# Properties

 Label 567.2.e.a Level $567$ Weight $2$ Character orbit 567.e Analytic conductor $4.528$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [567,2,Mod(163,567)]

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([0, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("567.163");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.e (of order $$3$$, degree $$2$$, 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: $$1$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6} - 2) q^{7} - 3 q^{8} +O(q^{10})$$ q - z * q^2 + (-z + 1) * q^4 + z * q^5 + (-z - 2) * q^7 - 3 * q^8 $$q - \zeta_{6} q^{2} + ( - \zeta_{6} + 1) q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6} - 2) q^{7} - 3 q^{8} + ( - \zeta_{6} + 1) q^{10} + (5 \zeta_{6} - 5) q^{11} - 5 q^{13} + (3 \zeta_{6} - 1) q^{14} + \zeta_{6} q^{16} + (3 \zeta_{6} - 3) q^{17} - \zeta_{6} q^{19} + q^{20} + 5 q^{22} - 3 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} + 5 \zeta_{6} q^{26} + (2 \zeta_{6} - 3) q^{28} - q^{29} + (5 \zeta_{6} - 5) q^{32} + 3 q^{34} + ( - 3 \zeta_{6} + 1) q^{35} - 3 \zeta_{6} q^{37} + (\zeta_{6} - 1) q^{38} - 3 \zeta_{6} q^{40} - 5 q^{41} - q^{43} + 5 \zeta_{6} q^{44} + (3 \zeta_{6} - 3) q^{46} + (5 \zeta_{6} + 3) q^{49} - 4 q^{50} + (5 \zeta_{6} - 5) q^{52} + ( - 9 \zeta_{6} + 9) q^{53} - 5 q^{55} + (3 \zeta_{6} + 6) q^{56} + \zeta_{6} q^{58} + 14 \zeta_{6} q^{61} + 7 q^{64} - 5 \zeta_{6} q^{65} + (4 \zeta_{6} - 4) q^{67} + 3 \zeta_{6} q^{68} + (2 \zeta_{6} - 3) q^{70} - 12 q^{71} + (3 \zeta_{6} - 3) q^{73} + (3 \zeta_{6} - 3) q^{74} - q^{76} + ( - 10 \zeta_{6} + 15) q^{77} - 8 \zeta_{6} q^{79} + (\zeta_{6} - 1) q^{80} + 5 \zeta_{6} q^{82} - 9 q^{83} - 3 q^{85} + \zeta_{6} q^{86} + ( - 15 \zeta_{6} + 15) q^{88} + 13 \zeta_{6} q^{89} + (5 \zeta_{6} + 10) q^{91} - 3 q^{92} + ( - \zeta_{6} + 1) q^{95} - 9 q^{97} + ( - 8 \zeta_{6} + 5) q^{98} +O(q^{100})$$ q - z * q^2 + (-z + 1) * q^4 + z * q^5 + (-z - 2) * q^7 - 3 * q^8 + (-z + 1) * q^10 + (5*z - 5) * q^11 - 5 * q^13 + (3*z - 1) * q^14 + z * q^16 + (3*z - 3) * q^17 - z * q^19 + q^20 + 5 * q^22 - 3*z * q^23 + (-4*z + 4) * q^25 + 5*z * q^26 + (2*z - 3) * q^28 - q^29 + (5*z - 5) * q^32 + 3 * q^34 + (-3*z + 1) * q^35 - 3*z * q^37 + (z - 1) * q^38 - 3*z * q^40 - 5 * q^41 - q^43 + 5*z * q^44 + (3*z - 3) * q^46 + (5*z + 3) * q^49 - 4 * q^50 + (5*z - 5) * q^52 + (-9*z + 9) * q^53 - 5 * q^55 + (3*z + 6) * q^56 + z * q^58 + 14*z * q^61 + 7 * q^64 - 5*z * q^65 + (4*z - 4) * q^67 + 3*z * q^68 + (2*z - 