# Properties

 Label 201.2.i.b Level 201 Weight 2 Character orbit 201.i Analytic conductor 1.605 Analytic rank 0 Dimension 70 CM no Inner twists 2

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## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 201.i (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.60499308063$$ Analytic rank: $$0$$ Dimension: $$70$$ Relative dimension: $$7$$ over $$\Q(\zeta_{11})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$70q - 2q^{2} - 7q^{3} - 12q^{4} - 2q^{5} + 9q^{6} - 10q^{7} + 15q^{8} - 7q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$70q - 2q^{2} - 7q^{3} - 12q^{4} - 2q^{5} + 9q^{6} - 10q^{7} + 15q^{8} - 7q^{9} + 17q^{10} + 3q^{11} - q^{12} - 20q^{13} - 20q^{14} + 9q^{15} - 30q^{16} + 3q^{17} - 2q^{18} - 12q^{19} - 36q^{20} - 10q^{21} - 25q^{22} - 8q^{23} - 18q^{24} - 7q^{25} - 42q^{26} - 7q^{27} - 3q^{28} + 40q^{29} + 6q^{30} - 12q^{31} + 73q^{32} + 14q^{33} - 30q^{34} - 24q^{35} - q^{36} + 48q^{37} - 56q^{38} - 20q^{39} + 75q^{40} + 12q^{41} + 2q^{42} - 19q^{43} - q^{44} - 2q^{45} + 31q^{46} + 26q^{47} - 30q^{48} - 39q^{49} - 47q^{50} + 3q^{51} + 72q^{52} - q^{53} - 2q^{54} - 49q^{55} + 2q^{56} + 54q^{57} + 28q^{58} - 13q^{59} + 63q^{60} - 22q^{61} - 24q^{62} + 34q^{63} - 63q^{64} + 22q^{65} + 8q^{66} - 52q^{67} + 434q^{68} - 8q^{69} - 48q^{70} + 6q^{71} + 4q^{72} + 82q^{73} - 72q^{74} - 7q^{75} - 34q^{76} - 18q^{77} - 20q^{78} + 43q^{79} - 86q^{80} - 7q^{81} + 6q^{82} - 50q^{83} + 74q^{84} - 30q^{85} - 60q^{86} - 37q^{87} - 28q^{88} + 16q^{89} + 17q^{90} - 26q^{91} - 48q^{92} - 23q^{93} - 54q^{94} - 89q^{95} - 48q^{96} - 28q^{97} - 109q^{98} - 8q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −2.50000 + 0.734065i −0.142315 0.989821i 4.02862 2.58904i −0.828554 + 1.81428i 1.08238 + 2.37008i −0.536019 + 0.157389i −4.75848 + 5.49158i −0.959493 + 0.281733i 0.739582 5.14391i
22.2 −1.76651 + 0.518694i −0.142315 0.989821i 1.16901 0.751277i 1.29267 2.83056i 0.764816 + 1.67471i −4.56976 + 1.34180i 0.735927 0.849305i −0.959493 + 0.281733i −0.815327 + 5.67072i
22.3 −1.10959 + 0.325806i −0.142315 0.989821i −0.557459 + 0.358257i −0.581489 + 1.27328i 0.480401 + 1.05193i 2.51048 0.737144i 2.01644 2.32710i −0.959493 + 0.281733i 0.230373 1.60228i
22.4 −0.168673 + 0.0495270i −0.142315 0.989821i −1.65651 + 1.06457i 0.879977 1.92688i 0.0730276 + 0.159908i 1.48853 0.437073i 0.456925 0.527320i −0.959493 + 0.281733i −0.0529961 + 0.368596i
22.5 0.313803 0.0921408i −0.142315 0.989821i −1.59252 + 1.02345i −1.18061 + 2.58518i −0.135862 0.297496i −3.96942 + 1.16553i −0.833782 + 0.962236i −0.959493 + 0.281733i −0.132279 + 0.920020i
22.6 1.80987 0.531424i −0.142315 0.989821i 1.31069 0.842331i 0.860755 1.88479i −0.783586 1.71581i −0.882654 + 0.259170i −0.545949 + 0.630059i −0.959493 + 0.281733i 0.556227 3.86864i
22.7 2.62393 0.770455i −0.142315 0.989821i 4.60890 2.96196i −1.23993 + 2.71506i −1.13604 2.48757i −2.41757 + 0.709862i 6.22967 7.18942i −0.959493 + 0.281733i −1.16165 + 8.07943i
