LMFDB - Embedded newform 4016.2.a.h.1.9

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 $ x^9 - x^8 - 11x^7 + 7x^6 + 40x^5 - 11x^4 - 53x^3 - 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$2.59002$ 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)

$f(q)$	$=$	$q+2.59002 q^{3} -1.92989 q^{5} +1.80522 q^{7} +3.70818 q^{9} +O(q^{10})$	$q+2.59002 q^{3} -1.92989 q^{5} +1.80522 q^{7} +3.70818 q^{9} -1.71997 q^{11} +0.374064 q^{13} -4.99846 q^{15} -4.23825 q^{17} -7.16485 q^{19} +4.67556 q^{21} -1.54199 q^{23} -1.27551 q^{25} +1.83421 q^{27} -3.54108 q^{29} -9.29782 q^{31} -4.45475 q^{33} -3.48389 q^{35} +0.384158 q^{37} +0.968831 q^{39} +4.15934 q^{41} -2.72439 q^{43} -7.15640 q^{45} +0.802404 q^{47} -3.74117 q^{49} -10.9771 q^{51} +2.16015 q^{53} +3.31936 q^{55} -18.5571 q^{57} +1.88478 q^{59} +2.18668 q^{61} +6.69410 q^{63} -0.721903 q^{65} +3.55803 q^{67} -3.99378 q^{69} +7.85944 q^{71} -9.55914 q^{73} -3.30360 q^{75} -3.10493 q^{77} +1.53948 q^{79} -6.37392 q^{81} +6.07124 q^{83} +8.17936 q^{85} -9.17145 q^{87} -8.62064 q^{89} +0.675268 q^{91} -24.0815 q^{93} +13.8274 q^{95} +8.66932 q^{97} -6.37797 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	2.59002	1.49535	0.747673	−	0.664067i	$-0.231171\pi$
$3$	2.59002	1.49535	0.747673	+	0.664067i	$0.231171\pi$
$4$	0	0
$5$	−1.92989	−0.863075	−0.431537	−	0.902095i	$-0.642029\pi$
$5$	−1.92989	−0.863075	−0.431537	+	0.902095i	$0.642029\pi$
$6$	0	0
$7$	1.80522	0.682310	0.341155	−	0.940007i	$-0.389182\pi$
$7$	1.80522	0.682310	0.341155	+	0.940007i	$0.389182\pi$
$8$	0	0
$9$	3.70818	1.23606
$10$	0	0
$11$	−1.71997	−0.518591	−0.259295	−	0.965798i	$-0.583490\pi$
$11$	−1.71997	−0.518591	−0.259295	+	0.965798i	$0.583490\pi$
$12$	0	0
$13$	0.374064	0.103747	0.0518733	−	0.998654i	$-0.483481\pi$
$13$	0.374064	0.103747	0.0518733	+	0.998654i	$0.483481\pi$
$14$	0	0
$15$	−4.99846	−1.29060
$16$	0	0
$17$	−4.23825	−1.02793	−0.513963	−	0.857812i	$-0.671823\pi$
$17$	−4.23825	−1.02793	−0.513963	+	0.857812i	$0.671823\pi$
$18$	0	0
$19$	−7.16485	−1.64373	−0.821864	−	0.569683i	$-0.807066\pi$
$19$	−7.16485	−1.64373	−0.821864	+	0.569683i	$0.807066\pi$
$20$	0	0
$21$	4.67556	1.02029
$22$	0	0
$23$	−1.54199	−0.321527	−0.160764	−	0.986993i	$-0.551396\pi$
$23$	−1.54199	−0.321527	−0.160764	+	0.986993i	$0.551396\pi$
$24$	0	0
$25$	−1.27551	−0.255102
$26$	0	0
$27$	1.83421	0.352993
$28$	0	0
$29$	−3.54108	−0.657561	−0.328781	−	0.944406i	$-0.606638\pi$
$29$	−3.54108	−0.657561	−0.328781	+	0.944406i	$0.606638\pi$
$30$	0	0
$31$	−9.29782	−1.66994	−0.834969	−	0.550297i	$-0.814514\pi$
$31$	−9.29782	−1.66994	−0.834969	+	0.550297i	$0.814514\pi$
$32$	0	0
$33$	−4.45475	−0.775473
$34$	0	0
$35$	−3.48389	−0.588885
$36$	0	0
$37$	0.384158	0.0631551	0.0315776	−	0.999501i	$-0.489947\pi$
$37$	0.384158	0.0631551	0.0315776	+	0.999501i	$0.489947\pi$
$38$	0	0
$39$	0.968831	0.155137
$40$	0	0
$41$	4.15934	0.649580	0.324790	−	0.945786i	$-0.394706\pi$
$41$	4.15934	0.649580	0.324790	+	0.945786i	$0.394706\pi$
$42$	0	0
$43$	−2.72439	−0.415466	−0.207733	−	0.978186i	$-0.566609\pi$
$43$	−2.72439	−0.415466	−0.207733	+	0.978186i	$0.566609\pi$
$44$	0	0
$45$	−7.15640	−1.06681
$46$	0	0
$47$	0.802404	0.117043	0.0585213	−	0.998286i	$-0.481361\pi$
$47$	0.802404	0.117043	0.0585213	+	0.998286i	$0.481361\pi$
$48$	0	0
$49$	−3.74117	−0.534453
$50$	0	0
$51$	−10.9771	−1.53710
$52$	0	0
$53$	2.16015	0.296720	0.148360	−	0.988933i	$-0.452601\pi$
$53$	2.16015	0.296720	0.148360	+	0.988933i	$0.452601\pi$
$54$	0	0
$55$	3.31936	0.447583
$56$	0	0
$57$	−18.5571	−2.45794
$58$	0	0
$59$	1.88478	0.245377	0.122689	−	0.992445i	$-0.460848\pi$
$59$	1.88478	0.245377	0.122689	+	0.992445i	$0.460848\pi$
$60$	0	0
$61$	2.18668	0.279976	0.139988	−	0.990153i	$-0.455294\pi$
$61$	2.18668	0.279976	0.139988	+	0.990153i	$0.455294\pi$
$62$	0	0
$63$	6.69410	0.843377
$64$	0	0
$65$	−0.721903	−0.0895410
$66$	0	0
$67$	3.55803	0.434683	0.217341	−	0.976096i	$-0.430262\pi$
$67$	3.55803	0.434683	0.217341	+	0.976096i	$0.430262\pi$
$68$	0	0
$69$	−3.99378	−0.480795
$70$	0	0
$71$	7.85944	0.932744	0.466372	−	0.884589i	$-0.345561\pi$
$71$	7.85944	0.932744	0.466372	+	0.884589i	$0.345561\pi$
$72$	0	0
$73$	−9.55914	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$73$	−9.55914	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$74$	0	0
$75$	−3.30360	−0.381466
$76$	0	0
$77$	−3.10493	−0.353840
$78$	0	0
$79$	1.53948	0.173205	0.0866023	−	0.996243i	$-0.472399\pi$
