LMFDB - Embedded newform 8016.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1300.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 10x - 10 $ x^3 - 10*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.42362$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.42362 q^{5} +3.72119 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.42362 q^{5} +3.72119 q^{7} +1.00000 q^{9} -0.721189 q^{11} -2.12606 q^{13} +3.42362 q^{15} -0.126056 q^{17} +6.72119 q^{19} +3.72119 q^{21} +8.72119 q^{23} +6.72119 q^{25} +1.00000 q^{27} +0.721189 q^{29} +1.00000 q^{31} -0.721189 q^{33} +12.7399 q^{35} +7.42362 q^{37} -2.12606 q^{39} -7.44238 q^{41} -10.1636 q^{43} +3.42362 q^{45} +3.72119 q^{47} +6.84724 q^{49} -0.126056 q^{51} -7.29757 q^{53} -2.46908 q^{55} +6.72119 q^{57} +0.828489 q^{59} -9.19027 q^{61} +3.72119 q^{63} -7.27881 q^{65} +3.17151 q^{67} +8.72119 q^{69} -5.31632 q^{71} -15.5684 q^{73} +6.72119 q^{75} -2.68368 q^{77} -9.44238 q^{79} +1.00000 q^{81} +5.85519 q^{83} -0.431567 q^{85} +0.721189 q^{87} +7.97330 q^{89} -7.91145 q^{91} +1.00000 q^{93} +23.0108 q^{95} -13.7587 q^{97} -0.721189 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} - q^{21} + 14 q^{23} + 8 q^{25} + 3 q^{27} - 10 q^{29} + 3 q^{31} + 10 q^{33} + 9 q^{35} + 15 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} + 2 q^{51} - 17 q^{53} + 8 q^{57} + 5 q^{59} - 8 q^{61} - q^{63} - 34 q^{65} + 7 q^{67} + 14 q^{69} + 6 q^{71} - 20 q^{73} + 8 q^{75} - 30 q^{77} - 4 q^{79} + 3 q^{81} + 37 q^{83} - 28 q^{85} - 10 q^{87} + 7 q^{89} + 8 q^{91} + 3 q^{93} + 18 q^{95} + 5 q^{97} + 10 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 - q^21 + 14 * q^23 + 8 * q^25 + 3 * q^27 - 10 * q^29 + 3 * q^31 + 10 * q^33 + 9 * q^35 + 15 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 + 3 * q^45 - q^47 + 6 * q^49 + 2 * q^51 - 17 * q^53 + 8 * q^57 + 5 * q^59 - 8 * q^61 - q^63 - 34 * q^65 + 7 * q^67 + 14 * q^69 + 6 * q^71 - 20 * q^73 + 8 * q^75 - 30 * q^77 - 4 * q^79 + 3 * q^81 + 37 * q^83 - 28 * q^85 - 10 * q^87 + 7 * q^89 + 8 * q^91 + 3 * q^93 + 18 * q^95 + 5 * q^97 + 10 * q^99

Display 100 coefficients 10 coefficients

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
$5$	3.42362	1.53109	0.765545	−	0.643382i	$-0.222469\pi$
$5$	3.42362	1.53109	0.765545	+	0.643382i	$0.222469\pi$
$6$	0	0
$7$	3.72119	1.40648	0.703239	−	0.710954i	$-0.251737\pi$
$7$	3.72119	1.40648	0.703239	+	0.710954i	$0.251737\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.721189	−0.217447	−0.108723	−	0.994072i	$-0.534676\pi$
$11$	−0.721189	−0.217447	−0.108723	+	0.994072i	$0.534676\pi$
$12$	0	0
$13$	−2.12606	−0.589662	−0.294831	−	0.955549i	$-0.595263\pi$
$13$	−2.12606	−0.589662	−0.294831	+	0.955549i	$0.595263\pi$
$14$	0	0
$15$	3.42362	0.883975
$16$	0	0
$17$	−0.126056	−0.0305730	−0.0152865	−	0.999883i	$-0.504866\pi$
$17$	−0.126056	−0.0305730	−0.0152865	+	0.999883i	$0.504866\pi$
$18$	0	0
$19$	6.72119	1.54195	0.770973	−	0.636868i	$-0.219770\pi$
$19$	6.72119	1.54195	0.770973	+	0.636868i	$0.219770\pi$
$20$	0	0
$21$	3.72119	0.812030
$22$	0	0
$23$	8.72119	1.81849	0.909247	−	0.416258i	$-0.136659\pi$
$23$	8.72119	1.81849	0.909247	+	0.416258i	$0.136659\pi$
$24$	0	0
$25$	6.72119	1.34424
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.721189	0.133921	0.0669607	−	0.997756i	$-0.478670\pi$
$29$	0.721189	0.133921	0.0669607	+	0.997756i	$0.478670\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	−0.721189	−0.125543
$34$	0	0
$35$	12.7399	2.15344
$36$	0	0
$37$	7.42362	1.22044	0.610218	−	0.792234i	$-0.291082\pi$
$37$	7.42362	1.22044	0.610218	+	0.792234i	$0.291082\pi$
$38$	0	0
$39$	−2.12606	−0.340441
$40$	0	0
$41$	−7.44238	−1.16230	−0.581152	−	0.813795i	$-0.697398\pi$
$41$	−7.44238	−1.16230	−0.581152	+	0.813795i	$0.697398\pi$
$42$	0	0
$43$	−10.1636	−1.54993	−0.774965	−	0.632005i	$-0.782232\pi$
$43$	−10.1636	−1.54993	−0.774965	+	0.632005i	$0.782232\pi$
$44$	0	0
$45$	3.42362	0.510363
$46$	0	0
$47$	3.72119	0.542791	0.271396	−	0.962468i	$-0.412515\pi$
$47$	3.72119	0.542791	0.271396	+	0.962468i	$0.412515\pi$
$48$	0	0
$49$	6.84724	0.978178
$50$	0	0
$51$	−0.126056	−0.0176513
$52$	0	0
$53$	−7.29757	−1.00240	−0.501199	−	0.865332i	$-0.667108\pi$
$53$	−7.29757	−1.00240	−0.501199	+	0.865332i	$0.667108\pi$
$54$	0	0
$55$	−2.46908	−0.332930
$56$	0	0
$57$	6.72119	0.890243
$58$	0	0
$59$	0.828489	0.107860	0.0539301	−	0.998545i	$-0.482825\pi$
$59$	0.828489	0.107860	0.0539301	+	0.998545i	$0.482825\pi$
$60$	0	0
$61$	−9.19027	−1.17669	−0.588346	−	0.808609i	$-0.700221\pi$
$61$	−9.19027	−1.17669	−0.588346	+	0.808609i	$0.700221\pi$
$62$	0	0
$63$	3.72119	0.468826
$64$	0	0
$65$	−7.27881	−0.902825
$66$	0	0
$67$	3.17151	0.387462	0.193731	−	0.981055i	$-0.437941\pi$
