LMFDB - Embedded newform 8036.2.a.m.1.1

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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 $ x^8 - 14x^6 - 4x^5 + 60x^4 + 31x^3 - 75x^2 - 60x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.86706$ 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)

$f(q)$	$=$	$q-2.86706 q^{3} +1.20206 q^{5} +5.22005 q^{9} +O(q^{10})$	$q-2.86706 q^{3} +1.20206 q^{5} +5.22005 q^{9} -0.847080 q^{11} -1.77566 q^{13} -3.44639 q^{15} +1.33039 q^{17} +5.12029 q^{19} -0.816312 q^{23} -3.55505 q^{25} -6.36501 q^{27} -8.14823 q^{29} +5.88904 q^{31} +2.42863 q^{33} +1.65896 q^{37} +5.09092 q^{39} +1.00000 q^{41} -3.82843 q^{43} +6.27482 q^{45} +4.67296 q^{47} -3.81431 q^{51} -3.64949 q^{53} -1.01824 q^{55} -14.6802 q^{57} -0.851417 q^{59} -9.13127 q^{61} -2.13445 q^{65} -8.39092 q^{67} +2.34042 q^{69} +1.66604 q^{71} +3.08663 q^{73} +10.1925 q^{75} +8.12201 q^{79} +2.58874 q^{81} +3.90385 q^{83} +1.59921 q^{85} +23.3615 q^{87} -9.52945 q^{89} -16.8843 q^{93} +6.15491 q^{95} +13.3723 q^{97} -4.42180 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * 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
$3$	−2.86706	−1.65530	−0.827650	−	0.561245i	$-0.810322\pi$
$3$	−2.86706	−1.65530	−0.827650	+	0.561245i	$0.810322\pi$
$4$	0	0
$5$	1.20206	0.537579	0.268789	−	0.963199i	$-0.413376\pi$
$5$	1.20206	0.537579	0.268789	+	0.963199i	$0.413376\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	5.22005	1.74002
$10$	0	0
$11$	−0.847080	−0.255404	−0.127702	−	0.991813i	$-0.540760\pi$
$11$	−0.847080	−0.255404	−0.127702	+	0.991813i	$0.540760\pi$
$12$	0	0
$13$	−1.77566	−0.492479	−0.246239	−	0.969209i	$-0.579195\pi$
$13$	−1.77566	−0.492479	−0.246239	+	0.969209i	$0.579195\pi$
$14$	0	0
$15$	−3.44639	−0.889853
$16$	0	0
$17$	1.33039	0.322667	0.161334	−	0.986900i	$-0.448420\pi$
$17$	1.33039	0.322667	0.161334	+	0.986900i	$0.448420\pi$
$18$	0	0
$19$	5.12029	1.17468	0.587338	−	0.809342i	$-0.300176\pi$
$19$	5.12029	1.17468	0.587338	+	0.809342i	$0.300176\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.816312	−0.170213	−0.0851064	−	0.996372i	$-0.527123\pi$
$23$	−0.816312	−0.170213	−0.0851064	+	0.996372i	$0.527123\pi$
$24$	0	0
$25$	−3.55505	−0.711009
$26$	0	0
$27$	−6.36501	−1.22495
$28$	0	0
$29$	−8.14823	−1.51309	−0.756544	−	0.653942i	$-0.773114\pi$
$29$	−8.14823	−1.51309	−0.756544	+	0.653942i	$0.773114\pi$
$30$	0	0
$31$	5.88904	1.05770	0.528852	−	0.848714i	$-0.322623\pi$
$31$	5.88904	1.05770	0.528852	+	0.848714i	$0.322623\pi$
$32$	0	0
$33$	2.42863	0.422770
$34$	0	0
$35$	0	0
$36$	0	0
$37$	1.65896	0.272732	0.136366	−	0.990659i	$-0.456458\pi$
$37$	1.65896	0.272732	0.136366	+	0.990659i	$0.456458\pi$
$38$	0	0
$39$	5.09092	0.815199
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−3.82843	−0.583831	−0.291915	−	0.956444i	$-0.594293\pi$
$43$	−3.82843	−0.583831	−0.291915	+	0.956444i	$0.594293\pi$
$44$	0	0
$45$	6.27482	0.935395
$46$	0	0
$47$	4.67296	0.681621	0.340810	−	0.940132i	$-0.389299\pi$
$47$	4.67296	0.681621	0.340810	+	0.940132i	$0.389299\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.81431	−0.534111
$52$	0	0
$53$	−3.64949	−0.501297	−0.250648	−	0.968078i	$-0.580644\pi$
$53$	−3.64949	−0.501297	−0.250648	+	0.968078i	$0.580644\pi$
$54$	0	0
$55$	−1.01824	−0.137300
$56$	0	0
$57$	−14.6802	−1.94444
$58$	0	0
$59$	−0.851417	−0.110845	−0.0554225	−	0.998463i	$-0.517651\pi$
$59$	−0.851417	−0.110845	−0.0554225	+	0.998463i	$0.517651\pi$
$60$	0	0
$61$	−9.13127	−1.16914	−0.584570	−	0.811343i	$-0.698737\pi$
$61$	−9.13127	−1.16914	−0.584570	+	0.811343i	$0.698737\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.13445	−0.264746
$66$	0	0
$67$	−8.39092	−1.02511	−0.512557	−	0.858653i	$-0.671302\pi$
$67$	−8.39092	−1.02511	−0.512557	+	0.858653i	$0.671302\pi$
$68$	0	0
$69$	2.34042	0.281753
$70$	0	0
$71$	1.66604	0.197723	0.0988615	−	0.995101i	$-0.468480\pi$
$71$	1.66604	0.197723	0.0988615	+	0.995101i	$0.468480\pi$
$72$	0	0
$73$	3.08663	0.361263	0.180631	−	0.983551i	$-0.442186\pi$
$73$	3.08663	0.361263	0.180631	+	0.983551i	$0.442186\pi$
$74$	0	0
$75$	10.1925	1.17693
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.12201	0.913797	0.456898	−	0.889519i	$-0.348960\pi$
$79$	8.12201	0.913797	0.456898	+	0.889519i	$0.348960\pi$
$80$	0	0
$81$	2.58874	0.287638
$82$	0	0
$83$	3.90385	0.428503	0.214251	−	0.976779i	$-0.431269\pi$
$83$	3.90385	0.428503	0.214251	+	0.976779i	$0.431269\pi$
