Properties

Label 567.2.h.a
Level $567$
Weight $2$
Character orbit 567.h
Analytic conductor $4.528$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [567,2,Mod(298,567)]
 
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([4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("567.298");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 567 = 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 567.h (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: \(2\)
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: $\mathrm{SU}(2)[C_{3}]$

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

\(f(q)\) \(=\) \( q - 2 q^{2} + 2 q^{4} + (2 \zeta_{6} - 2) q^{5} + ( - 2 \zeta_{6} + 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{2} + 2 q^{4} + (2 \zeta_{6} - 2) q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + ( - 4 \zeta_{6} + 4) q^{10} - 2 \zeta_{6} q^{11} - \zeta_{6} q^{13} + (4 \zeta_{6} - 6) q^{14} - 4 q^{16} - \zeta_{6} q^{19} + (4 \zeta_{6} - 4) q^{20} + 4 \zeta_{6} q^{22} + \zeta_{6} q^{25} + 2 \zeta_{6} q^{26} + ( - 4 \zeta_{6} + 6) q^{28} + ( - 4 \zeta_{6} + 4) q^{29} + 9 q^{31} + 8 q^{32} + (6 \zeta_{6} - 2) q^{35} - 3 \zeta_{6} q^{37} + 2 \zeta_{6} q^{38} - 10 \zeta_{6} q^{41} + (5 \zeta_{6} - 5) q^{43} - 4 \zeta_{6} q^{44} + 6 q^{47} + ( - 8 \zeta_{6} + 5) q^{49} - 2 \zeta_{6} q^{50} - 2 \zeta_{6} q^{52} + ( - 12 \zeta_{6} + 12) q^{53} + 4 q^{55} + (8 \zeta_{6} - 8) q^{58} + 12 q^{59} + 10 q^{61} - 18 q^{62} - 8 q^{64} + 2 q^{65} - 5 q^{67} + ( - 12 \zeta_{6} + 4) q^{70} + 6 q^{71} + ( - 3 \zeta_{6} + 3) q^{73} + 6 \zeta_{6} q^{74} - 2 \zeta_{6} q^{76} + ( - 2 \zeta_{6} - 4) q^{77} - q^{79} + ( - 8 \zeta_{6} + 8) q^{80} + 20 \zeta_{6} q^{82} + ( - 6 \zeta_{6} + 6) q^{83} + ( - 10 \zeta_{6} + 10) q^{86} + 16 \zeta_{6} q^{89} + ( - \zeta_{6} - 2) q^{91} - 12 q^{94} + 2 q^{95} + ( - 6 \zeta_{6} + 6) q^{97} + (16 \zeta_{6} - 10) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{2} + 4 q^{4} - 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{2} + 4 q^{4} - 2 q^{5} + 4 q^{7} + 4 q^{10} - 2 q^{11} - q^{13} - 8 q^{14} - 8 q^{16} - q^{19} - 4 q^{20} + 4 q^{22} + q^{25} + 2 q^{26} + 8 q^{28} + 4 q^{29} + 18 q^{31} + 16 q^{32} + 2 q^{35} - 3 q^{37} + 2 q^{38} - 10 q^{41} - 5 q^{43} - 4 q^{44} + 12 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} + 12 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 20 q^{61} - 36 q^{62} - 16 q^{64} + 4 q^{65} - 10 q^{67} - 4 q^{70} + 12 q^{71} + 3 q^{73} + 6 q^{74} - 2 q^{76} - 10 q^{77} - 2 q^{79} + 8 q^{80} + 20 q^{82} + 6 q^{83} + 10 q^{86} + 16 q^{89} - 5 q^{91} - 24 q^{94} + 4 q^{95} + 6 q^{97} - 4 q^{98}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
298.1
0.500000 + 0.866025i
0.500000 0.866025i
−2.00000 0 2.00000 −1.00000 + 1.73205i 0 2.00000 1.73205i 0 0 2.00000 3.46410i
352.1 −2.00000 0 2.00000 −1.00000 1.73205i 0 2.00000 + 1.73205i 0 0 2.00000 + 3.46410i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.h.a 2
3.b odd 2 1 567.2.h.f 2
7.c even 3 1 567.2.g.f 2
