# Properties

 Label 567.2.f.c 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - 3 q^{8}+O(q^{10})$$ q + (z - 1) * q^2 + z * q^4 + z * q^5 + (-z + 1) * q^7 - 3 * q^8 $$q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} + \zeta_{6} q^{5} + ( - \zeta_{6} + 1) q^{7} - 3 q^{8} - q^{10} + (2 \zeta_{6} - 2) q^{11} + 5 \zeta_{6} q^{13} + \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} - 3 q^{17} - 2 q^{19} + (\zeta_{6} - 1) q^{20} - 2 \zeta_{6} q^{22} + 6 \zeta_{6} q^{23} + ( - 4 \zeta_{6} + 4) q^{25} - 5 q^{26} + q^{28} + (5 \zeta_{6} - 5) q^{29} + 6 \zeta_{6} q^{31} - 5 \zeta_{6} q^{32} + ( - 3 \zeta_{6} + 3) q^{34} + q^{35} - 3 q^{37} + ( - 2 \zeta_{6} + 2) q^{38} - 3 \zeta_{6} q^{40} - 10 \zeta_{6} q^{41} + ( - 4 \zeta_{6} + 4) q^{43} - 2 q^{44} - 6 q^{46} + (6 \zeta_{6} - 6) q^{47} - \zeta_{6} q^{49} + 4 \zeta_{6} q^{50} + (5 \zeta_{6} - 5) q^{52} + 6 q^{53} - 2 q^{55} + (3 \zeta_{6} - 3) q^{56} - 5 \zeta_{6} q^{58} + 6 \zeta_{6} q^{59} + (7 \zeta_{6} - 7) q^{61} - 6 q^{62} + 7 q^{64} + (5 \zeta_{6} - 5) q^{65} + 2 \zeta_{6} q^{67} - 3 \zeta_{6} q^{68} + (\zeta_{6} - 1) q^{70} + 12 q^{71} - 15 q^{73} + ( - 3 \zeta_{6} + 3) q^{74} - 2 \zeta_{6} q^{76} + 2 \zeta_{6} q^{77} + (14 \zeta_{6} - 14) q^{79} + q^{80} + 10 q^{82} + ( - 18 \zeta_{6} + 18) q^{83} - 3 \zeta_{6} q^{85} + 4 \zeta_{6} q^{86} + ( - 6 \zeta_{6} + 6) q^{88} + 5 q^{89} + 5 q^{91} + (6 \zeta_{6} - 6) q^{92} - 6 \zeta_{6} q^{94} - 2 \zeta_{6} q^{95} + ( - 18 \zeta_{6} + 18) q^{97} + q^{98} +O(q^{100})$$ q + (z - 1) * q^2 + z * q^4 + z * q^5 + (-z + 1) * q^7 - 3 * q^8 - q^10 + (2*z - 2) * q^11 + 5*z * q^13 + z * q^14 + (-z + 1) * q^16 - 3 * q^17 - 2 * q^19 + (z - 1) * q^20 - 2*z * q^22 + 6*z * q^23 + (-4*z + 4) * q^25 - 5 * q^26 + q^28 + (5*z - 5) * q^29 + 6*z * q^31 - 5*z * q^32 + (-3*z + 3) * q^34 + q^35 - 3 * q^37 + (-2*z + 2) * q^38 - 3*z * q^40 - 10*z * q^41 + (-4*z + 4) * q^43 - 2 * q^44 - 6 * q^46 + (6*z - 6) * q^47 - z * q^49 + 4*z * q^50 + (5*z - 5) * q^52 + 6 * q^53 - 2 * q^55 + (3*z - 3) * q^56 - 5*z * q^58 + 6*z * q^59 + (7*z - 7) * q^61 - 6 * q^62 + 7 * q^64 + (5*z - 5) * q^65 + 2*z * q^67 - 3*z * q^68 + (z - 1) * q^70 + 12 * q^71 - 15 * q^73 + (-3*z + 3) * q^74 - 2*z * q^76 + 2*z * q^77 + (14*z - 14) * q^79 + q^80 + 10 * q^82 + (-18*z + 18) * q^83 - 3*z * q^85 + 4*z * q^86 + (-6*z + 6) * q^88 + 5 * q^89 + 5 * q^91 + (6*z - 6) * q^92 - 6*z * q^94 - 2*z * q^95 + (-18*z + 18) * q^97 + q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + q^{4} + q^{5} + q^{7} - 6 q^{8}+O(q^{10})$$ 2 * q - q^2 + q^4 + q^5 + q^7 - 6 * q^8 $$2 q - q^{2} + q^{4} + q^{5} + q^{7} - 6 q^{8} - 2 q^{10} - 2 q^{11} + 5 q^{13} + q^{14} + q^{16} - 6 q^{17} - 4 q^{19} - q^{20} - 2 q^{22} + 6 q^{23} + 4 q^{25} - 10 q^{26} + 2 q^{28} - 5 q^{29} + 6 q^{31} - 5 q^{32} + 3 q^{34} + 2 q^{35} - 6 q^{37} + 2 q^{38} - 3 q^{40} - 10 q^{41} + 4 q^{43} - 4 q^{44} - 12 q^{46} - 6 q^{47} - q^{49} + 4 q^{50} - 5 q^{52} + 12 q^{53} - 4 q^{55} - 3 q^{56} - 5 q^{58} + 6 q^{59} - 7 q^{61} - 12 q^{62} + 14 q^{64} - 5 q^{65} + 2 q^{67} - 3 q^{68} - q^{70} + 24 q^{71} - 30 q^{73} + 3 q^{74} - 2 q^{76} + 2 q^{77} - 14 q^{79} + 2 q^{80} + 20 q^{82} + 18 q^{83} - 3 q^{85} + 4 q^{86} + 6 q^{88} + 10 q^{89} + 10 q^{91} - 6 q^{92} - 6 q^{94} - 2 q^{95} + 18 q^{97} + 2 q^{98}+O(q^{100})$$ 2 * q - q^2 + q^4 + q^5 + q^7 - 6 * q^8 - 2 * q^10 - 2 * q^11 + 5 * q^13 + q^14 + q^16 - 6 * q^17 - 4 * q^19 - q^20 - 2 * q^22 + 6 * q^23 + 4 * q^25 - 10 * q^26 + 2 * q^28 - 5 * q^29 + 6 * q^31 - 5 * q^32 + 3 * q^34 + 2 * q^35 - 6 * q^37 + 2 * q^38 - 3 * q^40 - 10 * q^41 + 4 * q^43 - 4 * q^44 - 12 * q^46 - 6 * q^47 - q^49 + 4 * q^50 - 5 * q^52 + 12 * q^53 - 4 * q^55 - 3 * q^56 - 5 * q^58 + 6 * q^59 - 7 * q^61 - 12 * q^62 + 14 * q^64 - 5 * q^65 + 2 * q^67 - 3 * q^68 - q^70 + 24 * q^71 - 30 * q^73 + 3 * q^74 - 2 * q^76 + 2 * q^77 - 14 * q^79 + 2 * q^80 + 20 * q^82 + 18 * q^83 - 3 * q^85 + 4 * q^86 + 6 * q^88 + 10 * q^89 + 10 * q^91 - 6 * q^92 - 6 * q^94 - 2 * q^95 + 18 * q^97 + 2 * q^98

