# Properties

 Label 273.2.br.b Level $273$ Weight $2$ Character orbit 273.br Analytic conductor $2.180$ Analytic rank $0$ Dimension $64$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.br (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$64$$ Relative dimension: $$32$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 6q^{3} + 56q^{4} - 6q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 6q^{3} + 56q^{4} - 6q^{7} - 6q^{10} - 6q^{12} - 12q^{13} - 9q^{15} + 16q^{16} - 36q^{18} - 12q^{21} - 18q^{22} - 12q^{24} + 16q^{25} + 18q^{28} - 22q^{30} + 36q^{31} - 15q^{33} - 24q^{34} - 4q^{36} - 9q^{39} - 54q^{40} + 39q^{42} - 12q^{43} - 6q^{45} - 54q^{48} + 2q^{49} - 14q^{51} - 48q^{52} + 18q^{54} + 42q^{55} + 30q^{58} - 9q^{60} + 6q^{61} + 3q^{63} - 28q^{64} - 9q^{66} + 48q^{67} - 39q^{69} - 84q^{70} - 123q^{72} - 12q^{73} + 18q^{76} - 11q^{78} + 4q^{79} - 66q^{82} + 27q^{84} - 30q^{85} - 30q^{88} + 130q^{91} + 84q^{94} - 24q^{96} - 12q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 −2.61237 1.05832 + 1.37112i 4.82446 −0.648305 + 0.374299i −2.76471 3.58186i −2.10140 + 1.60752i −7.37851 −0.759929 + 2.90216i 1.69361 0.977806i
17.2 −2.61040 −1.58401 0.700656i 4.81421 1.65792 0.957199i 4.13490 + 1.82900i 0.549406 2.58808i −7.34624 2.01816 + 2.21969i −4.32784 + 2.49868i
17.3 −2.36811 −1.07288 + 1.35975i 3.60795 −0.0190083 + 0.0109745i 2.54070 3.22004i 2.35552 + 1.20478i −3.80781 −0.697847 2.91771i 0.0450138 0.0259887i
17.4 −2.28808 1.60560 0.649648i 3.23529 3.02328 1.74549i −3.67374 + 1.48644i 2.39720 + 1.11958i −2.82644 2.15591 2.08615i −6.91750 + 3.99382i
17.5 −2.13558 −1.59741 0.669547i 2.56071 −3.07271 + 1.77403i 3.41139 + 1.42987i −1.82045 + 1.91989i −1.19745 2.10341 + 2.13908i 6.56202 3.78859i
17.6 −1.97083 1.69172 0.371591i 1.88419 0.0906165 0.0523174i −3.33410 + 0.732343i −2.05714 1.66379i 0.228249 2.72384 1.25726i −0.178590 + 0.103109i
17.7 −1.76950 0.796103 + 1.53825i 1.13114 −1.27320 + 0.735080i −1.40871 2.72194i 1.84076 1.90042i 1.53745 −1.73244 + 2.44921i 2.25292 1.30072i
17.8 −1.59822 −0.201046 1.72034i 0.554302 −1.21627 + 0.702213i 0.321315 + 2.74948i 0.626391 2.57053i 2.31054 −2.91916 + 0.691736i 1.94386 1.12229i
17.9 −1.52138 −1.64324 + 0.547514i 0.314598 1.84777 1.06681i 2.49999 0.832977i −2.46764 + 0.954340i 2.56414 2.40046 1.79939i −2.81117 + 1.62303i
17.10 −1.27714 −0.939485 1.45512i −0.368921 2.29307 1.32390i 1.19985 + 1.85839i 1.96549 + 1.77112i 3.02544 −1.23474 + 2.73412i −2.92856 + 1.69081i
17.11 −1.27365 1.02504 1.39617i −0.377814 −1.93046 + 1.11455i −1.30554 + 1.77824i −0.0252117 + 2.64563i 3.02850 −0.898600 2.86226i 2.45873 1.41955i
17.12 −0.983060 −0.830478 + 1.51997i −1.03359 −1.05876 + 0.611275i 0.816410 1.49422i −1.76362 1.97222i 2.98220 −1.62061 2.52460i 1.04082 0.600920i
17.13 −0.732547 1.45763 + 0.935580i −1.46338 2.08780 1.20539i −1.06778 0.685356i −1.88359 + 1.85798i 2.53708 1.24938 + 2.72746i −1.52941 + 0.883004i
17.14 −0.419567 −1.72793 + 0.119408i −1.82396 −3.21070 + 1.85370i 0.724983 0.0500996i 2.36714 1.18181i 1.60441 2.97148 0.412657i 1.34710 0.777750i
17.15 −0.367418 −0.167841 + 1.72390i −1.86500 −0.333187 + 0.192366i 0.0616679 0.633392i 0.931377 + 2.47640i 1.42007 −2.94366 0.578683i 0.122419 0.0706787i
17.16 −0.0763024 0.399573 1.68533i −1.99418 3.58354 2.06896i −0.0304884 + 0.128595i −2.41424 1.08232i 0.304765 −2.68068 1.34683i −0.273433 + 0.157867i
17.17 0.0763024 1.65933 + 0.496625i −1.99418 −3.58354 + 2.06896i 0.126611 + 0.0378937i −2.41424 1.08232i −0.304765 2.50673 + 1.64813i −0.273433 + 0.157867i
17.18 0.367418 −1.57686 0.716595i −1.86500 0.333187 0.192366i −0.579367 0.263290i 0.931377 + 2.47640i −1.42007 1.97298 + 2.25994i 0.122419 0.0706787i
17.19 0.419567 −0.967375 + 1.43673i −1.82396 3.21070 1.85370i −0.405879 + 0.602804i 2.36714 1.18181i −1.60441 −1.12837 2.77971i 1.34710 0.777750i
17.20 0.732547 −0.0814203 1.73014i −1.46338 −2.08780 + 1.20539i −0.0596442 1.26741i −1.88359 + 1.85798i −2.53708 −2.98674 + 0.281736i −1.52941 + 0.883004i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 257.32 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
91.l odd 6 1 inner
273.br even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 273.2.br.b yes 64
3.b odd 2 1 inner 273.2.br.b yes 64
7.d odd 6 1 273.2.y.b 64
13.e even 6 1 273.2.y.b 64
21.g even 6 1 273.2.y.b 64
39.h odd 6 1 273.2.y.b 64
91.l odd 6 1 inner 273.2.br.b yes 64
273.br even 6 1 inner 273.2.br.b yes 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.y.b 64 7.d odd 6 1
273.2.y.b 64 13.e even 6 1
273.2.y.b 64 21.g even 6 1
273.2.y.b 64 39.h odd 6 1
273.2.br.b yes 64 1.a even 1 1 trivial
273.2.br.b yes 64 3.b odd 2 1 inner
273.2.br.b yes 64 91.l odd 6 1 inner
273.2.br.b yes 64 273.br even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{32} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.