# Properties

 Label 273.2.bz.a Level $273$ Weight $2$ Character orbit 273.bz Analytic conductor $2.180$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.bz (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q - 6q^{7} + 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q - 6q^{7} + 18q^{9} + 4q^{11} - 16q^{12} - 36q^{14} + 12q^{16} + 4q^{17} - 18q^{19} + 44q^{20} + 2q^{21} - 8q^{22} - 12q^{23} - 18q^{24} - 48q^{25} - 32q^{26} + 4q^{28} - 16q^{29} - 6q^{31} + 76q^{32} - 4q^{33} - 48q^{34} + 8q^{35} - 8q^{37} + 16q^{38} + 10q^{39} + 60q^{40} - 32q^{41} + 12q^{42} + 4q^{44} + 28q^{46} + 14q^{47} + 6q^{49} - 68q^{50} - 12q^{51} - 62q^{52} - 8q^{53} - 8q^{56} - 6q^{57} + 36q^{58} + 26q^{59} - 46q^{60} + 36q^{61} + 48q^{62} - 8q^{65} - 40q^{67} + 36q^{68} - 8q^{69} - 64q^{70} - 36q^{71} - 18q^{72} - 8q^{73} + 40q^{74} + 10q^{75} - 60q^{76} + 60q^{77} + 32q^{78} + 26q^{80} - 18q^{81} + 24q^{83} - 18q^{84} + 44q^{85} + 48q^{86} + 36q^{87} + 168q^{88} + 10q^{89} + 4q^{91} - 40q^{92} + 6q^{93} + 76q^{96} + 36q^{97} + 38q^{98} + 8q^{99} + O(q^{100})$$

