LMFDB - Embedded newform 6004.2.a.h.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6004,2,Mod(1,6004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6004, 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("6004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6004 = 2^{2} \cdot 19 \cdot 79 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6004.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:	$47.9421813736$
Analytic rank:	$0$
Dimension:	$31$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	6004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.75235 q^{3} -3.07293 q^{5} +4.09507 q^{7} +4.57545 q^{9} +O(q^{10})$	$q-2.75235 q^{3} -3.07293 q^{5} +4.09507 q^{7} +4.57545 q^{9} +0.690063 q^{11} -6.65909 q^{13} +8.45779 q^{15} -3.83540 q^{17} -1.00000 q^{19} -11.2711 q^{21} +8.12897 q^{23} +4.44290 q^{25} -4.33621 q^{27} -4.25611 q^{29} -0.724058 q^{31} -1.89930 q^{33} -12.5839 q^{35} -0.241235 q^{37} +18.3282 q^{39} -0.711554 q^{41} +4.10110 q^{43} -14.0600 q^{45} +9.79154 q^{47} +9.76961 q^{49} +10.5564 q^{51} -1.44935 q^{53} -2.12052 q^{55} +2.75235 q^{57} -4.85831 q^{59} -15.1909 q^{61} +18.7368 q^{63} +20.4629 q^{65} -1.89645 q^{67} -22.3738 q^{69} -6.49882 q^{71} -0.864300 q^{73} -12.2284 q^{75} +2.82586 q^{77} -1.00000 q^{79} -1.79159 q^{81} +7.78889 q^{83} +11.7859 q^{85} +11.7143 q^{87} -14.5560 q^{89} -27.2695 q^{91} +1.99286 q^{93} +3.07293 q^{95} -19.1175 q^{97} +3.15735 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	$ 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) $ 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * 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
$3$	−2.75235	−1.58907	−0.794536	−	0.607217i	$-0.792286\pi$
$3$	−2.75235	−1.58907	−0.794536	+	0.607217i	$0.792286\pi$
$4$	0	0
$5$	−3.07293	−1.37426	−0.687128	−	0.726536i	$-0.741129\pi$
$5$	−3.07293	−1.37426	−0.687128	+	0.726536i	$0.741129\pi$
$6$	0	0
$7$	4.09507	1.54779	0.773896	−	0.633313i	$-0.218305\pi$
$7$	4.09507	1.54779	0.773896	+	0.633313i	$0.218305\pi$
$8$	0	0
$9$	4.57545	1.52515
$10$	0	0
$11$	0.690063	0.208062	0.104031	−	0.994574i	$-0.466826\pi$
$11$	0.690063	0.208062	0.104031	+	0.994574i	$0.466826\pi$
$12$	0	0
$13$	−6.65909	−1.84690	−0.923450	−	0.383719i	$-0.874643\pi$
$13$	−6.65909	−1.84690	−0.923450	+	0.383719i	$0.874643\pi$
$14$	0	0
$15$	8.45779	2.18379
$16$	0	0
$17$	−3.83540	−0.930222	−0.465111	−	0.885252i	$-0.653986\pi$
$17$	−3.83540	−0.930222	−0.465111	+	0.885252i	$0.653986\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−11.2711	−2.45955
$22$	0	0
$23$	8.12897	1.69501	0.847504	−	0.530790i	$-0.178105\pi$
$23$	8.12897	1.69501	0.847504	+	0.530790i	$0.178105\pi$
$24$	0	0
$25$	4.44290	0.888580
$26$	0	0
$27$	−4.33621	−0.834503
$28$	0	0
$29$	−4.25611	−0.790341	−0.395170	−	0.918608i	$-0.629314\pi$
$29$	−4.25611	−0.790341	−0.395170	+	0.918608i	$0.629314\pi$
$30$	0	0
$31$	−0.724058	−0.130045	−0.0650223	−	0.997884i	$-0.520712\pi$
$31$	−0.724058	−0.130045	−0.0650223	+	0.997884i	$0.520712\pi$