9.c even 3 1 63.2.e.b 2
9.c even 3 1 567.2.g.f 2
9.d odd 6 1 21.2.e.a 2
9.d odd 6 1 567.2.g.a 2
21.h odd 6 1 567.2.g.a 2
36.f odd 6 1 1008.2.s.d 2
36.h even 6 1 336.2.q.f 2
45.h odd 6 1 525.2.i.e 2
45.l even 12 2 525.2.r.e 4
63.g even 3 1 63.2.e.b 2
63.h even 3 1 441.2.a.b 1
63.h even 3 1 inner 567.2.h.a 2
63.i even 6 1 147.2.a.b 1
63.j odd 6 1 147.2.a.c 1
63.j odd 6 1 567.2.h.f 2
63.k odd 6 1 441.2.e.e 2
63.l odd 6 1 441.2.e.e 2
63.n odd 6 1 21.2.e.a 2
63.o even 6 1 147.2.e.a 2
63.s even 6 1 147.2.e.a 2
63.t odd 6 1 441.2.a.a 1
72.j odd 6 1 1344.2.q.m 2
72.l even 6 1 1344.2.q.c 2
252.o even 6 1 336.2.q.f 2
252.r odd 6 1 2352.2.a.w 1
252.s odd 6 1 2352.2.q.c 2
252.u odd 6 1 7056.2.a.bp 1
252.bb even 6 1 2352.2.a.d 1
252.bj even 6 1 7056.2.a.m 1
252.bl odd 6 1 1008.2.s.d 2
252.bn odd 6 1 2352.2.q.c 2
315.v odd 6 1 525.2.i.e 2
315.bq even 6 1 3675.2.a.c 1
315.br odd 6 1 3675.2.a.a 1
315.bx even 12 2 525.2.r.e 4
504.bi odd 6 1 9408.2.a.bg 1
504.bt even 6 1 9408.2.a.cv 1
504.ca even 6 1 9408.2.a.bz 1
504.cm odd 6 1 9408.2.a.k 1
504.cy even 6 1 1344.2.q.c 2
504.db odd 6 1 1344.2.q.m 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 9.d odd 6 1
21.2.e.a 2 63.n odd 6 1
63.2.e.b 2 9.c even 3 1
63.2.e.b 2 63.g even 3 1
147.2.a.b 1 63.i even 6 1
147.2.a.c 1 63.j odd 6 1
147.2.e.a 2 63.o even 6 1
147.2.e.a 2 63.s even 6 1
336.2.q.f 2 36.h even 6 1
336.2.q.f 2 252.o even 6 1
441.2.a.a 1 63.t odd 6 1
441.2.a.b 1 63.h even 3 1
441.2.e.e 2 63.k odd 6 1
441.2.e.e 2 63.l odd 6 1
525.2.i.e 2 45.h odd 6 1
525.2.i.e 2 315.v odd 6 1
525.2.r.e 4 45.l even 12 2
525.2.r.e 4 315.bx even 12 2
567.2.g.a 2 9.d odd 6 1
567.2.g.a 2 21.h odd 6 1
567.2.g.f 2 7.c even 3 1
567.2.g.f 2 9.c even 3 1
567.2.h.a 2 1.a even 1 1 trivial
567.2.h.a 2 63.h even 3 1 inner
567.2.h.f 2 3.b odd 2 1
567.2.h.f 2 63.j odd 6 1
1008.2.s.d 2 36.f odd 6 1
1008.2.s.d 2 252.bl odd 6 1
1344.2.q.c 2 72.l even 6 1
1344.2.q.c 2 504.cy even 6 1
1344.2.q.m 2 72.j odd 6 1
1344.2.q.m 2 504.db odd 6 1
2352.2.a.d 1 252.bb even 6 1
2352.2.a.w 1 252.r odd 6 1
2352.2.q.c 2 252.s odd 6 1
2352.2.q.c 2 252.bn odd 6 1
3675.2.a.a 1 315.br odd 6 1
3675.2.a.c 1 315.bq even 6 1
7056.2.a.m 1 252.bj even 6 1
7056.2.a.bp 1 252.u odd 6 1
9408.2.a.k 1 504.cm odd 6 1
9408.2.a.bg 1 504.bi odd 6 1
9408.2.a.bz 1 504.ca even 6 1
9408.2.a.cv 1 504.bt even 6 1

Hecke kernels

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

\( T_{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{2} + T_{13} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$31$ \( (T - 9)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$47$ \( (T - 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$59$ \( (T - 12)^{2} \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( (T + 5)^{2} \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$79$ \( (T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$89$ \( T^{2} - 16T + 256 \) Copy content Toggle raw display
$97$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
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