LMFDB - Embedded newform 6800.2.a.cg.1.2

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)

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);

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")

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 = [5,0,3,0,0,0,1,0,6,0,6,0,-3] 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:	$5$
Coefficient field:	5.5.1981136.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 10x^{2} + 18x - 2 $ x^5 - 2x^4 - 8x^3 + 10x^2 + 18x - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3400)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.50269$ 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)

$f(q)$	$=$	$q-1.50269 q^{3} -4.25913 q^{7} -0.741913 q^{9} +5.60408 q^{11} -5.09704 q^{13} +1.00000 q^{17} +3.67930 q^{19} +6.40016 q^{21} -2.08252 q^{23} +5.62295 q^{27} -5.88207 q^{29} +8.50808 q^{31} -8.42121 q^{33} -2.48191 q^{37} +7.65929 q^{39} -10.5633 q^{41} -10.0848 q^{43} +1.28105 q^{47} +11.1402 q^{49} -1.50269 q^{51} -1.32070 q^{53} -5.52886 q^{57} -11.7607 q^{59} -12.0814 q^{61} +3.15990 q^{63} -4.52695 q^{67} +3.12939 q^{69} +0.0273119 q^{71} -7.67930 q^{73} -23.8685 q^{77} -7.94712 q^{79} -6.22383 q^{81} +12.8840 q^{83} +8.83895 q^{87} +12.3228 q^{89} +21.7089 q^{91} -12.7850 q^{93} +5.70487 q^{97} -4.15774 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{3} + q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + 5 q^{17} + 6 q^{19} + 7 q^{21} + 10 q^{23} + 15 q^{27} + 6 q^{29} + 11 q^{31} - 20 q^{33} - 2 q^{37} - 9 q^{39} + 4 q^{41} + 8 q^{43} + 10 q^{47}+ \cdots - 10 q^{99}+O(q^{100}) $ 5 * q + 3 * q^3 + q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + 5 * q^17 + 6 * q^19 + 7 * q^21 + 10 * q^23 + 15 * q^27 + 6 * q^29 + 11 * q^31 - 20 * q^33 - 2 * q^37 - 9 * q^39 + 4 * q^41 + 8 * q^43 + 10 * q^47 + 30 * q^49 + 3 * q^51 - 19 * q^53 + 4 * q^57 - 14 * q^61 + 22 * q^63 + 32 * q^69 + 5 * q^71 - 26 * q^73 + 4 * q^77 - 27 * q^79 - 11 * q^81 + 20 * q^83 + 12 * q^87 + 10 * q^89 + 63 * q^91 - 51 * q^93 + 24 * q^97 - 10 * 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
$3$	−1.50269	−0.867580	−0.433790	−	0.901014i	$-0.642824\pi$
$3$	−1.50269	−0.867580	−0.433790	+	0.901014i	$0.642824\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.25913	−1.60980	−0.804899	−	0.593411i	$-0.797781\pi$
$7$	−4.25913	−1.60980	−0.804899	+	0.593411i	$0.797781\pi$
$8$	0	0
$9$	−0.741913	−0.247304
$10$	0	0
$11$	5.60408	1.68969	0.844847	−	0.535008i	$-0.179691\pi$
$11$	5.60408	1.68969	0.844847	+	0.535008i	$0.179691\pi$
$12$	0	0
$13$	−5.09704	−1.41366	−0.706832	−	0.707381i	$-0.749876\pi$
$13$	−5.09704	−1.41366	−0.706832	+	0.707381i	$0.749876\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	3.67930	0.844089	0.422045	−	0.906575i	$-0.361313\pi$
$19$	3.67930	0.844089	0.422045	+	0.906575i	$0.361313\pi$
$20$	0	0
$21$	6.40016	1.39663
$22$	0	0
$23$	−2.08252	−0.434235	−0.217118	−	0.976145i	$-0.569666\pi$
$23$	−2.08252	−0.434235	−0.217118	+	0.976145i	$0.569666\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.62295	1.08214
$28$	0	0
$29$	−5.88207	−1.09227	−0.546137	−	0.837696i	$-0.683902\pi$
$29$	−5.88207	−1.09227	−0.546137	+	0.837696i	$0.683902\pi$
$30$	0	0
$31$	8.50808	1.52810	0.764048	−	0.645159i	$-0.223209\pi$
$31$	8.50808	1.52810	0.764048	+	0.645159i	$0.223209\pi$
$32$	0	0
$33$	−8.42121	−1.46595
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.48191	−0.408024	−0.204012	−	0.978968i	$-0.565398\pi$
$37$	−2.48191	−0.408024	−0.204012	+	0.978968i	$0.565398\pi$
$38$	0	0
$39$	7.65929	1.22647
$40$	0	0
$41$	−10.5633	−1.64971	−0.824854	−	0.565346i	$-0.808743\pi$
$41$	−10.5633	−1.64971	−0.824854	+	0.565346i	$0.808743\pi$
$42$	0	0
$43$	−10.0848	−1.53792	−0.768962	−	0.639294i	$-0.779227\pi$
$43$	−10.0848	−1.53792	−0.768962	+	0.639294i	$0.779227\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	1.28105	0.186861	0.0934303	−	0.995626i	$-0.470217\pi$
$47$	1.28105	0.186861	0.0934303	+	0.995626i	$0.470217\pi$
$48$	0	0
$49$	11.1402	1.59145
$50$	0	0
$51$	−1.50269	−0.210419
$52$	0	0
$53$	−1.32070	−0.181412	−0.0907060	−	0.995878i	$-0.528912\pi$
$53$	−1.32070	−0.181412	−0.0907060	+	0.995878i	$0.528912\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.52886	−0.732315
$58$	0	0
$59$	−11.7607	−1.53111	−0.765555	−	0.643371i	$-0.777535\pi$
$59$	−11.7607	−1.53111	−0.765555	+	0.643371i	$0.777535\pi$
$60$	0	0
$61$	−12.0814	−1.54686	−0.773431	−	0.633881i	$-0.781461\pi$
$61$	−12.0814	−1.54686	−0.773431	+	0.633881i	$0.781461\pi$
$62$	0	0
$63$	3.15990	0.398110
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−4.52695	−0.553055	−0.276527	−	0.961006i	$-0.589184\pi$
