# Properties

 Label 273.2.cg.b Level $273$ Weight $2$ Character orbit 273.cg Analytic conductor $2.180$ Analytic rank $0$ Dimension $40$ 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: $$40$$ Relative dimension: $$10$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{7} - 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 8q^{7} - 40q^{9} + 4q^{11} - 24q^{12} - 18q^{14} + 32q^{16} + 4q^{17} + 14q^{19} + 14q^{20} + 2q^{21} + 4q^{22} + 12q^{23} + 24q^{25} - 32q^{26} + 16q^{28} + 8q^{29} + 14q^{31} - 26q^{32} - 4q^{33} - 24q^{34} + 26q^{35} + 36q^{37} - 8q^{38} + 18q^{39} - 30q^{40} - 2q^{41} - 66q^{43} - 32q^{44} - 26q^{46} - 4q^{47} + 24q^{48} - 14q^{49} - 20q^{50} + 2q^{52} - 8q^{53} - 42q^{55} + 46q^{56} - 14q^{57} + 24q^{58} + 14q^{59} + 2q^{60} + 24q^{62} + 8q^{63} + 28q^{65} - 18q^{66} - 44q^{67} - 18q^{68} + 4q^{69} - 4q^{70} - 6q^{71} + 14q^{73} - 20q^{74} + 24q^{75} - 64q^{76} + 24q^{77} + 8q^{78} + 20q^{80} + 40q^{81} + 48q^{82} - 12q^{83} + 22q^{84} + 2q^{85} - 60q^{86} + 18q^{87} - 2q^{89} - 14q^{91} + 236q^{92} - 8q^{93} + 24q^{95} + 16q^{96} - 62q^{97} - 88q^{98} - 4q^{99} + O(q^{100})$$

## 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 −2.64088 0.707621i 1.00000i 4.74145 + 2.73748i −0.792066 + 0.212233i 0.707621 2.64088i −1.19502 2.36049i −6.71797 6.71797i −1.00000 2.24193
19.2 −2.14623 0.575080i 1.00000i 2.54352 + 1.46850i 3.44337 0.922649i 0.575080 2.14623i 2.25660 + 1.38122i −1.47218 1.47218i −1.00000 −7.92086
19.3 −1.49615 0.400893i 1.00000i 0.345704 + 0.199592i 0.481371 0.128983i 0.400893 1.49615i −2.58563 0.560827i 1.75331 + 1.75331i −1.00000 −0.771912
19.4 −1.46932 0.393703i 1.00000i 0.271846 + 0.156950i −3.59085 + 0.962166i 0.393703 1.46932i 2.64173 0.145891i 1.81360 + 1.81360i −1.00000 5.65491
19.5 0.251431 + 0.0673706i 1.00000i −1.67337 0.966122i 3.35960 0.900201i −0.0673706 + 0.251431i 0.422545 2.61179i −0.723769 0.723769i −1.00000 0.905353
19.6 0.446344 + 0.119598i 1.00000i −1.54713 0.893237i −2.09830 + 0.562238i −0.119598 + 0.446344i −1.26927 2.32141i −1.23722 1.23722i −1.00000 −1.00381
19.7 1.07087 + 0.286939i 1.00000i −0.667621 0.385451i −3.71307 + 0.994915i −0.286939 + 1.07087i −1.35644 + 2.27158i −2.17220 2.17220i −1.00000 −4.26170
19.8 1.41061 + 0.377973i 1.00000i 0.114915 + 0.0663464i 1.70489 0.456824i −0.377973 + 1.41061i 1.96341 + 1.77342i −1.92826 1.92826i −1.00000 2.57761
19.9 2.18205 + 0.584679i 1.00000i 2.68745 + 1.55160i 2.27456 0.609466i −0.584679 + 2.18205i −2.33957 + 1.23548i 1.76223 + 1.76223i −1.00000 5.31955
19.10 2.39126 + 0.640737i 1.00000i 3.57554 + 2.06434i −1.06950 + 0.286571i −0.640737 + 2.39126i 0.327682 2.62538i 3.72630 + 3.72630i −1.00000 −2.74106
115.1 −2.64088 + 0.707621i 1.00000i 4.74145 2.73748i −0.792066 0.212233i 0.707621 + 2.64088i −1.19502 + 2.36049i −6.71797 + 6.71797i −1.00000 2.24193
115.2 −2.14623 + 0.575080i 1.00000i 2.54352 1.46850i 3.44337 + 0.922649i 0.575080 + 2.14623i 2.25660 1.38122i −1.47218 + 1.47218i −1.00000 −7.92086
115.3 −1.49615 + 0.400893i 1.00000i 0.345704 0.199592i 0.481371 + 0.128983i 0.400893 + 1.49615i −2.58563 + 0.560827i 1.75331 1.75331i −1.00000 −0.771912
115.4 −1.46932 + 0.393703i 1.00000i 0.271846 0.156950i −3.59085 0.962166i 0.393703 + 1.46932i 2.64173 + 0.145891i 1.81360 1.81360i −1.00000 5.65491
115.5 0.251431 0.0673706i 1.00000i −1.67337 + 0.966122i 3.35960 + 0.900201i −0.0673706 0.251431i 0.422545 + 2.61179i −0.723769 + 0.723769i −1.00000 0.905353
115.6 0.446344 0.119598i 1.00000i −1.54713 + 0.893237i −2.09830 0.562238i −0.119598 0.446344i −1.26927 + 2.32141i −1.23722 + 1.23722i −1.00000 −1.00381
115.7 1.07087 0.286939i 1.00000i −0.667621 + 0.385451i −3.71307 0.994915i −0.286939 1.07087i −1.35644 2.27158i −2.17220 + 2.17220i −1.00000 −4.26170
115.8 1.41061 0.377973i 1.00000i 0.114915 0.0663464i 1.70489 + 0.456824i −0.377973 1.41061i 1.96341 1.77342i −1.92826 + 1.92826i −1.00000 2.57761
115.9 2.18205 0.584679i 1.00000i 2.68745 1.55160i 2.27456 + 0.609466i −0.584679 2.18205i −2.33957 1.23548i 1.76223 1.76223i −1.00000 5.31955
115.10 2.39126 0.640737i 1.00000i 3.57554 2.06434i −1.06950 0.286571i −0.640737 2.39126i 0.327682 + 2.62538i 3.72630 3.72630i −1.00000 −2.74106
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 262.10 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.b yes 40
3.b odd 2 1 819.2.gh.d 40
7.d odd 6 1 273.2.bt.b 40
13.f odd 12 1 273.2.bt.b 40
21.g even 6 1 819.2.et.d 40
39.k even 12 1 819.2.et.d 40
91.w even 12 1 inner 273.2.cg.b yes 40
273.ch odd 12 1 819.2.gh.d 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.bt.b 40 7.d odd 6 1
273.2.bt.b 40 13.f odd 12 1
273.2.cg.b yes 40 1.a even 1 1 trivial
273.2.cg.b yes 40 91.w even 12 1 inner
819.2.et.d 40 21.g even 6 1
819.2.et.d 40 39.k even 12 1
819.2.gh.d 40 3.b odd 2 1
819.2.gh.d 40 273.ch odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{40} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(273, [\chi])$$.