LMFDB - Embedded newform 4400.2.a.bg.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,-1,0,1,0,-2,0,8] 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:	$\Q(\zeta_{10})^+$
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 275)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-2.61803 q^{3} +2.85410 q^{7} +3.85410 q^{9} -1.00000 q^{11} +6.23607 q^{13} -0.618034 q^{17} +6.70820 q^{19} -7.47214 q^{21} +4.09017 q^{23} -2.23607 q^{27} -1.38197 q^{29} +3.00000 q^{31} +2.61803 q^{33} +10.2361 q^{37} -16.3262 q^{39} -3.00000 q^{41} +6.00000 q^{43} -11.9443 q^{47} +1.14590 q^{49} +1.61803 q^{51} +9.32624 q^{53} -17.5623 q^{57} -0.527864 q^{59} +0.0901699 q^{61} +11.0000 q^{63} -8.00000 q^{67} -10.7082 q^{69} -8.18034 q^{71} +10.3820 q^{73} -2.85410 q^{77} -5.85410 q^{79} -5.70820 q^{81} +10.1459 q^{83} +3.61803 q^{87} -6.90983 q^{89} +17.7984 q^{91} -7.85410 q^{93} +1.61803 q^{97} -3.85410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{3} - q^{7} + q^{9} - 2 q^{11} + 8 q^{13} + q^{17} - 6 q^{21} - 3 q^{23} - 5 q^{29} + 6 q^{31} + 3 q^{33} + 16 q^{37} - 17 q^{39} - 6 q^{41} + 12 q^{43} - 6 q^{47} + 9 q^{49} + q^{51} + 3 q^{53}+ \cdots - q^{99}+O(q^{100}) $ 2 * q - 3 * q^3 - q^7 + q^9 - 2 * q^11 + 8 * q^13 + q^17 - 6 * q^21 - 3 * q^23 - 5 * q^29 + 6 * q^31 + 3 * q^33 + 16 * q^37 - 17 * q^39 - 6 * q^41 + 12 * q^43 - 6 * q^47 + 9 * q^49 + q^51 + 3 * q^53 - 15 * q^57 - 10 * q^59 - 11 * q^61 + 22 * q^63 - 16 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + q^77 - 5 * q^79 + 2 * q^81 + 27 * q^83 + 5 * q^87 - 25 * q^89 + 11 * q^91 - 9 * q^93 + q^97 - 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
$3$	−2.61803	−1.51152	−0.755761	−	0.654847i	$-0.772733\pi$
$3$	−2.61803	−1.51152	−0.755761	+	0.654847i	$0.772733\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.85410	1.07875	0.539375	−	0.842066i	$-0.318661\pi$
$7$	2.85410	1.07875	0.539375	+	0.842066i	$0.318661\pi$
$8$	0	0
$9$	3.85410	1.28470
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	6.23607	1.72957	0.864787	−	0.502139i	$-0.167453\pi$
$13$	6.23607	1.72957	0.864787	+	0.502139i	$0.167453\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−0.618034	−0.149895	−0.0749476	−	0.997187i	$-0.523879\pi$
$17$	−0.618034	−0.149895	−0.0749476	+	0.997187i	$0.523879\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	0	0
$21$	−7.47214	−1.63055
$22$	0	0
$23$	4.09017	0.852859	0.426430	−	0.904521i	$-0.359771\pi$
$23$	4.09017	0.852859	0.426430	+	0.904521i	$0.359771\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	−1.38197	−0.256625	−0.128312	−	0.991734i	$-0.540956\pi$
$29$	−1.38197	−0.256625	−0.128312	+	0.991734i	$0.540956\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	2.61803	0.455741
$34$	0	0
$35$	0	0
$36$	0	0
$37$	10.2361	1.68280	0.841400	−	0.540413i	$-0.181732\pi$
$37$	10.2361	1.68280	0.841400	+	0.540413i	$0.181732\pi$
$38$	0	0
$39$	−16.3262	−2.61429
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−11.9443	−1.74225	−0.871126	−	0.491060i	$-0.836609\pi$
$47$	−11.9443	−1.74225	−0.871126	+	0.491060i	$0.836609\pi$
$48$	0	0
$49$	1.14590	0.163700
$50$	0	0
$51$	1.61803	0.226570
$52$	0	0
$53$	9.32624	1.28106	0.640529	−	0.767934i	$-0.278715\pi$
$53$	9.32624	1.28106	0.640529	+	0.767934i	$0.278715\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−17.5623	−2.32618
$58$	0	0
$59$	−0.527864	−0.0687220	−0.0343610	−	0.999409i	$-0.510940\pi$
$59$	−0.527864	−0.0687220	−0.0343610	+	0.999409i	$0.510940\pi$
$60$	0	0
$61$	0.0901699	0.0115451	0.00577254	−	0.999983i	$-0.498163\pi$
$61$	0.0901699	0.0115451	0.00577254	+	0.999983i	$0.498163\pi$
$62$	0	0
$63$	11.0000	1.38587
$64$	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
$69$	−10.7082	−1.28912
$70$	0	0
$71$	−8.18034	−0.970828	−0.485414	−	0.874284i	$-0.661331\pi$
$71$	−8.18034	−0.970828	−0.485414	+	0.874284i	$0.661331\pi$
$72$	0	0
$73$	10.3820	1.21512	0.607559	−	0.794275i	$-0.292149\pi$
$73$	10.3820	1.21512	0.607559	+	0.794275i	$0.292149\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.85410	−0.325255
$78$	0	0
$79$	−5.85410	−0.658638	−0.329319	−	0.944219i	$-0.606819\pi$
$79$	−5.85410	−0.658638	−0.329319	+	0.944219i	$0.606819\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	0	0
$83$	10.1459	1.11366	0.556828	−	0.830627i	$-0.312018\pi$
$83$	10.1459	1.11366	0.556828	+	0.830627i	$0.312018\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	3.61803	0.387894
$88$	0	0
$89$	−6.90983	−0.732441	−0.366220	−	0.930528i	$-0.619348\pi$
