LMFDB - Embedded newform 4400.2.a.bf.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:	$1$
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			$2.30278$ 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)

$f(q)$	$=$	$q-3.30278 q^{3} -4.30278 q^{7} +7.90833 q^{9} -1.00000 q^{11} -5.60555 q^{13} +0.697224 q^{17} +1.00000 q^{19} +14.2111 q^{21} +6.90833 q^{23} -16.2111 q^{27} -5.30278 q^{29} +5.60555 q^{31} +3.30278 q^{33} +0.394449 q^{37} +18.5139 q^{39} -6.21110 q^{41} +7.21110 q^{43} +1.60555 q^{47} +11.5139 q^{49} -2.30278 q^{51} +11.5139 q^{53} -3.30278 q^{57} -1.60555 q^{59} -0.302776 q^{61} -34.0278 q^{63} -8.00000 q^{67} -22.8167 q^{69} +4.60555 q^{71} +8.90833 q^{73} +4.30278 q^{77} -2.69722 q^{79} +29.8167 q^{81} +3.90833 q^{83} +17.5139 q^{87} -15.9083 q^{89} +24.1194 q^{91} -18.5139 q^{93} +18.1194 q^{97} -7.90833 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.30278	−1.90686	−0.953429	−	0.301617i	$-0.902474\pi$
$3$	−3.30278	−1.90686	−0.953429	+	0.301617i	$0.902474\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.30278	−1.62630	−0.813148	−	0.582057i	$-0.802248\pi$
$7$	−4.30278	−1.62630	−0.813148	+	0.582057i	$0.802248\pi$
$8$	0	0
$9$	7.90833	2.63611
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.60555	−1.55470	−0.777350	−	0.629068i	$-0.783437\pi$
$13$	−5.60555	−1.55470	−0.777350	+	0.629068i	$0.783437\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.697224	0.169102	0.0845509	−	0.996419i	$-0.473054\pi$
$17$	0.697224	0.169102	0.0845509	+	0.996419i	$0.473054\pi$
$18$	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
$21$	14.2111	3.10112
$22$	0	0
$23$	6.90833	1.44049	0.720243	−	0.693722i	$-0.244030\pi$
$23$	6.90833	1.44049	0.720243	+	0.693722i	$0.244030\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−16.2111	−3.11983
$28$	0	0
$29$	−5.30278	−0.984701	−0.492350	−	0.870397i	$-0.663862\pi$
$29$	−5.30278	−0.984701	−0.492350	+	0.870397i	$0.663862\pi$
$30$	0	0
$31$	5.60555	1.00679	0.503393	−	0.864057i	$-0.332085\pi$
$31$	5.60555	1.00679	0.503393	+	0.864057i	$0.332085\pi$
$32$	0	0
$33$	3.30278	0.574939
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.394449	0.0648470	0.0324235	−	0.999474i	$-0.489677\pi$
$37$	0.394449	0.0648470	0.0324235	+	0.999474i	$0.489677\pi$
$38$	0	0
$39$	18.5139	2.96459
$40$	0	0
$41$	−6.21110	−0.970011	−0.485006	−	0.874511i	$-0.661182\pi$
$41$	−6.21110	−0.970011	−0.485006	+	0.874511i	$0.661182\pi$
$42$	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
$47$	1.60555	0.234194	0.117097	−	0.993120i	$-0.462641\pi$
$47$	1.60555	0.234194	0.117097	+	0.993120i	$0.462641\pi$
$48$	0	0
$49$	11.5139	1.64484
$50$	0	0
$51$	−2.30278	−0.322453
$52$	0	0
$53$	11.5139	1.58155	0.790776	−	0.612105i	$-0.209677\pi$
$53$	11.5139	1.58155	0.790776	+	0.612105i	$0.209677\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.30278	−0.437463
$58$	0	0
$59$	−1.60555	−0.209025	−0.104512	−	0.994524i	$-0.533328\pi$
$59$	−1.60555	−0.209025	−0.104512	+	0.994524i	$0.533328\pi$
$60$	0	0
$61$	−0.302776	−0.0387664	−0.0193832	−	0.999812i	$-0.506170\pi$
$61$	−0.302776	−0.0387664	−0.0193832	+	0.999812i	$0.506170\pi$
$62$	0	0
$63$	−34.0278	−4.28709
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	−22.8167	−2.74680
$70$	0	0
$71$	4.60555	0.546578	0.273289	−	0.961932i	$-0.411888\pi$
$71$	4.60555	0.546578	0.273289	+	0.961932i	$0.411888\pi$
$72$	0	0
$73$	8.90833	1.04264	0.521320	−	0.853361i	$-0.325440\pi$
$73$	8.90833	1.04264	0.521320	+	0.853361i	$0.325440\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.30278	0.490347
$78$	0	0
$79$	−2.69722	−0.303461	−0.151731	−	0.988422i	$-0.548485\pi$
$79$	−2.69722	−0.303461	−0.151731	+	0.988422i	$0.548485\pi$
$80$	0	0
$81$	29.8167	3.31296
$82$	0	0
$83$	3.90833	0.428995	0.214497	−	0.976725i	$-0.431189\pi$
$83$	3.90833	0.428995	0.214497	+	0.976725i	$0.431189\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	17.5139	1.87768
$88$	0	0
$89$	−15.9083	−1.68628	−0.843140	−	0.537695i	$-0.819295\pi$
$89$	−15.9083	−1.68628	−0.843140	+	0.537695i	$0.819295\pi$
$90$	0	0
$91$	24.1194	2.52840
$92$	0	0
$93$	−18.5139	−1.91980
$94$	0	0
$95$	0	0
$96$	0	0
$97$	18.1194	1.83975	0.919875	−	0.392212i	$-0.128290\pi$
$97$	18.1194	1.83975	0.919875	+	0.392212i	$0.128290\pi$
$98$	0	0
$99$	−7.90833	−0.794817
$100$	0	0
$101$	11.5139	1.14567	0.572837	−	0.819669i	$-0.305843\pi$
