Properties

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

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

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
−1.41421
1.41421
−2.41421 −1.00000 3.82843 1.00000 2.41421 0 −4.41421 1.00000 −2.41421
1.2 0.414214 −1.00000 −1.82843 1.00000 −0.414214 0 −1.58579 1.00000 0.414214
\(n\): 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6027))\):

\( T_{2}^{2} + 2 T_{2} - 1 \)
\( T_{5} - 1 \)
\( T_{13}^{2} - 6 T_{13} + 7 \)