$67$	3.17151	0.387462	0.193731	+	0.981055i	$0.437941\pi$
$68$	0	0
$69$	8.72119	1.04991
$70$	0	0
$71$	−5.31632	−0.630931	−0.315466	−	0.948937i	$-0.602161\pi$
$71$	−5.31632	−0.630931	−0.315466	+	0.948937i	$0.602161\pi$
$72$	0	0
$73$	−15.5684	−1.82215	−0.911074	−	0.412244i	$-0.864745\pi$
$73$	−15.5684	−1.82215	−0.911074	+	0.412244i	$0.864745\pi$
$74$	0	0
$75$	6.72119	0.776096
$76$	0	0
$77$	−2.68368	−0.305834
$78$	0	0
$79$	−9.44238	−1.06235	−0.531175	−	0.847262i	$-0.678249\pi$
$79$	−9.44238	−1.06235	−0.531175	+	0.847262i	$0.678249\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.85519	0.642691	0.321345	−	0.946962i	$-0.395865\pi$
$83$	5.85519	0.642691	0.321345	+	0.946962i	$0.395865\pi$
$84$	0	0
$85$	−0.431567	−0.0468100
$86$	0	0
$87$	0.721189	0.0773195
$88$	0	0
$89$	7.97330	0.845168	0.422584	−	0.906324i	$-0.361123\pi$
$89$	7.97330	0.845168	0.422584	+	0.906324i	$0.361123\pi$
$90$	0	0
$91$	−7.91145	−0.829346
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	23.0108	2.36086
$96$	0	0
$97$	−13.7587	−1.39698	−0.698492	−	0.715618i	$-0.746145\pi$
$97$	−13.7587	−1.39698	−0.698492	+	0.715618i	$0.746145\pi$
$98$	0	0
$99$	−0.721189	−0.0724822
$100$	0	0
$101$	−9.58719	−0.953961	−0.476980	−	0.878914i	$-0.658269\pi$
$101$	−9.58719	−0.953961	−0.476980	+	0.878914i	$0.658269\pi$
$102$	0	0
$103$	5.15276	0.507716	0.253858	−	0.967241i	$-0.418300\pi$
$103$	5.15276	0.507716	0.253858	+	0.967241i	$0.418300\pi$
$104$	0	0
$105$	12.7399	1.24329
$106$	0	0
$107$	16.0375	1.55040	0.775202	−	0.631713i	$-0.217648\pi$
$107$	16.0375	1.55040	0.775202	+	0.631713i	$0.217648\pi$
$108$	0	0
$109$	15.5684	1.49119	0.745593	−	0.666402i	$-0.232166\pi$
$109$	15.5684	1.49119	0.745593	+	0.666402i	$0.232166\pi$
$110$	0	0
$111$	7.42362	0.704619
$112$	0	0
$113$	−9.44238	−0.888264	−0.444132	−	0.895961i	$-0.646488\pi$
$113$	−9.44238	−0.888264	−0.444132	+	0.895961i	$0.646488\pi$
$114$	0	0
$115$	29.8581	2.78428
$116$	0	0
$117$	−2.12606	−0.196554
$118$	0	0
$119$	−0.469077	−0.0430002
$120$	0	0
$121$	−10.4799	−0.952717
$122$	0	0
$123$	−7.44238	−0.671057
$124$	0	0
$125$	5.89270	0.527059
$126$	0	0
$127$	19.0375	1.68931	0.844653	−	0.535314i	$-0.179807\pi$
$127$	19.0375	1.68931	0.844653	+	0.535314i	$0.179807\pi$
$128$	0	0
$129$	−10.1636	−0.894852
$130$	0	0
$131$	−8.72119	−0.761974	−0.380987	−	0.924580i	$-0.624416\pi$
$131$	−8.72119	−0.761974	−0.380987	+	0.924580i	$0.624416\pi$
$132$	0	0
$133$	25.0108	2.16871
$134$	0	0
$135$	3.42362	0.294658
$136$	0	0
$137$	0.568433	0.0485645	0.0242822	−	0.999705i	$-0.492270\pi$
$137$	0.568433	0.0485645	0.0242822	+	0.999705i	$0.492270\pi$
$138$	0	0
$139$	−6.61389	−0.560983	−0.280491	−	0.959857i	$-0.590497\pi$
$139$	−6.61389	−0.560983	−0.280491	+	0.959857i	$0.590497\pi$
$140$	0	0
$141$	3.72119	0.313381
$142$	0	0
$143$	1.53329	0.128220
$144$	0	0
$145$	2.46908	0.205046
$146$	0	0
$147$	6.84724	0.564751
$148$	0	0
$149$	−6.95455	−0.569739	−0.284869	−	0.958566i	$-0.591950\pi$
$149$	−6.95455	−0.569739	−0.284869	+	0.958566i	$0.591950\pi$
$150$	0	0
$151$	−16.5951	−1.35049	−0.675246	−	0.737592i	$-0.735963\pi$
$151$	−16.5951	−1.35049	−0.675246	+	0.737592i	$0.735963\pi$
$152$	0	0
$153$	−0.126056	−0.0101910
$154$	0	0
$155$	3.42362	0.274992
$156$	0	0
$157$	12.0375	0.960698	0.480349	−	0.877077i	$-0.340510\pi$
$157$	12.0375	0.960698	0.480349	+	0.877077i	$0.340510\pi$
$158$	0	0
$159$	−7.29757	−0.578735
$160$	0	0
$161$	32.4532	2.55767
$162$	0	0
$163$	−15.3351	−1.20114	−0.600568	−	0.799574i	$-0.705059\pi$
$163$	−15.3351	−1.20114	−0.600568	+	0.799574i	$0.705059\pi$
$164$	0	0
$165$	−2.46908	−0.192217
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	−8.47989	−0.652299
$170$	0	0
$171$	6.72119	0.513982
$172$	0	0
$173$	−11.4049	−0.867096	−0.433548	−	0.901130i	$-0.642739\pi$
$173$	−11.4049	−0.867096	−0.433548	+	0.901130i	$0.642739\pi$
$174$	0	0
$175$	25.0108	1.89064
$176$	0	0
$177$	0.828489	0.0622731
$178$	0	0
$179$	−22.6678	−1.69427	−0.847135	−	0.531378i	$-0.821675\pi$
$179$	−22.6678	−1.69427	−0.847135	+	0.531378i	$0.821675\pi$
$180$	0	0
$181$	25.2629	1.87778	0.938889	−	0.344220i	$-0.111857\pi$
$181$	25.2629	1.87778	0.938889	+	0.344220i	$0.111857\pi$
$182$	0	0
$183$	−9.19027	−0.679364
$184$	0	0
$185$	25.4157	1.86860
$186$	0	0
$187$	0.0909099	0.00664799
$188$	0	0
$189$	3.72119	0.270677
$190$	0	0
$191$	8.74789	0.632975	0.316488	−	0.948597i	$-0.397496\pi$
$191$	8.74789	0.632975	0.316488	+	0.948597i	$0.397496\pi$
$192$	0	0
$193$	13.0642	0.940382	0.470191	−	0.882565i	$-0.344185\pi$
$193$	13.0642	0.940382	0.470191	+	0.882565i	$0.344185\pi$
$194$	0	0
$195$	−7.27881	−0.521247
$196$	0	0
$197$	11.3163	0.806255	0.403127	−	0.915144i	$-0.367923\pi$
$197$	11.3163	0.806255	0.403127	+	0.915144i	$0.367923\pi$
