LMFDB - Embedded newform 4016.2.a.e.1.5

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 = [5,0,1] 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:	$1$
Dimension:	$5$
Coefficient field:	5.5.242773.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 6x + 1 $ x^5 - x^4 - 7x^3 + 4x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 502)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.71702$ 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)

$f(q)$	$=$	$q+2.71702 q^{3} -2.20324 q^{5} -2.45833 q^{7} +4.38220 q^{9} -0.626744 q^{11} +2.36311 q^{13} -5.98625 q^{15} -1.70155 q^{17} -1.16481 q^{19} -6.67932 q^{21} +4.01908 q^{23} -0.145728 q^{25} +3.75545 q^{27} -5.57129 q^{29} -1.20191 q^{31} -1.70287 q^{33} +5.41629 q^{35} -10.8149 q^{37} +6.42063 q^{39} +3.14126 q^{41} +3.97210 q^{43} -9.65503 q^{45} -2.37859 q^{47} -0.956632 q^{49} -4.62314 q^{51} -13.2463 q^{53} +1.38087 q^{55} -3.16481 q^{57} +2.59425 q^{59} +0.891376 q^{61} -10.7729 q^{63} -5.20651 q^{65} -2.69273 q^{67} +10.9199 q^{69} -11.5924 q^{71} -6.11857 q^{73} -0.395946 q^{75} +1.54074 q^{77} -1.82465 q^{79} -2.94295 q^{81} +12.7422 q^{83} +3.74892 q^{85} -15.1373 q^{87} +8.93573 q^{89} -5.80931 q^{91} -3.26562 q^{93} +2.56636 q^{95} -4.22438 q^{97} -2.74651 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - 6 q^{5} + q^{7} + 4 q^{11} - 7 q^{13} + 5 q^{15} - 8 q^{17} - 3 q^{19} - 10 q^{21} + 17 q^{23} - q^{25} + 4 q^{27} - 15 q^{29} + q^{31} - 10 q^{33} + 7 q^{35} - 5 q^{37} + 8 q^{39} - 12 q^{41}+ \cdots - 28 q^{99}+O(q^{100}) $ 5 * q + q^3 - 6 * q^5 + q^7 + 4 * q^11 - 7 * q^13 + 5 * q^15 - 8 * q^17 - 3 * q^19 - 10 * q^21 + 17 * q^23 - q^25 + 4 * q^27 - 15 * q^29 + q^31 - 10 * q^33 + 7 * q^35 - 5 * q^37 + 8 * q^39 - 12 * q^41 - q^43 - 13 * q^45 + 19 * q^47 + 4 * q^49 - 7 * q^51 - 34 * q^53 - 17 * q^55 - 13 * q^57 + 10 * q^59 + 2 * q^63 + 6 * q^65 + 11 * q^67 + q^69 + q^71 + 3 * q^73 - 15 * q^75 - 30 * q^77 - 35 * q^79 - 3 * q^81 - 3 * q^83 - 13 * q^85 - 3 * q^87 + 15 * q^89 - 14 * q^91 + 15 * q^93 - 11 * q^95 - 2 * q^97 - 28 * 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
$3$	2.71702	1.56867	0.784336	−	0.620336i	$-0.213004\pi$
$3$	2.71702	1.56867	0.784336	+	0.620336i	$0.213004\pi$
$4$	0	0
$5$	−2.20324	−0.985319	−0.492660	−	0.870222i	$-0.663975\pi$
$5$	−2.20324	−0.985319	−0.492660	+	0.870222i	$0.663975\pi$
$6$	0	0
$7$	−2.45833	−0.929160	−0.464580	−	0.885531i	$-0.653795\pi$
$7$	−2.45833	−0.929160	−0.464580	+	0.885531i	$0.653795\pi$
$8$	0	0
$9$	4.38220	1.46073
$10$	0	0
$11$	−0.626744	−0.188970	−0.0944852	−	0.995526i	$-0.530120\pi$
$11$	−0.626744	−0.188970	−0.0944852	+	0.995526i	$0.530120\pi$
$12$	0	0
$13$	2.36311	0.655410	0.327705	−	0.944780i	$-0.393725\pi$
$13$	2.36311	0.655410	0.327705	+	0.944780i	$0.393725\pi$
$14$	0	0
$15$	−5.98625	−1.54564
$16$	0	0
$17$	−1.70155	−0.412686	−0.206343	−	0.978480i	$-0.566156\pi$
$17$	−1.70155	−0.412686	−0.206343	+	0.978480i	$0.566156\pi$
$18$	0	0
$19$	−1.16481	−0.267226	−0.133613	−	0.991034i	$-0.542658\pi$
$19$	−1.16481	−0.267226	−0.133613	+	0.991034i	$0.542658\pi$
$20$	0	0
$21$	−6.67932	−1.45755
$22$	0	0
$23$	4.01908	0.838036	0.419018	−	0.907978i	$-0.362374\pi$
$23$	4.01908	0.838036	0.419018	+	0.907978i	$0.362374\pi$
$24$	0	0
$25$	−0.145728	−0.0291456
$26$	0	0
$27$	3.75545	0.722737
$28$	0	0
$29$	−5.57129	−1.03456	−0.517281	−	0.855815i	$-0.673056\pi$
$29$	−5.57129	−1.03456	−0.517281	+	0.855815i	$0.673056\pi$
$30$	0	0
$31$	−1.20191	−0.215870	−0.107935	−	0.994158i	$-0.534424\pi$
$31$	−1.20191	−0.215870	−0.107935	+	0.994158i	$0.534424\pi$
$32$	0	0
$33$	−1.70287	−0.296432
$34$	0	0
$35$	5.41629	0.915519
$36$	0	0
$37$	−10.8149	−1.77796	−0.888980	−	0.457946i	$-0.848585\pi$
$37$	−10.8149	−1.77796	−0.888980	+	0.457946i	$0.848585\pi$
$38$	0	0
$39$	6.42063	1.02812
$40$	0	0
$41$	3.14126	0.490582	0.245291	−	0.969450i	$-0.421117\pi$
$41$	3.14126	0.490582	0.245291	+	0.969450i	$0.421117\pi$
$42$	0	0
$43$	3.97210	0.605740	0.302870	−	0.953032i	$-0.402055\pi$
$43$	3.97210	0.605740	0.302870	+	0.953032i	$0.402055\pi$
$44$	0	0
$45$	−9.65503	−1.43929
$46$	0	0
$47$	−2.37859	−0.346953	−0.173476	−	0.984838i	$-0.555500\pi$
$47$	−2.37859	−0.346953	−0.173476	+	0.984838i	$0.555500\pi$
$48$	0	0
$49$	−0.956632	−0.136662
$50$	0	0
$51$	−4.62314	−0.647368
$52$	0	0
$53$	−13.2463	−1.81952	−0.909758	−	0.415140i	$-0.863733\pi$
$53$	−13.2463	−1.81952	−0.909758	+	0.415140i	$0.863733\pi$
$54$	0	0
$55$	1.38087	0.186196
$56$	0	0
$57$	−3.16481	−0.419189
$58$	0	0
$59$	2.59425	0.337743	0.168871	−	0.985638i	$-0.445988\pi$
$59$	2.59425	0.337743	0.168871	+	0.985638i	$0.445988\pi$
$60$	0	0
$61$	0.891376	0.114129	0.0570645	−	0.998370i	$-0.481826\pi$