$89$	−6.90983	−0.732441	−0.366220	+	0.930528i	$0.619348\pi$
$90$	0	0
$91$	17.7984	1.86578
$92$	0	0
$93$	−7.85410	−0.814432
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.61803	0.164286	0.0821432	−	0.996621i	$-0.473824\pi$
$97$	1.61803	0.164286	0.0821432	+	0.996621i	$0.473824\pi$
$98$	0	0
$99$	−3.85410	−0.387352
$100$	0	0
$101$	−6.09017	−0.605995	−0.302997	−	0.952991i	$-0.597987\pi$
$101$	−6.09017	−0.605995	−0.302997	+	0.952991i	$0.597987\pi$
$102$	0	0
$103$	−5.38197	−0.530301	−0.265150	−	0.964207i	$-0.585422\pi$
$103$	−5.38197	−0.530301	−0.265150	+	0.964207i	$0.585422\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.23607	0.409516	0.204758	−	0.978813i	$-0.434359\pi$
$107$	4.23607	0.409516	0.204758	+	0.978813i	$0.434359\pi$
$108$	0	0
$109$	−3.09017	−0.295985	−0.147992	−	0.988989i	$-0.547281\pi$
$109$	−3.09017	−0.295985	−0.147992	+	0.988989i	$0.547281\pi$
$110$	0	0
$111$	−26.7984	−2.54359
$112$	0	0
$113$	−11.6525	−1.09617	−0.548086	−	0.836422i	$-0.684643\pi$
$113$	−11.6525	−1.09617	−0.548086	+	0.836422i	$0.684643\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	24.0344	2.22198
$118$	0	0
$119$	−1.76393	−0.161699
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.85410	0.708181
$124$	0	0
$125$	0	0
$126$	0	0
$127$	0.618034	0.0548416	0.0274208	−	0.999624i	$-0.491271\pi$
$127$	0.618034	0.0548416	0.0274208	+	0.999624i	$0.491271\pi$
$128$	0	0
$129$	−15.7082	−1.38303
$130$	0	0
$131$	−10.0902	−0.881582	−0.440791	−	0.897610i	$-0.645302\pi$
$131$	−10.0902	−0.881582	−0.440791	+	0.897610i	$0.645302\pi$
$132$	0	0
$133$	19.1459	1.66016
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.56231	0.475220	0.237610	−	0.971361i	$-0.423636\pi$
$137$	5.56231	0.475220	0.237610	+	0.971361i	$0.423636\pi$
$138$	0	0
$139$	3.29180	0.279206	0.139603	−	0.990208i	$-0.455417\pi$
$139$	3.29180	0.279206	0.139603	+	0.990208i	$0.455417\pi$
$140$	0	0
$141$	31.2705	2.63345
$142$	0	0
$143$	−6.23607	−0.521486
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	−8.94427	−0.732743	−0.366372	−	0.930469i	$-0.619400\pi$
$149$	−8.94427	−0.732743	−0.366372	+	0.930469i	$0.619400\pi$
$150$	0	0
$151$	3.00000	0.244137	0.122068	−	0.992522i	$-0.461047\pi$
$151$	3.00000	0.244137	0.122068	+	0.992522i	$0.461047\pi$
$152$	0	0
$153$	−2.38197	−0.192571
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.41641	−0.432276	−0.216138	−	0.976363i	$-0.569346\pi$
$157$	−5.41641	−0.432276	−0.216138	+	0.976363i	$0.569346\pi$
$158$	0	0
$159$	−24.4164	−1.93635
$160$	0	0
$161$	11.6738	0.920021
$162$	0	0
$163$	6.85410	0.536855	0.268427	−	0.963300i	$-0.413496\pi$
$163$	6.85410	0.536855	0.268427	+	0.963300i	$0.413496\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.29180	0.409491	0.204746	−	0.978815i	$-0.434363\pi$
$167$	5.29180	0.409491	0.204746	+	0.978815i	$0.434363\pi$
$168$	0	0
$169$	25.8885	1.99143
$170$	0	0
$171$	25.8541	1.97711
$172$	0	0
$173$	−5.47214	−0.416039	−0.208019	−	0.978125i	$-0.566702\pi$
$173$	−5.47214	−0.416039	−0.208019	+	0.978125i	$0.566702\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	1.38197	0.103875
$178$	0	0
$179$	13.6180	1.01786	0.508930	−	0.860808i	$-0.330041\pi$
$179$	13.6180	1.01786	0.508930	+	0.860808i	$0.330041\pi$
$180$	0	0
$181$	−6.09017	−0.452679	−0.226339	−	0.974049i	$-0.572676\pi$
$181$	−6.09017	−0.452679	−0.226339	+	0.974049i	$0.572676\pi$
$182$	0	0
$183$	−0.236068	−0.0174506
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0.618034	0.0451951
$188$	0	0
$189$	−6.38197	−0.464220
$190$	0	0
$191$	21.0902	1.52603	0.763016	−	0.646380i	$-0.223718\pi$
$191$	21.0902	1.52603	0.763016	+	0.646380i	$0.223718\pi$
$192$	0	0
$193$	12.9443	0.931749	0.465875	−	0.884851i	$-0.345740\pi$
$193$	12.9443	0.931749	0.465875	+	0.884851i	$0.345740\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−20.0902	−1.43137	−0.715683	−	0.698426i	$-0.753884\pi$
$197$	−20.0902	−1.43137	−0.715683	+	0.698426i	$0.753884\pi$
$198$	0	0
$199$	3.09017	0.219056	0.109528	−	0.993984i	$-0.465066\pi$
$199$	3.09017	0.219056	0.109528	+	0.993984i	$0.465066\pi$
$200$	0	0
$201$	20.9443	1.47730
$202$	0	0
$203$	−3.94427	−0.276834
$204$	0	0
$205$	0	0
$206$	0	0
$207$	15.7639	1.09567
$208$	0	0
$209$	−6.70820	−0.464016
$210$	0	0
$211$	−17.0000	−1.17033	−0.585164	−	0.810915i	$-0.698970\pi$
$211$	−17.0000	−1.17033	−0.585164	+	0.810915i	$0.698970\pi$
$212$	0	0
$213$	21.4164	1.46743
$214$	0	0
$215$	0	0
$216$	0	0
$217$	8.56231	0.581247
$218$	0	0
$219$	−27.1803	−1.83668
$220$	0	0
$221$	−3.85410	−0.259255
$222$	0	0
$223$	−16.8885	−1.13094	−0.565470	−	0.824769i	$-0.691305\pi$
