LMFDB - Embedded newform 9984.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9984,2,Mod(1,9984)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9984, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9984.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9984 = 2^{8} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9984.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$79.7226413780$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2496)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9984.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.41421 q^{5} +4.82843 q^{7} +1.00000 q^{9} -6.24264 q^{11} -1.00000 q^{13} +3.41421 q^{15} +4.82843 q^{17} +2.00000 q^{19} +4.82843 q^{21} -5.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} +0.828427 q^{29} -3.65685 q^{31} -6.24264 q^{33} +16.4853 q^{35} +3.17157 q^{37} -1.00000 q^{39} -3.41421 q^{41} +11.6569 q^{43} +3.41421 q^{45} -9.07107 q^{47} +16.3137 q^{49} +4.82843 q^{51} +7.17157 q^{53} -21.3137 q^{55} +2.00000 q^{57} +14.7279 q^{59} +7.65685 q^{61} +4.82843 q^{63} -3.41421 q^{65} -12.8284 q^{67} -5.65685 q^{69} -7.41421 q^{71} -10.4853 q^{73} +6.65685 q^{75} -30.1421 q^{77} +11.6569 q^{79} +1.00000 q^{81} +2.24264 q^{83} +16.4853 q^{85} +0.828427 q^{87} +0.585786 q^{89} -4.82843 q^{91} -3.65685 q^{93} +6.82843 q^{95} +3.17157 q^{97} -6.24264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 4 q^{31} - 4 q^{33} + 16 q^{35} + 12 q^{37} - 2 q^{39} - 4 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 + 4 * q^19 + 4 * q^21 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 4 * q^31 - 4 * q^33 + 16 * q^35 + 12 * q^37 - 2 * q^39 - 4 * q^41 + 12 * q^43 + 4 * q^45 - 4 * q^47 + 10 * q^49 + 4 * q^51 + 20 * q^53 - 20 * q^55 + 4 * q^57 + 4 * q^59 + 4 * q^61 + 4 * q^63 - 4 * q^65 - 20 * q^67 - 12 * q^71 - 4 * q^73 + 2 * q^75 - 32 * q^77 + 12 * q^79 + 2 * q^81 - 4 * q^83 + 16 * q^85 - 4 * q^87 + 4 * q^89 - 4 * q^91 + 4 * q^93 + 8 * q^95 + 12 * q^97 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.41421	1.52688	0.763441	−	0.645877i	$-0.223508\pi$
$5$	3.41421	1.52688	0.763441	+	0.645877i	$0.223508\pi$
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−6.24264	−1.88223	−0.941113	−	0.338091i	$-0.890219\pi$
$11$	−6.24264	−1.88223	−0.941113	+	0.338091i	$0.890219\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	3.41421	0.881546
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	4.82843	1.05365
$22$	0	0
$23$	−5.65685	−1.17954	−0.589768	−	0.807573i	$-0.700781\pi$
$23$	−5.65685	−1.17954	−0.589768	+	0.807573i	$0.700781\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0	0
$31$	−3.65685	−0.656790	−0.328395	−	0.944540i	$-0.606508\pi$
$31$	−3.65685	−0.656790	−0.328395	+	0.944540i	$0.606508\pi$
$32$	0	0
$33$	−6.24264	−1.08670
$34$	0	0
$35$	16.4853	2.78652
$36$	0	0
$37$	3.17157	0.521403	0.260702	−	0.965419i	$-0.416046\pi$
$37$	3.17157	0.521403	0.260702	+	0.965419i	$0.416046\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	−3.41421	−0.533211	−0.266605	−	0.963806i	$-0.585902\pi$
$41$	−3.41421	−0.533211	−0.266605	+	0.963806i	$0.585902\pi$
$42$	0	0
$43$	11.6569	1.77765	0.888827	−	0.458243i	$-0.151521\pi$
$43$	11.6569	1.77765	0.888827	+	0.458243i	$0.151521\pi$
$44$	0	0
$45$	3.41421	0.508961
$46$	0	0
$47$	−9.07107	−1.32315	−0.661576	−	0.749878i	$-0.730112\pi$
$47$	−9.07107	−1.32315	−0.661576	+	0.749878i	$0.730112\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	4.82843	0.676115
$52$	0	0
$53$	7.17157	0.985091	0.492546	−	0.870287i	$-0.336066\pi$
$53$	7.17157	0.985091	0.492546	+	0.870287i	$0.336066\pi$
$54$	0	0
$55$	−21.3137	−2.87394
$56$	0	0
$57$	2.00000	0.264906
$58$	0	0
$59$	14.7279	1.91741	0.958706	−	0.284399i	$-0.0917940\pi$
$59$	14.7279	1.91741	0.958706	+	0.284399i	$0.0917940\pi$
$60$	0	0
$61$	7.65685	0.980360	0.490180	−	0.871621i	$-0.336931\pi$
$61$	7.65685	0.980360	0.490180	+	0.871621i	$0.336931\pi$
$62$	0	0
$63$	4.82843	0.608325
$64$	0	0
$65$	−3.41421	−0.423481
$66$	0	0
$67$	−12.8284	−1.56724	−0.783621	−	0.621239i	$-0.786629\pi$
$67$	−12.8284	−1.56724	−0.783621	+	0.621239i	$0.786629\pi$
$68$	0	0
$69$	−5.65685	−0.681005
$70$	0	0
$71$	−7.41421	−0.879905	−0.439953	−	0.898021i	$-0.645005\pi$
$71$	−7.41421	−0.879905	−0.439953	+	0.898021i	$0.645005\pi$
$72$	0	0
$73$	−10.4853	−1.22721	−0.613605	−	0.789613i	$-0.710281\pi$
$73$	−10.4853	−1.22721	−0.613605	+	0.789613i	$0.710281\pi$
$74$	0	0
$75$	6.65685	0.768667
$76$	0	0
$77$	−30.1421	−3.43502
$78$	0	0
$79$	11.6569	1.31150	0.655749	−	0.754979i	$-0.272353\pi$
