LMFDB - Embedded newform 2300.4.a.l.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2300,4,Mod(1,2300)] 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("2300.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2300 = 2^{2} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2300.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,0,12,0,0,0,40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$135.704393013$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 259 x^{14} + 890 x^{13} + 26158 x^{12} - 73156 x^{11} - 1317747 x^{10} + \cdots + 2184881904 $ x^16 - 4x^15 - 259x^14 + 890x^13 + 26158x^12 - 73156x^11 - 1317747x^10 + 2789794x^9 + 35297191x^8 - 52654820x^7 - 490930868x^6 + 501233488x^5 + 3189821396x^4 - 2518334320x^3 - 7718875776x^2 + 6001900128*x + 2184881904
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{9}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 460)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$1.29215$ of defining polynomial
Character	$\chi$	$=$	2300.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.292148 q^{3} -7.10909 q^{7} -26.9146 q^{9} +4.55016 q^{11} -52.7407 q^{13} -38.6210 q^{17} -115.242 q^{19} +2.07691 q^{21} -23.0000 q^{23} +15.7511 q^{27} -4.77188 q^{29} -210.527 q^{31} -1.32932 q^{33} -258.451 q^{37} +15.4081 q^{39} +340.326 q^{41} +158.472 q^{43} +221.720 q^{47} -292.461 q^{49} +11.2831 q^{51} +228.760 q^{53} +33.6677 q^{57} -170.004 q^{59} +138.559 q^{61} +191.339 q^{63} -526.020 q^{67} +6.71941 q^{69} +43.6126 q^{71} -149.395 q^{73} -32.3475 q^{77} +51.5799 q^{79} +722.094 q^{81} +779.356 q^{83} +1.39410 q^{87} -438.859 q^{89} +374.939 q^{91} +61.5049 q^{93} -256.291 q^{97} -122.466 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{3} + 40 q^{7} + 110 q^{9} - 4 q^{11} + 104 q^{13} - 30 q^{17} - 72 q^{19} - 16 q^{21} - 368 q^{23} + 456 q^{27} + 38 q^{29} + 326 q^{31} + 590 q^{33} + 524 q^{37} + 564 q^{39} - 280 q^{41}+ \cdots - 976 q^{99}+O(q^{100}) $ 16 * q + 12 * q^3 + 40 * q^7 + 110 * q^9 - 4 * q^11 + 104 * q^13 - 30 * q^17 - 72 * q^19 - 16 * q^21 - 368 * q^23 + 456 * q^27 + 38 * q^29 + 326 * q^31 + 590 * q^33 + 524 * q^37 + 564 * q^39 - 280 * q^41 + 872 * q^43 + 568 * q^47 + 1112 * q^49 + 1596 * q^51 + 476 * q^53 + 408 * q^57 + 188 * q^59 + 500 * q^61 + 1082 * q^63 + 1514 * q^67 - 276 * q^69 - 366 * q^71 - 424 * q^73 + 1264 * q^77 + 544 * q^79 + 1080 * q^81 - 166 * q^83 + 4648 * q^87 - 1508 * q^89 - 48 * q^91 - 184 * q^93 + 986 * q^97 - 976 * 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
$3$	−0.292148	−0.0562239	−0.0281120	−	0.999605i	$-0.508949\pi$
$3$	−0.292148	−0.0562239	−0.0281120	+	0.999605i	$0.508949\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−7.10909	−0.383855	−0.191928	−	0.981409i	$-0.561474\pi$
$7$	−7.10909	−0.383855	−0.191928	+	0.981409i	$0.561474\pi$
$8$	0	0
$9$	−26.9146	−0.996839
$10$	0	0
$11$	4.55016	0.124720	0.0623602	−	0.998054i	$-0.480137\pi$
$11$	4.55016	0.124720	0.0623602	+	0.998054i	$0.480137\pi$
$12$	0	0
$13$	−52.7407	−1.12520	−0.562601	−	0.826728i	$-0.690199\pi$
$13$	−52.7407	−1.12520	−0.562601	+	0.826728i	$0.690199\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−38.6210	−0.550999	−0.275499	−	0.961301i	$-0.588843\pi$
$17$	−38.6210	−0.550999	−0.275499	+	0.961301i	$0.588843\pi$
$18$	0	0
$19$	−115.242	−1.39149	−0.695744	−	0.718290i	$-0.744925\pi$
$19$	−115.242	−1.39149	−0.695744	+	0.718290i	$0.744925\pi$
$20$	0	0
$21$	2.07691	0.0215818
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	15.7511	0.112270
$28$	0	0
$29$	−4.77188	−0.0305557	−0.0152779	−	0.999883i	$-0.504863\pi$
$29$	−4.77188	−0.0305557	−0.0152779	+	0.999883i	$0.504863\pi$
$30$	0	0
$31$	−210.527	−1.21973	−0.609866	−	0.792504i	$-0.708777\pi$
$31$	−210.527	−1.21973	−0.609866	+	0.792504i	$0.708777\pi$
$32$	0	0
$33$	−1.32932	−0.00701227
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−258.451	−1.14836	−0.574178	−	0.818731i	$-0.694678\pi$
$37$	−258.451	−1.14836	−0.574178	+	0.818731i	$0.694678\pi$
$38$	0	0
$39$	15.4081	0.0632633
$40$	0	0
$41$	340.326	1.29634	0.648170	−	0.761495i	$-0.275534\pi$
$41$	340.326	1.29634	0.648170	+	0.761495i	$0.275534\pi$
$42$	0	0
$43$	158.472	0.562018	0.281009	−	0.959705i	$-0.409331\pi$
$43$	158.472	0.562018	0.281009	+	0.959705i	$0.409331\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	221.720	0.688109	0.344054	−	0.938950i	$-0.388200\pi$
$47$	221.720	0.688109	0.344054	+	0.938950i	$0.388200\pi$
$48$	0	0
$49$	−292.461	−0.852655
$50$	0	0
$51$	11.2831	0.0309793
$52$	0	0
$53$	228.760	0.592880	0.296440	−	0.955051i	$-0.404200\pi$
$53$	228.760	0.592880	0.296440	+	0.955051i	$0.404200\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	33.6677	0.0782349
