LMFDB - Embedded newform 4400.2.a.bw.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 = [2,0,3,0,0,0,5,0,5,0,-2,0,4] 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:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			$-1.30278$ 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)

$f(q)$	$=$	$q-0.302776 q^{3} +0.697224 q^{7} -2.90833 q^{9} -1.00000 q^{11} -1.60555 q^{13} -4.30278 q^{17} +1.00000 q^{19} -0.211103 q^{21} +3.90833 q^{23} +1.78890 q^{27} -1.69722 q^{29} -1.60555 q^{31} +0.302776 q^{33} -7.60555 q^{37} +0.486122 q^{39} +8.21110 q^{41} +7.21110 q^{43} +5.60555 q^{47} -6.51388 q^{49} +1.30278 q^{51} +6.51388 q^{53} -0.302776 q^{57} +5.60555 q^{59} +3.30278 q^{61} -2.02776 q^{63} +8.00000 q^{67} -1.18335 q^{69} -2.60555 q^{71} +1.90833 q^{73} -0.697224 q^{77} -6.30278 q^{79} +8.18335 q^{81} +6.90833 q^{83} +0.513878 q^{87} -5.09167 q^{89} -1.11943 q^{91} +0.486122 q^{93} +7.11943 q^{97} +2.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{3} + 5 q^{7} + 5 q^{9} - 2 q^{11} + 4 q^{13} - 5 q^{17} + 2 q^{19} + 14 q^{21} - 3 q^{23} + 18 q^{27} - 7 q^{29} + 4 q^{31} - 3 q^{33} - 8 q^{37} + 19 q^{39} + 2 q^{41} + 4 q^{47} + 5 q^{49}+ \cdots - 5 q^{99}+O(q^{100}) $ 2 * q + 3 * q^3 + 5 * q^7 + 5 * q^9 - 2 * q^11 + 4 * q^13 - 5 * q^17 + 2 * q^19 + 14 * q^21 - 3 * q^23 + 18 * q^27 - 7 * q^29 + 4 * q^31 - 3 * q^33 - 8 * q^37 + 19 * q^39 + 2 * q^41 + 4 * q^47 + 5 * q^49 - q^51 - 5 * q^53 + 3 * q^57 + 4 * q^59 + 3 * q^61 + 32 * q^63 + 16 * q^67 - 24 * q^69 + 2 * q^71 - 7 * q^73 - 5 * q^77 - 9 * q^79 + 38 * q^81 + 3 * q^83 - 17 * q^87 - 21 * q^89 + 23 * q^91 + 19 * q^93 - 11 * 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
$3$	−0.302776	−0.174808	−0.0874038	−	0.996173i	$-0.527857\pi$
$3$	−0.302776	−0.174808	−0.0874038	+	0.996173i	$0.527857\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0.697224	0.263526	0.131763	−	0.991281i	$-0.457936\pi$
$7$	0.697224	0.263526	0.131763	+	0.991281i	$0.457936\pi$
$8$	0	0
$9$	−2.90833	−0.969442
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.60555	−0.445300	−0.222650	−	0.974898i	$-0.571471\pi$
$13$	−1.60555	−0.445300	−0.222650	+	0.974898i	$0.571471\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.30278	−1.04358	−0.521788	−	0.853075i	$-0.674735\pi$
$17$	−4.30278	−1.04358	−0.521788	+	0.853075i	$0.674735\pi$
$18$	0	0
$19$	1.00000	0.229416	0.114708	−	0.993399i	$-0.463407\pi$
$19$	1.00000	0.229416	0.114708	+	0.993399i	$0.463407\pi$
$20$	0	0
$21$	−0.211103	−0.0460664
$22$	0	0
$23$	3.90833	0.814942	0.407471	−	0.913218i	$-0.366411\pi$
$23$	3.90833	0.814942	0.407471	+	0.913218i	$0.366411\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.78890	0.344273
$28$	0	0
$29$	−1.69722	−0.315167	−0.157583	−	0.987506i	$-0.550370\pi$
$29$	−1.69722	−0.315167	−0.157583	+	0.987506i	$0.550370\pi$
$30$	0	0
$31$	−1.60555	−0.288366	−0.144183	−	0.989551i	$-0.546055\pi$
$31$	−1.60555	−0.288366	−0.144183	+	0.989551i	$0.546055\pi$
$32$	0	0
$33$	0.302776	0.0527065
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−7.60555	−1.25034	−0.625172	−	0.780487i	$-0.714971\pi$
$37$	−7.60555	−1.25034	−0.625172	+	0.780487i	$0.714971\pi$
$38$	0	0
$39$	0.486122	0.0778418
$40$	0	0
$41$	8.21110	1.28236	0.641179	−	0.767391i	$-0.278445\pi$
$41$	8.21110	1.28236	0.641179	+	0.767391i	$0.278445\pi$
$42$	0	0
$43$	7.21110	1.09968	0.549841	−	0.835269i	$-0.314688\pi$
$43$	7.21110	1.09968	0.549841	+	0.835269i	$0.314688\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.60555	0.817654	0.408827	−	0.912612i	$-0.365938\pi$
$47$	5.60555	0.817654	0.408827	+	0.912612i	$0.365938\pi$
$48$	0	0
$49$	−6.51388	−0.930554
$50$	0	0
$51$	1.30278	0.182425
$52$	0	0
$53$	6.51388	0.894750	0.447375	−	0.894346i	$-0.352359\pi$
$53$	6.51388	0.894750	0.447375	+	0.894346i	$0.352359\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−0.302776	−0.0401036
$58$	0	0
$59$	5.60555	0.729781	0.364890	−	0.931051i	$-0.381106\pi$
$59$	5.60555	0.729781	0.364890	+	0.931051i	$0.381106\pi$
$60$	0	0
$61$	3.30278	0.422877	0.211439	−	0.977391i	$-0.432185\pi$
$61$	3.30278	0.422877	0.211439	+	0.977391i	$0.432185\pi$
$62$	0	0
$63$	−2.02776	−0.255473
$64$	0	0
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−1.18335	−0.142458
$70$	0	0
$71$	−2.60555	−0.309222	−0.154611	−	0.987975i	$-0.549412\pi$
$71$	−2.60555	−0.309222	−0.154611	+	0.987975i	$0.549412\pi$
$72$	0	0
$73$	1.90833	0.223353	0.111676	−	0.993745i	$-0.464378\pi$
$73$	1.90833	0.223353	0.111676	+	0.993745i	$0.464378\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−0.697224	−0.0794561
$78$	0	0
