LMFDB - Embedded newform 4400.2.a.bp.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,0,0,-2,0,3,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ 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)

$f(q)$	$=$	$q-1.56155 q^{3} +3.12311 q^{7} -0.561553 q^{9} +1.00000 q^{11} +5.12311 q^{13} -2.00000 q^{17} +4.00000 q^{19} -4.87689 q^{21} +2.43845 q^{23} +5.56155 q^{27} -5.12311 q^{29} +5.56155 q^{31} -1.56155 q^{33} +7.56155 q^{37} -8.00000 q^{39} -1.12311 q^{41} -7.12311 q^{43} +8.00000 q^{47} +2.75379 q^{49} +3.12311 q^{51} -12.2462 q^{53} -6.24621 q^{57} -7.80776 q^{59} +1.12311 q^{61} -1.75379 q^{63} +9.56155 q^{67} -3.80776 q^{69} +8.68466 q^{71} -5.12311 q^{73} +3.12311 q^{77} +11.1231 q^{79} -7.00000 q^{81} +0.876894 q^{83} +8.00000 q^{87} +2.68466 q^{89} +16.0000 q^{91} -8.68466 q^{93} -15.5616 q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} - 18 q^{21} + 9 q^{23} + 7 q^{27} - 2 q^{29} + 7 q^{31} + q^{33} + 11 q^{37} - 16 q^{39} + 6 q^{41} - 6 q^{43} + 16 q^{47} + 22 q^{49}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q + q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 8 * q^19 - 18 * q^21 + 9 * q^23 + 7 * q^27 - 2 * q^29 + 7 * q^31 + q^33 + 11 * q^37 - 16 * q^39 + 6 * q^41 - 6 * q^43 + 16 * q^47 + 22 * q^49 - 2 * q^51 - 8 * q^53 + 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 15 * q^67 + 13 * q^69 + 5 * q^71 - 2 * q^73 - 2 * q^77 + 14 * q^79 - 14 * q^81 + 10 * q^83 + 16 * q^87 - 7 * q^89 + 32 * q^91 - 5 * q^93 - 27 * q^97 + 3 * 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
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.12311	1.18042	0.590211	−	0.807249i	$-0.299044\pi$
$7$	3.12311	1.18042	0.590211	+	0.807249i	$0.299044\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−4.87689	−1.06423
$22$	0	0
$23$	2.43845	0.508451	0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845	0.508451	0.254226	+	0.967145i	$0.418179\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	−5.12311	−0.951337	−0.475668	−	0.879625i	$-0.657794\pi$
$29$	−5.12311	−0.951337	−0.475668	+	0.879625i	$0.657794\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	−1.56155	−0.271831
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.56155	1.24311	0.621556	−	0.783370i	$-0.286501\pi$
$37$	7.56155	1.24311	0.621556	+	0.783370i	$0.286501\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	−7.12311	−1.08626	−0.543132	−	0.839648i	$-0.682762\pi$
$43$	−7.12311	−1.08626	−0.543132	+	0.839648i	$0.682762\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	3.12311	0.437322
$52$	0	0
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6.24621	−0.827331
$58$	0	0
$59$	−7.80776	−1.01648	−0.508242	−	0.861214i	$-0.669705\pi$
$59$	−7.80776	−1.01648	−0.508242	+	0.861214i	$0.669705\pi$
$60$	0	0
$61$	1.12311	0.143799	0.0718995	−	0.997412i	$-0.477094\pi$
$61$	1.12311	0.143799	0.0718995	+	0.997412i	$0.477094\pi$
$62$	0	0
$63$	−1.75379	−0.220957
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.56155	1.16813	0.584065	−	0.811707i	$-0.301461\pi$
$67$	9.56155	1.16813	0.584065	+	0.811707i	$0.301461\pi$
$68$	0	0
$69$	−3.80776	−0.458401
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	−5.12311	−0.599614	−0.299807	−	0.954000i	$-0.596922\pi$
$73$	−5.12311	−0.599614	−0.299807	+	0.954000i	$0.596922\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.12311	0.355911
$78$	0	0
$79$	11.1231	1.25145	0.625724	−	0.780045i	$-0.284804\pi$
$79$	11.1231	1.25145	0.625724	+	0.780045i	$0.284804\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	0.876894	0.0962517	0.0481258	−	0.998841i	$-0.484675\pi$
$83$	0.876894	0.0962517	0.0481258	+	0.998841i	$0.484675\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	2.68466	0.284573	0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466	0.284573	0.142287	+	0.989825i	$0.454555\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	0	0
$93$	−8.68466	−0.900557
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.5616	−1.58004	−0.790018	−	0.613083i	$-0.789929\pi$
$97$	−15.5616	−1.58004	−0.790018	+	0.613083i	$0.789929\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.3693	−1.29246	−0.646230	−	0.763142i	$-0.723655\pi$
$107$	−13.3693	−1.29246	−0.646230	+	0.763142i	$0.723655\pi$
$108$	0	0
$109$	12.2462	1.17297	0.586487	−	0.809959i	$-0.300510\pi$
$109$	12.2462	1.17297	0.586487	+	0.809959i	$0.300510\pi$
