Properties

Label 6003.2.a.a
Level $6003$
Weight $2$
Character orbit 6003.a
Self dual yes
Analytic conductor $47.934$
Analytic rank $1$
Dimension $1$
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: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{4} - 4q^{5} - 4q^{7} + O(q^{10}) \) \( q - 2q^{4} - 4q^{5} - 4q^{7} - 4q^{11} - 5q^{13} + 4q^{16} + 5q^{17} + 5q^{19} + 8q^{20} - q^{23} + 11q^{25} + 8q^{28} + q^{29} - 2q^{31} + 16q^{35} + 5q^{37} + 2q^{41} + q^{43} + 8q^{44} - 6q^{47} + 9q^{49} + 10q^{52} - 2q^{53} + 16q^{55} - 9q^{59} - 10q^{61} - 8q^{64} + 20q^{65} + 8q^{67} - 10q^{68} + 3q^{71} + 8q^{73} - 10q^{76} + 16q^{77} + 13q^{79} - 16q^{80} + 6q^{83} - 20q^{85} + 9q^{89} + 20q^{91} + 2q^{92} - 20q^{95} - 6q^{97} + O(q^{100}) \)

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
0
0 0 −2.00000 −4.00000 0 −4.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(23\) \(1\)
\(29\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6003.2.a.a 1
3.b odd 2 1 2001.2.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.b 1 3.b odd 2 1
6003.2.a.a 1 1.a even 1 1 trivial

Hecke kernels

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

\( T_{2} \)
\( T_{5} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 4 + T \)
$7$ \( 4 + T \)
$11$ \( 4 + T \)
$13$ \( 5 + T \)
$17$ \( -5 + T \)
$19$ \( -5 + T \)
$23$ \( 1 + T \)
$29$ \( -1 + T \)
$31$ \( 2 + T \)
$37$ \( -5 + T \)
$41$ \( -2 + T \)
$43$ \( -1 + T \)
$47$ \( 6 + T \)
$53$ \( 2 + T \)
$59$ \( 9 + T \)
$61$ \( 10 + T \)
$67$ \( -8 + T \)
$71$ \( -3 + T \)
$73$ \( -8 + T \)
$79$ \( -13 + T \)
$83$ \( -6 + T \)
$89$ \( -9 + T \)
$97$ \( 6 + T \)
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