$101$	11.5139	1.14567	0.572837	+	0.819669i	$0.305843\pi$
$102$	0	0
$103$	−8.69722	−0.856963	−0.428481	−	0.903551i	$-0.640951\pi$
$103$	−8.69722	−0.856963	−0.428481	+	0.903551i	$0.640951\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.2111	1.18049	0.590246	−	0.807223i	$-0.299031\pi$
$107$	12.2111	1.18049	0.590246	+	0.807223i	$0.299031\pi$
$108$	0	0
$109$	−1.90833	−0.182785	−0.0913923	−	0.995815i	$-0.529132\pi$
$109$	−1.90833	−0.182785	−0.0913923	+	0.995815i	$0.529132\pi$
$110$	0	0
$111$	−1.30278	−0.123654
$112$	0	0
$113$	1.60555	0.151038	0.0755188	−	0.997144i	$-0.475939\pi$
$113$	1.60555	0.151038	0.0755188	+	0.997144i	$0.475939\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−44.3305	−4.09836
$118$	0	0
$119$	−3.00000	−0.275010
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	20.5139	1.84967
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.09167	−0.363077	−0.181539	−	0.983384i	$-0.558108\pi$
$127$	−4.09167	−0.363077	−0.181539	+	0.983384i	$0.558108\pi$
$128$	0	0
$129$	−23.8167	−2.09694
$130$	0	0
$131$	8.30278	0.725417	0.362708	−	0.931903i	$-0.381852\pi$
$131$	8.30278	0.725417	0.362708	+	0.931903i	$0.381852\pi$
$132$	0	0
$133$	−4.30278	−0.373098
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.90833	0.846525	0.423263	−	0.906007i	$-0.360885\pi$
$137$	9.90833	0.846525	0.423263	+	0.906007i	$0.360885\pi$
$138$	0	0
$139$	−1.78890	−0.151732	−0.0758662	−	0.997118i	$-0.524172\pi$
$139$	−1.78890	−0.151732	−0.0758662	+	0.997118i	$0.524172\pi$
$140$	0	0
$141$	−5.30278	−0.446574
$142$	0	0
$143$	5.60555	0.468760
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−38.0278	−3.13648
$148$	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
$151$	−6.81665	−0.554731	−0.277366	−	0.960764i	$-0.589461\pi$
$151$	−6.81665	−0.554731	−0.277366	+	0.960764i	$0.589461\pi$
$152$	0	0
$153$	5.51388	0.445771
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−0.788897	−0.0629609	−0.0314804	−	0.999504i	$-0.510022\pi$
$157$	−0.788897	−0.0629609	−0.0314804	+	0.999504i	$0.510022\pi$
$158$	0	0
$159$	−38.0278	−3.01580
$160$	0	0
$161$	−29.7250	−2.34266
$162$	0	0
$163$	0.302776	0.0237152	0.0118576	−	0.999930i	$-0.496226\pi$
$163$	0.302776	0.0237152	0.0118576	+	0.999930i	$0.496226\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6333	−1.44189	−0.720944	−	0.692993i	$-0.756292\pi$
$167$	−18.6333	−1.44189	−0.720944	+	0.692993i	$0.756292\pi$
$168$	0	0
$169$	18.4222	1.41709
$170$	0	0
$171$	7.90833	0.604765
$172$	0	0
$173$	−24.2111	−1.84074	−0.920368	−	0.391053i	$-0.872111\pi$
$173$	−24.2111	−1.84074	−0.920368	+	0.391053i	$0.872111\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	5.30278	0.398581
$178$	0	0
$179$	−18.9083	−1.41327	−0.706637	−	0.707576i	$-0.749789\pi$
$179$	−18.9083	−1.41327	−0.706637	+	0.707576i	$0.749789\pi$
$180$	0	0
$181$	2.90833	0.216174	0.108087	−	0.994141i	$-0.465527\pi$
$181$	2.90833	0.216174	0.108087	+	0.994141i	$0.465527\pi$
$182$	0	0
$183$	1.00000	0.0739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−0.697224	−0.0509861
$188$	0	0
$189$	69.7527	5.07377
$190$	0	0
$191$	−2.09167	−0.151348	−0.0756741	−	0.997133i	$-0.524111\pi$
$191$	−2.09167	−0.151348	−0.0756741	+	0.997133i	$0.524111\pi$
$192$	0	0
$193$	26.4222	1.90191	0.950956	−	0.309326i	$-0.100104\pi$
$193$	26.4222	1.90191	0.950956	+	0.309326i	$0.100104\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−11.0917	−0.790249	−0.395124	−	0.918628i	$-0.629299\pi$
$197$	−11.0917	−0.790249	−0.395124	+	0.918628i	$0.629299\pi$
$198$	0	0
$199$	7.90833	0.560606	0.280303	−	0.959912i	$-0.409565\pi$
$199$	7.90833	0.560606	0.280303	+	0.959912i	$0.409565\pi$
$200$	0	0
$201$	26.4222	1.86368
$202$	0	0
$203$	22.8167	1.60142
$204$	0	0
$205$	0	0
$206$	0	0
$207$	54.6333	3.79728
$208$	0	0
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	14.3944	0.990955	0.495477	−	0.868621i	$-0.334993\pi$
$211$	14.3944	0.990955	0.495477	+	0.868621i	$0.334993\pi$
$212$	0	0
$213$	−15.2111	−1.04225
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−24.1194	−1.63733
$218$	0	0
$219$	−29.4222	−1.98817
$220$	0	0
$221$	−3.90833	−0.262903
$222$	0	0
$223$	−6.39445	−0.428204	−0.214102	−	0.976811i	$-0.568682\pi$
$223$	−6.39445	−0.428204	−0.214102	+	0.976811i	$0.568682\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−20.9361	−1.38958	−0.694788	−	0.719214i	$-0.744502\pi$
