LMFDB - Embedded newform 1368.4.a.p.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,4,Mod(1,1368)] 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("1368.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1368.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,16,0,-4,0,0,0,16] 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:	$80.7146128879$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 539x^{6} + 1968x^{5} + 81033x^{4} - 222055x^{3} - 4314559x^{2} + 4793814x + 61552224 $ x^8 - 3x^7 - 539x^6 + 1968x^5 + 81033x^4 - 222055x^3 - 4314559x^2 + 4793814*x + 61552224
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{9} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-7.09542$ of defining polynomial
Character	$\chi$	$=$	1368.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-11.1186 q^{5} -2.21338 q^{7} +16.4590 q^{11} +3.91157 q^{13} +89.9069 q^{17} +19.0000 q^{19} -75.3174 q^{23} -1.37617 q^{25} -197.642 q^{29} -43.0936 q^{31} +24.6098 q^{35} -214.099 q^{37} +282.522 q^{41} -518.945 q^{43} +486.843 q^{47} -338.101 q^{49} -513.121 q^{53} -183.001 q^{55} +230.881 q^{59} +578.922 q^{61} -43.4913 q^{65} +461.830 q^{67} +119.555 q^{71} +433.266 q^{73} -36.4301 q^{77} +445.729 q^{79} +1393.37 q^{83} -999.641 q^{85} +617.364 q^{89} -8.65780 q^{91} -211.254 q^{95} +1302.63 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{5} - 4 q^{7} + 16 q^{11} - 36 q^{13} - 48 q^{17} + 152 q^{19} + 88 q^{23} + 124 q^{25} + 48 q^{29} - 4 q^{31} + 72 q^{35} + 4 q^{37} + 376 q^{41} + 276 q^{43} + 384 q^{47} + 780 q^{49} + 1528 q^{53}+ \cdots + 840 q^{97}+O(q^{100}) $ 8 * q + 16 * q^5 - 4 * q^7 + 16 * q^11 - 36 * q^13 - 48 * q^17 + 152 * q^19 + 88 * q^23 + 124 * q^25 + 48 * q^29 - 4 * q^31 + 72 * q^35 + 4 * q^37 + 376 * q^41 + 276 * q^43 + 384 * q^47 + 780 * q^49 + 1528 * q^53 - 604 * q^55 + 664 * q^59 + 68 * q^61 + 1576 * q^65 - 184 * q^67 + 1704 * q^71 + 628 * q^73 + 1344 * q^77 - 1380 * q^79 + 1480 * q^83 + 796 * q^85 + 1800 * q^89 - 1864 * q^91 + 304 * q^95 + 840 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−11.1186	−0.994480	−0.497240	−	0.867613i	$-0.665653\pi$
$5$	−11.1186	−0.994480	−0.497240	+	0.867613i	$0.665653\pi$
$6$	0	0
$7$	−2.21338	−0.119511	−0.0597557	−	0.998213i	$-0.519032\pi$
$7$	−2.21338	−0.119511	−0.0597557	+	0.998213i	$0.519032\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	16.4590	0.451143	0.225572	−	0.974227i	$-0.427575\pi$
$11$	16.4590	0.451143	0.225572	+	0.974227i	$0.427575\pi$
$12$	0	0
$13$	3.91157	0.0834519	0.0417259	−	0.999129i	$-0.486714\pi$
$13$	3.91157	0.0834519	0.0417259	+	0.999129i	$0.486714\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	89.9069	1.28268	0.641342	−	0.767255i	$-0.278378\pi$
$17$	89.9069	1.28268	0.641342	+	0.767255i	$0.278378\pi$
$18$	0	0
$19$	19.0000	0.229416
$19$	19.0000	0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−75.3174	−0.682816	−0.341408	−	0.939915i	$-0.610904\pi$
$23$	−75.3174	−0.682816	−0.341408	+	0.939915i	$0.610904\pi$
$24$	0	0
$25$	−1.37617	−0.0110093
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−197.642	−1.26556	−0.632779	−	0.774333i	$-0.718086\pi$
$29$	−197.642	−1.26556	−0.632779	+	0.774333i	$0.718086\pi$
$30$	0	0
$31$	−43.0936	−0.249672	−0.124836	−	0.992177i	$-0.539840\pi$
$31$	−43.0936	−0.249672	−0.124836	+	0.992177i	$0.539840\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	24.6098	0.118852
$36$	0	0
$37$	−214.099	−0.951290	−0.475645	−	0.879637i	$-0.657785\pi$
$37$	−214.099	−0.951290	−0.475645	+	0.879637i	$0.657785\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	282.522	1.07616	0.538080	−	0.842894i	$-0.319150\pi$
$41$	282.522	1.07616	0.538080	+	0.842894i	$0.319150\pi$
$42$	0	0
$43$	−518.945	−1.84043	−0.920214	−	0.391416i	$-0.871985\pi$
$43$	−518.945	−1.84043	−0.920214	+	0.391416i	$0.871985\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	486.843	1.51092	0.755461	−	0.655194i	$-0.227413\pi$
$47$	486.843	1.51092	0.755461	+	0.655194i	$0.227413\pi$
$48$	0	0
$49$	−338.101	−0.985717
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−513.121	−1.32986	−0.664930	−	0.746906i	$-0.731539\pi$
$53$	−513.121	−1.32986	−0.664930	+	0.746906i	$0.731539\pi$
$54$	0	0
$55$	−183.001	−0.448653
$56$	0	0
$57$	0	0
$58$	0	0
$59$	230.881	0.509460	0.254730	−	0.967012i	$-0.418013\pi$
$59$	230.881	0.509460	0.254730	+	0.967012i	$0.418013\pi$
$60$	0	0
$61$	578.922	1.21514	0.607568	−	0.794267i	$-0.292145\pi$
$61$	578.922	1.21514	0.607568	+	0.794267i	$0.292145\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−43.4913	−0.0829912
