# Properties

 Label 2004.1.g.b Level 2004 Weight 1 Character orbit 2004.g Self dual yes Analytic conductor 1.000 Analytic rank 0 Dimension 1 Projective image $$D_{2}$$ CM/RM discs -167, -2004, 12 Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2004 = 2^{2} \cdot 3 \cdot 167$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2004.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.00012628532$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{2}$$ Projective field Galois closure of $$\Q(\sqrt{3}, \sqrt{-167})$$ Artin image $D_4$ Artin field Galois closure of 4.2.24048.2

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + q^{6} + q^{8} + q^{9} - 2q^{11} + q^{12} + q^{16} + q^{18} - 2q^{22} + q^{24} - q^{25} + q^{27} + q^{32} - 2q^{33} + q^{36} - 2q^{44} - 2q^{47} + q^{48} + q^{49} - q^{50} + q^{54} - 2q^{61} + q^{64} - 2q^{66} + q^{72} - q^{75} + q^{81} - 2q^{88} - 2q^{94} + q^{96} + 2q^{97} + q^{98} - 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2004\mathbb{Z}\right)^\times$$.

 $$n$$ $$673$$ $$1003$$ $$1337$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2003.1
 0
1.00000 1.00000 1.00000 0 1.00000 0 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
12.b even 2 1 RM by $$\Q(\sqrt{3})$$
167.b odd 2 1 CM by $$\Q(\sqrt{-167})$$
2004.g odd 2 1 CM by $$\Q(\sqrt{-501})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2004.1.g.b yes 1
3.b odd 2 1 2004.1.g.a 1
4.b odd 2 1 2004.1.g.a 1
12.b even 2 1 RM 2004.1.g.b yes 1
167.b odd 2 1 CM 2004.1.g.b yes 1
501.c even 2 1 2004.1.g.a 1
668.b even 2 1 2004.1.g.a 1
2004.g odd 2 1 CM 2004.1.g.b yes 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.1.g.a 1 3.b odd 2 1
2004.1.g.a 1 4.b odd 2 1
2004.1.g.a 1 501.c even 2 1
2004.1.g.a 1 668.b even 2 1
2004.1.g.b yes 1 1.a even 1 1 trivial
2004.1.g.b yes 1 12.b even 2 1 RM
2004.1.g.b yes 1 167.b odd 2 1 CM
2004.1.g.b yes 1 2004.g odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(2004, [\chi])$$:

 $$T_{5}$$ $$T_{7}$$ $$T_{11} + 2$$ $$T_{179} - 2$$

## Hecke characteristic polynomials

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