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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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2004,1,Mod(2003,2004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2004.2003");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
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

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.00012628532\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

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

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
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 \(S_{1}^{\mathrm{new}}(2004, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7}^{10} + 11T_{7}^{8} + 44T_{7}^{6} + 77T_{7}^{4} + 55T_{7}^{2} + 11 \) Copy content Toggle raw display
\( T_{11}^{5} + T_{11}^{4} - 4T_{11}^{3} - 3T_{11}^{2} + 3T_{11} + 1 \) Copy content Toggle raw display
\( T_{179}^{5} - T_{179}^{4} - 4T_{179}^{3} + 3T_{179}^{2} + 3T_{179} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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