$84$	0	0
$85$	1.59921	0.173459
$86$	0	0
$87$	23.3615	2.50461
$88$	0	0
$89$	−9.52945	−1.01012	−0.505060	−	0.863084i	$-0.668530\pi$
$89$	−9.52945	−1.01012	−0.505060	+	0.863084i	$0.668530\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−16.8843	−1.75082
$94$	0	0
$95$	6.15491	0.631481
$96$	0	0
$97$	13.3723	1.35776	0.678878	−	0.734251i	$-0.262466\pi$
$97$	13.3723	1.35776	0.678878	+	0.734251i	$0.262466\pi$
$98$	0	0
$99$	−4.42180	−0.444407
$100$	0	0
$101$	2.75641	0.274273	0.137136	−	0.990552i	$-0.456210\pi$
$101$	2.75641	0.274273	0.137136	+	0.990552i	$0.456210\pi$
$102$	0	0
$103$	−1.46151	−0.144007	−0.0720034	−	0.997404i	$-0.522939\pi$
$103$	−1.46151	−0.144007	−0.0720034	+	0.997404i	$0.522939\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.6228	1.22029	0.610147	−	0.792288i	$-0.291110\pi$
$107$	12.6228	1.22029	0.610147	+	0.792288i	$0.291110\pi$
$108$	0	0
$109$	−20.5779	−1.97101	−0.985504	−	0.169650i	$-0.945736\pi$
$109$	−20.5779	−1.97101	−0.985504	+	0.169650i	$0.945736\pi$
$110$	0	0
$111$	−4.75635	−0.451453
$112$	0	0
$113$	4.11456	0.387065	0.193532	−	0.981094i	$-0.438006\pi$
$113$	4.11456	0.387065	0.193532	+	0.981094i	$0.438006\pi$
$114$	0	0
$115$	−0.981258	−0.0915028
$116$	0	0
$117$	−9.26901	−0.856920
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−10.2825	−0.934769
$122$	0	0
$123$	−2.86706	−0.258514
$124$	0	0
$125$	−10.2837	−0.919802
$126$	0	0
$127$	1.96838	0.174665	0.0873327	−	0.996179i	$-0.472166\pi$
$127$	1.96838	0.174665	0.0873327	+	0.996179i	$0.472166\pi$
$128$	0	0
$129$	10.9764	0.966414
$130$	0	0
$131$	13.1656	1.15029	0.575143	−	0.818053i	$-0.304946\pi$
$131$	13.1656	1.15029	0.575143	+	0.818053i	$0.304946\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−7.65114	−0.658505
$136$	0	0
$137$	4.14489	0.354122	0.177061	−	0.984200i	$-0.443341\pi$
$137$	4.14489	0.354122	0.177061	+	0.984200i	$0.443341\pi$
$138$	0	0
$139$	−12.0961	−1.02598	−0.512990	−	0.858394i	$-0.671462\pi$
$139$	−12.0961	−1.02598	−0.512990	+	0.858394i	$0.671462\pi$
$140$	0	0
$141$	−13.3977	−1.12829
$142$	0	0
$143$	1.50412	0.125781
$144$	0	0
$145$	−9.79468	−0.813404
$146$	0	0
$147$	0	0
$148$	0	0
$149$	7.10279	0.581883	0.290942	−	0.956741i	$-0.406031\pi$
$149$	7.10279	0.581883	0.290942	+	0.956741i	$0.406031\pi$
$150$	0	0
$151$	−9.70079	−0.789439	−0.394720	−	0.918802i	$-0.629158\pi$
$151$	−9.70079	−0.789439	−0.394720	+	0.918802i	$0.629158\pi$
$152$	0	0
$153$	6.94470	0.561446
$154$	0	0
$155$	7.07900	0.568599
$156$	0	0
$157$	−2.92200	−0.233201	−0.116600	−	0.993179i	$-0.537200\pi$
$157$	−2.92200	−0.233201	−0.116600	+	0.993179i	$0.537200\pi$
$158$	0	0
$159$	10.4633	0.829796
$160$	0	0
$161$	0	0
$162$	0	0
$163$	19.5722	1.53301	0.766506	−	0.642238i	$-0.221994\pi$
$163$	19.5722	1.53301	0.766506	+	0.642238i	$0.221994\pi$
$164$	0	0
$165$	2.91937	0.227272
$166$	0	0
$167$	−14.8666	−1.15042	−0.575208	−	0.818007i	$-0.695079\pi$
$167$	−14.8666	−1.15042	−0.575208	+	0.818007i	$0.695079\pi$
$168$	0	0
$169$	−9.84704	−0.757465
$170$	0	0
$171$	26.7282	2.04395
$172$	0	0
$173$	16.4240	1.24869	0.624347	−	0.781147i	$-0.285365\pi$
$173$	16.4240	1.24869	0.624347	+	0.781147i	$0.285365\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	2.44107	0.183482
$178$	0	0
$179$	−25.7614	−1.92550	−0.962750	−	0.270393i	$-0.912846\pi$
$179$	−25.7614	−1.92550	−0.962750	+	0.270393i	$0.912846\pi$
$180$	0	0
$181$	2.11304	0.157061	0.0785305	−	0.996912i	$-0.474977\pi$
$181$	2.11304	0.157061	0.0785305	+	0.996912i	$0.474977\pi$
$182$	0	0
$183$	26.1799	1.93528
$184$	0	0
$185$	1.99418	0.146615
$186$	0	0
$187$	−1.12695	−0.0824106
$188$	0	0
$189$	0	0
$190$	0	0
$191$	18.5308	1.34084	0.670422	−	0.741980i	$-0.266113\pi$
$191$	18.5308	1.34084	0.670422	+	0.741980i	$0.266113\pi$
$192$	0	0
$193$	2.86073	0.205920	0.102960	−	0.994686i	$-0.467169\pi$
$193$	2.86073	0.205920	0.102960	+	0.994686i	$0.467169\pi$
$194$	0	0
$195$	6.11960	0.438234
$196$	0	0
$197$	8.05021	0.573553	0.286777	−	0.957997i	$-0.407416\pi$
$197$	8.05021	0.573553	0.286777	+	0.957997i	$0.407416\pi$
$198$	0	0
$199$	−0.790791	−0.0560577	−0.0280288	−	0.999607i	$-0.508923\pi$
$199$	−0.790791	−0.0560577	−0.0280288	+	0.999607i	$0.508923\pi$
$200$	0	0
$201$	24.0573	1.69687
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.20206	0.0839557
$206$	0	0
$207$	−4.26119	−0.296173
$208$	0	0
$209$	−4.33730	−0.300017
$210$	0	0
$211$	17.9976	1.23900	0.619502	−	0.784995i	$-0.287334\pi$
