LMFDB - Embedded newform 6003.2.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.5744.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} - 2x + 1 $ x^4 - 5x^2 - 2x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.751024$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.00000 q^{4} +1.58049 q^{5} -3.08254 q^{7} +O(q^{10})$	$q-2.00000 q^{4} +1.58049 q^{5} -3.08254 q^{7} -6.03291 q^{11} -2.36988 q^{13} +4.00000 q^{16} -6.85002 q^{17} -5.13736 q^{19} -3.16098 q^{20} -1.00000 q^{23} -2.50205 q^{25} +6.16508 q^{28} -1.00000 q^{29} +0.0784427 q^{31} -4.87193 q^{35} +5.42641 q^{37} -7.95446 q^{41} -2.05653 q^{43} +12.0658 q^{44} -1.58049 q^{47} +2.50205 q^{49} +4.73975 q^{52} +5.00410 q^{53} -9.53496 q^{55} +0.756479 q^{59} -3.79110 q^{61} -8.00000 q^{64} -3.74557 q^{65} +6.16508 q^{67} +13.7000 q^{68} +14.1154 q^{71} -2.91746 q^{73} +10.2747 q^{76} +18.5967 q^{77} -14.1415 q^{79} +6.32196 q^{80} -5.48123 q^{83} -10.8264 q^{85} +13.5378 q^{89} +7.30524 q^{91} +2.00000 q^{92} -8.11954 q^{95} +2.53086 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) $ 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7	$ 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7} + 16 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{23} - 4 q^{25} + 4 q^{28} - 4 q^{29} + 2 q^{31} - 4 q^{35} + 4 q^{37} - 6 q^{41} - 2 q^{47} + 4 q^{49} + 8 q^{53} - 8 q^{55} + 22 q^{59} - 16 q^{61} - 32 q^{64} + 10 q^{65} + 4 q^{67} + 4 q^{68} + 22 q^{71} - 22 q^{73} - 8 q^{76} + 8 q^{77} - 20 q^{79} + 8 q^{80} + 10 q^{83} - 2 q^{85} + 14 q^{89} - 26 q^{91} + 8 q^{92} + 14 q^{95} - 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7 + 16 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^23 - 4 * q^25 + 4 * q^28 - 4 * q^29 + 2 * q^31 - 4 * q^35 + 4 * q^37 - 6 * q^41 - 2 * q^47 + 4 * q^49 + 8 * q^53 - 8 * q^55 + 22 * q^59 - 16 * q^61 - 32 * q^64 + 10 * q^65 + 4 * q^67 + 4 * q^68 + 22 * q^71 - 22 * q^73 - 8 * q^76 + 8 * q^77 - 20 * q^79 + 8 * q^80 + 10 * q^83 - 2 * q^85 + 14 * q^89 - 26 * q^91 + 8 * q^92 + 14 * q^95 - 8 * q^97

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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0	0
$3$	0	0
$4$	−2.00000	−1.00000
$5$	1.58049	0.706817	0.353409	−	0.935469i	$-0.385023\pi$
$5$	1.58049	0.706817	0.353409	+	0.935469i	$0.385023\pi$
$6$	0	0
$7$	−3.08254	−1.16509	−0.582545	−	0.812798i	$-0.697943\pi$
$7$	−3.08254	−1.16509	−0.582545	+	0.812798i	$0.697943\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−6.03291	−1.81899	−0.909495	−	0.415715i	$-0.863531\pi$
$11$	−6.03291	−1.81899	−0.909495	+	0.415715i	$0.863531\pi$
$12$	0	0
$13$	−2.36988	−0.657286	−0.328643	−	0.944454i	$-0.606591\pi$
$13$	−2.36988	−0.657286	−0.328643	+	0.944454i	$0.606591\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−6.85002	−1.66137	−0.830687	−	0.556740i	$-0.812052\pi$
$17$	−6.85002	−1.66137	−0.830687	+	0.556740i	$0.812052\pi$
$18$	0	0
$19$	−5.13736	−1.17859	−0.589295	−	0.807918i	$-0.700594\pi$
$19$	−5.13736	−1.17859	−0.589295	+	0.807918i	$0.700594\pi$
$20$	−3.16098	−0.706817
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−2.50205	−0.500410
$26$	0	0
$27$	0	0
$28$	6.16508	1.16509
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	0.0784427	0.0140887	0.00704436	−	0.999975i	$-0.497758\pi$
$31$	0.0784427	0.0140887	0.00704436	+	0.999975i	$0.497758\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.87193	−0.823506
$36$	0	0
$37$	5.42641	0.892097	0.446048	−	0.895009i	$-0.352831\pi$
$37$	5.42641	0.892097	0.446048	+	0.895009i	$0.352831\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.95446	−1.24228	−0.621139	−	0.783700i	$-0.713330\pi$
$41$	−7.95446	−1.24228	−0.621139	+	0.783700i	$0.713330\pi$
$42$	0	0
$43$	−2.05653	−0.313619	−0.156809	−	0.987629i	$-0.550121\pi$
$43$	−2.05653	−0.313619	−0.156809	+	0.987629i	$0.550121\pi$
$44$	12.0658	1.81899
$45$	0	0
$46$	0	0
$47$	−1.58049	−0.230538	−0.115269	−	0.993334i	$-0.536773\pi$
$47$	−1.58049	−0.230538	−0.115269	+	0.993334i	$0.536773\pi$
$48$	0	0
$49$	2.50205	0.357435
$50$	0	0
$51$	0	0
$52$	4.73975	0.657286
$53$	5.00410	0.687366	0.343683	−	0.939086i	$-0.388326\pi$
$53$	5.00410	0.687366	0.343683	+	0.939086i	$0.388326\pi$
$54$	0	0
$55$	−9.53496	−1.28569
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.756479	0.0984852	0.0492426	−	0.998787i	$-0.484319\pi$
$59$	0.756479	0.0984852	0.0492426	+	0.998787i	$0.484319\pi$
$60$	0	0
$61$	−3.79110	−0.485401	−0.242701	−	0.970101i	$-0.578033\pi$
$61$	−3.79110	−0.485401	−0.242701	+	0.970101i	$0.578033\pi$
$62$	0	0
$63$	0	0
$64$	−8.00000	−1.00000
$65$	−3.74557	−0.464581
$66$	0	0
