Properties

Label 8034.2.a.ba
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 0
Dimension 14
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8034 = 2 \cdot 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8034.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1518129839\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \(- q^{3}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \(+ q^{6}\) \( -\beta_{7} q^{7} \) \(- q^{8}\) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(- q^{2}\) \(- q^{3}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \(+ q^{6}\) \( -\beta_{7} q^{7} \) \(- q^{8}\) \(+ q^{9}\) \( + \beta_{1} q^{10} \) \( + \beta_{11} q^{11} \) \(- q^{12}\) \(- q^{13}\) \( + \beta_{7} q^{14} \) \( + \beta_{1} q^{15} \) \(+ q^{16}\) \( + ( -\beta_{1} + \beta_{2} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{17} \) \(- q^{18}\) \( + ( -\beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{12} ) q^{19} \) \( -\beta_{1} q^{20} \) \( + \beta_{7} q^{21} \) \( -\beta_{11} q^{22} \) \( + ( 1 + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{12} - \beta_{13} ) q^{23} \) \(+ q^{24}\) \( + ( 2 + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{11} - \beta_{13} ) q^{25} \) \(+ q^{26}\) \(- q^{27}\) \( -\beta_{7} q^{28} \) \( + ( -\beta_{1} + \beta_{5} + \beta_{13} ) q^{29} \) \( -\beta_{1} q^{30} \) \( + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{9} + \beta_{10} ) q^{31} \) \(- q^{32}\) \( -\beta_{11} q^{33} \) \( + ( \beta_{1} - \beta_{2} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{34} \) \( + ( -2 - \beta_{1} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{35} \) \(+ q^{36}\) \( + ( -1 - \beta_{3} + \beta_{8} - \beta_{10} ) q^{37} \) \( + ( \beta_{1} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{12} ) q^{38} \) \(+ q^{39}\) \( + \beta_{1} q^{40} \) \( + ( 1 + \beta_{3} + \beta_{4} + \beta_{11} - \beta_{13} ) q^{41} \) \( -\beta_{7} q^{42} \) \( + ( \beta_{2} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{10} ) q^{43} \) \( + \beta_{11} q^{44} \) \( -\beta_{1} q^{45} \) \( + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{46} \) \( + ( -\beta_{2} - \beta_{5} - \beta_{9} + \beta_{10} - \beta_{13} ) q^{47} \) \(- q^{48}\) \( + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{8} + \beta_{10} ) q^{49} \) \( + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} ) q^{50} \) \( + ( \beta_{1} - \beta_{2} + \beta_{9} - \beta_{10} + \beta_{11} ) q^{51} \) \(- q^{52}\) \( + ( -3 - \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{9} + \beta_{11} ) q^{53} \) \(+ q^{54}\) \( + ( 1 - 2 \beta_{1} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{12} ) q^{55} \) \( + \beta_{7} q^{56} \) \( + ( \beta_{1} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{12} ) q^{57} \) \( + ( \beta_{1} - \beta_{5} - \beta_{13} ) q^{58} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{3} + \beta_{7} + \beta_{12} ) q^{59} \) \( + \beta_{1} q^{60} \) \( + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - \beta_{6} - \beta_{10} + \beta_{12} ) q^{61} \) \( + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{9} - \beta_{10} ) q^{62} \) \( -\beta_{7} q^{63} \) \(+ q^{64}\) \( + \beta_{1} q^{65} \) \( + \beta_{11} q^{66} \) \( + ( 2 \beta_{1} - \beta_{2} - \beta_{4} + \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{12} ) q^{67} \) \( + ( -\beta_{1} + \beta_{2} - \beta_{9} + \beta_{10} - \beta_{11} ) q^{68} \) \( + ( -1 