$198$	0	0
$199$	0.847244	0.0600596	0.0300298	−	0.999549i	$-0.490440\pi$
$199$	0.847244	0.0600596	0.0300298	+	0.999549i	$0.490440\pi$
$200$	0	0
$201$	3.17151	0.223701
$202$	0	0
$203$	2.68368	0.188357
$204$	0	0
$205$	−25.4799	−1.77959
$206$	0	0
$207$	8.72119	0.606165
$208$	0	0
$209$	−4.84724	−0.335291
$210$	0	0
$211$	3.69449	0.254339	0.127170	−	0.991881i	$-0.459411\pi$
$211$	3.69449	0.254339	0.127170	+	0.991881i	$0.459411\pi$
$212$	0	0
$213$	−5.31632	−0.364268
$214$	0	0
$215$	−34.7962	−2.37308
$216$	0	0
$217$	3.72119	0.252611
$218$	0	0
$219$	−15.5684	−1.05202
$220$	0	0
$221$	0.268001	0.0180277
$222$	0	0
$223$	−10.0108	−0.670373	−0.335187	−	0.942152i	$-0.608799\pi$
$223$	−10.0108	−0.670373	−0.335187	+	0.942152i	$0.608799\pi$
$224$	0	0
$225$	6.72119	0.448079
$226$	0	0
$227$	17.7132	1.17567	0.587835	−	0.808981i	$-0.299981\pi$
$227$	17.7132	1.17567	0.587835	+	0.808981i	$0.299981\pi$
$228$	0	0
$229$	27.3890	1.80991	0.904957	−	0.425502i	$-0.139903\pi$
$229$	27.3890	1.80991	0.904957	+	0.425502i	$0.139903\pi$
$230$	0	0
$231$	−2.68368	−0.176573
$232$	0	0
$233$	18.4049	1.20574	0.602871	−	0.797838i	$-0.294023\pi$
$233$	18.4049	1.20574	0.602871	+	0.797838i	$0.294023\pi$
$234$	0	0
$235$	12.7399	0.831062
$236$	0	0
$237$	−9.44238	−0.613348
$238$	0	0
$239$	18.5417	1.19936	0.599682	−	0.800238i	$-0.295294\pi$
$239$	18.5417	1.19936	0.599682	+	0.800238i	$0.295294\pi$
$240$	0	0
$241$	−22.4157	−1.44392	−0.721960	−	0.691934i	$-0.756759\pi$
$241$	−22.4157	−1.44392	−0.721960	+	0.691934i	$0.756759\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	23.4424	1.49768
$246$	0	0
$247$	−14.2896	−0.909227
$248$	0	0
$249$	5.85519	0.371058
$250$	0	0
$251$	−0.935790	−0.0590665	−0.0295333	−	0.999564i	$-0.509402\pi$
$251$	−0.935790	−0.0590665	−0.0295333	+	0.999564i	$0.509402\pi$
$252$	0	0
$253$	−6.28962	−0.395425
$254$	0	0
$255$	−0.431567	−0.0270258
$256$	0	0
$257$	−0.163566	−0.0102029	−0.00510147	−	0.999987i	$-0.501624\pi$
$257$	−0.163566	−0.0102029	−0.00510147	+	0.999987i	$0.501624\pi$
$258$	0	0
$259$	27.6247	1.71652
$260$	0	0
$261$	0.721189	0.0446405
$262$	0	0
$263$	−24.6059	−1.51727	−0.758634	−	0.651517i	$-0.774133\pi$
$263$	−24.6059	−1.51727	−0.758634	+	0.651517i	$0.774133\pi$
$264$	0	0
$265$	−24.9841	−1.53476
$266$	0	0
$267$	7.97330	0.487958
$268$	0	0
$269$	−1.67573	−0.102171	−0.0510856	−	0.998694i	$-0.516268\pi$
$269$	−1.67573	−0.102171	−0.0510856	+	0.998694i	$0.516268\pi$
$270$	0	0
$271$	−5.87394	−0.356817	−0.178408	−	0.983957i	$-0.557095\pi$
$271$	−5.87394	−0.356817	−0.178408	+	0.983957i	$0.557095\pi$
$272$	0	0
$273$	−7.91145	−0.478823
$274$	0	0
$275$	−4.84724	−0.292300
$276$	0	0
$277$	−12.1073	−0.727457	−0.363729	−	0.931505i	$-0.618496\pi$
$277$	−12.1073	−0.727457	−0.363729	+	0.931505i	$0.618496\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	−7.54173	−0.449902	−0.224951	−	0.974370i	$-0.572222\pi$
$281$	−7.54173	−0.449902	−0.224951	+	0.974370i	$0.572222\pi$
$282$	0	0
$283$	−31.7320	−1.88627	−0.943136	−	0.332408i	$-0.892139\pi$
$283$	−31.7320	−1.88627	−0.943136	+	0.332408i	$0.892139\pi$
$284$	0	0
$285$	23.0108	1.36304
$286$	0	0
$287$	−27.6945	−1.63475
$288$	0	0
$289$	−16.9841	−0.999065
$290$	0	0
$291$	−13.7587	−0.806549
$292$	0	0
$293$	−17.4424	−1.01899	−0.509497	−	0.860472i	$-0.670169\pi$
$293$	−17.4424	−1.01899	−0.509497	+	0.860472i	$0.670169\pi$
$294$	0	0
$295$	2.83643	0.165144
$296$	0	0
$297$	−0.721189	−0.0418476
$298$	0	0
$299$	−18.5417	−1.07230
$300$	0	0
$301$	−37.8205	−2.17994
$302$	0	0
$303$	−9.58719	−0.550770
$304$	0	0
$305$	−31.4640	−1.80162
$306$	0	0
$307$	−6.39692	−0.365092	−0.182546	−	0.983197i	$-0.558434\pi$
$307$	−6.39692	−0.365092	−0.182546	+	0.983197i	$0.558434\pi$
$308$	0	0
$309$	5.15276	0.293130
$310$	0	0
$311$	23.5951	1.33796	0.668979	−	0.743282i	$-0.266732\pi$
$311$	23.5951	1.33796	0.668979	+	0.743282i	$0.266732\pi$
$312$	0	0
$313$	5.01081	0.283228	0.141614	−	0.989922i	$-0.454771\pi$
$313$	5.01081	0.283228	0.141614	+	0.989922i	$0.454771\pi$
$314$	0	0
$315$	12.7399	0.717814
$316$	0	0
$317$	25.9115	1.45533	0.727666	−	0.685931i	$-0.240605\pi$
$317$	25.9115	1.45533	0.727666	+	0.685931i	$0.240605\pi$
$318$	0	0
$319$	−0.520113	−0.0291207
$320$	0	0
$321$	16.0375	0.895127
$322$	0	0
$323$	−0.847244	−0.0471419
$324$	0	0
$325$	−14.2896	−0.792646
$326$	0	0
$327$	15.5684	0.860936
$328$	0	0
$329$	13.8472	0.763423
$330$	0	0
$331$	21.9115	1.20436	0.602181	−	0.798359i	$-0.294298\pi$
$331$	21.9115	1.20436	0.602181	+	0.798359i	$0.294298\pi$
$332$	0	0
$333$	7.42362	0.406812
$334$	0	0
$335$	10.8581	0.593239
$336$	0	0
$337$	−27.3271	−1.48860	−0.744302	−	0.667843i	$-0.767218\pi$
$337$	−27.3271	−1.48860	−0.744302	+	0.667843i	$0.767218\pi$
$338$	0	0
$339$	−9.44238	−0.512840
