LMFDB - Embedded newform 8036.2.a.r.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 37 x^{13} + 39 x^{12} + 537 x^{11} - 616 x^{10} - 3853 x^{9} + 4929 x^{8} + \cdots + 3381 $ x^15 - x^14 - 37x^13 + 39x^12 + 537x^11 - 616x^10 - 3853x^9 + 4929x^8 + 13971x^7 - 20311x^6 - 22309x^5 + 38415x^4 + 8429x^3 - 22584x^2 - 399*x + 3381
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$1.40418$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.40418 q^{3} -3.61567 q^{5} -1.02828 q^{9} +O(q^{10})$	$q+1.40418 q^{3} -3.61567 q^{5} -1.02828 q^{9} -5.64929 q^{11} -6.54345 q^{13} -5.07705 q^{15} -3.39066 q^{17} -5.19738 q^{19} -2.81011 q^{23} +8.07310 q^{25} -5.65643 q^{27} +2.17600 q^{29} -6.26055 q^{31} -7.93262 q^{33} +0.641460 q^{37} -9.18818 q^{39} +1.00000 q^{41} -4.44365 q^{43} +3.71793 q^{45} -6.69641 q^{47} -4.76109 q^{51} +4.95277 q^{53} +20.4260 q^{55} -7.29805 q^{57} +11.0289 q^{59} +12.0389 q^{61} +23.6590 q^{65} +10.5842 q^{67} -3.94589 q^{69} -8.70411 q^{71} -0.422919 q^{73} +11.3361 q^{75} -14.2265 q^{79} -4.85779 q^{81} -5.06521 q^{83} +12.2595 q^{85} +3.05549 q^{87} +1.56457 q^{89} -8.79093 q^{93} +18.7920 q^{95} -16.4451 q^{97} +5.80907 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9	$ 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) $ 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * 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
$3$	1.40418	0.810703	0.405351	−	0.914161i	$-0.367149\pi$
$3$	1.40418	0.810703	0.405351	+	0.914161i	$0.367149\pi$
$4$	0	0
$5$	−3.61567	−1.61698	−0.808489	−	0.588511i	$-0.799714\pi$
$5$	−3.61567	−1.61698	−0.808489	+	0.588511i	$0.799714\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.02828	−0.342761
$10$	0	0
$11$	−5.64929	−1.70333	−0.851663	−	0.524090i	$-0.824406\pi$
$11$	−5.64929	−1.70333	−0.851663	+	0.524090i	$0.824406\pi$
$12$	0	0
$13$	−6.54345	−1.81483	−0.907414	−	0.420238i	$-0.861947\pi$
$13$	−6.54345	−1.81483	−0.907414	+	0.420238i	$0.861947\pi$
$14$	0	0
$15$	−5.07705	−1.31089
$16$	0	0
$17$	−3.39066	−0.822356	−0.411178	−	0.911555i	$-0.634882\pi$
$17$	−3.39066	−0.822356	−0.411178	+	0.911555i	$0.634882\pi$
$18$	0	0
$19$	−5.19738	−1.19236	−0.596181	−	0.802850i	$-0.703316\pi$
$19$	−5.19738	−1.19236	−0.596181	+	0.802850i	$0.703316\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.81011	−0.585948	−0.292974	−	0.956120i	$-0.594645\pi$
$23$	−2.81011	−0.585948	−0.292974	+	0.956120i	$0.594645\pi$
$24$	0	0
$25$	8.07310	1.61462
$26$	0	0
$27$	−5.65643	−1.08858
$28$	0	0
$29$	2.17600	0.404072	0.202036	−	0.979378i	$-0.435244\pi$
$29$	2.17600	0.404072	0.202036	+	0.979378i	$0.435244\pi$
$30$	0	0
$31$	−6.26055	−1.12443	−0.562214	−	0.826992i	$-0.690050\pi$
$31$	−6.26055	−1.12443	−0.562214	+	0.826992i	$0.690050\pi$
$32$	0	0
$33$	−7.93262	−1.38089
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.641460	0.105455	0.0527277	−	0.998609i	$-0.483208\pi$
$37$	0.641460	0.105455	0.0527277	+	0.998609i	$0.483208\pi$
$38$	0	0
$39$	−9.18818	−1.47129
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−4.44365	−0.677650	−0.338825	−	0.940849i	$-0.610029\pi$
$43$	−4.44365	−0.677650	−0.338825	+	0.940849i	$0.610029\pi$
$44$	0	0
$45$	3.71793	0.554237
$46$	0	0
$47$	−6.69641	−0.976772	−0.488386	−	0.872628i	$-0.662414\pi$
$47$	−6.69641	−0.976772	−0.488386	+	0.872628i	$0.662414\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−4.76109	−0.666686
$52$	0	0
$53$	4.95277	0.680316	0.340158	−	0.940368i	$-0.389519\pi$
$53$	4.95277	0.680316	0.340158	+	0.940368i	$0.389519\pi$
$54$	0	0
$55$	20.4260	2.75424
$56$	0	0
$57$	−7.29805	−0.966651
$58$	0	0
$59$	11.0289	1.43584	0.717922	−	0.696124i	$-0.245094\pi$
$59$	11.0289	1.43584	0.717922	+	0.696124i	$0.245094\pi$
$60$	0	0
$61$	12.0389	1.54143	0.770713	−	0.637182i	$-0.219900\pi$
$61$	12.0389	1.54143	0.770713	+	0.637182i	$0.219900\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23.6590	2.93454
$66$	0	0
$67$	10.5842	1.29306	0.646531	−	0.762888i	$-0.276219\pi$
$67$	10.5842	1.29306	0.646531	+	0.762888i	$0.276219\pi$
$68$	0	0
$69$	−3.94589	−0.475030
$70$	0	0
$71$	−8.70411	−1.03299	−0.516494	−	0.856291i	$-0.672763\pi$
$71$	−8.70411	−1.03299	−0.516494	+	0.856291i	$0.672763\pi$
$72$	0	0
$73$	−0.422919	−0.0494990	−0.0247495	−	0.999694i	$-0.507879\pi$
$73$	−0.422919	−0.0494990	−0.0247495	+	0.999694i	$0.507879\pi$
$74$	0	0
$75$	11.3361	1.30898
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−14.2265	−1.60061	−0.800305	−	0.599594i	$-0.795329\pi$
$79$	−14.2265	−1.60061	−0.800305	+	0.599594i	$0.795329\pi$
$80$	0	0
$81$	−4.85779	−0.539754
$82$	0	0