$58$	0	0
$59$	−170.004	−0.375129	−0.187564	−	0.982252i	$-0.560059\pi$
$59$	−170.004	−0.375129	−0.187564	+	0.982252i	$0.560059\pi$
$60$	0	0
$61$	138.559	0.290831	0.145415	−	0.989371i	$-0.453548\pi$
$61$	138.559	0.290831	0.145415	+	0.989371i	$0.453548\pi$
$62$	0	0
$63$	191.339	0.382642
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−526.020	−0.959158	−0.479579	−	0.877499i	$-0.659211\pi$
$67$	−526.020	−0.959158	−0.479579	+	0.877499i	$0.659211\pi$
$68$	0	0
$69$	6.71941	0.0117235
$70$	0	0
$71$	43.6126	0.0728995	0.0364498	−	0.999335i	$-0.488395\pi$
$71$	43.6126	0.0728995	0.0364498	+	0.999335i	$0.488395\pi$
$72$	0	0
$73$	−149.395	−0.239526	−0.119763	−	0.992803i	$-0.538213\pi$
$73$	−149.395	−0.239526	−0.119763	+	0.992803i	$0.538213\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−32.3475	−0.0478745
$78$	0	0
$79$	51.5799	0.0734581	0.0367290	−	0.999325i	$-0.488306\pi$
$79$	51.5799	0.0734581	0.0367290	+	0.999325i	$0.488306\pi$
$80$	0	0
$81$	722.094	0.990527
$82$	0	0
$83$	779.356	1.03067	0.515334	−	0.856989i	$-0.327668\pi$
$83$	779.356	1.03067	0.515334	+	0.856989i	$0.327668\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	1.39410	0.00171796
$88$	0	0
$89$	−438.859	−0.522685	−0.261343	−	0.965246i	$-0.584165\pi$
$89$	−438.859	−0.522685	−0.261343	+	0.965246i	$0.584165\pi$
$90$	0	0
$91$	374.939	0.431915
$92$	0	0
$93$	61.5049	0.0685781
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−256.291	−0.268272	−0.134136	−	0.990963i	$-0.542826\pi$
$97$	−256.291	−0.268272	−0.134136	+	0.990963i	$0.542826\pi$
$98$	0	0
$99$	−122.466	−0.124326
$100$	0	0
$101$	1021.09	1.00596	0.502981	−	0.864297i	$-0.332236\pi$
$101$	1021.09	1.00596	0.502981	+	0.864297i	$0.332236\pi$
$102$	0	0
$103$	−822.786	−0.787102	−0.393551	−	0.919303i	$-0.628754\pi$
$103$	−822.786	−0.787102	−0.393551	+	0.919303i	$0.628754\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	722.700	0.652954	0.326477	−	0.945205i	$-0.394138\pi$
$107$	722.700	0.652954	0.326477	+	0.945205i	$0.394138\pi$
$108$	0	0
$109$	−37.4248	−0.0328867	−0.0164433	−	0.999865i	$-0.505234\pi$
$109$	−37.4248	−0.0328867	−0.0164433	+	0.999865i	$0.505234\pi$
$110$	0	0
$111$	75.5061	0.0645651
$112$	0	0
$113$	1923.02	1.60091	0.800454	−	0.599394i	$-0.204592\pi$
$113$	1923.02	1.60091	0.800454	+	0.599394i	$0.204592\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1419.50	1.12165
$118$	0	0
$119$	274.561	0.211504
$120$	0	0
$121$	−1310.30	−0.984445
$122$	0	0
$123$	−99.4256	−0.0728854
$124$	0	0
$125$	0	0
$126$	0	0
$127$	251.538	0.175751	0.0878757	−	0.996131i	$-0.471992\pi$
$127$	251.538	0.175751	0.0878757	+	0.996131i	$0.471992\pi$
$128$	0	0
$129$	−46.2973	−0.0315988
$130$	0	0
$131$	−2279.44	−1.52027	−0.760136	−	0.649764i	$-0.774868\pi$
$131$	−2279.44	−1.52027	−0.760136	+	0.649764i	$0.774868\pi$
$132$	0	0
$133$	819.264	0.534130
$134$	0	0
$135$	0	0
$136$	0	0
$137$	22.0168	0.0137301	0.00686505	−	0.999976i	$-0.497815\pi$
$137$	22.0168	0.0137301	0.00686505	+	0.999976i	$0.497815\pi$
$138$	0	0
$139$	564.504	0.344465	0.172232	−	0.985056i	$-0.444902\pi$
$139$	564.504	0.344465	0.172232	+	0.985056i	$0.444902\pi$
$140$	0	0
$141$	−64.7749	−0.0386882
$142$	0	0
$143$	−239.978	−0.140336
$144$	0	0
$145$	0	0
$146$	0	0
$147$	85.4418	0.0479396
$148$	0	0
$149$	−1102.70	−0.606289	−0.303144	−	0.952945i	$-0.598036\pi$
$149$	−1102.70	−0.606289	−0.303144	+	0.952945i	$0.598036\pi$
$150$	0	0
$151$	−874.985	−0.471558	−0.235779	−	0.971807i	$-0.575764\pi$
$151$	−874.985	−0.471558	−0.235779	+	0.971807i	$0.575764\pi$
$152$	0	0
$153$	1039.47	0.549257
$154$	0	0
$155$	0	0
$156$	0	0
$157$	1362.77	0.692743	0.346372	−	0.938097i	$-0.387414\pi$
$157$	1362.77	0.692743	0.346372	+	0.938097i	$0.387414\pi$
$158$	0	0
$159$	−66.8319	−0.0333341
$160$	0	0
$161$	163.509	0.0800393
$162$	0	0
$163$	−1469.77	−0.706265	−0.353133	−	0.935573i	$-0.614884\pi$
$163$	−1469.77	−0.706265	−0.353133	+	0.935573i	$0.614884\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2391.58	1.10818	0.554090	−	0.832457i	$-0.313067\pi$
$167$	2391.58	1.10818	0.554090	+	0.832457i	$0.313067\pi$
$168$	0	0
$169$	584.580	0.266081
$170$	0	0
$171$	3101.69	1.38709
$172$	0	0
$173$	−1572.79	−0.691198	−0.345599	−	0.938382i	$-0.612324\pi$
$173$	−1572.79	−0.691198	−0.345599	+	0.938382i	$0.612324\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	49.6662	0.0210912
$178$	0	0
$179$	2625.98	1.09651	0.548254	−	0.836312i	$-0.315293\pi$
$179$	2625.98	1.09651	0.548254	+	0.836312i	$0.315293\pi$
$180$	0	0
$181$	703.386	0.288852	0.144426	−	0.989516i	$-0.453866\pi$
$181$	703.386	0.288852	0.144426	+	0.989516i	$0.453866\pi$
$182$	0	0
$183$	−40.4798	−0.0163516
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−175.732	−0.0687207