$32$	0	0
$33$	−1.89930	−0.330625
$34$	0	0
$35$	−12.5839	−2.12706
$36$	0	0
$37$	−0.241235	−0.0396587	−0.0198294	−	0.999803i	$-0.506312\pi$
$37$	−0.241235	−0.0396587	−0.0198294	+	0.999803i	$0.506312\pi$
$38$	0	0
$39$	18.3282	2.93486
$40$	0	0
$41$	−0.711554	−0.111126	−0.0555630	−	0.998455i	$-0.517695\pi$
$41$	−0.711554	−0.111126	−0.0555630	+	0.998455i	$0.517695\pi$
$42$	0	0
$43$	4.10110	0.625411	0.312706	−	0.949850i	$-0.398765\pi$
$43$	4.10110	0.625411	0.312706	+	0.949850i	$0.398765\pi$
$44$	0	0
$45$	−14.0600	−2.09595
$46$	0	0
$47$	9.79154	1.42824	0.714122	−	0.700022i	$-0.246826\pi$
$47$	9.79154	1.42824	0.714122	+	0.700022i	$0.246826\pi$
$48$	0	0
$49$	9.76961	1.39566
$50$	0	0
$51$	10.5564	1.47819
$52$	0	0
$53$	−1.44935	−0.199083	−0.0995416	−	0.995033i	$-0.531738\pi$
$53$	−1.44935	−0.199083	−0.0995416	+	0.995033i	$0.531738\pi$
$54$	0	0
$55$	−2.12052	−0.285930
$56$	0	0
$57$	2.75235	0.364558
$58$	0	0
$59$	−4.85831	−0.632499	−0.316249	−	0.948676i	$-0.602424\pi$
$59$	−4.85831	−0.632499	−0.316249	+	0.948676i	$0.602424\pi$
$60$	0	0
$61$	−15.1909	−1.94499	−0.972496	−	0.232920i	$-0.925172\pi$
$61$	−15.1909	−1.94499	−0.972496	+	0.232920i	$0.925172\pi$
$62$	0	0
$63$	18.7368	2.36062
$64$	0	0
$65$	20.4629	2.53811
$66$	0	0
$67$	−1.89645	−0.231688	−0.115844	−	0.993267i	$-0.536957\pi$
$67$	−1.89645	−0.231688	−0.115844	+	0.993267i	$0.536957\pi$
$68$	0	0
$69$	−22.3738	−2.69349
$70$	0	0
$71$	−6.49882	−0.771268	−0.385634	−	0.922652i	$-0.626017\pi$
$71$	−6.49882	−0.771268	−0.385634	+	0.922652i	$0.626017\pi$
$72$	0	0
$73$	−0.864300	−0.101159	−0.0505793	−	0.998720i	$-0.516107\pi$
$73$	−0.864300	−0.101159	−0.0505793	+	0.998720i	$0.516107\pi$
$74$	0	0
$75$	−12.2284	−1.41202
$76$	0	0
$77$	2.82586	0.322036
$78$	0	0
$79$	−1.00000	−0.112509
$79$	−1.00000	−0.112509
$80$	0	0
$81$	−1.79159	−0.199065
$82$	0	0
$83$	7.78889	0.854942	0.427471	−	0.904029i	$-0.359405\pi$
$83$	7.78889	0.854942	0.427471	+	0.904029i	$0.359405\pi$
$84$	0	0
$85$	11.7859	1.27836
$86$	0	0
$87$	11.7143	1.25591
$88$	0	0
$89$	−14.5560	−1.54293	−0.771466	−	0.636271i	$-0.780476\pi$
$89$	−14.5560	−1.54293	−0.771466	+	0.636271i	$0.780476\pi$
$90$	0	0
$91$	−27.2695	−2.85862
$92$	0	0
$93$	1.99286	0.206650
$94$	0	0
$95$	3.07293	0.315276
$96$	0	0
$97$	−19.1175	−1.94109	−0.970545	−	0.240918i	$-0.922551\pi$
$97$	−19.1175	−1.94109	−0.970545	+	0.240918i	$0.922551\pi$
$98$	0	0
$99$	3.15735	0.317326
$100$	0	0
$101$	−3.75498	−0.373634	−0.186817	−	0.982395i	$-0.559817\pi$
$101$	−3.75498	−0.373634	−0.186817	+	0.982395i	$0.559817\pi$
$102$	0	0
$103$	−4.09742	−0.403731	−0.201865	−	0.979413i	$-0.564700\pi$
$103$	−4.09742	−0.403731	−0.201865	+	0.979413i	$0.564700\pi$
$104$	0	0
$105$	34.6353	3.38005
$106$	0	0
$107$	1.03559	0.100114	0.0500569	−	0.998746i	$-0.484060\pi$
$107$	1.03559	0.100114	0.0500569	+	0.998746i	$0.484060\pi$
$108$	0	0
$109$	10.4871	1.00449	0.502243	−	0.864726i	$-0.332508\pi$
$109$	10.4871	1.00449	0.502243	+	0.864726i	$0.332508\pi$
$110$	0	0
$111$	0.663963	0.0630206
