# Properties

 Label 560.2.e.d Level 560 Weight 2 Character orbit 560.e Analytic conductor 4.472 Analytic rank 0 Dimension 8 CM no Inner twists 8

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.12745506816.5 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{5} q^{3} + \beta_{4} q^{5} + \beta_{1} q^{7} + q^{9} +O(q^{10})$$ $$q + \beta_{5} q^{3} + \beta_{4} q^{5} + \beta_{1} q^{7} + q^{9} + 2 \beta_{3} q^{11} + ( \beta_{4} - \beta_{6} ) q^{13} + ( -\beta_{1} - \beta_{3} ) q^{15} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{17} + ( 2 \beta_{2} + \beta_{5} ) q^{19} + ( \beta_{4} + \beta_{6} ) q^{21} + ( -2 - \beta_{7} ) q^{25} + 4 \beta_{5} q^{27} -6 q^{29} + ( 2 \beta_{4} - 2 \beta_{6} ) q^{33} + ( -\beta_{2} + 3 \beta_{5} ) q^{35} + 2 \beta_{7} q^{37} -2 \beta_{3} q^{39} + ( 2 \beta_{4} + 2 \beta_{6} ) q^{41} -2 \beta_{1} q^{43} + \beta_{4} q^{45} -2 \beta_{5} q^{47} + 7 q^{49} -4 \beta_{3} q^{51} -2 \beta_{7} q^{53} + ( -2 \beta_{2} - 4 \beta_{5} ) q^{55} + 2 \beta_{7} q^{57} + ( 2 \beta_{2} + \beta_{5} ) q^{59} + ( -3 \beta_{4} - 3 \beta_{6} ) q^{61} + \beta_{1} q^{63} + ( 3 - \beta_{7} ) q^{65} + 2 \beta_{1} q^{67} + ( -4 \beta_{4} + 4 \beta_{6} ) q^{73} + ( 2 \beta_{2} - \beta_{5} ) q^{75} + 2 \beta_{7} q^{77} + 4 \beta_{3} q^{79} -5 q^{81} + 7 \beta_{5} q^{83} + ( 6 - 2 \beta_{7} ) q^{85} -6 \beta_{5} q^{87} + ( -2 \beta_{4} - 2 \beta_{6} ) q^{89} + ( -2 \beta_{2} - \beta_{5} ) q^{91} + ( -3 \beta_{1} + 7 \beta_{3} ) q^{95} + ( -6 \beta_{4} + 6 \beta_{6} ) q^{97} + 2 \beta_{3} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 8q^{9} + O(q^{10})$$ $$8q + 8q^{9} - 16q^{25} - 48q^{29} + 56q^{49} + 24q^{65} - 40q^{81} + 48q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 8 x^{6} + 55 x^{4} - 72 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} - 148$$$$)/55$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{7} + 11 \nu^{5} - 88 \nu^{3} + 336 \nu$$$$)/99$$ $$\beta_{3}$$ $$=$$ $$($$$$16 \nu^{6} - 110 \nu^{4} + 880 \nu^{2} - 657$$$$)/495$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} - 5 \nu^{5} + 40 \nu^{3} + 48 \nu$$$$)/45$$ $$\beta_{5}$$ $$=$$ $$($$$$-8 \nu^{7} + 55 \nu^{5} - 341 \nu^{3} + 81 \nu$$$$)/297$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - 8 \nu^{5} + 55 \nu^{3} - 45 \nu$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$-\nu^{6} + 8 \nu^{4} - 46 \nu^{2} + 36$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{6} + \beta_{5} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + 4 \beta_{3} + \beta_{1} + 4$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{6} + 11 \beta_{5} + 5 \beta_{4}$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$($$$$8 \beta_{7} + 23 \beta_{3} - 8 \beta_{1} - 23$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-24 \beta_{6} + 24 \beta_{5} + 55 \beta_{4} - 31 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$-55 \beta_{1} - 148$$ $$\nu^{7}$$ $$=$$ $$($$$$-368 \beta_{6} - 368 \beta_{5} + 165 \beta_{4} - 203 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/560\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
559.1
 2.23256 − 1.28897i −1.00781 + 0.581861i −2.23256 − 1.28897i 1.00781 + 0.581861i −1.00781 − 0.581861i 2.23256 + 1.28897i 1.00781 − 0.581861i −2.23256 + 1.28897i
0 1.41421i 0 −1.22474 1.87083i 0 2.64575 0 1.00000 0
559.2 0 1.41421i 0 −1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
559.3 0 1.41421i 0 1.22474 1.87083i 0 2.64575 0 1.00000 0
559.4 0 1.41421i 0 1.22474 + 1.87083i 0 −2.64575 0 1.00000 0
559.5 0 1.41421i 0 −1.22474 1.87083i 0 −2.64575 0 1.00000 0
559.6 0 1.41421i 0 −1.22474 + 1.87083i 0 2.64575 0 1.00000 0
559.7 0 1.41421i 0 1.22474 1.87083i 0 −2.64575 0 1.00000 0
559.8 0 1.41421i 0 1.22474 + 1.87083i 0 2.64575 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 559.8 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
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.e.d 8
4.b odd 2 1 inner 560.2.e.d 8
5.b even 2 1 inner 560.2.e.d 8
5.c odd 4 2 2800.2.k.k 8
7.b odd 2 1 inner 560.2.e.d 8
8.b even 2 1 2240.2.e.e 8
8.d odd 2 1 2240.2.e.e 8
20.d odd 2 1 inner 560.2.e.d 8
20.e even 4 2 2800.2.k.k 8
28.d even 2 1 inner 560.2.e.d 8
35.c odd 2 1 inner 560.2.e.d 8
35.f even 4 2 2800.2.k.k 8
40.e odd 2 1 2240.2.e.e 8
40.f even 2 1 2240.2.e.e 8
56.e even 2 1 2240.2.e.e 8
56.h odd 2 1 2240.2.e.e 8
140.c even 2 1 inner 560.2.e.d 8
140.j odd 4 2 2800.2.k.k 8
280.c odd 2 1 2240.2.e.e 8
280.n even 2 1 2240.2.e.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.d 8 1.a even 1 1 trivial
560.2.e.d 8 4.b odd 2 1 inner
560.2.e.d 8 5.b even 2 1 inner
560.2.e.d 8 7.b odd 2 1 inner
560.2.e.d 8 20.d odd 2 1 inner
560.2.e.d 8 28.d even 2 1 inner
560.2.e.d 8 35.c odd 2 1 inner
560.2.e.d 8 140.c even 2 1 inner
2240.2.e.e 8 8.b even 2 1
2240.2.e.e 8 8.d odd 2 1
2240.2.e.e 8 40.e odd 2 1
2240.2.e.e 8 40.f even 2 1
2240.2.e.e 8 56.e even 2 1
2240.2.e.e 8 56.h odd 2 1
2240.2.e.e 8 280.c odd 2 1
2240.2.e.e 8 280.n even 2 1
2800.2.k.k 8 5.c odd 4 2
2800.2.k.k 8 20.e even 4 2
2800.2.k.k 8 35.f even 4 2
2800.2.k.k 8 140.j odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

