LMFDB - Embedded newform 8016.2.a.o.1.4

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:	$4$
Coefficient field:	4.4.2777.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + x + 2 $ x^4 - x^3 - 4*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.50848$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.69113 q^{5} +2.24216 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.69113 q^{5} +2.24216 q^{7} +1.00000 q^{9} +5.56799 q^{13} +3.69113 q^{15} -1.01696 q^{17} +2.65167 q^{19} +2.24216 q^{21} -2.65167 q^{23} +8.62441 q^{25} +1.00000 q^{27} -1.38225 q^{29} -4.89383 q^{31} +8.27608 q^{35} -1.97751 q^{37} +5.56799 q^{39} +4.28639 q^{41} -0.551031 q^{43} +3.69113 q^{45} +5.62441 q^{47} -1.97273 q^{49} -1.01696 q^{51} +1.03945 q^{53} +2.65167 q^{57} +2.42790 q^{59} +5.38225 q^{61} +2.24216 q^{63} +20.5522 q^{65} -14.8610 q^{67} -2.65167 q^{69} +8.68560 q^{71} -1.75374 q^{73} +8.62441 q^{75} -3.74754 q^{79} +1.00000 q^{81} +5.43867 q^{83} -3.75374 q^{85} -1.38225 q^{87} -6.27608 q^{89} +12.4843 q^{91} -4.89383 q^{93} +9.78766 q^{95} -11.0067 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

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
$5$	3.69113	1.65072	0.825361	−	0.564606i	$-0.190972\pi$
$5$	3.69113	1.65072	0.825361	+	0.564606i	$0.190972\pi$
$6$	0	0
$7$	2.24216	0.847456	0.423728	−	0.905790i	$-0.360721\pi$
$7$	2.24216	0.847456	0.423728	+	0.905790i	$0.360721\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	5.56799	1.54428	0.772142	−	0.635450i	$-0.219186\pi$
$13$	5.56799	1.54428	0.772142	+	0.635450i	$0.219186\pi$
$14$	0	0
$15$	3.69113	0.953045
$16$	0	0
$17$	−1.01696	−0.246650	−0.123325	−	0.992366i	$-0.539356\pi$
$17$	−1.01696	−0.246650	−0.123325	+	0.992366i	$0.539356\pi$
$18$	0	0
$19$	2.65167	0.608336	0.304168	−	0.952618i	$-0.401622\pi$
$19$	2.65167	0.608336	0.304168	+	0.952618i	$0.401622\pi$
$20$	0	0
$21$	2.24216	0.489279
$22$	0	0
$23$	−2.65167	−0.552912	−0.276456	−	0.961027i	$-0.589160\pi$
$23$	−2.65167	−0.552912	−0.276456	+	0.961027i	$0.589160\pi$
$24$	0	0
$25$	8.62441	1.72488
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−1.38225	−0.256678	−0.128339	−	0.991730i	$-0.540964\pi$
$29$	−1.38225	−0.256678	−0.128339	+	0.991730i	$0.540964\pi$
$30$	0	0
$31$	−4.89383	−0.878958	−0.439479	−	0.898253i	$-0.644837\pi$
$31$	−4.89383	−0.878958	−0.439479	+	0.898253i	$0.644837\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	8.27608	1.39891
$36$	0	0
$37$	−1.97751	−0.325101	−0.162550	−	0.986700i	$-0.551972\pi$
$37$	−1.97751	−0.325101	−0.162550	+	0.986700i	$0.551972\pi$
$38$	0	0
$39$	5.56799	0.891593
$40$	0	0
$41$	4.28639	0.669421	0.334710	−	0.942321i	$-0.391361\pi$
$41$	4.28639	0.669421	0.334710	+	0.942321i	$0.391361\pi$
$42$	0	0
$43$	−0.551031	−0.0840314	−0.0420157	−	0.999117i	$-0.513378\pi$
$43$	−0.551031	−0.0840314	−0.0420157	+	0.999117i	$0.513378\pi$
$44$	0	0
$45$	3.69113	0.550241
$46$	0	0
$47$	5.62441	0.820404	0.410202	−	0.911995i	$-0.365458\pi$
$47$	5.62441	0.820404	0.410202	+	0.911995i	$0.365458\pi$
$48$	0	0
$49$	−1.97273	−0.281819
$50$	0	0
$51$	−1.01696	−0.142403
$52$	0	0
$53$	1.03945	0.142780	0.0713898	−	0.997448i	$-0.477257\pi$
$53$	1.03945	0.142780	0.0713898	+	0.997448i	$0.477257\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.65167	0.351223
$58$	0	0
$59$	2.42790	0.316086	0.158043	−	0.987432i	$-0.449482\pi$
$59$	2.42790	0.316086	0.158043	+	0.987432i	$0.449482\pi$
$60$	0	0
$61$	5.38225	0.689127	0.344563	−	0.938763i	$-0.388027\pi$
$61$	5.38225	0.689127	0.344563	+	0.938763i	$0.388027\pi$
$62$	0	0
$63$	2.24216	0.282485
$64$	0	0
$65$	20.5522	2.54918
$66$	0	0
$67$	−14.8610	−1.81556	−0.907782	−	0.419442i	$-0.862226\pi$
$67$	−14.8610	−1.81556	−0.907782	+	0.419442i	$0.862226\pi$
$68$	0	0
$69$	−2.65167	−0.319224
$70$	0	0
$71$	8.68560	1.03079	0.515395	−	0.856952i	$-0.327645\pi$
$71$	8.68560	1.03079	0.515395	+	0.856952i	$0.327645\pi$
$72$	0	0
$73$	−1.75374	−0.205259	−0.102630	−	0.994720i	$-0.532726\pi$
$73$	−1.75374	−0.205259	−0.102630	+	0.994720i	$0.532726\pi$
$74$	0	0
$75$	8.62441	0.995861
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.74754	−0.421631	−0.210816	−	0.977526i	$-0.567612\pi$
$79$	−3.74754	−0.421631	−0.210816	+	0.977526i	$0.567612\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.43867	0.596971	0.298486	−	0.954414i	$-0.403519\pi$
$83$	5.43867	0.596971	0.298486	+	0.954414i	$0.403519\pi$
$84$	0	0
$85$	−3.75374	−0.407150
$86$	0	0
$87$	−1.38225	−0.148193
$88$	0	0
$89$	−6.27608	−0.665263	−0.332632	−	0.943057i	$-0.607937\pi$
