Properties

Label 8008.2.a.z
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 0
Dimension 15
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8008.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{11} q^{5} \) \(- q^{7}\) \( + ( 2 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( -\beta_{11} q^{5} \) \(- q^{7}\) \( + ( 2 + \beta_{2} ) q^{9} \) \(- q^{11}\) \(+ q^{13}\) \( + \beta_{3} q^{15} \) \( + ( 1 + \beta_{9} ) q^{17} \) \( + ( -1 + \beta_{6} - \beta_{11} ) q^{19} \) \( + \beta_{1} q^{21} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{13} ) q^{23} \) \( + ( 2 + \beta_{4} - \beta_{10} - \beta_{11} ) q^{25} \) \( + ( 1 - 2 \beta_{1} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{11} ) q^{27} \) \( + ( 1 + \beta_{1} + \beta_{8} + \beta_{12} ) q^{29} \) \( + ( -1 + \beta_{5} - \beta_{11} - \beta_{12} ) q^{31} \) \( + \beta_{1} q^{33} \) \( + \beta_{11} q^{35} \) \( + ( \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} - \beta_{9} + \beta_{10} - \beta_{11} - \beta_{14} ) q^{37} \) \( -\beta_{1} q^{39} \) \( + ( -\beta_{2} + \beta_{3} + \beta_{4} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} ) q^{41} \) \( + ( -1 + \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{43} \) \( + ( 1 + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} + \beta_{14} ) q^{45} \) \( + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} + \beta_{14} ) q^{47} \) \(+ q^{49}\) \( + ( -\beta_{1} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} ) q^{51} \) \( + ( 3 + \beta_{3} - \beta_{5} + \beta_{10} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{53} \) \( + \beta_{11} q^{55} \) \( + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{10} - \beta_{13} + 2 \beta_{14} ) q^{57} \) \( + ( -1 + \beta_{3} - \beta_{9} + \beta_{10} - \beta_{11} + \beta_{13} - \beta_{14} ) q^{59} \) \( + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{12} ) q^{61} \) \( + ( -2 - \beta_{2} ) q^{63} \) \( -\beta_{11} q^{65} \) \( + ( -\beta_{1} + \beta_{3} - \beta_{5} - \beta_{6} + \beta_{10} + \beta_{11} ) q^{67} \) \( + ( 3 - 3 \beta_{1} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} ) q^{69} \) \( + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{7} - \beta_{9} - \beta_{11} - \beta_{12} ) q^{71} \) \( + ( -\beta_{3} - \beta_{4} + \beta_{5} - \beta_{11} + \beta_{13} ) q^{73} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{4} + \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} - \beta_{14} ) q^{75} \) \(+ q^{77}\) \( + ( \beta_{1} - \beta_{3} + \beta_{6} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{14} ) q^{79} \) \( + ( 4 - \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{6} + \beta_{10} + 2 \beta_{11} + \beta_{12} + \beta_{14} ) q^{81} \) \( + ( -3 - \beta_{2} - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{14} ) q^{83} \) \( + ( 