# Properties

 Label 201.2.m.b Level 201 Weight 2 Character orbit 201.m Analytic conductor 1.605 Analytic rank 0 Dimension 120 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 201.m (of order $$33$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$120$$ Relative dimension: $$6$$ over $$\Q(\zeta_{33})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$120q + 2q^{2} - 12q^{3} + 8q^{4} + 2q^{5} - 9q^{6} + 5q^{7} - 33q^{8} - 12q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$120q + 2q^{2} - 12q^{3} + 8q^{4} + 2q^{5} - 9q^{6} + 5q^{7} - 33q^{8} - 12q^{9} - 29q^{10} - 15q^{11} - 3q^{12} + 6q^{13} - 10q^{14} - 9q^{15} + 20q^{16} + 42q^{17} + 2q^{18} - 2q^{19} + 5q^{21} - 41q^{22} - 10q^{23} + 30q^{25} + 27q^{26} - 12q^{27} + 82q^{28} - 67q^{29} - 18q^{30} - 5q^{31} - 124q^{32} + 40q^{33} + 12q^{34} - 33q^{35} - 3q^{36} - 80q^{37} - 22q^{38} + 6q^{39} - 177q^{40} - 24q^{41} - 32q^{42} - 7q^{43} + 115q^{44} + 2q^{45} - 85q^{46} - 44q^{47} + 20q^{48} + 81q^{49} + 14q^{50} - 24q^{51} - 84q^{52} + 16q^{53} + 2q^{54} - 5q^{55} + 121q^{56} - 24q^{57} - 40q^{58} + 34q^{59} + 198q^{60} + 8q^{61} - 66q^{62} - 28q^{63} - 37q^{64} + 170q^{65} + 58q^{66} + 15q^{67} - 14q^{68} + 23q^{69} + 84q^{70} + 147q^{71} + 44q^{72} - 106q^{73} - 33q^{74} + 30q^{75} + 254q^{76} - 18q^{77} + 71q^{78} - 31q^{79} + 95q^{80} - 12q^{81} - 108q^{82} - 13q^{83} - 83q^{84} - 99q^{85} - 111q^{86} + 10q^{87} - 194q^{88} - 79q^{89} - 29q^{90} - 110q^{91} + 156q^{92} + 6q^{93} - 150q^{94} + 53q^{95} - 3q^{96} - 46q^{97} - 71q^{98} - 4q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −2.46190 0.235083i 0.841254 + 0.540641i 4.04182 + 0.778997i −0.0802800 0.558360i −1.94398 1.52877i 1.37223 1.92702i −5.02158 1.47447i 0.415415 + 0.909632i 0.0663804 + 1.39350i
4.2 −1.48103 0.141421i 0.841254 + 0.540641i 0.209585 + 0.0403942i −0.0810398 0.563644i −1.16946 0.919675i −2.41465 + 3.39091i 2.55031 + 0.748839i 0.415415 + 0.909632i 0.0403110 + 0.846233i
4.3 −0.152873 0.0145976i 0.841254 + 0.540641i −1.94070 0.374039i −0.293313 2.04004i −0.120713 0.0949298i 2.10012 2.94920i 0.585917 + 0.172041i 0.415415 + 0.909632i 0.0150600 + 0.316149i
4.4 0.574545 + 0.0548624i 0.841254 + 0.540641i −1.63677 0.315461i 0.534139 + 3.71502i 0.453677 + 0.356776i −1.88652 + 2.64924i −2.03065 0.596252i 0.415415 + 0.909632i 0.103072 + 2.16375i
4.5 1.68971 + 0.161347i 0.841254 + 0.540641i 0.865219 + 0.166757i 0.104733 + 0.728437i 1.33424 + 1.04926i 0.850168 1.19389i −1.82221 0.535050i 0.415415 + 0.909632i 0.0594373 + 1.24774i
4.6 2.11489 + 0.201947i 0.841254 + 0.540641i 2.46811 + 0.475689i −0.349341 2.42972i 1.66998 + 1.31328i −0.760089 + 1.06740i 1.04680 + 0.307370i 0.415415 + 0.909632i −0.248142 5.20914i
10.1 −1.47812 1.40939i −0.142315 0.989821i 0.103311 + 2.16877i −0.00760418 + 0.0166508i −1.18468 + 1.66366i 0.879995 3.62739i 0.229013 0.264295i −0.959493 + 0.281733i 0.0347074 0.0138948i