$112$	0	0
$113$	18.4057	1.73147	0.865733	−	0.500506i	$-0.166853\pi$
$113$	18.4057	1.73147	0.865733	+	0.500506i	$0.166853\pi$
$114$	0	0
$115$	−24.9798	−2.32937
$116$	0	0
$117$	−30.4684	−2.81680
$118$	0	0
$119$	−15.7062	−1.43979
$120$	0	0
$121$	−10.5238	−0.956710
$122$	0	0
$123$	1.95845	0.176587
$124$	0	0
$125$	1.71194	0.153120
$126$	0	0
$127$	−14.0004	−1.24234	−0.621169	−	0.783676i	$-0.713342\pi$
$127$	−14.0004	−1.24234	−0.621169	+	0.783676i	$0.713342\pi$
$128$	0	0
$129$	−11.2877	−0.993824
$130$	0	0
$131$	−7.37449	−0.644312	−0.322156	−	0.946687i	$-0.604408\pi$
$131$	−7.37449	−0.644312	−0.322156	+	0.946687i	$0.604408\pi$
$132$	0	0
$133$	−4.09507	−0.355088
$134$	0	0
$135$	13.3249	1.14682
$136$	0	0
$137$	−3.73041	−0.318710	−0.159355	−	0.987221i	$-0.550941\pi$
$137$	−3.73041	−0.318710	−0.159355	+	0.987221i	$0.550941\pi$
$138$	0	0
$139$	15.7052	1.33210	0.666050	−	0.745907i	$-0.267984\pi$
$139$	15.7052	1.33210	0.666050	+	0.745907i	$0.267984\pi$
$140$	0	0
$141$	−26.9498	−2.26958
$142$	0	0
$143$	−4.59519	−0.384269
$144$	0	0
$145$	13.0787	1.08613
$146$	0	0
$147$	−26.8894	−2.21780
$148$	0	0
$149$	18.7748	1.53809	0.769047	−	0.639193i	$-0.220731\pi$
$149$	18.7748	1.53809	0.769047	+	0.639193i	$0.220731\pi$
$150$	0	0
$151$	2.09105	0.170167	0.0850835	−	0.996374i	$-0.472884\pi$
$151$	2.09105	0.170167	0.0850835	+	0.996374i	$0.472884\pi$
$152$	0	0
$153$	−17.5487	−1.41873
$154$	0	0
$155$	2.22498	0.178715
$156$	0	0
$157$	−19.7408	−1.57549	−0.787745	−	0.616001i	$-0.788752\pi$
$157$	−19.7408	−1.57549	−0.787745	+	0.616001i	$0.788752\pi$
$158$	0	0
$159$	3.98912	0.316358
$160$	0	0
$161$	33.2887	2.62352
$162$	0	0
$163$	−0.360526	−0.0282386	−0.0141193	−	0.999900i	$-0.504494\pi$
$163$	−0.360526	−0.0282386	−0.0141193	+	0.999900i	$0.504494\pi$
$164$	0	0
$165$	5.83641	0.454364
$166$	0	0
$167$	1.41597	0.109571	0.0547854	−	0.998498i	$-0.482553\pi$
$167$	1.41597	0.109571	0.0547854	+	0.998498i	$0.482553\pi$
$168$	0	0
$169$	31.3435	2.41104
$170$	0	0
$171$	−4.57545	−0.349894
$172$	0	0
$173$	−8.63074	−0.656183	−0.328092	−	0.944646i	$-0.606405\pi$
$173$	−8.63074	−0.656183	−0.328092	+	0.944646i	$0.606405\pi$
$174$	0	0
$175$	18.1940	1.37534
$176$	0	0
$177$	13.3718	1.00509
$178$	0	0
$179$	11.1681	0.834746	0.417373	−	0.908735i	$-0.362951\pi$
$179$	11.1681	0.834746	0.417373	+	0.908735i	$0.362951\pi$
$180$	0	0
$181$	18.6457	1.38593	0.692963	−	0.720973i	$-0.256305\pi$
$181$	18.6457	1.38593	0.692963	+	0.720973i	$0.256305\pi$
$182$	0	0
$183$	41.8107	3.09073
$184$	0	0
$185$	0.741297	0.0545012
$186$	0	0
$187$	−2.64667	−0.193544
$188$	0	0
$189$	−17.7571	−1.29164
$190$	0	0
$191$	4.70042	0.340111	0.170055	−	0.985435i	$-0.445605\pi$
$191$	4.70042	0.340111	0.170055	+	0.985435i	$0.445605\pi$
$192$	0	0
$193$	−19.9129	−1.43336	−0.716679	−	0.697403i	$-0.754339\pi$
$193$	−19.9129	−1.43336	−0.716679	+	0.697403i	$0.754339\pi$
$194$	0	0
$195$	−56.3212	−4.03325
$196$	0	0
$197$	5.23361	0.372879	0.186439	−	0.982466i	$-0.440305\pi$
$197$	5.23361	0.372879	0.186439	+	0.982466i	$0.440305\pi$
