Properties

Label 8.8.a.a
Level $8$
Weight $8$
Character orbit 8.a
Self dual yes
Analytic conductor $2.499$
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] = [8,8,Mod(1,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(2.49908020387\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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)
 
\(f(q)\) \(=\) \( q - 84 q^{3} - 82 q^{5} - 456 q^{7} + 4869 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 84 q^{3} - 82 q^{5} - 456 q^{7} + 4869 q^{9} - 2524 q^{11} - 10778 q^{13} + 6888 q^{15} - 11150 q^{17} + 4124 q^{19} + 38304 q^{21} + 81704 q^{23} - 71401 q^{25} - 225288 q^{27} + 99798 q^{29} - 40480 q^{31} + 212016 q^{33} + 37392 q^{35} - 419442 q^{37} + 905352 q^{39} + 141402 q^{41} - 690428 q^{43} - 399258 q^{45} - 682032 q^{47} - 615607 q^{49} + 936600 q^{51} + 1813118 q^{53} + 206968 q^{55} - 346416 q^{57} - 966028 q^{59} + 1887670 q^{61} - 2220264 q^{63} + 883796 q^{65} + 2965868 q^{67} - 6863136 q^{69} - 2548232 q^{71} - 1680326 q^{73} + 5997684 q^{75} + 1150944 q^{77} + 4038064 q^{79} + 8275689 q^{81} - 5385764 q^{83} + 914300 q^{85} - 8383032 q^{87} - 6473046 q^{89} + 4914768 q^{91} + 3400320 q^{93} - 338168 q^{95} - 6065758 q^{97} - 12289356 q^{99}+O(q^{100}) \) 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 −84.0000 0 −82.0000 0 −456.000 0 4869.00 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)

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 8.8.a.a 1
3.b odd 2 1 72.8.a.d 1
4.b odd 2 1 16.8.a.c 1
5.b even 2 1 200.8.a.i 1
5.c odd 4 2 200.8.c.a 2
7.b odd 2 1 392.8.a.d 1
8.b even 2 1 64.8.a.g 1
8.d odd 2 1 64.8.a.a 1
12.b even 2 1 144.8.a.g 1
16.e even 4 2 256.8.b.e 2
16.f odd 4 2 256.8.b.c 2
20.d odd 2 1 400.8.a.b 1
20.e even 4 2 400.8.c.b 2
24.f even 2 1 576.8.a.k 1
24.h odd 2 1 576.8.a.j 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 1.a even 1 1 trivial
16.8.a.c 1 4.b odd 2 1
64.8.a.a 1 8.d odd 2 1
64.8.a.g 1 8.b even 2 1
72.8.a.d 1 3.b odd 2 1
144.8.a.g 1 12.b even 2 1
200.8.a.i 1 5.b even 2 1
200.8.c.a 2 5.c odd 4 2
256.8.b.c 2 16.f odd 4 2
256.8.b.e 2 16.e even 4 2
392.8.a.d 1 7.b odd 2 1
400.8.a.b 1 20.d odd 2 1
400.8.c.b 2 20.e even 4 2
576.8.a.j 1 24.h odd 2 1
576.8.a.k 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} + 84 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(8))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 84 \) Copy content Toggle raw display
$5$ \( T + 82 \) Copy content Toggle raw display
$7$ \( T + 456 \) Copy content Toggle raw display
$11$ \( T + 2524 \) Copy content Toggle raw display
$13$ \( T + 10778 \) Copy content Toggle raw display
$17$ \( T + 11150 \) Copy content Toggle raw display
$19$ \( T - 4124 \) Copy content Toggle raw display
$23$ \( T - 81704 \) Copy content Toggle raw display
$29$ \( T - 99798 \) Copy content Toggle raw display
$31$ \( T + 40480 \) Copy content Toggle raw display
$37$ \( T + 419442 \) Copy content Toggle raw display
$41$ \( T - 141402 \) Copy content Toggle raw display
$43$ \( T + 690428 \) Copy content Toggle raw display
$47$ \( T + 682032 \) Copy content Toggle raw display
$53$ \( T - 1813118 \) Copy content Toggle raw display
$59$ \( T + 966028 \) Copy content Toggle raw display
$61$ \( T - 1887670 \) Copy content Toggle raw display
$67$ \( T - 2965868 \) Copy content Toggle raw display
$71$ \( T + 2548232 \) Copy content Toggle raw display
$73$ \( T + 1680326 \) Copy content Toggle raw display
$79$ \( T - 4038064 \) Copy content Toggle raw display
$83$ \( T + 5385764 \) Copy content Toggle raw display
$89$ \( T + 6473046 \) Copy content Toggle raw display
$97$ \( T + 6065758 \) Copy content Toggle raw display
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