$89$	−6.27608	−0.665263	−0.332632	+	0.943057i	$0.607937\pi$
$90$	0	0
$91$	12.4843	1.30871
$92$	0	0
$93$	−4.89383	−0.507467
$94$	0	0
$95$	9.78766	1.00419
$96$	0	0
$97$	−11.0067	−1.11756	−0.558778	−	0.829317i	$-0.688730\pi$
$97$	−11.0067	−1.11756	−0.558778	+	0.829317i	$0.688730\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.3114	1.52354	0.761772	−	0.647845i	$-0.224330\pi$
$101$	15.3114	1.52354	0.761772	+	0.647845i	$0.224330\pi$
$102$	0	0
$103$	13.2972	1.31021	0.655104	−	0.755539i	$-0.272625\pi$
$103$	13.2972	1.31021	0.655104	+	0.755539i	$0.272625\pi$
$104$	0	0
$105$	8.27608	0.807663
$106$	0	0
$107$	−18.3848	−1.77733	−0.888663	−	0.458561i	$-0.848365\pi$
$107$	−18.3848	−1.77733	−0.888663	+	0.458561i	$0.848365\pi$
$108$	0	0
$109$	−18.1651	−1.73990	−0.869952	−	0.493136i	$-0.835850\pi$
$109$	−18.1651	−1.73990	−0.869952	+	0.493136i	$0.835850\pi$
$110$	0	0
$111$	−1.97751	−0.187697
$112$	0	0
$113$	14.4331	1.35776	0.678878	−	0.734251i	$-0.262467\pi$
$113$	14.4331	1.35776	0.678878	+	0.734251i	$0.262467\pi$
$114$	0	0
$115$	−9.78766	−0.912704
$116$	0	0
$117$	5.56799	0.514761
$118$	0	0
$119$	−2.28019	−0.209025
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	4.28639	0.386490
$124$	0	0
$125$	13.3781	1.19658
$126$	0	0
$127$	14.4434	1.28165	0.640824	−	0.767688i	$-0.278593\pi$
$127$	14.4434	1.28165	0.640824	+	0.767688i	$0.278593\pi$
$128$	0	0
$129$	−0.551031	−0.0485156
$130$	0	0
$131$	−11.4346	−0.999042	−0.499521	−	0.866302i	$-0.666491\pi$
$131$	−11.4346	−0.999042	−0.499521	+	0.866302i	$0.666491\pi$
$132$	0	0
$133$	5.94547	0.515537
$134$	0	0
$135$	3.69113	0.317682
$136$	0	0
$137$	−17.6134	−1.50481	−0.752405	−	0.658701i	$-0.771106\pi$
$137$	−17.6134	−1.50481	−0.752405	+	0.658701i	$0.771106\pi$
$138$	0	0
$139$	14.9060	1.26431	0.632156	−	0.774841i	$-0.282170\pi$
$139$	14.9060	1.26431	0.632156	+	0.774841i	$0.282170\pi$
$140$	0	0
$141$	5.62441	0.473661
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.10206	−0.423703
$146$	0	0
$147$	−1.97273	−0.162708
$148$	0	0
$149$	0.308874	0.0253040	0.0126520	−	0.999920i	$-0.495973\pi$
$149$	0.308874	0.0253040	0.0126520	+	0.999920i	$0.495973\pi$
$150$	0	0
$151$	4.61155	0.375283	0.187641	−	0.982238i	$-0.439916\pi$
$151$	4.61155	0.375283	0.187641	+	0.982238i	$0.439916\pi$
$152$	0	0
$153$	−1.01696	−0.0822165
$154$	0	0
$155$	−18.0637	−1.45091
$156$	0	0
$157$	−7.30335	−0.582871	−0.291435	−	0.956591i	$-0.594133\pi$
$157$	−7.30335	−0.582871	−0.291435	+	0.956591i	$0.594133\pi$
$158$	0	0
$159$	1.03945	0.0824339
$160$	0	0
$161$	−5.94547	−0.468569
$162$	0	0
$163$	1.73610	0.135982	0.0679911	−	0.997686i	$-0.478341\pi$
$163$	1.73610	0.135982	0.0679911	+	0.997686i	$0.478341\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.00000	0.0773823
$167$	1.00000	0.0773823
$168$	0	0
$169$	18.0026	1.38481
$170$	0	0
$171$	2.65167	0.202779
$172$	0	0
$173$	−5.38225	−0.409205	−0.204602	−	0.978845i	$-0.565590\pi$
$173$	−5.38225	−0.409205	−0.204602	+	0.978845i	$0.565590\pi$
$174$	0	0
$175$	19.3373	1.46176
$176$	0	0
$177$	2.42790	0.182492
$178$	0	0
$179$	−1.30335	−0.0974168	−0.0487084	−	0.998813i	$-0.515510\pi$
$179$	−1.30335	−0.0974168	−0.0487084	+	0.998813i	$0.515510\pi$
$180$	0	0
$181$	−16.5632	−1.23113	−0.615567	−	0.788084i	$-0.711073\pi$
$181$	−16.5632	−1.23113	−0.615567	+	0.788084i	$0.711073\pi$
$182$	0	0
$183$	5.38225	0.397867
$184$	0	0
$185$	−7.29924	−0.536651
$186$	0	0
$187$	0	0
$188$	0	0
$189$	2.24216	0.163093
$190$	0	0
$191$	21.3332	1.54361	0.771807	−	0.635857i	$-0.219353\pi$
$191$	21.3332	1.54361	0.771807	+	0.635857i	$0.219353\pi$
$192$	0	0
$193$	−13.1360	−0.945549	−0.472775	−	0.881183i	$-0.656747\pi$
$193$	−13.1360	−0.945549	−0.472775	+	0.881183i	$0.656747\pi$
$194$	0	0
$195$	20.5522	1.47177
$196$	0	0
$197$	−6.55103	−0.466742	−0.233371	−	0.972388i	$-0.574976\pi$
$197$	−6.55103	−0.466742	−0.233371	+	0.972388i	$0.574976\pi$
$198$	0	0
$199$	−7.13599	−0.505857	−0.252928	−	0.967485i	$-0.581394\pi$
$199$	−7.13599	−0.505857	−0.252928	+	0.967485i	$0.581394\pi$
$200$	0	0
$201$	−14.8610	−1.04822
$202$	0	0
$203$	−3.09922	−0.217523
$204$	0	0
$205$	15.8216	1.10503
$206$	0	0
$207$	−2.65167	−0.184304
$208$	0	0
$209$	0	0
$210$	0	0
$211$	18.6517	1.28403	0.642017	−	0.766690i	$-0.278098\pi$
$211$	18.6517	1.28403	0.642017	+	0.766690i	$0.278098\pi$
$212$	0	0
$213$	8.68560	0.595127
$214$	0	0
$215$	−2.03392	−0.138713
$216$	0	0
$217$	−10.9727	−0.744878
$218$	0	0
$219$	−1.75374	−0.118506
$220$	0	0
$221$	−5.66244	−0.380897
$222$	0	0
$223$	−2.24216	−0.150146	−0.0750730	−	0.997178i	$-0.523919\pi$
