LMFDB - Embedded newform 2160.3.l.g.161.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2160,3,Mod(161,2160)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2160.161"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.l (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 135)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-0.618034i$ of defining polynomial
Character	$\chi$	$=$	2160.161
Dual form			2160.3.l.g.161.3

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{5} -3.70820 q^{7} +8.18034i q^{11} +7.70820 q^{13} +11.9443i q^{17} -5.58359 q^{19} -28.4164i q^{23} -5.00000 q^{25} -56.0689i q^{29} +31.2492 q^{31} +8.29180i q^{35} -53.4164 q^{37} +60.7639i q^{41} +71.3738 q^{43} +46.1378i q^{47} -35.2492 q^{49} +73.3607i q^{53} +18.2918 q^{55} +90.7639i q^{59} +19.0000 q^{61} -17.2361i q^{65} -0.334369 q^{67} -81.2624i q^{71} +50.7902 q^{73} -30.3344i q^{77} +48.1672 q^{79} -62.6656i q^{83} +26.7082 q^{85} -69.7082i q^{89} -28.5836 q^{91} +12.4853i q^{95} +159.416 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{7} + 4 q^{13} - 76 q^{19} - 20 q^{25} - 36 q^{31} - 160 q^{37} + 44 q^{43} + 20 q^{49} + 100 q^{55} + 76 q^{61} - 216 q^{67} - 92 q^{73} + 300 q^{79} + 80 q^{85} - 168 q^{91} + 584 q^{97}+O(q^{100}) $ 4 * q + 12 * q^7 + 4 * q^13 - 76 * q^19 - 20 * q^25 - 36 * q^31 - 160 * q^37 + 44 * q^43 + 20 * q^49 + 100 * q^55 + 76 * q^61 - 216 * q^67 - 92 * q^73 + 300 * q^79 + 80 * q^85 - 168 * q^91 + 584 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	−3.70820	−0.529743	−0.264872	−	0.964284i	$-0.585330\pi$
$7$	−3.70820	−0.529743	−0.264872	+	0.964284i	$0.585330\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	8.18034i	0.743667i	0.928299	+	0.371834i	$0.121271\pi$
$11$	8.18034i	0.743667i	−0.928299	+	0.371834i	$0.878729\pi$
$12$	0	0
$13$	7.70820	0.592939	0.296469	−	0.955042i	$-0.404191\pi$
$13$	7.70820	0.592939	0.296469	+	0.955042i	$0.404191\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	11.9443i	0.702604i	0.936262	+	0.351302i	$0.114261\pi$
$17$	11.9443i	0.702604i	−0.936262	+	0.351302i	$0.885739\pi$
$18$	0	0
$19$	−5.58359	−0.293873	−0.146937	−	0.989146i	$-0.546941\pi$
$19$	−5.58359	−0.293873	−0.146937	+	0.989146i	$0.546941\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 28.4164i	− 1.23550i	−0.786376	−	0.617748i	$-0.788045\pi$
$23$	− 28.4164i	− 1.23550i	0.786376	−	0.617748i	$-0.211955\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 56.0689i	− 1.93341i	−0.255894	−	0.966705i	$-0.582370\pi$
$29$	− 56.0689i	− 1.93341i	0.255894	−	0.966705i	$-0.417630\pi$
$30$	0	0
$31$	31.2492	1.00804	0.504020	−	0.863692i	$-0.331854\pi$
$31$	31.2492	1.00804	0.504020	+	0.863692i	$0.331854\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.29180i	0.236908i
$36$	0	0
$37$	−53.4164	−1.44369	−0.721843	−	0.692056i	$-0.756705\pi$
$37$	−53.4164	−1.44369	−0.721843	+	0.692056i	$0.756705\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	60.7639i	1.48205i	0.671479	+	0.741024i	$0.265659\pi$
$41$	60.7639i	1.48205i	−0.671479	+	0.741024i	$0.734341\pi$
$42$	0	0
$43$	71.3738	1.65986	0.829928	−	0.557870i	$-0.188381\pi$
$43$	71.3738	1.65986	0.829928	+	0.557870i	$0.188381\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	46.1378i	0.981655i	0.871257	+	0.490827i	$0.163305\pi$
$47$	46.1378i	0.981655i	−0.871257	+	0.490827i	$0.836695\pi$
$48$	0	0
$49$	−35.2492	−0.719372
$50$	0	0
$51$	0	0
$52$	0	0
$53$	73.3607i	1.38416i	0.721819	+	0.692082i	$0.243306\pi$
$53$	73.3607i	1.38416i	−0.721819	+	0.692082i	$0.756694\pi$
$54$	0	0
$55$	18.2918	0.332578
$56$	0	0
$57$	0	0
$58$	0	0
$59$	90.7639i	1.53837i	0.639025	+	0.769186i	$0.279338\pi$
$59$	90.7639i	1.53837i	−0.639025	+	0.769186i	$0.720662\pi$
$60$	0	0
$61$	19.0000	0.311475	0.155738	−	0.987798i	$-0.450225\pi$
$61$	19.0000	0.311475	0.155738	+	0.987798i	$0.450225\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 17.2361i	− 0.265170i
$66$	0	0
$67$	−0.334369	−0.00499058	−0.00249529	−	0.999997i	$-0.500794\pi$
$67$	−0.334369	−0.00499058	−0.00249529	+	0.999997i	$0.500794\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 81.2624i	− 1.14454i	−0.820065	−	0.572270i	$-0.806063\pi$
$71$	− 81.2624i	− 1.14454i	0.820065	−	0.572270i	$-0.193937\pi$
$72$	0	0
$73$	50.7902	0.695757	0.347878	−	0.937540i	$-0.386902\pi$
$73$	50.7902	0.695757	0.347878	+	0.937540i	$0.386902\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 30.3344i	− 0.393953i
$78$	0	0
$79$	48.1672	0.609711	0.304856	−	0.952399i	$-0.401392\pi$
$79$	48.1672	0.609711	0.304856	+	0.952399i	$0.401392\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 62.6656i	− 0.755008i	−0.926008	−	0.377504i	$-0.876783\pi$
