Properties

 Label 201.2.i.a Level 201 Weight 2 Character orbit 201.i Analytic conductor 1.605 Analytic rank 0 Dimension 50 CM no Inner twists 2

Related objects

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: $$50$$ Relative dimension: $$5$$ 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.

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

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
22.1 −1.87424 + 0.550325i 0.142315 + 0.989821i 1.52739 0.981596i −1.74459 + 3.82012i −0.811455 1.77684i −0.0581100 + 0.0170626i 0.235861 0.272198i −0.959493 + 0.281733i 1.16746 8.11989i
22.2 −1.59954 + 0.469669i 0.142315 + 0.989821i 0.655446 0.421230i 1.13272 2.48031i −0.692527 1.51642i 0.0374006 0.0109818i 1.33282 1.53816i −0.959493 + 0.281733i −0.646912 + 4.49937i
22.3 −0.385020 + 0.113052i 0.142315 + 0.989821i −1.54705 + 0.994227i 0.0716395 0.156869i −0.166695 0.365012i −2.67716 + 0.786085i 1.00880 1.16422i −0.959493 + 0.281733i −0.00984831 + 0.0684965i
22.4 0.907634 0.266505i 0.142315 + 0.989821i −0.929732 + 0.597503i 1.48348 3.24837i 0.392963 + 0.860468i 4.49742 1.32056i −1.92355 + 2.21990i −0.959493 + 0.281733i 0.480750 3.34369i
22.5 2.15399 0.632468i 0.142315 + 0.989821i 2.55715 1.64338i −0.146077 + 0.319865i 0.932575 + 2.04206i −0.489834 + 0.143828i 1.52846 1.76393i −0.959493 + 0.281733i −0.112345 + 0.781375i
25.1 −0.381059 2.65032i 0.654861 0.755750i −4.96003 + 1.45640i −2.85080 + 1.83210i −2.25252 1.44761i −0.558441 3.88404i 3.52537 + 7.71949i −0.142315 0.989821i 5.94199 + 6.85742i
25.2 −0.254047 1.76694i 0.654861 0.755750i −1.13854 + 0.334305i 1.11491 0.716511i −1.50173 0.965101i 0.110811 + 0.770710i −0.603181 1.32078i −0.142315 0.989821i −1.54927 1.78795i
25.3 0.0345454 + 0.240269i 0.654861 0.755750i 1.86245 0.546865i 0.0517218 0.0332396i 0.204205 + 0.131235i −0.587961 4.08936i 0.397409 + 0.870204i −0.142315 0.989821i 0.00977317 + 0.0112788i
25.4 0.0704781 + 0.490186i 0.654861 0.755750i 1.68367 0.494370i −0.688821 + 0.442678i 0.416611 + 0.267740i 0.350196 + 2.43567i 0.772445 + 1.69142i −0.142315 0.989821i −0.265542 0.306451i
25.5 0.290637 + 2.02142i 0.654861 0.755750i −2.08270 + 0.611536i 2.61244 1.67891i 1.71802 + 1.10410i −0.145436 1.01153i −0.144754 0.316968i −0.142315 0.989821i 4.15306 + 4.79289i
40.1 −1.51073 1.74347i −0.415415 + 0.909632i −0.472767 + 3.28817i −2.59924 0.763205i 2.21349 0.649941i 0.716487 + 0.826871i 2.56560 1.64881i −0.654861 0.755750i 2.59611 + 5.68469i
40.2 −0.371960 0.429265i −0.415415 + 0.909632i 0.238716 1.66030i −2.09548 0.615288i 0.544991 0.160024i −1.13239 1.30685i −1.75717 + 1.12926i −0.654861 0.755750i 0.515313 + 1.12838i
40.3 0.260342 + 0.300451i −0.415415 + 0.909632i 0.262137 1.82320i 3.42699 + 1.00626i −0.381450 + 0.112004i −1.73782 2.00555i 1.28492 0.825765i −0.654861 0.755750i 0.589861 + 1.29161i
40.4 1.22439 + 1.41303i −0.415415 + 0.909632i −0.212872 + 1.48056i 1.12534 + 0.330431i −1.79397 + 0.526756i 1.59786 + 1.84403i 0.793078 0.509681i −0.654861 0.755750i 0.910957 + 1.99472i
40.5 1.65462 + 1.90953i −0.415415 + 0.909632i −0.623918 + 4.33945i −1.11429 0.327184i −2.42432 + 0.711846i −1.12665 1.30022i −5.06751 + 3.25669i −0.654861 0.755750i −1.21895 2.66913i
64.1 −1.87424 0.550325i 0.142315 0.989821i 1.52739 + 0.981596i −1.74459 3.82012i −0.811455 + 1.77684i −0.0581100 0.0170626i 0.235861 + 0.272198i −0.959493 0.281733i 1.16746 + 8.11989i
64.2 −1.59954 0.469669i 0.142315 0.989821i 0.655446 + 0.421230i 1.13272 + 2.48031i −0.692527 + 1.51642i 0.0374006 + 0.0109818i 1.33282 + 1.53816i −0.959493 0.281733i −0.646912 4.49937i
64.3 −0.385020 0.113052i 0.142315 0.989821i −1.54705 0.994227i 0.0716395 + 0.156869i −0.166695 + 0.365012i −2.67716 0.786085i 1.00880 + 1.16422i −0.959493 0.281733i −0.00984831 0.0684965i
64.4 0.907634 + 0.266505i 0.142315 0.989821i −0.929732 0.597503i 1.48348 + 3.24837i 0.392963 0.860468i 4.49742 + 1.32056i −1.92355 2.21990i −0.959493 0.281733i 0.480750 + 3.34369i
64.5 2.15399 + 0.632468i 0.142315 0.989821i 2.55715 + 1.64338i −0.146077 0.319865i 0.932575 2.04206i −0.489834 0.143828i 1.52846 + 1.76393i −0.959493 0.281733i −0.112345 0.781375i
See all 50 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 196.5 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.a 50
3.b odd 2 1 603.2.u.d 50
67.e even 11 1 inner 201.2.i.a 50
201.k odd 22 1 603.2.u.d 50

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.i.a 50 1.a even 1 1 trivial
201.2.i.a 50 67.e even 11 1 inner
603.2.u.d 50 3.b odd 2 1
603.2.u.d 50 201.k odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{50} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(201, [\chi])$$.

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database