# Properties

 Label 201.3.l.a Level 201 Weight 3 Character orbit 201.l Analytic conductor 5.477 Analytic rank 0 Dimension 220 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 201.l (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.47685331364$$ Analytic rank: $$0$$ Dimension: $$220$$ Relative dimension: $$22$$ over $$\Q(\zeta_{22})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$220q + 52q^{4} + 66q^{8} + 66q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$220q + 52q^{4} + 66q^{8} + 66q^{9} - 162q^{10} - 72q^{14} + 54q^{15} - 116q^{16} + 14q^{17} - 84q^{19} - 48q^{21} + 78q^{22} + 82q^{23} - 36q^{24} + 62q^{25} + 100q^{26} - 154q^{28} + 52q^{29} - 88q^{31} + 770q^{32} - 36q^{33} + 180q^{35} + 42q^{36} - 304q^{37} + 72q^{39} - 1066q^{40} + 330q^{41} + 770q^{43} - 242q^{46} - 616q^{47} - 146q^{49} + 858q^{50} + 264q^{52} - 286q^{53} - 62q^{55} - 1484q^{56} - 660q^{57} - 352q^{58} - 634q^{59} - 504q^{60} - 352q^{61} + 124q^{62} - 132q^{63} - 1226q^{64} + 196q^{65} + 156q^{67} - 192q^{68} + 66q^{69} + 1144q^{70} + 280q^{71} + 264q^{72} + 552q^{73} + 352q^{74} + 396q^{75} + 400q^{76} - 176q^{77} + 528q^{78} + 682q^{79} + 1848q^{80} - 198q^{81} + 728q^{82} + 246q^{83} - 1320q^{84} - 1120q^{86} + 1120q^{88} + 448q^{89} + 486q^{90} - 128q^{91} + 604q^{92} + 72q^{93} + 704q^{94} - 462q^{95} + 144q^{96} - 176q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −3.38399 1.54542i 0.936417 + 1.45709i 6.44363 + 7.43634i 1.51116 + 0.217272i −0.917010 6.37794i 8.17740 + 3.73449i −6.12054 20.8446i −1.24625 + 2.72890i −4.77798 3.07062i
43.2 −3.31598 1.51436i −0.936417 1.45709i 6.08302 + 7.02018i −3.45931 0.497373i 0.898581 + 6.24977i 4.95524 + 2.26298i −5.43199 18.4997i −1.24625 + 2.72890i 10.7178 + 6.88791i
43.3 −2.57988 1.17819i −0.936417 1.45709i 2.64821 + 3.05620i 4.45790 + 0.640949i 0.699109 + 4.86241i −4.40917 2.01360i −0.0350994 0.119537i −1.24625 + 2.72890i −10.7457 6.90583i
43.4 −2.54688 1.16312i 0.936417 + 1.45709i 2.51429 + 2.90164i 1.62480 + 0.233611i −0.690165 4.80020i −7.97050 3.64000i 0.126668 + 0.431390i −1.24625 + 2.72890i −3.86645 2.48482i
43.5 −2.27098 1.03712i −0.936417 1.45709i 1.46229 + 1.68758i 2.84282 + 0.408736i 0.615402 + 4.28021i −0.295181 0.134805i 1.24287 + 4.23283i −1.24625 + 2.72890i −6.03208 3.87658i
43.6 −1.91347 0.873852i 0.936417 + 1.45709i 0.278305 + 0.321181i 9.57220 + 1.37628i −0.518521 3.60640i 5.09555 + 2.32706i 2.11871 + 7.21565i −1.24625 + 2.72890i −17.1135 10.9982i
43.7 −1.73436 0.792054i −0.936417 1.45709i −0.238801 0.275591i −8.70735 1.25193i 0.469984 + 3.26881i −9.89366 4.51828i 2.34455 + 7.98481i −1.24625 + 2.72890i 14.1101 + 9.06798i
