LMFDB - Embedded newform 4400.2.a.cb.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 = [4,0,-1,0,0,0,-1,0,7,0,-4,0,16] 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:	$0$
Dimension:	$4$
Coefficient field:	4.4.54764.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 9x^{2} + 3x + 2 $ x^4 - x^3 - 9x^2 + 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.36007$ 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)

$f(q)$	$=$	$q-3.36007 q^{3} -1.08258 q^{7} +8.29009 q^{9} -1.00000 q^{11} +4.00000 q^{13} -0.107866 q^{17} +6.61228 q^{19} +3.63756 q^{21} +5.97235 q^{23} -17.7751 q^{27} +7.80273 q^{29} -1.12492 q^{31} +3.36007 q^{33} -7.05494 q^{37} -13.4403 q^{39} +5.19045 q^{41} -5.52969 q^{43} +7.69486 q^{47} -5.82801 q^{49} +0.362439 q^{51} -4.77745 q^{53} -22.2177 q^{57} +0.677809 q^{59} +0.197271 q^{61} -8.97472 q^{63} +1.41737 q^{67} -20.0675 q^{69} +6.15020 q^{71} +6.16517 q^{73} +1.08258 q^{77} +9.35562 q^{79} +34.8553 q^{81} -13.7454 q^{83} -26.2177 q^{87} -1.29009 q^{89} -4.33034 q^{91} +3.77981 q^{93} -6.60782 q^{97} -8.29009 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{3} - q^{7} + 7 q^{9} - 4 q^{11} + 16 q^{13} + 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} + 15 q^{31} + q^{33} + 5 q^{37} - 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49}+ \cdots - 7 q^{99}+O(q^{100}) $ 4 * q - q^3 - q^7 + 7 * q^9 - 4 * q^11 + 16 * q^13 + 7 * q^17 + 9 * q^19 - 7 * q^21 - 6 * q^23 - 13 * q^27 + 3 * q^29 + 15 * q^31 + q^33 + 5 * q^37 - 4 * q^39 + 10 * q^41 - 8 * q^43 + 10 * q^47 + 9 * q^49 + 23 * q^51 + 5 * q^53 - 15 * q^57 - 6 * q^59 + 29 * q^61 - 40 * q^63 - 6 * q^67 - 30 * q^69 + q^71 + 18 * q^73 + q^77 + 20 * q^79 + 44 * q^81 - 26 * q^83 - 31 * q^87 + 21 * q^89 - 4 * q^91 + 25 * q^93 - 4 * q^97 - 7 * 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
$3$	−3.36007	−1.93994	−0.969969	−	0.243227i	$-0.921794\pi$
$3$	−3.36007	−1.93994	−0.969969	+	0.243227i	$0.921794\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.08258	−0.409178	−0.204589	−	0.978848i	$-0.565586\pi$
$7$	−1.08258	−0.409178	−0.204589	+	0.978848i	$0.565586\pi$
$8$	0	0
$9$	8.29009	2.76336
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.107866	−0.0261614	−0.0130807	−	0.999914i	$-0.504164\pi$
$17$	−0.107866	−0.0261614	−0.0130807	+	0.999914i	$0.504164\pi$
$18$	0	0
$19$	6.61228	1.51696	0.758480	−	0.651696i	$-0.225942\pi$
$19$	6.61228	1.51696	0.758480	+	0.651696i	$0.225942\pi$
$20$	0	0
$21$	3.63756	0.793781
$22$	0	0
$23$	5.97235	1.24532	0.622661	−	0.782492i	$-0.286052\pi$
$23$	5.97235	1.24532	0.622661	+	0.782492i	$0.286052\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−17.7751	−3.42082
$28$	0	0
$29$	7.80273	1.44893	0.724465	−	0.689311i	$-0.242087\pi$
$29$	7.80273	1.44893	0.724465	+	0.689311i	$0.242087\pi$
$30$	0	0
$31$	−1.12492	−0.202042	−0.101021	−	0.994884i	$-0.532211\pi$
$31$	−1.12492	−0.202042	−0.101021	+	0.994884i	$0.532211\pi$
$32$	0	0
$33$	3.36007	0.584914
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.05494	−1.15982	−0.579912	−	0.814679i	$-0.696913\pi$
$37$	−7.05494	−1.15982	−0.579912	+	0.814679i	$0.696913\pi$
$38$	0	0
$39$	−13.4403	−2.15217
$40$	0	0
$41$	5.19045	0.810612	0.405306	−	0.914181i	$-0.367165\pi$
$41$	5.19045	0.810612	0.405306	+	0.914181i	$0.367165\pi$
$42$	0	0
$43$	−5.52969	−0.843271	−0.421635	−	0.906766i	$-0.638544\pi$
$43$	−5.52969	−0.843271	−0.421635	+	0.906766i	$0.638544\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.69486	1.12241	0.561206	−	0.827676i	$-0.310338\pi$
$47$	7.69486	1.12241	0.561206	+	0.827676i	$0.310338\pi$
$48$	0	0
$49$	−5.82801	−0.832573
$50$	0	0
$51$	0.362439	0.0507516
$52$	0	0
$53$	−4.77745	−0.656233	−0.328116	−	0.944637i	$-0.606414\pi$
$53$	−4.77745	−0.656233	−0.328116	+	0.944637i	$0.606414\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−22.2177	−2.94281
$58$	0	0
$59$	0.677809	0.0882433	0.0441216	−	0.999026i	$-0.485951\pi$
$59$	0.677809	0.0882433	0.0441216	+	0.999026i	$0.485951\pi$
$60$	0	0
$61$	0.197271	0.0252579	0.0126290	−	0.999920i	$-0.495980\pi$
$61$	0.197271	0.0252579	0.0126290	+	0.999920i	$0.495980\pi$
$62$	0	0
$63$	−8.97472	−1.13071
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.41737	0.173160	0.0865799	−	0.996245i	$-0.472406\pi$
$67$	1.41737	0.173160	0.0865799	+	0.996245i	$0.472406\pi$
$68$	0	0
$69$	−20.0675	−2.41585
$70$	0	0
$71$	6.15020	0.729895	0.364947	−	0.931028i	$-0.381087\pi$
$71$	6.15020	0.729895	0.364947	+	0.931028i	$0.381087\pi$
$72$	0	0
$73$	6.16517	0.721578	0.360789	−	0.932647i	$-0.382507\pi$
$73$	6.16517	0.721578	0.360789	+	0.932647i	$0.382507\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.08258	0.123372
