# Properties

 Label 201.3.n.b Level 201 Weight 3 Character orbit 201.n Analytic conductor 5.477 Analytic rank 0 Dimension 240 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$3$$ Character orbit: $$[\chi]$$ = 201.n (of order $$66$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.47685331364$$ Analytic rank: $$0$$ Dimension: $$240$$ Relative dimension: $$12$$ over $$\Q(\zeta_{66})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{66}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$240q - 34q^{4} - 33q^{6} - 21q^{7} - 33q^{8} + 72q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$240q - 34q^{4} - 33q^{6} - 21q^{7} - 33q^{8} + 72q^{9} + 69q^{10} - 111q^{11} - 3q^{12} - 30q^{13} - 6q^{14} - 27q^{15} + 98q^{16} - 4q^{17} + 16q^{19} - 108q^{20} + 21q^{21} + 27q^{22} + 178q^{23} + 36q^{24} + 222q^{25} - 29q^{26} - 112q^{28} - 77q^{29} + 90q^{30} + 137q^{31} + 44q^{32} + 12q^{33} - 72q^{34} - 237q^{35} + 3q^{36} + 132q^{37} + 210q^{38} - 30q^{39} + 749q^{40} - 150q^{41} - 132q^{42} - 385q^{43} + 9q^{44} - 443q^{46} - 166q^{47} - 294q^{48} - 295q^{49} - 6q^{50} + 276q^{51} - 1804q^{52} + 176q^{53} + 199q^{55} - 1361q^{56} - 114q^{57} + 968q^{58} - 214q^{59} - 420q^{60} - 274q^{61} + 334q^{62} - 102q^{63} + 683q^{64} - 224q^{65} + 47q^{67} + 870q^{68} + 27q^{69} - 44q^{70} + 271q^{71} + 264q^{72} + 594q^{73} - 1289q^{74} + 396q^{75} + 494q^{76} + 1360q^{77} + 441q^{78} + 1023q^{79} + 15q^{80} - 216q^{81} - 316q^{82} - 225q^{83} + 1527q^{84} - 153q^{85} - 91q^{86} - 1676q^{88} + 871q^{89} - 207q^{90} - 692q^{91} - 488q^{92} - 390q^{93} + 440q^{94} - 531q^{95} - 33q^{96} + 84q^{97} + 85q^{98} + 36q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1 −1.79502 3.48186i −0.487975 + 1.66189i −6.58102 + 9.24175i 1.84877 + 1.60197i 6.66240 1.28407i −4.70180 0.223974i 28.4818 + 4.09506i −2.52376 1.62192i 2.25925 9.31274i
7.2 −1.35193 2.62237i −0.487975 + 1.66189i −2.72891 + 3.83222i −4.43909 3.84649i 5.01781 0.967103i 1.03058 + 0.0490928i 2.05754 + 0.295830i −2.52376 1.62192i −4.08562 + 16.8411i
7.3 −1.08640 2.10732i −0.487975 + 1.66189i −0.940314 + 1.32049i 6.32185 + 5.47792i 4.03228 0.777157i −2.91931 0.139064i −5.58274 0.802677i −2.52376 1.62192i 4.67567 19.2734i
7.4 −0.738773 1.43302i −0.487975 + 1.66189i 0.812467 1.14095i 1.60508 + 1.39081i 2.74203 0.528482i −7.37536 0.351332i −8.61857 1.23916i −2.52376 1.62192i 0.807267 3.32760i
7.5 −0.577269 1.11975i −0.487975 + 1.66189i 1.39964 1.96551i −2.24142 1.94220i 2.14259 0.412950i 11.5313 + 0.549306i −7.99671 1.14975i −2.52376 1.62192i −0.880870 + 3.63099i
7.6 −0.110765 0.214854i −0.487975 + 1.66189i 2.28633 3.21071i −3.52548 3.05485i 0.411115 0.0792359i −9.28062 0.442091i −1.90014 0.273199i −2.52376 1.62192i −0.265847 + 1.09584i
7.7 0.193811 + 0.375940i −0.487975 + 1.66189i 2.21646 3.11258i 4.88626 + 4.23397i −0.719346 + 0.138643i 1.38901 + 0.0661669i 3.27433 + 0.470777i −2.52376 1.62192i −0.644710 + 2.65753i