$61$	0.891376	0.114129	0.0570645	+	0.998370i	$0.481826\pi$
$62$	0	0
$63$	−10.7729	−1.35725
$64$	0	0
$65$	−5.20651	−0.645788
$66$	0	0
$67$	−2.69273	−0.328970	−0.164485	−	0.986380i	$-0.552596\pi$
$67$	−2.69273	−0.328970	−0.164485	+	0.986380i	$0.552596\pi$
$68$	0	0
$69$	10.9199	1.31460
$70$	0	0
$71$	−11.5924	−1.37576	−0.687880	−	0.725824i	$-0.741459\pi$
$71$	−11.5924	−1.37576	−0.687880	+	0.725824i	$0.741459\pi$
$72$	0	0
$73$	−6.11857	−0.716124	−0.358062	−	0.933698i	$-0.616562\pi$
$73$	−6.11857	−0.716124	−0.358062	+	0.933698i	$0.616562\pi$
$74$	0	0
$75$	−0.395946	−0.0457199
$76$	0	0
$77$	1.54074	0.175584
$78$	0	0
$79$	−1.82465	−0.205290	−0.102645	−	0.994718i	$-0.532731\pi$
$79$	−1.82465	−0.205290	−0.102645	+	0.994718i	$0.532731\pi$
$80$	0	0
$81$	−2.94295	−0.326994
$82$	0	0
$83$	12.7422	1.39863	0.699317	−	0.714812i	$-0.253488\pi$
$83$	12.7422	1.39863	0.699317	+	0.714812i	$0.253488\pi$
$84$	0	0
$85$	3.74892	0.406627
$86$	0	0
$87$	−15.1373	−1.62289
$88$	0	0
$89$	8.93573	0.947186	0.473593	−	0.880744i	$-0.342957\pi$
$89$	8.93573	0.947186	0.473593	+	0.880744i	$0.342957\pi$
$90$	0	0
$91$	−5.80931	−0.608981
$92$	0	0
$93$	−3.26562	−0.338629
$94$	0	0
$95$	2.56636	0.263303
$96$	0	0
$97$	−4.22438	−0.428921	−0.214461	−	0.976733i	$-0.568799\pi$
$97$	−4.22438	−0.428921	−0.214461	+	0.976733i	$0.568799\pi$
$98$	0	0
$99$	−2.74651	−0.276035
$100$	0	0
$101$	−12.4203	−1.23586	−0.617932	−	0.786231i	$-0.712029\pi$
$101$	−12.4203	−1.23586	−0.617932	+	0.786231i	$0.712029\pi$
$102$	0	0
$103$	9.45378	0.931509	0.465755	−	0.884914i	$-0.345783\pi$
$103$	9.45378	0.931509	0.465755	+	0.884914i	$0.345783\pi$
$104$	0	0
$105$	14.7162	1.43615
$106$	0	0
$107$	8.05091	0.778311	0.389155	−	0.921172i	$-0.372767\pi$
$107$	8.05091	0.778311	0.389155	+	0.921172i	$0.372767\pi$
$108$	0	0
$109$	8.57817	0.821640	0.410820	−	0.911717i	$-0.365243\pi$
$109$	8.57817	0.821640	0.410820	+	0.911717i	$0.365243\pi$
$110$	0	0
$111$	−29.3843	−2.78904
$112$	0	0
$113$	−3.06827	−0.288638	−0.144319	−	0.989531i	$-0.546099\pi$
$113$	−3.06827	−0.288638	−0.144319	+	0.989531i	$0.546099\pi$
$114$	0	0
$115$	−8.85501	−0.825734
$116$	0	0
$117$	10.3556	0.957378
$118$	0	0
$119$	4.18296	0.383451
$120$	0	0
$121$	−10.6072	−0.964290
$122$	0	0
$123$	8.53485	0.769562
$124$	0	0
$125$	11.3373	1.01404
$126$	0	0
$127$	−16.0241	−1.42191	−0.710954	−	0.703239i	$-0.751737\pi$
$127$	−16.0241	−1.42191	−0.710954	+	0.703239i	$0.751737\pi$
$128$	0	0
$129$	10.7923	0.950208
$130$	0	0
$131$	−11.3865	−0.994840	−0.497420	−	0.867510i	$-0.665719\pi$
$131$	−11.3865	−0.994840	−0.497420	+	0.867510i	$0.665719\pi$
$132$	0	0
$133$	2.86348	0.248295
$134$	0	0
$135$	−8.27417	−0.712127
$136$	0	0
$137$	11.3251	0.967567	0.483783	−	0.875188i	$-0.339262\pi$
$137$	11.3251	0.967567	0.483783	+	0.875188i	$0.339262\pi$
$138$	0	0
$139$	−10.7412	−0.911060	−0.455530	−	0.890220i	$-0.650550\pi$
$139$	−10.7412	−0.911060	−0.455530	+	0.890220i	$0.650550\pi$
$140$	0	0
$141$	−6.46267	−0.544255
$142$	0	0
$143$	−1.48107	−0.123853
$144$	0	0
$145$	12.2749	1.01937
$146$	0	0
$147$	−2.59919	−0.214377
$148$	0	0
$149$	−17.5934	−1.44131	−0.720655	−	0.693294i	$-0.756159\pi$
$149$	−17.5934	−1.44131	−0.720655	+	0.693294i	$0.756159\pi$
$150$	0	0
$151$	21.1563	1.72167	0.860836	−	0.508883i	$-0.169941\pi$
$151$	21.1563	1.72167	0.860836	+	0.508883i	$0.169941\pi$
$152$	0	0
$153$	−7.45651	−0.602823
$154$	0	0
$155$	2.64810	0.212701
$156$	0	0
$157$	−24.3583	−1.94400	−0.972002	−	0.234972i	$-0.924500\pi$
$157$	−24.3583	−1.94400	−0.972002	+	0.234972i	$0.924500\pi$
$158$	0	0
$159$	−35.9904	−2.85422
$160$	0	0
$161$	−9.88021	−0.778670
$162$	0	0
$163$	18.4677	1.44650	0.723249	−	0.690587i	$-0.242648\pi$
$163$	18.4677	1.44650	0.723249	+	0.690587i	$0.242648\pi$
$164$	0	0
$165$	3.75184	0.292081
$166$	0	0
$167$	−10.5586	−0.817049	−0.408524	−	0.912747i	$-0.633957\pi$
$167$	−10.5586	−0.817049	−0.408524	+	0.912747i	$0.633957\pi$
$168$	0	0
$169$	−7.41569	−0.570438
$170$	0	0
$171$	−5.10442	−0.390345
$172$	0	0
$173$	−14.9856	−1.13933	−0.569667	−	0.821876i	$-0.692928\pi$
$173$	−14.9856	−1.13933	−0.569667	+	0.821876i	$0.692928\pi$
$174$	0	0
$175$	0.358247	0.0270809
$176$	0	0
$177$	7.04863	0.529807
$178$	0	0
$179$	−4.59403	−0.343374	−0.171687	−	0.985152i	$-0.554922\pi$
$179$	−4.59403	−0.343374	−0.171687	+	0.985152i	$0.554922\pi$
$180$	0	0
$181$	2.46292	0.183068	0.0915338	−	0.995802i	$-0.470823\pi$
$181$	2.46292	0.183068	0.0915338	+	0.995802i	$0.470823\pi$
$182$	0	0
$183$	2.42189	0.179031
$184$	0	0
$185$	23.8278	1.75186
$186$	0	0
$187$	1.06643	0.0779853
$188$	0	0
$189$	−9.23213	−0.671538
$190$	0	0
$191$	13.9040	1.00606	0.503028	−	0.864270i	$-0.332219\pi$