25.1 −0.354867 2.46816i −0.654861 + 0.755750i −4.04687 + 1.18827i 3.46834 2.22897i 2.09770 + 1.34811i −0.450350 3.13225i 2.29723 + 5.03024i −0.142315 0.989821i −6.73224 7.76942i
25.2 −0.337801 2.34946i −0.654861 + 0.755750i −3.48686 + 1.02384i −1.73799 + 1.11694i 1.99682 + 1.28328i 0.698525 + 4.85835i 1.61125 + 3.52815i −0.142315 0.989821i 3.21130 + 3.70604i
25.3 −0.197156 1.37125i −0.654861 + 0.755750i 0.0775356 0.0227665i −1.53079 + 0.983778i 1.16543 + 0.748976i −0.598366 4.16172i −1.19749 2.62215i −0.142315 0.989821i 1.65081 + 1.90513i
25.4 −0.0977461 0.679839i −0.654861 + 0.755750i 1.46636 0.430562i 1.02094 0.656119i 0.577798 + 0.371328i 0.446563 + 3.10591i −1.00668 2.20433i −0.142315 0.989821i −0.545848 0.629943i
25.5 0.135223 + 0.940496i −0.654861 + 0.755750i 1.05274 0.309112i −3.06288 + 1.96839i −0.799331 0.513699i 0.0762863 + 0.530583i 1.22250 + 2.67690i −0.142315 0.989821i −2.26543 2.61445i
25.6 0.241886 + 1.68236i −0.654861 + 0.755750i −0.852828 + 0.250413i 2.42873 1.56085i −1.42984 0.918904i 0.394581 + 2.74437i 0.784554 + 1.71793i −0.142315 0.989821i 3.21338 + 3.70844i
25.7 0.371015 + 2.58047i −0.654861 + 0.755750i −4.60217 + 1.35132i −0.825801 + 0.530710i −2.19315 1.40945i 0.240532 + 1.67294i −3.02854 6.63157i −0.142315 0.989821i −1.67586 1.93405i
40.1 −1.78536 2.06042i 0.415415 0.909632i −0.773176 + 5.37756i 2.58917 + 0.760248i −2.61589 + 0.768095i 0.566803 + 0.654126i 7.87337 5.05991i −0.654861 0.755750i −3.05618 6.69209i
40.2 −1.24364 1.43523i 0.415415 0.909632i −0.228633 + 1.59018i −2.11340 0.620551i −1.82216 + 0.535034i −0.191393 0.220879i −0.628614 + 0.403985i −0.654861 0.755750i 1.73767 + 3.80497i
40.3 −0.310677 0.358540i 0.415415 0.909632i 0.252599 1.75686i 2.52830 + 0.742376i −0.455200 + 0.133659i 0.907367 + 1.04716i −1.50659 + 0.968228i −0.654861 0.755750i −0.519313 1.13714i
40.4 0.546699 + 0.630925i 0.415415 0.909632i 0.185444 1.28979i −3.51047 1.03077i 0.801016 0.235200i −2.02356 2.33531i 2.31976 1.49082i −0.654861 0.755750i −1.26884 2.77836i
40.5 0.824588 + 0.951626i 0.415415 0.909632i 0.0589842 0.410244i 0.464170 + 0.136293i 1.20818 0.354752i 0.666534 + 0.769221i 2.55762 1.64368i −0.654861 0.755750i 0.253050 + 0.554102i
40.6 1.56927 + 1.81104i 0.415415 0.909632i −0.532609 + 3.70438i −1.90976 0.560756i 2.29928 0.675129i 2.91688 + 3.36626i −3.51271 + 2.25748i −0.654861 0.755750i −1.98139 4.33863i
See all 70 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 196.7 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.e even 11 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.2.i.b 70
3.b odd 2 1 603.2.u.e 70
67.e even 11 1 inner 201.2.i.b 70
201.k odd 22 1 603.2.u.e 70

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.i.b 70 1.a even 1 1 trivial
201.2.i.b 70 67.e even 11 1 inner
603.2.u.e 70 3.b odd 2 1
603.2.u.e 70 201.k odd 22 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{70} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database