LMFDB - Embedded newform 6800.2.a.bs.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6800,2,Mod(1,6800)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6800, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6800.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6800 = 2^{4} \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6800.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,3,0,0,0,4,0,8,0,2,0,-5] 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:	$54.2982733745$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.940.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 7x - 4 $ x^3 - 7*x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 680)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.29240$ of defining polynomial
Character	$\chi$	$=$	6800.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.29240 q^{3} -1.54751 q^{7} +7.83991 q^{9} +3.54751 q^{11} -2.25511 q^{13} +1.00000 q^{17} -4.83991 q^{19} -5.09501 q^{21} +1.54751 q^{23} +15.9349 q^{27} -2.83991 q^{29} +7.87720 q^{31} +11.6798 q^{33} +0.584803 q^{37} -7.42471 q^{39} +9.09501 q^{41} +2.00000 q^{43} -6.83991 q^{47} -4.60522 q^{49} +3.29240 q^{51} +12.9145 q^{53} -15.9349 q^{57} +13.3501 q^{59} +3.74489 q^{61} -12.1323 q^{63} -10.2646 q^{67} +5.09501 q^{69} +4.78219 q^{71} +6.83991 q^{73} -5.48979 q^{77} -2.64252 q^{79} +28.9444 q^{81} +2.51021 q^{83} -9.35012 q^{87} +18.0095 q^{89} +3.48979 q^{91} +25.9349 q^{93} -11.7449 q^{97} +27.8121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 4 q^{7} + 8 q^{9} + 2 q^{11} - 5 q^{13} + 3 q^{17} + q^{19} + 2 q^{21} - 4 q^{23} + 15 q^{27} + 7 q^{29} + 3 q^{31} + 4 q^{33} - 12 q^{37} + 7 q^{39} + 10 q^{41} + 6 q^{43} - 5 q^{47} + 7 q^{49}+ \cdots + 30 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 4 * q^7 + 8 * q^9 + 2 * q^11 - 5 * q^13 + 3 * q^17 + q^19 + 2 * q^21 - 4 * q^23 + 15 * q^27 + 7 * q^29 + 3 * q^31 + 4 * q^33 - 12 * q^37 + 7 * q^39 + 10 * q^41 + 6 * q^43 - 5 * q^47 + 7 * q^49 + 3 * q^51 + 13 * q^53 - 15 * q^57 + 21 * q^59 + 13 * q^61 - 14 * q^63 + 14 * q^67 - 2 * q^69 + 11 * q^71 + 5 * q^73 - 20 * q^77 + 18 * q^79 + 11 * q^81 + 4 * q^83 - 9 * q^87 + 11 * q^89 + 14 * q^91 + 45 * q^93 - 37 * q^97 + 30 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.29240	1.90087	0.950434	−	0.310925i	$-0.100639\pi$
$3$	3.29240	1.90087	0.950434	+	0.310925i	$0.100639\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.54751	−0.584903	−0.292451	−	0.956280i	$-0.594471\pi$
$7$	−1.54751	−0.584903	−0.292451	+	0.956280i	$0.594471\pi$
$8$	0	0
$9$	7.83991	2.61330
$10$	0	0
$11$	3.54751	1.06961	0.534807	−	0.844974i	$-0.320384\pi$
$11$	3.54751	1.06961	0.534807	+	0.844974i	$0.320384\pi$
$12$	0	0
$13$	−2.25511	−0.625454	−0.312727	−	0.949843i	$-0.601242\pi$
$13$	−2.25511	−0.625454	−0.312727	+	0.949843i	$0.601242\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.83991	−1.11035	−0.555176	−	0.831733i	$-0.687349\pi$
$19$	−4.83991	−1.11035	−0.555176	+	0.831733i	$0.687349\pi$
$20$	0	0
$21$	−5.09501	−1.11182
$22$	0	0
$23$	1.54751	0.322677	0.161339	−	0.986899i	$-0.448419\pi$
$23$	1.54751	0.322677	0.161339	+	0.986899i	$0.448419\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	15.9349	3.06668
$28$	0	0
$29$	−2.83991	−0.527358	−0.263679	−	0.964611i	$-0.584936\pi$
$29$	−2.83991	−0.527358	−0.263679	+	0.964611i	$0.584936\pi$
$30$	0	0
$31$	7.87720	1.41479	0.707394	−	0.706820i	$-0.249871\pi$
$31$	7.87720	1.41479	0.707394	+	0.706820i	$0.249871\pi$
$32$	0	0
$33$	11.6798	2.03320
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.584803	0.0961410	0.0480705	−	0.998844i	$-0.484693\pi$
$37$	0.584803	0.0961410	0.0480705	+	0.998844i	$0.484693\pi$
$38$	0	0
$39$	−7.42471	−1.18891
$40$	0	0
$41$	9.09501	1.42040	0.710201	−	0.703999i	$-0.248604\pi$
$41$	9.09501	1.42040	0.710201	+	0.703999i	$0.248604\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−6.83991	−0.997703	−0.498852	−	0.866687i	$-0.666245\pi$
$47$	−6.83991	−0.997703	−0.498852	+	0.866687i	$0.666245\pi$
$48$	0	0
$49$	−4.60522	−0.657889
$50$	0	0
$51$	3.29240	0.461028
$52$	0	0
$53$	12.9145	1.77394	0.886972	−	0.461824i	$-0.152805\pi$
$53$	12.9145	1.77394	0.886972	+	0.461824i	$0.152805\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−15.9349	−2.11063
$58$	0	0
$59$	13.3501	1.73804	0.869019	−	0.494779i	$-0.164751\pi$
$59$	13.3501	1.73804	0.869019	+	0.494779i	$0.164751\pi$
$60$	0	0
$61$	3.74489	0.479485	0.239742	−	0.970837i	$-0.422937\pi$
$61$	3.74489	0.479485	0.239742	+	0.970837i	$0.422937\pi$
$62$	0	0
$63$	−12.1323	−1.52853
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−10.2646	−1.25402	−0.627011	−	0.779010i	$-0.715722\pi$
