Properties

Label 6027.2.a.r
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: 3.3.785.1
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.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 4\)

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
−2.38849
0.812716
2.57577
−2.38849 −1.00000 3.70488 −1.00000 2.38849 0 −4.07210 1.00000 2.38849
1.2 0.812716 −1.00000 −1.33949 −1.00000 −0.812716 0 −2.71406 1.00000 −0.812716
1.3 2.57577 −1.00000 4.63461 −1.00000 −2.57577 0 6.78616 1.00000 −2.57577
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{3} - T_{2}^{2} - 6 T_{2} + 5 \)
\( T_{5} + 1 \)
\( T_{13}^{3} + 8 T_{13}^{2} + 15 T_{13} + 5 \)