$110$	0	0
$111$	−11.8078	−1.12074
$112$	0	0
$113$	0.438447	0.0412456	0.0206228	−	0.999787i	$-0.493435\pi$
$113$	0.438447	0.0412456	0.0206228	+	0.999787i	$0.493435\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−2.87689	−0.265969
$118$	0	0
$119$	−6.24621	−0.572589
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.75379	0.158134
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	0	0
$129$	11.1231	0.979335
$130$	0	0
$131$	−13.3693	−1.16808	−0.584041	−	0.811724i	$-0.698529\pi$
$131$	−13.3693	−1.16808	−0.584041	+	0.811724i	$0.698529\pi$
$132$	0	0
$133$	12.4924	1.08323
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.43845	0.720945	0.360473	−	0.932770i	$-0.382615\pi$
$137$	8.43845	0.720945	0.360473	+	0.932770i	$0.382615\pi$
$138$	0	0
$139$	−15.1231	−1.28273	−0.641363	−	0.767238i	$-0.721631\pi$
$139$	−15.1231	−1.28273	−0.641363	+	0.767238i	$0.721631\pi$
$140$	0	0
$141$	−12.4924	−1.05205
$142$	0	0
$143$	5.12311	0.428416
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−4.30019	−0.354673
$148$	0	0
$149$	4.24621	0.347863	0.173932	−	0.984758i	$-0.444353\pi$
$149$	4.24621	0.347863	0.173932	+	0.984758i	$0.444353\pi$
$150$	0	0
$151$	9.36932	0.762464	0.381232	−	0.924479i	$-0.375500\pi$
$151$	9.36932	0.762464	0.381232	+	0.924479i	$0.375500\pi$
$152$	0	0
$153$	1.12311	0.0907977
$154$	0	0
$155$	0	0
$156$	0	0
$157$	4.43845	0.354227	0.177113	−	0.984190i	$-0.443324\pi$
$157$	4.43845	0.354227	0.177113	+	0.984190i	$0.443324\pi$
$158$	0	0
$159$	19.1231	1.51656
$160$	0	0
$161$	7.61553	0.600188
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	−2.24621	−0.171772
$172$	0	0
$173$	−12.2462	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$173$	−12.2462	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	12.1922	0.916425
$178$	0	0
$179$	6.43845	0.481232	0.240616	−	0.970620i	$-0.422651\pi$
$179$	6.43845	0.481232	0.240616	+	0.970620i	$0.422651\pi$
$180$	0	0
$181$	−1.31534	−0.0977686	−0.0488843	−	0.998804i	$-0.515567\pi$
$181$	−1.31534	−0.0977686	−0.0488843	+	0.998804i	$0.515567\pi$
$182$	0	0
$183$	−1.75379	−0.129644
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	17.3693	1.26343
$190$	0	0
$191$	10.4384	0.755300	0.377650	−	0.925949i	$-0.376732\pi$
$191$	10.4384	0.755300	0.377650	+	0.925949i	$0.376732\pi$
$192$	0	0
$193$	9.12311	0.656696	0.328348	−	0.944557i	$-0.393508\pi$
$193$	9.12311	0.656696	0.328348	+	0.944557i	$0.393508\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	14.4924	1.03254	0.516271	−	0.856425i	$-0.327320\pi$
$197$	14.4924	1.03254	0.516271	+	0.856425i	$0.327320\pi$
$198$	0	0
$199$	12.4924	0.885564	0.442782	−	0.896629i	$-0.353991\pi$
$199$	12.4924	0.885564	0.442782	+	0.896629i	$0.353991\pi$
$200$	0	0
$201$	−14.9309	−1.05314
$202$	0	0
$203$	−16.0000	−1.12298
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.36932	−0.0951741
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−8.49242	−0.584642	−0.292321	−	0.956320i	$-0.594428\pi$
$211$	−8.49242	−0.584642	−0.292321	+	0.956320i	$0.594428\pi$
$212$	0	0
$213$	−13.5616	−0.929222
$214$	0	0
$215$	0	0
$216$	0	0
$217$	17.3693	1.17911
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	−10.2462	−0.689235
$222$	0	0
$223$	−11.8078	−0.790706	−0.395353	−	0.918529i	$-0.629378\pi$
$223$	−11.8078	−0.790706	−0.395353	+	0.918529i	$0.629378\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.1231	1.53473	0.767367	−	0.641208i	$-0.221566\pi$
$227$	23.1231	1.53473	0.767367	+	0.641208i	$0.221566\pi$
$228$	0	0
$229$	14.6847	0.970390	0.485195	−	0.874406i	$-0.338749\pi$
$229$	14.6847	0.970390	0.485195	+	0.874406i	$0.338749\pi$
$230$	0	0
$231$	−4.87689	−0.320876
$232$	0	0
$233$	7.36932	0.482780	0.241390	−	0.970428i	$-0.422397\pi$
$233$	7.36932	0.482780	0.241390	+	0.970428i	$0.422397\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−17.3693	−1.12826
$238$	0	0
$239$	4.87689	0.315460	0.157730	−	0.987482i	$-0.449582\pi$
$239$	4.87689	0.315460	0.157730	+	0.987482i	$0.449582\pi$
$240$	0	0
$241$	29.1231	1.87598	0.937992	−	0.346657i	$-0.112683\pi$
$241$	29.1231	1.87598	0.937992	+	0.346657i	$0.112683\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.4924	1.30390
$248$	0	0
$249$	−1.36932	−0.0867769
$250$	0	0
$251$	−1.56155	−0.0985643	−0.0492822	−	0.998785i	$-0.515693\pi$
$251$	−1.56155	−0.0985643	−0.0492822	+	0.998785i	$0.515693\pi$
$252$	0	0
$253$	2.43845	0.153304