$83$	− 62.6656i	− 0.755008i	0.926008	−	0.377504i	$-0.123217\pi$
$84$	0	0
$85$	26.7082	0.314214
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 69.7082i	− 0.783238i	−0.920127	−	0.391619i	$-0.871915\pi$
$89$	− 69.7082i	− 0.783238i	0.920127	−	0.391619i	$-0.128085\pi$
$90$	0	0
$91$	−28.5836	−0.314105
$92$	0	0
$93$	0	0
$94$	0	0
$95$	12.4853i	0.131424i
$96$	0	0
$97$	159.416	1.64347	0.821734	−	0.569871i	$-0.193007\pi$
$97$	159.416	1.64347	0.821734	+	0.569871i	$0.193007\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	24.0000i	0.237624i	0.992917	+	0.118812i	$0.0379085\pi$
$101$	24.0000i	0.237624i	−0.992917	+	0.118812i	$0.962091\pi$
$102$	0	0
$103$	25.1672	0.244342	0.122171	−	0.992509i	$-0.461014\pi$
$103$	25.1672	0.244342	0.122171	+	0.992509i	$0.461014\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 47.6656i	− 0.445473i	−0.974879	−	0.222737i	$-0.928501\pi$
$107$	− 47.6656i	− 0.445473i	0.974879	−	0.222737i	$-0.0714990\pi$
$108$	0	0
$109$	164.915	1.51298	0.756490	−	0.654005i	$-0.226913\pi$
$109$	164.915	1.51298	0.756490	+	0.654005i	$0.226913\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 68.3870i	− 0.605195i	−0.953118	−	0.302597i	$-0.902146\pi$
$113$	− 68.3870i	− 0.605195i	0.953118	−	0.302597i	$-0.0978537\pi$
$114$	0	0
$115$	−63.5410	−0.552531
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 44.2918i	− 0.372200i
$120$	0	0
$121$	54.0820	0.446959
$122$	0	0
$123$	0	0
$124$	0	0
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	200.748	1.58069	0.790345	−	0.612662i	$-0.209901\pi$
$127$	200.748	1.58069	0.790345	+	0.612662i	$0.209901\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	8.16408i	0.0623212i	0.999514	+	0.0311606i	$0.00992033\pi$
$131$	8.16408i	0.0623212i	−0.999514	+	0.0311606i	$0.990080\pi$
$132$	0	0
$133$	20.7051	0.155677
$134$	0	0
$135$	0	0
$136$	0	0
$137$	50.3313i	0.367381i	0.982984	+	0.183691i	$0.0588045\pi$
$137$	50.3313i	0.367381i	−0.982984	+	0.183691i	$0.941196\pi$
$138$	0	0
$139$	126.498	0.910061	0.455030	−	0.890476i	$-0.349628\pi$
$139$	126.498	0.910061	0.455030	+	0.890476i	$0.349628\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	63.0557i	0.440949i
$144$	0	0
$145$	−125.374	−0.864647
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 15.8197i	− 0.106172i	−0.998590	−	0.0530861i	$-0.983094\pi$
$149$	− 15.8197i	− 0.106172i	0.998590	−	0.0530861i	$-0.0169058\pi$
$150$	0	0
$151$	138.997	0.920509	0.460255	−	0.887787i	$-0.347758\pi$
$151$	138.997	0.920509	0.460255	+	0.887787i	$0.347758\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 69.8754i	− 0.450809i
$156$	0	0
$157$	190.705	1.21468	0.607341	−	0.794441i	$-0.292236\pi$
$157$	190.705	1.21468	0.607341	+	0.794441i	$0.292236\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	105.374i	0.654496i
$162$	0	0
$163$	36.8328	0.225968	0.112984	−	0.993597i	$-0.463959\pi$
$163$	36.8328	0.225968	0.112984	+	0.993597i	$0.463959\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	258.580i	1.54839i	0.632950	+	0.774193i	$0.281844\pi$
$167$	258.580i	1.54839i	−0.632950	+	0.774193i	$0.718156\pi$
$168$	0	0
$169$	−109.584	−0.648424
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 182.331i	− 1.05394i	−0.849885	−	0.526969i	$-0.823328\pi$
$173$	− 182.331i	− 1.05394i	0.849885	−	0.526969i	$-0.176672\pi$
$174$	0	0
$175$	18.5410	0.105949
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 256.774i	− 1.43449i	−0.696820	−	0.717246i	$-0.745402\pi$
$179$	− 256.774i	− 1.43449i	0.696820	−	0.717246i	$-0.254598\pi$
$180$	0	0
$181$	180.082	0.994928	0.497464	−	0.867485i	$-0.334265\pi$
$181$	180.082	0.994928	0.497464	+	0.867485i	$0.334265\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	119.443i	0.645636i
$186$	0	0
$187$	−97.7082	−0.522504
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 128.387i	− 0.672183i	−0.941829	−	0.336092i	$-0.890895\pi$
$191$	− 128.387i	− 0.672183i	0.941829	−	0.336092i	$-0.109105\pi$
$192$	0	0
$193$	64.2918	0.333118	0.166559	−	0.986031i	$-0.446734\pi$
$193$	64.2918	0.333118	0.166559	+	0.986031i	$0.446734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 217.997i	− 1.10658i	−0.832988	−	0.553292i	$-0.813372\pi$
$197$	− 217.997i	− 1.10658i	0.832988	−	0.553292i	$-0.186628\pi$
$198$	0	0
$199$	−320.249	−1.60929	−0.804646	−	0.593754i	$-0.797645\pi$
$199$	−320.249	−1.60929	−0.804646	+	0.593754i	$0.797645\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	207.915i	1.02421i
$204$	0	0
$205$	135.872	0.662792
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 45.6757i	− 0.218544i
$210$	0	0
$211$	63.8328	0.302525	0.151263	−	0.988494i	$-0.451666\pi$
