# Properties

 Label 572.2.e.a Level $572$ Weight $2$ Character orbit 572.e Analytic conductor $4.567$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.56744299562$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.342102016.5 Defining polynomial: $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{4} q^{2} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{4} + ( -2 - \beta_{6} - \beta_{7} ) q^{5} + ( 3 \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q -\beta_{4} q^{2} + ( -\beta_{3} - \beta_{5} + \beta_{7} ) q^{4} + ( -2 - \beta_{6} - \beta_{7} ) q^{5} + ( 3 \beta_{1} - \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{7} + ( -2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{8} + 3 q^{9} + ( 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} ) q^{10} + ( -\beta_{1} + \beta_{3} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{11} + \beta_{2} q^{13} + ( 4 - 3 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} + 3 \beta_{7} ) q^{14} + ( -2 + \beta_{3} + \beta_{5} + 2 \beta_{6} - 3 \beta_{7} ) q^{16} + ( -2 \beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{17} -3 \beta_{4} q^{18} + ( 2 \beta_{1} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{19} + ( -2 + \beta_{3} + \beta_{5} - 2 \beta_{6} + \beta_{7} ) q^{20} + ( -1 + \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{22} + ( -\beta_{6} + \beta_{7} ) q^{23} + ( 3 + 3 \beta_{6} + 3 \beta_{7} ) q^{25} + \beta_{7} q^{26} + ( -6 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} - \beta_{4} - 3 \beta_{5} ) q^{28} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{4} ) q^{29} + ( 3 \beta_{3} + 3 \beta_{5} + 4 \beta_{6} - 4 \beta_{7} ) q^{31} + ( 2 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + 3 \beta_{5} ) q^{32} + ( -4 - 2 \beta_{3} - 2 \beta_{5} ) q^{34} + ( -\beta_{1} - 4 \beta_{3} + \beta_{4} + 4 \beta_{5} ) q^{35} + ( -3 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{36} + ( 2 + 4 \beta_{6} + 4 \beta_{7} ) q^{37} + ( -2 - 2 \beta_{3} - 2 \beta_{5} - 6 \beta_{6} + 2 \beta_{7} ) q^{38} + ( 2 \beta_{1} + 6 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{5} ) q^{40} + ( -\beta_{1} + 6 \beta_{2} - \beta_{4} ) q^{41} + ( -3 \beta_{1} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{43} + ( -2 + 2 \beta_{1} - 3 \beta_{3} - \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{44} + ( -6 - 3 \beta_{6} - 3 \beta_{7} ) q^{45} + ( 2 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{46} + ( -\beta_{3} - \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{47} + ( 11 - \beta_{6} - \beta_{7} ) q^{49} + ( -6 \beta_{2} + 3 \beta_{3} - 3 \beta_{5} ) q^{50} + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{52} + ( 6 + 4 \beta_{6} + 4 \beta_{7} ) q^{53} + ( 4 \beta_{3} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{55} + ( -6 - \beta_{3} - \beta_{5} + 6 \beta_{6} - \beta_{7} ) q^{56} + ( -6 - 3 \beta_{3} - 3 \beta_{5} + \beta_{7} ) q^{58} + ( \beta_{3} + \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{59} + ( 3 \beta_{1} + 8 \beta_{2} + 3 \beta_{4} ) q^{61} + ( 6 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{62} + ( 9 \beta_{1} - 3 \beta_{3} - 9 \beta_{4} + 3 \beta_{5} ) q^{63} + ( -2 - \beta_{3} - \beta_{5} - 6 \beta_{6} - \beta_{7} ) q^{64} + ( -\beta_{1} - 2 \beta_{2} - \beta_{4} ) q^{65} + ( 7 \beta_{3} + 7 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{67} + ( -4 \beta_{1} - 4 \beta_{2} + 4 \beta_{4} ) q^{68} + ( -10 + \beta_{3} + \beta_{5} - 8 \beta_{6} - \beta_{7} ) q^{70} + ( 3 \beta_{3} + 3 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{71} + ( -6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} ) q^{72} + ( 5 \beta_{1} - 2 \beta_{2} + 5 \beta_{4} ) q^{73} + ( -8 \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} ) q^{74} + ( -4 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} ) q^{76} + ( -5 - 13 \beta_{2} + \beta_{6} + \beta_{7} ) q^{77} + ( 2 \beta_{1} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{79} + ( 6 + 3 \beta_{3} + 3 \beta_{5} + 2 \beta_{6} + 3 \beta_{7} ) q^{80} + 9 q^{81} + ( -2 - \beta_{3} - \beta_{5} + 7 \beta_{7} ) q^{82} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{83} + ( 4 \beta_{1} + 12 \beta_{2} + 4 \beta_{4} ) q^{85} + ( -2 + 3 \beta_{3} + 3 \beta_{5} + 4 \beta_{6} - 3 \beta_{7} ) q^{86} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{88} -2 q^{89} + ( 6 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{90} + ( -\beta_{3} - \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{91} + ( 2 + \beta_{3} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{92} + ( -2 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} + 4 \beta_{4} - 4 \beta_{5} ) q^{94} + ( 4 \beta_{1} + 2 \beta_{3} - 4 \beta_{4} - 2 \beta_{5} ) q^{95} + ( 2 - 4 \beta_{6} - 4 \beta_{7} ) q^{97} + ( 2 \beta_{2} - \beta_{3} - 12 \beta_{4} + \beta_{5} ) q^{98} + ( -3 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 6 \beta_{6} - 6 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{4} - 12q^{5} + 24q^{9} + O(q^{10})$$ $$8q - 2q^{4} - 12q^{5} + 24q^{9} + 30q^{14} - 14q^{16} - 14q^{20} - 8q^{22} + 12q^{25} - 2q^{26} - 32q^{34} - 6q^{36} - 8q^{38} - 6q^{44} - 36q^{45} + 92q^{49} + 32q^{53} - 58q^{56} - 50q^{58} - 2q^{64} - 62q^{70} - 44q^{77} + 38q^{80} + 72q^{81} - 30q^{82} - 18q^{86} + 20q^{88} - 16q^{89} + 10q^{92} + 32q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{6} + 4 x^{4} + 4 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} + 2 \nu^{3}$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} - 3 \nu^{5} + 3 \nu^{4} - 2 \nu^{3} + 10 \nu^{2} + 8$$$$)/16$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + \nu^{5} + 4 \nu^{3} + 4 \nu$$$$)/8$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} + \nu^{6} + 3 \nu^{5} + 3 \nu^{4} + 2 \nu^{3} + 10 \nu^{2} + 8$$$$)/16$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} - 3 \nu^{4} - 2 \nu^{2} - 8$$$$)/8$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + \nu^{4} + 2 \nu^{2}$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$-\beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_{2} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} - 3 \beta_{6} - \beta_{5} - \beta_{3} - 2$$ $$\nu^{5}$$ $$=$$ $$3 \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + 2 \beta_{2} + \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-6 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} - 2$$ $$\nu^{7}$$ $$=$$ $$\beta_{5} + 2 \beta_{4} - \beta_{3} - 10 \beta_{2} - \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
131.1
 −1.17915 + 0.780776i −1.17915 − 0.780776i −0.599676 + 1.28078i −0.599676 − 1.28078i 0.599676 + 1.28078i 0.599676 − 1.28078i 1.17915 + 0.780776i 1.17915 − 0.780776i
−1.17915 0.780776i 0 0.780776 + 1.84130i −3.56155 0 −4.05444 0.516994 2.78078i 3.00000 4.19960 + 2.78078i
131.2 −1.17915 + 0.780776i 0 0.780776 1.84130i −3.56155 0 −4.05444 0.516994 + 2.78078i 3.00000 4.19960 2.78078i
131.3 −0.599676 1.28078i 0 −1.28078 + 1.53610i 0.561553 0 −4.53448 2.73546 + 0.719224i 3.00000 −0.336750 0.719224i
131.4 −0.599676 + 1.28078i 0 −1.28078 1.53610i 0.561553 0 −4.53448 2.73546 0.719224i 3.00000 −0.336750 + 0.719224i
131.5 0.599676 1.28078i 0 −1.28078 1.53610i 0.561553 0 4.53448 −2.73546 + 0.719224i 3.00000 0.336750 0.719224i
131.6 0.599676 + 1.28078i 0 −1.28078 + 1.53610i 0.561553 0 4.53448 −2.73546 0.719224i 3.00000 0.336750 + 0.719224i
131.7 1.17915 0.780776i 0 0.780776 1.84130i −3.56155 0 4.05444 −0.516994 2.78078i 3.00000 −4.19960 + 2.78078i
131.8 1.17915 + 0.780776i 0 0.780776 + 1.84130i −3.56155 0 4.05444 −0.516994 + 2.78078i 3.00000 −4.19960 2.78078i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 131.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
11.b odd 2 1 inner
44.c 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.e.a 8
4.b odd 2 1 inner 572.2.e.a 8
11.b odd 2 1 inner 572.2.e.a 8
44.c even 2 1 inner 572.2.e.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.e.a 8 1.a even 1 1 trivial
572.2.e.a 8 4.b odd 2 1 inner
572.2.e.a 8 11.b odd 2 1 inner
572.2.e.a 8 44.c even 2 1 inner