$223$	−16.8885	−1.13094	−0.565470	+	0.824769i	$0.691305\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	24.0344	1.59522	0.797611	−	0.603172i	$-0.206097\pi$
$227$	24.0344	1.59522	0.797611	+	0.603172i	$0.206097\pi$
$228$	0	0
$229$	12.0344	0.795258	0.397629	−	0.917546i	$-0.369833\pi$
$229$	12.0344	0.795258	0.397629	+	0.917546i	$0.369833\pi$
$230$	0	0
$231$	7.47214	0.491630
$232$	0	0
$233$	15.5066	1.01587	0.507935	−	0.861395i	$-0.330409\pi$
$233$	15.5066	1.01587	0.507935	+	0.861395i	$0.330409\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	15.3262	0.995546
$238$	0	0
$239$	−6.38197	−0.412815	−0.206408	−	0.978466i	$-0.566177\pi$
$239$	−6.38197	−0.412815	−0.206408	+	0.978466i	$0.566177\pi$
$240$	0	0
$241$	21.2705	1.37015	0.685077	−	0.728471i	$-0.259769\pi$
$241$	21.2705	1.37015	0.685077	+	0.728471i	$0.259769\pi$
$242$	0	0
$243$	21.6525	1.38901
$244$	0	0
$245$	0	0
$246$	0	0
$247$	41.8328	2.66176
$248$	0	0
$249$	−26.5623	−1.68332
$250$	0	0
$251$	27.2705	1.72130	0.860650	−	0.509198i	$-0.170058\pi$
$251$	27.2705	1.72130	0.860650	+	0.509198i	$0.170058\pi$
$252$	0	0
$253$	−4.09017	−0.257147
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−10.9443	−0.682685	−0.341342	−	0.939939i	$-0.610882\pi$
$257$	−10.9443	−0.682685	−0.341342	+	0.939939i	$0.610882\pi$
$258$	0	0
$259$	29.2148	1.81532
$260$	0	0
$261$	−5.32624	−0.329686
$262$	0	0
$263$	21.0000	1.29492	0.647458	−	0.762101i	$-0.275832\pi$
$263$	21.0000	1.29492	0.647458	+	0.762101i	$0.275832\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	18.0902	1.10710
$268$	0	0
$269$	30.3262	1.84902	0.924512	−	0.381154i	$-0.124473\pi$
$269$	30.3262	1.84902	0.924512	+	0.381154i	$0.124473\pi$
$270$	0	0
$271$	−13.1803	−0.800649	−0.400324	−	0.916374i	$-0.631103\pi$
$271$	−13.1803	−0.800649	−0.400324	+	0.916374i	$0.631103\pi$
$272$	0	0
$273$	−46.5967	−2.82016
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−11.4721	−0.689294	−0.344647	−	0.938732i	$-0.612001\pi$
$277$	−11.4721	−0.689294	−0.344647	+	0.938732i	$0.612001\pi$
$278$	0	0
$279$	11.5623	0.692217
$280$	0	0
$281$	−25.3607	−1.51289	−0.756446	−	0.654057i	$-0.773066\pi$
$281$	−25.3607	−1.51289	−0.756446	+	0.654057i	$0.773066\pi$
$282$	0	0
$283$	7.38197	0.438812	0.219406	−	0.975634i	$-0.429588\pi$
$283$	7.38197	0.438812	0.219406	+	0.975634i	$0.429588\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.56231	−0.505417
$288$	0	0
$289$	−16.6180	−0.977531
$290$	0	0
$291$	−4.23607	−0.248323
$292$	0	0
$293$	−23.8885	−1.39558	−0.697792	−	0.716301i	$-0.745834\pi$
$293$	−23.8885	−1.39558	−0.697792	+	0.716301i	$0.745834\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.23607	0.129750
$298$	0	0
$299$	25.5066	1.47508
$300$	0	0
$301$	17.1246	0.987046
$302$	0	0
$303$	15.9443	0.915974
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−33.4508	−1.90914	−0.954570	−	0.297985i	$-0.903685\pi$
$307$	−33.4508	−1.90914	−0.954570	+	0.297985i	$0.903685\pi$
$308$	0	0
$309$	14.0902	0.801562
$310$	0	0
$311$	19.1803	1.08762	0.543809	−	0.839209i	$-0.316982\pi$
$311$	19.1803	1.08762	0.543809	+	0.839209i	$0.316982\pi$
$312$	0	0
$313$	−3.23607	−0.182913	−0.0914567	−	0.995809i	$-0.529152\pi$
$313$	−3.23607	−0.182913	−0.0914567	+	0.995809i	$0.529152\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	16.6180	0.933362	0.466681	−	0.884426i	$-0.345450\pi$
$317$	16.6180	0.933362	0.466681	+	0.884426i	$0.345450\pi$
$318$	0	0
$319$	1.38197	0.0773752
$320$	0	0
$321$	−11.0902	−0.618993
$322$	0	0
$323$	−4.14590	−0.230684
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.09017	0.447387
$328$	0	0
$329$	−34.0902	−1.87945
$330$	0	0
$331$	19.1803	1.05425	0.527123	−	0.849789i	$-0.323271\pi$
$331$	19.1803	1.05425	0.527123	+	0.849789i	$0.323271\pi$
$332$	0	0
$333$	39.4508	2.16189
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.41641	0.0771567	0.0385783	−	0.999256i	$-0.487717\pi$
$337$	1.41641	0.0771567	0.0385783	+	0.999256i	$0.487717\pi$
$338$	0	0
$339$	30.5066	1.65689
$340$	0	0
$341$	−3.00000	−0.162459
$342$	0	0
$343$	−16.7082	−0.902158
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.5623	−1.10384	−0.551921	−	0.833896i	$-0.686105\pi$
$347$	−20.5623	−1.10384	−0.551921	+	0.833896i	$0.686105\pi$
$348$	0	0
$349$	21.8328	1.16868	0.584342	−	0.811508i	$-0.301353\pi$
$349$	21.8328	1.16868	0.584342	+	0.811508i	$0.301353\pi$
$350$	0	0
$351$	−13.9443	−0.744290
$352$	0	0
$353$	21.3607	1.13691	0.568457	−	0.822713i	$-0.307541\pi$
$353$	21.3607	1.13691	0.568457	+	0.822713i	$0.307541\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.61803	0.244412