$67$	−10.2646	−1.25402	−0.627011	+	0.779010i	$0.715722\pi$
$68$	0	0
$69$	5.09501	0.613368
$70$	0	0
$71$	4.78219	0.567542	0.283771	−	0.958892i	$-0.408414\pi$
$71$	4.78219	0.567542	0.283771	+	0.958892i	$0.408414\pi$
$72$	0	0
$73$	6.83991	0.800551	0.400275	−	0.916395i	$-0.368914\pi$
$73$	6.83991	0.800551	0.400275	+	0.916395i	$0.368914\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.48979	−0.625620
$78$	0	0
$79$	−2.64252	−0.297307	−0.148653	−	0.988889i	$-0.547494\pi$
$79$	−2.64252	−0.297307	−0.148653	+	0.988889i	$0.547494\pi$
$80$	0	0
$81$	28.9444	3.21605
$82$	0	0
$83$	2.51021	0.275531	0.137766	−	0.990465i	$-0.456008\pi$
$83$	2.51021	0.275531	0.137766	+	0.990465i	$0.456008\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−9.35012	−1.00244
$88$	0	0
$89$	18.0095	1.90900	0.954502	−	0.298203i	$-0.0963874\pi$
$89$	18.0095	1.90900	0.954502	+	0.298203i	$0.0963874\pi$
$90$	0	0
$91$	3.48979	0.365829
$92$	0	0
$93$	25.9349	2.68933
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.7449	−1.19251	−0.596257	−	0.802794i	$-0.703346\pi$
$97$	−11.7449	−1.19251	−0.596257	+	0.802794i	$0.703346\pi$
$98$	0	0
$99$	27.8121	2.79522
$100$	0	0
$101$	−10.5848	−1.05323	−0.526614	−	0.850105i	$-0.676539\pi$
$101$	−10.5848	−1.05323	−0.526614	+	0.850105i	$0.676539\pi$
$102$	0	0
$103$	−9.60522	−0.946431	−0.473215	−	0.880947i	$-0.656907\pi$
$103$	−9.60522	−0.946431	−0.473215	+	0.880947i	$0.656907\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	19.2273	1.85878	0.929388	−	0.369105i	$-0.120336\pi$
$107$	19.2273	1.85878	0.929388	+	0.369105i	$0.120336\pi$
$108$	0	0
$109$	9.93492	0.951593	0.475796	−	0.879555i	$-0.342160\pi$
$109$	9.93492	0.951593	0.475796	+	0.879555i	$0.342160\pi$
$110$	0	0
$111$	1.92541	0.182752
$112$	0	0
$113$	−12.0095	−1.12976	−0.564880	−	0.825173i	$-0.691078\pi$
$113$	−12.0095	−1.12976	−0.564880	+	0.825173i	$0.691078\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−17.6798	−1.63450
$118$	0	0
$119$	−1.54751	−0.141860
$120$	0	0
$121$	1.58480	0.144073
$122$	0	0
$123$	29.9444	2.70000
$124$	0	0
$125$	0	0
$126$	0	0
$127$	13.9349	1.23652	0.618262	−	0.785972i	$-0.287837\pi$
$127$	13.9349	1.23652	0.618262	+	0.785972i	$0.287837\pi$
$128$	0	0
$129$	6.58480	0.579760
$130$	0	0
$131$	17.2273	1.50516	0.752579	−	0.658502i	$-0.228810\pi$
$131$	17.2273	1.50516	0.752579	+	0.658502i	$0.228810\pi$
$132$	0	0
$133$	7.48979	0.649447
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.51021	0.727076	0.363538	−	0.931579i	$-0.381569\pi$
$137$	8.51021	0.727076	0.363538	+	0.931579i	$0.381569\pi$
$138$	0	0
$139$	0.452493	0.0383800	0.0191900	−	0.999816i	$-0.493891\pi$
$139$	0.452493	0.0383800	0.0191900	+	0.999816i	$0.493891\pi$
$140$	0	0
$141$	−22.5197	−1.89650
$142$	0	0
$143$	−8.00000	−0.668994
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−15.1622	−1.25056
$148$	0	0
$149$	−21.8698	−1.79165	−0.895824	−	0.444410i	$-0.853413\pi$
$149$	−21.8698	−1.79165	−0.895824	+	0.444410i	$0.853413\pi$
$150$	0	0
$151$	−10.5848	−0.861379	−0.430690	−	0.902500i	$-0.641730\pi$
$151$	−10.5848	−0.861379	−0.430690	+	0.902500i	$0.641730\pi$
$152$	0	0
$153$	7.83991	0.633819
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−11.1696	−0.891432	−0.445716	−	0.895175i	$-0.647051\pi$
$157$	−11.1696	−0.891432	−0.445716	+	0.895175i	$0.647051\pi$
$158$	0	0
$159$	42.5197	3.37203
$160$	0	0
$161$	−2.39478	−0.188735
$162$	0	0
$163$	−5.15273	−0.403593	−0.201796	−	0.979427i	$-0.564678\pi$
$163$	−5.15273	−0.403593	−0.201796	+	0.979427i	$0.564678\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−23.6221	−1.82793	−0.913966	−	0.405790i	$-0.866997\pi$
$167$	−23.6221	−1.82793	−0.913966	+	0.405790i	$0.866997\pi$
$168$	0	0
$169$	−7.91450	−0.608808
$170$	0	0
$171$	−37.9444	−2.90168
$172$	0	0
$173$	−19.2850	−1.46621	−0.733107	−	0.680113i	$-0.761931\pi$
$173$	−19.2850	−1.46621	−0.733107	+	0.680113i	$0.761931\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	43.9540	3.30378
$178$	0	0
$179$	4.90499	0.366616	0.183308	−	0.983056i	$-0.441319\pi$
$179$	4.90499	0.366616	0.183308	+	0.983056i	$0.441319\pi$
$180$	0	0
$181$	18.2646	1.35760	0.678799	−	0.734324i	$-0.262501\pi$
$181$	18.2646	1.35760	0.678799	+	0.734324i	$0.262501\pi$
$182$	0	0
$183$	12.3297	0.911438
$184$	0	0
$185$	0	0
$186$	0	0
$187$	3.54751	0.259419
$188$	0	0
$189$	−24.6594	−1.79371
$190$	0	0
$191$	25.1696	1.82121	0.910604	−	0.413279i	$-0.135617\pi$
$191$	25.1696	1.82121	0.910604	+	0.413279i	$0.135617\pi$
$192$	0	0
$193$	−4.07459	−0.293296	−0.146648	−	0.989189i	$-0.546848\pi$
$193$	−4.07459	−0.293296	−0.146648	+	0.989189i	$0.546848\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.7544	−1.54994	−0.774969	−	0.632000i	$-0.782234\pi$
