# Properties

 Label 160.3.b.a Level $160$ Weight $3$ Character orbit 160.b Analytic conductor $4.360$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 160.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35968422976$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( -\beta_{1} + 4 \beta_{2} ) q^{7} + ( -5 + 6 \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} -\beta_{3} q^{5} + ( -\beta_{1} + 4 \beta_{2} ) q^{7} + ( -5 + 6 \beta_{3} ) q^{9} + ( 3 \beta_{1} + \beta_{2} ) q^{11} + ( 16 + 2 \beta_{3} ) q^{13} + ( -2 \beta_{1} + \beta_{2} ) q^{15} + ( -10 + 4 \beta_{3} ) q^{17} + ( -3 \beta_{1} - 3 \beta_{2} ) q^{19} + ( -22 + 14 \beta_{3} ) q^{21} + ( -\beta_{1} + 10 \beta_{2} ) q^{23} + 5 q^{25} + ( 8 \beta_{1} - 6 \beta_{2} ) q^{27} + ( -18 - 12 \beta_{3} ) q^{29} + ( 2 \beta_{1} + 8 \beta_{2} ) q^{31} + ( 40 - 16 \beta_{3} ) q^{33} + ( -6 \beta_{1} - 7 \beta_{2} ) q^{35} + ( 28 - 2 \beta_{3} ) q^{37} + ( -12 \beta_{1} - 2 \beta_{2} ) q^{39} + ( 40 - 14 \beta_{3} ) q^{41} + ( -\beta_{1} - 8 \beta_{2} ) q^{43} + ( -30 + 5 \beta_{3} ) q^{45} + ( \beta_{1} - 14 \beta_{2} ) q^{47} + ( -77 - 10 \beta_{3} ) q^{49} + ( 18 \beta_{1} - 4 \beta_{2} ) q^{51} + ( 32 - 14 \beta_{3} ) q^{53} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{55} + ( -36 + 12 \beta_{3} ) q^{57} + ( -\beta_{1} - 29 \beta_{2} ) q^{59} + ( 16 - 22 \beta_{3} ) q^{61} + ( 41 \beta_{1} + 22 \beta_{2} ) q^{63} + ( -10 - 16 \beta_{3} ) q^{65} + ( -11 \beta_{1} + 28 \beta_{2} ) q^{67} + ( -34 + 26 \beta_{3} ) q^{69} + ( -16 \beta_{1} - 34 \beta_{2} ) q^{71} + ( -2 + 60 \beta_{3} ) q^{73} -5 \beta_{1} q^{75} + ( 40 - 48 \beta_{3} ) q^{77} + ( -24 \beta_{1} - 36 \beta_{2} ) q^{79} + ( 79 - 6 \beta_{3} ) q^{81} + ( 17 \beta_{1} + 2 \beta_{2} ) q^{83} + ( -20 + 10 \beta_{3} ) q^{85} + ( -6 \beta_{1} + 12 \beta_{2} ) q^{87} -30 q^{89} + ( -4 \beta_{1} + 78 \beta_{2} ) q^{91} + ( 12 + 4 \beta_{3} ) q^{93} + ( -3 \beta_{1} + 9 \beta_{2} ) q^{95} + ( 66 - 24 \beta_{3} ) q^{97} + ( -45 \beta_{1} + 25 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 20q^{9} + O(q^{10})$$ $$4q - 20q^{9} + 64q^{13} - 40q^{17} - 88q^{21} + 20q^{25} - 72q^{29} + 160q^{33} + 112q^{37} + 160q^{41} - 120q^{45} - 308q^{49} + 128q^{53} - 144q^{57} + 64q^{61} - 40q^{65} - 136q^{69} - 8q^{73} + 160q^{77} + 316q^{81} - 80q^{85} - 120q^{89} + 48q^{93} + 264q^{97} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu^{3} + 2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 6 \nu$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{2} + 3 \beta_{1}$$$$)/4$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
