LMFDB - Embedded newform 4016.2.a.k.1.10

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:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.84638$ 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)

$f(q)$	$=$	$q+0.508487 q^{3} +3.51373 q^{5} -0.924114 q^{7} -2.74144 q^{9} +O(q^{10})$	$q+0.508487 q^{3} +3.51373 q^{5} -0.924114 q^{7} -2.74144 q^{9} +4.08390 q^{11} +6.64512 q^{13} +1.78668 q^{15} +3.17487 q^{17} +2.91460 q^{19} -0.469900 q^{21} +5.07065 q^{23} +7.34628 q^{25} -2.91945 q^{27} -5.42809 q^{29} +2.09456 q^{31} +2.07661 q^{33} -3.24709 q^{35} +1.40862 q^{37} +3.37896 q^{39} -5.44976 q^{41} +3.35663 q^{43} -9.63268 q^{45} -12.5815 q^{47} -6.14601 q^{49} +1.61438 q^{51} -1.49034 q^{53} +14.3497 q^{55} +1.48204 q^{57} -4.39923 q^{59} -13.5559 q^{61} +2.53341 q^{63} +23.3491 q^{65} -7.34857 q^{67} +2.57836 q^{69} +4.29406 q^{71} +14.4732 q^{73} +3.73548 q^{75} -3.77399 q^{77} -14.3190 q^{79} +6.73983 q^{81} -5.39629 q^{83} +11.1556 q^{85} -2.76011 q^{87} -0.401440 q^{89} -6.14085 q^{91} +1.06505 q^{93} +10.2411 q^{95} +12.3454 q^{97} -11.1958 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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
$3$	0.508487	0.293575	0.146787	−	0.989168i	$-0.453107\pi$
$3$	0.508487	0.293575	0.146787	+	0.989168i	$0.453107\pi$
$4$	0	0
$5$	3.51373	1.57139	0.785693	−	0.618616i	$-0.212306\pi$
$5$	3.51373	1.57139	0.785693	+	0.618616i	$0.212306\pi$
$6$	0	0
$7$	−0.924114	−0.349282	−0.174641	−	0.984632i	$-0.555877\pi$
$7$	−0.924114	−0.349282	−0.174641	+	0.984632i	$0.555877\pi$
$8$	0	0
$9$	−2.74144	−0.913814
$10$	0	0
$11$	4.08390	1.23134	0.615671	−	0.788003i	$-0.288885\pi$
$11$	4.08390	1.23134	0.615671	+	0.788003i	$0.288885\pi$
$12$	0	0
$13$	6.64512	1.84303	0.921513	−	0.388348i	$-0.126954\pi$
$13$	6.64512	1.84303	0.921513	+	0.388348i	$0.126954\pi$
$14$	0	0
$15$	1.78668	0.461320
$16$	0	0
$17$	3.17487	0.770019	0.385009	−	0.922913i	$-0.374198\pi$
$17$	3.17487	0.770019	0.385009	+	0.922913i	$0.374198\pi$
$18$	0	0
$19$	2.91460	0.668655	0.334328	−	0.942457i	$-0.391491\pi$
$19$	2.91460	0.668655	0.334328	+	0.942457i	$0.391491\pi$
$20$	0	0
$21$	−0.469900	−0.102541
$22$	0	0
$23$	5.07065	1.05730	0.528651	−	0.848839i	$-0.322698\pi$
$23$	5.07065	1.05730	0.528651	+	0.848839i	$0.322698\pi$
$24$	0	0
$25$	7.34628	1.46926
$26$	0	0
$27$	−2.91945	−0.561848
$28$	0	0
$29$	−5.42809	−1.00797	−0.503985	−	0.863712i	$-0.668133\pi$
$29$	−5.42809	−1.00797	−0.503985	+	0.863712i	$0.668133\pi$
$30$	0	0
$31$	2.09456	0.376194	0.188097	−	0.982151i	$-0.439768\pi$
$31$	2.09456	0.376194	0.188097	+	0.982151i	$0.439768\pi$
$32$	0	0
$33$	2.07661	0.361491
$34$	0	0
$35$	−3.24709	−0.548858
$36$	0	0
$37$	1.40862	0.231576	0.115788	−	0.993274i	$-0.463061\pi$
$37$	1.40862	0.231576	0.115788	+	0.993274i	$0.463061\pi$
$38$	0	0
$39$	3.37896	0.541066
$40$	0	0
$41$	−5.44976	−0.851110	−0.425555	−	0.904933i	$-0.639921\pi$
$41$	−5.44976	−0.851110	−0.425555	+	0.904933i	$0.639921\pi$
$42$	0	0
$43$	3.35663	0.511881	0.255941	−	0.966693i	$-0.417615\pi$
$43$	3.35663	0.511881	0.255941	+	0.966693i	$0.417615\pi$
$44$	0	0
$45$	−9.63268	−1.43595
$46$	0	0
$47$	−12.5815	−1.83521	−0.917604	−	0.397497i	$-0.869879\pi$
$47$	−12.5815	−1.83521	−0.917604	+	0.397497i	$0.869879\pi$
$48$	0	0
$49$	−6.14601	−0.878002
$50$	0	0
$51$	1.61438	0.226058
$52$	0	0
$53$	−1.49034	−0.204713	−0.102357	−	0.994748i	$-0.532638\pi$
$53$	−1.49034	−0.204713	−0.102357	+	0.994748i	$0.532638\pi$
$54$	0	0
$55$	14.3497	1.93491
$56$	0	0
$57$	1.48204	0.196300
$58$	0	0
$59$	−4.39923	−0.572731	−0.286365	−	0.958121i	$-0.592447\pi$
$59$	−4.39923	−0.572731	−0.286365	+	0.958121i	$0.592447\pi$
$60$	0	0
$61$	−13.5559	−1.73565	−0.867827	−	0.496866i	$-0.834484\pi$
$61$	−13.5559	−1.73565	−0.867827	+	0.496866i	$0.834484\pi$
$62$	0	0
$63$	2.53341	0.319179
$64$	0	0
$65$	23.3491	2.89611
$66$	0	0
$67$	−7.34857	−0.897771	−0.448886	−	0.893589i	$-0.648179\pi$
$67$	−7.34857	−0.897771	−0.448886	+	0.893589i	$0.648179\pi$
$68$	0	0
$69$	2.57836	0.310397
$70$	0	0
$71$	4.29406	0.509611	0.254805	−	0.966992i	$-0.417989\pi$
$71$	4.29406	0.509611	0.254805	+	0.966992i	$0.417989\pi$
$72$	0	0
$73$	14.4732	1.69396	0.846979	−	0.531626i	$-0.178419\pi$
$73$	14.4732	1.69396	0.846979	+	0.531626i	$0.178419\pi$
$74$	0	0
$75$	3.73548	0.431336
$76$	0	0
$77$	−3.77399	−0.430086
$78$	0	0
$79$	−14.3190	−1.61101	−0.805505	−	0.592589i	$-0.798106\pi$
$79$	−14.3190	−1.61101	−0.805505	+	0.592589i	$0.798106\pi$
$80$	0	0
