# Properties

 Label 567.2.f.l Level $567$ Weight $2$ Character orbit 567.f Analytic conductor $4.528$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1156923.1 Defining polynomial: $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4$$ x^6 - 3*x^5 + 12*x^4 - 19*x^3 + 27*x^2 - 18*x + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{3} - \beta_1) q^{5} + (\beta_{3} - 1) q^{7} + (2 \beta_{2} - 3) q^{8}+O(q^{10})$$ q + b4 * q^2 + (b5 + b4 - 2*b3 + b2 - b1) * q^4 + (b5 - b3 - b1) * q^5 + (b3 - 1) * q^7 + (2*b2 - 3) * q^8 $$q + \beta_{4} q^{2} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{4} + (\beta_{5} - \beta_{3} - \beta_1) q^{5} + (\beta_{3} - 1) q^{7} + (2 \beta_{2} - 3) q^{8} + (2 \beta_{2} + \beta_1 + 1) q^{10} + (\beta_{5} + 2 \beta_{3} - 2) q^{11} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1) q^{13} + ( - \beta_{4} - \beta_{2}) q^{14} + ( - 3 \beta_{4} + 4 \beta_{3} - 4) q^{16} + ( - 2 \beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{2} - 2 \beta_1) q^{19} + ( - \beta_{5} + 7 \beta_{3} - 7) q^{20} + ( - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{22} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{23} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} - 2) q^{25} + (2 \beta_{2} - 3 \beta_1 - 9) q^{26} + ( - \beta_{2} + \beta_1 + 2) q^{28} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - 1) q^{29} - 2 \beta_{3} q^{31} + ( - 3 \beta_{5} - 3 \beta_{4} + 6 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{32} + (\beta_{5} + 4 \beta_{4} - 7 \beta_{3} + 7) q^{34} + (\beta_1 + 1) q^{35} + 5 q^{37} + ( - 4 \beta_{4} + 6 \beta_{3} - 6) q^{38} + ( - \beta_{5} - 4 \beta_{4} + \beta_{3} - 4 \beta_{2} + \beta_1) q^{40} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{41} + (\beta_{5} + 4 \beta_{4} + 4 \beta_{3} - 4) q^{43} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{44} + ( - 4 \beta_{2} - 2 \beta_1 - 2) q^{46} - 2 \beta_{4} q^{47} - \beta_{3} q^{49} + (3 \beta_{5} - \beta_{4} - 9 \beta_{3} - \beta_{2} - 3 \beta_1) q^{50} + ( - \beta_{5} - 10 \beta_{4} + 3 \beta_{3} - 3) q^{52} + ( - 2 \beta_{2} + \beta_1 + 4) q^{53} + ( - 2 \beta_{2} + 2 \beta_1 - 4) q^{55} + (2 \beta_{4} - 3 \beta_{3} + 3) q^{56} + (3 \beta_{5} - 9 \beta_{3} - 3 \beta_1) q^{58} + (4 \beta_{4} + 6 \beta_{3} + 4 \beta_{2}) q^{59} + (3 \beta_{5} - \beta_{3} + 1) q^{61} + 2 \beta_{2} q^{62} + ( - 6 \beta_{2} + 1) q^{64} + (\beta_{5} - 6 \beta_{4} - 7 \beta_{3} + 7) q^{65} + (3 \beta_{5} - 2 \beta_{3} - 3 \beta_1) q^{67} + (\beta_{5} + 8 \beta_{4} - 13 \beta_{3} + 8 \beta_{2} - \beta_1) q^{68} + ( - \beta_{5} + 2 \beta_{4} + \beta_{3} - 1) q^{70} - 3 \beta_1 q^{71} + (2 \beta_{2} + \beta_1 + 3) q^{73} + 5 \beta_{4} q^{74} + ( - 6 \beta_{4} + 16 \beta_{3} - 