2 - 2 \beta_{1} - \beta_{4} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{85} \) \( + ( -2 - 2 \beta_{2} - \beta_{5} + \beta_{11} + \beta_{12} - 2 \beta_{13} + \beta_{14} ) q^{87} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{7} - \beta_{9} - \beta_{11} ) q^{89} \) \(- q^{91}\) \( + ( -2 - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - 2 \beta_{9} - 3 \beta_{11} - \beta_{12} + \beta_{13} - \beta_{14} ) q^{93} \) \( + ( 5 + \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{9} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} ) q^{95} \) \( + ( 1 + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} ) q^{97} \) \( + ( -2 - \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(15q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(15q \) \(\mathstrut -\mathstrut q^{3} \) \(\mathstrut +\mathstrut 4q^{5} \) \(\mathstrut -\mathstrut 15q^{7} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut -\mathstrut 15q^{11} \) \(\mathstrut +\mathstrut 15q^{13} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 17q^{19} \) \(\mathstrut +\mathstrut q^{21} \) \(\mathstrut +\mathstrut 7q^{23} \) \(\mathstrut +\mathstrut 33q^{25} \) \(\mathstrut -\mathstrut 4q^{27} \) \(\mathstrut +\mathstrut 14q^{29} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut q^{39} \) \(\mathstrut -\mathstrut 13q^{43} \) \(\mathstrut +\mathstrut 20q^{45} \) \(\mathstrut +\mathstrut 6q^{47} \) \(\mathstrut +\mathstrut 15q^{49} \) \(\mathstrut +\mathstrut 8q^{51} \) \(\mathstrut +\mathstrut 38q^{53} \) \(\mathstrut -\mathstrut 4q^{55} \) \(\mathstrut +\mathstrut 24q^{57} \) \(\mathstrut -\mathstrut 18q^{59} \) \(\mathstrut +\mathstrut 23q^{61} \) \(\mathstrut -\mathstrut 26q^{63} \) \(\mathstrut +\mathstrut 4q^{65} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut +\mathstrut 43q^{69} \) \(\mathstrut -\mathstrut 12q^{71} \) \(\mathstrut +\mathstrut 11q^{73} \) \(\mathstrut +\mathstrut 12q^{75} \) \(\mathstrut +\mathstrut 15q^{77} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut +\mathstrut 51q^{81} \) \(\mathstrut -\mathstrut 16q^{83} \) \(\mathstrut +\mathstrut 13q^{85} \) \(\mathstrut -\mathstrut 25q^{87} \) \(\mathstrut +\mathstrut 28q^{89} \) \(\mathstrut -\mathstrut 15q^{91} \) \(\mathstrut -\mathstrut 14q^{93} \) \(\mathstrut +\mathstrut 49q^{95} \) \(\mathstrut +\mathstrut 30q^{97} \) \(\mathstrut -\mathstrut 26q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(x^{14}\mathstrut -\mathstrut \) \(35\) \(x^{13}\mathstrut +\mathstrut \) \(32\) \(x^{12}\mathstrut +\mathstrut \) \(477\) \(x^{11}\mathstrut -\mathstrut \) \(392\) \(x^{10}\mathstrut -\mathstrut \) \(3236\) \(x^{9}\mathstrut +\mathstrut \) \(2330\) \(x^{8}\mathstrut +\mathstrut \) \(11690\) \(x^{7}\mathstrut -\mathstrut \) \(7119\) \(x^{6}\mathstrut -\mathstrut \) \(22246\) \(x^{5}\mathstrut +\mathstrut \) \(11137\) \(x^{4}\mathstrut +\mathstrut \) \(20034\) \(x^{3}\mathstrut -\mathstrut \) \(8392\) \(x^{2}\mathstrut -\mathstrut \) \(6016\) \(x\mathstrut +\mathstrut \) \(2560\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(1523579\) \(\nu^{14}\mathstrut +\mathstrut \) \(213661034\) \(\nu^{13}\mathstrut -\mathstrut \) \(179139600\) \(\nu^{12}\mathstrut -\mathstrut \) \(6986479725\) \(\nu^{11}\mathstrut +\mathstrut \) \(4541129757\) \(\nu^{10}\mathstrut +\mathstrut \) \(85561543273\) \(\nu^{9}\mathstrut -\mathstrut \) \(48599699344\) \(\nu^{8}\mathstrut -\mathstrut \) \(488772294898\) \(\nu^{7}\mathstrut +\mathstrut \) \(252291530370\) \(\nu^{6}\mathstrut +\mathstrut \) \(1334713440565\) \(\nu^{5}\mathstrut -\mathstrut \) \(626492952357\) \(\nu^{4}\mathstrut -\mathstrut \) \(1586347082289\) \(\nu^{3}\mathstrut +\mathstrut \) \(678120858481\) \(\nu^{2}\mathstrut +\mathstrut \) \(568931122604\) \(\nu\mathstrut -\mathstrut \) \(253585799040\)\()/\)\(10828273438\)
\(\beta_{4}\)\(=\)\((\)\(204728997\) \(\nu^{14}\mathstrut -\mathstrut \) \(5797949397\) \(\nu^{13}\mathstrut +\mathstrut \) \(1200404289\) \(\nu^{12}\mathstrut +\mathstrut \) \(186669594656\) \(\nu^{11}\mathstrut -\mathstrut \) \(160969626719\) \(\nu^{10}\mathstrut -\mathstrut \) \(2248818957080\) \(\nu^{9}\mathstrut +\mathstrut \) \(2312689549068\) \(\nu^{8}\mathstrut +\mathstrut \) \(12635569411410\) \(\nu^{7}\mathstrut -\mathstrut \) \(13379380540814\) \(\nu^{6}\mathstrut -\mathstrut \) \(34174617664635\) \(\nu^{5}\mathstrut +\mathstrut \) \(34513885202850\) \(\nu^{4}\mathstrut +\mathstrut \) \(42027402294341\) \(\nu^{3}\mathstrut -\mathstrut \) \(37011471642118\) \(\nu^{2}\mathstrut -\mathstrut \) \(18367614659000\) \(\nu\mathstrut +\mathstrut \) \(11235942086336\)\()/\)\(173252375008\)
\(\beta_{5}\)\(=\)\((\)\(157141431\) \(\nu^{14}\mathstrut -\mathstrut \) \(1230552919\) \(\nu^{13}\mathstrut -\mathstrut \) \(3155705101\) \(\nu^{12}\mathstrut +\mathstrut \) \(40387483536\) \(\nu^{11}\mathstrut +\mathstrut \) \(1702785579\) \(\nu^{10}\mathstrut -\mathstrut \) \(504234185256\) \(\nu^{9}\mathstrut +\mathstrut \) \(342178471052\) \(\nu^{8}\mathstrut +\mathstrut \) \(3010380395486\) \(\nu^{7}\mathstrut -\mathstrut \) \(2722381354490\) \(\nu^{6}\mathstrut -\mathstrut \) \(8922472657697\) \(\nu^{5}\mathstrut +\mathstrut \) \(8079082533374\) \(\nu^{4}\mathstrut +\mathstrut \) \(12254264192927\) \(\nu^{3}\mathstrut -\mathstrut \) \(9827207416642\) \(\nu^{2}\mathstrut -\mathstrut \) \(5792078145608\) \(\nu\mathstrut +\mathstrut \) \(3811771653184\)\()/\)\(86626187504\)
