LMFDB - Embedded newform 4016.2.a.e.1.2

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:	$5$
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			1.2
Root			$-0.567497$ 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)

$f(q)$	$=$	$q-0.567497 q^{3} -2.51380 q^{5} -3.57054 q^{7} -2.67795 q^{9} +O(q^{10})$	$q-0.567497 q^{3} -2.51380 q^{5} -3.57054 q^{7} -2.67795 q^{9} +5.90017 q^{11} -3.72120 q^{13} +1.42657 q^{15} +5.80554 q^{17} +1.27592 q^{19} +2.02627 q^{21} +3.04325 q^{23} +1.31917 q^{25} +3.22222 q^{27} -3.75167 q^{29} +7.64007 q^{31} -3.34833 q^{33} +8.97562 q^{35} +11.9668 q^{37} +2.11177 q^{39} -9.65052 q^{41} +1.48925 q^{43} +6.73181 q^{45} -0.516843 q^{47} +5.74879 q^{49} -3.29463 q^{51} -10.5893 q^{53} -14.8318 q^{55} -0.724081 q^{57} -0.696653 q^{59} -4.04498 q^{61} +9.56173 q^{63} +9.35433 q^{65} +8.27304 q^{67} -1.72704 q^{69} -10.3096 q^{71} +0.498979 q^{73} -0.748625 q^{75} -21.0668 q^{77} -3.99695 q^{79} +6.20524 q^{81} -2.57663 q^{83} -14.5939 q^{85} +2.12906 q^{87} +10.1843 q^{89} +13.2867 q^{91} -4.33572 q^{93} -3.20740 q^{95} -10.0418 q^{97} -15.8003 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{3} - 6 q^{5} + q^{7}+O(q^{10}) $ 5 * q + q^3 - 6 * q^5 + q^7	$ 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}+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

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
$3$	−0.567497	−0.327645	−0.163822	−	0.986490i	$-0.552382\pi$
$3$	−0.567497	−0.327645	−0.163822	+	0.986490i	$0.552382\pi$
$4$	0	0
$5$	−2.51380	−1.12420	−0.562102	−	0.827068i	$-0.690007\pi$
$5$	−2.51380	−1.12420	−0.562102	+	0.827068i	$0.690007\pi$
$6$	0	0
$7$	−3.57054	−1.34954	−0.674769	−	0.738029i	$-0.735757\pi$
$7$	−3.57054	−1.34954	−0.674769	+	0.738029i	$0.735757\pi$
$8$	0	0
$9$	−2.67795	−0.892649
$10$	0	0
$11$	5.90017	1.77897	0.889483	−	0.456968i	$-0.151064\pi$
$11$	5.90017	1.77897	0.889483	+	0.456968i	$0.151064\pi$
$12$	0	0
$13$	−3.72120	−1.03207	−0.516037	−	0.856566i	$-0.672593\pi$
$13$	−3.72120	−1.03207	−0.516037	+	0.856566i	$0.672593\pi$
$14$	0	0
$15$	1.42657	0.368339
$16$	0	0
$17$	5.80554	1.40805	0.704025	−	0.710175i	$-0.251384\pi$
$17$	5.80554	1.40805	0.704025	+	0.710175i	$0.251384\pi$
$18$	0	0
$19$	1.27592	0.292716	0.146358	−	0.989232i	$-0.453245\pi$
$19$	1.27592	0.292716	0.146358	+	0.989232i	$0.453245\pi$
$20$	0	0
$21$	2.02627	0.442169
$22$	0	0
$23$	3.04325	0.634561	0.317281	−	0.948332i	$-0.397230\pi$
$23$	3.04325	0.634561	0.317281	+	0.948332i	$0.397230\pi$
$24$	0	0
$25$	1.31917	0.263834
$26$	0	0
$27$	3.22222	0.620116
$28$	0	0
$29$	−3.75167	−0.696668	−0.348334	−	0.937370i	$-0.613252\pi$
$29$	−3.75167	−0.696668	−0.348334	+	0.937370i	$0.613252\pi$
$30$	0	0
$31$	7.64007	1.37220	0.686098	−	0.727509i	$-0.259322\pi$
$31$	7.64007	1.37220	0.686098	+	0.727509i	$0.259322\pi$
$32$	0	0
$33$	−3.34833	−0.582869
$34$	0	0
$35$	8.97562	1.51716
$36$	0	0
$37$	11.9668	1.96733	0.983665	−	0.180007i	$-0.0576121\pi$
$37$	11.9668	1.96733	0.983665	+	0.180007i	$0.0576121\pi$
$38$	0	0
$39$	2.11177	0.338154
$40$	0	0
$41$	−9.65052	−1.50716	−0.753579	−	0.657357i	$-0.771674\pi$
$41$	−9.65052	−1.50716	−0.753579	+	0.657357i	$0.771674\pi$
$42$	0	0
$43$	1.48925	0.227109	0.113554	−	0.993532i	$-0.463776\pi$
$43$	1.48925	0.227109	0.113554	+	0.993532i	$0.463776\pi$
$44$	0	0
$45$	6.73181	1.00352
$46$	0	0
$47$	−0.516843	−0.0753893	−0.0376947	−	0.999289i	$-0.512001\pi$
$47$	−0.516843	−0.0753893	−0.0376947	+	0.999289i	$0.512001\pi$
$48$	0	0
$49$	5.74879	0.821255
$50$	0	0
$51$	−3.29463	−0.461340
$52$	0	0
$53$	−10.5893	−1.45455	−0.727273	−	0.686349i	$-0.759212\pi$
$53$	−10.5893	−1.45455	−0.727273	+	0.686349i	$0.759212\pi$
$54$	0	0
$55$	−14.8318	−1.99992
$56$	0	0
$57$	−0.724081	−0.0959068
$58$	0	0
$59$	−0.696653	−0.0906965	−0.0453482	−	0.998971i	$-0.514440\pi$
$59$	−0.696653	−0.0906965	−0.0453482	+	0.998971i	$0.514440\pi$
$60$	0	0
$61$	−4.04498	−0.517906	−0.258953	−	0.965890i	$-0.583378\pi$
$61$	−4.04498	−0.517906	−0.258953	+	0.965890i	$0.583378\pi$
$62$	0	0
$63$	9.56173	1.20466
$64$	0	0
$65$	9.35433	1.16026
$66$	0	0
$67$	8.27304	1.01071	0.505356	−	0.862911i	$-0.331361\pi$
$67$	8.27304	1.01071	0.505356	+	0.862911i	$0.331361\pi$
$68$	0	0
$69$	−1.72704	−0.207911
$70$	0	0
$71$	−10.3096	−1.22353	−0.611763	−	0.791041i	$-0.709539\pi$
$71$	−10.3096	−1.22353	−0.611763	+	0.791041i	$0.709539\pi$
$72$	0	0
$73$	0.498979	0.0584010	0.0292005	−	0.999574i	$-0.490704\pi$
$73$	0.498979	0.0584010	0.0292005	+	0.999574i	$0.490704\pi$
$74$	0	0
$75$	−0.748625	−0.0864437
$76$	0	0
$77$	−21.0668	−2.40078
$78$	0	0
