LMFDB - Embedded newform 8016.2.a.bb.1.5

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 $ x^9 - 29x^7 - 7x^6 + 266x^5 + 69x^4 - 901x^3 - 199x^2 + 875*x + 391
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-0.529979$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.52998 q^{5} +1.05249 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.52998 q^{5} +1.05249 q^{7} +1.00000 q^{9} -0.162063 q^{11} +3.63900 q^{13} -1.52998 q^{15} +4.89726 q^{17} -1.02632 q^{19} -1.05249 q^{21} +7.59021 q^{23} -2.65917 q^{25} -1.00000 q^{27} +9.59787 q^{29} -2.68185 q^{31} +0.162063 q^{33} +1.61029 q^{35} +4.12252 q^{37} -3.63900 q^{39} -1.08012 q^{41} -2.77829 q^{43} +1.52998 q^{45} +9.50274 q^{47} -5.89226 q^{49} -4.89726 q^{51} +4.80545 q^{53} -0.247952 q^{55} +1.02632 q^{57} -8.22708 q^{59} +0.510754 q^{61} +1.05249 q^{63} +5.56759 q^{65} +0.317219 q^{67} -7.59021 q^{69} -5.83623 q^{71} +11.2379 q^{73} +2.65917 q^{75} -0.170570 q^{77} -5.59029 q^{79} +1.00000 q^{81} -4.98299 q^{83} +7.49271 q^{85} -9.59787 q^{87} -0.199333 q^{89} +3.83003 q^{91} +2.68185 q^{93} -1.57025 q^{95} -10.9371 q^{97} -0.162063 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9} - 7 q^{11} + 6 q^{13} - 9 q^{15} + 7 q^{17} - 2 q^{19} + 2 q^{21} - 19 q^{23} + 22 q^{25} - 9 q^{27} + 13 q^{29} - 12 q^{31} + 7 q^{33} - 4 q^{35} + 15 q^{37} - 6 q^{39} + 18 q^{41} + 6 q^{43} + 9 q^{45} - 25 q^{47} + 19 q^{49} - 7 q^{51} + 17 q^{53} + 3 q^{55} + 2 q^{57} - 3 q^{59} + 14 q^{61} - 2 q^{63} + 14 q^{65} + 4 q^{67} + 19 q^{69} - 17 q^{71} - 20 q^{73} - 22 q^{75} + 14 q^{77} + 8 q^{79} + 9 q^{81} + q^{83} + 5 q^{85} - 13 q^{87} + 36 q^{89} + 41 q^{91} + 12 q^{93} - 5 q^{95} + 31 q^{97} - 7 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9 - 7 * q^11 + 6 * q^13 - 9 * q^15 + 7 * q^17 - 2 * q^19 + 2 * q^21 - 19 * q^23 + 22 * q^25 - 9 * q^27 + 13 * q^29 - 12 * q^31 + 7 * q^33 - 4 * q^35 + 15 * q^37 - 6 * q^39 + 18 * q^41 + 6 * q^43 + 9 * q^45 - 25 * q^47 + 19 * q^49 - 7 * q^51 + 17 * q^53 + 3 * q^55 + 2 * q^57 - 3 * q^59 + 14 * q^61 - 2 * q^63 + 14 * q^65 + 4 * q^67 + 19 * q^69 - 17 * q^71 - 20 * q^73 - 22 * q^75 + 14 * q^77 + 8 * q^79 + 9 * q^81 + q^83 + 5 * q^85 - 13 * q^87 + 36 * q^89 + 41 * q^91 + 12 * q^93 - 5 * q^95 + 31 * q^97 - 7 * 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
$5$	1.52998	0.684227	0.342114	−	0.939659i	$-0.388857\pi$
$5$	1.52998	0.684227	0.342114	+	0.939659i	$0.388857\pi$
$6$	0	0
$7$	1.05249	0.397806	0.198903	−	0.980019i	$-0.436262\pi$
$7$	1.05249	0.397806	0.198903	+	0.980019i	$0.436262\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.162063	−0.0488637	−0.0244318	−	0.999701i	$-0.507778\pi$
$11$	−0.162063	−0.0488637	−0.0244318	+	0.999701i	$0.507778\pi$
$12$	0	0
$13$	3.63900	1.00928	0.504638	−	0.863331i	$-0.331626\pi$
$13$	3.63900	1.00928	0.504638	+	0.863331i	$0.331626\pi$
$14$	0	0
$15$	−1.52998	−0.395039
$16$	0	0
$17$	4.89726	1.18776	0.593880	−	0.804553i	$-0.297595\pi$
$17$	4.89726	1.18776	0.593880	+	0.804553i	$0.297595\pi$
$18$	0	0
$19$	−1.02632	−0.235455	−0.117727	−	0.993046i	$-0.537561\pi$
$19$	−1.02632	−0.235455	−0.117727	+	0.993046i	$0.537561\pi$
$20$	0	0
$21$	−1.05249	−0.229673
$22$	0	0
$23$	7.59021	1.58267	0.791334	−	0.611384i	$-0.209387\pi$
$23$	7.59021	1.58267	0.791334	+	0.611384i	$0.209387\pi$
$24$	0	0
$25$	−2.65917	−0.531833
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	9.59787	1.78228	0.891140	−	0.453729i	$-0.149906\pi$
$29$	9.59787	1.78228	0.891140	+	0.453729i	$0.149906\pi$
$30$	0	0
$31$	−2.68185	−0.481675	−0.240837	−	0.970565i	$-0.577422\pi$
$31$	−2.68185	−0.481675	−0.240837	+	0.970565i	$0.577422\pi$
$32$	0	0
$33$	0.162063	0.0282115
$34$	0	0
$35$	1.61029	0.272189
$36$	0	0
$37$	4.12252	0.677738	0.338869	−	0.940834i	$-0.389956\pi$
$37$	4.12252	0.677738	0.338869	+	0.940834i	$0.389956\pi$
$38$	0	0
$39$	−3.63900	−0.582706
$40$	0	0
$41$	−1.08012	−0.168687	−0.0843433	−	0.996437i	$-0.526879\pi$
$41$	−1.08012	−0.168687	−0.0843433	+	0.996437i	$0.526879\pi$
$42$	0	0
$43$	−2.77829	−0.423686	−0.211843	−	0.977304i	$-0.567946\pi$
$43$	−2.77829	−0.423686	−0.211843	+	0.977304i	$0.567946\pi$
$44$	0	0
$45$	1.52998	0.228076
$46$	0	0
$47$	9.50274	1.38612	0.693059	−	0.720881i	$-0.256263\pi$
$47$	9.50274	1.38612	0.693059	+	0.720881i	$0.256263\pi$
$48$	0	0
$49$	−5.89226	−0.841751
$50$	0	0
$51$	−4.89726	−0.685754
$52$	0	0
$53$	4.80545	0.660080	0.330040	−	0.943967i	$-0.392938\pi$
$53$	4.80545	0.660080	0.330040	+	0.943967i	$0.392938\pi$
$54$	0	0
$55$	−0.247952	−0.0334339
$56$	0	0
$57$	1.02632	0.135940
$58$	0	0
$59$	−8.22708	−1.07107	−0.535537	−	0.844512i	$-0.679891\pi$
$59$	−8.22708	−1.07107	−0.535537	+	0.844512i	$0.679891\pi$
$60$	0	0
$61$	0.510754	0.0653953	0.0326976	−	0.999465i	$-0.489590\pi$