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−11.7538	−0.733181	−0.366591	−	0.930382i	$-0.619475\pi$
$257$	−11.7538	−0.733181	−0.366591	+	0.930382i	$0.619475\pi$
$258$	0	0
$259$	23.6155	1.46740
$260$	0	0
$261$	2.87689	0.178075
$262$	0	0
$263$	19.1231	1.17918	0.589591	−	0.807702i	$-0.299289\pi$
$263$	19.1231	1.17918	0.589591	+	0.807702i	$0.299289\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−4.19224	−0.256561
$268$	0	0
$269$	−20.7386	−1.26446	−0.632228	−	0.774782i	$-0.717860\pi$
$269$	−20.7386	−1.26446	−0.632228	+	0.774782i	$0.717860\pi$
$270$	0	0
$271$	28.4924	1.73079	0.865396	−	0.501089i	$-0.167067\pi$
$271$	28.4924	1.73079	0.865396	+	0.501089i	$0.167067\pi$
$272$	0	0
$273$	−24.9848	−1.51215
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	−3.12311	−0.186975
$280$	0	0
$281$	16.2462	0.969168	0.484584	−	0.874745i	$-0.338971\pi$
$281$	16.2462	0.969168	0.484584	+	0.874745i	$0.338971\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.50758	−0.207046
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	24.3002	1.42450
$292$	0	0
$293$	3.36932	0.196838	0.0984188	−	0.995145i	$-0.468622\pi$
$293$	3.36932	0.196838	0.0984188	+	0.995145i	$0.468622\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.56155	0.322714
$298$	0	0
$299$	12.4924	0.722455
$300$	0	0
$301$	−22.2462	−1.28225
$302$	0	0
$303$	3.12311	0.179418
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−32.4924	−1.85444	−0.927220	−	0.374516i	$-0.877809\pi$
$307$	−32.4924	−1.85444	−0.927220	+	0.374516i	$0.877809\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.75379	−0.553087	−0.276543	−	0.961001i	$-0.589189\pi$
$311$	−9.75379	−0.553087	−0.276543	+	0.961001i	$0.589189\pi$
$312$	0	0
$313$	9.80776	0.554368	0.277184	−	0.960817i	$-0.410599\pi$
$313$	9.80776	0.554368	0.277184	+	0.960817i	$0.410599\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.1922	0.797115	0.398558	−	0.917143i	$-0.369511\pi$
$317$	14.1922	0.797115	0.398558	+	0.917143i	$0.369511\pi$
$318$	0	0
$319$	−5.12311	−0.286839
$320$	0	0
$321$	20.8769	1.16523
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−19.1231	−1.05751
$328$	0	0
$329$	24.9848	1.37746
$330$	0	0
$331$	−34.9309	−1.91997	−0.959987	−	0.280044i	$-0.909651\pi$
$331$	−34.9309	−1.91997	−0.959987	+	0.280044i	$0.909651\pi$
$332$	0	0
$333$	−4.24621	−0.232691
$334$	0	0
$335$	0	0
$336$	0	0
$337$	16.7386	0.911811	0.455906	−	0.890028i	$-0.349315\pi$
$337$	16.7386	0.911811	0.455906	+	0.890028i	$0.349315\pi$
$338$	0	0
$339$	−0.684658	−0.0371855
$340$	0	0
$341$	5.56155	0.301175
$342$	0	0
$343$	−13.2614	−0.716046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	22.7386	1.22067	0.610337	−	0.792142i	$-0.291034\pi$
$347$	22.7386	1.22067	0.610337	+	0.792142i	$0.291034\pi$
$348$	0	0
$349$	−32.2462	−1.72610	−0.863050	−	0.505118i	$-0.831449\pi$
$349$	−32.2462	−1.72610	−0.863050	+	0.505118i	$0.831449\pi$
$350$	0	0
$351$	28.4924	1.52081
$352$	0	0
$353$	24.0540	1.28026	0.640132	−	0.768265i	$-0.278880\pi$
$353$	24.0540	1.28026	0.640132	+	0.768265i	$0.278880\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.75379	0.516225
$358$	0	0
$359$	−4.49242	−0.237101	−0.118550	−	0.992948i	$-0.537825\pi$
$359$	−4.49242	−0.237101	−0.118550	+	0.992948i	$0.537825\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	−1.56155	−0.0819603
$364$	0	0
$365$	0	0
$366$	0	0
$367$	22.9309	1.19698	0.598491	−	0.801130i	$-0.295767\pi$
$367$	22.9309	1.19698	0.598491	+	0.801130i	$0.295767\pi$
$368$	0	0
$369$	0.630683	0.0328321
$370$	0	0
$371$	−38.2462	−1.98564
$372$	0	0
$373$	8.24621	0.426973	0.213486	−	0.976946i	$-0.431518\pi$
$373$	8.24621	0.426973	0.213486	+	0.976946i	$0.431518\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−26.2462	−1.35175
$378$	0	0
$379$	−0.192236	−0.00987450	−0.00493725	−	0.999988i	$-0.501572\pi$
$379$	−0.192236	−0.00987450	−0.00493725	+	0.999988i	$0.501572\pi$
$380$	0	0
$381$	−9.75379	−0.499702
$382$	0	0
$383$	−2.05398	−0.104953	−0.0524766	−	0.998622i	$-0.516712\pi$
$383$	−2.05398	−0.104953	−0.0524766	+	0.998622i	$0.516712\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	3.56155	0.180578	0.0902889	−	0.995916i	$-0.471221\pi$
$389$	3.56155	0.180578	0.0902889	+	0.995916i	$0.471221\pi$
$390$	0	0
$391$	−4.87689	−0.246635
$392$	0	0
$393$	20.8769	1.05310
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−10.4924	−0.526600	−0.263300	−	0.964714i	$-0.584811\pi$
