LMFDB - Embedded newform 800.3.g.e.751.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [800,3,Mod(751,800)] 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("800.751"); 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, 0])) N = Newforms(chi, 3, names="a")

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

Newform invariants

comment:select newform

sage:traces = [6,0,-2,0,0,0,0,0,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$21.7984211488$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.189974000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 8x^{4} - 8x^{3} + 23x^{2} + 3x + 40 $ x^6 - 3x^5 + 8x^4 - 8x^3 + 23x^2 + 3*x + 40
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.2
Root			$0.198648 + 1.83244i$ of defining polynomial
Character	$\chi$	$=$	800.751
Dual form			800.3.g.e.751.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.03404 q^{3} +11.1194i q^{7} +7.27349 q^{9} -17.3075 q^{11} +6.70171i q^{13} +3.45185 q^{17} -0.787598 q^{19} -44.8561i q^{21} +38.1544i q^{23} +6.96480 q^{27} +37.7759i q^{29} -42.7225i q^{31} +69.8193 q^{33} -0.378525i q^{37} -27.0350i q^{39} -8.91793 q^{41} -9.21939 q^{43} -31.6031i q^{47} -74.6410 q^{49} -13.9249 q^{51} -23.9939i q^{53} +3.17720 q^{57} +47.9566 q^{59} -59.0166i q^{61} +80.8769i q^{63} -96.3852 q^{67} -153.916i q^{69} -41.1955i q^{71} -112.816 q^{73} -192.449i q^{77} -69.7575i q^{79} -93.5577 q^{81} +58.6785 q^{83} -152.389i q^{87} +44.9025 q^{89} -74.5190 q^{91} +172.344i q^{93} -126.268 q^{97} -125.886 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{3} - 8 q^{9} - 30 q^{11} - 2 q^{17} + 2 q^{19} - 62 q^{27} + 138 q^{33} - 70 q^{41} - 76 q^{43} - 138 q^{49} - 114 q^{51} - 78 q^{57} + 44 q^{59} - 18 q^{67} - 18 q^{73} - 142 q^{81} + 398 q^{83}+ \cdots - 304 q^{99}+O(q^{100}) $ 6 * q - 2 * q^3 - 8 * q^9 - 30 * q^11 - 2 * q^17 + 2 * q^19 - 62 * q^27 + 138 * q^33 - 70 * q^41 - 76 * q^43 - 138 * q^49 - 114 * q^51 - 78 * q^57 + 44 * q^59 - 18 * q^67 - 18 * q^73 - 142 * q^81 + 398 * q^83 + 98 * q^89 - 288 * q^91 - 76 * q^97 - 304 * 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
$3$	−4.03404	−1.34468	−0.672340	−	0.740242i	$-0.734711\pi$
$3$	−4.03404	−1.34468	−0.672340	+	0.740242i	$0.734711\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	11.1194i	1.58849i	0.607601	+	0.794243i	$0.292132\pi$
$7$	11.1194i	1.58849i	−0.607601	+	0.794243i	$0.707868\pi$
$8$	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
$13$	6.70171i	0.515517i	0.966209	+	0.257758i	$0.0829838\pi$
$13$	6.70171i	0.515517i	−0.966209	+	0.257758i	$0.917016\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.45185	0.203050	0.101525	−	0.994833i	$-0.467628\pi$
$17$	3.45185	0.203050	0.101525	+	0.994833i	$0.467628\pi$
$18$	0	0
$19$	−0.787598	−0.0414525	−0.0207263	−	0.999785i	$-0.506598\pi$
$19$	−0.787598	−0.0414525	−0.0207263	+	0.999785i	$0.506598\pi$
$20$	0	0
$21$	− 44.8561i	− 2.13601i
$22$	0	0
$23$	38.1544i	1.65889i	0.558591	+	0.829443i	$0.311342\pi$
$23$	38.1544i	1.65889i	−0.558591	+	0.829443i	$0.688658\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	6.96480	0.257956
$28$	0	0
$29$	37.7759i	1.30262i	0.758813	+	0.651308i	$0.225779\pi$
$29$	37.7759i	1.30262i	−0.758813	+	0.651308i	$0.774221\pi$
$30$	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
$33$	69.8193	2.11574
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 0.378525i	− 0.0102304i	−0.999987	−	0.00511520i	$-0.998372\pi$
$37$	− 0.378525i	− 0.0102304i	0.999987	−	0.00511520i	$-0.00162823\pi$
$38$	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
$43$	−9.21939	−0.214405	−0.107202	−	0.994237i	$-0.534189\pi$
$43$	−9.21939	−0.214405	−0.107202	+	0.994237i	$0.534189\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 31.6031i	− 0.672406i	−0.941790	−	0.336203i	$-0.890857\pi$
$47$	− 31.6031i	− 0.672406i	0.941790	−	0.336203i	$-0.109143\pi$
$48$	0	0
$49$	−74.6410	−1.52328
$50$	0	0
$51$	−13.9249	−0.273038
$52$	0	0
$53$	− 23.9939i	− 0.452715i	−0.974044	−	0.226358i	$-0.927318\pi$
$53$	− 23.9939i	− 0.452715i	0.974044	−	0.226358i	$-0.0726818\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.17720	0.0557404
$58$	0	0
$59$	47.9566	0.812825	0.406412	−	0.913690i	$-0.366780\pi$