$227$	−20.9361	−1.38958	−0.694788	+	0.719214i	$0.744502\pi$
$228$	0	0
$229$	0.880571	0.0581897	0.0290949	−	0.999577i	$-0.490738\pi$
$229$	0.880571	0.0581897	0.0290949	+	0.999577i	$0.490738\pi$
$230$	0	0
$231$	−14.2111	−0.935022
$232$	0	0
$233$	−24.9083	−1.63180	−0.815899	−	0.578194i	$-0.803758\pi$
$233$	−24.9083	−1.63180	−0.815899	+	0.578194i	$0.803758\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	8.90833	0.578658
$238$	0	0
$239$	25.3305	1.63850	0.819248	−	0.573439i	$-0.194391\pi$
$239$	25.3305	1.63850	0.819248	+	0.573439i	$0.194391\pi$
$240$	0	0
$241$	−4.69722	−0.302575	−0.151287	−	0.988490i	$-0.548342\pi$
$241$	−4.69722	−0.302575	−0.151287	+	0.988490i	$0.548342\pi$
$242$	0	0
$243$	−49.8444	−3.19752
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−5.60555	−0.356673
$248$	0	0
$249$	−12.9083	−0.818032
$250$	0	0
$251$	−16.3305	−1.03077	−0.515387	−	0.856958i	$-0.672352\pi$
$251$	−16.3305	−1.03077	−0.515387	+	0.856958i	$0.672352\pi$
$252$	0	0
$253$	−6.90833	−0.434323
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−0.422205	−0.0263364	−0.0131682	−	0.999913i	$-0.504192\pi$
$257$	−0.422205	−0.0263364	−0.0131682	+	0.999913i	$0.504192\pi$
$258$	0	0
$259$	−1.69722	−0.105460
$260$	0	0
$261$	−41.9361	−2.59578
$262$	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
$267$	52.5416	3.21550
$268$	0	0
$269$	−14.5139	−0.884927	−0.442463	−	0.896787i	$-0.645895\pi$
$269$	−14.5139	−0.884927	−0.442463	+	0.896787i	$0.645895\pi$
$270$	0	0
$271$	0.577795	0.0350985	0.0175493	−	0.999846i	$-0.494414\pi$
$271$	0.577795	0.0350985	0.0175493	+	0.999846i	$0.494414\pi$
$272$	0	0
$273$	−79.6611	−4.82131
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−15.7889	−0.948663	−0.474331	−	0.880346i	$-0.657310\pi$
$277$	−15.7889	−0.948663	−0.474331	+	0.880346i	$0.657310\pi$
$278$	0	0
$279$	44.3305	2.65400
$280$	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
$283$	−5.48612	−0.326116	−0.163058	−	0.986616i	$-0.552136\pi$
$283$	−5.48612	−0.326116	−0.163058	+	0.986616i	$0.552136\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26.7250	1.57753
$288$	0	0
$289$	−16.5139	−0.971405
$290$	0	0
$291$	−59.8444	−3.50814
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	16.2111	0.940664
$298$	0	0
$299$	−38.7250	−2.23952
$300$	0	0
$301$	−31.0278	−1.78841
$302$	0	0
$303$	−38.0278	−2.18464
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−24.3305	−1.38862	−0.694308	−	0.719678i	$-0.744290\pi$
$307$	−24.3305	−1.38862	−0.694308	+	0.719678i	$0.744290\pi$
$308$	0	0
$309$	28.7250	1.63411
$310$	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
$313$	15.3944	0.870146	0.435073	−	0.900395i	$-0.356723\pi$
$313$	15.3944	0.870146	0.435073	+	0.900395i	$0.356723\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−23.3028	−1.30881	−0.654407	−	0.756142i	$-0.727082\pi$
$317$	−23.3028	−1.30881	−0.654407	+	0.756142i	$0.727082\pi$
$318$	0	0
$319$	5.30278	0.296898
$320$	0	0
$321$	−40.3305	−2.25103
$322$	0	0
$323$	0.697224	0.0387946
$324$	0	0
$325$	0	0
$326$	0	0
$327$	6.30278	0.348544
$328$	0	0
$329$	−6.90833	−0.380868
$330$	0	0
$331$	28.6333	1.57383	0.786914	−	0.617062i	$-0.211677\pi$
$331$	28.6333	1.57383	0.786914	+	0.617062i	$0.211677\pi$
$332$	0	0
$333$	3.11943	0.170944
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−19.2111	−1.04650	−0.523248	−	0.852181i	$-0.675280\pi$
$337$	−19.2111	−1.04650	−0.523248	+	0.852181i	$0.675280\pi$
$338$	0	0
$339$	−5.30278	−0.288007
$340$	0	0
$341$	−5.60555	−0.303558
$342$	0	0
$343$	−19.4222	−1.04870
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−11.0917	−0.595432	−0.297716	−	0.954654i	$-0.596225\pi$
$347$	−11.0917	−0.595432	−0.297716	+	0.954654i	$0.596225\pi$
$348$	0	0
$349$	−25.4222	−1.36082	−0.680410	−	0.732832i	$-0.738198\pi$
$349$	−25.4222	−1.36082	−0.680410	+	0.732832i	$0.738198\pi$
$350$	0	0
$351$	90.8722	4.85040
$352$	0	0
$353$	12.2111	0.649931	0.324966	−	0.945726i	$-0.394647\pi$
$353$	12.2111	0.649931	0.324966	+	0.945726i	$0.394647\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	9.90833	0.524404
$358$	0	0
$359$	12.4222	0.655619	0.327809	−	0.944744i	$-0.393690\pi$
$359$	12.4222	0.655619	0.327809	+	0.944744i	$0.393690\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	−3.30278	−0.173351
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−29.6972	−1.55018	−0.775091	−	0.631849i	$-0.782296\pi$
