Properties

Label 8046.2.a.n
Level 8046
Weight 2
Character orbit 8046.a
Self dual Yes
Analytic conductor 64.248
Analytic rank 1
Dimension 12
CM No
Inner twists 1

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8046.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.2476334663\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( + ( -1 - \beta_{9} ) q^{7} \) \(+ q^{8}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( -\beta_{1} q^{5} \) \( + ( -1 - \beta_{9} ) q^{7} \) \(+ q^{8}\) \( -\beta_{1} q^{10} \) \( + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{11} \) \( + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{13} \) \( + ( -1 - \beta_{9} ) q^{14} \) \(+ q^{16}\) \( + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{17} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{19} \) \( -\beta_{1} q^{20} \) \( + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{22} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{23} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{25} \) \( + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{26} \) \( + ( -1 - \beta_{9} ) q^{28} \) \( + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{29} \) \( + ( -2 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{31} \) \(+ q^{32}\) \( + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{34} \) \( + ( -4 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{35} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} - \beta_{11} ) q^{37} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{38} \) \( -\beta_{1} q^{40} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{41} \) \( + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{43} \) \( + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{44} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{46} \) \( + ( \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{47} \) \( + ( 3 - \beta_{2} - 2 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{49} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} + \beta_{6} - \beta_{8} + \beta_{10} ) q^{50} \) \( + ( 1 + \beta_{1} - \beta_{4} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{11} ) q^{52} \) \( + ( -1 + \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{53} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} - \beta_{4} + 3 \beta_{5} + \beta_{6} + \beta_{9} + \beta_{10} - \beta_{11} ) q^{55} \) \( + ( -1 - \beta_{9} ) q^{56} \) \( + ( -2 + \beta_{1} + \beta_{3} + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{8} - 2 \beta_{10} ) q^{58} \) \( + ( -2 + \beta_{4} + 3 \beta_{7} + 2 \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{59} \) \( + ( -\beta_{2} - \beta_{4} + \beta_{6} - 4 \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} ) q^{61} \) \( + ( -2 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{8} + \beta_{11} ) q^{62} \) \(+ q^{64}\) \( + ( 1 + 2 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} - \beta_{8} + 3 \beta_{9} + \beta_{10} - \beta_{11} ) q^{65} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + \beta_{6} - 2 \beta_{8} + 2 \beta_{9} - 2 \beta_{11} ) q^{67} \) \( + ( -1 - \beta_{2} + \beta_{5} + \beta_{6} ) q^{68} \) \( + ( -4 + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{10} + \beta_{11} ) q^{70} \) \( + ( -1 - \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} ) q^{71} \) \( + ( 1 - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{6} - \beta_{9} - \beta_{10} - 2 \beta_{11} ) q^{73} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} - \beta_{11} ) q^{74} \) \( + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{11} ) q^{76} \) \( + ( -3 + 2 \beta_{2} - \beta_{3} + \beta_{4} - 3 \beta_{5} + \beta_{6} - 2 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + 3 \beta_{11} ) q^{77} \) \( + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} + \beta_{7} - \beta_{9} + \beta_{11} ) q^{79} \) \( -\beta_{1} q^{80} \) \( + ( -3 - 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + 2 \beta_{10} - \beta_{11} ) q^{82} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + 2 \beta_{7} + 3 \beta_{11} ) q^{83} \) \( + ( -2 - \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 2 \beta_{5} - 3 \beta_{6} - \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{85} \) \( + ( -4 + \beta_{1} + 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{9} + 3 \beta_{11} ) q^{86} \) \( + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{88} \) \( + ( -1 + 2 \beta_{2} + \beta_{5} - \beta_{6} + 3 \beta_{7} + 3 \beta_{9} + \beta_{10} ) q^{89} \) \( + ( -5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{7} + \beta_{8} - 5 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} ) q^{91} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} + \beta_{6} - \beta_{7} + \beta_{8} - \beta_{9} - \beta_{11} ) q^{92} \) \( + ( \beta_{1} - 2 \beta_{2} + \beta_{5} - \beta_{7} + \beta_{9} + \beta_{10} ) q^{94} \) \( + ( -2 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 2 \beta_{4} - \beta_{5} - \beta_{6} - 4 \beta_{7} - 2 \beta_{9} - 2 \beta_{10} + \beta_{11} ) q^{95} \) \( + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{9} - \beta_{10} ) q^{97} \) \( + ( 3 - \beta_{2} - 2 \beta_{4} - \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut -\mathstrut 3q^{5} \) \(\mathstrut -\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut -\mathstrut 3q^{10} \) \(\mathstrut -\mathstrut 14q^{11} \) \(\mathstrut -\mathstrut 3q^{13} \) \(\mathstrut -\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut -\mathstrut 8q^{17} \) \(\mathstrut -\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 3q^{20} \) \(\mathstrut -\mathstrut 14q^{22} \) \(\mathstrut -\mathstrut 13q^{23} \) \(\mathstrut +\mathstrut 11q^{25} \) \(\mathstrut -\mathstrut 3q^{26} \) \(\mathstrut -\mathstrut 6q^{28} \) \(\mathstrut -\mathstrut 23q^{29} \) \(\mathstrut -\mathstrut 14q^{31} \) \(\mathstrut +\mathstrut 12q^{32} \) \(\mathstrut -\mathstrut 8q^{34} \) \(\mathstrut -\mathstrut 32q^{35} \) \(\mathstrut -\mathstrut 19q^{37} \) \(\mathstrut -\mathstrut 4q^{38} \) \(\mathstrut -\mathstrut 3q^{40} \) \(\mathstrut -\mathstrut 30q^{41} \) \(\mathstrut -\mathstrut 15q^{43} \) \(\mathstrut -\mathstrut 14q^{44} \) \(\mathstrut -\mathstrut 13q^{46} \) \(\mathstrut +\mathstrut q^{47} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut +\mathstrut 11q^{50} \) \(\mathstrut -\mathstrut 3q^{52} \) \(\mathstrut -\mathstrut 16q^{53} \) \(\mathstrut -\mathstrut 7q^{55} \) \(\mathstrut -\mathstrut 