$83$	−5.06521	−0.555979	−0.277989	−	0.960584i	$-0.589668\pi$
$83$	−5.06521	−0.555979	−0.277989	+	0.960584i	$0.589668\pi$
$84$	0	0
$85$	12.2595	1.32973
$86$	0	0
$87$	3.05549	0.327583
$88$	0	0
$89$	1.56457	0.165844	0.0829218	−	0.996556i	$-0.473575\pi$
$89$	1.56457	0.165844	0.0829218	+	0.996556i	$0.473575\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.79093	−0.911577
$94$	0	0
$95$	18.7920	1.92802
$96$	0	0
$97$	−16.4451	−1.66975	−0.834873	−	0.550442i	$-0.814459\pi$
$97$	−16.4451	−1.66975	−0.834873	+	0.550442i	$0.814459\pi$
$98$	0	0
$99$	5.80907	0.583833
$100$	0	0
$101$	11.1405	1.10852	0.554260	−	0.832343i	$-0.313001\pi$
$101$	11.1405	1.10852	0.554260	+	0.832343i	$0.313001\pi$
$102$	0	0
$103$	5.00165	0.492827	0.246414	−	0.969165i	$-0.420748\pi$
$103$	5.00165	0.492827	0.246414	+	0.969165i	$0.420748\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.29287	−0.801702	−0.400851	−	0.916143i	$-0.631285\pi$
$107$	−8.29287	−0.801702	−0.400851	+	0.916143i	$0.631285\pi$
$108$	0	0
$109$	11.0279	1.05629	0.528143	−	0.849156i	$-0.322889\pi$
$109$	11.0279	1.05629	0.528143	+	0.849156i	$0.322889\pi$
$110$	0	0
$111$	0.900725	0.0854930
$112$	0	0
$113$	−8.26658	−0.777654	−0.388827	−	0.921311i	$-0.627120\pi$
$113$	−8.26658	−0.777654	−0.388827	+	0.921311i	$0.627120\pi$
$114$	0	0
$115$	10.1604	0.947465
$116$	0	0
$117$	6.72852	0.622052
$118$	0	0
$119$	0	0
$120$	0	0
$121$	20.9145	1.90132
$122$	0	0
$123$	1.40418	0.126611
$124$	0	0
$125$	−11.1113	−0.993828
$126$	0	0
$127$	1.12925	0.100205	0.0501024	−	0.998744i	$-0.484045\pi$
$127$	1.12925	0.100205	0.0501024	+	0.998744i	$0.484045\pi$
$128$	0	0
$129$	−6.23967	−0.549373
$130$	0	0
$131$	−18.9249	−1.65348	−0.826739	−	0.562586i	$-0.809806\pi$
$131$	−18.9249	−1.65348	−0.826739	+	0.562586i	$0.809806\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	20.4518	1.76021
$136$	0	0
$137$	8.36226	0.714436	0.357218	−	0.934021i	$-0.383725\pi$
$137$	8.36226	0.714436	0.357218	+	0.934021i	$0.383725\pi$
$138$	0	0
$139$	−18.1952	−1.54330	−0.771648	−	0.636050i	$-0.780567\pi$
$139$	−18.1952	−1.54330	−0.771648	+	0.636050i	$0.780567\pi$
$140$	0	0
$141$	−9.40296	−0.791872
$142$	0	0
$143$	36.9659	3.09124
$144$	0	0
$145$	−7.86769	−0.653376
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.0464	−0.986877	−0.493438	−	0.869781i	$-0.664260\pi$
$149$	−12.0464	−0.986877	−0.493438	+	0.869781i	$0.664260\pi$
$150$	0	0
$151$	5.44182	0.442849	0.221424	−	0.975178i	$-0.428929\pi$
$151$	5.44182	0.442849	0.221424	+	0.975178i	$0.428929\pi$
$152$	0	0
$153$	3.48656	0.281871
$154$	0	0
$155$	22.6361	1.81818
$156$	0	0
$157$	−20.9240	−1.66992	−0.834960	−	0.550311i	$-0.814509\pi$
$157$	−20.9240	−1.66992	−0.834960	+	0.550311i	$0.814509\pi$
$158$	0	0
$159$	6.95458	0.551534
$160$	0	0
$161$	0	0
$162$	0	0
$163$	17.0858	1.33826	0.669132	−	0.743144i	$-0.266666\pi$
$163$	17.0858	1.33826	0.669132	+	0.743144i	$0.266666\pi$
$164$	0	0
$165$	28.6818	2.23287
$166$	0	0
$167$	−13.0854	−1.01258	−0.506289	−	0.862364i	$-0.668983\pi$
$167$	−13.0854	−1.01258	−0.506289	+	0.862364i	$0.668983\pi$
$168$	0	0
$169$	29.8168	2.29360
$170$	0	0
$171$	5.34438	0.408695
$172$	0	0
$173$	−19.2270	−1.46180	−0.730902	−	0.682482i	$-0.760900\pi$
$173$	−19.2270	−1.46180	−0.730902	+	0.682482i	$0.760900\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.4866	1.16404
$178$	0	0
$179$	−8.73014	−0.652521	−0.326261	−	0.945280i	$-0.605789\pi$
$179$	−8.73014	−0.652521	−0.326261	+	0.945280i	$0.605789\pi$
$180$	0	0
$181$	11.2949	0.839544	0.419772	−	0.907630i	$-0.362110\pi$
$181$	11.2949	0.839544	0.419772	+	0.907630i	$0.362110\pi$
$182$	0	0
$183$	16.9048	1.24964
$184$	0	0
$185$	−2.31931	−0.170519
$186$	0	0
$187$	19.1548	1.40074
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−16.9084	−1.22345	−0.611724	−	0.791071i	$-0.709524\pi$
$191$	−16.9084	−1.22345	−0.611724	+	0.791071i	$0.709524\pi$
$192$	0	0
$193$	3.98688	0.286982	0.143491	−	0.989652i	$-0.454167\pi$
$193$	3.98688	0.286982	0.143491	+	0.989652i	$0.454167\pi$
$194$	0	0
$195$	33.2215	2.37904
$196$	0	0
$197$	−12.0163	−0.856124	−0.428062	−	0.903749i	$-0.640803\pi$
$197$	−12.0163	−0.856124	−0.428062	+	0.903749i	$0.640803\pi$
$198$	0	0
$199$	10.9253	0.774475	0.387238	−	0.921980i	$-0.373429\pi$
$199$	10.9253	0.774475	0.387238	+	0.921980i	$0.373429\pi$
$200$	0	0
$201$	14.8621	1.04829
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−3.61567	−0.252530
$206$	0	0
$207$	2.88958	0.200840
$208$	0	0
$209$	29.3615	2.03098
$210$	0	0
$211$	−7.98904	−0.549988	−0.274994	−	0.961446i	$-0.588676\pi$
$211$	−7.98904	−0.549988	−0.274994	+	0.961446i	$0.588676\pi$
$212$	0	0