$367$	−29.6972	−1.55018	−0.775091	+	0.631849i	$0.782296\pi$
$368$	0	0
$369$	−49.1194	−2.55706
$370$	0	0
$371$	−49.5416	−2.57207
$372$	0	0
$373$	22.0278	1.14055	0.570277	−	0.821452i	$-0.306836\pi$
$373$	22.0278	1.14055	0.570277	+	0.821452i	$0.306836\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	29.7250	1.53091
$378$	0	0
$379$	−14.2111	−0.729975	−0.364988	−	0.931012i	$-0.618927\pi$
$379$	−14.2111	−0.729975	−0.364988	+	0.931012i	$0.618927\pi$
$380$	0	0
$381$	13.5139	0.692337
$382$	0	0
$383$	−15.2111	−0.777251	−0.388626	−	0.921396i	$-0.627050\pi$
$383$	−15.2111	−0.777251	−0.388626	+	0.921396i	$0.627050\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	57.0278	2.89888
$388$	0	0
$389$	15.6333	0.792640	0.396320	−	0.918112i	$-0.370287\pi$
$389$	15.6333	0.792640	0.396320	+	0.918112i	$0.370287\pi$
$390$	0	0
$391$	4.81665	0.243589
$392$	0	0
$393$	−27.4222	−1.38327
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.5139	1.12994	0.564970	−	0.825112i	$-0.308888\pi$
$397$	22.5139	1.12994	0.564970	+	0.825112i	$0.308888\pi$
$398$	0	0
$399$	14.2111	0.711445
$400$	0	0
$401$	−3.21110	−0.160355	−0.0801774	−	0.996781i	$-0.525549\pi$
$401$	−3.21110	−0.160355	−0.0801774	+	0.996781i	$0.525549\pi$
$402$	0	0
$403$	−31.4222	−1.56525
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.394449	−0.0195521
$408$	0	0
$409$	−6.02776	−0.298053	−0.149027	−	0.988833i	$-0.547614\pi$
$409$	−6.02776	−0.298053	−0.149027	+	0.988833i	$0.547614\pi$
$410$	0	0
$411$	−32.7250	−1.61420
$412$	0	0
$413$	6.90833	0.339937
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.90833	0.289332
$418$	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
$421$	−16.9083	−0.824061	−0.412031	−	0.911170i	$-0.635180\pi$
$421$	−16.9083	−0.824061	−0.412031	+	0.911170i	$0.635180\pi$
$422$	0	0
$423$	12.6972	0.617360
$424$	0	0
$425$	0	0
$426$	0	0
$427$	1.30278	0.0630457
$428$	0	0
$429$	−18.5139	−0.893858
$430$	0	0
$431$	−18.2111	−0.877198	−0.438599	−	0.898683i	$-0.644525\pi$
$431$	−18.2111	−0.877198	−0.438599	+	0.898683i	$0.644525\pi$
$432$	0	0
$433$	29.0000	1.39365	0.696826	−	0.717241i	$-0.254595\pi$
$433$	29.0000	1.39365	0.696826	+	0.717241i	$0.254595\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	6.90833	0.330470
$438$	0	0
$439$	−17.9083	−0.854718	−0.427359	−	0.904082i	$-0.640556\pi$
$439$	−17.9083	−0.854718	−0.427359	+	0.904082i	$0.640556\pi$
$440$	0	0
$441$	91.0555	4.33598
$442$	0	0
$443$	−1.39445	−0.0662523	−0.0331261	−	0.999451i	$-0.510546\pi$
$443$	−1.39445	−0.0662523	−0.0331261	+	0.999451i	$0.510546\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	39.6333	1.87459
$448$	0	0
$449$	−12.9083	−0.609182	−0.304591	−	0.952483i	$-0.598520\pi$
$449$	−12.9083	−0.609182	−0.304591	+	0.952483i	$0.598520\pi$
$450$	0	0
$451$	6.21110	0.292469
$452$	0	0
$453$	22.5139	1.05779
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−19.1472	−0.895668	−0.447834	−	0.894117i	$-0.647804\pi$
$457$	−19.1472	−0.895668	−0.447834	+	0.894117i	$0.647804\pi$
$458$	0	0
$459$	−11.3028	−0.527568
$460$	0	0
$461$	−1.18335	−0.0551139	−0.0275570	−	0.999620i	$-0.508773\pi$
$461$	−1.18335	−0.0551139	−0.0275570	+	0.999620i	$0.508773\pi$
$462$	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
$467$	32.0278	1.48207	0.741034	−	0.671468i	$-0.234336\pi$
$467$	32.0278	1.48207	0.741034	+	0.671468i	$0.234336\pi$
$468$	0	0
$469$	34.4222	1.58947
$470$	0	0
$471$	2.60555	0.120057
$472$	0	0
$473$	−7.21110	−0.331567
$474$	0	0
$475$	0	0
$476$	0	0
$477$	91.0555	4.16915
$478$	0	0
$479$	−38.4500	−1.75682	−0.878412	−	0.477905i	$-0.841396\pi$
$479$	−38.4500	−1.75682	−0.878412	+	0.477905i	$0.841396\pi$
$480$	0	0
$481$	−2.21110	−0.100818
$482$	0	0
$483$	98.1749	4.46711
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−0.394449	−0.0178742	−0.00893709	−	0.999960i	$-0.502845\pi$
$487$	−0.394449	−0.0178742	−0.00893709	+	0.999960i	$0.502845\pi$
$488$	0	0
$489$	−1.00000	−0.0452216
$490$	0	0
$491$	30.6333	1.38246	0.691231	−	0.722634i	$-0.257069\pi$
$491$	30.6333	1.38246	0.691231	+	0.722634i	$0.257069\pi$
$492$	0	0
$493$	−3.69722	−0.166515
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−19.8167	−0.888898
$498$	0	0
$499$	21.0917	0.944193	0.472096	−	0.881547i	$-0.343497\pi$
$499$	21.0917	0.944193	0.472096	+	0.881547i	$0.343497\pi$
$500$	0	0
