LMFDB - Embedded newform 784.4.a.bf.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,4,Mod(1,784)] 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("784.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	784.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,40,0,-100] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$46.2574974445$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{65})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 35x^{2} + 36x + 194 $ x^4 - 2x^3 - 35x^2 + 36*x + 194
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2\cdot 7 $
Twist minimal:	no (minimal twist has level 49)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-4.94534$ of defining polynomial
Character	$\chi$	$=$	784.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+7.82220 q^{3} -2.07730 q^{5} +34.1868 q^{9} -49.1868 q^{11} -44.8559 q^{13} -16.2490 q^{15} +26.5179 q^{17} -77.7350 q^{19} -55.7510 q^{23} -120.685 q^{25} +56.2164 q^{27} +121.436 q^{29} -305.553 q^{31} -384.749 q^{33} +77.1868 q^{37} -350.872 q^{39} +248.720 q^{41} +147.179 q^{43} -71.0161 q^{45} -269.851 q^{47} +207.428 q^{51} -141.121 q^{53} +102.176 q^{55} -608.058 q^{57} +424.834 q^{59} -587.996 q^{61} +93.1790 q^{65} +179.634 q^{67} -436.095 q^{69} -674.872 q^{71} +237.489 q^{73} -944.021 q^{75} -495.852 q^{79} -483.307 q^{81} +24.4406 q^{83} -55.0855 q^{85} +949.895 q^{87} +1072.29 q^{89} -2390.09 q^{93} +161.479 q^{95} +1667.43 q^{97} -1681.54 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 40 q^{9} - 100 q^{11} + 64 q^{15} - 352 q^{23} - 128 q^{25} + 260 q^{29} + 212 q^{37} - 952 q^{39} - 540 q^{43} - 428 q^{51} + 16 q^{53} - 1884 q^{57} - 756 q^{65} + 1944 q^{67} - 2248 q^{71} + 1048 q^{79}+ \cdots - 3340 q^{99}+O(q^{100}) $ 4 * q + 40 * q^9 - 100 * q^11 + 64 * q^15 - 352 * q^23 - 128 * q^25 + 260 * q^29 + 212 * q^37 - 952 * q^39 - 540 * q^43 - 428 * q^51 + 16 * q^53 - 1884 * q^57 - 756 * q^65 + 1944 * q^67 - 2248 * q^71 + 1048 * q^79 - 1256 * q^81 - 3284 * q^85 - 5368 * q^93 - 2192 * q^95 - 3340 * 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
$3$	7.82220	1.50538	0.752691	−	0.658374i	$-0.228755\pi$
$3$	7.82220	1.50538	0.752691	+	0.658374i	$0.228755\pi$
$4$	0	0
$5$	−2.07730	−0.185799	−0.0928996	−	0.995675i	$-0.529614\pi$
$5$	−2.07730	−0.185799	−0.0928996	+	0.995675i	$0.529614\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	34.1868	1.26618
$10$	0	0
$11$	−49.1868	−1.34822	−0.674108	−	0.738633i	$-0.735472\pi$
$11$	−49.1868	−1.34822	−0.674108	+	0.738633i	$0.735472\pi$
$12$	0	0
$13$	−44.8559	−0.956983	−0.478492	−	0.878092i	$-0.658816\pi$
$13$	−44.8559	−0.956983	−0.478492	+	0.878092i	$0.658816\pi$
$14$	0	0
$15$	−16.2490	−0.279699
$16$	0	0
$17$	26.5179	0.378325	0.189163	−	0.981946i	$-0.439423\pi$
$17$	26.5179	0.378325	0.189163	+	0.981946i	$0.439423\pi$
$18$	0	0
$19$	−77.7350	−0.938612	−0.469306	−	0.883036i	$-0.655496\pi$
$19$	−77.7350	−0.938612	−0.469306	+	0.883036i	$0.655496\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−55.7510	−0.505430	−0.252715	−	0.967541i	$-0.581323\pi$
$23$	−55.7510	−0.505430	−0.252715	+	0.967541i	$0.581323\pi$
$24$	0	0
$25$	−120.685	−0.965479
$26$	0	0
$27$	56.2164	0.400698
$28$	0	0
$29$	121.436	0.777588	0.388794	−	0.921325i	$-0.372892\pi$
$29$	121.436	0.777588	0.388794	+	0.921325i	$0.372892\pi$
$30$	0	0
$31$	−305.553	−1.77029	−0.885143	−	0.465319i	$-0.845940\pi$
$31$	−305.553	−1.77029	−0.885143	+	0.465319i	$0.845940\pi$
$32$	0	0
$33$	−384.749	−2.02958
$34$	0	0
$35$	0	0
$36$	0	0
$37$	77.1868	0.342957	0.171479	−	0.985188i	$-0.445146\pi$
$37$	77.1868	0.342957	0.171479	+	0.985188i	$0.445146\pi$
$38$	0	0
$39$	−350.872	−1.44063
$40$	0	0
$41$	248.720	0.947403	0.473702	−	0.880685i	$-0.342917\pi$
$41$	248.720	0.947403	0.473702	+	0.880685i	$0.342917\pi$
$42$	0	0
$43$	147.179	0.521967	0.260984	−	0.965343i	$-0.415953\pi$
$43$	147.179	0.521967	0.260984	+	0.965343i	$0.415953\pi$
$44$	0	0
$45$	−71.0161	−0.235255
$46$	0	0
$47$	−269.851	−0.837484	−0.418742	−	0.908105i	$-0.637529\pi$
$47$	−269.851	−0.837484	−0.418742	+	0.908105i	$0.637529\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	207.428	0.569524
$52$	0	0
$53$	−141.121	−0.365744	−0.182872	−	0.983137i	$-0.558539\pi$
$53$	−141.121	−0.365744	−0.182872	+	0.983137i	$0.558539\pi$
$54$	0	0
$55$	102.176	0.250497
$56$	0	0
$57$	−608.058	−1.41297
$58$	0	0
$59$	424.834	0.937434	0.468717	−	0.883348i	$-0.344716\pi$
$59$	424.834	0.937434	0.468717	+	0.883348i	$0.344716\pi$
$60$	0	0
$61$	−587.996	−1.23418	−0.617092	−	0.786891i	$-0.711689\pi$
$61$	−587.996	−1.23418	−0.617092	+	0.786891i	$0.711689\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	93.1790	0.177807
$66$	0	0
$67$	179.634	0.327549	0.163775	−	0.986498i	$-0.447633\pi$
