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Properties

Label 1638.2.dg.a
Level $1638$
Weight $2$
Character orbit 1638.dg
Analytic conductor $13.079$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1638,2,Mod(17,1638)]
 
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, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.17");
 
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.dg (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: \(36\)
Relative dimension: \(18\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{2} + 36 q^{4} + 2 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 36 q^{2} + 36 q^{4} + 2 q^{7} - 36 q^{8} - 2 q^{13} - 2 q^{14} + 36 q^{16} + 14 q^{19} + 14 q^{25} + 2 q^{26} + 2 q^{28} + 4 q^{31} - 36 q^{32} - 14 q^{38} - 24 q^{41} + 14 q^{43} - 36 q^{47} + 18 q^{49} - 14 q^{50} - 2 q^{52} - 24 q^{53} + 24 q^{55} - 2 q^{56} + 30 q^{61} - 4 q^{62} + 36 q^{64} - 24 q^{65} + 12 q^{67} - 2 q^{73} + 14 q^{76} - 16 q^{79} + 24 q^{82} - 24 q^{85} - 14 q^{86} + 64 q^{91} + 36 q^{94} + 2 q^{97} - 18 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1 −1.00000 0 1.00000 −3.10625 + 1.79339i 0 −1.90544 + 1.83556i −1.00000 0 3.10625 − 1.79339i
17.2 −1.00000 0 1.00000 −3.02815 + 1.74830i 0 0.396688 + 2.61584i −1.00000 0 3.02815 − 1.74830i
17.3 −1.00000 0 1.00000 −2.50735 + 1.44762i 0 0.591115 − 2.57887i −1.00000 0 2.50735 − 1.44762i
17.4 −1.00000 0 1.00000 −2.16301 + 1.24882i 0 −0.373309 − 2.61928i −1.00000 0 2.16301 − 1.24882i
17.5 −1.00000 0 1.00000 −2.01146 + 1.16131i 0 −2.30554 + 1.29788i −1.00000 0 2.01146 − 1.16131i
17.6 −1.00000 0 1.00000 −1.63702 + 0.945133i 0 2.05437 + 1.66721i −1.00000 0 1.63702 − 0.945133i
17.7 −1.00000 0 1.00000 −1.44460 + 0.834040i 0 2.55431 − 0.689555i −1.00000 0 1.44460 − 0.834040i
17.8 −1.00000 0 1.00000 −0.298936 + 0.172591i 0 1.59119 − 2.11379i −1.00000 0 0.298936 − 0.172591i
17.9 −1.00000 0 1.00000 −0.0523532 + 0.0302261i 0 −1.41145 + 2.23781i −1.00000 0 0.0523532 − 0.0302261i
17.10 −1.00000 0 1.00000 0.737389 − 0.425732i 0 −2.62940 − 0.293686i −1.00000 0 −0.737389 + 0.425732i
17.11 −1.00000 0 1.00000 0.784010 − 0.452648i 0 −1.87742 − 1.86422i −1.00000 0 −0.784010 + 0.452648i
17.12 −1.00000 0 1.00000 0.935182 − 0.539928i 0 −2.56910 + 0.632223i −1.00000 0 −0.935182 + 0.539928i
17.13 −1.00000 0 1.00000 0.972403 − 0.561417i 0 2.49731 + 0.873758i −1.00000 0 −0.972403 + 0.561417i
17.14 −1.00000 0 1.00000 1.34689 − 0.777629i 0 0.979682 + 2.45769i −1.00000 0 −1.34689 + 0.777629i
17.15 −1.00000 0 1.00000 1.86587 − 1.07726i 0 1.67444 + 2.04847i −1.00000 0 −1.86587 + 1.07726i
17.16 −1.00000 0 1.00000 2.84052 − 1.63997i 0 2.40929 − 1.09331i −1.00000 0 −2.84052 + 1.63997i
17.17 −1.00000 0 1.00000 3.19176 − 1.84276i 0 1.84865 − 1.89275i −1.00000 0 −3.19176 + 1.84276i
17.18 −1.00000 0 1.00000 3.57510 − 2.06409i 0 −2.52539 − 0.788934i −1.00000 0 −3.57510 + 2.06409i
1349.1 −1.00000 0 1.00000 −3.10625 − 1.79339i 0 −1.90544 − 1.83556i −1.00000 0 3.10625 + 1.79339i
1349.2 −1.00000 0 1.00000 −3.02815 − 1.74830i 0 0.396688 − 2.61584i −1.00000 0 3.02815 + 1.74830i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.18
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
273.br 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.dg.a ✓ 36
3.b odd 2 1 1638.2.dg.b yes 36
7.d odd 6 1 1638.2.dw.b yes 36
13.e even 6 1 1638.2.dw.a yes 36
21.g even 6 1 1638.2.dw.a yes 36
39.h odd 6 1 1638.2.dw.b yes 36
91.l odd 6 1 1638.2.dg.b yes 36
273.br even 6 1 inner 1638.2.dg.a ✓ 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1638.2.dg.a ✓ 36 1.a even 1 1 trivial
1638.2.dg.a ✓ 36 273.br even 6 1 inner
1638.2.dg.b yes 36 3.b odd 2 1
1638.2.dg.b yes 36 91.l odd 6 1
1638.2.dw.a yes 36 13.e even 6 1
1638.2.dw.a yes 36 21.g even 6 1
1638.2.dw.b yes 36 7.d odd 6 1
1638.2.dw.b yes 36 39.h odd 6 1