# Properties

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

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.bj (of order $$4$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.11574317056.3 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + ( \beta_{2} - \beta_{6} ) q^{5} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + ( \beta_{2} - \beta_{6} ) q^{5} + ( -\beta_{4} + \beta_{6} ) q^{7} + ( -\beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{9} + ( 1 - \beta_{4} - \beta_{5} ) q^{11} -\beta_{1} q^{13} + ( -2 + \beta_{3} + \beta_{4} + \beta_{5} ) q^{15} + ( -\beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{17} + ( -\beta_{1} + \beta_{2} - 2 \beta_{7} ) q^{19} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{6} ) q^{21} + ( 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{23} + ( -2 + \beta_{3} - 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{25} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{27} + ( 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{29} + ( 3 \beta_{1} + 3 \beta_{2} - 4 \beta_{6} ) q^{31} + ( 5 \beta_{1} - \beta_{6} - \beta_{7} ) q^{33} + ( 2 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{35} + ( 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{37} + ( 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} ) q^{39} + ( \beta_{1} + \beta_{2} + 2 \beta_{6} ) q^{41} + ( 5 + 5 \beta_{3} ) q^{43} + ( -3 \beta_{1} - \beta_{2} + \beta_{6} - 2 \beta_{7} ) q^{45} + ( -5 \beta_{2} + 2 \beta_{6} - 2 \beta_{7} ) q^{47} + ( -\beta_{1} + \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + 4 \beta_{7} ) q^{49} + ( -1 - \beta_{4} - \beta_{5} ) q^{51} + ( 5 + 5 \beta_{3} ) q^{53} + ( -\beta_{1} + 4 \beta_{2} + \beta_{6} + \beta_{7} ) q^{55} + ( -5 + 5 \beta_{3} + 2 \beta_{4} - \beta_{6} - \beta_{7} ) q^{57} + ( \beta_{1} - \beta_{2} + 2 \beta_{7} ) q^{59} + ( -2 \beta_{1} - 2 \beta_{2} - 4 \beta_{6} ) q^{61} + ( -5 - \beta_{1} - 5 \beta_{3} + \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{63} + ( 2 - \beta_{3} - \beta_{4} - \beta_{5} ) q^{65} + ( -5 + 5 \beta_{3} ) q^{67} + ( -4 \beta_{1} + 4 \beta_{2} + 2 \beta_{7} ) q^{69} + ( -2 - 2 \beta_{4} - 2 \beta_{5} ) q^{71} + ( -2 \beta_{1} + \beta_{6} + \beta_{7} ) q^{73} + ( 2 \beta_{1} - 5 \beta_{2} - 2 \beta_{7} ) q^{75} + ( 5 + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{7} ) q^{77} + ( -3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} ) q^{79} + ( -1 - 2 \beta_{4} - 2 \beta_{5} ) q^{81} + ( 4 \beta_{1} + \beta_{6} + \beta_{7} ) q^{83} + ( 7 - 6 \beta_{3} + 3 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{85} + ( \beta_{2} - \beta_{6} + \beta_{7} ) q^{87} + ( 4 \beta_{1} - 4 \beta_{2} - 2 \beta_{7} ) q^{89} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{6} ) q^{91} + ( -5 - 5 \beta_{3} + 6 \beta_{5} + 3 \beta_{6} + 3 \beta_{7} ) q^{93} + ( 5 - 5 \beta_{3} + 2 \beta_{4} - 4 \beta_{5} - 3 \beta_{6} - 3 \beta_{7} ) q^{95} -3 \beta_{2} q^{97} + ( -10 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + 2q^{7} + O(q^{10})$$ $$8q + 2q^{7} + 12q^{11} - 20q^{15} - 8q^{21} - 4q^{23} - 12q^{25} + 14q^{35} - 4q^{37} + 40q^{43} - 4q^{51} + 40q^{53} - 44q^{57} - 42q^{63} + 20q^{65} - 40q^{67} - 8q^{71} + 44q^{77} + 48q^{85} + 8q^{91} - 52q^{93} + 44q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{7} + 47 \nu^{3}$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{6} - 47 \nu^{2}$$$$)/14$$ $$\beta_{4}$$ $$=$$ $$($$$$3 \nu^{6} + 2 \nu^{5} + 2 \nu^{4} + 127 \nu^{2} + 94 \nu + 38$$$$)/28$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{6} - 2 \nu^{5} + 2 \nu^{4} - 127 \nu^{2} - 94 \nu + 38$$$$)/28$$ $$\beta_{6}$$ $$=$$ $$($$$$7 \nu^{7} + 2 \nu^{5} + 315 \nu^{3} + 94 \nu$$$$)/28$$ $$\beta_{7}$$ $$=$$ $$($$$$-7 \nu^{7} + 2 \nu^{5} - 315 \nu^{3} + 94 \nu$$$$)/28$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - 3 \beta_{3}$$ $$\nu^{3}$$ $$=$$ $$\beta_{7} - \beta_{6} + 7 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$7 \beta_{5} + 7 \beta_{4} - 19$$ $$\nu^{5}$$ $$=$$ $$7 \beta_{7} + 7 \beta_{6} - 47 \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$-47 \beta_{7} - 47 \beta_{6} - 47 \beta_{5} + 47 \beta_{4} + 127 \beta_{3}$$ $$\nu^{7}$$ $$=$$ $$-47 \beta_{7} + 47 \beta_{6} - 315 \beta_{2}$$

