LMFDB - Embedded newform 4400.2.a.bo.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,0,0,-3,0,-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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2200)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.61803 q^{3} -2.61803 q^{7} -0.381966 q^{9} -1.00000 q^{11} +3.47214 q^{13} -6.09017 q^{17} +8.23607 q^{19} -4.23607 q^{21} +2.61803 q^{23} -5.47214 q^{27} -7.32624 q^{29} -4.70820 q^{31} -1.61803 q^{33} +4.23607 q^{37} +5.61803 q^{39} +2.70820 q^{41} -10.9443 q^{43} +2.23607 q^{47} -0.145898 q^{49} -9.85410 q^{51} -1.38197 q^{53} +13.3262 q^{57} -2.70820 q^{59} -13.0902 q^{61} +1.00000 q^{63} -12.0000 q^{67} +4.23607 q^{69} +14.1803 q^{71} -9.38197 q^{73} +2.61803 q^{77} +9.61803 q^{79} -7.70820 q^{81} +6.56231 q^{83} -11.8541 q^{87} +1.32624 q^{89} -9.09017 q^{91} -7.61803 q^{93} -12.6180 q^{97} +0.381966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 3 q^{7} - 3 q^{9} - 2 q^{11} - 2 q^{13} - q^{17} + 12 q^{19} - 4 q^{21} + 3 q^{23} - 2 q^{27} + q^{29} + 4 q^{31} - q^{33} + 4 q^{37} + 9 q^{39} - 8 q^{41} - 4 q^{43} - 7 q^{49} - 13 q^{51}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q + q^3 - 3 * q^7 - 3 * q^9 - 2 * q^11 - 2 * q^13 - q^17 + 12 * q^19 - 4 * q^21 + 3 * q^23 - 2 * q^27 + q^29 + 4 * q^31 - q^33 + 4 * q^37 + 9 * q^39 - 8 * q^41 - 4 * q^43 - 7 * q^49 - 13 * q^51 - 5 * q^53 + 11 * q^57 + 8 * q^59 - 15 * q^61 + 2 * q^63 - 24 * q^67 + 4 * q^69 + 6 * q^71 - 21 * q^73 + 3 * q^77 + 17 * q^79 - 2 * q^81 - 7 * q^83 - 17 * q^87 - 13 * q^89 - 7 * q^91 - 13 * q^93 - 23 * 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
$3$	1.61803	0.934172	0.467086	−	0.884212i	$-0.345304\pi$
$3$	1.61803	0.934172	0.467086	+	0.884212i	$0.345304\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.61803	−0.989524	−0.494762	−	0.869029i	$-0.664745\pi$
$7$	−2.61803	−0.989524	−0.494762	+	0.869029i	$0.664745\pi$
$8$	0	0
$9$	−0.381966	−0.127322
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.47214	0.962997	0.481499	−	0.876447i	$-0.340093\pi$
$13$	3.47214	0.962997	0.481499	+	0.876447i	$0.340093\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.09017	−1.47708	−0.738542	−	0.674208i	$-0.764485\pi$
$17$	−6.09017	−1.47708	−0.738542	+	0.674208i	$0.764485\pi$
$18$	0	0
$19$	8.23607	1.88948	0.944742	−	0.327815i	$-0.106312\pi$
$19$	8.23607	1.88948	0.944742	+	0.327815i	$0.106312\pi$
$20$	0	0
$21$	−4.23607	−0.924386
$22$	0	0
$23$	2.61803	0.545898	0.272949	−	0.962029i	$-0.412001\pi$
$23$	2.61803	0.545898	0.272949	+	0.962029i	$0.412001\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.47214	−1.05311
$28$	0	0
$29$	−7.32624	−1.36045	−0.680224	−	0.733004i	$-0.738118\pi$
$29$	−7.32624	−1.36045	−0.680224	+	0.733004i	$0.738118\pi$
$30$	0	0
$31$	−4.70820	−0.845618	−0.422809	−	0.906219i	$-0.638956\pi$
$31$	−4.70820	−0.845618	−0.422809	+	0.906219i	$0.638956\pi$
$32$	0	0
$33$	−1.61803	−0.281664
$34$	0	0
$35$	0	0
$36$	0	0
$37$	4.23607	0.696405	0.348203	−	0.937419i	$-0.386792\pi$
$37$	4.23607	0.696405	0.348203	+	0.937419i	$0.386792\pi$
$38$	0	0
$39$	5.61803	0.899605
$40$	0	0
$41$	2.70820	0.422950	0.211475	−	0.977383i	$-0.432173\pi$
$41$	2.70820	0.422950	0.211475	+	0.977383i	$0.432173\pi$
$42$	0	0
$43$	−10.9443	−1.66899	−0.834493	−	0.551019i	$-0.814239\pi$
$43$	−10.9443	−1.66899	−0.834493	+	0.551019i	$0.814239\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.23607	0.326164	0.163082	−	0.986613i	$-0.447856\pi$
$47$	2.23607	0.326164	0.163082	+	0.986613i	$0.447856\pi$
$48$	0	0
$49$	−0.145898	−0.0208426
$50$	0	0
$51$	−9.85410	−1.37985
$52$	0	0
$53$	−1.38197	−0.189828	−0.0949138	−	0.995485i	$-0.530258\pi$
$53$	−1.38197	−0.189828	−0.0949138	+	0.995485i	$0.530258\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	13.3262	1.76510
$58$	0	0
$59$	−2.70820	−0.352578	−0.176289	−	0.984338i	$-0.556409\pi$
$59$	−2.70820	−0.352578	−0.176289	+	0.984338i	$0.556409\pi$
$60$	0	0
$61$	−13.0902	−1.67602	−0.838012	−	0.545651i	$-0.816282\pi$
$61$	−13.0902	−1.67602	−0.838012	+	0.545651i	$0.816282\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	4.23607	0.509963
$70$	0	0
$71$	14.1803	1.68290	0.841448	−	0.540338i	$-0.181703\pi$
$71$	14.1803	1.68290	0.841448	+	0.540338i	$0.181703\pi$
$72$	0	0
$73$	−9.38197	−1.09808	−0.549038	−	0.835797i	$-0.685006\pi$
$73$	−9.38197	−1.09808	−0.549038	+	0.835797i	$0.685006\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.61803	0.298353
$78$	0	0
$79$	9.61803	1.08211	0.541057	−	0.840986i	$-0.318024\pi$
$79$	9.61803	1.08211	0.541057	+	0.840986i	$0.318024\pi$