$358$	0	0
$359$	4.47214	0.236030	0.118015	−	0.993012i	$-0.462347\pi$
$359$	4.47214	0.236030	0.118015	+	0.993012i	$0.462347\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	0	0
$363$	−2.61803	−0.137411
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−7.14590	−0.373013	−0.186506	−	0.982454i	$-0.559717\pi$
$367$	−7.14590	−0.373013	−0.186506	+	0.982454i	$0.559717\pi$
$368$	0	0
$369$	−11.5623	−0.601910
$370$	0	0
$371$	26.6180	1.38194
$372$	0	0
$373$	2.81966	0.145996	0.0729982	−	0.997332i	$-0.476743\pi$
$373$	2.81966	0.145996	0.0729982	+	0.997332i	$0.476743\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.61803	−0.443851
$378$	0	0
$379$	17.7639	0.912472	0.456236	−	0.889859i	$-0.349197\pi$
$379$	17.7639	0.912472	0.456236	+	0.889859i	$0.349197\pi$
$380$	0	0
$381$	−1.61803	−0.0828944
$382$	0	0
$383$	−5.05573	−0.258336	−0.129168	−	0.991623i	$-0.541231\pi$
$383$	−5.05573	−0.258336	−0.129168	+	0.991623i	$0.541231\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	23.1246	1.17549
$388$	0	0
$389$	5.52786	0.280274	0.140137	−	0.990132i	$-0.455246\pi$
$389$	5.52786	0.280274	0.140137	+	0.990132i	$0.455246\pi$
$390$	0	0
$391$	−2.52786	−0.127840
$392$	0	0
$393$	26.4164	1.33253
$394$	0	0
$395$	0	0
$396$	0	0
$397$	14.9098	0.748303	0.374151	−	0.927368i	$-0.377934\pi$
$397$	14.9098	0.748303	0.374151	+	0.927368i	$0.377934\pi$
$398$	0	0
$399$	−50.1246	−2.50937
$400$	0	0
$401$	34.3607	1.71589	0.857945	−	0.513741i	$-0.171741\pi$
$401$	34.3607	1.71589	0.857945	+	0.513741i	$0.171741\pi$
$402$	0	0
$403$	18.7082	0.931922
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.2361	−0.507383
$408$	0	0
$409$	20.1246	0.995098	0.497549	−	0.867436i	$-0.334233\pi$
$409$	20.1246	0.995098	0.497549	+	0.867436i	$0.334233\pi$
$410$	0	0
$411$	−14.5623	−0.718306
$412$	0	0
$413$	−1.50658	−0.0741338
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−8.61803	−0.422027
$418$	0	0
$419$	−21.1803	−1.03473	−0.517364	−	0.855766i	$-0.673087\pi$
$419$	−21.1803	−1.03473	−0.517364	+	0.855766i	$0.673087\pi$
$420$	0	0
$421$	−12.2705	−0.598028	−0.299014	−	0.954249i	$-0.596658\pi$
$421$	−12.2705	−0.598028	−0.299014	+	0.954249i	$0.596658\pi$
$422$	0	0
$423$	−46.0344	−2.23827
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.257354	0.0124542
$428$	0	0
$429$	16.3262	0.788238
$430$	0	0
$431$	−0.819660	−0.0394816	−0.0197408	−	0.999805i	$-0.506284\pi$
$431$	−0.819660	−0.0394816	−0.0197408	+	0.999805i	$0.506284\pi$
$432$	0	0
$433$	−18.8885	−0.907725	−0.453863	−	0.891072i	$-0.649954\pi$
$433$	−18.8885	−0.907725	−0.453863	+	0.891072i	$0.649954\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.4377	1.31252
$438$	0	0
$439$	0.729490	0.0348167	0.0174083	−	0.999848i	$-0.494458\pi$
$439$	0.729490	0.0348167	0.0174083	+	0.999848i	$0.494458\pi$
$440$	0	0
$441$	4.41641	0.210305
$442$	0	0
$443$	36.6525	1.74141	0.870706	−	0.491804i	$-0.163662\pi$
$443$	36.6525	1.74141	0.870706	+	0.491804i	$0.163662\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	23.4164	1.10756
$448$	0	0
$449$	−15.3262	−0.723290	−0.361645	−	0.932316i	$-0.617785\pi$
$449$	−15.3262	−0.723290	−0.361645	+	0.932316i	$0.617785\pi$
$450$	0	0
$451$	3.00000	0.141264
$452$	0	0
$453$	−7.85410	−0.369018
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.97871	−0.373228	−0.186614	−	0.982433i	$-0.559751\pi$
$457$	−7.97871	−0.373228	−0.186614	+	0.982433i	$0.559751\pi$
$458$	0	0
$459$	1.38197	0.0645046
$460$	0	0
$461$	−9.18034	−0.427571	−0.213786	−	0.976881i	$-0.568579\pi$
$461$	−9.18034	−0.427571	−0.213786	+	0.976881i	$0.568579\pi$
$462$	0	0
$463$	−11.3607	−0.527976	−0.263988	−	0.964526i	$-0.585038\pi$
$463$	−11.3607	−0.527976	−0.263988	+	0.964526i	$0.585038\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.4721	−1.73400	−0.867002	−	0.498305i	$-0.833956\pi$
$467$	−37.4721	−1.73400	−0.867002	+	0.498305i	$0.833956\pi$
$468$	0	0
$469$	−22.8328	−1.05432
$470$	0	0
$471$	14.1803	0.653396
$472$	0	0
$473$	−6.00000	−0.275880
$474$	0	0
$475$	0	0
$476$	0	0
$477$	35.9443	1.64578
$478$	0	0
$479$	−28.4164	−1.29838	−0.649189	−	0.760627i	$-0.724892\pi$
$479$	−28.4164	−1.29838	−0.649189	+	0.760627i	$0.724892\pi$
$480$	0	0
$481$	63.8328	2.91053
$482$	0	0
$483$	−30.5623	−1.39063
$484$	0	0
$485$	0	0
$486$	0	0
$487$	30.4164	1.37830	0.689150	−	0.724619i	$-0.257984\pi$
$487$	30.4164	1.37830	0.689150	+	0.724619i	$0.257984\pi$
$488$	0	0
$489$	−17.9443	−0.811468
$490$	0	0
$491$	6.81966	0.307767	0.153883	−	0.988089i	$-0.450822\pi$
