Properties

Label 2205.2.d.a
Level $2205$
Weight $2$
Character orbit 2205.d
Analytic conductor $17.607$
Analytic rank $0$
Dimension $2$
CM discriminant -15
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2205,2,Mod(1324,2205)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2205.1324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.d (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: \(17.6070136457\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 45)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - 3 q^{4} + \beta q^{5} - \beta q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - 3 q^{4} + \beta q^{5} - \beta q^{8} - 5 q^{10} - q^{16} + 2 \beta q^{17} + 4 q^{19} - 3 \beta q^{20} + 4 \beta q^{23} - 5 q^{25} - 8 q^{31} - 3 \beta q^{32} - 10 q^{34} + 4 \beta q^{38} + 5 q^{40} - 20 q^{46} - 4 \beta q^{47} - 5 \beta q^{50} - 2 \beta q^{53} - 2 q^{61} - 8 \beta q^{62} + 13 q^{64} - 6 \beta q^{68} - 12 q^{76} - 16 q^{79} - \beta q^{80} + 8 \beta q^{83} - 10 q^{85} - 12 \beta q^{92} + 20 q^{94} + 4 \beta q^{95} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{4} - 10 q^{10} - 2 q^{16} + 8 q^{19} - 10 q^{25} - 16 q^{31} - 20 q^{34} + 10 q^{40} - 40 q^{46} - 4 q^{61} + 26 q^{64} - 24 q^{76} - 32 q^{79} - 20 q^{85} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1324.1
2.23607i
2.23607i
2.23607i 0 −3.00000 2.23607i 0 0 2.23607i 0 −5.00000
1324.2 2.23607i 0 −3.00000 2.23607i 0 0 2.23607i 0 −5.00000
\(n\): 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 \(\Q(\sqrt{-15}) \)
3.b odd 2 1 inner
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2205.2.d.a 2
3.b odd 2 1 inner 2205.2.d.a 2
5.b even 2 1 inner 2205.2.d.a 2
7.b odd 2 1 45.2.b.a 2
15.d odd 2 1 CM 2205.2.d.a 2
21.c even 2 1 45.2.b.a 2
28.d even 2 1 720.2.f.d 2
35.c odd 2 1 45.2.b.a 2
35.f even 4 2 225.2.a.f 2
56.e even 2 1 2880.2.f.j 2
56.h odd 2 1 2880.2.f.k 2
63.l odd 6 2 405.2.j.c 4
63.o even 6 2 405.2.j.c 4
84.h odd 2 1 720.2.f.d 2
105.g even 2 1 45.2.b.a 2
105.k odd 4 2 225.2.a.f 2
140.c even 2 1 720.2.f.d 2
140.j odd 4 2 3600.2.a.bs 2
168.e odd 2 1 2880.2.f.j 2
168.i even 2 1 2880.2.f.k 2
280.c odd 2 1 2880.2.f.k 2
280.n even 2 1 2880.2.f.j 2
315.z even 6 2 405.2.j.c 4
315.bg odd 6 2 405.2.j.c 4
420.o odd 2 1 720.2.f.d 2
420.w even 4 2 3600.2.a.bs 2
840.b odd 2 1 2880.2.f.j 2
840.u even 2 1 2880.2.f.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.2.b.a 2 7.b odd 2 1
45.2.b.a 2 21.c even 2 1
45.2.b.a 2 35.c odd 2 1
45.2.b.a 2 105.g even 2 1
225.2.a.f 2 35.f even 4 2
225.2.a.f 2 105.k odd 4 2
405.2.j.c 4 63.l odd 6 2
405.2.j.c 4 63.o even 6 2
405.2.j.c 4 315.z even 6 2
405.2.j.c 4 315.bg odd 6 2
720.2.f.d 2 28.d even 2 1
720.2.f.d 2 84.h odd 2 1
720.2.f.d 2 140.c even 2 1
720.2.f.d 2 420.o odd 2 1
2205.2.d.a 2 1.a even 1 1 trivial
2205.2.d.a 2 3.b odd 2 1 inner
2205.2.d.a 2 5.b even 2 1 inner
2205.2.d.a 2 15.d odd 2 1 CM
2880.2.f.j 2 56.e even 2 1
2880.2.f.j 2 168.e odd 2 1
2880.2.f.j 2 280.n even 2 1
2880.2.f.j 2 840.b odd 2 1
2880.2.f.k 2 56.h odd 2 1
2880.2.f.k 2 168.i even 2 1
2880.2.f.k 2 280.c odd 2 1
2880.2.f.k 2 840.u even 2 1
3600.2.a.bs 2 140.j odd 4 2
3600.2.a.bs 2 420.w even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2205, [\chi])\):

\( T_{2}^{2} + 5 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{19} - 4 \) Copy content Toggle raw display
\( T_{29} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 5 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 5 \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 20 \) Copy content Toggle raw display
$19$ \( (T - 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 80 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) 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
$47$ \( T^{2} + 80 \) Copy content Toggle raw display
$53$ \( T^{2} + 20 \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( (T + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 320 \) 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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