$211$	63.8328	0.302525	0.151263	+	0.988494i	$0.451666\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 159.597i	− 0.742310i
$216$	0	0
$217$	−115.878	−0.534002
$218$	0	0
$219$	0	0
$220$	0	0
$221$	92.0689i	0.416601i
$222$	0	0
$223$	8.24922	0.0369920	0.0184960	−	0.999829i	$-0.494112\pi$
$223$	8.24922	0.0369920	0.0184960	+	0.999829i	$0.494112\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	339.525i	1.49570i	0.663866	+	0.747852i	$0.268915\pi$
$227$	339.525i	1.49570i	−0.663866	+	0.747852i	$0.731085\pi$
$228$	0	0
$229$	202.495	0.884259	0.442130	−	0.896951i	$-0.354223\pi$
$229$	202.495	0.884259	0.442130	+	0.896951i	$0.354223\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	19.0820i	0.0818972i	0.999161	+	0.0409486i	$0.0130380\pi$
$233$	19.0820i	0.0818972i	−0.999161	+	0.0409486i	$0.986962\pi$
$234$	0	0
$235$	103.167	0.439009
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 272.610i	− 1.14063i	−0.821427	−	0.570314i	$-0.806822\pi$
$239$	− 272.610i	− 1.14063i	0.821427	−	0.570314i	$-0.193178\pi$
$240$	0	0
$241$	232.580	0.965064	0.482532	−	0.875878i	$-0.339717\pi$
$241$	232.580	0.965064	0.482532	+	0.875878i	$0.339717\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	78.8197i	0.321713i
$246$	0	0
$247$	−43.0395	−0.174249
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 14.4195i	− 0.0574483i	−0.999587	−	0.0287241i	$-0.990856\pi$
$251$	− 14.4195i	− 0.0574483i	0.999587	−	0.0287241i	$-0.00914443\pi$
$252$	0	0
$253$	232.456	0.918798
$254$	0	0
$255$	0	0
$256$	0	0
$257$	165.748i	0.644933i	0.946581	+	0.322466i	$0.104512\pi$
$257$	165.748i	0.644933i	−0.946581	+	0.322466i	$0.895488\pi$
$258$	0	0
$259$	198.079	0.764784
$260$	0	0
$261$	0	0
$262$	0	0
$263$	520.026i	1.97729i	0.150282	+	0.988643i	$0.451982\pi$
$263$	520.026i	1.97729i	−0.150282	+	0.988643i	$0.548018\pi$
$264$	0	0
$265$	164.039	0.619017
$266$	0	0
$267$	0	0
$268$	0	0
$269$	87.3576i	0.324749i	0.986729	+	0.162375i	$0.0519153\pi$
$269$	87.3576i	0.324749i	−0.986729	+	0.162375i	$0.948085\pi$
$270$	0	0
$271$	−35.5836	−0.131305	−0.0656524	−	0.997843i	$-0.520913\pi$
$271$	−35.5836	−0.131305	−0.0656524	+	0.997843i	$0.520913\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 40.9017i	− 0.148733i
$276$	0	0
$277$	147.374	0.532036	0.266018	−	0.963968i	$-0.414292\pi$
$277$	147.374	0.532036	0.266018	+	0.963968i	$0.414292\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 190.869i	− 0.679250i	−0.940561	−	0.339625i	$-0.889700\pi$
$281$	− 190.869i	− 0.679250i	0.940561	−	0.339625i	$-0.110300\pi$
$282$	0	0
$283$	−117.708	−0.415930	−0.207965	−	0.978136i	$-0.566684\pi$
$283$	−117.708	−0.415930	−0.207965	+	0.978136i	$0.566684\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 225.325i	− 0.785105i
$288$	0	0
$289$	146.334	0.506347
$290$	0	0
$291$	0	0
$292$	0	0
$293$	413.243i	1.41039i	0.709016	+	0.705193i	$0.249140\pi$
$293$	413.243i	1.41039i	−0.709016	+	0.705193i	$0.750860\pi$
$294$	0	0
$295$	202.954	0.687981
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 219.039i	− 0.732573i
$300$	0	0
$301$	−264.669	−0.879298
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 42.4853i	− 0.139296i
$306$	0	0
$307$	458.158	1.49237	0.746185	−	0.665738i	$-0.231883\pi$
$307$	458.158	1.49237	0.746185	+	0.665738i	$0.231883\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	118.043i	0.379558i	0.981827	+	0.189779i	$0.0607772\pi$
$311$	118.043i	0.379558i	−0.981827	+	0.189779i	$0.939223\pi$
$312$	0	0
$313$	−412.827	−1.31893	−0.659467	−	0.751733i	$-0.729218\pi$
$313$	−412.827	−1.31893	−0.659467	+	0.751733i	$0.729218\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 269.944i	− 0.851559i	−0.904827	−	0.425780i	$-0.860000\pi$
$317$	− 269.944i	− 0.851559i	0.904827	−	0.425780i	$-0.140000\pi$
$318$	0	0
$319$	458.663	1.43781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 66.6919i	− 0.206477i
$324$	0	0
$325$	−38.5410	−0.118588
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 171.088i	− 0.520025i
$330$	0	0
$331$	−9.91486	−0.0299542	−0.0149771	−	0.999888i	$-0.504768\pi$
$331$	−9.91486	−0.0299542	−0.0149771	+	0.999888i	$0.504768\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.747671i	0.00223185i
$336$	0	0
$337$	−436.450	−1.29510	−0.647551	−	0.762022i	$-0.724207\pi$
$337$	−436.450	−1.29510	−0.647551	+	0.762022i	$0.724207\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	255.629i	0.749646i
$342$	0	0
$343$	312.413	0.910826
$344$	0	0
$345$	0	0
$346$	0	0
$347$	25.3839i	0.0731524i	0.999331	+	0.0365762i	$0.0116452\pi$
$347$	25.3839i	0.0731524i	−0.999331	+	0.0365762i	$0.988355\pi$
$348$	0	0
