Properties

Label 6021.2.a.k
Level 6021
Weight 2
Character orbit 6021.a
Self dual Yes
Analytic conductor 48.078
Analytic rank 1
Dimension 4
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6021.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.077927057\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{5} \) \(+ q^{7}\) \( -2 \beta_{2} q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{2} q^{2} \) \( + ( -\beta_{1} + \beta_{2} ) q^{5} \) \(+ q^{7}\) \( -2 \beta_{2} q^{8} \) \( + ( 1 - \beta_{3} ) q^{10} \) \( + ( -2 \beta_{1} + 2 \beta_{2} ) q^{11} \) \(+ q^{13}\) \( + \beta_{2} q^{14} \) \( -4 q^{16} \) \( + ( 2 \beta_{1} - 3 \beta_{2} ) q^{17} \) \( + 3 \beta_{3} q^{19} \) \( + ( 2 - 2 \beta_{3} ) q^{22} \) \( + ( 3 \beta_{1} - 4 \beta_{2} ) q^{23} \) \( + ( -2 - \beta_{3} ) q^{25} \) \( + \beta_{2} q^{26} \) \( + ( 3 \beta_{1} - 3 \beta_{2} ) q^{29} \) \( + ( -5 + \beta_{3} ) q^{31} \) \( + ( -4 + 2 \beta_{3} ) q^{34} \) \( + ( -\beta_{1} + \beta_{2} ) q^{35} \) \( -9 q^{37} \) \( + ( 6 \beta_{1} - 3 \beta_{2} ) q^{38} \) \( + ( -2 + 2 \beta_{3} ) q^{40} \) \( + ( \beta_{1} + 2 \beta_{2} ) q^{41} \) \( + ( 4 - 2 \beta_{3} ) q^{43} \) \( + ( -5 + 3 \beta_{3} ) q^{46} \) \( + ( -2 \beta_{1} + 8 \beta_{2} ) q^{47} \) \( -6 q^{49} \) \( + ( -2 \beta_{1} - \beta_{2} ) q^{50} \) \( + ( -\beta_{1} - 6 \beta_{2} ) q^{53} \) \( + ( 6 - 2 \beta_{3} ) q^{55} \) \( -2 \beta_{2} q^{56} \) \( + ( -3 + 3 \beta_{3} ) q^{58} \) \( + ( 6 \beta_{1} - 6 \beta_{2} ) q^{59} \) \( + ( -4 - 3 \beta_{3} ) q^{61} \) \( + ( 2 \beta_{1} - 6 \beta_{2} ) q^{62} \) \( + 8 q^{64} \) \( + ( -\beta_{1} + \beta_{2} ) q^{65} \) \( -9 q^{67} \) \( + ( 1 - \beta_{3} ) q^{70} \) \( + ( 6 \beta_{1} - 5 \beta_{2} ) q^{71} \) \( + ( 2 - 5 \beta_{3} ) q^{73} \) \( -9 \beta_{2} q^{74} \) \( + ( -2 \beta_{1} + 2 \beta_{2} ) q^{77} \) \( + ( 8 + \beta_{3} ) q^{79} \) \( + ( 4 \beta_{1} - 4 \beta_{2} ) q^{80} \) \( + ( 5 + \beta_{3} ) q^{82} \) \( + ( -5 \beta_{1} + 5 \beta_{2} ) q^{83} \) \( + ( -7 + 3 \beta_{3} ) q^{85} \) \( + ( -4 \beta_{1} + 6 \beta_{2} ) q^{86} \) \( + ( -4 + 4 \beta_{3} ) q^{88} \) \( + ( 2 \beta_{1} - 2 \beta_{2} ) q^{89} \) \(+ q^{91}\) \( + ( 14 - 2 \beta_{3} ) q^{94} \) \( + ( 3 \beta_{1} - 9 \beta_{2} ) q^{95} \) \( + ( -4 - 3 \beta_{3} ) q^{97} \) \( -6 \beta_{2} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 4q^{10} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 8q^{22} \) \(\mathstrut -\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 20q^{31} \) \(\mathstrut -\mathstrut 16q^{34} \) \(\mathstrut -\mathstrut 36q^{37} \) \(\mathstrut -\mathstrut 8q^{40} \) \(\mathstrut +\mathstrut 16q^{43} \) \(\mathstrut -\mathstrut 20q^{46} \) \(\mathstrut -\mathstrut 24q^{49} \) \(\mathstrut +\mathstrut 24q^{55} \) \(\mathstrut -\mathstrut 12q^{58} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut +\mathstrut 32q^{64} \) \(\mathstrut -\mathstrut 36q^{67} \) \(\mathstrut +\mathstrut 4q^{70} \) \(\mathstrut +\mathstrut 8q^{73} \) \(\mathstrut +\mathstrut 32q^{79} \) \(\mathstrut +\mathstrut 20q^{82} \) \(\mathstrut -\mathstrut 28q^{85} \) \(\mathstrut -\mathstrut 16q^{88} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut 56q^{94} \) \(\mathstrut -\mathstrut 16q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(6\) \(x^{2}\mathstrut +\mathstrut \) \(4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} - 4 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\( \nu^{2} - 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\)

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
0.874032
−2.28825
2.28825
−0.874032
−1.41421 0 0 −2.28825 0 1.00000 2.82843 0 3.23607
1.2 −1.41421 0 0 0.874032 0 1.00000 2.82843 0 −1.23607
1.3 1.41421 0 0 −0.874032 0 1.00000 −2.82843 0 −1.23607
1.4 1.41421 0 0 2.28825 0 1.00000 −2.82843 0 3.23607
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(223\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6021))\):

\(T_{2}^{2} \) \(\mathstrut -\mathstrut 2 \)
\(T_{5}^{4} \) \(\mathstrut -\mathstrut 6 T_{5}^{2} \) \(\mathstrut +\mathstrut 4 \)