LMFDB - Embedded newform 4332.2.a.t.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4332,2,Mod(1,4332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4332, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4332.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4332 = 2^{2} \cdot 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4332.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,3,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$34.5911941556$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.73227321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 18x^{4} + 27x^{3} + 96x^{2} - 48x - 127 $ x^6 - 3x^5 - 18x^4 + 27x^3 + 96x^2 - 48*x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 228)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.32474$ of defining polynomial
Character	$\chi$	$=$	4332.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +3.32474 q^{5} +1.19263 q^{7} +1.00000 q^{9} +1.29512 q^{11} +2.85683 q^{13} -3.32474 q^{15} -2.21885 q^{17} -1.19263 q^{21} +7.92884 q^{23} +6.05389 q^{25} -1.00000 q^{27} -4.48970 q^{29} +7.83427 q^{31} -1.29512 q^{33} +3.96517 q^{35} -7.86729 q^{37} -2.85683 q^{39} +8.06264 q^{41} -1.78889 q^{43} +3.32474 q^{45} +5.70881 q^{47} -5.57764 q^{49} +2.21885 q^{51} -6.31291 q^{53} +4.30594 q^{55} -7.89384 q^{59} +14.6331 q^{61} +1.19263 q^{63} +9.49821 q^{65} +4.58548 q^{67} -7.92884 q^{69} +4.18036 q^{71} +1.94679 q^{73} -6.05389 q^{75} +1.54460 q^{77} -6.20504 q^{79} +1.00000 q^{81} +7.83717 q^{83} -7.37709 q^{85} +4.48970 q^{87} -4.27972 q^{89} +3.40713 q^{91} -7.83427 q^{93} -2.53635 q^{97} +1.29512 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 9 q^{11} - 9 q^{13} - 3 q^{15} + 9 q^{17} - 9 q^{21} + 12 q^{23} + 15 q^{25} - 6 q^{27} - 9 q^{29} + 6 q^{31} - 9 q^{33} - 3 q^{35} + 6 q^{37} + 9 q^{39} + 18 q^{41}+ \cdots + 9 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9 + 9 * q^11 - 9 * q^13 - 3 * q^15 + 9 * q^17 - 9 * q^21 + 12 * q^23 + 15 * q^25 - 6 * q^27 - 9 * q^29 + 6 * q^31 - 9 * q^33 - 3 * q^35 + 6 * q^37 + 9 * q^39 + 18 * q^41 + 15 * q^43 + 3 * q^45 + 21 * q^47 + 21 * q^49 - 9 * q^51 - 6 * q^53 + 12 * q^55 - 15 * q^59 - 21 * q^61 + 9 * q^63 + 33 * q^65 - 18 * q^67 - 12 * q^69 + 15 * q^71 + 18 * q^73 - 15 * q^75 + 9 * q^79 + 6 * q^81 - 3 * q^83 + 6 * q^85 + 9 * q^87 - 36 * q^89 - 18 * q^91 - 6 * q^93 - 39 * q^97 + 9 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	3.32474	1.48687	0.743434	−	0.668809i	$-0.233196\pi$
$5$	3.32474	1.48687	0.743434	+	0.668809i	$0.233196\pi$
$6$	0	0
$7$	1.19263	0.450770	0.225385	−	0.974270i	$-0.427636\pi$
$7$	1.19263	0.450770	0.225385	+	0.974270i	$0.427636\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.29512	0.390494	0.195247	−	0.980754i	$-0.437449\pi$
$11$	1.29512	0.390494	0.195247	+	0.980754i	$0.437449\pi$
$12$	0	0
$13$	2.85683	0.792342	0.396171	−	0.918177i	$-0.370339\pi$
$13$	2.85683	0.792342	0.396171	+	0.918177i	$0.370339\pi$
$14$	0	0
$15$	−3.32474	−0.858444
$16$	0	0
$17$	−2.21885	−0.538150	−0.269075	−	0.963119i	$-0.586718\pi$
$17$	−2.21885	−0.538150	−0.269075	+	0.963119i	$0.586718\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.19263	−0.260252
$22$	0	0
$23$	7.92884	1.65328	0.826639	−	0.562733i	$-0.190250\pi$
$23$	7.92884	1.65328	0.826639	+	0.562733i	$0.190250\pi$
$24$	0	0
$25$	6.05389	1.21078
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−4.48970	−0.833716	−0.416858	−	0.908972i	$-0.636869\pi$
$29$	−4.48970	−0.833716	−0.416858	+	0.908972i	$0.636869\pi$
$30$	0	0
$31$	7.83427	1.40708	0.703538	−	0.710657i	$-0.251602\pi$
$31$	7.83427	1.40708	0.703538	+	0.710657i	$0.251602\pi$
$32$	0	0
$33$	−1.29512	−0.225452
$34$	0	0
$35$	3.96517	0.670236
$36$	0	0
$37$	−7.86729	−1.29337	−0.646687	−	0.762755i	$-0.723846\pi$
$37$	−7.86729	−1.29337	−0.646687	+	0.762755i	$0.723846\pi$
$38$	0	0
$39$	−2.85683	−0.457459
$40$	0	0
$41$	8.06264	1.25917	0.629586	−	0.776931i	$-0.283224\pi$
$41$	8.06264	1.25917	0.629586	+	0.776931i	$0.283224\pi$
$42$	0	0
$43$	−1.78889	−0.272804	−0.136402	−	0.990654i	$-0.543554\pi$
$43$	−1.78889	−0.272804	−0.136402	+	0.990654i	$0.543554\pi$
$44$	0	0
$45$	3.32474	0.495623
$46$	0	0
$47$	5.70881	0.832716	0.416358	−	0.909201i	$-0.363306\pi$
$47$	5.70881	0.832716	0.416358	+	0.909201i	$0.363306\pi$
$48$	0	0
$49$	−5.57764	−0.796806
$50$	0	0
$51$	2.21885	0.310701
$52$	0	0
$53$	−6.31291	−0.867145	−0.433573	−	0.901119i	$-0.642747\pi$
$53$	−6.31291	−0.867145	−0.433573	+	0.901119i	$0.642747\pi$
$54$	0	0
$55$	4.30594	0.580613
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.89384	−1.02769	−0.513845	−	0.857883i	$-0.671779\pi$
$59$	−7.89384	−1.02769	−0.513845	+	0.857883i	$0.671779\pi$
$60$	0	0
$61$	14.6331	1.87358	0.936790	−	0.349893i	$-0.113782\pi$
