Properties

Label 2205.1.e.b
Level $2205$
Weight $1$
Character orbit 2205.e
Analytic conductor $1.100$
Analytic rank $0$
Dimension $4$
Projective image $D_{8}$
CM discriminant -15
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,1,Mod(244,2205)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2205, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2205.244");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2205.e (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: \(1.10043835286\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2048.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.8338372875.1

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} + ( - \beta_{3} + \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} + ( - \beta_{3} + \beta_1) q^{8} - \beta_1 q^{10} + ( - \beta_{2} + 1) q^{16} - \beta_{2} q^{17} - \beta_1 q^{19} + (\beta_{2} - 1) q^{20} - \beta_{3} q^{23} + q^{25} + \beta_{3} q^{31} - \beta_1 q^{32} + (\beta_{3} - \beta_1) q^{34} + (\beta_{2} - 2) q^{38} + ( - \beta_{3} + \beta_1) q^{40} - \beta_{2} q^{46} + \beta_{2} q^{47} - \beta_1 q^{50} + \beta_{3} q^{53} + \beta_{3} q^{61} + \beta_{2} q^{62} - q^{64} + (\beta_{2} - 2) q^{68} + ( - \beta_{3} + 2 \beta_1) q^{76} + \beta_{2} q^{79} + ( - \beta_{2} + 1) q^{80} - \beta_{2} q^{83} - \beta_{2} q^{85} - \beta_1 q^{92} + ( - \beta_{3} + \beta_1) q^{94} - \beta_1 q^{95}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} + 4 q^{5} + 4 q^{16} - 4 q^{20} + 4 q^{25} - 8 q^{38} - 4 q^{64} - 8 q^{68} + 4 q^{80}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 4x^{2} + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(442\) \(1081\) \(1226\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1
1.84776i
0.765367i
0.765367i
1.84776i
1.84776i 0 −2.41421 1.00000 0 0 2.61313i 0 1.84776i
244.2 0.765367i 0 0.414214 1.00000 0 0 1.08239i 0 0.765367i
244.3 0.765367i 0 0.414214 1.00000 0 0 1.08239i 0 0.765367i
244.4 1.84776i 0 −2.41421 1.00000 0 0 2.61313i 0 1.84776i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
15.d odd 2 1 CM by \(\Q(\sqrt{-15}) \)
21.c even 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.1.e.b yes 4
3.b odd 2 1 2205.1.e.a 4
5.b even 2 1 2205.1.e.a 4
7.b odd 2 1 2205.1.e.a 4
7.c even 3 2 2205.1.bi.a 8
7.d odd 6 2 2205.1.bi.b 8
15.d odd 2 1 CM 2205.1.e.b yes 4
21.c even 2 1 inner 2205.1.e.b yes 4
21.g even 6 2 2205.1.bi.a 8
21.h odd 6 2 2205.1.bi.b 8
35.c odd 2 1 inner 2205.1.e.b yes 4
35.i odd 6 2 2205.1.bi.a 8
35.j even 6 2 2205.1.bi.b 8
105.g even 2 1 2205.1.e.a 4
105.o odd 6 2 2205.1.bi.a 8
105.p even 6 2 2205.1.bi.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2205.1.e.a 4 3.b odd 2 1
2205.1.e.a 4 5.b even 2 1
2205.1.e.a 4 7.b odd 2 1
2205.1.e.a 4 105.g even 2 1
2205.1.e.b yes 4 1.a even 1 1 trivial
2205.1.e.b yes 4 15.d odd 2 1 CM
2205.1.e.b yes 4 21.c even 2 1 inner
2205.1.e.b yes 4 35.c odd 2 1 inner
2205.1.bi.a 8 7.c even 3 2
2205.1.bi.a 8 21.g even 6 2
2205.1.bi.a 8 35.i odd 6 2
2205.1.bi.a 8 105.o odd 6 2
2205.1.bi.b 8 7.d odd 6 2
2205.1.bi.b 8 21.h odd 6 2
2205.1.bi.b 8 35.j even 6 2
2205.1.bi.b 8 105.p even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{167} + 2 \) acting on \(S_{1}^{\mathrm{new}}(2205, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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