6 \beta_{2}) q^{76} + ( - \beta_{5} - 2 \beta_{3} + \beta_1) q^{77} + ( - 3 \beta_{5} + 2 \beta_{3} - 2) q^{79} + ( - 6 \beta_{2} + \beta_1 + 1) q^{80} - 6 q^{82} + (2 \beta_{5} + 2 \beta_{4} + 4 \beta_{3} - 4) q^{83} + ( - \beta_{5} + 2 \beta_{4} - 5 \beta_{3} + 2 \beta_{2} + \beta_1) q^{85} + (3 \beta_{5} + \beta_{4} - 15 \beta_{3} + \beta_{2} - 3 \beta_1) q^{86} + ( - \beta_{5} + 2 \beta_{4} - 8 \beta_{3} + 8) q^{88} + (2 \beta_{2} - \beta_1 + 5) q^{89} + ( - 2 \beta_{2} - \beta_1 + 1) q^{91} + (2 \beta_{5} - 14 \beta_{3} + 14) q^{92} + ( - 2 \beta_{5} - 2 \beta_{4} + 8 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{94} + (2 \beta_{5} + 10 \beta_{3} - 2 \beta_1) q^{95} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 4) q^{97} + \beta_{2} q^{98}+O(q^{100})$$ q + b4 * q^2 + (b5 + b4 - 2*b3 + b2 - b1) * q^4 + (b5 - b3 - b1) * q^5 + (b3 - 1) * q^7 + (2*b2 - 3) * q^8 + (2*b2 + b1 + 1) * q^10 + (b5 + 2*b3 - 2) * q^11 + (-b5 + 2*b4 - b3 + 2*b2 + b1) * q^13 + (-b4 - b2) * q^14 + (-3*b4 + 4*b3 - 4) * q^16 + (-2*b2 + b1 + 1) * q^17 + (2*b2 - 2*b1) * q^19 + (-b5 + 7*b3 - 7) * q^20 + (-b5 - b4 + b3 - b2 + b1) * q^22 + (-2*b5 + 2*b3 + 2*b1) * q^23 + (-b5 + 2*b4 + 2*b3 - 2) * q^25 + (2*b2 - 3*b1 - 9) * q^26 + (-b2 + b1 + 2) * q^28 + (-b5 + 2*b4 + b3 - 1) * q^29 - 2*b3 * q^31 + (-3*b5 - 3*b4 + 6*b3 - 3*b2 + 3*b1) * q^32 + (b5 + 4*b4 - 7*b3 + 7) * q^34 + (b1 + 1) * q^35 + 5 * q^37 + (-4*b4 + 6*b3 - 6) * q^38 + (-b5 - 4*b4 + b3 - 4*b2 + b1) * q^40 + (2*b5 + 2*b4 + 4*b3 + 2*b2 - 2*b1) * q^41 + (b5 + 4*b4 + 4*b3 - 4) * q^43 + (-3*b2 + 2*b1 - 1) * q^44 + (-4*b2 - 2*b1 - 2) * q^46 - 2*b4 * q^47 - b3 * q^49 + (3*b5 - b4 - 9*b3 - b2 - 3*b1) * q^50 + (-b5 - 10*b4 + 3*b3 - 3) * q^52 + (-2*b2 + b1 + 4) * q^53 + (-2*b2 + 2*b1 - 4) * q^55 + (2*b4 - 3*b3 + 3) * q^56 + (3*b5 - 9*b3 - 3*b1) * q^58 + (4*b4 + 6*b3 + 4*b2) * q^59 + (3*b5 - b3 + 1) * q^61 + 2*b2 * q^62 + (-6*b2 + 1) * q^64 + (b5 - 6*b4 - 7*b3 + 7) * q^65 + (3*b5 - 2*b3 - 3*b1) * q^67 + (b5 + 8*b4 - 13*b3 + 8*b2 - b1) * q^68 + (-b5 + 2*b4 + b3 - 1) * q^70 - 3*b1 * q^71 + (2*b2 + b1 + 3) * q^73 + 5*b4 * q^74 + (-6*b4 + 16*b3 - 6*b2) * q^76 + (-b5 - 2*b3 + b1) * q^77 + (-3*b5 + 2*b3 - 2) * q^79 + (-6*b2 + b1 + 1) * q^80 - 6 * q^82 + (2*b5 + 2*b4 + 4*b3 - 4) * q^83 + (-b5 + 2*b4 - 5*b3 + 2*b2 + b1) * q^85 + (3*b5 + b4 - 15*b3 + b2 - 3*b1) * q^86 + (-b5 + 2*b4 - 8*b3 + 8) * q^88 + (2*b2 - b1 + 5) * q^89 + (-2*b2 - b1 + 1) * q^91 + (2*b5 - 14*b3 + 14) * q^92 + (-2*b5 - 2*b4 + 8*b3 - 2*b2 + 2*b1) * q^94 + (2*b5 + 10*b3 - 2*b1) * q^95 + (-2*b5 - 2*b4 + 4*b3 - 4) * q^97 + b2 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4} - 3 q^{5} - 3 q^{7} - 18 q^{8}+O(q^{10})$$ 6 * q - 6 * q^4 - 3 * q^5 - 3 * q^7 - 18 * q^8 $$6 q - 6 q^{4} - 3 