# Properties

 Label 567.2.f.g Level $567$ Weight $2$ Character orbit 567.f Analytic conductor $4.528$ Analytic rank $0$ 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(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 # Warning: the index may be different

gp: f = lf \\ 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: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + 3 q^{8}+O(q^{10})$$ q + (-z + 1) * q^2 + z * q^4 + 2*z * q^5 + (-z + 1) * q^7 + 3 * q^8 $$q + ( - \zeta_{6} + 1) q^{2} + \zeta_{6} q^{4} + 2 \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} + 3 q^{8} + 2 q^{10} + (4 \zeta_{6} - 4) q^{11} + 2 \zeta_{6} q^{13} - \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} - 6 q^{17} + 4 q^{19} + (2 \zeta_{6} - 2) q^{20} + 4 \zeta_{6} q^{22} + ( - \zeta_{6} + 1) q^{25} + 2 q^{26} + q^{28} + ( - 2 \zeta_{6} + 2) q^{29} + 5 \zeta_{6} q^{32} + (6 \zeta_{6} - 6) q^{34} + 2 q^{35} + 6 q^{37} + ( - 4 \zeta_{6} + 4) q^{38} + 6 \zeta_{6} q^{40} - 2 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 4 q^{44} - \zeta_{6} q^{49} - \zeta_{6} q^{50} + (2 \zeta_{6} - 2) q^{52} + 6 q^{53} - 8 q^{55} + ( - 3 \zeta_{6} + 3) q^{56} - 2 \zeta_{6} q^{58} - 12 \zeta_{6} q^{59} + ( - 2 \zeta_{6} + 2) q^{61} + 7 q^{64} + (4 \zeta_{6} - 4) q^{65} - 4 \zeta_{6} q^{67} - 6 \zeta_{6} q^{68} + ( - 2 \zeta_{6} + 2) q^{70} - 6 q^{73} + ( - 6 \zeta_{6} + 6) q^{74} + 4 \zeta_{6} q^{76} + 4 \zeta_{6} q^{77} + ( - 16 \zeta_{6} + 16) q^{79} + 2 q^{80} - 2 q^{82} + ( - 12 \zeta_{6} + 12) q^{83} - 12 \zeta_{6} q^{85} - 4 \zeta_{6} q^{86} + (12 \zeta_{6} - 12) q^{88} - 14 q^{89} + 2 q^{91} + 8 \zeta_{6} q^{95} + (18 \zeta_{6} - 18) q^{97} - q^{98} +O(q^{100})$$ q + (-z + 1) * q^2 + z * q^4 + 2*z * q^5 + (-z + 1) * q^7 + 3 * q^8 + 2 * q^10 + (4*z - 4) * q^11 + 2*z * q^13 - z * q^14 + (-z + 1) * q^16 - 6 * q^17 + 4 * q^19 + (2*z - 2) * q^20 + 4*z * q^22 + (-z + 1) * q^25 + 2 * q^26 + q^28 + (-2*z + 2) * q^29 + 5*z * q^32 + (6*z - 6) * q^34 + 2 * q^35 + 6 * q^37 + (-4*z + 4) * q^38 + 6*z * q^40 - 2*z * q^41 + (-4*z + 4) * q^43 - 4 * q^44 - z * q^49 - z * q^50 + (2*z - 2) * q^52 + 6 * q^53 - 8 * q^55 + (-3*z + 3) * q^56 - 2*z * q^58 - 12*z * q^59 + (-2*z + 2) * q^61 + 7 * q^64 + (4*z - 4) * q^65 - 4*z * q^67 - 6*z * q^68 + (-2*z + 2) * q^70 - 6 * q^73 + (-6*z + 6) * q^74 + 4*z * q^76 + 4*z * q^77 + (-16*z + 16) * q^79 + 2 * q^80 - 2 * q^82 + (-12*z + 12) * q^83 - 12*z * q^85 - 4*z * q^86 + (12*z - 