$79$	−3.99695	−0.449692	−0.224846	−	0.974394i	$-0.572188\pi$
$79$	−3.99695	−0.449692	−0.224846	+	0.974394i	$0.572188\pi$
$80$	0	0
$81$	6.20524	0.689471
$82$	0	0
$83$	−2.57663	−0.282822	−0.141411	−	0.989951i	$-0.545164\pi$
$83$	−2.57663	−0.282822	−0.141411	+	0.989951i	$0.545164\pi$
$84$	0	0
$85$	−14.5939	−1.58293
$86$	0	0
$87$	2.12906	0.228259
$88$	0	0
$89$	10.1843	1.07954	0.539769	−	0.841813i	$-0.318512\pi$
$89$	10.1843	1.07954	0.539769	+	0.841813i	$0.318512\pi$
$90$	0	0
$91$	13.2867	1.39282
$92$	0	0
$93$	−4.33572	−0.449593
$94$	0	0
$95$	−3.20740	−0.329072
$96$	0	0
$97$	−10.0418	−1.01960	−0.509798	−	0.860294i	$-0.670280\pi$
$97$	−10.0418	−1.01960	−0.509798	+	0.860294i	$0.670280\pi$
$98$	0	0
$99$	−15.8003	−1.58799
$100$	0	0
$101$	1.56157	0.155382	0.0776908	−	0.996978i	$-0.475245\pi$
$101$	1.56157	0.155382	0.0776908	+	0.996978i	$0.475245\pi$
$102$	0	0
$103$	−5.73092	−0.564685	−0.282342	−	0.959314i	$-0.591111\pi$
$103$	−5.73092	−0.564685	−0.282342	+	0.959314i	$0.591111\pi$
$104$	0	0
$105$	−5.09364	−0.497088
$106$	0	0
$107$	−12.9533	−1.25224	−0.626122	−	0.779725i	$-0.715359\pi$
$107$	−12.9533	−1.25224	−0.626122	+	0.779725i	$0.715359\pi$
$108$	0	0
$109$	−13.3112	−1.27499	−0.637493	−	0.770456i	$-0.720028\pi$
$109$	−13.3112	−1.27499	−0.637493	+	0.770456i	$0.720028\pi$
$110$	0	0
$111$	−6.79113	−0.644585
$112$	0	0
$113$	15.7215	1.47896	0.739478	−	0.673180i	$-0.235072\pi$
$113$	15.7215	1.47896	0.739478	+	0.673180i	$0.235072\pi$
$114$	0	0
$115$	−7.65011	−0.713376
$116$	0	0
$117$	9.96517	0.921280
$118$	0	0
$119$	−20.7289	−1.90022
$120$	0	0
$121$	23.8120	2.16472
$122$	0	0
$123$	5.47664	0.493812
$124$	0	0
$125$	9.25286	0.827601
$126$	0	0
$127$	0.459781	0.0407990	0.0203995	−	0.999792i	$-0.493506\pi$
$127$	0.459781	0.0407990	0.0203995	+	0.999792i	$0.493506\pi$
$128$	0	0
$129$	−0.845146	−0.0744110
$130$	0	0
$131$	8.51192	0.743690	0.371845	−	0.928295i	$-0.378725\pi$
$131$	8.51192	0.743690	0.371845	+	0.928295i	$0.378725\pi$
$132$	0	0
$133$	−4.55573	−0.395032
$134$	0	0
$135$	−8.10000	−0.697137
$136$	0	0
$137$	−9.85331	−0.841825	−0.420912	−	0.907101i	$-0.638290\pi$
$137$	−9.85331	−0.841825	−0.420912	+	0.907101i	$0.638290\pi$
$138$	0	0
$139$	−10.1679	−0.862432	−0.431216	−	0.902249i	$-0.641915\pi$
$139$	−10.1679	−0.862432	−0.431216	+	0.902249i	$0.641915\pi$
$140$	0	0
$141$	0.293307	0.0247009
$142$	0	0
$143$	−21.9557	−1.83603
$144$	0	0
$145$	9.43094	0.783197
$146$	0	0
$147$	−3.26242	−0.269080
$148$	0	0
$149$	−20.7603	−1.70075	−0.850373	−	0.526180i	$-0.823624\pi$
$149$	−20.7603	−1.70075	−0.850373	+	0.526180i	$0.823624\pi$
$150$	0	0
$151$	−16.2810	−1.32493	−0.662465	−	0.749093i	$-0.730490\pi$
$151$	−16.2810	−1.32493	−0.662465	+	0.749093i	$0.730490\pi$
$152$	0	0
$153$	−15.5469	−1.25689
$154$	0	0
$155$	−19.2056	−1.54263
$156$	0	0
$157$	2.72588	0.217549	0.108774	−	0.994066i	$-0.465307\pi$
$157$	2.72588	0.217549	0.108774	+	0.994066i	$0.465307\pi$
$158$	0	0
$159$	6.00937	0.476574
$160$	0	0
$161$	−10.8661	−0.856365
$162$	0	0
$163$	−6.34996	−0.497367	−0.248684	−	0.968585i	$-0.579998\pi$
$163$	−6.34996	−0.497367	−0.248684	+	0.968585i	$0.579998\pi$
$164$	0	0
$165$	8.41701	0.655263
$166$	0	0
$167$	−5.77575	−0.446941	−0.223470	−	0.974711i	$-0.571739\pi$
$167$	−5.77575	−0.446941	−0.223470	+	0.974711i	$0.571739\pi$
$168$	0	0
$169$	0.847305	0.0651773
$170$	0	0
$171$	−3.41684	−0.261293
$172$	0	0
$173$	9.21961	0.700954	0.350477	−	0.936571i	$-0.386019\pi$
$173$	9.21961	0.700954	0.350477	+	0.936571i	$0.386019\pi$
$174$	0	0
$175$	−4.71015	−0.356054
$176$	0	0
$177$	0.395348	0.0297162
$178$	0	0
$179$	−22.1368	−1.65458	−0.827291	−	0.561773i	$-0.810119\pi$
$179$	−22.1368	−1.65458	−0.827291	+	0.561773i	$0.810119\pi$
$180$	0	0
$181$	−2.14372	−0.159341	−0.0796706	−	0.996821i	$-0.525387\pi$
$181$	−2.14372	−0.159341	−0.0796706	+	0.996821i	$0.525387\pi$
$182$	0	0
$183$	2.29551	0.169689
$184$	0	0
$185$	−30.0821	−2.21168
$186$	0	0
$187$	34.2536	2.50487
$188$	0	0
$189$	−11.5051	−0.836871
$190$	0	0
$191$	13.1652	0.952598	0.476299	−	0.879283i	$-0.341978\pi$
$191$	13.1652	0.952598	0.476299	+	0.879283i	$0.341978\pi$
$192$	0	0
$193$	−0.207237	−0.0149173	−0.00745863	−	0.999972i	$-0.502374\pi$
$193$	−0.207237	−0.0149173	−0.00745863	+	0.999972i	$0.502374\pi$
$194$	0	0
$195$	−5.30855	−0.380153
$196$	0	0
$197$	−23.7876	−1.69480	−0.847399	−	0.530956i	$-0.821833\pi$
$197$	−23.7876	−1.69480	−0.847399	+	0.530956i	$0.821833\pi$
$198$	0	0
$199$	21.5227	1.52570	0.762852	−	0.646573i	$-0.223799\pi$
$199$	21.5227	1.52570	0.762852	+	0.646573i	$0.223799\pi$
$200$	0	0
$201$	−4.69492	−0.331154
$202$	0	0
$203$	13.3955	0.940181