$59$	47.9566	0.812825	0.406412	+	0.913690i	$0.366780\pi$
$60$	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
$63$	80.8769i	1.28376i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−96.3852	−1.43859	−0.719293	−	0.694707i	$-0.755534\pi$
$67$	−96.3852	−1.43859	−0.719293	+	0.694707i	$0.755534\pi$
$68$	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
$73$	−112.816	−1.54542	−0.772711	−	0.634758i	$-0.781100\pi$
$73$	−112.816	−1.54542	−0.772711	+	0.634758i	$0.781100\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 192.449i	− 2.49934i
$78$	0	0
$79$	− 69.7575i	− 0.883006i	−0.897260	−	0.441503i	$-0.854445\pi$
$79$	− 69.7575i	− 0.883006i	0.897260	−	0.441503i	$-0.145555\pi$
$80$	0	0
$81$	−93.5577	−1.15503
$82$	0	0
$83$	58.6785	0.706970	0.353485	−	0.935440i	$-0.384997\pi$
$83$	58.6785	0.706970	0.353485	+	0.935440i	$0.384997\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 152.389i	− 1.75160i
$88$	0	0
$89$	44.9025	0.504523	0.252262	−	0.967659i	$-0.418826\pi$
$89$	44.9025	0.504523	0.252262	+	0.967659i	$0.418826\pi$
$90$	0	0
$91$	−74.5190	−0.818890
$92$	0	0
$93$	172.344i	1.85316i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−126.268	−1.30173	−0.650864	−	0.759194i	$-0.725593\pi$
$97$	−126.268	−1.30173	−0.650864	+	0.759194i	$0.725593\pi$
$98$	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
$103$	− 111.422i	− 1.08177i	−0.841097	−	0.540884i	$-0.818090\pi$
$103$	− 111.422i	− 1.08177i	0.841097	−	0.540884i	$-0.181910\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−28.2816	−0.264314	−0.132157	−	0.991229i	$-0.542190\pi$
$107$	−28.2816	−0.264314	−0.132157	+	0.991229i	$0.542190\pi$
$108$	0	0
$109$	12.2679i	0.112549i	0.998415	+	0.0562746i	$0.0179222\pi$
$109$	12.2679i	0.112549i	−0.998415	+	0.0562746i	$0.982078\pi$
$110$	0	0
$111$	1.52698i	0.0137566i
$112$	0	0
$113$	73.6529	0.651795	0.325898	−	0.945405i	$-0.394334\pi$
$113$	73.6529	0.651795	0.325898	+	0.945405i	$0.394334\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	48.7449i	0.416623i
$118$	0	0
$119$	38.3825i	0.322542i
$120$	0	0
$121$	178.551	1.47563
$122$	0	0
$123$	35.9753	0.292482
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 55.8251i	− 0.439568i	−0.975549	−	0.219784i	$-0.929465\pi$
$127$	− 55.8251i	− 0.439568i	0.975549	−	0.219784i	$-0.0705352\pi$
$128$	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
$133$	− 8.75761i	− 0.0658467i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	93.0774	0.679397	0.339699	−	0.940534i	$-0.389675\pi$
$137$	93.0774	0.679397	0.339699	+	0.940534i	$0.389675\pi$
$138$	0	0
$139$	−98.2327	−0.706710	−0.353355	−	0.935489i	$-0.614959\pi$
$139$	−98.2327	−0.706710	−0.353355	+	0.935489i	$0.614959\pi$
$140$	0	0
$141$	127.488i	0.904171i
$142$	0	0
$143$	− 115.990i	− 0.811120i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	301.105	2.04833
$148$	0	0
$149$	29.2413i	0.196250i	0.995174	+	0.0981251i	$0.0312845\pi$
$149$	29.2413i	0.196250i	−0.995174	+	0.0981251i	$0.968715\pi$
$150$	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
$153$	25.1070	0.164098
$154$	0	0
$155$	0	0
$156$	0	0
$157$	230.982i	1.47122i	0.677403	+	0.735612i	$0.263105\pi$
$157$	230.982i	1.47122i	−0.677403	+	0.735612i	$0.736895\pi$
$158$	0	0
$159$	96.7925i	0.608758i
$160$	0	0
$161$	−424.254	−2.63512
$162$	0	0
$163$	139.882	0.858175	0.429087	−	0.903263i	$-0.358835\pi$
$163$	139.882	0.858175	0.429087	+	0.903263i	$0.358835\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	259.015i	1.55099i	0.631354	+	0.775495i	$0.282500\pi$
$167$	259.015i	1.55099i	−0.631354	+	0.775495i	$0.717500\pi$
$168$	0	0
$169$	124.087	0.734243
$170$	0	0
$171$	−5.72859	−0.0335005
$172$	0	0
$173$	− 45.2346i	− 0.261472i	−0.991417	−	0.130736i	$-0.958266\pi$
$173$	− 45.2346i	− 0.261472i	0.991417	−	0.130736i	$-0.0417340\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−193.459	−1.09299
$178$	0	0
$179$	39.3701	0.219944	0.109972	−	0.993935i	$-0.464924\pi$
$179$	39.3701	0.219944	0.109972	+	0.993935i	$0.464924\pi$
$180$	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
$183$	238.075i	1.30096i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−59.7431	−0.319482
$188$	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
