LMFDB - Embedded newform 220.4.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [220,4,Mod(1,220)] 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("220.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(220, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 220 = 2^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	220.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$12.9804202013$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	220.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+8.00000 q^{3} +5.00000 q^{5} +24.0000 q^{7} +37.0000 q^{9} -11.0000 q^{11} -22.0000 q^{13} +40.0000 q^{15} +22.0000 q^{17} -28.0000 q^{19} +192.000 q^{21} -44.0000 q^{23} +25.0000 q^{25} +80.0000 q^{27} +110.000 q^{29} -40.0000 q^{31} -88.0000 q^{33} +120.000 q^{35} -362.000 q^{37} -176.000 q^{39} +210.000 q^{41} +260.000 q^{43} +185.000 q^{45} -460.000 q^{47} +233.000 q^{49} +176.000 q^{51} +662.000 q^{53} -55.0000 q^{55} -224.000 q^{57} -68.0000 q^{59} +606.000 q^{61} +888.000 q^{63} -110.000 q^{65} -312.000 q^{67} -352.000 q^{69} +360.000 q^{71} -1042.00 q^{73} +200.000 q^{75} -264.000 q^{77} -552.000 q^{79} -359.000 q^{81} +268.000 q^{83} +110.000 q^{85} +880.000 q^{87} -966.000 q^{89} -528.000 q^{91} -320.000 q^{93} -140.000 q^{95} -1334.00 q^{97} -407.000 q^{99} +O(q^{100})$

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
$3$	8.00000	1.53960	0.769800	−	0.638285i	$-0.220356\pi$
$3$	8.00000	1.53960	0.769800	+	0.638285i	$0.220356\pi$
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	24.0000	1.29588	0.647939	−	0.761692i	$-0.275631\pi$
$7$	24.0000	1.29588	0.647939	+	0.761692i	$0.275631\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	−22.0000	−0.469362	−0.234681	−	0.972072i	$-0.575405\pi$
$13$	−22.0000	−0.469362	−0.234681	+	0.972072i	$0.575405\pi$
$14$	0	0
$15$	40.0000	0.688530
$16$	0	0
$17$	22.0000	0.313870	0.156935	−	0.987609i	$-0.449839\pi$
$17$	22.0000	0.313870	0.156935	+	0.987609i	$0.449839\pi$
$18$	0	0
$19$	−28.0000	−0.338086	−0.169043	−	0.985609i	$-0.554068\pi$
$19$	−28.0000	−0.338086	−0.169043	+	0.985609i	$0.554068\pi$
$20$	0	0
$21$	192.000	1.99513
$22$	0	0
$23$	−44.0000	−0.398897	−0.199449	−	0.979908i	$-0.563915\pi$
$23$	−44.0000	−0.398897	−0.199449	+	0.979908i	$0.563915\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	80.0000	0.570222
$28$	0	0
$29$	110.000	0.704362	0.352181	−	0.935932i	$-0.385440\pi$
$29$	110.000	0.704362	0.352181	+	0.935932i	$0.385440\pi$
$30$	0	0
$31$	−40.0000	−0.231749	−0.115874	−	0.993264i	$-0.536967\pi$
$31$	−40.0000	−0.231749	−0.115874	+	0.993264i	$0.536967\pi$
$32$	0	0
$33$	−88.0000	−0.464207
$34$	0	0
$35$	120.000	0.579534
$36$	0	0
$37$	−362.000	−1.60844	−0.804222	−	0.594329i	$-0.797418\pi$
$37$	−362.000	−1.60844	−0.804222	+	0.594329i	$0.797418\pi$
$38$	0	0
$39$	−176.000	−0.722630
$40$	0	0
$41$	210.000	0.799914	0.399957	−	0.916534i	$-0.369025\pi$
$41$	210.000	0.799914	0.399957	+	0.916534i	$0.369025\pi$
$42$	0	0
$43$	260.000	0.922084	0.461042	−	0.887378i	$-0.347476\pi$
$43$	260.000	0.922084	0.461042	+	0.887378i	$0.347476\pi$
$44$	0	0
$45$	185.000	0.612848
$46$	0	0
$47$	−460.000	−1.42761	−0.713807	−	0.700342i	$-0.753031\pi$
$47$	−460.000	−1.42761	−0.713807	+	0.700342i	$0.753031\pi$
$48$	0	0
$49$	233.000	0.679300
$50$	0	0
$51$	176.000	0.483234
$52$	0	0
$53$	662.000	1.71571	0.857856	−	0.513891i	$-0.171796\pi$
$53$	662.000	1.71571	0.857856	+	0.513891i	$0.171796\pi$
$54$	0	0
$55$	−55.0000	−0.134840
$56$	0	0
$57$	−224.000	−0.520518
$58$	0	0
$59$	−68.0000	−0.150048	−0.0750241	−	0.997182i	$-0.523903\pi$
$59$	−68.0000	−0.150048	−0.0750241	+	0.997182i	$0.523903\pi$
$60$	0	0
$61$	606.000	1.27197	0.635986	−	0.771700i	$-0.280593\pi$
$61$	606.000	1.27197	0.635986	+	0.771700i	$0.280593\pi$
$62$	0	0
$63$	888.000	1.77583
$64$	0	0
$65$	−110.000	−0.209905
$66$	0	0
$67$	−312.000	−0.568908	−0.284454	−	0.958690i	$-0.591812\pi$
$67$	−312.000	−0.568908	−0.284454	+	0.958690i	$0.591812\pi$
$68$	0	0
$69$	−352.000	−0.614142
$70$	0	0
$71$	360.000	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$72$	0	0
$73$	−1042.00	−1.67064	−0.835321	−	0.549762i	$-0.814718\pi$
$73$	−1042.00	−1.67064	−0.835321	+	0.549762i	$0.814718\pi$
$74$	0	0
$75$	200.000	0.307920
$76$	0	0
$77$	−264.000	−0.390722
$78$	0	0
$79$	−552.000	−0.786137	−0.393069	−	0.919509i	$-0.628587\pi$
