Properties

Label 6027.2.a.bf
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 12
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: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{12}\mathstrut -\mathstrut \) \(2\) \(x^{11}\mathstrut -\mathstrut \) \(15\) \(x^{10}\mathstrut +\mathstrut \) \(30\) \(x^{9}\mathstrut +\mathstrut \) \(74\) \(x^{8}\mathstrut -\mathstrut \) \(149\) \(x^{7}\mathstrut -\mathstrut \) \(140\) \(x^{6}\mathstrut +\mathstrut \) \(278\) \(x^{5}\mathstrut +\mathstrut \) \(126\) \(x^{4}\mathstrut -\mathstrut \) \(211\) \(x^{3}\mathstrut -\mathstrut \) \(64\) \(x^{2}\mathstrut +\mathstrut \) \(53\) \(x\mathstrut +\mathstrut \) \(18\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{6} - 2 \nu^{5} - 8 \nu^{4} + 14 \nu^{3} + 12 \nu^{2} - 13 \nu - 6 \)
\(\beta_{4}\)\(=\)\( \nu^{9} - 2 \nu^{8} - 13 \nu^{7} + 27 \nu^{6} + 47 \nu^{5} - 109 \nu^{4} - 30 \nu^{3} + 115 \nu^{2} - 9 \nu - 25 \)
\(\beta_{5}\)\(=\)\( \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} + 60 \nu^{7} - 121 \nu^{6} - 80 \nu^{5} + 158 \nu^{4} + 45 \nu^{3} - 61 \nu^{2} - 12 \nu + 3 \)
\(\beta_{6}\)\(=\)\( -\nu^{10} + 2 \nu^{9} + 14 \nu^{8} - 28 \nu^{7} - 60 \nu^{6} + 121 \nu^{5} + 81 \nu^{4} - 158 \nu^{3} - 53 \nu^{2} + 60 \nu + 23 \)
\(\beta_{7}\)\(=\)\( \nu^{11} - 2 \nu^{10} - 14 \nu^{9} + 28 \nu^{8} + 60 \nu^{7} - 122 \nu^{6} - 79 \nu^{5} + 167 \nu^{4} + 39 \nu^{3} - 79 \nu^{2} - 10 \nu + 11 \)
\(\beta_{8}\)\(=\)\( \nu^{11} - \nu^{10} - 16 \nu^{9} + 15 \nu^{8} + 87 \nu^{7} - 75 \nu^{6} - 188 \nu^{5} + 137 \nu^{4} + 154 \nu^{3} - 87 \nu^{2} - 35 \nu + 9 \)
\(\beta_{9}\)\(=\)\( \nu^{11} - 4 \nu^{10} - 11 \nu^{9} + 58 \nu^{8} + 18 \nu^{7} - 269 \nu^{6} + 104 \nu^{5} + 435 \nu^{4} - 207 \nu^{3} - 281 \nu^{2} + 87 \nu + 62 \)
\(\beta_{10}\)\(=\)\( -2 \nu^{11} + 4 \nu^{10} + 29 \nu^{9} - 59 \nu^{8} - 132 \nu^{7} + 283 \nu^{6} + 194 \nu^{5} - 484 \nu^{4} - 71 \nu^{3} + 309 \nu^{2} - 21 \nu - 54 \)
\(\beta_{11}\)\(=\)\( 3 \nu^{11} - 5 \nu^{10} - 45 \nu^{9} + 73 \nu^{8} + 220 \nu^{7} - 345 \nu^{6} - 394 \nu^{5} + 571 \nu^{4} + 267 \nu^{3} - 342 \nu^{2} - 44 \nu + 48 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(-\)\(8\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(9\) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut -\mathstrut \) \(9\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(20\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(-\)\(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut +\mathstrut \) \(12\) \(\beta_{8}\mathstrut +\mathstrut \) \(10\) \(\beta_{6}\mathstrut +\mathstrut \) \(14\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(46\) \(\beta_{2}\mathstrut -\mathstrut \) \(11\) \(\beta_{1}\mathstrut +\mathstrut \) \(88\)
\(\nu^{7}\)\(=\)\(-\)\(56\) \(\beta_{11}\mathstrut +\mathstrut \) \(15\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(71\) \(\beta_{8}\mathstrut +\mathstrut \) \(43\) \(\beta_{7}\mathstrut +\mathstrut \) \(13\) \(\beta_{6}\mathstrut +\mathstrut \) \(83\) \(\beta_{5}\mathstrut -\mathstrut \) \(70\) \(\beta_{4}\mathstrut -\mathstrut \) \(12\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(109\) \(\beta_{1}\mathstrut -\mathstrut \) \(4\)
