LMFDB - Embedded newform 4016.2.a.i.1.5

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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.5
Root			$1.57790$ 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)

$f(q)$	$=$	$q-1.57790 q^{3} -2.90606 q^{5} -2.41450 q^{7} -0.510222 q^{9} +O(q^{10})$	$q-1.57790 q^{3} -2.90606 q^{5} -2.41450 q^{7} -0.510222 q^{9} -2.86962 q^{11} +1.00500 q^{13} +4.58548 q^{15} +3.49998 q^{17} +3.36194 q^{19} +3.80985 q^{21} -1.20028 q^{23} +3.44516 q^{25} +5.53879 q^{27} +8.57021 q^{29} -0.396943 q^{31} +4.52798 q^{33} +7.01668 q^{35} +5.74464 q^{37} -1.58579 q^{39} -0.0521746 q^{41} -1.35229 q^{43} +1.48273 q^{45} -1.74343 q^{47} -1.17018 q^{49} -5.52264 q^{51} +7.34361 q^{53} +8.33927 q^{55} -5.30481 q^{57} -6.61102 q^{59} -1.18183 q^{61} +1.23193 q^{63} -2.92059 q^{65} -7.09191 q^{67} +1.89392 q^{69} +0.470260 q^{71} +3.15215 q^{73} -5.43613 q^{75} +6.92870 q^{77} -8.78103 q^{79} -7.20901 q^{81} +1.18887 q^{83} -10.1712 q^{85} -13.5230 q^{87} +10.0470 q^{89} -2.42657 q^{91} +0.626337 q^{93} -9.76997 q^{95} -14.4676 q^{97} +1.46414 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	−1.57790	−0.911003	−0.455501	−	0.890235i	$-0.650540\pi$
$3$	−1.57790	−0.911003	−0.455501	+	0.890235i	$0.650540\pi$
$4$	0	0
$5$	−2.90606	−1.29963	−0.649814	−	0.760093i	$-0.725153\pi$
$5$	−2.90606	−1.29963	−0.649814	+	0.760093i	$0.725153\pi$
$6$	0	0
$7$	−2.41450	−0.912596	−0.456298	−	0.889827i	$-0.650825\pi$
$7$	−2.41450	−0.912596	−0.456298	+	0.889827i	$0.650825\pi$
$8$	0	0
$9$	−0.510222	−0.170074
$10$	0	0
$11$	−2.86962	−0.865222	−0.432611	−	0.901581i	$-0.642408\pi$
$11$	−2.86962	−0.865222	−0.432611	+	0.901581i	$0.642408\pi$
$12$	0	0
$13$	1.00500	0.278737	0.139368	−	0.990241i	$-0.455493\pi$
$13$	1.00500	0.278737	0.139368	+	0.990241i	$0.455493\pi$
$14$	0	0
$15$	4.58548	1.18396
$16$	0	0
$17$	3.49998	0.848871	0.424435	−	0.905458i	$-0.360473\pi$
$17$	3.49998	0.848871	0.424435	+	0.905458i	$0.360473\pi$
$18$	0	0
$19$	3.36194	0.771281	0.385640	−	0.922649i	$-0.373981\pi$
$19$	3.36194	0.771281	0.385640	+	0.922649i	$0.373981\pi$
$20$	0	0
$21$	3.80985	0.831377
$22$	0	0
$23$	−1.20028	−0.250276	−0.125138	−	0.992139i	$-0.539937\pi$
$23$	−1.20028	−0.250276	−0.125138	+	0.992139i	$0.539937\pi$
$24$	0	0
$25$	3.44516	0.689033
$26$	0	0
$27$	5.53879	1.06594
$28$	0	0
$29$	8.57021	1.59145	0.795724	−	0.605659i	$-0.207090\pi$
$29$	8.57021	1.59145	0.795724	+	0.605659i	$0.207090\pi$
$30$	0	0
$31$	−0.396943	−0.0712930	−0.0356465	−	0.999364i	$-0.511349\pi$
$31$	−0.396943	−0.0712930	−0.0356465	+	0.999364i	$0.511349\pi$
$32$	0	0
$33$	4.52798	0.788220
$34$	0	0
$35$	7.01668	1.18603
$36$	0	0
$37$	5.74464	0.944413	0.472206	−	0.881488i	$-0.343458\pi$
$37$	5.74464	0.944413	0.472206	+	0.881488i	$0.343458\pi$
$38$	0	0
$39$	−1.58579	−0.253930
$40$	0	0
$41$	−0.0521746	−0.00814830	−0.00407415	−	0.999992i	$-0.501297\pi$
$41$	−0.0521746	−0.00814830	−0.00407415	+	0.999992i	$0.501297\pi$
$42$	0	0
$43$	−1.35229	−0.206223	−0.103111	−	0.994670i	$-0.532880\pi$
$43$	−1.35229	−0.206223	−0.103111	+	0.994670i	$0.532880\pi$
$44$	0	0
$45$	1.48273	0.221033
$46$	0	0
$47$	−1.74343	−0.254305	−0.127152	−	0.991883i	$-0.540584\pi$
$47$	−1.74343	−0.254305	−0.127152	+	0.991883i	$0.540584\pi$
$48$	0	0
$49$	−1.17018	−0.167169
$50$	0	0
$51$	−5.52264	−0.773324
$52$	0	0
$53$	7.34361	1.00872	0.504361	−	0.863493i	$-0.331728\pi$
$53$	7.34361	1.00872	0.504361	+	0.863493i	$0.331728\pi$
$54$	0	0
$55$	8.33927	1.12447
$56$	0	0
$57$	−5.30481	−0.702639
$58$	0	0
$59$	−6.61102	−0.860681	−0.430340	−	0.902667i	$-0.641606\pi$
$59$	−6.61102	−0.860681	−0.430340	+	0.902667i	$0.641606\pi$
$60$	0	0
$61$	−1.18183	−0.151318	−0.0756590	−	0.997134i	$-0.524106\pi$
$61$	−1.18183	−0.151318	−0.0756590	+	0.997134i	$0.524106\pi$
$62$	0	0
$63$	1.23193	0.155209
$64$	0	0
$65$	−2.92059	−0.362254
$66$	0	0
$67$	−7.09191	−0.866415	−0.433207	−	0.901294i	$-0.642618\pi$
$67$	−7.09191	−0.866415	−0.433207	+	0.901294i	$0.642618\pi$
$68$	0	0
$69$	1.89392	0.228002
$70$	0	0
$71$	0.470260	0.0558096	0.0279048	−	0.999611i	$-0.491116\pi$
$71$	0.470260	0.0558096	0.0279048	+	0.999611i	$0.491116\pi$
$72$	0	0
$73$	3.15215	0.368931	0.184466	−	0.982839i	$-0.440945\pi$
$73$	3.15215	0.368931	0.184466	+	0.982839i	$0.440945\pi$
$74$	0	0
$75$	−5.43613	−0.627711
$76$	0	0
$77$	6.92870	0.789598
$78$	0	0
$79$	−8.78103	−0.987943	−0.493972	−	0.869478i	$-0.664455\pi$
$79$	−8.78103	−0.987943	−0.493972	+	0.869478i	$0.664455\pi$
