LMFDB - Embedded newform 8036.2.a.o.1.6

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:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 18x^{8} + 32x^{7} + 110x^{6} - 154x^{5} - 282x^{4} + 256x^{3} + 253x^{2} - 126x - 49 $ x^10 - 2x^9 - 18x^8 + 32x^7 + 110x^6 - 154x^5 - 282x^4 + 256x^3 + 253x^2 - 126*x - 49
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-0.284914$ 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)

$f(q)$	$=$	$q+0.284914 q^{3} -2.38934 q^{5} -2.91882 q^{9} +O(q^{10})$	$q+0.284914 q^{3} -2.38934 q^{5} -2.91882 q^{9} +5.72114 q^{11} -5.63894 q^{13} -0.680755 q^{15} -0.225296 q^{17} +1.88313 q^{19} +1.32240 q^{23} +0.708922 q^{25} -1.68636 q^{27} +0.835045 q^{29} +4.68008 q^{31} +1.63003 q^{33} +5.80526 q^{37} -1.60661 q^{39} -1.00000 q^{41} -0.313719 q^{43} +6.97405 q^{45} +0.467063 q^{47} -0.0641900 q^{51} +11.1454 q^{53} -13.6697 q^{55} +0.536530 q^{57} -7.92358 q^{59} +4.62009 q^{61} +13.4733 q^{65} +0.187114 q^{67} +0.376771 q^{69} +2.39451 q^{71} -6.57946 q^{73} +0.201982 q^{75} -3.56412 q^{79} +8.27600 q^{81} -14.4593 q^{83} +0.538308 q^{85} +0.237916 q^{87} -8.14466 q^{89} +1.33342 q^{93} -4.49943 q^{95} +17.7258 q^{97} -16.6990 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9	$ 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9} - 6 q^{13} - 4 q^{15} - 12 q^{17} - 8 q^{19} + 4 q^{23} + 16 q^{25} - 8 q^{27} + 2 q^{29} - 8 q^{31} + 6 q^{33} - 2 q^{37} - 2 q^{39} - 10 q^{41} - 2 q^{43} - 44 q^{45} + 14 q^{47} + 14 q^{51} + 8 q^{53} - 8 q^{55} - 10 q^{57} - 24 q^{59} - 14 q^{61} + 2 q^{65} - 8 q^{67} - 16 q^{69} + 10 q^{71} - 44 q^{73} + 50 q^{75} + 10 q^{79} - 14 q^{81} - 20 q^{83} + 8 q^{85} - 20 q^{87} - 6 q^{89} + 8 q^{93} + 4 q^{95} - 46 q^{97} + 2 q^{99}+O(q^{100}) $ 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9 - 6 * q^13 - 4 * q^15 - 12 * q^17 - 8 * q^19 + 4 * q^23 + 16 * q^25 - 8 * q^27 + 2 * q^29 - 8 * q^31 + 6 * q^33 - 2 * q^37 - 2 * q^39 - 10 * q^41 - 2 * q^43 - 44 * q^45 + 14 * q^47 + 14 * q^51 + 8 * q^53 - 8 * q^55 - 10 * q^57 - 24 * q^59 - 14 * q^61 + 2 * q^65 - 8 * q^67 - 16 * q^69 + 10 * q^71 - 44 * q^73 + 50 * q^75 + 10 * q^79 - 14 * q^81 - 20 * q^83 + 8 * q^85 - 20 * q^87 - 6 * q^89 + 8 * q^93 + 4 * q^95 - 46 * q^97 + 2 * 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
$3$	0.284914	0.164495	0.0822476	−	0.996612i	$-0.473790\pi$
$3$	0.284914	0.164495	0.0822476	+	0.996612i	$0.473790\pi$
$4$	0	0
$5$	−2.38934	−1.06854	−0.534272	−	0.845313i	$-0.679414\pi$
$5$	−2.38934	−1.06854	−0.534272	+	0.845313i	$0.679414\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−2.91882	−0.972941
$10$	0	0
$11$	5.72114	1.72499	0.862494	−	0.506068i	$-0.168901\pi$
$11$	5.72114	1.72499	0.862494	+	0.506068i	$0.168901\pi$
$12$	0	0
$13$	−5.63894	−1.56396	−0.781980	−	0.623304i	$-0.785790\pi$
$13$	−5.63894	−1.56396	−0.781980	+	0.623304i	$0.785790\pi$
$14$	0	0
$15$	−0.680755	−0.175770
$16$	0	0
$17$	−0.225296	−0.0546423	−0.0273212	−	0.999627i	$-0.508698\pi$
$17$	−0.225296	−0.0546423	−0.0273212	+	0.999627i	$0.508698\pi$
$18$	0	0
$19$	1.88313	0.432019	0.216010	−	0.976391i	$-0.430696\pi$
$19$	1.88313	0.432019	0.216010	+	0.976391i	$0.430696\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.32240	0.275740	0.137870	−	0.990450i	$-0.455974\pi$
$23$	1.32240	0.275740	0.137870	+	0.990450i	$0.455974\pi$
$24$	0	0
$25$	0.708922	0.141784
$26$	0	0
$27$	−1.68636	−0.324540
$28$	0	0
$29$	0.835045	0.155064	0.0775320	−	0.996990i	$-0.475296\pi$
$29$	0.835045	0.155064	0.0775320	+	0.996990i	$0.475296\pi$
$30$	0	0
$31$	4.68008	0.840567	0.420283	−	0.907393i	$-0.361931\pi$
$31$	4.68008	0.840567	0.420283	+	0.907393i	$0.361931\pi$
$32$	0	0
$33$	1.63003	0.283752
$34$	0	0
$35$	0	0
$36$	0	0
$37$	5.80526	0.954380	0.477190	−	0.878800i	$-0.341655\pi$
$37$	5.80526	0.954380	0.477190	+	0.878800i	$0.341655\pi$
$38$	0	0
$39$	−1.60661	−0.257264
$40$	0	0
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−0.313719	−0.0478417	−0.0239209	−	0.999714i	$-0.507615\pi$
$43$	−0.313719	−0.0478417	−0.0239209	+	0.999714i	$0.507615\pi$
$44$	0	0
$45$	6.97405	1.03963
$46$	0	0
$47$	0.467063	0.0681282	0.0340641	−	0.999420i	$-0.489155\pi$
$47$	0.467063	0.0681282	0.0340641	+	0.999420i	$0.489155\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−0.0641900	−0.00898840
$52$	0	0
$53$	11.1454	1.53094	0.765468	−	0.643474i	$-0.222508\pi$
$53$	11.1454	1.53094	0.765468	+	0.643474i	$0.222508\pi$
$54$	0	0
$55$	−13.6697	−1.84322
$56$	0	0
$57$	0.536530	0.0710652
$58$	0	0
$59$	−7.92358	−1.03156	−0.515781	−	0.856720i	$-0.672498\pi$
$59$	−7.92358	−1.03156	−0.515781	+	0.856720i	$0.672498\pi$
$60$	0	0
$61$	4.62009	0.591542	0.295771	−	0.955259i	$-0.404424\pi$
