# Properties

 Label 83.1.b.a Level $83$ Weight $1$ Character orbit 83.b Self dual yes Analytic conductor $0.041$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -83 Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [83,1,Mod(82,83)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(83, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("83.82");

S:= CuspForms(chi, 1);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$83$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 83.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: $$0.0414223960485$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.83.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.83.1

## $q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{4} - q^{7}+O(q^{10})$$ q - q^3 + q^4 - q^7 $$q - q^{3} + q^{4} - q^{7} - q^{11} - q^{12} + q^{16} - q^{17} + q^{21} + 2 q^{23} + q^{25} + q^{27} - q^{28} - q^{29} - q^{31} + q^{33} - q^{37} + 2 q^{41} - q^{44} - q^{48} + q^{51} - q^{59} - q^{61} + q^{64} - q^{68} - 2 q^{69} - q^{75} + q^{77} - q^{81} + q^{83} + q^{84} + q^{87} + 2 q^{92} + q^{93}+O(q^{100})$$ q - q^3 + q^4 - q^7 - q^11 - q^12 + q^16 - q^17 + q^21 + 2 * q^23 + q^25 + q^27 - q^28 - q^29 - q^31 + q^33 - q^37 + 2 * q^41 - q^44 - q^48 + q^51 - q^59 - q^61 + q^64 - q^68 - 2 * q^69 - q^75 + q^77 - q^81 + q^83 + q^84 + q^87 + 2 * q^92 + q^93

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/83\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
82.1
 0
0 −1.00000 1.00000 0 0 −1.00000 0 0 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
83.b odd 2 1 CM by $$\Q(\sqrt{-83})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 83.1.b.a 1
3.b odd 2 1 747.1.c.a 1
4.b odd 2 1 1328.1.g.a 1
5.b even 2 1 2075.1.c.f 1
5.c odd 4 2 2075.1.d.c 2
83.b odd 2 1 CM 83.1.b.a 1
249.d even 2 1 747.1.c.a 1
332.b even 2 1 1328.1.g.a 1
415.d odd 2 1 2075.1.c.f 1
415.e even 4 2 2075.1.d.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.1.b.a 1 1.a even 1 1 trivial
83.1.b.a 1 83.b odd 2 1 CM
747.1.c.a 1 3.b odd 2 1
747.1.c.a 1 249.d even 2 1
1328.1.g.a 1 4.b odd 2 1
1328.1.g.a 1 332.b even 2 1
2075.1.c.f 1 5.b even 2 1
2075.1.c.f 1 415.d odd 2 1
2075.1.d.c 2 5.c odd 4 2
2075.1.d.c 2 415.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(83, [\chi])$$.

## Hecke characteristic polynomials

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