LMFDB - Embedded newform 4400.2.a.bx.1.2

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 = [3,0,-3,0,0,0,-3,0,6,0,3,0,-3] 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:	$3$
Coefficient field:	3.3.837.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.167449$ 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)

$f(q)$	$=$	$q-1.16745 q^{3} -4.97196 q^{7} -1.63706 q^{9} +1.00000 q^{11} -0.665102 q^{13} +6.77647 q^{17} -1.00000 q^{19} +5.80451 q^{21} -2.16745 q^{23} +5.41353 q^{27} +7.97196 q^{29} +8.94392 q^{31} -1.16745 q^{33} -0.139410 q^{37} +0.776472 q^{39} -1.80451 q^{41} -2.80451 q^{43} -0.530387 q^{47} +17.7204 q^{49} -7.91119 q^{51} -6.30216 q^{53} +1.16745 q^{57} -11.4696 q^{59} +10.5810 q^{61} +8.13941 q^{63} -9.60902 q^{67} +2.53039 q^{69} +9.80921 q^{71} -7.02804 q^{73} -4.97196 q^{77} -5.50235 q^{79} -1.40884 q^{81} -13.5810 q^{83} -9.30686 q^{87} -9.58098 q^{89} +3.30686 q^{91} -10.4416 q^{93} -14.7812 q^{97} -1.63706 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 3 q^{13} - 3 q^{17} - 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} + 3 q^{31} - 3 q^{33} + 12 q^{37} - 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 3 * q^13 - 3 * q^17 - 3 * q^19 + 6 * q^21 - 6 * q^23 - 18 * q^27 + 12 * q^29 + 3 * q^31 - 3 * q^33 + 12 * q^37 - 21 * q^39 + 6 * q^41 + 3 * q^43 - 12 * q^47 + 6 * q^49 + 9 * q^51 - 9 * q^53 + 3 * q^57 - 24 * q^59 - 3 * q^61 + 12 * q^63 - 6 * q^67 + 18 * q^69 + 15 * q^71 - 33 * q^73 - 3 * q^77 - 15 * q^79 + 27 * q^81 - 6 * q^83 - 15 * q^87 + 6 * q^89 - 3 * q^91 - 9 * q^93 - 18 * q^97 + 6 * 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
$3$	−1.16745	−0.674027	−0.337014	−	0.941500i	$-0.609417\pi$
$3$	−1.16745	−0.674027	−0.337014	+	0.941500i	$0.609417\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.97196	−1.87922	−0.939612	−	0.342241i	$-0.888814\pi$
$7$	−4.97196	−1.87922	−0.939612	+	0.342241i	$0.888814\pi$
$8$	0	0
$9$	−1.63706	−0.545687
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.665102	−0.184466	−0.0922330	−	0.995737i	$-0.529400\pi$
$13$	−0.665102	−0.184466	−0.0922330	+	0.995737i	$0.529400\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.77647	1.64354	0.821768	−	0.569822i	$-0.192988\pi$
$17$	6.77647	1.64354	0.821768	+	0.569822i	$0.192988\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	5.80451	1.26665
$22$	0	0
$23$	−2.16745	−0.451944	−0.225972	−	0.974134i	$-0.572556\pi$
$23$	−2.16745	−0.451944	−0.225972	+	0.974134i	$0.572556\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.41353	1.04184
$28$	0	0
$29$	7.97196	1.48036	0.740178	−	0.672411i	$-0.234741\pi$
$29$	7.97196	1.48036	0.740178	+	0.672411i	$0.234741\pi$
$30$	0	0
$31$	8.94392	1.60638	0.803188	−	0.595726i	$-0.203136\pi$
$31$	8.94392	1.60638	0.803188	+	0.595726i	$0.203136\pi$
$32$	0	0
$33$	−1.16745	−0.203227
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.139410	−0.0229189	−0.0114594	−	0.999934i	$-0.503648\pi$
$37$	−0.139410	−0.0229189	−0.0114594	+	0.999934i	$0.503648\pi$
$38$	0	0
$39$	0.776472	0.124335
$40$	0	0
$41$	−1.80451	−0.281817	−0.140909	−	0.990023i	$-0.545002\pi$
$41$	−1.80451	−0.281817	−0.140909	+	0.990023i	$0.545002\pi$
$42$	0	0
$43$	−2.80451	−0.427684	−0.213842	−	0.976868i	$-0.568598\pi$
$43$	−2.80451	−0.427684	−0.213842	+	0.976868i	$0.568598\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−0.530387	−0.0773649	−0.0386824	−	0.999252i	$-0.512316\pi$
$47$	−0.530387	−0.0773649	−0.0386824	+	0.999252i	$0.512316\pi$
$48$	0	0
$49$	17.7204	2.53148
$50$	0	0
$51$	−7.91119	−1.10779
$52$	0	0
$53$	−6.30216	−0.865669	−0.432834	−	0.901473i	$-0.642487\pi$
$53$	−6.30216	−0.865669	−0.432834	+	0.901473i	$0.642487\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1.16745	0.154632
$58$	0	0
$59$	−11.4696	−1.49322	−0.746608	−	0.665264i	$-0.768319\pi$
$59$	−11.4696	−1.49322	−0.746608	+	0.665264i	$0.768319\pi$
$60$	0	0
$61$	10.5810	1.35476	0.677378	−	0.735635i	$-0.263116\pi$
$61$	10.5810	1.35476	0.677378	+	0.735635i	$0.263116\pi$
$62$	0	0
$63$	8.13941	1.02547
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.60902	−1.17393	−0.586965	−	0.809613i	$-0.699677\pi$
$67$	−9.60902	−1.17393	−0.586965	+	0.809613i	$0.699677\pi$
$68$	0	0
$69$	2.53039	0.304623
$70$	0	0
$71$	9.80921	1.16414	0.582069	−	0.813139i	$-0.302243\pi$
$71$	9.80921	1.16414	0.582069	+	0.813139i	$0.302243\pi$
$72$	0	0
$73$	−7.02804	−0.822570	−0.411285	−	0.911507i	$-0.634920\pi$
$73$	−7.02804	−0.822570	−0.411285	+	0.911507i	$0.634920\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.97196	−0.566608
$78$	0	0