$211$	17.9976	1.23900	0.619502	+	0.784995i	$0.287334\pi$
$212$	0	0
$213$	−4.77665	−0.327291
$214$	0	0
$215$	−4.60202	−0.313855
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−8.84956	−0.597998
$220$	0	0
$221$	−2.36232	−0.158907
$222$	0	0
$223$	20.5976	1.37932	0.689659	−	0.724134i	$-0.257760\pi$
$223$	20.5976	1.37932	0.689659	+	0.724134i	$0.257760\pi$
$224$	0	0
$225$	−18.5575	−1.23717
$226$	0	0
$227$	−1.09041	−0.0723733	−0.0361867	−	0.999345i	$-0.511521\pi$
$227$	−1.09041	−0.0723733	−0.0361867	+	0.999345i	$0.511521\pi$
$228$	0	0
$229$	−13.5693	−0.896685	−0.448342	−	0.893862i	$-0.647985\pi$
$229$	−13.5693	−0.896685	−0.448342	+	0.893862i	$0.647985\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	17.4985	1.14636	0.573182	−	0.819428i	$-0.305709\pi$
$233$	17.4985	1.14636	0.573182	+	0.819428i	$0.305709\pi$
$234$	0	0
$235$	5.61718	0.366425
$236$	0	0
$237$	−23.2863	−1.51261
$238$	0	0
$239$	5.17257	0.334586	0.167293	−	0.985907i	$-0.446497\pi$
$239$	5.17257	0.334586	0.167293	+	0.985907i	$0.446497\pi$
$240$	0	0
$241$	23.4184	1.50851	0.754257	−	0.656580i	$-0.227997\pi$
$241$	23.4184	1.50851	0.754257	+	0.656580i	$0.227997\pi$
$242$	0	0
$243$	11.6729	0.748820
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−9.09188	−0.578503
$248$	0	0
$249$	−11.1926	−0.709300
$250$	0	0
$251$	−4.39194	−0.277217	−0.138608	−	0.990347i	$-0.544263\pi$
$251$	−4.39194	−0.277217	−0.138608	+	0.990347i	$0.544263\pi$
$252$	0	0
$253$	0.691482	0.0434731
$254$	0	0
$255$	−4.58504	−0.287127
$256$	0	0
$257$	−19.3302	−1.20578	−0.602892	−	0.797823i	$-0.705985\pi$
$257$	−19.3302	−1.20578	−0.602892	+	0.797823i	$0.705985\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−42.5341	−2.63280
$262$	0	0
$263$	4.28513	0.264233	0.132116	−	0.991234i	$-0.457823\pi$
$263$	4.28513	0.264233	0.132116	+	0.991234i	$0.457823\pi$
$264$	0	0
$265$	−4.38692	−0.269486
$266$	0	0
$267$	27.3215	1.67205
$268$	0	0
$269$	−10.1858	−0.621042	−0.310521	−	0.950567i	$-0.600503\pi$
$269$	−10.1858	−0.621042	−0.310521	+	0.950567i	$0.600503\pi$
$270$	0	0
$271$	−29.5946	−1.79775	−0.898873	−	0.438210i	$-0.855613\pi$
$271$	−29.5946	−1.79775	−0.898873	+	0.438210i	$0.855613\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.01141	0.181595
$276$	0	0
$277$	6.01059	0.361141	0.180571	−	0.983562i	$-0.442206\pi$
$277$	6.01059	0.361141	0.180571	+	0.983562i	$0.442206\pi$
$278$	0	0
$279$	30.7411	1.84042
$280$	0	0
$281$	−24.4365	−1.45776	−0.728881	−	0.684641i	$-0.759959\pi$
$281$	−24.4365	−1.45776	−0.728881	+	0.684641i	$0.759959\pi$
$282$	0	0
$283$	−0.391665	−0.0232821	−0.0116410	−	0.999932i	$-0.503706\pi$
$283$	−0.391665	−0.0232821	−0.0116410	+	0.999932i	$0.503706\pi$
$284$	0	0
$285$	−17.6465	−1.04529
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−15.2301	−0.895886
$290$	0	0
$291$	−38.3394	−2.24749
$292$	0	0
$293$	1.12533	0.0657427	0.0328713	−	0.999460i	$-0.489535\pi$
$293$	1.12533	0.0657427	0.0328713	+	0.999460i	$0.489535\pi$
$294$	0	0
$295$	−1.02346	−0.0595879
$296$	0	0
$297$	5.39167	0.312857
$298$	0	0
$299$	1.44949	0.0838262
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−7.90278	−0.454003
$304$	0	0
$305$	−10.9764	−0.628505
$306$	0	0
$307$	−10.8412	−0.618740	−0.309370	−	0.950942i	$-0.600118\pi$
$307$	−10.8412	−0.618740	−0.309370	+	0.950942i	$0.600118\pi$
$308$	0	0
$309$	4.19024	0.238374
$310$	0	0
$311$	−5.65270	−0.320535	−0.160268	−	0.987074i	$-0.551236\pi$
$311$	−5.65270	−0.320535	−0.160268	+	0.987074i	$0.551236\pi$
$312$	0	0
$313$	−5.41270	−0.305944	−0.152972	−	0.988231i	$-0.548884\pi$
$313$	−5.41270	−0.305944	−0.152972	+	0.988231i	$0.548884\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.7283	−1.38888	−0.694439	−	0.719551i	$-0.744347\pi$
$317$	−24.7283	−1.38888	−0.694439	+	0.719551i	$0.744347\pi$
$318$	0	0
$319$	6.90220	0.386449
$320$	0	0
$321$	−36.1904	−2.01995
$322$	0	0
$323$	6.81199	0.379029
$324$	0	0
$325$	6.31254	0.350157
$326$	0	0
$327$	58.9982	3.26261
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−17.8838	−0.982984	−0.491492	−	0.870882i	$-0.663548\pi$
$331$	−17.8838	−0.982984	−0.491492	+	0.870882i	$0.663548\pi$
$332$	0	0
$333$	8.65986	0.474558
$334$	0	0
$335$	−10.0864	−0.551080
$336$	0	0
$337$	−22.1922	−1.20888	−0.604442	−	0.796649i	$-0.706604\pi$
$337$	−22.1922	−1.20888	−0.604442	+	0.796649i	$0.706604\pi$
$338$	0	0
$339$	−11.7967	−0.640708
$340$	0	0
$341$	−4.98849	−0.270142
$342$	0	0
$343$	0	0
$344$	0	0
$345$	2.81333	0.151465
$346$	0	0
