# Properties

 Label 201.4.i.b Level 201 Weight 4 Character orbit 201.i Analytic conductor 11.859 Analytic rank 0 Dimension 170 CM no Inner twists 2

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.i (of order $$11$$, degree $$10$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$170$$ Relative dimension: $$17$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$170q + 2q^{2} + 51q^{3} - 74q^{4} - 4q^{5} + 27q^{6} + 12q^{7} - 237q^{8} - 153q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$170q + 2q^{2} + 51q^{3} - 74q^{4} - 4q^{5} + 27q^{6} + 12q^{7} - 237q^{8} - 153q^{9} - 103q^{10} - 144q^{11} + 57q^{12} + 124q^{13} - 46q^{14} - 54q^{15} - 186q^{16} - 30q^{17} + 18q^{18} - 80q^{19} - 252q^{20} - 36q^{21} + 85q^{22} - 802q^{23} + 18q^{24} - 613q^{25} + 240q^{26} + 459q^{27} - 1555q^{28} + 2192q^{29} - 450q^{30} - 58q^{31} + 1859q^{32} - 426q^{33} + 302q^{34} - 36q^{35} - 171q^{36} + 3954q^{37} - 784q^{38} - 372q^{39} + 3005q^{40} - 366q^{41} + 798q^{42} + 3050q^{43} + 1975q^{44} - 36q^{45} + 3333q^{46} + 328q^{47} + 558q^{48} - 3167q^{49} + 2759q^{50} + 486q^{51} - 3726q^{52} - 104q^{53} - 54q^{54} + 998q^{55} + 7120q^{56} - 255q^{57} - 1612q^{58} - 2324q^{59} + 2835q^{60} - 1876q^{61} - 2496q^{62} + 207q^{63} + 3083q^{64} - 7636q^{65} - 3060q^{66} - 659q^{67} - 1178q^{68} + 30q^{69} - 2112q^{70} - 8352q^{71} - 846q^{72} - 3925q^{73} + 2196q^{74} + 1839q^{75} + 4532q^{76} - 2118q^{77} + 732q^{78} - 4505q^{79} - 3364q^{80} - 1377q^{81} - 6904q^{82} + 3272q^{83} - 2166q^{84} + 6262q^{85} - 8196q^{86} - 174q^{87} - 11862q^{88} - 3226q^{89} - 927q^{90} - 7534q^{91} + 10236q^{92} - 3258q^{93} + 5012q^{94} - 1294q^{95} + 1188q^{96} - 1674q^{97} - 9839q^{98} + 288q^{99} + O(q^{100})$$