$193$	−152.406	−0.789666	−0.394833	−	0.918753i	$-0.629198\pi$
$193$	−152.406	−0.789666	−0.394833	+	0.918753i	$0.629198\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	43.0232i	0.218392i	0.994020	+	0.109196i	$0.0348276\pi$
$197$	43.0232i	0.218392i	−0.994020	+	0.109196i	$0.965172\pi$
$198$	0	0
$199$	57.1240i	0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$199$	57.1240i	0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$200$	0	0
$201$	388.822	1.93444
$202$	0	0
$203$	−420.045	−2.06919
$204$	0	0
$205$	0	0
$206$	0	0
$207$	277.516i	1.34066i
$208$	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
$213$	166.184i	0.780208i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	475.048	2.18916
$218$	0	0
$219$	455.104	2.07810
$220$	0	0
$221$	23.1333i	0.104676i
$222$	0	0
$223$	− 172.404i	− 0.773112i	−0.922266	−	0.386556i	$-0.873665\pi$
$223$	− 172.404i	− 0.773112i	0.922266	−	0.386556i	$-0.126335\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−159.295	−0.701738	−0.350869	−	0.936425i	$-0.614114\pi$
$227$	−159.295	−0.701738	−0.350869	+	0.936425i	$0.614114\pi$
$228$	0	0
$229$	249.350i	1.08887i	0.838804	+	0.544433i	$0.183255\pi$
$229$	249.350i	1.08887i	−0.838804	+	0.544433i	$0.816745\pi$
$230$	0	0
$231$	776.349i	3.36082i
$232$	0	0
$233$	−149.303	−0.640787	−0.320394	−	0.947285i	$-0.603815\pi$
$233$	−149.303	−0.640787	−0.320394	+	0.947285i	$0.603815\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	281.405i	1.18736i
$238$	0	0
$239$	309.119i	1.29338i	0.762751	+	0.646692i	$0.223848\pi$
$239$	309.119i	1.29338i	−0.762751	+	0.646692i	$0.776152\pi$
$240$	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
$243$	314.733	1.29520
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.27825i	− 0.0213695i
$248$	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
$253$	− 660.359i	− 2.61011i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	115.730	0.450311	0.225156	−	0.974323i	$-0.427711\pi$
$257$	115.730	0.450311	0.225156	+	0.974323i	$0.427711\pi$
$258$	0	0
$259$	4.20897	0.0162508
$260$	0	0
$261$	274.763i	1.05273i
$262$	0	0
$263$	− 207.074i	− 0.787353i	−0.919249	−	0.393677i	$-0.871203\pi$
$263$	− 207.074i	− 0.787353i	0.919249	−	0.393677i	$-0.128797\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−181.139	−0.678422
$268$	0	0
$269$	− 179.906i	− 0.668797i	−0.942432	−	0.334399i	$-0.891467\pi$
$269$	− 179.906i	− 0.668797i	0.942432	−	0.334399i	$-0.108533\pi$
$270$	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
$273$	300.613	1.10115
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 168.937i	− 0.609883i	−0.952371	−	0.304941i	$-0.901363\pi$
$277$	− 168.937i	− 0.609883i	0.952371	−	0.304941i	$-0.0986368\pi$
$278$	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
$283$	−326.668	−1.15430	−0.577151	−	0.816637i	$-0.695836\pi$
$283$	−326.668	−1.15430	−0.577151	+	0.816637i	$0.695836\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 99.1620i	− 0.345512i
$288$	0	0
$289$	−277.085	−0.958771
$290$	0	0
$291$	509.369	1.75041
$292$	0	0
$293$	− 363.495i	− 1.24060i	−0.784366	−	0.620298i	$-0.787012\pi$
$293$	− 363.495i	− 1.24060i	0.784366	−	0.620298i	$-0.212988\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−120.544	−0.405870
$298$	0	0
$299$	−255.700	−0.855183
$300$	0	0
$301$	− 102.514i	− 0.340578i
$302$	0	0
$303$	163.737i	0.540386i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	566.800	1.84626	0.923128	−	0.384494i	$-0.125624\pi$
$307$	566.800	1.84626	0.923128	+	0.384494i	$0.125624\pi$
$308$	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
$313$	24.2700	0.0775400	0.0387700	−	0.999248i	$-0.487656\pi$
$313$	24.2700	0.0775400	0.0387700	+	0.999248i	$0.487656\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 398.458i	− 1.25696i	−0.777824	−	0.628482i	$-0.783677\pi$
$317$	− 398.458i	− 1.25696i	0.777824	−	0.628482i	$-0.216323\pi$
$318$	0	0
$319$	− 653.807i	− 2.04955i
$320$	0	0
$321$	114.089	0.355419
$322$	0	0
$323$	−2.71867	−0.00841694
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 49.4890i	− 0.151343i
$328$	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
$333$	− 2.75320i	− 0.00826786i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−312.690	−0.927863	−0.463932	−	0.885871i	$-0.653562\pi$
