LMFDB - Embedded newform 4400.2.a.bi.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] 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("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,0,0,-5,0,5,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1100)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79129 q^{3} -0.208712 q^{7} +4.79129 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.791288 q^{17} -6.58258 q^{19} +0.582576 q^{21} -3.79129 q^{23} -5.00000 q^{27} +6.79129 q^{29} +8.58258 q^{31} -2.79129 q^{33} +2.58258 q^{37} -2.79129 q^{39} -1.41742 q^{41} -10.0000 q^{43} -1.41742 q^{47} -6.95644 q^{49} +2.20871 q^{51} -11.3739 q^{53} +18.3739 q^{57} +10.5826 q^{59} +4.20871 q^{61} -1.00000 q^{63} -4.00000 q^{67} +10.5826 q^{69} +10.7477 q^{71} +7.79129 q^{73} -0.208712 q^{77} +15.5390 q^{79} -0.417424 q^{81} -9.95644 q^{83} -18.9564 q^{87} -0.791288 q^{89} -0.208712 q^{91} -23.9564 q^{93} +6.20871 q^{97} +4.79129 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 2 q^{13} + 3 q^{17} - 4 q^{19} - 8 q^{21} - 3 q^{23} - 10 q^{27} + 9 q^{29} + 8 q^{31} - q^{33} - 4 q^{37} - q^{39} - 12 q^{41} - 20 q^{43} - 12 q^{47} + 9 q^{49}+ \cdots + 5 q^{99}+O(q^{100}) $ 2 * q - q^3 - 5 * q^7 + 5 * q^9 + 2 * q^11 + 2 * q^13 + 3 * q^17 - 4 * q^19 - 8 * q^21 - 3 * q^23 - 10 * q^27 + 9 * q^29 + 8 * q^31 - q^33 - 4 * q^37 - q^39 - 12 * q^41 - 20 * q^43 - 12 * q^47 + 9 * q^49 + 9 * q^51 - 9 * q^53 + 23 * q^57 + 12 * q^59 + 13 * q^61 - 2 * q^63 - 8 * q^67 + 12 * q^69 - 6 * q^71 + 11 * q^73 - 5 * q^77 - q^79 - 10 * q^81 + 3 * q^83 - 15 * q^87 + 3 * q^89 - 5 * q^91 - 25 * q^93 + 17 * q^97 + 5 * 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
$3$	−2.79129	−1.61155	−0.805775	−	0.592221i	$-0.798251\pi$
$3$	−2.79129	−1.61155	−0.805775	+	0.592221i	$0.798251\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.208712	−0.0788858	−0.0394429	−	0.999222i	$-0.512558\pi$
$7$	−0.208712	−0.0788858	−0.0394429	+	0.999222i	$0.512558\pi$
$8$	0	0
$9$	4.79129	1.59710
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.791288	−0.191915	−0.0959577	−	0.995385i	$-0.530591\pi$
$17$	−0.791288	−0.191915	−0.0959577	+	0.995385i	$0.530591\pi$
$18$	0	0
$19$	−6.58258	−1.51015	−0.755073	−	0.655640i	$-0.772399\pi$
$19$	−6.58258	−1.51015	−0.755073	+	0.655640i	$0.772399\pi$
$20$	0	0
$21$	0.582576	0.127128
$22$	0	0
$23$	−3.79129	−0.790538	−0.395269	−	0.918565i	$-0.629349\pi$
$23$	−3.79129	−0.790538	−0.395269	+	0.918565i	$0.629349\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	6.79129	1.26111	0.630555	−	0.776144i	$-0.282827\pi$
$29$	6.79129	1.26111	0.630555	+	0.776144i	$0.282827\pi$
$30$	0	0
$31$	8.58258	1.54148	0.770738	−	0.637152i	$-0.219888\pi$
$31$	8.58258	1.54148	0.770738	+	0.637152i	$0.219888\pi$
$32$	0	0
$33$	−2.79129	−0.485901
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.58258	0.424573	0.212286	−	0.977207i	$-0.431909\pi$
$37$	2.58258	0.424573	0.212286	+	0.977207i	$0.431909\pi$
$38$	0	0
$39$	−2.79129	−0.446964
$40$	0	0
$41$	−1.41742	−0.221364	−0.110682	−	0.993856i	$-0.535304\pi$
$41$	−1.41742	−0.221364	−0.110682	+	0.993856i	$0.535304\pi$
$42$	0	0
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.41742	−0.206753	−0.103376	−	0.994642i	$-0.532965\pi$
$47$	−1.41742	−0.206753	−0.103376	+	0.994642i	$0.532965\pi$
$48$	0	0
$49$	−6.95644	−0.993777
$50$	0	0
$51$	2.20871	0.309282
$52$	0	0
$53$	−11.3739	−1.56232	−0.781160	−	0.624331i	$-0.785372\pi$
$53$	−11.3739	−1.56232	−0.781160	+	0.624331i	$0.785372\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	18.3739	2.43368
$58$	0	0
$59$	10.5826	1.37773	0.688867	−	0.724888i	$-0.258108\pi$
$59$	10.5826	1.37773	0.688867	+	0.724888i	$0.258108\pi$
$60$	0	0
$61$	4.20871	0.538870	0.269435	−	0.963019i	$-0.413163\pi$
$61$	4.20871	0.538870	0.269435	+	0.963019i	$0.413163\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	10.5826	1.27399
$70$	0	0
$71$	10.7477	1.27552	0.637760	−	0.770235i	$-0.279861\pi$
$71$	10.7477	1.27552	0.637760	+	0.770235i	$0.279861\pi$
$72$	0	0
$73$	7.79129	0.911901	0.455951	−	0.890005i	$-0.349299\pi$
$73$	7.79129	0.911901	0.455951	+	0.890005i	$0.349299\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.208712	−0.0237850
$78$	0	0
$79$	15.5390	1.74828	0.874138	−	0.485678i	$-0.161427\pi$
$79$	15.5390	1.74828	0.874138	+	0.485678i	$0.161427\pi$
$80$	0	0
$81$	−0.417424	−0.0463805
$82$	0	0
$83$	−9.95644	−1.09286	−0.546431	−	0.837504i	$-0.684014\pi$
