Properties

Label 64.4.b.b
Level $64$
Weight $4$
Character orbit 64.b
Analytic conductor $3.776$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,4,Mod(33,64)]
 
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, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.33");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 64.b (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(3.77612224037\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + \beta_{3} q^{7} + 23 q^{9} + 21 \beta_1 q^{11} - 3 \beta_{2} q^{13} + \beta_{3} q^{15} - 6 q^{17} - 47 \beta_1 q^{19} + 4 \beta_{2} q^{21} - 5 \beta_{3} q^{23} - 67 q^{25} + 50 \beta_1 q^{27} + 17 \beta_{2} q^{29} - 4 \beta_{3} q^{31} - 84 q^{33} - 192 \beta_1 q^{35} - \beta_{2} q^{37} + 3 \beta_{3} q^{39} - 54 q^{41} + 221 \beta_1 q^{43} - 23 \beta_{2} q^{45} - 2 \beta_{3} q^{47} + 425 q^{49} - 6 \beta_1 q^{51} - 5 \beta_{2} q^{53} + 21 \beta_{3} q^{55} + 188 q^{57} + 69 \beta_1 q^{59} - 31 \beta_{2} q^{61} + 23 \beta_{3} q^{63} - 576 q^{65} + 89 \beta_1 q^{67} - 20 \beta_{2} q^{69} - 31 \beta_{3} q^{71} + 434 q^{73} - 67 \beta_1 q^{75} + 84 \beta_{2} q^{77} - 6 \beta_{3} q^{79} + 421 q^{81} - 135 \beta_1 q^{83} + 6 \beta_{2} q^{85} - 17 \beta_{3} q^{87} - 1182 q^{89} - 576 \beta_1 q^{91} - 16 \beta_{2} q^{93} - 47 \beta_{3} q^{95} - 1238 q^{97} + 483 \beta_1 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 92 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 92 q^{9} - 24 q^{17} - 268 q^{25} - 336 q^{33} - 216 q^{41} + 1700 q^{49} + 752 q^{57} - 2304 q^{65} + 1736 q^{73} + 1684 q^{81} - 4728 q^{89} - 4952 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 16\zeta_{12}^{2} - 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -16\zeta_{12}^{3} + 32\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + 8\beta_1 ) / 32 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
−0.866025 0.500000i
0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0 2.00000i 0 13.8564i 0 −27.7128 0 23.0000 0
33.2 0 2.00000i 0 13.8564i 0 27.7128 0 23.0000 0
33.3 0 2.00000i 0 13.8564i 0 27.7128 0 23.0000 0
33.4 0 2.00000i 0 13.8564i 0 −27.7128 0 23.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 64.4.b.b 4
3.b odd 2 1 576.4.d.e 4
4.b odd 2 1 inner 64.4.b.b 4
8.b even 2 1 inner 64.4.b.b 4
8.d odd 2 1 inner 64.4.b.b 4
12.b even 2 1 576.4.d.e 4
16.e even 4 1 256.4.a.i 2
16.e even 4 1 256.4.a.m 2
16.f odd 4 1 256.4.a.i 2
16.f odd 4 1 256.4.a.m 2
24.f even 2 1 576.4.d.e 4
24.h odd 2 1 576.4.d.e 4
48.i odd 4 1 2304.4.a.ba 2
48.i odd 4 1 2304.4.a.bi 2
48.k even 4 1 2304.4.a.ba 2
48.k even 4 1 2304.4.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
64.4.b.b 4 1.a even 1 1 trivial
64.4.b.b 4 4.b odd 2 1 inner
64.4.b.b 4 8.b even 2 1 inner
64.4.b.b 4 8.d odd 2 1 inner
256.4.a.i 2 16.e even 4 1
256.4.a.i 2 16.f odd 4 1
256.4.a.m 2 16.e even 4 1
256.4.a.m 2 16.f odd 4 1
576.4.d.e 4 3.b odd 2 1
576.4.d.e 4 12.b even 2 1
576.4.d.e 4 24.f even 2 1
576.4.d.e 4 24.h odd 2 1
2304.4.a.ba 2 48.i odd 4 1
2304.4.a.ba 2 48.k even 4 1
2304.4.a.bi 2 48.i odd 4 1
2304.4.a.bi 2 48.k even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) acting on \(S_{4}^{\mathrm{new}}(64, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 768)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + 1764)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 1728)^{2} \) Copy content Toggle raw display
$17$ \( (T + 6)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 8836)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 19200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 55488)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12288)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 192)^{2} \) Copy content Toggle raw display
$41$ \( (T + 54)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 195364)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 3072)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4800)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 19044)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 184512)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 31684)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 738048)^{2} \) Copy content Toggle raw display
$73$ \( (T - 434)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 27648)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 72900)^{2} \) Copy content Toggle raw display
$89$ \( (T + 1182)^{4} \) Copy content Toggle raw display
$97$ \( (T + 1238)^{4} \) Copy content Toggle raw display
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