$81$	6.73983	0.748869
$82$	0	0
$83$	−5.39629	−0.592320	−0.296160	−	0.955138i	$-0.595706\pi$
$83$	−5.39629	−0.592320	−0.296160	+	0.955138i	$0.595706\pi$
$84$	0	0
$85$	11.1556	1.21000
$86$	0	0
$87$	−2.76011	−0.295915
$88$	0	0
$89$	−0.401440	−0.0425525	−0.0212763	−	0.999774i	$-0.506773\pi$
$89$	−0.401440	−0.0425525	−0.0212763	+	0.999774i	$0.506773\pi$
$90$	0	0
$91$	−6.14085	−0.643736
$92$	0	0
$93$	1.06505	0.110441
$94$	0	0
$95$	10.2411	1.05072
$96$	0	0
$97$	12.3454	1.25348	0.626742	−	0.779227i	$-0.284388\pi$
$97$	12.3454	1.25348	0.626742	+	0.779227i	$0.284388\pi$
$98$	0	0
$99$	−11.1958	−1.12522
$100$	0	0
$101$	16.7125	1.66296	0.831479	−	0.555555i	$-0.187494\pi$
$101$	16.7125	1.66296	0.831479	+	0.555555i	$0.187494\pi$
$102$	0	0
$103$	12.7311	1.25443	0.627215	−	0.778846i	$-0.284195\pi$
$103$	12.7311	1.25443	0.627215	+	0.778846i	$0.284195\pi$
$104$	0	0
$105$	−1.65110	−0.161131
$106$	0	0
$107$	8.98767	0.868870	0.434435	−	0.900703i	$-0.356948\pi$
$107$	8.98767	0.868870	0.434435	+	0.900703i	$0.356948\pi$
$108$	0	0
$109$	5.87997	0.563199	0.281600	−	0.959532i	$-0.409135\pi$
$109$	5.87997	0.563199	0.281600	+	0.959532i	$0.409135\pi$
$110$	0	0
$111$	0.716265	0.0679849
$112$	0	0
$113$	1.10837	0.104267	0.0521334	−	0.998640i	$-0.483398\pi$
$113$	1.10837	0.104267	0.0521334	+	0.998640i	$0.483398\pi$
$114$	0	0
$115$	17.8169	1.66143
$116$	0	0
$117$	−18.2172	−1.68418
$118$	0	0
$119$	−2.93394	−0.268954
$120$	0	0
$121$	5.67823	0.516202
$122$	0	0
$123$	−2.77113	−0.249865
$124$	0	0
$125$	8.24418	0.737382
$126$	0	0
$127$	−0.966017	−0.0857201	−0.0428601	−	0.999081i	$-0.513647\pi$
$127$	−0.966017	−0.0857201	−0.0428601	+	0.999081i	$0.513647\pi$
$128$	0	0
$129$	1.70680	0.150275
$130$	0	0
$131$	3.99030	0.348634	0.174317	−	0.984690i	$-0.444228\pi$
$131$	3.99030	0.348634	0.174317	+	0.984690i	$0.444228\pi$
$132$	0	0
$133$	−2.69342	−0.233549
$134$	0	0
$135$	−10.2581	−0.882880
$136$	0	0
$137$	1.48886	0.127202	0.0636008	−	0.997975i	$-0.479742\pi$
$137$	1.48886	0.127202	0.0636008	+	0.997975i	$0.479742\pi$
$138$	0	0
$139$	−20.5355	−1.74179	−0.870897	−	0.491465i	$-0.836462\pi$
$139$	−20.5355	−1.74179	−0.870897	+	0.491465i	$0.836462\pi$
$140$	0	0
$141$	−6.39755	−0.538771
$142$	0	0
$143$	27.1380	2.26939
$144$	0	0
$145$	−19.0728	−1.58391
$146$	0	0
$147$	−3.12517	−0.257759
$148$	0	0
$149$	−13.3546	−1.09405	−0.547025	−	0.837116i	$-0.684240\pi$
$149$	−13.3546	−1.09405	−0.547025	+	0.837116i	$0.684240\pi$
$150$	0	0
$151$	14.2350	1.15843	0.579214	−	0.815175i	$-0.303359\pi$
$151$	14.2350	1.15843	0.579214	+	0.815175i	$0.303359\pi$
$152$	0	0
$153$	−8.70372	−0.703654
$154$	0	0
$155$	7.35970	0.591145
$156$	0	0
$157$	16.4064	1.30937	0.654686	−	0.755901i	$-0.272801\pi$
$157$	16.4064	1.30937	0.654686	+	0.755901i	$0.272801\pi$
$158$	0	0
$159$	−0.757816	−0.0600987
$160$	0	0
$161$	−4.68586	−0.369297
$162$	0	0
$163$	4.04187	0.316584	0.158292	−	0.987392i	$-0.449401\pi$
$163$	4.04187	0.316584	0.158292	+	0.987392i	$0.449401\pi$
$164$	0	0
$165$	7.29663	0.568042
$166$	0	0
$167$	5.67256	0.438956	0.219478	−	0.975617i	$-0.429565\pi$
$167$	5.67256	0.438956	0.219478	+	0.975617i	$0.429565\pi$
$168$	0	0
$169$	31.1577	2.39674
$170$	0	0
$171$	−7.99021	−0.611026
$172$	0	0
$173$	−1.56558	−0.119029	−0.0595145	−	0.998227i	$-0.518955\pi$
$173$	−1.56558	−0.119029	−0.0595145	+	0.998227i	$0.518955\pi$
$174$	0	0
$175$	−6.78880	−0.513185
$176$	0	0
$177$	−2.23695	−0.168139
$178$	0	0
$179$	6.55334	0.489819	0.244910	−	0.969546i	$-0.421242\pi$
$179$	6.55334	0.489819	0.244910	+	0.969546i	$0.421242\pi$
$180$	0	0
$181$	−13.2589	−0.985528	−0.492764	−	0.870163i	$-0.664013\pi$
$181$	−13.2589	−0.985528	−0.492764	+	0.870163i	$0.664013\pi$
$182$	0	0
$183$	−6.89299	−0.509545
$184$	0	0
$185$	4.94951	0.363895
$186$	0	0
$187$	12.9658	0.948156
$188$	0	0
$189$	2.69790	0.196243
$190$	0	0
$191$	25.2592	1.82769	0.913847	−	0.406059i	$-0.133097\pi$
$191$	25.2592	1.82769	0.913847	+	0.406059i	$0.133097\pi$
$192$	0	0
$193$	4.74401	0.341482	0.170741	−	0.985316i	$-0.445384\pi$
$193$	4.74401	0.341482	0.170741	+	0.985316i	$0.445384\pi$
$194$	0	0
$195$	11.8727	0.850224
$196$	0	0
$197$	−2.44186	−0.173975	−0.0869875	−	0.996209i	$-0.527724\pi$
$197$	−2.44186	−0.173975	−0.0869875	+	0.996209i	$0.527724\pi$
$198$	0	0
$199$	−26.4754	−1.87679	−0.938396	−	0.345562i	$-0.887688\pi$
$199$	−26.4754	−1.87679	−0.938396	+	0.345562i	$0.887688\pi$
$200$	0	0
$201$	−3.73665	−0.263563
$202$	0	0
$203$	5.01617	0.352066
$204$	0	0
$205$	−19.1490	−1.33742
$206$	0	0