\(\nu^{8}\)\(=\)\(-\)\(29\) \(\beta_{11}\mathstrut +\mathstrut \) \(82\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(112\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(81\) \(\beta_{6}\mathstrut +\mathstrut \) \(137\) \(\beta_{5}\mathstrut -\mathstrut \) \(110\) \(\beta_{4}\mathstrut -\mathstrut \) \(13\) \(\beta_{3}\mathstrut +\mathstrut \) \(304\) \(\beta_{2}\mathstrut -\mathstrut \) \(86\) \(\beta_{1}\mathstrut +\mathstrut \) \(549\)
\(\nu^{9}\)\(=\)\(-\)\(386\) \(\beta_{11}\mathstrut +\mathstrut \) \(151\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(539\) \(\beta_{8}\mathstrut +\mathstrut \) \(262\) \(\beta_{7}\mathstrut +\mathstrut \) \(123\) \(\beta_{6}\mathstrut +\mathstrut \) \(644\) \(\beta_{5}\mathstrut -\mathstrut \) \(521\) \(\beta_{4}\mathstrut -\mathstrut \) \(108\) \(\beta_{3}\mathstrut +\mathstrut \) \(149\) \(\beta_{2}\mathstrut +\mathstrut \) \(622\) \(\beta_{1}\mathstrut +\mathstrut \) \(32\)
\(\nu^{10}\)\(=\)\(-\)\(300\) \(\beta_{11}\mathstrut +\mathstrut \) \(632\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut +\mathstrut \) \(950\) \(\beta_{8}\mathstrut +\mathstrut \) \(23\) \(\beta_{7}\mathstrut +\mathstrut \) \(617\) \(\beta_{6}\mathstrut +\mathstrut \) \(1175\) \(\beta_{5}\mathstrut -\mathstrut \) \(914\) \(\beta_{4}\mathstrut -\mathstrut \) \(123\) \(\beta_{3}\mathstrut +\mathstrut \) \(2037\) \(\beta_{2}\mathstrut -\mathstrut \) \(585\) \(\beta_{1}\mathstrut +\mathstrut \) \(3540\)
\(\nu^{11}\)\(=\)\(-\)\(2669\) \(\beta_{11}\mathstrut +\mathstrut \) \(1314\) \(\beta_{10}\mathstrut +\mathstrut \) \(154\) \(\beta_{9}\mathstrut +\mathstrut \) \(4019\) \(\beta_{8}\mathstrut +\mathstrut \) \(1621\) \(\beta_{7}\mathstrut +\mathstrut \) \(1040\) \(\beta_{6}\mathstrut +\mathstrut \) \(4842\) \(\beta_{5}\mathstrut -\mathstrut \) \(3811\) \(\beta_{4}\mathstrut -\mathstrut \) \(875\) \(\beta_{3}\mathstrut +\mathstrut \) \(1409\) \(\beta_{2}\mathstrut +\mathstrut \) \(3665\) \(\beta_{1}\mathstrut +\mathstrut \) \(774\)

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
2.71780
2.11101
1.94261
1.22157
1.14271
0.837060
−0.420502
−0.467496
−1.05133
−1.09852
−2.44902
−2.48589
−2.71780 −1.00000 5.38643 −0.137890 2.71780 0 −9.20364 1.00000 0.374758
1.2 −2.11101 −1.00000 2.45637 2.44925 2.11101 0 −0.963413 1.00000 −5.17041
1.3 −1.94261 −1.00000 1.77372 −2.73840 1.94261 0 0.439573 1.00000 5.31964
1.4 −1.22157 −1.00000 −0.507765 −0.145411 1.22157 0 3.06341 1.00000 0.177630
1.5 −1.14271 −1.00000 −0.694209 −1.38626 1.14271 0 3.07871 1.00000 1.58409
1.6 −0.837060 −1.00000 −1.29933 −2.51468 0.837060 0 2.76174 1.00000 2.10494
1.7 0.420502 −1.00000 −1.82318 1.49913 −0.420502 0 −1.60765 1.00000 0.630387
1.8 0.467496 −1.00000 −1.78145 −2.48991 −0.467496 0 −1.76781 1.00000 −1.16402
1.9 1.05133 −1.00000 −0.894704 −4.26226 −1.05133 0 −3.04329 1.00000 −4.48105
1.10 1.09852 −1.00000 −0.793257 3.51189 −1.09852 0 −3.06844 1.00000 3.85788
1.11 2.44902 −1.00000 3.99771 −4.01828 −2.44902 0 4.89242 1.00000 −9.84085
1.12 2.48589 −1.00000 4.17966 −1.76717 −2.48589 0 5.41840 1.00000 −4.39300
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{12} + \cdots\)
\(T_{5}^{12} + \cdots\)
\(T_{13}^{12} + \cdots\)