$83$	−9.95644	−1.09286	−0.546431	+	0.837504i	$0.684014\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−18.9564	−2.03234
$88$	0	0
$89$	−0.791288	−0.0838763	−0.0419382	−	0.999120i	$-0.513353\pi$
$89$	−0.791288	−0.0838763	−0.0419382	+	0.999120i	$0.513353\pi$
$90$	0	0
$91$	−0.208712	−0.0218790
$92$	0	0
$93$	−23.9564	−2.48417
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.20871	0.630399	0.315200	−	0.949025i	$-0.397929\pi$
$97$	6.20871	0.630399	0.315200	+	0.949025i	$0.397929\pi$
$98$	0	0
$99$	4.79129	0.481543
$100$	0	0
$101$	−17.3739	−1.72876	−0.864382	−	0.502836i	$-0.832290\pi$
$101$	−17.3739	−1.72876	−0.864382	+	0.502836i	$0.832290\pi$
$102$	0	0
$103$	5.95644	0.586905	0.293453	−	0.955974i	$-0.405196\pi$
$103$	5.95644	0.586905	0.293453	+	0.955974i	$0.405196\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.7477	1.90908	0.954542	−	0.298075i	$-0.0963446\pi$
$107$	19.7477	1.90908	0.954542	+	0.298075i	$0.0963446\pi$
$108$	0	0
$109$	−4.79129	−0.458922	−0.229461	−	0.973318i	$-0.573696\pi$
$109$	−4.79129	−0.458922	−0.229461	+	0.973318i	$0.573696\pi$
$110$	0	0
$111$	−7.20871	−0.684221
$112$	0	0
$113$	−1.41742	−0.133340	−0.0666700	−	0.997775i	$-0.521237\pi$
$113$	−1.41742	−0.133340	−0.0666700	+	0.997775i	$0.521237\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.79129	0.442955
$118$	0	0
$119$	0.165151	0.0151394
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.95644	0.356740
$124$	0	0
$125$	0	0
$126$	0	0
$127$	16.3739	1.45295	0.726473	−	0.687195i	$-0.241158\pi$
$127$	16.3739	1.45295	0.726473	+	0.687195i	$0.241158\pi$
$128$	0	0
$129$	27.9129	2.45759
$130$	0	0
$131$	0.626136	0.0547058	0.0273529	−	0.999626i	$-0.491292\pi$
$131$	0.626136	0.0547058	0.0273529	+	0.999626i	$0.491292\pi$
$132$	0	0
$133$	1.37386	0.119129
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.5390	0.985845	0.492922	−	0.870073i	$-0.335929\pi$
$137$	11.5390	0.985845	0.492922	+	0.870073i	$0.335929\pi$
$138$	0	0
$139$	−9.74773	−0.826791	−0.413396	−	0.910551i	$-0.635657\pi$
$139$	−9.74773	−0.826791	−0.413396	+	0.910551i	$0.635657\pi$
$140$	0	0
$141$	3.95644	0.333192
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0	0
$146$	0	0
$147$	19.4174	1.60152
$148$	0	0
$149$	3.16515	0.259299	0.129650	−	0.991560i	$-0.458615\pi$
$149$	3.16515	0.259299	0.129650	+	0.991560i	$0.458615\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	0	0
$153$	−3.79129	−0.306507
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−17.1652	−1.36993	−0.684964	−	0.728577i	$-0.740182\pi$
$157$	−17.1652	−1.36993	−0.684964	+	0.728577i	$0.740182\pi$
$158$	0	0
$159$	31.7477	2.51776
$160$	0	0
$161$	0.791288	0.0623622
$162$	0	0
$163$	−21.3739	−1.67413	−0.837065	−	0.547103i	$-0.815730\pi$
$163$	−21.3739	−1.67413	−0.837065	+	0.547103i	$0.815730\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−16.5826	−1.28320	−0.641599	−	0.767040i	$-0.721729\pi$
$167$	−16.5826	−1.28320	−0.641599	+	0.767040i	$0.721729\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	−31.5390	−2.41185
$172$	0	0
$173$	13.7477	1.04522	0.522610	−	0.852572i	$-0.324958\pi$
$173$	13.7477	1.04522	0.522610	+	0.852572i	$0.324958\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−29.5390	−2.22029
$178$	0	0
$179$	−22.1216	−1.65345	−0.826723	−	0.562610i	$-0.809797\pi$
$179$	−22.1216	−1.65345	−0.826723	+	0.562610i	$0.809797\pi$
$180$	0	0
$181$	−3.37386	−0.250777	−0.125389	−	0.992108i	$-0.540018\pi$
$181$	−3.37386	−0.250777	−0.125389	+	0.992108i	$0.540018\pi$
$182$	0	0
$183$	−11.7477	−0.868417
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.791288	−0.0578647
$188$	0	0
$189$	1.04356	0.0759079
$190$	0	0
$191$	−17.2087	−1.24518	−0.622589	−	0.782549i	$-0.713919\pi$
$191$	−17.2087	−1.24518	−0.622589	+	0.782549i	$0.713919\pi$
$192$	0	0
$193$	7.16515	0.515759	0.257879	−	0.966177i	$-0.416976\pi$
$193$	7.16515	0.515759	0.257879	+	0.966177i	$0.416976\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.5390	−1.89083	−0.945413	−	0.325874i	$-0.894342\pi$
$197$	−26.5390	−1.89083	−0.945413	+	0.325874i	$0.894342\pi$
$198$	0	0
$199$	1.62614	0.115274	0.0576369	−	0.998338i	$-0.481643\pi$
$199$	1.62614	0.115274	0.0576369	+	0.998338i	$0.481643\pi$
$200$	0	0
$201$	11.1652	0.787529
$202$	0	0
$203$	−1.41742	−0.0994837
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−18.1652	−1.26257
$208$	0	0
$209$	−6.58258	−0.455326
$210$	0	0
$211$	−5.00000	−0.344214	−0.172107	−	0.985078i	$-0.555058\pi$
$211$	−5.00000	−0.344214	−0.172107	+	0.985078i	$0.555058\pi$