$197$	−21.7544	−1.54994	−0.774969	+	0.632000i	$0.782234\pi$
$198$	0	0
$199$	−19.9926	−1.41724	−0.708620	−	0.705590i	$-0.750682\pi$
$199$	−19.9926	−1.41724	−0.708620	+	0.705590i	$0.750682\pi$
$200$	0	0
$201$	−33.7952	−2.38373
$202$	0	0
$203$	4.39478	0.308453
$204$	0	0
$205$	0	0
$206$	0	0
$207$	12.1323	0.843254
$208$	0	0
$209$	−17.1696	−1.18765
$210$	0	0
$211$	1.11189	0.0765456	0.0382728	−	0.999267i	$-0.487814\pi$
$211$	1.11189	0.0765456	0.0382728	+	0.999267i	$0.487814\pi$
$212$	0	0
$213$	15.7449	1.07882
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−12.1900	−0.827513
$218$	0	0
$219$	22.5197	1.52174
$220$	0	0
$221$	−2.25511	−0.151695
$222$	0	0
$223$	5.78574	0.387441	0.193721	−	0.981057i	$-0.437944\pi$
$223$	5.78574	0.387441	0.193721	+	0.981057i	$0.437944\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−15.5570	−1.03256	−0.516278	−	0.856421i	$-0.672683\pi$
$227$	−15.5570	−1.03256	−0.516278	+	0.856421i	$0.672683\pi$
$228$	0	0
$229$	7.60522	0.502567	0.251284	−	0.967913i	$-0.419147\pi$
$229$	7.60522	0.502567	0.251284	+	0.967913i	$0.419147\pi$
$230$	0	0
$231$	−18.0746	−1.18922
$232$	0	0
$233$	−9.93492	−0.650858	−0.325429	−	0.945566i	$-0.605509\pi$
$233$	−9.93492	−0.650858	−0.325429	+	0.945566i	$0.605509\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−8.70024	−0.565141
$238$	0	0
$239$	8.51021	0.550480	0.275240	−	0.961376i	$-0.411243\pi$
$239$	8.51021	0.550480	0.275240	+	0.961376i	$0.411243\pi$
$240$	0	0
$241$	−29.6243	−1.90827	−0.954133	−	0.299383i	$-0.903219\pi$
$241$	−29.6243	−1.90827	−0.954133	+	0.299383i	$0.903219\pi$
$242$	0	0
$243$	47.4919	3.04661
$244$	0	0
$245$	0	0
$246$	0	0
$247$	10.9145	0.694473
$248$	0	0
$249$	8.26462	0.523749
$250$	0	0
$251$	2.19003	0.138233	0.0691166	−	0.997609i	$-0.477982\pi$
$251$	2.19003	0.138233	0.0691166	+	0.997609i	$0.477982\pi$
$252$	0	0
$253$	5.48979	0.345140
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.67982	0.104784	0.0523920	−	0.998627i	$-0.483315\pi$
$257$	1.67982	0.104784	0.0523920	+	0.998627i	$0.483315\pi$
$258$	0	0
$259$	−0.904987	−0.0562331
$260$	0	0
$261$	−22.2646	−1.37815
$262$	0	0
$263$	31.7639	1.95865	0.979324	−	0.202299i	$-0.0648414\pi$
$263$	31.7639	1.95865	0.979324	+	0.202299i	$0.0648414\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	59.2946	3.62877
$268$	0	0
$269$	−3.23468	−0.197222	−0.0986111	−	0.995126i	$-0.531440\pi$
$269$	−3.23468	−0.197222	−0.0986111	+	0.995126i	$0.531440\pi$
$270$	0	0
$271$	−10.8304	−0.657900	−0.328950	−	0.944347i	$-0.606695\pi$
$271$	−10.8304	−0.657900	−0.328950	+	0.944347i	$0.606695\pi$
$272$	0	0
$273$	11.4898	0.695394
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−23.1696	−1.39213	−0.696063	−	0.717980i	$-0.745067\pi$
$277$	−23.1696	−1.39213	−0.696063	+	0.717980i	$0.745067\pi$
$278$	0	0
$279$	61.7566	3.69727
$280$	0	0
$281$	−15.7449	−0.939262	−0.469631	−	0.882863i	$-0.655613\pi$
$281$	−15.7449	−0.939262	−0.469631	+	0.882863i	$0.655613\pi$
$282$	0	0
$283$	9.61259	0.571409	0.285704	−	0.958318i	$-0.407772\pi$
$283$	9.61259	0.571409	0.285704	+	0.958318i	$0.407772\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−14.0746	−0.830797
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−38.6689	−2.26681
$292$	0	0
$293$	5.78574	0.338006	0.169003	−	0.985616i	$-0.445945\pi$
$293$	5.78574	0.338006	0.169003	+	0.985616i	$0.445945\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	56.5292	3.28016
$298$	0	0
$299$	−3.48979	−0.201820
$300$	0	0
$301$	−3.09501	−0.178394
$302$	0	0
$303$	−34.8494	−2.00205
$304$	0	0
$305$	0	0
$306$	0	0
$307$	28.5848	1.63142	0.815710	−	0.578460i	$-0.196346\pi$
$307$	28.5848	1.63142	0.815710	+	0.578460i	$0.196346\pi$
$308$	0	0
$309$	−31.6243	−1.79904
$310$	0	0
$311$	25.7375	1.45944	0.729721	−	0.683745i	$-0.239650\pi$
$311$	25.7375	1.45944	0.729721	+	0.683745i	$0.239650\pi$
$312$	0	0
$313$	−17.2442	−0.974700	−0.487350	−	0.873207i	$-0.662036\pi$
$313$	−17.2442	−0.974700	−0.487350	+	0.873207i	$0.662036\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9.63898	−0.541379	−0.270689	−	0.962667i	$-0.587252\pi$
$317$	−9.63898	−0.541379	−0.270689	+	0.962667i	$0.587252\pi$
$318$	0	0
$319$	−10.0746	−0.564069
$320$	0	0
$321$	63.3041	3.53329
$322$	0	0
$323$	−4.83991	−0.269300
$324$	0	0
$325$	0	0
$326$	0	0
$327$	32.7098	1.80885
$328$	0	0
$329$	10.5848	0.583559
$330$	0	0
$331$	−13.4993	−0.741989	−0.370994	−	0.928635i	$-0.620983\pi$
$331$	−13.4993	−0.741989	−0.370994	+	0.928635i	$0.620983\pi$
$332$	0	0
$333$	4.58480	0.251246
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.72447	0.148412	0.0742058	−	0.997243i	$-0.476358\pi$
