# Properties

 Label 201.4.m.a Level 201 Weight 4 Character orbit 201.m Analytic conductor 11.859 Analytic rank 0 Dimension 320 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: $$320$$ Relative dimension: $$16$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$320q - 2q^{2} - 96q^{3} + 66q^{4} - 4q^{5} + 27q^{6} + 14q^{7} + 123q^{8} - 288q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$320q - 2q^{2} - 96q^{3} + 66q^{4} - 4q^{5} + 27q^{6} + 14q^{7} + 123q^{8} - 288q^{9} + 35q^{10} - 192q^{11} + 33q^{12} - 88q^{13} - 214q^{14} + 54q^{15} + 298q^{16} - 272q^{17} - 18q^{18} - 64q^{19} - 164q^{20} + 42q^{21} + 473q^{22} - 732q^{23} - 324q^{24} - 484q^{25} - 353q^{26} - 864q^{27} + 4338q^{28} + 1646q^{29} + 666q^{30} + 302q^{31} + 910q^{32} + 612q^{33} - 138q^{34} - 552q^{35} + 99q^{36} + 3388q^{37} + 194q^{38} - 264q^{39} + 2003q^{40} + 1152q^{41} + 18q^{42} - 2788q^{43} - 2439q^{44} - 36q^{45} - 477q^{46} - 512q^{47} + 894q^{48} + 5020q^{49} - 4098q^{50} - 90q^{51} - 2992q^{52} - 368q^{53} - 54q^{54} - 2492q^{55} + 2775q^{56} + 1194q^{57} + 2088q^{58} - 364q^{59} - 8874q^{60} + 14q^{61} + 1762q^{62} - 666q^{63} - 10293q^{64} + 5568q^{65} - 4290q^{66} - 1562q^{67} + 4726q^{68} - 1800q^{69} - 2498q^{70} + 6194q^{71} - 2556q^{72} - 2154q^{73} + 3221q^{74} - 1452q^{75} - 8038q^{76} + 1358q^{77} + 1845q^{78} + 4924q^{79} - 2143q^{80} - 2592q^{81} + 5080q^{82} + 2606q^{83} + 243q^{84} + 2786q^{85} + 6437q^{86} - 144q^{87} - 2884q^{88} + 3018q^{89} + 315q^{90} + 5506q^{91} - 15636q^{92} - 2526q^{93} + 2358q^{94} - 140q^{95} + 1575q^{96} + 2408q^{97} - 7221q^{98} - 144q^{99} + O(q^{100})$$