$78$	0	0
$79$	9.35562	1.05259	0.526295	−	0.850302i	$-0.323581\pi$
$79$	9.35562	1.05259	0.526295	+	0.850302i	$0.323581\pi$
$80$	0	0
$81$	34.8553	3.87281
$82$	0	0
$83$	−13.7454	−1.50876	−0.754378	−	0.656440i	$-0.772062\pi$
$83$	−13.7454	−1.50876	−0.754378	+	0.656440i	$0.772062\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−26.2177	−2.81084
$88$	0	0
$89$	−1.29009	−0.136749	−0.0683745	−	0.997660i	$-0.521781\pi$
$89$	−1.29009	−0.136749	−0.0683745	+	0.997660i	$0.521781\pi$
$90$	0	0
$91$	−4.33034	−0.453943
$92$	0	0
$93$	3.77981	0.391948
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.60782	−0.670923	−0.335461	−	0.942054i	$-0.608892\pi$
$97$	−6.60782	−0.670923	−0.335461	+	0.942054i	$0.608892\pi$
$98$	0	0
$99$	−8.29009	−0.833185
$100$	0	0
$101$	−15.4403	−1.53637	−0.768183	−	0.640230i	$-0.778839\pi$
$101$	−15.4403	−1.53637	−0.768183	+	0.640230i	$0.778839\pi$
$102$	0	0
$103$	5.74543	0.566114	0.283057	−	0.959103i	$-0.408651\pi$
$103$	5.74543	0.566114	0.283057	+	0.959103i	$0.408651\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−4.30514	−0.416193	−0.208097	−	0.978108i	$-0.566727\pi$
$107$	−4.30514	−0.416193	−0.208097	+	0.978108i	$0.566727\pi$
$108$	0	0
$109$	−2.33034	−0.223206	−0.111603	−	0.993753i	$-0.535598\pi$
$109$	−2.33034	−0.223206	−0.111603	+	0.993753i	$0.535598\pi$
$110$	0	0
$111$	23.7051	2.24999
$112$	0	0
$113$	10.9471	1.02981	0.514907	−	0.857246i	$-0.327827\pi$
$113$	10.9471	1.02981	0.514907	+	0.857246i	$0.327827\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	33.1604	3.06568
$118$	0	0
$119$	0.116774	0.0107047
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−17.4403	−1.57254
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−12.5802	−1.11631	−0.558155	−	0.829737i	$-0.688491\pi$
$127$	−12.5802	−1.11631	−0.558155	+	0.829737i	$0.688491\pi$
$128$	0	0
$129$	18.5802	1.63589
$130$	0	0
$131$	−15.2430	−1.33179	−0.665894	−	0.746046i	$-0.731950\pi$
$131$	−15.2430	−1.33179	−0.665894	+	0.746046i	$0.731950\pi$
$132$	0	0
$133$	−7.15835	−0.620707
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.6078	1.24803	0.624015	−	0.781412i	$-0.285500\pi$
$137$	14.6078	1.24803	0.624015	+	0.781412i	$0.285500\pi$
$138$	0	0
$139$	20.4150	1.73158	0.865789	−	0.500409i	$-0.166817\pi$
$139$	20.4150	1.73158	0.865789	+	0.500409i	$0.166817\pi$
$140$	0	0
$141$	−25.8553	−2.17741
$142$	0	0
$143$	−4.00000	−0.334497
$144$	0	0
$145$	0	0
$146$	0	0
$147$	19.5825	1.61514
$148$	0	0
$149$	−15.8874	−1.30155	−0.650773	−	0.759272i	$-0.725555\pi$
$149$	−15.8874	−1.30155	−0.650773	+	0.759272i	$0.725555\pi$
$150$	0	0
$151$	−6.58018	−0.535487	−0.267744	−	0.963490i	$-0.586278\pi$
$151$	−6.58018	−0.535487	−0.267744	+	0.963490i	$0.586278\pi$
$152$	0	0
$153$	−0.894222	−0.0722935
$154$	0	0
$155$	0	0
$156$	0	0
$157$	6.38535	0.509607	0.254803	−	0.966993i	$-0.417989\pi$
$157$	6.38535	0.509607	0.254803	+	0.966993i	$0.417989\pi$
$158$	0	0
$159$	16.0526	1.27305
$160$	0	0
$161$	−6.46557	−0.509559
$162$	0	0
$163$	18.3071	1.43393	0.716963	−	0.697111i	$-0.245532\pi$
$163$	18.3071	1.43393	0.716963	+	0.697111i	$0.245532\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.197271	−0.0152653	−0.00763263	−	0.999971i	$-0.502430\pi$
$167$	−0.197271	−0.0152653	−0.00763263	+	0.999971i	$0.502430\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	54.8164	4.19191
$172$	0	0
$173$	14.7201	1.11915	0.559576	−	0.828779i	$-0.310964\pi$
$173$	14.7201	1.11915	0.559576	+	0.828779i	$0.310964\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.27749	−0.171187
$178$	0	0
$179$	−11.8518	−0.885845	−0.442923	−	0.896560i	$-0.646058\pi$
$179$	−11.8518	−0.885845	−0.442923	+	0.896560i	$0.646058\pi$
$180$	0	0
$181$	−10.7625	−0.799969	−0.399984	−	0.916522i	$-0.630984\pi$
$181$	−10.7625	−0.799969	−0.399984	+	0.916522i	$0.630984\pi$
$182$	0	0
$183$	−0.662843	−0.0489988
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.107866	0.00788797
$188$	0	0
$189$	19.2430	1.39972
$190$	0	0
$191$	−1.70309	−0.123231	−0.0616157	−	0.998100i	$-0.519625\pi$
$191$	−1.70309	−0.123231	−0.0616157	+	0.998100i	$0.519625\pi$
$192$	0	0
$193$	10.8280	0.779417	0.389709	−	0.920938i	$-0.372576\pi$
$193$	10.8280	0.779417	0.389709	+	0.920938i	$0.372576\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−6.72015	−0.478791	−0.239395	−	0.970922i	$-0.576949\pi$
$197$	−6.72015	−0.478791	−0.239395	+	0.970922i	$0.576949\pi$
$198$	0	0
$199$	10.8280	0.767577	0.383789	−	0.923421i	$-0.374619\pi$
$199$	10.8280	0.767577	0.383789	+	0.923421i	$0.374619\pi$
$200$	0	0
$201$	−4.76248	−0.335920
$202$	0	0
$203$	−8.44711	−0.592871