$501$	61.5416	2.74948
$502$	0	0
$503$	−13.8167	−0.616054	−0.308027	−	0.951378i	$-0.599669\pi$
$503$	−13.8167	−0.616054	−0.308027	+	0.951378i	$0.599669\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−60.8444	−2.70220
$508$	0	0
$509$	9.69722	0.429822	0.214911	−	0.976634i	$-0.431054\pi$
$509$	9.69722	0.429822	0.214911	+	0.976634i	$0.431054\pi$
$510$	0	0
$511$	−38.3305	−1.69564
$512$	0	0
$513$	−16.2111	−0.715738
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−1.60555	−0.0706121
$518$	0	0
$519$	79.9638	3.51002
$520$	0	0
$521$	25.3944	1.11255	0.556275	−	0.830998i	$-0.312230\pi$
$521$	25.3944	1.11255	0.556275	+	0.830998i	$0.312230\pi$
$522$	0	0
$523$	1.63331	0.0714196	0.0357098	−	0.999362i	$-0.488631\pi$
$523$	1.63331	0.0714196	0.0357098	+	0.999362i	$0.488631\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	3.90833	0.170249
$528$	0	0
$529$	24.7250	1.07500
$530$	0	0
$531$	−12.6972	−0.551013
$532$	0	0
$533$	34.8167	1.50808
$534$	0	0
$535$	0	0
$536$	0	0
$537$	62.4500	2.69491
$538$	0	0
$539$	−11.5139	−0.495938
$540$	0	0
$541$	9.33053	0.401151	0.200575	−	0.979678i	$-0.435719\pi$
$541$	9.33053	0.401151	0.200575	+	0.979678i	$0.435719\pi$
$542$	0	0
$543$	−9.60555	−0.412214
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−32.9083	−1.40706	−0.703529	−	0.710666i	$-0.748394\pi$
$547$	−32.9083	−1.40706	−0.703529	+	0.710666i	$0.748394\pi$
$548$	0	0
$549$	−2.39445	−0.102193
$550$	0	0
$551$	−5.30278	−0.225906
$552$	0	0
$553$	11.6056	0.493518
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.18335	−0.304368	−0.152184	−	0.988352i	$-0.548631\pi$
$557$	−7.18335	−0.304368	−0.152184	+	0.988352i	$0.548631\pi$
$558$	0	0
$559$	−40.4222	−1.70968
$560$	0	0
$561$	2.30278	0.0972233
$562$	0	0
$563$	29.0917	1.22607	0.613034	−	0.790057i	$-0.289949\pi$
$563$	29.0917	1.22607	0.613034	+	0.790057i	$0.289949\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−128.294	−5.38786
$568$	0	0
$569$	30.3583	1.27269	0.636343	−	0.771406i	$-0.280446\pi$
$569$	30.3583	1.27269	0.636343	+	0.771406i	$0.280446\pi$
$570$	0	0
$571$	−37.9361	−1.58758	−0.793788	−	0.608195i	$-0.791894\pi$
$571$	−37.9361	−1.58758	−0.793788	+	0.608195i	$0.791894\pi$
$572$	0	0
$573$	6.90833	0.288599
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−3.72498	−0.155073	−0.0775365	−	0.996990i	$-0.524705\pi$
$577$	−3.72498	−0.155073	−0.0775365	+	0.996990i	$0.524705\pi$
$578$	0	0
$579$	−87.2666	−3.62668
$580$	0	0
$581$	−16.8167	−0.697672
$582$	0	0
$583$	−11.5139	−0.476856
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−0.275019	−0.0113513	−0.00567563	−	0.999984i	$-0.501807\pi$
$587$	−0.275019	−0.0113513	−0.00567563	+	0.999984i	$0.501807\pi$
$588$	0	0
$589$	5.60555	0.230973
$590$	0	0
$591$	36.6333	1.50689
$592$	0	0
$593$	11.7889	0.484112	0.242056	−	0.970262i	$-0.422178\pi$
$593$	11.7889	0.484112	0.242056	+	0.970262i	$0.422178\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−26.1194	−1.06900
$598$	0	0
$599$	12.2750	0.501544	0.250772	−	0.968046i	$-0.419316\pi$
$599$	12.2750	0.501544	0.250772	+	0.968046i	$0.419316\pi$
$600$	0	0
$601$	−27.5139	−1.12231	−0.561157	−	0.827709i	$-0.689644\pi$
$601$	−27.5139	−1.12231	−0.561157	+	0.827709i	$0.689644\pi$
$602$	0	0
$603$	−63.2666	−2.57642
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−20.4222	−0.828912	−0.414456	−	0.910069i	$-0.636028\pi$
$607$	−20.4222	−0.828912	−0.414456	+	0.910069i	$0.636028\pi$
$608$	0	0
$609$	−75.3583	−3.05367
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	24.3305	0.982701	0.491350	−	0.870962i	$-0.336503\pi$
$613$	24.3305	0.982701	0.491350	+	0.870962i	$0.336503\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	6.97224	0.280692	0.140346	−	0.990103i	$-0.455179\pi$
$617$	6.97224	0.280692	0.140346	+	0.990103i	$0.455179\pi$
$618$	0	0
$619$	−6.81665	−0.273984	−0.136992	−	0.990572i	$-0.543744\pi$
$619$	−6.81665	−0.273984	−0.136992	+	0.990572i	$0.543744\pi$
$620$	0	0
$621$	−111.992	−4.49407
$622$	0	0
$623$	68.4500	2.74239
$624$	0	0
$625$	0	0
$626$	0	0
$627$	3.30278	0.131900
$628$	0	0
$629$	0.275019	0.0109657
$630$	0	0
$631$	−20.4861	−0.815540	−0.407770	−	0.913085i	$-0.633693\pi$
$631$	−20.4861	−0.815540	−0.407770	+	0.913085i	$0.633693\pi$
$632$	0	0
$633$	−47.5416	−1.88961
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−64.5416	−2.55723
$638$	0	0
$639$	36.4222	1.44084