$347$	−27.3076	−1.46595	−0.732975	−	0.680256i	$-0.761869\pi$
$347$	−27.3076	−1.46595	−0.732975	+	0.680256i	$0.761869\pi$
$348$	0	0
$349$	23.0020	1.23127	0.615633	−	0.788033i	$-0.288900\pi$
$349$	23.0020	1.23127	0.615633	+	0.788033i	$0.288900\pi$
$350$	0	0
$351$	11.3021	0.603260
$352$	0	0
$353$	−23.2482	−1.23738	−0.618689	−	0.785636i	$-0.712336\pi$
$353$	−23.2482	−1.23738	−0.618689	+	0.785636i	$0.712336\pi$
$354$	0	0
$355$	2.00269	0.106292
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.4835	−1.08108	−0.540538	−	0.841319i	$-0.681779\pi$
$359$	−20.4835	−1.08108	−0.540538	+	0.841319i	$0.681779\pi$
$360$	0	0
$361$	7.21740	0.379863
$362$	0	0
$363$	29.4804	1.54732
$364$	0	0
$365$	3.71032	0.194207
$366$	0	0
$367$	0.405080	0.0211450	0.0105725	−	0.999944i	$-0.496635\pi$
$367$	0.405080	0.0211450	0.0105725	+	0.999944i	$0.496635\pi$
$368$	0	0
$369$	5.22005	0.271745
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.71385	−0.140518	−0.0702589	−	0.997529i	$-0.522383\pi$
$373$	−2.71385	−0.140518	−0.0702589	+	0.997529i	$0.522383\pi$
$374$	0	0
$375$	29.4840	1.52255
$376$	0	0
$377$	14.4685	0.745164
$378$	0	0
$379$	10.2270	0.525327	0.262663	−	0.964888i	$-0.415399\pi$
$379$	10.2270	0.525327	0.262663	+	0.964888i	$0.415399\pi$
$380$	0	0
$381$	−5.64346	−0.289123
$382$	0	0
$383$	33.0051	1.68648	0.843241	−	0.537536i	$-0.180645\pi$
$383$	33.0051	1.68648	0.843241	+	0.537536i	$0.180645\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−19.9846	−1.01587
$388$	0	0
$389$	27.3654	1.38748	0.693742	−	0.720224i	$-0.255961\pi$
$389$	27.3654	1.38748	0.693742	+	0.720224i	$0.255961\pi$
$390$	0	0
$391$	−1.08601	−0.0549221
$392$	0	0
$393$	−37.7467	−1.90407
$394$	0	0
$395$	9.76316	0.491238
$396$	0	0
$397$	−14.6964	−0.737592	−0.368796	−	0.929510i	$-0.620230\pi$
$397$	−14.6964	−0.737592	−0.368796	+	0.929510i	$0.620230\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.41753	−0.470289	−0.235145	−	0.971960i	$-0.575556\pi$
$401$	−9.41753	−0.470289	−0.235145	+	0.971960i	$0.575556\pi$
$402$	0	0
$403$	−10.4569	−0.520896
$404$	0	0
$405$	3.11183	0.154628
$406$	0	0
$407$	−1.40527	−0.0696569
$408$	0	0
$409$	−29.5757	−1.46243	−0.731213	−	0.682149i	$-0.761045\pi$
$409$	−29.5757	−1.46243	−0.731213	+	0.682149i	$0.761045\pi$
$410$	0	0
$411$	−11.8837	−0.586178
$412$	0	0
$413$	0	0
$414$	0	0
$415$	4.69267	0.230354
$416$	0	0
$417$	34.6804	1.69830
$418$	0	0
$419$	13.4890	0.658979	0.329489	−	0.944159i	$-0.393123\pi$
$419$	13.4890	0.658979	0.329489	+	0.944159i	$0.393123\pi$
$420$	0	0
$421$	13.6752	0.666489	0.333244	−	0.942841i	$-0.391857\pi$
$421$	13.6752	0.666489	0.333244	+	0.942841i	$0.391857\pi$
$422$	0	0
$423$	24.3930	1.18603
$424$	0	0
$425$	−4.72960	−0.229419
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−4.31242	−0.208205
$430$	0	0
$431$	−17.7367	−0.854345	−0.427173	−	0.904170i	$-0.640490\pi$
$431$	−17.7367	−0.854345	−0.427173	+	0.904170i	$0.640490\pi$
$432$	0	0
$433$	12.5658	0.603874	0.301937	−	0.953328i	$-0.402367\pi$
$433$	12.5658	0.603874	0.301937	+	0.953328i	$0.402367\pi$
$434$	0	0
$435$	28.0820	1.34643
$436$	0	0
$437$	−4.17976	−0.199945
$438$	0	0
$439$	0.553996	0.0264408	0.0132204	−	0.999913i	$-0.495792\pi$
$439$	0.553996	0.0264408	0.0132204	+	0.999913i	$0.495792\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−35.5905	−1.69095	−0.845477	−	0.534012i	$-0.820684\pi$
$443$	−35.5905	−1.69095	−0.845477	+	0.534012i	$0.820684\pi$
$444$	0	0
$445$	−11.4550	−0.543019
$446$	0	0
$447$	−20.3642	−0.963191
$448$	0	0
$449$	9.72773	0.459080	0.229540	−	0.973299i	$-0.426278\pi$
$449$	9.72773	0.459080	0.229540	+	0.973299i	$0.426278\pi$
$450$	0	0
$451$	−0.847080	−0.0398874
$452$	0	0
$453$	27.8128	1.30676
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−36.8027	−1.72156	−0.860778	−	0.508980i	$-0.830023\pi$
$457$	−36.8027	−1.72156	−0.860778	+	0.508980i	$0.830023\pi$
$458$	0	0
$459$	−8.46795	−0.395250
$460$	0	0
$461$	−33.6797	−1.56862	−0.784310	−	0.620369i	$-0.786983\pi$
$461$	−33.6797	−1.56862	−0.784310	+	0.620369i	$0.786983\pi$
$462$	0	0
$463$	3.14990	0.146388	0.0731942	−	0.997318i	$-0.476681\pi$
$463$	3.14990	0.146388	0.0731942	+	0.997318i	$0.476681\pi$
$464$	0	0
$465$	−20.2959	−0.941201
$466$	0	0
$467$	32.7863	1.51717	0.758584	−	0.651575i	$-0.225892\pi$
$467$	32.7863	1.51717	0.758584	+	0.651575i	$0.225892\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	8.37755	0.386017
$472$	0	0
$473$	3.24299	0.149113
$474$	0	0