$80$	0	0
$81$	−7.20901	−0.801001
$82$	0	0
$83$	1.18887	0.130495	0.0652476	−	0.997869i	$-0.479216\pi$
$83$	1.18887	0.130495	0.0652476	+	0.997869i	$0.479216\pi$
$84$	0	0
$85$	−10.1712	−1.10322
$86$	0	0
$87$	−13.5230	−1.44981
$88$	0	0
$89$	10.0470	1.06498	0.532490	−	0.846436i	$-0.321256\pi$
$89$	10.0470	1.06498	0.532490	+	0.846436i	$0.321256\pi$
$90$	0	0
$91$	−2.42657	−0.254374
$92$	0	0
$93$	0.626337	0.0649481
$94$	0	0
$95$	−9.76997	−1.00238
$96$	0	0
$97$	−14.4676	−1.46896	−0.734481	−	0.678630i	$-0.762574\pi$
$97$	−14.4676	−1.46896	−0.734481	+	0.678630i	$0.762574\pi$
$98$	0	0
$99$	1.46414	0.147152
$100$	0	0
$101$	−0.340708	−0.0339017	−0.0169508	−	0.999856i	$-0.505396\pi$
$101$	−0.340708	−0.0339017	−0.0169508	+	0.999856i	$0.505396\pi$
$102$	0	0
$103$	−4.18287	−0.412150	−0.206075	−	0.978536i	$-0.566069\pi$
$103$	−4.18287	−0.412150	−0.206075	+	0.978536i	$0.566069\pi$
$104$	0	0
$105$	−11.0716	−1.08048
$106$	0	0
$107$	−1.80618	−0.174610	−0.0873050	−	0.996182i	$-0.527825\pi$
$107$	−1.80618	−0.174610	−0.0873050	+	0.996182i	$0.527825\pi$
$108$	0	0
$109$	18.6559	1.78692	0.893458	−	0.449147i	$-0.148272\pi$
$109$	18.6559	1.78692	0.893458	+	0.449147i	$0.148272\pi$
$110$	0	0
$111$	−9.06448	−0.860363
$112$	0	0
$113$	−0.0814176	−0.00765912	−0.00382956	−	0.999993i	$-0.501219\pi$
$113$	−0.0814176	−0.00765912	−0.00382956	+	0.999993i	$0.501219\pi$
$114$	0	0
$115$	3.48808	0.325265
$116$	0	0
$117$	−0.512773	−0.0474059
$118$	0	0
$119$	−8.45072	−0.774676
$120$	0	0
$121$	−2.76529	−0.251390
$122$	0	0
$123$	0.0823265	0.00742313
$124$	0	0
$125$	4.51844	0.404142
$126$	0	0
$127$	−11.7010	−1.03829	−0.519146	−	0.854686i	$-0.673750\pi$
$127$	−11.7010	−1.03829	−0.519146	+	0.854686i	$0.673750\pi$
$128$	0	0
$129$	2.13379	0.187870
$130$	0	0
$131$	1.94247	0.169715	0.0848573	−	0.996393i	$-0.472957\pi$
$131$	1.94247	0.169715	0.0848573	+	0.996393i	$0.472957\pi$
$132$	0	0
$133$	−8.11740	−0.703868
$134$	0	0
$135$	−16.0960	−1.38533
$136$	0	0
$137$	−3.56970	−0.304980	−0.152490	−	0.988305i	$-0.548729\pi$
$137$	−3.56970	−0.304980	−0.152490	+	0.988305i	$0.548729\pi$
$138$	0	0
$139$	2.25346	0.191136	0.0955679	−	0.995423i	$-0.469533\pi$
$139$	2.25346	0.191136	0.0955679	+	0.995423i	$0.469533\pi$
$140$	0	0
$141$	2.75096	0.231673
$142$	0	0
$143$	−2.88397	−0.241169
$144$	0	0
$145$	−24.9055	−2.06829
$146$	0	0
$147$	1.84643	0.152291
$148$	0	0
$149$	7.31604	0.599354	0.299677	−	0.954041i	$-0.403121\pi$
$149$	7.31604	0.599354	0.299677	+	0.954041i	$0.403121\pi$
$150$	0	0
$151$	9.17475	0.746630	0.373315	−	0.927705i	$-0.378221\pi$
$151$	9.17475	0.746630	0.373315	+	0.927705i	$0.378221\pi$
$152$	0	0
$153$	−1.78577	−0.144371
$154$	0	0
$155$	1.15354	0.0926544
$156$	0	0
$157$	−0.358279	−0.0285938	−0.0142969	−	0.999898i	$-0.504551\pi$
$157$	−0.358279	−0.0285938	−0.0142969	+	0.999898i	$0.504551\pi$
$158$	0	0
$159$	−11.5875	−0.918949
$160$	0	0
$161$	2.89808	0.228400
$162$	0	0
$163$	10.5979	0.830094	0.415047	−	0.909800i	$-0.363765\pi$
$163$	10.5979	0.830094	0.415047	+	0.909800i	$0.363765\pi$
$164$	0	0
$165$	−13.1586	−1.02439
$166$	0	0
$167$	3.77713	0.292283	0.146142	−	0.989264i	$-0.453314\pi$
$167$	3.77713	0.292283	0.146142	+	0.989264i	$0.453314\pi$
$168$	0	0
$169$	−11.9900	−0.922306
$170$	0	0
$171$	−1.71533	−0.131175
$172$	0	0
$173$	−8.85366	−0.673131	−0.336565	−	0.941660i	$-0.609265\pi$
$173$	−8.85366	−0.673131	−0.336565	+	0.941660i	$0.609265\pi$
$174$	0	0
$175$	−8.31835	−0.628808
$176$	0	0
$177$	10.4315	0.784083
$178$	0	0
$179$	−10.3602	−0.774360	−0.387180	−	0.922004i	$-0.626551\pi$
$179$	−10.3602	−0.774360	−0.387180	+	0.922004i	$0.626551\pi$
$180$	0	0
$181$	−11.6511	−0.866022	−0.433011	−	0.901389i	$-0.642549\pi$
$181$	−11.6511	−0.866022	−0.433011	+	0.901389i	$0.642549\pi$
$182$	0	0
$183$	1.86482	0.137851
$184$	0	0
$185$	−16.6942	−1.22739
$186$	0	0
$187$	−10.0436	−0.734462
$188$	0	0
$189$	−13.3734	−0.972773
$190$	0	0
$191$	7.24124	0.523957	0.261979	−	0.965074i	$-0.415625\pi$
$191$	7.24124	0.523957	0.261979	+	0.965074i	$0.415625\pi$
$192$	0	0
$193$	−16.6324	−1.19722	−0.598611	−	0.801040i	$-0.704281\pi$
$193$	−16.6324	−1.19722	−0.598611	+	0.801040i	$0.704281\pi$
$194$	0	0
$195$	4.60840	0.330015
$196$	0	0
$197$	8.49503	0.605246	0.302623	−	0.953110i	$-0.402138\pi$
$197$	8.49503	0.605246	0.302623	+	0.953110i	$0.402138\pi$
$198$	0	0
$199$	5.59538	0.396646	0.198323	−	0.980137i	$-0.436451\pi$
$199$	5.59538	0.396646	0.198323	+	0.980137i	$0.436451\pi$
$200$	0	0
$201$	11.1903	0.789306
