LMFDB - Embedded newform 800.4.f.a.49.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,4,Mod(49,800)] 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("800.49"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0] 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:	no
Analytic conductor:	$47.2015280046$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.3
Root			$-1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	800.49
Dual form			800.4.f.a.49.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.29150 q^{3} -8.00000i q^{7} +1.00000 q^{9} +15.8745i q^{11} -52.9150 q^{13} +14.0000i q^{17} -37.0405i q^{19} -42.3320i q^{21} +152.000i q^{23} -137.579 q^{27} +158.745i q^{29} -224.000 q^{31} +84.0000i q^{33} +243.409 q^{37} -280.000 q^{39} -70.0000 q^{41} -439.195 q^{43} +336.000i q^{47} +279.000 q^{49} +74.0810i q^{51} -31.7490 q^{53} -196.000i q^{57} +534.442i q^{59} +95.2470i q^{61} -8.00000i q^{63} -174.620 q^{67} +804.308i q^{69} +72.0000 q^{71} -294.000i q^{73} +126.996 q^{77} -464.000 q^{79} -755.000 q^{81} -545.025 q^{83} +840.000i q^{87} -266.000 q^{89} +423.320i q^{91} -1185.30 q^{93} -994.000i q^{97} +15.8745i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{9} - 896 q^{31} - 1120 q^{39} - 280 q^{41} + 1116 q^{49} + 288 q^{71} - 1856 q^{79} - 3020 q^{81} - 1064 q^{89}+O(q^{100}) $ 4 * q + 4 * q^9 - 896 * q^31 - 1120 * q^39 - 280 * q^41 + 1116 * q^49 + 288 * q^71 - 1856 * q^79 - 3020 * q^81 - 1064 * q^89

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	5.29150	1.01835	0.509175	−	0.860663i	$-0.329951\pi$
$3$	5.29150	1.01835	0.509175	+	0.860663i	$0.329951\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 8.00000i	− 0.431959i	−0.976398	−	0.215980i	$-0.930705\pi$
$7$	− 8.00000i	− 0.431959i	0.976398	−	0.215980i	$-0.0692945\pi$
$8$	0	0
$9$	1.00000	0.0370370
$10$	0	0
$11$	15.8745i	0.435122i	0.976047	+	0.217561i	$0.0698101\pi$
$11$	15.8745i	0.435122i	−0.976047	+	0.217561i	$0.930190\pi$
$12$	0	0
$13$	−52.9150	−1.12892	−0.564461	−	0.825460i	$-0.690916\pi$
$13$	−52.9150	−1.12892	−0.564461	+	0.825460i	$0.690916\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	14.0000i	0.199735i	0.995001	+	0.0998676i	$0.0318419\pi$
$17$	14.0000i	0.199735i	−0.995001	+	0.0998676i	$0.968158\pi$
$18$	0	0
$19$	− 37.0405i	− 0.447246i	−0.974676	−	0.223623i	$-0.928212\pi$
$19$	− 37.0405i	− 0.447246i	0.974676	−	0.223623i	$-0.0717885\pi$
$20$	0	0
$21$	− 42.3320i	− 0.439886i
$22$	0	0
$23$	152.000i	1.37801i	0.724757	+	0.689004i	$0.241952\pi$
$23$	152.000i	1.37801i	−0.724757	+	0.689004i	$0.758048\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−137.579	−0.980633
$28$	0	0
$29$	158.745i	1.01649i	0.861212	+	0.508245i	$0.169706\pi$
$29$	158.745i	1.01649i	−0.861212	+	0.508245i	$0.830294\pi$
$30$	0	0
$31$	−224.000	−1.29779	−0.648897	−	0.760877i	$-0.724769\pi$
$31$	−224.000	−1.29779	−0.648897	+	0.760877i	$0.724769\pi$
$32$	0	0
$33$	84.0000i	0.443107i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	243.409	1.08152	0.540760	−	0.841177i	$-0.318137\pi$
$37$	243.409	1.08152	0.540760	+	0.841177i	$0.318137\pi$
$38$	0	0
$39$	−280.000	−1.14964
$40$	0	0
$41$	−70.0000	−0.266638	−0.133319	−	0.991073i	$-0.542564\pi$
$41$	−70.0000	−0.266638	−0.133319	+	0.991073i	$0.542564\pi$
$42$	0	0
$43$	−439.195	−1.55759	−0.778797	−	0.627276i	$-0.784170\pi$
$43$	−439.195	−1.55759	−0.778797	+	0.627276i	$0.784170\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	336.000i	1.04278i	0.853319	+	0.521390i	$0.174586\pi$
$47$	336.000i	1.04278i	−0.853319	+	0.521390i	$0.825414\pi$
$48$	0	0
$49$	279.000	0.813411
$50$	0	0
$51$	74.0810i	0.203400i
$52$	0	0
$53$	−31.7490	−0.0822842	−0.0411421	−	0.999153i	$-0.513100\pi$
$53$	−31.7490	−0.0822842	−0.0411421	+	0.999153i	$0.513100\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 196.000i	− 0.455453i
$58$	0	0
$59$	534.442i	1.17929i	0.807661	+	0.589647i	$0.200733\pi$
$59$	534.442i	1.17929i	−0.807661	+	0.589647i	$0.799267\pi$
$60$	0	0
$61$	95.2470i	0.199920i	0.994991	+	0.0999601i	$0.0318715\pi$
$61$	95.2470i	0.199920i	−0.994991	+	0.0999601i	$0.968128\pi$
$62$	0	0
$63$	− 8.00000i	− 0.0159985i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−174.620	−0.318406	−0.159203	−	0.987246i	$-0.550892\pi$
$67$	−174.620	−0.318406	−0.159203	+	0.987246i	$0.550892\pi$
$68$	0	0
$69$	804.308i	1.40329i
$70$	0	0
$71$	72.0000	0.120350	0.0601748	−	0.998188i	$-0.480834\pi$
$71$	72.0000	0.120350	0.0601748	+	0.998188i	$0.480834\pi$
$72$	0	0
$73$	− 294.000i	− 0.471371i	−0.971829	−	0.235686i	$-0.924266\pi$
$73$	− 294.000i	− 0.471371i	0.971829	−	0.235686i	$-0.0757336\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	126.996	0.187955
$78$	0	0
$79$	−464.000	−0.660811	−0.330406	−	0.943839i	$-0.607186\pi$
$79$	−464.000	−0.660811	−0.330406	+	0.943839i	$0.607186\pi$
$80$	0	0
