# Properties

 Label 567.2.h.c Level $567$ Weight $2$ Character orbit 567.h Analytic conductor $4.528$ Analytic rank $0$ 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.h (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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -2 q^{4} + ( -1 - 2 \zeta_{6} ) q^{7} +O(q^{10})$$ $$q -2 q^{4} + ( -1 - 2 \zeta_{6} ) q^{7} + 7 \zeta_{6} q^{13} + 4 q^{16} + 7 \zeta_{6} q^{19} + 5 \zeta_{6} q^{25} + ( 2 + 4 \zeta_{6} ) q^{28} -7 q^{31} + \zeta_{6} q^{37} + ( -5 + 5 \zeta_{6} ) q^{43} + ( -3 + 8 \zeta_{6} ) q^{49} -14 \zeta_{6} q^{52} + 14 q^{61} -8 q^{64} + 11 q^{67} + ( 7 - 7 \zeta_{6} ) q^{73} -14 \zeta_{6} q^{76} -13 q^{79} + ( 14 - 21 \zeta_{6} ) q^{91} + ( -14 + 14 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} - 4q^{7} + O(q^{10})$$ $$2q - 4q^{4} - 4q^{7} + 7q^{13} + 8q^{16} + 7q^{19} + 5q^{25} + 8q^{28} - 14q^{31} + q^{37} - 5q^{43} + 2q^{49} - 14q^{52} + 28q^{61} - 16q^{64} + 22q^{67} + 7q^{73} - 14q^{76} - 26q^{79} + 7q^{91} - 14q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
298.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 −2.00000 0 0 −2.00000 1.73205i 0 0 0
352.1 0 0 −2.00000 0 0 −2.00000 + 1.73205i 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.h even 3 1 inner
63.j odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.h.c 2
3.b odd 2 1 CM 567.2.h.c 2
7.c even 3 1 567.2.g.d 2
9.c even 3 1 63.2.e.a 2
9.c even 3 1 567.2.g.d 2
9.d odd 6 1 63.2.e.a 2
9.d odd 6 1 567.2.g.d 2
21.h odd 6 1 567.2.g.d 2
36.f odd 6 1 1008.2.s.j 2
36.h even 6 1 1008.2.s.j 2
63.g even 3 1 63.2.e.a 2
63.h even 3 1 441.2.a.d 1
63.h even 3 1 inner 567.2.h.c 2
63.i even 6 1 441.2.a.e 1
63.j odd 6 1 441.2.a.d 1
63.j odd 6 1 inner 567.2.h.c 2
63.k odd 6 1 441.2.e.c 2
63.l odd 6 1 441.2.e.c 2
63.n odd 6 1 63.2.e.a 2
63.o even 6 1 441.2.e.c 2
63.s even 6 1 441.2.e.c 2
63.t odd 6 1 441.2.a.e 1
252.o even 6 1 1008.2.s.j 2
252.r odd 6 1 7056.2.a.bf 1
252.u odd 6 1 7056.2.a.y 1
252.bb even 6 1 7056.2.a.y 1
252.bj even 6 1 7056.2.a.bf 1
252.bl odd 6 1 1008.2.s.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.e.a 2 9.c even 3 1
63.2.e.a 2 9.d odd 6 1
63.2.e.a 2 63.g even 3 1
63.2.e.a 2 63.n odd 6 1
441.2.a.d 1 63.h even 3 1
441.2.a.d 1 63.j odd 6 1
441.2.a.e 1 63.i even 6 1
441.2.a.e 1 63.t odd 6 1
441.2.e.c 2 63.k odd 6 1
441.2.e.c 2 63.l odd 6 1
441.2.e.c 2 63.o even 6 1
441.2.e.c 2 63.s even 6 1
567.2.g.d 2 7.c even 3 1
567.2.g.d 2 9.c even 3 1
567.2.g.d 2 9.d odd 6 1
567.2.g.d 2 21.h odd 6 1
567.2.h.c 2 1.a even 1 1 trivial
567.2.h.c 2 3.b odd 2 1 CM
567.2.h.c 2 63.h even 3 1 inner
567.2.h.c 2 63.j odd 6 1 inner
1008.2.s.j 2 36.f odd 6 1
1008.2.s.j 2 36.h even 6 1
1008.2.s.j 2 252.o even 6 1
1008.2.s.j 2 252.bl odd 6 1
7056.2.a.y 1 252.u odd 6 1
7056.2.a.y 1 252.bb even 6 1
7056.2.a.bf 1 252.r odd 6 1
7056.2.a.bf 1 252.bj even 6 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{13}^{2} - 7 T_{13} + 49$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$T^{2}$$
$13$ $$49 - 7 T + T^{2}$$
$17$ $$T^{2}$$
$19$ $$49 - 7 T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$T^{2}$$
$31$ $$( 7 + T )^{2}$$
$37$ $$1 - T + T^{2}$$
$41$ $$T^{2}$$
$43$ $$25 + 5 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2}$$
$59$ $$T^{2}$$
$61$ $$( -14 + T )^{2}$$
$67$ $$( -11 + T )^{2}$$
$71$ $$T^{2}$$
$73$ $$49 - 7 T + T^{2}$$
$79$ $$( 13 + T )^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2}$$
$97$ $$196 + 14 T + T^{2}$$