$79$	−5.50235	−0.619062	−0.309531	−	0.950889i	$-0.600172\pi$
$79$	−5.50235	−0.619062	−0.309531	+	0.950889i	$0.600172\pi$
$80$	0	0
$81$	−1.40884	−0.156538
$82$	0	0
$83$	−13.5810	−1.49071	−0.745353	−	0.666670i	$-0.767719\pi$
$83$	−13.5810	−1.49071	−0.745353	+	0.666670i	$0.767719\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.30686	−0.997800
$88$	0	0
$89$	−9.58098	−1.01558	−0.507791	−	0.861480i	$-0.669538\pi$
$89$	−9.58098	−1.01558	−0.507791	+	0.861480i	$0.669538\pi$
$90$	0	0
$91$	3.30686	0.346653
$92$	0	0
$93$	−10.4416	−1.08274
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−14.7812	−1.50080	−0.750400	−	0.660984i	$-0.770139\pi$
$97$	−14.7812	−1.50080	−0.750400	+	0.660984i	$0.770139\pi$
$98$	0	0
$99$	−1.63706	−0.164531
$100$	0	0
$101$	4.22353	0.420257	0.210128	−	0.977674i	$-0.432612\pi$
$101$	4.22353	0.420257	0.210128	+	0.977674i	$0.432612\pi$
$102$	0	0
$103$	16.9206	1.66723	0.833617	−	0.552343i	$-0.186266\pi$
$103$	16.9206	1.66723	0.833617	+	0.552343i	$0.186266\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	5.66980	0.548120	0.274060	−	0.961713i	$-0.411633\pi$
$107$	5.66980	0.548120	0.274060	+	0.961713i	$0.411633\pi$
$108$	0	0
$109$	−16.2508	−1.55654	−0.778271	−	0.627928i	$-0.783903\pi$
$109$	−16.2508	−1.55654	−0.778271	+	0.627928i	$0.783903\pi$
$110$	0	0
$111$	0.162754	0.0154479
$112$	0	0
$113$	−1.86059	−0.175030	−0.0875148	−	0.996163i	$-0.527893\pi$
$113$	−1.86059	−0.175030	−0.0875148	+	0.996163i	$0.527893\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.08881	0.100661
$118$	0	0
$119$	−33.6924	−3.08857
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	2.10668	0.189953
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−11.1674	−0.990951	−0.495475	−	0.868622i	$-0.665006\pi$
$127$	−11.1674	−0.990951	−0.495475	+	0.868622i	$0.665006\pi$
$128$	0	0
$129$	3.27412	0.288271
$130$	0	0
$131$	−14.7812	−1.29144	−0.645718	−	0.763576i	$-0.723442\pi$
$131$	−14.7812	−1.29144	−0.645718	+	0.763576i	$0.723442\pi$
$132$	0	0
$133$	4.97196	0.431124
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−7.63706	−0.652478	−0.326239	−	0.945287i	$-0.605781\pi$
$137$	−7.63706	−0.652478	−0.326239	+	0.945287i	$0.605781\pi$
$138$	0	0
$139$	5.27882	0.447744	0.223872	−	0.974619i	$-0.428130\pi$
$139$	5.27882	0.447744	0.223872	+	0.974619i	$0.428130\pi$
$140$	0	0
$141$	0.619200	0.0521460
$142$	0	0
$143$	−0.665102	−0.0556186
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−20.6877	−1.70629
$148$	0	0
$149$	5.60902	0.459509	0.229755	−	0.973249i	$-0.426208\pi$
$149$	5.60902	0.459509	0.229755	+	0.973249i	$0.426208\pi$
$150$	0	0
$151$	3.86059	0.314170	0.157085	−	0.987585i	$-0.449790\pi$
$151$	3.86059	0.314170	0.157085	+	0.987585i	$0.449790\pi$
$152$	0	0
$153$	−11.0935	−0.896857
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.5482	1.32069	0.660347	−	0.750961i	$-0.270409\pi$
$157$	16.5482	1.32069	0.660347	+	0.750961i	$0.270409\pi$
$158$	0	0
$159$	7.35746	0.583484
$160$	0	0
$161$	10.7765	0.849305
$162$	0	0
$163$	20.8598	1.63387	0.816933	−	0.576733i	$-0.195673\pi$
$163$	20.8598	1.63387	0.816933	+	0.576733i	$0.195673\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.0226	−1.62677	−0.813387	−	0.581723i	$-0.802379\pi$
$167$	−21.0226	−1.62677	−0.813387	+	0.581723i	$0.802379\pi$
$168$	0	0
$169$	−12.5576	−0.965972
$170$	0	0
$171$	1.63706	0.125189
$172$	0	0
$173$	4.33020	0.329219	0.164610	−	0.986359i	$-0.447364\pi$
$173$	4.33020	0.329219	0.164610	+	0.986359i	$0.447364\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.3902	1.00647
$178$	0	0
$179$	−14.6371	−1.09403	−0.547013	−	0.837124i	$-0.684235\pi$
$179$	−14.6371	−1.09403	−0.547013	+	0.837124i	$0.684235\pi$
$180$	0	0
$181$	11.9112	0.885352	0.442676	−	0.896682i	$-0.354029\pi$
$181$	11.9112	0.885352	0.442676	+	0.896682i	$0.354029\pi$
$182$	0	0
$183$	−12.3528	−0.913142
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.77647	0.495545
$188$	0	0
$189$	−26.9159	−1.95784
$190$	0	0
$191$	−20.2508	−1.46530	−0.732648	−	0.680608i	$-0.761716\pi$
$191$	−20.2508	−1.46530	−0.732648	+	0.680608i	$0.761716\pi$
$192$	0	0
$193$	−19.2180	−1.38335	−0.691673	−	0.722211i	$-0.743126\pi$
$193$	−19.2180	−1.38335	−0.691673	+	0.722211i	$0.743126\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.9945	1.49580	0.747899	−	0.663813i	$-0.231063\pi$
$197$	20.9945	1.49580	0.747899	+	0.663813i	$0.231063\pi$
$198$	0	0
$199$	−4.64176	−0.329045	−0.164523	−	0.986373i	$-0.552608\pi$
$199$	−4.64176	−0.329045	−0.164523	+	0.986373i	$0.552608\pi$
$200$	0	0
$201$	11.2180	0.791260
$202$	0	0
$203$	−39.6363	−2.78192
$204$	0	0
