Properties

Label 2004.1.g.f
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

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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.

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

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
−0.415415 0.909632i
−0.415415 + 0.909632i
0.654861 0.755750i
0.654861 + 0.755750i
0.959493 + 0.281733i
0.959493 0.281733i
0.142315 + 0.989821i
0.142315 0.989821i
−0.841254 + 0.540641i
−0.841254 0.540641i
−0.841254 0.540641i 0.654861 + 0.755750i 0.415415 + 0.909632i 0 −0.142315 0.989821i 0.563465i 0.142315 0.989821i −0.142315 + 0.989821i 0
2003.2 −0.841254 + 0.540641i 0.654861 0.755750i 0.415415 0.909632i 0 −0.142315 + 0.989821i 0.563465i 0.142315 + 0.989821i −0.142315 0.989821i 0
2003.3 −0.415415 0.909632i 0.142315 0.989821i −0.654861 + 0.755750i 0 −0.959493 + 0.281733i 1.08128i 0.959493 + 0.281733i −0.959493 0.281733i 0
2003.4 −0.415415 + 0.909632i 0.142315 + 0.989821i −0.654861 0.755750i 0 −0.959493 0.281733i 1.08128i 0.959493 0.281733i −0.959493 + 0.281733i 0
2003.5 0.142315 0.989821i −0.841254 + 0.540641i −0.959493 0.281733i 0 0.415415 + 0.909632i 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 0.415415 0.909632i 1.51150i −0.415415 0.909632i 0.415415 + 0.909632i 0
2003.7 0.654861 0.755750i 0.959493 + 0.281733i −0.142315 0.989821i 0 0.841254 0.540641i 1.81926i −0.841254 0.540641i 0.841254 + 0.540641i 0
2003.8 0.654861 + 0.755750i 0.959493 0.281733i −0.142315 + 0.989821i 0 0.841254 + 0.540641i 1.81926i −0.841254 + 0.540641i 0.841254 0.540641i 0
2003.9 0.959493 0.281733i −0.415415 0.909632i 0.841254 0.540641i 0 −0.654861 0.755750i 1.97964i 0.654861 0.755750i −0.654861 + 0.755750i 0
2003.10 0.959493 + 0.281733i −0.415415 + 0.909632i 0.841254 + 0.540641i 0 −0.654861 + 0.755750i 1.97964i 0.654861 + 0.755750i −0.654861 0.755750i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2003.10
Significant digits:
Format:

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

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

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