# Properties

 Label 560.2.k.a Level 560 Weight 2 Character orbit 560.k Analytic conductor 4.472 Analytic rank 0 Dimension 8 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.303595776.1 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.

 $$f(q)$$ $$=$$ $$q -\beta_{6} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{7} -\beta_{7} q^{9} +O(q^{10})$$ $$q -\beta_{6} q^{3} + \beta_{4} q^{5} + ( -\beta_{1} + \beta_{5} - \beta_{6} ) q^{7} -\beta_{7} q^{9} + \beta_{1} q^{11} + ( \beta_{2} + 3 \beta_{4} ) q^{13} -\beta_{1} q^{15} + ( \beta_{2} + \beta_{4} ) q^{17} + 2 \beta_{3} q^{19} + ( 3 + 2 \beta_{4} - \beta_{7} ) q^{21} + ( -2 \beta_{1} - 2 \beta_{5} ) q^{23} - q^{25} + ( -2 \beta_{3} + \beta_{6} ) q^{27} + ( -3 - \beta_{7} ) q^{29} + ( 4 \beta_{3} - 2 \beta_{6} ) q^{31} + ( \beta_{2} - 3 \beta_{4} ) q^{33} + ( -\beta_{1} - \beta_{3} + \beta_{6} ) q^{35} + ( 4 - 2 \beta_{7} ) q^{37} + ( -\beta_{1} + 2 \beta_{5} ) q^{39} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{41} -2 \beta_{5} q^{43} -\beta_{2} q^{45} + ( 6 \beta_{3} - \beta_{6} ) q^{47} + ( -1 + 4 \beta_{4} - 2 \beta_{7} ) q^{49} + ( \beta_{1} + 2 \beta_{5} ) q^{51} + ( 2 + 4 \beta_{7} ) q^{53} -\beta_{6} q^{55} + ( -2 + 2 \beta_{7} ) q^{57} + ( -2 \beta_{3} + 4 \beta_{6} ) q^{59} + ( 2 \beta_{2} - 4 \beta_{4} ) q^{61} + ( \beta_{1} - 2 \beta_{3} - 3 \beta_{5} - 2 \beta_{6} ) q^{63} + ( -3 - \beta_{7} ) q^{65} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{67} + ( -4 \beta_{2} + 8 \beta_{4} ) q^{69} + ( 4 \beta_{1} - 2 \beta_{5} ) q^{71} + ( 2 \beta_{2} - 2 \beta_{4} ) q^{73} + \beta_{6} q^{75} + ( 2 + \beta_{2} - 3 \beta_{4} ) q^{77} + ( -9 \beta_{1} + 4 \beta_{5} ) q^{79} + ( -1 + 2 \beta_{7} ) q^{81} -6 \beta_{3} q^{83} + ( -1 - \beta_{7} ) q^{85} + ( -2 \beta_{3} + \beta_{6} ) q^{87} + ( -4 \beta_{2} - 8 \beta_{4} ) q^{89} + ( -\beta_{1} - 6 \beta_{3} + 2 \beta_{5} + 4 \beta_{6} ) q^{91} + ( 2 + 2 \beta_{7} ) q^{93} + 2 \beta_{5} q^{95} + ( -3 \beta_{2} + 5 \beta_{4} ) q^{97} + ( 2 \beta_{1} + 2 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 4q^{9} + O(q^{10})$$ $$8q + 4q^{9} + 28q^{21} - 8q^{25} - 20q^{29} + 40q^{37} - 24q^{57} - 20q^{65} + 16q^{77} - 16q^{81} - 4q^{85} + 8q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 5 x^{6} + 16 x^{4} + 45 x^{2} + 81$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} + 20 \nu^{2} + 27$$$$)/36$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 8 \nu^{5} + 40 \nu^{3} + 165 \nu$$$$)/72$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 4 \nu^{5} + 2 \nu^{3} - 9 \nu$$$$)/54$$ $$\beta_{4}$$ $$=$$ $$($$$$5 \nu^{7} + 16 \nu^{5} + 8 \nu^{3} + 81 \nu$$$$)/216$$ $$\beta_{5}$$ $$=$$ $$($$$$5 \nu^{6} + 16 \nu^{4} + 80 \nu^{2} + 153$$$$)/72$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} + 5 \nu^{5} + 16 \nu^{3} + 18 \nu$$$$)/27$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{6} + 5 \nu^{4} + 7 \nu^{2} + 18$$$$)/9$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{6} + \beta_{4} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{7} + 2 \beta_{5} + \beta_{1} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{6} - 4 \beta_{4} - \beta_{3}$$ $$\nu^{4}$$ $$=$$ $$($$$$5 \beta_{7} - 6 \beta_{5} + 5 \beta_{1} - 1$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-\beta_{6} + 15 \beta_{4} + 16 \beta_{3} - \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$8 \beta_{5} - 16 \beta_{1} - 5$$ $$\nu^{7}$$ $$=$$ $$($$$$13 \beta_{6} + 35 \beta_{4} - 48 \beta_{3} - 13 \beta_{2}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
111.1
 −1.26217 + 1.18614i −1.26217 − 1.18614i −0.396143 − 1.68614i −0.396143 + 1.68614i 0.396143 − 1.68614i 0.396143 + 1.68614i 1.26217 + 1.18614i 1.26217 − 1.18614i
