LMFDB - Embedded newform 4016.2.a.l.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4016,2,Mod(1,4016)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4016.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4016, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.a (trivial)

Newform invariants

comment:select newform

sage:traces = [19,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 6 x^{18} - 21 x^{17} + 179 x^{16} + 90 x^{15} - 2109 x^{14} + 926 x^{13} + 12681 x^{12} + \cdots + 4 $ x^19 - 6x^18 - 21x^17 + 179x^16 + 90x^15 - 2109x^14 + 926x^13 + 12681x^12 - 10845x^11 - 41921x^10 + 43551x^9 + 76260x^8 - 80907x^7 - 72526x^6 + 64793x^5 + 33209x^4 - 13777x^3 - 7607x^2 - 747x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.31693$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31693 q^{3} -1.28443 q^{5} +1.02036 q^{7} -1.26569 q^{9} -3.40771 q^{11} -6.32304 q^{13} +1.69151 q^{15} +5.36124 q^{17} +2.29090 q^{19} -1.34374 q^{21} -6.25279 q^{23} -3.35023 q^{25} +5.61763 q^{27} -8.07720 q^{29} -1.34002 q^{31} +4.48773 q^{33} -1.31058 q^{35} -7.74614 q^{37} +8.32702 q^{39} +9.95371 q^{41} +1.55335 q^{43} +1.62569 q^{45} +11.5709 q^{47} -5.95887 q^{49} -7.06039 q^{51} -8.43397 q^{53} +4.37697 q^{55} -3.01697 q^{57} +12.9358 q^{59} +0.212306 q^{61} -1.29145 q^{63} +8.12152 q^{65} -1.62764 q^{67} +8.23450 q^{69} -11.1685 q^{71} -10.7073 q^{73} +4.41203 q^{75} -3.47708 q^{77} +14.1646 q^{79} -3.60098 q^{81} -1.94879 q^{83} -6.88615 q^{85} +10.6371 q^{87} -4.10583 q^{89} -6.45176 q^{91} +1.76472 q^{93} -2.94251 q^{95} +13.6029 q^{97} +4.31309 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37}+ \cdots + 74 q^{99}+O(q^{100}) $ 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.31693	−0.760332	−0.380166	−	0.924918i	$-0.624133\pi$
$3$	−1.31693	−0.760332	−0.380166	+	0.924918i	$0.624133\pi$
$4$	0	0
$5$	−1.28443	−0.574416	−0.287208	−	0.957868i	$-0.592727\pi$
$5$	−1.28443	−0.574416	−0.287208	+	0.957868i	$0.592727\pi$
$6$	0	0
$7$	1.02036	0.385659	0.192829	−	0.981232i	$-0.438234\pi$
$7$	1.02036	0.385659	0.192829	+	0.981232i	$0.438234\pi$
$8$	0	0
$9$	−1.26569	−0.421895
$10$	0	0
$11$	−3.40771	−1.02746	−0.513731	−	0.857951i	$-0.671737\pi$
$11$	−3.40771	−1.02746	−0.513731	+	0.857951i	$0.671737\pi$
$12$	0	0
$13$	−6.32304	−1.75370	−0.876848	−	0.480768i	$-0.840358\pi$
$13$	−6.32304	−1.75370	−0.876848	+	0.480768i	$0.840358\pi$
$14$	0	0
$15$	1.69151	0.436747
$16$	0	0
$17$	5.36124	1.30029	0.650145	−	0.759810i	$-0.274708\pi$
$17$	5.36124	1.30029	0.650145	+	0.759810i	$0.274708\pi$
$18$	0	0
$19$	2.29090	0.525569	0.262785	−	0.964855i	$-0.415359\pi$
$19$	2.29090	0.525569	0.262785	+	0.964855i	$0.415359\pi$
$20$	0	0
$21$	−1.34374	−0.293229
$22$	0	0
$23$	−6.25279	−1.30380	−0.651898	−	0.758307i	$-0.726027\pi$
$23$	−6.25279	−1.30380	−0.651898	+	0.758307i	$0.726027\pi$
$24$	0	0
$25$	−3.35023	−0.670047
$26$	0	0
$27$	5.61763	1.08111
$28$	0	0
$29$	−8.07720	−1.49990	−0.749949	−	0.661495i	$-0.769922\pi$
$29$	−8.07720	−1.49990	−0.749949	+	0.661495i	$0.769922\pi$
$30$	0	0
$31$	−1.34002	−0.240675	−0.120337	−	0.992733i	$-0.538398\pi$
$31$	−1.34002	−0.240675	−0.120337	+	0.992733i	$0.538398\pi$
$32$	0	0
$33$	4.48773	0.781213
$34$	0	0
$35$	−1.31058	−0.221528
$36$	0	0
$37$	−7.74614	−1.27346	−0.636729	−	0.771088i	$-0.719713\pi$
$37$	−7.74614	−1.27346	−0.636729	+	0.771088i	$0.719713\pi$
$38$	0	0
$39$	8.32702	1.33339
$40$	0	0
$41$	9.95371	1.55451	0.777254	−	0.629187i	$-0.216612\pi$
$41$	9.95371	1.55451	0.777254	+	0.629187i	$0.216612\pi$
$42$	0	0
$43$	1.55335	0.236884	0.118442	−	0.992961i	$-0.462210\pi$
$43$	1.55335	0.236884	0.118442	+	0.992961i	$0.462210\pi$
$44$	0	0
$45$	1.62569	0.242343
$46$	0	0
$47$	11.5709	1.68780	0.843898	−	0.536503i	$-0.180255\pi$
$47$	11.5709	1.68780	0.843898	+	0.536503i	$0.180255\pi$
$48$	0	0
$49$	−5.95887	−0.851267
$50$	0	0
$51$	−7.06039	−0.988653
$52$	0	0
$53$	−8.43397	−1.15850	−0.579248	−	0.815152i	$-0.696654\pi$
$53$	−8.43397	−1.15850	−0.579248	+	0.815152i	$0.696654\pi$
$54$	0	0
$55$	4.37697	0.590191
$56$	0	0
$57$	−3.01697	−0.399607
$58$	0	0
$59$	12.9358	1.68410	0.842051	−	0.539397i	$-0.181348\pi$
$59$	12.9358	1.68410	0.842051	+	0.539397i	$0.181348\pi$
$60$	0	0
$61$	0.212306	0.0271830	0.0135915	−	0.999908i	$-0.495674\pi$
$61$	0.212306	0.0271830	0.0135915	+	0.999908i	$0.495674\pi$
$62$	0	0
$63$	−1.29145	−0.162708
$64$	0	0
$65$	8.12152	1.00735
$66$	0	0
$67$	−1.62764	−0.198848	−0.0994240	−	0.995045i	$-0.531700\pi$
$67$	−1.62764	−0.198848	−0.0994240	+	0.995045i	$0.531700\pi$
$68$	0	0
$69$	8.23450	0.991318
$70$	0	0
$71$	−11.1685	−1.32546	−0.662729	−	0.748860i	$-0.730602\pi$
