# Properties

 Label 160.4.c.a Level $160$ Weight $4$ Character orbit 160.c Analytic conductor $9.440$ Analytic rank $0$ Dimension $2$ CM discriminant -4 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [160,4,Mod(129,160)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("160.129");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$9.44030560092$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field 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

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 11) q^{5} + 27 q^{9}+O(q^{10})$$ q + (b - 11) * q^5 + 27 * q^9 $$q + (\beta - 11) q^{5} + 27 q^{9} - 46 \beta q^{13} - 52 \beta q^{17} + ( - 22 \beta + 117) q^{25} - 130 q^{29} - 198 \beta q^{37} + 230 q^{41} + (27 \beta - 297) q^{45} + 343 q^{49} + 286 \beta q^{53} - 830 q^{61} + (506 \beta + 184) q^{65} + 296 \beta q^{73} + 729 q^{81} + (572 \beta + 208) q^{85} - 1670 q^{89} - 908 \beta q^{97} +O(q^{100})$$ q + (b - 11) * q^5 + 27 * q^9 - 46*b * q^13 - 52*b * q^17 + (-22*b + 117) * q^25 - 130 * q^29 - 198*b * q^37 + 230 * q^41 + (27*b - 297) * q^45 + 343 * q^49 + 286*b * q^53 - 830 * q^61 + (506*b + 184) * q^65 + 296*b * q^73 + 729 * q^81 + (572*b + 208) * q^85 - 1670 * q^89 - 908*b * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 22 q^{5} + 54 q^{9}+O(q^{10})$$ 2 * q - 22 * q^5 + 54 * q^9 $$2 q - 22 q^{5} + 54 q^{9} + 234 q^{25} - 260 q^{29} + 460 q^{41} - 594 q^{45} + 686 q^{49} - 1660 q^{61} + 368 q^{65} + 1458 q^{81} + 416 q^{85} - 3340 q^{89}+O(q^{100})$$ 2 * q - 22 * q^5 + 54 * q^9 + 234 * q^25 - 260 * q^29 + 460 * q^41 - 594 * q^45 + 686 * q^49 - 1660 * q^61 + 368 * q^65 + 1458 * q^81 + 416 * q^85 - 3340 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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 −11.0000 2.00000i 0 0 0 27.0000 0
129.2 0 0 0 −11.0000 + 2.00000i 0 0 0 27.0000 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 160.4.c.a 2
3.b odd 2 1 1440.4.f.e 2
4.b odd 2 1 CM 160.4.c.a 2
5.b even 2 1 inner 160.4.c.a 2
5.c odd 4 1 800.4.a.e 1
5.c odd 4 1 800.4.a.g 1
8.b even 2 1 320.4.c.e 2
8.d odd 2 1 320.4.c.e 2
12.b even 2 1 1440.4.f.e 2
15.d odd 2 1 1440.4.f.e 2
20.d odd 2 1 inner 160.4.c.a 2
20.e even 4 1 800.4.a.e 1
20.e even 4 1 800.4.a.g 1
40.e odd 2 1 320.4.c.e 2
40.f even 2 1 320.4.c.e 2
40.i odd 4 1 1600.4.a.z 1
40.i odd 4 1 1600.4.a.bb 1
40.k even 4 1 1600.4.a.z 1
40.k even 4 1 1600.4.a.bb 1
60.h even 2 1 1440.4.f.e 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 22T + 125$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} + 8464$$
$17$ $$T^{2} + 10816$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$(T + 130)^{2}$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 156816$$
$41$ $$(T - 230)^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2}$$
$53$ $$T^{2} + 327184$$
$59$ $$T^{2}$$
$61$ $$(T + 830)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2} + 350464$$
$79$ $$T^{2}$$
$83$ $$T^{2}$$
$89$ $$(T + 1670)^{2}$$
$97$ $$T^{2} + 3297856$$