LMFDB - Embedded newform 800.3.e.c.399.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,3,Mod(399,800)] 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("800.399"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	800.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$21.7984211488$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 7x^{10} + 44x^{8} - 156x^{6} + 704x^{4} - 1792x^{2} + 4096 $ x^12 - 7x^10 + 44x^8 - 156x^6 + 704x^4 - 1792*x^2 + 4096
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{22}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.2
Root			$1.83244 - 0.801352i$ of defining polynomial
Character	$\chi$	$=$	800.399
Dual form			800.3.e.c.399.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.03404i q^{3} +11.1194 q^{7} -7.27349 q^{9} -17.3075 q^{11} -6.70171 q^{13} -3.45185i q^{17} +0.787598 q^{19} -44.8561i q^{21} -38.1544 q^{23} -6.96480i q^{27} -37.7759i q^{29} -42.7225i q^{31} +69.8193i q^{33} -0.378525 q^{37} +27.0350i q^{39} -8.91793 q^{41} -9.21939i q^{43} -31.6031 q^{47} +74.6410 q^{49} -13.9249 q^{51} +23.9939 q^{53} -3.17720i q^{57} -47.9566 q^{59} -59.0166i q^{61} -80.8769 q^{63} +96.3852i q^{67} +153.916i q^{69} -41.1955i q^{71} -112.816i q^{73} -192.449 q^{77} +69.7575i q^{79} -93.5577 q^{81} +58.6785i q^{83} -152.389 q^{87} -44.9025 q^{89} -74.5190 q^{91} -172.344 q^{93} +126.268i q^{97} +125.886 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 16 q^{9} - 60 q^{11} - 4 q^{19} - 140 q^{41} + 276 q^{49} - 228 q^{51} - 88 q^{59} - 284 q^{81} - 196 q^{89} - 576 q^{91} + 608 q^{99}+O(q^{100}) $ 12 * q + 16 * q^9 - 60 * q^11 - 4 * q^19 - 140 * q^41 + 276 * q^49 - 228 * q^51 - 88 * q^59 - 284 * q^81 - 196 * q^89 - 576 * q^91 + 608 * q^99

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 4.03404i	− 1.34468i	−0.740242	−	0.672340i	$-0.765289\pi$
$3$	− 4.03404i	− 1.34468i	0.740242	−	0.672340i	$-0.234711\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	11.1194	1.58849	0.794243	−	0.607601i	$-0.207868\pi$
$7$	11.1194	1.58849	0.794243	+	0.607601i	$0.207868\pi$
$8$	0	0
$9$	−7.27349	−0.808166
$10$	0	0
$11$	−17.3075	−1.57341	−0.786706	−	0.617328i	$-0.788215\pi$
$11$	−17.3075	−1.57341	−0.786706	+	0.617328i	$0.788215\pi$
$12$	0	0
$13$	−6.70171	−0.515517	−0.257758	−	0.966209i	$-0.582984\pi$
$13$	−6.70171	−0.515517	−0.257758	+	0.966209i	$0.582984\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3.45185i	− 0.203050i	−0.994833	−	0.101525i	$-0.967628\pi$
$17$	− 3.45185i	− 0.203050i	0.994833	−	0.101525i	$-0.0323722\pi$
$18$	0	0
$19$	0.787598	0.0414525	0.0207263	−	0.999785i	$-0.493402\pi$
$19$	0.787598	0.0414525	0.0207263	+	0.999785i	$0.493402\pi$
$20$	0	0
$21$	− 44.8561i	− 2.13601i
$22$	0	0
$23$	−38.1544	−1.65889	−0.829443	−	0.558591i	$-0.811342\pi$
$23$	−38.1544	−1.65889	−0.829443	+	0.558591i	$0.811342\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 6.96480i	− 0.257956i
$28$	0	0
$29$	− 37.7759i	− 1.30262i	−0.758813	−	0.651308i	$-0.774221\pi$
$29$	− 37.7759i	− 1.30262i	0.758813	−	0.651308i	$-0.225779\pi$
$30$	0	0
$31$	− 42.7225i	− 1.37814i	−0.724693	−	0.689072i	$-0.758018\pi$
$31$	− 42.7225i	− 1.37814i	0.724693	−	0.689072i	$-0.241982\pi$
$32$	0	0
$33$	69.8193i	2.11574i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.378525	−0.0102304	−0.00511520	−	0.999987i	$-0.501628\pi$
$37$	−0.378525	−0.0102304	−0.00511520	+	0.999987i	$0.501628\pi$
$38$	0	0
$39$	27.0350i	0.693205i
$40$	0	0
$41$	−8.91793	−0.217511	−0.108755	−	0.994069i	$-0.534686\pi$
$41$	−8.91793	−0.217511	−0.108755	+	0.994069i	$0.534686\pi$
$42$	0	0
$43$	− 9.21939i	− 0.214405i	−0.994237	−	0.107202i	$-0.965811\pi$
$43$	− 9.21939i	− 0.214405i	0.994237	−	0.107202i	$-0.0341892\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−31.6031	−0.672406	−0.336203	−	0.941790i	$-0.609143\pi$
$47$	−31.6031	−0.672406	−0.336203	+	0.941790i	$0.609143\pi$
$48$	0	0
$49$	74.6410	1.52328
$50$	0	0
$51$	−13.9249	−0.273038
$52$	0	0
$53$	23.9939	0.452715	0.226358	−	0.974044i	$-0.427318\pi$
$53$	23.9939	0.452715	0.226358	+	0.974044i	$0.427318\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 3.17720i	− 0.0557404i
$58$	0	0
$59$	−47.9566	−0.812825	−0.406412	−	0.913690i	$-0.633220\pi$
$59$	−47.9566	−0.812825	−0.406412	+	0.913690i	$0.633220\pi$
$60$	0	0
$61$	− 59.0166i	− 0.967485i	−0.875210	−	0.483743i	$-0.839277\pi$
$61$	− 59.0166i	− 0.967485i	0.875210	−	0.483743i	$-0.160723\pi$
$62$	0	0
$63$	−80.8769	−1.28376
$64$	0	0
$65$	0	0
$66$	0	0
$67$	96.3852i	1.43859i	0.694707	+	0.719293i	$0.255534\pi$
$67$	96.3852i	1.43859i	−0.694707	+	0.719293i	$0.744466\pi$
$68$	0	0
$69$	153.916i	2.23067i
$70$	0	0
$71$	− 41.1955i	− 0.580218i	−0.956994	−	0.290109i	$-0.906308\pi$
$71$	− 41.1955i	− 0.580218i	0.956994	−	0.290109i	$-0.0936916\pi$