$67$	6.16508	0.753184	0.376592	−	0.926379i	$-0.377096\pi$
$67$	6.16508	0.753184	0.376592	+	0.926379i	$0.377096\pi$
$68$	13.7000	1.66137
$69$	0	0
$70$	0	0
$71$	14.1154	1.67520	0.837598	−	0.546288i	$-0.183959\pi$
$71$	14.1154	1.67520	0.837598	+	0.546288i	$0.183959\pi$
$72$	0	0
$73$	−2.91746	−0.341463	−0.170731	−	0.985318i	$-0.554613\pi$
$73$	−2.91746	−0.341463	−0.170731	+	0.985318i	$0.554613\pi$
$74$	0	0
$75$	0	0
$76$	10.2747	1.17859
$77$	18.5967	2.11929
$78$	0	0
$79$	−14.1415	−1.59104	−0.795519	−	0.605929i	$-0.792802\pi$
$79$	−14.1415	−1.59104	−0.795519	+	0.605929i	$0.792802\pi$
$80$	6.32196	0.706817
$81$	0	0
$82$	0	0
$83$	−5.48123	−0.601643	−0.300821	−	0.953680i	$-0.597261\pi$
$83$	−5.48123	−0.601643	−0.300821	+	0.953680i	$0.597261\pi$
$84$	0	0
$85$	−10.8264	−1.17429
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.5378	1.43500	0.717500	−	0.696559i	$-0.245286\pi$
$89$	13.5378	1.43500	0.717500	+	0.696559i	$0.245286\pi$
$90$	0	0
$91$	7.30524	0.765797
$92$	2.00000	0.208514
$93$	0	0
$94$	0	0
$95$	−8.11954	−0.833048
$96$	0	0
$97$	2.53086	0.256970	0.128485	−	0.991711i	$-0.458989\pi$
$97$	2.53086	0.256970	0.128485	+	0.991711i	$0.458989\pi$
$98$	0	0
$99$	0	0
$100$	5.00410	0.500410
$101$	9.11545	0.907021	0.453510	−	0.891251i	$-0.350171\pi$
$101$	9.11545	0.907021	0.453510	+	0.891251i	$0.350171\pi$
$102$	0	0
$103$	−5.74385	−0.565958	−0.282979	−	0.959126i	$-0.591323\pi$
$103$	−5.74385	−0.565958	−0.282979	+	0.959126i	$0.591323\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.8032	1.52775	0.763876	−	0.645363i	$-0.223294\pi$
$107$	15.8032	1.52775	0.763876	+	0.645363i	$0.223294\pi$
$108$	0	0
$109$	−5.81410	−0.556890	−0.278445	−	0.960452i	$-0.589819\pi$
$109$	−5.81410	−0.556890	−0.278445	+	0.960452i	$0.589819\pi$
$110$	0	0
$111$	0	0
$112$	−12.3302	−1.16509
$113$	−5.53496	−0.520685	−0.260342	−	0.965516i	$-0.583835\pi$
$113$	−5.53496	−0.520685	−0.260342	+	0.965516i	$0.583835\pi$
$114$	0	0
$115$	−1.58049	−0.147382
$116$	2.00000	0.185695
$117$	0	0
$118$	0	0
$119$	21.1154	1.93565
$120$	0	0
$121$	25.3960	2.30872
$122$	0	0
$123$	0	0
$124$	−0.156885	−0.0140887
$125$	−11.8569	−1.06052
$126$	0	0
$127$	−0.543487	−0.0482267	−0.0241133	−	0.999709i	$-0.507676\pi$
$127$	−0.543487	−0.0482267	−0.0241133	+	0.999709i	$0.507676\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−16.2227	−1.41738	−0.708692	−	0.705518i	$-0.750714\pi$
$131$	−16.2227	−1.41738	−0.708692	+	0.705518i	$0.750714\pi$
$132$	0	0
$133$	15.8361	1.37316
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.0548158	0.00468323	0.00234161	−	0.999997i	$-0.499255\pi$
$137$	0.0548158	0.00468323	0.00234161	+	0.999997i	$0.499255\pi$
$138$	0	0
$139$	17.5679	1.49009	0.745043	−	0.667016i	$-0.232429\pi$
$139$	17.5679	1.49009	0.745043	+	0.667016i	$0.232429\pi$
$140$	9.74385	0.823506
$141$	0	0
$142$	0	0
$143$	14.2972	1.19560
$144$	0	0
$145$	−1.58049	−0.131253
$146$	0	0
$147$	0	0
$148$	−10.8528	−0.892097
$149$	14.3740	1.17756	0.588781	−	0.808293i	$-0.299608\pi$
$149$	14.3740	1.17756	0.588781	+	0.808293i	$0.299608\pi$
$150$	0	0
$151$	−14.1939	−1.15508	−0.577541	−	0.816362i	$-0.695988\pi$
$151$	−14.1939	−1.15508	−0.577541	+	0.816362i	$0.695988\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.123978	0.00995815
$156$	0	0
$157$	−5.96546	−0.476096	−0.238048	−	0.971253i	$-0.576508\pi$
$157$	−5.96546	−0.476096	−0.238048	+	0.971253i	$0.576508\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.08254	0.242938
$162$	0	0
$163$	−22.0220	−1.72490	−0.862448	−	0.506146i	$-0.831070\pi$
$163$	−22.0220	−1.72490	−0.862448	+	0.506146i	$0.831070\pi$
$164$	15.9089	1.24228
$165$	0	0
$166$	0	0
$167$	−19.1333	−1.48058	−0.740291	−	0.672286i	$-0.765313\pi$
$167$	−19.1333	−1.48058	−0.740291	+	0.672286i	$0.765313\pi$
$168$	0	0
$169$	−7.38368	−0.567976
$170$	0	0
$171$	0	0
$172$	4.11307	0.313619
$173$	−5.03119	−0.382514	−0.191257	−	0.981540i	$-0.561256\pi$
$173$	−5.03119	−0.382514	−0.191257	+	0.981540i	$0.561256\pi$
$174$	0	0
$175$	7.71266	0.583022
$176$	−24.1316	−1.81899
$177$	0	0
$178$	0	0
$179$	−3.93946	−0.294449	−0.147224	−	0.989103i	$-0.547034\pi$
$179$	−3.93946	−0.294449	−0.147224	+	0.989103i	$0.547034\pi$
$180$	0	0
$181$	−3.32606	−0.247224	−0.123612	−	0.992331i	$-0.539448\pi$
$181$	−3.32606	−0.247224	−0.123612	+	0.992331i	$0.539448\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	8.57639	0.630549
$186$	0	0
$187$	41.3255	3.02202
$188$	3.16098	0.230538
$189$	0	0
$190$	0	0
$191$	10.5610	0.764164	0.382082	−	0.924128i	$-0.375207\pi$
$191$	10.5610	0.764164	0.382082	+	0.924128i	$0.375207\pi$
$192$	0	0
$193$	8.00819	0.576442	0.288221	−	0.957564i	$-0.406936\pi$
$193$	8.00819	0.576442	0.288221	+	0.957564i	$0.406936\pi$