$212$	0	0
$213$	−30.0000	−2.05557
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.79129	−0.121601
$218$	0	0
$219$	−21.7477	−1.46958
$220$	0	0
$221$	−0.791288	−0.0532278
$222$	0	0
$223$	−20.5826	−1.37831	−0.689156	−	0.724613i	$-0.742018\pi$
$223$	−20.5826	−1.37831	−0.689156	+	0.724613i	$0.742018\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.5390	0.765871	0.382936	−	0.923775i	$-0.374913\pi$
$227$	11.5390	0.765871	0.382936	+	0.923775i	$0.374913\pi$
$228$	0	0
$229$	−7.62614	−0.503949	−0.251975	−	0.967734i	$-0.581080\pi$
$229$	−7.62614	−0.503949	−0.251975	+	0.967734i	$0.581080\pi$
$230$	0	0
$231$	0.582576	0.0383307
$232$	0	0
$233$	−15.7913	−1.03452	−0.517261	−	0.855828i	$-0.673048\pi$
$233$	−15.7913	−1.03452	−0.517261	+	0.855828i	$0.673048\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−43.3739	−2.81744
$238$	0	0
$239$	2.20871	0.142870	0.0714349	−	0.997445i	$-0.477242\pi$
$239$	2.20871	0.142870	0.0714349	+	0.997445i	$0.477242\pi$
$240$	0	0
$241$	−23.1216	−1.48939	−0.744696	−	0.667404i	$-0.767406\pi$
$241$	−23.1216	−1.48939	−0.744696	+	0.667404i	$0.767406\pi$
$242$	0	0
$243$	16.1652	1.03699
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.58258	−0.418839
$248$	0	0
$249$	27.7913	1.76120
$250$	0	0
$251$	11.5390	0.728336	0.364168	−	0.931333i	$-0.381353\pi$
$251$	11.5390	0.728336	0.364168	+	0.931333i	$0.381353\pi$
$252$	0	0
$253$	−3.79129	−0.238356
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−0.539015	−0.0334928
$260$	0	0
$261$	32.5390	2.01411
$262$	0	0
$263$	24.1652	1.49009	0.745044	−	0.667016i	$-0.232429\pi$
$263$	24.1652	1.49009	0.745044	+	0.667016i	$0.232429\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.20871	0.135171
$268$	0	0
$269$	−20.7042	−1.26236	−0.631178	−	0.775638i	$-0.717428\pi$
$269$	−20.7042	−1.26236	−0.631178	+	0.775638i	$0.717428\pi$
$270$	0	0
$271$	−15.7477	−0.956606	−0.478303	−	0.878195i	$-0.658748\pi$
$271$	−15.7477	−0.956606	−0.478303	+	0.878195i	$0.658748\pi$
$272$	0	0
$273$	0.582576	0.0352591
$274$	0	0
$275$	0	0
$276$	0	0
$277$	8.58258	0.515677	0.257838	−	0.966188i	$-0.416990\pi$
$277$	8.58258	0.515677	0.257838	+	0.966188i	$0.416990\pi$
$278$	0	0
$279$	41.1216	2.46189
$280$	0	0
$281$	−13.7477	−0.820121	−0.410060	−	0.912058i	$-0.634492\pi$
$281$	−13.7477	−0.820121	−0.410060	+	0.912058i	$0.634492\pi$
$282$	0	0
$283$	−30.7042	−1.82517	−0.912587	−	0.408883i	$-0.865918\pi$
$283$	−30.7042	−1.82517	−0.912587	+	0.408883i	$0.865918\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.295834	0.0174625
$288$	0	0
$289$	−16.3739	−0.963168
$290$	0	0
$291$	−17.3303	−1.01592
$292$	0	0
$293$	2.83485	0.165614	0.0828068	−	0.996566i	$-0.473612\pi$
$293$	2.83485	0.165614	0.0828068	+	0.996566i	$0.473612\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.00000	−0.290129
$298$	0	0
$299$	−3.79129	−0.219256
$300$	0	0
$301$	2.08712	0.120300
$302$	0	0
$303$	48.4955	2.78599
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−27.2087	−1.55288	−0.776442	−	0.630189i	$-0.782977\pi$
$307$	−27.2087	−1.55288	−0.776442	+	0.630189i	$0.782977\pi$
$308$	0	0
$309$	−16.6261	−0.945828
$310$	0	0
$311$	5.83485	0.330864	0.165432	−	0.986221i	$-0.447098\pi$
$311$	5.83485	0.330864	0.165432	+	0.986221i	$0.447098\pi$
$312$	0	0
$313$	−6.74773	−0.381404	−0.190702	−	0.981648i	$-0.561076\pi$
$313$	−6.74773	−0.381404	−0.190702	+	0.981648i	$0.561076\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.37386	0.470323	0.235162	−	0.971956i	$-0.424438\pi$
$317$	8.37386	0.470323	0.235162	+	0.971956i	$0.424438\pi$
$318$	0	0
$319$	6.79129	0.380239
$320$	0	0
$321$	−55.1216	−3.07659
$322$	0	0
$323$	5.20871	0.289820
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.3739	0.739576
$328$	0	0
$329$	0.295834	0.0163098
$330$	0	0
$331$	−17.3303	−0.952560	−0.476280	−	0.879294i	$-0.658015\pi$
$331$	−17.3303	−0.952560	−0.476280	+	0.879294i	$0.658015\pi$
$332$	0	0
$333$	12.3739	0.678084
$334$	0	0
$335$	0	0
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	0	0
$339$	3.95644	0.214884
$340$	0	0
$341$	8.58258	0.464773
$342$	0	0
$343$	2.91288	0.157281
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.6261	0.677807	0.338903	−	0.940821i	$-0.389944\pi$
$347$	12.6261	0.677807	0.338903	+	0.940821i	$0.389944\pi$
$348$	0	0
$349$	−14.5826	−0.780587	−0.390294	−	0.920690i	$-0.627627\pi$
$349$	−14.5826	−0.780587	−0.390294	+	0.920690i	$0.627627\pi$
$350$	0	0
$351$	−5.00000	−0.266880
$352$	0	0
