Properties

 Label 560.2.e.b Level 560 Weight 2 Character orbit 560.e Analytic conductor 4.472 Analytic rank 0 Dimension 4 CM discriminant -20 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: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} + q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} + \beta_{3} q^{5} + ( \beta_{1} - 2 \beta_{2} ) q^{7} + q^{9} + ( 2 \beta_{1} - \beta_{2} ) q^{15} + ( -3 - \beta_{3} ) q^{21} + ( 6 \beta_{1} - 3 \beta_{2} ) q^{23} -5 q^{25} -4 \beta_{2} q^{27} + 6 q^{29} + ( 3 \beta_{1} + \beta_{2} ) q^{35} -2 \beta_{3} q^{41} + ( -2 \beta_{1} + \beta_{2} ) q^{43} + \beta_{3} q^{45} -7 \beta_{2} q^{47} + ( -2 - 3 \beta_{3} ) q^{49} + 6 \beta_{3} q^{61} + ( \beta_{1} - 2 \beta_{2} ) q^{63} + ( -10 \beta_{1} + 5 \beta_{2} ) q^{67} -6 \beta_{3} q^{69} + 5 \beta_{2} q^{75} -5 q^{81} + 11 \beta_{2} q^{83} -6 \beta_{2} q^{87} + 8 \beta_{3} q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{9} + O(q^{10})$$ $$4q + 4q^{9} - 12q^{21} - 20q^{25} + 24q^{29} - 8q^{49} - 20q^{81} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + 2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{2} + \beta_{1}$$

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
 −1.58114 + 0.707107i 1.58114 + 0.707107i 1.58114 − 0.707107i −1.58114 − 0.707107i
0 1.41421i 0 2.23607i 0 −1.58114 2.12132i 0 1.00000 0
559.2 0 1.41421i 0 2.23607i 0 1.58114 2.12132i 0 1.00000 0
559.3 0 1.41421i 0 2.23607i 0 1.58114 + 2.12132i 0 1.00000 0
559.4 0 1.41421i 0 2.23607i 0 −1.58114 + 2.12132i 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner
7.b 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.b 4
4.b odd 2 1 inner 560.2.e.b 4
5.b even 2 1 inner 560.2.e.b 4
5.c odd 4 2 2800.2.k.g 4
7.b odd 2 1 inner 560.2.e.b 4
8.b even 2 1 2240.2.e.c 4
8.d odd 2 1 2240.2.e.c 4
20.d odd 2 1 CM 560.2.e.b 4
20.e even 4 2 2800.2.k.g 4
28.d even 2 1 inner 560.2.e.b 4
35.c odd 2 1 inner 560.2.e.b 4
35.f even 4 2 2800.2.k.g 4
40.e odd 2 1 2240.2.e.c 4
40.f even 2 1 2240.2.e.c 4
56.e even 2 1 2240.2.e.c 4
56.h odd 2 1 2240.2.e.c 4
140.c even 2 1 inner 560.2.e.b 4
140.j odd 4 2 2800.2.k.g 4
280.c odd 2 1 2240.2.e.c 4
280.n even 2 1 2240.2.e.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.e.b 4 1.a even 1 1 trivial
560.2.e.b 4 4.b odd 2 1 inner
560.2.e.b 4 5.b even 2 1 inner
560.2.e.b 4 7.b odd 2 1 inner
560.2.e.b 4 20.d odd 2 1 CM
560.2.e.b 4 28.d even 2 1 inner
560.2.e.b 4 35.c odd 2 1 inner
560.2.e.b 4 140.c even 2 1 inner
2240.2.e.c 4 8.b even 2 1
2240.2.e.c 4 8.d odd 2 1
2240.2.e.c 4 40.e odd 2 1
2240.2.e.c 4 40.f even 2 1
2240.2.e.c 4 56.e even 2 1
2240.2.e.c 4 56.h odd 2 1
2240.2.e.c 4 280.c odd 2 1
2240.2.e.c 4 280.n even 2 1
2800.2.k.g 4 5.c odd 4 2
2800.2.k.g 4 20.e even 4 2
2800.2.k.g 4 35.f even 4 2
2800.2.k.g 4 140.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 $$S_{2}^{\mathrm{new}}(560, [\chi])$$:

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$( 1 - 4 T^{2} + 9 T^{4} )^{2}$$
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ $$1 + 4 T^{2} + 49 T^{4}$$
$11$ $$( 1 - 11 T^{2} )^{4}$$
$13$ $$( 1 + 13 T^{2} )^{4}$$
$17$ $$( 1 + 17 T^{2} )^{4}$$
$19$ $$( 1 + 19 T^{2} )^{4}$$
$23$ $$( 1 - 44 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{4}$$
$31$ $$( 1 + 31 T^{2} )^{4}$$
$37$ $$( 1 - 37 T^{2} )^{4}$$
$41$ $$( 1 - 12 T + 41 T^{2} )^{2}( 1 + 12 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 76 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 4 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 53 T^{2} )^{4}$$
$59$ $$( 1 + 59 T^{2} )^{4}$$
$61$ $$( 1 - 8 T + 61 T^{2} )^{2}( 1 + 8 T + 61 T^{2} )^{2}$$
$67$ $$( 1 - 116 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 71 T^{2} )^{4}$$
$73$ $$( 1 + 73 T^{2} )^{4}$$
$79$ $$( 1 - 79 T^{2} )^{4}$$
$83$ $$( 1 + 76 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$( 1 + 97 T^{2} )^{4}$$