$204$	0	0
$205$	0	0
$206$	0	0
$207$	49.5113	3.44127
$208$	0	0
$209$	−6.61228	−0.457381
$210$	0	0
$211$	−0.447111	−0.0307804	−0.0153902	−	0.999882i	$-0.504899\pi$
$211$	−0.447111	−0.0307804	−0.0153902	+	0.999882i	$0.504899\pi$
$212$	0	0
$213$	−20.6651	−1.41595
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.21782	0.0826710
$218$	0	0
$219$	−20.7154	−1.39982
$220$	0	0
$221$	−0.431465	−0.0290235
$222$	0	0
$223$	17.8577	1.19584	0.597922	−	0.801555i	$-0.295993\pi$
$223$	17.8577	1.19584	0.597922	+	0.801555i	$0.295993\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.31396	−0.0872107	−0.0436054	−	0.999049i	$-0.513884\pi$
$227$	−1.31396	−0.0872107	−0.0436054	+	0.999049i	$0.513884\pi$
$228$	0	0
$229$	4.84298	0.320033	0.160016	−	0.987114i	$-0.448845\pi$
$229$	4.84298	0.320033	0.160016	+	0.987114i	$0.448845\pi$
$230$	0	0
$231$	−3.63756	−0.239334
$232$	0	0
$233$	−20.3829	−1.33533	−0.667664	−	0.744462i	$-0.732706\pi$
$233$	−20.3829	−1.33533	−0.667664	+	0.744462i	$0.732706\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−31.4356	−2.04196
$238$	0	0
$239$	9.35562	0.605165	0.302582	−	0.953123i	$-0.402151\pi$
$239$	9.35562	0.605165	0.302582	+	0.953123i	$0.402151\pi$
$240$	0	0
$241$	22.3004	1.43650	0.718248	−	0.695788i	$-0.244944\pi$
$241$	22.3004	1.43650	0.718248	+	0.695788i	$0.244944\pi$
$242$	0	0
$243$	−63.7911	−4.09220
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26.4491	1.68292
$248$	0	0
$249$	46.1856	2.92690
$250$	0	0
$251$	10.9276	0.689747	0.344874	−	0.938649i	$-0.387922\pi$
$251$	10.9276	0.689747	0.344874	+	0.938649i	$0.387922\pi$
$252$	0	0
$253$	−5.97235	−0.375479
$254$	0	0
$255$	0	0
$256$	0	0
$257$	21.4403	1.33741	0.668704	−	0.743528i	$-0.266849\pi$
$257$	21.4403	1.33741	0.668704	+	0.743528i	$0.266849\pi$
$258$	0	0
$259$	7.63756	0.474575
$260$	0	0
$261$	64.6853	4.00392
$262$	0	0
$263$	6.52287	0.402218	0.201109	−	0.979569i	$-0.435546\pi$
$263$	6.52287	0.402218	0.201109	+	0.979569i	$0.435546\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	4.33479	0.265285
$268$	0	0
$269$	5.60546	0.341771	0.170885	−	0.985291i	$-0.445337\pi$
$269$	5.60546	0.341771	0.170885	+	0.985291i	$0.445337\pi$
$270$	0	0
$271$	28.9446	1.75826	0.879130	−	0.476582i	$-0.158124\pi$
$271$	28.9446	1.75826	0.879130	+	0.476582i	$0.158124\pi$
$272$	0	0
$273$	14.5502	0.880621
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.3850	1.52524	0.762618	−	0.646849i	$-0.223914\pi$
$277$	25.3850	1.52524	0.762618	+	0.646849i	$0.223914\pi$
$278$	0	0
$279$	−9.32569	−0.558314
$280$	0	0
$281$	−27.1604	−1.62025	−0.810125	−	0.586257i	$-0.800601\pi$
$281$	−27.1604	−1.62025	−0.810125	+	0.586257i	$0.800601\pi$
$282$	0	0
$283$	22.0205	1.30898	0.654490	−	0.756070i	$-0.272883\pi$
$283$	22.0205	1.30898	0.654490	+	0.756070i	$0.272883\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.61910	−0.331685
$288$	0	0
$289$	−16.9884	−0.999316
$290$	0	0
$291$	22.2028	1.30155
$292$	0	0
$293$	1.44029	0.0841427	0.0420713	−	0.999115i	$-0.486604\pi$
$293$	1.44029	0.0841427	0.0420713	+	0.999115i	$0.486604\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	17.7751	1.03141
$298$	0	0
$299$	23.8894	1.38156
$300$	0	0
$301$	5.98636	0.345048
$302$	0	0
$303$	51.8805	2.98046
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.80955	0.160349	0.0801747	−	0.996781i	$-0.474452\pi$
$307$	2.80955	0.160349	0.0801747	+	0.996781i	$0.474452\pi$
$308$	0	0
$309$	−19.3051	−1.09823
$310$	0	0
$311$	11.5529	0.655104	0.327552	−	0.944833i	$-0.393776\pi$
$311$	11.5529	0.655104	0.327552	+	0.944833i	$0.393776\pi$
$312$	0	0
$313$	26.1580	1.47854	0.739268	−	0.673411i	$-0.235171\pi$
$313$	26.1580	1.47854	0.739268	+	0.673411i	$0.235171\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.3805	1.65018	0.825088	−	0.565005i	$-0.191126\pi$
$317$	29.3805	1.65018	0.825088	+	0.565005i	$0.191126\pi$
$318$	0	0
$319$	−7.80273	−0.436869
$320$	0	0
$321$	14.4656	0.807390
$322$	0	0
$323$	−0.713242	−0.0396859
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.83010	0.433005
$328$	0	0
$329$	−8.33034	−0.459266
$330$	0	0
$331$	−12.2833	−0.675149	−0.337575	−	0.941299i	$-0.609607\pi$
$331$	−12.2833	−0.675149	−0.337575	+	0.941299i	$0.609607\pi$
$332$	0	0
$333$	−58.4860	−3.20502
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−21.3324	−1.16205	−0.581026	−	0.813885i	$-0.697348\pi$
$337$	−21.3324	−1.16205	−0.581026	+	0.813885i	$0.697348\pi$
$338$	0	0
$339$	−36.7829	−1.99778
$340$	0	0
$341$	1.12492	0.0609178
$342$	0	0
$343$	13.8874	0.749849
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−10.4150	−0.559107	−0.279553	−	0.960130i	$-0.590186\pi$