$61$	4.62009	0.591542	0.295771	+	0.955259i	$0.404424\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.4733	1.67116
$66$	0	0
$67$	0.187114	0.0228596	0.0114298	−	0.999935i	$-0.496362\pi$
$67$	0.187114	0.0228596	0.0114298	+	0.999935i	$0.496362\pi$
$68$	0	0
$69$	0.376771	0.0453579
$70$	0	0
$71$	2.39451	0.284176	0.142088	−	0.989854i	$-0.454618\pi$
$71$	2.39451	0.284176	0.142088	+	0.989854i	$0.454618\pi$
$72$	0	0
$73$	−6.57946	−0.770068	−0.385034	−	0.922902i	$-0.625810\pi$
$73$	−6.57946	−0.770068	−0.385034	+	0.922902i	$0.625810\pi$
$74$	0	0
$75$	0.201982	0.0233229
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−3.56412	−0.400995	−0.200498	−	0.979694i	$-0.564256\pi$
$79$	−3.56412	−0.400995	−0.200498	+	0.979694i	$0.564256\pi$
$80$	0	0
$81$	8.27600	0.919556
$82$	0	0
$83$	−14.4593	−1.58712	−0.793558	−	0.608495i	$-0.791774\pi$
$83$	−14.4593	−1.58712	−0.793558	+	0.608495i	$0.791774\pi$
$84$	0	0
$85$	0.538308	0.0583877
$86$	0	0
$87$	0.237916	0.0255073
$88$	0	0
$89$	−8.14466	−0.863332	−0.431666	−	0.902033i	$-0.642074\pi$
$89$	−8.14466	−0.863332	−0.431666	+	0.902033i	$0.642074\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.33342	0.138269
$94$	0	0
$95$	−4.49943	−0.461631
$96$	0	0
$97$	17.7258	1.79979	0.899893	−	0.436112i	$-0.143645\pi$
$97$	17.7258	1.79979	0.899893	+	0.436112i	$0.143645\pi$
$98$	0	0
$99$	−16.6990	−1.67831
$100$	0	0
$101$	−9.59821	−0.955057	−0.477529	−	0.878616i	$-0.658467\pi$
$101$	−9.59821	−0.955057	−0.477529	+	0.878616i	$0.658467\pi$
$102$	0	0
$103$	6.43285	0.633847	0.316924	−	0.948451i	$-0.397350\pi$
$103$	6.43285	0.633847	0.316924	+	0.948451i	$0.397350\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.6742	−1.32194	−0.660969	−	0.750413i	$-0.729855\pi$
$107$	−13.6742	−1.32194	−0.660969	+	0.750413i	$0.729855\pi$
$108$	0	0
$109$	−13.7242	−1.31454	−0.657270	−	0.753655i	$-0.728289\pi$
$109$	−13.7242	−1.31454	−0.657270	+	0.753655i	$0.728289\pi$
$110$	0	0
$111$	1.65400	0.156991
$112$	0	0
$113$	−18.4103	−1.73189	−0.865947	−	0.500137i	$-0.833283\pi$
$113$	−18.4103	−1.73189	−0.865947	+	0.500137i	$0.833283\pi$
$114$	0	0
$115$	−3.15966	−0.294640
$116$	0	0
$117$	16.4591	1.52164
$118$	0	0
$119$	0	0
$120$	0	0
$121$	21.7314	1.97558
$122$	0	0
$123$	−0.284914	−0.0256898
$124$	0	0
$125$	10.2528	0.917040
$126$	0	0
$127$	−3.04644	−0.270327	−0.135164	−	0.990823i	$-0.543156\pi$
$127$	−3.04644	−0.270327	−0.135164	+	0.990823i	$0.543156\pi$
$128$	0	0
$129$	−0.0893831	−0.00786974
$130$	0	0
$131$	9.24397	0.807649	0.403825	−	0.914836i	$-0.367681\pi$
$131$	9.24397	0.807649	0.403825	+	0.914836i	$0.367681\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	4.02927	0.346784
$136$	0	0
$137$	12.2287	1.04477	0.522386	−	0.852709i	$-0.325042\pi$
$137$	12.2287	1.04477	0.522386	+	0.852709i	$0.325042\pi$
$138$	0	0
$139$	−7.60223	−0.644813	−0.322407	−	0.946601i	$-0.604492\pi$
$139$	−7.60223	−0.644813	−0.322407	+	0.946601i	$0.604492\pi$
$140$	0	0
$141$	0.133073	0.0112068
$142$	0	0
$143$	−32.2611	−2.69781
$144$	0	0
$145$	−1.99520	−0.165693
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.38712	−0.523253	−0.261627	−	0.965169i	$-0.584259\pi$
$149$	−6.38712	−0.523253	−0.261627	+	0.965169i	$0.584259\pi$
$150$	0	0
$151$	−5.05067	−0.411018	−0.205509	−	0.978655i	$-0.565885\pi$
$151$	−5.05067	−0.411018	−0.205509	+	0.978655i	$0.565885\pi$
$152$	0	0
$153$	0.657599	0.0531638
$154$	0	0
$155$	−11.1823	−0.898182
$156$	0	0
$157$	−5.41932	−0.432509	−0.216255	−	0.976337i	$-0.569384\pi$
$157$	−5.41932	−0.432509	−0.216255	+	0.976337i	$0.569384\pi$
$158$	0	0
$159$	3.17548	0.251832
$160$	0	0
$161$	0	0
$162$	0	0
$163$	7.39830	0.579480	0.289740	−	0.957105i	$-0.406431\pi$
$163$	7.39830	0.579480	0.289740	+	0.957105i	$0.406431\pi$
$164$	0	0
$165$	−3.89469	−0.303202
$166$	0	0
$167$	−7.05784	−0.546152	−0.273076	−	0.961992i	$-0.588041\pi$
$167$	−7.05784	−0.546152	−0.273076	+	0.961992i	$0.588041\pi$
$168$	0	0
$169$	18.7976	1.44597
$170$	0	0
$171$	−5.49652	−0.420330
$172$	0	0
$173$	−8.90963	−0.677387	−0.338693	−	0.940897i	$-0.609985\pi$
$173$	−8.90963	−0.677387	−0.338693	+	0.940897i	$0.609985\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.25754	−0.169687
$178$	0	0
$179$	−7.91862	−0.591865	−0.295933	−	0.955209i	$-0.595630\pi$
$179$	−7.91862	−0.591865	−0.295933	+	0.955209i	$0.595630\pi$
$180$	0	0
$181$	−0.182128	−0.0135375	−0.00676874	−	0.999977i	$-0.502155\pi$
$181$	−0.182128	−0.0135375	−0.00676874	+	0.999977i	$0.502155\pi$
$182$	0	0
$183$	1.31633	0.0973058
$184$	0	0
$185$	−13.8707	−1.01980
$186$	0	0
$187$	−1.28895	−0.0942573
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.7058	−1.13643	−0.568215	−	0.822880i	$-0.692366\pi$
$191$	−15.7058	−1.13643	−0.568215	+	0.822880i	$0.692366\pi$