$349$	−72.0820	−0.206539	−0.103269	−	0.994653i	$-0.532930\pi$
$349$	−72.0820	−0.206539	−0.103269	+	0.994653i	$0.532930\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	392.164i	1.11095i	0.831534	+	0.555473i	$0.187463\pi$
$353$	392.164i	1.11095i	−0.831534	+	0.555473i	$0.812537\pi$
$354$	0	0
$355$	−181.708	−0.511854
$356$	0	0
$357$	0	0
$358$	0	0
$359$	399.994i	1.11419i	0.830449	+	0.557094i	$0.188084\pi$
$359$	399.994i	1.11419i	−0.830449	+	0.557094i	$0.811916\pi$
$360$	0	0
$361$	−329.823	−0.913639
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 113.570i	− 0.311152i
$366$	0	0
$367$	−206.371	−0.562318	−0.281159	−	0.959661i	$-0.590719\pi$
$367$	−206.371	−0.562318	−0.281159	+	0.959661i	$0.590719\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 272.036i	− 0.733252i
$372$	0	0
$373$	354.833	0.951294	0.475647	−	0.879636i	$-0.342214\pi$
$373$	354.833	0.951294	0.475647	+	0.879636i	$0.342214\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 432.190i	− 1.14639i
$378$	0	0
$379$	423.997	1.11873	0.559363	−	0.828923i	$-0.311046\pi$
$379$	423.997	1.11873	0.559363	+	0.828923i	$0.311046\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	207.158i	0.540882i	0.962737	+	0.270441i	$0.0871695\pi$
$383$	207.158i	0.540882i	−0.962737	+	0.270441i	$0.912830\pi$
$384$	0	0
$385$	−67.8297	−0.176181
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 439.957i	− 1.13100i	−0.824750	−	0.565498i	$-0.808684\pi$
$389$	− 439.957i	− 1.13100i	0.824750	−	0.565498i	$-0.191316\pi$
$390$	0	0
$391$	339.413	0.868065
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 107.705i	− 0.272671i
$396$	0	0
$397$	453.872	1.14326	0.571628	−	0.820513i	$-0.306312\pi$
$397$	453.872	1.14326	0.571628	+	0.820513i	$0.306312\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	556.407i	1.38755i	0.720192	+	0.693774i	$0.244054\pi$
$401$	556.407i	1.38755i	−0.720192	+	0.693774i	$0.755946\pi$
$402$	0	0
$403$	240.875	0.597706
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 436.964i	− 1.07362i
$408$	0	0
$409$	75.6625	0.184994	0.0924970	−	0.995713i	$-0.470515\pi$
$409$	75.6625	0.184994	0.0924970	+	0.995713i	$0.470515\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 336.571i	− 0.814942i
$414$	0	0
$415$	−140.125	−0.337650
$416$	0	0
$417$	0	0
$418$	0	0
$419$	521.060i	1.24358i	0.783184	+	0.621789i	$0.213594\pi$
$419$	521.060i	1.24358i	−0.783184	+	0.621789i	$0.786406\pi$
$420$	0	0
$421$	114.745	0.272552	0.136276	−	0.990671i	$-0.456487\pi$
$421$	114.745	0.272552	0.136276	+	0.990671i	$0.456487\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 59.7214i	− 0.140521i
$426$	0	0
$427$	−70.4559	−0.165002
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 233.204i	− 0.541076i	−0.962709	−	0.270538i	$-0.912799\pi$
$431$	− 233.204i	− 0.541076i	0.962709	−	0.270538i	$-0.0872015\pi$
$432$	0	0
$433$	−501.702	−1.15867	−0.579333	−	0.815091i	$-0.696687\pi$
$433$	−501.702	−1.15867	−0.579333	+	0.815091i	$0.696687\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	158.666i	0.363079i
$438$	0	0
$439$	−365.997	−0.833706	−0.416853	−	0.908974i	$-0.636867\pi$
$439$	−365.997	−0.833706	−0.416853	+	0.908974i	$0.636867\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 632.902i	− 1.42867i	−0.699802	−	0.714337i	$-0.746728\pi$
$443$	− 632.902i	− 1.42867i	0.699802	−	0.714337i	$-0.253272\pi$
$444$	0	0
$445$	−155.872	−0.350275
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 796.344i	− 1.77360i	−0.462158	−	0.886798i	$-0.652925\pi$
$449$	− 796.344i	− 1.77360i	0.462158	−	0.886798i	$-0.347075\pi$
$450$	0	0
$451$	−497.070	−1.10215
$452$	0	0
$453$	0	0
$454$	0	0
$455$	63.9149i	0.140472i
$456$	0	0
$457$	−202.577	−0.443277	−0.221638	−	0.975129i	$-0.571140\pi$
$457$	−202.577	−0.443277	−0.221638	+	0.975129i	$0.571140\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	727.909i	1.57898i	0.613765	+	0.789489i	$0.289654\pi$
$461$	727.909i	1.57898i	−0.613765	+	0.789489i	$0.710346\pi$
$462$	0	0
$463$	−493.003	−1.06480	−0.532401	−	0.846492i	$-0.678710\pi$
$463$	−493.003	−1.06480	−0.532401	+	0.846492i	$0.678710\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	453.190i	0.970429i	0.874395	+	0.485215i	$0.161259\pi$
$467$	453.190i	0.970429i	−0.874395	+	0.485215i	$0.838741\pi$
$468$	0	0
$469$	1.23991	0.00264372
$470$	0	0
$471$	0	0
$472$	0	0
$473$	583.862i	1.23438i
$474$	0	0
$475$	27.9180	0.0587747
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 319.544i	− 0.667107i	−0.942731	−	0.333553i	$-0.891752\pi$
$479$	− 319.544i	− 0.667107i	0.942731	−	0.333553i	$-0.108248\pi$
$480$	0	0
$481$	−411.745	−0.856018
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 356.466i	− 0.734981i