$67$	−4.52695	−0.553055	−0.276527	+	0.961006i	$0.589184\pi$
$68$	0	0
$69$	3.12939	0.376734
$70$	0	0
$71$	0.0273119	0.00324132	0.00162066	−	0.999999i	$-0.499484\pi$
$71$	0.0273119	0.00324132	0.00162066	+	0.999999i	$0.499484\pi$
$72$	0	0
$73$	−7.67930	−0.898794	−0.449397	−	0.893332i	$-0.648361\pi$
$73$	−7.67930	−0.898794	−0.449397	+	0.893332i	$0.648361\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−23.8685	−2.72007
$78$	0	0
$79$	−7.94712	−0.894121	−0.447060	−	0.894504i	$-0.647529\pi$
$79$	−7.94712	−0.894121	−0.447060	+	0.894504i	$0.647529\pi$
$80$	0	0
$81$	−6.22383	−0.691536
$82$	0	0
$83$	12.8840	1.41420	0.707101	−	0.707113i	$-0.250003\pi$
$83$	12.8840	1.41420	0.707101	+	0.707113i	$0.250003\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.83895	0.947635
$88$	0	0
$89$	12.3228	1.30621	0.653106	−	0.757267i	$-0.273466\pi$
$89$	12.3228	1.30621	0.653106	+	0.757267i	$0.273466\pi$
$90$	0	0
$91$	21.7089	2.27572
$92$	0	0
$93$	−12.7850	−1.32575
$94$	0	0
$95$	0	0
$96$	0	0
$97$	5.70487	0.579242	0.289621	−	0.957141i	$-0.406471\pi$
$97$	5.70487	0.579242	0.289621	+	0.957141i	$0.406471\pi$
$98$	0	0
$99$	−4.15774	−0.417869
$100$	0	0
$101$	5.06800	0.504285	0.252142	−	0.967690i	$-0.418865\pi$
$101$	5.06800	0.504285	0.252142	+	0.967690i	$0.418865\pi$
$102$	0	0
$103$	−5.35860	−0.527999	−0.263999	−	0.964523i	$-0.585042\pi$
$103$	−5.35860	−0.527999	−0.263999	+	0.964523i	$0.585042\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.2955	1.76869	0.884345	−	0.466834i	$-0.154606\pi$
$107$	18.2955	1.76869	0.884345	+	0.466834i	$0.154606\pi$
$108$	0	0
$109$	−10.4090	−0.997003	−0.498502	−	0.866889i	$-0.666116\pi$
$109$	−10.4090	−0.997003	−0.498502	+	0.866889i	$0.666116\pi$
$110$	0	0
$111$	3.72955	0.353994
$112$	0	0
$113$	13.7555	1.29400	0.647002	−	0.762488i	$-0.276022\pi$
$113$	13.7555	1.29400	0.647002	+	0.762488i	$0.276022\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	3.78156	0.349605
$118$	0	0
$119$	−4.25913	−0.390434
$120$	0	0
$121$	20.4057	1.85506
$122$	0	0
$123$	15.8734	1.43125
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.2014	−0.905226	−0.452613	−	0.891707i	$-0.649508\pi$
$127$	−10.2014	−0.905226	−0.452613	+	0.891707i	$0.649508\pi$
$128$	0	0
$129$	15.1544	1.33427
$130$	0	0
$131$	−9.30955	−0.813379	−0.406689	−	0.913566i	$-0.633317\pi$
$131$	−9.30955	−0.813379	−0.406689	+	0.913566i	$0.633317\pi$
$132$	0	0
$133$	−15.6706	−1.35881
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.49181	0.298326	0.149163	−	0.988813i	$-0.452342\pi$
$137$	3.49181	0.298326	0.149163	+	0.988813i	$0.452342\pi$
$138$	0	0
$139$	−12.0244	−1.01990	−0.509949	−	0.860205i	$-0.670336\pi$
$139$	−12.0244	−1.01990	−0.509949	+	0.860205i	$0.670336\pi$
$140$	0	0
$141$	−1.92503	−0.162117
$142$	0	0
$143$	−28.5642	−2.38866
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−16.7402	−1.38071
$148$	0	0
$149$	−9.01077	−0.738191	−0.369096	−	0.929391i	$-0.620333\pi$
$149$	−9.01077	−0.738191	−0.369096	+	0.929391i	$0.620333\pi$
$150$	0	0
$151$	−7.60293	−0.618718	−0.309359	−	0.950945i	$-0.600114\pi$
$151$	−7.60293	−0.618718	−0.309359	+	0.950945i	$0.600114\pi$
$152$	0	0
$153$	−0.741913	−0.0599801
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.0989	1.28483	0.642417	−	0.766355i	$-0.277932\pi$
$157$	16.0989	1.28483	0.642417	+	0.766355i	$0.277932\pi$
$158$	0	0
$159$	1.98461	0.157390
$160$	0	0
$161$	8.86972	0.699031
$162$	0	0
$163$	13.0190	1.01973	0.509865	−	0.860255i	$-0.329695\pi$
$163$	13.0190	1.01973	0.509865	+	0.860255i	$0.329695\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	21.8272	1.68904	0.844520	−	0.535524i	$-0.179886\pi$
$167$	21.8272	1.68904	0.844520	+	0.535524i	$0.179886\pi$
$168$	0	0
$169$	12.9798	0.998448
$170$	0	0
$171$	−2.72972	−0.208747
$172$	0	0
$173$	12.7245	0.967426	0.483713	−	0.875227i	$-0.339288\pi$
$173$	12.7245	0.967426	0.483713	+	0.875227i	$0.339288\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	17.6727	1.32836
$178$	0	0
$179$	15.1085	1.12926	0.564631	−	0.825343i	$-0.309018\pi$
$179$	15.1085	1.12926	0.564631	+	0.825343i	$0.309018\pi$
$180$	0	0
$181$	25.6085	1.90346	0.951732	−	0.306930i	$-0.0993019\pi$
$181$	25.6085	1.90346	0.951732	+	0.306930i	$0.0993019\pi$
$182$	0	0
$183$	18.1546	1.34203
$184$	0	0
$185$	0	0
$186$	0	0
$187$	5.60408	0.409811
$188$	0	0
$189$	−23.9488	−1.74202
$190$	0	0
$191$	13.0708	0.945768	0.472884	−	0.881125i	$-0.343213\pi$
$191$	13.0708	0.945768	0.472884	+	0.881125i	$0.343213\pi$
$192$	0	0
$193$	−3.24242	−0.233395	−0.116697	−	0.993168i	$-0.537231\pi$