$79$	−6.30278	−0.709118	−0.354559	−	0.935034i	$-0.615369\pi$
$79$	−6.30278	−0.709118	−0.354559	+	0.935034i	$0.615369\pi$
$80$	0	0
$81$	8.18335	0.909261
$82$	0	0
$83$	6.90833	0.758287	0.379144	−	0.925338i	$-0.376219\pi$
$83$	6.90833	0.758287	0.379144	+	0.925338i	$0.376219\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0.513878	0.0550935
$88$	0	0
$89$	−5.09167	−0.539716	−0.269858	−	0.962900i	$-0.586977\pi$
$89$	−5.09167	−0.539716	−0.269858	+	0.962900i	$0.586977\pi$
$90$	0	0
$91$	−1.11943	−0.117348
$92$	0	0
$93$	0.486122	0.0504085
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.11943	0.722869	0.361434	−	0.932398i	$-0.382287\pi$
$97$	7.11943	0.722869	0.361434	+	0.932398i	$0.382287\pi$
$98$	0	0
$99$	2.90833	0.292298
$100$	0	0
$101$	−6.51388	−0.648155	−0.324078	−	0.946031i	$-0.605054\pi$
$101$	−6.51388	−0.648155	−0.324078	+	0.946031i	$0.605054\pi$
$102$	0	0
$103$	12.3028	1.21223	0.606114	−	0.795378i	$-0.292727\pi$
$103$	12.3028	1.21223	0.606114	+	0.795378i	$0.292727\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.21110	0.213755	0.106878	−	0.994272i	$-0.465915\pi$
$107$	2.21110	0.213755	0.106878	+	0.994272i	$0.465915\pi$
$108$	0	0
$109$	8.90833	0.853263	0.426631	−	0.904426i	$-0.359700\pi$
$109$	8.90833	0.853263	0.426631	+	0.904426i	$0.359700\pi$
$110$	0	0
$111$	2.30278	0.218570
$112$	0	0
$113$	5.60555	0.527326	0.263663	−	0.964615i	$-0.415069\pi$
$113$	5.60555	0.527326	0.263663	+	0.964615i	$0.415069\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.66947	0.431692
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−2.48612	−0.224166
$124$	0	0
$125$	0	0
$126$	0	0
$127$	14.9083	1.32290	0.661450	−	0.749989i	$-0.269941\pi$
$127$	14.9083	1.32290	0.661450	+	0.749989i	$0.269941\pi$
$128$	0	0
$129$	−2.18335	−0.192233
$130$	0	0
$131$	4.69722	0.410398	0.205199	−	0.978720i	$-0.434216\pi$
$131$	4.69722	0.410398	0.205199	+	0.978720i	$0.434216\pi$
$132$	0	0
$133$	0.697224	0.0604570
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.908327	0.0776036	0.0388018	−	0.999247i	$-0.487646\pi$
$137$	0.908327	0.0776036	0.0388018	+	0.999247i	$0.487646\pi$
$138$	0	0
$139$	−16.2111	−1.37501	−0.687504	−	0.726181i	$-0.741294\pi$
$139$	−16.2111	−1.37501	−0.687504	+	0.726181i	$0.741294\pi$
$140$	0	0
$141$	−1.69722	−0.142932
$142$	0	0
$143$	1.60555	0.134263
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.97224	0.162668
$148$	0	0
$149$	−12.0000	−0.983078	−0.491539	−	0.870855i	$-0.663566\pi$
$149$	−12.0000	−0.983078	−0.491539	+	0.870855i	$0.663566\pi$
$150$	0	0
$151$	14.8167	1.20576	0.602881	−	0.797831i	$-0.294019\pi$
$151$	14.8167	1.20576	0.602881	+	0.797831i	$0.294019\pi$
$152$	0	0
$153$	12.5139	1.01169
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.2111	1.21398	0.606989	−	0.794710i	$-0.292377\pi$
$157$	15.2111	1.21398	0.606989	+	0.794710i	$0.292377\pi$
$158$	0	0
$159$	−1.97224	−0.156409
$160$	0	0
$161$	2.72498	0.214759
$162$	0	0
$163$	3.30278	0.258693	0.129347	−	0.991599i	$-0.458712\pi$
$163$	3.30278	0.258693	0.129347	+	0.991599i	$0.458712\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−24.6333	−1.90618	−0.953091	−	0.302683i	$-0.902118\pi$
$167$	−24.6333	−1.90618	−0.953091	+	0.302683i	$0.902118\pi$
$168$	0	0
$169$	−10.4222	−0.801708
$170$	0	0
$171$	−2.90833	−0.222405
$172$	0	0
$173$	9.78890	0.744236	0.372118	−	0.928185i	$-0.378632\pi$
$173$	9.78890	0.744236	0.372118	+	0.928185i	$0.378632\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1.69722	−0.127571
$178$	0	0
$179$	−8.09167	−0.604800	−0.302400	−	0.953181i	$-0.597788\pi$
$179$	−8.09167	−0.604800	−0.302400	+	0.953181i	$0.597788\pi$
$180$	0	0
$181$	−7.90833	−0.587821	−0.293911	−	0.955833i	$-0.594957\pi$
$181$	−7.90833	−0.587821	−0.293911	+	0.955833i	$0.594957\pi$
$182$	0	0
$183$	−1.00000	−0.0739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	4.30278	0.314650
$188$	0	0
$189$	1.24726	0.0907250
$190$	0	0
$191$	−12.9083	−0.934014	−0.467007	−	0.884254i	$-0.654668\pi$
$191$	−12.9083	−0.934014	−0.467007	+	0.884254i	$0.654668\pi$
$192$	0	0
$193$	2.42221	0.174354	0.0871771	−	0.996193i	$-0.472215\pi$
$193$	2.42221	0.174354	0.0871771	+	0.996193i	$0.472215\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.9083	1.56090	0.780452	−	0.625216i	$-0.214989\pi$
$197$	21.9083	1.56090	0.780452	+	0.625216i	$0.214989\pi$
$198$	0	0
$199$	−2.90833	−0.206166	−0.103083	−	0.994673i	$-0.532871\pi$
$199$	−2.90833	−0.206166	−0.103083	+	0.994673i	$0.532871\pi$
$200$	0	0
$201$	−2.42221	−0.170849
$202$	0	0
$203$	−1.18335	−0.0830546
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11.3667	−0.790040