$486$	0	0
$487$	−792.869	−1.62807	−0.814034	−	0.580817i	$-0.802733\pi$
$487$	−792.869	−1.62807	−0.814034	+	0.580817i	$0.802733\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	369.437i	0.752417i	0.926535	+	0.376208i	$0.122772\pi$
$491$	369.437i	0.752417i	−0.926535	+	0.376208i	$0.877228\pi$
$492$	0	0
$493$	669.702	1.35842
$494$	0	0
$495$	0	0
$496$	0	0
$497$	301.337i	0.606313i
$498$	0	0
$499$	−609.912	−1.22227	−0.611134	−	0.791527i	$-0.709286\pi$
$499$	−609.912	−1.22227	−0.611134	+	0.791527i	$0.709286\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 305.308i	− 0.606974i	−0.952836	−	0.303487i	$-0.901849\pi$
$503$	− 305.308i	− 0.606974i	0.952836	−	0.303487i	$-0.0981509\pi$
$504$	0	0
$505$	53.6656	0.106269
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 204.478i	− 0.401726i	−0.979619	−	0.200863i	$-0.935625\pi$
$509$	− 204.478i	− 0.401726i	0.979619	−	0.200863i	$-0.0643745\pi$
$510$	0	0
$511$	−188.341	−0.368573
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 56.2755i	− 0.109273i
$516$	0	0
$517$	−377.423	−0.730024
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 590.227i	− 1.13287i	−0.824105	−	0.566436i	$-0.808322\pi$
$521$	− 590.227i	− 1.13287i	0.824105	−	0.566436i	$-0.191678\pi$
$522$	0	0
$523$	19.7933	0.0378458	0.0189229	−	0.999821i	$-0.493976\pi$
$523$	19.7933	0.0378458	0.0189229	+	0.999821i	$0.493976\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	373.249i	0.708253i
$528$	0	0
$529$	−278.492	−0.526450
$530$	0	0
$531$	0	0
$532$	0	0
$533$	468.381i	0.878763i
$534$	0	0
$535$	−106.584	−0.199222
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 288.351i	− 0.534973i
$540$	0	0
$541$	428.164	0.791431	0.395715	−	0.918373i	$-0.370497\pi$
$541$	428.164	0.791431	0.395715	+	0.918373i	$0.370497\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 368.761i	− 0.676625i
$546$	0	0
$547$	982.705	1.79654	0.898268	−	0.439448i	$-0.144826\pi$
$547$	982.705	1.79654	0.898268	+	0.439448i	$0.144826\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	313.066i	0.568177i
$552$	0	0
$553$	−178.614	−0.322990
$554$	0	0
$555$	0	0
$556$	0	0
$557$	310.839i	0.558059i	0.960282	+	0.279030i	$0.0900128\pi$
$557$	310.839i	0.558059i	−0.960282	+	0.279030i	$0.909987\pi$
$558$	0	0
$559$	550.164	0.984193
$560$	0	0
$561$	0	0
$562$	0	0
$563$	850.997i	1.51154i	0.654837	+	0.755770i	$0.272737\pi$
$563$	850.997i	1.51154i	−0.654837	+	0.755770i	$0.727263\pi$
$564$	0	0
$565$	−152.918	−0.270651
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 292.377i	− 0.513843i	−0.966432	−	0.256922i	$-0.917292\pi$
$569$	− 292.377i	− 0.513843i	0.966432	−	0.256922i	$-0.0827083\pi$
$570$	0	0
$571$	357.754	0.626539	0.313270	−	0.949664i	$-0.398576\pi$
$571$	357.754	0.626539	0.313270	+	0.949664i	$0.398576\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	142.082i	0.247099i
$576$	0	0
$577$	216.456	0.375140	0.187570	−	0.982251i	$-0.439939\pi$
$577$	216.456	0.375140	0.187570	+	0.982251i	$0.439939\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	232.377i	0.399960i
$582$	0	0
$583$	−600.115	−1.02936
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 578.489i	− 0.985501i	−0.870171	−	0.492751i	$-0.835992\pi$
$587$	− 578.489i	− 0.985501i	0.870171	−	0.492751i	$-0.164008\pi$
$588$	0	0
$589$	−174.483	−0.296236
$590$	0	0
$591$	0	0
$592$	0	0
$593$	159.971i	0.269765i	0.990862	+	0.134882i	$0.0430657\pi$
$593$	159.971i	0.269765i	−0.990862	+	0.134882i	$0.956934\pi$
$594$	0	0
$595$	−99.0395	−0.166453
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 453.755i	− 0.757520i	−0.925495	−	0.378760i	$-0.876351\pi$
$599$	− 453.755i	− 0.757520i	0.925495	−	0.378760i	$-0.123649\pi$
$600$	0	0
$601$	−653.584	−1.08749	−0.543747	−	0.839249i	$-0.682995\pi$
$601$	−653.584	−1.08749	−0.543747	+	0.839249i	$0.682995\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 120.931i	− 0.199886i
$606$	0	0
$607$	184.626	0.304162	0.152081	−	0.988368i	$-0.451403\pi$
$607$	184.626	0.304162	0.152081	+	0.988368i	$0.451403\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	355.639i	0.582061i
$612$	0	0
$613$	−903.416	−1.47376	−0.736881	−	0.676022i	$-0.763702\pi$
$613$	−903.416	−1.47376	−0.736881	+	0.676022i	$0.763702\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	386.489i	0.626401i	0.949687	+	0.313200i	$0.101401\pi$
$617$	386.489i	0.626401i	−0.949687	+	0.313200i	$0.898599\pi$
$618$	0	0
$619$	−1165.90	−1.88353	−0.941763	−	0.336278i	$-0.890832\pi$
$619$	−1165.90	−1.88353	−0.941763	+	0.336278i	$0.890832\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	258.492i	0.414915i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 638.020i	− 1.01434i
