# Properties

 Label 160.4.c.c Level $160$ Weight $4$ Character orbit 160.c Analytic conductor $9.440$ Analytic rank $0$ Dimension $4$ CM discriminant -20 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$160 = 2^{5} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 160.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.44030560092$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{1} + 2 \beta_{3} ) q^{3} -5 \beta_{2} q^{5} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{7} + ( -27 - 14 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -\beta_{1} + 2 \beta_{3} ) q^{3} -5 \beta_{2} q^{5} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{7} + ( -27 - 14 \beta_{2} ) q^{9} + ( -20 \beta_{1} - 15 \beta_{3} ) q^{15} + ( 118 + 86 \beta_{2} ) q^{21} + ( -64 \beta_{1} + \beta_{3} ) q^{23} + 125 q^{25} + ( -56 \beta_{1} - 42 \beta_{3} ) q^{27} -306 q^{29} + ( -35 \beta_{1} + 80 \beta_{3} ) q^{35} -206 \beta_{2} q^{41} + ( -15 \beta_{1} - 104 \beta_{3} ) q^{43} + ( 350 + 135 \beta_{2} ) q^{45} + ( 190 \beta_{1} - 37 \beta_{3} ) q^{47} + ( -343 - 198 \beta_{2} ) q^{49} -18 \beta_{2} q^{61} + ( 64 \beta_{1} + 359 \beta_{3} ) q^{63} + ( 321 \beta_{1} - 76 \beta_{3} ) q^{67} + ( -154 + 374 \beta_{2} ) q^{69} + ( -125 \beta_{1} + 250 \beta_{3} ) q^{75} + ( 251 + 378 \beta_{2} ) q^{81} + ( 377 \beta_{1} + 100 \beta_{3} ) q^{83} + ( 306 \beta_{1} - 612 \beta_{3} ) q^{87} + 1386 q^{89} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 108q^{9} + O(q^{10})$$ $$4q - 108q^{9} + 472q^{21} + 500q^{25} - 1224q^{29} + 1400q^{45} - 1372q^{49} - 616q^{69} + 1004q^{81} + 5544q^{89} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
129.1
 − 0.618034i 1.61803i − 1.61803i 0.618034i
0 9.23607i 0 −11.1803 0 33.5967i 0 −58.3050 0
129.2 0 4.76393i 0 11.1803 0 15.5967i 0 4.30495 0
129.3 0 4.76393i 0 11.1803 0 15.5967i 0 4.30495 0
129.4 0 9.23607i 0 −11.1803 0 33.5967i 0 −58.3050 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
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$
4.b odd 2 1 inner
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 160.4.c.c 4
3.b odd 2 1 1440.4.f.g 4
4.b odd 2 1 inner 160.4.c.c 4
5.b even 2 1 inner 160.4.c.c 4
5.c odd 4 1 800.4.a.l 2
5.c odd 4 1 800.4.a.t 2
8.b even 2 1 320.4.c.f 4
8.d odd 2 1 320.4.c.f 4
12.b even 2 1 1440.4.f.g 4
15.d odd 2 1 1440.4.f.g 4
20.d odd 2 1 CM 160.4.c.c 4
20.e even 4 1 800.4.a.l 2
20.e even 4 1 800.4.a.t 2
40.e odd 2 1 320.4.c.f 4
40.f even 2 1 320.4.c.f 4
40.i odd 4 1 1600.4.a.cb 2
40.i odd 4 1 1600.4.a.cp 2
40.k even 4 1 1600.4.a.cb 2
40.k even 4 1 1600.4.a.cp 2
60.h even 2 1 1440.4.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.c.c 4 1.a even 1 1 trivial
160.4.c.c 4 4.b odd 2 1 inner
160.4.c.c 4 5.b even 2 1 inner
160.4.c.c 4 20.d odd 2 1 CM
320.4.c.f 4 8.b even 2 1
320.4.c.f 4 8.d odd 2 1
320.4.c.f 4 40.e odd 2 1
320.4.c.f 4 40.f even 2 1
800.4.a.l 2 5.c odd 4 1
800.4.a.l 2 20.e even 4 1
800.4.a.t 2 5.c odd 4 1
800.4.a.t 2 20.e even 4 1
1440.4.f.g 4 3.b odd 2 1
1440.4.f.g 4 12.b even 2 1
1440.4.f.g 4 15.d odd 2 1
1440.4.f.g 4 60.h even 2 1
1600.4.a.cb 2 40.i odd 4 1
1600.4.a.cb 2 40.k even 4 1
1600.4.a.cp 2 40.i odd 4 1
1600.4.a.cp 2 40.k even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 108 T_{3}^{2} + 1936$$ acting on $$S_{4}^{\mathrm{new}}(160, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$1936 + 108 T^{2} + T^{4}$$
$5$ $$( -125 + T^{2} )^{2}$$
$7$ $$274576 + 1372 T^{2} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4}$$
$19$ $$T^{4}$$
$23$ $$235806736 + 48668 T^{2} + T^{4}$$
$29$ $$( 306 + T )^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$( -212180 + T^{2} )^{2}$$
$43$ $$302899216 + 318028 T^{2} + T^{4}$$
$47$ $$699285136 + 415292 T^{2} + T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -1620 + T^{2} )^{2}$$
$67$ $$1628176 + 1203052 T^{2} + T^{4}$$
$71$ $$T^{4}$$
$73$ $$T^{4}$$
$79$ $$T^{4}$$
$83$ $$1280781104656 + 2287148 T^{2} + T^{4}$$
$89$ $$( -1386 + T )^{4}$$
$97$ $$T^{4}$$