$191$	13.9040	1.00606	0.503028	+	0.864270i	$0.332219\pi$
$192$	0	0
$193$	−0.786222	−0.0565935	−0.0282967	−	0.999600i	$-0.509008\pi$
$193$	−0.786222	−0.0565935	−0.0282967	+	0.999600i	$0.509008\pi$
$194$	0	0
$195$	−14.1462	−1.01303
$196$	0	0
$197$	5.95082	0.423978	0.211989	−	0.977272i	$-0.432006\pi$
$197$	5.95082	0.423978	0.211989	+	0.977272i	$0.432006\pi$
$198$	0	0
$199$	−15.0310	−1.06552	−0.532758	−	0.846267i	$-0.678845\pi$
$199$	−15.0310	−1.06552	−0.532758	+	0.846267i	$0.678845\pi$
$200$	0	0
$201$	−7.31621	−0.516045
$202$	0	0
$203$	13.6961	0.961274
$204$	0	0
$205$	−6.92094	−0.483380
$206$	0	0
$207$	17.6124	1.22415
$208$	0	0
$209$	0.730037	0.0504977
$210$	0	0
$211$	7.12711	0.490651	0.245325	−	0.969441i	$-0.421105\pi$
$211$	7.12711	0.490651	0.245325	+	0.969441i	$0.421105\pi$
$212$	0	0
$213$	−31.4967	−2.15812
$214$	0	0
$215$	−8.75150	−0.596848
$216$	0	0
$217$	2.95470	0.200578
$218$	0	0
$219$	−16.6243	−1.12336
$220$	0	0
$221$	−4.02095	−0.270478
$222$	0	0
$223$	−20.1880	−1.35189	−0.675945	−	0.736952i	$-0.736264\pi$
$223$	−20.1880	−1.35189	−0.675945	+	0.736952i	$0.736264\pi$
$224$	0	0
$225$	−0.638608	−0.0425739
$226$	0	0
$227$	−0.351443	−0.0233261	−0.0116630	−	0.999932i	$-0.503713\pi$
$227$	−0.351443	−0.0233261	−0.0116630	+	0.999932i	$0.503713\pi$
$228$	0	0
$229$	−11.1399	−0.736147	−0.368074	−	0.929797i	$-0.619983\pi$
$229$	−11.1399	−0.736147	−0.368074	+	0.929797i	$0.619983\pi$
$230$	0	0
$231$	4.18622	0.275433
$232$	0	0
$233$	−1.93861	−0.127002	−0.0635012	−	0.997982i	$-0.520227\pi$
$233$	−1.93861	−0.127002	−0.0635012	+	0.997982i	$0.520227\pi$
$234$	0	0
$235$	5.24060	0.341859
$236$	0	0
$237$	−4.95762	−0.322032
$238$	0	0
$239$	27.9507	1.80798	0.903989	−	0.427555i	$-0.140625\pi$
$239$	27.9507	1.80798	0.903989	+	0.427555i	$0.140625\pi$
$240$	0	0
$241$	1.52706	0.0983665	0.0491832	−	0.998790i	$-0.484338\pi$
$241$	1.52706	0.0983665	0.0491832	+	0.998790i	$0.484338\pi$
$242$	0	0
$243$	−19.2624	−1.23568
$244$	0	0
$245$	2.10769	0.134655
$246$	0	0
$247$	−2.75258	−0.175142
$248$	0	0
$249$	34.6207	2.19400
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−2.51893	−0.158364
$254$	0	0
$255$	10.1859	0.637865
$256$	0	0
$257$	−17.1502	−1.06980	−0.534900	−	0.844915i	$-0.679651\pi$
$257$	−17.1502	−1.06980	−0.534900	+	0.844915i	$0.679651\pi$
$258$	0	0
$259$	26.5866	1.65201
$260$	0	0
$261$	−24.4145	−1.51122
$262$	0	0
$263$	−8.67331	−0.534819	−0.267410	−	0.963583i	$-0.586168\pi$
$263$	−8.67331	−0.534819	−0.267410	+	0.963583i	$0.586168\pi$
$264$	0	0
$265$	29.1847	1.79280
$266$	0	0
$267$	24.2786	1.48582
$268$	0	0
$269$	17.5263	1.06860	0.534299	−	0.845296i	$-0.320576\pi$
$269$	17.5263	1.06860	0.534299	+	0.845296i	$0.320576\pi$
$270$	0	0
$271$	−21.3676	−1.29799	−0.648994	−	0.760793i	$-0.724810\pi$
$271$	−21.3676	−1.29799	−0.648994	+	0.760793i	$0.724810\pi$
$272$	0	0
$273$	−15.7840	−0.955291
$274$	0	0
$275$	0.0913341	0.00550765
$276$	0	0
$277$	−2.52213	−0.151540	−0.0757700	−	0.997125i	$-0.524141\pi$
$277$	−2.52213	−0.151540	−0.0757700	+	0.997125i	$0.524141\pi$
$278$	0	0
$279$	−5.26702	−0.315328
$280$	0	0
$281$	3.12510	0.186428	0.0932139	−	0.995646i	$-0.470286\pi$
$281$	3.12510	0.186428	0.0932139	+	0.995646i	$0.470286\pi$
$282$	0	0
$283$	−5.34841	−0.317930	−0.158965	−	0.987284i	$-0.550816\pi$
$283$	−5.34841	−0.317930	−0.158965	+	0.987284i	$0.550816\pi$
$284$	0	0
$285$	6.97284	0.413035
$286$	0	0
$287$	−7.72223	−0.455829
$288$	0	0
$289$	−14.1047	−0.829691
$290$	0	0
$291$	−11.4777	−0.672837
$292$	0	0
$293$	16.8829	0.986310	0.493155	−	0.869941i	$-0.335844\pi$
$293$	16.8829	0.986310	0.493155	+	0.869941i	$0.335844\pi$
$294$	0	0
$295$	−5.71576	−0.332784
$296$	0	0
$297$	−2.35371	−0.136576
$298$	0	0
$299$	9.49755	0.549257
$300$	0	0
$301$	−9.76473	−0.562830
$302$	0	0
$303$	−33.7462	−1.93867
$304$	0	0
$305$	−1.96392	−0.112454
$306$	0	0
$307$	1.42984	0.0816051	0.0408025	−	0.999167i	$-0.487009\pi$
$307$	1.42984	0.0816051	0.0408025	+	0.999167i	$0.487009\pi$
$308$	0	0
$309$	25.6861	1.46123
$310$	0	0
$311$	22.8314	1.29465	0.647324	−	0.762215i	$-0.275888\pi$
$311$	22.8314	1.29465	0.647324	+	0.762215i	$0.275888\pi$
$312$	0	0
$313$	17.5505	0.992011	0.496006	−	0.868319i	$-0.334800\pi$
$313$	17.5505	0.992011	0.496006	+	0.868319i	$0.334800\pi$
$314$	0	0
$315$	23.7352	1.33733
$316$	0	0
$317$	25.4561	1.42976	0.714879	−	0.699248i	$-0.246482\pi$
$317$	25.4561	1.42976	0.714879	+	0.699248i	$0.246482\pi$
$318$	0	0
$319$	3.49177	0.195502
$320$	0	0
$321$	21.8745	1.22091
$322$	0	0
$323$	1.98198	0.110280
$324$	0	0
$325$	−0.344372	−0.0191023
$326$	0	0
$327$	23.3071	1.28888
$328$	0	0
$329$	5.84734	0.322374
$330$	0	0
$331$	4.40015	0.241854	0.120927	−	0.992661i	$-0.461413\pi$
$331$	4.40015	0.241854	0.120927	+	0.992661i	$0.461413\pi$