$61$	0.510754	0.0653953	0.0326976	+	0.999465i	$0.489590\pi$
$62$	0	0
$63$	1.05249	0.132602
$64$	0	0
$65$	5.56759	0.690575
$66$	0	0
$67$	0.317219	0.0387544	0.0193772	−	0.999812i	$-0.493832\pi$
$67$	0.317219	0.0387544	0.0193772	+	0.999812i	$0.493832\pi$
$68$	0	0
$69$	−7.59021	−0.913754
$70$	0	0
$71$	−5.83623	−0.692633	−0.346316	−	0.938118i	$-0.612568\pi$
$71$	−5.83623	−0.692633	−0.346316	+	0.938118i	$0.612568\pi$
$72$	0	0
$73$	11.2379	1.31529	0.657647	−	0.753326i	$-0.271552\pi$
$73$	11.2379	1.31529	0.657647	+	0.753326i	$0.271552\pi$
$74$	0	0
$75$	2.65917	0.307054
$76$	0	0
$77$	−0.170570	−0.0194382
$78$	0	0
$79$	−5.59029	−0.628957	−0.314478	−	0.949265i	$-0.601830\pi$
$79$	−5.59029	−0.628957	−0.314478	+	0.949265i	$0.601830\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−4.98299	−0.546954	−0.273477	−	0.961879i	$-0.588174\pi$
$83$	−4.98299	−0.546954	−0.273477	+	0.961879i	$0.588174\pi$
$84$	0	0
$85$	7.49271	0.812698
$86$	0	0
$87$	−9.59787	−1.02900
$88$	0	0
$89$	−0.199333	−0.0211292	−0.0105646	−	0.999944i	$-0.503363\pi$
$89$	−0.199333	−0.0211292	−0.0105646	+	0.999944i	$0.503363\pi$
$90$	0	0
$91$	3.83003	0.401496
$92$	0	0
$93$	2.68185	0.278095
$94$	0	0
$95$	−1.57025	−0.161104
$96$	0	0
$97$	−10.9371	−1.11049	−0.555247	−	0.831686i	$-0.687376\pi$
$97$	−10.9371	−1.11049	−0.555247	+	0.831686i	$0.687376\pi$
$98$	0	0
$99$	−0.162063	−0.0162879
$100$	0	0
$101$	11.8811	1.18221	0.591107	−	0.806593i	$-0.298691\pi$
$101$	11.8811	1.18221	0.591107	+	0.806593i	$0.298691\pi$
$102$	0	0
$103$	1.54707	0.152437	0.0762187	−	0.997091i	$-0.475715\pi$
$103$	1.54707	0.152437	0.0762187	+	0.997091i	$0.475715\pi$
$104$	0	0
$105$	−1.61029	−0.157149
$106$	0	0
$107$	2.89728	0.280091	0.140045	−	0.990145i	$-0.455275\pi$
$107$	2.89728	0.280091	0.140045	+	0.990145i	$0.455275\pi$
$108$	0	0
$109$	8.16825	0.782377	0.391188	−	0.920311i	$-0.372064\pi$
$109$	8.16825	0.782377	0.391188	+	0.920311i	$0.372064\pi$
$110$	0	0
$111$	−4.12252	−0.391292
$112$	0	0
$113$	−6.73714	−0.633777	−0.316889	−	0.948463i	$-0.602638\pi$
$113$	−6.73714	−0.633777	−0.316889	+	0.948463i	$0.602638\pi$
$114$	0	0
$115$	11.6129	1.08290
$116$	0	0
$117$	3.63900	0.336426
$118$	0	0
$119$	5.15434	0.472498
$120$	0	0
$121$	−10.9737	−0.997612
$122$	0	0
$123$	1.08012	0.0973912
$124$	0	0
$125$	−11.7184	−1.04812
$126$	0	0
$127$	15.8812	1.40923	0.704616	−	0.709588i	$-0.251119\pi$
$127$	15.8812	1.40923	0.704616	+	0.709588i	$0.251119\pi$
$128$	0	0
$129$	2.77829	0.244615
$130$	0	0
$131$	−9.81474	−0.857517	−0.428759	−	0.903419i	$-0.641049\pi$
$131$	−9.81474	−0.857517	−0.428759	+	0.903419i	$0.641049\pi$
$132$	0	0
$133$	−1.08020	−0.0936651
$134$	0	0
$135$	−1.52998	−0.131680
$136$	0	0
$137$	−2.22904	−0.190440	−0.0952199	−	0.995456i	$-0.530355\pi$
$137$	−2.22904	−0.190440	−0.0952199	+	0.995456i	$0.530355\pi$
$138$	0	0
$139$	7.71321	0.654226	0.327113	−	0.944985i	$-0.393924\pi$
$139$	7.71321	0.654226	0.327113	+	0.944985i	$0.393924\pi$
$140$	0	0
$141$	−9.50274	−0.800275
$142$	0	0
$143$	−0.589745	−0.0493170
$144$	0	0
$145$	14.6845	1.21948
$146$	0	0
$147$	5.89226	0.485985
$148$	0	0
$149$	−19.1731	−1.57072	−0.785360	−	0.619039i	$-0.787522\pi$
$149$	−19.1731	−1.57072	−0.785360	+	0.619039i	$0.787522\pi$
$150$	0	0
$151$	8.05210	0.655271	0.327635	−	0.944804i	$-0.393748\pi$
$151$	8.05210	0.655271	0.327635	+	0.944804i	$0.393748\pi$
$152$	0	0
$153$	4.89726	0.395920
$154$	0	0
$155$	−4.10318	−0.329575
$156$	0	0
$157$	10.1858	0.812914	0.406457	−	0.913670i	$-0.366764\pi$
$157$	10.1858	0.812914	0.406457	+	0.913670i	$0.366764\pi$
$158$	0	0
$159$	−4.80545	−0.381097
$160$	0	0
$161$	7.98866	0.629594
$162$	0	0
$163$	−0.432916	−0.0339086	−0.0169543	−	0.999856i	$-0.505397\pi$
$163$	−0.432916	−0.0339086	−0.0169543	+	0.999856i	$0.505397\pi$
$164$	0	0
$165$	0.247952	0.0193031
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	0.242320	0.0186400
$170$	0	0
$171$	−1.02632	−0.0784848
$172$	0	0
$173$	17.2653	1.31266	0.656329	−	0.754475i	$-0.272108\pi$
$173$	17.2653	1.31266	0.656329	+	0.754475i	$0.272108\pi$
$174$	0	0
$175$	−2.79876	−0.211566
$176$	0	0
$177$	8.22708	0.618385
$178$	0	0
$179$	−7.05820	−0.527554	−0.263777	−	0.964584i	$-0.584968\pi$
$179$	−7.05820	−0.527554	−0.263777	+	0.964584i	$0.584968\pi$
$180$	0	0
$181$	−14.7420	−1.09577	−0.547883	−	0.836555i	$-0.684566\pi$
$181$	−14.7420	−1.09577	−0.547883	+	0.836555i	$0.684566\pi$
$182$	0	0
$183$	−0.510754	−0.0377560
$184$	0	0
$185$	6.30737	0.463727
$186$	0	0
$187$	−0.793663	−0.0580384
$188$	0	0
$189$	−1.05249	−0.0765577
$190$	0	0
$191$	−22.5477	−1.63149	−0.815747	−	0.578409i	$-0.803674\pi$
$191$	−22.5477	−1.63149	−0.815747	+	0.578409i	$0.803674\pi$
$192$	0	0
$193$	3.60996	0.259850	0.129925	−	0.991524i	$-0.458526\pi$
$193$	3.60996	0.259850	0.129925	+	0.991524i	$0.458526\pi$