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{12} + \beta_{13} ) q^{69} \) \( + ( 2 + \beta_{1} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{13} ) q^{70} \) \( + ( -\beta_{3} - \beta_{4} - \beta_{10} - \beta_{11} - \beta_{12} + \beta_{13} ) q^{71} \) \(- q^{72}\) \( + ( -\beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} + \beta_{13} ) q^{73} \) \( + ( 1 + \beta_{3} - \beta_{8} + \beta_{10} ) q^{74} \) \( + ( -2 - \beta_{3} - \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{11} + \beta_{13} ) q^{75} \) \( + ( -\beta_{1} + \beta_{5} - \beta_{7} + \beta_{8} + \beta_{12} ) q^{76} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{8} - 2 \beta_{10} + 2 \beta_{11} ) q^{77} \) \(- q^{78}\) \( + ( 2 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{11} - \beta_{12} ) q^{79} \) \( -\beta_{1} q^{80} \) \(+ q^{81}\) \( + ( -1 - \beta_{3} - \beta_{4} - \beta_{11} + \beta_{13} ) q^{82} \) \( + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{8} ) q^{83} \) \( + \beta_{7} q^{84} \) \( + ( 2 \beta_{1} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{7} - \beta_{8} + \beta_{9} + \beta_{10} - \beta_{12} ) q^{85} \) \( + ( -\beta_{2} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{10} ) q^{86} \) \( + ( \beta_{1} - \beta_{5} - \beta_{13} ) q^{87} \) \( -\beta_{11} q^{88} \) \( + ( -2 \beta_{1} - \beta_{4} - \beta_{6} - 2 \beta_{7} - \beta_{9} + 2 \beta_{10} + \beta_{12} ) q^{89} \) \( + \beta_{1} q^{90} \) \( + \beta_{7} q^{91} \) \( + ( 1 + \beta_{3} + \beta_{4} + \beta_{6} - \beta_{12} - \beta_{13} ) q^{92} \) \( + ( -3 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{9} - \beta_{10} ) q^{93} \) \( + ( \beta_{2} + \beta_{5} + \beta_{9} - \beta_{10} + \beta_{13} ) q^{94} \) \( + ( 4 - 4 \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{6} + \beta_{11} ) q^{95} \) \(+ q^{96}\) \( + ( 2 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} - \beta_{9} + \beta_{11} ) q^{97} \) \( + ( -2 - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{8} - \beta_{10} ) q^{98} \) \( + \beta_{11} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(14q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 14q^{6} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut 14q^{8} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(14q \) \(\mathstrut -\mathstrut 14q^{2} \) \(\mathstrut -\mathstrut 14q^{3} \) \(\mathstrut +\mathstrut 14q^{4} \) \(\mathstrut -\mathstrut q^{5} \) \(\mathstrut +\mathstrut 14q^{6} \) \(\mathstrut +\mathstrut 5q^{7} \) \(\mathstrut -\mathstrut 14q^{8} \) \(\mathstrut +\mathstrut 14q^{9} \) \(\mathstrut +\mathstrut q^{10} \) \(\mathstrut -\mathstrut q^{11} \) \(\mathstrut -\mathstrut 14q^{12} \) \(\mathstrut -\mathstrut 14q^{13} \) \(\mathstrut -\mathstrut 5q^{14} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 14q^{16} \) \(\mathstrut -\mathstrut 12q^{17} \) \(\mathstrut -\mathstrut 14q^{18} \) \(\mathstrut +\mathstrut 10q^{19} \) \(\mathstrut -\mathstrut q^{20} \) \(\mathstrut -\mathstrut 5q^{21} \) \(\mathstrut +\mathstrut q^{22} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut 14q^{24} \) \(\mathstrut +\mathstrut 19q^{25} \) \(\mathstrut +\mathstrut 14q^{26} \) \(\mathstrut -\mathstrut 14q^{27} \) \(\mathstrut +\mathstrut 5q^{28} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut q^{30} \) \(\mathstrut +\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 