$337$	−312.690	−0.927863	−0.463932	+	0.885871i	$0.653562\pi$
$338$	0	0
$339$	−297.119	−0.876456
$340$	0	0
$341$	739.420i	2.16839i
$342$	0	0
$343$	− 285.112i	− 0.831230i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−429.418	−1.23751	−0.618757	−	0.785582i	$-0.712364\pi$
$347$	−429.418	−1.23751	−0.618757	+	0.785582i	$0.712364\pi$
$348$	0	0
$349$	− 650.147i	− 1.86288i	−0.363890	−	0.931442i	$-0.618551\pi$
$349$	− 650.147i	− 1.86288i	0.363890	−	0.931442i	$-0.381449\pi$
$350$	0	0
$351$	46.6761i	0.132980i
$352$	0	0
$353$	383.224	1.08562	0.542810	−	0.839856i	$-0.317360\pi$
$353$	383.224	1.08562	0.542810	+	0.839856i	$0.317360\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 154.837i	− 0.433716i
$358$	0	0
$359$	656.475i	1.82862i	0.405014	+	0.914310i	$0.367267\pi$
$359$	656.475i	1.82862i	−0.405014	+	0.914310i	$0.632733\pi$
$360$	0	0
$361$	−360.380	−0.998282
$362$	0	0
$363$	−720.281	−1.98425
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.05782i	0.00288235i	0.999999	+	0.00144117i	$0.000458740\pi$
$367$	1.05782i	0.00288235i	−0.999999	+	0.00144117i	$0.999541\pi$
$368$	0	0
$369$	−64.8645	−0.175785
$370$	0	0
$371$	266.798	0.719132
$372$	0	0
$373$	− 0.816797i	− 0.00218980i	−0.999999	−	0.00109490i	$-0.999651\pi$
$373$	− 0.816797i	− 0.00218980i	0.999999	−	0.00109490i	$-0.000348518\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−253.163	−0.671520
$378$	0	0
$379$	630.860	1.66454	0.832269	−	0.554372i	$-0.187041\pi$
$379$	630.860	1.66454	0.832269	+	0.554372i	$0.187041\pi$
$380$	0	0
$381$	225.201i	0.591078i
$382$	0	0
$383$	392.809i	1.02561i	0.858505	+	0.512805i	$0.171394\pi$
$383$	392.809i	1.02561i	−0.858505	+	0.512805i	$0.828606\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−67.0572	−0.173274
$388$	0	0
$389$	292.218i	0.751203i	0.926781	+	0.375602i	$0.122564\pi$
$389$	292.218i	0.751203i	−0.926781	+	0.375602i	$0.877436\pi$
$390$	0	0
$391$	131.703i	0.336837i
$392$	0	0
$393$	−649.467	−1.65259
$394$	0	0
$395$	0	0
$396$	0	0
$397$	676.730i	1.70461i	0.523045	+	0.852305i	$0.324796\pi$
$397$	676.730i	1.70461i	−0.523045	+	0.852305i	$0.675204\pi$
$398$	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
$403$	286.314	0.710456
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.55133i	0.0160966i
$408$	0	0
$409$	424.927	1.03894	0.519471	−	0.854488i	$-0.326129\pi$
$409$	424.927	1.03894	0.519471	+	0.854488i	$0.326129\pi$
$410$	0	0
$411$	−375.478	−0.913572
$412$	0	0
$413$	533.249i	1.29116i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	396.275	0.950300
$418$	0	0
$419$	−448.621	−1.07069	−0.535347	−	0.844632i	$-0.679819\pi$
$419$	−448.621	−1.07069	−0.535347	+	0.844632i	$0.679819\pi$
$420$	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
$423$	− 229.865i	− 0.543415i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	656.229	1.53684
$428$	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
$433$	−290.705	−0.671375	−0.335687	−	0.941973i	$-0.608969\pi$
$433$	−290.705	−0.671375	−0.335687	+	0.941973i	$0.608969\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 30.0503i	− 0.0687650i
$438$	0	0
$439$	− 331.332i	− 0.754742i	−0.926062	−	0.377371i	$-0.876828\pi$
$439$	− 331.332i	− 0.754742i	0.926062	−	0.377371i	$-0.123172\pi$
$440$	0	0
$441$	−542.901	−1.23107
$442$	0	0
$443$	533.782	1.20493	0.602463	−	0.798147i	$-0.294186\pi$
$443$	533.782	1.20493	0.602463	+	0.798147i	$0.294186\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 117.961i	− 0.263894i
$448$	0	0
$449$	431.419	0.960844	0.480422	−	0.877038i	$-0.340484\pi$
$449$	431.419	0.960844	0.480422	+	0.877038i	$0.340484\pi$
$450$	0	0
$451$	154.347	0.342234
$452$	0	0
$453$	960.406i	2.12010i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−142.066	−0.310865	−0.155433	−	0.987846i	$-0.549677\pi$
$457$	−142.066	−0.310865	−0.155433	+	0.987846i	$0.549677\pi$
$458$	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
$463$	− 594.339i	− 1.28367i	−0.766843	−	0.641835i	$-0.778173\pi$
$463$	− 594.339i	− 1.28367i	0.766843	−	0.641835i	$-0.221827\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−305.531	−0.654242	−0.327121	−	0.944982i	$-0.606078\pi$