$79$	11.6569	1.31150	0.655749	+	0.754979i	$0.272353\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.24264	0.246162	0.123081	−	0.992397i	$-0.460723\pi$
$83$	2.24264	0.246162	0.123081	+	0.992397i	$0.460723\pi$
$84$	0	0
$85$	16.4853	1.78808
$86$	0	0
$87$	0.828427	0.0888167
$88$	0	0
$89$	0.585786	0.0620932	0.0310466	−	0.999518i	$-0.490116\pi$
$89$	0.585786	0.0620932	0.0310466	+	0.999518i	$0.490116\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	0	0
$93$	−3.65685	−0.379198
$94$	0	0
$95$	6.82843	0.700582
$96$	0	0
$97$	3.17157	0.322024	0.161012	−	0.986952i	$-0.448524\pi$
$97$	3.17157	0.322024	0.161012	+	0.986952i	$0.448524\pi$
$98$	0	0
$99$	−6.24264	−0.627409
$100$	0	0
$101$	10.0000	0.995037	0.497519	−	0.867453i	$-0.334245\pi$
$101$	10.0000	0.995037	0.497519	+	0.867453i	$0.334245\pi$
$102$	0	0
$103$	11.3137	1.11477	0.557386	−	0.830253i	$-0.311804\pi$
$103$	11.3137	1.11477	0.557386	+	0.830253i	$0.311804\pi$
$104$	0	0
$105$	16.4853	1.60880
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	0	0
$109$	11.6569	1.11652	0.558262	−	0.829665i	$-0.311468\pi$
$109$	11.6569	1.11652	0.558262	+	0.829665i	$0.311468\pi$
$110$	0	0
$111$	3.17157	0.301032
$112$	0	0
$113$	−3.65685	−0.344008	−0.172004	−	0.985096i	$-0.555024\pi$
$113$	−3.65685	−0.344008	−0.172004	+	0.985096i	$0.555024\pi$
$114$	0	0
$115$	−19.3137	−1.80101
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	23.3137	2.13716
$120$	0	0
$121$	27.9706	2.54278
$122$	0	0
$123$	−3.41421	−0.307849
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	8.34315	0.740334	0.370167	−	0.928965i	$-0.379300\pi$
$127$	8.34315	0.740334	0.370167	+	0.928965i	$0.379300\pi$
$128$	0	0
$129$	11.6569	1.02633
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	9.65685	0.837355
$134$	0	0
$135$	3.41421	0.293849
$136$	0	0
$137$	−7.89949	−0.674899	−0.337450	−	0.941344i	$-0.609564\pi$
$137$	−7.89949	−0.674899	−0.337450	+	0.941344i	$0.609564\pi$
$138$	0	0
$139$	7.65685	0.649446	0.324723	−	0.945809i	$-0.394729\pi$
$139$	7.65685	0.649446	0.324723	+	0.945809i	$0.394729\pi$
$140$	0	0
$141$	−9.07107	−0.763922
$142$	0	0
$143$	6.24264	0.522036
$144$	0	0
$145$	2.82843	0.234888
$146$	0	0
$147$	16.3137	1.34553
$148$	0	0
$149$	−3.89949	−0.319459	−0.159730	−	0.987161i	$-0.551062\pi$
$149$	−3.89949	−0.319459	−0.159730	+	0.987161i	$0.551062\pi$
$150$	0	0
$151$	−11.6569	−0.948621	−0.474311	−	0.880358i	$-0.657303\pi$
$151$	−11.6569	−0.948621	−0.474311	+	0.880358i	$0.657303\pi$
$152$	0	0
$153$	4.82843	0.390355
$154$	0	0
$155$	−12.4853	−1.00284
$156$	0	0
$157$	2.34315	0.187003	0.0935017	−	0.995619i	$-0.470194\pi$
$157$	2.34315	0.187003	0.0935017	+	0.995619i	$0.470194\pi$
$158$	0	0
$159$	7.17157	0.568743
$160$	0	0
$161$	−27.3137	−2.15262
$162$	0	0
$163$	11.6569	0.913035	0.456518	−	0.889714i	$-0.349097\pi$
$163$	11.6569	0.913035	0.456518	+	0.889714i	$0.349097\pi$
$164$	0	0
$165$	−21.3137	−1.65927
$166$	0	0
$167$	−4.58579	−0.354859	−0.177429	−	0.984134i	$-0.556778\pi$
$167$	−4.58579	−0.354859	−0.177429	+	0.984134i	$0.556778\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.00000	0.152944
$172$	0	0
$173$	7.65685	0.582140	0.291070	−	0.956702i	$-0.405989\pi$
$173$	7.65685	0.582140	0.291070	+	0.956702i	$0.405989\pi$
$174$	0	0
$175$	32.1421	2.42972
$176$	0	0
$177$	14.7279	1.10702
$178$	0	0
$179$	−8.48528	−0.634220	−0.317110	−	0.948389i	$-0.602712\pi$
$179$	−8.48528	−0.634220	−0.317110	+	0.948389i	$0.602712\pi$
$180$	0	0
$181$	19.3137	1.43558	0.717788	−	0.696261i	$-0.245155\pi$
$181$	19.3137	1.43558	0.717788	+	0.696261i	$0.245155\pi$
$182$	0	0
$183$	7.65685	0.566011
$184$	0	0
$185$	10.8284	0.796122
$186$	0	0
$187$	−30.1421	−2.20421
$188$	0	0
$189$	4.82843	0.351216
$190$	0	0
$191$	10.1421	0.733859	0.366930	−	0.930249i	$-0.380409\pi$
$191$	10.1421	0.733859	0.366930	+	0.930249i	$0.380409\pi$
$192$	0	0
$193$	−7.65685	−0.551152	−0.275576	−	0.961279i	$-0.588869\pi$
$193$	−7.65685	−0.551152	−0.275576	+	0.961279i	$0.588869\pi$
$194$	0	0
$195$	−3.41421	−0.244497
$196$	0	0
$197$	−6.72792	−0.479345	−0.239672	−	0.970854i	$-0.577040\pi$
$197$	−6.72792	−0.479345	−0.239672	+	0.970854i	$0.577040\pi$
$198$	0	0
$199$	−21.3137	−1.51089	−0.755444	−	0.655213i	$-0.772579\pi$
$199$	−21.3137	−1.51089	−0.755444	+	0.655213i	$0.772579\pi$
$200$	0	0
$201$	−12.8284	−0.904847
$202$	0	0
$203$	4.00000	0.280745
$204$	0	0
$205$	−11.6569	−0.814150
$206$	0	0
$207$	−5.65685	−0.393179
$208$	0	0
$209$	−12.4853	−0.863625
$210$	0	0
$211$	1.65685	0.114063	0.0570313	−	0.998372i	$-0.481837\pi$
$211$	1.65685	0.114063	0.0570313	+	0.998372i	$0.481837\pi$