$340$	0	0
$341$	−0.721189	−0.0390545
$342$	0	0
$343$	−0.568433	−0.0306925
$344$	0	0
$345$	29.8581	1.60750
$346$	0	0
$347$	10.0188	0.537835	0.268917	−	0.963163i	$-0.413334\pi$
$347$	10.0188	0.537835	0.268917	+	0.963163i	$0.413334\pi$
$348$	0	0
$349$	−5.11811	−0.273966	−0.136983	−	0.990573i	$-0.543741\pi$
$349$	−5.11811	−0.273966	−0.136983	+	0.990573i	$0.543741\pi$
$350$	0	0
$351$	−2.12606	−0.113480
$352$	0	0
$353$	8.44238	0.449342	0.224671	−	0.974435i	$-0.427869\pi$
$353$	8.44238	0.449342	0.224671	+	0.974435i	$0.427869\pi$
$354$	0	0
$355$	−18.2011	−0.966013
$356$	0	0
$357$	−0.469077	−0.0248262
$358$	0	0
$359$	−14.5417	−0.767483	−0.383742	−	0.923440i	$-0.625365\pi$
$359$	−14.5417	−0.767483	−0.383742	+	0.923440i	$0.625365\pi$
$360$	0	0
$361$	26.1744	1.37760
$362$	0	0
$363$	−10.4799	−0.550051
$364$	0	0
$365$	−53.3004	−2.78987
$366$	0	0
$367$	17.3515	0.905739	0.452870	−	0.891577i	$-0.350400\pi$
$367$	17.3515	0.905739	0.452870	+	0.891577i	$0.350400\pi$
$368$	0	0
$369$	−7.44238	−0.387435
$370$	0	0
$371$	−27.1556	−1.40985
$372$	0	0
$373$	0.702434	0.0363706	0.0181853	−	0.999835i	$-0.494211\pi$
$373$	0.702434	0.0363706	0.0181853	+	0.999835i	$0.494211\pi$
$374$	0	0
$375$	5.89270	0.304298
$376$	0	0
$377$	−1.53329	−0.0789683
$378$	0	0
$379$	−1.29757	−0.0666515	−0.0333258	−	0.999445i	$-0.510610\pi$
$379$	−1.29757	−0.0666515	−0.0333258	+	0.999445i	$0.510610\pi$
$380$	0	0
$381$	19.0375	0.975321
$382$	0	0
$383$	−18.7938	−0.960321	−0.480160	−	0.877181i	$-0.659422\pi$
$383$	−18.7938	−0.960321	−0.480160	+	0.877181i	$0.659422\pi$
$384$	0	0
$385$	−9.18790	−0.468259
$386$	0	0
$387$	−10.1636	−0.516643
$388$	0	0
$389$	−3.91145	−0.198319	−0.0991593	−	0.995072i	$-0.531615\pi$
$389$	−3.91145	−0.198319	−0.0991593	+	0.995072i	$0.531615\pi$
$390$	0	0
$391$	−1.09936	−0.0555968
$392$	0	0
$393$	−8.72119	−0.439926
$394$	0	0
$395$	−32.3271	−1.62655
$396$	0	0
$397$	1.56843	0.0787174	0.0393587	−	0.999225i	$-0.487469\pi$
$397$	1.56843	0.0787174	0.0393587	+	0.999225i	$0.487469\pi$
$398$	0	0
$399$	25.0108	1.25211
$400$	0	0
$401$	−9.53329	−0.476070	−0.238035	−	0.971257i	$-0.576503\pi$
$401$	−9.53329	−0.476070	−0.238035	+	0.971257i	$0.576503\pi$
$402$	0	0
$403$	−2.12606	−0.105906
$404$	0	0
$405$	3.42362	0.170121
$406$	0	0
$407$	−5.35383	−0.265380
$408$	0	0
$409$	26.0750	1.28933	0.644663	−	0.764467i	$-0.276998\pi$
$409$	26.0750	1.28933	0.644663	+	0.764467i	$0.276998\pi$
$410$	0	0
$411$	0.568433	0.0280387
$412$	0	0
$413$	3.08296	0.151703
$414$	0	0
$415$	20.0460	0.984017
$416$	0	0
$417$	−6.61389	−0.323883
$418$	0	0
$419$	20.5792	1.00536	0.502681	−	0.864472i	$-0.332347\pi$
$419$	20.5792	1.00536	0.502681	+	0.864472i	$0.332347\pi$
$420$	0	0
$421$	10.0727	0.490911	0.245456	−	0.969408i	$-0.421062\pi$
$421$	10.0727	0.490911	0.245456	+	0.969408i	$0.421062\pi$
$422$	0	0
$423$	3.72119	0.180930
$424$	0	0
$425$	−0.847244	−0.0410974
$426$	0	0
$427$	−34.1987	−1.65499
$428$	0	0
$429$	1.53329	0.0740278
$430$	0	0
$431$	−14.1903	−0.683521	−0.341761	−	0.939787i	$-0.611023\pi$
$431$	−14.1903	−0.683521	−0.341761	+	0.939787i	$0.611023\pi$
$432$	0	0
$433$	3.54173	0.170205	0.0851024	−	0.996372i	$-0.472878\pi$
$433$	3.54173	0.170205	0.0851024	+	0.996372i	$0.472878\pi$
$434$	0	0
$435$	2.46908	0.118383
$436$	0	0
$437$	58.6168	2.80402
$438$	0	0
$439$	−26.2896	−1.25474	−0.627368	−	0.778723i	$-0.715868\pi$
$439$	−26.2896	−1.25474	−0.627368	+	0.778723i	$0.715868\pi$
$440$	0	0
$441$	6.84724	0.326059
$442$	0	0
$443$	−7.71324	−0.366467	−0.183234	−	0.983069i	$-0.558656\pi$
$443$	−7.71324	−0.366467	−0.183234	+	0.983069i	$0.558656\pi$
$444$	0	0
$445$	27.2976	1.29403
$446$	0	0
$447$	−6.95455	−0.328939
$448$	0	0
$449$	41.9949	1.98186	0.990931	−	0.134369i	$-0.0429007\pi$
$449$	41.9949	1.98186	0.990931	+	0.134369i	$0.0429007\pi$
$450$	0	0
$451$	5.36736	0.252739
$452$	0	0
$453$	−16.5951	−0.779707
$454$	0	0
$455$	−27.0858	−1.26980
$456$	0	0
$457$	14.3271	0.670195	0.335097	−	0.942183i	$-0.391231\pi$
$457$	14.3271	0.670195	0.335097	+	0.942183i	$0.391231\pi$
$458$	0	0
$459$	−0.126056	−0.00588378
$460$	0	0
$461$	−17.5684	−0.818243	−0.409122	−	0.912480i	$-0.634165\pi$
$461$	−17.5684	−0.818243	−0.409122	+	0.912480i	$0.634165\pi$
$462$	0	0
$463$	19.6945	0.915281	0.457640	−	0.889137i	$-0.348695\pi$
$463$	19.6945	0.915281	0.457640	+	0.889137i	$0.348695\pi$
$464$	0	0
$465$	3.42362	0.158767
$466$	0	0
$467$	0.809734	0.0374700	0.0187350	−	0.999824i	$-0.494036\pi$
$467$	0.809734	0.0374700	0.0187350	+	0.999824i	$0.494036\pi$
$468$	0	0
$469$	11.8018	0.544956
$470$	0	0
$471$	12.0375	0.554659
$472$	0	0
$473$	7.32985	0.337027
$474$	0	0
$475$	45.1744	2.07274
$476$	0	0
$477$	−7.29757	−0.334133
$478$	0	0
$479$	13.7854	0.629871	0.314935	−	0.949113i	$-0.398017\pi$