10.2 −0.865960 0.825691i −0.142315 0.989821i −0.0270430 0.567703i −1.60788 + 3.52076i −0.694048 + 0.974654i −0.961624 + 3.96387i −2.01243 + 2.32247i −0.959493 + 0.281733i 4.29942 1.72123i
10.3 −0.601743 0.573760i −0.142315 0.989821i −0.0622707 1.30722i 1.55758 3.41063i −0.482283 + 0.677272i −0.343913 + 1.41763i −1.80152 + 2.07906i −0.959493 + 0.281733i −2.89415 + 1.15864i
10.4 0.453484 + 0.432396i −0.142315 0.989821i −0.0764825 1.60557i −0.976375 + 2.13796i 0.363457 0.510404i 1.16821 4.81544i 1.48021 1.70826i −0.959493 + 0.281733i −1.36722 + 0.547351i
10.5 1.07016 + 1.02039i −0.142315 0.989821i 0.00887090 + 0.186223i 0.810859 1.77553i 0.857707 1.20448i −0.301535 + 1.24295i 1.75610 2.02665i −0.959493 + 0.281733i 2.67949 1.07270i
10.6 2.02349 + 1.92939i −0.142315 0.989821i 0.276787 + 5.81046i 0.419287 0.918112i 1.62178 2.27747i 0.404612 1.66783i −6.98874 + 8.06543i −0.959493 + 0.281733i 2.61982 1.04882i
16.1 −2.69052 0.518556i 0.415415 + 0.909632i 5.11329 + 2.04705i 4.04867 1.18880i −0.645989 2.66280i 0.854798 + 2.46978i −8.08577 5.19641i −0.654861 + 0.755750i −11.5095 + 1.09902i
16.2 −2.13211 0.410930i 0.415415 + 0.909632i 2.52028 + 1.00897i −2.10182 + 0.617149i −0.511914 2.11014i −1.23081 3.55620i −1.30558 0.839048i −0.654861 + 0.755750i 4.73490 0.452128i
16.3 −0.774865 0.149343i 0.415415 + 0.909632i −1.27862 0.511883i 1.13149 0.332236i −0.186043 0.766882i 0.783813 + 2.26468i 2.24202 + 1.44086i −0.654861 + 0.755750i −0.926370 + 0.0884577i
16.4 0.102091 + 0.0196764i 0.415415 + 0.909632i −1.84670 0.739307i −3.73486 + 1.09665i 0.0245118 + 0.101039i −0.828185 2.39288i −0.348914 0.224234i −0.654861 + 0.755750i −0.402874 + 0.0384698i
16.5 1.62962 + 0.314083i 0.415415 + 0.909632i 0.700266 + 0.280344i 2.51331 0.737974i 0.391267 + 1.61283i −0.464313 1.34155i −1.73919 1.11771i −0.654861 + 0.755750i 4.32752 0.413228i
16.6 1.98148 + 0.381899i 0.415415 + 0.909632i 1.92369 + 0.770128i −1.22915 + 0.360910i 0.475749 + 1.96107i 0.930934 + 2.68976i 0.122425 + 0.0786779i −0.654861 + 0.755750i −2.57336 + 0.245726i
19.1 −2.09771 1.08145i −0.959493 + 0.281733i 2.07076 + 2.90797i −0.870318 1.00440i 2.31742 + 0.446646i 0.0874147 + 1.83506i −0.527294 3.66741i 0.841254 0.540641i 0.739472 + 3.04814i
19.2 −0.925403 0.477079i −0.959493 + 0.281733i −0.531347 0.746172i −0.201746 0.232827i 1.02233 + 0.197037i −0.0460665 0.967055i 0.432067 + 3.00509i 0.841254 0.540641i 0.0756194 + 0.311708i
See next 80 embeddings (of 120 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.6 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.g even 33 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.m.b 120
3.b odd 2 1 603.2.z.d 120
67.g even 33 1 inner 201.2.m.b 120
201.o odd 66 1 603.2.z.d 120

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.m.b 120 1.a even 1 1 trivial
201.2.m.b 120 67.g even 33 1 inner
603.2.z.d 120 3.b odd 2 1
603.2.z.d 120 201.o odd 66 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{120} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database