$67$	179.634	0.327549	0.163775	+	0.986498i	$0.447633\pi$
$68$	0	0
$69$	−436.095	−0.760865
$70$	0	0
$71$	−674.872	−1.12806	−0.564032	−	0.825753i	$-0.690750\pi$
$71$	−674.872	−1.12806	−0.564032	+	0.825753i	$0.690750\pi$
$72$	0	0
$73$	237.489	0.380767	0.190383	−	0.981710i	$-0.439027\pi$
$73$	237.489	0.380767	0.190383	+	0.981710i	$0.439027\pi$
$74$	0	0
$75$	−944.021	−1.45341
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−495.852	−0.706174	−0.353087	−	0.935591i	$-0.614868\pi$
$79$	−495.852	−0.706174	−0.353087	+	0.935591i	$0.614868\pi$
$80$	0	0
$81$	−483.307	−0.662973
$82$	0	0
$83$	24.4406	0.0323217	0.0161609	−	0.999869i	$-0.494856\pi$
$83$	24.4406	0.0323217	0.0161609	+	0.999869i	$0.494856\pi$
$84$	0	0
$85$	−55.0855	−0.0702925
$86$	0	0
$87$	949.895	1.17057
$88$	0	0
$89$	1072.29	1.27710	0.638552	−	0.769579i	$-0.279534\pi$
$89$	1072.29	1.27710	0.638552	+	0.769579i	$0.279534\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−2390.09	−2.66496
$94$	0	0
$95$	161.479	0.174393
$96$	0	0
$97$	1667.43	1.74538	0.872690	−	0.488275i	$-0.162374\pi$
$97$	1667.43	1.74538	0.872690	+	0.488275i	$0.162374\pi$
$98$	0	0
$99$	−1681.54	−1.70708
$100$	0	0
$101$	−77.1187	−0.0759762	−0.0379881	−	0.999278i	$-0.512095\pi$
$101$	−77.1187	−0.0759762	−0.0379881	+	0.999278i	$0.512095\pi$
$102$	0	0
$103$	−164.693	−0.157550	−0.0787749	−	0.996892i	$-0.525101\pi$
$103$	−164.693	−0.157550	−0.0787749	+	0.996892i	$0.525101\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1022.62	−0.923931	−0.461966	−	0.886898i	$-0.652856\pi$
$107$	−1022.62	−0.923931	−0.461966	+	0.886898i	$0.652856\pi$
$108$	0	0
$109$	1362.52	1.19730	0.598649	−	0.801011i	$-0.295704\pi$
$109$	1362.52	1.19730	0.598649	+	0.801011i	$0.295704\pi$
$110$	0	0
$111$	603.770	0.516282
$112$	0	0
$113$	−1538.41	−1.28072	−0.640360	−	0.768075i	$-0.721215\pi$
$113$	−1538.41	−1.28072	−0.640360	+	0.768075i	$0.721215\pi$
$114$	0	0
$115$	115.811	0.0939084
$116$	0	0
$117$	−1533.48	−1.21171
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1088.34	0.817685
$122$	0	0
$123$	1945.54	1.42620
$124$	0	0
$125$	510.360	0.365184
$126$	0	0
$127$	170.358	0.119030	0.0595151	−	0.998227i	$-0.481045\pi$
$127$	170.358	0.119030	0.0595151	+	0.998227i	$0.481045\pi$
$128$	0	0
$129$	1151.26	0.785760
$130$	0	0
$131$	751.935	0.501503	0.250751	−	0.968051i	$-0.419322\pi$
$131$	751.935	0.501503	0.250751	+	0.968051i	$0.419322\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−116.778	−0.0744493
$136$	0	0
$137$	518.623	0.323423	0.161711	−	0.986838i	$-0.448299\pi$
$137$	518.623	0.323423	0.161711	+	0.986838i	$0.448299\pi$
$138$	0	0
$139$	2975.72	1.81581	0.907905	−	0.419177i	$-0.137681\pi$
$139$	2975.72	1.81581	0.907905	+	0.419177i	$0.137681\pi$
$140$	0	0
$141$	−2110.83	−1.26073
$142$	0	0
$143$	2206.32	1.29022
$144$	0	0
$145$	−252.258	−0.144475
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2717.94	−1.49438	−0.747188	−	0.664612i	$-0.768597\pi$
$149$	−2717.94	−1.49438	−0.747188	+	0.664612i	$0.768597\pi$
$150$	0	0
$151$	−707.650	−0.381376	−0.190688	−	0.981651i	$-0.561072\pi$
$151$	−707.650	−0.381376	−0.190688	+	0.981651i	$0.561072\pi$
$152$	0	0
$153$	906.561	0.479027
$154$	0	0
$155$	634.724	0.328918
$156$	0	0
$157$	3117.91	1.58495	0.792473	−	0.609906i	$-0.208793\pi$
$157$	3117.91	1.58495	0.792473	+	0.609906i	$0.208793\pi$
$158$	0	0
$159$	−1103.87	−0.550584
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−1808.77	−0.869167	−0.434583	−	0.900632i	$-0.643104\pi$
$163$	−1808.77	−0.869167	−0.434583	+	0.900632i	$0.643104\pi$
$164$	0	0
$165$	799.237	0.377094
$166$	0	0
$167$	−3147.38	−1.45839	−0.729197	−	0.684303i	$-0.760106\pi$
$167$	−3147.38	−1.45839	−0.729197	+	0.684303i	$0.760106\pi$
$168$	0	0
$169$	−184.949	−0.0841827
$170$	0	0
$171$	−2657.51	−1.18845
$172$	0	0
$173$	−3284.36	−1.44338	−0.721691	−	0.692215i	$-0.756635\pi$
$173$	−3284.36	−1.44338	−0.721691	+	0.692215i	$0.756635\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3323.13	1.41120
$178$	0	0
$179$	−2798.83	−1.16868	−0.584341	−	0.811508i	$-0.698647\pi$
$179$	−2798.83	−1.16868	−0.584341	+	0.811508i	$0.698647\pi$
$180$	0	0
$181$	−3723.04	−1.52890	−0.764451	−	0.644682i	$-0.776990\pi$
$181$	−3723.04	−1.52890	−0.764451	+	0.644682i	$0.776990\pi$
$182$	0	0
$183$	−4599.42	−1.85792
$184$	0	0
$185$	−160.340	−0.0637212
$186$	0	0
$187$	−1304.33	−0.510064
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−959.650	−0.363549	−0.181774	−	0.983340i	$-0.558184\pi$
$191$	−959.650	−0.363549	−0.181774	+	0.983340i	$0.558184\pi$
$192$	0	0
$193$	−3790.25	−1.41362	−0.706808	−	0.707406i	$-0.749865\pi$
$193$	−3790.25	−1.41362	−0.706808	+	0.707406i	$0.749865\pi$
$194$	0	0
$195$	728.865	0.267667
$196$	0	0
