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Properties

Label 273.2.bz.b
Level $273$
Weight $2$
Character orbit 273.bz
Analytic conductor $2.180$
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] = [273,2,Mod(31,273)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(273, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([0, 2, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("273.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 273 = 3 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 273.bz (of order \(12\), degree \(4\), 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: \(2.17991597518\)
Analytic rank: \(0\)
Dimension: \(36\)
Relative dimension: \(9\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 36 q + 4 q^{7} + 18 q^{9} + 4 q^{11} + 16 q^{12} - 36 q^{14} + 12 q^{16} - 4 q^{17} - 12 q^{19} - 44 q^{20} + 6 q^{21} - 8 q^{22} + 12 q^{23} - 18 q^{24} + 48 q^{25} - 28 q^{26} + 12 q^{28} - 16 q^{29} - 56 q^{32} + 4 q^{33} + 48 q^{34} + 8 q^{35} + 22 q^{37} - 16 q^{38} - 8 q^{39} + 60 q^{40} + 32 q^{41} + 12 q^{42} + 4 q^{44} - 44 q^{46} - 14 q^{47} - 6 q^{49} - 68 q^{50} + 12 q^{51} - 82 q^{52} - 8 q^{53} + 8 q^{56} - 6 q^{57} - 84 q^{58} + 70 q^{59} + 2 q^{60} + 36 q^{61} - 48 q^{62} + 2 q^{63} - 8 q^{65} + 38 q^{67} + 36 q^{68} + 8 q^{69} - 40 q^{70} - 36 q^{71} + 18 q^{72} - 46 q^{73} + 40 q^{74} - 10 q^{75} + 60 q^{76} - 60 q^{77} + 32 q^{78} - 38 q^{80} - 18 q^{81} - 24 q^{83} + 38 q^{84} + 44 q^{85} - 24 q^{86} + 36 q^{87} - 168 q^{88} + 38 q^{89} - 14 q^{91} - 40 q^{92} + 56 q^{96} - 36 q^{97} + 58 q^{98} + 8 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 −2.62411 − 0.703128i 0.866025 + 0.500000i 4.65951 + 2.69017i −0.276925 + 1.03350i −1.92098 − 1.92098i 2.15331 − 1.53729i −6.49357 − 6.49357i 0.500000 + 0.866025i 1.45336 − 2.51730i
31.2 −2.02552 − 0.542736i 0.866025 + 0.500000i 2.07611 + 1.19864i 0.220234 − 0.821926i −1.48278 − 1.48278i 0.498246 + 2.59841i −0.589092 − 0.589092i 0.500000 + 0.866025i −0.892178 + 1.54530i
31.3 −1.28847 − 0.345245i 0.866025 + 0.500000i −0.191081 − 0.110321i 0.306956 − 1.14557i −0.943228 − 0.943228i 1.14734 − 2.38403i 2.09457 + 2.09457i 0.500000 + 0.866025i −0.791009 + 1.37007i
31.4 −0.263033 − 0.0704795i 0.866025 + 0.500000i −1.66783 − 0.962923i −0.650540 + 2.42785i −0.192554 − 0.192554i −2.34911 + 1.21724i 0.755936 + 0.755936i 0.500000 + 0.866025i 0.342227 − 0.592755i
31.5 −0.170094 − 0.0455765i 0.866025 + 0.500000i −1.70520 − 0.984495i −0.0851485 + 0.317778i −0.124517 − 0.124517i 2.06944 + 1.64846i 0.494209 + 0.494209i 0.500000 + 0.866025i 0.0289665 − 0.0501714i
31.6 0.612438 + 0.164102i 0.866025 + 0.500000i −1.38390 − 0.798995i 0.690032 − 2.57523i 0.448336 + 0.448336i −1.89836 − 1.84289i −1.61311 − 1.61311i 0.500000 + 0.866025i 0.845203 − 1.46393i
31.7 1.71164 + 0.458633i 0.866025 + 0.500000i 0.987324 + 0.570032i 0.586631 − 2.18934i 1.25301 + 1.25301i 2.59066 − 0.537086i −1.07751 − 1.07751i 0.500000 + 0.866025i 2.00820 − 3.47831i
31.8 1.93173 + 0.517606i 0.866025 + 0.500000i 1.73162 + 0.999754i −0.960415 + 3.58432i 1.41413 + 1.41413i −0.534135 − 2.59127i −0.000695828 0 0.000695828i 0.500000 + 0.866025i −3.71053 + 6.42683i
