Properties

Label 8046.2.a.o
Level 8046
Weight 2
Character orbit 8046.a
Self dual Yes
Analytic conductor 64.248
Analytic rank 0
Dimension 12
CM No
Inner twists 1

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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: \(0\)
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.

\(f(q)\) \(=\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \( + ( 1 + \beta_{6} ) q^{7} \) \(+ q^{8}\) \(+O(q^{10})\) \( q\) \(+ q^{2}\) \(+ q^{4}\) \( + \beta_{1} q^{5} \) \( + ( 1 + \beta_{6} ) q^{7} \) \(+ q^{8}\) \( + \beta_{1} q^{10} \) \( + ( 1 - \beta_{5} ) q^{11} \) \( + ( 1 - \beta_{5} + \beta_{9} ) q^{13} \) \( + ( 1 + \beta_{6} ) q^{14} \) \(+ q^{16}\) \( + ( 1 + \beta_{7} ) q^{17} \) \( -\beta_{11} q^{19} \) \( + \beta_{1} q^{20} \) \( + ( 1 - \beta_{5} ) q^{22} \) \( + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{23} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{25} \) \( + ( 1 - \beta_{5} + \beta_{9} ) q^{26} \) \( + ( 1 + \beta_{6} ) q^{28} \) \( + ( 2 - \beta_{8} ) q^{29} \) \( + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} ) q^{31} \) \(+ q^{32}\) \( + ( 1 + \beta_{7} ) q^{34} \) \( + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{35} \) \( + ( 1 + \beta_{7} - \beta_{9} + \beta_{11} ) q^{37} \) \( -\beta_{11} q^{38} \) \( + \beta_{1} q^{40} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{41} \) \( + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{43} \) \( + ( 1 - \beta_{5} ) q^{44} \) \( + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{46} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{47} \) \( + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{49} \) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{11} ) q^{50} \) \( + ( 1 - \beta_{5} + \beta_{9} ) q^{52} \) \( + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{10} - \beta_{11} ) q^{53} \) \( + ( \beta_{1} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{11} ) q^{55} \) \( + ( 1 + \beta_{6} ) q^{56} \) \( + ( 2 - \beta_{8} ) q^{58} \) \( + ( 2 - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + \beta_{9} ) q^{59} \) \( + ( \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{61} \) \( + ( 1 + \beta_{1} - \beta_{4} - \beta_{6} - \beta_{8} + \beta_{11} ) q^{62} \) \(+ q^{64}\) \( + ( 1 - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{65} \) \( + ( 1 - \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} ) q^{67} \) \( + ( 1 + \beta_{7} ) q^{68} \) \( + ( 1 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} - \beta_{10} - \beta_{11} ) q^{70} \) \( + ( 4 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{8} - 2 \beta_{9} + \beta_{10} + 2 \beta_{11} ) q^{71} \) \( + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} - \beta_{7} ) q^{73} \) \( + ( 1 + \beta_{7} - \beta_{9} + \beta_{11} ) q^{74} \) \( -\beta_{11} q^{76} \) \( + ( 1 + 2 \beta_{3} - \beta_{5} + \beta_{6} + \beta_{8} - \beta_{9} + 2 \beta_{11} ) q^{77} \) \( + ( 1 + \beta_{2} + 2 \beta_{3} - \beta_{4} + \beta_{5} - \beta_{9} - \beta_{10} ) q^{79} \) \( + \beta_{1} q^{80} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{9} - \beta_{10} ) q^{82} \) \( + ( 2 - 2 \beta_{3} + \beta_{5} - \beta_{6} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{83} \) \( + ( 2 \beta_{1} - \beta_{3} - \beta_{6} - \beta_{8} - \beta_{9} + \beta_{10} ) q^{85} \) \( + ( \beta_{1} + \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{86} \) \( + ( 1 - \beta_{5} ) q^{88} \) \( + ( 3 - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{11} ) q^{89} \) \( + ( 1 - \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + 2 \beta_{6} - 2 \beta_{7} + 2 \beta_{9} - \beta_{10} - \beta_{11} ) q^{91} \) \( + ( -\beta_{2} - \beta_{7} + \beta_{8} + \beta_{10} ) q^{92} \) \( + ( 3 - 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{6} + \beta_{7} + \beta_{9} - \beta_{10} ) q^{94} \) \( + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{4} + \beta_{5} + \beta_{7} - \beta_{8} ) q^{95} \) \( + ( \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} ) q^{97} \) \( + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{10} + \beta_{11} ) q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 12q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 3q^{5} \) \(\mathstrut +\mathstrut 6q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut +\mathstrut 3q^{10} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 5q^{13} \) \(\mathstrut +\mathstrut 6q^{14} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut +\mathstrut 8q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 3q^{20} \) \(\mathstrut +\mathstrut 10q^{22} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 7q^{25} \) \(\mathstrut +\mathstrut 5q^{26} \) \(\mathstrut +\mathstrut 6q^{28} \) \(\mathstrut +\mathstrut 19q^{29} \) \(\mathstrut +\mathstrut 10q^{31} \) \(\mathstrut +\mathstrut 12q^{32} \) \(\mathstrut +\mathstrut 8q^{34} \) \(\mathstrut +\mathstrut 20q^{35} \) \(\mathstrut +\mathstrut 11q^{37} \) \(\mathstrut +\mathstrut 2q^{38} \) \(\mathstrut +\mathstrut 3q^{40} \) \(\mathstrut +\mathstrut 8q^{41} \) \(\mathstrut +\mathstrut 13q^{43} \) \(\mathstrut +\mathstrut 10q^{44} \) \(\mathstrut +\mathstrut 9q^{46} \) \(\mathstrut +\mathstrut 11q^{47} \) \(\mathstrut +\mathstrut 2q^{49} \) \(\mathstrut +\mathstrut 7q^{50} \) \(\mathstrut +\mathstrut 5q^{52} \) \(\mathstrut +\mathstrut 24q^{53} \) \(\mathstrut +\mathstrut 3q^{55} \) \(\mathstrut +\mathstrut 6q^{56} \) \(\mathstrut +\mathstrut 19q^{58} \) \(\mathstrut +\mathstrut 10q^{59} \) \(\mathstrut +\mathstrut 10q^{62} \) \(\mathstrut +\mathstrut 12q^{64} \) \(\mathstrut +\mathstrut 28q^{65} \) \(\mathstrut +\mathstrut 21q^{67} \) \(\mathstrut +\mathstrut 8q^{68} \) \(\mathstrut +\mathstrut 20q^{70} \) \(\mathstrut +\mathstrut 37q^{71} \) \(\mathstrut -\mathstrut 2q^{73} \) \(\mathstrut +\mathstrut 11q^{74} \) \(\mathstrut +\mathstrut 2q^{76} \) \(\mathstrut +\mathstrut 2q^{77} \) \(\mathstrut +\mathstrut 7q^{79} \) \(\mathstrut +\mathstrut 3q^{80} \) \(\mathstrut +\mathstrut 8q^{82} \) \(\mathstrut +\mathstrut 22q^{83} \) \(\mathstrut +\mathstrut 15q^{85} \) \(\mathstrut +\mathstrut 13q^{86} \) \(\mathstrut +\mathstrut 10q^{88} \) \(\mathstrut +\mathstrut 40q^{89} \) \(\mathstrut +\mathstrut q^{91} \) \(\mathstrut +\mathstrut 9q^{92} \) \(\mathstrut +\mathstrut 11q^{94} \) \(\mathstrut +\mathstrut 11q^{95} \) \(\mathstrut +\mathstrut 7q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(3\) \(x^{11}\mathstrut -\mathstrut \) \(29\) \(x^{10}\mathstrut +\mathstrut \) \(76\) \(x^{9}\mathstrut +\mathstrut \) \(320\) \(x^{8}\mathstrut -\mathstrut \) \(724\) \(x^{7}\mathstrut -\mathstrut \) \(1643\) \(x^{6}\mathstrut +\mathstrut \) \(3265\) \(x^{5}\mathstrut +\mathstrut \) \(3921\) \(x^{4}\mathstrut -\mathstrut \) \(6927\) \(x^{3}\mathstrut -\mathstrut \) \(3639\) \(x^{2}\mathstrut +\mathstrut \) \(5508\) \(x\mathstrut +\mathstrut \) \(423\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(1293103\) \(\nu^{11}\mathstrut +\mathstrut \) \(6001945\) \(\nu^{10}\mathstrut +\mathstrut \) \(23580251\) \(\nu^{9}\mathstrut -\mathstrut \) \(131132508\) \(\nu^{8}\mathstrut -\mathstrut \) \(90139960\) \(\nu^{7}\mathstrut +\mathstrut \) \(1024446702\) \(\nu^{6}\mathstrut -\mathstrut \) \(573136895\) \(\nu^{5}\mathstrut -\mathstrut \) \(3491861196\) \(\nu^{4}\mathstrut +\mathstrut \) \(4215813919\) \(\nu^{3}\mathstrut +\mathstrut \) \(4581336765\) \(\nu^{2}\mathstrut -\mathstrut \) \(6050385741\) \(\nu\mathstrut -\mathstrut \) \(735740835\)\()/\)\(174942609\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(1915090\) \(\nu^{11}\mathstrut -\mathstrut \) \(1046362\) \(\nu^{10}\mathstrut +\mathstrut \) \(74238453\) \(\nu^{9}\mathstrut +\mathstrut \) \(35401589\) \(\nu^{8}\mathstrut -\mathstrut \) \(1005031345\) \(\nu^{7}\mathstrut -\mathstrut \) \(425190469\) \(\nu^{6}\mathstrut +\mathstrut \) \(5944816403\) \(\nu^{5}\mathstrut +\mathstrut \) \(1979959133\) \(\nu^{4}\mathstrut -\mathstrut \) \(15609041924\) \(\nu^{3}\mathstrut -\mathstrut \) \(3485888106\) \(\nu^{2}\mathstrut +\mathstrut \) \(14934980991\) \(\nu\mathstrut +\mathstrut \) \(1318582776\)\()/\)\(174942609\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(4181218\) \(\nu^{11}\mathstrut +\mathstrut \) \(5656554\) \(\nu^{10}\mathstrut +\mathstrut \) \(132467602\) \(\nu^{9}\mathstrut -\mathstrut \) \(109619327\) \(\nu^{8}\mathstrut -\mathstrut \) \(1543743823\) \(\nu^{7}\mathstrut +\mathstrut \) \(649333723\) \(\nu^{6}\mathstrut +\mathstrut \) \(8064151794\) \(\nu^{5}\mathstrut -\mathstrut \) \(1288637477\) \(\nu^{4}\mathstrut -\mathstrut \) \(19088618612\) \(\nu^{3}\mathstrut -\mathstrut \) \(59111379\) \(\nu^{2}\mathstrut +\mathstrut \) \(16748632692\) \(\nu\mathstrut +\mathstrut \) \(1376131821\)\()/\)\(174942609\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(4389972\) \(\nu^{11}\mathstrut +\mathstrut \) \(4775951\) \(\nu^{10}\mathstrut +\mathstrut \) \(138192352\) \(\nu^{9}\mathstrut -\mathstrut \) \(72357264\) \(\nu^{8}\mathstrut -\mathstrut \) \(1596499871\) \(\nu^{7}\mathstrut +\mathstrut \) \(185639731\) \(\nu^{6}\mathstrut +\mathstrut \) \(8187073056\) \(\nu^{5}\mathstrut +\mathstrut \) \(784010463\) \(\nu^{4}\mathstrut -\mathstrut \) \(18702206507\) \(\nu^{3}\mathstrut -\mathstrut \) \(2790478839\) \(\nu^{2}\mathstrut +\mathstrut \) \(15392930346\) \(\nu\mathstrut +\mathstrut \) \(1169371344\)\()/\)\(174942609\)