$81$	−755.000	−1.03567
$82$	0	0
$83$	−545.025	−0.720774	−0.360387	−	0.932803i	$-0.617355\pi$
$83$	−545.025	−0.720774	−0.360387	+	0.932803i	$0.617355\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	840.000i	1.03514i
$88$	0	0
$89$	−266.000	−0.316808	−0.158404	−	0.987374i	$-0.550635\pi$
$89$	−266.000	−0.316808	−0.158404	+	0.987374i	$0.550635\pi$
$90$	0	0
$91$	423.320i	0.487649i
$92$	0	0
$93$	−1185.30	−1.32161
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 994.000i	− 1.04047i	−0.854024	−	0.520234i	$-0.825845\pi$
$97$	− 994.000i	− 1.04047i	0.854024	−	0.520234i	$-0.174155\pi$
$98$	0	0
$99$	15.8745i	0.0161156i
$100$	0	0
$101$	751.393i	0.740262i	0.928980	+	0.370131i	$0.120687\pi$
$101$	751.393i	0.740262i	−0.928980	+	0.370131i	$0.879313\pi$
$102$	0	0
$103$	− 1176.00i	− 1.12500i	−0.826798	−	0.562499i	$-0.809840\pi$
$103$	− 1176.00i	− 1.12500i	0.826798	−	0.562499i	$-0.190160\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	269.867	0.243822	0.121911	−	0.992541i	$-0.461098\pi$
$107$	269.867	0.243822	0.121911	+	0.992541i	$0.461098\pi$
$108$	0	0
$109$	1894.36i	1.66465i	0.554290	+	0.832324i	$0.312990\pi$
$109$	1894.36i	1.66465i	−0.554290	+	0.832324i	$0.687010\pi$
$110$	0	0
$111$	1288.00	1.10137
$112$	0	0
$113$	− 1710.00i	− 1.42357i	−0.702398	−	0.711784i	$-0.747887\pi$
$113$	− 1710.00i	− 1.42357i	0.702398	−	0.711784i	$-0.252113\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−52.9150	−0.0418119
$118$	0	0
$119$	112.000	0.0862775
$120$	0	0
$121$	1079.00	0.810669
$122$	0	0
$123$	−370.405	−0.271531
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 1664.00i	− 1.16265i	−0.813673	−	0.581323i	$-0.802535\pi$
$127$	− 1664.00i	− 1.16265i	0.813673	−	0.581323i	$-0.197465\pi$
$128$	0	0
$129$	−2324.00	−1.58618
$130$	0	0
$131$	672.021i	0.448204i	0.974566	+	0.224102i	$0.0719449\pi$
$131$	672.021i	0.448204i	−0.974566	+	0.224102i	$0.928055\pi$
$132$	0	0
$133$	−296.324	−0.193192
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1062.00i	0.662283i	0.943581	+	0.331142i	$0.107434\pi$
$137$	1062.00i	0.662283i	−0.943581	+	0.331142i	$0.892566\pi$
$138$	0	0
$139$	2693.37i	1.64352i	0.569835	+	0.821759i	$0.307007\pi$
$139$	2693.37i	1.64352i	−0.569835	+	0.821759i	$0.692993\pi$
$140$	0	0
$141$	1777.94i	1.06191i
$142$	0	0
$143$	− 840.000i	− 0.491219i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1476.33	0.828337
$148$	0	0
$149$	− 793.725i	− 0.436406i	−0.975903	−	0.218203i	$-0.929980\pi$
$149$	− 793.725i	− 0.436406i	0.975903	−	0.218203i	$-0.0700195\pi$
$150$	0	0
$151$	−744.000	−0.400966	−0.200483	−	0.979697i	$-0.564251\pi$
$151$	−744.000	−0.400966	−0.200483	+	0.979697i	$0.564251\pi$
$152$	0	0
$153$	14.0000i	0.00739760i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	179.911	0.0914552	0.0457276	−	0.998954i	$-0.485439\pi$
$157$	179.911	0.0914552	0.0457276	+	0.998954i	$0.485439\pi$
$158$	0	0
$159$	−168.000	−0.0837941
$160$	0	0
$161$	1216.00	0.595244
$162$	0	0
$163$	−1772.65	−0.851809	−0.425905	−	0.904768i	$-0.640044\pi$
$163$	−1772.65	−0.851809	−0.425905	+	0.904768i	$0.640044\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1960.00i	− 0.908200i	−0.890951	−	0.454100i	$-0.849961\pi$
$167$	− 1960.00i	− 0.908200i	0.890951	−	0.454100i	$-0.150039\pi$
$168$	0	0
$169$	603.000	0.274465
$170$	0	0
$171$	− 37.0405i	− 0.0165647i
$172$	0	0
$173$	−2000.19	−0.879026	−0.439513	−	0.898236i	$-0.644849\pi$
$173$	−2000.19	−0.879026	−0.439513	+	0.898236i	$0.644849\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2828.00i	1.20094i
$178$	0	0
$179$	3264.86i	1.36328i	0.731688	+	0.681639i	$0.238733\pi$
$179$	3264.86i	1.36328i	−0.731688	+	0.681639i	$0.761267\pi$
$180$	0	0
$181$	− 2338.84i	− 0.960469i	−0.877140	−	0.480235i	$-0.840552\pi$
$181$	− 2338.84i	− 0.960469i	0.877140	−	0.480235i	$-0.159448\pi$
$182$	0	0
$183$	504.000i	0.203589i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−222.243	−0.0869092
$188$	0	0
$189$	1100.63i	0.423594i
$190$	0	0
$191$	−3904.00	−1.47897	−0.739486	−	0.673172i	$-0.764931\pi$
$191$	−3904.00	−1.47897	−0.739486	+	0.673172i	$0.764931\pi$
$192$	0	0
$193$	3330.00i	1.24196i	0.783826	+	0.620981i	$0.213266\pi$
$193$	3330.00i	1.24196i	−0.783826	+	0.620981i	$0.786734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1195.88	−0.432502	−0.216251	−	0.976338i	$-0.569383\pi$
$197$	−1195.88	−0.432502	−0.216251	+	0.976338i	$0.569383\pi$
$198$	0	0
$199$	−1736.00	−0.618401	−0.309200	−	0.950997i	$-0.600061\pi$
$199$	−1736.00	−0.618401	−0.309200	+	0.950997i	$0.600061\pi$
$200$	0	0
$201$	−924.000	−0.324248
$202$	0	0
$203$	1269.96	0.439083
$204$	0	0
$205$	0	0
$206$	0	0
$207$	152.000i	0.0510373i
$208$	0	0
$209$	588.000	0.194607
$210$	0	0
$211$	2915.62i	0.951277i	0.879641	+	0.475638i	$0.157783\pi$
$211$	2915.62i	0.951277i	−0.879641	+	0.475638i	$0.842217\pi$
$212$	0	0
$213$	380.988	0.122558
