Properties

Label 8008.2.a.y
Level 8008
Weight 2
Character orbit 8008.a
Self dual Yes
Analytic conductor 63.944
Analytic rank 1
Dimension 14
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: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
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.

\(f(q)\) \(=\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{10} q^{5} \) \(+ q^{7}\) \( + ( 2 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \( -\beta_{1} q^{3} \) \( + \beta_{10} q^{5} \) \(+ q^{7}\) \( + ( 2 + \beta_{2} ) q^{9} \) \(- q^{11}\) \(- q^{13}\) \( + ( -3 + 2 \beta_{1} - \beta_{2} + \beta_{7} - \beta_{9} - \beta_{10} + \beta_{11} - \beta_{12} + \beta_{13} ) q^{15} \) \( -\beta_{8} q^{17} \) \( + ( -1 - \beta_{6} ) q^{19} \) \( -\beta_{1} q^{21} \) \( + ( \beta_{9} + \beta_{12} ) q^{23} \) \( + ( 1 + \beta_{1} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} ) q^{25} \) \( + ( -3 \beta_{1} - \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} - \beta_{13} ) q^{27} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{11} - \beta_{13} ) q^{29} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{8} - \beta_{10} ) q^{31} \) \( + \beta_{1} q^{33} \) \( + \beta_{10} q^{35} \) \( + ( 1 + \beta_{2} - \beta_{7} - \beta_{13} ) q^{37} \) \( + \beta_{1} q^{39} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + \beta_{9} - \beta_{11} ) q^{41} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} + \beta_{9} ) q^{43} \) \( + ( -2 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{5} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - \beta_{11} + \beta_{12} ) q^{45} \) \( + ( -1 + \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{10} + \beta_{12} - \beta_{13} ) q^{47} \) \(+ q^{49}\) \( + ( -4 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{6} + 2 \beta_{8} + \beta_{11} - 2 \beta_{12} + \beta_{13} ) q^{51} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{12} ) q^{53} \) \( -\beta_{10} q^{55} \) \( + ( -3 + 4 \beta_{1} - 2 \beta_{2} + \beta_{4} + 2 \beta_{5} + \beta_{7} + \beta_{8} + \beta_{13} ) q^{57} \) \( + ( -5 - \beta_{3} - \beta_{6} + 2 \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{12} ) q^{59} \) \( + ( -2 \beta_{1} + \beta_{2} - \beta_{5} - \beta_{7} - \beta_{8} ) q^{61} \) \( + ( 2 + \beta_{2} ) q^{63} \) \( -\beta_{10} q^{65} \) \( + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{9} - \beta_{12} + \beta_{13} ) q^{67} \) \( + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{9} - \beta_{10} - \beta_{12} ) q^{69} \) \( + ( -3 + 3 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{12} + 2 \beta_{13} ) q^{71} \) \( + ( 3 - 5 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{12} - \beta_{13} ) q^{73} \) \( + ( 3 - 8 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} - 2 \beta_{7} - 3 \beta_{8} + \beta_{9} - 2 \beta_{11} + 3 \beta_{12} - 3 \beta_{13} ) q^{75} \) \(- q^{77}\) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} - 2 \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{79} \) \( + ( 3 + 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - 2 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{81} \) \( + ( -5 + \beta_{1} - 2 \beta_{2} + \beta_{5} + \beta_{8} + \beta_{12} ) q^{83} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{10} + \beta_{11} + \beta_{12} - \beta_{13} ) q^{85} \) \( + ( 2 - 4 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{7} + \beta_{9} - 2 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{13} ) q^{87} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{89} \) \(- q^{91}\) \( + ( 1 + \beta_{1} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} + 2 \beta_{12} - \beta_{13} ) q^{93} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{10} - \beta_{11} + \beta_{12} ) q^{95} \) \( + ( -2 - \beta_{5} - \beta_{9} + \beta_{11} + \beta_{13} ) q^{97} \) \( + ( -2 - \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 3q^{3} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 14q^{7} \) \(\mathstrut +\mathstrut 21q^{9} \) \(\mathstrut -\mathstrut 14q^{11} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut -\mathstrut 6q^{15} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut -\mathstrut 13q^{19} \) \(\mathstrut -\mathstrut 3q^{21} \) \(\mathstrut -\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 22q^{25} \) \(\mathstrut -\mathstrut 18q^{27} \) \(\mathstrut +\mathstrut 2q^{29} \) \(\mathstrut -\mathstrut 2q^{31} \) \(\mathstrut +\mathstrut 3q^{33} \) \(\mathstrut -\mathstrut 6q^{35} \) \(\mathstrut -\mathstrut q^{37} \) \(\mathstrut +\mathstrut 3q^{39} \) \(\mathstrut -\mathstrut 16q^{41} \) \(\mathstrut -\mathstrut 15q^{43} \) \(\mathstrut -\mathstrut 44q^{45} \) \(\mathstrut -\mathstrut 8q^{47} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut -\mathstrut 14q^{51} \) \(\mathstrut -\mathstrut 6q^{53} \) \(\mathstrut +\mathstrut 6q^{55} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut -\mathstrut 36q^{59} \) \(\mathstrut -\mathstrut 19q^{61} \) \(\mathstrut +\mathstrut 21q^{63} \) \(\mathstrut +\mathstrut 6q^{65} \) \(\mathstrut -\mathstrut 34q^{67} \) \(\mathstrut -\mathstrut q^{69} \) \(\mathstrut -\mathstrut 10q^{71} \) \(\mathstrut +\mathstrut 9q^{73} \) \(\mathstrut -\mathstrut 44q^{75} \) \(\mathstrut -\mathstrut 14q^{77} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut +\mathstrut 42q^{81} \) \(\mathstrut -\mathstrut 56q^{83} \) \(\mathstrut +\mathstrut 21q^{85} \) \(\mathstrut -\mathstrut 5q^{87} \) \(\mathstrut -\mathstrut 14q^{89} \) \(\mathstrut -\mathstrut 14q^{91} \) \(\mathstrut -\mathstrut 20q^{93} \) \(\mathstrut +\mathstrut q^{95} \) \(\mathstrut -\mathstrut 14q^{97} \) \(\mathstrut -\mathstrut 21q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(3\) \(x^{13}\mathstrut -\mathstrut \) \(27\) \(x^{12}\mathstrut +\mathstrut \) \(78\) \(x^{11}\mathstrut +\mathstrut \) \(273\) \(x^{10}\mathstrut -\mathstrut \) \(750\) \(x^{9}\mathstrut -\mathstrut \) \(1306\) \(x^{8}\mathstrut +\mathstrut \) \(3378\) \(x^{7}\mathstrut +\mathstrut \) \(2996\) \(x^{6}\mathstrut -\mathstrut \) \(7275\) \(x^{5}\mathstrut -\mathstrut \) \(2804\) \(x^{4}\mathstrut +\mathstrut \) \(6417\) \(x^{3}\mathstrut +\mathstrut \) \(538\) \(x^{2}\mathstrut -\mathstrut \) \(1032\) \(x\mathstrut -\mathstrut \) \(128\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 5 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(11074521\) \(\nu^{13}\mathstrut +\mathstrut \) \(106614016\) \(\nu^{12}\mathstrut +\mathstrut \) \(202633696\) \(\nu^{11}\mathstrut -\mathstrut \) \(2843591948\) \(\nu^{10}\mathstrut -\mathstrut \) \(1105297503\) \(\nu^{9}\mathstrut +\mathstrut \) \(27555851639\) \(\nu^{8}\mathstrut +\mathstrut \) \(3080027798\) \(\nu^{7}\mathstrut -\mathstrut \) \(120227610741\) \(\nu^{6}\mathstrut -\mathstrut \) \(11970806879\) \(\nu^{5}\mathstrut +\mathstrut \) \(239059287029\) \(\nu^{4}\mathstrut +\mathstrut \) \(81567084\) \(\nu^{3}\mathstrut -\mathstrut \) \(180363225628\) \(\nu^{2}\mathstrut +\mathstrut \) \(99775185884\) \(\nu\mathstrut -\mathstrut \) \(16683138390\)\()/\)\(15572760974\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(58724453\) \(\nu^{13}\mathstrut +\mathstrut \) \(415732687\) \(\nu^{12}\mathstrut +\mathstrut \) \(429387489\) \(\nu^{11}\mathstrut -\mathstrut \) \(9589273014\) \(\nu^{10}\mathstrut +\mathstrut \) \(12008354815\) \(\nu^{9}\mathstrut +\mathstrut \) \(76649830082\) \(\nu^{8}\mathstrut -\mathstrut \) \(161921785412\) \(\nu^{7}\mathstrut -\mathstrut \) \(265081043224\) \(\nu^{6}\mathstrut +\mathstrut \) \(685142582580\) \(\nu^{5}\mathstrut +\mathstrut \) \(431622615285\) \(\nu^{4}\mathstrut -\mathstrut \) \(1032089287458\) \(\nu^{3}\mathstrut -\mathstrut \) \(437413157891\) \(\nu^{2}\mathstrut +\mathstrut \) \(297904178158\) \(\nu\mathstrut +\mathstrut \) \(264264384876\)\()/\)\(31145521948\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(84573421\) \(\nu^{13}\mathstrut +\mathstrut \) \(56007919\) \(\nu^{12}\mathstrut +\mathstrut \) \(1990802275\) \(\nu^{11}\mathstrut +\mathstrut \) \(1134603604\) \(\nu^{10}\mathstrut -\mathstrut \) \(16590727105\) \(\nu^{9}\mathstrut -\mathstrut \) \(46010891430\) \(\nu^{8}\mathstrut +\mathstrut \) \(64820773162\) \(\nu^{7}\mathstrut +\mathstrut \) \(415744840748\) \(\nu^{6}\mathstrut -\mathstrut \) \(158509912178\) \(\nu^{5}\mathstrut -\mathstrut \) \(1451593421773\) \(\nu^{4}\mathstrut +\mathstrut \) \(342029081890\) \(\nu^{3}\mathstrut +\mathstrut \) \(1859229248461\) \(\nu^{2}\mathstrut -\mathstrut \) \(443394945668\) \(\nu\mathstrut -\mathstrut \) \(403244409496\)\()/\)\(31145521948\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(86197394\) \(\nu^{13}\mathstrut -\mathstrut \) \(519312690\) \(\nu^{12}\mathstrut +\mathstrut \) \(4042783649\) \(\nu^{11}\mathstrut +\mathstrut \) \(14135363599\) \(\nu^{10}\mathstrut -\mathstrut \) \(62980268748\) \(\nu^{9}\mathstrut -\mathstrut \) \(143598407974\) \(\nu^{8}\mathstrut +\mathstrut \) \(424769570879\) \(\nu^{7}\mathstrut +\mathstrut \) \(681805991838\) \(\nu^{6}\mathstrut -\mathstrut \) \(1291874241299\) \(\nu^{5}\mathstrut -\mathstrut \) \(1505481346651\) \(\nu^{4}\mathstrut +\mathstrut \) \(1580032475418\) \(\nu^{3}\mathstrut +\mathstrut \) \(1262115684952\) \(\nu^{2}\mathstrut -\mathstrut \) \(467654431587\) \(\nu\mathstrut -\mathstrut \) \(148105342534\)\()/\)\(15572760974\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(100221529\) \(\nu^{13}\mathstrut +\mathstrut \) \(20088978\) \(\nu^{12}\mathstrut +\mathstrut \) \(3433412919\) \(\nu^{11}\mathstrut -\mathstrut \) \(872022317\) \(\nu^{10}\mathstrut -\mathstrut \) \(43636307733\) \(\nu^{9}\mathstrut +\mathstrut \) \(14040705209\) \(\nu^{8}\mathstrut +\mathstrut \) \(252001600285\) \(\nu^{7}\mathstrut -\mathstrut \) \(104822338105\) \(\nu^{6}\mathstrut -\mathstrut \) \(640067051370\) \(\nu^{5}\mathstrut +\mathstrut \) \(362130604330\) \(\nu^{4}\mathstrut +\mathstrut \) \(492483408208\) \(\nu^{3}\mathstrut -\mathstrut \) \(465460641758\) \(\nu^{2}\mathstrut +\mathstrut \) \(207142479617\) \(\nu\mathstrut +\mathstrut \) \(51792914190\)\()/\)\(15572760974\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(204529955\) \(\nu^{13}\mathstrut +\mathstrut \) \(624487815\) \(\nu^{12}\mathstrut +\mathstrut \) \(5370665871\) \(\nu^{11}\mathstrut -\mathstrut \) \(15034911330\) \(\nu^{10}\mathstrut -\mathstrut \) \(53506698243\) \(\nu^{9}\mathstrut +\mathstrut \) \(127831458804\) \(\nu^{8}\mathstrut +\mathstrut \) \(259840610114\) \(\nu^{7}\mathstrut -\mathstrut \) \(471825438700\) \(\nu^{6}\mathstrut -\mathstrut \) \(618802526998\) \(\nu^{5}\mathstrut +\mathstrut \) \(721519658405\) \(\nu^{4}\mathstrut +\mathstrut \) \(562295462232\) \(\nu^{3}\mathstrut -\mathstrut \) \(319648593677\) \(\nu^{2}\mathstrut -\mathstrut \) \(32398485750\) \(\nu\mathstrut -\mathstrut \) \(12295329000\)\()/\)\(31145521948\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(217441493\) \(\nu^{13}\mathstrut +\mathstrut \) \(681733073\) \(\nu^{12}\mathstrut +\mathstrut \) \(6654223007\) \(\nu^{11}\mathstrut -\mathstrut \) \(21109853824\) \(\nu^{10}\mathstrut -\mathstrut \) \(75443571221\) \(\nu^{9}\mathstrut +\mathstrut \) \(245864340184\) \(\nu^{8}\mathstrut +\mathstrut \) \(385128276528\) \(\nu^{7}\mathstrut -\mathstrut \) \(1330673265748\) \(\nu^{6}\mathstrut -\mathstrut \) \(832031771460\) \(\nu^{5}\mathstrut +\mathstrut \) \(3310520475965\) \(\nu^{4}\mathstrut +\mathstrut \) \(436856651496\) \(\nu^{3}\mathstrut -\mathstrut \) \(3075140922419\) \(\nu^{2}\mathstrut +\mathstrut \) \(331065135556\) \(\nu\mathstrut +\mathstrut \) \(306825120252\)\()/\)\(31145521948\)