$194$	0	0
$195$	−5.56759	−0.398704
$196$	0	0
$197$	12.4286	0.885502	0.442751	−	0.896645i	$-0.354003\pi$
$197$	12.4286	0.885502	0.442751	+	0.896645i	$0.354003\pi$
$198$	0	0
$199$	−6.92918	−0.491197	−0.245598	−	0.969372i	$-0.578984\pi$
$199$	−6.92918	−0.491197	−0.245598	+	0.969372i	$0.578984\pi$
$200$	0	0
$201$	−0.317219	−0.0223749
$202$	0	0
$203$	10.1017	0.709001
$204$	0	0
$205$	−1.65256	−0.115420
$206$	0	0
$207$	7.59021	0.527556
$208$	0	0
$209$	0.166328	0.0115052
$210$	0	0
$211$	20.3772	1.40283	0.701414	−	0.712754i	$-0.252553\pi$
$211$	20.3772	1.40283	0.701414	+	0.712754i	$0.252553\pi$
$212$	0	0
$213$	5.83623	0.399892
$214$	0	0
$215$	−4.25073	−0.289897
$216$	0	0
$217$	−2.82264	−0.191613
$218$	0	0
$219$	−11.2379	−0.759385
$220$	0	0
$221$	17.8211	1.19878
$222$	0	0
$223$	20.2826	1.35823	0.679113	−	0.734033i	$-0.262364\pi$
$223$	20.2826	1.35823	0.679113	+	0.734033i	$0.262364\pi$
$224$	0	0
$225$	−2.65917	−0.177278
$226$	0	0
$227$	−7.39895	−0.491085	−0.245543	−	0.969386i	$-0.578966\pi$
$227$	−7.39895	−0.491085	−0.245543	+	0.969386i	$0.578966\pi$
$228$	0	0
$229$	−10.7782	−0.712247	−0.356123	−	0.934439i	$-0.615902\pi$
$229$	−10.7782	−0.712247	−0.356123	+	0.934439i	$0.615902\pi$
$230$	0	0
$231$	0.170570	0.0112227
$232$	0	0
$233$	−7.03570	−0.460924	−0.230462	−	0.973081i	$-0.574024\pi$
$233$	−7.03570	−0.460924	−0.230462	+	0.973081i	$0.574024\pi$
$234$	0	0
$235$	14.5390	0.948420
$236$	0	0
$237$	5.59029	0.363128
$238$	0	0
$239$	−7.21369	−0.466615	−0.233307	−	0.972403i	$-0.574955\pi$
$239$	−7.21369	−0.466615	−0.233307	+	0.972403i	$0.574955\pi$
$240$	0	0
$241$	−6.07039	−0.391028	−0.195514	−	0.980701i	$-0.562638\pi$
$241$	−6.07039	−0.391028	−0.195514	+	0.980701i	$0.562638\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.01502	−0.575949
$246$	0	0
$247$	−3.73479	−0.237639
$248$	0	0
$249$	4.98299	0.315784
$250$	0	0
$251$	1.20193	0.0758652	0.0379326	−	0.999280i	$-0.487923\pi$
$251$	1.20193	0.0758652	0.0379326	+	0.999280i	$0.487923\pi$
$252$	0	0
$253$	−1.23009	−0.0773350
$254$	0	0
$255$	−7.49271	−0.469212
$256$	0	0
$257$	−14.5799	−0.909471	−0.454735	−	0.890627i	$-0.650266\pi$
$257$	−14.5799	−0.909471	−0.454735	+	0.890627i	$0.650266\pi$
$258$	0	0
$259$	4.33893	0.269608
$260$	0	0
$261$	9.59787	0.594093
$262$	0	0
$263$	−16.6235	−1.02505	−0.512524	−	0.858673i	$-0.671289\pi$
$263$	−16.6235	−1.02505	−0.512524	+	0.858673i	$0.671289\pi$
$264$	0	0
$265$	7.35224	0.451644
$266$	0	0
$267$	0.199333	0.0121990
$268$	0	0
$269$	−8.32925	−0.507843	−0.253922	−	0.967225i	$-0.581721\pi$
$269$	−8.32925	−0.507843	−0.253922	+	0.967225i	$0.581721\pi$
$270$	0	0
$271$	26.8577	1.63149	0.815744	−	0.578414i	$-0.196328\pi$
$271$	26.8577	1.63149	0.815744	+	0.578414i	$0.196328\pi$
$272$	0	0
$273$	−3.83003	−0.231804
$274$	0	0
$275$	0.430951	0.0259873
$276$	0	0
$277$	7.42647	0.446213	0.223107	−	0.974794i	$-0.428380\pi$
$277$	7.42647	0.446213	0.223107	+	0.974794i	$0.428380\pi$
$278$	0	0
$279$	−2.68185	−0.160558
$280$	0	0
$281$	2.07711	0.123910	0.0619551	−	0.998079i	$-0.480266\pi$
$281$	2.07711	0.123910	0.0619551	+	0.998079i	$0.480266\pi$
$282$	0	0
$283$	22.8081	1.35580	0.677900	−	0.735154i	$-0.262890\pi$
$283$	22.8081	1.35580	0.677900	+	0.735154i	$0.262890\pi$
$284$	0	0
$285$	1.57025	0.0930137
$286$	0	0
$287$	−1.13682	−0.0671044
$288$	0	0
$289$	6.98318	0.410775
$290$	0	0
$291$	10.9371	0.641144
$292$	0	0
$293$	−19.3193	−1.12864	−0.564322	−	0.825555i	$-0.690862\pi$
$293$	−19.3193	−1.12864	−0.564322	+	0.825555i	$0.690862\pi$
$294$	0	0
$295$	−12.5873	−0.732858
$296$	0	0
$297$	0.162063	0.00940382
$298$	0	0
$299$	27.6208	1.59735
$300$	0	0
$301$	−2.92414	−0.168544
$302$	0	0
$303$	−11.8811	−0.682552
$304$	0	0
$305$	0.781442	0.0447452
$306$	0	0
$307$	9.57084	0.546237	0.273118	−	0.961980i	$-0.411945\pi$
$307$	9.57084	0.546237	0.273118	+	0.961980i	$0.411945\pi$
$308$	0	0
$309$	−1.54707	−0.0880097
$310$	0	0
$311$	8.69478	0.493036	0.246518	−	0.969138i	$-0.420714\pi$
$311$	8.69478	0.493036	0.246518	+	0.969138i	$0.420714\pi$
$312$	0	0
$313$	−24.7334	−1.39802	−0.699009	−	0.715113i	$-0.746375\pi$
$313$	−24.7334	−1.39802	−0.699009	+	0.715113i	$0.746375\pi$
$314$	0	0
$315$	1.61029	0.0907298
$316$	0	0
$317$	30.3559	1.70496	0.852479	−	0.522762i	$-0.175098\pi$
$317$	30.3559	1.70496	0.852479	+	0.522762i	$0.175098\pi$
$318$	0	0
$319$	−1.55546	−0.0870888
$320$	0	0
$321$	−2.89728	−0.161710
$322$	0	0
$323$	−5.02617	−0.279664
$324$	0	0
$325$	−9.67670	−0.536767
$326$	0	0
$327$	−8.16825	−0.451705
$328$	0	0
$329$	10.0016	0.551405
$330$	0	0
$331$	−30.2944	−1.66513	−0.832566	−	0.553926i	$-0.813129\pi$
$331$	−30.2944	−1.66513	−0.832566	+	0.553926i	$0.813129\pi$
$332$	0	0
$333$	4.12252	0.225913
$334$	0	0
$335$	0.485338	0.0265168
$336$	0	0