$193$	−3.24242	−0.233395	−0.116697	+	0.993168i	$0.537231\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.32956	−0.379715	−0.189858	−	0.981812i	$-0.560803\pi$
$197$	−5.32956	−0.379715	−0.189858	+	0.981812i	$0.560803\pi$
$198$	0	0
$199$	−23.8505	−1.69072	−0.845358	−	0.534200i	$-0.820613\pi$
$199$	−23.8505	−1.69072	−0.845358	+	0.534200i	$0.820613\pi$
$200$	0	0
$201$	6.80261	0.479819
$202$	0	0
$203$	25.0525	1.75834
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.54505	0.107388
$208$	0	0
$209$	20.6191	1.42625
$210$	0	0
$211$	24.6612	1.69775	0.848874	−	0.528596i	$-0.177281\pi$
$211$	24.6612	1.69775	0.848874	+	0.528596i	$0.177281\pi$
$212$	0	0
$213$	−0.0410414	−0.00281211
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−36.2370	−2.45993
$218$	0	0
$219$	11.5396	0.779776
$220$	0	0
$221$	−5.09704	−0.342864
$222$	0	0
$223$	3.05389	0.204504	0.102252	−	0.994759i	$-0.467395\pi$
$223$	3.05389	0.204504	0.102252	+	0.994759i	$0.467395\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.46614	−0.495545	−0.247773	−	0.968818i	$-0.579699\pi$
$227$	−7.46614	−0.495545	−0.247773	+	0.968818i	$0.579699\pi$
$228$	0	0
$229$	14.7096	0.972035	0.486018	−	0.873949i	$-0.338449\pi$
$229$	14.7096	0.972035	0.486018	+	0.873949i	$0.338449\pi$
$230$	0	0
$231$	35.8670	2.35988
$232$	0	0
$233$	−15.1718	−0.993939	−0.496969	−	0.867768i	$-0.665554\pi$
$233$	−15.1718	−0.993939	−0.496969	+	0.867768i	$0.665554\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.9421	0.775722
$238$	0	0
$239$	7.88068	0.509759	0.254879	−	0.966973i	$-0.417964\pi$
$239$	7.88068	0.509759	0.254879	+	0.966973i	$0.417964\pi$
$240$	0	0
$241$	−25.9292	−1.67025	−0.835123	−	0.550063i	$-0.814604\pi$
$241$	−25.9292	−1.67025	−0.835123	+	0.550063i	$0.814604\pi$
$242$	0	0
$243$	−7.51634	−0.482173
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−18.7535	−1.19326
$248$	0	0
$249$	−19.3607	−1.22693
$250$	0	0
$251$	26.3969	1.66615	0.833077	−	0.553157i	$-0.186577\pi$
$251$	26.3969	1.66615	0.833077	+	0.553157i	$0.186577\pi$
$252$	0	0
$253$	−11.6706	−0.733725
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−7.58765	−0.473304	−0.236652	−	0.971594i	$-0.576050\pi$
$257$	−7.58765	−0.473304	−0.236652	+	0.971594i	$0.576050\pi$
$258$	0	0
$259$	10.5708	0.656836
$260$	0	0
$261$	4.36399	0.270124
$262$	0	0
$263$	12.7210	0.784412	0.392206	−	0.919877i	$-0.371712\pi$
$263$	12.7210	0.784412	0.392206	+	0.919877i	$0.371712\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−18.5174	−1.13324
$268$	0	0
$269$	4.20764	0.256544	0.128272	−	0.991739i	$-0.459057\pi$
$269$	4.20764	0.256544	0.128272	+	0.991739i	$0.459057\pi$
$270$	0	0
$271$	−18.4090	−1.11827	−0.559134	−	0.829077i	$-0.688866\pi$
$271$	−18.4090	−1.11827	−0.559134	+	0.829077i	$0.688866\pi$
$272$	0	0
$273$	−32.6219	−1.97437
$274$	0	0
$275$	0	0
$276$	0	0
$277$	23.7386	1.42631	0.713157	−	0.701005i	$-0.247265\pi$
$277$	23.7386	1.42631	0.713157	+	0.701005i	$0.247265\pi$
$278$	0	0
$279$	−6.31225	−0.377905
$280$	0	0
$281$	5.92784	0.353625	0.176813	−	0.984245i	$-0.443421\pi$
$281$	5.92784	0.353625	0.176813	+	0.984245i	$0.443421\pi$
$282$	0	0
$283$	11.2070	0.666188	0.333094	−	0.942894i	$-0.391907\pi$
$283$	11.2070	0.666188	0.333094	+	0.942894i	$0.391907\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	44.9904	2.65570
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−8.57267	−0.502539
$292$	0	0
$293$	−13.0722	−0.763684	−0.381842	−	0.924228i	$-0.624710\pi$
$293$	−13.0722	−0.763684	−0.381842	+	0.924228i	$0.624710\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	31.5114	1.82848
$298$	0	0
$299$	10.6147	0.613863
$300$	0	0
$301$	42.9526	2.47575
$302$	0	0
$303$	−7.61565	−0.437508
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−17.0804	−0.974828	−0.487414	−	0.873171i	$-0.662060\pi$
$307$	−17.0804	−0.974828	−0.487414	+	0.873171i	$0.662060\pi$
$308$	0	0
$309$	8.05233	0.458081
$310$	0	0
$311$	10.2710	0.582418	0.291209	−	0.956660i	$-0.405943\pi$
$311$	10.2710	0.582418	0.291209	+	0.956660i	$0.405943\pi$
$312$	0	0
$313$	−4.09096	−0.231235	−0.115617	−	0.993294i	$-0.536885\pi$
$313$	−4.09096	−0.231235	−0.115617	+	0.993294i	$0.536885\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−5.82279	−0.327041	−0.163520	−	0.986540i	$-0.552285\pi$
$317$	−5.82279	−0.327041	−0.163520	+	0.986540i	$0.552285\pi$
$318$	0	0
$319$	−32.9636	−1.84561
$320$	0	0
$321$	−27.4925	−1.53448
$322$	0	0
$323$	3.67930	0.204722
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.6416	0.864981
$328$	0	0
$329$	−5.45616	−0.300808
$330$	0	0
$331$	21.9561	1.20682	0.603410	−	0.797431i	$-0.293808\pi$
$331$	21.9561	1.20682	0.603410	+	0.797431i	$0.293808\pi$
$332$	0	0
$333$	1.84136	0.100906
$334$	0	0