$205$	0	0
$206$	0	0
$207$	3.54825	0.246620
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	3.19079	0.219663	0.109832	−	0.993950i	$-0.464969\pi$
$211$	3.19079	0.219663	0.109832	+	0.993950i	$0.464969\pi$
$212$	0	0
$213$	−11.4518	−0.784661
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−44.4688	−3.01874
$218$	0	0
$219$	8.20488	0.554434
$220$	0	0
$221$	−4.50704	−0.303177
$222$	0	0
$223$	7.46961	0.500202	0.250101	−	0.968220i	$-0.419536\pi$
$223$	7.46961	0.500202	0.250101	+	0.968220i	$0.419536\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	22.1947	1.47311	0.736557	−	0.676375i	$-0.236450\pi$
$227$	22.1947	1.47311	0.736557	+	0.676375i	$0.236450\pi$
$228$	0	0
$229$	3.50235	0.231442	0.115721	−	0.993282i	$-0.463082\pi$
$229$	3.50235	0.231442	0.115721	+	0.993282i	$0.463082\pi$
$230$	0	0
$231$	5.80451	0.381909
$232$	0	0
$233$	0.637062	0.0417353	0.0208677	−	0.999782i	$-0.493357\pi$
$233$	0.637062	0.0417353	0.0208677	+	0.999782i	$0.493357\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	6.42371	0.417265
$238$	0	0
$239$	8.35824	0.540650	0.270325	−	0.962769i	$-0.412869\pi$
$239$	8.35824	0.540650	0.270325	+	0.962769i	$0.412869\pi$
$240$	0	0
$241$	10.0506	0.647416	0.323708	−	0.946157i	$-0.395070\pi$
$241$	10.0506	0.647416	0.323708	+	0.946157i	$0.395070\pi$
$242$	0	0
$243$	−14.5959	−0.936325
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.665102	0.0423194
$248$	0	0
$249$	15.8551	1.00478
$250$	0	0
$251$	3.44627	0.217527	0.108763	−	0.994068i	$-0.465311\pi$
$251$	3.44627	0.217527	0.108763	+	0.994068i	$0.465311\pi$
$252$	0	0
$253$	−2.16745	−0.136266
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.4743	−0.715748	−0.357874	−	0.933770i	$-0.616498\pi$
$257$	−11.4743	−0.715748	−0.357874	+	0.933770i	$0.616498\pi$
$258$	0	0
$259$	0.693141	0.0430697
$260$	0	0
$261$	−13.0506	−0.807812
$262$	0	0
$263$	4.41823	0.272440	0.136220	−	0.990679i	$-0.456505\pi$
$263$	4.41823	0.272440	0.136220	+	0.990679i	$0.456505\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.1853	0.684530
$268$	0	0
$269$	6.11607	0.372903	0.186452	−	0.982464i	$-0.440301\pi$
$269$	6.11607	0.372903	0.186452	+	0.982464i	$0.440301\pi$
$270$	0	0
$271$	27.3014	1.65844	0.829220	−	0.558922i	$-0.188785\pi$
$271$	27.3014	1.65844	0.829220	+	0.558922i	$0.188785\pi$
$272$	0	0
$273$	−3.86059	−0.233654
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.14411	0.0687426	0.0343713	−	0.999409i	$-0.489057\pi$
$277$	1.14411	0.0687426	0.0343713	+	0.999409i	$0.489057\pi$
$278$	0	0
$279$	−14.6418	−0.876579
$280$	0	0
$281$	−30.6410	−1.82789	−0.913944	−	0.405841i	$-0.866979\pi$
$281$	−30.6410	−1.82789	−0.913944	+	0.405841i	$0.866979\pi$
$282$	0	0
$283$	−4.16275	−0.247450	−0.123725	−	0.992317i	$-0.539484\pi$
$283$	−4.16275	−0.247450	−0.123725	+	0.992317i	$0.539484\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.97196	0.529598
$288$	0	0
$289$	28.9206	1.70121
$290$	0	0
$291$	17.2563	1.01158
$292$	0	0
$293$	−25.7757	−1.50583	−0.752916	−	0.658117i	$-0.771353\pi$
$293$	−25.7757	−1.50583	−0.752916	+	0.658117i	$0.771353\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.41353	0.314125
$298$	0	0
$299$	1.44157	0.0833684
$300$	0	0
$301$	13.9439	0.803714
$302$	0	0
$303$	−4.93075	−0.283264
$304$	0	0
$305$	0	0
$306$	0	0
$307$	12.7859	0.729728	0.364864	−	0.931061i	$-0.381116\pi$
$307$	12.7859	0.729728	0.364864	+	0.931061i	$0.381116\pi$
$308$	0	0
$309$	−19.7539	−1.12376
$310$	0	0
$311$	−3.39098	−0.192285	−0.0961423	−	0.995368i	$-0.530650\pi$
$311$	−3.39098	−0.192285	−0.0961423	+	0.995368i	$0.530650\pi$
$312$	0	0
$313$	−6.32551	−0.357539	−0.178769	−	0.983891i	$-0.557212\pi$
$313$	−6.32551	−0.357539	−0.178769	+	0.983891i	$0.557212\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−18.4510	−1.03631	−0.518155	−	0.855287i	$-0.673381\pi$
$317$	−18.4510	−1.03631	−0.518155	+	0.855287i	$0.673381\pi$
$318$	0	0
$319$	7.97196	0.446344
$320$	0	0
$321$	−6.61920	−0.369448
$322$	0	0
$323$	−6.77647	−0.377053
$324$	0	0
$325$	0	0
$326$	0	0
$327$	18.9720	1.04915
$328$	0	0
$329$	2.63706	0.145386
$330$	0	0
$331$	2.19549	0.120675	0.0603375	−	0.998178i	$-0.480782\pi$
$331$	2.19549	0.120675	0.0603375	+	0.998178i	$0.480782\pi$
$332$	0	0
$333$	0.228223	0.0125065
$334$	0	0
$335$	0	0
$336$	0	0
$337$	17.8878	0.974413	0.487206	−	0.873287i	$-0.338016\pi$
$337$	17.8878	0.974413	0.487206	+	0.873287i	$0.338016\pi$
$338$	0	0
$339$	2.17214	0.117975
$340$	0	0
$341$	8.94392	0.484341
$342$	0	0
$343$	−53.3014	−2.87800
$344$	0	0
$345$	0	0
$346$	0	0
$347$	19.0280	1.02148	0.510739	−	0.859736i	$-0.329372\pi$
