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Properties

Label 2100.2.ce.d
Level $2100$
Weight $2$
Character orbit 2100.ce
Analytic conductor $16.769$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2100,2,Mod(157,2100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2100, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2100.157");
 
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.ce (of order \(12\), degree \(4\), 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: \(6\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{11} + 8 q^{21} - 12 q^{31} - 8 q^{51} - 48 q^{61} + 64 q^{71} + 12 q^{81} + 116 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1 0 −0.965926 + 0.258819i 0 0 0 2.25190 + 1.38886i 0 0.866025 − 0.500000i 0
157.2 0 −0.965926 + 0.258819i 0 0 0 −0.728357 − 2.54352i 0 0.866025 − 0.500000i 0
157.3 0 −0.965926 + 0.258819i 0 0 0 −2.48947 + 0.895840i 0 0.866025 − 0.500000i 0
157.4 0 0.965926 − 0.258819i 0 0 0 0.728357 + 2.54352i 0 0.866025 − 0.500000i 0
157.5 0 0.965926 − 0.258819i 0 0 0 2.48947 − 0.895840i 0 0.866025 − 0.500000i 0
157.6 0 0.965926 − 0.258819i 0 0 0 −2.25190 − 1.38886i 0 0.866025 − 0.500000i 0
493.1 0 −0.258819 − 0.965926i 0 0 0 1.38886 − 2.25190i 0 −0.866025 + 0.500000i 0
493.2 0 −0.258819 − 0.965926i 0 0 0 −2.54352 + 0.728357i 0 −0.866025 + 0.500000i 0
493.3 0 −0.258819 − 0.965926i 0 0 0 0.895840 + 2.48947i 0 −0.866025 + 0.500000i 0
493.4 0 0.258819 + 0.965926i 0 0 0 2.54352 − 0.728357i 0 −0.866025 + 0.500000i 0
493.5 0 0.258819 + 0.965926i 0 0 0 −1.38886 + 2.25190i 0 −0.866025 + 0.500000i 0
493.6 0 0.258819 + 0.965926i 0 0 0 −0.895840 − 2.48947i 0 −0.866025 + 0.500000i 0
1657.1 0 −0.258819 + 0.965926i 0 0 0 1.38886 + 2.25190i 0 −0.866025 − 0.500000i 0
1657.2 0 −0.258819 + 0.965926i 0 0 0 −2.54352 − 0.728357i 0 −0.866025 − 0.500000i 0
1657.3 0 −0.258819 + 0.965926i 0 0 0 0.895840 − 2.48947i 0 −0.866025 − 0.500000i 0
1657.4 0 0.258819 − 0.965926i 0 0 0 2.54352 + 0.728357i 0 −0.866025 − 0.500000i 0
1657.5 0 0.258819 − 0.965926i 0 0 0 −1.38886 − 2.25190i 0 −0.866025 − 0.500000i 0
1657.6 0 0.258819 − 0.965926i 0 0 0 −0.895840 + 2.48947i 0 −0.866025 − 0.500000i 0
1993.1 0 −0.965926 − 0.258819i 0 0 0 2.25190 − 1.38886i 0 0.866025 + 0.500000i 0
1993.2 0 −0.965926 − 0.258819i 0 0 0 −0.728357 + 2.54352i 0 0.866025 + 0.500000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 157.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
7.d odd 6 1 inner
35.i odd 6 1 inner
35.k even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2100.2.ce.d ✓ 24
5.b even 2 1 inner 2100.2.ce.d ✓ 24
5.c odd 4 2 inner 2100.2.ce.d ✓ 24
7.d odd 6 1 inner 2100.2.ce.d ✓ 24
35.i odd 6 1 inner 2100.2.ce.d ✓ 24
35.k even 12 2 inner 2100.2.ce.d ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2100.2.ce.d ✓ 24 1.a even 1 1 trivial
2100.2.ce.d ✓ 24 5.b even 2 1 inner
2100.2.ce.d ✓ 24 5.c odd 4 2 inner
2100.2.ce.d ✓ 24 7.d odd 6 1 inner
2100.2.ce.d ✓ 24 35.i odd 6 1 inner
2100.2.ce.d ✓ 24 35.k even 12 2 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2100, [\chi])\):

\( T_{11}^{6} + 2T_{11}^{5} + 24T_{11}^{4} + 20T_{11}^{3} + 460T_{11}^{2} + 600T_{11} + 900 \) Copy content Toggle raw display
\( T_{17}^{24} - 392 T_{17}^{20} + 111408 T_{17}^{16} - 14944352 T_{17}^{12} + 1468049536 T_{17}^{8} + \cdots + 656100000000 \) Copy content Toggle raw display