## Character values

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

 $$n$$ $$241$$ $$337$$ $$351$$ $$421$$ $$\chi(n)$$ $$-1$$ $$-\beta_{3}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
97.1
 −1.83051 − 1.83051i −0.386289 − 0.386289i 0.386289 + 0.386289i 1.83051 + 1.83051i −1.83051 + 1.83051i −0.386289 + 0.386289i 0.386289 − 0.386289i 1.83051 − 1.83051i
0 −1.83051 1.83051i 0 1.83051 1.28422i 0 2.12393 + 1.57763i 0 3.70156i 0
97.2 0 −0.386289 0.386289i 0 0.386289 + 2.20245i 0 −0.0564123 2.64515i 0 2.70156i 0
97.3 0 0.386289 + 0.386289i 0 −0.386289 2.20245i 0 −2.64515 0.0564123i 0 2.70156i 0
97.4 0 1.83051 + 1.83051i 0 −1.83051 + 1.28422i 0 1.57763 + 2.12393i 0 3.70156i 0
433.1 0 −1.83051 + 1.83051i 0 1.83051 + 1.28422i 0 2.12393 1.57763i 0 3.70156i 0
433.2 0 −0.386289 + 0.386289i 0 0.386289 2.20245i 0 −0.0564123 + 2.64515i 0 2.70156i 0
433.3 0 0.386289 0.386289i 0 −0.386289 + 2.20245i 0 −2.64515 + 0.0564123i 0 2.70156i 0
433.4 0 1.83051 1.83051i 0 −1.83051 1.28422i 0 1.57763 2.12393i 0 3.70156i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 433.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.bj.b 8
4.b odd 2 1 140.2.m.a 8
5.c odd 4 1 inner 560.2.bj.b 8
7.b odd 2 1 inner 560.2.bj.b 8
12.b even 2 1 1260.2.ba.a 8
20.d odd 2 1 700.2.m.c 8
20.e even 4 1 140.2.m.a 8
20.e even 4 1 700.2.m.c 8
28.d even 2 1 140.2.m.a 8
28.f even 6 2 980.2.v.b 16
28.g odd 6 2 980.2.v.b 16
35.f even 4 1 inner 560.2.bj.b 8
60.l odd 4 1 1260.2.ba.a 8
84.h odd 2 1 1260.2.ba.a 8
140.c even 2 1 700.2.m.c 8
140.j odd 4 1 140.2.m.a 8
140.j odd 4 1 700.2.m.c 8
140.w even 12 2 980.2.v.b 16
140.x odd 12 2 980.2.v.b 16
420.w even 4 1 1260.2.ba.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.m.a 8 4.b odd 2 1
140.2.m.a 8 20.e even 4 1
140.2.m.a 8 28.d even 2 1
140.2.m.a 8 140.j odd 4 1
560.2.bj.b 8 1.a even 1 1 trivial
560.2.bj.b 8 5.c odd 4 1 inner
560.2.bj.b 8 7.b odd 2 1 inner
560.2.bj.b 8 35.f even 4 1 inner
700.2.m.c 8 20.d odd 2 1
700.2.m.c 8 20.e even 4 1
700.2.m.c 8 140.c even 2 1
700.2.m.c 8 140.j odd 4 1
980.2.v.b 16 28.f even 6 2
980.2.v.b 16 28.g odd 6 2
980.2.v.b 16 140.w even 12 2