$223$	−2.24216	−0.150146	−0.0750730	+	0.997178i	$0.523919\pi$
$224$	0	0
$225$	8.62441	0.574961
$226$	0	0
$227$	18.3829	1.22012	0.610059	−	0.792356i	$-0.291146\pi$
$227$	18.3829	1.22012	0.610059	+	0.792356i	$0.291146\pi$
$228$	0	0
$229$	−11.2149	−0.741101	−0.370550	−	0.928812i	$-0.620831\pi$
$229$	−11.2149	−0.741101	−0.370550	+	0.928812i	$0.620831\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−17.5249	−1.14809	−0.574047	−	0.818822i	$-0.694627\pi$
$233$	−17.5249	−1.14809	−0.574047	+	0.818822i	$0.694627\pi$
$234$	0	0
$235$	20.7604	1.35426
$236$	0	0
$237$	−3.74754	−0.243429
$238$	0	0
$239$	7.13599	0.461589	0.230794	−	0.973003i	$-0.425868\pi$
$239$	7.13599	0.461589	0.230794	+	0.973003i	$0.425868\pi$
$240$	0	0
$241$	−5.86656	−0.377899	−0.188949	−	0.981987i	$-0.560508\pi$
$241$	−5.86656	−0.377899	−0.188949	+	0.981987i	$0.560508\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−7.28161	−0.465205
$246$	0	0
$247$	14.7645	0.939443
$248$	0	0
$249$	5.43867	0.344661
$250$	0	0
$251$	−17.0815	−1.07817	−0.539086	−	0.842251i	$-0.681230\pi$
$251$	−17.0815	−1.07817	−0.539086	+	0.842251i	$0.681230\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−3.75374	−0.235068
$256$	0	0
$257$	11.0959	0.692141	0.346071	−	0.938208i	$-0.387516\pi$
$257$	11.0959	0.692141	0.346071	+	0.938208i	$0.387516\pi$
$258$	0	0
$259$	−4.43389	−0.275508
$260$	0	0
$261$	−1.38225	−0.0855592
$262$	0	0
$263$	−23.7836	−1.46656	−0.733278	−	0.679929i	$-0.762011\pi$
$263$	−23.7836	−1.46656	−0.733278	+	0.679929i	$0.762011\pi$
$264$	0	0
$265$	3.83675	0.235689
$266$	0	0
$267$	−6.27608	−0.384090
$268$	0	0
$269$	−24.8404	−1.51455	−0.757274	−	0.653097i	$-0.773469\pi$
$269$	−24.8404	−1.51455	−0.757274	+	0.653097i	$0.773469\pi$
$270$	0	0
$271$	16.6918	1.01395	0.506977	−	0.861959i	$-0.330763\pi$
$271$	16.6918	1.01395	0.506977	+	0.861959i	$0.330763\pi$
$272$	0	0
$273$	12.4843	0.755585
$274$	0	0
$275$	0	0
$276$	0	0
$277$	30.8888	1.85593	0.927963	−	0.372672i	$-0.121558\pi$
$277$	30.8888	1.85593	0.927963	+	0.372672i	$0.121558\pi$
$278$	0	0
$279$	−4.89383	−0.292986
$280$	0	0
$281$	25.0406	1.49380	0.746898	−	0.664939i	$-0.231542\pi$
$281$	25.0406	1.49380	0.746898	+	0.664939i	$0.231542\pi$
$282$	0	0
$283$	−27.6964	−1.64638	−0.823189	−	0.567767i	$-0.807807\pi$
$283$	−27.6964	−1.64638	−0.823189	+	0.567767i	$0.807807\pi$
$284$	0	0
$285$	9.78766	0.579771
$286$	0	0
$287$	9.61075	0.567304
$288$	0	0
$289$	−15.9658	−0.939164
$290$	0	0
$291$	−11.0067	−0.645222
$292$	0	0
$293$	21.3823	1.24916	0.624582	−	0.780959i	$-0.285269\pi$
$293$	21.3823	1.24916	0.624582	+	0.780959i	$0.285269\pi$
$294$	0	0
$295$	8.96168	0.521769
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−14.7645	−0.853853
$300$	0	0
$301$	−1.23550	−0.0712129
$302$	0	0
$303$	15.3114	0.879618
$304$	0	0
$305$	19.8666	1.13756
$306$	0	0
$307$	−5.35640	−0.305706	−0.152853	−	0.988249i	$-0.548846\pi$
$307$	−5.35640	−0.305706	−0.152853	+	0.988249i	$0.548846\pi$
$308$	0	0
$309$	13.2972	0.756449
$310$	0	0
$311$	−11.0980	−0.629307	−0.314654	−	0.949207i	$-0.601888\pi$
$311$	−11.0980	−0.629307	−0.314654	+	0.949207i	$0.601888\pi$
$312$	0	0
$313$	−20.4527	−1.15605	−0.578026	−	0.816018i	$-0.696177\pi$
$313$	−20.4527	−1.15605	−0.578026	+	0.816018i	$0.696177\pi$
$314$	0	0
$315$	8.27608	0.466304
$316$	0	0
$317$	0.889723	0.0499718	0.0249859	−	0.999688i	$-0.492046\pi$
$317$	0.889723	0.0499718	0.0249859	+	0.999688i	$0.492046\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−18.3848	−1.02614
$322$	0	0
$323$	−2.69665	−0.150046
$324$	0	0
$325$	48.0206	2.66371
$326$	0	0
$327$	−18.1651	−1.00453
$328$	0	0
$329$	12.6108	0.695256
$330$	0	0
$331$	−7.31553	−0.402098	−0.201049	−	0.979581i	$-0.564435\pi$
$331$	−7.31553	−0.402098	−0.201049	+	0.979581i	$0.564435\pi$
$332$	0	0
$333$	−1.97751	−0.108367
$334$	0	0
$335$	−54.8540	−2.99699
$336$	0	0
$337$	6.11827	0.333284	0.166642	−	0.986017i	$-0.446708\pi$
$337$	6.11827	0.333284	0.166642	+	0.986017i	$0.446708\pi$
$338$	0	0
$339$	14.4331	0.783900
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−20.1183	−1.08628
$344$	0	0
$345$	−9.78766	−0.526950
$346$	0	0
$347$	−17.6400	−0.946962	−0.473481	−	0.880804i	$-0.657003\pi$
$347$	−17.6400	−0.946962	−0.473481	+	0.880804i	$0.657003\pi$
$348$	0	0
$349$	12.7420	0.682064	0.341032	−	0.940052i	$-0.389223\pi$
$349$	12.7420	0.682064	0.341032	+	0.940052i	$0.389223\pi$
$350$	0	0
$351$	5.56799	0.297198
$352$	0	0
$353$	33.0406	1.75857	0.879286	−	0.476293i	$-0.158020\pi$
$353$	33.0406	1.75857	0.879286	+	0.476293i	$0.158020\pi$
$354$	0	0
