# Properties

 Label 2004.1.g.e Level 2004 Weight 1 Character orbit 2004.g Analytic conductor 1.000 Analytic rank 0 Dimension 10 Projective image $$D_{22}$$ CM discriminant -167 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: no Analytic conductor: $$1.00012628532$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\Q(\zeta_{22})$$ Defining polynomial: $$x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image $$D_{22}$$ Projective field Galois closure of $$\mathbb{Q}[x]/(x^{22} + \cdots)$$

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{22}^{6} q^{2} + \zeta_{22}^{2} q^{3} -\zeta_{22} q^{4} + \zeta_{22}^{8} q^{6} + ( \zeta_{22}^{3} + \zeta_{22}^{8} ) q^{7} -\zeta_{22}^{7} q^{8} + \zeta_{22}^{4} q^{9} +O(q^{10})$$ $$q + \zeta_{22}^{6} q^{2} + \zeta_{22}^{2} q^{3} -\zeta_{22} q^{4} + \zeta_{22}^{8} q^{6} + ( \zeta_{22}^{3} + \zeta_{22}^{8} ) q^{7} -\zeta_{22}^{7} q^{8} + \zeta_{22}^{4} q^{9} + ( \zeta_{22} - \zeta_{22}^{10} ) q^{11} -\zeta_{22}^{3} q^{12} + ( -\zeta_{22}^{3} + \zeta_{22}^{9} ) q^{14} + \zeta_{22}^{2} q^{16} + \zeta_{22}^{10} q^{18} + ( -\zeta_{22}^{4} - \zeta_{22}^{7} ) q^{19} + ( \zeta_{22}^{5} + \zeta_{22}^{10} ) q^{21} + ( \zeta_{22}^{5} + \zeta_{22}^{7} ) q^{22} -\zeta_{22}^{9} q^{24} - q^{25} + \zeta_{22}^{6} q^{27} + ( -\zeta_{22}^{4} - \zeta_{22}^{9} ) q^{28} + ( \zeta_{22}^{3} + \zeta_{22}^{8} ) q^{29} + ( -\zeta_{22}^{5} - \zeta_{22}^{6} ) q^{31} + \zeta_{22}^{8} q^{32} + ( \zeta_{22} + \zeta_{22}^{3} ) q^{33} -\zeta_{22}^{5} q^{36} + ( \zeta_{22}^{2} - \zeta_{22}^{10} ) q^{38} + ( -1 - \zeta_{22}^{5} ) q^{42} + ( -1 - \zeta_{22}^{2} ) q^{44} + ( -\zeta_{22}^{4} + \zeta_{22}^{7} ) q^{47} + \zeta_{22}^{4} q^{48} + ( -1 - \zeta_{22}^{5} + \zeta_{22}^{6} ) q^{49} -\zeta_{22}^{6} q^{50} -\zeta_{22} q^{54} + ( \zeta_{22}^{4} - \zeta_{22}^{10} ) q^{56} + ( -\zeta_{22}^{6} - \zeta_{22}^{9} ) q^{57} + ( -\zeta_{22}^{3} + \zeta_{22}^{9} ) q^{58} + ( -\zeta_{22}^{2} + \zeta_{22}^{9} ) q^{61} + ( 1 + \zeta_{22} ) q^{62} + ( -\zeta_{22} + \zeta_{22}^{7} ) q^{63} -\zeta_{22}^{3} q^{64} + ( \zeta_{22}^{7} + \zeta_{22}^{9} ) q^{66} + q^{72} -\zeta_{22}^{2} q^{75} + ( \zeta_{22}^{5} + \zeta_{22}^{8} ) q^{76} + ( \zeta_{22}^{2} + \zeta_{22}^{4} + \zeta_{22}^{7} + \zeta_{22}^{9} ) q^{77} + \zeta_{22}^{8} q^{81} + ( 1 - \zeta_{22}^{6} ) q^{84} + ( \zeta_{22}^{5} + \zeta_{22}^{10} ) q^{87} + ( -\zeta_{22}^{6} - \zeta_{22}^{8} ) q^{88} + ( -\zeta_{22}^{2} - \zeta_{22}^{9} ) q^{89} + ( -\zeta_{22}^{7} - \zeta_{22}^{8} ) q^{93} + ( -\zeta_{22}^{2} - \zeta_{22}^{10} ) q^{94} + \zeta_{22}^{10} q^{96} + ( -\zeta_{22} + \zeta_{22}^{10} ) q^{97} + ( 1 - \zeta_{22} - \zeta_{22}^{6} ) q^{98} + ( \zeta_{22}^{3} + \zeta_{22}^{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10q - q^{2} - q^{3} - q^{4} - q^{6} - q^{8} - q^{9} + O(q^{10})$$ $$10q - q^{2} - q^{3} - q^{4} - q^{6} - q^{8} - q^{9} + 2q^{11} - q^{12} - q^{16} - q^{18} + 2q^{22} - q^{24} - 10q^{25} - q^{27} - q^{32} + 2q^{33} - q^{36} - 11q^{42} - 9q^{44} + 2q^{47} - q^{48} - 12q^{49} + q^{50} - q^{54} + 2q^{61} + 11q^{62} - q^{64} + 2q^{66} + 10q^{72} + q^{75} - q^{81} + 11q^{84} + 2q^{88} + 2q^{94} - q^{96} - 2q^{97} + 10q^{98} + 2q^{99} + O(q^{100})$$