\(\beta_{10}\)\(=\)\((\)\(208926672\) \(\nu^{13}\mathstrut -\mathstrut \) \(491561333\) \(\nu^{12}\mathstrut -\mathstrut \) \(5735162063\) \(\nu^{11}\mathstrut +\mathstrut \) \(12048495611\) \(\nu^{10}\mathstrut +\mathstrut \) \(58834255396\) \(\nu^{9}\mathstrut -\mathstrut \) \(105792814085\) \(\nu^{8}\mathstrut -\mathstrut \) \(282276492982\) \(\nu^{7}\mathstrut +\mathstrut \) \(415135249248\) \(\nu^{6}\mathstrut +\mathstrut \) \(630297354346\) \(\nu^{5}\mathstrut -\mathstrut \) \(721447644136\) \(\nu^{4}\mathstrut -\mathstrut \) \(522691012619\) \(\nu^{3}\mathstrut +\mathstrut \) \(434273622092\) \(\nu^{2}\mathstrut +\mathstrut \) \(10035841931\) \(\nu\mathstrut -\mathstrut \) \(9943373676\)\()/\)\(15572760974\)
\(\beta_{11}\)\(=\)\((\)\(488425337\) \(\nu^{13}\mathstrut -\mathstrut \) \(1340879233\) \(\nu^{12}\mathstrut -\mathstrut \) \(12295355455\) \(\nu^{11}\mathstrut +\mathstrut \) \(31978291166\) \(\nu^{10}\mathstrut +\mathstrut \) \(110106082525\) \(\nu^{9}\mathstrut -\mathstrut \) \(271168142180\) \(\nu^{8}\mathstrut -\mathstrut \) \(418235692408\) \(\nu^{7}\mathstrut +\mathstrut \) \(1033264260798\) \(\nu^{6}\mathstrut +\mathstrut \) \(540080988466\) \(\nu^{5}\mathstrut -\mathstrut \) \(1837139952593\) \(\nu^{4}\mathstrut +\mathstrut \) \(267466107302\) \(\nu^{3}\mathstrut +\mathstrut \) \(1348441940209\) \(\nu^{2}\mathstrut -\mathstrut \) \(668143324930\) \(\nu\mathstrut -\mathstrut \) \(141640358616\)\()/\)\(31145521948\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(282478821\) \(\nu^{13}\mathstrut +\mathstrut \) \(525449102\) \(\nu^{12}\mathstrut +\mathstrut \) \(7980519403\) \(\nu^{11}\mathstrut -\mathstrut \) \(11978040794\) \(\nu^{10}\mathstrut -\mathstrut \) \(86063793263\) \(\nu^{9}\mathstrut +\mathstrut \) \(93824334855\) \(\nu^{8}\mathstrut +\mathstrut \) \(449176063003\) \(\nu^{7}\mathstrut -\mathstrut \) \(309731889602\) \(\nu^{6}\mathstrut -\mathstrut \) \(1152974583333\) \(\nu^{5}\mathstrut +\mathstrut \) \(424114073096\) \(\nu^{4}\mathstrut +\mathstrut \) \(1239699101363\) \(\nu^{3}\mathstrut -\mathstrut \) \(211254743701\) \(\nu^{2}\mathstrut -\mathstrut \) \(245962834998\) \(\nu\mathstrut +\mathstrut \) \(816058514\)\()/\)\(15572760974\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(461482718\) \(\nu^{13}\mathstrut +\mathstrut \) \(1119246863\) \(\nu^{12}\mathstrut +\mathstrut \) \(12534518457\) \(\nu^{11}\mathstrut -\mathstrut \) \(27398869301\) \(\nu^{10}\mathstrut -\mathstrut \) \(127270246922\) \(\nu^{9}\mathstrut +\mathstrut \) \(241925316093\) \(\nu^{8}\mathstrut +\mathstrut \) \(607199002972\) \(\nu^{7}\mathstrut -\mathstrut \) \(976096110556\) \(\nu^{6}\mathstrut -\mathstrut \) \(1367160452244\) \(\nu^{5}\mathstrut +\mathstrut \) \(1851226663240\) \(\nu^{4}\mathstrut +\mathstrut \) \(1215628784765\) \(\nu^{3}\mathstrut -\mathstrut \) \(1405372442586\) \(\nu^{2}\mathstrut -\mathstrut \) \(180279716217\) \(\nu\mathstrut +\mathstrut \) \(105424374118\)\()/\)\(15572760974\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(39\)