$202$	0	0
$203$	−20.6928	−1.45235
$204$	0	0
$205$	0.151622	0.0105898
$206$	0	0
$207$	0.612409	0.0425653
$208$	0	0
$209$	−9.64747	−0.667329
$210$	0	0
$211$	17.3024	1.19114	0.595572	−	0.803302i	$-0.296925\pi$
$211$	17.3024	1.19114	0.595572	+	0.803302i	$0.296925\pi$
$212$	0	0
$213$	−0.742024	−0.0508427
$214$	0	0
$215$	3.92984	0.268013
$216$	0	0
$217$	0.958419	0.0650617
$218$	0	0
$219$	−4.97379	−0.336098
$220$	0	0
$221$	3.51748	0.236612
$222$	0	0
$223$	20.4044	1.36638	0.683189	−	0.730241i	$-0.260592\pi$
$223$	20.4044	1.36638	0.683189	+	0.730241i	$0.260592\pi$
$224$	0	0
$225$	−1.75780	−0.117186
$226$	0	0
$227$	3.29322	0.218579	0.109289	−	0.994010i	$-0.465142\pi$
$227$	3.29322	0.218579	0.109289	+	0.994010i	$0.465142\pi$
$228$	0	0
$229$	−11.1842	−0.739071	−0.369536	−	0.929217i	$-0.620483\pi$
$229$	−11.1842	−0.739071	−0.369536	+	0.929217i	$0.620483\pi$
$230$	0	0
$231$	−10.9328	−0.719326
$232$	0	0
$233$	13.6418	0.893706	0.446853	−	0.894607i	$-0.352545\pi$
$233$	13.6418	0.893706	0.446853	+	0.894607i	$0.352545\pi$
$234$	0	0
$235$	5.06650	0.330502
$236$	0	0
$237$	13.8556	0.900019
$238$	0	0
$239$	0.550808	0.0356288	0.0178144	−	0.999841i	$-0.494329\pi$
$239$	0.550808	0.0356288	0.0178144	+	0.999841i	$0.494329\pi$
$240$	0	0
$241$	−18.9461	−1.22043	−0.610214	−	0.792237i	$-0.708917\pi$
$241$	−18.9461	−1.22043	−0.610214	+	0.792237i	$0.708917\pi$
$242$	0	0
$243$	−5.24125	−0.336227
$244$	0	0
$245$	3.40062	0.217257
$246$	0	0
$247$	3.37875	0.214984
$248$	0	0
$249$	−1.87592	−0.118881
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	3.44434	0.216544
$254$	0	0
$255$	16.0491	1.00503
$256$	0	0
$257$	−21.4665	−1.33904	−0.669520	−	0.742794i	$-0.733500\pi$
$257$	−21.4665	−1.33904	−0.669520	+	0.742794i	$0.733500\pi$
$258$	0	0
$259$	−13.8704	−0.861867
$260$	0	0
$261$	−4.37271	−0.270664
$262$	0	0
$263$	10.3666	0.639229	0.319615	−	0.947548i	$-0.396447\pi$
$263$	10.3666	0.639229	0.319615	+	0.947548i	$0.396447\pi$
$264$	0	0
$265$	−21.3410	−1.31096
$266$	0	0
$267$	−15.8532	−0.970199
$268$	0	0
$269$	16.7949	1.02401	0.512003	−	0.858984i	$-0.328904\pi$
$269$	16.7949	1.02401	0.512003	+	0.858984i	$0.328904\pi$
$270$	0	0
$271$	3.60012	0.218692	0.109346	−	0.994004i	$-0.465124\pi$
$271$	3.60012	0.218692	0.109346	+	0.994004i	$0.465124\pi$
$272$	0	0
$273$	3.82890	0.231736
$274$	0	0
$275$	−9.88630	−0.596166
$276$	0	0
$277$	−13.5402	−0.813555	−0.406777	−	0.913527i	$-0.633347\pi$
$277$	−13.5402	−0.813555	−0.406777	+	0.913527i	$0.633347\pi$
$278$	0	0
$279$	0.202529	0.0121251
$280$	0	0
$281$	25.5613	1.52486	0.762429	−	0.647072i	$-0.224007\pi$
$281$	25.5613	1.52486	0.762429	+	0.647072i	$0.224007\pi$
$282$	0	0
$283$	−20.2346	−1.20282	−0.601412	−	0.798939i	$-0.705395\pi$
$283$	−20.2346	−1.20282	−0.601412	+	0.798939i	$0.705395\pi$
$284$	0	0
$285$	15.4161	0.913169
$286$	0	0
$287$	0.125976	0.00743611
$288$	0	0
$289$	−4.75011	−0.279418
$290$	0	0
$291$	22.8285	1.33823
$292$	0	0
$293$	12.7240	0.743343	0.371671	−	0.928364i	$-0.378785\pi$
$293$	12.7240	0.743343	0.371671	+	0.928364i	$0.378785\pi$
$294$	0	0
$295$	19.2120	1.11856
$296$	0	0
$297$	−15.8942	−0.922276
$298$	0	0
$299$	−1.20628	−0.0697610
$300$	0	0
$301$	3.26511	0.188198
$302$	0	0
$303$	0.537603	0.0308845
$304$	0	0
$305$	3.43447	0.196657
$306$	0	0
$307$	8.09305	0.461895	0.230948	−	0.972966i	$-0.425817\pi$
$307$	8.09305	0.461895	0.230948	+	0.972966i	$0.425817\pi$
$308$	0	0
$309$	6.60016	0.375470
$310$	0	0
$311$	−19.5684	−1.10962	−0.554812	−	0.831976i	$-0.687210\pi$
$311$	−19.5684	−1.10962	−0.554812	+	0.831976i	$0.687210\pi$
$312$	0	0
$313$	−7.08081	−0.400231	−0.200115	−	0.979772i	$-0.564132\pi$
$313$	−7.08081	−0.400231	−0.200115	+	0.979772i	$0.564132\pi$
$314$	0	0
$315$	−3.58006	−0.201714
$316$	0	0
$317$	−3.07453	−0.172683	−0.0863413	−	0.996266i	$-0.527518\pi$
$317$	−3.07453	−0.172683	−0.0863413	+	0.996266i	$0.527518\pi$
$318$	0	0
$319$	−24.5932	−1.37696
$320$	0	0
$321$	2.84998	0.159070
$322$	0	0
$323$	11.7667	0.654718
$324$	0	0
$325$	3.46239	0.192059
$326$	0	0
$327$	−29.4373	−1.62788
$328$	0	0
$329$	4.20951	0.232078
$330$	0	0
$331$	−8.88752	−0.488502	−0.244251	−	0.969712i	$-0.578542\pi$
$331$	−8.88752	−0.488502	−0.244251	+	0.969712i	$0.578542\pi$
$332$	0	0
$333$	−2.93104	−0.160620
$334$	0	0
$335$	20.6095	1.12602
$336$	0	0
$337$	7.67293	0.417971	0.208986	−	0.977919i	$-0.432984\pi$
$337$	7.67293	0.417971	0.208986	+	0.977919i	$0.432984\pi$
$338$	0	0
$339$	0.128469	0.00697748
$340$	0	0
$341$	1.13907	0.0616843
$342$	0	0