$204$	0	0
$205$	24.2594	1.69435
$206$	0	0
$207$	−8.14966	−0.566441
$208$	0	0
$209$	7.52814	0.520732
$210$	0	0
$211$	−0.734696	−0.0505786	−0.0252893	−	0.999680i	$-0.508051\pi$
$211$	−0.734696	−0.0505786	−0.0252893	+	0.999680i	$0.508051\pi$
$212$	0	0
$213$	5.85067	0.400881
$214$	0	0
$215$	−3.74367	−0.255316
$216$	0	0
$217$	−27.2792	−1.85183
$218$	0	0
$219$	−0.283169	−0.0191348
$220$	0	0
$221$	−21.6035	−1.45321
$222$	0	0
$223$	17.8155	1.19301	0.596506	−	0.802609i	$-0.296555\pi$
$223$	17.8155	1.19301	0.596506	+	0.802609i	$0.296555\pi$
$224$	0	0
$225$	−3.53266	−0.235511
$226$	0	0
$227$	27.2295	1.80729	0.903643	−	0.428286i	$-0.140882\pi$
$227$	27.2295	1.80729	0.903643	+	0.428286i	$0.140882\pi$
$228$	0	0
$229$	10.8044	0.713974	0.356987	−	0.934109i	$-0.383804\pi$
$229$	10.8044	0.713974	0.356987	+	0.934109i	$0.383804\pi$
$230$	0	0
$231$	11.9553	0.786604
$232$	0	0
$233$	−0.658609	−0.0431469	−0.0215735	−	0.999767i	$-0.506868\pi$
$233$	−0.658609	−0.0431469	−0.0215735	+	0.999767i	$0.506868\pi$
$234$	0	0
$235$	1.29924	0.0847530
$236$	0	0
$237$	2.26826	0.147339
$238$	0	0
$239$	−0.416275	−0.0269266	−0.0134633	−	0.999909i	$-0.504286\pi$
$239$	−0.416275	−0.0269266	−0.0134633	+	0.999909i	$0.504286\pi$
$240$	0	0
$241$	−23.5233	−1.51527	−0.757633	−	0.652681i	$-0.773644\pi$
$241$	−23.5233	−1.51527	−0.757633	+	0.652681i	$0.773644\pi$
$242$	0	0
$243$	−13.1881	−0.846018
$244$	0	0
$245$	−14.4513	−0.923258
$246$	0	0
$247$	−4.74795	−0.302105
$248$	0	0
$249$	1.46223	0.0926651
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	17.9557	1.12886
$254$	0	0
$255$	8.28201	0.518640
$256$	0	0
$257$	9.22863	0.575666	0.287833	−	0.957681i	$-0.407065\pi$
$257$	9.22863	0.575666	0.287833	+	0.957681i	$0.407065\pi$
$258$	0	0
$259$	−42.7280	−2.65499
$260$	0	0
$261$	10.0468	0.621880
$262$	0	0
$263$	10.9896	0.677649	0.338824	−	0.940850i	$-0.389971\pi$
$263$	10.9896	0.677649	0.338824	+	0.940850i	$0.389971\pi$
$264$	0	0
$265$	26.6192	1.63521
$266$	0	0
$267$	−5.77958	−0.353705
$268$	0	0
$269$	15.7124	0.957999	0.479000	−	0.877815i	$-0.340999\pi$
$269$	15.7124	0.957999	0.479000	+	0.877815i	$0.340999\pi$
$270$	0	0
$271$	9.63066	0.585021	0.292510	−	0.956262i	$-0.405509\pi$
$271$	9.63066	0.585021	0.292510	+	0.956262i	$0.405509\pi$
$272$	0	0
$273$	−7.54016	−0.456351
$274$	0	0
$275$	7.78332	0.469352
$276$	0	0
$277$	−9.75848	−0.586330	−0.293165	−	0.956062i	$-0.594709\pi$
$277$	−9.75848	−0.586330	−0.293165	+	0.956062i	$0.594709\pi$
$278$	0	0
$279$	−20.4597	−1.22489
$280$	0	0
$281$	14.3172	0.854091	0.427045	−	0.904230i	$-0.359554\pi$
$281$	14.3172	0.854091	0.427045	+	0.904230i	$0.359554\pi$
$282$	0	0
$283$	−15.3547	−0.912745	−0.456373	−	0.889789i	$-0.650852\pi$
$283$	−15.3547	−0.912745	−0.456373	+	0.889789i	$0.650852\pi$
$284$	0	0
$285$	1.82019	0.107819
$286$	0	0
$287$	34.4576	2.03397
$288$	0	0
$289$	16.7043	0.982603
$290$	0	0
$291$	5.69872	0.334065
$292$	0	0
$293$	−29.3915	−1.71707	−0.858536	−	0.512753i	$-0.828626\pi$
$293$	−29.3915	−1.71707	−0.858536	+	0.512753i	$0.828626\pi$
$294$	0	0
$295$	1.75124	0.101961
$296$	0	0
$297$	19.0116	1.10317
$298$	0	0
$299$	−11.3245	−0.654914
$300$	0	0
$301$	−5.31744	−0.306492
$302$	0	0
$303$	−0.886184	−0.0509099
$304$	0	0
$305$	10.1683	0.582232
$306$	0	0
$307$	−11.7631	−0.671357	−0.335679	−	0.941977i	$-0.608966\pi$
$307$	−11.7631	−0.671357	−0.335679	+	0.941977i	$0.608966\pi$
$308$	0	0
$309$	3.25228	0.185016
$310$	0	0
$311$	3.75176	0.212743	0.106372	−	0.994326i	$-0.466077\pi$
$311$	3.75176	0.212743	0.106372	+	0.994326i	$0.466077\pi$
$312$	0	0
$313$	−18.8975	−1.06815	−0.534074	−	0.845437i	$-0.679340\pi$
$313$	−18.8975	−1.06815	−0.534074	+	0.845437i	$0.679340\pi$
$314$	0	0
$315$	−24.0362	−1.35429
$316$	0	0
$317$	−26.0953	−1.46566	−0.732829	−	0.680413i	$-0.761801\pi$
$317$	−26.0953	−1.46566	−0.732829	+	0.680413i	$0.761801\pi$
$318$	0	0
$319$	−22.1355	−1.23935
$320$	0	0
$321$	7.35096	0.410291
$322$	0	0
$323$	7.40740	0.412159
$324$	0	0
$325$	−4.90889	−0.272296
$326$	0	0
$327$	7.55409	0.417742
$328$	0	0
$329$	1.84541	0.101741
$330$	0	0
$331$	−13.0555	−0.717593	−0.358797	−	0.933416i	$-0.616813\pi$
$331$	−13.0555	−0.717593	−0.358797	+	0.933416i	$0.616813\pi$
$332$	0	0
$333$	−32.0465	−1.75614
$334$	0	0
$335$	−20.7967	−1.13625
$336$	0	0
$337$	−5.36186	−0.292079	−0.146039	−	0.989279i	$-0.546653\pi$
$337$	−5.36186	−0.292079	−0.146039	+	0.989279i	$0.546653\pi$
$338$	0	0
$339$	−8.92192	−0.484572
$340$	0	0
$341$	45.0777	2.44109
$342$	0	0
$343$	4.46751	0.241223
$344$	0	0
$345$	4.34141	0.233734
$346$	0	0
$347$	2.05799	0.110479	0.0552393	−	0.998473i	$-0.482408\pi$