$337$	2.72447	0.148412	0.0742058	+	0.997243i	$0.476358\pi$
$338$	0	0
$339$	−39.5401	−2.14753
$340$	0	0
$341$	27.9444	1.51328
$342$	0	0
$343$	17.9592	0.969703
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.4620	1.09846	0.549229	−	0.835672i	$-0.314921\pi$
$347$	20.4620	1.09846	0.549229	+	0.835672i	$0.314921\pi$
$348$	0	0
$349$	−8.70024	−0.465713	−0.232856	−	0.972511i	$-0.574807\pi$
$349$	−8.70024	−0.465713	−0.232856	+	0.972511i	$0.574807\pi$
$350$	0	0
$351$	−35.9349	−1.91806
$352$	0	0
$353$	14.5292	0.773313	0.386657	−	0.922224i	$-0.373630\pi$
$353$	14.5292	0.773313	0.386657	+	0.922224i	$0.373630\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.09501	−0.269657
$358$	0	0
$359$	4.24559	0.224074	0.112037	−	0.993704i	$-0.464263\pi$
$359$	4.24559	0.224074	0.112037	+	0.993704i	$0.464263\pi$
$360$	0	0
$361$	4.42471	0.232880
$362$	0	0
$363$	5.21781	0.273864
$364$	0	0
$365$	0	0
$366$	0	0
$367$	12.6425	0.659934	0.329967	−	0.943992i	$-0.392962\pi$
$367$	12.6425	0.659934	0.329967	+	0.943992i	$0.392962\pi$
$368$	0	0
$369$	71.3041	3.71194
$370$	0	0
$371$	−19.9853	−1.03758
$372$	0	0
$373$	−20.0190	−1.03655	−0.518273	−	0.855215i	$-0.673425\pi$
$373$	−20.0190	−1.03655	−0.518273	+	0.855215i	$0.673425\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	6.40429	0.329838
$378$	0	0
$379$	−12.4525	−0.639642	−0.319821	−	0.947478i	$-0.603623\pi$
$379$	−12.4525	−0.639642	−0.319821	+	0.947478i	$0.603623\pi$
$380$	0	0
$381$	45.8794	2.35047
$382$	0	0
$383$	6.95534	0.355401	0.177701	−	0.984085i	$-0.443134\pi$
$383$	6.95534	0.355401	0.177701	+	0.984085i	$0.443134\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.6798	0.797050
$388$	0	0
$389$	31.0950	1.57658	0.788290	−	0.615304i	$-0.210967\pi$
$389$	31.0950	1.57658	0.788290	+	0.615304i	$0.210967\pi$
$390$	0	0
$391$	1.54751	0.0782608
$392$	0	0
$393$	56.7193	2.86111
$394$	0	0
$395$	0	0
$396$	0	0
$397$	7.53063	0.377951	0.188976	−	0.981982i	$-0.439483\pi$
$397$	7.53063	0.377951	0.188976	+	0.981982i	$0.439483\pi$
$398$	0	0
$399$	24.6594	1.23451
$400$	0	0
$401$	−6.65940	−0.332554	−0.166277	−	0.986079i	$-0.553175\pi$
$401$	−6.65940	−0.332554	−0.166277	+	0.986079i	$0.553175\pi$
$402$	0	0
$403$	−17.7639	−0.884884
$404$	0	0
$405$	0	0
$406$	0	0
$407$	2.07459	0.102834
$408$	0	0
$409$	5.93492	0.293463	0.146731	−	0.989176i	$-0.453125\pi$
$409$	5.93492	0.293463	0.146731	+	0.989176i	$0.453125\pi$
$410$	0	0
$411$	28.0190	1.38208
$412$	0	0
$413$	−20.6594	−1.01658
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1.48979	0.0729553
$418$	0	0
$419$	−9.07814	−0.443496	−0.221748	−	0.975104i	$-0.571176\pi$
$419$	−9.07814	−0.443496	−0.221748	+	0.975104i	$0.571176\pi$
$420$	0	0
$421$	7.73538	0.376999	0.188500	−	0.982073i	$-0.439638\pi$
$421$	7.73538	0.376999	0.188500	+	0.982073i	$0.439638\pi$
$422$	0	0
$423$	−53.6243	−2.60730
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−5.79525	−0.280452
$428$	0	0
$429$	−26.3392	−1.27167
$430$	0	0
$431$	−5.62210	−0.270807	−0.135404	−	0.990791i	$-0.543233\pi$
$431$	−5.62210	−0.270807	−0.135404	+	0.990791i	$0.543233\pi$
$432$	0	0
$433$	−6.84942	−0.329162	−0.164581	−	0.986364i	$-0.552627\pi$
$433$	−6.84942	−0.329162	−0.164581	+	0.986364i	$0.552627\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.48979	−0.358285
$438$	0	0
$439$	−7.43207	−0.354714	−0.177357	−	0.984147i	$-0.556755\pi$
$439$	−7.43207	−0.354714	−0.177357	+	0.984147i	$0.556755\pi$
$440$	0	0
$441$	−36.1045	−1.71926
$442$	0	0
$443$	−15.5644	−0.739486	−0.369743	−	0.929134i	$-0.620554\pi$
$443$	−15.5644	−0.739486	−0.369743	+	0.929134i	$0.620554\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−72.0043	−3.40569
$448$	0	0
$449$	11.4152	0.538716	0.269358	−	0.963040i	$-0.413188\pi$
$449$	11.4152	0.538716	0.269358	+	0.963040i	$0.413188\pi$
$450$	0	0
$451$	32.2646	1.51928
$452$	0	0
$453$	−34.8494	−1.63737
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−7.63898	−0.357336	−0.178668	−	0.983909i	$-0.557179\pi$
$457$	−7.63898	−0.357336	−0.178668	+	0.983909i	$0.557179\pi$
$458$	0	0
$459$	15.9349	0.743778
$460$	0	0
$461$	−1.85081	−0.0862010	−0.0431005	−	0.999071i	$-0.513724\pi$
$461$	−1.85081	−0.0862010	−0.0431005	+	0.999071i	$0.513724\pi$
$462$	0	0
$463$	−22.1995	−1.03170	−0.515850	−	0.856679i	$-0.672524\pi$
$463$	−22.1995	−1.03170	−0.515850	+	0.856679i	$0.672524\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.5102	−0.486355	−0.243177	−	0.969982i	$-0.578190\pi$
$467$	−10.5102	−0.486355	−0.243177	+	0.969982i	$0.578190\pi$
$468$	0	0
$469$	15.8846	0.733481
$470$	0	0
$471$	−36.7748	−1.69449
$472$	0	0
$473$	7.09501	0.326229
$474$	0	0
$475$	0	0
$476$	0	0
$477$	101.249	4.63585
$478$	0	0