$397$	−10.4924	−0.526600	−0.263300	+	0.964714i	$0.584811\pi$
$398$	0	0
$399$	−19.5076	−0.976600
$400$	0	0
$401$	30.4924	1.52272	0.761359	−	0.648330i	$-0.224532\pi$
$401$	30.4924	1.52272	0.761359	+	0.648330i	$0.224532\pi$
$402$	0	0
$403$	28.4924	1.41931
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.56155	0.374812
$408$	0	0
$409$	22.4924	1.11218	0.556089	−	0.831123i	$-0.312301\pi$
$409$	22.4924	1.11218	0.556089	+	0.831123i	$0.312301\pi$
$410$	0	0
$411$	−13.1771	−0.649977
$412$	0	0
$413$	−24.3845	−1.19988
$414$	0	0
$415$	0	0
$416$	0	0
$417$	23.6155	1.15646
$418$	0	0
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	0	0
$421$	2.49242	0.121473	0.0607366	−	0.998154i	$-0.480655\pi$
$421$	2.49242	0.121473	0.0607366	+	0.998154i	$0.480655\pi$
$422$	0	0
$423$	−4.49242	−0.218429
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.50758	0.169744
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−27.1231	−1.30647	−0.653237	−	0.757153i	$-0.726589\pi$
$431$	−27.1231	−1.30647	−0.653237	+	0.757153i	$0.726589\pi$
$432$	0	0
$433$	22.6847	1.09016	0.545078	−	0.838386i	$-0.316500\pi$
$433$	22.6847	1.09016	0.545078	+	0.838386i	$0.316500\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	9.75379	0.466587
$438$	0	0
$439$	−4.49242	−0.214412	−0.107206	−	0.994237i	$-0.534190\pi$
$439$	−4.49242	−0.214412	−0.107206	+	0.994237i	$0.534190\pi$
$440$	0	0
$441$	−1.54640	−0.0736380
$442$	0	0
$443$	11.3153	0.537608	0.268804	−	0.963195i	$-0.413372\pi$
$443$	11.3153	0.537608	0.268804	+	0.963195i	$0.413372\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−6.63068	−0.313621
$448$	0	0
$449$	−36.5464	−1.72473	−0.862366	−	0.506286i	$-0.831018\pi$
$449$	−36.5464	−1.72473	−0.862366	+	0.506286i	$0.831018\pi$
$450$	0	0
$451$	−1.12311	−0.0528850
$452$	0	0
$453$	−14.6307	−0.687409
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.8617	−1.11621	−0.558103	−	0.829772i	$-0.688470\pi$
$457$	−23.8617	−1.11621	−0.558103	+	0.829772i	$0.688470\pi$
$458$	0	0
$459$	−11.1231	−0.519182
$460$	0	0
$461$	1.12311	0.0523082	0.0261541	−	0.999658i	$-0.491674\pi$
$461$	1.12311	0.0523082	0.0261541	+	0.999658i	$0.491674\pi$
$462$	0	0
$463$	15.3153	0.711764	0.355882	−	0.934531i	$-0.384180\pi$
$463$	15.3153	0.711764	0.355882	+	0.934531i	$0.384180\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	28.3002	1.30958	0.654788	−	0.755812i	$-0.272758\pi$
$467$	28.3002	1.30958	0.654788	+	0.755812i	$0.272758\pi$
$468$	0	0
$469$	29.8617	1.37889
$470$	0	0
$471$	−6.93087	−0.319358
$472$	0	0
$473$	−7.12311	−0.327521
$474$	0	0
$475$	0	0
$476$	0	0
$477$	6.87689	0.314871
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	38.7386	1.76633
$482$	0	0
$483$	−11.8920	−0.541107
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.9309	0.676582	0.338291	−	0.941041i	$-0.390151\pi$
$487$	14.9309	0.676582	0.338291	+	0.941041i	$0.390151\pi$
$488$	0	0
$489$	6.24621	0.282463
$490$	0	0
$491$	−13.7538	−0.620700	−0.310350	−	0.950622i	$-0.600446\pi$
$491$	−13.7538	−0.620700	−0.310350	+	0.950622i	$0.600446\pi$
$492$	0	0
$493$	10.2462	0.461466
$494$	0	0
$495$	0	0
$496$	0	0
$497$	27.1231	1.21664
$498$	0	0
$499$	28.9848	1.29754	0.648770	−	0.760985i	$-0.275284\pi$
$499$	28.9848	1.29754	0.648770	+	0.760985i	$0.275284\pi$
$500$	0	0
$501$	−12.4924	−0.558120
$502$	0	0
$503$	−31.6155	−1.40967	−0.704833	−	0.709373i	$-0.748978\pi$
$503$	−31.6155	−1.40967	−0.704833	+	0.709373i	$0.748978\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−20.6847	−0.918638
$508$	0	0
$509$	−18.3002	−0.811142	−0.405571	−	0.914064i	$-0.632927\pi$
$509$	−18.3002	−0.811142	−0.405571	+	0.914064i	$0.632927\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	22.2462	0.982194
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	19.1231	0.839411
$520$	0	0
$521$	1.31534	0.0576262	0.0288131	−	0.999585i	$-0.490827\pi$
$521$	1.31534	0.0576262	0.0288131	+	0.999585i	$0.490827\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−11.1231	−0.484530
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	4.38447	0.190270
$532$	0	0
$533$	−5.75379	−0.249224
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−10.0540	−0.433861
$538$	0	0
$539$	2.75379	0.118614
$540$	0	0
$541$	−23.8617	−1.02590	−0.512948	−	0.858420i	$-0.671447\pi$
$541$	−23.8617	−1.02590	−0.512948	+	0.858420i	$0.671447\pi$
$542$	0	0
$543$	2.05398	0.0881445
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−42.2462	−1.80632	−0.903159	−	0.429307i	$-0.858758\pi$