$491$	6.81966	0.307767	0.153883	+	0.988089i	$0.450822\pi$
$492$	0	0
$493$	0.854102	0.0384668
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−23.3475	−1.04728
$498$	0	0
$499$	−29.7984	−1.33396	−0.666979	−	0.745076i	$-0.732413\pi$
$499$	−29.7984	−1.33396	−0.666979	+	0.745076i	$0.732413\pi$
$500$	0	0
$501$	−13.8541	−0.618956
$502$	0	0
$503$	6.65248	0.296619	0.148310	−	0.988941i	$-0.452617\pi$
$503$	6.65248	0.296619	0.148310	+	0.988941i	$0.452617\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−67.7771	−3.01009
$508$	0	0
$509$	16.3820	0.726118	0.363059	−	0.931766i	$-0.381732\pi$
$509$	16.3820	0.726118	0.363059	+	0.931766i	$0.381732\pi$
$510$	0	0
$511$	29.6312	1.31081
$512$	0	0
$513$	−15.0000	−0.662266
$514$	0	0
$515$	0	0
$516$	0	0
$517$	11.9443	0.525308
$518$	0	0
$519$	14.3262	0.628852
$520$	0	0
$521$	−1.81966	−0.0797208	−0.0398604	−	0.999205i	$-0.512691\pi$
$521$	−1.81966	−0.0797208	−0.0398604	+	0.999205i	$0.512691\pi$
$522$	0	0
$523$	−22.9443	−1.00328	−0.501641	−	0.865076i	$-0.667270\pi$
$523$	−22.9443	−1.00328	−0.501641	+	0.865076i	$0.667270\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.85410	−0.0807660
$528$	0	0
$529$	−6.27051	−0.272631
$530$	0	0
$531$	−2.03444	−0.0882873
$532$	0	0
$533$	−18.7082	−0.810342
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−35.6525	−1.53852
$538$	0	0
$539$	−1.14590	−0.0493573
$540$	0	0
$541$	−42.2705	−1.81735	−0.908676	−	0.417503i	$-0.862905\pi$
$541$	−42.2705	−1.81735	−0.908676	+	0.417503i	$0.862905\pi$
$542$	0	0
$543$	15.9443	0.684234
$544$	0	0
$545$	0	0
$546$	0	0
$547$	21.1459	0.904133	0.452067	−	0.891984i	$-0.350687\pi$
$547$	21.1459	0.904133	0.452067	+	0.891984i	$0.350687\pi$
$548$	0	0
$549$	0.347524	0.0148320
$550$	0	0
$551$	−9.27051	−0.394937
$552$	0	0
$553$	−16.7082	−0.710505
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.76393	−0.413711	−0.206856	−	0.978371i	$-0.566323\pi$
$557$	−9.76393	−0.413711	−0.206856	+	0.978371i	$0.566323\pi$
$558$	0	0
$559$	37.4164	1.58255
$560$	0	0
$561$	−1.61803	−0.0683134
$562$	0	0
$563$	13.0344	0.549336	0.274668	−	0.961539i	$-0.411432\pi$
$563$	13.0344	0.549336	0.274668	+	0.961539i	$0.411432\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−16.2918	−0.684191
$568$	0	0
$569$	26.3820	1.10599	0.552995	−	0.833185i	$-0.313485\pi$
$569$	26.3820	1.10599	0.552995	+	0.833185i	$0.313485\pi$
$570$	0	0
$571$	−36.2705	−1.51787	−0.758937	−	0.651164i	$-0.774281\pi$
$571$	−36.2705	−1.51787	−0.758937	+	0.651164i	$0.774281\pi$
$572$	0	0
$573$	−55.2148	−2.30663
$574$	0	0
$575$	0	0
$576$	0	0
$577$	15.5623	0.647867	0.323934	−	0.946080i	$-0.394995\pi$
$577$	15.5623	0.647867	0.323934	+	0.946080i	$0.394995\pi$
$578$	0	0
$579$	−33.8885	−1.40836
$580$	0	0
$581$	28.9574	1.20136
$582$	0	0
$583$	−9.32624	−0.386253
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4.03444	0.166519	0.0832596	−	0.996528i	$-0.473467\pi$
$587$	4.03444	0.166519	0.0832596	+	0.996528i	$0.473467\pi$
$588$	0	0
$589$	20.1246	0.829220
$590$	0	0
$591$	52.5967	2.16354
$592$	0	0
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.09017	−0.331109
$598$	0	0
$599$	−0.326238	−0.0133297	−0.00666486	−	0.999978i	$-0.502122\pi$
$599$	−0.326238	−0.0133297	−0.00666486	+	0.999978i	$0.502122\pi$
$600$	0	0
$601$	−22.2705	−0.908433	−0.454217	−	0.890891i	$-0.650081\pi$
$601$	−22.2705	−0.908433	−0.454217	+	0.890891i	$0.650081\pi$
$602$	0	0
$603$	−30.8328	−1.25561
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.52786	−0.143192	−0.0715958	−	0.997434i	$-0.522809\pi$
$607$	−3.52786	−0.143192	−0.0715958	+	0.997434i	$0.522809\pi$
$608$	0	0
$609$	10.3262	0.418440
$610$	0	0
$611$	−74.4853	−3.01335
$612$	0	0
$613$	6.43769	0.260016	0.130008	−	0.991513i	$-0.458500\pi$
$613$	6.43769	0.260016	0.130008	+	0.991513i	$0.458500\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−38.1803	−1.53708	−0.768541	−	0.639800i	$-0.779017\pi$
$617$	−38.1803	−1.53708	−0.768541	+	0.639800i	$0.779017\pi$
$618$	0	0
$619$	22.8885	0.919968	0.459984	−	0.887927i	$-0.347855\pi$
$619$	22.8885	0.919968	0.459984	+	0.887927i	$0.347855\pi$
$620$	0	0
$621$	−9.14590	−0.367012
$622$	0	0
$623$	−19.7214	−0.790120
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17.5623	0.701371
$628$	0	0
$629$	−6.32624	−0.252244
$630$	0	0
$631$	−36.2705	−1.44391	−0.721953	−	0.691942i	$-0.756755\pi$
$631$	−36.2705	−1.44391	−0.721953	+	0.691942i	$0.756755\pi$
$632$	0	0
$633$	44.5066	1.76898
$634$	0	0
$635$	0	0
$636$	0	0
$637$	7.14590	0.283131
$638$	0	0
$639$	−31.5279	−1.24722
$640$	0	0
$641$	−3.00000	−0.118493	−0.0592464	−	0.998243i	$-0.518870\pi$