$355$	32.0596	1.70155
$356$	0	0
$357$	−2.28019	−0.120680
$358$	0	0
$359$	10.4054	0.549177	0.274588	−	0.961562i	$-0.411458\pi$
$359$	10.4054	0.549177	0.274588	+	0.961562i	$0.411458\pi$
$360$	0	0
$361$	−11.9686	−0.629928
$362$	0	0
$363$	−11.0000	−0.577350
$364$	0	0
$365$	−6.47326	−0.338826
$366$	0	0
$367$	−6.89794	−0.360069	−0.180035	−	0.983660i	$-0.557621\pi$
$367$	−6.89794	−0.360069	−0.180035	+	0.983660i	$0.557621\pi$
$368$	0	0
$369$	4.28639	0.223140
$370$	0	0
$371$	2.33061	0.120999
$372$	0	0
$373$	9.15026	0.473783	0.236891	−	0.971536i	$-0.423871\pi$
$373$	9.15026	0.473783	0.236891	+	0.971536i	$0.423871\pi$
$374$	0	0
$375$	13.3781	0.690844
$376$	0	0
$377$	−7.69636	−0.396383
$378$	0	0
$379$	−0.229971	−0.0118128	−0.00590641	−	0.999983i	$-0.501880\pi$
$379$	−0.229971	−0.0118128	−0.00590641	+	0.999983i	$0.501880\pi$
$380$	0	0
$381$	14.4434	0.739960
$382$	0	0
$383$	−21.6992	−1.10878	−0.554389	−	0.832258i	$-0.687048\pi$
$383$	−21.6992	−1.10878	−0.554389	+	0.832258i	$0.687048\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.551031	−0.0280105
$388$	0	0
$389$	−15.1917	−0.770247	−0.385124	−	0.922865i	$-0.625841\pi$
$389$	−15.1917	−0.770247	−0.385124	+	0.922865i	$0.625841\pi$
$390$	0	0
$391$	2.69665	0.136376
$392$	0	0
$393$	−11.4346	−0.576797
$394$	0	0
$395$	−13.8326	−0.695996
$396$	0	0
$397$	8.45039	0.424113	0.212056	−	0.977257i	$-0.431984\pi$
$397$	8.45039	0.424113	0.212056	+	0.977257i	$0.431984\pi$
$398$	0	0
$399$	5.94547	0.297646
$400$	0	0
$401$	−2.56657	−0.128169	−0.0640843	−	0.997944i	$-0.520413\pi$
$401$	−2.56657	−0.128169	−0.0640843	+	0.997944i	$0.520413\pi$
$402$	0	0
$403$	−27.2488	−1.35736
$404$	0	0
$405$	3.69113	0.183414
$406$	0	0
$407$	0	0
$408$	0	0
$409$	18.6650	0.922924	0.461462	−	0.887160i	$-0.347325\pi$
$409$	18.6650	0.922924	0.461462	+	0.887160i	$0.347325\pi$
$410$	0	0
$411$	−17.6134	−0.868803
$412$	0	0
$413$	5.44373	0.267868
$414$	0	0
$415$	20.0748	0.985433
$416$	0	0
$417$	14.9060	0.729951
$418$	0	0
$419$	1.83264	0.0895303	0.0447651	−	0.998998i	$-0.485746\pi$
$419$	1.83264	0.0895303	0.0447651	+	0.998998i	$0.485746\pi$
$420$	0	0
$421$	−38.2174	−1.86260	−0.931302	−	0.364248i	$-0.881326\pi$
$421$	−38.2174	−1.86260	−0.931302	+	0.364248i	$0.881326\pi$
$422$	0	0
$423$	5.62441	0.273468
$424$	0	0
$425$	−8.77070	−0.425441
$426$	0	0
$427$	12.0678	0.584004
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.31029	0.0631146	0.0315573	−	0.999502i	$-0.489953\pi$
$431$	1.31029	0.0631146	0.0315573	+	0.999502i	$0.489953\pi$
$432$	0	0
$433$	23.0434	1.10740	0.553698	−	0.832717i	$-0.313216\pi$
$433$	23.0434	1.10740	0.553698	+	0.832717i	$0.313216\pi$
$434$	0	0
$435$	−5.10206	−0.244625
$436$	0	0
$437$	−7.03137	−0.336356
$438$	0	0
$439$	−38.4154	−1.83347	−0.916733	−	0.399501i	$-0.869183\pi$
$439$	−38.4154	−1.83347	−0.916733	+	0.399501i	$0.869183\pi$
$440$	0	0
$441$	−1.97273	−0.0939397
$442$	0	0
$443$	36.8127	1.74902	0.874512	−	0.485004i	$-0.161182\pi$
$443$	36.8127	1.74902	0.874512	+	0.485004i	$0.161182\pi$
$444$	0	0
$445$	−23.1658	−1.09816
$446$	0	0
$447$	0.308874	0.0146093
$448$	0	0
$449$	19.8257	0.935632	0.467816	−	0.883826i	$-0.345041\pi$
$449$	19.8257	0.935632	0.467816	+	0.883826i	$0.345041\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	4.61155	0.216670
$454$	0	0
$455$	46.0812	2.16032
$456$	0	0
$457$	−3.25987	−0.152490	−0.0762451	−	0.997089i	$-0.524293\pi$
$457$	−3.25987	−0.152490	−0.0762451	+	0.997089i	$0.524293\pi$
$458$	0	0
$459$	−1.01696	−0.0474677
$460$	0	0
$461$	−6.97684	−0.324944	−0.162472	−	0.986713i	$-0.551947\pi$
$461$	−6.97684	−0.324944	−0.162472	+	0.986713i	$0.551947\pi$
$462$	0	0
$463$	−37.8943	−1.76110	−0.880549	−	0.473956i	$-0.842826\pi$
$463$	−37.8943	−1.76110	−0.880549	+	0.473956i	$0.842826\pi$
$464$	0	0
$465$	−18.0637	−0.837686
$466$	0	0
$467$	−4.32651	−0.200207	−0.100103	−	0.994977i	$-0.531917\pi$
$467$	−4.32651	−0.200207	−0.100103	+	0.994977i	$0.531917\pi$
$468$	0	0
$469$	−33.3208	−1.53861
$470$	0	0
$471$	−7.30335	−0.336520
$472$	0	0
$473$	0	0
$474$	0	0
$475$	22.8691	1.04931
$476$	0	0
$477$	1.03945	0.0475932
$478$	0	0
$479$	41.4323	1.89309	0.946546	−	0.322569i	$-0.104547\pi$
$479$	41.4323	1.89309	0.946546	+	0.322569i	$0.104547\pi$
$480$	0	0
$481$	−11.0108	−0.502048
$482$	0	0
$483$	−5.94547	−0.270528
$484$	0	0
$485$	−40.6270	−1.84478
$486$	0	0
$487$	16.2319	0.735535	0.367768	−	0.929918i	$-0.380122\pi$
$487$	16.2319	0.735535	0.367768	+	0.929918i	$0.380122\pi$
$488$	0	0
$489$	1.73610	0.0785093
$490$	0	0
$491$	−17.6868	−0.798195	−0.399097	−	0.916909i	$-0.630676\pi$