14q^{32} \) \(\mathstrut +\mathstrut q^{33} \) \(\mathstrut +\mathstrut 12q^{34} \) \(\mathstrut -\mathstrut 16q^{35} \) \(\mathstrut +\mathstrut 14q^{36} \) \(\mathstrut -\mathstrut 3q^{37} \) \(\mathstrut -\mathstrut 10q^{38} \) \(\mathstrut +\mathstrut 14q^{39} \) \(\mathstrut +\mathstrut q^{40} \) \(\mathstrut +\mathstrut q^{41} \) \(\mathstrut +\mathstrut 5q^{42} \) \(\mathstrut +\mathstrut 6q^{43} \) \(\mathstrut -\mathstrut q^{44} \) \(\mathstrut -\mathstrut q^{45} \) \(\mathstrut +\mathstrut q^{46} \) \(\mathstrut -\mathstrut 13q^{47} \) \(\mathstrut -\mathstrut 14q^{48} \) \(\mathstrut +\mathstrut 9q^{49} \) \(\mathstrut -\mathstrut 19q^{50} \) \(\mathstrut +\mathstrut 12q^{51} \) \(\mathstrut -\mathstrut 14q^{52} \) \(\mathstrut -\mathstrut 27q^{53} \) \(\mathstrut +\mathstrut 14q^{54} \) \(\mathstrut +\mathstrut 10q^{55} \) \(\mathstrut -\mathstrut 5q^{56} \) \(\mathstrut -\mathstrut 10q^{57} \) \(\mathstrut -\mathstrut 6q^{58} \) \(\mathstrut -\mathstrut 6q^{59} \) \(\mathstrut +\mathstrut q^{60} \) \(\mathstrut -\mathstrut 4q^{61} \) \(\mathstrut -\mathstrut 20q^{62} \) \(\mathstrut +\mathstrut 5q^{63} \) \(\mathstrut +\mathstrut 14q^{64} \) \(\mathstrut +\mathstrut q^{65} \) \(\mathstrut -\mathstrut q^{66} \) \(\mathstrut +\mathstrut 13q^{67} \) \(\mathstrut -\mathstrut 12q^{68} \) \(\mathstrut +\mathstrut q^{69} \) \(\mathstrut +\mathstrut 16q^{70} \) \(\mathstrut +\mathstrut 18q^{71} \) \(\mathstrut -\mathstrut 14q^{72} \) \(\mathstrut +\mathstrut 11q^{73} \) \(\mathstrut +\mathstrut 3q^{74} \) \(\mathstrut -\mathstrut 19q^{75} \) \(\mathstrut +\mathstrut 10q^{76} \) \(\mathstrut -\mathstrut 15q^{77} \) \(\mathstrut -\mathstrut 14q^{78} \) \(\mathstrut +\mathstrut 33q^{79} \) \(\mathstrut -\mathstrut q^{80} \) \(\mathstrut +\mathstrut 14q^{81} \) \(\mathstrut -\mathstrut q^{82} \) \(\mathstrut -\mathstrut 25q^{83} \) \(\mathstrut -\mathstrut 5q^{84} \) \(\mathstrut +\mathstrut 25q^{85} \) \(\mathstrut -\mathstrut 6q^{86} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut +\mathstrut q^{88} \) \(\mathstrut +\mathstrut 3q^{89} \) \(\mathstrut +\mathstrut q^{90} \) \(\mathstrut -\mathstrut 5q^{91} \) \(\mathstrut -\mathstrut q^{92} \) \(\mathstrut -\mathstrut 20q^{93} \) \(\mathstrut +\mathstrut 13q^{94} \) \(\mathstrut +\mathstrut 30q^{95} \) \(\mathstrut +\mathstrut 14q^{96} \) \(\mathstrut +\mathstrut 11q^{97} \) \(\mathstrut -\mathstrut 9q^{98} \) \(\mathstrut -\mathstrut q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{14}\mathstrut -\mathstrut \) \(x^{13}\mathstrut -\mathstrut \) \(44\) \(x^{12}\mathstrut +\mathstrut \) \(36\) \(x^{11}\mathstrut +\mathstrut \) \(722\) \(x^{10}\mathstrut -\mathstrut \) \(451\) \(x^{9}\mathstrut -\mathstrut \) \(5438\) \(x^{8}\mathstrut +\mathstrut \) \(2268\) \(x^{7}\mathstrut +\mathstrut \) \(18441\) \(x^{6}\mathstrut -\mathstrut \) \(4126\) \(x^{5}\mathstrut -\mathstrut \) \(22759\) \(x^{4}\mathstrut +\mathstrut \) \(2997\) \(x^{3}\mathstrut +\mathstrut \) \(10504\) \(x^{2}\mathstrut -\mathstrut \) \(506\) \(x\mathstrut -\mathstrut \) \(1492\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(266306853678383\) \(\nu^{13}\mathstrut +\mathstrut \) \(134001969718133\) \(\nu^{12}\mathstrut +\mathstrut \) \(11730896555981242\) \(\nu^{11}\mathstrut -\mathstrut \) \(3739404291058554\) \(\nu^{10}\mathstrut -\mathstrut \) \(191585597830484336\) \(\nu^{9}\mathstrut +\mathstrut \) \(24886922325561149\) \(\nu^{8}\mathstrut +\mathstrut \) \(1414782771710524552\) \(\nu^{7}\mathstrut +\mathstrut \) \(84321828569428448\) \(\nu^{6}\mathstrut -\mathstrut \) \(4492827496115707927\) \(\nu^{5}\mathstrut -\mathstrut \) \(915363310465441330\) \(\nu^{4}\mathstrut +\mathstrut \) \(4255301279762091841\) \(\nu^{3}\mathstrut +\mathstrut \) \(422048253089054251\) \(\nu^{2}\mathstrut -\mathstrut \) \(1047631892432145166\) \(\nu\mathstrut +\mathstrut \) \(11371595281753640\)\()/\)\(43692742328669662\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(276561632426969\) \(\nu^{13}\mathstrut +\mathstrut \) \(421045515071387\) \(\nu^{12}\mathstrut +\mathstrut \) \(11831212064366610\) \(\nu^{11}\mathstrut -\mathstrut \) \(15979394388187493\) \(\nu^{10}\mathstrut -\mathstrut \) \(186676696579320143\) \(\nu^{9}\mathstrut +\mathstrut \) \(217017383487373922\) \(\nu^{8}\mathstrut +\mathstrut \) \(1324468395061602794\) \(\nu^{7}\mathstrut -\mathstrut \) \(1262761014021014640\) \(\nu^{6}\mathstrut -\mathstrut \) \(4033414366634615670\) \(\nu^{5}\mathstrut +\mathstrut \) \(3051387240952734794\) \(\nu^{4}\mathstrut +\mathstrut \) \(3669499430064327162\) \(\nu^{3}\mathstrut -\mathstrut \) \(2701606611286975207\) \(\nu^{2}\mathstrut -\mathstrut \) \(780973362285615831\) \(\nu\mathstrut +\mathstrut \) \(588284015133280532\)\()/\)\(21846371164334831\)
\(\beta_{4}\)\(=\)\((\)\(281008689100883\) \(\nu^{13}\mathstrut -\mathstrut \) \(281957981628135\) \(\nu^{12}\mathstrut -\mathstrut \) \(12211701038963858\) \(\nu^{11}\mathstrut +\mathstrut \) \(10147274836518033\) \(\nu^{10}\mathstrut +\mathstrut \) \(196191030805468054\) \(\nu^{9}\mathstrut -\mathstrut \) \(127532267940051610\) \(\nu^{8}\mathstrut -\mathstrut \) \(1419834757376023830\) \(\nu^{7}\mathstrut +\mathstrut \) \(652528796048380961\) \(\nu^{6}\mathstrut +\mathstrut \) \(4397320407290358181\) \(\nu^{5}\mathstrut -\mathstrut \) \(1303030165431025011\) \(\nu^{4}\mathstrut -\mathstrut \) \(3995402229552693528\) \(\nu^{3}\mathstrut +\mathstrut \) \(1346052708231791049\) \(\nu^{2}\mathstrut +\mathstrut \) \(908907497419418757\) \(\nu\mathstrut -\mathstrut \) \(376805384643590757\)\()/\)\(21846371164334831\)
\(\beta_{5}\)\(=\)\((\)\(563276920787713\) \(\nu^{13}\mathstrut -\mathstrut \) \(153049032331183\) \(\nu^{12}\mathstrut -\mathstrut \) \(24789913096510800\) \(\nu^{11}\mathstrut +\mathstrut \) \(2179601153394382\) \(\nu^{10}\mathstrut +\mathstrut \) \(403199471683301528\) \(\nu^{9}\mathstrut +\mathstrut \) \(41321895819192391\) \(\nu^{8}\mathstrut -\mathstrut \) \(2942272434616681480\) \(\nu^{7}\mathstrut -\mathstrut \) \(888965469465066868\) \(\nu^{6}\mathstrut +\mathstrut \) \(9011134325268674427\) \(\nu^{5}\mathstrut +\mathstrut \) \(4368493534082155588\) \(\nu^{4}\mathstrut -\mathstrut \) \(7216360830653724145\) \(\nu^{3}\mathstrut -\mathstrut \) \(3783713813055134269\) \(\nu^{2}\mathstrut +\mathstrut \) \(1120836385553257082\) \(\nu\mathstrut +\mathstrut \) \(727260622566264690\)\()/\)\(43692742328669662\)