$640$	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
$643$	−20.4222	−0.805373	−0.402687	−	0.915338i	$-0.631924\pi$
$643$	−20.4222	−0.805373	−0.402687	+	0.915338i	$0.631924\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	33.0000	1.29736	0.648682	−	0.761060i	$-0.275321\pi$
$647$	33.0000	1.29736	0.648682	+	0.761060i	$0.275321\pi$
$648$	0	0
$649$	1.60555	0.0630234
$650$	0	0
$651$	79.6611	3.12216
$652$	0	0
$653$	−7.11943	−0.278605	−0.139302	−	0.990250i	$-0.544486\pi$
$653$	−7.11943	−0.278605	−0.139302	+	0.990250i	$0.544486\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	70.4500	2.74851
$658$	0	0
$659$	39.3583	1.53318	0.766591	−	0.642136i	$-0.221952\pi$
$659$	39.3583	1.53318	0.766591	+	0.642136i	$0.221952\pi$
$660$	0	0
$661$	0.816654	0.0317642	0.0158821	−	0.999874i	$-0.494944\pi$
$661$	0.816654	0.0317642	0.0158821	+	0.999874i	$0.494944\pi$
$662$	0	0
$663$	12.9083	0.501318
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−36.6333	−1.41845
$668$	0	0
$669$	21.1194	0.816524
$670$	0	0
$671$	0.302776	0.0116885
$672$	0	0
$673$	35.0000	1.34915	0.674575	−	0.738206i	$-0.264327\pi$
$673$	35.0000	1.34915	0.674575	+	0.738206i	$0.264327\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−26.2389	−1.00844	−0.504221	−	0.863575i	$-0.668220\pi$
$677$	−26.2389	−1.00844	−0.504221	+	0.863575i	$0.668220\pi$
$678$	0	0
$679$	−77.9638	−2.99198
$680$	0	0
$681$	69.1472	2.64973
$682$	0	0
$683$	−1.60555	−0.0614347	−0.0307174	−	0.999528i	$-0.509779\pi$
$683$	−1.60555	−0.0614347	−0.0307174	+	0.999528i	$0.509779\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−2.90833	−0.110960
$688$	0	0
$689$	−64.5416	−2.45884
$690$	0	0
$691$	−25.5139	−0.970594	−0.485297	−	0.874349i	$-0.661288\pi$
$691$	−25.5139	−0.970594	−0.485297	+	0.874349i	$0.661288\pi$
$692$	0	0
$693$	34.0278	1.29261
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−4.33053	−0.164031
$698$	0	0
$699$	82.2666	3.11161
$700$	0	0
$701$	9.21110	0.347899	0.173949	−	0.984755i	$-0.444347\pi$
$701$	9.21110	0.347899	0.173949	+	0.984755i	$0.444347\pi$
$702$	0	0
$703$	0.394449	0.0148769
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−49.5416	−1.86320
$708$	0	0
$709$	−7.63331	−0.286675	−0.143337	−	0.989674i	$-0.545783\pi$
$709$	−7.63331	−0.286675	−0.143337	+	0.989674i	$0.545783\pi$
$710$	0	0
$711$	−21.3305	−0.799957
$712$	0	0
$713$	38.7250	1.45026
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−83.6611	−3.12438
$718$	0	0
$719$	45.4222	1.69396	0.846981	−	0.531623i	$-0.178418\pi$
$719$	45.4222	1.69396	0.846981	+	0.531623i	$0.178418\pi$
$720$	0	0
$721$	37.4222	1.39368
$722$	0	0
$723$	15.5139	0.576967
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−16.0917	−0.596807	−0.298404	−	0.954440i	$-0.596454\pi$
$727$	−16.0917	−0.596807	−0.298404	+	0.954440i	$0.596454\pi$
$728$	0	0
$729$	75.1749	2.78426
$730$	0	0
$731$	5.02776	0.185958
$732$	0	0
$733$	−19.4222	−0.717376	−0.358688	−	0.933458i	$-0.616776\pi$
$733$	−19.4222	−0.717376	−0.358688	+	0.933458i	$0.616776\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	26.1194	0.960819	0.480409	−	0.877044i	$-0.340488\pi$
$739$	26.1194	0.960819	0.480409	+	0.877044i	$0.340488\pi$
$740$	0	0
$741$	18.5139	0.680124
$742$	0	0
$743$	0.908327	0.0333233	0.0166616	−	0.999861i	$-0.494696\pi$
$743$	0.908327	0.0333233	0.0166616	+	0.999861i	$0.494696\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	30.9083	1.13088
$748$	0	0
$749$	−52.5416	−1.91983
$750$	0	0
$751$	−53.3583	−1.94707	−0.973536	−	0.228535i	$-0.926607\pi$
$751$	−53.3583	−1.94707	−0.973536	+	0.228535i	$0.926607\pi$
$752$	0	0
$753$	53.9361	1.96554
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−25.0000	−0.908640	−0.454320	−	0.890838i	$-0.650118\pi$
$757$	−25.0000	−0.908640	−0.454320	+	0.890838i	$0.650118\pi$
$758$	0	0
$759$	22.8167	0.828192
$760$	0	0
$761$	43.2666	1.56841	0.784207	−	0.620500i	$-0.213070\pi$
$761$	43.2666	1.56841	0.784207	+	0.620500i	$0.213070\pi$
$762$	0	0
$763$	8.21110	0.297262
$764$	0	0
$765$	0	0
$766$	0	0
$767$	9.00000	0.324971
$768$	0	0
$769$	44.6333	1.60952	0.804759	−	0.593602i	$-0.202294\pi$
$769$	44.6333	1.60952	0.804759	+	0.593602i	$0.202294\pi$
$770$	0	0
$771$	1.39445	0.0502198
$772$	0	0
$773$	−31.3305	−1.12688	−0.563440	−	0.826157i	$-0.690523\pi$
$773$	−31.3305	−1.12688	−0.563440	+	0.826157i	$0.690523\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	5.60555	0.201098