$332$	0	0
$333$	−47.3930	−2.59712
$334$	0	0
$335$	5.93274	0.324140
$336$	0	0
$337$	6.63055	0.361189	0.180595	−	0.983558i	$-0.442198\pi$
$337$	6.63055	0.361189	0.180595	+	0.983558i	$0.442198\pi$
$338$	0	0
$339$	−8.33655	−0.452779
$340$	0	0
$341$	0.753291	0.0407930
$342$	0	0
$343$	19.5600	1.05614
$344$	0	0
$345$	−24.0592	−1.29531
$346$	0	0
$347$	14.4024	0.773159	0.386580	−	0.922256i	$-0.373656\pi$
$347$	14.4024	0.773159	0.386580	+	0.922256i	$0.373656\pi$
$348$	0	0
$349$	16.4842	0.882380	0.441190	−	0.897414i	$-0.354557\pi$
$349$	16.4842	0.882380	0.441190	+	0.897414i	$0.354557\pi$
$350$	0	0
$351$	8.87456	0.473689
$352$	0	0
$353$	9.46606	0.503827	0.251914	−	0.967750i	$-0.418940\pi$
$353$	9.46606	0.503827	0.251914	+	0.967750i	$0.418940\pi$
$354$	0	0
$355$	25.5408	1.35556
$356$	0	0
$357$	11.3652	0.601509
$358$	0	0
$359$	27.7828	1.46632	0.733160	−	0.680057i	$-0.238045\pi$
$359$	27.7828	1.46632	0.733160	+	0.680057i	$0.238045\pi$
$360$	0	0
$361$	−17.6432	−0.928591
$362$	0	0
$363$	−28.8199	−1.51266
$364$	0	0
$365$	13.4807	0.705611
$366$	0	0
$367$	−20.7430	−1.08277	−0.541387	−	0.840774i	$-0.682100\pi$
$367$	−20.7430	−1.08277	−0.541387	+	0.840774i	$0.682100\pi$
$368$	0	0
$369$	13.7656	0.716608
$370$	0	0
$371$	32.5637	1.69062
$372$	0	0
$373$	8.38553	0.434187	0.217093	−	0.976151i	$-0.430342\pi$
$373$	8.38553	0.434187	0.217093	+	0.976151i	$0.430342\pi$
$374$	0	0
$375$	30.8036	1.59069
$376$	0	0
$377$	−13.1656	−0.678063
$378$	0	0
$379$	−1.80599	−0.0927672	−0.0463836	−	0.998924i	$-0.514770\pi$
$379$	−1.80599	−0.0927672	−0.0463836	+	0.998924i	$0.514770\pi$
$380$	0	0
$381$	−43.5378	−2.23051
$382$	0	0
$383$	16.0809	0.821697	0.410848	−	0.911704i	$-0.365233\pi$
$383$	16.0809	0.821697	0.410848	+	0.911704i	$0.365233\pi$
$384$	0	0
$385$	−3.39462	−0.173006
$386$	0	0
$387$	17.4065	0.884824
$388$	0	0
$389$	37.4083	1.89668	0.948339	−	0.317259i	$-0.102762\pi$
$389$	37.4083	1.89668	0.948339	+	0.317259i	$0.102762\pi$
$390$	0	0
$391$	−6.83865	−0.345846
$392$	0	0
$393$	−30.9373	−1.56058
$394$	0	0
$395$	4.02015	0.202276
$396$	0	0
$397$	23.8883	1.19892	0.599461	−	0.800404i	$-0.295382\pi$
$397$	23.8883	1.19892	0.599461	+	0.800404i	$0.295382\pi$
$398$	0	0
$399$	7.78013	0.389494
$400$	0	0
$401$	32.6539	1.63066	0.815329	−	0.578997i	$-0.196556\pi$
$401$	32.6539	1.63066	0.815329	+	0.578997i	$0.196556\pi$
$402$	0	0
$403$	−2.84026	−0.141483
$404$	0	0
$405$	6.48403	0.322194
$406$	0	0
$407$	6.77817	0.335982
$408$	0	0
$409$	−0.586640	−0.0290075	−0.0145037	−	0.999895i	$-0.504617\pi$
$409$	−0.586640	−0.0290075	−0.0145037	+	0.999895i	$0.504617\pi$
$410$	0	0
$411$	30.7705	1.51779
$412$	0	0
$413$	−6.37752	−0.313817
$414$	0	0
$415$	−28.0741	−1.37810
$416$	0	0
$417$	−29.1842	−1.42915
$418$	0	0
$419$	−24.6938	−1.20637	−0.603185	−	0.797601i	$-0.706102\pi$
$419$	−24.6938	−1.20637	−0.603185	+	0.797601i	$0.706102\pi$
$420$	0	0
$421$	38.3992	1.87146	0.935731	−	0.352714i	$-0.114741\pi$
$421$	38.3992	1.87146	0.935731	+	0.352714i	$0.114741\pi$
$422$	0	0
$423$	−10.4234	−0.506805
$424$	0	0
$425$	0.247963	0.0120280
$426$	0	0
$427$	−2.19129	−0.106044
$428$	0	0
$429$	−4.02409	−0.194285
$430$	0	0
$431$	12.1175	0.583679	0.291840	−	0.956467i	$-0.405733\pi$
$431$	12.1175	0.583679	0.291840	+	0.956467i	$0.405733\pi$
$432$	0	0
$433$	9.18656	0.441478	0.220739	−	0.975333i	$-0.429153\pi$
$433$	9.18656	0.441478	0.220739	+	0.975333i	$0.429153\pi$
$434$	0	0
$435$	33.3511	1.59906
$436$	0	0
$437$	−4.68146	−0.223945
$438$	0	0
$439$	16.5692	0.790806	0.395403	−	0.918508i	$-0.370605\pi$
$439$	16.5692	0.790806	0.395403	+	0.918508i	$0.370605\pi$
$440$	0	0
$441$	−4.19215	−0.199626
$442$	0	0
$443$	−24.6364	−1.17051	−0.585255	−	0.810849i	$-0.699006\pi$
$443$	−24.6364	−1.17051	−0.585255	+	0.810849i	$0.699006\pi$
$444$	0	0
$445$	−19.6876	−0.933281
$446$	0	0
$447$	−47.8017	−2.26094
$448$	0	0
$449$	30.9412	1.46021	0.730103	−	0.683337i	$-0.239472\pi$
$449$	30.9412	1.46021	0.730103	+	0.683337i	$0.239472\pi$
$450$	0	0
$451$	−1.96876	−0.0927054
$452$	0	0
$453$	57.4820	2.70074
$454$	0	0
$455$	12.7993	0.600041
$456$	0	0
$457$	30.8381	1.44255	0.721273	−	0.692651i	$-0.243557\pi$
$457$	30.8381	1.44255	0.721273	+	0.692651i	$0.243557\pi$
$458$	0	0
$459$	−6.39008	−0.298263
$460$	0	0
$461$	−30.6619	−1.42807	−0.714033	−	0.700112i	$-0.753133\pi$
$461$	−30.6619	−1.42807	−0.714033	+	0.700112i	$0.753133\pi$
$462$	0	0
$463$	40.9205	1.90173	0.950867	−	0.309598i	$-0.100195\pi$
$463$	40.9205	1.90173	0.950867	+	0.309598i	$0.100195\pi$
$464$	0	0
$465$	7.19495	0.333658
$466$	0	0
$467$	−25.5509	−1.18236	−0.591178	−	0.806541i	$-0.701337\pi$
$467$	−25.5509	−1.18236	−0.591178	+	0.806541i	$0.701337\pi$
$468$	0	0
