LMFDB - Embedded newform 800.4.a.u.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,4,Mod(1,800)] 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("800.1"); 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, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,-5,0,0,0,2,0,18,0,-31,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$47.2015280046$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.5685.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 11x + 6 $ x^3 - x^2 - 11*x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.12873$ of defining polynomial
Character	$\chi$	$=$	800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.25747 q^{3} +9.15590 q^{7} +0.640965 q^{9} -11.8984 q^{11} -41.9826 q^{13} -75.7604 q^{17} +2.49047 q^{19} +48.1368 q^{21} -17.8566 q^{23} -138.582 q^{27} -143.538 q^{29} -88.6094 q^{31} -62.5556 q^{33} -351.059 q^{37} -220.722 q^{39} +195.922 q^{41} +366.067 q^{43} +58.5931 q^{47} -259.169 q^{49} -398.308 q^{51} +0.374917 q^{53} +13.0936 q^{57} +318.952 q^{59} +446.597 q^{61} +5.86861 q^{63} -709.343 q^{67} -93.8805 q^{69} -1137.43 q^{71} +85.3855 q^{73} -108.941 q^{77} -1249.83 q^{79} -745.895 q^{81} +926.261 q^{83} -754.648 q^{87} +973.552 q^{89} -384.389 q^{91} -465.861 q^{93} +1158.55 q^{97} -7.62648 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 5 q^{3} + 2 q^{7} + 18 q^{9} - 31 q^{11} - 8 q^{13} + 89 q^{17} - 87 q^{19} - 70 q^{21} + 122 q^{23} - 143 q^{27} + 84 q^{29} - 294 q^{31} + 209 q^{33} + 94 q^{37} + 32 q^{39} - 345 q^{41} + 412 q^{43}+ \cdots - 1954 q^{99}+O(q^{100}) $ 3 * q - 5 * q^3 + 2 * q^7 + 18 * q^9 - 31 * q^11 - 8 * q^13 + 89 * q^17 - 87 * q^19 - 70 * q^21 + 122 * q^23 - 143 * q^27 + 84 * q^29 - 294 * q^31 + 209 * q^33 + 94 * q^37 + 32 * q^39 - 345 * q^41 + 412 * q^43 - 824 * q^47 + 359 * q^49 - 1351 * q^51 - 74 * q^53 + 913 * q^57 - 1040 * q^59 + 482 * q^61 + 1532 * q^63 - 735 * q^67 - 614 * q^69 - 2048 * q^71 + 15 * q^73 - 1474 * q^77 - 998 * q^79 - 549 * q^81 + 1221 * q^83 - 2564 * q^87 + 1189 * q^89 - 4032 * q^91 - 454 * q^93 + 1010 * q^97 - 1954 * q^99

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
$3$	5.25747	1.01180	0.505900	−	0.862592i	$-0.331160\pi$
$3$	5.25747	1.01180	0.505900	+	0.862592i	$0.331160\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	9.15590	0.494372	0.247186	−	0.968968i	$-0.420494\pi$
$7$	9.15590	0.494372	0.247186	+	0.968968i	$0.420494\pi$
$8$	0	0
$9$	0.640965	0.0237394
$10$	0	0
$11$	−11.8984	−0.326137	−0.163069	−	0.986615i	$-0.552139\pi$
$11$	−11.8984	−0.326137	−0.163069	+	0.986615i	$0.552139\pi$
$12$	0	0
$13$	−41.9826	−0.895684	−0.447842	−	0.894113i	$-0.647807\pi$
$13$	−41.9826	−0.895684	−0.447842	+	0.894113i	$0.647807\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−75.7604	−1.08086	−0.540430	−	0.841389i	$-0.681738\pi$
$17$	−75.7604	−1.08086	−0.540430	+	0.841389i	$0.681738\pi$
$18$	0	0
$19$	2.49047	0.0300712	0.0150356	−	0.999887i	$-0.495214\pi$
$19$	2.49047	0.0300712	0.0150356	+	0.999887i	$0.495214\pi$
$20$	0	0
$21$	48.1368	0.500206
$22$	0	0
$23$	−17.8566	−0.161885	−0.0809426	−	0.996719i	$-0.525793\pi$
$23$	−17.8566	−0.161885	−0.0809426	+	0.996719i	$0.525793\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−138.582	−0.987781
$28$	0	0
$29$	−143.538	−0.919117	−0.459558	−	0.888148i	$-0.651992\pi$
$29$	−143.538	−0.919117	−0.459558	+	0.888148i	$0.651992\pi$
$30$	0	0
$31$	−88.6094	−0.513378	−0.256689	−	0.966494i	$-0.582632\pi$
$31$	−88.6094	−0.513378	−0.256689	+	0.966494i	$0.582632\pi$
$32$	0	0
$33$	−62.5556	−0.329986
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−351.059	−1.55983	−0.779916	−	0.625884i	$-0.784738\pi$
$37$	−351.059	−1.55983	−0.779916	+	0.625884i	$0.784738\pi$
$38$	0	0
$39$	−220.722	−0.906253
$40$	0	0
$41$	195.922	0.746291	0.373145	−	0.927773i	$-0.378279\pi$
$41$	195.922	0.746291	0.373145	+	0.927773i	$0.378279\pi$
$42$	0	0
$43$	366.067	1.29825	0.649125	−	0.760682i	$-0.275135\pi$
$43$	366.067	1.29825	0.649125	+	0.760682i	$0.275135\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	58.5931	0.181844	0.0909222	−	0.995858i	$-0.471019\pi$
$47$	58.5931	0.181844	0.0909222	+	0.995858i	$0.471019\pi$
$48$	0	0
$49$	−259.169	−0.755596
$50$	0	0
$51$	−398.308	−1.09361
$52$	0	0
$53$	0.374917	0.000971675 0	0.000485837	−	1.00000i	$-0.499845\pi$
$53$	0.374917	0.000971675 0	0.000485837		1.00000i	$0.499845\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.0936	0.0304261
$58$	0	0
$59$	318.952	0.703796	0.351898	−	0.936038i	$-0.385536\pi$
$59$	318.952	0.703796	0.351898	+	0.936038i	$0.385536\pi$
$60$	0	0
$61$	446.597	0.937391	0.468695	−	0.883360i	$-0.344724\pi$
$61$	446.597	0.937391	0.468695	+	0.883360i	$0.344724\pi$
$62$	0	0
$63$	5.86861	0.0117361