$79$	1.53948	0.173205	0.0866023	+	0.996243i	$0.472399\pi$
$80$	0	0
$81$	−6.37392	−0.708214
$82$	0	0
$83$	6.07124	0.666405	0.333202	−	0.942855i	$-0.391871\pi$
$83$	6.07124	0.666405	0.333202	+	0.942855i	$0.391871\pi$
$84$	0	0
$85$	8.17936	0.887176
$86$	0	0
$87$	−9.17145	−0.983282
$88$	0	0
$89$	−8.62064	−0.913786	−0.456893	−	0.889522i	$-0.651038\pi$
$89$	−8.62064	−0.913786	−0.456893	+	0.889522i	$0.651038\pi$
$90$	0	0
$91$	0.675268	0.0707873
$92$	0	0
$93$	−24.0815	−2.49714
$94$	0	0
$95$	13.8274	1.41866
$96$	0	0
$97$	8.66932	0.880236	0.440118	−	0.897940i	$-0.354937\pi$
$97$	8.66932	0.880236	0.440118	+	0.897940i	$0.354937\pi$
$98$	0	0
$99$	−6.37797	−0.641010
$100$	0	0
$101$	5.05335	0.502827	0.251414	−	0.967880i	$-0.419105\pi$
$101$	5.05335	0.502827	0.251414	+	0.967880i	$0.419105\pi$
$102$	0	0
$103$	0.0988808	0.00974302	0.00487151	−	0.999988i	$-0.498449\pi$
$103$	0.0988808	0.00974302	0.00487151	+	0.999988i	$0.498449\pi$
$104$	0	0
$105$	−9.02333	−0.880587
$106$	0	0
$107$	9.76036	0.943570	0.471785	−	0.881714i	$-0.343610\pi$
$107$	9.76036	0.943570	0.471785	+	0.881714i	$0.343610\pi$
$108$	0	0
$109$	5.44751	0.521777	0.260888	−	0.965369i	$-0.415985\pi$
$109$	5.44751	0.521777	0.260888	+	0.965369i	$0.415985\pi$
$110$	0	0
$111$	0.994974	0.0944388
$112$	0	0
$113$	−8.74335	−0.822505	−0.411252	−	0.911522i	$-0.634909\pi$
$113$	−8.74335	−0.822505	−0.411252	+	0.911522i	$0.634909\pi$
$114$	0	0
$115$	2.97588	0.277502
$116$	0	0
$117$	1.38710	0.128237
$118$	0	0
$119$	−7.65098	−0.701364
$120$	0	0
$121$	−8.04170	−0.731063
$122$	0	0
$123$	10.7728	0.971347
$124$	0	0
$125$	12.1111	1.08325
$126$	0	0
$127$	15.2599	1.35410	0.677048	−	0.735939i	$-0.263259\pi$
$127$	15.2599	1.35410	0.677048	+	0.735939i	$0.263259\pi$
$128$	0	0
$129$	−7.05623	−0.621266
$130$	0	0
$131$	3.02367	0.264179	0.132090	−	0.991238i	$-0.457831\pi$
$131$	3.02367	0.264179	0.132090	+	0.991238i	$0.457831\pi$
$132$	0	0
$133$	−12.9342	−1.12153
$134$	0	0
$135$	−3.53983	−0.304660
$136$	0	0
$137$	−8.18863	−0.699602	−0.349801	−	0.936824i	$-0.613751\pi$
$137$	−8.18863	−0.699602	−0.349801	+	0.936824i	$0.613751\pi$
$138$	0	0
$139$	11.8333	1.00368	0.501842	−	0.864959i	$-0.332656\pi$
$139$	11.8333	1.00368	0.501842	+	0.864959i	$0.332656\pi$
$140$	0	0
$141$	2.07824	0.175019
$142$	0	0
$143$	−0.643379	−0.0538020
$144$	0	0
$145$	6.83390	0.567525
$146$	0	0
$147$	−9.68969	−0.799192
$148$	0	0
$149$	15.6474	1.28189	0.640944	−	0.767587i	$-0.278543\pi$
$149$	15.6474	1.28189	0.640944	+	0.767587i	$0.278543\pi$
$150$	0	0
$151$	−10.8430	−0.882393	−0.441196	−	0.897411i	$-0.645446\pi$
$151$	−10.8430	−0.882393	−0.441196	+	0.897411i	$0.645446\pi$
$152$	0	0
$153$	−15.7162	−1.27058
$154$	0	0
$155$	17.9438	1.44128
$156$	0	0
$157$	−15.8121	−1.26194	−0.630970	−	0.775807i	$-0.717343\pi$
$157$	−15.8121	−1.26194	−0.630970	+	0.775807i	$0.717343\pi$
$158$	0	0
$159$	5.59483	0.443699
$160$	0	0
$161$	−2.78364	−0.219381
$162$	0	0
$163$	−15.8337	−1.24019	−0.620095	−	0.784527i	$-0.712906\pi$
$163$	−15.8337	−1.24019	−0.620095	+	0.784527i	$0.712906\pi$
$164$	0	0
$165$	8.59720	0.669291
$166$	0	0
$167$	11.5871	0.896638	0.448319	−	0.893874i	$-0.352023\pi$
$167$	11.5871	0.896638	0.448319	+	0.893874i	$0.352023\pi$
$168$	0	0
$169$	−12.8601	−0.989237
$170$	0	0
$171$	−26.5686	−2.03175
$172$	0	0
$173$	6.04423	0.459534	0.229767	−	0.973246i	$-0.426204\pi$
$173$	6.04423	0.459534	0.229767	+	0.973246i	$0.426204\pi$
$174$	0	0
$175$	−2.30258	−0.174059
$176$	0	0
$177$	4.88161	0.366924
$178$	0	0
$179$	−2.21377	−0.165465	−0.0827323	−	0.996572i	$-0.526365\pi$
$179$	−2.21377	−0.165465	−0.0827323	+	0.996572i	$0.526365\pi$
$180$	0	0
$181$	−9.81657	−0.729660	−0.364830	−	0.931074i	$-0.618873\pi$
$181$	−9.81657	−0.729660	−0.364830	+	0.931074i	$0.618873\pi$
$182$	0	0
$183$	5.66353	0.418660
$184$	0	0
$185$	−0.741383	−0.0545076
$186$	0	0
$187$	7.28966	0.533073
$188$	0	0
$189$	3.31115	0.240851
$190$	0	0
$191$	0.0691517	0.00500364	0.00250182	−	0.999997i	$-0.499204\pi$
$191$	0.0691517	0.00500364	0.00250182	+	0.999997i	$0.499204\pi$
$192$	0	0
$193$	−14.8888	−1.07172	−0.535859	−	0.844308i	$-0.680012\pi$
$193$	−14.8888	−1.07172	−0.535859	+	0.844308i	$0.680012\pi$
$194$	0	0
$195$	−1.86974	−0.133895
$196$	0	0
$197$	5.19393	0.370053	0.185026	−	0.982734i	$-0.440763\pi$
$197$	5.19393	0.370053	0.185026	+	0.982734i	$0.440763\pi$
$198$	0	0
$199$	−14.3303	−1.01585	−0.507926	−	0.861401i	$-0.669588\pi$
$199$	−14.3303	−1.01585	−0.507926	+	0.861401i	$0.669588\pi$
$200$	0	0
$201$	9.21535	0.650001