$207$	−13.9009	−0.966178
$208$	0	0
$209$	11.9029	0.823343
$210$	0	0
$211$	6.42757	0.442492	0.221246	−	0.975218i	$-0.428988\pi$
$211$	6.42757	0.442492	0.221246	+	0.975218i	$0.428988\pi$
$212$	0	0
$213$	2.18347	0.149609
$214$	0	0
$215$	11.7943	0.804363
$216$	0	0
$217$	−1.93561	−0.131398
$218$	0	0
$219$	7.35942	0.497304
$220$	0	0
$221$	21.0974	1.41916
$222$	0	0
$223$	−4.16943	−0.279206	−0.139603	−	0.990208i	$-0.544583\pi$
$223$	−4.16943	−0.279206	−0.139603	+	0.990208i	$0.544583\pi$
$224$	0	0
$225$	−20.1394	−1.34263
$226$	0	0
$227$	−20.5579	−1.36447	−0.682237	−	0.731131i	$-0.738993\pi$
$227$	−20.5579	−1.36447	−0.682237	+	0.731131i	$0.738993\pi$
$228$	0	0
$229$	−22.2049	−1.46734	−0.733669	−	0.679507i	$-0.762194\pi$
$229$	−22.2049	−1.46734	−0.733669	+	0.679507i	$0.762194\pi$
$230$	0	0
$231$	−1.91902	−0.126262
$232$	0	0
$233$	−22.3018	−1.46104	−0.730518	−	0.682893i	$-0.760722\pi$
$233$	−22.3018	−1.46104	−0.730518	+	0.682893i	$0.760722\pi$
$234$	0	0
$235$	−44.2081	−2.88382
$236$	0	0
$237$	−7.28100	−0.472952
$238$	0	0
$239$	22.2657	1.44025	0.720124	−	0.693845i	$-0.244085\pi$
$239$	22.2657	1.44025	0.720124	+	0.693845i	$0.244085\pi$
$240$	0	0
$241$	−13.0787	−0.842473	−0.421237	−	0.906951i	$-0.638404\pi$
$241$	−13.0787	−0.842473	−0.421237	+	0.906951i	$0.638404\pi$
$242$	0	0
$243$	12.1854	0.781697
$244$	0	0
$245$	−21.5954	−1.37968
$246$	0	0
$247$	19.3679	1.23235
$248$	0	0
$249$	−2.74394	−0.173890
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	20.7080	1.30190
$254$	0	0
$255$	5.67248	0.355225
$256$	0	0
$257$	−6.21894	−0.387927	−0.193964	−	0.981009i	$-0.562134\pi$
$257$	−6.21894	−0.387927	−0.193964	+	0.981009i	$0.562134\pi$
$258$	0	0
$259$	−1.30173	−0.0808854
$260$	0	0
$261$	14.8808	0.921097
$262$	0	0
$263$	−11.5351	−0.711284	−0.355642	−	0.934622i	$-0.615738\pi$
$263$	−11.5351	−0.711284	−0.355642	+	0.934622i	$0.615738\pi$
$264$	0	0
$265$	−5.23663	−0.321684
$266$	0	0
$267$	−0.204127	−0.0124924
$268$	0	0
$269$	−1.51373	−0.0922935	−0.0461467	−	0.998935i	$-0.514694\pi$
$269$	−1.51373	−0.0922935	−0.0461467	+	0.998935i	$0.514694\pi$
$270$	0	0
$271$	−16.1398	−0.980423	−0.490212	−	0.871603i	$-0.663080\pi$
$271$	−16.1398	−0.980423	−0.490212	+	0.871603i	$0.663080\pi$
$272$	0	0
$273$	−3.12254	−0.188985
$274$	0	0
$275$	30.0015	1.80916
$276$	0	0
$277$	−1.84917	−0.111106	−0.0555530	−	0.998456i	$-0.517692\pi$
$277$	−1.84917	−0.111106	−0.0555530	+	0.998456i	$0.517692\pi$
$278$	0	0
$279$	−5.74210	−0.343771
$280$	0	0
$281$	2.61357	0.155913	0.0779563	−	0.996957i	$-0.475161\pi$
$281$	2.61357	0.155913	0.0779563	+	0.996957i	$0.475161\pi$
$282$	0	0
$283$	6.95366	0.413352	0.206676	−	0.978409i	$-0.433735\pi$
$283$	6.95366	0.413352	0.206676	+	0.978409i	$0.433735\pi$
$284$	0	0
$285$	5.20747	0.308464
$286$	0	0
$287$	5.03620	0.297278
$288$	0	0
$289$	−6.92021	−0.407071
$290$	0	0
$291$	6.27746	0.367991
$292$	0	0
$293$	−5.89067	−0.344137	−0.172068	−	0.985085i	$-0.555045\pi$
$293$	−5.89067	−0.344137	−0.172068	+	0.985085i	$0.555045\pi$
$294$	0	0
$295$	−15.4577	−0.899981
$296$	0	0
$297$	−11.9227	−0.691826
$298$	0	0
$299$	33.6951	1.94864
$300$	0	0
$301$	−3.10191	−0.178791
$302$	0	0
$303$	8.49810	0.488203
$304$	0	0
$305$	−47.6317	−2.72738
$306$	0	0
$307$	2.53032	0.144413	0.0722066	−	0.997390i	$-0.476996\pi$
$307$	2.53032	0.144413	0.0722066	+	0.997390i	$0.476996\pi$
$308$	0	0
$309$	6.47358	0.368269
$310$	0	0
$311$	9.42804	0.534615	0.267308	−	0.963611i	$-0.413866\pi$
$311$	9.42804	0.534615	0.267308	+	0.963611i	$0.413866\pi$
$312$	0	0
$313$	3.79894	0.214729	0.107364	−	0.994220i	$-0.465759\pi$
$313$	3.79894	0.214729	0.107364	+	0.994220i	$0.465759\pi$
$314$	0	0
$315$	8.90169	0.501554
$316$	0	0
$317$	17.0835	0.959507	0.479753	−	0.877403i	$-0.340726\pi$
$317$	17.0835	0.959507	0.479753	+	0.877403i	$0.340726\pi$
$318$	0	0
$319$	−22.1678	−1.24116
$320$	0	0
$321$	4.57011	0.255079
$322$	0	0
$323$	9.25347	0.514877
$324$	0	0
$325$	48.8169	2.70788
$326$	0	0
$327$	2.98989	0.165341
$328$	0	0
$329$	11.6268	0.641005
$330$	0	0
$331$	−35.8579	−1.97093	−0.985465	−	0.169880i	$-0.945662\pi$
$331$	−35.8579	−1.97093	−0.985465	+	0.169880i	$0.945662\pi$
$332$	0	0
$333$	−3.86165	−0.211617
$334$	0	0
$335$	−25.8209	−1.41075
$336$	0	0
$337$	−26.6556	−1.45202	−0.726012	−	0.687682i	$-0.758628\pi$
$337$	−26.6556	−1.45202	−0.726012	+	0.687682i	$0.758628\pi$
$338$	0	0
$339$	0.563592	0.0306101
$340$	0	0
$341$	8.55396	0.463223
$342$	0	0
$343$	12.1484	0.655953
$344$	0	0
$345$	9.05964	0.487754
$346$	0	0