$72$	0	0
$73$	− 112.816i	− 1.54542i	−0.634758	−	0.772711i	$-0.718900\pi$
$73$	− 112.816i	− 1.54542i	0.634758	−	0.772711i	$-0.281100\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−192.449	−2.49934
$78$	0	0
$79$	69.7575i	0.883006i	0.897260	+	0.441503i	$0.145555\pi$
$79$	69.7575i	0.883006i	−0.897260	+	0.441503i	$0.854445\pi$
$80$	0	0
$81$	−93.5577	−1.15503
$82$	0	0
$83$	58.6785i	0.706970i	0.935440	+	0.353485i	$0.115003\pi$
$83$	58.6785i	0.706970i	−0.935440	+	0.353485i	$0.884997\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−152.389	−1.75160
$88$	0	0
$89$	−44.9025	−0.504523	−0.252262	−	0.967659i	$-0.581174\pi$
$89$	−44.9025	−0.504523	−0.252262	+	0.967659i	$0.581174\pi$
$90$	0	0
$91$	−74.5190	−0.818890
$92$	0	0
$93$	−172.344	−1.85316
$94$	0	0
$95$	0	0
$96$	0	0
$97$	126.268i	1.30173i	0.759194	+	0.650864i	$0.225593\pi$
$97$	126.268i	1.30173i	−0.759194	+	0.650864i	$0.774407\pi$
$98$	0	0
$99$	125.886	1.27158
$100$	0	0
$101$	− 40.5888i	− 0.401869i	−0.979605	−	0.200935i	$-0.935602\pi$
$101$	− 40.5888i	− 0.401869i	0.979605	−	0.200935i	$-0.0643979\pi$
$102$	0	0
$103$	111.422	1.08177	0.540884	−	0.841097i	$-0.318090\pi$
$103$	111.422	1.08177	0.540884	+	0.841097i	$0.318090\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	28.2816i	0.264314i	0.991229	+	0.132157i	$0.0421904\pi$
$107$	28.2816i	0.264314i	−0.991229	+	0.132157i	$0.957810\pi$
$108$	0	0
$109$	− 12.2679i	− 0.112549i	−0.998415	−	0.0562746i	$-0.982078\pi$
$109$	− 12.2679i	− 0.112549i	0.998415	−	0.0562746i	$-0.0179222\pi$
$110$	0	0
$111$	1.52698i	0.0137566i
$112$	0	0
$113$	73.6529i	0.651795i	0.945405	+	0.325898i	$0.105666\pi$
$113$	73.6529i	0.651795i	−0.945405	+	0.325898i	$0.894334\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	48.7449	0.416623
$118$	0	0
$119$	− 38.3825i	− 0.322542i
$120$	0	0
$121$	178.551	1.47563
$122$	0	0
$123$	35.9753i	0.292482i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−55.8251	−0.439568	−0.219784	−	0.975549i	$-0.570535\pi$
$127$	−55.8251	−0.439568	−0.219784	+	0.975549i	$0.570535\pi$
$128$	0	0
$129$	−37.1914	−0.288306
$130$	0	0
$131$	160.997	1.22898	0.614491	−	0.788924i	$-0.289362\pi$
$131$	160.997	1.22898	0.614491	+	0.788924i	$0.289362\pi$
$132$	0	0
$133$	8.75761	0.0658467
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 93.0774i	− 0.679397i	−0.940534	−	0.339699i	$-0.889675\pi$
$137$	− 93.0774i	− 0.679397i	0.940534	−	0.339699i	$-0.110325\pi$
$138$	0	0
$139$	98.2327	0.706710	0.353355	−	0.935489i	$-0.385041\pi$
$139$	98.2327	0.706710	0.353355	+	0.935489i	$0.385041\pi$
$140$	0	0
$141$	127.488i	0.904171i
$142$	0	0
$143$	115.990	0.811120
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 301.105i	− 2.04833i
$148$	0	0
$149$	− 29.2413i	− 0.196250i	−0.995174	−	0.0981251i	$-0.968715\pi$
$149$	− 29.2413i	− 0.196250i	0.995174	−	0.0981251i	$-0.0312845\pi$
$150$	0	0
$151$	− 238.075i	− 1.57666i	−0.615254	−	0.788329i	$-0.710946\pi$
$151$	− 238.075i	− 1.57666i	0.615254	−	0.788329i	$-0.289054\pi$
$152$	0	0
$153$	25.1070i	0.164098i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	230.982	1.47122	0.735612	−	0.677403i	$-0.236895\pi$
$157$	230.982	1.47122	0.735612	+	0.677403i	$0.236895\pi$
$158$	0	0
$159$	− 96.7925i	− 0.608758i
$160$	0	0
$161$	−424.254	−2.63512
$162$	0	0
$163$	139.882i	0.858175i	0.903263	+	0.429087i	$0.141165\pi$
$163$	139.882i	0.858175i	−0.903263	+	0.429087i	$0.858835\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	259.015	1.55099	0.775495	−	0.631354i	$-0.217500\pi$
$167$	259.015	1.55099	0.775495	+	0.631354i	$0.217500\pi$
$168$	0	0
$169$	−124.087	−0.734243
$170$	0	0
$171$	−5.72859	−0.0335005
$172$	0	0
$173$	45.2346	0.261472	0.130736	−	0.991417i	$-0.458266\pi$
$173$	45.2346	0.261472	0.130736	+	0.991417i	$0.458266\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	193.459i	1.09299i
$178$	0	0
$179$	−39.3701	−0.219944	−0.109972	−	0.993935i	$-0.535076\pi$
$179$	−39.3701	−0.219944	−0.109972	+	0.993935i	$0.535076\pi$
$180$	0	0
$181$	2.81294i	0.0155411i	0.999970	+	0.00777056i	$0.00247347\pi$
$181$	2.81294i	0.0155411i	−0.999970	+	0.00777056i	$0.997527\pi$
$182$	0	0
$183$	−238.075	−1.30096
$184$	0	0
$185$	0	0
$186$	0	0
$187$	59.7431i	0.319482i
$188$	0	0
$189$	− 77.4444i	− 0.409759i
$190$	0	0
$191$	− 22.9361i	− 0.120084i	−0.998196	−	0.0600421i	$-0.980876\pi$
$191$	− 22.9361i	− 0.120084i	0.998196	−	0.0600421i	$-0.0191235\pi$
$192$	0	0
$193$	− 152.406i	− 0.789666i	−0.918753	−	0.394833i	$-0.870802\pi$
$193$	− 152.406i	− 0.789666i	0.918753	−	0.394833i	$-0.129198\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	43.0232	0.218392	0.109196	−	0.994020i	$-0.465172\pi$
$197$	43.0232	0.218392	0.109196	+	0.994020i	$0.465172\pi$
$198$	0	0
$199$	− 57.1240i	− 0.287055i	−0.989646	−	0.143528i	$-0.954155\pi$