q^{5} - 3 q^{7} - 18 q^{8} + 6 q^{10} - 6 q^{11} - 3 q^{13} - 12 q^{16} + 6 q^{17} - 21 q^{20} + 3 q^{22} + 6 q^{23} - 6 q^{25} - 54 q^{26} + 12 q^{28} - 3 q^{29} - 6 q^{31} + 18 q^{32} + 21 q^{34} + 6 q^{35} + 30 q^{37} - 18 q^{38} + 3 q^{40} + 12 q^{41} - 12 q^{43} - 6 q^{44} - 12 q^{46} - 3 q^{49} - 27 q^{50} - 9 q^{52} + 24 q^{53} - 24 q^{55} + 9 q^{56} - 27 q^{58} + 18 q^{59} + 3 q^{61} + 6 q^{64} + 21 q^{65} - 6 q^{67} - 39 q^{68} - 3 q^{70} + 18 q^{73} + 48 q^{76} - 6 q^{77} - 6 q^{79} + 6 q^{80} - 36 q^{82} - 12 q^{83} - 15 q^{85} - 45 q^{86} + 24 q^{88} + 30 q^{89} + 6 q^{91} + 42 q^{92} + 24 q^{94} + 30 q^{95} - 12 q^{97}+O(q^{100})$$ 6 * q - 6 * q^4 - 3 * q^5 - 3 * q^7 - 18 * q^8 + 6 * q^10 - 6 * q^11 - 3 * q^13 - 12 * q^16 + 6 * q^17 - 21 * q^20 + 3 * q^22 + 6 * q^23 - 6 * q^25 - 54 * q^26 + 12 * q^28 - 3 * q^29 - 6 * q^31 + 18 * q^32 + 21 * q^34 + 6 * q^35 + 30 * q^37 - 18 * q^38 + 3 * q^40 + 12 * q^41 - 12 * q^43 - 6 * q^44 - 12 * q^46 - 3 * q^49 - 27 * q^50 - 9 * q^52 + 24 * q^53 - 24 * q^55 + 9 * q^56 - 27 * q^58 + 18 * q^59 + 3 * q^61 + 6 * q^64 + 21 * q^65 - 6 * q^67 - 39 * q^68 - 3 * q^70 + 18 * q^73 + 48 * q^76 - 6 * q^77 - 6 * q^79 + 6 * q^80 - 36 * q^82 - 12 * q^83 - 15 * q^85 - 45 * q^86 + 24 * q^88 + 30 * q^89 + 6 * q^91 + 42 * q^92 + 24 * q^94 + 30 * q^95 - 12 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 3x^{5} + 12x^{4} - 19x^{3} + 27x^{2} - 18x + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu^{2} - \nu + 3$$ v^2 - v + 3 $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 2\nu^{3} + 8\nu^{2} - 7\nu + 6 ) / 2$$ (v^4 - 2*v^3 + 8*v^2 - 7*v + 6) / 2 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - 5\nu^{4} + 22\nu^{3} - 28\nu^{2} + 43\nu - 16 ) / 2$$ (2*v^5 - 5*v^4 + 22*v^3 - 28*v^2 + 43*v - 16) / 2 $$\beta_{4}$$ $$=$$ $$( 5\nu^{5} - 13\nu^{4} + 54\nu^{3} - 71\nu^{2} + 99\nu - 40 ) / 2$$ (5*v^5 - 13*v^4 + 54*v^3 - 71*v^2 + 99*v - 40) / 2 $$\beta_{5}$$ $$=$$ $$( -6\nu^{5} + 15\nu^{4} - 64\nu^{3} + 82\nu^{2} - 121\nu + 50 ) / 2$$ (-6*v^5 + 15*v^4 - 64*v^3 + 82*v^2 - 121*v + 50) / 2
 $$\nu$$ $$=$$ $$( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 2 ) / 3$$ (-2*b5 - 2*b4 - b3 - b2 + b1 + 2) / 3 $$\nu^{2}$$ $$=$$ $$( -2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 4\beta _1 - 7 ) / 3$$ (-2*b5 - 2*b4 - b3 - b2 + 4*b1 - 7) / 3 $$\nu^{3}$$ $$=$$ $$3\beta_{5} + 2\beta_{4} + 4\beta_{3} + \beta_{2} - 6$$ 3*b5 + 2*b4 + 4*b3 + b2 - 6 $$\nu^{4}$$ $$=$$ $$( 20\beta_{5} + 14\beta_{4} + 25\beta_{3} + 13\beta_{2} - 25\beta _1 + 16 ) / 3$$ (20*b5 + 14*b4 + 25*b3 + 13*b2 - 25*b1 + 16) / 3 $$\nu^{5}$$ $$=$$ $$( -34\beta_{5} - 16\beta_{4} - 59\beta_{3} + 7\beta_{2} - 28\beta _1 + 121 ) / 3$$ (-34*b5 - 16*b4 - 59*b3 + 7*b2 - 28*b1 + 121) / 3

