Properties

Label 1008.4.a.h
Level $1008$
Weight $4$
Character orbit 1008.a
Self dual yes
Analytic conductor $59.474$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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 84)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 6q^{5} - 7q^{7} + O(q^{10}) \) \( q - 6q^{5} - 7q^{7} + 36q^{11} + 62q^{13} - 114q^{17} + 76q^{19} - 24q^{23} - 89q^{25} - 54q^{29} + 112q^{31} + 42q^{35} - 178q^{37} - 378q^{41} + 172q^{43} - 192q^{47} + 49q^{49} + 402q^{53} - 216q^{55} + 396q^{59} + 254q^{61} - 372q^{65} + 1012q^{67} + 840q^{71} + 890q^{73} - 252q^{77} - 80q^{79} - 108q^{83} + 684q^{85} + 1638q^{89} - 434q^{91} - 456q^{95} + 1010q^{97} + O(q^{100}) \)

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.h 1
3.b odd 2 1 336.4.a.k 1
4.b odd 2 1 252.4.a.b 1
12.b even 2 1 84.4.a.a 1
21.c even 2 1 2352.4.a.d 1
24.f even 2 1 1344.4.a.q 1
24.h odd 2 1 1344.4.a.d 1
28.d even 2 1 1764.4.a.j 1
28.f even 6 2 1764.4.k.f 2
28.g odd 6 2 1764.4.k.l 2
60.h even 2 1 2100.4.a.l 1
60.l odd 4 2 2100.4.k.j 2
84.h odd 2 1 588.4.a.d 1
84.j odd 6 2 588.4.i.c 2
84.n even 6 2 588.4.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.a.a 1 12.b even 2 1
252.4.a.b 1 4.b odd 2 1
336.4.a.k 1 3.b odd 2 1
588.4.a.d 1 84.h odd 2 1
588.4.i.c 2 84.j odd 6 2
588.4.i.f 2 84.n even 6 2
1008.4.a.h 1 1.a even 1 1 trivial
1344.4.a.d 1 24.h odd 2 1
1344.4.a.q 1 24.f even 2 1
1764.4.a.j 1 28.d even 2 1
1764.4.k.f 2 28.f even 6 2
1764.4.k.l 2 28.g odd 6 2
2100.4.a.l 1 60.h even 2 1
2100.4.k.j 2 60.l odd 4 2
2352.4.a.d 1 21.c even 2 1

Hecke kernels

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

\( T_{5} + 6 \)
\( T_{11} - 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 6 + T \)
$7$ \( 7 + T \)
$11$ \( -36 + T \)
$13$ \( -62 + T \)
$17$ \( 114 + T \)
$19$ \( -76 + T \)
$23$ \( 24 + T \)
$29$ \( 54 + T \)
$31$ \( -112 + T \)
$37$ \( 178 + T \)
$41$ \( 378 + T \)
$43$ \( -172 + T \)
$47$ \( 192 + T \)
$53$ \( -402 + T \)
$59$ \( -396 + T \)
$61$ \( -254 + T \)
$67$ \( -1012 + T \)
$71$ \( -840 + T \)
$73$ \( -890 + T \)
$79$ \( 80 + T \)
$83$ \( 108 + T \)
$89$ \( -1638 + T \)
$97$ \( -1010 + T \)
show more
show less