$347$	20.6852	1.11044	0.555220	−	0.831703i	$-0.312634\pi$
$347$	20.6852	1.11044	0.555220	+	0.831703i	$0.312634\pi$
$348$	0	0
$349$	17.2621	0.924019	0.462009	−	0.886875i	$-0.347129\pi$
$349$	17.2621	0.924019	0.462009	+	0.886875i	$0.347129\pi$
$350$	0	0
$351$	−19.4001	−1.03550
$352$	0	0
$353$	4.66285	0.248178	0.124089	−	0.992271i	$-0.460399\pi$
$353$	4.66285	0.248178	0.124089	+	0.992271i	$0.460399\pi$
$354$	0	0
$355$	15.0881	0.800795
$356$	0	0
$357$	−1.49187	−0.0789581
$358$	0	0
$359$	−20.4518	−1.07940	−0.539702	−	0.841856i	$-0.681463\pi$
$359$	−20.4518	−1.07940	−0.539702	+	0.841856i	$0.681463\pi$
$360$	0	0
$361$	−10.5051	−0.552900
$362$	0	0
$363$	2.88730	0.151544
$364$	0	0
$365$	50.8548	2.66186
$366$	0	0
$367$	−7.56007	−0.394632	−0.197316	−	0.980340i	$-0.563223\pi$
$367$	−7.56007	−0.394632	−0.197316	+	0.980340i	$0.563223\pi$
$368$	0	0
$369$	14.9402	0.777756
$370$	0	0
$371$	1.37724	0.0715028
$372$	0	0
$373$	19.8335	1.02694	0.513471	−	0.858107i	$-0.328360\pi$
$373$	19.8335	1.02694	0.513471	+	0.858107i	$0.328360\pi$
$374$	0	0
$375$	4.19205	0.216477
$376$	0	0
$377$	−36.0703	−1.85772
$378$	0	0
$379$	−6.51686	−0.334749	−0.167374	−	0.985893i	$-0.553529\pi$
$379$	−6.51686	−0.334749	−0.167374	+	0.985893i	$0.553529\pi$
$380$	0	0
$381$	−0.491207	−0.0251653
$382$	0	0
$383$	−8.57561	−0.438193	−0.219097	−	0.975703i	$-0.570311\pi$
$383$	−8.57561	−0.438193	−0.219097	+	0.975703i	$0.570311\pi$
$384$	0	0
$385$	−13.2608	−0.675831
$386$	0	0
$387$	−9.20200	−0.467764
$388$	0	0
$389$	−2.98143	−0.151164	−0.0755822	−	0.997140i	$-0.524082\pi$
$389$	−2.98143	−0.151164	−0.0755822	+	0.997140i	$0.524082\pi$
$390$	0	0
$391$	16.0986	0.814143
$392$	0	0
$393$	2.02901	0.102350
$394$	0	0
$395$	−50.3129	−2.53152
$396$	0	0
$397$	26.7890	1.34450	0.672250	−	0.740324i	$-0.265328\pi$
$397$	26.7890	1.34450	0.672250	+	0.740324i	$0.265328\pi$
$398$	0	0
$399$	−1.36957	−0.0685643
$400$	0	0
$401$	−11.7496	−0.586746	−0.293373	−	0.955998i	$-0.594778\pi$
$401$	−11.7496	−0.586746	−0.293373	+	0.955998i	$0.594778\pi$
$402$	0	0
$403$	13.9186	0.693334
$404$	0	0
$405$	23.6819	1.17676
$406$	0	0
$407$	5.75266	0.285149
$408$	0	0
$409$	−9.91593	−0.490311	−0.245156	−	0.969484i	$-0.578839\pi$
$409$	−9.91593	−0.490311	−0.245156	+	0.969484i	$0.578839\pi$
$410$	0	0
$411$	0.757063	0.0373432
$412$	0	0
$413$	4.06539	0.200045
$414$	0	0
$415$	−18.9611	−0.930764
$416$	0	0
$417$	−10.4420	−0.511347
$418$	0	0
$419$	9.02076	0.440693	0.220346	−	0.975422i	$-0.429281\pi$
$419$	9.02076	0.440693	0.220346	+	0.975422i	$0.429281\pi$
$420$	0	0
$421$	18.8082	0.916653	0.458327	−	0.888784i	$-0.348449\pi$
$421$	18.8082	0.916653	0.458327	+	0.888784i	$0.348449\pi$
$422$	0	0
$423$	34.4916	1.67704
$424$	0	0
$425$	23.3235	1.13135
$426$	0	0
$427$	12.5272	0.606234
$428$	0	0
$429$	13.7993	0.666237
$430$	0	0
$431$	17.6639	0.850839	0.425420	−	0.904996i	$-0.360126\pi$
$431$	17.6639	0.850839	0.425420	+	0.904996i	$0.360126\pi$
$432$	0	0
$433$	10.5780	0.508346	0.254173	−	0.967159i	$-0.418197\pi$
$433$	10.5780	0.508346	0.254173	+	0.967159i	$0.418197\pi$
$434$	0	0
$435$	−9.69827	−0.464996
$436$	0	0
$437$	14.7789	0.706971
$438$	0	0
$439$	32.7373	1.56246	0.781232	−	0.624241i	$-0.214592\pi$
$439$	32.7373	1.56246	0.781232	+	0.624241i	$0.214592\pi$
$440$	0	0
$441$	16.8489	0.802330
$442$	0	0
$443$	6.45369	0.306624	0.153312	−	0.988178i	$-0.451006\pi$
$443$	6.45369	0.306624	0.153312	+	0.988178i	$0.451006\pi$
$444$	0	0
$445$	−1.41055	−0.0668665
$446$	0	0
$447$	−6.79062	−0.321185
$448$	0	0
$449$	−9.38049	−0.442693	−0.221346	−	0.975195i	$-0.571045\pi$
$449$	−9.38049	−0.442693	−0.221346	+	0.975195i	$0.571045\pi$
$450$	0	0
$451$	−22.2563	−1.04801
$452$	0	0
$453$	7.23831	0.340086
$454$	0	0
$455$	−21.5773	−1.01156
$456$	0	0
$457$	−41.2228	−1.92832	−0.964161	−	0.265316i	$-0.914524\pi$
$457$	−41.2228	−1.92832	−0.964161	+	0.265316i	$0.914524\pi$
$458$	0	0
$459$	−9.26886	−0.432633
$460$	0	0
$461$	−40.7035	−1.89575	−0.947875	−	0.318641i	$-0.896774\pi$
$461$	−40.7035	−1.89575	−0.947875	+	0.318641i	$0.896774\pi$
$462$	0	0
$463$	8.24450	0.383154	0.191577	−	0.981478i	$-0.438640\pi$
$463$	8.24450	0.383154	0.191577	+	0.981478i	$0.438640\pi$
$464$	0	0
$465$	3.74231	0.173545
$466$	0	0
$467$	4.09916	0.189687	0.0948434	−	0.995492i	$-0.469765\pi$
$467$	4.09916	0.189687	0.0948434	+	0.995492i	$0.469765\pi$
$468$	0	0
$469$	6.79092	0.313576
$470$	0	0
$471$	8.34243	0.384399
$472$	0	0
$473$	13.7081	0.630301
$474$	0	0
$475$	21.4115	0.982425
$476$	0	0