$199$	− 57.1240i	− 0.287055i	0.989646	−	0.143528i	$-0.0458446\pi$
$200$	0	0
$201$	388.822	1.93444
$202$	0	0
$203$	− 420.045i	− 2.06919i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	277.516	1.34066
$208$	0	0
$209$	−13.6314	−0.0652219
$210$	0	0
$211$	−149.294	−0.707553	−0.353776	−	0.935330i	$-0.615103\pi$
$211$	−149.294	−0.707553	−0.353776	+	0.935330i	$0.615103\pi$
$212$	0	0
$213$	−166.184	−0.780208
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 475.048i	− 2.18916i
$218$	0	0
$219$	−455.104	−2.07810
$220$	0	0
$221$	23.1333i	0.104676i
$222$	0	0
$223$	172.404	0.773112	0.386556	−	0.922266i	$-0.373665\pi$
$223$	172.404	0.773112	0.386556	+	0.922266i	$0.373665\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	159.295i	0.701738i	0.936425	+	0.350869i	$0.114114\pi$
$227$	159.295i	0.701738i	−0.936425	+	0.350869i	$0.885886\pi$
$228$	0	0
$229$	− 249.350i	− 1.08887i	−0.838804	−	0.544433i	$-0.816745\pi$
$229$	− 249.350i	− 1.08887i	0.838804	−	0.544433i	$-0.183255\pi$
$230$	0	0
$231$	776.349i	3.36082i
$232$	0	0
$233$	− 149.303i	− 0.640787i	−0.947285	−	0.320394i	$-0.896185\pi$
$233$	− 149.303i	− 0.640787i	0.947285	−	0.320394i	$-0.103815\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	281.405	1.18736
$238$	0	0
$239$	− 309.119i	− 1.29338i	−0.762751	−	0.646692i	$-0.776152\pi$
$239$	− 309.119i	− 1.29338i	0.762751	−	0.646692i	$-0.223848\pi$
$240$	0	0
$241$	3.56700	0.0148008	0.00740042	−	0.999973i	$-0.497644\pi$
$241$	3.56700	0.0148008	0.00740042	+	0.999973i	$0.497644\pi$
$242$	0	0
$243$	314.733i	1.29520i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.27825	−0.0213695
$248$	0	0
$249$	236.711	0.950648
$250$	0	0
$251$	67.0372	0.267080	0.133540	−	0.991043i	$-0.457365\pi$
$251$	67.0372	0.267080	0.133540	+	0.991043i	$0.457365\pi$
$252$	0	0
$253$	660.359	2.61011
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 115.730i	− 0.450311i	−0.974323	−	0.225156i	$-0.927711\pi$
$257$	− 115.730i	− 0.450311i	0.974323	−	0.225156i	$-0.0722890\pi$
$258$	0	0
$259$	−4.20897	−0.0162508
$260$	0	0
$261$	274.763i	1.05273i
$262$	0	0
$263$	207.074	0.787353	0.393677	−	0.919249i	$-0.371203\pi$
$263$	207.074	0.787353	0.393677	+	0.919249i	$0.371203\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	181.139i	0.678422i
$268$	0	0
$269$	179.906i	0.668797i	0.942432	+	0.334399i	$0.108533\pi$
$269$	179.906i	0.668797i	−0.942432	+	0.334399i	$0.891467\pi$
$270$	0	0
$271$	410.106i	1.51331i	0.653817	+	0.756653i	$0.273167\pi$
$271$	410.106i	1.51331i	−0.653817	+	0.756653i	$0.726833\pi$
$272$	0	0
$273$	300.613i	1.10115i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−168.937	−0.609883	−0.304941	−	0.952371i	$-0.598637\pi$
$277$	−168.937	−0.609883	−0.304941	+	0.952371i	$0.598637\pi$
$278$	0	0
$279$	310.742i	1.11377i
$280$	0	0
$281$	419.569	1.49313	0.746564	−	0.665314i	$-0.231702\pi$
$281$	419.569	1.49313	0.746564	+	0.665314i	$0.231702\pi$
$282$	0	0
$283$	− 326.668i	− 1.15430i	−0.816637	−	0.577151i	$-0.804164\pi$
$283$	− 326.668i	− 1.15430i	0.816637	−	0.577151i	$-0.195836\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−99.1620	−0.345512
$288$	0	0
$289$	277.085	0.958771
$290$	0	0
$291$	509.369	1.75041
$292$	0	0
$293$	363.495	1.24060	0.620298	−	0.784366i	$-0.287012\pi$
$293$	363.495	1.24060	0.620298	+	0.784366i	$0.287012\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	120.544i	0.405870i
$298$	0	0
$299$	255.700	0.855183
$300$	0	0
$301$	− 102.514i	− 0.340578i
$302$	0	0
$303$	−163.737	−0.540386
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 566.800i	− 1.84626i	−0.384494	−	0.923128i	$-0.625624\pi$
$307$	− 566.800i	− 1.84626i	0.384494	−	0.923128i	$-0.374376\pi$
$308$	0	0
$309$	− 449.481i	− 1.45463i
$310$	0	0
$311$	− 258.199i	− 0.830220i	−0.909771	−	0.415110i	$-0.863743\pi$
$311$	− 258.199i	− 0.830220i	0.909771	−	0.415110i	$-0.136257\pi$
$312$	0	0
$313$	24.2700i	0.0775400i	0.999248	+	0.0387700i	$0.0123440\pi$
$313$	24.2700i	0.0775400i	−0.999248	+	0.0387700i	$0.987656\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−398.458	−1.25696	−0.628482	−	0.777824i	$-0.716323\pi$
$317$	−398.458	−1.25696	−0.628482	+	0.777824i	$0.716323\pi$
$318$	0	0
$319$	653.807i	2.04955i
$320$	0	0
$321$	114.089	0.355419
$322$	0	0
$323$	− 2.71867i	− 0.00841694i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−49.4890	−0.151343
$328$	0	0
$329$	−351.407	−1.06811
$330$	0	0
$331$	−412.040	−1.24483	−0.622417	−	0.782686i	$-0.713849\pi$
$331$	−412.040	−1.24483	−0.622417	+	0.782686i	$0.713849\pi$
$332$	0	0
$333$	2.75320	0.00826786
$334$	0	0
$335$	0	0
$336$	0	0
$337$	312.690i	0.927863i	0.885871	+	0.463932i	$0.153562\pi$
$337$	312.690i	0.927863i	−0.885871	+	0.463932i	$0.846438\pi$
$338$	0	0
$339$	297.119	0.876456
$340$	0	0
$341$	739.420i	2.16839i
$342$	0	0