$188$	0	0
$189$	−111.976	−0.0430954
$190$	0	0
$191$	−630.137	−0.238718	−0.119359	−	0.992851i	$-0.538084\pi$
$191$	−630.137	−0.238718	−0.119359	+	0.992851i	$0.538084\pi$
$192$	0	0
$193$	2741.33	1.02241	0.511206	−	0.859458i	$-0.329199\pi$
$193$	2741.33	1.02241	0.511206	+	0.859458i	$0.329199\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5491.23	1.98596	0.992980	−	0.118286i	$-0.0377401\pi$
$197$	5491.23	1.98596	0.992980	+	0.118286i	$0.0377401\pi$
$198$	0	0
$199$	−2266.88	−0.807512	−0.403756	−	0.914867i	$-0.632295\pi$
$199$	−2266.88	−0.807512	−0.403756	+	0.914867i	$0.632295\pi$
$200$	0	0
$201$	153.676	0.0539276
$202$	0	0
$203$	33.9238	0.0117290
$204$	0	0
$205$	0	0
$206$	0	0
$207$	619.037	0.207855
$208$	0	0
$209$	−524.368	−0.173547
$210$	0	0
$211$	−4189.84	−1.36701	−0.683507	−	0.729944i	$-0.739546\pi$
$211$	−4189.84	−1.36701	−0.683507	+	0.729944i	$0.739546\pi$
$212$	0	0
$213$	−12.7413	−0.00409870
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1496.65	0.468200
$218$	0	0
$219$	43.6455	0.0134671
$220$	0	0
$221$	2036.90	0.619985
$222$	0	0
$223$	−1537.44	−0.461681	−0.230841	−	0.972992i	$-0.574148\pi$
$223$	−1537.44	−0.461681	−0.230841	+	0.972992i	$0.574148\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2750.64	0.804256	0.402128	−	0.915583i	$-0.368271\pi$
$227$	2750.64	0.804256	0.402128	+	0.915583i	$0.368271\pi$
$228$	0	0
$229$	4271.72	1.23268	0.616339	−	0.787481i	$-0.288615\pi$
$229$	4271.72	1.23268	0.616339	+	0.787481i	$0.288615\pi$
$230$	0	0
$231$	9.45026	0.00269169
$232$	0	0
$233$	−4520.82	−1.27111	−0.635556	−	0.772055i	$-0.719229\pi$
$233$	−4520.82	−1.27111	−0.635556	+	0.772055i	$0.719229\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−15.0690	−0.00413010
$238$	0	0
$239$	5908.41	1.59909	0.799547	−	0.600604i	$-0.205073\pi$
$239$	5908.41	1.59909	0.799547	+	0.600604i	$0.205073\pi$
$240$	0	0
$241$	−3386.52	−0.905166	−0.452583	−	0.891722i	$-0.649497\pi$
$241$	−3386.52	−0.905166	−0.452583	+	0.891722i	$0.649497\pi$
$242$	0	0
$243$	−636.237	−0.167961
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6077.93	1.56571
$248$	0	0
$249$	−227.687	−0.0579482
$250$	0	0
$251$	−4366.62	−1.09808	−0.549041	−	0.835795i	$-0.685007\pi$
$251$	−4366.62	−1.09808	−0.549041	+	0.835795i	$0.685007\pi$
$252$	0	0
$253$	−104.654	−0.0260060
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2833.34	0.687700	0.343850	−	0.939025i	$-0.388269\pi$
$257$	2833.34	0.687700	0.343850	+	0.939025i	$0.388269\pi$
$258$	0	0
$259$	1837.36	0.440802
$260$	0	0
$261$	128.434	0.0304591
$262$	0	0
$263$	−4963.55	−1.16375	−0.581874	−	0.813279i	$-0.697680\pi$
$263$	−4963.55	−1.16375	−0.581874	+	0.813279i	$0.697680\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	128.212	0.0293874
$268$	0	0
$269$	−5273.22	−1.19522	−0.597610	−	0.801787i	$-0.703883\pi$
$269$	−5273.22	−1.19522	−0.597610	+	0.801787i	$0.703883\pi$
$270$	0	0
$271$	865.945	0.194105	0.0970524	−	0.995279i	$-0.469059\pi$
$271$	865.945	0.194105	0.0970524	+	0.995279i	$0.469059\pi$
$272$	0	0
$273$	−109.538	−0.0242839
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−5414.27	−1.17441	−0.587206	−	0.809437i	$-0.699772\pi$
$277$	−5414.27	−1.17441	−0.587206	+	0.809437i	$0.699772\pi$
$278$	0	0
$279$	5666.25	1.21588
$280$	0	0
$281$	−1644.75	−0.349173	−0.174586	−	0.984642i	$-0.555859\pi$
$281$	−1644.75	−0.349173	−0.174586	+	0.984642i	$0.555859\pi$
$282$	0	0
$283$	5421.79	1.13884	0.569420	−	0.822047i	$-0.307168\pi$
$283$	5421.79	1.13884	0.569420	+	0.822047i	$0.307168\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2419.41	−0.497607
$288$	0	0
$289$	−3421.42	−0.696401
$290$	0	0
$291$	74.8749	0.0150833
$292$	0	0
$293$	−2284.99	−0.455598	−0.227799	−	0.973708i	$-0.573153\pi$
$293$	−2284.99	−0.455598	−0.227799	+	0.973708i	$0.573153\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	71.6698	0.0140024
$298$	0	0
$299$	1213.04	0.234621
$300$	0	0
$301$	−1126.59	−0.215733
$302$	0	0
$303$	−298.309	−0.0565592
$304$	0	0
$305$	0	0
$306$	0	0
$307$	5671.70	1.05440	0.527200	−	0.849741i	$-0.323242\pi$
$307$	5671.70	1.05440	0.527200	+	0.849741i	$0.323242\pi$
$308$	0	0
$309$	240.375	0.0442540
$310$	0	0
$311$	4958.58	0.904100	0.452050	−	0.891993i	$-0.350693\pi$
$311$	4958.58	0.904100	0.452050	+	0.891993i	$0.350693\pi$
$312$	0	0
$313$	8754.65	1.58097	0.790483	−	0.612485i	$-0.209830\pi$
$313$	8754.65	1.58097	0.790483	+	0.612485i	$0.209830\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3419.56	0.605872	0.302936	−	0.953011i	$-0.402033\pi$
$317$	3419.56	0.605872	0.302936	+	0.953011i	$0.402033\pi$
$318$	0	0
$319$	−21.7128	−0.00381092
$320$	0	0
$321$	−211.136	−0.0367116
$322$	0	0
$323$	4450.75	0.766708
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.9336	0.00184902