## 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
 −0.841254 − 0.540641i −0.841254 + 0.540641i 0.142315 − 0.989821i 0.142315 + 0.989821i 0.959493 − 0.281733i 0.959493 + 0.281733i 0.654861 + 0.755750i 0.654861 − 0.755750i −0.415415 + 0.909632i −0.415415 − 0.909632i
−0.959493 0.281733i 0.415415 + 0.909632i 0.841254 + 0.540641i 0 −0.142315 0.989821i 1.97964i −0.654861 0.755750i −0.654861 + 0.755750i 0
2003.2 −0.959493 + 0.281733i 0.415415 0.909632i 0.841254 0.540641i 0 −0.142315 + 0.989821i 1.97964i −0.654861 + 0.755750i −0.654861 0.755750i 0
2003.3 −0.654861 0.755750i −0.959493 0.281733i −0.142315 + 0.989821i 0 0.415415 + 0.909632i 1.81926i 0.841254 0.540641i 0.841254 + 0.540641i 0
2003.4 −0.654861 + 0.755750i −0.959493 + 0.281733i −0.142315 0.989821i 0 0.415415 0.909632i 1.81926i 0.841254 + 0.540641i 0.841254 0.540641i 0
2003.5 −0.142315 0.989821i 0.841254 0.540641i −0.959493 + 0.281733i 0 −0.654861 0.755750i 1.51150i 0.415415 + 0.909632i 0.415415 0.909632i 0
2003.6 −0.142315 + 0.989821i 0.841254 + 0.540641i −0.959493 0.281733i 0 −0.654861 + 0.755750i 1.51150i 0.415415 0.909632i 0.415415 + 0.909632i 0
2003.7 0.415415 0.909632i −0.142315 + 0.989821i −0.654861 0.755750i 0 0.841254 + 0.540641i 1.08128i −0.959493 + 0.281733i −0.959493 0.281733i 0
2003.8 0.415415 + 0.909632i −0.142315 0.989821i −0.654861 + 0.755750i 0 0.841254 0.540641i 1.08128i −0.959493 0.281733i −0.959493 + 0.281733i 0
2003.9 0.841254 0.540641i −0.654861 0.755750i 0.415415 0.909632i 0 −0.959493 0.281733i 0.563465i −0.142315 0.989821i −0.142315 + 0.989821i 0
2003.10 0.841254 + 0.540641i −0.654861 + 0.755750i 0.415415 + 0.909632i 0 −0.959493 + 0.281733i 0.563465i −0.142315 + 0.989821i −0.142315 0.989821i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2003.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
167.b odd 2 1 CM by $$\Q(\sqrt{-167})$$
12.b even 2 1 inner
2004.g 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.e 10
3.b odd 2 1 2004.1.g.f yes 10
4.b odd 2 1 2004.1.g.f yes 10
12.b even 2 1 inner 2004.1.g.e 10
167.b odd 2 1 CM 2004.1.g.e 10
501.c even 2 1 2004.1.g.f yes 10
668.b even 2 1 2004.1.g.f yes 10
2004.g odd 2 1 inner 2004.1.g.e 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2004.1.g.e 10 1.a even 1 1 trivial
2004.1.g.e 10 12.b even 2 1 inner
2004.1.g.e 10 167.b odd 2 1 CM
2004.1.g.e 10 2004.g odd 2 1 inner
2004.1.g.f yes 10 3.b odd 2 1
2004.1.g.f yes 10 4.b odd 2 1
2004.1.g.f yes 10 501.c even 2 1
2004.1.g.f yes 10 668.b even 2 1

## Hecke kernels

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

 $$T_{5}$$ $$T_{7}^{10} + 11 T_{7}^{8} + 44 T_{7}^{6} + 77 T_{7}^{4} + 55 T_{7}^{2} + 11$$ $$T_{11}^{5} - T_{11}^{4} - 4 T_{11}^{3} + 3 T_{11}^{2} + 3 T_{11} - 1$$ $$T_{179}^{5} + T_{179}^{4} - 4 T_{179}^{3} - 3 T_{179}^{2} + 3 T_{179} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$$
$3$ $$1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10}$$
$5$ $$( 1 + T^{2} )^{10}$$
$7$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
$11$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$13$ $$( 1 - T )^{10}( 1 + T )^{10}$$
$17$ $$( 1 + T^{2} )^{10}$$
$19$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
$23$ $$( 1 - T )^{10}( 1 + T )^{10}$$
$29$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
$31$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
$37$ $$( 1 - T )^{10}( 1 + T )^{10}$$
$41$ $$( 1 + T^{2} )^{10}$$
$43$ $$( 1 + T^{2} )^{10}$$
$47$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$53$ $$( 1 + T^{2} )^{10}$$
$59$ $$( 1 - T )^{10}( 1 + T )^{10}$$
$61$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )^{2}$$
$67$ $$( 1 + T^{2} )^{10}$$
$71$ $$( 1 - T )^{10}( 1 + T )^{10}$$
$73$ $$( 1 - T )^{10}( 1 + T )^{10}$$
$79$ $$( 1 + T^{2} )^{10}$$
$83$ $$( 1 - T )^{10}( 1 + T )^{10}$$
$89$ $$( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} - T^{7} + T^{8} - T^{9} + T^{10} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )$$
$97$ $$( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} + T^{7} + T^{8} + T^{9} + T^{10} )^{2}$$