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Properties

Label 1638.2.dw.b
Level $1638$
Weight $2$
Character orbit 1638.dw
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(647,1638)]
 
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, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1638.647");
 
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.dw (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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 18 q^{2} - 18 q^{4} - 2 q^{7} - 36 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 18 q^{2} - 18 q^{4} - 2 q^{7} - 36 q^{8} + 2 q^{13} + 2 q^{14} - 18 q^{16} + 28 q^{19} + 12 q^{23} + 14 q^{25} + 4 q^{26} + 4 q^{28} - 4 q^{31} + 18 q^{32} + 24 q^{35} - 30 q^{37} + 14 q^{38} + 24 q^{41} + 14 q^{43} + 12 q^{46} - 36 q^{47} + 6 q^{49} - 14 q^{50} + 2 q^{52} + 24 q^{53} - 24 q^{55} + 2 q^{56} - 12 q^{59} + 4 q^{62} + 36 q^{64} + 12 q^{65} + 2 q^{73} - 30 q^{74} - 14 q^{76} - 16 q^{79} - 24 q^{85} - 14 q^{86} + 16 q^{91} + 12 q^{95} - 2 q^{97} + 24 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
647.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −3.10625 + 1.79339i 0 −2.54236 − 0.732384i −1.00000 0 3.58679i
647.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −3.02815 + 1.74830i 0 −2.06704 + 1.65146i −1.00000 0 3.49661i
647.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.50735 + 1.44762i 0 2.52893 − 0.777515i −1.00000 0 2.89524i
647.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.16301 + 1.24882i 0 2.08171 − 1.63294i −1.00000 0 2.49763i
647.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.01146 + 1.16131i 0 −2.27676 − 1.34772i −1.00000 0 2.32263i
647.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.63702 + 0.945133i 0 −0.416659 + 2.61274i −1.00000 0 1.89027i
647.7 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.44460 + 0.834040i 0 1.87433 + 1.86732i −1.00000 0 1.66808i
647.8 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.298936 + 0.172591i 0 2.62619 + 0.321120i −1.00000 0 0.345181i
647.9 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.0523532 + 0.0302261i 0 −2.64373 − 0.103446i −1.00000 0 0.0604523i
647.10 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.737389 − 0.425732i 0 −1.06036 − 2.42397i −1.00000 0 − 0.851463i
647.11 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.784010 − 0.452648i 0 0.675752 − 2.55800i −1.00000 0 − 0.905296i
647.12 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.935182 − 0.539928i 0 −1.83207 − 1.90880i −1.00000 0 − 1.07986i
647.13 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.972403 − 0.561417i 0 0.491957 + 2.59961i −1.00000 0 − 1.12283i
647.14 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.34689 − 0.777629i 0 −1.63858 + 2.07727i −1.00000 0 − 1.55526i
647.15 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.86587 − 1.07726i 0 −0.936807 + 2.47435i −1.00000 0 − 2.15452i
647.16 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.84052 − 1.63997i 0 2.15148 + 1.53985i −1.00000 0 − 3.27995i
647.17 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 3.19176 − 1.84276i 0 2.56349 + 0.654608i −1.00000 0 − 3.68552i
647.18 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 3.57510 − 2.06409i 0 −0.579457 − 2.58152i −1.00000 0 − 4.12817i
719.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.10625 − 1.79339i 0 −2.54236 + 0.732384i −1.00000 0 − 3.58679i
719.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.02815 − 1.74830i 0 −2.06704 − 1.65146i −1.00000 0 − 3.49661i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 647.18
Significant digits:
Format:

Inner twists

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