Properties

Label 2005.1.g.b
Level $2005$
Weight $1$
Character orbit 2005.g
Analytic conductor $1.001$
Analytic rank $0$
Dimension $8$
Projective image $D_{20}$
RM discriminant 401
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2005,1,Mod(1202,2005)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2005, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2005.1202");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2005 = 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2005.g (of order \(4\), degree \(2\), 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: \(1.00062535033\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{20})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{20}\)
Projective field: Galois closure of 20.0.8181810539473601738391143798828125.1

$q$-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.

\(f(q)\) \(=\) \( q + (\zeta_{20}^{4} - \zeta_{20}) q^{2} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{4} - \zeta_{20}^{6} q^{5} + ( - \zeta_{20}^{3} + \zeta_{20}^{2}) q^{7} + ( - \zeta_{20}^{9} + \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{8} + \zeta_{20}^{5} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{20}^{4} - \zeta_{20}) q^{2} + (\zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{4} - \zeta_{20}^{6} q^{5} + ( - \zeta_{20}^{3} + \zeta_{20}^{2}) q^{7} + ( - \zeta_{20}^{9} + \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2}) q^{8} + \zeta_{20}^{5} q^{9} + (\zeta_{20}^{7} + 1) q^{10} + ( - \zeta_{20}^{7} + \zeta_{20}^{3}) q^{11} + ( - \zeta_{20}^{7} + \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{3}) q^{14} + ( - \zeta_{20}^{7} - \zeta_{20}^{6} + \zeta_{20}^{4} - \zeta_{20}^{3} + 1) q^{16} + (\zeta_{20}^{9} - \zeta_{20}^{6}) q^{18} + ( - \zeta_{20}^{8} + \zeta_{20}^{4} - \zeta_{20}) q^{20} + (\zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{4} + \zeta_{20}) q^{22} - \zeta_{20}^{2} q^{25} + (\zeta_{20}^{8} - \zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20} - 1) q^{28} + ( - \zeta_{20}^{9} - \zeta_{20}) q^{29} + (\zeta_{20}^{8} + \zeta_{20}^{7} - \zeta_{20}^{5} + \zeta_{20}^{4} - \zeta_{20} + 1) q^{32} + (\zeta_{20}^{9} - \zeta_{20}^{8}) q^{35} + (\zeta_{20}^{7} - \zeta_{20}^{3} + 1) q^{36} + (\zeta_{20}^{9} + \zeta_{20}^{8} - \zeta_{20}^{5} + \zeta_{20}^{2}) q^{40} + ( - \zeta_{20}^{9} + \zeta_{20}) q^{41} + ( - \zeta_{20}^{8} + \zeta_{20}^{7}) q^{43} + ( - \zeta_{20}^{9} - \zeta_{20}^{8} + \zeta_{20}^{5} - \zeta_{20}^{2} - \zeta_{20}) q^{44} + \zeta_{20} q^{45} + ( - \zeta_{20}^{4} + \zeta_{20}) q^{47} + (\zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{4}) q^{49} + ( - \zeta_{20}^{6} + \zeta_{20}^{3}) q^{50} + ( - \zeta_{20}^{9} - \zeta_{20}^{3}) q^{55} + ( - \zeta_{20}^{9} + \zeta_{20}^{8} + \zeta_{20}^{6} - \zeta_{20}^{4} - \zeta_{20}^{2} + \zeta_{20}) q^{56} + ( - \zeta_{20}^{5} + \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{58} + ( - \zeta_{20}^{8} + \zeta_{20}^{7}) q^{63} + ( - \zeta_{20}^{9} - \zeta_{20}^{8} + \zeta_{20}^{6} + \zeta_{20}^{5} + \zeta_{20}^{4} + \zeta_{20}^{2} + \zeta_{20}) q^{64} + (\zeta_{20}^{9} - \zeta_{20}^{3} + \zeta_{20}^{2} + 1) q^{70} + ( - \zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{4} - \zeta_{20}) q^{72} + ( - \zeta_{20}^{9} + \zeta_{20}^{6}) q^{73} + ( - \zeta_{20}^{9} - \zeta_{20}^{6} + \zeta_{20}^{5} - 1) q^{77} + ( - \zeta_{20}^{9} + \zeta_{20}^{6} - \zeta_{20}^{3} - \zeta_{20}^{2} + 1) q^{80} - q^{81} + (\zeta_{20}^{5} + \zeta_{20}^{3} - \zeta_{20}^{2} - 1) q^{82} + (\zeta_{20}^{5} + 1) q^{83} + (\zeta_{20}^{9} - \zeta_{20}^{8} + \zeta_{20}^{2} - \zeta_{20}) q^{86} + (2 \zeta_{20}^{9} - \zeta_{20}^{6} - \zeta_{20}^{5} + \zeta_{20}^{3} + \zeta_{20}^{2} - 1) q^{88} + ( - \zeta_{20}^{8} - \zeta_{20}^{2}) q^{89} + (\zeta_{20}^{5} - \zeta_{20}^{2}) q^{90} + ( - \zeta_{20}^{8} + 2 \zeta_{20}^{5} - \zeta_{20}^{2}) q^{94} + ( - \zeta_{20}^{9} + \zeta_{20}^{8} - \zeta_{20}^{7} + \zeta_{20}^{6} - \zeta_{20}^{5} - 1) q^{98} + (\zeta_{20}^{8} + \zeta_{20}^{2}) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{5} + 2 q^{7} + 8 q^{10} - 12 q^{16} - 2 q^{18} - 2 q^{25} - 12 q^{28} + 8 q^{32} + 2 q^{35} + 8 q^{36} + 2 q^{43} + 2 q^{47} - 2 q^{50} - 6 q^{58} + 2 q^{63} + 10 q^{70} + 2 q^{73} - 10 q^{77} + 8 q^{80} - 8 q^{81} - 10 q^{82} + 8 q^{83} + 4 q^{86} - 10 q^{88} - 2 q^{90} - 8 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2005\mathbb{Z}\right)^\times\).