$71$	−11.1685	−1.32546	−0.662729	+	0.748860i	$0.730602\pi$
$72$	0	0
$73$	−10.7073	−1.25319	−0.626597	−	0.779343i	$-0.715553\pi$
$73$	−10.7073	−1.25319	−0.626597	+	0.779343i	$0.715553\pi$
$74$	0	0
$75$	4.41203	0.509458
$76$	0	0
$77$	−3.47708	−0.396250
$78$	0	0
$79$	14.1646	1.59365	0.796824	−	0.604212i	$-0.206512\pi$
$79$	14.1646	1.59365	0.796824	+	0.604212i	$0.206512\pi$
$80$	0	0
$81$	−3.60098	−0.400109
$82$	0	0
$83$	−1.94879	−0.213907	−0.106954	−	0.994264i	$-0.534110\pi$
$83$	−1.94879	−0.213907	−0.106954	+	0.994264i	$0.534110\pi$
$84$	0	0
$85$	−6.88615	−0.746907
$86$	0	0
$87$	10.6371	1.14042
$88$	0	0
$89$	−4.10583	−0.435217	−0.217608	−	0.976036i	$-0.569826\pi$
$89$	−4.10583	−0.435217	−0.217608	+	0.976036i	$0.569826\pi$
$90$	0	0
$91$	−6.45176	−0.676328
$92$	0	0
$93$	1.76472	0.182993
$94$	0	0
$95$	−2.94251	−0.301895
$96$	0	0
$97$	13.6029	1.38117	0.690583	−	0.723253i	$-0.257354\pi$
$97$	13.6029	1.38117	0.690583	+	0.723253i	$0.257354\pi$
$98$	0	0
$99$	4.31309	0.433482
$100$	0	0
$101$	−0.708875	−0.0705357	−0.0352679	−	0.999378i	$-0.511228\pi$
$101$	−0.708875	−0.0705357	−0.0352679	+	0.999378i	$0.511228\pi$
$102$	0	0
$103$	−0.913943	−0.0900535	−0.0450267	−	0.998986i	$-0.514337\pi$
$103$	−0.913943	−0.0900535	−0.0450267	+	0.998986i	$0.514337\pi$
$104$	0	0
$105$	1.72595	0.168435
$106$	0	0
$107$	−3.00141	−0.290157	−0.145078	−	0.989420i	$-0.546343\pi$
$107$	−3.00141	−0.290157	−0.145078	+	0.989420i	$0.546343\pi$
$108$	0	0
$109$	12.7818	1.22427	0.612137	−	0.790752i	$-0.290310\pi$
$109$	12.7818	1.22427	0.612137	+	0.790752i	$0.290310\pi$
$110$	0	0
$111$	10.2011	0.968250
$112$	0	0
$113$	−3.91713	−0.368493	−0.184246	−	0.982880i	$-0.558984\pi$
$113$	−3.91713	−0.368493	−0.184246	+	0.982880i	$0.558984\pi$
$114$	0	0
$115$	8.03128	0.748921
$116$	0	0
$117$	8.00298	0.739876
$118$	0	0
$119$	5.47038	0.501468
$120$	0	0
$121$	0.612476	0.0556796
$122$	0	0
$123$	−13.1084	−1.18194
$124$	0	0
$125$	10.7253	0.959301
$126$	0	0
$127$	8.76931	0.778150	0.389075	−	0.921206i	$-0.372795\pi$
$127$	8.76931	0.778150	0.389075	+	0.921206i	$0.372795\pi$
$128$	0	0
$129$	−2.04566	−0.180110
$130$	0	0
$131$	−3.48973	−0.304899	−0.152450	−	0.988311i	$-0.548716\pi$
$131$	−3.48973	−0.304899	−0.152450	+	0.988311i	$0.548716\pi$
$132$	0	0
$133$	2.33754	0.202690
$134$	0	0
$135$	−7.21546	−0.621008
$136$	0	0
$137$	3.10500	0.265278	0.132639	−	0.991164i	$-0.457655\pi$
$137$	3.10500	0.265278	0.132639	+	0.991164i	$0.457655\pi$
$138$	0	0
$139$	3.59477	0.304904	0.152452	−	0.988311i	$-0.451283\pi$
$139$	3.59477	0.304904	0.152452	+	0.988311i	$0.451283\pi$
$140$	0	0
$141$	−15.2382	−1.28329
$142$	0	0
$143$	21.5471	1.80186
$144$	0	0
$145$	10.3746	0.861565
$146$	0	0
$147$	7.84744	0.647246
$148$	0	0
$149$	−14.2901	−1.17069	−0.585344	−	0.810785i	$-0.699041\pi$
$149$	−14.2901	−1.17069	−0.585344	+	0.810785i	$0.699041\pi$
$150$	0	0
$151$	4.83424	0.393405	0.196702	−	0.980463i	$-0.436977\pi$
$151$	4.83424	0.393405	0.196702	+	0.980463i	$0.436977\pi$
$152$	0	0
$153$	−6.78564	−0.548587
$154$	0	0
$155$	1.72117	0.138247
$156$	0	0
$157$	10.3653	0.827240	0.413620	−	0.910450i	$-0.364264\pi$
$157$	10.3653	0.827240	0.413620	+	0.910450i	$0.364264\pi$
$158$	0	0
$159$	11.1070	0.880841
$160$	0	0
$161$	−6.38007	−0.502820
$162$	0	0
$163$	5.74307	0.449832	0.224916	−	0.974378i	$-0.427789\pi$
$163$	5.74307	0.449832	0.224916	+	0.974378i	$0.427789\pi$
$164$	0	0
$165$	−5.76418	−0.448741
$166$	0	0
$167$	−7.01822	−0.543086	−0.271543	−	0.962426i	$-0.587534\pi$
$167$	−7.01822	−0.543086	−0.271543	+	0.962426i	$0.587534\pi$
$168$	0	0
$169$	26.9808	2.07545
$170$	0	0
$171$	−2.89956	−0.221735
$172$	0	0
$173$	3.76197	0.286018	0.143009	−	0.989721i	$-0.454322\pi$
$173$	3.76197	0.286018	0.143009	+	0.989721i	$0.454322\pi$
$174$	0	0
$175$	−3.41843	−0.258409
$176$	0	0
$177$	−17.0356	−1.28048
$178$	0	0
$179$	−19.7663	−1.47740	−0.738700	−	0.674035i	$-0.764560\pi$
$179$	−19.7663	−1.47740	−0.738700	+	0.674035i	$0.764560\pi$
$180$	0	0
$181$	14.9611	1.11205	0.556024	−	0.831167i	$-0.312326\pi$
$181$	14.9611	1.11205	0.556024	+	0.831167i	$0.312326\pi$
$182$	0	0
$183$	−0.279593	−0.0206681
$184$	0	0
$185$	9.94939	0.731494
$186$	0	0
$187$	−18.2695	−1.33600
$188$	0	0
$189$	5.73198	0.416940
$190$	0	0
$191$	10.6702	0.772071	0.386036	−	0.922484i	$-0.373844\pi$
$191$	10.6702	0.772071	0.386036	+	0.922484i	$0.373844\pi$
$192$	0	0
$193$	16.3570	1.17740	0.588702	−	0.808350i	$-0.299639\pi$
$193$	16.3570	1.17740	0.588702	+	0.808350i	$0.299639\pi$
$194$	0	0
$195$	−10.6955	−0.765921
$196$	0	0
$197$	−5.80928	−0.413894	−0.206947	−	0.978352i	$-0.566353\pi$
$197$	−5.80928	−0.413894	−0.206947	+	0.978352i	$0.566353\pi$