$343$	285.112	0.831230
$344$	0	0
$345$	0	0
$346$	0	0
$347$	429.418i	1.23751i	0.785582	+	0.618757i	$0.212364\pi$
$347$	429.418i	1.23751i	−0.785582	+	0.618757i	$0.787636\pi$
$348$	0	0
$349$	650.147i	1.86288i	0.363890	+	0.931442i	$0.381449\pi$
$349$	650.147i	1.86288i	−0.363890	+	0.931442i	$0.618551\pi$
$350$	0	0
$351$	46.6761i	0.132980i
$352$	0	0
$353$	383.224i	1.08562i	0.839856	+	0.542810i	$0.182640\pi$
$353$	383.224i	1.08562i	−0.839856	+	0.542810i	$0.817360\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−154.837	−0.433716
$358$	0	0
$359$	− 656.475i	− 1.82862i	−0.405014	−	0.914310i	$-0.632733\pi$
$359$	− 656.475i	− 1.82862i	0.405014	−	0.914310i	$-0.367267\pi$
$360$	0	0
$361$	−360.380	−0.998282
$362$	0	0
$363$	− 720.281i	− 1.98425i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.05782	0.00288235	0.00144117	−	0.999999i	$-0.499541\pi$
$367$	1.05782	0.00288235	0.00144117	+	0.999999i	$0.499541\pi$
$368$	0	0
$369$	64.8645	0.175785
$370$	0	0
$371$	266.798	0.719132
$372$	0	0
$373$	0.816797	0.00218980	0.00109490	−	0.999999i	$-0.499651\pi$
$373$	0.816797	0.00218980	0.00109490	+	0.999999i	$0.499651\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	253.163i	0.671520i
$378$	0	0
$379$	−630.860	−1.66454	−0.832269	−	0.554372i	$-0.812959\pi$
$379$	−630.860	−1.66454	−0.832269	+	0.554372i	$0.812959\pi$
$380$	0	0
$381$	225.201i	0.591078i
$382$	0	0
$383$	−392.809	−1.02561	−0.512805	−	0.858505i	$-0.671394\pi$
$383$	−392.809	−1.02561	−0.512805	+	0.858505i	$0.671394\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	67.0572i	0.173274i
$388$	0	0
$389$	− 292.218i	− 0.751203i	−0.926781	−	0.375602i	$-0.877436\pi$
$389$	− 292.218i	− 0.751203i	0.926781	−	0.375602i	$-0.122564\pi$
$390$	0	0
$391$	131.703i	0.336837i
$392$	0	0
$393$	− 649.467i	− 1.65259i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	676.730	1.70461	0.852305	−	0.523045i	$-0.175204\pi$
$397$	676.730	1.70461	0.852305	+	0.523045i	$0.175204\pi$
$398$	0	0
$399$	− 35.3286i	− 0.0885428i
$400$	0	0
$401$	195.666	0.487945	0.243972	−	0.969782i	$-0.421549\pi$
$401$	195.666	0.487945	0.243972	+	0.969782i	$0.421549\pi$
$402$	0	0
$403$	286.314i	0.710456i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.55133	0.0160966
$408$	0	0
$409$	−424.927	−1.03894	−0.519471	−	0.854488i	$-0.673871\pi$
$409$	−424.927	−1.03894	−0.519471	+	0.854488i	$0.673871\pi$
$410$	0	0
$411$	−375.478	−0.913572
$412$	0	0
$413$	−533.249	−1.29116
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 396.275i	− 0.950300i
$418$	0	0
$419$	448.621	1.07069	0.535347	−	0.844632i	$-0.320181\pi$
$419$	448.621	1.07069	0.535347	+	0.844632i	$0.320181\pi$
$420$	0	0
$421$	− 762.020i	− 1.81002i	−0.425387	−	0.905012i	$-0.639862\pi$
$421$	− 762.020i	− 1.81002i	0.425387	−	0.905012i	$-0.360138\pi$
$422$	0	0
$423$	229.865	0.543415
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 656.229i	− 1.53684i
$428$	0	0
$429$	− 467.909i	− 1.09070i
$430$	0	0
$431$	− 103.559i	− 0.240276i	−0.992757	−	0.120138i	$-0.961666\pi$
$431$	− 103.559i	− 0.240276i	0.992757	−	0.120138i	$-0.0383337\pi$
$432$	0	0
$433$	− 290.705i	− 0.671375i	−0.941973	−	0.335687i	$-0.891031\pi$
$433$	− 290.705i	− 0.671375i	0.941973	−	0.335687i	$-0.108969\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−30.0503	−0.0687650
$438$	0	0
$439$	331.332i	0.754742i	0.926062	+	0.377371i	$0.123172\pi$
$439$	331.332i	0.754742i	−0.926062	+	0.377371i	$0.876828\pi$
$440$	0	0
$441$	−542.901	−1.23107
$442$	0	0
$443$	533.782i	1.20493i	0.798147	+	0.602463i	$0.205814\pi$
$443$	533.782i	1.20493i	−0.798147	+	0.602463i	$0.794186\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−117.961	−0.263894
$448$	0	0
$449$	−431.419	−0.960844	−0.480422	−	0.877038i	$-0.659516\pi$
$449$	−431.419	−0.960844	−0.480422	+	0.877038i	$0.659516\pi$
$450$	0	0
$451$	154.347	0.342234
$452$	0	0
$453$	−960.406	−2.12010
$454$	0	0
$455$	0	0
$456$	0	0
$457$	142.066i	0.310865i	0.987846	+	0.155433i	$0.0496772\pi$
$457$	142.066i	0.310865i	−0.987846	+	0.155433i	$0.950323\pi$
$458$	0	0
$459$	−24.0415	−0.0523779
$460$	0	0
$461$	− 594.329i	− 1.28922i	−0.764513	−	0.644609i	$-0.777020\pi$
$461$	− 594.329i	− 1.28922i	0.764513	−	0.644609i	$-0.222980\pi$
$462$	0	0
$463$	594.339	1.28367	0.641835	−	0.766843i	$-0.278173\pi$
$463$	594.339	1.28367	0.641835	+	0.766843i	$0.278173\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	305.531i	0.654242i	0.944982	+	0.327121i	$0.106078\pi$
$467$	305.531i	0.654242i	−0.944982	+	0.327121i	$0.893922\pi$
$468$	0	0
$469$	1071.75i	2.28517i
$470$	0	0
$471$	− 931.792i	− 1.97833i
$472$	0	0
$473$	159.565i	0.337347i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−174.520	−0.365869
$478$	0	0
$479$	− 726.619i	− 1.51695i	−0.651703	−	0.758475i	$-0.725945\pi$