$347$	−10.4150	−0.559107	−0.279553	+	0.960130i	$0.590186\pi$
$348$	0	0
$349$	−13.4909	−0.722149	−0.361074	−	0.932537i	$-0.617590\pi$
$349$	−13.4909	−0.722149	−0.361074	+	0.932537i	$0.617590\pi$
$350$	0	0
$351$	−71.1003	−3.79505
$352$	0	0
$353$	−12.7177	−0.676895	−0.338447	−	0.940985i	$-0.609902\pi$
$353$	−12.7177	−0.676895	−0.338447	+	0.940985i	$0.609902\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−0.392370	−0.0207664
$358$	0	0
$359$	19.8212	1.04612	0.523061	−	0.852295i	$-0.324790\pi$
$359$	19.8212	1.04612	0.523061	+	0.852295i	$0.324790\pi$
$360$	0	0
$361$	24.7222	1.30117
$362$	0	0
$363$	−3.36007	−0.176358
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.3027	−0.537796	−0.268898	−	0.963169i	$-0.586659\pi$
$367$	−10.3027	−0.537796	−0.268898	+	0.963169i	$0.586659\pi$
$368$	0	0
$369$	43.0293	2.24002
$370$	0	0
$371$	5.17199	0.268516
$372$	0	0
$373$	23.3344	1.20821	0.604105	−	0.796904i	$-0.293531\pi$
$373$	23.3344	1.20821	0.604105	+	0.796904i	$0.293531\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.2109	1.60744
$378$	0	0
$379$	21.2074	1.08935	0.544676	−	0.838647i	$-0.316653\pi$
$379$	21.2074	1.08935	0.544676	+	0.838647i	$0.316653\pi$
$380$	0	0
$381$	42.2703	2.16557
$382$	0	0
$383$	0.972434	0.0496891	0.0248445	−	0.999691i	$-0.492091\pi$
$383$	0.972434	0.0496891	0.0248445	+	0.999691i	$0.492091\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−45.8417	−2.33026
$388$	0	0
$389$	2.76248	0.140063	0.0700317	−	0.997545i	$-0.477690\pi$
$389$	2.76248	0.140063	0.0700317	+	0.997545i	$0.477690\pi$
$390$	0	0
$391$	−0.644216	−0.0325794
$392$	0	0
$393$	51.2177	2.58359
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−11.2751	−0.565882	−0.282941	−	0.959137i	$-0.591310\pi$
$397$	−11.2751	−0.565882	−0.282941	+	0.959137i	$0.591310\pi$
$398$	0	0
$399$	24.0526	1.20413
$400$	0	0
$401$	−10.4471	−0.521704	−0.260852	−	0.965379i	$-0.584003\pi$
$401$	−10.4471	−0.521704	−0.260852	+	0.965379i	$0.584003\pi$
$402$	0	0
$403$	−4.49968	−0.224145
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.05494	0.349700
$408$	0	0
$409$	−1.27512	−0.0630507	−0.0315254	−	0.999503i	$-0.510037\pi$
$409$	−1.27512	−0.0630507	−0.0315254	+	0.999503i	$0.510037\pi$
$410$	0	0
$411$	−49.0834	−2.42110
$412$	0	0
$413$	−0.733786	−0.0361072
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−68.5959	−3.35916
$418$	0	0
$419$	−18.7795	−0.917436	−0.458718	−	0.888582i	$-0.651691\pi$
$419$	−18.7795	−0.917436	−0.458718	+	0.888582i	$0.651691\pi$
$420$	0	0
$421$	−1.27512	−0.0621457	−0.0310728	−	0.999517i	$-0.509892\pi$
$421$	−1.27512	−0.0621457	−0.0310728	+	0.999517i	$0.509892\pi$
$422$	0	0
$423$	63.7911	3.10163
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−0.213562	−0.0103350
$428$	0	0
$429$	13.4403	0.648903
$430$	0	0
$431$	−1.63556	−0.0787820	−0.0393910	−	0.999224i	$-0.512542\pi$
$431$	−1.63556	−0.0787820	−0.0393910	+	0.999224i	$0.512542\pi$
$432$	0	0
$433$	−26.4932	−1.27318	−0.636591	−	0.771201i	$-0.719656\pi$
$433$	−26.4932	−1.27318	−0.636591	+	0.771201i	$0.719656\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	39.4909	1.88910
$438$	0	0
$439$	−31.9358	−1.52421	−0.762106	−	0.647452i	$-0.775835\pi$
$439$	−31.9358	−1.52421	−0.762106	+	0.647452i	$0.775835\pi$
$440$	0	0
$441$	−48.3147	−2.30070
$442$	0	0
$443$	2.02765	0.0963365	0.0481682	−	0.998839i	$-0.484662\pi$
$443$	2.02765	0.0963365	0.0481682	+	0.998839i	$0.484662\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	53.3828	2.52492
$448$	0	0
$449$	−10.0675	−0.475116	−0.237558	−	0.971373i	$-0.576347\pi$
$449$	−10.0675	−0.475116	−0.237558	+	0.971373i	$0.576347\pi$
$450$	0	0
$451$	−5.19045	−0.244409
$452$	0	0
$453$	22.1099	1.03881
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.04839	−0.282932	−0.141466	−	0.989943i	$-0.545182\pi$
$457$	−6.04839	−0.282932	−0.141466	+	0.989943i	$0.545182\pi$
$458$	0	0
$459$	1.91733	0.0894934
$460$	0	0
$461$	31.8397	1.48292	0.741460	−	0.670997i	$-0.234134\pi$
$461$	31.8397	1.48292	0.741460	+	0.670997i	$0.234134\pi$
$462$	0	0
$463$	−30.2474	−1.40572	−0.702858	−	0.711330i	$-0.748093\pi$
$463$	−30.2474	−1.40572	−0.702858	+	0.711330i	$0.748093\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	31.3417	1.45032	0.725160	−	0.688580i	$-0.241766\pi$
$467$	31.3417	1.45032	0.725160	+	0.688580i	$0.241766\pi$
$468$	0	0
$469$	−1.53443	−0.0708533
$470$	0	0
$471$	−21.4553	−0.988606
$472$	0	0
$473$	5.52969	0.254256
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−39.6055	−1.81341
$478$	0	0
$479$	−6.59663	−0.301408	−0.150704	−	0.988579i	$-0.548154\pi$
$479$	−6.59663	−0.301408	−0.150704	+	0.988579i	$0.548154\pi$