$343$	19.7269	1.06515
$344$	0	0
$345$	−5.50385	−0.296317
$346$	0	0
$347$	6.48651	0.348214	0.174107	−	0.984727i	$-0.444296\pi$
$347$	6.48651	0.348214	0.174107	+	0.984727i	$0.444296\pi$
$348$	0	0
$349$	−22.2909	−1.19321	−0.596603	−	0.802537i	$-0.703483\pi$
$349$	−22.2909	−1.19321	−0.596603	+	0.802537i	$0.703483\pi$
$350$	0	0
$351$	5.56648	0.297117
$352$	0	0
$353$	−26.9650	−1.43520	−0.717602	−	0.696453i	$-0.754760\pi$
$353$	−26.9650	−1.43520	−0.717602	+	0.696453i	$0.754760\pi$
$354$	0	0
$355$	−1.36660	−0.0725317
$356$	0	0
$357$	13.3344	0.705732
$358$	0	0
$359$	−36.0904	−1.90478	−0.952388	−	0.304888i	$-0.901381\pi$
$359$	−36.0904	−1.90478	−0.952388	+	0.304888i	$0.901381\pi$
$360$	0	0
$361$	−7.69739	−0.405126
$362$	0	0
$363$	4.36336	0.229017
$364$	0	0
$365$	−9.16033	−0.479474
$366$	0	0
$367$	20.4410	1.06701	0.533506	−	0.845797i	$-0.320874\pi$
$367$	20.4410	1.06701	0.533506	+	0.845797i	$0.320874\pi$
$368$	0	0
$369$	0.0266206	0.00138581
$370$	0	0
$371$	−17.7312	−0.920556
$372$	0	0
$373$	−8.95958	−0.463909	−0.231955	−	0.972727i	$-0.574512\pi$
$373$	−8.95958	−0.463909	−0.231955	+	0.972727i	$0.574512\pi$
$374$	0	0
$375$	−7.12967	−0.368174
$376$	0	0
$377$	8.61307	0.443595
$378$	0	0
$379$	−34.7838	−1.78673	−0.893363	−	0.449336i	$-0.851661\pi$
$379$	−34.7838	−1.78673	−0.893363	+	0.449336i	$0.851661\pi$
$380$	0	0
$381$	18.4630	0.945887
$382$	0	0
$383$	−3.85343	−0.196901	−0.0984506	−	0.995142i	$-0.531389\pi$
$383$	−3.85343	−0.196901	−0.0984506	+	0.995142i	$0.531389\pi$
$384$	0	0
$385$	−20.1352	−1.02618
$386$	0	0
$387$	0.689970	0.0350731
$388$	0	0
$389$	−21.6515	−1.09777	−0.548886	−	0.835897i	$-0.684948\pi$
$389$	−21.6515	−1.09777	−0.548886	+	0.835897i	$0.684948\pi$
$390$	0	0
$391$	−4.20096	−0.212452
$392$	0	0
$393$	−3.06503	−0.154610
$394$	0	0
$395$	25.5182	1.28396
$396$	0	0
$397$	15.4664	0.776235	0.388117	−	0.921610i	$-0.373125\pi$
$397$	15.4664	0.776235	0.388117	+	0.921610i	$0.373125\pi$
$398$	0	0
$399$	12.8085	0.641225
$400$	0	0
$401$	−17.0694	−0.852404	−0.426202	−	0.904628i	$-0.640149\pi$
$401$	−17.0694	−0.852404	−0.426202	+	0.904628i	$0.640149\pi$
$402$	0	0
$403$	−0.398928	−0.0198720
$404$	0	0
$405$	20.9498	1.04100
$406$	0	0
$407$	−16.4849	−0.817127
$408$	0	0
$409$	13.9964	0.692079	0.346040	−	0.938220i	$-0.387526\pi$
$409$	13.9964	0.692079	0.346040	+	0.938220i	$0.387526\pi$
$410$	0	0
$411$	5.63264	0.277838
$412$	0	0
$413$	15.9623	0.785454
$414$	0	0
$415$	−3.45492	−0.169595
$416$	0	0
$417$	−3.55574	−0.174125
$418$	0	0
$419$	35.7647	1.74722	0.873611	−	0.486625i	$-0.161772\pi$
$419$	35.7647	1.74722	0.873611	+	0.486625i	$0.161772\pi$
$420$	0	0
$421$	−15.4326	−0.752138	−0.376069	−	0.926592i	$-0.622724\pi$
$421$	−15.4326	−0.752138	−0.376069	+	0.926592i	$0.622724\pi$
$422$	0	0
$423$	0.889534	0.0432506
$424$	0	0
$425$	12.0580	0.584900
$426$	0	0
$427$	2.85353	0.138092
$428$	0	0
$429$	4.55062	0.219706
$430$	0	0
$431$	−29.9024	−1.44035	−0.720174	−	0.693794i	$-0.755938\pi$
$431$	−29.9024	−1.44035	−0.720174	+	0.693794i	$0.755938\pi$
$432$	0	0
$433$	12.0764	0.580354	0.290177	−	0.956973i	$-0.406286\pi$
$433$	12.0764	0.580354	0.290177	+	0.956973i	$0.406286\pi$
$434$	0	0
$435$	39.2985	1.88422
$436$	0	0
$437$	−4.03526	−0.193033
$438$	0	0
$439$	−35.9118	−1.71398	−0.856988	−	0.515337i	$-0.827667\pi$
$439$	−35.9118	−1.71398	−0.856988	+	0.515337i	$0.827667\pi$
$440$	0	0
$441$	0.597053	0.0284311
$442$	0	0
$443$	14.0956	0.669701	0.334850	−	0.942271i	$-0.391314\pi$
$443$	14.0956	0.669701	0.334850	+	0.942271i	$0.391314\pi$
$444$	0	0
$445$	−29.1971	−1.38408
$446$	0	0
$447$	−11.5440	−0.546013
$448$	0	0
$449$	15.1010	0.712661	0.356330	−	0.934360i	$-0.384028\pi$
$449$	15.1010	0.712661	0.356330	+	0.934360i	$0.384028\pi$
$450$	0	0
$451$	0.149721	0.00705010
$452$	0	0
$453$	−14.4769	−0.680182
$454$	0	0
$455$	7.05176	0.330592
$456$	0	0
$457$	−0.00130019	−6.08205e−5 0	−3.04103e−5	−	1.00000i	$-0.500010\pi$
$457$	−0.00130019	−6.08205e−5 0	−3.04103e−5		1.00000i	$0.500010\pi$
$458$	0	0
$459$	19.3857	0.904846
$460$	0	0
$461$	−4.75737	−0.221573	−0.110786	−	0.993844i	$-0.535337\pi$
$461$	−4.75737	−0.221573	−0.110786	+	0.993844i	$0.535337\pi$
$462$	0	0
$463$	18.9396	0.880197	0.440098	−	0.897950i	$-0.354944\pi$
$463$	18.9396	0.880197	0.440098	+	0.897950i	$0.354944\pi$
$464$	0	0
$465$	−1.82017	−0.0844084
$466$	0	0
$467$	4.84927	0.224397	0.112199	−	0.993686i	$-0.464211\pi$
$467$	4.84927	0.224397	0.112199	+	0.993686i	$0.464211\pi$
$468$	0	0
$469$	17.1234	0.790687
