↑

Properties

Label 666.2.br.a
Level $666$
Weight $2$
Character orbit 666.br
Analytic conductor $5.318$
Analytic rank $0$
Dimension $456$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [666,2,Mod(59,666)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(666, base_ring=CyclotomicField(36))
 
chi = DirichletCharacter(H, H._module([30, 31]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("666.59");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 666 = 2 \cdot 3^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 666.br (of order \(36\), degree \(12\), 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: \(5.31803677462\)
Analytic rank: \(0\)
Dimension: \(456\)
Relative dimension: \(38\) over \(\Q(\zeta_{36})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{36}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 456 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 456 q - 24 q^{9} + 24 q^{12} + 12 q^{15} + 72 q^{21} - 108 q^{23} + 12 q^{28} - 36 q^{31} + 108 q^{35} + 12 q^{37} + 12 q^{42} + 144 q^{45} + 12 q^{49} + 36 q^{54} - 36 q^{57} + 108 q^{62} - 36 q^{63} + 72 q^{67} + 12 q^{69} - 144 q^{71} - 36 q^{75} + 96 q^{78} - 36 q^{80} + 24 q^{81} - 108 q^{83} + 36 q^{87} - 84 q^{91} - 48 q^{93} - 144 q^{95} - 24 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
59.1 −0.573576 + 0.819152i −1.73197 + 0.0162271i −0.342020 − 0.939693i 3.39614 − 0.297124i 0.980127 − 1.42806i 3.24525 − 1.18117i 0.965926 + 0.258819i 2.99947 − 0.0562099i −1.70456 + 2.95238i
59.2 −0.573576 + 0.819152i −1.61395 + 0.628629i −0.342020 − 0.939693i −0.369704 + 0.0323449i 0.410780 − 1.68263i 0.441018 − 0.160518i 0.965926 + 0.258819i 2.20965 − 2.02915i 0.185558 − 0.321396i
59.3 −0.573576 + 0.819152i −1.55753 − 0.757699i −0.342020 − 0.939693i −1.31757 + 0.115272i 1.51403 − 0.841254i −3.18428 + 1.15898i 0.965926 + 0.258819i 1.85179 + 2.36027i 0.661300 − 1.14541i
59.4 −0.573576 + 0.819152i −1.52824 − 0.815154i −0.342020 − 0.939693i −3.87422 + 0.338950i 1.54430 − 0.784310i 2.54757 − 0.927241i 0.965926 + 0.258819i 1.67105 + 2.49150i 1.94451 − 3.36799i
59.5 −0.573576 + 0.819152i −1.33942 + 1.09816i −0.342020 − 0.939693i 2.70467 − 0.236628i −0.131295 − 1.72707i −4.37564 + 1.59260i 0.965926 + 0.258819i 0.588108 − 2.94179i −1.35750 + 2.35126i
59.6 −0.573576 + 0.819152i −0.901422 − 1.47900i −0.342020 − 0.939693i 3.11371 − 0.272415i 1.72856 + 0.109917i −0.502752 + 0.182987i 0.965926 + 0.258819i −1.37488 + 2.66640i −1.56280 + 2.70685i
59.7 −0.573576 + 0.819152i −0.814880 + 1.52839i −0.342020 − 0.939693i −0.137103 + 0.0119949i −0.784586 − 1.54416i 0.0331629 − 0.0120703i 0.965926 + 0.258819i −1.67194 − 2.49091i 0.0688132 − 0.119188i
59.8 −0.573576 + 0.819152i −0.393058 + 1.68686i −0.342020 − 0.939693i −1.34807 + 0.117941i −1.15635 − 1.28952i 2.30784 − 0.839987i 0.965926 + 0.258819i −2.69101 − 1.32607i 0.676612 − 1.17193i