$475$	−18.2029	−0.835205
$476$	0	0
$477$	−19.0505	−0.872264
$478$	0	0
$479$	−15.8283	−0.723214	−0.361607	−	0.932331i	$-0.617772\pi$
$479$	−15.8283	−0.723214	−0.361607	+	0.932331i	$0.617772\pi$
$480$	0	0
$481$	−2.94575	−0.134315
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.0744	0.729901
$486$	0	0
$487$	11.7043	0.530373	0.265186	−	0.964197i	$-0.414567\pi$
$487$	11.7043	0.530373	0.265186	+	0.964197i	$0.414567\pi$
$488$	0	0
$489$	−56.1146	−2.53759
$490$	0	0
$491$	−25.0759	−1.13166	−0.565830	−	0.824522i	$-0.691444\pi$
$491$	−25.0759	−1.13166	−0.565830	+	0.824522i	$0.691444\pi$
$492$	0	0
$493$	−10.8403	−0.488224
$494$	0	0
$495$	−5.31527	−0.238904
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−32.1419	−1.43887	−0.719435	−	0.694560i	$-0.755599\pi$
$499$	−32.1419	−1.43887	−0.719435	+	0.694560i	$0.755599\pi$
$500$	0	0
$501$	42.6236	1.90428
$502$	0	0
$503$	−20.8480	−0.929567	−0.464784	−	0.885424i	$-0.653868\pi$
$503$	−20.8480	−0.929567	−0.464784	+	0.885424i	$0.653868\pi$
$504$	0	0
$505$	3.31337	0.147443
$506$	0	0
$507$	28.2321	1.25383
$508$	0	0
$509$	−28.7098	−1.27254	−0.636270	−	0.771466i	$-0.719524\pi$
$509$	−28.7098	−1.27254	−0.636270	+	0.771466i	$0.719524\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−32.5907	−1.43892
$514$	0	0
$515$	−1.75683	−0.0774150
$516$	0	0
$517$	−3.95837	−0.174089
$518$	0	0
$519$	−47.0886	−2.06696
$520$	0	0
$521$	0.650229	0.0284870	0.0142435	−	0.999899i	$-0.495466\pi$
$521$	0.650229	0.0284870	0.0142435	+	0.999899i	$0.495466\pi$
$522$	0	0
$523$	6.66254	0.291332	0.145666	−	0.989334i	$-0.453467\pi$
$523$	6.66254	0.291332	0.145666	+	0.989334i	$0.453467\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.83473	0.341286
$528$	0	0
$529$	−22.3336	−0.971028
$530$	0	0
$531$	−4.44444	−0.192872
$532$	0	0
$533$	−1.77566	−0.0769122
$534$	0	0
$535$	15.1734	0.656004
$536$	0	0
$537$	73.8596	3.18728
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−33.8319	−1.45455	−0.727274	−	0.686347i	$-0.759213\pi$
$541$	−33.8319	−1.45455	−0.727274	+	0.686347i	$0.759213\pi$
$542$	0	0
$543$	−6.05821	−0.259983
$544$	0	0
$545$	−24.7360	−1.05957
$546$	0	0
$547$	−31.0612	−1.32808	−0.664040	−	0.747697i	$-0.731160\pi$
$547$	−31.0612	−1.32808	−0.664040	+	0.747697i	$0.731160\pi$
$548$	0	0
$549$	−47.6657	−2.03432
$550$	0	0
$551$	−41.7213	−1.77739
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−5.71743	−0.242691
$556$	0	0
$557$	−26.2296	−1.11138	−0.555692	−	0.831388i	$-0.687547\pi$
$557$	−26.2296	−1.11138	−0.555692	+	0.831388i	$0.687547\pi$
$558$	0	0
$559$	6.79798	0.287524
$560$	0	0
$561$	3.23103	0.136414
$562$	0	0
$563$	15.1985	0.640539	0.320269	−	0.947327i	$-0.396227\pi$
$563$	15.1985	0.640539	0.320269	+	0.947327i	$0.396227\pi$
$564$	0	0
$565$	4.94595	0.208078
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.1611	1.18058	0.590288	−	0.807193i	$-0.299014\pi$
$569$	28.1611	1.18058	0.590288	+	0.807193i	$0.299014\pi$
$570$	0	0
$571$	−29.9765	−1.25448	−0.627239	−	0.778827i	$-0.715815\pi$
$571$	−29.9765	−1.25448	−0.627239	+	0.778827i	$0.715815\pi$
$572$	0	0
$573$	−53.1291	−2.21950
$574$	0	0
$575$	2.90203	0.121023
$576$	0	0
$577$	23.2322	0.967171	0.483585	−	0.875297i	$-0.339334\pi$
$577$	23.2322	0.967171	0.483585	+	0.875297i	$0.339334\pi$
$578$	0	0
$579$	−8.20189	−0.340859
$580$	0	0
$581$	0	0
$582$	0	0
$583$	3.09141	0.128033
$584$	0	0
$585$	−11.1419	−0.460662
$586$	0	0
$587$	−29.8264	−1.23107	−0.615533	−	0.788111i	$-0.711059\pi$
$587$	−29.8264	−1.23107	−0.615533	+	0.788111i	$0.711059\pi$
$588$	0	0
$589$	30.1536	1.24246
$590$	0	0
$591$	−23.0804	−0.949402
$592$	0	0
$593$	−2.00085	−0.0821651	−0.0410825	−	0.999156i	$-0.513081\pi$
$593$	−2.00085	−0.0821651	−0.0410825	+	0.999156i	$0.513081\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	2.26725	0.0927922
$598$	0	0
$599$	35.3457	1.44419	0.722094	−	0.691795i	$-0.243180\pi$
$599$	35.3457	1.44419	0.722094	+	0.691795i	$0.243180\pi$
$600$	0	0
$601$	−11.2751	−0.459920	−0.229960	−	0.973200i	$-0.573860\pi$
$601$	−11.2751	−0.459920	−0.229960	+	0.973200i	$0.573860\pi$
$602$	0	0
$603$	−43.8010	−1.78371
$604$	0	0
$605$	−12.3602	−0.502512
$606$	0	0
$607$	11.2062	0.454847	0.227424	−	0.973796i	$-0.426970\pi$
$607$	11.2062	0.454847	0.227424	+	0.973796i	$0.426970\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−8.29757	−0.335684
$612$	0	0
$613$	−7.78051	−0.314252	−0.157126	−	0.987579i	$-0.550223\pi$
$613$	−7.78051	−0.314252	−0.157126	+	0.987579i	$0.550223\pi$
