# Properties

 Label 1083.1.b.a Level $1083$ Weight $1$ Character orbit 1083.b Self dual yes Analytic conductor $0.540$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -3 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1083,1,Mod(362,1083)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1083, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1083.362");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1083 = 3 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1083.b (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: yes Analytic conductor: $$0.540487408682$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 57) Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.1083.1 Artin image: $D_6$ Artin field: Galois closure of 6.0.22284891.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{4} - q^{7} + q^{9}+O(q^{10})$$ q - q^3 + q^4 - q^7 + q^9 $$q - q^{3} + q^{4} - q^{7} + q^{9} - q^{12} + q^{13} + q^{16} + q^{21} + q^{25} - q^{27} - q^{28} + q^{31} + q^{36} + q^{37} - q^{39} - q^{43} - q^{48} + q^{52} - q^{61} - q^{63} + q^{64} + q^{67} - q^{73} - q^{75} + q^{79} + q^{81} + q^{84} - q^{91} - q^{93} - 2 q^{97}+O(q^{100})$$ q - q^3 + q^4 - q^7 + q^9 - q^12 + q^13 + q^16 + q^21 + q^25 - q^27 - q^28 + q^31 + q^36 + q^37 - q^39 - q^43 - q^48 + q^52 - q^61 - q^63 + q^64 + q^67 - q^73 - q^75 + q^79 + q^81 + q^84 - q^91 - q^93 - 2 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1083\mathbb{Z}\right)^\times$$.

 $$n$$ $$362$$ $$724$$ $$\chi(n)$$ $$1$$ $$0$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
362.1
 0
0 −1.00000 1.00000 0 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1083.1.b.a 1
3.b odd 2 1 CM 1083.1.b.a 1
19.b odd 2 1 1083.1.b.b 1
19.c even 3 2 1083.1.h.a 2
19.d odd 6 2 57.1.h.a 2
19.e even 9 6 1083.1.l.b 6
19.f odd 18 6 1083.1.l.a 6
57.d even 2 1 1083.1.b.b 1
57.f even 6 2 57.1.h.a 2
57.h odd 6 2 1083.1.h.a 2
57.j even 18 6 1083.1.l.a 6
57.l odd 18 6 1083.1.l.b 6
76.f even 6 2 912.1.bl.a 2
95.h odd 6 2 1425.1.t.a 2
95.l even 12 4 1425.1.o.a 4
133.i even 6 2 2793.1.n.b 2
133.j odd 6 2 2793.1.n.a 2
133.n odd 6 2 2793.1.bi.b 2
133.p even 6 2 2793.1.bf.a 2
133.s even 6 2 2793.1.bi.a 2
152.l odd 6 2 3648.1.bl.b 2
152.o even 6 2 3648.1.bl.a 2
171.i odd 6 2 1539.1.j.a 2
171.k even 6 2 1539.1.n.a 2
171.s odd 6 2 1539.1.n.a 2
171.t even 6 2 1539.1.j.a 2
228.n odd 6 2 912.1.bl.a 2
285.q even 6 2 1425.1.t.a 2
285.w odd 12 4 1425.1.o.a 4
399.q odd 6 2 2793.1.bf.a 2
399.r odd 6 2 2793.1.bi.a 2
399.x even 6 2 2793.1.bi.b 2
399.bm even 6 2 2793.1.n.a 2
399.bn odd 6 2 2793.1.n.b 2
456.s odd 6 2 3648.1.bl.a 2
456.v even 6 2 3648.1.bl.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.1.h.a 2 19.d odd 6 2
57.1.h.a 2 57.f even 6 2
912.1.bl.a 2 76.f even 6 2
912.1.bl.a 2 228.n odd 6 2
1083.1.b.a 1 1.a even 1 1 trivial
1083.1.b.a 1 3.b odd 2 1 CM
1083.1.b.b 1 19.b odd 2 1
1083.1.b.b 1 57.d even 2 1
1083.1.h.a 2 19.c even 3 2
1083.1.h.a 2 57.h odd 6 2
1083.1.l.a 6 19.f odd 18 6
1083.1.l.a 6 57.j even 18 6
1083.1.l.b 6 19.e even 9 6
1083.1.l.b 6 57.l odd 18 6
1425.1.o.a 4 95.l even 12 4
1425.1.o.a 4 285.w odd 12 4
1425.1.t.a 2 95.h odd 6 2
1425.1.t.a 2 285.q even 6 2
1539.1.j.a 2 171.i odd 6 2
1539.1.j.a 2 171.t even 6 2
1539.1.n.a 2 171.k even 6 2
1539.1.n.a 2 171.s odd 6 2
2793.1.n.a 2 133.j odd 6 2
2793.1.n.a 2 399.bm even 6 2
2793.1.n.b 2 133.i even 6 2
2793.1.n.b 2 399.bn odd 6 2
2793.1.bf.a 2 133.p even 6 2
2793.1.bf.a 2 399.q odd 6 2
2793.1.bi.a 2 133.s even 6 2
2793.1.bi.a 2 399.r odd 6 2
2793.1.bi.b 2 133.n odd 6 2
2793.1.bi.b 2 399.x even 6 2
3648.1.bl.a 2 152.o even 6 2
3648.1.bl.a 2 456.s odd 6 2
3648.1.bl.b 2 152.l odd 6 2
3648.1.bl.b 2 456.v even 6 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1083, [\chi])$$.

## Hecke characteristic polynomials

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