Properties

Label 504.4.a.g
Level $504$
Weight $4$
Character orbit 504.a
Self dual yes
Analytic conductor $29.737$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,4,Mod(1,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 504.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(29.7369626429\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + 16 q^{5} - 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 16 q^{5} - 7 q^{7} - 24 q^{11} - 68 q^{13} - 54 q^{17} - 46 q^{19} - 176 q^{23} + 131 q^{25} + 174 q^{29} - 116 q^{31} - 112 q^{35} + 74 q^{37} + 10 q^{41} - 480 q^{43} + 572 q^{47} + 49 q^{49} + 162 q^{53} - 384 q^{55} + 86 q^{59} - 904 q^{61} - 1088 q^{65} + 660 q^{67} - 1024 q^{71} + 770 q^{73} + 168 q^{77} - 904 q^{79} - 682 q^{83} - 864 q^{85} + 102 q^{89} + 476 q^{91} - 736 q^{95} - 218 q^{97}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 16.0000 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 504.4.a.g 1
3.b odd 2 1 56.4.a.a 1
4.b odd 2 1 1008.4.a.u 1
12.b even 2 1 112.4.a.d 1
15.d odd 2 1 1400.4.a.f 1
15.e even 4 2 1400.4.g.f 2
21.c even 2 1 392.4.a.c 1
21.g even 6 2 392.4.i.d 2
21.h odd 6 2 392.4.i.e 2
24.f even 2 1 448.4.a.h 1
24.h odd 2 1 448.4.a.l 1
84.h odd 2 1 784.4.a.i 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.4.a.a 1 3.b odd 2 1
112.4.a.d 1 12.b even 2 1
392.4.a.c 1 21.c even 2 1
392.4.i.d 2 21.g even 6 2
392.4.i.e 2 21.h odd 6 2
448.4.a.h 1 24.f even 2 1
448.4.a.l 1 24.h odd 2 1
504.4.a.g 1 1.a even 1 1 trivial
784.4.a.i 1 84.h odd 2 1
1008.4.a.u 1 4.b odd 2 1
1400.4.a.f 1 15.d odd 2 1
1400.4.g.f 2 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(504))\):

\( T_{5} - 16 \) Copy content Toggle raw display
\( T_{11} + 24 \) 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
$5$ \( T - 16 \) Copy content Toggle raw display
$7$ \( T + 7 \) Copy content Toggle raw display
$11$ \( T + 24 \) Copy content Toggle raw display
$13$ \( T + 68 \) Copy content Toggle raw display
$17$ \( T + 54 \) Copy content Toggle raw display
$19$ \( T + 46 \) Copy content Toggle raw display
$23$ \( T + 176 \) Copy content Toggle raw display
$29$ \( T - 174 \) Copy content Toggle raw display
$31$ \( T + 116 \) Copy content Toggle raw display
$37$ \( T - 74 \) Copy content Toggle raw display
$41$ \( T - 10 \) Copy content Toggle raw display
$43$ \( T + 480 \) Copy content Toggle raw display
$47$ \( T - 572 \) Copy content Toggle raw display
$53$ \( T - 162 \) Copy content Toggle raw display
$59$ \( T - 86 \) Copy content Toggle raw display
$61$ \( T + 904 \) Copy content Toggle raw display
$67$ \( T - 660 \) Copy content Toggle raw display
$71$ \( T + 1024 \) Copy content Toggle raw display
$73$ \( T - 770 \) Copy content Toggle raw display
$79$ \( T + 904 \) Copy content Toggle raw display
$83$ \( T + 682 \) Copy content Toggle raw display
$89$ \( T - 102 \) Copy content Toggle raw display
$97$ \( T + 218 \) Copy content Toggle raw display
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