Properties

Label 6027.2.a.q
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 3
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: \(3\)
Coefficient field: 3.3.1620.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} -2 q^{4} + \beta_{1} q^{5} + q^{9} +O(q^{10})\) \( q + q^{3} -2 q^{4} + \beta_{1} q^{5} + q^{9} + ( -\beta_{1} + \beta_{2} ) q^{11} -2 q^{12} + ( 1 + \beta_{1} + \beta_{2} ) q^{13} + \beta_{1} q^{15} + 4 q^{16} + 2 q^{17} + ( 3 + \beta_{1} ) q^{19} -2 \beta_{1} q^{20} + ( 2 + \beta_{1} ) q^{23} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{25} + q^{27} + 2 \beta_{1} q^{29} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{31} + ( -\beta_{1} + \beta_{2} ) q^{33} -2 q^{36} - q^{37} + ( 1 + \beta_{1} + \beta_{2} ) q^{39} + q^{41} + ( -7 - \beta_{2} ) q^{43} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{44} + \beta_{1} q^{45} + ( \beta_{1} + 3 \beta_{2} ) q^{47} + 4 q^{48} + 2 q^{51} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{52} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{53} + ( -10 - 2 \beta_{1} - 3 \beta_{2} ) q^{55} + ( 3 + \beta_{1} ) q^{57} + ( -6 - 2 \beta_{1} - 2 \beta_{2} ) q^{59} -2 \beta_{1} q^{60} + ( -2 - 2 \beta_{1} - 3 \beta_{2} ) q^{61} -8 q^{64} + ( 6 + 3 \beta_{1} - \beta_{2} ) q^{65} + ( 1 - \beta_{1} - \beta_{2} ) q^{67} -4 q^{68} + ( 2 + \beta_{1} ) q^{69} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( 7 + \beta_{2} ) q^{73} + ( 3 + 2 \beta_{1} + \beta_{2} ) q^{75} + ( -6 - 2 \beta_{1} ) q^{76} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{79} + 4 \beta_{1} q^{80} + q^{81} + ( 2 + \beta_{1} + 4 \beta_{2} ) q^{83} + 2 \beta_{1} q^{85} + 2 \beta_{1} q^{87} + ( -4 - \beta_{1} - 3 \beta_{2} ) q^{89} + ( -4 - 2 \beta_{1} ) q^{92} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{93} + ( 8 + 5 \beta_{1} + \beta_{2} ) q^{95} + ( -6 + 4 \beta_{1} ) q^{97} + ( -\beta_{1} + \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{3} - 6q^{4} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{3} - 6q^{4} + 3q^{9} - 6q^{12} + 3q^{13} + 12q^{16} + 6q^{17} + 9q^{19} + 6q^{23} + 9q^{25} + 3q^{27} - 3q^{31} - 6q^{36} - 3q^{37} + 3q^{39} + 3q^{41} - 21q^{43} + 12q^{48} + 6q^{51} - 6q^{52} - 6q^{53} - 30q^{55} + 9q^{57} - 18q^{59} - 6q^{61} - 24q^{64} + 18q^{65} + 3q^{67} - 12q^{68} + 6q^{69} + 21q^{73} + 9q^{75} - 18q^{76} + 21q^{79} + 3q^{81} + 6q^{83} - 12q^{89} - 12q^{92} - 3q^{93} + 24q^{95} - 18q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3} - 12 x - 14\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 2 \nu - 8 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 2 \beta_{1} + 8\)

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
−2.55247
−1.39091
3.94338
0 1.00000 −2.00000 −2.55247 0 0 0 1.00000 0
1.2 0 1.00000 −2.00000 −1.39091 0 0 0 1.00000 0
1.3 0 1.00000 −2.00000 3.94338 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2} \)
\( T_{5}^{3} - 12 T_{5} - 14 \)
\( T_{13}^{3} - 3 T_{13}^{2} - 15 T_{13} + 35 \)