7.8 0.547626 + 1.06225i −0.487975 + 1.66189i 1.49175 2.09488i −5.31200 4.60288i −2.03256 + 0.391745i −3.76569 0.179382i 7.77394 + 1.11772i −2.52376 1.62192i 1.98040 8.16331i
7.9 0.860039 + 1.66824i −0.487975 + 1.66189i 0.276863 0.388799i 0.779784 + 0.675687i −3.19211 + 0.615229i 7.09141 + 0.337805i 8.31785 + 1.19593i −2.52376 1.62192i −0.456565 + 1.88199i
7.10 1.46078 + 2.83351i −0.487975 + 1.66189i −3.57469 + 5.01994i 1.03546 + 0.897230i −5.42181 + 1.04497i −11.9148 0.567571i −6.82411 0.981159i −2.52376 1.62192i −1.02974 + 4.24463i
7.11 1.51795 + 2.94440i −0.487975 + 1.66189i −4.04513 + 5.68059i 5.90107 + 5.11330i −5.63400 + 1.08586i 7.02729 + 0.334751i −9.75051 1.40191i −2.52376 1.62192i −6.09813 + 25.1368i
7.12 1.77257 + 3.43831i −0.487975 + 1.66189i −6.35974 + 8.93100i −6.98484 6.05240i −6.57907 + 1.26801i 6.92425 + 0.329843i −26.6648 3.83382i −2.52376 1.62192i 8.42889 34.7444i
13.1 −2.23952 2.84778i −1.30900 1.13425i −2.15136 + 8.86804i −3.98657 6.20322i −0.298577 + 6.26791i −2.90854 + 7.26517i 16.8902 7.71351i 0.426945 + 2.96946i −8.73741 + 25.2451i
13.2 −2.22017 2.82317i −1.30900 1.13425i −2.09813 + 8.64859i 3.60424 + 5.60831i −0.295998 + 6.21375i 3.22377 8.05259i 16.0066 7.30997i 0.426945 + 2.96946i 7.83121 22.6268i
13.3 −1.77293 2.25446i −1.30900 1.13425i −0.996290 + 4.10676i 0.859686 + 1.33770i −0.236371 + 4.96203i −1.21984 + 3.04701i 0.589306 0.269127i 0.426945 + 2.96946i 1.49163 4.30977i
13.4 −1.37068 1.74296i −1.30900 1.13425i −0.216110 + 0.890817i −2.17329 3.38170i −0.182742 + 3.83622i 4.27625 10.6815i −6.21903 + 2.84014i 0.426945 + 2.96946i −2.91529 + 8.42317i
13.5 −0.827200 1.05187i −1.30900 1.13425i 0.520864 2.14703i 3.07093 + 4.77845i −0.110284 + 2.31515i −2.67312 + 6.67713i −7.55822 + 3.45172i 0.426945 + 2.96946i 2.48604 7.18295i
13.6 −0.333348 0.423887i −1.30900 1.13425i 0.874477 3.60464i −3.59519 5.59422i −0.0444427 + 0.932967i 1.22122 3.05045i −3.78158 + 1.72699i 0.426945 + 2.96946i −1.17287 + 3.38877i
13.7 0.127437 + 0.162049i −1.30900 1.13425i 0.933016 3.84594i 2.68870 + 4.18370i 0.0169901 0.356667i 1.06344 2.65636i 1.49223 0.681481i 0.426945 + 2.96946i −0.335325 + 0.968859i
13.8 0.774122 + 0.984377i −1.30900 1.13425i 0.573304 2.36319i −3.17987 4.94798i 0.103208 2.16660i −3.85408 + 9.62704i 7.32662 3.34595i 0.426945 + 2.96946i 2.40906 6.96053i
See next 80 embeddings (of 240 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 184.12 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.h odd 66 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.3.n.b 240
67.h odd 66 1 inner 201.3.n.b 240

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.3.n.b 240 1.a even 1 1 trivial
201.3.n.b 240 67.h odd 66 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{240} + \cdots$$ acting on $$S_{3}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database