$192$	0	0
$193$	−8.44563	−0.607930	−0.303965	−	0.952683i	$-0.598311\pi$
$193$	−8.44563	−0.607930	−0.303965	+	0.952683i	$0.598311\pi$
$194$	0	0
$195$	3.83874	0.274898
$196$	0	0
$197$	16.6247	1.18446	0.592230	−	0.805769i	$-0.298248\pi$
$197$	16.6247	1.18446	0.592230	+	0.805769i	$0.298248\pi$
$198$	0	0
$199$	−23.5536	−1.66967	−0.834835	−	0.550500i	$-0.814437\pi$
$199$	−23.5536	−1.66967	−0.834835	+	0.550500i	$0.814437\pi$
$200$	0	0
$201$	0.0533113	0.00376029
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.38934	0.166878
$206$	0	0
$207$	−3.85986	−0.268279
$208$	0	0
$209$	10.7736	0.745228
$210$	0	0
$211$	−8.47900	−0.583719	−0.291859	−	0.956461i	$-0.594274\pi$
$211$	−8.47900	−0.583719	−0.291859	+	0.956461i	$0.594274\pi$
$212$	0	0
$213$	0.682229	0.0467456
$214$	0	0
$215$	0.749580	0.0511210
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−1.87458	−0.126673
$220$	0	0
$221$	1.27043	0.0854584
$222$	0	0
$223$	−19.0449	−1.27534	−0.637672	−	0.770308i	$-0.720102\pi$
$223$	−19.0449	−1.27534	−0.637672	+	0.770308i	$0.720102\pi$
$224$	0	0
$225$	−2.06922	−0.137948
$226$	0	0
$227$	22.4620	1.49085	0.745427	−	0.666588i	$-0.232246\pi$
$227$	22.4620	1.49085	0.745427	+	0.666588i	$0.232246\pi$
$228$	0	0
$229$	13.4786	0.890689	0.445345	−	0.895359i	$-0.353081\pi$
$229$	13.4786	0.890689	0.445345	+	0.895359i	$0.353081\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.63269	−0.237986	−0.118993	−	0.992895i	$-0.537967\pi$
$233$	−3.63269	−0.237986	−0.118993	+	0.992895i	$0.537967\pi$
$234$	0	0
$235$	−1.11597	−0.0727979
$236$	0	0
$237$	−1.01547	−0.0659618
$238$	0	0
$239$	11.8940	0.769358	0.384679	−	0.923050i	$-0.374312\pi$
$239$	11.8940	0.769358	0.384679	+	0.923050i	$0.374312\pi$
$240$	0	0
$241$	−19.4290	−1.25153	−0.625766	−	0.780011i	$-0.715213\pi$
$241$	−19.4290	−1.25153	−0.625766	+	0.780011i	$0.715213\pi$
$242$	0	0
$243$	7.41702	0.475802
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−10.6188	−0.675661
$248$	0	0
$249$	−4.11966	−0.261073
$250$	0	0
$251$	8.54770	0.539526	0.269763	−	0.962927i	$-0.413055\pi$
$251$	8.54770	0.539526	0.269763	+	0.962927i	$0.413055\pi$
$252$	0	0
$253$	7.56564	0.475648
$254$	0	0
$255$	0.153372	0.00960450
$256$	0	0
$257$	−14.1942	−0.885411	−0.442706	−	0.896667i	$-0.645981\pi$
$257$	−14.1942	−0.885411	−0.442706	+	0.896667i	$0.645981\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−2.43735	−0.150868
$262$	0	0
$263$	23.1055	1.42475	0.712374	−	0.701800i	$-0.247620\pi$
$263$	23.1055	1.42475	0.712374	+	0.701800i	$0.247620\pi$
$264$	0	0
$265$	−26.6300	−1.63587
$266$	0	0
$267$	−2.32053	−0.142014
$268$	0	0
$269$	−8.03904	−0.490149	−0.245075	−	0.969504i	$-0.578812\pi$
$269$	−8.03904	−0.490149	−0.245075	+	0.969504i	$0.578812\pi$
$270$	0	0
$271$	14.0642	0.854339	0.427169	−	0.904172i	$-0.359511\pi$
$271$	14.0642	0.854339	0.427169	+	0.904172i	$0.359511\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.05584	0.244576
$276$	0	0
$277$	2.91125	0.174920	0.0874601	−	0.996168i	$-0.472125\pi$
$277$	2.91125	0.174920	0.0874601	+	0.996168i	$0.472125\pi$
$278$	0	0
$279$	−13.6603	−0.817822
$280$	0	0
$281$	−8.63718	−0.515251	−0.257625	−	0.966245i	$-0.582940\pi$
$281$	−8.63718	−0.515251	−0.257625	+	0.966245i	$0.582940\pi$
$282$	0	0
$283$	13.6433	0.811010	0.405505	−	0.914093i	$-0.367096\pi$
$283$	13.6433	0.811010	0.405505	+	0.914093i	$0.367096\pi$
$284$	0	0
$285$	−1.28195	−0.0759362
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.9492	−0.997014
$290$	0	0
$291$	5.05034	0.296056
$292$	0	0
$293$	−1.58934	−0.0928503	−0.0464252	−	0.998922i	$-0.514783\pi$
$293$	−1.58934	−0.0928503	−0.0464252	+	0.998922i	$0.514783\pi$
$294$	0	0
$295$	18.9321	1.10227
$296$	0	0
$297$	−9.64788	−0.559827
$298$	0	0
$299$	−7.45694	−0.431246
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.73467	−0.157102
$304$	0	0
$305$	−11.0389	−0.632088
$306$	0	0
$307$	−23.7023	−1.35276	−0.676382	−	0.736551i	$-0.736453\pi$
$307$	−23.7023	−1.35276	−0.676382	+	0.736551i	$0.736453\pi$
$308$	0	0
$309$	1.83281	0.104265
$310$	0	0
$311$	−28.2042	−1.59931	−0.799657	−	0.600457i	$-0.794985\pi$
$311$	−28.2042	−1.59931	−0.799657	+	0.600457i	$0.794985\pi$
$312$	0	0
$313$	−13.5469	−0.765715	−0.382858	−	0.923807i	$-0.625060\pi$
$313$	−13.5469	−0.765715	−0.382858	+	0.923807i	$0.625060\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4.07490	0.228869	0.114435	−	0.993431i	$-0.463494\pi$
$317$	4.07490	0.228869	0.114435	+	0.993431i	$0.463494\pi$
$318$	0	0
$319$	4.77741	0.267484
$320$	0	0
$321$	−3.89598	−0.217453
$322$	0	0
$323$	−0.424262	−0.0236065
$324$	0	0
$325$	−3.99757	−0.221745
$326$	0	0
$327$	−3.91022	−0.216236
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−29.8518	−1.64080	−0.820401	−	0.571788i	$-0.806250\pi$
