# 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.f (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ 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

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.

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

## 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.

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

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{5}^{2} - 2 T_{5} + 4$$

## Hecke characteristic polynomials

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