6q^{56} \) \(\mathstrut -\mathstrut 23q^{58} \) \(\mathstrut -\mathstrut 26q^{59} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 14q^{62} \) \(\mathstrut +\mathstrut 12q^{64} \) \(\mathstrut -\mathstrut 8q^{65} \) \(\mathstrut -\mathstrut 39q^{67} \) \(\mathstrut -\mathstrut 8q^{68} \) \(\mathstrut -\mathstrut 32q^{70} \) \(\mathstrut -\mathstrut 15q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut -\mathstrut 19q^{74} \) \(\mathstrut -\mathstrut 4q^{76} \) \(\mathstrut -\mathstrut 34q^{77} \) \(\mathstrut -\mathstrut 13q^{79} \) \(\mathstrut -\mathstrut 3q^{80} \) \(\mathstrut -\mathstrut 30q^{82} \) \(\mathstrut -\mathstrut 6q^{83} \) \(\mathstrut -\mathstrut 11q^{85} \) \(\mathstrut -\mathstrut 15q^{86} \) \(\mathstrut -\mathstrut 14q^{88} \) \(\mathstrut -\mathstrut 18q^{89} \) \(\mathstrut -\mathstrut 35q^{91} \) \(\mathstrut -\mathstrut 13q^{92} \) \(\mathstrut +\mathstrut q^{94} \) \(\mathstrut -\mathstrut 51q^{95} \) \(\mathstrut +\mathstrut 19q^{97} \) \(\mathstrut +\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut -\mathstrut \) \(31\) \(x^{10}\mathstrut +\mathstrut \) \(82\) \(x^{9}\mathstrut +\mathstrut \) \(334\) \(x^{8}\mathstrut -\mathstrut \) \(684\) \(x^{7}\mathstrut -\mathstrut \) \(1561\) \(x^{6}\mathstrut +\mathstrut \) \(1551\) \(x^{5}\mathstrut +\mathstrut \) \(3573\) \(x^{4}\mathstrut +\mathstrut \) \(345\) \(x^{3}\mathstrut -\mathstrut \) \(1607\) \(x^{2}\mathstrut -\mathstrut \) \(594\) \(x\mathstrut -\mathstrut \) \(3\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(24748733\) \(\nu^{11}\mathstrut +\mathstrut \) \(373555153\) \(\nu^{10}\mathstrut -\mathstrut \) \(2862765831\) \(\nu^{9}\mathstrut -\mathstrut \) \(8579109025\) \(\nu^{8}\mathstrut +\mathstrut \) \(62879646690\) \(\nu^{7}\mathstrut +\mathstrut \) \(50310595815\) \(\nu^{6}\mathstrut -\mathstrut \) \(465585432656\) \(\nu^{5}\mathstrut -\mathstrut \) \(38007995419\) \(\nu^{4}\mathstrut +\mathstrut \) \(994832840887\) \(\nu^{3}\mathstrut +\mathstrut \) \(294511464920\) \(\nu^{2}\mathstrut -\mathstrut \) \(403274368788\) \(\nu\mathstrut -\mathstrut \) \(135751602546\)\()/\)\(24466366011\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(142183724\) \(\nu^{11}\mathstrut +\mathstrut \) \(838314338\) \(\nu^{10}\mathstrut +\mathstrut \) \(2550850410\) \(\nu^{9}\mathstrut -\mathstrut \) \(22317146501\) \(\nu^{8}\mathstrut +\mathstrut \) \(3959704842\) \(\nu^{7}\mathstrut +\mathstrut \) \(175851726345\) \(\nu^{6}\mathstrut -\mathstrut \) \(208981623823\) \(\nu^{5}\mathstrut -\mathstrut \) \(377751239522\) \(\nu^{4}\mathstrut +\mathstrut \) \(495900208592\) \(\nu^{3}\mathstrut +\mathstrut \) \(403759068040\) \(\nu^{2}\mathstrut -\mathstrut \) \(184654740606\) \(\nu\mathstrut -\mathstrut \) \(100388062134\)\()/\)\(24466366011\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(743676686\) \(\nu^{11}\mathstrut +\mathstrut \) \(3071881331\) \(\nu^{10}\mathstrut +\mathstrut \) \(19932549972\) \(\nu^{9}\mathstrut -\mathstrut \) \(83789564729\) \(\nu^{8}\mathstrut -\mathstrut \) \(164381452443\) \(\nu^{7}\mathstrut +\mathstrut \) \(697617356013\) \(\nu^{6}\mathstrut +\mathstrut \) \(479743706735\) \(\nu^{5}\mathstrut -\mathstrut \) \(1672929464267\) \(\nu^{4}\mathstrut -\mathstrut \) \(1150914398347\) \(\nu^{3}\mathstrut +\mathstrut \) \(745303063093\) \(\nu^{2}\mathstrut +\mathstrut \) \(651960038514\) \(\nu\mathstrut +\mathstrut \) \(65302848768\)\()/\)\(24466366011\)
