Properties

Label 2005.1.g.a
Level $2005$
Weight $1$
Character orbit 2005.g
Analytic conductor $1.001$
Analytic rank $0$
Dimension $2$
Projective image $D_{4}$
RM discriminant 401
Inner twists $4$

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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: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.0.20100125.1

$q$-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.

\(f(q)\) \(=\) \( q + ( - i + 1) q^{2} - i q^{4} + q^{5} + (i - 1) q^{7} + q^{8} + i q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - i + 1) q^{2} - i q^{4} + q^{5} + (i - 1) q^{7} + q^{8} + i q^{9} + ( - i + 1) q^{10} + (2 i + 1) q^{14} + q^{16} + (i + 1) q^{18} - i q^{20} + q^{25} + (i + 1) q^{28} - i q^{29} + ( - i + 1) q^{32} + (i - 1) q^{35} + q^{36} + ( - i - 1) q^{43} + i q^{45} + (i - 1) q^{47} - i q^{49} + ( - i + 1) q^{50} + ( - 2 i - 2) q^{58} + ( - i - 1) q^{63} - i q^{64} + (2 i + 1) q^{70} + ( - i - 1) q^{73} + q^{80} - q^{81} + (i + 1) q^{83} - q^{86} + (i + 1) q^{90} + (2 i + 1) q^{94} + ( - i - 1) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{5} - 2 q^{7} + 2 q^{10} + 2 q^{16} + 2 q^{18} + 2 q^{25} + 2 q^{28} + 2 q^{32} - 2 q^{35} + 2 q^{36} - 2 q^{43} - 2 q^{47} + 2 q^{50} - 4 q^{58} - 2 q^{63} - 2 q^{73} + 2 q^{80} - 2 q^{81} + 2 q^{83} - 4 q^{86} + 2 q^{90} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(402\) \(1206\)
\(\chi(n)\) \(-i\) \(-1\)

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
1.00000i
1.00000i
1.00000 + 1.00000i 0 1.00000i 1.00000 0 −1.00000 1.00000i 0 1.00000i 1.00000 + 1.00000i
1603.1 1.00000 1.00000i 0 1.00000i 1.00000 0 −1.00000 + 1.00000i 0 1.00000i 1.00000 1.00000i
\(n\): e.g. 2-40 or 990-1000
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.a 2
5.c odd 4 1 inner 2005.1.g.a 2
401.b even 2 1 RM 2005.1.g.a 2
2005.g odd 4 1 inner 2005.1.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2005.1.g.a 2 1.a even 1 1 trivial
2005.1.g.a 2 5.c odd 4 1 inner
2005.1.g.a 2 401.b even 2 1 RM
2005.1.g.a 2 2005.g odd 4 1 inner

Hecke kernels

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

Hecke characteristic polynomials

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