LMFDB - Embedded newform 8016.2.a.bb.1.3

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 29x^{7} - 7x^{6} + 266x^{5} + 69x^{4} - 901x^{3} - 199x^{2} + 875x + 391 $ x^9 - 29x^7 - 7x^6 + 266x^5 + 69x^4 - 901x^3 - 199x^2 + 875*x + 391
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.79204$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.792043 q^{5} -3.80237 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.792043 q^{5} -3.80237 q^{7} +1.00000 q^{9} -5.89024 q^{11} -0.555473 q^{13} +0.792043 q^{15} -1.29236 q^{17} +0.877845 q^{19} +3.80237 q^{21} -1.89018 q^{23} -4.37267 q^{25} -1.00000 q^{27} +4.21505 q^{29} -6.06288 q^{31} +5.89024 q^{33} +3.01164 q^{35} -8.81590 q^{37} +0.555473 q^{39} +0.966875 q^{41} -1.48271 q^{43} -0.792043 q^{45} -4.12328 q^{47} +7.45799 q^{49} +1.29236 q^{51} -13.1556 q^{53} +4.66532 q^{55} -0.877845 q^{57} -10.0336 q^{59} -7.38784 q^{61} -3.80237 q^{63} +0.439959 q^{65} +6.38640 q^{67} +1.89018 q^{69} -12.6943 q^{71} -11.0478 q^{73} +4.37267 q^{75} +22.3968 q^{77} -0.414580 q^{79} +1.00000 q^{81} +2.57849 q^{83} +1.02360 q^{85} -4.21505 q^{87} +0.384059 q^{89} +2.11211 q^{91} +6.06288 q^{93} -0.695291 q^{95} -3.26789 q^{97} -5.89024 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) $ 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9	$ 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9} - 7 q^{11} + 6 q^{13} - 9 q^{15} + 7 q^{17} - 2 q^{19} + 2 q^{21} - 19 q^{23} + 22 q^{25} - 9 q^{27} + 13 q^{29} - 12 q^{31} + 7 q^{33} - 4 q^{35} + 15 q^{37} - 6 q^{39} + 18 q^{41} + 6 q^{43} + 9 q^{45} - 25 q^{47} + 19 q^{49} - 7 q^{51} + 17 q^{53} + 3 q^{55} + 2 q^{57} - 3 q^{59} + 14 q^{61} - 2 q^{63} + 14 q^{65} + 4 q^{67} + 19 q^{69} - 17 q^{71} - 20 q^{73} - 22 q^{75} + 14 q^{77} + 8 q^{79} + 9 q^{81} + q^{83} + 5 q^{85} - 13 q^{87} + 36 q^{89} + 41 q^{91} + 12 q^{93} - 5 q^{95} + 31 q^{97} - 7 q^{99}+O(q^{100}) $ 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9 - 7 * q^11 + 6 * q^13 - 9 * q^15 + 7 * q^17 - 2 * q^19 + 2 * q^21 - 19 * q^23 + 22 * q^25 - 9 * q^27 + 13 * q^29 - 12 * q^31 + 7 * q^33 - 4 * q^35 + 15 * q^37 - 6 * q^39 + 18 * q^41 + 6 * q^43 + 9 * q^45 - 25 * q^47 + 19 * q^49 - 7 * q^51 + 17 * q^53 + 3 * q^55 + 2 * q^57 - 3 * q^59 + 14 * q^61 - 2 * q^63 + 14 * q^65 + 4 * q^67 + 19 * q^69 - 17 * q^71 - 20 * q^73 - 22 * q^75 + 14 * q^77 + 8 * q^79 + 9 * q^81 + q^83 + 5 * q^85 - 13 * q^87 + 36 * q^89 + 41 * q^91 + 12 * q^93 - 5 * q^95 + 31 * q^97 - 7 * 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
$5$	−0.792043	−0.354212	−0.177106	−	0.984192i	$-0.556674\pi$
$5$	−0.792043	−0.354212	−0.177106	+	0.984192i	$0.556674\pi$
$6$	0	0
$7$	−3.80237	−1.43716	−0.718580	−	0.695445i	$-0.755207\pi$
$7$	−3.80237	−1.43716	−0.718580	+	0.695445i	$0.755207\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.89024	−1.77597	−0.887987	−	0.459869i	$-0.847896\pi$
$11$	−5.89024	−1.77597	−0.887987	+	0.459869i	$0.847896\pi$
$12$	0	0
$13$	−0.555473	−0.154061	−0.0770303	−	0.997029i	$-0.524544\pi$
$13$	−0.555473	−0.154061	−0.0770303	+	0.997029i	$0.524544\pi$
$14$	0	0
$15$	0.792043	0.204505
$16$	0	0
$17$	−1.29236	−0.313443	−0.156721	−	0.987643i	$-0.550092\pi$
$17$	−1.29236	−0.313443	−0.156721	+	0.987643i	$0.550092\pi$
$18$	0	0
$19$	0.877845	0.201391	0.100696	−	0.994917i	$-0.467893\pi$
$19$	0.877845	0.201391	0.100696	+	0.994917i	$0.467893\pi$
$20$	0	0
$21$	3.80237	0.829744
$22$	0	0
$23$	−1.89018	−0.394131	−0.197065	−	0.980390i	$-0.563141\pi$
$23$	−1.89018	−0.394131	−0.197065	+	0.980390i	$0.563141\pi$
$24$	0	0
$25$	−4.37267	−0.874534
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.21505	0.782715	0.391358	−	0.920239i	$-0.372006\pi$
$29$	4.21505	0.782715	0.391358	+	0.920239i	$0.372006\pi$
$30$	0	0
$31$	−6.06288	−1.08893	−0.544463	−	0.838785i	$-0.683266\pi$
$31$	−6.06288	−1.08893	−0.544463	+	0.838785i	$0.683266\pi$
$32$	0	0
$33$	5.89024	1.02536
$34$	0	0
$35$	3.01164	0.509060
$36$	0	0
$37$	−8.81590	−1.44932	−0.724662	−	0.689104i	$-0.758004\pi$
$37$	−8.81590	−1.44932	−0.724662	+	0.689104i	$0.758004\pi$
$38$	0	0
$39$	0.555473	0.0889469
$40$	0	0
$41$	0.966875	0.151001	0.0755003	−	0.997146i	$-0.475945\pi$
$41$	0.966875	0.151001	0.0755003	+	0.997146i	$0.475945\pi$
$42$	0	0
$43$	−1.48271	−0.226111	−0.113056	−	0.993589i	$-0.536064\pi$
$43$	−1.48271	−0.226111	−0.113056	+	0.993589i	$0.536064\pi$
$44$	0	0
$45$	−0.792043	−0.118071
$46$	0	0
$47$	−4.12328	−0.601442	−0.300721	−	0.953712i	$-0.597227\pi$
$47$	−4.12328	−0.601442	−0.300721	+	0.953712i	$0.597227\pi$
$48$	0	0
$49$	7.45799	1.06543
$50$	0	0
$51$	1.29236	0.180966
$52$	0	0
$53$	−13.1556	−1.80706	−0.903532	−	0.428521i	$-0.859035\pi$
$53$	−13.1556	−1.80706	−0.903532	+	0.428521i	$0.859035\pi$
$54$	0	0
$55$	4.66532	0.629072
$56$	0	0
$57$	−0.877845	−0.116273