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−709.343	−1.29343	−0.646717	−	0.762730i	$-0.723858\pi$
$67$	−709.343	−1.29343	−0.646717	+	0.762730i	$0.723858\pi$
$68$	0	0
$69$	−93.8805	−0.163795
$70$	0	0
$71$	−1137.43	−1.90124	−0.950621	−	0.310354i	$-0.899552\pi$
$71$	−1137.43	−1.90124	−0.950621	+	0.310354i	$0.899552\pi$
$72$	0	0
$73$	85.3855	0.136899	0.0684495	−	0.997655i	$-0.478195\pi$
$73$	85.3855	0.136899	0.0684495	+	0.997655i	$0.478195\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−108.941	−0.161233
$78$	0	0
$79$	−1249.83	−1.77996	−0.889981	−	0.455997i	$-0.849283\pi$
$79$	−1249.83	−1.77996	−0.889981	+	0.455997i	$0.849283\pi$
$80$	0	0
$81$	−745.895	−1.02318
$82$	0	0
$83$	926.261	1.22494	0.612472	−	0.790492i	$-0.290175\pi$
$83$	926.261	1.22494	0.612472	+	0.790492i	$0.290175\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−754.648	−0.929962
$88$	0	0
$89$	973.552	1.15951	0.579755	−	0.814791i	$-0.303148\pi$
$89$	973.552	1.15951	0.579755	+	0.814791i	$0.303148\pi$
$90$	0	0
$91$	−384.389	−0.442801
$92$	0	0
$93$	−465.861	−0.519436
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1158.55	1.21271	0.606355	−	0.795194i	$-0.292631\pi$
$97$	1158.55	1.21271	0.606355	+	0.795194i	$0.292631\pi$
$98$	0	0
$99$	−7.62648	−0.00774232
$100$	0	0
$101$	−466.816	−0.459900	−0.229950	−	0.973202i	$-0.573856\pi$
$101$	−466.816	−0.459900	−0.229950	+	0.973202i	$0.573856\pi$
$102$	0	0
$103$	963.139	0.921368	0.460684	−	0.887564i	$-0.347604\pi$
$103$	963.139	0.921368	0.460684	+	0.887564i	$0.347604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1066.94	−0.963971	−0.481986	−	0.876179i	$-0.660084\pi$
$107$	−1066.94	−0.963971	−0.481986	+	0.876179i	$0.660084\pi$
$108$	0	0
$109$	−217.521	−0.191145	−0.0955723	−	0.995422i	$-0.530468\pi$
$109$	−217.521	−0.191145	−0.0955723	+	0.995422i	$0.530468\pi$
$110$	0	0
$111$	−1845.68	−1.57824
$112$	0	0
$113$	1160.71	0.966286	0.483143	−	0.875542i	$-0.339495\pi$
$113$	1160.71	0.966286	0.483143	+	0.875542i	$0.339495\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−26.9094	−0.0212630
$118$	0	0
$119$	−693.655	−0.534347
$120$	0	0
$121$	−1189.43	−0.893634
$122$	0	0
$123$	1030.06	0.755097
$124$	0	0
$125$	0	0
$126$	0	0
$127$	34.2804	0.0239519	0.0119759	−	0.999928i	$-0.496188\pi$
$127$	34.2804	0.0239519	0.0119759	+	0.999928i	$0.496188\pi$
$128$	0	0
$129$	1924.59	1.31357
$130$	0	0
$131$	−2364.02	−1.57669	−0.788343	−	0.615236i	$-0.789061\pi$
$131$	−2364.02	−1.57669	−0.788343	+	0.615236i	$0.789061\pi$
$132$	0	0
$133$	22.8025	0.0148664
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2991.88	1.86579	0.932897	−	0.360144i	$-0.117272\pi$
$137$	2991.88	1.86579	0.932897	+	0.360144i	$0.117272\pi$
$138$	0	0
$139$	−132.246	−0.0806974	−0.0403487	−	0.999186i	$-0.512847\pi$
$139$	−132.246	−0.0806974	−0.0403487	+	0.999186i	$0.512847\pi$
$140$	0	0
$141$	308.052	0.183990
$142$	0	0
$143$	499.528	0.292116
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1362.58	−0.764512
$148$	0	0
$149$	−788.190	−0.433363	−0.216681	−	0.976242i	$-0.569523\pi$
$149$	−788.190	−0.433363	−0.216681	+	0.976242i	$0.569523\pi$
$150$	0	0
$151$	−2200.74	−1.18605	−0.593025	−	0.805184i	$-0.702066\pi$
$151$	−2200.74	−1.18605	−0.593025	+	0.805184i	$0.702066\pi$
$152$	0	0
$153$	−48.5598	−0.0256590
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−1702.05	−0.865212	−0.432606	−	0.901583i	$-0.642406\pi$
$157$	−1702.05	−0.865212	−0.432606	+	0.901583i	$0.642406\pi$
$158$	0	0
$159$	1.97111	0.000983140 0
$160$	0	0
$161$	−163.493	−0.0800315
$162$	0	0
$163$	822.009	0.394998	0.197499	−	0.980303i	$-0.436718\pi$
$163$	822.009	0.394998	0.197499	+	0.980303i	$0.436718\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2545.32	−1.17942	−0.589710	−	0.807615i	$-0.700758\pi$
$167$	−2545.32	−1.17942	−0.589710	+	0.807615i	$0.700758\pi$
$168$	0	0
$169$	−434.458	−0.197751
$170$	0	0
$171$	1.59631	0.000713874 0
$172$	0	0
$173$	868.181	0.381541	0.190770	−	0.981635i	$-0.438901\pi$
$173$	868.181	0.381541	0.190770	+	0.981635i	$0.438901\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1676.88	0.712101
$178$	0	0
$179$	−3255.65	−1.35944	−0.679718	−	0.733474i	$-0.737898\pi$
$179$	−3255.65	−1.35944	−0.679718	+	0.733474i	$0.737898\pi$
$180$	0	0
$181$	3482.49	1.43012	0.715059	−	0.699065i	$-0.246400\pi$
$181$	3482.49	1.43012	0.715059	+	0.699065i	$0.246400\pi$
$182$	0	0
$183$	2347.97	0.948452
$184$	0	0
$185$	0	0
$186$	0	0
$187$	901.431	0.352509
$188$	0	0
$189$	−1268.84	−0.488331
$190$	0	0
$191$	2981.37	1.12945	0.564724	−	0.825280i	$-0.308983\pi$
$191$	2981.37	1.12945	0.564724	+	0.825280i	$0.308983\pi$
$192$	0	0
$193$	1666.53	0.621552	0.310776	−	0.950483i	$-0.399411\pi$