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1792.00i	0.560594i
$218$	0	0
$219$	− 1555.70i	− 0.480021i
$220$	0	0
$221$	− 740.810i	− 0.225486i
$222$	0	0
$223$	− 1568.00i	− 0.470857i	−0.971892	−	0.235428i	$-0.924351\pi$
$223$	− 1568.00i	− 0.470857i	0.971892	−	0.235428i	$-0.0756493\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1264.67	0.369775	0.184888	−	0.982760i	$-0.440808\pi$
$227$	1264.67	0.369775	0.184888	+	0.982760i	$0.440808\pi$
$228$	0	0
$229$	− 5153.92i	− 1.48725i	−0.668595	−	0.743626i	$-0.733104\pi$
$229$	− 5153.92i	− 1.48725i	0.668595	−	0.743626i	$-0.266896\pi$
$230$	0	0
$231$	672.000	0.191404
$232$	0	0
$233$	− 838.000i	− 0.235619i	−0.993036	−	0.117809i	$-0.962413\pi$
$233$	− 838.000i	− 0.235619i	0.993036	−	0.117809i	$-0.0375872\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2455.26	−0.672937
$238$	0	0
$239$	6288.00	1.70183	0.850914	−	0.525305i	$-0.176049\pi$
$239$	6288.00	1.70183	0.850914	+	0.525305i	$0.176049\pi$
$240$	0	0
$241$	−2926.00	−0.782076	−0.391038	−	0.920375i	$-0.627884\pi$
$241$	−2926.00	−0.782076	−0.391038	+	0.920375i	$0.627884\pi$
$242$	0	0
$243$	−280.450	−0.0740364
$244$	0	0
$245$	0	0
$246$	0	0
$247$	1960.00i	0.504906i
$248$	0	0
$249$	−2884.00	−0.734000
$250$	0	0
$251$	− 5444.96i	− 1.36925i	−0.728894	−	0.684627i	$-0.759965\pi$
$251$	− 5444.96i	− 1.36925i	0.728894	−	0.684627i	$-0.240035\pi$
$252$	0	0
$253$	−2412.93	−0.599602
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2562.00i	− 0.621841i	−0.950436	−	0.310921i	$-0.899363\pi$
$257$	− 2562.00i	− 0.621841i	0.950436	−	0.310921i	$-0.100637\pi$
$258$	0	0
$259$	− 1947.27i	− 0.467172i
$260$	0	0
$261$	158.745i	0.0376478i
$262$	0	0
$263$	5896.00i	1.38237i	0.722679	+	0.691184i	$0.242911\pi$
$263$	5896.00i	1.38237i	−0.722679	+	0.691184i	$0.757089\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1407.54	−0.322622
$268$	0	0
$269$	5365.58i	1.21615i	0.793878	+	0.608077i	$0.208059\pi$
$269$	5365.58i	1.21615i	−0.793878	+	0.608077i	$0.791941\pi$
$270$	0	0
$271$	1680.00	0.376578	0.188289	−	0.982114i	$-0.439706\pi$
$271$	1680.00	0.376578	0.188289	+	0.982114i	$0.439706\pi$
$272$	0	0
$273$	2240.00i	0.496597i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1576.87	−0.342039	−0.171019	−	0.985268i	$-0.554706\pi$
$277$	−1576.87	−0.342039	−0.171019	+	0.985268i	$0.554706\pi$
$278$	0	0
$279$	−224.000	−0.0480664
$280$	0	0
$281$	−2742.00	−0.582114	−0.291057	−	0.956706i	$-0.594007\pi$
$281$	−2742.00	−0.582114	−0.291057	+	0.956706i	$0.594007\pi$
$282$	0	0
$283$	2989.70	0.627983	0.313991	−	0.949426i	$-0.398334\pi$
$283$	2989.70	0.627983	0.313991	+	0.949426i	$0.398334\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	560.000i	0.115177i
$288$	0	0
$289$	4717.00	0.960106
$290$	0	0
$291$	− 5259.75i	− 1.05956i
$292$	0	0
$293$	9238.96	1.84214	0.921068	−	0.389401i	$-0.127318\pi$
$293$	9238.96	1.84214	0.921068	+	0.389401i	$0.127318\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2184.00i	− 0.426695i
$298$	0	0
$299$	− 8043.08i	− 1.55566i
$300$	0	0
$301$	3513.56i	0.672818i
$302$	0	0
$303$	3976.00i	0.753846i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−2587.54	−0.481039	−0.240520	−	0.970644i	$-0.577318\pi$
$307$	−2587.54	−0.481039	−0.240520	+	0.970644i	$0.577318\pi$
$308$	0	0
$309$	− 6222.81i	− 1.14564i
$310$	0	0
$311$	2744.00	0.500315	0.250157	−	0.968205i	$-0.419518\pi$
$311$	2744.00	0.500315	0.250157	+	0.968205i	$0.419518\pi$
$312$	0	0
$313$	2282.00i	0.412097i	0.978542	+	0.206048i	$0.0660604\pi$
$313$	2282.00i	0.412097i	−0.978542	+	0.206048i	$0.933940\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9577.62	1.69695	0.848474	−	0.529237i	$-0.177522\pi$
$317$	9577.62	1.69695	0.848474	+	0.529237i	$0.177522\pi$
$318$	0	0
$319$	−2520.00	−0.442298
$320$	0	0
$321$	1428.00	0.248297
$322$	0	0
$323$	518.567	0.0893308
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10024.0i	1.69519i
$328$	0	0
$329$	2688.00	0.450438
$330$	0	0
$331$	4249.08i	0.705590i	0.935701	+	0.352795i	$0.114769\pi$
$331$	4249.08i	0.705590i	−0.935701	+	0.352795i	$0.885231\pi$
$332$	0	0
$333$	243.409	0.0400563
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 6130.00i	− 0.990868i	−0.868646	−	0.495434i	$-0.835009\pi$
$337$	− 6130.00i	− 0.990868i	0.868646	−	0.495434i	$-0.164991\pi$
$338$	0	0
$339$	− 9048.47i	− 1.44969i
$340$	0	0
$341$	− 3555.89i	− 0.564699i
$342$	0	0
$343$	− 4976.00i	− 0.783320i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2481.71	−0.383935	−0.191967	−	0.981401i	$-0.561487\pi$
$347$	−2481.71	−0.383935	−0.191967	+	0.981401i	$0.561487\pi$
$348$	0	0
$349$	328.073i	0.0503191i	0.999683	+	0.0251595i	$0.00800937\pi$
$349$	328.073i	0.0503191i	−0.999683	+	0.0251595i	$0.991991\pi$
$350$	0	0
$351$	7280.00	1.10706
$352$	0	0
$353$	− 10206.0i	− 1.53884i	−0.638743	−	0.769420i	$-0.720545\pi$
