# Properties

 Label 560.2.k.b 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: 8.0.116319195136.7 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{7} q^{3} + \beta_{3} q^{5} + ( \beta_{4} - \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{7} q^{3} + \beta_{3} q^{5} + ( \beta_{4} - \beta_{6} ) q^{7} + ( 1 + \beta_{2} ) q^{9} + ( \beta_{1} + \beta_{4} + \beta_{6} ) q^{11} -\beta_{5} q^{13} + \beta_{6} q^{15} + \beta_{5} q^{17} + ( -2 \beta_{1} + 2 \beta_{4} + 2 \beta_{7} ) q^{19} + ( -1 - 3 \beta_{3} + \beta_{5} ) q^{21} + ( \beta_{1} + \beta_{4} - 2 \beta_{6} ) q^{23} - q^{25} + ( -\beta_{1} + \beta_{4} + 3 \beta_{7} ) q^{27} -\beta_{2} q^{29} + 2 \beta_{7} q^{31} + ( 6 \beta_{3} - \beta_{5} ) q^{33} + ( -\beta_{1} + \beta_{7} ) q^{35} + ( -2 - 2 \beta_{2} ) q^{37} + ( -\beta_{1} - \beta_{4} + 5 \beta_{6} ) q^{39} -6 \beta_{3} q^{41} + ( 3 \beta_{1} + 3 \beta_{4} ) q^{43} + ( \beta_{3} - \beta_{5} ) q^{45} + ( 2 \beta_{1} - 2 \beta_{4} - \beta_{7} ) q^{47} + ( -2 - \beta_{2} - 3 \beta_{3} - \beta_{5} ) q^{49} + ( \beta_{1} + \beta_{4} - 5 \beta_{6} ) q^{51} -6 q^{53} + ( -\beta_{1} + \beta_{4} - \beta_{7} ) q^{55} + ( 4 + 2 \beta_{2} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{4} - 2 \beta_{7} ) q^{59} + ( 6 \beta_{3} + 2 \beta_{5} ) q^{61} + ( \beta_{1} - 2 \beta_{4} - 5 \beta_{6} - \beta_{7} ) q^{63} -\beta_{2} q^{65} + ( \beta_{1} + \beta_{4} + 4 \beta_{6} ) q^{67} + ( -6 \beta_{3} + 2 \beta_{5} ) q^{69} + ( -2 \beta_{1} - 2 \beta_{4} - 2 \beta_{6} ) q^{71} + 6 \beta_{3} q^{73} -\beta_{7} q^{75} + ( 2 \beta_{2} - 6 \beta_{3} - \beta_{5} ) q^{77} + ( -3 \beta_{1} - 3 \beta_{4} + 3 \beta_{6} ) q^{79} + 7 q^{81} + ( 3 \beta_{1} - 3 \beta_{4} ) q^{83} + \beta_{2} q^{85} + ( \beta_{1} - \beta_{4} - 5 \beta_{7} ) q^{87} + 12 \beta_{3} q^{89} + ( 3 \beta_{1} + \beta_{4} - \beta_{6} + 4 \beta_{7} ) q^{91} + ( 8 + 2 \beta_{2} ) q^{93} + ( -2 \beta_{1} - 2 \beta_{4} + 2 \beta_{6} ) q^{95} + ( -12 \beta_{3} + \beta_{5} ) q^{97} + ( -4 \beta_{1} - 4 \beta_{4} + 8 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 12q^{9} + O(q^{10})$$ $$8q + 12q^{9} - 8q^{21} - 8q^{25} - 4q^{29} - 24q^{37} - 20q^{49} - 48q^{53} + 40q^{57} - 4q^{65} + 8q^{77} + 56q^{81} + 4q^{85} + 72q^{93} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 77 x^{4} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{4} + 43$$$$)/9$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{6} + 79 \nu^{2}$$$$)/18$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{7} + 79 \nu^{3}$$$$)/18$$ $$\beta_{5}$$ $$=$$ $$($$$$-5 \nu^{6} - 377 \nu^{2}$$$$)/18$$ $$\beta_{6}$$ $$=$$ $$($$$$9 \nu^{7} + 2 \nu^{5} + 693 \nu^{3} + 158 \nu$$$$)/36$$ $$\beta_{7}$$ $$=$$ $$($$$$-9 \nu^{7} + 2 \nu^{5} - 693 \nu^{3} + 158 \nu$$$$)/36$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{5} + 5 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + 9 \beta_{4}$$ $$\nu^{4}$$ $$=$$ $$9 \beta_{2} - 43$$ $$\nu^{5}$$ $$=$$ $$9 \beta_{7} + 9 \beta_{6} - 79 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-79 \beta_{5} - 377 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$-79 \beta_{7} + 79 \beta_{6} - 693 \beta_{4}$$