## 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
−0.500000 0.866025i 0 0.500000 0.866025i 0.500000 0.866025i 0 0.500000 + 0.866025i −3.00000 0 −1.00000
379.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i 0.500000 + 0.866025i 0 0.500000 0.866025i −3.00000 0 −1.00000
 $$n$$: 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.c 2
3.b odd 2 1 567.2.f.f 2
9.c even 3 1 567.2.a.b yes 1
9.c even 3 1 inner 567.2.f.c 2
9.d odd 6 1 567.2.a.a 1
9.d odd 6 1 567.2.f.f 2
36.f odd 6 1 9072.2.a.j 1
36.h even 6 1 9072.2.a.q 1
63.l odd 6 1 3969.2.a.e 1
63.o even 6 1 3969.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
567.2.a.a 1 9.d odd 6 1
567.2.a.b yes 1 9.c even 3 1
567.2.f.c 2 1.a even 1 1 trivial
567.2.f.c 2 9.c even 3 1 inner
567.2.f.f 2 3.b odd 2 1
567.2.f.f 2 9.d odd 6 1
3969.2.a.b 1 63.o even 6 1
3969.2.a.e 1 63.l odd 6 1
9072.2.a.j 1 36.f odd 6 1
9072.2.a.q 1 36.h even 6 1

## Hecke kernels

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

 $$T_{2}^{2} + T_{2} + 1$$ T2^2 + T2 + 1 $$T_{5}^{2} - T_{5} + 1$$ T5^2 - T5 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + T + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} - T + 1$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} + 2T + 4$$
$13$ $$T^{2} - 5T + 25$$
$17$ $$(T + 3)^{2}$$
$19$ $$(T + 2)^{2}$$
$23$ $$T^{2} - 6T + 36$$
$29$ $$T^{2} + 5T + 25$$
$31$ $$T^{2} - 6T + 36$$
$37$ $$(T + 3)^{2}$$
$41$ $$T^{2} + 10T + 100$$
$43$ $$T^{2} - 4T + 16$$
$47$ $$T^{2} + 6T + 36$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 6T + 36$$
$61$ $$T^{2} + 7T + 49$$
$67$ $$T^{2} - 2T + 4$$
$71$ $$(T - 12)^{2}$$
$73$ $$(T + 15)^{2}$$
$79$ $$T^{2} + 14T + 196$$
$83$ $$T^{2} - 18T + 324$$
$89$ $$(T - 5)^{2}$$
$97$ $$T^{2} - 18T + 324$$