LMFDB - Embedded newform 8016.2.a.p.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.36497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 $ x^5 - 2x^4 - 3x^3 + 5*x^2 + x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 501)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.31801$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.48658 q^{5} -3.13190 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.48658 q^{5} -3.13190 q^{7} +1.00000 q^{9} +5.14945 q^{11} -2.47676 q^{13} +3.48658 q^{15} -7.36613 q^{17} +1.17562 q^{19} +3.13190 q^{21} -4.38769 q^{23} +7.15622 q^{25} -1.00000 q^{27} -8.41200 q^{29} +4.26508 q^{31} -5.14945 q^{33} +10.9196 q^{35} -6.00097 q^{37} +2.47676 q^{39} +10.6359 q^{41} -1.62326 q^{43} -3.48658 q^{45} -8.29637 q^{47} +2.80882 q^{49} +7.36613 q^{51} -5.10653 q^{53} -17.9540 q^{55} -1.17562 q^{57} -0.146919 q^{59} -7.55907 q^{61} -3.13190 q^{63} +8.63543 q^{65} -12.8674 q^{67} +4.38769 q^{69} -13.9141 q^{71} -2.23198 q^{73} -7.15622 q^{75} -16.1276 q^{77} -3.03137 q^{79} +1.00000 q^{81} +6.08068 q^{83} +25.6826 q^{85} +8.41200 q^{87} -10.2591 q^{89} +7.75699 q^{91} -4.26508 q^{93} -4.09889 q^{95} -7.33303 q^{97} +5.14945 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9} + 15 q^{11} + 9 q^{15} - 11 q^{17} + 16 q^{19} - 4 q^{21} + 9 q^{23} + 6 q^{25} - 5 q^{27} - q^{29} + 18 q^{31} - 15 q^{33} + 4 q^{35} + 7 q^{37} - 10 q^{41} - 6 q^{43} - 9 q^{45} + 7 q^{47} + 11 q^{49} + 11 q^{51} - 9 q^{53} - 17 q^{55} - 16 q^{57} + 37 q^{59} - 2 q^{61} + 4 q^{63} - 16 q^{65} - 9 q^{69} - 13 q^{71} - 6 q^{73} - 6 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 5 q^{83} + 29 q^{85} + q^{87} - 30 q^{89} + 33 q^{91} - 18 q^{93} - 43 q^{95} - 9 q^{97} + 15 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9 + 15 * q^11 + 9 * q^15 - 11 * q^17 + 16 * q^19 - 4 * q^21 + 9 * q^23 + 6 * q^25 - 5 * q^27 - q^29 + 18 * q^31 - 15 * q^33 + 4 * q^35 + 7 * q^37 - 10 * q^41 - 6 * q^43 - 9 * q^45 + 7 * q^47 + 11 * q^49 + 11 * q^51 - 9 * q^53 - 17 * q^55 - 16 * q^57 + 37 * q^59 - 2 * q^61 + 4 * q^63 - 16 * q^65 - 9 * q^69 - 13 * q^71 - 6 * q^73 - 6 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 5 * q^83 + 29 * q^85 + q^87 - 30 * q^89 + 33 * q^91 - 18 * q^93 - 43 * q^95 - 9 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.48658	−1.55924	−0.779622	−	0.626250i	$-0.784589\pi$
$5$	−3.48658	−1.55924	−0.779622	+	0.626250i	$0.784589\pi$
$6$	0	0
$7$	−3.13190	−1.18375	−0.591874	−	0.806030i	$-0.701612\pi$
$7$	−3.13190	−1.18375	−0.591874	+	0.806030i	$0.701612\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.14945	1.55262	0.776309	−	0.630353i	$-0.217090\pi$
$11$	5.14945	1.55262	0.776309	+	0.630353i	$0.217090\pi$
$12$	0	0
$13$	−2.47676	−0.686931	−0.343465	−	0.939165i	$-0.611601\pi$
$13$	−2.47676	−0.686931	−0.343465	+	0.939165i	$0.611601\pi$
$14$	0	0
$15$	3.48658	0.900230
$16$	0	0
$17$	−7.36613	−1.78655	−0.893275	−	0.449512i	$-0.851598\pi$
$17$	−7.36613	−1.78655	−0.893275	+	0.449512i	$0.851598\pi$
$18$	0	0
$19$	1.17562	0.269706	0.134853	−	0.990866i	$-0.456944\pi$
$19$	1.17562	0.269706	0.134853	+	0.990866i	$0.456944\pi$
$20$	0	0
$21$	3.13190	0.683438
$22$	0	0
$23$	−4.38769	−0.914896	−0.457448	−	0.889236i	$-0.651236\pi$
$23$	−4.38769	−0.914896	−0.457448	+	0.889236i	$0.651236\pi$
$24$	0	0
$25$	7.15622	1.43124
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.41200	−1.56207	−0.781035	−	0.624487i	$-0.785308\pi$
$29$	−8.41200	−1.56207	−0.781035	+	0.624487i	$0.785308\pi$
$30$	0	0
$31$	4.26508	0.766032	0.383016	−	0.923742i	$-0.374885\pi$
$31$	4.26508	0.766032	0.383016	+	0.923742i	$0.374885\pi$
$32$	0	0
$33$	−5.14945	−0.896404
$34$	0	0
$35$	10.9196	1.84575
$36$	0	0
$37$	−6.00097	−0.986553	−0.493277	−	0.869873i	$-0.664201\pi$
$37$	−6.00097	−0.986553	−0.493277	+	0.869873i	$0.664201\pi$
$38$	0	0
$39$	2.47676	0.396600
$40$	0	0
$41$	10.6359	1.66105	0.830527	−	0.556978i	$-0.188039\pi$
$41$	10.6359	1.66105	0.830527	+	0.556978i	$0.188039\pi$
$42$	0	0
$43$	−1.62326	−0.247544	−0.123772	−	0.992311i	$-0.539499\pi$
$43$	−1.62326	−0.247544	−0.123772	+	0.992311i	$0.539499\pi$
$44$	0	0
$45$	−3.48658	−0.519748
$46$	0	0
$47$	−8.29637	−1.21015	−0.605075	−	0.796169i	$-0.706857\pi$
$47$	−8.29637	−1.21015	−0.605075	+	0.796169i	$0.706857\pi$
$48$	0	0
$49$	2.80882	0.401261
$50$	0	0
$51$	7.36613	1.03146
$52$	0	0
$53$	−5.10653	−0.701436	−0.350718	−	0.936481i	$-0.614063\pi$
$53$	−5.10653	−0.701436	−0.350718	+	0.936481i	$0.614063\pi$
$54$	0	0
$55$	−17.9540	−2.42091
$56$	0	0
$57$	−1.17562	−0.155715
$58$	0	0
$59$	−0.146919	−0.0191272	−0.00956362	−	0.999954i	$-0.503044\pi$
$59$	−0.146919	−0.0191272	−0.00956362	+	0.999954i	$0.503044\pi$
$60$	0	0