$197$	5117.99	1.85097	0.925487	−	0.378779i	$-0.123656\pi$
$197$	5117.99	1.85097	0.925487	+	0.378779i	$0.123656\pi$
$198$	0	0
$199$	−864.855	−0.308080	−0.154040	−	0.988065i	$-0.549228\pi$
$199$	−864.855	−0.308080	−0.154040	+	0.988065i	$0.549228\pi$
$200$	0	0
$201$	1405.13	0.493087
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−516.665	−0.176027
$206$	0	0
$207$	−1905.95	−0.639963
$208$	0	0
$209$	3823.53	1.26545
$210$	0	0
$211$	1344.61	0.438707	0.219353	−	0.975645i	$-0.429605\pi$
$211$	1344.61	0.438707	0.219353	+	0.975645i	$0.429605\pi$
$212$	0	0
$213$	−5278.98	−1.69817
$214$	0	0
$215$	−305.735	−0.0969811
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1857.68	0.573200
$220$	0	0
$221$	−1189.48	−0.362051
$222$	0	0
$223$	864.916	0.259727	0.129863	−	0.991532i	$-0.458546\pi$
$223$	864.916	0.259727	0.129863	+	0.991532i	$0.458546\pi$
$224$	0	0
$225$	−4125.83	−1.22247
$226$	0	0
$227$	−1715.34	−0.501548	−0.250774	−	0.968046i	$-0.580685\pi$
$227$	−1715.34	−0.501548	−0.250774	+	0.968046i	$0.580685\pi$
$228$	0	0
$229$	1045.46	0.301685	0.150842	−	0.988558i	$-0.451801\pi$
$229$	1045.46	0.301685	0.150842	+	0.988558i	$0.451801\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1448.67	0.407320	0.203660	−	0.979042i	$-0.434716\pi$
$233$	1448.67	0.407320	0.203660	+	0.979042i	$0.434716\pi$
$234$	0	0
$235$	560.560	0.155604
$236$	0	0
$237$	−3878.65	−1.06306
$238$	0	0
$239$	3153.12	0.853383	0.426691	−	0.904397i	$-0.359679\pi$
$239$	3153.12	0.853383	0.426691	+	0.904397i	$0.359679\pi$
$240$	0	0
$241$	381.012	0.101839	0.0509194	−	0.998703i	$-0.483785\pi$
$241$	381.012	0.101839	0.0509194	+	0.998703i	$0.483785\pi$
$242$	0	0
$243$	−5298.37	−1.39873
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3486.87	0.898236
$248$	0	0
$249$	191.179	0.0486565
$250$	0	0
$251$	3776.23	0.949617	0.474808	−	0.880089i	$-0.342517\pi$
$251$	3776.23	0.949617	0.474808	+	0.880089i	$0.342517\pi$
$252$	0	0
$253$	2742.21	0.681428
$254$	0	0
$255$	−430.890	−0.105817
$256$	0	0
$257$	−4258.42	−1.03359	−0.516795	−	0.856109i	$-0.672875\pi$
$257$	−4258.42	−1.03359	−0.516795	+	0.856109i	$0.672875\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4151.50	0.984564
$262$	0	0
$263$	−4198.83	−0.984451	−0.492226	−	0.870468i	$-0.663816\pi$
$263$	−4198.83	−0.984451	−0.492226	+	0.870468i	$0.663816\pi$
$264$	0	0
$265$	293.150	0.0679548
$266$	0	0
$267$	8387.65	1.92253
$268$	0	0
$269$	−3740.59	−0.847835	−0.423917	−	0.905701i	$-0.639345\pi$
$269$	−3740.59	−0.847835	−0.423917	+	0.905701i	$0.639345\pi$
$270$	0	0
$271$	4356.30	0.976480	0.488240	−	0.872709i	$-0.337639\pi$
$271$	4356.30	0.976480	0.488240	+	0.872709i	$0.337639\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5936.10	1.30167
$276$	0	0
$277$	−1344.30	−0.291593	−0.145797	−	0.989315i	$-0.546575\pi$
$277$	−1344.30	−0.291593	−0.145797	+	0.989315i	$0.546575\pi$
$278$	0	0
$279$	−10445.9	−2.24150
$280$	0	0
$281$	4205.54	0.892817	0.446408	−	0.894829i	$-0.352703\pi$
$281$	4205.54	0.892817	0.446408	+	0.894829i	$0.352703\pi$
$282$	0	0
$283$	4752.03	0.998159	0.499079	−	0.866556i	$-0.333672\pi$
$283$	4752.03	0.998159	0.499079	+	0.866556i	$0.333672\pi$
$284$	0	0
$285$	1263.12	0.262529
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−4209.80	−0.856870
$290$	0	0
$291$	13043.0	2.62746
$292$	0	0
$293$	4961.17	0.989196	0.494598	−	0.869122i	$-0.335315\pi$
$293$	4961.17	0.989196	0.494598	+	0.869122i	$0.335315\pi$
$294$	0	0
$295$	−882.506	−0.174174
$296$	0	0
$297$	−2765.10	−0.540227
$298$	0	0
$299$	2500.76	0.483688
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−603.237	−0.114373
$304$	0	0
$305$	1221.44	0.229310
$306$	0	0
$307$	−4234.00	−0.787124	−0.393562	−	0.919298i	$-0.628757\pi$
$307$	−4234.00	−0.787124	−0.393562	+	0.919298i	$0.628757\pi$
$308$	0	0
$309$	−1288.26	−0.237173
$310$	0	0
$311$	−684.700	−0.124842	−0.0624209	−	0.998050i	$-0.519882\pi$
$311$	−684.700	−0.124842	−0.0624209	+	0.998050i	$0.519882\pi$
$312$	0	0
$313$	5944.07	1.07341	0.536707	−	0.843768i	$-0.319668\pi$
$313$	5944.07	1.07341	0.536707	+	0.843768i	$0.319668\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−2823.89	−0.500333	−0.250166	−	0.968203i	$-0.580485\pi$
$317$	−2823.89	−0.500333	−0.250166	+	0.968203i	$0.580485\pi$
$318$	0	0
$319$	−5973.04	−1.04836
$320$	0	0
$321$	−7999.16	−1.39087
$322$	0	0
$323$	−2061.37	−0.355101
$324$	0	0
$325$	5413.43	0.923947
$326$	0	0
$327$	10657.9	1.80239
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2812.97	0.467114	0.233557	−	0.972343i	$-0.424963\pi$
$331$	2812.97	0.467114	0.233557	+	0.972343i	$0.424963\pi$
$332$	0	0
$333$	2638.77	0.434245
$334$	0	0
$335$	−373.154	−0.0608584
$336$	0	0
$337$	4260.10	0.688612	0.344306	−	0.938857i	$-0.388114\pi$