\(\beta_{6}\)\(=\)\((\)\(6533895\) \(\nu^{11}\mathstrut -\mathstrut \) \(11760004\) \(\nu^{10}\mathstrut -\mathstrut \) \(202559541\) \(\nu^{9}\mathstrut +\mathstrut \) \(251802455\) \(\nu^{8}\mathstrut +\mathstrut \) \(2345517923\) \(\nu^{7}\mathstrut -\mathstrut \) \(1816569674\) \(\nu^{6}\mathstrut -\mathstrut \) \(12292593347\) \(\nu^{5}\mathstrut +\mathstrut \) \(5268511082\) \(\nu^{4}\mathstrut +\mathstrut \) \(28827085403\) \(\nu^{3}\mathstrut -\mathstrut \) \(4670185662\) \(\nu^{2}\mathstrut -\mathstrut \) \(24116897136\) \(\nu\mathstrut -\mathstrut \) \(1422840759\)\()/\)\(174942609\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(6869077\) \(\nu^{11}\mathstrut +\mathstrut \) \(5839142\) \(\nu^{10}\mathstrut +\mathstrut \) \(230548446\) \(\nu^{9}\mathstrut -\mathstrut \) \(91270315\) \(\nu^{8}\mathstrut -\mathstrut \) \(2856540142\) \(\nu^{7}\mathstrut +\mathstrut \) \(274308218\) \(\nu^{6}\mathstrut +\mathstrut \) \(15985954088\) \(\nu^{5}\mathstrut +\mathstrut \) \(676618097\) \(\nu^{4}\mathstrut -\mathstrut \) \(40668047126\) \(\nu^{3}\mathstrut -\mathstrut \) \(3142259250\) \(\nu^{2}\mathstrut +\mathstrut \) \(37843096842\) \(\nu\mathstrut +\mathstrut \) \(2092764192\)\()/\)\(174942609\)
\(\beta_{8}\)\(=\)\((\)\(9158236\) \(\nu^{11}\mathstrut -\mathstrut \) \(12464738\) \(\nu^{10}\mathstrut -\mathstrut \) \(290204756\) \(\nu^{9}\mathstrut +\mathstrut \) \(242859302\) \(\nu^{8}\mathstrut +\mathstrut \) \(3388516467\) \(\nu^{7}\mathstrut -\mathstrut \) \(1488120542\) \(\nu^{6}\mathstrut -\mathstrut \) \(17686820067\) \(\nu^{5}\mathstrut +\mathstrut \) \(3338509649\) \(\nu^{4}\mathstrut +\mathstrut \) \(41281617595\) \(\nu^{3}\mathstrut -\mathstrut \) \(819687051\) \(\nu^{2}\mathstrut -\mathstrut \) \(34744019517\) \(\nu\mathstrut -\mathstrut \) \(3556778949\)\()/\)\(174942609\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(10768077\) \(\nu^{11}\mathstrut +\mathstrut \) \(13709156\) \(\nu^{10}\mathstrut +\mathstrut \) \(343745338\) \(\nu^{9}\mathstrut -\mathstrut \) \(248488722\) \(\nu^{8}\mathstrut -\mathstrut \) \(4077208325\) \(\nu^{7}\mathstrut +\mathstrut \) \(1283221540\) \(\nu^{6}\mathstrut +\mathstrut \) \(21816188427\) \(\nu^{5}\mathstrut -\mathstrut \) \(1622126526\) \(\nu^{4}\mathstrut -\mathstrut \) \(52604128721\) \(\nu^{3}\mathstrut -\mathstrut \) \(2624930826\) \(\nu^{2}\mathstrut +\mathstrut \) \(45986566413\) \(\nu\mathstrut +\mathstrut \) \(4392624474\)\()/\)\(174942609\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(11759125\) \(\nu^{11}\mathstrut +\mathstrut \) \(19783265\) \(\nu^{10}\mathstrut +\mathstrut \) \(355999031\) \(\nu^{9}\mathstrut -\mathstrut \) \(376860878\) \(\nu^{8}\mathstrut -\mathstrut \) \(4047108810\) \(\nu^{7}\mathstrut +\mathstrut \) \(2253401432\) \(\nu^{6}\mathstrut +\mathstrut \) \(20885745918\) \(\nu^{5}\mathstrut -\mathstrut \) \(4769542871\) \(\nu^{4}\mathstrut -\mathstrut \) \(48828823276\) \(\nu^{3}\mathstrut +\mathstrut \) \(1245833994\) \(\nu^{2}\mathstrut +\mathstrut \) \(41813328450\) \(\nu\mathstrut +\mathstrut \) \(3637207113\)\()/\)\(174942609\)