$66$	0	0
$67$	461.830	0.842113	0.421056	−	0.907034i	$-0.361659\pi$
$67$	461.830	0.842113	0.421056	+	0.907034i	$0.361659\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	119.555	0.199839	0.0999197	−	0.994995i	$-0.468141\pi$
$71$	119.555	0.199839	0.0999197	+	0.994995i	$0.468141\pi$
$72$	0	0
$73$	433.266	0.694656	0.347328	−	0.937744i	$-0.387089\pi$
$73$	433.266	0.694656	0.347328	+	0.937744i	$0.387089\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−36.4301	−0.0539168
$78$	0	0
$79$	445.729	0.634791	0.317395	−	0.948293i	$-0.397192\pi$
$79$	445.729	0.634791	0.317395	+	0.948293i	$0.397192\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1393.37	1.84268	0.921338	−	0.388762i	$-0.127097\pi$
$83$	1393.37	1.84268	0.921338	+	0.388762i	$0.127097\pi$
$84$	0	0
$85$	−999.641	−1.27560
$86$	0	0
$87$	0	0
$88$	0	0
$89$	617.364	0.735286	0.367643	−	0.929967i	$-0.380165\pi$
$89$	617.364	0.735286	0.367643	+	0.929967i	$0.380165\pi$
$90$	0	0
$91$	−8.65780	−0.00997345
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−211.254	−0.228149
$96$	0	0
$97$	1302.63	1.36353	0.681764	−	0.731572i	$-0.261213\pi$
$97$	1302.63	1.36353	0.681764	+	0.731572i	$0.261213\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−829.641	−0.817350	−0.408675	−	0.912680i	$-0.634009\pi$
$101$	−829.641	−0.817350	−0.408675	+	0.912680i	$0.634009\pi$
$102$	0	0
$103$	−1842.95	−1.76302	−0.881510	−	0.472164i	$-0.843473\pi$
$103$	−1842.95	−1.76302	−0.881510	+	0.472164i	$0.843473\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1382.22	1.24883	0.624413	−	0.781094i	$-0.285338\pi$
$107$	1382.22	1.24883	0.624413	+	0.781094i	$0.285338\pi$
$108$	0	0
$109$	1058.81	0.930422	0.465211	−	0.885200i	$-0.345978\pi$
$109$	1058.81	0.930422	0.465211	+	0.885200i	$0.345978\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	823.297	0.685392	0.342696	−	0.939446i	$-0.388660\pi$
$113$	823.297	0.685392	0.342696	+	0.939446i	$0.388660\pi$
$114$	0	0
$115$	837.426	0.679047
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−198.998	−0.153295
$120$	0	0
$121$	−1060.10	−0.796470
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1405.13	1.00543
$126$	0	0
$127$	1718.04	1.20040	0.600202	−	0.799849i	$-0.295087\pi$
$127$	1718.04	1.20040	0.600202	+	0.799849i	$0.295087\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−539.899	−0.360085	−0.180043	−	0.983659i	$-0.557624\pi$
$131$	−539.899	−0.360085	−0.180043	+	0.983659i	$0.557624\pi$
$132$	0	0
$133$	−42.0543	−0.0274178
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1550.46	0.966894	0.483447	−	0.875374i	$-0.339385\pi$
$137$	1550.46	0.966894	0.483447	+	0.875374i	$0.339385\pi$
$138$	0	0
$139$	636.392	0.388331	0.194166	−	0.980969i	$-0.437800\pi$
$139$	636.392	0.388331	0.194166	+	0.980969i	$0.437800\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	64.3805	0.0376488
$144$	0	0
$145$	2197.51	1.25857
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1991.41	1.09492	0.547458	−	0.836833i	$-0.315596\pi$
$149$	1991.41	1.09492	0.547458	+	0.836833i	$0.315596\pi$
$150$	0	0
$151$	−1558.28	−0.839809	−0.419904	−	0.907568i	$-0.637936\pi$
$151$	−1558.28	−0.839809	−0.419904	+	0.907568i	$0.637936\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	479.141	0.248294
$156$	0	0
$157$	2036.07	1.03501	0.517503	−	0.855681i	$-0.326862\pi$
$157$	2036.07	1.03501	0.517503	+	0.855681i	$0.326862\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	166.706	0.0816042
$162$	0	0
$163$	−2196.49	−1.05547	−0.527736	−	0.849408i	$-0.676959\pi$
$163$	−2196.49	−1.05547	−0.527736	+	0.849408i	$0.676959\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1130.26	0.523727	0.261863	−	0.965105i	$-0.415663\pi$
$167$	1130.26	0.523727	0.261863	+	0.965105i	$0.415663\pi$
$168$	0	0
$169$	−2181.70	−0.993036
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1199.05	0.526950	0.263475	−	0.964666i	$-0.415131\pi$
$173$	1199.05	0.526950	0.263475	+	0.964666i	$0.415131\pi$
$174$	0	0
$175$	3.04598	0.00131574
$176$	0	0
$177$	0	0
$178$	0	0
$179$	546.261	0.228097	0.114049	−	0.993475i	$-0.463618\pi$
$179$	546.261	0.228097	0.114049	+	0.993475i	$0.463618\pi$
$180$	0	0
$181$	3018.45	1.23956	0.619778	−	0.784777i	$-0.287223\pi$
$181$	3018.45	1.23956	0.619778	+	0.784777i	$0.287223\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2380.49	0.946039
$186$	0	0
$187$	1479.78	0.578674
$188$	0	0
$189$	0	0
$190$	0	0
$191$	3093.00	1.17173	0.585867	−	0.810407i	$-0.300754\pi$
$191$	3093.00	1.17173	0.585867	+	0.810407i	$0.300754\pi$
$192$	0	0
$193$	−495.699	−0.184877	−0.0924383	−	0.995718i	$-0.529466\pi$