$58$	0	0
$59$	−10.0336	−1.30627	−0.653133	−	0.757243i	$-0.726546\pi$
$59$	−10.0336	−1.30627	−0.653133	+	0.757243i	$0.726546\pi$
$60$	0	0
$61$	−7.38784	−0.945916	−0.472958	−	0.881085i	$-0.656814\pi$
$61$	−7.38784	−0.945916	−0.472958	+	0.881085i	$0.656814\pi$
$62$	0	0
$63$	−3.80237	−0.479053
$64$	0	0
$65$	0.439959	0.0545702
$66$	0	0
$67$	6.38640	0.780223	0.390111	−	0.920768i	$-0.372437\pi$
$67$	6.38640	0.780223	0.390111	+	0.920768i	$0.372437\pi$
$68$	0	0
$69$	1.89018	0.227551
$70$	0	0
$71$	−12.6943	−1.50654	−0.753270	−	0.657712i	$-0.771525\pi$
$71$	−12.6943	−1.50654	−0.753270	+	0.657712i	$0.771525\pi$
$72$	0	0
$73$	−11.0478	−1.29305	−0.646523	−	0.762895i	$-0.723778\pi$
$73$	−11.0478	−1.29305	−0.646523	+	0.762895i	$0.723778\pi$
$74$	0	0
$75$	4.37267	0.504912
$76$	0	0
$77$	22.3968	2.55236
$78$	0	0
$79$	−0.414580	−0.0466439	−0.0233219	−	0.999728i	$-0.507424\pi$
$79$	−0.414580	−0.0466439	−0.0233219	+	0.999728i	$0.507424\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.57849	0.283026	0.141513	−	0.989936i	$-0.454803\pi$
$83$	2.57849	0.283026	0.141513	+	0.989936i	$0.454803\pi$
$84$	0	0
$85$	1.02360	0.111025
$86$	0	0
$87$	−4.21505	−0.451901
$88$	0	0
$89$	0.384059	0.0407102	0.0203551	−	0.999793i	$-0.493520\pi$
$89$	0.384059	0.0407102	0.0203551	+	0.999793i	$0.493520\pi$
$90$	0	0
$91$	2.11211	0.221410
$92$	0	0
$93$	6.06288	0.628691
$94$	0	0
$95$	−0.695291	−0.0713353
$96$	0	0
$97$	−3.26789	−0.331804	−0.165902	−	0.986142i	$-0.553054\pi$
$97$	−3.26789	−0.331804	−0.165902	+	0.986142i	$0.553054\pi$
$98$	0	0
$99$	−5.89024	−0.591991
$100$	0	0
$101$	8.49046	0.844832	0.422416	−	0.906402i	$-0.361182\pi$
$101$	8.49046	0.844832	0.422416	+	0.906402i	$0.361182\pi$
$102$	0	0
$103$	−14.0009	−1.37955	−0.689776	−	0.724023i	$-0.742291\pi$
$103$	−14.0009	−1.37955	−0.689776	+	0.724023i	$0.742291\pi$
$104$	0	0
$105$	−3.01164	−0.293906
$106$	0	0
$107$	−15.4921	−1.49767	−0.748837	−	0.662754i	$-0.769387\pi$
$107$	−15.4921	−1.49767	−0.748837	+	0.662754i	$0.769387\pi$
$108$	0	0
$109$	11.8458	1.13462	0.567311	−	0.823504i	$-0.307984\pi$
$109$	11.8458	1.13462	0.567311	+	0.823504i	$0.307984\pi$
$110$	0	0
$111$	8.81590	0.836768
$112$	0	0
$113$	−13.1420	−1.23630	−0.618148	−	0.786062i	$-0.712117\pi$
$113$	−13.1420	−1.23630	−0.618148	+	0.786062i	$0.712117\pi$
$114$	0	0
$115$	1.49711	0.139606
$116$	0	0
$117$	−0.555473	−0.0513535
$118$	0	0
$119$	4.91401	0.450467
$120$	0	0
$121$	23.6949	2.15408
$122$	0	0
$123$	−0.966875	−0.0871802
$124$	0	0
$125$	7.42355	0.663983
$126$	0	0
$127$	1.48314	0.131607	0.0658037	−	0.997833i	$-0.479039\pi$
$127$	1.48314	0.131607	0.0658037	+	0.997833i	$0.479039\pi$
$128$	0	0
$129$	1.48271	0.130545
$130$	0	0
$131$	−0.619826	−0.0541545	−0.0270772	−	0.999633i	$-0.508620\pi$
$131$	−0.619826	−0.0541545	−0.0270772	+	0.999633i	$0.508620\pi$
$132$	0	0
$133$	−3.33789	−0.289432
$134$	0	0
$135$	0.792043	0.0681682
$136$	0	0
$137$	10.3559	0.884760	0.442380	−	0.896828i	$-0.354134\pi$
$137$	10.3559	0.884760	0.442380	+	0.896828i	$0.354134\pi$
$138$	0	0
$139$	4.31897	0.366330	0.183165	−	0.983082i	$-0.441366\pi$
$139$	4.31897	0.366330	0.183165	+	0.983082i	$0.441366\pi$
$140$	0	0
$141$	4.12328	0.347243
$142$	0	0
$143$	3.27187	0.273608
$144$	0	0
$145$	−3.33850	−0.277247
$146$	0	0
$147$	−7.45799	−0.615125
$148$	0	0
$149$	0.416459	0.0341177	0.0170588	−	0.999854i	$-0.494570\pi$
$149$	0.416459	0.0341177	0.0170588	+	0.999854i	$0.494570\pi$
$150$	0	0
$151$	10.4266	0.848501	0.424250	−	0.905545i	$-0.360538\pi$
$151$	10.4266	0.848501	0.424250	+	0.905545i	$0.360538\pi$
$152$	0	0
$153$	−1.29236	−0.104481
$154$	0	0
$155$	4.80206	0.385711
$156$	0	0
$157$	10.6346	0.848736	0.424368	−	0.905490i	$-0.360496\pi$
$157$	10.6346	0.848736	0.424368	+	0.905490i	$0.360496\pi$
$158$	0	0
$159$	13.1556	1.04331
$160$	0	0
$161$	7.18717	0.566429
$162$	0	0
$163$	10.6617	0.835092	0.417546	−	0.908656i	$-0.362890\pi$
$163$	10.6617	0.835092	0.417546	+	0.908656i	$0.362890\pi$
$164$	0	0
$165$	−4.66532	−0.363195
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−12.6914	−0.976265
$170$	0	0
$171$	0.877845	0.0671305
$172$	0	0
$173$	−9.69755	−0.737291	−0.368645	−	0.929570i	$-0.620178\pi$
$173$	−9.69755	−0.737291	−0.368645	+	0.929570i	$0.620178\pi$
$174$	0	0
$175$	16.6265	1.25684
$176$	0	0
$177$	10.0336	0.754173
$178$	0	0
$179$	−9.09640	−0.679897	−0.339948	−	0.940444i	$-0.610410\pi$
$179$	−9.09640	−0.679897	−0.339948	+	0.940444i	$0.610410\pi$
$180$	0	0
$181$	3.40901	0.253390	0.126695	−	0.991942i	$-0.459563\pi$
$181$	3.40901	0.253390	0.126695	+	0.991942i	$0.459563\pi$
$182$	0	0
$183$	7.38784	0.546125
$184$	0	0
$185$	6.98257	0.513369
$186$	0	0
$187$	7.61229	0.556666
$188$	0	0
$189$	3.80237	0.276581
$190$	0	0
$191$	7.76442	0.561814	0.280907	−	0.959735i	$-0.409365\pi$
$191$	7.76442	0.561814	0.280907	+	0.959735i	$0.409365\pi$