$479$	−14.3319	−0.654839	−0.327419	−	0.944879i	$-0.606179\pi$
$479$	−14.3319	−0.654839	−0.327419	+	0.944879i	$0.606179\pi$
$480$	0	0
$481$	−1.31879	−0.0601318
$482$	0	0
$483$	−7.88457	−0.358760
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−26.8663	−1.21743	−0.608714	−	0.793390i	$-0.708314\pi$
$487$	−26.8663	−1.21743	−0.608714	+	0.793390i	$0.708314\pi$
$488$	0	0
$489$	−16.9649	−0.767177
$490$	0	0
$491$	8.69072	0.392207	0.196103	−	0.980583i	$-0.437171\pi$
$491$	8.69072	0.392207	0.196103	+	0.980583i	$0.437171\pi$
$492$	0	0
$493$	−2.83991	−0.127903
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.40047	−0.331957
$498$	0	0
$499$	−11.4173	−0.511111	−0.255555	−	0.966794i	$-0.582258\pi$
$499$	−11.4173	−0.511111	−0.255555	+	0.966794i	$0.582258\pi$
$500$	0	0
$501$	−77.7734	−3.47466
$502$	0	0
$503$	2.81352	0.125449	0.0627243	−	0.998031i	$-0.480021\pi$
$503$	2.81352	0.125449	0.0627243	+	0.998031i	$0.480021\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−26.0577	−1.15726
$508$	0	0
$509$	9.48979	0.420628	0.210314	−	0.977634i	$-0.432551\pi$
$509$	9.48979	0.420628	0.210314	+	0.977634i	$0.432551\pi$
$510$	0	0
$511$	−10.5848	−0.468244
$512$	0	0
$513$	−77.1236	−3.40509
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−24.2646	−1.06716
$518$	0	0
$519$	−63.4941	−2.78708
$520$	0	0
$521$	−27.8290	−1.21921	−0.609605	−	0.792705i	$-0.708672\pi$
$521$	−27.8290	−1.21921	−0.609605	+	0.792705i	$0.708672\pi$
$522$	0	0
$523$	−19.4342	−0.849799	−0.424900	−	0.905240i	$-0.639691\pi$
$523$	−19.4342	−0.849799	−0.424900	+	0.905240i	$0.639691\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.87720	0.343136
$528$	0	0
$529$	−20.6052	−0.895879
$530$	0	0
$531$	104.664	4.54202
$532$	0	0
$533$	−20.5102	−0.888396
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.1492	0.696889
$538$	0	0
$539$	−16.3371	−0.703687
$540$	0	0
$541$	3.28504	0.141235	0.0706174	−	0.997503i	$-0.477503\pi$
$541$	3.28504	0.141235	0.0706174	+	0.997503i	$0.477503\pi$
$542$	0	0
$543$	60.1345	2.58062
$544$	0	0
$545$	0	0
$546$	0	0
$547$	14.2720	0.610226	0.305113	−	0.952316i	$-0.401306\pi$
$547$	14.2720	0.610226	0.305113	+	0.952316i	$0.401306\pi$
$548$	0	0
$549$	29.3596	1.25304
$550$	0	0
$551$	13.7449	0.585552
$552$	0	0
$553$	4.08932	0.173895
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8.44513	−0.357832	−0.178916	−	0.983864i	$-0.557259\pi$
$557$	−8.44513	−0.357832	−0.178916	+	0.983864i	$0.557259\pi$
$558$	0	0
$559$	−4.51021	−0.190762
$560$	0	0
$561$	11.6798	0.493122
$562$	0	0
$563$	−7.16961	−0.302163	−0.151081	−	0.988521i	$-0.548276\pi$
$563$	−7.16961	−0.302163	−0.151081	+	0.988521i	$0.548276\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−44.7917	−1.88107
$568$	0	0
$569$	21.5739	0.904425	0.452212	−	0.891910i	$-0.350635\pi$
$569$	21.5739	0.904425	0.452212	+	0.891910i	$0.350635\pi$
$570$	0	0
$571$	−32.7361	−1.36996	−0.684982	−	0.728560i	$-0.740190\pi$
$571$	−32.7361	−1.36996	−0.684982	+	0.728560i	$0.740190\pi$
$572$	0	0
$573$	82.8685	3.46188
$574$	0	0
$575$	0	0
$576$	0	0
$577$	44.5292	1.85378	0.926888	−	0.375337i	$-0.122473\pi$
$577$	44.5292	1.85378	0.926888	+	0.375337i	$0.122473\pi$
$578$	0	0
$579$	−13.4152	−0.557517
$580$	0	0
$581$	−3.88457	−0.161159
$582$	0	0
$583$	45.8143	1.89743
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−18.0000	−0.742940	−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000	−0.742940	−0.371470	+	0.928445i	$0.621146\pi$
$588$	0	0
$589$	−38.1249	−1.57091
$590$	0	0
$591$	−71.6243	−2.94623
$592$	0	0
$593$	−20.9796	−0.861528	−0.430764	−	0.902465i	$-0.641756\pi$
$593$	−20.9796	−0.861528	−0.430764	+	0.902465i	$0.641756\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−65.8238	−2.69399
$598$	0	0
$599$	22.7002	0.927507	0.463753	−	0.885964i	$-0.346502\pi$
$599$	22.7002	0.927507	0.463753	+	0.885964i	$0.346502\pi$
$600$	0	0
$601$	−33.1140	−1.35075	−0.675375	−	0.737474i	$-0.736018\pi$
$601$	−33.1140	−1.35075	−0.675375	+	0.737474i	$0.736018\pi$
$602$	0	0
$603$	−80.4737	−3.27714
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3.58835	−0.145647	−0.0728233	−	0.997345i	$-0.523201\pi$
$607$	−3.58835	−0.145647	−0.0728233	+	0.997345i	$0.523201\pi$
$608$	0	0
$609$	14.4694	0.586328
$610$	0	0
$611$	15.4247	0.624017
$612$	0	0
$613$	32.7843	1.32415	0.662074	−	0.749439i	$-0.269677\pi$
$613$	32.7843	1.32415	0.662074	+	0.749439i	$0.269677\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22.0504	−0.887714	−0.443857	−	0.896098i	$-0.646390\pi$
$617$	−22.0504	−0.887714	−0.443857	+	0.896098i	$0.646390\pi$
$618$	0	0
$619$	−19.4511	−0.781806	−0.390903	−	0.920432i	$-0.627837\pi$
