# Properties

 Label 1984.1.e.b Level $1984$ Weight $1$ Character orbit 1984.e Self dual yes Analytic conductor $0.990$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -31 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1984,1,Mod(1921,1984)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1984, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1984.1921");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1984 = 2^{6} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1984.e (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: $$0.990144985064$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 31) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.31.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.492032.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q + q^{5} + q^{7} + q^{9}+O(q^{10})$$ q + q^5 + q^7 + q^9 $$q + q^{5} + q^{7} + q^{9} - q^{19} - q^{31} + q^{35} - q^{41} + q^{45} - 2 q^{47} - q^{59} + q^{63} + 2 q^{67} + q^{71} + q^{81} - q^{95} - q^{97}+O(q^{100})$$ q + q^5 + q^7 + q^9 - q^19 - q^31 + q^35 - q^41 + q^45 - 2 * q^47 - q^59 + q^63 + 2 * q^67 + q^71 + q^81 - q^95 - q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1984\mathbb{Z}\right)^\times$$.

 $$n$$ $$63$$ $$65$$ $$1861$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1921.1
 0
0 0 0 1.00000 0 1.00000 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
31.b odd 2 1 CM by $$\Q(\sqrt{-31})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1984.1.e.b 1
4.b odd 2 1 1984.1.e.a 1
8.b even 2 1 496.1.e.a 1
8.d odd 2 1 31.1.b.a 1
24.f even 2 1 279.1.d.b 1
31.b odd 2 1 CM 1984.1.e.b 1
40.e odd 2 1 775.1.d.b 1
40.k even 4 2 775.1.c.a 2
56.e even 2 1 1519.1.c.a 1
56.k odd 6 2 1519.1.n.b 2
56.m even 6 2 1519.1.n.a 2
72.l even 6 2 2511.1.m.a 2
72.p odd 6 2 2511.1.m.e 2
88.g even 2 1 3751.1.d.b 1
88.k even 10 4 3751.1.t.a 4
88.l odd 10 4 3751.1.t.c 4
124.d even 2 1 1984.1.e.a 1
248.b even 2 1 31.1.b.a 1
248.g odd 2 1 496.1.e.a 1
248.m odd 6 2 961.1.e.a 2
248.q even 6 2 961.1.e.a 2
248.s odd 10 4 961.1.f.a 4
248.v even 10 4 961.1.f.a 4
248.bb even 30 8 961.1.h.a 8
248.be odd 30 8 961.1.h.a 8
744.m odd 2 1 279.1.d.b 1
1240.o even 2 1 775.1.d.b 1
1240.s odd 4 2 775.1.c.a 2
1736.n odd 2 1 1519.1.c.a 1
1736.bh odd 6 2 1519.1.n.a 2
1736.ct even 6 2 1519.1.n.b 2
2232.bp odd 6 2 2511.1.m.a 2
2232.cq even 6 2 2511.1.m.e 2
2728.e odd 2 1 3751.1.d.b 1
2728.eb odd 10 4 3751.1.t.a 4
2728.ef even 10 4 3751.1.t.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 8.d odd 2 1
31.1.b.a 1 248.b even 2 1
279.1.d.b 1 24.f even 2 1
279.1.d.b 1 744.m odd 2 1
496.1.e.a 1 8.b even 2 1
496.1.e.a 1 248.g odd 2 1
775.1.c.a 2 40.k even 4 2
775.1.c.a 2 1240.s odd 4 2
775.1.d.b 1 40.e odd 2 1
775.1.d.b 1 1240.o even 2 1
961.1.e.a 2 248.m odd 6 2
961.1.e.a 2 248.q even 6 2
961.1.f.a 4 248.s odd 10 4
961.1.f.a 4 248.v even 10 4
961.1.h.a 8 248.bb even 30 8
961.1.h.a 8 248.be odd 30 8
1519.1.c.a 1 56.e even 2 1
1519.1.c.a 1 1736.n odd 2 1
1519.1.n.a 2 56.m even 6 2
1519.1.n.a 2 1736.bh odd 6 2
1519.1.n.b 2 56.k odd 6 2
1519.1.n.b 2 1736.ct even 6 2
1984.1.e.a 1 4.b odd 2 1
1984.1.e.a 1 124.d even 2 1
1984.1.e.b 1 1.a even 1 1 trivial
1984.1.e.b 1 31.b odd 2 1 CM
2511.1.m.a 2 72.l even 6 2
2511.1.m.a 2 2232.bp odd 6 2
2511.1.m.e 2 72.p odd 6 2
2511.1.m.e 2 2232.cq even 6 2
3751.1.d.b 1 88.g even 2 1
3751.1.d.b 1 2728.e odd 2 1
3751.1.t.a 4 88.k even 10 4
3751.1.t.a 4 2728.eb odd 10 4
3751.1.t.c 4 88.l odd 10 4
3751.1.t.c 4 2728.ef even 10 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1984, [\chi])$$.

## Hecke characteristic polynomials

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