Properties

Label 6018.2.a.u
Level 6018
Weight 2
Character orbit 6018.a
Self dual Yes
Analytic conductor 48.054
Analytic rank 1
Dimension 9
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6018.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.0539719364\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{9}\mathstrut -\mathstrut \) \(4\) \(x^{8}\mathstrut -\mathstrut \) \(16\) \(x^{7}\mathstrut +\mathstrut \) \(37\) \(x^{6}\mathstrut +\mathstrut \) \(97\) \(x^{5}\mathstrut -\mathstrut \) \(72\) \(x^{4}\mathstrut -\mathstrut \) \(182\) \(x^{3}\mathstrut +\mathstrut \) \(24\) \(x^{2}\mathstrut +\mathstrut \) \(70\) \(x\mathstrut -\mathstrut \) \(19\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -7 \nu^{8} - 595 \nu^{7} + 3093 \nu^{6} + 7114 \nu^{5} - 28461 \nu^{4} - 34541 \nu^{3} + 63957 \nu^{2} + 48357 \nu - 30285 \)\()/1472\)
\(\beta_{3}\)\(=\)\((\)\( 77 \nu^{8} - 79 \nu^{7} - 2375 \nu^{6} + 498 \nu^{5} + 17199 \nu^{4} + 6063 \nu^{3} - 33031 \nu^{2} - 17463 \nu + 10767 \)\()/736\)
\(\beta_{4}\)\(=\)\((\)\( 163 \nu^{8} - 865 \nu^{7} - 1577 \nu^{6} + 8462 \nu^{5} + 6433 \nu^{4} - 23583 \nu^{3} - 7401 \nu^{2} + 17927 \nu - 3455 \)\()/1472\)
\(\beta_{5}\)\(=\)\((\)\( 211 \nu^{8} + 271 \nu^{7} - 8697 \nu^{6} - 5202 \nu^{5} + 68273 \nu^{4} + 47985 \nu^{3} - 136633 \nu^{2} - 84873 \nu + 49233 \)\()/1472\)
\(\beta_{6}\)\(=\)\((\)\( -205 \nu^{8} + 975 \nu^{7} + 2471 \nu^{6} - 9202 \nu^{5} - 11599 \nu^{4} + 20945 \nu^{3} + 14311 \nu^{2} - 10409 \nu + 2065 \)\()/736\)
\(\beta_{7}\)\(=\)\((\)\( 705 \nu^{8} - 3371 \nu^{7} - 8067 \nu^{6} + 29402 \nu^{5} + 39979 \nu^{4} - 56341 \nu^{3} - 59843 \nu^{2} + 10701 \nu + 3723 \)\()/1472\)
\(\beta_{8}\)\(=\)\((\)\( -511 \nu^{8} + 2197 \nu^{7} + 7197 \nu^{6} - 19430 \nu^{5} - 40405 \nu^{4} + 33899 \nu^{3} + 67389 \nu^{2} + 941 \nu - 13109 \)\()/736\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(4\) \(\beta_{8}\mathstrut +\mathstrut \) \(4\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(3\) \(\beta_{4}\mathstrut +\mathstrut \) \(5\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(17\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(24\) \(\beta_{8}\mathstrut +\mathstrut \) \(23\) \(\beta_{7}\mathstrut -\mathstrut \) \(7\) \(\beta_{6}\mathstrut -\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(21\) \(\beta_{4}\mathstrut +\mathstrut \) \(26\) \(\beta_{3}\mathstrut -\mathstrut \) \(18\) \(\beta_{2}\mathstrut +\mathstrut \) \(79\) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)
\(\nu^{5}\)\(=\)\(120\) \(\beta_{8}\mathstrut +\mathstrut \) \(113\) \(\beta_{7}\mathstrut -\mathstrut \) \(48\) \(\beta_{6}\mathstrut -\mathstrut \) \(63\) \(\beta_{5}\mathstrut +\mathstrut \) \(94\) \(\beta_{4}\mathstrut +\mathstrut \) \(133\) \(\beta_{3}\mathstrut -\mathstrut \) \(112\) \(\beta_{2}\mathstrut +\mathstrut \) \(404\) \(\beta_{1}\mathstrut +\mathstrut \) \(64\)
\(\nu^{6}\)\(=\)\(635\) \(\beta_{8}\mathstrut +\mathstrut \) \(592\) \(\beta_{7}\mathstrut -\mathstrut \) \(268\) \(\beta_{6}\mathstrut -\mathstrut \) \(326\) \(\beta_{5}\mathstrut +\mathstrut \) \(506\) \(\beta_{4}\mathstrut +\mathstrut \) \(674\) \(\beta_{3}\mathstrut -\mathstrut \) \(606\) \(\beta_{2}\mathstrut +\mathstrut \) \(2035\) \(\beta_{1}\mathstrut +\mathstrut \) \(328\)
\(\nu^{7}\)\(=\)\(3264\) \(\beta_{8}\mathstrut +\mathstrut \) \(3020\) \(\beta_{7}\mathstrut -\mathstrut \) \(1482\) \(\beta_{6}\mathstrut -\mathstrut \) \(1747\) \(\beta_{5}\mathstrut +\mathstrut \) \(2529\) \(\beta_{4}\mathstrut +\mathstrut \) \(3459\) \(\beta_{3}\mathstrut -\mathstrut \) \(3256\) \(\beta_{2}\mathstrut +\mathstrut \) \(10419\) \(\beta_{1}\mathstrut +\mathstrut \) \(1431\)
\(\nu^{8}\)\(=\)\(16912\) \(\beta_{8}\mathstrut +\mathstrut \) \(15604\) \(\beta_{7}\mathstrut -\mathstrut \) \(7834\) \(\beta_{6}\mathstrut -\mathstrut \) \(9049\) \(\beta_{5}\mathstrut +\mathstrut \) \(13096\) \(\beta_{4}\mathstrut +\mathstrut \) \(17715\) \(\beta_{3}\mathstrut -\mathstrut \) \(17051\) \(\beta_{2}\mathstrut +\mathstrut \) \(53374\) \(\beta_{1}\mathstrut +\mathstrut \) \(7168\)

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
−1.12576
0.443000
2.85528
0.342793
−2.14979
−1.32747
1.53430
−1.72522
5.15287
−1.00000 −1.00000 1.00000 −3.14251 1.00000 −1.86106 −1.00000 1.00000 3.14251
1.2 −1.00000 −1.00000 1.00000 −2.03963 1.00000 −4.77480 −1.00000 1.00000 2.03963
1.3 −1.00000 −1.00000 1.00000 −1.56975 1.00000 1.56796 −1.00000 1.00000 1.56975
1.4 −1.00000 −1.00000 1.00000 −1.11819 1.00000 −1.49540 −1.00000 1.00000 1.11819
1.5 −1.00000 −1.00000 1.00000 −1.03642 1.00000 2.58814 −1.00000 1.00000 1.03642
1.6 −1.00000 −1.00000 1.00000 2.06239 1.00000 1.71252 −1.00000 1.00000 −2.06239
1.7 −1.00000 −1.00000 1.00000 2.21366 1.00000 −2.01100 −1.00000 1.00000 −2.21366
1.8 −1.00000 −1.00000 1.00000 3.17259 1.00000 −4.61193 −1.00000 1.00000 −3.17259
1.9 −1.00000 −1.00000 1.00000 3.45787 1.00000 3.88559 −1.00000 1.00000 −3.45787
\(n\): 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
\(2\) \(1\)
\(3\) \(1\)
\(17\) \(-1\)
\(59\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6018))\):

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