\(\beta_{11}\)\(=\)\((\)\(13139198\) \(\nu^{11}\mathstrut -\mathstrut \) \(18649271\) \(\nu^{10}\mathstrut -\mathstrut \) \(411865851\) \(\nu^{9}\mathstrut +\mathstrut \) \(355642798\) \(\nu^{8}\mathstrut +\mathstrut \) \(4791452746\) \(\nu^{7}\mathstrut -\mathstrut \) \(2137541739\) \(\nu^{6}\mathstrut -\mathstrut \) \(25035485466\) \(\nu^{5}\mathstrut +\mathstrut \) \(4865115271\) \(\nu^{4}\mathstrut +\mathstrut \) \(58645232190\) \(\nu^{3}\mathstrut -\mathstrut \) \(2492818554\) \(\nu^{2}\mathstrut -\mathstrut \) \(49613155317\) \(\nu\mathstrut -\mathstrut \) \(2837900049\)\()/\)\(174942609\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(-\)\(3\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(18\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(6\) \(\beta_{9}\mathstrut +\mathstrut \) \(16\) \(\beta_{8}\mathstrut -\mathstrut \) \(19\) \(\beta_{7}\mathstrut +\mathstrut \) \(4\) \(\beta_{6}\mathstrut -\mathstrut \) \(5\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{4}\mathstrut -\mathstrut \) \(3\) \(\beta_{3}\mathstrut -\mathstrut \) \(17\) \(\beta_{2}\mathstrut +\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(27\)
\(\nu^{5}\)\(=\)\(-\)\(62\) \(\beta_{11}\mathstrut -\mathstrut \) \(20\) \(\beta_{10}\mathstrut +\mathstrut \) \(42\) \(\beta_{9}\mathstrut +\mathstrut \) \(47\) \(\beta_{8}\mathstrut -\mathstrut \) \(61\) \(\beta_{7}\mathstrut +\mathstrut \) \(23\) \(\beta_{6}\mathstrut -\mathstrut \) \(26\) \(\beta_{5}\mathstrut +\mathstrut \) \(53\) \(\beta_{4}\mathstrut -\mathstrut \) \(35\) \(\beta_{3}\mathstrut -\mathstrut \) \(56\) \(\beta_{2}\mathstrut +\mathstrut \) \(122\) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)
\(\nu^{6}\)\(=\)\(-\)\(296\) \(\beta_{11}\mathstrut -\mathstrut \) \(61\) \(\beta_{10}\mathstrut +\mathstrut \) \(153\) \(\beta_{9}\mathstrut +\mathstrut \) \(256\) \(\beta_{8}\mathstrut -\mathstrut \) \(304\) \(\beta_{7}\mathstrut +\mathstrut \) \(94\) \(\beta_{6}\mathstrut -\mathstrut \) \(119\) \(\beta_{5}\mathstrut +\mathstrut \) \(295\) \(\beta_{4}\mathstrut -\mathstrut \) \(81\) \(\beta_{3}\mathstrut -\mathstrut \) \(254\) \(\beta_{2}\mathstrut +\mathstrut \) \(336\) \(\beta_{1}\mathstrut +\mathstrut \) \(248\)
\(\nu^{7}\)\(=\)\(-\)\(1096\) \(\beta_{11}\mathstrut -\mathstrut \) \(361\) \(\beta_{10}\mathstrut +\mathstrut \) \(776\) \(\beta_{9}\mathstrut +\mathstrut \) \(897\) \(\beta_{8}\mathstrut -\mathstrut \) \(1076\) \(\beta_{7}\mathstrut +\mathstrut \) \(428\) \(\beta_{6}\mathstrut -\mathstrut \) \(516\) \(\beta_{5}\mathstrut +\mathstrut \) \(1042\) \(\beta_{4}\mathstrut -\mathstrut \) \(538\) \(\beta_{3}\mathstrut -\mathstrut \) \(905\) \(\beta_{2}\mathstrut +\mathstrut \) \(1656\) \(\beta_{1}\mathstrut +\mathstrut \) \(333\)