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	21.6056	1.48739	0.743694	−	0.668520i	$-0.233072\pi$
$211$	21.6056	1.48739	0.743694	+	0.668520i	$0.233072\pi$
$212$	0	0
$213$	0.788897	0.0540544
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−1.11943	−0.0759918
$218$	0	0
$219$	−0.577795	−0.0390438
$220$	0	0
$221$	6.90833	0.464704
$222$	0	0
$223$	13.6056	0.911095	0.455548	−	0.890211i	$-0.349443\pi$
$223$	13.6056	0.911095	0.455548	+	0.890211i	$0.349443\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−25.9361	−1.72144	−0.860719	−	0.509080i	$-0.829986\pi$
$227$	−25.9361	−1.72144	−0.860719	+	0.509080i	$0.829986\pi$
$228$	0	0
$229$	26.1194	1.72602	0.863010	−	0.505186i	$-0.168576\pi$
$229$	26.1194	1.72602	0.863010	+	0.505186i	$0.168576\pi$
$230$	0	0
$231$	0.211103	0.0138895
$232$	0	0
$233$	14.0917	0.923176	0.461588	−	0.887094i	$-0.347280\pi$
$233$	14.0917	0.923176	0.461588	+	0.887094i	$0.347280\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.90833	0.123959
$238$	0	0
$239$	−14.3305	−0.926965	−0.463483	−	0.886106i	$-0.653400\pi$
$239$	−14.3305	−0.926965	−0.463483	+	0.886106i	$0.653400\pi$
$240$	0	0
$241$	−8.30278	−0.534829	−0.267414	−	0.963582i	$-0.586169\pi$
$241$	−8.30278	−0.534829	−0.267414	+	0.963582i	$0.586169\pi$
$242$	0	0
$243$	−7.84441	−0.503219
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−1.60555	−0.102159
$248$	0	0
$249$	−2.09167	−0.132554
$250$	0	0
$251$	23.3305	1.47261	0.736305	−	0.676650i	$-0.236569\pi$
$251$	23.3305	1.47261	0.736305	+	0.676650i	$0.236569\pi$
$252$	0	0
$253$	−3.90833	−0.245714
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.4222	−1.77293	−0.886464	−	0.462797i	$-0.846846\pi$
$257$	−28.4222	−1.77293	−0.886464	+	0.462797i	$0.846846\pi$
$258$	0	0
$259$	−5.30278	−0.329498
$260$	0	0
$261$	4.93608	0.305536
$262$	0	0
$263$	−10.8167	−0.666983	−0.333492	−	0.942753i	$-0.608227\pi$
$263$	−10.8167	−0.666983	−0.333492	+	0.942753i	$0.608227\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.54163	0.0943465
$268$	0	0
$269$	3.51388	0.214245	0.107122	−	0.994246i	$-0.465836\pi$
$269$	3.51388	0.214245	0.107122	+	0.994246i	$0.465836\pi$
$270$	0	0
$271$	29.4222	1.78727	0.893636	−	0.448793i	$-0.148146\pi$
$271$	29.4222	1.78727	0.893636	+	0.448793i	$0.148146\pi$
$272$	0	0
$273$	0.338936	0.0205133
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.2111	1.81521	0.907605	−	0.419826i	$-0.137909\pi$
$277$	30.2111	1.81521	0.907605	+	0.419826i	$0.137909\pi$
$278$	0	0
$279$	4.66947	0.279554
$280$	0	0
$281$	21.0000	1.25275	0.626377	−	0.779520i	$-0.284537\pi$
$281$	21.0000	1.25275	0.626377	+	0.779520i	$0.284537\pi$
$282$	0	0
$283$	23.5139	1.39775	0.698877	−	0.715241i	$-0.253683\pi$
$283$	23.5139	1.39775	0.698877	+	0.715241i	$0.253683\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.72498	0.337935
$288$	0	0
$289$	1.51388	0.0890517
$290$	0	0
$291$	−2.15559	−0.126363
$292$	0	0
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.78890	−0.103802
$298$	0	0
$299$	−6.27502	−0.362894
$300$	0	0
$301$	5.02776	0.289795
$302$	0	0
$303$	1.97224	0.113302
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−15.3305	−0.874960	−0.437480	−	0.899228i	$-0.644129\pi$
$307$	−15.3305	−0.874960	−0.437480	+	0.899228i	$0.644129\pi$
$308$	0	0
$309$	−3.72498	−0.211907
$310$	0	0
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	−22.6056	−1.27774	−0.638871	−	0.769314i	$-0.720598\pi$
$313$	−22.6056	−1.27774	−0.638871	+	0.769314i	$0.720598\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	19.6972	1.10631	0.553153	−	0.833080i	$-0.313424\pi$
$317$	19.6972	1.10631	0.553153	+	0.833080i	$0.313424\pi$
$318$	0	0
$319$	1.69722	0.0950263
$320$	0	0
$321$	−0.669468	−0.0373661
$322$	0	0
$323$	−4.30278	−0.239413
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−2.69722	−0.149157
$328$	0	0
$329$	3.90833	0.215473
$330$	0	0
$331$	−14.6333	−0.804319	−0.402160	−	0.915570i	$-0.631740\pi$
$331$	−14.6333	−0.804319	−0.402160	+	0.915570i	$0.631740\pi$
$332$	0	0
$333$	22.1194	1.21214
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4.78890	0.260868	0.130434	−	0.991457i	$-0.458363\pi$
$337$	4.78890	0.260868	0.130434	+	0.991457i	$0.458363\pi$
$338$	0	0
$339$	−1.69722	−0.0921806
$340$	0	0
$341$	1.60555	0.0869455
$342$	0	0
$343$	−9.42221	−0.508751
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.9083	1.17610	0.588050	−	0.808824i	$-0.299896\pi$
$347$	21.9083	1.17610	0.588050	+	0.808824i	$0.299896\pi$
$348$	0	0
$349$	3.42221	0.183186	0.0915932	−	0.995797i	$-0.470804\pi$
$349$	3.42221	0.183186	0.0915932	+	0.995797i	$0.470804\pi$
$350$	0	0
$351$	−2.87217	−0.153305
$352$	0	0