$491$	−17.6868	−0.798195	−0.399097	+	0.916909i	$0.630676\pi$
$492$	0	0
$493$	1.40570	0.0633094
$494$	0	0
$495$	0	0
$496$	0	0
$497$	19.4745	0.873549
$498$	0	0
$499$	4.88575	0.218716	0.109358	−	0.994002i	$-0.465120\pi$
$499$	4.88575	0.218716	0.109358	+	0.994002i	$0.465120\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−36.4637	−1.62584	−0.812918	−	0.582378i	$-0.802122\pi$
$503$	−36.4637	−1.62584	−0.812918	+	0.582378i	$0.802122\pi$
$504$	0	0
$505$	56.5164	2.51495
$506$	0	0
$507$	18.0026	0.799521
$508$	0	0
$509$	−7.71981	−0.342175	−0.171087	−	0.985256i	$-0.554728\pi$
$509$	−7.71981	−0.342175	−0.171087	+	0.985256i	$0.554728\pi$
$510$	0	0
$511$	−3.93215	−0.173948
$512$	0	0
$513$	2.65167	0.117074
$514$	0	0
$515$	49.0815	2.16279
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−5.38225	−0.236255
$520$	0	0
$521$	32.2658	1.41359	0.706795	−	0.707419i	$-0.250140\pi$
$521$	32.2658	1.41359	0.706795	+	0.707419i	$0.250140\pi$
$522$	0	0
$523$	−18.9782	−0.829858	−0.414929	−	0.909854i	$-0.636194\pi$
$523$	−18.9782	−0.829858	−0.414929	+	0.909854i	$0.636194\pi$
$524$	0	0
$525$	19.3373	0.843948
$526$	0	0
$527$	4.97684	0.216795
$528$	0	0
$529$	−15.9686	−0.694288
$530$	0	0
$531$	2.42790	0.105362
$532$	0	0
$533$	23.8666	1.03378
$534$	0	0
$535$	−67.8606	−2.93387
$536$	0	0
$537$	−1.30335	−0.0562436
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−41.9037	−1.80158	−0.900791	−	0.434254i	$-0.857012\pi$
$541$	−41.9037	−1.80158	−0.900791	+	0.434254i	$0.857012\pi$
$542$	0	0
$543$	−16.5632	−0.710796
$544$	0	0
$545$	−67.0498	−2.87210
$546$	0	0
$547$	−2.27211	−0.0971484	−0.0485742	−	0.998820i	$-0.515468\pi$
$547$	−2.27211	−0.0971484	−0.0485742	+	0.998820i	$0.515468\pi$
$548$	0	0
$549$	5.38225	0.229709
$550$	0	0
$551$	−3.66528	−0.156146
$552$	0	0
$553$	−8.40257	−0.357314
$554$	0	0
$555$	−7.29924	−0.309835
$556$	0	0
$557$	1.58354	0.0670966	0.0335483	−	0.999437i	$-0.489319\pi$
$557$	1.58354	0.0670966	0.0335483	+	0.999437i	$0.489319\pi$
$558$	0	0
$559$	−3.06814	−0.129768
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−8.85580	−0.373227	−0.186614	−	0.982433i	$-0.559751\pi$
$563$	−8.85580	−0.373227	−0.186614	+	0.982433i	$0.559751\pi$
$564$	0	0
$565$	53.2745	2.24128
$566$	0	0
$567$	2.24216	0.0941617
$568$	0	0
$569$	44.1552	1.85108	0.925541	−	0.378646i	$-0.123610\pi$
$569$	44.1552	1.85108	0.925541	+	0.378646i	$0.123610\pi$
$570$	0	0
$571$	29.3114	1.22664	0.613322	−	0.789833i	$-0.289833\pi$
$571$	29.3114	1.22664	0.613322	+	0.789833i	$0.289833\pi$
$572$	0	0
$573$	21.3332	0.891206
$574$	0	0
$575$	−22.8691	−0.953708
$576$	0	0
$577$	7.78221	0.323978	0.161989	−	0.986793i	$-0.448209\pi$
$577$	7.78221	0.323978	0.161989	+	0.986793i	$0.448209\pi$
$578$	0	0
$579$	−13.1360	−0.545913
$580$	0	0
$581$	12.1943	0.505906
$582$	0	0
$583$	0	0
$584$	0	0
$585$	20.5522	0.849727
$586$	0	0
$587$	25.7895	1.06445	0.532224	−	0.846603i	$-0.321356\pi$
$587$	25.7895	1.06445	0.532224	+	0.846603i	$0.321356\pi$
$588$	0	0
$589$	−12.9768	−0.534701
$590$	0	0
$591$	−6.55103	−0.269473
$592$	0	0
$593$	12.9720	0.532696	0.266348	−	0.963877i	$-0.414183\pi$
$593$	12.9720	0.532696	0.266348	+	0.963877i	$0.414183\pi$
$594$	0	0
$595$	−8.41646	−0.345041
$596$	0	0
$597$	−7.13599	−0.292057
$598$	0	0
$599$	−23.0406	−0.941413	−0.470706	−	0.882290i	$-0.656001\pi$
$599$	−23.0406	−0.941413	−0.470706	+	0.882290i	$0.656001\pi$
$600$	0	0
$601$	19.0884	0.778632	0.389316	−	0.921104i	$-0.372711\pi$
$601$	19.0884	0.778632	0.389316	+	0.921104i	$0.372711\pi$
$602$	0	0
$603$	−14.8610	−0.605188
$604$	0	0
$605$	−40.6024	−1.65072
$606$	0	0
$607$	30.3704	1.23270	0.616348	−	0.787474i	$-0.288611\pi$
$607$	30.3704	1.23270	0.616348	+	0.787474i	$0.288611\pi$
$608$	0	0
$609$	−3.09922	−0.125587
$610$	0	0
$611$	31.3167	1.26694
$612$	0	0
$613$	36.6678	1.48100	0.740500	−	0.672057i	$-0.234589\pi$
$613$	36.6678	1.48100	0.740500	+	0.672057i	$0.234589\pi$
$614$	0	0
$615$	15.8216	0.637988
$616$	0	0
$617$	27.1331	1.09234	0.546170	−	0.837675i	$-0.316085\pi$
$617$	27.1331	1.09234	0.546170	+	0.837675i	$0.316085\pi$
$618$	0	0
$619$	31.2867	1.25752	0.628760	−	0.777600i	$-0.283563\pi$
$619$	31.2867	1.25752	0.628760	+	0.777600i	$0.283563\pi$
$620$	0	0
$621$	−2.65167	−0.106408
$622$	0	0
$623$	−14.0720	−0.563781
$624$	0	0
$625$	6.25837	0.250335
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.01105	0.0801860
$630$	0	0
$631$	−2.37559	−0.0945708	−0.0472854	−	0.998881i	$-0.515057\pi$
$631$	−2.37559	−0.0945708	−0.0472854	+	0.998881i	$0.515057\pi$
$632$	0	0
$633$	18.6517	0.741337
