# Properties

 Label 572.2.i.a Level $572$ Weight $2$ Character orbit 572.i 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.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{3} - q^{5} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{3} - q^{5} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{9} + \zeta_{12}^{2} q^{11} + ( -3 \zeta_{12} - 2 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{13} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{15} + ( -2 + 3 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{17} + ( 1 + \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{19} + ( -4 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{21} + ( -2 \zeta_{12} - 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{23} -4 q^{25} -4 q^{27} + ( -\zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{29} + 6 q^{31} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{35} -7 \zeta_{12}^{2} q^{37} + ( 5 - 5 \zeta_{12} + \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{39} + ( -\zeta_{12} - 4 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{41} + ( 4 + 4 \zeta_{12} - 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{43} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{45} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{47} + ( 2 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{49} + ( -11 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{51} + ( -1 - 12 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} -\zeta_{12}^{2} q^{55} -2 q^{57} + ( 7 - \zeta_{12} - 7 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{59} + ( -10 - \zeta_{12} + 10 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{61} + ( 3 \zeta_{12} - 7 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{63} + ( 3 \zeta_{12} + 2 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{65} + ( 5 \zeta_{12} + 7 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{67} + ( -2 - 2 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{69} + ( -6 + 6 \zeta_{12}^{2} ) q^{71} + ( 2 + 10 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{73} + ( 4 \zeta_{12} - 4 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{75} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{77} + ( 11 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{79} + ( -2 \zeta_{12} - \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{81} + ( 3 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{83} + ( 2 - 3 \zeta_{12} - 2 \zeta_{12}^{2} + 6 \zeta_{12}^{3} ) q^{85} + ( -5 + 3 \zeta_{12} + 5 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{87} + ( -8 \zeta_{12} + 4 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{89} + ( -1 - 4 \zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{91} + ( -6 \zeta_{12} + 6 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{93} + ( -1 - \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{95} + ( 16 - 16 \zeta_{12}^{2} ) q^{97} + ( -1 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} - 4q^{5} - 2q^{7} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{3} - 4q^{5} - 2q^{7} - 2q^{9} + 2q^{11} - 4q^{13} - 2q^{15} - 4q^{17} + 2q^{19} - 16q^{21} - 8q^{23} - 16q^{25} - 16q^{27} + 4q^{29} + 24q^{31} - 2q^{33} + 2q^{35} - 14q^{37} + 22q^{39} - 8q^{41} + 8q^{43} + 2q^{45} + 20q^{47} + 6q^{49} - 44q^{51} - 4q^{53} - 2q^{55} - 8q^{57} + 14q^{59} - 20q^{61} - 14q^{63} + 4q^{65} + 14q^{67} - 4q^{69} - 12q^{71} + 8q^{73} - 8q^{75} - 4q^{77} + 44q^{79} - 2q^{81} + 12q^{83} + 4q^{85} - 10q^{87} + 8q^{89} + 8q^{91} + 12q^{93} - 2q^{95} + 32q^{97} - 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$287$$ $$353$$ $$365$$ $$\chi(n)$$ $$1$$ $$-1 + \zeta_{12}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
133.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.366025 0.633975i 0 −1.00000 0 0.366025 0.633975i 0 1.23205 2.13397i 0
133.2 0 1.36603 + 2.36603i 0 −1.00000 0 −1.36603 + 2.36603i 0 −2.23205 + 3.86603i 0
529.1 0 −0.366025 + 0.633975i 0 −1.00000 0 0.366025 + 0.633975i 0 1.23205 + 2.13397i 0
529.2 0 1.36603 2.36603i 0 −1.00000 0 −1.36603 2.36603i 0 −2.23205 3.86603i 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 572.2.i.a 4
13.c even 3 1 inner 572.2.i.a 4
13.c even 3 1 7436.2.a.f 2
13.e even 6 1 7436.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.i.a 4 1.a even 1 1 trivial
572.2.i.a 4 13.c even 3 1 inner
7436.2.a.f 2 13.c even 3 1
7436.2.a.g 2 13.e even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 2 T_{3}^{3} + 6 T_{3}^{2} + 4 T_{3} + 4$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4}$$
$11$ $$( 1 - T + T^{2} )^{2}$$
$13$ $$169 + 52 T + 3 T^{2} + 4 T^{3} + T^{4}$$
$17$ $$529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4}$$
$19$ $$4 + 4 T + 6 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$16 + 32 T + 60 T^{2} + 8 T^{3} + T^{4}$$
$29$ $$1 - 4 T + 15 T^{2} - 4 T^{3} + T^{4}$$
$31$ $$( -6 + T )^{4}$$
$37$ $$( 49 + 7 T + T^{2} )^{2}$$
$41$ $$169 + 104 T + 51 T^{2} + 8 T^{3} + T^{4}$$
$43$ $$1024 + 256 T + 96 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$( 22 - 10 T + T^{2} )^{2}$$
$53$ $$( -107 + 2 T + T^{2} )^{2}$$
$59$ $$2116 - 644 T + 150 T^{2} - 14 T^{3} + T^{4}$$
$61$ $$9409 + 1940 T + 303 T^{2} + 20 T^{3} + T^{4}$$
$67$ $$676 + 364 T + 222 T^{2} - 14 T^{3} + T^{4}$$
$71$ $$( 36 + 6 T + T^{2} )^{2}$$
$73$ $$( -71 - 4 T + T^{2} )^{2}$$
$79$ $$( 94 - 22 T + T^{2} )^{2}$$
$83$ $$( -18 - 6 T + T^{2} )^{2}$$
$89$ $$30976 + 1408 T + 240 T^{2} - 8 T^{3} + T^{4}$$
$97$ $$( 256 - 16 T + T^{2} )^{2}$$