$202$	0	0
$203$	−6.39243	−0.448661
$204$	0	0
$205$	−8.02708	−0.560636
$206$	0	0
$207$	−5.71799	−0.397428
$208$	0	0
$209$	12.3233	0.852423
$210$	0	0
$211$	−2.75100	−0.189387	−0.0946933	−	0.995506i	$-0.530187\pi$
$211$	−2.75100	−0.189387	−0.0946933	+	0.995506i	$0.530187\pi$
$212$	0	0
$213$	20.3561	1.39478
$214$	0	0
$215$	5.25779	0.358578
$216$	0	0
$217$	−16.7846	−1.13942
$218$	0	0
$219$	−24.7583	−1.67301
$220$	0	0
$221$	−1.58537	−0.106644
$222$	0	0
$223$	1.69621	0.113587	0.0567933	−	0.998386i	$-0.481912\pi$
$223$	1.69621	0.113587	0.0567933	+	0.998386i	$0.481912\pi$
$224$	0	0
$225$	−4.72983	−0.315322
$226$	0	0
$227$	−4.98493	−0.330861	−0.165431	−	0.986221i	$-0.552901\pi$
$227$	−4.98493	−0.330861	−0.165431	+	0.986221i	$0.552901\pi$
$228$	0	0
$229$	25.0550	1.65568	0.827841	−	0.560963i	$-0.189569\pi$
$229$	25.0550	1.65568	0.827841	+	0.560963i	$0.189569\pi$
$230$	0	0
$231$	−8.04183	−0.529113
$232$	0	0
$233$	−10.1650	−0.665932	−0.332966	−	0.942939i	$-0.608049\pi$
$233$	−10.1650	−0.665932	−0.332966	+	0.942939i	$0.608049\pi$
$234$	0	0
$235$	−1.54855	−0.101017
$236$	0	0
$237$	3.98727	0.259001
$238$	0	0
$239$	3.06612	0.198331	0.0991654	−	0.995071i	$-0.468383\pi$
$239$	3.06612	0.198331	0.0991654	+	0.995071i	$0.468383\pi$
$240$	0	0
$241$	−17.7915	−1.14605	−0.573025	−	0.819538i	$-0.694230\pi$
$241$	−17.7915	−1.14605	−0.573025	+	0.819538i	$0.694230\pi$
$242$	0	0
$243$	−22.0112	−1.41202
$244$	0	0
$245$	7.22006	0.461272
$246$	0	0
$247$	−2.68011	−0.170531
$248$	0	0
$249$	15.7246	0.996506
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	2.65218	0.166741
$254$	0	0
$255$	21.1847	1.32664
$256$	0	0
$257$	−14.4178	−0.899359	−0.449679	−	0.893190i	$-0.648462\pi$
$257$	−14.4178	−0.899359	−0.449679	+	0.893190i	$0.648462\pi$
$258$	0	0
$259$	0.693490	0.0430914
$260$	0	0
$261$	−13.1310	−0.812786
$262$	0	0
$263$	17.2426	1.06323	0.531614	−	0.846987i	$-0.321586\pi$
$263$	17.2426	1.06323	0.531614	+	0.846987i	$0.321586\pi$
$264$	0	0
$265$	−4.16886	−0.256091
$266$	0	0
$267$	−22.3276	−1.36643
$268$	0	0
$269$	9.23304	0.562948	0.281474	−	0.959569i	$-0.409177\pi$
$269$	9.23304	0.562948	0.281474	+	0.959569i	$0.409177\pi$
$270$	0	0
$271$	−1.03078	−0.0626152	−0.0313076	−	0.999510i	$-0.509967\pi$
$271$	−1.03078	−0.0626152	−0.0313076	+	0.999510i	$0.509967\pi$
$272$	0	0
$273$	1.74896	0.105852
$274$	0	0
$275$	2.19384	0.132294
$276$	0	0
$277$	−13.4554	−0.808454	−0.404227	−	0.914659i	$-0.632459\pi$
$277$	−13.4554	−0.808454	−0.404227	+	0.914659i	$0.632459\pi$
$278$	0	0
$279$	−34.4780	−2.06415
$280$	0	0
$281$	−31.3775	−1.87183	−0.935913	−	0.352231i	$-0.885423\pi$
$281$	−31.3775	−1.87183	−0.935913	+	0.352231i	$0.885423\pi$
$282$	0	0
$283$	−24.0833	−1.43161	−0.715803	−	0.698302i	$-0.753939\pi$
$283$	−24.0833	−1.43161	−0.715803	+	0.698302i	$0.753939\pi$
$284$	0	0
$285$	35.8132	2.12139
$286$	0	0
$287$	7.50854	0.443215
$288$	0	0
$289$	0.962726	0.0566309
$290$	0	0
$291$	22.4537	1.31626
$292$	0	0
$293$	−21.3046	−1.24463	−0.622315	−	0.782767i	$-0.713808\pi$
$293$	−21.3046	−1.24463	−0.622315	+	0.782767i	$0.713808\pi$
$294$	0	0
$295$	−3.63742	−0.211779
$296$	0	0
$297$	−3.15478	−0.183059
$298$	0	0
$299$	−0.576803	−0.0333574
$300$	0	0
$301$	−4.91814	−0.283477
$302$	0	0
$303$	13.0883	0.751901
$304$	0	0
$305$	−4.22006	−0.241640
$306$	0	0
$307$	−25.7102	−1.46736	−0.733679	−	0.679497i	$-0.762198\pi$
$307$	−25.7102	−1.46736	−0.733679	+	0.679497i	$0.762198\pi$
$308$	0	0
$309$	0.256103	0.0145692
$310$	0	0
$311$	10.0291	0.568700	0.284350	−	0.958721i	$-0.408222\pi$
$311$	10.0291	0.568700	0.284350	+	0.958721i	$0.408222\pi$
$312$	0	0
$313$	23.8474	1.34793	0.673967	−	0.738761i	$-0.264589\pi$
$313$	23.8474	1.34793	0.673967	+	0.738761i	$0.264589\pi$
$314$	0	0
$315$	−12.9189	−0.727898
$316$	0	0
$317$	−14.2235	−0.798874	−0.399437	−	0.916761i	$-0.630794\pi$
$317$	−14.2235	−0.798874	−0.399437	+	0.916761i	$0.630794\pi$
$318$	0	0
$319$	6.09055	0.341005
$320$	0	0
$321$	25.2795	1.41096
$322$	0	0
$323$	30.3664	1.68963
$324$	0	0
$325$	−0.477122	−0.0264660
$326$	0	0
$327$	14.1091	0.780237
$328$	0	0
$329$	1.44852	0.0798594
$330$	0	0
$331$	24.5228	1.34789	0.673947	−	0.738780i	$-0.264598\pi$
$331$	24.5228	1.34789	0.673947	+	0.738780i	$0.264598\pi$
$332$	0	0
$333$	1.42453	0.0780636
$334$	0	0
$335$	−6.86662	−0.375163
$336$	0	0
$337$	13.6880	0.745630	0.372815	−	0.927906i	$-0.378393\pi$
$337$	13.6880	0.745630	0.372815	+	0.927906i	$0.378393\pi$
$338$	0	0
$339$	−22.6454	−1.22993
$340$	0	0
$341$	15.9920	0.866014