$61$	−7.55907	−0.967840	−0.483920	−	0.875112i	$-0.660787\pi$
$61$	−7.55907	−0.967840	−0.483920	+	0.875112i	$0.660787\pi$
$62$	0	0
$63$	−3.13190	−0.394583
$64$	0	0
$65$	8.63543	1.07109
$66$	0	0
$67$	−12.8674	−1.57200	−0.786002	−	0.618224i	$-0.787853\pi$
$67$	−12.8674	−1.57200	−0.786002	+	0.618224i	$0.787853\pi$
$68$	0	0
$69$	4.38769	0.528216
$70$	0	0
$71$	−13.9141	−1.65130	−0.825652	−	0.564180i	$-0.809193\pi$
$71$	−13.9141	−1.65130	−0.825652	+	0.564180i	$0.809193\pi$
$72$	0	0
$73$	−2.23198	−0.261234	−0.130617	−	0.991433i	$-0.541696\pi$
$73$	−2.23198	−0.261234	−0.130617	+	0.991433i	$0.541696\pi$
$74$	0	0
$75$	−7.15622	−0.826329
$76$	0	0
$77$	−16.1276	−1.83791
$78$	0	0
$79$	−3.03137	−0.341056	−0.170528	−	0.985353i	$-0.554547\pi$
$79$	−3.03137	−0.341056	−0.170528	+	0.985353i	$0.554547\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	6.08068	0.667441	0.333721	−	0.942672i	$-0.391696\pi$
$83$	6.08068	0.667441	0.333721	+	0.942672i	$0.391696\pi$
$84$	0	0
$85$	25.6826	2.78567
$86$	0	0
$87$	8.41200	0.901861
$88$	0	0
$89$	−10.2591	−1.08746	−0.543732	−	0.839259i	$-0.682989\pi$
$89$	−10.2591	−1.08746	−0.543732	+	0.839259i	$0.682989\pi$
$90$	0	0
$91$	7.75699	0.813153
$92$	0	0
$93$	−4.26508	−0.442269
$94$	0	0
$95$	−4.09889	−0.420537
$96$	0	0
$97$	−7.33303	−0.744556	−0.372278	−	0.928121i	$-0.621423\pi$
$97$	−7.33303	−0.744556	−0.372278	+	0.928121i	$0.621423\pi$
$98$	0	0
$99$	5.14945	0.517539
$100$	0	0
$101$	1.72959	0.172101	0.0860503	−	0.996291i	$-0.472575\pi$
$101$	1.72959	0.172101	0.0860503	+	0.996291i	$0.472575\pi$
$102$	0	0
$103$	9.84238	0.969799	0.484899	−	0.874570i	$-0.338856\pi$
$103$	9.84238	0.969799	0.484899	+	0.874570i	$0.338856\pi$
$104$	0	0
$105$	−10.9196	−1.06565
$106$	0	0
$107$	−0.708156	−0.0684600	−0.0342300	−	0.999414i	$-0.510898\pi$
$107$	−0.708156	−0.0684600	−0.0342300	+	0.999414i	$0.510898\pi$
$108$	0	0
$109$	8.02849	0.768990	0.384495	−	0.923127i	$-0.374376\pi$
$109$	8.02849	0.768990	0.384495	+	0.923127i	$0.374376\pi$
$110$	0	0
$111$	6.00097	0.569587
$112$	0	0
$113$	−8.65810	−0.814485	−0.407243	−	0.913320i	$-0.633510\pi$
$113$	−8.65810	−0.814485	−0.407243	+	0.913320i	$0.633510\pi$
$114$	0	0
$115$	15.2980	1.42655
$116$	0	0
$117$	−2.47676	−0.228977
$118$	0	0
$119$	23.0700	2.11482
$120$	0	0
$121$	15.5168	1.41062
$122$	0	0
$123$	−10.6359	−0.959010
$124$	0	0
$125$	−7.51782	−0.672415
$126$	0	0
$127$	−2.86053	−0.253831	−0.126915	−	0.991914i	$-0.540508\pi$
$127$	−2.86053	−0.253831	−0.126915	+	0.991914i	$0.540508\pi$
$128$	0	0
$129$	1.62326	0.142920
$130$	0	0
$131$	6.98528	0.610306	0.305153	−	0.952303i	$-0.401292\pi$
$131$	6.98528	0.610306	0.305153	+	0.952303i	$0.401292\pi$
$132$	0	0
$133$	−3.68193	−0.319264
$134$	0	0
$135$	3.48658	0.300077
$136$	0	0
$137$	−22.9270	−1.95879	−0.979395	−	0.201955i	$-0.935270\pi$
$137$	−22.9270	−1.95879	−0.979395	+	0.201955i	$0.935270\pi$
$138$	0	0
$139$	−9.08870	−0.770894	−0.385447	−	0.922730i	$-0.625953\pi$
$139$	−9.08870	−0.770894	−0.385447	+	0.922730i	$0.625953\pi$
$140$	0	0
$141$	8.29637	0.698680
$142$	0	0
$143$	−12.7540	−1.06654
$144$	0	0
$145$	29.3291	2.43565
$146$	0	0
$147$	−2.80882	−0.231668
$148$	0	0
$149$	−1.09136	−0.0894074	−0.0447037	−	0.999000i	$-0.514234\pi$
$149$	−1.09136	−0.0894074	−0.0447037	+	0.999000i	$0.514234\pi$
$150$	0	0
$151$	−1.31108	−0.106694	−0.0533471	−	0.998576i	$-0.516989\pi$
$151$	−1.31108	−0.106694	−0.0533471	+	0.998576i	$0.516989\pi$
$152$	0	0
$153$	−7.36613	−0.595516
$154$	0	0
$155$	−14.8705	−1.19443
$156$	0	0
$157$	−16.8200	−1.34238	−0.671190	−	0.741285i	$-0.734217\pi$
$157$	−16.8200	−1.34238	−0.671190	+	0.741285i	$0.734217\pi$
$158$	0	0
$159$	5.10653	0.404975
$160$	0	0
$161$	13.7418	1.08301
$162$	0	0
$163$	8.09630	0.634151	0.317076	−	0.948400i	$-0.397299\pi$
$163$	8.09630	0.634151	0.317076	+	0.948400i	$0.397299\pi$
$164$	0	0
$165$	17.9540	1.39771
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−6.86564	−0.528126
$170$	0	0
$171$	1.17562	0.0899019
$172$	0	0
$173$	11.5024	0.874513	0.437257	−	0.899337i	$-0.355950\pi$
$173$	11.5024	0.874513	0.437257	+	0.899337i	$0.355950\pi$
$174$	0	0
$175$	−22.4126	−1.69423
$176$	0	0
$177$	0.146919	0.0110431
$178$	0	0
$179$	14.7851	1.10509	0.552545	−	0.833483i	$-0.313657\pi$
$179$	14.7851	1.10509	0.552545	+	0.833483i	$0.313657\pi$
$180$	0	0
$181$	−7.88729	−0.586257	−0.293129	−	0.956073i	$-0.594696\pi$
$181$	−7.88729	−0.586257	−0.293129	+	0.956073i	$0.594696\pi$
$182$	0	0
$183$	7.55907	0.558783
$184$	0	0
$185$	20.9228	1.53828
$186$	0	0
$187$	−37.9315	−2.77383
$188$	0	0
$189$	3.13190	0.227813
$190$	0	0
$191$	−22.6900	−1.64179	−0.820894	−	0.571080i	$-0.806525\pi$
$191$	−22.6900	−1.64179	−0.820894	+	0.571080i	$0.806525\pi$
$192$	0	0
$193$	−23.0432	−1.65869	−0.829343	−	0.558740i	$-0.811285\pi$