$198$	0	0
$199$	5.49860	0.389785	0.194893	−	0.980825i	$-0.437564\pi$
$199$	5.49860	0.389785	0.194893	+	0.980825i	$0.437564\pi$
$200$	0	0
$201$	2.14350	0.151191
$202$	0	0
$203$	−8.24163	−0.578449
$204$	0	0
$205$	−12.7849	−0.892934
$206$	0	0
$207$	7.91406	0.550065
$208$	0	0
$209$	−7.80673	−0.540003
$210$	0	0
$211$	−9.32109	−0.641690	−0.320845	−	0.947132i	$-0.603967\pi$
$211$	−9.32109	−0.641690	−0.320845	+	0.947132i	$0.603967\pi$
$212$	0	0
$213$	14.7082	1.00779
$214$	0	0
$215$	−1.99517	−0.136070
$216$	0	0
$217$	−1.36730	−0.0928184
$218$	0	0
$219$	14.1008	0.952843
$220$	0	0
$221$	−33.8993	−2.28031
$222$	0	0
$223$	−22.9589	−1.53744	−0.768721	−	0.639584i	$-0.779106\pi$
$223$	−22.9589	−1.53744	−0.768721	+	0.639584i	$0.779106\pi$
$224$	0	0
$225$	4.24034	0.282690
$226$	0	0
$227$	26.8349	1.78109	0.890546	−	0.454893i	$-0.150323\pi$
$227$	26.8349	1.78109	0.890546	+	0.454893i	$0.150323\pi$
$228$	0	0
$229$	23.7770	1.57123	0.785613	−	0.618718i	$-0.212348\pi$
$229$	23.7770	1.57123	0.785613	+	0.618718i	$0.212348\pi$
$230$	0	0
$231$	4.57908	0.301282
$232$	0	0
$233$	6.34838	0.415896	0.207948	−	0.978140i	$-0.433321\pi$
$233$	6.34838	0.415896	0.207948	+	0.978140i	$0.433321\pi$
$234$	0	0
$235$	−14.8621	−0.969497
$236$	0	0
$237$	−18.6539	−1.21170
$238$	0	0
$239$	−20.5726	−1.33073	−0.665366	−	0.746518i	$-0.731724\pi$
$239$	−20.5726	−1.33073	−0.665366	+	0.746518i	$0.731724\pi$
$240$	0	0
$241$	−17.1473	−1.10455	−0.552277	−	0.833661i	$-0.686241\pi$
$241$	−17.1473	−1.10455	−0.552277	+	0.833661i	$0.686241\pi$
$242$	0	0
$243$	−12.1106	−0.776897
$244$	0	0
$245$	7.65377	0.488981
$246$	0	0
$247$	−14.4855	−0.921689
$248$	0	0
$249$	2.56643	0.162641
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	21.3077	1.33960
$254$	0	0
$255$	9.06860	0.567898
$256$	0	0
$257$	19.0889	1.19073	0.595367	−	0.803454i	$-0.297007\pi$
$257$	19.0889	1.19073	0.595367	+	0.803454i	$0.297007\pi$
$258$	0	0
$259$	−7.90383	−0.491120
$260$	0	0
$261$	10.2232	0.632800
$262$	0	0
$263$	26.8459	1.65539	0.827695	−	0.561178i	$-0.189652\pi$
$263$	26.8459	1.65539	0.827695	+	0.561178i	$0.189652\pi$
$264$	0	0
$265$	10.8329	0.665458
$266$	0	0
$267$	5.40710	0.330909
$268$	0	0
$269$	−17.8198	−1.08649	−0.543246	−	0.839573i	$-0.682805\pi$
$269$	−17.8198	−1.08649	−0.543246	+	0.839573i	$0.682805\pi$
$270$	0	0
$271$	25.9178	1.57440	0.787198	−	0.616700i	$-0.211531\pi$
$271$	25.9178	1.57440	0.787198	+	0.616700i	$0.211531\pi$
$272$	0	0
$273$	8.49654	0.514234
$274$	0	0
$275$	11.4166	0.688448
$276$	0	0
$277$	0.255098	0.0153273	0.00766367	−	0.999971i	$-0.497561\pi$
$277$	0.255098	0.0153273	0.00766367	+	0.999971i	$0.497561\pi$
$278$	0	0
$279$	1.69605	0.101540
$280$	0	0
$281$	24.1427	1.44023	0.720115	−	0.693855i	$-0.244089\pi$
$281$	24.1427	1.44023	0.720115	+	0.693855i	$0.244089\pi$
$282$	0	0
$283$	30.9395	1.83916	0.919581	−	0.392899i	$-0.128528\pi$
$283$	30.9395	1.83916	0.919581	+	0.392899i	$0.128528\pi$
$284$	0	0
$285$	3.87509	0.229541
$286$	0	0
$287$	10.1563	0.599510
$288$	0	0
$289$	11.7429	0.690756
$290$	0	0
$291$	−17.9141	−1.05014
$292$	0	0
$293$	−23.8506	−1.39337	−0.696684	−	0.717378i	$-0.745342\pi$
$293$	−23.8506	−1.39337	−0.696684	+	0.717378i	$0.745342\pi$
$294$	0	0
$295$	−16.6152	−0.967375
$296$	0	0
$297$	−19.1432	−1.11080
$298$	0	0
$299$	39.5366	2.28646
$300$	0	0
$301$	1.58497	0.0913563
$302$	0	0
$303$	0.933542	0.0536306
$304$	0	0
$305$	−0.272693	−0.0156143
$306$	0	0
$307$	16.2007	0.924622	0.462311	−	0.886718i	$-0.347020\pi$
$307$	16.2007	0.924622	0.462311	+	0.886718i	$0.347020\pi$
$308$	0	0
$309$	1.20360	0.0684705
$310$	0	0
$311$	−6.09690	−0.345724	−0.172862	−	0.984946i	$-0.555301\pi$
$311$	−6.09690	−0.345724	−0.172862	+	0.984946i	$0.555301\pi$
$312$	0	0
$313$	4.43223	0.250525	0.125262	−	0.992124i	$-0.460023\pi$
$313$	4.43223	0.250525	0.125262	+	0.992124i	$0.460023\pi$
$314$	0	0
$315$	1.65878	0.0934618
$316$	0	0
$317$	−8.00439	−0.449571	−0.224786	−	0.974408i	$-0.572168\pi$
$317$	−8.00439	−0.449571	−0.224786	+	0.974408i	$0.572168\pi$
$318$	0	0
$319$	27.5247	1.54109
$320$	0	0
$321$	3.95265	0.220616
$322$	0	0
$323$	12.2821	0.683393
$324$	0	0
$325$	21.1837	1.17506
$326$	0	0
$327$	−16.8328	−0.930854
$328$	0	0
$329$	11.8065	0.650913
$330$	0	0
$331$	30.8762	1.69711	0.848554	−	0.529109i	$-0.177474\pi$
$331$	30.8762	1.69711	0.848554	+	0.529109i	$0.177474\pi$
$332$	0	0
$333$	9.80418	0.537266
$334$	0	0
$335$	2.09060	0.114221
$336$	0	0
$337$	25.1121	1.36794	0.683971	−	0.729509i	$-0.260252\pi$
$337$	25.1121	1.36794	0.683971	+	0.729509i	$0.260252\pi$
$338$	0	0
$339$	5.15860	0.280177
$340$	0	0
$341$	4.56640	0.247285
$342$	0	0
