# Properties

 Label 567.2.i.c Level $567$ Weight $2$ Character orbit 567.i Analytic conductor $4.528$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.52751779461$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 5 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{2} -3 q^{4} + ( 2 + \beta_{2} ) q^{7} -\beta_{3} q^{8} +O(q^{10})$$ $$q + \beta_{3} q^{2} -3 q^{4} + ( 2 + \beta_{2} ) q^{7} -\beta_{3} q^{8} + 2 \beta_{1} q^{11} + ( -1 - \beta_{2} ) q^{13} + ( -\beta_{1} + 3 \beta_{3} ) q^{14} - q^{16} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{17} + ( -2 - 2 \beta_{2} ) q^{19} + ( -10 + 10 \beta_{2} ) q^{22} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{23} + ( 5 - 5 \beta_{2} ) q^{25} + ( \beta_{1} - 2 \beta_{3} ) q^{26} + ( -6 - 3 \beta_{2} ) q^{28} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{29} + ( 1 - 2 \beta_{2} ) q^{31} -3 \beta_{3} q^{32} + ( -20 + 10 \beta_{2} ) q^{34} + ( -5 + 5 \beta_{2} ) q^{37} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{38} + ( 2 \beta_{1} - 4 \beta_{3} ) q^{41} + 7 \beta_{2} q^{43} -6 \beta_{1} q^{44} -10 \beta_{2} q^{46} + ( -4 \beta_{1} + 2 \beta_{3} ) q^{47} + ( 3 + 5 \beta_{2} ) q^{49} + 5 \beta_{1} q^{50} + ( 3 + 3 \beta_{2} ) q^{52} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{53} + ( \beta_{1} - 3 \beta_{3} ) q^{56} -10 \beta_{2} q^{58} + ( 4 \beta_{1} - 2 \beta_{3} ) q^{59} + ( 5 - 10 \beta_{2} ) q^{61} + ( 2 \beta_{1} - \beta_{3} ) q^{62} + 13 q^{64} - q^{67} + ( -6 \beta_{1} - 6 \beta_{3} ) q^{68} + 4 \beta_{3} q^{71} + ( 8 - 4 \beta_{2} ) q^{73} -5 \beta_{1} q^{74} + ( 6 + 6 \beta_{2} ) q^{76} + ( 4 \beta_{1} + 2 \beta_{3} ) q^{77} + 11 q^{79} + ( 10 + 10 \beta_{2} ) q^{82} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{83} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{86} + ( 10 - 10 \beta_{2} ) q^{88} + ( 4 \beta_{1} - 8 \beta_{3} ) q^{89} + ( -1 - 4 \beta_{2} ) q^{91} + ( 6 \beta_{1} - 6 \beta_{3} ) q^{92} + ( 10 - 20 \beta_{2} ) q^{94} + ( -2 + \beta_{2} ) q^{97} + ( -5 \beta_{1} + 8 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 12q^{4} + 10q^{7} + O(q^{10})$$ $$4q - 12q^{4} + 10q^{7} - 6q^{13} - 4q^{16} - 12q^{19} - 20q^{22} + 10q^{25} - 30q^{28} - 60q^{34} - 10q^{37} + 14q^{43} - 20q^{46} + 22q^{49} + 18q^{52} - 20q^{58} + 52q^{64} - 4q^{67} + 24q^{73} + 36q^{76} + 44q^{79} + 60q^{82} + 20q^{88} - 12q^{91} - 6q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$5 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{3}$$

## Character values

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

 $$n$$ $$325$$ $$407$$ $$\chi(n)$$ $$1 - \beta_{2}$$ $$1 - \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
215.1
 −1.93649 − 1.11803i 1.93649 + 1.11803i 1.93649 − 1.11803i −1.93649 + 1.11803i
2.23607i 0 −3.00000 0 0 2.50000 + 0.866025i 2.23607i 0 0
215.2 2.23607i 0 −3.00000 0 0 2.50000 + 0.866025i 2.23607i 0 0
269.1 2.23607i 0 −3.00000 0 0 2.50000 0.866025i 2.23607i 0 0
269.2 2.23607i 0 −3.00000 0 0 2.50000 0.866025i 2.23607i 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
3.b odd 2 1 inner
63.i even 6 1 inner
63.t odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.i.c 4
3.b odd 2 1 inner 567.2.i.c 4
7.d odd 6 1 567.2.s.e 4
9.c even 3 1 189.2.p.c 4
9.c even 3 1 567.2.s.e 4
9.d odd 6 1 189.2.p.c 4
9.d odd 6 1 567.2.s.e 4
21.g even 6 1 567.2.s.e 4
63.h even 3 1 1323.2.c.b 4
63.i even 6 1 inner 567.2.i.c 4
63.i even 6 1 1323.2.c.b 4
63.j odd 6 1 1323.2.c.b 4
63.k odd 6 1 189.2.p.c 4
63.s even 6 1 189.2.p.c 4
63.t odd 6 1 inner 567.2.i.c 4
63.t odd 6 1 1323.2.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.2.p.c 4 9.c even 3 1
189.2.p.c 4 9.d odd 6 1
189.2.p.c 4 63.k odd 6 1
189.2.p.c 4 63.s even 6 1
567.2.i.c 4 1.a even 1 1 trivial
567.2.i.c 4 3.b odd 2 1 inner
567.2.i.c 4 63.i even 6 1 inner
567.2.i.c 4 63.t odd 6 1 inner
567.2.s.e 4 7.d odd 6 1
567.2.s.e 4 9.c even 3 1
567.2.s.e 4 9.d odd 6 1
567.2.s.e 4 21.g even 6 1
1323.2.c.b 4 63.h even 3 1
1323.2.c.b 4 63.i even 6 1
1323.2.c.b 4 63.j odd 6 1
1323.2.c.b 4 63.t odd 6 1

## Hecke kernels

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

 $$T_{2}^{2} + 5$$ $$T_{11}^{4} - 20 T_{11}^{2} + 400$$ $$T_{13}^{2} + 3 T_{13} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 5 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 7 - 5 T + T^{2} )^{2}$$
$11$ $$400 - 20 T^{2} + T^{4}$$
$13$ $$( 3 + 3 T + T^{2} )^{2}$$
$17$ $$3600 + 60 T^{2} + T^{4}$$
$19$ $$( 12 + 6 T + T^{2} )^{2}$$
$23$ $$400 - 20 T^{2} + T^{4}$$
$29$ $$400 - 20 T^{2} + T^{4}$$
$31$ $$( 3 + T^{2} )^{2}$$
$37$ $$( 25 + 5 T + T^{2} )^{2}$$
$41$ $$3600 + 60 T^{2} + T^{4}$$
$43$ $$( 49 - 7 T + T^{2} )^{2}$$
$47$ $$( -60 + T^{2} )^{2}$$
$53$ $$400 - 20 T^{2} + T^{4}$$
$59$ $$( -60 + T^{2} )^{2}$$
$61$ $$( 75 + T^{2} )^{2}$$
$67$ $$( 1 + T )^{4}$$
$71$ $$( 80 + T^{2} )^{2}$$
$73$ $$( 48 - 12 T + T^{2} )^{2}$$
$79$ $$( -11 + T )^{4}$$
$83$ $$3600 + 60 T^{2} + T^{4}$$
$89$ $$57600 + 240 T^{2} + T^{4}$$
$97$ $$( 3 + 3 T + T^{2} )^{2}$$