$193$	1666.53	0.621552	0.310776	+	0.950483i	$0.399411\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4355.01	−1.57503	−0.787516	−	0.616294i	$-0.788633\pi$
$197$	−4355.01	−1.57503	−0.787516	+	0.616294i	$0.788633\pi$
$198$	0	0
$199$	820.282	0.292202	0.146101	−	0.989270i	$-0.453328\pi$
$199$	820.282	0.292202	0.146101	+	0.989270i	$0.453328\pi$
$200$	0	0
$201$	−3729.35	−1.30870
$202$	0	0
$203$	−1314.22	−0.454386
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11.4455	−0.00384306
$208$	0	0
$209$	−29.6327	−0.00980736
$210$	0	0
$211$	−4678.00	−1.52629	−0.763144	−	0.646229i	$-0.776345\pi$
$211$	−4678.00	−1.52629	−0.763144	+	0.646229i	$0.776345\pi$
$212$	0	0
$213$	−5980.00	−1.92368
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−811.299	−0.253800
$218$	0	0
$219$	448.912	0.138514
$220$	0	0
$221$	3180.62	0.968108
$222$	0	0
$223$	−2238.14	−0.672094	−0.336047	−	0.941845i	$-0.609090\pi$
$223$	−2238.14	−0.672094	−0.336047	+	0.941845i	$0.609090\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1036.89	0.303175	0.151588	−	0.988444i	$-0.451561\pi$
$227$	1036.89	0.303175	0.151588	+	0.988444i	$0.451561\pi$
$228$	0	0
$229$	−3888.44	−1.12208	−0.561038	−	0.827790i	$-0.689598\pi$
$229$	−3888.44	−1.12208	−0.561038	+	0.827790i	$0.689598\pi$
$230$	0	0
$231$	−572.753	−0.163136
$232$	0	0
$233$	130.042	0.0365638	0.0182819	−	0.999833i	$-0.494180\pi$
$233$	130.042	0.0365638	0.0182819	+	0.999833i	$0.494180\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6570.95	−1.80097
$238$	0	0
$239$	5724.03	1.54919	0.774595	−	0.632458i	$-0.217954\pi$
$239$	5724.03	1.54919	0.774595	+	0.632458i	$0.217954\pi$
$240$	0	0
$241$	1818.75	0.486123	0.243062	−	0.970011i	$-0.421848\pi$
$241$	1818.75	0.486123	0.243062	+	0.970011i	$0.421848\pi$
$242$	0	0
$243$	−179.812	−0.0474689
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−104.557	−0.0269343
$248$	0	0
$249$	4869.79	1.23940
$250$	0	0
$251$	3637.78	0.914800	0.457400	−	0.889261i	$-0.348781\pi$
$251$	3637.78	0.914800	0.457400	+	0.889261i	$0.348781\pi$
$252$	0	0
$253$	212.466	0.0527968
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2969.73	0.720804	0.360402	−	0.932797i	$-0.382640\pi$
$257$	2969.73	0.720804	0.360402	+	0.932797i	$0.382640\pi$
$258$	0	0
$259$	−3214.26	−0.771137
$260$	0	0
$261$	−92.0030	−0.0218193
$262$	0	0
$263$	3035.72	0.711752	0.355876	−	0.934533i	$-0.384183\pi$
$263$	3035.72	0.711752	0.355876	+	0.934533i	$0.384183\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	5118.42	1.17319
$268$	0	0
$269$	−6888.55	−1.56135	−0.780673	−	0.624939i	$-0.785124\pi$
$269$	−6888.55	−1.56135	−0.780673	+	0.624939i	$0.785124\pi$
$270$	0	0
$271$	6173.52	1.38382	0.691909	−	0.721985i	$-0.256770\pi$
$271$	6173.52	1.38382	0.691909	+	0.721985i	$0.256770\pi$
$272$	0	0
$273$	−2020.91	−0.448026
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−3273.30	−0.710013	−0.355007	−	0.934864i	$-0.615521\pi$
$277$	−3273.30	−0.710013	−0.355007	+	0.934864i	$0.615521\pi$
$278$	0	0
$279$	−56.7955	−0.0121873
$280$	0	0
$281$	458.400	0.0973161	0.0486581	−	0.998815i	$-0.484506\pi$
$281$	458.400	0.0973161	0.0486581	+	0.998815i	$0.484506\pi$
$282$	0	0
$283$	−8064.66	−1.69397	−0.846987	−	0.531614i	$-0.821586\pi$
$283$	−8064.66	−1.69397	−0.846987	+	0.531614i	$0.821586\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1793.84	0.368945
$288$	0	0
$289$	826.645	0.168257
$290$	0	0
$291$	6091.03	1.22702
$292$	0	0
$293$	−812.863	−0.162075	−0.0810375	−	0.996711i	$-0.525823\pi$
$293$	−812.863	−0.162075	−0.0810375	+	0.996711i	$0.525823\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1648.91	0.322152
$298$	0	0
$299$	749.667	0.144998
$300$	0	0
$301$	3351.68	0.641819
$302$	0	0
$303$	−2454.27	−0.465327
$304$	0	0
$305$	0	0
$306$	0	0
$307$	383.469	0.0712891	0.0356445	−	0.999365i	$-0.488652\pi$
$307$	383.469	0.0712891	0.0356445	+	0.999365i	$0.488652\pi$
$308$	0	0
$309$	5063.67	0.932240
$310$	0	0
$311$	1665.18	0.303613	0.151807	−	0.988410i	$-0.451491\pi$
$311$	1665.18	0.303613	0.151807	+	0.988410i	$0.451491\pi$
$312$	0	0
$313$	3769.02	0.680632	0.340316	−	0.940311i	$-0.389466\pi$
$313$	3769.02	0.680632	0.340316	+	0.940311i	$0.389466\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1859.70	0.329498	0.164749	−	0.986336i	$-0.447319\pi$
$317$	1859.70	0.329498	0.164749	+	0.986336i	$0.447319\pi$
$318$	0	0
$319$	1707.88	0.299758
$320$	0	0
$321$	−5609.40	−0.975346
$322$	0	0
$323$	−188.679	−0.0325028
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1143.61	−0.193400
$328$	0	0
$329$	536.473	0.0898988
$330$	0	0
$331$	7235.06	1.20144	0.600718	−	0.799461i	$-0.294882\pi$
$331$	7235.06	1.20144	0.600718	+	0.799461i	$0.294882\pi$
$332$	0	0
$333$	−225.017	−0.0370295
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9890.08	1.59866	0.799328	−	0.600895i	$-0.205189\pi$