$212$	0	0
$213$	−7.41421	−0.508014
$214$	0	0
$215$	39.7990	2.71427
$216$	0	0
$217$	−17.6569	−1.19863
$218$	0	0
$219$	−10.4853	−0.708530
$220$	0	0
$221$	−4.82843	−0.324795
$222$	0	0
$223$	6.00000	0.401790	0.200895	−	0.979613i	$-0.435615\pi$
$223$	6.00000	0.401790	0.200895	+	0.979613i	$0.435615\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	0	0
$227$	−2.72792	−0.181059	−0.0905293	−	0.995894i	$-0.528856\pi$
$227$	−2.72792	−0.181059	−0.0905293	+	0.995894i	$0.528856\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	−30.1421	−1.98321
$232$	0	0
$233$	−9.51472	−0.623330	−0.311665	−	0.950192i	$-0.600887\pi$
$233$	−9.51472	−0.623330	−0.311665	+	0.950192i	$0.600887\pi$
$234$	0	0
$235$	−30.9706	−2.02030
$236$	0	0
$237$	11.6569	0.757194
$238$	0	0
$239$	−2.92893	−0.189457	−0.0947284	−	0.995503i	$-0.530198\pi$
$239$	−2.92893	−0.189457	−0.0947284	+	0.995503i	$0.530198\pi$
$240$	0	0
$241$	14.4853	0.933079	0.466539	−	0.884500i	$-0.345501\pi$
$241$	14.4853	0.933079	0.466539	+	0.884500i	$0.345501\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	55.6985	3.55845
$246$	0	0
$247$	−2.00000	−0.127257
$248$	0	0
$249$	2.24264	0.142122
$250$	0	0
$251$	5.17157	0.326427	0.163213	−	0.986591i	$-0.447814\pi$
$251$	5.17157	0.326427	0.163213	+	0.986591i	$0.447814\pi$
$252$	0	0
$253$	35.3137	2.22015
$254$	0	0
$255$	16.4853	1.03235
$256$	0	0
$257$	11.1716	0.696864	0.348432	−	0.937334i	$-0.386714\pi$
$257$	11.1716	0.696864	0.348432	+	0.937334i	$0.386714\pi$
$258$	0	0
$259$	15.3137	0.951548
$260$	0	0
$261$	0.828427	0.0512784
$262$	0	0
$263$	−30.1421	−1.85864	−0.929322	−	0.369271i	$-0.879607\pi$
$263$	−30.1421	−1.85864	−0.929322	+	0.369271i	$0.879607\pi$
$264$	0	0
$265$	24.4853	1.50412
$266$	0	0
$267$	0.585786	0.0358495
$268$	0	0
$269$	18.4853	1.12707	0.563534	−	0.826093i	$-0.309441\pi$
$269$	18.4853	1.12707	0.563534	+	0.826093i	$0.309441\pi$
$270$	0	0
$271$	−4.14214	−0.251617	−0.125808	−	0.992055i	$-0.540152\pi$
$271$	−4.14214	−0.251617	−0.125808	+	0.992055i	$0.540152\pi$
$272$	0	0
$273$	−4.82843	−0.292230
$274$	0	0
$275$	−41.5563	−2.50594
$276$	0	0
$277$	−5.65685	−0.339887	−0.169944	−	0.985454i	$-0.554359\pi$
$277$	−5.65685	−0.339887	−0.169944	+	0.985454i	$0.554359\pi$
$278$	0	0
$279$	−3.65685	−0.218930
$280$	0	0
$281$	4.58579	0.273565	0.136783	−	0.990601i	$-0.456324\pi$
$281$	4.58579	0.273565	0.136783	+	0.990601i	$0.456324\pi$
$282$	0	0
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	0	0
$285$	6.82843	0.404481
$286$	0	0
$287$	−16.4853	−0.973095
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	3.17157	0.185921
$292$	0	0
$293$	1.75736	0.102666	0.0513330	−	0.998682i	$-0.483653\pi$
$293$	1.75736	0.102666	0.0513330	+	0.998682i	$0.483653\pi$
$294$	0	0
$295$	50.2843	2.92766
$296$	0	0
$297$	−6.24264	−0.362235
$298$	0	0
$299$	5.65685	0.327144
$300$	0	0
$301$	56.2843	3.24417
$302$	0	0
$303$	10.0000	0.574485
$304$	0	0
$305$	26.1421	1.49689
$306$	0	0
$307$	−25.7990	−1.47243	−0.736213	−	0.676750i	$-0.763388\pi$
$307$	−25.7990	−1.47243	−0.736213	+	0.676750i	$0.763388\pi$
$308$	0	0
$309$	11.3137	0.643614
$310$	0	0
$311$	−11.3137	−0.641542	−0.320771	−	0.947157i	$-0.603942\pi$
$311$	−11.3137	−0.641542	−0.320771	+	0.947157i	$0.603942\pi$
$312$	0	0
$313$	−24.2843	−1.37263	−0.686314	−	0.727305i	$-0.740772\pi$
$313$	−24.2843	−1.37263	−0.686314	+	0.727305i	$0.740772\pi$
$314$	0	0
$315$	16.4853	0.928840
$316$	0	0
$317$	−22.7279	−1.27653	−0.638264	−	0.769818i	$-0.720347\pi$
$317$	−22.7279	−1.27653	−0.638264	+	0.769818i	$0.720347\pi$
$318$	0	0
$319$	−5.17157	−0.289552
$320$	0	0
$321$	8.00000	0.446516
$322$	0	0
$323$	9.65685	0.537322
$324$	0	0
$325$	−6.65685	−0.369256
$326$	0	0
$327$	11.6569	0.644626
$328$	0	0
$329$	−43.7990	−2.41472
$330$	0	0
$331$	−24.8284	−1.36469	−0.682347	−	0.731029i	$-0.739041\pi$
$331$	−24.8284	−1.36469	−0.682347	+	0.731029i	$0.739041\pi$
$332$	0	0
$333$	3.17157	0.173801
$334$	0	0
$335$	−43.7990	−2.39299
$336$	0	0
$337$	12.0000	0.653682	0.326841	−	0.945079i	$-0.394016\pi$
$337$	12.0000	0.653682	0.326841	+	0.945079i	$0.394016\pi$
$338$	0	0
$339$	−3.65685	−0.198613
$340$	0	0
$341$	22.8284	1.23623
$342$	0	0
$343$	44.9706	2.42818
$344$	0	0
$345$	−19.3137	−1.03982
$346$	0	0
$347$	5.65685	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$347$	5.65685	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$348$	0	0
$349$	−21.7990	−1.16687	−0.583437	−	0.812159i	$-0.698292\pi$
$349$	−21.7990	−1.16687	−0.583437	+	0.812159i	$0.698292\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	−11.4142	−0.607517	−0.303759	−	0.952749i	$-0.598242\pi$