$213$	−12.2221	−0.837446
$214$	0	0
$215$	16.0668	1.09575
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−0.593854	−0.0401290
$220$	0	0
$221$	22.1866	1.49243
$222$	0	0
$223$	−27.5999	−1.84823	−0.924115	−	0.382116i	$-0.875196\pi$
$223$	−27.5999	−1.84823	−0.924115	+	0.382116i	$0.875196\pi$
$224$	0	0
$225$	−8.30143	−0.553429
$226$	0	0
$227$	24.3315	1.61494	0.807468	−	0.589911i	$-0.200837\pi$
$227$	24.3315	1.61494	0.807468	+	0.589911i	$0.200837\pi$
$228$	0	0
$229$	−0.109735	−0.00725149	−0.00362574	−	0.999993i	$-0.501154\pi$
$229$	−0.109735	−0.00725149	−0.00362574	+	0.999993i	$0.501154\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−8.22953	−0.539134	−0.269567	−	0.962982i	$-0.586881\pi$
$233$	−8.22953	−0.539134	−0.269567	+	0.962982i	$0.586881\pi$
$234$	0	0
$235$	24.2120	1.57942
$236$	0	0
$237$	−19.9766	−1.29762
$238$	0	0
$239$	7.13997	0.461846	0.230923	−	0.972972i	$-0.425825\pi$
$239$	7.13997	0.461846	0.230923	+	0.972972i	$0.425825\pi$
$240$	0	0
$241$	−12.3191	−0.793543	−0.396771	−	0.917917i	$-0.629869\pi$
$241$	−12.3191	−0.793543	−0.396771	+	0.917917i	$0.629869\pi$
$242$	0	0
$243$	10.1481	0.651000
$244$	0	0
$245$	0	0
$246$	0	0
$247$	34.0088	2.16393
$248$	0	0
$249$	−7.11245	−0.450734
$250$	0	0
$251$	6.03854	0.381149	0.190575	−	0.981673i	$-0.438965\pi$
$251$	6.03854	0.381149	0.190575	+	0.981673i	$0.438965\pi$
$252$	0	0
$253$	15.8751	0.998060
$254$	0	0
$255$	17.2146	1.07802
$256$	0	0
$257$	−28.5716	−1.78225	−0.891123	−	0.453763i	$-0.850081\pi$
$257$	−28.5716	−1.78225	−0.891123	+	0.453763i	$0.850081\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.23754	−0.138500
$262$	0	0
$263$	−30.7054	−1.89338	−0.946690	−	0.322147i	$-0.895595\pi$
$263$	−30.7054	−1.89338	−0.946690	+	0.322147i	$0.895595\pi$
$264$	0	0
$265$	−17.9076	−1.10006
$266$	0	0
$267$	2.19693	0.134450
$268$	0	0
$269$	−12.1607	−0.741451	−0.370726	−	0.928742i	$-0.620891\pi$
$269$	−12.1607	−0.741451	−0.370726	+	0.928742i	$0.620891\pi$
$270$	0	0
$271$	−5.84160	−0.354852	−0.177426	−	0.984134i	$-0.556777\pi$
$271$	−5.84160	−0.354852	−0.177426	+	0.984134i	$0.556777\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−45.6073	−2.75022
$276$	0	0
$277$	−22.7835	−1.36893	−0.684463	−	0.729047i	$-0.739963\pi$
$277$	−22.7835	−1.36893	−0.684463	+	0.729047i	$0.739963\pi$
$278$	0	0
$279$	6.43761	0.385410
$280$	0	0
$281$	−2.36022	−0.140799	−0.0703995	−	0.997519i	$-0.522427\pi$
$281$	−2.36022	−0.140799	−0.0703995	+	0.997519i	$0.522427\pi$
$282$	0	0
$283$	−0.0967397	−0.00575058	−0.00287529	−	0.999996i	$-0.500915\pi$
$283$	−0.0967397	−0.00575058	−0.00287529	+	0.999996i	$0.500915\pi$
$284$	0	0
$285$	26.3874	1.56305
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−5.50343	−0.323731
$290$	0	0
$291$	−23.0919	−1.35367
$292$	0	0
$293$	−24.8023	−1.44897	−0.724484	−	0.689292i	$-0.757922\pi$
$293$	−24.8023	−1.44897	−0.724484	+	0.689292i	$0.757922\pi$
$294$	0	0
$295$	−39.8770	−2.32173
$296$	0	0
$297$	31.9548	1.85421
$298$	0	0
$299$	18.3878	1.06339
$300$	0	0
$301$	0	0
$302$	0	0
$303$	15.6432	0.898681
$304$	0	0
$305$	−43.5288	−2.49245
$306$	0	0
$307$	3.46391	0.197696	0.0988480	−	0.995103i	$-0.468484\pi$
$307$	3.46391	0.197696	0.0988480	+	0.995103i	$0.468484\pi$
$308$	0	0
$309$	7.02321	0.399537
$310$	0	0
$311$	−26.0188	−1.47539	−0.737697	−	0.675132i	$-0.764087\pi$
$311$	−26.0188	−1.47539	−0.737697	+	0.675132i	$0.764087\pi$
$312$	0	0
$313$	−4.95821	−0.280255	−0.140127	−	0.990133i	$-0.544751\pi$
$313$	−4.95821	−0.280255	−0.140127	+	0.990133i	$0.544751\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	17.0035	0.955012	0.477506	−	0.878629i	$-0.341541\pi$
$317$	17.0035	0.955012	0.477506	+	0.878629i	$0.341541\pi$
$318$	0	0
$319$	−12.2928	−0.688267
$320$	0	0
$321$	−11.6447	−0.649942
$322$	0	0
$323$	17.6226	0.980545
$324$	0	0
$325$	−52.8260	−2.93026
$326$	0	0
$327$	15.4852	0.856334
$328$	0	0
$329$	0	0
$330$	0	0
$331$	11.2694	0.619424	0.309712	−	0.950830i	$-0.399767\pi$
$331$	11.2694	0.619424	0.309712	+	0.950830i	$0.399767\pi$
$332$	0	0
$333$	−0.659602	−0.0361460
$334$	0	0
$335$	−38.2689	−2.09085
$336$	0	0
$337$	29.7342	1.61972	0.809862	−	0.586620i	$-0.199542\pi$
$337$	29.7342	1.61972	0.809862	+	0.586620i	$0.199542\pi$
$338$	0	0
$339$	−11.6077	−0.630446
$340$	0	0
$341$	35.3677	1.91527
$342$	0	0
$343$	0	0
$344$	0	0
$345$	14.2671	0.768113
$346$	0	0
$347$	13.6670	0.733683	0.366841	−	0.930283i	$-0.380439\pi$
$347$	13.6670	0.733683	0.366841	+	0.930283i	$0.380439\pi$
$348$	0	0
$349$	21.4570	1.14857	0.574284	−	0.818656i	$-0.305280\pi$
$349$	21.4570	1.14857	0.574284	+	0.818656i	$0.305280\pi$
$350$	0	0