$331$	−29.8518	−1.64080	−0.820401	+	0.571788i	$0.806250\pi$
$332$	0	0
$333$	−16.9445	−0.928555
$334$	0	0
$335$	−0.447077	−0.0244264
$336$	0	0
$337$	1.23104	0.0670590	0.0335295	−	0.999438i	$-0.489325\pi$
$337$	1.23104	0.0670590	0.0335295	+	0.999438i	$0.489325\pi$
$338$	0	0
$339$	−5.24535	−0.284888
$340$	0	0
$341$	26.7754	1.44997
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−0.900232	−0.0484669
$346$	0	0
$347$	9.00756	0.483551	0.241775	−	0.970332i	$-0.422270\pi$
$347$	9.00756	0.483551	0.241775	+	0.970332i	$0.422270\pi$
$348$	0	0
$349$	−11.0534	−0.591676	−0.295838	−	0.955238i	$-0.595599\pi$
$349$	−11.0534	−0.591676	−0.295838	+	0.955238i	$0.595599\pi$
$350$	0	0
$351$	9.50926	0.507567
$352$	0	0
$353$	−24.0375	−1.27939	−0.639695	−	0.768629i	$-0.720939\pi$
$353$	−24.0375	−1.27939	−0.639695	+	0.768629i	$0.720939\pi$
$354$	0	0
$355$	−5.72128	−0.303654
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−15.3867	−0.812080	−0.406040	−	0.913855i	$-0.633091\pi$
$359$	−15.3867	−0.812080	−0.406040	+	0.913855i	$0.633091\pi$
$360$	0	0
$361$	−15.4538	−0.813359
$362$	0	0
$363$	6.19158	0.324974
$364$	0	0
$365$	15.7205	0.822851
$366$	0	0
$367$	19.0729	0.995599	0.497799	−	0.867292i	$-0.334142\pi$
$367$	19.0729	0.995599	0.497799	+	0.867292i	$0.334142\pi$
$368$	0	0
$369$	2.91882	0.151948
$370$	0	0
$371$	0	0
$372$	0	0
$373$	13.7210	0.710447	0.355223	−	0.934781i	$-0.384405\pi$
$373$	13.7210	0.710447	0.355223	+	0.934781i	$0.384405\pi$
$374$	0	0
$375$	2.92117	0.150849
$376$	0	0
$377$	−4.70877	−0.242514
$378$	0	0
$379$	−19.5626	−1.00486	−0.502432	−	0.864617i	$-0.667561\pi$
$379$	−19.5626	−1.00486	−0.502432	+	0.864617i	$0.667561\pi$
$380$	0	0
$381$	−0.867973	−0.0444676
$382$	0	0
$383$	3.84968	0.196709	0.0983547	−	0.995151i	$-0.468642\pi$
$383$	3.84968	0.196709	0.0983547	+	0.995151i	$0.468642\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.915691	0.0465472
$388$	0	0
$389$	11.6658	0.591478	0.295739	−	0.955269i	$-0.404434\pi$
$389$	11.6658	0.591478	0.295739	+	0.955269i	$0.404434\pi$
$390$	0	0
$391$	−0.297932	−0.0150671
$392$	0	0
$393$	2.63374	0.132855
$394$	0	0
$395$	8.51588	0.428480
$396$	0	0
$397$	−15.5657	−0.781218	−0.390609	−	0.920557i	$-0.627736\pi$
$397$	−15.5657	−0.781218	−0.390609	+	0.920557i	$0.627736\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−29.0383	−1.45010	−0.725051	−	0.688696i	$-0.758184\pi$
$401$	−29.0383	−1.45010	−0.725051	+	0.688696i	$0.758184\pi$
$402$	0	0
$403$	−26.3907	−1.31461
$404$	0	0
$405$	−19.7741	−0.982585
$406$	0	0
$407$	33.2127	1.64629
$408$	0	0
$409$	10.5338	0.520863	0.260432	−	0.965492i	$-0.416135\pi$
$409$	10.5338	0.520863	0.260432	+	0.965492i	$0.416135\pi$
$410$	0	0
$411$	3.48414	0.171860
$412$	0	0
$413$	0	0
$414$	0	0
$415$	34.5481	1.69590
$416$	0	0
$417$	−2.16598	−0.106069
$418$	0	0
$419$	−32.6004	−1.59263	−0.796316	−	0.604880i	$-0.793221\pi$
$419$	−32.6004	−1.59263	−0.796316	+	0.604880i	$0.793221\pi$
$420$	0	0
$421$	−0.624956	−0.0304585	−0.0152293	−	0.999884i	$-0.504848\pi$
$421$	−0.624956	−0.0304585	−0.0152293	+	0.999884i	$0.504848\pi$
$422$	0	0
$423$	−1.36328	−0.0662847
$424$	0	0
$425$	−0.159717	−0.00774743
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.19165	−0.443777
$430$	0	0
$431$	7.11125	0.342537	0.171268	−	0.985224i	$-0.445213\pi$
$431$	7.11125	0.342537	0.171268	+	0.985224i	$0.445213\pi$
$432$	0	0
$433$	−39.2606	−1.88675	−0.943373	−	0.331734i	$-0.892366\pi$
$433$	−39.2606	−1.88675	−0.943373	+	0.331734i	$0.892366\pi$
$434$	0	0
$435$	−0.568462	−0.0272557
$436$	0	0
$437$	2.49025	0.119125
$438$	0	0
$439$	38.2853	1.82726	0.913630	−	0.406548i	$-0.133267\pi$
$439$	38.2853	1.82726	0.913630	+	0.406548i	$0.133267\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	12.5341	0.595515	0.297758	−	0.954642i	$-0.403761\pi$
$443$	12.5341	0.595515	0.297758	+	0.954642i	$0.403761\pi$
$444$	0	0
$445$	19.4603	0.922508
$446$	0	0
$447$	−1.81978	−0.0860727
$448$	0	0
$449$	35.8296	1.69090	0.845451	−	0.534053i	$-0.179332\pi$
$449$	35.8296	1.69090	0.845451	+	0.534053i	$0.179332\pi$
$450$	0	0
$451$	−5.72114	−0.269398
$452$	0	0
$453$	−1.43901	−0.0676105
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.8056	0.552245	0.276122	−	0.961122i	$-0.410950\pi$
$457$	11.8056	0.552245	0.276122	+	0.961122i	$0.410950\pi$
$458$	0	0
$459$	0.379930	0.0177336
$460$	0	0
$461$	−34.3509	−1.59988	−0.799941	−	0.600079i	$-0.795136\pi$
$461$	−34.3509	−1.59988	−0.799941	+	0.600079i	$0.795136\pi$
$462$	0	0
$463$	−29.1253	−1.35357	−0.676784	−	0.736181i	$-0.736627\pi$
$463$	−29.1253	−1.35357	−0.676784	+	0.736181i	$0.736627\pi$
$464$	0	0
$465$	−3.18599	−0.147747
$466$	0	0
$467$	−10.6140	−0.491159	−0.245579	−	0.969376i	$-0.578978\pi$