$480$	0	0
$481$	−28.2197	−1.28671
$482$	0	0
$483$	21.7248	0.988512
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1.13343	0.0513605	0.0256802	−	0.999670i	$-0.491825\pi$
$487$	1.13343	0.0513605	0.0256802	+	0.999670i	$0.491825\pi$
$488$	0	0
$489$	−61.5133	−2.78173
$490$	0	0
$491$	−43.5569	−1.96570	−0.982848	−	0.184419i	$-0.940960\pi$
$491$	−43.5569	−1.96570	−0.982848	+	0.184419i	$0.940960\pi$
$492$	0	0
$493$	−0.841652	−0.0379061
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−6.65811	−0.298657
$498$	0	0
$499$	34.5502	1.54668	0.773341	−	0.633991i	$-0.218584\pi$
$499$	34.5502	1.54668	0.773341	+	0.633991i	$0.218584\pi$
$500$	0	0
$501$	0.662843	0.0296137
$502$	0	0
$503$	−5.92424	−0.264149	−0.132074	−	0.991240i	$-0.542164\pi$
$503$	−5.92424	−0.264149	−0.132074	+	0.991240i	$0.542164\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−10.0802	−0.447678
$508$	0	0
$509$	−42.3679	−1.87793	−0.938963	−	0.344018i	$-0.888212\pi$
$509$	−42.3679	−1.87793	−0.938963	+	0.344018i	$0.888212\pi$
$510$	0	0
$511$	−6.67431	−0.295254
$512$	0	0
$513$	−117.534	−5.18924
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−7.69486	−0.338420
$518$	0	0
$519$	−49.4608	−2.17109
$520$	0	0
$521$	21.2116	0.929297	0.464648	−	0.885495i	$-0.346181\pi$
$521$	21.2116	0.929297	0.464648	+	0.885495i	$0.346181\pi$
$522$	0	0
$523$	5.19045	0.226963	0.113481	−	0.993540i	$-0.463800\pi$
$523$	5.19045	0.226963	0.113481	+	0.993540i	$0.463800\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.121341	0.00528570
$528$	0	0
$529$	12.6690	0.550825
$530$	0	0
$531$	5.61910	0.243848
$532$	0	0
$533$	20.7618	0.899293
$534$	0	0
$535$	0	0
$536$	0	0
$537$	39.8229	1.71849
$538$	0	0
$539$	5.82801	0.251030
$540$	0	0
$541$	−24.0185	−1.03263	−0.516317	−	0.856397i	$-0.672697\pi$
$541$	−24.0185	−1.03263	−0.516317	+	0.856397i	$0.672697\pi$
$542$	0	0
$543$	36.1627	1.55189
$544$	0	0
$545$	0	0
$546$	0	0
$547$	12.8601	0.549859	0.274929	−	0.961464i	$-0.411346\pi$
$547$	12.8601	0.549859	0.274929	+	0.961464i	$0.411346\pi$
$548$	0	0
$549$	1.63539	0.0697968
$550$	0	0
$551$	51.5938	2.19797
$552$	0	0
$553$	−10.1282	−0.430697
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−31.3344	−1.32768	−0.663841	−	0.747874i	$-0.731075\pi$
$557$	−31.3344	−1.32768	−0.663841	+	0.747874i	$0.731075\pi$
$558$	0	0
$559$	−22.1188	−0.935525
$560$	0	0
$561$	−0.362439	−0.0153022
$562$	0	0
$563$	17.6901	0.745550	0.372775	−	0.927922i	$-0.378406\pi$
$563$	17.6901	0.745550	0.372775	+	0.927922i	$0.378406\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−37.7338	−1.58467
$568$	0	0
$569$	26.3004	1.10257	0.551285	−	0.834317i	$-0.314138\pi$
$569$	26.3004	1.10257	0.551285	+	0.834317i	$0.314138\pi$
$570$	0	0
$571$	35.9884	1.50607	0.753033	−	0.657983i	$-0.228590\pi$
$571$	35.9884	1.50607	0.753033	+	0.657983i	$0.228590\pi$
$572$	0	0
$573$	5.72251	0.239061
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−32.6031	−1.35728	−0.678642	−	0.734469i	$-0.737431\pi$
$577$	−32.6031	−1.35728	−0.678642	+	0.734469i	$0.737431\pi$
$578$	0	0
$579$	−36.3829	−1.51202
$580$	0	0
$581$	14.8806	0.617351
$582$	0	0
$583$	4.77745	0.197862
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−17.6287	−0.727612	−0.363806	−	0.931475i	$-0.618523\pi$
$587$	−17.6287	−0.727612	−0.363806	+	0.931475i	$0.618523\pi$
$588$	0	0
$589$	−7.43829	−0.306489
$590$	0	0
$591$	22.5802	0.928824
$592$	0	0
$593$	13.2246	0.543068	0.271534	−	0.962429i	$-0.412469\pi$
$593$	13.2246	0.543068	0.271534	+	0.962429i	$0.412469\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−36.3829	−1.48905
$598$	0	0
$599$	−37.2771	−1.52310	−0.761551	−	0.648105i	$-0.775562\pi$
$599$	−37.2771	−1.52310	−0.761551	+	0.648105i	$0.775562\pi$
$600$	0	0
$601$	17.0594	0.695867	0.347934	−	0.937519i	$-0.386883\pi$
$601$	17.0594	0.695867	0.347934	+	0.937519i	$0.386883\pi$
$602$	0	0
$603$	11.7502	0.478503
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−7.92624	−0.321716	−0.160858	−	0.986978i	$-0.551426\pi$
$607$	−7.92624	−0.321716	−0.160858	+	0.986978i	$0.551426\pi$
$608$	0	0
$609$	28.3829	1.15013
$610$	0	0
$611$	30.7795	1.24520
$612$	0	0
$613$	9.93596	0.401310	0.200655	−	0.979662i	$-0.435693\pi$
$613$	9.93596	0.401310	0.200655	+	0.979662i	$0.435693\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	38.1010	1.53389	0.766944	−	0.641715i	$-0.221777\pi$
$617$	38.1010	1.53389	0.766944	+	0.641715i	$0.221777\pi$
$618$	0	0
$619$	−5.70309	−0.229227	−0.114613	−	0.993410i	$-0.536563\pi$
$619$	−5.70309	−0.229227	−0.114613	+	0.993410i	$0.536563\pi$
$620$	0	0
$621$	−106.159	−4.26002
$622$	0	0
$623$	1.39663	0.0559548
$624$	0	0
$625$	0	0
$626$	0	0
