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Properties

Label 483.2.m.a
Level $483$
Weight $2$
Character orbit 483.m
Analytic conductor $3.857$
Analytic rank $0$
Dimension $120$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [483,2,Mod(137,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.137");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 483 = 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 483.m (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: \(3.85677441763\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(60\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 120 q - 2 q^{3} + 52 q^{4} - 24 q^{6} + 2 q^{9} + 14 q^{12} - 16 q^{13} - 60 q^{16} - 10 q^{18} - 24 q^{24} - 40 q^{25} - 20 q^{27} - 4 q^{31} + 64 q^{36} - 12 q^{39} - 20 q^{46} + 140 q^{48} - 48 q^{49}+ \cdots + 72 q^{96}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
137.1 −2.35341 + 1.35874i 1.22585 + 1.22364i 2.69235 − 4.66329i −1.88300 − 3.26145i −4.54753 − 1.21410i −0.393331 − 2.61635i 9.19786i 0.00542953 + 3.00000i 8.86291 + 5.11701i
137.2 −2.35341 + 1.35874i 1.22585 + 1.22364i 2.69235 − 4.66329i 1.88300 + 3.26145i −4.54753 − 1.21410i 0.393331 + 2.61635i 9.19786i 0.00542953 + 3.00000i −8.86291 − 5.11701i
137.3 −2.20025 + 1.27031i −1.66604 − 0.473602i 2.22739 − 3.85796i −0.823724 − 1.42673i 4.26733 − 1.07436i −2.24449 + 1.40082i 6.23669i 2.55140 + 1.57808i 3.62479 + 2.09277i
137.4 −2.20025 + 1.27031i −1.66604 − 0.473602i 2.22739 − 3.85796i 0.823724 + 1.42673i 4.26733 − 1.07436i 2.24449 − 1.40082i 6.23669i 2.55140 + 1.57808i −3.62479 − 2.09277i
137.5 −2.14419 + 1.23795i −0.991594 + 1.42012i 2.06503 − 3.57673i −0.969003 − 1.67836i 0.368129 − 4.27255i 0.562535 + 2.58526i 5.27380i −1.03348 − 2.81637i 4.15545 + 2.39915i
137.6 −2.14419 + 1.23795i −0.991594 + 1.42012i 2.06503 − 3.57673i 0.969003 + 1.67836i 0.368129 − 4.27255i −0.562535 − 2.58526i 5.27380i −1.03348 − 2.81637i −4.15545 − 2.39915i
137.7 −2.05926 + 1.18892i 1.51759 − 0.834823i 1.82704 − 3.16453i −0.178457 − 0.309097i −2.13257 + 3.52340i 2.62376 + 0.340385i 3.93312i 1.60614 − 2.53383i 0.734981 + 0.424342i
137.8 −2.05926 + 1.18892i 1.51759 − 0.834823i 1.82704 − 3.16453i 0.178457 + 0.309097i −2.13257 + 3.52340i −2.62376 − 0.340385i 3.93312i 1.60614 − 2.53383i −0.734981 − 0.424342i
137.9 −1.78161 + 1.02861i 0.0211372 − 1.73192i 1.11608 − 1.93311i −1.27213 − 2.20340i 1.74382 + 3.10735i −0.668238 − 2.55997i 0.477615i −2.99911 − 0.0732160i 4.53288 + 2.61706i
137.10 −1.78161 + 1.02861i 0.0211372 − 1.73192i 1.11608 − 1.93311i 1.27213 + 2.20340i 1.74382 + 3.10735i 0.668238 + 2.55997i 0.477615i −2.99911 − 0.0732160i −4.53288 − 2.61706i
