# Properties

 Label 572.2.f.b Level $572$ Weight $2$ Character orbit 572.f Analytic conductor $4.567$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$572 = 2^{2} \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 572.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( 3 - \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{3} ) q^{3} + ( \beta_{1} - \beta_{2} ) q^{7} + ( 3 - \beta_{3} ) q^{9} -\beta_{2} q^{11} + ( 2 - 3 \beta_{2} ) q^{13} + ( -4 + 2 \beta_{3} ) q^{17} + ( -\beta_{1} + 4 \beta_{2} ) q^{19} + ( 2 \beta_{1} - 5 \beta_{2} ) q^{21} + ( 2 - \beta_{3} ) q^{23} + 5 q^{25} + 5 q^{27} + ( 2 + 2 \beta_{3} ) q^{29} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{31} + \beta_{1} q^{33} + ( -4 \beta_{1} - 2 \beta_{2} ) q^{37} + ( 2 + 3 \beta_{1} - 2 \beta_{3} ) q^{39} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{41} + 2 q^{43} + 6 \beta_{2} q^{47} + ( -2 + 3 \beta_{3} ) q^{49} + ( -14 + 4 \beta_{3} ) q^{51} + ( -5 + \beta_{3} ) q^{53} + ( -5 \beta_{1} + 5 \beta_{2} ) q^{57} -6 \beta_{2} q^{59} + 10 q^{61} + ( 4 \beta_{1} - 7 \beta_{2} ) q^{63} + ( -4 \beta_{1} + 4 \beta_{2} ) q^{67} + ( 7 - 2 \beta_{3} ) q^{69} + ( 6 \beta_{1} + 6 \beta_{2} ) q^{71} + ( 5 \beta_{1} + 4 \beta_{2} ) q^{73} + ( 5 - 5 \beta_{3} ) q^{75} + ( -2 + \beta_{3} ) q^{77} -8 q^{79} + ( -4 - 2 \beta_{3} ) q^{81} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{83} + ( -8 - 2 \beta_{3} ) q^{87} -6 \beta_{2} q^{89} + ( -6 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{91} + ( 2 \beta_{1} + 10 \beta_{2} ) q^{93} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{97} + ( \beta_{1} - 2 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 10q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 10q^{9} + 8q^{13} - 12q^{17} + 6q^{23} + 20q^{25} + 20q^{27} + 12q^{29} + 4q^{39} + 8q^{43} - 2q^{49} - 48q^{51} - 18q^{53} + 40q^{61} + 24q^{69} + 10q^{75} - 6q^{77} - 32q^{79} - 20q^{81} - 36q^{87} - 18q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 6 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ $$\nu^{3}$$ $$=$$ $$5 \beta_{2} - 6 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/572\mathbb{Z}\right)^\times$$.

 $$n$$ $$287$$ $$353$$ $$365$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
441.1
 − 1.79129i 1.79129i − 2.79129i 2.79129i
0 −1.79129 0 0 0 0.791288i 0 0.208712 0
441.2 0 −1.79129 0 0 0 0.791288i 0 0.208712 0
441.3 0 2.79129 0 0 0 3.79129i 0 4.79129 0
441.4 0 2.79129 0 0 0 3.79129i 0 4.79129 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.f.b 4
3.b odd 2 1 5148.2.e.b 4
4.b odd 2 1 2288.2.j.f 4
13.b even 2 1 inner 572.2.f.b 4
13.d odd 4 1 7436.2.a.i 2
13.d odd 4 1 7436.2.a.j 2
39.d odd 2 1 5148.2.e.b 4
52.b odd 2 1 2288.2.j.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.f.b 4 1.a even 1 1 trivial
572.2.f.b 4 13.b even 2 1 inner
2288.2.j.f 4 4.b odd 2 1
2288.2.j.f 4 52.b odd 2 1
5148.2.e.b 4 3.b odd 2 1
5148.2.e.b 4 39.d odd 2 1
7436.2.a.i 2 13.d odd 4 1
7436.2.a.j 2 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - T_{3} - 5$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -5 - T + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$9 + 15 T^{2} + T^{4}$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$( 13 - 4 T + T^{2} )^{2}$$
$17$ $$( -12 + 6 T + T^{2} )^{2}$$
$19$ $$225 + 51 T^{2} + T^{4}$$
$23$ $$( -3 - 3 T + T^{2} )^{2}$$
$29$ $$( -12 - 6 T + T^{2} )^{2}$$
$31$ $$144 + 60 T^{2} + T^{4}$$
$37$ $$( 84 + T^{2} )^{2}$$
$41$ $$2025 + 99 T^{2} + T^{4}$$
$43$ $$( -2 + T )^{4}$$
$47$ $$( 36 + T^{2} )^{2}$$
$53$ $$( 15 + 9 T + T^{2} )^{2}$$
$59$ $$( 36 + T^{2} )^{2}$$
$61$ $$( -10 + T )^{4}$$
$67$ $$2304 + 240 T^{2} + T^{4}$$
$71$ $$32400 + 396 T^{2} + T^{4}$$
$73$ $$16641 + 267 T^{2} + T^{4}$$
$79$ $$( 8 + T )^{4}$$
$83$ $$729 + 135 T^{2} + T^{4}$$
$89$ $$( 36 + T^{2} )^{2}$$
$97$ $$144 + 60 T^{2} + T^{4}$$