$342$	0	0
$343$	−19.3902	−1.04697
$344$	0	0
$345$	7.70757	0.414962
$346$	0	0
$347$	−0.544927	−0.0292532	−0.0146266	−	0.999893i	$-0.504656\pi$
$347$	−0.544927	−0.0292532	−0.0146266	+	0.999893i	$0.504656\pi$
$348$	0	0
$349$	19.6127	1.04985	0.524923	−	0.851150i	$-0.324094\pi$
$349$	19.6127	1.04985	0.524923	+	0.851150i	$0.324094\pi$
$350$	0	0
$351$	0.686110	0.0366219
$352$	0	0
$353$	21.0762	1.12177	0.560887	−	0.827892i	$-0.310460\pi$
$353$	21.0762	1.12177	0.560887	+	0.827892i	$0.310460\pi$
$354$	0	0
$355$	−15.1679	−0.805027
$356$	0	0
$357$	−19.8162	−1.04878
$358$	0	0
$359$	9.36411	0.494219	0.247109	−	0.968988i	$-0.420519\pi$
$359$	9.36411	0.494219	0.247109	+	0.968988i	$0.420519\pi$
$360$	0	0
$361$	32.3350	1.70184
$362$	0	0
$363$	−20.8281	−1.09319
$364$	0	0
$365$	18.4481	0.965618
$366$	0	0
$367$	18.8366	0.983261	0.491631	−	0.870804i	$-0.336401\pi$
$367$	18.8366	0.983261	0.491631	+	0.870804i	$0.336401\pi$
$368$	0	0
$369$	15.4236	0.802921
$370$	0	0
$371$	3.89956	0.202455
$372$	0	0
$373$	5.86190	0.303518	0.151759	−	0.988418i	$-0.451506\pi$
$373$	5.86190	0.303518	0.151759	+	0.988418i	$0.451506\pi$
$374$	0	0
$375$	31.3679	1.61983
$376$	0	0
$377$	−1.32459	−0.0682197
$378$	0	0
$379$	12.9288	0.664106	0.332053	−	0.943261i	$-0.392259\pi$
$379$	12.9288	0.664106	0.332053	+	0.943261i	$0.392259\pi$
$380$	0	0
$381$	39.5233	2.02484
$382$	0	0
$383$	28.3845	1.45038	0.725190	−	0.688548i	$-0.241752\pi$
$383$	28.3845	1.45038	0.725190	+	0.688548i	$0.241752\pi$
$384$	0	0
$385$	5.99219	0.305390
$386$	0	0
$387$	−10.1026	−0.513542
$388$	0	0
$389$	−16.7839	−0.850975	−0.425488	−	0.904964i	$-0.639897\pi$
$389$	−16.7839	−0.850975	−0.425488	+	0.904964i	$0.639897\pi$
$390$	0	0
$391$	6.53534	0.330506
$392$	0	0
$393$	7.83136	0.395040
$394$	0	0
$395$	−2.97102	−0.149488
$396$	0	0
$397$	11.0036	0.552253	0.276127	−	0.961121i	$-0.410949\pi$
$397$	11.0036	0.552253	0.276127	+	0.961121i	$0.410949\pi$
$398$	0	0
$399$	−33.4997	−1.67708
$400$	0	0
$401$	35.5126	1.77342	0.886708	−	0.462329i	$-0.152986\pi$
$401$	35.5126	1.77342	0.886708	+	0.462329i	$0.152986\pi$
$402$	0	0
$403$	−3.47798	−0.173250
$404$	0	0
$405$	12.3010	0.611241
$406$	0	0
$407$	−0.660740	−0.0327517
$408$	0	0
$409$	1.59566	0.0789005	0.0394502	−	0.999222i	$-0.487439\pi$
$409$	1.59566	0.0789005	0.0394502	+	0.999222i	$0.487439\pi$
$410$	0	0
$411$	−21.2087	−1.04615
$412$	0	0
$413$	3.40245	0.167423
$414$	0	0
$415$	−11.7168	−0.575157
$416$	0	0
$417$	30.6484	1.50086
$418$	0	0
$419$	35.7280	1.74543	0.872714	−	0.488232i	$-0.162358\pi$
$419$	35.7280	1.74543	0.872714	+	0.488232i	$0.162358\pi$
$420$	0	0
$421$	−30.1600	−1.46991	−0.734955	−	0.678116i	$-0.762797\pi$
$421$	−30.1600	−1.46991	−0.734955	+	0.678116i	$0.762797\pi$
$422$	0	0
$423$	2.97546	0.144672
$424$	0	0
$425$	5.40593	0.262226
$426$	0	0
$427$	3.94744	0.191030
$428$	0	0
$429$	−1.66636	−0.0804527
$430$	0	0
$431$	−26.2899	−1.26634	−0.633170	−	0.774013i	$-0.718246\pi$
$431$	−26.2899	−1.26634	−0.633170	+	0.774013i	$0.718246\pi$
$432$	0	0
$433$	−13.8139	−0.663852	−0.331926	−	0.943305i	$-0.607698\pi$
$433$	−13.8139	−0.663852	−0.331926	+	0.943305i	$0.607698\pi$
$434$	0	0
$435$	17.6999	0.848646
$436$	0	0
$437$	11.0481	0.528504
$438$	0	0
$439$	6.57503	0.313809	0.156904	−	0.987614i	$-0.449849\pi$
$439$	6.57503	0.313809	0.156904	+	0.987614i	$0.449849\pi$
$440$	0	0
$441$	−13.8729	−0.660616
$442$	0	0
$443$	2.88621	0.137128	0.0685640	−	0.997647i	$-0.478158\pi$
$443$	2.88621	0.137128	0.0685640	+	0.997647i	$0.478158\pi$
$444$	0	0
$445$	16.6369	0.788665
$446$	0	0
$447$	40.5271	1.91687
$448$	0	0
$449$	2.26057	0.106683	0.0533415	−	0.998576i	$-0.483013\pi$
$449$	2.26057	0.106683	0.0533415	+	0.998576i	$0.483013\pi$
$450$	0	0
$451$	−7.15395	−0.336866
$452$	0	0
$453$	−28.0836	−1.31948
$454$	0	0
$455$	−1.30320	−0.0610948
$456$	0	0
$457$	−9.34162	−0.436983	−0.218491	−	0.975839i	$-0.570114\pi$
$457$	−9.34162	−0.436983	−0.218491	+	0.975839i	$0.570114\pi$
$458$	0	0
$459$	−7.77382	−0.362851
$460$	0	0
$461$	−15.2983	−0.712514	−0.356257	−	0.934388i	$-0.615947\pi$
$461$	−15.2983	−0.712514	−0.356257	+	0.934388i	$0.615947\pi$
$462$	0	0
$463$	−37.5529	−1.74523	−0.872615	−	0.488408i	$-0.837578\pi$
$463$	−37.5529	−1.74523	−0.872615	+	0.488408i	$0.837578\pi$
$464$	0	0
$465$	46.4747	2.15521
$466$	0	0
$467$	16.4302	0.760301	0.380150	−	0.924925i	$-0.375872\pi$
$467$	16.4302	0.760301	0.380150	+	0.924925i	$0.375872\pi$
$468$	0	0
$469$	6.42304	0.296588
$470$	0	0
$471$	−40.9535	−1.88704