$470$	0	0
$471$	0.565329	0.0260490
$472$	0	0
$473$	3.88057	0.178429
$474$	0	0
$475$	11.5824	0.531438
$476$	0	0
$477$	−3.74687	−0.171557
$478$	0	0
$479$	4.44581	0.203134	0.101567	−	0.994829i	$-0.467614\pi$
$479$	4.44581	0.203134	0.101567	+	0.994829i	$0.467614\pi$
$480$	0	0
$481$	5.77336	0.263243
$482$	0	0
$483$	−4.57288	−0.208073
$484$	0	0
$485$	42.0436	1.90910
$486$	0	0
$487$	8.81272	0.399342	0.199671	−	0.979863i	$-0.436013\pi$
$487$	8.81272	0.399342	0.199671	+	0.979863i	$0.436013\pi$
$488$	0	0
$489$	−16.7225	−0.756218
$490$	0	0
$491$	2.13680	0.0964324	0.0482162	−	0.998837i	$-0.484646\pi$
$491$	2.13680	0.0964324	0.0482162	+	0.998837i	$0.484646\pi$
$492$	0	0
$493$	29.9956	1.35093
$494$	0	0
$495$	−4.25488	−0.191243
$496$	0	0
$497$	−1.13544	−0.0509316
$498$	0	0
$499$	−26.6439	−1.19275	−0.596373	−	0.802708i	$-0.703392\pi$
$499$	−26.6439	−1.19275	−0.596373	+	0.802708i	$0.703392\pi$
$500$	0	0
$501$	−5.95995	−0.266271
$502$	0	0
$503$	27.5947	1.23039	0.615194	−	0.788376i	$-0.289078\pi$
$503$	27.5947	1.23039	0.615194	+	0.788376i	$0.289078\pi$
$504$	0	0
$505$	0.990115	0.0440596
$506$	0	0
$507$	18.9190	0.840223
$508$	0	0
$509$	16.2577	0.720612	0.360306	−	0.932834i	$-0.382672\pi$
$509$	16.2577	0.720612	0.360306	+	0.932834i	$0.382672\pi$
$510$	0	0
$511$	−7.61087	−0.336685
$512$	0	0
$513$	18.6211	0.822140
$514$	0	0
$515$	12.1556	0.535642
$516$	0	0
$517$	5.00297	0.220030
$518$	0	0
$519$	13.9702	0.613224
$520$	0	0
$521$	−19.2056	−0.841415	−0.420707	−	0.907196i	$-0.638218\pi$
$521$	−19.2056	−0.841415	−0.420707	+	0.907196i	$0.638218\pi$
$522$	0	0
$523$	−18.2517	−0.798090	−0.399045	−	0.916931i	$-0.630658\pi$
$523$	−18.2517	−0.798090	−0.399045	+	0.916931i	$0.630658\pi$
$524$	0	0
$525$	13.1256	0.572846
$526$	0	0
$527$	−1.38929	−0.0605186
$528$	0	0
$529$	−21.5593	−0.937362
$530$	0	0
$531$	3.37308	0.146379
$532$	0	0
$533$	−0.0524355	−0.00227123
$534$	0	0
$535$	5.24886	0.226928
$536$	0	0
$537$	16.3474	0.705444
$538$	0	0
$539$	3.35798	0.144638
$540$	0	0
$541$	−43.5633	−1.87293	−0.936467	−	0.350756i	$-0.885925\pi$
$541$	−43.5633	−1.87293	−0.936467	+	0.350756i	$0.885925\pi$
$542$	0	0
$543$	18.3844	0.788949
$544$	0	0
$545$	−54.2152	−2.32233
$546$	0	0
$547$	27.2247	1.16404	0.582022	−	0.813173i	$-0.302262\pi$
$547$	27.2247	1.16404	0.582022	+	0.813173i	$0.302262\pi$
$548$	0	0
$549$	0.602996	0.0257352
$550$	0	0
$551$	28.8125	1.22745
$552$	0	0
$553$	21.2018	0.901593
$554$	0	0
$555$	26.3419	1.11815
$556$	0	0
$557$	−8.03832	−0.340594	−0.170297	−	0.985393i	$-0.554473\pi$
$557$	−8.03832	−0.340594	−0.170297	+	0.985393i	$0.554473\pi$
$558$	0	0
$559$	−1.35906	−0.0574819
$560$	0	0
$561$	15.8479	0.669097
$562$	0	0
$563$	−8.87016	−0.373833	−0.186916	−	0.982376i	$-0.559849\pi$
$563$	−8.87016	−0.373833	−0.186916	+	0.982376i	$0.559849\pi$
$564$	0	0
$565$	0.236604	0.00995401
$566$	0	0
$567$	17.4062	0.730990
$568$	0	0
$569$	−8.44073	−0.353854	−0.176927	−	0.984224i	$-0.556616\pi$
$569$	−8.44073	−0.353854	−0.176927	+	0.984224i	$0.556616\pi$
$570$	0	0
$571$	−22.9046	−0.958527	−0.479263	−	0.877671i	$-0.659096\pi$
$571$	−22.9046	−0.958527	−0.479263	+	0.877671i	$0.659096\pi$
$572$	0	0
$573$	−11.4260	−0.477327
$574$	0	0
$575$	−4.13516	−0.172448
$576$	0	0
$577$	19.5190	0.812587	0.406294	−	0.913743i	$-0.366821\pi$
$577$	19.5190	0.812587	0.406294	+	0.913743i	$0.366821\pi$
$578$	0	0
$579$	26.2442	1.09067
$580$	0	0
$581$	−2.87052	−0.119089
$582$	0	0
$583$	−21.0734	−0.872770
$584$	0	0
$585$	1.49015	0.0616100
$586$	0	0
$587$	15.3948	0.635411	0.317705	−	0.948189i	$-0.397088\pi$
$587$	15.3948	0.635411	0.317705	+	0.948189i	$0.397088\pi$
$588$	0	0
$589$	−1.33450	−0.0549869
$590$	0	0
$591$	−13.4043	−0.551381
$592$	0	0
$593$	−10.8617	−0.446035	−0.223018	−	0.974814i	$-0.571591\pi$
$593$	−10.8617	−0.446035	−0.223018	+	0.974814i	$0.571591\pi$
$594$	0	0
$595$	24.5583	1.00679
$596$	0	0
$597$	−8.82897	−0.361346
$598$	0	0
$599$	9.72197	0.397229	0.198615	−	0.980078i	$-0.436356\pi$
$599$	9.72197	0.397229	0.198615	+	0.980078i	$0.436356\pi$
$600$	0	0
$601$	−20.3822	−0.831405	−0.415703	−	0.909501i	$-0.636464\pi$
$601$	−20.3822	−0.831405	−0.415703	+	0.909501i	$0.636464\pi$
$602$	0	0
$603$	3.61845	0.147355
$604$	0	0
$605$	8.03610	0.326714
$606$	0	0
$607$	6.68760	0.271441	0.135721	−	0.990747i	$-0.456665\pi$
$607$	6.68760	0.271441	0.135721	+	0.990747i	$0.456665\pi$
$608$	0	0
$609$	32.6512	1.32309
$610$	0	0
$611$	−1.75214	−0.0708842
$612$	0	0
$613$	21.4545	0.866539	0.433269	−	0.901264i	$-0.357360\pi$
