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Properties

Label 588.2.a.f
Level $588$
Weight $2$
Character orbit 588.a
Self dual yes
Analytic conductor $4.695$
Analytic rank $0$
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: \(0\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{3} + 2 q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + 2 q^{5} + q^{9} + 2 q^{11} + 3 q^{13} + 2 q^{15} - 8 q^{17} + q^{19} + 8 q^{23} - q^{25} + q^{27} + 4 q^{29} - 3 q^{31} + 2 q^{33} - q^{37} + 3 q^{39} - 6 q^{41} + 11 q^{43} + 2 q^{45} - 6 q^{47} - 8 q^{51} - 12 q^{53} + 4 q^{55} + q^{57} - 4 q^{59} + 6 q^{61} + 6 q^{65} + 13 q^{67} + 8 q^{69} - 10 q^{71} + 11 q^{73} - q^{75} - 3 q^{79} + q^{81} - 2 q^{83} - 16 q^{85} + 4 q^{87} - 3 q^{93} + 2 q^{95} - 10 q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 2.00000 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.f 1
3.b odd 2 1 1764.2.a.c 1
4.b odd 2 1 2352.2.a.k 1
7.b odd 2 1 588.2.a.a 1
7.c even 3 2 588.2.i.b 2
7.d odd 6 2 84.2.i.a ✓ 2
8.b even 2 1 9408.2.a.i 1
8.d odd 2 1 9408.2.a.bx 1
12.b even 2 1 7056.2.a.o 1
21.c even 2 1 1764.2.a.h 1
21.g even 6 2 252.2.k.a 2
21.h odd 6 2 1764.2.k.j 2
28.d even 2 1 2352.2.a.o 1
28.f even 6 2 336.2.q.c 2
28.g odd 6 2 2352.2.q.q 2
35.i odd 6 2 2100.2.q.b 2
35.k even 12 4 2100.2.bc.a 4
56.e even 2 1 9408.2.a.bi 1
56.h odd 2 1 9408.2.a.cx 1
56.j odd 6 2 1344.2.q.b 2
56.m even 6 2 1344.2.q.n 2
63.i even 6 2 2268.2.i.b 2
63.k odd 6 2 2268.2.l.b 2
63.s even 6 2 2268.2.l.g 2
63.t odd 6 2 2268.2.i.g 2
84.h odd 2 1 7056.2.a.bs 1
84.j odd 6 2 1008.2.s.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.i.a ✓ 2 7.d odd 6 2
252.2.k.a 2 21.g even 6 2
336.2.q.c 2 28.f even 6 2
588.2.a.a 1 7.b odd 2 1
588.2.a.f 1 1.a even 1 1 trivial
588.2.i.b 2 7.c even 3 2
1008.2.s.c 2 84.j odd 6 2
1344.2.q.b 2 56.j odd 6 2
1344.2.q.n 2 56.m even 6 2
1764.2.a.c 1 3.b odd 2 1
1764.2.a.h 1 21.c even 2 1
1764.2.k.j 2 21.h odd 6 2
2100.2.q.b 2 35.i odd 6 2
2100.2.bc.a 4 35.k even 12 4
2268.2.i.b 2 63.i even 6 2
2268.2.i.g 2 63.t odd 6 2
2268.2.l.b 2 63.k odd 6 2
2268.2.l.g 2 63.s even 6 2
2352.2.a.k 1 4.b odd 2 1
2352.2.a.o 1 28.d even 2 1
2352.2.q.q 2 28.g odd 6 2
7056.2.a.o 1 12.b even 2 1
7056.2.a.bs 1 84.h odd 2 1
9408.2.a.i 1 8.b even 2 1
9408.2.a.bi 1 56.e even 2 1
9408.2.a.bx 1 8.d odd 2 1
9408.2.a.cx 1 56.h odd 2 1

Hecke kernels

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

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