$472$	0	0
$473$	4.68588	0.215457
$474$	0	0
$475$	9.13885	0.419319
$476$	0	0
$477$	8.01024	0.366764
$478$	0	0
$479$	−13.3764	−0.611183	−0.305591	−	0.952163i	$-0.598854\pi$
$479$	−13.3764	−0.611183	−0.305591	+	0.952163i	$0.598854\pi$
$480$	0	0
$481$	0.143699	0.00655213
$482$	0	0
$483$	−7.20967	−0.328051
$484$	0	0
$485$	−16.7309	−0.759709
$486$	0	0
$487$	−21.4221	−0.970727	−0.485363	−	0.874312i	$-0.661313\pi$
$487$	−21.4221	−0.970727	−0.485363	+	0.874312i	$0.661313\pi$
$488$	0	0
$489$	−41.0095	−1.85451
$490$	0	0
$491$	−6.56785	−0.296403	−0.148202	−	0.988957i	$-0.547348\pi$
$491$	−6.56785	−0.296403	−0.148202	+	0.988957i	$0.547348\pi$
$492$	0	0
$493$	15.0080	0.675924
$494$	0	0
$495$	12.3088	0.553240
$496$	0	0
$497$	14.1880	0.636421
$498$	0	0
$499$	−11.8788	−0.531767	−0.265884	−	0.964005i	$-0.585664\pi$
$499$	−11.8788	−0.531767	−0.265884	+	0.964005i	$0.585664\pi$
$500$	0	0
$501$	30.0108	1.34078
$502$	0	0
$503$	5.34097	0.238142	0.119071	−	0.992886i	$-0.462008\pi$
$503$	5.34097	0.238142	0.119071	+	0.992886i	$0.462008\pi$
$504$	0	0
$505$	−9.75243	−0.433977
$506$	0	0
$507$	−33.3078	−1.47925
$508$	0	0
$509$	3.87273	0.171656	0.0858278	−	0.996310i	$-0.472647\pi$
$509$	3.87273	0.171656	0.0858278	+	0.996310i	$0.472647\pi$
$510$	0	0
$511$	−17.2564	−0.763377
$512$	0	0
$513$	−13.1418	−0.580225
$514$	0	0
$515$	−0.190829	−0.00840895
$516$	0	0
$517$	−1.38011	−0.0606972
$518$	0	0
$519$	15.6546	0.687163
$520$	0	0
$521$	−17.0308	−0.746134	−0.373067	−	0.927804i	$-0.621694\pi$
$521$	−17.0308	−0.746134	−0.373067	+	0.927804i	$0.621694\pi$
$522$	0	0
$523$	8.35122	0.365173	0.182587	−	0.983190i	$-0.441553\pi$
$523$	8.35122	0.365173	0.182587	+	0.983190i	$0.441553\pi$
$524$	0	0
$525$	−5.96373	−0.260278
$526$	0	0
$527$	39.4064	1.71657
$528$	0	0
$529$	−20.6223	−0.896620
$530$	0	0
$531$	6.98911	0.303301
$532$	0	0
$533$	1.55586	0.0673917
$534$	0	0
$535$	−18.8365	−0.814371
$536$	0	0
$537$	−5.73369	−0.247427
$538$	0	0
$539$	6.43470	0.277162
$540$	0	0
$541$	−9.89386	−0.425370	−0.212685	−	0.977121i	$-0.568221\pi$
$541$	−9.89386	−0.425370	−0.212685	+	0.977121i	$0.568221\pi$
$542$	0	0
$543$	−25.4251	−1.09109
$544$	0	0
$545$	−10.5131	−0.450332
$546$	0	0
$547$	34.8195	1.48878	0.744388	−	0.667748i	$-0.232741\pi$
$547$	34.8195	1.48878	0.744388	+	0.667748i	$0.232741\pi$
$548$	0	0
$549$	8.10861	0.346067
$550$	0	0
$551$	25.3713	1.08085
$552$	0	0
$553$	2.77910	0.118179
$554$	0	0
$555$	−1.92019	−0.0815077
$556$	0	0
$557$	−33.4141	−1.41580	−0.707900	−	0.706313i	$-0.750357\pi$
$557$	−33.4141	−1.41580	−0.707900	+	0.706313i	$0.750357\pi$
$558$	0	0
$559$	−1.01910	−0.0431032
$560$	0	0
$561$	18.8803	0.797129
$562$	0	0
$563$	−6.88248	−0.290062	−0.145031	−	0.989427i	$-0.546328\pi$
$563$	−6.88248	−0.290062	−0.145031	+	0.989427i	$0.546328\pi$
$564$	0	0
$565$	16.8737	0.709883
$566$	0	0
$567$	−11.5064	−0.483222
$568$	0	0
$569$	−12.7343	−0.533849	−0.266925	−	0.963717i	$-0.586008\pi$
$569$	−12.7343	−0.533849	−0.266925	+	0.963717i	$0.586008\pi$
$570$	0	0
$571$	−27.1224	−1.13504	−0.567520	−	0.823360i	$-0.692097\pi$
$571$	−27.1224	−1.13504	−0.567520	+	0.823360i	$0.692097\pi$
$572$	0	0
$573$	0.179104	0.00748218
$574$	0	0
$575$	1.96683	0.0820224
$576$	0	0
$577$	25.0219	1.04168	0.520839	−	0.853655i	$-0.325619\pi$
$577$	25.0219	1.04168	0.520839	+	0.853655i	$0.325619\pi$
$578$	0	0
$579$	−38.5622	−1.60259
$580$	0	0
$581$	10.9599	0.454695
$582$	0	0
$583$	−3.71540	−0.153876
$584$	0	0
$585$	−2.67695	−0.110678
$586$	0	0
$587$	−19.0449	−0.786065	−0.393033	−	0.919525i	$-0.628574\pi$
$587$	−19.0449	−0.786065	−0.393033	+	0.919525i	$0.628574\pi$
$588$	0	0
$589$	66.6175	2.74492
$590$	0	0
$591$	13.4524	0.553357
$592$	0	0
$593$	13.9954	0.574722	0.287361	−	0.957822i	$-0.407222\pi$
$593$	13.9954	0.574722	0.287361	+	0.957822i	$0.407222\pi$
$594$	0	0
$595$	14.7656	0.605330
$596$	0	0
$597$	−37.1158	−1.51905
$598$	0	0
$599$	38.6886	1.58077	0.790386	−	0.612609i	$-0.209880\pi$
$599$	38.6886	1.58077	0.790386	+	0.612609i	$0.209880\pi$
$600$	0	0
$601$	−2.71351	−0.110686	−0.0553432	−	0.998467i	$-0.517625\pi$
$601$	−2.71351	−0.110686	−0.0553432	+	0.998467i	$0.517625\pi$
$602$	0	0
$603$	13.1938	0.537294
$604$	0	0
$605$	15.5196	0.630962
$606$	0	0
$607$	−15.3770	−0.624134	−0.312067	−	0.950060i	$-0.601021\pi$
$607$	−15.3770	−0.624134	−0.312067	+	0.950060i	$0.601021\pi$
$608$	0	0
$609$	−16.5565	−0.670904
$610$	0	0
$611$	0.300150	0.0121428
$612$	0	0
$613$	1.93486	0.0781483	0.0390742	−	0.999236i	$-0.487559\pi$