$353$	−11.4142	−0.607517	−0.303759	+	0.952749i	$0.598242\pi$
$354$	0	0
$355$	−25.3137	−1.34351
$356$	0	0
$357$	23.3137	1.23389
$358$	0	0
$359$	26.2426	1.38503	0.692517	−	0.721402i	$-0.256502\pi$
$359$	26.2426	1.38503	0.692517	+	0.721402i	$0.256502\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	27.9706	1.46807
$364$	0	0
$365$	−35.7990	−1.87380
$366$	0	0
$367$	−4.00000	−0.208798	−0.104399	−	0.994535i	$-0.533292\pi$
$367$	−4.00000	−0.208798	−0.104399	+	0.994535i	$0.533292\pi$
$368$	0	0
$369$	−3.41421	−0.177737
$370$	0	0
$371$	34.6274	1.79777
$372$	0	0
$373$	−23.6569	−1.22491	−0.612453	−	0.790507i	$-0.709817\pi$
$373$	−23.6569	−1.22491	−0.612453	+	0.790507i	$0.709817\pi$
$374$	0	0
$375$	5.65685	0.292119
$376$	0	0
$377$	−0.828427	−0.0426662
$378$	0	0
$379$	−26.2843	−1.35013	−0.675066	−	0.737757i	$-0.735885\pi$
$379$	−26.2843	−1.35013	−0.675066	+	0.737757i	$0.735885\pi$
$380$	0	0
$381$	8.34315	0.427432
$382$	0	0
$383$	−35.4142	−1.80958	−0.904791	−	0.425856i	$-0.859973\pi$
$383$	−35.4142	−1.80958	−0.904791	+	0.425856i	$0.859973\pi$
$384$	0	0
$385$	−102.912	−5.24487
$386$	0	0
$387$	11.6569	0.592551
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	−27.3137	−1.38131
$392$	0	0
$393$	4.00000	0.201773
$394$	0	0
$395$	39.7990	2.00250
$396$	0	0
$397$	−19.4558	−0.976461	−0.488230	−	0.872715i	$-0.662357\pi$
$397$	−19.4558	−0.976461	−0.488230	+	0.872715i	$0.662357\pi$
$398$	0	0
$399$	9.65685	0.483447
$400$	0	0
$401$	−9.75736	−0.487259	−0.243630	−	0.969868i	$-0.578338\pi$
$401$	−9.75736	−0.487259	−0.243630	+	0.969868i	$0.578338\pi$
$402$	0	0
$403$	3.65685	0.182161
$404$	0	0
$405$	3.41421	0.169654
$406$	0	0
$407$	−19.7990	−0.981399
$408$	0	0
$409$	9.79899	0.484529	0.242264	−	0.970210i	$-0.422110\pi$
$409$	9.79899	0.484529	0.242264	+	0.970210i	$0.422110\pi$
$410$	0	0
$411$	−7.89949	−0.389653
$412$	0	0
$413$	71.1127	3.49923
$414$	0	0
$415$	7.65685	0.375860
$416$	0	0
$417$	7.65685	0.374958
$418$	0	0
$419$	−14.3431	−0.700709	−0.350354	−	0.936617i	$-0.613939\pi$
$419$	−14.3431	−0.700709	−0.350354	+	0.936617i	$0.613939\pi$
$420$	0	0
$421$	−22.9706	−1.11952	−0.559758	−	0.828656i	$-0.689106\pi$
$421$	−22.9706	−1.11952	−0.559758	+	0.828656i	$0.689106\pi$
$422$	0	0
$423$	−9.07107	−0.441050
$424$	0	0
$425$	32.1421	1.55912
$426$	0	0
$427$	36.9706	1.78913
$428$	0	0
$429$	6.24264	0.301398
$430$	0	0
$431$	−27.4142	−1.32050	−0.660248	−	0.751048i	$-0.729549\pi$
$431$	−27.4142	−1.32050	−0.660248	+	0.751048i	$0.729549\pi$
$432$	0	0
$433$	8.00000	0.384455	0.192228	−	0.981350i	$-0.438429\pi$
$433$	8.00000	0.384455	0.192228	+	0.981350i	$0.438429\pi$
$434$	0	0
$435$	2.82843	0.135613
$436$	0	0
$437$	−11.3137	−0.541208
$438$	0	0
$439$	−0.970563	−0.0463224	−0.0231612	−	0.999732i	$-0.507373\pi$
$439$	−0.970563	−0.0463224	−0.0231612	+	0.999732i	$0.507373\pi$
$440$	0	0
$441$	16.3137	0.776843
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	−3.89949	−0.184440
$448$	0	0
$449$	−26.7279	−1.26137	−0.630684	−	0.776039i	$-0.717226\pi$
$449$	−26.7279	−1.26137	−0.630684	+	0.776039i	$0.717226\pi$
$450$	0	0
$451$	21.3137	1.00362
$452$	0	0
$453$	−11.6569	−0.547687
$454$	0	0
$455$	−16.4853	−0.772842
$456$	0	0
$457$	−20.1421	−0.942209	−0.471105	−	0.882077i	$-0.656145\pi$
$457$	−20.1421	−0.942209	−0.471105	+	0.882077i	$0.656145\pi$
$458$	0	0
$459$	4.82843	0.225372
$460$	0	0
$461$	−17.0711	−0.795079	−0.397539	−	0.917585i	$-0.630136\pi$
$461$	−17.0711	−0.795079	−0.397539	+	0.917585i	$0.630136\pi$
$462$	0	0
$463$	24.1421	1.12198	0.560990	−	0.827823i	$-0.310421\pi$
$463$	24.1421	1.12198	0.560990	+	0.827823i	$0.310421\pi$
$464$	0	0
$465$	−12.4853	−0.578991
$466$	0	0
$467$	−25.4558	−1.17796	−0.588978	−	0.808149i	$-0.700470\pi$
$467$	−25.4558	−1.17796	−0.588978	+	0.808149i	$0.700470\pi$
$468$	0	0
$469$	−61.9411	−2.86018
$470$	0	0
$471$	2.34315	0.107966
$472$	0	0
$473$	−72.7696	−3.34595
$474$	0	0
$475$	13.3137	0.610875
$476$	0	0
$477$	7.17157	0.328364
$478$	0	0
$479$	−13.2721	−0.606417	−0.303208	−	0.952924i	$-0.598058\pi$
$479$	−13.2721	−0.606417	−0.303208	+	0.952924i	$0.598058\pi$
$480$	0	0
$481$	−3.17157	−0.144611
$482$	0	0
$483$	−27.3137	−1.24282
$484$	0	0
$485$	10.8284	0.491694
$486$	0	0
$487$	−23.6569	−1.07199	−0.535997	−	0.844220i	$-0.680064\pi$
$487$	−23.6569	−1.07199	−0.535997	+	0.844220i	$0.680064\pi$
$488$	0	0
$489$	11.6569	0.527141
$490$	0	0
$491$	−1.17157	−0.0528723	−0.0264362	−	0.999651i	$-0.508416\pi$
$491$	−1.17157	−0.0528723	−0.0264362	+	0.999651i	$0.508416\pi$
