Properties

Label 64.4.a.c
Level $64$
Weight $4$
Character orbit 64.a
Self dual yes
Analytic conductor $3.776$
Analytic rank $1$
Dimension $1$
CM discriminant -4
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,4,Mod(1,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 64.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: \(3.77612224037\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 22 q^{5} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 22 q^{5} - 27 q^{9} + 18 q^{13} - 94 q^{17} + 359 q^{25} + 130 q^{29} - 214 q^{37} - 230 q^{41} + 594 q^{45} - 343 q^{49} - 518 q^{53} - 830 q^{61} - 396 q^{65} + 1098 q^{73} + 729 q^{81} + 2068 q^{85} - 1670 q^{89} + 594 q^{97}+O(q^{100}) \) 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 0 0 −22.0000 0 0 0 −27.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.4.a.c 1
3.b odd 2 1 576.4.a.y 1
4.b odd 2 1 CM 64.4.a.c 1
5.b even 2 1 1600.4.a.ba 1
8.b even 2 1 32.4.a.b 1
8.d odd 2 1 32.4.a.b 1
12.b even 2 1 576.4.a.y 1
16.e even 4 2 256.4.b.d 2
16.f odd 4 2 256.4.b.d 2
20.d odd 2 1 1600.4.a.ba 1
24.f even 2 1 288.4.a.a 1
24.h odd 2 1 288.4.a.a 1
40.e odd 2 1 800.4.a.f 1
40.f even 2 1 800.4.a.f 1
40.i odd 4 2 800.4.c.g 2
40.k even 4 2 800.4.c.g 2
56.e even 2 1 1568.4.a.g 1
56.h odd 2 1 1568.4.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.4.a.b 1 8.b even 2 1
32.4.a.b 1 8.d odd 2 1
64.4.a.c 1 1.a even 1 1 trivial
64.4.a.c 1 4.b odd 2 1 CM
256.4.b.d 2 16.e even 4 2
256.4.b.d 2 16.f odd 4 2
288.4.a.a 1 24.f even 2 1
288.4.a.a 1 24.h odd 2 1
576.4.a.y 1 3.b odd 2 1
576.4.a.y 1 12.b even 2 1
800.4.a.f 1 40.e odd 2 1
800.4.a.f 1 40.f even 2 1
800.4.c.g 2 40.i odd 4 2
800.4.c.g 2 40.k even 4 2
1568.4.a.g 1 56.e even 2 1
1568.4.a.g 1 56.h odd 2 1
1600.4.a.ba 1 5.b even 2 1
1600.4.a.ba 1 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(64))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T + 22 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T - 18 \) Copy content Toggle raw display
$17$ \( T + 94 \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 130 \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T + 214 \) Copy content Toggle raw display
$41$ \( T + 230 \) Copy content Toggle raw display
$43$ \( T \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 518 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 830 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 1098 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T + 1670 \) Copy content Toggle raw display
$97$ \( T - 594 \) Copy content Toggle raw display
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