# Properties

 Label 187.2.r.a Level 187 Weight 2 Character orbit 187.r Analytic conductor 1.493 Analytic rank 0 Dimension 256 CM No

# Related objects

## Newspace parameters

 Level: $$N$$ = $$187 = 11 \cdot 17$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 187.r (of order $$40$$ and degree $$16$$)

## Newform invariants

 Self dual: No Analytic conductor: $$1.4932025178$$ Analytic rank: $$0$$ Dimension: $$256$$ Relative dimension: $$16$$ over $$\Q(\zeta_{40})$$ Sato-Tate group: $\mathrm{SU}(2)[C_{40}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$256q - 12q^{2} - 12q^{3} - 20q^{5} - 12q^{6} - 12q^{7} - 28q^{8} - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$256q - 12q^{2} - 12q^{3} - 20q^{5} - 12q^{6} - 12q^{7} - 28q^{8} - 36q^{9} - 32q^{10} - 16q^{11} - 32q^{12} - 12q^{14} + 12q^{15} + 16q^{16} + 12q^{17} - 16q^{18} - 12q^{19} - 44q^{20} + 88q^{22} - 48q^{23} - 80q^{24} - 4q^{25} - 12q^{26} - 48q^{27} - 28q^{28} - 12q^{29} + 44q^{31} - 8q^{32} - 56q^{33} - 64q^{34} - 88q^{35} + 56q^{36} - 28q^{37} + 12q^{39} + 120q^{40} - 48q^{41} + 44q^{42} + 8q^{43} - 16q^{44} - 32q^{45} - 44q^{46} + 60q^{48} + 64q^{49} + 32q^{50} - 28q^{51} - 232q^{52} - 20q^{53} + 48q^{54} - 64q^{56} + 128q^{57} + 124q^{58} + 104q^{59} + 4q^{60} + 64q^{61} - 52q^{62} - 12q^{63} - 88q^{65} - 208q^{66} - 96q^{67} + 44q^{68} + 48q^{69} + 92q^{70} - 44q^{71} + 28q^{73} - 12q^{74} + 104q^{75} + 176q^{76} - 148q^{77} - 12q^{79} + 32q^{80} - 72q^{82} - 16q^{83} + 216q^{84} + 80q^{85} - 24q^{86} - 128q^{87} - 32q^{88} - 28q^{90} - 108q^{91} + 76q^{92} + 164q^{93} - 88q^{94} - 32q^{95} - 44q^{96} + 128q^{97} + 144q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −1.22225 2.39880i 0.924642 0.221987i −3.08479 + 4.24585i 3.50496 + 0.275847i −1.66265 1.94671i 0.704149 2.93299i 8.63717 + 1.36799i −1.86733 + 0.951454i −3.62225 8.74487i
9.2 −1.12869 2.21518i −2.41487 + 0.579760i −2.45751 + 3.38248i −2.80097 0.220442i 4.00992 + 4.69501i 0.508363 2.11748i 5.35549 + 0.848226i 2.82247 1.43812i 2.67312 + 6.45348i
9.3 −0.963667 1.89130i −1.12315 + 0.269645i −1.47280 + 2.02714i 0.737536 + 0.0580454i 1.59233 + 1.86437i −1.14657 + 4.77579i 1.06017 + 0.167914i −1.48425 + 0.756265i −0.600958 1.45084i
9.4 −0.766568 1.50447i 2.49633 0.599317i −0.500247 + 0.688531i 1.08364 + 0.0852843i −2.81527 3.29625i −0.0461198 + 0.192103i −1.91609 0.303480i 3.19948 1.63022i −0.702375 1.69568i
9.5 −0.569376 1.11746i −2.08050 + 0.499483i 0.251036 0.345522i 0.846510 + 0.0666218i 1.74274 + 2.04048i 0.546289 2.27546i −3.00648 0.476179i 1.40596 0.716372i −0.407535 0.983876i
9.6 −0.528719 1.03767i 1.63739 0.393102i 0.378355 0.520761i −3.04927 0.239983i −1.27363 1.49123i 0.541420 2.25518i −3.04095 0.481640i −0.146513 + 0.0746520i 1.36319 + 3.29102i
