Properties

Label 6027.2.a.s
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 1
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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)

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
−1.81361
0.470683
2.34292
−1.81361 1.00000 1.28917 1.10278 −1.81361 0 1.28917 1.00000 −2.00000
1.2 0.470683 1.00000 −1.77846 −4.24914 0.470683 0 −1.77846 1.00000 −2.00000
1.3 2.34292 1.00000 3.48929 −0.853635 2.34292 0 3.48929 1.00000 −2.00000
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{3} - T_{2}^{2} - 4 T_{2} + 2 \)
\( T_{5}^{3} + 4 T_{5}^{2} - 2 T_{5} - 4 \)
\( T_{13}^{3} + 8 T_{13}^{2} + 14 T_{13} - 4 \)