$335$	0	0
$336$	0	0
$337$	7.15965	0.390011	0.195006	−	0.980802i	$-0.437528\pi$
$337$	7.15965	0.390011	0.195006	+	0.980802i	$0.437528\pi$
$338$	0	0
$339$	−20.6702	−1.12265
$340$	0	0
$341$	47.6800	2.58201
$342$	0	0
$343$	−17.6335	−0.952118
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.3561	1.09277	0.546387	−	0.837533i	$-0.316003\pi$
$347$	20.3561	1.09277	0.546387	+	0.837533i	$0.316003\pi$
$348$	0	0
$349$	−18.7735	−1.00492	−0.502462	−	0.864599i	$-0.667572\pi$
$349$	−18.7735	−1.00492	−0.502462	+	0.864599i	$0.667572\pi$
$350$	0	0
$351$	−28.6604	−1.52978
$352$	0	0
$353$	−31.7716	−1.69103	−0.845516	−	0.533950i	$-0.820707\pi$
$353$	−31.7716	−1.69103	−0.845516	+	0.533950i	$0.820707\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	6.40016	0.338732
$358$	0	0
$359$	6.52347	0.344296	0.172148	−	0.985071i	$-0.444929\pi$
$359$	6.52347	0.344296	0.172148	+	0.985071i	$0.444929\pi$
$360$	0	0
$361$	−5.46275	−0.287513
$362$	0	0
$363$	−30.6635	−1.60942
$364$	0	0
$365$	0	0
$366$	0	0
$367$	17.3578	0.906070	0.453035	−	0.891493i	$-0.350341\pi$
$367$	17.3578	0.906070	0.453035	+	0.891493i	$0.350341\pi$
$368$	0	0
$369$	7.83704	0.407980
$370$	0	0
$371$	5.62503	0.292037
$372$	0	0
$373$	−7.63878	−0.395521	−0.197761	−	0.980250i	$-0.563367\pi$
$373$	−7.63878	−0.395521	−0.197761	+	0.980250i	$0.563367\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	29.9812	1.54411
$378$	0	0
$379$	22.0927	1.13483	0.567413	−	0.823434i	$-0.307944\pi$
$379$	22.0927	1.13483	0.567413	+	0.823434i	$0.307944\pi$
$380$	0	0
$381$	15.3295	0.785356
$382$	0	0
$383$	12.9203	0.660198	0.330099	−	0.943946i	$-0.392918\pi$
$383$	12.9203	0.660198	0.330099	+	0.943946i	$0.392918\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.48208	0.380335
$388$	0	0
$389$	1.35513	0.0687077	0.0343538	−	0.999410i	$-0.489063\pi$
$389$	1.35513	0.0687077	0.0343538	+	0.999410i	$0.489063\pi$
$390$	0	0
$391$	−2.08252	−0.105318
$392$	0	0
$393$	13.9894	0.705672
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.68991	−0.235380	−0.117690	−	0.993050i	$-0.537549\pi$
$397$	−4.68991	−0.235380	−0.117690	+	0.993050i	$0.537549\pi$
$398$	0	0
$399$	23.5481	1.17888
$400$	0	0
$401$	18.6534	0.931504	0.465752	−	0.884915i	$-0.345784\pi$
$401$	18.6534	0.931504	0.465752	+	0.884915i	$0.345784\pi$
$402$	0	0
$403$	−43.3660	−2.16022
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−13.9088	−0.689436
$408$	0	0
$409$	21.0675	1.04172	0.520859	−	0.853642i	$-0.325612\pi$
$409$	21.0675	1.04172	0.520859	+	0.853642i	$0.325612\pi$
$410$	0	0
$411$	−5.24713	−0.258822
$412$	0	0
$413$	50.0902	2.46478
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.0690	0.884843
$418$	0	0
$419$	33.9507	1.65860	0.829300	−	0.558804i	$-0.188740\pi$
$419$	33.9507	1.65860	0.829300	+	0.558804i	$0.188740\pi$
$420$	0	0
$421$	−20.6895	−1.00835	−0.504173	−	0.863603i	$-0.668203\pi$
$421$	−20.6895	−1.00835	−0.504173	+	0.863603i	$0.668203\pi$
$422$	0	0
$423$	−0.950429	−0.0462115
$424$	0	0
$425$	0	0
$426$	0	0
$427$	51.4561	2.49014
$428$	0	0
$429$	42.9233	2.07235
$430$	0	0
$431$	21.2599	1.02405	0.512027	−	0.858970i	$-0.328895\pi$
$431$	21.2599	1.02405	0.512027	+	0.858970i	$0.328895\pi$
$432$	0	0
$433$	22.7709	1.09430	0.547151	−	0.837034i	$-0.315713\pi$
$433$	22.7709	1.09430	0.547151	+	0.837034i	$0.315713\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7.66222	−0.366533
$438$	0	0
$439$	−24.6182	−1.17496	−0.587482	−	0.809237i	$-0.699881\pi$
$439$	−24.6182	−1.17496	−0.587482	+	0.809237i	$0.699881\pi$
$440$	0	0
$441$	−8.26503	−0.393573
$442$	0	0
$443$	−9.80661	−0.465926	−0.232963	−	0.972486i	$-0.574842\pi$
$443$	−9.80661	−0.465926	−0.232963	+	0.972486i	$0.574842\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	13.5404	0.640440
$448$	0	0
$449$	7.96832	0.376048	0.188024	−	0.982164i	$-0.439792\pi$
$449$	7.96832	0.376048	0.188024	+	0.982164i	$0.439792\pi$
$450$	0	0
$451$	−59.1975	−2.78750
$452$	0	0
$453$	11.4249	0.536788
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−22.6640	−1.06018	−0.530088	−	0.847943i	$-0.677841\pi$
$457$	−22.6640	−1.06018	−0.530088	+	0.847943i	$0.677841\pi$
$458$	0	0
$459$	5.62295	0.262457
$460$	0	0
$461$	−7.03235	−0.327529	−0.163765	−	0.986499i	$-0.552364\pi$
$461$	−7.03235	−0.327529	−0.163765	+	0.986499i	$0.552364\pi$
$462$	0	0
$463$	−15.3351	−0.712683	−0.356342	−	0.934356i	$-0.615976\pi$
$463$	−15.3351	−0.712683	−0.356342	+	0.934356i	$0.615976\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11.0414	−0.510934	−0.255467	−	0.966818i	$-0.582229\pi$
$467$	−11.0414	−0.510934	−0.255467	+	0.966818i	$0.582229\pi$
$468$	0	0
$469$	19.2808	0.890307
$470$	0	0
$471$	−24.1918	−1.11470