$613$	1.93486	0.0781483	0.0390742	+	0.999236i	$0.487559\pi$
$614$	0	0
$615$	−20.7903	−0.838345
$616$	0	0
$617$	−13.9249	−0.560596	−0.280298	−	0.959913i	$-0.590433\pi$
$617$	−13.9249	−0.560596	−0.280298	+	0.959913i	$0.590433\pi$
$618$	0	0
$619$	−4.98650	−0.200424	−0.100212	−	0.994966i	$-0.531952\pi$
$619$	−4.98650	−0.200424	−0.100212	+	0.994966i	$0.531952\pi$
$620$	0	0
$621$	−2.82833	−0.113497
$622$	0	0
$623$	−15.5622	−0.623485
$624$	0	0
$625$	−16.9955	−0.679820
$626$	0	0
$627$	31.9176	1.27467
$628$	0	0
$629$	−1.62815	−0.0649188
$630$	0	0
$631$	22.4678	0.894429	0.447215	−	0.894427i	$-0.352416\pi$
$631$	22.4678	0.894429	0.447215	+	0.894427i	$0.352416\pi$
$632$	0	0
$633$	−7.12513	−0.283198
$634$	0	0
$635$	−29.4499	−1.16869
$636$	0	0
$637$	−1.39943	−0.0554476
$638$	0	0
$639$	29.1442	1.15293
$640$	0	0
$641$	−4.11043	−0.162352	−0.0811761	−	0.996700i	$-0.525868\pi$
$641$	−4.11043	−0.162352	−0.0811761	+	0.996700i	$0.525868\pi$
$642$	0	0
$643$	36.4350	1.43686	0.718429	−	0.695601i	$-0.244862\pi$
$643$	36.4350	1.43686	0.718429	+	0.695601i	$0.244862\pi$
$644$	0	0
$645$	13.6178	0.536199
$646$	0	0
$647$	12.7115	0.499741	0.249870	−	0.968279i	$-0.419612\pi$
$647$	12.7115	0.499741	0.249870	+	0.968279i	$0.419612\pi$
$648$	0	0
$649$	−3.24177	−0.127250
$650$	0	0
$651$	−43.4725	−1.70382
$652$	0	0
$653$	−4.17541	−0.163396	−0.0816981	−	0.996657i	$-0.526034\pi$
$653$	−4.17541	−0.163396	−0.0816981	+	0.996657i	$0.526034\pi$
$654$	0	0
$655$	−5.83536	−0.228007
$656$	0	0
$657$	−35.4470	−1.38292
$658$	0	0
$659$	36.4179	1.41864	0.709319	−	0.704887i	$-0.249002\pi$
$659$	36.4179	1.41864	0.709319	+	0.704887i	$0.249002\pi$
$660$	0	0
$661$	4.32320	0.168153	0.0840764	−	0.996459i	$-0.473206\pi$
$661$	4.32320	0.168153	0.0840764	+	0.996459i	$0.473206\pi$
$662$	0	0
$663$	−4.10614	−0.159469
$664$	0	0
$665$	24.9615	0.967967
$666$	0	0
$667$	5.46031	0.211424
$668$	0	0
$669$	4.39321	0.169851
$670$	0	0
$671$	−3.76102	−0.145193
$672$	0	0
$673$	−36.6004	−1.41084	−0.705421	−	0.708788i	$-0.749242\pi$
$673$	−36.6004	−1.41084	−0.705421	+	0.708788i	$0.749242\pi$
$674$	0	0
$675$	−2.33955	−0.0900494
$676$	0	0
$677$	−43.2549	−1.66242	−0.831211	−	0.555957i	$-0.812352\pi$
$677$	−43.2549	−1.66242	−0.831211	+	0.555957i	$0.812352\pi$
$678$	0	0
$679$	15.6501	0.600594
$680$	0	0
$681$	−12.9110	−0.494752
$682$	0	0
$683$	25.6169	0.980202	0.490101	−	0.871666i	$-0.336960\pi$
$683$	25.6169	0.980202	0.490101	+	0.871666i	$0.336960\pi$
$684$	0	0
$685$	15.8032	0.603809
$686$	0	0
$687$	64.8929	2.47582
$688$	0	0
$689$	0.808034	0.0307836
$690$	0	0
$691$	21.1381	0.804131	0.402065	−	0.915611i	$-0.368292\pi$
$691$	21.1381	0.804131	0.402065	+	0.915611i	$0.368292\pi$
$692$	0	0
$693$	−11.5137	−0.437368
$694$	0	0
$695$	−22.8369	−0.866255
$696$	0	0
$697$	−17.6283	−0.667720
$698$	0	0
$699$	−26.3275	−0.995799
$700$	0	0
$701$	−40.3180	−1.52279	−0.761396	−	0.648288i	$-0.775485\pi$
$701$	−40.3180	−1.52279	−0.761396	+	0.648288i	$0.775485\pi$
$702$	0	0
$703$	−2.75243	−0.103810
$704$	0	0
$705$	−4.01078	−0.151055
$706$	0	0
$707$	9.12243	0.343084
$708$	0	0
$709$	−42.6151	−1.60044	−0.800222	−	0.599704i	$-0.795285\pi$
$709$	−42.6151	−1.60044	−0.800222	+	0.599704i	$0.795285\pi$
$710$	0	0
$711$	5.70866	0.214091
$712$	0	0
$713$	14.3372	0.536931
$714$	0	0
$715$	1.24165	0.0464352
$716$	0	0
$717$	7.94130	0.296573
$718$	0	0
$719$	−12.8224	−0.478193	−0.239097	−	0.970996i	$-0.576851\pi$
$719$	−12.8224	−0.478193	−0.239097	+	0.970996i	$0.576851\pi$
$720$	0	0
$721$	0.178502	0.00664776
$722$	0	0
$723$	−46.0802	−1.71374
$724$	0	0
$725$	4.51668	0.167745
$726$	0	0
$727$	−36.8887	−1.36812	−0.684062	−	0.729424i	$-0.739788\pi$
$727$	−36.8887	−1.36812	−0.684062	+	0.729424i	$0.739788\pi$
$728$	0	0
$729$	−37.8876	−1.40324
$730$	0	0
$731$	11.5467	0.427068
$732$	0	0
$733$	11.4159	0.421657	0.210829	−	0.977523i	$-0.432384\pi$
$733$	11.4159	0.421657	0.210829	+	0.977523i	$0.432384\pi$
$734$	0	0
$735$	18.7001	0.689762
$736$	0	0
$737$	−6.11971	−0.225422
$738$	0	0
$739$	43.0415	1.58331	0.791654	−	0.610970i	$-0.209220\pi$
$739$	43.0415	1.58331	0.791654	+	0.610970i	$0.209220\pi$
$740$	0	0
$741$	−6.94152	−0.255003
$742$	0	0
$743$	−31.8141	−1.16715	−0.583573	−	0.812061i	$-0.698346\pi$
$743$	−31.8141	−1.16715	−0.583573	+	0.812061i	$0.698346\pi$
$744$	0	0
$745$	−30.1979	−1.10637
$746$	0	0
$747$	22.5133	0.823717
$748$	0	0
$749$	17.6196	0.643807
$750$	0	0
$751$	0.793237	0.0289456	0.0144728	−	0.999895i	$-0.495393\pi$