$328$	0	0
$329$	−1576.22	−0.264134
$330$	0	0
$331$	3963.29	0.658134	0.329067	−	0.944307i	$-0.393266\pi$
$331$	3963.29	0.658134	0.329067	+	0.944307i	$0.393266\pi$
$332$	0	0
$333$	6956.13	1.14473
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3948.95	0.638318	0.319159	−	0.947701i	$-0.396600\pi$
$337$	3948.95	0.638318	0.319159	+	0.947701i	$0.396600\pi$
$338$	0	0
$339$	−561.807	−0.0900094
$340$	0	0
$341$	−957.929	−0.152125
$342$	0	0
$343$	4517.55	0.711151
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12097.0	1.87147	0.935734	−	0.352705i	$-0.114738\pi$
$347$	12097.0	1.87147	0.935734	+	0.352705i	$0.114738\pi$
$348$	0	0
$349$	−9421.29	−1.44501	−0.722507	−	0.691363i	$-0.757010\pi$
$349$	−9421.29	−1.44501	−0.722507	+	0.691363i	$0.757010\pi$
$350$	0	0
$351$	−830.722	−0.126327
$352$	0	0
$353$	8940.60	1.34805	0.674023	−	0.738710i	$-0.264565\pi$
$353$	8940.60	1.34805	0.674023	+	0.738710i	$0.264565\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−80.2123	−0.0118916
$358$	0	0
$359$	1326.09	0.194953	0.0974765	−	0.995238i	$-0.468923\pi$
$359$	1326.09	0.194953	0.0974765	+	0.995238i	$0.468923\pi$
$360$	0	0
$361$	6421.66	0.936239
$362$	0	0
$363$	382.800	0.0553493
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11844.4	1.68467	0.842335	−	0.538954i	$-0.181180\pi$
$367$	11844.4	1.68467	0.842335	+	0.538954i	$0.181180\pi$
$368$	0	0
$369$	−9159.75	−1.29224
$370$	0	0
$371$	−1626.28	−0.227580
$372$	0	0
$373$	−6222.18	−0.863732	−0.431866	−	0.901938i	$-0.642145\pi$
$373$	−6222.18	−0.863732	−0.431866	+	0.901938i	$0.642145\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	251.672	0.0343814
$378$	0	0
$379$	−13623.2	−1.84638	−0.923189	−	0.384347i	$-0.874427\pi$
$379$	−13623.2	−1.84638	−0.923189	+	0.384347i	$0.874427\pi$
$380$	0	0
$381$	−73.4865	−0.00988143
$382$	0	0
$383$	9284.02	1.23862	0.619310	−	0.785147i	$-0.287413\pi$
$383$	9284.02	1.23862	0.619310	+	0.785147i	$0.287413\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−4265.22	−0.560241
$388$	0	0
$389$	5668.23	0.738794	0.369397	−	0.929272i	$-0.379564\pi$
$389$	5668.23	0.738794	0.369397	+	0.929272i	$0.379564\pi$
$390$	0	0
$391$	888.284	0.114891
$392$	0	0
$393$	665.934	0.0854756
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−3253.29	−0.411279	−0.205640	−	0.978628i	$-0.565927\pi$
$397$	−3253.29	−0.411279	−0.205640	+	0.978628i	$0.565927\pi$
$398$	0	0
$399$	−239.347	−0.0300309
$400$	0	0
$401$	2295.32	0.285843	0.142921	−	0.989734i	$-0.454350\pi$
$401$	2295.32	0.285843	0.142921	+	0.989734i	$0.454350\pi$
$402$	0	0
$403$	11103.3	1.37245
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1175.99	−0.143223
$408$	0	0
$409$	−6436.06	−0.778099	−0.389050	−	0.921217i	$-0.627197\pi$
$409$	−6436.06	−0.778099	−0.389050	+	0.921217i	$0.627197\pi$
$410$	0	0
$411$	−6.43217	−0.000771960 0
$412$	0	0
$413$	1208.57	0.143995
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−164.919	−0.0193672
$418$	0	0
$419$	14369.5	1.67541	0.837705	−	0.546123i	$-0.183897\pi$
$419$	14369.5	1.67541	0.837705	+	0.546123i	$0.183897\pi$
$420$	0	0
$421$	6204.78	0.718296	0.359148	−	0.933281i	$-0.383067\pi$
$421$	6204.78	0.718296	0.359148	+	0.933281i	$0.383067\pi$
$422$	0	0
$423$	−5967.50	−0.685934
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−985.029	−0.111637
$428$	0	0
$429$	70.1092	0.00789022
$430$	0	0
$431$	−3516.78	−0.393033	−0.196516	−	0.980501i	$-0.562963\pi$
$431$	−3516.78	−0.393033	−0.196516	+	0.980501i	$0.562963\pi$
$432$	0	0
$433$	−4169.13	−0.462715	−0.231358	−	0.972869i	$-0.574317\pi$
$433$	−4169.13	−0.462715	−0.231358	+	0.972869i	$0.574317\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2650.56	0.290145
$438$	0	0
$439$	−5241.82	−0.569883	−0.284941	−	0.958545i	$-0.591974\pi$
$439$	−5241.82	−0.569883	−0.284941	+	0.958545i	$0.591974\pi$
$440$	0	0
$441$	7871.48	0.849960
$442$	0	0
$443$	−4385.53	−0.470345	−0.235172	−	0.971954i	$-0.575565\pi$
$443$	−4385.53	−0.470345	−0.235172	+	0.971954i	$0.575565\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	322.153	0.0340879
$448$	0	0
$449$	5002.42	0.525788	0.262894	−	0.964825i	$-0.415323\pi$
$449$	5002.42	0.525788	0.262894	+	0.964825i	$0.415323\pi$
$450$	0	0
$451$	1548.54	0.161680
$452$	0	0
$453$	255.625	0.0265128
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10397.3	−1.06426	−0.532130	−	0.846662i	$-0.678608\pi$
$457$	−10397.3	−1.06426	−0.532130	+	0.846662i	$0.678608\pi$
$458$	0	0
$459$	−608.322	−0.0618607
$460$	0	0
$461$	−846.002	−0.0854712	−0.0427356	−	0.999086i	$-0.513607\pi$
$461$	−846.002	−0.0854712	−0.0427356	+	0.999086i	$0.513607\pi$
$462$	0	0
$463$	9863.22	0.990027	0.495014	−	0.868885i	$-0.335163\pi$