$61$	14.6331	1.87358	0.936790	+	0.349893i	$0.113782\pi$
$62$	0	0
$63$	1.19263	0.150257
$64$	0	0
$65$	9.49821	1.17811
$66$	0	0
$67$	4.58548	0.560205	0.280103	−	0.959970i	$-0.409632\pi$
$67$	4.58548	0.560205	0.280103	+	0.959970i	$0.409632\pi$
$68$	0	0
$69$	−7.92884	−0.954520
$70$	0	0
$71$	4.18036	0.496117	0.248058	−	0.968745i	$-0.420207\pi$
$71$	4.18036	0.496117	0.248058	+	0.968745i	$0.420207\pi$
$72$	0	0
$73$	1.94679	0.227854	0.113927	−	0.993489i	$-0.463657\pi$
$73$	1.94679	0.227854	0.113927	+	0.993489i	$0.463657\pi$
$74$	0	0
$75$	−6.05389	−0.699043
$76$	0	0
$77$	1.54460	0.176023
$78$	0	0
$79$	−6.20504	−0.698121	−0.349061	−	0.937100i	$-0.613499\pi$
$79$	−6.20504	−0.698121	−0.349061	+	0.937100i	$0.613499\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.83717	0.860241	0.430120	−	0.902772i	$-0.358471\pi$
$83$	7.83717	0.860241	0.430120	+	0.902772i	$0.358471\pi$
$84$	0	0
$85$	−7.37709	−0.800158
$86$	0	0
$87$	4.48970	0.481346
$88$	0	0
$89$	−4.27972	−0.453650	−0.226825	−	0.973936i	$-0.572834\pi$
$89$	−4.27972	−0.453650	−0.226825	+	0.973936i	$0.572834\pi$
$90$	0	0
$91$	3.40713	0.357164
$92$	0	0
$93$	−7.83427	−0.812376
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.53635	−0.257527	−0.128764	−	0.991675i	$-0.541101\pi$
$97$	−2.53635	−0.257527	−0.128764	+	0.991675i	$0.541101\pi$
$98$	0	0
$99$	1.29512	0.130165
$100$	0	0
$101$	12.4763	1.24144	0.620718	−	0.784034i	$-0.286841\pi$
$101$	12.4763	1.24144	0.620718	+	0.784034i	$0.286841\pi$
$102$	0	0
$103$	17.5778	1.73199	0.865997	−	0.500049i	$-0.166685\pi$
$103$	17.5778	1.73199	0.865997	+	0.500049i	$0.166685\pi$
$104$	0	0
$105$	−3.96517	−0.386961
$106$	0	0
$107$	−8.71759	−0.842761	−0.421381	−	0.906884i	$-0.638454\pi$
$107$	−8.71759	−0.842761	−0.421381	+	0.906884i	$0.638454\pi$
$108$	0	0
$109$	−19.5311	−1.87074	−0.935372	−	0.353667i	$-0.884935\pi$
$109$	−19.5311	−1.87074	−0.935372	+	0.353667i	$0.884935\pi$
$110$	0	0
$111$	7.86729	0.746730
$112$	0	0
$113$	0.595683	0.0560372	0.0280186	−	0.999607i	$-0.491080\pi$
$113$	0.595683	0.0560372	0.0280186	+	0.999607i	$0.491080\pi$
$114$	0	0
$115$	26.3613	2.45821
$116$	0	0
$117$	2.85683	0.264114
$118$	0	0
$119$	−2.64626	−0.242582
$120$	0	0
$121$	−9.32266	−0.847515
$122$	0	0
$123$	−8.06264	−0.726983
$124$	0	0
$125$	3.50391	0.313400
$126$	0	0
$127$	9.78123	0.867944	0.433972	−	0.900926i	$-0.357112\pi$
$127$	9.78123	0.867944	0.433972	+	0.900926i	$0.357112\pi$
$128$	0	0
$129$	1.78889	0.157503
$130$	0	0
$131$	−15.6461	−1.36700	−0.683502	−	0.729949i	$-0.739544\pi$
$131$	−15.6461	−1.36700	−0.683502	+	0.729949i	$0.739544\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−3.32474	−0.286148
$136$	0	0
$137$	−15.4793	−1.32249	−0.661243	−	0.750171i	$-0.729971\pi$
$137$	−15.4793	−1.32249	−0.661243	+	0.750171i	$0.729971\pi$
$138$	0	0
$139$	11.5448	0.979214	0.489607	−	0.871943i	$-0.337140\pi$
$139$	11.5448	0.979214	0.489607	+	0.871943i	$0.337140\pi$
$140$	0	0
$141$	−5.70881	−0.480769
$142$	0	0
$143$	3.69994	0.309404
$144$	0	0
$145$	−14.9271	−1.23963
$146$	0	0
$147$	5.57764	0.460036
$148$	0	0
$149$	−23.3235	−1.91073	−0.955366	−	0.295425i	$-0.904539\pi$
$149$	−23.3235	−1.91073	−0.955366	+	0.295425i	$0.904539\pi$
$150$	0	0
$151$	−3.39770	−0.276501	−0.138250	−	0.990397i	$-0.544148\pi$
$151$	−3.39770	−0.276501	−0.138250	+	0.990397i	$0.544148\pi$
$152$	0	0
$153$	−2.21885	−0.179383
$154$	0	0
$155$	26.0469	2.09214
$156$	0	0
$157$	−22.7640	−1.81676	−0.908381	−	0.418143i	$-0.862681\pi$
$157$	−22.7640	−1.81676	−0.908381	+	0.418143i	$0.862681\pi$
$158$	0	0
$159$	6.31291	0.500646
$160$	0	0
$161$	9.45614	0.745249
$162$	0	0
$163$	−13.0725	−1.02392	−0.511960	−	0.859009i	$-0.671080\pi$
$163$	−13.0725	−1.02392	−0.511960	+	0.859009i	$0.671080\pi$
$164$	0	0
$165$	−4.30594	−0.335217
$166$	0	0
$167$	−15.1710	−1.17396	−0.586982	−	0.809600i	$-0.699684\pi$
$167$	−15.1710	−1.17396	−0.586982	+	0.809600i	$0.699684\pi$
$168$	0	0
$169$	−4.83853	−0.372195
$170$	0	0
$171$	0	0
$172$	0	0
$173$	16.7006	1.26972	0.634861	−	0.772626i	$-0.281057\pi$
$173$	16.7006	1.26972	0.634861	+	0.772626i	$0.281057\pi$
$174$	0	0
$175$	7.22003	0.545783
$176$	0	0
$177$	7.89384	0.593337
$178$	0	0
$179$	13.6942	1.02356	0.511778	−	0.859118i	$-0.328987\pi$
$179$	13.6942	1.02356	0.511778	+	0.859118i	$0.328987\pi$
$180$	0	0
$181$	20.3687	1.51400	0.756998	−	0.653417i	$-0.226665\pi$
$181$	20.3687	1.51400	0.756998	+	0.653417i	$0.226665\pi$
$182$	0	0
$183$	−14.6331	−1.08171
$184$	0	0
$185$	−26.1567	−1.92308
$186$	0	0
$187$	−2.87368	−0.210144
$188$	0	0
$189$	−1.19263	−0.0867508
$190$	0	0
$191$	21.2065	1.53445	0.767224	−	0.641380i	$-0.221638\pi$
$191$	21.2065	1.53445	0.767224	+	0.641380i	$0.221638\pi$
$192$	0	0
$193$	14.0188	1.00910	0.504549	−	0.863383i	$-0.331659\pi$