$353$	− 10206.0i	− 1.53884i	0.638743	−	0.769420i	$-0.279455\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	592.648	0.0878607
$358$	0	0
$359$	−3176.00	−0.466916	−0.233458	−	0.972367i	$-0.575004\pi$
$359$	−3176.00	−0.466916	−0.233458	+	0.972367i	$0.575004\pi$
$360$	0	0
$361$	5487.00	0.799971
$362$	0	0
$363$	5709.53	0.825545
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 11760.0i	− 1.67266i	−0.548225	−	0.836331i	$-0.684696\pi$
$367$	− 11760.0i	− 1.67266i	0.548225	−	0.836331i	$-0.315304\pi$
$368$	0	0
$369$	−70.0000	−0.00987549
$370$	0	0
$371$	253.992i	0.0355434i
$372$	0	0
$373$	10974.6	1.52344	0.761719	−	0.647908i	$-0.224356\pi$
$373$	10974.6	1.52344	0.761719	+	0.647908i	$0.224356\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8400.00i	− 1.14754i
$378$	0	0
$379$	3074.36i	0.416674i	0.978057	+	0.208337i	$0.0668051\pi$
$379$	3074.36i	0.416674i	−0.978057	+	0.208337i	$0.933195\pi$
$380$	0	0
$381$	− 8805.06i	− 1.18398i
$382$	0	0
$383$	− 2688.00i	− 0.358617i	−0.983793	−	0.179309i	$-0.942614\pi$
$383$	− 2688.00i	− 0.358617i	0.983793	−	0.179309i	$-0.0573861\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−439.195	−0.0576887
$388$	0	0
$389$	− 10487.8i	− 1.36697i	−0.729966	−	0.683484i	$-0.760464\pi$
$389$	− 10487.8i	− 1.36697i	0.729966	−	0.683484i	$-0.239536\pi$
$390$	0	0
$391$	−2128.00	−0.275237
$392$	0	0
$393$	3556.00i	0.456429i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−5704.24	−0.721127	−0.360564	−	0.932735i	$-0.617416\pi$
$397$	−5704.24	−0.721127	−0.360564	+	0.932735i	$0.617416\pi$
$398$	0	0
$399$	−1568.00	−0.196737
$400$	0	0
$401$	12402.0	1.54445	0.772227	−	0.635346i	$-0.219143\pi$
$401$	12402.0	1.54445	0.772227	+	0.635346i	$0.219143\pi$
$402$	0	0
$403$	11853.0	1.46511
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3864.00i	0.470593i
$408$	0	0
$409$	12278.0	1.48437	0.742186	−	0.670194i	$-0.233789\pi$
$409$	12278.0	1.48437	0.742186	+	0.670194i	$0.233789\pi$
$410$	0	0
$411$	5619.58i	0.674436i
$412$	0	0
$413$	4275.53	0.509407
$414$	0	0
$415$	0	0
$416$	0	0
$417$	14252.0i	1.67368i
$418$	0	0
$419$	− 8207.12i	− 0.956907i	−0.878113	−	0.478454i	$-0.841198\pi$
$419$	− 8207.12i	− 0.956907i	0.878113	−	0.478454i	$-0.158802\pi$
$420$	0	0
$421$	− 1449.87i	− 0.167844i	−0.996472	−	0.0839221i	$-0.973255\pi$
$421$	− 1449.87i	− 0.167844i	0.996472	−	0.0839221i	$-0.0267447\pi$
$422$	0	0
$423$	336.000i	0.0386215i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	761.976	0.0863574
$428$	0	0
$429$	− 4444.86i	− 0.500233i
$430$	0	0
$431$	−7632.00	−0.852948	−0.426474	−	0.904500i	$-0.640244\pi$
$431$	−7632.00	−0.852948	−0.426474	+	0.904500i	$0.640244\pi$
$432$	0	0
$433$	3794.00i	0.421081i	0.977585	+	0.210540i	$0.0675224\pi$
$433$	3794.00i	0.421081i	−0.977585	+	0.210540i	$0.932478\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5630.16	0.616309
$438$	0	0
$439$	−1848.00	−0.200912	−0.100456	−	0.994942i	$-0.532030\pi$
$439$	−1848.00	−0.200912	−0.100456	+	0.994942i	$0.532030\pi$
$440$	0	0
$441$	279.000	0.0301263
$442$	0	0
$443$	−12334.5	−1.32287	−0.661433	−	0.750004i	$-0.730051\pi$
$443$	−12334.5	−1.32287	−0.661433	+	0.750004i	$0.730051\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4200.00i	− 0.444414i
$448$	0	0
$449$	3582.00	0.376492	0.188246	−	0.982122i	$-0.439720\pi$
$449$	3582.00	0.376492	0.188246	+	0.982122i	$0.439720\pi$
$450$	0	0
$451$	− 1111.22i	− 0.116020i
$452$	0	0
$453$	−3936.88	−0.408324
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2714.00i	− 0.277802i	−0.990306	−	0.138901i	$-0.955643\pi$
$457$	− 2714.00i	− 0.277802i	0.990306	−	0.138901i	$-0.0443570\pi$
$458$	0	0
$459$	− 1926.11i	− 0.195867i
$460$	0	0
$461$	8349.99i	0.843596i	0.906690	+	0.421798i	$0.138601\pi$
$461$	8349.99i	0.843596i	−0.906690	+	0.421798i	$0.861399\pi$
$462$	0	0
$463$	− 2224.00i	− 0.223236i	−0.993751	−	0.111618i	$-0.964397\pi$
$463$	− 2224.00i	− 0.223236i	0.993751	−	0.111618i	$-0.0356032\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10292.0	−1.01982	−0.509910	−	0.860228i	$-0.670321\pi$
$467$	−10292.0	−1.01982	−0.509910	+	0.860228i	$0.670321\pi$
$468$	0	0
$469$	1396.96i	0.137538i
$470$	0	0
$471$	952.000	0.0931334
$472$	0	0
$473$	− 6972.00i	− 0.677744i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−31.7490	−0.00304756
$478$	0	0
$479$	17696.0	1.68800	0.843999	−	0.536345i	$-0.180195\pi$
$479$	17696.0	1.68800	0.843999	+	0.536345i	$0.180195\pi$
$480$	0	0
$481$	−12880.0	−1.22095
$482$	0	0
$483$	6434.47	0.606166
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1304.00i	0.121334i	0.998158	+	0.0606672i	$0.0193228\pi$
$487$	1304.00i	0.121334i	−0.998158	+	0.0606672i	$0.980677\pi$
$488$	0	0
$489$	−9380.00	−0.867440
$490$	0	0
$491$	− 16662.9i	− 1.53154i	−0.643112	−	0.765772i	$-0.722357\pi$
$491$	− 16662.9i	− 1.53154i	0.643112	−	0.765772i	$-0.277643\pi$
$492$	0	0