$194$	0	0
$195$	0	0
$196$	−5.00410	−0.357435
$197$	−18.8443	−1.34260	−0.671300	−	0.741186i	$-0.734264\pi$
$197$	−18.8443	−1.34260	−0.671300	+	0.741186i	$0.734264\pi$
$198$	0	0
$199$	−0.807829	−0.0572655	−0.0286327	−	0.999590i	$-0.509115\pi$
$199$	−0.807829	−0.0572655	−0.0286327	+	0.999590i	$0.509115\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	3.08254	0.216352
$204$	0	0
$205$	−12.5720	−0.878064
$206$	0	0
$207$	0	0
$208$	−9.47951	−0.657286
$209$	30.9932	2.14384
$210$	0	0
$211$	−16.2517	−1.11881	−0.559407	−	0.828893i	$-0.688971\pi$
$211$	−16.2517	−1.11881	−0.559407	+	0.828893i	$0.688971\pi$
$212$	−10.0082	−0.687366
$213$	0	0
$214$	0	0
$215$	−3.25033	−0.221671
$216$	0	0
$217$	−0.241803	−0.0164146
$218$	0	0
$219$	0	0
$220$	19.0699	1.28569
$221$	16.2337	1.09200
$222$	0	0
$223$	9.39869	0.629383	0.314691	−	0.949194i	$-0.398099\pi$
$223$	9.39869	0.629383	0.314691	+	0.949194i	$0.398099\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−19.9145	−1.32177	−0.660887	−	0.750485i	$-0.729820\pi$
$227$	−19.9145	−1.32177	−0.660887	+	0.750485i	$0.729820\pi$
$228$	0	0
$229$	13.9682	0.923043	0.461522	−	0.887129i	$-0.347304\pi$
$229$	13.9682	0.923043	0.461522	+	0.887129i	$0.347304\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	24.1813	1.58417	0.792084	−	0.610413i	$-0.208996\pi$
$233$	24.1813	1.58417	0.792084	+	0.610413i	$0.208996\pi$
$234$	0	0
$235$	−2.49795	−0.162948
$236$	−1.51296	−0.0984852
$237$	0	0
$238$	0	0
$239$	6.94066	0.448954	0.224477	−	0.974479i	$-0.427933\pi$
$239$	6.94066	0.448954	0.224477	+	0.974479i	$0.427933\pi$
$240$	0	0
$241$	−8.10963	−0.522387	−0.261194	−	0.965286i	$-0.584116\pi$
$241$	−8.10963	−0.522387	−0.261194	+	0.965286i	$0.584116\pi$
$242$	0	0
$243$	0	0
$244$	7.58221	0.485401
$245$	3.95446	0.252641
$246$	0	0
$247$	12.1749	0.774670
$248$	0	0
$249$	0	0
$250$	0	0
$251$	16.7178	1.05522	0.527611	−	0.849486i	$-0.323088\pi$
$251$	16.7178	1.05522	0.527611	+	0.849486i	$0.323088\pi$
$252$	0	0
$253$	6.03291	0.379286
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−5.06410	−0.315890	−0.157945	−	0.987448i	$-0.550487\pi$
$257$	−5.06410	−0.315890	−0.157945	+	0.987448i	$0.550487\pi$
$258$	0	0
$259$	−16.7271	−1.03937
$260$	7.49114	0.464581
$261$	0	0
$262$	0	0
$263$	−7.51585	−0.463447	−0.231724	−	0.972782i	$-0.574437\pi$
$263$	−7.51585	−0.463447	−0.231724	+	0.972782i	$0.574437\pi$
$264$	0	0
$265$	7.90893	0.485842
$266$	0	0
$267$	0	0
$268$	−12.3302	−0.753184
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	−30.4016	−1.84676	−0.923382	−	0.383882i	$-0.874587\pi$
$271$	−30.4016	−1.84676	−0.923382	+	0.383882i	$0.874587\pi$
$272$	−27.4001	−1.66137
$273$	0	0
$274$	0	0
$275$	15.0946	0.910240
$276$	0	0
$277$	−1.57196	−0.0944499	−0.0472250	−	0.998884i	$-0.515038\pi$
$277$	−1.57196	−0.0944499	−0.0472250	+	0.998884i	$0.515038\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.0179	0.895893	0.447946	−	0.894060i	$-0.352156\pi$
$281$	15.0179	0.895893	0.447946	+	0.894060i	$0.352156\pi$
$282$	0	0
$283$	0.0866358	0.00514997	0.00257498	−	0.999997i	$-0.499180\pi$
$283$	0.0866358	0.00514997	0.00257498	+	0.999997i	$0.499180\pi$
$284$	−28.2309	−1.67520
$285$	0	0
$286$	0	0
$287$	24.5199	1.44737
$288$	0	0
$289$	29.9227	1.76016
$290$	0	0
$291$	0	0
$292$	5.83492	0.341463
$293$	23.0809	1.34840	0.674201	−	0.738548i	$-0.264488\pi$
$293$	23.0809	1.34840	0.674201	+	0.738548i	$0.264488\pi$
$294$	0	0
$295$	1.19561	0.0696110
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.36988	0.137054
$300$	0	0
$301$	6.33935	0.365394
$302$	0	0
$303$	0	0
$304$	−20.5494	−1.17859
$305$	−5.99181	−0.343090
$306$	0	0
$307$	−20.8956	−1.19258	−0.596289	−	0.802770i	$-0.703359\pi$
$307$	−20.8956	−1.19258	−0.596289	+	0.802770i	$0.703359\pi$
$308$	−37.1933	−2.11929
$309$	0	0
$310$	0	0
$311$	4.85282	0.275178	0.137589	−	0.990489i	$-0.456065\pi$
$311$	4.85282	0.275178	0.137589	+	0.990489i	$0.456065\pi$
$312$	0	0
$313$	29.2004	1.65050	0.825251	−	0.564766i	$-0.191034\pi$
$313$	29.2004	1.65050	0.825251	+	0.564766i	$0.191034\pi$
$314$	0	0
$315$	0	0
$316$	28.2829	1.59104
$317$	−22.2512	−1.24975	−0.624875	−	0.780725i	$-0.714850\pi$
$317$	−22.2512	−1.24975	−0.624875	+	0.780725i	$0.714850\pi$
$318$	0	0
$319$	6.03291	0.337778
$320$	−12.6439	−0.706817
$321$	0	0
$322$	0	0
$323$	35.1910	1.95808
$324$	0	0
$325$	5.92955	0.328912
$326$	0	0
$327$	0	0
$328$	0	0
$329$	4.87193	0.268598
$330$	0	0
$331$	−17.4001	−0.956394	−0.478197	−	0.878253i	$-0.658710\pi$
$331$	−17.4001	−0.956394	−0.478197	+	0.878253i	$0.658710\pi$
$332$	10.9625	0.601643
$333$	0	0
$334$	0	0
$335$	9.74385	0.532363
$336$	0	0
$337$	−34.6141	−1.88555	−0.942776	−	0.333426i	$-0.891795\pi$