$351$	37.0126	1.97559
$352$	0	0
$353$	−20.1025	−1.06995	−0.534974	−	0.844868i	$-0.679679\pi$
$353$	−20.1025	−1.06995	−0.534974	+	0.844868i	$0.679679\pi$
$354$	0	0
$355$	31.4712	1.67032
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−13.7108	−0.723629	−0.361814	−	0.932250i	$-0.617843\pi$
$359$	−13.7108	−0.723629	−0.361814	+	0.932250i	$0.617843\pi$
$360$	0	0
$361$	8.01278	0.421725
$362$	0	0
$363$	29.3677	1.54140
$364$	0	0
$365$	1.52914	0.0800388
$366$	0	0
$367$	−27.1307	−1.41621	−0.708105	−	0.706107i	$-0.750450\pi$
$367$	−27.1307	−1.41621	−0.708105	+	0.706107i	$0.750450\pi$
$368$	0	0
$369$	−1.02828	−0.0535302
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−10.5573	−0.546636	−0.273318	−	0.961924i	$-0.588121\pi$
$373$	−10.5573	−0.546636	−0.273318	+	0.961924i	$0.588121\pi$
$374$	0	0
$375$	−15.6023	−0.805699
$376$	0	0
$377$	−14.2385	−0.733322
$378$	0	0
$379$	24.9019	1.27913	0.639563	−	0.768739i	$-0.279115\pi$
$379$	24.9019	1.27913	0.639563	+	0.768739i	$0.279115\pi$
$380$	0	0
$381$	1.58567	0.0812364
$382$	0	0
$383$	−8.74847	−0.447026	−0.223513	−	0.974701i	$-0.571752\pi$
$383$	−8.74847	−0.447026	−0.223513	+	0.974701i	$0.571752\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.56932	0.232272
$388$	0	0
$389$	−5.28198	−0.267807	−0.133903	−	0.990994i	$-0.542751\pi$
$389$	−5.28198	−0.267807	−0.133903	+	0.990994i	$0.542751\pi$
$390$	0	0
$391$	9.52812	0.481858
$392$	0	0
$393$	−26.5740	−1.34048
$394$	0	0
$395$	51.4385	2.58815
$396$	0	0
$397$	8.19379	0.411235	0.205617	−	0.978632i	$-0.434080\pi$
$397$	8.19379	0.411235	0.205617	+	0.978632i	$0.434080\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.7598	0.587256	0.293628	−	0.955920i	$-0.405137\pi$
$401$	11.7598	0.587256	0.293628	+	0.955920i	$0.405137\pi$
$402$	0	0
$403$	40.9656	2.04064
$404$	0	0
$405$	17.5642	0.872771
$406$	0	0
$407$	−3.62380	−0.179625
$408$	0	0
$409$	14.1419	0.699274	0.349637	−	0.936885i	$-0.386305\pi$
$409$	14.1419	0.699274	0.349637	+	0.936885i	$0.386305\pi$
$410$	0	0
$411$	11.7421	0.579196
$412$	0	0
$413$	0	0
$414$	0	0
$415$	18.3141	0.899006
$416$	0	0
$417$	−25.5493	−1.25115
$418$	0	0
$419$	3.94153	0.192556	0.0962781	−	0.995354i	$-0.469306\pi$
$419$	3.94153	0.192556	0.0962781	+	0.995354i	$0.469306\pi$
$420$	0	0
$421$	−28.9273	−1.40983	−0.704914	−	0.709292i	$-0.749015\pi$
$421$	−28.9273	−1.40983	−0.704914	+	0.709292i	$0.749015\pi$
$422$	0	0
$423$	6.88580	0.334799
$424$	0	0
$425$	−27.3731	−1.32779
$426$	0	0
$427$	0	0
$428$	0	0
$429$	51.9067	2.50608
$430$	0	0
$431$	4.29158	0.206718	0.103359	−	0.994644i	$-0.467041\pi$
$431$	4.29158	0.206718	0.103359	+	0.994644i	$0.467041\pi$
$432$	0	0
$433$	−1.69228	−0.0813259	−0.0406629	−	0.999173i	$-0.512947\pi$
$433$	−1.69228	−0.0813259	−0.0406629	+	0.999173i	$0.512947\pi$
$434$	0	0
$435$	−11.0476	−0.529694
$436$	0	0
$437$	14.6052	0.698661
$438$	0	0
$439$	4.55266	0.217287	0.108643	−	0.994081i	$-0.465349\pi$
$439$	4.55266	0.217287	0.108643	+	0.994081i	$0.465349\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4.59951	0.218529	0.109265	−	0.994013i	$-0.465150\pi$
$443$	4.59951	0.218529	0.109265	+	0.994013i	$0.465150\pi$
$444$	0	0
$445$	−5.65696	−0.268166
$446$	0	0
$447$	−16.9153	−0.800064
$448$	0	0
$449$	4.72931	0.223190	0.111595	−	0.993754i	$-0.464404\pi$
$449$	4.72931	0.223190	0.111595	+	0.993754i	$0.464404\pi$
$450$	0	0
$451$	−5.64929	−0.266015
$452$	0	0
$453$	7.64128	0.359019
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.5087	0.912581	0.456290	−	0.889831i	$-0.349178\pi$
$457$	19.5087	0.912581	0.456290	+	0.889831i	$0.349178\pi$
$458$	0	0
$459$	19.1790	0.895200
$460$	0	0
$461$	−1.57849	−0.0735178	−0.0367589	−	0.999324i	$-0.511703\pi$
$461$	−1.57849	−0.0735178	−0.0367589	+	0.999324i	$0.511703\pi$
$462$	0	0
$463$	4.94122	0.229638	0.114819	−	0.993386i	$-0.463371\pi$
$463$	4.94122	0.229638	0.114819	+	0.993386i	$0.463371\pi$
$464$	0	0
$465$	31.7851	1.47400
$466$	0	0
$467$	−23.2577	−1.07624	−0.538120	−	0.842868i	$-0.680865\pi$
$467$	−23.2577	−1.07624	−0.538120	+	0.842868i	$0.680865\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−29.3811	−1.35381
$472$	0	0
$473$	25.1035	1.15426
$474$	0	0
$475$	−41.9590	−1.92521
$476$	0	0
$477$	−5.09285	−0.233186
$478$	0	0
$479$	−15.5270	−0.709449	−0.354724	−	0.934971i	$-0.615425\pi$
$479$	−15.5270	−0.709449	−0.354724	+	0.934971i	$0.615425\pi$
$480$	0	0
$481$	−4.19737	−0.191383
$482$	0	0
$483$	0	0
$484$	0	0
$485$	59.4601	2.69995
$486$	0	0
$487$	15.9614	0.723279	0.361640	−	0.932318i	$-0.382217\pi$
$487$	15.9614	0.723279	0.361640	+	0.932318i	$0.382217\pi$
$488$	0	0
$489$	23.9915	1.08493
$490$	0	0
