# Properties

 Label 160.4.a.a Level $160$ Weight $4$ Character orbit 160.a Self dual yes Analytic conductor $9.440$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.44030560092$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2q^{3} - 5q^{5} + 6q^{7} - 23q^{9} + O(q^{10})$$ $$q - 2q^{3} - 5q^{5} + 6q^{7} - 23q^{9} + 60q^{11} + 50q^{13} + 10q^{15} - 30q^{17} + 40q^{19} - 12q^{21} + 178q^{23} + 25q^{25} + 100q^{27} + 166q^{29} + 20q^{31} - 120q^{33} - 30q^{35} + 10q^{37} - 100q^{39} - 250q^{41} + 142q^{43} + 115q^{45} + 214q^{47} - 307q^{49} + 60q^{51} + 490q^{53} - 300q^{55} - 80q^{57} - 800q^{59} + 250q^{61} - 138q^{63} - 250q^{65} - 774q^{67} - 356q^{69} + 100q^{71} - 230q^{73} - 50q^{75} + 360q^{77} - 1320q^{79} + 421q^{81} + 982q^{83} + 150q^{85} - 332q^{87} + 874q^{89} + 300q^{91} - 40q^{93} - 200q^{95} - 310q^{97} - 1380q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −2.00000 0 −5.00000 0 6.00000 0 −23.0000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.a.a 1
3.b odd 2 1 1440.4.a.o 1
4.b odd 2 1 160.4.a.b yes 1
5.b even 2 1 800.4.a.h 1
5.c odd 4 2 800.4.c.f 2
8.b even 2 1 320.4.a.i 1
8.d odd 2 1 320.4.a.f 1
12.b even 2 1 1440.4.a.n 1
16.e even 4 2 1280.4.d.f 2
16.f odd 4 2 1280.4.d.k 2
20.d odd 2 1 800.4.a.d 1
20.e even 4 2 800.4.c.e 2
40.e odd 2 1 1600.4.a.bj 1
40.f even 2 1 1600.4.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 1.a even 1 1 trivial
160.4.a.b yes 1 4.b odd 2 1
320.4.a.f 1 8.d odd 2 1
320.4.a.i 1 8.b even 2 1
800.4.a.d 1 20.d odd 2 1
800.4.a.h 1 5.b even 2 1
800.4.c.e 2 20.e even 4 2
800.4.c.f 2 5.c odd 4 2
1280.4.d.f 2 16.e even 4 2
1280.4.d.k 2 16.f odd 4 2
1440.4.a.n 1 12.b even 2 1
1440.4.a.o 1 3.b odd 2 1
1600.4.a.r 1 40.f even 2 1
1600.4.a.bj 1 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(160))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$5 + T$$
$7$ $$-6 + T$$
$11$ $$-60 + T$$
$13$ $$-50 + T$$
$17$ $$30 + T$$
$19$ $$-40 + T$$
$23$ $$-178 + T$$
$29$ $$-166 + T$$
$31$ $$-20 + T$$
$37$ $$-10 + T$$
$41$ $$250 + T$$
$43$ $$-142 + T$$
$47$ $$-214 + T$$
$53$ $$-490 + T$$
$59$ $$800 + T$$
$61$ $$-250 + T$$
$67$ $$774 + T$$
$71$ $$-100 + T$$
$73$ $$230 + T$$
$79$ $$1320 + T$$
$83$ $$-982 + T$$
$89$ $$-874 + T$$
$97$ $$310 + T$$