$492$	0	0
$493$	4.00000	0.180151
$494$	0	0
$495$	−21.3137	−0.957980
$496$	0	0
$497$	−35.7990	−1.60580
$498$	0	0
$499$	−31.4558	−1.40816	−0.704078	−	0.710122i	$-0.748640\pi$
$499$	−31.4558	−1.40816	−0.704078	+	0.710122i	$0.748640\pi$
$500$	0	0
$501$	−4.58579	−0.204878
$502$	0	0
$503$	22.1421	0.987269	0.493635	−	0.869669i	$-0.335668\pi$
$503$	22.1421	0.987269	0.493635	+	0.869669i	$0.335668\pi$
$504$	0	0
$505$	34.1421	1.51931
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−3.21320	−0.142423	−0.0712114	−	0.997461i	$-0.522686\pi$
$509$	−3.21320	−0.142423	−0.0712114	+	0.997461i	$0.522686\pi$
$510$	0	0
$511$	−50.6274	−2.23963
$512$	0	0
$513$	2.00000	0.0883022
$514$	0	0
$515$	38.6274	1.70213
$516$	0	0
$517$	56.6274	2.49047
$518$	0	0
$519$	7.65685	0.336099
$520$	0	0
$521$	25.3137	1.10901	0.554507	−	0.832179i	$-0.312907\pi$
$521$	25.3137	1.10901	0.554507	+	0.832179i	$0.312907\pi$
$522$	0	0
$523$	16.3431	0.714636	0.357318	−	0.933983i	$-0.383691\pi$
$523$	16.3431	0.714636	0.357318	+	0.933983i	$0.383691\pi$
$524$	0	0
$525$	32.1421	1.40280
$526$	0	0
$527$	−17.6569	−0.769145
$528$	0	0
$529$	9.00000	0.391304
$530$	0	0
$531$	14.7279	0.639137
$532$	0	0
$533$	3.41421	0.147886
$534$	0	0
$535$	27.3137	1.18087
$536$	0	0
$537$	−8.48528	−0.366167
$538$	0	0
$539$	−101.841	−4.38659
$540$	0	0
$541$	42.0000	1.80572	0.902861	−	0.429934i	$-0.141463\pi$
$541$	42.0000	1.80572	0.902861	+	0.429934i	$0.141463\pi$
$542$	0	0
$543$	19.3137	0.828831
$544$	0	0
$545$	39.7990	1.70480
$546$	0	0
$547$	−39.9411	−1.70776	−0.853880	−	0.520471i	$-0.825757\pi$
$547$	−39.9411	−1.70776	−0.853880	+	0.520471i	$0.825757\pi$
$548$	0	0
$549$	7.65685	0.326787
$550$	0	0
$551$	1.65685	0.0705844
$552$	0	0
$553$	56.2843	2.39345
$554$	0	0
$555$	10.8284	0.459641
$556$	0	0
$557$	20.8701	0.884293	0.442146	−	0.896943i	$-0.354217\pi$
$557$	20.8701	0.884293	0.442146	+	0.896943i	$0.354217\pi$
$558$	0	0
$559$	−11.6569	−0.493032
$560$	0	0
$561$	−30.1421	−1.27260
$562$	0	0
$563$	3.79899	0.160108	0.0800542	−	0.996791i	$-0.474491\pi$
$563$	3.79899	0.160108	0.0800542	+	0.996791i	$0.474491\pi$
$564$	0	0
$565$	−12.4853	−0.525260
$566$	0	0
$567$	4.82843	0.202775
$568$	0	0
$569$	−13.5147	−0.566566	−0.283283	−	0.959036i	$-0.591424\pi$
$569$	−13.5147	−0.566566	−0.283283	+	0.959036i	$0.591424\pi$
$570$	0	0
$571$	−2.34315	−0.0980576	−0.0490288	−	0.998797i	$-0.515613\pi$
$571$	−2.34315	−0.0980576	−0.0490288	+	0.998797i	$0.515613\pi$
$572$	0	0
$573$	10.1421	0.423694
$574$	0	0
$575$	−37.6569	−1.57040
$576$	0	0
$577$	−10.6863	−0.444876	−0.222438	−	0.974947i	$-0.571402\pi$
$577$	−10.6863	−0.444876	−0.222438	+	0.974947i	$0.571402\pi$
$578$	0	0
$579$	−7.65685	−0.318208
$580$	0	0
$581$	10.8284	0.449239
$582$	0	0
$583$	−44.7696	−1.85417
$584$	0	0
$585$	−3.41421	−0.141160
$586$	0	0
$587$	14.9289	0.616183	0.308091	−	0.951357i	$-0.400310\pi$
$587$	14.9289	0.616183	0.308091	+	0.951357i	$0.400310\pi$
$588$	0	0
$589$	−7.31371	−0.301356
$590$	0	0
$591$	−6.72792	−0.276750
$592$	0	0
$593$	14.0416	0.576621	0.288310	−	0.957537i	$-0.406907\pi$
$593$	14.0416	0.576621	0.288310	+	0.957537i	$0.406907\pi$
$594$	0	0
$595$	79.5980	3.26320
$596$	0	0
$597$	−21.3137	−0.872312
$598$	0	0
$599$	−21.4558	−0.876662	−0.438331	−	0.898814i	$-0.644430\pi$
$599$	−21.4558	−0.876662	−0.438331	+	0.898814i	$0.644430\pi$
$600$	0	0
$601$	−1.65685	−0.0675845	−0.0337922	−	0.999429i	$-0.510758\pi$
$601$	−1.65685	−0.0675845	−0.0337922	+	0.999429i	$0.510758\pi$
$602$	0	0
$603$	−12.8284	−0.522414
$604$	0	0
$605$	95.4975	3.88252
$606$	0	0
$607$	1.65685	0.0672496	0.0336248	−	0.999435i	$-0.489295\pi$
$607$	1.65685	0.0672496	0.0336248	+	0.999435i	$0.489295\pi$
$608$	0	0
$609$	4.00000	0.162088
$610$	0	0
$611$	9.07107	0.366976
$612$	0	0
$613$	30.7696	1.24277	0.621385	−	0.783505i	$-0.286570\pi$
$613$	30.7696	1.24277	0.621385	+	0.783505i	$0.286570\pi$
$614$	0	0
$615$	−11.6569	−0.470050
$616$	0	0
$617$	−5.75736	−0.231783	−0.115891	−	0.993262i	$-0.536972\pi$
$617$	−5.75736	−0.231783	−0.115891	+	0.993262i	$0.536972\pi$
$618$	0	0
$619$	17.5147	0.703976	0.351988	−	0.936005i	$-0.385506\pi$
$619$	17.5147	0.703976	0.351988	+	0.936005i	$0.385506\pi$
$620$	0	0
$621$	−5.65685	−0.227002
$622$	0	0
$623$	2.82843	0.113319
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	−12.4853	−0.498614
$628$	0	0
$629$	15.3137	0.610598
$630$	0	0
$631$	30.9706	1.23292	0.616459	−	0.787387i	$-0.288566\pi$
$631$	30.9706	1.23292	0.616459	+	0.787387i	$0.288566\pi$
$632$	0	0
$633$	1.65685	0.0658540
$634$	0	0
$635$	28.4853	1.13040
$636$	0	0
