# 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 $2$

# Learn more

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 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.

 $$f(q)$$ $$=$$ $$q - 2 q^{2} + 2 q^{4} + (2 \zeta_{6} - 2) q^{5} + ( - 2 \zeta_{6} + 3) q^{7}+O(q^{10})$$ q - 2 * q^2 + 2 * q^4 + (2*z - 2) * q^5 + (-2*z + 3) * q^7 $$q - 2 q^{2} + 2 q^{4} + (2 \zeta_{6} - 2) q^{5} + ( - 2 \zeta_{6} + 3) q^{7} + ( - 4 \zeta_{6} + 4) q^{10} - 2 \zeta_{6} q^{11} - \zeta_{6} q^{13} + (4 \zeta_{6} - 6) q^{14} - 4 q^{16} - \zeta_{6} q^{19} + (4 \zeta_{6} - 4) q^{20} + 4 \zeta_{6} q^{22} + \zeta_{6} q^{25} + 2 \zeta_{6} q^{26} + ( - 4 \zeta_{6} + 6) q^{28} + ( - 4 \zeta_{6} + 4) q^{29} + 9 q^{31} + 8 q^{32} + (6 \zeta_{6} - 2) q^{35} - 3 \zeta_{6} q^{37} + 2 \zeta_{6} q^{38} - 10 \zeta_{6} q^{41} + (5 \zeta_{6} - 5) q^{43} - 4 \zeta_{6} q^{44} + 6 q^{47} + ( - 8 \zeta_{6} + 5) q^{49} - 2 \zeta_{6} q^{50} - 2 \zeta_{6} q^{52} + ( - 12 \zeta_{6} + 12) q^{53} + 4 q^{55} + (8 \zeta_{6} - 8) q^{58} + 12 q^{59} + 10 q^{61} - 18 q^{62} - 8 q^{64} + 2 q^{65} - 5 q^{67} + ( - 12 \zeta_{6} + 4) q^{70} + 6 q^{71} + ( - 3 \zeta_{6} + 3) q^{73} + 6 \zeta_{6} q^{74} - 2 \zeta_{6} q^{76} + ( - 2 \zeta_{6} - 4) q^{77} - q^{79} + ( - 8 \zeta_{6} + 8) q^{80} + 20 \zeta_{6} q^{82} + ( - 6 \zeta_{6} + 6) q^{83} + ( - 10 \zeta_{6} + 10) q^{86} + 16 \zeta_{6} q^{89} + ( - \zeta_{6} - 2) q^{91} - 12 q^{94} + 2 q^{95} + ( - 6 \zeta_{6} + 6) q^{97} + (16 \zeta_{6} - 10) q^{98} +O(q^{100})$$ q - 2 * q^2 + 2 * q^4 + (2*z - 2) * q^5 + (-2*z + 3) * q^7 + (-4*z + 4) * q^10 - 2*z * q^11 - z * q^13 + (4*z - 6) * q^14 - 4 * q^16 - z * q^19 + (4*z - 4) * q^20 + 4*z * q^22 + z * q^25 + 2*z * q^26 + (-4*z + 6) * q^28 + (-4*z + 4) * q^29 + 9 * q^31 + 8 * q^32 + (6*z - 2) * q^35 - 3*z * q^37 + 2*z * q^38 - 10*z * q^41 + (5*z - 5) * q^43 - 4*z * q^44 + 6 * q^47 + (-8*z + 5) * q^49 - 2*z * q^50 - 2*z * q^52 + (-12*z + 12) * q^53 + 4 * q^55 + (8*z - 8) * q^58 + 12 * q^59 + 10 * q^61 - 18 * q^62 - 8 * q^64 + 2 * q^65 - 5 * q^67 + (-12*z + 4) * q^70 + 6 * q^71 + (-3*z + 3) * q^73 + 6*z * q^74 - 2*z * q^76 + (-2*z - 4) * q^77 - q^79 + (-8*z + 8) * q^80 + 20*z * q^82 + (-6*z + 6) * q^83 + (-10*z + 10) * q^86 + 16*z * q^89 + (-z - 2) * q^91 - 12 * q^94 + 2 * q^95 + (-6*z + 6) * q^97 + (16*z - 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{2} + 4 q^{4} - 2 q^{5} + 4 q^{7}+O(q^{10})$$ 2 * q - 4 * q^2 + 4 * q^4 - 2 * q^5 + 4 * q^7 $$2 q - 4 q^{2} + 4 q^{4} - 2 q^{5} + 4 q^{7} + 4 q^{10} - 2 q^{11} - q^{13} - 8 q^{14} - 8 q^{16} - q^{19} - 4 q^{20} + 4 q^{22} + q^{25} + 2 q^{26} + 8 q^{28} + 4 q^{29} + 18 q^{31} + 16 q^{32} + 2 q^{35} - 3 q^{37} + 2 q^{38} - 10 q^{41} - 5 q^{43} - 4 q^{44} + 12 q^{47} + 2 q^{49} - 2 q^{50} - 2 q^{52} + 12 q^{53} + 8 q^{55} - 8 q^{58} + 24 q^{59} + 20 q^{61} - 36 q^{62} - 16 q^{64} + 4 q^{65} - 10 q^{67} - 4 q^{70} + 12 q^{71} + 3 q^{73} + 6 q^{74} - 2 q^{76} - 10 q^{77} - 2 q^{79} + 8 q^{80} + 20 q^{82} + 6 q^{83} + 10 q^{86} + 16 q^{89} - 5 q^{91} - 24 q^{94} + 4 q^{95} + 6 q^{97} - 4 q^{98}+O(q^{100})$$ 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

## 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
−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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(567, [\chi])$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{13}^{2} + T_{13} + 1$$ T13^2 + T13 + 1

## Hecke characteristic polynomials

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