# Properties

 Label 567.2.h.c Level $567$ Weight $2$ Character orbit 567.h Analytic conductor $4.528$ Analytic rank $0$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

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: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - 2 q^{4} + ( - 2 \zeta_{6} - 1) q^{7}+O(q^{10})$$ q - 2 * q^4 + (-2*z - 1) * q^7 $$q - 2 q^{4} + ( - 2 \zeta_{6} - 1) q^{7} + 7 \zeta_{6} q^{13} + 4 q^{16} + 7 \zeta_{6} q^{19} + 5 \zeta_{6} q^{25} + (4 \zeta_{6} + 2) q^{28} - 7 q^{31} + \zeta_{6} q^{37} + (5 \zeta_{6} - 5) q^{43} + (8 \zeta_{6} - 3) q^{49} - 14 \zeta_{6} q^{52} + 14 q^{61} - 8 q^{64} + 11 q^{67} + ( - 7 \zeta_{6} + 7) q^{73} - 14 \zeta_{6} q^{76} - 13 q^{79} + ( - 21 \zeta_{6} + 14) q^{91} + (14 \zeta_{6} - 14) q^{97} +O(q^{100})$$ q - 2 * q^4 + (-2*z - 1) * q^7 + 7*z * q^13 + 4 * q^16 + 7*z * q^19 + 5*z * q^25 + (4*z + 2) * q^28 - 7 * q^31 + z * q^37 + (5*z - 5) * q^43 + (8*z - 3) * q^49 - 14*z * q^52 + 14 * q^61 - 8 * q^64 + 11 * q^67 + (-7*z + 7) * q^73 - 14*z * q^76 - 13 * q^79 + (-21*z + 14) * q^91 + (14*z - 14) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} - 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^4 - 4 * q^7 $$2 q - 4 q^{4} - 4 q^{7} + 7 q^{13} + 8 q^{16} + 7 q^{19} + 5 q^{25} + 8 q^{28} - 14 q^{31} + q^{37} - 5 q^{43} + 2 q^{49} - 14 q^{52} + 28 q^{61} - 16 q^{64} + 22 q^{67} + 7 q^{73} - 14 q^{76} - 26 q^{79} + 7 q^{91} - 14 q^{97}+O(q^{100})$$ 2 * q - 4 * q^4 - 4 * q^7 + 7 * q^13 + 8 * q^16 + 7 * q^19 + 5 * q^25 + 8 * q^28 - 14 * q^31 + q^37 - 5 * q^43 + 2 * q^49 - 14 * q^52 + 28 * q^61 - 16 * q^64 + 22 * q^67 + 7 * q^73 - 14 * q^76 - 26 * q^79 + 7 * q^91 - 14 * q^97

## 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
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 −2.00000 0 0 −2.00000 1.73205i 0 0 0
352.1 0 0 −2.00000 0 0 −2.00000 + 1.73205i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
63.h even 3 1 inner
63.j 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.h.c 2
3.b odd 2 1 CM 567.2.h.c 2
7.c even 3 1 567.2.g.d 2
9.c even 3 1 63.2.e.a 2
9.c even 3 1 567.2.g.d 2
9.d odd 6 1 63.2.e.a 2
9.d odd 6 1 567.2.g.d 2
21.h odd 6 1 567.2.g.d 2
36.f odd 6 1 1008.2.s.j 2
36.h even 6 1 1008.2.s.j 2
63.g even 3 1 63.2.e.a 2
63.h even 3 1 441.2.a.d 1
63.h even 3 1 inner 567.2.h.c 2
63.i even 6 1 441.2.a.e 1
63.j odd 6 1 441.2.a.d 1
63.j odd 6 1 inner 567.2.h.c 2
63.k odd 6 1 441.2.e.c 2
63.l odd 6 1 441.2.e.c 2
63.n odd 6 1 63.2.e.a 2
63.o even 6 1 441.2.e.c 2
63.s even 6 1 441.2.e.c 2
63.t odd 6 1 441.2.a.e 1
252.o even 6 1 1008.2.s.j 2
252.r odd 6 1 7056.2.a.bf 1
252.u odd 6 1 7056.2.a.y 1
252.bb even 6 1 7056.2.a.y 1
252.bj even 6 1 7056.2.a.bf 1
252.bl odd 6 1 1008.2.s.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 9.c even 3 1
63.2.e.a 2 9.d odd 6 1
63.2.e.a 2 63.g even 3 1
63.2.e.a 2 63.n odd 6 1
441.2.a.d 1 63.h even 3 1
441.2.a.d 1 63.j odd 6 1
441.2.a.e 1 63.i even 6 1
441.2.a.e 1 63.t odd 6 1
441.2.e.c 2 63.k odd 6 1
441.2.e.c 2 63.l odd 6 1
441.2.e.c 2 63.o even 6 1
441.2.e.c 2 63.s even 6 1
567.2.g.d 2 7.c even 3 1
567.2.g.d 2 9.c even 3 1
567.2.g.d 2 9.d odd 6 1
567.2.g.d 2 21.h odd 6 1
567.2.h.c 2 1.a even 1 1 trivial
567.2.h.c 2 3.b odd 2 1 CM
567.2.h.c 2 63.h even 3 1 inner
567.2.h.c 2 63.j odd 6 1 inner
1008.2.s.j 2 36.f odd 6 1
1008.2.s.j 2 36.h even 6 1
1008.2.s.j 2 252.o even 6 1
1008.2.s.j 2 252.bl odd 6 1
7056.2.a.y 1 252.u odd 6 1
7056.2.a.y 1 252.bb even 6 1
7056.2.a.bf 1 252.r odd 6 1
7056.2.a.bf 1 252.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 $$S_{2}^{\mathrm{new}}(567, [\chi])$$:

 $$T_{2}$$ T2 $$T_{13}^{2} - 7T_{13} + 49$$ T13^2 - 7*T13 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4T + 7$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 7T + 49$$
$17$ $$T^{2}$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$(T + 7)^{2}$$
$37$ $$T^{2} - T + 1$$
$41$ $$T^{2}$$
$43$ $$T^{2} + 5T + 25$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$(T - 14)^{2}$$
$67$ $$(T - 11)^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} - 7T + 49$$
$79$ $$(T + 13)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 14T + 196$$