$192$	0	0
$193$	1.67335	0.120450	0.0602250	−	0.998185i	$-0.480818\pi$
$193$	1.67335	0.120450	0.0602250	+	0.998185i	$0.480818\pi$
$194$	0	0
$195$	−0.439959	−0.0315061
$196$	0	0
$197$	−9.01551	−0.642329	−0.321164	−	0.947023i	$-0.604074\pi$
$197$	−9.01551	−0.642329	−0.321164	+	0.947023i	$0.604074\pi$
$198$	0	0
$199$	−7.27504	−0.515714	−0.257857	−	0.966183i	$-0.583016\pi$
$199$	−7.27504	−0.515714	−0.257857	+	0.966183i	$0.583016\pi$
$200$	0	0
$201$	−6.38640	−0.450462
$202$	0	0
$203$	−16.0272	−1.12489
$204$	0	0
$205$	−0.765807	−0.0534863
$206$	0	0
$207$	−1.89018	−0.131377
$208$	0	0
$209$	−5.17072	−0.357666
$210$	0	0
$211$	2.15224	0.148166	0.0740831	−	0.997252i	$-0.476397\pi$
$211$	2.15224	0.148166	0.0740831	+	0.997252i	$0.476397\pi$
$212$	0	0
$213$	12.6943	0.869801
$214$	0	0
$215$	1.17437	0.0800914
$216$	0	0
$217$	23.0533	1.56496
$218$	0	0
$219$	11.0478	0.746540
$220$	0	0
$221$	0.717870	0.0482892
$222$	0	0
$223$	14.5871	0.976823	0.488412	−	0.872613i	$-0.337577\pi$
$223$	14.5871	0.976823	0.488412	+	0.872613i	$0.337577\pi$
$224$	0	0
$225$	−4.37267	−0.291511
$226$	0	0
$227$	19.2690	1.27893	0.639466	−	0.768819i	$-0.279156\pi$
$227$	19.2690	1.27893	0.639466	+	0.768819i	$0.279156\pi$
$228$	0	0
$229$	−14.8221	−0.979469	−0.489735	−	0.871872i	$-0.662906\pi$
$229$	−14.8221	−0.979469	−0.489735	+	0.871872i	$0.662906\pi$
$230$	0	0
$231$	−22.3968	−1.47360
$232$	0	0
$233$	−7.73508	−0.506742	−0.253371	−	0.967369i	$-0.581539\pi$
$233$	−7.73508	−0.506742	−0.253371	+	0.967369i	$0.581539\pi$
$234$	0	0
$235$	3.26581	0.213038
$236$	0	0
$237$	0.414580	0.0269299
$238$	0	0
$239$	11.7199	0.758096	0.379048	−	0.925377i	$-0.376252\pi$
$239$	11.7199	0.758096	0.379048	+	0.925377i	$0.376252\pi$
$240$	0	0
$241$	−0.783620	−0.0504774	−0.0252387	−	0.999681i	$-0.508035\pi$
$241$	−0.783620	−0.0504774	−0.0252387	+	0.999681i	$0.508035\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−5.90705	−0.377387
$246$	0	0
$247$	−0.487620	−0.0310265
$248$	0	0
$249$	−2.57849	−0.163405
$250$	0	0
$251$	1.99474	0.125907	0.0629533	−	0.998016i	$-0.479948\pi$
$251$	1.99474	0.125907	0.0629533	+	0.998016i	$0.479948\pi$
$252$	0	0
$253$	11.1336	0.699966
$254$	0	0
$255$	−1.02360	−0.0641004
$256$	0	0
$257$	−12.9694	−0.809012	−0.404506	−	0.914535i	$-0.632556\pi$
$257$	−12.9694	−0.809012	−0.404506	+	0.914535i	$0.632556\pi$
$258$	0	0
$259$	33.5213	2.08291
$260$	0	0
$261$	4.21505	0.260905
$262$	0	0
$263$	1.41225	0.0870832	0.0435416	−	0.999052i	$-0.486136\pi$
$263$	1.41225	0.0870832	0.0435416	+	0.999052i	$0.486136\pi$
$264$	0	0
$265$	10.4198	0.640084
$266$	0	0
$267$	−0.384059	−0.0235040
$268$	0	0
$269$	−0.320518	−0.0195423	−0.00977116	−	0.999952i	$-0.503110\pi$
$269$	−0.320518	−0.0195423	−0.00977116	+	0.999952i	$0.503110\pi$
$270$	0	0
$271$	−16.4920	−1.00182	−0.500908	−	0.865501i	$-0.667000\pi$
$271$	−16.4920	−1.00182	−0.500908	+	0.865501i	$0.667000\pi$
$272$	0	0
$273$	−2.11211	−0.127831
$274$	0	0
$275$	25.7561	1.55315
$276$	0	0
$277$	21.5960	1.29758	0.648788	−	0.760969i	$-0.275276\pi$
$277$	21.5960	1.29758	0.648788	+	0.760969i	$0.275276\pi$
$278$	0	0
$279$	−6.06288	−0.362975
$280$	0	0
$281$	18.5310	1.10547	0.552734	−	0.833358i	$-0.313585\pi$
$281$	18.5310	1.10547	0.552734	+	0.833358i	$0.313585\pi$
$282$	0	0
$283$	17.7434	1.05473	0.527367	−	0.849637i	$-0.323179\pi$
$283$	17.7434	1.05473	0.527367	+	0.849637i	$0.323179\pi$
$284$	0	0
$285$	0.695291	0.0411855
$286$	0	0
$287$	−3.67641	−0.217012
$288$	0	0
$289$	−15.3298	−0.901754
$290$	0	0
$291$	3.26789	0.191567
$292$	0	0
$293$	10.7896	0.630337	0.315169	−	0.949036i	$-0.397939\pi$
$293$	10.7896	0.630337	0.315169	+	0.949036i	$0.397939\pi$
$294$	0	0
$295$	7.94706	0.462696
$296$	0	0
$297$	5.89024	0.341786
$298$	0	0
$299$	1.04995	0.0607200
$300$	0	0
$301$	5.63781	0.324958
$302$	0	0
$303$	−8.49046	−0.487764
$304$	0	0
$305$	5.85149	0.335055
$306$	0	0
$307$	4.09400	0.233657	0.116829	−	0.993152i	$-0.462727\pi$
$307$	4.09400	0.233657	0.116829	+	0.993152i	$0.462727\pi$
$308$	0	0
$309$	14.0009	0.796485
$310$	0	0
$311$	3.47522	0.197062	0.0985309	−	0.995134i	$-0.468586\pi$
$311$	3.47522	0.197062	0.0985309	+	0.995134i	$0.468586\pi$
$312$	0	0
$313$	−7.74695	−0.437883	−0.218942	−	0.975738i	$-0.570260\pi$
$313$	−7.74695	−0.437883	−0.218942	+	0.975738i	$0.570260\pi$
$314$	0	0
$315$	3.01164	0.169687
$316$	0	0
$317$	−19.6818	−1.10544	−0.552720	−	0.833367i	$-0.686410\pi$
$317$	−19.6818	−1.10544	−0.552720	+	0.833367i	$0.686410\pi$
$318$	0	0
$319$	−24.8277	−1.39008
$320$	0	0
$321$	15.4921	0.864683
$322$	0	0
$323$	−1.13449	−0.0631247
$324$	0	0
$325$	2.42890	0.134731
$326$	0	0
$327$	−11.8458	−0.655074
$328$	0	0
$329$	15.6782	0.864368
$330$	0	0
$331$	33.7021	1.85244	0.926219	−	0.376987i	$-0.123040\pi$
$331$	33.7021	1.85244	0.926219	+	0.376987i	$0.123040\pi$
$332$	0	0