$467$	−305.531	−0.654242	−0.327121	+	0.944982i	$0.606078\pi$
$468$	0	0
$469$	− 1071.75i	− 2.28517i
$470$	0	0
$471$	− 931.792i	− 1.97833i
$472$	0	0
$473$	159.565	0.337347
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 174.520i	− 0.365869i
$478$	0	0
$479$	726.619i	1.51695i	0.651703	+	0.758475i	$0.274055\pi$
$479$	726.619i	1.51695i	−0.651703	+	0.758475i	$0.725945\pi$
$480$	0	0
$481$	2.53676	0.00527394
$482$	0	0
$483$	1711.46	3.54339
$484$	0	0
$485$	0	0
$486$	0	0
$487$	331.827i	0.681369i	0.940178	+	0.340685i	$0.110659\pi$
$487$	331.827i	0.681369i	−0.940178	+	0.340685i	$0.889341\pi$
$488$	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
$493$	130.397i	0.264497i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	458.069	0.921668
$498$	0	0
$499$	−992.291	−1.98856	−0.994279	−	0.106812i	$-0.965936\pi$
$499$	−992.291	−1.98856	−0.994279	+	0.106812i	$0.965936\pi$
$500$	0	0
$501$	− 1044.88i	− 2.08559i
$502$	0	0
$503$	949.823i	1.88832i	0.329492	+	0.944158i	$0.393123\pi$
$503$	949.823i	1.88832i	−0.329492	+	0.944158i	$0.606877\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−500.572	−0.987322
$508$	0	0
$509$	283.683i	0.557335i	0.960388	+	0.278667i	$0.0898927\pi$
$509$	283.683i	0.557335i	−0.960388	+	0.278667i	$0.910107\pi$
$510$	0	0
$511$	− 1254.44i	− 2.45488i
$512$	0	0
$513$	−5.48546	−0.0106929
$514$	0	0
$515$	0	0
$516$	0	0
$517$	546.971i	1.05797i
$518$	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
$523$	−681.592	−1.30324	−0.651618	−	0.758548i	$-0.725909\pi$
$523$	−681.592	−1.30324	−0.651618	+	0.758548i	$0.725909\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 147.472i	− 0.279832i
$528$	0	0
$529$	−926.758	−1.75190
$530$	0	0
$531$	348.812	0.656897
$532$	0	0
$533$	− 59.7654i	− 0.112130i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−158.820	−0.295755
$538$	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
$543$	− 11.3475i	− 0.0208979i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	377.177	0.689538	0.344769	−	0.938688i	$-0.387957\pi$
$547$	377.177	0.689538	0.344769	+	0.938688i	$0.387957\pi$
$548$	0	0
$549$	− 429.257i	− 0.781888i
$550$	0	0
$551$	− 29.7522i	− 0.0539967i
$552$	0	0
$553$	775.661	1.40264
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 729.906i	− 1.31042i	−0.755445	−	0.655212i	$-0.772579\pi$
$557$	− 729.906i	− 1.31042i	0.755445	−	0.655212i	$-0.227421\pi$
$558$	0	0
$559$	− 61.7858i	− 0.110529i
$560$	0	0
$561$	241.006	0.429601
$562$	0	0
$563$	93.4375	0.165964	0.0829818	−	0.996551i	$-0.473556\pi$
$563$	93.4375	0.165964	0.0829818	+	0.996551i	$0.473556\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1040.31i	− 1.83475i
$568$	0	0
$569$	−152.601	−0.268191	−0.134096	−	0.990968i	$-0.542813\pi$
$569$	−152.601	−0.268191	−0.134096	+	0.990968i	$0.542813\pi$
$570$	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
$573$	92.5252i	0.161475i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−105.556	−0.182939	−0.0914693	−	0.995808i	$-0.529156\pi$
$577$	−105.556	−0.182939	−0.0914693	+	0.995808i	$0.529156\pi$
$578$	0	0
$579$	614.810	1.06185
$580$	0	0
$581$	652.469i	1.12301i
$582$	0	0
$583$	415.276i	0.712308i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	140.879	0.239998	0.119999	−	0.992774i	$-0.461711\pi$
$587$	140.879	0.239998	0.119999	+	0.992774i	$0.461711\pi$
$588$	0	0
$589$	33.6481i	0.0571275i
$590$	0	0
$591$	− 173.558i	− 0.293668i
$592$	0	0
$593$	−830.224	−1.40004	−0.700020	−	0.714123i	$-0.746826\pi$
$593$	−830.224	−1.40004	−0.700020	+	0.714123i	$0.746826\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 230.440i	− 0.385997i
$598$	0	0
$599$	− 708.746i	− 1.18321i	−0.806226	−	0.591607i	$-0.798494\pi$
$599$	− 708.746i	− 1.18321i	0.806226	−	0.591607i	$-0.201506\pi$
$600$	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
$603$	−701.057	−1.16262
$604$	0	0
$605$	0	0
$606$	0	0
$607$	177.574i	0.292543i	0.989244	+	0.146271i	$0.0467273\pi$
$607$	177.574i	0.292543i	−0.989244	+	0.146271i	$0.953273\pi$
$608$	0	0
$609$	1694.48	2.78239
$610$	0	0
$611$	211.795	0.346636
$612$	0	0
$613$	− 332.413i	− 0.542272i	−0.962541	−	0.271136i	$-0.912601\pi$
$613$	− 332.413i	− 0.542272i	0.962541	−	0.271136i	$-0.0873993\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−726.357	−1.17724	−0.588620	−	0.808410i	$-0.700329\pi$