$630$	0	0
$631$	−50.8297	−0.0805542	−0.0402771	−	0.999189i	$-0.512824\pi$
$631$	−50.8297	−0.0805542	−0.0402771	+	0.999189i	$0.512824\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 448.885i	− 0.706906i
$636$	0	0
$637$	−271.708	−0.426543
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 548.754i	− 0.856090i	−0.903757	−	0.428045i	$-0.859202\pi$
$641$	− 548.754i	− 0.856090i	0.903757	−	0.428045i	$-0.140798\pi$
$642$	0	0
$643$	−526.711	−0.819147	−0.409573	−	0.912277i	$-0.634322\pi$
$643$	−526.711	−0.819147	−0.409573	+	0.912277i	$0.634322\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 694.607i	− 1.07358i	−0.843716	−	0.536790i	$-0.819637\pi$
$647$	− 694.607i	− 1.07358i	0.843716	−	0.536790i	$-0.180363\pi$
$648$	0	0
$649$	−742.480	−1.14404
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1122.29i	1.71867i	0.511412	+	0.859336i	$0.329123\pi$
$653$	1122.29i	1.71867i	−0.511412	+	0.859336i	$0.670877\pi$
$654$	0	0
$655$	18.2554	0.0278709
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 670.488i	− 1.01743i	−0.860934	−	0.508717i	$-0.830120\pi$
$659$	− 670.488i	− 1.01743i	0.860934	−	0.508717i	$-0.169880\pi$
$660$	0	0
$661$	1088.15	1.64622	0.823110	−	0.567882i	$-0.192237\pi$
$661$	1088.15	1.64622	0.823110	+	0.567882i	$0.192237\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 46.2980i	− 0.0696211i
$666$	0	0
$667$	−1593.28	−2.38872
$668$	0	0
$669$	0	0
$670$	0	0
$671$	155.426i	0.231634i
$672$	0	0
$673$	−638.371	−0.948545	−0.474272	−	0.880378i	$-0.657289\pi$
$673$	−638.371	−0.948545	−0.474272	+	0.880378i	$0.657289\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 798.158i	− 1.17896i	−0.807782	−	0.589481i	$-0.799332\pi$
$677$	− 798.158i	− 1.17896i	0.807782	−	0.589481i	$-0.200668\pi$
$678$	0	0
$679$	−591.149	−0.870616
$680$	0	0
$681$	0	0
$682$	0	0
$683$	757.505i	1.10908i	0.832156	+	0.554542i	$0.187107\pi$
$683$	757.505i	1.10908i	−0.832156	+	0.554542i	$0.812893\pi$
$684$	0	0
$685$	112.544	0.164298
$686$	0	0
$687$	0	0
$688$	0	0
$689$	565.479i	0.820724i
$690$	0	0
$691$	−702.331	−1.01640	−0.508199	−	0.861240i	$-0.669689\pi$
$691$	−702.331	−1.01640	−0.508199	+	0.861240i	$0.669689\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 282.859i	− 0.406992i
$696$	0	0
$697$	−725.781	−1.04129
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 406.616i	− 0.580052i	−0.957019	−	0.290026i	$-0.906336\pi$
$701$	− 406.616i	− 0.580052i	0.957019	−	0.290026i	$-0.0936639\pi$
$702$	0	0
$703$	298.255	0.424261
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 88.9969i	− 0.125880i
$708$	0	0
$709$	564.322	0.795941	0.397970	−	0.917398i	$-0.369715\pi$
$709$	564.322	0.795941	0.397970	+	0.917398i	$0.369715\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 887.991i	− 1.24543i
$714$	0	0
$715$	140.997	0.197198
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1340.07i	− 1.86379i	−0.362725	−	0.931896i	$-0.618154\pi$
$719$	− 1340.07i	− 1.86379i	0.362725	−	0.931896i	$-0.381846\pi$
$720$	0	0
$721$	−93.3251	−0.129438
$722$	0	0
$723$	0	0
$724$	0	0
$725$	280.344i	0.386682i
$726$	0	0
$727$	−130.991	−0.180180	−0.0900899	−	0.995934i	$-0.528715\pi$
$727$	−130.991	−0.180180	−0.0900899	+	0.995934i	$0.528715\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	852.508i	1.16622i
$732$	0	0
$733$	1414.48	1.92971	0.964857	−	0.262777i	$-0.0846383\pi$
$733$	1414.48	1.92971	0.964857	+	0.262777i	$0.0846383\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.73525i	− 0.00371133i
$738$	0	0
$739$	−10.5805	−0.0143173	−0.00715865	−	0.999974i	$-0.502279\pi$
$739$	−10.5805	−0.0143173	−0.00715865	+	0.999974i	$0.502279\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	794.053i	1.06871i	0.845260	+	0.534356i	$0.179446\pi$
$743$	794.053i	1.06871i	−0.845260	+	0.534356i	$0.820554\pi$
$744$	0	0
$745$	−35.3738	−0.0474817
$746$	0	0
$747$	0	0
$748$	0	0
$749$	176.754i	0.235986i
$750$	0	0
$751$	53.5109	0.0712528	0.0356264	−	0.999365i	$-0.488657\pi$
$751$	53.5109	0.0712528	0.0356264	+	0.999365i	$0.488657\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 310.807i	− 0.411664i
$756$	0	0
$757$	921.745	1.21763	0.608814	−	0.793313i	$-0.291646\pi$
$757$	921.745	1.21763	0.608814	+	0.793313i	$0.291646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	578.115i	0.759678i	0.925053	+	0.379839i	$0.124021\pi$
$761$	578.115i	0.759678i	−0.925053	+	0.379839i	$0.875979\pi$
$762$	0	0
$763$	−611.538	−0.801491
$764$	0	0
$765$	0	0
$766$	0	0
$767$	699.627i	0.912160i
$768$	0	0
$769$	−651.656	−0.847407	−0.423704	−	0.905801i	$-0.639270\pi$