$637$	−16.3137	−0.646373
$638$	0	0
$639$	−7.41421	−0.293302
$640$	0	0
$641$	−5.79899	−0.229046	−0.114523	−	0.993421i	$-0.536534\pi$
$641$	−5.79899	−0.229046	−0.114523	+	0.993421i	$0.536534\pi$
$642$	0	0
$643$	−8.14214	−0.321094	−0.160547	−	0.987028i	$-0.551326\pi$
$643$	−8.14214	−0.321094	−0.160547	+	0.987028i	$0.551326\pi$
$644$	0	0
$645$	39.7990	1.56708
$646$	0	0
$647$	−7.02944	−0.276356	−0.138178	−	0.990407i	$-0.544125\pi$
$647$	−7.02944	−0.276356	−0.138178	+	0.990407i	$0.544125\pi$
$648$	0	0
$649$	−91.9411	−3.60900
$650$	0	0
$651$	−17.6569	−0.692027
$652$	0	0
$653$	−9.79899	−0.383464	−0.191732	−	0.981447i	$-0.561410\pi$
$653$	−9.79899	−0.383464	−0.191732	+	0.981447i	$0.561410\pi$
$654$	0	0
$655$	13.6569	0.533617
$656$	0	0
$657$	−10.4853	−0.409070
$658$	0	0
$659$	−2.62742	−0.102350	−0.0511748	−	0.998690i	$-0.516297\pi$
$659$	−2.62742	−0.102350	−0.0511748	+	0.998690i	$0.516297\pi$
$660$	0	0
$661$	−32.1421	−1.25018	−0.625092	−	0.780551i	$-0.714939\pi$
$661$	−32.1421	−1.25018	−0.625092	+	0.780551i	$0.714939\pi$
$662$	0	0
$663$	−4.82843	−0.187521
$664$	0	0
$665$	32.9706	1.27854
$666$	0	0
$667$	−4.68629	−0.181454
$668$	0	0
$669$	6.00000	0.231973
$670$	0	0
$671$	−47.7990	−1.84526
$672$	0	0
$673$	18.0000	0.693849	0.346925	−	0.937893i	$-0.387226\pi$
$673$	18.0000	0.693849	0.346925	+	0.937893i	$0.387226\pi$
$674$	0	0
$675$	6.65685	0.256222
$676$	0	0
$677$	13.7990	0.530338	0.265169	−	0.964202i	$-0.414572\pi$
$677$	13.7990	0.530338	0.265169	+	0.964202i	$0.414572\pi$
$678$	0	0
$679$	15.3137	0.587686
$680$	0	0
$681$	−2.72792	−0.104534
$682$	0	0
$683$	−50.7279	−1.94105	−0.970525	−	0.241000i	$-0.922524\pi$
$683$	−50.7279	−1.94105	−0.970525	+	0.241000i	$0.922524\pi$
$684$	0	0
$685$	−26.9706	−1.03049
$686$	0	0
$687$	6.00000	0.228914
$688$	0	0
$689$	−7.17157	−0.273215
$690$	0	0
$691$	−2.20101	−0.0837304	−0.0418652	−	0.999123i	$-0.513330\pi$
$691$	−2.20101	−0.0837304	−0.0418652	+	0.999123i	$0.513330\pi$
$692$	0	0
$693$	−30.1421	−1.14501
$694$	0	0
$695$	26.1421	0.991628
$696$	0	0
$697$	−16.4853	−0.624425
$698$	0	0
$699$	−9.51472	−0.359880
$700$	0	0
$701$	−31.9411	−1.20640	−0.603200	−	0.797590i	$-0.706108\pi$
$701$	−31.9411	−1.20640	−0.603200	+	0.797590i	$0.706108\pi$
$702$	0	0
$703$	6.34315	0.239236
$704$	0	0
$705$	−30.9706	−1.16642
$706$	0	0
$707$	48.2843	1.81592
$708$	0	0
$709$	34.0000	1.27690	0.638448	−	0.769665i	$-0.279577\pi$
$709$	34.0000	1.27690	0.638448	+	0.769665i	$0.279577\pi$
$710$	0	0
$711$	11.6569	0.437166
$712$	0	0
$713$	20.6863	0.774708
$714$	0	0
$715$	21.3137	0.797088
$716$	0	0
$717$	−2.92893	−0.109383
$718$	0	0
$719$	−21.6569	−0.807664	−0.403832	−	0.914833i	$-0.632322\pi$
$719$	−21.6569	−0.807664	−0.403832	+	0.914833i	$0.632322\pi$
$720$	0	0
$721$	54.6274	2.03443
$722$	0	0
$723$	14.4853	0.538713
$724$	0	0
$725$	5.51472	0.204812
$726$	0	0
$727$	−10.9706	−0.406876	−0.203438	−	0.979088i	$-0.565211\pi$
$727$	−10.9706	−0.406876	−0.203438	+	0.979088i	$0.565211\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	56.2843	2.08175
$732$	0	0
$733$	−18.2843	−0.675345	−0.337672	−	0.941264i	$-0.609640\pi$
$733$	−18.2843	−0.675345	−0.337672	+	0.941264i	$0.609640\pi$
$734$	0	0
$735$	55.6985	2.05447
$736$	0	0
$737$	80.0833	2.94990
$738$	0	0
$739$	−9.51472	−0.350005	−0.175002	−	0.984568i	$-0.555993\pi$
$739$	−9.51472	−0.350005	−0.175002	+	0.984568i	$0.555993\pi$
$740$	0	0
$741$	−2.00000	−0.0734718
$742$	0	0
$743$	16.3848	0.601099	0.300550	−	0.953766i	$-0.402830\pi$
$743$	16.3848	0.601099	0.300550	+	0.953766i	$0.402830\pi$
$744$	0	0
$745$	−13.3137	−0.487777
$746$	0	0
$747$	2.24264	0.0820539
$748$	0	0
$749$	38.6274	1.41142
$750$	0	0
$751$	8.97056	0.327340	0.163670	−	0.986515i	$-0.447667\pi$
$751$	8.97056	0.327340	0.163670	+	0.986515i	$0.447667\pi$
$752$	0	0
$753$	5.17157	0.188463
$754$	0	0
$755$	−39.7990	−1.44843
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	35.3137	1.28181
$760$	0	0
$761$	42.0416	1.52401	0.762004	−	0.647572i	$-0.224216\pi$
$761$	42.0416	1.52401	0.762004	+	0.647572i	$0.224216\pi$
$762$	0	0
$763$	56.2843	2.03763
$764$	0	0
$765$	16.4853	0.596027
$766$	0	0
$767$	−14.7279	−0.531794
$768$	0	0
$769$	24.6274	0.888087	0.444044	−	0.896005i	$-0.353543\pi$
$769$	24.6274	0.888087	0.444044	+	0.896005i	$0.353543\pi$
$770$	0	0
$771$	11.1716	0.402334
$772$	0	0
$773$	26.5269	0.954107	0.477053	−	0.878874i	$-0.341705\pi$
$773$	26.5269	0.954107	0.477053	+	0.878874i	$0.341705\pi$
$774$	0	0
$775$	−24.3431	−0.874432
$776$	0	0
$777$	15.3137	0.549376
$778$	0	0
$779$	−6.82843	−0.244654
$780$	0	0
$781$	46.2843	1.65618