0 5.23607i 0 2.23607 0 10.1803i 0 −18.4164 0
31.2 0 0.763932i 0 −2.23607 0 12.1803i 0 8.41641 0
31.3 0 0.763932i 0 −2.23607 0 12.1803i 0 8.41641 0
31.4 0 5.23607i 0 2.23607 0 10.1803i 0 −18.4164 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.3.b.a 4
3.b odd 2 1 1440.3.e.b 4
4.b odd 2 1 inner 160.3.b.a 4
5.b even 2 1 800.3.b.d 4
5.c odd 4 1 800.3.h.c 4
5.c odd 4 1 800.3.h.j 4
8.b even 2 1 320.3.b.b 4
8.d odd 2 1 320.3.b.b 4
12.b even 2 1 1440.3.e.b 4
16.e even 4 1 1280.3.g.a 4
16.e even 4 1 1280.3.g.d 4
16.f odd 4 1 1280.3.g.a 4
16.f odd 4 1 1280.3.g.d 4
20.d odd 2 1 800.3.b.d 4
20.e even 4 1 800.3.h.c 4
20.e even 4 1 800.3.h.j 4
24.f even 2 1 2880.3.e.a 4
24.h odd 2 1 2880.3.e.a 4
40.e odd 2 1 1600.3.b.n 4
40.f even 2 1 1600.3.b.n 4
40.i odd 4 1 1600.3.h.d 4
40.i odd 4 1 1600.3.h.m 4
40.k even 4 1 1600.3.h.d 4
40.k even 4 1 1600.3.h.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.3.b.a 4 1.a even 1 1 trivial
160.3.b.a 4 4.b odd 2 1 inner
320.3.b.b 4 8.b even 2 1
320.3.b.b 4 8.d odd 2 1
800.3.b.d 4 5.b even 2 1
800.3.b.d 4 20.d odd 2 1
800.3.h.c 4 5.c odd 4 1
800.3.h.c 4 20.e even 4 1
800.3.h.j 4 5.c odd 4 1
800.3.h.j 4 20.e even 4 1
1280.3.g.a 4 16.e even 4 1
1280.3.g.a 4 16.f odd 4 1
1280.3.g.d 4 16.e even 4 1
1280.3.g.d 4 16.f odd 4 1
1440.3.e.b 4 3.b odd 2 1
1440.3.e.b 4 12.b even 2 1
1600.3.b.n 4 40.e odd 2 1
1600.3.b.n 4 40.f even 2 1
1600.3.h.d 4 40.i odd 4 1
1600.3.h.d 4 40.k even 4 1
1600.3.h.m 4 40.i odd 4 1
1600.3.h.m 4 40.k even 4 1
2880.3.e.a 4 24.f even 2 1
2880.3.e.a 4 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 28 T_{3}^{2} + 16$$ acting on $$S_{3}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 28 T^{2} + T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$15376 + 252 T^{2} + T^{4}$$
$11$ $$6400 + 240 T^{2} + T^{4}$$
$13$ $$( 236 - 32 T + T^{2} )^{2}$$
$17$ $$( 20 + 20 T + T^{2} )^{2}$$
$19$ $$( 144 + T^{2} )^{2}$$
$23$ $$309136 + 1308 T^{2} + T^{4}$$
$29$ $$( -396 + 36 T + T^{2} )^{2}$$
$31$ $$256 + 752 T^{2} + T^{4}$$
$37$ $$( 764 - 56 T + T^{2} )^{2}$$
$41$ $$( 620 - 80 T + T^{2} )^{2}$$
$43$ $$15376 + 732 T^{2} + T^{4}$$
$47$ $$1008016 + 2492 T^{2} + T^{4}$$
$53$ $$( 44 - 64 T + T^{2} )^{2}$$
$59$ $$8386816 + 9888 T^{2} + T^{4}$$
$61$ $$( -2164 - 32 T + T^{2} )^{2}$$
$67$ $$57456400 + 15260 T^{2} + T^{4}$$
$71$ $$26050816 + 16688 T^{2} + T^{4}$$
$73$ $$( -17996 + 4 T + T^{2} )^{2}$$
$79$ $$119771136 + 24768 T^{2} + T^{4}$$
$83$ $$2835856 + 7868 T^{2} + T^{4}$$
$89$ $$( 30 + T )^{4}$$
$97$ $$( 1476 - 132 T + T^{2} )^{2}$$