$477$	4.08567	0.187070
$478$	0	0
$479$	−22.6507	−1.03494	−0.517468	−	0.855703i	$-0.673125\pi$
$479$	−22.6507	−1.03494	−0.517468	+	0.855703i	$0.673125\pi$
$480$	0	0
$481$	9.36046	0.426800
$482$	0	0
$483$	−2.38269	−0.108416
$484$	0	0
$485$	43.3783	1.96971
$486$	0	0
$487$	29.4194	1.33312	0.666561	−	0.745451i	$-0.267766\pi$
$487$	29.4194	1.33312	0.666561	+	0.745451i	$0.267766\pi$
$488$	0	0
$489$	2.05524	0.0929411
$490$	0	0
$491$	21.7774	0.982799	0.491399	−	0.870934i	$-0.336486\pi$
$491$	21.7774	0.982799	0.491399	+	0.870934i	$0.336486\pi$
$492$	0	0
$493$	−17.2335	−0.776156
$494$	0	0
$495$	−39.3389	−1.76815
$496$	0	0
$497$	−3.96820	−0.177998
$498$	0	0
$499$	9.61718	0.430524	0.215262	−	0.976556i	$-0.430939\pi$
$499$	9.61718	0.430524	0.215262	+	0.976556i	$0.430939\pi$
$500$	0	0
$501$	2.88442	0.128866
$502$	0	0
$503$	11.0738	0.493758	0.246879	−	0.969046i	$-0.420595\pi$
$503$	11.0738	0.493758	0.246879	+	0.969046i	$0.420595\pi$
$504$	0	0
$505$	58.7233	2.61315
$506$	0	0
$507$	15.8433	0.703624
$508$	0	0
$509$	−33.7145	−1.49437	−0.747184	−	0.664617i	$-0.768595\pi$
$509$	−33.7145	−1.49437	−0.747184	+	0.664617i	$0.768595\pi$
$510$	0	0
$511$	−13.3749	−0.591670
$512$	0	0
$513$	−8.50902	−0.375682
$514$	0	0
$515$	44.7335	1.97119
$516$	0	0
$517$	−51.3818	−2.25977
$518$	0	0
$519$	−0.796077	−0.0349439
$520$	0	0
$521$	−2.44241	−0.107004	−0.0535019	−	0.998568i	$-0.517038\pi$
$521$	−2.44241	−0.107004	−0.0535019	+	0.998568i	$0.517038\pi$
$522$	0	0
$523$	25.0351	1.09471	0.547354	−	0.836901i	$-0.315635\pi$
$523$	25.0351	1.09471	0.547354	+	0.836901i	$0.315635\pi$
$524$	0	0
$525$	−3.45201	−0.150658
$526$	0	0
$527$	6.64994	0.289676
$528$	0	0
$529$	2.71144	0.117889
$530$	0	0
$531$	12.0602	0.523369
$532$	0	0
$533$	−36.2144	−1.56862
$534$	0	0
$535$	31.5802	1.36533
$536$	0	0
$537$	3.33228	0.143799
$538$	0	0
$539$	−25.0997	−1.08112
$540$	0	0
$541$	34.7261	1.49299	0.746496	−	0.665390i	$-0.231735\pi$
$541$	34.7261	1.49299	0.746496	+	0.665390i	$0.231735\pi$
$542$	0	0
$543$	−6.74198	−0.289326
$544$	0	0
$545$	20.6606	0.885003
$546$	0	0
$547$	−18.1096	−0.774310	−0.387155	−	0.922015i	$-0.626542\pi$
$547$	−18.1096	−0.774310	−0.387155	+	0.922015i	$0.626542\pi$
$548$	0	0
$549$	37.1627	1.58607
$550$	0	0
$551$	−15.8207	−0.673985
$552$	0	0
$553$	13.2324	0.562697
$554$	0	0
$555$	2.51676	0.106830
$556$	0	0
$557$	10.2659	0.434981	0.217491	−	0.976062i	$-0.430213\pi$
$557$	10.2659	0.434981	0.217491	+	0.976062i	$0.430213\pi$
$558$	0	0
$559$	22.3052	0.943410
$560$	0	0
$561$	6.59296	0.278355
$562$	0	0
$563$	−34.6695	−1.46115	−0.730573	−	0.682835i	$-0.760747\pi$
$563$	−34.6695	−1.46115	−0.730573	+	0.682835i	$0.760747\pi$
$564$	0	0
$565$	3.89452	0.163844
$566$	0	0
$567$	−6.22837	−0.261567
$568$	0	0
$569$	9.22288	0.386643	0.193322	−	0.981135i	$-0.438074\pi$
$569$	9.22288	0.386643	0.193322	+	0.981135i	$0.438074\pi$
$570$	0	0
$571$	−32.5615	−1.36266	−0.681328	−	0.731978i	$-0.738597\pi$
$571$	−32.5615	−1.36266	−0.681328	+	0.731978i	$0.738597\pi$
$572$	0	0
$573$	12.8440	0.536565
$574$	0	0
$575$	37.2504	1.55345
$576$	0	0
$577$	38.6305	1.60821	0.804104	−	0.594489i	$-0.202646\pi$
$577$	38.6305	1.60821	0.804104	+	0.594489i	$0.202646\pi$
$578$	0	0
$579$	2.41227	0.100250
$580$	0	0
$581$	4.98679	0.206887
$582$	0	0
$583$	−6.08638	−0.252072
$584$	0	0
$585$	−64.0103	−2.64650
$586$	0	0
$587$	−47.8352	−1.97437	−0.987184	−	0.159584i	$-0.948985\pi$
$587$	−47.8352	−1.97437	−0.987184	+	0.159584i	$0.948985\pi$
$588$	0	0
$589$	6.10480	0.251544
$590$	0	0
$591$	−1.24165	−0.0510747
$592$	0	0
$593$	10.4933	0.430910	0.215455	−	0.976514i	$-0.430877\pi$
$593$	10.4933	0.430910	0.215455	+	0.976514i	$0.430877\pi$
$594$	0	0
$595$	−10.3091	−0.422631
$596$	0	0
$597$	−13.4624	−0.550979
$598$	0	0
$599$	34.9392	1.42758	0.713789	−	0.700360i	$-0.246977\pi$
$599$	34.9392	1.42758	0.713789	+	0.700360i	$0.246977\pi$
$600$	0	0
$601$	−19.0447	−0.776850	−0.388425	−	0.921480i	$-0.626981\pi$
$601$	−19.0447	−0.776850	−0.388425	+	0.921480i	$0.626981\pi$
$602$	0	0
$603$	20.1457	0.820396
$604$	0	0
$605$	19.9517	0.811153
$606$	0	0
$607$	−40.3053	−1.63594	−0.817971	−	0.575260i	$-0.804901\pi$
$607$	−40.3053	−1.63594	−0.817971	+	0.575260i	$0.804901\pi$
$608$	0	0
$609$	2.55066	0.103358
$610$	0	0
$611$	−83.6059	−3.38233
$612$	0	0
$613$	1.32821	0.0536458	0.0268229	−	0.999640i	$-0.491461\pi$
$613$	1.32821	0.0536458	0.0268229	+	0.999640i	$0.491461\pi$
$614$	0	0
$615$	−9.73700	−0.392634
$616$	0	0
$617$	−6.42999	−0.258862	−0.129431	−	0.991588i	$-0.541315\pi$
