Properties

Label 1008.4.a.g
Level $1008$
Weight $4$
Character orbit 1008.a
Self dual yes
Analytic conductor $59.474$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1008.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(59.4739252858\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 126)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 6q^{5} - 7q^{7} + O(q^{10}) \) \( q - 6q^{5} - 7q^{7} + 30q^{11} + 2q^{13} - 66q^{17} + 52q^{19} + 114q^{23} - 89q^{25} - 72q^{29} + 196q^{31} + 42q^{35} - 286q^{37} + 378q^{41} - 164q^{43} - 228q^{47} + 49q^{49} + 348q^{53} - 180q^{55} - 348q^{59} - 106q^{61} - 12q^{65} - 596q^{67} + 630q^{71} - 1042q^{73} - 210q^{77} + 88q^{79} - 1440q^{83} + 396q^{85} - 1374q^{89} - 14q^{91} - 312q^{95} - 34q^{97} + O(q^{100}) \)

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 0 −6.00000 0 −7.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

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 1008.4.a.g 1
3.b odd 2 1 1008.4.a.n 1
4.b odd 2 1 126.4.a.b 1
12.b even 2 1 126.4.a.g yes 1
28.d even 2 1 882.4.a.e 1
28.f even 6 2 882.4.g.q 2
28.g odd 6 2 882.4.g.t 2
84.h odd 2 1 882.4.a.m 1
84.j odd 6 2 882.4.g.h 2
84.n even 6 2 882.4.g.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.b 1 4.b odd 2 1
126.4.a.g yes 1 12.b even 2 1
882.4.a.e 1 28.d even 2 1
882.4.a.m 1 84.h odd 2 1
882.4.g.e 2 84.n even 6 2
882.4.g.h 2 84.j odd 6 2
882.4.g.q 2 28.f even 6 2
882.4.g.t 2 28.g odd 6 2
1008.4.a.g 1 1.a even 1 1 trivial
1008.4.a.n 1 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5} + 6 \)
\( T_{11} - 30 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 6 + T \)
$7$ \( 7 + T \)
$11$ \( -30 + T \)
$13$ \( -2 + T \)
$17$ \( 66 + T \)
$19$ \( -52 + T \)
$23$ \( -114 + T \)
$29$ \( 72 + T \)
$31$ \( -196 + T \)
$37$ \( 286 + T \)
$41$ \( -378 + T \)
$43$ \( 164 + T \)
$47$ \( 228 + T \)
$53$ \( -348 + T \)
$59$ \( 348 + T \)
$61$ \( 106 + T \)
$67$ \( 596 + T \)
$71$ \( -630 + T \)
$73$ \( 1042 + T \)
$79$ \( -88 + T \)
$83$ \( 1440 + T \)
$89$ \( 1374 + T \)
$97$ \( 34 + T \)
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