$769$	−651.656	−0.847407	−0.423704	+	0.905801i	$0.639270\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	389.754i	0.504209i	0.967700	+	0.252105i	$0.0811227\pi$
$773$	389.754i	0.504209i	−0.967700	+	0.252105i	$0.918877\pi$
$774$	0	0
$775$	−156.246	−0.201608
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 339.281i	− 0.435534i
$780$	0	0
$781$	664.754	0.851157
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 426.430i	− 0.543222i
$786$	0	0
$787$	36.4559	0.0463226	0.0231613	−	0.999732i	$-0.492627\pi$
$787$	36.4559	0.0463226	0.0231613	+	0.999732i	$0.492627\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	253.593i	0.320598i
$792$	0	0
$793$	146.456	0.184686
$794$	0	0
$795$	0	0
$796$	0	0
$797$	731.466i	0.917774i	0.888495	+	0.458887i	$0.151752\pi$
$797$	731.466i	0.917774i	−0.888495	+	0.458887i	$0.848248\pi$
$798$	0	0
$799$	−551.082	−0.689715
$800$	0	0
$801$	0	0
$802$	0	0
$803$	415.481i	0.517412i
$804$	0	0
$805$	235.623	0.292699
$806$	0	0
$807$	0	0
$808$	0	0
$809$	595.767i	0.736424i	0.929742	+	0.368212i	$0.120030\pi$
$809$	595.767i	0.736424i	−0.929742	+	0.368212i	$0.879970\pi$
$810$	0	0
$811$	−180.177	−0.222166	−0.111083	−	0.993811i	$-0.535432\pi$
$811$	−180.177	−0.222166	−0.111083	+	0.993811i	$0.535432\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 82.3607i	− 0.101056i
$816$	0	0
$817$	−398.522	−0.487787
$818$	0	0
$819$	0	0
$820$	0	0
$821$	31.3375i	0.0381699i	0.999818	+	0.0190849i	$0.00607529\pi$
$821$	31.3375i	0.0381699i	−0.999818	+	0.0190849i	$0.993925\pi$
$822$	0	0
$823$	234.803	0.285301	0.142650	−	0.989773i	$-0.454438\pi$
$823$	234.803	0.285301	0.142650	+	0.989773i	$0.454438\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	47.8003i	0.0577996i	0.999582	+	0.0288998i	$0.00920038\pi$
$827$	47.8003i	0.0577996i	−0.999582	+	0.0288998i	$0.990800\pi$
$828$	0	0
$829$	−292.760	−0.353148	−0.176574	−	0.984287i	$-0.556502\pi$
$829$	−292.760	−0.353148	−0.176574	+	0.984287i	$0.556502\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 421.026i	− 0.505434i
$834$	0	0
$835$	578.204	0.692459
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 494.387i	− 0.589257i	−0.955612	−	0.294629i	$-0.904804\pi$
$839$	− 494.387i	− 0.589257i	0.955612	−	0.294629i	$-0.0951960\pi$
$840$	0	0
$841$	−2302.72	−2.73807
$842$	0	0
$843$	0	0
$844$	0	0
$845$	245.036i	0.289984i
$846$	0	0
$847$	−200.547	−0.236774
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1517.90i	1.78367i
$852$	0	0
$853$	−1151.65	−1.35011	−0.675057	−	0.737766i	$-0.735881\pi$
$853$	−1151.65	−1.35011	−0.675057	+	0.737766i	$0.735881\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1098.02i	1.28124i	0.767858	+	0.640620i	$0.221323\pi$
$857$	1098.02i	1.28124i	−0.767858	+	0.640620i	$0.778677\pi$
$858$	0	0
$859$	290.757	0.338483	0.169242	−	0.985575i	$-0.445868\pi$
$859$	290.757	0.338483	0.169242	+	0.985575i	$0.445868\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1069.28i	1.23903i	0.784985	+	0.619514i	$0.212670\pi$
$863$	1069.28i	1.23903i	−0.784985	+	0.619514i	$0.787330\pi$
$864$	0	0
$865$	−407.705	−0.471335
$866$	0	0
$867$	0	0
$868$	0	0
$869$	394.024i	0.453422i
$870$	0	0
$871$	−2.57738	−0.00295911
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 41.4590i	− 0.0473817i
$876$	0	0
$877$	−160.043	−0.182489	−0.0912443	−	0.995829i	$-0.529084\pi$
$877$	−160.043	−0.182489	−0.0912443	+	0.995829i	$0.529084\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 712.502i	− 0.808743i	−0.914595	−	0.404371i	$-0.867490\pi$
$881$	− 712.502i	− 0.808743i	0.914595	−	0.404371i	$-0.132510\pi$
$882$	0	0
$883$	1244.12	1.40897	0.704486	−	0.709718i	$-0.251178\pi$
$883$	1244.12	1.40897	0.704486	+	0.709718i	$0.251178\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	429.827i	0.484585i	0.970203	+	0.242292i	$0.0778993\pi$
$887$	429.827i	0.484585i	−0.970203	+	0.242292i	$0.922101\pi$
$888$	0	0
$889$	−744.413	−0.837360
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 257.614i	− 0.288482i
$894$	0	0
$895$	−574.164	−0.641524
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1752.11i	− 1.94895i
$900$	0	0
$901$	−876.240	−0.972519
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 402.676i	− 0.444946i
$906$	0	0
$907$	553.240	0.609967	0.304983	−	0.952358i	$-0.401349\pi$
$907$	553.240	0.609967	0.304983	+	0.952358i	$0.401349\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	667.275i	0.732464i	0.930524	+	0.366232i	$0.119352\pi$
$911$	667.275i	0.732464i	−0.930524	+	0.366232i	$0.880648\pi$
$912$	0	0
$913$	512.626	0.561474
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 30.2741i	− 0.0330143i
$918$	0	0
$919$	338.000	0.367791	0.183896	−	0.982946i	$-0.441129\pi$
