# Properties

 Label 567.2.g.d Level 567 Weight 2 Character orbit 567.g 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.g (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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

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

## Character values

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

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

## 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
0 0 1.00000 + 1.73205i 0 0 2.50000 0.866025i 0 0 0
541.1 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.g even 3 1 inner
63.n 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.g.d 2
3.b odd 2 1 CM 567.2.g.d 2
7.c even 3 1 567.2.h.c 2
9.c even 3 1 63.2.e.a 2
9.c even 3 1 567.2.h.c 2
9.d odd 6 1 63.2.e.a 2
9.d odd 6 1 567.2.h.c 2
21.h odd 6 1 567.2.h.c 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 1 441.2.a.d 1
63.g even 3 1 inner 567.2.g.d 2
63.h even 3 1 63.2.e.a 2
63.i even 6 1 441.2.e.c 2
63.j odd 6 1 63.2.e.a 2
63.k odd 6 1 441.2.a.e 1
63.l odd 6 1 441.2.e.c 2
63.n odd 6 1 441.2.a.d 1
63.n odd 6 1 inner 567.2.g.d 2
63.o even 6 1 441.2.e.c 2
63.s even 6 1 441.2.a.e 1
63.t odd 6 1 441.2.e.c 2
252.n even 6 1 7056.2.a.bf 1
252.o even 6 1 7056.2.a.y 1
252.u odd 6 1 1008.2.s.j 2
252.bb even 6 1 1008.2.s.j 2
252.bl odd 6 1 7056.2.a.y 1
252.bn odd 6 1 7056.2.a.bf 1

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.h even 3 1
63.2.e.a 2 63.j odd 6 1
441.2.a.d 1 63.g even 3 1
441.2.a.d 1 63.n odd 6 1
441.2.a.e 1 63.k odd 6 1
441.2.a.e 1 63.s even 6 1
441.2.e.c 2 63.i even 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.t odd 6 1
567.2.g.d 2 1.a even 1 1 trivial
567.2.g.d 2 3.b odd 2 1 CM
567.2.g.d 2 63.g even 3 1 inner
567.2.g.d 2 63.n odd 6 1 inner
567.2.h.c 2 7.c even 3 1
567.2.h.c 2 9.c even 3 1
567.2.h.c 2 9.d odd 6 1
567.2.h.c 2 21.h odd 6 1
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.u odd 6 1
1008.2.s.j 2 252.bb even 6 1
7056.2.a.y 1 252.o even 6 1
7056.2.a.y 1 252.bl odd 6 1
7056.2.a.bf 1 252.n even 6 1
7056.2.a.bf 1 252.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 $$S_{2}^{\mathrm{new}}(567, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$1 - 5 T + 7 T^{2}$$
$11$ $$( 1 + 11 T^{2} )^{2}$$
$13$ $$( 1 - 5 T + 13 T^{2} )( 1 - 2 T + 13 T^{2} )$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + T + 19 T^{2} )$$
$23$ $$( 1 + 23 T^{2} )^{2}$$
$29$ $$1 - 29 T^{2} + 841 T^{4}$$
$31$ $$( 1 - 11 T + 31 T^{2} )( 1 + 4 T + 31 T^{2} )$$
$37$ $$( 1 - 11 T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$1 - 41 T^{2} + 1681 T^{4}$$
$43$ $$( 1 - 8 T + 43 T^{2} )( 1 + 13 T + 43 T^{2} )$$
$47$ $$1 - 47 T^{2} + 2209 T^{4}$$
$53$ $$1 - 53 T^{2} + 2809 T^{4}$$
$59$ $$1 - 59 T^{2} + 3481 T^{4}$$
$61$ $$( 1 + T + 61 T^{2} )( 1 + 13 T + 61 T^{2} )$$
$67$ $$( 1 - 5 T + 67 T^{2} )( 1 + 16 T + 67 T^{2} )$$
$71$ $$( 1 + 71 T^{2} )^{2}$$
$73$ $$( 1 - 17 T + 73 T^{2} )( 1 + 10 T + 73 T^{2} )$$
$79$ $$( 1 - 17 T + 79 T^{2} )( 1 + 4 T + 79 T^{2} )$$
$83$ $$1 - 83 T^{2} + 6889 T^{4}$$
$89$ $$1 - 89 T^{2} + 7921 T^{4}$$
$97$ $$( 1 - 5 T + 97 T^{2} )( 1 + 19 T + 97 T^{2} )$$