$463$	9863.22	0.990027	0.495014	+	0.868885i	$0.335163\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	9411.05	0.932530	0.466265	−	0.884645i	$-0.345599\pi$
$467$	9411.05	0.932530	0.466265	+	0.884645i	$0.345599\pi$
$468$	0	0
$469$	3739.53	0.368178
$470$	0	0
$471$	−398.130	−0.0389487
$472$	0	0
$473$	721.073	0.0700950
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6157.01	−0.591006
$478$	0	0
$479$	−3950.76	−0.376857	−0.188429	−	0.982087i	$-0.560339\pi$
$479$	−3950.76	−0.376857	−0.188429	+	0.982087i	$0.560339\pi$
$480$	0	0
$481$	13630.9	1.29213
$482$	0	0
$483$	−47.7689	−0.00450012
$484$	0	0
$485$	0	0
$486$	0	0
$487$	10452.7	0.972598	0.486299	−	0.873792i	$-0.338347\pi$
$487$	10452.7	0.972598	0.486299	+	0.873792i	$0.338347\pi$
$488$	0	0
$489$	429.390	0.0397090
$490$	0	0
$491$	−1735.44	−0.159510	−0.0797550	−	0.996814i	$-0.525414\pi$
$491$	−1735.44	−0.159510	−0.0797550	+	0.996814i	$0.525414\pi$
$492$	0	0
$493$	184.295	0.0168362
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−310.046	−0.0279828
$498$	0	0
$499$	−3870.88	−0.347263	−0.173632	−	0.984811i	$-0.555550\pi$
$499$	−3870.88	−0.347263	−0.173632	+	0.984811i	$0.555550\pi$
$500$	0	0
$501$	−698.695	−0.0623062
$502$	0	0
$503$	−5227.23	−0.463361	−0.231681	−	0.972792i	$-0.574422\pi$
$503$	−5227.23	−0.463361	−0.231681	+	0.972792i	$0.574422\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−170.784	−0.0149601
$508$	0	0
$509$	21902.4	1.90729	0.953643	−	0.300941i	$-0.0973007\pi$
$509$	21902.4	1.90729	0.953643	+	0.300941i	$0.0973007\pi$
$510$	0	0
$511$	1062.06	0.0919431
$512$	0	0
$513$	−1815.18	−0.156222
$514$	0	0
$515$	0	0
$516$	0	0
$517$	1008.86	0.0858212
$518$	0	0
$519$	459.488	0.0388619
$520$	0	0
$521$	−10699.6	−0.899725	−0.449863	−	0.893098i	$-0.648527\pi$
$521$	−10699.6	−0.899725	−0.449863	+	0.893098i	$0.648527\pi$
$522$	0	0
$523$	−217.089	−0.0181504	−0.00907518	−	0.999959i	$-0.502889\pi$
$523$	−217.089	−0.0181504	−0.00907518	+	0.999959i	$0.502889\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8130.76	0.672071
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	4575.59	0.373943
$532$	0	0
$533$	−17949.0	−1.45865
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−767.175	−0.0616500
$538$	0	0
$539$	−1330.74	−0.106343
$540$	0	0
$541$	−21517.8	−1.71002	−0.855010	−	0.518612i	$-0.826449\pi$
$541$	−21517.8	−1.71002	−0.855010	+	0.518612i	$0.826449\pi$
$542$	0	0
$543$	−205.493	−0.0162404
$544$	0	0
$545$	0	0
$546$	0	0
$547$	24453.9	1.91147	0.955734	−	0.294233i	$-0.0950640\pi$
$547$	24453.9	1.91147	0.955734	+	0.294233i	$0.0950640\pi$
$548$	0	0
$549$	−3729.27	−0.289911
$550$	0	0
$551$	549.920	0.0425179
$552$	0	0
$553$	−366.686	−0.0281973
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8782.67	−0.668103	−0.334052	−	0.942555i	$-0.608416\pi$
$557$	−8782.67	−0.668103	−0.334052	+	0.942555i	$0.608416\pi$
$558$	0	0
$559$	−8357.93	−0.632384
$560$	0	0
$561$	51.3397	0.00386375
$562$	0	0
$563$	7498.30	0.561306	0.280653	−	0.959809i	$-0.409449\pi$
$563$	7498.30	0.561306	0.280653	+	0.959809i	$0.409449\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−5133.43	−0.380219
$568$	0	0
$569$	−3244.49	−0.239044	−0.119522	−	0.992832i	$-0.538136\pi$
$569$	−3244.49	−0.239044	−0.119522	+	0.992832i	$0.538136\pi$
$570$	0	0
$571$	13388.3	0.981230	0.490615	−	0.871376i	$-0.336772\pi$
$571$	13388.3	0.981230	0.490615	+	0.871376i	$0.336772\pi$
$572$	0	0
$573$	184.093	0.0134217
$574$	0	0
$575$	0	0
$576$	0	0
$577$	11148.2	0.804344	0.402172	−	0.915564i	$-0.368255\pi$
$577$	11148.2	0.804344	0.402172	+	0.915564i	$0.368255\pi$
$578$	0	0
$579$	−800.875	−0.0574840
$580$	0	0
$581$	−5540.52	−0.395627
$582$	0	0
$583$	1040.90	0.0739442
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2580.08	0.181416	0.0907082	−	0.995878i	$-0.471087\pi$
$587$	2580.08	0.181416	0.0907082	+	0.995878i	$0.471087\pi$
$588$	0	0
$589$	24261.5	1.69724
$590$	0	0
$591$	−1604.25	−0.111658
$592$	0	0
$593$	−3256.27	−0.225496	−0.112748	−	0.993624i	$-0.535965\pi$
$593$	−3256.27	−0.225496	−0.112748	+	0.993624i	$0.535965\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	662.264	0.0454015
$598$	0	0
$599$	−12464.2	−0.850208	−0.425104	−	0.905145i	$-0.639762\pi$
$599$	−12464.2	−0.850208	−0.425104	+	0.905145i	$0.639762\pi$
$600$	0	0
$601$	17644.6	1.19757	0.598785	−	0.800910i	$-0.295650\pi$
$601$	17644.6	1.19757	0.598785	+	0.800910i	$0.295650\pi$
$602$	0	0
$603$	14157.6	0.956126
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−28761.4	−1.92321	−0.961606	−	0.274433i	$-0.911510\pi$
$607$	−28761.4	−1.92321	−0.961606	+	0.274433i	$0.911510\pi$
$608$	0	0
$609$	−9.91076	−0.000659449 0
$610$	0	0
$611$	−11693.6	−0.774262
$612$	0	0
$613$	−20506.0	−1.35111	−0.675555	−	0.737310i	$-0.736096\pi$