$79$	−552.000	−0.786137	−0.393069	+	0.919509i	$0.628587\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	268.000	0.354420	0.177210	−	0.984173i	$-0.443293\pi$
$83$	268.000	0.354420	0.177210	+	0.984173i	$0.443293\pi$
$84$	0	0
$85$	110.000	0.140367
$86$	0	0
$87$	880.000	1.08444
$88$	0	0
$89$	−966.000	−1.15051	−0.575257	−	0.817973i	$-0.695098\pi$
$89$	−966.000	−1.15051	−0.575257	+	0.817973i	$0.695098\pi$
$90$	0	0
$91$	−528.000	−0.608236
$92$	0	0
$93$	−320.000	−0.356801
$94$	0	0
$95$	−140.000	−0.151197
$96$	0	0
$97$	−1334.00	−1.39636	−0.698181	−	0.715921i	$-0.746007\pi$
$97$	−1334.00	−1.39636	−0.698181	+	0.715921i	$0.746007\pi$
$98$	0	0
$99$	−407.000	−0.413182
$100$	0	0
$101$	−1362.00	−1.34182	−0.670911	−	0.741538i	$-0.734097\pi$
$101$	−1362.00	−1.34182	−0.670911	+	0.741538i	$0.734097\pi$
$102$	0	0
$103$	1564.00	1.49617	0.748085	−	0.663603i	$-0.230974\pi$
$103$	1564.00	1.49617	0.748085	+	0.663603i	$0.230974\pi$
$104$	0	0
$105$	960.000	0.892251
$106$	0	0
$107$	−1108.00	−1.00107	−0.500535	−	0.865717i	$-0.666863\pi$
$107$	−1108.00	−1.00107	−0.500535	+	0.865717i	$0.666863\pi$
$108$	0	0
$109$	−1482.00	−1.30229	−0.651146	−	0.758952i	$-0.725712\pi$
$109$	−1482.00	−1.30229	−0.651146	+	0.758952i	$0.725712\pi$
$110$	0	0
$111$	−2896.00	−2.47636
$112$	0	0
$113$	−1030.00	−0.857471	−0.428736	−	0.903430i	$-0.641041\pi$
$113$	−1030.00	−0.857471	−0.428736	+	0.903430i	$0.641041\pi$
$114$	0	0
$115$	−220.000	−0.178392
$116$	0	0
$117$	−814.000	−0.643199
$118$	0	0
$119$	528.000	0.406737
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	1680.00	1.23155
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1928.00	1.34711	0.673553	−	0.739139i	$-0.264768\pi$
$127$	1928.00	1.34711	0.673553	+	0.739139i	$0.264768\pi$
$128$	0	0
$129$	2080.00	1.41964
$130$	0	0
$131$	−996.000	−0.664282	−0.332141	−	0.943230i	$-0.607771\pi$
$131$	−996.000	−0.664282	−0.332141	+	0.943230i	$0.607771\pi$
$132$	0	0
$133$	−672.000	−0.438119
$134$	0	0
$135$	400.000	0.255011
$136$	0	0
$137$	578.000	0.360452	0.180226	−	0.983625i	$-0.442317\pi$
$137$	578.000	0.360452	0.180226	+	0.983625i	$0.442317\pi$
$138$	0	0
$139$	−1484.00	−0.905548	−0.452774	−	0.891625i	$-0.649566\pi$
$139$	−1484.00	−0.905548	−0.452774	+	0.891625i	$0.649566\pi$
$140$	0	0
$141$	−3680.00	−2.19796
$142$	0	0
$143$	242.000	0.141518
$144$	0	0
$145$	550.000	0.315000
$146$	0	0
$147$	1864.00	1.04585
$148$	0	0
$149$	3214.00	1.76712	0.883561	−	0.468316i	$-0.155139\pi$
$149$	3214.00	1.76712	0.883561	+	0.468316i	$0.155139\pi$
$150$	0	0
$151$	1960.00	1.05631	0.528154	−	0.849149i	$-0.322884\pi$
$151$	1960.00	1.05631	0.528154	+	0.849149i	$0.322884\pi$
$152$	0	0
$153$	814.000	0.430118
$154$	0	0
$155$	−200.000	−0.103641
$156$	0	0
$157$	−2810.00	−1.42842	−0.714212	−	0.699930i	$-0.753215\pi$
$157$	−2810.00	−1.42842	−0.714212	+	0.699930i	$0.753215\pi$
$158$	0	0
$159$	5296.00	2.64151
$160$	0	0
$161$	−1056.00	−0.516922
$162$	0	0
$163$	1688.00	0.811131	0.405566	−	0.914066i	$-0.367075\pi$
$163$	1688.00	0.811131	0.405566	+	0.914066i	$0.367075\pi$
$164$	0	0
$165$	−440.000	−0.207600
$166$	0	0
$167$	3616.00	1.67554	0.837768	−	0.546026i	$-0.183860\pi$
$167$	3616.00	1.67554	0.837768	+	0.546026i	$0.183860\pi$
$168$	0	0
$169$	−1713.00	−0.779700
$170$	0	0
$171$	−1036.00	−0.463304
$172$	0	0
$173$	3714.00	1.63220	0.816099	−	0.577912i	$-0.196132\pi$
$173$	3714.00	1.63220	0.816099	+	0.577912i	$0.196132\pi$
$174$	0	0
$175$	600.000	0.259176
$176$	0	0
$177$	−544.000	−0.231014
$178$	0	0
$179$	444.000	0.185397	0.0926987	−	0.995694i	$-0.470451\pi$
$179$	444.000	0.185397	0.0926987	+	0.995694i	$0.470451\pi$
$180$	0	0
$181$	−4658.00	−1.91285	−0.956427	−	0.291973i	$-0.905688\pi$
$181$	−4658.00	−1.91285	−0.956427	+	0.291973i	$0.905688\pi$
$182$	0	0
$183$	4848.00	1.95833
$184$	0	0
$185$	−1810.00	−0.719318
$186$	0	0
$187$	−242.000	−0.0946353
$188$	0	0
$189$	1920.00	0.738939
$190$	0	0
$191$	−216.000	−0.0818283	−0.0409142	−	0.999163i	$-0.513027\pi$
$191$	−216.000	−0.0818283	−0.0409142	+	0.999163i	$0.513027\pi$
$192$	0	0
$193$	470.000	0.175292	0.0876460	−	0.996152i	$-0.472066\pi$
$193$	470.000	0.175292	0.0876460	+	0.996152i	$0.472066\pi$
$194$	0	0
$195$	−880.000	−0.323170
$196$	0	0
$197$	2362.00	0.854241	0.427121	−	0.904195i	$-0.359528\pi$
$197$	2362.00	0.854241	0.427121	+	0.904195i	$0.359528\pi$
$198$	0	0
$199$	−1728.00	−0.615551	−0.307776	−	0.951459i	$-0.599585\pi$
$199$	−1728.00	−0.615551	−0.307776	+	0.951459i	$0.599585\pi$
$200$	0	0
