Properties

Label 588.6.a.e
Level $588$
Weight $6$
Character orbit 588.a
Self dual yes
Analytic conductor $94.306$
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] = [588,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("588.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(94.3056860500\)
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)
 
\(f(q)\) \(=\) \( q + 9 q^{3} - 6 q^{5} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} - 6 q^{5} + 81 q^{9} - 108 q^{11} + 346 q^{13} - 54 q^{15} + 1398 q^{17} + 1012 q^{19} - 1536 q^{23} - 3089 q^{25} + 729 q^{27} - 3762 q^{29} + 736 q^{31} - 972 q^{33} + 2054 q^{37} + 3114 q^{39} + 15534 q^{41} + 11036 q^{43} - 486 q^{45} - 4560 q^{47} + 12582 q^{51} - 7962 q^{53} + 648 q^{55} + 9108 q^{57} + 7020 q^{59} - 26870 q^{61} - 2076 q^{65} + 52148 q^{67} - 13824 q^{69} - 2544 q^{71} + 9766 q^{73} - 27801 q^{75} + 68672 q^{79} + 6561 q^{81} + 61668 q^{83} - 8388 q^{85} - 33858 q^{87} + 41454 q^{89} + 6624 q^{93} - 6072 q^{95} + 111262 q^{97} - 8748 q^{99}+O(q^{100}) \) 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 9.00000 0 −6.00000 0 0 0 81.0000 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 588.6.a.e 1
7.b odd 2 1 84.6.a.a 1
7.c even 3 2 588.6.i.b 2
7.d odd 6 2 588.6.i.f 2
21.c even 2 1 252.6.a.b 1
28.d even 2 1 336.6.a.n 1
84.h odd 2 1 1008.6.a.o 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.a 1 7.b odd 2 1
252.6.a.b 1 21.c even 2 1
336.6.a.n 1 28.d even 2 1
588.6.a.e 1 1.a even 1 1 trivial
588.6.i.b 2 7.c even 3 2
588.6.i.f 2 7.d odd 6 2
1008.6.a.o 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 6 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(588))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T - 9 \) Copy content Toggle raw display
$5$ \( T + 6 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 108 \) Copy content Toggle raw display
$13$ \( T - 346 \) Copy content Toggle raw display
$17$ \( T - 1398 \) Copy content Toggle raw display
$19$ \( T - 1012 \) Copy content Toggle raw display
$23$ \( T + 1536 \) Copy content Toggle raw display
$29$ \( T + 3762 \) Copy content Toggle raw display
$31$ \( T - 736 \) Copy content Toggle raw display
$37$ \( T - 2054 \) Copy content Toggle raw display
$41$ \( T - 15534 \) Copy content Toggle raw display
$43$ \( T - 11036 \) Copy content Toggle raw display
$47$ \( T + 4560 \) Copy content Toggle raw display
$53$ \( T + 7962 \) Copy content Toggle raw display
$59$ \( T - 7020 \) Copy content Toggle raw display
$61$ \( T + 26870 \) Copy content Toggle raw display
$67$ \( T - 52148 \) Copy content Toggle raw display
$71$ \( T + 2544 \) Copy content Toggle raw display
$73$ \( T - 9766 \) Copy content Toggle raw display
$79$ \( T - 68672 \) Copy content Toggle raw display
$83$ \( T - 61668 \) Copy content Toggle raw display
$89$ \( T - 41454 \) Copy content Toggle raw display
$97$ \( T - 111262 \) Copy content Toggle raw display
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