LMFDB - Embedded newform 1600.4.a.cu.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1600,4,Mod(1,1600)] 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("1600.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,76,0,0,0,-208] 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:	$94.4030560092$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.37485.2
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 8x^{2} + 12x - 3 $ x^4 - x^3 - 8x^2 + 12x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.320221$ of defining polynomial
Character	$\chi$	$=$	1600.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.30169 q^{3} +28.3162 q^{7} -8.49545 q^{9} -65.2358 q^{11} -33.6697 q^{13} -73.3212 q^{17} -134.063 q^{19} -121.808 q^{21} -14.7007 q^{23} +152.690 q^{27} +224.642 q^{29} +68.8271 q^{31} +280.624 q^{33} -196.312 q^{37} +144.837 q^{39} -143.147 q^{41} -15.0755 q^{43} +134.399 q^{47} +458.808 q^{49} +315.405 q^{51} +262.955 q^{53} +576.697 q^{57} +119.698 q^{59} -16.5409 q^{61} -240.559 q^{63} -545.565 q^{67} +63.2379 q^{69} -199.299 q^{71} -43.2667 q^{73} -1847.23 q^{77} +438.694 q^{79} -427.450 q^{81} -1220.89 q^{83} -966.342 q^{87} +723.212 q^{89} -953.398 q^{91} -296.073 q^{93} +1136.00 q^{97} +554.208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 76 q^{9} - 208 q^{13} + 136 q^{21} + 312 q^{29} + 96 q^{33} - 272 q^{37} - 96 q^{41} + 1212 q^{49} - 48 q^{53} + 3040 q^{57} - 1056 q^{61} + 1976 q^{69} + 1440 q^{73} - 4896 q^{77} - 500 q^{81} - 40 q^{89}+ \cdots + 4544 q^{97}+O(q^{100}) $ 4 * q + 76 * q^9 - 208 * q^13 + 136 * q^21 + 312 * q^29 + 96 * q^33 - 272 * q^37 - 96 * q^41 + 1212 * q^49 - 48 * q^53 + 3040 * q^57 - 1056 * q^61 + 1976 * q^69 + 1440 * q^73 - 4896 * q^77 - 500 * q^81 - 40 * q^89 - 2944 * q^93 + 4544 * q^97

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
$3$	−4.30169	−0.827861	−0.413930	−	0.910309i	$-0.635844\pi$
$3$	−4.30169	−0.827861	−0.413930	+	0.910309i	$0.635844\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	28.3162	1.52893	0.764466	−	0.644664i	$-0.223003\pi$
$7$	28.3162	1.52893	0.764466	+	0.644664i	$0.223003\pi$
$8$	0	0
$9$	−8.49545	−0.314646
$10$	0	0
$11$	−65.2358	−1.78812	−0.894061	−	0.447946i	$-0.852156\pi$
$11$	−65.2358	−1.78812	−0.894061	+	0.447946i	$0.852156\pi$
$12$	0	0
$13$	−33.6697	−0.718330	−0.359165	−	0.933274i	$-0.616939\pi$
$13$	−33.6697	−0.718330	−0.359165	+	0.933274i	$0.616939\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−73.3212	−1.04606	−0.523030	−	0.852315i	$-0.675198\pi$
$17$	−73.3212	−1.04606	−0.523030	+	0.852315i	$0.675198\pi$
$18$	0	0
$19$	−134.063	−1.61874	−0.809372	−	0.587297i	$-0.800192\pi$
$19$	−134.063	−1.61874	−0.809372	+	0.587297i	$0.800192\pi$
$20$	0	0
$21$	−121.808	−1.26574
$22$	0	0
$23$	−14.7007	−0.133274	−0.0666371	−	0.997777i	$-0.521227\pi$
$23$	−14.7007	−0.133274	−0.0666371	+	0.997777i	$0.521227\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	152.690	1.08834
$28$	0	0
$29$	224.642	1.43845	0.719225	−	0.694777i	$-0.244497\pi$
$29$	224.642	1.43845	0.719225	+	0.694777i	$0.244497\pi$
$30$	0	0
$31$	68.8271	0.398765	0.199382	−	0.979922i	$-0.436106\pi$
$31$	68.8271	0.398765	0.199382	+	0.979922i	$0.436106\pi$
$32$	0	0
$33$	280.624	1.48032
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−196.312	−0.872257	−0.436129	−	0.899884i	$-0.643651\pi$
$37$	−196.312	−0.872257	−0.436129	+	0.899884i	$0.643651\pi$
$38$	0	0
$39$	144.837	0.594678
$40$	0	0
$41$	−143.147	−0.545263	−0.272632	−	0.962118i	$-0.587894\pi$
$41$	−143.147	−0.545263	−0.272632	+	0.962118i	$0.587894\pi$
$42$	0	0
$43$	−15.0755	−0.0534648	−0.0267324	−	0.999643i	$-0.508510\pi$
$43$	−15.0755	−0.0534648	−0.0267324	+	0.999643i	$0.508510\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	134.399	0.417107	0.208554	−	0.978011i	$-0.433124\pi$
$47$	134.399	0.417107	0.208554	+	0.978011i	$0.433124\pi$
$48$	0	0
$49$	458.808	1.33763
$50$	0	0
$51$	315.405	0.865991
$52$	0	0
$53$	262.955	0.681502	0.340751	−	0.940154i	$-0.389319\pi$
$53$	262.955	0.681502	0.340751	+	0.940154i	$0.389319\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	576.697	1.34009
$58$	0	0
$59$	119.698	0.264124	0.132062	−	0.991241i	$-0.457840\pi$
$59$	119.698	0.264124	0.132062	+	0.991241i	$0.457840\pi$
$60$	0	0
$61$	−16.5409	−0.0347188	−0.0173594	−	0.999849i	$-0.505526\pi$
$61$	−16.5409	−0.0347188	−0.0173594	+	0.999849i	$0.505526\pi$
$62$	0	0
$63$	−240.559	−0.481073
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−545.565	−0.994797	−0.497399	−	0.867522i	$-0.665711\pi$
$67$	−545.565	−0.994797	−0.497399	+	0.867522i	$0.665711\pi$
$68$	0	0
$69$	63.2379	0.110333
$70$	0	0
$71$	−199.299	−0.333132	−0.166566	−	0.986030i	$-0.553268\pi$
