Properties

Label 6027.2.a.u
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 4
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: \(4\)
Coefficient field: 4.4.8468.1
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

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

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 4 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{3} + 4 \beta_{1} + 1\)

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
−1.89122
−0.704624
1.31743
2.27841
−1.89122 1.00000 1.57671 1.57671 −1.89122 0 0.800530 1.00000 −2.98191
1.2 −0.704624 1.00000 −1.50350 −1.50350 −0.704624 0 2.46865 1.00000 1.05941
1.3 1.31743 1.00000 −0.264377 −0.264377 1.31743 0 −2.98316 1.00000 −0.348298
1.4 2.27841 1.00000 3.19117 3.19117 2.27841 0 2.71397 1.00000 7.27080
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{4} - T_{2}^{3} - 5 T_{2}^{2} + 3 T_{2} + 4 \)
\( T_{5}^{4} - 3 T_{5}^{3} - 3 T_{5}^{2} + 7 T_{5} + 2 \)
\( T_{13}^{4} - 5 T_{13}^{3} - 13 T_{13}^{2} + 47 T_{13} + 88 \)