$80$	0	0
$81$	−7.70820	−0.856467
$82$	0	0
$83$	6.56231	0.720307	0.360153	−	0.932893i	$-0.382724\pi$
$83$	6.56231	0.720307	0.360153	+	0.932893i	$0.382724\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−11.8541	−1.27089
$88$	0	0
$89$	1.32624	0.140581	0.0702905	−	0.997527i	$-0.477607\pi$
$89$	1.32624	0.140581	0.0702905	+	0.997527i	$0.477607\pi$
$90$	0	0
$91$	−9.09017	−0.952909
$92$	0	0
$93$	−7.61803	−0.789953
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−12.6180	−1.28117	−0.640584	−	0.767888i	$-0.721308\pi$
$97$	−12.6180	−1.28117	−0.640584	+	0.767888i	$0.721308\pi$
$98$	0	0
$99$	0.381966	0.0383890
$100$	0	0
$101$	−11.3820	−1.13255	−0.566274	−	0.824217i	$-0.691616\pi$
$101$	−11.3820	−1.13255	−0.566274	+	0.824217i	$0.691616\pi$
$102$	0	0
$103$	3.32624	0.327744	0.163872	−	0.986482i	$-0.447602\pi$
$103$	3.32624	0.327744	0.163872	+	0.986482i	$0.447602\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	9.18034	0.887497	0.443748	−	0.896151i	$-0.353648\pi$
$107$	9.18034	0.887497	0.443748	+	0.896151i	$0.353648\pi$
$108$	0	0
$109$	−16.5623	−1.58638	−0.793191	−	0.608973i	$-0.791582\pi$
$109$	−16.5623	−1.58638	−0.793191	+	0.608973i	$0.791582\pi$
$110$	0	0
$111$	6.85410	0.650563
$112$	0	0
$113$	−3.76393	−0.354081	−0.177040	−	0.984204i	$-0.556652\pi$
$113$	−3.76393	−0.354081	−0.177040	+	0.984204i	$0.556652\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1.32624	−0.122611
$118$	0	0
$119$	15.9443	1.46161
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	4.38197	0.395109
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−8.85410	−0.785675	−0.392837	−	0.919608i	$-0.628506\pi$
$127$	−8.85410	−0.785675	−0.392837	+	0.919608i	$0.628506\pi$
$128$	0	0
$129$	−17.7082	−1.55912
$130$	0	0
$131$	−7.56231	−0.660722	−0.330361	−	0.943855i	$-0.607170\pi$
$131$	−7.56231	−0.660722	−0.330361	+	0.943855i	$0.607170\pi$
$132$	0	0
$133$	−21.5623	−1.86969
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.5623	−0.902399	−0.451199	−	0.892423i	$-0.649004\pi$
$137$	−10.5623	−0.902399	−0.451199	+	0.892423i	$0.649004\pi$
$138$	0	0
$139$	−11.1803	−0.948304	−0.474152	−	0.880443i	$-0.657245\pi$
$139$	−11.1803	−0.948304	−0.474152	+	0.880443i	$0.657245\pi$
$140$	0	0
$141$	3.61803	0.304693
$142$	0	0
$143$	−3.47214	−0.290355
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.236068	−0.0194706
$148$	0	0
$149$	19.4164	1.59065	0.795327	−	0.606181i	$-0.207299\pi$
$149$	19.4164	1.59065	0.795327	+	0.606181i	$0.207299\pi$
$150$	0	0
$151$	1.94427	0.158223	0.0791113	−	0.996866i	$-0.474792\pi$
$151$	1.94427	0.158223	0.0791113	+	0.996866i	$0.474792\pi$
$152$	0	0
$153$	2.32624	0.188065
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−4.47214	−0.356915	−0.178458	−	0.983948i	$-0.557111\pi$
$157$	−4.47214	−0.356915	−0.178458	+	0.983948i	$0.557111\pi$
$158$	0	0
$159$	−2.23607	−0.177332
$160$	0	0
$161$	−6.85410	−0.540179
$162$	0	0
$163$	−8.61803	−0.675017	−0.337508	−	0.941323i	$-0.609584\pi$
$163$	−8.61803	−0.675017	−0.337508	+	0.941323i	$0.609584\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.70820	−0.519096	−0.259548	−	0.965730i	$-0.583574\pi$
$167$	−6.70820	−0.519096	−0.259548	+	0.965730i	$0.583574\pi$
$168$	0	0
$169$	−0.944272	−0.0726363
$170$	0	0
$171$	−3.14590	−0.240573
$172$	0	0
$173$	−24.7082	−1.87853	−0.939265	−	0.343193i	$-0.888492\pi$
$173$	−24.7082	−1.87853	−0.939265	+	0.343193i	$0.888492\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−4.38197	−0.329369
$178$	0	0
$179$	9.09017	0.679431	0.339716	−	0.940528i	$-0.389669\pi$
$179$	9.09017	0.679431	0.339716	+	0.940528i	$0.389669\pi$
$180$	0	0
$181$	6.61803	0.491915	0.245957	−	0.969281i	$-0.420898\pi$
$181$	6.61803	0.491915	0.245957	+	0.969281i	$0.420898\pi$
$182$	0	0
$183$	−21.1803	−1.56570
$184$	0	0
$185$	0	0
$186$	0	0
$187$	6.09017	0.445357
$188$	0	0
$189$	14.3262	1.04208
$190$	0	0
$191$	−17.0344	−1.23257	−0.616284	−	0.787524i	$-0.711363\pi$
$191$	−17.0344	−1.23257	−0.616284	+	0.787524i	$0.711363\pi$
$192$	0	0
$193$	−18.4721	−1.32965	−0.664827	−	0.746998i	$-0.731495\pi$
$193$	−18.4721	−1.32965	−0.664827	+	0.746998i	$0.731495\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	18.0344	1.28490	0.642450	−	0.766327i	$-0.277918\pi$
$197$	18.0344	1.28490	0.642450	+	0.766327i	$0.277918\pi$
$198$	0	0
$199$	12.3820	0.877734	0.438867	−	0.898552i	$-0.355380\pi$
$199$	12.3820	0.877734	0.438867	+	0.898552i	$0.355380\pi$
$200$	0	0
$201$	−19.4164	−1.36953
$202$	0	0
$203$	19.1803	1.34620
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−8.23607	−0.569701
$210$	0	0