$479$	− 726.619i	− 1.51695i	0.651703	−	0.758475i	$-0.274055\pi$
$480$	0	0
$481$	2.53676	0.00527394
$482$	0	0
$483$	1711.46i	3.54339i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	331.827	0.681369	0.340685	−	0.940178i	$-0.389341\pi$
$487$	331.827	0.681369	0.340685	+	0.940178i	$0.389341\pi$
$488$	0	0
$489$	564.292	1.15397
$490$	0	0
$491$	−431.486	−0.878790	−0.439395	−	0.898294i	$-0.644807\pi$
$491$	−431.486	−0.878790	−0.439395	+	0.898294i	$0.644807\pi$
$492$	0	0
$493$	−130.397	−0.264497
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 458.069i	− 0.921668i
$498$	0	0
$499$	992.291	1.98856	0.994279	−	0.106812i	$-0.0340643\pi$
$499$	992.291	1.98856	0.994279	+	0.106812i	$0.0340643\pi$
$500$	0	0
$501$	− 1044.88i	− 2.08559i
$502$	0	0
$503$	−949.823	−1.88832	−0.944158	−	0.329492i	$-0.893123\pi$
$503$	−949.823	−1.88832	−0.944158	+	0.329492i	$0.893123\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	500.572i	0.987322i
$508$	0	0
$509$	− 283.683i	− 0.557335i	−0.960388	−	0.278667i	$-0.910107\pi$
$509$	− 283.683i	− 0.557335i	0.960388	−	0.278667i	$-0.0898927\pi$
$510$	0	0
$511$	− 1254.44i	− 2.45488i
$512$	0	0
$513$	− 5.48546i	− 0.0106929i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	546.971	1.05797
$518$	0	0
$519$	− 182.478i	− 0.351596i
$520$	0	0
$521$	−605.738	−1.16264	−0.581322	−	0.813673i	$-0.697464\pi$
$521$	−605.738	−1.16264	−0.581322	+	0.813673i	$0.697464\pi$
$522$	0	0
$523$	− 681.592i	− 1.30324i	−0.758548	−	0.651618i	$-0.774091\pi$
$523$	− 681.592i	− 1.30324i	0.758548	−	0.651618i	$-0.225909\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−147.472	−0.279832
$528$	0	0
$529$	926.758	1.75190
$530$	0	0
$531$	348.812	0.656897
$532$	0	0
$533$	59.7654	0.112130
$534$	0	0
$535$	0	0
$536$	0	0
$537$	158.820i	0.295755i
$538$	0	0
$539$	−1291.85	−2.39676
$540$	0	0
$541$	244.206i	0.451398i	0.974197	+	0.225699i	$0.0724666\pi$
$541$	244.206i	0.451398i	−0.974197	+	0.225699i	$0.927533\pi$
$542$	0	0
$543$	11.3475	0.0208979
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 377.177i	− 0.689538i	−0.938688	−	0.344769i	$-0.887957\pi$
$547$	− 377.177i	− 0.689538i	0.938688	−	0.344769i	$-0.112043\pi$
$548$	0	0
$549$	429.257i	0.781888i
$550$	0	0
$551$	− 29.7522i	− 0.0539967i
$552$	0	0
$553$	775.661i	1.40264i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−729.906	−1.31042	−0.655212	−	0.755445i	$-0.727421\pi$
$557$	−729.906	−1.31042	−0.655212	+	0.755445i	$0.727421\pi$
$558$	0	0
$559$	61.7858i	0.110529i
$560$	0	0
$561$	241.006	0.429601
$562$	0	0
$563$	93.4375i	0.165964i	0.996551	+	0.0829818i	$0.0264443\pi$
$563$	93.4375i	0.165964i	−0.996551	+	0.0829818i	$0.973556\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−1040.31	−1.83475
$568$	0	0
$569$	152.601	0.268191	0.134096	−	0.990968i	$-0.457187\pi$
$569$	152.601	0.268191	0.134096	+	0.990968i	$0.457187\pi$
$570$	0	0
$571$	197.023	0.345049	0.172524	−	0.985005i	$-0.444808\pi$
$571$	197.023	0.345049	0.172524	+	0.985005i	$0.444808\pi$
$572$	0	0
$573$	−92.5252	−0.161475
$574$	0	0
$575$	0	0
$576$	0	0
$577$	105.556i	0.182939i	0.995808	+	0.0914693i	$0.0291563\pi$
$577$	105.556i	0.182939i	−0.995808	+	0.0914693i	$0.970844\pi$
$578$	0	0
$579$	−614.810	−1.06185
$580$	0	0
$581$	652.469i	1.12301i
$582$	0	0
$583$	−415.276	−0.712308
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 140.879i	− 0.239998i	−0.992774	−	0.119999i	$-0.961711\pi$
$587$	− 140.879i	− 0.239998i	0.992774	−	0.119999i	$-0.0382892\pi$
$588$	0	0
$589$	− 33.6481i	− 0.0571275i
$590$	0	0
$591$	− 173.558i	− 0.293668i
$592$	0	0
$593$	− 830.224i	− 1.40004i	−0.714123	−	0.700020i	$-0.753174\pi$
$593$	− 830.224i	− 1.40004i	0.714123	−	0.700020i	$-0.246826\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−230.440	−0.385997
$598$	0	0
$599$	708.746i	1.18321i	0.806226	+	0.591607i	$0.201506\pi$
$599$	708.746i	1.18321i	−0.806226	+	0.591607i	$0.798494\pi$
$600$	0	0
$601$	−877.605	−1.46024	−0.730120	−	0.683319i	$-0.760536\pi$
$601$	−877.605	−1.46024	−0.730120	+	0.683319i	$0.760536\pi$
$602$	0	0
$603$	− 701.057i	− 1.16262i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	177.574	0.292543	0.146271	−	0.989244i	$-0.453273\pi$
$607$	177.574	0.292543	0.146271	+	0.989244i	$0.453273\pi$
$608$	0	0
$609$	−1694.48	−2.78239
$610$	0	0
$611$	211.795	0.346636
$612$	0	0
$613$	332.413	0.542272	0.271136	−	0.962541i	$-0.412601\pi$
$613$	332.413	0.542272	0.271136	+	0.962541i	$0.412601\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	726.357i	1.17724i	0.808410	+	0.588620i	$0.200329\pi$
$617$	726.357i	1.17724i	−0.808410	+	0.588620i	$0.799671\pi$
$618$	0	0
$619$	944.637	1.52607	0.763035	−	0.646358i	$-0.223709\pi$
$619$	944.637	1.52607	0.763035	+	0.646358i	$0.223709\pi$
$620$	0	0
$621$	265.738i	0.427919i
$622$	0	0
$623$	−499.289	−0.801427
$624$	0	0
$625$	0	0