\(\nu^{5}\)\(=\)\(14\) \(\beta_{13}\mathstrut -\mathstrut \) \(15\) \(\beta_{12}\mathstrut +\mathstrut \) \(16\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut +\mathstrut \) \(17\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(90\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(-\)\(15\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(18\) \(\beta_{11}\mathstrut -\mathstrut \) \(38\) \(\beta_{10}\mathstrut -\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(15\) \(\beta_{5}\mathstrut +\mathstrut \) \(29\) \(\beta_{4}\mathstrut -\mathstrut \) \(19\) \(\beta_{3}\mathstrut +\mathstrut \) \(131\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(353\)
\(\nu^{7}\)\(=\)\(167\) \(\beta_{13}\mathstrut -\mathstrut \) \(186\) \(\beta_{12}\mathstrut +\mathstrut \) \(205\) \(\beta_{11}\mathstrut +\mathstrut \) \(77\) \(\beta_{10}\mathstrut -\mathstrut \) \(43\) \(\beta_{9}\mathstrut +\mathstrut \) \(139\) \(\beta_{8}\mathstrut +\mathstrut \) \(215\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\) \(\beta_{6}\mathstrut +\mathstrut \) \(156\) \(\beta_{5}\mathstrut +\mathstrut \) \(181\) \(\beta_{4}\mathstrut +\mathstrut \) \(16\) \(\beta_{3}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(932\) \(\beta_{1}\mathstrut -\mathstrut \) \(16\)
\(\nu^{8}\)\(=\)\(-\)\(160\) \(\beta_{13}\mathstrut +\mathstrut \) \(116\) \(\beta_{12}\mathstrut +\mathstrut \) \(257\) \(\beta_{11}\mathstrut -\mathstrut \) \(531\) \(\beta_{10}\mathstrut -\mathstrut \) \(80\) \(\beta_{9}\mathstrut -\mathstrut \) \(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(76\) \(\beta_{7}\mathstrut -\mathstrut \) \(42\) \(\beta_{6}\mathstrut -\mathstrut \) \(187\) \(\beta_{5}\mathstrut +\mathstrut \) \(345\) \(\beta_{4}\mathstrut -\mathstrut \) \(261\) \(\beta_{3}\mathstrut +\mathstrut \) \(1403\) \(\beta_{2}\mathstrut +\mathstrut \) \(106\) \(\beta_{1}\mathstrut +\mathstrut \) \(3423\)
\(\nu^{9}\)\(=\)\(1906\) \(\beta_{13}\mathstrut -\mathstrut \) \(2198\) \(\beta_{12}\mathstrut +\mathstrut \) \(2435\) \(\beta_{11}\mathstrut +\mathstrut \) \(482\) \(\beta_{10}\mathstrut -\mathstrut \) \(667\) \(\beta_{9}\mathstrut +\mathstrut \) \(1449\) \(\beta_{8}\mathstrut +\mathstrut \) \(2498\) \(\beta_{7}\mathstrut -\mathstrut \) \(238\) \(\beta_{6}\mathstrut +\mathstrut \) \(1627\) \(\beta_{5}\mathstrut +\mathstrut \) \(2054\) \(\beta_{4}\mathstrut +\mathstrut \) \(185\) \(\beta_{3}\mathstrut +\mathstrut \) \(160\) \(\beta_{2}\mathstrut +\mathstrut \) \(9828\) \(\beta_{1}\mathstrut -\mathstrut \) \(200\)
\(\nu^{10}\)\(=\)\(-\)\(1452\) \(\beta_{13}\mathstrut +\mathstrut \) \(725\) \(\beta_{12}\mathstrut +\mathstrut \) \(3377\) \(\beta_{11}\mathstrut -\mathstrut \) \(6685\) \(\beta_{10}\mathstrut -\mathstrut \) \(1435\) \(\beta_{9}\mathstrut -\mathstrut \) \(153\) \(\beta_{8}\mathstrut +\mathstrut \) \(1309\) \(\beta_{7}\mathstrut -\mathstrut \) \(611\) \(\beta_{6}\mathstrut -\mathstrut \) \(2214\) \(\beta_{5}\mathstrut +\mathstrut \) \(3921\) \(\beta_{4}\mathstrut -\mathstrut \) \(3166\) \(\beta_{3}\mathstrut +\mathstrut \) \(14968\) \(\beta_{2}\mathstrut +\mathstrut \) \(1990\) \(\beta_{1}\mathstrut +\mathstrut \) \(34307\)
