Properties

 Label 273.2.cc.a Level $273$ Weight $2$ Character orbit 273.cc Analytic conductor $2.180$ Analytic rank $0$ Dimension $112$ CM no Inner twists $4$

Related objects

Newspace parameters

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

Newform invariants

 Self dual: no Analytic conductor: $$2.17991597518$$ Analytic rank: $$0$$ Dimension: $$112$$ Relative dimension: $$28$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$112q - 12q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$112q - 12q^{6} - 48q^{10} - 32q^{13} + 4q^{15} + 40q^{16} - 16q^{18} - 8q^{19} - 4q^{21} - 16q^{22} - 88q^{24} + 24q^{27} - 72q^{30} + 16q^{31} + 48q^{34} + 12q^{36} + 48q^{37} + 56q^{39} + 32q^{40} - 28q^{45} + 72q^{46} + 24q^{48} - 144q^{52} - 108q^{54} - 28q^{57} - 120q^{58} - 116q^{60} - 48q^{61} + 16q^{63} + 40q^{66} - 16q^{67} + 72q^{69} + 48q^{70} + 52q^{72} - 16q^{73} + 60q^{75} + 16q^{76} - 4q^{78} + 16q^{79} - 20q^{81} + 120q^{82} + 72q^{84} - 40q^{85} - 24q^{87} + 72q^{88} + 16q^{91} + 92q^{93} - 96q^{94} + 28q^{96} + 96q^{97} - 144q^{99} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
50.1 −2.59589 0.695566i −1.18735 1.26103i 4.52278 + 2.61123i 0.855518 0.855518i 2.20509 + 4.09938i 0.258819 + 0.965926i −6.12370 6.12370i −0.180409 + 2.99457i −2.81590 + 1.62576i
50.2 −2.46820 0.661352i 1.55028 0.772424i 3.92257 + 2.26470i 2.12589 2.12589i −4.33724 + 0.881217i −0.258819 0.965926i −4.57023 4.57023i 1.80672 2.39494i −6.65309 + 3.84116i
50.3 −2.24002 0.600211i 1.07699 1.35650i 2.92537 + 1.68896i −2.36227 + 2.36227i −3.22666 + 2.39216i 0.258819 + 0.965926i −2.25953 2.25953i −0.680177 2.92188i 6.70937 3.87366i
50.4 −2.14995 0.576078i 1.44893 + 0.949002i 2.55837 + 1.47708i −1.66934 + 1.66934i −2.56843 2.87500i −0.258819 0.965926i −1.50172 1.50172i 1.19879 + 2.75007i 4.55066 2.62732i
50.5 −2.03579 0.545489i −0.604204 + 1.62325i 2.11484 + 1.22100i 2.41109 2.41109i 2.11550 2.97501i −0.258819 0.965926i −0.658724 0.658724i −2.26987 1.96155i −6.22370 + 3.59326i
50.6 −1.84038 0.493129i −1.11254 1.32750i 1.41178 + 0.815092i −1.68431 + 1.68431i 1.39288 + 2.99173i −0.258819 0.965926i 0.498237 + 0.498237i −0.524491 + 2.95380i 3.93035 2.26919i
50.7 −1.78824 0.479158i −1.54238 + 0.788082i 1.23617 + 0.713702i 0.0341444 0.0341444i 3.13576 0.670240i 0.258819 + 0.965926i 0.749576 + 0.749576i 1.75785 2.43104i −0.0774191 + 0.0446979i
50.8 −1.74208 0.466790i 0.804463 + 1.53390i 1.08492 + 0.626376i 0.450139 0.450139i −0.685436 3.04769i 0.258819 + 0.965926i 0.952961 + 0.952961i −1.70568 + 2.46793i −0.994301 + 0.574060i
50.9 −1.06694 0.285886i 1.72116 0.193962i −0.675418 0.389953i 1.84315 1.84315i −1.89182 0.285109i 0.258819 + 0.965926i 2.17126 + 2.17126i 2.92476 0.667676i −2.49346 + 1.43960i
50.10 −0.790296 0.211759i −0.427528 + 1.67846i −1.15232 0.665295i −2.01842 + 2.01842i 0.693303 1.23595i −0.258819 0.965926i 1.92687 + 1.92687i −2.63444 1.43518i 2.02257 1.16773i
50.11 −0.751420 0.201342i −1.18266 1.26543i −1.20796 0.697414i 2.48091 2.48091i 0.633888 + 1.18899i −0.258819 0.965926i 1.86742 + 1.86742i −0.202642 + 2.99315i −2.36372 + 1.36469i
50.12 −0.609155 0.163223i −1.67612 0.436610i −1.38762 0.801144i −1.27442 + 1.27442i 0.949752 + 0.539544i 0.258819 + 0.965926i 1.60638 + 1.60638i 2.61874 + 1.46362i 0.984333 0.568305i
50.13 −0.397883 0.106613i −1.64214 + 0.550805i −1.58511 0.915161i 0.406846 0.406846i 0.712102 0.0440836i −0.258819 0.965926i 1.11566 + 1.11566i 2.39323 1.80899i −0.205252 + 0.118502i
50.14 −0.307973 0.0825212i 1.72958 + 0.0924370i −1.64401 0.949171i −2.80070 + 2.80070i −0.525037 0.171195i 0.258819 + 0.965926i 0.878890 + 0.878890i 2.98291 + 0.319755i 1.09366 0.631424i
50.15 0.307973 + 0.0825212i −0.784738 + 1.54408i −1.64401 0.949171i 2.80070 2.80070i −0.369098 + 0.410778i 0.258819 + 0.965926i −0.878890 0.878890i −1.76837 2.42340i 1.09366 0.631424i
50.16 0.397883 + 0.106613i 1.29808 1.14673i −1.58511 0.915161i −0.406846 + 0.406846i 0.638740 0.317873i −0.258819 0.965926i −1.11566 1.11566i 0.370020 2.97709i −0.205252 + 0.118502i
50.17 0.609155 + 0.163223i 0.459944 1.66987i −1.38762 0.801144i 1.27442 1.27442i 0.552737 0.942134i 0.258819 + 0.965926i −1.60638 1.60638i −2.57690 1.53609i 0.984333 0.568305i
50.18 0.751420 + 0.201342i −0.504568 1.65693i −1.20796 0.697414i −2.48091 + 2.48091i −0.0455330 1.34664i −0.258819 0.965926i −1.86742 1.86742i −2.49082 + 1.67207i −2.36372 + 1.36469i
50.19 0.790296 + 0.211759i 1.66735 + 0.468978i −1.15232 0.665295i 2.01842 2.01842i 1.21839 + 0.723709i −0.258819 0.965926i −1.92687 1.92687i 2.56012 + 1.56390i 2.02257 1.16773i
50.20 1.06694 + 0.285886i −1.02855 + 1.39358i −0.675418 0.389953i −1.84315 + 1.84315i −1.49581 + 1.19282i 0.258819 + 0.965926i −2.17126 2.17126i −0.884154 2.86675i −2.49346 + 1.43960i
See next 80 embeddings (of 112 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 197.28 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
13.f odd 12 1 inner
39.k 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.cc.a 112
3.b odd 2 1 inner 273.2.cc.a 112
13.f odd 12 1 inner 273.2.cc.a 112
39.k even 12 1 inner 273.2.cc.a 112

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
273.2.cc.a 112 1.a even 1 1 trivial
273.2.cc.a 112 3.b odd 2 1 inner
273.2.cc.a 112 13.f odd 12 1 inner
273.2.cc.a 112 39.k even 12 1 inner

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(273, [\chi])$$.