$617$	−6.42999	−0.258862	−0.129431	+	0.991588i	$0.541315\pi$
$618$	0	0
$619$	−17.1506	−0.689341	−0.344671	−	0.938724i	$-0.612009\pi$
$619$	−17.1506	−0.689341	−0.344671	+	0.938724i	$0.612009\pi$
$620$	0	0
$621$	−14.8035	−0.594043
$622$	0	0
$623$	0.370976	0.0148629
$624$	0	0
$625$	−7.76360	−0.310544
$626$	0	0
$627$	6.05248	0.241713
$628$	0	0
$629$	4.47219	0.178318
$630$	0	0
$631$	9.34803	0.372139	0.186070	−	0.982537i	$-0.440425\pi$
$631$	9.34803	0.372139	0.186070	+	0.982537i	$0.440425\pi$
$632$	0	0
$633$	3.26834	0.129905
$634$	0	0
$635$	−3.39432	−0.134699
$636$	0	0
$637$	−40.8410	−1.61818
$638$	0	0
$639$	−11.7719	−0.465689
$640$	0	0
$641$	−29.5636	−1.16769	−0.583846	−	0.811865i	$-0.698453\pi$
$641$	−29.5636	−1.16769	−0.583846	+	0.811865i	$0.698453\pi$
$642$	0	0
$643$	−17.9222	−0.706782	−0.353391	−	0.935476i	$-0.614971\pi$
$643$	−17.9222	−0.706782	−0.353391	+	0.935476i	$0.614971\pi$
$644$	0	0
$645$	5.99723	0.236141
$646$	0	0
$647$	−16.7138	−0.657085	−0.328543	−	0.944489i	$-0.606557\pi$
$647$	−16.7138	−0.657085	−0.328543	+	0.944489i	$0.606557\pi$
$648$	0	0
$649$	−17.9660	−0.705227
$650$	0	0
$651$	−0.984232	−0.0385751
$652$	0	0
$653$	−16.1132	−0.630557	−0.315278	−	0.948999i	$-0.602098\pi$
$653$	−16.1132	−0.630557	−0.315278	+	0.948999i	$0.602098\pi$
$654$	0	0
$655$	14.0208	0.547839
$656$	0	0
$657$	−39.6774	−1.54796
$658$	0	0
$659$	−44.3385	−1.72718	−0.863592	−	0.504192i	$-0.831790\pi$
$659$	−44.3385	−1.72718	−0.863592	+	0.504192i	$0.831790\pi$
$660$	0	0
$661$	38.2157	1.48642	0.743210	−	0.669058i	$-0.233302\pi$
$661$	38.2157	1.48642	0.743210	+	0.669058i	$0.233302\pi$
$662$	0	0
$663$	10.7277	0.416631
$664$	0	0
$665$	−9.46396	−0.366997
$666$	0	0
$667$	−27.5239	−1.06573
$668$	0	0
$669$	−2.12010	−0.0819678
$670$	0	0
$671$	−55.3609	−2.13718
$672$	0	0
$673$	−25.5896	−0.986406	−0.493203	−	0.869914i	$-0.664174\pi$
$673$	−25.5896	−0.986406	−0.493203	+	0.869914i	$0.664174\pi$
$674$	0	0
$675$	−21.4471	−0.825498
$676$	0	0
$677$	−7.27438	−0.279577	−0.139789	−	0.990181i	$-0.544642\pi$
$677$	−7.27438	−0.279577	−0.139789	+	0.990181i	$0.544642\pi$
$678$	0	0
$679$	−11.4085	−0.437820
$680$	0	0
$681$	−10.4534	−0.400575
$682$	0	0
$683$	13.3023	0.508997	0.254499	−	0.967073i	$-0.418090\pi$
$683$	13.3023	0.508997	0.254499	+	0.967073i	$0.418090\pi$
$684$	0	0
$685$	5.23143	0.199883
$686$	0	0
$687$	−11.2909	−0.430774
$688$	0	0
$689$	−9.90347	−0.377292
$690$	0	0
$691$	16.0461	0.610422	0.305211	−	0.952285i	$-0.401273\pi$
$691$	16.0461	0.610422	0.305211	+	0.952285i	$0.401273\pi$
$692$	0	0
$693$	10.3462	0.393018
$694$	0	0
$695$	−72.1560	−2.73703
$696$	0	0
$697$	−17.3023	−0.655371
$698$	0	0
$699$	−11.3401	−0.428924
$700$	0	0
$701$	16.9390	0.639777	0.319889	−	0.947455i	$-0.396354\pi$
$701$	16.9390	0.639777	0.319889	+	0.947455i	$0.396354\pi$
$702$	0	0
$703$	4.10557	0.154844
$704$	0	0
$705$	−22.4792	−0.846617
$706$	0	0
$707$	−15.4443	−0.580842
$708$	0	0
$709$	−27.7952	−1.04387	−0.521936	−	0.852985i	$-0.674790\pi$
$709$	−27.7952	−1.04387	−0.521936	+	0.852985i	$0.674790\pi$
$710$	0	0
$711$	39.2546	1.47216
$712$	0	0
$713$	10.6208	0.397750
$714$	0	0
$715$	95.3555	3.56610
$716$	0	0
$717$	11.3218	0.422821
$718$	0	0
$719$	36.2441	1.35168	0.675838	−	0.737050i	$-0.263782\pi$
$719$	36.2441	1.35168	0.675838	+	0.737050i	$0.263782\pi$
$720$	0	0
$721$	−11.7650	−0.438150
$722$	0	0
$723$	−6.65034	−0.247329
$724$	0	0
$725$	−39.8762	−1.48097
$726$	0	0
$727$	24.4270	0.905946	0.452973	−	0.891524i	$-0.350363\pi$
$727$	24.4270	0.905946	0.452973	+	0.891524i	$0.350363\pi$
$728$	0	0
$729$	−14.0233	−0.519383
$730$	0	0
$731$	10.6569	0.394158
$732$	0	0
$733$	−21.8141	−0.805724	−0.402862	−	0.915261i	$-0.631985\pi$
$733$	−21.8141	−0.805724	−0.402862	+	0.915261i	$0.631985\pi$
$734$	0	0
$735$	−10.9810	−0.405039
$736$	0	0
$737$	−30.0108	−1.10546
$738$	0	0
$739$	−12.9714	−0.477160	−0.238580	−	0.971123i	$-0.576682\pi$
$739$	−12.9714	−0.477160	−0.238580	+	0.971123i	$0.576682\pi$
$740$	0	0
$741$	9.84831	0.361787
$742$	0	0
$743$	−22.5997	−0.829104	−0.414552	−	0.910026i	$-0.636062\pi$
$743$	−22.5997	−0.829104	−0.414552	+	0.910026i	$0.636062\pi$
$744$	0	0
$745$	−46.9243	−1.71917
$746$	0	0
$747$	14.7936	0.541270
$748$	0	0
$749$	−8.30563	−0.303481
$750$	0	0
$751$	−44.2023	−1.61296	−0.806482	−	0.591258i	$-0.798631\pi$
$751$	−44.2023	−1.61296	−0.806482	+	0.591258i	$0.798631\pi$
$752$	0	0
$753$	−0.508487	−0.0185303
$754$	0	0
$755$	50.0180	1.82034
$756$	0	0
$757$	−19.2146	−0.698367	−0.349184	−	0.937054i	$-0.613541\pi$