$751$	0.793237	0.0289456	0.0144728	+	0.999895i	$0.495393\pi$
$752$	0	0
$753$	−2.59002	−0.0943854
$754$	0	0
$755$	20.9259	0.761571
$756$	0	0
$757$	−31.6564	−1.15057	−0.575286	−	0.817952i	$-0.695109\pi$
$757$	−31.6564	−1.15057	−0.575286	+	0.817952i	$0.695109\pi$
$758$	0	0
$759$	6.86919	0.249336
$760$	0	0
$761$	−0.512420	−0.0185752	−0.00928760	−	0.999957i	$-0.502956\pi$
$761$	−0.512420	−0.0185752	−0.00928760	+	0.999957i	$0.502956\pi$
$762$	0	0
$763$	9.83397	0.356014
$764$	0	0
$765$	30.3306	1.09660
$766$	0	0
$767$	0.705027	0.0254571
$768$	0	0
$769$	43.1196	1.55493	0.777466	−	0.628925i	$-0.216505\pi$
$769$	43.1196	1.55493	0.777466	+	0.628925i	$0.216505\pi$
$770$	0	0
$771$	−37.3424	−1.34485
$772$	0	0
$773$	24.2776	0.873203	0.436602	−	0.899655i	$-0.356182\pi$
$773$	24.2776	0.873203	0.436602	+	0.899655i	$0.356182\pi$
$774$	0	0
$775$	11.8595	0.426005
$776$	0	0
$777$	1.79615	0.0644365
$778$	0	0
$779$	−29.8010	−1.06773
$780$	0	0
$781$	−13.5180	−0.483712
$782$	0	0
$783$	−6.49507	−0.232115
$784$	0	0
$785$	30.5156	1.08915
$786$	0	0
$787$	−4.18900	−0.149322	−0.0746609	−	0.997209i	$-0.523787\pi$
$787$	−4.18900	−0.149322	−0.0746609	+	0.997209i	$0.523787\pi$
$788$	0	0
$789$	44.6587	1.58989
$790$	0	0
$791$	−15.7837	−0.561203
$792$	0	0
$793$	0.817957	0.0290465
$794$	0	0
$795$	−10.7974	−0.382945
$796$	0	0
$797$	−32.9316	−1.16650	−0.583248	−	0.812294i	$-0.698218\pi$
$797$	−32.9316	−1.16650	−0.583248	+	0.812294i	$0.698218\pi$
$798$	0	0
$799$	−3.40078	−0.120311
$800$	0	0
$801$	−31.9669	−1.12949
$802$	0	0
$803$	16.4414	0.580206
$804$	0	0
$805$	5.37212	0.189343
$806$	0	0
$807$	23.9137	0.841803
$808$	0	0
$809$	−45.7440	−1.60827	−0.804136	−	0.594445i	$-0.797372\pi$
$809$	−45.7440	−1.60827	−0.804136	+	0.594445i	$0.797372\pi$
$810$	0	0
$811$	−28.2035	−0.990358	−0.495179	−	0.868791i	$-0.664898\pi$
$811$	−28.2035	−0.990358	−0.495179	+	0.868791i	$0.664898\pi$
$812$	0	0
$813$	−2.66973	−0.0936314
$814$	0	0
$815$	30.5573	1.07038
$816$	0	0
$817$	19.5199	0.682914
$818$	0	0
$819$	2.50402	0.0874975
$820$	0	0
$821$	25.5799	0.892744	0.446372	−	0.894847i	$-0.352716\pi$
$821$	25.5799	0.892744	0.446372	+	0.894847i	$0.352716\pi$
$822$	0	0
$823$	−48.1446	−1.67822	−0.839108	−	0.543964i	$-0.816923\pi$
$823$	−48.1446	−1.67822	−0.839108	+	0.543964i	$0.816923\pi$
$824$	0	0
$825$	5.68209	0.197825
$826$	0	0
$827$	33.7768	1.17453	0.587267	−	0.809393i	$-0.300204\pi$
$827$	33.7768	1.17453	0.587267	+	0.809393i	$0.300204\pi$
$828$	0	0
$829$	19.6942	0.684008	0.342004	−	0.939698i	$-0.388894\pi$
$829$	19.6942	0.684008	0.342004	+	0.939698i	$0.388894\pi$
$830$	0	0
$831$	−34.8496	−1.20892
$832$	0	0
$833$	15.8560	0.549378
$834$	0	0
$835$	−22.3619	−0.773865
$836$	0	0
$837$	−17.0541	−0.589477
$838$	0	0
$839$	−16.4005	−0.566209	−0.283104	−	0.959089i	$-0.591364\pi$
$839$	−16.4005	−0.566209	−0.283104	+	0.959089i	$0.591364\pi$
$840$	0	0
$841$	−16.4608	−0.567613
$842$	0	0
$843$	−81.2683	−2.79903
$844$	0	0
$845$	24.8186	0.853785
$846$	0	0
$847$	−14.5171	−0.498812
$848$	0	0
$849$	−62.3762	−2.14075
$850$	0	0
$851$	−0.592367	−0.0203061
$852$	0	0
$853$	3.76233	0.128820	0.0644098	−	0.997924i	$-0.479484\pi$
$853$	3.76233	0.128820	0.0644098	+	0.997924i	$0.479484\pi$
$854$	0	0
$855$	51.2745	1.75355
$856$	0	0
$857$	−1.71368	−0.0585382	−0.0292691	−	0.999572i	$-0.509318\pi$
$857$	−1.71368	−0.0585382	−0.0292691	+	0.999572i	$0.509318\pi$
$858$	0	0
$859$	33.7551	1.15171	0.575855	−	0.817552i	$-0.304669\pi$
$859$	33.7551	1.15171	0.575855	+	0.817552i	$0.304669\pi$
$860$	0	0
$861$	19.4472	0.662760
$862$	0	0
$863$	16.3956	0.558113	0.279056	−	0.960275i	$-0.409978\pi$
$863$	16.3956	0.558113	0.279056	+	0.960275i	$0.409978\pi$
$864$	0	0
$865$	−11.6647	−0.396612
$866$	0	0
$867$	2.49348	0.0846829
$868$	0	0
$869$	−2.64785	−0.0898223
$870$	0	0
$871$	1.33093	0.0450968
$872$	0	0
$873$	32.1474	1.08803
$874$	0	0
$875$	21.8632	0.739111
$876$	0	0
$877$	3.50085	0.118215	0.0591076	−	0.998252i	$-0.481174\pi$
$877$	3.50085	0.118215	0.0591076	+	0.998252i	$0.481174\pi$
$878$	0	0
$879$	−55.1793	−1.86115
$880$	0	0
$881$	−53.2976	−1.79564	−0.897821	−	0.440360i	$-0.854851\pi$
$881$	−53.2976	−1.79564	−0.897821	+	0.440360i	$0.854851\pi$
$882$	0	0
$883$	−23.5606	−0.792877	−0.396438	−	0.918061i	$-0.629754\pi$
$883$	−23.5606	−0.792877	−0.396438	+	0.918061i	$0.629754\pi$
$884$	0	0
$885$	−9.42098	−0.316683
$886$	0	0
$887$	48.8312	1.63959	0.819796	−	0.572656i	$-0.194087\pi$
$887$	48.8312	1.63959	0.819796	+	0.572656i	$0.194087\pi$
$888$	0	0
$889$	27.5475	0.923913