$337$	9890.08	1.59866	0.799328	+	0.600895i	$0.205189\pi$
$338$	0	0
$339$	6102.39	0.977688
$340$	0	0
$341$	1054.31	0.167432
$342$	0	0
$343$	−5513.40	−0.867918
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6269.88	0.969984	0.484992	−	0.874519i	$-0.338822\pi$
$347$	6269.88	0.969984	0.484992	+	0.874519i	$0.338822\pi$
$348$	0	0
$349$	−2436.94	−0.373772	−0.186886	−	0.982382i	$-0.559840\pi$
$349$	−2436.94	−0.373772	−0.186886	+	0.982382i	$0.559840\pi$
$350$	0	0
$351$	5818.03	0.884739
$352$	0	0
$353$	10712.4	1.61519	0.807595	−	0.589738i	$-0.200769\pi$
$353$	10712.4	1.61519	0.807595	+	0.589738i	$0.200769\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−3646.87	−0.540652
$358$	0	0
$359$	3540.39	0.520487	0.260243	−	0.965543i	$-0.416197\pi$
$359$	3540.39	0.520487	0.260243	+	0.965543i	$0.416197\pi$
$360$	0	0
$361$	−6852.80	−0.999096
$362$	0	0
$363$	−6253.38	−0.904179
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12208.3	1.73642	0.868212	−	0.496193i	$-0.165269\pi$
$367$	12208.3	1.73642	0.868212	+	0.496193i	$0.165269\pi$
$368$	0	0
$369$	125.579	0.0177165
$370$	0	0
$371$	3.43270	0.000480369 0
$372$	0	0
$373$	3056.65	0.424309	0.212154	−	0.977236i	$-0.431952\pi$
$373$	3056.65	0.424309	0.212154	+	0.977236i	$0.431952\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6026.11	0.823238
$378$	0	0
$379$	1476.91	0.200168	0.100084	−	0.994979i	$-0.468089\pi$
$379$	1476.91	0.200168	0.100084	+	0.994979i	$0.468089\pi$
$380$	0	0
$381$	180.228	0.0242345
$382$	0	0
$383$	−5567.95	−0.742843	−0.371422	−	0.928464i	$-0.621130\pi$
$383$	−5567.95	−0.742843	−0.371422	+	0.928464i	$0.621130\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	234.636	0.0308197
$388$	0	0
$389$	5401.03	0.703966	0.351983	−	0.936006i	$-0.385507\pi$
$389$	5401.03	0.703966	0.351983	+	0.936006i	$0.385507\pi$
$390$	0	0
$391$	1352.82	0.174975
$392$	0	0
$393$	−12428.8	−1.59529
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8592.52	1.08626	0.543131	−	0.839648i	$-0.317239\pi$
$397$	8592.52	1.08626	0.543131	+	0.839648i	$0.317239\pi$
$398$	0	0
$399$	119.884	0.0150418
$400$	0	0
$401$	−1794.62	−0.223489	−0.111745	−	0.993737i	$-0.535644\pi$
$401$	−1794.62	−0.223489	−0.111745	+	0.993737i	$0.535644\pi$
$402$	0	0
$403$	3720.06	0.459824
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4177.05	0.508720
$408$	0	0
$409$	11342.3	1.37125	0.685623	−	0.727957i	$-0.259530\pi$
$409$	11342.3	1.37125	0.685623	+	0.727957i	$0.259530\pi$
$410$	0	0
$411$	15729.7	1.88781
$412$	0	0
$413$	2920.29	0.347937
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−695.278	−0.0816496
$418$	0	0
$419$	8568.54	0.999047	0.499524	−	0.866300i	$-0.333508\pi$
$419$	8568.54	0.999047	0.499524	+	0.866300i	$0.333508\pi$
$420$	0	0
$421$	−7705.53	−0.892030	−0.446015	−	0.895026i	$-0.647157\pi$
$421$	−7705.53	−0.892030	−0.446015	+	0.895026i	$0.647157\pi$
$422$	0	0
$423$	37.5561	0.00431688
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4088.99	0.463420
$428$	0	0
$429$	2626.25	0.295563
$430$	0	0
$431$	12316.2	1.37645	0.688226	−	0.725496i	$-0.258390\pi$
$431$	12316.2	1.37645	0.688226	+	0.725496i	$0.258390\pi$
$432$	0	0
$433$	7105.69	0.788632	0.394316	−	0.918975i	$-0.370981\pi$
$433$	7105.69	0.788632	0.394316	+	0.918975i	$0.370981\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−44.4714	−0.00486809
$438$	0	0
$439$	−17585.8	−1.91190	−0.955951	−	0.293527i	$-0.905171\pi$
$439$	−17585.8	−1.91190	−0.955951	+	0.293527i	$0.905171\pi$
$440$	0	0
$441$	−166.119	−0.0179374
$442$	0	0
$443$	2154.24	0.231041	0.115520	−	0.993305i	$-0.463146\pi$
$443$	2154.24	0.231041	0.115520	+	0.993305i	$0.463146\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−4143.88	−0.438476
$448$	0	0
$449$	−6853.94	−0.720395	−0.360197	−	0.932876i	$-0.617291\pi$
$449$	−6853.94	−0.720395	−0.360197	+	0.932876i	$0.617291\pi$
$450$	0	0
$451$	−2331.17	−0.243393
$452$	0	0
$453$	−11570.3	−1.20005
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−1822.27	−0.186526	−0.0932629	−	0.995642i	$-0.529730\pi$
$457$	−1822.27	−0.186526	−0.0932629	+	0.995642i	$0.529730\pi$
$458$	0	0
$459$	10499.0	1.06765
$460$	0	0
$461$	3510.72	0.354687	0.177343	−	0.984149i	$-0.443250\pi$
$461$	3510.72	0.354687	0.177343	+	0.984149i	$0.443250\pi$
$462$	0	0
$463$	−13454.1	−1.35047	−0.675235	−	0.737603i	$-0.735958\pi$
$463$	−13454.1	−1.35047	−0.675235	+	0.737603i	$0.735958\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−12549.1	−1.24347	−0.621736	−	0.783227i	$-0.713572\pi$
$467$	−12549.1	−1.24347	−0.621736	+	0.783227i	$0.713572\pi$
$468$	0	0
$469$	−6494.67	−0.639438
$470$	0	0
$471$	−8948.46	−0.875421
$472$	0	0
$473$	−4355.63	−0.423408
