Properties

Label 567.2.f.n
Level $567$
Weight $2$
Character orbit 567.f
Analytic conductor $4.528$
Analytic rank $0$
Dimension $8$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(190,567)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(567, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.190");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.f (of order \(3\), degree \(2\), not 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: \(4.52751779461\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.49787136.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{4} q^{4} - 2 \beta_{7} q^{5} + ( - \beta_{5} + 1) q^{7} + ( - \beta_{7} - \beta_{3} + \cdots + \beta_1) q^{8} + 2 q^{10} + (\beta_{3} + \beta_1) q^{11} - 4 \beta_{5} q^{13}+ \cdots + ( - \beta_{3} + \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} + 4 q^{7} + 16 q^{10} - 16 q^{13} - 2 q^{16} + 8 q^{19} - 14 q^{22} - 20 q^{25} - 4 q^{28} - 12 q^{34} + 24 q^{37} + 12 q^{40} + 16 q^{43} - 24 q^{46} - 4 q^{49} - 8 q^{52} + 56 q^{55} + 40 q^{58}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} - 5\nu^{3} + 28\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 5\nu^{5} + 15\nu^{3} + 22\nu ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} + 5\nu^{3} + 24\nu ) / 40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 5\nu^{4} + 5\nu^{2} - 2 ) / 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{6} - 5\nu^{4} - 15\nu^{2} - 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 3\nu^{4} + \nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{7} + 15\nu^{5} + 25\nu^{3} + 88\nu ) / 40 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{3} - \beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{6} - 4\beta_{5} + \beta_{4} - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + \beta_{3} + 4\beta_{2} - 4\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{7} - 11\beta_{3} + 7\beta_{2} + 11\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 5\beta_{6} - 5\beta_{5} - 10\beta_{4} - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 13\beta_{7} - 4\beta_{3} - 13\beta_{2} - 17\beta_1 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(1\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
190.1
−1.09445 0.895644i
−0.228425 1.39564i
0.228425 + 1.39564i
1.09445 + 0.895644i
−1.09445 + 0.895644i
−0.228425 + 1.39564i
0.228425 1.39564i
1.09445 0.895644i
−1.09445 1.89564i 0 −1.39564 + 2.41733i −0.456850 + 0.791288i 0 0.500000 + 0.866025i 1.73205 0 2.00000
190.2 −0.228425 0.395644i 0 0.895644 1.55130i −2.18890 + 3.79129i 0 0.500000 + 0.866025i −1.73205 0 2.00000
190.3 0.228425 + 0.395644i 0 0.895644 1.55130i 2.18890 3.79129i 0 0.500000 + 0.866025i 1.73205 0 2.00000
190.4 1.09445 + 1.89564i 0 −1.39564 + 2.41733i 0.456850 0.791288i 0 0.500000 + 0.866025i −1.73205 0 2.00000
379.1 −1.09445 + 1.89564i 0 −1.39564 2.41733i −0.456850 0.791288i 0 0.500000 0.866025i 1.73205 0 2.00000
379.2 −0.228425 + 0.395644i 0 0.895644 + 1.55130i −2.18890 3.79129i 0 0.500000 0.866025i −1.73205 0 2.00000
379.3 0.228425 0.395644i 0 0.895644 + 1.55130i 2.18890 + 3.79129i 0 0.500000 0.866025i 1.73205 0 2.00000
379.4 1.09445 1.89564i 0 −1.39564 2.41733i 0.456850 + 0.791288i 0 0.500000 0.866025i −1.73205 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 190.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.f.n 8
3.b odd 2 1 inner 567.2.f.n 8
9.c even 3 1 567.2.a.i 4
9.c even 3 1 inner 567.2.f.n 8
9.d odd 6 1 567.2.a.i 4
9.d odd 6 1 inner 567.2.f.n 8
36.f odd 6 1 9072.2.a.ci 4
36.h even 6 1 9072.2.a.ci 4
63.l odd 6 1 3969.2.a.u 4
63.o even 6 1 3969.2.a.u 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.a.i 4 9.c even 3 1
567.2.a.i 4 9.d odd 6 1
567.2.f.n 8 1.a even 1 1 trivial
567.2.f.n 8 3.b odd 2 1 inner
567.2.f.n 8 9.c even 3 1 inner
567.2.f.n 8 9.d odd 6 1 inner
3969.2.a.u 4 63.l odd 6 1
3969.2.a.u 4 63.o even 6 1
9072.2.a.ci 4 36.f odd 6 1
9072.2.a.ci 4 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(567, [\chi])\):

\( T_{2}^{8} + 5T_{2}^{6} + 24T_{2}^{4} + 5T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{8} + 20T_{5}^{6} + 384T_{5}^{4} + 320T_{5}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 5 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 20 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 7 T^{2} + 49)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 4 T + 16)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 2 T - 20)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 80 T^{6} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( (T^{4} + 84 T^{2} + 7056)^{2} \) Copy content Toggle raw display
$37$ \( (T - 3)^{8} \) Copy content Toggle raw display
$41$ \( T^{8} + 20 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$43$ \( (T^{4} - 8 T^{3} + 69 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 180 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$53$ \( (T^{2} - 75)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 12 T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{3} + \cdots + 784)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 8 T^{3} + 69 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 150 T^{2} + 2601)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 12 T - 48)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 8 T^{3} + 69 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 132 T^{6} + \cdots + 12960000 \) Copy content Toggle raw display
$89$ \( (T^{4} - 80 T^{2} + 256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 6 T^{3} + \cdots + 144)^{2} \) Copy content Toggle raw display
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