$890$	0	0
$891$	10.9630	0.367273
$892$	0	0
$893$	−5.74910	−0.192386
$894$	0	0
$895$	4.27233	0.142808
$896$	0	0
$897$	−1.49393	−0.0498808
$898$	0	0
$899$	32.9243	1.09809
$900$	0	0
$901$	−9.15525	−0.305006
$902$	0	0
$903$	−12.7381	−0.423896
$904$	0	0
$905$	18.9449	0.629751
$906$	0	0
$907$	38.4195	1.27570	0.637850	−	0.770161i	$-0.279824\pi$
$907$	38.4195	1.27570	0.637850	+	0.770161i	$0.279824\pi$
$908$	0	0
$909$	18.7388	0.621525
$910$	0	0
$911$	−28.6116	−0.947944	−0.473972	−	0.880540i	$-0.657180\pi$
$911$	−28.6116	−0.947944	−0.473972	+	0.880540i	$0.657180\pi$
$912$	0	0
$913$	−10.4424	−0.345592
$914$	0	0
$915$	−10.9300	−0.361335
$916$	0	0
$917$	5.45840	0.180252
$918$	0	0
$919$	14.0853	0.464632	0.232316	−	0.972640i	$-0.425370\pi$
$919$	14.0853	0.464632	0.232316	+	0.972640i	$0.425370\pi$
$920$	0	0
$921$	−66.5898	−2.19421
$922$	0	0
$923$	2.93993	0.0967690
$924$	0	0
$925$	−0.489997	−0.0161110
$926$	0	0
$927$	0.366668	0.0120430
$928$	0	0
$929$	13.9389	0.457321	0.228660	−	0.973506i	$-0.426566\pi$
$929$	13.9389	0.457321	0.228660	+	0.973506i	$0.426566\pi$
$930$	0	0
$931$	26.8049	0.878495
$932$	0	0
$933$	25.9756	0.850404
$934$	0	0
$935$	−14.0683	−0.460082
$936$	0	0
$937$	−3.21360	−0.104984	−0.0524919	−	0.998621i	$-0.516716\pi$
$937$	−3.21360	−0.104984	−0.0524919	+	0.998621i	$0.516716\pi$
$938$	0	0
$939$	61.7651	2.01563
$940$	0	0
$941$	37.3161	1.21647	0.608235	−	0.793757i	$-0.291878\pi$
$941$	37.3161	1.21647	0.608235	+	0.793757i	$0.291878\pi$
$942$	0	0
$943$	−6.41367	−0.208858
$944$	0	0
$945$	−6.39018	−0.207872
$946$	0	0
$947$	−52.8465	−1.71728	−0.858641	−	0.512578i	$-0.828691\pi$
$947$	−52.8465	−1.71728	−0.858641	+	0.512578i	$0.828691\pi$
$948$	0	0
$949$	−3.57572	−0.116073
$950$	0	0
$951$	−36.8392	−1.19459
$952$	0	0
$953$	−18.7238	−0.606523	−0.303261	−	0.952907i	$-0.598076\pi$
$953$	−18.7238	−0.606523	−0.303261	+	0.952907i	$0.598076\pi$
$954$	0	0
$955$	−0.133455	−0.00431852
$956$	0	0
$957$	15.7746	0.509921
$958$	0	0
$959$	−14.7823	−0.477346
$960$	0	0
$961$	55.4495	1.78869
$962$	0	0
$963$	36.1932	1.16631
$964$	0	0
$965$	28.7338	0.924972
$966$	0	0
$967$	46.5600	1.49727	0.748634	−	0.662984i	$-0.230710\pi$
$967$	46.5600	1.49727	0.748634	+	0.662984i	$0.230710\pi$
$968$	0	0
$969$	78.6494	2.52658
$970$	0	0
$971$	25.9137	0.831611	0.415805	−	0.909454i	$-0.363500\pi$
$971$	25.9137	0.831611	0.415805	+	0.909454i	$0.363500\pi$
$972$	0	0
$973$	21.3617	0.684824
$974$	0	0
$975$	−1.23575	−0.0395758
$976$	0	0
$977$	15.9711	0.510962	0.255481	−	0.966814i	$-0.417766\pi$
$977$	15.9711	0.510962	0.255481	+	0.966814i	$0.417766\pi$
$978$	0	0
$979$	14.8272	0.473881
$980$	0	0
$981$	20.2004	0.644948
$982$	0	0
$983$	−14.0538	−0.448248	−0.224124	−	0.974561i	$-0.571952\pi$
$983$	−14.0538	−0.448248	−0.224124	+	0.974561i	$0.571952\pi$
$984$	0	0
$985$	−10.0237	−0.319383
$986$	0	0
$987$	3.75169	0.119417
$988$	0	0
$989$	4.20099	0.133584
$990$	0	0
$991$	−29.8745	−0.948994	−0.474497	−	0.880257i	$-0.657370\pi$
$991$	−29.8745	−0.948994	−0.474497	+	0.880257i	$0.657370\pi$
$992$	0	0
$993$	63.5144	2.01557
$994$	0	0
$995$	27.6560	0.876755
$996$	0	0
$997$	−22.9128	−0.725657	−0.362829	−	0.931856i	$-0.618189\pi$
$997$	−22.9128	−0.725657	−0.362829	+	0.931856i	$0.618189\pi$
$998$	0	0
$999$	0.704625	0.0222933

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.9		9
4.3	odd	2		2008.2.a.a.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.1	✓	9	4.3	odd	2
4016.2.a.h.1.9		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.59002 q^{3} -1.92989 q^{5} +1.80522 q^{7} +3.70818 q^{9} +O(q^{10})\)	\(q+2.59002 q^{3} -1.92989 q^{5} +1.80522 q^{7} +3.70818 q^{9} -1.71997 q^{11} +0.374064 q^{13} -4.99846 q^{15} -4.23825 q^{17} -7.16485 q^{19} +4.67556 q^{21} -1.54199 q^{23} -1.27551 q^{25} +1.83421 q^{27} -3.54108 q^{29} -9.29782 q^{31} -4.45475 q^{33} -3.48389 q^{35} +0.384158 q^{37} +0.968831 q^{39} +4.15934 q^{41} -2.72439 q^{43} -7.15640 q^{45} +0.802404 q^{47} -3.74117 q^{49} -10.9771 q^{51} +2.16015 q^{53} +3.31936 q^{55} -18.5571 q^{57} +1.88478 q^{59} +2.18668 q^{61} +6.69410 q^{63} -0.721903 q^{65} +3.55803 q^{67} -3.99378 q^{69} +7.85944 q^{71} -9.55914 q^{73} -3.30360 q^{75} -3.10493 q^{77} +1.53948 q^{79} -6.37392 q^{81} +6.07124 q^{83} +8.17936 q^{85} -9.17145 q^{87} -8.62064 q^{89} +0.675268 q^{91} -24.0815 q^{93} +13.8274 q^{95} +8.66932 q^{97} -6.37797 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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