$469$	6.61961	0.305665
$470$	0	0
$471$	−66.1820	−3.04950
$472$	0	0
$473$	−2.48949	−0.114467
$474$	0	0
$475$	0.169745	0.00778845
$476$	0	0
$477$	−58.0477	−2.65782
$478$	0	0
$479$	11.8475	0.541324	0.270662	−	0.962674i	$-0.412757\pi$
$479$	11.8475	0.541324	0.270662	+	0.962674i	$0.412757\pi$
$480$	0	0
$481$	−25.5569	−1.16529
$482$	0	0
$483$	−26.8447	−1.22148
$484$	0	0
$485$	9.30734	0.422624
$486$	0	0
$487$	5.07419	0.229933	0.114967	−	0.993369i	$-0.463324\pi$
$487$	5.07419	0.229933	0.114967	+	0.993369i	$0.463324\pi$
$488$	0	0
$489$	50.1770	2.26908
$490$	0	0
$491$	−41.5821	−1.87658	−0.938288	−	0.345856i	$-0.887589\pi$
$491$	−41.5821	−1.87658	−0.938288	+	0.345856i	$0.887589\pi$
$492$	0	0
$493$	9.47981	0.426949
$494$	0	0
$495$	6.05123	0.271983
$496$	0	0
$497$	28.4978	1.27830
$498$	0	0
$499$	−34.6802	−1.55250	−0.776250	−	0.630425i	$-0.782881\pi$
$499$	−34.6802	−1.55250	−0.776250	+	0.630425i	$0.782881\pi$
$500$	0	0
$501$	−28.6879	−1.28168
$502$	0	0
$503$	−28.2335	−1.25887	−0.629435	−	0.777053i	$-0.716714\pi$
$503$	−28.2335	−1.25887	−0.629435	+	0.777053i	$0.716714\pi$
$504$	0	0
$505$	27.3649	1.21772
$506$	0	0
$507$	−20.1486	−0.894830
$508$	0	0
$509$	11.3844	0.504604	0.252302	−	0.967649i	$-0.418812\pi$
$509$	11.3844	0.504604	0.252302	+	0.967649i	$0.418812\pi$
$510$	0	0
$511$	15.0414	0.665394
$512$	0	0
$513$	−4.37438	−0.193134
$514$	0	0
$515$	−20.8290	−0.917834
$516$	0	0
$517$	1.49076	0.0655637
$518$	0	0
$519$	−40.7161	−1.78724
$520$	0	0
$521$	−13.3861	−0.586456	−0.293228	−	0.956043i	$-0.594729\pi$
$521$	−13.3861	−0.586456	−0.293228	+	0.956043i	$0.594729\pi$
$522$	0	0
$523$	28.2928	1.23716	0.618580	−	0.785722i	$-0.287708\pi$
$523$	28.2928	1.23716	0.618580	+	0.785722i	$0.287708\pi$
$524$	0	0
$525$	0.973364	0.0424811
$526$	0	0
$527$	2.04511	0.0890865
$528$	0	0
$529$	−6.84699	−0.297695
$530$	0	0
$531$	11.3685	0.493351
$532$	0	0
$533$	7.42314	0.321532
$534$	0	0
$535$	−17.7381	−0.766885
$536$	0	0
$537$	−12.4821	−0.538641
$538$	0	0
$539$	0.599563	0.0258250
$540$	0	0
$541$	−43.1435	−1.85488	−0.927442	−	0.373967i	$-0.877997\pi$
$541$	−43.1435	−1.85488	−0.927442	+	0.373967i	$0.877997\pi$
$542$	0	0
$543$	6.69181	0.287173
$544$	0	0
$545$	−18.8998	−0.809577
$546$	0	0
$547$	3.56283	0.152336	0.0761679	−	0.997095i	$-0.475731\pi$
$547$	3.56283	0.152336	0.0761679	+	0.997095i	$0.475731\pi$
$548$	0	0
$549$	3.90618	0.166712
$550$	0	0
$551$	6.48949	0.276462
$552$	0	0
$553$	4.48560	0.190747
$554$	0	0
$555$	64.7407	2.74809
$556$	0	0
$557$	−41.3898	−1.75374	−0.876872	−	0.480724i	$-0.840374\pi$
$557$	−41.3898	−1.75374	−0.876872	+	0.480724i	$0.840374\pi$
$558$	0	0
$559$	9.38654	0.397008
$560$	0	0
$561$	2.89752	0.122333
$562$	0	0
$563$	2.18917	0.0922625	0.0461313	−	0.998935i	$-0.485311\pi$
$563$	2.18917	0.0922625	0.0461313	+	0.998935i	$0.485311\pi$
$564$	0	0
$565$	6.76014	0.284401
$566$	0	0
$567$	7.23473	0.303830
$568$	0	0
$569$	11.6584	0.488747	0.244373	−	0.969681i	$-0.421418\pi$
$569$	11.6584	0.488747	0.244373	+	0.969681i	$0.421418\pi$
$570$	0	0
$571$	−14.0137	−0.586456	−0.293228	−	0.956043i	$-0.594729\pi$
$571$	−14.0137	−0.586456	−0.293228	+	0.956043i	$0.594729\pi$
$572$	0	0
$573$	37.7773	1.57817
$574$	0	0
$575$	−0.585692	−0.0244251
$576$	0	0
$577$	−30.4746	−1.26867	−0.634337	−	0.773057i	$-0.718727\pi$
$577$	−30.4746	−1.26867	−0.634337	+	0.773057i	$0.718727\pi$
$578$	0	0
$579$	−2.13618	−0.0887766
$580$	0	0
$581$	−31.3244	−1.29956
$582$	0	0
$583$	8.30202	0.343834
$584$	0	0
$585$	−22.8159	−0.943323
$586$	0	0
$587$	−6.45165	−0.266288	−0.133144	−	0.991097i	$-0.542507\pi$
$587$	−6.45165	−0.266288	−0.133144	+	0.991097i	$0.542507\pi$
$588$	0	0
$589$	1.40000	0.0576860
$590$	0	0
$591$	16.1685	0.665083
$592$	0	0
$593$	−1.47035	−0.0603802	−0.0301901	−	0.999544i	$-0.509611\pi$
$593$	−1.47035	−0.0603802	−0.0301901	+	0.999544i	$0.509611\pi$
$594$	0	0
$595$	−9.21606	−0.377822
$596$	0	0
$597$	−40.8394	−1.67145
$598$	0	0
$599$	37.8677	1.54723	0.773616	−	0.633655i	$-0.218446\pi$
$599$	37.8677	1.54723	0.773616	+	0.633655i	$0.218446\pi$
$600$	0	0
$601$	−40.6304	−1.65735	−0.828675	−	0.559730i	$-0.810905\pi$
$601$	−40.6304	−1.65735	−0.828675	+	0.559730i	$0.810905\pi$
$602$	0	0
$603$	−11.8001	−0.480536
$604$	0	0
$605$	23.3702	0.950134
$606$	0	0
$607$	40.8417	1.65771	0.828857	−	0.559460i	$-0.188992\pi$
$607$	40.8417	1.65771	0.828857	+	0.559460i	$0.188992\pi$
$608$	0	0
$609$	37.2124	1.50792
$610$	0	0
$611$	−5.62087	−0.227396
$612$	0	0
$613$	20.9266	0.845216	0.422608	−	0.906313i	$-0.361115\pi$
$613$	20.9266	0.845216	0.422608	+	0.906313i	$0.361115\pi$
$614$	0	0
$615$	−18.8043	−0.758264
$616$	0	0
$617$	−12.4425	−0.500917	−0.250458	−	0.968127i	$-0.580581\pi$
