LMFDB - Embedded newform 4016.2.a.m.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$23$
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	$q-2.67736 q^{3} +4.29688 q^{5} +4.01755 q^{7} +4.16827 q^{9} +O(q^{10})$	$q-2.67736 q^{3} +4.29688 q^{5} +4.01755 q^{7} +4.16827 q^{9} -4.88107 q^{11} -3.92210 q^{13} -11.5043 q^{15} +0.345632 q^{17} -3.26874 q^{19} -10.7564 q^{21} -7.48325 q^{23} +13.4632 q^{25} -3.12789 q^{27} -10.5555 q^{29} +9.70877 q^{31} +13.0684 q^{33} +17.2630 q^{35} +4.92845 q^{37} +10.5009 q^{39} +7.39986 q^{41} +4.90581 q^{43} +17.9106 q^{45} -1.34062 q^{47} +9.14073 q^{49} -0.925381 q^{51} +1.36208 q^{53} -20.9734 q^{55} +8.75160 q^{57} +0.330955 q^{59} +9.87403 q^{61} +16.7463 q^{63} -16.8528 q^{65} +11.0820 q^{67} +20.0354 q^{69} +10.1094 q^{71} +13.6687 q^{73} -36.0459 q^{75} -19.6100 q^{77} +8.91873 q^{79} -4.13032 q^{81} -3.12340 q^{83} +1.48514 q^{85} +28.2609 q^{87} -7.28030 q^{89} -15.7573 q^{91} -25.9939 q^{93} -14.0454 q^{95} +2.84471 q^{97} -20.3456 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * 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
$3$	−2.67736	−1.54578	−0.772888	−	0.634542i	$-0.781189\pi$
$3$	−2.67736	−1.54578	−0.772888	+	0.634542i	$0.781189\pi$
$4$	0	0
$5$	4.29688	1.92162	0.960812	−	0.277200i	$-0.0894063\pi$
$5$	4.29688	1.92162	0.960812	+	0.277200i	$0.0894063\pi$
$6$	0	0
$7$	4.01755	1.51849	0.759246	−	0.650804i	$-0.225568\pi$
$7$	4.01755	1.51849	0.759246	+	0.650804i	$0.225568\pi$
$8$	0	0
$9$	4.16827	1.38942
$10$	0	0
$11$	−4.88107	−1.47170	−0.735849	−	0.677146i	$-0.763217\pi$
$11$	−4.88107	−1.47170	−0.735849	+	0.677146i	$0.763217\pi$
$12$	0	0
$13$	−3.92210	−1.08780	−0.543898	−	0.839151i	$-0.683052\pi$
$13$	−3.92210	−1.08780	−0.543898	+	0.839151i	$0.683052\pi$
$14$	0	0
$15$	−11.5043	−2.97040
$16$	0	0
$17$	0.345632	0.0838280	0.0419140	−	0.999121i	$-0.486654\pi$
$17$	0.345632	0.0838280	0.0419140	+	0.999121i	$0.486654\pi$
$18$	0	0
$19$	−3.26874	−0.749900	−0.374950	−	0.927045i	$-0.622340\pi$
$19$	−3.26874	−0.749900	−0.374950	+	0.927045i	$0.622340\pi$
$20$	0	0
$21$	−10.7564	−2.34725
$22$	0	0
$23$	−7.48325	−1.56037	−0.780183	−	0.625551i	$-0.784874\pi$
$23$	−7.48325	−1.56037	−0.780183	+	0.625551i	$0.784874\pi$
$24$	0	0
$25$	13.4632	2.69264
$26$	0	0
$27$	−3.12789	−0.601962
$28$	0	0
$29$	−10.5555	−1.96011	−0.980053	−	0.198735i	$-0.936317\pi$
$29$	−10.5555	−1.96011	−0.980053	+	0.198735i	$0.936317\pi$
$30$	0	0
$31$	9.70877	1.74375	0.871873	−	0.489731i	$-0.162905\pi$
$31$	9.70877	1.74375	0.871873	+	0.489731i	$0.162905\pi$
$32$	0	0
$33$	13.0684	2.27492
$34$	0	0
$35$	17.2630	2.91797
$36$	0	0
$37$	4.92845	0.810232	0.405116	−	0.914265i	$-0.367231\pi$
$37$	4.92845	0.810232	0.405116	+	0.914265i	$0.367231\pi$
$38$	0	0
$39$	10.5009	1.68149
$40$	0	0
$41$	7.39986	1.15566	0.577832	−	0.816156i	$-0.303899\pi$
$41$	7.39986	1.15566	0.577832	+	0.816156i	$0.303899\pi$
$42$	0	0
$43$	4.90581	0.748129	0.374064	−	0.927403i	$-0.377964\pi$
$43$	4.90581	0.748129	0.374064	+	0.927403i	$0.377964\pi$
$44$	0	0
$45$	17.9106	2.66995
$46$	0	0
$47$	−1.34062	−0.195550	−0.0977750	−	0.995209i	$-0.531173\pi$
$47$	−1.34062	−0.195550	−0.0977750	+	0.995209i	$0.531173\pi$
$48$	0	0
$49$	9.14073	1.30582
$50$	0	0
$51$	−0.925381	−0.129579
$52$	0	0
$53$	1.36208	0.187096	0.0935479	−	0.995615i	$-0.470179\pi$
$53$	1.36208	0.187096	0.0935479	+	0.995615i	$0.470179\pi$
$54$	0	0
$55$	−20.9734	−2.82805
$56$	0	0
$57$	8.75160	1.15918
$58$	0	0
$59$	0.330955	0.0430867	0.0215434	−	0.999768i	$-0.493142\pi$
$59$	0.330955	0.0430867	0.0215434	+	0.999768i	$0.493142\pi$
$60$	0	0
$61$	9.87403	1.26424	0.632120	−	0.774871i	$-0.282185\pi$
$61$	9.87403	1.26424	0.632120	+	0.774871i	$0.282185\pi$
$62$	0	0
$63$	16.7463	2.10983
$64$	0	0
$65$	−16.8528	−2.09034
$66$	0	0
$67$	11.0820	1.35389	0.676943	−	0.736035i	$-0.263304\pi$
$67$	11.0820	1.35389	0.676943	+	0.736035i	$0.263304\pi$
$68$	0	0
$69$	20.0354	2.41198
$70$	0	0
$71$	10.1094	1.19977	0.599884	−	0.800087i	$-0.295213\pi$
$71$	10.1094	1.19977	0.599884	+	0.800087i	$0.295213\pi$
$72$	0	0
$73$	13.6687	1.59980	0.799900	−	0.600133i	$-0.204886\pi$
$73$	13.6687	1.59980	0.799900	+	0.600133i	$0.204886\pi$
$74$	0	0
$75$	−36.0459	−4.16222
$76$	0	0
$77$	−19.6100	−2.23476
$78$	0	0
$79$	8.91873	1.00344	0.501718	−	0.865031i	$-0.332702\pi$
$79$	8.91873	1.00344	0.501718	+	0.865031i	$0.332702\pi$
$80$	0	0
$81$	−4.13032	−0.458925
$82$	0	0
