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.

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

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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6026))\):

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