$211$	1.00000	0.0688428	0.0344214	−	0.999407i	$-0.489041\pi$
$211$	1.00000	0.0688428	0.0344214	+	0.999407i	$0.489041\pi$
$212$	0	0
$213$	22.9443	1.57212
$214$	0	0
$215$	0	0
$216$	0	0
$217$	12.3262	0.836760
$218$	0	0
$219$	−15.1803	−1.02579
$220$	0	0
$221$	−21.1459	−1.42243
$222$	0	0
$223$	5.65248	0.378518	0.189259	−	0.981927i	$-0.439391\pi$
$223$	5.65248	0.378518	0.189259	+	0.981927i	$0.439391\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	7.14590	0.474290	0.237145	−	0.971474i	$-0.423788\pi$
$227$	7.14590	0.474290	0.237145	+	0.971474i	$0.423788\pi$
$228$	0	0
$229$	10.3820	0.686060	0.343030	−	0.939325i	$-0.388547\pi$
$229$	10.3820	0.686060	0.343030	+	0.939325i	$0.388547\pi$
$230$	0	0
$231$	4.23607	0.278713
$232$	0	0
$233$	−3.56231	−0.233374	−0.116687	−	0.993169i	$-0.537228\pi$
$233$	−3.56231	−0.233374	−0.116687	+	0.993169i	$0.537228\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	15.5623	1.01088
$238$	0	0
$239$	24.6180	1.59241	0.796204	−	0.605028i	$-0.206838\pi$
$239$	24.6180	1.59241	0.796204	+	0.605028i	$0.206838\pi$
$240$	0	0
$241$	6.85410	0.441512	0.220756	−	0.975329i	$-0.429148\pi$
$241$	6.85410	0.441512	0.220756	+	0.975329i	$0.429148\pi$
$242$	0	0
$243$	3.94427	0.253025
$244$	0	0
$245$	0	0
$246$	0	0
$247$	28.5967	1.81957
$248$	0	0
$249$	10.6180	0.672891
$250$	0	0
$251$	19.5066	1.23124	0.615622	−	0.788041i	$-0.288905\pi$
$251$	19.5066	1.23124	0.615622	+	0.788041i	$0.288905\pi$
$252$	0	0
$253$	−2.61803	−0.164594
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.9443	−1.43122	−0.715612	−	0.698498i	$-0.753852\pi$
$257$	−22.9443	−1.43122	−0.715612	+	0.698498i	$0.753852\pi$
$258$	0	0
$259$	−11.0902	−0.689110
$260$	0	0
$261$	2.79837	0.173215
$262$	0	0
$263$	−6.52786	−0.402525	−0.201263	−	0.979537i	$-0.564504\pi$
$263$	−6.52786	−0.402525	−0.201263	+	0.979537i	$0.564504\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	2.14590	0.131327
$268$	0	0
$269$	−0.381966	−0.0232889	−0.0116444	−	0.999932i	$-0.503707\pi$
$269$	−0.381966	−0.0232889	−0.0116444	+	0.999932i	$0.503707\pi$
$270$	0	0
$271$	0.708204	0.0430203	0.0215102	−	0.999769i	$-0.493153\pi$
$271$	0.708204	0.0430203	0.0215102	+	0.999769i	$0.493153\pi$
$272$	0	0
$273$	−14.7082	−0.890181
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−14.1246	−0.848666	−0.424333	−	0.905506i	$-0.639491\pi$
$277$	−14.1246	−0.848666	−0.424333	+	0.905506i	$0.639491\pi$
$278$	0	0
$279$	1.79837	0.107666
$280$	0	0
$281$	−0.236068	−0.0140826	−0.00704132	−	0.999975i	$-0.502241\pi$
$281$	−0.236068	−0.0140826	−0.00704132	+	0.999975i	$0.502241\pi$
$282$	0	0
$283$	−28.4508	−1.69123	−0.845614	−	0.533795i	$-0.820765\pi$
$283$	−28.4508	−1.69123	−0.845614	+	0.533795i	$0.820765\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−7.09017	−0.418519
$288$	0	0
$289$	20.0902	1.18177
$290$	0	0
$291$	−20.4164	−1.19683
$292$	0	0
$293$	20.4721	1.19599	0.597997	−	0.801498i	$-0.295963\pi$
$293$	20.4721	1.19599	0.597997	+	0.801498i	$0.295963\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.47214	0.317526
$298$	0	0
$299$	9.09017	0.525698
$300$	0	0
$301$	28.6525	1.65150
$302$	0	0
$303$	−18.4164	−1.05799
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−17.1459	−0.978568	−0.489284	−	0.872124i	$-0.662742\pi$
$307$	−17.1459	−0.978568	−0.489284	+	0.872124i	$0.662742\pi$
$308$	0	0
$309$	5.38197	0.306169
$310$	0	0
$311$	11.9443	0.677298	0.338649	−	0.940913i	$-0.390030\pi$
$311$	11.9443	0.677298	0.338649	+	0.940913i	$0.390030\pi$
$312$	0	0
$313$	−20.6525	−1.16735	−0.583673	−	0.811988i	$-0.698385\pi$
$313$	−20.6525	−1.16735	−0.583673	+	0.811988i	$0.698385\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.61803	−0.315540	−0.157770	−	0.987476i	$-0.550430\pi$
$317$	−5.61803	−0.315540	−0.157770	+	0.987476i	$0.550430\pi$
$318$	0	0
$319$	7.32624	0.410191
$320$	0	0
$321$	14.8541	0.829075
$322$	0	0
$323$	−50.1591	−2.79092
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−26.7984	−1.48195
$328$	0	0
$329$	−5.85410	−0.322747
$330$	0	0
$331$	31.3607	1.72374	0.861869	−	0.507130i	$-0.169294\pi$
$331$	31.3607	1.72374	0.861869	+	0.507130i	$0.169294\pi$
$332$	0	0
$333$	−1.61803	−0.0886677
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−0.111456	−0.00607140	−0.00303570	−	0.999995i	$-0.500966\pi$
$337$	−0.111456	−0.00607140	−0.00303570	+	0.999995i	$0.500966\pi$
$338$	0	0
$339$	−6.09017	−0.330773
$340$	0	0
$341$	4.70820	0.254964
$342$	0	0
$343$	18.7082	1.01015
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.38197	0.0741878	0.0370939	−	0.999312i	$-0.488190\pi$