## 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.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
31.1 −2.19079 0.587020i −0.866025 0.500000i 2.72291 + 1.57208i −0.477944 + 1.78371i 1.60377 + 1.60377i 0.565048 2.58471i −1.83495 1.83495i 0.500000 + 0.866025i 2.09415 3.62718i
31.2 −2.05139 0.549668i −0.866025 0.500000i 2.17401 + 1.25516i 1.02157 3.81253i 1.50172 + 1.50172i 2.63560 0.231538i −0.766371 0.766371i 0.500000 + 0.866025i −4.19125 + 7.25946i
31.3 −1.27474 0.341564i −0.866025 0.500000i −0.223767 0.129192i −0.0726797 + 0.271244i 0.933171 + 0.933171i −1.12096 + 2.39655i 2.10746 + 2.10746i 0.500000 + 0.866025i 0.185295 0.320940i
31.4 −0.812358 0.217671i −0.866025 0.500000i −1.11951 0.646347i 0.429239 1.60194i 0.594687 + 0.594687i −2.64477 + 0.0720542i 1.95812 + 1.95812i 0.500000 + 0.866025i −0.697391 + 1.20792i
31.5 −0.227099 0.0608509i −0.866025 0.500000i −1.68418 0.972362i −0.890829 + 3.32462i 0.166248 + 0.166248i 1.43872 2.22038i 0.655801 + 0.655801i 0.500000 + 0.866025i 0.404612 0.700809i
31.6 0.905729 + 0.242689i −0.866025 0.500000i −0.970604 0.560379i −0.812951 + 3.03397i −0.663040 0.663040i −0.856042 + 2.50344i −2.06919 2.06919i 0.500000 + 0.866025i −1.47263 + 2.55066i
31.7 0.918377 + 0.246078i −0.866025 0.500000i −0.949189 0.548015i 0.213650 0.797354i −0.672298 0.672298i −1.42430 2.22965i −2.08146 2.08146i 0.500000 + 0.866025i 0.392423 0.679697i
31.8 2.20302 + 0.590298i −0.866025 0.500000i 2.77280 + 1.60088i −0.460631 + 1.71910i −1.61272 1.61272i 2.30189 + 1.30433i 1.93810 + 1.93810i 0.500000 + 0.866025i −2.02956 + 3.51530i
31.9 2.52924 + 0.677708i −0.866025 0.500000i 4.20573 + 2.42818i 1.05058 3.92082i −1.85153 1.85153i −2.39518 + 1.12389i 5.28864 + 5.28864i 0.500000 + 0.866025i 5.31435 9.20472i
73.1 −0.566824 + 2.11542i 0.866025 + 0.500000i −2.42164 1.39814i 0.631372 + 0.169176i −1.54859 + 1.54859i 1.92845 + 1.81137i 1.23310 1.23310i 0.500000 + 0.866025i −0.715753 + 1.23972i
73.2 −0.517606 + 1.93173i 0.866025 + 0.500000i −1.73162 0.999754i −3.58432 0.960415i −1.41413 + 1.41413i −2.59127 + 0.534135i −0.000695828 0 0.000695828i 0.500000 + 0.866025i 3.71053 6.42683i
73.3 −0.458633 + 1.71164i 0.866025 + 0.500000i −0.987324 0.570032i 2.18934 + 0.586631i −1.25301 + 1.25301i −0.537086 2.59066i −1.07751 + 1.07751i 0.500000 + 0.866025i −2.00820 + 3.47831i
73.4 −0.164102 + 0.612438i 0.866025 + 0.500000i 1.38390 + 0.798995i 2.57523 + 0.690032i −0.448336 + 0.448336i −1.84289 + 1.89836i −1.61311 + 1.61311i 0.500000 + 0.866025i −0.845203 + 1.46393i
73.5 0.0455765 0.170094i 0.866025 + 0.500000i 1.70520 + 0.984495i −0.317778 0.0851485i 0.124517 0.124517i 1.64846 2.06944i 0.494209 0.494209i 0.500000 + 0.866025i −0.0289665 + 0.0501714i
73.6 0.0704795 0.263033i 0.866025 + 0.500000i 1.66783 + 0.962923i −2.42785 0.650540i 0.192554 0.192554i 1.21724 + 2.34911i 0.755936 0.755936i 0.500000 + 0.866025i −0.342227 + 0.592755i
73.7 0.345245 1.28847i 0.866025 + 0.500000i 0.191081 + 0.110321i 1.14557 + 0.306956i 0.943228 0.943228i −2.38403 1.14734i 2.09457 2.09457i 0.500000 + 0.866025i 0.791009 1.37007i
73.8 0.542736 2.02552i 0.866025 + 0.500000i −2.07611 1.19864i 0.821926 + 0.220234i 1.48278 1.48278i 2.59841 0.498246i −0.589092 + 0.589092i 0.500000 + 0.866025i 0.892178 1.54530i
73.9 0.703128 2.62411i 0.866025 + 0.500000i −4.65951 2.69017i −1.03350 0.276925i 1.92098 1.92098i −1.53729 2.15331i −6.49357 + 6.49357i 0.500000 + 0.866025i −1.45336 + 2.51730i
187.1 −0.566824 2.11542i 0.866025 0.500000i −2.42164 + 1.39814i 0.631372 0.169176i −1.54859 1.54859i 1.92845 1.81137i 1.23310 + 1.23310i 0.500000 0.866025i −0.715753 1.23972i
187.2 −0.517606 1.93173i 0.866025 0.500000i −1.73162 + 0.999754i −3.58432 + 0.960415i −1.41413 1.41413i −2.59127 0.534135i −0.000695828 0 0.000695828i 0.500000 0.866025i 3.71053 + 6.42683i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.9 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.a 36
3.b odd 2 1 819.2.fn.f 36
7.d odd 6 1 273.2.bz.b yes 36
13.d odd 4 1 273.2.bz.b yes 36
21.g even 6 1 819.2.fn.g 36
39.f even 4 1 819.2.fn.g 36
91.bb even 12 1 inner 273.2.bz.a 36
273.cb odd 12 1 819.2.fn.f 36

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{36} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.