## 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

$p$ $F_p(T)$
$2$ $$16 + 4 T^{2} + 4 T^{4} + T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$( -2 + 3 T + T^{2} )^{4}$$
$7$ $$( 338 - 37 T^{2} + T^{4} )^{2}$$
$11$ $$14641 + 3630 T^{2} + 450 T^{4} + 30 T^{6} + T^{8}$$
$13$ $$( 1 + T^{2} )^{4}$$
$17$ $$( 256 + 36 T^{2} + T^{4} )^{2}$$
$19$ $$( 512 - 46 T^{2} + T^{4} )^{2}$$
$23$ $$( 8 + 7 T^{2} + T^{4} )^{2}$$
$29$ $$( 1444 + 77 T^{2} + T^{4} )^{2}$$
$31$ $$( 8 + 58 T^{2} + T^{4} )^{2}$$
$37$ $$( -68 + T^{2} )^{4}$$
$41$ $$( 1444 + 93 T^{2} + T^{4} )^{2}$$
$43$ $$( 32 - 31 T^{2} + T^{4} )^{2}$$
$47$ $$( 1352 + 74 T^{2} + T^{4} )^{2}$$
$53$ $$( -52 - 8 T + T^{2} )^{4}$$
$59$ $$( 338 + 37 T^{2} + T^{4} )^{2}$$
$61$ $$( 16 + 161 T^{2} + T^{4} )^{2}$$
$67$ $$( 20402 + 301 T^{2} + T^{4} )^{2}$$
$71$ $$( 32 + 190 T^{2} + T^{4} )^{2}$$
$73$ $$( 7396 + 253 T^{2} + T^{4} )^{2}$$
$79$ $$( 32 - 20 T^{2} + T^{4} )^{2}$$
$83$ $$( 32 - 14 T^{2} + T^{4} )^{2}$$
$89$ $$( 2 + T )^{8}$$
$97$ $$( -52 - 8 T + T^{2} )^{4}$$