# Properties

 Label 201.4.m.b Level 201 Weight 4 Character orbit 201.m Analytic conductor 11.859 Analytic rank 0 Dimension 360 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$201 = 3 \cdot 67$$ Weight: $$k$$ = $$4$$ Character orbit: $$[\chi]$$ = 201.m (of order $$33$$, degree $$20$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.8593839112$$ Analytic rank: $$0$$ Dimension: $$360$$ Relative dimension: $$18$$ over $$\Q(\zeta_{33})$$ Coefficient ring index: multiple of None Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{33}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$360q - 2q^{2} + 108q^{3} + 90q^{4} + 4q^{5} - 27q^{6} - 22q^{7} + 183q^{8} - 324q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$360q - 2q^{2} + 108q^{3} + 90q^{4} + 4q^{5} - 27q^{6} - 22q^{7} + 183q^{8} - 324q^{9} + 19q^{10} + 192q^{11} - 105q^{12} + 46q^{13} - 14q^{14} + 54q^{15} + 346q^{16} + 228q^{17} - 18q^{18} + 88q^{19} + 180q^{20} + 66q^{21} - 247q^{22} + 676q^{23} + 144q^{24} - 1120q^{25} + 333q^{26} + 972q^{27} - 3894q^{28} - 2210q^{29} + 702q^{30} + 96q^{31} - 1970q^{32} - 708q^{33} - 26q^{34} + 1272q^{35} + 315q^{36} - 3616q^{37} - 20q^{38} - 138q^{39} - 5357q^{40} + 288q^{41} - 618q^{42} - 1680q^{43} + 2951q^{44} + 36q^{45} - 1125q^{46} - 40q^{47} - 1038q^{48} - 4954q^{49} - 3590q^{50} - 90q^{51} + 9120q^{52} + 3896q^{53} + 54q^{54} - 1076q^{55} + 3557q^{56} + 660q^{57} + 3232q^{58} + 116q^{59} + 3618q^{60} - 672q^{61} + 3882q^{62} - 1584q^{63} - 11581q^{64} - 15380q^{65} - 5166q^{66} - 288q^{67} + 13898q^{68} - 1632q^{69} - 5058q^{70} - 15102q^{71} - 2016q^{72} - 146q^{73} - 3531q^{74} + 3360q^{75} + 10770q^{76} + 3594q^{77} + 1905q^{78} + 2852q^{79} + 27997q^{80} - 2916q^{81} + 8824q^{82} + 5278q^{83} + 6765q^{84} - 5290q^{85} + 5973q^{86} + 228q^{87} + 13872q^{88} + 1462q^{89} + 171q^{90} + 74q^{91} - 3144q^{92} + 3144q^{93} + 3262q^{94} + 1600q^{95} - 855q^{96} + 6626q^{97} + 15491q^{98} + 144q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1 −5.53116 0.528162i −2.52376 1.62192i 22.4593 + 4.32868i 0.468128 + 3.25590i 13.1027 + 10.3041i −14.8464 + 20.8489i −79.2899 23.2816i 3.73874 + 8.18669i −0.869647 18.2561i
4.2 −4.34497 0.414894i −2.52376 1.62192i 10.8512 + 2.09139i 0.231938 + 1.61316i 10.2927 + 8.09430i 0.778258 1.09291i −12.7769 3.75164i 3.73874 + 8.18669i −0.338470 7.10537i
4.3 −4.30544 0.411120i −2.52376 1.62192i 10.5124 + 2.02609i 2.02353 + 14.0740i 10.1991 + 8.02066i 12.0391 16.9066i −11.2287 3.29705i 3.73874 + 8.18669i −2.92611 61.4266i
4.4 −4.10740 0.392210i −2.52376 1.62192i 8.86152 + 1.70792i −1.30857 9.10131i 9.72997 + 7.65174i 6.79116 9.53685i −4.05636 1.19105i 3.73874 + 8.18669i 1.80521 + 37.8960i
4.5 −3.31558 0.316600i −2.52376 1.62192i 3.03741 + 0.585414i −2.57817 17.9316i 7.85423 + 6.17664i −18.7844 + 26.3790i 15.6805 + 4.60422i 3.73874 + 8.18669i 2.87100 + 60.2698i
4.6 −2.29557 0.219201i −2.52376 1.62192i −2.63381 0.507626i 2.82242 + 19.6304i 5.43795 + 4.27645i −16.2624 + 22.8374i 23.6357 + 6.94007i 3.73874 + 8.18669i −2.17608 45.6816i