$343$	−13.2227	−0.713957
$344$	0	0
$345$	−10.5767	−0.569428
$346$	0	0
$347$	21.6440	1.16191	0.580956	−	0.813935i	$-0.302679\pi$
$347$	21.6440	1.16191	0.580956	+	0.813935i	$0.302679\pi$
$348$	0	0
$349$	31.8202	1.70330	0.851648	−	0.524113i	$-0.175603\pi$
$349$	31.8202	1.70330	0.851648	+	0.524113i	$0.175603\pi$
$350$	0	0
$351$	−35.5205	−1.89594
$352$	0	0
$353$	−18.9369	−1.00791	−0.503954	−	0.863730i	$-0.668122\pi$
$353$	−18.9369	−1.00791	−0.503954	+	0.863730i	$0.668122\pi$
$354$	0	0
$355$	14.3452	0.761364
$356$	0	0
$357$	−7.20412	−0.381282
$358$	0	0
$359$	−16.4551	−0.868468	−0.434234	−	0.900800i	$-0.642981\pi$
$359$	−16.4551	−0.868468	−0.434234	+	0.900800i	$0.642981\pi$
$360$	0	0
$361$	−13.7518	−0.723777
$362$	0	0
$363$	−0.806590	−0.0423350
$364$	0	0
$365$	13.7528	0.719854
$366$	0	0
$367$	−7.04730	−0.367866	−0.183933	−	0.982939i	$-0.558883\pi$
$367$	−7.04730	−0.367866	−0.183933	+	0.982939i	$0.558883\pi$
$368$	0	0
$369$	−12.5983	−0.655840
$370$	0	0
$371$	−8.60566	−0.446784
$372$	0	0
$373$	−10.9354	−0.566211	−0.283106	−	0.959089i	$-0.591365\pi$
$373$	−10.9354	−0.566211	−0.283106	+	0.959089i	$0.591365\pi$
$374$	0	0
$375$	−14.1245	−0.729387
$376$	0	0
$377$	51.0725	2.63037
$378$	0	0
$379$	−3.20641	−0.164702	−0.0823511	−	0.996603i	$-0.526243\pi$
$379$	−3.20641	−0.164702	−0.0823511	+	0.996603i	$0.526243\pi$
$380$	0	0
$381$	−11.5486	−0.591652
$382$	0	0
$383$	25.5125	1.30363	0.651814	−	0.758379i	$-0.274008\pi$
$383$	25.5125	1.30363	0.651814	+	0.758379i	$0.274008\pi$
$384$	0	0
$385$	4.46607	0.227612
$386$	0	0
$387$	−1.96605	−0.0999402
$388$	0	0
$389$	19.7967	1.00373	0.501866	−	0.864945i	$-0.332647\pi$
$389$	19.7967	1.00373	0.501866	+	0.864945i	$0.332647\pi$
$390$	0	0
$391$	−33.5227	−1.69531
$392$	0	0
$393$	4.59574	0.231824
$394$	0	0
$395$	−18.1935	−0.915416
$396$	0	0
$397$	−20.8522	−1.04654	−0.523270	−	0.852167i	$-0.675288\pi$
$397$	−20.8522	−1.04654	−0.523270	+	0.852167i	$0.675288\pi$
$398$	0	0
$399$	−3.07838	−0.154112
$400$	0	0
$401$	−30.4625	−1.52122	−0.760612	−	0.649207i	$-0.775101\pi$
$401$	−30.4625	−1.52122	−0.760612	+	0.649207i	$0.775101\pi$
$402$	0	0
$403$	8.47301	0.422071
$404$	0	0
$405$	4.62522	0.229829
$406$	0	0
$407$	26.3966	1.30843
$408$	0	0
$409$	21.9664	1.08617	0.543085	−	0.839678i	$-0.317256\pi$
$409$	21.9664	1.08617	0.543085	+	0.839678i	$0.317256\pi$
$410$	0	0
$411$	−4.08907	−0.201699
$412$	0	0
$413$	13.1992	0.649489
$414$	0	0
$415$	2.50309	0.122872
$416$	0	0
$417$	−4.73407	−0.231828
$418$	0	0
$419$	−6.11485	−0.298730	−0.149365	−	0.988782i	$-0.547723\pi$
$419$	−6.11485	−0.298730	−0.149365	+	0.988782i	$0.547723\pi$
$420$	0	0
$421$	−7.04861	−0.343528	−0.171764	−	0.985138i	$-0.554947\pi$
$421$	−7.04861	−0.343528	−0.171764	+	0.985138i	$0.554947\pi$
$422$	0	0
$423$	−14.6452	−0.712073
$424$	0	0
$425$	−17.9614	−0.871255
$426$	0	0
$427$	0.216628	0.0104834
$428$	0	0
$429$	−28.3761	−1.37001
$430$	0	0
$431$	1.56669	0.0754646	0.0377323	−	0.999288i	$-0.487987\pi$
$431$	1.56669	0.0754646	0.0377323	+	0.999288i	$0.487987\pi$
$432$	0	0
$433$	22.5996	1.08607	0.543034	−	0.839711i	$-0.317276\pi$
$433$	22.5996	1.08607	0.543034	+	0.839711i	$0.317276\pi$
$434$	0	0
$435$	−13.6627	−0.655076
$436$	0	0
$437$	−14.3245	−0.685235
$438$	0	0
$439$	−26.3805	−1.25907	−0.629537	−	0.776970i	$-0.716756\pi$
$439$	−26.3805	−1.25907	−0.629537	+	0.776970i	$0.716756\pi$
$440$	0	0
$441$	7.54206	0.359146
$442$	0	0
$443$	−31.1676	−1.48082	−0.740409	−	0.672157i	$-0.765368\pi$
$443$	−31.1676	−1.48082	−0.740409	+	0.672157i	$0.765368\pi$
$444$	0	0
$445$	5.27366	0.249995
$446$	0	0
$447$	18.8191	0.890111
$448$	0	0
$449$	−10.6847	−0.504242	−0.252121	−	0.967696i	$-0.581128\pi$
$449$	−10.6847	−0.504242	−0.252121	+	0.967696i	$0.581128\pi$
$450$	0	0
$451$	−33.9193	−1.59720
$452$	0	0
$453$	−6.36637	−0.299118
$454$	0	0
$455$	8.28685	0.388494
$456$	0	0
$457$	−2.95521	−0.138239	−0.0691194	−	0.997608i	$-0.522019\pi$
$457$	−2.95521	−0.138239	−0.0691194	+	0.997608i	$0.522019\pi$
$458$	0	0
$459$	30.1174	1.40576
$460$	0	0
$461$	40.6933	1.89528	0.947639	−	0.319344i	$-0.103463\pi$
$461$	40.6933	1.89528	0.947639	+	0.319344i	$0.103463\pi$
$462$	0	0
$463$	−14.1764	−0.658834	−0.329417	−	0.944185i	$-0.606852\pi$
$463$	−14.1764	−0.658834	−0.329417	+	0.944185i	$0.606852\pi$
$464$	0	0
$465$	−2.26666	−0.105114
$466$	0	0
$467$	6.85700	0.317304	0.158652	−	0.987335i	$-0.449285\pi$
$467$	6.85700	0.317304	0.158652	+	0.987335i	$0.449285\pi$
$468$	0	0
$469$	−1.66078	−0.0766875
$470$	0	0
$471$	−13.6504	−0.628977
$472$	0	0
$473$	−5.29337	−0.243389
$474$	0	0
$475$	−7.67506	−0.352156
$476$	0	0
$477$	10.6748	0.488764
$478$	0	0