$193$	−495.699	−0.184877	−0.0924383	+	0.995718i	$0.529466\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3927.80	−1.42053	−0.710265	−	0.703934i	$-0.751425\pi$
$197$	−3927.80	−1.42053	−0.710265	+	0.703934i	$0.751425\pi$
$198$	0	0
$199$	1332.22	0.474565	0.237283	−	0.971441i	$-0.423743\pi$
$199$	1332.22	0.474565	0.237283	+	0.971441i	$0.423743\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	437.457	0.151249
$204$	0	0
$205$	−3141.26	−1.07022
$206$	0	0
$207$	0	0
$208$	0	0
$209$	312.721	0.103499
$210$	0	0
$211$	−3819.48	−1.24618	−0.623090	−	0.782150i	$-0.714123\pi$
$211$	−3819.48	−1.24618	−0.623090	+	0.782150i	$0.714123\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5769.96	1.83027
$216$	0	0
$217$	95.3825	0.0298386
$218$	0	0
$219$	0	0
$220$	0	0
$221$	351.677	0.107042
$222$	0	0
$223$	−6289.80	−1.88877	−0.944386	−	0.328838i	$-0.893343\pi$
$223$	−6289.80	−1.88877	−0.944386	+	0.328838i	$0.893343\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1075.57	0.314484	0.157242	−	0.987560i	$-0.449740\pi$
$227$	1075.57	0.314484	0.157242	+	0.987560i	$0.449740\pi$
$228$	0	0
$229$	1609.62	0.464483	0.232241	−	0.972658i	$-0.425394\pi$
$229$	1609.62	0.464483	0.232241	+	0.972658i	$0.425394\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4212.84	1.18452	0.592258	−	0.805748i	$-0.298236\pi$
$233$	4212.84	1.18452	0.592258	+	0.805748i	$0.298236\pi$
$234$	0	0
$235$	−5413.02	−1.50258
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4036.11	−1.09236	−0.546181	−	0.837667i	$-0.683919\pi$
$239$	−4036.11	−1.09236	−0.546181	+	0.837667i	$0.683919\pi$
$240$	0	0
$241$	2664.41	0.712158	0.356079	−	0.934456i	$-0.384113\pi$
$241$	2664.41	0.712158	0.356079	+	0.934456i	$0.384113\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3759.22	0.980276
$246$	0	0
$247$	74.3198	0.0191452
$248$	0	0
$249$	0	0
$250$	0	0
$251$	491.421	0.123579	0.0617893	−	0.998089i	$-0.480319\pi$
$251$	491.421	0.123579	0.0617893	+	0.998089i	$0.480319\pi$
$252$	0	0
$253$	−1239.65	−0.308048
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3630.31	0.881139	0.440569	−	0.897719i	$-0.354777\pi$
$257$	3630.31	0.881139	0.440569	+	0.897719i	$0.354777\pi$
$258$	0	0
$259$	473.884	0.113690
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2179.88	−0.511091	−0.255546	−	0.966797i	$-0.582255\pi$
$263$	−2179.88	−0.511091	−0.255546	+	0.966797i	$0.582255\pi$
$264$	0	0
$265$	5705.20	1.32252
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1493.24	0.338455	0.169227	−	0.985577i	$-0.445873\pi$
$269$	1493.24	0.338455	0.169227	+	0.985577i	$0.445873\pi$
$270$	0	0
$271$	3622.17	0.811923	0.405961	−	0.913890i	$-0.366937\pi$
$271$	3622.17	0.811923	0.405961	+	0.913890i	$0.366937\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.6503	−0.00496678
$276$	0	0
$277$	−6022.42	−1.30632	−0.653162	−	0.757218i	$-0.726558\pi$
$277$	−6022.42	−1.30632	−0.653162	+	0.757218i	$0.726558\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6510.42	1.38213	0.691066	−	0.722791i	$-0.257141\pi$
$281$	6510.42	1.38213	0.691066	+	0.722791i	$0.257141\pi$
$282$	0	0
$283$	−4767.16	−1.00134	−0.500668	−	0.865639i	$-0.666912\pi$
$283$	−4767.16	−1.00134	−0.500668	+	0.865639i	$0.666912\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−625.330	−0.128613
$288$	0	0
$289$	3170.24	0.645277
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4027.26	0.802986	0.401493	−	0.915862i	$-0.368491\pi$
$293$	4027.26	0.802986	0.401493	+	0.915862i	$0.368491\pi$
$294$	0	0
$295$	−2567.08	−0.506648
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−294.609	−0.0569823
$300$	0	0
$301$	1148.62	0.219952
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6436.81	−1.20843
$306$	0	0
$307$	3668.66	0.682025	0.341012	−	0.940059i	$-0.389230\pi$
$307$	3668.66	0.682025	0.341012	+	0.940059i	$0.389230\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4956.65	0.903749	0.451875	−	0.892081i	$-0.350755\pi$
$311$	4956.65	0.903749	0.451875	+	0.892081i	$0.350755\pi$
$312$	0	0
$313$	7017.04	1.26718	0.633589	−	0.773670i	$-0.281581\pi$
$313$	7017.04	1.26718	0.633589	+	0.773670i	$0.281581\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−531.473	−0.0941656	−0.0470828	−	0.998891i	$-0.514992\pi$
$317$	−531.473	−0.0941656	−0.0470828	+	0.998891i	$0.514992\pi$
$318$	0	0
$319$	−3252.99	−0.570948
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1708.23	0.294268
$324$	0	0
$325$	−5.38297	−0.000918749 0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1077.57	−0.180572
$330$	0	0