$617$	−12.4425	−0.500917	−0.250458	+	0.968127i	$0.580581\pi$
$618$	0	0
$619$	4.47560	0.179890	0.0899448	−	0.995947i	$-0.471331\pi$
$619$	4.47560	0.179890	0.0899448	+	0.995947i	$0.471331\pi$
$620$	0	0
$621$	15.0935	0.605680
$622$	0	0
$623$	−21.9669	−0.880087
$624$	0	0
$625$	−24.2501	−0.970005
$626$	0	0
$627$	1.98352	0.0792143
$628$	0	0
$629$	18.4021	0.733738
$630$	0	0
$631$	−4.47879	−0.178298	−0.0891488	−	0.996018i	$-0.528415\pi$
$631$	−4.47879	−0.178298	−0.0891488	+	0.996018i	$0.528415\pi$
$632$	0	0
$633$	19.3645	0.769670
$634$	0	0
$635$	35.3049	1.40103
$636$	0	0
$637$	−2.26063	−0.0895694
$638$	0	0
$639$	−50.8000	−2.00962
$640$	0	0
$641$	−3.20664	−0.126655	−0.0633274	−	0.997993i	$-0.520171\pi$
$641$	−3.20664	−0.126655	−0.0633274	+	0.997993i	$0.520171\pi$
$642$	0	0
$643$	27.8300	1.09751	0.548753	−	0.835984i	$-0.315103\pi$
$643$	27.8300	1.09751	0.548753	+	0.835984i	$0.315103\pi$
$644$	0	0
$645$	−23.7780	−0.936258
$646$	0	0
$647$	−6.44508	−0.253382	−0.126691	−	0.991942i	$-0.540436\pi$
$647$	−6.44508	−0.253382	−0.126691	+	0.991942i	$0.540436\pi$
$648$	0	0
$649$	−1.62593	−0.0638233
$650$	0	0
$651$	8.02796	0.314641
$652$	0	0
$653$	21.4765	0.840442	0.420221	−	0.907422i	$-0.361953\pi$
$653$	21.4765	0.840442	0.420221	+	0.907422i	$0.361953\pi$
$654$	0	0
$655$	25.0871	0.980236
$656$	0	0
$657$	−26.8128	−1.04607
$658$	0	0
$659$	4.90000	0.190877	0.0954385	−	0.995435i	$-0.469575\pi$
$659$	4.90000	0.190877	0.0954385	+	0.995435i	$0.469575\pi$
$660$	0	0
$661$	5.00081	0.194509	0.0972546	−	0.995260i	$-0.468994\pi$
$661$	5.00081	0.194509	0.0972546	+	0.995260i	$0.468994\pi$
$662$	0	0
$663$	−10.9250	−0.424292
$664$	0	0
$665$	−6.30894	−0.244650
$666$	0	0
$667$	−22.3915	−0.867001
$668$	0	0
$669$	−54.8512	−2.12067
$670$	0	0
$671$	−0.558664	−0.0215670
$672$	0	0
$673$	−32.9370	−1.26963	−0.634813	−	0.772665i	$-0.718923\pi$
$673$	−32.9370	−1.26963	−0.634813	+	0.772665i	$0.718923\pi$
$674$	0	0
$675$	−0.547274	−0.0210646
$676$	0	0
$677$	26.4198	1.01539	0.507697	−	0.861536i	$-0.330497\pi$
$677$	26.4198	1.01539	0.507697	+	0.861536i	$0.330497\pi$
$678$	0	0
$679$	10.3849	0.398536
$680$	0	0
$681$	−0.954877	−0.0365910
$682$	0	0
$683$	−24.0624	−0.920723	−0.460362	−	0.887731i	$-0.652280\pi$
$683$	−24.0624	−0.920723	−0.460362	+	0.887731i	$0.652280\pi$
$684$	0	0
$685$	−24.9519	−0.953362
$686$	0	0
$687$	−30.2674	−1.15477
$688$	0	0
$689$	−31.3025	−1.19253
$690$	0	0
$691$	26.3594	1.00276	0.501379	−	0.865228i	$-0.332826\pi$
$691$	26.3594	1.00276	0.501379	+	0.865228i	$0.332826\pi$
$692$	0	0
$693$	6.75182	0.256481
$694$	0	0
$695$	23.6655	0.897685
$696$	0	0
$697$	−5.34499	−0.202456
$698$	0	0
$699$	−5.26724	−0.199225
$700$	0	0
$701$	−28.6977	−1.08390	−0.541948	−	0.840412i	$-0.682313\pi$
$701$	−28.6977	−1.08390	−0.541948	+	0.840412i	$0.682313\pi$
$702$	0	0
$703$	12.5973	0.475116
$704$	0	0
$705$	14.2388	0.536265
$706$	0	0
$707$	30.5331	1.14832
$708$	0	0
$709$	−13.4638	−0.505644	−0.252822	−	0.967513i	$-0.581359\pi$
$709$	−13.4638	−0.505644	−0.252822	+	0.967513i	$0.581359\pi$
$710$	0	0
$711$	−7.99599	−0.299873
$712$	0	0
$713$	−4.83059	−0.180907
$714$	0	0
$715$	3.26315	0.122035
$716$	0	0
$717$	75.9425	2.83613
$718$	0	0
$719$	20.9723	0.782135	0.391067	−	0.920362i	$-0.372106\pi$
$719$	20.9723	0.782135	0.391067	+	0.920362i	$0.372106\pi$
$720$	0	0
$721$	−23.2405	−0.865521
$722$	0	0
$723$	4.14905	0.154305
$724$	0	0
$725$	0.811893	0.0301529
$726$	0	0
$727$	−24.9166	−0.924104	−0.462052	−	0.886853i	$-0.652887\pi$
$727$	−24.9166	−0.924104	−0.462052	+	0.886853i	$0.652887\pi$
$728$	0	0
$729$	−43.5075	−1.61139
$730$	0	0
$731$	−6.75872	−0.249980
$732$	0	0
$733$	−27.5799	−1.01869	−0.509343	−	0.860564i	$-0.670112\pi$
$733$	−27.5799	−1.01869	−0.509343	+	0.860564i	$0.670112\pi$
$734$	0	0
$735$	5.72664	0.211230
$736$	0	0
$737$	1.68765	0.0621655
$738$	0	0
$739$	−28.3870	−1.04423	−0.522116	−	0.852874i	$-0.674857\pi$
$739$	−28.3870	−1.04423	−0.522116	+	0.852874i	$0.674857\pi$
$740$	0	0
$741$	−7.47881	−0.274741
$742$	0	0
$743$	−4.74424	−0.174049	−0.0870246	−	0.996206i	$-0.527736\pi$
$743$	−4.74424	−0.174049	−0.0870246	+	0.996206i	$0.527736\pi$
$744$	0	0
$745$	38.7626	1.42015
$746$	0	0
$747$	55.8387	2.04303
$748$	0	0
$749$	−19.7918	−0.723175
$750$	0	0
$751$	3.57930	0.130611	0.0653053	−	0.997865i	$-0.479198\pi$
$751$	3.57930	0.130611	0.0653053	+	0.997865i	$0.479198\pi$
$752$	0	0
$753$	2.71702	0.0990137
$754$	0	0
$755$	−46.6123	−1.69640
$756$	0	0
$757$	−27.2557	−0.990626	−0.495313	−	0.868714i	$-0.664947\pi$
$757$	−27.2557	−0.990626	−0.495313	+	0.868714i	$0.664947\pi$
$758$	0	0
$759$	−6.84399	−0.248421
$760$	0	0
$761$	−33.4975	−1.21428	−0.607141	−	0.794594i	$-0.707684\pi$
$761$	−33.4975	−1.21428	−0.607141	+	0.794594i	$0.707684\pi$