$353$	−18.1652	−0.966833	−0.483417	−	0.875390i	$-0.660604\pi$
$353$	−18.1652	−0.966833	−0.483417	+	0.875390i	$0.660604\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.460985	−0.0243979
$358$	0	0
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	24.3303	1.28054
$362$	0	0
$363$	−2.79129	−0.146505
$364$	0	0
$365$	0	0
$366$	0	0
$367$	8.62614	0.450281	0.225140	−	0.974326i	$-0.427716\pi$
$367$	8.62614	0.450281	0.225140	+	0.974326i	$0.427716\pi$
$368$	0	0
$369$	−6.79129	−0.353540
$370$	0	0
$371$	2.37386	0.123245
$372$	0	0
$373$	19.3303	1.00089	0.500443	−	0.865770i	$-0.333171\pi$
$373$	19.3303	1.00089	0.500443	+	0.865770i	$0.333171\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.79129	0.349769
$378$	0	0
$379$	31.0000	1.59236	0.796182	−	0.605058i	$-0.206850\pi$
$379$	31.0000	1.59236	0.796182	+	0.605058i	$0.206850\pi$
$380$	0	0
$381$	−45.7042	−2.34150
$382$	0	0
$383$	−18.0000	−0.919757	−0.459879	−	0.887982i	$-0.652107\pi$
$383$	−18.0000	−0.919757	−0.459879	+	0.887982i	$0.652107\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−47.9129	−2.43555
$388$	0	0
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	3.00000	0.151717
$392$	0	0
$393$	−1.74773	−0.0881612
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−7.20871	−0.361795	−0.180897	−	0.983502i	$-0.557900\pi$
$397$	−7.20871	−0.361795	−0.180897	+	0.983502i	$0.557900\pi$
$398$	0	0
$399$	−3.83485	−0.191983
$400$	0	0
$401$	−14.8348	−0.740817	−0.370408	−	0.928869i	$-0.620782\pi$
$401$	−14.8348	−0.740817	−0.370408	+	0.928869i	$0.620782\pi$
$402$	0	0
$403$	8.58258	0.427529
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.58258	0.128014
$408$	0	0
$409$	17.0000	0.840596	0.420298	−	0.907386i	$-0.361926\pi$
$409$	17.0000	0.840596	0.420298	+	0.907386i	$0.361926\pi$
$410$	0	0
$411$	−32.2087	−1.58874
$412$	0	0
$413$	−2.20871	−0.108684
$414$	0	0
$415$	0	0
$416$	0	0
$417$	27.2087	1.33242
$418$	0	0
$419$	−27.0000	−1.31904	−0.659518	−	0.751689i	$-0.729240\pi$
$419$	−27.0000	−1.31904	−0.659518	+	0.751689i	$0.729240\pi$
$420$	0	0
$421$	25.3739	1.23665	0.618323	−	0.785924i	$-0.287812\pi$
$421$	25.3739	1.23665	0.618323	+	0.785924i	$0.287812\pi$
$422$	0	0
$423$	−6.79129	−0.330204
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.878409	−0.0425092
$428$	0	0
$429$	−2.79129	−0.134765
$430$	0	0
$431$	−10.9129	−0.525655	−0.262827	−	0.964843i	$-0.584655\pi$
$431$	−10.9129	−0.525655	−0.262827	+	0.964843i	$0.584655\pi$
$432$	0	0
$433$	13.3303	0.640613	0.320307	−	0.947314i	$-0.396214\pi$
$433$	13.3303	0.640613	0.320307	+	0.947314i	$0.396214\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	24.9564	1.19383
$438$	0	0
$439$	−11.9564	−0.570650	−0.285325	−	0.958431i	$-0.592101\pi$
$439$	−11.9564	−0.570650	−0.285325	+	0.958431i	$0.592101\pi$
$440$	0	0
$441$	−33.3303	−1.58716
$442$	0	0
$443$	22.4174	1.06508	0.532542	−	0.846404i	$-0.321237\pi$
$443$	22.4174	1.06508	0.532542	+	0.846404i	$0.321237\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−8.83485	−0.417874
$448$	0	0
$449$	26.2087	1.23687	0.618433	−	0.785838i	$-0.287768\pi$
$449$	26.2087	1.23687	0.618433	+	0.785838i	$0.287768\pi$
$450$	0	0
$451$	−1.41742	−0.0667439
$452$	0	0
$453$	13.9564	0.655731
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−13.3739	−0.625603	−0.312801	−	0.949819i	$-0.601267\pi$
$457$	−13.3739	−0.625603	−0.312801	+	0.949819i	$0.601267\pi$
$458$	0	0
$459$	3.95644	0.184671
$460$	0	0
$461$	−24.4955	−1.14087	−0.570434	−	0.821344i	$-0.693225\pi$
$461$	−24.4955	−1.14087	−0.570434	+	0.821344i	$0.693225\pi$
$462$	0	0
$463$	−2.25227	−0.104672	−0.0523360	−	0.998630i	$-0.516667\pi$
$463$	−2.25227	−0.104672	−0.0523360	+	0.998630i	$0.516667\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.5826	0.489703	0.244852	−	0.969561i	$-0.421261\pi$
$467$	10.5826	0.489703	0.244852	+	0.969561i	$0.421261\pi$
$468$	0	0
$469$	0.834849	0.0385497
$470$	0	0
$471$	47.9129	2.20771
$472$	0	0
$473$	−10.0000	−0.459800
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−54.4955	−2.49518
$478$	0	0
$479$	−0.165151	−0.00754596	−0.00377298	−	0.999993i	$-0.501201\pi$
$479$	−0.165151	−0.00754596	−0.00377298	+	0.999993i	$0.501201\pi$
$480$	0	0
$481$	2.58258	0.117755
$482$	0	0
$483$	−2.20871	−0.100500
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−29.7477	−1.34800	−0.673999	−	0.738732i	$-0.735425\pi$
$487$	−29.7477	−1.34800	−0.673999	+	0.738732i	$0.735425\pi$
$488$	0	0
$489$	59.6606	2.69795
$490$	0	0
$491$	7.41742	0.334744	0.167372	−	0.985894i	$-0.446472\pi$