$347$	19.0280	1.02148	0.510739	+	0.859736i	$0.329372\pi$
$348$	0	0
$349$	−28.6090	−1.53141	−0.765703	−	0.643194i	$-0.777609\pi$
$349$	−28.6090	−1.53141	−0.765703	+	0.643194i	$0.777609\pi$
$350$	0	0
$351$	−3.60055	−0.192183
$352$	0	0
$353$	19.6698	1.04692	0.523459	−	0.852051i	$-0.324641\pi$
$353$	19.6698	1.04692	0.523459	+	0.852051i	$0.324641\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	39.3341	2.08178
$358$	0	0
$359$	10.8271	0.571431	0.285715	−	0.958315i	$-0.407769\pi$
$359$	10.8271	0.571431	0.285715	+	0.958315i	$0.407769\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−1.16745	−0.0612752
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−6.50235	−0.339420	−0.169710	−	0.985494i	$-0.554283\pi$
$367$	−6.50235	−0.339420	−0.169710	+	0.985494i	$0.554283\pi$
$368$	0	0
$369$	2.95410	0.153784
$370$	0	0
$371$	31.3341	1.62679
$372$	0	0
$373$	−8.14880	−0.421929	−0.210964	−	0.977494i	$-0.567660\pi$
$373$	−8.14880	−0.421929	−0.210964	+	0.977494i	$0.567660\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−5.30216	−0.273075
$378$	0	0
$379$	−1.14411	−0.0587687	−0.0293844	−	0.999568i	$-0.509355\pi$
$379$	−1.14411	−0.0587687	−0.0293844	+	0.999568i	$0.509355\pi$
$380$	0	0
$381$	13.0374	0.667928
$382$	0	0
$383$	2.53508	0.129537	0.0647683	−	0.997900i	$-0.479369\pi$
$383$	2.53508	0.129537	0.0647683	+	0.997900i	$0.479369\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.59116	0.233382
$388$	0	0
$389$	−12.6698	−0.642384	−0.321192	−	0.947014i	$-0.604084\pi$
$389$	−12.6698	−0.642384	−0.321192	+	0.947014i	$0.604084\pi$
$390$	0	0
$391$	−14.6877	−0.742787
$392$	0	0
$393$	17.2563	0.870463
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−31.9945	−1.60576	−0.802879	−	0.596141i	$-0.796700\pi$
$397$	−31.9945	−1.60576	−0.802879	+	0.596141i	$0.796700\pi$
$398$	0	0
$399$	−5.80451	−0.290589
$400$	0	0
$401$	9.19549	0.459201	0.229600	−	0.973285i	$-0.426258\pi$
$401$	9.19549	0.459201	0.229600	+	0.973285i	$0.426258\pi$
$402$	0	0
$403$	−5.94862	−0.296322
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.139410	−0.00691030
$408$	0	0
$409$	−25.4790	−1.25986	−0.629928	−	0.776654i	$-0.716916\pi$
$409$	−25.4790	−1.25986	−0.629928	+	0.776654i	$0.716916\pi$
$410$	0	0
$411$	8.91588	0.439788
$412$	0	0
$413$	57.0265	2.80609
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.16275	−0.301791
$418$	0	0
$419$	25.0132	1.22197	0.610987	−	0.791641i	$-0.290773\pi$
$419$	25.0132	1.22197	0.610987	+	0.791641i	$0.290773\pi$
$420$	0	0
$421$	−31.8084	−1.55025	−0.775124	−	0.631809i	$-0.782313\pi$
$421$	−31.8084	−1.55025	−0.775124	+	0.631809i	$0.782313\pi$
$422$	0	0
$423$	0.868276	0.0422170
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−52.6082	−2.54589
$428$	0	0
$429$	0.776472	0.0374884
$430$	0	0
$431$	−29.9712	−1.44366	−0.721830	−	0.692070i	$-0.756699\pi$
$431$	−29.9712	−1.44366	−0.721830	+	0.692070i	$0.756699\pi$
$432$	0	0
$433$	−39.8271	−1.91397	−0.956983	−	0.290143i	$-0.906297\pi$
$433$	−39.8271	−1.91397	−0.956983	+	0.290143i	$0.906297\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.16745	0.103683
$438$	0	0
$439$	17.4369	0.832217	0.416108	−	0.909315i	$-0.363394\pi$
$439$	17.4369	0.832217	0.416108	+	0.909315i	$0.363394\pi$
$440$	0	0
$441$	−29.0094	−1.38140
$442$	0	0
$443$	4.21335	0.200182	0.100091	−	0.994978i	$-0.468087\pi$
$443$	4.21335	0.200182	0.100091	+	0.994978i	$0.468087\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.54825	−0.309722
$448$	0	0
$449$	−13.6643	−0.644859	−0.322429	−	0.946593i	$-0.604500\pi$
$449$	−13.6643	−0.644859	−0.322429	+	0.946593i	$0.604500\pi$
$450$	0	0
$451$	−1.80451	−0.0849711
$452$	0	0
$453$	−4.50704	−0.211759
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.25078	0.245621	0.122811	−	0.992430i	$-0.460809\pi$
$457$	5.25078	0.245621	0.122811	+	0.992430i	$0.460809\pi$
$458$	0	0
$459$	36.6847	1.71229
$460$	0	0
$461$	−5.74843	−0.267731	−0.133866	−	0.990999i	$-0.542739\pi$
$461$	−5.74843	−0.267731	−0.133866	+	0.990999i	$0.542739\pi$
$462$	0	0
$463$	−27.7710	−1.29063	−0.645314	−	0.763918i	$-0.723273\pi$
$463$	−27.7710	−1.29063	−0.645314	+	0.763918i	$0.723273\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−13.3575	−0.618109	−0.309055	−	0.951044i	$-0.600013\pi$
$467$	−13.3575	−0.618109	−0.309055	+	0.951044i	$0.600013\pi$
$468$	0	0
$469$	47.7757	2.20608
$470$	0	0
$471$	−19.3192	−0.890184
$472$	0	0
$473$	−2.80451	−0.128952
$474$	0	0
$475$	0	0
$476$	0	0
$477$	10.3170	0.472385
$478$	0	0
$479$	−11.7578	−0.537229	−0.268614	−	0.963248i	$-0.586566\pi$
$479$	−11.7578	−0.537229	−0.268614	+	0.963248i	$0.586566\pi$
$480$	0	0