$353$	2.21110	0.117685	0.0588426	−	0.998267i	$-0.481259\pi$
$353$	2.21110	0.117685	0.0588426	+	0.998267i	$0.481259\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.908327	0.0480738
$358$	0	0
$359$	−16.4222	−0.866731	−0.433365	−	0.901218i	$-0.642674\pi$
$359$	−16.4222	−0.866731	−0.433365	+	0.901218i	$0.642674\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−0.302776	−0.0158916
$364$	0	0
$365$	0	0
$366$	0	0
$367$	33.3028	1.73839	0.869195	−	0.494469i	$-0.164637\pi$
$367$	33.3028	1.73839	0.869195	+	0.494469i	$0.164637\pi$
$368$	0	0
$369$	−23.8806	−1.24317
$370$	0	0
$371$	4.54163	0.235790
$372$	0	0
$373$	14.0278	0.726330	0.363165	−	0.931725i	$-0.381696\pi$
$373$	14.0278	0.726330	0.363165	+	0.931725i	$0.381696\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.72498	0.140344
$378$	0	0
$379$	0.211103	0.0108436	0.00542180	−	0.999985i	$-0.498274\pi$
$379$	0.211103	0.0108436	0.00542180	+	0.999985i	$0.498274\pi$
$380$	0	0
$381$	−4.51388	−0.231253
$382$	0	0
$383$	0.788897	0.0403108	0.0201554	−	0.999797i	$-0.493584\pi$
$383$	0.788897	0.0403108	0.0201554	+	0.999797i	$0.493584\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−20.9722	−1.06608
$388$	0	0
$389$	−27.6333	−1.40106	−0.700532	−	0.713621i	$-0.747054\pi$
$389$	−27.6333	−1.40106	−0.700532	+	0.713621i	$0.747054\pi$
$390$	0	0
$391$	−16.8167	−0.850455
$392$	0	0
$393$	−1.42221	−0.0717408
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.48612	−0.225152	−0.112576	−	0.993643i	$-0.535910\pi$
$397$	−4.48612	−0.225152	−0.112576	+	0.993643i	$0.535910\pi$
$398$	0	0
$399$	−0.211103	−0.0105683
$400$	0	0
$401$	11.2111	0.559856	0.279928	−	0.960021i	$-0.409689\pi$
$401$	11.2111	0.559856	0.279928	+	0.960021i	$0.409689\pi$
$402$	0	0
$403$	2.57779	0.128409
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.60555	0.376993
$408$	0	0
$409$	30.0278	1.48478	0.742388	−	0.669970i	$-0.233693\pi$
$409$	30.0278	1.48478	0.742388	+	0.669970i	$0.233693\pi$
$410$	0	0
$411$	−0.275019	−0.0135657
$412$	0	0
$413$	3.90833	0.192316
$414$	0	0
$415$	0	0
$416$	0	0
$417$	4.90833	0.240362
$418$	0	0
$419$	21.0000	1.02592	0.512959	−	0.858413i	$-0.328549\pi$
$419$	21.0000	1.02592	0.512959	+	0.858413i	$0.328549\pi$
$420$	0	0
$421$	−6.09167	−0.296890	−0.148445	−	0.988921i	$-0.547427\pi$
$421$	−6.09167	−0.296890	−0.148445	+	0.988921i	$0.547427\pi$
$422$	0	0
$423$	−16.3028	−0.792668
$424$	0	0
$425$	0	0
$426$	0	0
$427$	2.30278	0.111439
$428$	0	0
$429$	−0.486122	−0.0234702
$430$	0	0
$431$	−3.78890	−0.182505	−0.0912524	−	0.995828i	$-0.529087\pi$
$431$	−3.78890	−0.182505	−0.0912524	+	0.995828i	$0.529087\pi$
$432$	0	0
$433$	−29.0000	−1.39365	−0.696826	−	0.717241i	$-0.745405\pi$
$433$	−29.0000	−1.39365	−0.696826	+	0.717241i	$0.745405\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	3.90833	0.186961
$438$	0	0
$439$	−7.09167	−0.338467	−0.169234	−	0.985576i	$-0.554129\pi$
$439$	−7.09167	−0.338467	−0.169234	+	0.985576i	$0.554129\pi$
$440$	0	0
$441$	18.9445	0.902118
$442$	0	0
$443$	8.60555	0.408862	0.204431	−	0.978881i	$-0.434466\pi$
$443$	8.60555	0.408862	0.204431	+	0.978881i	$0.434466\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	3.63331	0.171850
$448$	0	0
$449$	−2.09167	−0.0987122	−0.0493561	−	0.998781i	$-0.515717\pi$
$449$	−2.09167	−0.0987122	−0.0493561	+	0.998781i	$0.515717\pi$
$450$	0	0
$451$	−8.21110	−0.386646
$452$	0	0
$453$	−4.48612	−0.210776
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−42.1472	−1.97156	−0.985781	−	0.168035i	$-0.946258\pi$
$457$	−42.1472	−1.97156	−0.985781	+	0.168035i	$0.946258\pi$
$458$	0	0
$459$	−7.69722	−0.359276
$460$	0	0
$461$	−22.8167	−1.06268	−0.531339	−	0.847159i	$-0.678311\pi$
$461$	−22.8167	−1.06268	−0.531339	+	0.847159i	$0.678311\pi$
$462$	0	0
$463$	39.6611	1.84321	0.921603	−	0.388134i	$-0.126880\pi$
$463$	39.6611	1.84321	0.921603	+	0.388134i	$0.126880\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	4.02776	0.186382	0.0931912	−	0.995648i	$-0.470293\pi$
$467$	4.02776	0.186382	0.0931912	+	0.995648i	$0.470293\pi$
$468$	0	0
$469$	5.57779	0.257559
$470$	0	0
$471$	−4.60555	−0.212213
$472$	0	0
$473$	−7.21110	−0.331567
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−18.9445	−0.867408
$478$	0	0
$479$	26.4500	1.20853	0.604265	−	0.796784i	$-0.293467\pi$
$479$	26.4500	1.20853	0.604265	+	0.796784i	$0.293467\pi$
$480$	0	0
$481$	12.2111	0.556778
$482$	0	0
$483$	−0.825058	−0.0375414
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.60555	0.344640	0.172320	−	0.985041i	$-0.444874\pi$
$487$	7.60555	0.344640	0.172320	+	0.985041i	$0.444874\pi$
$488$	0	0
