# Properties

 Label 567.2.g.a Level $567$ Weight $2$ Character orbit 567.g Analytic conductor $4.528$ Analytic rank $1$ 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.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} -2 q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + ( -2 + 2 \zeta_{6} ) q^{2} -2 \zeta_{6} q^{4} -2 q^{5} + ( -1 + 3 \zeta_{6} ) q^{7} + ( 4 - 4 \zeta_{6} ) q^{10} -2 q^{11} + ( -1 + \zeta_{6} ) q^{13} + ( -4 - 2 \zeta_{6} ) q^{14} + ( 4 - 4 \zeta_{6} ) q^{16} -\zeta_{6} q^{19} + 4 \zeta_{6} q^{20} + ( 4 - 4 \zeta_{6} ) q^{22} - q^{25} -2 \zeta_{6} q^{26} + ( 6 - 4 \zeta_{6} ) q^{28} -4 \zeta_{6} q^{29} -9 \zeta_{6} q^{31} + 8 \zeta_{6} q^{32} + ( 2 - 6 \zeta_{6} ) q^{35} -3 \zeta_{6} q^{37} + 2 q^{38} + ( 10 - 10 \zeta_{6} ) q^{41} -5 \zeta_{6} q^{43} + 4 \zeta_{6} q^{44} + ( 6 - 6 \zeta_{6} ) q^{47} + ( -8 + 3 \zeta_{6} ) q^{49} + ( 2 - 2 \zeta_{6} ) q^{50} + 2 q^{52} + ( -12 + 12 \zeta_{6} ) q^{53} + 4 q^{55} + 8 q^{58} + 12 \zeta_{6} q^{59} + ( -10 + 10 \zeta_{6} ) q^{61} + 18 q^{62} -8 q^{64} + ( 2 - 2 \zeta_{6} ) q^{65} + 5 \zeta_{6} q^{67} + ( 8 + 4 \zeta_{6} ) q^{70} -6 q^{71} + ( 3 - 3 \zeta_{6} ) q^{73} + 6 q^{74} + ( -2 + 2 \zeta_{6} ) q^{76} + ( 2 - 6 \zeta_{6} ) q^{77} + ( 1 - \zeta_{6} ) q^{79} + ( -8 + 8 \zeta_{6} ) q^{80} + 20 \zeta_{6} q^{82} -6 \zeta_{6} q^{83} + 10 q^{86} -16 \zeta_{6} q^{89} + ( -2 - \zeta_{6} ) q^{91} + 12 \zeta_{6} q^{94} + 2 \zeta_{6} q^{95} + 6 \zeta_{6} q^{97} + ( 10 - 16 \zeta_{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{4} - 4q^{5} + q^{7} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{4} - 4q^{5} + q^{7} + 4q^{10} - 4q^{11} - q^{13} - 10q^{14} + 4q^{16} - q^{19} + 4q^{20} + 4q^{22} - 2q^{25} - 2q^{26} + 8q^{28} - 4q^{29} - 9q^{31} + 8q^{32} - 2q^{35} - 3q^{37} + 4q^{38} + 10q^{41} - 5q^{43} + 4q^{44} + 6q^{47} - 13q^{49} + 2q^{50} + 4q^{52} - 12q^{53} + 8q^{55} + 16q^{58} + 12q^{59} - 10q^{61} + 36q^{62} - 16q^{64} + 2q^{65} + 5q^{67} + 20q^{70} - 12q^{71} + 3q^{73} + 12q^{74} - 2q^{76} - 2q^{77} + q^{79} - 8q^{80} + 20q^{82} - 6q^{83} + 20q^{86} - 16q^{89} - 5q^{91} + 12q^{94} + 2q^{95} + 6q^{97} + 4q^{98} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1
 0.5 + 0.866025i 0.5 − 0.866025i
−1.00000 + 1.73205i 0 −1.00000 1.73205i −2.00000 0 0.500000 + 2.59808i 0 0 2.00000 3.46410i
541.1 −1.00000 1.73205i 0 −1.00000 + 1.73205i −2.00000 0 0.500000 2.59808i 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.g 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.g.a 2
