690398991875395764\) \(\nu^{2}\mathstrut +\mathstrut \) \(7117140897075219296\) \(\nu\mathstrut -\mathstrut \) \(1192615697745108912\)\()/\)\(226960549473601184\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(-\)\(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(2\) \(\beta_{13}\mathstrut +\mathstrut \) \(4\) \(\beta_{12}\mathstrut -\mathstrut \) \(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(11\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(22\) \(\beta_{7}\mathstrut +\mathstrut \) \(40\) \(\beta_{6}\mathstrut -\mathstrut \) \(20\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(36\) \(\beta_{1}\mathstrut +\mathstrut \) \(81\)
\(\nu^{5}\)\(=\)\(-\)\(48\) \(\beta_{13}\mathstrut -\mathstrut \) \(20\) \(\beta_{12}\mathstrut +\mathstrut \) \(43\) \(\beta_{10}\mathstrut -\mathstrut \) \(27\) \(\beta_{9}\mathstrut +\mathstrut \) \(60\) \(\beta_{8}\mathstrut +\mathstrut \) \(75\) \(\beta_{7}\mathstrut +\mathstrut \) \(61\) \(\beta_{6}\mathstrut -\mathstrut \) \(30\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut -\mathstrut \) \(21\) \(\beta_{3}\mathstrut +\mathstrut \) \(41\) \(\beta_{2}\mathstrut +\mathstrut \) \(200\) \(\beta_{1}\mathstrut -\mathstrut \) \(46\)
\(\nu^{6}\)\(=\)\(-\)\(56\) \(\beta_{13}\mathstrut +\mathstrut \) \(111\) \(\beta_{12}\mathstrut -\mathstrut \) \(247\) \(\beta_{11}\mathstrut +\mathstrut \) \(130\) \(\beta_{10}\mathstrut +\mathstrut \) \(140\) \(\beta_{9}\mathstrut +\mathstrut \) \(98\) \(\beta_{8}\mathstrut +\mathstrut \) \(410\) \(\beta_{7}\mathstrut +\mathstrut \) \(752\) \(\beta_{6}\mathstrut -\mathstrut \) \(362\) \(\beta_{5}\mathstrut +\mathstrut \) \(83\) \(\beta_{4}\mathstrut +\mathstrut \) \(35\) \(\beta_{3}\mathstrut +\mathstrut \) \(88\) \(\beta_{2}\mathstrut +\mathstrut \) \(648\) \(\beta_{1}\mathstrut +\mathstrut \) \(1069\)
\(\nu^{7}\)\(=\)\(-\)\(944\) \(\beta_{13}\mathstrut -\mathstrut \) \(311\) \(\beta_{12}\mathstrut -\mathstrut \) \(39\) \(\beta_{11}\mathstrut +\mathstrut \) \(813\) \(\beta_{10}\mathstrut -\mathstrut \) \(579\) \(\beta_{9}\mathstrut +\mathstrut \) \(1349\) \(\beta_{8}\mathstrut +\mathstrut \) \(1528\) \(\beta_{7}\mathstrut +\mathstrut \) \(1452\) \(\beta_{6}\mathstrut -\mathstrut \) \(750\) \(\beta_{5}\mathstrut +\mathstrut \) \(337\) \(\beta_{4}\mathstrut -\mathstrut \) \(387\) \(\beta_{3}\mathstrut +\mathstrut \) \(749\) \(\beta_{2}\mathstrut +\mathstrut \) \(3343\) \(\beta_{1}\mathstrut -\mathstrut \) \(732\)
\(\nu^{8}\)\(=\)\(-\)\(1342\) \(\beta_{13}\mathstrut +\mathstrut \) \(2361\) \(\beta_{12}\mathstrut -\mathstrut \) \(3882\) \(\beta_{11}\mathstrut +\mathstrut \) \(2724\) \(\beta_{10}\mathstrut +\mathstrut \) \(1917\) \(\beta_{9}\mathstrut +\mathstrut \) \(2439\) \(\beta_{8}\mathstrut +\mathstrut \) \(7496\) \(\beta_{7}\mathstrut +\mathstrut \) \(13953\) \(\beta_{6}\mathstrut -\mathstrut \) \(6525\) \(\beta_{5}\mathstrut +\mathstrut \) \(2162\) \(\beta_{4}\mathstrut +\mathstrut \) \(390\) \(\beta_{3}\mathstrut +\mathstrut \) \(2096\) \(\beta_{2}\mathstrut +\mathstrut \) \(11959\) \(\beta_{1}\mathstrut +\mathstrut \) \(15161\)
\(\nu^{9}\)\(=\)\(-\)\(17722\) \(\beta_{13}\mathstrut -\mathstrut \) \(4265\) \(\beta_{12}\mathstrut -\mathstrut \) \(1739\) \(\beta_{11}\mathstrut +\mathstrut \) \(14966\) \(\beta_{10}\mathstrut -\mathstrut \) \(11226\) \(\beta_{9}\mathstrut +\mathstrut \) \(27283\) \(\beta_{8}\mathstrut +\mathstrut \) \(29643\) \(\beta_{7}\mathstrut +\mathstrut \) \(31805\) \(\beta_{6}\mathstrut -\mathstrut \) \(16974\) \(\beta_{5}\mathstrut +\mathstrut \) \(7210\) \(\beta_{4}\mathstrut -\mathstrut \) \(6882\) \(\beta_{3}\mathstrut +\mathstrut \) \(13690\) \(\beta_{2}\mathstrut +\mathstrut \) \(58882\) \(\beta_{1}\mathstrut -\mathstrut \) \(9586\)