$193$	14.0188	1.00910	0.504549	+	0.863383i	$0.331659\pi$
$194$	0	0
$195$	−9.49821	−0.680181
$196$	0	0
$197$	5.68277	0.404880	0.202440	−	0.979295i	$-0.435113\pi$
$197$	5.68277	0.404880	0.202440	+	0.979295i	$0.435113\pi$
$198$	0	0
$199$	10.7421	0.761491	0.380745	−	0.924680i	$-0.375667\pi$
$199$	10.7421	0.761491	0.380745	+	0.924680i	$0.375667\pi$
$200$	0	0
$201$	−4.58548	−0.323435
$202$	0	0
$203$	−5.35453	−0.375814
$204$	0	0
$205$	26.8062	1.87222
$206$	0	0
$207$	7.92884	0.551092
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−20.1260	−1.38553	−0.692767	−	0.721162i	$-0.743609\pi$
$211$	−20.1260	−1.38553	−0.692767	+	0.721162i	$0.743609\pi$
$212$	0	0
$213$	−4.18036	−0.286433
$214$	0	0
$215$	−5.94760	−0.405623
$216$	0	0
$217$	9.34336	0.634269
$218$	0	0
$219$	−1.94679	−0.131552
$220$	0	0
$221$	−6.33887	−0.426398
$222$	0	0
$223$	3.39571	0.227394	0.113697	−	0.993515i	$-0.463731\pi$
$223$	3.39571	0.227394	0.113697	+	0.993515i	$0.463731\pi$
$224$	0	0
$225$	6.05389	0.403593
$226$	0	0
$227$	11.7293	0.778503	0.389251	−	0.921132i	$-0.372734\pi$
$227$	11.7293	0.778503	0.389251	+	0.921132i	$0.372734\pi$
$228$	0	0
$229$	7.83401	0.517686	0.258843	−	0.965919i	$-0.416659\pi$
$229$	7.83401	0.517686	0.258843	+	0.965919i	$0.416659\pi$
$230$	0	0
$231$	−1.54460	−0.101627
$232$	0	0
$233$	−2.74354	−0.179735	−0.0898677	−	0.995954i	$-0.528644\pi$
$233$	−2.74354	−0.179735	−0.0898677	+	0.995954i	$0.528644\pi$
$234$	0	0
$235$	18.9803	1.23814
$236$	0	0
$237$	6.20504	0.403060
$238$	0	0
$239$	17.0871	1.10527	0.552636	−	0.833423i	$-0.313622\pi$
$239$	17.0871	1.10527	0.552636	+	0.833423i	$0.313622\pi$
$240$	0	0
$241$	3.23596	0.208447	0.104223	−	0.994554i	$-0.466764\pi$
$241$	3.23596	0.208447	0.104223	+	0.994554i	$0.466764\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−18.5442	−1.18475
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−7.83717	−0.496660
$250$	0	0
$251$	6.61629	0.417616	0.208808	−	0.977957i	$-0.433042\pi$
$251$	6.61629	0.417616	0.208808	+	0.977957i	$0.433042\pi$
$252$	0	0
$253$	10.2688	0.645594
$254$	0	0
$255$	7.37709	0.461971
$256$	0	0
$257$	14.5749	0.909159	0.454579	−	0.890706i	$-0.349790\pi$
$257$	14.5749	0.909159	0.454579	+	0.890706i	$0.349790\pi$
$258$	0	0
$259$	−9.38274	−0.583015
$260$	0	0
$261$	−4.48970	−0.277905
$262$	0	0
$263$	9.82446	0.605802	0.302901	−	0.953022i	$-0.402045\pi$
$263$	9.82446	0.605802	0.302901	+	0.953022i	$0.402045\pi$
$264$	0	0
$265$	−20.9888	−1.28933
$266$	0	0
$267$	4.27972	0.261915
$268$	0	0
$269$	21.6388	1.31934	0.659669	−	0.751556i	$-0.270696\pi$
$269$	21.6388	1.31934	0.659669	+	0.751556i	$0.270696\pi$
$270$	0	0
$271$	−6.09081	−0.369990	−0.184995	−	0.982739i	$-0.559227\pi$
$271$	−6.09081	−0.369990	−0.184995	+	0.982739i	$0.559227\pi$
$272$	0	0
$273$	−3.40713	−0.206209
$274$	0	0
$275$	7.84052	0.472801
$276$	0	0
$277$	16.2276	0.975023	0.487512	−	0.873116i	$-0.337905\pi$
$277$	16.2276	0.975023	0.487512	+	0.873116i	$0.337905\pi$
$278$	0	0
$279$	7.83427	0.469026
$280$	0	0
$281$	4.11893	0.245715	0.122857	−	0.992424i	$-0.460794\pi$
$281$	4.11893	0.245715	0.122857	+	0.992424i	$0.460794\pi$
$282$	0	0
$283$	26.9812	1.60387	0.801933	−	0.597414i	$-0.203805\pi$
$283$	26.9812	1.60387	0.801933	+	0.597414i	$0.203805\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.61571	0.567598
$288$	0	0
$289$	−12.0767	−0.710395
$290$	0	0
$291$	2.53635	0.148683
$292$	0	0
$293$	6.24794	0.365009	0.182504	−	0.983205i	$-0.441580\pi$
$293$	6.24794	0.365009	0.182504	+	0.983205i	$0.441580\pi$
$294$	0	0
$295$	−26.2450	−1.52804
$296$	0	0
$297$	−1.29512	−0.0751505
$298$	0	0
$299$	22.6513	1.30996
$300$	0	0
$301$	−2.13348	−0.122972
$302$	0	0
$303$	−12.4763	−0.716743
$304$	0	0
$305$	48.6513	2.78577
$306$	0	0
$307$	−11.9806	−0.683769	−0.341885	−	0.939742i	$-0.611065\pi$
$307$	−11.9806	−0.683769	−0.341885	+	0.939742i	$0.611065\pi$
$308$	0	0
$309$	−17.5778	−0.999967
$310$	0	0
$311$	−9.08216	−0.515002	−0.257501	−	0.966278i	$-0.582899\pi$
$311$	−9.08216	−0.515002	−0.257501	+	0.966278i	$0.582899\pi$
$312$	0	0
$313$	−16.5092	−0.933155	−0.466577	−	0.884480i	$-0.654513\pi$
$313$	−16.5092	−0.933155	−0.466577	+	0.884480i	$0.654513\pi$
$314$	0	0
$315$	3.96517	0.223412
$316$	0	0
$317$	19.4219	1.09084	0.545422	−	0.838161i	$-0.316369\pi$
$317$	19.4219	1.09084	0.545422	+	0.838161i	$0.316369\pi$
$318$	0	0
$319$	−5.81470	−0.325561
$320$	0	0
$321$	8.71759	0.486568
$322$	0	0
$323$	0	0
$324$	0	0
$325$	17.2949	0.959350
$326$	0	0
$327$	19.5311	1.08007
$328$	0	0
$329$	6.80848	0.375364
$330$	0	0
$331$	−14.9691	−0.822775	−0.411387	−	0.911461i	$-0.634956\pi$
$331$	−14.9691	−0.822775	−0.411387	+	0.911461i	$0.634956\pi$
$332$	0	0
$333$	−7.86729	−0.431125
$334$	0	0
$335$	15.2455	0.832951
