# Properties

 Label 567.2.o.e Level 567 Weight 2 Character orbit 567.o Analytic conductor 4.528 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.o (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 189) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{24}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{24} - \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{24} - \zeta_{24}^{7} ) q^{2} + ( -\zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{5} + ( -\zeta_{24} - 2 \zeta_{24}^{3} - \zeta_{24}^{4} + 2 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( -2 \zeta_{24} + 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} ) q^{8} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{10} + ( \zeta_{24} - \zeta_{24}^{7} ) q^{11} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{13} + ( -2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{14} + 4 \zeta_{24}^{4} q^{16} + ( -6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{17} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{19} + ( 2 - 2 \zeta_{24}^{4} ) q^{22} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{23} + 2 \zeta_{24}^{4} q^{25} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{26} + ( -5 \zeta_{24} + 5 \zeta_{24}^{7} ) q^{29} + ( 2 \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{31} + ( -3 \zeta_{24} - 6 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{34} + ( 3 \zeta_{24} + 2 \zeta_{24}^{2} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} - \zeta_{24}^{6} ) q^{35} + 5 q^{37} + ( 6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{38} + ( -4 \zeta_{24} - 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{40} + ( 5 \zeta_{24}^{2} - 10 \zeta_{24}^{6} ) q^{41} -5 \zeta_{24}^{4} q^{43} + 4 q^{46} + ( -5 \zeta_{24}^{2} - 5 \zeta_{24}^{6} ) q^{47} + ( 5 + 4 \zeta_{24} + 2 \zeta_{24}^{3} - 5 \zeta_{24}^{4} - 2 \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{49} + ( 2 \zeta_{24}^{3} + 2 \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{50} + ( 8 \zeta_{24} - 8 \zeta_{24}^{3} - 8 \zeta_{24}^{5} ) q^{53} + ( \zeta_{24} - \zeta_{24}^{3} + \zeta_{24}^{5} + 2 \zeta_{24}^{7} ) q^{55} + ( 2 \zeta_{24} - 4 \zeta_{24}^{2} - 4 \zeta_{24}^{6} - 2 \zeta_{24}^{7} ) q^{56} + ( -10 + 10 \zeta_{24}^{4} ) q^{58} + ( 5 \zeta_{24}^{2} - 10 \zeta_{24}^{6} ) q^{59} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{61} + ( 4 \zeta_{24}^{2} - 2 \zeta_{24}^{6} ) q^{62} -8 q^{64} + ( -3 \zeta_{24} + 3 \zeta_{24}^{7} ) q^{65} + ( -2 + 2 \zeta_{24}^{4} ) q^{67} + ( \zeta_{24} + 2 \zeta_{24}^{3} - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{5} - \zeta_{24}^{7} ) q^{70} + ( -10 \zeta_{24} + 10 \zeta_{24}^{3} + 10 \zeta_{24}^{5} ) q^{71} + ( 5 \zeta_{24} - 5 \zeta_{24}^{7} ) q^{74} + ( -2 \zeta_{24}^{2} - \zeta_{24}^{3} - \zeta_{24}^{5} + 4 \zeta_{24}^{6} + \zeta_{24}^{7} ) q^{77} + 13 \zeta_{24}^{4} q^{79} + ( -8 \zeta_{24}^{2} + 4 \zeta_{24}^{6} ) q^{80} + ( -5 \zeta_{24} + 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 10 \zeta_{24}^{7} ) q^{82} + ( \zeta_{24}^{2} + \zeta_{24}^{6} ) q^{83} + ( 9 - 9 \zeta_{24}^{4} ) q^{85} + ( -5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{86} + 4 \zeta_{24}^{4} q^{88} + ( 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{89} + ( -6 - \zeta_{24} + \zeta_{24}^{3} - \zeta_{24}^{5} - 2 \zeta_{24}^{7} ) q^{91} + ( -10 \zeta_{24} - 5 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 5 \zeta_{24}^{7} ) q^{94} + ( -9 \zeta_{24}^{3} - 9 \zeta_{24}^{5} + 9 \zeta_{24}^{7} ) q^{95} + ( 7 \zeta_{24} + 14 \zeta_{24}^{3} - 14 \zeta_{24}^{5} - 7 \zeta_{24}^{7} ) q^{97} + ( 5 \zeta_{24} + 8 \zeta_{24}^{2} - 5 \zeta_{24}^{3} - 5 \zeta_{24}^{5} - 4 \zeta_{24}^{6} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 4q^{7} + O(q^{10})$$ $$8q - 4q^{7} + 16q^{16} + 8q^{22} + 8q^{25} + 40q^{37} - 20q^{43} + 32q^{46} + 20q^{49} - 40q^{58} - 64q^{64} - 8q^{67} - 24q^{70} + 52q^{79} + 36q^{85} + 16q^{88} - 48q^{91} + O(q^{100})$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{24}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
188.1
 −0.965926 + 0.258819i −0.258819 − 0.965926i 0.965926 − 0.258819i 0.258819 + 0.965926i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.965926 + 0.258819i 0.258819 − 0.965926i
−1.22474 0.707107i 0 0 −0.866025 1.50000i 0 1.62132 + 2.09077i 2.82843i 0 2.44949i
188.2 −1.22474 0.707107i 0 0 0.866025 + 1.50000i 0 −2.62132 0.358719i 2.82843i 0 2.44949i
188.3 1.22474 + 0.707107i 0 0 −0.866025 1.50000i 0 −2.62132 0.358719i 2.82843i 0 2.44949i
188.4 1.22474 + 0.707107i 0 0 0.866025 + 1.50000i 0 1.62132 + 2.09077i 2.82843i 0 2.44949i
377.1 −1.22474 + 0.707107i 0 0 −0.866025 + 1.50000i 0 1.62132 2.09077i 2.82843i 0 2.44949i
377.2 −1.22474 + 0.707107i 0 0 0.866025 1.50000i 0 −2.62132 + 0.358719i 2.82843i 0 2.44949i
377.3 1.22474 0.707107i 0 0 −0.866025 + 1.50000i 0 −2.62132 + 0.358719i 2.82843i 0 2.44949i
377.4 1.22474 0.707107i 0 0 0.866025 1.50000i 0 1.62132 2.09077i 2.82843i 0 2.44949i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 377.4 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 inner
7.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
21.c even 2 1 inner
63.l odd 6 1 inner
63.o even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.o.e 8
3.b odd 2 1 inner 567.2.o.e 8
7.b odd 2 1 inner 567.2.o.e 8
9.c even 3 1 189.2.c.c 4
9.c even 3 1 inner 567.2.o.e 8
9.d odd 6 1 189.2.c.c 4
9.d odd 6 1 inner 567.2.o.e 8
21.c even 2 1 inner 567.2.o.e 8
36.f odd 6 1 3024.2.k.g 4
36.h even 6 1 3024.2.k.g 4
63.l odd 6 1 189.2.c.c 4
63.l odd 6 1 inner 567.2.o.e 8
63.o even 6 1 189.2.c.c 4
63.o even 6 1 inner 567.2.o.e 8
252.s odd 6 1 3024.2.k.g 4
252.bi even 6 1 3024.2.k.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.c.c 4 9.c even 3 1
189.2.c.c 4 9.d odd 6 1
189.2.c.c 4 63.l odd 6 1
189.2.c.c 4 63.o even 6 1
567.2.o.e 8 1.a even 1 1 trivial
567.2.o.e 8 3.b odd 2 1 inner
567.2.o.e 8 7.b odd 2 1 inner
567.2.o.e 8 9.c even 3 1 inner
567.2.o.e 8 9.d odd 6 1 inner
567.2.o.e 8 21.c even 2 1 inner
567.2.o.e 8 63.l odd 6 1 inner
567.2.o.e 8 63.o even 6 1 inner
3024.2.k.g 4 36.f odd 6 1
3024.2.k.g 4 36.h even 6 1
3024.2.k.g 4 252.s odd 6 1
3024.2.k.g 4 252.bi even 6 1

