# Properties

 Label 572.2.f.c Level $572$ Weight $2$ Character orbit 572.f Analytic conductor $4.567$ Analytic rank $0$ Dimension $8$ 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ Defining polynomial: $$x^{8} + 21 x^{6} + 136 x^{4} + 309 x^{2} + 225$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( 2 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + ( \beta_{1} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{5} + ( \beta_{1} + \beta_{4} ) q^{7} + ( 2 + \beta_{3} - \beta_{5} + \beta_{6} ) q^{9} + \beta_{4} q^{11} + ( -1 - \beta_{1} - \beta_{3} - \beta_{6} ) q^{13} + ( \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{15} + ( 2 - 2 \beta_{2} - \beta_{5} + \beta_{6} ) q^{17} + ( -2 \beta_{1} - \beta_{7} ) q^{19} + ( -\beta_{1} - 5 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{21} + ( 1 + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{23} + ( -7 - \beta_{2} - \beta_{3} - \beta_{5} + \beta_{6} ) q^{25} + ( -3 + 2 \beta_{5} - 2 \beta_{6} ) q^{27} + ( -2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{29} + ( \beta_{1} - 6 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{31} -\beta_{1} q^{33} + ( -2 - 2 \beta_{5} + 2 \beta_{6} ) q^{35} + ( -\beta_{1} + 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{37} + ( 1 + 2 \beta_{2} - \beta_{3} + 3 \beta_{4} + \beta_{6} ) q^{39} + ( \beta_{1} + \beta_{4} + \beta_{5} + \beta_{6} + 2 \beta_{7} ) q^{41} + ( -2 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{43} + ( -10 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} - 2 \beta_{7} ) q^{45} + ( -2 \beta_{1} + 4 \beta_{4} ) q^{47} + ( 1 - 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} ) q^{49} + ( 6 - 2 \beta_{2} + \beta_{5} - \beta_{6} ) q^{51} + ( 4 - \beta_{3} - \beta_{5} + \beta_{6} ) q^{53} + ( -\beta_{2} + \beta_{3} - \beta_{5} + \beta_{6} ) q^{55} + ( \beta_{1} + 9 \beta_{4} - \beta_{5} - \beta_{6} ) q^{57} + ( -\beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} + 3 \beta_{7} ) q^{59} + ( -4 - 2 \beta_{2} + 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{61} + ( 3 \beta_{1} + 5 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} + \beta_{7} ) q^{63} + ( 5 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} + 7 \beta_{4} - 3 \beta_{6} + \beta_{7} ) q^{65} + ( 3 \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{67} + ( -7 - 2 \beta_{3} ) q^{69} + ( -\beta_{1} - 4 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{71} + ( -2 \beta_{1} + 6 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{73} + ( 8 \beta_{2} - 3 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} ) q^{75} + ( -1 - \beta_{2} ) q^{77} + ( 4 - 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{79} + ( 2 + 3 \beta_{2} + \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{81} + ( \beta_{1} + 3 \beta_{4} - \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{83} + ( -2 \beta_{1} - 10 \beta_{4} - \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{85} + ( 8 - 2 \beta_{2} + 4 \beta_{3} - \beta_{5} + \beta_{6} ) q^{87} + ( 5 \beta_{1} - 4 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} ) q^{89} + ( 3 - 2 \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{91} + ( 7 \beta_{1} - 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{93} + ( 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{5} - 3 \beta_{6} ) q^{95} + ( -\beta_{1} - 8 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{97} + ( 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{3} + 18q^{9} + O(q^{10})$$ $$8q - 2q^{3} + 18q^{9} - 8q^{13} + 16q^{17} + 10q^{23} - 52q^{25} - 32q^{27} - 4q^{29} - 8q^{35} + 16q^{39} - 20q^{43} + 2q^{49} + 40q^{51} + 38q^{53} - 36q^{61} + 36q^{65} - 52q^{69} + 10q^{75} - 10q^{77} + 40q^{79} + 32q^{81} + 56q^{87} + 22q^{91} - 12q^{95} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 21 x^{6} + 136 x^{4} + 309 x^{2} + 225$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{6} - 18 \nu^{4} - 88 \nu^{2} - 105$$$$)/6$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 20 \nu^{4} - 112 \nu^{2} - 149$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$-7 \nu^{7} - 132 \nu^{5} - 682 \nu^{3} - 843 \nu$$$$)/90$$ $$\beta_{5}$$ $$=$$ $$($$$$-7 \nu^{7} - 15 \nu^{6} - 132 \nu^{5} - 300 \nu^{4} - 652 \nu^{3} - 1620 \nu^{2} - 663 \nu - 1935$$$$)/120$$ $$\beta_{6}$$ $$=$$ $$($$$$-7 \nu^{7} + 15 \nu^{6} - 132 \nu^{5} + 300 \nu^{4} - 652 \nu^{3} + 1620 \nu^{2} - 663 \nu + 1935$$$$)/120$$ $$\beta_{7}$$ $$=$$ $$($$$$19 \nu^{7} + 384 \nu^{5} + 2224 \nu^{3} + 3291 \nu$$$$)/180$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{6} + \beta_{5} - \beta_{3} - 5$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 6 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$12 \beta_{6} - 12 \beta_{5} + 10 \beta_{3} + 3 \beta_{2} + 38$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{7} - 29 \beta_{6} - 29 \beta_{5} + 53 \beta_{4} + 48 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-128 \beta_{6} + 128 \beta_{5} - 92 \beta_{3} - 60 \beta_{2} - 349$$ $$\nu^{7}$$ $$=$$ $$-132 \beta_{7} + 352 \beta_{6} + 352 \beta_{5} - 720 \beta_{4} - 441 \beta_{1}$$

## 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
 3.32727i − 3.32727i − 1.43491i 1.43491i 1.25126i − 1.25126i − 2.51091i 2.51091i
0 −3.32727 0 2.74343i 0 4.32727i 0 8.07070 0
441.2 0 −3.32727 0 2.74343i 0 4.32727i 0 8.07070 0
441.3 0 −1.43491 0 4.37595i 0 2.43491i 0 −0.941037 0
441.4 0 −1.43491 0 4.37595i 0 2.43491i 0 −0.941037 0
441.5 0 1.25126 0 2.18309i 0 0.251260i 0 −1.43435 0
441.6 0 1.25126 0 2.18309i 0 0.251260i 0 −1.43435 0
441.7 0 2.51091 0 3.81560i 0 1.51091i 0 3.30469 0
441.8 0 2.51091 0 3.81560i 0 1.51091i 0 3.30469 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 441.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.c 8
3.b odd 2 1 5148.2.e.c 8
4.b odd 2 1 2288.2.j.i 8
13.b even 2 1 inner 572.2.f.c 8
13.d odd 4 1 7436.2.a.o 4
13.d odd 4 1 7436.2.a.p 4
39.d odd 2 1 5148.2.e.c 8
52.b odd 2 1 2288.2.j.i 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
572.2.f.c 8 1.a even 1 1 trivial
572.2.f.c 8 13.b even 2 1 inner
2288.2.j.i 8 4.b odd 2 1
2288.2.j.i 8 52.b odd 2 1
5148.2.e.c 8 3.b odd 2 1
5148.2.e.c 8 39.d odd 2 1
7436.2.a.o 4 13.d odd 4 1
7436.2.a.p 4 13.d odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + T_{3}^{3} - 10 T_{3}^{2} - 3 T_{3} + 15$$ acting on $$S_{2}^{\mathrm{new}}(572, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$( 15 - 3 T - 10 T^{2} + T^{3} + T^{4} )^{2}$$
$5$ $$10000 + 4636 T^{2} + 729 T^{4} + 46 T^{6} + T^{8}$$
$7$ $$16 + 264 T^{2} + 169 T^{4} + 27 T^{6} + T^{8}$$
$11$ $$( 1 + T^{2} )^{4}$$
$13$ $$28561 + 17576 T + 3042 T^{2} + 104 T^{3} + 2 T^{4} + 8 T^{5} + 18 T^{6} + 8 T^{7} + T^{8}$$
$17$ $$( -352 + 216 T - 14 T^{2} - 8 T^{3} + T^{4} )^{2}$$
$19$ $$106276 + 42656 T^{2} + 3641 T^{4} + 107 T^{6} + T^{8}$$
$23$ $$( -111 + 123 T - 28 T^{2} - 5 T^{3} + T^{4} )^{2}$$
$29$ $$( -344 - 340 T - 70 T^{2} + 2 T^{3} + T^{4} )^{2}$$
$31$ $$283024 + 133372 T^{2} + 10377 T^{4} + 190 T^{6} + T^{8}$$
$37$ $$309136 + 67784 T^{2} + 4817 T^{4} + 122 T^{6} + T^{8}$$
$41$ $$9326916 + 880896 T^{2} + 25297 T^{4} + 275 T^{6} + T^{8}$$
$43$ $$( -1272 - 564 T - 34 T^{2} + 10 T^{3} + T^{4} )^{2}$$
$47$ $$30976 + 29760 T^{2} + 4480 T^{4} + 132 T^{6} + T^{8}$$
$53$ $$( 216 + 288 T + 59 T^{2} - 19 T^{3} + T^{4} )^{2}$$
$59$ $$156816 + 176328 T^{2} + 15937 T^{4} + 266 T^{6} + T^{8}$$
$61$ $$( -2440 - 548 T + 50 T^{2} + 18 T^{3} + T^{4} )^{2}$$
$67$ $$5776 + 176844 T^{2} + 12889 T^{4} + 222 T^{6} + T^{8}$$
$71$ $$17424 + 225372 T^{2} + 21385 T^{4} + 318 T^{6} + T^{8}$$
$73$ $$22429696 + 1499392 T^{2} + 34569 T^{4} + 319 T^{6} + T^{8}$$
$79$ $$( 2456 + 1008 T - 2 T^{2} - 20 T^{3} + T^{4} )^{2}$$
$83$ $$1119364 + 215832 T^{2} + 11401 T^{4} + 207 T^{6} + T^{8}$$
$89$ $$23931664 + 1784300 T^{2} + 45881 T^{4} + 446 T^{6} + T^{8}$$
$97$ $$7683984 + 841212 T^{2} + 27961 T^{4} + 302 T^{6} + T^{8}$$