9.7 −0.254464 0.499414i −0.115650 + 0.0277651i 0.990908 1.36387i 2.58249 + 0.203246i 0.0432950 + 0.0506919i −0.0105709 + 0.0440308i −2.04049 0.323183i −2.66042 + 1.35555i −0.555647 1.34145i
9.8 −0.188923 0.370783i −1.09784 + 0.263568i 1.07378 1.47793i −4.10355 0.322956i 0.305134 + 0.357266i −0.175147 + 0.729539i −1.57289 0.249121i −1.53724 + 0.783262i 0.655510 + 1.58254i
9.9 0.0402376 + 0.0789707i 2.17340 0.521788i 1.17095 1.61168i 0.213343 + 0.0167905i 0.128658 + 0.150640i −0.980687 + 4.08486i 0.349471 + 0.0553508i 1.77840 0.906138i 0.00725846 + 0.0175235i
9.10 0.135448 + 0.265831i −3.01634 + 0.724159i 1.12325 1.54602i 1.25823 + 0.0990246i −0.601061 0.703752i −0.700911 + 2.91950i 1.15248 + 0.182534i 5.90088 3.00665i 0.144100 + 0.347889i
9.11 0.437887 + 0.859402i 0.611686 0.146853i 0.628744 0.865392i 0.107631 + 0.00847074i 0.394055 + 0.461379i 1.06873 4.45158i 2.92435 + 0.463171i −2.32043 + 1.18232i 0.0398504 + 0.0962074i
9.12 0.574486 + 1.12749i 1.00067 0.240239i 0.234367 0.322578i −0.448655 0.0353099i 0.845737 + 0.990230i −0.348136 + 1.45009i 2.99801 + 0.474839i −1.72940 + 0.881173i −0.217934 0.526140i
9.13 0.808667 + 1.58710i −1.31588 + 0.315915i −0.689370 + 0.948836i 3.89226 + 0.306327i −1.56550 1.83296i −0.115270 + 0.480136i 1.45526 + 0.230490i −1.04128 + 0.530561i 2.66137 + 6.42512i
9.14 0.864466 + 1.69661i −1.81207 + 0.435039i −0.955613 + 1.31529i −3.43817 0.270590i −2.30456 2.69830i −0.552442 + 2.30109i 0.703788 + 0.111469i 0.421312 0.214669i −2.51310 6.06716i
9.15 1.03095 + 2.02335i 2.48157 0.595772i −1.85553 + 2.55392i −2.24954 0.177043i 3.76383 + 4.40688i 0.153522 0.639467i −2.59463 0.410949i 3.13023 1.59493i −1.96094 4.73414i
9.16 1.21934 + 2.39309i −0.796569 + 0.191239i −3.06452 + 4.21795i 0.157430 + 0.0123900i −1.42894 1.67308i 0.144534 0.602029i −8.52512 1.35025i −2.07507 + 1.05730i 0.162310 + 0.391851i
15.1 −0.404781 2.55569i −0.751908 + 0.0591764i −4.46558 + 1.45095i −0.925538 + 1.51034i 0.455595 + 1.89769i −0.251290 + 3.19294i 3.16633 + 6.21427i −2.40120 + 0.380313i 4.23460 + 1.75403i
15.2 −0.368659 2.32762i −0.913416 + 0.0718874i −3.37979 + 1.09816i 1.97221 3.21836i 0.504065 + 2.09958i 0.151479 1.92472i 1.66231 + 3.26246i −2.13390 + 0.337977i −8.21820 3.40409i
15.3 −0.294511 1.85947i 1.75686 0.138268i −1.46877 + 0.477232i 0.295816 0.482728i −0.774520 3.22611i −0.00222850 + 0.0283158i −0.389442 0.764323i 0.104385 0.0165330i −0.984737 0.407891i
15.4 −0.270768 1.70956i −3.36059 + 0.264484i −0.947178 + 0.307757i −0.735432 + 1.20012i 1.36209 + 5.67353i −0.137628 + 1.74873i −0.789004 1.54851i 8.26056 1.30834i 2.25081 + 0.932314i
See next 80 embeddings (of 256 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 185.16 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not have CM; other inner twists have not been computed.

## Hecke kernels

There are no other newforms in $$S_{2}^{\mathrm{new}}(187, [\chi])$$.