$479$	13.7854	0.629871	0.314935	+	0.949113i	$0.398017\pi$
$480$	0	0
$481$	−15.7830	−0.719644
$482$	0	0
$483$	32.4532	1.47667
$484$	0	0
$485$	−47.1046	−2.13891
$486$	0	0
$487$	2.16357	0.0980405	0.0490203	−	0.998798i	$-0.484390\pi$
$487$	2.16357	0.0980405	0.0490203	+	0.998798i	$0.484390\pi$
$488$	0	0
$489$	−15.3351	−0.693476
$490$	0	0
$491$	21.4424	0.967681	0.483840	−	0.875156i	$-0.339241\pi$
$491$	21.4424	0.967681	0.483840	+	0.875156i	$0.339241\pi$
$492$	0	0
$493$	−0.0909099	−0.00409438
$494$	0	0
$495$	−2.46908	−0.110977
$496$	0	0
$497$	−19.7830	−0.887390
$498$	0	0
$499$	−30.4157	−1.36159	−0.680796	−	0.732473i	$-0.738366\pi$
$499$	−30.4157	−1.36159	−0.680796	+	0.732473i	$0.738366\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	17.1744	0.765768	0.382884	−	0.923797i	$-0.374931\pi$
$503$	17.1744	0.765768	0.382884	+	0.923797i	$0.374931\pi$
$504$	0	0
$505$	−32.8229	−1.46060
$506$	0	0
$507$	−8.47989	−0.376605
$508$	0	0
$509$	−17.4265	−0.772415	−0.386208	−	0.922412i	$-0.626215\pi$
$509$	−17.4265	−0.772415	−0.386208	+	0.922412i	$0.626215\pi$
$510$	0	0
$511$	−57.9331	−2.56281
$512$	0	0
$513$	6.72119	0.296748
$514$	0	0
$515$	17.6411	0.777359
$516$	0	0
$517$	−2.68368	−0.118028
$518$	0	0
$519$	−11.4049	−0.500618
$520$	0	0
$521$	5.02670	0.220224	0.110112	−	0.993919i	$-0.464879\pi$
$521$	5.02670	0.220224	0.110112	+	0.993919i	$0.464879\pi$
$522$	0	0
$523$	34.5066	1.50887	0.754434	−	0.656376i	$-0.227912\pi$
$523$	34.5066	1.50887	0.754434	+	0.656376i	$0.227912\pi$
$524$	0	0
$525$	25.0108	1.09156
$526$	0	0
$527$	−0.126056	−0.00549107
$528$	0	0
$529$	53.0591	2.30692
$530$	0	0
$531$	0.828489	0.0359534
$532$	0	0
$533$	15.8229	0.685366
$534$	0	0
$535$	54.9064	2.37381
$536$	0	0
$537$	−22.6678	−0.978187
$538$	0	0
$539$	−4.93815	−0.212701
$540$	0	0
$541$	−30.0914	−1.29373	−0.646865	−	0.762604i	$-0.723920\pi$
$541$	−30.0914	−1.29373	−0.646865	+	0.762604i	$0.723920\pi$
$542$	0	0
$543$	25.2629	1.08414
$544$	0	0
$545$	53.3004	2.28314
$546$	0	0
$547$	−33.1931	−1.41924	−0.709618	−	0.704587i	$-0.751132\pi$
$547$	−33.1931	−1.41924	−0.709618	+	0.704587i	$0.751132\pi$
$548$	0	0
$549$	−9.19027	−0.392231
$550$	0	0
$551$	4.84724	0.206500
$552$	0	0
$553$	−35.1369	−1.49417
$554$	0	0
$555$	25.4157	1.07884
$556$	0	0
$557$	−0.378167	−0.0160235	−0.00801173	−	0.999968i	$-0.502550\pi$
$557$	−0.378167	−0.0160235	−0.00801173	+	0.999968i	$0.502550\pi$
$558$	0	0
$559$	21.6083	0.913934
$560$	0	0
$561$	0.0909099	0.00383822
$562$	0	0
$563$	27.6593	1.16570	0.582851	−	0.812579i	$-0.301937\pi$
$563$	27.6593	1.16570	0.582851	+	0.812579i	$0.301937\pi$
$564$	0	0
$565$	−32.3271	−1.36001
$566$	0	0
$567$	3.72119	0.156275
$568$	0	0
$569$	−23.0108	−0.964663	−0.482332	−	0.875989i	$-0.660210\pi$
$569$	−23.0108	−0.964663	−0.482332	+	0.875989i	$0.660210\pi$
$570$	0	0
$571$	3.38848	0.141803	0.0709017	−	0.997483i	$-0.477412\pi$
$571$	3.38848	0.141803	0.0709017	+	0.997483i	$0.477412\pi$
$572$	0	0
$573$	8.74789	0.365448
$574$	0	0
$575$	58.6168	2.44449
$576$	0	0
$577$	−10.5309	−0.438408	−0.219204	−	0.975679i	$-0.570346\pi$
$577$	−10.5309	−0.438408	−0.219204	+	0.975679i	$0.570346\pi$
$578$	0	0
$579$	13.0642	0.542930
$580$	0	0
$581$	21.7883	0.903929
$582$	0	0
$583$	5.26292	0.217968
$584$	0	0
$585$	−7.27881	−0.300942
$586$	0	0
$587$	18.4878	0.763074	0.381537	−	0.924353i	$-0.375395\pi$
$587$	18.4878	0.763074	0.381537	+	0.924353i	$0.375395\pi$
$588$	0	0
$589$	6.72119	0.276942
$590$	0	0
$591$	11.3163	0.465491
$592$	0	0
$593$	−0.415677	−0.0170698	−0.00853491	−	0.999964i	$-0.502717\pi$
$593$	−0.415677	−0.0170698	−0.00853491	+	0.999964i	$0.502717\pi$
$594$	0	0
$595$	−1.60594	−0.0658372
$596$	0	0
$597$	0.847244	0.0346754
$598$	0	0
$599$	18.2629	0.746203	0.373101	−	0.927791i	$-0.378294\pi$
$599$	18.2629	0.746203	0.373101	+	0.927791i	$0.378294\pi$
$600$	0	0
$601$	28.6059	1.16686	0.583430	−	0.812163i	$-0.301710\pi$
$601$	28.6059	1.16686	0.583430	+	0.812163i	$0.301710\pi$
$602$	0	0
$603$	3.17151	0.129154
$604$	0	0
$605$	−35.8792	−1.45870
$606$	0	0
$607$	−25.2278	−1.02396	−0.511982	−	0.858996i	$-0.671089\pi$
$607$	−25.2278	−1.02396	−0.511982	+	0.858996i	$0.671089\pi$
$608$	0	0
$609$	2.68368	0.108748
$610$	0	0
$611$	−7.91145	−0.320063
$612$	0	0
$613$	30.1261	1.21678	0.608390	−	0.793638i	$-0.291816\pi$
$613$	30.1261	1.21678	0.608390	+	0.793638i	$0.291816\pi$
$614$	0	0
$615$	−25.4799	−1.02745
$616$	0	0
$617$	10.5951	0.426544	0.213272	−	0.976993i	$-0.431588\pi$
$617$	10.5951	0.426544	0.213272	+	0.976993i	$0.431588\pi$
$618$	0	0
$619$	−23.9490	−0.962590	−0.481295	−	0.876559i	$-0.659833\pi$
$619$	−23.9490	−0.962590	−0.481295	+	0.876559i	$0.659833\pi$
$620$	0	0
$621$	8.72119	0.349969
$622$	0	0
$623$	29.6702	1.18871
$624$	0	0