$193$	−23.0432	−1.65869	−0.829343	+	0.558740i	$0.811285\pi$
$194$	0	0
$195$	−8.63543	−0.618396
$196$	0	0
$197$	1.54673	0.110200	0.0550998	−	0.998481i	$-0.482452\pi$
$197$	1.54673	0.110200	0.0550998	+	0.998481i	$0.482452\pi$
$198$	0	0
$199$	17.9970	1.27577	0.637887	−	0.770130i	$-0.279809\pi$
$199$	17.9970	1.27577	0.637887	+	0.770130i	$0.279809\pi$
$200$	0	0
$201$	12.8674	0.907597
$202$	0	0
$203$	26.3456	1.84910
$204$	0	0
$205$	−37.0830	−2.58999
$206$	0	0
$207$	−4.38769	−0.304965
$208$	0	0
$209$	6.05379	0.418750
$210$	0	0
$211$	−18.2144	−1.25393	−0.626964	−	0.779048i	$-0.715703\pi$
$211$	−18.2144	−1.25393	−0.626964	+	0.779048i	$0.715703\pi$
$212$	0	0
$213$	13.9141	0.953381
$214$	0	0
$215$	5.65961	0.385982
$216$	0	0
$217$	−13.3578	−0.906789
$218$	0	0
$219$	2.23198	0.150823
$220$	0	0
$221$	18.2442	1.22724
$222$	0	0
$223$	27.1693	1.81939	0.909695	−	0.415277i	$-0.136315\pi$
$223$	27.1693	1.81939	0.909695	+	0.415277i	$0.136315\pi$
$224$	0	0
$225$	7.15622	0.477081
$226$	0	0
$227$	−2.40896	−0.159888	−0.0799441	−	0.996799i	$-0.525474\pi$
$227$	−2.40896	−0.159888	−0.0799441	+	0.996799i	$0.525474\pi$
$228$	0	0
$229$	19.0913	1.26159	0.630796	−	0.775949i	$-0.282729\pi$
$229$	19.0913	1.26159	0.630796	+	0.775949i	$0.282729\pi$
$230$	0	0
$231$	16.1276	1.06112
$232$	0	0
$233$	−11.8734	−0.777851	−0.388925	−	0.921269i	$-0.627154\pi$
$233$	−11.8734	−0.777851	−0.388925	+	0.921269i	$0.627154\pi$
$234$	0	0
$235$	28.9259	1.88692
$236$	0	0
$237$	3.03137	0.196909
$238$	0	0
$239$	5.68491	0.367726	0.183863	−	0.982952i	$-0.441140\pi$
$239$	5.68491	0.367726	0.183863	+	0.982952i	$0.441140\pi$
$240$	0	0
$241$	2.95547	0.190379	0.0951894	−	0.995459i	$-0.469654\pi$
$241$	2.95547	0.190379	0.0951894	+	0.995459i	$0.469654\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−9.79318	−0.625663
$246$	0	0
$247$	−2.91173	−0.185269
$248$	0	0
$249$	−6.08068	−0.385347
$250$	0	0
$251$	13.0979	0.826730	0.413365	−	0.910565i	$-0.364353\pi$
$251$	13.0979	0.826730	0.413365	+	0.910565i	$0.364353\pi$
$252$	0	0
$253$	−22.5942	−1.42048
$254$	0	0
$255$	−25.6826	−1.60831
$256$	0	0
$257$	27.2262	1.69832	0.849161	−	0.528135i	$-0.177108\pi$
$257$	27.2262	1.69832	0.849161	+	0.528135i	$0.177108\pi$
$258$	0	0
$259$	18.7945	1.16783
$260$	0	0
$261$	−8.41200	−0.520690
$262$	0	0
$263$	7.21748	0.445049	0.222524	−	0.974927i	$-0.428570\pi$
$263$	7.21748	0.445049	0.222524	+	0.974927i	$0.428570\pi$
$264$	0	0
$265$	17.8043	1.09371
$266$	0	0
$267$	10.2591	0.627848
$268$	0	0
$269$	−0.488664	−0.0297944	−0.0148972	−	0.999889i	$-0.504742\pi$
$269$	−0.488664	−0.0297944	−0.0148972	+	0.999889i	$0.504742\pi$
$270$	0	0
$271$	7.69604	0.467501	0.233751	−	0.972297i	$-0.424900\pi$
$271$	7.69604	0.467501	0.233751	+	0.972297i	$0.424900\pi$
$272$	0	0
$273$	−7.75699	−0.469474
$274$	0	0
$275$	36.8506	2.22217
$276$	0	0
$277$	20.7186	1.24486	0.622431	−	0.782675i	$-0.286145\pi$
$277$	20.7186	1.24486	0.622431	+	0.782675i	$0.286145\pi$
$278$	0	0
$279$	4.26508	0.255344
$280$	0	0
$281$	−31.1451	−1.85796	−0.928982	−	0.370126i	$-0.879315\pi$
$281$	−31.1451	−1.85796	−0.928982	+	0.370126i	$0.879315\pi$
$282$	0	0
$283$	22.6017	1.34353	0.671765	−	0.740765i	$-0.265537\pi$
$283$	22.6017	1.34353	0.671765	+	0.740765i	$0.265537\pi$
$284$	0	0
$285$	4.09889	0.242797
$286$	0	0
$287$	−33.3107	−1.96627
$288$	0	0
$289$	37.2599	2.19176
$290$	0	0
$291$	7.33303	0.429870
$292$	0	0
$293$	11.1767	0.652951	0.326476	−	0.945206i	$-0.394139\pi$
$293$	11.1767	0.652951	0.326476	+	0.945206i	$0.394139\pi$
$294$	0	0
$295$	0.512245	0.0298240
$296$	0	0
$297$	−5.14945	−0.298801
$298$	0	0
$299$	10.8673	0.628470
$300$	0	0
$301$	5.08388	0.293030
$302$	0	0
$303$	−1.72959	−0.0993624
$304$	0	0
$305$	26.3553	1.50910
$306$	0	0
$307$	20.6660	1.17947	0.589735	−	0.807597i	$-0.299232\pi$
$307$	20.6660	1.17947	0.589735	+	0.807597i	$0.299232\pi$
$308$	0	0
$309$	−9.84238	−0.559913
$310$	0	0
$311$	−29.1465	−1.65274	−0.826372	−	0.563125i	$-0.809599\pi$
$311$	−29.1465	−1.65274	−0.826372	+	0.563125i	$0.809599\pi$
$312$	0	0
$313$	30.0446	1.69822	0.849110	−	0.528216i	$-0.177139\pi$
$313$	30.0446	1.69822	0.849110	+	0.528216i	$0.177139\pi$
$314$	0	0
$315$	10.9196	0.615251
$316$	0	0
$317$	13.1392	0.737970	0.368985	−	0.929435i	$-0.379705\pi$
$317$	13.1392	0.737970	0.368985	+	0.929435i	$0.379705\pi$
$318$	0	0
$319$	−43.3172	−2.42530
$320$	0	0
$321$	0.708156	0.0395254
$322$	0	0
$323$	−8.65977	−0.481842
$324$	0	0
$325$	−17.7243	−0.983165
$326$	0	0
$327$	−8.02849	−0.443977
$328$	0	0
$329$	25.9834	1.43251
$330$	0	0
$331$	9.37504	0.515299	0.257649	−	0.966238i	$-0.417052\pi$
$331$	9.37504	0.515299	0.257649	+	0.966238i	$0.417052\pi$
$332$	0	0
$333$	−6.00097	−0.328851
$334$	0	0
$335$	44.8632	2.45114
$336$	0	0