$491$	7.41742	0.334744	0.167372	+	0.985894i	$0.446472\pi$
$492$	0	0
$493$	−5.37386	−0.242027
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2.24318	−0.100620
$498$	0	0
$499$	−14.9564	−0.669542	−0.334771	−	0.942299i	$-0.608659\pi$
$499$	−14.9564	−0.669542	−0.334771	+	0.942299i	$0.608659\pi$
$500$	0	0
$501$	46.2867	2.06794
$502$	0	0
$503$	16.4174	0.732017	0.366008	−	0.930612i	$-0.380724\pi$
$503$	16.4174	0.732017	0.366008	+	0.930612i	$0.380724\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	33.4955	1.48759
$508$	0	0
$509$	10.1216	0.448632	0.224316	−	0.974517i	$-0.427985\pi$
$509$	10.1216	0.448632	0.224316	+	0.974517i	$0.427985\pi$
$510$	0	0
$511$	−1.62614	−0.0719360
$512$	0	0
$513$	32.9129	1.45314
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.41742	−0.0623382
$518$	0	0
$519$	−38.3739	−1.68443
$520$	0	0
$521$	1.58258	0.0693339	0.0346670	−	0.999399i	$-0.488963\pi$
$521$	1.58258	0.0693339	0.0346670	+	0.999399i	$0.488963\pi$
$522$	0	0
$523$	−25.1652	−1.10040	−0.550198	−	0.835034i	$-0.685448\pi$
$523$	−25.1652	−1.10040	−0.550198	+	0.835034i	$0.685448\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−6.79129	−0.295833
$528$	0	0
$529$	−8.62614	−0.375049
$530$	0	0
$531$	50.7042	2.20037
$532$	0	0
$533$	−1.41742	−0.0613955
$534$	0	0
$535$	0	0
$536$	0	0
$537$	61.7477	2.66461
$538$	0	0
$539$	−6.95644	−0.299635
$540$	0	0
$541$	−10.6261	−0.456853	−0.228427	−	0.973561i	$-0.573358\pi$
$541$	−10.6261	−0.456853	−0.228427	+	0.973561i	$0.573358\pi$
$542$	0	0
$543$	9.41742	0.404140
$544$	0	0
$545$	0	0
$546$	0	0
$547$	7.37386	0.315284	0.157642	−	0.987496i	$-0.449611\pi$
$547$	7.37386	0.315284	0.157642	+	0.987496i	$0.449611\pi$
$548$	0	0
$549$	20.1652	0.860628
$550$	0	0
$551$	−44.7042	−1.90446
$552$	0	0
$553$	−3.24318	−0.137914
$554$	0	0
$555$	0	0
$556$	0	0
$557$	45.6606	1.93470	0.967351	−	0.253441i	$-0.0815622\pi$
$557$	45.6606	1.93470	0.967351	+	0.253441i	$0.0815622\pi$
$558$	0	0
$559$	−10.0000	−0.422955
$560$	0	0
$561$	2.20871	0.0932519
$562$	0	0
$563$	−40.1216	−1.69092	−0.845462	−	0.534036i	$-0.820675\pi$
$563$	−40.1216	−1.69092	−0.845462	+	0.534036i	$0.820675\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0.0871215	0.00365876
$568$	0	0
$569$	18.9564	0.794695	0.397348	−	0.917668i	$-0.369931\pi$
$569$	18.9564	0.794695	0.397348	+	0.917668i	$0.369931\pi$
$570$	0	0
$571$	−27.2867	−1.14191	−0.570957	−	0.820980i	$-0.693428\pi$
$571$	−27.2867	−1.14191	−0.570957	+	0.820980i	$0.693428\pi$
$572$	0	0
$573$	48.0345	2.00667
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.9564	−0.497753	−0.248877	−	0.968535i	$-0.580061\pi$
$577$	−11.9564	−0.497753	−0.248877	+	0.968535i	$0.580061\pi$
$578$	0	0
$579$	−20.0000	−0.831172
$580$	0	0
$581$	2.07803	0.0862112
$582$	0	0
$583$	−11.3739	−0.471057
$584$	0	0
$585$	0	0
$586$	0	0
$587$	15.9564	0.658593	0.329296	−	0.944227i	$-0.393188\pi$
$587$	15.9564	0.658593	0.329296	+	0.944227i	$0.393188\pi$
$588$	0	0
$589$	−56.4955	−2.32785
$590$	0	0
$591$	74.0780	3.04716
$592$	0	0
$593$	22.5826	0.927355	0.463678	−	0.886004i	$-0.346530\pi$
$593$	22.5826	0.927355	0.463678	+	0.886004i	$0.346530\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−4.53901	−0.185770
$598$	0	0
$599$	−44.7042	−1.82656	−0.913281	−	0.407329i	$-0.866460\pi$
$599$	−44.7042	−1.82656	−0.913281	+	0.407329i	$0.866460\pi$
$600$	0	0
$601$	28.5390	1.16413	0.582065	−	0.813142i	$-0.302245\pi$
$601$	28.5390	1.16413	0.582065	+	0.813142i	$0.302245\pi$
$602$	0	0
$603$	−19.1652	−0.780465
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	0	0
$609$	3.95644	0.160323
$610$	0	0
$611$	−1.41742	−0.0573428
$612$	0	0
$613$	−33.4519	−1.35111	−0.675555	−	0.737310i	$-0.736096\pi$
$613$	−33.4519	−1.35111	−0.675555	+	0.737310i	$0.736096\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	28.7477	1.15734	0.578670	−	0.815562i	$-0.303572\pi$
$617$	28.7477	1.15734	0.578670	+	0.815562i	$0.303572\pi$
$618$	0	0
$619$	−6.25227	−0.251300	−0.125650	−	0.992075i	$-0.540102\pi$
$619$	−6.25227	−0.251300	−0.125650	+	0.992075i	$0.540102\pi$
$620$	0	0
$621$	18.9564	0.760696
$622$	0	0
$623$	0.165151	0.00661665
$624$	0	0
$625$	0	0
$626$	0	0
$627$	18.3739	0.733781
$628$	0	0
$629$	−2.04356	−0.0814821
$630$	0	0
$631$	−18.1216	−0.721409	−0.360705	−	0.932680i	$-0.617464\pi$
$631$	−18.1216	−0.721409	−0.360705	+	0.932680i	$0.617464\pi$
$632$	0	0
$633$	13.9564	0.554719
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−6.95644	−0.275624
$638$	0	0