$472$	0	0
$473$	−56.5163	−2.59862
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0.979844	0.0448640
$478$	0	0
$479$	−1.02113	−0.0466566	−0.0233283	−	0.999728i	$-0.507426\pi$
$479$	−1.02113	−0.0466566	−0.0233283	+	0.999728i	$0.507426\pi$
$480$	0	0
$481$	12.6504	0.576809
$482$	0	0
$483$	−13.3285	−0.606466
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−25.2759	−1.14536	−0.572680	−	0.819779i	$-0.694096\pi$
$487$	−25.2759	−1.14536	−0.572680	+	0.819779i	$0.694096\pi$
$488$	0	0
$489$	−19.5636	−0.884697
$490$	0	0
$491$	−4.07984	−0.184121	−0.0920603	−	0.995753i	$-0.529345\pi$
$491$	−4.07984	−0.184121	−0.0920603	+	0.995753i	$0.529345\pi$
$492$	0	0
$493$	−5.88207	−0.264915
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−0.116325	−0.00521788
$498$	0	0
$499$	16.8340	0.753593	0.376797	−	0.926296i	$-0.377026\pi$
$499$	16.8340	0.753593	0.376797	+	0.926296i	$0.377026\pi$
$500$	0	0
$501$	−32.7996	−1.46538
$502$	0	0
$503$	23.1652	1.03289	0.516444	−	0.856321i	$-0.327256\pi$
$503$	23.1652	1.03289	0.516444	+	0.856321i	$0.327256\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−19.5047	−0.866234
$508$	0	0
$509$	1.05059	0.0465665	0.0232832	−	0.999729i	$-0.492588\pi$
$509$	1.05059	0.0465665	0.0232832	+	0.999729i	$0.492588\pi$
$510$	0	0
$511$	32.7071	1.44688
$512$	0	0
$513$	20.6885	0.913420
$514$	0	0
$515$	0	0
$516$	0	0
$517$	7.17912	0.315737
$518$	0	0
$519$	−19.1210	−0.839320
$520$	0	0
$521$	−14.9831	−0.656421	−0.328210	−	0.944605i	$-0.606445\pi$
$521$	−14.9831	−0.656421	−0.328210	+	0.944605i	$0.606445\pi$
$522$	0	0
$523$	2.71233	0.118602	0.0593010	−	0.998240i	$-0.481113\pi$
$523$	2.71233	0.118602	0.0593010	+	0.998240i	$0.481113\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.50808	0.370618
$528$	0	0
$529$	−18.6631	−0.811440
$530$	0	0
$531$	8.72540	0.378650
$532$	0	0
$533$	53.8415	2.33213
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−22.7034	−0.979726
$538$	0	0
$539$	62.4304	2.68907
$540$	0	0
$541$	28.7748	1.23712	0.618562	−	0.785736i	$-0.287716\pi$
$541$	28.7748	1.23712	0.618562	+	0.785736i	$0.287716\pi$
$542$	0	0
$543$	−38.4817	−1.65141
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−18.0706	−0.772643	−0.386322	−	0.922364i	$-0.626254\pi$
$547$	−18.0706	−0.772643	−0.386322	+	0.922364i	$0.626254\pi$
$548$	0	0
$549$	8.96333	0.382546
$550$	0	0
$551$	−21.6419	−0.921977
$552$	0	0
$553$	33.8478	1.43935
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−11.3000	−0.478797	−0.239399	−	0.970921i	$-0.576950\pi$
$557$	−11.3000	−0.478797	−0.239399	+	0.970921i	$0.576950\pi$
$558$	0	0
$559$	51.4029	2.17411
$560$	0	0
$561$	−8.42121	−0.355544
$562$	0	0
$563$	−32.8640	−1.38505	−0.692525	−	0.721394i	$-0.743502\pi$
$563$	−32.8640	−1.38505	−0.692525	+	0.721394i	$0.743502\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	26.5081	1.11323
$568$	0	0
$569$	23.5100	0.985592	0.492796	−	0.870145i	$-0.335975\pi$
$569$	23.5100	0.985592	0.492796	+	0.870145i	$0.335975\pi$
$570$	0	0
$571$	−22.3428	−0.935018	−0.467509	−	0.883988i	$-0.654849\pi$
$571$	−22.3428	−0.935018	−0.467509	+	0.883988i	$0.654849\pi$
$572$	0	0
$573$	−19.6414	−0.820529
$574$	0	0
$575$	0	0
$576$	0	0
$577$	26.1662	1.08931	0.544657	−	0.838659i	$-0.316660\pi$
$577$	26.1662	1.08931	0.544657	+	0.838659i	$0.316660\pi$
$578$	0	0
$579$	4.87236	0.202489
$580$	0	0
$581$	−54.8745	−2.27658
$582$	0	0
$583$	−7.40131	−0.306531
$584$	0	0
$585$	0	0
$586$	0	0
$587$	2.67859	0.110557	0.0552785	−	0.998471i	$-0.482395\pi$
$587$	2.67859	0.110557	0.0552785	+	0.998471i	$0.482395\pi$
$588$	0	0
$589$	31.3038	1.28985
$590$	0	0
$591$	8.00869	0.329434
$592$	0	0
$593$	26.8528	1.10271	0.551357	−	0.834270i	$-0.314110\pi$
$593$	26.8528	1.10271	0.551357	+	0.834270i	$0.314110\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	35.8400	1.46683
$598$	0	0
$599$	−33.0060	−1.34859	−0.674295	−	0.738462i	$-0.735552\pi$
$599$	−33.0060	−1.34859	−0.674295	+	0.738462i	$0.735552\pi$
$600$	0	0
$601$	−17.1268	−0.698616	−0.349308	−	0.937008i	$-0.613583\pi$
$601$	−17.1268	−0.698616	−0.349308	+	0.937008i	$0.613583\pi$
$602$	0	0
$603$	3.35860	0.136773
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.3785	1.67950	0.839750	−	0.542973i	$-0.182702\pi$
$607$	41.3785	1.67950	0.839750	+	0.542973i	$0.182702\pi$
$608$	0	0
$609$	−37.6462	−1.52550
$610$	0	0
$611$	−6.52957	−0.264158
$612$	0	0
$613$	30.1547	1.21794	0.608969	−	0.793194i	$-0.291583\pi$
$613$	30.1547	1.21794	0.608969	+	0.793194i	$0.291583\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.8901	1.28385	0.641923	−	0.766769i	$-0.278137\pi$
$617$	31.8901	1.28385	0.641923	+	0.766769i	$0.278137\pi$