$331$	742.863	0.123358	0.0616789	−	0.998096i	$-0.480355\pi$
$331$	742.863	0.123358	0.0616789	+	0.998096i	$0.480355\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−5134.92	−0.837465
$336$	0	0
$337$	−1094.10	−0.176853	−0.0884263	−	0.996083i	$-0.528184\pi$
$337$	−1094.10	−0.176853	−0.0884263	+	0.996083i	$0.528184\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−709.277	−0.112638
$342$	0	0
$343$	1507.54	0.237316
$344$	0	0
$345$	0	0
$346$	0	0
$347$	488.290	0.0755411	0.0377706	−	0.999286i	$-0.487974\pi$
$347$	488.290	0.0755411	0.0377706	+	0.999286i	$0.487974\pi$
$348$	0	0
$349$	9492.76	1.45598	0.727989	−	0.685589i	$-0.240455\pi$
$349$	9492.76	1.45598	0.727989	+	0.685589i	$0.240455\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10883.0	1.64092	0.820460	−	0.571704i	$-0.193718\pi$
$353$	10883.0	1.64092	0.820460	+	0.571704i	$0.193718\pi$
$354$	0	0
$355$	−1329.29	−0.198736
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5459.16	0.802572	0.401286	−	0.915953i	$-0.368563\pi$
$359$	5459.16	0.802572	0.401286	+	0.915953i	$0.368563\pi$
$360$	0	0
$361$	361.000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4817.32	−0.690822
$366$	0	0
$367$	−8963.69	−1.27493	−0.637467	−	0.770478i	$-0.720018\pi$
$367$	−8963.69	−1.27493	−0.637467	+	0.770478i	$0.720018\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1135.73	0.158933
$372$	0	0
$373$	6843.50	0.949982	0.474991	−	0.879991i	$-0.342451\pi$
$373$	6843.50	0.949982	0.474991	+	0.879991i	$0.342451\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−773.090	−0.105613
$378$	0	0
$379$	972.369	0.131787	0.0658935	−	0.997827i	$-0.479010\pi$
$379$	972.369	0.131787	0.0658935	+	0.997827i	$0.479010\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−7483.04	−0.998343	−0.499171	−	0.866503i	$-0.666362\pi$
$383$	−7483.04	−0.998343	−0.499171	+	0.866503i	$0.666362\pi$
$384$	0	0
$385$	405.052	0.0536191
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−4421.52	−0.576298	−0.288149	−	0.957586i	$-0.593040\pi$
$389$	−4421.52	−0.576298	−0.288149	+	0.957586i	$0.593040\pi$
$390$	0	0
$391$	−6771.55	−0.875836
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4955.90	−0.631287
$396$	0	0
$397$	4188.60	0.529520	0.264760	−	0.964314i	$-0.414707\pi$
$397$	4188.60	0.529520	0.264760	+	0.964314i	$0.414707\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−6690.78	−0.833221	−0.416611	−	0.909085i	$-0.636782\pi$
$401$	−6690.78	−0.833221	−0.416611	+	0.909085i	$0.636782\pi$
$402$	0	0
$403$	−168.564	−0.0208356
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−3523.86	−0.429168
$408$	0	0
$409$	−5839.59	−0.705988	−0.352994	−	0.935626i	$-0.614836\pi$
$409$	−5839.59	−0.705988	−0.352994	+	0.935626i	$0.614836\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−511.028	−0.0608863
$414$	0	0
$415$	−15492.4	−1.83251
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12578.9	1.46663	0.733316	−	0.679887i	$-0.237971\pi$
$419$	12578.9	1.46663	0.733316	+	0.679887i	$0.237971\pi$
$420$	0	0
$421$	10382.2	1.20189	0.600946	−	0.799290i	$-0.294791\pi$
$421$	10382.2	1.20189	0.600946	+	0.799290i	$0.294791\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−123.727	−0.0141215
$426$	0	0
$427$	−1281.37	−0.145223
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7149.99	0.799079	0.399540	−	0.916716i	$-0.369170\pi$
$431$	7149.99	0.799079	0.399540	+	0.916716i	$0.369170\pi$
$432$	0	0
$433$	9644.23	1.07037	0.535187	−	0.844734i	$-0.320241\pi$
$433$	9644.23	1.07037	0.535187	+	0.844734i	$0.320241\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1431.03	−0.156649
$438$	0	0
$439$	1763.20	0.191692	0.0958462	−	0.995396i	$-0.469444\pi$
$439$	1763.20	0.191692	0.0958462	+	0.995396i	$0.469444\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10013.8	−1.07397	−0.536985	−	0.843592i	$-0.680437\pi$
$443$	−10013.8	−1.07397	−0.536985	+	0.843592i	$0.680437\pi$
$444$	0	0
$445$	−6864.24	−0.731227
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11595.3	1.21875	0.609374	−	0.792883i	$-0.291421\pi$
$449$	11595.3	1.21875	0.609374	+	0.792883i	$0.291421\pi$
$450$	0	0
$451$	4650.04	0.485503
$452$	0	0
$453$	0	0
$454$	0	0
$455$	96.2628	0.00991840
$456$	0	0
$457$	19174.4	1.96267	0.981336	−	0.192303i	$-0.0615955\pi$
$457$	19174.4	1.96267	0.981336	+	0.192303i	$0.0615955\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−13708.5	−1.38497	−0.692483	−	0.721435i	$-0.743483\pi$
$461$	−13708.5	−1.38497	−0.692483	+	0.721435i	$0.743483\pi$
$462$	0	0
$463$	17084.3	1.71485	0.857423	−	0.514612i	$-0.172064\pi$
