Properties

Label 8034.2.a.r
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 1
Dimension 9
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(4\) \(x^{8}\mathstrut -\mathstrut \) \(9\) \(x^{7}\mathstrut +\mathstrut \) \(45\) \(x^{6}\mathstrut +\mathstrut \) \(7\) \(x^{5}\mathstrut -\mathstrut \) \(123\) \(x^{4}\mathstrut +\mathstrut \) \(37\) \(x^{3}\mathstrut +\mathstrut \) \(87\) \(x^{2}\mathstrut -\mathstrut \) \(54\) \(x\mathstrut +\mathstrut \) \(8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 55 \nu^{8} - 272 \nu^{7} - 307 \nu^{6} + 2869 \nu^{5} - 1267 \nu^{4} - 6789 \nu^{3} + 3935 \nu^{2} + 4811 \nu - 2412 \)\()/634\)
\(\beta_{3}\)\(=\)\((\)\( 129 \nu^{8} - 442 \nu^{7} - 1331 \nu^{6} + 4781 \nu^{5} + 2815 \nu^{4} - 11785 \nu^{3} - 223 \nu^{2} + 7065 \nu - 3444 \)\()/634\)
\(\beta_{4}\)\(=\)\((\)\( -131 \nu^{8} + 498 \nu^{7} + 1273 \nu^{6} - 5381 \nu^{5} - 2377 \nu^{4} + 12931 \nu^{3} + 1175 \nu^{2} - 5995 \nu + 1964 \)\()/634\)
\(\beta_{5}\)\(=\)\((\)\( 92 \nu^{8} - 357 \nu^{7} - 819 \nu^{6} + 3825 \nu^{5} + 774 \nu^{4} - 9287 \nu^{3} + 1539 \nu^{2} + 5938 \nu - 1977 \)\()/317\)
\(\beta_{6}\)\(=\)\((\)\( -359 \nu^{8} + 1176 \nu^{7} + 4171 \nu^{6} - 13551 \nu^{5} - 12675 \nu^{4} + 38965 \nu^{3} + 11433 \nu^{2} - 29201 \nu + 5692 \)\()/634\)
\(\beta_{7}\)\(=\)\((\)\( -415 \nu^{8} + 1476 \nu^{7} + 4449 \nu^{6} - 17037 \nu^{5} - 10555 \nu^{4} + 49497 \nu^{3} + 3219 \nu^{2} - 39183 \nu + 10534 \)\()/634\)
\(\beta_{8}\)\(=\)\((\)\( 261 \nu^{8} - 968 \nu^{7} - 2575 \nu^{6} + 10779 \nu^{5} + 4656 \nu^{4} - 28776 \nu^{3} + 28 \nu^{2} + 20133 \nu - 5619 \)\()/317\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(7\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(21\)
\(\nu^{5}\)\(=\)\(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut -\mathstrut \) \(27\) \(\beta_{5}\mathstrut -\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(54\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{6}\)\(=\)\(34\) \(\beta_{8}\mathstrut +\mathstrut \) \(27\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut -\mathstrut \) \(114\) \(\beta_{5}\mathstrut +\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(73\) \(\beta_{3}\mathstrut +\mathstrut \) \(86\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(177\)
\(\nu^{7}\)\(=\)\(165\) \(\beta_{8}\mathstrut +\mathstrut \) \(45\) \(\beta_{7}\mathstrut +\mathstrut \) \(86\) \(\beta_{6}\mathstrut -\mathstrut \) \(305\) \(\beta_{5}\mathstrut -\mathstrut \) \(25\) \(\beta_{4}\mathstrut +\mathstrut \) \(141\) \(\beta_{3}\mathstrut -\mathstrut \) \(35\) \(\beta_{2}\mathstrut +\mathstrut \) \(434\) \(\beta_{1}\mathstrut +\mathstrut \) \(213\)
\(\nu^{8}\)\(=\)\(445\) \(\beta_{8}\mathstrut +\mathstrut \) \(315\) \(\beta_{7}\mathstrut -\mathstrut \) \(35\) \(\beta_{6}\mathstrut -\mathstrut \) \(1165\) \(\beta_{5}\mathstrut +\mathstrut \) \(236\) \(\beta_{4}\mathstrut +\mathstrut \) \(715\) \(\beta_{3}\mathstrut +\mathstrut \) \(719\) \(\beta_{2}\mathstrut +\mathstrut \) \(224\) \(\beta_{1}\mathstrut +\mathstrut \) \(1591\)

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
3.20969
2.80812
1.90297
0.840786
0.435443
0.254984
−1.25914
−1.44298
−2.74988
1.00000 −1.00000 1.00000 −3.20969 −1.00000 −2.02210 1.00000 1.00000 −3.20969
1.2 1.00000 −1.00000 1.00000 −2.80812 −1.00000 −1.72587 1.00000 1.00000 −2.80812
1.3 1.00000 −1.00000 1.00000 −1.90297 −1.00000 −5.10863 1.00000 1.00000 −1.90297
1.4 1.00000 −1.00000 1.00000 −0.840786 −1.00000 2.87036 1.00000 1.00000 −0.840786
1.5 1.00000 −1.00000 1.00000 −0.435443 −1.00000 1.91967 1.00000 1.00000 −0.435443
1.6 1.00000 −1.00000 1.00000 −0.254984 −1.00000 2.89551 1.00000 1.00000 −0.254984
1.7 1.00000 −1.00000 1.00000 1.25914 −1.00000 −0.797445 1.00000 1.00000 1.25914
1.8 1.00000 −1.00000 1.00000 1.44298 −1.00000 0.367530 1.00000 1.00000 1.44298
1.9 1.00000 −1.00000 1.00000 2.74988 −1.00000 −2.39902 1.00000 1.00000 2.74988
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{9} + \cdots\)
\(T_{7}^{9} + \cdots\)