$493$	−2222.43	−0.203029
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 576.000i	− 0.0519862i
$498$	0	0
$499$	3095.53i	0.277705i	0.990313	+	0.138853i	$0.0443414\pi$
$499$	3095.53i	0.277705i	−0.990313	+	0.138853i	$0.955659\pi$
$500$	0	0
$501$	− 10371.3i	− 0.924865i
$502$	0	0
$503$	19320.0i	1.71260i	0.516481	+	0.856298i	$0.327242\pi$
$503$	19320.0i	1.71260i	−0.516481	+	0.856298i	$0.672758\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3190.78	0.279502
$508$	0	0
$509$	4476.61i	0.389828i	0.980820	+	0.194914i	$0.0624427\pi$
$509$	4476.61i	0.389828i	−0.980820	+	0.194914i	$0.937557\pi$
$510$	0	0
$511$	−2352.00	−0.203613
$512$	0	0
$513$	5096.00i	0.438585i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5333.83	−0.453737
$518$	0	0
$519$	−10584.0	−0.895156
$520$	0	0
$521$	−2982.00	−0.250756	−0.125378	−	0.992109i	$-0.540014\pi$
$521$	−2982.00	−0.250756	−0.125378	+	0.992109i	$0.540014\pi$
$522$	0	0
$523$	2016.06	0.168559	0.0842794	−	0.996442i	$-0.473141\pi$
$523$	2016.06	0.168559	0.0842794	+	0.996442i	$0.473141\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 3136.00i	− 0.259215i
$528$	0	0
$529$	−10937.0	−0.898907
$530$	0	0
$531$	534.442i	0.0436776i
$532$	0	0
$533$	3704.05	0.301014
$534$	0	0
$535$	0	0
$536$	0	0
$537$	17276.0i	1.38830i
$538$	0	0
$539$	4428.99i	0.353933i
$540$	0	0
$541$	15419.4i	1.22539i	0.790321	+	0.612693i	$0.209914\pi$
$541$	15419.4i	1.22539i	−0.790321	+	0.612693i	$0.790086\pi$
$542$	0	0
$543$	− 12376.0i	− 0.978094i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12609.7	0.985649	0.492824	−	0.870129i	$-0.335965\pi$
$547$	12609.7	0.985649	0.492824	+	0.870129i	$0.335965\pi$
$548$	0	0
$549$	95.2470i	0.00740445i
$550$	0	0
$551$	5880.00	0.454621
$552$	0	0
$553$	3712.00i	0.285444i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7143.53	−0.543413	−0.271706	−	0.962380i	$-0.587588\pi$
$557$	−7143.53	−0.543413	−0.271706	+	0.962380i	$0.587588\pi$
$558$	0	0
$559$	23240.0	1.75840
$560$	0	0
$561$	−1176.00	−0.0885040
$562$	0	0
$563$	−7572.14	−0.566834	−0.283417	−	0.958997i	$-0.591468\pi$
$563$	−7572.14	−0.566834	−0.283417	+	0.958997i	$0.591468\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	6040.00i	0.447365i
$568$	0	0
$569$	−15594.0	−1.14892	−0.574459	−	0.818533i	$-0.694788\pi$
$569$	−15594.0	−1.14892	−0.574459	+	0.818533i	$0.694788\pi$
$570$	0	0
$571$	16737.0i	1.22666i	0.789827	+	0.613330i	$0.210170\pi$
$571$	16737.0i	1.22666i	−0.789827	+	0.613330i	$0.789830\pi$
$572$	0	0
$573$	−20658.0	−1.50611
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 6594.00i	− 0.475757i	−0.971295	−	0.237879i	$-0.923548\pi$
$577$	− 6594.00i	− 0.475757i	0.971295	−	0.237879i	$-0.0764520\pi$
$578$	0	0
$579$	17620.7i	1.26475i
$580$	0	0
$581$	4360.20i	0.311345i
$582$	0	0
$583$	− 504.000i	− 0.0358037i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23213.8	1.63226	0.816130	−	0.577868i	$-0.196115\pi$
$587$	23213.8	1.63226	0.816130	+	0.577868i	$0.196115\pi$
$588$	0	0
$589$	8297.08i	0.580433i
$590$	0	0
$591$	−6328.00	−0.440438
$592$	0	0
$593$	14322.0i	0.991794i	0.868381	+	0.495897i	$0.165161\pi$
$593$	14322.0i	0.991794i	−0.868381	+	0.495897i	$0.834839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−9186.05	−0.629749
$598$	0	0
$599$	−16088.0	−1.09739	−0.548696	−	0.836022i	$-0.684876\pi$
$599$	−16088.0	−1.09739	−0.548696	+	0.836022i	$0.684876\pi$
$600$	0	0
$601$	−21238.0	−1.44146	−0.720729	−	0.693217i	$-0.756193\pi$
$601$	−21238.0	−1.44146	−0.720729	+	0.693217i	$0.756193\pi$
$602$	0	0
$603$	−174.620	−0.0117928
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 13664.0i	− 0.913681i	−0.889549	−	0.456841i	$-0.848981\pi$
$607$	− 13664.0i	− 0.913681i	0.889549	−	0.456841i	$-0.151019\pi$
$608$	0	0
$609$	6720.00	0.447140
$610$	0	0
$611$	− 17779.4i	− 1.17722i
$612$	0	0
$613$	−20393.5	−1.34369	−0.671846	−	0.740690i	$-0.734499\pi$
$613$	−20393.5	−1.34369	−0.671846	+	0.740690i	$0.734499\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	3782.00i	0.246771i	0.992359	+	0.123385i	$0.0393751\pi$
$617$	3782.00i	0.246771i	−0.992359	+	0.123385i	$0.960625\pi$
$618$	0	0
$619$	5825.94i	0.378295i	0.981949	+	0.189147i	$0.0605724\pi$
$619$	5825.94i	0.378295i	−0.981949	+	0.189147i	$0.939428\pi$
$620$	0	0
$621$	− 20912.0i	− 1.35132i
$622$	0	0
$623$	2128.00i	0.136848i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3111.40	0.198178
$628$	0	0
$629$	3407.73i	0.216017i
$630$	0	0
$631$	−2056.00	−0.129712	−0.0648558	−	0.997895i	$-0.520659\pi$
$631$	−2056.00	−0.129712	−0.0648558	+	0.997895i	$0.520659\pi$
$632$	0	0
$633$	15428.0i	0.968733i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−14763.3	−0.918278
$638$	0	0
$639$	72.0000	0.00445740
$640$	0	0
$641$	11842.0	0.729689	0.364845	−	0.931068i	$-0.381122\pi$
$641$	11842.0	0.729689	0.364845	+	0.931068i	$0.381122\pi$
$642$	0	0