$625$	−13.4316	−0.537263
$626$	0	0
$627$	−4.84724	−0.193580
$628$	0	0
$629$	−0.935790	−0.0373124
$630$	0	0
$631$	10.9841	0.437271	0.218635	−	0.975807i	$-0.429840\pi$
$631$	10.9841	0.437271	0.218635	+	0.975807i	$0.429840\pi$
$632$	0	0
$633$	3.69449	0.146843
$634$	0	0
$635$	65.1772	2.58648
$636$	0	0
$637$	−14.5576	−0.576794
$638$	0	0
$639$	−5.31632	−0.210310
$640$	0	0
$641$	21.3890	0.844814	0.422407	−	0.906406i	$-0.361185\pi$
$641$	21.3890	0.844814	0.422407	+	0.906406i	$0.361185\pi$
$642$	0	0
$643$	2.66779	0.105207	0.0526037	−	0.998615i	$-0.483248\pi$
$643$	2.66779	0.105207	0.0526037	+	0.998615i	$0.483248\pi$
$644$	0	0
$645$	−34.7962	−1.37010
$646$	0	0
$647$	8.84724	0.347821	0.173911	−	0.984761i	$-0.444360\pi$
$647$	8.84724	0.347821	0.173911	+	0.984761i	$0.444360\pi$
$648$	0	0
$649$	−0.597497	−0.0234538
$650$	0	0
$651$	3.72119	0.145845
$652$	0	0
$653$	−37.3356	−1.46105	−0.730527	−	0.682884i	$-0.760726\pi$
$653$	−37.3356	−1.46105	−0.730527	+	0.682884i	$0.760726\pi$
$654$	0	0
$655$	−29.8581	−1.16665
$656$	0	0
$657$	−15.5684	−0.607382
$658$	0	0
$659$	21.4963	0.837376	0.418688	−	0.908130i	$-0.362490\pi$
$659$	21.4963	0.837376	0.418688	+	0.908130i	$0.362490\pi$
$660$	0	0
$661$	−4.68368	−0.182174	−0.0910870	−	0.995843i	$-0.529034\pi$
$661$	−4.68368	−0.182174	−0.0910870	+	0.995843i	$0.529034\pi$
$662$	0	0
$663$	0.268001	0.0104083
$664$	0	0
$665$	85.6276	3.32049
$666$	0	0
$667$	6.28962	0.243535
$668$	0	0
$669$	−10.0108	−0.387040
$670$	0	0
$671$	6.62791	0.255868
$672$	0	0
$673$	6.66779	0.257024	0.128512	−	0.991708i	$-0.458980\pi$
$673$	6.66779	0.257024	0.128512	+	0.991708i	$0.458980\pi$
$674$	0	0
$675$	6.72119	0.258699
$676$	0	0
$677$	37.2095	1.43008	0.715039	−	0.699085i	$-0.246409\pi$
$677$	37.2095	1.43008	0.715039	+	0.699085i	$0.246409\pi$
$678$	0	0
$679$	−51.1987	−1.96483
$680$	0	0
$681$	17.7132	0.678773
$682$	0	0
$683$	−22.5066	−0.861191	−0.430595	−	0.902545i	$-0.641696\pi$
$683$	−22.5066	−0.861191	−0.430595	+	0.902545i	$0.641696\pi$
$684$	0	0
$685$	1.94610	0.0743566
$686$	0	0
$687$	27.3890	1.04495
$688$	0	0
$689$	15.5150	0.591076
$690$	0	0
$691$	20.4691	0.778680	0.389340	−	0.921094i	$-0.372703\pi$
$691$	20.4691	0.778680	0.389340	+	0.921094i	$0.372703\pi$
$692$	0	0
$693$	−2.68368	−0.101945
$694$	0	0
$695$	−22.6435	−0.858915
$696$	0	0
$697$	0.938154	0.0355351
$698$	0	0
$699$	18.4049	0.696136
$700$	0	0
$701$	−42.9598	−1.62257	−0.811284	−	0.584652i	$-0.801231\pi$
$701$	−42.9598	−1.62257	−0.811284	+	0.584652i	$0.801231\pi$
$702$	0	0
$703$	49.8956	1.88185
$704$	0	0
$705$	12.7399	0.479814
$706$	0	0
$707$	−35.6757	−1.34172
$708$	0	0
$709$	−37.2817	−1.40014	−0.700071	−	0.714073i	$-0.746849\pi$
$709$	−37.2817	−1.40014	−0.700071	+	0.714073i	$0.746849\pi$
$710$	0	0
$711$	−9.44238	−0.354117
$712$	0	0
$713$	8.72119	0.326611
$714$	0	0
$715$	5.24940	0.196316
$716$	0	0
$717$	18.5417	0.692454
$718$	0	0
$719$	−18.8848	−0.704282	−0.352141	−	0.935947i	$-0.614546\pi$
$719$	−18.8848	−0.704282	−0.352141	+	0.935947i	$0.614546\pi$
$720$	0	0
$721$	19.1744	0.714091
$722$	0	0
$723$	−22.4157	−0.833648
$724$	0	0
$725$	4.84724	0.180022
$726$	0	0
$727$	0.126056	0.00467515	0.00233757	−	0.999997i	$-0.499256\pi$
$727$	0.126056	0.00467515	0.00233757	+	0.999997i	$0.499256\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.28118	0.0473860
$732$	0	0
$733$	2.50659	0.0925829	0.0462915	−	0.998928i	$-0.485260\pi$
$733$	2.50659	0.0925829	0.0462915	+	0.998928i	$0.485260\pi$
$734$	0	0
$735$	23.4424	0.864685
$736$	0	0
$737$	−2.28726	−0.0842522
$738$	0	0
$739$	45.0137	1.65585	0.827927	−	0.560835i	$-0.189520\pi$
$739$	45.0137	1.65585	0.827927	+	0.560835i	$0.189520\pi$
$740$	0	0
$741$	−14.2896	−0.524942
$742$	0	0
$743$	−4.33221	−0.158933	−0.0794667	−	0.996838i	$-0.525322\pi$
$743$	−4.33221	−0.158933	−0.0794667	+	0.996838i	$0.525322\pi$
$744$	0	0
$745$	−23.8097	−0.872321
$746$	0	0
$747$	5.85519	0.214230
$748$	0	0
$749$	59.6786	2.18061
$750$	0	0
$751$	−14.9199	−0.544435	−0.272217	−	0.962236i	$-0.587757\pi$
$751$	−14.9199	−0.544435	−0.272217	+	0.962236i	$0.587757\pi$
$752$	0	0
$753$	−0.935790	−0.0341021
$754$	0	0
$755$	−56.8155	−2.06773
$756$	0	0
$757$	0.630279	0.0229079	0.0114539	−	0.999934i	$-0.496354\pi$
$757$	0.630279	0.0229079	0.0114539	+	0.999934i	$0.496354\pi$
$758$	0	0
$759$	−6.28962	−0.228299
$760$	0	0
$761$	−23.3055	−0.844824	−0.422412	−	0.906404i	$-0.638816\pi$
$761$	−23.3055	−0.844824	−0.422412	+	0.906404i	$0.638816\pi$
$762$	0	0
$763$	57.9331	2.09732
$764$	0	0
$765$	−0.431567	−0.0156033
$766$	0	0
$767$	−1.76141	−0.0636010
$768$	0	0
$769$	31.8364	1.14805	0.574026	−	0.818837i	$-0.305381\pi$
$769$	31.8364	1.14805	0.574026	+	0.818837i	$0.305381\pi$
$770$	0	0
$771$	−0.163566	−0.00589067
$772$	0	0