$337$	14.7899	0.805659	0.402830	−	0.915275i	$-0.368027\pi$
$337$	14.7899	0.805659	0.402830	+	0.915275i	$0.368027\pi$
$338$	0	0
$339$	8.65810	0.470243
$340$	0	0
$341$	21.9628	1.18935
$342$	0	0
$343$	13.1264	0.708757
$344$	0	0
$345$	−15.2980	−0.823617
$346$	0	0
$347$	−4.40910	−0.236693	−0.118347	−	0.992972i	$-0.537759\pi$
$347$	−4.40910	−0.236693	−0.118347	+	0.992972i	$0.537759\pi$
$348$	0	0
$349$	−13.3569	−0.714977	−0.357488	−	0.933918i	$-0.616367\pi$
$349$	−13.3569	−0.714977	−0.357488	+	0.933918i	$0.616367\pi$
$350$	0	0
$351$	2.47676	0.132200
$352$	0	0
$353$	−20.4993	−1.09107	−0.545535	−	0.838088i	$-0.683673\pi$
$353$	−20.4993	−1.09107	−0.545535	+	0.838088i	$0.683673\pi$
$354$	0	0
$355$	48.5127	2.57479
$356$	0	0
$357$	−23.0700	−1.22099
$358$	0	0
$359$	−0.219373	−0.0115780	−0.00578902	−	0.999983i	$-0.501843\pi$
$359$	−0.219373	−0.0115780	−0.00578902	+	0.999983i	$0.501843\pi$
$360$	0	0
$361$	−17.6179	−0.927259
$362$	0	0
$363$	−15.5168	−0.814423
$364$	0	0
$365$	7.78198	0.407327
$366$	0	0
$367$	−21.8828	−1.14227	−0.571137	−	0.820855i	$-0.693497\pi$
$367$	−21.8828	−1.14227	−0.571137	+	0.820855i	$0.693497\pi$
$368$	0	0
$369$	10.6359	0.553685
$370$	0	0
$371$	15.9932	0.830324
$372$	0	0
$373$	32.7568	1.69608	0.848041	−	0.529930i	$-0.177782\pi$
$373$	32.7568	1.69608	0.848041	+	0.529930i	$0.177782\pi$
$374$	0	0
$375$	7.51782	0.388219
$376$	0	0
$377$	20.8345	1.07303
$378$	0	0
$379$	26.3586	1.35395	0.676974	−	0.736007i	$-0.263291\pi$
$379$	26.3586	1.35395	0.676974	+	0.736007i	$0.263291\pi$
$380$	0	0
$381$	2.86053	0.146549
$382$	0	0
$383$	13.0910	0.668919	0.334460	−	0.942410i	$-0.391446\pi$
$383$	13.0910	0.668919	0.334460	+	0.942410i	$0.391446\pi$
$384$	0	0
$385$	56.2301	2.86575
$386$	0	0
$387$	−1.62326	−0.0825147
$388$	0	0
$389$	8.62641	0.437377	0.218688	−	0.975795i	$-0.429822\pi$
$389$	8.62641	0.437377	0.218688	+	0.975795i	$0.429822\pi$
$390$	0	0
$391$	32.3203	1.63451
$392$	0	0
$393$	−6.98528	−0.352361
$394$	0	0
$395$	10.5691	0.531790
$396$	0	0
$397$	−1.96271	−0.0985057	−0.0492529	−	0.998786i	$-0.515684\pi$
$397$	−1.96271	−0.0985057	−0.0492529	+	0.998786i	$0.515684\pi$
$398$	0	0
$399$	3.68193	0.184327
$400$	0	0
$401$	−32.5639	−1.62617	−0.813083	−	0.582148i	$-0.802212\pi$
$401$	−32.5639	−1.62617	−0.813083	+	0.582148i	$0.802212\pi$
$402$	0	0
$403$	−10.5636	−0.526211
$404$	0	0
$405$	−3.48658	−0.173249
$406$	0	0
$407$	−30.9017	−1.53174
$408$	0	0
$409$	−13.8736	−0.686007	−0.343003	−	0.939334i	$-0.611444\pi$
$409$	−13.8736	−0.686007	−0.343003	+	0.939334i	$0.611444\pi$
$410$	0	0
$411$	22.9270	1.13091
$412$	0	0
$413$	0.460136	0.0226418
$414$	0	0
$415$	−21.2008	−1.04070
$416$	0	0
$417$	9.08870	0.445076
$418$	0	0
$419$	−9.55415	−0.466751	−0.233375	−	0.972387i	$-0.574977\pi$
$419$	−9.55415	−0.466751	−0.233375	+	0.972387i	$0.574977\pi$
$420$	0	0
$421$	2.21491	0.107948	0.0539741	−	0.998542i	$-0.482811\pi$
$421$	2.21491	0.107948	0.0539741	+	0.998542i	$0.482811\pi$
$422$	0	0
$423$	−8.29637	−0.403383
$424$	0	0
$425$	−52.7136	−2.55699
$426$	0	0
$427$	23.6743	1.14568
$428$	0	0
$429$	12.7540	0.615768
$430$	0	0
$431$	−22.7614	−1.09638	−0.548189	−	0.836355i	$-0.684682\pi$
$431$	−22.7614	−1.09638	−0.548189	+	0.836355i	$0.684682\pi$
$432$	0	0
$433$	−14.8738	−0.714791	−0.357396	−	0.933953i	$-0.616335\pi$
$433$	−14.8738	−0.714791	−0.357396	+	0.933953i	$0.616335\pi$
$434$	0	0
$435$	−29.3291	−1.40622
$436$	0	0
$437$	−5.15825	−0.246753
$438$	0	0
$439$	−1.84773	−0.0881872	−0.0440936	−	0.999027i	$-0.514040\pi$
$439$	−1.84773	−0.0881872	−0.0440936	+	0.999027i	$0.514040\pi$
$440$	0	0
$441$	2.80882	0.133754
$442$	0	0
$443$	−11.7435	−0.557950	−0.278975	−	0.960298i	$-0.589995\pi$
$443$	−11.7435	−0.557950	−0.278975	+	0.960298i	$0.589995\pi$
$444$	0	0
$445$	35.7692	1.69562
$446$	0	0
$447$	1.09136	0.0516194
$448$	0	0
$449$	−11.7259	−0.553378	−0.276689	−	0.960960i	$-0.589237\pi$
$449$	−11.7259	−0.553378	−0.276689	+	0.960960i	$0.589237\pi$
$450$	0	0
$451$	54.7692	2.57898
$452$	0	0
$453$	1.31108	0.0616000
$454$	0	0
$455$	−27.0453	−1.26790
$456$	0	0
$457$	42.2015	1.97410	0.987051	−	0.160404i	$-0.0512798\pi$
$457$	42.2015	1.97410	0.987051	+	0.160404i	$0.0512798\pi$
$458$	0	0
$459$	7.36613	0.343822
$460$	0	0
$461$	24.5098	1.14153	0.570767	−	0.821112i	$-0.306646\pi$
$461$	24.5098	1.14153	0.570767	+	0.821112i	$0.306646\pi$
$462$	0	0
$463$	3.45658	0.160641	0.0803205	−	0.996769i	$-0.474406\pi$
$463$	3.45658	0.160641	0.0803205	+	0.996769i	$0.474406\pi$
$464$	0	0
$465$	14.8705	0.689605
$466$	0	0
$467$	−23.6624	−1.09496	−0.547482	−	0.836818i	$-0.684413\pi$
$467$	−23.6624	−1.09496	−0.547482	+	0.836818i	$0.684413\pi$
$468$	0	0
$469$	40.2995	1.86086
$470$	0	0
$471$	16.8200	0.775023
$472$	0	0
$473$	−8.35888	−0.384342
$474$	0	0
