# Properties

 Label 567.2.s.a Level $567$ Weight $2$ Character orbit 567.s Analytic conductor $4.528$ Analytic rank $1$ Dimension $2$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.s (of order $$6$$, 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: $\mathrm{U}(1)[D_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 \zeta_{6} q^{4} + ( -3 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -2 \zeta_{6} q^{4} + ( -3 + \zeta_{6} ) q^{7} + ( -1 - \zeta_{6} ) q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} + ( -6 + 3 \zeta_{6} ) q^{19} -5 q^{25} + ( 2 + 4 \zeta_{6} ) q^{28} + ( -10 + 5 \zeta_{6} ) q^{31} -\zeta_{6} q^{37} + 5 \zeta_{6} q^{43} + ( 8 - 5 \zeta_{6} ) q^{49} + ( -2 + 4 \zeta_{6} ) q^{52} + ( -4 - 4 \zeta_{6} ) q^{61} + 8 q^{64} -11 \zeta_{6} q^{67} + ( -9 - 9 \zeta_{6} ) q^{73} + ( 6 + 6 \zeta_{6} ) q^{76} + ( 13 - 13 \zeta_{6} ) q^{79} + ( 4 + \zeta_{6} ) q^{91} + ( 16 - 8 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} - 5q^{7} + O(q^{10})$$ $$2q - 2q^{4} - 5q^{7} - 3q^{13} - 4q^{16} - 9q^{19} - 10q^{25} + 8q^{28} - 15q^{31} - q^{37} + 5q^{43} + 11q^{49} - 12q^{61} + 16q^{64} - 11q^{67} - 27q^{73} + 18q^{76} + 13q^{79} + 9q^{91} + 24q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 −1.00000 + 1.73205i 0 0 −2.50000 0.866025i 0 0 0
458.1 0 0 −1.00000 1.73205i 0 0 −2.50000 + 0.866025i 0 0 0
 $$n$$: 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
63.k odd 6 1 inner
63.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.s.a 2
3.b odd 2 1 CM 567.2.s.a 2
7.d odd 6 1 567.2.i.b 2
9.c even 3 1 21.2.g.a 2
9.c even 3 1 567.2.i.b 2
9.d odd 6 1 21.2.g.a 2
9.d odd 6 1 567.2.i.b 2
21.g even 6 1 567.2.i.b 2
36.f odd 6 1 336.2.bc.c 2
36.h even 6 1 336.2.bc.c 2
45.h odd 6 1 525.2.t.c 2
45.j even 6 1 525.2.t.c 2
45.k odd 12 2 525.2.q.d 4
45.l even 12 2 525.2.q.d 4
63.g even 3 1 147.2.c.a 2
63.h even 3 1 147.2.g.a 2
63.i even 6 1 21.2.g.a 2
63.j odd 6 1 147.2.g.a 2
63.k odd 6 1 147.2.c.a 2
63.k odd 6 1 inner 567.2.s.a 2
63.l odd 6 1 147.2.g.a 2
63.n odd 6 1 147.2.c.a 2
63.o even 6 1 147.2.g.a 2
63.s even 6 1 147.2.c.a 2
63.s even 6 1 inner 567.2.s.a 2
63.t odd 6 1 21.2.g.a 2
252.n even 6 1 2352.2.k.c 2
252.o even 6 1 2352.2.k.c 2
252.r odd 6 1 336.2.bc.c 2
252.bj even 6 1 336.2.bc.c 2
252.bl odd 6 1 2352.2.k.c 2
252.bn odd 6 1 2352.2.k.c 2
315.q odd 6 1 525.2.t.c 2
315.bq even 6 1 525.2.t.c 2
315.bs even 12 2 525.2.q.d 4
315.bu odd 12 2 525.2.q.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.2.g.a 2 9.c even 3 1
21.2.g.a 2 9.d odd 6 1
21.2.g.a 2 63.i even 6 1
21.2.g.a 2 63.t odd 6 1
147.2.c.a 2 63.g even 3 1
147.2.c.a 2 63.k odd 6 1
147.2.c.a 2 63.n odd 6 1
147.2.c.a 2 63.s even 6 1
147.2.g.a 2 63.h even 3 1
147.2.g.a 2 63.j odd 6 1
147.2.g.a 2 63.l odd 6 1
147.2.g.a 2 63.o even 6 1
336.2.bc.c 2 36.f odd 6 1
336.2.bc.c 2 36.h even 6 1
336.2.bc.c 2 252.r odd 6 1
336.2.bc.c 2 252.bj even 6 1
525.2.q.d 4 45.k odd 12 2
525.2.q.d 4 45.l even 12 2
525.2.q.d 4 315.bs even 12 2
525.2.q.d 4 315.bu odd 12 2
525.2.t.c 2 45.h odd 6 1
525.2.t.c 2 45.j even 6 1
525.2.t.c 2 315.q odd 6 1
525.2.t.c 2 315.bq even 6 1
567.2.i.b 2 7.d odd 6 1
567.2.i.b 2 9.c even 3 1
567.2.i.b 2 9.d odd 6 1
567.2.i.b 2 21.g even 6 1
567.2.s.a 2 1.a even 1 1 trivial
567.2.s.a 2 3.b odd 2 1 CM
567.2.s.a 2 63.k odd 6 1 inner
567.2.s.a 2 63.s even 6 1 inner
2352.2.k.c 2 252.n even 6 1
2352.2.k.c 2 252.o even 6 1
2352.2.k.c 2 252.bl odd 6 1
2352.2.k.c 2 252.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 $$S_{2}^{\mathrm{new}}(567, [\chi])$$:

 $$T_{2}$$ $$T_{11}$$ $$T_{13}^{2} + 3 T_{13} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$3 + 3 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$27 + 9 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$75 + 15 T + T^{2}$$
$37$ $$1 + T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$25 - 5 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$48 + 12 T + T^{2}$$
$67$ $$121 + 11 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$243 + 27 T + T^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$192 - 24 T + T^{2}$$