$201$	−2496.00	−0.875892
$202$	0	0
$203$	2640.00	0.912767
$204$	0	0
$205$	1050.00	0.357733
$206$	0	0
$207$	−1628.00	−0.546637
$208$	0	0
$209$	308.000	0.101937
$210$	0	0
$211$	1740.00	0.567709	0.283854	−	0.958867i	$-0.408387\pi$
$211$	1740.00	0.567709	0.283854	+	0.958867i	$0.408387\pi$
$212$	0	0
$213$	2880.00	0.926452
$214$	0	0
$215$	1300.00	0.412369
$216$	0	0
$217$	−960.000	−0.300318
$218$	0	0
$219$	−8336.00	−2.57212
$220$	0	0
$221$	−484.000	−0.147318
$222$	0	0
$223$	−964.000	−0.289481	−0.144740	−	0.989470i	$-0.546235\pi$
$223$	−964.000	−0.289481	−0.144740	+	0.989470i	$0.546235\pi$
$224$	0	0
$225$	925.000	0.274074
$226$	0	0
$227$	1932.00	0.564896	0.282448	−	0.959283i	$-0.408854\pi$
$227$	1932.00	0.564896	0.282448	+	0.959283i	$0.408854\pi$
$228$	0	0
$229$	4718.00	1.36146	0.680730	−	0.732535i	$-0.261663\pi$
$229$	4718.00	1.36146	0.680730	+	0.732535i	$0.261663\pi$
$230$	0	0
$231$	−2112.00	−0.601556
$232$	0	0
$233$	198.000	0.0556713	0.0278356	−	0.999613i	$-0.491138\pi$
$233$	198.000	0.0556713	0.0278356	+	0.999613i	$0.491138\pi$
$234$	0	0
$235$	−2300.00	−0.638449
$236$	0	0
$237$	−4416.00	−1.21034
$238$	0	0
$239$	−4136.00	−1.11940	−0.559698	−	0.828697i	$-0.689083\pi$
$239$	−4136.00	−1.11940	−0.559698	+	0.828697i	$0.689083\pi$
$240$	0	0
$241$	−3222.00	−0.861192	−0.430596	−	0.902545i	$-0.641697\pi$
$241$	−3222.00	−0.861192	−0.430596	+	0.902545i	$0.641697\pi$
$242$	0	0
$243$	−5032.00	−1.32841
$244$	0	0
$245$	1165.00	0.303792
$246$	0	0
$247$	616.000	0.158685
$248$	0	0
$249$	2144.00	0.545665
$250$	0	0
$251$	6916.00	1.73918	0.869590	−	0.493775i	$-0.164383\pi$
$251$	6916.00	1.73918	0.869590	+	0.493775i	$0.164383\pi$
$252$	0	0
$253$	484.000	0.120272
$254$	0	0
$255$	880.000	0.216109
$256$	0	0
$257$	5714.00	1.38689	0.693443	−	0.720512i	$-0.256093\pi$
$257$	5714.00	1.38689	0.693443	+	0.720512i	$0.256093\pi$
$258$	0	0
$259$	−8688.00	−2.08435
$260$	0	0
$261$	4070.00	0.965236
$262$	0	0
$263$	4464.00	1.04662	0.523312	−	0.852141i	$-0.324696\pi$
$263$	4464.00	1.04662	0.523312	+	0.852141i	$0.324696\pi$
$264$	0	0
$265$	3310.00	0.767289
$266$	0	0
$267$	−7728.00	−1.77133
$268$	0	0
$269$	4886.00	1.10745	0.553726	−	0.832699i	$-0.313205\pi$
$269$	4886.00	1.10745	0.553726	+	0.832699i	$0.313205\pi$
$270$	0	0
$271$	5040.00	1.12974	0.564868	−	0.825182i	$-0.308927\pi$
$271$	5040.00	1.12974	0.564868	+	0.825182i	$0.308927\pi$
$272$	0	0
$273$	−4224.00	−0.936440
$274$	0	0
$275$	−275.000	−0.0603023
$276$	0	0
$277$	3970.00	0.861134	0.430567	−	0.902559i	$-0.358314\pi$
$277$	3970.00	0.861134	0.430567	+	0.902559i	$0.358314\pi$
$278$	0	0
$279$	−1480.00	−0.317582
$280$	0	0
$281$	2882.00	0.611835	0.305918	−	0.952058i	$-0.401037\pi$
$281$	2882.00	0.611835	0.305918	+	0.952058i	$0.401037\pi$
$282$	0	0
$283$	2308.00	0.484793	0.242396	−	0.970177i	$-0.422067\pi$
$283$	2308.00	0.484793	0.242396	+	0.970177i	$0.422067\pi$
$284$	0	0
$285$	−1120.00	−0.232783
$286$	0	0
$287$	5040.00	1.03659
$288$	0	0
$289$	−4429.00	−0.901486
$290$	0	0
$291$	−10672.0	−2.14984
$292$	0	0
$293$	8610.00	1.71673	0.858364	−	0.513040i	$-0.171481\pi$
$293$	8610.00	1.71673	0.858364	+	0.513040i	$0.171481\pi$
$294$	0	0
$295$	−340.000	−0.0671036
$296$	0	0
$297$	−880.000	−0.171929
$298$	0	0
$299$	968.000	0.187227
$300$	0	0
$301$	6240.00	1.19491
$302$	0	0
$303$	−10896.0	−2.06587
$304$	0	0
$305$	3030.00	0.568844
$306$	0	0
$307$	−5532.00	−1.02843	−0.514215	−	0.857661i	$-0.671917\pi$
$307$	−5532.00	−1.02843	−0.514215	+	0.857661i	$0.671917\pi$
$308$	0	0
$309$	12512.0	2.30350
$310$	0	0
$311$	−1688.00	−0.307774	−0.153887	−	0.988088i	$-0.549179\pi$
$311$	−1688.00	−0.307774	−0.153887	+	0.988088i	$0.549179\pi$
$312$	0	0
$313$	−678.000	−0.122437	−0.0612186	−	0.998124i	$-0.519499\pi$
$313$	−678.000	−0.122437	−0.0612186	+	0.998124i	$0.519499\pi$
$314$	0	0
$315$	4440.00	0.794177
$316$	0	0
$317$	−10794.0	−1.91247	−0.956233	−	0.292608i	$-0.905477\pi$
$317$	−10794.0	−1.91247	−0.956233	+	0.292608i	$0.905477\pi$
$318$	0	0
$319$	−1210.00	−0.212373
$320$	0	0
$321$	−8864.00	−1.54125
$322$	0	0
$323$	−616.000	−0.106115
$324$	0	0
$325$	−550.000	−0.0938723
$326$	0	0
$327$	−11856.0	−2.00501
$328$	0	0
$329$	−11040.0	−1.85001
$330$	0	0
$331$	9228.00	1.53238	0.766188	−	0.642616i	$-0.222151\pi$
$331$	9228.00	1.53238	0.766188	+	0.642616i	$0.222151\pi$
$332$	0	0
$333$	−13394.0	−2.20416
$334$	0	0
$335$	−1560.00	−0.254424
$336$	0	0
$337$	−7946.00	−1.28441	−0.642205	−	0.766533i	$-0.721980\pi$