$626$	0	0
$627$	54.9895i	0.0877026i
$628$	0	0
$629$	1.30661i	0.00207728i
$630$	0	0
$631$	373.587i	0.592056i	0.955179	+	0.296028i	$0.0956621\pi$
$631$	373.587i	0.592056i	−0.955179	+	0.296028i	$0.904338\pi$
$632$	0	0
$633$	602.257i	0.951432i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−500.222	−0.785279
$638$	0	0
$639$	299.635i	0.468912i
$640$	0	0
$641$	218.071	0.340205	0.170102	−	0.985426i	$-0.445590\pi$
$641$	218.071	0.340205	0.170102	+	0.985426i	$0.445590\pi$
$642$	0	0
$643$	− 180.700i	− 0.281027i	−0.990079	−	0.140513i	$-0.955125\pi$
$643$	− 180.700i	− 0.281027i	0.990079	−	0.140513i	$-0.0448753\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−286.652	−0.443048	−0.221524	−	0.975155i	$-0.571103\pi$
$647$	−286.652	−0.443048	−0.221524	+	0.975155i	$0.571103\pi$
$648$	0	0
$649$	830.011	1.27891
$650$	0	0
$651$	−1916.36	−2.94372
$652$	0	0
$653$	−863.087	−1.32173	−0.660863	−	0.750506i	$-0.729810\pi$
$653$	−863.087	−1.32173	−0.660863	+	0.750506i	$0.729810\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	820.565i	1.24896i
$658$	0	0
$659$	1119.80	1.69924	0.849618	−	0.527398i	$-0.176832\pi$
$659$	1119.80	1.69924	0.849618	+	0.527398i	$0.176832\pi$
$660$	0	0
$661$	− 805.904i	− 1.21922i	−0.792702	−	0.609609i	$-0.791326\pi$
$661$	− 805.904i	− 1.21922i	0.792702	−	0.609609i	$-0.208674\pi$
$662$	0	0
$663$	93.3209	0.140755
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1441.32i	2.16089i
$668$	0	0
$669$	− 695.485i	− 1.03959i
$670$	0	0
$671$	1021.43i	1.52225i
$672$	0	0
$673$	− 531.504i	− 0.789753i	−0.918734	−	0.394876i	$-0.870787\pi$
$673$	− 531.504i	− 0.789753i	0.918734	−	0.394876i	$-0.129213\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−520.567	−0.768931	−0.384466	−	0.923139i	$-0.625614\pi$
$677$	−520.567	−0.768931	−0.384466	+	0.923139i	$0.625614\pi$
$678$	0	0
$679$	1404.02i	2.06778i
$680$	0	0
$681$	642.601	0.943613
$682$	0	0
$683$	− 870.065i	− 1.27389i	−0.770910	−	0.636944i	$-0.780198\pi$
$683$	− 870.065i	− 1.27389i	0.770910	−	0.636944i	$-0.219802\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1005.89	−1.46418
$688$	0	0
$689$	−160.800	−0.233382
$690$	0	0
$691$	713.215	1.03215	0.516074	−	0.856544i	$-0.327393\pi$
$691$	713.215	1.03215	0.516074	+	0.856544i	$0.327393\pi$
$692$	0	0
$693$	1399.78	2.01988
$694$	0	0
$695$	0	0
$696$	0	0
$697$	30.7834i	0.0441656i
$698$	0	0
$699$	−602.296	−0.861654
$700$	0	0
$701$	− 1002.11i	− 1.42954i	−0.699360	−	0.714769i	$-0.746532\pi$
$701$	− 1002.11i	− 1.42954i	0.699360	−	0.714769i	$-0.253468\pi$
$702$	0	0
$703$	−0.298125	−0.000424076 0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 451.323i	− 0.638364i
$708$	0	0
$709$	1003.90i	1.41594i	0.706241	+	0.707971i	$0.250389\pi$
$709$	1003.90i	1.41594i	−0.706241	+	0.707971i	$0.749611\pi$
$710$	0	0
$711$	− 507.380i	− 0.713615i
$712$	0	0
$713$	1630.05i	2.28618i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1247.00	−1.73919
$718$	0	0
$719$	1207.30i	1.67914i	0.543252	+	0.839569i	$0.317193\pi$
$719$	1207.30i	1.67914i	−0.543252	+	0.839569i	$0.682807\pi$
$720$	0	0
$721$	1238.95	1.71837
$722$	0	0
$723$	− 14.3894i	− 0.0199024i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	587.117	0.807588	0.403794	−	0.914850i	$-0.367691\pi$
$727$	587.117	0.807588	0.403794	+	0.914850i	$0.367691\pi$
$728$	0	0
$729$	427.625	0.586591
$730$	0	0
$731$	−31.8240	−0.0435349
$732$	0	0
$733$	−434.408	−0.592644	−0.296322	−	0.955088i	$-0.595760\pi$
$733$	−434.408	−0.592644	−0.296322	+	0.955088i	$0.595760\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1668.19i	− 2.26349i
$738$	0	0
$739$	−310.341	−0.419948	−0.209974	−	0.977707i	$-0.567338\pi$
$739$	−310.341	−0.419948	−0.209974	+	0.977707i	$0.567338\pi$
$740$	0	0
$741$	21.2927i	0.0287351i
$742$	0	0
$743$	610.976	0.822310	0.411155	−	0.911566i	$-0.365126\pi$
$743$	610.976	0.822310	0.411155	+	0.911566i	$0.365126\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 426.798i	− 0.571349i
$748$	0	0
$749$	314.475i	0.419860i
$750$	0	0
$751$	622.287i	0.828611i	0.910138	+	0.414306i	$0.135976\pi$
$751$	622.287i	0.828611i	−0.910138	+	0.414306i	$0.864024\pi$
$752$	0	0
$753$	− 270.431i	− 0.359138i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	796.999	1.05284	0.526419	−	0.850225i	$-0.323534\pi$
$757$	796.999	1.05284	0.526419	+	0.850225i	$0.323534\pi$
$758$	0	0
$759$	− 2663.91i	− 3.50977i
$760$	0	0
$761$	129.738	0.170484	0.0852419	−	0.996360i	$-0.472834\pi$
$761$	129.738	0.170484	0.0852419	+	0.996360i	$0.472834\pi$
$762$	0	0
$763$	− 136.411i	− 0.178783i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	321.392	0.419024
$768$	0	0
$769$	814.433	1.05908	0.529540	−	0.848285i	$-0.322364\pi$
$769$	814.433	1.05908	0.529540	+	0.848285i	$0.322364\pi$
$770$	0	0
$771$	−466.860	−0.605525
$772$	0	0