3) * q^70 - 12 * q^71 + (3*z - 3) * q^73 + (3*z - 3) * q^74 - q^76 + (-10*z + 15) * q^77 - 8*z * q^79 + (z - 1) * q^80 + 5*z * q^82 - 9 * q^83 - 3 * q^85 + z * q^86 + (-15*z + 15) * q^88 + 13*z * q^89 + (5*z + 10) * q^91 - 3 * q^92 + (-z + 1) * q^95 - 9 * q^97 + (-8*z + 5) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} + q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 + q^5 - 5 * q^7 - 6 * q^8 $$2 q - q^{2} + q^{4} + q^{5} - 5 q^{7} - 6 q^{8} + q^{10} - 5 q^{11} - 10 q^{13} + q^{14} + q^{16} - 3 q^{17} - q^{19} + 2 q^{20} + 10 q^{22} - 3 q^{23} + 4 q^{25} + 5 q^{26} - 4 q^{28} - 2 q^{29} - 5 q^{32} + 6 q^{34} - q^{35} - 3 q^{37} - q^{38} - 3 q^{40} - 10 q^{41} - 2 q^{43} + 5 q^{44} - 3 q^{46} + 11 q^{49} - 8 q^{50} - 5 q^{52} + 9 q^{53} - 10 q^{55} + 15 q^{56} + q^{58} + 14 q^{61} + 14 q^{64} - 5 q^{65} - 4 q^{67} + 3 q^{68} - 4 q^{70} - 24 q^{71} - 3 q^{73} - 3 q^{74} - 2 q^{76} + 20 q^{77} - 8 q^{79} - q^{80} + 5 q^{82} - 18 q^{83} - 6 q^{85} + q^{86} + 15 q^{88} + 13 q^{89} + 25 q^{91} - 6 q^{92} + q^{95} - 18 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 + q^4 + q^5 - 5 * q^7 - 6 * q^8 + q^10 - 5 * q^11 - 10 * q^13 + q^14 + q^16 - 3 * q^17 - q^19 + 2 * q^20 + 10 * q^22 - 3 * q^23 + 4 * q^25 + 5 * q^26 - 4 * q^28 - 2 * q^29 - 5 * q^32 + 6 * q^34 - q^35 - 3 * q^37 - q^38 - 3 * q^40 - 10 * q^41 - 2 * q^43 + 5 * q^44 - 3 * q^46 + 11 * q^49 - 8 * q^50 - 5 * q^52 + 9 * q^53 - 10 * q^55 + 15 * q^56 + q^58 + 14 * q^61 + 14 * q^64 - 5 * q^65 - 4 * q^67 + 3 * q^68 - 4 * q^70 - 24 * q^71 - 3 * q^73 - 3 * q^74 - 2 * q^76 + 20 * q^77 - 8 * q^79 - q^80 + 5 * q^82 - 18 * q^83 - 6 * q^85 + q^86 + 15 * q^88 + 13 * q^89 + 25 * q^91 - 6 * q^92 + q^95 - 18 * q^97 + 2 * q^98

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
163.1
 0.5 − 0.866025i 0.5 + 0.866025i
−0.500000 + 0.866025i 0 0.500000 + 0.866025i 0.500000 0.866025i 0 −2.50000 + 0.866025i −3.00000 0 0.500000 + 0.866025i
487.1 −0.500000 0.866025i 0 0.500000 0.866025i 0.500000 + 0.866025i 0 −2.50000 0.866025i −3.00000 0 0.500000 0.866025i
 $$n$$: 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
7.c 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.e.a 2
3.b odd 2 1 567.2.e.b 2
7.c even 3 1 inner 567.2.e.a 2
7.c even 3 1 3969.2.a.d 1
7.d odd 6 1 3969.2.a.f 1
9.c even 3 1 63.2.g.a 2
9.c even 3 1 63.2.h.a yes 2
9.d odd 6 1 189.2.g.a 2
9.d odd 6 1 189.2.h.a 2
21.g even 6 1 3969.2.a.a 1
21.h odd 6 1 567.2.e.b 2
21.h odd 6 1 3969.2.a.c 1
36.f odd 6 1 1008.2.q.c 2
36.f odd 6 1 1008.2.t.d 2