$613$	−20506.0	−1.35111	−0.675555	+	0.737310i	$0.736096\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	7492.32	0.488864	0.244432	−	0.969666i	$-0.421398\pi$
$617$	7492.32	0.488864	0.244432	+	0.969666i	$0.421398\pi$
$618$	0	0
$619$	−10829.2	−0.703169	−0.351584	−	0.936156i	$-0.614357\pi$
$619$	−10829.2	−0.703169	−0.351584	+	0.936156i	$0.614357\pi$
$620$	0	0
$621$	−362.274	−0.0234099
$622$	0	0
$623$	3119.89	0.200635
$624$	0	0
$625$	0	0
$626$	0	0
$627$	153.193	0.00975748
$628$	0	0
$629$	9981.66	0.632742
$630$	0	0
$631$	−9017.61	−0.568915	−0.284457	−	0.958689i	$-0.591813\pi$
$631$	−9017.61	−0.568915	−0.284457	+	0.958689i	$0.591813\pi$
$632$	0	0
$633$	1224.05	0.0768589
$634$	0	0
$635$	0	0
$636$	0	0
$637$	15424.6	0.959410
$638$	0	0
$639$	−1173.82	−0.0726691
$640$	0	0
$641$	−2967.74	−0.182869	−0.0914343	−	0.995811i	$-0.529145\pi$
$641$	−2967.74	−0.182869	−0.0914343	+	0.995811i	$0.529145\pi$
$642$	0	0
$643$	4524.25	0.277479	0.138740	−	0.990329i	$-0.455695\pi$
$643$	4524.25	0.277479	0.138740	+	0.990329i	$0.455695\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−4930.54	−0.299598	−0.149799	−	0.988717i	$-0.547863\pi$
$647$	−4930.54	−0.299598	−0.149799	+	0.988717i	$0.547863\pi$
$648$	0	0
$649$	−773.543	−0.0467862
$650$	0	0
$651$	−437.244	−0.0263241
$652$	0	0
$653$	−23814.8	−1.42717	−0.713587	−	0.700567i	$-0.752930\pi$
$653$	−23814.8	−1.42717	−0.713587	+	0.700567i	$0.752930\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	4020.92	0.238769
$658$	0	0
$659$	19071.3	1.12733	0.563667	−	0.826002i	$-0.309390\pi$
$659$	19071.3	1.12733	0.563667	+	0.826002i	$0.309390\pi$
$660$	0	0
$661$	−12620.8	−0.742653	−0.371327	−	0.928502i	$-0.621097\pi$
$661$	−12620.8	−0.742653	−0.371327	+	0.928502i	$0.621097\pi$
$662$	0	0
$663$	−595.076	−0.0348580
$664$	0	0
$665$	0	0
$666$	0	0
$667$	109.753	0.00637131
$668$	0	0
$669$	449.161	0.0259575
$670$	0	0
$671$	630.465	0.0362725
$672$	0	0
$673$	18422.4	1.05517	0.527586	−	0.849501i	$-0.323097\pi$
$673$	18422.4	1.05517	0.527586	+	0.849501i	$0.323097\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20904.7	1.18675	0.593376	−	0.804925i	$-0.297795\pi$
$677$	20904.7	1.18675	0.593376	+	0.804925i	$0.297795\pi$
$678$	0	0
$679$	1822.00	0.102978
$680$	0	0
$681$	−803.593	−0.0452184
$682$	0	0
$683$	−27298.4	−1.52935	−0.764673	−	0.644419i	$-0.777099\pi$
$683$	−27298.4	−1.52935	−0.764673	+	0.644419i	$0.777099\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1247.98	−0.0693060
$688$	0	0
$689$	−12065.0	−0.667111
$690$	0	0
$691$	−34979.1	−1.92572	−0.962858	−	0.270009i	$-0.912973\pi$
$691$	−34979.1	−1.92572	−0.962858	+	0.270009i	$0.912973\pi$
$692$	0	0
$693$	870.621	0.0477232
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−13143.7	−0.714282
$698$	0	0
$699$	1320.75	0.0714669
$700$	0	0
$701$	−25369.2	−1.36688	−0.683438	−	0.730009i	$-0.739516\pi$
$701$	−25369.2	−1.36688	−0.683438	+	0.730009i	$0.739516\pi$
$702$	0	0
$703$	29784.4	1.59792
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7259.03	−0.386144
$708$	0	0
$709$	18331.2	0.971008	0.485504	−	0.874235i	$-0.338636\pi$
$709$	18331.2	0.971008	0.485504	+	0.874235i	$0.338636\pi$
$710$	0	0
$711$	−1388.25	−0.0732259
$712$	0	0
$713$	4842.11	0.254332
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1726.13	−0.0899073
$718$	0	0
$719$	32517.9	1.68667	0.843334	−	0.537390i	$-0.180590\pi$
$719$	32517.9	1.68667	0.843334	+	0.537390i	$0.180590\pi$
$720$	0	0
$721$	5849.26	0.302133
$722$	0	0
$723$	989.365	0.0508920
$724$	0	0
$725$	0	0
$726$	0	0
$727$	9191.71	0.468916	0.234458	−	0.972126i	$-0.424669\pi$
$727$	9191.71	0.468916	0.234458	+	0.972126i	$0.424669\pi$
$728$	0	0
$729$	−19310.7	−0.981083
$730$	0	0
$731$	−6120.35	−0.309671
$732$	0	0
$733$	−31847.3	−1.60478	−0.802392	−	0.596797i	$-0.796440\pi$
$733$	−31847.3	−1.60478	−0.802392	+	0.596797i	$0.796440\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−2393.47	−0.119626
$738$	0	0
$739$	4227.58	0.210439	0.105219	−	0.994449i	$-0.466446\pi$
$739$	4227.58	0.210439	0.105219	+	0.994449i	$0.466446\pi$
$740$	0	0
$741$	−1775.66	−0.0880301
$742$	0	0
$743$	−12086.4	−0.596781	−0.298391	−	0.954444i	$-0.596450\pi$
$743$	−12086.4	−0.596781	−0.298391	+	0.954444i	$0.596450\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−20976.1	−1.02741
$748$	0	0
$749$	−5137.74	−0.250640
$750$	0	0
$751$	21471.6	1.04329	0.521643	−	0.853164i	$-0.325319\pi$
$751$	21471.6	1.04329	0.521643	+	0.853164i	$0.325319\pi$
$752$	0	0
$753$	1275.70	0.0617385
$754$	0	0
$755$	0	0
$756$	0	0
$757$	4325.57	0.207682	0.103841	−	0.994594i	$-0.466887\pi$
$757$	4325.57	0.207682	0.103841	+	0.994594i	$0.466887\pi$
$758$	0	0
$759$	30.5743	0.00146216
$760$	0	0
$761$	3652.81	0.174000	0.0870001	−	0.996208i	$-0.472272\pi$
