# Properties

 Label 160.2.f.a Level $160$ Weight $2$ Character orbit 160.f Analytic conductor $1.278$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 160.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.27760643234$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 40) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} - q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( -\beta_{1} - \beta_{2} ) q^{5} + \beta_{3} q^{7} - q^{9} -2 \beta_{2} q^{11} + ( 2 - \beta_{3} ) q^{15} + 2 \beta_{3} q^{17} + 2 \beta_{2} q^{19} + 2 \beta_{2} q^{21} -\beta_{3} q^{23} + ( -1 - 2 \beta_{3} ) q^{25} + 4 \beta_{1} q^{27} -4 q^{31} -2 \beta_{3} q^{33} + ( -3 \beta_{1} + 2 \beta_{2} ) q^{35} + 6 \beta_{1} q^{37} + 3 \beta_{1} q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + 3 \beta_{3} q^{47} + q^{49} + 4 \beta_{2} q^{51} -4 \beta_{1} q^{53} + ( -6 - 2 \beta_{3} ) q^{55} + 2 \beta_{3} q^{57} -6 \beta_{2} q^{59} -2 \beta_{2} q^{61} -\beta_{3} q^{63} -3 \beta_{1} q^{67} -2 \beta_{2} q^{69} + 12 q^{71} + 2 \beta_{3} q^{73} + ( \beta_{1} - 4 \beta_{2} ) q^{75} -6 \beta_{1} q^{77} + 4 q^{79} -5 q^{81} -7 \beta_{1} q^{83} + ( -6 \beta_{1} + 4 \beta_{2} ) q^{85} + 6 q^{89} + 4 \beta_{1} q^{93} + ( 6 + 2 \beta_{3} ) q^{95} -2 \beta_{3} q^{97} + 2 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} + 8q^{15} - 4q^{25} - 16q^{31} + 4q^{49} - 24q^{55} + 48q^{71} + 16q^{79} - 20q^{81} + 24q^{89} + 24q^{95} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/2$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 4 \nu$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.707107 − 1.22474i −0.707107 + 1.22474i 0.707107 + 1.22474i 0.707107 − 1.22474i
0 −1.41421 0 −1.41421 1.73205i 0 2.44949i 0 −1.00000 0
49.2 0 −1.41421 0 −1.41421 + 1.73205i 0 2.44949i 0 −1.00000 0
49.3 0 1.41421 0 1.41421 1.73205i 0 2.44949i 0 −1.00000 0
49.4 0 1.41421 0 1.41421 + 1.73205i 0 2.44949i 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
5.b even 2 1 inner
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.2.f.a 4
3.b odd 2 1 1440.2.d.c 4
4.b odd 2 1 40.2.f.a 4
5.b even 2 1 inner 160.2.f.a 4
5.c odd 4 2 800.2.d.f 4
8.b even 2 1 inner 160.2.f.a 4
8.d odd 2 1 40.2.f.a 4
12.b even 2 1 360.2.d.b 4
15.d odd 2 1 1440.2.d.c 4
15.e even 4 2 7200.2.k.l 4
16.e even 4 2 1280.2.c.k 4
16.f odd 4 2 1280.2.c.i 4
20.d odd 2 1 40.2.f.a 4
20.e even 4 2 200.2.d.e 4
24.f even 2 1 360.2.d.b 4
24.h odd 2 1 1440.2.d.c 4
40.e odd 2 1 40.2.f.a 4
40.f even 2 1 inner 160.2.f.a 4
40.i odd 4 2 800.2.d.f 4
40.k even 4 2 200.2.d.e 4
60.h even 2 1 360.2.d.b 4
60.l odd 4 2 1800.2.k.m 4
80.i odd 4 2 6400.2.a.cm 4
80.j even 4 2 6400.2.a.co 4
80.k odd 4 2 1280.2.c.i 4
80.q even 4 2 1280.2.c.k 4
80.s even 4 2 6400.2.a.co 4
80.t odd 4 2 6400.2.a.cm 4
120.i odd 2 1 1440.2.d.c 4
120.m even 2 1 360.2.d.b 4
120.q odd 4 2 1800.2.k.m 4
120.w even 4 2 7200.2.k.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.2.f.a 4 4.b odd 2 1
40.2.f.a 4 8.d odd 2 1
40.2.f.a 4 20.d odd 2 1
40.2.f.a 4 40.e odd 2 1
160.2.f.a 4 1.a even 1 1 trivial
160.2.f.a 4 5.b even 2 1 inner
160.2.f.a 4 8.b even 2 1 inner
160.2.f.a 4 40.f even 2 1 inner
200.2.d.e 4 20.e even 4 2
200.2.d.e 4 40.k even 4 2
360.2.d.b 4 12.b even 2 1
360.2.d.b 4 24.f even 2 1
360.2.d.b 4 60.h even 2 1
360.2.d.b 4 120.m even 2 1
800.2.d.f 4 5.c odd 4 2
800.2.d.f 4 40.i odd 4 2
1280.2.c.i 4 16.f odd 4 2
1280.2.c.i 4 80.k odd 4 2
1280.2.c.k 4 16.e even 4 2
1280.2.c.k 4 80.q even 4 2
1440.2.d.c 4 3.b odd 2 1
1440.2.d.c 4 15.d odd 2 1
1440.2.d.c 4 24.h odd 2 1
1440.2.d.c 4 120.i odd 2 1
1800.2.k.m 4 60.l odd 4 2
1800.2.k.m 4 120.q odd 4 2
6400.2.a.cm 4 80.i odd 4 2
6400.2.a.cm 4 80.t odd 4 2
6400.2.a.co 4 80.j even 4 2
6400.2.a.co 4 80.s even 4 2
7200.2.k.l 4 15.e even 4 2
7200.2.k.l 4 120.w even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$25 + 2 T^{2} + T^{4}$$
$7$ $$( 6 + T^{2} )^{2}$$
$11$ $$( 12 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 24 + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$( 6 + T^{2} )^{2}$$
$29$ $$T^{4}$$
$31$ $$( 4 + T )^{4}$$
$37$ $$( -72 + T^{2} )^{2}$$
$41$ $$T^{4}$$
$43$ $$( -18 + T^{2} )^{2}$$
$47$ $$( 54 + T^{2} )^{2}$$
$53$ $$( -32 + T^{2} )^{2}$$
$59$ $$( 108 + T^{2} )^{2}$$
$61$ $$( 12 + T^{2} )^{2}$$
$67$ $$( -18 + T^{2} )^{2}$$
$71$ $$( -12 + T )^{4}$$
$73$ $$( 24 + T^{2} )^{2}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( -98 + T^{2} )^{2}$$
$89$ $$( -6 + T )^{4}$$
$97$ $$( 24 + T^{2} )^{2}$$