Properties

Label 6027.2.a.v
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
Dimension 5
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: \(5\)
Coefficient field: 5.5.981328.1
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{4} - 2 \nu^{3} - 5 \nu^{2} + 6 \nu + 5 \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 3 \nu^{3} - 4 \nu^{2} + 10 \nu + 5 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(-\beta_{4} + \beta_{3} + \beta_{2} + 5 \beta_{1} + 3\)
\(\nu^{4}\)\(=\)\(-2 \beta_{4} + 3 \beta_{3} + 7 \beta_{2} + 9 \beta_{1} + 16\)

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
−1.74401
−0.844325
−0.243417
1.90925
2.92250
−2.74401 1.00000 5.52961 0.811635 −2.74401 0 −9.68531 1.00000 −2.22714
1.2 −1.84433 1.00000 1.40154 1.91836 −1.84433 0 1.10376 1.00000 −3.53808
1.3 −1.24342 1.00000 −0.453913 −3.27559 −1.24342 0 3.05124 1.00000 4.07293
1.4 0.909252 1.00000 −1.17326 2.40229 0.909252 0 −2.88529 1.00000 2.18428
1.5 1.92250 1.00000 1.69602 −2.85669 1.92250 0 −0.584397 1.00000 −5.49199
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{5} + 3 T_{2}^{4} - 4 T_{2}^{3} - 14 T_{2}^{2} + T_{2} + 11 \)
\( T_{5}^{5} + T_{5}^{4} - 14 T_{5}^{3} - 2 T_{5}^{2} + 53 T_{5} - 35 \)
\( T_{13}^{5} + 3 T_{13}^{4} - 18 T_{13}^{3} - 26 T_{13}^{2} + 91 T_{13} - 1 \)