$347$	2.05799	0.110479	0.0552393	+	0.998473i	$0.482408\pi$
$348$	0	0
$349$	−8.41924	−0.450671	−0.225336	−	0.974281i	$-0.572348\pi$
$349$	−8.41924	−0.450671	−0.225336	+	0.974281i	$0.572348\pi$
$350$	0	0
$351$	−11.9905	−0.640006
$352$	0	0
$353$	18.3454	0.976425	0.488212	−	0.872725i	$-0.337649\pi$
$353$	18.3454	0.976425	0.488212	+	0.872725i	$0.337649\pi$
$354$	0	0
$355$	25.9162	1.37549
$356$	0	0
$357$	11.7636	0.622596
$358$	0	0
$359$	4.65228	0.245538	0.122769	−	0.992435i	$-0.460823\pi$
$359$	4.65228	0.245538	0.122769	+	0.992435i	$0.460823\pi$
$360$	0	0
$361$	−17.3720	−0.914317
$362$	0	0
$363$	−13.5132	−0.709260
$364$	0	0
$365$	−1.25433	−0.0656546
$366$	0	0
$367$	19.2614	1.00544	0.502719	−	0.864450i	$-0.332333\pi$
$367$	19.2614	1.00544	0.502719	+	0.864450i	$0.332333\pi$
$368$	0	0
$369$	25.8436	1.34536
$370$	0	0
$371$	37.8094	1.96297
$372$	0	0
$373$	−33.0082	−1.70910	−0.854549	−	0.519370i	$-0.826167\pi$
$373$	−33.0082	−1.70910	−0.854549	+	0.519370i	$0.826167\pi$
$374$	0	0
$375$	−5.25097	−0.271159
$376$	0	0
$377$	13.9607	0.719013
$378$	0	0
$379$	27.9552	1.43596	0.717982	−	0.696061i	$-0.245066\pi$
$379$	27.9552	1.43596	0.717982	+	0.696061i	$0.245066\pi$
$380$	0	0
$381$	−0.260924	−0.0133676
$382$	0	0
$383$	6.45874	0.330026	0.165013	−	0.986291i	$-0.447233\pi$
$383$	6.45874	0.330026	0.165013	+	0.986291i	$0.447233\pi$
$384$	0	0
$385$	52.9576	2.69897
$386$	0	0
$387$	−3.98814	−0.202728
$388$	0	0
$389$	−29.7590	−1.50884	−0.754419	−	0.656393i	$-0.772081\pi$
$389$	−29.7590	−1.50884	−0.754419	+	0.656393i	$0.772081\pi$
$390$	0	0
$391$	17.6677	0.893494
$392$	0	0
$393$	−4.83049	−0.243666
$394$	0	0
$395$	10.0475	0.505546
$396$	0	0
$397$	20.2985	1.01875	0.509376	−	0.860544i	$-0.329876\pi$
$397$	20.2985	1.01875	0.509376	+	0.860544i	$0.329876\pi$
$398$	0	0
$399$	2.58536	0.129430
$400$	0	0
$401$	38.5884	1.92701	0.963507	−	0.267683i	$-0.0862578\pi$
$401$	38.5884	1.92701	0.963507	+	0.267683i	$0.0862578\pi$
$402$	0	0
$403$	−28.4302	−1.41621
$404$	0	0
$405$	−15.5987	−0.775106
$406$	0	0
$407$	70.6061	3.49982
$408$	0	0
$409$	−20.0353	−0.990683	−0.495341	−	0.868698i	$-0.664957\pi$
$409$	−20.0353	−0.990683	−0.495341	+	0.868698i	$0.664957\pi$
$410$	0	0
$411$	5.59172	0.275819
$412$	0	0
$413$	2.48743	0.122398
$414$	0	0
$415$	6.47712	0.317949
$416$	0	0
$417$	5.77027	0.282571
$418$	0	0
$419$	−38.7442	−1.89278	−0.946388	−	0.323032i	$-0.895298\pi$
$419$	−38.7442	−1.89278	−0.946388	+	0.323032i	$0.895298\pi$
$420$	0	0
$421$	−39.3462	−1.91761	−0.958807	−	0.284058i	$-0.908319\pi$
$421$	−39.3462	−1.91761	−0.958807	+	0.284058i	$0.908319\pi$
$422$	0	0
$423$	1.38408	0.0672962
$424$	0	0
$425$	7.65848	0.371491
$426$	0	0
$427$	14.4428	0.698935
$428$	0	0
$429$	12.4598	0.601564
$430$	0	0
$431$	−1.50325	−0.0724091	−0.0362045	−	0.999344i	$-0.511527\pi$
$431$	−1.50325	−0.0724091	−0.0362045	+	0.999344i	$0.511527\pi$
$432$	0	0
$433$	26.6287	1.27969	0.639846	−	0.768503i	$-0.278998\pi$
$433$	26.6287	1.27969	0.639846	+	0.768503i	$0.278998\pi$
$434$	0	0
$435$	−5.35203	−0.256610
$436$	0	0
$437$	3.88294	0.185746
$438$	0	0
$439$	−27.9739	−1.33512	−0.667562	−	0.744554i	$-0.732662\pi$
$439$	−27.9739	−1.33512	−0.667562	+	0.744554i	$0.732662\pi$
$440$	0	0
$441$	−15.3950	−0.733093
$442$	0	0
$443$	−22.4192	−1.06517	−0.532585	−	0.846377i	$-0.678779\pi$
$443$	−22.4192	−1.06517	−0.532585	+	0.846377i	$0.678779\pi$
$444$	0	0
$445$	−25.6013	−1.21362
$446$	0	0
$447$	11.7814	0.557240
$448$	0	0
$449$	−38.2434	−1.80482	−0.902408	−	0.430882i	$-0.858202\pi$
$449$	−38.2434	−1.80482	−0.902408	+	0.430882i	$0.858202\pi$
$450$	0	0
$451$	−56.9397	−2.68118
$452$	0	0
$453$	9.23943	0.434106
$454$	0	0
$455$	−33.4000	−1.56582
$456$	0	0
$457$	17.4924	0.818261	0.409131	−	0.912476i	$-0.365832\pi$
$457$	17.4924	0.818261	0.409131	+	0.912476i	$0.365832\pi$
$458$	0	0
$459$	18.7067	0.873154
$460$	0	0
$461$	21.4690	0.999910	0.499955	−	0.866051i	$-0.333350\pi$
$461$	21.4690	0.999910	0.499955	+	0.866051i	$0.333350\pi$
$462$	0	0
$463$	1.43858	0.0668567	0.0334283	−	0.999441i	$-0.489357\pi$
$463$	1.43858	0.0668567	0.0334283	+	0.999441i	$0.489357\pi$
$464$	0	0
$465$	10.8991	0.505434
$466$	0	0
$467$	−4.85042	−0.224451	−0.112225	−	0.993683i	$-0.535798\pi$
$467$	−4.85042	−0.224451	−0.112225	+	0.993683i	$0.535798\pi$
$468$	0	0
$469$	−29.5392	−1.36400
$470$	0	0
$471$	−1.54693	−0.0712787
$472$	0	0
$473$	8.78683	0.404019
$474$	0	0
$475$	1.68315	0.0772284
$476$	0	0
$477$	28.3575	1.29840
$478$	0	0
$479$	−36.1860	−1.65338	−0.826691	−	0.562656i	$-0.809780\pi$
$479$	−36.1860	−1.65338	−0.826691	+	0.562656i	$0.809780\pi$
$480$	0	0
$481$	−44.5308	−2.03043