$347$	1.38197	0.0741878	0.0370939	+	0.999312i	$0.488190\pi$
$348$	0	0
$349$	32.1246	1.71959	0.859796	−	0.510638i	$-0.170591\pi$
$349$	32.1246	1.71959	0.859796	+	0.510638i	$0.170591\pi$
$350$	0	0
$351$	−19.0000	−1.01414
$352$	0	0
$353$	25.3607	1.34981	0.674906	−	0.737903i	$-0.264184\pi$
$353$	25.3607	1.34981	0.674906	+	0.737903i	$0.264184\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	25.7984	1.36539
$358$	0	0
$359$	3.88854	0.205229	0.102615	−	0.994721i	$-0.467279\pi$
$359$	3.88854	0.205229	0.102615	+	0.994721i	$0.467279\pi$
$360$	0	0
$361$	48.8328	2.57015
$362$	0	0
$363$	1.61803	0.0849248
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−10.9787	−0.573084	−0.286542	−	0.958068i	$-0.592506\pi$
$367$	−10.9787	−0.573084	−0.286542	+	0.958068i	$0.592506\pi$
$368$	0	0
$369$	−1.03444	−0.0538509
$370$	0	0
$371$	3.61803	0.187839
$372$	0	0
$373$	12.4164	0.642897	0.321449	−	0.946927i	$-0.395830\pi$
$373$	12.4164	0.642897	0.321449	+	0.946927i	$0.395830\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−25.4377	−1.31011
$378$	0	0
$379$	−20.5279	−1.05445	−0.527223	−	0.849727i	$-0.676767\pi$
$379$	−20.5279	−1.05445	−0.527223	+	0.849727i	$0.676767\pi$
$380$	0	0
$381$	−14.3262	−0.733955
$382$	0	0
$383$	−1.05573	−0.0539452	−0.0269726	−	0.999636i	$-0.508587\pi$
$383$	−1.05573	−0.0539452	−0.0269726	+	0.999636i	$0.508587\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.18034	0.212499
$388$	0	0
$389$	15.0557	0.763356	0.381678	−	0.924295i	$-0.375346\pi$
$389$	15.0557	0.763356	0.381678	+	0.924295i	$0.375346\pi$
$390$	0	0
$391$	−15.9443	−0.806336
$392$	0	0
$393$	−12.2361	−0.617228
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0.201626	0.0101193	0.00505966	−	0.999987i	$-0.498389\pi$
$397$	0.201626	0.0101193	0.00505966	+	0.999987i	$0.498389\pi$
$398$	0	0
$399$	−34.8885	−1.74661
$400$	0	0
$401$	−8.47214	−0.423078	−0.211539	−	0.977370i	$-0.567848\pi$
$401$	−8.47214	−0.423078	−0.211539	+	0.977370i	$0.567848\pi$
$402$	0	0
$403$	−16.3475	−0.814328
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−4.23607	−0.209974
$408$	0	0
$409$	−29.4721	−1.45730	−0.728652	−	0.684884i	$-0.759853\pi$
$409$	−29.4721	−1.45730	−0.728652	+	0.684884i	$0.759853\pi$
$410$	0	0
$411$	−17.0902	−0.842996
$412$	0	0
$413$	7.09017	0.348884
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−18.0902	−0.885879
$418$	0	0
$419$	21.4721	1.04898	0.524491	−	0.851416i	$-0.324256\pi$
$419$	21.4721	1.04898	0.524491	+	0.851416i	$0.324256\pi$
$420$	0	0
$421$	20.3262	0.990640	0.495320	−	0.868711i	$-0.335051\pi$
$421$	20.3262	0.990640	0.495320	+	0.868711i	$0.335051\pi$
$422$	0	0
$423$	−0.854102	−0.0415279
$424$	0	0
$425$	0	0
$426$	0	0
$427$	34.2705	1.65847
$428$	0	0
$429$	−5.61803	−0.271241
$430$	0	0
$431$	10.2361	0.493054	0.246527	−	0.969136i	$-0.420711\pi$
$431$	10.2361	0.493054	0.246527	+	0.969136i	$0.420711\pi$
$432$	0	0
$433$	24.8885	1.19607	0.598034	−	0.801471i	$-0.295949\pi$
$433$	24.8885	1.19607	0.598034	+	0.801471i	$0.295949\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21.5623	1.03146
$438$	0	0
$439$	40.9230	1.95315	0.976574	−	0.215183i	$-0.0690348\pi$
$439$	40.9230	1.95315	0.976574	+	0.215183i	$0.0690348\pi$
$440$	0	0
$441$	0.0557281	0.00265372
$442$	0	0
$443$	−37.7082	−1.79157	−0.895785	−	0.444487i	$-0.853386\pi$
$443$	−37.7082	−1.79157	−0.895785	+	0.444487i	$0.853386\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	31.4164	1.48595
$448$	0	0
$449$	41.2705	1.94768	0.973838	−	0.227244i	$-0.0729714\pi$
$449$	41.2705	1.94768	0.973838	+	0.227244i	$0.0729714\pi$
$450$	0	0
$451$	−2.70820	−0.127524
$452$	0	0
$453$	3.14590	0.147807
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−25.8541	−1.20940	−0.604702	−	0.796452i	$-0.706708\pi$
$457$	−25.8541	−1.20940	−0.604702	+	0.796452i	$0.706708\pi$
$458$	0	0
$459$	33.3262	1.55554
$460$	0	0
$461$	−28.8885	−1.34547	−0.672737	−	0.739882i	$-0.734881\pi$
$461$	−28.8885	−1.34547	−0.672737	+	0.739882i	$0.734881\pi$
$462$	0	0
$463$	31.7639	1.47620	0.738098	−	0.674694i	$-0.235724\pi$
$463$	31.7639	1.47620	0.738098	+	0.674694i	$0.235724\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.2918	0.985267	0.492633	−	0.870237i	$-0.336034\pi$
$467$	21.2918	0.985267	0.492633	+	0.870237i	$0.336034\pi$
$468$	0	0
$469$	31.4164	1.45067
$470$	0	0
$471$	−7.23607	−0.333420
$472$	0	0
$473$	10.9443	0.503218
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.527864	0.0241692
$478$	0	0
$479$	−22.4164	−1.02423	−0.512116	−	0.858916i	$-0.671138\pi$
$479$	−22.4164	−1.02423	−0.512116	+	0.858916i	$0.671138\pi$
$480$	0	0
$481$	14.7082	0.670636