$71$	−199.299	−0.333132	−0.166566	+	0.986030i	$0.553268\pi$
$72$	0	0
$73$	−43.2667	−0.0693696	−0.0346848	−	0.999398i	$-0.511043\pi$
$73$	−43.2667	−0.0693696	−0.0346848	+	0.999398i	$0.511043\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1847.23	−2.73391
$78$	0	0
$79$	438.694	0.624772	0.312386	−	0.949955i	$-0.398872\pi$
$79$	438.694	0.624772	0.312386	+	0.949955i	$0.398872\pi$
$80$	0	0
$81$	−427.450	−0.586351
$82$	0	0
$83$	−1220.89	−1.61458	−0.807291	−	0.590154i	$-0.799067\pi$
$83$	−1220.89	−1.61458	−0.807291	+	0.590154i	$0.799067\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−966.342	−1.19084
$88$	0	0
$89$	723.212	0.861352	0.430676	−	0.902507i	$-0.358275\pi$
$89$	723.212	0.861352	0.430676	+	0.902507i	$0.358275\pi$
$90$	0	0
$91$	−953.398	−1.09828
$92$	0	0
$93$	−296.073	−0.330122
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1136.00	1.18911	0.594553	−	0.804056i	$-0.297329\pi$
$97$	1136.00	1.18911	0.594553	+	0.804056i	$0.297329\pi$
$98$	0	0
$99$	554.208	0.562626
$100$	0	0
$101$	1175.21	1.15780	0.578901	−	0.815398i	$-0.303482\pi$
$101$	1175.21	1.15780	0.578901	+	0.815398i	$0.303482\pi$
$102$	0	0
$103$	−1752.43	−1.67643	−0.838213	−	0.545344i	$-0.816399\pi$
$103$	−1752.43	−1.67643	−0.838213	+	0.545344i	$0.816399\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−94.5981	−0.0854686	−0.0427343	−	0.999086i	$-0.513607\pi$
$107$	−94.5981	−0.0854686	−0.0427343	+	0.999086i	$0.513607\pi$
$108$	0	0
$109$	1349.68	1.18602	0.593009	−	0.805196i	$-0.297940\pi$
$109$	1349.68	1.18602	0.593009	+	0.805196i	$0.297940\pi$
$110$	0	0
$111$	844.474	0.722108
$112$	0	0
$113$	529.248	0.440597	0.220299	−	0.975432i	$-0.429297\pi$
$113$	529.248	0.440597	0.220299	+	0.975432i	$0.429297\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	286.039	0.226020
$118$	0	0
$119$	−2076.18	−1.59935
$120$	0	0
$121$	2924.71	2.19738
$122$	0	0
$123$	615.774	0.451402
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−814.066	−0.568793	−0.284396	−	0.958707i	$-0.591793\pi$
$127$	−814.066	−0.568793	−0.284396	+	0.958707i	$0.591793\pi$
$128$	0	0
$129$	64.8500	0.0442614
$130$	0	0
$131$	1329.85	0.886946	0.443473	−	0.896288i	$-0.353746\pi$
$131$	1329.85	0.886946	0.443473	+	0.896288i	$0.353746\pi$
$132$	0	0
$133$	−3796.15	−2.47495
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2906.98	1.81284	0.906422	−	0.422373i	$-0.138803\pi$
$137$	2906.98	1.81284	0.906422	+	0.422373i	$0.138803\pi$
$138$	0	0
$139$	−921.077	−0.562048	−0.281024	−	0.959701i	$-0.590674\pi$
$139$	−921.077	−0.562048	−0.281024	+	0.959701i	$0.590674\pi$
$140$	0	0
$141$	−578.141	−0.345307
$142$	0	0
$143$	2196.47	1.28446
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1973.65	−1.10737
$148$	0	0
$149$	−265.259	−0.145845	−0.0729224	−	0.997338i	$-0.523233\pi$
$149$	−265.259	−0.145845	−0.0729224	+	0.997338i	$0.523233\pi$
$150$	0	0
$151$	−2507.69	−1.35148	−0.675738	−	0.737142i	$-0.736175\pi$
$151$	−2507.69	−1.35148	−0.675738	+	0.737142i	$0.736175\pi$
$152$	0	0
$153$	622.897	0.329139
$154$	0	0
$155$	0	0
$156$	0	0
$157$	14.8818	0.00756495	0.00378248	−	0.999993i	$-0.498796\pi$
$157$	14.8818	0.00756495	0.00378248	+	0.999993i	$0.498796\pi$
$158$	0	0
$159$	−1131.15	−0.564188
$160$	0	0
$161$	−416.268	−0.203767
$162$	0	0
$163$	−1184.23	−0.569055	−0.284528	−	0.958668i	$-0.591837\pi$
$163$	−1184.23	−0.569055	−0.284528	+	0.958668i	$0.591837\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2688.78	1.24589	0.622945	−	0.782266i	$-0.285936\pi$
$167$	2688.78	1.24589	0.622945	+	0.782266i	$0.285936\pi$
$168$	0	0
$169$	−1063.35	−0.484002
$170$	0	0
$171$	1138.92	0.509332
$172$	0	0
$173$	1940.72	0.852891	0.426445	−	0.904513i	$-0.359766\pi$
$173$	1940.72	0.852891	0.426445	+	0.904513i	$0.359766\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−514.903	−0.218658
$178$	0	0
$179$	1580.62	0.660005	0.330002	−	0.943980i	$-0.392950\pi$
$179$	1580.62	0.660005	0.330002	+	0.943980i	$0.392950\pi$
$180$	0	0
$181$	2561.93	1.05208	0.526040	−	0.850460i	$-0.323676\pi$
$181$	2561.93	1.05208	0.526040	+	0.850460i	$0.323676\pi$
$182$	0	0
$183$	71.1539	0.0287423
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4783.17	1.87048
$188$	0	0
$189$	4323.62	1.66400
$190$	0	0
$191$	−137.654	−0.0521482	−0.0260741	−	0.999660i	$-0.508301\pi$
$191$	−137.654	−0.0521482	−0.0260741	+	0.999660i	$0.508301\pi$
$192$	0	0
$193$	845.503	0.315340	0.157670	−	0.987492i	$-0.449602\pi$
$193$	845.503	0.315340	0.157670	+	0.987492i	$0.449602\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2733.63	−0.988646	−0.494323	−	0.869278i	$-0.664584\pi$
$197$	−2733.63	−0.988646	−0.494323	+	0.869278i	$0.664584\pi$
$198$	0	0
$199$	4649.70	1.65632	0.828162	−	0.560489i	$-0.189387\pi$