$481$	0.0927218	0.00422775
$482$	0	0
$483$	−12.5810	−0.572455
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−18.6651	−0.845796	−0.422898	−	0.906177i	$-0.638987\pi$
$487$	−18.6651	−0.845796	−0.422898	+	0.906177i	$0.638987\pi$
$488$	0	0
$489$	−24.3528	−1.10127
$490$	0	0
$491$	−32.4969	−1.46656	−0.733282	−	0.679925i	$-0.762012\pi$
$491$	−32.4969	−1.46656	−0.733282	+	0.679925i	$0.762012\pi$
$492$	0	0
$493$	54.0218	2.43302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−48.7710	−2.18768
$498$	0	0
$499$	−11.6884	−0.523247	−0.261623	−	0.965170i	$-0.584258\pi$
$499$	−11.6884	−0.523247	−0.261623	+	0.965170i	$0.584258\pi$
$500$	0	0
$501$	24.5428	1.09649
$502$	0	0
$503$	5.67449	0.253013	0.126507	−	0.991966i	$-0.459624\pi$
$503$	5.67449	0.253013	0.126507	+	0.991966i	$0.459624\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.6604	0.651092
$508$	0	0
$509$	−17.5857	−0.779472	−0.389736	−	0.920927i	$-0.627434\pi$
$509$	−17.5857	−0.779472	−0.389736	+	0.920927i	$0.627434\pi$
$510$	0	0
$511$	34.9431	1.54579
$512$	0	0
$513$	−5.41353	−0.239013
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−0.530387	−0.0233264
$518$	0	0
$519$	−5.05529	−0.221903
$520$	0	0
$521$	13.1300	0.575237	0.287618	−	0.957745i	$-0.407137\pi$
$521$	13.1300	0.575237	0.287618	+	0.957745i	$0.407137\pi$
$522$	0	0
$523$	42.0226	1.83752	0.918759	−	0.394819i	$-0.129193\pi$
$523$	42.0226	1.83752	0.918759	+	0.394819i	$0.129193\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	60.6082	2.64014
$528$	0	0
$529$	−18.3022	−0.795746
$530$	0	0
$531$	18.7765	0.814829
$532$	0	0
$533$	1.20018	0.0519857
$534$	0	0
$535$	0	0
$536$	0	0
$537$	17.0880	0.737403
$538$	0	0
$539$	17.7204	0.763271
$540$	0	0
$541$	−27.9992	−1.20378	−0.601890	−	0.798579i	$-0.705585\pi$
$541$	−27.9992	−1.20378	−0.601890	+	0.798579i	$0.705585\pi$
$542$	0	0
$543$	−13.9057	−0.596751
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−9.05999	−0.387377	−0.193689	−	0.981063i	$-0.562045\pi$
$547$	−9.05999	−0.387377	−0.193689	+	0.981063i	$0.562045\pi$
$548$	0	0
$549$	−17.3217	−0.739273
$550$	0	0
$551$	−7.97196	−0.339617
$552$	0	0
$553$	27.3575	1.16336
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.8224	1.30599	0.652993	−	0.757364i	$-0.273513\pi$
$557$	30.8224	1.30599	0.652993	+	0.757364i	$0.273513\pi$
$558$	0	0
$559$	1.86529	0.0788932
$560$	0	0
$561$	−7.91119	−0.334011
$562$	0	0
$563$	9.13862	0.385147	0.192574	−	0.981283i	$-0.438317\pi$
$563$	9.13862	0.385147	0.192574	+	0.981283i	$0.438317\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	7.00470	0.294170
$568$	0	0
$569$	−24.9720	−1.04688	−0.523440	−	0.852063i	$-0.675351\pi$
$569$	−24.9720	−1.04688	−0.523440	+	0.852063i	$0.675351\pi$
$570$	0	0
$571$	17.9898	0.752851	0.376425	−	0.926447i	$-0.377153\pi$
$571$	17.9898	0.752851	0.376425	+	0.926447i	$0.377153\pi$
$572$	0	0
$573$	23.6418	0.987649
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−39.7812	−1.65611	−0.828056	−	0.560646i	$-0.810553\pi$
$577$	−39.7812	−1.65611	−0.828056	+	0.560646i	$0.810553\pi$
$578$	0	0
$579$	22.4361	0.932412
$580$	0	0
$581$	67.5241	2.80137
$582$	0	0
$583$	−6.30216	−0.261009
$584$	0	0
$585$	0	0
$586$	0	0
$587$	3.72509	0.153751	0.0768754	−	0.997041i	$-0.475506\pi$
$587$	3.72509	0.153751	0.0768754	+	0.997041i	$0.475506\pi$
$588$	0	0
$589$	−8.94392	−0.368528
$590$	0	0
$591$	−24.5100	−1.00821
$592$	0	0
$593$	−21.7017	−0.891184	−0.445592	−	0.895236i	$-0.647007\pi$
$593$	−21.7017	−0.891184	−0.445592	+	0.895236i	$0.647007\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	5.41902	0.221786
$598$	0	0
$599$	−25.7157	−1.05071	−0.525357	−	0.850882i	$-0.676068\pi$
$599$	−25.7157	−1.05071	−0.525357	+	0.850882i	$0.676068\pi$
$600$	0	0
$601$	−37.8551	−1.54414	−0.772071	−	0.635536i	$-0.780779\pi$
$601$	−37.8551	−1.54414	−0.772071	+	0.635536i	$0.780779\pi$
$602$	0	0
$603$	15.7306	0.640598
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16.4361	0.667120	0.333560	−	0.942729i	$-0.391750\pi$
$607$	16.4361	0.667120	0.333560	+	0.942729i	$0.391750\pi$
$608$	0	0
$609$	46.2733	1.87509
$610$	0	0
$611$	0.352761	0.0142712
$612$	0	0
$613$	21.1674	0.854945	0.427473	−	0.904028i	$-0.359404\pi$
$613$	21.1674	0.854945	0.427473	+	0.904028i	$0.359404\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−46.3575	−1.86628	−0.933140	−	0.359512i	$-0.882943\pi$
$617$	−46.3575	−1.86628	−0.933140	+	0.359512i	$0.882943\pi$
$618$	0	0
$619$	6.52100	0.262101	0.131050	−	0.991376i	$-0.458165\pi$
$619$	6.52100	0.262101	0.131050	+	0.991376i	$0.458165\pi$
$620$	0	0
$621$	−11.7336	−0.470852
$622$	0	0
$623$	47.6363	1.90851