$463$	17084.3	1.71485	0.857423	+	0.514612i	$0.172064\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−7790.86	−0.771987	−0.385994	−	0.922501i	$-0.626141\pi$
$467$	−7790.86	−0.771987	−0.385994	+	0.922501i	$0.626141\pi$
$468$	0	0
$469$	−1022.21	−0.100642
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8541.32	−0.830296
$474$	0	0
$475$	−26.1472	−0.00252571
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2542.50	0.242526	0.121263	−	0.992620i	$-0.461306\pi$
$479$	2542.50	0.242526	0.121263	+	0.992620i	$0.461306\pi$
$480$	0	0
$481$	−837.465	−0.0793870
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−14483.5	−1.35600
$486$	0	0
$487$	13824.9	1.28638	0.643190	−	0.765707i	$-0.277610\pi$
$487$	13824.9	1.28638	0.643190	+	0.765707i	$0.277610\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6966.95	−0.640355	−0.320177	−	0.947358i	$-0.603743\pi$
$491$	−6966.95	−0.640355	−0.320177	+	0.947358i	$0.603743\pi$
$492$	0	0
$493$	−17769.4	−1.62331
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−264.622	−0.0238831
$498$	0	0
$499$	−13824.1	−1.24019	−0.620094	−	0.784528i	$-0.712905\pi$
$499$	−13824.1	−1.24019	−0.620094	+	0.784528i	$0.712905\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	8840.27	0.783634	0.391817	−	0.920043i	$-0.371847\pi$
$503$	8840.27	0.783634	0.391817	+	0.920043i	$0.371847\pi$
$504$	0	0
$505$	9224.46	0.812838
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−15711.9	−1.36821	−0.684105	−	0.729383i	$-0.739807\pi$
$509$	−15711.9	−1.36821	−0.684105	+	0.729383i	$0.739807\pi$
$510$	0	0
$511$	−958.982	−0.0830193
$512$	0	0
$513$	0	0
$514$	0	0
$515$	20491.1	1.75329
$516$	0	0
$517$	8012.94	0.681642
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−9815.02	−0.825343	−0.412672	−	0.910880i	$-0.635404\pi$
$521$	−9815.02	−0.825343	−0.412672	+	0.910880i	$0.635404\pi$
$522$	0	0
$523$	−12004.3	−1.00365	−0.501825	−	0.864969i	$-0.667338\pi$
$523$	−12004.3	−1.00365	−0.501825	+	0.864969i	$0.667338\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−3874.41	−0.320250
$528$	0	0
$529$	−6494.29	−0.533763
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1105.11	0.0898076
$534$	0	0
$535$	−15368.4	−1.24193
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−5564.80	−0.444700
$540$	0	0
$541$	11639.1	0.924958	0.462479	−	0.886630i	$-0.346960\pi$
$541$	11639.1	0.924958	0.462479	+	0.886630i	$0.346960\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−11772.6	−0.925286
$546$	0	0
$547$	−22896.1	−1.78970	−0.894850	−	0.446368i	$-0.852717\pi$
$547$	−22896.1	−1.78970	−0.894850	+	0.446368i	$0.852717\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3755.20	−0.290339
$552$	0	0
$553$	−986.569	−0.0758647
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2521.60	0.191819	0.0959097	−	0.995390i	$-0.469424\pi$
$557$	2521.60	0.191819	0.0959097	+	0.995390i	$0.469424\pi$
$558$	0	0
$559$	−2029.89	−0.153587
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−18316.7	−1.37115	−0.685574	−	0.728003i	$-0.740448\pi$
$563$	−18316.7	−1.37115	−0.685574	+	0.728003i	$0.740448\pi$
$564$	0	0
$565$	−9153.93	−0.681608
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−2736.24	−0.201598	−0.100799	−	0.994907i	$-0.532140\pi$
$569$	−2736.24	−0.201598	−0.100799	+	0.994907i	$0.532140\pi$
$570$	0	0
$571$	−8179.03	−0.599443	−0.299721	−	0.954027i	$-0.596894\pi$
$571$	−8179.03	−0.599443	−0.299721	+	0.954027i	$0.596894\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	103.649	0.00751734
$576$	0	0
$577$	11705.7	0.844565	0.422282	−	0.906464i	$-0.361229\pi$
$577$	11705.7	0.844565	0.422282	+	0.906464i	$0.361229\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3084.06	−0.220221
$582$	0	0
$583$	−8445.45	−0.599957
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6703.30	−0.471337	−0.235668	−	0.971834i	$-0.575728\pi$
$587$	−6703.30	−0.471337	−0.235668	+	0.971834i	$0.575728\pi$
$588$	0	0
$589$	−818.778	−0.0572787
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−5086.61	−0.352246	−0.176123	−	0.984368i	$-0.556356\pi$
$593$	−5086.61	−0.352246	−0.176123	+	0.984368i	$0.556356\pi$
$594$	0	0
$595$	2212.59	0.152449
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6549.05	−0.446723	−0.223361	−	0.974736i	$-0.571703\pi$
$599$	−6549.05	−0.446723	−0.223361	+	0.974736i	$0.571703\pi$
$600$	0	0
$601$	2076.22	0.140916	0.0704582	−	0.997515i	$-0.477554\pi$
$601$	2076.22	0.140916	0.0704582	+	0.997515i	$0.477554\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11786.9	0.792073
$606$	0	0
$607$	−7533.61	−0.503756	−0.251878	−	0.967759i	$-0.581048\pi$