$773$	650.087	0.840992	0.420496	−	0.907294i	$-0.361856\pi$
$773$	650.087	0.840992	0.420496	+	0.907294i	$0.361856\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.9791i	0.0218522i
$778$	0	0
$779$	−7.02374	−0.00901636
$780$	0	0
$781$	712.992i	0.912922i
$782$	0	0
$783$	−263.101	−0.336017
$784$	0	0
$785$	0	0
$786$	0	0
$787$	20.6114i	0.0261898i	0.999914	+	0.0130949i	$0.00416835\pi$
$787$	20.6114i	0.0261898i	−0.999914	+	0.0130949i	$0.995832\pi$
$788$	0	0
$789$	− 835.345i	− 1.05874i
$790$	0	0
$791$	818.975i	1.03537i
$792$	0	0
$793$	395.512i	0.498754i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	805.629	1.01083	0.505413	−	0.862877i	$-0.331340\pi$
$797$	805.629	1.01083	0.505413	+	0.862877i	$0.331340\pi$
$798$	0	0
$799$	109.089i	0.136532i
$800$	0	0
$801$	326.598	0.407738
$802$	0	0
$803$	1952.56i	2.43159i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	725.750	0.899319
$808$	0	0
$809$	292.924	0.362081	0.181041	−	0.983476i	$-0.442053\pi$
$809$	292.924	0.362081	0.181041	+	0.983476i	$0.442053\pi$
$810$	0	0
$811$	827.255	1.02004	0.510021	−	0.860162i	$-0.329637\pi$
$811$	827.255	1.02004	0.510021	+	0.860162i	$0.329637\pi$
$812$	0	0
$813$	1654.38	2.03491
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 7.26117i	− 0.00888761i
$818$	0	0
$819$	542.014	0.661799
$820$	0	0
$821$	1205.34i	1.46814i	0.679075	+	0.734069i	$0.262381\pi$
$821$	1205.34i	1.46814i	−0.679075	+	0.734069i	$0.737619\pi$
$822$	0	0
$823$	667.377	0.810908	0.405454	−	0.914116i	$-0.367114\pi$
$823$	667.377	0.810908	0.405454	+	0.914116i	$0.367114\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	513.257i	0.620625i	0.950635	+	0.310313i	$0.100434\pi$
$827$	513.257i	0.620625i	−0.950635	+	0.310313i	$0.899566\pi$
$828$	0	0
$829$	− 749.848i	− 0.904521i	−0.891886	−	0.452260i	$-0.850618\pi$
$829$	− 749.848i	− 0.904521i	0.891886	−	0.452260i	$-0.149382\pi$
$830$	0	0
$831$	681.501i	0.820097i
$832$	0	0
$833$	− 257.650i	− 0.309303i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−297.553	−0.355500
$838$	0	0
$839$	48.2988i	0.0575672i	0.999586	+	0.0287836i	$0.00916336\pi$
$839$	48.2988i	0.0575672i	−0.999586	+	0.0287836i	$0.990837\pi$
$840$	0	0
$841$	−586.016	−0.696809
$842$	0	0
$843$	− 1692.56i	− 2.00778i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1985.38	2.34401
$848$	0	0
$849$	−1317.79	−1.55217
$850$	0	0
$851$	14.4424	0.0169711
$852$	0	0
$853$	−925.590	−1.08510	−0.542550	−	0.840023i	$-0.682541\pi$
$853$	−925.590	−1.08510	−0.542550	+	0.840023i	$0.682541\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23.8375i	0.0278151i	0.999903	+	0.0139076i	$0.00442705\pi$
$857$	23.8375i	0.0278151i	−0.999903	+	0.0139076i	$0.995573\pi$
$858$	0	0
$859$	−1227.62	−1.42913	−0.714563	−	0.699571i	$-0.753374\pi$
$859$	−1227.62	−1.42913	−0.714563	+	0.699571i	$0.753374\pi$
$860$	0	0
$861$	400.024i	0.464604i
$862$	0	0
$863$	−698.456	−0.809335	−0.404667	−	0.914464i	$-0.632613\pi$
$863$	−698.456	−0.809335	−0.404667	+	0.914464i	$0.632613\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1117.77i	− 1.28924i
$868$	0	0
$869$	− 1207.33i	− 1.38933i
$870$	0	0
$871$	− 645.946i	− 0.741614i
$872$	0	0
$873$	− 918.407i	− 1.05201i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1098.23	1.25225	0.626126	−	0.779722i	$-0.284640\pi$
$877$	1098.23	1.25225	0.626126	+	0.779722i	$0.284640\pi$
$878$	0	0
$879$	− 1466.35i	− 1.66821i
$880$	0	0
$881$	435.466	0.494286	0.247143	−	0.968979i	$-0.420508\pi$
$881$	435.466	0.494286	0.247143	+	0.968979i	$0.420508\pi$
$882$	0	0
$883$	146.771i	0.166219i	0.996540	+	0.0831094i	$0.0264851\pi$
$883$	146.771i	0.166219i	−0.996540	+	0.0831094i	$0.973515\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−95.2031	−0.107332	−0.0536658	−	0.998559i	$-0.517091\pi$
$887$	−95.2031	−0.107332	−0.0536658	+	0.998559i	$0.517091\pi$
$888$	0	0
$889$	−620.742	−0.698247
$890$	0	0
$891$	1619.25	1.81734
$892$	0	0
$893$	−24.8905	−0.0278729
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1031.50i	− 1.14995i
$898$	0	0
$899$	−1613.88	−1.79519
$900$	0	0
$901$	− 82.8235i	− 0.0919240i
$902$	0	0
$903$	−413.546	−0.457969
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 496.182i	− 0.547058i	−0.961864	−	0.273529i	$-0.911809\pi$
$907$	− 496.182i	− 0.547058i	0.961864	−	0.273529i	$-0.0881909\pi$
$908$	0	0
$909$	295.222i	0.324777i
$910$	0	0
$911$	− 1431.36i	− 1.57120i	−0.618736	−	0.785599i	$-0.712355\pi$
$911$	− 1431.36i	− 1.57120i	0.618736	−	0.785599i	$-0.287645\pi$
$912$	0	0
$913$	− 1015.58i	− 1.11235i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1790.18	1.95222
$918$	0	0
$919$	496.178i	0.539911i	0.962873	+	0.269955i	$0.0870090\pi$
$919$	496.178i	0.539911i	−0.962873	+	0.269955i	$0.912991\pi$
$920$	0	0
$921$	−2286.50	−2.48262
$922$	0	0