$333$	−8.81590	−0.483108
$334$	0	0
$335$	−5.05830	−0.276364
$336$	0	0
$337$	9.85356	0.536758	0.268379	−	0.963313i	$-0.413512\pi$
$337$	9.85356	0.536758	0.268379	+	0.963313i	$0.413512\pi$
$338$	0	0
$339$	13.1420	0.713776
$340$	0	0
$341$	35.7118	1.93390
$342$	0	0
$343$	−1.74145	−0.0940295
$344$	0	0
$345$	−1.49711	−0.0806015
$346$	0	0
$347$	−5.82717	−0.312819	−0.156409	−	0.987692i	$-0.549992\pi$
$347$	−5.82717	−0.312819	−0.156409	+	0.987692i	$0.549992\pi$
$348$	0	0
$349$	−4.76718	−0.255181	−0.127591	−	0.991827i	$-0.540724\pi$
$349$	−4.76718	−0.255181	−0.127591	+	0.991827i	$0.540724\pi$
$350$	0	0
$351$	0.555473	0.0296490
$352$	0	0
$353$	−8.19037	−0.435930	−0.217965	−	0.975957i	$-0.569942\pi$
$353$	−8.19037	−0.435930	−0.217965	+	0.975957i	$0.569942\pi$
$354$	0	0
$355$	10.0545	0.533635
$356$	0	0
$357$	−4.91401	−0.260077
$358$	0	0
$359$	−4.07201	−0.214912	−0.107456	−	0.994210i	$-0.534271\pi$
$359$	−4.07201	−0.214912	−0.107456	+	0.994210i	$0.534271\pi$
$360$	0	0
$361$	−18.2294	−0.959441
$362$	0	0
$363$	−23.6949	−1.24366
$364$	0	0
$365$	8.75032	0.458013
$366$	0	0
$367$	−32.3570	−1.68902	−0.844511	−	0.535539i	$-0.820109\pi$
$367$	−32.3570	−1.68902	−0.844511	+	0.535539i	$0.820109\pi$
$368$	0	0
$369$	0.966875	0.0503335
$370$	0	0
$371$	50.0225	2.59704
$372$	0	0
$373$	−36.8166	−1.90629	−0.953147	−	0.302509i	$-0.902176\pi$
$373$	−36.8166	−1.90629	−0.953147	+	0.302509i	$0.902176\pi$
$374$	0	0
$375$	−7.42355	−0.383351
$376$	0	0
$377$	−2.34135	−0.120586
$378$	0	0
$379$	32.7057	1.67998	0.839989	−	0.542604i	$-0.182561\pi$
$379$	32.7057	1.67998	0.839989	+	0.542604i	$0.182561\pi$
$380$	0	0
$381$	−1.48314	−0.0759836
$382$	0	0
$383$	−13.0798	−0.668346	−0.334173	−	0.942512i	$-0.608457\pi$
$383$	−13.0798	−0.668346	−0.334173	+	0.942512i	$0.608457\pi$
$384$	0	0
$385$	−17.7393	−0.904076
$386$	0	0
$387$	−1.48271	−0.0753705
$388$	0	0
$389$	−30.1800	−1.53019	−0.765094	−	0.643919i	$-0.777308\pi$
$389$	−30.1800	−1.53019	−0.765094	+	0.643919i	$0.777308\pi$
$390$	0	0
$391$	2.44279	0.123537
$392$	0	0
$393$	0.619826	0.0312661
$394$	0	0
$395$	0.328365	0.0165218
$396$	0	0
$397$	−33.9012	−1.70145	−0.850727	−	0.525608i	$-0.823838\pi$
$397$	−33.9012	−1.70145	−0.850727	+	0.525608i	$0.823838\pi$
$398$	0	0
$399$	3.33789	0.167103
$400$	0	0
$401$	−10.9947	−0.549047	−0.274524	−	0.961580i	$-0.588520\pi$
$401$	−10.9947	−0.549047	−0.274524	+	0.961580i	$0.588520\pi$
$402$	0	0
$403$	3.36777	0.167761
$404$	0	0
$405$	−0.792043	−0.0393569
$406$	0	0
$407$	51.9278	2.57396
$408$	0	0
$409$	−38.5602	−1.90668	−0.953340	−	0.301899i	$-0.902380\pi$
$409$	−38.5602	−1.90668	−0.953340	+	0.301899i	$0.902380\pi$
$410$	0	0
$411$	−10.3559	−0.510817
$412$	0	0
$413$	38.1515	1.87731
$414$	0	0
$415$	−2.04227	−0.100251
$416$	0	0
$417$	−4.31897	−0.211501
$418$	0	0
$419$	−15.4499	−0.754777	−0.377388	−	0.926055i	$-0.623178\pi$
$419$	−15.4499	−0.754777	−0.377388	+	0.926055i	$0.623178\pi$
$420$	0	0
$421$	32.2837	1.57341	0.786706	−	0.617328i	$-0.211785\pi$
$421$	32.2837	1.57341	0.786706	+	0.617328i	$0.211785\pi$
$422$	0	0
$423$	−4.12328	−0.200481
$424$	0	0
$425$	5.65105	0.274116
$426$	0	0
$427$	28.0913	1.35943
$428$	0	0
$429$	−3.27187	−0.157967
$430$	0	0
$431$	−6.55545	−0.315765	−0.157882	−	0.987458i	$-0.550467\pi$
$431$	−6.55545	−0.315765	−0.157882	+	0.987458i	$0.550467\pi$
$432$	0	0
$433$	29.1380	1.40028	0.700140	−	0.714005i	$-0.253121\pi$
$433$	29.1380	1.40028	0.700140	+	0.714005i	$0.253121\pi$
$434$	0	0
$435$	3.33850	0.160069
$436$	0	0
$437$	−1.65929	−0.0793746
$438$	0	0
$439$	−29.3375	−1.40020	−0.700102	−	0.714043i	$-0.746862\pi$
$439$	−29.3375	−1.40020	−0.700102	+	0.714043i	$0.746862\pi$
$440$	0	0
$441$	7.45799	0.355142
$442$	0	0
$443$	−18.6272	−0.885004	−0.442502	−	0.896768i	$-0.645909\pi$
$443$	−18.6272	−0.885004	−0.442502	+	0.896768i	$0.645909\pi$
$444$	0	0
$445$	−0.304191	−0.0144200
$446$	0	0
$447$	−0.416459	−0.0196978
$448$	0	0
$449$	4.79727	0.226397	0.113199	−	0.993572i	$-0.463890\pi$
$449$	4.79727	0.226397	0.113199	+	0.993572i	$0.463890\pi$
$450$	0	0
$451$	−5.69513	−0.268173
$452$	0	0
$453$	−10.4266	−0.489882
$454$	0	0
$455$	−1.67288	−0.0784260
$456$	0	0
$457$	14.7293	0.689009	0.344505	−	0.938785i	$-0.388047\pi$
$457$	14.7293	0.689009	0.344505	+	0.938785i	$0.388047\pi$
$458$	0	0
$459$	1.29236	0.0603221
$460$	0	0
$461$	28.1180	1.30959	0.654793	−	0.755808i	$-0.272756\pi$
$461$	28.1180	1.30959	0.654793	+	0.755808i	$0.272756\pi$
$462$	0	0
$463$	−5.00233	−0.232478	−0.116239	−	0.993221i	$-0.537084\pi$
$463$	−5.00233	−0.232478	−0.116239	+	0.993221i	$0.537084\pi$
$464$	0	0
$465$	−4.80206	−0.222690
$466$	0	0
$467$	28.7693	1.33128	0.665642	−	0.746271i	$-0.268158\pi$
$467$	28.7693	1.33128	0.665642	+	0.746271i	$0.268158\pi$
$468$	0	0
$469$	−24.2834	−1.12130
$470$	0	0
$471$	−10.6346	−0.490018
$472$	0	0
