# Properties

 Label 2004.1.g.d Level 2004 Weight 1 Character orbit 2004.g Self dual yes Analytic conductor 1.000 Analytic rank 0 Dimension 2 Projective image $$D_{4}$$ CM discriminant -2004 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{4}$$ Projective field Galois closure of 4.0.334668.2 Artin image $D_8$ Artin field Galois closure of 8.2.96577152768.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} -\beta q^{5} - q^{6} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} -\beta q^{5} - q^{6} + q^{8} + q^{9} -\beta q^{10} - q^{12} + \beta q^{15} + q^{16} + \beta q^{17} + q^{18} -\beta q^{20} - q^{24} + q^{25} - q^{27} + \beta q^{30} + q^{32} + \beta q^{34} + q^{36} -\beta q^{40} + \beta q^{41} + \beta q^{43} -\beta q^{45} - q^{48} + q^{49} + q^{50} -\beta q^{51} + \beta q^{53} - q^{54} + \beta q^{60} + q^{64} -\beta q^{67} + \beta q^{68} + q^{72} - q^{75} -\beta q^{79} -\beta q^{80} + q^{81} + \beta q^{82} -2 q^{85} + \beta q^{86} -\beta q^{90} - q^{96} -2 q^{97} + q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q + 2q^{2} - 2q^{3} + 2q^{4} - 2q^{6} + 2q^{8} + 2q^{9} - 2q^{12} + 2q^{16} + 2q^{18} - 2q^{24} + 2q^{25} - 2q^{27} + 2q^{32} + 2q^{36} - 2q^{48} + 2q^{49} + 2q^{50} - 2q^{54} + 2q^{64} + 2q^{72} - 2q^{75} + 2q^{81} - 4q^{85} - 2q^{96} - 4q^{97} + 2q^{98} + O(q^{100})$$

## 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
 1.41421 −1.41421
1.00000 −1.00000 1.00000 −1.41421 −1.00000 0 1.00000 1.00000 −1.41421
2003.2 1.00000 −1.00000 1.00000 1.41421 −1.00000 0 1.00000 1.00000 1.41421
 $$n$$: 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
2004.g odd 2 1 CM by $$\Q(\sqrt{-501})$$
12.b even 2 1 inner
167.b odd 2 1 inner

## Twists

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.1.g.c 2 3.b odd 2 1
2004.1.g.c 2 4.b odd 2 1
2004.1.g.c 2 501.c even 2 1
2004.1.g.c 2 668.b even 2 1
2004.1.g.d yes 2 1.a even 1 1 trivial
2004.1.g.d yes 2 12.b even 2 1 inner
2004.1.g.d yes 2 167.b odd 2 1 inner
2004.1.g.d yes 2 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 $$S_{1}^{\mathrm{new}}(2004, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + T^{4}$$
$7$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$11$ $$( 1 + T^{2} )^{2}$$
$13$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$17$ $$1 + T^{4}$$
$19$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$23$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$29$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$31$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$37$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$41$ $$1 + T^{4}$$
$43$ $$1 + T^{4}$$
$47$ $$( 1 + T^{2} )^{2}$$
$53$ $$1 + T^{4}$$
$59$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$61$ $$( 1 + T^{2} )^{2}$$
$67$ $$1 + T^{4}$$
$71$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$73$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$79$ $$1 + T^{4}$$
$83$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$89$ $$( 1 - T )^{2}( 1 + T )^{2}$$
$97$ $$( 1 + T )^{4}$$