## 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
 −0.337637 + 0.337637i −0.337637 − 0.337637i −2.09428 + 2.09428i −2.09428 − 2.09428i 2.09428 − 2.09428i 2.09428 + 2.09428i 0.337637 − 0.337637i 0.337637 + 0.337637i
0 −2.96176 0 1.00000i 0 0.337637 2.62412i 0 5.77200 0
111.2 0 −2.96176 0 1.00000i 0 0.337637 + 2.62412i 0 5.77200 0
111.3 0 −0.477491 0 1.00000i 0 2.09428 + 1.61679i 0 −2.77200 0
111.4 0 −0.477491 0 1.00000i 0 2.09428 1.61679i 0 −2.77200 0
111.5 0 0.477491 0 1.00000i 0 −2.09428 1.61679i 0 −2.77200 0
111.6 0 0.477491 0 1.00000i 0 −2.09428 + 1.61679i 0 −2.77200 0
111.7 0 2.96176 0 1.00000i 0 −0.337637 + 2.62412i 0 5.77200 0
111.8 0 2.96176 0 1.00000i 0 −0.337637 2.62412i 0 5.77200 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.b 8
3.b odd 2 1 5040.2.d.d 8
4.b odd 2 1 inner 560.2.k.b 8
5.b even 2 1 2800.2.k.m 8
5.c odd 4 1 2800.2.e.g 8
5.c odd 4 1 2800.2.e.h 8
7.b odd 2 1 inner 560.2.k.b 8
8.b even 2 1 2240.2.k.d 8
8.d odd 2 1 2240.2.k.d 8
12.b even 2 1 5040.2.d.d 8
20.d odd 2 1 2800.2.k.m 8
20.e even 4 1 2800.2.e.g 8
20.e even 4 1 2800.2.e.h 8
21.c even 2 1 5040.2.d.d 8
28.d even 2 1 inner 560.2.k.b 8
35.c odd 2 1 2800.2.k.m 8
35.f even 4 1 2800.2.e.g 8
35.f even 4 1 2800.2.e.h 8
56.e even 2 1 2240.2.k.d 8
56.h odd 2 1 2240.2.k.d 8
84.h odd 2 1 5040.2.d.d 8
140.c even 2 1 2800.2.k.m 8
140.j odd 4 1 2800.2.e.g 8
140.j odd 4 1 2800.2.e.h 8

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 9 T_{3}^{2} + 2$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 + 3 T^{2} + 2 T^{4} + 27 T^{6} + 81 T^{8} )^{2}$$
$5$ $$( 1 + T^{2} )^{4}$$
$7$ $$1 + 10 T^{2} + 50 T^{4} + 490 T^{6} + 2401 T^{8}$$
$11$ $$( 1 - 9 T^{2} + 244 T^{4} - 1089 T^{6} + 14641 T^{8} )^{2}$$
$13$ $$( 1 - 15 T^{2} + 376 T^{4} - 2535 T^{6} + 28561 T^{8} )^{2}$$
$17$ $$( 1 - 31 T^{2} + 800 T^{4} - 8959 T^{6} + 83521 T^{8} )^{2}$$
$19$ $$( 1 + 430 T^{4} + 130321 T^{8} )^{2}$$
$23$ $$( 1 - 54 T^{2} + 1714 T^{4} - 28566 T^{6} + 279841 T^{8} )^{2}$$
$29$ $$( 1 + T + 40 T^{2} + 29 T^{3} + 841 T^{4} )^{4}$$
$31$ $$( 1 + 88 T^{2} + 3566 T^{4} + 84568 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 6 T + 10 T^{2} + 222 T^{3} + 1369 T^{4} )^{4}$$
$41$ $$( 1 - 46 T^{2} + 1681 T^{4} )^{4}$$
$43$ $$( 1 - 10 T^{2} - 2190 T^{4} - 18490 T^{6} + 3418801 T^{8} )^{2}$$
$47$ $$( 1 + 123 T^{2} + 7306 T^{4} + 271707 T^{6} + 4879681 T^{8} )^{2}$$
$53$ $$( 1 + 6 T + 53 T^{2} )^{8}$$
$59$ $$( 1 + 96 T^{2} + 8974 T^{4} + 334176 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 48 T^{2} + 718 T^{4} - 178608 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 - 74 T^{2} + 6770 T^{4} - 332186 T^{6} + 20151121 T^{8} )^{2}$$
$71$ $$( 1 - 144 T^{2} + 14974 T^{4} - 725904 T^{6} + 25411681 T^{8} )^{2}$$
$73$ $$( 1 - 16 T + 73 T^{2} )^{4}( 1 + 16 T + 73 T^{2} )^{4}$$
$79$ $$( 1 - 145 T^{2} + 16260 T^{4} - 904945 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$( 1 + 170 T^{2} + 15090 T^{4} + 1171130 T^{6} + 47458321 T^{8} )^{2}$$
$89$ $$( 1 - 34 T^{2} + 7921 T^{4} )^{4}$$
$97$ $$( 1 - 39 T^{2} + 7792 T^{4} - 366951 T^{6} + 88529281 T^{8} )^{2}$$