# Properties

 Label 273.2.by.c Level $273$ Weight $2$ Character orbit 273.by Analytic conductor $2.180$ Analytic rank $0$ Dimension $32$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$273 = 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 273.by (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$32$$ Relative dimension: $$8$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 2q^{2} + 6q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 16q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 2q^{2} + 6q^{4} - 2q^{5} + 2q^{6} - 2q^{7} + 2q^{8} + 16q^{9} - 2q^{10} - 4q^{11} - 32q^{12} - 6q^{13} + 34q^{14} + 4q^{15} + 14q^{16} + 8q^{17} + 2q^{18} - 2q^{19} - 44q^{20} + 2q^{21} - 4q^{22} - 18q^{23} - 4q^{24} + 28q^{26} - 18q^{28} - 18q^{29} + 14q^{31} - 8q^{32} - 4q^{33} + 66q^{34} - 20q^{35} + 6q^{36} - 24q^{37} - 24q^{38} + 8q^{39} + 16q^{42} - 6q^{43} - 20q^{44} - 4q^{45} - 58q^{46} + 28q^{47} + 60q^{48} + 10q^{49} + 70q^{50} - 28q^{52} - 80q^{53} + 4q^{54} - 60q^{55} - 120q^{56} + 16q^{57} - 4q^{58} + 42q^{59} - 58q^{60} - 36q^{61} - 52q^{62} + 2q^{63} + 14q^{65} + 26q^{67} + 72q^{68} - 2q^{69} + 68q^{70} - 4q^{71} + 4q^{72} - 12q^{73} - 18q^{74} - 16q^{75} + 48q^{76} - 28q^{77} - 14q^{78} - 4q^{79} + 98q^{80} - 16q^{81} - 20q^{82} + 36q^{83} + 32q^{84} - 10q^{85} - 40q^{86} + 96q^{88} + 54q^{89} - 4q^{90} - 54q^{91} - 4q^{92} + 2q^{93} + 60q^{95} - 22q^{96} + 40q^{97} + 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1 −0.706652 + 2.63726i 0.866025 0.500000i −4.72373 2.72725i 2.18431 2.18431i 0.706652 + 2.63726i −1.85732 1.88424i 6.66928 6.66928i 0.500000 0.866025i 4.21705 + 7.30414i
76.2 −0.500642 + 1.86842i 0.866025 0.500000i −1.50830 0.870817i −2.44068 + 2.44068i 0.500642 + 1.86842i −2.60515 0.461729i −0.353388 + 0.353388i 0.500000 0.866025i −3.33831 5.78213i
76.3 −0.281068 + 1.04896i 0.866025 0.500000i 0.710736 + 0.410344i 1.02734 1.02734i 0.281068 + 1.04896i −1.22381 + 2.34570i −2.16598 + 2.16598i 0.500000 0.866025i 0.788887 + 1.36639i
76.4 −0.189683 + 0.707908i 0.866025 0.500000i 1.26690 + 0.731443i 1.23329 1.23329i 0.189683 + 0.707908i 2.32720 1.25862i −1.79455 + 1.79455i 0.500000 0.866025i 0.639122 + 1.10699i
76.5 0.0473445 0.176692i 0.866025 0.500000i 1.70307 + 0.983269i −2.80040 + 2.80040i −0.0473445 0.176692i 1.70697 + 2.02145i 0.513062 0.513062i 0.500000 0.866025i 0.362225 + 0.627392i
76.6 0.0578232 0.215799i 0.866025 0.500000i 1.68883 + 0.975044i −1.08760 + 1.08760i −0.0578232 0.215799i −0.643420 2.56632i 0.624019 0.624019i 0.500000 0.866025i 0.171815 + 0.297592i
76.7 0.385482 1.43864i 0.866025 0.500000i −0.189037 0.109141i 1.07205 1.07205i −0.385482 1.43864i −0.612570 + 2.57386i 1.87643 1.87643i 0.500000 0.866025i −1.12904 1.95556i
76.8 0.687394 2.56539i 0.866025 0.500000i −4.37666 2.52687i 1.17771 1.17771i −0.687394 2.56539i 2.40809 + 1.09594i −5.73490 + 5.73490i 0.500000 0.866025i −2.21174 3.83085i