$613$	21.4545	0.866539	0.433269	+	0.901264i	$0.357360\pi$
$614$	0	0
$615$	−0.239245	−0.00964730
$616$	0	0
$617$	−36.5365	−1.47091	−0.735453	−	0.677576i	$-0.763030\pi$
$617$	−36.5365	−1.47091	−0.735453	+	0.677576i	$0.763030\pi$
$618$	0	0
$619$	5.26759	0.211722	0.105861	−	0.994381i	$-0.466240\pi$
$619$	5.26759	0.211722	0.105861	+	0.994381i	$0.466240\pi$
$620$	0	0
$621$	−6.64809	−0.266779
$622$	0	0
$623$	−24.2585	−0.971896
$624$	0	0
$625$	−30.3567	−1.21427
$626$	0	0
$627$	15.2228	0.607939
$628$	0	0
$629$	20.1061	0.801685
$630$	0	0
$631$	38.5714	1.53550	0.767752	−	0.640748i	$-0.221376\pi$
$631$	38.5714	1.53550	0.767752	+	0.640748i	$0.221376\pi$
$632$	0	0
$633$	−27.3015	−1.08514
$634$	0	0
$635$	34.0036	1.34939
$636$	0	0
$637$	−1.17603	−0.0465961
$638$	0	0
$639$	−0.239937	−0.00949175
$640$	0	0
$641$	−41.3892	−1.63477	−0.817387	−	0.576089i	$-0.804578\pi$
$641$	−41.3892	−1.63477	−0.817387	+	0.576089i	$0.804578\pi$
$642$	0	0
$643$	29.6627	1.16978	0.584892	−	0.811111i	$-0.301137\pi$
$643$	29.6627	1.16978	0.584892	+	0.811111i	$0.301137\pi$
$644$	0	0
$645$	−6.20091	−0.244161
$646$	0	0
$647$	20.9286	0.822789	0.411394	−	0.911457i	$-0.365042\pi$
$647$	20.9286	0.822789	0.411394	+	0.911457i	$0.365042\pi$
$648$	0	0
$649$	18.9711	0.744680
$650$	0	0
$651$	−1.51229	−0.0592714
$652$	0	0
$653$	−22.9975	−0.899961	−0.449981	−	0.893038i	$-0.648569\pi$
$653$	−22.9975	−0.899961	−0.449981	+	0.893038i	$0.648569\pi$
$654$	0	0
$655$	−5.64493	−0.220566
$656$	0	0
$657$	−1.60830	−0.0627456
$658$	0	0
$659$	3.01002	0.117254	0.0586269	−	0.998280i	$-0.481328\pi$
$659$	3.01002	0.117254	0.0586269	+	0.998280i	$0.481328\pi$
$660$	0	0
$661$	26.0763	1.01425	0.507125	−	0.861873i	$-0.330708\pi$
$661$	26.0763	1.01425	0.507125	+	0.861873i	$0.330708\pi$
$662$	0	0
$663$	−5.55025	−0.215554
$664$	0	0
$665$	23.5896	0.914766
$666$	0	0
$667$	−10.2866	−0.398301
$668$	0	0
$669$	−32.1961	−1.24477
$670$	0	0
$671$	3.39140	0.130924
$672$	0	0
$673$	23.7917	0.917104	0.458552	−	0.888668i	$-0.348368\pi$
$673$	23.7917	0.917104	0.458552	+	0.888668i	$0.348368\pi$
$674$	0	0
$675$	19.0820	0.734468
$676$	0	0
$677$	29.2621	1.12463	0.562317	−	0.826922i	$-0.309910\pi$
$677$	29.2621	1.12463	0.562317	+	0.826922i	$0.309910\pi$
$678$	0	0
$679$	34.9320	1.34057
$680$	0	0
$681$	−5.19639	−0.199126
$682$	0	0
$683$	−27.5689	−1.05489	−0.527447	−	0.849588i	$-0.676851\pi$
$683$	−27.5689	−1.05489	−0.527447	+	0.849588i	$0.676851\pi$
$684$	0	0
$685$	10.3738	0.396361
$686$	0	0
$687$	17.6475	0.673296
$688$	0	0
$689$	7.38033	0.281168
$690$	0	0
$691$	−24.1566	−0.918960	−0.459480	−	0.888188i	$-0.651964\pi$
$691$	−24.1566	−0.918960	−0.459480	+	0.888188i	$0.651964\pi$
$692$	0	0
$693$	−3.53517	−0.134290
$694$	0	0
$695$	−6.54867	−0.248405
$696$	0	0
$697$	−0.182610	−0.00691686
$698$	0	0
$699$	−21.5255	−0.814169
$700$	0	0
$701$	−9.76588	−0.368852	−0.184426	−	0.982846i	$-0.559043\pi$
$701$	−9.76588	−0.368852	−0.184426	+	0.982846i	$0.559043\pi$
$702$	0	0
$703$	19.3131	0.728408
$704$	0	0
$705$	−7.99444	−0.301088
$706$	0	0
$707$	0.822639	0.0309385
$708$	0	0
$709$	−27.0498	−1.01588	−0.507939	−	0.861393i	$-0.669592\pi$
$709$	−27.0498	−1.01588	−0.507939	+	0.861393i	$0.669592\pi$
$710$	0	0
$711$	4.48027	0.168023
$712$	0	0
$713$	0.476442	0.0178429
$714$	0	0
$715$	8.38097	0.313430
$716$	0	0
$717$	−0.869122	−0.0324580
$718$	0	0
$719$	−34.7327	−1.29531	−0.647655	−	0.761934i	$-0.724250\pi$
$719$	−34.7327	−1.29531	−0.647655	+	0.761934i	$0.724250\pi$
$720$	0	0
$721$	10.0995	0.376126
$722$	0	0
$723$	29.8952	1.11181
$724$	0	0
$725$	29.5258	1.09656
$726$	0	0
$727$	−16.8376	−0.624471	−0.312235	−	0.950005i	$-0.601078\pi$
$727$	−16.8376	−0.624471	−0.312235	+	0.950005i	$0.601078\pi$
$728$	0	0
$729$	29.8972	1.10730
$730$	0	0
$731$	−4.73301	−0.175057
$732$	0	0
$733$	−16.7230	−0.617679	−0.308839	−	0.951114i	$-0.599941\pi$
$733$	−16.7230	−0.617679	−0.308839	+	0.951114i	$0.599941\pi$
$734$	0	0
$735$	−5.36584	−0.197922
$736$	0	0
$737$	20.3511	0.749642
$738$	0	0
$739$	22.6614	0.833611	0.416806	−	0.908996i	$-0.363150\pi$
$739$	22.6614	0.833611	0.416806	+	0.908996i	$0.363150\pi$
$740$	0	0
$741$	−5.33133	−0.195851
$742$	0	0
$743$	−37.5491	−1.37754	−0.688772	−	0.724978i	$-0.741850\pi$
$743$	−37.5491	−1.37754	−0.688772	+	0.724978i	$0.741850\pi$
$744$	0	0
$745$	−21.2608	−0.778937
$746$	0	0
$747$	−0.606586	−0.0221938
$748$	0	0
$749$	4.36102	0.159348
$750$	0	0
$751$	14.1278	0.515530	0.257765	−	0.966208i	$-0.417014\pi$
