Properties

Label 6003.2.a.g
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 0
Dimension 4
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6003.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(47.9341963334\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.5744.1
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

\(f(q)\) \(=\) \( q\) \( -2 q^{4} \) \( + \beta_{1} q^{5} \) \( + ( -1 - \beta_{2} + \beta_{3} ) q^{7} \) \(+O(q^{10})\) \( q\) \( -2 q^{4} \) \( + \beta_{1} q^{5} \) \( + ( -1 - \beta_{2} + \beta_{3} ) q^{7} \) \( + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{11} \) \( + ( 1 + 2 \beta_{2} ) q^{13} \) \( + 4 q^{16} \) \( + ( -1 - \beta_{2} + 2 \beta_{3} ) q^{17} \) \( + ( 2 + 2 \beta_{2} + \beta_{3} ) q^{19} \) \( -2 \beta_{1} q^{20} \) \(- q^{23}\) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} \) \( + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{28} \) \(- q^{29}\) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{31} \) \( + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{35} \) \( + ( -1 + 3 \beta_{1} - \beta_{2} ) q^{37} \) \( + ( -2 + \beta_{1} + 2 \beta_{3} ) q^{41} \) \( + ( 1 - 3 \beta_{1} - \beta_{2} ) q^{43} \) \( + ( -2 + 2 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{44} \) \( -\beta_{1} q^{47} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{49} \) \( + ( -2 - 4 \beta_{2} ) q^{52} \) \( + ( 4 - 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{53} \) \( + ( -2 + 2 \beta_{3} ) q^{55} \) \( + ( 6 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{59} \) \( + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{61} \) \( -8 q^{64} \) \( + ( 2 - \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{65} \) \( + ( 2 + 2 \beta_{2} - 2 \beta_{3} ) q^{67} \) \( + ( 2 + 2 \beta_{2} - 4 \beta_{3} ) q^{68} \) \( + ( 5 + \beta_{1} - 2 \beta_{3} ) q^{71} \) \( + ( -5 + \beta_{2} - \beta_{3} ) q^{73} \) \( + ( -4 - 4 \beta_{2} - 2 \beta_{3} ) q^{76} \) \( + ( -2 + 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{77} \) \( + ( -6 + 2 \beta_{1} + 3 \beta_{3} ) q^{79} \) \( + 4 \beta_{1} q^{80} \) \( + ( 6 - 3 \beta_{1} + 4 \beta_{2} ) q^{83} \) \( + ( -1 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{85} \) \( + ( -1 + 6 \beta_{1} - 3 \beta_{2} ) q^{89} \) \( + ( -11 + 6 \beta_{1} - 3 \beta_{2} - \beta_{3} ) q^{91} \) \( + 2 q^{92} \) \( + ( 2 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{95} \) \( + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut +\mathstrut 2q^{5} \) \(\mathstrut -\mathstrut 2q^{7} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut -\mathstrut 2q^{17} \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 4q^{20} \) \(\mathstrut -\mathstrut 4q^{23} \) \(\mathstrut -\mathstrut 4q^{25} \) \(\mathstrut +\mathstrut 4q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 2q^{31} \) \(\mathstrut -\mathstrut 4q^{35} \) \(\mathstrut +\mathstrut 4q^{37} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 2q^{47} \) \(\mathstrut +\mathstrut 4q^{49} \) \(\mathstrut +\mathstrut 8q^{53} \) \(\mathstrut -\mathstrut 8q^{55} \) \(\mathstrut +\mathstrut 22q^{59} \) \(\mathstrut -\mathstrut 16q^{61} \) \(\mathstrut -\mathstrut 32q^{64} \) \(\mathstrut +\mathstrut 10q^{65} \) \(\mathstrut +\mathstrut 4q^{67} \) \(\mathstrut +\mathstrut 4q^{68} \) \(\mathstrut +\mathstrut 22q^{71} \) \(\mathstrut -\mathstrut 22q^{73} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 8q^{77} \) \(\mathstrut -\mathstrut 20q^{79} \) \(\mathstrut +\mathstrut 8q^{80} \) \(\mathstrut +\mathstrut 10q^{83} \) \(\mathstrut -\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 14q^{89} \) \(\mathstrut -\mathstrut 26q^{91} \) \(\mathstrut +\mathstrut 8q^{92} \) \(\mathstrut +\mathstrut 14q^{95} \) \(\mathstrut -\mathstrut 8q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4}\mathstrut -\mathstrut \) \(5\) \(x^{2}\mathstrut -\mathstrut \) \(2\) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} - 4 \nu - 1 \)
\(\beta_{2}\)\(=\)\( \nu^{2} - \nu - 3 \)
\(\beta_{3}\)\(=\)\( -\nu^{3} + \nu^{2} + 5 \nu - 1 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(5\)\()/2\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(3\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)

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
0.291367
−1.92022
−0.751024
2.37988
0 0 −2.00000 −2.14073 0 2.72347 0 0 0
1.2 0 0 −2.00000 −0.399447 0 −3.44099 0 0 0
1.3 0 0 −2.00000 1.58049 0 −3.08254 0 0 0
1.4 0 0 −2.00000 2.95969 0 1.80007 0 0 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
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\(T_{2} \)
\(T_{5}^{4} \) \(\mathstrut -\mathstrut 2 T_{5}^{3} \) \(\mathstrut -\mathstrut 6 T_{5}^{2} \) \(\mathstrut +\mathstrut 8 T_{5} \) \(\mathstrut +\mathstrut 4 \)