$619$	−19.4511	−0.781806	−0.390903	+	0.920432i	$0.627837\pi$
$620$	0	0
$621$	24.6594	0.989547
$622$	0	0
$623$	−27.8698	−1.11658
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−56.5292	−2.25756
$628$	0	0
$629$	0.584803	0.0233176
$630$	0	0
$631$	−11.0950	−0.441686	−0.220843	−	0.975309i	$-0.570881\pi$
$631$	−11.0950	−0.441686	−0.220843	+	0.975309i	$0.570881\pi$
$632$	0	0
$633$	3.66079	0.145503
$634$	0	0
$635$	0	0
$636$	0	0
$637$	10.3853	0.411479
$638$	0	0
$639$	37.4919	1.48316
$640$	0	0
$641$	−11.1359	−0.439840	−0.219920	−	0.975518i	$-0.570580\pi$
$641$	−11.1359	−0.439840	−0.219920	+	0.975518i	$0.570580\pi$
$642$	0	0
$643$	30.7361	1.21212	0.606058	−	0.795421i	$-0.292750\pi$
$643$	30.7361	1.21212	0.606058	+	0.795421i	$0.292750\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	19.3691	0.761480	0.380740	−	0.924682i	$-0.375669\pi$
$647$	19.3691	0.761480	0.380740	+	0.924682i	$0.375669\pi$
$648$	0	0
$649$	47.3596	1.85903
$650$	0	0
$651$	−40.1345	−1.57299
$652$	0	0
$653$	−17.7354	−0.694039	−0.347020	−	0.937858i	$-0.612806\pi$
$653$	−17.7354	−0.694039	−0.347020	+	0.937858i	$0.612806\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	53.6243	2.09208
$658$	0	0
$659$	−18.7843	−0.731734	−0.365867	−	0.930667i	$-0.619228\pi$
$659$	−18.7843	−0.731734	−0.365867	+	0.930667i	$0.619228\pi$
$660$	0	0
$661$	−40.2091	−1.56395	−0.781976	−	0.623309i	$-0.785788\pi$
$661$	−40.2091	−1.56395	−0.781976	+	0.623309i	$0.785788\pi$
$662$	0	0
$663$	−7.42471	−0.288352
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.39478	−0.170166
$668$	0	0
$669$	19.0490	0.736475
$670$	0	0
$671$	13.2850	0.512863
$672$	0	0
$673$	−1.01091	−0.0389675	−0.0194838	−	0.999810i	$-0.506202\pi$
$673$	−1.01091	−0.0389675	−0.0194838	+	0.999810i	$0.506202\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.85081	0.0711326	0.0355663	−	0.999367i	$-0.488677\pi$
$677$	1.85081	0.0711326	0.0355663	+	0.999367i	$0.488677\pi$
$678$	0	0
$679$	18.1753	0.697504
$680$	0	0
$681$	−51.2200	−1.96275
$682$	0	0
$683$	36.6112	1.40089	0.700444	−	0.713707i	$-0.252985\pi$
$683$	36.6112	1.40089	0.700444	+	0.713707i	$0.252985\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	25.0394	0.955315
$688$	0	0
$689$	−29.1236	−1.10952
$690$	0	0
$691$	29.8867	1.13694	0.568472	−	0.822702i	$-0.307535\pi$
$691$	29.8867	1.13694	0.568472	+	0.822702i	$0.307535\pi$
$692$	0	0
$693$	−43.0394	−1.63493
$694$	0	0
$695$	0	0
$696$	0	0
$697$	9.09501	0.344498
$698$	0	0
$699$	−32.7098	−1.23720
$700$	0	0
$701$	−15.2442	−0.575765	−0.287883	−	0.957666i	$-0.592951\pi$
$701$	−15.2442	−0.575765	−0.287883	+	0.957666i	$0.592951\pi$
$702$	0	0
$703$	−2.83039	−0.106750
$704$	0	0
$705$	0	0
$706$	0	0
$707$	16.3801	0.616035
$708$	0	0
$709$	34.5606	1.29795	0.648975	−	0.760810i	$-0.275198\pi$
$709$	34.5606	1.29795	0.648975	+	0.760810i	$0.275198\pi$
$710$	0	0
$711$	−20.7171	−0.776952
$712$	0	0
$713$	12.1900	0.456520
$714$	0	0
$715$	0	0
$716$	0	0
$717$	28.0190	1.04639
$718$	0	0
$719$	20.1418	0.751163	0.375582	−	0.926789i	$-0.377443\pi$
$719$	20.1418	0.751163	0.375582	+	0.926789i	$0.377443\pi$
$720$	0	0
$721$	14.8641	0.553570
$722$	0	0
$723$	−97.5349	−3.62736
$724$	0	0
$725$	0	0
$726$	0	0
$727$	36.8998	1.36854	0.684268	−	0.729230i	$-0.260122\pi$
$727$	36.8998	1.36854	0.684268	+	0.729230i	$0.260122\pi$
$728$	0	0
$729$	69.5292	2.57516
$730$	0	0
$731$	2.00000	0.0739727
$732$	0	0
$733$	42.6784	1.57636	0.788182	−	0.615442i	$-0.211023\pi$
$733$	42.6784	1.57636	0.788182	+	0.615442i	$0.211023\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−36.4138	−1.34132
$738$	0	0
$739$	−28.7098	−1.05611	−0.528053	−	0.849212i	$-0.677078\pi$
$739$	−28.7098	−1.05611	−0.528053	+	0.849212i	$0.677078\pi$
$740$	0	0
$741$	35.9349	1.32010
$742$	0	0
$743$	18.4525	0.676956	0.338478	−	0.940974i	$-0.390088\pi$
$743$	18.4525	0.676956	0.338478	+	0.940974i	$0.390088\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	19.6798	0.720047
$748$	0	0
$749$	−29.7544	−1.08720
$750$	0	0
$751$	0.271981	0.00992474	0.00496237	−	0.999988i	$-0.498420\pi$
$751$	0.271981	0.00992474	0.00496237	+	0.999988i	$0.498420\pi$
$752$	0	0
$753$	7.21045	0.262763
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−30.7843	−1.11888	−0.559438	−	0.828872i	$-0.688983\pi$
$757$	−30.7843	−1.11888	−0.559438	+	0.828872i	$0.688983\pi$
$758$	0	0
$759$	18.0746	0.656066
$760$	0	0
$761$	−39.9853	−1.44947	−0.724733	−	0.689030i	$-0.758037\pi$
$761$	−39.9853	−1.44947	−0.724733	+	0.689030i	$0.758037\pi$
$762$	0	0
$763$	−15.3744	−0.556589
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−30.1059	−1.08706
$768$	0	0
$769$	−19.8195	−0.714709	−0.357355	−	0.933969i	$-0.616321\pi$