$473$	8.73353	0.401568
$474$	0	0
$475$	−3.83853	−0.176124
$476$	0	0
$477$	−13.1556	−0.602354
$478$	0	0
$479$	−18.3805	−0.839825	−0.419912	−	0.907565i	$-0.637939\pi$
$479$	−18.3805	−0.839825	−0.419912	+	0.907565i	$0.637939\pi$
$480$	0	0
$481$	4.89700	0.223284
$482$	0	0
$483$	−7.18717	−0.327028
$484$	0	0
$485$	2.58831	0.117529
$486$	0	0
$487$	−6.59228	−0.298725	−0.149362	−	0.988783i	$-0.547722\pi$
$487$	−6.59228	−0.298725	−0.149362	+	0.988783i	$0.547722\pi$
$488$	0	0
$489$	−10.6617	−0.482141
$490$	0	0
$491$	−1.84351	−0.0831966	−0.0415983	−	0.999134i	$-0.513245\pi$
$491$	−1.84351	−0.0831966	−0.0415983	+	0.999134i	$0.513245\pi$
$492$	0	0
$493$	−5.44735	−0.245336
$494$	0	0
$495$	4.66532	0.209691
$496$	0	0
$497$	48.2685	2.16514
$498$	0	0
$499$	5.85805	0.262242	0.131121	−	0.991366i	$-0.458142\pi$
$499$	5.85805	0.262242	0.131121	+	0.991366i	$0.458142\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	21.3257	0.950868	0.475434	−	0.879751i	$-0.342291\pi$
$503$	21.3257	0.950868	0.475434	+	0.879751i	$0.342291\pi$
$504$	0	0
$505$	−6.72481	−0.299250
$506$	0	0
$507$	12.6914	0.563647
$508$	0	0
$509$	30.5669	1.35485	0.677426	−	0.735591i	$-0.263095\pi$
$509$	30.5669	1.35485	0.677426	+	0.735591i	$0.263095\pi$
$510$	0	0
$511$	42.0077	1.85831
$512$	0	0
$513$	−0.877845	−0.0387578
$514$	0	0
$515$	11.0893	0.488654
$516$	0	0
$517$	24.2871	1.06815
$518$	0	0
$519$	9.69755	0.425675
$520$	0	0
$521$	27.9492	1.22448	0.612238	−	0.790674i	$-0.290269\pi$
$521$	27.9492	1.22448	0.612238	+	0.790674i	$0.290269\pi$
$522$	0	0
$523$	45.2812	1.98001	0.990005	−	0.141035i	$-0.0450430\pi$
$523$	45.2812	1.98001	0.990005	+	0.141035i	$0.0450430\pi$
$524$	0	0
$525$	−16.6265	−0.725639
$526$	0	0
$527$	7.83541	0.341316
$528$	0	0
$529$	−19.4272	−0.844661
$530$	0	0
$531$	−10.0336	−0.435422
$532$	0	0
$533$	−0.537074	−0.0232632
$534$	0	0
$535$	12.2704	0.530495
$536$	0	0
$537$	9.09640	0.392538
$538$	0	0
$539$	−43.9294	−1.89217
$540$	0	0
$541$	−39.2895	−1.68919	−0.844594	−	0.535407i	$-0.820158\pi$
$541$	−39.2895	−1.68919	−0.844594	+	0.535407i	$0.820158\pi$
$542$	0	0
$543$	−3.40901	−0.146295
$544$	0	0
$545$	−9.38237	−0.401897
$546$	0	0
$547$	35.2105	1.50549	0.752745	−	0.658312i	$-0.228729\pi$
$547$	35.2105	1.50549	0.752745	+	0.658312i	$0.228729\pi$
$548$	0	0
$549$	−7.38784	−0.315305
$550$	0	0
$551$	3.70016	0.157632
$552$	0	0
$553$	1.57639	0.0670347
$554$	0	0
$555$	−6.98257	−0.296394
$556$	0	0
$557$	−43.6784	−1.85071	−0.925356	−	0.379099i	$-0.876234\pi$
$557$	−43.6784	−1.85071	−0.925356	+	0.379099i	$0.876234\pi$
$558$	0	0
$559$	0.823607	0.0348349
$560$	0	0
$561$	−7.61229	−0.321391
$562$	0	0
$563$	−45.6949	−1.92581	−0.962905	−	0.269839i	$-0.913030\pi$
$563$	−45.6949	−1.92581	−0.962905	+	0.269839i	$0.913030\pi$
$564$	0	0
$565$	10.4090	0.437911
$566$	0	0
$567$	−3.80237	−0.159684
$568$	0	0
$569$	−39.9641	−1.67538	−0.837690	−	0.546145i	$-0.816095\pi$
$569$	−39.9641	−1.67538	−0.837690	+	0.546145i	$0.816095\pi$
$570$	0	0
$571$	−14.3873	−0.602091	−0.301045	−	0.953610i	$-0.597336\pi$
$571$	−14.3873	−0.602091	−0.301045	+	0.953610i	$0.597336\pi$
$572$	0	0
$573$	−7.76442	−0.324363
$574$	0	0
$575$	8.26515	0.344681
$576$	0	0
$577$	9.63964	0.401303	0.200652	−	0.979663i	$-0.435694\pi$
$577$	9.63964	0.401303	0.200652	+	0.979663i	$0.435694\pi$
$578$	0	0
$579$	−1.67335	−0.0695419
$580$	0	0
$581$	−9.80435	−0.406753
$582$	0	0
$583$	77.4897	3.20930
$584$	0	0
$585$	0.439959	0.0181901
$586$	0	0
$587$	−25.9969	−1.07301	−0.536504	−	0.843898i	$-0.680255\pi$
$587$	−25.9969	−1.07301	−0.536504	+	0.843898i	$0.680255\pi$
$588$	0	0
$589$	−5.32227	−0.219300
$590$	0	0
$591$	9.01551	0.370849
$592$	0	0
$593$	−25.5448	−1.04900	−0.524499	−	0.851411i	$-0.675747\pi$
$593$	−25.5448	−1.04900	−0.524499	+	0.851411i	$0.675747\pi$
$594$	0	0
$595$	−3.89211	−0.159561
$596$	0	0
$597$	7.27504	0.297748
$598$	0	0
$599$	22.2348	0.908491	0.454245	−	0.890877i	$-0.349909\pi$
$599$	22.2348	0.908491	0.454245	+	0.890877i	$0.349909\pi$
$600$	0	0
$601$	−4.94235	−0.201602	−0.100801	−	0.994907i	$-0.532141\pi$
$601$	−4.94235	−0.201602	−0.100801	+	0.994907i	$0.532141\pi$
$602$	0	0
$603$	6.38640	0.260074
$604$	0	0
$605$	−18.7674	−0.763003
$606$	0	0
$607$	−16.2877	−0.661095	−0.330548	−	0.943789i	$-0.607233\pi$
$607$	−16.2877	−0.661095	−0.330548	+	0.943789i	$0.607233\pi$
$608$	0	0
$609$	16.0272	0.649454
$610$	0	0
$611$	2.29037	0.0926585
$612$	0	0
$613$	21.2986	0.860240	0.430120	−	0.902772i	$-0.358471\pi$
$613$	21.2986	0.860240	0.430120	+	0.902772i	$0.358471\pi$
$614$	0	0
$615$	0.765807	0.0308803
$616$	0	0
$617$	2.94116	0.118406	0.0592032	−	0.998246i	$-0.481144\pi$
$617$	2.94116	0.118406	0.0592032	+	0.998246i	$0.481144\pi$
$618$	0	0
$619$	15.9182	0.639805	0.319903	−	0.947450i	$-0.396350\pi$
$619$	15.9182	0.639805	0.319903	+	0.947450i	$0.396350\pi$
$620$	0	0