$83$	−3.12340	−0.342838	−0.171419	−	0.985198i	$-0.554835\pi$
$83$	−3.12340	−0.342838	−0.171419	+	0.985198i	$0.554835\pi$
$84$	0	0
$85$	1.48514	0.161086
$86$	0	0
$87$	28.2609	3.02989
$88$	0	0
$89$	−7.28030	−0.771710	−0.385855	−	0.922559i	$-0.626094\pi$
$89$	−7.28030	−0.771710	−0.385855	+	0.922559i	$0.626094\pi$
$90$	0	0
$91$	−15.7573	−1.65181
$92$	0	0
$93$	−25.9939	−2.69544
$94$	0	0
$95$	−14.0454	−1.44103
$96$	0	0
$97$	2.84471	0.288837	0.144418	−	0.989517i	$-0.453869\pi$
$97$	2.84471	0.288837	0.144418	+	0.989517i	$0.453869\pi$
$98$	0	0
$99$	−20.3456	−2.04481
$100$	0	0
$101$	2.85666	0.284248	0.142124	−	0.989849i	$-0.454607\pi$
$101$	2.85666	0.284248	0.142124	+	0.989849i	$0.454607\pi$
$102$	0	0
$103$	2.89586	0.285338	0.142669	−	0.989770i	$-0.454432\pi$
$103$	2.89586	0.285338	0.142669	+	0.989770i	$0.454432\pi$
$104$	0	0
$105$	−46.2192	−4.51053
$106$	0	0
$107$	12.3609	1.19498	0.597488	−	0.801878i	$-0.296166\pi$
$107$	12.3609	1.19498	0.597488	+	0.801878i	$0.296166\pi$
$108$	0	0
$109$	2.64043	0.252908	0.126454	−	0.991973i	$-0.459640\pi$
$109$	2.64043	0.252908	0.126454	+	0.991973i	$0.459640\pi$
$110$	0	0
$111$	−13.1953	−1.25244
$112$	0	0
$113$	−2.02036	−0.190060	−0.0950299	−	0.995474i	$-0.530295\pi$
$113$	−2.02036	−0.190060	−0.0950299	+	0.995474i	$0.530295\pi$
$114$	0	0
$115$	−32.1547	−2.99844
$116$	0	0
$117$	−16.3484	−1.51141
$118$	0	0
$119$	1.38859	0.127292
$120$	0	0
$121$	12.8248	1.16589
$122$	0	0
$123$	−19.8121	−1.78640
$124$	0	0
$125$	36.3654	3.25262
$126$	0	0
$127$	1.08326	0.0961240	0.0480620	−	0.998844i	$-0.484695\pi$
$127$	1.08326	0.0961240	0.0480620	+	0.998844i	$0.484695\pi$
$128$	0	0
$129$	−13.1346	−1.15644
$130$	0	0
$131$	10.8410	0.947181	0.473590	−	0.880745i	$-0.342958\pi$
$131$	10.8410	0.947181	0.473590	+	0.880745i	$0.342958\pi$
$132$	0	0
$133$	−13.1323	−1.13872
$134$	0	0
$135$	−13.4402	−1.15675
$136$	0	0
$137$	−4.67905	−0.399758	−0.199879	−	0.979821i	$-0.564055\pi$
$137$	−4.67905	−0.399758	−0.199879	+	0.979821i	$0.564055\pi$
$138$	0	0
$139$	8.19070	0.694726	0.347363	−	0.937731i	$-0.387077\pi$
$139$	8.19070	0.694726	0.347363	+	0.937731i	$0.387077\pi$
$140$	0	0
$141$	3.58933	0.302276
$142$	0	0
$143$	19.1441	1.60091
$144$	0	0
$145$	−45.3557	−3.76659
$146$	0	0
$147$	−24.4730	−2.01850
$148$	0	0
$149$	−11.2822	−0.924273	−0.462136	−	0.886809i	$-0.652917\pi$
$149$	−11.2822	−0.924273	−0.462136	+	0.886809i	$0.652917\pi$
$150$	0	0
$151$	2.55491	0.207916	0.103958	−	0.994582i	$-0.466849\pi$
$151$	2.55491	0.207916	0.103958	+	0.994582i	$0.466849\pi$
$152$	0	0
$153$	1.44069	0.116473
$154$	0	0
$155$	41.7175	3.35083
$156$	0	0
$157$	12.5622	1.00257	0.501286	−	0.865281i	$-0.332860\pi$
$157$	12.5622	1.00257	0.501286	+	0.865281i	$0.332860\pi$
$158$	0	0
$159$	−3.64678	−0.289208
$160$	0	0
$161$	−30.0644	−2.36940
$162$	0	0
$163$	6.55995	0.513815	0.256908	−	0.966436i	$-0.417296\pi$
$163$	6.55995	0.513815	0.256908	+	0.966436i	$0.417296\pi$
$164$	0	0
$165$	56.1534	4.37153
$166$	0	0
$167$	21.9818	1.70100	0.850502	−	0.525972i	$-0.176298\pi$
$167$	21.9818	1.70100	0.850502	+	0.525972i	$0.176298\pi$
$168$	0	0
$169$	2.38290	0.183300
$170$	0	0
$171$	−13.6250	−1.04193
$172$	0	0
$173$	11.5338	0.876900	0.438450	−	0.898756i	$-0.355528\pi$
$173$	11.5338	0.876900	0.438450	+	0.898756i	$0.355528\pi$
$174$	0	0
$175$	54.0891	4.08875
$176$	0	0
$177$	−0.886087	−0.0666024
$178$	0	0
$179$	12.1386	0.907278	0.453639	−	0.891185i	$-0.350126\pi$
$179$	12.1386	0.907278	0.453639	+	0.891185i	$0.350126\pi$
$180$	0	0
$181$	5.83347	0.433598	0.216799	−	0.976216i	$-0.430438\pi$
$181$	5.83347	0.433598	0.216799	+	0.976216i	$0.430438\pi$
$182$	0	0
$183$	−26.4364	−1.95423
$184$	0	0
$185$	21.1770	1.55696
$186$	0	0
$187$	−1.68705	−0.123369
$188$	0	0
$189$	−12.5665	−0.914075
$190$	0	0
$191$	−12.2293	−0.884879	−0.442440	−	0.896798i	$-0.645887\pi$
$191$	−12.2293	−0.884879	−0.442440	+	0.896798i	$0.645887\pi$
$192$	0	0
$193$	−5.50148	−0.396005	−0.198003	−	0.980201i	$-0.563445\pi$
$193$	−5.50148	−0.396005	−0.198003	+	0.980201i	$0.563445\pi$
$194$	0	0
$195$	45.1211	3.23119
$196$	0	0
$197$	−9.94394	−0.708477	−0.354238	−	0.935155i	$-0.615260\pi$
$197$	−9.94394	−0.708477	−0.354238	+	0.935155i	$0.615260\pi$
$198$	0	0
$199$	−17.1814	−1.21796	−0.608979	−	0.793187i	$-0.708421\pi$
$199$	−17.1814	−1.21796	−0.608979	+	0.793187i	$0.708421\pi$
$200$	0	0
$201$	−29.6706	−2.09281
$202$	0	0
$203$	−42.4073	−2.97641
$204$	0	0
$205$	31.7963	2.22075
$206$	0	0
$207$	−31.1922	−2.16801
