Properties

Label 6024.2.a.j
Level 6024
Weight 2
Character orbit 6024.a
Self dual Yes
Analytic conductor 48.102
Analytic rank 1
Dimension 3
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1018821776\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + \beta_{2} q^{5} \) \( -3 q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + \beta_{2} q^{5} \) \( -3 q^{7} \) \(+ q^{9}\) \( + ( \beta_{1} + \beta_{2} ) q^{11} \) \( + ( -2 + \beta_{1} - \beta_{2} ) q^{13} \) \( + \beta_{2} q^{15} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{17} \) \( + ( 1 - \beta_{2} ) q^{19} \) \( -3 q^{21} \) \( -\beta_{2} q^{23} \) \( + ( 2 - \beta_{1} - \beta_{2} ) q^{25} \) \(+ q^{27}\) \( + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{29} \) \( + ( -1 - 2 \beta_{1} ) q^{31} \) \( + ( \beta_{1} + \beta_{2} ) q^{33} \) \( -3 \beta_{2} q^{35} \) \( + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{37} \) \( + ( -2 + \beta_{1} - \beta_{2} ) q^{39} \) \( + ( -2 + 3 \beta_{1} ) q^{41} \) \( + ( 5 - 2 \beta_{1} + \beta_{2} ) q^{43} \) \( + \beta_{2} q^{45} \) \( + ( 4 - 3 \beta_{1} - \beta_{2} ) q^{47} \) \( + 2 q^{49} \) \( + ( 2 - 2 \beta_{1} - \beta_{2} ) q^{51} \) \( + ( -7 + 2 \beta_{1} + \beta_{2} ) q^{53} \) \( + ( 6 + \beta_{1} - \beta_{2} ) q^{55} \) \( + ( 1 - \beta_{2} ) q^{57} \) \( + ( -5 - 2 \beta_{1} - \beta_{2} ) q^{59} \) \( + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{61} \) \( -3 q^{63} \) \( + ( -8 + 3 \beta_{1} - \beta_{2} ) q^{65} \) \( + ( 1 + 5 \beta_{1} - 3 \beta_{2} ) q^{67} \) \( -\beta_{2} q^{69} \) \( + ( -11 - \beta_{2} ) q^{71} \) \( + ( -7 - 4 \beta_{1} + 2 \beta_{2} ) q^{73} \) \( + ( 2 - \beta_{1} - \beta_{2} ) q^{75} \) \( + ( -3 \beta_{1} - 3 \beta_{2} ) q^{77} \) \( + ( -1 + \beta_{1} + \beta_{2} ) q^{79} \) \(+ q^{81}\) \( + ( -2 \beta_{1} - \beta_{2} ) q^{83} \) \( + ( -5 - 3 \beta_{1} + 3 \beta_{2} ) q^{85} \) \( + ( -1 - \beta_{1} - 2 \beta_{2} ) q^{87} \) \( + ( -2 + 3 \beta_{1} - 3 \beta_{2} ) q^{89} \) \( + ( 6 - 3 \beta_{1} + 3 \beta_{2} ) q^{91} \) \( + ( -1 - 2 \beta_{1} ) q^{93} \) \( + ( -7 + \beta_{1} + 2 \beta_{2} ) q^{95} \) \( + ( -3 + 6 \beta_{1} + \beta_{2} ) q^{97} \) \( + ( \beta_{1} + \beta_{2} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(3q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(3q \) \(\mathstrut +\mathstrut 3q^{3} \) \(\mathstrut +\mathstrut q^{5} \) \(\mathstrut -\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 3q^{9} \) \(\mathstrut +\mathstrut 2q^{11} \) \(\mathstrut -\mathstrut 6q^{13} \) \(\mathstrut +\mathstrut q^{15} \) \(\mathstrut +\mathstrut 3q^{17} \) \(\mathstrut +\mathstrut 2q^{19} \) \(\mathstrut -\mathstrut 9q^{21} \) \(\mathstrut -\mathstrut q^{23} \) \(\mathstrut +\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 3q^{27} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 5q^{31} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut -\mathstrut 3q^{35} \) \(\mathstrut -\mathstrut 6q^{37} \) \(\mathstrut -\mathstrut 6q^{39} \) \(\mathstrut -\mathstrut 3q^{41} \) \(\mathstrut +\mathstrut 14q^{43} \) \(\mathstrut +\mathstrut q^{45} \) \(\mathstrut +\mathstrut 8q^{47} \) \(\mathstrut +\mathstrut 6q^{49} \) \(\mathstrut +\mathstrut 3q^{51} \) \(\mathstrut -\mathstrut 18q^{53} \) \(\mathstrut +\mathstrut 18q^{55} \) \(\mathstrut +\mathstrut 2q^{57} \) \(\mathstrut -\mathstrut 18q^{59} \) \(\mathstrut -\mathstrut 9q^{63} \) \(\mathstrut -\mathstrut 22q^{65} \) \(\mathstrut +\mathstrut 5q^{67} \) \(\mathstrut -\mathstrut q^{69} \) \(\mathstrut -\mathstrut 34q^{71} \) \(\mathstrut -\mathstrut 23q^{73} \) \(\mathstrut +\mathstrut 4q^{75} \) \(\mathstrut -\mathstrut 6q^{77} \) \(\mathstrut -\mathstrut q^{79} \) \(\mathstrut +\mathstrut 3q^{81} \) \(\mathstrut -\mathstrut 3q^{83} \) \(\mathstrut -\mathstrut 15q^{85} \) \(\mathstrut -\mathstrut 6q^{87} \) \(\mathstrut -\mathstrut 6q^{89} \) \(\mathstrut +\mathstrut 18q^{91} \) \(\mathstrut -\mathstrut 5q^{93} \) \(\mathstrut -\mathstrut 18q^{95} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut +\mathstrut 2q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{3}\mathstrut -\mathstrut \) \(x^{2}\mathstrut -\mathstrut \) \(3\) \(x\mathstrut +\mathstrut \) \(1\):

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

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
−1.48119
2.17009
0.311108
0 1.00000 0 −3.15633 0 −3.00000 0 1.00000 0
1.2 0 1.00000 0 1.63090 0 −3.00000 0 1.00000 0
1.3 0 1.00000 0 2.52543 0 −3.00000 0 1.00000 0
\(n\): 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
\(2\) \(1\)
\(3\) \(-1\)
\(251\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

\(T_{5}^{3} \) \(\mathstrut -\mathstrut T_{5}^{2} \) \(\mathstrut -\mathstrut 9 T_{5} \) \(\mathstrut +\mathstrut 13 \)
\(T_{7} \) \(\mathstrut +\mathstrut 3 \)