0 −2.52434 0 1.00000i 0 −2.52434 + 0.792287i 0 3.37228 0
111.2 0 −2.52434 0 1.00000i 0 −2.52434 0.792287i 0 3.37228 0
111.3 0 −0.792287 0 1.00000i 0 −0.792287 + 2.52434i 0 −2.37228 0
111.4 0 −0.792287 0 1.00000i 0 −0.792287 2.52434i 0 −2.37228 0
111.5 0 0.792287 0 1.00000i 0 0.792287 2.52434i 0 −2.37228 0
111.6 0 0.792287 0 1.00000i 0 0.792287 + 2.52434i 0 −2.37228 0
111.7 0 2.52434 0 1.00000i 0 2.52434 0.792287i 0 3.37228 0
111.8 0 2.52434 0 1.00000i 0 2.52434 + 0.792287i 0 3.37228 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 111.8 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
7.b odd 2 1 inner
28.d 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.k.a 8
3.b odd 2 1 5040.2.d.e 8
4.b odd 2 1 inner 560.2.k.a 8
5.b even 2 1 2800.2.k.l 8
5.c odd 4 1 2800.2.e.i 8
5.c odd 4 1 2800.2.e.j 8
7.b odd 2 1 inner 560.2.k.a 8
8.b even 2 1 2240.2.k.c 8
8.d odd 2 1 2240.2.k.c 8
12.b even 2 1 5040.2.d.e 8
20.d odd 2 1 2800.2.k.l 8
20.e even 4 1 2800.2.e.i 8
20.e even 4 1 2800.2.e.j 8
21.c even 2 1 5040.2.d.e 8
28.d even 2 1 inner 560.2.k.a 8
35.c odd 2 1 2800.2.k.l 8
35.f even 4 1 2800.2.e.i 8
35.f even 4 1 2800.2.e.j 8
56.e even 2 1 2240.2.k.c 8
56.h odd 2 1 2240.2.k.c 8
84.h odd 2 1 5040.2.d.e 8
140.c even 2 1 2800.2.k.l 8
140.j odd 4 1 2800.2.e.i 8
140.j odd 4 1 2800.2.e.j 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.k.a 8 1.a even 1 1 trivial
560.2.k.a 8 4.b odd 2 1 inner
560.2.k.a 8 7.b odd 2 1 inner
560.2.k.a 8 28.d even 2 1 inner
2240.2.k.c 8 8.b even 2 1
2240.2.k.c 8 8.d odd 2 1
2240.2.k.c 8 56.e even 2 1
2240.2.k.c 8 56.h odd 2 1
2800.2.e.i 8 5.c odd 4 1
2800.2.e.i 8 20.e even 4 1
2800.2.e.i 8 35.f even 4 1
2800.2.e.i 8 140.j odd 4 1
2800.2.e.j 8 5.c odd 4 1
2800.2.e.j 8 20.e even 4 1
2800.2.e.j 8 35.f even 4 1
2800.2.e.j 8 140.j odd 4 1
2800.2.k.l 8 5.b even 2 1
2800.2.k.l 8 20.d odd 2 1
2800.2.k.l 8 35.c odd 2 1
2800.2.k.l 8 140.c even 2 1
5040.2.d.e 8 3.b odd 2 1
5040.2.d.e 8 12.b even 2 1
5040.2.d.e 8 21.c even 2 1
5040.2.d.e 8 84.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 7 T_{3}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 5 T^{2} + 16 T^{4} + 45 T^{6} + 81 T^{8} )^{2}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$1 - 34 T^{4} + 2401 T^{8}$$
$11$ $$( 1 - 37 T^{2} + 576 T^{4} - 4477 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 23 T^{2} + 264 T^{4} - 3887 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 51 T^{2} + 1220 T^{4} - 14739 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 26 T^{2} + 361 T^{4} )^{4}$$
$23$ $$( 1 - 16 T^{2} - 66 T^{4} - 8464 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + 5 T + 56 T^{2} + 145 T^{3} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 48 T^{2} + 1310 T^{4} + 46128 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 - 10 T + 66 T^{2} - 370 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 80 T^{2} + 3774 T^{4} - 134480 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 74 T^{2} + 1849 T^{4} )^{4}$$
$47$ $$( 1 + T^{2} + 3420 T^{4} + 2209 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 - 26 T^{2} + 2809 T^{4} )^{4}$$
$59$ $$( 1 + 74 T^{2} + 3481 T^{4} )^{4}$$
$61$ $$( 1 - 128 T^{2} + 8238 T^{4} - 476288 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 90 T^{2} + 4489 T^{4} )^{4}$$
$71$ $$( 1 - 98 T^{2} + 5041 T^{4} )^{4}$$
$73$ $$( 1 - 208 T^{2} + 20286 T^{4} - 1108432 T^{6} + 28398241 T^{8} )^{2}$$
$79$ $$( 1 + 131 T^{2} + 16104 T^{4} + 817571 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 58 T^{2} + 6889 T^{4} )^{4}$$
$89$ $$( 1 - 20 T^{2} - 3066 T^{4} - 158420 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$( 1 - 155 T^{2} + 12276 T^{4} - 1458395 T^{6} + 88529281 T^{8} )^{2}$$