$199$	4649.70	1.65632	0.828162	+	0.560489i	$0.189387\pi$
$200$	0	0
$201$	2346.85	0.823553
$202$	0	0
$203$	6361.02	2.19929
$204$	0	0
$205$	0	0
$206$	0	0
$207$	124.889	0.0419343
$208$	0	0
$209$	8745.70	2.89451
$210$	0	0
$211$	−3845.91	−1.25480	−0.627402	−	0.778696i	$-0.715882\pi$
$211$	−3845.91	−1.25480	−0.627402	+	0.778696i	$0.715882\pi$
$212$	0	0
$213$	857.321	0.275787
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1948.92	0.609684
$218$	0	0
$219$	186.120	0.0574284
$220$	0	0
$221$	2468.70	0.751416
$222$	0	0
$223$	1460.74	0.438648	0.219324	−	0.975652i	$-0.429615\pi$
$223$	1460.74	0.438648	0.219324	+	0.975652i	$0.429615\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3435.16	1.00440	0.502202	−	0.864750i	$-0.332523\pi$
$227$	3435.16	1.00440	0.502202	+	0.864750i	$0.332523\pi$
$228$	0	0
$229$	595.358	0.171801	0.0859003	−	0.996304i	$-0.472623\pi$
$229$	595.358	0.171801	0.0859003	+	0.996304i	$0.472623\pi$
$230$	0	0
$231$	7946.21	2.26330
$232$	0	0
$233$	−5432.62	−1.52748	−0.763740	−	0.645523i	$-0.776639\pi$
$233$	−5432.62	−1.52748	−0.763740	+	0.645523i	$0.776639\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−1887.13	−0.517224
$238$	0	0
$239$	3931.51	1.06405	0.532026	−	0.846728i	$-0.321431\pi$
$239$	3931.51	1.06405	0.532026	+	0.846728i	$0.321431\pi$
$240$	0	0
$241$	−2587.88	−0.691701	−0.345851	−	0.938290i	$-0.612410\pi$
$241$	−2587.88	−0.691701	−0.345851	+	0.938290i	$0.612410\pi$
$242$	0	0
$243$	−2283.89	−0.602927
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4513.86	1.16279
$248$	0	0
$249$	5251.90	1.33665
$250$	0	0
$251$	2231.79	0.561232	0.280616	−	0.959820i	$-0.409461\pi$
$251$	2231.79	0.561232	0.280616	+	0.959820i	$0.409461\pi$
$252$	0	0
$253$	959.012	0.238311
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2477.07	−0.601226	−0.300613	−	0.953746i	$-0.597191\pi$
$257$	−2477.07	−0.601226	−0.300613	+	0.953746i	$0.597191\pi$
$258$	0	0
$259$	−5558.81	−1.33362
$260$	0	0
$261$	−1908.44	−0.452603
$262$	0	0
$263$	4865.27	1.14071	0.570353	−	0.821400i	$-0.306806\pi$
$263$	4865.27	1.14071	0.570353	+	0.821400i	$0.306806\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3111.04	−0.713080
$268$	0	0
$269$	6753.93	1.53084	0.765418	−	0.643534i	$-0.222532\pi$
$269$	6753.93	1.53084	0.765418	+	0.643534i	$0.222532\pi$
$270$	0	0
$271$	−4315.74	−0.967390	−0.483695	−	0.875237i	$-0.660706\pi$
$271$	−4315.74	−0.967390	−0.483695	+	0.875237i	$0.660706\pi$
$272$	0	0
$273$	4101.22	0.909221
$274$	0	0
$275$	0	0
$276$	0	0
$277$	5799.07	1.25788	0.628939	−	0.777454i	$-0.283489\pi$
$277$	5799.07	1.25788	0.628939	+	0.777454i	$0.283489\pi$
$278$	0	0
$279$	−584.717	−0.125470
$280$	0	0
$281$	−538.674	−0.114358	−0.0571790	−	0.998364i	$-0.518211\pi$
$281$	−538.674	−0.114358	−0.0571790	+	0.998364i	$0.518211\pi$
$282$	0	0
$283$	−1648.50	−0.346265	−0.173133	−	0.984899i	$-0.555389\pi$
$283$	−1648.50	−0.346265	−0.173133	+	0.984899i	$0.555389\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4053.38	−0.833670
$288$	0	0
$289$	463.000	0.0942398
$290$	0	0
$291$	−4886.72	−0.984415
$292$	0	0
$293$	4963.73	0.989707	0.494854	−	0.868976i	$-0.335222\pi$
$293$	4963.73	0.989707	0.494854	+	0.868976i	$0.335222\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−9960.88	−1.94609
$298$	0	0
$299$	494.968	0.0957350
$300$	0	0
$301$	−426.880	−0.0817441
$302$	0	0
$303$	−5055.40	−0.958499
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−9906.61	−1.84169	−0.920847	−	0.389925i	$-0.872501\pi$
$307$	−9906.61	−1.84169	−0.920847	+	0.389925i	$0.872501\pi$
$308$	0	0
$309$	7538.40	1.38785
$310$	0	0
$311$	−3477.27	−0.634012	−0.317006	−	0.948424i	$-0.602677\pi$
$311$	−3477.27	−0.634012	−0.317006	+	0.948424i	$0.602677\pi$
$312$	0	0
$313$	3958.27	0.714808	0.357404	−	0.933950i	$-0.383662\pi$
$313$	3958.27	0.714808	0.357404	+	0.933950i	$0.383662\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3296.86	0.584132	0.292066	−	0.956398i	$-0.405657\pi$
$317$	3296.86	0.584132	0.292066	+	0.956398i	$0.405657\pi$
$318$	0	0
$319$	−14654.7	−2.57212
$320$	0	0
$321$	406.932	0.0707561
$322$	0	0
$323$	9829.65	1.69330
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5805.91	−0.981858
$328$	0	0
$329$	3805.66	0.637728
$330$	0	0
$331$	6048.38	1.00438	0.502189	−	0.864758i	$-0.332528\pi$
$331$	6048.38	1.00438	0.502189	+	0.864758i	$0.332528\pi$
$332$	0	0
$333$	1667.76	0.274453
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8539.53	−1.38035	−0.690175	−	0.723643i	$-0.742466\pi$
$337$	−8539.53	−1.38035	−0.690175	+	0.723643i	$0.742466\pi$
$338$	0	0
$339$	−2276.66	−0.364753
$340$	0	0
$341$	−4489.99	−0.713040
$342$	0	0
$343$	3279.23	0.516215
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6610.79	1.02273	0.511363	−	0.859365i	$-0.329141\pi$