$607$	−7533.61	−0.503756	−0.251878	+	0.967759i	$0.581048\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1904.32	0.126089
$612$	0	0
$613$	−17312.7	−1.14071	−0.570353	−	0.821399i	$-0.693194\pi$
$613$	−17312.7	−1.14071	−0.570353	+	0.821399i	$0.693194\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	15551.9	1.01475	0.507373	−	0.861727i	$-0.330617\pi$
$617$	15551.9	1.01475	0.507373	+	0.861727i	$0.330617\pi$
$618$	0	0
$619$	324.437	0.0210666	0.0105333	−	0.999945i	$-0.496647\pi$
$619$	324.437	0.0210666	0.0105333	+	0.999945i	$0.496647\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1366.46	−0.0878750
$624$	0	0
$625$	−15451.1	−0.988869
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−19249.0	−1.22020
$630$	0	0
$631$	−4269.40	−0.269354	−0.134677	−	0.990890i	$-0.543000\pi$
$631$	−4269.40	−0.269354	−0.134677	+	0.990890i	$0.543000\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−19102.2	−1.19378
$636$	0	0
$637$	−1322.51	−0.0822599
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−10325.0	−0.636212	−0.318106	−	0.948055i	$-0.603047\pi$
$641$	−10325.0	−0.636212	−0.318106	+	0.948055i	$0.603047\pi$
$642$	0	0
$643$	14537.6	0.891611	0.445805	−	0.895130i	$-0.352917\pi$
$643$	14537.6	0.891611	0.445805	+	0.895130i	$0.352917\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−27202.4	−1.65292	−0.826459	−	0.562997i	$-0.809648\pi$
$647$	−27202.4	−1.65292	−0.826459	+	0.562997i	$0.809648\pi$
$648$	0	0
$649$	3800.07	0.229839
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17304.6	−1.03703	−0.518514	−	0.855069i	$-0.673515\pi$
$653$	−17304.6	−1.03703	−0.518514	+	0.855069i	$0.673515\pi$
$654$	0	0
$655$	6002.93	0.358098
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4284.87	0.253285	0.126642	−	0.991948i	$-0.459580\pi$
$659$	4284.87	0.253285	0.126642	+	0.991948i	$0.459580\pi$
$660$	0	0
$661$	−27503.0	−1.61837	−0.809185	−	0.587554i	$-0.800091\pi$
$661$	−27503.0	−1.61837	−0.809185	+	0.587554i	$0.800091\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	467.585	0.0272664
$666$	0	0
$667$	14885.9	0.864143
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9528.47	0.548201
$672$	0	0
$673$	29127.5	1.66832	0.834162	−	0.551519i	$-0.185952\pi$
$673$	29127.5	1.66832	0.834162	+	0.551519i	$0.185952\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	32358.5	1.83698	0.918491	−	0.395443i	$-0.129409\pi$
$677$	32358.5	1.83698	0.918491	+	0.395443i	$0.129409\pi$
$678$	0	0
$679$	−2883.22	−0.162957
$680$	0	0
$681$	0	0
$682$	0	0
$683$	3021.59	0.169279	0.0846397	−	0.996412i	$-0.473026\pi$
$683$	3021.59	0.169279	0.0846397	+	0.996412i	$0.473026\pi$
$684$	0	0
$685$	−17239.0	−0.961557
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−2007.11	−0.110979
$690$	0	0
$691$	28057.2	1.54464	0.772320	−	0.635234i	$-0.219096\pi$
$691$	28057.2	1.54464	0.772320	+	0.635234i	$0.219096\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−7075.81	−0.386188
$696$	0	0
$697$	25400.7	1.38037
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−13934.2	−0.750768	−0.375384	−	0.926869i	$-0.622489\pi$
$701$	−13934.2	−0.750768	−0.375384	+	0.926869i	$0.622489\pi$
$702$	0	0
$703$	−4067.89	−0.218241
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1836.31	0.0976826
$708$	0	0
$709$	29781.7	1.57754	0.788770	−	0.614689i	$-0.210718\pi$
$709$	29781.7	1.57754	0.788770	+	0.614689i	$0.210718\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3245.69	0.170480
$714$	0	0
$715$	−715.823	−0.0374409
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−17984.2	−0.932821	−0.466410	−	0.884568i	$-0.654453\pi$
$719$	−17984.2	−0.932821	−0.466410	+	0.884568i	$0.654453\pi$
$720$	0	0
$721$	4079.15	0.210701
$722$	0	0
$723$	0	0
$724$	0	0
$725$	271.988	0.0139329
$726$	0	0
$727$	3735.98	0.190591	0.0952957	−	0.995449i	$-0.469620\pi$
$727$	3735.98	0.190591	0.0952957	+	0.995449i	$0.469620\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−46656.7	−2.36069
$732$	0	0
$733$	22378.8	1.12767	0.563833	−	0.825889i	$-0.309326\pi$
$733$	22378.8	1.12767	0.563833	+	0.825889i	$0.309326\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7601.27	0.379914
$738$	0	0
$739$	−33239.3	−1.65457	−0.827285	−	0.561782i	$-0.810116\pi$
$739$	−33239.3	−1.65457	−0.827285	+	0.561782i	$0.810116\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−24372.3	−1.20341	−0.601705	−	0.798718i	$-0.705512\pi$
$743$	−24372.3	−1.20341	−0.601705	+	0.798718i	$0.705512\pi$
$744$	0	0
$745$	−22141.7	−1.08887
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3059.38	−0.149249
$750$	0	0
$751$	−23175.9	−1.12610	−0.563051	−	0.826422i	$-0.690372\pi$