$621$	1.89018	0.0758505
$622$	0	0
$623$	−1.46033	−0.0585070
$624$	0	0
$625$	15.9836	0.639343
$626$	0	0
$627$	5.17072	0.206499
$628$	0	0
$629$	11.3933	0.454280
$630$	0	0
$631$	24.0346	0.956801	0.478400	−	0.878142i	$-0.341217\pi$
$631$	24.0346	0.956801	0.478400	+	0.878142i	$0.341217\pi$
$632$	0	0
$633$	−2.15224	−0.0855438
$634$	0	0
$635$	−1.17471	−0.0466170
$636$	0	0
$637$	−4.14272	−0.164140
$638$	0	0
$639$	−12.6943	−0.502180
$640$	0	0
$641$	−22.1395	−0.874460	−0.437230	−	0.899350i	$-0.644040\pi$
$641$	−22.1395	−0.874460	−0.437230	+	0.899350i	$0.644040\pi$
$642$	0	0
$643$	29.2492	1.15348	0.576738	−	0.816929i	$-0.304325\pi$
$643$	29.2492	1.15348	0.576738	+	0.816929i	$0.304325\pi$
$644$	0	0
$645$	−1.17437	−0.0462408
$646$	0	0
$647$	−38.1852	−1.50121	−0.750607	−	0.660749i	$-0.770239\pi$
$647$	−38.1852	−1.50121	−0.750607	+	0.660749i	$0.770239\pi$
$648$	0	0
$649$	59.1004	2.31990
$650$	0	0
$651$	−23.0533	−0.903530
$652$	0	0
$653$	−48.3317	−1.89137	−0.945684	−	0.325088i	$-0.894606\pi$
$653$	−48.3317	−1.89137	−0.945684	+	0.325088i	$0.894606\pi$
$654$	0	0
$655$	0.490929	0.0191822
$656$	0	0
$657$	−11.0478	−0.431015
$658$	0	0
$659$	25.6331	0.998525	0.499262	−	0.866451i	$-0.333604\pi$
$659$	25.6331	0.998525	0.499262	+	0.866451i	$0.333604\pi$
$660$	0	0
$661$	−14.4566	−0.562297	−0.281149	−	0.959664i	$-0.590715\pi$
$661$	−14.4566	−0.562297	−0.281149	+	0.959664i	$0.590715\pi$
$662$	0	0
$663$	−0.717870	−0.0278798
$664$	0	0
$665$	2.64375	0.102520
$666$	0	0
$667$	−7.96722	−0.308492
$668$	0	0
$669$	−14.5871	−0.563969
$670$	0	0
$671$	43.5162	1.67992
$672$	0	0
$673$	46.7998	1.80400	0.902000	−	0.431735i	$-0.142098\pi$
$673$	46.7998	1.80400	0.902000	+	0.431735i	$0.142098\pi$
$674$	0	0
$675$	4.37267	0.168304
$676$	0	0
$677$	5.07773	0.195153	0.0975765	−	0.995228i	$-0.468891\pi$
$677$	5.07773	0.195153	0.0975765	+	0.995228i	$0.468891\pi$
$678$	0	0
$679$	12.4257	0.476855
$680$	0	0
$681$	−19.2690	−0.738392
$682$	0	0
$683$	11.3830	0.435557	0.217779	−	0.975998i	$-0.430119\pi$
$683$	11.3830	0.435557	0.217779	+	0.975998i	$0.430119\pi$
$684$	0	0
$685$	−8.20228	−0.313393
$686$	0	0
$687$	14.8221	0.565497
$688$	0	0
$689$	7.30760	0.278397
$690$	0	0
$691$	23.9443	0.910883	0.455441	−	0.890266i	$-0.349481\pi$
$691$	23.9443	0.910883	0.455441	+	0.890266i	$0.349481\pi$
$692$	0	0
$693$	22.3968	0.850786
$694$	0	0
$695$	−3.42081	−0.129759
$696$	0	0
$697$	−1.24955	−0.0473300
$698$	0	0
$699$	7.73508	0.292568
$700$	0	0
$701$	−28.5154	−1.07701	−0.538506	−	0.842622i	$-0.681011\pi$
$701$	−28.5154	−1.07701	−0.538506	+	0.842622i	$0.681011\pi$
$702$	0	0
$703$	−7.73899	−0.291882
$704$	0	0
$705$	−3.26581	−0.122998
$706$	0	0
$707$	−32.2838	−1.21416
$708$	0	0
$709$	21.7312	0.816132	0.408066	−	0.912953i	$-0.366203\pi$
$709$	21.7312	0.816132	0.408066	+	0.912953i	$0.366203\pi$
$710$	0	0
$711$	−0.414580	−0.0155480
$712$	0	0
$713$	11.4600	0.429179
$714$	0	0
$715$	−2.59146	−0.0969152
$716$	0	0
$717$	−11.7199	−0.437687
$718$	0	0
$719$	−16.3856	−0.611078	−0.305539	−	0.952180i	$-0.598837\pi$
$719$	−16.3856	−0.611078	−0.305539	+	0.952180i	$0.598837\pi$
$720$	0	0
$721$	53.2367	1.98264
$722$	0	0
$723$	0.783620	0.0291431
$724$	0	0
$725$	−18.4310	−0.684511
$726$	0	0
$727$	−2.36008	−0.0875304	−0.0437652	−	0.999042i	$-0.513935\pi$
$727$	−2.36008	−0.0875304	−0.0437652	+	0.999042i	$0.513935\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.91619	0.0708730
$732$	0	0
$733$	19.6007	0.723967	0.361983	−	0.932185i	$-0.382100\pi$
$733$	19.6007	0.723967	0.361983	+	0.932185i	$0.382100\pi$
$734$	0	0
$735$	5.90705	0.217885
$736$	0	0
$737$	−37.6174	−1.38565
$738$	0	0
$739$	24.5195	0.901965	0.450982	−	0.892533i	$-0.351074\pi$
$739$	24.5195	0.901965	0.450982	+	0.892533i	$0.351074\pi$
$740$	0	0
$741$	0.487620	0.0179132
$742$	0	0
$743$	−17.3096	−0.635028	−0.317514	−	0.948254i	$-0.602848\pi$
$743$	−17.3096	−0.635028	−0.317514	+	0.948254i	$0.602848\pi$
$744$	0	0
$745$	−0.329853	−0.0120849
$746$	0	0
$747$	2.57849	0.0943419
$748$	0	0
$749$	58.9065	2.15240
$750$	0	0
$751$	−14.8116	−0.540481	−0.270241	−	0.962793i	$-0.587103\pi$
$751$	−14.8116	−0.540481	−0.270241	+	0.962793i	$0.587103\pi$
$752$	0	0
$753$	−1.99474	−0.0726922
$754$	0	0
$755$	−8.25828	−0.300549
$756$	0	0
$757$	39.6516	1.44116	0.720580	−	0.693371i	$-0.243875\pi$
$757$	39.6516	1.44116	0.720580	+	0.693371i	$0.243875\pi$
$758$	0	0
$759$	−11.1336	−0.404125
$760$	0	0
$761$	−44.3103	−1.60625	−0.803124	−	0.595812i	$-0.796830\pi$
$761$	−44.3103	−1.60625	−0.803124	+	0.595812i	$0.796830\pi$
$762$	0	0
$763$	−45.0420	−1.63063
$764$	0	0
$765$	1.02360	0.0370084
$766$	0	0
$767$	5.57341	0.201244
$768$	0	0
$769$	−1.55817	−0.0561891	−0.0280946	−	0.999605i	$-0.508944\pi$
$769$	−1.55817	−0.0561891	−0.0280946	+	0.999605i	$0.508944\pi$
$770$	0	0
$771$	12.9694	0.467083
$772$	0	0