$208$	0	0
$209$	15.9549	1.10363
$210$	0	0
$211$	3.36612	0.231733	0.115867	−	0.993265i	$-0.463035\pi$
$211$	3.36612	0.231733	0.115867	+	0.993265i	$0.463035\pi$
$212$	0	0
$213$	−27.0666	−1.85457
$214$	0	0
$215$	21.0797	1.43762
$216$	0	0
$217$	39.0055	2.64787
$218$	0	0
$219$	−36.5961	−2.47293
$220$	0	0
$221$	−1.35560	−0.0911877
$222$	0	0
$223$	−6.42756	−0.430421	−0.215210	−	0.976568i	$-0.569044\pi$
$223$	−6.42756	−0.430421	−0.215210	+	0.976568i	$0.569044\pi$
$224$	0	0
$225$	56.1183	3.74122
$226$	0	0
$227$	−6.11613	−0.405942	−0.202971	−	0.979185i	$-0.565060\pi$
$227$	−6.11613	−0.405942	−0.202971	+	0.979185i	$0.565060\pi$
$228$	0	0
$229$	−18.0264	−1.19122	−0.595608	−	0.803275i	$-0.703089\pi$
$229$	−18.0264	−1.19122	−0.595608	+	0.803275i	$0.703089\pi$
$230$	0	0
$231$	52.5030	3.45444
$232$	0	0
$233$	−13.2752	−0.869689	−0.434845	−	0.900506i	$-0.643197\pi$
$233$	−13.2752	−0.869689	−0.434845	+	0.900506i	$0.643197\pi$
$234$	0	0
$235$	−5.76050	−0.375774
$236$	0	0
$237$	−23.8787	−1.55109
$238$	0	0
$239$	4.87379	0.315259	0.157630	−	0.987498i	$-0.449615\pi$
$239$	4.87379	0.315259	0.157630	+	0.987498i	$0.449615\pi$
$240$	0	0
$241$	14.7989	0.953280	0.476640	−	0.879099i	$-0.341854\pi$
$241$	14.7989	0.953280	0.476640	+	0.879099i	$0.341854\pi$
$242$	0	0
$243$	20.4420	1.31136
$244$	0	0
$245$	39.2766	2.50929
$246$	0	0
$247$	12.8203	0.815739
$248$	0	0
$249$	8.36248	0.529950
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	36.5263	2.29639
$254$	0	0
$255$	−3.97626	−0.249003
$256$	0	0
$257$	−0.551631	−0.0344098	−0.0172049	−	0.999852i	$-0.505477\pi$
$257$	−0.551631	−0.0344098	−0.0172049	+	0.999852i	$0.505477\pi$
$258$	0	0
$259$	19.8003	1.23033
$260$	0	0
$261$	−43.9982	−2.72342
$262$	0	0
$263$	−25.8398	−1.59335	−0.796676	−	0.604407i	$-0.793410\pi$
$263$	−25.8398	−1.59335	−0.796676	+	0.604407i	$0.793410\pi$
$264$	0	0
$265$	5.85269	0.359528
$266$	0	0
$267$	19.4920	1.19289
$268$	0	0
$269$	17.6689	1.07729	0.538647	−	0.842532i	$-0.318936\pi$
$269$	17.6689	1.07729	0.538647	+	0.842532i	$0.318936\pi$
$270$	0	0
$271$	7.60636	0.462053	0.231027	−	0.972947i	$-0.425792\pi$
$271$	7.60636	0.462053	0.231027	+	0.972947i	$0.425792\pi$
$272$	0	0
$273$	42.1879	2.55333
$274$	0	0
$275$	−65.7148	−3.96275
$276$	0	0
$277$	−21.6033	−1.29802	−0.649009	−	0.760781i	$-0.724816\pi$
$277$	−21.6033	−1.29802	−0.649009	+	0.760781i	$0.724816\pi$
$278$	0	0
$279$	40.4688	2.42280
$280$	0	0
$281$	−22.6953	−1.35389	−0.676943	−	0.736035i	$-0.736696\pi$
$281$	−22.6953	−1.35389	−0.676943	+	0.736035i	$0.736696\pi$
$282$	0	0
$283$	−23.1542	−1.37638	−0.688188	−	0.725533i	$-0.741593\pi$
$283$	−23.1542	−1.37638	−0.688188	+	0.725533i	$0.741593\pi$
$284$	0	0
$285$	37.6046	2.22751
$286$	0	0
$287$	29.7293	1.75487
$288$	0	0
$289$	−16.8805	−0.992973
$290$	0	0
$291$	−7.61633	−0.446477
$292$	0	0
$293$	19.7530	1.15398	0.576991	−	0.816751i	$-0.304227\pi$
$293$	19.7530	1.15398	0.576991	+	0.816751i	$0.304227\pi$
$294$	0	0
$295$	1.42208	0.0827965
$296$	0	0
$297$	15.2674	0.885907
$298$	0	0
$299$	29.3501	1.69736
$300$	0	0
$301$	19.7093	1.13603
$302$	0	0
$303$	−7.64831	−0.439384
$304$	0	0
$305$	42.4275	2.42939
$306$	0	0
$307$	4.29451	0.245101	0.122550	−	0.992462i	$-0.460893\pi$
$307$	4.29451	0.245101	0.122550	+	0.992462i	$0.460893\pi$
$308$	0	0
$309$	−7.75327	−0.441068
$310$	0	0
$311$	15.9165	0.902540	0.451270	−	0.892387i	$-0.350971\pi$
$311$	15.9165	0.902540	0.451270	+	0.892387i	$0.350971\pi$
$312$	0	0
$313$	21.1655	1.19635	0.598173	−	0.801367i	$-0.295893\pi$
$313$	21.1655	1.19635	0.598173	+	0.801367i	$0.295893\pi$
$314$	0	0
$315$	71.9567	4.05430
$316$	0	0
$317$	14.6269	0.821530	0.410765	−	0.911741i	$-0.365262\pi$
$317$	14.6269	0.821530	0.410765	+	0.911741i	$0.365262\pi$
$318$	0	0
$319$	51.5221	2.88468
$320$	0	0
$321$	−33.0947	−1.84716
$322$	0	0
$323$	−1.12978	−0.0628626
$324$	0	0
$325$	−52.8041	−2.92904
$326$	0	0
$327$	−7.06939	−0.390938
$328$	0	0
$329$	−5.38602	−0.296941
$330$	0	0
$331$	−10.2268	−0.562114	−0.281057	−	0.959691i	$-0.590685\pi$
$331$	−10.2268	−0.562114	−0.281057	+	0.959691i	$0.590685\pi$
$332$	0	0
$333$	20.5431	1.12576
$334$	0	0
$335$	47.6182	2.60166
$336$	0	0
$337$	19.9429	1.08636	0.543181	−	0.839616i	$-0.317220\pi$
$337$	19.9429	1.08636	0.543181	+	0.839616i	$0.317220\pi$
$338$	0	0
$339$	5.40925	0.293790
$340$	0	0
$341$	−47.3892	−2.56627
$342$	0	0
$343$	8.60049	0.464383
$344$	0	0
$345$	86.0897	4.63491
$346$	0	0