$757$	−19.2146	−0.698367	−0.349184	+	0.937054i	$0.613541\pi$
$758$	0	0
$759$	10.5297	0.382205
$760$	0	0
$761$	23.6826	0.858493	0.429246	−	0.903187i	$-0.358779\pi$
$761$	23.6826	0.858493	0.429246	+	0.903187i	$0.358779\pi$
$762$	0	0
$763$	−5.43376	−0.196715
$764$	0	0
$765$	−30.5825	−1.10571
$766$	0	0
$767$	−29.2334	−1.05556
$768$	0	0
$769$	5.59488	0.201757	0.100878	−	0.994899i	$-0.467835\pi$
$769$	5.59488	0.201757	0.100878	+	0.994899i	$0.467835\pi$
$770$	0	0
$771$	−3.16225	−0.113886
$772$	0	0
$773$	−13.6726	−0.491768	−0.245884	−	0.969299i	$-0.579078\pi$
$773$	−13.6726	−0.491768	−0.245884	+	0.969299i	$0.579078\pi$
$774$	0	0
$775$	15.3872	0.552724
$776$	0	0
$777$	−0.661911	−0.0237459
$778$	0	0
$779$	−15.8839	−0.569099
$780$	0	0
$781$	17.5365	0.627505
$782$	0	0
$783$	15.8470	0.566326
$784$	0	0
$785$	57.6476	2.05753
$786$	0	0
$787$	36.2874	1.29351	0.646753	−	0.762700i	$-0.276127\pi$
$787$	36.2874	1.29351	0.646753	+	0.762700i	$0.276127\pi$
$788$	0	0
$789$	−5.86544	−0.208815
$790$	0	0
$791$	−1.02426	−0.0364186
$792$	0	0
$793$	−90.0806	−3.19886
$794$	0	0
$795$	−2.66276	−0.0944383
$796$	0	0
$797$	9.28096	0.328748	0.164374	−	0.986398i	$-0.447440\pi$
$797$	9.28096	0.328748	0.164374	+	0.986398i	$0.447440\pi$
$798$	0	0
$799$	−39.9448	−1.41314
$800$	0	0
$801$	1.10052	0.0388851
$802$	0	0
$803$	59.1070	2.08584
$804$	0	0
$805$	−16.4648	−0.580309
$806$	0	0
$807$	−0.769709	−0.0270950
$808$	0	0
$809$	−22.5865	−0.794099	−0.397050	−	0.917797i	$-0.629966\pi$
$809$	−22.5865	−0.794099	−0.397050	+	0.917797i	$0.629966\pi$
$810$	0	0
$811$	−45.3964	−1.59408	−0.797042	−	0.603924i	$-0.793603\pi$
$811$	−45.3964	−1.59408	−0.797042	+	0.603924i	$0.793603\pi$
$812$	0	0
$813$	−8.20687	−0.287828
$814$	0	0
$815$	14.2020	0.497476
$816$	0	0
$817$	9.78323	0.342272
$818$	0	0
$819$	16.8348	0.588255
$820$	0	0
$821$	27.0419	0.943768	0.471884	−	0.881661i	$-0.343574\pi$
$821$	27.0419	0.943768	0.471884	+	0.881661i	$0.343574\pi$
$822$	0	0
$823$	16.8476	0.587270	0.293635	−	0.955918i	$-0.405135\pi$
$823$	16.8476	0.587270	0.293635	+	0.955918i	$0.405135\pi$
$824$	0	0
$825$	15.2553	0.531123
$826$	0	0
$827$	−32.6423	−1.13508	−0.567541	−	0.823345i	$-0.692105\pi$
$827$	−32.6423	−1.13508	−0.567541	+	0.823345i	$0.692105\pi$
$828$	0	0
$829$	41.6463	1.44644	0.723218	−	0.690620i	$-0.242662\pi$
$829$	41.6463	1.44644	0.723218	+	0.690620i	$0.242662\pi$
$830$	0	0
$831$	−0.940279	−0.0326179
$832$	0	0
$833$	−19.5128	−0.676078
$834$	0	0
$835$	19.9318	0.689769
$836$	0	0
$837$	−6.11495	−0.211363
$838$	0	0
$839$	20.6409	0.712604	0.356302	−	0.934371i	$-0.384037\pi$
$839$	20.6409	0.712604	0.356302	+	0.934371i	$0.384037\pi$
$840$	0	0
$841$	0.464120	0.0160041
$842$	0	0
$843$	1.32897	0.0457720
$844$	0	0
$845$	109.480	3.76621
$846$	0	0
$847$	−5.24733	−0.180300
$848$	0	0
$849$	3.53584	0.121350
$850$	0	0
$851$	7.14262	0.244846
$852$	0	0
$853$	21.9922	0.753000	0.376500	−	0.926417i	$-0.377127\pi$
$853$	21.9922	0.753000	0.376500	+	0.926417i	$0.377127\pi$
$854$	0	0
$855$	−28.0754	−0.960159
$856$	0	0
$857$	29.9500	1.02307	0.511537	−	0.859261i	$-0.329076\pi$
$857$	29.9500	1.02307	0.511537	+	0.859261i	$0.329076\pi$
$858$	0	0
$859$	−1.42729	−0.0486984	−0.0243492	−	0.999704i	$-0.507751\pi$
$859$	−1.42729	−0.0486984	−0.0243492	+	0.999704i	$0.507751\pi$
$860$	0	0
$861$	2.56084	0.0872733
$862$	0	0
$863$	−39.6839	−1.35086	−0.675428	−	0.737426i	$-0.736041\pi$
$863$	−39.6839	−1.35086	−0.675428	+	0.737426i	$0.736041\pi$
$864$	0	0
$865$	−5.50102	−0.187040
$866$	0	0
$867$	−3.51883	−0.119506
$868$	0	0
$869$	−58.4772	−1.98370
$870$	0	0
$871$	−48.8322	−1.65462
$872$	0	0
$873$	−33.8441	−1.14545
$874$	0	0
$875$	−7.61856	−0.257554
$876$	0	0
$877$	−32.8536	−1.10939	−0.554694	−	0.832054i	$-0.687165\pi$
$877$	−32.8536	−1.10939	−0.554694	+	0.832054i	$0.687165\pi$
$878$	0	0
$879$	−2.99533	−0.101030
$880$	0	0
$881$	−11.7696	−0.396527	−0.198264	−	0.980149i	$-0.563530\pi$
$881$	−11.7696	−0.396527	−0.198264	+	0.980149i	$0.563530\pi$
$882$	0	0
$883$	−0.0708481	−0.00238423	−0.00119211	−	0.999999i	$-0.500379\pi$
$883$	−0.0708481	−0.00238423	−0.00119211	+	0.999999i	$0.500379\pi$
$884$	0	0
$885$	−7.86002	−0.264212
$886$	0	0
$887$	33.0756	1.11057	0.555284	−	0.831661i	$-0.312609\pi$
$887$	33.0756	1.11057	0.555284	+	0.831661i	$0.312609\pi$
$888$	0	0
$889$	0.892710	0.0299405
$890$	0	0
$891$	27.5248	0.922114
$892$	0	0
$893$	−36.6702	−1.22712
$894$	0	0
$895$	23.0266	0.769695
$896$	0	0
$897$	17.1335	0.572070
