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Properties

Label 1638.2.bq.a
Level $1638$
Weight $2$
Character orbit 1638.bq
Analytic conductor $13.079$
Analytic rank $0$
Dimension $72$
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] = [1638,2,Mod(971,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 5, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.971");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.bq (of order \(6\), 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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(72\)
Relative dimension: \(36\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 72 q + 36 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 72 q + 36 q^{4} - 4 q^{7} - 4 q^{13} - 36 q^{16} - 28 q^{25} - 8 q^{28} + 20 q^{37} + 12 q^{43} + 8 q^{46} - 20 q^{49} + 4 q^{52} - 48 q^{55} - 32 q^{58} - 72 q^{64} - 128 q^{67} + 108 q^{73} - 36 q^{76} + 64 q^{79} - 64 q^{85} - 48 q^{91} + 108 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
971.1 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.60938 + 2.78753i 0 −0.0209287 − 2.64567i 1.00000i 0 − 3.21876i
971.2 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.50876 + 2.61325i 0 −0.469438 − 2.60377i 1.00000i 0 − 3.01752i
971.3 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.24423 + 2.15508i 0 2.34035 + 1.23401i 1.00000i 0 − 2.48847i
971.4 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.02424 − 1.77403i 0 −2.04115 − 1.68336i 1.00000i 0 2.04847i
971.5 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.902939 − 1.56394i 0 −2.64566 + 0.0225266i 1.00000i 0 1.80588i
971.6 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.967738 − 1.67617i 0 1.57453 + 2.12623i 1.00000i 0 1.93548i
971.7 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.860385 + 1.49023i 0 −0.590359 + 2.57905i 1.00000i 0 − 1.72077i
971.8 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −0.367496 + 0.636523i 0 −2.39816 + 1.11752i 1.00000i 0 − 0.734993i
971.9 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.377027 − 0.653029i 0 2.26423 − 1.36866i 1.00000i 0 0.754053i
971.10 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.436307 − 0.755706i 0 −1.00297 + 2.44828i 1.00000i 0 0.872614i
971.11 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.485658 − 0.841184i 0 2.05124 − 1.67105i 1.00000i 0 0.971315i
971.12 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 0.831839 − 1.44079i 0 2.54731 + 0.714992i 1.00000i 0 1.66368i
971.13 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.08916 + 1.88647i 0 1.89938 − 1.84183i 1.00000i 0 − 2.17831i
971.14 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.29153 − 2.23700i 0 −0.899284 − 2.48823i 1.00000i 0 2.58307i
971.15 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.60179 + 2.77439i 0 1.77018 + 1.96633i 1.00000i 0 − 3.20359i
971.16 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.76190 − 3.05170i 0 −1.38950 − 2.25151i 1.00000i 0 3.52380i
971.17 −0.866025 + 0.500000i 0 0.500000 − 0.866025i −1.64601 + 2.85096i 0 −2.62279 + 0.347849i 1.00000i 0 − 3.29201i
971.18 −0.866025 + 0.500000i 0 0.500000 − 0.866025i 1.84803 − 3.20089i 0 −1.36699 + 2.26525i 1.00000i 0 3.69607i
971.19 0.866025 − 0.500000i 0 0.500000 − 0.866025i −1.84803 + 3.20089i 0 −1.36699 + 2.26525i − 1.00000i 0 3.69607i
971.20 0.866025 − 0.500000i 0 0.500000 − 0.866025i 1.64601 − 2.85096i 0 −2.62279 + 0.347849i − 1.00000i 0 − 3.29201i
See all 72 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 971.36
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.m odd 6 1 inner
273.bf even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1638.2.bq.a ✓ 72
3.b odd 2 1 inner 1638.2.bq.a ✓ 72
7.d odd 6 1 1638.2.cm.a yes 72
13.c even 3 1 1638.2.cm.a yes 72
21.g even 6 1 1638.2.cm.a yes 72
39.i odd 6 1 1638.2.cm.a yes 72
91.m odd 6 1 inner 1638.2.bq.a ✓ 72
273.bf even 6 1 inner 1638.2.bq.a ✓ 72
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.bq.a ✓ 72 1.a even 1 1 trivial
1638.2.bq.a ✓ 72 3.b odd 2 1 inner
1638.2.bq.a ✓ 72 91.m odd 6 1 inner
1638.2.bq.a ✓ 72 273.bf even 6 1 inner
1638.2.cm.a yes 72 7.d odd 6 1
1638.2.cm.a yes 72 13.c even 3 1
1638.2.cm.a yes 72 21.g even 6 1
1638.2.cm.a yes 72 39.i odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(1638, [\chi])\).