\(\nu^{11}\)\(=\)\(21440\) \(\beta_{13}\mathstrut -\mathstrut \) \(25547\) \(\beta_{12}\mathstrut +\mathstrut \) \(28043\) \(\beta_{11}\mathstrut +\mathstrut \) \(1700\) \(\beta_{10}\mathstrut -\mathstrut \) \(9180\) \(\beta_{9}\mathstrut +\mathstrut \) \(15259\) \(\beta_{8}\mathstrut +\mathstrut \) \(28129\) \(\beta_{7}\mathstrut -\mathstrut \) \(2822\) \(\beta_{6}\mathstrut +\mathstrut \) \(16505\) \(\beta_{5}\mathstrut +\mathstrut \) \(22781\) \(\beta_{4}\mathstrut +\mathstrut \) \(1878\) \(\beta_{3}\mathstrut +\mathstrut \) \(2896\) \(\beta_{2}\mathstrut +\mathstrut \) \(104779\) \(\beta_{1}\mathstrut -\mathstrut \) \(2231\)
\(\nu^{12}\)\(=\)\(-\)\(11353\) \(\beta_{13}\mathstrut +\mathstrut \) \(597\) \(\beta_{12}\mathstrut +\mathstrut \) \(42620\) \(\beta_{11}\mathstrut -\mathstrut \) \(80507\) \(\beta_{10}\mathstrut -\mathstrut \) \(21831\) \(\beta_{9}\mathstrut -\mathstrut \) \(402\) \(\beta_{8}\mathstrut +\mathstrut \) \(19277\) \(\beta_{7}\mathstrut -\mathstrut \) \(7711\) \(\beta_{6}\mathstrut -\mathstrut \) \(25713\) \(\beta_{5}\mathstrut +\mathstrut \) \(44080\) \(\beta_{4}\mathstrut -\mathstrut \) \(36193\) \(\beta_{3}\mathstrut +\mathstrut \) \(159733\) \(\beta_{2}\mathstrut +\mathstrut \) \(32051\) \(\beta_{1}\mathstrut +\mathstrut \) \(349673\)
\(\nu^{13}\)\(=\)\(240060\) \(\beta_{13}\mathstrut -\mathstrut \) \(294913\) \(\beta_{12}\mathstrut +\mathstrut \) \(318651\) \(\beta_{11}\mathstrut -\mathstrut \) \(16807\) \(\beta_{10}\mathstrut -\mathstrut \) \(119388\) \(\beta_{9}\mathstrut +\mathstrut \) \(163166\) \(\beta_{8}\mathstrut +\mathstrut \) \(312802\) \(\beta_{7}\mathstrut -\mathstrut \) \(31947\) \(\beta_{6}\mathstrut +\mathstrut \) \(164782\) \(\beta_{5}\mathstrut +\mathstrut \) \(250307\) \(\beta_{4}\mathstrut +\mathstrut \) \(17619\) \(\beta_{3}\mathstrut +\mathstrut \) \(44674\) \(\beta_{2}\mathstrut +\mathstrut \) \(1125192\) \(\beta_{1}\mathstrut -\mathstrut \) \(23048\)

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
3.38813
3.22236
2.52493
1.97367
1.53687
1.36889
0.528665
−0.129238
−0.374242
−1.44419
−1.85932
−2.05490
−2.46105
−3.22058
0 −3.38813 0 −4.16385 0 1.00000 0 8.47945 0
1.2 0 −3.22236 0 1.94503 0 1.00000 0 7.38359 0
1.3 0 −2.52493 0 3.43935 0 1.00000 0 3.37528 0
1.4 0 −1.97367 0 −1.63177 0 1.00000 0 0.895375 0
1.5 0 −1.53687 0 −0.216819 0 1.00000 0 −0.638026 0
1.6 0 −1.36889 0 −3.92349 0 1.00000 0 −1.12615 0
1.7 0 −0.528665 0 0.933973 0 1.00000 0 −2.72051 0
1.8 0 0.129238 0 −0.197815 0 1.00000 0 −2.98330 0
1.9 0 0.374242 0 3.66821 0 1.00000 0 −2.85994 0
1.10 0 1.44419 0 −3.20437 0 1.00000 0 −0.914318 0
1.11 0 1.85932 0 1.13919 0 1.00000 0 0.457073 0
1.12 0 2.05490 0 0.929320 0 1.00000 0 1.22261 0
1.13 0 2.46105 0 −1.12857 0 1.00000 0 3.05676 0
1.14 0 3.22058 0 −3.58838 0 1.00000 0 7.37212 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
\(2\) \(-1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

Hecke kernels

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

\(T_{3}^{14} + \cdots\)
\(T_{5}^{14} + \cdots\)
\(T_{17}^{14} + \cdots\)