$337$	−0.467044	−0.0254415	−0.0127208	−	0.999919i	$-0.504049\pi$
$337$	−0.467044	−0.0254415	−0.0127208	+	0.999919i	$0.504049\pi$
$338$	0	0
$339$	6.73714	0.365911
$340$	0	0
$341$	0.434628	0.0235364
$342$	0	0
$343$	−13.5690	−0.732659
$344$	0	0
$345$	−11.6129	−0.625215
$346$	0	0
$347$	2.81839	0.151299	0.0756496	−	0.997134i	$-0.475897\pi$
$347$	2.81839	0.151299	0.0756496	+	0.997134i	$0.475897\pi$
$348$	0	0
$349$	37.0275	1.98204	0.991018	−	0.133726i	$-0.0426943\pi$
$349$	37.0275	1.98204	0.991018	+	0.133726i	$0.0426943\pi$
$350$	0	0
$351$	−3.63900	−0.194235
$352$	0	0
$353$	−4.85955	−0.258648	−0.129324	−	0.991602i	$-0.541281\pi$
$353$	−4.85955	−0.258648	−0.129324	+	0.991602i	$0.541281\pi$
$354$	0	0
$355$	−8.92930	−0.473918
$356$	0	0
$357$	−5.15434	−0.272797
$358$	0	0
$359$	0.0868793	0.00458531	0.00229266	−	0.999997i	$-0.499270\pi$
$359$	0.0868793	0.00458531	0.00229266	+	0.999997i	$0.499270\pi$
$360$	0	0
$361$	−17.9467	−0.944561
$362$	0	0
$363$	10.9737	0.575972
$364$	0	0
$365$	17.1937	0.899960
$366$	0	0
$367$	15.3217	0.799787	0.399894	−	0.916562i	$-0.369047\pi$
$367$	15.3217	0.799787	0.399894	+	0.916562i	$0.369047\pi$
$368$	0	0
$369$	−1.08012	−0.0562288
$370$	0	0
$371$	5.05771	0.262583
$372$	0	0
$373$	0.349987	0.0181216	0.00906081	−	0.999959i	$-0.497116\pi$
$373$	0.349987	0.0181216	0.00906081	+	0.999959i	$0.497116\pi$
$374$	0	0
$375$	11.7184	0.605133
$376$	0	0
$377$	34.9267	1.79881
$378$	0	0
$379$	29.6312	1.52206	0.761028	−	0.648720i	$-0.224695\pi$
$379$	29.6312	1.52206	0.761028	+	0.648720i	$0.224695\pi$
$380$	0	0
$381$	−15.8812	−0.813621
$382$	0	0
$383$	28.4092	1.45164	0.725821	−	0.687884i	$-0.241460\pi$
$383$	28.4092	1.45164	0.725821	+	0.687884i	$0.241460\pi$
$384$	0	0
$385$	−0.260968	−0.0133002
$386$	0	0
$387$	−2.77829	−0.141229
$388$	0	0
$389$	18.4318	0.934530	0.467265	−	0.884117i	$-0.345239\pi$
$389$	18.4318	0.934530	0.467265	+	0.884117i	$0.345239\pi$
$390$	0	0
$391$	37.1713	1.87983
$392$	0	0
$393$	9.81474	0.495088
$394$	0	0
$395$	−8.55302	−0.430349
$396$	0	0
$397$	0.947842	0.0475708	0.0237854	−	0.999717i	$-0.492428\pi$
$397$	0.947842	0.0475708	0.0237854	+	0.999717i	$0.492428\pi$
$398$	0	0
$399$	1.08020	0.0540776
$400$	0	0
$401$	1.29454	0.0646464	0.0323232	−	0.999477i	$-0.489709\pi$
$401$	1.29454	0.0646464	0.0323232	+	0.999477i	$0.489709\pi$
$402$	0	0
$403$	−9.75926	−0.486143
$404$	0	0
$405$	1.52998	0.0760253
$406$	0	0
$407$	−0.668106	−0.0331168
$408$	0	0
$409$	−29.1328	−1.44053	−0.720263	−	0.693701i	$-0.755979\pi$
$409$	−29.1328	−1.44053	−0.720263	+	0.693701i	$0.755979\pi$
$410$	0	0
$411$	2.22904	0.109951
$412$	0	0
$413$	−8.65896	−0.426079
$414$	0	0
$415$	−7.62386	−0.374241
$416$	0	0
$417$	−7.71321	−0.377717
$418$	0	0
$419$	−15.3595	−0.750362	−0.375181	−	0.926952i	$-0.622419\pi$
$419$	−15.3595	−0.750362	−0.375181	+	0.926952i	$0.622419\pi$
$420$	0	0
$421$	−22.9588	−1.11894	−0.559472	−	0.828849i	$-0.688996\pi$
$421$	−22.9588	−1.11894	−0.559472	+	0.828849i	$0.688996\pi$
$422$	0	0
$423$	9.50274	0.462039
$424$	0	0
$425$	−13.0226	−0.631690
$426$	0	0
$427$	0.537565	0.0260146
$428$	0	0
$429$	0.589745	0.0284732
$430$	0	0
$431$	23.4208	1.12814	0.564070	−	0.825727i	$-0.309235\pi$
$431$	23.4208	1.12814	0.564070	+	0.825727i	$0.309235\pi$
$432$	0	0
$433$	−30.2786	−1.45510	−0.727549	−	0.686056i	$-0.759340\pi$
$433$	−30.2786	−1.45510	−0.727549	+	0.686056i	$0.759340\pi$
$434$	0	0
$435$	−14.6845	−0.704070
$436$	0	0
$437$	−7.79000	−0.372646
$438$	0	0
$439$	20.4125	0.974236	0.487118	−	0.873336i	$-0.338048\pi$
$439$	20.4125	0.974236	0.487118	+	0.873336i	$0.338048\pi$
$440$	0	0
$441$	−5.89226	−0.280584
$442$	0	0
$443$	30.1725	1.43354	0.716770	−	0.697310i	$-0.245620\pi$
$443$	30.1725	1.43354	0.716770	+	0.697310i	$0.245620\pi$
$444$	0	0
$445$	−0.304975	−0.0144572
$446$	0	0
$447$	19.1731	0.906856
$448$	0	0
$449$	−14.7721	−0.697137	−0.348568	−	0.937283i	$-0.613332\pi$
$449$	−14.7721	−0.697137	−0.348568	+	0.937283i	$0.613332\pi$
$450$	0	0
$451$	0.175047	0.00824265
$452$	0	0
$453$	−8.05210	−0.378321
$454$	0	0
$455$	5.85986	0.274715
$456$	0	0
$457$	33.0643	1.54668	0.773342	−	0.633989i	$-0.218583\pi$
$457$	33.0643	1.54668	0.773342	+	0.633989i	$0.218583\pi$
$458$	0	0
$459$	−4.89726	−0.228585
$460$	0	0
$461$	−20.9924	−0.977714	−0.488857	−	0.872364i	$-0.662586\pi$
$461$	−20.9924	−0.977714	−0.488857	+	0.872364i	$0.662586\pi$
$462$	0	0
$463$	38.8607	1.80601	0.903005	−	0.429631i	$-0.141356\pi$
$463$	38.8607	1.80601	0.903005	+	0.429631i	$0.141356\pi$
$464$	0	0
$465$	4.10318	0.190280
$466$	0	0
$467$	−30.7010	−1.42067	−0.710335	−	0.703863i	$-0.751457\pi$
$467$	−30.7010	−1.42067	−0.710335	+	0.703863i	$0.751457\pi$
$468$	0	0
$469$	0.333871	0.0154167
$470$	0	0
$471$	−10.1858	−0.469336
$472$	0	0
$473$	0.450257	0.0207028
$474$	0	0