\(\beta_{6}\)\(=\)\((\)\(587894636532799\) \(\nu^{13}\mathstrut -\mathstrut \) \(401515966880495\) \(\nu^{12}\mathstrut -\mathstrut \) \(25559574153488160\) \(\nu^{11}\mathstrut +\mathstrut \) \(12469830127322176\) \(\nu^{10}\mathstrut +\mathstrut \) \(410367492349937474\) \(\nu^{9}\mathstrut -\mathstrut \) \(114850747220457015\) \(\nu^{8}\mathstrut -\mathstrut \) \(2960764032107237352\) \(\nu^{7}\mathstrut +\mathstrut \) \(169914683015283554\) \(\nu^{6}\mathstrut +\mathstrut \) \(9072106839066945823\) \(\nu^{5}\mathstrut +\mathstrut \) \(1281274510944306808\) \(\nu^{4}\mathstrut -\mathstrut \) \(7818651825966159249\) \(\nu^{3}\mathstrut -\mathstrut \) \(1046735607854569041\) \(\nu^{2}\mathstrut +\mathstrut \) \(1635447715064563388\) \(\nu\mathstrut +\mathstrut \) \(69082076903089780\)\()/\)\(43692742328669662\)
\(\beta_{7}\)\(=\)\((\)\(600547475775077\) \(\nu^{13}\mathstrut +\mathstrut \) \(160676916331065\) \(\nu^{12}\mathstrut -\mathstrut \) \(26573482465291666\) \(\nu^{11}\mathstrut -\mathstrut \) \(12031687706537128\) \(\nu^{10}\mathstrut +\mathstrut \) \(434206046573977704\) \(\nu^{9}\mathstrut +\mathstrut \) \(279533615334483141\) \(\nu^{8}\mathstrut -\mathstrut \) \(3175735116060119102\) \(\nu^{7}\mathstrut -\mathstrut \) \(2683158768977673228\) \(\nu^{6}\mathstrut +\mathstrut \) \(9656661921220973037\) \(\nu^{5}\mathstrut +\mathstrut \) \(10041227588618067172\) \(\nu^{4}\mathstrut -\mathstrut \) \(7220568332423014109\) \(\nu^{3}\mathstrut -\mathstrut \) \(8543202845942904737\) \(\nu^{2}\mathstrut +\mathstrut \) \(983162040010572610\) \(\nu\mathstrut +\mathstrut \) \(1584010040979027426\)\()/\)\(43692742328669662\)
\(\beta_{8}\)\(=\)\((\)\(323027709062960\) \(\nu^{13}\mathstrut -\mathstrut \) \(172381882120881\) \(\nu^{12}\mathstrut -\mathstrut \) \(14134878235621520\) \(\nu^{11}\mathstrut +\mathstrut \) \(4846129448665246\) \(\nu^{10}\mathstrut +\mathstrut \) \(228696579415282775\) \(\nu^{9}\mathstrut -\mathstrut \) \(32681850003000513\) \(\nu^{8}\mathstrut -\mathstrut \) \(1664502571125875459\) \(\nu^{7}\mathstrut -\mathstrut \) \(111735197101452200\) \(\nu^{6}\mathstrut +\mathstrut \) \(5139130808556126422\) \(\nu^{5}\mathstrut +\mathstrut \) \(1294647822855345817\) \(\nu^{4}\mathstrut -\mathstrut \) \(4402465041131927787\) \(\nu^{3}\mathstrut -\mathstrut \) \(1059281835457961059\) \(\nu^{2}\mathstrut +\mathstrut \) \(828635442318437489\) \(\nu\mathstrut +\mathstrut \) \(87259893240249591\)\()/\)\(21846371164334831\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(682482407227411\) \(\nu^{13}\mathstrut -\mathstrut \) \(152764996823\) \(\nu^{12}\mathstrut +\mathstrut \) \(30122063309684406\) \(\nu^{11}\mathstrut +\mathstrut \) \(5643504601298036\) \(\nu^{10}\mathstrut -\mathstrut \) \(491662813166917868\) \(\nu^{9}\mathstrut -\mathstrut \) \(187136585825562229\) \(\nu^{8}\mathstrut +\mathstrut \) \(3607458160860328818\) \(\nu^{7}\mathstrut +\mathstrut \) \(2101336739470417196\) \(\nu^{6}\mathstrut -\mathstrut \) \(11175426298123444609\) \(\nu^{5}\mathstrut -\mathstrut \) \(8583170474917069942\) \(\nu^{4}\mathstrut +\mathstrut \) \(9392486941861559115\) \(\nu^{3}\mathstrut +\mathstrut \) \(7853101744503182057\) \(\nu^{2}\mathstrut -\mathstrut \) \(1800196355230650124\) \(\nu\mathstrut -\mathstrut \) \(1698473816153287560\)\()/\)\(43692742328669662\)
\(\beta_{10}\)\(=\)\((\)\(352781396910762\) \(\nu^{13}\mathstrut -\mathstrut \) \(166497424213521\) \(\nu^{12}\mathstrut -\mathstrut \) \(15413816930355473\) \(\nu^{11}\mathstrut +\mathstrut \) \(4333005041742547\) \(\nu^{10}\mathstrut +\mathstrut \) \(248673668063896890\) \(\nu^{9}\mathstrut -\mathstrut \) \(20208645124353222\) \(\nu^{8}\mathstrut -\mathstrut \) \(1799297386675220078\) \(\nu^{7}\mathstrut -\mathstrut \) \(231617478407447300\) \(\nu^{6}\mathstrut +\mathstrut \) \(5476590592887655990\) \(\nu^{5}\mathstrut +\mathstrut \) \(1713977797830856848\) \(\nu^{4}\mathstrut -\mathstrut \) \(4440659955237449610\) \(\nu^{3}\mathstrut -\mathstrut \) \(1259504021300032236\) \(\nu^{2}\mathstrut +\mathstrut \) \(849290650220779131\) \(\nu\mathstrut +\mathstrut \) \(91607361939384532\)\()/\)\(21846371164334831\)
