# Properties

 Label 1984.1.e.a 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 6.2.492032.2

## $q$-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.a 1
4.b odd 2 1 1984.1.e.b 1
8.b even 2 1 31.1.b.a 1
8.d odd 2 1 496.1.e.a 1
24.h odd 2 1 279.1.d.b 1
31.b odd 2 1 CM 1984.1.e.a 1
40.f even 2 1 775.1.d.b 1
40.i odd 4 2 775.1.c.a 2
56.h odd 2 1 1519.1.c.a 1
56.j odd 6 2 1519.1.n.a 2
56.p even 6 2 1519.1.n.b 2
72.j odd 6 2 2511.1.m.a 2
72.n even 6 2 2511.1.m.e 2
88.b odd 2 1 3751.1.d.b 1
88.o even 10 4 3751.1.t.c 4
88.p odd 10 4 3751.1.t.a 4
124.d even 2 1 1984.1.e.b 1
248.b even 2 1 496.1.e.a 1
248.g odd 2 1 31.1.b.a 1
248.l odd 6 2 961.1.e.a 2
248.p even 6 2 961.1.e.a 2
248.r odd 10 4 961.1.f.a 4
248.u even 10 4 961.1.f.a 4
248.bc even 30 8 961.1.h.a 8
248.bf odd 30 8 961.1.h.a 8
744.o even 2 1 279.1.d.b 1
1240.e odd 2 1 775.1.d.b 1
1240.y even 4 2 775.1.c.a 2
1736.h even 2 1 1519.1.c.a 1
1736.br odd 6 2 1519.1.n.b 2
1736.ca even 6 2 1519.1.n.a 2
2232.bi even 6 2 2511.1.m.a 2
2232.ca odd 6 2 2511.1.m.e 2
2728.k even 2 1 3751.1.d.b 1
2728.co odd 10 4 3751.1.t.c 4
2728.cw even 10 4 3751.1.t.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.1.b.a 1 8.b even 2 1
31.1.b.a 1 248.g odd 2 1
279.1.d.b 1 24.h odd 2 1
279.1.d.b 1 744.o even 2 1
496.1.e.a 1 8.d odd 2 1
496.1.e.a 1 248.b even 2 1
775.1.c.a 2 40.i odd 4 2
775.1.c.a 2 1240.y even 4 2
775.1.d.b 1 40.f even 2 1
775.1.d.b 1 1240.e odd 2 1
961.1.e.a 2 248.l odd 6 2
961.1.e.a 2 248.p even 6 2
961.1.f.a 4 248.r odd 10 4
961.1.f.a 4 248.u even 10 4
961.1.h.a 8 248.bc even 30 8
961.1.h.a 8 248.bf odd 30 8
1519.1.c.a 1 56.h odd 2 1
1519.1.c.a 1 1736.h even 2 1
1519.1.n.a 2 56.j odd 6 2
1519.1.n.a 2 1736.ca even 6 2
1519.1.n.b 2 56.p even 6 2
1519.1.n.b 2 1736.br odd 6 2
1984.1.e.a 1 1.a even 1 1 trivial
1984.1.e.a 1 31.b odd 2 1 CM
1984.1.e.b 1 4.b odd 2 1
1984.1.e.b 1 124.d even 2 1
2511.1.m.a 2 72.j odd 6 2
2511.1.m.a 2 2232.bi even 6 2
2511.1.m.e 2 72.n even 6 2
2511.1.m.e 2 2232.ca odd 6 2
3751.1.d.b 1 88.b odd 2 1
3751.1.d.b 1 2728.k even 2 1
3751.1.t.a 4 88.p odd 10 4
3751.1.t.a 4 2728.cw even 10 4
3751.1.t.c 4 88.o even 10 4
3751.1.t.c 4 2728.co odd 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

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