$751$	14.1278	0.515530	0.257765	+	0.966208i	$0.417014\pi$
$752$	0	0
$753$	1.57790	0.0575020
$754$	0	0
$755$	−26.6623	−0.970342
$756$	0	0
$757$	−32.0239	−1.16393	−0.581965	−	0.813214i	$-0.697716\pi$
$757$	−32.0239	−1.16393	−0.581965	+	0.813214i	$0.697716\pi$
$758$	0	0
$759$	−5.43484	−0.197272
$760$	0	0
$761$	26.8084	0.971804	0.485902	−	0.874013i	$-0.338491\pi$
$761$	26.8084	0.971804	0.485902	+	0.874013i	$0.338491\pi$
$762$	0	0
$763$	−45.0448	−1.63073
$764$	0	0
$765$	5.18954	0.187628
$766$	0	0
$767$	−6.64407	−0.239904
$768$	0	0
$769$	−25.8846	−0.933424	−0.466712	−	0.884409i	$-0.654561\pi$
$769$	−25.8846	−0.933424	−0.466712	+	0.884409i	$0.654561\pi$
$770$	0	0
$771$	33.8720	1.21987
$772$	0	0
$773$	25.8647	0.930289	0.465144	−	0.885235i	$-0.346002\pi$
$773$	25.8647	0.930289	0.465144	+	0.885235i	$0.346002\pi$
$774$	0	0
$775$	−1.36753	−0.0491232
$776$	0	0
$777$	21.8862	0.785163
$778$	0	0
$779$	−0.175408	−0.00628463
$780$	0	0
$781$	−1.34947	−0.0482877
$782$	0	0
$783$	47.4686	1.69639
$784$	0	0
$785$	1.04118	0.0371613
$786$	0	0
$787$	−7.68090	−0.273795	−0.136897	−	0.990585i	$-0.543713\pi$
$787$	−7.68090	−0.273795	−0.136897	+	0.990585i	$0.543713\pi$
$788$	0	0
$789$	−16.3574	−0.582340
$790$	0	0
$791$	0.196583	0.00698969
$792$	0	0
$793$	−1.18774	−0.0421779
$794$	0	0
$795$	33.6740	1.19429
$796$	0	0
$797$	−6.42225	−0.227488	−0.113744	−	0.993510i	$-0.536284\pi$
$797$	−6.42225	−0.227488	−0.113744	+	0.993510i	$0.536284\pi$
$798$	0	0
$799$	−6.10197	−0.215872
$800$	0	0
$801$	−5.12620	−0.181125
$802$	0	0
$803$	−9.04547	−0.319208
$804$	0	0
$805$	−8.42197	−0.296835
$806$	0	0
$807$	−26.5008	−0.932872
$808$	0	0
$809$	−16.0016	−0.562585	−0.281293	−	0.959622i	$-0.590763\pi$
$809$	−16.0016	−0.562585	−0.281293	+	0.959622i	$0.590763\pi$
$810$	0	0
$811$	−8.33890	−0.292818	−0.146409	−	0.989224i	$-0.546772\pi$
$811$	−8.33890	−0.292818	−0.146409	+	0.989224i	$0.546772\pi$
$812$	0	0
$813$	−5.68064	−0.199229
$814$	0	0
$815$	−30.7982	−1.07881
$816$	0	0
$817$	−4.54632	−0.159056
$818$	0	0
$819$	1.23809	0.0432624
$820$	0	0
$821$	39.3489	1.37329	0.686643	−	0.726995i	$-0.259084\pi$
$821$	39.3489	1.37329	0.686643	+	0.726995i	$0.259084\pi$
$822$	0	0
$823$	−50.1460	−1.74798	−0.873990	−	0.485943i	$-0.838476\pi$
$823$	−50.1460	−1.74798	−0.873990	+	0.485943i	$0.838476\pi$
$824$	0	0
$825$	15.5996	0.543109
$826$	0	0
$827$	20.1903	0.702087	0.351044	−	0.936359i	$-0.385827\pi$
$827$	20.1903	0.702087	0.351044	+	0.936359i	$0.385827\pi$
$828$	0	0
$829$	−6.90820	−0.239932	−0.119966	−	0.992778i	$-0.538279\pi$
$829$	−6.90820	−0.239932	−0.119966	+	0.992778i	$0.538279\pi$
$830$	0	0
$831$	21.3652	0.741150
$832$	0	0
$833$	−4.09562	−0.141905
$834$	0	0
$835$	−10.9766	−0.379860
$836$	0	0
$837$	−2.19858	−0.0759941
$838$	0	0
$839$	−0.814716	−0.0281271	−0.0140636	−	0.999901i	$-0.504477\pi$
$839$	−0.814716	−0.0281271	−0.0140636	+	0.999901i	$0.504477\pi$
$840$	0	0
$841$	44.4486	1.53271
$842$	0	0
$843$	−40.3332	−1.38915
$844$	0	0
$845$	34.8435	1.19865
$846$	0	0
$847$	6.67680	0.229418
$848$	0	0
$849$	31.9283	1.09578
$850$	0	0
$851$	−6.89517	−0.236363
$852$	0	0
$853$	34.2404	1.17237	0.586184	−	0.810178i	$-0.300630\pi$
$853$	34.2404	1.17237	0.586184	+	0.810178i	$0.300630\pi$
$854$	0	0
$855$	4.98485	0.170478
$856$	0	0
$857$	50.4219	1.72238	0.861190	−	0.508283i	$-0.169719\pi$
$857$	50.4219	1.72238	0.861190	+	0.508283i	$0.169719\pi$
$858$	0	0
$859$	−32.0842	−1.09470	−0.547350	−	0.836904i	$-0.684363\pi$
$859$	−32.0842	−1.09470	−0.547350	+	0.836904i	$0.684363\pi$
$860$	0	0
$861$	−0.198777	−0.00677432
$862$	0	0
$863$	−21.1541	−0.720092	−0.360046	−	0.932934i	$-0.617239\pi$
$863$	−21.1541	−0.720092	−0.360046	+	0.932934i	$0.617239\pi$
$864$	0	0
$865$	25.7292	0.874820
$866$	0	0
$867$	7.49522	0.254551
$868$	0	0
$869$	25.1982	0.854791
$870$	0	0
$871$	−7.12737	−0.241502
$872$	0	0
$873$	7.38168	0.249832
$874$	0	0
$875$	−10.9098	−0.368818
$876$	0	0
$877$	48.9827	1.65403	0.827014	−	0.562181i	$-0.190038\pi$
$877$	48.9827	1.65403	0.827014	+	0.562181i	$0.190038\pi$
$878$	0	0
$879$	−20.0772	−0.677187
$880$	0	0
$881$	19.0001	0.640129	0.320065	−	0.947396i	$-0.396295\pi$
$881$	19.0001	0.640129	0.320065	+	0.947396i	$0.396295\pi$
$882$	0	0
$883$	−48.5984	−1.63547	−0.817734	−	0.575597i	$-0.804770\pi$
$883$	−48.5984	−1.63547	−0.817734	+	0.575597i	$0.804770\pi$
$884$	0	0
$885$	−30.3146	−1.01902
$886$	0	0
$887$	58.2955	1.95737	0.978685	−	0.205365i	$-0.0658381\pi$
$887$	58.2955	1.95737	0.978685	+	0.205365i	$0.0658381\pi$
$888$	0	0