980.2.v.b 16 140.x odd 12 2
1260.2.ba.a 8 12.b even 2 1
1260.2.ba.a 8 60.l odd 4 1
1260.2.ba.a 8 84.h odd 2 1
1260.2.ba.a 8 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 45 T_{3}^{4} + 4$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - 3 T^{4} - 92 T^{8} - 243 T^{12} + 6561 T^{16}$$
$5$ $$1 + 6 T^{2} + 18 T^{4} + 150 T^{6} + 625 T^{8}$$
$7$ $$1 - 2 T + 2 T^{2} + 26 T^{3} - 62 T^{4} + 182 T^{5} + 98 T^{6} - 686 T^{7} + 2401 T^{8}$$
$11$ $$( 1 - 3 T + 14 T^{2} - 33 T^{3} + 121 T^{4} )^{4}$$
$13$ $$1 + 357 T^{4} + 68228 T^{8} + 10196277 T^{12} + 815730721 T^{16}$$
$17$ $$1 + 69 T^{4} - 53260 T^{8} + 5762949 T^{12} + 6975757441 T^{16}$$
$19$ $$( 1 + 18 T^{2} + 762 T^{4} + 6498 T^{6} + 130321 T^{8} )^{2}$$
$23$ $$( 1 + 2 T + 2 T^{2} + 6 T^{3} - 382 T^{4} + 138 T^{5} + 1058 T^{6} + 24334 T^{7} + 279841 T^{8} )^{2}$$
$29$ $$( 1 - 83 T^{2} + 3148 T^{4} - 69803 T^{6} + 707281 T^{8} )^{2}$$
$31$ $$( 1 + 18 T^{2} + 1962 T^{4} + 17298 T^{6} + 923521 T^{8} )^{2}$$
$37$ $$( 1 + 2 T + 2 T^{2} + 34 T^{3} + 178 T^{4} + 1258 T^{5} + 2738 T^{6} + 101306 T^{7} + 1874161 T^{8} )^{2}$$
$41$ $$( 1 - 106 T^{2} + 6130 T^{4} - 178186 T^{6} + 2825761 T^{8} )^{2}$$
$43$ $$( 1 - 10 T + 50 T^{2} - 430 T^{3} + 1849 T^{4} )^{4}$$
$47$ $$1 - 2227 T^{4} + 6943924 T^{8} - 10867049587 T^{12} + 23811286661761 T^{16}$$
$53$ $$( 1 - 14 T + 53 T^{2} )^{4}( 1 + 4 T + 53 T^{2} )^{4}$$
$59$ $$( 1 + 178 T^{2} + 14842 T^{4} + 619618 T^{6} + 12117361 T^{8} )^{2}$$
$61$ $$( 1 - 12 T^{2} + 6822 T^{4} - 44652 T^{6} + 13845841 T^{8} )^{2}$$
$67$ $$( 1 + 10 T + 50 T^{2} + 670 T^{3} + 4489 T^{4} )^{4}$$
$71$ $$( 1 + 2 T + 102 T^{2} + 142 T^{3} + 5041 T^{4} )^{4}$$
$73$ $$1 + 14144 T^{4} + 103908670 T^{8} + 401664720704 T^{12} + 806460091894081 T^{16}$$
$79$ $$( 1 - 271 T^{2} + 30340 T^{4} - 1691311 T^{6} + 38950081 T^{8} )^{2}$$
$83$ $$1 - 8400 T^{4} + 51732158 T^{8} - 398649896400 T^{12} + 2252292232139041 T^{16}$$
$89$ $$( 1 + 168 T^{2} + 14862 T^{4} + 1330728 T^{6} + 62742241 T^{8} )^{2}$$
$97$ $$1 + 16837 T^{4} + 160234548 T^{8} + 1490567504197 T^{12} + 7837433594376961 T^{16}$$
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