$751$	−23175.9	−1.12610	−0.563051	+	0.826422i	$0.690372\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17325.9	0.835173
$756$	0	0
$757$	−9698.83	−0.465667	−0.232834	−	0.972517i	$-0.574800\pi$
$757$	−9698.83	−0.465667	−0.232834	+	0.972517i	$0.574800\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−5800.47	−0.276303	−0.138152	−	0.990411i	$-0.544116\pi$
$761$	−5800.47	−0.276303	−0.138152	+	0.990411i	$0.544116\pi$
$762$	0	0
$763$	−2343.56	−0.111196
$764$	0	0
$765$	0	0
$766$	0	0
$767$	903.107	0.0425154
$768$	0	0
$769$	27306.3	1.28048	0.640241	−	0.768174i	$-0.278834\pi$
$769$	27306.3	1.28048	0.640241	+	0.768174i	$0.278834\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	15985.2	0.743787	0.371893	−	0.928275i	$-0.378709\pi$
$773$	15985.2	0.743787	0.371893	+	0.928275i	$0.378709\pi$
$774$	0	0
$775$	59.3039	0.00274872
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5367.93	0.246888
$780$	0	0
$781$	1967.76	0.0901562
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−22638.3	−1.02929
$786$	0	0
$787$	−4817.65	−0.218209	−0.109105	−	0.994030i	$-0.534798\pi$
$787$	−4817.65	−0.218209	−0.109105	+	0.994030i	$0.534798\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1822.27	−0.0819121
$792$	0	0
$793$	2264.49	0.101405
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−6536.97	−0.290529	−0.145264	−	0.989393i	$-0.546403\pi$
$797$	−6536.97	−0.290529	−0.145264	+	0.989393i	$0.546403\pi$
$798$	0	0
$799$	43770.5	1.93803
$800$	0	0
$801$	0	0
$802$	0	0
$803$	7131.12	0.313389
$804$	0	0
$805$	−1853.54	−0.0811538
$806$	0	0
$807$	0	0
$808$	0	0
$809$	78.4071	0.00340747	0.00170374	−	0.999999i	$-0.499458\pi$
$809$	78.4071	0.00340747	0.00170374	+	0.999999i	$0.499458\pi$
$810$	0	0
$811$	19724.2	0.854019	0.427010	−	0.904247i	$-0.359567\pi$
$811$	19724.2	0.854019	0.427010	+	0.904247i	$0.359567\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	24421.9	1.04965
$816$	0	0
$817$	−9859.96	−0.422223
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2282.56	0.0970305	0.0485152	−	0.998822i	$-0.484551\pi$
$821$	2282.56	0.0970305	0.0485152	+	0.998822i	$0.484551\pi$
$822$	0	0
$823$	21187.9	0.897406	0.448703	−	0.893681i	$-0.351886\pi$
$823$	21187.9	0.897406	0.448703	+	0.893681i	$0.351886\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11000.1	−0.462529	−0.231264	−	0.972891i	$-0.574286\pi$
$827$	−11000.1	−0.462529	−0.231264	+	0.972891i	$0.574286\pi$
$828$	0	0
$829$	9371.45	0.392622	0.196311	−	0.980542i	$-0.437104\pi$
$829$	9371.45	0.392622	0.196311	+	0.980542i	$0.437104\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−30397.6	−1.26436
$834$	0	0
$835$	−12567.0	−0.520836
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−34312.7	−1.41192	−0.705962	−	0.708250i	$-0.749485\pi$
$839$	−34312.7	−1.41192	−0.705962	+	0.708250i	$0.749485\pi$
$840$	0	0
$841$	14673.3	0.601637
$842$	0	0
$843$	0	0
$844$	0	0
$845$	24257.5	0.987554
$846$	0	0
$847$	2346.41	0.0951872
$848$	0	0
$849$	0	0
$850$	0	0
$851$	16125.4	0.649556
$852$	0	0
$853$	−12008.4	−0.482017	−0.241008	−	0.970523i	$-0.577478\pi$
$853$	−12008.4	−0.482017	−0.241008	+	0.970523i	$0.577478\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30381.8	1.21099	0.605497	−	0.795848i	$-0.292974\pi$
$857$	30381.8	1.21099	0.605497	+	0.795848i	$0.292974\pi$
$858$	0	0
$859$	15306.0	0.607955	0.303978	−	0.952679i	$-0.401685\pi$
$859$	15306.0	0.607955	0.303978	+	0.952679i	$0.401685\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44221.6	1.74429	0.872143	−	0.489250i	$-0.162730\pi$
$863$	44221.6	1.74429	0.872143	+	0.489250i	$0.162730\pi$
$864$	0	0
$865$	−13331.8	−0.524041
$866$	0	0
$867$	0	0
$868$	0	0
$869$	7336.26	0.286381
$870$	0	0
$871$	1806.48	0.0702759
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3110.09	−0.120160
$876$	0	0
$877$	−31182.3	−1.20063	−0.600315	−	0.799764i	$-0.704958\pi$
$877$	−31182.3	−1.20063	−0.600315	+	0.799764i	$0.704958\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−17093.9	−0.653698	−0.326849	−	0.945077i	$-0.605987\pi$
$881$	−17093.9	−0.653698	−0.326849	+	0.945077i	$0.605987\pi$
$882$	0	0
$883$	47467.7	1.80908	0.904539	−	0.426392i	$-0.140215\pi$
$883$	47467.7	1.80908	0.904539	+	0.426392i	$0.140215\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32829.5	−1.24274	−0.621369	−	0.783518i	$-0.713423\pi$
$887$	−32829.5	−1.24274	−0.621369	+	0.783518i	$0.713423\pi$
$888$	0	0
$889$	−3802.67	−0.143462
$890$	0	0
$891$	0	0
$892$	0	0
$893$	9250.01	0.346629
$894$	0	0
$895$	−6073.67	−0.226838
$896$	0	0
$897$	0	0
$898$	0	0
$899$	8517.09	0.315974
$900$	0	0
