Properties

Label 588.2.a.c
Level $588$
Weight $2$
Character orbit 588.a
Self dual yes
Analytic conductor $4.695$
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] = [588,2,Mod(1,588)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(588, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.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: \(4.69520363885\)
Analytic rank: \(1\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{9} - 6 q^{11} - 2 q^{13} + 4 q^{19} - 6 q^{23} - 5 q^{25} - q^{27} + 6 q^{29} - 8 q^{31} + 6 q^{33} + 2 q^{37} + 2 q^{39} - 12 q^{41} - 4 q^{43} - 12 q^{47} - 6 q^{53} - 4 q^{57} + 10 q^{61} + 8 q^{67} + 6 q^{69} + 6 q^{71} + 10 q^{73} + 5 q^{75} - 4 q^{79} + q^{81} + 12 q^{83} - 6 q^{87} - 12 q^{89} + 8 q^{93} + 10 q^{97} - 6 q^{99}+O(q^{100}) \) 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 −1.00000 0 0 0 0 0 1.00000 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 588.2.a.c 1
3.b odd 2 1 1764.2.a.g 1
4.b odd 2 1 2352.2.a.s 1
7.b odd 2 1 84.2.a.b 1
7.c even 3 2 588.2.i.f 2
7.d odd 6 2 588.2.i.c 2
8.b even 2 1 9408.2.a.co 1
8.d odd 2 1 9408.2.a.r 1
12.b even 2 1 7056.2.a.x 1
21.c even 2 1 252.2.a.b 1
21.g even 6 2 1764.2.k.e 2
21.h odd 6 2 1764.2.k.d 2
28.d even 2 1 336.2.a.b 1
28.f even 6 2 2352.2.q.s 2
28.g odd 6 2 2352.2.q.g 2
35.c odd 2 1 2100.2.a.a 1
35.f even 4 2 2100.2.k.a 2
56.e even 2 1 1344.2.a.o 1
56.h odd 2 1 1344.2.a.f 1
63.l odd 6 2 2268.2.j.i 2
63.o even 6 2 2268.2.j.f 2
84.h odd 2 1 1008.2.a.g 1
105.g even 2 1 6300.2.a.p 1
105.k odd 4 2 6300.2.k.r 2
112.j even 4 2 5376.2.c.x 2
112.l odd 4 2 5376.2.c.i 2
140.c even 2 1 8400.2.a.ct 1
168.e odd 2 1 4032.2.a.t 1
168.i even 2 1 4032.2.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.a.b 1 7.b odd 2 1
252.2.a.b 1 21.c even 2 1
336.2.a.b 1 28.d even 2 1
588.2.a.c 1 1.a even 1 1 trivial
588.2.i.c 2 7.d odd 6 2
588.2.i.f 2 7.c even 3 2
1008.2.a.g 1 84.h odd 2 1
1344.2.a.f 1 56.h odd 2 1
1344.2.a.o 1 56.e even 2 1
1764.2.a.g 1 3.b odd 2 1
1764.2.k.d 2 21.h odd 6 2
1764.2.k.e 2 21.g even 6 2
2100.2.a.a 1 35.c odd 2 1
2100.2.k.a 2 35.f even 4 2
2268.2.j.f 2 63.o even 6 2
2268.2.j.i 2 63.l odd 6 2
2352.2.a.s 1 4.b odd 2 1
2352.2.q.g 2 28.g odd 6 2
2352.2.q.s 2 28.f even 6 2
4032.2.a.t 1 168.e odd 2 1
4032.2.a.u 1 168.i even 2 1
5376.2.c.i 2 112.l odd 4 2
5376.2.c.x 2 112.j even 4 2
6300.2.a.p 1 105.g even 2 1
6300.2.k.r 2 105.k odd 4 2
7056.2.a.x 1 12.b even 2 1
8400.2.a.ct 1 140.c even 2 1
9408.2.a.r 1 8.d odd 2 1
9408.2.a.co 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(588))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 6 \) Copy content Toggle raw display
$13$ \( T + 2 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 4 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T + 12 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T + 12 \) Copy content Toggle raw display
$53$ \( T + 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T - 10 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 10 \) Copy content Toggle raw display
$79$ \( T + 4 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 12 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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