$923$	276.080i	0.299112i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−810.428	−0.874248
$928$	0	0
$929$	832.634	0.896269	0.448135	−	0.893966i	$-0.352089\pi$
$929$	832.634	0.896269	0.448135	+	0.893966i	$0.352089\pi$
$930$	0	0
$931$	58.7870	0.0631440
$932$	0	0
$933$	−1041.58	−1.11638
$934$	0	0
$935$	0	0
$936$	0	0
$937$	146.334i	0.156173i	0.996947	+	0.0780865i	$0.0248810\pi$
$937$	146.334i	0.156173i	−0.996947	+	0.0780865i	$0.975119\pi$
$938$	0	0
$939$	97.9063	0.104267
$940$	0	0
$941$	672.850i	0.715037i	0.933906	+	0.357519i	$0.116377\pi$
$941$	672.850i	0.715037i	−0.933906	+	0.357519i	$0.883623\pi$
$942$	0	0
$943$	340.258	0.360825
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 445.927i	− 0.470883i	−0.971888	−	0.235442i	$-0.924346\pi$
$947$	− 445.927i	− 0.470883i	0.971888	−	0.235442i	$-0.0756537\pi$
$948$	0	0
$949$	756.060i	0.796691i
$950$	0	0
$951$	1607.39i	1.69022i
$952$	0	0
$953$	− 1490.29i	− 1.56379i	−0.623411	−	0.781894i	$-0.714254\pi$
$953$	− 1490.29i	− 1.56379i	0.623411	−	0.781894i	$-0.285746\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2637.49	2.75599
$958$	0	0
$959$	− 1034.96i	− 1.07921i
$960$	0	0
$961$	−864.209	−0.899281
$962$	0	0
$963$	− 205.706i	− 0.213610i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−165.170	−0.170807	−0.0854034	−	0.996346i	$-0.527218\pi$
$967$	−165.170	−0.170807	−0.0854034	+	0.996346i	$0.527218\pi$
$968$	0	0
$969$	−10.9672	−0.0113181
$970$	0	0
$971$	−1252.88	−1.29030	−0.645151	−	0.764055i	$-0.723205\pi$
$971$	−1252.88	−1.29030	−0.645151	+	0.764055i	$0.723205\pi$
$972$	0	0
$973$	1092.29	1.12260
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1446.44i	1.48049i	0.672336	+	0.740246i	$0.265291\pi$
$977$	1446.44i	1.48049i	−0.672336	+	0.740246i	$0.734709\pi$
$978$	0	0
$979$	777.152	0.793823
$980$	0	0
$981$	89.2302i	0.0909584i
$982$	0	0
$983$	762.431	0.775617	0.387808	−	0.921740i	$-0.373232\pi$
$983$	762.431	0.775617	0.387808	+	0.921740i	$0.373232\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1417.59i	1.43626i
$988$	0	0
$989$	351.760i	0.355673i
$990$	0	0
$991$	− 191.767i	− 0.193509i	−0.995308	−	0.0967545i	$-0.969154\pi$
$991$	− 191.767i	− 0.193509i	0.995308	−	0.0967545i	$-0.0308462\pi$
$992$	0	0
$993$	1662.19i	1.67390i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−409.797	−0.411030	−0.205515	−	0.978654i	$-0.565887\pi$
$997$	−409.797	−0.411030	−0.205515	+	0.978654i	$0.565887\pi$
$998$	0	0
$999$	2.63635i	0.00263899i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.3.e.c.399.2		12
4.3	odd	2		200.3.e.c.99.12		12
5.2	odd	4		800.3.g.e.751.2		6
5.3	odd	4		800.3.g.f.751.5		6
5.4	even	2	inner	800.3.e.c.399.11		12
8.3	odd	2	inner	800.3.e.c.399.1		12
8.5	even	2		200.3.e.c.99.2		12
20.3	even	4		200.3.g.f.51.3	yes	6
20.7	even	4		200.3.g.e.51.4	yes	6
20.19	odd	2		200.3.e.c.99.1		12
40.3	even	4		800.3.g.f.751.6		6
40.13	odd	4		200.3.g.f.51.4	yes	6
40.19	odd	2	inner	800.3.e.c.399.12		12
40.27	even	4		800.3.g.e.751.1		6
40.29	even	2		200.3.e.c.99.11		12
40.37	odd	4		200.3.g.e.51.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.3.e.c.99.1		12	20.19	odd	2
200.3.e.c.99.2		12	8.5	even	2
200.3.e.c.99.11		12	40.29	even	2
200.3.e.c.99.12		12	4.3	odd	2
200.3.g.e.51.3	✓	6	40.37	odd	4
200.3.g.e.51.4	yes	6	20.7	even	4
200.3.g.f.51.3	yes	6	20.3	even	4
200.3.g.f.51.4	yes	6	40.13	odd	4
800.3.e.c.399.1		12	8.3	odd	2	inner
800.3.e.c.399.2		12	1.1	even	1	trivial
800.3.e.c.399.11		12	5.4	even	2	inner
800.3.e.c.399.12		12	40.19	odd	2	inner
800.3.g.e.751.1		6	40.27	even	4
800.3.g.e.751.2		6	5.2	odd	4
800.3.g.f.751.5		6	5.3	odd	4
800.3.g.f.751.6		6	40.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	800.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-4.03404i q^{3} +11.1194 q^{7} -7.27349 q^{9} -17.3075 q^{11} -6.70171 q^{13} -3.45185i q^{17} +0.787598 q^{19} -44.8561i q^{21} -38.1544 q^{23} -6.96480i q^{27} -37.7759i q^{29} -42.7225i q^{31} +69.8193i q^{33} -0.378525 q^{37} +27.0350i q^{39} -8.91793 q^{41} -9.21939i q^{43} -31.6031 q^{47} +74.6410 q^{49} -13.9249 q^{51} +23.9939 q^{53} -3.17720i q^{57} -47.9566 q^{59} -59.0166i q^{61} -80.8769 q^{63} +96.3852i q^{67} +153.916i q^{69} -41.1955i q^{71} -112.816i q^{73} -192.449 q^{77} +69.7575i q^{79} -93.5577 q^{81} +58.6785i q^{83} -152.389 q^{87} -44.9025 q^{89} -74.5190 q^{91} -172.344 q^{93} +126.268i q^{97} +125.886 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 16 q^{9} - 60 q^{11} - 4 q^{19} - 140 q^{41} + 276 q^{49} - 228 q^{51} - 88 q^{59} - 284 q^{81} - 196 q^{89} - 576 q^{91} + 608 q^{99}+O(q^{100}) \) 12 * q + 16 * q^9 - 60 * q^11 - 4 * q^19 - 140 * q^41 + 276 * q^49 - 228 * q^51 - 88 * q^59 - 284 * q^81 - 196 * q^89 - 576 * q^91 + 608 * q^99

Introduction

L-functions

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