$337$	4260.10	0.688612	0.344306	+	0.938857i	$0.388114\pi$
$338$	0	0
$339$	−12033.7	−1.92797
$340$	0	0
$341$	15029.2	2.38673
$342$	0	0
$343$	0	0
$344$	0	0
$345$	905.899	0.141368
$346$	0	0
$347$	−36.0584	−0.00557843	−0.00278922	−	0.999996i	$-0.500888\pi$
$347$	−36.0584	−0.00557843	−0.00278922	+	0.999996i	$0.500888\pi$
$348$	0	0
$349$	242.692	0.0372236	0.0186118	−	0.999827i	$-0.494075\pi$
$349$	242.692	0.0372236	0.0186118	+	0.999827i	$0.494075\pi$
$350$	0	0
$351$	−2521.63	−0.383461
$352$	0	0
$353$	109.990	0.0165840	0.00829201	−	0.999966i	$-0.497361\pi$
$353$	109.990	0.0165840	0.00829201	+	0.999966i	$0.497361\pi$
$354$	0	0
$355$	1401.91	0.209593
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−12404.5	−1.82363	−0.911814	−	0.410604i	$-0.865318\pi$
$359$	−12404.5	−1.82363	−0.911814	+	0.410604i	$0.865318\pi$
$360$	0	0
$361$	−816.273	−0.119008
$362$	0	0
$363$	8513.20	1.23093
$364$	0	0
$365$	−493.335	−0.0707461
$366$	0	0
$367$	13859.6	1.97130	0.985649	−	0.168807i	$-0.0539913\pi$
$367$	13859.6	1.97130	0.985649	+	0.168807i	$0.0539913\pi$
$368$	0	0
$369$	8502.93	1.19958
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4898.06	0.679925	0.339963	−	0.940439i	$-0.389586\pi$
$373$	4898.06	0.679925	0.339963	+	0.940439i	$0.389586\pi$
$374$	0	0
$375$	3992.14	0.549742
$376$	0	0
$377$	−5447.11	−0.744139
$378$	0	0
$379$	9806.25	1.32906	0.664530	−	0.747262i	$-0.268632\pi$
$379$	9806.25	1.32906	0.664530	+	0.747262i	$0.268632\pi$
$380$	0	0
$381$	1332.57	0.179186
$382$	0	0
$383$	10729.7	1.43149	0.715746	−	0.698361i	$-0.246087\pi$
$383$	10729.7	1.43149	0.715746	+	0.698361i	$0.246087\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	5031.58	0.660903
$388$	0	0
$389$	5264.05	0.686113	0.343057	−	0.939315i	$-0.388538\pi$
$389$	5264.05	0.686113	0.343057	+	0.939315i	$0.388538\pi$
$390$	0	0
$391$	−1478.40	−0.191217
$392$	0	0
$393$	5881.79	0.754954
$394$	0	0
$395$	1030.03	0.131206
$396$	0	0
$397$	−1214.90	−0.153587	−0.0767935	−	0.997047i	$-0.524468\pi$
$397$	−1214.90	−0.153587	−0.0767935	+	0.997047i	$0.524468\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2295.45	0.285859	0.142929	−	0.989733i	$-0.454348\pi$
$401$	2295.45	0.285859	0.142929	+	0.989733i	$0.454348\pi$
$402$	0	0
$403$	13705.8	1.69413
$404$	0	0
$405$	1003.97	0.123180
$406$	0	0
$407$	−3796.57	−0.462381
$408$	0	0
$409$	−4646.54	−0.561753	−0.280876	−	0.959744i	$-0.590625\pi$
$409$	−4646.54	−0.561753	−0.280876	+	0.959744i	$0.590625\pi$
$410$	0	0
$411$	4056.77	0.486875
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−50.7703	−0.00600535
$416$	0	0
$417$	23276.7	2.73349
$418$	0	0
$419$	7541.24	0.879269	0.439634	−	0.898177i	$-0.355108\pi$
$419$	7541.24	0.879269	0.439634	+	0.898177i	$0.355108\pi$
$420$	0	0
$421$	−6243.63	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$421$	−6243.63	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$422$	0	0
$423$	−9225.32	−1.06040
$424$	0	0
$425$	−3200.31	−0.365265
$426$	0	0
$427$	0	0
$428$	0	0
$429$	17258.2	1.94227
$430$	0	0
$431$	−11465.8	−1.28141	−0.640706	−	0.767786i	$-0.721358\pi$
$431$	−11465.8	−1.28141	−0.640706	+	0.767786i	$0.721358\pi$
$432$	0	0
$433$	−5156.40	−0.572289	−0.286144	−	0.958187i	$-0.592374\pi$
$433$	−5156.40	−0.572289	−0.286144	+	0.958187i	$0.592374\pi$
$434$	0	0
$435$	−1973.21	−0.217491
$436$	0	0
$437$	4333.80	0.474402
$438$	0	0
$439$	−5064.25	−0.550577	−0.275289	−	0.961362i	$-0.588773\pi$
$439$	−5064.25	−0.550577	−0.275289	+	0.961362i	$0.588773\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12703.6	−1.36246	−0.681228	−	0.732071i	$-0.738554\pi$
$443$	−12703.6	−1.36246	−0.681228	+	0.732071i	$0.738554\pi$
$444$	0	0
$445$	−2227.46	−0.237285
$446$	0	0
$447$	−21260.2	−2.24961
$448$	0	0
$449$	13942.2	1.46542	0.732709	−	0.680542i	$-0.238256\pi$
$449$	13942.2	1.46542	0.732709	+	0.680542i	$0.238256\pi$
$450$	0	0
$451$	−12233.7	−1.27730
$452$	0	0
$453$	−5535.38	−0.574116
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−15214.0	−1.55729	−0.778646	−	0.627464i	$-0.784093\pi$
$457$	−15214.0	−1.55729	−0.778646	+	0.627464i	$0.784093\pi$
$458$	0	0
$459$	1490.74	0.151594
$460$	0	0
$461$	11430.2	1.15479	0.577394	−	0.816465i	$-0.304070\pi$
$461$	11430.2	1.15479	0.577394	+	0.816465i	$0.304070\pi$
$462$	0	0
$463$	9347.88	0.938300	0.469150	−	0.883119i	$-0.344560\pi$
$463$	9347.88	0.938300	0.469150	+	0.883119i	$0.344560\pi$
$464$	0	0
$465$	4964.94	0.495147
$466$	0	0
$467$	3630.84	0.359776	0.179888	−	0.983687i	$-0.442427\pi$
$467$	3630.84	0.359776	0.179888	+	0.983687i	$0.442427\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	24388.9	2.38595
$472$	0	0
$473$	−7239.26	−0.703724
$474$	0	0
$475$	9381.43	0.906210
$476$	0	0
$477$	−4824.46	−0.463096
$478$	0	0
$479$	6521.25	0.622053	0.311027	−	0.950401i	$-0.399327\pi$