12) * q^88 - 14 * q^89 + 2 * q^91 + 8*z * q^95 + (18*z - 18) * q^97 - q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8}+O(q^{10})$$ 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8 $$2 q + q^{2} + q^{4} + 2 q^{5} + q^{7} + 6 q^{8} + 4 q^{10} - 4 q^{11} + 2 q^{13} - q^{14} + q^{16} - 12 q^{17} + 8 q^{19} - 2 q^{20} + 4 q^{22} + q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{29} + 5 q^{32} - 6 q^{34} + 4 q^{35} + 12 q^{37} + 4 q^{38} + 6 q^{40} - 2 q^{41} + 4 q^{43} - 8 q^{44} - q^{49} - q^{50} - 2 q^{52} + 12 q^{53} - 16 q^{55} + 3 q^{56} - 2 q^{58} - 12 q^{59} + 2 q^{61} + 14 q^{64} - 4 q^{65} - 4 q^{67} - 6 q^{68} + 2 q^{70} - 12 q^{73} + 6 q^{74} + 4 q^{76} + 4 q^{77} + 16 q^{79} + 4 q^{80} - 4 q^{82} + 12 q^{83} - 12 q^{85} - 4 q^{86} - 12 q^{88} - 28 q^{89} + 4 q^{91} + 8 q^{95} - 18 q^{97} - 2 q^{98}+O(q^{100})$$ 2 * q + q^2 + q^4 + 2 * q^5 + q^7 + 6 * q^8 + 4 * q^10 - 4 * q^11 + 2 * q^13 - q^14 + q^16 - 12 * q^17 + 8 * q^19 - 2 * q^20 + 4 * q^22 + q^25 + 4 * q^26 + 2 * q^28 + 2 * q^29 + 5 * q^32 - 6 * q^34 + 4 * q^35 + 12 * q^37 + 4 * q^38 + 6 * q^40 - 2 * q^41 + 4 * q^43 - 8 * q^44 - q^49 - q^50 - 2 * q^52 + 12 * q^53 - 16 * q^55 + 3 * q^56 - 2 * q^58 - 12 * q^59 + 2 * q^61 + 14 * q^64 - 4 * q^65 - 4 * q^67 - 6 * q^68 + 2 * q^70 - 12 * q^73 + 6 * q^74 + 4 * q^76 + 4 * q^77 + 16 * q^79 + 4 * q^80 - 4 * q^82 + 12 * q^83 - 12 * q^85 - 4 * q^86 - 12 * q^88 - 28 * q^89 + 4 * q^91 + 8 * q^95 - 18 * q^97 - 2 * q^98

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
190.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 1.73205i 0 0.500000 + 0.866025i 3.00000 0 2.00000
379.1 0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 + 1.73205i 0 0.500000 0.866025i 3.00000 0 2.00000
 $$n$$: 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
9.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.f.g 2
3.b odd 2 1 567.2.f.b 2
9.c even 3 1 21.2.a.a 1
9.c even 3 1 inner 567.2.f.g 2
9.d odd 6 1 63.2.a.a 1
9.d odd 6 1 567.2.f.b 2
36.f odd 6 1 336.2.a.a 1
36.h even 6 1 1008.2.a.l 1
45.h odd 6 1 1575.2.a.c 1
45.j even 6 1 525.2.a.d 1
45.k odd 12 2 525.2.d.a 2
45.l even 12 2 1575.2.d.a 2
63.g even 3 1 147.2.e.b 2
63.h even 3 1 147.2.e.b 2
63.i even 6 1 441.2.e.b 2
63.j odd 6 1 441.2.e.a 2
63.k odd 6 1 147.2.e.c 2
63.l odd 6 1 147.2.a.a 1
63.n odd 6 1 441.2.e.a 2
63.o even 6 1 441.2.a.f 1
63.s even 6 1 441.2.e.b 2
63.t odd 6 1 147.2.e.c 2
72.j odd 6 1 4032.2.a.h 1
72.l even 6 1 4032.2.a.k 1