## 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 −4.76164 + 1.39814i 0.426945 + 2.96946i 13.9884 8.98979i 6.28403 13.7601i −6.18469 13.5426i −26.7243 + 7.84696i −28.0398 + 32.3597i −8.63544 + 2.53559i −10.6837 + 74.3067i
22.2 −4.74745 + 1.39398i 0.426945 + 2.96946i 13.8651 8.91057i −4.28777 + 9.38890i −6.16627 13.5022i 10.9343 3.21061i −27.4815 + 31.7153i −8.63544 + 2.53559i 7.26805 50.5504i
22.3 −4.08747 + 1.20019i 0.426945 + 2.96946i 8.53694 5.48636i −2.24994 + 4.92667i −5.30904 11.6252i −22.8945 + 6.72243i −5.99199 + 6.91513i −8.63544 + 2.53559i 3.28361 22.8380i
22.4 −3.30674 + 0.970945i 0.426945 + 2.96946i 3.26174 2.09619i 3.42747 7.50511i −4.29498 9.40469i 18.8853 5.54523i 9.30455 10.7380i −8.63544 + 2.53559i −4.04668 + 28.1453i
22.5 −2.46694 + 0.724359i 0.426945 + 2.96946i −1.16894 + 0.751230i −2.43114 + 5.32346i −3.20420 7.01623i 16.7988 4.93256i 15.8092 18.2447i −8.63544 + 2.53559i 2.14139 14.8937i
22.6 −2.29706 + 0.674478i 0.426945 + 2.96946i −1.90846 + 1.22649i 5.88081 12.8772i −2.98356 6.53308i −8.81845 + 2.58933i 16.0987 18.5789i −8.63544 + 2.53559i −4.82321 + 33.5462i
22.7 −1.26486 + 0.371395i 0.426945 + 2.96946i −5.26810 + 3.38560i −4.40899 + 9.65433i −1.64287 3.59738i −28.6135 + 8.40167i 12.3122 14.2090i −8.63544 + 2.53559i 1.99116 13.8488i
22.8 −0.497304 + 0.146022i 0.426945 + 2.96946i −6.50404 + 4.17989i 3.14929 6.89598i −0.645927 1.41438i 18.8432 5.53285i 5.33944 6.16204i −8.63544 + 2.53559i −0.559192 + 3.88926i
22.9 −0.416363 + 0.122255i 0.426945 + 2.96946i −6.57162 + 4.22332i −8.47768 + 18.5635i −0.540797 1.18418i 21.7231 6.37847i 4.49322 5.18546i −8.63544 + 2.53559i 1.26030 8.76561i
22.10 0.736610 0.216288i 0.426945 + 2.96946i −6.23421 + 4.00649i 6.86250 15.0268i 0.956752 + 2.09500i −4.51523 + 1.32579i −7.74757 + 8.94117i −8.63544 + 2.53559i 1.80487 12.5532i
22.11 1.28293 0.376701i 0.426945 + 2.96946i −5.22603 + 3.35857i −3.87080 + 8.47587i 1.66634 + 3.64877i −3.40266 + 0.999110i −12.4443 + 14.3615i −8.63544 + 2.53559i −1.77308 + 12.3321i
22.12 2.63676 0.774222i 0.426945 + 2.96946i −0.376956 + 0.242255i −2.47257 + 5.41416i 3.42477 + 7.49921i 15.4808 4.54556i −15.2032 + 17.5455i −8.63544 + 2.53559i −2.32779 + 16.1902i
22.13 3.06465 0.899863i 0.426945 + 2.96946i 1.85231 1.19040i 1.33631 2.92610i 3.98055 + 8.71618i −27.3642 + 8.03485i −12.1277 + 13.9961i −8.63544 + 2.53559i 1.46222 10.1700i
22.14 3.34038 0.980825i 0.426945 + 2.96946i 3.46611 2.22754i −0.109121 + 0.238942i 4.33868 + 9.50039i 26.3727 7.74373i −8.84536 + 10.2081i −8.63544 + 2.53559i −0.130146 + 0.905187i
22.15 4.03324 1.18427i 0.426945 + 2.96946i 8.13449 5.22772i 8.01749 17.5559i 5.23860 + 11.4709i 5.21241 1.53050i 4.59562 5.30363i −8.63544 + 2.53559i 11.5457 80.3018i
22.16 4.37690 1.28517i 0.426945 + 2.96946i 10.7756 6.92503i −8.92036 + 19.5329i 5.68497 + 12.4484i −12.4208 + 3.64706i 14.3655 16.5787i −8.63544 + 2.53559i −13.9404 + 96.9576i
22.17 5.17153 1.51850i 0.426945 + 2.96946i 17.7089 11.3808i 0.676102 1.48046i 6.71709 + 14.7084i 12.3215 3.61792i 46.0635 53.1601i −8.63544 + 2.53559i 1.24841 8.68289i
25.1 −0.715229 4.97453i 1.96458 2.26725i −16.5584 + 4.86199i 10.1409 6.51716i −12.6836 8.15127i −4.69471 32.6525i 19.3272 + 42.3208i −1.28083 8.90839i −39.6729 45.7849i
25.2 −0.708000 4.92425i 1.96458 2.26725i −16.0710 + 4.71887i −2.26976 + 1.45869i −12.5554 8.06888i 2.15164 + 14.9650i 18.0820 + 39.5942i −1.28083 8.90839i 8.78993 + 10.1441i
25.3 −0.619824 4.31097i 1.96458 2.26725i −10.5243 + 3.09022i 11.4827 7.37951i −10.9917 7.06396i 0.851874 + 5.92491i 5.37102 + 11.7609i −1.28083 8.90839i −38.9301 44.9277i
See next 80 embeddings (of 170 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 196.17 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 201.4.i.b 170
67.e even 11 1 inner 201.4.i.b 170

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.i.b 170 1.a even 1 1 trivial
201.4.i.b 170 67.e even 11 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{170} - \cdots$$ acting on $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database