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.16745	0.0466234
$628$	0	0
$629$	−0.944708	−0.0376680
$630$	0	0
$631$	−23.8645	−0.950031	−0.475015	−	0.879978i	$-0.657557\pi$
$631$	−23.8645	−0.950031	−0.475015	+	0.879978i	$0.657557\pi$
$632$	0	0
$633$	−3.72509	−0.148059
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−11.7859	−0.466973
$638$	0	0
$639$	−16.0583	−0.635256
$640$	0	0
$641$	38.1153	1.50546	0.752732	−	0.658328i	$-0.228736\pi$
$641$	38.1153	1.50546	0.752732	+	0.658328i	$0.228736\pi$
$642$	0	0
$643$	−30.2694	−1.19371	−0.596855	−	0.802349i	$-0.703583\pi$
$643$	−30.2694	−1.19371	−0.596855	+	0.802349i	$0.703583\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.18610	0.164572	0.0822862	−	0.996609i	$-0.473778\pi$
$647$	4.18610	0.164572	0.0822862	+	0.996609i	$0.473778\pi$
$648$	0	0
$649$	−11.4696	−0.450222
$650$	0	0
$651$	51.9151	2.03471
$652$	0	0
$653$	−18.3949	−0.719848	−0.359924	−	0.932982i	$-0.617197\pi$
$653$	−18.3949	−0.719848	−0.359924	+	0.932982i	$0.617197\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11.5053	0.448866
$658$	0	0
$659$	2.92058	0.113770	0.0568848	−	0.998381i	$-0.481883\pi$
$659$	2.92058	0.113770	0.0568848	+	0.998381i	$0.481883\pi$
$660$	0	0
$661$	−24.5304	−0.954121	−0.477061	−	0.878870i	$-0.658298\pi$
$661$	−24.5304	−0.954121	−0.477061	+	0.878870i	$0.658298\pi$
$662$	0	0
$663$	5.26174	0.204349
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−17.2788	−0.669039
$668$	0	0
$669$	−8.72039	−0.337150
$670$	0	0
$671$	10.5810	0.408474
$672$	0	0
$673$	−5.91667	−0.228071	−0.114035	−	0.993477i	$-0.536378\pi$
$673$	−5.91667	−0.228071	−0.114035	+	0.993477i	$0.536378\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3.80921	−0.146400	−0.0731999	−	0.997317i	$-0.523321\pi$
$677$	−3.80921	−0.146400	−0.0731999	+	0.997317i	$0.523321\pi$
$678$	0	0
$679$	73.4914	2.82034
$680$	0	0
$681$	−25.9112	−0.992919
$682$	0	0
$683$	31.5241	1.20624	0.603118	−	0.797652i	$-0.293925\pi$
$683$	31.5241	1.20624	0.603118	+	0.797652i	$0.293925\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.08881	−0.155998
$688$	0	0
$689$	4.19158	0.159686
$690$	0	0
$691$	12.9253	0.491701	0.245850	−	0.969308i	$-0.420933\pi$
$691$	12.9253	0.491701	0.245850	+	0.969308i	$0.420933\pi$
$692$	0	0
$693$	8.13941	0.309191
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.2282	−0.463177
$698$	0	0
$699$	−0.743738	−0.0281308
$700$	0	0
$701$	37.6504	1.42203	0.711017	−	0.703175i	$-0.248235\pi$
$701$	37.6504	1.42203	0.711017	+	0.703175i	$0.248235\pi$
$702$	0	0
$703$	0.139410	0.00525795
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−20.9992	−0.789757
$708$	0	0
$709$	−35.1059	−1.31843	−0.659215	−	0.751955i	$-0.729111\pi$
$709$	−35.1059	−1.31843	−0.659215	+	0.751955i	$0.729111\pi$
$710$	0	0
$711$	9.00769	0.337815
$712$	0	0
$713$	−19.3855	−0.725993
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.75782	−0.364413
$718$	0	0
$719$	−38.0833	−1.42027	−0.710134	−	0.704066i	$-0.751366\pi$
$719$	−38.0833	−1.42027	−0.710134	+	0.704066i	$0.751366\pi$
$720$	0	0
$721$	−84.1284	−3.13311
$722$	0	0
$723$	−11.7336	−0.436376
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.6465	0.876998	0.438499	−	0.898732i	$-0.355510\pi$
$727$	23.6465	0.876998	0.438499	+	0.898732i	$0.355510\pi$
$728$	0	0
$729$	21.2664	0.787646
$730$	0	0
$731$	−19.0047	−0.702914
$732$	0	0
$733$	44.3847	1.63939	0.819693	−	0.572803i	$-0.194144\pi$
$733$	44.3847	1.63939	0.819693	+	0.572803i	$0.194144\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−9.60902	−0.353953
$738$	0	0
$739$	−43.2959	−1.59267	−0.796333	−	0.604859i	$-0.793229\pi$
$739$	−43.2959	−1.59267	−0.796333	+	0.604859i	$0.793229\pi$
$740$	0	0
$741$	−0.776472	−0.0285244
$742$	0	0
$743$	−0.0327344	−0.00120091	−0.000600454	−	1.00000i	$-0.500191\pi$
$743$	−0.0327344	−0.00120091	−0.000600454		1.00000i	$0.500191\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22.2329	0.813460
$748$	0	0
$749$	−28.1900	−1.03004
$750$	0	0
$751$	−42.1292	−1.53732	−0.768659	−	0.639659i	$-0.779076\pi$
$751$	−42.1292	−1.53732	−0.768659	+	0.639659i	$0.779076\pi$
$752$	0	0
$753$	−4.02334	−0.146619
$754$	0	0
$755$	0	0
$756$	0	0
$757$	29.5257	1.07313	0.536565	−	0.843859i	$-0.319722\pi$
$757$	29.5257	1.07313	0.536565	+	0.843859i	$0.319722\pi$
$758$	0	0
$759$	2.53039	0.0918472
$760$	0	0
$761$	2.66041	0.0964397	0.0482198	−	0.998837i	$-0.484645\pi$
$761$	2.66041	0.0964397	0.0482198	+	0.998837i	$0.484645\pi$
$762$	0	0
$763$	80.7982	2.92509
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.62846	0.275448
$768$	0	0
$769$	45.3108	1.63395	0.816974	−	0.576674i	$-0.195650\pi$