$336$	0	0
$337$	16.1502	0.879759	0.439879	−	0.898057i	$-0.355021\pi$
$337$	16.1502	0.879759	0.439879	+	0.898057i	$0.355021\pi$
$338$	0	0
$339$	−0.595683	−0.0323531
$340$	0	0
$341$	10.1463	0.549454
$342$	0	0
$343$	−15.0004	−0.809947
$344$	0	0
$345$	−26.3613	−1.41925
$346$	0	0
$347$	31.8046	1.70736	0.853680	−	0.520798i	$-0.174366\pi$
$347$	31.8046	1.70736	0.853680	+	0.520798i	$0.174366\pi$
$348$	0	0
$349$	−3.55421	−0.190252	−0.0951261	−	0.995465i	$-0.530325\pi$
$349$	−3.55421	−0.190252	−0.0951261	+	0.995465i	$0.530325\pi$
$350$	0	0
$351$	−2.85683	−0.152486
$352$	0	0
$353$	−31.4019	−1.67136	−0.835678	−	0.549219i	$-0.814925\pi$
$353$	−31.4019	−1.67136	−0.835678	+	0.549219i	$0.814925\pi$
$354$	0	0
$355$	13.8986	0.737661
$356$	0	0
$357$	2.64626	0.140055
$358$	0	0
$359$	−16.9204	−0.893027	−0.446513	−	0.894777i	$-0.647335\pi$
$359$	−16.9204	−0.893027	−0.446513	+	0.894777i	$0.647335\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	9.32266	0.489313
$364$	0	0
$365$	6.47256	0.338789
$366$	0	0
$367$	9.50220	0.496011	0.248005	−	0.968759i	$-0.420225\pi$
$367$	9.50220	0.496011	0.248005	+	0.968759i	$0.420225\pi$
$368$	0	0
$369$	8.06264	0.419724
$370$	0	0
$371$	−7.52895	−0.390883
$372$	0	0
$373$	−23.3929	−1.21124	−0.605619	−	0.795755i	$-0.707074\pi$
$373$	−23.3929	−1.21124	−0.605619	+	0.795755i	$0.707074\pi$
$374$	0	0
$375$	−3.50391	−0.180941
$376$	0	0
$377$	−12.8263	−0.660587
$378$	0	0
$379$	−22.1013	−1.13527	−0.567634	−	0.823281i	$-0.692141\pi$
$379$	−22.1013	−1.13527	−0.567634	+	0.823281i	$0.692141\pi$
$380$	0	0
$381$	−9.78123	−0.501108
$382$	0	0
$383$	18.0824	0.923970	0.461985	−	0.886888i	$-0.347137\pi$
$383$	18.0824	0.923970	0.461985	+	0.886888i	$0.347137\pi$
$384$	0	0
$385$	5.13538	0.261723
$386$	0	0
$387$	−1.78889	−0.0909346
$388$	0	0
$389$	33.8958	1.71859	0.859293	−	0.511484i	$-0.170904\pi$
$389$	33.8958	1.71859	0.859293	+	0.511484i	$0.170904\pi$
$390$	0	0
$391$	−17.5929	−0.889711
$392$	0	0
$393$	15.6461	0.789240
$394$	0	0
$395$	−20.6301	−1.03801
$396$	0	0
$397$	−17.4561	−0.876096	−0.438048	−	0.898952i	$-0.644330\pi$
$397$	−17.4561	−0.876096	−0.438048	+	0.898952i	$0.644330\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−32.9012	−1.64301	−0.821503	−	0.570205i	$-0.806864\pi$
$401$	−32.9012	−1.64301	−0.821503	+	0.570205i	$0.806864\pi$
$402$	0	0
$403$	22.3812	1.11489
$404$	0	0
$405$	3.32474	0.165208
$406$	0	0
$407$	−10.1891	−0.505055
$408$	0	0
$409$	7.73471	0.382457	0.191228	−	0.981546i	$-0.438753\pi$
$409$	7.73471	0.382457	0.191228	+	0.981546i	$0.438753\pi$
$410$	0	0
$411$	15.4793	0.763538
$412$	0	0
$413$	−9.41440	−0.463252
$414$	0	0
$415$	26.0565	1.27907
$416$	0	0
$417$	−11.5448	−0.565350
$418$	0	0
$419$	−26.4903	−1.29414	−0.647068	−	0.762432i	$-0.724005\pi$
$419$	−26.4903	−1.29414	−0.647068	+	0.762432i	$0.724005\pi$
$420$	0	0
$421$	10.5701	0.515153	0.257577	−	0.966258i	$-0.417076\pi$
$421$	10.5701	0.515153	0.257577	+	0.966258i	$0.417076\pi$
$422$	0	0
$423$	5.70881	0.277572
$424$	0	0
$425$	−13.4327	−0.651580
$426$	0	0
$427$	17.4518	0.844554
$428$	0	0
$429$	−3.69994	−0.178635
$430$	0	0
$431$	15.0543	0.725140	0.362570	−	0.931957i	$-0.381899\pi$
$431$	15.0543	0.725140	0.362570	+	0.931957i	$0.381899\pi$
$432$	0	0
$433$	−38.8432	−1.86668	−0.933342	−	0.358989i	$-0.883122\pi$
$433$	−38.8432	−1.86668	−0.933342	+	0.358989i	$0.883122\pi$
$434$	0	0
$435$	14.9271	0.715698
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.01644	0.0485120	0.0242560	−	0.999706i	$-0.492278\pi$
$439$	1.01644	0.0485120	0.0242560	+	0.999706i	$0.492278\pi$
$440$	0	0
$441$	−5.57764	−0.265602
$442$	0	0
$443$	−11.9863	−0.569485	−0.284742	−	0.958604i	$-0.591908\pi$
$443$	−11.9863	−0.569485	−0.284742	+	0.958604i	$0.591908\pi$
$444$	0	0
$445$	−14.2290	−0.674517
$446$	0	0
$447$	23.3235	1.10316
$448$	0	0
$449$	31.4582	1.48460	0.742301	−	0.670066i	$-0.233734\pi$
$449$	31.4582	1.48460	0.742301	+	0.670066i	$0.233734\pi$
$450$	0	0
$451$	10.4421	0.491699
$452$	0	0
$453$	3.39770	0.159638
$454$	0	0
$455$	11.3278	0.531056
$456$	0	0
$457$	−5.25795	−0.245956	−0.122978	−	0.992409i	$-0.539245\pi$
$457$	−5.25795	−0.245956	−0.122978	+	0.992409i	$0.539245\pi$
$458$	0	0
$459$	2.21885	0.103567
$460$	0	0
$461$	16.7782	0.781438	0.390719	−	0.920510i	$-0.372226\pi$
$461$	16.7782	0.781438	0.390719	+	0.920510i	$0.372226\pi$
$462$	0	0
$463$	9.46597	0.439921	0.219961	−	0.975509i	$-0.429407\pi$
$463$	9.46597	0.439921	0.219961	+	0.975509i	$0.429407\pi$
$464$	0	0
$465$	−26.0469	−1.20790
$466$	0	0
$467$	−18.5583	−0.858775	−0.429387	−	0.903120i	$-0.641270\pi$
$467$	−18.5583	−0.858775	−0.429387	+	0.903120i	$0.641270\pi$
$468$	0	0
$469$	5.46876	0.252524
$470$	0	0
$471$	22.7640	1.04891
$472$	0	0
$473$	−2.31683	−0.106528