$773$	−29.8336	−1.07304	−0.536520	−	0.843888i	$-0.680261\pi$
$773$	−29.8336	−1.07304	−0.536520	+	0.843888i	$0.680261\pi$
$774$	0	0
$775$	26.5110	0.952302
$776$	0	0
$777$	−33.5213	−1.20257
$778$	0	0
$779$	0.848767	0.0304102
$780$	0	0
$781$	74.7726	2.67558
$782$	0	0
$783$	−4.21505	−0.150634
$784$	0	0
$785$	−8.42308	−0.300633
$786$	0	0
$787$	−31.9000	−1.13711	−0.568556	−	0.822644i	$-0.692498\pi$
$787$	−31.9000	−1.13711	−0.568556	+	0.822644i	$0.692498\pi$
$788$	0	0
$789$	−1.41225	−0.0502775
$790$	0	0
$791$	49.9707	1.77676
$792$	0	0
$793$	4.10375	0.145728
$794$	0	0
$795$	−10.4198	−0.369553
$796$	0	0
$797$	−41.7901	−1.48028	−0.740140	−	0.672453i	$-0.765241\pi$
$797$	−41.7901	−1.48028	−0.740140	+	0.672453i	$0.765241\pi$
$798$	0	0
$799$	5.32875	0.188518
$800$	0	0
$801$	0.384059	0.0135701
$802$	0	0
$803$	65.0741	2.29642
$804$	0	0
$805$	−5.69255	−0.200636
$806$	0	0
$807$	0.320518	0.0112828
$808$	0	0
$809$	20.3756	0.716366	0.358183	−	0.933651i	$-0.383396\pi$
$809$	20.3756	0.716366	0.358183	+	0.933651i	$0.383396\pi$
$810$	0	0
$811$	20.4734	0.718917	0.359458	−	0.933161i	$-0.382961\pi$
$811$	20.4734	0.718917	0.359458	+	0.933161i	$0.382961\pi$
$812$	0	0
$813$	16.4920	0.578398
$814$	0	0
$815$	−8.44456	−0.295800
$816$	0	0
$817$	−1.30159	−0.0455369
$818$	0	0
$819$	2.11211	0.0738032
$820$	0	0
$821$	26.8617	0.937481	0.468741	−	0.883336i	$-0.344708\pi$
$821$	26.8617	0.937481	0.468741	+	0.883336i	$0.344708\pi$
$822$	0	0
$823$	−24.2379	−0.844879	−0.422440	−	0.906391i	$-0.638826\pi$
$823$	−24.2379	−0.844879	−0.422440	+	0.906391i	$0.638826\pi$
$824$	0	0
$825$	−25.7561	−0.896711
$826$	0	0
$827$	−23.7526	−0.825958	−0.412979	−	0.910740i	$-0.635512\pi$
$827$	−23.7526	−0.825958	−0.412979	+	0.910740i	$0.635512\pi$
$828$	0	0
$829$	41.3698	1.43683	0.718417	−	0.695613i	$-0.244867\pi$
$829$	41.3698	1.43683	0.718417	+	0.695613i	$0.244867\pi$
$830$	0	0
$831$	−21.5960	−0.749156
$832$	0	0
$833$	−9.63839	−0.333950
$834$	0	0
$835$	0.792043	0.0274098
$836$	0	0
$837$	6.06288	0.209564
$838$	0	0
$839$	−3.11634	−0.107588	−0.0537940	−	0.998552i	$-0.517131\pi$
$839$	−3.11634	−0.107588	−0.0537940	+	0.998552i	$0.517131\pi$
$840$	0	0
$841$	−11.2333	−0.387357
$842$	0	0
$843$	−18.5310	−0.638242
$844$	0	0
$845$	10.0522	0.345805
$846$	0	0
$847$	−90.0968	−3.09576
$848$	0	0
$849$	−17.7434	−0.608951
$850$	0	0
$851$	16.6637	0.571223
$852$	0	0
$853$	−17.5784	−0.601874	−0.300937	−	0.953644i	$-0.597299\pi$
$853$	−17.5784	−0.601874	−0.300937	+	0.953644i	$0.597299\pi$
$854$	0	0
$855$	−0.695291	−0.0237784
$856$	0	0
$857$	39.7135	1.35659	0.678294	−	0.734790i	$-0.262719\pi$
$857$	39.7135	1.35659	0.678294	+	0.734790i	$0.262719\pi$
$858$	0	0
$859$	−20.4277	−0.696985	−0.348492	−	0.937312i	$-0.613306\pi$
$859$	−20.4277	−0.696985	−0.348492	+	0.937312i	$0.613306\pi$
$860$	0	0
$861$	3.67641	0.125292
$862$	0	0
$863$	−33.9054	−1.15415	−0.577076	−	0.816690i	$-0.695806\pi$
$863$	−33.9054	−1.15415	−0.577076	+	0.816690i	$0.695806\pi$
$864$	0	0
$865$	7.68087	0.261157
$866$	0	0
$867$	15.3298	0.520628
$868$	0	0
$869$	2.44198	0.0828383
$870$	0	0
$871$	−3.54747	−0.120202
$872$	0	0
$873$	−3.26789	−0.110601
$874$	0	0
$875$	−28.2271	−0.954249
$876$	0	0
$877$	26.8020	0.905040	0.452520	−	0.891754i	$-0.350525\pi$
$877$	26.8020	0.905040	0.452520	+	0.891754i	$0.350525\pi$
$878$	0	0
$879$	−10.7896	−0.363925
$880$	0	0
$881$	50.1242	1.68873	0.844364	−	0.535770i	$-0.179978\pi$
$881$	50.1242	1.68873	0.844364	+	0.535770i	$0.179978\pi$
$882$	0	0
$883$	5.39331	0.181499	0.0907496	−	0.995874i	$-0.471074\pi$
$883$	5.39331	0.181499	0.0907496	+	0.995874i	$0.471074\pi$
$884$	0	0
$885$	−7.94706	−0.267137
$886$	0	0
$887$	−15.0636	−0.505786	−0.252893	−	0.967494i	$-0.581382\pi$
$887$	−15.0636	−0.505786	−0.252893	+	0.967494i	$0.581382\pi$
$888$	0	0
$889$	−5.63944	−0.189141
$890$	0	0
$891$	−5.89024	−0.197330
$892$	0	0
$893$	−3.61960	−0.121125
$894$	0	0
$895$	7.20473	0.240828
$896$	0	0
$897$	−1.04995	−0.0350567
$898$	0	0
$899$	−25.5554	−0.852319
$900$	0	0
$901$	17.0018	0.566411
$902$	0	0
$903$	−5.63781	−0.187615
$904$	0	0
$905$	−2.70008	−0.0897537
$906$	0	0
$907$	−49.2144	−1.63414	−0.817069	−	0.576540i	$-0.804402\pi$
$907$	−49.2144	−1.63414	−0.817069	+	0.576540i	$0.804402\pi$
$908$	0	0
$909$	8.49046	0.281611
$910$	0	0
$911$	6.92136	0.229315	0.114657	−	0.993405i	$-0.463423\pi$
$911$	6.92136	0.229315	0.114657	+	0.993405i	$0.463423\pi$
$912$	0	0
$913$	−15.1879	−0.502646
$914$	0	0
$915$	−5.85149	−0.193444
$916$	0	0
$917$	2.35681	0.0778286
$918$	0	0
$919$	−13.2852	−0.438238	−0.219119	−	0.975698i	$-0.570318\pi$
$919$	−13.2852	−0.438238	−0.219119	+	0.975698i	$0.570318\pi$
$920$	0	0
$921$	−4.09400	−0.134902
$922$	0	0
$923$	7.05136	0.232098
$924$	0	0
$925$	38.5490	1.26748
$926$	0	0
$927$	−14.0009	−0.459851
$928$	0	0
$929$	−4.83070	−0.158490	−0.0792451	−	0.996855i	$-0.525251\pi$
