# Properties

 Label 2023.1.c.b Level $2023$ Weight $1$ Character orbit 2023.c Self dual yes Analytic conductor $1.010$ Analytic rank $0$ Dimension $1$ Projective image $D_{3}$ CM discriminant -7 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2023 = 7 \cdot 17^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2023.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.00960852056$$ 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 3.1.2023.1 Artin image: $S_3$ Artin field: Galois closure of 3.1.2023.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{7} + q^{8} + q^{9} + 2q^{11} - q^{14} - q^{16} - q^{18} - 2q^{22} - q^{23} + q^{25} - q^{29} - q^{37} - q^{43} + q^{46} + q^{49} - q^{50} - q^{53} + q^{56} + q^{58} + q^{63} + q^{64} + 2q^{67} - q^{71} + q^{72} + q^{74} + 2q^{77} - q^{79} + q^{81} + q^{86} + 2q^{88} - q^{98} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2023\mathbb{Z}\right)^\times$$.

 $$n$$ $$290$$ $$1737$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1735.1
 0
−1.00000 0 0 0 0 1.00000 1.00000 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2023.1.c.b yes 1
7.b odd 2 1 CM 2023.1.c.b yes 1
17.b even 2 1 2023.1.c.a 1
17.c even 4 2 2023.1.d.a 2
17.d even 8 4 2023.1.f.a 4
17.e odd 16 8 2023.1.l.a 8
119.d odd 2 1 2023.1.c.a 1
119.f odd 4 2 2023.1.d.a 2
119.l odd 8 4 2023.1.f.a 4
119.p even 16 8 2023.1.l.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2023.1.c.a 1 17.b even 2 1
2023.1.c.a 1 119.d odd 2 1
2023.1.c.b yes 1 1.a even 1 1 trivial
2023.1.c.b yes 1 7.b odd 2 1 CM
2023.1.d.a 2 17.c even 4 2
2023.1.d.a 2 119.f odd 4 2
2023.1.f.a 4 17.d even 8 4
2023.1.f.a 4 119.l odd 8 4
2023.1.l.a 8 17.e odd 16 8
2023.1.l.a 8 119.p even 16 8

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2023, [\chi])$$:

 $$T_{2} + 1$$ $$T_{11} - 2$$

## Hecke characteristic polynomials

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