$482$	0	0
$483$	−11.0902	−0.504620
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−2.81966	−0.127771	−0.0638855	−	0.997957i	$-0.520349\pi$
$487$	−2.81966	−0.127771	−0.0638855	+	0.997957i	$0.520349\pi$
$488$	0	0
$489$	−13.9443	−0.630582
$490$	0	0
$491$	−23.6525	−1.06742	−0.533711	−	0.845667i	$-0.679203\pi$
$491$	−23.6525	−1.06742	−0.533711	+	0.845667i	$0.679203\pi$
$492$	0	0
$493$	44.6180	2.00950
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−37.1246	−1.66527
$498$	0	0
$499$	−20.4377	−0.914917	−0.457458	−	0.889231i	$-0.651240\pi$
$499$	−20.4377	−0.914917	−0.457458	+	0.889231i	$0.651240\pi$
$500$	0	0
$501$	−10.8541	−0.484926
$502$	0	0
$503$	−5.81966	−0.259486	−0.129743	−	0.991548i	$-0.541415\pi$
$503$	−5.81966	−0.259486	−0.129743	+	0.991548i	$0.541415\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.52786	−0.0678548
$508$	0	0
$509$	20.1459	0.892951	0.446476	−	0.894796i	$-0.352679\pi$
$509$	20.1459	0.892951	0.446476	+	0.894796i	$0.352679\pi$
$510$	0	0
$511$	24.5623	1.08657
$512$	0	0
$513$	−45.0689	−1.98984
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−2.23607	−0.0983422
$518$	0	0
$519$	−39.9787	−1.75487
$520$	0	0
$521$	7.23607	0.317018	0.158509	−	0.987358i	$-0.449331\pi$
$521$	7.23607	0.317018	0.158509	+	0.987358i	$0.449331\pi$
$522$	0	0
$523$	9.41641	0.411751	0.205875	−	0.978578i	$-0.433996\pi$
$523$	9.41641	0.411751	0.205875	+	0.978578i	$0.433996\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	28.6738	1.24905
$528$	0	0
$529$	−16.1459	−0.701996
$530$	0	0
$531$	1.03444	0.0448910
$532$	0	0
$533$	9.40325	0.407300
$534$	0	0
$535$	0	0
$536$	0	0
$537$	14.7082	0.634706
$538$	0	0
$539$	0.145898	0.00628427
$540$	0	0
$541$	−15.4508	−0.664284	−0.332142	−	0.943229i	$-0.607771\pi$
$541$	−15.4508	−0.664284	−0.332142	+	0.943229i	$0.607771\pi$
$542$	0	0
$543$	10.7082	0.459533
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.3262	−0.783573	−0.391787	−	0.920056i	$-0.628143\pi$
$547$	−18.3262	−0.783573	−0.391787	+	0.920056i	$0.628143\pi$
$548$	0	0
$549$	5.00000	0.213395
$550$	0	0
$551$	−60.3394	−2.57054
$552$	0	0
$553$	−25.1803	−1.07078
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−1.47214	−0.0623764	−0.0311882	−	0.999514i	$-0.509929\pi$
$557$	−1.47214	−0.0623764	−0.0311882	+	0.999514i	$0.509929\pi$
$558$	0	0
$559$	−38.0000	−1.60723
$560$	0	0
$561$	9.85410	0.416041
$562$	0	0
$563$	28.0344	1.18151	0.590755	−	0.806851i	$-0.298830\pi$
$563$	28.0344	1.18151	0.590755	+	0.806851i	$0.298830\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	20.1803	0.847495
$568$	0	0
$569$	−44.7984	−1.87805	−0.939023	−	0.343855i	$-0.888267\pi$
$569$	−44.7984	−1.87805	−0.939023	+	0.343855i	$0.888267\pi$
$570$	0	0
$571$	−35.1591	−1.47136	−0.735680	−	0.677329i	$-0.763137\pi$
$571$	−35.1591	−1.47136	−0.735680	+	0.677329i	$0.763137\pi$
$572$	0	0
$573$	−27.5623	−1.15143
$574$	0	0
$575$	0	0
$576$	0	0
$577$	12.7426	0.530483	0.265242	−	0.964182i	$-0.414548\pi$
$577$	12.7426	0.530483	0.265242	+	0.964182i	$0.414548\pi$
$578$	0	0
$579$	−29.8885	−1.24213
$580$	0	0
$581$	−17.1803	−0.712761
$582$	0	0
$583$	1.38197	0.0572352
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.7984	1.31246	0.656230	−	0.754561i	$-0.272150\pi$
$587$	31.7984	1.31246	0.656230	+	0.754561i	$0.272150\pi$
$588$	0	0
$589$	−38.7771	−1.59778
$590$	0	0
$591$	29.1803	1.20032
$592$	0	0
$593$	47.6525	1.95685	0.978426	−	0.206596i	$-0.0662386\pi$
$593$	47.6525	1.95685	0.978426	+	0.206596i	$0.0662386\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	20.0344	0.819955
$598$	0	0
$599$	−39.3951	−1.60964	−0.804821	−	0.593518i	$-0.797738\pi$
$599$	−39.3951	−1.60964	−0.804821	+	0.593518i	$0.797738\pi$
$600$	0	0
$601$	20.3262	0.829125	0.414562	−	0.910021i	$-0.363935\pi$
$601$	20.3262	0.829125	0.414562	+	0.910021i	$0.363935\pi$
$602$	0	0
$603$	4.58359	0.186658
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−44.8328	−1.81971	−0.909854	−	0.414929i	$-0.863807\pi$
$607$	−44.8328	−1.81971	−0.909854	+	0.414929i	$0.863807\pi$
$608$	0	0
$609$	31.0344	1.25758
$610$	0	0
$611$	7.76393	0.314095
$612$	0	0
$613$	−30.6738	−1.23890	−0.619451	−	0.785035i	$-0.712645\pi$
$613$	−30.6738	−1.23890	−0.619451	+	0.785035i	$0.712645\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	8.76393	0.352823	0.176411	−	0.984317i	$-0.443551\pi$
$617$	8.76393	0.352823	0.176411	+	0.984317i	$0.443551\pi$
$618$	0	0
$619$	−27.2918	−1.09695	−0.548475	−	0.836167i	$-0.684791\pi$
$619$	−27.2918	−1.09695	−0.548475	+	0.836167i	$0.684791\pi$
$620$	0	0
$621$	−14.3262	−0.574892
$622$	0	0
$623$	−3.47214	−0.139108