$901$	−46133.1	−1.70579
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−33561.0	−1.23271
$906$	0	0
$907$	−21485.1	−0.786550	−0.393275	−	0.919421i	$-0.628658\pi$
$907$	−21485.1	−0.786550	−0.393275	+	0.919421i	$0.628658\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35721.6	1.29913	0.649566	−	0.760305i	$-0.274951\pi$
$911$	35721.6	1.29913	0.649566	+	0.760305i	$0.274951\pi$
$912$	0	0
$913$	22933.5	0.831311
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1195.00	0.0430343
$918$	0	0
$919$	−8821.16	−0.316630	−0.158315	−	0.987389i	$-0.550606\pi$
$919$	−8821.16	−0.316630	−0.158315	+	0.987389i	$0.550606\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	467.649	0.0166770
$924$	0	0
$925$	294.636	0.0104731
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29112.9	1.02816	0.514082	−	0.857741i	$-0.328133\pi$
$929$	29112.9	1.02816	0.514082	+	0.857741i	$0.328133\pi$
$930$	0	0
$931$	−6423.92	−0.226139
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−16453.1	−0.575480
$936$	0	0
$937$	−26746.1	−0.932506	−0.466253	−	0.884651i	$-0.654396\pi$
$937$	−26746.1	−0.932506	−0.466253	+	0.884651i	$0.654396\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	15834.9	0.548568	0.274284	−	0.961649i	$-0.411559\pi$
$941$	15834.9	0.548568	0.274284	+	0.961649i	$0.411559\pi$
$942$	0	0
$943$	−21278.9	−0.734819
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20454.0	−0.701863	−0.350932	−	0.936401i	$-0.614135\pi$
$947$	−20454.0	−0.701863	−0.350932	+	0.936401i	$0.614135\pi$
$948$	0	0
$949$	1694.75	0.0579704
$950$	0	0
$951$	0	0
$952$	0	0
$953$	298.716	0.0101536	0.00507679	−	0.999987i	$-0.498384\pi$
$953$	298.716	0.0101536	0.00507679	+	0.999987i	$0.498384\pi$
$954$	0	0
$955$	−34389.9	−1.16527
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3431.75	−0.115555
$960$	0	0
$961$	−27933.9	−0.937664
$962$	0	0
$963$	0	0
$964$	0	0
$965$	5511.49	0.183856
$966$	0	0
$967$	35204.9	1.17075	0.585373	−	0.810764i	$-0.300948\pi$
$967$	35204.9	1.17075	0.585373	+	0.810764i	$0.300948\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	24873.7	0.822075	0.411037	−	0.911619i	$-0.365167\pi$
$971$	24873.7	0.822075	0.411037	+	0.911619i	$0.365167\pi$
$972$	0	0
$973$	−1408.58	−0.0464100
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27947.2	−0.915158	−0.457579	−	0.889169i	$-0.651283\pi$
$977$	−27947.2	−0.915158	−0.457579	+	0.889169i	$0.651283\pi$
$978$	0	0
$979$	10161.2	0.331719
$980$	0	0
$981$	0	0
$982$	0	0
$983$	46047.2	1.49408	0.747038	−	0.664782i	$-0.231475\pi$
$983$	46047.2	1.49408	0.747038	+	0.664782i	$0.231475\pi$
$984$	0	0
$985$	43671.8	1.41269
$986$	0	0
$987$	0	0
$988$	0	0
$989$	39085.6	1.25667
$990$	0	0
$991$	18890.5	0.605528	0.302764	−	0.953066i	$-0.402091\pi$
$991$	18890.5	0.605528	0.302764	+	0.953066i	$0.402091\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14812.4	−0.471946
$996$	0	0
$997$	−34796.2	−1.10532	−0.552661	−	0.833406i	$-0.686388\pi$
$997$	−34796.2	−1.10532	−0.552661	+	0.833406i	$0.686388\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.4.a.p.1.2	yes	8
3.2	odd	2		1368.4.a.o.1.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1368.4.a.o.1.7	✓	8	3.2	odd	2
1368.4.a.p.1.2	yes	8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)

\(f(q)\)	\(=\)	\(q-11.1186 q^{5} -2.21338 q^{7} +16.4590 q^{11} +3.91157 q^{13} +89.9069 q^{17} +19.0000 q^{19} -75.3174 q^{23} -1.37617 q^{25} -197.642 q^{29} -43.0936 q^{31} +24.6098 q^{35} -214.099 q^{37} +282.522 q^{41} -518.945 q^{43} +486.843 q^{47} -338.101 q^{49} -513.121 q^{53} -183.001 q^{55} +230.881 q^{59} +578.922 q^{61} -43.4913 q^{65} +461.830 q^{67} +119.555 q^{71} +433.266 q^{73} -36.4301 q^{77} +445.729 q^{79} +1393.37 q^{83} -999.641 q^{85} +617.364 q^{89} -8.65780 q^{91} -211.254 q^{95} +1302.63 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{5} - 4 q^{7} + 16 q^{11} - 36 q^{13} - 48 q^{17} + 152 q^{19} + 88 q^{23} + 124 q^{25} + 48 q^{29} - 4 q^{31} + 72 q^{35} + 4 q^{37} + 376 q^{41} + 276 q^{43} + 384 q^{47} + 780 q^{49} + 1528 q^{53}+ \cdots + 840 q^{97}+O(q^{100}) \) 8 * q + 16 * q^5 - 4 * q^7 + 16 * q^11 - 36 * q^13 - 48 * q^17 + 152 * q^19 + 88 * q^23 + 124 * q^25 + 48 * q^29 - 4 * q^31 + 72 * q^35 + 4 * q^37 + 376 * q^41 + 276 * q^43 + 384 * q^47 + 780 * q^49 + 1528 * q^53 - 604 * q^55 + 664 * q^59 + 68 * q^61 + 1576 * q^65 - 184 * q^67 + 1704 * q^71 + 628 * q^73 + 1344 * q^77 - 1380 * q^79 + 1480 * q^83 + 796 * q^85 + 1800 * q^89 - 1864 * q^91 + 304 * q^95 + 840 * q^97

Introduction

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