$769$	45.3108	1.63395	0.816974	+	0.576674i	$0.195650\pi$
$770$	0	0
$771$	13.3957	0.482433
$772$	0	0
$773$	−7.52021	−0.270483	−0.135242	−	0.990813i	$-0.543181\pi$
$773$	−7.52021	−0.270483	−0.135242	+	0.990813i	$0.543181\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−0.809207	−0.0290301
$778$	0	0
$779$	1.80451	0.0646533
$780$	0	0
$781$	9.80921	0.351001
$782$	0	0
$783$	43.1565	1.54229
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−2.30765	−0.0822587	−0.0411293	−	0.999154i	$-0.513096\pi$
$787$	−2.30765	−0.0822587	−0.0411293	+	0.999154i	$0.513096\pi$
$788$	0	0
$789$	−5.15806	−0.183632
$790$	0	0
$791$	9.25078	0.328920
$792$	0	0
$793$	−7.03743	−0.249906
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−21.3902	−0.757679	−0.378840	−	0.925462i	$-0.623677\pi$
$797$	−21.3902	−0.757679	−0.378840	+	0.925462i	$0.623677\pi$
$798$	0	0
$799$	−3.59415	−0.127152
$800$	0	0
$801$	15.6847	0.554191
$802$	0	0
$803$	−7.02804	−0.248014
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−7.14020	−0.251347
$808$	0	0
$809$	38.8177	1.36476	0.682378	−	0.730999i	$-0.260946\pi$
$809$	38.8177	1.36476	0.682378	+	0.730999i	$0.260946\pi$
$810$	0	0
$811$	−42.6877	−1.49897	−0.749483	−	0.662023i	$-0.769698\pi$
$811$	−42.6877	−1.49897	−0.749483	+	0.662023i	$0.769698\pi$
$812$	0	0
$813$	−31.8730	−1.11783
$814$	0	0
$815$	0	0
$816$	0	0
$817$	2.80451	0.0981174
$818$	0	0
$819$	−5.41353	−0.189164
$820$	0	0
$821$	31.4649	1.09813	0.549067	−	0.835779i	$-0.314983\pi$
$821$	31.4649	1.09813	0.549067	+	0.835779i	$0.314983\pi$
$822$	0	0
$823$	12.1122	0.422203	0.211102	−	0.977464i	$-0.432295\pi$
$823$	12.1122	0.422203	0.211102	+	0.977464i	$0.432295\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	52.2359	1.81642	0.908210	−	0.418514i	$-0.137449\pi$
$827$	52.2359	1.81642	0.908210	+	0.418514i	$0.137449\pi$
$828$	0	0
$829$	45.3808	1.57614	0.788070	−	0.615585i	$-0.211080\pi$
$829$	45.3808	1.57614	0.788070	+	0.615585i	$0.211080\pi$
$830$	0	0
$831$	−1.33568	−0.0463344
$832$	0	0
$833$	120.082	4.16059
$834$	0	0
$835$	0	0
$836$	0	0
$837$	48.4182	1.67358
$838$	0	0
$839$	−9.01865	−0.311358	−0.155679	−	0.987808i	$-0.549757\pi$
$839$	−9.01865	−0.311358	−0.155679	+	0.987808i	$0.549757\pi$
$840$	0	0
$841$	34.5522	1.19145
$842$	0	0
$843$	35.7718	1.23205
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.97196	−0.170839
$848$	0	0
$849$	4.85980	0.166788
$850$	0	0
$851$	0.302164	0.0103580
$852$	0	0
$853$	−19.2780	−0.660067	−0.330034	−	0.943969i	$-0.607060\pi$
$853$	−19.2780	−0.660067	−0.330034	+	0.943969i	$0.607060\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−21.6269	−0.738760	−0.369380	−	0.929278i	$-0.620430\pi$
$857$	−21.6269	−0.738760	−0.369380	+	0.929278i	$0.620430\pi$
$858$	0	0
$859$	17.6557	0.602405	0.301203	−	0.953560i	$-0.402612\pi$
$859$	17.6557	0.602405	0.301203	+	0.953560i	$0.402612\pi$
$860$	0	0
$861$	−10.4743	−0.356963
$862$	0	0
$863$	−4.21805	−0.143584	−0.0717920	−	0.997420i	$-0.522872\pi$
$863$	−4.21805	−0.143584	−0.0717920	+	0.997420i	$0.522872\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−33.7633	−1.14666
$868$	0	0
$869$	−5.50235	−0.186654
$870$	0	0
$871$	6.39098	0.216550
$872$	0	0
$873$	24.1977	0.818968
$874$	0	0
$875$	0	0
$876$	0	0
$877$	47.7982	1.61403	0.807016	−	0.590530i	$-0.201081\pi$
$877$	47.7982	1.61403	0.807016	+	0.590530i	$0.201081\pi$
$878$	0	0
$879$	30.0918	1.01497
$880$	0	0
$881$	−23.3761	−0.787561	−0.393780	−	0.919205i	$-0.628833\pi$
$881$	−23.3761	−0.787561	−0.393780	+	0.919205i	$0.628833\pi$
$882$	0	0
$883$	22.1955	0.746938	0.373469	−	0.927643i	$-0.378168\pi$
$883$	22.1955	0.746938	0.373469	+	0.927643i	$0.378168\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.46961	−0.183652	−0.0918258	−	0.995775i	$-0.529270\pi$
$887$	−5.46961	−0.183652	−0.0918258	+	0.995775i	$0.529270\pi$
$888$	0	0
$889$	55.5241	1.86222
$890$	0	0
$891$	−1.40884	−0.0471979
$892$	0	0
$893$	0.530387	0.0177487
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.68296	−0.0561925
$898$	0	0
$899$	71.3006	2.37801
$900$	0	0
$901$	−42.7064	−1.42276
$902$	0	0
$903$	−16.2788	−0.541725
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−42.9486	−1.42609	−0.713043	−	0.701121i	$-0.752683\pi$
$907$	−42.9486	−1.42609	−0.713043	+	0.701121i	$0.752683\pi$
$908$	0	0
$909$	−6.91418	−0.229329
$910$	0	0
$911$	−30.1573	−0.999155	−0.499578	−	0.866269i	$-0.666511\pi$
$911$	−30.1573	−0.999155	−0.499578	+	0.866269i	$0.666511\pi$
$912$	0	0
$913$	−13.5810	−0.449465
$914$	0	0
$915$	0	0
$916$	0	0
$917$	73.4914	2.42690
$918$	0	0
