Properties

Label 6027.2.a.be
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 10
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6027.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1258372982\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10}\mathstrut -\mathstrut \) \(4\) \(x^{9}\mathstrut -\mathstrut \) \(11\) \(x^{8}\mathstrut +\mathstrut \) \(56\) \(x^{7}\mathstrut +\mathstrut \) \(26\) \(x^{6}\mathstrut -\mathstrut \) \(266\) \(x^{5}\mathstrut +\mathstrut \) \(52\) \(x^{4}\mathstrut +\mathstrut \) \(526\) \(x^{3}\mathstrut -\mathstrut \) \(255\) \(x^{2}\mathstrut -\mathstrut \) \(372\) \(x\mathstrut +\mathstrut \) \(239\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\( \nu^{8} - 4 \nu^{7} - 8 \nu^{6} + 46 \nu^{5} - 2 \nu^{4} - 152 \nu^{3} + 90 \nu^{2} + 154 \nu - 121 \)\()/2\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{9} - 4 \nu^{8} - 8 \nu^{7} + 46 \nu^{6} - 4 \nu^{5} - 148 \nu^{4} + 110 \nu^{3} + 120 \nu^{2} - 163 \nu + 52 \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{9} + 5 \nu^{8} + 4 \nu^{7} - 54 \nu^{6} + 50 \nu^{5} + 148 \nu^{4} - 262 \nu^{3} - 46 \nu^{2} + 315 \nu - 153 \)\()/2\)
\(\beta_{6}\)\(=\)\( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 47 \nu^{5} - 9 \nu^{4} + 160 \nu^{3} - 59 \nu^{2} - 165 \nu + 102 \)
\(\beta_{7}\)\(=\)\( \nu^{8} - 4 \nu^{7} - 9 \nu^{6} + 48 \nu^{5} + 8 \nu^{4} - 169 \nu^{3} + 67 \nu^{2} + 180 \nu - 114 \)
\(\beta_{8}\)\(=\)\( -\nu^{8} + 4 \nu^{7} + 9 \nu^{6} - 48 \nu^{5} - 8 \nu^{4} + 170 \nu^{3} - 68 \nu^{2} - 185 \nu + 117 \)
\(\beta_{9}\)\(=\)\((\)\( -2 \nu^{9} + 9 \nu^{8} + 14 \nu^{7} - 104 \nu^{6} + 28 \nu^{5} + 342 \nu^{4} - 268 \nu^{3} - 318 \nu^{2} + 348 \nu - 57 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(22\)
\(\nu^{5}\)\(=\)\(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(10\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(31\) \(\beta_{1}\mathstrut +\mathstrut \) \(11\)
\(\nu^{6}\)\(=\)\(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(12\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(58\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(140\)
\(\nu^{7}\)\(=\)\(\beta_{9}\mathstrut +\mathstrut \) \(67\) \(\beta_{8}\mathstrut +\mathstrut \) \(82\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(15\) \(\beta_{4}\mathstrut -\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(73\) \(\beta_{2}\mathstrut +\mathstrut \) \(211\) \(\beta_{1}\mathstrut +\mathstrut \) \(95\)
\(\nu^{8}\)\(=\)\(4\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut +\mathstrut \) \(36\) \(\beta_{7}\mathstrut +\mathstrut \) \(38\) \(\beta_{6}\mathstrut +\mathstrut \) \(104\) \(\beta_{5}\mathstrut +\mathstrut \) \(112\) \(\beta_{4}\mathstrut -\mathstrut \) \(74\) \(\beta_{3}\mathstrut +\mathstrut \) \(420\) \(\beta_{2}\mathstrut +\mathstrut \) \(130\) \(\beta_{1}\mathstrut +\mathstrut \) \(951\)
\(\nu^{9}\)\(=\)\(24\) \(\beta_{9}\mathstrut +\mathstrut \) \(472\) \(\beta_{8}\mathstrut +\mathstrut \) \(638\) \(\beta_{7}\mathstrut +\mathstrut \) \(200\) \(\beta_{6}\mathstrut +\mathstrut \) \(120\) \(\beta_{5}\mathstrut +\mathstrut \) \(170\) \(\beta_{4}\mathstrut -\mathstrut \) \(68\) \(\beta_{3}\mathstrut +\mathstrut \) \(586\) \(\beta_{2}\mathstrut +\mathstrut \) \(1495\) \(\beta_{1}\mathstrut +\mathstrut \) \(782\)

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
−2.58399
−1.69562
−1.49626
−1.44855
0.775610
1.32624
1.55584
2.13183
2.62834
2.80657
−2.58399 1.00000 4.67700 0.798812 −2.58399 0 −6.91733 1.00000 −2.06412
1.2 −1.69562 1.00000 0.875138 −0.871357 −1.69562 0 1.90734 1.00000 1.47749
1.3 −1.49626 1.00000 0.238800 −0.660548 −1.49626 0 2.63522 1.00000 0.988352
1.4 −1.44855 1.00000 0.0982847 3.93238 −1.44855 0 2.75472 1.00000 −5.69623
1.5 0.775610 1.00000 −1.39843 2.32372 0.775610 0 −2.63586 1.00000 1.80230
1.6 1.32624 1.00000 −0.241096 −0.903630 1.32624 0 −2.97222 1.00000 −1.19843
1.7 1.55584 1.00000 0.420641 −3.26436 1.55584 0 −2.45723 1.00000 −5.07882
1.8 2.13183 1.00000 2.54468 2.36072 2.13183 0 1.16116 1.00000 5.03264
1.9 2.62834 1.00000 4.90817 −1.82844 2.62834 0 7.64366 1.00000 −4.80575
1.10 2.80657 1.00000 5.87681 4.11270 2.80657 0 10.8805 1.00000 11.5426
\(n\): 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
\(3\) \(-1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{10} - \cdots\)
\(T_{5}^{10} - \cdots\)
\(T_{13}^{10} - \cdots\)