$337$	−34.6141	−1.88555	−0.942776	+	0.333426i	$0.891795\pi$
$338$	0	0
$339$	0	0
$340$	21.6528	1.17429
$341$	−0.473237	−0.0256272
$342$	0	0
$343$	13.8651	0.748646
$344$	0	0
$345$	0	0
$346$	0	0
$347$	6.08254	0.326528	0.163264	−	0.986582i	$-0.447798\pi$
$347$	6.08254	0.326528	0.163264	+	0.986582i	$0.447798\pi$
$348$	0	0
$349$	−20.4911	−1.09687	−0.548433	−	0.836195i	$-0.684775\pi$
$349$	−20.4911	−1.09687	−0.548433	+	0.836195i	$0.684775\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	16.4623	0.876201	0.438101	−	0.898926i	$-0.355651\pi$
$353$	16.4623	0.876201	0.438101	+	0.898926i	$0.355651\pi$
$354$	0	0
$355$	22.3093	1.18406
$356$	−27.0755	−1.43500
$357$	0	0
$358$	0	0
$359$	31.0398	1.63822	0.819109	−	0.573638i	$-0.194468\pi$
$359$	31.0398	1.63822	0.819109	+	0.573638i	$0.194468\pi$
$360$	0	0
$361$	7.39242	0.389075
$362$	0	0
$363$	0	0
$364$	−14.6105	−0.765797
$365$	−4.61102	−0.241352
$366$	0	0
$367$	6.90483	0.360429	0.180215	−	0.983627i	$-0.442321\pi$
$367$	6.90483	0.360429	0.180215	+	0.983627i	$0.442321\pi$
$368$	−4.00000	−0.208514
$369$	0	0
$370$	0	0
$371$	−15.4253	−0.800843
$372$	0	0
$373$	20.3093	1.05158	0.525789	−	0.850615i	$-0.323770\pi$
$373$	20.3093	1.05158	0.525789	+	0.850615i	$0.323770\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.36988	0.122055
$378$	0	0
$379$	−12.4414	−0.639073	−0.319536	−	0.947574i	$-0.603527\pi$
$379$	−12.4414	−0.639073	−0.319536	+	0.947574i	$0.603527\pi$
$380$	16.2391	0.833048
$381$	0	0
$382$	0	0
$383$	13.0535	0.667004	0.333502	−	0.942749i	$-0.391770\pi$
$383$	13.0535	0.667004	0.333502	+	0.942749i	$0.391770\pi$
$384$	0	0
$385$	29.3919	1.49795
$386$	0	0
$387$	0	0
$388$	−5.06172	−0.256970
$389$	−5.61231	−0.284555	−0.142278	−	0.989827i	$-0.545443\pi$
$389$	−5.61231	−0.284555	−0.142278	+	0.989827i	$0.545443\pi$
$390$	0	0
$391$	6.85002	0.346420
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−22.3504	−1.12457
$396$	0	0
$397$	3.27935	0.164586	0.0822929	−	0.996608i	$-0.473776\pi$
$397$	3.27935	0.164586	0.0822929	+	0.996608i	$0.473776\pi$
$398$	0	0
$399$	0	0
$400$	−10.0082	−0.500410
$401$	−21.9644	−1.09685	−0.548424	−	0.836200i	$-0.684772\pi$
$401$	−21.9644	−1.09685	−0.548424	+	0.836200i	$0.684772\pi$
$402$	0	0
$403$	−0.185900	−0.00926032
$404$	−18.2309	−0.907021
$405$	0	0
$406$	0	0
$407$	−32.7370	−1.62271
$408$	0	0
$409$	32.7409	1.61893	0.809467	−	0.587165i	$-0.199756\pi$
$409$	32.7409	1.61893	0.809467	+	0.587165i	$0.199756\pi$
$410$	0	0
$411$	0	0
$412$	11.4877	0.565958
$413$	−2.33188	−0.114744
$414$	0	0
$415$	−8.66303	−0.425251
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.1131	0.494056	0.247028	−	0.969008i	$-0.420546\pi$
$419$	10.1131	0.494056	0.247028	+	0.969008i	$0.420546\pi$
$420$	0	0
$421$	−19.5169	−0.951197	−0.475599	−	0.879662i	$-0.657769\pi$
$421$	−19.5169	−0.951197	−0.475599	+	0.879662i	$0.657769\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	17.1391	0.831367
$426$	0	0
$427$	11.6862	0.565536
$428$	−31.6064	−1.52775
$429$	0	0
$430$	0	0
$431$	25.2432	1.21592	0.607961	−	0.793967i	$-0.291988\pi$
$431$	25.2432	1.21592	0.607961	+	0.793967i	$0.291988\pi$
$432$	0	0
$433$	19.4330	0.933889	0.466945	−	0.884287i	$-0.345355\pi$
$433$	19.4330	0.933889	0.466945	+	0.884287i	$0.345355\pi$
$434$	0	0
$435$	0	0
$436$	11.6282	0.556890
$437$	5.13736	0.245753
$438$	0	0
$439$	10.0808	0.481131	0.240566	−	0.970633i	$-0.422667\pi$
$439$	10.0808	0.481131	0.240566	+	0.970633i	$0.422667\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−24.1277	−1.14634	−0.573172	−	0.819435i	$-0.694287\pi$
$443$	−24.1277	−1.14634	−0.573172	+	0.819435i	$0.694287\pi$
$444$	0	0
$445$	21.3963	1.01428
$446$	0	0
$447$	0	0
$448$	24.6603	1.16509
$449$	13.9042	0.656179	0.328089	−	0.944647i	$-0.393595\pi$
$449$	13.9042	0.656179	0.328089	+	0.944647i	$0.393595\pi$
$450$	0	0
$451$	47.9885	2.25969
$452$	11.0699	0.520685
$453$	0	0
$454$	0	0
$455$	11.5459	0.541279
$456$	0	0
$457$	−2.45242	−0.114719	−0.0573596	−	0.998354i	$-0.518268\pi$
$457$	−2.45242	−0.114719	−0.0573596	+	0.998354i	$0.518268\pi$
$458$	0	0
$459$	0	0
$460$	3.16098	0.147382
$461$	9.63983	0.448972	0.224486	−	0.974477i	$-0.427930\pi$
$461$	9.63983	0.448972	0.224486	+	0.974477i	$0.427930\pi$
$462$	0	0
$463$	−36.7467	−1.70777	−0.853883	−	0.520465i	$-0.825759\pi$
$463$	−36.7467	−1.70777	−0.853883	+	0.520465i	$0.825759\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	0	0
$467$	15.9885	0.739858	0.369929	−	0.929060i	$-0.379382\pi$
$467$	15.9885	0.739858	0.369929	+	0.929060i	$0.379382\pi$
$468$	0	0
$469$	−19.0041	−0.877528
$470$	0	0
$471$	0	0
$472$	0	0
$473$	12.4069	0.570469
$474$	0	0
$475$	12.8539	0.589778