$614$	0	0
$615$	−3.44639	−0.138972
$616$	0	0
$617$	−40.9869	−1.65007	−0.825035	−	0.565082i	$-0.808845\pi$
$617$	−40.9869	−1.65007	−0.825035	+	0.565082i	$0.808845\pi$
$618$	0	0
$619$	−15.3119	−0.615438	−0.307719	−	0.951477i	$-0.599566\pi$
$619$	−15.3119	−0.615438	−0.307719	+	0.951477i	$0.599566\pi$
$620$	0	0
$621$	5.19584	0.208502
$622$	0	0
$623$	0	0
$624$	0	0
$625$	5.41358	0.216543
$626$	0	0
$627$	12.4353	0.496618
$628$	0	0
$629$	2.20707	0.0880016
$630$	0	0
$631$	−20.4649	−0.814696	−0.407348	−	0.913273i	$-0.633546\pi$
$631$	−20.4649	−0.814696	−0.407348	+	0.913273i	$0.633546\pi$
$632$	0	0
$633$	−51.6002	−2.05092
$634$	0	0
$635$	2.36611	0.0938963
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.69683	0.344041
$640$	0	0
$641$	0.716490	0.0282997	0.0141498	−	0.999900i	$-0.495496\pi$
$641$	0.716490	0.0282997	0.0141498	+	0.999900i	$0.495496\pi$
$642$	0	0
$643$	−43.2936	−1.70733	−0.853667	−	0.520819i	$-0.825627\pi$
$643$	−43.2936	−1.70733	−0.853667	+	0.520819i	$0.825627\pi$
$644$	0	0
$645$	13.1943	0.519524
$646$	0	0
$647$	0.742902	0.0292065	0.0146032	−	0.999893i	$-0.495351\pi$
$647$	0.742902	0.0292065	0.0146032	+	0.999893i	$0.495351\pi$
$648$	0	0
$649$	0.721218	0.0283103
$650$	0	0
$651$	0	0
$652$	0	0
$653$	50.1326	1.96184	0.980919	−	0.194415i	$-0.0622808\pi$
$653$	50.1326	1.96184	0.980919	+	0.194415i	$0.0622808\pi$
$654$	0	0
$655$	15.8259	0.618370
$656$	0	0
$657$	16.1123	0.628602
$658$	0	0
$659$	14.7410	0.574226	0.287113	−	0.957897i	$-0.407304\pi$
$659$	14.7410	0.574226	0.287113	+	0.957897i	$0.407304\pi$
$660$	0	0
$661$	15.5836	0.606132	0.303066	−	0.952970i	$-0.401990\pi$
$661$	15.5836	0.606132	0.303066	+	0.952970i	$0.401990\pi$
$662$	0	0
$663$	6.77291	0.263038
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6.65150	0.257547
$668$	0	0
$669$	−59.0546	−2.28318
$670$	0	0
$671$	7.73492	0.298603
$672$	0	0
$673$	8.12856	0.313333	0.156666	−	0.987652i	$-0.449925\pi$
$673$	8.12856	0.313333	0.156666	+	0.987652i	$0.449925\pi$
$674$	0	0
$675$	22.6279	0.870948
$676$	0	0
$677$	−45.5185	−1.74942	−0.874709	−	0.484648i	$-0.838948\pi$
$677$	−45.5185	−1.74942	−0.874709	+	0.484648i	$0.838948\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	3.12628	0.119799
$682$	0	0
$683$	−19.5323	−0.747382	−0.373691	−	0.927553i	$-0.621908\pi$
$683$	−19.5323	−0.747382	−0.373691	+	0.927553i	$0.621908\pi$
$684$	0	0
$685$	4.98242	0.190368
$686$	0	0
$687$	38.9040	1.48428
$688$	0	0
$689$	6.48025	0.246878
$690$	0	0
$691$	−23.0362	−0.876337	−0.438168	−	0.898893i	$-0.644373\pi$
$691$	−23.0362	−0.876337	−0.438168	+	0.898893i	$0.644373\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−14.5403	−0.551545
$696$	0	0
$697$	1.33039	0.0503921
$698$	0	0
$699$	−50.1692	−1.89758
$700$	0	0
$701$	13.2873	0.501854	0.250927	−	0.968006i	$-0.419265\pi$
$701$	13.2873	0.501854	0.250927	+	0.968006i	$0.419265\pi$
$702$	0	0
$703$	8.49438	0.320372
$704$	0	0
$705$	−16.1048	−0.606543
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−25.0874	−0.942177	−0.471089	−	0.882086i	$-0.656139\pi$
$709$	−25.0874	−0.942177	−0.471089	+	0.882086i	$0.656139\pi$
$710$	0	0
$711$	42.3972	1.59002
$712$	0	0
$713$	−4.80730	−0.180035
$714$	0	0
$715$	1.80805	0.0676172
$716$	0	0
$717$	−14.8301	−0.553840
$718$	0	0
$719$	−14.4513	−0.538943	−0.269471	−	0.963008i	$-0.586849\pi$
$719$	−14.4513	−0.538943	−0.269471	+	0.963008i	$0.586849\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−67.1421	−2.49704
$724$	0	0
$725$	28.9673	1.07582
$726$	0	0
$727$	−13.3781	−0.496167	−0.248084	−	0.968739i	$-0.579801\pi$
$727$	−13.3781	−0.496167	−0.248084	+	0.968739i	$0.579801\pi$
$728$	0	0
$729$	−41.2333	−1.52716
$730$	0	0
$731$	−5.09331	−0.188383
$732$	0	0
$733$	33.9657	1.25455	0.627275	−	0.778798i	$-0.284170\pi$
$733$	33.9657	1.25455	0.627275	+	0.778798i	$0.284170\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	7.10778	0.261819
$738$	0	0
$739$	7.60323	0.279689	0.139845	−	0.990173i	$-0.455340\pi$
$739$	7.60323	0.279689	0.139845	+	0.990173i	$0.455340\pi$
$740$	0	0
$741$	26.0670	0.957595
$742$	0	0
$743$	8.92565	0.327450	0.163725	−	0.986506i	$-0.447649\pi$
$743$	8.92565	0.327450	0.163725	+	0.986506i	$0.447649\pi$
$744$	0	0
$745$	8.53800	0.312808
$746$	0	0
$747$	20.3783	0.745601
$748$	0	0
$749$	0	0
$750$	0	0
$751$	11.0932	0.404798	0.202399	−	0.979303i	$-0.435126\pi$
$751$	11.0932	0.404798	0.202399	+	0.979303i	$0.435126\pi$
$752$	0	0
$753$	12.5920	0.458877