$643$	−16250.2	−0.996649	−0.498325	−	0.866991i	$-0.666051\pi$
$643$	−16250.2	−0.996649	−0.498325	+	0.866991i	$0.666051\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19320.0i	1.17395i	0.809604	+	0.586976i	$0.199682\pi$
$647$	19320.0i	1.17395i	−0.809604	+	0.586976i	$0.800318\pi$
$648$	0	0
$649$	−8484.00	−0.513137
$650$	0	0
$651$	9482.37i	0.570881i
$652$	0	0
$653$	2317.68	0.138894	0.0694470	−	0.997586i	$-0.477877\pi$
$653$	2317.68	0.138894	0.0694470	+	0.997586i	$0.477877\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 294.000i	− 0.0174582i
$658$	0	0
$659$	27732.8i	1.63932i	0.572847	+	0.819662i	$0.305839\pi$
$659$	27732.8i	1.63932i	−0.572847	+	0.819662i	$0.694161\pi$
$660$	0	0
$661$	22467.7i	1.32208i	0.750352	+	0.661039i	$0.229884\pi$
$661$	22467.7i	1.32208i	−0.750352	+	0.661039i	$0.770116\pi$
$662$	0	0
$663$	− 3920.00i	− 0.229623i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−24129.3	−1.40073
$668$	0	0
$669$	− 8297.08i	− 0.479497i
$670$	0	0
$671$	−1512.00	−0.0869897
$672$	0	0
$673$	− 10078.0i	− 0.577234i	−0.957445	−	0.288617i	$-0.906805\pi$
$673$	− 10078.0i	− 0.577234i	0.957445	−	0.288617i	$-0.0931954\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16160.2	0.917413	0.458707	−	0.888588i	$-0.348313\pi$
$677$	16160.2	0.917413	0.458707	+	0.888588i	$0.348313\pi$
$678$	0	0
$679$	−7952.00	−0.449440
$680$	0	0
$681$	6692.00	0.376561
$682$	0	0
$683$	−16356.0	−0.916320	−0.458160	−	0.888870i	$-0.651491\pi$
$683$	−16356.0	−0.916320	−0.458160	+	0.888870i	$0.651491\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 27272.0i	− 1.51454i
$688$	0	0
$689$	1680.00	0.0928925
$690$	0	0
$691$	29246.1i	1.61009i	0.593211	+	0.805047i	$0.297860\pi$
$691$	29246.1i	1.61009i	−0.593211	+	0.805047i	$0.702140\pi$
$692$	0	0
$693$	126.996	0.00696130
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 980.000i	− 0.0532570i
$698$	0	0
$699$	− 4434.28i	− 0.239943i
$700$	0	0
$701$	2465.84i	0.132858i	0.997791	+	0.0664290i	$0.0211606\pi$
$701$	2465.84i	0.132858i	−0.997791	+	0.0664290i	$0.978839\pi$
$702$	0	0
$703$	− 9016.00i	− 0.483705i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6011.15	0.319763
$708$	0	0
$709$	31674.9i	1.67782i	0.544267	+	0.838912i	$0.316808\pi$
$709$	31674.9i	1.67782i	−0.544267	+	0.838912i	$0.683192\pi$
$710$	0	0
$711$	−464.000	−0.0244745
$712$	0	0
$713$	− 34048.0i	− 1.78837i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33273.0	1.73306
$718$	0	0
$719$	−9296.00	−0.482173	−0.241086	−	0.970504i	$-0.577504\pi$
$719$	−9296.00	−0.482173	−0.241086	+	0.970504i	$0.577504\pi$
$720$	0	0
$721$	−9408.00	−0.485953
$722$	0	0
$723$	−15482.9	−0.796427
$724$	0	0
$725$	0	0
$726$	0	0
$727$	21672.0i	1.10560i	0.833315	+	0.552799i	$0.186440\pi$
$727$	21672.0i	1.10560i	−0.833315	+	0.552799i	$0.813560\pi$
$728$	0	0
$729$	18901.0	0.960270
$730$	0	0
$731$	− 6148.73i	− 0.311106i
$732$	0	0
$733$	9471.79	0.477283	0.238642	−	0.971108i	$-0.423298\pi$
$733$	9471.79	0.477283	0.238642	+	0.971108i	$0.423298\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2772.00i	− 0.138545i
$738$	0	0
$739$	6863.08i	0.341627i	0.985303	+	0.170814i	$0.0546396\pi$
$739$	6863.08i	0.341627i	−0.985303	+	0.170814i	$0.945360\pi$
$740$	0	0
$741$	10371.3i	0.514171i
$742$	0	0
$743$	− 17432.0i	− 0.860724i	−0.902656	−	0.430362i	$-0.858386\pi$
$743$	− 17432.0i	− 0.860724i	0.902656	−	0.430362i	$-0.141614\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−545.025	−0.0266953
$748$	0	0
$749$	− 2158.93i	− 0.105321i
$750$	0	0
$751$	11632.0	0.565190	0.282595	−	0.959239i	$-0.408805\pi$
$751$	11632.0	0.565190	0.282595	+	0.959239i	$0.408805\pi$
$752$	0	0
$753$	− 28812.0i	− 1.39438i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−16731.7	−0.803336	−0.401668	−	0.915785i	$-0.631569\pi$
$757$	−16731.7	−0.803336	−0.401668	+	0.915785i	$0.631569\pi$
$758$	0	0
$759$	−12768.0	−0.610605
$760$	0	0
$761$	39466.0	1.87995	0.939975	−	0.341244i	$-0.110848\pi$
$761$	39466.0	1.87995	0.939975	+	0.341244i	$0.110848\pi$
$762$	0	0
$763$	15154.9	0.719060
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 28280.0i	− 1.33133i
$768$	0	0
$769$	−35266.0	−1.65374	−0.826869	−	0.562395i	$-0.809880\pi$
$769$	−35266.0	−1.65374	−0.826869	+	0.562395i	$0.809880\pi$
$770$	0	0
$771$	− 13556.8i	− 0.633252i
$772$	0	0
$773$	−16244.9	−0.755872	−0.377936	−	0.925832i	$-0.623366\pi$
$773$	−16244.9	−0.755872	−0.377936	+	0.925832i	$0.623366\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 10304.0i	− 0.475745i
$778$	0	0
$779$	2592.84i	0.119253i
$780$	0	0
$781$	1142.96i	0.0523668i
$782$	0	0
$783$	− 21840.0i	− 0.996805i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−34844.5	−1.57824	−0.789119	−	0.614240i	$-0.789463\pi$
$787$	−34844.5	−1.57824	−0.789119	+	0.614240i	$0.789463\pi$
$788$	0	0
$789$	31198.7i	1.40774i
$790$	0	0
$791$	−13680.0	−0.614924