43.8 −1.59211 0.727091i 0.936417 + 1.45709i −0.613300 0.707786i −8.19855 1.17877i −0.431437 3.00071i 10.0493 + 4.58935i 2.43425 + 8.29030i −1.24625 + 2.72890i 12.1959 + 7.83782i
43.9 −1.00045 0.456891i 0.936417 + 1.45709i −1.82729 2.10880i −2.48696 0.357570i −0.271107 1.88559i −2.35904 1.07734i 2.10407 + 7.16579i −1.24625 + 2.72890i 2.32471 + 1.49400i
43.10 −0.736337 0.336274i −0.936417 1.45709i −2.19033 2.52778i 5.35256 + 0.769582i 0.199536 + 1.38780i 8.86602 + 4.04898i 1.67503 + 5.70464i −1.24625 + 2.72890i −3.68250 2.36660i
43.11 −0.480619 0.219491i 0.936417 + 1.45709i −2.43662 2.81201i −0.667348 0.0959501i −0.130240 0.905842i −2.12406 0.970026i 1.14931 + 3.91418i −1.24625 + 2.72890i 0.299680 + 0.192592i
43.12 −0.139433 0.0636767i −0.936417 1.45709i −2.60406 3.00524i −4.85688 0.698314i 0.0377841 + 0.262794i 4.72724 + 2.15886i 0.344467 + 1.17315i −1.24625 + 2.72890i 0.632741 + 0.406638i
43.13 −0.132389 0.0604598i −0.936417 1.45709i −2.60557 3.00699i 5.16768 + 0.743001i 0.0358753 + 0.249518i −10.9487 5.00010i 0.327160 + 1.11420i −1.24625 + 2.72890i −0.639220 0.410802i
43.14 0.191268 + 0.0873493i 0.936417 + 1.45709i −2.59049 2.98958i 8.50765 + 1.22322i 0.0518308 + 0.360491i −5.30827 2.42421i −0.471300 1.60510i −1.24625 + 2.72890i 1.52040 + 0.977100i
43.15 1.40253 + 0.640516i 0.936417 + 1.45709i −1.06260 1.22631i −7.67747 1.10385i 0.380065 + 2.64341i −5.69858 2.60245i −2.44245 8.31820i −1.24625 + 2.72890i −10.0609 6.46573i
43.16 1.43499 + 0.655336i 0.936417 + 1.45709i −0.989723 1.14220i −0.383091 0.0550801i 0.388860 + 2.70458i 12.2137 + 5.57779i −2.44950 8.34222i −1.24625 + 2.72890i −0.513634 0.330092i
43.17 1.62013 + 0.739889i −0.936417 1.45709i −0.542056 0.625565i 0.289414 + 0.0416115i −0.439031 3.05353i −6.47623 2.95760i −2.42251 8.25031i −1.24625 + 2.72890i 0.438101 + 0.281550i
43.18 2.16194 + 0.987325i −0.936417 1.45709i 1.07973 + 1.24607i −7.15134 1.02821i −0.585853 4.07470i −3.35568 1.53249i −1.57437 5.36180i −1.24625 + 2.72890i −14.4456 9.28362i
43.19 2.26882 + 1.03614i −0.936417 1.45709i 1.45454 + 1.67863i 6.52672 + 0.938401i −0.614817 4.27615i 2.05853 + 0.940098i −1.25001 4.25713i −1.24625 + 2.72890i 13.8357 + 8.89165i
43.20 2.70381 + 1.23479i 0.936417 + 1.45709i 3.16643 + 3.65425i 4.86060 + 0.698849i 0.732691 + 5.09598i −0.313460 0.143152i 0.699474 + 2.38219i −1.24625 + 2.72890i 12.2792 + 7.89136i
See next 80 embeddings (of 220 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 187.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
67.f odd 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.l.a 220
67.f odd 22 1 inner 201.3.l.a 220

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.l.a 220 1.a even 1 1 trivial
201.3.l.a 220 67.f odd 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database