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.240308	2.30670e−5 0
$478$	0	0
$479$	6582.21	0.627868	0.313934	−	0.949445i	$-0.398353\pi$
$479$	6582.21	0.627868	0.313934	+	0.949445i	$0.398353\pi$
$480$	0	0
$481$	14738.4	1.39712
$482$	0	0
$483$	−859.561	−0.0809759
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10288.5	0.957326	0.478663	−	0.877999i	$-0.341122\pi$
$487$	10288.5	0.957326	0.478663	+	0.877999i	$0.341122\pi$
$488$	0	0
$489$	4321.68	0.399659
$490$	0	0
$491$	−3423.20	−0.314637	−0.157319	−	0.987548i	$-0.550285\pi$
$491$	−3423.20	−0.314637	−0.157319	+	0.987548i	$0.550285\pi$
$492$	0	0
$493$	10874.5	0.993436
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−10414.2	−0.939921
$498$	0	0
$499$	−8210.85	−0.736609	−0.368305	−	0.929705i	$-0.620062\pi$
$499$	−8210.85	−0.736609	−0.368305	+	0.929705i	$0.620062\pi$
$500$	0	0
$501$	−13382.0	−1.19334
$502$	0	0
$503$	−7377.71	−0.653988	−0.326994	−	0.945026i	$-0.606036\pi$
$503$	−7377.71	−0.653988	−0.326994	+	0.945026i	$0.606036\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2284.15	−0.200084
$508$	0	0
$509$	−456.652	−0.0397657	−0.0198829	−	0.999802i	$-0.506329\pi$
$509$	−456.652	−0.0397657	−0.0198829	+	0.999802i	$0.506329\pi$
$510$	0	0
$511$	781.781	0.0676790
$512$	0	0
$513$	−345.134	−0.0297038
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−697.166	−0.0593063
$518$	0	0
$519$	4564.43	0.386043
$520$	0	0
$521$	3000.32	0.252297	0.126148	−	0.992011i	$-0.459738\pi$
$521$	3000.32	0.252297	0.126148	+	0.992011i	$0.459738\pi$
$522$	0	0
$523$	7731.71	0.646433	0.323216	−	0.946325i	$-0.395236\pi$
$523$	7731.71	0.646433	0.323216	+	0.946325i	$0.395236\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6713.09	0.554889
$528$	0	0
$529$	−11848.1	−0.973793
$530$	0	0
$531$	204.437	0.0167077
$532$	0	0
$533$	−8225.33	−0.668440
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−17116.5	−1.37548
$538$	0	0
$539$	3083.71	0.246428
$540$	0	0
$541$	13614.4	1.08194	0.540971	−	0.841041i	$-0.318057\pi$
$541$	13614.4	1.08194	0.540971	+	0.841041i	$0.318057\pi$
$542$	0	0
$543$	18309.1	1.44699
$544$	0	0
$545$	0	0
$546$	0	0
$547$	3198.43	0.250009	0.125005	−	0.992156i	$-0.460105\pi$
$547$	3198.43	0.250009	0.125005	+	0.992156i	$0.460105\pi$
$548$	0	0
$549$	286.253	0.0222531
$550$	0	0
$551$	−357.478	−0.0276390
$552$	0	0
$553$	−11443.3	−0.879964
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−178.506	−0.0135790	−0.00678952	−	0.999977i	$-0.502161\pi$
$557$	−178.506	−0.0135790	−0.00678952	+	0.999977i	$0.502161\pi$
$558$	0	0
$559$	−15368.5	−1.16282
$560$	0	0
$561$	4739.24	0.356668
$562$	0	0
$563$	10610.5	0.794280	0.397140	−	0.917758i	$-0.370003\pi$
$563$	10610.5	0.794280	0.397140	+	0.917758i	$0.370003\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−6829.34	−0.505830
$568$	0	0
$569$	−21286.8	−1.56834	−0.784171	−	0.620544i	$-0.786912\pi$
$569$	−21286.8	−1.56834	−0.784171	+	0.620544i	$0.786912\pi$
$570$	0	0
$571$	−12064.5	−0.884209	−0.442104	−	0.896964i	$-0.645768\pi$
$571$	−12064.5	−0.884209	−0.442104	+	0.896964i	$0.645768\pi$
$572$	0	0
$573$	15674.5	1.14278
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−8292.57	−0.598309	−0.299155	−	0.954205i	$-0.596705\pi$
$577$	−8292.57	−0.598309	−0.299155	+	0.954205i	$0.596705\pi$
$578$	0	0
$579$	8761.74	0.628887
$580$	0	0
$581$	8480.75	0.605578
$582$	0	0
$583$	−4.46092	−0.000316900 0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15928.8	−1.12002	−0.560011	−	0.828485i	$-0.689203\pi$
$587$	−15928.8	−1.12002	−0.560011	+	0.828485i	$0.689203\pi$
$588$	0	0
$589$	−220.679	−0.0154379
$590$	0	0
$591$	−22896.3	−1.59362
$592$	0	0
$593$	13330.3	0.923116	0.461558	−	0.887110i	$-0.347291\pi$
$593$	13330.3	0.923116	0.461558	+	0.887110i	$0.347291\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4312.61	0.295650
$598$	0	0
$599$	−3311.48	−0.225882	−0.112941	−	0.993602i	$-0.536027\pi$
$599$	−3311.48	−0.225882	−0.112941	+	0.993602i	$0.536027\pi$
$600$	0	0
$601$	−6082.46	−0.412826	−0.206413	−	0.978465i	$-0.566179\pi$
$601$	−6082.46	−0.412826	−0.206413	+	0.978465i	$0.566179\pi$
$602$	0	0
$603$	−454.664	−0.0307054
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−6162.45	−0.412069	−0.206035	−	0.978545i	$-0.566056\pi$
$607$	−6162.45	−0.412069	−0.206035	+	0.978545i	$0.566056\pi$
$608$	0	0
$609$	−6909.48	−0.459747
$610$	0	0
$611$	−2459.89	−0.162875
$612$	0	0
$613$	−8628.65	−0.568529	−0.284264	−	0.958746i	$-0.591749\pi$
$613$	−8628.65	−0.568529	−0.284264	+	0.958746i	$0.591749\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−16309.0	−1.06414	−0.532070	−	0.846700i	$-0.678586\pi$
$617$	−16309.0	−1.06414	−0.532070	+	0.846700i	$0.678586\pi$
$618$	0	0
$619$	−25185.0	−1.63533	−0.817665	−	0.575694i	$-0.804732\pi$
