# Properties

 Label 2003.1.b.a Level 2003 Weight 1 Character orbit 2003.b Self dual yes Analytic conductor 1.000 Analytic rank 0 Dimension 1 Projective image $$D_{3}$$ CM discriminant -2003 Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$2003$$ Weight: $$k$$ = $$1$$ Character orbit: $$[\chi]$$ = 2003.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.999627220304$$ 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 3.1.2003.1 Artin image $S_3$ Artin field Galois closure of 3.1.2003.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{4} + O(q^{10})$$ $$q - q^{3} + q^{4} - q^{12} - q^{13} + q^{16} + 2q^{19} + q^{25} + q^{27} + q^{39} - q^{47} - q^{48} + q^{49} - q^{52} + 2q^{53} - 2q^{57} - q^{59} + q^{64} - q^{73} - q^{75} + 2q^{76} - q^{79} - q^{81} + 2q^{89} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2003\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2002.1
 0
0 −1.00000 1.00000 0 0 0 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
2003.b odd 2 1 CM by $$\Q(\sqrt{-2003})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2003.1.b.a 1
2003.b odd 2 1 CM 2003.1.b.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2003.1.b.a 1 1.a even 1 1 trivial
2003.1.b.a 1 2003.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(2003, [\chi])$$.

## Hecke Characteristic Polynomials

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