## 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 −4.94279 0.471979i 2.52376 + 1.62192i 16.3530 + 3.15178i −1.82165 12.6699i −11.7089 9.20798i −8.77874 + 12.3280i −41.2286 12.1058i 3.73874 + 8.18669i 3.02413 + 63.4843i
4.2 −4.56923 0.436309i 2.52376 + 1.62192i 12.8321 + 2.47318i 0.0429643 + 0.298823i −10.8240 8.51208i 4.15175 5.83032i −22.3209 6.55402i 3.73874 + 8.18669i −0.0659346 1.38414i
4.3 −4.11427 0.392866i 2.52376 + 1.62192i 8.91749 + 1.71870i 2.22379 + 15.4668i −9.74625 7.66453i −16.1259 + 22.6457i −4.28917 1.25941i 3.73874 + 8.18669i −3.07291 64.5082i
4.4 −3.06689 0.292852i 2.52376 + 1.62192i 1.46462 + 0.282282i 1.97722 + 13.7518i −7.26511 5.71335i 4.23013 5.94039i 19.2392 + 5.64914i 3.73874 + 8.18669i −2.03664 42.7544i
4.5 −3.05732 0.291939i 2.52376 + 1.62192i 1.40655 + 0.271090i −1.09359 7.60610i −7.24244 5.69552i 20.0509 28.1576i 19.3534 + 5.68268i 3.73874 + 8.18669i 1.12295 + 23.5735i
4.6 −1.70394 0.162706i 2.52376 + 1.62192i −4.97850 0.959527i −0.668012 4.64612i −4.03643 3.17429i −17.4462 + 24.4998i 21.4658 + 6.30291i 3.73874 + 8.18669i 0.382297 + 8.02539i
4.7 −1.46775 0.140153i 2.52376 + 1.62192i −5.72077 1.10259i −0.719139 5.00172i −3.47694 2.73430i −1.20398 + 1.69075i 19.5598 + 5.74327i 3.73874 + 8.18669i 0.354510 + 7.44208i
4.8 −0.0857546 0.00818857i 2.52376 + 1.62192i −7.84814 1.51261i 2.32615 + 16.1787i −0.203143 0.159753i 3.76989 5.29407i 1.32187 + 0.388136i 3.73874 + 8.18669i −0.0669974 1.40645i
4.9 −0.0473779 0.00452404i 2.52376 + 1.62192i −7.85321 1.51358i −2.96060 20.5914i −0.112233 0.0882609i −2.98860 + 4.19691i 0.730545 + 0.214507i 3.73874 + 8.18669i 0.0471106 + 0.988972i
4.10 0.993416 + 0.0948597i 2.52376 + 1.62192i −6.87755 1.32554i 0.448481 + 3.11925i 2.35329 + 1.85065i 10.9985 15.4453i −14.3666 4.21842i 3.73874 + 8.18669i 0.149636 + 3.14126i
4.11 2.23941 + 0.213838i 2.52376 + 1.62192i −2.88618 0.556265i 0.118431 + 0.823704i 5.30492 + 4.17183i −8.75517 + 12.2949i −23.6122 6.93317i 3.73874 + 8.18669i 0.0890762 + 1.86994i
4.12 2.82303 + 0.269566i 2.52376 + 1.62192i 0.0413796 + 0.00797527i −2.49944 17.3840i 6.68743 + 5.25905i −2.70946 + 3.80491i −21.6533 6.35798i 3.73874 + 8.18669i −2.36984 49.7491i
4.13 2.84022 + 0.271208i 2.52376 + 1.62192i 0.137874 + 0.0265731i 1.10231 + 7.66675i 6.72816 + 5.29108i −17.6977 + 24.8529i −21.5162 6.31771i 3.73874 + 8.18669i 1.05152 + 22.0742i
4.14 4.04250 + 0.386012i 2.52376 + 1.62192i 8.33734 + 1.60689i 2.93548 + 20.4167i 9.57621 + 7.53082i 11.8752 16.6763i 1.91228 + 0.561495i 3.73874 + 8.18669i 3.98557 + 83.6675i
4.15 4.76735 + 0.455226i 2.52376 + 1.62192i 14.6649 + 2.82643i −1.49068 10.3679i 11.2933 + 8.88115i 14.0800 19.7726i 31.8658 + 9.35663i 3.73874 + 8.18669i −2.38685 50.1061i
4.16 5.06607 + 0.483751i 2.52376 + 1.62192i 17.5756 + 3.38742i 0.408501 + 2.84119i 12.0009 + 9.43764i −9.17020 + 12.8778i 48.3368 + 14.1930i 3.73874 + 8.18669i 0.695068 + 14.5913i
10.1 −3.91689 3.73474i −0.426945 2.96946i 1.01303 + 21.2662i −4.66810 + 10.2217i −9.41789 + 13.2256i 7.68968 31.6973i 47.1027 54.3594i −8.63544 + 2.53559i 56.4599 22.6031i
10.2 −3.69235 3.52065i −0.426945 2.96946i 0.857825 + 18.0080i 5.73558 12.5592i −8.87801 + 12.4674i −1.77652 + 7.32293i 33.5045 38.6663i −8.63544 + 2.53559i −65.3941 + 26.1799i
10.3 −3.14869 3.00227i −0.426945 2.96946i 0.519968 + 10.9155i −2.82594 + 6.18794i −7.57081 + 10.6317i −4.75860 + 19.6152i 8.34160 9.62671i −8.63544 + 2.53559i 27.4759 10.9997i
10.4 −2.42146 2.30886i −0.426945 2.96946i 0.151991 + 3.19068i 5.31778 11.6443i −5.82224 + 8.17619i 3.18602 13.1329i −10.5294 + 12.1516i −8.63544 + 2.53559i −39.7619 + 15.9183i
See next 80 embeddings (of 320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 199.16 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.a 320
67.g even 33 1 inner 201.4.m.a 320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.4.m.a 320 1.a even 1 1 trivial
201.4.m.a 320 67.g even 33 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{320} + \cdots$$ acting on $$S_{4}^{\mathrm{new}}(201, [\chi])$$.

## Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database