Properties

Label 6027.2.a.z
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 8
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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8}\mathstrut -\mathstrut \) \(2\) \(x^{7}\mathstrut -\mathstrut \) \(8\) \(x^{6}\mathstrut +\mathstrut \) \(14\) \(x^{5}\mathstrut +\mathstrut \) \(18\) \(x^{4}\mathstrut -\mathstrut \) \(24\) \(x^{3}\mathstrut -\mathstrut \) \(10\) \(x^{2}\mathstrut +\mathstrut \) \(10\) \(x\mathstrut -\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - \nu^{2} - 4 \nu + 2 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 6 \nu^{2} - \nu + 4 \)
\(\beta_{5}\)\(=\)\( \nu^{5} - 6 \nu^{3} - \nu^{2} + 5 \nu \)
\(\beta_{6}\)\(=\)\( \nu^{7} - \nu^{6} - 9 \nu^{5} + 6 \nu^{4} + 24 \nu^{3} - 7 \nu^{2} - 17 \nu + 2 \)
\(\beta_{7}\)\(=\)\( \nu^{7} - 2 \nu^{6} - 8 \nu^{5} + 13 \nu^{4} + 18 \nu^{3} - 18 \nu^{2} - 9 \nu + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(\beta_{5}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(9\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(32\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(70\)
\(\nu^{7}\)\(=\)\(-\)\(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(10\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(30\) \(\beta_{3}\mathstrut +\mathstrut \) \(42\) \(\beta_{2}\mathstrut +\mathstrut \) \(96\) \(\beta_{1}\mathstrut +\mathstrut \) \(62\)

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
2.39346
2.28873
1.17091
0.487949
0.117246
−0.896239
−1.41849
−2.14356
−2.39346 −1.00000 3.72866 −2.51820 2.39346 0 −4.13748 1.00000 6.02722
1.2 −2.28873 −1.00000 3.23826 2.92705 2.28873 0 −2.83405 1.00000 −6.69922
1.3 −1.17091 −1.00000 −0.628975 2.94759 1.17091 0 3.07829 1.00000 −3.45135
1.4 −0.487949 −1.00000 −1.76191 −1.53225 0.487949 0 1.83562 1.00000 0.747657
1.5 −0.117246 −1.00000 −1.98625 −0.562834 0.117246 0 0.467371 1.00000 0.0659898
1.6 0.896239 −1.00000 −1.19676 1.54330 −0.896239 0 −2.86506 1.00000 1.38317
1.7 1.41849 −1.00000 0.0121162 −2.27752 −1.41849 0 −2.81979 1.00000 −3.23064
1.8 2.14356 −1.00000 2.59485 1.47286 −2.14356 0 1.27510 1.00000 3.15717
\(n\): 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
\(3\) \(1\)
\(7\) \(-1\)
\(41\) \(-1\)

Hecke kernels

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

\(T_{2}^{8} + \cdots\)
\(T_{5}^{8} - \cdots\)
\(T_{13}^{8} - \cdots\)