$639$	51.4955	2.03713
$640$	0	0
$641$	−23.8348	−0.941420	−0.470710	−	0.882288i	$-0.656002\pi$
$641$	−23.8348	−0.941420	−0.470710	+	0.882288i	$0.656002\pi$
$642$	0	0
$643$	−43.4955	−1.71529	−0.857647	−	0.514239i	$-0.828074\pi$
$643$	−43.4955	−1.71529	−0.857647	+	0.514239i	$0.828074\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	21.3303	0.838581	0.419290	−	0.907852i	$-0.362279\pi$
$647$	21.3303	0.838581	0.419290	+	0.907852i	$0.362279\pi$
$648$	0	0
$649$	10.5826	0.415402
$650$	0	0
$651$	5.00000	0.195965
$652$	0	0
$653$	−26.7042	−1.04501	−0.522507	−	0.852635i	$-0.675003\pi$
$653$	−26.7042	−1.04501	−0.522507	+	0.852635i	$0.675003\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	37.3303	1.45639
$658$	0	0
$659$	−0.460985	−0.0179574	−0.00897871	−	0.999960i	$-0.502858\pi$
$659$	−0.460985	−0.0179574	−0.00897871	+	0.999960i	$0.502858\pi$
$660$	0	0
$661$	30.5826	1.18952	0.594762	−	0.803902i	$-0.297246\pi$
$661$	30.5826	1.18952	0.594762	+	0.803902i	$0.297246\pi$
$662$	0	0
$663$	2.20871	0.0857793
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−25.7477	−0.996956
$668$	0	0
$669$	57.4519	2.22122
$670$	0	0
$671$	4.20871	0.162476
$672$	0	0
$673$	38.9129	1.49998	0.749991	−	0.661448i	$-0.230058\pi$
$673$	38.9129	1.49998	0.749991	+	0.661448i	$0.230058\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.58258	0.0608233	0.0304117	−	0.999537i	$-0.490318\pi$
$677$	1.58258	0.0608233	0.0304117	+	0.999537i	$0.490318\pi$
$678$	0	0
$679$	−1.29583	−0.0497295
$680$	0	0
$681$	−32.2087	−1.23424
$682$	0	0
$683$	34.9129	1.33590	0.667952	−	0.744204i	$-0.267171\pi$
$683$	34.9129	1.33590	0.667952	+	0.744204i	$0.267171\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	21.2867	0.812140
$688$	0	0
$689$	−11.3739	−0.433310
$690$	0	0
$691$	32.1216	1.22196	0.610981	−	0.791645i	$-0.290775\pi$
$691$	32.1216	1.22196	0.610981	+	0.791645i	$0.290775\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.12159	0.0424833
$698$	0	0
$699$	44.0780	1.66718
$700$	0	0
$701$	20.8348	0.786921	0.393461	−	0.919341i	$-0.371278\pi$
$701$	20.8348	0.786921	0.393461	+	0.919341i	$0.371278\pi$
$702$	0	0
$703$	−17.0000	−0.641167
$704$	0	0
$705$	0	0
$706$	0	0
$707$	3.62614	0.136375
$708$	0	0
$709$	29.4955	1.10773	0.553863	−	0.832608i	$-0.313153\pi$
$709$	29.4955	1.10773	0.553863	+	0.832608i	$0.313153\pi$
$710$	0	0
$711$	74.4519	2.79216
$712$	0	0
$713$	−32.5390	−1.21860
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−6.16515	−0.230242
$718$	0	0
$719$	−6.49545	−0.242240	−0.121120	−	0.992638i	$-0.538649\pi$
$719$	−6.49545	−0.242240	−0.121120	+	0.992638i	$0.538649\pi$
$720$	0	0
$721$	−1.24318	−0.0462985
$722$	0	0
$723$	64.5390	2.40023
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23.1216	−0.857532	−0.428766	−	0.903416i	$-0.641052\pi$
$727$	−23.1216	−0.857532	−0.428766	+	0.903416i	$0.641052\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	0	0
$731$	7.91288	0.292668
$732$	0	0
$733$	−31.2432	−1.15399	−0.576997	−	0.816747i	$-0.695775\pi$
$733$	−31.2432	−1.15399	−0.576997	+	0.816747i	$0.695775\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−42.1216	−1.54947	−0.774734	−	0.632287i	$-0.782116\pi$
$739$	−42.1216	−1.54947	−0.774734	+	0.632287i	$0.782116\pi$
$740$	0	0
$741$	18.3739	0.674981
$742$	0	0
$743$	39.7913	1.45980	0.729900	−	0.683554i	$-0.239566\pi$
$743$	39.7913	1.45980	0.729900	+	0.683554i	$0.239566\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−47.7042	−1.74540
$748$	0	0
$749$	−4.12159	−0.150600
$750$	0	0
$751$	−13.2087	−0.481993	−0.240996	−	0.970526i	$-0.577474\pi$
$751$	−13.2087	−0.481993	−0.240996	+	0.970526i	$0.577474\pi$
$752$	0	0
$753$	−32.2087	−1.17375
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.33030	0.266424	0.133212	−	0.991088i	$-0.457471\pi$
$757$	7.33030	0.266424	0.133212	+	0.991088i	$0.457471\pi$
$758$	0	0
$759$	10.5826	0.384123
$760$	0	0
$761$	30.3303	1.09947	0.549736	−	0.835338i	$-0.314728\pi$
$761$	30.3303	1.09947	0.549736	+	0.835338i	$0.314728\pi$
$762$	0	0
$763$	1.00000	0.0362024
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.5826	0.382115
$768$	0	0
$769$	−41.7477	−1.50546	−0.752731	−	0.658328i	$-0.771264\pi$
$769$	−41.7477	−1.50546	−0.752731	+	0.658328i	$0.771264\pi$
$770$	0	0
$771$	50.2432	1.80946
$772$	0	0
$773$	−26.2087	−0.942662	−0.471331	−	0.881956i	$-0.656226\pi$
$773$	−26.2087	−0.942662	−0.471331	+	0.881956i	$0.656226\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.50455	0.0539753