$347$	−36.2758	−1.94739	−0.973693	−	0.227864i	$-0.926826\pi$
$347$	−36.2758	−1.94739	−0.973693	+	0.227864i	$0.926826\pi$
$348$	0	0
$349$	−32.9190	−1.76211	−0.881056	−	0.473011i	$-0.843167\pi$
$349$	−32.9190	−1.76211	−0.881056	+	0.473011i	$0.843167\pi$
$350$	0	0
$351$	12.2679	0.654812
$352$	0	0
$353$	−8.34949	−0.444398	−0.222199	−	0.975001i	$-0.571324\pi$
$353$	−8.34949	−0.444398	−0.222199	+	0.975001i	$0.571324\pi$
$354$	0	0
$355$	43.4390	2.30550
$356$	0	0
$357$	−3.71777	−0.196765
$358$	0	0
$359$	−23.3855	−1.23424	−0.617119	−	0.786870i	$-0.711700\pi$
$359$	−23.3855	−1.23424	−0.617119	+	0.786870i	$0.711700\pi$
$360$	0	0
$361$	−8.31534	−0.437649
$362$	0	0
$363$	−34.3367	−1.80221
$364$	0	0
$365$	58.7328	3.07422
$366$	0	0
$367$	−9.95074	−0.519424	−0.259712	−	0.965686i	$-0.583628\pi$
$367$	−9.95074	−0.519424	−0.259712	+	0.965686i	$0.583628\pi$
$368$	0	0
$369$	30.8446	1.60571
$370$	0	0
$371$	5.47222	0.284103
$372$	0	0
$373$	−16.1591	−0.836688	−0.418344	−	0.908289i	$-0.637389\pi$
$373$	−16.1591	−0.836688	−0.418344	+	0.908289i	$0.637389\pi$
$374$	0	0
$375$	−97.3634	−5.02782
$376$	0	0
$377$	41.3997	2.13220
$378$	0	0
$379$	−15.0285	−0.771962	−0.385981	−	0.922507i	$-0.626137\pi$
$379$	−15.0285	−0.771962	−0.385981	+	0.922507i	$0.626137\pi$
$380$	0	0
$381$	−2.90029	−0.148586
$382$	0	0
$383$	−15.6944	−0.801944	−0.400972	−	0.916090i	$-0.631328\pi$
$383$	−15.6944	−0.801944	−0.400972	+	0.916090i	$0.631328\pi$
$384$	0	0
$385$	−84.2617	−4.29437
$386$	0	0
$387$	20.4487	1.03947
$388$	0	0
$389$	3.70628	0.187916	0.0939580	−	0.995576i	$-0.470048\pi$
$389$	3.70628	0.187916	0.0939580	+	0.995576i	$0.470048\pi$
$390$	0	0
$391$	−2.58645	−0.130802
$392$	0	0
$393$	−29.0252	−1.46413
$394$	0	0
$395$	38.3227	1.92823
$396$	0	0
$397$	0.520696	0.0261330	0.0130665	−	0.999915i	$-0.495841\pi$
$397$	0.520696	0.0261330	0.0130665	+	0.999915i	$0.495841\pi$
$398$	0	0
$399$	35.1600	1.76020
$400$	0	0
$401$	−1.47427	−0.0736217	−0.0368109	−	0.999322i	$-0.511720\pi$
$401$	−1.47427	−0.0736217	−0.0368109	+	0.999322i	$0.511720\pi$
$402$	0	0
$403$	−38.0788	−1.89684
$404$	0	0
$405$	−17.7475	−0.881882
$406$	0	0
$407$	−24.0561	−1.19242
$408$	0	0
$409$	23.3081	1.15251	0.576256	−	0.817269i	$-0.304513\pi$
$409$	23.3081	1.15251	0.576256	+	0.817269i	$0.304513\pi$
$410$	0	0
$411$	12.5275	0.617937
$412$	0	0
$413$	1.32963	0.0654268
$414$	0	0
$415$	−13.4209	−0.658806
$416$	0	0
$417$	−21.9295	−1.07389
$418$	0	0
$419$	16.5740	0.809691	0.404846	−	0.914385i	$-0.367325\pi$
$419$	16.5740	0.809691	0.404846	+	0.914385i	$0.367325\pi$
$420$	0	0
$421$	−15.7926	−0.769686	−0.384843	−	0.922982i	$-0.625744\pi$
$421$	−15.7926	−0.769686	−0.384843	+	0.922982i	$0.625744\pi$
$422$	0	0
$423$	−5.58808	−0.271702
$424$	0	0
$425$	4.65331	0.225719
$426$	0	0
$427$	39.6694	1.91974
$428$	0	0
$429$	−51.2556	−2.47464
$430$	0	0
$431$	9.47799	0.456539	0.228269	−	0.973598i	$-0.426693\pi$
$431$	9.47799	0.456539	0.228269	+	0.973598i	$0.426693\pi$
$432$	0	0
$433$	−31.5164	−1.51458	−0.757291	−	0.653078i	$-0.773477\pi$
$433$	−31.5164	−1.51458	−0.757291	+	0.653078i	$0.773477\pi$
$434$	0	0
$435$	121.434	5.82230
$436$	0	0
$437$	24.4608	1.17012
$438$	0	0
$439$	31.6164	1.50897	0.754485	−	0.656318i	$-0.227887\pi$
$439$	31.6164	1.50897	0.754485	+	0.656318i	$0.227887\pi$
$440$	0	0
$441$	38.1010	1.81434
$442$	0	0
$443$	−6.80425	−0.323279	−0.161640	−	0.986850i	$-0.551678\pi$
$443$	−6.80425	−0.323279	−0.161640	+	0.986850i	$0.551678\pi$
$444$	0	0
$445$	−31.2826	−1.48294
$446$	0	0
$447$	30.2065	1.42872
$448$	0	0
$449$	32.7833	1.54714	0.773571	−	0.633710i	$-0.218469\pi$
$449$	32.7833	1.54714	0.773571	+	0.633710i	$0.218469\pi$
$450$	0	0
$451$	−36.1192	−1.70079
$452$	0	0
$453$	−6.84043	−0.321391
$454$	0	0
$455$	−67.7071	−3.17416
$456$	0	0
$457$	−16.4517	−0.769578	−0.384789	−	0.923004i	$-0.625726\pi$
$457$	−16.4517	−0.769578	−0.384789	+	0.923004i	$0.625726\pi$
$458$	0	0
$459$	−1.08110	−0.0504613
$460$	0	0
$461$	16.4167	0.764602	0.382301	−	0.924038i	$-0.375132\pi$
$461$	16.4167	0.764602	0.382301	+	0.924038i	$0.375132\pi$
$462$	0	0
$463$	4.17134	0.193859	0.0969293	−	0.995291i	$-0.469098\pi$
$463$	4.17134	0.193859	0.0969293	+	0.995291i	$0.469098\pi$
$464$	0	0
$465$	−111.693	−5.17963
$466$	0	0
$467$	13.6901	0.633501	0.316751	−	0.948509i	$-0.397408\pi$
$467$	13.6901	0.633501	0.316751	+	0.948509i	$0.397408\pi$
$468$	0	0
$469$	44.5227	2.05587
$470$	0	0
$471$	−33.6336	−1.54975
$472$	0	0
$473$	−23.9456	−1.10102
$474$	0	0