$475$	8.41299	0.386015
$476$	0	0
$477$	−5.10653	−0.233812
$478$	0	0
$479$	30.2044	1.38007	0.690037	−	0.723774i	$-0.257594\pi$
$479$	30.2044	1.38007	0.690037	+	0.723774i	$0.257594\pi$
$480$	0	0
$481$	14.8630	0.677694
$482$	0	0
$483$	−13.7418	−0.625274
$484$	0	0
$485$	25.5672	1.16095
$486$	0	0
$487$	−8.65596	−0.392239	−0.196119	−	0.980580i	$-0.562834\pi$
$487$	−8.65596	−0.392239	−0.196119	+	0.980580i	$0.562834\pi$
$488$	0	0
$489$	−8.09630	−0.366127
$490$	0	0
$491$	−28.6729	−1.29399	−0.646995	−	0.762495i	$-0.723974\pi$
$491$	−28.6729	−1.29399	−0.646995	+	0.762495i	$0.723974\pi$
$492$	0	0
$493$	61.9639	2.79071
$494$	0	0
$495$	−17.9540	−0.806970
$496$	0	0
$497$	43.5777	1.95473
$498$	0	0
$499$	17.4394	0.780693	0.390346	−	0.920668i	$-0.372355\pi$
$499$	17.4394	0.780693	0.390346	+	0.920668i	$0.372355\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	25.1536	1.12154	0.560772	−	0.827970i	$-0.310504\pi$
$503$	25.1536	1.12154	0.560772	+	0.827970i	$0.310504\pi$
$504$	0	0
$505$	−6.03035	−0.268347
$506$	0	0
$507$	6.86564	0.304914
$508$	0	0
$509$	−24.4858	−1.08531	−0.542657	−	0.839954i	$-0.682582\pi$
$509$	−24.4858	−1.08531	−0.542657	+	0.839954i	$0.682582\pi$
$510$	0	0
$511$	6.99035	0.309235
$512$	0	0
$513$	−1.17562	−0.0519049
$514$	0	0
$515$	−34.3162	−1.51215
$516$	0	0
$517$	−42.7217	−1.87890
$518$	0	0
$519$	−11.5024	−0.504900
$520$	0	0
$521$	18.0239	0.789639	0.394820	−	0.918759i	$-0.370807\pi$
$521$	18.0239	0.789639	0.394820	+	0.918759i	$0.370807\pi$
$522$	0	0
$523$	−12.7597	−0.557943	−0.278971	−	0.960299i	$-0.589993\pi$
$523$	−12.7597	−0.557943	−0.278971	+	0.960299i	$0.589993\pi$
$524$	0	0
$525$	22.4126	0.978166
$526$	0	0
$527$	−31.4172	−1.36855
$528$	0	0
$529$	−3.74819	−0.162965
$530$	0	0
$531$	−0.146919	−0.00637574
$532$	0	0
$533$	−26.3427	−1.14103
$534$	0	0
$535$	2.46904	0.106746
$536$	0	0
$537$	−14.7851	−0.638024
$538$	0	0
$539$	14.4639	0.623004
$540$	0	0
$541$	29.8187	1.28201	0.641003	−	0.767538i	$-0.278519\pi$
$541$	29.8187	1.28201	0.641003	+	0.767538i	$0.278519\pi$
$542$	0	0
$543$	7.88729	0.338476
$544$	0	0
$545$	−27.9919	−1.19904
$546$	0	0
$547$	24.4152	1.04392	0.521958	−	0.852971i	$-0.325202\pi$
$547$	24.4152	1.04392	0.521958	+	0.852971i	$0.325202\pi$
$548$	0	0
$549$	−7.55907	−0.322613
$550$	0	0
$551$	−9.88932	−0.421299
$552$	0	0
$553$	9.49396	0.403724
$554$	0	0
$555$	−20.9228	−0.888125
$556$	0	0
$557$	−14.0494	−0.595291	−0.297645	−	0.954676i	$-0.596201\pi$
$557$	−14.0494	−0.595291	−0.297645	+	0.954676i	$0.596201\pi$
$558$	0	0
$559$	4.02042	0.170046
$560$	0	0
$561$	37.9315	1.60147
$562$	0	0
$563$	31.8744	1.34335	0.671673	−	0.740848i	$-0.265576\pi$
$563$	31.8744	1.34335	0.671673	+	0.740848i	$0.265576\pi$
$564$	0	0
$565$	30.1871	1.26998
$566$	0	0
$567$	−3.13190	−0.131528
$568$	0	0
$569$	35.9946	1.50897	0.754487	−	0.656315i	$-0.227886\pi$
$569$	35.9946	1.50897	0.754487	+	0.656315i	$0.227886\pi$
$570$	0	0
$571$	−0.742474	−0.0310716	−0.0155358	−	0.999879i	$-0.504945\pi$
$571$	−0.742474	−0.0310716	−0.0155358	+	0.999879i	$0.504945\pi$
$572$	0	0
$573$	22.6900	0.947887
$574$	0	0
$575$	−31.3993	−1.30944
$576$	0	0
$577$	−24.8825	−1.03587	−0.517936	−	0.855419i	$-0.673300\pi$
$577$	−24.8825	−1.03587	−0.517936	+	0.855419i	$0.673300\pi$
$578$	0	0
$579$	23.0432	0.957643
$580$	0	0
$581$	−19.0441	−0.790082
$582$	0	0
$583$	−26.2958	−1.08906
$584$	0	0
$585$	8.63543	0.357031
$586$	0	0
$587$	−40.7068	−1.68015	−0.840075	−	0.542471i	$-0.817489\pi$
$587$	−40.7068	−1.68015	−0.840075	+	0.542471i	$0.817489\pi$
$588$	0	0
$589$	5.01412	0.206603
$590$	0	0
$591$	−1.54673	−0.0636238
$592$	0	0
$593$	−15.9843	−0.656398	−0.328199	−	0.944609i	$-0.606442\pi$
$593$	−15.9843	−0.656398	−0.328199	+	0.944609i	$0.606442\pi$
$594$	0	0
$595$	−80.4354	−3.29753
$596$	0	0
$597$	−17.9970	−0.736568
$598$	0	0
$599$	−47.2868	−1.93209	−0.966043	−	0.258381i	$-0.916811\pi$
$599$	−47.2868	−1.93209	−0.966043	+	0.258381i	$0.916811\pi$
$600$	0	0
$601$	10.1748	0.415038	0.207519	−	0.978231i	$-0.433461\pi$
$601$	10.1748	0.415038	0.207519	+	0.978231i	$0.433461\pi$
$602$	0	0
$603$	−12.8674	−0.524001
$604$	0	0
$605$	−54.1006	−2.19950
$606$	0	0
$607$	−14.4349	−0.585895	−0.292947	−	0.956129i	$-0.594636\pi$
$607$	−14.4349	−0.585895	−0.292947	+	0.956129i	$0.594636\pi$
$608$	0	0
$609$	−26.3456	−1.06758
$610$	0	0
$611$	20.5481	0.831289
$612$	0	0
$613$	31.5888	1.27586	0.637931	−	0.770094i	$-0.279791\pi$
$613$	31.5888	1.27586	0.637931	+	0.770094i	$0.279791\pi$
$614$	0	0
$615$	37.0830	1.49533
$616$	0	0
$617$	−43.1763	−1.73821	−0.869106	−	0.494626i	$-0.835305\pi$
$617$	−43.1763	−1.73821	−0.869106	+	0.494626i	$0.835305\pi$
$618$	0	0
$619$	30.1013	1.20987	0.604937	−	0.796273i	$-0.293198\pi$
$619$	30.1013	1.20987	0.604937	+	0.796273i	$0.293198\pi$