$491$	−4.11148	−0.185548	−0.0927742	−	0.995687i	$-0.529573\pi$
$491$	−4.11148	−0.185548	−0.0927742	+	0.995687i	$0.529573\pi$
$492$	0	0
$493$	−7.37806	−0.332291
$494$	0	0
$495$	−21.0037	−0.944046
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−7.15438	−0.320274	−0.160137	−	0.987095i	$-0.551194\pi$
$499$	−7.15438	−0.320274	−0.160137	+	0.987095i	$0.551194\pi$
$500$	0	0
$501$	−18.3742	−0.820901
$502$	0	0
$503$	−4.71388	−0.210182	−0.105091	−	0.994463i	$-0.533513\pi$
$503$	−4.71388	−0.210182	−0.105091	+	0.994463i	$0.533513\pi$
$504$	0	0
$505$	−40.2804	−1.79245
$506$	0	0
$507$	41.8681	1.85943
$508$	0	0
$509$	18.2184	0.807516	0.403758	−	0.914866i	$-0.367704\pi$
$509$	18.2184	0.807516	0.403758	+	0.914866i	$0.367704\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	29.3986	1.29798
$514$	0	0
$515$	−18.0843	−0.796891
$516$	0	0
$517$	37.8300	1.66376
$518$	0	0
$519$	−26.9982	−1.18509
$520$	0	0
$521$	−10.9519	−0.479811	−0.239905	−	0.970796i	$-0.577116\pi$
$521$	−10.9519	−0.479811	−0.239905	+	0.970796i	$0.577116\pi$
$522$	0	0
$523$	−19.6211	−0.857972	−0.428986	−	0.903311i	$-0.641129\pi$
$523$	−19.6211	−0.857972	−0.428986	+	0.903311i	$0.641129\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	21.2274	0.924679
$528$	0	0
$529$	−15.1033	−0.656665
$530$	0	0
$531$	−11.3408	−0.492151
$532$	0	0
$533$	−6.54345	−0.283428
$534$	0	0
$535$	29.9843	1.29633
$536$	0	0
$537$	−12.2587	−0.529001
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.4723	0.966160	0.483080	−	0.875576i	$-0.339518\pi$
$541$	22.4723	0.966160	0.483080	+	0.875576i	$0.339518\pi$
$542$	0	0
$543$	15.8601	0.680621
$544$	0	0
$545$	−39.8734	−1.70799
$546$	0	0
$547$	−5.78852	−0.247499	−0.123750	−	0.992313i	$-0.539492\pi$
$547$	−5.78852	−0.247499	−0.123750	+	0.992313i	$0.539492\pi$
$548$	0	0
$549$	−12.3794	−0.528340
$550$	0	0
$551$	−11.3095	−0.481800
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.25673	−0.138240
$556$	0	0
$557$	8.22507	0.348507	0.174254	−	0.984701i	$-0.444249\pi$
$557$	8.22507	0.348507	0.174254	+	0.984701i	$0.444249\pi$
$558$	0	0
$559$	29.0768	1.22982
$560$	0	0
$561$	26.8968	1.13558
$562$	0	0
$563$	20.5297	0.865223	0.432611	−	0.901580i	$-0.357592\pi$
$563$	20.5297	0.865223	0.432611	+	0.901580i	$0.357592\pi$
$564$	0	0
$565$	29.8892	1.25745
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−23.9665	−1.00473	−0.502364	−	0.864656i	$-0.667536\pi$
$569$	−23.9665	−1.00473	−0.502364	+	0.864656i	$0.667536\pi$
$570$	0	0
$571$	34.4864	1.44321	0.721605	−	0.692306i	$-0.243405\pi$
$571$	34.4864	1.44321	0.721605	+	0.692306i	$0.243405\pi$
$572$	0	0
$573$	−23.7424	−0.991853
$574$	0	0
$575$	−22.6863	−0.946083
$576$	0	0
$577$	41.0357	1.70834	0.854168	−	0.519996i	$-0.174067\pi$
$577$	41.0357	1.70834	0.854168	+	0.519996i	$0.174067\pi$
$578$	0	0
$579$	5.59829	0.232657
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−27.9797	−1.15880
$584$	0	0
$585$	−24.3281	−1.00584
$586$	0	0
$587$	−6.21910	−0.256690	−0.128345	−	0.991730i	$-0.540966\pi$
$587$	−6.21910	−0.256690	−0.128345	+	0.991730i	$0.540966\pi$
$588$	0	0
$589$	32.5385	1.34072
$590$	0	0
$591$	−16.8730	−0.694062
$592$	0	0
$593$	2.92318	0.120041	0.0600204	−	0.998197i	$-0.480883\pi$
$593$	2.92318	0.120041	0.0600204	+	0.998197i	$0.480883\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	15.3411	0.627869
$598$	0	0
$599$	−13.4830	−0.550901	−0.275450	−	0.961315i	$-0.588827\pi$
$599$	−13.4830	−0.550901	−0.275450	+	0.961315i	$0.588827\pi$
$600$	0	0
$601$	35.1493	1.43377	0.716886	−	0.697191i	$-0.245567\pi$
$601$	35.1493	1.43377	0.716886	+	0.697191i	$0.245567\pi$
$602$	0	0
$603$	−10.8835	−0.443211
$604$	0	0
$605$	−75.6201	−3.07439
$606$	0	0
$607$	−31.7046	−1.28685	−0.643425	−	0.765509i	$-0.722487\pi$
$607$	−31.7046	−1.28685	−0.643425	+	0.765509i	$0.722487\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	43.8177	1.77267
$612$	0	0
$613$	0.747674	0.0301983	0.0150991	−	0.999886i	$-0.495194\pi$
$613$	0.747674	0.0301983	0.0150991	+	0.999886i	$0.495194\pi$
$614$	0	0
$615$	−5.07705	−0.204727
$616$	0	0
$617$	16.6618	0.670780	0.335390	−	0.942079i	$-0.391132\pi$
$617$	16.6618	0.670780	0.335390	+	0.942079i	$0.391132\pi$
$618$	0	0
$619$	27.0691	1.08800	0.543999	−	0.839086i	$-0.316909\pi$
$619$	27.0691	1.08800	0.543999	+	0.839086i	$0.316909\pi$
$620$	0	0
$621$	15.8952	0.637851
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−0.190547	−0.00762189
$626$	0	0
$627$	41.2288	1.64652
$628$	0	0
$629$	−2.17497	−0.0867219
$630$	0	0
$631$	−23.1258	−0.920625	−0.460312	−	0.887757i	$-0.652263\pi$
$631$	−23.1258	−0.920625	−0.460312	+	0.887757i	$0.652263\pi$