$773$	30.6865	1.10372	0.551859	−	0.833937i	$-0.313919\pi$
$773$	30.6865	1.10372	0.551859	+	0.833937i	$0.313919\pi$
$774$	0	0
$775$	6.72119	0.241432
$776$	0	0
$777$	27.6247	0.991031
$778$	0	0
$779$	−50.0216	−1.79221
$780$	0	0
$781$	3.83407	0.137194
$782$	0	0
$783$	0.721189	0.0257732
$784$	0	0
$785$	41.2119	1.47092
$786$	0	0
$787$	15.2788	0.544631	0.272315	−	0.962208i	$-0.412211\pi$
$787$	15.2788	0.544631	0.272315	+	0.962208i	$0.412211\pi$
$788$	0	0
$789$	−24.6059	−0.875995
$790$	0	0
$791$	−35.1369	−1.24932
$792$	0	0
$793$	19.5390	0.693851
$794$	0	0
$795$	−24.9841	−0.886095
$796$	0	0
$797$	53.5901	1.89826	0.949129	−	0.314889i	$-0.101967\pi$
$797$	53.5901	1.89826	0.949129	+	0.314889i	$0.101967\pi$
$798$	0	0
$799$	−0.469077	−0.0165948
$800$	0	0
$801$	7.97330	0.281723
$802$	0	0
$803$	11.2278	0.396220
$804$	0	0
$805$	111.107	3.91602
$806$	0	0
$807$	−1.67573	−0.0589886
$808$	0	0
$809$	1.68368	0.0591950	0.0295975	−	0.999562i	$-0.490577\pi$
$809$	1.68368	0.0591950	0.0295975	+	0.999562i	$0.490577\pi$
$810$	0	0
$811$	−27.0647	−0.950371	−0.475185	−	0.879886i	$-0.657619\pi$
$811$	−27.0647	−0.950371	−0.475185	+	0.879886i	$0.657619\pi$
$812$	0	0
$813$	−5.87394	−0.206008
$814$	0	0
$815$	−52.5015	−1.83905
$816$	0	0
$817$	−68.3112	−2.38991
$818$	0	0
$819$	−7.91145	−0.276449
$820$	0	0
$821$	−50.0943	−1.74830	−0.874151	−	0.485654i	$-0.838582\pi$
$821$	−50.0943	−1.74830	−0.874151	+	0.485654i	$0.838582\pi$
$822$	0	0
$823$	−11.0483	−0.385120	−0.192560	−	0.981285i	$-0.561679\pi$
$823$	−11.0483	−0.385120	−0.192560	+	0.981285i	$0.561679\pi$
$824$	0	0
$825$	−4.84724	−0.168759
$826$	0	0
$827$	3.78303	0.131549	0.0657745	−	0.997835i	$-0.479048\pi$
$827$	3.78303	0.131549	0.0657745	+	0.997835i	$0.479048\pi$
$828$	0	0
$829$	−6.18232	−0.214721	−0.107360	−	0.994220i	$-0.534240\pi$
$829$	−6.18232	−0.214721	−0.107360	+	0.994220i	$0.534240\pi$
$830$	0	0
$831$	−12.1073	−0.419998
$832$	0	0
$833$	−0.863134	−0.0299058
$834$	0	0
$835$	3.42362	0.118479
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−11.8631	−0.409561	−0.204780	−	0.978808i	$-0.565648\pi$
$839$	−11.8631	−0.409561	−0.204780	+	0.978808i	$0.565648\pi$
$840$	0	0
$841$	−28.4799	−0.982065
$842$	0	0
$843$	−7.54173	−0.259751
$844$	0	0
$845$	−29.0319	−0.998729
$846$	0	0
$847$	−38.9976	−1.33997
$848$	0	0
$849$	−31.7320	−1.08904
$850$	0	0
$851$	64.7428	2.21935
$852$	0	0
$853$	−20.0750	−0.687356	−0.343678	−	0.939088i	$-0.611673\pi$
$853$	−20.0750	−0.687356	−0.343678	+	0.939088i	$0.611673\pi$
$854$	0	0
$855$	23.0108	0.786953
$856$	0	0
$857$	−7.05340	−0.240940	−0.120470	−	0.992717i	$-0.538440\pi$
$857$	−7.05340	−0.240940	−0.120470	+	0.992717i	$0.538440\pi$
$858$	0	0
$859$	38.9949	1.33049	0.665245	−	0.746625i	$-0.268327\pi$
$859$	38.9949	1.33049	0.665245	+	0.746625i	$0.268327\pi$
$860$	0	0
$861$	−27.6945	−0.943826
$862$	0	0
$863$	−24.0750	−0.819523	−0.409762	−	0.912193i	$-0.634388\pi$
$863$	−24.0750	−0.819523	−0.409762	+	0.912193i	$0.634388\pi$
$864$	0	0
$865$	−39.0460	−1.32760
$866$	0	0
$867$	−16.9841	−0.576811
$868$	0	0
$869$	6.80973	0.231004
$870$	0	0
$871$	−6.74281	−0.228471
$872$	0	0
$873$	−13.7587	−0.465661
$874$	0	0
$875$	21.9278	0.741296
$876$	0	0
$877$	−43.2470	−1.46035	−0.730174	−	0.683261i	$-0.760561\pi$
$877$	−43.2470	−1.46035	−0.730174	+	0.683261i	$0.760561\pi$
$878$	0	0
$879$	−17.4424	−0.588317
$880$	0	0
$881$	−26.8689	−0.905235	−0.452618	−	0.891705i	$-0.649510\pi$
$881$	−26.8689	−0.905235	−0.452618	+	0.891705i	$0.649510\pi$
$882$	0	0
$883$	14.3248	0.482067	0.241033	−	0.970517i	$-0.422514\pi$
$883$	14.3248	0.482067	0.241033	+	0.970517i	$0.422514\pi$
$884$	0	0
$885$	2.83643	0.0953457
$886$	0	0
$887$	55.9307	1.87797	0.938985	−	0.343959i	$-0.111768\pi$
$887$	55.9307	1.87797	0.938985	+	0.343959i	$0.111768\pi$
$888$	0	0
$889$	70.8422	2.37597
$890$	0	0
$891$	−0.721189	−0.0241607
$892$	0	0
$893$	25.0108	0.836955
$894$	0	0
$895$	−77.6059	−2.59408
$896$	0	0
$897$	−18.5417	−0.619090
$898$	0	0
$899$	0.721189	0.0240530
$900$	0	0
$901$	0.919900	0.0306463
$902$	0	0
$903$	−37.8205	−1.25859
$904$	0	0
$905$	86.4907	2.87505
$906$	0	0
$907$	−49.7344	−1.65140	−0.825701	−	0.564108i	$-0.809220\pi$
$907$	−49.7344	−1.65140	−0.825701	+	0.564108i	$0.809220\pi$
$908$	0	0
$909$	−9.58719	−0.317987
$910$	0	0
$911$	41.8314	1.38593	0.692967	−	0.720969i	$-0.256303\pi$
$911$	41.8314	1.38593	0.692967	+	0.720969i	$0.256303\pi$
$912$	0	0
$913$	−4.22270	−0.139751
$914$	0	0
$915$	−31.4640	−1.04017
$916$	0	0
$917$	−32.4532	−1.07170
$918$	0	0
$919$	−52.1392	−1.71991	−0.859957	−	0.510366i	$-0.829510\pi$
$919$	−52.1392	−1.71991	−0.859957	+	0.510366i	$0.829510\pi$
$920$	0	0
$921$	−6.39692	−0.210786
$922$	0	0
$923$	11.3028	0.372036
$924$	0	0
$925$	49.8956	1.64056
$926$	0	0