$792$	0	0
$793$	− 5040.00i	− 0.225694i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2550.50	0.113354	0.0566772	−	0.998393i	$-0.481949\pi$
$797$	2550.50	0.113354	0.0566772	+	0.998393i	$0.481949\pi$
$798$	0	0
$799$	−4704.00	−0.208280
$800$	0	0
$801$	−266.000	−0.0117336
$802$	0	0
$803$	4667.11	0.205104
$804$	0	0
$805$	0	0
$806$	0	0
$807$	28392.0i	1.23847i
$808$	0	0
$809$	24390.0	1.05996	0.529979	−	0.848010i	$-0.322200\pi$
$809$	24390.0	1.05996	0.529979	+	0.848010i	$0.322200\pi$
$810$	0	0
$811$	9582.91i	0.414922i	0.978243	+	0.207461i	$0.0665200\pi$
$811$	9582.91i	0.414922i	−0.978243	+	0.207461i	$0.933480\pi$
$812$	0	0
$813$	8889.72	0.383489
$814$	0	0
$815$	0	0
$816$	0	0
$817$	16268.0i	0.696628i
$818$	0	0
$819$	423.320i	0.0180611i
$820$	0	0
$821$	− 8773.31i	− 0.372948i	−0.982460	−	0.186474i	$-0.940294\pi$
$821$	− 8773.31i	− 0.372948i	0.982460	−	0.186474i	$-0.0597061\pi$
$822$	0	0
$823$	21688.0i	0.918586i	0.888285	+	0.459293i	$0.151897\pi$
$823$	21688.0i	0.918586i	−0.888285	+	0.459293i	$0.848103\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19446.3	0.817670	0.408835	−	0.912608i	$-0.365935\pi$
$827$	19446.3	0.817670	0.408835	+	0.912608i	$0.365935\pi$
$828$	0	0
$829$	19546.8i	0.818925i	0.912327	+	0.409462i	$0.134284\pi$
$829$	19546.8i	0.818925i	−0.912327	+	0.409462i	$0.865716\pi$
$830$	0	0
$831$	−8344.00	−0.348315
$832$	0	0
$833$	3906.00i	0.162467i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30817.7	1.27266
$838$	0	0
$839$	−18760.0	−0.771951	−0.385976	−	0.922509i	$-0.626135\pi$
$839$	−18760.0	−0.771951	−0.385976	+	0.922509i	$0.626135\pi$
$840$	0	0
$841$	−811.000	−0.0332527
$842$	0	0
$843$	−14509.3	−0.592796
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 8632.00i	− 0.350176i
$848$	0	0
$849$	15820.0	0.639506
$850$	0	0
$851$	36998.2i	1.49034i
$852$	0	0
$853$	−28732.9	−1.15333	−0.576667	−	0.816979i	$-0.695647\pi$
$853$	−28732.9	−1.15333	−0.576667	+	0.816979i	$0.695647\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 8778.00i	− 0.349884i	−0.984579	−	0.174942i	$-0.944026\pi$
$857$	− 8778.00i	− 0.349884i	0.984579	−	0.174942i	$-0.0559738\pi$
$858$	0	0
$859$	− 5646.03i	− 0.224261i	−0.993693	−	0.112130i	$-0.964233\pi$
$859$	− 5646.03i	− 0.224261i	0.993693	−	0.112130i	$-0.0357675\pi$
$860$	0	0
$861$	2963.24i	0.117290i
$862$	0	0
$863$	9312.00i	0.367305i	0.982991	+	0.183652i	$0.0587921\pi$
$863$	9312.00i	0.367305i	−0.982991	+	0.183652i	$0.941208\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	24960.0	0.977724
$868$	0	0
$869$	− 7365.77i	− 0.287534i
$870$	0	0
$871$	9240.00	0.359455
$872$	0	0
$873$	− 994.000i	− 0.0385359i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	137.579	0.00529728	0.00264864	−	0.999996i	$-0.499157\pi$
$877$	137.579	0.00529728	0.00264864	+	0.999996i	$0.499157\pi$
$878$	0	0
$879$	48888.0	1.87594
$880$	0	0
$881$	−31150.0	−1.19123	−0.595613	−	0.803272i	$-0.703091\pi$
$881$	−31150.0	−1.19123	−0.595613	+	0.803272i	$0.703091\pi$
$882$	0	0
$883$	12577.9	0.479366	0.239683	−	0.970851i	$-0.422957\pi$
$883$	12577.9	0.479366	0.239683	+	0.970851i	$0.422957\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37128.0i	1.40545i	0.711460	+	0.702726i	$0.248034\pi$
$887$	37128.0i	1.40545i	−0.711460	+	0.702726i	$0.751966\pi$
$888$	0	0
$889$	−13312.0	−0.502216
$890$	0	0
$891$	− 11985.3i	− 0.450641i
$892$	0	0
$893$	12445.6	0.466379
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 42560.0i	− 1.58421i
$898$	0	0
$899$	− 35558.9i	− 1.31919i
$900$	0	0
$901$	− 444.486i	− 0.0164351i
$902$	0	0
$903$	18592.0i	0.685164i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−35204.4	−1.28880	−0.644400	−	0.764688i	$-0.722893\pi$
$907$	−35204.4	−1.28880	−0.644400	+	0.764688i	$0.722893\pi$
$908$	0	0
$909$	751.393i	0.0274171i
$910$	0	0
$911$	10512.0	0.382303	0.191152	−	0.981561i	$-0.438778\pi$
$911$	10512.0	0.382303	0.191152	+	0.981561i	$0.438778\pi$
$912$	0	0
$913$	− 8652.00i	− 0.313625i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5376.17	0.193606
$918$	0	0
$919$	−46104.0	−1.65488	−0.827438	−	0.561557i	$-0.810202\pi$
$919$	−46104.0	−1.65488	−0.827438	+	0.561557i	$0.810202\pi$
$920$	0	0
$921$	−13692.0	−0.489866
$922$	0	0
$923$	−3809.88	−0.135865
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 1176.00i	− 0.0416666i
$928$	0	0
$929$	5726.00	0.202222	0.101111	−	0.994875i	$-0.467760\pi$
$929$	5726.00	0.202222	0.101111	+	0.994875i	$0.467760\pi$
$930$	0	0
$931$	− 10334.3i	− 0.363795i
$932$	0	0
$933$	14519.9	0.509496
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1274.00i	− 0.0444181i	−0.999753	−	0.0222091i	$-0.992930\pi$
$937$	− 1274.00i	− 0.0444181i	0.999753	−	0.0222091i	$-0.00706994\pi$
$938$	0	0
$939$	12075.2i	0.419659i
$940$	0	0
$941$	− 26446.9i	− 0.916201i	−0.888900	−	0.458101i	$-0.848530\pi$
$941$	− 26446.9i	− 0.916201i	0.888900	−	0.458101i	$-0.151470\pi$