$754$	0	0
$755$	−11.6610	−0.424386
$756$	0	0
$757$	3.86787	0.140580	0.0702900	−	0.997527i	$-0.477608\pi$
$757$	3.86787	0.140580	0.0702900	+	0.997527i	$0.477608\pi$
$758$	0	0
$759$	−1.98252	−0.0719610
$760$	0	0
$761$	−49.8422	−1.80678	−0.903390	−	0.428820i	$-0.858929\pi$
$761$	−49.8422	−1.80678	−0.903390	+	0.428820i	$0.858929\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	8.34796	0.301821
$766$	0	0
$767$	1.51182	0.0545888
$768$	0	0
$769$	48.6248	1.75345	0.876727	−	0.480988i	$-0.159722\pi$
$769$	48.6248	1.75345	0.876727	+	0.480988i	$0.159722\pi$
$770$	0	0
$771$	55.4208	1.99593
$772$	0	0
$773$	−23.7886	−0.855615	−0.427808	−	0.903870i	$-0.640714\pi$
$773$	−23.7886	−0.855615	−0.427808	+	0.903870i	$0.640714\pi$
$774$	0	0
$775$	−20.9358	−0.752037
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.12029	0.183454
$780$	0	0
$781$	−1.41127	−0.0504993
$782$	0	0
$783$	51.8636	1.85345
$784$	0	0
$785$	−3.51242	−0.125364
$786$	0	0
$787$	14.2106	0.506554	0.253277	−	0.967394i	$-0.418492\pi$
$787$	14.2106	0.506554	0.253277	+	0.967394i	$0.418492\pi$
$788$	0	0
$789$	−12.2857	−0.437384
$790$	0	0
$791$	0	0
$792$	0	0
$793$	16.2140	0.575776
$794$	0	0
$795$	12.5776	0.446081
$796$	0	0
$797$	−52.4563	−1.85810	−0.929049	−	0.369956i	$-0.879373\pi$
$797$	−52.4563	−1.85810	−0.929049	+	0.369956i	$0.879373\pi$
$798$	0	0
$799$	6.21686	0.219937
$800$	0	0
$801$	−49.7442	−1.75762
$802$	0	0
$803$	−2.61462	−0.0922680
$804$	0	0
$805$	0	0
$806$	0	0
$807$	29.2034	1.02801
$808$	0	0
$809$	36.2765	1.27541	0.637707	−	0.770279i	$-0.279883\pi$
$809$	36.2765	1.27541	0.637707	+	0.770279i	$0.279883\pi$
$810$	0	0
$811$	−25.3129	−0.888857	−0.444428	−	0.895814i	$-0.646593\pi$
$811$	−25.3129	−0.888857	−0.444428	+	0.895814i	$0.646593\pi$
$812$	0	0
$813$	84.8496	2.97581
$814$	0	0
$815$	23.5270	0.824114
$816$	0	0
$817$	−19.6027	−0.685812
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−46.5314	−1.62396	−0.811979	−	0.583687i	$-0.801610\pi$
$821$	−46.5314	−1.62396	−0.811979	+	0.583687i	$0.801610\pi$
$822$	0	0
$823$	−26.9254	−0.938561	−0.469280	−	0.883049i	$-0.655487\pi$
$823$	−26.9254	−0.938561	−0.469280	+	0.883049i	$0.655487\pi$
$824$	0	0
$825$	−8.63389	−0.300594
$826$	0	0
$827$	−2.99334	−0.104089	−0.0520444	−	0.998645i	$-0.516574\pi$
$827$	−2.99334	−0.104089	−0.0520444	+	0.998645i	$0.516574\pi$
$828$	0	0
$829$	24.7194	0.858541	0.429270	−	0.903176i	$-0.358771\pi$
$829$	24.7194	0.858541	0.429270	+	0.903176i	$0.358771\pi$
$830$	0	0
$831$	−17.2327	−0.597797
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−17.8706	−0.618439
$836$	0	0
$837$	−37.4838	−1.29563
$838$	0	0
$839$	−6.82342	−0.235571	−0.117785	−	0.993039i	$-0.537579\pi$
$839$	−6.82342	−0.235571	−0.117785	+	0.993039i	$0.537579\pi$
$840$	0	0
$841$	37.3937	1.28944
$842$	0	0
$843$	70.0611	2.41303
$844$	0	0
$845$	−11.8368	−0.407197
$846$	0	0
$847$	0	0
$848$	0	0
$849$	1.12293	0.0385388
$850$	0	0
$851$	−1.35423	−0.0464225
$852$	0	0
$853$	−7.11780	−0.243709	−0.121854	−	0.992548i	$-0.538884\pi$
$853$	−7.11780	−0.243709	−0.121854	+	0.992548i	$0.538884\pi$
$854$	0	0
$855$	32.1289	1.09879
$856$	0	0
$857$	−29.2432	−0.998928	−0.499464	−	0.866335i	$-0.666470\pi$
$857$	−29.2432	−0.998928	−0.499464	+	0.866335i	$0.666470\pi$
$858$	0	0
$859$	−52.6341	−1.79585	−0.897927	−	0.440145i	$-0.854927\pi$
$859$	−52.6341	−1.79585	−0.897927	+	0.440145i	$0.854927\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	21.2880	0.724652	0.362326	−	0.932051i	$-0.381983\pi$
$863$	21.2880	0.724652	0.362326	+	0.932051i	$0.381983\pi$
$864$	0	0
$865$	19.7427	0.671271
$866$	0	0
$867$	43.6655	1.48296
$868$	0	0
$869$	−6.87999	−0.233388
$870$	0	0
$871$	14.8994	0.504847
$872$	0	0
$873$	69.8043	2.36252
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.4150	1.39848	0.699242	−	0.714885i	$-0.253521\pi$
$877$	41.4150	1.39848	0.699242	+	0.714885i	$0.253521\pi$
$878$	0	0
$879$	−3.22640	−0.108824
$880$	0	0
$881$	28.0441	0.944831	0.472415	−	0.881376i	$-0.343382\pi$
$881$	28.0441	0.944831	0.472415	+	0.881376i	$0.343382\pi$
$882$	0	0
$883$	42.7112	1.43735	0.718674	−	0.695347i	$-0.244750\pi$
$883$	42.7112	1.43735	0.718674	+	0.695347i	$0.244750\pi$
$884$	0	0
$885$	2.93431	0.0986359
$886$	0	0
$887$	−27.4023	−0.920080	−0.460040	−	0.887898i	$-0.652165\pi$
$887$	−27.4023	−0.920080	−0.460040	+	0.887898i	$0.652165\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−2.19287	−0.0734639
$892$	0	0
$893$	23.9269	0.800683
$894$	0	0