\(\beta_{11}\)\(=\)\((\)\(756784517059657\) \(\nu^{13}\mathstrut -\mathstrut \) \(144783307200521\) \(\nu^{12}\mathstrut -\mathstrut \) \(33253610912031224\) \(\nu^{11}\mathstrut -\mathstrut \) \(62661878222990\) \(\nu^{10}\mathstrut +\mathstrut \) \(540215049753776070\) \(\nu^{9}\mathstrut +\mathstrut \) \(110071620789847963\) \(\nu^{8}\mathstrut -\mathstrut \) \(3944695232805112230\) \(\nu^{7}\mathstrut -\mathstrut \) \(1643994001829427568\) \(\nu^{6}\mathstrut +\mathstrut \) \(12173804052853690563\) \(\nu^{5}\mathstrut +\mathstrut \) \(7459946733255625440\) \(\nu^{4}\mathstrut -\mathstrut \) \(10254865555632034835\) \(\nu^{3}\mathstrut -\mathstrut \) \(6870285231682939387\) \(\nu^{2}\mathstrut +\mathstrut \) \(2137547859976077336\) \(\nu\mathstrut +\mathstrut \) \(1424281961283445274\)\()/\)\(43692742328669662\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(461379618788969\) \(\nu^{13}\mathstrut +\mathstrut \) \(101629522210970\) \(\nu^{12}\mathstrut +\mathstrut \) \(20312989919152389\) \(\nu^{11}\mathstrut -\mathstrut \) \(683299742724623\) \(\nu^{10}\mathstrut -\mathstrut \) \(330994820970097058\) \(\nu^{9}\mathstrut -\mathstrut \) \(52645645511792452\) \(\nu^{8}\mathstrut +\mathstrut \) \(2429698868506903576\) \(\nu^{7}\mathstrut +\mathstrut \) \(870810312435771415\) \(\nu^{6}\mathstrut -\mathstrut \) \(7583560590369499978\) \(\nu^{5}\mathstrut -\mathstrut \) \(4027119627133068275\) \(\nu^{4}\mathstrut +\mathstrut \) \(6656963138830352527\) \(\nu^{3}\mathstrut +\mathstrut \) \(3488559562581261272\) \(\nu^{2}\mathstrut -\mathstrut \) \(1371444613064346176\) \(\nu\mathstrut -\mathstrut \) \(641231492639617501\)\()/\)\(21846371164334831\)
\(\beta_{13}\)\(=\)\((\)\(1399081209291127\) \(\nu^{13}\mathstrut -\mathstrut \) \(773564887767339\) \(\nu^{12}\mathstrut -\mathstrut \) \(61270437020665254\) \(\nu^{11}\mathstrut +\mathstrut \) \(22466875749627886\) \(\nu^{10}\mathstrut +\mathstrut \) \(992798322812597212\) \(\nu^{9}\mathstrut -\mathstrut \) \(170706210676448595\) \(\nu^{8}\mathstrut -\mathstrut \) \(7249462015732823470\) \(\nu^{7}\mathstrut -\mathstrut \) \(234855379984642544\) \(\nu^{6}\mathstrut +\mathstrut \) \(22595322669123401215\) \(\nu^{5}\mathstrut +\mathstrut \) \(4786003452335976276\) \(\nu^{4}\mathstrut -\mathstrut \) \(20309684730415768281\) \(\nu^{3}\mathstrut -\mathstrut \) \(4247182212949563787\) \(\nu^{2}\mathstrut +\mathstrut \) \(4702972689934548944\) \(\nu\mathstrut +\mathstrut \) \(812680240968073994\)\()/\)\(43692742328669662\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(16\) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(18\) \(\beta_{11}\mathstrut -\mathstrut \) \(3\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(13\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(82\)
\(\nu^{5}\)\(=\)\(-\)\(23\) \(\beta_{13}\mathstrut -\mathstrut \) \(18\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\) \(\beta_{11}\mathstrut +\mathstrut \) \(6\) \(\beta_{10}\mathstrut -\mathstrut \) \(19\) \(\beta_{9}\mathstrut -\mathstrut \) \(4\) \(\beta_{8}\mathstrut -\mathstrut \) \(28\) \(\beta_{7}\mathstrut +\mathstrut \) \(23\) \(\beta_{6}\mathstrut +\mathstrut \) \(21\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(4\) \(\beta_{2}\mathstrut +\mathstrut \) \(159\) \(\beta_{1}\mathstrut +\mathstrut \) \(25\)