$347$	6610.79	1.02273	0.511363	+	0.859365i	$0.329141\pi$
$348$	0	0
$349$	2491.07	0.382074	0.191037	−	0.981583i	$-0.438815\pi$
$349$	2491.07	0.382074	0.191037	+	0.981583i	$0.438815\pi$
$350$	0	0
$351$	−5141.04	−0.781791
$352$	0	0
$353$	−3383.35	−0.510134	−0.255067	−	0.966923i	$-0.582098\pi$
$353$	−3383.35	−0.510134	−0.255067	+	0.966923i	$0.582098\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8931.08	1.32404
$358$	0	0
$359$	−1303.53	−0.191637	−0.0958185	−	0.995399i	$-0.530547\pi$
$359$	−1303.53	−0.191637	−0.0958185	+	0.995399i	$0.530547\pi$
$360$	0	0
$361$	11113.8	1.62033
$362$	0	0
$363$	−12581.2	−1.81912
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6672.42	−0.949040	−0.474520	−	0.880245i	$-0.657378\pi$
$367$	−6672.42	−0.949040	−0.474520	+	0.880245i	$0.657378\pi$
$368$	0	0
$369$	1216.10	0.171565
$370$	0	0
$371$	7445.88	1.04197
$372$	0	0
$373$	−5945.42	−0.825314	−0.412657	−	0.910886i	$-0.635399\pi$
$373$	−5945.42	−0.825314	−0.412657	+	0.910886i	$0.635399\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−7563.64	−1.03328
$378$	0	0
$379$	12553.4	1.70139	0.850693	−	0.525663i	$-0.176183\pi$
$379$	12553.4	1.70139	0.850693	+	0.525663i	$0.176183\pi$
$380$	0	0
$381$	3501.86	0.470881
$382$	0	0
$383$	−7602.57	−1.01429	−0.507145	−	0.861861i	$-0.669299\pi$
$383$	−7602.57	−1.01429	−0.507145	+	0.861861i	$0.669299\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	128.073	0.0168225
$388$	0	0
$389$	484.765	0.0631840	0.0315920	−	0.999501i	$-0.489942\pi$
$389$	484.765	0.0631840	0.0315920	+	0.999501i	$0.489942\pi$
$390$	0	0
$391$	1077.87	0.139413
$392$	0	0
$393$	−5720.62	−0.734268
$394$	0	0
$395$	0	0
$396$	0	0
$397$	9367.85	1.18428	0.592140	−	0.805835i	$-0.298283\pi$
$397$	9367.85	1.18428	0.592140	+	0.805835i	$0.298283\pi$
$398$	0	0
$399$	16329.9	2.04891
$400$	0	0
$401$	−6650.48	−0.828203	−0.414101	−	0.910231i	$-0.635904\pi$
$401$	−6650.48	−0.828203	−0.414101	+	0.910231i	$0.635904\pi$
$402$	0	0
$403$	−2317.39	−0.286445
$404$	0	0
$405$	0	0
$406$	0	0
$407$	12806.6	1.55970
$408$	0	0
$409$	−1615.31	−0.195286	−0.0976432	−	0.995221i	$-0.531130\pi$
$409$	−1615.31	−0.195286	−0.0976432	+	0.995221i	$0.531130\pi$
$410$	0	0
$411$	−12504.9	−1.50078
$412$	0	0
$413$	3389.39	0.403828
$414$	0	0
$415$	0	0
$416$	0	0
$417$	3962.19	0.465298
$418$	0	0
$419$	10181.0	1.18705	0.593526	−	0.804815i	$-0.297736\pi$
$419$	10181.0	1.18705	0.593526	+	0.804815i	$0.297736\pi$
$420$	0	0
$421$	4145.13	0.479860	0.239930	−	0.970790i	$-0.422876\pi$
$421$	4145.13	0.479860	0.239930	+	0.970790i	$0.422876\pi$
$422$	0	0
$423$	−1141.78	−0.131241
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−468.376	−0.0530827
$428$	0	0
$429$	−9448.53	−1.06336
$430$	0	0
$431$	14283.0	1.59627	0.798133	−	0.602482i	$-0.205821\pi$
$431$	14283.0	1.59627	0.798133	+	0.602482i	$0.205821\pi$
$432$	0	0
$433$	−7773.88	−0.862792	−0.431396	−	0.902163i	$-0.641979\pi$
$433$	−7773.88	−0.862792	−0.431396	+	0.902163i	$0.641979\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1970.82	0.215737
$438$	0	0
$439$	−3841.14	−0.417602	−0.208801	−	0.977958i	$-0.566956\pi$
$439$	−3841.14	−0.417602	−0.208801	+	0.977958i	$0.566956\pi$
$440$	0	0
$441$	−3897.78	−0.420881
$442$	0	0
$443$	−2955.34	−0.316958	−0.158479	−	0.987362i	$-0.550659\pi$
$443$	−2955.34	−0.316958	−0.158479	+	0.987362i	$0.550659\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1141.06	0.120739
$448$	0	0
$449$	−10786.9	−1.13377	−0.566886	−	0.823796i	$-0.691852\pi$
$449$	−10786.9	−1.13377	−0.566886	+	0.823796i	$0.691852\pi$
$450$	0	0
$451$	9338.31	0.974997
$452$	0	0
$453$	10787.3	1.11883
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15384.9	1.57479	0.787393	−	0.616451i	$-0.211430\pi$
$457$	15384.9	1.57479	0.787393	+	0.616451i	$0.211430\pi$
$458$	0	0
$459$	−11195.5	−1.13847
$460$	0	0
$461$	14590.9	1.47411	0.737057	−	0.675830i	$-0.236215\pi$
$461$	14590.9	1.47411	0.737057	+	0.675830i	$0.236215\pi$
$462$	0	0
$463$	−8828.72	−0.886188	−0.443094	−	0.896475i	$-0.646119\pi$
$463$	−8828.72	−0.886188	−0.443094	+	0.896475i	$0.646119\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−4110.64	−0.407319	−0.203659	−	0.979042i	$-0.565284\pi$
$467$	−4110.64	−0.407319	−0.203659	+	0.979042i	$0.565284\pi$
$468$	0	0
$469$	−15448.3	−1.52098
$470$	0	0
$471$	−64.0169	−0.00626273
$472$	0	0
$473$	983.461	0.0956016
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2233.92	−0.214432
$478$	0	0
$479$	1852.96	0.176751	0.0883757	−	0.996087i	$-0.471832\pi$
$479$	1852.96	0.176751	0.0883757	+	0.996087i	$0.471832\pi$
$480$	0	0
$481$	6609.77	0.626569
$482$	0	0
$483$	1790.66	0.168691
$484$	0	0
$485$	0	0
$486$	0	0