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	−12.6333	−0.570133	−0.285067	−	0.958508i	$-0.592016\pi$
$491$	−12.6333	−0.570133	−0.285067	+	0.958508i	$0.592016\pi$
$492$	0	0
$493$	7.30278	0.328900
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.81665	−0.0814881
$498$	0	0
$499$	31.9083	1.42841	0.714206	−	0.699935i	$-0.246788\pi$
$499$	31.9083	1.42841	0.714206	+	0.699935i	$0.246788\pi$
$500$	0	0
$501$	7.45837	0.333215
$502$	0	0
$503$	−7.81665	−0.348527	−0.174264	−	0.984699i	$-0.555755\pi$
$503$	−7.81665	−0.348527	−0.174264	+	0.984699i	$0.555755\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.15559	0.140145
$508$	0	0
$509$	13.3028	0.589635	0.294818	−	0.955554i	$-0.404741\pi$
$509$	13.3028	0.589635	0.294818	+	0.955554i	$0.404741\pi$
$510$	0	0
$511$	1.33053	0.0588593
$512$	0	0
$513$	1.78890	0.0789818
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−5.60555	−0.246532
$518$	0	0
$519$	−2.96384	−0.130098
$520$	0	0
$521$	32.6056	1.42848	0.714238	−	0.699903i	$-0.246774\pi$
$521$	32.6056	1.42848	0.714238	+	0.699903i	$0.246774\pi$
$522$	0	0
$523$	41.6333	1.82050	0.910249	−	0.414062i	$-0.135890\pi$
$523$	41.6333	1.82050	0.910249	+	0.414062i	$0.135890\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.90833	0.300931
$528$	0	0
$529$	−7.72498	−0.335869
$530$	0	0
$531$	−16.3028	−0.707480
$532$	0	0
$533$	−13.1833	−0.571034
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2.44996	0.105724
$538$	0	0
$539$	6.51388	0.280573
$540$	0	0
$541$	−30.3305	−1.30401	−0.652006	−	0.758214i	$-0.726072\pi$
$541$	−30.3305	−1.30401	−0.652006	+	0.758214i	$0.726072\pi$
$542$	0	0
$543$	2.39445	0.102756
$544$	0	0
$545$	0	0
$546$	0	0
$547$	22.0917	0.944572	0.472286	−	0.881445i	$-0.343429\pi$
$547$	22.0917	0.944572	0.472286	+	0.881445i	$0.343429\pi$
$548$	0	0
$549$	−9.60555	−0.409955
$550$	0	0
$551$	−1.69722	−0.0723042
$552$	0	0
$553$	−4.39445	−0.186871
$554$	0	0
$555$	0	0
$556$	0	0
$557$	28.8167	1.22100	0.610500	−	0.792016i	$-0.290968\pi$
$557$	28.8167	1.22100	0.610500	+	0.792016i	$0.290968\pi$
$558$	0	0
$559$	−11.5778	−0.489689
$560$	0	0
$561$	−1.30278	−0.0550032
$562$	0	0
$563$	−39.9083	−1.68194	−0.840968	−	0.541085i	$-0.818014\pi$
$563$	−39.9083	−1.68194	−0.840968	+	0.541085i	$0.818014\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	5.70563	0.239614
$568$	0	0
$569$	−45.3583	−1.90152	−0.950759	−	0.309931i	$-0.899694\pi$
$569$	−45.3583	−1.90152	−0.950759	+	0.309931i	$0.899694\pi$
$570$	0	0
$571$	8.93608	0.373963	0.186982	−	0.982363i	$-0.440129\pi$
$571$	8.93608	0.373963	0.186982	+	0.982363i	$0.440129\pi$
$572$	0	0
$573$	3.90833	0.163273
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−28.7250	−1.19584	−0.597918	−	0.801557i	$-0.704005\pi$
$577$	−28.7250	−1.19584	−0.597918	+	0.801557i	$0.704005\pi$
$578$	0	0
$579$	−0.733385	−0.0304784
$580$	0	0
$581$	4.81665	0.199828
$582$	0	0
$583$	−6.51388	−0.269777
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.7250	1.35070	0.675352	−	0.737495i	$-0.263992\pi$
$587$	32.7250	1.35070	0.675352	+	0.737495i	$0.263992\pi$
$588$	0	0
$589$	−1.60555	−0.0661556
$590$	0	0
$591$	−6.63331	−0.272858
$592$	0	0
$593$	−26.2111	−1.07636	−0.538180	−	0.842830i	$-0.680888\pi$
$593$	−26.2111	−1.07636	−0.538180	+	0.842830i	$0.680888\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0.880571	0.0360394
$598$	0	0
$599$	44.7250	1.82741	0.913707	−	0.406375i	$-0.133207\pi$
$599$	44.7250	1.82741	0.913707	+	0.406375i	$0.133207\pi$
$600$	0	0
$601$	−9.48612	−0.386947	−0.193473	−	0.981106i	$-0.561975\pi$
$601$	−9.48612	−0.386947	−0.193473	+	0.981106i	$0.561975\pi$
$602$	0	0
$603$	−23.2666	−0.947490
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−8.42221	−0.341847	−0.170923	−	0.985284i	$-0.554675\pi$
$607$	−8.42221	−0.341847	−0.170923	+	0.985284i	$0.554675\pi$
$608$	0	0
$609$	0.358288	0.0145186
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	15.3305	0.619194	0.309597	−	0.950868i	$-0.399806\pi$
$613$	15.3305	0.619194	0.309597	+	0.950868i	$0.399806\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−43.0278	−1.73223	−0.866116	−	0.499843i	$-0.833391\pi$
$617$	−43.0278	−1.73223	−0.866116	+	0.499843i	$0.833391\pi$
$618$	0	0
$619$	14.8167	0.595532	0.297766	−	0.954639i	$-0.403759\pi$
$619$	14.8167	0.595532	0.297766	+	0.954639i	$0.403759\pi$
$620$	0	0
$621$	6.99160	0.280563
$622$	0	0
$623$	−3.55004	−0.142229
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0.302776	0.0120917
$628$	0	0
$629$	32.7250	1.30483
$630$	0	0
$631$	−38.5139	−1.53321	−0.766607	−	0.642117i	$-0.778056\pi$
$631$	−38.5139	−1.53321	−0.766607	+	0.642117i	$0.778056\pi$
$632$	0	0