$475$	−44.0077	−2.01921
$476$	0	0
$477$	5.67751	0.259955
$478$	0	0
$479$	15.4423	0.705578	0.352789	−	0.935703i	$-0.385233\pi$
$479$	15.4423	0.705578	0.352789	+	0.935703i	$0.385233\pi$
$480$	0	0
$481$	−19.3299	−0.881367
$482$	0	0
$483$	80.4932	3.66257
$484$	0	0
$485$	12.2234	0.555036
$486$	0	0
$487$	27.7327	1.25669	0.628343	−	0.777936i	$-0.283733\pi$
$487$	27.7327	1.25669	0.628343	+	0.777936i	$0.283733\pi$
$488$	0	0
$489$	−17.5634	−0.794243
$490$	0	0
$491$	34.8181	1.57132	0.785660	−	0.618658i	$-0.212323\pi$
$491$	34.8181	1.57132	0.785660	+	0.618658i	$0.212323\pi$
$492$	0	0
$493$	−3.64831	−0.164312
$494$	0	0
$495$	−87.4228	−3.92936
$496$	0	0
$497$	40.6152	1.82184
$498$	0	0
$499$	−7.95889	−0.356289	−0.178144	−	0.984004i	$-0.557009\pi$
$499$	−7.95889	−0.356289	−0.178144	+	0.984004i	$0.557009\pi$
$500$	0	0
$501$	−58.8533	−2.62937
$502$	0	0
$503$	3.50633	0.156339	0.0781697	−	0.996940i	$-0.475092\pi$
$503$	3.50633	0.156339	0.0781697	+	0.996940i	$0.475092\pi$
$504$	0	0
$505$	12.2747	0.546218
$506$	0	0
$507$	−6.37988	−0.283340
$508$	0	0
$509$	−26.4551	−1.17260	−0.586300	−	0.810094i	$-0.699416\pi$
$509$	−26.4551	−1.17260	−0.586300	+	0.810094i	$0.699416\pi$
$510$	0	0
$511$	54.9147	2.42928
$512$	0	0
$513$	10.2243	0.451412
$514$	0	0
$515$	12.4432	0.548312
$516$	0	0
$517$	6.54367	0.287790
$518$	0	0
$519$	−30.8802	−1.35549
$520$	0	0
$521$	−32.3080	−1.41544	−0.707720	−	0.706493i	$-0.750276\pi$
$521$	−32.3080	−1.41544	−0.707720	+	0.706493i	$0.750276\pi$
$522$	0	0
$523$	1.08738	0.0475479	0.0237739	−	0.999717i	$-0.492432\pi$
$523$	1.08738	0.0475479	0.0237739	+	0.999717i	$0.492432\pi$
$524$	0	0
$525$	−144.816	−6.32030
$526$	0	0
$527$	3.35566	0.146175
$528$	0	0
$529$	32.9991	1.43474
$530$	0	0
$531$	1.37951	0.0598657
$532$	0	0
$533$	−29.0230	−1.25713
$534$	0	0
$535$	53.1134	2.29629
$536$	0	0
$537$	−32.4993	−1.40245
$538$	0	0
$539$	−44.6165	−1.92177
$540$	0	0
$541$	−14.5224	−0.624369	−0.312184	−	0.950022i	$-0.601061\pi$
$541$	−14.5224	−0.624369	−0.312184	+	0.950022i	$0.601061\pi$
$542$	0	0
$543$	−15.6183	−0.670246
$544$	0	0
$545$	11.3456	0.485993
$546$	0	0
$547$	26.7522	1.14384	0.571920	−	0.820309i	$-0.306199\pi$
$547$	26.7522	1.14384	0.571920	+	0.820309i	$0.306199\pi$
$548$	0	0
$549$	41.1576	1.75657
$550$	0	0
$551$	34.5032	1.46988
$552$	0	0
$553$	35.8315	1.52371
$554$	0	0
$555$	−56.6985	−2.40672
$556$	0	0
$557$	17.5837	0.745047	0.372523	−	0.928023i	$-0.378493\pi$
$557$	17.5837	0.745047	0.372523	+	0.928023i	$0.378493\pi$
$558$	0	0
$559$	−19.2411	−0.813811
$560$	0	0
$561$	4.51685	0.190702
$562$	0	0
$563$	−33.8164	−1.42519	−0.712595	−	0.701575i	$-0.752480\pi$
$563$	−33.8164	−1.42519	−0.712595	+	0.701575i	$0.752480\pi$
$564$	0	0
$565$	−8.68127	−0.365224
$566$	0	0
$567$	−16.5938	−0.696874
$568$	0	0
$569$	16.5754	0.694877	0.347439	−	0.937703i	$-0.387051\pi$
$569$	16.5754	0.694877	0.347439	+	0.937703i	$0.387051\pi$
$570$	0	0
$571$	−19.8103	−0.829034	−0.414517	−	0.910041i	$-0.636050\pi$
$571$	−19.8103	−0.829034	−0.414517	+	0.910041i	$0.636050\pi$
$572$	0	0
$573$	32.7422	1.36783
$574$	0	0
$575$	−100.749	−4.20151
$576$	0	0
$577$	−4.26144	−0.177406	−0.0887031	−	0.996058i	$-0.528272\pi$
$577$	−4.26144	−0.177406	−0.0887031	+	0.996058i	$0.528272\pi$
$578$	0	0
$579$	14.7295	0.612135
$580$	0	0
$581$	−12.5484	−0.520596
$582$	0	0
$583$	−6.64840	−0.275348
$584$	0	0
$585$	−70.2471	−2.90436
$586$	0	0
$587$	−20.7936	−0.858244	−0.429122	−	0.903247i	$-0.641177\pi$
$587$	−20.7936	−0.858244	−0.429122	+	0.903247i	$0.641177\pi$
$588$	0	0
$589$	−31.7355	−1.30764
$590$	0	0
$591$	26.6235	1.09515
$592$	0	0
$593$	−34.8668	−1.43181	−0.715904	−	0.698199i	$-0.753985\pi$
$593$	−34.8668	−1.43181	−0.715904	+	0.698199i	$0.753985\pi$
$594$	0	0
$595$	5.96662	0.244608
$596$	0	0
$597$	46.0009	1.88269
$598$	0	0
$599$	−14.5394	−0.594064	−0.297032	−	0.954868i	$-0.595997\pi$
$599$	−14.5394	−0.594064	−0.297032	+	0.954868i	$0.595997\pi$
$600$	0	0
$601$	22.9360	0.935579	0.467790	−	0.883840i	$-0.345051\pi$
$601$	22.9360	0.935579	0.467790	+	0.883840i	$0.345051\pi$
$602$	0	0
$603$	46.1929	1.88112
$604$	0	0
$605$	55.1068	2.24041
$606$	0	0
$607$	−38.8454	−1.57669	−0.788343	−	0.615235i	$-0.789061\pi$
$607$	−38.8454	−1.57669	−0.788343	+	0.615235i	$0.789061\pi$
$608$	0	0
$609$	113.540	4.60086
$610$	0	0
$611$	5.25806	0.212718
$612$	0	0
$613$	−3.80742	−0.153780	−0.0768901	−	0.997040i	$-0.524499\pi$
$613$	−3.80742	−0.153780	−0.0768901	+	0.997040i	$0.524499\pi$