$467$	−10.6140	−0.491159	−0.245579	+	0.969376i	$0.578978\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−1.54404	−0.0711457
$472$	0	0
$473$	−1.79483	−0.0825264
$474$	0	0
$475$	1.33499	0.0612536
$476$	0	0
$477$	−32.5314	−1.48951
$478$	0	0
$479$	−8.03739	−0.367238	−0.183619	−	0.982998i	$-0.558781\pi$
$479$	−8.03739	−0.367238	−0.183619	+	0.982998i	$0.558781\pi$
$480$	0	0
$481$	−32.7355	−1.49261
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−42.3529	−1.92315
$486$	0	0
$487$	22.3245	1.01162	0.505811	−	0.862645i	$-0.331193\pi$
$487$	22.3245	1.01162	0.505811	+	0.862645i	$0.331193\pi$
$488$	0	0
$489$	2.10788	0.0953217
$490$	0	0
$491$	−16.5660	−0.747611	−0.373806	−	0.927507i	$-0.621947\pi$
$491$	−16.5660	−0.747611	−0.373806	+	0.927507i	$0.621947\pi$
$492$	0	0
$493$	−0.188132	−0.00847306
$494$	0	0
$495$	39.8995	1.79335
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−10.8446	−0.485472	−0.242736	−	0.970092i	$-0.578045\pi$
$499$	−10.8446	−0.485472	−0.242736	+	0.970092i	$0.578045\pi$
$500$	0	0
$501$	−2.01088	−0.0898394
$502$	0	0
$503$	−34.6678	−1.54576	−0.772881	−	0.634552i	$-0.781185\pi$
$503$	−34.6678	−1.54576	−0.772881	+	0.634552i	$0.781185\pi$
$504$	0	0
$505$	22.9333	1.02052
$506$	0	0
$507$	5.35571	0.237855
$508$	0	0
$509$	10.1641	0.450515	0.225258	−	0.974299i	$-0.427678\pi$
$509$	10.1641	0.450515	0.225258	+	0.974299i	$0.427678\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.17563	−0.140207
$514$	0	0
$515$	−15.3702	−0.677293
$516$	0	0
$517$	2.67213	0.117520
$518$	0	0
$519$	−2.53848	−0.111427
$520$	0	0
$521$	−32.3278	−1.41630	−0.708152	−	0.706060i	$-0.750471\pi$
$521$	−32.3278	−1.41630	−0.708152	+	0.706060i	$0.750471\pi$
$522$	0	0
$523$	13.4973	0.590197	0.295099	−	0.955467i	$-0.404647\pi$
$523$	13.4973	0.590197	0.295099	+	0.955467i	$0.404647\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.05440	−0.0459305
$528$	0	0
$529$	−21.2513	−0.923968
$530$	0	0
$531$	23.1275	1.00365
$532$	0	0
$533$	5.63894	0.244250
$534$	0	0
$535$	32.6723	1.41255
$536$	0	0
$537$	−2.25613	−0.0973590
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21.0207	0.903749	0.451874	−	0.892082i	$-0.350755\pi$
$541$	21.0207	0.903749	0.451874	+	0.892082i	$0.350755\pi$
$542$	0	0
$543$	−0.0518909	−0.00222685
$544$	0	0
$545$	32.7917	1.40464
$546$	0	0
$547$	−37.0787	−1.58537	−0.792685	−	0.609631i	$-0.791318\pi$
$547$	−37.0787	−1.58537	−0.792685	+	0.609631i	$0.791318\pi$
$548$	0	0
$549$	−13.4852	−0.575535
$550$	0	0
$551$	1.57250	0.0669907
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−3.95197	−0.167752
$556$	0	0
$557$	−7.78331	−0.329789	−0.164895	−	0.986311i	$-0.552728\pi$
$557$	−7.78331	−0.329789	−0.164895	+	0.986311i	$0.552728\pi$
$558$	0	0
$559$	1.76904	0.0748226
$560$	0	0
$561$	−0.367240	−0.0155049
$562$	0	0
$563$	16.9855	0.715854	0.357927	−	0.933750i	$-0.383484\pi$
$563$	16.9855	0.715854	0.357927	+	0.933750i	$0.383484\pi$
$564$	0	0
$565$	43.9883	1.85060
$566$	0	0
$567$	0	0
$568$	0	0
$569$	46.1734	1.93569	0.967844	−	0.251551i	$-0.0809407\pi$
$569$	46.1734	1.93569	0.967844	+	0.251551i	$0.0809407\pi$
$570$	0	0
$571$	−32.7866	−1.37208	−0.686038	−	0.727566i	$-0.740652\pi$
$571$	−32.7866	−1.37208	−0.686038	+	0.727566i	$0.740652\pi$
$572$	0	0
$573$	−4.47480	−0.186937
$574$	0	0
$575$	0.937479	0.0390956
$576$	0	0
$577$	−15.9785	−0.665192	−0.332596	−	0.943069i	$-0.607925\pi$
$577$	−15.9785	−0.665192	−0.332596	+	0.943069i	$0.607925\pi$
$578$	0	0
$579$	−2.40628	−0.100002
$580$	0	0
$581$	0	0
$582$	0	0
$583$	63.7642	2.64084
$584$	0	0
$585$	−39.3262	−1.62594
$586$	0	0
$587$	12.4963	0.515777	0.257888	−	0.966175i	$-0.416973\pi$
$587$	12.4963	0.515777	0.257888	+	0.966175i	$0.416973\pi$
$588$	0	0
$589$	8.81319	0.363141
$590$	0	0
$591$	4.73661	0.194838
$592$	0	0
$593$	12.3518	0.507229	0.253614	−	0.967305i	$-0.418381\pi$
$593$	12.3518	0.507229	0.253614	+	0.967305i	$0.418381\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−6.71076	−0.274653
$598$	0	0
$599$	−31.1090	−1.27108	−0.635541	−	0.772068i	$-0.719223\pi$
$599$	−31.1090	−1.27108	−0.635541	+	0.772068i	$0.719223\pi$
$600$	0	0
$601$	18.9355	0.772393	0.386197	−	0.922416i	$-0.373789\pi$
$601$	18.9355	0.772393	0.386197	+	0.922416i	$0.373789\pi$
$602$	0	0
$603$	−0.546152	−0.0222410
$604$	0	0
$605$	−51.9236	−2.11099
$606$	0	0
$607$	10.0928	0.409655	0.204828	−	0.978798i	$-0.434337\pi$
$607$	10.0928	0.409655	0.204828	+	0.978798i	$0.434337\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−2.63374	−0.106550
$612$	0	0
$613$	14.2143	0.574111	0.287055	−	0.957914i	$-0.407324\pi$
$613$	14.2143	0.574111	0.287055	+	0.957914i	$0.407324\pi$
$614$	0	0
$615$	0.680755	0.0274507
$616$	0	0
$617$	0.265395	0.0106844	0.00534220	−	0.999986i	$-0.498300\pi$