$761$	3652.81	0.174000	0.0870001	+	0.996208i	$0.472272\pi$
$762$	0	0
$763$	266.057	0.0126237
$764$	0	0
$765$	0	0
$766$	0	0
$767$	8966.11	0.422096
$768$	0	0
$769$	−22832.8	−1.07071	−0.535353	−	0.844628i	$-0.679821\pi$
$769$	−22832.8	−1.07071	−0.535353	+	0.844628i	$0.679821\pi$
$770$	0	0
$771$	−827.755	−0.0386652
$772$	0	0
$773$	−13646.1	−0.634950	−0.317475	−	0.948267i	$-0.602835\pi$
$773$	−13646.1	−0.634950	−0.317475	+	0.948267i	$0.602835\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−536.780	−0.0247836
$778$	0	0
$779$	−39219.8	−1.80384
$780$	0	0
$781$	198.444	0.00909205
$782$	0	0
$783$	−75.1622	−0.00343049
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−36872.5	−1.67009	−0.835045	−	0.550181i	$-0.814559\pi$
$787$	−36872.5	−1.67009	−0.835045	+	0.550181i	$0.814559\pi$
$788$	0	0
$789$	1450.09	0.0654305
$790$	0	0
$791$	−13670.9	−0.614517
$792$	0	0
$793$	−7307.70	−0.327243
$794$	0	0
$795$	0	0
$796$	0	0
$797$	36139.5	1.60618	0.803091	−	0.595856i	$-0.203187\pi$
$797$	36139.5	1.60618	0.803091	+	0.595856i	$0.203187\pi$
$798$	0	0
$799$	−8563.04	−0.379147
$800$	0	0
$801$	11811.7	0.521033
$802$	0	0
$803$	−679.771	−0.0298737
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1540.56	0.0671999
$808$	0	0
$809$	3087.69	0.134187	0.0670936	−	0.997747i	$-0.478627\pi$
$809$	3087.69	0.134187	0.0670936	+	0.997747i	$0.478627\pi$
$810$	0	0
$811$	28559.2	1.23656	0.618279	−	0.785959i	$-0.287830\pi$
$811$	28559.2	1.23656	0.618279	+	0.785959i	$0.287830\pi$
$812$	0	0
$813$	−252.984	−0.0109133
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−18262.6	−0.782041
$818$	0	0
$819$	−10091.3	−0.430549
$820$	0	0
$821$	31651.7	1.34549	0.672747	−	0.739872i	$-0.265114\pi$
$821$	31651.7	1.34549	0.672747	+	0.739872i	$0.265114\pi$
$822$	0	0
$823$	37841.7	1.60277	0.801385	−	0.598150i	$-0.204097\pi$
$823$	37841.7	1.60277	0.801385	+	0.598150i	$0.204097\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13854.8	−0.582562	−0.291281	−	0.956638i	$-0.594081\pi$
$827$	−13854.8	−0.582562	−0.291281	+	0.956638i	$0.594081\pi$
$828$	0	0
$829$	41784.1	1.75057	0.875285	−	0.483607i	$-0.160674\pi$
$829$	41784.1	1.75057	0.875285	+	0.483607i	$0.160674\pi$
$830$	0	0
$831$	1581.77	0.0660301
$832$	0	0
$833$	11295.1	0.469812
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−3316.02	−0.136939
$838$	0	0
$839$	1734.45	0.0713706	0.0356853	−	0.999363i	$-0.488639\pi$
$839$	1734.45	0.0713706	0.0356853	+	0.999363i	$0.488639\pi$
$840$	0	0
$841$	−24366.2	−0.999066
$842$	0	0
$843$	480.511	0.0196319
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9315.02	0.377884
$848$	0	0
$849$	−1583.96	−0.0640300
$850$	0	0
$851$	5944.38	0.239449
$852$	0	0
$853$	−3088.04	−0.123954	−0.0619768	−	0.998078i	$-0.519740\pi$
$853$	−3088.04	−0.123954	−0.0619768	+	0.998078i	$0.519740\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−43691.8	−1.74152	−0.870761	−	0.491707i	$-0.836373\pi$
$857$	−43691.8	−1.74152	−0.870761	+	0.491707i	$0.836373\pi$
$858$	0	0
$859$	21184.0	0.841430	0.420715	−	0.907193i	$-0.361779\pi$
$859$	21184.0	0.841430	0.420715	+	0.907193i	$0.361779\pi$
$860$	0	0
$861$	706.826	0.0279774
$862$	0	0
$863$	1805.30	0.0712086	0.0356043	−	0.999366i	$-0.488664\pi$
$863$	1805.30	0.0712086	0.0356043	+	0.999366i	$0.488664\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	999.560	0.0391544
$868$	0	0
$869$	234.696	0.00916172
$870$	0	0
$871$	27742.7	1.07925
$872$	0	0
$873$	6897.98	0.267424
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−20531.8	−0.790547	−0.395273	−	0.918564i	$-0.629350\pi$
$877$	−20531.8	−0.790547	−0.395273	+	0.918564i	$0.629350\pi$
$878$	0	0
$879$	667.554	0.0256155
$880$	0	0
$881$	−10449.3	−0.399599	−0.199800	−	0.979837i	$-0.564029\pi$
$881$	−10449.3	−0.399599	−0.199800	+	0.979837i	$0.564029\pi$
$882$	0	0
$883$	32086.4	1.22287	0.611434	−	0.791295i	$-0.290593\pi$
$883$	32086.4	1.22287	0.611434	+	0.791295i	$0.290593\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−25748.0	−0.974672	−0.487336	−	0.873215i	$-0.662031\pi$
$887$	−25748.0	−0.974672	−0.487336	+	0.873215i	$0.662031\pi$
$888$	0	0
$889$	−1788.21	−0.0674631
$890$	0	0
$891$	3285.64	0.123539
$892$	0	0
$893$	−25551.3	−0.957495
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−354.386	−0.0131913
$898$	0	0
$899$	1004.61	0.0372698
$900$	0	0
$901$	−8834.96	−0.326676
$902$	0	0
$903$	329.132	0.0121294
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−24981.3	−0.914544	−0.457272	−	0.889327i	$-0.651173\pi$
$907$	−24981.3	−0.914544	−0.457272	+	0.889327i	$0.651173\pi$
$908$	0	0
$909$	−27482.3	−1.00278
$910$	0	0
$911$	−7923.44	−0.288162	−0.144081	−	0.989566i	$-0.546023\pi$
$911$	−7923.44	−0.288162	−0.144081	+	0.989566i	$0.546023\pi$
$912$	0	0
