# Properties

 Label 560.3.f.a Level $560$ Weight $3$ Character orbit 560.f Analytic conductor $15.259$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 560.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.2588948042$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ x^2 + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{3} - \beta q^{5} - 7 q^{7} - 11 q^{9} +O(q^{10})$$ q + 2*b * q^3 - b * q^5 - 7 * q^7 - 11 * q^9 $$q + 2 \beta q^{3} - \beta q^{5} - 7 q^{7} - 11 q^{9} - 2 q^{11} + 6 \beta q^{13} + 10 q^{15} - 12 \beta q^{17} - 6 \beta q^{19} - 14 \beta q^{21} - 26 q^{23} - 5 q^{25} - 4 \beta q^{27} - 22 q^{29} - 24 \beta q^{31} - 4 \beta q^{33} + 7 \beta q^{35} + 14 q^{37} - 60 q^{39} + 12 \beta q^{41} + 34 q^{43} + 11 \beta q^{45} - 12 \beta q^{47} + 49 q^{49} + 120 q^{51} - 34 q^{53} + 2 \beta q^{55} + 60 q^{57} - 18 \beta q^{59} - 42 \beta q^{61} + 77 q^{63} + 30 q^{65} - 14 q^{67} - 52 \beta q^{69} - 62 q^{71} + 24 \beta q^{73} - 10 \beta q^{75} + 14 q^{77} - 38 q^{79} - 59 q^{81} + 18 \beta q^{83} - 60 q^{85} - 44 \beta q^{87} + 12 \beta q^{89} - 42 \beta q^{91} + 240 q^{93} - 30 q^{95} + 12 \beta q^{97} + 22 q^{99} +O(q^{100})$$ q + 2*b * q^3 - b * q^5 - 7 * q^7 - 11 * q^9 - 2 * q^11 + 6*b * q^13 + 10 * q^15 - 12*b * q^17 - 6*b * q^19 - 14*b * q^21 - 26 * q^23 - 5 * q^25 - 4*b * q^27 - 22 * q^29 - 24*b * q^31 - 4*b * q^33 + 7*b * q^35 + 14 * q^37 - 60 * q^39 + 12*b * q^41 + 34 * q^43 + 11*b * q^45 - 12*b * q^47 + 49 * q^49 + 120 * q^51 - 34 * q^53 + 2*b * q^55 + 60 * q^57 - 18*b * q^59 - 42*b * q^61 + 77 * q^63 + 30 * q^65 - 14 * q^67 - 52*b * q^69 - 62 * q^71 + 24*b * q^73 - 10*b * q^75 + 14 * q^77 - 38 * q^79 - 59 * q^81 + 18*b * q^83 - 60 * q^85 - 44*b * q^87 + 12*b * q^89 - 42*b * q^91 + 240 * q^93 - 30 * q^95 + 12*b * q^97 + 22 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 14 q^{7} - 22 q^{9}+O(q^{10})$$ 2 * q - 14 * q^7 - 22 * q^9 $$2 q - 14 q^{7} - 22 q^{9} - 4 q^{11} + 20 q^{15} - 52 q^{23} - 10 q^{25} - 44 q^{29} + 28 q^{37} - 120 q^{39} + 68 q^{43} + 98 q^{49} + 240 q^{51} - 68 q^{53} + 120 q^{57} + 154 q^{63} + 60 q^{65} - 28 q^{67} - 124 q^{71} + 28 q^{77} - 76 q^{79} - 118 q^{81} - 120 q^{85} + 480 q^{93} - 60 q^{95} + 44 q^{99}+O(q^{100})$$ 2 * q - 14 * q^7 - 22 * q^9 - 4 * q^11 + 20 * q^15 - 52 * q^23 - 10 * q^25 - 44 * q^29 + 28 * q^37 - 120 * q^39 + 68 * q^43 + 98 * q^49 + 240 * q^51 - 68 * q^53 + 120 * q^57 + 154 * q^63 + 60 * q^65 - 28 * q^67 - 124 * q^71 + 28 * q^77 - 76 * q^79 - 118 * q^81 - 120 * q^85 + 480 * q^93 - 60 * q^95 + 44 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
321.1
 − 2.23607i 2.23607i
0 4.47214i 0 2.23607i 0 −7.00000 0 −11.0000 0
321.2 0 4.47214i 0 2.23607i 0 −7.00000 0 −11.0000 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
7.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.3.f.a 2
4.b odd 2 1 35.3.d.a 2
7.b odd 2 1 inner 560.3.f.a 2
12.b even 2 1 315.3.h.b 2
20.d odd 2 1 175.3.d.f 2
20.e even 4 2 175.3.c.d 4
28.d even 2 1 35.3.d.a 2
28.f even 6 2 245.3.h.b 4
28.g odd 6 2 245.3.h.b 4
84.h odd 2 1 315.3.h.b 2
140.c even 2 1 175.3.d.f 2
140.j odd 4 2 175.3.c.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.3.d.a 2 4.b odd 2 1
35.3.d.a 2 28.d even 2 1
175.3.c.d 4 20.e even 4 2
175.3.c.d 4 140.j odd 4 2
175.3.d.f 2 20.d odd 2 1
175.3.d.f 2 140.c even 2 1
245.3.h.b 4 28.f even 6 2
245.3.h.b 4 28.g odd 6 2
315.3.h.b 2 12.b even 2 1
315.3.h.b 2 84.h odd 2 1
560.3.f.a 2 1.a even 1 1 trivial
560.3.f.a 2 7.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 20$$ acting on $$S_{3}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 20$$
$5$ $$T^{2} + 5$$
$7$ $$(T + 7)^{2}$$
$11$ $$(T + 2)^{2}$$
$13$ $$T^{2} + 180$$
$17$ $$T^{2} + 720$$
$19$ $$T^{2} + 180$$
$23$ $$(T + 26)^{2}$$
$29$ $$(T + 22)^{2}$$
$31$ $$T^{2} + 2880$$
$37$ $$(T - 14)^{2}$$
$41$ $$T^{2} + 720$$
$43$ $$(T - 34)^{2}$$
$47$ $$T^{2} + 720$$
$53$ $$(T + 34)^{2}$$
$59$ $$T^{2} + 1620$$
$61$ $$T^{2} + 8820$$
$67$ $$(T + 14)^{2}$$
$71$ $$(T + 62)^{2}$$
$73$ $$T^{2} + 2880$$
$79$ $$(T + 38)^{2}$$
$83$ $$T^{2} + 1620$$
$89$ $$T^{2} + 720$$
$97$ $$T^{2} + 720$$