$632$	0	0
$633$	−11.2180	−0.445877
$634$	0	0
$635$	−4.08301	−0.162029
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.95028	0.354068
$640$	0	0
$641$	13.8551	0.547244	0.273622	−	0.961837i	$-0.411778\pi$
$641$	13.8551	0.547244	0.273622	+	0.961837i	$0.411778\pi$
$642$	0	0
$643$	6.61525	0.260880	0.130440	−	0.991456i	$-0.458361\pi$
$643$	6.61525	0.260880	0.130440	+	0.991456i	$0.458361\pi$
$644$	0	0
$645$	22.5606	0.888324
$646$	0	0
$647$	−5.04073	−0.198172	−0.0990858	−	0.995079i	$-0.531592\pi$
$647$	−5.04073	−0.198172	−0.0990858	+	0.995079i	$0.531592\pi$
$648$	0	0
$649$	−62.3056	−2.44571
$650$	0	0
$651$	0	0
$652$	0	0
$653$	15.9694	0.624931	0.312465	−	0.949929i	$-0.398845\pi$
$653$	15.9694	0.624931	0.312465	+	0.949929i	$0.398845\pi$
$654$	0	0
$655$	68.4263	2.67364
$656$	0	0
$657$	0.434881	0.0169663
$658$	0	0
$659$	4.81563	0.187590	0.0937952	−	0.995592i	$-0.470100\pi$
$659$	4.81563	0.187590	0.0937952	+	0.995592i	$0.470100\pi$
$660$	0	0
$661$	22.6400	0.880593	0.440296	−	0.897853i	$-0.354873\pi$
$661$	22.6400	0.880593	0.440296	+	0.897853i	$0.354873\pi$
$662$	0	0
$663$	31.1540	1.20992
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.11478	−0.236765
$668$	0	0
$669$	−38.7553	−1.49836
$670$	0	0
$671$	−68.0114	−2.62555
$672$	0	0
$673$	−15.0840	−0.581446	−0.290723	−	0.956807i	$-0.593896\pi$
$673$	−15.0840	−0.581446	−0.290723	+	0.956807i	$0.593896\pi$
$674$	0	0
$675$	−45.6649	−1.75764
$676$	0	0
$677$	−29.4473	−1.13175	−0.565876	−	0.824490i	$-0.691462\pi$
$677$	−29.4473	−1.13175	−0.565876	+	0.824490i	$0.691462\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	34.1657	1.30923
$682$	0	0
$683$	0.575409	0.0220174	0.0110087	−	0.999939i	$-0.496496\pi$
$683$	0.575409	0.0220174	0.0110087	+	0.999939i	$0.496496\pi$
$684$	0	0
$685$	−30.2352	−1.15523
$686$	0	0
$687$	−0.154087	−0.00587880
$688$	0	0
$689$	−32.4082	−1.23466
$690$	0	0
$691$	37.5440	1.42824	0.714120	−	0.700023i	$-0.246827\pi$
$691$	37.5440	1.42824	0.714120	+	0.700023i	$0.246827\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	65.7879	2.49548
$696$	0	0
$697$	−3.39066	−0.128430
$698$	0	0
$699$	−11.5557	−0.437078
$700$	0	0
$701$	−31.1046	−1.17480	−0.587402	−	0.809295i	$-0.699849\pi$
$701$	−31.1046	−1.17480	−0.587402	+	0.809295i	$0.699849\pi$
$702$	0	0
$703$	−3.33391	−0.125741
$704$	0	0
$705$	33.9980	1.28044
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−5.85434	−0.219864	−0.109932	−	0.993939i	$-0.535063\pi$
$709$	−5.85434	−0.219864	−0.109932	+	0.993939i	$0.535063\pi$
$710$	0	0
$711$	14.6289	0.548626
$712$	0	0
$713$	17.5928	0.658856
$714$	0	0
$715$	−133.657	−4.99847
$716$	0	0
$717$	10.0258	0.374420
$718$	0	0
$719$	−27.7937	−1.03653	−0.518264	−	0.855220i	$-0.673422\pi$
$719$	−27.7937	−1.03653	−0.518264	+	0.855220i	$0.673422\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−17.2982	−0.643328
$724$	0	0
$725$	17.5670	0.652423
$726$	0	0
$727$	−15.5974	−0.578474	−0.289237	−	0.957257i	$-0.593402\pi$
$727$	−15.5974	−0.578474	−0.289237	+	0.957257i	$0.593402\pi$
$728$	0	0
$729$	28.8231	1.06752
$730$	0	0
$731$	15.0669	0.557269
$732$	0	0
$733$	39.6572	1.46477	0.732386	−	0.680889i	$-0.238406\pi$
$733$	39.6572	1.46477	0.732386	+	0.680889i	$0.238406\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−59.7930	−2.20250
$738$	0	0
$739$	−12.9478	−0.476292	−0.238146	−	0.971229i	$-0.576540\pi$
$739$	−12.9478	−0.476292	−0.238146	+	0.971229i	$0.576540\pi$
$740$	0	0
$741$	47.7545	1.75430
$742$	0	0
$743$	−5.12957	−0.188186	−0.0940929	−	0.995563i	$-0.529995\pi$
$743$	−5.12957	−0.188186	−0.0940929	+	0.995563i	$0.529995\pi$
$744$	0	0
$745$	43.5557	1.59576
$746$	0	0
$747$	5.20846	0.190568
$748$	0	0
$749$	0	0
$750$	0	0
$751$	19.0378	0.694697	0.347349	−	0.937736i	$-0.387082\pi$
$751$	19.0378	0.694697	0.347349	+	0.937736i	$0.387082\pi$
$752$	0	0
$753$	8.47919	0.308999
$754$	0	0
$755$	−19.6758	−0.716077
$756$	0	0
$757$	32.8052	1.19233	0.596163	−	0.802864i	$-0.296691\pi$
$757$	32.8052	1.19233	0.596163	+	0.802864i	$0.296691\pi$
$758$	0	0
$759$	22.2915	0.809130
$760$	0	0
$761$	−41.2671	−1.49593	−0.747966	−	0.663737i	$-0.768969\pi$
$761$	−41.2671	−1.49593	−0.747966	+	0.663737i	$0.768969\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−12.6062	−0.455780
$766$	0	0
$767$	−72.1672	−2.60581
$768$	0	0
$769$	−36.2060	−1.30562	−0.652811	−	0.757520i	$-0.726411\pi$
$769$	−36.2060	−1.30562	−0.652811	+	0.757520i	$0.726411\pi$
$770$	0	0
$771$	−40.1196	−1.44487
$772$	0	0
$773$	23.6296	0.849898	0.424949	−	0.905217i	$-0.360292\pi$
$773$	23.6296	0.849898	0.424949	+	0.905217i	$0.360292\pi$
$774$	0	0