$617$	0.265395	0.0106844	0.00534220	+	0.999986i	$0.498300\pi$
$618$	0	0
$619$	−39.5133	−1.58817	−0.794086	−	0.607806i	$-0.792050\pi$
$619$	−39.5133	−1.58817	−0.794086	+	0.607806i	$0.792050\pi$
$620$	0	0
$621$	−2.23004	−0.0894885
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−28.0420	−1.12168
$626$	0	0
$627$	3.06956	0.122587
$628$	0	0
$629$	−1.30790	−0.0521495
$630$	0	0
$631$	−20.7060	−0.824293	−0.412146	−	0.911118i	$-0.635221\pi$
$631$	−20.7060	−0.824293	−0.412146	+	0.911118i	$0.635221\pi$
$632$	0	0
$633$	−2.41579	−0.0960190
$634$	0	0
$635$	7.27895	0.288857
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−6.98915	−0.276486
$640$	0	0
$641$	25.7099	1.01548	0.507741	−	0.861510i	$-0.330481\pi$
$641$	25.7099	1.01548	0.507741	+	0.861510i	$0.330481\pi$
$642$	0	0
$643$	22.1357	0.872947	0.436474	−	0.899717i	$-0.356227\pi$
$643$	22.1357	0.872947	0.436474	+	0.899717i	$0.356227\pi$
$644$	0	0
$645$	0.213566	0.00840916
$646$	0	0
$647$	2.96781	0.116677	0.0583384	−	0.998297i	$-0.481420\pi$
$647$	2.96781	0.116677	0.0583384	+	0.998297i	$0.481420\pi$
$648$	0	0
$649$	−45.3319	−1.77943
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.5485	0.686725	0.343362	−	0.939203i	$-0.388434\pi$
$653$	17.5485	0.686725	0.343362	+	0.939203i	$0.388434\pi$
$654$	0	0
$655$	−22.0869	−0.863008
$656$	0	0
$657$	19.2043	0.749231
$658$	0	0
$659$	−42.4886	−1.65512	−0.827560	−	0.561378i	$-0.810272\pi$
$659$	−42.4886	−1.65512	−0.827560	+	0.561378i	$0.810272\pi$
$660$	0	0
$661$	6.53462	0.254167	0.127084	−	0.991892i	$-0.459438\pi$
$661$	6.53462	0.254167	0.127084	+	0.991892i	$0.459438\pi$
$662$	0	0
$663$	0.361964	0.0140575
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.10427	0.0427573
$668$	0	0
$669$	−5.42617	−0.209788
$670$	0	0
$671$	26.4322	1.02040
$672$	0	0
$673$	23.0722	0.889367	0.444684	−	0.895688i	$-0.353316\pi$
$673$	23.0722	0.889367	0.444684	+	0.895688i	$0.353316\pi$
$674$	0	0
$675$	−1.19550	−0.0460146
$676$	0	0
$677$	−4.35184	−0.167255	−0.0836274	−	0.996497i	$-0.526651\pi$
$677$	−4.35184	−0.167255	−0.0836274	+	0.996497i	$0.526651\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	6.39973	0.245238
$682$	0	0
$683$	47.1520	1.80422	0.902110	−	0.431507i	$-0.142018\pi$
$683$	47.1520	1.80422	0.902110	+	0.431507i	$0.142018\pi$
$684$	0	0
$685$	−29.2185	−1.11638
$686$	0	0
$687$	3.84024	0.146514
$688$	0	0
$689$	−62.8481	−2.39432
$690$	0	0
$691$	−15.4736	−0.588642	−0.294321	−	0.955707i	$-0.595093\pi$
$691$	−15.4736	−0.588642	−0.294321	+	0.955707i	$0.595093\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	18.1643	0.689011
$696$	0	0
$697$	0.225296	0.00853370
$698$	0	0
$699$	−1.03501	−0.0391475
$700$	0	0
$701$	−20.7147	−0.782385	−0.391192	−	0.920309i	$-0.627937\pi$
$701$	−20.7147	−0.782385	−0.391192	+	0.920309i	$0.627937\pi$
$702$	0	0
$703$	10.9321	0.412311
$704$	0	0
$705$	−0.317956	−0.0119749
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−18.0362	−0.677365	−0.338683	−	0.940901i	$-0.609981\pi$
$709$	−18.0362	−0.677365	−0.338683	+	0.940901i	$0.609981\pi$
$710$	0	0
$711$	10.4030	0.390145
$712$	0	0
$713$	6.18894	0.231778
$714$	0	0
$715$	77.0827	2.88273
$716$	0	0
$717$	3.38877	0.126556
$718$	0	0
$719$	22.5833	0.842214	0.421107	−	0.907011i	$-0.361642\pi$
$719$	22.5833	0.842214	0.421107	+	0.907011i	$0.361642\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−5.53560	−0.205871
$724$	0	0
$725$	0.591982	0.0219857
$726$	0	0
$727$	15.3118	0.567882	0.283941	−	0.958842i	$-0.408358\pi$
$727$	15.3118	0.567882	0.283941	+	0.958842i	$0.408358\pi$
$728$	0	0
$729$	−22.7148	−0.841289
$730$	0	0
$731$	0.0706797	0.00261418
$732$	0	0
$733$	−18.2825	−0.675279	−0.337639	−	0.941276i	$-0.609628\pi$
$733$	−18.2825	−0.675279	−0.337639	+	0.941276i	$0.609628\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.07050	0.0394325
$738$	0	0
$739$	8.46264	0.311303	0.155652	−	0.987812i	$-0.450252\pi$
$739$	8.46264	0.311303	0.155652	+	0.987812i	$0.450252\pi$
$740$	0	0
$741$	−3.02546	−0.111143
$742$	0	0
$743$	28.7597	1.05509	0.527545	−	0.849527i	$-0.323113\pi$
$743$	28.7597	1.05509	0.527545	+	0.849527i	$0.323113\pi$
$744$	0	0
$745$	15.2610	0.559119
$746$	0	0
$747$	42.2042	1.54417
$748$	0	0
$749$	0	0
$750$	0	0
$751$	33.0211	1.20496	0.602478	−	0.798135i	$-0.294180\pi$
$751$	33.0211	1.20496	0.602478	+	0.798135i	$0.294180\pi$
$752$	0	0
$753$	2.43536	0.0887495
$754$	0	0
$755$	12.0677	0.439190
$756$	0	0
$757$	−9.70050	−0.352571	−0.176285	−	0.984339i	$-0.556408\pi$
$757$	−9.70050	−0.352571	−0.176285	+	0.984339i	$0.556408\pi$
$758$	0	0
$759$	2.15556	0.0782418
$760$	0	0
$761$	45.3326	1.64331	0.821653	−	0.569988i	$-0.193052\pi$
$761$	45.3326	1.64331	0.821653	+	0.569988i	$0.193052\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−1.57123	−0.0568078
