Properties

Label 1008.4.a.a
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

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 126)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 22q^{5} + 7q^{7} + O(q^{10}) \) \( q - 22q^{5} + 7q^{7} + 26q^{11} - 54q^{13} - 74q^{17} - 116q^{19} - 58q^{23} + 359q^{25} - 208q^{29} + 252q^{31} - 154q^{35} + 50q^{37} - 126q^{41} - 164q^{43} - 444q^{47} + 49q^{49} - 12q^{53} - 572q^{55} + 124q^{59} - 162q^{61} + 1188q^{65} + 860q^{67} - 238q^{71} - 146q^{73} + 182q^{77} + 984q^{79} + 656q^{83} + 1628q^{85} + 954q^{89} - 378q^{91} + 2552q^{95} + 526q^{97} + O(q^{100}) \)

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 −22.0000 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.a 1
3.b odd 2 1 1008.4.a.w 1
4.b odd 2 1 126.4.a.f yes 1
12.b even 2 1 126.4.a.e 1
28.d even 2 1 882.4.a.s 1
28.f even 6 2 882.4.g.a 2
28.g odd 6 2 882.4.g.m 2
84.h odd 2 1 882.4.a.a 1
84.j odd 6 2 882.4.g.x 2
84.n even 6 2 882.4.g.n 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.a.e 1 12.b even 2 1
126.4.a.f yes 1 4.b odd 2 1
882.4.a.a 1 84.h odd 2 1
882.4.a.s 1 28.d even 2 1
882.4.g.a 2 28.f even 6 2
882.4.g.m 2 28.g odd 6 2
882.4.g.n 2 84.n even 6 2
882.4.g.x 2 84.j odd 6 2
1008.4.a.a 1 1.a even 1 1 trivial
1008.4.a.w 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(1008))\):

\( T_{5} + 22 \)
\( T_{11} - 26 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( 22 + T \)
$7$ \( -7 + T \)
$11$ \( -26 + T \)
$13$ \( 54 + T \)
$17$ \( 74 + T \)
$19$ \( 116 + T \)
$23$ \( 58 + T \)
$29$ \( 208 + T \)
$31$ \( -252 + T \)
$37$ \( -50 + T \)
$41$ \( 126 + T \)
$43$ \( 164 + T \)
$47$ \( 444 + T \)
$53$ \( 12 + T \)
$59$ \( -124 + T \)
$61$ \( 162 + T \)
$67$ \( -860 + T \)
$71$ \( 238 + T \)
$73$ \( 146 + T \)
$79$ \( -984 + T \)
$83$ \( -656 + T \)
$89$ \( -954 + T \)
$97$ \( -526 + T \)
show more
show less