$479$	6521.25	0.622053	0.311027	+	0.950401i	$0.399327\pi$
$480$	0	0
$481$	−3462.28	−0.328205
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3463.75	−0.324290
$486$	0	0
$487$	3666.29	0.341140	0.170570	−	0.985346i	$-0.445439\pi$
$487$	3666.29	0.341140	0.170570	+	0.985346i	$0.445439\pi$
$488$	0	0
$489$	−14148.6	−1.30843
$490$	0	0
$491$	12470.7	1.14623	0.573113	−	0.819476i	$-0.305736\pi$
$491$	12470.7	1.14623	0.573113	+	0.819476i	$0.305736\pi$
$492$	0	0
$493$	3220.22	0.294181
$494$	0	0
$495$	3493.05	0.317174
$496$	0	0
$497$	0	0
$498$	0	0
$499$	2303.93	0.206690	0.103345	−	0.994646i	$-0.467045\pi$
$499$	2303.93	0.206690	0.103345	+	0.994646i	$0.467045\pi$
$500$	0	0
$501$	−24619.5	−2.19544
$502$	0	0
$503$	−10520.4	−0.932570	−0.466285	−	0.884635i	$-0.654408\pi$
$503$	−10520.4	−0.932570	−0.466285	+	0.884635i	$0.654408\pi$
$504$	0	0
$505$	160.198	0.0141163
$506$	0	0
$507$	−1446.71	−0.126727
$508$	0	0
$509$	9662.22	0.841395	0.420698	−	0.907201i	$-0.361785\pi$
$509$	9662.22	0.841395	0.420698	+	0.907201i	$0.361785\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4369.98	−0.376100
$514$	0	0
$515$	342.115	0.0292726
$516$	0	0
$517$	13273.1	1.12911
$518$	0	0
$519$	−25690.9	−2.17284
$520$	0	0
$521$	−8607.81	−0.723829	−0.361914	−	0.932211i	$-0.617877\pi$
$521$	−8607.81	−0.723829	−0.361914	+	0.932211i	$0.617877\pi$
$522$	0	0
$523$	−10482.7	−0.876439	−0.438219	−	0.898868i	$-0.644391\pi$
$523$	−10482.7	−0.876439	−0.438219	+	0.898868i	$0.644391\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8102.61	−0.669744
$528$	0	0
$529$	−9058.83	−0.744541
$530$	0	0
$531$	14523.7	1.18696
$532$	0	0
$533$	−11156.6	−0.906649
$534$	0	0
$535$	2124.29	0.171666
$536$	0	0
$537$	−21893.0	−1.75931
$538$	0	0
$539$	0	0
$540$	0	0
$541$	20722.6	1.64683	0.823416	−	0.567438i	$-0.192066\pi$
$541$	20722.6	1.64683	0.823416	+	0.567438i	$0.192066\pi$
$542$	0	0
$543$	−29122.3	−2.30158
$544$	0	0
$545$	−2830.35	−0.222457
$546$	0	0
$547$	4175.09	0.326351	0.163176	−	0.986597i	$-0.447826\pi$
$547$	4175.09	0.326351	0.163176	+	0.986597i	$0.447826\pi$
$548$	0	0
$549$	−20101.7	−1.56270
$550$	0	0
$551$	−9439.81	−0.729854
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1254.21	−0.0959248
$556$	0	0
$557$	−10161.7	−0.773011	−0.386505	−	0.922287i	$-0.626318\pi$
$557$	−10161.7	−0.773011	−0.386505	+	0.922287i	$0.626318\pi$
$558$	0	0
$559$	−6601.85	−0.499514
$560$	0	0
$561$	−10202.7	−0.767841
$562$	0	0
$563$	−17104.4	−1.28040	−0.640201	−	0.768208i	$-0.721149\pi$
$563$	−17104.4	−1.28040	−0.640201	+	0.768208i	$0.721149\pi$
$564$	0	0
$565$	3195.73	0.237957
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18257.6	−1.34516	−0.672581	−	0.740023i	$-0.734815\pi$
$569$	−18257.6	−1.34516	−0.672581	+	0.740023i	$0.734815\pi$
$570$	0	0
$571$	−13630.5	−0.998982	−0.499491	−	0.866319i	$-0.666480\pi$
$571$	−13630.5	−0.998982	−0.499491	+	0.866319i	$0.666480\pi$
$572$	0	0
$573$	−7506.57	−0.547280
$574$	0	0
$575$	6728.30	0.487981
$576$	0	0
$577$	−4442.08	−0.320496	−0.160248	−	0.987077i	$-0.551229\pi$
$577$	−4442.08	−0.320496	−0.160248	+	0.987077i	$0.551229\pi$
$578$	0	0
$579$	−29648.0	−2.12803
$580$	0	0
$581$	0	0
$582$	0	0
$583$	6941.27	0.493101
$584$	0	0
$585$	3185.49	0.225135
$586$	0	0
$587$	3103.38	0.218211	0.109106	−	0.994030i	$-0.465201\pi$
$587$	3103.38	0.218211	0.109106	+	0.994030i	$0.465201\pi$
$588$	0	0
$589$	23752.1	1.66161
$590$	0	0
$591$	40033.9	2.78642
$592$	0	0
$593$	−5937.71	−0.411185	−0.205592	−	0.978638i	$-0.565912\pi$
$593$	−5937.71	−0.411185	−0.205592	+	0.978638i	$0.565912\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6765.07	−0.463778
$598$	0	0
$599$	2600.33	0.177373	0.0886866	−	0.996060i	$-0.471733\pi$
$599$	2600.33	0.177373	0.0886866	+	0.996060i	$0.471733\pi$
$600$	0	0
$601$	13881.4	0.942156	0.471078	−	0.882092i	$-0.343865\pi$
$601$	13881.4	0.942156	0.471078	+	0.882092i	$0.343865\pi$
$602$	0	0
$603$	6141.11	0.414735
$604$	0	0
$605$	−2260.80	−0.151925
$606$	0	0
$607$	−12284.6	−0.821442	−0.410721	−	0.911761i	$-0.634723\pi$
$607$	−12284.6	−0.821442	−0.410721	+	0.911761i	$0.634723\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12104.4	0.801459
$612$	0	0
$613$	22062.0	1.45363	0.726815	−	0.686833i	$-0.241000\pi$
$613$	22062.0	1.45363	0.726815	+	0.686833i	$0.241000\pi$
$614$	0	0
$615$	−4041.46	−0.264988
$616$	0	0
$617$	−12182.2	−0.794871	−0.397436	−	0.917630i	$-0.630100\pi$
$617$	−12182.2	−0.794871	−0.397436	+	0.917630i	$0.630100\pi$
$618$	0	0
$619$	23248.6	1.50960	0.754799	−	0.655956i	$-0.227734\pi$
$619$	23248.6	1.50960	0.754799	+	0.655956i	$0.227734\pi$
$620$	0	0
$621$	−3134.12	−0.202525
$622$	0	0
$623$	0	0
$624$	0	0
$625$	14025.4	0.897628
$626$	0	0
$627$	29908.4	1.90499
$628$	0	0