\(n\) \(402\) \(1206\)
\(\chi(n)\) \(-\zeta_{20}^{5}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1202.1
0.587785 + 0.809017i
0.951057 0.309017i
−0.587785 + 0.809017i
−0.951057 0.309017i
0.587785 0.809017i
0.951057 + 0.309017i
−0.587785 0.809017i
−0.951057 + 0.309017i
−1.39680 1.39680i 0 2.90211i −0.809017 + 0.587785i 0 0.642040 + 0.642040i 2.65688 2.65688i 1.00000i 1.95106 + 0.309017i
1202.2 −0.642040 0.642040i 0 0.175571i 0.309017 + 0.951057i 0 0.221232 + 0.221232i −0.754763 + 0.754763i 1.00000i 0.412215 0.809017i
1202.3 −0.221232 0.221232i 0 0.902113i −0.809017 0.587785i 0 −1.26007 1.26007i −0.420808 + 0.420808i 1.00000i 0.0489435 + 0.309017i
1202.4 1.26007 + 1.26007i 0 2.17557i 0.309017 0.951057i 0 1.39680 + 1.39680i −1.48131 + 1.48131i 1.00000i 1.58779 0.809017i
1603.1 −1.39680 + 1.39680i 0 2.90211i −0.809017 0.587785i 0 0.642040 0.642040i 2.65688 + 2.65688i 1.00000i 1.95106 0.309017i
1603.2 −0.642040 + 0.642040i 0 0.175571i 0.309017 0.951057i 0 0.221232 0.221232i −0.754763 0.754763i 1.00000i 0.412215 + 0.809017i
1603.3 −0.221232 + 0.221232i 0 0.902113i −0.809017 + 0.587785i 0 −1.26007 + 1.26007i −0.420808 0.420808i 1.00000i 0.0489435 0.309017i
1603.4 1.26007 1.26007i 0 2.17557i 0.309017 + 0.951057i 0 1.39680 1.39680i −1.48131 1.48131i 1.00000i 1.58779 + 0.809017i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1202.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
401.b even 2 1 RM by \(\Q(\sqrt{401}) \)
5.c odd 4 1 inner
2005.g odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2005.1.g.b 8
5.c odd 4 1 inner 2005.1.g.b 8
401.b even 2 1 RM 2005.1.g.b 8
2005.g odd 4 1 inner 2005.1.g.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2005.1.g.b 8 1.a even 1 1 trivial
2005.1.g.b 8 5.c odd 4 1 inner
2005.1.g.b 8 401.b even 2 1 RM
2005.1.g.b 8 2005.g odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 2T_{2}^{7} + 2T_{2}^{6} + 11T_{2}^{4} + 20T_{2}^{3} + 18T_{2}^{2} + 6T_{2} + 1 \) acting on \(S_{1}^{\mathrm{new}}(2005, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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