Properties

Label 567.1.d.c
Level $567$
Weight $1$
Character orbit 567.d
Self dual yes
Analytic conductor $0.283$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -7
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,1,Mod(244,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.244");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 567.d (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: yes
Analytic conductor: \(0.282969862163\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.6751269.1

$q$-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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{2} + 2 q^{4} - q^{7} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{2} + 2 q^{4} - q^{7} - \beta q^{8} + \beta q^{11} + \beta q^{14} + q^{16} - 3 q^{22} + q^{25} - 2 q^{28} + q^{37} + q^{43} + 2 \beta q^{44} + q^{49} - \beta q^{50} + \beta q^{53} + \beta q^{56} - q^{64} - q^{67} - \beta q^{71} - \beta q^{74} - \beta q^{77} - q^{79} - \beta q^{86} - 3 q^{88} - \beta q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} - 2 q^{7} + 2 q^{16} - 6 q^{22} + 2 q^{25} - 4 q^{28} + 2 q^{37} + 2 q^{43} + 2 q^{49} - 2 q^{64} - 2 q^{67} - 2 q^{79} - 6 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/567\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(407\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
1.73205
−1.73205
−1.73205 0 2.00000 0 0 −1.00000 −1.73205 0 0
244.2 1.73205 0 2.00000 0 0 −1.00000 1.73205 0 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
7.b odd 2 1 CM by \(\Q(\sqrt{-7}) \)
3.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.1.d.c 2
3.b odd 2 1 inner 567.1.d.c 2
7.b odd 2 1 CM 567.1.d.c 2
7.c even 3 2 3969.1.m.c 4
7.d odd 6 2 3969.1.m.c 4
9.c even 3 2 567.1.l.d 4
9.d odd 6 2 567.1.l.d 4
21.c even 2 1 inner 567.1.d.c 2
21.g even 6 2 3969.1.m.c 4
21.h odd 6 2 3969.1.m.c 4
63.g even 3 2 3969.1.k.e 4
63.h even 3 2 3969.1.t.e 4
63.i even 6 2 3969.1.t.e 4
63.j odd 6 2 3969.1.t.e 4
63.k odd 6 2 3969.1.k.e 4
63.l odd 6 2 567.1.l.d 4
63.n odd 6 2 3969.1.k.e 4
63.o even 6 2 567.1.l.d 4
63.s even 6 2 3969.1.k.e 4
63.t odd 6 2 3969.1.t.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.1.d.c 2 1.a even 1 1 trivial
567.1.d.c 2 3.b odd 2 1 inner
567.1.d.c 2 7.b odd 2 1 CM
567.1.d.c 2 21.c even 2 1 inner
567.1.l.d 4 9.c even 3 2
567.1.l.d 4 9.d odd 6 2
567.1.l.d 4 63.l odd 6 2
567.1.l.d 4 63.o even 6 2
3969.1.k.e 4 63.g even 3 2
3969.1.k.e 4 63.k odd 6 2
3969.1.k.e 4 63.n odd 6 2
3969.1.k.e 4 63.s even 6 2
3969.1.m.c 4 7.c even 3 2
3969.1.m.c 4 7.d odd 6 2
3969.1.m.c 4 21.g even 6 2
3969.1.m.c 4 21.h odd 6 2
3969.1.t.e 4 63.h even 3 2
3969.1.t.e 4 63.i even 6 2
3969.1.t.e 4 63.j odd 6 2
3969.1.t.e 4 63.t odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(567, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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