Properties

Label 3264.1.cw.d
Level $3264$
Weight $1$
Character orbit 3264.cw
Analytic conductor $1.629$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -8
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3264,1,Mod(161,3264)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3264, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 4, 4, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3264.161");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3264 = 2^{6} \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3264.cw (of order \(8\), degree \(4\), 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: \(1.62894820123\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.0.136140523573248.1

$q$-expansion

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

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{3} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + q^{9} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{11} + \zeta_{8}^{3} q^{17} + \zeta_{8} q^{25} + q^{27} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{33} + (\zeta_{8}^{2} - \zeta_{8}) q^{41} + ( - \zeta_{8}^{2} + 1) q^{43} - \zeta_{8}^{3} q^{49} + \zeta_{8}^{3} q^{51} + ( - \zeta_{8}^{2} - 1) q^{59} + ( - \zeta_{8}^{3} - \zeta_{8}) q^{67} + (\zeta_{8}^{3} + \zeta_{8}^{2}) q^{73} + \zeta_{8} q^{75} + q^{81} + (\zeta_{8}^{2} - 1) q^{83} + (\zeta_{8}^{3} - \zeta_{8}) q^{89} + ( - \zeta_{8} + 1) q^{97} + ( - \zeta_{8}^{2} + \zeta_{8}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{3} + 4 q^{9} + 4 q^{27} + 4 q^{43} - 4 q^{59} + 4 q^{81} - 4 q^{83} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3264\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(2177\) \(2245\) \(2689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{8}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 1.00000 0 0 0 0 0 1.00000 0
1505.1 0 1.00000 0 0 0 0 0 1.00000 0
1889.1 0 1.00000 0 0 0 0 0 1.00000 0
3041.1 0 1.00000 0 0 0 0 0 1.00000 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
204.p even 8 1 inner
408.be odd 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3264.1.cw.d yes 4
3.b odd 2 1 3264.1.cw.c yes 4
4.b odd 2 1 3264.1.cw.a 4
8.b even 2 1 3264.1.cw.a 4
8.d odd 2 1 CM 3264.1.cw.d yes 4
12.b even 2 1 3264.1.cw.b yes 4
17.d even 8 1 3264.1.cw.b yes 4
24.f even 2 1 3264.1.cw.c yes 4
24.h odd 2 1 3264.1.cw.b yes 4
51.g odd 8 1 3264.1.cw.a 4
68.g odd 8 1 3264.1.cw.c yes 4
136.o even 8 1 3264.1.cw.c yes 4
136.p odd 8 1 3264.1.cw.b yes 4
204.p even 8 1 inner 3264.1.cw.d yes 4
408.bd even 8 1 3264.1.cw.a 4
408.be odd 8 1 inner 3264.1.cw.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3264.1.cw.a 4 4.b odd 2 1
3264.1.cw.a 4 8.b even 2 1
3264.1.cw.a 4 51.g odd 8 1
3264.1.cw.a 4 408.bd even 8 1
3264.1.cw.b yes 4 12.b even 2 1
3264.1.cw.b yes 4 17.d even 8 1
3264.1.cw.b yes 4 24.h odd 2 1
3264.1.cw.b yes 4 136.p odd 8 1
3264.1.cw.c yes 4 3.b odd 2 1
3264.1.cw.c yes 4 24.f even 2 1
3264.1.cw.c yes 4 68.g odd 8 1
3264.1.cw.c yes 4 136.o even 8 1
3264.1.cw.d yes 4 1.a even 1 1 trivial
3264.1.cw.d yes 4 8.d odd 2 1 CM
3264.1.cw.d yes 4 204.p even 8 1 inner
3264.1.cw.d yes 4 408.be odd 8 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(3264, [\chi])\):

\( T_{11}^{4} + 2T_{11}^{2} - 4T_{11} + 2 \) Copy content Toggle raw display
\( T_{41}^{4} + 2T_{41}^{2} + 4T_{41} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T - 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 1 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} + 2 T^{2} + 4 T + 2 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 2 T^{2} - 4 T + 2 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 4 T^{3} + 6 T^{2} - 4 T + 2 \) Copy content Toggle raw display
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