Properties

Label 1100.1.f.b
Level $1100$
Weight $1$
Character orbit 1100.f
Self dual yes
Analytic conductor $0.549$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,1,Mod(901,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1100.901");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1100.f (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: \(0.548971513896\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 220)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.242000.1

$q$-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 \(\beta = \sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{3} + 2 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{3} + 2 q^{9} - q^{11} + \beta q^{23} - \beta q^{27} + q^{31} + \beta q^{33} + \beta q^{37} + q^{49} + q^{59} + \beta q^{67} - 3 q^{69} - q^{71} + q^{81} - q^{89} - \beta q^{93} - \beta q^{97} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{9} - 2 q^{11} + 2 q^{31} + 2 q^{49} + 2 q^{59} - 6 q^{69} - 2 q^{71} + 2 q^{81} - 2 q^{89} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
1.73205
−1.73205
0 −1.73205 0 0 0 0 0 2.00000 0
901.2 0 1.73205 0 0 0 0 0 2.00000 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
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
5.b even 2 1 inner
55.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.1.f.b 2
5.b even 2 1 inner 1100.1.f.b 2
5.c odd 4 2 220.1.e.a 2
11.b odd 2 1 CM 1100.1.f.b 2
15.e even 4 2 1980.1.p.a 2
20.e even 4 2 880.1.i.b 2
40.i odd 4 2 3520.1.i.d 2
40.k even 4 2 3520.1.i.c 2
55.d odd 2 1 inner 1100.1.f.b 2
55.e even 4 2 220.1.e.a 2
55.k odd 20 8 2420.1.q.a 8
55.l even 20 8 2420.1.q.a 8
165.l odd 4 2 1980.1.p.a 2
220.i odd 4 2 880.1.i.b 2
440.t even 4 2 3520.1.i.d 2
440.w odd 4 2 3520.1.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.1.e.a 2 5.c odd 4 2
220.1.e.a 2 55.e even 4 2
880.1.i.b 2 20.e even 4 2
880.1.i.b 2 220.i odd 4 2
1100.1.f.b 2 1.a even 1 1 trivial
1100.1.f.b 2 5.b even 2 1 inner
1100.1.f.b 2 11.b odd 2 1 CM
1100.1.f.b 2 55.d odd 2 1 inner
1980.1.p.a 2 15.e even 4 2
1980.1.p.a 2 165.l odd 4 2
2420.1.q.a 8 55.k odd 20 8
2420.1.q.a 8 55.l even 20 8
3520.1.i.c 2 40.k even 4 2
3520.1.i.c 2 440.w odd 4 2
3520.1.i.d 2 40.i odd 4 2
3520.1.i.d 2 440.t even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 3 \) acting on \(S_{1}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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