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.31291	−0.289048
$478$	0	0
$479$	−7.21580	−0.329698	−0.164849	−	0.986319i	$-0.552714\pi$
$479$	−7.21580	−0.329698	−0.164849	+	0.986319i	$0.552714\pi$
$480$	0	0
$481$	−22.4755	−1.02479
$482$	0	0
$483$	−9.45614	−0.430269
$484$	0	0
$485$	−8.43270	−0.382909
$486$	0	0
$487$	4.65410	0.210897	0.105449	−	0.994425i	$-0.466372\pi$
$487$	4.65410	0.210897	0.105449	+	0.994425i	$0.466372\pi$
$488$	0	0
$489$	13.0725	0.591160
$490$	0	0
$491$	3.97523	0.179399	0.0896997	−	0.995969i	$-0.471409\pi$
$491$	3.97523	0.179399	0.0896997	+	0.995969i	$0.471409\pi$
$492$	0	0
$493$	9.96195	0.448664
$494$	0	0
$495$	4.30594	0.193538
$496$	0	0
$497$	4.98560	0.223635
$498$	0	0
$499$	0.281073	0.0125826	0.00629129	−	0.999980i	$-0.497997\pi$
$499$	0.281073	0.0125826	0.00629129	+	0.999980i	$0.497997\pi$
$500$	0	0
$501$	15.1710	0.677788
$502$	0	0
$503$	36.8744	1.64415	0.822074	−	0.569380i	$-0.192817\pi$
$503$	36.8744	1.64415	0.822074	+	0.569380i	$0.192817\pi$
$504$	0	0
$505$	41.4803	1.84585
$506$	0	0
$507$	4.83853	0.214887
$508$	0	0
$509$	26.0449	1.15442	0.577210	−	0.816596i	$-0.304141\pi$
$509$	26.0449	1.15442	0.577210	+	0.816596i	$0.304141\pi$
$510$	0	0
$511$	2.32179	0.102710
$512$	0	0
$513$	0	0
$514$	0	0
$515$	58.4417	2.57525
$516$	0	0
$517$	7.39360	0.325170
$518$	0	0
$519$	−16.7006	−0.733074
$520$	0	0
$521$	9.94224	0.435577	0.217789	−	0.975996i	$-0.430116\pi$
$521$	9.94224	0.435577	0.217789	+	0.975996i	$0.430116\pi$
$522$	0	0
$523$	−37.5695	−1.64280	−0.821399	−	0.570354i	$-0.806806\pi$
$523$	−37.5695	−1.64280	−0.821399	+	0.570354i	$0.806806\pi$
$524$	0	0
$525$	−7.22003	−0.315108
$526$	0	0
$527$	−17.3831	−0.757218
$528$	0	0
$529$	39.8665	1.73333
$530$	0	0
$531$	−7.89384	−0.342563
$532$	0	0
$533$	23.0336	0.997695
$534$	0	0
$535$	−28.9837	−1.25308
$536$	0	0
$537$	−13.6942	−0.590950
$538$	0	0
$539$	−7.22372	−0.311148
$540$	0	0
$541$	−2.66625	−0.114631	−0.0573155	−	0.998356i	$-0.518254\pi$
$541$	−2.66625	−0.114631	−0.0573155	+	0.998356i	$0.518254\pi$
$542$	0	0
$543$	−20.3687	−0.874106
$544$	0	0
$545$	−64.9359	−2.78155
$546$	0	0
$547$	−22.9448	−0.981048	−0.490524	−	0.871428i	$-0.663195\pi$
$547$	−22.9448	−0.981048	−0.490524	+	0.871428i	$0.663195\pi$
$548$	0	0
$549$	14.6331	0.624526
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−7.40029	−0.314692
$554$	0	0
$555$	26.1567	1.11029
$556$	0	0
$557$	−7.30091	−0.309350	−0.154675	−	0.987965i	$-0.549433\pi$
$557$	−7.30091	−0.309350	−0.154675	+	0.987965i	$0.549433\pi$
$558$	0	0
$559$	−5.11056	−0.216154
$560$	0	0
$561$	2.87368	0.121327
$562$	0	0
$563$	42.2689	1.78142	0.890712	−	0.454569i	$-0.150207\pi$
$563$	42.2689	1.78142	0.890712	+	0.454569i	$0.150207\pi$
$564$	0	0
$565$	1.98049	0.0833199
$566$	0	0
$567$	1.19263	0.0500856
$568$	0	0
$569$	15.0782	0.632112	0.316056	−	0.948741i	$-0.397641\pi$
$569$	15.0782	0.632112	0.316056	+	0.948741i	$0.397641\pi$
$570$	0	0
$571$	−42.7731	−1.79000	−0.894998	−	0.446070i	$-0.852823\pi$
$571$	−42.7731	−1.79000	−0.894998	+	0.446070i	$0.852823\pi$
$572$	0	0
$573$	−21.2065	−0.885914
$574$	0	0
$575$	48.0003	2.00175
$576$	0	0
$577$	23.2799	0.969153	0.484576	−	0.874749i	$-0.338974\pi$
$577$	23.2799	0.969153	0.484576	+	0.874749i	$0.338974\pi$
$578$	0	0
$579$	−14.0188	−0.582603
$580$	0	0
$581$	9.34681	0.387771
$582$	0	0
$583$	−8.17598	−0.338615
$584$	0	0
$585$	9.49821	0.392703
$586$	0	0
$587$	21.8768	0.902951	0.451475	−	0.892284i	$-0.350898\pi$
$587$	21.8768	0.902951	0.451475	+	0.892284i	$0.350898\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−5.68277	−0.233758
$592$	0	0
$593$	−32.8159	−1.34759	−0.673794	−	0.738919i	$-0.735337\pi$
$593$	−32.8159	−1.34759	−0.673794	+	0.738919i	$0.735337\pi$
$594$	0	0
$595$	−8.79811	−0.360687
$596$	0	0
$597$	−10.7421	−0.439647
$598$	0	0
$599$	−6.91197	−0.282415	−0.141208	−	0.989980i	$-0.545099\pi$
$599$	−6.91197	−0.282415	−0.141208	+	0.989980i	$0.545099\pi$
$600$	0	0
$601$	−34.0010	−1.38693	−0.693465	−	0.720491i	$-0.743917\pi$
$601$	−34.0010	−1.38693	−0.693465	+	0.720491i	$0.743917\pi$
$602$	0	0
$603$	4.58548	0.186735
$604$	0	0
$605$	−30.9954	−1.26014
$606$	0	0
$607$	21.1493	0.858424	0.429212	−	0.903204i	$-0.358791\pi$
$607$	21.1493	0.858424	0.429212	+	0.903204i	$0.358791\pi$
$608$	0	0
$609$	5.35453	0.216977
$610$	0	0
$611$	16.3091	0.659795
$612$	0	0
$613$	−5.12714	−0.207083	−0.103542	−	0.994625i	$-0.533018\pi$
$613$	−5.12714	−0.207083	−0.103542	+	0.994625i	$0.533018\pi$
$614$	0	0
$615$	−26.8062	−1.08093
$616$	0	0
$617$	−27.2061	−1.09528	−0.547639	−	0.836715i	$-0.684473\pi$
$617$	−27.2061	−1.09528	−0.547639	+	0.836715i	$0.684473\pi$
$618$	0	0
$619$	−11.4844	−0.461599	−0.230799	−	0.973001i	$-0.574134\pi$
$619$	−11.4844	−0.461599	−0.230799	+	0.973001i	$0.574134\pi$
$620$	0	0
