# Properties

 Label 201.2.j.a Level 201 Weight 2 Character orbit 201.j Analytic conductor 1.605 Analytic rank 0 Dimension 200 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 201.j (of order $$22$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$200$$ Relative dimension: $$20$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$200q - 11q^{3} - 34q^{4} - 7q^{6} - 22q^{7} + 3q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$200q - 11q^{3} - 34q^{4} - 7q^{6} - 22q^{7} + 3q^{9} - 10q^{10} - 44q^{12} - 22q^{13} - 13q^{15} - 34q^{16} - 11q^{18} - 24q^{19} + 43q^{21} - 82q^{22} + 53q^{24} - 18q^{25} - 11q^{27} - 110q^{28} + 22q^{31} - 32q^{33} - 22q^{34} + 33q^{36} - 68q^{37} - 69q^{39} + 10q^{40} - 11q^{42} - 44q^{43} + 99q^{45} + 66q^{46} + 99q^{48} + 26q^{49} - 11q^{51} + 176q^{52} - 128q^{54} + 30q^{55} - 11q^{57} + 66q^{58} + 5q^{60} - 110q^{61} - 11q^{63} + 170q^{64} - 32q^{67} - 11q^{69} - 66q^{70} - 121q^{72} + 150q^{73} - 22q^{75} - 94q^{76} - 11q^{78} + 132q^{79} + 63q^{81} + 76q^{82} - 101q^{84} - 22q^{85} + 88q^{87} - 114q^{88} - 85q^{90} - 174q^{91} - 75q^{93} + 22q^{94} - 250q^{96} - 66q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1 −1.80413 + 2.08208i −1.15463 1.29106i −0.795528 5.53302i −0.470328 + 0.138101i 4.77118 0.0747868i −0.367215 0.318193i 8.32012 + 5.34702i −0.333667 + 2.98139i 0.560996 1.22841i
5.2 −1.48703 + 1.71613i −0.582539 + 1.63115i −0.449198 3.12424i 1.31922 0.387358i −1.93301 3.42529i −3.29666 2.85657i 2.20900 + 1.41964i −2.32130 1.90042i −1.29697 + 2.83997i
5.3 −1.45043 + 1.67388i 1.46447 + 0.924840i −0.413514 2.87605i 1.41694 0.416050i −3.67218 + 1.10994i 1.28570 + 1.11407i 1.68743 + 1.08444i 1.28934 + 2.70880i −1.35875 + 2.97524i
5.4 −1.36124 + 1.57095i 0.885763 1.48843i −0.330294 2.29725i 1.00399 0.294799i 1.13252 + 3.41760i −0.415510 0.360042i 0.561099 + 0.360596i −1.43085 2.63679i −0.903560 + 1.97852i
5.5 −1.16364 + 1.34291i −1.67608 + 0.436744i −0.164727 1.14570i −2.73659 + 0.803536i 1.36385 2.75905i 0.246444 + 0.213545i −1.25944 0.809393i 2.61851 1.46404i 2.10533 4.61004i
5.6 −0.706711 + 0.815588i −1.69171 0.371647i 0.118886 + 0.826873i 3.89728 1.14434i 1.49866 1.11709i −1.64135 1.42224i −2.57413 1.65429i 2.72376 + 1.25744i −1.82094 + 3.98729i
5.7 −0.634589 + 0.732355i 1.17689 + 1.27080i 0.150989 + 1.05015i −2.77562 + 0.814995i −1.67752 + 0.0554604i 0.0640238 + 0.0554769i −2.49533 1.60365i −0.229880 + 2.99118i 1.16451 2.54992i
5.8 −0.422797 + 0.487934i −0.589489 + 1.62865i 0.225308 + 1.56705i 1.17382 0.344663i −0.545439 0.976220i 2.44605 + 2.11952i −1.94615 1.25071i −2.30500 1.92014i −0.328113 + 0.718466i