4.7 −1.73979 0.166129i −2.52376 1.62192i −4.85617 0.935951i −1.62576 11.3074i 4.12135 + 3.24107i 11.0364 15.4984i 21.7085 + 6.37418i 3.73874 + 8.18669i 0.949981 + 19.9426i
4.8 −1.65101 0.157652i −2.52376 1.62192i −5.15445 0.993439i 0.854289 + 5.94171i 3.91105 + 3.07569i 2.57806 3.62038i 21.0841 + 6.19086i 3.73874 + 8.18669i −0.473715 9.94449i
4.9 0.315936 + 0.0301682i −2.52376 1.62192i −7.75652 1.49495i 0.375986 + 2.61504i −0.748416 0.588561i −0.540924 + 0.759621i −4.84160 1.42162i 3.73874 + 8.18669i 0.0398963 + 0.837527i
4.10 0.598865 + 0.0571847i −2.52376 1.62192i −7.50006 1.44552i −2.98516 20.7622i −1.41864 1.11563i 11.0160 15.4698i −9.02663 2.65046i 3.73874 + 8.18669i −0.600425 12.6045i
4.11 0.793373 + 0.0757579i −2.52376 1.62192i −7.23173 1.39380i −1.23463 8.58703i −1.87941 1.47798i −17.8196 + 25.0241i −11.7495 3.44995i 3.73874 + 8.18669i −0.328985 6.90625i
4.12 1.55324 + 0.148316i −2.52376 1.62192i −5.46489 1.05327i 1.80350 + 12.5436i −3.67944 2.89354i 18.8939 26.5327i −20.3088 5.96321i 3.73874 + 8.18669i 0.940843 + 19.7507i
4.13 1.84819 + 0.176481i −2.52376 1.62192i −4.47077 0.861670i 1.71935 + 11.9584i −4.37815 3.44302i 1.67021 2.34548i −22.3619 6.56604i 3.73874 + 8.18669i 1.06727 + 22.4048i
4.14 3.24171 + 0.309546i −2.52376 1.62192i 2.55743 + 0.492904i −1.28041 8.90545i −7.67924 6.03902i −6.81145 + 9.56535i −16.8585 4.95010i 3.73874 + 8.18669i −1.39407 29.2652i
4.15 4.06355 + 0.388022i −2.52376 1.62192i 8.50643 + 1.63948i 1.57648 + 10.9647i −9.62608 7.57003i −13.4370 + 18.8697i 2.59669 + 0.762458i 3.73874 + 8.18669i 2.15158 + 45.1672i
4.16 4.21003 + 0.402009i −2.52376 1.62192i 9.70733 + 1.87093i −0.265215 1.84461i −9.97308 7.84292i 17.7472 24.9224i 7.65306 + 2.24714i 3.73874 + 8.18669i −0.375013 7.87250i
4.17 5.11914 + 0.488818i −2.52376 1.62192i 18.1112 + 3.49064i 1.77055 + 12.3144i −12.1266 9.53650i −2.95569 + 4.15068i 51.5344 + 15.1319i 3.73874 + 8.18669i 3.04415 + 63.9046i
4.18 5.26356 + 0.502609i −2.52376 1.62192i 19.5970 + 3.77701i −2.69847 18.7683i −12.4688 9.80555i −5.28570 + 7.42273i 60.6650 + 17.8129i 3.73874 + 8.18669i −4.77045 100.144i
10.1 −4.01070 3.82419i 0.426945 + 2.96946i 1.08059 + 22.6844i −5.58319 + 12.2255i 9.64345 13.5423i −8.14672 + 33.5812i 53.3833 61.6077i −8.63544 + 2.53559i 69.1451 27.6815i
10.2 −3.39327 3.23548i 0.426945 + 2.96946i 0.665321 + 13.9668i −1.46096 + 3.19905i 8.15890 11.4576i 3.19725 13.1792i 18.3688 21.1988i −8.63544 + 2.53559i 15.3079 6.12835i
See next 80 embeddings (of 360 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.18 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.g even 33 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 201.4.m.b 360
67.g even 33 1 inner 201.4.m.b 360

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.m.b 360 1.a even 1 1 trivial
201.4.m.b 360 67.g even 33 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{360} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database