$487$	5285.78	0.491831	0.245915	−	0.969291i	$-0.420912\pi$
$487$	5285.78	0.491831	0.245915	+	0.969291i	$0.420912\pi$
$488$	0	0
$489$	5094.19	0.471099
$490$	0	0
$491$	−12149.4	−1.11669	−0.558347	−	0.829608i	$-0.688564\pi$
$491$	−12149.4	−1.11669	−0.558347	+	0.829608i	$0.688564\pi$
$492$	0	0
$493$	−16471.1	−1.50470
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5643.38	−0.509337
$498$	0	0
$499$	3587.97	0.321883	0.160941	−	0.986964i	$-0.448547\pi$
$499$	3587.97	0.321883	0.160941	+	0.986964i	$0.448547\pi$
$500$	0	0
$501$	−11566.3	−1.03142
$502$	0	0
$503$	18351.9	1.62678	0.813389	−	0.581720i	$-0.197620\pi$
$503$	18351.9	1.62678	0.813389	+	0.581720i	$0.197620\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	4574.21	0.400686
$508$	0	0
$509$	−992.521	−0.0864297	−0.0432149	−	0.999066i	$-0.513760\pi$
$509$	−992.521	−0.0864297	−0.0432149	+	0.999066i	$0.513760\pi$
$510$	0	0
$511$	−1225.15	−0.106061
$512$	0	0
$513$	−20470.1	−1.76175
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−8767.59	−0.745838
$518$	0	0
$519$	−8348.37	−0.706075
$520$	0	0
$521$	11192.3	0.941155	0.470577	−	0.882359i	$-0.344046\pi$
$521$	11192.3	0.941155	0.470577	+	0.882359i	$0.344046\pi$
$522$	0	0
$523$	4959.46	0.414650	0.207325	−	0.978272i	$-0.433524\pi$
$523$	4959.46	0.414650	0.207325	+	0.978272i	$0.433524\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5046.48	−0.417131
$528$	0	0
$529$	−11950.9	−0.982238
$530$	0	0
$531$	−1016.89	−0.0831057
$532$	0	0
$533$	4819.72	0.391679
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−6799.33	−0.546392
$538$	0	0
$539$	−29930.7	−2.39185
$540$	0	0
$541$	18555.9	1.47464	0.737319	−	0.675545i	$-0.236092\pi$
$541$	18555.9	1.47464	0.737319	+	0.675545i	$0.236092\pi$
$542$	0	0
$543$	−11020.6	−0.870976
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24604.8	1.92326	0.961631	−	0.274345i	$-0.0884610\pi$
$547$	24604.8	1.92326	0.961631	+	0.274345i	$0.0884610\pi$
$548$	0	0
$549$	140.523	0.0109241
$550$	0	0
$551$	−30116.2	−2.32848
$552$	0	0
$553$	12422.2	0.955233
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13010.2	0.989694	0.494847	−	0.868980i	$-0.335224\pi$
$557$	13010.2	0.989694	0.494847	+	0.868980i	$0.335224\pi$
$558$	0	0
$559$	507.587	0.0384054
$560$	0	0
$561$	−20575.7	−1.54850
$562$	0	0
$563$	−9751.95	−0.730010	−0.365005	−	0.931006i	$-0.618933\pi$
$563$	−9751.95	−0.730010	−0.365005	+	0.931006i	$0.618933\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−12103.8	−0.896491
$568$	0	0
$569$	5645.90	0.415973	0.207986	−	0.978132i	$-0.433309\pi$
$569$	5645.90	0.415973	0.207986	+	0.978132i	$0.433309\pi$
$570$	0	0
$571$	17188.1	1.25972	0.629860	−	0.776709i	$-0.283112\pi$
$571$	17188.1	1.25972	0.629860	+	0.776709i	$0.283112\pi$
$572$	0	0
$573$	592.145	0.0431714
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23294.0	1.68066	0.840330	−	0.542076i	$-0.182361\pi$
$577$	23294.0	1.68066	0.840330	+	0.542076i	$0.182361\pi$
$578$	0	0
$579$	−3637.09	−0.261058
$580$	0	0
$581$	−34571.0	−2.46858
$582$	0	0
$583$	−17154.0	−1.21861
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15182.5	−1.06754	−0.533771	−	0.845629i	$-0.679226\pi$
$587$	−15182.5	−1.06754	−0.533771	+	0.845629i	$0.679226\pi$
$588$	0	0
$589$	−9227.15	−0.645498
$590$	0	0
$591$	11759.2	0.818461
$592$	0	0
$593$	26734.6	1.85137	0.925683	−	0.378301i	$-0.123492\pi$
$593$	26734.6	1.85137	0.925683	+	0.378301i	$0.123492\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−20001.6	−1.37121
$598$	0	0
$599$	2868.56	0.195670	0.0978350	−	0.995203i	$-0.468808\pi$
$599$	2868.56	0.195670	0.0978350	+	0.995203i	$0.468808\pi$
$600$	0	0
$601$	−4242.60	−0.287952	−0.143976	−	0.989581i	$-0.545989\pi$
$601$	−4242.60	−0.287952	−0.143976	+	0.989581i	$0.545989\pi$
$602$	0	0
$603$	4634.82	0.313009
$604$	0	0
$605$	0	0
$606$	0	0
$607$	2058.87	0.137672	0.0688361	−	0.997628i	$-0.478071\pi$
$607$	2058.87	0.137672	0.0688361	+	0.997628i	$0.478071\pi$
$608$	0	0
$609$	−27363.1	−1.82071
$610$	0	0
$611$	−4525.16	−0.299621
$612$	0	0
$613$	−16492.5	−1.08667	−0.543333	−	0.839518i	$-0.682838\pi$
$613$	−16492.5	−1.08667	−0.543333	+	0.839518i	$0.682838\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−12562.1	−0.819662	−0.409831	−	0.912161i	$-0.634412\pi$
$617$	−12562.1	−0.819662	−0.409831	+	0.912161i	$0.634412\pi$
$618$	0	0
$619$	18297.2	1.18809	0.594043	−	0.804434i	$-0.297531\pi$
$619$	18297.2	1.18809	0.594043	+	0.804434i	$0.297531\pi$
$620$	0	0
$621$	−2244.66	−0.145048
$622$	0	0
$623$	20478.6	1.31695
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−37621.3	−2.39625
$628$	0	0
$629$	14393.8	0.912433
$630$	0	0
$631$	25681.4	1.62022	0.810112	−	0.586275i	$-0.199406\pi$
$631$	25681.4	1.62022	0.810112	+	0.586275i	$0.199406\pi$
$632$	0	0