$942$	0	0
$943$	− 10640.0i	− 0.367430i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23922.9	−0.820897	−0.410448	−	0.911884i	$-0.634628\pi$
$947$	−23922.9	−0.820897	−0.410448	+	0.911884i	$0.634628\pi$
$948$	0	0
$949$	15557.0i	0.532141i
$950$	0	0
$951$	50680.0	1.72809
$952$	0	0
$953$	38250.0i	1.30015i	0.759872	+	0.650073i	$0.225262\pi$
$953$	38250.0i	1.30015i	−0.759872	+	0.650073i	$0.774738\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−13334.6	−0.450414
$958$	0	0
$959$	8496.00	0.286079
$960$	0	0
$961$	20385.0	0.684267
$962$	0	0
$963$	269.867	0.00903046
$964$	0	0
$965$	0	0
$966$	0	0
$967$	4664.00i	0.155103i	0.996988	+	0.0775513i	$0.0247101\pi$
$967$	4664.00i	0.155103i	−0.996988	+	0.0775513i	$0.975290\pi$
$968$	0	0
$969$	2744.00	0.0909701
$970$	0	0
$971$	− 30971.2i	− 1.02360i	−0.859106	−	0.511798i	$-0.828980\pi$
$971$	− 30971.2i	− 1.02360i	0.859106	−	0.511798i	$-0.171020\pi$
$972$	0	0
$973$	21547.0	0.709933
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4814.00i	0.157639i	0.996889	+	0.0788196i	$0.0251151\pi$
$977$	4814.00i	0.157639i	−0.996889	+	0.0788196i	$0.974885\pi$
$978$	0	0
$979$	− 4222.62i	− 0.137850i
$980$	0	0
$981$	1894.36i	0.0616536i
$982$	0	0
$983$	12376.0i	0.401560i	0.979636	+	0.200780i	$0.0643476\pi$
$983$	12376.0i	0.401560i	−0.979636	+	0.200780i	$0.935652\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	14223.6	0.458704
$988$	0	0
$989$	− 66757.6i	− 2.14638i
$990$	0	0
$991$	−45344.0	−1.45348	−0.726740	−	0.686912i	$-0.758966\pi$
$991$	−45344.0	−1.45348	−0.726740	+	0.686912i	$0.758966\pi$
$992$	0	0
$993$	22484.0i	0.718538i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26002.4	−0.825984	−0.412992	−	0.910735i	$-0.635516\pi$
$997$	−26002.4	−0.825984	−0.412992	+	0.910735i	$0.635516\pi$
$998$	0	0
$999$	−33488.0	−1.06057

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.f.a.49.3		4
4.3	odd	2		200.4.f.a.149.2		4
5.2	odd	4		32.4.b.a.17.1		2
5.3	odd	4		800.4.d.a.401.2		2
5.4	even	2	inner	800.4.f.a.49.2		4
8.3	odd	2		200.4.f.a.149.4		4
8.5	even	2	inner	800.4.f.a.49.1		4
15.2	even	4		288.4.d.a.145.2		2
20.3	even	4		200.4.d.a.101.2		2
20.7	even	4		8.4.b.a.5.1	✓	2
20.19	odd	2		200.4.f.a.149.3		4
40.3	even	4		200.4.d.a.101.1		2
40.13	odd	4		800.4.d.a.401.1		2
40.19	odd	2		200.4.f.a.149.1		4
40.27	even	4		8.4.b.a.5.2	yes	2
40.29	even	2	inner	800.4.f.a.49.4		4
40.37	odd	4		32.4.b.a.17.2		2
60.47	odd	4		72.4.d.b.37.2		2
80.27	even	4		256.4.a.l.1.2		2
80.37	odd	4		256.4.a.j.1.1		2
80.67	even	4		256.4.a.l.1.1		2
80.77	odd	4		256.4.a.j.1.2		2
120.77	even	4		288.4.d.a.145.1		2
120.107	odd	4		72.4.d.b.37.1		2
240.77	even	4		2304.4.a.v.1.2		2
240.107	odd	4		2304.4.a.bn.1.1		2
240.197	even	4		2304.4.a.v.1.1		2
240.227	odd	4		2304.4.a.bn.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.b.a.5.1	✓	2	20.7	even	4
8.4.b.a.5.2	yes	2	40.27	even	4
32.4.b.a.17.1		2	5.2	odd	4
32.4.b.a.17.2		2	40.37	odd	4
72.4.d.b.37.1		2	120.107	odd	4
72.4.d.b.37.2		2	60.47	odd	4
200.4.d.a.101.1		2	40.3	even	4
200.4.d.a.101.2		2	20.3	even	4
200.4.f.a.149.1		4	40.19	odd	2
200.4.f.a.149.2		4	4.3	odd	2
200.4.f.a.149.3		4	20.19	odd	2
200.4.f.a.149.4		4	8.3	odd	2
256.4.a.j.1.1		2	80.37	odd	4
256.4.a.j.1.2		2	80.77	odd	4
256.4.a.l.1.1		2	80.67	even	4
256.4.a.l.1.2		2	80.27	even	4
288.4.d.a.145.1		2	120.77	even	4
288.4.d.a.145.2		2	15.2	even	4
800.4.d.a.401.1		2	40.13	odd	4
800.4.d.a.401.2		2	5.3	odd	4
800.4.f.a.49.1		4	8.5	even	2	inner
800.4.f.a.49.2		4	5.4	even	2	inner
800.4.f.a.49.3		4	1.1	even	1	trivial
800.4.f.a.49.4		4	40.29	even	2	inner
2304.4.a.v.1.1		2	240.197	even	4
2304.4.a.v.1.2		2	240.77	even	4
2304.4.a.bn.1.1		2	240.107	odd	4
2304.4.a.bn.1.2		2	240.227	odd	4

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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+5.29150 q^{3} -8.00000i q^{7} +1.00000 q^{9} +15.8745i q^{11} -52.9150 q^{13} +14.0000i q^{17} -37.0405i q^{19} -42.3320i q^{21} +152.000i q^{23} -137.579 q^{27} +158.745i q^{29} -224.000 q^{31} +84.0000i q^{33} +243.409 q^{37} -280.000 q^{39} -70.0000 q^{41} -439.195 q^{43} +336.000i q^{47} +279.000 q^{49} +74.0810i q^{51} -31.7490 q^{53} -196.000i q^{57} +534.442i q^{59} +95.2470i q^{61} -8.00000i q^{63} -174.620 q^{67} +804.308i q^{69} +72.0000 q^{71} -294.000i q^{73} +126.996 q^{77} -464.000 q^{79} -755.000 q^{81} -545.025 q^{83} +840.000i q^{87} -266.000 q^{89} +423.320i q^{91} -1185.30 q^{93} -994.000i q^{97} +15.8745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{9} - 896 q^{31} - 1120 q^{39} - 280 q^{41} + 1116 q^{49} + 288 q^{71} - 1856 q^{79} - 3020 q^{81} - 1064 q^{89}+O(q^{100}) \) 4 * q + 4 * q^9 - 896 * q^31 - 1120 * q^39 - 280 * q^41 + 1116 * q^49 + 288 * q^71 - 1856 * q^79 - 3020 * q^81 - 1064 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more