36.h even 6 1 3024.2.q.b 2
36.h even 6 1 3024.2.t.d 2
63.g even 3 1 63.2.h.a yes 2
63.g even 3 1 441.2.f.b 2
63.h even 3 1 63.2.g.a 2
63.h even 3 1 441.2.f.b 2
63.i even 6 1 1323.2.f.b 2
63.i even 6 1 1323.2.g.a 2
63.j odd 6 1 189.2.g.a 2
63.j odd 6 1 1323.2.f.a 2
63.k odd 6 1 441.2.f.a 2
63.k odd 6 1 441.2.h.a 2
63.l odd 6 1 441.2.g.a 2
63.l odd 6 1 441.2.h.a 2
63.n odd 6 1 189.2.h.a 2
63.n odd 6 1 1323.2.f.a 2
63.o even 6 1 1323.2.g.a 2
63.o even 6 1 1323.2.h.a 2
63.s even 6 1 1323.2.f.b 2
63.s even 6 1 1323.2.h.a 2
63.t odd 6 1 441.2.f.a 2
63.t odd 6 1 441.2.g.a 2
252.o even 6 1 3024.2.q.b 2
252.u odd 6 1 1008.2.t.d 2
252.bb even 6 1 3024.2.t.d 2
252.bl odd 6 1 1008.2.q.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.a 2 9.c even 3 1
63.2.g.a 2 63.h even 3 1
63.2.h.a yes 2 9.c even 3 1
63.2.h.a yes 2 63.g even 3 1
189.2.g.a 2 9.d odd 6 1
189.2.g.a 2 63.j odd 6 1
189.2.h.a 2 9.d odd 6 1
189.2.h.a 2 63.n odd 6 1
441.2.f.a 2 63.k odd 6 1
441.2.f.a 2 63.t odd 6 1
441.2.f.b 2 63.g even 3 1
441.2.f.b 2 63.h even 3 1
441.2.g.a 2 63.l odd 6 1
441.2.g.a 2 63.t odd 6 1
441.2.h.a 2 63.k odd 6 1
441.2.h.a 2 63.l odd 6 1
567.2.e.a 2 1.a even 1 1 trivial
567.2.e.a 2 7.c even 3 1 inner
567.2.e.b 2 3.b odd 2 1
567.2.e.b 2 21.h odd 6 1
1008.2.q.c 2 36.f odd 6 1
1008.2.q.c 2 252.bl odd 6 1
1008.2.t.d 2 36.f odd 6 1
1008.2.t.d 2 252.u odd 6 1
1323.2.f.a 2 63.j odd 6 1
1323.2.f.a 2 63.n odd 6 1
1323.2.f.b 2 63.i even 6 1
1323.2.f.b 2 63.s even 6 1
1323.2.g.a 2 63.i even 6 1
1323.2.g.a 2 63.o even 6 1
1323.2.h.a 2 63.o even 6 1
1323.2.h.a 2 63.s even 6 1
3024.2.q.b 2 36.h even 6 1
3024.2.q.b 2 252.o even 6 1
3024.2.t.d 2 36.h even 6 1
3024.2.t.d 2 252.bb even 6 1
3969.2.a.a 1 21.g even 6 1
3969.2.a.c 1 21.h odd 6 1
3969.2.a.d 1 7.c even 3 1
3969.2.a.f 1 7.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(567, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} + 5T + 7$$
$11$ $$T^{2} + 5T + 25$$
$13$ $$(T + 5)^{2}$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + T + 1$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$(T + 1)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 3T + 9$$
$41$ $$(T + 5)^{2}$$
$43$ $$(T + 1)^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} - 9T + 81$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 14T + 196$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$(T + 12)^{2}$$
$73$ $$T^{2} + 3T + 9$$
$79$ $$T^{2} + 8T + 64$$
$83$ $$(T + 9)^{2}$$
$89$ $$T^{2} - 13T + 169$$
$97$ $$(T + 9)^{2}$$