$619$	−25185.0	−1.63533	−0.817665	+	0.575694i	$0.804732\pi$
$620$	0	0
$621$	2474.60	0.159907
$622$	0	0
$623$	8913.75	0.573229
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−155.793	−0.00992309
$628$	0	0
$629$	26596.4	1.68596
$630$	0	0
$631$	−13579.0	−0.856689	−0.428344	−	0.903616i	$-0.640903\pi$
$631$	−13579.0	−0.856689	−0.428344	+	0.903616i	$0.640903\pi$
$632$	0	0
$633$	−24594.4	−1.54430
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10880.6	0.676775
$638$	0	0
$639$	−729.053	−0.0451344
$640$	0	0
$641$	26562.1	1.63672	0.818362	−	0.574703i	$-0.194883\pi$
$641$	26562.1	1.63672	0.818362	+	0.574703i	$0.194883\pi$
$642$	0	0
$643$	−10349.9	−0.634773	−0.317387	−	0.948296i	$-0.602805\pi$
$643$	−10349.9	−0.634773	−0.317387	+	0.948296i	$0.602805\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−30738.0	−1.86775	−0.933875	−	0.357598i	$-0.883596\pi$
$647$	−30738.0	−1.86775	−0.933875	+	0.357598i	$0.883596\pi$
$648$	0	0
$649$	−3795.02	−0.229534
$650$	0	0
$651$	−4265.38	−0.256795
$652$	0	0
$653$	6912.35	0.414244	0.207122	−	0.978315i	$-0.433590\pi$
$653$	6912.35	0.414244	0.207122	+	0.978315i	$0.433590\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	54.7291	0.00324990
$658$	0	0
$659$	−3793.27	−0.224226	−0.112113	−	0.993695i	$-0.535762\pi$
$659$	−3793.27	−0.224226	−0.112113	+	0.993695i	$0.535762\pi$
$660$	0	0
$661$	−30232.2	−1.77897	−0.889483	−	0.456968i	$-0.848935\pi$
$661$	−30232.2	−1.77897	−0.889483	+	0.456968i	$0.848935\pi$
$662$	0	0
$663$	16722.0	0.979532
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2563.11	0.148791
$668$	0	0
$669$	−11767.0	−0.680025
$670$	0	0
$671$	−5313.80	−0.305718
$672$	0	0
$673$	−8389.60	−0.480528	−0.240264	−	0.970708i	$-0.577234\pi$
$673$	−8389.60	−0.480528	−0.240264	+	0.970708i	$0.577234\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8879.70	0.504099	0.252049	−	0.967714i	$-0.418895\pi$
$677$	8879.70	0.504099	0.252049	+	0.967714i	$0.418895\pi$
$678$	0	0
$679$	10607.6	0.599530
$680$	0	0
$681$	5451.42	0.306753
$682$	0	0
$683$	12555.2	0.703384	0.351692	−	0.936116i	$-0.385606\pi$
$683$	12555.2	0.703384	0.351692	+	0.936116i	$0.385606\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−20443.3	−1.13532
$688$	0	0
$689$	−15.7400	−0.000870313 0
$690$	0	0
$691$	−30372.7	−1.67211	−0.836057	−	0.548642i	$-0.815145\pi$
$691$	−30372.7	−1.67211	−0.836057	+	0.548642i	$0.815145\pi$
$692$	0	0
$693$	−69.8272	−0.00382759
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−14843.2	−0.806635
$698$	0	0
$699$	683.694	0.0369953
$700$	0	0
$701$	−33508.8	−1.80543	−0.902716	−	0.430237i	$-0.858430\pi$
$701$	−33508.8	−1.80543	−0.902716	+	0.430237i	$0.858430\pi$
$702$	0	0
$703$	−874.303	−0.0469061
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4274.12	−0.227362
$708$	0	0
$709$	29743.4	1.57551	0.787756	−	0.615987i	$-0.211243\pi$
$709$	29743.4	1.57551	0.787756	+	0.615987i	$0.211243\pi$
$710$	0	0
$711$	−801.098	−0.0422553
$712$	0	0
$713$	1582.26	0.0831083
$714$	0	0
$715$	0	0
$716$	0	0
$717$	30093.9	1.56747
$718$	0	0
$719$	−404.733	−0.0209930	−0.0104965	−	0.999945i	$-0.503341\pi$
$719$	−404.733	−0.0209930	−0.0104965	+	0.999945i	$0.503341\pi$
$720$	0	0
$721$	8818.40	0.455499
$722$	0	0
$723$	9561.99	0.491859
$724$	0	0
$725$	0	0
$726$	0	0
$727$	19096.9	0.974230	0.487115	−	0.873338i	$-0.338049\pi$
$727$	19096.9	0.974230	0.487115	+	0.873338i	$0.338049\pi$
$728$	0	0
$729$	19193.8	0.975147
$730$	0	0
$731$	−27733.4	−1.40323
$732$	0	0
$733$	28856.0	1.45405	0.727026	−	0.686610i	$-0.240902\pi$
$733$	28856.0	1.45405	0.727026	+	0.686610i	$0.240902\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8440.07	0.421837
$738$	0	0
$739$	−36005.6	−1.79227	−0.896134	−	0.443784i	$-0.853636\pi$
$739$	−36005.6	−1.79227	−0.896134	+	0.443784i	$0.853636\pi$
$740$	0	0
$741$	−549.703	−0.0272522
$742$	0	0
$743$	36423.7	1.79846	0.899230	−	0.437475i	$-0.144127\pi$
$743$	36423.7	1.79846	0.899230	+	0.437475i	$0.144127\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	593.701	0.0290795
$748$	0	0
$749$	−9768.79	−0.476560
$750$	0	0
$751$	−25845.4	−1.25581	−0.627905	−	0.778290i	$-0.716088\pi$
$751$	−25845.4	−1.25581	−0.627905	+	0.778290i	$0.716088\pi$
$752$	0	0
$753$	19125.5	0.925594
$754$	0	0
$755$	0	0
$756$	0	0
$757$	17862.6	0.857634	0.428817	−	0.903391i	$-0.358931\pi$
$757$	17862.6	0.857634	0.428817	+	0.903391i	$0.358931\pi$
$758$	0	0
$759$	1117.03	0.0534198
$760$	0	0
$761$	−22733.0	−1.08288	−0.541439	−	0.840740i	$-0.682120\pi$
$761$	−22733.0	−1.08288	−0.541439	+	0.840740i	$0.682120\pi$
$762$	0	0
$763$	−1991.60	−0.0944965
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13390.4	−0.630379
$768$	0	0
$769$	−23153.6	−1.08575	−0.542873	−	0.839815i	$-0.682664\pi$
$769$	−23153.6	−1.08575	−0.542873	+	0.839815i	$0.682664\pi$
