Properties

Label 8034.2.a.p
Level 8034
Weight 2
Character orbit 8034.a
Self dual Yes
Analytic conductor 64.152
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(x^{7}\mathstrut -\mathstrut \) \(12\) \(x^{6}\mathstrut +\mathstrut \) \(12\) \(x^{5}\mathstrut +\mathstrut \) \(43\) \(x^{4}\mathstrut -\mathstrut \) \(38\) \(x^{3}\mathstrut -\mathstrut \) \(49\) \(x^{2}\mathstrut +\mathstrut \) \(23\) \(x\mathstrut +\mathstrut \) \(20\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{6} - 11 \nu^{4} + \nu^{3} + 32 \nu^{2} - 5 \nu - 17 \)
\(\beta_{3}\)\(=\)\( 3 \nu^{7} + \nu^{6} - 32 \nu^{5} - 6 \nu^{4} + 92 \nu^{3} + 7 \nu^{2} - 53 \nu - 11 \)
\(\beta_{4}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 21 \nu^{5} - 18 \nu^{4} + 60 \nu^{3} + 43 \nu^{2} - 37 \nu - 25 \)
\(\beta_{5}\)\(=\)\( 2 \nu^{7} + 2 \nu^{6} - 21 \nu^{5} - 18 \nu^{4} + 60 \nu^{3} + 44 \nu^{2} - 37 \nu - 28 \)
\(\beta_{6}\)\(=\)\( \nu^{7} - 3 \nu^{6} - 11 \nu^{5} + 33 \nu^{4} + 29 \nu^{3} - 95 \nu^{2} - 2 \nu + 46 \)
\(\beta_{7}\)\(=\)\( 2 \nu^{7} + 3 \nu^{6} - 21 \nu^{5} - 29 \nu^{4} + 62 \nu^{3} + 76 \nu^{2} - 47 \nu - 45 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(6\) \(\beta_{5}\mathstrut -\mathstrut \) \(6\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(13\)
\(\nu^{5}\)\(=\)\(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut -\mathstrut \) \(4\) \(\beta_{3}\mathstrut -\mathstrut \) \(10\) \(\beta_{2}\mathstrut +\mathstrut \) \(27\) \(\beta_{1}\mathstrut -\mathstrut \) \(2\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{7}\mathstrut -\mathstrut \) \(11\) \(\beta_{6}\mathstrut +\mathstrut \) \(35\) \(\beta_{5}\mathstrut -\mathstrut \) \(34\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(64\)
\(\nu^{7}\)\(=\)\(78\) \(\beta_{7}\mathstrut +\mathstrut \) \(23\) \(\beta_{6}\mathstrut -\mathstrut \) \(46\) \(\beta_{5}\mathstrut +\mathstrut \) \(23\) \(\beta_{4}\mathstrut -\mathstrut \) \(44\) \(\beta_{3}\mathstrut -\mathstrut \) \(75\) \(\beta_{2}\mathstrut +\mathstrut \) \(154\) \(\beta_{1}\mathstrut -\mathstrut \) \(20\)

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
1.82602
−0.685988
0.965971
2.41269
−2.01106
−0.735807
−2.57192
1.80009
1.00000 1.00000 1.00000 −3.53629 1.00000 0.738654 1.00000 1.00000 −3.53629
1.2 1.00000 1.00000 1.00000 −3.08319 1.00000 1.60046 1.00000 1.00000 −3.08319
1.3 1.00000 1.00000 1.00000 −2.24957 1.00000 3.57309 1.00000 1.00000 −2.24957
1.4 1.00000 1.00000 1.00000 −1.15050 1.00000 −2.62035 1.00000 1.00000 −1.15050
1.5 1.00000 1.00000 1.00000 −0.612396 1.00000 −3.48353 1.00000 1.00000 −0.612396
1.6 1.00000 1.00000 1.00000 −0.216275 1.00000 −0.843680 1.00000 1.00000 −0.216275
1.7 1.00000 1.00000 1.00000 −0.176895 1.00000 −2.27129 1.00000 1.00000 −0.176895
1.8 1.00000 1.00000 1.00000 3.02511 1.00000 −2.69336 1.00000 1.00000 3.02511
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8034))\):

\(T_{5}^{8} + \cdots\)
\(T_{7}^{8} + \cdots\)