5.9 −0.401105 + 0.462899i −0.467910 1.66765i 0.231239 + 1.60830i −2.18653 + 0.642023i 0.959636 + 0.452308i −3.07269 2.66251i −1.86777 1.20035i −2.56212 + 1.56062i 0.579835 1.26966i
5.10 −0.333078 + 0.384392i 1.47238 0.912186i 0.247813 + 1.72358i 2.02247 0.593851i −0.139781 + 0.869802i 0.616666 + 0.534344i −1.60083 1.02879i 1.33583 2.68618i −0.445368 + 0.975220i
5.11 0.333078 0.384392i 1.65359 0.515399i 0.247813 + 1.72358i −2.02247 + 0.593851i 0.352659 0.807295i 0.616666 + 0.534344i 1.60083 + 1.02879i 2.46873 1.70452i −0.445368 + 0.975220i
5.12 0.401105 0.462899i 0.953911 + 1.44570i 0.231239 + 1.60830i 2.18653 0.642023i 1.05183 + 0.138313i −3.07269 2.66251i 1.86777 + 1.20035i −1.18011 + 2.75814i 0.579835 1.26966i
5.13 0.422797 0.487934i −1.61689 0.621033i 0.225308 + 1.56705i −1.17382 + 0.344663i −0.986637 + 0.526362i 2.44605 + 2.11952i 1.94615 + 1.25071i 2.22864 + 2.00828i −0.328113 + 0.718466i
5.14 0.634589 0.732355i −0.189712 1.72163i 0.150989 + 1.05015i 2.77562 0.814995i −1.38123 0.953590i 0.0640238 + 0.0554769i 2.49533 + 1.60365i −2.92802 + 0.653229i 1.16451 2.54992i
5.15 0.706711 0.815588i −0.826962 + 1.52189i 0.118886 + 0.826873i −3.89728 + 1.14434i 0.656808 + 1.74999i −1.64135 1.42224i 2.57413 + 1.65429i −1.63227 2.51708i −1.82094 + 3.98729i
5.16 1.16364 1.34291i −1.42767 + 0.980693i −0.164727 1.14570i 2.73659 0.803536i −0.344310 + 3.05841i 0.246444 + 0.213545i 1.25944 + 0.809393i 1.07648 2.80021i 2.10533 4.61004i
5.17 1.36124 1.57095i 1.70493 + 0.305299i −0.330294 2.29725i −1.00399 + 0.294799i 2.80043 2.26278i −0.415510 0.360042i −0.561099 0.360596i 2.81358 + 1.04103i −0.903560 + 1.97852i
5.18 1.45043 1.67388i 0.260076 1.71241i −0.413514 2.87605i −1.41694 + 0.416050i −2.48916 2.91907i 1.28570 + 1.11407i −1.68743 1.08444i −2.86472 0.890717i −1.35875 + 2.97524i
5.19 1.48703 1.71613i −1.61422 0.627922i −0.449198 3.12424i −1.31922 + 0.387358i −3.47800 + 1.83647i −3.29666 2.85657i −2.20900 1.41964i 2.21143 + 2.02721i −1.29697 + 2.83997i
5.20 1.80413 2.08208i 0.219597 + 1.71807i −0.795528 5.53302i 0.470328 0.138101i 3.97334 + 2.64241i −0.367215 0.318193i −8.32012 5.34702i −2.90355 + 0.754566i 0.560996 1.22841i
See next 80 embeddings (of 200 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 179.20 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
67.f odd 22 1 inner
201.j even 22 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.j.a 200
3.b odd 2 1 inner 201.2.j.a 200
67.f odd 22 1 inner 201.2.j.a 200
201.j even 22 1 inner 201.2.j.a 200

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.j.a 200 1.a even 1 1 trivial
201.2.j.a 200 3.b odd 2 1 inner
201.2.j.a 200 67.f odd 22 1 inner
201.2.j.a 200 201.j even 22 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database