\(\nu^{6}\)\(=\)\(-\)\(252\) \(\beta_{13}\mathstrut +\mathstrut \) \(35\) \(\beta_{12}\mathstrut +\mathstrut \) \(292\) \(\beta_{11}\mathstrut -\mathstrut \) \(69\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(186\) \(\beta_{8}\mathstrut -\mathstrut \) \(181\) \(\beta_{7}\mathstrut +\mathstrut \) \(190\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(268\) \(\beta_{4}\mathstrut +\mathstrut \) \(178\) \(\beta_{3}\mathstrut -\mathstrut \) \(76\) \(\beta_{2}\mathstrut +\mathstrut \) \(90\) \(\beta_{1}\mathstrut +\mathstrut \) \(1070\)
\(\nu^{7}\)\(=\)\(-\)\(454\) \(\beta_{13}\mathstrut -\mathstrut \) \(291\) \(\beta_{12}\mathstrut -\mathstrut \) \(23\) \(\beta_{11}\mathstrut +\mathstrut \) \(114\) \(\beta_{10}\mathstrut -\mathstrut \) \(290\) \(\beta_{9}\mathstrut -\mathstrut \) \(91\) \(\beta_{8}\mathstrut -\mathstrut \) \(537\) \(\beta_{7}\mathstrut +\mathstrut \) \(413\) \(\beta_{6}\mathstrut +\mathstrut \) \(379\) \(\beta_{5}\mathstrut +\mathstrut \) \(116\) \(\beta_{4}\mathstrut +\mathstrut \) \(123\) \(\beta_{3}\mathstrut -\mathstrut \) \(126\) \(\beta_{2}\mathstrut +\mathstrut \) \(2248\) \(\beta_{1}\mathstrut +\mathstrut \) \(493\)
\(\nu^{8}\)\(=\)\(-\)\(3992\) \(\beta_{13}\mathstrut +\mathstrut \) \(446\) \(\beta_{12}\mathstrut +\mathstrut \) \(4637\) \(\beta_{11}\mathstrut -\mathstrut \) \(1265\) \(\beta_{10}\mathstrut -\mathstrut \) \(143\) \(\beta_{9}\mathstrut +\mathstrut \) \(2477\) \(\beta_{8}\mathstrut -\mathstrut \) \(2713\) \(\beta_{7}\mathstrut +\mathstrut \) \(3001\) \(\beta_{6}\mathstrut +\mathstrut \) \(451\) \(\beta_{5}\mathstrut +\mathstrut \) \(4163\) \(\beta_{4}\mathstrut +\mathstrut \) \(2583\) \(\beta_{3}\mathstrut -\mathstrut \) \(1435\) \(\beta_{2}\mathstrut +\mathstrut \) \(2047\) \(\beta_{1}\mathstrut +\mathstrut \) \(14763\)
\(\nu^{9}\)\(=\)\(-\)\(8449\) \(\beta_{13}\mathstrut -\mathstrut \) \(4639\) \(\beta_{12}\mathstrut +\mathstrut \) \(1814\) \(\beta_{11}\mathstrut +\mathstrut \) \(1546\) \(\beta_{10}\mathstrut -\mathstrut \) \(4099\) \(\beta_{9}\mathstrut -\mathstrut \) \(1550\) \(\beta_{8}\mathstrut -\mathstrut \) \(9328\) \(\beta_{7}\mathstrut +\mathstrut \) \(7012\) \(\beta_{6}\mathstrut +\mathstrut \) \(6459\) \(\beta_{5}\mathstrut +\mathstrut \) \(3029\) \(\beta_{4}\mathstrut +\mathstrut \) \(2427\) \(\beta_{3}\mathstrut -\mathstrut \) \(2861\) \(\beta_{2}\mathstrut +\mathstrut \) \(33152\) \(\beta_{1}\mathstrut +\mathstrut \) \(9432\)
\(\nu^{10}\)\(=\)\(-\)\(63639\) \(\beta_{13}\mathstrut +\mathstrut \) \(4544\) \(\beta_{12}\mathstrut +\mathstrut \) \(73258\) \(\beta_{11}\mathstrut -\mathstrut \) \(21545\) \(\beta_{10}\mathstrut -\mathstrut \) \(620\) \(\beta_{9}\mathstrut +\mathstrut \) \(33285\) \(\beta_{8}\mathstrut -\mathstrut \) \(42376\) \(\beta_{7}\mathstrut +\mathstrut \) \(48823\) \(\beta_{6}\mathstrut +\mathstrut \) \(11248\) \(\beta_{5}\mathstrut +\mathstrut \) \(64765\) \(\beta_{4}\mathstrut +\mathstrut \) \(38846\) \(\beta_{3}\mathstrut -\mathstrut \) \(24636\) \(\beta_{2}\mathstrut +\mathstrut \) \(41751\) \(\beta_{1}\mathstrut +\mathstrut \) \(211587\)