$475$	2.72916	0.125222
$476$	0	0
$477$	4.80545	0.220027
$478$	0	0
$479$	4.10481	0.187554	0.0937768	−	0.995593i	$-0.470106\pi$
$479$	4.10481	0.187554	0.0937768	+	0.995593i	$0.470106\pi$
$480$	0	0
$481$	15.0019	0.684026
$482$	0	0
$483$	−7.98866	−0.363496
$484$	0	0
$485$	−16.7335	−0.759830
$486$	0	0
$487$	−10.0473	−0.455289	−0.227644	−	0.973744i	$-0.573102\pi$
$487$	−10.0473	−0.455289	−0.227644	+	0.973744i	$0.573102\pi$
$488$	0	0
$489$	0.432916	0.0195771
$490$	0	0
$491$	16.4195	0.741001	0.370500	−	0.928832i	$-0.379186\pi$
$491$	16.4195	0.741001	0.370500	+	0.928832i	$0.379186\pi$
$492$	0	0
$493$	47.0033	2.11692
$494$	0	0
$495$	−0.247952	−0.0111446
$496$	0	0
$497$	−6.14260	−0.275533
$498$	0	0
$499$	−11.1636	−0.499753	−0.249877	−	0.968278i	$-0.580390\pi$
$499$	−11.1636	−0.499753	−0.249877	+	0.968278i	$0.580390\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	22.6818	1.01133	0.505666	−	0.862729i	$-0.331247\pi$
$503$	22.6818	1.01133	0.505666	+	0.862729i	$0.331247\pi$
$504$	0	0
$505$	18.1778	0.808903
$506$	0	0
$507$	−0.242320	−0.0107618
$508$	0	0
$509$	−2.92089	−0.129466	−0.0647331	−	0.997903i	$-0.520620\pi$
$509$	−2.92089	−0.129466	−0.0647331	+	0.997903i	$0.520620\pi$
$510$	0	0
$511$	11.8278	0.523231
$512$	0	0
$513$	1.02632	0.0453132
$514$	0	0
$515$	2.36698	0.104302
$516$	0	0
$517$	−1.54004	−0.0677308
$518$	0	0
$519$	−17.2653	−0.757864
$520$	0	0
$521$	28.6454	1.25498	0.627490	−	0.778625i	$-0.284082\pi$
$521$	28.6454	1.25498	0.627490	+	0.778625i	$0.284082\pi$
$522$	0	0
$523$	5.60009	0.244875	0.122437	−	0.992476i	$-0.460929\pi$
$523$	5.60009	0.244875	0.122437	+	0.992476i	$0.460929\pi$
$524$	0	0
$525$	2.79876	0.122148
$526$	0	0
$527$	−13.1337	−0.572114
$528$	0	0
$529$	34.6113	1.50484
$530$	0	0
$531$	−8.22708	−0.357025
$532$	0	0
$533$	−3.93056	−0.170251
$534$	0	0
$535$	4.43278	0.191646
$536$	0	0
$537$	7.05820	0.304584
$538$	0	0
$539$	0.954914	0.0411310
$540$	0	0
$541$	−37.7504	−1.62302	−0.811509	−	0.584341i	$-0.801353\pi$
$541$	−37.7504	−1.62302	−0.811509	+	0.584341i	$0.801353\pi$
$542$	0	0
$543$	14.7420	0.632641
$544$	0	0
$545$	12.4973	0.535323
$546$	0	0
$547$	8.04129	0.343821	0.171910	−	0.985113i	$-0.445006\pi$
$547$	8.04129	0.343821	0.171910	+	0.985113i	$0.445006\pi$
$548$	0	0
$549$	0.510754	0.0217984
$550$	0	0
$551$	−9.85051	−0.419646
$552$	0	0
$553$	−5.88375	−0.250202
$554$	0	0
$555$	−6.30737	−0.267733
$556$	0	0
$557$	17.6075	0.746052	0.373026	−	0.927821i	$-0.378320\pi$
$557$	17.6075	0.746052	0.373026	+	0.927821i	$0.378320\pi$
$558$	0	0
$559$	−10.1102	−0.427616
$560$	0	0
$561$	0.793663	0.0335085
$562$	0	0
$563$	40.8654	1.72227	0.861135	−	0.508376i	$-0.169754\pi$
$563$	40.8654	1.72227	0.861135	+	0.508376i	$0.169754\pi$
$564$	0	0
$565$	−10.3077	−0.433648
$566$	0	0
$567$	1.05249	0.0442006
$568$	0	0
$569$	−28.5716	−1.19779	−0.598893	−	0.800829i	$-0.704392\pi$
$569$	−28.5716	−1.19779	−0.598893	+	0.800829i	$0.704392\pi$
$570$	0	0
$571$	15.5883	0.652352	0.326176	−	0.945309i	$-0.394240\pi$
$571$	15.5883	0.652352	0.326176	+	0.945309i	$0.394240\pi$
$572$	0	0
$573$	22.5477	0.941943
$574$	0	0
$575$	−20.1836	−0.841715
$576$	0	0
$577$	−24.1993	−1.00743	−0.503715	−	0.863870i	$-0.668034\pi$
$577$	−24.1993	−1.00743	−0.503715	+	0.863870i	$0.668034\pi$
$578$	0	0
$579$	−3.60996	−0.150025
$580$	0	0
$581$	−5.24457	−0.217581
$582$	0	0
$583$	−0.778784	−0.0322539
$584$	0	0
$585$	5.56759	0.230192
$586$	0	0
$587$	29.3193	1.21014	0.605069	−	0.796173i	$-0.293145\pi$
$587$	29.3193	1.21014	0.605069	+	0.796173i	$0.293145\pi$
$588$	0	0
$589$	2.75245	0.113413
$590$	0	0
$591$	−12.4286	−0.511245
$592$	0	0
$593$	−21.3422	−0.876418	−0.438209	−	0.898873i	$-0.644387\pi$
$593$	−21.3422	−0.876418	−0.438209	+	0.898873i	$0.644387\pi$
$594$	0	0
$595$	7.88603	0.323296
$596$	0	0
$597$	6.92918	0.283593
$598$	0	0
$599$	22.3543	0.913373	0.456686	−	0.889628i	$-0.349036\pi$
$599$	22.3543	0.913373	0.456686	+	0.889628i	$0.349036\pi$
$600$	0	0
$601$	41.2799	1.68384	0.841921	−	0.539602i	$-0.181425\pi$
$601$	41.2799	1.68384	0.841921	+	0.539602i	$0.181425\pi$
$602$	0	0
$603$	0.317219	0.0129181
$604$	0	0
$605$	−16.7896	−0.682594
$606$	0	0
$607$	−6.06522	−0.246180	−0.123090	−	0.992396i	$-0.539280\pi$
$607$	−6.06522	−0.246180	−0.123090	+	0.992396i	$0.539280\pi$
$608$	0	0
$609$	−10.1017	−0.409342
$610$	0	0
$611$	34.5805	1.39898
$612$	0	0
$613$	−11.1383	−0.449871	−0.224935	−	0.974374i	$-0.572217\pi$
$613$	−11.1383	−0.449871	−0.224935	+	0.974374i	$0.572217\pi$
$614$	0	0
$615$	1.65256	0.0666377
$616$	0	0
$617$	19.1170	0.769621	0.384810	−	0.922996i	$-0.374267\pi$
$617$	19.1170	0.769621	0.384810	+	0.922996i	$0.374267\pi$
$618$	0	0
$619$	−30.9654	−1.24460	−0.622302	−	0.782777i	$-0.713803\pi$
$619$	−30.9654	−1.24460	−0.622302	+	0.782777i	$0.713803\pi$