 $$T_{3}^{2} + 2$$ $$T_{11}^{2} + 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 4 T^{2} + 9 T^{4} )^{4}$$
$5$ $$( 1 + 4 T^{2} + 25 T^{4} )^{2}$$
$7$ $$( 1 - 7 T^{2} )^{4}$$
$11$ $$( 1 - 10 T^{2} + 121 T^{4} )^{4}$$
$13$ $$( 1 + 20 T^{2} + 169 T^{4} )^{4}$$
$17$ $$( 1 + 10 T^{2} + 289 T^{4} )^{4}$$
$19$ $$( 1 - 4 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 + 23 T^{2} )^{8}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{8}$$
$31$ $$( 1 + 31 T^{2} )^{8}$$
$37$ $$( 1 - 8 T + 37 T^{2} )^{4}( 1 + 8 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 26 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 + 58 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 - 86 T^{2} + 2209 T^{4} )^{4}$$
$53$ $$( 1 - 22 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 76 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 + 4 T^{2} + 3721 T^{4} )^{4}$$
$67$ $$( 1 + 106 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 71 T^{2} )^{8}$$
$73$ $$( 1 + 50 T^{2} + 5329 T^{4} )^{4}$$
$79$ $$( 1 - 110 T^{2} + 6241 T^{4} )^{4}$$
$83$ $$( 1 - 68 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 - 122 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 22 T^{2} + 9409 T^{4} )^{4}$$