$482$	0	0
$483$	6.16646	0.280583
$484$	0	0
$485$	25.2432	1.14623
$486$	0	0
$487$	37.2705	1.68889	0.844443	−	0.535645i	$-0.179931\pi$
$487$	37.2705	1.68889	0.844443	+	0.535645i	$0.179931\pi$
$488$	0	0
$489$	3.60359	0.162960
$490$	0	0
$491$	16.0763	0.725514	0.362757	−	0.931884i	$-0.381836\pi$
$491$	16.0763	0.725514	0.362757	+	0.931884i	$0.381836\pi$
$492$	0	0
$493$	−21.7805	−0.980943
$494$	0	0
$495$	39.7188	1.78523
$496$	0	0
$497$	36.8109	1.65119
$498$	0	0
$499$	−31.7957	−1.42337	−0.711686	−	0.702498i	$-0.752068\pi$
$499$	−31.7957	−1.42337	−0.711686	+	0.702498i	$0.752068\pi$
$500$	0	0
$501$	3.27772	0.146438
$502$	0	0
$503$	−17.6432	−0.786672	−0.393336	−	0.919395i	$-0.628679\pi$
$503$	−17.6432	−0.786672	−0.393336	+	0.919395i	$0.628679\pi$
$504$	0	0
$505$	−3.92546	−0.174681
$506$	0	0
$507$	−0.480843	−0.0213550
$508$	0	0
$509$	−6.23876	−0.276528	−0.138264	−	0.990395i	$-0.544152\pi$
$509$	−6.23876	−0.276528	−0.138264	+	0.990395i	$0.544152\pi$
$510$	0	0
$511$	−1.78163	−0.0788145
$512$	0	0
$513$	4.11129	0.181518
$514$	0	0
$515$	14.4064	0.634821
$516$	0	0
$517$	−3.04946	−0.134115
$518$	0	0
$519$	−5.23210	−0.229664
$520$	0	0
$521$	−40.1251	−1.75791	−0.878957	−	0.476902i	$-0.841760\pi$
$521$	−40.1251	−1.75791	−0.878957	+	0.476902i	$0.841760\pi$
$522$	0	0
$523$	23.8114	1.04120	0.520600	−	0.853801i	$-0.325708\pi$
$523$	23.8114	1.04120	0.520600	+	0.853801i	$0.325708\pi$
$524$	0	0
$525$	2.67300	0.116659
$526$	0	0
$527$	44.3547	1.93212
$528$	0	0
$529$	−13.7386	−0.597332
$530$	0	0
$531$	1.86560	0.0809601
$532$	0	0
$533$	35.9115	1.55550
$534$	0	0
$535$	32.5620	1.40778
$536$	0	0
$537$	12.5626	0.542115
$538$	0	0
$539$	33.9188	1.46099
$540$	0	0
$541$	−34.0138	−1.46237	−0.731183	−	0.682182i	$-0.761031\pi$
$541$	−34.0138	−1.46237	−0.731183	+	0.682182i	$0.761031\pi$
$542$	0	0
$543$	1.21655	0.0522073
$544$	0	0
$545$	33.4617	1.43334
$546$	0	0
$547$	−37.0413	−1.58377	−0.791885	−	0.610670i	$-0.790900\pi$
$547$	−37.0413	−1.58377	−0.791885	+	0.610670i	$0.790900\pi$
$548$	0	0
$549$	10.8322	0.462309
$550$	0	0
$551$	−4.78683	−0.203926
$552$	0	0
$553$	14.2713	0.606877
$554$	0	0
$555$	17.0715	0.724645
$556$	0	0
$557$	1.66769	0.0706623	0.0353311	−	0.999376i	$-0.488751\pi$
$557$	1.66769	0.0706623	0.0353311	+	0.999376i	$0.488751\pi$
$558$	0	0
$559$	−5.54180	−0.234393
$560$	0	0
$561$	−19.4388	−0.820708
$562$	0	0
$563$	−35.1527	−1.48151	−0.740754	−	0.671776i	$-0.765532\pi$
$563$	−35.1527	−1.48151	−0.740754	+	0.671776i	$0.765532\pi$
$564$	0	0
$565$	−39.5207	−1.66265
$566$	0	0
$567$	−22.1561	−0.930468
$568$	0	0
$569$	10.9069	0.457239	0.228620	−	0.973516i	$-0.426579\pi$
$569$	10.9069	0.457239	0.228620	+	0.973516i	$0.426579\pi$
$570$	0	0
$571$	−3.19869	−0.133861	−0.0669305	−	0.997758i	$-0.521321\pi$
$571$	−3.19869	−0.133861	−0.0669305	+	0.997758i	$0.521321\pi$
$572$	0	0
$573$	−7.47119	−0.312114
$574$	0	0
$575$	4.01456	0.167419
$576$	0	0
$577$	36.4286	1.51654	0.758271	−	0.651939i	$-0.226044\pi$
$577$	36.4286	1.51654	0.758271	+	0.651939i	$0.226044\pi$
$578$	0	0
$579$	0.117607	0.00488756
$580$	0	0
$581$	9.19997	0.381679
$582$	0	0
$583$	−62.4783	−2.58759
$584$	0	0
$585$	−25.0504	−1.03571
$586$	0	0
$587$	31.9944	1.32055	0.660276	−	0.751023i	$-0.270439\pi$
$587$	31.9944	1.32055	0.660276	+	0.751023i	$0.270439\pi$
$588$	0	0
$589$	9.74811	0.401664
$590$	0	0
$591$	13.4994	0.555291
$592$	0	0
$593$	−13.1856	−0.541470	−0.270735	−	0.962654i	$-0.587267\pi$
$593$	−13.1856	−0.541470	−0.270735	+	0.962654i	$0.587267\pi$
$594$	0	0
$595$	52.1083	2.13623
$596$	0	0
$597$	−12.2141	−0.499888
$598$	0	0
$599$	44.3812	1.81337	0.906683	−	0.421812i	$-0.138606\pi$
$599$	44.3812	1.81337	0.906683	+	0.421812i	$0.138606\pi$
$600$	0	0
$601$	6.36845	0.259775	0.129887	−	0.991529i	$-0.458538\pi$
$601$	6.36845	0.259775	0.129887	+	0.991529i	$0.458538\pi$
$602$	0	0
$603$	−22.1548	−0.902211
$604$	0	0
$605$	−59.8584	−2.43359
$606$	0	0
$607$	−47.1334	−1.91308	−0.956542	−	0.291594i	$-0.905814\pi$
$607$	−47.1334	−1.91308	−0.956542	+	0.291594i	$0.905814\pi$
$608$	0	0
$609$	−7.60191	−0.308045
$610$	0	0
$611$	1.92328	0.0778074
$612$	0	0
$613$	−6.63075	−0.267814	−0.133907	−	0.990994i	$-0.542752\pi$
$613$	−6.63075	−0.267814	−0.133907	+	0.990994i	$0.542752\pi$
$614$	0	0
$615$	−13.7672	−0.555145
$616$	0	0
$617$	0.340829	0.0137213	0.00686063	−	0.999976i	$-0.497816\pi$
$617$	0.340829	0.0137213	0.00686063	+	0.999976i	$0.497816\pi$
$618$	0	0
$619$	0.278583	0.0111972	0.00559859	−	0.999984i	$-0.498218\pi$
$619$	0.278583	0.0111972	0.00559859	+	0.999984i	$0.498218\pi$
$620$	0	0
$621$	9.80601	0.393502
$622$	0	0
$623$	−36.3636	−1.45688