$633$	−6.54163	−0.260007
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.4584	0.414376
$638$	0	0
$639$	7.57779	0.299773
$640$	0	0
$641$	21.0000	0.829450	0.414725	−	0.909947i	$-0.363878\pi$
$641$	21.0000	0.829450	0.414725	+	0.909947i	$0.363878\pi$
$642$	0	0
$643$	−8.42221	−0.332139	−0.166070	−	0.986114i	$-0.553108\pi$
$643$	−8.42221	−0.332139	−0.166070	+	0.986114i	$0.553108\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−33.0000	−1.29736	−0.648682	−	0.761060i	$-0.724679\pi$
$647$	−33.0000	−1.29736	−0.648682	+	0.761060i	$0.724679\pi$
$648$	0	0
$649$	−5.60555	−0.220037
$650$	0	0
$651$	0.338936	0.0132839
$652$	0	0
$653$	−18.1194	−0.709068	−0.354534	−	0.935043i	$-0.615360\pi$
$653$	−18.1194	−0.709068	−0.354534	+	0.935043i	$0.615360\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−5.55004	−0.216528
$658$	0	0
$659$	−36.3583	−1.41632	−0.708159	−	0.706053i	$-0.750474\pi$
$659$	−36.3583	−1.41632	−0.708159	+	0.706053i	$0.750474\pi$
$660$	0	0
$661$	−20.8167	−0.809674	−0.404837	−	0.914389i	$-0.632672\pi$
$661$	−20.8167	−0.809674	−0.404837	+	0.914389i	$0.632672\pi$
$662$	0	0
$663$	−2.09167	−0.0812339
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.63331	−0.256843
$668$	0	0
$669$	−4.11943	−0.159266
$670$	0	0
$671$	−3.30278	−0.127502
$672$	0	0
$673$	−35.0000	−1.34915	−0.674575	−	0.738206i	$-0.735673\pi$
$673$	−35.0000	−1.34915	−0.674575	+	0.738206i	$0.735673\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.2389	−0.931575	−0.465788	−	0.884897i	$-0.654229\pi$
$677$	−24.2389	−0.931575	−0.465788	+	0.884897i	$0.654229\pi$
$678$	0	0
$679$	4.96384	0.190495
$680$	0	0
$681$	7.85281	0.300920
$682$	0	0
$683$	−5.60555	−0.214490	−0.107245	−	0.994233i	$-0.534203\pi$
$683$	−5.60555	−0.214490	−0.107245	+	0.994233i	$0.534203\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−7.90833	−0.301721
$688$	0	0
$689$	−10.4584	−0.398432
$690$	0	0
$691$	−7.48612	−0.284785	−0.142393	−	0.989810i	$-0.545480\pi$
$691$	−7.48612	−0.284785	−0.142393	+	0.989810i	$0.545480\pi$
$692$	0	0
$693$	2.02776	0.0770281
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−35.3305	−1.33824
$698$	0	0
$699$	−4.26662	−0.161378
$700$	0	0
$701$	−5.21110	−0.196821	−0.0984103	−	0.995146i	$-0.531376\pi$
$701$	−5.21110	−0.196821	−0.0984103	+	0.995146i	$0.531376\pi$
$702$	0	0
$703$	−7.60555	−0.286849
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4.54163	−0.170806
$708$	0	0
$709$	35.6333	1.33824	0.669118	−	0.743156i	$-0.266672\pi$
$709$	35.6333	1.33824	0.669118	+	0.743156i	$0.266672\pi$
$710$	0	0
$711$	18.3305	0.687449
$712$	0	0
$713$	−6.27502	−0.235001
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.33894	0.162041
$718$	0	0
$719$	16.5778	0.618247	0.309124	−	0.951022i	$-0.399964\pi$
$719$	16.5778	0.618247	0.309124	+	0.951022i	$0.399964\pi$
$720$	0	0
$721$	8.57779	0.319454
$722$	0	0
$723$	2.51388	0.0934921
$724$	0	0
$725$	0	0
$726$	0	0
$727$	26.9083	0.997975	0.498987	−	0.866609i	$-0.333705\pi$
$727$	26.9083	0.997975	0.498987	+	0.866609i	$0.333705\pi$
$728$	0	0
$729$	−22.1749	−0.821294
$730$	0	0
$731$	−31.0278	−1.14760
$732$	0	0
$733$	−9.42221	−0.348017	−0.174009	−	0.984744i	$-0.555672\pi$
$733$	−9.42221	−0.348017	−0.174009	+	0.984744i	$0.555672\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	0.880571	0.0323923	0.0161962	−	0.999869i	$-0.494844\pi$
$739$	0.880571	0.0323923	0.0161962	+	0.999869i	$0.494844\pi$
$740$	0	0
$741$	0.486122	0.0178581
$742$	0	0
$743$	9.90833	0.363501	0.181751	−	0.983345i	$-0.441824\pi$
$743$	9.90833	0.363501	0.181751	+	0.983345i	$0.441824\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−20.0917	−0.735116
$748$	0	0
$749$	1.54163	0.0563301
$750$	0	0
$751$	22.3583	0.815866	0.407933	−	0.913012i	$-0.366250\pi$
$751$	22.3583	0.815866	0.407933	+	0.913012i	$0.366250\pi$
$752$	0	0
$753$	−7.06392	−0.257423
$754$	0	0
$755$	0	0
$756$	0	0
$757$	25.0000	0.908640	0.454320	−	0.890838i	$-0.349882\pi$
$757$	25.0000	0.908640	0.454320	+	0.890838i	$0.349882\pi$
$758$	0	0
$759$	1.18335	0.0429527
$760$	0	0
$761$	−43.2666	−1.56841	−0.784207	−	0.620500i	$-0.786930\pi$
$761$	−43.2666	−1.56841	−0.784207	+	0.620500i	$0.786930\pi$
$762$	0	0
$763$	6.21110	0.224857
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.00000	−0.324971
$768$	0	0
$769$	1.36669	0.0492842	0.0246421	−	0.999696i	$-0.492155\pi$
$769$	1.36669	0.0492842	0.0246421	+	0.999696i	$0.492155\pi$
$770$	0	0
$771$	8.60555	0.309921
$772$	0	0
$773$	−8.33053	−0.299628	−0.149814	−	0.988714i	$-0.547868\pi$
$773$	−8.33053	−0.299628	−0.149814	+	0.988714i	$0.547868\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.60555	0.0575988