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−13.3262	−0.532199
$628$	0	0
$629$	−25.7984	−1.02865
$630$	0	0
$631$	17.4508	0.694707	0.347354	−	0.937734i	$-0.387080\pi$
$631$	17.4508	0.694707	0.347354	+	0.937734i	$0.387080\pi$
$632$	0	0
$633$	1.61803	0.0643111
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−0.506578	−0.0200713
$638$	0	0
$639$	−5.41641	−0.214270
$640$	0	0
$641$	16.4164	0.648409	0.324205	−	0.945987i	$-0.394903\pi$
$641$	16.4164	0.648409	0.324205	+	0.945987i	$0.394903\pi$
$642$	0	0
$643$	−1.41641	−0.0558577	−0.0279288	−	0.999610i	$-0.508891\pi$
$643$	−1.41641	−0.0558577	−0.0279288	+	0.999610i	$0.508891\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.4164	−0.409511	−0.204756	−	0.978813i	$-0.565640\pi$
$647$	−10.4164	−0.409511	−0.204756	+	0.978813i	$0.565640\pi$
$648$	0	0
$649$	2.70820	0.106306
$650$	0	0
$651$	19.9443	0.781678
$652$	0	0
$653$	49.6180	1.94170	0.970852	−	0.239680i	$-0.0770426\pi$
$653$	49.6180	1.94170	0.970852	+	0.239680i	$0.0770426\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	3.58359	0.139809
$658$	0	0
$659$	−22.7426	−0.885928	−0.442964	−	0.896539i	$-0.646073\pi$
$659$	−22.7426	−0.885928	−0.442964	+	0.896539i	$0.646073\pi$
$660$	0	0
$661$	−22.5967	−0.878912	−0.439456	−	0.898264i	$-0.644829\pi$
$661$	−22.5967	−0.878912	−0.439456	+	0.898264i	$0.644829\pi$
$662$	0	0
$663$	−34.2148	−1.32879
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−19.1803	−0.742666
$668$	0	0
$669$	9.14590	0.353601
$670$	0	0
$671$	13.0902	0.505340
$672$	0	0
$673$	30.5967	1.17942	0.589709	−	0.807616i	$-0.299242\pi$
$673$	30.5967	1.17942	0.589709	+	0.807616i	$0.299242\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−47.5967	−1.82929	−0.914646	−	0.404256i	$-0.867530\pi$
$677$	−47.5967	−1.82929	−0.914646	+	0.404256i	$0.867530\pi$
$678$	0	0
$679$	33.0344	1.26775
$680$	0	0
$681$	11.5623	0.443069
$682$	0	0
$683$	21.1803	0.810443	0.405222	−	0.914218i	$-0.367194\pi$
$683$	21.1803	0.810443	0.405222	+	0.914218i	$0.367194\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	16.7984	0.640898
$688$	0	0
$689$	−4.79837	−0.182803
$690$	0	0
$691$	37.6312	1.43156	0.715779	−	0.698327i	$-0.246072\pi$
$691$	37.6312	1.43156	0.715779	+	0.698327i	$0.246072\pi$
$692$	0	0
$693$	−1.00000	−0.0379869
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−16.4934	−0.624733
$698$	0	0
$699$	−5.76393	−0.218012
$700$	0	0
$701$	−46.4721	−1.75523	−0.877614	−	0.479368i	$-0.840866\pi$
$701$	−46.4721	−1.75523	−0.877614	+	0.479368i	$0.840866\pi$
$702$	0	0
$703$	34.8885	1.31585
$704$	0	0
$705$	0	0
$706$	0	0
$707$	29.7984	1.12068
$708$	0	0
$709$	20.3607	0.764661	0.382331	−	0.924026i	$-0.375122\pi$
$709$	20.3607	0.764661	0.382331	+	0.924026i	$0.375122\pi$
$710$	0	0
$711$	−3.67376	−0.137777
$712$	0	0
$713$	−12.3262	−0.461621
$714$	0	0
$715$	0	0
$716$	0	0
$717$	39.8328	1.48758
$718$	0	0
$719$	3.83282	0.142940	0.0714700	−	0.997443i	$-0.477231\pi$
$719$	3.83282	0.142940	0.0714700	+	0.997443i	$0.477231\pi$
$720$	0	0
$721$	−8.70820	−0.324310
$722$	0	0
$723$	11.0902	0.412448
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.90983	−0.0708317	−0.0354158	−	0.999373i	$-0.511276\pi$
$727$	−1.90983	−0.0708317	−0.0354158	+	0.999373i	$0.511276\pi$
$728$	0	0
$729$	29.5066	1.09284
$730$	0	0
$731$	66.6525	2.46523
$732$	0	0
$733$	36.1246	1.33429	0.667146	−	0.744927i	$-0.267516\pi$
$733$	36.1246	1.33429	0.667146	+	0.744927i	$0.267516\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.0000	0.442026
$738$	0	0
$739$	−3.85410	−0.141775	−0.0708877	−	0.997484i	$-0.522583\pi$
$739$	−3.85410	−0.141775	−0.0708877	+	0.997484i	$0.522583\pi$
$740$	0	0
$741$	46.2705	1.69979
$742$	0	0
$743$	4.21478	0.154625	0.0773127	−	0.997007i	$-0.475366\pi$
$743$	4.21478	0.154625	0.0773127	+	0.997007i	$0.475366\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−2.50658	−0.0917109
$748$	0	0
$749$	−24.0344	−0.878199
$750$	0	0
$751$	−29.0344	−1.05948	−0.529741	−	0.848160i	$-0.677711\pi$
$751$	−29.0344	−1.05948	−0.529741	+	0.848160i	$0.677711\pi$
$752$	0	0
$753$	31.5623	1.15019
$754$	0	0
$755$	0	0
$756$	0	0
$757$	30.8885	1.12266	0.561332	−	0.827591i	$-0.310289\pi$
$757$	30.8885	1.12266	0.561332	+	0.827591i	$0.310289\pi$
$758$	0	0
$759$	−4.23607	−0.153760
$760$	0	0
$761$	8.00000	0.290000	0.145000	−	0.989432i	$-0.453682\pi$
$761$	8.00000	0.290000	0.145000	+	0.989432i	$0.453682\pi$
$762$	0	0
$763$	43.3607	1.56976
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.40325	−0.339532
$768$	0	0
$769$	−10.3475	−0.373141	−0.186571	−	0.982442i	$-0.559737\pi$