\(\nu^{11}\)\(=\)\(-\)\(152268\) \(\beta_{13}\mathstrut -\mathstrut \) \(73921\) \(\beta_{12}\mathstrut +\mathstrut \) \(60922\) \(\beta_{11}\mathstrut +\mathstrut \) \(17182\) \(\beta_{10}\mathstrut -\mathstrut \) \(56254\) \(\beta_{9}\mathstrut -\mathstrut \) \(23521\) \(\beta_{8}\mathstrut -\mathstrut \) \(156735\) \(\beta_{7}\mathstrut +\mathstrut \) \(117792\) \(\beta_{6}\mathstrut +\mathstrut \) \(107248\) \(\beta_{5}\mathstrut +\mathstrut \) \(67121\) \(\beta_{4}\mathstrut +\mathstrut \) \(45133\) \(\beta_{3}\mathstrut -\mathstrut \) \(56731\) \(\beta_{2}\mathstrut +\mathstrut \) \(502445\) \(\beta_{1}\mathstrut +\mathstrut \) \(179584\)
\(\nu^{12}\)\(=\)\(-\)\(1020459\) \(\beta_{13}\mathstrut +\mathstrut \) \(30075\) \(\beta_{12}\mathstrut +\mathstrut \) \(1158028\) \(\beta_{11}\mathstrut -\mathstrut \) \(355420\) \(\beta_{10}\mathstrut +\mathstrut \) \(12256\) \(\beta_{9}\mathstrut +\mathstrut \) \(452268\) \(\beta_{8}\mathstrut -\mathstrut \) \(676595\) \(\beta_{7}\mathstrut +\mathstrut \) \(800959\) \(\beta_{6}\mathstrut +\mathstrut \) \(236684\) \(\beta_{5}\mathstrut +\mathstrut \) \(1013938\) \(\beta_{4}\mathstrut +\mathstrut \) \(596937\) \(\beta_{3}\mathstrut -\mathstrut \) \(407361\) \(\beta_{2}\mathstrut +\mathstrut \) \(802077\) \(\beta_{1}\mathstrut +\mathstrut \) \(3117311\)
\(\nu^{13}\)\(=\)\(-\)\(2688292\) \(\beta_{13}\mathstrut -\mathstrut \) \(1179924\) \(\beta_{12}\mathstrut +\mathstrut \) \(1428695\) \(\beta_{11}\mathstrut +\mathstrut \) \(142999\) \(\beta_{10}\mathstrut -\mathstrut \) \(765007\) \(\beta_{9}\mathstrut -\mathstrut \) \(332940\) \(\beta_{8}\mathstrut -\mathstrut \) \(2603136\) \(\beta_{7}\mathstrut +\mathstrut \) \(1981261\) \(\beta_{6}\mathstrut +\mathstrut \) \(1759863\) \(\beta_{5}\mathstrut +\mathstrut \) \(1361075\) \(\beta_{4}\mathstrut +\mathstrut \) \(820748\) \(\beta_{3}\mathstrut -\mathstrut \) \(1049738\) \(\beta_{2}\mathstrut +\mathstrut \) \(7757649\) \(\beta_{1}\mathstrut +\mathstrut \) \(3383769\)

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
4.07159
3.43788
3.09495
2.71343
0.924862
0.891518
0.503883
−0.522326
−0.643572
−1.05309
−2.46518
−2.56958
−3.59906
−3.78531
−1.00000 −1.00000 1.00000 −4.07159 1.00000 2.58120 −1.00000 1.00000 4.07159
1.2 −1.00000 −1.00000 1.00000 −3.43788 1.00000 −0.875663 −1.00000 1.00000 3.43788
1.3 −1.00000 −1.00000 1.00000 −3.09495 1.00000 3.97310 −1.00000 1.00000 3.09495
1.4 −1.00000 −1.00000 1.00000 −2.71343 1.00000 −2.34175 −1.00000 1.00000 2.71343
1.5 −1.00000 −1.00000 1.00000 −0.924862 1.00000 −4.24556 −1.00000 1.00000 0.924862
1.6 −1.00000 −1.00000 1.00000 −0.891518 1.00000 4.07845 −1.00000 1.00000 0.891518
1.7 −1.00000 −1.00000 1.00000 −0.503883 1.00000 2.75905 −1.00000 1.00000 0.503883
1.8 −1.00000 −1.00000 1.00000 0.522326 1.00000 −2.74868 −1.00000 1.00000 −0.522326
1.9 −1.00000 −1.00000 1.00000 0.643572 1.00000 1.17705 −1.00000 1.00000 −0.643572
1.10 −1.00000 −1.00000 1.00000 1.05309 1.00000 −0.189425 −1.00000 1.00000 −1.05309
1.11 −1.00000 −1.00000 1.00000 2.46518 1.00000 −0.952790 −1.00000 1.00000 −2.46518
1.12 −1.00000 −1.00000 1.00000 2.56958 1.00000 3.96287 −1.00000 1.00000 −2.56958
1.13 −1.00000 −1.00000 1.00000 3.59906 1.00000 0.919823 −1.00000 1.00000 −3.59906
1.14 −1.00000 −1.00000 1.00000 3.78531 1.00000 −3.09768 −1.00000 1.00000 −3.78531
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(13\) \(1\)
\(103\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

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