\(\beta_{5}\)\(=\)\((\)\(260555422\) \(\nu^{11}\mathstrut -\mathstrut \) \(1008357826\) \(\nu^{10}\mathstrut -\mathstrut \) \(7145245871\) \(\nu^{9}\mathstrut +\mathstrut \) \(27363214940\) \(\nu^{8}\mathstrut +\mathstrut \) \(61692870961\) \(\nu^{7}\mathstrut -\mathstrut \) \(226011640501\) \(\nu^{6}\mathstrut -\mathstrut \) \(196533632656\) \(\nu^{5}\mathstrut +\mathstrut \) \(528630702028\) \(\nu^{4}\mathstrut +\mathstrut \) \(427946138875\) \(\nu^{3}\mathstrut -\mathstrut \) \(193999453699\) \(\nu^{2}\mathstrut -\mathstrut \) \(177941248499\) \(\nu\mathstrut +\mathstrut \) \(915421136\)\()/\)\(8155455337\)
\(\beta_{6}\)\(=\)\((\)\(791457988\) \(\nu^{11}\mathstrut -\mathstrut \) \(2873219941\) \(\nu^{10}\mathstrut -\mathstrut \) \(22855402146\) \(\nu^{9}\mathstrut +\mathstrut \) \(79587640096\) \(\nu^{8}\mathstrut +\mathstrut \) \(218268434445\) \(\nu^{7}\mathstrut -\mathstrut \) \(686060598954\) \(\nu^{6}\mathstrut -\mathstrut \) \(843237849712\) \(\nu^{5}\mathstrut +\mathstrut \) \(1806674571358\) \(\nu^{4}\mathstrut +\mathstrut \) \(1819129337198\) \(\nu^{3}\mathstrut -\mathstrut \) \(899515205951\) \(\nu^{2}\mathstrut -\mathstrut \) \(830466305577\) \(\nu\mathstrut -\mathstrut \) \(29055232476\)\()/\)\(24466366011\)
\(\beta_{7}\)\(=\)\((\)\(905897063\) \(\nu^{11}\mathstrut -\mathstrut \) \(3376570442\) \(\nu^{10}\mathstrut -\mathstrut \) \(25430053989\) \(\nu^{9}\mathstrut +\mathstrut \) \(92407418951\) \(\nu^{8}\mathstrut +\mathstrut \) \(230188534851\) \(\nu^{7}\mathstrut -\mathstrut \) \(779842057185\) \(\nu^{6}\mathstrut -\mathstrut \) \(806208285203\) \(\nu^{5}\mathstrut +\mathstrut \) \(1958545237151\) \(\nu^{4}\mathstrut +\mathstrut \) \(1735102707136\) \(\nu^{3}\mathstrut -\mathstrut \) \(954029651272\) \(\nu^{2}\mathstrut -\mathstrut \) \(857897373063\) \(\nu\mathstrut -\mathstrut \) \(7334114235\)\()/\)\(24466366011\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(303077175\) \(\nu^{11}\mathstrut +\mathstrut \) \(1063333243\) \(\nu^{10}\mathstrut +\mathstrut \) \(8694135210\) \(\nu^{9}\mathstrut -\mathstrut \) \(28833755646\) \(\nu^{8}\mathstrut -\mathstrut \) \(82150526984\) \(\nu^{7}\mathstrut +\mathstrut \) \(238485257763\) \(\nu^{6}\mathstrut +\mathstrut \) \(313671881091\) \(\nu^{5}\mathstrut -\mathstrut \) \(556795852562\) \(\nu^{4}\mathstrut -\mathstrut \) \(689122894818\) \(\nu^{3}\mathstrut +\mathstrut \) \(140774506663\) \(\nu^{2}\mathstrut +\mathstrut \) \(293198015341\) \(\nu\mathstrut +\mathstrut \) \(44724727668\)\()/\)\(8155455337\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(329344029\) \(\nu^{11}\mathstrut +\mathstrut \) \(1140734420\) \(\nu^{10}\mathstrut +\mathstrut \) \(9678326681\) \(\nu^{9}\mathstrut -\mathstrut \) \(31578991963\) \(\nu^{8}\mathstrut -\mathstrut \) \(94978496783\) \(\nu^{7}\mathstrut +\mathstrut \) \(271887620614\) \(\nu^{6}\mathstrut +\mathstrut \) \(378855759913\) \(\nu^{5}\mathstrut -\mathstrut \) \(709008838650\) \(\nu^{4}\mathstrut -\mathstrut \) \(783228359697\) \(\nu^{3}\mathstrut +\mathstrut \) \(297129010822\) \(\nu^{2}\mathstrut +\mathstrut \) \(318441901306\) \(\nu\mathstrut +\mathstrut \) \(17837198367\)\()/\)\(8155455337\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(353735657\) \(\nu^{11}\mathstrut +\mathstrut \) \(1292153510\) \(\nu^{10}\mathstrut +\mathstrut \) \(10017640191\) \(\nu^{9}\mathstrut -\mathstrut \) \(35438802905\) \(\nu^{8}\mathstrut -\mathstrut \) \(91893899224\) \(\nu^{7}\mathstrut +\mathstrut \) \(299777725695\) \(\nu^{6}\mathstrut +\mathstrut \) \(328556990398\) \(\nu^{5}\mathstrut -\mathstrut \) \(756307087494\) \(\nu^{4}\mathstrut -\mathstrut \) \(702253132145\) \(\nu^{3}\mathstrut +\mathstrut \) \(389355266298\) \(\nu^{2}\mathstrut +\mathstrut \) \(322371833162\) \(\nu\mathstrut -\mathstrut \) \(27070193104\)\()/\)\(8155455337\)
\(\beta_{11}\)\(=\)\((\)\(701406468\) \(\nu^{11}\mathstrut -\mathstrut \) \(2905422747\) \(\nu^{10}\mathstrut -\mathstrut \) \(18475689520\) \(\nu^{9}\mathstrut +\mathstrut \) \(78645513257\) \(\nu^{8}\mathstrut +\mathstrut \) \(146244224743\) \(\nu^{7}\mathstrut -\mathstrut \) \(647726374616\) \(\nu^{6}\mathstrut -\mathstrut \) \(374724551769\) \(\nu^{5}\mathstrut +\mathstrut \) \(1522325564529\) \(\nu^{4}\mathstrut +\mathstrut \) \(836181281764\) \(\nu^{3}\mathstrut -\mathstrut \) \(700613773249\) \(\nu^{2}\mathstrut -\mathstrut \) \(384310270718\) \(\nu\mathstrut -\mathstrut \) \(1717939244\)\()/\)\(8155455337\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(6\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(-\)\(3\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut +\mathstrut \) \(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(20\) \(\beta_{6}\mathstrut -\mathstrut \) \(6\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(16\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(68\)
\(\nu^{5}\)\(=\)\(-\)\(19\) \(\beta_{11}\mathstrut +\mathstrut \) \(43\) \(\beta_{10}\mathstrut +\mathstrut \) \(23\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(50\) \(\beta_{7}\mathstrut +\mathstrut \) \(35\) \(\beta_{6}\mathstrut +\mathstrut \) \(45\) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{4}\mathstrut -\mathstrut \) \(24\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(127\) \(\beta_{1}\mathstrut +\mathstrut \) \(31\)
\(\nu^{6}\)\(=\)\(-\)\(62\) \(\beta_{11}\mathstrut +\mathstrut \) \(258\) \(\beta_{10}\mathstrut +\mathstrut \) \(85\) \(\beta_{9}\mathstrut -\mathstrut \) \(184\) \(\beta_{8}\mathstrut +\mathstrut \) \(55\) \(\beta_{7}\mathstrut +\mathstrut \) \(335\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(41\) \(\beta_{4}\mathstrut -\mathstrut \) \(239\) \(\beta_{3}\mathstrut -\mathstrut \) \(26\) \(\beta_{2}\mathstrut +\mathstrut \) \(238\) \(\beta_{1}\mathstrut +\mathstrut \) \(846\)
\(\nu^{7}\)\(=\)\(-\)\(313\) \(\beta_{11}\mathstrut +\mathstrut \) \(786\) \(\beta_{10}\mathstrut +\mathstrut \) \(465\) \(\beta_{9}\mathstrut -\mathstrut \) \(93\) \(\beta_{8}\mathstrut +\mathstrut \) \(795\) \(\beta_{7}\mathstrut +\mathstrut \) \(780\) \(\beta_{6}\mathstrut +\mathstrut \) \(809\) \(\beta_{5}\mathstrut +\mathstrut \) \(225\) \(\beta_{4}\mathstrut -\mathstrut \) \(459\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(1763\) \(\beta_{1}\mathstrut +\mathstrut \) \(576\)
