Properties

Label 2200.1.d.c
Level $2200$
Weight $1$
Character orbit 2200.d
Self dual yes
Analytic conductor $1.098$
Analytic rank $0$
Dimension $1$
Projective image $D_{3}$
CM discriminant -88
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2200,1,Mod(901,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2200.901");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2200.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: yes
Analytic conductor: \(1.09794302779\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.2200.1
Artin image: $D_6$
Artin field: Galois closure of 6.2.193600000.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{4} + q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{4} + q^{8} + q^{9} - q^{11} - q^{13} + q^{16} + q^{18} + q^{19} - q^{22} + q^{23} - q^{26} + q^{29} - q^{31} + q^{32} + q^{36} + q^{38} - q^{43} - q^{44} + q^{46} - 2 q^{47} + q^{49} - q^{52} + q^{58} - 2 q^{61} - q^{62} + q^{64} - q^{71} + q^{72} + q^{76} + q^{81} - q^{83} - q^{86} - q^{88} - q^{89} + q^{92} - 2 q^{94} + q^{97} + q^{98} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(551\) \(1101\) \(1201\)
\(\chi(n)\) \(0\) \(0\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0
1.00000 0 1.00000 0 0 0 1.00000 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
88.b odd 2 1 CM by \(\Q(\sqrt{-22}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2200.1.d.c yes 1
5.b even 2 1 2200.1.d.a 1
5.c odd 4 2 2200.1.o.a 2
8.b even 2 1 2200.1.d.b yes 1
11.b odd 2 1 2200.1.d.b yes 1
40.f even 2 1 2200.1.d.d yes 1
40.i odd 4 2 2200.1.o.b 2
55.d odd 2 1 2200.1.d.d yes 1
55.e even 4 2 2200.1.o.b 2
88.b odd 2 1 CM 2200.1.d.c yes 1
440.o odd 2 1 2200.1.d.a 1
440.t even 4 2 2200.1.o.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2200.1.d.a 1 5.b even 2 1
2200.1.d.a 1 440.o odd 2 1
2200.1.d.b yes 1 8.b even 2 1
2200.1.d.b yes 1 11.b odd 2 1
2200.1.d.c yes 1 1.a even 1 1 trivial
2200.1.d.c yes 1 88.b odd 2 1 CM
2200.1.d.d yes 1 40.f even 2 1
2200.1.d.d yes 1 55.d odd 2 1
2200.1.o.a 2 5.c odd 4 2
2200.1.o.a 2 440.t even 4 2
2200.1.o.b 2 40.i odd 4 2
2200.1.o.b 2 55.e even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2200, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{13} + 1 \) Copy content Toggle raw display
\( T_{19} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

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