$547$	−42.2462	−1.80632	−0.903159	+	0.429307i	$0.858758\pi$
$548$	0	0
$549$	−0.630683	−0.0269169
$550$	0	0
$551$	−20.4924	−0.873007
$552$	0	0
$553$	34.7386	1.47724
$554$	0	0
$555$	0	0
$556$	0	0
$557$	3.75379	0.159053	0.0795266	−	0.996833i	$-0.474659\pi$
$557$	3.75379	0.159053	0.0795266	+	0.996833i	$0.474659\pi$
$558$	0	0
$559$	−36.4924	−1.54347
$560$	0	0
$561$	3.12311	0.131858
$562$	0	0
$563$	24.4924	1.03223	0.516116	−	0.856519i	$-0.327377\pi$
$563$	24.4924	1.03223	0.516116	+	0.856519i	$0.327377\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−21.8617	−0.918107
$568$	0	0
$569$	−26.8769	−1.12674	−0.563369	−	0.826205i	$-0.690495\pi$
$569$	−26.8769	−1.12674	−0.563369	+	0.826205i	$0.690495\pi$
$570$	0	0
$571$	−16.4924	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$571$	−16.4924	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$572$	0	0
$573$	−16.3002	−0.680950
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−15.5616	−0.647836	−0.323918	−	0.946085i	$-0.605000\pi$
$577$	−15.5616	−0.647836	−0.323918	+	0.946085i	$0.605000\pi$
$578$	0	0
$579$	−14.2462	−0.592052
$580$	0	0
$581$	2.73863	0.113618
$582$	0	0
$583$	−12.2462	−0.507186
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.4924	−1.01091	−0.505455	−	0.862853i	$-0.668675\pi$
$587$	−24.4924	−1.01091	−0.505455	+	0.862853i	$0.668675\pi$
$588$	0	0
$589$	22.2462	0.916639
$590$	0	0
$591$	−22.6307	−0.930902
$592$	0	0
$593$	−3.36932	−0.138361	−0.0691806	−	0.997604i	$-0.522038\pi$
$593$	−3.36932	−0.138361	−0.0691806	+	0.997604i	$0.522038\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−19.5076	−0.798392
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	3.75379	0.153120	0.0765601	−	0.997065i	$-0.475606\pi$
$601$	3.75379	0.153120	0.0765601	+	0.997065i	$0.475606\pi$
$602$	0	0
$603$	−5.36932	−0.218655
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−45.8617	−1.86147	−0.930735	−	0.365694i	$-0.880832\pi$
$607$	−45.8617	−1.86147	−0.930735	+	0.365694i	$0.880832\pi$
$608$	0	0
$609$	24.9848	1.01244
$610$	0	0
$611$	40.9848	1.65807
$612$	0	0
$613$	−11.8617	−0.479091	−0.239546	−	0.970885i	$-0.576998\pi$
$613$	−11.8617	−0.479091	−0.239546	+	0.970885i	$0.576998\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.49242	0.100341	0.0501706	−	0.998741i	$-0.484024\pi$
$617$	2.49242	0.100341	0.0501706	+	0.998741i	$0.484024\pi$
$618$	0	0
$619$	−18.9309	−0.760896	−0.380448	−	0.924802i	$-0.624230\pi$
$619$	−18.9309	−0.760896	−0.380448	+	0.924802i	$0.624230\pi$
$620$	0	0
$621$	13.5616	0.544206
$622$	0	0
$623$	8.38447	0.335917
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6.24621	−0.249450
$628$	0	0
$629$	−15.1231	−0.602998
$630$	0	0
$631$	42.0540	1.67414	0.837071	−	0.547094i	$-0.184266\pi$
$631$	42.0540	1.67414	0.837071	+	0.547094i	$0.184266\pi$
$632$	0	0
$633$	13.2614	0.527092
$634$	0	0
$635$	0	0
$636$	0	0
$637$	14.1080	0.558977
$638$	0	0
$639$	−4.87689	−0.192927
$640$	0	0
$641$	−46.3002	−1.82875	−0.914374	−	0.404871i	$-0.867316\pi$
$641$	−46.3002	−1.82875	−0.914374	+	0.404871i	$0.867316\pi$
$642$	0	0
$643$	−9.17708	−0.361909	−0.180954	−	0.983491i	$-0.557919\pi$
$643$	−9.17708	−0.361909	−0.180954	+	0.983491i	$0.557919\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.5616	0.533160	0.266580	−	0.963813i	$-0.414106\pi$
$647$	13.5616	0.533160	0.266580	+	0.963813i	$0.414106\pi$
$648$	0	0
$649$	−7.80776	−0.306482
$650$	0	0
$651$	−27.1231	−1.06304
$652$	0	0
$653$	−35.1771	−1.37659	−0.688293	−	0.725433i	$-0.741640\pi$
$653$	−35.1771	−1.37659	−0.688293	+	0.725433i	$0.741640\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	2.87689	0.112238
$658$	0	0
$659$	11.6155	0.452477	0.226238	−	0.974072i	$-0.427357\pi$
$659$	11.6155	0.452477	0.226238	+	0.974072i	$0.427357\pi$
$660$	0	0
$661$	41.8078	1.62613	0.813067	−	0.582170i	$-0.197796\pi$
$661$	41.8078	1.62613	0.813067	+	0.582170i	$0.197796\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.4924	−0.483709
$668$	0	0
$669$	18.4384	0.712872
$670$	0	0
$671$	1.12311	0.0433570
$672$	0	0
$673$	−33.2311	−1.28096	−0.640482	−	0.767974i	$-0.721265\pi$
$673$	−33.2311	−1.28096	−0.640482	+	0.767974i	$0.721265\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.7386	0.797050	0.398525	−	0.917157i	$-0.369522\pi$
$677$	20.7386	0.797050	0.398525	+	0.917157i	$0.369522\pi$
$678$	0	0
$679$	−48.6004	−1.86511