$620$	0	0
$621$	−7.59021	−0.304585
$622$	0	0
$623$	−0.209797	−0.00840533
$624$	0	0
$625$	−4.63301	−0.185321
$626$	0	0
$627$	−0.166328	−0.00664252
$628$	0	0
$629$	20.1891	0.804991
$630$	0	0
$631$	−24.1070	−0.959686	−0.479843	−	0.877354i	$-0.659306\pi$
$631$	−24.1070	−0.959686	−0.479843	+	0.877354i	$0.659306\pi$
$632$	0	0
$633$	−20.3772	−0.809923
$634$	0	0
$635$	24.2980	0.964235
$636$	0	0
$637$	−21.4419	−0.849560
$638$	0	0
$639$	−5.83623	−0.230878
$640$	0	0
$641$	−32.5154	−1.28428	−0.642141	−	0.766586i	$-0.721954\pi$
$641$	−32.5154	−1.28428	−0.642141	+	0.766586i	$0.721954\pi$
$642$	0	0
$643$	15.8759	0.626085	0.313042	−	0.949739i	$-0.398652\pi$
$643$	15.8759	0.626085	0.313042	+	0.949739i	$0.398652\pi$
$644$	0	0
$645$	4.25073	0.167372
$646$	0	0
$647$	−25.0625	−0.985310	−0.492655	−	0.870225i	$-0.663974\pi$
$647$	−25.0625	−0.985310	−0.492655	+	0.870225i	$0.663974\pi$
$648$	0	0
$649$	1.33330	0.0523366
$650$	0	0
$651$	2.82264	0.110628
$652$	0	0
$653$	−5.74801	−0.224937	−0.112469	−	0.993655i	$-0.535876\pi$
$653$	−5.74801	−0.224937	−0.112469	+	0.993655i	$0.535876\pi$
$654$	0	0
$655$	−15.0163	−0.586737
$656$	0	0
$657$	11.2379	0.438431
$658$	0	0
$659$	−11.0446	−0.430238	−0.215119	−	0.976588i	$-0.569014\pi$
$659$	−11.0446	−0.430238	−0.215119	+	0.976588i	$0.569014\pi$
$660$	0	0
$661$	27.4709	1.06849	0.534247	−	0.845329i	$-0.320595\pi$
$661$	27.4709	1.06849	0.534247	+	0.845329i	$0.320595\pi$
$662$	0	0
$663$	−17.8211	−0.692116
$664$	0	0
$665$	−1.65268	−0.0640882
$666$	0	0
$667$	72.8499	2.82076
$668$	0	0
$669$	−20.2826	−0.784172
$670$	0	0
$671$	−0.0827740	−0.00319545
$672$	0	0
$673$	13.7360	0.529485	0.264743	−	0.964319i	$-0.414713\pi$
$673$	13.7360	0.529485	0.264743	+	0.964319i	$0.414713\pi$
$674$	0	0
$675$	2.65917	0.102351
$676$	0	0
$677$	35.5759	1.36729	0.683646	−	0.729814i	$-0.260393\pi$
$677$	35.5759	1.36729	0.683646	+	0.729814i	$0.260393\pi$
$678$	0	0
$679$	−11.5112	−0.441760
$680$	0	0
$681$	7.39895	0.283528
$682$	0	0
$683$	−4.58516	−0.175446	−0.0877231	−	0.996145i	$-0.527959\pi$
$683$	−4.58516	−0.175446	−0.0877231	+	0.996145i	$0.527959\pi$
$684$	0	0
$685$	−3.41039	−0.130304
$686$	0	0
$687$	10.7782	0.411216
$688$	0	0
$689$	17.4870	0.666203
$690$	0	0
$691$	−8.54129	−0.324926	−0.162463	−	0.986715i	$-0.551944\pi$
$691$	−8.54129	−0.324926	−0.162463	+	0.986715i	$0.551944\pi$
$692$	0	0
$693$	−0.170570	−0.00647942
$694$	0	0
$695$	11.8010	0.447639
$696$	0	0
$697$	−5.28964	−0.200359
$698$	0	0
$699$	7.03570	0.266115
$700$	0	0
$701$	46.3748	1.75155	0.875776	−	0.482718i	$-0.160350\pi$
$701$	46.3748	1.75155	0.875776	+	0.482718i	$0.160350\pi$
$702$	0	0
$703$	−4.23104	−0.159577
$704$	0	0
$705$	−14.5390	−0.547570
$706$	0	0
$707$	12.5048	0.470292
$708$	0	0
$709$	46.4820	1.74567	0.872835	−	0.488016i	$-0.162279\pi$
$709$	46.4820	1.74567	0.872835	+	0.488016i	$0.162279\pi$
$710$	0	0
$711$	−5.59029	−0.209652
$712$	0	0
$713$	−20.3558	−0.762332
$714$	0	0
$715$	−0.902298	−0.0337440
$716$	0	0
$717$	7.21369	0.269400
$718$	0	0
$719$	−27.2461	−1.01611	−0.508053	−	0.861326i	$-0.669635\pi$
$719$	−27.2461	−1.01611	−0.508053	+	0.861326i	$0.669635\pi$
$720$	0	0
$721$	1.62828	0.0606404
$722$	0	0
$723$	6.07039	0.225760
$724$	0	0
$725$	−25.5223	−0.947875
$726$	0	0
$727$	21.9595	0.814432	0.407216	−	0.913332i	$-0.366500\pi$
$727$	21.9595	0.814432	0.407216	+	0.913332i	$0.366500\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.6060	−0.503237
$732$	0	0
$733$	−17.3357	−0.640309	−0.320154	−	0.947365i	$-0.603735\pi$
$733$	−17.3357	−0.640309	−0.320154	+	0.947365i	$0.603735\pi$
$734$	0	0
$735$	9.01502	0.332524
$736$	0	0
$737$	−0.0514092	−0.00189368
$738$	0	0
$739$	−14.7105	−0.541135	−0.270568	−	0.962701i	$-0.587211\pi$
$739$	−14.7105	−0.541135	−0.270568	+	0.962701i	$0.587211\pi$
$740$	0	0
$741$	3.73479	0.137201
$742$	0	0
$743$	−16.0012	−0.587029	−0.293514	−	0.955955i	$-0.594825\pi$
$743$	−16.0012	−0.587029	−0.293514	+	0.955955i	$0.594825\pi$
$744$	0	0
$745$	−29.3344	−1.07473
$746$	0	0
$747$	−4.98299	−0.182318
$748$	0	0
$749$	3.04937	0.111422
$750$	0	0
$751$	−43.2600	−1.57858	−0.789289	−	0.614022i	$-0.789551\pi$
$751$	−43.2600	−1.57858	−0.789289	+	0.614022i	$0.789551\pi$
$752$	0	0
$753$	−1.20193	−0.0438008
$754$	0	0
$755$	12.3195	0.448354
$756$	0	0
$757$	−9.81138	−0.356601	−0.178300	−	0.983976i	$-0.557060\pi$
$757$	−9.81138	−0.356601	−0.178300	+	0.983976i	$0.557060\pi$
$758$	0	0
$759$	1.23009	0.0446494
$760$	0	0
$761$	15.1070	0.547627	0.273814	−	0.961783i	$-0.411715\pi$
$761$	15.1070	0.547627	0.273814	+	0.961783i	$0.411715\pi$
$762$	0	0
$763$	8.59704	0.311234
$764$	0	0
$765$	7.49271	0.270899
$766$	0	0
$767$	−29.9383	−1.08101
$768$	0	0
$769$	20.1675	0.727258	0.363629	−	0.931544i	$-0.381538\pi$
$769$	20.1675	0.727258	0.363629	+	0.931544i	$0.381538\pi$