$337$	−7946.00	−1.28441	−0.642205	+	0.766533i	$0.721980\pi$
$338$	0	0
$339$	−8240.00	−1.32016
$340$	0	0
$341$	440.000	0.0698749
$342$	0	0
$343$	−2640.00	−0.415588
$344$	0	0
$345$	−1760.00	−0.274653
$346$	0	0
$347$	−4676.00	−0.723403	−0.361701	−	0.932294i	$-0.617804\pi$
$347$	−4676.00	−0.723403	−0.361701	+	0.932294i	$0.617804\pi$
$348$	0	0
$349$	606.000	0.0929468	0.0464734	−	0.998920i	$-0.485202\pi$
$349$	606.000	0.0929468	0.0464734	+	0.998920i	$0.485202\pi$
$350$	0	0
$351$	−1760.00	−0.267641
$352$	0	0
$353$	−238.000	−0.0358852	−0.0179426	−	0.999839i	$-0.505712\pi$
$353$	−238.000	−0.0358852	−0.0179426	+	0.999839i	$0.505712\pi$
$354$	0	0
$355$	1800.00	0.269110
$356$	0	0
$357$	4224.00	0.626212
$358$	0	0
$359$	9856.00	1.44897	0.724484	−	0.689291i	$-0.242078\pi$
$359$	9856.00	1.44897	0.724484	+	0.689291i	$0.242078\pi$
$360$	0	0
$361$	−6075.00	−0.885698
$362$	0	0
$363$	968.000	0.139964
$364$	0	0
$365$	−5210.00	−0.747134
$366$	0	0
$367$	772.000	0.109804	0.0549020	−	0.998492i	$-0.482515\pi$
$367$	772.000	0.109804	0.0549020	+	0.998492i	$0.482515\pi$
$368$	0	0
$369$	7770.00	1.09618
$370$	0	0
$371$	15888.0	2.22335
$372$	0	0
$373$	−7990.00	−1.10913	−0.554566	−	0.832139i	$-0.687116\pi$
$373$	−7990.00	−1.10913	−0.554566	+	0.832139i	$0.687116\pi$
$374$	0	0
$375$	1000.00	0.137706
$376$	0	0
$377$	−2420.00	−0.330600
$378$	0	0
$379$	−3444.00	−0.466771	−0.233386	−	0.972384i	$-0.574980\pi$
$379$	−3444.00	−0.466771	−0.233386	+	0.972384i	$0.574980\pi$
$380$	0	0
$381$	15424.0	2.07400
$382$	0	0
$383$	1764.00	0.235343	0.117671	−	0.993053i	$-0.462457\pi$
$383$	1764.00	0.235343	0.117671	+	0.993053i	$0.462457\pi$
$384$	0	0
$385$	−1320.00	−0.174736
$386$	0	0
$387$	9620.00	1.26360
$388$	0	0
$389$	7262.00	0.946524	0.473262	−	0.880922i	$-0.343076\pi$
$389$	7262.00	0.946524	0.473262	+	0.880922i	$0.343076\pi$
$390$	0	0
$391$	−968.000	−0.125202
$392$	0	0
$393$	−7968.00	−1.02273
$394$	0	0
$395$	−2760.00	−0.351571
$396$	0	0
$397$	1702.00	0.215166	0.107583	−	0.994196i	$-0.465689\pi$
$397$	1702.00	0.215166	0.107583	+	0.994196i	$0.465689\pi$
$398$	0	0
$399$	−5376.00	−0.674528
$400$	0	0
$401$	1698.00	0.211457	0.105728	−	0.994395i	$-0.466283\pi$
$401$	1698.00	0.211457	0.105728	+	0.994395i	$0.466283\pi$
$402$	0	0
$403$	880.000	0.108774
$404$	0	0
$405$	−1795.00	−0.220233
$406$	0	0
$407$	3982.00	0.484964
$408$	0	0
$409$	5258.00	0.635676	0.317838	−	0.948145i	$-0.397043\pi$
$409$	5258.00	0.635676	0.317838	+	0.948145i	$0.397043\pi$
$410$	0	0
$411$	4624.00	0.554952
$412$	0	0
$413$	−1632.00	−0.194444
$414$	0	0
$415$	1340.00	0.158501
$416$	0	0
$417$	−11872.0	−1.39418
$418$	0	0
$419$	−4772.00	−0.556390	−0.278195	−	0.960525i	$-0.589736\pi$
$419$	−4772.00	−0.556390	−0.278195	+	0.960525i	$0.589736\pi$
$420$	0	0
$421$	12830.0	1.48526	0.742632	−	0.669700i	$-0.233577\pi$
$421$	12830.0	1.48526	0.742632	+	0.669700i	$0.233577\pi$
$422$	0	0
$423$	−17020.0	−1.95636
$424$	0	0
$425$	550.000	0.0627739
$426$	0	0
$427$	14544.0	1.64832
$428$	0	0
$429$	1936.00	0.217881
$430$	0	0
$431$	−10872.0	−1.21505	−0.607524	−	0.794301i	$-0.707837\pi$
$431$	−10872.0	−1.21505	−0.607524	+	0.794301i	$0.707837\pi$
$432$	0	0
$433$	14282.0	1.58510	0.792551	−	0.609806i	$-0.208753\pi$
$433$	14282.0	1.58510	0.792551	+	0.609806i	$0.208753\pi$
$434$	0	0
$435$	4400.00	0.484974
$436$	0	0
$437$	1232.00	0.134862
$438$	0	0
$439$	−3928.00	−0.427046	−0.213523	−	0.976938i	$-0.568494\pi$
$439$	−3928.00	−0.427046	−0.213523	+	0.976938i	$0.568494\pi$
$440$	0	0
$441$	8621.00	0.930893
$442$	0	0
$443$	8192.00	0.878586	0.439293	−	0.898344i	$-0.355229\pi$
$443$	8192.00	0.878586	0.439293	+	0.898344i	$0.355229\pi$
$444$	0	0
$445$	−4830.00	−0.514526
$446$	0	0
$447$	25712.0	2.72066
$448$	0	0
$449$	−3518.00	−0.369765	−0.184883	−	0.982761i	$-0.559191\pi$
$449$	−3518.00	−0.369765	−0.184883	+	0.982761i	$0.559191\pi$
$450$	0	0
$451$	−2310.00	−0.241183
$452$	0	0
$453$	15680.0	1.62629
$454$	0	0
$455$	−2640.00	−0.272011
$456$	0	0
$457$	−4346.00	−0.444852	−0.222426	−	0.974950i	$-0.571398\pi$
$457$	−4346.00	−0.444852	−0.222426	+	0.974950i	$0.571398\pi$
$458$	0	0
$459$	1760.00	0.178976
$460$	0	0
$461$	1062.00	0.107293	0.0536467	−	0.998560i	$-0.482916\pi$
$461$	1062.00	0.107293	0.0536467	+	0.998560i	$0.482916\pi$
$462$	0	0
$463$	5084.00	0.510310	0.255155	−	0.966900i	$-0.417874\pi$
$463$	5084.00	0.510310	0.255155	+	0.966900i	$0.417874\pi$
$464$	0	0