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1$$ $$-\beta_{3}$$

## 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 − 0.0585812i 0.5 + 2.43956i 0.5 − 1.51496i 0.5 + 0.0585812i 0.5 − 2.43956i 0.5 + 1.51496i
−1.07255 1.85771i 0 −1.30073 + 2.25294i −1.87328 + 3.24462i 0 −0.500000 0.866025i 1.29021 0 8.03677
190.2 −0.261988 0.453777i 0 0.862724 1.49428i 1.10074 1.90653i 0 −0.500000 0.866025i −1.95205 0 −1.15352
190.3 1.33454 + 2.31149i 0 −2.56199 + 4.43750i −0.727452 + 1.25998i 0 −0.500000 0.866025i −8.33816 0 −3.88325
379.1 −1.07255 + 1.85771i 0 −1.30073 2.25294i −1.87328 3.24462i 0 −0.500000 + 0.866025i 1.29021 0 8.03677
379.2 −0.261988 + 0.453777i 0 0.862724 + 1.49428i 1.10074 + 1.90653i 0 −0.500000 + 0.866025i −1.95205 0 −1.15352
379.3 1.33454 2.31149i 0 −2.56199 4.43750i −0.727452 1.25998i 0 −0.500000 + 0.866025i −8.33816 0 −3.88325
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 379.3 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.l 6
3.b odd 2 1 567.2.f.m 6
9.c even 3 1 567.2.a.f yes 3
9.c even 3 1 inner 567.2.f.l 6
9.d odd 6 1 567.2.a.e 3
9.d odd 6 1 567.2.f.m 6
36.f odd 6 1 9072.2.a.cb 3
36.h even 6 1 9072.2.a.bu 3
63.l odd 6 1 3969.2.a.n 3
63.o even 6 1 3969.2.a.o 3

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

## Hecke kernels

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

 $$T_{2}^{6} + 6T_{2}^{4} + 6T_{2}^{3} + 36T_{2}^{2} + 18T_{2} + 9$$ T2^6 + 6*T2^4 + 6*T2^3 + 36*T2^2 + 18*T2 + 9 $$T_{5}^{6} + 3T_{5}^{5} + 15T_{5}^{4} + 6T_{5}^{3} + 72T_{5}^{2} + 72T_{5} + 144$$ T5^6 + 3*T5^5 + 15*T5^4 + 6*T5^3 + 72*T5^2 + 72*T5 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 6 T^{4} + 6 T^{3} + 36 T^{2} + \cdots + 9$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 3 T^{5} + 15 T^{4} + 6 T^{3} + \cdots + 144$$
$7$ $$(T^{2} + T + 1)^{3}$$
$11$ $$T^{6} + 6 T^{5} + 33 T^{4} + 30 T^{3} + \cdots + 36$$
$13$ $$T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 12544$$
$17$ $$(T^{3} - 3 T^{2} - 24 T - 12)^{2}$$
$19$ $$(T^{3} - 48 T - 56)^{2}$$
$23$ $$T^{6} - 6 T^{5} + 60 T^{4} + \cdots + 9216$$
$29$ $$T^{6} + 3 T^{5} + 45 T^{4} + \cdots + 1296$$
$31$ $$(T^{2} + 2 T + 4)^{3}$$
$37$ $$(T - 5)^{6}$$
$41$ $$T^{6} - 12 T^{5} + 144 T^{4} + \cdots + 5184$$
$43$ $$T^{6} + 12 T^{5} + 189 T^{4} + \cdots + 425104$$
$47$ $$T^{6} + 24 T^{4} - 48 T^{3} + \cdots + 576$$
$53$ $$(T^{3} - 12 T^{2} + 21 T + 6)^{2}$$
$59$ $$T^{6} - 18 T^{5} + 312 T^{4} + \cdots + 304704$$
$61$ $$T^{6} - 3 T^{5} + 87 T^{4} + \cdots + 35344$$
$67$ $$T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 68644$$
$71$ $$(T^{3} - 81 T - 108)^{2}$$
$73$ $$(T^{3} - 9 T^{2} - 12 T + 16)^{2}$$
$79$ $$T^{6} + 6 T^{5} + 105 T^{4} + \cdots + 68644$$
$83$ $$T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 5184$$
$89$ $$(T^{3} - 15 T^{2} + 48 T + 48)^{2}$$
$97$ $$T^{6} + 12 T^{5} + 144 T^{4} + \cdots + 33856$$