$641$	−3.00000	−0.118493	−0.0592464	+	0.998243i	$0.518870\pi$
$642$	0	0
$643$	−10.5836	−0.417376	−0.208688	−	0.977982i	$-0.566919\pi$
$643$	−10.5836	−0.417376	−0.208688	+	0.977982i	$0.566919\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−12.3475	−0.485431	−0.242716	−	0.970097i	$-0.578038\pi$
$647$	−12.3475	−0.485431	−0.242716	+	0.970097i	$0.578038\pi$
$648$	0	0
$649$	0.527864	0.0207205
$650$	0	0
$651$	−22.4164	−0.878568
$652$	0	0
$653$	7.74265	0.302993	0.151497	−	0.988458i	$-0.451591\pi$
$653$	7.74265	0.302993	0.151497	+	0.988458i	$0.451591\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	40.0132	1.56106
$658$	0	0
$659$	−9.27051	−0.361128	−0.180564	−	0.983563i	$-0.557792\pi$
$659$	−9.27051	−0.361128	−0.180564	+	0.983563i	$0.557792\pi$
$660$	0	0
$661$	−49.1803	−1.91289	−0.956447	−	0.291907i	$-0.905710\pi$
$661$	−49.1803	−1.91289	−0.956447	+	0.291907i	$0.905710\pi$
$662$	0	0
$663$	10.0902	0.391870
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.65248	−0.218865
$668$	0	0
$669$	44.2148	1.70944
$670$	0	0
$671$	−0.0901699	−0.00348097
$672$	0	0
$673$	2.41641	0.0931457	0.0465728	−	0.998915i	$-0.485170\pi$
$673$	2.41641	0.0931457	0.0465728	+	0.998915i	$0.485170\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.6525	1.10120	0.550602	−	0.834768i	$-0.314398\pi$
$677$	28.6525	1.10120	0.550602	+	0.834768i	$0.314398\pi$
$678$	0	0
$679$	4.61803	0.177224
$680$	0	0
$681$	−62.9230	−2.41121
$682$	0	0
$683$	43.3607	1.65915	0.829575	−	0.558395i	$-0.188583\pi$
$683$	43.3607	1.65915	0.829575	+	0.558395i	$0.188583\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−31.5066	−1.20205
$688$	0	0
$689$	58.1591	2.21568
$690$	0	0
$691$	−30.0902	−1.14468	−0.572342	−	0.820015i	$-0.693965\pi$
$691$	−30.0902	−1.14468	−0.572342	+	0.820015i	$0.693965\pi$
$692$	0	0
$693$	−11.0000	−0.417855
$694$	0	0
$695$	0	0
$696$	0	0
$697$	1.85410	0.0702291
$698$	0	0
$699$	−40.5967	−1.53551
$700$	0	0
$701$	−0.360680	−0.0136227	−0.00681134	−	0.999977i	$-0.502168\pi$
$701$	−0.360680	−0.0136227	−0.00681134	+	0.999977i	$0.502168\pi$
$702$	0	0
$703$	68.6656	2.58977
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.3820	−0.653716
$708$	0	0
$709$	−41.3050	−1.55124	−0.775620	−	0.631200i	$-0.782563\pi$
$709$	−41.3050	−1.55124	−0.775620	+	0.631200i	$0.782563\pi$
$710$	0	0
$711$	−22.5623	−0.846153
$712$	0	0
$713$	12.2705	0.459534
$714$	0	0
$715$	0	0
$716$	0	0
$717$	16.7082	0.623979
$718$	0	0
$719$	15.6525	0.583739	0.291869	−	0.956458i	$-0.405723\pi$
$719$	15.6525	0.583739	0.291869	+	0.956458i	$0.405723\pi$
$720$	0	0
$721$	−15.3607	−0.572062
$722$	0	0
$723$	−55.6869	−2.07102
$724$	0	0
$725$	0	0
$726$	0	0
$727$	41.2705	1.53064	0.765319	−	0.643651i	$-0.222581\pi$
$727$	41.2705	1.53064	0.765319	+	0.643651i	$0.222581\pi$
$728$	0	0
$729$	−39.5623	−1.46527
$730$	0	0
$731$	−3.70820	−0.137153
$732$	0	0
$733$	45.8328	1.69287	0.846437	−	0.532489i	$-0.178743\pi$
$733$	45.8328	1.69287	0.846437	+	0.532489i	$0.178743\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	8.00000	0.294684
$738$	0	0
$739$	29.1459	1.07215	0.536075	−	0.844171i	$-0.319907\pi$
$739$	29.1459	1.07215	0.536075	+	0.844171i	$0.319907\pi$
$740$	0	0
$741$	−109.520	−4.02331
$742$	0	0
$743$	−18.6738	−0.685074	−0.342537	−	0.939504i	$-0.611286\pi$
$743$	−18.6738	−0.685074	−0.342537	+	0.939504i	$0.611286\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	39.1033	1.43072
$748$	0	0
$749$	12.0902	0.441765
$750$	0	0
$751$	−11.2705	−0.411267	−0.205633	−	0.978629i	$-0.565925\pi$
$751$	−11.2705	−0.411267	−0.205633	+	0.978629i	$0.565925\pi$
$752$	0	0
$753$	−71.3951	−2.60178
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−21.4721	−0.780418	−0.390209	−	0.920726i	$-0.627597\pi$
$757$	−21.4721	−0.780418	−0.390209	+	0.920726i	$0.627597\pi$
$758$	0	0
$759$	10.7082	0.388683
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	0	0
$763$	−8.81966	−0.319293
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−3.29180	−0.118860
$768$	0	0
$769$	−5.00000	−0.180305	−0.0901523	−	0.995928i	$-0.528735\pi$
$769$	−5.00000	−0.180305	−0.0901523	+	0.995928i	$0.528735\pi$
$770$	0	0
$771$	28.6525	1.03189
$772$	0	0
$773$	33.7984	1.21564	0.607822	−	0.794074i	$-0.292044\pi$
$773$	33.7984	1.21564	0.607822	+	0.794074i	$0.292044\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−76.4853	−2.74389
$778$	0	0
$779$	−20.1246	−0.721039
$780$	0	0
$781$	8.18034	0.292716
$782$	0	0
$783$	3.09017	0.110434
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−21.2918	−0.758971	−0.379485	−	0.925198i	$-0.623899\pi$