$919$	43.6635	1.44033	0.720163	−	0.693804i	$-0.244067\pi$
$919$	43.6635	1.44033	0.720163	+	0.693804i	$0.244067\pi$
$920$	0	0
$921$	−14.9268	−0.491856
$922$	0	0
$923$	−6.52412	−0.214744
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−27.7000	−0.909789
$928$	0	0
$929$	28.8318	0.945940	0.472970	−	0.881079i	$-0.343182\pi$
$929$	28.8318	0.945940	0.472970	+	0.881079i	$0.343182\pi$
$930$	0	0
$931$	−17.7204	−0.580762
$932$	0	0
$933$	3.95879	0.129605
$934$	0	0
$935$	0	0
$936$	0	0
$937$	15.5063	0.506567	0.253284	−	0.967392i	$-0.418489\pi$
$937$	15.5063	0.506567	0.253284	+	0.967392i	$0.418489\pi$
$938$	0	0
$939$	7.38471	0.240991
$940$	0	0
$941$	32.4743	1.05863	0.529316	−	0.848425i	$-0.322449\pi$
$941$	32.4743	1.05863	0.529316	+	0.848425i	$0.322449\pi$
$942$	0	0
$943$	3.91119	0.127366
$944$	0	0
$945$	0	0
$946$	0	0
$947$	53.3575	1.73388	0.866942	−	0.498409i	$-0.166082\pi$
$947$	53.3575	1.73388	0.866942	+	0.498409i	$0.166082\pi$
$948$	0	0
$949$	4.67436	0.151736
$950$	0	0
$951$	21.5406	0.698501
$952$	0	0
$953$	−4.54825	−0.147332	−0.0736661	−	0.997283i	$-0.523470\pi$
$953$	−4.54825	−0.147332	−0.0736661	+	0.997283i	$0.523470\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−9.30686	−0.300848
$958$	0	0
$959$	37.9712	1.22615
$960$	0	0
$961$	48.9937	1.58044
$962$	0	0
$963$	−9.28181	−0.299102
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−40.7765	−1.31128	−0.655641	−	0.755072i	$-0.727602\pi$
$967$	−40.7765	−1.31128	−0.655641	+	0.755072i	$0.727602\pi$
$968$	0	0
$969$	7.91119	0.254144
$970$	0	0
$971$	−47.4641	−1.52320	−0.761598	−	0.648049i	$-0.775585\pi$
$971$	−47.4641	−1.52320	−0.761598	+	0.648049i	$0.775585\pi$
$972$	0	0
$973$	−26.2461	−0.841411
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.5755	1.29813	0.649063	−	0.760735i	$-0.275161\pi$
$977$	40.5755	1.29813	0.649063	+	0.760735i	$0.275161\pi$
$978$	0	0
$979$	−9.58098	−0.306210
$980$	0	0
$981$	26.6035	0.849386
$982$	0	0
$983$	−33.3622	−1.06409	−0.532044	−	0.846717i	$-0.678576\pi$
$983$	−33.3622	−1.06409	−0.532044	+	0.846717i	$0.678576\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.07864	−0.0979941
$988$	0	0
$989$	6.07864	0.193289
$990$	0	0
$991$	20.6082	0.654642	0.327321	−	0.944913i	$-0.393854\pi$
$991$	20.6082	0.654642	0.327321	+	0.944913i	$0.393854\pi$
$992$	0	0
$993$	−2.56312	−0.0813382
$994$	0	0
$995$	0	0
$996$	0	0
$997$	13.9027	0.440303	0.220152	−	0.975466i	$-0.429345\pi$
$997$	13.9027	0.440303	0.220152	+	0.975466i	$0.429345\pi$
$998$	0	0
$999$	−0.754701	−0.0238777

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bx.1.2		3
4.3	odd	2		2200.2.a.w.1.2	yes	3
5.2	odd	4		4400.2.b.bc.4049.4		6
5.3	odd	4		4400.2.b.bc.4049.3		6
5.4	even	2		4400.2.a.ca.1.2		3
20.3	even	4		2200.2.b.l.1849.4		6
20.7	even	4		2200.2.b.l.1849.3		6
20.19	odd	2		2200.2.a.t.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.t.1.2	✓	3	20.19	odd	2
2200.2.a.w.1.2	yes	3	4.3	odd	2
2200.2.b.l.1849.3		6	20.7	even	4
2200.2.b.l.1849.4		6	20.3	even	4
4400.2.a.bx.1.2		3	1.1	even	1	trivial
4400.2.a.ca.1.2		3	5.4	even	2
4400.2.b.bc.4049.3		6	5.3	odd	4
4400.2.b.bc.4049.4		6	5.2	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)

\(f(q)\)	\(=\)	\(q-1.16745 q^{3} -4.97196 q^{7} -1.63706 q^{9} +1.00000 q^{11} -0.665102 q^{13} +6.77647 q^{17} -1.00000 q^{19} +5.80451 q^{21} -2.16745 q^{23} +5.41353 q^{27} +7.97196 q^{29} +8.94392 q^{31} -1.16745 q^{33} -0.139410 q^{37} +0.776472 q^{39} -1.80451 q^{41} -2.80451 q^{43} -0.530387 q^{47} +17.7204 q^{49} -7.91119 q^{51} -6.30216 q^{53} +1.16745 q^{57} -11.4696 q^{59} +10.5810 q^{61} +8.13941 q^{63} -9.60902 q^{67} +2.53039 q^{69} +9.80921 q^{71} -7.02804 q^{73} -4.97196 q^{77} -5.50235 q^{79} -1.40884 q^{81} -13.5810 q^{83} -9.30686 q^{87} -9.58098 q^{89} +3.30686 q^{91} -10.4416 q^{93} -14.7812 q^{97} -1.63706 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{7} + 6 q^{9} + 3 q^{11} - 3 q^{13} - 3 q^{17} - 3 q^{19} + 6 q^{21} - 6 q^{23} - 18 q^{27} + 12 q^{29} + 3 q^{31} - 3 q^{33} + 12 q^{37} - 21 q^{39} + 6 q^{41} + 3 q^{43} - 12 q^{47}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^7 + 6 * q^9 + 3 * q^11 - 3 * q^13 - 3 * q^17 - 3 * q^19 + 6 * q^21 - 6 * q^23 - 18 * q^27 + 12 * q^29 + 3 * q^31 - 3 * q^33 + 12 * q^37 - 21 * q^39 + 6 * q^41 + 3 * q^43 - 12 * q^47 + 6 * q^49 + 9 * q^51 - 9 * q^53 + 3 * q^57 - 24 * q^59 - 3 * q^61 + 12 * q^63 - 6 * q^67 + 18 * q^69 + 15 * q^71 - 33 * q^73 - 3 * q^77 - 15 * q^79 + 27 * q^81 - 6 * q^83 - 15 * q^87 + 6 * q^89 - 3 * q^91 - 9 * q^93 - 18 * q^97 + 6 * q^99

Introduction

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