Properties

Label 6026.2.a.e
Level 6026
Weight 2
Character orbit 6026.a
Self dual Yes
Analytic conductor 48.118
Analytic rank 0
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6026 = 2 \cdot 23 \cdot 131 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6026.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.117852258\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} + 2 q^{7} + q^{8} + ( 1 + 2 \beta ) q^{9} +O(q^{10})\) \( q + q^{2} + ( 1 + \beta ) q^{3} + q^{4} + \beta q^{5} + ( 1 + \beta ) q^{6} + 2 q^{7} + q^{8} + ( 1 + 2 \beta ) q^{9} + \beta q^{10} + \beta q^{11} + ( 1 + \beta ) q^{12} + ( -1 - \beta ) q^{13} + 2 q^{14} + ( 3 + \beta ) q^{15} + q^{16} -\beta q^{17} + ( 1 + 2 \beta ) q^{18} + ( 5 - \beta ) q^{19} + \beta q^{20} + ( 2 + 2 \beta ) q^{21} + \beta q^{22} - q^{23} + ( 1 + \beta ) q^{24} -2 q^{25} + ( -1 - \beta ) q^{26} + 4 q^{27} + 2 q^{28} + ( -6 + 2 \beta ) q^{29} + ( 3 + \beta ) q^{30} + 8 q^{31} + q^{32} + ( 3 + \beta ) q^{33} -\beta q^{34} + 2 \beta q^{35} + ( 1 + 2 \beta ) q^{36} + ( -4 - 2 \beta ) q^{37} + ( 5 - \beta ) q^{38} + ( -4 - 2 \beta ) q^{39} + \beta q^{40} + ( 3 + 2 \beta ) q^{41} + ( 2 + 2 \beta ) q^{42} + ( -4 + 3 \beta ) q^{43} + \beta q^{44} + ( 6 + \beta ) q^{45} - q^{46} + ( 3 - 5 \beta ) q^{47} + ( 1 + \beta ) q^{48} -3 q^{49} -2 q^{50} + ( -3 - \beta ) q^{51} + ( -1 - \beta ) q^{52} + 2 \beta q^{53} + 4 q^{54} + 3 q^{55} + 2 q^{56} + ( 2 + 4 \beta ) q^{57} + ( -6 + 2 \beta ) q^{58} + ( 9 + \beta ) q^{59} + ( 3 + \beta ) q^{60} + ( 2 + 3 \beta ) q^{61} + 8 q^{62} + ( 2 + 4 \beta ) q^{63} + q^{64} + ( -3 - \beta ) q^{65} + ( 3 + \beta ) q^{66} + ( -1 - 7 \beta ) q^{67} -\beta q^{68} + ( -1 - \beta ) q^{69} + 2 \beta q^{70} + ( 9 + \beta ) q^{71} + ( 1 + 2 \beta ) q^{72} + ( -1 - 3 \beta ) q^{73} + ( -4 - 2 \beta ) q^{74} + ( -2 - 2 \beta ) q^{75} + ( 5 - \beta ) q^{76} + 2 \beta q^{77} + ( -4 - 2 \beta ) q^{78} + ( -10 - 2 \beta ) q^{79} + \beta q^{80} + ( 1 - 2 \beta ) q^{81} + ( 3 + 2 \beta ) q^{82} + ( 3 + 3 \beta ) q^{83} + ( 2 + 2 \beta ) q^{84} -3 q^{85} + ( -4 + 3 \beta ) q^{86} -4 \beta q^{87} + \beta q^{88} + ( 6 - 2 \beta ) q^{89} + ( 6 + \beta ) q^{90} + ( -2 - 2 \beta ) q^{91} - q^{92} + ( 8 + 8 \beta ) q^{93} + ( 3 - 5 \beta ) q^{94} + ( -3 + 5 \beta ) q^{95} + ( 1 + \beta ) q^{96} + ( -4 - 2 \beta ) q^{97} -3 q^{98} + ( 6 + \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + 2q^{12} - 2q^{13} + 4q^{14} + 6q^{15} + 2q^{16} + 2q^{18} + 10q^{19} + 4q^{21} - 2q^{23} + 2q^{24} - 4q^{25} - 2q^{26} + 8q^{27} + 4q^{28} - 12q^{29} + 6q^{30} + 16q^{31} + 2q^{32} + 6q^{33} + 2q^{36} - 8q^{37} + 10q^{38} - 8q^{39} + 6q^{41} + 4q^{42} - 8q^{43} + 12q^{45} - 2q^{46} + 6q^{47} + 2q^{48} - 6q^{49} - 4q^{50} - 6q^{51} - 2q^{52} + 8q^{54} + 6q^{55} + 4q^{56} + 4q^{57} - 12q^{58} + 18q^{59} + 6q^{60} + 4q^{61} + 16q^{62} + 4q^{63} + 2q^{64} - 6q^{65} + 6q^{66} - 2q^{67} - 2q^{69} + 18q^{71} + 2q^{72} - 2q^{73} - 8q^{74} - 4q^{75} + 10q^{76} - 8q^{78} - 20q^{79} + 2q^{81} + 6q^{82} + 6q^{83} + 4q^{84} - 6q^{85} - 8q^{86} + 12q^{89} + 12q^{90} - 4q^{91} - 2q^{92} + 16q^{93} + 6q^{94} - 6q^{95} + 2q^{96} - 8q^{97} - 6q^{98} + 12q^{99} + O(q^{100}) \)

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
−1.73205
1.73205
1.00000 −0.732051 1.00000 −1.73205 −0.732051 2.00000 1.00000 −2.46410 −1.73205
1.2 1.00000 2.73205 1.00000 1.73205 2.73205 2.00000 1.00000 4.46410 1.73205
\(n\): 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
\(2\) \(-1\)
\(23\) \(1\)
\(131\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6026))\):

\( T_{3}^{2} - 2 T_{3} - 2 \)
\( T_{5}^{2} - 3 \)