$479$	−20.6396	−0.943048	−0.471524	−	0.881853i	$-0.656296\pi$
$479$	−20.6396	−0.943048	−0.471524	+	0.881853i	$0.656296\pi$
$480$	0	0
$481$	48.9791	2.23326
$482$	0	0
$483$	8.40213	0.382310
$484$	0	0
$485$	−17.4720	−0.793363
$486$	0	0
$487$	33.9244	1.53726	0.768631	−	0.639693i	$-0.220938\pi$
$487$	33.9244	1.53726	0.768631	+	0.639693i	$0.220938\pi$
$488$	0	0
$489$	−7.56325	−0.342022
$490$	0	0
$491$	−29.3239	−1.32337	−0.661685	−	0.749782i	$-0.730158\pi$
$491$	−29.3239	−1.32337	−0.661685	+	0.749782i	$0.730158\pi$
$492$	0	0
$493$	−43.3038	−1.95030
$494$	0	0
$495$	−5.53987	−0.248999
$496$	0	0
$497$	−11.3959	−0.511174
$498$	0	0
$499$	8.74198	0.391345	0.195672	−	0.980669i	$-0.437311\pi$
$499$	8.74198	0.391345	0.195672	+	0.980669i	$0.437311\pi$
$500$	0	0
$501$	9.24253	0.412926
$502$	0	0
$503$	−6.56731	−0.292822	−0.146411	−	0.989224i	$-0.546772\pi$
$503$	−6.56731	−0.292822	−0.146411	+	0.989224i	$0.546772\pi$
$504$	0	0
$505$	0.910503	0.0405168
$506$	0	0
$507$	−35.5320	−1.57803
$508$	0	0
$509$	34.1486	1.51361	0.756804	−	0.653642i	$-0.226760\pi$
$509$	34.1486	1.51361	0.756804	+	0.653642i	$0.226760\pi$
$510$	0	0
$511$	−10.9253	−0.483305
$512$	0	0
$513$	12.8694	0.568199
$514$	0	0
$515$	1.17390	0.0517281
$516$	0	0
$517$	−39.4304	−1.73415
$518$	0	0
$519$	−4.95427	−0.217468
$520$	0	0
$521$	−18.7659	−0.822150	−0.411075	−	0.911601i	$-0.634847\pi$
$521$	−18.7659	−0.822150	−0.411075	+	0.911601i	$0.634847\pi$
$522$	0	0
$523$	−25.5983	−1.11934	−0.559669	−	0.828716i	$-0.689072\pi$
$523$	−25.5983	−1.11934	−0.559669	+	0.828716i	$0.689072\pi$
$524$	0	0
$525$	4.50185	0.196477
$526$	0	0
$527$	−7.18417	−0.312947
$528$	0	0
$529$	16.0973	0.699884
$530$	0	0
$531$	−16.3727	−0.710515
$532$	0	0
$533$	−62.9377	−2.72613
$534$	0	0
$535$	3.85510	0.166671
$536$	0	0
$537$	26.0308	1.12331
$538$	0	0
$539$	20.3061	0.874645
$540$	0	0
$541$	−25.6658	−1.10346	−0.551729	−	0.834023i	$-0.686032\pi$
$541$	−25.6658	−1.10346	−0.551729	+	0.834023i	$0.686032\pi$
$542$	0	0
$543$	−19.7027	−0.845525
$544$	0	0
$545$	−16.4174	−0.703242
$546$	0	0
$547$	−20.9323	−0.895000	−0.447500	−	0.894284i	$-0.647686\pi$
$547$	−20.9323	−0.895000	−0.447500	+	0.894284i	$0.647686\pi$
$548$	0	0
$549$	−0.268713	−0.0114684
$550$	0	0
$551$	−18.5041	−0.788301
$552$	0	0
$553$	14.4530	0.614604
$554$	0	0
$555$	−13.1027	−0.556178
$556$	0	0
$557$	−36.1674	−1.53246	−0.766230	−	0.642566i	$-0.777870\pi$
$557$	−36.1674	−1.53246	−0.766230	+	0.642566i	$0.777870\pi$
$558$	0	0
$559$	−9.82190	−0.415422
$560$	0	0
$561$	24.0598	1.01580
$562$	0	0
$563$	−41.5029	−1.74914	−0.874569	−	0.484901i	$-0.838856\pi$
$563$	−41.5029	−1.74914	−0.874569	+	0.484901i	$0.838856\pi$
$564$	0	0
$565$	5.03129	0.211668
$566$	0	0
$567$	−3.67429	−0.154306
$568$	0	0
$569$	−19.7499	−0.827958	−0.413979	−	0.910286i	$-0.635861\pi$
$569$	−19.7499	−0.827958	−0.413979	+	0.910286i	$0.635861\pi$
$570$	0	0
$571$	1.83271	0.0766964	0.0383482	−	0.999264i	$-0.487790\pi$
$571$	1.83271	0.0766964	0.0383482	+	0.999264i	$0.487790\pi$
$572$	0	0
$573$	−14.0520	−0.587030
$574$	0	0
$575$	20.9483	0.873604
$576$	0	0
$577$	39.2668	1.63470	0.817349	−	0.576143i	$-0.195443\pi$
$577$	39.2668	1.63470	0.817349	+	0.576143i	$0.195443\pi$
$578$	0	0
$579$	−21.5411	−0.895217
$580$	0	0
$581$	−1.98846	−0.0824952
$582$	0	0
$583$	28.7405	1.19031
$584$	0	0
$585$	−10.2793	−0.424996
$586$	0	0
$587$	−33.6019	−1.38690	−0.693449	−	0.720506i	$-0.743910\pi$
$587$	−33.6019	−1.38690	−0.693449	+	0.720506i	$0.743910\pi$
$588$	0	0
$589$	−3.06986	−0.126491
$590$	0	0
$591$	7.65044	0.314697
$592$	0	0
$593$	−26.0943	−1.07157	−0.535783	−	0.844356i	$-0.679984\pi$
$593$	−26.0943	−1.07157	−0.535783	+	0.844356i	$0.679984\pi$
$594$	0	0
$595$	−7.02633	−0.288051
$596$	0	0
$597$	−7.24129	−0.296366
$598$	0	0
$599$	35.7789	1.46189	0.730944	−	0.682438i	$-0.239080\pi$
$599$	35.7789	1.46189	0.730944	+	0.682438i	$0.239080\pi$
$600$	0	0
$601$	4.13530	0.168682	0.0843411	−	0.996437i	$-0.473121\pi$
$601$	4.13530	0.168682	0.0843411	+	0.996437i	$0.473121\pi$
$602$	0	0
$603$	2.06008	0.0838931
$604$	0	0
$605$	−0.786684	−0.0319833
$606$	0	0
$607$	−35.4600	−1.43928	−0.719638	−	0.694349i	$-0.755692\pi$
$607$	−35.4600	−1.43928	−0.719638	+	0.694349i	$0.755692\pi$
$608$	0	0
$609$	10.8537	0.439813
$610$	0	0
$611$	−73.1636	−2.95988
$612$	0	0
$613$	−12.4080	−0.501156	−0.250578	−	0.968096i	$-0.580621\pi$
$613$	−12.4080	−0.501156	−0.250578	+	0.968096i	$0.580621\pi$
$614$	0	0
$615$	16.8368	0.678926
$616$	0	0
$617$	−10.4736	−0.421653	−0.210826	−	0.977524i	$-0.567615\pi$
$617$	−10.4736	−0.421653	−0.210826	+	0.977524i	$0.567615\pi$
$618$	0	0
$619$	31.7351	1.27554	0.637770	−	0.770227i	$-0.279857\pi$