$775$	−50.5420	−1.81552
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−5.19738	−0.186216
$780$	0	0
$781$	49.1721	1.75951
$782$	0	0
$783$	−12.3084	−0.439865
$784$	0	0
$785$	75.6545	2.70022
$786$	0	0
$787$	−9.23570	−0.329217	−0.164609	−	0.986359i	$-0.552636\pi$
$787$	−9.23570	−0.329217	−0.164609	+	0.986359i	$0.552636\pi$
$788$	0	0
$789$	−43.1159	−1.53497
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−78.7761	−2.79742
$794$	0	0
$795$	−25.1455	−0.891819
$796$	0	0
$797$	−41.1963	−1.45925	−0.729624	−	0.683848i	$-0.760305\pi$
$797$	−41.1963	−1.45925	−0.729624	+	0.683848i	$0.760305\pi$
$798$	0	0
$799$	22.7052	0.803254
$800$	0	0
$801$	−1.60881	−0.0568447
$802$	0	0
$803$	2.38920	0.0843129
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−17.0758	−0.601097
$808$	0	0
$809$	4.01458	0.141145	0.0705725	−	0.997507i	$-0.477517\pi$
$809$	4.01458	0.141145	0.0705725	+	0.997507i	$0.477517\pi$
$810$	0	0
$811$	−26.1749	−0.919124	−0.459562	−	0.888146i	$-0.651994\pi$
$811$	−26.1749	−0.919124	−0.459562	+	0.888146i	$0.651994\pi$
$812$	0	0
$813$	−8.20264	−0.287679
$814$	0	0
$815$	−61.7767	−2.16394
$816$	0	0
$817$	23.0953	0.808003
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40.0484	−1.39770	−0.698849	−	0.715269i	$-0.746304\pi$
$821$	−40.0484	−1.39770	−0.698849	+	0.715269i	$0.746304\pi$
$822$	0	0
$823$	17.6884	0.616577	0.308289	−	0.951293i	$-0.400244\pi$
$823$	17.6884	0.616577	0.308289	+	0.951293i	$0.400244\pi$
$824$	0	0
$825$	−64.0408	−2.22961
$826$	0	0
$827$	27.9098	0.970520	0.485260	−	0.874370i	$-0.338725\pi$
$827$	27.9098	0.970520	0.485260	+	0.874370i	$0.338725\pi$
$828$	0	0
$829$	−55.4075	−1.92438	−0.962191	−	0.272377i	$-0.912190\pi$
$829$	−55.4075	−1.92438	−0.962191	+	0.272377i	$0.912190\pi$
$830$	0	0
$831$	−31.9921	−1.10979
$832$	0	0
$833$	0	0
$834$	0	0
$835$	47.3126	1.63732
$836$	0	0
$837$	35.4123	1.22403
$838$	0	0
$839$	46.1671	1.59386	0.796932	−	0.604069i	$-0.206455\pi$
$839$	46.1671	1.59386	0.796932	+	0.604069i	$0.206455\pi$
$840$	0	0
$841$	−24.2650	−0.836726
$842$	0	0
$843$	−3.31417	−0.114146
$844$	0	0
$845$	−107.808	−3.70870
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−0.135840	−0.00466201
$850$	0	0
$851$	−1.80257	−0.0617914
$852$	0	0
$853$	46.7110	1.59935	0.799677	−	0.600430i	$-0.205004\pi$
$853$	46.7110	1.59935	0.799677	+	0.600430i	$0.205004\pi$
$854$	0	0
$855$	−19.3235	−0.660851
$856$	0	0
$857$	−28.0091	−0.956773	−0.478387	−	0.878149i	$-0.658778\pi$
$857$	−28.0091	−0.956773	−0.478387	+	0.878149i	$0.658778\pi$
$858$	0	0
$859$	−9.73604	−0.332189	−0.166095	−	0.986110i	$-0.553116\pi$
$859$	−9.73604	−0.332189	−0.166095	+	0.986110i	$0.553116\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	10.8036	0.367759	0.183880	−	0.982949i	$-0.441134\pi$
$863$	10.8036	0.367759	0.183880	+	0.982949i	$0.441134\pi$
$864$	0	0
$865$	69.5187	2.36371
$866$	0	0
$867$	−7.72780	−0.262450
$868$	0	0
$869$	80.3698	2.72636
$870$	0	0
$871$	−69.2570	−2.34668
$872$	0	0
$873$	16.9102	0.572324
$874$	0	0
$875$	0	0
$876$	0	0
$877$	14.1267	0.477026	0.238513	−	0.971139i	$-0.423340\pi$
$877$	14.1267	0.477026	0.238513	+	0.971139i	$0.423340\pi$
$878$	0	0
$879$	−34.8269	−1.17468
$880$	0	0
$881$	37.6737	1.26926	0.634630	−	0.772816i	$-0.281153\pi$
$881$	37.6737	1.26926	0.634630	+	0.772816i	$0.281153\pi$
$882$	0	0
$883$	20.1033	0.676531	0.338266	−	0.941051i	$-0.390160\pi$
$883$	20.1033	0.676531	0.338266	+	0.941051i	$0.390160\pi$
$884$	0	0
$885$	−55.9944	−1.88223
$886$	0	0
$887$	44.1332	1.48185	0.740924	−	0.671589i	$-0.234388\pi$
$887$	44.1332	1.48185	0.740924	+	0.671589i	$0.234388\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	27.4431	0.919377
$892$	0	0
$893$	34.8038	1.16466
$894$	0	0
$895$	31.5653	1.05511
$896$	0	0
$897$	25.8198	0.862097
$898$	0	0
$899$	−13.6229	−0.454350
$900$	0	0
$901$	−16.7932	−0.559462
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−40.8387	−1.35753
$906$	0	0
$907$	1.85749	0.0616770	0.0308385	−	0.999524i	$-0.490182\pi$
$907$	1.85749	0.0616770	0.0308385	+	0.999524i	$0.490182\pi$
$908$	0	0
$909$	−11.4556	−0.379957
$910$	0	0
$911$	14.5018	0.480466	0.240233	−	0.970715i	$-0.422776\pi$
$911$	14.5018	0.480466	0.240233	+	0.970715i	$0.422776\pi$
$912$	0	0
$913$	28.6148	0.947013
$914$	0	0
$915$	−61.1222	−2.02064
$916$	0	0
$917$	0	0
$918$	0	0
$919$	21.1624	0.698082	0.349041	−	0.937107i	$-0.386507\pi$
$919$	21.1624	0.698082	0.349041	+	0.937107i	$0.386507\pi$
$920$	0	0
$921$	4.86395	0.160273
$922$	0	0
$923$	56.9549	1.87469
$924$	0	0
$925$	5.17857	0.170270
$926$	0	0
$927$	−5.14311	−0.168922
$928$	0	0
$929$	59.7145	1.95917	0.979585	−	0.201029i	$-0.0644286\pi$