$634$	0	0
$635$	53.3126	2.11564
$636$	0	0
$637$	−10.9842	−0.435209
$638$	0	0
$639$	8.68560	0.343597
$640$	0	0
$641$	−1.27144	−0.0502189	−0.0251095	−	0.999685i	$-0.507993\pi$
$641$	−1.27144	−0.0502189	−0.0251095	+	0.999685i	$0.507993\pi$
$642$	0	0
$643$	0.789080	0.0311183	0.0155591	−	0.999879i	$-0.495047\pi$
$643$	0.789080	0.0311183	0.0155591	+	0.999879i	$0.495047\pi$
$644$	0	0
$645$	−2.03392	−0.0800857
$646$	0	0
$647$	20.6774	0.812912	0.406456	−	0.913670i	$-0.366764\pi$
$647$	20.6774	0.812912	0.406456	+	0.913670i	$0.366764\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−10.9727	−0.430055
$652$	0	0
$653$	−22.8434	−0.893932	−0.446966	−	0.894551i	$-0.647495\pi$
$653$	−22.8434	−0.893932	−0.446966	+	0.894551i	$0.647495\pi$
$654$	0	0
$655$	−42.2064	−1.64914
$656$	0	0
$657$	−1.75374	−0.0684198
$658$	0	0
$659$	10.5040	0.409176	0.204588	−	0.978848i	$-0.434414\pi$
$659$	10.5040	0.409176	0.204588	+	0.978848i	$0.434414\pi$
$660$	0	0
$661$	15.0168	0.584084	0.292042	−	0.956405i	$-0.405665\pi$
$661$	15.0168	0.584084	0.292042	+	0.956405i	$0.405665\pi$
$662$	0	0
$663$	−5.66244	−0.219911
$664$	0	0
$665$	21.9455	0.851009
$666$	0	0
$667$	3.66528	0.141920
$668$	0	0
$669$	−2.24216	−0.0866868
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−16.9370	−0.652872	−0.326436	−	0.945219i	$-0.605848\pi$
$673$	−16.9370	−0.652872	−0.326436	+	0.945219i	$0.605848\pi$
$674$	0	0
$675$	8.62441	0.331954
$676$	0	0
$677$	−34.3509	−1.32021	−0.660106	−	0.751173i	$-0.729489\pi$
$677$	−34.3509	−1.32021	−0.660106	+	0.751173i	$0.729489\pi$
$678$	0	0
$679$	−24.6787	−0.947080
$680$	0	0
$681$	18.3829	0.704435
$682$	0	0
$683$	−3.52452	−0.134862	−0.0674309	−	0.997724i	$-0.521480\pi$
$683$	−3.52452	−0.134862	−0.0674309	+	0.997724i	$0.521480\pi$
$684$	0	0
$685$	−65.0131	−2.48402
$686$	0	0
$687$	−11.2149	−0.427875
$688$	0	0
$689$	5.78766	0.220492
$690$	0	0
$691$	−2.91968	−0.111070	−0.0555349	−	0.998457i	$-0.517686\pi$
$691$	−2.91968	−0.111070	−0.0555349	+	0.998457i	$0.517686\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	55.0200	2.08703
$696$	0	0
$697$	−4.35909	−0.165112
$698$	0	0
$699$	−17.5249	−0.662852
$700$	0	0
$701$	42.3944	1.60121	0.800606	−	0.599191i	$-0.204511\pi$
$701$	42.3944	1.60121	0.800606	+	0.599191i	$0.204511\pi$
$702$	0	0
$703$	−5.24371	−0.197770
$704$	0	0
$705$	20.7604	0.781882
$706$	0	0
$707$	34.3306	1.29114
$708$	0	0
$709$	−48.2207	−1.81096	−0.905482	−	0.424384i	$-0.860491\pi$
$709$	−48.2207	−1.81096	−0.905482	+	0.424384i	$0.860491\pi$
$710$	0	0
$711$	−3.74754	−0.140544
$712$	0	0
$713$	12.9768	0.485987
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.13599	0.266498
$718$	0	0
$719$	−27.9344	−1.04178	−0.520889	−	0.853624i	$-0.674399\pi$
$719$	−27.9344	−1.04178	−0.520889	+	0.853624i	$0.674399\pi$
$720$	0	0
$721$	29.8143	1.11034
$722$	0	0
$723$	−5.86656	−0.218180
$724$	0	0
$725$	−11.9211	−0.442738
$726$	0	0
$727$	−4.65653	−0.172701	−0.0863506	−	0.996265i	$-0.527521\pi$
$727$	−4.65653	−0.172701	−0.0863506	+	0.996265i	$0.527521\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0.560378	0.0207263
$732$	0	0
$733$	3.63957	0.134431	0.0672153	−	0.997738i	$-0.478589\pi$
$733$	3.63957	0.134431	0.0672153	+	0.997738i	$0.478589\pi$
$734$	0	0
$735$	−7.28161	−0.268586
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−40.1562	−1.47717	−0.738584	−	0.674161i	$-0.764505\pi$
$739$	−40.1562	−1.47717	−0.738584	+	0.674161i	$0.764505\pi$
$740$	0	0
$741$	14.7645	0.542388
$742$	0	0
$743$	23.5167	0.862743	0.431372	−	0.902174i	$-0.358030\pi$
$743$	23.5167	0.862743	0.431372	+	0.902174i	$0.358030\pi$
$744$	0	0
$745$	1.14009	0.0417698
$746$	0	0
$747$	5.43867	0.198990
$748$	0	0
$749$	−41.2216	−1.50620
$750$	0	0
$751$	9.49844	0.346603	0.173301	−	0.984869i	$-0.444557\pi$
$751$	9.49844	0.346603	0.173301	+	0.984869i	$0.444557\pi$
$752$	0	0
$753$	−17.0815	−0.622483
$754$	0	0
$755$	17.0218	0.619487
$756$	0	0
$757$	40.6663	1.47804	0.739021	−	0.673682i	$-0.235288\pi$
$757$	40.6663	1.47804	0.739021	+	0.673682i	$0.235288\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−16.3018	−0.590939	−0.295470	−	0.955352i	$-0.595476\pi$
$761$	−16.3018	−0.590939	−0.295470	+	0.955352i	$0.595476\pi$
$762$	0	0
$763$	−40.7291	−1.47449
$764$	0	0
$765$	−3.75374	−0.135717
$766$	0	0
$767$	13.5185	0.488126
$768$	0	0
$769$	−28.9452	−1.04379	−0.521895	−	0.853010i	$-0.674775\pi$
$769$	−28.9452	−1.04379	−0.521895	+	0.853010i	$0.674775\pi$
$770$	0	0
$771$	11.0959	0.399608
$772$	0	0
$773$	25.6351	0.922030	0.461015	−	0.887392i	$-0.347485\pi$