\(\beta_{6}\)\(=\)\((\)\(413408383\) \(\nu^{14}\mathstrut +\mathstrut \) \(1544421129\) \(\nu^{13}\mathstrut -\mathstrut \) \(15746220005\) \(\nu^{12}\mathstrut -\mathstrut \) \(49653568744\) \(\nu^{11}\mathstrut +\mathstrut \) \(228419477571\) \(\nu^{10}\mathstrut +\mathstrut \) \(588397861744\) \(\nu^{9}\mathstrut -\mathstrut \) \(1564264168284\) \(\nu^{8}\mathstrut -\mathstrut \) \(3156065445642\) \(\nu^{7}\mathstrut +\mathstrut \) \(5007460939446\) \(\nu^{6}\mathstrut +\mathstrut \) \(7618648102207\) \(\nu^{5}\mathstrut -\mathstrut \) \(5948270274850\) \(\nu^{4}\mathstrut -\mathstrut \) \(6815219183521\) \(\nu^{3}\mathstrut +\mathstrut \) \(347842922614\) \(\nu^{2}\mathstrut +\mathstrut \) \(98512228088\) \(\nu\mathstrut +\mathstrut \) \(545872471392\)\()/\)\(173252375008\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(742937461\) \(\nu^{14}\mathstrut -\mathstrut \) \(1826318179\) \(\nu^{13}\mathstrut +\mathstrut \) \(30835167495\) \(\nu^{12}\mathstrut +\mathstrut \) \(59173290264\) \(\nu^{11}\mathstrut -\mathstrut \) \(502014930785\) \(\nu^{10}\mathstrut -\mathstrut \) \(714535204112\) \(\nu^{9}\mathstrut +\mathstrut \) \(4061739875940\) \(\nu^{8}\mathstrut +\mathstrut \) \(3991707328670\) \(\nu^{7}\mathstrut -\mathstrut \) \(17068026173042\) \(\nu^{6}\mathstrut -\mathstrut \) \(10549349624229\) \(\nu^{5}\mathstrut +\mathstrut \) \(35541387263846\) \(\nu^{4}\mathstrut +\mathstrut \) \(12283735275291\) \(\nu^{3}\mathstrut -\mathstrut \) \(31984400742130\) \(\nu^{2}\mathstrut -\mathstrut \) \(5198752858728\) \(\nu\mathstrut +\mathstrut \) \(8430631658656\)\()/\)\(173252375008\)
\(\beta_{8}\)\(=\)\((\)\(1890437133\) \(\nu^{14}\mathstrut -\mathstrut \) \(1073055941\) \(\nu^{13}\mathstrut -\mathstrut \) \(66111397935\) \(\nu^{12}\mathstrut +\mathstrut \) \(31796527976\) \(\nu^{11}\mathstrut +\mathstrut \) \(904394485593\) \(\nu^{10}\mathstrut -\mathstrut \) \(346151330752\) \(\nu^{9}\mathstrut -\mathstrut \) \(6238137460212\) \(\nu^{8}\mathstrut +\mathstrut \) \(1668210629874\) \(\nu^{7}\mathstrut +\mathstrut \) \(23572513434290\) \(\nu^{6}\mathstrut -\mathstrut \) \(3218674662259\) \(\nu^{5}\mathstrut -\mathstrut \) \(49012994686182\) \(\nu^{4}\mathstrut +\mathstrut \) \(934503342013\) \(\nu^{3}\mathstrut +\mathstrut \) \(49331444492578\) \(\nu^{2}\mathstrut +\mathstrut \) \(1799763702344\) \(\nu\mathstrut -\mathstrut \) \(14280187531456\)\()/\)\(346504750016\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(1938443511\) \(\nu^{14}\mathstrut +\mathstrut \) \(1484217919\) \(\nu^{13}\mathstrut +\mathstrut \) \(68173131037\) \(\nu^{12}\mathstrut -\mathstrut \) \(44865881400\) \(\nu^{11}\mathstrut -\mathstrut \) \(931234396411\) \(\nu^{10}\mathstrut +\mathstrut \) \(498828247584\) \(\nu^{9}\mathstrut +\mathstrut \) \(6276390141532\) \(\nu^{8}\mathstrut -\mathstrut \) \(2491001622102\) \(\nu^{7}\mathstrut -\mathstrut \) \(21963128480886\) \(\nu^{6}\mathstrut +\mathstrut \) \(5412559613929\) \(\nu^{5}\mathstrut +\mathstrut \) \(37978280261778\) \(\nu^{4}\mathstrut -\mathstrut \) \(3729507797895\) \(\nu^{3}\mathstrut -\mathstrut \) \(26682788237798\) \(\nu^{2}\mathstrut -\mathstrut \) \(727396710904\) \(\nu\mathstrut +\mathstrut \) \(4190172143104\)\()/\)\(346504750016\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(1184655527\) \(\nu^{14}\mathstrut +\mathstrut \) \(40599471\) \(\nu^{13}\mathstrut +\mathstrut \) \(42697917709\) \(\nu^{12}\mathstrut +\mathstrut \) \(530650360\) \(\nu^{11}\mathstrut -\mathstrut \) \(604962621531\) \(\nu^{10}\mathstrut -\mathstrut \) \(33239466432\) \(\nu^{9}\mathstrut +\mathstrut \) \(4319125883612\) \(\nu^{8}\mathstrut +\mathstrut \) \(423236823962\) \(\nu^{7}\mathstrut -\mathstrut \) \(16606302188950\) \(\nu^{6}\mathstrut -\mathstrut \) \(2490295041815\) \(\nu^{5}\mathstrut +\mathstrut \) \(33684876907586\) \(\nu^{4}\mathstrut +\mathstrut \) \(7018914045929\) \(\nu^{3}\mathstrut -\mathstrut \) \(31804427646502\) \(\nu^{2}\mathstrut -\mathstrut \) \(7104276001064\) \(\nu\mathstrut +\mathstrut \) \(9057739992992\)\()/\)\(173252375008\)
\(\beta_{11}\)\(=\)\((\)\(396227811\) \(\nu^{14}\mathstrut -\mathstrut \) \(390133495\) \(\nu^{13}\mathstrut -\mathstrut \) \(13013329249\) \(\nu^{12}\mathstrut +\mathstrut \) \(11962731552\) \(\nu^{11}\mathstrut +\mathstrut \) \(161054746947\) \(\nu^{10}\mathstrut -\mathstrut \) \(137156782884\) \(\nu^{9}\mathstrut -\mathstrut \) \(939947023304\) \(\nu^{8}\mathstrut +\mathstrut \) \(728812002254\) \(\nu^{7}\mathstrut +\mathstrut \) \(2676813930998\) \(\nu^{6}\mathstrut -\mathstrut \) \(1811579665029\) \(\nu^{5}\mathstrut -\mathstrut \) \(3475630121246\) \(\nu^{4}\mathstrut +\mathstrut \) \(1906817321679\) \(\nu^{3}\mathstrut +\mathstrut \) \(1592639636418\) \(\nu^{2}\mathstrut -\mathstrut \) \(612660355988\) \(\nu\mathstrut -\mathstrut \) \(107982020560\)\()/\)\(43313093752\)
\(\beta_{12}\)\(=\)\((\)\(2388875051\) \(\nu^{14}\mathstrut -\mathstrut \) \(8261584387\) \(\nu^{13}\mathstrut -\mathstrut \) \(71865979993\) \(\nu^{12}\mathstrut +\mathstrut \) \(259474132888\) \(\nu^{11}\mathstrut +\mathstrut \) \(775871452191\) \(\nu^{10}\mathstrut -\mathstrut \) \(3032322946288\) \(\nu^{9}\mathstrut -\mathstrut \) \(3545121573980\) \(\nu^{8}\mathstrut +\mathstrut \) \(16349472328158\) \(\nu^{7}\mathstrut +\mathstrut \) \(5728168812302\) \(\nu^{6}\mathstrut -\mathstrut \) \(41488112322565\) \(\nu^{5}\mathstrut +\mathstrut \) \(2209043811558\) \(\nu^{4}\mathstrut +\mathstrut \) \(45742051602491\) \(\nu^{3}\mathstrut -\mathstrut \) \(12659651595602\) \(\nu^{2}\mathstrut -\mathstrut \) \(15947603806024\) \(\nu\mathstrut +\mathstrut \) \(7139114332064\)\()/\)\(173252375008\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(2665656839\) \(\nu^{14}\mathstrut +\mathstrut \) \(4559868439\) \(\nu^{13}\mathstrut +\mathstrut \) \(84068282789\) \(\nu^{12}\mathstrut -\mathstrut \) \(139774279616\) \(\nu^{11}\mathstrut -\mathstrut \) \(980978775387\) \(\nu^{10}\mathstrut +\mathstrut \) \(1587508125288\) \(\nu^{9}\mathstrut +\mathstrut \) \(5227482261084\) \(\nu^{8}\mathstrut -\mathstrut \) \(8262828912134\) \(\nu^{7}\mathstrut -\mathstrut \) \(12866431836518\) \(\nu^{6}\mathstrut +\mathstrut \) \(20039589961337\) \(\nu^{5}\mathstrut +\mathstrut \) \(13182923217098\) \(\nu^{4}\mathstrut -\mathstrut \) \(21042920256135\) \(\nu^{3}\mathstrut -\mathstrut \) \(3528585941854\) \(\nu^{2}\mathstrut +\mathstrut \) \(7004343553608\) \(\nu\mathstrut -\mathstrut \) \(486592259136\)\()/\)\(173252375008\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(3526798371\) \(\nu^{14}\mathstrut +\mathstrut \) \(4709151755\) \(\nu^{13}\mathstrut +\mathstrut \) \(117063051249\) \(\nu^{12}\mathstrut -\mathstrut \) \(143804690696\) \(\nu^{11}\mathstrut -\mathstrut \) \(1480208505991\) \(\nu^{10}\mathstrut +\mathstrut \) \(1622561315744\) \(\nu^{9}\mathstrut +\mathstrut \) \(9029091617148\) \(\nu^{8}\mathstrut -\mathstrut \) \(8308488511342\) \(\nu^{7}\mathstrut -\mathstrut \) \(28251013727598\) \(\nu^{6}\mathstrut +\mathstrut \) \(19095872506589\) \(\nu^{5}\mathstrut +\mathstrut \) \(45139985518970\) \(\nu^{4}\mathstrut -\mathstrut \) \(16157472133315\) \(\nu^{3}\mathstrut -\mathstrut \) \(33181232975262\) \(\nu^{2}\mathstrut +\mathstrut \) \(592595901752\) \(\nu\mathstrut +\mathstrut \) \(7092433199008\)\()/\)\(173252375008\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(40\)
\(\nu^{5}\)\(=\)\(-\)\(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(15\) \(\beta_{11}\mathstrut -\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(14\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(74\) \(\beta_{1}\mathstrut -\mathstrut \) \(14\)
\(\nu^{6}\)\(=\)\(18\) \(\beta_{14}\mathstrut -\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(16\) \(\beta_{12}\mathstrut +\mathstrut \) \(31\) \(\beta_{11}\mathstrut +\mathstrut \) \(14\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(33\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(129\) \(\beta_{2}\mathstrut -\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(374\)
\(\nu^{7}\)\(=\)\(-\)\(18\) \(\beta_{14}\mathstrut -\mathstrut \) \(21\) \(\beta_{13}\mathstrut -\mathstrut \) \(21\) \(\beta_{12}\mathstrut -\mathstrut \) \(184\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(167\) \(\beta_{9}\mathstrut +\mathstrut \) \(129\) \(\beta_{8}\mathstrut +\mathstrut \) \(161\) \(\beta_{7}\mathstrut +\mathstrut \) \(144\) \(\beta_{6}\mathstrut +\mathstrut \) \(18\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut -\mathstrut \) \(17\) \(\beta_{3}\mathstrut +\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(733\) \(\beta_{1}\mathstrut -\mathstrut \) \(157\)