$633$	16543.9	1.03880
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−15447.9	−0.960861
$638$	0	0
$639$	1693.13	0.104819
$640$	0	0
$641$	12994.7	0.800720	0.400360	−	0.916358i	$-0.368885\pi$
$641$	12994.7	0.800720	0.400360	+	0.916358i	$0.368885\pi$
$642$	0	0
$643$	−4625.71	−0.283701	−0.141851	−	0.989888i	$-0.545305\pi$
$643$	−4625.71	−0.283701	−0.141851	+	0.989888i	$0.545305\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8771.72	0.533002	0.266501	−	0.963835i	$-0.414132\pi$
$647$	8771.72	0.533002	0.266501	+	0.963835i	$0.414132\pi$
$648$	0	0
$649$	−7808.58	−0.472286
$650$	0	0
$651$	−8383.66	−0.504733
$652$	0	0
$653$	2239.49	0.134209	0.0671043	−	0.997746i	$-0.478624\pi$
$653$	2239.49	0.134209	0.0671043	+	0.997746i	$0.478624\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	367.570	0.0218269
$658$	0	0
$659$	−4539.62	−0.268344	−0.134172	−	0.990958i	$-0.542837\pi$
$659$	−4539.62	−0.268344	−0.134172	+	0.990958i	$0.542837\pi$
$660$	0	0
$661$	−28228.4	−1.66105	−0.830526	−	0.556980i	$-0.811960\pi$
$661$	−28228.4	−1.66105	−0.830526	+	0.556980i	$0.811960\pi$
$662$	0	0
$663$	−10619.6	−0.622068
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3302.40	−0.191708
$668$	0	0
$669$	−6283.65	−0.363139
$670$	0	0
$671$	1079.06	0.0620814
$672$	0	0
$673$	7649.89	0.438160	0.219080	−	0.975707i	$-0.429694\pi$
$673$	7649.89	0.438160	0.219080	+	0.975707i	$0.429694\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−12087.9	−0.686228	−0.343114	−	0.939294i	$-0.611482\pi$
$677$	−12087.9	−0.686228	−0.343114	+	0.939294i	$0.611482\pi$
$678$	0	0
$679$	32167.2	1.81806
$680$	0	0
$681$	−14777.0	−0.831506
$682$	0	0
$683$	−8827.56	−0.494549	−0.247275	−	0.968945i	$-0.579535\pi$
$683$	−8827.56	−0.494549	−0.247275	+	0.968945i	$0.579535\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2561.04	−0.142227
$688$	0	0
$689$	−8853.60	−0.489543
$690$	0	0
$691$	−12913.1	−0.710908	−0.355454	−	0.934694i	$-0.615674\pi$
$691$	−12913.1	−0.710908	−0.355454	+	0.934694i	$0.615674\pi$
$692$	0	0
$693$	15693.1	0.860217
$694$	0	0
$695$	0	0
$696$	0	0
$697$	10495.7	0.570378
$698$	0	0
$699$	23369.5	1.26454
$700$	0	0
$701$	12297.2	0.662567	0.331283	−	0.943531i	$-0.392518\pi$
$701$	12297.2	0.662567	0.331283	+	0.943531i	$0.392518\pi$
$702$	0	0
$703$	26318.2	1.41196
$704$	0	0
$705$	0	0
$706$	0	0
$707$	33277.5	1.77020
$708$	0	0
$709$	−34308.2	−1.81731	−0.908653	−	0.417552i	$-0.862888\pi$
$709$	−34308.2	−1.81731	−0.908653	+	0.417552i	$0.862888\pi$
$710$	0	0
$711$	−3726.91	−0.196582
$712$	0	0
$713$	−1011.81	−0.0531451
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16912.1	−0.880887
$718$	0	0
$719$	−21631.4	−1.12200	−0.560999	−	0.827817i	$-0.689583\pi$
$719$	−21631.4	−1.12200	−0.560999	+	0.827817i	$0.689583\pi$
$720$	0	0
$721$	−49622.1	−2.56314
$722$	0	0
$723$	11132.3	0.572632
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−12049.8	−0.614723	−0.307361	−	0.951593i	$-0.599446\pi$
$727$	−12049.8	−0.614723	−0.307361	+	0.951593i	$0.599446\pi$
$728$	0	0
$729$	21365.7	1.08549
$730$	0	0
$731$	1105.35	0.0559274
$732$	0	0
$733$	−15221.6	−0.767017	−0.383508	−	0.923537i	$-0.625284\pi$
$733$	−15221.6	−0.767017	−0.383508	+	0.923537i	$0.625284\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	35590.4	1.77882
$738$	0	0
$739$	−9845.80	−0.490099	−0.245050	−	0.969511i	$-0.578804\pi$
$739$	−9845.80	−0.490099	−0.245050	+	0.969511i	$0.578804\pi$
$740$	0	0
$741$	−19417.2	−0.962630
$742$	0	0
$743$	−35081.5	−1.73219	−0.866094	−	0.499881i	$-0.833377\pi$
$743$	−35081.5	−1.73219	−0.866094	+	0.499881i	$0.833377\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	10372.0	0.508022
$748$	0	0
$749$	−2678.66	−0.130676
$750$	0	0
$751$	−20007.7	−0.972158	−0.486079	−	0.873915i	$-0.661573\pi$
$751$	−20007.7	−0.972158	−0.486079	+	0.873915i	$0.661573\pi$
$752$	0	0
$753$	−9600.47	−0.464622
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−33478.7	−1.60741	−0.803703	−	0.595031i	$-0.797140\pi$
$757$	−33478.7	−1.60741	−0.803703	+	0.595031i	$0.797140\pi$
$758$	0	0
$759$	−4125.37	−0.197288
$760$	0	0
$761$	12728.1	0.606298	0.303149	−	0.952943i	$-0.401962\pi$
$761$	12728.1	0.606298	0.303149	+	0.952943i	$0.401962\pi$
$762$	0	0
$763$	38217.8	1.81334
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4030.19	−0.189728
$768$	0	0
$769$	−20061.4	−0.940747	−0.470373	−	0.882468i	$-0.655881\pi$
$769$	−20061.4	−0.940747	−0.470373	+	0.882468i	$0.655881\pi$
$770$	0	0
$771$	10655.6	0.497732
$772$	0	0
$773$	672.445	0.0312887	0.0156444	−	0.999878i	$-0.495020\pi$
$773$	672.445	0.0312887	0.0156444	+	0.999878i	$0.495020\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	23912.3	1.10405
$778$	0	0
$779$	19190.7	0.882642
$780$	0	0