$619$	31.7351	1.27554	0.637770	+	0.770227i	$0.279857\pi$
$620$	0	0
$621$	−35.1258	−1.40955
$622$	0	0
$623$	−4.18941	−0.167845
$624$	0	0
$625$	2.97522	0.119009
$626$	0	0
$627$	10.2809	0.410581
$628$	0	0
$629$	−41.5289	−1.65586
$630$	0	0
$631$	47.7008	1.89894	0.949470	−	0.313857i	$-0.101621\pi$
$631$	47.7008	1.89894	0.949470	+	0.313857i	$0.101621\pi$
$632$	0	0
$633$	12.2753	0.487898
$634$	0	0
$635$	−11.2636	−0.446982
$636$	0	0
$637$	37.6782	1.49286
$638$	0	0
$639$	14.1358	0.559204
$640$	0	0
$641$	29.2583	1.15563	0.577817	−	0.816166i	$-0.303905\pi$
$641$	29.2583	1.15563	0.577817	+	0.816166i	$0.303905\pi$
$642$	0	0
$643$	21.3885	0.843481	0.421740	−	0.906717i	$-0.361419\pi$
$643$	21.3885	0.843481	0.421740	+	0.906717i	$0.361419\pi$
$644$	0	0
$645$	2.62751	0.103458
$646$	0	0
$647$	−21.9187	−0.861715	−0.430857	−	0.902420i	$-0.641789\pi$
$647$	−21.9187	−0.861715	−0.430857	+	0.902420i	$0.641789\pi$
$648$	0	0
$649$	−44.0816	−1.73035
$650$	0	0
$651$	1.80064	0.0705728
$652$	0	0
$653$	13.0864	0.512110	0.256055	−	0.966662i	$-0.417577\pi$
$653$	13.0864	0.512110	0.256055	+	0.966662i	$0.417577\pi$
$654$	0	0
$655$	4.48232	0.175139
$656$	0	0
$657$	13.5521	0.528717
$658$	0	0
$659$	−1.04943	−0.0408800	−0.0204400	−	0.999791i	$-0.506507\pi$
$659$	−1.04943	−0.0408800	−0.0204400	+	0.999791i	$0.506507\pi$
$660$	0	0
$661$	7.93829	0.308764	0.154382	−	0.988011i	$-0.450661\pi$
$661$	7.93829	0.308764	0.154382	+	0.988011i	$0.450661\pi$
$662$	0	0
$663$	44.6431	1.73380
$664$	0	0
$665$	−3.00241	−0.116429
$666$	0	0
$667$	50.5050	1.95556
$668$	0	0
$669$	30.2353	1.16897
$670$	0	0
$671$	−0.723477	−0.0279295
$672$	0	0
$673$	−31.2158	−1.20328	−0.601640	−	0.798767i	$-0.705486\pi$
$673$	−31.2158	−1.20328	−0.601640	+	0.798767i	$0.705486\pi$
$674$	0	0
$675$	−18.8204	−0.724396
$676$	0	0
$677$	21.1051	0.811135	0.405568	−	0.914065i	$-0.367074\pi$
$677$	21.1051	0.811135	0.405568	+	0.914065i	$0.367074\pi$
$678$	0	0
$679$	13.8798	0.532659
$680$	0	0
$681$	−35.3397	−1.35422
$682$	0	0
$683$	15.8340	0.605872	0.302936	−	0.953011i	$-0.402033\pi$
$683$	15.8340	0.605872	0.302936	+	0.953011i	$0.402033\pi$
$684$	0	0
$685$	−3.98816	−0.152380
$686$	0	0
$687$	−31.3127	−1.19465
$688$	0	0
$689$	53.3284	2.03165
$690$	0	0
$691$	36.7200	1.39689	0.698447	−	0.715662i	$-0.253875\pi$
$691$	36.7200	1.39689	0.698447	+	0.715662i	$0.253875\pi$
$692$	0	0
$693$	4.40089	0.167176
$694$	0	0
$695$	−4.61724	−0.175142
$696$	0	0
$697$	53.3642	2.02131
$698$	0	0
$699$	−8.36039	−0.316219
$700$	0	0
$701$	17.9385	0.677528	0.338764	−	0.940871i	$-0.389991\pi$
$701$	17.9385	0.677528	0.338764	+	0.940871i	$0.389991\pi$
$702$	0	0
$703$	−17.7457	−0.669290
$704$	0	0
$705$	19.5724	0.737139
$706$	0	0
$707$	−0.723306	−0.0272027
$708$	0	0
$709$	−32.2917	−1.21274	−0.606371	−	0.795182i	$-0.707375\pi$
$709$	−32.2917	−1.21274	−0.606371	+	0.795182i	$0.707375\pi$
$710$	0	0
$711$	−17.9280	−0.672352
$712$	0	0
$713$	8.37887	0.313791
$714$	0	0
$715$	−27.6758	−1.03502
$716$	0	0
$717$	27.0928	1.01180
$718$	0	0
$719$	11.5719	0.431560	0.215780	−	0.976442i	$-0.430771\pi$
$719$	11.5719	0.431560	0.215780	+	0.976442i	$0.430771\pi$
$720$	0	0
$721$	−0.932548	−0.0347299
$722$	0	0
$723$	22.5819	0.839828
$724$	0	0
$725$	27.0605	1.00500
$726$	0	0
$727$	−6.59189	−0.244480	−0.122240	−	0.992501i	$-0.539008\pi$
$727$	−6.59189	−0.244480	−0.122240	+	0.992501i	$0.539008\pi$
$728$	0	0
$729$	26.7518	0.990808
$730$	0	0
$731$	8.32788	0.308018
$732$	0	0
$733$	−6.34073	−0.234200	−0.117100	−	0.993120i	$-0.537360\pi$
$733$	−6.34073	−0.234200	−0.117100	+	0.993120i	$0.537360\pi$
$734$	0	0
$735$	−10.0795	−0.371788
$736$	0	0
$737$	5.54653	0.204309
$738$	0	0
$739$	−23.9608	−0.881411	−0.440705	−	0.897652i	$-0.645272\pi$
$739$	−23.9608	−0.881411	−0.440705	+	0.897652i	$0.645272\pi$
$740$	0	0
$741$	19.0764	0.700789
$742$	0	0
$743$	−40.3846	−1.48157	−0.740784	−	0.671743i	$-0.765546\pi$
$743$	−40.3846	−1.48157	−0.740784	+	0.671743i	$0.765546\pi$
$744$	0	0
$745$	18.3546	0.672462
$746$	0	0
$747$	2.46655	0.0902465
$748$	0	0
$749$	−3.06251	−0.111902
$750$	0	0
$751$	−14.5929	−0.532503	−0.266251	−	0.963904i	$-0.585785\pi$
$751$	−14.5929	−0.532503	−0.266251	+	0.963904i	$0.585785\pi$
$752$	0	0
$753$	−1.31693	−0.0479917
$754$	0	0
$755$	−6.20925	−0.225978
$756$	0	0
$757$	−15.6202	−0.567727	−0.283863	−	0.958865i	$-0.591616\pi$
$757$	−15.6202	−0.567727	−0.283863	+	0.958865i	$0.591616\pi$
$758$	0	0
$759$	−28.0608	−1.01854
$760$	0	0
$761$	40.6289	1.47280	0.736399	−	0.676548i	$-0.236525\pi$
$761$	40.6289	1.47280	0.736399	+	0.676548i	$0.236525\pi$
$762$	0	0
$763$	13.0420	0.472152
$764$	0	0
$765$	8.71570	0.315117
$766$	0	0
