Properties

Label 2205.1.ba.a
Level $2205$
Weight $1$
Character orbit 2205.ba
Analytic conductor $1.100$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -35
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,1,Mod(344,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([1, 3, 0]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.344");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2205.ba (of order \(6\), 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: \(1.10043835286\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.168781725.1

$q$-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.

\(f(q)\) \(=\) \( q - q^{3} - \zeta_{6}^{2} q^{4} - \zeta_{6}^{2} q^{5} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} - \zeta_{6}^{2} q^{4} - \zeta_{6}^{2} q^{5} + q^{9} + (\zeta_{6}^{2} - 1) q^{11} + \zeta_{6}^{2} q^{12} + ( - \zeta_{6} - 1) q^{13} + \zeta_{6}^{2} q^{15} - \zeta_{6} q^{16} - q^{17} - \zeta_{6} q^{20} - \zeta_{6} q^{25} - q^{27} + ( - \zeta_{6}^{2} + 1) q^{33} - \zeta_{6}^{2} q^{36} + (\zeta_{6} + 1) q^{39} + (\zeta_{6}^{2} + \zeta_{6}) q^{44} - \zeta_{6}^{2} q^{45} + \zeta_{6} q^{47} + \zeta_{6} q^{48} + q^{51} + (\zeta_{6}^{2} - 1) q^{52} + (\zeta_{6}^{2} + \zeta_{6}) q^{55} + \zeta_{6} q^{60} - q^{64} + (\zeta_{6}^{2} - 1) q^{65} + \zeta_{6}^{2} q^{68} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{71} + ( - \zeta_{6}^{2} - \zeta_{6}) q^{73} + \zeta_{6} q^{75} - \zeta_{6} q^{79} - q^{80} + q^{81} + \zeta_{6} q^{83} + \zeta_{6}^{2} q^{85} + ( - \zeta_{6}^{2} + 1) q^{97} + (\zeta_{6}^{2} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + q^{4} + q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + q^{4} + q^{5} + 2 q^{9} - 3 q^{11} - q^{12} - 3 q^{13} - q^{15} - q^{16} - 2 q^{17} - q^{20} - q^{25} - 2 q^{27} + 3 q^{33} + q^{36} + 3 q^{39} + q^{45} + q^{47} + q^{48} + 2 q^{51} - 3 q^{52} + q^{60} - 2 q^{64} - 3 q^{65} - q^{68} + q^{75} - q^{79} - 2 q^{80} + 2 q^{81} + q^{83} - q^{85} + 3 q^{97} - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times\).

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(1\) \(-\zeta_{6}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
344.1
0.500000 0.866025i
0.500000 + 0.866025i
0 −1.00000 0.500000 + 0.866025i 0.500000 + 0.866025i 0 0 0 1.00000 0
1814.1 0 −1.00000 0.500000 0.866025i 0.500000 0.866025i 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
35.c odd 2 1 CM by \(\Q(\sqrt{-35}) \)
45.h odd 6 1 inner
63.o even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.1.ba.a 2
5.b even 2 1 2205.1.ba.b yes 2
7.b odd 2 1 2205.1.ba.b yes 2
7.c even 3 1 2205.1.v.b 2
7.c even 3 1 2205.1.br.b 2
7.d odd 6 1 2205.1.v.a 2
7.d odd 6 1 2205.1.br.a 2
9.d odd 6 1 2205.1.ba.b yes 2
35.c odd 2 1 CM 2205.1.ba.a 2
35.i odd 6 1 2205.1.v.b 2
35.i odd 6 1 2205.1.br.b 2
35.j even 6 1 2205.1.v.a 2
35.j even 6 1 2205.1.br.a 2
45.h odd 6 1 inner 2205.1.ba.a 2
63.i even 6 1 2205.1.v.b 2
63.j odd 6 1 2205.1.v.a 2
63.n odd 6 1 2205.1.br.a 2
63.o even 6 1 inner 2205.1.ba.a 2
63.s even 6 1 2205.1.br.b 2
315.u even 6 1 2205.1.br.a 2
315.v odd 6 1 2205.1.br.b 2
315.z even 6 1 2205.1.ba.b yes 2
315.bq even 6 1 2205.1.v.a 2
315.br odd 6 1 2205.1.v.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2205.1.v.a 2 7.d odd 6 1
2205.1.v.a 2 35.j even 6 1
2205.1.v.a 2 63.j odd 6 1
2205.1.v.a 2 315.bq even 6 1
2205.1.v.b 2 7.c even 3 1
2205.1.v.b 2 35.i odd 6 1
2205.1.v.b 2 63.i even 6 1
2205.1.v.b 2 315.br odd 6 1
2205.1.ba.a 2 1.a even 1 1 trivial
2205.1.ba.a 2 35.c odd 2 1 CM
2205.1.ba.a 2 45.h odd 6 1 inner
2205.1.ba.a 2 63.o even 6 1 inner
2205.1.ba.b yes 2 5.b even 2 1
2205.1.ba.b yes 2 7.b odd 2 1
2205.1.ba.b yes 2 9.d odd 6 1
2205.1.ba.b yes 2 315.z even 6 1
2205.1.br.a 2 7.d odd 6 1
2205.1.br.a 2 35.j even 6 1
2205.1.br.a 2 63.n odd 6 1
2205.1.br.a 2 315.u even 6 1
2205.1.br.b 2 7.c even 3 1
2205.1.br.b 2 35.i odd 6 1
2205.1.br.b 2 63.s even 6 1
2205.1.br.b 2 315.v odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{13}^{2} + 3T_{13} + 3 \) acting on \(S_{1}^{\mathrm{new}}(2205, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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