$621$	−7.92884	−0.318173
$622$	0	0
$623$	−5.10411	−0.204492
$624$	0	0
$625$	−18.6199	−0.744794
$626$	0	0
$627$	0	0
$628$	0	0
$629$	17.4563	0.696029
$630$	0	0
$631$	3.58436	0.142691	0.0713456	−	0.997452i	$-0.477271\pi$
$631$	3.58436	0.142691	0.0713456	+	0.997452i	$0.477271\pi$
$632$	0	0
$633$	20.1260	0.799938
$634$	0	0
$635$	32.5201	1.29052
$636$	0	0
$637$	−15.9344	−0.631343
$638$	0	0
$639$	4.18036	0.165372
$640$	0	0
$641$	4.32078	0.170661	0.0853303	−	0.996353i	$-0.472805\pi$
$641$	4.32078	0.170661	0.0853303	+	0.996353i	$0.472805\pi$
$642$	0	0
$643$	−18.2228	−0.718638	−0.359319	−	0.933215i	$-0.616991\pi$
$643$	−18.2228	−0.718638	−0.359319	+	0.933215i	$0.616991\pi$
$644$	0	0
$645$	5.94760	0.234187
$646$	0	0
$647$	−26.4490	−1.03982	−0.519910	−	0.854221i	$-0.674034\pi$
$647$	−26.4490	−1.03982	−0.519910	+	0.854221i	$0.674034\pi$
$648$	0	0
$649$	−10.2235	−0.401307
$650$	0	0
$651$	−9.34336	−0.366195
$652$	0	0
$653$	−0.870759	−0.0340754	−0.0170377	−	0.999855i	$-0.505424\pi$
$653$	−0.870759	−0.0340754	−0.0170377	+	0.999855i	$0.505424\pi$
$654$	0	0
$655$	−52.0191	−2.03255
$656$	0	0
$657$	1.94679	0.0759514
$658$	0	0
$659$	27.1262	1.05669	0.528343	−	0.849031i	$-0.322814\pi$
$659$	27.1262	1.05669	0.528343	+	0.849031i	$0.322814\pi$
$660$	0	0
$661$	3.16622	0.123152	0.0615759	−	0.998102i	$-0.480387\pi$
$661$	3.16622	0.123152	0.0615759	+	0.998102i	$0.480387\pi$
$662$	0	0
$663$	6.33887	0.246181
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−35.5981	−1.37836
$668$	0	0
$669$	−3.39571	−0.131286
$670$	0	0
$671$	18.9517	0.731621
$672$	0	0
$673$	−9.85403	−0.379845	−0.189922	−	0.981799i	$-0.560824\pi$
$673$	−9.85403	−0.379845	−0.189922	+	0.981799i	$0.560824\pi$
$674$	0	0
$675$	−6.05389	−0.233014
$676$	0	0
$677$	0.495225	0.0190331	0.00951653	−	0.999955i	$-0.496971\pi$
$677$	0.495225	0.0190331	0.00951653	+	0.999955i	$0.496971\pi$
$678$	0	0
$679$	−3.02492	−0.116086
$680$	0	0
$681$	−11.7293	−0.449469
$682$	0	0
$683$	11.8365	0.452910	0.226455	−	0.974022i	$-0.427286\pi$
$683$	11.8365	0.452910	0.226455	+	0.974022i	$0.427286\pi$
$684$	0	0
$685$	−51.4647	−1.96636
$686$	0	0
$687$	−7.83401	−0.298886
$688$	0	0
$689$	−18.0349	−0.687075
$690$	0	0
$691$	−18.2263	−0.693361	−0.346680	−	0.937983i	$-0.612691\pi$
$691$	−18.2263	−0.693361	−0.346680	+	0.937983i	$0.612691\pi$
$692$	0	0
$693$	1.54460	0.0586743
$694$	0	0
$695$	38.3833	1.45596
$696$	0	0
$697$	−17.8898	−0.677623
$698$	0	0
$699$	2.74354	0.103770
$700$	0	0
$701$	16.5291	0.624297	0.312149	−	0.950033i	$-0.398951\pi$
$701$	16.5291	0.624297	0.312149	+	0.950033i	$0.398951\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−18.9803	−0.714840
$706$	0	0
$707$	14.8795	0.559602
$708$	0	0
$709$	−18.8344	−0.707339	−0.353670	−	0.935370i	$-0.615066\pi$
$709$	−18.8344	−0.707339	−0.353670	+	0.935370i	$0.615066\pi$
$710$	0	0
$711$	−6.20504	−0.232707
$712$	0	0
$713$	62.1167	2.32629
$714$	0	0
$715$	12.3013	0.460044
$716$	0	0
$717$	−17.0871	−0.638129
$718$	0	0
$719$	−35.5596	−1.32615	−0.663075	−	0.748553i	$-0.730749\pi$
$719$	−35.5596	−1.32615	−0.663075	+	0.748553i	$0.730749\pi$
$720$	0	0
$721$	20.9638	0.780732
$722$	0	0
$723$	−3.23596	−0.120347
$724$	0	0
$725$	−27.1801	−1.00944
$726$	0	0
$727$	−9.17253	−0.340190	−0.170095	−	0.985428i	$-0.554408\pi$
$727$	−9.17253	−0.340190	−0.170095	+	0.985428i	$0.554408\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.96928	0.146809
$732$	0	0
$733$	11.5404	0.426256	0.213128	−	0.977024i	$-0.431635\pi$
$733$	11.5404	0.426256	0.213128	+	0.977024i	$0.431635\pi$
$734$	0	0
$735$	18.5442	0.684013
$736$	0	0
$737$	5.93875	0.218757
$738$	0	0
$739$	−9.67212	−0.355795	−0.177897	−	0.984049i	$-0.556929\pi$
$739$	−9.67212	−0.355795	−0.177897	+	0.984049i	$0.556929\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.4040	−0.858609	−0.429304	−	0.903160i	$-0.641241\pi$
$743$	−23.4040	−0.858609	−0.429304	+	0.903160i	$0.641241\pi$
$744$	0	0
$745$	−77.5444	−2.84101
$746$	0	0
$747$	7.83717	0.286747
$748$	0	0
$749$	−10.3968	−0.379892
$750$	0	0
$751$	−2.76855	−0.101026	−0.0505129	−	0.998723i	$-0.516086\pi$
$751$	−2.76855	−0.101026	−0.0505129	+	0.998723i	$0.516086\pi$
$752$	0	0
$753$	−6.61629	−0.241111
$754$	0	0
$755$	−11.2965	−0.411120
$756$	0	0
$757$	−22.3062	−0.810734	−0.405367	−	0.914154i	$-0.632856\pi$
$757$	−22.3062	−0.810734	−0.405367	+	0.914154i	$0.632856\pi$
$758$	0	0
$759$	−10.2688	−0.372734
$760$	0	0
$761$	−0.666605	−0.0241644	−0.0120822	−	0.999927i	$-0.503846\pi$
$761$	−0.666605	−0.0241644	−0.0120822	+	0.999927i	$0.503846\pi$
$762$	0	0
$763$	−23.2933	−0.843276
$764$	0	0
$765$	−7.37709	−0.266719
$766$	0	0
$767$	−22.5513	−0.814282
$768$	0	0
$769$	−19.6883	−0.709979	−0.354990	−	0.934870i	$-0.615516\pi$
$769$	−19.6883	−0.709979	−0.354990	+	0.934870i	$0.615516\pi$