$919$	338.000	0.367791	0.183896	+	0.982946i	$0.441129\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 626.387i	− 0.678642i
$924$	0	0
$925$	267.082	0.288737
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 216.829i	− 0.233400i	−0.993167	−	0.116700i	$-0.962768\pi$
$929$	− 216.829i	− 0.233400i	0.993167	−	0.116700i	$-0.0372317\pi$
$930$	0	0
$931$	196.817	0.211404
$932$	0	0
$933$	0	0
$934$	0	0
$935$	218.482i	0.233671i
$936$	0	0
$937$	−851.647	−0.908908	−0.454454	−	0.890770i	$-0.650166\pi$
$937$	−851.647	−0.908908	−0.454454	+	0.890770i	$0.650166\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1073.42i	− 1.14073i	−0.821392	−	0.570363i	$-0.806802\pi$
$941$	− 1073.42i	− 1.14073i	0.821392	−	0.570363i	$-0.193198\pi$
$942$	0	0
$943$	1726.69	1.83106
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1645.82i	− 1.73793i	−0.494873	−	0.868965i	$-0.664785\pi$
$947$	− 1645.82i	− 1.73793i	0.494873	−	0.868965i	$-0.335215\pi$
$948$	0	0
$949$	391.502	0.412541
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1139.03i	1.19520i	0.801793	+	0.597602i	$0.203880\pi$
$953$	1139.03i	1.19520i	−0.801793	+	0.597602i	$0.796120\pi$
$954$	0	0
$955$	−287.082	−0.300609
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 186.639i	− 0.194618i
$960$	0	0
$961$	15.5140	0.0161436
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 143.761i	− 0.148975i
$966$	0	0
$967$	357.489	0.369689	0.184844	−	0.982768i	$-0.440822\pi$
$967$	357.489	0.369689	0.184844	+	0.982768i	$0.440822\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	46.5836i	0.0479749i	0.999712	+	0.0239874i	$0.00763617\pi$
$971$	46.5836i	0.0479749i	−0.999712	+	0.0239874i	$0.992364\pi$
$972$	0	0
$973$	−469.082	−0.482099
$974$	0	0
$975$	0	0
$976$	0	0
$977$	346.616i	0.354776i	0.984141	+	0.177388i	$0.0567647\pi$
$977$	346.616i	0.354776i	−0.984141	+	0.177388i	$0.943235\pi$
$978$	0	0
$979$	570.237	0.582469
$980$	0	0
$981$	0	0
$982$	0	0
$983$	846.975i	0.861623i	0.902442	+	0.430811i	$0.141773\pi$
$983$	846.975i	0.861623i	−0.902442	+	0.430811i	$0.858227\pi$
$984$	0	0
$985$	−487.456	−0.494879
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 2028.19i	− 2.05075i
$990$	0	0
$991$	−1304.33	−1.31617	−0.658085	−	0.752943i	$-0.728633\pi$
$991$	−1304.33	−1.31617	−0.658085	+	0.752943i	$0.728633\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	716.099i	0.719698i
$996$	0	0
$997$	1835.77	1.84130	0.920649	−	0.390391i	$-0.127660\pi$
$997$	1835.77	1.84130	0.920649	+	0.390391i	$0.127660\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.l.g.161.1		4
3.2	odd	2	inner	2160.3.l.g.161.3		4
4.3	odd	2		135.3.c.c.26.3	yes	4
12.11	even	2		135.3.c.c.26.2	✓	4
20.3	even	4		675.3.d.i.674.1		4
20.7	even	4		675.3.d.e.674.4		4
20.19	odd	2		675.3.c.p.26.2		4
36.7	odd	6		405.3.i.c.296.3		8
36.11	even	6		405.3.i.c.296.2		8
36.23	even	6		405.3.i.c.26.3		8
36.31	odd	6		405.3.i.c.26.2		8
60.23	odd	4		675.3.d.e.674.3		4
60.47	odd	4		675.3.d.i.674.2		4
60.59	even	2		675.3.c.p.26.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.c.c.26.2	✓	4	12.11	even	2
135.3.c.c.26.3	yes	4	4.3	odd	2
405.3.i.c.26.2		8	36.31	odd	6
405.3.i.c.26.3		8	36.23	even	6
405.3.i.c.296.2		8	36.11	even	6
405.3.i.c.296.3		8	36.7	odd	6
675.3.c.p.26.2		4	20.19	odd	2
675.3.c.p.26.3		4	60.59	even	2
675.3.d.e.674.3		4	60.23	odd	4
675.3.d.e.674.4		4	20.7	even	4
675.3.d.i.674.1		4	20.3	even	4
675.3.d.i.674.2		4	60.47	odd	4
2160.3.l.g.161.1		4	1.1	even	1	trivial
2160.3.l.g.161.3		4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.l (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{5} -3.70820 q^{7} +8.18034i q^{11} +7.70820 q^{13} +11.9443i q^{17} -5.58359 q^{19} -28.4164i q^{23} -5.00000 q^{25} -56.0689i q^{29} +31.2492 q^{31} +8.29180i q^{35} -53.4164 q^{37} +60.7639i q^{41} +71.3738 q^{43} +46.1378i q^{47} -35.2492 q^{49} +73.3607i q^{53} +18.2918 q^{55} +90.7639i q^{59} +19.0000 q^{61} -17.2361i q^{65} -0.334369 q^{67} -81.2624i q^{71} +50.7902 q^{73} -30.3344i q^{77} +48.1672 q^{79} -62.6656i q^{83} +26.7082 q^{85} -69.7082i q^{89} -28.5836 q^{91} +12.4853i q^{95} +159.416 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{7} + 4 q^{13} - 76 q^{19} - 20 q^{25} - 36 q^{31} - 160 q^{37} + 44 q^{43} + 20 q^{49} + 100 q^{55} + 76 q^{61} - 216 q^{67} - 92 q^{73} + 300 q^{79} + 80 q^{85} - 168 q^{91} + 584 q^{97}+O(q^{100}) \) 4 * q + 12 * q^7 + 4 * q^13 - 76 * q^19 - 20 * q^25 - 36 * q^31 - 160 * q^37 + 44 * q^43 + 20 * q^49 + 100 * q^55 + 76 * q^61 - 216 * q^67 - 92 * q^73 + 300 * q^79 + 80 * q^85 - 168 * q^91 + 584 * q^97

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