# Properties

 Label 320.1.h.a Level $320$ Weight $1$ Character orbit 320.h Self dual yes Analytic conductor $0.160$ Analytic rank $0$ Dimension $1$ Projective image $D_{2}$ CM/RM discs -4, -20, 5 Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [320,1,Mod(319,320)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(320, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("320.319");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$320 = 2^{6} \cdot 5$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 320.h (of order $$2$$, degree $$1$$, not 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: yes Analytic conductor: $$0.159700804043$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 80) Projective image: $$D_{2}$$ Projective field: Galois closure of $$\Q(i, \sqrt{5})$$ Artin image: $D_4$ Artin field: Galois closure of 4.0.1280.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{5} - q^{9}+O(q^{10})$$ q + q^5 - q^9 $$q + q^{5} - q^{9} + q^{25} - 2 q^{29} - 2 q^{41} - q^{45} - q^{49} + 2 q^{61} + q^{81} + 2 q^{89}+O(q^{100})$$ q + q^5 - q^9 + q^25 - 2 * q^29 - 2 * q^41 - q^45 - q^49 + 2 * q^61 + q^81 + 2 * q^89

## Expression as an eta quotient

$$f(z) = \dfrac{\eta(8z)^{3}\eta(40z)^{3}}{\eta(4z)\eta(16z)\eta(20z)\eta(80z)}=q\prod_{n=1}^\infty(1 - q^{4n})^{-1}(1 - q^{8n})^{3}(1 - q^{16n})^{-1}(1 - q^{20n})^{-1}(1 - q^{40n})^{3}(1 - q^{80n})^{-1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/320\mathbb{Z}\right)^\times$$.

 $$n$$ $$191$$ $$257$$ $$261$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
319.1
 0
0 0 0 1.00000 0 0 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
5.b even 2 1 RM by $$\Q(\sqrt{5})$$
20.d odd 2 1 CM by $$\Q(\sqrt{-5})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.1.h.a 1
3.b odd 2 1 2880.1.j.a 1
4.b odd 2 1 CM 320.1.h.a 1
5.b even 2 1 RM 320.1.h.a 1
5.c odd 4 2 1600.1.b.a 1
8.b even 2 1 80.1.h.a 1
8.d odd 2 1 80.1.h.a 1
12.b even 2 1 2880.1.j.a 1
15.d odd 2 1 2880.1.j.a 1
16.e even 4 2 1280.1.e.a 2
16.f odd 4 2 1280.1.e.a 2
20.d odd 2 1 CM 320.1.h.a 1
20.e even 4 2 1600.1.b.a 1
24.f even 2 1 720.1.j.a 1
24.h odd 2 1 720.1.j.a 1
40.e odd 2 1 80.1.h.a 1
40.f even 2 1 80.1.h.a 1
40.i odd 4 2 400.1.b.a 1
40.k even 4 2 400.1.b.a 1
56.e even 2 1 3920.1.j.a 1
56.h odd 2 1 3920.1.j.a 1
56.j odd 6 2 3920.1.bt.a 2
56.k odd 6 2 3920.1.bt.b 2
56.m even 6 2 3920.1.bt.a 2
56.p even 6 2 3920.1.bt.b 2
60.h even 2 1 2880.1.j.a 1
80.k odd 4 2 1280.1.e.a 2
80.q even 4 2 1280.1.e.a 2
120.i odd 2 1 720.1.j.a 1
120.m even 2 1 720.1.j.a 1
120.q odd 4 2 3600.1.e.a 1
120.w even 4 2 3600.1.e.a 1
280.c odd 2 1 3920.1.j.a 1
280.n even 2 1 3920.1.j.a 1
280.ba even 6 2 3920.1.bt.a 2
280.bf even 6 2 3920.1.bt.b 2
280.bi odd 6 2 3920.1.bt.b 2
280.bk odd 6 2 3920.1.bt.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.1.h.a 1 8.b even 2 1
80.1.h.a 1 8.d odd 2 1
80.1.h.a 1 40.e odd 2 1
80.1.h.a 1 40.f even 2 1
320.1.h.a 1 1.a even 1 1 trivial
320.1.h.a 1 4.b odd 2 1 CM
320.1.h.a 1 5.b even 2 1 RM
320.1.h.a 1 20.d odd 2 1 CM
400.1.b.a 1 40.i odd 4 2
400.1.b.a 1 40.k even 4 2
720.1.j.a 1 24.f even 2 1
720.1.j.a 1 24.h odd 2 1
720.1.j.a 1 120.i odd 2 1
720.1.j.a 1 120.m even 2 1
1280.1.e.a 2 16.e even 4 2
1280.1.e.a 2 16.f odd 4 2
1280.1.e.a 2 80.k odd 4 2
1280.1.e.a 2 80.q even 4 2
1600.1.b.a 1 5.c odd 4 2
1600.1.b.a 1 20.e even 4 2
2880.1.j.a 1 3.b odd 2 1
2880.1.j.a 1 12.b even 2 1
2880.1.j.a 1 15.d odd 2 1
2880.1.j.a 1 60.h even 2 1
3600.1.e.a 1 120.q odd 4 2
3600.1.e.a 1 120.w even 4 2
3920.1.j.a 1 56.e even 2 1
3920.1.j.a 1 56.h odd 2 1
3920.1.j.a 1 280.c odd 2 1
3920.1.j.a 1 280.n even 2 1
3920.1.bt.a 2 56.j odd 6 2
3920.1.bt.a 2 56.m even 6 2
3920.1.bt.a 2 280.ba even 6 2
3920.1.bt.a 2 280.bk odd 6 2
3920.1.bt.b 2 56.k odd 6 2
3920.1.bt.b 2 56.p even 6 2
3920.1.bt.b 2 280.bf even 6 2
3920.1.bt.b 2 280.bi odd 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(320, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T$$
$29$ $$T + 2$$
$31$ $$T$$
$37$ $$T$$
$41$ $$T + 2$$
$43$ $$T$$
$47$ $$T$$
$53$ $$T$$
$59$ $$T$$
$61$ $$T - 2$$
$67$ $$T$$
$71$ $$T$$
$73$ $$T$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T - 2$$
$97$ $$T$$