$465$	−1600.00	−0.159566
$466$	0	0
$467$	18456.0	1.82878	0.914392	−	0.404831i	$-0.132670\pi$
$467$	18456.0	1.82878	0.914392	+	0.404831i	$0.132670\pi$
$468$	0	0
$469$	−7488.00	−0.737236
$470$	0	0
$471$	−22480.0	−2.19920
$472$	0	0
$473$	−2860.00	−0.278019
$474$	0	0
$475$	−700.000	−0.0676173
$476$	0	0
$477$	24494.0	2.35116
$478$	0	0
$479$	−6648.00	−0.634144	−0.317072	−	0.948402i	$-0.602700\pi$
$479$	−6648.00	−0.634144	−0.317072	+	0.948402i	$0.602700\pi$
$480$	0	0
$481$	7964.00	0.754942
$482$	0	0
$483$	−8448.00	−0.795854
$484$	0	0
$485$	−6670.00	−0.624472
$486$	0	0
$487$	−1492.00	−0.138827	−0.0694137	−	0.997588i	$-0.522113\pi$
$487$	−1492.00	−0.138827	−0.0694137	+	0.997588i	$0.522113\pi$
$488$	0	0
$489$	13504.0	1.24882
$490$	0	0
$491$	20364.0	1.87172	0.935860	−	0.352372i	$-0.114625\pi$
$491$	20364.0	1.87172	0.935860	+	0.352372i	$0.114625\pi$
$492$	0	0
$493$	2420.00	0.221078
$494$	0	0
$495$	−2035.00	−0.184781
$496$	0	0
$497$	8640.00	0.779793
$498$	0	0
$499$	1444.00	0.129544	0.0647719	−	0.997900i	$-0.479368\pi$
$499$	1444.00	0.129544	0.0647719	+	0.997900i	$0.479368\pi$
$500$	0	0
$501$	28928.0	2.57966
$502$	0	0
$503$	600.000	0.0531862	0.0265931	−	0.999646i	$-0.491534\pi$
$503$	600.000	0.0531862	0.0265931	+	0.999646i	$0.491534\pi$
$504$	0	0
$505$	−6810.00	−0.600081
$506$	0	0
$507$	−13704.0	−1.20043
$508$	0	0
$509$	−3274.00	−0.285103	−0.142552	−	0.989787i	$-0.545531\pi$
$509$	−3274.00	−0.285103	−0.142552	+	0.989787i	$0.545531\pi$
$510$	0	0
$511$	−25008.0	−2.16495
$512$	0	0
$513$	−2240.00	−0.192784
$514$	0	0
$515$	7820.00	0.669108
$516$	0	0
$517$	5060.00	0.430442
$518$	0	0
$519$	29712.0	2.51293
$520$	0	0
$521$	−11702.0	−0.984019	−0.492010	−	0.870590i	$-0.663737\pi$
$521$	−11702.0	−0.984019	−0.492010	+	0.870590i	$0.663737\pi$
$522$	0	0
$523$	−17316.0	−1.44775	−0.723877	−	0.689929i	$-0.757642\pi$
$523$	−17316.0	−1.44775	−0.723877	+	0.689929i	$0.757642\pi$
$524$	0	0
$525$	4800.00	0.399027
$526$	0	0
$527$	−880.000	−0.0727389
$528$	0	0
$529$	−10231.0	−0.840881
$530$	0	0
$531$	−2516.00	−0.205622
$532$	0	0
$533$	−4620.00	−0.375449
$534$	0	0
$535$	−5540.00	−0.447692
$536$	0	0
$537$	3552.00	0.285438
$538$	0	0
$539$	−2563.00	−0.204817
$540$	0	0
$541$	−19066.0	−1.51518	−0.757589	−	0.652732i	$-0.773623\pi$
$541$	−19066.0	−1.51518	−0.757589	+	0.652732i	$0.773623\pi$
$542$	0	0
$543$	−37264.0	−2.94503
$544$	0	0
$545$	−7410.00	−0.582403
$546$	0	0
$547$	20396.0	1.59428	0.797139	−	0.603796i	$-0.206346\pi$
$547$	20396.0	1.59428	0.797139	+	0.603796i	$0.206346\pi$
$548$	0	0
$549$	22422.0	1.74307
$550$	0	0
$551$	−3080.00	−0.238135
$552$	0	0
$553$	−13248.0	−1.01874
$554$	0	0
$555$	−14480.0	−1.10746
$556$	0	0
$557$	−21166.0	−1.61011	−0.805056	−	0.593199i	$-0.797865\pi$
$557$	−21166.0	−1.61011	−0.805056	+	0.593199i	$0.797865\pi$
$558$	0	0
$559$	−5720.00	−0.432791
$560$	0	0
$561$	−1936.00	−0.145701
$562$	0	0
$563$	17572.0	1.31540	0.657701	−	0.753279i	$-0.271529\pi$
$563$	17572.0	1.31540	0.657701	+	0.753279i	$0.271529\pi$
$564$	0	0
$565$	−5150.00	−0.383473
$566$	0	0
$567$	−8616.00	−0.638162
$568$	0	0
$569$	−14230.0	−1.04842	−0.524211	−	0.851588i	$-0.675640\pi$
$569$	−14230.0	−1.04842	−0.524211	+	0.851588i	$0.675640\pi$
$570$	0	0
$571$	−15316.0	−1.12251	−0.561256	−	0.827642i	$-0.689682\pi$
$571$	−15316.0	−1.12251	−0.561256	+	0.827642i	$0.689682\pi$
$572$	0	0
$573$	−1728.00	−0.125983
$574$	0	0
$575$	−1100.00	−0.0797794
$576$	0	0
$577$	22794.0	1.64459	0.822293	−	0.569064i	$-0.192694\pi$
$577$	22794.0	1.64459	0.822293	+	0.569064i	$0.192694\pi$
$578$	0	0
$579$	3760.00	0.269880
$580$	0	0
$581$	6432.00	0.459285
$582$	0	0
$583$	−7282.00	−0.517306
$584$	0	0
$585$	−4070.00	−0.287648
$586$	0	0
$587$	−12144.0	−0.853895	−0.426948	−	0.904276i	$-0.640411\pi$
$587$	−12144.0	−0.853895	−0.426948	+	0.904276i	$0.640411\pi$
$588$	0	0
$589$	1120.00	0.0783511
$590$	0	0
$591$	18896.0	1.31519
$592$	0	0
$593$	−698.000	−0.0483363	−0.0241681	−	0.999708i	$-0.507694\pi$
$593$	−698.000	−0.0483363	−0.0241681	+	0.999708i	$0.507694\pi$
$594$	0	0
$595$	2640.00	0.181898
$596$	0	0
$597$	−13824.0	−0.947703
$598$	0	0
$599$	6600.00	0.450198	0.225099	−	0.974336i	$-0.427729\pi$
$599$	6600.00	0.450198	0.225099	+	0.974336i	$0.427729\pi$
$600$	0	0
$601$	−2254.00	−0.152983	−0.0764913	−	0.997070i	$-0.524372\pi$
$601$	−2254.00	−0.152983	−0.0764913	+	0.997070i	$0.524372\pi$
$602$	0	0
$603$	−11544.0	−0.779615
$604$	0	0