\(\nu^{8}\)\(=\)\(252\) \(\beta_{14}\mathstrut -\mathstrut \) \(71\) \(\beta_{13}\mathstrut +\mathstrut \) \(202\) \(\beta_{12}\mathstrut +\mathstrut \) \(375\) \(\beta_{11}\mathstrut +\mathstrut \) \(148\) \(\beta_{10}\mathstrut +\mathstrut \) \(55\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(154\) \(\beta_{6}\mathstrut -\mathstrut \) \(21\) \(\beta_{5}\mathstrut -\mathstrut \) \(423\) \(\beta_{4}\mathstrut -\mathstrut \) \(266\) \(\beta_{3}\mathstrut +\mathstrut \) \(1362\) \(\beta_{2}\mathstrut -\mathstrut \) \(123\) \(\beta_{1}\mathstrut +\mathstrut \) \(3719\)
\(\nu^{9}\)\(=\)\(-\)\(231\) \(\beta_{14}\mathstrut -\mathstrut \) \(310\) \(\beta_{13}\mathstrut -\mathstrut \) \(314\) \(\beta_{12}\mathstrut -\mathstrut \) \(2079\) \(\beta_{11}\mathstrut +\mathstrut \) \(70\) \(\beta_{10}\mathstrut -\mathstrut \) \(1908\) \(\beta_{9}\mathstrut +\mathstrut \) \(1362\) \(\beta_{8}\mathstrut +\mathstrut \) \(1762\) \(\beta_{7}\mathstrut +\mathstrut \) \(1520\) \(\beta_{6}\mathstrut +\mathstrut \) \(228\) \(\beta_{5}\mathstrut -\mathstrut \) \(172\) \(\beta_{4}\mathstrut -\mathstrut \) \(222\) \(\beta_{3}\mathstrut +\mathstrut \) \(222\) \(\beta_{2}\mathstrut +\mathstrut \) \(7485\) \(\beta_{1}\mathstrut -\mathstrut \) \(1611\)
\(\nu^{10}\)\(=\)\(3231\) \(\beta_{14}\mathstrut -\mathstrut \) \(1178\) \(\beta_{13}\mathstrut +\mathstrut \) \(2365\) \(\beta_{12}\mathstrut +\mathstrut \) \(4177\) \(\beta_{11}\mathstrut +\mathstrut \) \(1396\) \(\beta_{10}\mathstrut +\mathstrut \) \(696\) \(\beta_{9}\mathstrut +\mathstrut \) \(222\) \(\beta_{8}\mathstrut +\mathstrut \) \(61\) \(\beta_{7}\mathstrut -\mathstrut \) \(1578\) \(\beta_{6}\mathstrut -\mathstrut \) \(304\) \(\beta_{5}\mathstrut -\mathstrut \) \(4992\) \(\beta_{4}\mathstrut -\mathstrut \) \(3330\) \(\beta_{3}\mathstrut +\mathstrut \) \(14343\) \(\beta_{2}\mathstrut -\mathstrut \) \(968\) \(\beta_{1}\mathstrut +\mathstrut \) \(37990\)
\(\nu^{11}\)\(=\)\(-\)\(2559\) \(\beta_{14}\mathstrut -\mathstrut \) \(4056\) \(\beta_{13}\mathstrut -\mathstrut \) \(4140\) \(\beta_{12}\mathstrut -\mathstrut \) \(22536\) \(\beta_{11}\mathstrut +\mathstrut \) \(1102\) \(\beta_{10}\mathstrut -\mathstrut \) \(21413\) \(\beta_{9}\mathstrut +\mathstrut \) \(14343\) \(\beta_{8}\mathstrut +\mathstrut \) \(18970\) \(\beta_{7}\mathstrut +\mathstrut \) \(15716\) \(\beta_{6}\mathstrut +\mathstrut \) \(2557\) \(\beta_{5}\mathstrut -\mathstrut \) \(1838\) \(\beta_{4}\mathstrut -\mathstrut \) \(2713\) \(\beta_{3}\mathstrut +\mathstrut \) \(2681\) \(\beta_{2}\mathstrut +\mathstrut \) \(77567\) \(\beta_{1}\mathstrut -\mathstrut \) \(15851\)
\(\nu^{12}\)\(=\)\(39686\) \(\beta_{14}\mathstrut -\mathstrut \) \(16983\) \(\beta_{13}\mathstrut +\mathstrut \) \(26769\) \(\beta_{12}\mathstrut +\mathstrut \) \(44927\) \(\beta_{11}\mathstrut +\mathstrut \) \(12188\) \(\beta_{10}\mathstrut +\mathstrut \) \(7551\) \(\beta_{9}\mathstrut +\mathstrut \) \(2681\) \(\beta_{8}\mathstrut +\mathstrut \) \(1224\) \(\beta_{7}\mathstrut -\mathstrut \) \(15767\) \(\beta_{6}\mathstrut -\mathstrut \) \(3816\) \(\beta_{5}\mathstrut -\mathstrut \) \(56875\) \(\beta_{4}\mathstrut -\mathstrut \) \(39609\) \(\beta_{3}\mathstrut +\mathstrut \) \(151136\) \(\beta_{2}\mathstrut -\mathstrut \) \(5951\) \(\beta_{1}\mathstrut +\mathstrut \) \(393237\)