$629$	2046.83	0.129749
$630$	0	0
$631$	−19184.4	−1.21033	−0.605165	−	0.796100i	$-0.706893\pi$
$631$	−19184.4	−1.21033	−0.605165	+	0.796100i	$0.706893\pi$
$632$	0	0
$633$	10517.8	0.660421
$634$	0	0
$635$	−353.884	−0.0221157
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−23071.7	−1.42833
$640$	0	0
$641$	−19433.4	−1.19746	−0.598730	−	0.800951i	$-0.704328\pi$
$641$	−19433.4	−1.19746	−0.598730	+	0.800951i	$0.704328\pi$
$642$	0	0
$643$	5777.47	0.354341	0.177170	−	0.984180i	$-0.443306\pi$
$643$	5777.47	0.354341	0.177170	+	0.984180i	$0.443306\pi$
$644$	0	0
$645$	−2391.52	−0.145994
$646$	0	0
$647$	−29231.5	−1.77621	−0.888106	−	0.459640i	$-0.847979\pi$
$647$	−29231.5	−1.77621	−0.888106	+	0.459640i	$0.847979\pi$
$648$	0	0
$649$	−20896.2	−1.26386
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7093.27	−0.425086	−0.212543	−	0.977152i	$-0.568174\pi$
$653$	−7093.27	−0.425086	−0.212543	+	0.977152i	$0.568174\pi$
$654$	0	0
$655$	−1561.99	−0.0931788
$656$	0	0
$657$	8118.98	0.482118
$658$	0	0
$659$	−19014.2	−1.12396	−0.561980	−	0.827151i	$-0.689960\pi$
$659$	−19014.2	−1.12396	−0.561980	+	0.827151i	$0.689960\pi$
$660$	0	0
$661$	21058.4	1.23915	0.619573	−	0.784939i	$-0.287306\pi$
$661$	21058.4	1.23915	0.619573	+	0.784939i	$0.287306\pi$
$662$	0	0
$663$	−9304.37	−0.545025
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6770.16	−0.393016
$668$	0	0
$669$	6765.54	0.390988
$670$	0	0
$671$	28921.6	1.66395
$672$	0	0
$673$	9634.87	0.551853	0.275926	−	0.961179i	$-0.411015\pi$
$673$	9634.87	0.551853	0.275926	+	0.961179i	$0.411015\pi$
$674$	0	0
$675$	−6784.46	−0.386865
$676$	0	0
$677$	−8371.31	−0.475237	−0.237619	−	0.971359i	$-0.576367\pi$
$677$	−8371.31	−0.475237	−0.237619	+	0.971359i	$0.576367\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−13417.8	−0.755022
$682$	0	0
$683$	12068.8	0.676137	0.338069	−	0.941121i	$-0.390226\pi$
$683$	12068.8	0.676137	0.338069	+	0.941121i	$0.390226\pi$
$684$	0	0
$685$	−1077.33	−0.0600917
$686$	0	0
$687$	8177.78	0.454151
$688$	0	0
$689$	6330.09	0.350011
$690$	0	0
$691$	2981.29	0.164130	0.0820648	−	0.996627i	$-0.473849\pi$
$691$	2981.29	0.164130	0.0820648	+	0.996627i	$0.473849\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6181.46	−0.337376
$696$	0	0
$697$	6595.53	0.358427
$698$	0	0
$699$	11331.8	0.613172
$700$	0	0
$701$	−28978.0	−1.56132	−0.780660	−	0.624956i	$-0.785117\pi$
$701$	−28978.0	−1.56132	−0.780660	+	0.624956i	$0.785117\pi$
$702$	0	0
$703$	−6000.11	−0.321904
$704$	0	0
$705$	4384.81	0.234243
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−16372.4	−0.867249	−0.433625	−	0.901094i	$-0.642766\pi$
$709$	−16372.4	−0.867249	−0.433625	+	0.901094i	$0.642766\pi$
$710$	0	0
$711$	−16951.6	−0.894141
$712$	0	0
$713$	17034.9	0.894755
$714$	0	0
$715$	−4583.18	−0.239722
$716$	0	0
$717$	24664.3	1.28467
$718$	0	0
$719$	23010.5	1.19353	0.596765	−	0.802416i	$-0.296453\pi$
$719$	23010.5	1.19353	0.596765	+	0.802416i	$0.296453\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	2980.35	0.153306
$724$	0	0
$725$	−14655.5	−0.750745
$726$	0	0
$727$	−24636.8	−1.25685	−0.628423	−	0.777872i	$-0.716299\pi$
$727$	−24636.8	−1.25685	−0.628423	+	0.777872i	$0.716299\pi$
$728$	0	0
$729$	−28395.6	−1.44264
$730$	0	0
$731$	3902.87	0.197473
$732$	0	0
$733$	−6904.76	−0.347931	−0.173965	−	0.984752i	$-0.555658\pi$
$733$	−6904.76	−0.347931	−0.173965	+	0.984752i	$0.555658\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8835.63	−0.441607
$738$	0	0
$739$	9234.89	0.459690	0.229845	−	0.973227i	$-0.426178\pi$
$739$	9234.89	0.459690	0.229845	+	0.973227i	$0.426178\pi$
$740$	0	0
$741$	27275.0	1.35219
$742$	0	0
$743$	−20216.9	−0.998232	−0.499116	−	0.866535i	$-0.666342\pi$
$743$	−20216.9	−0.998232	−0.499116	+	0.866535i	$0.666342\pi$
$744$	0	0
$745$	5645.97	0.277654
$746$	0	0
$747$	835.544	0.0409250
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−24054.9	−1.16881	−0.584405	−	0.811462i	$-0.698672\pi$
$751$	−24054.9	−1.16881	−0.584405	+	0.811462i	$0.698672\pi$
$752$	0	0
$753$	29538.4	1.42954
$754$	0	0
$755$	1470.00	0.0708593
$756$	0	0
$757$	−30328.2	−1.45614	−0.728069	−	0.685504i	$-0.759582\pi$
$757$	−30328.2	−1.45614	−0.728069	+	0.685504i	$0.759582\pi$
$758$	0	0
$759$	21450.1	1.02581
$760$	0	0
$761$	33834.1	1.61168	0.805839	−	0.592135i	$-0.201715\pi$
$761$	33834.1	1.61168	0.805839	+	0.592135i	$0.201715\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1883.20	−0.0890027
$766$	0	0
$767$	−19056.3	−0.897109
$768$	0	0
$769$	−31738.1	−1.48830	−0.744151	−	0.668011i	$-0.767146\pi$
$769$	−31738.1	−1.48830	−0.744151	+	0.668011i	$0.767146\pi$
$770$	0	0
$771$	−33310.2	−1.55595
$772$	0	0
$773$	−27494.2	−1.27930	−0.639650	−	0.768667i	$-0.720920\pi$
$773$	−27494.2	−1.27930	−0.639650	+	0.768667i	$0.720920\pi$