3.b odd 2 1 567.2.g.f 2
7.c even 3 1 567.2.h.f 2
9.c even 3 1 21.2.e.a 2
9.c even 3 1 567.2.h.f 2
9.d odd 6 1 63.2.e.b 2
9.d odd 6 1 567.2.h.a 2
21.h odd 6 1 567.2.h.a 2
36.f odd 6 1 336.2.q.f 2
36.h even 6 1 1008.2.s.d 2
45.j even 6 1 525.2.i.e 2
45.k odd 12 2 525.2.r.e 4
63.g even 3 1 147.2.a.c 1
63.g even 3 1 inner 567.2.g.a 2
63.h even 3 1 21.2.e.a 2
63.i even 6 1 441.2.e.e 2
63.j odd 6 1 63.2.e.b 2
63.k odd 6 1 147.2.a.b 1
63.l odd 6 1 147.2.e.a 2
63.n odd 6 1 441.2.a.b 1
63.n odd 6 1 567.2.g.f 2
63.o even 6 1 441.2.e.e 2
63.s even 6 1 441.2.a.a 1
63.t odd 6 1 147.2.e.a 2
72.n even 6 1 1344.2.q.m 2
72.p odd 6 1 1344.2.q.c 2
252.n even 6 1 2352.2.a.w 1
252.o even 6 1 7056.2.a.bp 1
252.u odd 6 1 336.2.q.f 2
252.bb even 6 1 1008.2.s.d 2
252.bi even 6 1 2352.2.q.c 2
252.bj even 6 1 2352.2.q.c 2
252.bl odd 6 1 2352.2.a.d 1
252.bn odd 6 1 7056.2.a.m 1
315.r even 6 1 525.2.i.e 2
315.bn odd 6 1 3675.2.a.c 1
315.bo even 6 1 3675.2.a.a 1
315.bt odd 12 2 525.2.r.e 4
504.w even 6 1 9408.2.a.bg 1
504.ba odd 6 1 9408.2.a.cv 1
504.ce odd 6 1 1344.2.q.c 2
504.cq even 6 1 1344.2.q.m 2
504.cw odd 6 1 9408.2.a.bz 1
504.cz even 6 1 9408.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.e.a 2 9.c even 3 1
21.2.e.a 2 63.h even 3 1
63.2.e.b 2 9.d odd 6 1
63.2.e.b 2 63.j odd 6 1
147.2.a.b 1 63.k odd 6 1
147.2.a.c 1 63.g even 3 1
147.2.e.a 2 63.l odd 6 1
147.2.e.a 2 63.t odd 6 1
336.2.q.f 2 36.f odd 6 1
336.2.q.f 2 252.u odd 6 1
441.2.a.a 1 63.s even 6 1
441.2.a.b 1 63.n odd 6 1
441.2.e.e 2 63.i even 6 1
441.2.e.e 2 63.o even 6 1
525.2.i.e 2 45.j even 6 1
525.2.i.e 2 315.r even 6 1
525.2.r.e 4 45.k odd 12 2
525.2.r.e 4 315.bt odd 12 2
567.2.g.a 2 1.a even 1 1 trivial
567.2.g.a 2 63.g even 3 1 inner
567.2.g.f 2 3.b odd 2 1
567.2.g.f 2 63.n odd 6 1
567.2.h.a 2 9.d odd 6 1
567.2.h.a 2 21.h odd 6 1
567.2.h.f 2 7.c even 3 1
567.2.h.f 2 9.c even 3 1
1008.2.s.d 2 36.h even 6 1
1008.2.s.d 2 252.bb even 6 1
1344.2.q.c 2 72.p odd 6 1
1344.2.q.c 2 504.ce odd 6 1
1344.2.q.m 2 72.n even 6 1
1344.2.q.m 2 504.cq even 6 1
2352.2.a.d 1 252.bl odd 6 1
2352.2.a.w 1 252.n even 6 1
2352.2.q.c 2 252.bi even 6 1
2352.2.q.c 2 252.bj even 6 1
3675.2.a.a 1 315.bo even 6 1
3675.2.a.c 1 315.bn odd 6 1
7056.2.a.m 1 252.bn odd 6 1
7056.2.a.bp 1 252.o even 6 1
9408.2.a.k 1 504.cz even 6 1
9408.2.a.bg 1 504.w even 6 1
9408.2.a.bz 1 504.cw odd 6 1
9408.2.a.cv 1 504.ba odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 2 T_{2} + 4$$ $$T_{13}^{2} + T_{13} + 1$$