$680$	0	0
$681$	−36.1080	−1.38366
$682$	0	0
$683$	6.73863	0.257847	0.128923	−	0.991655i	$-0.458848\pi$
$683$	6.73863	0.257847	0.128923	+	0.991655i	$0.458848\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.9309	−0.874867
$688$	0	0
$689$	−62.7386	−2.39015
$690$	0	0
$691$	9.94602	0.378365	0.189182	−	0.981942i	$-0.439416\pi$
$691$	9.94602	0.378365	0.189182	+	0.981942i	$0.439416\pi$
$692$	0	0
$693$	−1.75379	−0.0666209
$694$	0	0
$695$	0	0
$696$	0	0
$697$	2.24621	0.0850813
$698$	0	0
$699$	−11.5076	−0.435257
$700$	0	0
$701$	50.4924	1.90707	0.953536	−	0.301278i	$-0.0974133\pi$
$701$	50.4924	1.90707	0.953536	+	0.301278i	$0.0974133\pi$
$702$	0	0
$703$	30.2462	1.14076
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6.24621	−0.234913
$708$	0	0
$709$	2.19224	0.0823311	0.0411656	−	0.999152i	$-0.486893\pi$
$709$	2.19224	0.0823311	0.0411656	+	0.999152i	$0.486893\pi$
$710$	0	0
$711$	−6.24621	−0.234251
$712$	0	0
$713$	13.5616	0.507884
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−7.61553	−0.284407
$718$	0	0
$719$	−35.4233	−1.32107	−0.660533	−	0.750797i	$-0.729670\pi$
$719$	−35.4233	−1.32107	−0.660533	+	0.750797i	$0.729670\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−45.4773	−1.69132
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−23.3153	−0.864718	−0.432359	−	0.901702i	$-0.642319\pi$
$727$	−23.3153	−0.864718	−0.432359	+	0.901702i	$0.642319\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	14.2462	0.526915
$732$	0	0
$733$	−1.12311	−0.0414829	−0.0207414	−	0.999785i	$-0.506603\pi$
$733$	−1.12311	−0.0414829	−0.0207414	+	0.999785i	$0.506603\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.56155	0.352204
$738$	0	0
$739$	2.63068	0.0967712	0.0483856	−	0.998829i	$-0.484592\pi$
$739$	2.63068	0.0967712	0.0483856	+	0.998829i	$0.484592\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	0	0
$743$	10.7386	0.393962	0.196981	−	0.980407i	$-0.436886\pi$
$743$	10.7386	0.393962	0.196981	+	0.980407i	$0.436886\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−0.492423	−0.0180168
$748$	0	0
$749$	−41.7538	−1.52565
$750$	0	0
$751$	−5.56155	−0.202944	−0.101472	−	0.994838i	$-0.532355\pi$
$751$	−5.56155	−0.202944	−0.101472	+	0.994838i	$0.532355\pi$
$752$	0	0
$753$	2.43845	0.0888620
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−15.7538	−0.572581	−0.286291	−	0.958143i	$-0.592422\pi$
$757$	−15.7538	−0.572581	−0.286291	+	0.958143i	$0.592422\pi$
$758$	0	0
$759$	−3.80776	−0.138213
$760$	0	0
$761$	5.12311	0.185712	0.0928562	−	0.995680i	$-0.470400\pi$
$761$	5.12311	0.185712	0.0928562	+	0.995680i	$0.470400\pi$
$762$	0	0
$763$	38.2462	1.38461
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	25.6155	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$769$	25.6155	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$770$	0	0
$771$	18.3542	0.661009
$772$	0	0
$773$	−40.7386	−1.46527	−0.732633	−	0.680623i	$-0.761709\pi$
$773$	−40.7386	−1.46527	−0.732633	+	0.680623i	$0.761709\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−36.8769	−1.32295
$778$	0	0
$779$	−4.49242	−0.160958
$780$	0	0
$781$	8.68466	0.310762
$782$	0	0
$783$	−28.4924	−1.01824
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−29.7538	−1.06061	−0.530304	−	0.847808i	$-0.677922\pi$
$787$	−29.7538	−1.06061	−0.530304	+	0.847808i	$0.677922\pi$
$788$	0	0
$789$	−29.8617	−1.06311
$790$	0	0
$791$	1.36932	0.0486873
$792$	0	0
$793$	5.75379	0.204323
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.1922	0.502715	0.251357	−	0.967894i	$-0.419123\pi$
$797$	14.1922	0.502715	0.251357	+	0.967894i	$0.419123\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−1.50758	−0.0532676
$802$	0	0
$803$	−5.12311	−0.180790
$804$	0	0
$805$	0	0
$806$	0	0
$807$	32.3845	1.13999
$808$	0	0
$809$	−45.6155	−1.60376	−0.801878	−	0.597487i	$-0.796166\pi$
$809$	−45.6155	−1.60376	−0.801878	+	0.597487i	$0.796166\pi$
$810$	0	0
$811$	7.12311	0.250126	0.125063	−	0.992149i	$-0.460087\pi$
$811$	7.12311	0.250126	0.125063	+	0.992149i	$0.460087\pi$
$812$	0	0
$813$	−44.4924	−1.56042
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−28.4924	−0.996824
$818$	0	0
$819$	−8.98485	−0.313956
$820$	0	0
$821$	−42.9848	−1.50018	−0.750091	−	0.661335i	$-0.769990\pi$
$821$	−42.9848	−1.50018	−0.750091	+	0.661335i	$0.769990\pi$
$822$	0	0
$823$	−54.5464	−1.90137	−0.950684	−	0.310161i	$-0.899617\pi$
