Properties

Label 6027.2.a.j
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.

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

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 −0.561553 2.56155 0 −6.56155 1.00000 1.43845
1.2 1.56155 −1.00000 0.438447 3.56155 −1.56155 0 −2.43845 1.00000 5.56155
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

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