# Properties

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

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## Newspace parameters

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{7} - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{7} - 36q^{9} + 4q^{11} + 16q^{12} + 42q^{14} + 12q^{16} - 4q^{17} - 24q^{19} - 14q^{20} + 4q^{22} - 12q^{23} - 24q^{25} - 28q^{26} - 12q^{28} + 8q^{29} - 6q^{31} + 46q^{32} + 4q^{33} + 24q^{34} - 10q^{35} - 20q^{37} + 8q^{38} - 2q^{39} - 30q^{40} - 34q^{41} + 24q^{42} + 30q^{43} - 32q^{44} - 26q^{46} + 4q^{47} - 24q^{48} - 20q^{50} + 24q^{51} + 98q^{52} - 8q^{53} + 30q^{55} - 10q^{56} - 24q^{57} - 96q^{58} - 14q^{59} - 46q^{60} + 48q^{62} - 4q^{63} + 28q^{65} + 18q^{66} + 62q^{67} - 54q^{68} - 4q^{69} - 148q^{70} + 42q^{71} - 52q^{73} - 20q^{74} - 10q^{75} - 12q^{76} - 24q^{77} - 16q^{78} + 76q^{80} + 36q^{81} + 48q^{82} + 60q^{83} + 50q^{84} + 2q^{85} + 12q^{86} + 18q^{87} + 50q^{89} + 40q^{91} - 100q^{92} - 6q^{93} + 24q^{95} - 4q^{96} - 36q^{97} + 16q^{98} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −2.38279 0.638467i 1.00000i 3.53801 + 2.04267i 0.488495 0.130892i −0.638467 + 2.38279i −2.15177 + 1.53945i −3.63751 3.63751i −1.00000 −1.24755
19.2 −1.53002 0.409967i 1.00000i 0.440828 + 0.254512i −3.63307 + 0.973479i −0.409967 + 1.53002i −1.31124 2.29797i 1.66997 + 1.66997i −1.00000 5.95776
19.3 −1.50246 0.402582i 1.00000i 0.363252 + 0.209723i 2.78312 0.745735i −0.402582 + 1.50246i 0.599883 + 2.57685i 1.73841 + 1.73841i −1.00000 −4.48173
19.4 −0.759625 0.203541i 1.00000i −1.19645 0.690770i 2.02645 0.542987i −0.203541 + 0.759625i 1.65876 2.06120i 1.88042 + 1.88042i −1.00000 −1.64986
19.5 −0.511829 0.137144i 1.00000i −1.48889 0.859611i −2.03763 + 0.545981i −0.137144 + 0.511829i 0.917679 + 2.48150i 1.39354 + 1.39354i −1.00000 1.11780
19.6 0.562494 + 0.150720i 1.00000i −1.43837 0.830442i −0.672922 + 0.180309i 0.150720 0.562494i −2.49458 0.881516i −1.50746 1.50746i −1.00000 −0.405691
19.7 1.64664 + 0.441217i 1.00000i 0.784712 + 0.453054i 1.35835 0.363968i 0.441217 1.64664i 0.501823 2.59772i −1.31861 1.31861i −1.00000 2.39731
19.8 2.02536 + 0.542694i 1.00000i 2.07553 + 1.19831i 1.89270 0.507149i 0.542694 2.02536i −1.04247 + 2.43172i 0.588043 + 0.588043i −1.00000 4.10864
19.9 2.45222 + 0.657070i 1.00000i 3.84958 + 2.22256i −2.20549 + 0.590961i 0.657070 2.45222i 2.58986 + 0.540936i 4.38935 + 4.38935i −1.00000 −5.79666
115.1 −2.38279 + 0.638467i 1.00000i 3.53801 2.04267i 0.488495 + 0.130892i −0.638467 2.38279i −2.15177 1.53945i −3.63751 + 3.63751i −1.00000 −1.24755
115.2 −1.53002 + 0.409967i 1.00000i 0.440828 0.254512i −3.63307 0.973479i −0.409967 1.53002i −1.31124 + 2.29797i 1.66997 1.66997i −1.00000 5.95776
115.3 −1.50246 + 0.402582i 1.00000i 0.363252 0.209723i 2.78312 + 0.745735i −0.402582 1.50246i 0.599883 2.57685i 1.73841 1.73841i −1.00000 −4.48173
115.4 −0.759625 + 0.203541i 1.00000i −1.19645 + 0.690770i 2.02645 + 0.542987i −0.203541 0.759625i 1.65876 + 2.06120i 1.88042 1.88042i −1.00000 −1.64986
115.5 −0.511829 + 0.137144i 1.00000i −1.48889 + 0.859611i −2.03763 0.545981i −0.137144 0.511829i 0.917679 2.48150i 1.39354 1.39354i −1.00000 1.11780
115.6 0.562494 0.150720i 1.00000i −1.43837 + 0.830442i −0.672922 0.180309i 0.150720 + 0.562494i −2.49458 + 0.881516i −1.50746 + 1.50746i −1.00000 −0.405691
115.7 1.64664 0.441217i 1.00000i 0.784712 0.453054i 1.35835 + 0.363968i 0.441217 + 1.64664i 0.501823 + 2.59772i −1.31861 + 1.31861i −1.00000 2.39731
115.8 2.02536 0.542694i 1.00000i 2.07553 1.19831i 1.89270 + 0.507149i 0.542694 + 2.02536i −1.04247 2.43172i 0.588043 0.588043i −1.00000 4.10864
115.9 2.45222 0.657070i 1.00000i 3.84958 2.22256i −2.20549 0.590961i 0.657070 + 2.45222i 2.58986 0.540936i 4.38935 4.38935i −1.00000 −5.79666
124.1 −0.639011 + 2.38482i 1.00000i −3.54699 2.04786i −0.746344 2.78539i −2.38482 0.639011i −2.52355 0.794791i 3.65872 3.65872i −1.00000 7.11959
124.2 −0.478662 + 1.78639i 1.00000i −1.23003 0.710156i −0.0199621 0.0744995i −1.78639 0.478662i 1.85948 + 1.88211i −0.758075 + 0.758075i −1.00000 0.142640
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 262.9 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.w 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.cg.a yes 36
3.b odd 2 1 819.2.gh.c 36
7.d odd 6 1 273.2.bt.a 36
13.f odd 12 1 273.2.bt.a 36
21.g even 6 1 819.2.et.c 36
39.k even 12 1 819.2.et.c 36
91.w even 12 1 inner 273.2.cg.a yes 36
273.ch odd 12 1 819.2.gh.c 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.a 36 7.d odd 6 1
273.2.bt.a 36 13.f odd 12 1
273.2.cg.a yes 36 1.a even 1 1 trivial
273.2.cg.a yes 36 91.w even 12 1 inner
819.2.et.c 36 21.g even 6 1
819.2.et.c 36 39.k even 12 1
819.2.gh.c 36 3.b odd 2 1
819.2.gh.c 36 273.ch odd 12 1

## Hecke kernels

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