Properties

Label 64.4.a.d
Level $64$
Weight $4$
Character orbit 64.a
Self dual yes
Analytic conductor $3.776$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
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)
 
\(f(q)\) \(=\) \( q + 4 q^{3} + 2 q^{5} + 24 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{3} + 2 q^{5} + 24 q^{7} - 11 q^{9} + 44 q^{11} - 22 q^{13} + 8 q^{15} + 50 q^{17} - 44 q^{19} + 96 q^{21} - 56 q^{23} - 121 q^{25} - 152 q^{27} - 198 q^{29} - 160 q^{31} + 176 q^{33} + 48 q^{35} + 162 q^{37} - 88 q^{39} - 198 q^{41} - 52 q^{43} - 22 q^{45} + 528 q^{47} + 233 q^{49} + 200 q^{51} + 242 q^{53} + 88 q^{55} - 176 q^{57} + 668 q^{59} - 550 q^{61} - 264 q^{63} - 44 q^{65} - 188 q^{67} - 224 q^{69} + 728 q^{71} + 154 q^{73} - 484 q^{75} + 1056 q^{77} - 656 q^{79} - 311 q^{81} - 236 q^{83} + 100 q^{85} - 792 q^{87} + 714 q^{89} - 528 q^{91} - 640 q^{93} - 88 q^{95} - 478 q^{97} - 484 q^{99}+O(q^{100}) \) 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 4.00000 0 2.00000 0 24.0000 0 −11.0000 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 64.4.a.d 1
3.b odd 2 1 576.4.a.k 1
4.b odd 2 1 64.4.a.b 1
5.b even 2 1 1600.4.a.o 1
8.b even 2 1 8.4.a.a 1
8.d odd 2 1 16.4.a.a 1
12.b even 2 1 576.4.a.j 1
16.e even 4 2 256.4.b.a 2
16.f odd 4 2 256.4.b.g 2
20.d odd 2 1 1600.4.a.bm 1
24.f even 2 1 144.4.a.e 1
24.h odd 2 1 72.4.a.c 1
40.e odd 2 1 400.4.a.g 1
40.f even 2 1 200.4.a.g 1
40.i odd 4 2 200.4.c.e 2
40.k even 4 2 400.4.c.i 2
56.e even 2 1 784.4.a.e 1
56.h odd 2 1 392.4.a.e 1
56.j odd 6 2 392.4.i.b 2
56.p even 6 2 392.4.i.g 2
72.j odd 6 2 648.4.i.e 2
72.n even 6 2 648.4.i.h 2
88.b odd 2 1 968.4.a.a 1
88.g even 2 1 1936.4.a.l 1
104.e even 2 1 1352.4.a.a 1
120.i odd 2 1 1800.4.a.d 1
120.w even 4 2 1800.4.f.u 2
136.h even 2 1 2312.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.4.a.a 1 8.b even 2 1
16.4.a.a 1 8.d odd 2 1
64.4.a.b 1 4.b odd 2 1
64.4.a.d 1 1.a even 1 1 trivial
72.4.a.c 1 24.h odd 2 1
144.4.a.e 1 24.f even 2 1
200.4.a.g 1 40.f even 2 1
200.4.c.e 2 40.i odd 4 2
256.4.b.a 2 16.e even 4 2
256.4.b.g 2 16.f odd 4 2
392.4.a.e 1 56.h odd 2 1
392.4.i.b 2 56.j odd 6 2
392.4.i.g 2 56.p even 6 2
400.4.a.g 1 40.e odd 2 1
400.4.c.i 2 40.k even 4 2
576.4.a.j 1 12.b even 2 1
576.4.a.k 1 3.b odd 2 1
648.4.i.e 2 72.j odd 6 2
648.4.i.h 2 72.n even 6 2
784.4.a.e 1 56.e even 2 1
968.4.a.a 1 88.b odd 2 1
1352.4.a.a 1 104.e even 2 1
1600.4.a.o 1 5.b even 2 1
1600.4.a.bm 1 20.d odd 2 1
1800.4.a.d 1 120.i odd 2 1
1800.4.f.u 2 120.w even 4 2
1936.4.a.l 1 88.g even 2 1
2312.4.a.a 1 136.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 4 \) 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
$3$ \( T - 4 \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 24 \) Copy content Toggle raw display
$11$ \( T - 44 \) Copy content Toggle raw display
$13$ \( T + 22 \) Copy content Toggle raw display
$17$ \( T - 50 \) Copy content Toggle raw display
$19$ \( T + 44 \) Copy content Toggle raw display
$23$ \( T + 56 \) Copy content Toggle raw display
$29$ \( T + 198 \) Copy content Toggle raw display
$31$ \( T + 160 \) Copy content Toggle raw display
$37$ \( T - 162 \) Copy content Toggle raw display
$41$ \( T + 198 \) Copy content Toggle raw display
$43$ \( T + 52 \) Copy content Toggle raw display
$47$ \( T - 528 \) Copy content Toggle raw display
$53$ \( T - 242 \) Copy content Toggle raw display
$59$ \( T - 668 \) Copy content Toggle raw display
$61$ \( T + 550 \) Copy content Toggle raw display
$67$ \( T + 188 \) Copy content Toggle raw display
$71$ \( T - 728 \) Copy content Toggle raw display
$73$ \( T - 154 \) Copy content Toggle raw display
$79$ \( T + 656 \) Copy content Toggle raw display
$83$ \( T + 236 \) Copy content Toggle raw display
$89$ \( T - 714 \) Copy content Toggle raw display
$97$ \( T + 478 \) Copy content Toggle raw display
show more
show less