$605$	605.000	0.0406558
$606$	0	0
$607$	17848.0	1.19346	0.596728	−	0.802443i	$-0.296467\pi$
$607$	17848.0	1.19346	0.596728	+	0.802443i	$0.296467\pi$
$608$	0	0
$609$	21120.0	1.40530
$610$	0	0
$611$	10120.0	0.670068
$612$	0	0
$613$	2266.00	0.149303	0.0746516	−	0.997210i	$-0.476216\pi$
$613$	2266.00	0.149303	0.0746516	+	0.997210i	$0.476216\pi$
$614$	0	0
$615$	8400.00	0.550765
$616$	0	0
$617$	−19374.0	−1.26413	−0.632065	−	0.774916i	$-0.717792\pi$
$617$	−19374.0	−1.26413	−0.632065	+	0.774916i	$0.717792\pi$
$618$	0	0
$619$	−3692.00	−0.239732	−0.119866	−	0.992790i	$-0.538246\pi$
$619$	−3692.00	−0.239732	−0.119866	+	0.992790i	$0.538246\pi$
$620$	0	0
$621$	−3520.00	−0.227460
$622$	0	0
$623$	−23184.0	−1.49093
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	2464.00	0.156942
$628$	0	0
$629$	−7964.00	−0.504842
$630$	0	0
$631$	−2808.00	−0.177155	−0.0885774	−	0.996069i	$-0.528232\pi$
$631$	−2808.00	−0.177155	−0.0885774	+	0.996069i	$0.528232\pi$
$632$	0	0
$633$	13920.0	0.874045
$634$	0	0
$635$	9640.00	0.602444
$636$	0	0
$637$	−5126.00	−0.318838
$638$	0	0
$639$	13320.0	0.824618
$640$	0	0
$641$	7074.00	0.435891	0.217946	−	0.975961i	$-0.430064\pi$
$641$	7074.00	0.435891	0.217946	+	0.975961i	$0.430064\pi$
$642$	0	0
$643$	16696.0	1.02399	0.511995	−	0.858988i	$-0.328907\pi$
$643$	16696.0	1.02399	0.511995	+	0.858988i	$0.328907\pi$
$644$	0	0
$645$	10400.0	0.634883
$646$	0	0
$647$	4756.00	0.288992	0.144496	−	0.989505i	$-0.453844\pi$
$647$	4756.00	0.288992	0.144496	+	0.989505i	$0.453844\pi$
$648$	0	0
$649$	748.000	0.0452412
$650$	0	0
$651$	−7680.00	−0.462370
$652$	0	0
$653$	7614.00	0.456292	0.228146	−	0.973627i	$-0.426734\pi$
$653$	7614.00	0.456292	0.228146	+	0.973627i	$0.426734\pi$
$654$	0	0
$655$	−4980.00	−0.297076
$656$	0	0
$657$	−38554.0	−2.28940
$658$	0	0
$659$	27772.0	1.64164	0.820822	−	0.571184i	$-0.193516\pi$
$659$	27772.0	1.64164	0.820822	+	0.571184i	$0.193516\pi$
$660$	0	0
$661$	−17938.0	−1.05553	−0.527767	−	0.849389i	$-0.676970\pi$
$661$	−17938.0	−1.05553	−0.527767	+	0.849389i	$0.676970\pi$
$662$	0	0
$663$	−3872.00	−0.226811
$664$	0	0
$665$	−3360.00	−0.195933
$666$	0	0
$667$	−4840.00	−0.280968
$668$	0	0
$669$	−7712.00	−0.445685
$670$	0	0
$671$	−6666.00	−0.383514
$672$	0	0
$673$	−42.0000	−0.00240562	−0.00120281	−	0.999999i	$-0.500383\pi$
$673$	−42.0000	−0.00240562	−0.00120281	+	0.999999i	$0.500383\pi$
$674$	0	0
$675$	2000.00	0.114044
$676$	0	0
$677$	6226.00	0.353448	0.176724	−	0.984260i	$-0.443450\pi$
$677$	6226.00	0.353448	0.176724	+	0.984260i	$0.443450\pi$
$678$	0	0
$679$	−32016.0	−1.80952
$680$	0	0
$681$	15456.0	0.869714
$682$	0	0
$683$	−23176.0	−1.29840	−0.649198	−	0.760619i	$-0.724896\pi$
$683$	−23176.0	−1.29840	−0.649198	+	0.760619i	$0.724896\pi$
$684$	0	0
$685$	2890.00	0.161199
$686$	0	0
$687$	37744.0	2.09610
$688$	0	0
$689$	−14564.0	−0.805289
$690$	0	0
$691$	13004.0	0.715912	0.357956	−	0.933738i	$-0.383474\pi$
$691$	13004.0	0.715912	0.357956	+	0.933738i	$0.383474\pi$
$692$	0	0
$693$	−9768.00	−0.535434
$694$	0	0
$695$	−7420.00	−0.404974
$696$	0	0
$697$	4620.00	0.251069
$698$	0	0
$699$	1584.00	0.0857116
$700$	0	0
$701$	21758.0	1.17231	0.586154	−	0.810199i	$-0.300641\pi$
$701$	21758.0	1.17231	0.586154	+	0.810199i	$0.300641\pi$
$702$	0	0
$703$	10136.0	0.543793
$704$	0	0
$705$	−18400.0	−0.982956
$706$	0	0
$707$	−32688.0	−1.73884
$708$	0	0
$709$	−18578.0	−0.984078	−0.492039	−	0.870573i	$-0.663748\pi$
$709$	−18578.0	−0.984078	−0.492039	+	0.870573i	$0.663748\pi$
$710$	0	0
$711$	−20424.0	−1.07730
$712$	0	0
$713$	1760.00	0.0924439
$714$	0	0
$715$	1210.00	0.0632887
$716$	0	0
$717$	−33088.0	−1.72342
$718$	0	0
$719$	5760.00	0.298765	0.149382	−	0.988780i	$-0.452271\pi$
$719$	5760.00	0.298765	0.149382	+	0.988780i	$0.452271\pi$
$720$	0	0
$721$	37536.0	1.93885
$722$	0	0
$723$	−25776.0	−1.32589
$724$	0	0
$725$	2750.00	0.140872
$726$	0	0
$727$	15764.0	0.804201	0.402101	−	0.915595i	$-0.368280\pi$
$727$	15764.0	0.804201	0.402101	+	0.915595i	$0.368280\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	5720.00	0.289414
$732$	0	0
$733$	18898.0	0.952270	0.476135	−	0.879372i	$-0.342037\pi$
$733$	18898.0	0.952270	0.476135	+	0.879372i	$0.342037\pi$
$734$	0	0
$735$	9320.00	0.467719
$736$	0	0
$737$	3432.00	0.171532
$738$	0	0
$739$	−26916.0	−1.33981	−0.669906	−	0.742446i	$-0.733666\pi$
$739$	−26916.0	−1.33981	−0.669906	+	0.742446i	$0.733666\pi$
$740$	0	0
$741$	4928.00	0.244311
$742$	0	0