$614$	0	0
$615$	−85.1303	−3.43279
$616$	0	0
$617$	4.05826	0.163380	0.0816898	−	0.996658i	$-0.473968\pi$
$617$	4.05826	0.163380	0.0816898	+	0.996658i	$0.473968\pi$
$618$	0	0
$619$	−18.6601	−0.750014	−0.375007	−	0.927022i	$-0.622360\pi$
$619$	−18.6601	−0.750014	−0.375007	+	0.927022i	$0.622360\pi$
$620$	0	0
$621$	23.4068	0.939282
$622$	0	0
$623$	−29.2490	−1.17184
$624$	0	0
$625$	88.9419	3.55768
$626$	0	0
$627$	−42.7172	−1.70596
$628$	0	0
$629$	1.70343	0.0679202
$630$	0	0
$631$	1.64043	0.0653045	0.0326523	−	0.999467i	$-0.489605\pi$
$631$	1.64043	0.0653045	0.0326523	+	0.999467i	$0.489605\pi$
$632$	0	0
$633$	−9.01233	−0.358208
$634$	0	0
$635$	4.65465	0.184714
$636$	0	0
$637$	−35.8509	−1.42046
$638$	0	0
$639$	42.1388	1.66699
$640$	0	0
$641$	−15.0740	−0.595386	−0.297693	−	0.954662i	$-0.596217\pi$
$641$	−15.0740	−0.595386	−0.297693	+	0.954662i	$0.596217\pi$
$642$	0	0
$643$	−34.7735	−1.37133	−0.685666	−	0.727916i	$-0.740489\pi$
$643$	−34.7735	−1.37133	−0.685666	+	0.727916i	$0.740489\pi$
$644$	0	0
$645$	−56.4380	−2.22224
$646$	0	0
$647$	30.2866	1.19069	0.595344	−	0.803471i	$-0.297016\pi$
$647$	30.2866	1.19069	0.595344	+	0.803471i	$0.297016\pi$
$648$	0	0
$649$	−1.61542	−0.0634106
$650$	0	0
$651$	−104.432	−4.09301
$652$	0	0
$653$	15.7825	0.617618	0.308809	−	0.951124i	$-0.400070\pi$
$653$	15.7825	0.617618	0.308809	+	0.951124i	$0.400070\pi$
$654$	0	0
$655$	46.5824	1.82013
$656$	0	0
$657$	56.9749	2.22280
$658$	0	0
$659$	5.44667	0.212172	0.106086	−	0.994357i	$-0.466168\pi$
$659$	5.44667	0.212172	0.106086	+	0.994357i	$0.466168\pi$
$660$	0	0
$661$	12.6197	0.490851	0.245425	−	0.969416i	$-0.421072\pi$
$661$	12.6197	0.490851	0.245425	+	0.969416i	$0.421072\pi$
$662$	0	0
$663$	3.62944	0.140956
$664$	0	0
$665$	−56.4281	−2.18819
$666$	0	0
$667$	78.9894	3.05848
$668$	0	0
$669$	17.2089	0.665334
$670$	0	0
$671$	−48.1958	−1.86058
$672$	0	0
$673$	−2.25268	−0.0868345	−0.0434172	−	0.999057i	$-0.513824\pi$
$673$	−2.25268	−0.0868345	−0.0434172	+	0.999057i	$0.513824\pi$
$674$	0	0
$675$	−42.1114	−1.62087
$676$	0	0
$677$	−3.09785	−0.119060	−0.0595299	−	0.998227i	$-0.518960\pi$
$677$	−3.09785	−0.119060	−0.0595299	+	0.998227i	$0.518960\pi$
$678$	0	0
$679$	11.4288	0.438596
$680$	0	0
$681$	16.3751	0.627495
$682$	0	0
$683$	−12.1587	−0.465239	−0.232620	−	0.972568i	$-0.574730\pi$
$683$	−12.1587	−0.465239	−0.232620	+	0.972568i	$0.574730\pi$
$684$	0	0
$685$	−20.1053	−0.768185
$686$	0	0
$687$	48.2632	1.84135
$688$	0	0
$689$	−5.34221	−0.203522
$690$	0	0
$691$	−6.64379	−0.252742	−0.126371	−	0.991983i	$-0.540333\pi$
$691$	−6.64379	−0.252742	−0.126371	+	0.991983i	$0.540333\pi$
$692$	0	0
$693$	−81.7396	−3.10503
$694$	0	0
$695$	35.1945	1.33500
$696$	0	0
$697$	2.55763	0.0968770
$698$	0	0
$699$	35.5426	1.34434
$700$	0	0
$701$	−11.3048	−0.426976	−0.213488	−	0.976946i	$-0.568482\pi$
$701$	−11.3048	−0.426976	−0.213488	+	0.976946i	$0.568482\pi$
$702$	0	0
$703$	−16.1098	−0.607594
$704$	0	0
$705$	15.4229	0.580862
$706$	0	0
$707$	11.4768	0.431629
$708$	0	0
$709$	26.9207	1.01103	0.505514	−	0.862818i	$-0.331303\pi$
$709$	26.9207	1.01103	0.505514	+	0.862818i	$0.331303\pi$
$710$	0	0
$711$	37.1757	1.39420
$712$	0	0
$713$	−72.6532	−2.72088
$714$	0	0
$715$	82.2598	3.07634
$716$	0	0
$717$	−13.0489	−0.487320
$718$	0	0
$719$	28.7292	1.07142	0.535709	−	0.844403i	$-0.320045\pi$
$719$	28.7292	1.07142	0.535709	+	0.844403i	$0.320045\pi$
$720$	0	0
$721$	11.6343	0.433283
$722$	0	0
$723$	−39.6220	−1.47356
$724$	0	0
$725$	−142.111	−5.27786
$726$	0	0
$727$	−33.9674	−1.25978	−0.629891	−	0.776684i	$-0.716900\pi$
$727$	−33.9674	−1.25978	−0.629891	+	0.776684i	$0.716900\pi$
$728$	0	0
$729$	−42.3398	−1.56814
$730$	0	0
$731$	1.69560	0.0627141
$732$	0	0
$733$	9.35368	0.345486	0.172743	−	0.984967i	$-0.444737\pi$
$733$	9.35368	0.345486	0.172743	+	0.984967i	$0.444737\pi$
$734$	0	0
$735$	−105.158	−3.87880
$736$	0	0
$737$	−54.0922	−1.99251
$738$	0	0
$739$	3.87995	0.142726	0.0713632	−	0.997450i	$-0.477265\pi$
$739$	3.87995	0.142726	0.0713632	+	0.997450i	$0.477265\pi$
$740$	0	0
$741$	−34.3247	−1.26095
$742$	0	0
$743$	1.00715	0.0369487	0.0184744	−	0.999829i	$-0.494119\pi$
$743$	1.00715	0.0369487	0.0184744	+	0.999829i	$0.494119\pi$
$744$	0	0
$745$	−48.4782	−1.77611
$746$	0	0
$747$	−13.0192	−0.476347
$748$	0	0
$749$	49.6607	1.81456
$750$	0	0
$751$	7.10756	0.259359	0.129679	−	0.991556i	$-0.458605\pi$
$751$	7.10756	0.259359	0.129679	+	0.991556i	$0.458605\pi$
$752$	0	0