$770$	0	0
$771$	−14.5749	−0.524903
$772$	0	0
$773$	−49.3759	−1.77593	−0.887964	−	0.459913i	$-0.847881\pi$
$773$	−49.3759	−1.77593	−0.887964	+	0.459913i	$0.847881\pi$
$774$	0	0
$775$	47.4278	1.70366
$776$	0	0
$777$	9.38274	0.336604
$778$	0	0
$779$	0	0
$780$	0	0
$781$	5.41407	0.193731
$782$	0	0
$783$	4.48970	0.160449
$784$	0	0
$785$	−75.6842	−2.70129
$786$	0	0
$787$	−41.8546	−1.49196	−0.745978	−	0.665971i	$-0.768018\pi$
$787$	−41.8546	−1.49196	−0.745978	+	0.665971i	$0.768018\pi$
$788$	0	0
$789$	−9.82446	−0.349760
$790$	0	0
$791$	0.710428	0.0252599
$792$	0	0
$793$	41.8043	1.48451
$794$	0	0
$795$	20.9888	0.744396
$796$	0	0
$797$	50.2358	1.77944	0.889722	−	0.456503i	$-0.150898\pi$
$797$	50.2358	1.77944	0.889722	+	0.456503i	$0.150898\pi$
$798$	0	0
$799$	−12.6670	−0.448126
$800$	0	0
$801$	−4.27972	−0.151217
$802$	0	0
$803$	2.52133	0.0889757
$804$	0	0
$805$	31.4392	1.10809
$806$	0	0
$807$	−21.6388	−0.761720
$808$	0	0
$809$	9.83965	0.345944	0.172972	−	0.984927i	$-0.444663\pi$
$809$	9.83965	0.345944	0.172972	+	0.984927i	$0.444663\pi$
$810$	0	0
$811$	−28.6378	−1.00561	−0.502804	−	0.864400i	$-0.667698\pi$
$811$	−28.6378	−1.00561	−0.502804	+	0.864400i	$0.667698\pi$
$812$	0	0
$813$	6.09081	0.213614
$814$	0	0
$815$	−43.4628	−1.52243
$816$	0	0
$817$	0	0
$818$	0	0
$819$	3.40713	0.119055
$820$	0	0
$821$	−47.4389	−1.65563	−0.827814	−	0.561002i	$-0.810416\pi$
$821$	−47.4389	−1.65563	−0.827814	+	0.561002i	$0.810416\pi$
$822$	0	0
$823$	40.5906	1.41490	0.707450	−	0.706764i	$-0.249846\pi$
$823$	40.5906	1.41490	0.707450	+	0.706764i	$0.249846\pi$
$824$	0	0
$825$	−7.84052	−0.272972
$826$	0	0
$827$	−1.85364	−0.0644572	−0.0322286	−	0.999481i	$-0.510260\pi$
$827$	−1.85364	−0.0644572	−0.0322286	+	0.999481i	$0.510260\pi$
$828$	0	0
$829$	−30.9535	−1.07506	−0.537529	−	0.843245i	$-0.680642\pi$
$829$	−30.9535	−1.07506	−0.537529	+	0.843245i	$0.680642\pi$
$830$	0	0
$831$	−16.2276	−0.562930
$832$	0	0
$833$	12.3759	0.428801
$834$	0	0
$835$	−50.4395	−1.74553
$836$	0	0
$837$	−7.83427	−0.270792
$838$	0	0
$839$	31.3791	1.08333	0.541663	−	0.840596i	$-0.317795\pi$
$839$	31.3791	1.08333	0.541663	+	0.840596i	$0.317795\pi$
$840$	0	0
$841$	−8.84263	−0.304918
$842$	0	0
$843$	−4.11893	−0.141863
$844$	0	0
$845$	−16.0869	−0.553405
$846$	0	0
$847$	−11.1185	−0.382035
$848$	0	0
$849$	−26.9812	−0.925992
$850$	0	0
$851$	−62.3785	−2.13831
$852$	0	0
$853$	28.3884	0.972000	0.486000	−	0.873959i	$-0.338455\pi$
$853$	28.3884	0.972000	0.486000	+	0.873959i	$0.338455\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.50306	−0.222140	−0.111070	−	0.993813i	$-0.535428\pi$
$857$	−6.50306	−0.222140	−0.111070	+	0.993813i	$0.535428\pi$
$858$	0	0
$859$	−16.3745	−0.558690	−0.279345	−	0.960191i	$-0.590117\pi$
$859$	−16.3745	−0.558690	−0.279345	+	0.960191i	$0.590117\pi$
$860$	0	0
$861$	−9.61571	−0.327703
$862$	0	0
$863$	−49.1283	−1.67235	−0.836174	−	0.548465i	$-0.815212\pi$
$863$	−49.1283	−1.67235	−0.836174	+	0.548465i	$0.815212\pi$
$864$	0	0
$865$	55.5251	1.88791
$866$	0	0
$867$	12.0767	0.410147
$868$	0	0
$869$	−8.03627	−0.272612
$870$	0	0
$871$	13.0999	0.443874
$872$	0	0
$873$	−2.53635	−0.0858425
$874$	0	0
$875$	4.17886	0.141271
$876$	0	0
$877$	−12.9036	−0.435725	−0.217862	−	0.975979i	$-0.569908\pi$
$877$	−12.9036	−0.435725	−0.217862	+	0.975979i	$0.569908\pi$
$878$	0	0
$879$	−6.24794	−0.210738
$880$	0	0
$881$	−37.2780	−1.25593	−0.627965	−	0.778242i	$-0.716112\pi$
$881$	−37.2780	−1.25593	−0.627965	+	0.778242i	$0.716112\pi$
$882$	0	0
$883$	7.76349	0.261262	0.130631	−	0.991431i	$-0.458300\pi$
$883$	7.76349	0.261262	0.130631	+	0.991431i	$0.458300\pi$
$884$	0	0
$885$	26.2450	0.882215
$886$	0	0
$887$	−41.1272	−1.38092	−0.690459	−	0.723372i	$-0.742591\pi$
$887$	−41.1272	−1.38092	−0.690459	+	0.723372i	$0.742591\pi$
$888$	0	0
$889$	11.6654	0.391244
$890$	0	0
$891$	1.29512	0.0433882
$892$	0	0
$893$	0	0
$894$	0	0
$895$	45.5298	1.52189
$896$	0	0
$897$	−22.6513	−0.756306
$898$	0	0
$899$	−35.1735	−1.17310
$900$	0	0
$901$	14.0074	0.466654
$902$	0	0
$903$	2.13348	0.0709978
$904$	0	0
$905$	67.7207	2.25111
$906$	0	0
$907$	−33.3465	−1.10725	−0.553627	−	0.832765i	$-0.686757\pi$
$907$	−33.3465	−1.10725	−0.553627	+	0.832765i	$0.686757\pi$
$908$	0	0
$909$	12.4763	0.413812
$910$	0	0
$911$	26.6721	0.883687	0.441844	−	0.897092i	$-0.354325\pi$
$911$	26.6721	0.883687	0.441844	+	0.897092i	$0.354325\pi$
$912$	0	0
$913$	10.1501	0.335919
$914$	0	0
$915$	−48.6513	−1.60836
$916$	0	0
$917$	−18.6599	−0.616205
$918$	0	0
$919$	−2.99901	−0.0989283	−0.0494642	−	0.998776i	$-0.515751\pi$
$919$	−2.99901	−0.0989283	−0.0494642	+	0.998776i	$0.515751\pi$
$920$	0	0
$921$	11.9806	0.394774
$922$	0	0
$923$	11.9426	0.393094
$924$	0	0
$925$	−47.6277	−1.56599
$926$	0	0