$823$	−54.5464	−1.90137	−0.950684	+	0.310161i	$0.899617\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	38.7386	1.34707	0.673537	−	0.739153i	$-0.264774\pi$
$827$	38.7386	1.34707	0.673537	+	0.739153i	$0.264774\pi$
$828$	0	0
$829$	15.0691	0.523373	0.261686	−	0.965153i	$-0.415721\pi$
$829$	15.0691	0.523373	0.261686	+	0.965153i	$0.415721\pi$
$830$	0	0
$831$	−28.1080	−0.975054
$832$	0	0
$833$	−5.50758	−0.190826
$834$	0	0
$835$	0	0
$836$	0	0
$837$	30.9309	1.06913
$838$	0	0
$839$	19.8078	0.683840	0.341920	−	0.939729i	$-0.388923\pi$
$839$	19.8078	0.683840	0.341920	+	0.939729i	$0.388923\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	0	0
$843$	−25.3693	−0.873766
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.12311	0.107311
$848$	0	0
$849$	−31.2311	−1.07185
$850$	0	0
$851$	18.4384	0.632062
$852$	0	0
$853$	46.4924	1.59187	0.795935	−	0.605382i	$-0.206980\pi$
$853$	46.4924	1.59187	0.795935	+	0.605382i	$0.206980\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.1080	−1.02847	−0.514234	−	0.857650i	$-0.671924\pi$
$857$	−30.1080	−1.02847	−0.514234	+	0.857650i	$0.671924\pi$
$858$	0	0
$859$	−30.0540	−1.02543	−0.512714	−	0.858559i	$-0.671360\pi$
$859$	−30.0540	−1.02543	−0.512714	+	0.858559i	$0.671360\pi$
$860$	0	0
$861$	5.47727	0.186665
$862$	0	0
$863$	−36.4924	−1.24222	−0.621108	−	0.783725i	$-0.713317\pi$
$863$	−36.4924	−1.24222	−0.621108	+	0.783725i	$0.713317\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	20.3002	0.689430
$868$	0	0
$869$	11.1231	0.377326
$870$	0	0
$871$	48.9848	1.65979
$872$	0	0
$873$	8.73863	0.295758
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−55.3693	−1.86969	−0.934844	−	0.355057i	$-0.884461\pi$
$877$	−55.3693	−1.86969	−0.934844	+	0.355057i	$0.884461\pi$
$878$	0	0
$879$	−5.26137	−0.177461
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	0	0
$883$	8.49242	0.285793	0.142896	−	0.989738i	$-0.454358\pi$
$883$	8.49242	0.285793	0.142896	+	0.989738i	$0.454358\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	31.6155	1.06155	0.530773	−	0.847514i	$-0.321902\pi$
$887$	31.6155	1.06155	0.530773	+	0.847514i	$0.321902\pi$
$888$	0	0
$889$	19.5076	0.654263
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−19.5076	−0.651339
$898$	0	0
$899$	−28.4924	−0.950275
$900$	0	0
$901$	24.4924	0.815961
$902$	0	0
$903$	34.7386	1.15603
$904$	0	0
$905$	0	0
$906$	0	0
$907$	16.4924	0.547622	0.273811	−	0.961784i	$-0.411716\pi$
$907$	16.4924	0.547622	0.273811	+	0.961784i	$0.411716\pi$
$908$	0	0
$909$	1.12311	0.0372511
$910$	0	0
$911$	26.7386	0.885890	0.442945	−	0.896549i	$-0.353934\pi$
$911$	26.7386	0.885890	0.442945	+	0.896549i	$0.353934\pi$
$912$	0	0
$913$	0.876894	0.0290210
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−41.7538	−1.37883
$918$	0	0
$919$	−6.63068	−0.218726	−0.109363	−	0.994002i	$-0.534881\pi$
$919$	−6.63068	−0.218726	−0.109363	+	0.994002i	$0.534881\pi$
$920$	0	0
$921$	50.7386	1.67189
$922$	0	0
$923$	44.4924	1.46449
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.4924	1.52537	0.762683	−	0.646772i	$-0.223881\pi$
$929$	46.4924	1.52537	0.762683	+	0.646772i	$0.223881\pi$
$930$	0	0
$931$	11.0152	0.361007
$932$	0	0
$933$	15.2311	0.498642
$934$	0	0
$935$	0	0
$936$	0	0
$937$	42.1080	1.37561	0.687803	−	0.725897i	$-0.258575\pi$
$937$	42.1080	1.37561	0.687803	+	0.725897i	$0.258575\pi$
$938$	0	0
$939$	−15.3153	−0.499797
$940$	0	0
$941$	−32.2462	−1.05120	−0.525598	−	0.850733i	$-0.676158\pi$
$941$	−32.2462	−1.05120	−0.525598	+	0.850733i	$0.676158\pi$
$942$	0	0
$943$	−2.73863	−0.0891822
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−12.6847	−0.412196	−0.206098	−	0.978531i	$-0.566077\pi$
$947$	−12.6847	−0.412196	−0.206098	+	0.978531i	$0.566077\pi$
$948$	0	0
$949$	−26.2462	−0.851988
$950$	0	0
$951$	−22.1619	−0.718650
$952$	0	0
$953$	−0.246211	−0.00797556	−0.00398778	−	0.999992i	$-0.501269\pi$
$953$	−0.246211	−0.00797556	−0.00398778	+	0.999992i	$0.501269\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.00000	0.258603
$958$	0	0
$959$	26.3542	0.851020
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	7.50758	0.241928
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	12.4924	0.401314
$970$	0	0
$971$	34.5464	1.10865	0.554323	−	0.832301i	$-0.312977\pi$
$971$	34.5464	1.10865	0.554323	+	0.832301i	$0.312977\pi$
$972$	0	0
$973$	−47.2311	−1.51416
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53.8078	−1.72146	−0.860731	−	0.509059i	$-0.829993\pi$