$620$	0	0
$621$	4.38769	0.176072
$622$	0	0
$623$	32.1306	1.28728
$624$	0	0
$625$	−9.56962	−0.382785
$626$	0	0
$627$	−6.05379	−0.241765
$628$	0	0
$629$	44.2039	1.76253
$630$	0	0
$631$	14.7748	0.588176	0.294088	−	0.955778i	$-0.404984\pi$
$631$	14.7748	0.588176	0.294088	+	0.955778i	$0.404984\pi$
$632$	0	0
$633$	18.2144	0.723956
$634$	0	0
$635$	9.97345	0.395784
$636$	0	0
$637$	−6.95679	−0.275638
$638$	0	0
$639$	−13.9141	−0.550435
$640$	0	0
$641$	5.91157	0.233493	0.116747	−	0.993162i	$-0.462753\pi$
$641$	5.91157	0.233493	0.116747	+	0.993162i	$0.462753\pi$
$642$	0	0
$643$	−19.8852	−0.784197	−0.392098	−	0.919923i	$-0.628251\pi$
$643$	−19.8852	−0.784197	−0.392098	+	0.919923i	$0.628251\pi$
$644$	0	0
$645$	−5.65961	−0.222847
$646$	0	0
$647$	40.7214	1.60092	0.800462	−	0.599383i	$-0.204587\pi$
$647$	40.7214	1.60092	0.800462	+	0.599383i	$0.204587\pi$
$648$	0	0
$649$	−0.756552	−0.0296973
$650$	0	0
$651$	13.3578	0.523535
$652$	0	0
$653$	39.1398	1.53166	0.765829	−	0.643044i	$-0.222329\pi$
$653$	39.1398	1.53166	0.765829	+	0.643044i	$0.222329\pi$
$654$	0	0
$655$	−24.3547	−0.951617
$656$	0	0
$657$	−2.23198	−0.0870779
$658$	0	0
$659$	27.6703	1.07788	0.538941	−	0.842343i	$-0.318824\pi$
$659$	27.6703	1.07788	0.538941	+	0.842343i	$0.318824\pi$
$660$	0	0
$661$	3.89239	0.151396	0.0756982	−	0.997131i	$-0.475881\pi$
$661$	3.89239	0.151396	0.0756982	+	0.997131i	$0.475881\pi$
$662$	0	0
$663$	−18.2442	−0.708545
$664$	0	0
$665$	12.8373	0.497810
$666$	0	0
$667$	36.9092	1.42913
$668$	0	0
$669$	−27.1693	−1.05043
$670$	0	0
$671$	−38.9251	−1.50269
$672$	0	0
$673$	−16.1127	−0.621097	−0.310549	−	0.950557i	$-0.600513\pi$
$673$	−16.1127	−0.621097	−0.310549	+	0.950557i	$0.600513\pi$
$674$	0	0
$675$	−7.15622	−0.275443
$676$	0	0
$677$	4.73990	0.182169	0.0910845	−	0.995843i	$-0.470967\pi$
$677$	4.73990	0.182169	0.0910845	+	0.995843i	$0.470967\pi$
$678$	0	0
$679$	22.9663	0.881367
$680$	0	0
$681$	2.40896	0.0923115
$682$	0	0
$683$	−21.1490	−0.809243	−0.404622	−	0.914484i	$-0.632597\pi$
$683$	−21.1490	−0.809243	−0.404622	+	0.914484i	$0.632597\pi$
$684$	0	0
$685$	79.9369	3.05423
$686$	0	0
$687$	−19.0913	−0.728380
$688$	0	0
$689$	12.6477	0.481838
$690$	0	0
$691$	−20.3791	−0.775256	−0.387628	−	0.921816i	$-0.626706\pi$
$691$	−20.3791	−0.775256	−0.387628	+	0.921816i	$0.626706\pi$
$692$	0	0
$693$	−16.1276	−0.612636
$694$	0	0
$695$	31.6885	1.20201
$696$	0	0
$697$	−78.3457	−2.96756
$698$	0	0
$699$	11.8734	0.449092
$700$	0	0
$701$	0.365935	0.0138212	0.00691058	−	0.999976i	$-0.497800\pi$
$701$	0.365935	0.0138212	0.00691058	+	0.999976i	$0.497800\pi$
$702$	0	0
$703$	−7.05486	−0.266079
$704$	0	0
$705$	−28.9259	−1.08941
$706$	0	0
$707$	−5.41691	−0.203724
$708$	0	0
$709$	5.88589	0.221049	0.110525	−	0.993873i	$-0.464747\pi$
$709$	5.88589	0.221049	0.110525	+	0.993873i	$0.464747\pi$
$710$	0	0
$711$	−3.03137	−0.113685
$712$	0	0
$713$	−18.7139	−0.700840
$714$	0	0
$715$	44.4677	1.66300
$716$	0	0
$717$	−5.68491	−0.212307
$718$	0	0
$719$	−15.8978	−0.592886	−0.296443	−	0.955051i	$-0.595801\pi$
$719$	−15.8978	−0.592886	−0.296443	+	0.955051i	$0.595801\pi$
$720$	0	0
$721$	−30.8254	−1.14800
$722$	0	0
$723$	−2.95547	−0.109915
$724$	0	0
$725$	−60.1981	−2.23570
$726$	0	0
$727$	−18.9730	−0.703670	−0.351835	−	0.936062i	$-0.614442\pi$
$727$	−18.9730	−0.703670	−0.351835	+	0.936062i	$0.614442\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.9571	0.442250
$732$	0	0
$733$	5.90174	0.217986	0.108993	−	0.994043i	$-0.465237\pi$
$733$	5.90174	0.217986	0.108993	+	0.994043i	$0.465237\pi$
$734$	0	0
$735$	9.79318	0.361227
$736$	0	0
$737$	−66.2601	−2.44072
$738$	0	0
$739$	−3.94346	−0.145063	−0.0725313	−	0.997366i	$-0.523108\pi$
$739$	−3.94346	−0.145063	−0.0725313	+	0.997366i	$0.523108\pi$
$740$	0	0
$741$	2.91173	0.106965
$742$	0	0
$743$	−46.4212	−1.70303	−0.851514	−	0.524332i	$-0.824315\pi$
$743$	−46.4212	−1.70303	−0.851514	+	0.524332i	$0.824315\pi$
$744$	0	0
$745$	3.80510	0.139408
$746$	0	0
$747$	6.08068	0.222480
$748$	0	0
$749$	2.21788	0.0810395
$750$	0	0
$751$	−12.3421	−0.450370	−0.225185	−	0.974316i	$-0.572299\pi$
$751$	−12.3421	−0.450370	−0.225185	+	0.974316i	$0.572299\pi$
$752$	0	0
$753$	−13.0979	−0.477313
$754$	0	0
$755$	4.57119	0.166362
$756$	0	0
$757$	49.3731	1.79450	0.897248	−	0.441526i	$-0.145563\pi$
$757$	49.3731	1.79450	0.897248	+	0.441526i	$0.145563\pi$
$758$	0	0
$759$	22.5942	0.820117
$760$	0	0
$761$	−23.7202	−0.859856	−0.429928	−	0.902863i	$-0.641461\pi$
$761$	−23.7202	−0.859856	−0.429928	+	0.902863i	$0.641461\pi$
$762$	0	0
$763$	−25.1445	−0.910291
$764$	0	0
$765$	25.6826	0.928556
$766$	0	0
$767$	0.363884	0.0131391
$768$	0	0
$769$	−25.8993	−0.933952	−0.466976	−	0.884270i	$-0.654656\pi$