$476$	−42.2309	−1.93565
$477$	0	0
$478$	0	0
$479$	−38.7174	−1.76904	−0.884521	−	0.466500i	$-0.845515\pi$
$479$	−38.7174	−1.76904	−0.884521	+	0.466500i	$0.845515\pi$
$480$	0	0
$481$	−12.8599	−0.586362
$482$	0	0
$483$	0	0
$484$	−50.7919	−2.30872
$485$	4.00000	0.181631
$486$	0	0
$487$	−34.2084	−1.55013	−0.775064	−	0.631882i	$-0.782283\pi$
$487$	−34.2084	−1.55013	−0.775064	+	0.631882i	$0.782283\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−15.2723	−0.689231	−0.344615	−	0.938744i	$-0.611991\pi$
$491$	−15.2723	−0.689231	−0.344615	+	0.938744i	$0.611991\pi$
$492$	0	0
$493$	6.85002	0.308509
$494$	0	0
$495$	0	0
$496$	0.313771	0.0140887
$497$	−43.5114	−1.95175
$498$	0	0
$499$	−32.6891	−1.46337	−0.731683	−	0.681645i	$-0.761265\pi$
$499$	−32.6891	−1.46337	−0.731683	+	0.681645i	$0.761265\pi$
$500$	23.7138	1.06052
$501$	0	0
$502$	0	0
$503$	−19.2651	−0.858988	−0.429494	−	0.903070i	$-0.641308\pi$
$503$	−19.2651	−0.858988	−0.429494	+	0.903070i	$0.641308\pi$
$504$	0	0
$505$	14.4069	0.641098
$506$	0	0
$507$	0	0
$508$	1.08697	0.0482267
$509$	3.58802	0.159036	0.0795182	−	0.996833i	$-0.474662\pi$
$509$	3.58802	0.159036	0.0795182	+	0.996833i	$0.474662\pi$
$510$	0	0
$511$	8.99319	0.397835
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.07810	−0.400029
$516$	0	0
$517$	9.53496	0.419347
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.9113	0.916141	0.458071	−	0.888916i	$-0.348541\pi$
$521$	20.9113	0.916141	0.458071	+	0.888916i	$0.348541\pi$
$522$	0	0
$523$	15.6607	0.684793	0.342396	−	0.939556i	$-0.388761\pi$
$523$	15.6607	0.684793	0.342396	+	0.939556i	$0.388761\pi$
$524$	32.4454	1.41738
$525$	0	0
$526$	0	0
$527$	−0.537334	−0.0234066
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	−31.6722	−1.37316
$533$	18.8511	0.816532
$534$	0	0
$535$	24.9768	1.07984
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−15.0946	−0.650172
$540$	0	0
$541$	−43.8213	−1.88402	−0.942012	−	0.335578i	$-0.891068\pi$
$541$	−43.8213	−1.88402	−0.942012	+	0.335578i	$0.891068\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−9.18913	−0.393619
$546$	0	0
$547$	−0.650741	−0.0278237	−0.0139118	−	0.999903i	$-0.504428\pi$
$547$	−0.650741	−0.0278237	−0.0139118	+	0.999903i	$0.504428\pi$
$548$	−0.109632	−0.00468323
$549$	0	0
$550$	0	0
$551$	5.13736	0.218859
$552$	0	0
$553$	43.5916	1.85370
$554$	0	0
$555$	0	0
$556$	−35.1357	−1.49009
$557$	20.1750	0.854842	0.427421	−	0.904053i	$-0.359422\pi$
$557$	20.1750	0.854842	0.427421	+	0.904053i	$0.359422\pi$
$558$	0	0
$559$	4.87373	0.206137
$560$	−19.4877	−0.823506
$561$	0	0
$562$	0	0
$563$	39.0644	1.64637	0.823184	−	0.567775i	$-0.192196\pi$
$563$	39.0644	1.64637	0.823184	+	0.567775i	$0.192196\pi$
$564$	0	0
$565$	−8.74795	−0.368029
$566$	0	0
$567$	0	0
$568$	0	0
$569$	17.9474	0.752396	0.376198	−	0.926539i	$-0.377231\pi$
$569$	17.9474	0.752396	0.376198	+	0.926539i	$0.377231\pi$
$570$	0	0
$571$	4.81820	0.201635	0.100818	−	0.994905i	$-0.467854\pi$
$571$	4.81820	0.201635	0.100818	+	0.994905i	$0.467854\pi$
$572$	−28.5945	−1.19560
$573$	0	0
$574$	0	0
$575$	2.50205	0.104343
$576$	0	0
$577$	29.4011	1.22398	0.611991	−	0.790865i	$-0.290369\pi$
$577$	29.4011	1.22398	0.611991	+	0.790865i	$0.290369\pi$
$578$	0	0
$579$	0	0
$580$	3.16098	0.131253
$581$	16.8961	0.700968
$582$	0	0
$583$	−30.1893	−1.25031
$584$	0	0
$585$	0	0
$586$	0	0
$587$	37.8697	1.56305	0.781524	−	0.623875i	$-0.214443\pi$
$587$	37.8697	1.56305	0.781524	+	0.623875i	$0.214443\pi$
$588$	0	0
$589$	−0.402988	−0.0166048
$590$	0	0
$591$	0	0
$592$	21.7056	0.892097
$593$	34.4867	1.41620	0.708100	−	0.706113i	$-0.249553\pi$
$593$	34.4867	1.41620	0.708100	+	0.706113i	$0.249553\pi$
$594$	0	0
$595$	33.3728	1.36815
$596$	−28.7479	−1.17756
$597$	0	0
$598$	0	0
$599$	35.1675	1.43690	0.718452	−	0.695577i	$-0.244851\pi$
$599$	35.1675	1.43690	0.718452	+	0.695577i	$0.244851\pi$
$600$	0	0
$601$	22.0714	0.900312	0.450156	−	0.892950i	$-0.351368\pi$
$601$	22.0714	0.900312	0.450156	+	0.892950i	$0.351368\pi$
$602$	0	0
$603$	0	0
$604$	28.3878	1.15508
$605$	40.1381	1.63185
$606$	0	0
$607$	−34.0532	−1.38218	−0.691088	−	0.722771i	$-0.742868\pi$
$607$	−34.0532	−1.38218	−0.691088	+	0.722771i	$0.742868\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.74557	0.151530
$612$	0	0
$613$	28.2447	1.14079	0.570396	−	0.821370i	$-0.306790\pi$
$613$	28.2447	1.14079	0.570396	+	0.821370i	$0.306790\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	23.4226	0.942959	0.471479	−	0.881877i	$-0.343720\pi$
$617$	23.4226	0.942959	0.471479	+	0.881877i	$0.343720\pi$
$618$	0	0
$619$	−5.66876	−0.227847	−0.113923	−	0.993490i	$-0.536342\pi$
$619$	−5.66876	−0.227847	−0.113923	+	0.993490i	$0.536342\pi$