$769$	−10.3475	−0.373141	−0.186571	+	0.982442i	$0.559737\pi$
$770$	0	0
$771$	−37.1246	−1.33701
$772$	0	0
$773$	−29.6312	−1.06576	−0.532880	−	0.846191i	$-0.678890\pi$
$773$	−29.6312	−1.06576	−0.532880	+	0.846191i	$0.678890\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−17.9443	−0.643747
$778$	0	0
$779$	22.3050	0.799158
$780$	0	0
$781$	−14.1803	−0.507412
$782$	0	0
$783$	40.0902	1.43271
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−6.23607	−0.222292	−0.111146	−	0.993804i	$-0.535452\pi$
$787$	−6.23607	−0.222292	−0.111146	+	0.993804i	$0.535452\pi$
$788$	0	0
$789$	−10.5623	−0.376028
$790$	0	0
$791$	9.85410	0.350372
$792$	0	0
$793$	−45.4508	−1.61401
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15.4377	0.546831	0.273416	−	0.961896i	$-0.411847\pi$
$797$	15.4377	0.546831	0.273416	+	0.961896i	$0.411847\pi$
$798$	0	0
$799$	−13.6180	−0.481771
$800$	0	0
$801$	−0.506578	−0.0178990
$802$	0	0
$803$	9.38197	0.331082
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.618034	−0.0217558
$808$	0	0
$809$	30.2492	1.06351	0.531753	−	0.846899i	$-0.321533\pi$
$809$	30.2492	1.06351	0.531753	+	0.846899i	$0.321533\pi$
$810$	0	0
$811$	−3.94427	−0.138502	−0.0692511	−	0.997599i	$-0.522061\pi$
$811$	−3.94427	−0.138502	−0.0692511	+	0.997599i	$0.522061\pi$
$812$	0	0
$813$	1.14590	0.0401884
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−90.1378	−3.15352
$818$	0	0
$819$	3.47214	0.121326
$820$	0	0
$821$	36.7214	1.28158	0.640792	−	0.767714i	$-0.278606\pi$
$821$	36.7214	1.28158	0.640792	+	0.767714i	$0.278606\pi$
$822$	0	0
$823$	−38.8328	−1.35363	−0.676813	−	0.736155i	$-0.736640\pi$
$823$	−38.8328	−1.35363	−0.676813	+	0.736155i	$0.736640\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4.76393	0.165658	0.0828291	−	0.996564i	$-0.473604\pi$
$827$	4.76393	0.165658	0.0828291	+	0.996564i	$0.473604\pi$
$828$	0	0
$829$	46.1591	1.60317	0.801585	−	0.597881i	$-0.203990\pi$
$829$	46.1591	1.60317	0.801585	+	0.597881i	$0.203990\pi$
$830$	0	0
$831$	−22.8541	−0.792800
$832$	0	0
$833$	0.888544	0.0307862
$834$	0	0
$835$	0	0
$836$	0	0
$837$	25.7639	0.890532
$838$	0	0
$839$	2.14590	0.0740846	0.0370423	−	0.999314i	$-0.488206\pi$
$839$	2.14590	0.0740846	0.0370423	+	0.999314i	$0.488206\pi$
$840$	0	0
$841$	24.6738	0.850819
$842$	0	0
$843$	−0.381966	−0.0131556
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.61803	−0.0899567
$848$	0	0
$849$	−46.0344	−1.57990
$850$	0	0
$851$	11.0902	0.380166
$852$	0	0
$853$	−8.56231	−0.293168	−0.146584	−	0.989198i	$-0.546828\pi$
$853$	−8.56231	−0.293168	−0.146584	+	0.989198i	$0.546828\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−41.8328	−1.42898	−0.714491	−	0.699645i	$-0.753342\pi$
$857$	−41.8328	−1.42898	−0.714491	+	0.699645i	$0.753342\pi$
$858$	0	0
$859$	8.29180	0.282912	0.141456	−	0.989945i	$-0.454822\pi$
$859$	8.29180	0.282912	0.141456	+	0.989945i	$0.454822\pi$
$860$	0	0
$861$	−11.4721	−0.390969
$862$	0	0
$863$	25.5836	0.870876	0.435438	−	0.900219i	$-0.356594\pi$
$863$	25.5836	0.870876	0.435438	+	0.900219i	$0.356594\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	32.5066	1.10398
$868$	0	0
$869$	−9.61803	−0.326269
$870$	0	0
$871$	−41.6656	−1.41179
$872$	0	0
$873$	4.81966	0.163121
$874$	0	0
$875$	0	0
$876$	0	0
$877$	10.1115	0.341440	0.170720	−	0.985320i	$-0.445391\pi$
$877$	10.1115	0.341440	0.170720	+	0.985320i	$0.445391\pi$
$878$	0	0
$879$	33.1246	1.11727
$880$	0	0
$881$	−41.4508	−1.39651	−0.698257	−	0.715847i	$-0.746041\pi$
$881$	−41.4508	−1.39651	−0.698257	+	0.715847i	$0.746041\pi$
$882$	0	0
$883$	27.8328	0.936649	0.468324	−	0.883557i	$-0.344858\pi$
$883$	27.8328	0.936649	0.468324	+	0.883557i	$0.344858\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	17.0000	0.570804	0.285402	−	0.958408i	$-0.407873\pi$
$887$	17.0000	0.570804	0.285402	+	0.958408i	$0.407873\pi$
$888$	0	0
$889$	23.1803	0.777444
$890$	0	0
$891$	7.70820	0.258235
$892$	0	0
$893$	18.4164	0.616282
$894$	0	0
$895$	0	0
$896$	0	0
$897$	14.7082	0.491093
$898$	0	0
$899$	34.4934	1.15042
$900$	0	0
$901$	8.41641	0.280391
$902$	0	0
$903$	46.3607	1.54279
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−47.7771	−1.58641	−0.793206	−	0.608953i	$-0.791590\pi$
$907$	−47.7771	−1.58641	−0.793206	+	0.608953i	$0.791590\pi$
$908$	0	0
$909$	4.34752	0.144198
$910$	0	0
$911$	30.9443	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$911$	30.9443	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$912$	0	0
$913$	−6.56231	−0.217181
$914$	0	0
$915$	0	0
$916$	0	0
$917$	19.7984	0.653800
$918$	0	0
$919$	7.63932	0.251998	0.125999	−	0.992030i	$-0.459786\pi$