$769$	−25.8993	−0.933952	−0.466976	+	0.884270i	$0.654656\pi$
$770$	0	0
$771$	−27.2262	−0.980526
$772$	0	0
$773$	−43.2023	−1.55388	−0.776940	−	0.629575i	$-0.783229\pi$
$773$	−43.2023	−1.55388	−0.776940	+	0.629575i	$0.783229\pi$
$774$	0	0
$775$	30.5219	1.09638
$776$	0	0
$777$	−18.7945	−0.674247
$778$	0	0
$779$	12.5038	0.447996
$780$	0	0
$781$	−71.6502	−2.56384
$782$	0	0
$783$	8.41200	0.300620
$784$	0	0
$785$	58.6441	2.09310
$786$	0	0
$787$	1.87347	0.0667818	0.0333909	−	0.999442i	$-0.489369\pi$
$787$	1.87347	0.0667818	0.0333909	+	0.999442i	$0.489369\pi$
$788$	0	0
$789$	−7.21748	−0.256949
$790$	0	0
$791$	27.1163	0.964146
$792$	0	0
$793$	18.7220	0.664839
$794$	0	0
$795$	−17.8043	−0.631454
$796$	0	0
$797$	−27.4642	−0.972832	−0.486416	−	0.873727i	$-0.661696\pi$
$797$	−27.4642	−0.972832	−0.486416	+	0.873727i	$0.661696\pi$
$798$	0	0
$799$	61.1121	2.16199
$800$	0	0
$801$	−10.2591	−0.362488
$802$	0	0
$803$	−11.4935	−0.405596
$804$	0	0
$805$	−47.9119	−1.68867
$806$	0	0
$807$	0.488664	0.0172018
$808$	0	0
$809$	−22.1768	−0.779695	−0.389848	−	0.920879i	$-0.627472\pi$
$809$	−22.1768	−0.779695	−0.389848	+	0.920879i	$0.627472\pi$
$810$	0	0
$811$	17.7911	0.624729	0.312364	−	0.949962i	$-0.398879\pi$
$811$	17.7911	0.624729	0.312364	+	0.949962i	$0.398879\pi$
$812$	0	0
$813$	−7.69604	−0.269912
$814$	0	0
$815$	−28.2284	−0.988797
$816$	0	0
$817$	−1.90833	−0.0667641
$818$	0	0
$819$	7.75699	0.271051
$820$	0	0
$821$	−42.6849	−1.48971	−0.744857	−	0.667225i	$-0.767482\pi$
$821$	−42.6849	−1.48971	−0.744857	+	0.667225i	$0.767482\pi$
$822$	0	0
$823$	−11.7120	−0.408254	−0.204127	−	0.978944i	$-0.565436\pi$
$823$	−11.7120	−0.408254	−0.204127	+	0.978944i	$0.565436\pi$
$824$	0	0
$825$	−36.8506	−1.28297
$826$	0	0
$827$	−31.2241	−1.08577	−0.542884	−	0.839808i	$-0.682668\pi$
$827$	−31.2241	−1.08577	−0.542884	+	0.839808i	$0.682668\pi$
$828$	0	0
$829$	−6.77217	−0.235207	−0.117604	−	0.993061i	$-0.537521\pi$
$829$	−6.77217	−0.235207	−0.117604	+	0.993061i	$0.537521\pi$
$830$	0	0
$831$	−20.7186	−0.718722
$832$	0	0
$833$	−20.6902	−0.716872
$834$	0	0
$835$	3.48658	0.120658
$836$	0	0
$837$	−4.26508	−0.147423
$838$	0	0
$839$	25.6588	0.885840	0.442920	−	0.896561i	$-0.353943\pi$
$839$	25.6588	0.885840	0.442920	+	0.896561i	$0.353943\pi$
$840$	0	0
$841$	41.7618	1.44006
$842$	0	0
$843$	31.1451	1.07270
$844$	0	0
$845$	23.9376	0.823478
$846$	0	0
$847$	−48.5972	−1.66982
$848$	0	0
$849$	−22.6017	−0.775687
$850$	0	0
$851$	26.3304	0.902594
$852$	0	0
$853$	−41.1575	−1.40921	−0.704603	−	0.709602i	$-0.748875\pi$
$853$	−41.1575	−1.40921	−0.704603	+	0.709602i	$0.748875\pi$
$854$	0	0
$855$	−4.09889	−0.140179
$856$	0	0
$857$	16.2850	0.556284	0.278142	−	0.960540i	$-0.410281\pi$
$857$	16.2850	0.556284	0.278142	+	0.960540i	$0.410281\pi$
$858$	0	0
$859$	−4.34497	−0.148248	−0.0741242	−	0.997249i	$-0.523616\pi$
$859$	−4.34497	−0.148248	−0.0741242	+	0.997249i	$0.523616\pi$
$860$	0	0
$861$	33.3107	1.13523
$862$	0	0
$863$	−35.5748	−1.21098	−0.605490	−	0.795853i	$-0.707023\pi$
$863$	−35.5748	−1.21098	−0.605490	+	0.795853i	$0.707023\pi$
$864$	0	0
$865$	−40.1041	−1.36358
$866$	0	0
$867$	−37.2599	−1.26541
$868$	0	0
$869$	−15.6099	−0.529529
$870$	0	0
$871$	31.8695	1.07986
$872$	0	0
$873$	−7.33303	−0.248185
$874$	0	0
$875$	23.5451	0.795970
$876$	0	0
$877$	34.3006	1.15825	0.579125	−	0.815239i	$-0.303394\pi$
$877$	34.3006	1.15825	0.579125	+	0.815239i	$0.303394\pi$
$878$	0	0
$879$	−11.1767	−0.376981
$880$	0	0
$881$	−11.3825	−0.383485	−0.191742	−	0.981445i	$-0.561414\pi$
$881$	−11.3825	−0.383485	−0.191742	+	0.981445i	$0.561414\pi$
$882$	0	0
$883$	3.47050	0.116792	0.0583959	−	0.998294i	$-0.481401\pi$
$883$	3.47050	0.116792	0.0583959	+	0.998294i	$0.481401\pi$
$884$	0	0
$885$	−0.512245	−0.0172189
$886$	0	0
$887$	44.3988	1.49077	0.745383	−	0.666636i	$-0.232266\pi$
$887$	44.3988	1.49077	0.745383	+	0.666636i	$0.232266\pi$
$888$	0	0
$889$	8.95889	0.300472
$890$	0	0
$891$	5.14945	0.172513
$892$	0	0
$893$	−9.75337	−0.326384
$894$	0	0
$895$	−51.5494	−1.72311
$896$	0	0
$897$	−10.8673	−0.362847
$898$	0	0
$899$	−35.8779	−1.19660
$900$	0	0
$901$	37.6154	1.25315
$902$	0	0
$903$	−5.08388	−0.169181
$904$	0	0
$905$	27.4996	0.914119
$906$	0	0
$907$	−34.6801	−1.15153	−0.575767	−	0.817614i	$-0.695296\pi$
$907$	−34.6801	−1.15153	−0.575767	+	0.817614i	$0.695296\pi$
$908$	0	0
$909$	1.72959	0.0573669
$910$	0	0
$911$	−23.0692	−0.764316	−0.382158	−	0.924097i	$-0.624819\pi$
$911$	−23.0692	−0.764316	−0.382158	+	0.924097i	$0.624819\pi$
$912$	0	0
$913$	31.3121	1.03628
$914$	0	0
$915$	−26.3553	−0.871279
$916$	0	0
$917$	−21.8772	−0.722449
$918$	0	0
$919$	50.5957	1.66900	0.834500	−	0.551008i	$-0.185757\pi$
$919$	50.5957	1.66900	0.834500	+	0.551008i	$0.185757\pi$
$920$	0	0
$921$	−20.6660	−0.680968