$620$	−0.247956	−0.00995815
$621$	0	0
$622$	0	0
$623$	−41.7307	−1.67190
$624$	0	0
$625$	−6.22951	−0.249181
$626$	0	0
$627$	0	0
$628$	11.9309	0.476096
$629$	−37.1710	−1.48211
$630$	0	0
$631$	−17.9480	−0.714498	−0.357249	−	0.934009i	$-0.616285\pi$
$631$	−17.9480	−0.714498	−0.357249	+	0.934009i	$0.616285\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−0.858976	−0.0340874
$636$	0	0
$637$	−5.92955	−0.234937
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−37.0502	−1.46339	−0.731697	−	0.681630i	$-0.761271\pi$
$641$	−37.0502	−1.46339	−0.731697	+	0.681630i	$0.761271\pi$
$642$	0	0
$643$	15.5952	0.615013	0.307507	−	0.951546i	$-0.400505\pi$
$643$	15.5952	0.615013	0.307507	+	0.951546i	$0.400505\pi$
$644$	−6.16508	−0.242938
$645$	0	0
$646$	0	0
$647$	−0.957902	−0.0376590	−0.0188295	−	0.999823i	$-0.505994\pi$
$647$	−0.957902	−0.0376590	−0.0188295	+	0.999823i	$0.505994\pi$
$648$	0	0
$649$	−4.56377	−0.179144
$650$	0	0
$651$	0	0
$652$	44.0440	1.72490
$653$	14.8709	0.581944	0.290972	−	0.956732i	$-0.406021\pi$
$653$	14.8709	0.581944	0.290972	+	0.956732i	$0.406021\pi$
$654$	0	0
$655$	−25.6398	−1.00183
$656$	−31.8179	−1.24228
$657$	0	0
$658$	0	0
$659$	5.99937	0.233702	0.116851	−	0.993149i	$-0.462720\pi$
$659$	5.99937	0.233702	0.116851	+	0.993149i	$0.462720\pi$
$660$	0	0
$661$	22.1339	0.860909	0.430454	−	0.902612i	$-0.358353\pi$
$661$	22.1339	0.860909	0.430454	+	0.902612i	$0.358353\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	25.0288	0.970576
$666$	0	0
$667$	1.00000	0.0387202
$668$	38.2667	1.48058
$669$	0	0
$670$	0	0
$671$	22.8714	0.882940
$672$	0	0
$673$	−41.2309	−1.58933	−0.794667	−	0.607046i	$-0.792355\pi$
$673$	−41.2309	−1.58933	−0.794667	+	0.607046i	$0.792355\pi$
$674$	0	0
$675$	0	0
$676$	14.7674	0.567976
$677$	11.5761	0.444904	0.222452	−	0.974944i	$-0.428594\pi$
$677$	11.5761	0.444904	0.222452	+	0.974944i	$0.428594\pi$
$678$	0	0
$679$	−7.80147	−0.299393
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−43.3966	−1.66053	−0.830263	−	0.557372i	$-0.811810\pi$
$683$	−43.3966	−1.66053	−0.830263	+	0.557372i	$0.811810\pi$
$684$	0	0
$685$	0.0866358	0.00331019
$686$	0	0
$687$	0	0
$688$	−8.22614	−0.313619
$689$	−11.8591	−0.451796
$690$	0	0
$691$	−7.28754	−0.277231	−0.138616	−	0.990346i	$-0.544265\pi$
$691$	−7.28754	−0.277231	−0.138616	+	0.990346i	$0.544265\pi$
$692$	10.0624	0.382514
$693$	0	0
$694$	0	0
$695$	27.7658	1.05322
$696$	0	0
$697$	54.4882	2.06389
$698$	0	0
$699$	0	0
$700$	−15.4253	−0.583022
$701$	−50.4266	−1.90459	−0.952294	−	0.305184i	$-0.901282\pi$
$701$	−50.4266	−1.90459	−0.952294	+	0.305184i	$0.901282\pi$
$702$	0	0
$703$	−27.8774	−1.05142
$704$	48.2633	1.81899
$705$	0	0
$706$	0	0
$707$	−28.0987	−1.05676
$708$	0	0
$709$	50.9442	1.91325	0.956625	−	0.291322i	$-0.0940951\pi$
$709$	50.9442	1.91325	0.956625	+	0.291322i	$0.0940951\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.0784427	−0.00293770
$714$	0	0
$715$	22.5967	0.845068
$716$	7.87892	0.294449
$717$	0	0
$718$	0	0
$719$	−13.7468	−0.512668	−0.256334	−	0.966588i	$-0.582515\pi$
$719$	−13.7468	−0.512668	−0.256334	+	0.966588i	$0.582515\pi$
$720$	0	0
$721$	17.7056	0.659393
$722$	0	0
$723$	0	0
$724$	6.65212	0.247224
$725$	2.50205	0.0929237
$726$	0	0
$727$	−44.6736	−1.65685	−0.828426	−	0.560099i	$-0.810763\pi$
$727$	−44.6736	−1.65685	−0.828426	+	0.560099i	$0.810763\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	14.0873	0.521037
$732$	0	0
$733$	−8.27471	−0.305633	−0.152817	−	0.988255i	$-0.548834\pi$
$733$	−8.27471	−0.305633	−0.152817	+	0.988255i	$0.548834\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−37.1933	−1.37003
$738$	0	0
$739$	−43.6284	−1.60490	−0.802448	−	0.596722i	$-0.796469\pi$
$739$	−43.6284	−1.60490	−0.802448	+	0.596722i	$0.796469\pi$
$740$	−17.1528	−0.630549
$741$	0	0
$742$	0	0
$743$	−16.3815	−0.600980	−0.300490	−	0.953785i	$-0.597150\pi$
$743$	−16.3815	−0.600980	−0.300490	+	0.953785i	$0.597150\pi$
$744$	0	0
$745$	22.7179	0.832321
$746$	0	0
$747$	0	0
$748$	−82.6510	−3.02202
$749$	−48.7140	−1.77997
$750$	0	0
$751$	−16.7772	−0.612208	−0.306104	−	0.951998i	$-0.599026\pi$
$751$	−16.7772	−0.612208	−0.306104	+	0.951998i	$0.599026\pi$
$752$	−6.32196	−0.230538
$753$	0	0
$754$	0	0
$755$	−22.4333	−0.816432
$756$	0	0
$757$	30.6060	1.11239	0.556196	−	0.831051i	$-0.312260\pi$
$757$	30.6060	1.11239	0.556196	+	0.831051i	$0.312260\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−49.8457	−1.80690	−0.903452	−	0.428689i	$-0.858976\pi$
$761$	−49.8457	−1.80690	−0.903452	+	0.428689i	$0.858976\pi$
$762$	0	0
$763$	17.9222	0.648827
$764$	−21.1219	−0.764164
$765$	0	0
$766$	0	0
$767$	−1.79276	−0.0647329
$768$	0	0
