Properties

Label 588.6.a.b
Level $588$
Weight $6$
Character orbit 588.a
Self dual yes
Analytic conductor $94.306$
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] = [588,6,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: \(1\)
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)
 
\(f(q)\) \(=\) \( q - 9 q^{3} + 34 q^{5} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 9 q^{3} + 34 q^{5} + 81 q^{9} - 332 q^{11} + 1026 q^{13} - 306 q^{15} - 922 q^{17} - 452 q^{19} - 3776 q^{23} - 1969 q^{25} - 729 q^{27} + 1166 q^{29} + 9792 q^{31} + 2988 q^{33} + 2390 q^{37} - 9234 q^{39} + 7230 q^{41} + 4652 q^{43} + 2754 q^{45} - 24672 q^{47} + 8298 q^{51} + 1110 q^{53} - 11288 q^{55} + 4068 q^{57} - 46892 q^{59} + 9762 q^{61} + 34884 q^{65} - 26252 q^{67} + 33984 q^{69} + 65440 q^{71} + 5606 q^{73} + 17721 q^{75} - 9840 q^{79} + 6561 q^{81} - 61108 q^{83} - 31348 q^{85} - 10494 q^{87} + 62958 q^{89} - 88128 q^{93} - 15368 q^{95} + 37838 q^{97} - 26892 q^{99}+O(q^{100}) \) 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 34.0000 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.b 1
7.b odd 2 1 84.6.a.b 1
7.c even 3 2 588.6.i.e 2
7.d odd 6 2 588.6.i.c 2
21.c even 2 1 252.6.a.c 1
28.d even 2 1 336.6.a.e 1
84.h odd 2 1 1008.6.a.u 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.6.a.b 1 7.b odd 2 1
252.6.a.c 1 21.c even 2 1
336.6.a.e 1 28.d even 2 1
588.6.a.b 1 1.a even 1 1 trivial
588.6.i.c 2 7.d odd 6 2
588.6.i.e 2 7.c even 3 2
1008.6.a.u 1 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 34 \) 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
$5$ \( T - 34 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T + 332 \) Copy content Toggle raw display
$13$ \( T - 1026 \) Copy content Toggle raw display
$17$ \( T + 922 \) Copy content Toggle raw display
$19$ \( T + 452 \) Copy content Toggle raw display
$23$ \( T + 3776 \) Copy content Toggle raw display
$29$ \( T - 1166 \) Copy content Toggle raw display
$31$ \( T - 9792 \) Copy content Toggle raw display
$37$ \( T - 2390 \) Copy content Toggle raw display
$41$ \( T - 7230 \) Copy content Toggle raw display
$43$ \( T - 4652 \) Copy content Toggle raw display
$47$ \( T + 24672 \) Copy content Toggle raw display
$53$ \( T - 1110 \) Copy content Toggle raw display
$59$ \( T + 46892 \) Copy content Toggle raw display
$61$ \( T - 9762 \) Copy content Toggle raw display
$67$ \( T + 26252 \) Copy content Toggle raw display
$71$ \( T - 65440 \) Copy content Toggle raw display
$73$ \( T - 5606 \) Copy content Toggle raw display
$79$ \( T + 9840 \) Copy content Toggle raw display
$83$ \( T + 61108 \) Copy content Toggle raw display
$89$ \( T - 62958 \) Copy content Toggle raw display
$97$ \( T - 37838 \) Copy content Toggle raw display
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