# 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

## Newspace parameters

 Level: $$N$$ $$=$$ $$83$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 83.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 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

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

## 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.

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

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