$778$	0	0
$779$	9.33030	0.334293
$780$	0	0
$781$	10.7477	0.384584
$782$	0	0
$783$	−33.9564	−1.21350
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−17.7477	−0.632638	−0.316319	−	0.948653i	$-0.602447\pi$
$787$	−17.7477	−0.632638	−0.316319	+	0.948653i	$0.602447\pi$
$788$	0	0
$789$	−67.4519	−2.40135
$790$	0	0
$791$	0.295834	0.0105186
$792$	0	0
$793$	4.20871	0.149456
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−34.2867	−1.21450	−0.607249	−	0.794511i	$-0.707727\pi$
$797$	−34.2867	−1.21450	−0.607249	+	0.794511i	$0.707727\pi$
$798$	0	0
$799$	1.12159	0.0396790
$800$	0	0
$801$	−3.79129	−0.133959
$802$	0	0
$803$	7.79129	0.274949
$804$	0	0
$805$	0	0
$806$	0	0
$807$	57.7913	2.03435
$808$	0	0
$809$	−45.4955	−1.59953	−0.799767	−	0.600310i	$-0.795044\pi$
$809$	−45.4955	−1.59953	−0.799767	+	0.600310i	$0.795044\pi$
$810$	0	0
$811$	55.6606	1.95451	0.977254	−	0.212072i	$-0.0680210\pi$
$811$	55.6606	1.95451	0.977254	+	0.212072i	$0.0680210\pi$
$812$	0	0
$813$	43.9564	1.54162
$814$	0	0
$815$	0	0
$816$	0	0
$817$	65.8258	2.30295
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	48.6606	1.69827	0.849133	−	0.528178i	$-0.177125\pi$
$821$	48.6606	1.69827	0.849133	+	0.528178i	$0.177125\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.4174	−0.362249	−0.181125	−	0.983460i	$-0.557974\pi$
$827$	−10.4174	−0.362249	−0.181125	+	0.983460i	$0.557974\pi$
$828$	0	0
$829$	−37.9564	−1.31828	−0.659141	−	0.752020i	$-0.729080\pi$
$829$	−37.9564	−1.31828	−0.659141	+	0.752020i	$0.729080\pi$
$830$	0	0
$831$	−23.9564	−0.831040
$832$	0	0
$833$	5.50455	0.190721
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−42.9129	−1.48329
$838$	0	0
$839$	23.7042	0.818359	0.409179	−	0.912454i	$-0.365815\pi$
$839$	23.7042	0.818359	0.409179	+	0.912454i	$0.365815\pi$
$840$	0	0
$841$	17.1216	0.590400
$842$	0	0
$843$	38.3739	1.32167
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−0.208712	−0.00717143
$848$	0	0
$849$	85.7042	2.94136
$850$	0	0
$851$	−9.79129	−0.335641
$852$	0	0
$853$	−3.12159	−0.106881	−0.0534406	−	0.998571i	$-0.517019\pi$
$853$	−3.12159	−0.106881	−0.0534406	+	0.998571i	$0.517019\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.4955	1.04170	0.520852	−	0.853647i	$-0.325614\pi$
$857$	30.4955	1.04170	0.520852	+	0.853647i	$0.325614\pi$
$858$	0	0
$859$	2.08712	0.0712117	0.0356058	−	0.999366i	$-0.488664\pi$
$859$	2.08712	0.0712117	0.0356058	+	0.999366i	$0.488664\pi$
$860$	0	0
$861$	−0.825757	−0.0281417
$862$	0	0
$863$	−27.3303	−0.930334	−0.465167	−	0.885223i	$-0.654006\pi$
$863$	−27.3303	−0.930334	−0.465167	+	0.885223i	$0.654006\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	45.7042	1.55219
$868$	0	0
$869$	15.5390	0.527125
$870$	0	0
$871$	−4.00000	−0.135535
$872$	0	0
$873$	29.7477	1.00681
$874$	0	0
$875$	0	0
$876$	0	0
$877$	16.6606	0.562589	0.281294	−	0.959622i	$-0.409236\pi$
$877$	16.6606	0.562589	0.281294	+	0.959622i	$0.409236\pi$
$878$	0	0
$879$	−7.91288	−0.266895
$880$	0	0
$881$	1.87841	0.0632852	0.0316426	−	0.999499i	$-0.489926\pi$
$881$	1.87841	0.0632852	0.0316426	+	0.999499i	$0.489926\pi$
$882$	0	0
$883$	26.1652	0.880527	0.440264	−	0.897869i	$-0.354885\pi$
$883$	26.1652	0.880527	0.440264	+	0.897869i	$0.354885\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−39.3303	−1.32058	−0.660291	−	0.751010i	$-0.729567\pi$
$887$	−39.3303	−1.32058	−0.660291	+	0.751010i	$0.729567\pi$
$888$	0	0
$889$	−3.41742	−0.114617
$890$	0	0
$891$	−0.417424	−0.0139842
$892$	0	0
$893$	9.33030	0.312227
$894$	0	0
$895$	0	0
$896$	0	0
$897$	10.5826	0.353342
$898$	0	0
$899$	58.2867	1.94397
$900$	0	0
$901$	9.00000	0.299833
$902$	0	0
$903$	−5.82576	−0.193869
$904$	0	0
$905$	0	0
$906$	0	0
$907$	32.6606	1.08448	0.542239	−	0.840224i	$-0.317577\pi$
$907$	32.6606	1.08448	0.542239	+	0.840224i	$0.317577\pi$
$908$	0	0
$909$	−83.2432	−2.76100
$910$	0	0
$911$	12.3303	0.408521	0.204261	−	0.978917i	$-0.434521\pi$
$911$	12.3303	0.408521	0.204261	+	0.978917i	$0.434521\pi$
$912$	0	0
$913$	−9.95644	−0.329510
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.130682	−0.00431551
$918$	0	0
$919$	−28.8348	−0.951174	−0.475587	−	0.879669i	$-0.657764\pi$
$919$	−28.8348	−0.951174	−0.475587	+	0.879669i	$0.657764\pi$
$920$	0	0
$921$	75.9473	2.50255
$922$	0	0
$923$	10.7477	0.353766
$924$	0	0
$925$	0	0
$926$	0	0
$927$	28.5390	0.937344
$928$	0	0
$929$	13.7477	0.451048	0.225524	−	0.974238i	$-0.427591\pi$
$929$	13.7477	0.451048	0.225524	+	0.974238i	$0.427591\pi$
$930$	0	0