$624$	0	0
$625$	−29.8556	−1.19423
$626$	0	0
$627$	−4.27219	−0.170615
$628$	0	0
$629$	69.4737	2.77010
$630$	0	0
$631$	−39.3043	−1.56468	−0.782340	−	0.622851i	$-0.785974\pi$
$631$	−39.3043	−1.56468	−0.782340	+	0.622851i	$0.785974\pi$
$632$	0	0
$633$	0.416938	0.0165718
$634$	0	0
$635$	−1.15580	−0.0458663
$636$	0	0
$637$	−21.3924	−0.847597
$638$	0	0
$639$	27.6086	1.09218
$640$	0	0
$641$	−7.84087	−0.309696	−0.154848	−	0.987938i	$-0.549489\pi$
$641$	−7.84087	−0.309696	−0.154848	+	0.987938i	$0.549489\pi$
$642$	0	0
$643$	−5.37222	−0.211860	−0.105930	−	0.994374i	$-0.533782\pi$
$643$	−5.37222	−0.211860	−0.105930	+	0.994374i	$0.533782\pi$
$644$	0	0
$645$	2.12452	0.0836531
$646$	0	0
$647$	36.5528	1.43704	0.718520	−	0.695506i	$-0.244820\pi$
$647$	36.5528	1.43704	0.718520	+	0.695506i	$0.244820\pi$
$648$	0	0
$649$	−4.11037	−0.161346
$650$	0	0
$651$	15.4809	0.606743
$652$	0	0
$653$	−19.3446	−0.757011	−0.378506	−	0.925599i	$-0.623562\pi$
$653$	−19.3446	−0.757011	−0.378506	+	0.925599i	$0.623562\pi$
$654$	0	0
$655$	−21.3972	−0.836059
$656$	0	0
$657$	−1.33624	−0.0521316
$658$	0	0
$659$	−32.0544	−1.24866	−0.624331	−	0.781160i	$-0.714628\pi$
$659$	−32.0544	−1.24866	−0.624331	+	0.781160i	$0.714628\pi$
$660$	0	0
$661$	−4.01053	−0.155992	−0.0779958	−	0.996954i	$-0.524852\pi$
$661$	−4.01053	−0.155992	−0.0779958	+	0.996954i	$0.524852\pi$
$662$	0	0
$663$	12.2599	0.476137
$664$	0	0
$665$	11.4522	0.444096
$666$	0	0
$667$	−11.4173	−0.442079
$668$	0	0
$669$	−10.1102	−0.390884
$670$	0	0
$671$	−23.8660	−0.921338
$672$	0	0
$673$	−22.6938	−0.874783	−0.437391	−	0.899271i	$-0.644098\pi$
$673$	−22.6938	−0.874783	−0.437391	+	0.899271i	$0.644098\pi$
$674$	0	0
$675$	4.25065	0.163608
$676$	0	0
$677$	14.8406	0.570370	0.285185	−	0.958472i	$-0.407945\pi$
$677$	14.8406	0.570370	0.285185	+	0.958472i	$0.407945\pi$
$678$	0	0
$679$	35.8549	1.37598
$680$	0	0
$681$	−15.4527	−0.592148
$682$	0	0
$683$	33.4346	1.27934	0.639670	−	0.768650i	$-0.279071\pi$
$683$	33.4346	1.27934	0.639670	+	0.768650i	$0.279071\pi$
$684$	0	0
$685$	24.7692	0.946383
$686$	0	0
$687$	−6.13146	−0.233930
$688$	0	0
$689$	39.4047	1.50120
$690$	0	0
$691$	−5.03320	−0.191472	−0.0957359	−	0.995407i	$-0.530520\pi$
$691$	−5.03320	−0.191472	−0.0957359	+	0.995407i	$0.530520\pi$
$692$	0	0
$693$	56.4158	2.14306
$694$	0	0
$695$	25.5601	0.969549
$696$	0	0
$697$	−56.0264	−2.12215
$698$	0	0
$699$	0.373759	0.0141369
$700$	0	0
$701$	5.29981	0.200171	0.100086	−	0.994979i	$-0.468088\pi$
$701$	5.29981	0.200171	0.100086	+	0.994979i	$0.468088\pi$
$702$	0	0
$703$	15.2687	0.575869
$704$	0	0
$705$	−0.737314	−0.0277688
$706$	0	0
$707$	−5.57564	−0.209694
$708$	0	0
$709$	−27.9360	−1.04916	−0.524580	−	0.851361i	$-0.675778\pi$
$709$	−27.9360	−1.04916	−0.524580	+	0.851361i	$0.675778\pi$
$710$	0	0
$711$	10.7036	0.401417
$712$	0	0
$713$	23.2506	0.870743
$714$	0	0
$715$	55.1921	2.06407
$716$	0	0
$717$	0.236235	0.00882235
$718$	0	0
$719$	−18.0185	−0.671975	−0.335988	−	0.941866i	$-0.609070\pi$
$719$	−18.0185	−0.671975	−0.335988	+	0.941866i	$0.609070\pi$
$720$	0	0
$721$	20.4625	0.762064
$722$	0	0
$723$	13.3494	0.496469
$724$	0	0
$725$	−4.94909	−0.183805
$726$	0	0
$727$	48.3645	1.79374	0.896870	−	0.442295i	$-0.145836\pi$
$727$	48.3645	1.79374	0.896870	+	0.442295i	$0.145836\pi$
$728$	0	0
$729$	−11.1315	−0.412278
$730$	0	0
$731$	8.64591	0.319780
$732$	0	0
$733$	−23.0668	−0.851991	−0.425995	−	0.904725i	$-0.640076\pi$
$733$	−23.0668	−0.851991	−0.425995	+	0.904725i	$0.640076\pi$
$734$	0	0
$735$	8.20106	0.302501
$736$	0	0
$737$	48.8123	1.79802
$738$	0	0
$739$	−3.53230	−0.129938	−0.0649689	−	0.997887i	$-0.520695\pi$
$739$	−3.53230	−0.129938	−0.0649689	+	0.997887i	$0.520695\pi$
$740$	0	0
$741$	2.69445	0.0989829
$742$	0	0
$743$	15.4034	0.565097	0.282548	−	0.959253i	$-0.408820\pi$
$743$	15.4034	0.565097	0.282548	+	0.959253i	$0.408820\pi$
$744$	0	0
$745$	52.1870	1.91199
$746$	0	0
$747$	6.90008	0.252461
$748$	0	0
$749$	46.2504	1.68995
$750$	0	0
$751$	2.91811	0.106483	0.0532417	−	0.998582i	$-0.483045\pi$
$751$	2.91811	0.106483	0.0532417	+	0.998582i	$0.483045\pi$
$752$	0	0
$753$	−0.567497	−0.0206807
$754$	0	0
$755$	40.9271	1.48949
$756$	0	0
$757$	18.3769	0.667919	0.333959	−	0.942587i	$-0.391615\pi$
$757$	18.3769	0.667919	0.333959	+	0.942587i	$0.391615\pi$
$758$	0	0
$759$	−10.1898	−0.369866
$760$	0	0
$761$	−32.1376	−1.16499	−0.582493	−	0.812836i	$-0.697923\pi$
$761$	−32.1376	−1.16499	−0.582493	+	0.812836i	$0.697923\pi$
$762$	0	0
$763$	47.5284	1.72064
$764$	0	0
$765$	39.0818	1.41300
$766$	0	0
$767$	2.59238	0.0936055
$768$	0	0
$769$	−34.3458	−1.23854	−0.619270	−	0.785178i	$-0.712572\pi$