$753$	−2.67736	−0.0975685
$754$	0	0
$755$	10.9782	0.399536
$756$	0	0
$757$	−9.48536	−0.344751	−0.172376	−	0.985031i	$-0.555144\pi$
$757$	−9.48536	−0.344751	−0.172376	+	0.985031i	$0.555144\pi$
$758$	0	0
$759$	−97.7941	−3.54970
$760$	0	0
$761$	−13.9039	−0.504017	−0.252008	−	0.967725i	$-0.581091\pi$
$761$	−13.9039	−0.504017	−0.252008	+	0.967725i	$0.581091\pi$
$762$	0	0
$763$	10.6081	0.384038
$764$	0	0
$765$	6.19046	0.223817
$766$	0	0
$767$	−1.29804	−0.0468695
$768$	0	0
$769$	39.6544	1.42997	0.714987	−	0.699138i	$-0.246433\pi$
$769$	39.6544	1.42997	0.714987	+	0.699138i	$0.246433\pi$
$770$	0	0
$771$	1.47692	0.0531898
$772$	0	0
$773$	42.2169	1.51844	0.759218	−	0.650836i	$-0.225582\pi$
$773$	42.2169	1.51844	0.759218	+	0.650836i	$0.225582\pi$
$774$	0	0
$775$	130.711	4.69529
$776$	0	0
$777$	−53.0126	−1.90182
$778$	0	0
$779$	−24.1882	−0.866633
$780$	0	0
$781$	−49.3448	−1.76570
$782$	0	0
$783$	33.0164	1.17991
$784$	0	0
$785$	53.9783	1.92657
$786$	0	0
$787$	−24.7090	−0.880782	−0.440391	−	0.897806i	$-0.645160\pi$
$787$	−24.7090	−0.880782	−0.440391	+	0.897806i	$0.645160\pi$
$788$	0	0
$789$	69.1826	2.46296
$790$	0	0
$791$	−8.11692	−0.288604
$792$	0	0
$793$	−38.7270	−1.37523
$794$	0	0
$795$	−15.6698	−0.555750
$796$	0	0
$797$	13.7654	0.487595	0.243798	−	0.969826i	$-0.421607\pi$
$797$	13.7654	0.487595	0.243798	+	0.969826i	$0.421607\pi$
$798$	0	0
$799$	−0.463362	−0.0163926
$800$	0	0
$801$	−30.3463	−1.07223
$802$	0	0
$803$	−66.7179	−2.35442
$804$	0	0
$805$	−129.183	−4.55310
$806$	0	0
$807$	−47.3061	−1.66526
$808$	0	0
$809$	22.0960	0.776854	0.388427	−	0.921479i	$-0.373019\pi$
$809$	22.0960	0.776854	0.388427	+	0.921479i	$0.373019\pi$
$810$	0	0
$811$	14.1728	0.497676	0.248838	−	0.968545i	$-0.419951\pi$
$811$	14.1728	0.497676	0.248838	+	0.968545i	$0.419951\pi$
$812$	0	0
$813$	−20.3650	−0.714231
$814$	0	0
$815$	28.1874	0.987360
$816$	0	0
$817$	−16.0358	−0.561022
$818$	0	0
$819$	−65.6805	−2.29506
$820$	0	0
$821$	31.8108	1.11020	0.555102	−	0.831782i	$-0.312679\pi$
$821$	31.8108	1.11020	0.555102	+	0.831782i	$0.312679\pi$
$822$	0	0
$823$	−45.4343	−1.58374	−0.791870	−	0.610690i	$-0.790892\pi$
$823$	−45.4343	−1.58374	−0.791870	+	0.610690i	$0.790892\pi$
$824$	0	0
$825$	175.942	6.12553
$826$	0	0
$827$	33.6944	1.17167	0.585835	−	0.810430i	$-0.300767\pi$
$827$	33.6944	1.17167	0.585835	+	0.810430i	$0.300767\pi$
$828$	0	0
$829$	−30.1188	−1.04607	−0.523035	−	0.852311i	$-0.675200\pi$
$829$	−30.1188	−1.04607	−0.523035	+	0.852311i	$0.675200\pi$
$830$	0	0
$831$	57.8399	2.00644
$832$	0	0
$833$	3.15932	0.109464
$834$	0	0
$835$	94.4533	3.26869
$836$	0	0
$837$	−30.3680	−1.04967
$838$	0	0
$839$	−30.3559	−1.04800	−0.524001	−	0.851718i	$-0.675561\pi$
$839$	−30.3559	−1.04800	−0.524001	+	0.851718i	$0.675561\pi$
$840$	0	0
$841$	82.4185	2.84202
$842$	0	0
$843$	60.7635	2.09281
$844$	0	0
$845$	10.2390	0.352233
$846$	0	0
$847$	51.5245	1.77040
$848$	0	0
$849$	61.9922	2.12757
$850$	0	0
$851$	−36.8809	−1.26426
$852$	0	0
$853$	−26.6812	−0.913546	−0.456773	−	0.889583i	$-0.650995\pi$
$853$	−26.6812	−0.913546	−0.456773	+	0.889583i	$0.650995\pi$
$854$	0	0
$855$	−58.5450	−2.00220
$856$	0	0
$857$	23.6142	0.806646	0.403323	−	0.915058i	$-0.367855\pi$
$857$	23.6142	0.806646	0.403323	+	0.915058i	$0.367855\pi$
$858$	0	0
$859$	29.0244	0.990301	0.495151	−	0.868807i	$-0.335113\pi$
$859$	29.0244	0.990301	0.495151	+	0.868807i	$0.335113\pi$
$860$	0	0
$861$	−79.5962	−2.71263
$862$	0	0
$863$	−48.4293	−1.64855	−0.824275	−	0.566189i	$-0.808417\pi$
$863$	−48.4293	−1.64855	−0.824275	+	0.566189i	$0.808417\pi$
$864$	0	0
$865$	49.5595	1.68507
$866$	0	0
$867$	45.1953	1.53491
$868$	0	0
$869$	−43.5329	−1.47675
$870$	0	0
$871$	−43.4649	−1.47275
$872$	0	0
$873$	11.8575	0.401317
$874$	0	0
$875$	146.100	4.93908
$876$	0	0
$877$	−34.3861	−1.16114	−0.580569	−	0.814211i	$-0.697170\pi$
$877$	−34.3861	−1.16114	−0.580569	+	0.814211i	$0.697170\pi$
$878$	0	0
$879$	−52.8859	−1.78380
$880$	0	0
$881$	−56.7982	−1.91358	−0.956790	−	0.290780i	$-0.906085\pi$
$881$	−56.7982	−1.91358	−0.956790	+	0.290780i	$0.906085\pi$
$882$	0	0
$883$	1.65446	0.0556770	0.0278385	−	0.999612i	$-0.491138\pi$
$883$	1.65446	0.0556770	0.0278385	+	0.999612i	$0.491138\pi$
$884$	0	0
$885$	−3.80741	−0.127985
$886$	0	0
$887$	43.2107	1.45087	0.725436	−	0.688289i	$-0.241638\pi$
$887$	43.2107	1.45087	0.725436	+	0.688289i	$0.241638\pi$
$888$	0	0
$889$	4.35207	0.145964
$890$	0	0
$891$	20.1604	0.675399
$892$	0	0