$929$	−4.83070	−0.158490	−0.0792451	+	0.996855i	$0.525251\pi$
$930$	0	0
$931$	6.54696	0.214568
$932$	0	0
$933$	−3.47522	−0.113774
$934$	0	0
$935$	−6.02926	−0.197178
$936$	0	0
$937$	19.8142	0.647303	0.323651	−	0.946176i	$-0.395089\pi$
$937$	19.8142	0.647303	0.323651	+	0.946176i	$0.395089\pi$
$938$	0	0
$939$	7.74695	0.252812
$940$	0	0
$941$	36.3964	1.18649	0.593245	−	0.805022i	$-0.297846\pi$
$941$	36.3964	1.18649	0.593245	+	0.805022i	$0.297846\pi$
$942$	0	0
$943$	−1.82757	−0.0595140
$944$	0	0
$945$	−3.01164	−0.0979686
$946$	0	0
$947$	−14.7560	−0.479505	−0.239753	−	0.970834i	$-0.577066\pi$
$947$	−14.7560	−0.479505	−0.239753	+	0.970834i	$0.577066\pi$
$948$	0	0
$949$	6.13675	0.199207
$950$	0	0
$951$	19.6818	0.638226
$952$	0	0
$953$	−1.59837	−0.0517764	−0.0258882	−	0.999665i	$-0.508241\pi$
$953$	−1.59837	−0.0517764	−0.0258882	+	0.999665i	$0.508241\pi$
$954$	0	0
$955$	−6.14975	−0.199001
$956$	0	0
$957$	24.8277	0.802564
$958$	0	0
$959$	−39.3768	−1.27154
$960$	0	0
$961$	5.75852	0.185759
$962$	0	0
$963$	−15.4921	−0.499225
$964$	0	0
$965$	−1.32536	−0.0426649
$966$	0	0
$967$	−4.37787	−0.140783	−0.0703914	−	0.997519i	$-0.522425\pi$
$967$	−4.37787	−0.140783	−0.0703914	+	0.997519i	$0.522425\pi$
$968$	0	0
$969$	1.13449	0.0364450
$970$	0	0
$971$	−7.68220	−0.246534	−0.123267	−	0.992374i	$-0.539337\pi$
$971$	−7.68220	−0.246534	−0.123267	+	0.992374i	$0.539337\pi$
$972$	0	0
$973$	−16.4223	−0.526475
$974$	0	0
$975$	−2.42890	−0.0777871
$976$	0	0
$977$	−8.62319	−0.275880	−0.137940	−	0.990441i	$-0.544048\pi$
$977$	−8.62319	−0.275880	−0.137940	+	0.990441i	$0.544048\pi$
$978$	0	0
$979$	−2.26220	−0.0723002
$980$	0	0
$981$	11.8458	0.378207
$982$	0	0
$983$	−39.6553	−1.26481	−0.632404	−	0.774638i	$-0.717932\pi$
$983$	−39.6553	−1.26481	−0.632404	+	0.774638i	$0.717932\pi$
$984$	0	0
$985$	7.14067	0.227521
$986$	0	0
$987$	−15.6782	−0.499043
$988$	0	0
$989$	2.80260	0.0891174
$990$	0	0
$991$	−21.3832	−0.679259	−0.339629	−	0.940559i	$-0.610302\pi$
$991$	−21.3832	−0.679259	−0.339629	+	0.940559i	$0.610302\pi$
$992$	0	0
$993$	−33.7021	−1.06951
$994$	0	0
$995$	5.76214	0.182672
$996$	0	0
$997$	30.8641	0.977477	0.488738	−	0.872430i	$-0.337457\pi$
$997$	30.8641	0.977477	0.488738	+	0.872430i	$0.337457\pi$
$998$	0	0
$999$	8.81590	0.278923

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.bb.1.3		9
4.3	odd	2		2004.2.a.d.1.3	✓	9
12.11	even	2		6012.2.a.h.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.d.1.3	✓	9	4.3	odd	2
6012.2.a.h.1.7		9	12.11	even	2
8016.2.a.bb.1.3		9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.792043 q^{5} -3.80237 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.792043 q^{5} -3.80237 q^{7} +1.00000 q^{9} -5.89024 q^{11} -0.555473 q^{13} +0.792043 q^{15} -1.29236 q^{17} +0.877845 q^{19} +3.80237 q^{21} -1.89018 q^{23} -4.37267 q^{25} -1.00000 q^{27} +4.21505 q^{29} -6.06288 q^{31} +5.89024 q^{33} +3.01164 q^{35} -8.81590 q^{37} +0.555473 q^{39} +0.966875 q^{41} -1.48271 q^{43} -0.792043 q^{45} -4.12328 q^{47} +7.45799 q^{49} +1.29236 q^{51} -13.1556 q^{53} +4.66532 q^{55} -0.877845 q^{57} -10.0336 q^{59} -7.38784 q^{61} -3.80237 q^{63} +0.439959 q^{65} +6.38640 q^{67} +1.89018 q^{69} -12.6943 q^{71} -11.0478 q^{73} +4.37267 q^{75} +22.3968 q^{77} -0.414580 q^{79} +1.00000 q^{81} +2.57849 q^{83} +1.02360 q^{85} -4.21505 q^{87} +0.384059 q^{89} +2.11211 q^{91} +6.06288 q^{93} -0.695291 q^{95} -3.26789 q^{97} -5.89024 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9}+O(q^{10}) \) 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9	\( 9 q - 9 q^{3} + 9 q^{5} - 2 q^{7} + 9 q^{9} - 7 q^{11} + 6 q^{13} - 9 q^{15} + 7 q^{17} - 2 q^{19} + 2 q^{21} - 19 q^{23} + 22 q^{25} - 9 q^{27} + 13 q^{29} - 12 q^{31} + 7 q^{33} - 4 q^{35} + 15 q^{37} - 6 q^{39} + 18 q^{41} + 6 q^{43} + 9 q^{45} - 25 q^{47} + 19 q^{49} - 7 q^{51} + 17 q^{53} + 3 q^{55} + 2 q^{57} - 3 q^{59} + 14 q^{61} - 2 q^{63} + 14 q^{65} + 4 q^{67} + 19 q^{69} - 17 q^{71} - 20 q^{73} - 22 q^{75} + 14 q^{77} + 8 q^{79} + 9 q^{81} + q^{83} + 5 q^{85} - 13 q^{87} + 36 q^{89} + 41 q^{91} + 12 q^{93} - 5 q^{95} + 31 q^{97} - 7 q^{99}+O(q^{100}) \) 9 * q - 9 * q^3 + 9 * q^5 - 2 * q^7 + 9 * q^9 - 7 * q^11 + 6 * q^13 - 9 * q^15 + 7 * q^17 - 2 * q^19 + 2 * q^21 - 19 * q^23 + 22 * q^25 - 9 * q^27 + 13 * q^29 - 12 * q^31 + 7 * q^33 - 4 * q^35 + 15 * q^37 - 6 * q^39 + 18 * q^41 + 6 * q^43 + 9 * q^45 - 25 * q^47 + 19 * q^49 - 7 * q^51 + 17 * q^53 + 3 * q^55 + 2 * q^57 - 3 * q^59 + 14 * q^61 - 2 * q^63 + 14 * q^65 + 4 * q^67 + 19 * q^69 - 17 * q^71 - 20 * q^73 - 22 * q^75 + 14 * q^77 + 8 * q^79 + 9 * q^81 + q^83 + 5 * q^85 - 13 * q^87 + 36 * q^89 + 41 * q^91 + 12 * q^93 - 5 * q^95 + 31 * q^97 - 7 * q^99

Introduction

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