# Properties

 Label 560.5.p.a Level $560$ Weight $5$ Character orbit 560.p Self dual yes Analytic conductor $57.887$ Analytic rank $0$ Dimension $1$ CM discriminant -35 Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$57.8871793270$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 35) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 17q^{3} - 25q^{5} + 49q^{7} + 208q^{9} + O(q^{10})$$ $$q - 17q^{3} - 25q^{5} + 49q^{7} + 208q^{9} + 73q^{11} - 23q^{13} + 425q^{15} - 263q^{17} - 833q^{21} + 625q^{25} - 2159q^{27} - 1153q^{29} - 1241q^{33} - 1225q^{35} + 391q^{39} - 5200q^{45} - 3457q^{47} + 2401q^{49} + 4471q^{51} - 1825q^{55} + 10192q^{63} + 575q^{65} + 10078q^{71} + 9502q^{73} - 10625q^{75} + 3577q^{77} - 12167q^{79} + 19855q^{81} - 6382q^{83} + 6575q^{85} + 19601q^{87} - 1127q^{91} - 3383q^{97} + 15184q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1
 0
0 −17.0000 0 −25.0000 0 49.0000 0 208.000 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
35.c odd 2 1 CM by $$\Q(\sqrt{-35})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.5.p.a 1
4.b odd 2 1 35.5.c.b yes 1
5.b even 2 1 560.5.p.b 1
7.b odd 2 1 560.5.p.b 1
12.b even 2 1 315.5.e.b 1
20.d odd 2 1 35.5.c.a 1
20.e even 4 2 175.5.d.c 2
28.d even 2 1 35.5.c.a 1
35.c odd 2 1 CM 560.5.p.a 1
60.h even 2 1 315.5.e.a 1
84.h odd 2 1 315.5.e.a 1
140.c even 2 1 35.5.c.b yes 1
140.j odd 4 2 175.5.d.c 2
420.o odd 2 1 315.5.e.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.5.c.a 1 20.d odd 2 1
35.5.c.a 1 28.d even 2 1
35.5.c.b yes 1 4.b odd 2 1
35.5.c.b yes 1 140.c even 2 1
175.5.d.c 2 20.e even 4 2
175.5.d.c 2 140.j odd 4 2
315.5.e.a 1 60.h even 2 1
315.5.e.a 1 84.h odd 2 1
315.5.e.b 1 12.b even 2 1
315.5.e.b 1 420.o odd 2 1
560.5.p.a 1 1.a even 1 1 trivial
560.5.p.a 1 35.c odd 2 1 CM
560.5.p.b 1 5.b even 2 1
560.5.p.b 1 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 17$$ acting on $$S_{5}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$17 + T$$
$5$ $$25 + T$$
$7$ $$-49 + T$$
$11$ $$-73 + T$$
$13$ $$23 + T$$
$17$ $$263 + T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$1153 + T$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T$$
$43$ $$T$$
$47$ $$3457 + T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T$$
$67$ $$T$$
$71$ $$-10078 + T$$
$73$ $$-9502 + T$$
$79$ $$12167 + T$$
$83$ $$6382 + T$$
$89$ $$T$$
$97$ $$3383 + T$$