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Properties

Label 2100.1.m.a
Level $2100$
Weight $1$
Character orbit 2100.m
Self dual yes
Analytic conductor $1.048$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -20, -84, 105
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,1,Mod(251,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("2100.251");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2100.m (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: \(1.04803652653\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-5}, \sqrt{-21})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.42000.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} + q^{4} + q^{6} + q^{7} - q^{8} + q^{9} - q^{12} - q^{14} + q^{16} - q^{18} - q^{21} + 2 q^{23} + q^{24} - q^{27} + q^{28} - q^{32} + q^{36} - 2 q^{41} + q^{42} - 2 q^{46} - q^{48} + q^{49} + q^{54} - q^{56} + q^{63} + q^{64} - 2 q^{69} - q^{72} + q^{81} + 2 q^{82} - q^{84} + 2 q^{89} + 2 q^{92} + q^{96} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2100\mathbb{Z}\right)^\times\).

\(n\) \(701\) \(1051\) \(1177\) \(1501\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
0
−1.00000 −1.00000 1.00000 0 1.00000 1.00000 −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
20.d odd 2 1 CM by \(\Q(\sqrt{-5}) \)
84.h odd 2 1 CM by \(\Q(\sqrt{-21}) \)
105.g even 2 1 RM by \(\Q(\sqrt{105}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.1.m.a 1
3.b odd 2 1 2100.1.m.c 1
4.b odd 2 1 2100.1.m.d 1
5.b even 2 1 2100.1.m.d 1
5.c odd 4 2 420.1.o.b yes 2
7.b odd 2 1 2100.1.m.b 1
12.b even 2 1 2100.1.m.b 1
15.d odd 2 1 2100.1.m.b 1
15.e even 4 2 420.1.o.a ✓ 2
20.d odd 2 1 CM 2100.1.m.a 1
20.e even 4 2 420.1.o.b yes 2
21.c even 2 1 2100.1.m.d 1
28.d even 2 1 2100.1.m.c 1
35.c odd 2 1 2100.1.m.c 1
35.f even 4 2 420.1.o.a ✓ 2
35.k even 12 4 2940.1.be.c 4
35.l odd 12 4 2940.1.be.b 4
60.h even 2 1 2100.1.m.c 1
60.l odd 4 2 420.1.o.a ✓ 2
84.h odd 2 1 CM 2100.1.m.a 1
105.g even 2 1 RM 2100.1.m.a 1
105.k odd 4 2 420.1.o.b yes 2
105.w odd 12 4 2940.1.be.b 4
105.x even 12 4 2940.1.be.c 4
140.c even 2 1 2100.1.m.b 1
140.j odd 4 2 420.1.o.a ✓ 2
140.w even 12 4 2940.1.be.b 4
140.x odd 12 4 2940.1.be.c 4
420.o odd 2 1 2100.1.m.d 1
420.w even 4 2 420.1.o.b yes 2
420.bp odd 12 4 2940.1.be.c 4
420.br even 12 4 2940.1.be.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.1.o.a ✓ 2 15.e even 4 2
420.1.o.a ✓ 2 35.f even 4 2
420.1.o.a ✓ 2 60.l odd 4 2
420.1.o.a ✓ 2 140.j odd 4 2
420.1.o.b yes 2 5.c odd 4 2
420.1.o.b yes 2 20.e even 4 2
420.1.o.b yes 2 105.k odd 4 2
420.1.o.b yes 2 420.w even 4 2
2100.1.m.a 1 1.a even 1 1 trivial
2100.1.m.a 1 20.d odd 2 1 CM
2100.1.m.a 1 84.h odd 2 1 CM
2100.1.m.a 1 105.g even 2 1 RM
2100.1.m.b 1 7.b odd 2 1
2100.1.m.b 1 12.b even 2 1
2100.1.m.b 1 15.d odd 2 1
2100.1.m.b 1 140.c even 2 1
2100.1.m.c 1 3.b odd 2 1
2100.1.m.c 1 28.d even 2 1
2100.1.m.c 1 35.c odd 2 1
2100.1.m.c 1 60.h even 2 1
2100.1.m.d 1 4.b odd 2 1
2100.1.m.d 1 5.b even 2 1
2100.1.m.d 1 21.c even 2 1
2100.1.m.d 1 420.o odd 2 1
2940.1.be.b 4 35.l odd 12 4
2940.1.be.b 4 105.w odd 12 4
2940.1.be.b 4 140.w even 12 4
2940.1.be.b 4 420.br even 12 4
2940.1.be.c 4 35.k even 12 4
2940.1.be.c 4 105.x even 12 4
2940.1.be.c 4 140.x odd 12 4
2940.1.be.c 4 420.bp odd 12 4

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2100, [\chi])\):

\( T_{23} - 2 \) Copy content Toggle raw display
\( T_{41} + 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

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