$770$	0	0
$771$	14.5799	0.525083
$772$	0	0
$773$	26.4289	0.950582	0.475291	−	0.879829i	$-0.342343\pi$
$773$	26.4289	0.950582	0.475291	+	0.879829i	$0.342343\pi$
$774$	0	0
$775$	7.13149	0.256171
$776$	0	0
$777$	−4.33893	−0.155658
$778$	0	0
$779$	1.10855	0.0397180
$780$	0	0
$781$	0.945833	0.0338446
$782$	0	0
$783$	−9.59787	−0.343000
$784$	0	0
$785$	15.5840	0.556218
$786$	0	0
$787$	23.9094	0.852280	0.426140	−	0.904657i	$-0.359873\pi$
$787$	23.9094	0.852280	0.426140	+	0.904657i	$0.359873\pi$
$788$	0	0
$789$	16.6235	0.591811
$790$	0	0
$791$	−7.09081	−0.252120
$792$	0	0
$793$	1.85863	0.0660020
$794$	0	0
$795$	−7.35224	−0.260757
$796$	0	0
$797$	26.2713	0.930578	0.465289	−	0.885159i	$-0.345950\pi$
$797$	26.2713	0.930578	0.465289	+	0.885159i	$0.345950\pi$
$798$	0	0
$799$	46.5374	1.64638
$800$	0	0
$801$	−0.199333	−0.00704308
$802$	0	0
$803$	−1.82124	−0.0642701
$804$	0	0
$805$	12.2225	0.430786
$806$	0	0
$807$	8.32925	0.293203
$808$	0	0
$809$	−8.70973	−0.306218	−0.153109	−	0.988209i	$-0.548929\pi$
$809$	−8.70973	−0.306218	−0.153109	+	0.988209i	$0.548929\pi$
$810$	0	0
$811$	26.7512	0.939363	0.469682	−	0.882836i	$-0.344369\pi$
$811$	26.7512	0.939363	0.469682	+	0.882836i	$0.344369\pi$
$812$	0	0
$813$	−26.8577	−0.941940
$814$	0	0
$815$	−0.662352	−0.0232012
$816$	0	0
$817$	2.85142	0.0997587
$818$	0	0
$819$	3.83003	0.133832
$820$	0	0
$821$	4.36277	0.152262	0.0761309	−	0.997098i	$-0.475743\pi$
$821$	4.36277	0.152262	0.0761309	+	0.997098i	$0.475743\pi$
$822$	0	0
$823$	8.76960	0.305689	0.152844	−	0.988250i	$-0.451157\pi$
$823$	8.76960	0.305689	0.152844	+	0.988250i	$0.451157\pi$
$824$	0	0
$825$	−0.430951	−0.0150038
$826$	0	0
$827$	2.60546	0.0906009	0.0453004	−	0.998973i	$-0.485575\pi$
$827$	2.60546	0.0906009	0.0453004	+	0.998973i	$0.485575\pi$
$828$	0	0
$829$	−21.2663	−0.738608	−0.369304	−	0.929309i	$-0.620404\pi$
$829$	−21.2663	−0.738608	−0.369304	+	0.929309i	$0.620404\pi$
$830$	0	0
$831$	−7.42647	−0.257621
$832$	0	0
$833$	−28.8559	−0.999798
$834$	0	0
$835$	−1.52998	−0.0529471
$836$	0	0
$837$	2.68185	0.0926984
$838$	0	0
$839$	29.8458	1.03039	0.515195	−	0.857073i	$-0.327719\pi$
$839$	29.8458	1.03039	0.515195	+	0.857073i	$0.327719\pi$
$840$	0	0
$841$	63.1191	2.17652
$842$	0	0
$843$	−2.07711	−0.0715396
$844$	0	0
$845$	0.370744	0.0127540
$846$	0	0
$847$	−11.5498	−0.396856
$848$	0	0
$849$	−22.8081	−0.782771
$850$	0	0
$851$	31.2908	1.07263
$852$	0	0
$853$	7.85903	0.269088	0.134544	−	0.990908i	$-0.457043\pi$
$853$	7.85903	0.269088	0.134544	+	0.990908i	$0.457043\pi$
$854$	0	0
$855$	−1.57025	−0.0537015
$856$	0	0
$857$	−27.2600	−0.931183	−0.465591	−	0.885000i	$-0.654158\pi$
$857$	−27.2600	−0.931183	−0.465591	+	0.885000i	$0.654158\pi$
$858$	0	0
$859$	−1.16871	−0.0398758	−0.0199379	−	0.999801i	$-0.506347\pi$
$859$	−1.16871	−0.0398758	−0.0199379	+	0.999801i	$0.506347\pi$
$860$	0	0
$861$	1.13682	0.0387428
$862$	0	0
$863$	−57.0280	−1.94126	−0.970628	−	0.240587i	$-0.922660\pi$
$863$	−57.0280	−1.94126	−0.970628	+	0.240587i	$0.922660\pi$
$864$	0	0
$865$	26.4156	0.898157
$866$	0	0
$867$	−6.98318	−0.237161
$868$	0	0
$869$	0.905976	0.0307331
$870$	0	0
$871$	1.15436	0.0391139
$872$	0	0
$873$	−10.9371	−0.370164
$874$	0	0
$875$	−12.3335	−0.416949
$876$	0	0
$877$	25.3813	0.857065	0.428533	−	0.903526i	$-0.359031\pi$
$877$	25.3813	0.857065	0.428533	+	0.903526i	$0.359031\pi$
$878$	0	0
$879$	19.3193	0.651623
$880$	0	0
$881$	−22.2777	−0.750554	−0.375277	−	0.926913i	$-0.622452\pi$
$881$	−22.2777	−0.750554	−0.375277	+	0.926913i	$0.622452\pi$
$882$	0	0
$883$	−21.8081	−0.733900	−0.366950	−	0.930241i	$-0.619598\pi$
$883$	−21.8081	−0.733900	−0.366950	+	0.930241i	$0.619598\pi$
$884$	0	0
$885$	12.5873	0.423116
$886$	0	0
$887$	−31.0024	−1.04096	−0.520480	−	0.853874i	$-0.674247\pi$
$887$	−31.0024	−1.04096	−0.520480	+	0.853874i	$0.674247\pi$
$888$	0	0
$889$	16.7149	0.560601
$890$	0	0
$891$	−0.162063	−0.00542930
$892$	0	0
$893$	−9.75288	−0.326368
$894$	0	0
$895$	−10.7989	−0.360967
$896$	0	0
$897$	−27.6208	−0.922231
$898$	0	0
$899$	−25.7401	−0.858479
$900$	0	0
$901$	23.5336	0.784017
$902$	0	0
$903$	2.92414	0.0973092
$904$	0	0
$905$	−22.5550	−0.749753
$906$	0	0
$907$	49.9506	1.65858	0.829291	−	0.558817i	$-0.188745\pi$
$907$	49.9506	1.65858	0.829291	+	0.558817i	$0.188745\pi$
$908$	0	0
$909$	11.8811	0.394072
$910$	0	0
$911$	−31.1409	−1.03174	−0.515872	−	0.856666i	$-0.672532\pi$
$911$	−31.1409	−1.03174	−0.515872	+	0.856666i	$0.672532\pi$
$912$	0	0
$913$	0.807555	0.0267262
$914$	0	0
$915$	−0.781442	−0.0258337
$916$	0	0
$917$	−10.3300	−0.341125
$918$	0	0
$919$	50.0648	1.65149	0.825743	−	0.564046i	$-0.190756\pi$
$919$	50.0648	1.65149	0.825743	+	0.564046i	$0.190756\pi$
$920$	0	0
$921$	−9.57084	−0.315370
$922$	0	0
$923$	−21.2380	−0.699058
$924$	0	0