$778$	0	0
$779$	8.21110	0.294193
$780$	0	0
$781$	2.60555	0.0932340
$782$	0	0
$783$	−3.03616	−0.108504
$784$	0	0
$785$	0	0
$786$	0	0
$787$	25.8444	0.921254	0.460627	−	0.887594i	$-0.347625\pi$
$787$	25.8444	0.921254	0.460627	+	0.887594i	$0.347625\pi$
$788$	0	0
$789$	3.27502	0.116594
$790$	0	0
$791$	3.90833	0.138964
$792$	0	0
$793$	−5.30278	−0.188307
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−5.09167	−0.180356	−0.0901782	−	0.995926i	$-0.528744\pi$
$797$	−5.09167	−0.180356	−0.0901782	+	0.995926i	$0.528744\pi$
$798$	0	0
$799$	−24.1194	−0.853284
$800$	0	0
$801$	14.8082	0.523224
$802$	0	0
$803$	−1.90833	−0.0673434
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1.06392	−0.0374516
$808$	0	0
$809$	−3.63331	−0.127740	−0.0638701	−	0.997958i	$-0.520344\pi$
$809$	−3.63331	−0.127740	−0.0638701	+	0.997958i	$0.520344\pi$
$810$	0	0
$811$	9.60555	0.337297	0.168648	−	0.985676i	$-0.446060\pi$
$811$	9.60555	0.337297	0.168648	+	0.985676i	$0.446060\pi$
$812$	0	0
$813$	−8.90833	−0.312429
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.21110	0.252285
$818$	0	0
$819$	3.25567	0.113762
$820$	0	0
$821$	−3.63331	−0.126803	−0.0634017	−	0.997988i	$-0.520195\pi$
$821$	−3.63331	−0.126803	−0.0634017	+	0.997988i	$0.520195\pi$
$822$	0	0
$823$	−31.6333	−1.10267	−0.551334	−	0.834285i	$-0.685881\pi$
$823$	−31.6333	−1.10267	−0.551334	+	0.834285i	$0.685881\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.6056	−1.55109	−0.775543	−	0.631294i	$-0.782524\pi$
$827$	−44.6056	−1.55109	−0.775543	+	0.631294i	$0.782524\pi$
$828$	0	0
$829$	26.1194	0.907165	0.453583	−	0.891214i	$-0.350146\pi$
$829$	26.1194	0.907165	0.453583	+	0.891214i	$0.350146\pi$
$830$	0	0
$831$	−9.14719	−0.317312
$832$	0	0
$833$	28.0278	0.971104
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2.87217	−0.0992766
$838$	0	0
$839$	−26.0917	−0.900785	−0.450392	−	0.892831i	$-0.648716\pi$
$839$	−26.0917	−0.900785	−0.450392	+	0.892831i	$0.648716\pi$
$840$	0	0
$841$	−26.1194	−0.900670
$842$	0	0
$843$	−6.35829	−0.218991
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0.697224	0.0239569
$848$	0	0
$849$	−7.11943	−0.244338
$850$	0	0
$851$	−29.7250	−1.01896
$852$	0	0
$853$	−0.302776	−0.0103668	−0.00518342	−	0.999987i	$-0.501650\pi$
$853$	−0.302776	−0.0103668	−0.00518342	+	0.999987i	$0.501650\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−4.02776	−0.137586	−0.0687928	−	0.997631i	$-0.521915\pi$
$857$	−4.02776	−0.137586	−0.0687928	+	0.997631i	$0.521915\pi$
$858$	0	0
$859$	−10.6056	−0.361857	−0.180928	−	0.983496i	$-0.557910\pi$
$859$	−10.6056	−0.361857	−0.180928	+	0.983496i	$0.557910\pi$
$860$	0	0
$861$	−1.73338	−0.0590736
$862$	0	0
$863$	−52.2666	−1.77918	−0.889588	−	0.456764i	$-0.849009\pi$
$863$	−52.2666	−1.77918	−0.889588	+	0.456764i	$0.849009\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−0.458365	−0.0155669
$868$	0	0
$869$	6.30278	0.213807
$870$	0	0
$871$	−12.8444	−0.435216
$872$	0	0
$873$	−20.7056	−0.700779
$874$	0	0
$875$	0	0
$876$	0	0
$877$	31.6333	1.06818	0.534090	−	0.845427i	$-0.320654\pi$
$877$	31.6333	1.06818	0.534090	+	0.845427i	$0.320654\pi$
$878$	0	0
$879$	−5.44996	−0.183823
$880$	0	0
$881$	12.7527	0.429651	0.214825	−	0.976652i	$-0.431082\pi$
$881$	12.7527	0.429651	0.214825	+	0.976652i	$0.431082\pi$
$882$	0	0
$883$	−18.2111	−0.612852	−0.306426	−	0.951894i	$-0.599133\pi$
$883$	−18.2111	−0.612852	−0.306426	+	0.951894i	$0.599133\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−11.6056	−0.389676	−0.194838	−	0.980835i	$-0.562418\pi$
$887$	−11.6056	−0.389676	−0.194838	+	0.980835i	$0.562418\pi$
$888$	0	0
$889$	10.3944	0.348619
$890$	0	0
$891$	−8.18335	−0.274152
$892$	0	0
$893$	5.60555	0.187583
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.89992	0.0634366
$898$	0	0
$899$	2.72498	0.0908832
$900$	0	0
$901$	−28.0278	−0.933740
$902$	0	0
$903$	−1.52228	−0.0506584
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−5.57779	−0.185208	−0.0926038	−	0.995703i	$-0.529519\pi$
$907$	−5.57779	−0.185208	−0.0926038	+	0.995703i	$0.529519\pi$
$908$	0	0
$909$	18.9445	0.628349
$910$	0	0
$911$	33.6333	1.11432	0.557161	−	0.830405i	$-0.311891\pi$
$911$	33.6333	1.11432	0.557161	+	0.830405i	$0.311891\pi$
$912$	0	0
$913$	−6.90833	−0.228632
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.27502	0.108151
$918$	0	0
$919$	42.0555	1.38728	0.693642	−	0.720320i	$-0.256005\pi$
$919$	42.0555	1.38728	0.693642	+	0.720320i	$0.256005\pi$
$920$	0	0
$921$	4.64171	0.152950
$922$	0	0
$923$	4.18335	0.137697
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−35.7805	−1.17519
$928$	0	0