$617$	−726.357	−1.17724	−0.588620	+	0.808410i	$0.700329\pi$
$618$	0	0
$619$	−944.637	−1.52607	−0.763035	−	0.646358i	$-0.776291\pi$
$619$	−944.637	−1.52607	−0.763035	+	0.646358i	$0.776291\pi$
$620$	0	0
$621$	265.738i	0.427919i
$622$	0	0
$623$	499.289i	0.801427i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−54.9895	−0.0877026
$628$	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
$633$	602.257	0.951432
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 500.222i	− 0.785279i
$638$	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
$643$	−180.700	−0.281027	−0.140513	−	0.990079i	$-0.544875\pi$
$643$	−180.700	−0.281027	−0.140513	+	0.990079i	$0.544875\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 286.652i	− 0.443048i	−0.975155	−	0.221524i	$-0.928897\pi$
$647$	− 286.652i	− 0.443048i	0.975155	−	0.221524i	$-0.0711031\pi$
$648$	0	0
$649$	−830.011	−1.27891
$650$	0	0
$651$	−1916.36	−2.94372
$652$	0	0
$653$	863.087i	1.32173i	0.750506	+	0.660863i	$0.229810\pi$
$653$	863.087i	1.32173i	−0.750506	+	0.660863i	$0.770190\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−820.565	−1.24896
$658$	0	0
$659$	−1119.80	−1.69924	−0.849618	−	0.527398i	$-0.823168\pi$
$659$	−1119.80	−1.69924	−0.849618	+	0.527398i	$0.823168\pi$
$660$	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
$663$	− 93.3209i	− 0.140755i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1441.32	−2.16089
$668$	0	0
$669$	695.485i	1.03959i
$670$	0	0
$671$	1021.43i	1.52225i
$672$	0	0
$673$	−531.504	−0.789753	−0.394876	−	0.918734i	$-0.629213\pi$
$673$	−531.504	−0.789753	−0.394876	+	0.918734i	$0.629213\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 520.567i	− 0.768931i	−0.923139	−	0.384466i	$-0.874386\pi$
$677$	− 520.567i	− 0.768931i	0.923139	−	0.384466i	$-0.125614\pi$
$678$	0	0
$679$	− 1404.02i	− 2.06778i
$680$	0	0
$681$	642.601	0.943613
$682$	0	0
$683$	−870.065	−1.27389	−0.636944	−	0.770910i	$-0.719802\pi$
$683$	−870.065	−1.27389	−0.636944	+	0.770910i	$0.719802\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 1005.89i	− 1.46418i
$688$	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
$693$	− 1399.78i	− 2.01988i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−30.7834	−0.0441656
$698$	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
$703$	0.298125i	0 0.000424076i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	451.323	0.638364
$708$	0	0
$709$	− 1003.90i	− 1.41594i	−0.706241	−	0.707971i	$-0.749611\pi$
$709$	− 1003.90i	− 1.41594i	0.706241	−	0.707971i	$-0.250389\pi$
$710$	0	0
$711$	− 507.380i	− 0.713615i
$712$	0	0
$713$	1630.05	2.28618
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1247.00i	− 1.73919i
$718$	0	0
$719$	− 1207.30i	− 1.67914i	−0.543252	−	0.839569i	$-0.682807\pi$
$719$	− 1207.30i	− 1.67914i	0.543252	−	0.839569i	$-0.317193\pi$
$720$	0	0
$721$	1238.95	1.71837
$722$	0	0
$723$	−14.3894	−0.0199024
$724$	0	0
$725$	0	0
$726$	0	0
$727$	587.117i	0.807588i	0.914850	+	0.403794i	$0.132309\pi$
$727$	587.117i	0.807588i	−0.914850	+	0.403794i	$0.867691\pi$
$728$	0	0
$729$	−427.625	−0.586591
$730$	0	0
$731$	−31.8240	−0.0435349
$732$	0	0
$733$	434.408i	0.592644i	0.955088	+	0.296322i	$0.0957602\pi$
$733$	434.408i	0.592644i	−0.955088	+	0.296322i	$0.904240\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1668.19	2.26349
$738$	0	0
$739$	310.341	0.419948	0.209974	−	0.977707i	$-0.432662\pi$
$739$	310.341	0.419948	0.209974	+	0.977707i	$0.432662\pi$
$740$	0	0
$741$	21.2927i	0.0287351i
$742$	0	0
$743$	− 610.976i	− 0.822310i	−0.911566	−	0.411155i	$-0.865126\pi$
$743$	− 610.976i	− 0.822310i	0.911566	−	0.411155i	$-0.134874\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	426.798	0.571349
$748$	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
$753$	−270.431	−0.359138
$754$	0	0
$755$	0	0
$756$	0	0
$757$	796.999i	1.05284i	0.850225	+	0.526419i	$0.176466\pi$
$757$	796.999i	1.05284i	−0.850225	+	0.526419i	$0.823534\pi$
$758$	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
$763$	−136.411	−0.178783
$764$	0	0
$765$	0	0
$766$	0	0
$767$	321.392i	0.419024i
$768$	0	0
$769$	−814.433	−1.05908	−0.529540	−	0.848285i	$-0.677636\pi$