59.9 −0.573576 + 0.819152i −0.349422 − 1.69644i −0.342020 − 0.939693i −0.217156 + 0.0189986i 1.59006 + 0.686807i −1.03202 + 0.375623i 0.965926 + 0.258819i −2.75581 + 1.18555i 0.108993 − 0.188781i
59.10 −0.573576 + 0.819152i 0.0393754 − 1.73160i −0.342020 − 0.939693i −3.52228 + 0.308159i 1.39586 + 1.02546i 2.86629 − 1.04324i 0.965926 + 0.258819i −2.99690 − 0.136365i 1.76787 − 3.06203i
59.11 −0.573576 + 0.819152i 0.168508 + 1.72383i −0.342020 − 0.939693i 3.46637 − 0.303268i −1.50873 − 0.850717i 1.17422 − 0.427382i 0.965926 + 0.258819i −2.94321 + 0.580960i −1.73980 + 3.01343i
59.12 −0.573576 + 0.819152i 0.324665 + 1.70135i −0.342020 − 0.939693i −4.44102 + 0.388539i −1.57988 − 0.709904i −2.00251 + 0.728854i 0.965926 + 0.258819i −2.78919 + 1.10474i 2.22899 − 3.86073i
59.13 −0.573576 + 0.819152i 0.509851 − 1.65531i −0.342020 − 0.939693i 0.961323 − 0.0841048i 1.06351 + 1.36709i 1.98241 − 0.721536i 0.965926 + 0.258819i −2.48010 − 1.68792i −0.482497 + 0.835710i
59.14 −0.573576 + 0.819152i 1.09508 − 1.34194i −0.342020 − 0.939693i −1.72857 + 0.151231i 0.471138 + 1.66674i −3.93447 + 1.43203i 0.965926 + 0.258819i −0.601593 − 2.93906i 0.867588 − 1.50271i
59.15 −0.573576 + 0.819152i 1.33169 + 1.10752i −0.342020 − 0.939693i 0.677577 − 0.0592803i −1.67105 + 0.455608i 2.48877 − 0.905837i 0.965926 + 0.258819i 0.546793 + 2.94975i −0.340082 + 0.589040i
59.16 −0.573576 + 0.819152i 1.48178 + 0.896847i −0.342020 − 0.939693i −0.00217913 0.000190649i −1.58457 + 0.699391i −3.24095 + 1.17961i 0.965926 + 0.258819i 1.39133 + 2.65786i 0.00109373 − 0.00189439i
59.17 −0.573576 + 0.819152i 1.54546 − 0.782021i −0.342020 − 0.939693i 1.96664 − 0.172059i −0.245844 + 1.71451i 3.49159 − 1.27083i 0.965926 + 0.258819i 1.77689 − 2.41716i −0.987077 + 1.70967i
59.18 −0.573576 + 0.819152i 1.67682 − 0.433887i −0.342020 − 0.939693i −2.40694 + 0.210580i −0.606368 + 1.62244i −2.04756 + 0.745251i 0.965926 + 0.258819i 2.62348 − 1.45511i 1.20807 − 2.09243i
59.19 −0.573576 + 0.819152i 1.71465 + 0.244926i −0.342020 − 0.939693i 3.97155 − 0.347466i −1.18411 + 1.26407i −2.10877 + 0.767528i 0.965926 + 0.258819i 2.88002 + 0.839923i −1.99336 + 3.45260i
59.20 0.573576 − 0.819152i −1.72993 + 0.0857246i −0.342020 − 0.939693i 1.06358 − 0.0930510i −0.922025 + 1.46624i −1.39083 + 0.506221i −0.965926 − 0.258819i 2.98530 − 0.296595i 0.533820 − 0.924604i
See next 80 embeddings (of 456 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 59.38
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
333.bs even 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 666.2.br.a ✓ 456
9.d odd 6 1 666.2.bv.a yes 456
37.i odd 36 1 666.2.bv.a yes 456
333.bs even 36 1 inner 666.2.br.a ✓ 456
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
666.2.br.a ✓ 456 1.a even 1 1 trivial
666.2.br.a ✓ 456 333.bs even 36 1 inner
666.2.bv.a yes 456 9.d odd 6 1
666.2.bv.a yes 456 37.i odd 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(666, [\chi])\).