$781$	13001.4	0.595681
$782$	0	0
$783$	34300.8	1.56553
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7928.23	−0.359099	−0.179550	−	0.983749i	$-0.557464\pi$
$787$	−7928.23	−0.359099	−0.179550	+	0.983749i	$0.557464\pi$
$788$	0	0
$789$	−20928.9	−0.944346
$790$	0	0
$791$	14986.3	0.673643
$792$	0	0
$793$	556.928	0.0249396
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3196.35	−0.142059	−0.0710293	−	0.997474i	$-0.522628\pi$
$797$	−3196.35	−0.142059	−0.0710293	+	0.997474i	$0.522628\pi$
$798$	0	0
$799$	−9854.26	−0.436319
$800$	0	0
$801$	−6144.02	−0.271021
$802$	0	0
$803$	2822.54	0.124041
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−29053.3	−1.26732
$808$	0	0
$809$	−15278.6	−0.663987	−0.331993	−	0.943282i	$-0.607721\pi$
$809$	−15278.6	−0.663987	−0.331993	+	0.943282i	$0.607721\pi$
$810$	0	0
$811$	27275.2	1.18096	0.590482	−	0.807051i	$-0.298938\pi$
$811$	27275.2	1.18096	0.590482	+	0.807051i	$0.298938\pi$
$812$	0	0
$813$	18565.0	0.800864
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2021.06	0.0865459
$818$	0	0
$819$	8099.55	0.345569
$820$	0	0
$821$	11047.6	0.469627	0.234814	−	0.972040i	$-0.424552\pi$
$821$	11047.6	0.469627	0.234814	+	0.972040i	$0.424552\pi$
$822$	0	0
$823$	46589.2	1.97327	0.986633	−	0.162959i	$-0.0521038\pi$
$823$	46589.2	1.97327	0.986633	+	0.162959i	$0.0521038\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−7622.29	−0.320499	−0.160250	−	0.987077i	$-0.551230\pi$
$827$	−7622.29	−0.320499	−0.160250	+	0.987077i	$0.551230\pi$
$828$	0	0
$829$	637.729	0.0267180	0.0133590	−	0.999911i	$-0.495748\pi$
$829$	637.729	0.0267180	0.0133590	+	0.999911i	$0.495748\pi$
$830$	0	0
$831$	−24945.8	−1.04135
$832$	0	0
$833$	−33640.3	−1.39924
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10509.2	0.433993
$838$	0	0
$839$	−38563.3	−1.58683	−0.793417	−	0.608679i	$-0.791700\pi$
$839$	−38563.3	−1.58683	−0.793417	+	0.608679i	$0.791700\pi$
$840$	0	0
$841$	26075.2	1.06914
$842$	0	0
$843$	2317.21	0.0946725
$844$	0	0
$845$	0	0
$846$	0	0
$847$	82816.7	3.35964
$848$	0	0
$849$	7091.33	0.286660
$850$	0	0
$851$	2885.93	0.116249
$852$	0	0
$853$	−21077.3	−0.846040	−0.423020	−	0.906120i	$-0.639030\pi$
$853$	−21077.3	−0.846040	−0.423020	+	0.906120i	$0.639030\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−24424.6	−0.973546	−0.486773	−	0.873528i	$-0.661826\pi$
$857$	−24424.6	−0.973546	−0.486773	+	0.873528i	$0.661826\pi$
$858$	0	0
$859$	−25649.1	−1.01878	−0.509392	−	0.860534i	$-0.670130\pi$
$859$	−25649.1	−1.01878	−0.509392	+	0.860534i	$0.670130\pi$
$860$	0	0
$861$	17436.4	0.690163
$862$	0	0
$863$	−31095.6	−1.22654	−0.613272	−	0.789872i	$-0.710147\pi$
$863$	−31095.6	−1.22654	−0.613272	+	0.789872i	$0.710147\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1991.68	−0.0780174
$868$	0	0
$869$	−28618.6	−1.11717
$870$	0	0
$871$	18369.0	0.714593
$872$	0	0
$873$	−9650.84	−0.374148
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26626.0	−1.02519	−0.512597	−	0.858630i	$-0.671316\pi$
$877$	−26626.0	−1.02519	−0.512597	+	0.858630i	$0.671316\pi$
$878$	0	0
$879$	−21352.4	−0.819340
$880$	0	0
$881$	−18538.6	−0.708945	−0.354472	−	0.935066i	$-0.615340\pi$
$881$	−18538.6	−0.708945	−0.354472	+	0.935066i	$0.615340\pi$
$882$	0	0
$883$	−16994.0	−0.647672	−0.323836	−	0.946113i	$-0.604973\pi$
$883$	−16994.0	−0.647672	−0.323836	+	0.946113i	$0.604973\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	19948.2	0.755126	0.377563	−	0.925984i	$-0.376762\pi$
$887$	19948.2	0.755126	0.377563	+	0.925984i	$0.376762\pi$
$888$	0	0
$889$	−23051.3	−0.869645
$890$	0	0
$891$	27885.0	1.04847
$892$	0	0
$893$	−18017.8	−0.675190
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2129.20	−0.0792552
$898$	0	0
$899$	15461.5	0.573603
$900$	0	0
$901$	−19280.1	−0.712891
$902$	0	0
$903$	1836.31	0.0676727
$904$	0	0
$905$	0	0
$906$	0	0
$907$	31481.3	1.15250	0.576251	−	0.817273i	$-0.304515\pi$
$907$	31481.3	1.15250	0.576251	+	0.817273i	$0.304515\pi$
$908$	0	0
$909$	−9983.96	−0.364298
$910$	0	0
$911$	−29685.8	−1.07962	−0.539811	−	0.841787i	$-0.681504\pi$
$911$	−29685.8	−1.07962	−0.539811	+	0.841787i	$0.681504\pi$
$912$	0	0
$913$	79645.8	2.88707
$914$	0	0
$915$	0	0
$916$	0	0
$917$	37656.4	1.35608
$918$	0	0
$919$	21804.9	0.782673	0.391336	−	0.920248i	$-0.372013\pi$
$919$	21804.9	0.782673	0.391336	+	0.920248i	$0.372013\pi$
$920$	0	0
$921$	42615.2	1.52467
$922$	0	0
$923$	6710.33	0.239299
$924$	0	0
$925$	0	0
$926$	0	0
$927$	14887.7	0.527481
$928$	0	0
$929$	−49150.1	−1.73580	−0.867902	−	0.496735i	$-0.834532\pi$
$929$	−49150.1	−1.73580	−0.867902	+	0.496735i	$0.834532\pi$
$930$	0	0
$931$	−61509.1	−2.16528
$932$	0	0
$933$	14958.1	0.524873
$934$	0	0
$935$	0	0
$936$	0	0
