# Properties

 Label 160.3.b.b 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{1} q^{7} + ( 3 + 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + \beta_{2} q^{5} + \beta_{1} q^{7} + ( 3 + 2 \beta_{2} ) q^{9} + ( 5 \beta_{1} + \beta_{3} ) q^{11} + ( -8 + 6 \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{3} ) q^{15} + ( -2 + 4 \beta_{2} ) q^{17} + ( 3 \beta_{1} + 5 \beta_{3} ) q^{19} + ( -6 + 2 \beta_{2} ) q^{21} + ( -7 \beta_{1} - 2 \beta_{3} ) q^{23} + 5 q^{25} + ( 8 \beta_{1} + 2 \beta_{3} ) q^{27} + ( 30 - 4 \beta_{2} ) q^{29} + ( 6 \beta_{1} - 8 \beta_{3} ) q^{31} + ( -32 + 8 \beta_{2} ) q^{33} + ( -2 \beta_{1} + \beta_{3} ) q^{35} + ( 12 - 14 \beta_{2} ) q^{37} + ( -20 \beta_{1} + 6 \beta_{3} ) q^{39} -26 \beta_{2} q^{41} + ( 17 \beta_{1} - 16 \beta_{3} ) q^{43} + ( 10 + 3 \beta_{2} ) q^{45} + ( -9 \beta_{1} - 10 \beta_{3} ) q^{47} + ( 43 + 2 \beta_{2} ) q^{49} + ( -10 \beta_{1} + 4 \beta_{3} ) q^{51} + ( -40 - 26 \beta_{2} ) q^{53} + ( -9 \beta_{1} + 7 \beta_{3} ) q^{55} + ( -28 - 4 \beta_{2} ) q^{57} + ( -31 \beta_{1} + 11 \beta_{3} ) q^{59} + ( -48 + 22 \beta_{2} ) q^{61} + ( -\beta_{1} + 2 \beta_{3} ) q^{63} + ( 30 - 8 \beta_{2} ) q^{65} + ( -21 \beta_{1} + 4 \beta_{3} ) q^{67} + ( 46 - 10 \beta_{2} ) q^{69} + ( 8 \beta_{1} - 18 \beta_{3} ) q^{71} + ( -26 - 20 \beta_{2} ) q^{73} + 5 \beta_{1} q^{75} + ( -32 + 8 \beta_{2} ) q^{77} + ( 16 \beta_{1} + 20 \beta_{3} ) q^{79} + ( -25 + 30 \beta_{2} ) q^{81} + ( 31 \beta_{1} - 38 \beta_{3} ) q^{83} + ( 20 - 2 \beta_{2} ) q^{85} + ( 38 \beta_{1} - 4 \beta_{3} ) q^{87} + ( 82 - 16 \beta_{2} ) q^{89} + ( -20 \beta_{1} + 6 \beta_{3} ) q^{91} + ( -20 + 28 \beta_{2} ) q^{93} + ( -\beta_{1} + 13 \beta_{3} ) q^{95} + ( 26 + 48 \beta_{2} ) q^{97} + ( -3 \beta_{1} + 17 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 12q^{9} + O(q^{10})$$ $$4q + 12q^{9} - 32q^{13} - 8q^{17} - 24q^{21} + 20q^{25} + 120q^{29} - 128q^{33} + 48q^{37} + 40q^{45} + 172q^{49} - 160q^{53} - 112q^{57} - 192q^{61} + 120q^{65} + 184q^{69} - 104q^{73} - 128q^{77} - 100q^{81} + 80q^{85} + 328q^{89} - 80q^{93} + 104q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} + 3$$ $$\beta_{3}$$ $$=$$ $$4 \nu^{3} + 10 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{2} - 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{3} - 5 \beta_{1}$$$$)/4$$

## 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 3.23607i 0 −2.23607 0 3.23607i 0 −1.47214 0
31.2 0 1.23607i 0 2.23607 0 1.23607i 0 7.47214 0
31.3 0 1.23607i 0 2.23607 0 1.23607i 0 7.47214 0
31.4 0 3.23607i 0 −2.23607 0 3.23607i 0 −1.47214 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.b 4
3.b odd 2 1 1440.3.e.a 4
4.b odd 2 1 inner 160.3.b.b 4
5.b even 2 1 800.3.b.g 4
5.c odd 4 1 800.3.h.e 4
5.c odd 4 1 800.3.h.h 4
8.b even 2 1 320.3.b.d 4
8.d odd 2 1 320.3.b.d 4
12.b even 2 1 1440.3.e.a 4
16.e even 4 1 1280.3.g.b 4
16.e even 4 1 1280.3.g.c 4
16.f odd 4 1 1280.3.g.b 4
16.f odd 4 1 1280.3.g.c 4
20.d odd 2 1 800.3.b.g 4
20.e even 4 1 800.3.h.e 4
20.e even 4 1 800.3.h.h 4
24.f even 2 1 2880.3.e.h 4
24.h odd 2 1 2880.3.e.h 4
40.e odd 2 1 1600.3.b.u 4
40.f even 2 1 1600.3.b.u 4
40.i odd 4 1 1600.3.h.f 4
40.i odd 4 1 1600.3.h.k 4
40.k even 4 1 1600.3.h.f 4
40.k even 4 1 1600.3.h.k 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 12 T_{3}^{2} + 16$$ acting on $$S_{3}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$16 + 12 T^{2} + T^{4}$$
$5$ $$( -5 + T^{2} )^{2}$$
$7$ $$16 + 12 T^{2} + T^{4}$$
$11$ $$30976 + 368 T^{2} + T^{4}$$
$13$ $$( -116 + 16 T + T^{2} )^{2}$$
$17$ $$( -76 + 4 T + T^{2} )^{2}$$
$19$ $$30976 + 928 T^{2} + T^{4}$$
$23$ $$163216 + 812 T^{2} + T^{4}$$
$29$ $$( 820 - 60 T + T^{2} )^{2}$$
$31$ $$774400 + 1840 T^{2} + T^{4}$$
$37$ $$( -836 - 24 T + T^{2} )^{2}$$
$41$ $$( -3380 + T^{2} )^{2}$$
$43$ $$17808400 + 8460 T^{2} + T^{4}$$
$47$ $$1860496 + 4492 T^{2} + T^{4}$$
$53$ $$( -1780 + 80 T + T^{2} )^{2}$$
$59$ $$4393216 + 12192 T^{2} + T^{4}$$
$61$ $$( -116 + 96 T + T^{2} )^{2}$$
$67$ $$126736 + 5068 T^{2} + T^{4}$$
$71$ $$11182336 + 8688 T^{2} + T^{4}$$
$73$ $$( -1324 + 52 T + T^{2} )^{2}$$
$79$ $$20647936 + 16832 T^{2} + T^{4}$$
$83$ $$431808400 + 42540 T^{2} + T^{4}$$
$89$ $$( 5444 - 164 T + T^{2} )^{2}$$
$97$ $$( -10844 - 52 T + T^{2} )^{2}$$