\(\nu^{10}\)\(=\)\(-\)\(30286\) \(\beta_{13}\mathstrut +\mathstrut \) \(45634\) \(\beta_{12}\mathstrut -\mathstrut \) \(62908\) \(\beta_{11}\mathstrut +\mathstrut \) \(53784\) \(\beta_{10}\mathstrut +\mathstrut \) \(26730\) \(\beta_{9}\mathstrut +\mathstrut \) \(55035\) \(\beta_{8}\mathstrut +\mathstrut \) \(138103\) \(\beta_{7}\mathstrut +\mathstrut \) \(258397\) \(\beta_{6}\mathstrut -\mathstrut \) \(118842\) \(\beta_{5}\mathstrut +\mathstrut \) \(47957\) \(\beta_{4}\mathstrut +\mathstrut \) \(1959\) \(\beta_{3}\mathstrut +\mathstrut \) \(45930\) \(\beta_{2}\mathstrut +\mathstrut \) \(224791\) \(\beta_{1}\mathstrut +\mathstrut \) \(226411\)
\(\nu^{11}\)\(=\)\(-\)\(328124\) \(\beta_{13}\mathstrut -\mathstrut \) \(51235\) \(\beta_{12}\mathstrut -\mathstrut \) \(52212\) \(\beta_{11}\mathstrut +\mathstrut \) \(275611\) \(\beta_{10}\mathstrut -\mathstrut \) \(206710\) \(\beta_{9}\mathstrut +\mathstrut \) \(527032\) \(\beta_{8}\mathstrut +\mathstrut \) \(567670\) \(\beta_{7}\mathstrut +\mathstrut \) \(668909\) \(\beta_{6}\mathstrut -\mathstrut \) \(362055\) \(\beta_{5}\mathstrut +\mathstrut \) \(153928\) \(\beta_{4}\mathstrut -\mathstrut \) \(121168\) \(\beta_{3}\mathstrut +\mathstrut \) \(254071\) \(\beta_{2}\mathstrut +\mathstrut \) \(1071832\) \(\beta_{1}\mathstrut -\mathstrut \) \(97646\)
\(\nu^{12}\)\(=\)\(-\)\(658066\) \(\beta_{13}\mathstrut +\mathstrut \) \(844213\) \(\beta_{12}\mathstrut -\mathstrut \) \(1051148\) \(\beta_{11}\mathstrut +\mathstrut \) \(1042435\) \(\beta_{10}\mathstrut +\mathstrut \) \(365834\) \(\beta_{9}\mathstrut +\mathstrut \) \(1183883\) \(\beta_{8}\mathstrut +\mathstrut \) \(2573085\) \(\beta_{7}\mathstrut +\mathstrut \) \(4794828\) \(\beta_{6}\mathstrut -\mathstrut \) \(2190757\) \(\beta_{5}\mathstrut +\mathstrut \) \(988293\) \(\beta_{4}\mathstrut -\mathstrut \) \(49511\) \(\beta_{3}\mathstrut +\mathstrut \) \(958867\) \(\beta_{2}\mathstrut +\mathstrut \) \(4272168\) \(\beta_{1}\mathstrut +\mathstrut \) \(3524267\)
\(\nu^{13}\)\(=\)\(-\)\(6053854\) \(\beta_{13}\mathstrut -\mathstrut \) \(475034\) \(\beta_{12}\mathstrut -\mathstrut \) \(1323481\) \(\beta_{11}\mathstrut +\mathstrut \) \(5115726\) \(\beta_{10}\mathstrut -\mathstrut \) \(3700443\) \(\beta_{9}\mathstrut +\mathstrut \) \(9971909\) \(\beta_{8}\mathstrut +\mathstrut \) \(10844156\) \(\beta_{7}\mathstrut +\mathstrut \) \(13731109\) \(\beta_{6}\mathstrut -\mathstrut \) \(7445569\) \(\beta_{5}\mathstrut +\mathstrut \) \(3215275\) \(\beta_{4}\mathstrut -\mathstrut \) \(2135973\) \(\beta_{3}\mathstrut +\mathstrut \) \(4778588\) \(\beta_{2}\mathstrut +\mathstrut \) \(19911980\) \(\beta_{1}\mathstrut -\mathstrut \) \(397476\)

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
4.38301
3.42255
3.31199
2.85865
1.95823
0.787442
0.155434
−0.356687
−0.549450
−0.902324
−1.74358
−2.82032
−3.58563
−3.91931
0 −1.00000 0 −4.38301 0 0.798768 0 1.00000 0
1.2 0 −1.00000 0 −3.42255 0 −3.90388 0 1.00000 0
1.3 0 −1.00000 0 −3.31199 0 3.66743 0 1.00000 0
1.4 0 −1.00000 0 −2.85865 0 3.46115 0 1.00000 0
1.5 0 −1.00000 0 −1.95823 0 3.64071 0 1.00000 0
1.6 0 −1.00000 0 −0.787442 0 1.19810 0 1.00000 0
1.7 0 −1.00000 0 −0.155434 0 −3.91825 0 1.00000 0
1.8 0 −1.00000 0 0.356687 0 4.08866 0 1.00000 0
1.9 0 −1.00000 0 0.549450 0 1.28524 0 1.00000 0
1.10 0 −1.00000 0 0.902324 0 −3.04389 0 1.00000 0
1.11 0 −1.00000 0 1.74358 0 2.16730 0 1.00000 0
1.12 0 −1.00000 0 2.82032 0 1.34486 0 1.00000 0
1.13 0 −1.00000 0 3.58563 0 −1.47988 0 1.00000 0
1.14 0 −1.00000 0 3.91931 0 −2.30632 0 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(251\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

\(T_{5}^{14} + \cdots\)
\(T_{7}^{14} - \cdots\)