$627$	22.2177	0.887291
$628$	0	0
$629$	0.760990	0.0303427
$630$	0	0
$631$	−17.3242	−0.689665	−0.344833	−	0.938664i	$-0.612064\pi$
$631$	−17.3242	−0.689665	−0.344833	+	0.938664i	$0.612064\pi$
$632$	0	0
$633$	1.50232	0.0597120
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−23.3120	−0.923657
$638$	0	0
$639$	50.9857	2.01696
$640$	0	0
$641$	−22.3523	−0.882863	−0.441431	−	0.897295i	$-0.645529\pi$
$641$	−22.3523	−0.882863	−0.441431	+	0.897295i	$0.645529\pi$
$642$	0	0
$643$	25.2911	0.997385	0.498693	−	0.866779i	$-0.333814\pi$
$643$	25.2911	0.997385	0.498693	+	0.866779i	$0.333814\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−22.4262	−0.881665	−0.440832	−	0.897589i	$-0.645317\pi$
$647$	−22.4262	−0.881665	−0.440832	+	0.897589i	$0.645317\pi$
$648$	0	0
$649$	−0.677809	−0.0266063
$650$	0	0
$651$	−4.09197	−0.160377
$652$	0	0
$653$	11.8391	0.463301	0.231650	−	0.972799i	$-0.425587\pi$
$653$	11.8391	0.463301	0.231650	+	0.972799i	$0.425587\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	51.1098	1.99398
$658$	0	0
$659$	15.7522	0.613617	0.306809	−	0.951771i	$-0.400739\pi$
$659$	15.7522	0.613617	0.306809	+	0.951771i	$0.400739\pi$
$660$	0	0
$661$	−7.15702	−0.278376	−0.139188	−	0.990266i	$-0.544449\pi$
$661$	−7.15702	−0.278376	−0.139188	+	0.990266i	$0.544449\pi$
$662$	0	0
$663$	1.44976	0.0563038
$664$	0	0
$665$	0	0
$666$	0	0
$667$	46.6006	1.80438
$668$	0	0
$669$	−60.0033	−2.31986
$670$	0	0
$671$	−0.197271	−0.00761555
$672$	0	0
$673$	−11.4976	−0.443200	−0.221600	−	0.975138i	$-0.571128\pi$
$673$	−11.4976	−0.443200	−0.221600	+	0.975138i	$0.571128\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−21.9953	−0.845347	−0.422673	−	0.906282i	$-0.638908\pi$
$677$	−21.9953	−0.845347	−0.422673	+	0.906282i	$0.638908\pi$
$678$	0	0
$679$	7.15353	0.274527
$680$	0	0
$681$	4.41501	0.169183
$682$	0	0
$683$	−0.468300	−0.0179190	−0.00895950	−	0.999960i	$-0.502852\pi$
$683$	−0.468300	−0.0179190	−0.00895950	+	0.999960i	$0.502852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−16.2728	−0.620844
$688$	0	0
$689$	−19.1098	−0.728025
$690$	0	0
$691$	36.9140	1.40428	0.702138	−	0.712041i	$-0.252229\pi$
$691$	36.9140	1.40428	0.702138	+	0.712041i	$0.252229\pi$
$692$	0	0
$693$	8.97472	0.340921
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−0.559875	−0.0212068
$698$	0	0
$699$	68.4880	2.59046
$700$	0	0
$701$	18.6628	0.704886	0.352443	−	0.935833i	$-0.385351\pi$
$701$	18.6628	0.704886	0.352443	+	0.935833i	$0.385351\pi$
$702$	0	0
$703$	−46.6492	−1.75941
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.7154	0.628648
$708$	0	0
$709$	6.66135	0.250172	0.125086	−	0.992146i	$-0.460079\pi$
$709$	6.66135	0.250172	0.125086	+	0.992146i	$0.460079\pi$
$710$	0	0
$711$	77.5589	2.90869
$712$	0	0
$713$	−6.71842	−0.251607
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−31.4356	−1.17398
$718$	0	0
$719$	3.11128	0.116031	0.0580156	−	0.998316i	$-0.481523\pi$
$719$	3.11128	0.116031	0.0580156	+	0.998316i	$0.481523\pi$
$720$	0	0
$721$	−6.21991	−0.231641
$722$	0	0
$723$	−74.9310	−2.78671
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−33.9540	−1.25928	−0.629642	−	0.776886i	$-0.716798\pi$
$727$	−33.9540	−1.25928	−0.629642	+	0.776886i	$0.716798\pi$
$728$	0	0
$729$	109.777	4.06581
$730$	0	0
$731$	0.596468	0.0220612
$732$	0	0
$733$	21.0457	0.777342	0.388671	−	0.921377i	$-0.372934\pi$
$733$	21.0457	0.777342	0.388671	+	0.921377i	$0.372934\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.41737	−0.0522097
$738$	0	0
$739$	17.0253	0.626285	0.313143	−	0.949706i	$-0.398618\pi$
$739$	17.0253	0.626285	0.313143	+	0.949706i	$0.398618\pi$
$740$	0	0
$741$	−88.8710	−3.26476
$742$	0	0
$743$	−23.7563	−0.871536	−0.435768	−	0.900059i	$-0.643523\pi$
$743$	−23.7563	−0.871536	−0.435768	+	0.900059i	$0.643523\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−113.951	−4.16924
$748$	0	0
$749$	4.66067	0.170297
$750$	0	0
$751$	44.8654	1.63716	0.818582	−	0.574390i	$-0.194761\pi$
$751$	44.8654	1.63716	0.818582	+	0.574390i	$0.194761\pi$
$752$	0	0
$753$	−36.7177	−1.33807
$754$	0	0
$755$	0	0
$756$	0	0
$757$	41.3761	1.50384	0.751920	−	0.659255i	$-0.229128\pi$
$757$	41.3761	1.50384	0.751920	+	0.659255i	$0.229128\pi$
$758$	0	0
$759$	20.0675	0.728405
$760$	0	0
$761$	16.3002	0.590883	0.295442	−	0.955361i	$-0.404533\pi$
$761$	16.3002	0.590883	0.295442	+	0.955361i	$0.404533\pi$
$762$	0	0
$763$	2.52279	0.0913310
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.71124	0.0978971
$768$	0	0
$769$	41.1262	1.48305	0.741525	−	0.670925i	$-0.234103\pi$
$769$	41.1262	1.48305	0.741525	+	0.670925i	$0.234103\pi$
$770$	0	0
$771$	−72.0409	−2.59449