$977$	−53.8078	−1.72146	−0.860731	+	0.509059i	$0.829993\pi$
$978$	0	0
$979$	2.68466	0.0858021
$980$	0	0
$981$	−6.87689	−0.219562
$982$	0	0
$983$	30.9309	0.986542	0.493271	−	0.869876i	$-0.335801\pi$
$983$	30.9309	0.986542	0.493271	+	0.869876i	$0.335801\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−39.0152	−1.24187
$988$	0	0
$989$	−17.3693	−0.552312
$990$	0	0
$991$	4.49242	0.142707	0.0713533	−	0.997451i	$-0.477268\pi$
$991$	4.49242	0.142707	0.0713533	+	0.997451i	$0.477268\pi$
$992$	0	0
$993$	54.5464	1.73098
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−52.2462	−1.65465	−0.827327	−	0.561721i	$-0.810140\pi$
$997$	−52.2462	−1.65465	−0.827327	+	0.561721i	$0.810140\pi$
$998$	0	0
$999$	42.0540	1.33053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bp.1.1		2
4.3	odd	2		2200.2.a.o.1.2		2
5.2	odd	4		4400.2.b.v.4049.3		4
5.3	odd	4		4400.2.b.v.4049.2		4
5.4	even	2		176.2.a.d.1.2		2
15.14	odd	2		1584.2.a.t.1.1		2
20.3	even	4		2200.2.b.g.1849.3		4
20.7	even	4		2200.2.b.g.1849.2		4
20.19	odd	2		88.2.a.b.1.1	✓	2
35.34	odd	2		8624.2.a.cb.1.1		2
40.19	odd	2		704.2.a.m.1.2		2
40.29	even	2		704.2.a.p.1.1		2
55.54	odd	2		1936.2.a.r.1.2		2
60.59	even	2		792.2.a.h.1.1		2
80.19	odd	4		2816.2.c.w.1409.2		4
80.29	even	4		2816.2.c.p.1409.3		4
80.59	odd	4		2816.2.c.w.1409.3		4
80.69	even	4		2816.2.c.p.1409.2		4
120.29	odd	2		6336.2.a.cx.1.2		2
120.59	even	2		6336.2.a.cu.1.2		2
140.139	even	2		4312.2.a.n.1.2		2
220.19	even	10		968.2.i.q.9.2		8
220.39	even	10		968.2.i.q.729.1		8
220.59	odd	10		968.2.i.r.753.2		8
220.79	even	10		968.2.i.q.81.1		8
220.119	odd	10		968.2.i.r.81.1		8
220.139	even	10		968.2.i.q.753.2		8
220.159	odd	10		968.2.i.r.729.1		8
220.179	odd	10		968.2.i.r.9.2		8
220.219	even	2		968.2.a.j.1.1		2
440.109	odd	2		7744.2.a.cl.1.1		2
440.219	even	2		7744.2.a.by.1.2		2
660.659	odd	2		8712.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.b.1.1	✓	2	20.19	odd	2
176.2.a.d.1.2		2	5.4	even	2
704.2.a.m.1.2		2	40.19	odd	2
704.2.a.p.1.1		2	40.29	even	2
792.2.a.h.1.1		2	60.59	even	2
968.2.a.j.1.1		2	220.219	even	2
968.2.i.q.9.2		8	220.19	even	10
968.2.i.q.81.1		8	220.79	even	10
968.2.i.q.729.1		8	220.39	even	10
968.2.i.q.753.2		8	220.139	even	10
968.2.i.r.9.2		8	220.179	odd	10
968.2.i.r.81.1		8	220.119	odd	10
968.2.i.r.729.1		8	220.159	odd	10
968.2.i.r.753.2		8	220.59	odd	10
1584.2.a.t.1.1		2	15.14	odd	2
1936.2.a.r.1.2		2	55.54	odd	2
2200.2.a.o.1.2		2	4.3	odd	2
2200.2.b.g.1849.2		4	20.7	even	4
2200.2.b.g.1849.3		4	20.3	even	4
2816.2.c.p.1409.2		4	80.69	even	4
2816.2.c.p.1409.3		4	80.29	even	4
2816.2.c.w.1409.2		4	80.19	odd	4
2816.2.c.w.1409.3		4	80.59	odd	4
4312.2.a.n.1.2		2	140.139	even	2
4400.2.a.bp.1.1		2	1.1	even	1	trivial
4400.2.b.v.4049.2		4	5.3	odd	4
4400.2.b.v.4049.3		4	5.2	odd	4
6336.2.a.cu.1.2		2	120.59	even	2
6336.2.a.cx.1.2		2	120.29	odd	2
7744.2.a.by.1.2		2	440.219	even	2
7744.2.a.cl.1.1		2	440.109	odd	2
8624.2.a.cb.1.1		2	35.34	odd	2
8712.2.a.bb.1.1		2	660.659	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} +3.12311 q^{7} -0.561553 q^{9} +1.00000 q^{11} +5.12311 q^{13} -2.00000 q^{17} +4.00000 q^{19} -4.87689 q^{21} +2.43845 q^{23} +5.56155 q^{27} -5.12311 q^{29} +5.56155 q^{31} -1.56155 q^{33} +7.56155 q^{37} -8.00000 q^{39} -1.12311 q^{41} -7.12311 q^{43} +8.00000 q^{47} +2.75379 q^{49} +3.12311 q^{51} -12.2462 q^{53} -6.24621 q^{57} -7.80776 q^{59} +1.12311 q^{61} -1.75379 q^{63} +9.56155 q^{67} -3.80776 q^{69} +8.68466 q^{71} -5.12311 q^{73} +3.12311 q^{77} +11.1231 q^{79} -7.00000 q^{81} +0.876894 q^{83} +8.00000 q^{87} +2.68466 q^{89} +16.0000 q^{91} -8.68466 q^{93} -15.5616 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} - 18 q^{21} + 9 q^{23} + 7 q^{27} - 2 q^{29} + 7 q^{31} + q^{33} + 11 q^{37} - 16 q^{39} + 6 q^{41} - 6 q^{43} + 16 q^{47} + 22 q^{49}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q + q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 8 * q^19 - 18 * q^21 + 9 * q^23 + 7 * q^27 - 2 * q^29 + 7 * q^31 + q^33 + 11 * q^37 - 16 * q^39 + 6 * q^41 - 6 * q^43 + 16 * q^47 + 22 * q^49 - 2 * q^51 - 8 * q^53 + 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 15 * q^67 + 13 * q^69 + 5 * q^71 - 2 * q^73 - 2 * q^77 + 14 * q^79 - 14 * q^81 + 10 * q^83 + 16 * q^87 - 7 * q^89 + 32 * q^91 - 5 * q^93 - 27 * q^97 + 3 * q^99

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