Properties

Label 567.2.i.b
Level $567$
Weight $2$
Character orbit 567.i
Analytic conductor $4.528$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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(215,567)]
 
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([5, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.215");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.i (of order \(6\), 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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{4} + (2 \zeta_{6} + 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{4} + (2 \zeta_{6} + 1) q^{7} + ( - \zeta_{6} + 2) q^{13} + 4 q^{16} + (3 \zeta_{6} - 6) q^{19} + 5 \zeta_{6} q^{25} + (4 \zeta_{6} + 2) q^{28} + ( - 10 \zeta_{6} + 5) q^{31} - \zeta_{6} q^{37} + ( - 5 \zeta_{6} + 5) q^{43} + (8 \zeta_{6} - 3) q^{49} + ( - 2 \zeta_{6} + 4) q^{52} + (8 \zeta_{6} - 4) q^{61} + 8 q^{64} + 11 q^{67} + ( - 9 \zeta_{6} - 9) q^{73} + (6 \zeta_{6} - 12) q^{76} - 13 q^{79} + (\zeta_{6} + 4) q^{91} + ( - 8 \zeta_{6} - 8) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{4} + 4 q^{7} + 3 q^{13} + 8 q^{16} - 9 q^{19} + 5 q^{25} + 8 q^{28} - q^{37} + 5 q^{43} + 2 q^{49} + 6 q^{52} + 16 q^{64} + 22 q^{67} - 27 q^{73} - 18 q^{76} - 26 q^{79} + 9 q^{91} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(407\)
\(\chi(n)\) \(\zeta_{6}\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 2.00000 0 0 2.00000 1.73205i 0 0 0
269.1 0 0 2.00000 0 0 2.00000 + 1.73205i 0 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
63.i even 6 1 inner
63.t 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.i.b 2
3.b odd 2 1 CM 567.2.i.b 2
7.d odd 6 1 567.2.s.a 2
9.c even 3 1 21.2.g.a 2
9.c even 3 1 567.2.s.a 2
9.d odd 6 1 21.2.g.a 2
9.d odd 6 1 567.2.s.a 2
21.g even 6 1 567.2.s.a 2
36.f odd 6 1 336.2.bc.c 2
36.h even 6 1 336.2.bc.c 2
45.h odd 6 1 525.2.t.c 2
45.j even 6 1 525.2.t.c 2
45.k odd 12 2 525.2.q.d 4
45.l even 12 2 525.2.q.d 4
63.g even 3 1 147.2.g.a 2
63.h even 3 1 147.2.c.a 2
63.i even 6 1 147.2.c.a 2
63.i even 6 1 inner 567.2.i.b 2
63.j odd 6 1 147.2.c.a 2
63.k odd 6 1 21.2.g.a 2
63.l odd 6 1 147.2.g.a 2
63.n odd 6 1 147.2.g.a 2
63.o even 6 1 147.2.g.a 2
63.s even 6 1 21.2.g.a 2
63.t odd 6 1 147.2.c.a 2
63.t odd 6 1 inner 567.2.i.b 2
252.n even 6 1 336.2.bc.c 2
252.r odd 6 1 2352.2.k.c 2
252.u odd 6 1 2352.2.k.c 2
252.bb even 6 1 2352.2.k.c 2
252.bj even 6 1 2352.2.k.c 2
252.bn odd 6 1 336.2.bc.c 2
315.u even 6 1 525.2.t.c 2
315.bn odd 6 1 525.2.t.c 2
315.bw odd 12 2 525.2.q.d 4
315.cg even 12 2 525.2.q.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.g.a 2 9.c even 3 1
21.2.g.a 2 9.d odd 6 1
21.2.g.a 2 63.k odd 6 1
21.2.g.a 2 63.s even 6 1
147.2.c.a 2 63.h even 3 1
147.2.c.a 2 63.i even 6 1
147.2.c.a 2 63.j odd 6 1
147.2.c.a 2 63.t odd 6 1
147.2.g.a 2 63.g even 3 1
147.2.g.a 2 63.l odd 6 1
147.2.g.a 2 63.n odd 6 1
147.2.g.a 2 63.o even 6 1
336.2.bc.c 2 36.f odd 6 1
336.2.bc.c 2 36.h even 6 1
336.2.bc.c 2 252.n even 6 1
336.2.bc.c 2 252.bn odd 6 1
525.2.q.d 4 45.k odd 12 2
525.2.q.d 4 45.l even 12 2
525.2.q.d 4 315.bw odd 12 2
525.2.q.d 4 315.cg even 12 2
525.2.t.c 2 45.h odd 6 1
525.2.t.c 2 45.j even 6 1
525.2.t.c 2 315.u even 6 1
525.2.t.c 2 315.bn odd 6 1
567.2.i.b 2 1.a even 1 1 trivial
567.2.i.b 2 3.b odd 2 1 CM
567.2.i.b 2 63.i even 6 1 inner
567.2.i.b 2 63.t odd 6 1 inner
567.2.s.a 2 7.d odd 6 1
567.2.s.a 2 9.c even 3 1
567.2.s.a 2 9.d odd 6 1
567.2.s.a 2 21.g even 6 1
2352.2.k.c 2 252.r odd 6 1
2352.2.k.c 2 252.u odd 6 1
2352.2.k.c 2 252.bb even 6 1
2352.2.k.c 2 252.bj even 6 1

Hecke kernels

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

\( T_{2} \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} + 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 3T + 3 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 9T + 27 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 75 \) Copy content Toggle raw display
$37$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 48 \) Copy content Toggle raw display
$67$ \( (T - 11)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 27T + 243 \) Copy content Toggle raw display
$79$ \( (T + 13)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
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