$743$	2176.00	0.107442	0.0537212	−	0.998556i	$-0.482892\pi$
$743$	2176.00	0.107442	0.0537212	+	0.998556i	$0.482892\pi$
$744$	0	0
$745$	16070.0	0.790281
$746$	0	0
$747$	9916.00	0.485686
$748$	0	0
$749$	−26592.0	−1.29726
$750$	0	0
$751$	12800.0	0.621942	0.310971	−	0.950419i	$-0.399346\pi$
$751$	12800.0	0.621942	0.310971	+	0.950419i	$0.399346\pi$
$752$	0	0
$753$	55328.0	2.67764
$754$	0	0
$755$	9800.00	0.472395
$756$	0	0
$757$	36926.0	1.77292	0.886459	−	0.462808i	$-0.153158\pi$
$757$	36926.0	1.77292	0.886459	+	0.462808i	$0.153158\pi$
$758$	0	0
$759$	3872.00	0.185171
$760$	0	0
$761$	−34374.0	−1.63739	−0.818697	−	0.574226i	$-0.805303\pi$
$761$	−34374.0	−1.63739	−0.818697	+	0.574226i	$0.805303\pi$
$762$	0	0
$763$	−35568.0	−1.68761
$764$	0	0
$765$	4070.00	0.192354
$766$	0	0
$767$	1496.00	0.0704269
$768$	0	0
$769$	−7094.00	−0.332661	−0.166330	−	0.986070i	$-0.553192\pi$
$769$	−7094.00	−0.332661	−0.166330	+	0.986070i	$0.553192\pi$
$770$	0	0
$771$	45712.0	2.13525
$772$	0	0
$773$	−2234.00	−0.103947	−0.0519737	−	0.998648i	$-0.516551\pi$
$773$	−2234.00	−0.103947	−0.0519737	+	0.998648i	$0.516551\pi$
$774$	0	0
$775$	−1000.00	−0.0463498
$776$	0	0
$777$	−69504.0	−3.20906
$778$	0	0
$779$	−5880.00	−0.270440
$780$	0	0
$781$	−3960.00	−0.181434
$782$	0	0
$783$	8800.00	0.401643
$784$	0	0
$785$	−14050.0	−0.638810
$786$	0	0
$787$	−12956.0	−0.586825	−0.293413	−	0.955986i	$-0.594791\pi$
$787$	−12956.0	−0.586825	−0.293413	+	0.955986i	$0.594791\pi$
$788$	0	0
$789$	35712.0	1.61138
$790$	0	0
$791$	−24720.0	−1.11118
$792$	0	0
$793$	−13332.0	−0.597015
$794$	0	0
$795$	26480.0	1.18132
$796$	0	0
$797$	−6042.00	−0.268530	−0.134265	−	0.990945i	$-0.542867\pi$
$797$	−6042.00	−0.268530	−0.134265	+	0.990945i	$0.542867\pi$
$798$	0	0
$799$	−10120.0	−0.448085
$800$	0	0
$801$	−35742.0	−1.57663
$802$	0	0
$803$	11462.0	0.503718
$804$	0	0
$805$	−5280.00	−0.231175
$806$	0	0
$807$	39088.0	1.70503
$808$	0	0
$809$	−13566.0	−0.589561	−0.294781	−	0.955565i	$-0.595247\pi$
$809$	−13566.0	−0.589561	−0.294781	+	0.955565i	$0.595247\pi$
$810$	0	0
$811$	28348.0	1.22741	0.613707	−	0.789534i	$-0.289678\pi$
$811$	28348.0	1.22741	0.613707	+	0.789534i	$0.289678\pi$
$812$	0	0
$813$	40320.0	1.73934
$814$	0	0
$815$	8440.00	0.362749
$816$	0	0
$817$	−7280.00	−0.311744
$818$	0	0
$819$	−19536.0	−0.833508
$820$	0	0
$821$	−26810.0	−1.13968	−0.569839	−	0.821756i	$-0.692994\pi$
$821$	−26810.0	−1.13968	−0.569839	+	0.821756i	$0.692994\pi$
$822$	0	0
$823$	37596.0	1.59236	0.796181	−	0.605058i	$-0.206850\pi$
$823$	37596.0	1.59236	0.796181	+	0.605058i	$0.206850\pi$
$824$	0	0
$825$	−2200.00	−0.0928414
$826$	0	0
$827$	−852.000	−0.0358246	−0.0179123	−	0.999840i	$-0.505702\pi$
$827$	−852.000	−0.0358246	−0.0179123	+	0.999840i	$0.505702\pi$
$828$	0	0
$829$	28294.0	1.18539	0.592697	−	0.805426i	$-0.298063\pi$
$829$	28294.0	1.18539	0.592697	+	0.805426i	$0.298063\pi$
$830$	0	0
$831$	31760.0	1.32580
$832$	0	0
$833$	5126.00	0.213212
$834$	0	0
$835$	18080.0	0.749322
$836$	0	0
$837$	−3200.00	−0.132148
$838$	0	0
$839$	−10176.0	−0.418730	−0.209365	−	0.977838i	$-0.567140\pi$
$839$	−10176.0	−0.418730	−0.209365	+	0.977838i	$0.567140\pi$
$840$	0	0
$841$	−12289.0	−0.503875
$842$	0	0
$843$	23056.0	0.941982
$844$	0	0
$845$	−8565.00	−0.348692
$846$	0	0
$847$	2904.00	0.117807
$848$	0	0
$849$	18464.0	0.746387
$850$	0	0
$851$	15928.0	0.641604
$852$	0	0
$853$	20954.0	0.841092	0.420546	−	0.907271i	$-0.361839\pi$
$853$	20954.0	0.841092	0.420546	+	0.907271i	$0.361839\pi$
$854$	0	0
$855$	−5180.00	−0.207196
$856$	0	0
$857$	−36402.0	−1.45096	−0.725478	−	0.688246i	$-0.758381\pi$
$857$	−36402.0	−1.45096	−0.725478	+	0.688246i	$0.758381\pi$
$858$	0	0
$859$	−6140.00	−0.243881	−0.121941	−	0.992537i	$-0.538912\pi$
$859$	−6140.00	−0.243881	−0.121941	+	0.992537i	$0.538912\pi$
$860$	0	0
$861$	40320.0	1.59594
$862$	0	0
$863$	17524.0	0.691221	0.345611	−	0.938378i	$-0.387672\pi$
$863$	17524.0	0.691221	0.345611	+	0.938378i	$0.387672\pi$
$864$	0	0
$865$	18570.0	0.729941
$866$	0	0
$867$	−35432.0	−1.38793
$868$	0	0
$869$	6072.00	0.237029
$870$	0	0
$871$	6864.00	0.267024
$872$	0	0
$873$	−49358.0	−1.91353
$874$	0	0
$875$	3000.00	0.115907
$876$	0	0
$877$	9266.00	0.356774	0.178387	−	0.983960i	$-0.442912\pi$
$877$	9266.00	0.356774	0.178387	+	0.983960i	$0.442912\pi$
$878$	0	0
$879$	68880.0	2.64308
$880$	0	0
$881$	4978.00	0.190367	0.0951834	−	0.995460i	$-0.469656\pi$