$772$	0	0
$773$	9.77280	0.351503	0.175752	−	0.984435i	$-0.443764\pi$
$773$	9.77280	0.351503	0.175752	+	0.984435i	$0.443764\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−25.6628	−0.920646
$778$	0	0
$779$	34.3207	1.22967
$780$	0	0
$781$	−6.15020	−0.220072
$782$	0	0
$783$	−138.694	−4.95652
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5.25466	−0.187308	−0.0936541	−	0.995605i	$-0.529855\pi$
$787$	−5.25466	−0.187308	−0.0936541	+	0.995605i	$0.529855\pi$
$788$	0	0
$789$	−21.9173	−0.780278
$790$	0	0
$791$	−11.8511	−0.421377
$792$	0	0
$793$	0.789082	0.0280211
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−14.2133	−0.503460	−0.251730	−	0.967797i	$-0.581000\pi$
$797$	−14.2133	−0.503460	−0.251730	+	0.967797i	$0.581000\pi$
$798$	0	0
$799$	−0.830017	−0.0293639
$800$	0	0
$801$	−10.6949	−0.377887
$802$	0	0
$803$	−6.16517	−0.217564
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−18.8347	−0.663015
$808$	0	0
$809$	−12.8641	−0.452279	−0.226139	−	0.974095i	$-0.572610\pi$
$809$	−12.8641	−0.452279	−0.226139	+	0.974095i	$0.572610\pi$
$810$	0	0
$811$	−4.31605	−0.151557	−0.0757785	−	0.997125i	$-0.524144\pi$
$811$	−4.31605	−0.151557	−0.0757785	+	0.997125i	$0.524144\pi$
$812$	0	0
$813$	−97.2560	−3.41092
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−36.5639	−1.27921
$818$	0	0
$819$	−35.8989	−1.25441
$820$	0	0
$821$	−42.9994	−1.50069	−0.750344	−	0.661048i	$-0.770112\pi$
$821$	−42.9994	−1.50069	−0.750344	+	0.661048i	$0.770112\pi$
$822$	0	0
$823$	48.1834	1.67957	0.839783	−	0.542922i	$-0.182682\pi$
$823$	48.1834	1.67957	0.839783	+	0.542922i	$0.182682\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.2868	−1.12272	−0.561360	−	0.827571i	$-0.689722\pi$
$827$	−32.2868	−1.12272	−0.561360	+	0.827571i	$0.689722\pi$
$828$	0	0
$829$	54.9345	1.90795	0.953977	−	0.299881i	$-0.0969470\pi$
$829$	54.9345	1.90795	0.953977	+	0.299881i	$0.0969470\pi$
$830$	0	0
$831$	−85.2954	−2.95887
$832$	0	0
$833$	0.628646	0.0217813
$834$	0	0
$835$	0	0
$836$	0	0
$837$	19.9955	0.691147
$838$	0	0
$839$	−11.2281	−0.387635	−0.193818	−	0.981038i	$-0.562087\pi$
$839$	−11.2281	−0.387635	−0.193818	+	0.981038i	$0.562087\pi$
$840$	0	0
$841$	31.8826	1.09940
$842$	0	0
$843$	91.2608	3.14319
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.08258	−0.0371980
$848$	0	0
$849$	−73.9904	−2.53934
$850$	0	0
$851$	−42.1346	−1.44435
$852$	0	0
$853$	2.40328	0.0822869	0.0411434	−	0.999153i	$-0.486900\pi$
$853$	2.40328	0.0822869	0.0411434	+	0.999153i	$0.486900\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.8280	1.18970	0.594851	−	0.803836i	$-0.297211\pi$
$857$	34.8280	1.18970	0.594851	+	0.803836i	$0.297211\pi$
$858$	0	0
$859$	−27.0627	−0.923368	−0.461684	−	0.887044i	$-0.652755\pi$
$859$	−27.0627	−0.923368	−0.461684	+	0.887044i	$0.652755\pi$
$860$	0	0
$861$	18.8806	0.643448
$862$	0	0
$863$	−40.0157	−1.36215	−0.681076	−	0.732213i	$-0.738488\pi$
$863$	−40.0157	−1.36215	−0.681076	+	0.732213i	$0.738488\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	57.0821	1.93861
$868$	0	0
$869$	−9.35562	−0.317368
$870$	0	0
$871$	5.66950	0.192104
$872$	0	0
$873$	−54.7795	−1.85400
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−18.3167	−0.618511	−0.309255	−	0.950979i	$-0.600080\pi$
$877$	−18.3167	−0.618511	−0.309255	+	0.950979i	$0.600080\pi$
$878$	0	0
$879$	−4.83948	−0.163232
$880$	0	0
$881$	11.8560	0.399438	0.199719	−	0.979853i	$-0.435997\pi$
$881$	11.8560	0.399438	0.199719	+	0.979853i	$0.435997\pi$
$882$	0	0
$883$	−15.2478	−0.513128	−0.256564	−	0.966527i	$-0.582590\pi$
$883$	−15.2478	−0.513128	−0.256564	+	0.966527i	$0.582590\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−3.29631	−0.110679	−0.0553397	−	0.998468i	$-0.517624\pi$
$887$	−3.29631	−0.110679	−0.0553397	+	0.998468i	$0.517624\pi$
$888$	0	0
$889$	13.6191	0.456770
$890$	0	0
$891$	−34.8553	−1.16770
$892$	0	0
$893$	50.8806	1.70265
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−80.2701	−2.68014
$898$	0	0
$899$	−8.77745	−0.292744
$900$	0	0
$901$	0.515326	0.0171680
$902$	0	0
$903$	−20.1146	−0.669372
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.3577	0.543149	0.271574	−	0.962417i	$-0.412456\pi$
$907$	16.3577	0.543149	0.271574	+	0.962417i	$0.412456\pi$
$908$	0	0
$909$	−128.001	−4.24554
$910$	0	0
$911$	8.34598	0.276515	0.138257	−	0.990396i	$-0.455850\pi$
$911$	8.34598	0.276515	0.138257	+	0.990396i	$0.455850\pi$
$912$	0	0
$913$	13.7454	0.454907
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.5019	0.544939
$918$	0	0
$919$	12.9611	0.427546	0.213773	−	0.976883i	$-0.431425\pi$
$919$	12.9611	0.427546	0.213773	+	0.976883i	$0.431425\pi$