$618$	0	0
$619$	14.8764	0.597935	0.298967	−	0.954263i	$-0.403358\pi$
$619$	14.8764	0.597935	0.298967	+	0.954263i	$0.403358\pi$
$620$	0	0
$621$	−11.7099	−0.469902
$622$	0	0
$623$	−52.4843	−2.10274
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−30.9842	−1.23739
$628$	0	0
$629$	−2.48191	−0.0989604
$630$	0	0
$631$	9.33949	0.371799	0.185900	−	0.982569i	$-0.440480\pi$
$631$	9.33949	0.371799	0.185900	+	0.982569i	$0.440480\pi$
$632$	0	0
$633$	−37.0582	−1.47293
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−56.7818	−2.24978
$638$	0	0
$639$	−0.0202630	−0.000801593 0
$640$	0	0
$641$	−3.28606	−0.129792	−0.0648958	−	0.997892i	$-0.520672\pi$
$641$	−3.28606	−0.129792	−0.0648958	+	0.997892i	$0.520672\pi$
$642$	0	0
$643$	1.78820	0.0705196	0.0352598	−	0.999378i	$-0.488774\pi$
$643$	1.78820	0.0705196	0.0352598	+	0.999378i	$0.488774\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.04558	0.0804200	0.0402100	−	0.999191i	$-0.487197\pi$
$647$	2.04558	0.0804200	0.0402100	+	0.999191i	$0.487197\pi$
$648$	0	0
$649$	−65.9078	−2.58711
$650$	0	0
$651$	54.4531	2.13418
$652$	0	0
$653$	2.36051	0.0923740	0.0461870	−	0.998933i	$-0.485293\pi$
$653$	2.36051	0.0923740	0.0461870	+	0.998933i	$0.485293\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.69737	0.222276
$658$	0	0
$659$	−3.25145	−0.126658	−0.0633292	−	0.997993i	$-0.520172\pi$
$659$	−3.25145	−0.126658	−0.0633292	+	0.997993i	$0.520172\pi$
$660$	0	0
$661$	−2.78242	−0.108223	−0.0541117	−	0.998535i	$-0.517233\pi$
$661$	−2.78242	−0.108223	−0.0541117	+	0.998535i	$0.517233\pi$
$662$	0	0
$663$	7.65929	0.297462
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12.2495	0.474304
$668$	0	0
$669$	−4.58907	−0.177424
$670$	0	0
$671$	−67.7050	−2.61372
$672$	0	0
$673$	−17.4485	−0.672589	−0.336294	−	0.941757i	$-0.609174\pi$
$673$	−17.4485	−0.672589	−0.336294	+	0.941757i	$0.609174\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	5.57095	0.214109	0.107054	−	0.994253i	$-0.465858\pi$
$677$	5.57095	0.214109	0.107054	+	0.994253i	$0.465858\pi$
$678$	0	0
$679$	−24.2978	−0.932462
$680$	0	0
$681$	11.2193	0.429925
$682$	0	0
$683$	32.0094	1.22480	0.612402	−	0.790546i	$-0.290203\pi$
$683$	32.0094	1.22480	0.612402	+	0.790546i	$0.290203\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−22.1040	−0.843319
$688$	0	0
$689$	6.73166	0.256456
$690$	0	0
$691$	30.0817	1.14436	0.572181	−	0.820127i	$-0.306097\pi$
$691$	30.0817	1.14436	0.572181	+	0.820127i	$0.306097\pi$
$692$	0	0
$693$	17.7083	0.672684
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−10.5633	−0.400113
$698$	0	0
$699$	22.7986	0.862322
$700$	0	0
$701$	8.62366	0.325711	0.162856	−	0.986650i	$-0.447930\pi$
$701$	8.62366	0.325711	0.162856	+	0.986650i	$0.447930\pi$
$702$	0	0
$703$	−9.13170	−0.344409
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−21.5853	−0.811797
$708$	0	0
$709$	−11.9495	−0.448774	−0.224387	−	0.974500i	$-0.572038\pi$
$709$	−11.9495	−0.448774	−0.224387	+	0.974500i	$0.572038\pi$
$710$	0	0
$711$	5.89607	0.221120
$712$	0	0
$713$	−17.7182	−0.663553
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−11.8422	−0.442257
$718$	0	0
$719$	−0.453808	−0.0169242	−0.00846209	−	0.999964i	$-0.502694\pi$
$719$	−0.453808	−0.0169242	−0.00846209	+	0.999964i	$0.502694\pi$
$720$	0	0
$721$	22.8230	0.849971
$722$	0	0
$723$	38.9636	1.44907
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−33.8668	−1.25605	−0.628024	−	0.778194i	$-0.716136\pi$
$727$	−33.8668	−1.25605	−0.628024	+	0.778194i	$0.716136\pi$
$728$	0	0
$729$	29.9662	1.10986
$730$	0	0
$731$	−10.0848	−0.373002
$732$	0	0
$733$	−12.1335	−0.448162	−0.224081	−	0.974571i	$-0.571938\pi$
$733$	−12.1335	−0.448162	−0.224081	+	0.974571i	$0.571938\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.3694	−0.934493
$738$	0	0
$739$	22.0480	0.811049	0.405525	−	0.914084i	$-0.367089\pi$
$739$	22.0480	0.811049	0.405525	+	0.914084i	$0.367089\pi$
$740$	0	0
$741$	28.1808	1.03525
$742$	0	0
$743$	53.5023	1.96281	0.981404	−	0.191954i	$-0.0614823\pi$
$743$	53.5023	1.96281	0.981404	+	0.191954i	$0.0614823\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−9.55880	−0.349738
$748$	0	0
$749$	−77.9227	−2.84723
$750$	0	0
$751$	27.4545	1.00183	0.500914	−	0.865497i	$-0.332997\pi$
$751$	27.4545	1.00183	0.500914	+	0.865497i	$0.332997\pi$
$752$	0	0
$753$	−39.6664	−1.44552
$754$	0	0
$755$	0	0
$756$	0	0
$757$	35.2758	1.28212	0.641061	−	0.767490i	$-0.278495\pi$
$757$	35.2758	1.28212	0.641061	+	0.767490i	$0.278495\pi$
$758$	0	0
$759$	17.5373	0.636565
$760$	0	0
$761$	29.2373	1.05985	0.529926	−	0.848044i	$-0.322220\pi$
$761$	29.2373	1.05985	0.529926	+	0.848044i	$0.322220\pi$