## Hecke kernels

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

 $$T_{2}^{4} - 2 T_{2}^{2} + 4$$ $$T_{13}^{4} - 6 T_{13}^{2} + 36$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{4}( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ 1
$5$ $$( 1 - 7 T^{2} + 24 T^{4} - 175 T^{6} + 625 T^{8} )^{2}$$
$7$ $$( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 + 20 T^{2} + 279 T^{4} + 2420 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 + 20 T^{2} + 231 T^{4} + 3380 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 + 7 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 + 16 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 38 T^{2} + 915 T^{4} + 20102 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 8 T^{2} - 777 T^{4} + 6728 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 56 T^{2} + 2175 T^{4} + 53816 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 5 T + 37 T^{2} )^{8}$$
$41$ $$( 1 - 7 T^{2} - 1632 T^{4} - 11767 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 8 T + 43 T^{2} )^{4}( 1 + 13 T + 43 T^{2} )^{4}$$
$47$ $$( 1 - 19 T^{2} - 1848 T^{4} - 41971 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 22 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 + 116 T^{2} + 9735 T^{4} + 431636 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 + 58 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 73 T^{2} )^{8}$$
$79$ $$( 1 - 17 T + 79 T^{2} )^{4}( 1 + 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 163 T^{2} + 19680 T^{4} - 1122907 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 + 70 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 100 T^{2} + 591 T^{4} - 940900 T^{6} + 88529281 T^{8} )^{2}$$