137.11 −1.55658 + 0.898694i 0.279659 + 1.70932i 0.615303 − 1.06574i −0.620314 − 1.07442i −1.97147 − 2.40938i −2.49407 + 0.882968i − 1.38290i −2.84358 + 0.956058i 1.93114 + 1.11495i
137.12 −1.55658 + 0.898694i 0.279659 + 1.70932i 0.615303 − 1.06574i 0.620314 + 1.07442i −1.97147 − 2.40938i 2.49407 − 0.882968i − 1.38290i −2.84358 + 0.956058i −1.93114 − 1.11495i
137.13 −1.44748 + 0.835704i 1.59432 + 0.676852i 0.396802 − 0.687281i −1.25144 − 2.16756i −2.87340 + 0.352653i 2.40151 + 1.11029i − 2.01638i 2.08374 + 2.15824i 3.62287 + 2.09167i
137.14 −1.44748 + 0.835704i 1.59432 + 0.676852i 0.396802 − 0.687281i 1.25144 + 2.16756i −2.87340 + 0.352653i −2.40151 − 1.11029i − 2.01638i 2.08374 + 2.15824i −3.62287 − 2.09167i
137.15 −1.28014 + 0.739089i −1.73064 + 0.0699156i 0.0925045 − 0.160222i −0.974027 − 1.68706i 2.16379 − 1.36860i −2.08240 − 1.63205i − 2.68288i 2.99022 − 0.241997i 2.49378 + 1.43978i
137.16 −1.28014 + 0.739089i −1.73064 + 0.0699156i 0.0925045 − 0.160222i 0.974027 + 1.68706i 2.16379 − 1.36860i 2.08240 + 1.63205i − 2.68288i 2.99022 − 0.241997i −2.49378 − 1.43978i
137.17 −1.19221 + 0.688323i 1.08950 − 1.34647i −0.0524225 + 0.0907985i −2.03563 − 3.52582i −0.372109 + 2.35521i −1.00571 + 2.44715i − 2.89763i −0.625968 − 2.93397i 4.85381 + 2.80235i
137.18 −1.19221 + 0.688323i 1.08950 − 1.34647i −0.0524225 + 0.0907985i 2.03563 + 3.52582i −0.372109 + 2.35521i 1.00571 − 2.44715i − 2.89763i −0.625968 − 2.93397i −4.85381 − 2.80235i
137.19 −0.935072 + 0.539864i −0.991443 − 1.42023i −0.417093 + 0.722427i −0.281343 − 0.487300i 1.69380 + 0.792769i −1.64560 + 2.07172i − 3.06015i −1.03408 + 2.81615i 0.526152 + 0.303774i
137.20 −0.935072 + 0.539864i −0.991443 − 1.42023i −0.417093 + 0.722427i 0.281343 + 0.487300i 1.69380 + 0.792769i 1.64560 − 2.07172i − 3.06015i −1.03408 + 2.81615i −0.526152 − 0.303774i
See next 80 embeddings (of 120 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 137.60
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.c even 3 1 inner
21.h odd 6 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner
161.f odd 6 1 inner
483.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 483.2.m.a ✓ 120
3.b odd 2 1 inner 483.2.m.a ✓ 120
7.c even 3 1 inner 483.2.m.a ✓ 120
21.h odd 6 1 inner 483.2.m.a ✓ 120
23.b odd 2 1 inner 483.2.m.a ✓ 120
69.c even 2 1 inner 483.2.m.a ✓ 120
161.f odd 6 1 inner 483.2.m.a ✓ 120
483.m even 6 1 inner 483.2.m.a ✓ 120
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.2.m.a ✓ 120 1.a even 1 1 trivial
483.2.m.a ✓ 120 3.b odd 2 1 inner
483.2.m.a ✓ 120 7.c even 3 1 inner
483.2.m.a ✓ 120 21.h odd 6 1 inner
483.2.m.a ✓ 120 23.b odd 2 1 inner
483.2.m.a ✓ 120 69.c even 2 1 inner
483.2.m.a ✓ 120 161.f odd 6 1 inner
483.2.m.a ✓ 120 483.m even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(483, [\chi])\).