$762$	0	0
$763$	44.3333	1.60497
$764$	0	0
$765$	0	0
$766$	0	0
$767$	59.9446	2.16448
$768$	0	0
$769$	−22.0766	−0.796104	−0.398052	−	0.917363i	$-0.630314\pi$
$769$	−22.0766	−0.796104	−0.398052	+	0.917363i	$0.630314\pi$
$770$	0	0
$771$	11.4019	0.410630
$772$	0	0
$773$	29.2667	1.05265	0.526326	−	0.850283i	$-0.323569\pi$
$773$	29.2667	1.05265	0.526326	+	0.850283i	$0.323569\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−15.8846	−0.569858
$778$	0	0
$779$	−38.8655	−1.39250
$780$	0	0
$781$	0.153058	0.00547684
$782$	0	0
$783$	−33.0746	−1.18199
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−7.40215	−0.263858	−0.131929	−	0.991259i	$-0.542117\pi$
$787$	−7.40215	−0.263858	−0.131929	+	0.991259i	$0.542117\pi$
$788$	0	0
$789$	−19.1158	−0.680541
$790$	0	0
$791$	−58.5862	−2.08309
$792$	0	0
$793$	61.5793	2.18674
$794$	0	0
$795$	0	0
$796$	0	0
$797$	51.5831	1.82717	0.913583	−	0.406652i	$-0.133304\pi$
$797$	51.5831	1.82717	0.913583	+	0.406652i	$0.133304\pi$
$798$	0	0
$799$	1.28105	0.0453204
$800$	0	0
$801$	−9.14243	−0.323032
$802$	0	0
$803$	−43.0354	−1.51869
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6.32279	−0.222573
$808$	0	0
$809$	−30.8840	−1.08582	−0.542912	−	0.839790i	$-0.682678\pi$
$809$	−30.8840	−1.08582	−0.542912	+	0.839790i	$0.682678\pi$
$810$	0	0
$811$	−1.66218	−0.0583670	−0.0291835	−	0.999574i	$-0.509291\pi$
$811$	−1.66218	−0.0583670	−0.0291835	+	0.999574i	$0.509291\pi$
$812$	0	0
$813$	27.6631	0.970188
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−37.1052	−1.29815
$818$	0	0
$819$	−16.1061	−0.562794
$820$	0	0
$821$	39.9351	1.39374	0.696872	−	0.717195i	$-0.254574\pi$
$821$	39.9351	1.39374	0.696872	+	0.717195i	$0.254574\pi$
$822$	0	0
$823$	2.20540	0.0768754	0.0384377	−	0.999261i	$-0.487762\pi$
$823$	2.20540	0.0768754	0.0384377	+	0.999261i	$0.487762\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.1644	−0.492546	−0.246273	−	0.969201i	$-0.579206\pi$
$827$	−14.1644	−0.492546	−0.246273	+	0.969201i	$0.579206\pi$
$828$	0	0
$829$	34.9292	1.21314	0.606570	−	0.795030i	$-0.292545\pi$
$829$	34.9292	1.21314	0.606570	+	0.795030i	$0.292545\pi$
$830$	0	0
$831$	−35.6718	−1.23744
$832$	0	0
$833$	11.1402	0.385984
$834$	0	0
$835$	0	0
$836$	0	0
$837$	47.8405	1.65361
$838$	0	0
$839$	10.6389	0.367294	0.183647	−	0.982992i	$-0.441210\pi$
$839$	10.6389	0.367294	0.183647	+	0.982992i	$0.441210\pi$
$840$	0	0
$841$	5.59880	0.193062
$842$	0	0
$843$	−8.90772	−0.306798
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−86.9105	−2.98628
$848$	0	0
$849$	−16.8407	−0.577972
$850$	0	0
$851$	5.16863	0.177178
$852$	0	0
$853$	4.53305	0.155209	0.0776043	−	0.996984i	$-0.475273\pi$
$853$	4.53305	0.155209	0.0776043	+	0.996984i	$0.475273\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26.8119	0.915878	0.457939	−	0.888983i	$-0.348588\pi$
$857$	26.8119	0.915878	0.457939	+	0.888983i	$0.348588\pi$
$858$	0	0
$859$	−2.54555	−0.0868530	−0.0434265	−	0.999057i	$-0.513827\pi$
$859$	−2.54555	−0.0868530	−0.0434265	+	0.999057i	$0.513827\pi$
$860$	0	0
$861$	−67.6067	−2.30403
$862$	0	0
$863$	−21.3673	−0.727351	−0.363676	−	0.931526i	$-0.618478\pi$
$863$	−21.3673	−0.727351	−0.363676	+	0.931526i	$0.618478\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.50269	−0.0510341
$868$	0	0
$869$	−44.5363	−1.51079
$870$	0	0
$871$	23.0740	0.781834
$872$	0	0
$873$	−4.23252	−0.143249
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−19.1228	−0.645730	−0.322865	−	0.946445i	$-0.604646\pi$
$877$	−19.1228	−0.645730	−0.322865	+	0.946445i	$0.604646\pi$
$878$	0	0
$879$	19.6434	0.662557
$880$	0	0
$881$	33.8416	1.14015	0.570075	−	0.821592i	$-0.306914\pi$
$881$	33.8416	1.14015	0.570075	+	0.821592i	$0.306914\pi$
$882$	0	0
$883$	22.7756	0.766461	0.383231	−	0.923653i	$-0.374811\pi$
$883$	22.7756	0.766461	0.383231	+	0.923653i	$0.374811\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.7568	0.529062	0.264531	−	0.964377i	$-0.414783\pi$
$887$	15.7568	0.529062	0.264531	+	0.964377i	$0.414783\pi$
$888$	0	0
$889$	43.4490	1.45723
$890$	0	0
$891$	−34.8788	−1.16848
$892$	0	0
$893$	4.71338	0.157727
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−15.9506	−0.532576
$898$	0	0
$899$	−50.0452	−1.66910
$900$	0	0
$901$	−1.32070	−0.0439989
$902$	0	0
$903$	−64.5447	−2.14791
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−28.3574	−0.941593	−0.470796	−	0.882242i	$-0.656033\pi$
$907$	−28.3574	−0.941593	−0.470796	+	0.882242i	$0.656033\pi$
$908$	0	0
$909$	−3.76001	−0.124712
$910$	0	0
$911$	−4.46190	−0.147829	−0.0739147	−	0.997265i	$-0.523549\pi$
$911$	−4.46190	−0.147829	−0.0739147	+	0.997265i	$0.523549\pi$
$912$	0	0