\(\nu^{13}\)\(=\)\(-\)\(25891\) \(\beta_{14}\mathstrut -\mathstrut \) \(50465\) \(\beta_{13}\mathstrut -\mathstrut \) \(51380\) \(\beta_{12}\mathstrut -\mathstrut \) \(238942\) \(\beta_{11}\mathstrut +\mathstrut \) \(14779\) \(\beta_{10}\mathstrut -\mathstrut \) \(238140\) \(\beta_{9}\mathstrut +\mathstrut \) \(151136\) \(\beta_{8}\mathstrut +\mathstrut \) \(203010\) \(\beta_{7}\mathstrut +\mathstrut \) \(160838\) \(\beta_{6}\mathstrut +\mathstrut \) \(27223\) \(\beta_{5}\mathstrut -\mathstrut \) \(19447\) \(\beta_{4}\mathstrut -\mathstrut \) \(32565\) \(\beta_{3}\mathstrut +\mathstrut \) \(31400\) \(\beta_{2}\mathstrut +\mathstrut \) \(810362\) \(\beta_{1}\mathstrut -\mathstrut \) \(152360\)
\(\nu^{14}\)\(=\)\(475419\) \(\beta_{14}\mathstrut -\mathstrut \) \(227599\) \(\beta_{13}\mathstrut +\mathstrut \) \(297511\) \(\beta_{12}\mathstrut +\mathstrut \) \(475676\) \(\beta_{11}\mathstrut +\mathstrut \) \(98135\) \(\beta_{10}\mathstrut +\mathstrut \) \(74828\) \(\beta_{9}\mathstrut +\mathstrut \) \(31400\) \(\beta_{8}\mathstrut +\mathstrut \) \(20288\) \(\beta_{7}\mathstrut -\mathstrut \) \(156157\) \(\beta_{6}\mathstrut -\mathstrut \) \(44832\) \(\beta_{5}\mathstrut -\mathstrut \) \(637087\) \(\beta_{4}\mathstrut -\mathstrut \) \(459067\) \(\beta_{3}\mathstrut +\mathstrut \) \(1594630\) \(\beta_{2}\mathstrut -\mathstrut \) \(14591\) \(\beta_{1}\mathstrut +\mathstrut \) \(4100536\)

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
3.34164
3.21278
2.38300
2.04397
1.74947
1.40500
0.619952
0.455829
−0.804723
−1.32542
−1.58213
−1.69218
−2.42593
−3.11300
−3.26824
0 −3.34164 0 1.86211 0 −1.00000 0 8.16655 0
1.2 0 −3.21278 0 0.0641301 0 −1.00000 0 7.32193 0
1.3 0 −2.38300 0 −4.12953 0 −1.00000 0 2.67869 0
1.4 0 −2.04397 0 4.25466 0 −1.00000 0 1.17780 0
1.5 0 −1.74947 0 −0.876486 0 −1.00000 0 0.0606454 0
1.6 0 −1.40500 0 3.56716 0 −1.00000 0 −1.02598 0
1.7 0 −0.619952 0 −1.29231 0 −1.00000 0 −2.61566 0
1.8 0 −0.455829 0 0.836884 0 −1.00000 0 −2.79222 0
1.9 0 0.804723 0 −0.337817 0 −1.00000 0 −2.35242 0
1.10 0 1.32542 0 2.70237 0 −1.00000 0 −1.24326 0
1.11 0 1.58213 0 −3.58630 0 −1.00000 0 −0.496866 0
1.12 0 1.69218 0 −2.62665 0 −1.00000 0 −0.136527 0
1.13 0 2.42593 0 3.43967 0 −1.00000 0 2.88513 0
1.14 0 3.11300 0 −2.63092 0 −1.00000 0 6.69079 0
1.15 0 3.26824 0 2.75302 0 −1.00000 0 7.68141 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(7\) \(1\)
\(11\) \(1\)
\(13\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\(T_{3}^{15} + \cdots\)
\(T_{5}^{15} - \cdots\)
\(T_{17}^{15} - \cdots\)