97.1 −0.706652 2.63726i 0.866025 + 0.500000i −4.72373 + 2.72725i 2.18431 + 2.18431i 0.706652 2.63726i −1.85732 + 1.88424i 6.66928 + 6.66928i 0.500000 + 0.866025i 4.21705 7.30414i
97.2 −0.500642 1.86842i 0.866025 + 0.500000i −1.50830 + 0.870817i −2.44068 2.44068i 0.500642 1.86842i −2.60515 + 0.461729i −0.353388 0.353388i 0.500000 + 0.866025i −3.33831 + 5.78213i
97.3 −0.281068 1.04896i 0.866025 + 0.500000i 0.710736 0.410344i 1.02734 + 1.02734i 0.281068 1.04896i −1.22381 2.34570i −2.16598 2.16598i 0.500000 + 0.866025i 0.788887 1.36639i
97.4 −0.189683 0.707908i 0.866025 + 0.500000i 1.26690 0.731443i 1.23329 + 1.23329i 0.189683 0.707908i 2.32720 + 1.25862i −1.79455 1.79455i 0.500000 + 0.866025i 0.639122 1.10699i
97.5 0.0473445 + 0.176692i 0.866025 + 0.500000i 1.70307 0.983269i −2.80040 2.80040i −0.0473445 + 0.176692i 1.70697 2.02145i 0.513062 + 0.513062i 0.500000 + 0.866025i 0.362225 0.627392i
97.6 0.0578232 + 0.215799i 0.866025 + 0.500000i 1.68883 0.975044i −1.08760 1.08760i −0.0578232 + 0.215799i −0.643420 + 2.56632i 0.624019 + 0.624019i 0.500000 + 0.866025i 0.171815 0.297592i
97.7 0.385482 + 1.43864i 0.866025 + 0.500000i −0.189037 + 0.109141i 1.07205 + 1.07205i −0.385482 + 1.43864i −0.612570 2.57386i 1.87643 + 1.87643i 0.500000 + 0.866025i −1.12904 + 1.95556i
97.8 0.687394 + 2.56539i 0.866025 + 0.500000i −4.37666 + 2.52687i 1.17771 + 1.17771i −0.687394 + 2.56539i 2.40809 1.09594i −5.73490 5.73490i 0.500000 + 0.866025i −2.21174 + 3.83085i
202.1 −2.16866 + 0.581092i −0.866025 0.500000i 2.63339 1.52039i −1.87790 + 1.87790i 2.16866 + 0.581092i 2.25300 1.38707i −1.65230 + 1.65230i 0.500000 + 0.866025i 2.98130 5.16377i
202.2 −2.11902 + 0.567791i −0.866025 0.500000i 2.43582 1.40632i 3.00219 3.00219i 2.11902 + 0.567791i −2.49394 0.883331i −1.26060 + 1.26060i 0.500000 + 0.866025i −4.65710 + 8.06633i
202.3 −1.38083 + 0.369991i −0.866025 0.500000i 0.0377371 0.0217876i −0.512287 + 0.512287i 1.38083 + 0.369991i −1.54136 2.15040i 1.97762 1.97762i 0.500000 + 0.866025i 0.517838 0.896921i
202.4 −1.11595 + 0.299019i −0.866025 0.500000i −0.576113 + 0.332619i −0.549341 + 0.549341i 1.11595 + 0.299019i 0.347225 + 2.62287i 2.17732 2.17732i 0.500000 + 0.866025i 0.448776 0.777302i
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.8 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.bc 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.by.c 32
3.b odd 2 1 819.2.fm.f 32
7.b odd 2 1 273.2.by.d yes 32
13.f odd 12 1 273.2.by.d yes 32
21.c even 2 1 819.2.fm.e 32
39.k even 12 1 819.2.fm.e 32
91.bc even 12 1 inner 273.2.by.c 32
273.ca odd 12 1 819.2.fm.f 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.by.c 32 1.a even 1 1 trivial
273.2.by.c 32 91.bc even 12 1 inner
273.2.by.d yes 32 7.b odd 2 1
273.2.by.d yes 32 13.f odd 12 1
819.2.fm.e 32 21.c even 2 1
819.2.fm.e 32 39.k even 12 1
819.2.fm.f 32 3.b odd 2 1
819.2.fm.f 32 273.ca odd 12 1

## Hecke kernels

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

 $$T_{2}^{32} + \cdots$$ $$T_{5}^{32} + \cdots$$