$766$	0	0
$767$	44.6806	1.61332
$768$	0	0
$769$	−54.9681	−1.98220	−0.991100	−	0.133121i	$-0.957500\pi$
$769$	−54.9681	−1.98220	−0.991100	+	0.133121i	$0.957500\pi$
$770$	0	0
$771$	−4.04413	−0.145646
$772$	0	0
$773$	33.7153	1.21265	0.606327	−	0.795215i	$-0.292642\pi$
$773$	33.7153	1.21265	0.606327	+	0.795215i	$0.292642\pi$
$774$	0	0
$775$	3.31781	0.119179
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−1.88313	−0.0674701
$780$	0	0
$781$	13.6993	0.490200
$782$	0	0
$783$	−1.40818	−0.0503244
$784$	0	0
$785$	12.9486	0.462155
$786$	0	0
$787$	30.6525	1.09264	0.546322	−	0.837575i	$-0.316027\pi$
$787$	30.6525	1.09264	0.546322	+	0.837575i	$0.316027\pi$
$788$	0	0
$789$	6.58309	0.234364
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−26.0524	−0.925148
$794$	0	0
$795$	−7.58728	−0.269093
$796$	0	0
$797$	33.9994	1.20432	0.602160	−	0.798375i	$-0.294307\pi$
$797$	33.9994	1.20432	0.602160	+	0.798375i	$0.294307\pi$
$798$	0	0
$799$	−0.105227	−0.00372268
$800$	0	0
$801$	23.7728	0.839971
$802$	0	0
$803$	−37.6420	−1.32836
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.29044	−0.0806272
$808$	0	0
$809$	20.7327	0.728924	0.364462	−	0.931218i	$-0.381253\pi$
$809$	20.7327	0.728924	0.364462	+	0.931218i	$0.381253\pi$
$810$	0	0
$811$	−10.6026	−0.372308	−0.186154	−	0.982521i	$-0.559602\pi$
$811$	−10.6026	−0.372308	−0.186154	+	0.982521i	$0.559602\pi$
$812$	0	0
$813$	4.00709	0.140535
$814$	0	0
$815$	−17.6770	−0.619199
$816$	0	0
$817$	−0.590774	−0.0206686
$818$	0	0
$819$	0	0
$820$	0	0
$821$	17.4066	0.607495	0.303748	−	0.952752i	$-0.401762\pi$
$821$	17.4066	0.607495	0.303748	+	0.952752i	$0.401762\pi$
$822$	0	0
$823$	15.4934	0.540068	0.270034	−	0.962851i	$-0.412965\pi$
$823$	15.4934	0.540068	0.270034	+	0.962851i	$0.412965\pi$
$824$	0	0
$825$	1.15557	0.0402316
$826$	0	0
$827$	−31.9844	−1.11221	−0.556103	−	0.831113i	$-0.687704\pi$
$827$	−31.9844	−1.11221	−0.556103	+	0.831113i	$0.687704\pi$
$828$	0	0
$829$	−52.1808	−1.81231	−0.906157	−	0.422942i	$-0.860998\pi$
$829$	−52.1808	−1.81231	−0.906157	+	0.422942i	$0.860998\pi$
$830$	0	0
$831$	0.829457	0.0287735
$832$	0	0
$833$	0	0
$834$	0	0
$835$	16.8635	0.583587
$836$	0	0
$837$	−7.89228	−0.272797
$838$	0	0
$839$	24.0140	0.829054	0.414527	−	0.910037i	$-0.363947\pi$
$839$	24.0140	0.829054	0.414527	+	0.910037i	$0.363947\pi$
$840$	0	0
$841$	−28.3027	−0.975955
$842$	0	0
$843$	−2.46085	−0.0847563
$844$	0	0
$845$	−44.9138	−1.54508
$846$	0	0
$847$	0	0
$848$	0	0
$849$	3.88717	0.133407
$850$	0	0
$851$	7.67689	0.263160
$852$	0	0
$853$	−20.5132	−0.702359	−0.351179	−	0.936308i	$-0.614219\pi$
$853$	−20.5132	−0.702359	−0.351179	+	0.936308i	$0.614219\pi$
$854$	0	0
$855$	13.1330	0.449140
$856$	0	0
$857$	4.73663	0.161800	0.0809001	−	0.996722i	$-0.474221\pi$
$857$	4.73663	0.161800	0.0809001	+	0.996722i	$0.474221\pi$
$858$	0	0
$859$	28.8070	0.982881	0.491441	−	0.870911i	$-0.336470\pi$
$859$	28.8070	0.982881	0.491441	+	0.870911i	$0.336470\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−27.5865	−0.939056	−0.469528	−	0.882918i	$-0.655576\pi$
$863$	−27.5865	−0.939056	−0.469528	+	0.882918i	$0.655576\pi$
$864$	0	0
$865$	21.2881	0.723817
$866$	0	0
$867$	−4.82908	−0.164004
$868$	0	0
$869$	−20.3908	−0.691711
$870$	0	0
$871$	−1.05512	−0.0357514
$872$	0	0
$873$	−51.7386	−1.75109
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.4167	1.39854	0.699271	−	0.714857i	$-0.253508\pi$
$877$	41.4167	1.39854	0.699271	+	0.714857i	$0.253508\pi$
$878$	0	0
$879$	−0.452826	−0.0152734
$880$	0	0
$881$	16.5018	0.555960	0.277980	−	0.960587i	$-0.410335\pi$
$881$	16.5018	0.555960	0.277980	+	0.960587i	$0.410335\pi$
$882$	0	0
$883$	−54.4787	−1.83335	−0.916676	−	0.399630i	$-0.869138\pi$
$883$	−54.4787	−1.83335	−0.916676	+	0.399630i	$0.869138\pi$
$884$	0	0
$885$	5.39402	0.181318
$886$	0	0
$887$	17.5866	0.590499	0.295250	−	0.955420i	$-0.404597\pi$
$887$	17.5866	0.590499	0.295250	+	0.955420i	$0.404597\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	47.3481	1.58622
$892$	0	0
$893$	0.879540	0.0294327
$894$	0	0
$895$	18.9202	0.632434
$896$	0	0
$897$	−2.12459	−0.0709379
$898$	0	0
$899$	3.90808	0.130342
$900$	0	0
$901$	−2.51101	−0.0836539
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0.435165	0.0144654
$906$	0	0
$907$	−27.1953	−0.903004	−0.451502	−	0.892270i	$-0.649112\pi$
$907$	−27.1953	−0.903004	−0.451502	+	0.892270i	$0.649112\pi$
$908$	0	0
$909$	28.0155	0.929215
$910$	0	0
$911$	25.7790	0.854097	0.427048	−	0.904229i	$-0.359553\pi$
$911$	25.7790	0.854097	0.427048	+	0.904229i	$0.359553\pi$
$912$	0	0
$913$	−82.7237	−2.73775
$914$	0	0
$915$	−3.14515	−0.103975
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−19.6407	−0.647885	−0.323943	−	0.946077i	$-0.605009\pi$