$898$	0	0
$899$	−11.3694	−0.379192
$900$	0	0
$901$	−4.73162	−0.157633
$902$	0	0
$903$	−1.57728	−0.0524886
$904$	0	0
$905$	−46.5882	−1.54865
$906$	0	0
$907$	−40.9755	−1.36057	−0.680284	−	0.732949i	$-0.738144\pi$
$907$	−40.9755	−1.36057	−0.680284	+	0.732949i	$0.738144\pi$
$908$	0	0
$909$	−45.8164	−1.51963
$910$	0	0
$911$	−21.6253	−0.716477	−0.358239	−	0.933630i	$-0.616623\pi$
$911$	−21.6253	−0.716477	−0.358239	+	0.933630i	$0.616623\pi$
$912$	0	0
$913$	−22.0379	−0.729349
$914$	0	0
$915$	−24.2201	−0.800692
$916$	0	0
$917$	−3.68749	−0.121772
$918$	0	0
$919$	−24.0541	−0.793473	−0.396737	−	0.917933i	$-0.629857\pi$
$919$	−24.0541	−0.793473	−0.396737	+	0.917933i	$0.629857\pi$
$920$	0	0
$921$	1.28664	0.0423961
$922$	0	0
$923$	28.5345	0.939225
$924$	0	0
$925$	10.3481	0.340244
$926$	0	0
$927$	−34.9015	−1.14632
$928$	0	0
$929$	−45.0584	−1.47832	−0.739159	−	0.673531i	$-0.764777\pi$
$929$	−45.0584	−1.47832	−0.739159	+	0.673531i	$0.764777\pi$
$930$	0	0
$931$	−17.9132	−0.587081
$932$	0	0
$933$	4.79403	0.156950
$934$	0	0
$935$	45.5584	1.48992
$936$	0	0
$937$	48.8513	1.59590	0.797951	−	0.602722i	$-0.205917\pi$
$937$	48.8513	1.59590	0.797951	+	0.602722i	$0.205917\pi$
$938$	0	0
$939$	1.93171	0.0630390
$940$	0	0
$941$	−30.0033	−0.978079	−0.489039	−	0.872262i	$-0.662653\pi$
$941$	−30.0033	−0.978079	−0.489039	+	0.872262i	$0.662653\pi$
$942$	0	0
$943$	−27.6338	−0.899881
$944$	0	0
$945$	9.47969	0.308374
$946$	0	0
$947$	−20.0365	−0.651098	−0.325549	−	0.945525i	$-0.605549\pi$
$947$	−20.0365	−0.651098	−0.325549	+	0.945525i	$0.605549\pi$
$948$	0	0
$949$	96.1761	3.12201
$950$	0	0
$951$	8.68675	0.281687
$952$	0	0
$953$	14.8016	0.479472	0.239736	−	0.970838i	$-0.422939\pi$
$953$	14.8016	0.479472	0.239736	+	0.970838i	$0.422939\pi$
$954$	0	0
$955$	88.7541	2.87201
$956$	0	0
$957$	−11.2720	−0.364372
$958$	0	0
$959$	−1.37587	−0.0444292
$960$	0	0
$961$	−26.6128	−0.858478
$962$	0	0
$963$	−24.6392	−0.793986
$964$	0	0
$965$	16.6692	0.536599
$966$	0	0
$967$	44.7916	1.44040	0.720200	−	0.693766i	$-0.244050\pi$
$967$	44.7916	1.44040	0.720200	+	0.693766i	$0.244050\pi$
$968$	0	0
$969$	4.70527	0.151155
$970$	0	0
$971$	21.8504	0.701212	0.350606	−	0.936523i	$-0.385976\pi$
$971$	21.8504	0.701212	0.350606	+	0.936523i	$0.385976\pi$
$972$	0	0
$973$	18.9771	0.608378
$974$	0	0
$975$	24.8227	0.794964
$976$	0	0
$977$	6.24280	0.199725	0.0998625	−	0.995001i	$-0.468160\pi$
$977$	6.24280	0.199725	0.0998625	+	0.995001i	$0.468160\pi$
$978$	0	0
$979$	−1.63944	−0.0523967
$980$	0	0
$981$	−16.1196	−0.514659
$982$	0	0
$983$	38.9962	1.24379	0.621893	−	0.783102i	$-0.286364\pi$
$983$	38.9962	1.24379	0.621893	+	0.783102i	$0.286364\pi$
$984$	0	0
$985$	−8.58002	−0.273382
$986$	0	0
$987$	5.91206	0.188183
$988$	0	0
$989$	17.0203	0.541213
$990$	0	0
$991$	41.2192	1.30937	0.654685	−	0.755902i	$-0.272801\pi$
$991$	41.2192	1.30937	0.654685	+	0.755902i	$0.272801\pi$
$992$	0	0
$993$	−18.2333	−0.578615
$994$	0	0
$995$	−93.0273	−2.94917
$996$	0	0
$997$	36.6076	1.15937	0.579687	−	0.814839i	$-0.303175\pi$
$997$	36.6076	1.15937	0.579687	+	0.814839i	$0.303175\pi$
$998$	0	0
$999$	−4.11239	−0.130110

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.10		17
4.3	odd	2		251.2.a.b.1.13	✓	17
12.11	even	2		2259.2.a.k.1.5		17
20.19	odd	2		6275.2.a.e.1.5		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.13	✓	17	4.3	odd	2
2259.2.a.k.1.5		17	12.11	even	2
4016.2.a.k.1.10		17	1.1	even	1	trivial
6275.2.a.e.1.5		17	20.19	odd	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.508487 q^{3} +3.51373 q^{5} -0.924114 q^{7} -2.74144 q^{9} +O(q^{10})\)	\(q+0.508487 q^{3} +3.51373 q^{5} -0.924114 q^{7} -2.74144 q^{9} +4.08390 q^{11} +6.64512 q^{13} +1.78668 q^{15} +3.17487 q^{17} +2.91460 q^{19} -0.469900 q^{21} +5.07065 q^{23} +7.34628 q^{25} -2.91945 q^{27} -5.42809 q^{29} +2.09456 q^{31} +2.07661 q^{33} -3.24709 q^{35} +1.40862 q^{37} +3.37896 q^{39} -5.44976 q^{41} +3.35663 q^{43} -9.63268 q^{45} -12.5815 q^{47} -6.14601 q^{49} +1.61438 q^{51} -1.49034 q^{53} +14.3497 q^{55} +1.48204 q^{57} -4.39923 q^{59} -13.5559 q^{61} +2.53341 q^{63} +23.3491 q^{65} -7.34857 q^{67} +2.57836 q^{69} +4.29406 q^{71} +14.4732 q^{73} +3.73548 q^{75} -3.77399 q^{77} -14.3190 q^{79} +6.73983 q^{81} -5.39629 q^{83} +11.1556 q^{85} -2.76011 q^{87} -0.401440 q^{89} -6.14085 q^{91} +1.06505 q^{93} +10.2411 q^{95} +12.3454 q^{97} -11.1958 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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