# Properties

 Label 273.2.u.c Level $273$ Weight $2$ Character orbit 273.u Analytic conductor $2.180$ Analytic rank $0$ Dimension $64$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.u (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.

 $$\operatorname{Tr}(f)(q) =$$ $$64q - 40q^{4} - 6q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$64q - 40q^{4} - 6q^{9} + 18q^{15} - 32q^{16} + 24q^{22} - 80q^{25} + 12q^{28} + 20q^{30} + 50q^{36} - 36q^{37} - 60q^{39} + 48q^{43} - 84q^{46} - 16q^{49} + 52q^{51} - 48q^{58} - 36q^{63} + 176q^{64} + 60q^{67} - 66q^{72} + 166q^{78} - 104q^{79} - 30q^{81} + 60q^{84} + 132q^{85} - 8q^{91} - 162q^{93} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1 −1.35301 2.34347i −1.31833 + 1.12339i −2.66125 + 4.60942i 0.967216i 4.41634 + 1.56953i −2.52716 + 0.783246i 8.99072 0.476002 2.96200i 2.26665 1.30865i
62.2 −1.35301 2.34347i 1.31833 1.12339i −2.66125 + 4.60942i 0.967216i −4.41634 1.56953i −0.585267 2.58021i 8.99072 0.476002 2.96200i −2.26665 + 1.30865i
62.3 −1.25961 2.18170i −0.786041 1.54342i −2.17322 + 3.76413i 2.08918i −2.37718 + 3.65901i 2.30363 + 1.30126i 5.91120 −1.76428 + 2.42638i 4.55798 2.63155i
62.4 −1.25961 2.18170i 0.786041 + 1.54342i −2.17322 + 3.76413i 2.08918i 2.37718 3.65901i 2.27874 + 1.34438i 5.91120 −1.76428 + 2.42638i −4.55798 + 2.63155i
62.5 −1.03726 1.79658i −1.69497 + 0.356466i −1.15180 + 1.99498i 4.16126i 2.39854 + 2.67541i 1.87191 1.86974i 0.629840 2.74586 1.20840i −7.47605 + 4.31630i
62.6 −1.03726 1.79658i 1.69497 0.356466i −1.15180 + 1.99498i 4.16126i −2.39854 2.67541i −0.683288 + 2.55600i 0.629840 2.74586 1.20840i 7.47605 4.31630i
62.7 −0.879183 1.52279i −1.61457 0.627032i −0.545926 + 0.945572i 0.0518379i 0.464663 + 3.00992i −1.90128 + 1.83988i −1.59686 2.21366 + 2.02477i −0.0789383 + 0.0455750i
62.8 −0.879183 1.52279i 1.61457 + 0.627032i −0.545926 + 0.945572i 0.0518379i −0.464663 3.00992i 0.642741 2.56649i −1.59686 2.21366 + 2.02477i 0.0789383 0.0455750i
62.9 −0.835434 1.44701i −0.710596 + 1.57957i −0.395901 + 0.685720i 2.15210i 2.87932 0.291388i −0.338674 2.62399i −2.01874 −1.99011 2.24488i 3.11411 1.79793i
62.10 −0.835434 1.44701i 0.710596 1.57957i −0.395901 + 0.685720i 2.15210i −2.87932 + 0.291388i −2.44178 + 1.01869i −2.01874 −1.99011 2.24488i −3.11411 + 1.79793i
62.11 −0.584057 1.01162i −0.340045 1.69834i 0.317754 0.550366i 3.55045i −1.51947 + 1.33593i −1.36289 2.26771i −3.07858 −2.76874 + 1.15503i 3.59170 2.07367i
62.12 −0.584057 1.01162i 0.340045 + 1.69834i 0.317754 0.550366i 3.55045i 1.51947 1.33593i −2.64534 0.0464413i −3.07858 −2.76874 + 1.15503i −3.59170 + 2.07367i
62.13 −0.376715 0.652489i −1.73035 0.0766697i 0.716172 1.24045i 2.14750i 0.601824 + 1.15792i 2.55544 + 0.685379i −2.58603 2.98824 + 0.265331i 1.40122 0.808996i
62.14 −0.376715 0.652489i 1.73035 + 0.0766697i 0.716172 1.24045i 2.14750i −0.601824 1.15792i 1.87127 + 1.87038i −2.58603 2.98824 + 0.265331i −1.40122 + 0.808996i
62.15 −0.230023 0.398412i −0.688459 1.58935i 0.894179 1.54876i 2.35202i −0.474854 + 0.639877i 1.77437 1.96255i −1.74282 −2.05205 + 2.18840i −0.937075 + 0.541020i
62.16 −0.230023 0.398412i 0.688459 + 1.58935i 0.894179 1.54876i 2.35202i 0.474854 0.639877i −0.812435 + 2.51793i −1.74282 −2.05205 + 2.18840i 0.937075 0.541020i
62.17 0.230023 + 0.398412i −1.03219 1.39090i 0.894179 1.54876i 2.35202i 0.316723 0.731174i −0.812435 + 2.51793i 1.74282 −0.869185 + 2.87133i 0.937075 0.541020i
62.18 0.230023 + 0.398412i 1.03219 + 1.39090i 0.894179 1.54876i 2.35202i −0.316723 + 0.731174i 1.77437 1.96255i 1.74282 −0.869185 + 2.87133i −0.937075 + 0.541020i
62.19 0.376715 + 0.652489i −0.798779 + 1.53686i 0.716172 1.24045i 2.14750i −1.30370 + 0.0577652i 2.55544 + 0.685379i 2.58603 −1.72391 2.45523i 1.40122 0.808996i
62.20 0.376715 + 0.652489i 0.798779 1.53686i 0.716172 1.24045i 2.14750i 1.30370 0.0577652i 1.87127 + 1.87038i 2.58603 −1.72391 2.45523i −1.40122 + 0.808996i
See all 64 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 251.32 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
7.b odd 2 1 inner
13.e even 6 1 inner
21.c even 2 1 inner
39.h odd 6 1 inner
91.t odd 6 1 inner
273.u 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.u.c 64
3.b odd 2 1 inner 273.2.u.c 64
7.b odd 2 1 inner 273.2.u.c 64
13.e even 6 1 inner 273.2.u.c 64
21.c even 2 1 inner 273.2.u.c 64
39.h odd 6 1 inner 273.2.u.c 64
91.t odd 6 1 inner 273.2.u.c 64
273.u even 6 1 inner 273.2.u.c 64

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.u.c 64 1.a even 1 1 trivial
273.2.u.c 64 3.b odd 2 1 inner
273.2.u.c 64 7.b odd 2 1 inner
273.2.u.c 64 13.e even 6 1 inner
273.2.u.c 64 21.c even 2 1 inner
273.2.u.c 64 39.h odd 6 1 inner
273.2.u.c 64 91.t odd 6 1 inner
273.2.u.c 64 273.u even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$:

 $$T_{2}^{32} + \cdots$$ $$31\!\cdots\!33$$$$T_{19}^{12} +$$$$11\!\cdots\!80$$$$T_{19}^{10} +$$$$27\!\cdots\!24$$$$T_{19}^{8} +$$$$28\!\cdots\!56$$$$T_{19}^{6} +$$$$21\!\cdots\!28$$$$T_{19}^{4} +$$$$50\!\cdots\!76$$$$T_{19}^{2} + 937519681536$$">$$T_{19}^{32} + \cdots$$