$773$	25.6351	0.922030	0.461015	+	0.887392i	$0.347485\pi$
$774$	0	0
$775$	−42.2064	−1.51610
$776$	0	0
$777$	−4.43389	−0.159065
$778$	0	0
$779$	11.3661	0.407233
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−1.38225	−0.0493976
$784$	0	0
$785$	−26.9576	−0.962157
$786$	0	0
$787$	−12.1428	−0.432843	−0.216422	−	0.976300i	$-0.569439\pi$
$787$	−12.1428	−0.432843	−0.216422	+	0.976300i	$0.569439\pi$
$788$	0	0
$789$	−23.7836	−0.846717
$790$	0	0
$791$	32.3614	1.15064
$792$	0	0
$793$	29.9683	1.06421
$794$	0	0
$795$	3.83675	0.136075
$796$	0	0
$797$	15.6163	0.553159	0.276579	−	0.960991i	$-0.410799\pi$
$797$	15.6163	0.553159	0.276579	+	0.960991i	$0.410799\pi$
$798$	0	0
$799$	−5.71981	−0.202352
$800$	0	0
$801$	−6.27608	−0.221754
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−21.9455	−0.773476
$806$	0	0
$807$	−24.8404	−0.874425
$808$	0	0
$809$	−2.93881	−0.103323	−0.0516615	−	0.998665i	$-0.516452\pi$
$809$	−2.93881	−0.103323	−0.0516615	+	0.998665i	$0.516452\pi$
$810$	0	0
$811$	38.3561	1.34687	0.673433	−	0.739249i	$-0.264819\pi$
$811$	38.3561	1.34687	0.673433	+	0.739249i	$0.264819\pi$
$812$	0	0
$813$	16.6918	0.585407
$814$	0	0
$815$	6.40818	0.224469
$816$	0	0
$817$	−1.46115	−0.0511193
$818$	0	0
$819$	12.4843	0.436237
$820$	0	0
$821$	−16.7552	−0.584759	−0.292379	−	0.956302i	$-0.594447\pi$
$821$	−16.7552	−0.584759	−0.292379	+	0.956302i	$0.594447\pi$
$822$	0	0
$823$	13.9827	0.487408	0.243704	−	0.969850i	$-0.421637\pi$
$823$	13.9827	0.487408	0.243704	+	0.969850i	$0.421637\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−0.468770	−0.0163007	−0.00815037	−	0.999967i	$-0.502594\pi$
$827$	−0.468770	−0.0163007	−0.00815037	+	0.999967i	$0.502594\pi$
$828$	0	0
$829$	54.6669	1.89866	0.949329	−	0.314283i	$-0.101764\pi$
$829$	54.6669	1.89866	0.949329	+	0.314283i	$0.101764\pi$
$830$	0	0
$831$	30.8888	1.07152
$832$	0	0
$833$	2.00620	0.0695106
$834$	0	0
$835$	3.69113	0.127737
$836$	0	0
$837$	−4.89383	−0.169156
$838$	0	0
$839$	31.9399	1.10269	0.551343	−	0.834279i	$-0.314115\pi$
$839$	31.9399	1.10269	0.551343	+	0.834279i	$0.314115\pi$
$840$	0	0
$841$	−27.0894	−0.934117
$842$	0	0
$843$	25.0406	0.862444
$844$	0	0
$845$	66.4497	2.28594
$846$	0	0
$847$	−24.6637	−0.847456
$848$	0	0
$849$	−27.6964	−0.950537
$850$	0	0
$851$	5.24371	0.179752
$852$	0	0
$853$	13.0108	0.445480	0.222740	−	0.974878i	$-0.428500\pi$
$853$	13.0108	0.445480	0.222740	+	0.974878i	$0.428500\pi$
$854$	0	0
$855$	9.78766	0.334731
$856$	0	0
$857$	16.1331	0.551096	0.275548	−	0.961287i	$-0.411141\pi$
$857$	16.1331	0.551096	0.275548	+	0.961287i	$0.411141\pi$
$858$	0	0
$859$	6.77689	0.231225	0.115612	−	0.993294i	$-0.463117\pi$
$859$	6.77689	0.231225	0.115612	+	0.993294i	$0.463117\pi$
$860$	0	0
$861$	9.61075	0.327533
$862$	0	0
$863$	16.4517	0.560023	0.280012	−	0.959997i	$-0.409662\pi$
$863$	16.4517	0.560023	0.280012	+	0.959997i	$0.409662\pi$
$864$	0	0
$865$	−19.8666	−0.675483
$866$	0	0
$867$	−15.9658	−0.542227
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−82.7462	−2.80375
$872$	0	0
$873$	−11.0067	−0.372519
$874$	0	0
$875$	29.9959	1.01405
$876$	0	0
$877$	3.08145	0.104053	0.0520267	−	0.998646i	$-0.483432\pi$
$877$	3.08145	0.104053	0.0520267	+	0.998646i	$0.483432\pi$
$878$	0	0
$879$	21.3823	0.721205
$880$	0	0
$881$	26.5909	0.895872	0.447936	−	0.894066i	$-0.352159\pi$
$881$	26.5909	0.895872	0.447936	+	0.894066i	$0.352159\pi$
$882$	0	0
$883$	40.9997	1.37975	0.689875	−	0.723928i	$-0.257665\pi$
$883$	40.9997	1.37975	0.689875	+	0.723928i	$0.257665\pi$
$884$	0	0
$885$	8.96168	0.301244
$886$	0	0
$887$	−24.5233	−0.823413	−0.411707	−	0.911316i	$-0.635067\pi$
$887$	−24.5233	−0.823413	−0.411707	+	0.911316i	$0.635067\pi$
$888$	0	0
$889$	32.3845	1.08614
$890$	0	0
$891$	0	0
$892$	0	0
$893$	14.9141	0.499081
$894$	0	0
$895$	−4.81082	−0.160808
$896$	0	0
$897$	−14.7645	−0.492972
$898$	0	0
$899$	6.76450	0.225609
$900$	0	0
$901$	−1.05708	−0.0352166
$902$	0	0
$903$	−1.23550	−0.0411148
$904$	0	0
$905$	−61.1369	−2.03226
$906$	0	0
$907$	−8.73058	−0.289894	−0.144947	−	0.989439i	$-0.546301\pi$
$907$	−8.73058	−0.289894	−0.144947	+	0.989439i	$0.546301\pi$
$908$	0	0
$909$	15.3114	0.507848
$910$	0	0
$911$	−43.0841	−1.42744	−0.713719	−	0.700432i	$-0.752991\pi$
$911$	−43.0841	−1.42744	−0.713719	+	0.700432i	$0.752991\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	19.8666	0.656768
$916$	0	0
$917$	−25.6381	−0.846644
$918$	0	0
$919$	40.3233	1.33014	0.665072	−	0.746779i	$-0.268401\pi$
$919$	40.3233	1.33014	0.665072	+	0.746779i	$0.268401\pi$
$920$	0	0
$921$	−5.35640	−0.176500
$922$	0	0