\(\nu^{8}\)\(=\)\(-\)\(4821\) \(\beta_{11}\mathstrut -\mathstrut \) \(1304\) \(\beta_{10}\mathstrut +\mathstrut \) \(3055\) \(\beta_{9}\mathstrut +\mathstrut \) \(4200\) \(\beta_{8}\mathstrut -\mathstrut \) \(4846\) \(\beta_{7}\mathstrut +\mathstrut \) \(1764\) \(\beta_{6}\mathstrut -\mathstrut \) \(2258\) \(\beta_{5}\mathstrut +\mathstrut \) \(4857\) \(\beta_{4}\mathstrut -\mathstrut \) \(1606\) \(\beta_{3}\mathstrut -\mathstrut \) \(3827\) \(\beta_{2}\mathstrut +\mathstrut \) \(5403\) \(\beta_{1}\mathstrut +\mathstrut \) \(2803\)
\(\nu^{9}\)\(=\)\(-\)\(18627\) \(\beta_{11}\mathstrut -\mathstrut \) \(6354\) \(\beta_{10}\mathstrut +\mathstrut \) \(13806\) \(\beta_{9}\mathstrut +\mathstrut \) \(15966\) \(\beta_{8}\mathstrut -\mathstrut \) \(18280\) \(\beta_{7}\mathstrut +\mathstrut \) \(7495\) \(\beta_{6}\mathstrut -\mathstrut \) \(9404\) \(\beta_{5}\mathstrut +\mathstrut \) \(18623\) \(\beta_{4}\mathstrut -\mathstrut \) \(8326\) \(\beta_{3}\mathstrut -\mathstrut \) \(14361\) \(\beta_{2}\mathstrut +\mathstrut \) \(24022\) \(\beta_{1}\mathstrut +\mathstrut \) \(6252\)
\(\nu^{10}\)\(=\)\(-\)\(78834\) \(\beta_{11}\mathstrut -\mathstrut \) \(24603\) \(\beta_{10}\mathstrut +\mathstrut \) \(55891\) \(\beta_{9}\mathstrut +\mathstrut \) \(69732\) \(\beta_{8}\mathstrut -\mathstrut \) \(78265\) \(\beta_{7}\mathstrut +\mathstrut \) \(30911\) \(\beta_{6}\mathstrut -\mathstrut \) \(40046\) \(\beta_{5}\mathstrut +\mathstrut \) \(80628\) \(\beta_{4}\mathstrut -\mathstrut \) \(28744\) \(\beta_{3}\mathstrut -\mathstrut \) \(59098\) \(\beta_{2}\mathstrut +\mathstrut \) \(86165\) \(\beta_{1}\mathstrut +\mathstrut \) \(36254\)
\(\nu^{11}\)\(=\)\(-\)\(312025\) \(\beta_{11}\mathstrut -\mathstrut \) \(109982\) \(\beta_{10}\mathstrut +\mathstrut \) \(240193\) \(\beta_{9}\mathstrut +\mathstrut \) \(275200\) \(\beta_{8}\mathstrut -\mathstrut \) \(305996\) \(\beta_{7}\mathstrut +\mathstrut \) \(128162\) \(\beta_{6}\mathstrut -\mathstrut \) \(164919\) \(\beta_{5}\mathstrut +\mathstrut \) \(320329\) \(\beta_{4}\mathstrut -\mathstrut \) \(131955\) \(\beta_{3}\mathstrut -\mathstrut \) \(228964\) \(\beta_{2}\mathstrut +\mathstrut \) \(364213\) \(\beta_{1}\mathstrut +\mathstrut \) \(106224\)

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
−2.89416
−2.72416
−2.41916
−1.76721
−1.46524
−0.0737297
1.41673
1.51858
1.84210
2.15566
3.34249
4.06809
1.00000 0 1.00000 −2.89416 0 0.647278 1.00000 0 −2.89416
1.2 1.00000 0 1.00000 −2.72416 0 −2.55489 1.00000 0 −2.72416
1.3 1.00000 0 1.00000 −2.41916 0 0.521062 1.00000 0 −2.41916
1.4 1.00000 0 1.00000 −1.76721 0 2.53166 1.00000 0 −1.76721
1.5 1.00000 0 1.00000 −1.46524 0 −3.53999 1.00000 0 −1.46524
1.6 1.00000 0 1.00000 −0.0737297 0 2.82077 1.00000 0 −0.0737297
1.7 1.00000 0 1.00000 1.41673 0 −1.27536 1.00000 0 1.41673
1.8 1.00000 0 1.00000 1.51858 0 4.35237 1.00000 0 1.51858
1.9 1.00000 0 1.00000 1.84210 0 2.73011 1.00000 0 1.84210
1.10 1.00000 0 1.00000 2.15566 0 −3.96511 1.00000 0 2.15566
1.11 1.00000 0 1.00000 3.34249 0 1.09959 1.00000 0 3.34249
1.12 1.00000 0 1.00000 4.06809 0 2.63250 1.00000 0 4.06809
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8046))\):

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