$769$	−34.3458	−1.23854	−0.619270	+	0.785178i	$0.712572\pi$
$770$	0	0
$771$	−5.23722	−0.188614
$772$	0	0
$773$	−7.38522	−0.265628	−0.132814	−	0.991141i	$-0.542401\pi$
$773$	−7.38522	−0.265628	−0.132814	+	0.991141i	$0.542401\pi$
$774$	0	0
$775$	10.0785	0.362032
$776$	0	0
$777$	24.2480	0.869893
$778$	0	0
$779$	−12.3133	−0.441169
$780$	0	0
$781$	−60.8284	−2.17661
$782$	0	0
$783$	−12.0887	−0.432015
$784$	0	0
$785$	−6.85231	−0.244569
$786$	0	0
$787$	−15.5659	−0.554863	−0.277432	−	0.960745i	$-0.589483\pi$
$787$	−15.5659	−0.554863	−0.277432	+	0.960745i	$0.589483\pi$
$788$	0	0
$789$	−6.23658	−0.222028
$790$	0	0
$791$	−56.1344	−1.99591
$792$	0	0
$793$	15.0522	0.534518
$794$	0	0
$795$	−15.1063	−0.535766
$796$	0	0
$797$	−10.5659	−0.374262	−0.187131	−	0.982335i	$-0.559919\pi$
$797$	−10.5659	−0.374262	−0.187131	+	0.982335i	$0.559919\pi$
$798$	0	0
$799$	−3.00055	−0.106152
$800$	0	0
$801$	−27.2731	−0.963648
$802$	0	0
$803$	2.94406	0.103893
$804$	0	0
$805$	27.3151	0.962729
$806$	0	0
$807$	−8.91672	−0.313883
$808$	0	0
$809$	35.5466	1.24975	0.624877	−	0.780723i	$-0.285149\pi$
$809$	35.5466	1.24975	0.624877	+	0.780723i	$0.285149\pi$
$810$	0	0
$811$	−9.55645	−0.335572	−0.167786	−	0.985823i	$-0.553662\pi$
$811$	−9.55645	−0.335572	−0.167786	+	0.985823i	$0.553662\pi$
$812$	0	0
$813$	−5.46537	−0.191679
$814$	0	0
$815$	15.9625	0.559142
$816$	0	0
$817$	1.90017	0.0664784
$818$	0	0
$819$	−35.5811	−1.24330
$820$	0	0
$821$	24.8908	0.868693	0.434347	−	0.900746i	$-0.356979\pi$
$821$	24.8908	0.868693	0.434347	+	0.900746i	$0.356979\pi$
$822$	0	0
$823$	28.0259	0.976920	0.488460	−	0.872586i	$-0.337559\pi$
$823$	28.0259	0.976920	0.488460	+	0.872586i	$0.337559\pi$
$824$	0	0
$825$	−4.41701	−0.153780
$826$	0	0
$827$	35.9253	1.24924	0.624622	−	0.780927i	$-0.285253\pi$
$827$	35.9253	1.24924	0.624622	+	0.780927i	$0.285253\pi$
$828$	0	0
$829$	28.3202	0.983603	0.491801	−	0.870707i	$-0.336339\pi$
$829$	28.3202	0.983603	0.491801	+	0.870707i	$0.336339\pi$
$830$	0	0
$831$	5.53791	0.192108
$832$	0	0
$833$	33.3748	1.15637
$834$	0	0
$835$	14.5190	0.502452
$836$	0	0
$837$	24.6180	0.850921
$838$	0	0
$839$	14.3939	0.496932	0.248466	−	0.968641i	$-0.420074\pi$
$839$	14.3939	0.496932	0.248466	+	0.968641i	$0.420074\pi$
$840$	0	0
$841$	−14.9250	−0.514654
$842$	0	0
$843$	−8.12495	−0.279838
$844$	0	0
$845$	−2.12995	−0.0732726
$846$	0	0
$847$	−85.0216	−2.92138
$848$	0	0
$849$	8.71377	0.299056
$850$	0	0
$851$	36.4180	1.24839
$852$	0	0
$853$	−18.4618	−0.632119	−0.316060	−	0.948739i	$-0.602360\pi$
$853$	−18.4618	−0.632119	−0.316060	+	0.948739i	$0.602360\pi$
$854$	0	0
$855$	8.58925	0.293746
$856$	0	0
$857$	−2.82668	−0.0965576	−0.0482788	−	0.998834i	$-0.515374\pi$
$857$	−2.82668	−0.0965576	−0.0482788	+	0.998834i	$0.515374\pi$
$858$	0	0
$859$	−9.07219	−0.309539	−0.154770	−	0.987951i	$-0.549464\pi$
$859$	−9.07219	−0.309539	−0.154770	+	0.987951i	$0.549464\pi$
$860$	0	0
$861$	−19.5546	−0.666419
$862$	0	0
$863$	35.2858	1.20114	0.600572	−	0.799571i	$-0.294940\pi$
$863$	35.2858	1.20114	0.600572	+	0.799571i	$0.294940\pi$
$864$	0	0
$865$	−23.1762	−0.788015
$866$	0	0
$867$	−9.47962	−0.321945
$868$	0	0
$869$	−23.5827	−0.799988
$870$	0	0
$871$	−30.7856	−1.04313
$872$	0	0
$873$	26.8915	0.910141
$874$	0	0
$875$	−33.0377	−1.11688
$876$	0	0
$877$	36.5408	1.23390	0.616948	−	0.787004i	$-0.288369\pi$
$877$	36.5408	1.23390	0.616948	+	0.787004i	$0.288369\pi$
$878$	0	0
$879$	16.6796	0.562589
$880$	0	0
$881$	4.28437	0.144344	0.0721720	−	0.997392i	$-0.477007\pi$
$881$	4.28437	0.144344	0.0721720	+	0.997392i	$0.477007\pi$
$882$	0	0
$883$	49.0464	1.65054	0.825271	−	0.564737i	$-0.191022\pi$
$883$	49.0464	1.65054	0.825271	+	0.564737i	$0.191022\pi$
$884$	0	0
$885$	−0.993825	−0.0334071
$886$	0	0
$887$	−39.6151	−1.33014	−0.665072	−	0.746779i	$-0.731599\pi$
$887$	−39.6151	−1.33014	−0.665072	+	0.746779i	$0.731599\pi$
$888$	0	0
$889$	−1.64167	−0.0550598
$890$	0	0
$891$	36.6120	1.22655
$892$	0	0
$893$	−0.659450	−0.0220677
$894$	0	0
$895$	55.6474	1.86009
$896$	0	0
$897$	6.42664	0.214579
$898$	0	0
$899$	−28.6630	−0.955965
$900$	0	0
$901$	−61.4763	−2.04807
$902$	0	0
$903$	3.01763	0.100420
$904$	0	0
$905$	5.38887	0.179132
$906$	0	0
$907$	−4.28678	−0.142340	−0.0711701	−	0.997464i	$-0.522673\pi$
$907$	−4.28678	−0.142340	−0.0711701	+	0.997464i	$0.522673\pi$
$908$	0	0
$909$	−4.18179	−0.138701
$910$	0	0
$911$	7.75597	0.256967	0.128483	−	0.991712i	$-0.458989\pi$
$911$	7.75597	0.256967	0.128483	+	0.991712i	$0.458989\pi$
$912$	0	0
$913$	−15.2025	−0.503131
$914$	0	0
$915$	−5.77045	−0.190765
$916$	0	0
$917$	−30.3922	−1.00364