$787$	−21.2918	−0.758971	−0.379485	+	0.925198i	$0.623899\pi$
$788$	0	0
$789$	−54.9787	−1.95729
$790$	0	0
$791$	−33.2574	−1.18250
$792$	0	0
$793$	0.562306	0.0199681
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.20163	0.113407	0.0567037	−	0.998391i	$-0.481941\pi$
$797$	3.20163	0.113407	0.0567037	+	0.998391i	$0.481941\pi$
$798$	0	0
$799$	7.38197	0.261155
$800$	0	0
$801$	−26.6312	−0.940967
$802$	0	0
$803$	−10.3820	−0.366372
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−79.3951	−2.79484
$808$	0	0
$809$	−16.5836	−0.583048	−0.291524	−	0.956564i	$-0.594162\pi$
$809$	−16.5836	−0.583048	−0.291524	+	0.956564i	$0.594162\pi$
$810$	0	0
$811$	−17.0000	−0.596951	−0.298475	−	0.954417i	$-0.596478\pi$
$811$	−17.0000	−0.596951	−0.298475	+	0.954417i	$0.596478\pi$
$812$	0	0
$813$	34.5066	1.21020
$814$	0	0
$815$	0	0
$816$	0	0
$817$	40.2492	1.40814
$818$	0	0
$819$	68.5967	2.39696
$820$	0	0
$821$	4.36068	0.152189	0.0760944	−	0.997101i	$-0.475755\pi$
$821$	4.36068	0.152189	0.0760944	+	0.997101i	$0.475755\pi$
$822$	0	0
$823$	−27.4164	−0.955676	−0.477838	−	0.878448i	$-0.658579\pi$
$823$	−27.4164	−0.955676	−0.477838	+	0.878448i	$0.658579\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−32.0689	−1.11514	−0.557572	−	0.830128i	$-0.688267\pi$
$827$	−32.0689	−1.11514	−0.557572	+	0.830128i	$0.688267\pi$
$828$	0	0
$829$	−36.1033	−1.25392	−0.626960	−	0.779051i	$-0.715701\pi$
$829$	−36.1033	−1.25392	−0.626960	+	0.779051i	$0.715701\pi$
$830$	0	0
$831$	30.0344	1.04188
$832$	0	0
$833$	−0.708204	−0.0245378
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−6.70820	−0.231869
$838$	0	0
$839$	48.3394	1.66886	0.834431	−	0.551113i	$-0.185797\pi$
$839$	48.3394	1.66886	0.834431	+	0.551113i	$0.185797\pi$
$840$	0	0
$841$	−27.0902	−0.934144
$842$	0	0
$843$	66.3951	2.28677
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.85410	0.0980681
$848$	0	0
$849$	−19.3262	−0.663275
$850$	0	0
$851$	41.8673	1.43519
$852$	0	0
$853$	49.8541	1.70697	0.853486	−	0.521116i	$-0.174484\pi$
$853$	49.8541	1.70697	0.853486	+	0.521116i	$0.174484\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−53.1803	−1.81661	−0.908303	−	0.418313i	$-0.862621\pi$
$857$	−53.1803	−1.81661	−0.908303	+	0.418313i	$0.862621\pi$
$858$	0	0
$859$	−16.1803	−0.552066	−0.276033	−	0.961148i	$-0.589020\pi$
$859$	−16.1803	−0.552066	−0.276033	+	0.961148i	$0.589020\pi$
$860$	0	0
$861$	22.4164	0.763949
$862$	0	0
$863$	0.596748	0.0203135	0.0101568	−	0.999948i	$-0.496767\pi$
$863$	0.596748	0.0203135	0.0101568	+	0.999948i	$0.496767\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	43.5066	1.47756
$868$	0	0
$869$	5.85410	0.198587
$870$	0	0
$871$	−49.8885	−1.69041
$872$	0	0
$873$	6.23607	0.211059
$874$	0	0
$875$	0	0
$876$	0	0
$877$	4.58359	0.154777	0.0773885	−	0.997001i	$-0.475342\pi$
$877$	4.58359	0.154777	0.0773885	+	0.997001i	$0.475342\pi$
$878$	0	0
$879$	62.5410	2.10946
$880$	0	0
$881$	10.0902	0.339946	0.169973	−	0.985449i	$-0.445632\pi$
$881$	10.0902	0.339946	0.169973	+	0.985449i	$0.445632\pi$
$882$	0	0
$883$	31.6525	1.06519	0.532595	−	0.846370i	$-0.321217\pi$
$883$	31.6525	1.06519	0.532595	+	0.846370i	$0.321217\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.7771	−0.966240	−0.483120	−	0.875554i	$-0.660497\pi$
$887$	−28.7771	−0.966240	−0.483120	+	0.875554i	$0.660497\pi$
$888$	0	0
$889$	1.76393	0.0591604
$890$	0	0
$891$	5.70820	0.191232
$892$	0	0
$893$	−80.1246	−2.68127
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−66.7771	−2.22962
$898$	0	0
$899$	−4.14590	−0.138273
$900$	0	0
$901$	−5.76393	−0.192024
$902$	0	0
$903$	−44.8328	−1.49194
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−14.8328	−0.492516	−0.246258	−	0.969204i	$-0.579201\pi$
$907$	−14.8328	−0.492516	−0.246258	+	0.969204i	$0.579201\pi$
$908$	0	0
$909$	−23.4721	−0.778522
$910$	0	0
$911$	18.0000	0.596367	0.298183	−	0.954509i	$-0.403619\pi$
$911$	18.0000	0.596367	0.298183	+	0.954509i	$0.403619\pi$
$912$	0	0
$913$	−10.1459	−0.335780
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−28.7984	−0.951006
$918$	0	0
$919$	−23.4164	−0.772436	−0.386218	−	0.922408i	$-0.626219\pi$
$919$	−23.4164	−0.772436	−0.386218	+	0.922408i	$0.626219\pi$
$920$	0	0
$921$	87.5755	2.88571
$922$	0	0
$923$	−51.0132	−1.67912
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−20.7426	−0.681278
$928$	0	0
$929$	54.5967	1.79126	0.895631	−	0.444799i	$-0.146725\pi$
$929$	54.5967	1.79126	0.895631	+	0.444799i	$0.146725\pi$
$930$	0	0
$931$	7.68692	0.251929
$932$	0	0
