Properties

Label 2100.2.s.b
Level $2100$
Weight $2$
Character orbit 2100.s
Analytic conductor $16.769$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(1457,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 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("2100.1457");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2100 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2100.s (of order \(4\), degree \(2\), 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: \(16.7685844245\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 420)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{3} - 24 q^{13} + 4 q^{21} - 8 q^{27} - 16 q^{31} + 20 q^{33} - 32 q^{37} + 8 q^{43} + 52 q^{51} + 28 q^{57} - 8 q^{63} + 24 q^{67} - 12 q^{81} + 20 q^{87} - 24 q^{91} - 20 q^{93} + 104 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1457.1 0 −1.71916 0.210923i 0 0 0 0.707107 + 0.707107i 0 2.91102 + 0.725222i 0
1457.2 0 −1.61532 + 0.625101i 0 0 0 −0.707107 0.707107i 0 2.21850 2.01947i 0
1457.3 0 −1.16966 1.27746i 0 0 0 −0.707107 0.707107i 0 −0.263792 + 2.98838i 0
1457.4 0 −0.625101 + 1.61532i 0 0 0 −0.707107 0.707107i 0 −2.21850 2.01947i 0
1457.5 0 −0.587996 1.62919i 0 0 0 0.707107 + 0.707107i 0 −2.30852 + 1.91591i 0
1457.6 0 0.210923 + 1.71916i 0 0 0 0.707107 + 0.707107i 0 −2.91102 + 0.725222i 0
1457.7 0 0.522921 1.65123i 0 0 0 0.707107 + 0.707107i 0 −2.45311 1.72692i 0
1457.8 0 1.04182 1.38370i 0 0 0 −0.707107 0.707107i 0 −0.829228 2.88312i 0
1457.9 0 1.27746 + 1.16966i 0 0 0 −0.707107 0.707107i 0 0.263792 + 2.98838i 0
1457.10 0 1.38370 1.04182i 0 0 0 −0.707107 0.707107i 0 0.829228 2.88312i 0
1457.11 0 1.62919 + 0.587996i 0 0 0 0.707107 + 0.707107i 0 2.30852 + 1.91591i 0
1457.12 0 1.65123 0.522921i 0 0 0 0.707107 + 0.707107i 0 2.45311 1.72692i 0
1793.1 0 −1.71916 + 0.210923i 0 0 0 0.707107 0.707107i 0 2.91102 0.725222i 0
1793.2 0 −1.61532 0.625101i 0 0 0 −0.707107 + 0.707107i 0 2.21850 + 2.01947i 0
1793.3 0 −1.16966 + 1.27746i 0 0 0 −0.707107 + 0.707107i 0 −0.263792 2.98838i 0
1793.4 0 −0.625101 1.61532i 0 0 0 −0.707107 + 0.707107i 0 −2.21850 + 2.01947i 0
1793.5 0 −0.587996 + 1.62919i 0 0 0 0.707107 0.707107i 0 −2.30852 1.91591i 0
1793.6 0 0.210923 1.71916i 0 0 0 0.707107 0.707107i 0 −2.91102 0.725222i 0
1793.7 0 0.522921 + 1.65123i 0 0 0 0.707107 0.707107i 0 −2.45311 + 1.72692i 0
1793.8 0 1.04182 + 1.38370i 0 0 0 −0.707107 + 0.707107i 0 −0.829228 + 2.88312i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1457.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.s.b 24
3.b odd 2 1 inner 2100.2.s.b 24
5.b even 2 1 420.2.s.a 24
5.c odd 4 1 420.2.s.a 24
5.c odd 4 1 inner 2100.2.s.b 24
15.d odd 2 1 420.2.s.a 24
15.e even 4 1 420.2.s.a 24
15.e even 4 1 inner 2100.2.s.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.s.a 24 5.b even 2 1
420.2.s.a 24 5.c odd 4 1
420.2.s.a 24 15.d odd 2 1
420.2.s.a 24 15.e even 4 1
2100.2.s.b 24 1.a even 1 1 trivial
2100.2.s.b 24 3.b odd 2 1 inner
2100.2.s.b 24 5.c odd 4 1 inner
2100.2.s.b 24 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{12} + 70T_{11}^{10} + 1489T_{11}^{8} + 14112T_{11}^{6} + 65032T_{11}^{4} + 137504T_{11}^{2} + 99856 \) acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\). Copy content Toggle raw display