\(\nu^{8}\)\(=\)\(-\)\(1051\) \(\beta_{11}\mathstrut +\mathstrut \) \(4141\) \(\beta_{10}\mathstrut +\mathstrut \) \(1808\) \(\beta_{9}\mathstrut -\mathstrut \) \(2429\) \(\beta_{8}\mathstrut +\mathstrut \) \(1441\) \(\beta_{7}\mathstrut +\mathstrut \) \(5432\) \(\beta_{6}\mathstrut +\mathstrut \) \(1061\) \(\beta_{5}\mathstrut +\mathstrut \) \(1011\) \(\beta_{4}\mathstrut -\mathstrut \) \(3568\) \(\beta_{3}\mathstrut -\mathstrut \) \(225\) \(\beta_{2}\mathstrut +\mathstrut \) \(3995\) \(\beta_{1}\mathstrut +\mathstrut \) \(11067\)
\(\nu^{9}\)\(=\)\(-\)\(4900\) \(\beta_{11}\mathstrut +\mathstrut \) \(13689\) \(\beta_{10}\mathstrut +\mathstrut \) \(8697\) \(\beta_{9}\mathstrut -\mathstrut \) \(2068\) \(\beta_{8}\mathstrut +\mathstrut \) \(12789\) \(\beta_{7}\mathstrut +\mathstrut \) \(14973\) \(\beta_{6}\mathstrut +\mathstrut \) \(13495\) \(\beta_{5}\mathstrut +\mathstrut \) \(4850\) \(\beta_{4}\mathstrut -\mathstrut \) \(8212\) \(\beta_{3}\mathstrut +\mathstrut \) \(193\) \(\beta_{2}\mathstrut +\mathstrut \) \(25829\) \(\beta_{1}\mathstrut +\mathstrut \) \(11256\)
\(\nu^{10}\)\(=\)\(-\)\(16949\) \(\beta_{11}\mathstrut +\mathstrut \) \(66601\) \(\beta_{10}\mathstrut +\mathstrut \) \(34473\) \(\beta_{9}\mathstrut -\mathstrut \) \(32858\) \(\beta_{8}\mathstrut +\mathstrut \) \(30539\) \(\beta_{7}\mathstrut +\mathstrut \) \(87444\) \(\beta_{6}\mathstrut +\mathstrut \) \(27494\) \(\beta_{5}\mathstrut +\mathstrut \) \(20849\) \(\beta_{4}\mathstrut -\mathstrut \) \(54040\) \(\beta_{3}\mathstrut -\mathstrut \) \(667\) \(\beta_{2}\mathstrut +\mathstrut \) \(68339\) \(\beta_{1}\mathstrut +\mathstrut \) \(151126\)
\(\nu^{11}\)\(=\)\(-\)\(75864\) \(\beta_{11}\mathstrut +\mathstrut \) \(233383\) \(\beta_{10}\mathstrut +\mathstrut \) \(155287\) \(\beta_{9}\mathstrut -\mathstrut \) \(40971\) \(\beta_{8}\mathstrut +\mathstrut \) \(207800\) \(\beta_{7}\mathstrut +\mathstrut \) \(269383\) \(\beta_{6}\mathstrut +\mathstrut \) \(219090\) \(\beta_{5}\mathstrut +\mathstrut \) \(93810\) \(\beta_{4}\mathstrut -\mathstrut \) \(143006\) \(\beta_{3}\mathstrut +\mathstrut \) \(9222\) \(\beta_{2}\mathstrut +\mathstrut \) \(392614\) \(\beta_{1}\mathstrut +\mathstrut \) \(215351\)

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
4.06732
3.27568
2.95427
2.60206
0.761360
−0.00512154
−0.539392
−0.815716
−1.11280
−1.28984
−3.43402
−3.46379
1.00000 0 1.00000 −4.06732 0 1.57160 1.00000 0 −4.06732
1.2 1.00000 0 1.00000 −3.27568 0 4.39969 1.00000 0 −3.27568
1.3 1.00000 0 1.00000 −2.95427 0 0.794893 1.00000 0 −2.95427
1.4 1.00000 0 1.00000 −2.60206 0 −4.35416 1.00000 0 −2.60206
1.5 1.00000 0 1.00000 −0.761360 0 1.23946 1.00000 0 −0.761360
1.6 1.00000 0 1.00000 0.00512154 0 −2.98814 1.00000 0 0.00512154
1.7 1.00000 0 1.00000 0.539392 0 0.739124 1.00000 0 0.539392
1.8 1.00000 0 1.00000 0.815716 0 −4.13057 1.00000 0 0.815716
1.9 1.00000 0 1.00000 1.11280 0 −0.851837 1.00000 0 1.11280
1.10 1.00000 0 1.00000 1.28984 0 3.20787 1.00000 0 1.28984
1.11 1.00000 0 1.00000 3.43402 0 −1.90397 1.00000 0 3.43402
1.12 1.00000 0 1.00000 3.46379 0 −3.72396 1.00000 0 3.46379
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(149\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

\(T_{5}^{12} + \cdots\)
\(T_{11}^{12} + \cdots\)