$769$	−19.8195	−0.714709	−0.357355	+	0.933969i	$0.616321\pi$
$770$	0	0
$771$	5.53063	0.199181
$772$	0	0
$773$	−16.5102	−0.593831	−0.296915	−	0.954904i	$-0.595958\pi$
$773$	−16.5102	−0.593831	−0.296915	+	0.954904i	$0.595958\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−2.97958	−0.106892
$778$	0	0
$779$	−44.0190	−1.57715
$780$	0	0
$781$	16.9649	0.607050
$782$	0	0
$783$	−45.2537	−1.61724
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−33.1214	−1.18065	−0.590325	−	0.807165i	$-0.701001\pi$
$787$	−33.1214	−1.18065	−0.590325	+	0.807165i	$0.701001\pi$
$788$	0	0
$789$	104.580	3.72313
$790$	0	0
$791$	18.5848	0.660800
$792$	0	0
$793$	−8.44513	−0.299895
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−52.5701	−1.86213	−0.931064	−	0.364856i	$-0.881118\pi$
$797$	−52.5701	−1.86213	−0.931064	+	0.364856i	$0.881118\pi$
$798$	0	0
$799$	−6.83991	−0.241979
$800$	0	0
$801$	141.193	4.98881
$802$	0	0
$803$	24.2646	0.856280
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−10.6499	−0.374894
$808$	0	0
$809$	−0.979580	−0.0344402	−0.0172201	−	0.999852i	$-0.505482\pi$
$809$	−0.979580	−0.0344402	−0.0172201	+	0.999852i	$0.505482\pi$
$810$	0	0
$811$	−43.7157	−1.53507	−0.767533	−	0.641009i	$-0.778516\pi$
$811$	−43.7157	−1.53507	−0.767533	+	0.641009i	$0.778516\pi$
$812$	0	0
$813$	−35.6580	−1.25058
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−9.67982	−0.338654
$818$	0	0
$819$	27.3596	0.956023
$820$	0	0
$821$	−44.9335	−1.56819	−0.784096	−	0.620640i	$-0.786873\pi$
$821$	−44.9335	−1.56819	−0.784096	+	0.620640i	$0.786873\pi$
$822$	0	0
$823$	18.0767	0.630116	0.315058	−	0.949072i	$-0.397976\pi$
$823$	18.0767	0.630116	0.315058	+	0.949072i	$0.397976\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−30.7361	−1.06880	−0.534400	−	0.845232i	$-0.679462\pi$
$827$	−30.7361	−1.06880	−0.534400	+	0.845232i	$0.679462\pi$
$828$	0	0
$829$	−16.8494	−0.585205	−0.292602	−	0.956234i	$-0.594521\pi$
$829$	−16.8494	−0.585205	−0.292602	+	0.956234i	$0.594521\pi$
$830$	0	0
$831$	−76.2836	−2.64625
$832$	0	0
$833$	−4.60522	−0.159562
$834$	0	0
$835$	0	0
$836$	0	0
$837$	125.523	4.33870
$838$	0	0
$839$	41.5570	1.43471	0.717354	−	0.696709i	$-0.245353\pi$
$839$	41.5570	1.43471	0.717354	+	0.696709i	$0.245353\pi$
$840$	0	0
$841$	−20.9349	−0.721894
$842$	0	0
$843$	−51.8385	−1.78541
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.45249	−0.0842687
$848$	0	0
$849$	31.6485	1.08617
$850$	0	0
$851$	0.904987	0.0310225
$852$	0	0
$853$	30.6447	1.04925	0.524627	−	0.851332i	$-0.324205\pi$
$853$	30.6447	1.04925	0.524627	+	0.851332i	$0.324205\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	51.8794	1.77217	0.886083	−	0.463527i	$-0.153416\pi$
$857$	51.8794	1.77217	0.886083	+	0.463527i	$0.153416\pi$
$858$	0	0
$859$	−10.5007	−0.358279	−0.179140	−	0.983824i	$-0.557331\pi$
$859$	−10.5007	−0.358279	−0.179140	+	0.983824i	$0.557331\pi$
$860$	0	0
$861$	−46.3392	−1.57924
$862$	0	0
$863$	5.22517	0.177867	0.0889334	−	0.996038i	$-0.471654\pi$
$863$	5.22517	0.177867	0.0889334	+	0.996038i	$0.471654\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.29240	0.111816
$868$	0	0
$869$	−9.37436	−0.318003
$870$	0	0
$871$	23.1478	0.784333
$872$	0	0
$873$	−92.0789	−3.11640
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.38144	0.114183	0.0570916	−	0.998369i	$-0.481817\pi$
$877$	3.38144	0.114183	0.0570916	+	0.998369i	$0.481817\pi$
$878$	0	0
$879$	19.0490	0.642506
$880$	0	0
$881$	−38.6447	−1.30197	−0.650986	−	0.759090i	$-0.725644\pi$
$881$	−38.6447	−1.30197	−0.650986	+	0.759090i	$0.725644\pi$
$882$	0	0
$883$	18.4138	0.619674	0.309837	−	0.950790i	$-0.399726\pi$
$883$	18.4138	0.619674	0.309837	+	0.950790i	$0.399726\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−30.7171	−1.03138	−0.515690	−	0.856775i	$-0.672464\pi$
$887$	−30.7171	−1.03138	−0.515690	+	0.856775i	$0.672464\pi$
$888$	0	0
$889$	−21.5644	−0.723246
$890$	0	0
$891$	102.681	3.43993
$892$	0	0
$893$	33.1045	1.10780
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−11.4898	−0.383633
$898$	0	0
$899$	−22.3705	−0.746099
$900$	0	0
$901$	12.9145	0.430244
$902$	0	0
$903$	−10.1900	−0.339103
$904$	0	0
$905$	0	0
$906$	0	0
$907$	26.5219	0.880644	0.440322	−	0.897840i	$-0.354864\pi$
$907$	26.5219	0.880644	0.440322	+	0.897840i	$0.354864\pi$
$908$	0	0
$909$	−82.9839	−2.75240
$910$	0	0
$911$	−23.6967	−0.785106	−0.392553	−	0.919729i	$-0.628408\pi$
$911$	−23.6967	−0.785106	−0.392553	+	0.919729i	$0.628408\pi$
$912$	0	0
$913$	8.90499	0.294712
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−26.6594	−0.880371
$918$	0	0
$919$	27.2295	0.898218	0.449109	−	0.893477i	$-0.351742\pi$
$919$	27.2295	0.898218	0.449109	+	0.893477i	$0.351742\pi$
