# Properties

 Label 567.2.f.e 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: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 189) 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.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} +O(q^{10})$$ $$q + 2 \zeta_{6} q^{4} + 3 \zeta_{6} q^{5} + ( -1 + \zeta_{6} ) q^{7} + ( 6 - 6 \zeta_{6} ) q^{11} + 4 \zeta_{6} q^{13} + ( -4 + 4 \zeta_{6} ) q^{16} -3 q^{17} + 2 q^{19} + ( -6 + 6 \zeta_{6} ) q^{20} -6 \zeta_{6} q^{23} + ( -4 + 4 \zeta_{6} ) q^{25} -2 q^{28} + ( -6 + 6 \zeta_{6} ) q^{29} + 4 \zeta_{6} q^{31} -3 q^{35} -7 q^{37} -3 \zeta_{6} q^{41} + ( 1 - \zeta_{6} ) q^{43} + 12 q^{44} + ( 9 - 9 \zeta_{6} ) q^{47} -\zeta_{6} q^{49} + ( -8 + 8 \zeta_{6} ) q^{52} + 6 q^{53} + 18 q^{55} + 9 \zeta_{6} q^{59} + ( 10 - 10 \zeta_{6} ) q^{61} -8 q^{64} + ( -12 + 12 \zeta_{6} ) q^{65} + 4 \zeta_{6} q^{67} -6 \zeta_{6} q^{68} + 2 q^{73} + 4 \zeta_{6} q^{76} + 6 \zeta_{6} q^{77} + ( 1 - \zeta_{6} ) q^{79} -12 q^{80} + ( 3 - 3 \zeta_{6} ) q^{83} -9 \zeta_{6} q^{85} -6 q^{89} -4 q^{91} + ( 12 - 12 \zeta_{6} ) q^{92} + 6 \zeta_{6} q^{95} + ( 10 - 10 \zeta_{6} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 3q^{5} - q^{7} + O(q^{10})$$ $$2q + 2q^{4} + 3q^{5} - q^{7} + 6q^{11} + 4q^{13} - 4q^{16} - 6q^{17} + 4q^{19} - 6q^{20} - 6q^{23} - 4q^{25} - 4q^{28} - 6q^{29} + 4q^{31} - 6q^{35} - 14q^{37} - 3q^{41} + q^{43} + 24q^{44} + 9q^{47} - q^{49} - 8q^{52} + 12q^{53} + 36q^{55} + 9q^{59} + 10q^{61} - 16q^{64} - 12q^{65} + 4q^{67} - 6q^{68} + 4q^{73} + 4q^{76} + 6q^{77} + q^{79} - 24q^{80} + 3q^{83} - 9q^{85} - 12q^{89} - 8q^{91} + 12q^{92} + 6q^{95} + 10q^{97} + O(q^{100})$$

## Character values

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

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

## 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 0 1.00000 1.73205i 1.50000 2.59808i 0 −0.500000 0.866025i 0 0 0
379.1 0 0 1.00000 + 1.73205i 1.50000 + 2.59808i 0 −0.500000 + 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
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.e 2
3.b odd 2 1 567.2.f.d 2
9.c even 3 1 189.2.a.b 1
9.c even 3 1 inner 567.2.f.e 2
9.d odd 6 1 189.2.a.c yes 1
9.d odd 6 1 567.2.f.d 2
36.f odd 6 1 3024.2.a.f 1
36.h even 6 1 3024.2.a.y 1
45.h odd 6 1 4725.2.a.k 1
45.j even 6 1 4725.2.a.i 1
63.l odd 6 1 1323.2.a.l 1
63.o even 6 1 1323.2.a.h 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.a.b 1 9.c even 3 1
189.2.a.c yes 1 9.d odd 6 1
567.2.f.d 2 3.b odd 2 1
567.2.f.d 2 9.d odd 6 1
567.2.f.e 2 1.a even 1 1 trivial
567.2.f.e 2 9.c even 3 1 inner
1323.2.a.h 1 63.o even 6 1
1323.2.a.l 1 63.l odd 6 1
3024.2.a.f 1 36.f odd 6 1
3024.2.a.y 1 36.h even 6 1
4725.2.a.i 1 45.j even 6 1
4725.2.a.k 1 45.h odd 6 1

## Hecke kernels

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

 $$T_{2}$$ $$T_{5}^{2} - 3 T_{5} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$1 + T + T^{2}$$
$11$ $$36 - 6 T + T^{2}$$
$13$ $$16 - 4 T + T^{2}$$
$17$ $$( 3 + T )^{2}$$
$19$ $$( -2 + T )^{2}$$
$23$ $$36 + 6 T + T^{2}$$
$29$ $$36 + 6 T + T^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$( 7 + T )^{2}$$
$41$ $$9 + 3 T + T^{2}$$
$43$ $$1 - T + T^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$81 - 9 T + T^{2}$$
$61$ $$100 - 10 T + T^{2}$$
$67$ $$16 - 4 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$( -2 + T )^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$9 - 3 T + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$100 - 10 T + T^{2}$$