$895$	−30.9668	−1.03511
$896$	0	0
$897$	−4.15578	−0.138757
$898$	0	0
$899$	−47.9853	−1.60040
$900$	0	0
$901$	−4.85525	−0.161752
$902$	0	0
$903$	0	0
$904$	0	0
$905$	2.54000	0.0844326
$906$	0	0
$907$	−37.8924	−1.25820	−0.629099	−	0.777325i	$-0.716576\pi$
$907$	−37.8924	−1.25820	−0.629099	+	0.777325i	$0.716576\pi$
$908$	0	0
$909$	14.3886	0.477238
$910$	0	0
$911$	32.0315	1.06125	0.530626	−	0.847606i	$-0.321957\pi$
$911$	32.0315	1.06125	0.530626	+	0.847606i	$0.321957\pi$
$912$	0	0
$913$	−3.30687	−0.109441
$914$	0	0
$915$	31.4699	1.04036
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−11.1059	−0.366351	−0.183175	−	0.983080i	$-0.558638\pi$
$919$	−11.1059	−0.366351	−0.183175	+	0.983080i	$0.558638\pi$
$920$	0	0
$921$	31.0824	1.02420
$922$	0	0
$923$	−2.95832	−0.0973744
$924$	0	0
$925$	−5.89769	−0.193915
$926$	0	0
$927$	−7.62915	−0.250574
$928$	0	0
$929$	−10.9477	−0.359181	−0.179591	−	0.983741i	$-0.557477\pi$
$929$	−10.9477	−0.359181	−0.179591	+	0.983741i	$0.557477\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	16.2067	0.530582
$934$	0	0
$935$	−1.35466	−0.0443022
$936$	0	0
$937$	31.2474	1.02081	0.510404	−	0.859935i	$-0.329496\pi$
$937$	31.2474	1.02081	0.510404	+	0.859935i	$0.329496\pi$
$938$	0	0
$939$	15.5185	0.506429
$940$	0	0
$941$	60.0721	1.95829	0.979147	−	0.203152i	$-0.0651185\pi$
$941$	60.0721	1.95829	0.979147	+	0.203152i	$0.0651185\pi$
$942$	0	0
$943$	−0.816312	−0.0265828
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−28.4262	−0.923727	−0.461864	−	0.886951i	$-0.652819\pi$
$947$	−28.4262	−0.923727	−0.461864	+	0.886951i	$0.652819\pi$
$948$	0	0
$949$	−5.48079	−0.177914
$950$	0	0
$951$	70.8975	2.29901
$952$	0	0
$953$	8.26316	0.267670	0.133835	−	0.991004i	$-0.457271\pi$
$953$	8.26316	0.267670	0.133835	+	0.991004i	$0.457271\pi$
$954$	0	0
$955$	22.2752	0.720809
$956$	0	0
$957$	−19.7890	−0.639689
$958$	0	0
$959$	0	0
$960$	0	0
$961$	3.68083	0.118737
$962$	0	0
$963$	65.8917	2.12333
$964$	0	0
$965$	3.43878	0.110698
$966$	0	0
$967$	−2.49199	−0.0801369	−0.0400684	−	0.999197i	$-0.512758\pi$
$967$	−2.49199	−0.0801369	−0.0400684	+	0.999197i	$0.512758\pi$
$968$	0	0
$969$	−19.5304	−0.627407
$970$	0	0
$971$	58.5403	1.87865	0.939323	−	0.343033i	$-0.111454\pi$
$971$	58.5403	1.87865	0.939323	+	0.343033i	$0.111454\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−18.0985	−0.579614
$976$	0	0
$977$	−36.5234	−1.16849	−0.584244	−	0.811578i	$-0.698609\pi$
$977$	−36.5234	−1.16849	−0.584244	+	0.811578i	$0.698609\pi$
$978$	0	0
$979$	8.07221	0.257989
$980$	0	0
$981$	−107.418	−3.42959
$982$	0	0
$983$	52.7139	1.68131	0.840657	−	0.541569i	$-0.182169\pi$
$983$	52.7139	1.68131	0.840657	+	0.541569i	$0.182169\pi$
$984$	0	0
$985$	9.67685	0.308330
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.12520	0.0993755
$990$	0	0
$991$	−14.0151	−0.445206	−0.222603	−	0.974909i	$-0.571455\pi$
$991$	−14.0151	−0.445206	−0.222603	+	0.974909i	$0.571455\pi$
$992$	0	0
$993$	51.2741	1.62713
$994$	0	0
$995$	−0.950580	−0.0301354
$996$	0	0
$997$	−58.2075	−1.84345	−0.921725	−	0.387844i	$-0.873220\pi$
$997$	−58.2075	−1.84345	−0.921725	+	0.387844i	$0.873220\pi$
$998$	0	0
$999$	−10.5593	−0.334082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.m.1.1		8
7.2	even	3		1148.2.i.d.165.8	✓	16
7.4	even	3		1148.2.i.d.821.8	yes	16
7.6	odd	2		8036.2.a.n.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.d.165.8	✓	16	7.2	even	3
1148.2.i.d.821.8	yes	16	7.4	even	3
8036.2.a.m.1.1		8	1.1	even	1	trivial
8036.2.a.n.1.8		8	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-2.86706 q^{3} +1.20206 q^{5} +5.22005 q^{9} +O(q^{10})\)	\(q-2.86706 q^{3} +1.20206 q^{5} +5.22005 q^{9} -0.847080 q^{11} -1.77566 q^{13} -3.44639 q^{15} +1.33039 q^{17} +5.12029 q^{19} -0.816312 q^{23} -3.55505 q^{25} -6.36501 q^{27} -8.14823 q^{29} +5.88904 q^{31} +2.42863 q^{33} +1.65896 q^{37} +5.09092 q^{39} +1.00000 q^{41} -3.82843 q^{43} +6.27482 q^{45} +4.67296 q^{47} -3.81431 q^{51} -3.64949 q^{53} -1.01824 q^{55} -14.6802 q^{57} -0.851417 q^{59} -9.13127 q^{61} -2.13445 q^{65} -8.39092 q^{67} +2.34042 q^{69} +1.66604 q^{71} +3.08663 q^{73} +10.1925 q^{75} +8.12201 q^{79} +2.58874 q^{81} +3.90385 q^{83} +1.59921 q^{85} +23.3615 q^{87} -9.52945 q^{89} -16.8843 q^{93} +6.15491 q^{95} +13.3723 q^{97} -4.42180 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * q^99

Introduction

L-functions

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