$881$	4978.00	0.190367	0.0951834	+	0.995460i	$0.469656\pi$
$882$	0	0
$883$	42328.0	1.61319	0.806597	−	0.591102i	$-0.201307\pi$
$883$	42328.0	1.61319	0.806597	+	0.591102i	$0.201307\pi$
$884$	0	0
$885$	−2720.00	−0.103313
$886$	0	0
$887$	−20480.0	−0.775255	−0.387627	−	0.921816i	$-0.626705\pi$
$887$	−20480.0	−0.775255	−0.387627	+	0.921816i	$0.626705\pi$
$888$	0	0
$889$	46272.0	1.74568
$890$	0	0
$891$	3949.00	0.148481
$892$	0	0
$893$	12880.0	0.482657
$894$	0	0
$895$	2220.00	0.0829122
$896$	0	0
$897$	7744.00	0.288255
$898$	0	0
$899$	−4400.00	−0.163235
$900$	0	0
$901$	14564.0	0.538510
$902$	0	0
$903$	49920.0	1.83968
$904$	0	0
$905$	−23290.0	−0.855454
$906$	0	0
$907$	27232.0	0.996939	0.498470	−	0.866907i	$-0.333896\pi$
$907$	27232.0	0.996939	0.498470	+	0.866907i	$0.333896\pi$
$908$	0	0
$909$	−50394.0	−1.83879
$910$	0	0
$911$	17992.0	0.654338	0.327169	−	0.944966i	$-0.393905\pi$
$911$	17992.0	0.654338	0.327169	+	0.944966i	$0.393905\pi$
$912$	0	0
$913$	−2948.00	−0.106862
$914$	0	0
$915$	24240.0	0.875792
$916$	0	0
$917$	−23904.0	−0.860828
$918$	0	0
$919$	1064.00	0.0381916	0.0190958	−	0.999818i	$-0.493921\pi$
$919$	1064.00	0.0381916	0.0190958	+	0.999818i	$0.493921\pi$
$920$	0	0
$921$	−44256.0	−1.58337
$922$	0	0
$923$	−7920.00	−0.282438
$924$	0	0
$925$	−9050.00	−0.321689
$926$	0	0
$927$	57868.0	2.05031
$928$	0	0
$929$	32882.0	1.16127	0.580637	−	0.814163i	$-0.302804\pi$
$929$	32882.0	1.16127	0.580637	+	0.814163i	$0.302804\pi$
$930$	0	0
$931$	−6524.00	−0.229662
$932$	0	0
$933$	−13504.0	−0.473849
$934$	0	0
$935$	−1210.00	−0.0423222
$936$	0	0
$937$	9414.00	0.328220	0.164110	−	0.986442i	$-0.447525\pi$
$937$	9414.00	0.328220	0.164110	+	0.986442i	$0.447525\pi$
$938$	0	0
$939$	−5424.00	−0.188504
$940$	0	0
$941$	11270.0	0.390427	0.195213	−	0.980761i	$-0.437460\pi$
$941$	11270.0	0.390427	0.195213	+	0.980761i	$0.437460\pi$
$942$	0	0
$943$	−9240.00	−0.319084
$944$	0	0
$945$	9600.00	0.330464
$946$	0	0
$947$	−5576.00	−0.191336	−0.0956682	−	0.995413i	$-0.530499\pi$
$947$	−5576.00	−0.191336	−0.0956682	+	0.995413i	$0.530499\pi$
$948$	0	0
$949$	22924.0	0.784135
$950$	0	0
$951$	−86352.0	−2.94443
$952$	0	0
$953$	13526.0	0.459759	0.229879	−	0.973219i	$-0.426167\pi$
$953$	13526.0	0.459759	0.229879	+	0.973219i	$0.426167\pi$
$954$	0	0
$955$	−1080.00	−0.0365947
$956$	0	0
$957$	−9680.00	−0.326970
$958$	0	0
$959$	13872.0	0.467101
$960$	0	0
$961$	−28191.0	−0.946293
$962$	0	0
$963$	−40996.0	−1.37184
$964$	0	0
$965$	2350.00	0.0783929
$966$	0	0
$967$	14128.0	0.469830	0.234915	−	0.972016i	$-0.424519\pi$
$967$	14128.0	0.469830	0.234915	+	0.972016i	$0.424519\pi$
$968$	0	0
$969$	−4928.00	−0.163375
$970$	0	0
$971$	−3540.00	−0.116997	−0.0584985	−	0.998287i	$-0.518631\pi$
$971$	−3540.00	−0.116997	−0.0584985	+	0.998287i	$0.518631\pi$
$972$	0	0
$973$	−35616.0	−1.17348
$974$	0	0
$975$	−4400.00	−0.144526
$976$	0	0
$977$	11050.0	0.361843	0.180922	−	0.983498i	$-0.442092\pi$
$977$	11050.0	0.361843	0.180922	+	0.983498i	$0.442092\pi$
$978$	0	0
$979$	10626.0	0.346893
$980$	0	0
$981$	−54834.0	−1.78462
$982$	0	0
$983$	−19460.0	−0.631412	−0.315706	−	0.948857i	$-0.602241\pi$
$983$	−19460.0	−0.631412	−0.315706	+	0.948857i	$0.602241\pi$
$984$	0	0
$985$	11810.0	0.382028
$986$	0	0
$987$	−88320.0	−2.84828
$988$	0	0
$989$	−11440.0	−0.367817
$990$	0	0
$991$	−30776.0	−0.986510	−0.493255	−	0.869885i	$-0.664193\pi$
$991$	−30776.0	−0.986510	−0.493255	+	0.869885i	$0.664193\pi$
$992$	0	0
$993$	73824.0	2.35925
$994$	0	0
$995$	−8640.00	−0.275283
$996$	0	0
$997$	31370.0	0.996487	0.498244	−	0.867037i	$-0.333979\pi$
$997$	31370.0	0.996487	0.498244	+	0.867037i	$0.333979\pi$
$998$	0	0
$999$	−28960.0	−0.917171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	220.4.a.c.1.1	✓	1
3.2	odd	2		1980.4.a.c.1.1		1
4.3	odd	2		880.4.a.a.1.1		1
5.2	odd	4		1100.4.b.a.749.1		2
5.3	odd	4		1100.4.b.a.749.2		2
5.4	even	2		1100.4.a.a.1.1		1
11.10	odd	2		2420.4.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
220.4.a.c.1.1	✓	1	1.1	even	1	trivial
880.4.a.a.1.1		1	4.3	odd	2
1100.4.a.a.1.1		1	5.4	even	2
1100.4.b.a.749.1		2	5.2	odd	4
1100.4.b.a.749.2		2	5.3	odd	4
1980.4.a.c.1.1		1	3.2	odd	2
2420.4.a.f.1.1		1	11.10	odd	2

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:	\( N \)	\(=\)	\( 220 = 2^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	220.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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