Properties

Label 6027.2.a.k
Level 6027
Weight 2
Character orbit 6027.a
Self dual Yes
Analytic conductor 48.126
Analytic rank 0
Dimension 2
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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.

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

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
2.56155
−1.56155
−2.56155 1.00000 4.56155 −2.00000 −2.56155 0 −6.56155 1.00000 5.12311
1.2 1.56155 1.00000 0.438447 −2.00000 1.56155 0 −2.43845 1.00000 −3.12311
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{2} + T_{2} - 4 \)
\( T_{5} + 2 \)
\( T_{13}^{2} + 2 T_{13} - 16 \)