$782$	0	0
$783$	0.828427	0.0296056
$784$	0	0
$785$	8.00000	0.285532
$786$	0	0
$787$	0.142136	0.00506659	0.00253329	−	0.999997i	$-0.499194\pi$
$787$	0.142136	0.00506659	0.00253329	+	0.999997i	$0.499194\pi$
$788$	0	0
$789$	−30.1421	−1.07309
$790$	0	0
$791$	−17.6569	−0.627805
$792$	0	0
$793$	−7.65685	−0.271903
$794$	0	0
$795$	24.4853	0.868403
$796$	0	0
$797$	9.31371	0.329908	0.164954	−	0.986301i	$-0.447252\pi$
$797$	9.31371	0.329908	0.164954	+	0.986301i	$0.447252\pi$
$798$	0	0
$799$	−43.7990	−1.54950
$800$	0	0
$801$	0.585786	0.0206977
$802$	0	0
$803$	65.4558	2.30989
$804$	0	0
$805$	−93.2548	−3.28680
$806$	0	0
$807$	18.4853	0.650713
$808$	0	0
$809$	−4.62742	−0.162691	−0.0813457	−	0.996686i	$-0.525922\pi$
$809$	−4.62742	−0.162691	−0.0813457	+	0.996686i	$0.525922\pi$
$810$	0	0
$811$	14.9706	0.525688	0.262844	−	0.964838i	$-0.415340\pi$
$811$	14.9706	0.525688	0.262844	+	0.964838i	$0.415340\pi$
$812$	0	0
$813$	−4.14214	−0.145271
$814$	0	0
$815$	39.7990	1.39410
$816$	0	0
$817$	23.3137	0.815643
$818$	0	0
$819$	−4.82843	−0.168719
$820$	0	0
$821$	−34.5269	−1.20500	−0.602499	−	0.798120i	$-0.705828\pi$
$821$	−34.5269	−1.20500	−0.602499	+	0.798120i	$0.705828\pi$
$822$	0	0
$823$	30.6274	1.06760	0.533802	−	0.845609i	$-0.320763\pi$
$823$	30.6274	1.06760	0.533802	+	0.845609i	$0.320763\pi$
$824$	0	0
$825$	−41.5563	−1.44681
$826$	0	0
$827$	35.8995	1.24835	0.624174	−	0.781285i	$-0.285436\pi$
$827$	35.8995	1.24835	0.624174	+	0.781285i	$0.285436\pi$
$828$	0	0
$829$	10.6863	0.371150	0.185575	−	0.982630i	$-0.440585\pi$
$829$	10.6863	0.371150	0.185575	+	0.982630i	$0.440585\pi$
$830$	0	0
$831$	−5.65685	−0.196234
$832$	0	0
$833$	78.7696	2.72920
$834$	0	0
$835$	−15.6569	−0.541828
$836$	0	0
$837$	−3.65685	−0.126399
$838$	0	0
$839$	−50.7279	−1.75132	−0.875661	−	0.482926i	$-0.839574\pi$
$839$	−50.7279	−1.75132	−0.875661	+	0.482926i	$0.839574\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	4.58579	0.157943
$844$	0	0
$845$	3.41421	0.117453
$846$	0	0
$847$	135.054	4.64050
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−17.9411	−0.615014
$852$	0	0
$853$	44.1421	1.51140	0.755699	−	0.654919i	$-0.227297\pi$
$853$	44.1421	1.51140	0.755699	+	0.654919i	$0.227297\pi$
$854$	0	0
$855$	6.82843	0.233527
$856$	0	0
$857$	−20.8284	−0.711486	−0.355743	−	0.934584i	$-0.615772\pi$
$857$	−20.8284	−0.711486	−0.355743	+	0.934584i	$0.615772\pi$
$858$	0	0
$859$	−36.9706	−1.26142	−0.630710	−	0.776019i	$-0.717236\pi$
$859$	−36.9706	−1.26142	−0.630710	+	0.776019i	$0.717236\pi$
$860$	0	0
$861$	−16.4853	−0.561817
$862$	0	0
$863$	51.0122	1.73648	0.868238	−	0.496149i	$-0.165253\pi$
$863$	51.0122	1.73648	0.868238	+	0.496149i	$0.165253\pi$
$864$	0	0
$865$	26.1421	0.888859
$866$	0	0
$867$	6.31371	0.214425
$868$	0	0
$869$	−72.7696	−2.46854
$870$	0	0
$871$	12.8284	0.434675
$872$	0	0
$873$	3.17157	0.107341
$874$	0	0
$875$	27.3137	0.923372
$876$	0	0
$877$	0.828427	0.0279740	0.0139870	−	0.999902i	$-0.495548\pi$
$877$	0.828427	0.0279740	0.0139870	+	0.999902i	$0.495548\pi$
$878$	0	0
$879$	1.75736	0.0592743
$880$	0	0
$881$	−3.45584	−0.116430	−0.0582152	−	0.998304i	$-0.518541\pi$
$881$	−3.45584	−0.116430	−0.0582152	+	0.998304i	$0.518541\pi$
$882$	0	0
$883$	4.68629	0.157706	0.0788531	−	0.996886i	$-0.474874\pi$
$883$	4.68629	0.157706	0.0788531	+	0.996886i	$0.474874\pi$
$884$	0	0
$885$	50.2843	1.69029
$886$	0	0
$887$	−21.1716	−0.710872	−0.355436	−	0.934701i	$-0.615668\pi$
$887$	−21.1716	−0.710872	−0.355436	+	0.934701i	$0.615668\pi$
$888$	0	0
$889$	40.2843	1.35109
$890$	0	0
$891$	−6.24264	−0.209136
$892$	0	0
$893$	−18.1421	−0.607103
$894$	0	0
$895$	−28.9706	−0.968379
$896$	0	0
$897$	5.65685	0.188877
$898$	0	0
$899$	−3.02944	−0.101037
$900$	0	0
$901$	34.6274	1.15361
$902$	0	0
$903$	56.2843	1.87302
$904$	0	0
$905$	65.9411	2.19196
$906$	0	0
$907$	−51.3137	−1.70384	−0.851922	−	0.523669i	$-0.824563\pi$
$907$	−51.3137	−1.70384	−0.851922	+	0.523669i	$0.824563\pi$
$908$	0	0
$909$	10.0000	0.331679
$910$	0	0
$911$	17.4558	0.578338	0.289169	−	0.957278i	$-0.406621\pi$
$911$	17.4558	0.578338	0.289169	+	0.957278i	$0.406621\pi$
$912$	0	0
$913$	−14.0000	−0.463332
$914$	0	0
$915$	26.1421	0.864232
$916$	0	0
$917$	19.3137	0.637795
$918$	0	0
$919$	46.0000	1.51740	0.758700	−	0.651440i	$-0.225835\pi$
$919$	46.0000	1.51740	0.758700	+	0.651440i	$0.225835\pi$
$920$	0	0
$921$	−25.7990	−0.850106
$922$	0	0
$923$	7.41421	0.244042
$924$	0	0
$925$	21.1127	0.694181
$926$	0	0
$927$	11.3137	0.371591
$928$	0	0
$929$	−37.3553	−1.22559	−0.612794	−	0.790242i	$-0.709955\pi$