$767$	−81.7938	−2.95340
$768$	0	0
$769$	−13.1395	−0.473824	−0.236912	−	0.971531i	$-0.576135\pi$
$769$	−13.1395	−0.473824	−0.236912	+	0.971531i	$0.576135\pi$
$770$	0	0
$771$	−25.1388	−0.905353
$772$	0	0
$773$	−40.3078	−1.44977	−0.724886	−	0.688869i	$-0.758108\pi$
$773$	−40.3078	−1.44977	−0.724886	+	0.688869i	$0.758108\pi$
$774$	0	0
$775$	4.48938	0.161263
$776$	0	0
$777$	10.4088	0.373414
$778$	0	0
$779$	22.8030	0.817002
$780$	0	0
$781$	38.0590	1.36186
$782$	0	0
$783$	−45.3747	−1.62156
$784$	0	0
$785$	−13.3135	−0.475179
$786$	0	0
$787$	12.4714	0.444557	0.222279	−	0.974983i	$-0.428651\pi$
$787$	12.4714	0.444557	0.222279	+	0.974983i	$0.428651\pi$
$788$	0	0
$789$	−35.3543	−1.25865
$790$	0	0
$791$	−3.99687	−0.142112
$792$	0	0
$793$	−1.34242	−0.0476707
$794$	0	0
$795$	−14.2662	−0.505969
$796$	0	0
$797$	−14.5834	−0.516570	−0.258285	−	0.966069i	$-0.583157\pi$
$797$	−14.5834	−0.516570	−0.258285	+	0.966069i	$0.583157\pi$
$798$	0	0
$799$	62.0346	2.19463
$800$	0	0
$801$	5.19669	0.183616
$802$	0	0
$803$	36.4873	1.28761
$804$	0	0
$805$	8.19478	0.288828
$806$	0	0
$807$	23.4675	0.826095
$808$	0	0
$809$	−36.7020	−1.29037	−0.645186	−	0.764025i	$-0.723220\pi$
$809$	−36.7020	−1.29037	−0.645186	+	0.764025i	$0.723220\pi$
$810$	0	0
$811$	−8.80062	−0.309032	−0.154516	−	0.987990i	$-0.549382\pi$
$811$	−8.80062	−0.309032	−0.154516	+	0.987990i	$0.549382\pi$
$812$	0	0
$813$	−34.1321	−1.19706
$814$	0	0
$815$	−7.37659	−0.258391
$816$	0	0
$817$	3.55858	0.124499
$818$	0	0
$819$	8.16590	0.285340
$820$	0	0
$821$	−14.9979	−0.523432	−0.261716	−	0.965145i	$-0.584288\pi$
$821$	−14.9979	−0.523432	−0.261716	+	0.965145i	$0.584288\pi$
$822$	0	0
$823$	8.68961	0.302901	0.151450	−	0.988465i	$-0.451606\pi$
$823$	8.68961	0.302901	0.151450	+	0.988465i	$0.451606\pi$
$824$	0	0
$825$	−15.0349	−0.523449
$826$	0	0
$827$	12.9115	0.448976	0.224488	−	0.974477i	$-0.427929\pi$
$827$	12.9115	0.448976	0.224488	+	0.974477i	$0.427929\pi$
$828$	0	0
$829$	−51.2578	−1.78026	−0.890128	−	0.455711i	$-0.849385\pi$
$829$	−51.2578	−1.78026	−0.890128	+	0.455711i	$0.849385\pi$
$830$	0	0
$831$	−0.335947	−0.0116539
$832$	0	0
$833$	−31.9469	−1.10690
$834$	0	0
$835$	9.01443	0.311957
$836$	0	0
$837$	−7.52774	−0.260197
$838$	0	0
$839$	8.57114	0.295909	0.147954	−	0.988994i	$-0.452731\pi$
$839$	8.57114	0.295909	0.147954	+	0.988994i	$0.452731\pi$
$840$	0	0
$841$	36.2412	1.24970
$842$	0	0
$843$	−31.7943	−1.09505
$844$	0	0
$845$	−34.6551	−1.19217
$846$	0	0
$847$	0.624944	0.0214733
$848$	0	0
$849$	−40.7453	−1.39837
$850$	0	0
$851$	48.4349	1.66033
$852$	0	0
$853$	−45.0161	−1.54132	−0.770661	−	0.637245i	$-0.780074\pi$
$853$	−45.0161	−1.54132	−0.770661	+	0.637245i	$0.780074\pi$
$854$	0	0
$855$	3.72430	0.127368
$856$	0	0
$857$	−44.6140	−1.52398	−0.761992	−	0.647587i	$-0.775778\pi$
$857$	−44.6140	−1.52398	−0.761992	+	0.647587i	$0.775778\pi$
$858$	0	0
$859$	−37.0606	−1.26449	−0.632245	−	0.774768i	$-0.717866\pi$
$859$	−37.0606	−1.26449	−0.632245	+	0.774768i	$0.717866\pi$
$860$	0	0
$861$	−13.3752	−0.455826
$862$	0	0
$863$	15.7527	0.536229	0.268114	−	0.963387i	$-0.413599\pi$
$863$	15.7527	0.536229	0.268114	+	0.963387i	$0.413599\pi$
$864$	0	0
$865$	−4.83200	−0.164293
$866$	0	0
$867$	−15.4646	−0.525204
$868$	0	0
$869$	−48.2690	−1.63741
$870$	0	0
$871$	10.2916	0.348719
$872$	0	0
$873$	−17.2170	−0.582707
$874$	0	0
$875$	10.9436	0.369963
$876$	0	0
$877$	6.05982	0.204626	0.102313	−	0.994752i	$-0.467376\pi$
$877$	6.05982	0.204626	0.102313	+	0.994752i	$0.467376\pi$
$878$	0	0
$879$	31.4097	1.05942
$880$	0	0
$881$	−37.4775	−1.26265	−0.631324	−	0.775519i	$-0.717488\pi$
$881$	−37.4775	−1.26265	−0.631324	+	0.775519i	$0.717488\pi$
$882$	0	0
$883$	−9.74089	−0.327807	−0.163903	−	0.986476i	$-0.552409\pi$
$883$	−9.74089	−0.327807	−0.163903	+	0.986476i	$0.552409\pi$
$884$	0	0
$885$	21.8811	0.735526
$886$	0	0
$887$	27.8151	0.933940	0.466970	−	0.884273i	$-0.345346\pi$
$887$	27.8151	0.933940	0.466970	+	0.884273i	$0.345346\pi$
$888$	0	0
$889$	8.94783	0.300100
$890$	0	0
$891$	12.2711	0.411097
$892$	0	0
$893$	26.5079	0.887054
$894$	0	0
$895$	25.3884	0.848641
$896$	0	0
$897$	−52.0671	−1.73847
$898$	0	0
$899$	10.8236	0.360988
$900$	0	0
$901$	−45.2165	−1.50638
$902$	0	0
$903$	−2.08730	−0.0694611
$904$	0	0
$905$	−19.2165	−0.638777
$906$	0	0
$907$	−18.9267	−0.628452	−0.314226	−	0.949348i	$-0.601745\pi$
$907$	−18.9267	−0.628452	−0.314226	+	0.949348i	$0.601745\pi$
$908$	0	0
$909$	0.897214	0.0297587
$910$	0	0
$911$	7.37167	0.244234	0.122117	−	0.992516i	$-0.461032\pi$
$911$	7.37167	0.244234	0.122117	+	0.992516i	$0.461032\pi$
$912$	0	0
$913$	6.64090	0.219782
$914$	0	0
$915$	0.359118	0.0118721
