Properties

Label 6003.2.a.e
Level 6003
Weight 2
Character orbit 6003.a
Self dual Yes
Analytic conductor 47.934
Analytic rank 1
Dimension 2
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 q^{4} + \beta q^{5} + ( -2 + \beta ) q^{7} +O(q^{10})\) \( q -2 q^{4} + \beta q^{5} + ( -2 + \beta ) q^{7} + q^{13} + 4 q^{16} + ( -1 - \beta ) q^{17} - q^{19} -2 \beta q^{20} + q^{23} + q^{25} + ( 4 - 2 \beta ) q^{28} - q^{29} + ( -8 + \beta ) q^{31} + ( 6 - 2 \beta ) q^{35} + ( 5 - 2 \beta ) q^{37} + ( 2 - 3 \beta ) q^{41} + ( -5 - 2 \beta ) q^{43} + ( 10 - \beta ) q^{47} + ( 3 - 4 \beta ) q^{49} -2 q^{52} + 2 q^{53} + ( 3 + \beta ) q^{59} + ( -8 - 2 \beta ) q^{61} -8 q^{64} + \beta q^{65} + ( -4 + 2 \beta ) q^{67} + ( 2 + 2 \beta ) q^{68} + ( 5 - 3 \beta ) q^{71} + ( -2 + 5 \beta ) q^{73} + 2 q^{76} + ( 1 - 2 \beta ) q^{79} + 4 \beta q^{80} + ( 10 + \beta ) q^{83} + ( -6 - \beta ) q^{85} + ( -3 + \beta ) q^{89} + ( -2 + \beta ) q^{91} -2 q^{92} -\beta q^{95} + ( 12 - 2 \beta ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 4q^{4} - 4q^{7} + O(q^{10}) \) \( 2q - 4q^{4} - 4q^{7} + 2q^{13} + 8q^{16} - 2q^{17} - 2q^{19} + 2q^{23} + 2q^{25} + 8q^{28} - 2q^{29} - 16q^{31} + 12q^{35} + 10q^{37} + 4q^{41} - 10q^{43} + 20q^{47} + 6q^{49} - 4q^{52} + 4q^{53} + 6q^{59} - 16q^{61} - 16q^{64} - 8q^{67} + 4q^{68} + 10q^{71} - 4q^{73} + 4q^{76} + 2q^{79} + 20q^{83} - 12q^{85} - 6q^{89} - 4q^{91} - 4q^{92} + 24q^{97} + O(q^{100}) \)

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
−2.44949
2.44949
0 0 −2.00000 −2.44949 0 −4.44949 0 0 0
1.2 0 0 −2.00000 2.44949 0 0.449490 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

\( T_{2} \)
\( T_{5}^{2} - 6 \)