Properties

Label 6027.2.a.y
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 7
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: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7}\mathstrut -\mathstrut \) \(3\) \(x^{6}\mathstrut -\mathstrut \) \(6\) \(x^{5}\mathstrut +\mathstrut \) \(16\) \(x^{4}\mathstrut +\mathstrut \) \(14\) \(x^{3}\mathstrut -\mathstrut \) \(20\) \(x^{2}\mathstrut -\mathstrut \) \(10\) \(x\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 2 \nu^{2} - 3 \nu + 3 \)
\(\beta_{4}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 4 \nu^{4} + 12 \nu^{3} + 4 \nu^{2} - 8 \nu \)\()/2\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{6} - 3 \nu^{5} - 6 \nu^{4} + 16 \nu^{3} + 12 \nu^{2} - 16 \nu - 4 \)\()/2\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{6} + 5 \nu^{5} - 2 \nu^{4} - 18 \nu^{3} + 14 \nu^{2} + 14 \nu - 6 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(10\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(\beta_{6}\mathstrut -\mathstrut \) \(3\) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(21\) \(\beta_{2}\mathstrut +\mathstrut \) \(33\) \(\beta_{1}\mathstrut +\mathstrut \) \(33\)
\(\nu^{6}\)\(=\)\(3\) \(\beta_{6}\mathstrut -\mathstrut \) \(13\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(23\) \(\beta_{3}\mathstrut +\mathstrut \) \(67\) \(\beta_{2}\mathstrut +\mathstrut \) \(83\) \(\beta_{1}\mathstrut +\mathstrut \) \(115\)

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
2.95889
2.35311
1.19009
0.281557
−0.762978
−1.32604
−1.69463
−1.95889 −1.00000 1.83724 0.513172 1.95889 0 0.318823 1.00000 −1.00525
1.2 −1.35311 −1.00000 −0.169102 −1.31916 1.35311 0 2.93503 1.00000 1.78497
1.3 −0.190092 −1.00000 −1.96387 −2.28332 0.190092 0 0.753499 1.00000 0.434041
1.4 0.718443 −1.00000 −1.48384 3.61951 −0.718443 0 −2.50294 1.00000 2.60041
1.5 1.76298 −1.00000 1.10809 −3.51322 −1.76298 0 −1.57241 1.00000 −6.19372
1.6 2.32604 −1.00000 3.41045 −0.0978008 −2.32604 0 3.28076 1.00000 −0.227488
1.7 2.69463 −1.00000 5.26102 2.08082 −2.69463 0 8.78724 1.00000 5.60704
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\(T_{2}^{7} \) \(\mathstrut -\mathstrut 4 T_{2}^{6} \) \(\mathstrut -\mathstrut 3 T_{2}^{5} \) \(\mathstrut +\mathstrut 24 T_{2}^{4} \) \(\mathstrut -\mathstrut 7 T_{2}^{3} \) \(\mathstrut -\mathstrut 34 T_{2}^{2} \) \(\mathstrut +\mathstrut 15 T_{2} \) \(\mathstrut +\mathstrut 4 \)
\(T_{5}^{7} \) \(\mathstrut +\mathstrut T_{5}^{6} \) \(\mathstrut -\mathstrut 18 T_{5}^{5} \) \(\mathstrut -\mathstrut 18 T_{5}^{4} \) \(\mathstrut +\mathstrut 69 T_{5}^{3} \) \(\mathstrut +\mathstrut 57 T_{5}^{2} \) \(\mathstrut -\mathstrut 36 T_{5} \) \(\mathstrut -\mathstrut 4 \)
\(T_{13}^{7} \) \(\mathstrut -\mathstrut 7 T_{13}^{6} \) \(\mathstrut -\mathstrut 14 T_{13}^{5} \) \(\mathstrut +\mathstrut 114 T_{13}^{4} \) \(\mathstrut -\mathstrut 83 T_{13}^{3} \) \(\mathstrut -\mathstrut 223 T_{13}^{2} \) \(\mathstrut +\mathstrut 294 T_{13} \) \(\mathstrut -\mathstrut 72 \)