72.n even 6 1 1344.2.a.g 1
72.p odd 6 1 1344.2.a.s 1
99.g even 6 1 7623.2.a.g 1
99.h odd 6 1 2541.2.a.j 1
117.t even 6 1 3549.2.a.c 1
144.v odd 12 2 5376.2.c.l 2
144.x even 12 2 5376.2.c.r 2
153.h even 6 1 6069.2.a.b 1
171.o odd 6 1 7581.2.a.d 1
180.p odd 6 1 8400.2.a.bn 1
252.n even 6 1 2352.2.q.e 2
252.s odd 6 1 7056.2.a.p 1
252.u odd 6 1 2352.2.q.x 2
252.bi even 6 1 2352.2.a.v 1
252.bj even 6 1 2352.2.q.e 2
252.bl odd 6 1 2352.2.q.x 2
315.bg odd 6 1 3675.2.a.n 1
504.be even 6 1 9408.2.a.m 1
504.bn odd 6 1 9408.2.a.bv 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.a.a 1 9.c even 3 1
63.2.a.a 1 9.d odd 6 1
147.2.a.a 1 63.l odd 6 1
147.2.e.b 2 63.g even 3 1
147.2.e.b 2 63.h even 3 1
147.2.e.c 2 63.k odd 6 1
147.2.e.c 2 63.t odd 6 1
336.2.a.a 1 36.f odd 6 1
441.2.a.f 1 63.o even 6 1
441.2.e.a 2 63.j odd 6 1
441.2.e.a 2 63.n odd 6 1
441.2.e.b 2 63.i even 6 1
441.2.e.b 2 63.s even 6 1
525.2.a.d 1 45.j even 6 1
525.2.d.a 2 45.k odd 12 2
567.2.f.b 2 3.b odd 2 1
567.2.f.b 2 9.d odd 6 1
567.2.f.g 2 1.a even 1 1 trivial
567.2.f.g 2 9.c even 3 1 inner
1008.2.a.l 1 36.h even 6 1
1344.2.a.g 1 72.n even 6 1
1344.2.a.s 1 72.p odd 6 1
1575.2.a.c 1 45.h odd 6 1
1575.2.d.a 2 45.l even 12 2
2352.2.a.v 1 252.bi even 6 1
2352.2.q.e 2 252.n even 6 1
2352.2.q.e 2 252.bj even 6 1
2352.2.q.x 2 252.u odd 6 1
2352.2.q.x 2 252.bl odd 6 1
2541.2.a.j 1 99.h odd 6 1
3549.2.a.c 1 117.t even 6 1
3675.2.a.n 1 315.bg odd 6 1
4032.2.a.h 1 72.j odd 6 1
4032.2.a.k 1 72.l even 6 1
5376.2.c.l 2 144.v odd 12 2
5376.2.c.r 2 144.x even 12 2
6069.2.a.b 1 153.h even 6 1
7056.2.a.p 1 252.s odd 6 1
7581.2.a.d 1 171.o odd 6 1
7623.2.a.g 1 99.g even 6 1
8400.2.a.bn 1 180.p odd 6 1
9408.2.a.m 1 504.be even 6 1
9408.2.a.bv 1 504.bn odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} - T_{2} + 1$$ T2^2 - T2 + 1 $$T_{5}^{2} - 2T_{5} + 4$$ T5^2 - 2*T5 + 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 4$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 4T + 16$$
$13$ $$T^{2} - 2T + 4$$
$17$ $$(T + 6)^{2}$$
$19$ $$(T - 4)^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2} - 2T + 4$$
$31$ $$T^{2}$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} + 2T + 4$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 12T + 144$$
$61$ $$T^{2} - 2T + 4$$
$67$ $$T^{2} + 4T + 16$$
$71$ $$T^{2}$$
$73$ $$(T + 6)^{2}$$
$79$ $$T^{2} - 16T + 256$$
$83$ $$T^{2} - 12T + 144$$
$89$ $$(T + 14)^{2}$$
$97$ $$T^{2} + 18T + 324$$