$920$	0	0
$921$	94.1127	3.10112
$922$	0	0
$923$	−10.7843	−0.354971
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−75.3041	−2.47331
$928$	0	0
$929$	1.09501	0.0359262	0.0179631	−	0.999839i	$-0.494282\pi$
$929$	1.09501	0.0359262	0.0179631	+	0.999839i	$0.494282\pi$
$930$	0	0
$931$	22.2889	0.730488
$932$	0	0
$933$	84.7383	2.77421
$934$	0	0
$935$	0	0
$936$	0	0
$937$	19.6798	0.642912	0.321456	−	0.946925i	$-0.395828\pi$
$937$	19.6798	0.642912	0.321456	+	0.946925i	$0.395828\pi$
$938$	0	0
$939$	−56.7748	−1.85278
$940$	0	0
$941$	42.5606	1.38743	0.693717	−	0.720247i	$-0.255972\pi$
$941$	42.5606	1.38743	0.693717	+	0.720247i	$0.255972\pi$
$942$	0	0
$943$	14.0746	0.458332
$944$	0	0
$945$	0	0
$946$	0	0
$947$	5.72802	0.186136	0.0930678	−	0.995660i	$-0.470333\pi$
$947$	5.72802	0.186136	0.0930678	+	0.995660i	$0.470333\pi$
$948$	0	0
$949$	−15.4247	−0.500707
$950$	0	0
$951$	−31.7354	−1.02909
$952$	0	0
$953$	−7.85081	−0.254313	−0.127156	−	0.991883i	$-0.540585\pi$
$953$	−7.85081	−0.254313	−0.127156	+	0.991883i	$0.540585\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−33.1696	−1.07222
$958$	0	0
$959$	−13.1696	−0.425269
$960$	0	0
$961$	31.0504	1.00162
$962$	0	0
$963$	150.740	4.85754
$964$	0	0
$965$	0	0
$966$	0	0
$967$	12.3202	0.396190	0.198095	−	0.980183i	$-0.436524\pi$
$967$	12.3202	0.396190	0.198095	+	0.980183i	$0.436524\pi$
$968$	0	0
$969$	−15.9349	−0.511903
$970$	0	0
$971$	−4.69072	−0.150532	−0.0752662	−	0.997163i	$-0.523981\pi$
$971$	−4.69072	−0.150532	−0.0752662	+	0.997163i	$0.523981\pi$
$972$	0	0
$973$	−0.700237	−0.0224486
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−19.8290	−0.634386	−0.317193	−	0.948361i	$-0.602740\pi$
$977$	−19.8290	−0.634386	−0.317193	+	0.948361i	$0.602740\pi$
$978$	0	0
$979$	63.8889	2.04190
$980$	0	0
$981$	77.8889	2.48680
$982$	0	0
$983$	2.86630	0.0914207	0.0457104	−	0.998955i	$-0.485445\pi$
$983$	2.86630	0.0914207	0.0457104	+	0.998955i	$0.485445\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	34.8494	1.10927
$988$	0	0
$989$	3.09501	0.0984157
$990$	0	0
$991$	38.3319	1.21765	0.608826	−	0.793304i	$-0.291641\pi$
$991$	38.3319	1.21765	0.608826	+	0.793304i	$0.291641\pi$
$992$	0	0
$993$	−44.4451	−1.41042
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−20.3055	−0.643080	−0.321540	−	0.946896i	$-0.604201\pi$
$997$	−20.3055	−0.643080	−0.321540	+	0.946896i	$0.604201\pi$
$998$	0	0
$999$	9.31879	0.294834

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.bs.1.3		3
4.3	odd	2		3400.2.a.k.1.1		3
5.4	even	2		1360.2.a.p.1.1		3
20.3	even	4		3400.2.e.i.2449.1		6
20.7	even	4		3400.2.e.i.2449.6		6
20.19	odd	2		680.2.a.h.1.3	✓	3
40.19	odd	2		5440.2.a.bn.1.1		3
40.29	even	2		5440.2.a.bu.1.3		3
60.59	even	2		6120.2.a.bq.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
680.2.a.h.1.3	✓	3	20.19	odd	2
1360.2.a.p.1.1		3	5.4	even	2
3400.2.a.k.1.1		3	4.3	odd	2
3400.2.e.i.2449.1		6	20.3	even	4
3400.2.e.i.2449.6		6	20.7	even	4
5440.2.a.bn.1.1		3	40.19	odd	2
5440.2.a.bu.1.3		3	40.29	even	2
6120.2.a.bq.1.1		3	60.59	even	2
6800.2.a.bs.1.3		3	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 6800 = 2^{4} \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6800.a (trivial)

\(f(q)\)	\(=\)	\(q+3.29240 q^{3} -1.54751 q^{7} +7.83991 q^{9} +3.54751 q^{11} -2.25511 q^{13} +1.00000 q^{17} -4.83991 q^{19} -5.09501 q^{21} +1.54751 q^{23} +15.9349 q^{27} -2.83991 q^{29} +7.87720 q^{31} +11.6798 q^{33} +0.584803 q^{37} -7.42471 q^{39} +9.09501 q^{41} +2.00000 q^{43} -6.83991 q^{47} -4.60522 q^{49} +3.29240 q^{51} +12.9145 q^{53} -15.9349 q^{57} +13.3501 q^{59} +3.74489 q^{61} -12.1323 q^{63} -10.2646 q^{67} +5.09501 q^{69} +4.78219 q^{71} +6.83991 q^{73} -5.48979 q^{77} -2.64252 q^{79} +28.9444 q^{81} +2.51021 q^{83} -9.35012 q^{87} +18.0095 q^{89} +3.48979 q^{91} +25.9349 q^{93} -11.7449 q^{97} +27.8121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 4 q^{7} + 8 q^{9} + 2 q^{11} - 5 q^{13} + 3 q^{17} + q^{19} + 2 q^{21} - 4 q^{23} + 15 q^{27} + 7 q^{29} + 3 q^{31} + 4 q^{33} - 12 q^{37} + 7 q^{39} + 10 q^{41} + 6 q^{43} - 5 q^{47} + 7 q^{49}+ \cdots + 30 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 4 * q^7 + 8 * q^9 + 2 * q^11 - 5 * q^13 + 3 * q^17 + q^19 + 2 * q^21 - 4 * q^23 + 15 * q^27 + 7 * q^29 + 3 * q^31 + 4 * q^33 - 12 * q^37 + 7 * q^39 + 10 * q^41 + 6 * q^43 - 5 * q^47 + 7 * q^49 + 3 * q^51 + 13 * q^53 - 15 * q^57 + 21 * q^59 + 13 * q^61 - 14 * q^63 + 14 * q^67 - 2 * q^69 + 11 * q^71 + 5 * q^73 - 20 * q^77 + 18 * q^79 + 11 * q^81 + 4 * q^83 - 9 * q^87 + 11 * q^89 + 14 * q^91 + 45 * q^93 - 37 * q^97 + 30 * q^99

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