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 disc. -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 \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.00012628532\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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.

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

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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
167.b Odd 1 CM by \(\Q(\sqrt{-167}) \) yes
12.b Even 1 yes
2004.g Odd 1 yes

Hecke kernels

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

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