$762$	0	0
$763$	−21.0879	−0.763435
$764$	0	0
$765$	16.4285	0.593973
$766$	0	0
$767$	6.13051	0.221360
$768$	0	0
$769$	−48.5722	−1.75156	−0.875778	−	0.482713i	$-0.839651\pi$
$769$	−48.5722	−1.75156	−0.875778	+	0.482713i	$0.839651\pi$
$770$	0	0
$771$	−46.5974	−1.67816
$772$	0	0
$773$	13.2684	0.477230	0.238615	−	0.971114i	$-0.423307\pi$
$773$	13.2684	0.477230	0.238615	+	0.971114i	$0.423307\pi$
$774$	0	0
$775$	0.175152	0.00629166
$776$	0	0
$777$	72.2362	2.59146
$778$	0	0
$779$	−3.65896	−0.131096
$780$	0	0
$781$	7.26544	0.259978
$782$	0	0
$783$	−20.9227	−0.747717
$784$	0	0
$785$	53.6672	1.91547
$786$	0	0
$787$	−3.60077	−0.128354	−0.0641768	−	0.997939i	$-0.520442\pi$
$787$	−3.60077	−0.128354	−0.0641768	+	0.997939i	$0.520442\pi$
$788$	0	0
$789$	−23.5656	−0.838956
$790$	0	0
$791$	7.54281	0.268191
$792$	0	0
$793$	2.10642	0.0748013
$794$	0	0
$795$	79.2955	2.81232
$796$	0	0
$797$	42.2720	1.49735	0.748675	−	0.662937i	$-0.230690\pi$
$797$	42.2720	1.49735	0.748675	+	0.662937i	$0.230690\pi$
$798$	0	0
$799$	4.04728	0.143182
$800$	0	0
$801$	39.1581	1.38358
$802$	0	0
$803$	3.83477	0.135326
$804$	0	0
$805$	21.7685	0.767239
$806$	0	0
$807$	47.6193	1.67628
$808$	0	0
$809$	43.6696	1.53534	0.767671	−	0.640844i	$-0.221415\pi$
$809$	43.6696	1.53534	0.767671	+	0.640844i	$0.221415\pi$
$810$	0	0
$811$	−7.28785	−0.255911	−0.127955	−	0.991780i	$-0.540841\pi$
$811$	−7.28785	−0.255911	−0.127955	+	0.991780i	$0.540841\pi$
$812$	0	0
$813$	−58.0561	−2.03612
$814$	0	0
$815$	−40.6887	−1.42526
$816$	0	0
$817$	−4.62674	−0.161869
$818$	0	0
$819$	−25.4575	−0.889557
$820$	0	0
$821$	−3.79241	−0.132356	−0.0661780	−	0.997808i	$-0.521081\pi$
$821$	−3.79241	−0.132356	−0.0661780	+	0.997808i	$0.521081\pi$
$822$	0	0
$823$	28.5848	0.996404	0.498202	−	0.867061i	$-0.333994\pi$
$823$	28.5848	0.996404	0.498202	+	0.867061i	$0.333994\pi$
$824$	0	0
$825$	0.248156	0.00863970
$826$	0	0
$827$	6.25556	0.217527	0.108763	−	0.994068i	$-0.465311\pi$
$827$	6.25556	0.217527	0.108763	+	0.994068i	$0.465311\pi$
$828$	0	0
$829$	5.99888	0.208350	0.104175	−	0.994559i	$-0.466780\pi$
$829$	5.99888	0.208350	0.104175	+	0.994559i	$0.466780\pi$
$830$	0	0
$831$	−6.85267	−0.237717
$832$	0	0
$833$	1.62775	0.0563983
$834$	0	0
$835$	23.2631	0.805054
$836$	0	0
$837$	−4.51373	−0.156017
$838$	0	0
$839$	−12.5373	−0.432837	−0.216419	−	0.976301i	$-0.569438\pi$
$839$	−12.5373	−0.432837	−0.216419	+	0.976301i	$0.569438\pi$
$840$	0	0
$841$	2.03929	0.0703204
$842$	0	0
$843$	8.49096	0.292444
$844$	0	0
$845$	16.3386	0.562063
$846$	0	0
$847$	26.0759	0.895980
$848$	0	0
$849$	−14.5317	−0.498728
$850$	0	0
$851$	−43.4660	−1.48999
$852$	0	0
$853$	−30.2563	−1.03595	−0.517977	−	0.855394i	$-0.673315\pi$
$853$	−30.2563	−1.03595	−0.517977	+	0.855394i	$0.673315\pi$
$854$	0	0
$855$	11.2463	0.384614
$856$	0	0
$857$	13.7501	0.469695	0.234848	−	0.972032i	$-0.424541\pi$
$857$	13.7501	0.469695	0.234848	+	0.972032i	$0.424541\pi$
$858$	0	0
$859$	−26.5314	−0.905238	−0.452619	−	0.891704i	$-0.649510\pi$
$859$	−26.5314	−0.905238	−0.452619	+	0.891704i	$0.649510\pi$
$860$	0	0
$861$	−20.9815	−0.715046
$862$	0	0
$863$	2.34872	0.0799514	0.0399757	−	0.999201i	$-0.487272\pi$
$863$	2.34872	0.0799514	0.0399757	+	0.999201i	$0.487272\pi$
$864$	0	0
$865$	33.0169	1.12261
$866$	0	0
$867$	−38.3229	−1.30151
$868$	0	0
$869$	1.14359	0.0387936
$870$	0	0
$871$	−6.36323	−0.215610
$872$	0	0
$873$	−18.5121	−0.626539
$874$	0	0
$875$	−27.8707	−0.942203
$876$	0	0
$877$	−30.0685	−1.01534	−0.507670	−	0.861552i	$-0.669493\pi$
$877$	−30.0685	−1.01534	−0.507670	+	0.861552i	$0.669493\pi$
$878$	0	0
$879$	45.8712	1.54720
$880$	0	0
$881$	48.8148	1.64461	0.822306	−	0.569046i	$-0.192687\pi$
$881$	48.8148	1.64461	0.822306	+	0.569046i	$0.192687\pi$
$882$	0	0
$883$	−25.5925	−0.861257	−0.430628	−	0.902529i	$-0.641708\pi$
$883$	−25.5925	−0.861257	−0.430628	+	0.902529i	$0.641708\pi$
$884$	0	0
$885$	−15.5298	−0.522030
$886$	0	0
$887$	35.1440	1.18002	0.590010	−	0.807396i	$-0.299124\pi$
$887$	35.1440	1.18002	0.590010	+	0.807396i	$0.299124\pi$
$888$	0	0
$889$	39.3924	1.32118
$890$	0	0
$891$	1.84448	0.0617922
$892$	0	0
$893$	2.77060	0.0927146
$894$	0	0
$895$	10.1218	0.338333
$896$	0	0
$897$	25.8050	0.861605
$898$	0	0
$899$	6.69621	0.223331
$900$	0	0
$901$	22.5391	0.750888
$902$	0	0
$903$	−26.5310	−0.882895
$904$	0	0
$905$	−5.42642	−0.180380
$906$	0	0
$907$	16.1004	0.534605	0.267303	−	0.963613i	$-0.413868\pi$
$907$	16.1004	0.534605	0.267303	+	0.963613i	$0.413868\pi$
$908$	0	0
$909$	−54.4281	−1.80527
$910$	0	0
$911$	43.4451	1.43940	0.719700	−	0.694285i	$-0.244279\pi$
$911$	43.4451	1.43940	0.719700	+	0.694285i	$0.244279\pi$
$912$	0	0
$913$	−7.98607	−0.264300