$774$	0	0
$775$	36875.6	1.70917
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−19334.2	−0.889244
$780$	0	0
$781$	33194.8	1.52087
$782$	0	0
$783$	6826.68	0.311578
$784$	0	0
$785$	−6476.84	−0.294482
$786$	0	0
$787$	−468.356	−0.0212136	−0.0106068	−	0.999944i	$-0.503376\pi$
$787$	−468.356	−0.0212136	−0.0106068	+	0.999944i	$0.503376\pi$
$788$	0	0
$789$	−32844.0	−1.48198
$790$	0	0
$791$	0	0
$792$	0	0
$793$	26375.1	1.18109
$794$	0	0
$795$	2293.07	0.102298
$796$	0	0
$797$	−37723.8	−1.67659	−0.838297	−	0.545214i	$-0.816449\pi$
$797$	−37723.8	−1.67659	−0.838297	+	0.545214i	$0.816449\pi$
$798$	0	0
$799$	−7155.87	−0.316841
$800$	0	0
$801$	36658.1	1.61704
$802$	0	0
$803$	−11681.3	−0.513356
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−29259.6	−1.27632
$808$	0	0
$809$	−7797.13	−0.338854	−0.169427	−	0.985543i	$-0.554192\pi$
$809$	−7797.13	−0.338854	−0.169427	+	0.985543i	$0.554192\pi$
$810$	0	0
$811$	16925.9	0.732860	0.366430	−	0.930446i	$-0.380580\pi$
$811$	16925.9	0.732860	0.366430	+	0.930446i	$0.380580\pi$
$812$	0	0
$813$	34075.8	1.46998
$814$	0	0
$815$	3757.36	0.161490
$816$	0	0
$817$	−11441.0	−0.489925
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−30009.3	−1.27568	−0.637840	−	0.770169i	$-0.720172\pi$
$821$	−30009.3	−1.27568	−0.637840	+	0.770169i	$0.720172\pi$
$822$	0	0
$823$	23385.6	0.990486	0.495243	−	0.868754i	$-0.335079\pi$
$823$	23385.6	0.990486	0.495243	+	0.868754i	$0.335079\pi$
$824$	0	0
$825$	46433.3	1.95952
$826$	0	0
$827$	37325.9	1.56947	0.784734	−	0.619833i	$-0.212800\pi$
$827$	37325.9	1.56947	0.784734	+	0.619833i	$0.212800\pi$
$828$	0	0
$829$	24671.3	1.03362	0.516809	−	0.856100i	$-0.327120\pi$
$829$	24671.3	1.03362	0.516809	+	0.856100i	$0.327120\pi$
$830$	0	0
$831$	−10515.4	−0.438960
$832$	0	0
$833$	0	0
$834$	0	0
$835$	6538.05	0.270969
$836$	0	0
$837$	−17177.1	−0.709350
$838$	0	0
$839$	14147.4	0.582147	0.291074	−	0.956701i	$-0.405987\pi$
$839$	14147.4	0.582147	0.291074	+	0.956701i	$0.405987\pi$
$840$	0	0
$841$	−9642.35	−0.395356
$842$	0	0
$843$	32896.6	1.34403
$844$	0	0
$845$	384.195	0.0156411
$846$	0	0
$847$	0	0
$848$	0	0
$849$	37171.3	1.50261
$850$	0	0
$851$	−4303.24	−0.173341
$852$	0	0
$853$	−27963.6	−1.12246	−0.561229	−	0.827661i	$-0.689671\pi$
$853$	−27963.6	−1.12246	−0.561229	+	0.827661i	$0.689671\pi$
$854$	0	0
$855$	5520.44	0.220813
$856$	0	0
$857$	−33855.8	−1.34947	−0.674734	−	0.738061i	$-0.735741\pi$
$857$	−33855.8	−1.34947	−0.674734	+	0.738061i	$0.735741\pi$
$858$	0	0
$859$	−24282.7	−0.964511	−0.482255	−	0.876031i	$-0.660182\pi$
$859$	−24282.7	−0.964511	−0.482255	+	0.876031i	$0.660182\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19667.0	0.775750	0.387875	−	0.921712i	$-0.373209\pi$
$863$	19667.0	0.775750	0.387875	+	0.921712i	$0.373209\pi$
$864$	0	0
$865$	6822.59	0.268179
$866$	0	0
$867$	−32929.9	−1.28992
$868$	0	0
$869$	24389.4	0.952075
$870$	0	0
$871$	−8057.65	−0.313459
$872$	0	0
$873$	57004.0	2.20996
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−36061.0	−1.38848	−0.694238	−	0.719745i	$-0.744259\pi$
$877$	−36061.0	−1.38848	−0.694238	+	0.719745i	$0.744259\pi$
$878$	0	0
$879$	38807.2	1.48912
$880$	0	0
$881$	15889.7	0.607646	0.303823	−	0.952728i	$-0.401737\pi$
$881$	15889.7	0.607646	0.303823	+	0.952728i	$0.401737\pi$
$882$	0	0
$883$	−14861.3	−0.566390	−0.283195	−	0.959062i	$-0.591394\pi$
$883$	−14861.3	−0.566390	−0.283195	+	0.959062i	$0.591394\pi$
$884$	0	0
$885$	−6903.13	−0.262199
$886$	0	0
$887$	−38189.9	−1.44565	−0.722824	−	0.691032i	$-0.757156\pi$
$887$	−38189.9	−1.44565	−0.722824	+	0.691032i	$0.757156\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	23772.3	0.893831
$892$	0	0
$893$	20976.8	0.786073
$894$	0	0
$895$	5813.99	0.217140
$896$	0	0
$897$	19561.4	0.728135
$898$	0	0
$899$	−37105.0	−1.37655
$900$	0	0
$901$	−3742.22	−0.138370
$902$	0	0
$903$	0	0
$904$	0	0
$905$	7733.86	0.284069
$906$	0	0
$907$	−16865.5	−0.617431	−0.308715	−	0.951154i	$-0.599899\pi$
$907$	−16865.5	−0.617431	−0.308715	+	0.951154i	$0.599899\pi$
$908$	0	0
$909$	−2636.44	−0.0961993
$910$	0	0
$911$	−26754.1	−0.973000	−0.486500	−	0.873681i	$-0.661727\pi$
$911$	−26754.1	−0.973000	−0.486500	+	0.873681i	$0.661727\pi$
$912$	0	0
$913$	−1202.15	−0.0435766
$914$	0	0
$915$	9554.37	0.345200
$916$	0	0
$917$	0	0
$918$	0	0
$919$	41527.5	1.49061	0.745303	−	0.666726i	$-0.232305\pi$
$919$	41527.5	1.49061	0.745303	+	0.666726i	$0.232305\pi$
$920$	0	0
$921$	−33119.2	−1.18492
$922$	0	0
$923$	30272.0	1.07954
$924$	0	0
$925$	−9315.27	−0.331118
$926$	0	0
$927$	−5630.31	−0.199486
$928$	0	0
$929$	−4584.68	−0.161914	−0.0809572	−	0.996718i	$-0.525798\pi$
$929$	−4584.68	−0.161914	−0.0809572	+	0.996718i	$0.525798\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−5355.86	−0.187935