$933$	−50.2148	−1.64396
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−38.8328	−1.26861	−0.634306	−	0.773082i	$-0.718714\pi$
$937$	−38.8328	−1.26861	−0.634306	+	0.773082i	$0.718714\pi$
$938$	0	0
$939$	8.47214	0.276478
$940$	0	0
$941$	41.7214	1.36008	0.680039	−	0.733176i	$-0.261963\pi$
$941$	41.7214	1.36008	0.680039	+	0.733176i	$0.261963\pi$
$942$	0	0
$943$	−12.2705	−0.399583
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.3607	1.27905	0.639525	−	0.768770i	$-0.279131\pi$
$947$	39.3607	1.27905	0.639525	+	0.768770i	$0.279131\pi$
$948$	0	0
$949$	64.7426	2.10164
$950$	0	0
$951$	−43.5066	−1.41080
$952$	0	0
$953$	8.47214	0.274439	0.137220	−	0.990541i	$-0.456183\pi$
$953$	8.47214	0.274439	0.137220	+	0.990541i	$0.456183\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.61803	−0.116954
$958$	0	0
$959$	15.8754	0.512643
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	16.3262	0.526106
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−37.2705	−1.19854	−0.599269	−	0.800547i	$-0.704542\pi$
$967$	−37.2705	−1.19854	−0.599269	+	0.800547i	$0.704542\pi$
$968$	0	0
$969$	10.8541	0.348684
$970$	0	0
$971$	−23.9098	−0.767303	−0.383651	−	0.923478i	$-0.625334\pi$
$971$	−23.9098	−0.767303	−0.383651	+	0.923478i	$0.625334\pi$
$972$	0	0
$973$	9.39512	0.301194
$974$	0	0
$975$	0	0
$976$	0	0
$977$	57.0689	1.82580	0.912898	−	0.408188i	$-0.133839\pi$
$977$	57.0689	1.82580	0.912898	+	0.408188i	$0.133839\pi$
$978$	0	0
$979$	6.90983	0.220839
$980$	0	0
$981$	−11.9098	−0.380252
$982$	0	0
$983$	1.52786	0.0487313	0.0243656	−	0.999703i	$-0.492243\pi$
$983$	1.52786	0.0487313	0.0243656	+	0.999703i	$0.492243\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	89.2492	2.84083
$988$	0	0
$989$	24.5410	0.780359
$990$	0	0
$991$	−21.2705	−0.675680	−0.337840	−	0.941204i	$-0.609696\pi$
$991$	−21.2705	−0.675680	−0.337840	+	0.941204i	$0.609696\pi$
$992$	0	0
$993$	−50.2148	−1.59352
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32.1459	1.01807	0.509035	−	0.860746i	$-0.330002\pi$
$997$	32.1459	1.01807	0.509035	+	0.860746i	$0.330002\pi$
$998$	0	0
$999$	−22.8885	−0.724161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bg.1.1		2
4.3	odd	2		275.2.a.g.1.2	yes	2
5.2	odd	4		4400.2.b.x.4049.4		4
5.3	odd	4		4400.2.b.x.4049.1		4
5.4	even	2		4400.2.a.bv.1.2		2
12.11	even	2		2475.2.a.n.1.1		2
20.3	even	4		275.2.b.e.199.1		4
20.7	even	4		275.2.b.e.199.4		4
20.19	odd	2		275.2.a.d.1.1	✓	2
44.43	even	2		3025.2.a.i.1.1		2
60.23	odd	4		2475.2.c.p.199.4		4
60.47	odd	4		2475.2.c.p.199.1		4
60.59	even	2		2475.2.a.s.1.2		2
220.219	even	2		3025.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
275.2.a.d.1.1	✓	2	20.19	odd	2
275.2.a.g.1.2	yes	2	4.3	odd	2
275.2.b.e.199.1		4	20.3	even	4
275.2.b.e.199.4		4	20.7	even	4
2475.2.a.n.1.1		2	12.11	even	2
2475.2.a.s.1.2		2	60.59	even	2
2475.2.c.p.199.1		4	60.47	odd	4
2475.2.c.p.199.4		4	60.23	odd	4
3025.2.a.i.1.1		2	44.43	even	2
3025.2.a.m.1.2		2	220.219	even	2
4400.2.a.bg.1.1		2	1.1	even	1	trivial
4400.2.a.bv.1.2		2	5.4	even	2
4400.2.b.x.4049.1		4	5.3	odd	4
4400.2.b.x.4049.4		4	5.2	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)

\(f(q)\)	\(=\)	\(q-2.61803 q^{3} +2.85410 q^{7} +3.85410 q^{9} -1.00000 q^{11} +6.23607 q^{13} -0.618034 q^{17} +6.70820 q^{19} -7.47214 q^{21} +4.09017 q^{23} -2.23607 q^{27} -1.38197 q^{29} +3.00000 q^{31} +2.61803 q^{33} +10.2361 q^{37} -16.3262 q^{39} -3.00000 q^{41} +6.00000 q^{43} -11.9443 q^{47} +1.14590 q^{49} +1.61803 q^{51} +9.32624 q^{53} -17.5623 q^{57} -0.527864 q^{59} +0.0901699 q^{61} +11.0000 q^{63} -8.00000 q^{67} -10.7082 q^{69} -8.18034 q^{71} +10.3820 q^{73} -2.85410 q^{77} -5.85410 q^{79} -5.70820 q^{81} +10.1459 q^{83} +3.61803 q^{87} -6.90983 q^{89} +17.7984 q^{91} -7.85410 q^{93} +1.61803 q^{97} -3.85410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{3} - q^{7} + q^{9} - 2 q^{11} + 8 q^{13} + q^{17} - 6 q^{21} - 3 q^{23} - 5 q^{29} + 6 q^{31} + 3 q^{33} + 16 q^{37} - 17 q^{39} - 6 q^{41} + 12 q^{43} - 6 q^{47} + 9 q^{49} + q^{51} + 3 q^{53}+ \cdots - q^{99}+O(q^{100}) \) 2 * q - 3 * q^3 - q^7 + q^9 - 2 * q^11 + 8 * q^13 + q^17 - 6 * q^21 - 3 * q^23 - 5 * q^29 + 6 * q^31 + 3 * q^33 + 16 * q^37 - 17 * q^39 - 6 * q^41 + 12 * q^43 - 6 * q^47 + 9 * q^49 + q^51 + 3 * q^53 - 15 * q^57 - 10 * q^59 - 11 * q^61 + 22 * q^63 - 16 * q^67 - 8 * q^69 + 6 * q^71 + 23 * q^73 + q^77 - 5 * q^79 + 2 * q^81 + 27 * q^83 + 5 * q^87 - 25 * q^89 + 11 * q^91 - 9 * q^93 + q^97 - q^99

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