$919$	7.63932	0.251998	0.125999	+	0.992030i	$0.459786\pi$
$920$	0	0
$921$	−27.7426	−0.914151
$922$	0	0
$923$	49.2361	1.62062
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.27051	−0.0417290
$928$	0	0
$929$	−59.1803	−1.94164	−0.970822	−	0.239801i	$-0.922918\pi$
$929$	−59.1803	−1.94164	−0.970822	+	0.239801i	$0.922918\pi$
$930$	0	0
$931$	−1.20163	−0.0393817
$932$	0	0
$933$	19.3262	0.632713
$934$	0	0
$935$	0	0
$936$	0	0
$937$	27.4164	0.895655	0.447828	−	0.894120i	$-0.352198\pi$
$937$	27.4164	0.895655	0.447828	+	0.894120i	$0.352198\pi$
$938$	0	0
$939$	−33.4164	−1.09050
$940$	0	0
$941$	−31.1803	−1.01645	−0.508225	−	0.861224i	$-0.669698\pi$
$941$	−31.1803	−1.01645	−0.508225	+	0.861224i	$0.669698\pi$
$942$	0	0
$943$	7.09017	0.230888
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.1246	−1.04391	−0.521955	−	0.852973i	$-0.674797\pi$
$947$	−32.1246	−1.04391	−0.521955	+	0.852973i	$0.674797\pi$
$948$	0	0
$949$	−32.5755	−1.05744
$950$	0	0
$951$	−9.09017	−0.294769
$952$	0	0
$953$	35.8885	1.16254	0.581272	−	0.813709i	$-0.302555\pi$
$953$	35.8885	1.16254	0.581272	+	0.813709i	$0.302555\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	11.8541	0.383189
$958$	0	0
$959$	27.6525	0.892945
$960$	0	0
$961$	−8.83282	−0.284930
$962$	0	0
$963$	−3.50658	−0.112998
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−36.2705	−1.16638	−0.583190	−	0.812335i	$-0.698196\pi$
$967$	−36.2705	−1.16638	−0.583190	+	0.812335i	$0.698196\pi$
$968$	0	0
$969$	−81.1591	−2.60720
$970$	0	0
$971$	2.61803	0.0840167	0.0420084	−	0.999117i	$-0.486624\pi$
$971$	2.61803	0.0840167	0.0420084	+	0.999117i	$0.486624\pi$
$972$	0	0
$973$	29.2705	0.938369
$974$	0	0
$975$	0	0
$976$	0	0
$977$	33.5410	1.07307	0.536536	−	0.843877i	$-0.319733\pi$
$977$	33.5410	1.07307	0.536536	+	0.843877i	$0.319733\pi$
$978$	0	0
$979$	−1.32624	−0.0423867
$980$	0	0
$981$	6.32624	0.201981
$982$	0	0
$983$	−23.0557	−0.735364	−0.367682	−	0.929952i	$-0.619848\pi$
$983$	−23.0557	−0.735364	−0.367682	+	0.929952i	$0.619848\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−9.47214	−0.301501
$988$	0	0
$989$	−28.6525	−0.911096
$990$	0	0
$991$	37.9098	1.20425	0.602123	−	0.798404i	$-0.294322\pi$
$991$	37.9098	1.20425	0.602123	+	0.798404i	$0.294322\pi$
$992$	0	0
$993$	50.7426	1.61027
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−26.9656	−0.854008	−0.427004	−	0.904250i	$-0.640431\pi$
$997$	−26.9656	−0.854008	−0.427004	+	0.904250i	$0.640431\pi$
$998$	0	0
$999$	−23.1803	−0.733393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bo.1.2		2
4.3	odd	2		2200.2.a.p.1.1	✓	2
5.2	odd	4		4400.2.b.z.4049.1		4
5.3	odd	4		4400.2.b.z.4049.4		4
5.4	even	2		4400.2.a.bm.1.1		2
20.3	even	4		2200.2.b.k.1849.1		4
20.7	even	4		2200.2.b.k.1849.4		4
20.19	odd	2		2200.2.a.q.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.2.a.p.1.1	✓	2	4.3	odd	2
2200.2.a.q.1.2	yes	2	20.19	odd	2
2200.2.b.k.1849.1		4	20.3	even	4
2200.2.b.k.1849.4		4	20.7	even	4
4400.2.a.bm.1.1		2	5.4	even	2
4400.2.a.bo.1.2		2	1.1	even	1	trivial
4400.2.b.z.4049.1		4	5.2	odd	4
4400.2.b.z.4049.4		4	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.61803 q^{3} -2.61803 q^{7} -0.381966 q^{9} -1.00000 q^{11} +3.47214 q^{13} -6.09017 q^{17} +8.23607 q^{19} -4.23607 q^{21} +2.61803 q^{23} -5.47214 q^{27} -7.32624 q^{29} -4.70820 q^{31} -1.61803 q^{33} +4.23607 q^{37} +5.61803 q^{39} +2.70820 q^{41} -10.9443 q^{43} +2.23607 q^{47} -0.145898 q^{49} -9.85410 q^{51} -1.38197 q^{53} +13.3262 q^{57} -2.70820 q^{59} -13.0902 q^{61} +1.00000 q^{63} -12.0000 q^{67} +4.23607 q^{69} +14.1803 q^{71} -9.38197 q^{73} +2.61803 q^{77} +9.61803 q^{79} -7.70820 q^{81} +6.56231 q^{83} -11.8541 q^{87} +1.32624 q^{89} -9.09017 q^{91} -7.61803 q^{93} -12.6180 q^{97} +0.381966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 3 q^{7} - 3 q^{9} - 2 q^{11} - 2 q^{13} - q^{17} + 12 q^{19} - 4 q^{21} + 3 q^{23} - 2 q^{27} + q^{29} + 4 q^{31} - q^{33} + 4 q^{37} + 9 q^{39} - 8 q^{41} - 4 q^{43} - 7 q^{49} - 13 q^{51}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q + q^3 - 3 * q^7 - 3 * q^9 - 2 * q^11 - 2 * q^13 - q^17 + 12 * q^19 - 4 * q^21 + 3 * q^23 - 2 * q^27 + q^29 + 4 * q^31 - q^33 + 4 * q^37 + 9 * q^39 - 8 * q^41 - 4 * q^43 - 7 * q^49 - 13 * q^51 - 5 * q^53 + 11 * q^57 + 8 * q^59 - 15 * q^61 + 2 * q^63 - 24 * q^67 + 4 * q^69 + 6 * q^71 - 21 * q^73 + 3 * q^77 + 17 * q^79 - 2 * q^81 - 7 * q^83 - 17 * q^87 - 13 * q^89 - 7 * q^91 - 13 * q^93 - 23 * q^97 + 3 * q^99

Introduction

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