$920$	0	0
$921$	−9.44029	−0.311068
$922$	0	0
$923$	24.6008	0.809746
$924$	0	0
$925$	0	0
$926$	0	0
$927$	47.6301	1.56438
$928$	0	0
$929$	37.2635	1.22258	0.611288	−	0.791408i	$-0.290652\pi$
$929$	37.2635	1.22258	0.611288	+	0.791408i	$0.290652\pi$
$930$	0	0
$931$	−38.5364	−1.26298
$932$	0	0
$933$	−38.8185	−1.27086
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−4.28395	−0.139950	−0.0699752	−	0.997549i	$-0.522292\pi$
$937$	−4.28395	−0.139950	−0.0699752	+	0.997549i	$0.522292\pi$
$938$	0	0
$939$	−87.8927	−2.86827
$940$	0	0
$941$	23.2089	0.756589	0.378294	−	0.925685i	$-0.376511\pi$
$941$	23.2089	0.756589	0.378294	+	0.925685i	$0.376511\pi$
$942$	0	0
$943$	30.9992	1.00947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−20.8646	−0.678007	−0.339004	−	0.940785i	$-0.610090\pi$
$947$	−20.8646	−0.678007	−0.339004	+	0.940785i	$0.610090\pi$
$948$	0	0
$949$	24.6607	0.800519
$950$	0	0
$951$	−98.7207	−3.20124
$952$	0	0
$953$	45.0061	1.45789	0.728945	−	0.684572i	$-0.240011\pi$
$953$	45.0061	1.45789	0.728945	+	0.684572i	$0.240011\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	26.2177	0.847499
$958$	0	0
$959$	−15.8142	−0.510667
$960$	0	0
$961$	−29.7346	−0.959179
$962$	0	0
$963$	−35.6900	−1.15009
$964$	0	0
$965$	0	0
$966$	0	0
$967$	21.9679	0.706440	0.353220	−	0.935540i	$-0.385087\pi$
$967$	21.9679	0.706440	0.353220	+	0.935540i	$0.385087\pi$
$968$	0	0
$969$	2.39655	0.0769882
$970$	0	0
$971$	16.5837	0.532195	0.266098	−	0.963946i	$-0.414266\pi$
$971$	16.5837	0.532195	0.266098	+	0.963946i	$0.414266\pi$
$972$	0	0
$973$	−22.1010	−0.708524
$974$	0	0
$975$	0	0
$976$	0	0
$977$	19.8189	0.634063	0.317032	−	0.948415i	$-0.397314\pi$
$977$	19.8189	0.634063	0.317032	+	0.948415i	$0.397314\pi$
$978$	0	0
$979$	1.29009	0.0412314
$980$	0	0
$981$	−19.3187	−0.616798
$982$	0	0
$983$	13.9634	0.445365	0.222682	−	0.974891i	$-0.428519\pi$
$983$	13.9634	0.445365	0.222682	+	0.974891i	$0.428519\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	27.9905	0.890949
$988$	0	0
$989$	−33.0253	−1.05014
$990$	0	0
$991$	−4.10113	−0.130277	−0.0651383	−	0.997876i	$-0.520749\pi$
$991$	−4.10113	−0.130277	−0.0651383	+	0.997876i	$0.520749\pi$
$992$	0	0
$993$	41.2727	1.30975
$994$	0	0
$995$	0	0
$996$	0	0
$997$	33.3208	1.05528	0.527640	−	0.849468i	$-0.323077\pi$
$997$	33.3208	1.05528	0.527640	+	0.849468i	$0.323077\pi$
$998$	0	0
$999$	125.402	3.96755

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.cb.1.1		4
4.3	odd	2		2200.2.a.y.1.4		4
5.2	odd	4		880.2.b.j.529.8		8
5.3	odd	4		880.2.b.j.529.1		8
5.4	even	2		4400.2.a.ce.1.4		4
20.3	even	4		440.2.b.d.89.8	yes	8
20.7	even	4		440.2.b.d.89.1	✓	8
20.19	odd	2		2200.2.a.x.1.1		4
60.23	odd	4		3960.2.d.f.3169.5		8
60.47	odd	4		3960.2.d.f.3169.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.b.d.89.1	✓	8	20.7	even	4
440.2.b.d.89.8	yes	8	20.3	even	4
880.2.b.j.529.1		8	5.3	odd	4
880.2.b.j.529.8		8	5.2	odd	4
2200.2.a.x.1.1		4	20.19	odd	2
2200.2.a.y.1.4		4	4.3	odd	2
3960.2.d.f.3169.5		8	60.23	odd	4
3960.2.d.f.3169.6		8	60.47	odd	4
4400.2.a.cb.1.1		4	1.1	even	1	trivial
4400.2.a.ce.1.4		4	5.4	even	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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q-3.36007 q^{3} -1.08258 q^{7} +8.29009 q^{9} -1.00000 q^{11} +4.00000 q^{13} -0.107866 q^{17} +6.61228 q^{19} +3.63756 q^{21} +5.97235 q^{23} -17.7751 q^{27} +7.80273 q^{29} -1.12492 q^{31} +3.36007 q^{33} -7.05494 q^{37} -13.4403 q^{39} +5.19045 q^{41} -5.52969 q^{43} +7.69486 q^{47} -5.82801 q^{49} +0.362439 q^{51} -4.77745 q^{53} -22.2177 q^{57} +0.677809 q^{59} +0.197271 q^{61} -8.97472 q^{63} +1.41737 q^{67} -20.0675 q^{69} +6.15020 q^{71} +6.16517 q^{73} +1.08258 q^{77} +9.35562 q^{79} +34.8553 q^{81} -13.7454 q^{83} -26.2177 q^{87} -1.29009 q^{89} -4.33034 q^{91} +3.77981 q^{93} -6.60782 q^{97} -8.29009 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{3} - q^{7} + 7 q^{9} - 4 q^{11} + 16 q^{13} + 7 q^{17} + 9 q^{19} - 7 q^{21} - 6 q^{23} - 13 q^{27} + 3 q^{29} + 15 q^{31} + q^{33} + 5 q^{37} - 4 q^{39} + 10 q^{41} - 8 q^{43} + 10 q^{47} + 9 q^{49}+ \cdots - 7 q^{99}+O(q^{100}) \) 4 * q - q^3 - q^7 + 7 * q^9 - 4 * q^11 + 16 * q^13 + 7 * q^17 + 9 * q^19 - 7 * q^21 - 6 * q^23 - 13 * q^27 + 3 * q^29 + 15 * q^31 + q^33 + 5 * q^37 - 4 * q^39 + 10 * q^41 - 8 * q^43 + 10 * q^47 + 9 * q^49 + 23 * q^51 + 5 * q^53 - 15 * q^57 - 6 * q^59 + 29 * q^61 - 40 * q^63 - 6 * q^67 - 30 * q^69 + q^71 + 18 * q^73 + q^77 + 20 * q^79 + 44 * q^81 - 26 * q^83 - 31 * q^87 + 21 * q^89 - 4 * q^91 + 25 * q^93 - 4 * q^97 - 7 * q^99

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