$769$	−814.433	−1.05908	−0.529540	+	0.848285i	$0.677636\pi$
$770$	0	0
$771$	−466.860	−0.605525
$772$	0	0
$773$	− 650.087i	− 0.840992i	−0.907294	−	0.420496i	$-0.861856\pi$
$773$	− 650.087i	− 0.840992i	0.907294	−	0.420496i	$-0.138144\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−16.9791	−0.0218522
$778$	0	0
$779$	7.02374	0.00901636
$780$	0	0
$781$	712.992i	0.912922i
$782$	0	0
$783$	263.101i	0.336017i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−20.6114	−0.0261898	−0.0130949	−	0.999914i	$-0.504168\pi$
$787$	−20.6114	−0.0261898	−0.0130949	+	0.999914i	$0.504168\pi$
$788$	0	0
$789$	835.345i	1.05874i
$790$	0	0
$791$	818.975i	1.03537i
$792$	0	0
$793$	395.512	0.498754
$794$	0	0
$795$	0	0
$796$	0	0
$797$	805.629i	1.01083i	0.862877	+	0.505413i	$0.168660\pi$
$797$	805.629i	1.01083i	−0.862877	+	0.505413i	$0.831340\pi$
$798$	0	0
$799$	− 109.089i	− 0.136532i
$800$	0	0
$801$	326.598	0.407738
$802$	0	0
$803$	1952.56	2.43159
$804$	0	0
$805$	0	0
$806$	0	0
$807$	725.750i	0.899319i
$808$	0	0
$809$	−292.924	−0.362081	−0.181041	−	0.983476i	$-0.557947\pi$
$809$	−292.924	−0.362081	−0.181041	+	0.983476i	$0.557947\pi$
$810$	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
$813$	− 1654.38i	− 2.03491i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.26117	0.00888761
$818$	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
$823$	− 667.377i	− 0.810908i	−0.914116	−	0.405454i	$-0.867114\pi$
$823$	− 667.377i	− 0.810908i	0.914116	−	0.405454i	$-0.132886\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−513.257	−0.620625	−0.310313	−	0.950635i	$-0.600434\pi$
$827$	−513.257	−0.620625	−0.310313	+	0.950635i	$0.600434\pi$
$828$	0	0
$829$	749.848i	0.904521i	0.891886	+	0.452260i	$0.149382\pi$
$829$	749.848i	0.904521i	−0.891886	+	0.452260i	$0.850618\pi$
$830$	0	0
$831$	681.501i	0.820097i
$832$	0	0
$833$	−257.650	−0.309303
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 297.553i	− 0.355500i
$838$	0	0
$839$	− 48.2988i	− 0.0575672i	−0.999586	−	0.0287836i	$-0.990837\pi$
$839$	− 48.2988i	− 0.0575672i	0.999586	−	0.0287836i	$-0.00916336\pi$
$840$	0	0
$841$	−586.016	−0.696809
$842$	0	0
$843$	−1692.56	−2.00778
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1985.38i	2.34401i
$848$	0	0
$849$	1317.79	1.55217
$850$	0	0
$851$	14.4424	0.0169711
$852$	0	0
$853$	925.590i	1.08510i	0.840023	+	0.542550i	$0.182541\pi$
$853$	925.590i	1.08510i	−0.840023	+	0.542550i	$0.817459\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−23.8375	−0.0278151	−0.0139076	−	0.999903i	$-0.504427\pi$
$857$	−23.8375	−0.0278151	−0.0139076	+	0.999903i	$0.504427\pi$
$858$	0	0
$859$	1227.62	1.42913	0.714563	−	0.699571i	$-0.246626\pi$
$859$	1227.62	1.42913	0.714563	+	0.699571i	$0.246626\pi$
$860$	0	0
$861$	400.024i	0.464604i
$862$	0	0
$863$	698.456i	0.809335i	0.914464	+	0.404667i	$0.132613\pi$
$863$	698.456i	0.809335i	−0.914464	+	0.404667i	$0.867387\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1117.77	1.28924
$868$	0	0
$869$	1207.33i	1.38933i
$870$	0	0
$871$	− 645.946i	− 0.741614i
$872$	0	0
$873$	−918.407	−1.05201
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1098.23i	1.25225i	0.779722	+	0.626126i	$0.215360\pi$
$877$	1098.23i	1.25225i	−0.779722	+	0.626126i	$0.784640\pi$
$878$	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
$883$	146.771	0.166219	0.0831094	−	0.996540i	$-0.473515\pi$
$883$	146.771	0.166219	0.0831094	+	0.996540i	$0.473515\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 95.2031i	− 0.107332i	−0.998559	−	0.0536658i	$-0.982909\pi$
$887$	− 95.2031i	− 0.107332i	0.998559	−	0.0536658i	$-0.0170906\pi$
$888$	0	0
$889$	620.742	0.698247
$890$	0	0
$891$	1619.25	1.81734
$892$	0	0
$893$	24.8905i	0.0278729i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1031.50	1.14995
$898$	0	0
$899$	1613.88	1.79519
$900$	0	0
$901$	− 82.8235i	− 0.0919240i
$902$	0	0
$903$	413.546i	0.457969i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	496.182	0.547058	0.273529	−	0.961864i	$-0.411809\pi$
$907$	496.182	0.547058	0.273529	+	0.961864i	$0.411809\pi$
$908$	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
$913$	−1015.58	−1.11235
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1790.18i	1.95222i