$927$	17.5778	0.577331
$928$	0	0
$929$	−6.72762	−0.220726	−0.110363	−	0.993891i	$-0.535201\pi$
$929$	−6.72762	−0.220726	−0.110363	+	0.993891i	$0.535201\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	9.08216	0.297336
$934$	0	0
$935$	−9.55422	−0.312457
$936$	0	0
$937$	−15.7325	−0.513959	−0.256980	−	0.966417i	$-0.582727\pi$
$937$	−15.7325	−0.513959	−0.256980	+	0.966417i	$0.582727\pi$
$938$	0	0
$939$	16.5092	0.538757
$940$	0	0
$941$	4.78821	0.156091	0.0780456	−	0.996950i	$-0.475132\pi$
$941$	4.78821	0.156091	0.0780456	+	0.996950i	$0.475132\pi$
$942$	0	0
$943$	63.9274	2.08176
$944$	0	0
$945$	−3.96517	−0.128987
$946$	0	0
$947$	−29.8160	−0.968889	−0.484444	−	0.874822i	$-0.660978\pi$
$947$	−29.8160	−0.968889	−0.484444	+	0.874822i	$0.660978\pi$
$948$	0	0
$949$	5.56164	0.180538
$950$	0	0
$951$	−19.4219	−0.629799
$952$	0	0
$953$	−4.19741	−0.135968	−0.0679838	−	0.997686i	$-0.521657\pi$
$953$	−4.19741	−0.135968	−0.0679838	+	0.997686i	$0.521657\pi$
$954$	0	0
$955$	70.5060	2.28152
$956$	0	0
$957$	5.81470	0.187963
$958$	0	0
$959$	−18.4610	−0.596138
$960$	0	0
$961$	30.3758	0.979865
$962$	0	0
$963$	−8.71759	−0.280920
$964$	0	0
$965$	46.6089	1.50040
$966$	0	0
$967$	8.25806	0.265561	0.132781	−	0.991145i	$-0.457609\pi$
$967$	8.25806	0.265561	0.132781	+	0.991145i	$0.457609\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.1186	0.388904	0.194452	−	0.980912i	$-0.437707\pi$
$971$	12.1186	0.388904	0.194452	+	0.980912i	$0.437707\pi$
$972$	0	0
$973$	13.7686	0.441401
$974$	0	0
$975$	−17.2949	−0.553881
$976$	0	0
$977$	29.6313	0.947989	0.473994	−	0.880528i	$-0.342812\pi$
$977$	29.6313	0.947989	0.473994	+	0.880528i	$0.342812\pi$
$978$	0	0
$979$	−5.54276	−0.177147
$980$	0	0
$981$	−19.5311	−0.623581
$982$	0	0
$983$	35.8389	1.14308	0.571541	−	0.820574i	$-0.306346\pi$
$983$	35.8389	1.14308	0.571541	+	0.820574i	$0.306346\pi$
$984$	0	0
$985$	18.8937	0.602004
$986$	0	0
$987$	−6.80848	−0.216716
$988$	0	0
$989$	−14.1838	−0.451020
$990$	0	0
$991$	−14.3032	−0.454355	−0.227178	−	0.973853i	$-0.572950\pi$
$991$	−14.3032	−0.454355	−0.227178	+	0.973853i	$0.572950\pi$
$992$	0	0
$993$	14.9691	0.475029
$994$	0	0
$995$	35.7148	1.13224
$996$	0	0
$997$	−2.60633	−0.0825434	−0.0412717	−	0.999148i	$-0.513141\pi$
$997$	−2.60633	−0.0825434	−0.0412717	+	0.999148i	$0.513141\pi$
$998$	0	0
$999$	7.86729	0.248910

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4332.2.a.t.1.5		6
19.9	even	9		228.2.q.b.157.2	yes	12
19.17	even	9		228.2.q.b.61.2	✓	12
19.18	odd	2		4332.2.a.u.1.5		6
57.17	odd	18		684.2.bo.e.289.1		12
57.47	odd	18		684.2.bo.e.613.1		12
76.47	odd	18		912.2.bo.g.385.2		12
76.55	odd	18		912.2.bo.g.289.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
228.2.q.b.61.2	✓	12	19.17	even	9
228.2.q.b.157.2	yes	12	19.9	even	9
684.2.bo.e.289.1		12	57.17	odd	18
684.2.bo.e.613.1		12	57.47	odd	18
912.2.bo.g.289.2		12	76.55	odd	18
912.2.bo.g.385.2		12	76.47	odd	18
4332.2.a.t.1.5		6	1.1	even	1	trivial
4332.2.a.u.1.5		6	19.18	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:	\( N \)	\(=\)	\( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4332.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +3.32474 q^{5} +1.19263 q^{7} +1.00000 q^{9} +1.29512 q^{11} +2.85683 q^{13} -3.32474 q^{15} -2.21885 q^{17} -1.19263 q^{21} +7.92884 q^{23} +6.05389 q^{25} -1.00000 q^{27} -4.48970 q^{29} +7.83427 q^{31} -1.29512 q^{33} +3.96517 q^{35} -7.86729 q^{37} -2.85683 q^{39} +8.06264 q^{41} -1.78889 q^{43} +3.32474 q^{45} +5.70881 q^{47} -5.57764 q^{49} +2.21885 q^{51} -6.31291 q^{53} +4.30594 q^{55} -7.89384 q^{59} +14.6331 q^{61} +1.19263 q^{63} +9.49821 q^{65} +4.58548 q^{67} -7.92884 q^{69} +4.18036 q^{71} +1.94679 q^{73} -6.05389 q^{75} +1.54460 q^{77} -6.20504 q^{79} +1.00000 q^{81} +7.83717 q^{83} -7.37709 q^{85} +4.48970 q^{87} -4.27972 q^{89} +3.40713 q^{91} -7.83427 q^{93} -2.53635 q^{97} +1.29512 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 9 q^{11} - 9 q^{13} - 3 q^{15} + 9 q^{17} - 9 q^{21} + 12 q^{23} + 15 q^{25} - 6 q^{27} - 9 q^{29} + 6 q^{31} - 9 q^{33} - 3 q^{35} + 6 q^{37} + 9 q^{39} + 18 q^{41}+ \cdots + 9 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 3 * q^5 + 9 * q^7 + 6 * q^9 + 9 * q^11 - 9 * q^13 - 3 * q^15 + 9 * q^17 - 9 * q^21 + 12 * q^23 + 15 * q^25 - 6 * q^27 - 9 * q^29 + 6 * q^31 - 9 * q^33 - 3 * q^35 + 6 * q^37 + 9 * q^39 + 18 * q^41 + 15 * q^43 + 3 * q^45 + 21 * q^47 + 21 * q^49 - 9 * q^51 - 6 * q^53 + 12 * q^55 - 15 * q^59 - 21 * q^61 + 9 * q^63 + 33 * q^65 - 18 * q^67 - 12 * q^69 + 15 * q^71 + 18 * q^73 - 15 * q^75 + 9 * q^79 + 6 * q^81 - 3 * q^83 + 6 * q^85 + 9 * q^87 - 36 * q^89 - 18 * q^91 - 6 * q^93 - 39 * q^97 + 9 * q^99

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