# Properties

 Label 160.2.c.a Level $160$ Weight $2$ Character orbit 160.c Analytic conductor $1.278$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 160.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.27760643234$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{5} + 3 q^{9}+O(q^{10})$$ q + (b + 1) * q^5 + 3 * q^9 $$q + (\beta + 1) q^{5} + 3 q^{9} + 2 \beta q^{13} - 4 \beta q^{17} + (2 \beta - 3) q^{25} - 10 q^{29} - 6 \beta q^{37} - 10 q^{41} + (3 \beta + 3) q^{45} + 7 q^{49} - 2 \beta q^{53} + 10 q^{61} + (2 \beta - 8) q^{65} + 8 \beta q^{73} + 9 q^{81} + ( - 4 \beta + 16) q^{85} + 10 q^{89} + 4 \beta q^{97} +O(q^{100})$$ q + (b + 1) * q^5 + 3 * q^9 + 2*b * q^13 - 4*b * q^17 + (2*b - 3) * q^25 - 10 * q^29 - 6*b * q^37 - 10 * q^41 + (3*b + 3) * q^45 + 7 * q^49 - 2*b * q^53 + 10 * q^61 + (2*b - 8) * q^65 + 8*b * q^73 + 9 * q^81 + (-4*b + 16) * q^85 + 10 * q^89 + 4*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^5 + 6 * q^9 $$2 q + 2 q^{5} + 6 q^{9} - 6 q^{25} - 20 q^{29} - 20 q^{41} + 6 q^{45} + 14 q^{49} + 20 q^{61} - 16 q^{65} + 18 q^{81} + 32 q^{85} + 20 q^{89}+O(q^{100})$$ 2 * q + 2 * q^5 + 6 * q^9 - 6 * q^25 - 20 * q^29 - 20 * q^41 + 6 * q^45 + 14 * q^49 + 20 * q^61 - 16 * q^65 + 18 * q^81 + 32 * q^85 + 20 * q^89

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/160\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$97$$ $$101$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 1.00000i 1.00000i
0 0 0 1.00000 2.00000i 0 0 0 3.00000 0
129.2 0 0 0 1.00000 + 2.00000i 0 0 0 3.00000 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 inner
20.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.2.c.a 2
3.b odd 2 1 1440.2.f.c 2
4.b odd 2 1 CM 160.2.c.a 2
5.b even 2 1 inner 160.2.c.a 2
5.c odd 4 1 800.2.a.e 1
5.c odd 4 1 800.2.a.f 1
8.b even 2 1 320.2.c.a 2
8.d odd 2 1 320.2.c.a 2
12.b even 2 1 1440.2.f.c 2
15.d odd 2 1 1440.2.f.c 2
15.e even 4 1 7200.2.a.y 1
15.e even 4 1 7200.2.a.bb 1
16.e even 4 1 1280.2.f.c 2
16.e even 4 1 1280.2.f.d 2
16.f odd 4 1 1280.2.f.c 2
16.f odd 4 1 1280.2.f.d 2
20.d odd 2 1 inner 160.2.c.a 2
20.e even 4 1 800.2.a.e 1
20.e even 4 1 800.2.a.f 1
24.f even 2 1 2880.2.f.n 2
24.h odd 2 1 2880.2.f.n 2
40.e odd 2 1 320.2.c.a 2
40.f even 2 1 320.2.c.a 2
40.i odd 4 1 1600.2.a.l 1
40.i odd 4 1 1600.2.a.m 1
40.k even 4 1 1600.2.a.l 1
40.k even 4 1 1600.2.a.m 1
60.h even 2 1 1440.2.f.c 2
60.l odd 4 1 7200.2.a.y 1
60.l odd 4 1 7200.2.a.bb 1
80.k odd 4 1 1280.2.f.c 2
80.k odd 4 1 1280.2.f.d 2
80.q even 4 1 1280.2.f.c 2
80.q even 4 1 1280.2.f.d 2
120.i odd 2 1 2880.2.f.n 2
120.m even 2 1 2880.2.f.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.2.c.a 2 1.a even 1 1 trivial
160.2.c.a 2 4.b odd 2 1 CM
160.2.c.a 2 5.b even 2 1 inner
160.2.c.a 2 20.d odd 2 1 inner
320.2.c.a 2 8.b even 2 1
320.2.c.a 2 8.d odd 2 1
320.2.c.a 2 40.e odd 2 1
320.2.c.a 2 40.f even 2 1
800.2.a.e 1 5.c odd 4 1
800.2.a.e 1 20.e even 4 1
800.2.a.f 1 5.c odd 4 1
800.2.a.f 1 20.e even 4 1
1280.2.f.c 2 16.e even 4 1
1280.2.f.c 2 16.f odd 4 1
1280.2.f.c 2 80.k odd 4 1
1280.2.f.c 2 80.q even 4 1
1280.2.f.d 2 16.e even 4 1
1280.2.f.d 2 16.f odd 4 1
1280.2.f.d 2 80.k odd 4 1
1280.2.f.d 2 80.q even 4 1
1440.2.f.c 2 3.b odd 2 1
1440.2.f.c 2 12.b even 2 1
1440.2.f.c 2 15.d odd 2 1
1440.2.f.c 2 60.h even 2 1
1600.2.a.l 1 40.i odd 4 1
1600.2.a.l 1 40.k even 4 1
1600.2.a.m 1 40.i odd 4 1
1600.2.a.m 1 40.k even 4 1
2880.2.f.n 2 24.f even 2 1
2880.2.f.n 2 24.h odd 2 1
2880.2.f.n 2 120.i odd 2 1
2880.2.f.n 2 120.m even 2 1
7200.2.a.y 1 15.e even 4 1
7200.2.a.y 1 60.l odd 4 1
7200.2.a.bb 1 15.e even 4 1
7200.2.a.bb 1 60.l odd 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 2T + 5$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 64$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 10)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 144$$
$41$ $$(T + 10)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 16$$
$59$ $$T^{2}$$
$61$ $$(T - 10)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 256$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 64$$