$913$	3546.19	0.128545
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16204.8	0.583564
$918$	0	0
$919$	−7186.40	−0.257951	−0.128976	−	0.991648i	$-0.541169\pi$
$919$	−7186.40	−0.257951	−0.128976	+	0.991648i	$0.541169\pi$
$920$	0	0
$921$	−1656.98	−0.0592825
$922$	0	0
$923$	−2300.16	−0.0820267
$924$	0	0
$925$	0	0
$926$	0	0
$927$	22145.0	0.784614
$928$	0	0
$929$	30627.5	1.08165	0.540827	−	0.841134i	$-0.318111\pi$
$929$	30627.5	1.08165	0.540827	+	0.841134i	$0.318111\pi$
$930$	0	0
$931$	33703.7	1.18646
$932$	0	0
$933$	−1448.64	−0.0508320
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−42609.2	−1.48557	−0.742786	−	0.669529i	$-0.766496\pi$
$937$	−42609.2	−1.48557	−0.742786	+	0.669529i	$0.766496\pi$
$938$	0	0
$939$	−2557.65	−0.0888881
$940$	0	0
$941$	−17437.3	−0.604079	−0.302040	−	0.953295i	$-0.597667\pi$
$941$	−17437.3	−0.604079	−0.302040	+	0.953295i	$0.597667\pi$
$942$	0	0
$943$	−7827.50	−0.270306
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−42729.5	−1.46623	−0.733116	−	0.680103i	$-0.761935\pi$
$947$	−42729.5	−1.46623	−0.733116	+	0.680103i	$0.761935\pi$
$948$	0	0
$949$	7879.20	0.269515
$950$	0	0
$951$	−999.017	−0.0340645
$952$	0	0
$953$	−52838.9	−1.79603	−0.898016	−	0.439962i	$-0.854992\pi$
$953$	−52838.9	−1.79603	−0.898016	+	0.439962i	$0.854992\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	6.34335	0.000214265 0
$958$	0	0
$959$	−156.520	−0.00527037
$960$	0	0
$961$	14530.5	0.487747
$962$	0	0
$963$	−19451.2	−0.650890
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21922.2	−0.729028	−0.364514	−	0.931198i	$-0.618765\pi$
$967$	−21922.2	−0.729028	−0.364514	+	0.931198i	$0.618765\pi$
$968$	0	0
$969$	−1300.28	−0.0431073
$970$	0	0
$971$	−12207.4	−0.403455	−0.201728	−	0.979442i	$-0.564656\pi$
$971$	−12207.4	−0.403455	−0.201728	+	0.979442i	$0.564656\pi$
$972$	0	0
$973$	−4013.11	−0.132224
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3582.46	−0.117311	−0.0586556	−	0.998278i	$-0.518681\pi$
$977$	−3582.46	−0.117311	−0.0586556	+	0.998278i	$0.518681\pi$
$978$	0	0
$979$	−1996.88	−0.0651895
$980$	0	0
$981$	1007.28	0.0327827
$982$	0	0
$983$	33495.5	1.08682	0.543409	−	0.839468i	$-0.317133\pi$
$983$	33495.5	1.08682	0.543409	+	0.839468i	$0.317133\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	460.491	0.0148507
$988$	0	0
$989$	−3644.86	−0.117189
$990$	0	0
$991$	44737.7	1.43405	0.717024	−	0.697049i	$-0.245504\pi$
$991$	44737.7	1.43405	0.717024	+	0.697049i	$0.245504\pi$
$992$	0	0
$993$	−1157.87	−0.0370029
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−27119.5	−0.861467	−0.430734	−	0.902479i	$-0.641745\pi$
$997$	−27119.5	−0.861467	−0.430734	+	0.902479i	$0.641745\pi$
$998$	0	0
$999$	−4070.88	−0.128926

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2300.4.a.l.1.8		16
5.2	odd	4		460.4.c.a.369.17	yes	32
5.3	odd	4		460.4.c.a.369.16	✓	32
5.4	even	2		2300.4.a.k.1.9		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
460.4.c.a.369.16	✓	32	5.3	odd	4
460.4.c.a.369.17	yes	32	5.2	odd	4
2300.4.a.k.1.9		16	5.4	even	2
2300.4.a.l.1.8		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2300 = 2^{2} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2300.a (trivial)

\(f(q)\)	\(=\)	\(q-0.292148 q^{3} -7.10909 q^{7} -26.9146 q^{9} +4.55016 q^{11} -52.7407 q^{13} -38.6210 q^{17} -115.242 q^{19} +2.07691 q^{21} -23.0000 q^{23} +15.7511 q^{27} -4.77188 q^{29} -210.527 q^{31} -1.32932 q^{33} -258.451 q^{37} +15.4081 q^{39} +340.326 q^{41} +158.472 q^{43} +221.720 q^{47} -292.461 q^{49} +11.2831 q^{51} +228.760 q^{53} +33.6677 q^{57} -170.004 q^{59} +138.559 q^{61} +191.339 q^{63} -526.020 q^{67} +6.71941 q^{69} +43.6126 q^{71} -149.395 q^{73} -32.3475 q^{77} +51.5799 q^{79} +722.094 q^{81} +779.356 q^{83} +1.39410 q^{87} -438.859 q^{89} +374.939 q^{91} +61.5049 q^{93} -256.291 q^{97} -122.466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{3} + 40 q^{7} + 110 q^{9} - 4 q^{11} + 104 q^{13} - 30 q^{17} - 72 q^{19} - 16 q^{21} - 368 q^{23} + 456 q^{27} + 38 q^{29} + 326 q^{31} + 590 q^{33} + 524 q^{37} + 564 q^{39} - 280 q^{41}+ \cdots - 976 q^{99}+O(q^{100}) \) 16 * q + 12 * q^3 + 40 * q^7 + 110 * q^9 - 4 * q^11 + 104 * q^13 - 30 * q^17 - 72 * q^19 - 16 * q^21 - 368 * q^23 + 456 * q^27 + 38 * q^29 + 326 * q^31 + 590 * q^33 + 524 * q^37 + 564 * q^39 - 280 * q^41 + 872 * q^43 + 568 * q^47 + 1112 * q^49 + 1596 * q^51 + 476 * q^53 + 408 * q^57 + 188 * q^59 + 500 * q^61 + 1082 * q^63 + 1514 * q^67 - 276 * q^69 - 366 * q^71 - 424 * q^73 + 1264 * q^77 + 544 * q^79 + 1080 * q^81 - 166 * q^83 + 4648 * q^87 - 1508 * q^89 - 48 * q^91 - 184 * q^93 + 986 * q^97 - 976 * q^99

Introduction

L-functions

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