$922$	0	0
$923$	34.4620	1.13433
$924$	0	0
$925$	−42.9442	−1.41200
$926$	0	0
$927$	9.84238	0.323266
$928$	0	0
$929$	−28.3578	−0.930389	−0.465195	−	0.885208i	$-0.654016\pi$
$929$	−28.3578	−0.930389	−0.465195	+	0.885208i	$0.654016\pi$
$930$	0	0
$931$	3.30211	0.108222
$932$	0	0
$933$	29.1465	0.954212
$934$	0	0
$935$	132.251	4.32508
$936$	0	0
$937$	25.3720	0.828869	0.414434	−	0.910079i	$-0.363979\pi$
$937$	25.3720	0.828869	0.414434	+	0.910079i	$0.363979\pi$
$938$	0	0
$939$	−30.0446	−0.980468
$940$	0	0
$941$	−20.3430	−0.663163	−0.331581	−	0.943427i	$-0.607582\pi$
$941$	−20.3430	−0.663163	−0.331581	+	0.943427i	$0.607582\pi$
$942$	0	0
$943$	−46.6672	−1.51969
$944$	0	0
$945$	−10.9196	−0.355215
$946$	0	0
$947$	36.4928	1.18586	0.592928	−	0.805255i	$-0.297972\pi$
$947$	36.4928	1.18586	0.592928	+	0.805255i	$0.297972\pi$
$948$	0	0
$949$	5.52809	0.179449
$950$	0	0
$951$	−13.1392	−0.426067
$952$	0	0
$953$	38.5803	1.24974	0.624869	−	0.780730i	$-0.285152\pi$
$953$	38.5803	1.24974	0.624869	+	0.780730i	$0.285152\pi$
$954$	0	0
$955$	79.1103	2.55995
$956$	0	0
$957$	43.3172	1.40025
$958$	0	0
$959$	71.8053	2.31871
$960$	0	0
$961$	−12.8091	−0.413195
$962$	0	0
$963$	−0.708156	−0.0228200
$964$	0	0
$965$	80.3419	2.58630
$966$	0	0
$967$	4.38572	0.141035	0.0705176	−	0.997511i	$-0.477535\pi$
$967$	4.38572	0.141035	0.0705176	+	0.997511i	$0.477535\pi$
$968$	0	0
$969$	8.65977	0.278192
$970$	0	0
$971$	29.8005	0.956343	0.478172	−	0.878266i	$-0.341300\pi$
$971$	29.8005	0.956343	0.478172	+	0.878266i	$0.341300\pi$
$972$	0	0
$973$	28.4649	0.912544
$974$	0	0
$975$	17.7243	0.567631
$976$	0	0
$977$	28.7521	0.919862	0.459931	−	0.887955i	$-0.347874\pi$
$977$	28.7521	0.919862	0.459931	+	0.887955i	$0.347874\pi$
$978$	0	0
$979$	−52.8288	−1.68842
$980$	0	0
$981$	8.02849	0.256330
$982$	0	0
$983$	24.7559	0.789590	0.394795	−	0.918769i	$-0.370816\pi$
$983$	24.7559	0.789590	0.394795	+	0.918769i	$0.370816\pi$
$984$	0	0
$985$	−5.39278	−0.171828
$986$	0	0
$987$	−25.9834	−0.827062
$988$	0	0
$989$	7.12234	0.226477
$990$	0	0
$991$	47.1674	1.49832	0.749161	−	0.662388i	$-0.230457\pi$
$991$	47.1674	1.49832	0.749161	+	0.662388i	$0.230457\pi$
$992$	0	0
$993$	−9.37504	−0.297508
$994$	0	0
$995$	−62.7479	−1.98924
$996$	0	0
$997$	−32.9072	−1.04218	−0.521091	−	0.853501i	$-0.674475\pi$
$997$	−32.9072	−1.04218	−0.521091	+	0.853501i	$0.674475\pi$
$998$	0	0
$999$	6.00097	0.189862

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.p.1.1		5
4.3	odd	2		501.2.a.b.1.4	✓	5
12.11	even	2		1503.2.a.d.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
501.2.a.b.1.4	✓	5	4.3	odd	2
1503.2.a.d.1.2		5	12.11	even	2
8016.2.a.p.1.1		5	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.48658 q^{5} -3.13190 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.48658 q^{5} -3.13190 q^{7} +1.00000 q^{9} +5.14945 q^{11} -2.47676 q^{13} +3.48658 q^{15} -7.36613 q^{17} +1.17562 q^{19} +3.13190 q^{21} -4.38769 q^{23} +7.15622 q^{25} -1.00000 q^{27} -8.41200 q^{29} +4.26508 q^{31} -5.14945 q^{33} +10.9196 q^{35} -6.00097 q^{37} +2.47676 q^{39} +10.6359 q^{41} -1.62326 q^{43} -3.48658 q^{45} -8.29637 q^{47} +2.80882 q^{49} +7.36613 q^{51} -5.10653 q^{53} -17.9540 q^{55} -1.17562 q^{57} -0.146919 q^{59} -7.55907 q^{61} -3.13190 q^{63} +8.63543 q^{65} -12.8674 q^{67} +4.38769 q^{69} -13.9141 q^{71} -2.23198 q^{73} -7.15622 q^{75} -16.1276 q^{77} -3.03137 q^{79} +1.00000 q^{81} +6.08068 q^{83} +25.6826 q^{85} +8.41200 q^{87} -10.2591 q^{89} +7.75699 q^{91} -4.26508 q^{93} -4.09889 q^{95} -7.33303 q^{97} +5.14945 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9} + 15 q^{11} + 9 q^{15} - 11 q^{17} + 16 q^{19} - 4 q^{21} + 9 q^{23} + 6 q^{25} - 5 q^{27} - q^{29} + 18 q^{31} - 15 q^{33} + 4 q^{35} + 7 q^{37} - 10 q^{41} - 6 q^{43} - 9 q^{45} + 7 q^{47} + 11 q^{49} + 11 q^{51} - 9 q^{53} - 17 q^{55} - 16 q^{57} + 37 q^{59} - 2 q^{61} + 4 q^{63} - 16 q^{65} - 9 q^{69} - 13 q^{71} - 6 q^{73} - 6 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 5 q^{83} + 29 q^{85} + q^{87} - 30 q^{89} + 33 q^{91} - 18 q^{93} - 43 q^{95} - 9 q^{97} + 15 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 - 9 * q^5 + 4 * q^7 + 5 * q^9 + 15 * q^11 + 9 * q^15 - 11 * q^17 + 16 * q^19 - 4 * q^21 + 9 * q^23 + 6 * q^25 - 5 * q^27 - q^29 + 18 * q^31 - 15 * q^33 + 4 * q^35 + 7 * q^37 - 10 * q^41 - 6 * q^43 - 9 * q^45 + 7 * q^47 + 11 * q^49 + 11 * q^51 - 9 * q^53 - 17 * q^55 - 16 * q^57 + 37 * q^59 - 2 * q^61 + 4 * q^63 - 16 * q^65 - 9 * q^69 - 13 * q^71 - 6 * q^73 - 6 * q^75 - 2 * q^77 - 14 * q^79 + 5 * q^81 + 5 * q^83 + 29 * q^85 + q^87 - 30 * q^89 + 33 * q^91 - 18 * q^93 - 43 * q^95 - 9 * q^97 + 15 * q^99

Introduction

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