$914$	0	0
$915$	−5.33600	−0.176403
$916$	0	0
$917$	27.9917	0.924366
$918$	0	0
$919$	−1.79550	−0.0592281	−0.0296140	−	0.999561i	$-0.509428\pi$
$919$	−1.79550	−0.0592281	−0.0296140	+	0.999561i	$0.509428\pi$
$920$	0	0
$921$	3.88489	0.128012
$922$	0	0
$923$	−27.3941	−0.901687
$924$	0	0
$925$	1.57603	0.0518197
$926$	0	0
$927$	41.4283	1.36068
$928$	0	0
$929$	7.80958	0.256224	0.128112	−	0.991760i	$-0.459108\pi$
$929$	7.80958	0.256224	0.128112	+	0.991760i	$0.459108\pi$
$930$	0	0
$931$	1.11429	0.0365195
$932$	0	0
$933$	62.0333	2.03088
$934$	0	0
$935$	−2.34961	−0.0768405
$936$	0	0
$937$	25.0439	0.818148	0.409074	−	0.912501i	$-0.365852\pi$
$937$	25.0439	0.818148	0.409074	+	0.912501i	$0.365852\pi$
$938$	0	0
$939$	47.6850	1.55614
$940$	0	0
$941$	−36.1353	−1.17798	−0.588988	−	0.808142i	$-0.700474\pi$
$941$	−36.1353	−1.17798	−0.588988	+	0.808142i	$0.700474\pi$
$942$	0	0
$943$	12.6250	0.411125
$944$	0	0
$945$	20.3406	0.661680
$946$	0	0
$947$	−24.5097	−0.796457	−0.398229	−	0.917286i	$-0.630375\pi$
$947$	−24.5097	−0.796457	−0.398229	+	0.917286i	$0.630375\pi$
$948$	0	0
$949$	−14.4589	−0.469355
$950$	0	0
$951$	69.1648	2.24282
$952$	0	0
$953$	40.9614	1.32687	0.663435	−	0.748234i	$-0.269098\pi$
$953$	40.9614	1.32687	0.663435	+	0.748234i	$0.269098\pi$
$954$	0	0
$955$	−30.6338	−0.991286
$956$	0	0
$957$	9.48721	0.306678
$958$	0	0
$959$	−27.8407	−0.899024
$960$	0	0
$961$	−29.5554	−0.953400
$962$	0	0
$963$	35.2807	1.13690
$964$	0	0
$965$	1.73224	0.0557627
$966$	0	0
$967$	−39.7541	−1.27841	−0.639203	−	0.769038i	$-0.720735\pi$
$967$	−39.7541	−1.27841	−0.639203	+	0.769038i	$0.720735\pi$
$968$	0	0
$969$	5.38507	0.172993
$970$	0	0
$971$	24.1119	0.773788	0.386894	−	0.922124i	$-0.373548\pi$
$971$	24.1119	0.773788	0.386894	+	0.922124i	$0.373548\pi$
$972$	0	0
$973$	26.4055	0.846520
$974$	0	0
$975$	−0.935665	−0.0299653
$976$	0	0
$977$	4.53652	0.145136	0.0725680	−	0.997363i	$-0.476881\pi$
$977$	4.53652	0.145136	0.0725680	+	0.997363i	$0.476881\pi$
$978$	0	0
$979$	−5.60041	−0.178990
$980$	0	0
$981$	37.5912	1.20020
$982$	0	0
$983$	−5.03811	−0.160691	−0.0803453	−	0.996767i	$-0.525602\pi$
$983$	−5.03811	−0.160691	−0.0803453	+	0.996767i	$0.525602\pi$
$984$	0	0
$985$	−13.1111	−0.417754
$986$	0	0
$987$	15.8873	0.505700
$988$	0	0
$989$	15.9642	0.507632
$990$	0	0
$991$	−11.4826	−0.364757	−0.182379	−	0.983228i	$-0.558380\pi$
$991$	−11.4826	−0.364757	−0.182379	+	0.983228i	$0.558380\pi$
$992$	0	0
$993$	11.9553	0.379390
$994$	0	0
$995$	33.1168	1.04987
$996$	0	0
$997$	5.91042	0.187185	0.0935925	−	0.995611i	$-0.470165\pi$
$997$	5.91042	0.187185	0.0935925	+	0.995611i	$0.470165\pi$
$998$	0	0
$999$	−40.6149	−1.28500

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.e.1.5		5
4.3	odd	2		502.2.a.c.1.1	✓	5
12.11	even	2		4518.2.a.v.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.c.1.1	✓	5	4.3	odd	2
4016.2.a.e.1.5		5	1.1	even	1	trivial
4518.2.a.v.1.3		5	12.11	even	2

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)

\(f(q)\)	\(=\)	\(q+2.71702 q^{3} -2.20324 q^{5} -2.45833 q^{7} +4.38220 q^{9} -0.626744 q^{11} +2.36311 q^{13} -5.98625 q^{15} -1.70155 q^{17} -1.16481 q^{19} -6.67932 q^{21} +4.01908 q^{23} -0.145728 q^{25} +3.75545 q^{27} -5.57129 q^{29} -1.20191 q^{31} -1.70287 q^{33} +5.41629 q^{35} -10.8149 q^{37} +6.42063 q^{39} +3.14126 q^{41} +3.97210 q^{43} -9.65503 q^{45} -2.37859 q^{47} -0.956632 q^{49} -4.62314 q^{51} -13.2463 q^{53} +1.38087 q^{55} -3.16481 q^{57} +2.59425 q^{59} +0.891376 q^{61} -10.7729 q^{63} -5.20651 q^{65} -2.69273 q^{67} +10.9199 q^{69} -11.5924 q^{71} -6.11857 q^{73} -0.395946 q^{75} +1.54074 q^{77} -1.82465 q^{79} -2.94295 q^{81} +12.7422 q^{83} +3.74892 q^{85} -15.1373 q^{87} +8.93573 q^{89} -5.80931 q^{91} -3.26562 q^{93} +2.56636 q^{95} -4.22438 q^{97} -2.74651 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - 6 q^{5} + q^{7} + 4 q^{11} - 7 q^{13} + 5 q^{15} - 8 q^{17} - 3 q^{19} - 10 q^{21} + 17 q^{23} - q^{25} + 4 q^{27} - 15 q^{29} + q^{31} - 10 q^{33} + 7 q^{35} - 5 q^{37} + 8 q^{39} - 12 q^{41}+ \cdots - 28 q^{99}+O(q^{100}) \) 5 * q + q^3 - 6 * q^5 + q^7 + 4 * q^11 - 7 * q^13 + 5 * q^15 - 8 * q^17 - 3 * q^19 - 10 * q^21 + 17 * q^23 - q^25 + 4 * q^27 - 15 * q^29 + q^31 - 10 * q^33 + 7 * q^35 - 5 * q^37 + 8 * q^39 - 12 * q^41 - q^43 - 13 * q^45 + 19 * q^47 + 4 * q^49 - 7 * q^51 - 34 * q^53 - 17 * q^55 - 13 * q^57 + 10 * q^59 + 2 * q^63 + 6 * q^65 + 11 * q^67 + q^69 + q^71 + 3 * q^73 - 15 * q^75 - 30 * q^77 - 35 * q^79 - 3 * q^81 - 3 * q^83 - 13 * q^85 - 3 * q^87 + 15 * q^89 - 14 * q^91 + 15 * q^93 - 11 * q^95 - 2 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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