$769$	37.8744	1.36579	0.682893	−	0.730519i	$-0.260722\pi$
$769$	37.8744	1.36579	0.682893	+	0.730519i	$0.260722\pi$
$770$	0	0
$771$	0	0
$772$	−16.0164	−0.576442
$773$	−23.3083	−0.838340	−0.419170	−	0.907908i	$-0.637679\pi$
$773$	−23.3083	−0.838340	−0.419170	+	0.907908i	$0.637679\pi$
$774$	0	0
$775$	−0.196267	−0.00705013
$776$	0	0
$777$	0	0
$778$	0	0
$779$	40.8649	1.46414
$780$	0	0
$781$	−85.1572	−3.04716
$782$	0	0
$783$	0	0
$784$	10.0082	0.357435
$785$	−9.42836	−0.336513
$786$	0	0
$787$	38.9212	1.38739	0.693696	−	0.720268i	$-0.255981\pi$
$787$	38.9212	1.38739	0.693696	+	0.720268i	$0.255981\pi$
$788$	37.6886	1.34260
$789$	0	0
$790$	0	0
$791$	17.0617	0.606645
$792$	0	0
$793$	8.98445	0.319047
$794$	0	0
$795$	0	0
$796$	1.61566	0.0572655
$797$	−28.8892	−1.02331	−0.511653	−	0.859192i	$-0.670967\pi$
$797$	−28.8892	−1.02331	−0.511653	+	0.859192i	$0.670967\pi$
$798$	0	0
$799$	10.8264	0.383010
$800$	0	0
$801$	0	0
$802$	0	0
$803$	17.6008	0.621118
$804$	0	0
$805$	4.87193	0.171713
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−31.3008	−1.10048	−0.550239	−	0.835007i	$-0.685463\pi$
$809$	−31.3008	−1.10048	−0.550239	+	0.835007i	$0.685463\pi$
$810$	0	0
$811$	−39.8370	−1.39886	−0.699432	−	0.714699i	$-0.746564\pi$
$811$	−39.8370	−1.39886	−0.699432	+	0.714699i	$0.746564\pi$
$812$	−6.16508	−0.216352
$813$	0	0
$814$	0	0
$815$	−34.8056	−1.21919
$816$	0	0
$817$	10.5651	0.369628
$818$	0	0
$819$	0	0
$820$	25.1439	0.878064
$821$	−34.1067	−1.19033	−0.595166	−	0.803603i	$-0.702914\pi$
$821$	−34.1067	−1.19033	−0.595166	+	0.803603i	$0.702914\pi$
$822$	0	0
$823$	−4.34152	−0.151336	−0.0756680	−	0.997133i	$-0.524109\pi$
$823$	−4.34152	−0.151336	−0.0756680	+	0.997133i	$0.524109\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.3533	−0.464339	−0.232169	−	0.972675i	$-0.574582\pi$
$827$	−13.3533	−0.464339	−0.232169	+	0.972675i	$0.574582\pi$
$828$	0	0
$829$	13.5973	0.472255	0.236127	−	0.971722i	$-0.424122\pi$
$829$	13.5973	0.472255	0.236127	+	0.971722i	$0.424122\pi$
$830$	0	0
$831$	0	0
$832$	18.9590	0.657286
$833$	−17.1391	−0.593834
$834$	0	0
$835$	−30.2401	−1.04650
$836$	−61.9864	−2.14384
$837$	0	0
$838$	0	0
$839$	34.0788	1.17653	0.588265	−	0.808668i	$-0.299811\pi$
$839$	34.0788	1.17653	0.588265	+	0.808668i	$0.299811\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	32.5034	1.11881
$845$	−11.6698	−0.401455
$846$	0	0
$847$	−78.2841	−2.68987
$848$	20.0164	0.687366
$849$	0	0
$850$	0	0
$851$	−5.42641	−0.186015
$852$	0	0
$853$	−34.8414	−1.19295	−0.596473	−	0.802633i	$-0.703432\pi$
$853$	−34.8414	−1.19295	−0.596473	+	0.802633i	$0.703432\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	12.3185	0.420793	0.210396	−	0.977616i	$-0.432525\pi$
$857$	12.3185	0.420793	0.210396	+	0.977616i	$0.432525\pi$
$858$	0	0
$859$	32.9580	1.12451	0.562257	−	0.826963i	$-0.309933\pi$
$859$	32.9580	1.12451	0.562257	+	0.826963i	$0.309933\pi$
$860$	6.50067	0.221671
$861$	0	0
$862$	0	0
$863$	−0.00343710	−0.000117000 0	−5.85001e−5	−	1.00000i	$-0.500019\pi$
$863$	−0.00343710	−0.000117000 0	−5.85001e−5		1.00000i	$0.500019\pi$
$864$	0	0
$865$	−7.95175	−0.270368
$866$	0	0
$867$	0	0
$868$	0.483605	0.0164146
$869$	85.3141	2.89408
$870$	0	0
$871$	−14.6105	−0.495057
$872$	0	0
$873$	0	0
$874$	0	0
$875$	36.5494	1.23560
$876$	0	0
$877$	56.4139	1.90496	0.952481	−	0.304599i	$-0.0985226\pi$
$877$	56.4139	1.90496	0.952481	+	0.304599i	$0.0985226\pi$
$878$	0	0
$879$	0	0
$880$	−38.1398	−1.28569
$881$	33.4554	1.12714	0.563571	−	0.826068i	$-0.309427\pi$
$881$	33.4554	1.12714	0.563571	+	0.826068i	$0.309427\pi$
$882$	0	0
$883$	−16.1830	−0.544601	−0.272300	−	0.962212i	$-0.587784\pi$
$883$	−16.1830	−0.544601	−0.272300	+	0.962212i	$0.587784\pi$
$884$	−32.4674	−1.09200
$885$	0	0
$886$	0	0
$887$	7.86901	0.264215	0.132108	−	0.991235i	$-0.457826\pi$
$887$	7.86901	0.264215	0.132108	+	0.991235i	$0.457826\pi$
$888$	0	0
$889$	1.67532	0.0561884
$890$	0	0
$891$	0	0
$892$	−18.7974	−0.629383
$893$	8.11954	0.271710
$894$	0	0
$895$	−6.22628	−0.208122
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.0784427	−0.00261621
$900$	0	0
$901$	−34.2781	−1.14197
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.25681	−0.174742
$906$	0	0
$907$	−56.0273	−1.86036	−0.930178	−	0.367109i	$-0.880348\pi$
$907$	−56.0273	−1.86036	−0.930178	+	0.367109i	$0.880348\pi$
$908$	39.8291	1.32177
$909$	0	0
$910$	0	0
$911$	6.10153	0.202153	0.101076	−	0.994879i	$-0.467771\pi$
$911$	6.10153	0.202153	0.101076	+	0.994879i	$0.467771\pi$
$912$	0	0
$913$	33.0677	1.09438
$914$	0	0
$915$	0	0
$916$	−27.9364	−0.923043
$917$	50.0071	1.65138
$918$	0	0
$919$	−22.9048	−0.755561	−0.377780	−	0.925895i	$-0.623313\pi$
