# Properties

 Label 567.2.a.d Level $567$ Weight $2$ Character orbit 567.a Self dual yes Analytic conductor $4.528$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$567 = 3^{4} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 567.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.52751779461$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.321.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 63) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 4 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.46050 0.239123 −1.69963
−2.46050 0 4.05408 −2.59358 0 −1.00000 −5.05408 0 6.38151
1.2 −0.239123 0 −1.94282 1.18194 0 −1.00000 0.942820 0 −0.282630
1.3 1.69963 0 0.888736 −3.58836 0 −1.00000 −1.88874 0 −6.09888
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$7$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 567.2.a.d 3
3.b odd 2 1 567.2.a.g 3
4.b odd 2 1 9072.2.a.bq 3
7.b odd 2 1 3969.2.a.m 3
9.c even 3 2 63.2.f.b 6
9.d odd 6 2 189.2.f.a 6
12.b even 2 1 9072.2.a.cd 3
21.c even 2 1 3969.2.a.p 3
36.f odd 6 2 1008.2.r.k 6
36.h even 6 2 3024.2.r.g 6
63.g even 3 2 441.2.g.e 6
63.h even 3 2 441.2.h.c 6
63.i even 6 2 1323.2.h.e 6
63.j odd 6 2 1323.2.h.d 6
63.k odd 6 2 441.2.g.d 6
63.l odd 6 2 441.2.f.d 6
63.n odd 6 2 1323.2.g.c 6
63.o even 6 2 1323.2.f.c 6
63.s even 6 2 1323.2.g.b 6
63.t odd 6 2 441.2.h.b 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.f.b 6 9.c even 3 2
189.2.f.a 6 9.d odd 6 2
441.2.f.d 6 63.l odd 6 2
441.2.g.d 6 63.k odd 6 2
441.2.g.e 6 63.g even 3 2
441.2.h.b 6 63.t odd 6 2
441.2.h.c 6 63.h even 3 2
567.2.a.d 3 1.a even 1 1 trivial
567.2.a.g 3 3.b odd 2 1
1008.2.r.k 6 36.f odd 6 2
1323.2.f.c 6 63.o even 6 2
1323.2.g.b 6 63.s even 6 2
1323.2.g.c 6 63.n odd 6 2
1323.2.h.d 6 63.j odd 6 2
1323.2.h.e 6 63.i even 6 2
3024.2.r.g 6 36.h even 6 2
3969.2.a.m 3 7.b odd 2 1
3969.2.a.p 3 21.c even 2 1
9072.2.a.bq 3 4.b odd 2 1
9072.2.a.cd 3 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} + T_{2}^{2} - 4 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(567))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 - 4 T + T^{2} + T^{3}$$
$3$ $$T^{3}$$
$5$ $$-11 + 2 T + 5 T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$-47 - 19 T + 2 T^{2} + T^{3}$$
$13$ $$( -1 + T )^{3}$$
$17$ $$27 + 39 T + 12 T^{2} + T^{3}$$
$19$ $$-7 - 6 T + 3 T^{2} + T^{3}$$
$23$ $$-9 - 33 T + T^{3}$$
$29$ $$1 - 4 T - T^{2} + T^{3}$$
$31$ $$27 - 24 T + 3 T^{2} + T^{3}$$
$37$ $$81 - 54 T + 3 T^{2} + T^{3}$$
$41$ $$353 + 155 T + 22 T^{2} + T^{3}$$
$43$ $$121 - 66 T + 3 T^{2} + T^{3}$$
$47$ $$-189 - 54 T + 9 T^{2} + T^{3}$$
$53$ $$9 + 75 T + 18 T^{2} + T^{3}$$
$59$ $$-63 - 6 T + 9 T^{2} + T^{3}$$
$61$ $$-67 - 21 T + 6 T^{2} + T^{3}$$
$67$ $$683 - 207 T + T^{3}$$
$71$ $$81 - 6 T - 9 T^{2} + T^{3}$$
$73$ $$-243 - 168 T - 3 T^{2} + T^{3}$$
$79$ $$769 - 48 T - 15 T^{2} + T^{3}$$
$83$ $$27 + 39 T + 12 T^{2} + T^{3}$$
$89$ $$379 - 151 T + 2 T^{2} + T^{3}$$
$97$ $$603 - 114 T - 3 T^{2} + T^{3}$$