$916$	0	0
$917$	−3.56077	−0.117587
$918$	0	0
$919$	29.9676	0.988540	0.494270	−	0.869308i	$-0.335435\pi$
$919$	29.9676	0.988540	0.494270	+	0.869308i	$0.335435\pi$
$920$	0	0
$921$	−21.3352	−0.703020
$922$	0	0
$923$	70.6189	2.32445
$924$	0	0
$925$	25.9514	0.853276
$926$	0	0
$927$	1.15677	0.0379932
$928$	0	0
$929$	9.27346	0.304252	0.152126	−	0.988361i	$-0.451388\pi$
$929$	9.27346	0.304252	0.152126	+	0.988361i	$0.451388\pi$
$930$	0	0
$931$	−13.6512	−0.447400
$932$	0	0
$933$	8.02922	0.262865
$934$	0	0
$935$	23.4660	0.767420
$936$	0	0
$937$	9.16311	0.299346	0.149673	−	0.988736i	$-0.452178\pi$
$937$	9.16311	0.299346	0.149673	+	0.988736i	$0.452178\pi$
$938$	0	0
$939$	−5.83696	−0.190482
$940$	0	0
$941$	48.4655	1.57993	0.789965	−	0.613152i	$-0.210099\pi$
$941$	48.4655	1.57993	0.789965	+	0.613152i	$0.210099\pi$
$942$	0	0
$943$	−62.2384	−2.02676
$944$	0	0
$945$	−7.36235	−0.239497
$946$	0	0
$947$	−16.2316	−0.527455	−0.263727	−	0.964597i	$-0.584952\pi$
$947$	−16.2316	−0.527455	−0.263727	+	0.964597i	$0.584952\pi$
$948$	0	0
$949$	67.7026	2.19772
$950$	0	0
$951$	10.5412	0.341823
$952$	0	0
$953$	−37.7780	−1.22375	−0.611874	−	0.790955i	$-0.709584\pi$
$953$	−37.7780	−1.22375	−0.611874	+	0.790955i	$0.709584\pi$
$954$	0	0
$955$	−13.7052	−0.443490
$956$	0	0
$957$	−36.2483	−1.17174
$958$	0	0
$959$	3.16821	0.102307
$960$	0	0
$961$	−29.2043	−0.942076
$962$	0	0
$963$	3.79884	0.122416
$964$	0	0
$965$	−21.0095	−0.676319
$966$	0	0
$967$	−11.6664	−0.375167	−0.187583	−	0.982249i	$-0.560066\pi$
$967$	−11.6664	−0.375167	−0.187583	+	0.982249i	$0.560066\pi$
$968$	0	0
$969$	−16.1747	−0.519605
$970$	0	0
$971$	56.7485	1.82115	0.910574	−	0.413347i	$-0.135640\pi$
$971$	56.7485	1.82115	0.910574	+	0.413347i	$0.135640\pi$
$972$	0	0
$973$	3.66795	0.117589
$974$	0	0
$975$	−27.8975	−0.893434
$976$	0	0
$977$	−15.8276	−0.506371	−0.253185	−	0.967418i	$-0.581478\pi$
$977$	−15.8276	−0.506371	−0.253185	+	0.967418i	$0.581478\pi$
$978$	0	0
$979$	13.9915	0.447169
$980$	0	0
$981$	−16.1777	−0.516515
$982$	0	0
$983$	−22.0625	−0.703685	−0.351842	−	0.936059i	$-0.614445\pi$
$983$	−22.0625	−0.703685	−0.351842	+	0.936059i	$0.614445\pi$
$984$	0	0
$985$	7.46163	0.237747
$986$	0	0
$987$	−15.5484	−0.494910
$988$	0	0
$989$	−9.71277	−0.308848
$990$	0	0
$991$	12.2127	0.387949	0.193974	−	0.981007i	$-0.437862\pi$
$991$	12.2127	0.387949	0.193974	+	0.981007i	$0.437862\pi$
$992$	0	0
$993$	−40.6618	−1.29036
$994$	0	0
$995$	−7.06258	−0.223899
$996$	0	0
$997$	−36.0107	−1.14047	−0.570236	−	0.821481i	$-0.693148\pi$
$997$	−36.0107	−1.14047	−0.570236	+	0.821481i	$0.693148\pi$
$998$	0	0
$999$	−43.5149	−1.37675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.l.1.6		19
4.3	odd	2		2008.2.a.c.1.14	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.c.1.14	✓	19	4.3	odd	2
4016.2.a.l.1.6		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31693 q^{3} -1.28443 q^{5} +1.02036 q^{7} -1.26569 q^{9} -3.40771 q^{11} -6.32304 q^{13} +1.69151 q^{15} +5.36124 q^{17} +2.29090 q^{19} -1.34374 q^{21} -6.25279 q^{23} -3.35023 q^{25} +5.61763 q^{27} -8.07720 q^{29} -1.34002 q^{31} +4.48773 q^{33} -1.31058 q^{35} -7.74614 q^{37} +8.32702 q^{39} +9.95371 q^{41} +1.55335 q^{43} +1.62569 q^{45} +11.5709 q^{47} -5.95887 q^{49} -7.06039 q^{51} -8.43397 q^{53} +4.37697 q^{55} -3.01697 q^{57} +12.9358 q^{59} +0.212306 q^{61} -1.29145 q^{63} +8.12152 q^{65} -1.62764 q^{67} +8.23450 q^{69} -11.1685 q^{71} -10.7073 q^{73} +4.41203 q^{75} -3.47708 q^{77} +14.1646 q^{79} -3.60098 q^{81} -1.94879 q^{83} -6.88615 q^{85} +10.6371 q^{87} -4.10583 q^{89} -6.45176 q^{91} +1.76472 q^{93} -2.94251 q^{95} +13.6029 q^{97} +4.31309 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q + 6 q^{3} - 8 q^{5} + 11 q^{7} + 21 q^{9} + 15 q^{11} - 8 q^{13} + 17 q^{15} - 4 q^{17} + 14 q^{19} - 9 q^{21} + 28 q^{23} + 25 q^{25} + 21 q^{27} - 13 q^{29} + 20 q^{31} - 6 q^{33} + 32 q^{35} - 16 q^{37}+ \cdots + 74 q^{99}+O(q^{100}) \) 19 * q + 6 * q^3 - 8 * q^5 + 11 * q^7 + 21 * q^9 + 15 * q^11 - 8 * q^13 + 17 * q^15 - 4 * q^17 + 14 * q^19 - 9 * q^21 + 28 * q^23 + 25 * q^25 + 21 * q^27 - 13 * q^29 + 20 * q^31 - 6 * q^33 + 32 * q^35 - 16 * q^37 + 27 * q^39 + 2 * q^41 + 28 * q^43 - 29 * q^45 + 37 * q^47 + 36 * q^49 + 35 * q^51 - 37 * q^53 + 24 * q^55 - 11 * q^57 + 32 * q^59 - 7 * q^61 + 45 * q^63 + q^65 + 45 * q^67 - 12 * q^69 + 49 * q^71 + 16 * q^73 + 35 * q^75 - 40 * q^77 + 33 * q^79 + 15 * q^81 + 43 * q^83 - 28 * q^85 + 48 * q^87 + 3 * q^89 + 56 * q^91 - 48 * q^93 + 43 * q^95 + 8 * q^97 + 74 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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