$923$	48.3614	1.59183
$924$	0	0
$925$	−17.0549	−0.560760
$926$	0	0
$927$	13.2972	0.436736
$928$	0	0
$929$	−6.18762	−0.203009	−0.101505	−	0.994835i	$-0.532366\pi$
$929$	−6.18762	−0.203009	−0.101505	+	0.994835i	$0.532366\pi$
$930$	0	0
$931$	−5.23105	−0.171441
$932$	0	0
$933$	−11.0980	−0.363331
$934$	0	0
$935$	0	0
$936$	0	0
$937$	6.88688	0.224985	0.112492	−	0.993653i	$-0.464117\pi$
$937$	6.88688	0.224985	0.112492	+	0.993653i	$0.464117\pi$
$938$	0	0
$939$	−20.4527	−0.667447
$940$	0	0
$941$	48.3203	1.57520	0.787599	−	0.616188i	$-0.211324\pi$
$941$	48.3203	1.57520	0.787599	+	0.616188i	$0.211324\pi$
$942$	0	0
$943$	−11.3661	−0.370131
$944$	0	0
$945$	8.27608	0.269221
$946$	0	0
$947$	13.9867	0.454506	0.227253	−	0.973836i	$-0.427026\pi$
$947$	13.9867	0.454506	0.227253	+	0.973836i	$0.427026\pi$
$948$	0	0
$949$	−9.76479	−0.316978
$950$	0	0
$951$	0.889723	0.0288512
$952$	0	0
$953$	49.2220	1.59446	0.797229	−	0.603677i	$-0.206299\pi$
$953$	49.2220	1.59446	0.797229	+	0.603677i	$0.206299\pi$
$954$	0	0
$955$	78.7434	2.54808
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−39.4919	−1.27526
$960$	0	0
$961$	−7.05042	−0.227433
$962$	0	0
$963$	−18.3848	−0.592442
$964$	0	0
$965$	−48.4866	−1.56084
$966$	0	0
$967$	−5.91692	−0.190275	−0.0951376	−	0.995464i	$-0.530329\pi$
$967$	−5.91692	−0.190275	−0.0951376	+	0.995464i	$0.530329\pi$
$968$	0	0
$969$	−2.69665	−0.0866290
$970$	0	0
$971$	27.8006	0.892164	0.446082	−	0.894992i	$-0.352819\pi$
$971$	27.8006	0.892164	0.446082	+	0.894992i	$0.352819\pi$
$972$	0	0
$973$	33.4216	1.07145
$974$	0	0
$975$	48.0206	1.53789
$976$	0	0
$977$	−34.4493	−1.10213	−0.551065	−	0.834462i	$-0.685778\pi$
$977$	−34.4493	−1.10213	−0.551065	+	0.834462i	$0.685778\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−18.1651	−0.579968
$982$	0	0
$983$	7.58504	0.241925	0.120963	−	0.992657i	$-0.461402\pi$
$983$	7.58504	0.241925	0.120963	+	0.992657i	$0.461402\pi$
$984$	0	0
$985$	−24.1807	−0.770460
$986$	0	0
$987$	12.6108	0.401406
$988$	0	0
$989$	1.46115	0.0464620
$990$	0	0
$991$	30.1199	0.956792	0.478396	−	0.878144i	$-0.341218\pi$
$991$	30.1199	0.956792	0.478396	+	0.878144i	$0.341218\pi$
$992$	0	0
$993$	−7.31553	−0.232151
$994$	0	0
$995$	−26.3398	−0.835029
$996$	0	0
$997$	−17.5318	−0.555239	−0.277620	−	0.960691i	$-0.589545\pi$
$997$	−17.5318	−0.555239	−0.277620	+	0.960691i	$0.589545\pi$
$998$	0	0
$999$	−1.97751	−0.0625657

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.o.1.4		4
4.3	odd	2		1002.2.a.i.1.4	✓	4
12.11	even	2		3006.2.a.s.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.i.1.4	✓	4	4.3	odd	2
3006.2.a.s.1.1		4	12.11	even	2
8016.2.a.o.1.4		4	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.69113 q^{5} +2.24216 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.69113 q^{5} +2.24216 q^{7} +1.00000 q^{9} +5.56799 q^{13} +3.69113 q^{15} -1.01696 q^{17} +2.65167 q^{19} +2.24216 q^{21} -2.65167 q^{23} +8.62441 q^{25} +1.00000 q^{27} -1.38225 q^{29} -4.89383 q^{31} +8.27608 q^{35} -1.97751 q^{37} +5.56799 q^{39} +4.28639 q^{41} -0.551031 q^{43} +3.69113 q^{45} +5.62441 q^{47} -1.97273 q^{49} -1.01696 q^{51} +1.03945 q^{53} +2.65167 q^{57} +2.42790 q^{59} +5.38225 q^{61} +2.24216 q^{63} +20.5522 q^{65} -14.8610 q^{67} -2.65167 q^{69} +8.68560 q^{71} -1.75374 q^{73} +8.62441 q^{75} -3.74754 q^{79} +1.00000 q^{81} +5.43867 q^{83} -3.75374 q^{85} -1.38225 q^{87} -6.27608 q^{89} +12.4843 q^{91} -4.89383 q^{93} +9.78766 q^{95} -11.0067 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} - q^{7} + 4 q^{9} + 8 q^{13} + 5 q^{15} + 10 q^{17} + 2 q^{19} - q^{21} - 2 q^{23} + 5 q^{25} + 4 q^{27} + 14 q^{29} - q^{31} - 5 q^{35} + 5 q^{37} + 8 q^{39} + 14 q^{41} - 2 q^{43} + 5 q^{45} - 7 q^{47} + 13 q^{49} + 10 q^{51} + 3 q^{53} + 2 q^{57} + 5 q^{59} + 2 q^{61} - q^{63} + 6 q^{65} + 7 q^{67} - 2 q^{69} - 2 q^{71} + 2 q^{73} + 5 q^{75} + 10 q^{79} + 4 q^{81} - 13 q^{83} - 6 q^{85} + 14 q^{87} + 13 q^{89} + 30 q^{91} - q^{93} + 2 q^{95} + 5 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 - q^7 + 4 * q^9 + 8 * q^13 + 5 * q^15 + 10 * q^17 + 2 * q^19 - q^21 - 2 * q^23 + 5 * q^25 + 4 * q^27 + 14 * q^29 - q^31 - 5 * q^35 + 5 * q^37 + 8 * q^39 + 14 * q^41 - 2 * q^43 + 5 * q^45 - 7 * q^47 + 13 * q^49 + 10 * q^51 + 3 * q^53 + 2 * q^57 + 5 * q^59 + 2 * q^61 - q^63 + 6 * q^65 + 7 * q^67 - 2 * q^69 - 2 * q^71 + 2 * q^73 + 5 * q^75 + 10 * q^79 + 4 * q^81 - 13 * q^83 - 6 * q^85 + 14 * q^87 + 13 * q^89 + 30 * q^91 - q^93 + 2 * q^95 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more