$778$	0	0
$779$	−6.21110	−0.222536
$780$	0	0
$781$	−4.60555	−0.164800
$782$	0	0
$783$	85.9638	3.07210
$784$	0	0
$785$	0	0
$786$	0	0
$787$	31.8444	1.13513	0.567565	−	0.823328i	$-0.307885\pi$
$787$	31.8444	1.13513	0.567565	+	0.823328i	$0.307885\pi$
$788$	0	0
$789$	35.7250	1.27184
$790$	0	0
$791$	−6.90833	−0.245632
$792$	0	0
$793$	1.69722	0.0602702
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.9083	0.563502	0.281751	−	0.959488i	$-0.409085\pi$
$797$	15.9083	0.563502	0.281751	+	0.959488i	$0.409085\pi$
$798$	0	0
$799$	1.11943	0.0396026
$800$	0	0
$801$	−125.808	−4.44522
$802$	0	0
$803$	−8.90833	−0.314368
$804$	0	0
$805$	0	0
$806$	0	0
$807$	47.9361	1.68743
$808$	0	0
$809$	39.6333	1.39343	0.696716	−	0.717347i	$-0.254644\pi$
$809$	39.6333	1.39343	0.696716	+	0.717347i	$0.254644\pi$
$810$	0	0
$811$	2.39445	0.0840805	0.0420402	−	0.999116i	$-0.486614\pi$
$811$	2.39445	0.0840805	0.0420402	+	0.999116i	$0.486614\pi$
$812$	0	0
$813$	−1.90833	−0.0669279
$814$	0	0
$815$	0	0
$816$	0	0
$817$	7.21110	0.252285
$818$	0	0
$819$	190.744	6.66515
$820$	0	0
$821$	39.6333	1.38321	0.691606	−	0.722275i	$-0.256903\pi$
$821$	39.6333	1.38321	0.691606	+	0.722275i	$0.256903\pi$
$822$	0	0
$823$	−11.6333	−0.405512	−0.202756	−	0.979229i	$-0.564990\pi$
$823$	−11.6333	−0.405512	−0.202756	+	0.979229i	$0.564990\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.3944	1.30033	0.650166	−	0.759792i	$-0.274699\pi$
$827$	37.3944	1.30033	0.650166	+	0.759792i	$0.274699\pi$
$828$	0	0
$829$	0.880571	0.0305835	0.0152917	−	0.999883i	$-0.495132\pi$
$829$	0.880571	0.0305835	0.0152917	+	0.999883i	$0.495132\pi$
$830$	0	0
$831$	52.1472	1.80897
$832$	0	0
$833$	8.02776	0.278145
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−90.8722	−3.14100
$838$	0	0
$839$	−36.9083	−1.27422	−0.637108	−	0.770774i	$-0.719870\pi$
$839$	−36.9083	−1.27422	−0.637108	+	0.770774i	$0.719870\pi$
$840$	0	0
$841$	−0.880571	−0.0303645
$842$	0	0
$843$	−69.3583	−2.38883
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.30278	−0.147845
$848$	0	0
$849$	18.1194	0.621857
$850$	0	0
$851$	2.72498	0.0934111
$852$	0	0
$853$	−3.30278	−0.113085	−0.0565424	−	0.998400i	$-0.518008\pi$
$853$	−3.30278	−0.113085	−0.0565424	+	0.998400i	$0.518008\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32.0278	−1.09405	−0.547024	−	0.837117i	$-0.684239\pi$
$857$	−32.0278	−1.09405	−0.547024	+	0.837117i	$0.684239\pi$
$858$	0	0
$859$	−3.39445	−0.115817	−0.0579085	−	0.998322i	$-0.518443\pi$
$859$	−3.39445	−0.115817	−0.0579085	+	0.998322i	$0.518443\pi$
$860$	0	0
$861$	−88.2666	−3.00812
$862$	0	0
$863$	−34.2666	−1.16645	−0.583225	−	0.812311i	$-0.698209\pi$
$863$	−34.2666	−1.16645	−0.583225	+	0.812311i	$0.698209\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	54.5416	1.85233
$868$	0	0
$869$	2.69722	0.0914971
$870$	0	0
$871$	44.8444	1.51949
$872$	0	0
$873$	143.294	4.84978
$874$	0	0
$875$	0	0
$876$	0	0
$877$	11.6333	0.392829	0.196414	−	0.980521i	$-0.437070\pi$
$877$	11.6333	0.392829	0.196414	+	0.980521i	$0.437070\pi$
$878$	0	0
$879$	59.4500	2.00520
$880$	0	0
$881$	−55.7527	−1.87836	−0.939179	−	0.343429i	$-0.888412\pi$
$881$	−55.7527	−1.87836	−0.939179	+	0.343429i	$0.888412\pi$
$882$	0	0
$883$	3.78890	0.127507	0.0637533	−	0.997966i	$-0.479693\pi$
$883$	3.78890	0.127507	0.0637533	+	0.997966i	$0.479693\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.39445	0.147551	0.0737756	−	0.997275i	$-0.476495\pi$
$887$	4.39445	0.147551	0.0737756	+	0.997275i	$0.476495\pi$
$888$	0	0
$889$	17.6056	0.590471
$890$	0	0
$891$	−29.8167	−0.998895
$892$	0	0
$893$	1.60555	0.0537277
$894$	0	0
$895$	0	0
$896$	0	0
$897$	127.900	4.27045
$898$	0	0
$899$	−29.7250	−0.991384
$900$	0	0
$901$	8.02776	0.267443
$902$	0	0
$903$	102.478	3.41024
$904$	0	0
$905$	0	0
$906$	0	0
$907$	34.4222	1.14297	0.571485	−	0.820612i	$-0.306367\pi$
$907$	34.4222	1.14297	0.571485	+	0.820612i	$0.306367\pi$
$908$	0	0
$909$	91.0555	3.02012
$910$	0	0
$911$	−9.63331	−0.319166	−0.159583	−	0.987185i	$-0.551015\pi$
$911$	−9.63331	−0.319166	−0.159583	+	0.987185i	$0.551015\pi$
$912$	0	0
$913$	−3.90833	−0.129347
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−35.7250	−1.17974
$918$	0	0
$919$	−30.0555	−0.991440	−0.495720	−	0.868482i	$-0.665096\pi$
$919$	−30.0555	−0.991440	−0.495720	+	0.868482i	$0.665096\pi$
$920$	0	0