$919$	−22.9048	−0.755561	−0.377780	+	0.925895i	$0.623313\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−33.4519	−1.10108
$924$	0	0
$925$	−13.5771	−0.446414
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−38.8265	−1.27386	−0.636928	−	0.770923i	$-0.719795\pi$
$929$	−38.8265	−1.27386	−0.636928	+	0.770923i	$0.719795\pi$
$930$	0	0
$931$	−12.8539	−0.421270
$932$	−48.3625	−1.58417
$933$	0	0
$934$	0	0
$935$	65.3146	2.13602
$936$	0	0
$937$	−19.2384	−0.628492	−0.314246	−	0.949342i	$-0.601752\pi$
$937$	−19.2384	−0.628492	−0.314246	+	0.949342i	$0.601752\pi$
$938$	0	0
$939$	0	0
$940$	4.99590	0.162948
$941$	35.9529	1.17203	0.586016	−	0.810300i	$-0.300696\pi$
$941$	35.9529	1.17203	0.586016	+	0.810300i	$0.300696\pi$
$942$	0	0
$943$	7.95446	0.259033
$944$	3.02592	0.0984852
$945$	0	0
$946$	0	0
$947$	55.9193	1.81713	0.908567	−	0.417740i	$-0.137178\pi$
$947$	55.9193	1.81713	0.908567	+	0.417740i	$0.137178\pi$
$948$	0	0
$949$	6.91402	0.224439
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.8943	−0.385293	−0.192646	−	0.981268i	$-0.561707\pi$
$953$	−11.8943	−0.385293	−0.192646	+	0.981268i	$0.561707\pi$
$954$	0	0
$955$	16.6915	0.540124
$956$	−13.8813	−0.448954
$957$	0	0
$958$	0	0
$959$	−0.168972	−0.00545638
$960$	0	0
$961$	−30.9938	−0.999802
$962$	0	0
$963$	0	0
$964$	16.2193	0.522387
$965$	12.6569	0.407439
$966$	0	0
$967$	−6.93946	−0.223158	−0.111579	−	0.993756i	$-0.535591\pi$
$967$	−6.93946	−0.223158	−0.111579	+	0.993756i	$0.535591\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−46.0353	−1.47734	−0.738671	−	0.674066i	$-0.764546\pi$
$971$	−46.0353	−1.47734	−0.738671	+	0.674066i	$0.764546\pi$
$972$	0	0
$973$	−54.1536	−1.73609
$974$	0	0
$975$	0	0
$976$	−15.1644	−0.485401
$977$	32.8489	1.05093	0.525465	−	0.850815i	$-0.323891\pi$
$977$	32.8489	1.05093	0.525465	+	0.850815i	$0.323891\pi$
$978$	0	0
$979$	−81.6721	−2.61025
$980$	−7.90893	−0.252641
$981$	0	0
$982$	0	0
$983$	−31.6616	−1.00985	−0.504924	−	0.863164i	$-0.668480\pi$
$983$	−31.6616	−1.00985	−0.504924	+	0.863164i	$0.668480\pi$
$984$	0	0
$985$	−29.7832	−0.948973
$986$	0	0
$987$	0	0
$988$	−24.3498	−0.774670
$989$	2.05653	0.0653940
$990$	0	0
$991$	40.8796	1.29858	0.649291	−	0.760540i	$-0.275066\pi$
$991$	40.8796	1.29858	0.649291	+	0.760540i	$0.275066\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.27677	−0.0404762
$996$	0	0
$997$	18.9650	0.600629	0.300314	−	0.953840i	$-0.402908\pi$
$997$	18.9650	0.600629	0.300314	+	0.953840i	$0.402908\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.g.1.3		4
3.2	odd	2		2001.2.a.g.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.g.1.2	✓	4	3.2	odd	2
6003.2.a.g.1.3		4	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.00000 q^{4} +1.58049 q^{5} -3.08254 q^{7} +O(q^{10})\)	\(q-2.00000 q^{4} +1.58049 q^{5} -3.08254 q^{7} -6.03291 q^{11} -2.36988 q^{13} +4.00000 q^{16} -6.85002 q^{17} -5.13736 q^{19} -3.16098 q^{20} -1.00000 q^{23} -2.50205 q^{25} +6.16508 q^{28} -1.00000 q^{29} +0.0784427 q^{31} -4.87193 q^{35} +5.42641 q^{37} -7.95446 q^{41} -2.05653 q^{43} +12.0658 q^{44} -1.58049 q^{47} +2.50205 q^{49} +4.73975 q^{52} +5.00410 q^{53} -9.53496 q^{55} +0.756479 q^{59} -3.79110 q^{61} -8.00000 q^{64} -3.74557 q^{65} +6.16508 q^{67} +13.7000 q^{68} +14.1154 q^{71} -2.91746 q^{73} +10.2747 q^{76} +18.5967 q^{77} -14.1415 q^{79} +6.32196 q^{80} -5.48123 q^{83} -10.8264 q^{85} +13.5378 q^{89} +7.30524 q^{91} +2.00000 q^{92} -8.11954 q^{95} +2.53086 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7}+O(q^{10}) \) 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7	\( 4 q - 8 q^{4} + 2 q^{5} - 2 q^{7} + 16 q^{16} - 2 q^{17} + 4 q^{19} - 4 q^{20} - 4 q^{23} - 4 q^{25} + 4 q^{28} - 4 q^{29} + 2 q^{31} - 4 q^{35} + 4 q^{37} - 6 q^{41} - 2 q^{47} + 4 q^{49} + 8 q^{53} - 8 q^{55} + 22 q^{59} - 16 q^{61} - 32 q^{64} + 10 q^{65} + 4 q^{67} + 4 q^{68} + 22 q^{71} - 22 q^{73} - 8 q^{76} + 8 q^{77} - 20 q^{79} + 8 q^{80} + 10 q^{83} - 2 q^{85} + 14 q^{89} - 26 q^{91} + 8 q^{92} + 14 q^{95} - 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^4 + 2 * q^5 - 2 * q^7 + 16 * q^16 - 2 * q^17 + 4 * q^19 - 4 * q^20 - 4 * q^23 - 4 * q^25 + 4 * q^28 - 4 * q^29 + 2 * q^31 - 4 * q^35 + 4 * q^37 - 6 * q^41 - 2 * q^47 + 4 * q^49 + 8 * q^53 - 8 * q^55 + 22 * q^59 - 16 * q^61 - 32 * q^64 + 10 * q^65 + 4 * q^67 + 4 * q^68 + 22 * q^71 - 22 * q^73 - 8 * q^76 + 8 * q^77 - 20 * q^79 + 8 * q^80 + 10 * q^83 - 2 * q^85 + 14 * q^89 - 26 * q^91 + 8 * q^92 + 14 * q^95 - 8 * q^97

Introduction

L-functions

Modular forms

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Database

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