$770$	0	0
$771$	15613.3	0.729309
$772$	0	0
$773$	−26456.0	−1.23099	−0.615494	−	0.788141i	$-0.711044\pi$
$773$	−26456.0	−1.23099	−0.615494	+	0.788141i	$0.711044\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−16898.9	−0.780237
$778$	0	0
$779$	487.939	0.0224419
$780$	0	0
$781$	13533.6	0.620066
$782$	0	0
$783$	19891.8	0.907886
$784$	0	0
$785$	0	0
$786$	0	0
$787$	13415.1	0.607619	0.303809	−	0.952733i	$-0.401741\pi$
$787$	13415.1	0.607619	0.303809	+	0.952733i	$0.401741\pi$
$788$	0	0
$789$	15960.2	0.720151
$790$	0	0
$791$	10627.3	0.477705
$792$	0	0
$793$	−18749.3	−0.839605
$794$	0	0
$795$	0	0
$796$	0	0
$797$	31375.6	1.39445	0.697227	−	0.716850i	$-0.254417\pi$
$797$	31375.6	1.39445	0.697227	+	0.716850i	$0.254417\pi$
$798$	0	0
$799$	−4439.04	−0.196548
$800$	0	0
$801$	624.013	0.0275261
$802$	0	0
$803$	−1015.95	−0.0446479
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−36216.3	−1.57977
$808$	0	0
$809$	28335.1	1.23141	0.615705	−	0.787977i	$-0.288871\pi$
$809$	28335.1	1.23141	0.615705	+	0.787977i	$0.288871\pi$
$810$	0	0
$811$	−34538.1	−1.49543	−0.747717	−	0.664018i	$-0.768850\pi$
$811$	−34538.1	−1.49543	−0.747717	+	0.664018i	$0.768850\pi$
$812$	0	0
$813$	32457.1	1.40015
$814$	0	0
$815$	0	0
$816$	0	0
$817$	911.681	0.0390400
$818$	0	0
$819$	−246.380	−0.0105118
$820$	0	0
$821$	−29834.1	−1.26823	−0.634115	−	0.773239i	$-0.718635\pi$
$821$	−29834.1	−1.26823	−0.634115	+	0.773239i	$0.718635\pi$
$822$	0	0
$823$	−36246.8	−1.53522	−0.767608	−	0.640919i	$-0.778553\pi$
$823$	−36246.8	−1.53522	−0.767608	+	0.640919i	$0.778553\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−21831.3	−0.917955	−0.458978	−	0.888448i	$-0.651784\pi$
$827$	−21831.3	−0.917955	−0.458978	+	0.888448i	$0.651784\pi$
$828$	0	0
$829$	12273.0	0.514186	0.257093	−	0.966387i	$-0.417235\pi$
$829$	12273.0	0.514186	0.257093	+	0.966387i	$0.417235\pi$
$830$	0	0
$831$	−17209.3	−0.718391
$832$	0	0
$833$	19634.8	0.816693
$834$	0	0
$835$	0	0
$836$	0	0
$837$	12279.6	0.507105
$838$	0	0
$839$	37441.0	1.54065	0.770326	−	0.637650i	$-0.220094\pi$
$839$	37441.0	1.54065	0.770326	+	0.637650i	$0.220094\pi$
$840$	0	0
$841$	−3785.77	−0.155224
$842$	0	0
$843$	2410.02	0.0984644
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−10890.3	−0.441788
$848$	0	0
$849$	−42399.7	−1.71396
$850$	0	0
$851$	6268.72	0.252514
$852$	0	0
$853$	−1247.81	−0.0500871	−0.0250436	−	0.999686i	$-0.507972\pi$
$853$	−1247.81	−0.0500871	−0.0250436	+	0.999686i	$0.507972\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2927.40	0.116684	0.0583419	−	0.998297i	$-0.481419\pi$
$857$	2927.40	0.116684	0.0583419	+	0.998297i	$0.481419\pi$
$858$	0	0
$859$	−30892.1	−1.22704	−0.613518	−	0.789681i	$-0.710246\pi$
$859$	−30892.1	−1.22704	−0.613518	+	0.789681i	$0.710246\pi$
$860$	0	0
$861$	9431.08	0.373299
$862$	0	0
$863$	5815.96	0.229406	0.114703	−	0.993400i	$-0.463408\pi$
$863$	5815.96	0.229406	0.114703	+	0.993400i	$0.463408\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	4346.06	0.170242
$868$	0	0
$869$	14871.0	0.580513
$870$	0	0
$871$	29780.1	1.15851
$872$	0	0
$873$	742.589	0.0287890
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4732.26	0.182209	0.0911044	−	0.995841i	$-0.470960\pi$
$877$	4732.26	0.182209	0.0911044	+	0.995841i	$0.470960\pi$
$878$	0	0
$879$	−4273.60	−0.163988
$880$	0	0
$881$	43833.1	1.67625	0.838124	−	0.545480i	$-0.183653\pi$
$881$	43833.1	1.67625	0.838124	+	0.545480i	$0.183653\pi$
$882$	0	0
$883$	−37311.7	−1.42201	−0.711007	−	0.703185i	$-0.751761\pi$
$883$	−37311.7	−1.42201	−0.711007	+	0.703185i	$0.751761\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	44940.5	1.70119	0.850595	−	0.525822i	$-0.176242\pi$
$887$	44940.5	1.70119	0.850595	+	0.525822i	$0.176242\pi$
$888$	0	0
$889$	313.867	0.0118411
$890$	0	0
$891$	8874.98	0.333696
$892$	0	0
$893$	145.925	0.00546829
$894$	0	0
$895$	0	0
$896$	0	0
$897$	3941.35	0.146709
$898$	0	0
$899$	12718.8	0.471854
$900$	0	0
$901$	−28.4038	−0.00105024
$902$	0	0
$903$	17621.3	0.649392
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16241.6	−0.594591	−0.297295	−	0.954786i	$-0.596085\pi$
$907$	−16241.6	−0.594591	−0.297295	+	0.954786i	$0.596085\pi$
$908$	0	0
$909$	−299.213	−0.0109178
$910$	0	0
$911$	2088.21	0.0759444	0.0379722	−	0.999279i	$-0.487910\pi$
$911$	2088.21	0.0759444	0.0379722	+	0.999279i	$0.487910\pi$
$912$	0	0
$913$	−11021.1	−0.399500
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−21644.8	−0.779469
$918$	0	0
$919$	−40755.3	−1.46289	−0.731443	−	0.681903i	$-0.761153\pi$
$919$	−40755.3	−1.46289	−0.731443	+	0.681903i	$0.761153\pi$
$920$	0	0
$921$	2016.08	0.0721303
$922$	0	0
$923$	47752.3	1.70291
$924$	0	0
$925$	0	0
$926$	0	0
$927$	617.338	0.0218728
$928$	0	0