$913$	72.2029	2.38957
$914$	0	0
$915$	0	0
$916$	0	0
$917$	39.6505	1.30938
$918$	0	0
$919$	−12.2144	−0.402917	−0.201459	−	0.979497i	$-0.564568\pi$
$919$	−12.2144	−0.402917	−0.201459	+	0.979497i	$0.564568\pi$
$920$	0	0
$921$	25.6665	0.845741
$922$	0	0
$923$	−0.139210	−0.00458214
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.97562	0.130576
$928$	0	0
$929$	−24.0052	−0.787586	−0.393793	−	0.919199i	$-0.628837\pi$
$929$	−24.0052	−0.787586	−0.393793	+	0.919199i	$0.628837\pi$
$930$	0	0
$931$	40.9880	1.34333
$932$	0	0
$933$	−15.4342	−0.505294
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−46.3819	−1.51523	−0.757615	−	0.652702i	$-0.773635\pi$
$937$	−46.3819	−1.51523	−0.757615	+	0.652702i	$0.773635\pi$
$938$	0	0
$939$	6.14746	0.200615
$940$	0	0
$941$	0.530210	0.0172843	0.00864217	−	0.999963i	$-0.497249\pi$
$941$	0.530210	0.0172843	0.00864217	+	0.999963i	$0.497249\pi$
$942$	0	0
$943$	21.9983	0.716362
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.3865	0.889942	0.444971	−	0.895545i	$-0.353214\pi$
$947$	27.3865	0.889942	0.444971	+	0.895545i	$0.353214\pi$
$948$	0	0
$949$	39.1417	1.27059
$950$	0	0
$951$	8.74987	0.283734
$952$	0	0
$953$	−39.8826	−1.29192	−0.645962	−	0.763370i	$-0.723543\pi$
$953$	−39.8826	−1.29192	−0.645962	+	0.763370i	$0.723543\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	49.5342	1.60121
$958$	0	0
$959$	−14.8721	−0.480245
$960$	0	0
$961$	41.3874	1.33508
$962$	0	0
$963$	−13.5736	−0.437405
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−33.1843	−1.06714	−0.533568	−	0.845757i	$-0.679149\pi$
$967$	−33.1843	−1.06714	−0.533568	+	0.845757i	$0.679149\pi$
$968$	0	0
$969$	−5.52886	−0.177613
$970$	0	0
$971$	−1.88467	−0.0604818	−0.0302409	−	0.999543i	$-0.509627\pi$
$971$	−1.88467	−0.0604818	−0.0302409	+	0.999543i	$0.509627\pi$
$972$	0	0
$973$	51.2135	1.64183
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−53.8166	−1.72175	−0.860874	−	0.508819i	$-0.830082\pi$
$977$	−53.8166	−1.72175	−0.860874	+	0.508819i	$0.830082\pi$
$978$	0	0
$979$	69.0578	2.20710
$980$	0	0
$981$	7.72259	0.246563
$982$	0	0
$983$	−41.8680	−1.33538	−0.667690	−	0.744439i	$-0.732717\pi$
$983$	−41.8680	−1.33538	−0.667690	+	0.744439i	$0.732717\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	8.19894	0.260975
$988$	0	0
$989$	21.0019	0.667821
$990$	0	0
$991$	−29.6430	−0.941642	−0.470821	−	0.882229i	$-0.656042\pi$
$991$	−29.6430	−0.941642	−0.470821	+	0.882229i	$0.656042\pi$
$992$	0	0
$993$	−32.9934	−1.04701
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−28.3132	−0.896688	−0.448344	−	0.893861i	$-0.647986\pi$
$997$	−28.3132	−0.896688	−0.448344	+	0.893861i	$0.647986\pi$
$998$	0	0
$999$	−13.9557	−0.441538

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6800.2.a.cg.1.2		5
4.3	odd	2		3400.2.a.r.1.4	✓	5
5.4	even	2		6800.2.a.bx.1.4		5
20.3	even	4		3400.2.e.n.2449.7		10
20.7	even	4		3400.2.e.n.2449.4		10
20.19	odd	2		3400.2.a.w.1.2	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3400.2.a.r.1.4	✓	5	4.3	odd	2
3400.2.a.w.1.2	yes	5	20.19	odd	2
3400.2.e.n.2449.4		10	20.7	even	4
3400.2.e.n.2449.7		10	20.3	even	4
6800.2.a.bx.1.4		5	5.4	even	2
6800.2.a.cg.1.2		5	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)

\(f(q)\)	\(=\)	\(q-1.50269 q^{3} -4.25913 q^{7} -0.741913 q^{9} +5.60408 q^{11} -5.09704 q^{13} +1.00000 q^{17} +3.67930 q^{19} +6.40016 q^{21} -2.08252 q^{23} +5.62295 q^{27} -5.88207 q^{29} +8.50808 q^{31} -8.42121 q^{33} -2.48191 q^{37} +7.65929 q^{39} -10.5633 q^{41} -10.0848 q^{43} +1.28105 q^{47} +11.1402 q^{49} -1.50269 q^{51} -1.32070 q^{53} -5.52886 q^{57} -11.7607 q^{59} -12.0814 q^{61} +3.15990 q^{63} -4.52695 q^{67} +3.12939 q^{69} +0.0273119 q^{71} -7.67930 q^{73} -23.8685 q^{77} -7.94712 q^{79} -6.22383 q^{81} +12.8840 q^{83} +8.83895 q^{87} +12.3228 q^{89} +21.7089 q^{91} -12.7850 q^{93} +5.70487 q^{97} -4.15774 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{3} + q^{7} + 6 q^{9} + 6 q^{11} - 3 q^{13} + 5 q^{17} + 6 q^{19} + 7 q^{21} + 10 q^{23} + 15 q^{27} + 6 q^{29} + 11 q^{31} - 20 q^{33} - 2 q^{37} - 9 q^{39} + 4 q^{41} + 8 q^{43} + 10 q^{47}+ \cdots - 10 q^{99}+O(q^{100}) \) 5 * q + 3 * q^3 + q^7 + 6 * q^9 + 6 * q^11 - 3 * q^13 + 5 * q^17 + 6 * q^19 + 7 * q^21 + 10 * q^23 + 15 * q^27 + 6 * q^29 + 11 * q^31 - 20 * q^33 - 2 * q^37 - 9 * q^39 + 4 * q^41 + 8 * q^43 + 10 * q^47 + 30 * q^49 + 3 * q^51 - 19 * q^53 + 4 * q^57 - 14 * q^61 + 22 * q^63 + 32 * q^69 + 5 * q^71 - 26 * q^73 + 4 * q^77 - 27 * q^79 - 11 * q^81 + 20 * q^83 + 12 * q^87 + 10 * q^89 + 63 * q^91 - 51 * q^93 + 24 * q^97 - 10 * q^99

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