# Properties

 Label 567.2.f.d 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: $$x^{2} - x + 1$$ 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.

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

## 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.d 2
3.b odd 2 1 567.2.f.e 2
9.c even 3 1 189.2.a.c yes 1
9.c even 3 1 inner 567.2.f.d 2
9.d odd 6 1 189.2.a.b 1
9.d odd 6 1 567.2.f.e 2
36.f odd 6 1 3024.2.a.y 1
36.h even 6 1 3024.2.a.f 1
45.h odd 6 1 4725.2.a.i 1
45.j even 6 1 4725.2.a.k 1
63.l odd 6 1 1323.2.a.h 1
63.o even 6 1 1323.2.a.l 1

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

## Hecke kernels

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

 $$T_{2}$$ T2 $$T_{5}^{2} + 3T_{5} + 9$$ T5^2 + 3*T5 + 9

## Hecke characteristic polynomials

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