$929$	59.7145	1.95917	0.979585	+	0.201029i	$0.0644286\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−36.5351	−1.19611
$934$	0	0
$935$	−69.2576	−2.26497
$936$	0	0
$937$	12.1335	0.396384	0.198192	−	0.980163i	$-0.436493\pi$
$937$	12.1335	0.396384	0.198192	+	0.980163i	$0.436493\pi$
$938$	0	0
$939$	−6.96221	−0.227203
$940$	0	0
$941$	−12.8135	−0.417710	−0.208855	−	0.977947i	$-0.566974\pi$
$941$	−12.8135	−0.417710	−0.208855	+	0.977947i	$0.566974\pi$
$942$	0	0
$943$	−2.81011	−0.0915097
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−17.2252	−0.559744	−0.279872	−	0.960037i	$-0.590292\pi$
$947$	−17.2252	−0.559744	−0.279872	+	0.960037i	$0.590292\pi$
$948$	0	0
$949$	2.76735	0.0898321
$950$	0	0
$951$	23.8760	0.774231
$952$	0	0
$953$	15.2346	0.493496	0.246748	−	0.969080i	$-0.420638\pi$
$953$	15.2346	0.493496	0.246748	+	0.969080i	$0.420638\pi$
$954$	0	0
$955$	61.1352	1.97829
$956$	0	0
$957$	−17.2613	−0.557980
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.19445	0.264337
$962$	0	0
$963$	8.52741	0.274792
$964$	0	0
$965$	−14.4152	−0.464043
$966$	0	0
$967$	−27.5678	−0.886522	−0.443261	−	0.896393i	$-0.646179\pi$
$967$	−27.5678	−0.886522	−0.443261	+	0.896393i	$0.646179\pi$
$968$	0	0
$969$	24.7452	0.794931
$970$	0	0
$971$	10.8096	0.346896	0.173448	−	0.984843i	$-0.444509\pi$
$971$	10.8096	0.346896	0.173448	+	0.984843i	$0.444509\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−74.1771	−2.37557
$976$	0	0
$977$	−42.4703	−1.35875	−0.679373	−	0.733793i	$-0.737748\pi$
$977$	−42.4703	−1.35875	−0.679373	+	0.733793i	$0.737748\pi$
$978$	0	0
$979$	−8.83869	−0.282486
$980$	0	0
$981$	−11.3398	−0.362053
$982$	0	0
$983$	−61.6165	−1.96526	−0.982630	−	0.185576i	$-0.940585\pi$
$983$	−61.6165	−1.96526	−0.982630	+	0.185576i	$0.940585\pi$
$984$	0	0
$985$	43.4469	1.38433
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12.4871	0.397067
$990$	0	0
$991$	42.2933	1.34349	0.671745	−	0.740782i	$-0.265545\pi$
$991$	42.2933	1.34349	0.671745	+	0.740782i	$0.265545\pi$
$992$	0	0
$993$	15.8243	0.502169
$994$	0	0
$995$	−39.5024	−1.25231
$996$	0	0
$997$	−25.3094	−0.801557	−0.400778	−	0.916175i	$-0.631260\pi$
$997$	−25.3094	−0.801557	−0.400778	+	0.916175i	$0.631260\pi$
$998$	0	0
$999$	−3.62837	−0.114797

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.r.1.10		15
7.3	odd	6		1148.2.i.e.821.10	yes	30
7.5	odd	6		1148.2.i.e.165.10	✓	30
7.6	odd	2		8036.2.a.q.1.6		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.e.165.10	✓	30	7.5	odd	6
1148.2.i.e.821.10	yes	30	7.3	odd	6
8036.2.a.q.1.6		15	7.6	odd	2
8036.2.a.r.1.10		15	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q+1.40418 q^{3} -3.61567 q^{5} -1.02828 q^{9} +O(q^{10})\)	\(q+1.40418 q^{3} -3.61567 q^{5} -1.02828 q^{9} -5.64929 q^{11} -6.54345 q^{13} -5.07705 q^{15} -3.39066 q^{17} -5.19738 q^{19} -2.81011 q^{23} +8.07310 q^{25} -5.65643 q^{27} +2.17600 q^{29} -6.26055 q^{31} -7.93262 q^{33} +0.641460 q^{37} -9.18818 q^{39} +1.00000 q^{41} -4.44365 q^{43} +3.71793 q^{45} -6.69641 q^{47} -4.76109 q^{51} +4.95277 q^{53} +20.4260 q^{55} -7.29805 q^{57} +11.0289 q^{59} +12.0389 q^{61} +23.6590 q^{65} +10.5842 q^{67} -3.94589 q^{69} -8.70411 q^{71} -0.422919 q^{73} +11.3361 q^{75} -14.2265 q^{79} -4.85779 q^{81} -5.06521 q^{83} +12.2595 q^{85} +3.05549 q^{87} +1.56457 q^{89} -8.79093 q^{93} +18.7920 q^{95} -16.4451 q^{97} +5.80907 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9}+O(q^{10}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9	\( 15 q + q^{3} - 3 q^{5} + 30 q^{9} + 9 q^{11} - 7 q^{13} + 2 q^{15} - 3 q^{17} - 7 q^{19} - q^{23} + 32 q^{25} - 11 q^{27} + 18 q^{29} - 30 q^{31} + 16 q^{33} + 23 q^{37} + 5 q^{39} + 15 q^{41} + 12 q^{43} + 13 q^{45} + 16 q^{47} + 29 q^{51} + 33 q^{53} - 37 q^{55} + 16 q^{57} + 10 q^{59} - q^{61} + 16 q^{65} + 20 q^{67} - 21 q^{69} + 5 q^{71} + 3 q^{73} + 51 q^{75} + 25 q^{79} + 43 q^{81} - 18 q^{83} + 36 q^{85} + 53 q^{87} + 11 q^{89} + 65 q^{93} - 30 q^{95} - 16 q^{97} - 18 q^{99}+O(q^{100}) \) 15 * q + q^3 - 3 * q^5 + 30 * q^9 + 9 * q^11 - 7 * q^13 + 2 * q^15 - 3 * q^17 - 7 * q^19 - q^23 + 32 * q^25 - 11 * q^27 + 18 * q^29 - 30 * q^31 + 16 * q^33 + 23 * q^37 + 5 * q^39 + 15 * q^41 + 12 * q^43 + 13 * q^45 + 16 * q^47 + 29 * q^51 + 33 * q^53 - 37 * q^55 + 16 * q^57 + 10 * q^59 - q^61 + 16 * q^65 + 20 * q^67 - 21 * q^69 + 5 * q^71 + 3 * q^73 + 51 * q^75 + 25 * q^79 + 43 * q^81 - 18 * q^83 + 36 * q^85 + 53 * q^87 + 11 * q^89 + 65 * q^93 - 30 * q^95 - 16 * q^97 - 18 * q^99

Introduction

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