$929$	31.6611	1.03877	0.519383	−	0.854542i	$-0.326162\pi$
$929$	31.6611	1.03877	0.519383	+	0.854542i	$0.326162\pi$
$930$	0	0
$931$	−6.51388	−0.213484
$932$	0	0
$933$	−2.72498	−0.0892119
$934$	0	0
$935$	0	0
$936$	0	0
$937$	16.0000	0.522697	0.261349	−	0.965244i	$-0.415833\pi$
$937$	16.0000	0.522697	0.261349	+	0.965244i	$0.415833\pi$
$938$	0	0
$939$	6.84441	0.223359
$940$	0	0
$941$	4.57779	0.149232	0.0746159	−	0.997212i	$-0.476227\pi$
$941$	4.57779	0.149232	0.0746159	+	0.997212i	$0.476227\pi$
$942$	0	0
$943$	32.0917	1.04505
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38.4500	−1.24946	−0.624728	−	0.780843i	$-0.714790\pi$
$947$	−38.4500	−1.24946	−0.624728	+	0.780843i	$0.714790\pi$
$948$	0	0
$949$	−3.06392	−0.0994589
$950$	0	0
$951$	−5.96384	−0.193391
$952$	0	0
$953$	33.6333	1.08949	0.544745	−	0.838602i	$-0.316627\pi$
$953$	33.6333	1.08949	0.544745	+	0.838602i	$0.316627\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−0.513878	−0.0166113
$958$	0	0
$959$	0.633308	0.0204506
$960$	0	0
$961$	−28.4222	−0.916845
$962$	0	0
$963$	−6.43061	−0.207223
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−17.9361	−0.576786	−0.288393	−	0.957512i	$-0.593121\pi$
$967$	−17.9361	−0.576786	−0.288393	+	0.957512i	$0.593121\pi$
$968$	0	0
$969$	1.30278	0.0418512
$970$	0	0
$971$	51.9083	1.66582	0.832909	−	0.553410i	$-0.186674\pi$
$971$	51.9083	1.66582	0.832909	+	0.553410i	$0.186674\pi$
$972$	0	0
$973$	−11.3028	−0.362350
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−16.8167	−0.538012	−0.269006	−	0.963138i	$-0.586695\pi$
$977$	−16.8167	−0.538012	−0.269006	+	0.963138i	$0.586695\pi$
$978$	0	0
$979$	5.09167	0.162731
$980$	0	0
$981$	−25.9083	−0.827189
$982$	0	0
$983$	−39.6333	−1.26411	−0.632053	−	0.774925i	$-0.717788\pi$
$983$	−39.6333	−1.26411	−0.632053	+	0.774925i	$0.717788\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.18335	−0.0376663
$988$	0	0
$989$	28.1833	0.896178
$990$	0	0
$991$	−35.7527	−1.13572	−0.567862	−	0.823124i	$-0.692229\pi$
$991$	−35.7527	−1.13572	−0.567862	+	0.823124i	$0.692229\pi$
$992$	0	0
$993$	4.43061	0.140601
$994$	0	0
$995$	0	0
$996$	0	0
$997$	50.7805	1.60823	0.804117	−	0.594471i	$-0.202638\pi$
$997$	50.7805	1.60823	0.804117	+	0.594471i	$0.202638\pi$
$998$	0	0
$999$	−13.6056	−0.430461

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bw.1.1		2
4.3	odd	2		1100.2.a.f.1.2	✓	2
5.2	odd	4		4400.2.b.r.4049.3		4
5.3	odd	4		4400.2.b.r.4049.2		4
5.4	even	2		4400.2.a.bf.1.2		2
12.11	even	2		9900.2.a.bg.1.2		2
20.3	even	4		1100.2.b.e.749.3		4
20.7	even	4		1100.2.b.e.749.2		4
20.19	odd	2		1100.2.a.i.1.1	yes	2
60.23	odd	4		9900.2.c.r.5149.3		4
60.47	odd	4		9900.2.c.r.5149.2		4
60.59	even	2		9900.2.a.by.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.f.1.2	✓	2	4.3	odd	2
1100.2.a.i.1.1	yes	2	20.19	odd	2
1100.2.b.e.749.2		4	20.7	even	4
1100.2.b.e.749.3		4	20.3	even	4
4400.2.a.bf.1.2		2	5.4	even	2
4400.2.a.bw.1.1		2	1.1	even	1	trivial
4400.2.b.r.4049.2		4	5.3	odd	4
4400.2.b.r.4049.3		4	5.2	odd	4
9900.2.a.bg.1.2		2	12.11	even	2
9900.2.a.by.1.1		2	60.59	even	2
9900.2.c.r.5149.2		4	60.47	odd	4
9900.2.c.r.5149.3		4	60.23	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)

\(f(q)\)	\(=\)	\(q-0.302776 q^{3} +0.697224 q^{7} -2.90833 q^{9} -1.00000 q^{11} -1.60555 q^{13} -4.30278 q^{17} +1.00000 q^{19} -0.211103 q^{21} +3.90833 q^{23} +1.78890 q^{27} -1.69722 q^{29} -1.60555 q^{31} +0.302776 q^{33} -7.60555 q^{37} +0.486122 q^{39} +8.21110 q^{41} +7.21110 q^{43} +5.60555 q^{47} -6.51388 q^{49} +1.30278 q^{51} +6.51388 q^{53} -0.302776 q^{57} +5.60555 q^{59} +3.30278 q^{61} -2.02776 q^{63} +8.00000 q^{67} -1.18335 q^{69} -2.60555 q^{71} +1.90833 q^{73} -0.697224 q^{77} -6.30278 q^{79} +8.18335 q^{81} +6.90833 q^{83} +0.513878 q^{87} -5.09167 q^{89} -1.11943 q^{91} +0.486122 q^{93} +7.11943 q^{97} +2.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{3} + 5 q^{7} + 5 q^{9} - 2 q^{11} + 4 q^{13} - 5 q^{17} + 2 q^{19} + 14 q^{21} - 3 q^{23} + 18 q^{27} - 7 q^{29} + 4 q^{31} - 3 q^{33} - 8 q^{37} + 19 q^{39} + 2 q^{41} + 4 q^{47} + 5 q^{49}+ \cdots - 5 q^{99}+O(q^{100}) \) 2 * q + 3 * q^3 + 5 * q^7 + 5 * q^9 - 2 * q^11 + 4 * q^13 - 5 * q^17 + 2 * q^19 + 14 * q^21 - 3 * q^23 + 18 * q^27 - 7 * q^29 + 4 * q^31 - 3 * q^33 - 8 * q^37 + 19 * q^39 + 2 * q^41 + 4 * q^47 + 5 * q^49 - q^51 - 5 * q^53 + 3 * q^57 + 4 * q^59 + 3 * q^61 + 32 * q^63 + 16 * q^67 - 24 * q^69 + 2 * q^71 - 7 * q^73 - 5 * q^77 - 9 * q^79 + 38 * q^81 + 3 * q^83 - 17 * q^87 - 21 * q^89 + 23 * q^91 + 19 * q^93 - 11 * q^97 - 5 * q^99

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