$921$	80.3583	2.64790
$922$	0	0
$923$	−25.8167	−0.849766
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−68.7805	−2.25905
$928$	0	0
$929$	−47.6611	−1.56371	−0.781854	−	0.623461i	$-0.785726\pi$
$929$	−47.6611	−1.56371	−0.781854	+	0.623461i	$0.785726\pi$
$930$	0	0
$931$	11.5139	0.377352
$932$	0	0
$933$	−29.7250	−0.973152
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.0000	−0.522697	−0.261349	−	0.965244i	$-0.584167\pi$
$937$	−16.0000	−0.522697	−0.261349	+	0.965244i	$0.584167\pi$
$938$	0	0
$939$	−50.8444	−1.65924
$940$	0	0
$941$	33.4222	1.08953	0.544766	−	0.838588i	$-0.316618\pi$
$941$	33.4222	1.08953	0.544766	+	0.838588i	$0.316618\pi$
$942$	0	0
$943$	−42.9083	−1.39729
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−26.4500	−0.859508	−0.429754	−	0.902946i	$-0.641400\pi$
$947$	−26.4500	−0.859508	−0.429754	+	0.902946i	$0.641400\pi$
$948$	0	0
$949$	−49.9361	−1.62099
$950$	0	0
$951$	76.9638	2.49572
$952$	0	0
$953$	9.63331	0.312053	0.156027	−	0.987753i	$-0.450131\pi$
$953$	9.63331	0.312053	0.156027	+	0.987753i	$0.450131\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−17.5139	−0.566143
$958$	0	0
$959$	−42.6333	−1.37670
$960$	0	0
$961$	0.422205	0.0136195
$962$	0	0
$963$	96.5694	3.11191
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−28.9361	−0.930522	−0.465261	−	0.885174i	$-0.654039\pi$
$967$	−28.9361	−0.930522	−0.465261	+	0.885174i	$0.654039\pi$
$968$	0	0
$969$	−2.30278	−0.0739758
$970$	0	0
$971$	41.0917	1.31869	0.659347	−	0.751839i	$-0.270833\pi$
$971$	41.0917	1.31869	0.659347	+	0.751839i	$0.270833\pi$
$972$	0	0
$973$	7.69722	0.246762
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.81665	−0.154098	−0.0770492	−	0.997027i	$-0.524550\pi$
$977$	−4.81665	−0.154098	−0.0770492	+	0.997027i	$0.524550\pi$
$978$	0	0
$979$	15.9083	0.508432
$980$	0	0
$981$	−15.0917	−0.481840
$982$	0	0
$983$	−3.63331	−0.115885	−0.0579423	−	0.998320i	$-0.518454\pi$
$983$	−3.63331	−0.115885	−0.0579423	+	0.998320i	$0.518454\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	22.8167	0.726262
$988$	0	0
$989$	49.8167	1.58408
$990$	0	0
$991$	32.7527	1.04042	0.520212	−	0.854037i	$-0.325853\pi$
$991$	32.7527	1.04042	0.520212	+	0.854037i	$0.325853\pi$
$992$	0	0
$993$	−94.5694	−3.00107
$994$	0	0
$995$	0	0
$996$	0	0
$997$	53.7805	1.70325	0.851623	−	0.524155i	$-0.175619\pi$
$997$	53.7805	1.70325	0.851623	+	0.524155i	$0.175619\pi$
$998$	0	0
$999$	−6.39445	−0.202311

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bf.1.1		2
4.3	odd	2		1100.2.a.i.1.2	yes	2
5.2	odd	4		4400.2.b.r.4049.4		4
5.3	odd	4		4400.2.b.r.4049.1		4
5.4	even	2		4400.2.a.bw.1.2		2
12.11	even	2		9900.2.a.by.1.2		2
20.3	even	4		1100.2.b.e.749.4		4
20.7	even	4		1100.2.b.e.749.1		4
20.19	odd	2		1100.2.a.f.1.1	✓	2
60.23	odd	4		9900.2.c.r.5149.1		4
60.47	odd	4		9900.2.c.r.5149.4		4
60.59	even	2		9900.2.a.bg.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1100.2.a.f.1.1	✓	2	20.19	odd	2
1100.2.a.i.1.2	yes	2	4.3	odd	2
1100.2.b.e.749.1		4	20.7	even	4
1100.2.b.e.749.4		4	20.3	even	4
4400.2.a.bf.1.1		2	1.1	even	1	trivial
4400.2.a.bw.1.2		2	5.4	even	2
4400.2.b.r.4049.1		4	5.3	odd	4
4400.2.b.r.4049.4		4	5.2	odd	4
9900.2.a.bg.1.1		2	60.59	even	2
9900.2.a.by.1.2		2	12.11	even	2
9900.2.c.r.5149.1		4	60.23	odd	4
9900.2.c.r.5149.4		4	60.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.30278 q^{3} -4.30278 q^{7} +7.90833 q^{9} -1.00000 q^{11} -5.60555 q^{13} +0.697224 q^{17} +1.00000 q^{19} +14.2111 q^{21} +6.90833 q^{23} -16.2111 q^{27} -5.30278 q^{29} +5.60555 q^{31} +3.30278 q^{33} +0.394449 q^{37} +18.5139 q^{39} -6.21110 q^{41} +7.21110 q^{43} +1.60555 q^{47} +11.5139 q^{49} -2.30278 q^{51} +11.5139 q^{53} -3.30278 q^{57} -1.60555 q^{59} -0.302776 q^{61} -34.0278 q^{63} -8.00000 q^{67} -22.8167 q^{69} +4.60555 q^{71} +8.90833 q^{73} +4.30278 q^{77} -2.69722 q^{79} +29.8167 q^{81} +3.90833 q^{83} +17.5139 q^{87} -15.9083 q^{89} +24.1194 q^{91} -18.5139 q^{93} +18.1194 q^{97} -7.90833 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - 5 q^{7} + 5 q^{9} - 2 q^{11} - 4 q^{13} + 5 q^{17} + 2 q^{19} + 14 q^{21} + 3 q^{23} - 18 q^{27} - 7 q^{29} + 4 q^{31} + 3 q^{33} + 8 q^{37} + 19 q^{39} + 2 q^{41} - 4 q^{47} + 5 q^{49}+ \cdots - 5 q^{99}+O(q^{100}) \) 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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