$918$	0	0
$919$	27.1298	0.894928	0.447464	−	0.894302i	$-0.352327\pi$
$919$	27.1298	0.894928	0.447464	+	0.894302i	$0.352327\pi$
$920$	0	0
$921$	6.67554	0.219967
$922$	0	0
$923$	38.3641	1.26277
$924$	0	0
$925$	15.7862	0.519048
$926$	0	0
$927$	15.3471	0.504065
$928$	0	0
$929$	−6.58348	−0.215997	−0.107999	−	0.994151i	$-0.534444\pi$
$929$	−6.58348	−0.215997	−0.107999	+	0.994151i	$0.534444\pi$
$930$	0	0
$931$	7.33499	0.240395
$932$	0	0
$933$	−2.12912	−0.0697041
$934$	0	0
$935$	−86.1066	−2.81599
$936$	0	0
$937$	2.36078	0.0771234	0.0385617	−	0.999256i	$-0.487722\pi$
$937$	2.36078	0.0771234	0.0385617	+	0.999256i	$0.487722\pi$
$938$	0	0
$939$	10.7243	0.349973
$940$	0	0
$941$	0.871631	0.0284144	0.0142072	−	0.999899i	$-0.495478\pi$
$941$	0.871631	0.0284144	0.0142072	+	0.999899i	$0.495478\pi$
$942$	0	0
$943$	−29.3689	−0.956384
$944$	0	0
$945$	28.9214	0.940813
$946$	0	0
$947$	37.0497	1.20395	0.601976	−	0.798514i	$-0.294380\pi$
$947$	37.0497	1.20395	0.601976	+	0.798514i	$0.294380\pi$
$948$	0	0
$949$	−1.85680	−0.0602742
$950$	0	0
$951$	14.8090	0.480215
$952$	0	0
$953$	39.5816	1.28217	0.641087	−	0.767468i	$-0.278484\pi$
$953$	39.5816	1.28217	0.641087	+	0.767468i	$0.278484\pi$
$954$	0	0
$955$	−33.0945	−1.07091
$956$	0	0
$957$	12.5618	0.406066
$958$	0	0
$959$	35.1817	1.13608
$960$	0	0
$961$	27.3706	0.882924
$962$	0	0
$963$	34.6883	1.11781
$964$	0	0
$965$	0.520952	0.0167700
$966$	0	0
$967$	−11.8405	−0.380764	−0.190382	−	0.981710i	$-0.560973\pi$
$967$	−11.8405	−0.380764	−0.190382	+	0.981710i	$0.560973\pi$
$968$	0	0
$969$	−4.20368	−0.135042
$970$	0	0
$971$	16.7305	0.536907	0.268453	−	0.963293i	$-0.413487\pi$
$971$	16.7305	0.536907	0.268453	+	0.963293i	$0.413487\pi$
$972$	0	0
$973$	36.3050	1.16389
$974$	0	0
$975$	2.78578	0.0892163
$976$	0	0
$977$	20.3038	0.649577	0.324789	−	0.945787i	$-0.394707\pi$
$977$	20.3038	0.649577	0.324789	+	0.945787i	$0.394707\pi$
$978$	0	0
$979$	60.0893	1.92046
$980$	0	0
$981$	35.6468	1.13811
$982$	0	0
$983$	−25.1022	−0.800637	−0.400318	−	0.916376i	$-0.631100\pi$
$983$	−25.1022	−0.800637	−0.400318	+	0.916376i	$0.631100\pi$
$984$	0	0
$985$	59.7972	1.90530
$986$	0	0
$987$	−1.04727	−0.0333348
$988$	0	0
$989$	4.53216	0.144114
$990$	0	0
$991$	−20.2761	−0.644091	−0.322046	−	0.946724i	$-0.604370\pi$
$991$	−20.2761	−0.644091	−0.322046	+	0.946724i	$0.604370\pi$
$992$	0	0
$993$	7.40894	0.235115
$994$	0	0
$995$	−54.1037	−1.71520
$996$	0	0
$997$	42.4434	1.34420	0.672098	−	0.740462i	$-0.265393\pi$
$997$	42.4434	1.34420	0.672098	+	0.740462i	$0.265393\pi$
$998$	0	0
$999$	38.5597	1.21997

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.e.1.2		5
4.3	odd	2		502.2.a.c.1.4	✓	5
12.11	even	2		4518.2.a.v.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
502.2.a.c.1.4	✓	5	4.3	odd	2
4016.2.a.e.1.2		5	1.1	even	1	trivial
4518.2.a.v.1.4		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.567497 q^{3} -2.51380 q^{5} -3.57054 q^{7} -2.67795 q^{9} +O(q^{10})\)	\(q-0.567497 q^{3} -2.51380 q^{5} -3.57054 q^{7} -2.67795 q^{9} +5.90017 q^{11} -3.72120 q^{13} +1.42657 q^{15} +5.80554 q^{17} +1.27592 q^{19} +2.02627 q^{21} +3.04325 q^{23} +1.31917 q^{25} +3.22222 q^{27} -3.75167 q^{29} +7.64007 q^{31} -3.34833 q^{33} +8.97562 q^{35} +11.9668 q^{37} +2.11177 q^{39} -9.65052 q^{41} +1.48925 q^{43} +6.73181 q^{45} -0.516843 q^{47} +5.74879 q^{49} -3.29463 q^{51} -10.5893 q^{53} -14.8318 q^{55} -0.724081 q^{57} -0.696653 q^{59} -4.04498 q^{61} +9.56173 q^{63} +9.35433 q^{65} +8.27304 q^{67} -1.72704 q^{69} -10.3096 q^{71} +0.498979 q^{73} -0.748625 q^{75} -21.0668 q^{77} -3.99695 q^{79} +6.20524 q^{81} -2.57663 q^{83} -14.5939 q^{85} +2.12906 q^{87} +10.1843 q^{89} +13.2867 q^{91} -4.33572 q^{93} -3.20740 q^{95} -10.0418 q^{97} -15.8003 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{3} - 6 q^{5} + q^{7}+O(q^{10}) \) 5 * q + q^3 - 6 * q^5 + q^7	\( 5 q + q^{3} - 6 q^{5} + q^{7} + 4 q^{11} - 7 q^{13} + 5 q^{15} - 8 q^{17} - 3 q^{19} - 10 q^{21} + 17 q^{23} - q^{25} + 4 q^{27} - 15 q^{29} + q^{31} - 10 q^{33} + 7 q^{35} - 5 q^{37} + 8 q^{39} - 12 q^{41} - q^{43} - 13 q^{45} + 19 q^{47} + 4 q^{49} - 7 q^{51} - 34 q^{53} - 17 q^{55} - 13 q^{57} + 10 q^{59} + 2 q^{63} + 6 q^{65} + 11 q^{67} + q^{69} + q^{71} + 3 q^{73} - 15 q^{75} - 30 q^{77} - 35 q^{79} - 3 q^{81} - 3 q^{83} - 13 q^{85} - 3 q^{87} + 15 q^{89} - 14 q^{91} + 15 q^{93} - 11 q^{95} - 2 q^{97} - 28 q^{99}+O(q^{100}) \) 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

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Database

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