$934$	0	0
$935$	2709.48	0.0947694
$936$	0	0
$937$	6928.18	0.241552	0.120776	−	0.992680i	$-0.461462\pi$
$937$	6928.18	0.241552	0.120776	+	0.992680i	$0.461462\pi$
$938$	0	0
$939$	46495.7	1.61590
$940$	0	0
$941$	−20944.9	−0.725596	−0.362798	−	0.931868i	$-0.618178\pi$
$941$	−20944.9	−0.725596	−0.362798	+	0.931868i	$0.618178\pi$
$942$	0	0
$943$	−13866.4	−0.478846
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−29278.9	−1.00468	−0.502342	−	0.864669i	$-0.667528\pi$
$947$	−29278.9	−1.00468	−0.502342	+	0.864669i	$0.667528\pi$
$948$	0	0
$949$	−10652.8	−0.364387
$950$	0	0
$951$	−22089.0	−0.753192
$952$	0	0
$953$	2136.81	0.0726316	0.0363158	−	0.999340i	$-0.488438\pi$
$953$	2136.81	0.0726316	0.0363158	+	0.999340i	$0.488438\pi$
$954$	0	0
$955$	1993.48	0.0675470
$956$	0	0
$957$	−46722.3	−1.57818
$958$	0	0
$959$	0	0
$960$	0	0
$961$	63571.4	2.13391
$962$	0	0
$963$	−34960.2	−1.16986
$964$	0	0
$965$	7873.47	0.262649
$966$	0	0
$967$	3921.32	0.130405	0.0652023	−	0.997872i	$-0.479231\pi$
$967$	3921.32	0.130405	0.0652023	+	0.997872i	$0.479231\pi$
$968$	0	0
$969$	−16124.4	−0.534562
$970$	0	0
$971$	47809.1	1.58009	0.790045	−	0.613049i	$-0.210057\pi$
$971$	47809.1	1.58009	0.790045	+	0.613049i	$0.210057\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	42344.9	1.39089
$976$	0	0
$977$	−51329.1	−1.68082	−0.840411	−	0.541949i	$-0.817686\pi$
$977$	−51329.1	−1.68082	−0.840411	+	0.541949i	$0.817686\pi$
$978$	0	0
$979$	−52742.4	−1.72181
$980$	0	0
$981$	46580.1	1.51599
$982$	0	0
$983$	−16326.3	−0.529734	−0.264867	−	0.964285i	$-0.585328\pi$
$983$	−16326.3	−0.529734	−0.264867	+	0.964285i	$0.585328\pi$
$984$	0	0
$985$	−10631.6	−0.343909
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8205.37	−0.263818
$990$	0	0
$991$	−33770.0	−1.08248	−0.541242	−	0.840867i	$-0.682046\pi$
$991$	−33770.0	−1.08248	−0.541242	+	0.840867i	$0.682046\pi$
$992$	0	0
$993$	22003.6	0.703185
$994$	0	0
$995$	1796.56	0.0572410
$996$	0	0
$997$	−50695.4	−1.61037	−0.805185	−	0.593023i	$-0.797934\pi$
$997$	−50695.4	−1.61037	−0.805185	+	0.593023i	$0.797934\pi$
$998$	0	0
$999$	4339.16	0.137422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.4.a.bf.1.4		4
4.3	odd	2		49.4.a.e.1.1	✓	4
7.6	odd	2	inner	784.4.a.bf.1.1		4
12.11	even	2		441.4.a.u.1.4		4
20.19	odd	2		1225.4.a.bb.1.4		4
28.3	even	6		49.4.c.e.30.3		8
28.11	odd	6		49.4.c.e.30.4		8
28.19	even	6		49.4.c.e.18.3		8
28.23	odd	6		49.4.c.e.18.4		8
28.27	even	2		49.4.a.e.1.2	yes	4
84.11	even	6		441.4.e.y.226.1		8
84.23	even	6		441.4.e.y.361.1		8
84.47	odd	6		441.4.e.y.361.2		8
84.59	odd	6		441.4.e.y.226.2		8
84.83	odd	2		441.4.a.u.1.3		4
140.139	even	2		1225.4.a.bb.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
49.4.a.e.1.1	✓	4	4.3	odd	2
49.4.a.e.1.2	yes	4	28.27	even	2
49.4.c.e.18.3		8	28.19	even	6
49.4.c.e.18.4		8	28.23	odd	6
49.4.c.e.30.3		8	28.3	even	6
49.4.c.e.30.4		8	28.11	odd	6
441.4.a.u.1.3		4	84.83	odd	2
441.4.a.u.1.4		4	12.11	even	2
441.4.e.y.226.1		8	84.11	even	6
441.4.e.y.226.2		8	84.59	odd	6
441.4.e.y.361.1		8	84.23	even	6
441.4.e.y.361.2		8	84.47	odd	6
784.4.a.bf.1.1		4	7.6	odd	2	inner
784.4.a.bf.1.4		4	1.1	even	1	trivial
1225.4.a.bb.1.3		4	140.139	even	2
1225.4.a.bb.1.4		4	20.19	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:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	784.a (trivial)

\(f(q)\)	\(=\)	\(q+7.82220 q^{3} -2.07730 q^{5} +34.1868 q^{9} -49.1868 q^{11} -44.8559 q^{13} -16.2490 q^{15} +26.5179 q^{17} -77.7350 q^{19} -55.7510 q^{23} -120.685 q^{25} +56.2164 q^{27} +121.436 q^{29} -305.553 q^{31} -384.749 q^{33} +77.1868 q^{37} -350.872 q^{39} +248.720 q^{41} +147.179 q^{43} -71.0161 q^{45} -269.851 q^{47} +207.428 q^{51} -141.121 q^{53} +102.176 q^{55} -608.058 q^{57} +424.834 q^{59} -587.996 q^{61} +93.1790 q^{65} +179.634 q^{67} -436.095 q^{69} -674.872 q^{71} +237.489 q^{73} -944.021 q^{75} -495.852 q^{79} -483.307 q^{81} +24.4406 q^{83} -55.0855 q^{85} +949.895 q^{87} +1072.29 q^{89} -2390.09 q^{93} +161.479 q^{95} +1667.43 q^{97} -1681.54 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 40 q^{9} - 100 q^{11} + 64 q^{15} - 352 q^{23} - 128 q^{25} + 260 q^{29} + 212 q^{37} - 952 q^{39} - 540 q^{43} - 428 q^{51} + 16 q^{53} - 1884 q^{57} - 756 q^{65} + 1944 q^{67} - 2248 q^{71} + 1048 q^{79}+ \cdots - 3340 q^{99}+O(q^{100}) \) 4 * q + 40 * q^9 - 100 * q^11 + 64 * q^15 - 352 * q^23 - 128 * q^25 + 260 * q^29 + 212 * q^37 - 952 * q^39 - 540 * q^43 - 428 * q^51 + 16 * q^53 - 1884 * q^57 - 756 * q^65 + 1944 * q^67 - 2248 * q^71 + 1048 * q^79 - 1256 * q^81 - 3284 * q^85 - 5368 * q^93 - 2192 * q^95 - 3340 * q^99

Introduction

L-functions

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