# 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 $2$

# 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{7} + 18q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{7} + 18q^{9} + 4q^{11} + 16q^{12} - 36q^{14} + 12q^{16} - 4q^{17} - 12q^{19} - 44q^{20} + 6q^{21} - 8q^{22} + 12q^{23} - 18q^{24} + 48q^{25} - 28q^{26} + 12q^{28} - 16q^{29} - 56q^{32} + 4q^{33} + 48q^{34} + 8q^{35} + 22q^{37} - 16q^{38} - 8q^{39} + 60q^{40} + 32q^{41} + 12q^{42} + 4q^{44} - 44q^{46} - 14q^{47} - 6q^{49} - 68q^{50} + 12q^{51} - 82q^{52} - 8q^{53} + 8q^{56} - 6q^{57} - 84q^{58} + 70q^{59} + 2q^{60} + 36q^{61} - 48q^{62} + 2q^{63} - 8q^{65} + 38q^{67} + 36q^{68} + 8q^{69} - 40q^{70} - 36q^{71} + 18q^{72} - 46q^{73} + 40q^{74} - 10q^{75} + 60q^{76} - 60q^{77} + 32q^{78} - 38q^{80} - 18q^{81} - 24q^{83} + 38q^{84} + 44q^{85} - 24q^{86} + 36q^{87} - 168q^{88} + 38q^{89} - 14q^{91} - 40q^{92} + 56q^{96} - 36q^{97} + 58q^{98} + 8q^{99} + O(q^{100})$$

## 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 −2.62411 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 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.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 $$T_{2}^{36} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.