$937$	33165.6	1.15632	0.578161	−	0.815922i	$-0.303770\pi$
$937$	33165.6	1.15632	0.578161	+	0.815922i	$0.303770\pi$
$938$	0	0
$939$	−17027.3	−0.591761
$940$	0	0
$941$	15184.8	0.526048	0.263024	−	0.964789i	$-0.415280\pi$
$941$	15184.8	0.526048	0.263024	+	0.964789i	$0.415280\pi$
$942$	0	0
$943$	2104.36	0.0726696
$944$	0	0
$945$	0	0
$946$	0	0
$947$	44341.2	1.52154	0.760769	−	0.649023i	$-0.224822\pi$
$947$	44341.2	1.52154	0.760769	+	0.649023i	$0.224822\pi$
$948$	0	0
$949$	1456.78	0.0498303
$950$	0	0
$951$	−14182.1	−0.483580
$952$	0	0
$953$	−31056.8	−1.05564	−0.527821	−	0.849356i	$-0.676991\pi$
$953$	−31056.8	−1.05564	−0.527821	+	0.849356i	$0.676991\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	63040.1	2.12936
$958$	0	0
$959$	82314.5	2.77171
$960$	0	0
$961$	−25053.8	−0.840987
$962$	0	0
$963$	803.654	0.0268924
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40508.8	−1.34713	−0.673564	−	0.739128i	$-0.735238\pi$
$967$	−40508.8	−1.34713	−0.673564	+	0.739128i	$0.735238\pi$
$968$	0	0
$969$	−42284.1	−1.40182
$970$	0	0
$971$	−36986.9	−1.22242	−0.611208	−	0.791470i	$-0.709316\pi$
$971$	−36986.9	−1.22242	−0.611208	+	0.791470i	$0.709316\pi$
$972$	0	0
$973$	−26081.4	−0.859333
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−28979.9	−0.948975	−0.474488	−	0.880262i	$-0.657367\pi$
$977$	−28979.9	−0.948975	−0.474488	+	0.880262i	$0.657367\pi$
$978$	0	0
$979$	−47179.3	−1.54020
$980$	0	0
$981$	−11466.1	−0.373176
$982$	0	0
$983$	−31114.6	−1.00956	−0.504782	−	0.863247i	$-0.668427\pi$
$983$	−31114.6	−1.00956	−0.504782	+	0.863247i	$0.668427\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−16370.8	−0.527950
$988$	0	0
$989$	221.620	0.00712549
$990$	0	0
$991$	44477.8	1.42571	0.712857	−	0.701309i	$-0.247401\pi$
$991$	44477.8	1.42571	0.712857	+	0.701309i	$0.247401\pi$
$992$	0	0
$993$	−26018.3	−0.831485
$994$	0	0
$995$	0	0
$996$	0	0
$997$	57489.4	1.82619	0.913093	−	0.407752i	$-0.133687\pi$
$997$	57489.4	1.82619	0.913093	+	0.407752i	$0.133687\pi$
$998$	0	0
$999$	−29975.0	−0.949316

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.cu.1.2		4
4.3	odd	2	inner	1600.4.a.cu.1.3		4
5.2	odd	4		320.4.c.j.129.6		8
5.3	odd	4		320.4.c.j.129.3		8
5.4	even	2		1600.4.a.cv.1.3		4
8.3	odd	2		800.4.a.z.1.2		4
8.5	even	2		800.4.a.z.1.3		4
20.3	even	4		320.4.c.j.129.5		8
20.7	even	4		320.4.c.j.129.4		8
20.19	odd	2		1600.4.a.cv.1.2		4
40.3	even	4		160.4.c.d.129.4	yes	8
40.13	odd	4		160.4.c.d.129.6	yes	8
40.19	odd	2		800.4.a.y.1.3		4
40.27	even	4		160.4.c.d.129.5	yes	8
40.29	even	2		800.4.a.y.1.2		4
40.37	odd	4		160.4.c.d.129.3	✓	8
120.53	even	4		1440.4.f.k.289.1		8
120.77	even	4		1440.4.f.k.289.4		8
120.83	odd	4		1440.4.f.k.289.2		8
120.107	odd	4		1440.4.f.k.289.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.c.d.129.3	✓	8	40.37	odd	4
160.4.c.d.129.4	yes	8	40.3	even	4
160.4.c.d.129.5	yes	8	40.27	even	4
160.4.c.d.129.6	yes	8	40.13	odd	4
320.4.c.j.129.3		8	5.3	odd	4
320.4.c.j.129.4		8	20.7	even	4
320.4.c.j.129.5		8	20.3	even	4
320.4.c.j.129.6		8	5.2	odd	4
800.4.a.y.1.2		4	40.29	even	2
800.4.a.y.1.3		4	40.19	odd	2
800.4.a.z.1.2		4	8.3	odd	2
800.4.a.z.1.3		4	8.5	even	2
1440.4.f.k.289.1		8	120.53	even	4
1440.4.f.k.289.2		8	120.83	odd	4
1440.4.f.k.289.3		8	120.107	odd	4
1440.4.f.k.289.4		8	120.77	even	4
1600.4.a.cu.1.2		4	1.1	even	1	trivial
1600.4.a.cu.1.3		4	4.3	odd	2	inner
1600.4.a.cv.1.2		4	20.19	odd	2
1600.4.a.cv.1.3		4	5.4	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1600.a (trivial)

\(f(q)\)	\(=\)	\(q-4.30169 q^{3} +28.3162 q^{7} -8.49545 q^{9} -65.2358 q^{11} -33.6697 q^{13} -73.3212 q^{17} -134.063 q^{19} -121.808 q^{21} -14.7007 q^{23} +152.690 q^{27} +224.642 q^{29} +68.8271 q^{31} +280.624 q^{33} -196.312 q^{37} +144.837 q^{39} -143.147 q^{41} -15.0755 q^{43} +134.399 q^{47} +458.808 q^{49} +315.405 q^{51} +262.955 q^{53} +576.697 q^{57} +119.698 q^{59} -16.5409 q^{61} -240.559 q^{63} -545.565 q^{67} +63.2379 q^{69} -199.299 q^{71} -43.2667 q^{73} -1847.23 q^{77} +438.694 q^{79} -427.450 q^{81} -1220.89 q^{83} -966.342 q^{87} +723.212 q^{89} -953.398 q^{91} -296.073 q^{93} +1136.00 q^{97} +554.208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 76 q^{9} - 208 q^{13} + 136 q^{21} + 312 q^{29} + 96 q^{33} - 272 q^{37} - 96 q^{41} + 1212 q^{49} - 48 q^{53} + 3040 q^{57} - 1056 q^{61} + 1976 q^{69} + 1440 q^{73} - 4896 q^{77} - 500 q^{81} - 40 q^{89}+ \cdots + 4544 q^{97}+O(q^{100}) \) 4 * q + 76 * q^9 - 208 * q^13 + 136 * q^21 + 312 * q^29 + 96 * q^33 - 272 * q^37 - 96 * q^41 + 1212 * q^49 - 48 * q^53 + 3040 * q^57 - 1056 * q^61 + 1976 * q^69 + 1440 * q^73 - 4896 * q^77 - 500 * q^81 - 40 * q^89 - 2944 * q^93 + 4544 * q^97

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