$919$	−19.6407	−0.647885	−0.323943	+	0.946077i	$0.605009\pi$
$920$	0	0
$921$	−6.75313	−0.222523
$922$	0	0
$923$	−13.5025	−0.444440
$924$	0	0
$925$	4.11548	0.135316
$926$	0	0
$927$	−18.7764	−0.616696
$928$	0	0
$929$	−30.4274	−0.998290	−0.499145	−	0.866519i	$-0.666352\pi$
$929$	−30.4274	−0.998290	−0.499145	+	0.866519i	$0.666352\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−8.03578	−0.263080
$934$	0	0
$935$	3.07973	0.100718
$936$	0	0
$937$	29.4086	0.960737	0.480368	−	0.877067i	$-0.340503\pi$
$937$	29.4086	0.960737	0.480368	+	0.877067i	$0.340503\pi$
$938$	0	0
$939$	−3.85970	−0.125957
$940$	0	0
$941$	32.3930	1.05598	0.527991	−	0.849250i	$-0.322945\pi$
$941$	32.3930	1.05598	0.527991	+	0.849250i	$0.322945\pi$
$942$	0	0
$943$	−1.32240	−0.0430633
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.3938	−1.05266	−0.526329	−	0.850281i	$-0.676432\pi$
$947$	−32.3938	−1.05266	−0.526329	+	0.850281i	$0.676432\pi$
$948$	0	0
$949$	37.1012	1.20436
$950$	0	0
$951$	1.16100	0.0376479
$952$	0	0
$953$	−19.3888	−0.628065	−0.314033	−	0.949412i	$-0.601680\pi$
$953$	−19.3888	−0.628065	−0.314033	+	0.949412i	$0.601680\pi$
$954$	0	0
$955$	37.5263	1.21432
$956$	0	0
$957$	1.36115	0.0439998
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−9.09688	−0.293448
$962$	0	0
$963$	39.9127	1.28617
$964$	0	0
$965$	20.1794	0.649599
$966$	0	0
$967$	18.8317	0.605587	0.302794	−	0.953056i	$-0.402081\pi$
$967$	18.8317	0.605587	0.302794	+	0.953056i	$0.402081\pi$
$968$	0	0
$969$	−0.120878	−0.00388316
$970$	0	0
$971$	−27.3721	−0.878414	−0.439207	−	0.898386i	$-0.644741\pi$
$971$	−27.3721	−0.878414	−0.439207	+	0.898386i	$0.644741\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.13896	−0.0364760
$976$	0	0
$977$	26.5113	0.848172	0.424086	−	0.905622i	$-0.360595\pi$
$977$	26.5113	0.848172	0.424086	+	0.905622i	$0.360595\pi$
$978$	0	0
$979$	−46.5967	−1.48924
$980$	0	0
$981$	40.0586	1.27897
$982$	0	0
$983$	37.4952	1.19591	0.597956	−	0.801529i	$-0.295980\pi$
$983$	37.4952	1.19591	0.597956	+	0.801529i	$0.295980\pi$
$984$	0	0
$985$	−39.7219	−1.26565
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.414863	−0.0131919
$990$	0	0
$991$	28.4033	0.902262	0.451131	−	0.892458i	$-0.351021\pi$
$991$	28.4033	0.902262	0.451131	+	0.892458i	$0.351021\pi$
$992$	0	0
$993$	−8.50520	−0.269904
$994$	0	0
$995$	56.2775	1.78412
$996$	0	0
$997$	30.3006	0.959629	0.479815	−	0.877370i	$-0.340704\pi$
$997$	30.3006	0.959629	0.479815	+	0.877370i	$0.340704\pi$
$998$	0	0
$999$	−9.78975	−0.309734

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.o.1.6	✓	10
7.6	odd	2		8036.2.a.p.1.5	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8036.2.a.o.1.6	✓	10	1.1	even	1	trivial
8036.2.a.p.1.5	yes	10	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)

\(f(q)\)	\(=\)	\(q+0.284914 q^{3} -2.38934 q^{5} -2.91882 q^{9} +O(q^{10})\)	\(q+0.284914 q^{3} -2.38934 q^{5} -2.91882 q^{9} +5.72114 q^{11} -5.63894 q^{13} -0.680755 q^{15} -0.225296 q^{17} +1.88313 q^{19} +1.32240 q^{23} +0.708922 q^{25} -1.68636 q^{27} +0.835045 q^{29} +4.68008 q^{31} +1.63003 q^{33} +5.80526 q^{37} -1.60661 q^{39} -1.00000 q^{41} -0.313719 q^{43} +6.97405 q^{45} +0.467063 q^{47} -0.0641900 q^{51} +11.1454 q^{53} -13.6697 q^{55} +0.536530 q^{57} -7.92358 q^{59} +4.62009 q^{61} +13.4733 q^{65} +0.187114 q^{67} +0.376771 q^{69} +2.39451 q^{71} -6.57946 q^{73} +0.201982 q^{75} -3.56412 q^{79} +8.27600 q^{81} -14.4593 q^{83} +0.538308 q^{85} +0.237916 q^{87} -8.14466 q^{89} +1.33342 q^{93} -4.49943 q^{95} +17.7258 q^{97} -16.6990 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9	\( 10 q - 2 q^{3} - 4 q^{5} + 10 q^{9} - 6 q^{13} - 4 q^{15} - 12 q^{17} - 8 q^{19} + 4 q^{23} + 16 q^{25} - 8 q^{27} + 2 q^{29} - 8 q^{31} + 6 q^{33} - 2 q^{37} - 2 q^{39} - 10 q^{41} - 2 q^{43} - 44 q^{45} + 14 q^{47} + 14 q^{51} + 8 q^{53} - 8 q^{55} - 10 q^{57} - 24 q^{59} - 14 q^{61} + 2 q^{65} - 8 q^{67} - 16 q^{69} + 10 q^{71} - 44 q^{73} + 50 q^{75} + 10 q^{79} - 14 q^{81} - 20 q^{83} + 8 q^{85} - 20 q^{87} - 6 q^{89} + 8 q^{93} + 4 q^{95} - 46 q^{97} + 2 q^{99}+O(q^{100}) \) 10 * q - 2 * q^3 - 4 * q^5 + 10 * q^9 - 6 * q^13 - 4 * q^15 - 12 * q^17 - 8 * q^19 + 4 * q^23 + 16 * q^25 - 8 * q^27 + 2 * q^29 - 8 * q^31 + 6 * q^33 - 2 * q^37 - 2 * q^39 - 10 * q^41 - 2 * q^43 - 44 * q^45 + 14 * q^47 + 14 * q^51 + 8 * q^53 - 8 * q^55 - 10 * q^57 - 24 * q^59 - 14 * q^61 + 2 * q^65 - 8 * q^67 - 16 * q^69 + 10 * q^71 - 44 * q^73 + 50 * q^75 + 10 * q^79 - 14 * q^81 - 20 * q^83 + 8 * q^85 - 20 * q^87 - 6 * q^89 + 8 * q^93 + 4 * q^95 - 46 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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