Properties

Label 1008.4.a.f
Level $1008$
Weight $4$
Character orbit 1008.a
Self dual yes
Analytic conductor $59.474$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 6 q^{5} - 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{5} - 7 q^{7} - 12 q^{11} - 82 q^{13} + 30 q^{17} - 68 q^{19} + 216 q^{23} - 89 q^{25} - 246 q^{29} + 112 q^{31} + 42 q^{35} + 110 q^{37} + 246 q^{41} + 172 q^{43} + 192 q^{47} + 49 q^{49} - 558 q^{53} + 72 q^{55} + 540 q^{59} + 110 q^{61} + 492 q^{65} - 140 q^{67} - 840 q^{71} - 550 q^{73} + 84 q^{77} + 208 q^{79} + 516 q^{83} - 180 q^{85} + 1398 q^{89} + 574 q^{91} + 408 q^{95} + 1586 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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.f 1
3.b odd 2 1 112.4.a.c 1
4.b odd 2 1 252.4.a.c 1
12.b even 2 1 28.4.a.b 1
21.c even 2 1 784.4.a.n 1
24.f even 2 1 448.4.a.d 1
24.h odd 2 1 448.4.a.m 1
28.d even 2 1 1764.4.a.k 1
28.f even 6 2 1764.4.k.e 2
28.g odd 6 2 1764.4.k.k 2
60.h even 2 1 700.4.a.e 1
60.l odd 4 2 700.4.e.f 2
84.h odd 2 1 196.4.a.b 1
84.j odd 6 2 196.4.e.d 2
84.n even 6 2 196.4.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.b 1 12.b even 2 1
112.4.a.c 1 3.b odd 2 1
196.4.a.b 1 84.h odd 2 1
196.4.e.c 2 84.n even 6 2
196.4.e.d 2 84.j odd 6 2
252.4.a.c 1 4.b odd 2 1
448.4.a.d 1 24.f even 2 1
448.4.a.m 1 24.h odd 2 1
700.4.a.e 1 60.h even 2 1
700.4.e.f 2 60.l odd 4 2
784.4.a.n 1 21.c even 2 1
1008.4.a.f 1 1.a even 1 1 trivial
1764.4.a.k 1 28.d even 2 1
1764.4.k.e 2 28.f even 6 2
1764.4.k.k 2 28.g odd 6 2

Hecke kernels

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

\( T_{5} + 6 \) Copy content Toggle raw display
\( T_{11} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 12 \) Copy content Toggle raw display
$13$ \( T + 82 \) Copy content Toggle raw display
$17$ \( T - 30 \) Copy content Toggle raw display
$19$ \( T + 68 \) Copy content Toggle raw display
$23$ \( T - 216 \) Copy content Toggle raw display
$29$ \( T + 246 \) Copy content Toggle raw display
$31$ \( T - 112 \) Copy content Toggle raw display
$37$ \( T - 110 \) Copy content Toggle raw display
$41$ \( T - 246 \) Copy content Toggle raw display
$43$ \( T - 172 \) Copy content Toggle raw display
$47$ \( T - 192 \) Copy content Toggle raw display
$53$ \( T + 558 \) Copy content Toggle raw display
$59$ \( T - 540 \) Copy content Toggle raw display
$61$ \( T - 110 \) Copy content Toggle raw display
$67$ \( T + 140 \) Copy content Toggle raw display
$71$ \( T + 840 \) Copy content Toggle raw display
$73$ \( T + 550 \) Copy content Toggle raw display
$79$ \( T - 208 \) Copy content Toggle raw display
$83$ \( T - 516 \) Copy content Toggle raw display
$89$ \( T - 1398 \) Copy content Toggle raw display
$97$ \( T - 1586 \) Copy content Toggle raw display
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