$889$	28.2520	0.947541
$890$	0	0
$891$	20.6871	0.693044
$892$	0	0
$893$	−5.86129	−0.196141
$894$	0	0
$895$	30.1074	1.00638
$896$	0	0
$897$	1.90339	0.0635525
$898$	0	0
$899$	−3.40188	−0.113459
$900$	0	0
$901$	25.7025	0.856275
$902$	0	0
$903$	−5.15203	−0.171449
$904$	0	0
$905$	33.8589	1.12551
$906$	0	0
$907$	−42.2596	−1.40321	−0.701604	−	0.712567i	$-0.747532\pi$
$907$	−42.2596	−1.40321	−0.701604	+	0.712567i	$0.747532\pi$
$908$	0	0
$909$	0.173836	0.00576579
$910$	0	0
$911$	−8.01825	−0.265657	−0.132828	−	0.991139i	$-0.542406\pi$
$911$	−8.01825	−0.265657	−0.132828	+	0.991139i	$0.542406\pi$
$912$	0	0
$913$	−3.41160	−0.112907
$914$	0	0
$915$	−5.41926	−0.179155
$916$	0	0
$917$	−4.69010	−0.154881
$918$	0	0
$919$	−38.0786	−1.25610	−0.628048	−	0.778175i	$-0.716146\pi$
$919$	−38.0786	−1.25610	−0.628048	+	0.778175i	$0.716146\pi$
$920$	0	0
$921$	−12.7701	−0.420788
$922$	0	0
$923$	0.472611	0.0155562
$924$	0	0
$925$	19.7912	0.650731
$926$	0	0
$927$	2.13419	0.0700960
$928$	0	0
$929$	−20.3143	−0.666492	−0.333246	−	0.942840i	$-0.608144\pi$
$929$	−20.3143	−0.666492	−0.333246	+	0.942840i	$0.608144\pi$
$930$	0	0
$931$	−3.93408	−0.128934
$932$	0	0
$933$	30.8771	1.01087
$934$	0	0
$935$	29.1873	0.954527
$936$	0	0
$937$	−5.48853	−0.179302	−0.0896512	−	0.995973i	$-0.528575\pi$
$937$	−5.48853	−0.179302	−0.0896512	+	0.995973i	$0.528575\pi$
$938$	0	0
$939$	11.1728	0.364611
$940$	0	0
$941$	50.9988	1.66251	0.831257	−	0.555889i	$-0.187622\pi$
$941$	50.9988	1.66251	0.831257	+	0.555889i	$0.187622\pi$
$942$	0	0
$943$	0.0626241	0.00203932
$944$	0	0
$945$	38.8639	1.26424
$946$	0	0
$947$	37.1191	1.20621	0.603104	−	0.797663i	$-0.293930\pi$
$947$	37.1191	1.20621	0.603104	+	0.797663i	$0.293930\pi$
$948$	0	0
$949$	3.16791	0.102835
$950$	0	0
$951$	4.85131	0.157314
$952$	0	0
$953$	7.07414	0.229154	0.114577	−	0.993414i	$-0.463449\pi$
$953$	7.07414	0.229154	0.114577	+	0.993414i	$0.463449\pi$
$954$	0	0
$955$	−21.0434	−0.680950
$956$	0	0
$957$	38.8057	1.25441
$958$	0	0
$959$	8.61905	0.278324
$960$	0	0
$961$	−30.8424	−0.994917
$962$	0	0
$963$	0.921552	0.0296966
$964$	0	0
$965$	48.3345	1.55594
$966$	0	0
$967$	−19.8551	−0.638496	−0.319248	−	0.947671i	$-0.603430\pi$
$967$	−19.8551	−0.638496	−0.319248	+	0.947671i	$0.603430\pi$
$968$	0	0
$969$	−18.5667	−0.596450
$970$	0	0
$971$	−44.1204	−1.41589	−0.707946	−	0.706267i	$-0.750378\pi$
$971$	−44.1204	−1.41589	−0.707946	+	0.706267i	$0.750378\pi$
$972$	0	0
$973$	−5.44098	−0.174430
$974$	0	0
$975$	−5.46332	−0.174966
$976$	0	0
$977$	−20.3185	−0.650047	−0.325023	−	0.945706i	$-0.605372\pi$
$977$	−20.3185	−0.650047	−0.325023	+	0.945706i	$0.605372\pi$
$978$	0	0
$979$	−28.8310	−0.921444
$980$	0	0
$981$	−9.51867	−0.303908
$982$	0	0
$983$	14.2515	0.454553	0.227276	−	0.973830i	$-0.427018\pi$
$983$	14.2515	0.454553	0.227276	+	0.973830i	$0.427018\pi$
$984$	0	0
$985$	−24.6870	−0.786594
$986$	0	0
$987$	−6.64219	−0.211423
$988$	0	0
$989$	1.62313	0.0516125
$990$	0	0
$991$	13.6361	0.433167	0.216583	−	0.976264i	$-0.430509\pi$
$991$	13.6361	0.433167	0.216583	+	0.976264i	$0.430509\pi$
$992$	0	0
$993$	14.0236	0.445027
$994$	0	0
$995$	−16.2605	−0.515492
$996$	0	0
$997$	48.0129	1.52058	0.760291	−	0.649582i	$-0.225056\pi$
$997$	48.0129	1.52058	0.760291	+	0.649582i	$0.225056\pi$
$998$	0	0
$999$	31.8183	1.00669

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.5		12
4.3	odd	2		2008.2.a.b.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.8	✓	12	4.3	odd	2
4016.2.a.i.1.5		12	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)

\(f(q)\)	\(=\)	\(q-1.57790 q^{3} -2.90606 q^{5} -2.41450 q^{7} -0.510222 q^{9} +O(q^{10})\)	\(q-1.57790 q^{3} -2.90606 q^{5} -2.41450 q^{7} -0.510222 q^{9} -2.86962 q^{11} +1.00500 q^{13} +4.58548 q^{15} +3.49998 q^{17} +3.36194 q^{19} +3.80985 q^{21} -1.20028 q^{23} +3.44516 q^{25} +5.53879 q^{27} +8.57021 q^{29} -0.396943 q^{31} +4.52798 q^{33} +7.01668 q^{35} +5.74464 q^{37} -1.58579 q^{39} -0.0521746 q^{41} -1.35229 q^{43} +1.48273 q^{45} -1.74343 q^{47} -1.17018 q^{49} -5.52264 q^{51} +7.34361 q^{53} +8.33927 q^{55} -5.30481 q^{57} -6.61102 q^{59} -1.18183 q^{61} +1.23193 q^{63} -2.92059 q^{65} -7.09191 q^{67} +1.89392 q^{69} +0.470260 q^{71} +3.15215 q^{73} -5.43613 q^{75} +6.92870 q^{77} -8.78103 q^{79} -7.20901 q^{81} +1.18887 q^{83} -10.1712 q^{85} -13.5230 q^{87} +10.0470 q^{89} -2.42657 q^{91} +0.626337 q^{93} -9.76997 q^{95} -14.4676 q^{97} +1.46414 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more