$893$	4.38215	0.146643
$894$	0	0
$895$	52.1580	1.74345
$896$	0	0
$897$	−78.5809	−2.62374
$898$	0	0
$899$	−102.481	−3.41793
$900$	0	0
$901$	0.470777	0.0156839
$902$	0	0
$903$	−52.7691	−1.75604
$904$	0	0
$905$	25.0657	0.833213
$906$	0	0
$907$	−45.0162	−1.49474	−0.747369	−	0.664409i	$-0.768683\pi$
$907$	−45.0162	−1.49474	−0.747369	+	0.664409i	$0.768683\pi$
$908$	0	0
$909$	11.9073	0.394941
$910$	0	0
$911$	17.9780	0.595638	0.297819	−	0.954622i	$-0.403741\pi$
$911$	17.9780	0.595638	0.297819	+	0.954622i	$0.403741\pi$
$912$	0	0
$913$	15.2455	0.504554
$914$	0	0
$915$	−113.594	−3.75530
$916$	0	0
$917$	43.5542	1.43829
$918$	0	0
$919$	−3.36987	−0.111162	−0.0555809	−	0.998454i	$-0.517701\pi$
$919$	−3.36987	−0.111162	−0.0555809	+	0.998454i	$0.517701\pi$
$920$	0	0
$921$	−11.4980	−0.378871
$922$	0	0
$923$	−39.6502	−1.30510
$924$	0	0
$925$	66.3528	2.18167
$926$	0	0
$927$	12.0707	0.396455
$928$	0	0
$929$	42.8213	1.40492	0.702460	−	0.711723i	$-0.252085\pi$
$929$	42.8213	1.40492	0.702460	+	0.711723i	$0.252085\pi$
$930$	0	0
$931$	−29.8787	−0.979234
$932$	0	0
$933$	−42.6142	−1.39513
$934$	0	0
$935$	−7.24907	−0.237070
$936$	0	0
$937$	−6.08754	−0.198871	−0.0994356	−	0.995044i	$-0.531704\pi$
$937$	−6.08754	−0.198871	−0.0994356	+	0.995044i	$0.531704\pi$
$938$	0	0
$939$	−56.6678	−1.84928
$940$	0	0
$941$	49.2223	1.60460	0.802300	−	0.596921i	$-0.203609\pi$
$941$	49.2223	1.60460	0.802300	+	0.596921i	$0.203609\pi$
$942$	0	0
$943$	−55.3750	−1.80326
$944$	0	0
$945$	−53.9966	−1.75651
$946$	0	0
$947$	−1.53886	−0.0500062	−0.0250031	−	0.999687i	$-0.507960\pi$
$947$	−1.53886	−0.0500062	−0.0250031	+	0.999687i	$0.507960\pi$
$948$	0	0
$949$	−53.6100	−1.74026
$950$	0	0
$951$	−39.1616	−1.26990
$952$	0	0
$953$	36.0563	1.16798	0.583989	−	0.811761i	$-0.301491\pi$
$953$	36.0563	1.16798	0.583989	+	0.811761i	$0.301491\pi$
$954$	0	0
$955$	−52.5478	−1.70041
$956$	0	0
$957$	−137.943	−4.45908
$958$	0	0
$959$	−18.7983	−0.607030
$960$	0	0
$961$	63.2603	2.04065
$962$	0	0
$963$	51.5237	1.66033
$964$	0	0
$965$	−23.6392	−0.760973
$966$	0	0
$967$	32.9184	1.05858	0.529292	−	0.848440i	$-0.322458\pi$
$967$	32.9184	1.05858	0.529292	+	0.848440i	$0.322458\pi$
$968$	0	0
$969$	3.02483	0.0971716
$970$	0	0
$971$	−30.3151	−0.972858	−0.486429	−	0.873720i	$-0.661701\pi$
$971$	−30.3151	−0.972858	−0.486429	+	0.873720i	$0.661701\pi$
$972$	0	0
$973$	32.9066	1.05494
$974$	0	0
$975$	141.376	4.52765
$976$	0	0
$977$	−40.0269	−1.28057	−0.640287	−	0.768136i	$-0.721185\pi$
$977$	−40.0269	−1.28057	−0.640287	+	0.768136i	$0.721185\pi$
$978$	0	0
$979$	35.5356	1.13572
$980$	0	0
$981$	11.0060	0.351396
$982$	0	0
$983$	−5.34084	−0.170346	−0.0851732	−	0.996366i	$-0.527144\pi$
$983$	−5.34084	−0.170346	−0.0851732	+	0.996366i	$0.527144\pi$
$984$	0	0
$985$	−42.7280	−1.36143
$986$	0	0
$987$	14.4203	0.459004
$988$	0	0
$989$	−36.7114	−1.16735
$990$	0	0
$991$	32.3871	1.02881	0.514405	−	0.857547i	$-0.328013\pi$
$991$	32.3871	1.02881	0.514405	+	0.857547i	$0.328013\pi$
$992$	0	0
$993$	27.3808	0.868902
$994$	0	0
$995$	−73.8265	−2.34046
$996$	0	0
$997$	26.0628	0.825416	0.412708	−	0.910863i	$-0.364583\pi$
$997$	26.0628	0.825416	0.412708	+	0.910863i	$0.364583\pi$
$998$	0	0
$999$	−15.4156	−0.487729

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.m.1.4		23
4.3	odd	2		2008.2.a.d.1.20	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.20	✓	23	4.3	odd	2
4016.2.a.m.1.4		23	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.67736 q^{3} +4.29688 q^{5} +4.01755 q^{7} +4.16827 q^{9} +O(q^{10})\)	\(q-2.67736 q^{3} +4.29688 q^{5} +4.01755 q^{7} +4.16827 q^{9} -4.88107 q^{11} -3.92210 q^{13} -11.5043 q^{15} +0.345632 q^{17} -3.26874 q^{19} -10.7564 q^{21} -7.48325 q^{23} +13.4632 q^{25} -3.12789 q^{27} -10.5555 q^{29} +9.70877 q^{31} +13.0684 q^{33} +17.2630 q^{35} +4.92845 q^{37} +10.5009 q^{39} +7.39986 q^{41} +4.90581 q^{43} +17.9106 q^{45} -1.34062 q^{47} +9.14073 q^{49} -0.925381 q^{51} +1.36208 q^{53} -20.9734 q^{55} +8.75160 q^{57} +0.330955 q^{59} +9.87403 q^{61} +16.7463 q^{63} -16.8528 q^{65} +11.0820 q^{67} +20.0354 q^{69} +10.1094 q^{71} +13.6687 q^{73} -36.0459 q^{75} -19.6100 q^{77} +8.91873 q^{79} -4.13032 q^{81} -3.12340 q^{83} +1.48514 q^{85} +28.2609 q^{87} -7.28030 q^{89} -15.7573 q^{91} -25.9939 q^{93} -14.0454 q^{95} +2.84471 q^{97} -20.3456 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

Introduction

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