$929$	−37.3553	−1.22559	−0.612794	+	0.790242i	$0.709955\pi$
$930$	0	0
$931$	32.6274	1.06932
$932$	0	0
$933$	−11.3137	−0.370394
$934$	0	0
$935$	−102.912	−3.36557
$936$	0	0
$937$	31.3137	1.02297	0.511487	−	0.859291i	$-0.329095\pi$
$937$	31.3137	1.02297	0.511487	+	0.859291i	$0.329095\pi$
$938$	0	0
$939$	−24.2843	−0.792487
$940$	0	0
$941$	−4.10051	−0.133673	−0.0668363	−	0.997764i	$-0.521291\pi$
$941$	−4.10051	−0.133673	−0.0668363	+	0.997764i	$0.521291\pi$
$942$	0	0
$943$	19.3137	0.628941
$944$	0	0
$945$	16.4853	0.536266
$946$	0	0
$947$	21.5563	0.700487	0.350244	−	0.936659i	$-0.386099\pi$
$947$	21.5563	0.700487	0.350244	+	0.936659i	$0.386099\pi$
$948$	0	0
$949$	10.4853	0.340367
$950$	0	0
$951$	−22.7279	−0.737003
$952$	0	0
$953$	45.1127	1.46134	0.730672	−	0.682729i	$-0.239207\pi$
$953$	45.1127	1.46134	0.730672	+	0.682729i	$0.239207\pi$
$954$	0	0
$955$	34.6274	1.12052
$956$	0	0
$957$	−5.17157	−0.167173
$958$	0	0
$959$	−38.1421	−1.23167
$960$	0	0
$961$	−17.6274	−0.568626
$962$	0	0
$963$	8.00000	0.257796
$964$	0	0
$965$	−26.1421	−0.841545
$966$	0	0
$967$	−22.0000	−0.707472	−0.353736	−	0.935345i	$-0.615089\pi$
$967$	−22.0000	−0.707472	−0.353736	+	0.935345i	$0.615089\pi$
$968$	0	0
$969$	9.65685	0.310223
$970$	0	0
$971$	−33.6569	−1.08010	−0.540050	−	0.841633i	$-0.681595\pi$
$971$	−33.6569	−1.08010	−0.540050	+	0.841633i	$0.681595\pi$
$972$	0	0
$973$	36.9706	1.18522
$974$	0	0
$975$	−6.65685	−0.213190
$976$	0	0
$977$	−60.8701	−1.94741	−0.973703	−	0.227822i	$-0.926840\pi$
$977$	−60.8701	−1.94741	−0.973703	+	0.227822i	$0.926840\pi$
$978$	0	0
$979$	−3.65685	−0.116874
$980$	0	0
$981$	11.6569	0.372175
$982$	0	0
$983$	34.5269	1.10124	0.550619	−	0.834757i	$-0.314392\pi$
$983$	34.5269	1.10124	0.550619	+	0.834757i	$0.314392\pi$
$984$	0	0
$985$	−22.9706	−0.731903
$986$	0	0
$987$	−43.7990	−1.39414
$988$	0	0
$989$	−65.9411	−2.09681
$990$	0	0
$991$	−27.5980	−0.876679	−0.438339	−	0.898810i	$-0.644433\pi$
$991$	−27.5980	−0.876679	−0.438339	+	0.898810i	$0.644433\pi$
$992$	0	0
$993$	−24.8284	−0.787906
$994$	0	0
$995$	−72.7696	−2.30695
$996$	0	0
$997$	56.0000	1.77354	0.886769	−	0.462213i	$-0.152944\pi$
$997$	56.0000	1.77354	0.886769	+	0.462213i	$0.152944\pi$
$998$	0	0
$999$	3.17157	0.100344

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9984.2.a.p.1.2		2
4.3	odd	2		9984.2.a.h.1.2		2
8.3	odd	2		9984.2.a.i.1.1		2
8.5	even	2		9984.2.a.a.1.1		2
16.3	odd	4		2496.2.g.c.1249.1	yes	4
16.5	even	4		2496.2.g.b.1249.2	✓	4
16.11	odd	4		2496.2.g.c.1249.4	yes	4
16.13	even	4		2496.2.g.b.1249.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2496.2.g.b.1249.2	✓	4	16.5	even	4
2496.2.g.b.1249.3	yes	4	16.13	even	4
2496.2.g.c.1249.1	yes	4	16.3	odd	4
2496.2.g.c.1249.4	yes	4	16.11	odd	4
9984.2.a.a.1.1		2	8.5	even	2
9984.2.a.h.1.2		2	4.3	odd	2
9984.2.a.i.1.1		2	8.3	odd	2
9984.2.a.p.1.2		2	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 9984 = 2^{8} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9984.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.41421 q^{5} +4.82843 q^{7} +1.00000 q^{9} -6.24264 q^{11} -1.00000 q^{13} +3.41421 q^{15} +4.82843 q^{17} +2.00000 q^{19} +4.82843 q^{21} -5.65685 q^{23} +6.65685 q^{25} +1.00000 q^{27} +0.828427 q^{29} -3.65685 q^{31} -6.24264 q^{33} +16.4853 q^{35} +3.17157 q^{37} -1.00000 q^{39} -3.41421 q^{41} +11.6569 q^{43} +3.41421 q^{45} -9.07107 q^{47} +16.3137 q^{49} +4.82843 q^{51} +7.17157 q^{53} -21.3137 q^{55} +2.00000 q^{57} +14.7279 q^{59} +7.65685 q^{61} +4.82843 q^{63} -3.41421 q^{65} -12.8284 q^{67} -5.65685 q^{69} -7.41421 q^{71} -10.4853 q^{73} +6.65685 q^{75} -30.1421 q^{77} +11.6569 q^{79} +1.00000 q^{81} +2.24264 q^{83} +16.4853 q^{85} +0.828427 q^{87} +0.585786 q^{89} -4.82843 q^{91} -3.65685 q^{93} +6.82843 q^{95} +3.17157 q^{97} -6.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} + 4 q^{15} + 4 q^{17} + 4 q^{19} + 4 q^{21} + 2 q^{25} + 2 q^{27} - 4 q^{29} + 4 q^{31} - 4 q^{33} + 16 q^{35} + 12 q^{37} - 2 q^{39} - 4 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 + 4 * q^15 + 4 * q^17 + 4 * q^19 + 4 * q^21 + 2 * q^25 + 2 * q^27 - 4 * q^29 + 4 * q^31 - 4 * q^33 + 16 * q^35 + 12 * q^37 - 2 * q^39 - 4 * q^41 + 12 * q^43 + 4 * q^45 - 4 * q^47 + 10 * q^49 + 4 * q^51 + 20 * q^53 - 20 * q^55 + 4 * q^57 + 4 * q^59 + 4 * q^61 + 4 * q^63 - 4 * q^65 - 20 * q^67 - 12 * q^71 - 4 * q^73 + 2 * q^75 - 32 * q^77 + 12 * q^79 + 2 * q^81 - 4 * q^83 + 16 * q^85 - 4 * q^87 + 4 * q^89 - 4 * q^91 + 4 * q^93 + 8 * q^95 + 12 * q^97 - 4 * q^99

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