31.9 2.11542 + 0.566824i 0.866025 + 0.500000i 2.42164 + 1.39814i 0.169176 − 0.631372i 1.54859 + 1.54859i −1.81137 + 1.92845i 1.23310 + 1.23310i 0.500000 + 0.866025i 0.715753 − 1.23972i
73.1 −0.677708 + 2.52924i −0.866025 − 0.500000i −4.20573 − 2.42818i 3.92082 + 1.05058i 1.85153 − 1.85153i 1.12389 + 2.39518i 5.28864 − 5.28864i 0.500000 + 0.866025i −5.31435 + 9.20472i
73.2 −0.590298 + 2.20302i −0.866025 − 0.500000i −2.77280 − 1.60088i −1.71910 − 0.460631i 1.61272 − 1.61272i 1.30433 − 2.30189i 1.93810 − 1.93810i 0.500000 + 0.866025i 2.02956 − 3.51530i
73.3 −0.246078 + 0.918377i −0.866025 − 0.500000i 0.949189 + 0.548015i 0.797354 + 0.213650i 0.672298 − 0.672298i −2.22965 + 1.42430i −2.08146 + 2.08146i 0.500000 + 0.866025i −0.392423 + 0.679697i
73.4 −0.242689 + 0.905729i −0.866025 − 0.500000i 0.970604 + 0.560379i −3.03397 − 0.812951i 0.663040 − 0.663040i 2.50344 + 0.856042i −2.06919 + 2.06919i 0.500000 + 0.866025i 1.47263 − 2.55066i
73.5 0.0608509 − 0.227099i −0.866025 − 0.500000i 1.68418 + 0.972362i −3.32462 − 0.890829i −0.166248 + 0.166248i −2.22038 − 1.43872i 0.655801 − 0.655801i 0.500000 + 0.866025i −0.404612 + 0.700809i
73.6 0.217671 − 0.812358i −0.866025 − 0.500000i 1.11951 + 0.646347i 1.60194 + 0.429239i −0.594687 + 0.594687i 0.0720542 + 2.64477i 1.95812 − 1.95812i 0.500000 + 0.866025i 0.697391 − 1.20792i
73.7 0.341564 − 1.27474i −0.866025 − 0.500000i 0.223767 + 0.129192i −0.271244 − 0.0726797i −0.933171 + 0.933171i 2.39655 + 1.12096i 2.10746 − 2.10746i 0.500000 + 0.866025i −0.185295 + 0.320940i
73.8 0.549668 − 2.05139i −0.866025 − 0.500000i −2.17401 − 1.25516i 3.81253 + 1.02157i −1.50172 + 1.50172i −0.231538 − 2.63560i −0.766371 + 0.766371i 0.500000 + 0.866025i 4.19125 − 7.25946i
73.9 0.587020 − 2.19079i −0.866025 − 0.500000i −2.72291 − 1.57208i −1.78371 − 0.477944i −1.60377 + 1.60377i −2.58471 − 0.565048i −1.83495 + 1.83495i 0.500000 + 0.866025i −2.09415 + 3.62718i
187.1 −0.677708 − 2.52924i −0.866025 + 0.500000i −4.20573 + 2.42818i 3.92082 − 1.05058i 1.85153 + 1.85153i 1.12389 − 2.39518i 5.28864 + 5.28864i 0.500000 − 0.866025i −5.31435 − 9.20472i
187.2 −0.590298 − 2.20302i −0.866025 + 0.500000i −2.77280 + 1.60088i −1.71910 + 0.460631i 1.61272 + 1.61272i 1.30433 + 2.30189i 1.93810 + 1.93810i 0.500000 − 0.866025i 2.02956 + 3.51530i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.9
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.bb even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.bz.b yes 36
3.b odd 2 1 819.2.fn.g 36
7.d odd 6 1 273.2.bz.a ✓ 36
13.d odd 4 1 273.2.bz.a ✓ 36
21.g even 6 1 819.2.fn.f 36
39.f even 4 1 819.2.fn.f 36
91.bb even 12 1 inner 273.2.bz.b yes 36
273.cb odd 12 1 819.2.fn.g 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bz.a ✓ 36 7.d odd 6 1
273.2.bz.a ✓ 36 13.d odd 4 1
273.2.bz.b yes 36 1.a even 1 1 trivial
273.2.bz.b yes 36 91.bb even 12 1 inner
819.2.fn.f 36 21.g even 6 1
819.2.fn.f 36 39.f even 4 1
819.2.fn.g 36 3.b odd 2 1
819.2.fn.g 36 273.cb odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{36} - 63 T_{2}^{32} + 40 T_{2}^{31} - 244 T_{2}^{29} + 3022 T_{2}^{28} - 2840 T_{2}^{27} + \cdots + 144 \) acting on \(S_{2}^{\mathrm{new}}(273, [\chi])\). Copy content Toggle raw display