Properties

Label 8034.2.a.s
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 1
Dimension 10
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(4\) \(x^{9}\mathstrut -\mathstrut \) \(15\) \(x^{8}\mathstrut +\mathstrut \) \(72\) \(x^{7}\mathstrut -\mathstrut \) \(27\) \(x^{6}\mathstrut -\mathstrut \) \(115\) \(x^{5}\mathstrut +\mathstrut \) \(54\) \(x^{4}\mathstrut +\mathstrut \) \(68\) \(x^{3}\mathstrut -\mathstrut \) \(15\) \(x^{2}\mathstrut -\mathstrut \) \(15\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 20 \nu^{9} - 168 \nu^{8} - 115 \nu^{7} + 3250 \nu^{6} - 3919 \nu^{5} - 8561 \nu^{4} + 7648 \nu^{3} + 7872 \nu^{2} - 3054 \nu - 1728 \)\()/163\)
\(\beta_{3}\)\(=\)\((\)\( -54 \nu^{9} - 68 \nu^{8} + 1696 \nu^{7} + 1005 \nu^{6} - 14211 \nu^{5} + 2593 \nu^{4} + 18177 \nu^{3} - 1890 \nu^{2} - 5055 \nu - 94 \)\()/163\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{9} + 369 \nu^{8} - 804 \nu^{7} - 6760 \nu^{6} + 17110 \nu^{5} + 11815 \nu^{4} - 27696 \nu^{3} - 9932 \nu^{2} + 10486 \nu + 3027 \)\()/163\)
\(\beta_{5}\)\(=\)\((\)\( 114 \nu^{9} - 599 \nu^{8} - 1226 \nu^{7} + 10538 \nu^{6} - 11564 \nu^{5} - 12628 \nu^{4} + 16340 \nu^{3} + 6598 \nu^{2} - 4922 \nu - 1830 \)\()/163\)
\(\beta_{6}\)\(=\)\((\)\( 159 \nu^{9} - 651 \nu^{8} - 2259 \nu^{7} + 11412 \nu^{6} - 6323 \nu^{5} - 13349 \nu^{4} + 7957 \nu^{3} + 5402 \nu^{2} - 1280 \nu - 665 \)\()/163\)
\(\beta_{7}\)\(=\)\((\)\( 251 \nu^{9} - 1163 \nu^{8} - 3114 \nu^{7} + 20331 \nu^{6} - 18189 \nu^{5} - 22542 \nu^{4} + 26903 \nu^{3} + 9111 \nu^{2} - 9167 \nu - 2322 \)\()/163\)
\(\beta_{8}\)\(=\)\((\)\( -338 \nu^{9} + 1307 \nu^{8} + 5122 \nu^{7} - 23303 \nu^{6} + 8252 \nu^{5} + 33629 \nu^{4} - 17531 \nu^{3} - 14601 \nu^{2} + 6103 \nu + 1591 \)\()/163\)
\(\beta_{9}\)\(=\)\((\)\( 481 \nu^{9} - 1791 \nu^{8} - 7452 \nu^{7} + 31789 \nu^{6} - 8734 \nu^{5} - 43813 \nu^{4} + 18685 \nu^{3} + 17813 \nu^{2} - 5657 \nu - 2145 \)\()/163\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\)
\(\nu^{3}\)\(=\)\(-\)\(3\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(12\) \(\beta_{1}\mathstrut -\mathstrut \) \(5\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(6\) \(\beta_{6}\mathstrut +\mathstrut \) \(29\) \(\beta_{5}\mathstrut -\mathstrut \) \(21\) \(\beta_{4}\mathstrut -\mathstrut \) \(18\) \(\beta_{3}\mathstrut -\mathstrut \) \(38\) \(\beta_{2}\mathstrut -\mathstrut \) \(6\) \(\beta_{1}\mathstrut +\mathstrut \) \(67\)