$927$	5.15276	0.169239
$928$	0	0
$929$	−39.0375	−1.28078	−0.640390	−	0.768050i	$-0.721227\pi$
$929$	−39.0375	−1.28078	−0.640390	+	0.768050i	$0.721227\pi$
$930$	0	0
$931$	46.0216	1.50830
$932$	0	0
$933$	23.5951	0.772470
$934$	0	0
$935$	0.311241	0.0101787
$936$	0	0
$937$	−33.0108	−1.07842	−0.539208	−	0.842173i	$-0.681276\pi$
$937$	−33.0108	−1.07842	−0.539208	+	0.842173i	$0.681276\pi$
$938$	0	0
$939$	5.01081	0.163522
$940$	0	0
$941$	−26.9574	−0.878786	−0.439393	−	0.898295i	$-0.644807\pi$
$941$	−26.9574	−0.878786	−0.439393	+	0.898295i	$0.644807\pi$
$942$	0	0
$943$	−64.9064	−2.11364
$944$	0	0
$945$	12.7399	0.414430
$946$	0	0
$947$	43.5174	1.41413	0.707063	−	0.707151i	$-0.250020\pi$
$947$	43.5174	1.41413	0.707063	+	0.707151i	$0.250020\pi$
$948$	0	0
$949$	33.0994	1.07445
$950$	0	0
$951$	25.9115	0.840236
$952$	0	0
$953$	13.3697	0.433088	0.216544	−	0.976273i	$-0.430522\pi$
$953$	13.3697	0.433088	0.216544	+	0.976273i	$0.430522\pi$
$954$	0	0
$955$	29.9495	0.969142
$956$	0	0
$957$	−0.520113	−0.0168129
$958$	0	0
$959$	2.11525	0.0683048
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	16.0375	0.516802
$964$	0	0
$965$	44.7269	1.43981
$966$	0	0
$967$	−35.3890	−1.13803	−0.569016	−	0.822326i	$-0.692676\pi$
$967$	−35.3890	−1.13803	−0.569016	+	0.822326i	$0.692676\pi$
$968$	0	0
$969$	−0.847244	−0.0272174
$970$	0	0
$971$	−31.5713	−1.01317	−0.506586	−	0.862190i	$-0.669093\pi$
$971$	−31.5713	−1.01317	−0.506586	+	0.862190i	$0.669093\pi$
$972$	0	0
$973$	−24.6115	−0.789009
$974$	0	0
$975$	−14.2896	−0.457634
$976$	0	0
$977$	36.9733	1.18288	0.591440	−	0.806349i	$-0.298560\pi$
$977$	36.9733	1.18288	0.591440	+	0.806349i	$0.298560\pi$
$978$	0	0
$979$	−5.75025	−0.183779
$980$	0	0
$981$	15.5684	0.497062
$982$	0	0
$983$	−38.3646	−1.22364	−0.611821	−	0.790996i	$-0.709563\pi$
$983$	−38.3646	−1.22364	−0.611821	+	0.790996i	$0.709563\pi$
$984$	0	0
$985$	38.7428	1.23445
$986$	0	0
$987$	13.8472	0.440763
$988$	0	0
$989$	−88.6384	−2.81854
$990$	0	0
$991$	−10.1044	−0.320978	−0.160489	−	0.987038i	$-0.551307\pi$
$991$	−10.1044	−0.320978	−0.160489	+	0.987038i	$0.551307\pi$
$992$	0	0
$993$	21.9115	0.695339
$994$	0	0
$995$	2.90064	0.0919566
$996$	0	0
$997$	9.78303	0.309832	0.154916	−	0.987928i	$-0.450489\pi$
$997$	9.78303	0.309832	0.154916	+	0.987928i	$0.450489\pi$
$998$	0	0
$999$	7.42362	0.234873

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.m.1.3		3
4.3	odd	2		1002.2.a.g.1.3	✓	3
12.11	even	2		3006.2.a.q.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.g.1.3	✓	3	4.3	odd	2
3006.2.a.q.1.1		3	12.11	even	2
8016.2.a.m.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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.42362 q^{5} +3.72119 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.42362 q^{5} +3.72119 q^{7} +1.00000 q^{9} -0.721189 q^{11} -2.12606 q^{13} +3.42362 q^{15} -0.126056 q^{17} +6.72119 q^{19} +3.72119 q^{21} +8.72119 q^{23} +6.72119 q^{25} +1.00000 q^{27} +0.721189 q^{29} +1.00000 q^{31} -0.721189 q^{33} +12.7399 q^{35} +7.42362 q^{37} -2.12606 q^{39} -7.44238 q^{41} -10.1636 q^{43} +3.42362 q^{45} +3.72119 q^{47} +6.84724 q^{49} -0.126056 q^{51} -7.29757 q^{53} -2.46908 q^{55} +6.72119 q^{57} +0.828489 q^{59} -9.19027 q^{61} +3.72119 q^{63} -7.27881 q^{65} +3.17151 q^{67} +8.72119 q^{69} -5.31632 q^{71} -15.5684 q^{73} +6.72119 q^{75} -2.68368 q^{77} -9.44238 q^{79} +1.00000 q^{81} +5.85519 q^{83} -0.431567 q^{85} +0.721189 q^{87} +7.97330 q^{89} -7.91145 q^{91} +1.00000 q^{93} +23.0108 q^{95} -13.7587 q^{97} -0.721189 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} - q^{7} + 3 q^{9} + 10 q^{11} - 4 q^{13} + 3 q^{15} + 2 q^{17} + 8 q^{19} - q^{21} + 14 q^{23} + 8 q^{25} + 3 q^{27} - 10 q^{29} + 3 q^{31} + 10 q^{33} + 9 q^{35} + 15 q^{37} - 4 q^{39} + 2 q^{41} + 6 q^{43} + 3 q^{45} - q^{47} + 6 q^{49} + 2 q^{51} - 17 q^{53} + 8 q^{57} + 5 q^{59} - 8 q^{61} - q^{63} - 34 q^{65} + 7 q^{67} + 14 q^{69} + 6 q^{71} - 20 q^{73} + 8 q^{75} - 30 q^{77} - 4 q^{79} + 3 q^{81} + 37 q^{83} - 28 q^{85} - 10 q^{87} + 7 q^{89} + 8 q^{91} + 3 q^{93} + 18 q^{95} + 5 q^{97} + 10 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 - q^7 + 3 * q^9 + 10 * q^11 - 4 * q^13 + 3 * q^15 + 2 * q^17 + 8 * q^19 - q^21 + 14 * q^23 + 8 * q^25 + 3 * q^27 - 10 * q^29 + 3 * q^31 + 10 * q^33 + 9 * q^35 + 15 * q^37 - 4 * q^39 + 2 * q^41 + 6 * q^43 + 3 * q^45 - q^47 + 6 * q^49 + 2 * q^51 - 17 * q^53 + 8 * q^57 + 5 * q^59 - 8 * q^61 - q^63 - 34 * q^65 + 7 * q^67 + 14 * q^69 + 6 * q^71 - 20 * q^73 + 8 * q^75 - 30 * q^77 - 4 * q^79 + 3 * q^81 + 37 * q^83 - 28 * q^85 - 10 * q^87 + 7 * q^89 + 8 * q^91 + 3 * q^93 + 18 * q^95 + 5 * q^97 + 10 * q^99

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