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
Defining polynomial: \(x^{4} - 5 x^{2} - 2 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2001)
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.

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

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

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

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:

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.g 4
3.b odd 2 1 2001.2.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2001.2.a.g 4 3.b odd 2 1
6003.2.a.g 4 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6003))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( 4 + 8 T - 6 T^{2} - 2 T^{3} + T^{4} \)
$7$ \( 52 - 16 T - 14 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( -16 + 48 T - 28 T^{2} + T^{4} \)
$13$ \( 125 - 24 T - 38 T^{2} + T^{4} \)
$17$ \( 355 - 22 T - 44 T^{2} + 2 T^{3} + T^{4} \)
$19$ \( 835 + 116 T - 56 T^{2} - 4 T^{3} + T^{4} \)
$23$ \( ( 1 + T )^{4} \)
$29$ \( ( 1 + T )^{4} \)
$31$ \( 4 - 48 T - 38 T^{2} - 2 T^{3} + T^{4} \)
$37$ \( 835 + 108 T - 56 T^{2} - 4 T^{3} + T^{4} \)
$41$ \( -364 - 288 T - 46 T^{2} + 6 T^{3} + T^{4} \)
$43$ \( -73 - 216 T - 92 T^{2} + T^{4} \)
$47$ \( 4 - 8 T - 6 T^{2} + 2 T^{3} + T^{4} \)
$53$ \( -304 + 416 T - 56 T^{2} - 8 T^{3} + T^{4} \)
$59$ \( 133 - 262 T + 130 T^{2} - 22 T^{3} + T^{4} \)
$61$ \( -2240 - 736 T + 8 T^{2} + 16 T^{3} + T^{4} \)
$67$ \( 832 + 128 T - 56 T^{2} - 4 T^{3} + T^{4} \)
$71$ \( 197 - 278 T + 130 T^{2} - 22 T^{3} + T^{4} \)
$73$ \( 508 + 496 T + 166 T^{2} + 22 T^{3} + T^{4} \)
$79$ \( -7133 - 1676 T + 20 T^{3} + T^{4} \)
$83$ \( -68 - 216 T - 122 T^{2} - 10 T^{3} + T^{4} \)
$89$ \( 10207 + 1930 T - 192 T^{2} - 14 T^{3} + T^{4} \)
$97$ \( 64 - 32 T - 24 T^{2} + 8 T^{3} + T^{4} \)
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