\(\nu^{5}\)\(=\)\(-\)\(50\) \(\beta_{9}\mathstrut -\mathstrut \) \(47\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(70\) \(\beta_{6}\mathstrut -\mathstrut \) \(35\) \(\beta_{5}\mathstrut -\mathstrut \) \(47\) \(\beta_{4}\mathstrut -\mathstrut \) \(38\) \(\beta_{3}\mathstrut -\mathstrut \) \(60\) \(\beta_{2}\mathstrut +\mathstrut \) \(182\) \(\beta_{1}\mathstrut -\mathstrut \) \(106\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{9}\mathstrut -\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(299\) \(\beta_{7}\mathstrut +\mathstrut \) \(157\) \(\beta_{6}\mathstrut +\mathstrut \) \(419\) \(\beta_{5}\mathstrut -\mathstrut \) \(388\) \(\beta_{4}\mathstrut -\mathstrut \) \(309\) \(\beta_{3}\mathstrut -\mathstrut \) \(655\) \(\beta_{2}\mathstrut -\mathstrut \) \(129\) \(\beta_{1}\mathstrut +\mathstrut \) \(1040\)
\(\nu^{7}\)\(=\)\(-\)\(807\) \(\beta_{9}\mathstrut -\mathstrut \) \(728\) \(\beta_{8}\mathstrut -\mathstrut \) \(31\) \(\beta_{7}\mathstrut +\mathstrut \) \(1326\) \(\beta_{6}\mathstrut -\mathstrut \) \(734\) \(\beta_{5}\mathstrut -\mathstrut \) \(910\) \(\beta_{4}\mathstrut -\mathstrut \) \(674\) \(\beta_{3}\mathstrut -\mathstrut \) \(1093\) \(\beta_{2}\mathstrut +\mathstrut \) \(2916\) \(\beta_{1}\mathstrut -\mathstrut \) \(1863\)
\(\nu^{8}\)\(=\)\(-\)\(176\) \(\beta_{9}\mathstrut +\mathstrut \) \(60\) \(\beta_{8}\mathstrut -\mathstrut \) \(4970\) \(\beta_{7}\mathstrut +\mathstrut \) \(3126\) \(\beta_{6}\mathstrut +\mathstrut \) \(6429\) \(\beta_{5}\mathstrut -\mathstrut \) \(6818\) \(\beta_{4}\mathstrut -\mathstrut \) \(5256\) \(\beta_{3}\mathstrut -\mathstrut \) \(11136\) \(\beta_{2}\mathstrut -\mathstrut \) \(2220\) \(\beta_{1}\mathstrut +\mathstrut \) \(16758\)
\(\nu^{9}\)\(=\)\(-\)\(13247\) \(\beta_{9}\mathstrut -\mathstrut \) \(11685\) \(\beta_{8}\mathstrut -\mathstrut \) \(846\) \(\beta_{7}\mathstrut +\mathstrut \) \(23508\) \(\beta_{6}\mathstrut -\mathstrut \) \(13154\) \(\beta_{5}\mathstrut -\mathstrut \) \(16494\) \(\beta_{4}\mathstrut -\mathstrut \) \(11806\) \(\beta_{3}\mathstrut -\mathstrut \) \(19470\) \(\beta_{2}\mathstrut +\mathstrut \) \(47740\) \(\beta_{1}\mathstrut -\mathstrut \) \(31006\)

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
−0.297898
4.13381
−0.671383
0.757747
−0.207472
1.27214
−0.894486
2.73145
1.22324
−4.04714
−1.00000 −1.00000 1.00000 −4.09900 1.00000 −3.00074 −1.00000 1.00000 4.09900
1.2 −1.00000 −1.00000 1.00000 −2.24210 1.00000 −4.80624 −1.00000 1.00000 2.24210
1.3 −1.00000 −1.00000 1.00000 −1.82988 1.00000 1.45707 −1.00000 1.00000 1.82988
1.4 −1.00000 −1.00000 1.00000 −0.284726 1.00000 −1.57147 −1.00000 1.00000 0.284726
1.5 −1.00000 −1.00000 1.00000 0.448250 1.00000 1.73996 −1.00000 1.00000 −0.448250
1.6 −1.00000 −1.00000 1.00000 1.40775 1.00000 −4.52824 −1.00000 1.00000 −1.40775
1.7 −1.00000 −1.00000 1.00000 2.08098 1.00000 1.85452 −1.00000 1.00000 −2.08098
1.8 −1.00000 −1.00000 1.00000 2.66497 1.00000 2.36859 −1.00000 1.00000 −2.66497
1.9 −1.00000 −1.00000 1.00000 3.54892 1.00000 0.812393 −1.00000 1.00000 −3.54892
1.10 −1.00000 −1.00000 1.00000 4.30484 1.00000 −3.32584 −1.00000 1.00000 −4.30484
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{10} - \cdots\)
\(T_{7}^{10} + \cdots\)