$929$	−24239.7	−0.856058	−0.428029	−	0.903765i	$-0.640792\pi$
$929$	−24239.7	−0.856058	−0.428029	+	0.903765i	$0.640792\pi$
$930$	0	0
$931$	−645.455	−0.0227217
$932$	0	0
$933$	8754.64	0.307196
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−37308.1	−1.30075	−0.650374	−	0.759614i	$-0.725388\pi$
$937$	−37308.1	−1.30075	−0.650374	+	0.759614i	$0.725388\pi$
$938$	0	0
$939$	19815.5	0.688663
$940$	0	0
$941$	109.276	0.00378566	0.00189283	−	0.999998i	$-0.499397\pi$
$941$	109.276	0.00378566	0.00189283	+	0.999998i	$0.499397\pi$
$942$	0	0
$943$	−3498.51	−0.120813
$944$	0	0
$945$	0	0
$946$	0	0
$947$	13256.9	0.454902	0.227451	−	0.973790i	$-0.426961\pi$
$947$	13256.9	0.454902	0.227451	+	0.973790i	$0.426961\pi$
$948$	0	0
$949$	−3584.71	−0.122618
$950$	0	0
$951$	9777.29	0.333386
$952$	0	0
$953$	−56535.2	−1.92167	−0.960837	−	0.277112i	$-0.910623\pi$
$953$	−56535.2	−1.92167	−0.960837	+	0.277112i	$0.910623\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8979.12	0.303296
$958$	0	0
$959$	27393.4	0.922396
$960$	0	0
$961$	−21939.4	−0.736443
$962$	0	0
$963$	−683.870	−0.0228841
$964$	0	0
$965$	0	0
$966$	0	0
$967$	32548.6	1.08241	0.541206	−	0.840890i	$-0.317968\pi$
$967$	32548.6	1.08241	0.541206	+	0.840890i	$0.317968\pi$
$968$	0	0
$969$	−991.976	−0.0328863
$970$	0	0
$971$	1604.29	0.0530216	0.0265108	−	0.999649i	$-0.491560\pi$
$971$	1604.29	0.0530216	0.0265108	+	0.999649i	$0.491560\pi$
$972$	0	0
$973$	−1210.83	−0.0398945
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−11269.0	−0.369014	−0.184507	−	0.982831i	$-0.559069\pi$
$977$	−11269.0	−0.369014	−0.184507	+	0.982831i	$0.559069\pi$
$978$	0	0
$979$	−11583.7	−0.378159
$980$	0	0
$981$	−139.423	−0.00453766
$982$	0	0
$983$	−48373.3	−1.56955	−0.784775	−	0.619781i	$-0.787221\pi$
$983$	−48373.3	−1.56955	−0.784775	+	0.619781i	$0.787221\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2820.49	0.0909596
$988$	0	0
$989$	−6536.72	−0.210167
$990$	0	0
$991$	792.959	0.0254179	0.0127090	−	0.999919i	$-0.495954\pi$
$991$	792.959	0.0254179	0.0127090	+	0.999919i	$0.495954\pi$
$992$	0	0
$993$	38038.1	1.21561
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−52187.1	−1.65775	−0.828877	−	0.559430i	$-0.811020\pi$
$997$	−52187.1	−1.65775	−0.828877	+	0.559430i	$0.811020\pi$
$998$	0	0
$999$	48650.4	1.54077

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.4.a.u.1.3	✓	3
4.3	odd	2		800.4.a.x.1.1	yes	3
5.2	odd	4		800.4.c.m.449.2		6
5.3	odd	4		800.4.c.m.449.5		6
5.4	even	2		800.4.a.w.1.1	yes	3
8.3	odd	2		1600.4.a.cq.1.3		3
8.5	even	2		1600.4.a.ct.1.1		3
20.3	even	4		800.4.c.n.449.2		6
20.7	even	4		800.4.c.n.449.5		6
20.19	odd	2		800.4.a.v.1.3	yes	3
40.19	odd	2		1600.4.a.cs.1.1		3
40.29	even	2		1600.4.a.cr.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
800.4.a.u.1.3	✓	3	1.1	even	1	trivial
800.4.a.v.1.3	yes	3	20.19	odd	2
800.4.a.w.1.1	yes	3	5.4	even	2
800.4.a.x.1.1	yes	3	4.3	odd	2
800.4.c.m.449.2		6	5.2	odd	4
800.4.c.m.449.5		6	5.3	odd	4
800.4.c.n.449.2		6	20.3	even	4
800.4.c.n.449.5		6	20.7	even	4
1600.4.a.cq.1.3		3	8.3	odd	2
1600.4.a.cr.1.3		3	40.29	even	2
1600.4.a.cs.1.1		3	40.19	odd	2
1600.4.a.ct.1.1		3	8.5	even	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	800.a (trivial)

\(f(q)\)	\(=\)	\(q+5.25747 q^{3} +9.15590 q^{7} +0.640965 q^{9} -11.8984 q^{11} -41.9826 q^{13} -75.7604 q^{17} +2.49047 q^{19} +48.1368 q^{21} -17.8566 q^{23} -138.582 q^{27} -143.538 q^{29} -88.6094 q^{31} -62.5556 q^{33} -351.059 q^{37} -220.722 q^{39} +195.922 q^{41} +366.067 q^{43} +58.5931 q^{47} -259.169 q^{49} -398.308 q^{51} +0.374917 q^{53} +13.0936 q^{57} +318.952 q^{59} +446.597 q^{61} +5.86861 q^{63} -709.343 q^{67} -93.8805 q^{69} -1137.43 q^{71} +85.3855 q^{73} -108.941 q^{77} -1249.83 q^{79} -745.895 q^{81} +926.261 q^{83} -754.648 q^{87} +973.552 q^{89} -384.389 q^{91} -465.861 q^{93} +1158.55 q^{97} -7.62648 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 5 q^{3} + 2 q^{7} + 18 q^{9} - 31 q^{11} - 8 q^{13} + 89 q^{17} - 87 q^{19} - 70 q^{21} + 122 q^{23} - 143 q^{27} + 84 q^{29} - 294 q^{31} + 209 q^{33} + 94 q^{37} + 32 q^{39} - 345 q^{41} + 412 q^{43}+ \cdots - 1954 q^{99}+O(q^{100}) \) 3 * q - 5 * q^3 + 2 * q^7 + 18 * q^9 - 31 * q^11 - 8 * q^13 + 89 * q^17 - 87 * q^19 - 70 * q^21 + 122 * q^23 - 143 * q^27 + 84 * q^29 - 294 * q^31 + 209 * q^33 + 94 * q^37 + 32 * q^39 - 345 * q^41 + 412 * q^43 - 824 * q^47 + 359 * q^49 - 1351 * q^51 - 74 * q^53 + 913 * q^57 - 1040 * q^59 + 482 * q^61 + 1532 * q^63 - 735 * q^67 - 614 * q^69 - 2048 * q^71 + 15 * q^73 - 1474 * q^77 - 998 * q^79 - 549 * q^81 + 1221 * q^83 - 2564 * q^87 + 1189 * q^89 - 4032 * q^91 - 454 * q^93 + 1010 * q^97 - 1954 * q^99

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