$931$	45.7913	1.50075
$932$	0	0
$933$	−16.2867	−0.533204
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−35.1652	−1.14880	−0.574398	−	0.818576i	$-0.694764\pi$
$937$	−35.1652	−1.14880	−0.574398	+	0.818576i	$0.694764\pi$
$938$	0	0
$939$	18.8348	0.614652
$940$	0	0
$941$	−50.0780	−1.63250	−0.816249	−	0.577701i	$-0.803950\pi$
$941$	−50.0780	−1.63250	−0.816249	+	0.577701i	$0.803950\pi$
$942$	0	0
$943$	5.37386	0.174997
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4.91288	−0.159647	−0.0798235	−	0.996809i	$-0.525436\pi$
$947$	−4.91288	−0.159647	−0.0798235	+	0.996809i	$0.525436\pi$
$948$	0	0
$949$	7.79129	0.252916
$950$	0	0
$951$	−23.3739	−0.757949
$952$	0	0
$953$	42.0000	1.36051	0.680257	−	0.732974i	$-0.261868\pi$
$953$	42.0000	1.36051	0.680257	+	0.732974i	$0.261868\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−18.9564	−0.612775
$958$	0	0
$959$	−2.40833	−0.0777691
$960$	0	0
$961$	42.6606	1.37615
$962$	0	0
$963$	94.6170	3.04899
$964$	0	0
$965$	0	0
$966$	0	0
$967$	43.5390	1.40012	0.700060	−	0.714084i	$-0.253156\pi$
$967$	43.5390	1.40012	0.700060	+	0.714084i	$0.253156\pi$
$968$	0	0
$969$	−14.5390	−0.467060
$970$	0	0
$971$	−18.9564	−0.608341	−0.304171	−	0.952618i	$-0.598379\pi$
$971$	−18.9564	−0.608341	−0.304171	+	0.952618i	$0.598379\pi$
$972$	0	0
$973$	2.03447	0.0652221
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.7477	0.823743	0.411871	−	0.911242i	$-0.364875\pi$
$977$	25.7477	0.823743	0.411871	+	0.911242i	$0.364875\pi$
$978$	0	0
$979$	−0.791288	−0.0252897
$980$	0	0
$981$	−22.9564	−0.732943
$982$	0	0
$983$	−48.6606	−1.55203	−0.776016	−	0.630713i	$-0.782763\pi$
$983$	−48.6606	−1.55203	−0.776016	+	0.630713i	$0.782763\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−0.825757	−0.0262841
$988$	0	0
$989$	37.9129	1.20556
$990$	0	0
$991$	−55.8693	−1.77475	−0.887374	−	0.461051i	$-0.847473\pi$
$991$	−55.8693	−1.77475	−0.887374	+	0.461051i	$0.847473\pi$
$992$	0	0
$993$	48.3739	1.53510
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−40.3739	−1.27865	−0.639327	−	0.768935i	$-0.720787\pi$
$997$	−40.3739	−1.27865	−0.639327	+	0.768935i	$0.720787\pi$
$998$	0	0
$999$	−12.9129	−0.408545

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bi.1.1		2
4.3	odd	2		1100.2.a.h.1.2	yes	2
5.2	odd	4		4400.2.b.s.4049.4		4
5.3	odd	4		4400.2.b.s.4049.1		4
5.4	even	2		4400.2.a.bu.1.2		2
12.11	even	2		9900.2.a.bz.1.1		2
20.3	even	4		1100.2.b.d.749.4		4
20.7	even	4		1100.2.b.d.749.1		4
20.19	odd	2		1100.2.a.g.1.1	✓	2
60.23	odd	4		9900.2.c.x.5149.2		4
60.47	odd	4		9900.2.c.x.5149.3		4
60.59	even	2		9900.2.a.bh.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.g.1.1	✓	2	20.19	odd	2
1100.2.a.h.1.2	yes	2	4.3	odd	2
1100.2.b.d.749.1		4	20.7	even	4
1100.2.b.d.749.4		4	20.3	even	4
4400.2.a.bi.1.1		2	1.1	even	1	trivial
4400.2.a.bu.1.2		2	5.4	even	2
4400.2.b.s.4049.1		4	5.3	odd	4
4400.2.b.s.4049.4		4	5.2	odd	4
9900.2.a.bh.1.2		2	60.59	even	2
9900.2.a.bz.1.1		2	12.11	even	2
9900.2.c.x.5149.2		4	60.23	odd	4
9900.2.c.x.5149.3		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79129 q^{3} -0.208712 q^{7} +4.79129 q^{9} +1.00000 q^{11} +1.00000 q^{13} -0.791288 q^{17} -6.58258 q^{19} +0.582576 q^{21} -3.79129 q^{23} -5.00000 q^{27} +6.79129 q^{29} +8.58258 q^{31} -2.79129 q^{33} +2.58258 q^{37} -2.79129 q^{39} -1.41742 q^{41} -10.0000 q^{43} -1.41742 q^{47} -6.95644 q^{49} +2.20871 q^{51} -11.3739 q^{53} +18.3739 q^{57} +10.5826 q^{59} +4.20871 q^{61} -1.00000 q^{63} -4.00000 q^{67} +10.5826 q^{69} +10.7477 q^{71} +7.79129 q^{73} -0.208712 q^{77} +15.5390 q^{79} -0.417424 q^{81} -9.95644 q^{83} -18.9564 q^{87} -0.791288 q^{89} -0.208712 q^{91} -23.9564 q^{93} +6.20871 q^{97} +4.79129 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 5 q^{7} + 5 q^{9} + 2 q^{11} + 2 q^{13} + 3 q^{17} - 4 q^{19} - 8 q^{21} - 3 q^{23} - 10 q^{27} + 9 q^{29} + 8 q^{31} - q^{33} - 4 q^{37} - q^{39} - 12 q^{41} - 20 q^{43} - 12 q^{47} + 9 q^{49}+ \cdots + 5 q^{99}+O(q^{100}) \) 2 * q - q^3 - 5 * q^7 + 5 * q^9 + 2 * q^11 + 2 * q^13 + 3 * q^17 - 4 * q^19 - 8 * q^21 - 3 * q^23 - 10 * q^27 + 9 * q^29 + 8 * q^31 - q^33 - 4 * q^37 - q^39 - 12 * q^41 - 20 * q^43 - 12 * q^47 + 9 * q^49 + 9 * q^51 - 9 * q^53 + 23 * q^57 + 12 * q^59 + 13 * q^61 - 2 * q^63 - 8 * q^67 + 12 * q^69 - 6 * q^71 + 11 * q^73 - 5 * q^77 - q^79 - 10 * q^81 + 3 * q^83 - 15 * q^87 + 3 * q^89 - 5 * q^91 - 25 * q^93 + 17 * q^97 + 5 * q^99

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