$918$	0	0
$919$	− 496.178i	− 0.539911i	−0.962873	−	0.269955i	$-0.912991\pi$
$919$	− 496.178i	− 0.539911i	0.962873	−	0.269955i	$-0.0870090\pi$
$920$	0	0
$921$	−2286.50	−2.48262
$922$	0	0
$923$	276.080	0.299112
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 810.428i	− 0.874248i
$928$	0	0
$929$	−832.634	−0.896269	−0.448135	−	0.893966i	$-0.647911\pi$
$929$	−832.634	−0.896269	−0.448135	+	0.893966i	$0.647911\pi$
$930$	0	0
$931$	58.7870	0.0631440
$932$	0	0
$933$	1041.58i	1.11638i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−146.334	−0.156173	−0.0780865	−	0.996947i	$-0.524881\pi$
$937$	−146.334	−0.156173	−0.0780865	+	0.996947i	$0.524881\pi$
$938$	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
$943$	− 340.258i	− 0.360825i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	445.927	0.470883	0.235442	−	0.971888i	$-0.424346\pi$
$947$	445.927	0.470883	0.235442	+	0.971888i	$0.424346\pi$
$948$	0	0
$949$	− 756.060i	− 0.796691i
$950$	0	0
$951$	1607.39i	1.69022i
$952$	0	0
$953$	−1490.29	−1.56379	−0.781894	−	0.623411i	$-0.785746\pi$
$953$	−1490.29	−1.56379	−0.781894	+	0.623411i	$0.785746\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2637.49i	2.75599i
$958$	0	0
$959$	1034.96i	1.07921i
$960$	0	0
$961$	−864.209	−0.899281
$962$	0	0
$963$	−205.706	−0.213610
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 165.170i	− 0.170807i	−0.996346	−	0.0854034i	$-0.972782\pi$
$967$	− 165.170i	− 0.170807i	0.996346	−	0.0854034i	$-0.0272179\pi$
$968$	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
$973$	− 1092.29i	− 1.12260i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1446.44	−1.48049	−0.740246	−	0.672336i	$-0.765291\pi$
$977$	−1446.44	−1.48049	−0.740246	+	0.672336i	$0.765291\pi$
$978$	0	0
$979$	−777.152	−0.793823
$980$	0	0
$981$	89.2302i	0.0909584i
$982$	0	0
$983$	− 762.431i	− 0.775617i	−0.921740	−	0.387808i	$-0.873232\pi$
$983$	− 762.431i	− 0.775617i	0.921740	−	0.387808i	$-0.126768\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1417.59	−1.43626
$988$	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
$993$	1662.19	1.67390
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 409.797i	− 0.411030i	−0.978654	−	0.205515i	$-0.934113\pi$
$997$	− 409.797i	− 0.411030i	0.978654	−	0.205515i	$-0.0658869\pi$
$998$	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.g.e.751.2		6
4.3	odd	2		200.3.g.e.51.4	yes	6
5.2	odd	4		800.3.e.c.399.11		12
5.3	odd	4		800.3.e.c.399.2		12
5.4	even	2		800.3.g.f.751.5		6
8.3	odd	2	inner	800.3.g.e.751.1		6
8.5	even	2		200.3.g.e.51.3	✓	6
20.3	even	4		200.3.e.c.99.12		12
20.7	even	4		200.3.e.c.99.1		12
20.19	odd	2		200.3.g.f.51.3	yes	6
40.3	even	4		800.3.e.c.399.1		12
40.13	odd	4		200.3.e.c.99.2		12
40.19	odd	2		800.3.g.f.751.6		6
40.27	even	4		800.3.e.c.399.12		12
40.29	even	2		200.3.g.f.51.4	yes	6
40.37	odd	4		200.3.e.c.99.11		12

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-4.03404 q^{3} +11.1194i q^{7} +7.27349 q^{9} -17.3075 q^{11} +6.70171i q^{13} +3.45185 q^{17} -0.787598 q^{19} -44.8561i q^{21} +38.1544i q^{23} +6.96480 q^{27} +37.7759i q^{29} -42.7225i q^{31} +69.8193 q^{33} -0.378525i q^{37} -27.0350i q^{39} -8.91793 q^{41} -9.21939 q^{43} -31.6031i q^{47} -74.6410 q^{49} -13.9249 q^{51} -23.9939i q^{53} +3.17720 q^{57} +47.9566 q^{59} -59.0166i q^{61} +80.8769i q^{63} -96.3852 q^{67} -153.916i q^{69} -41.1955i q^{71} -112.816 q^{73} -192.449i q^{77} -69.7575i q^{79} -93.5577 q^{81} +58.6785 q^{83} -152.389i q^{87} +44.9025 q^{89} -74.5190 q^{91} +172.344i q^{93} -126.268 q^{97} -125.886 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{3} - 8 q^{9} - 30 q^{11} - 2 q^{17} + 2 q^{19} - 62 q^{27} + 138 q^{33} - 70 q^{41} - 76 q^{43} - 138 q^{49} - 114 q^{51} - 78 q^{57} + 44 q^{59} - 18 q^{67} - 18 q^{73} - 142 q^{81} + 398 q^{83}+ \cdots - 304 q^{99}+O(q^{100}) \) 6 * q - 2 * q^3 - 8 * q^9 - 30 * q^11 - 2 * q^17 + 2 * q^19 - 62 * q^27 + 138 * q^33 - 70 * q^41 - 76 * q^43 - 138 * q^49 - 114 * q^51 - 78 * q^57 + 44 * q^59 - 18 * q^67 - 18 * q^73 - 142 * q^81 + 398 * q^83 + 98 * q^89 - 288 * q^91 - 76 * q^97 - 304 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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