# Properties

 Label 572.2.f.a Level $572$ Weight $2$ Character orbit 572.f Analytic conductor $4.567$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 i q^{7} -3 q^{9} +O(q^{10})$$ $$q + 4 i q^{7} -3 q^{9} -i q^{11} + ( -3 + 2 i ) q^{13} -2 q^{17} + 4 i q^{19} -4 q^{23} + 5 q^{25} -6 q^{29} -4 i q^{31} + 8 i q^{37} + 12 i q^{41} + 12 q^{43} -4 i q^{47} -9 q^{49} + 6 q^{53} + 4 i q^{59} -10 q^{61} -12 i q^{63} + 4 i q^{67} -4 i q^{71} + 4 i q^{73} + 4 q^{77} -8 q^{79} + 9 q^{81} -12 i q^{83} -16 i q^{89} + ( -8 - 12 i ) q^{91} + 8 i q^{97} + 3 i q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 6q^{9} + O(q^{10})$$ $$2q - 6q^{9} - 6q^{13} - 4q^{17} - 8q^{23} + 10q^{25} - 12q^{29} + 24q^{43} - 18q^{49} + 12q^{53} - 20q^{61} + 8q^{77} - 16q^{79} + 18q^{81} - 16q^{91} + O(q^{100})$$

## 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
 − 1.00000i 1.00000i
0 0 0 0 0 4.00000i 0 −3.00000 0
441.2 0 0 0 0 0 4.00000i 0 −3.00000 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.a 2
3.b odd 2 1 5148.2.e.a 2
4.b odd 2 1 2288.2.j.b 2
13.b even 2 1 inner 572.2.f.a 2
13.d odd 4 1 7436.2.a.a 1
13.d odd 4 1 7436.2.a.b 1
39.d odd 2 1 5148.2.e.a 2
52.b odd 2 1 2288.2.j.b 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$1 + T^{2}$$
$13$ $$13 + 6 T + T^{2}$$
$17$ $$( 2 + T )^{2}$$
$19$ $$16 + T^{2}$$
$23$ $$( 4 + T )^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$16 + T^{2}$$
$37$ $$64 + T^{2}$$
$41$ $$144 + T^{2}$$
$43$ $$( -12 + T )^{2}$$
$47$ $$16 + T^{2}$$
$53$ $$( -6 + T )^{2}$$
$59$ $$16 + T^{2}$$
$61$ $$( 10 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$16 + T^{2}$$
$73$ $$16 + T^{2}$$
$79$ $$( 8 + T )^{2}$$
$83$ $$144 + T^{2}$$
$89$ $$256 + T^{2}$$
$97$ $$64 + T^{2}$$