$925$	−10.9625	−0.360444
$926$	0	0
$927$	1.54707	0.0508124
$928$	0	0
$929$	28.2739	0.927638	0.463819	−	0.885930i	$-0.346479\pi$
$929$	28.2739	0.927638	0.463819	+	0.885930i	$0.346479\pi$
$930$	0	0
$931$	6.04735	0.198194
$932$	0	0
$933$	−8.69478	−0.284654
$934$	0	0
$935$	−1.21429	−0.0397114
$936$	0	0
$937$	38.8173	1.26811	0.634053	−	0.773290i	$-0.281390\pi$
$937$	38.8173	1.26811	0.634053	+	0.773290i	$0.281390\pi$
$938$	0	0
$939$	24.7334	0.807146
$940$	0	0
$941$	−28.4498	−0.927437	−0.463719	−	0.885983i	$-0.653485\pi$
$941$	−28.4498	−0.927437	−0.463719	+	0.885983i	$0.653485\pi$
$942$	0	0
$943$	−8.19834	−0.266975
$944$	0	0
$945$	−1.61029	−0.0523829
$946$	0	0
$947$	−8.80156	−0.286012	−0.143006	−	0.989722i	$-0.545677\pi$
$947$	−8.80156	−0.286012	−0.143006	+	0.989722i	$0.545677\pi$
$948$	0	0
$949$	40.8946	1.32750
$950$	0	0
$951$	−30.3559	−0.984358
$952$	0	0
$953$	−11.0564	−0.358151	−0.179075	−	0.983835i	$-0.557311\pi$
$953$	−11.0564	−0.358151	−0.179075	+	0.983835i	$0.557311\pi$
$954$	0	0
$955$	−34.4975	−1.11631
$956$	0	0
$957$	1.55546	0.0502807
$958$	0	0
$959$	−2.34605	−0.0757581
$960$	0	0
$961$	−23.8077	−0.767989
$962$	0	0
$963$	2.89728	0.0933636
$964$	0	0
$965$	5.52316	0.177797
$966$	0	0
$967$	−24.2114	−0.778586	−0.389293	−	0.921114i	$-0.627281\pi$
$967$	−24.2114	−0.778586	−0.389293	+	0.921114i	$0.627281\pi$
$968$	0	0
$969$	5.02617	0.161464
$970$	0	0
$971$	−7.68129	−0.246504	−0.123252	−	0.992375i	$-0.539332\pi$
$971$	−7.68129	−0.246504	−0.123252	+	0.992375i	$0.539332\pi$
$972$	0	0
$973$	8.11811	0.260255
$974$	0	0
$975$	9.67670	0.309902
$976$	0	0
$977$	−36.4901	−1.16742	−0.583711	−	0.811961i	$-0.698400\pi$
$977$	−36.4901	−1.16742	−0.583711	+	0.811961i	$0.698400\pi$
$978$	0	0
$979$	0.0323044	0.00103245
$980$	0	0
$981$	8.16825	0.260792
$982$	0	0
$983$	27.9487	0.891424	0.445712	−	0.895176i	$-0.352951\pi$
$983$	27.9487	0.891424	0.445712	+	0.895176i	$0.352951\pi$
$984$	0	0
$985$	19.0155	0.605885
$986$	0	0
$987$	−10.0016	−0.318354
$988$	0	0
$989$	−21.0878	−0.670554
$990$	0	0
$991$	−1.69672	−0.0538980	−0.0269490	−	0.999637i	$-0.508579\pi$
$991$	−1.69672	−0.0538980	−0.0269490	+	0.999637i	$0.508579\pi$
$992$	0	0
$993$	30.2944	0.961365
$994$	0	0
$995$	−10.6015	−0.336090
$996$	0	0
$997$	20.9519	0.663553	0.331777	−	0.943358i	$-0.392352\pi$
$997$	20.9519	0.663553	0.331777	+	0.943358i	$0.392352\pi$
$998$	0	0
$999$	−4.12252	−0.130431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bb.1.5		9
4.3	odd	2		2004.2.a.d.1.5	✓	9
12.11	even	2		6012.2.a.h.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.5	✓	9	4.3	odd	2
6012.2.a.h.1.5		9	12.11	even	2
8016.2.a.bb.1.5		9	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.52998 q^{5} +1.05249 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.52998 q^{5} +1.05249 q^{7} +1.00000 q^{9} -0.162063 q^{11} +3.63900 q^{13} -1.52998 q^{15} +4.89726 q^{17} -1.02632 q^{19} -1.05249 q^{21} +7.59021 q^{23} -2.65917 q^{25} -1.00000 q^{27} +9.59787 q^{29} -2.68185 q^{31} +0.162063 q^{33} +1.61029 q^{35} +4.12252 q^{37} -3.63900 q^{39} -1.08012 q^{41} -2.77829 q^{43} +1.52998 q^{45} +9.50274 q^{47} -5.89226 q^{49} -4.89726 q^{51} +4.80545 q^{53} -0.247952 q^{55} +1.02632 q^{57} -8.22708 q^{59} +0.510754 q^{61} +1.05249 q^{63} +5.56759 q^{65} +0.317219 q^{67} -7.59021 q^{69} -5.83623 q^{71} +11.2379 q^{73} +2.65917 q^{75} -0.170570 q^{77} -5.59029 q^{79} +1.00000 q^{81} -4.98299 q^{83} +7.49271 q^{85} -9.59787 q^{87} -0.199333 q^{89} +3.83003 q^{91} +2.68185 q^{93} -1.57025 q^{95} -10.9371 q^{97} -0.162063 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9} - 7 q^{11} + 6 q^{13} - 9 q^{15} + 7 q^{17} - 2 q^{19} + 2 q^{21} - 19 q^{23} + 22 q^{25} - 9 q^{27} + 13 q^{29} - 12 q^{31} + 7 q^{33} - 4 q^{35} + 15 q^{37} - 6 q^{39} + 18 q^{41} + 6 q^{43} + 9 q^{45} - 25 q^{47} + 19 q^{49} - 7 q^{51} + 17 q^{53} + 3 q^{55} + 2 q^{57} - 3 q^{59} + 14 q^{61} - 2 q^{63} + 14 q^{65} + 4 q^{67} + 19 q^{69} - 17 q^{71} - 20 q^{73} - 22 q^{75} + 14 q^{77} + 8 q^{79} + 9 q^{81} + q^{83} + 5 q^{85} - 13 q^{87} + 36 q^{89} + 41 q^{91} + 12 q^{93} - 5 q^{95} + 31 q^{97} - 7 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9 - 7 * q^11 + 6 * q^13 - 9 * q^15 + 7 * q^17 - 2 * q^19 + 2 * q^21 - 19 * q^23 + 22 * q^25 - 9 * q^27 + 13 * q^29 - 12 * q^31 + 7 * q^33 - 4 * q^35 + 15 * q^37 - 6 * q^39 + 18 * q^41 + 6 * q^43 + 9 * q^45 - 25 * q^47 + 19 * q^49 - 7 * q^51 + 17 * q^53 + 3 * q^55 + 2 * q^57 - 3 * q^59 + 14 * q^61 - 2 * q^63 + 14 * q^65 + 4 * q^67 + 19 * q^69 - 17 * q^71 - 20 * q^73 - 22 * q^75 + 14 * q^77 + 8 * q^79 + 9 * q^81 + q^83 + 5 * q^85 - 13 * q^87 + 36 * q^89 + 41 * q^91 + 12 * q^93 - 5 * q^95 + 31 * q^97 - 7 * q^99

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