$198$	0	0
$199$	−4.76529	−0.337802	−0.168901	−	0.985633i	$-0.554022\pi$
$199$	−4.76529	−0.337802	−0.168901	+	0.985633i	$0.554022\pi$
$200$	0	0
$201$	5.21971	0.368170
$202$	0	0
$203$	−17.4291	−1.22328
$204$	0	0
$205$	2.18655	0.152716
$206$	0	0
$207$	37.1937	2.58514
$208$	0	0
$209$	−0.690063	−0.0477327
$210$	0	0
$211$	−13.5491	−0.932760	−0.466380	−	0.884584i	$-0.654442\pi$
$211$	−13.5491	−0.932760	−0.466380	+	0.884584i	$0.654442\pi$
$212$	0	0
$213$	17.8871	1.22560
$214$	0	0
$215$	−12.6024	−0.859475
$216$	0	0
$217$	−2.96507	−0.201282
$218$	0	0
$219$	2.37886	0.160748
$220$	0	0
$221$	25.5403	1.71803
$222$	0	0
$223$	19.4517	1.30258	0.651289	−	0.758829i	$-0.274228\pi$
$223$	19.4517	1.30258	0.651289	+	0.758829i	$0.274228\pi$
$224$	0	0
$225$	20.3283	1.35522
$226$	0	0
$227$	14.0654	0.933553	0.466777	−	0.884375i	$-0.345415\pi$
$227$	14.0654	0.933553	0.466777	+	0.884375i	$0.345415\pi$
$228$	0	0
$229$	−11.0421	−0.729685	−0.364842	−	0.931069i	$-0.618877\pi$
$229$	−11.0421	−0.729685	−0.364842	+	0.931069i	$0.618877\pi$
$230$	0	0
$231$	−7.77776	−0.511739
$232$	0	0
$233$	−0.919220	−0.0602201	−0.0301101	−	0.999547i	$-0.509586\pi$
$233$	−0.919220	−0.0602201	−0.0301101	+	0.999547i	$0.509586\pi$
$234$	0	0
$235$	−30.0887	−1.96277
$236$	0	0
$237$	2.75235	0.178785
$238$	0	0
$239$	0.666605	0.0431191	0.0215596	−	0.999768i	$-0.493137\pi$
$239$	0.666605	0.0431191	0.0215596	+	0.999768i	$0.493137\pi$
$240$	0	0
$241$	−2.65346	−0.170924	−0.0854620	−	0.996341i	$-0.527237\pi$
$241$	−2.65346	−0.170924	−0.0854620	+	0.996341i	$0.527237\pi$
$242$	0	0
$243$	17.9397	1.15083
$244$	0	0
$245$	−30.0213	−1.91799
$246$	0	0
$247$	6.65909	0.423708
$248$	0	0
$249$	−21.4378	−1.35856
$250$	0	0
$251$	23.6154	1.49059	0.745294	−	0.666736i	$-0.232309\pi$
$251$	23.6154	1.49059	0.745294	+	0.666736i	$0.232309\pi$
$252$	0	0
$253$	5.60950	0.352666
$254$	0	0
$255$	−32.4390	−2.03141
$256$	0	0
$257$	11.9883	0.747808	0.373904	−	0.927467i	$-0.378019\pi$
$257$	11.9883	0.747808	0.373904	+	0.927467i	$0.378019\pi$
$258$	0	0
$259$	−0.987873	−0.0613834
$260$	0	0
$261$	−19.4737	−1.20539
$262$	0	0
$263$	−31.5129	−1.94317	−0.971585	−	0.236690i	$-0.923938\pi$
$263$	−31.5129	−1.94317	−0.971585	+	0.236690i	$0.923938\pi$
$264$	0	0
$265$	4.45374	0.273591
$266$	0	0
$267$	40.0632	2.45183
$268$	0	0
$269$	23.9743	1.46174	0.730871	−	0.682516i	$-0.239114\pi$
$269$	23.9743	1.46174	0.730871	+	0.682516i	$0.239114\pi$
$270$	0	0
$271$	15.5585	0.945109	0.472555	−	0.881301i	$-0.343332\pi$
$271$	15.5585	0.945109	0.472555	+	0.881301i	$0.343332\pi$
$272$	0	0
$273$	75.0552	4.54255
$274$	0	0
$275$	3.06588	0.184880
$276$	0	0
$277$	−8.88080	−0.533596	−0.266798	−	0.963753i	$-0.585966\pi$
$277$	−8.88080	−0.533596	−0.266798	+	0.963753i	$0.585966\pi$
$278$	0	0
$279$	−3.31289	−0.198338
$280$	0	0
$281$	24.2114	1.44433	0.722165	−	0.691721i	$-0.243147\pi$
$281$	24.2114	1.44433	0.722165	+	0.691721i	$0.243147\pi$
$282$	0	0
$283$	9.80990	0.583138	0.291569	−	0.956550i	$-0.405823\pi$
$283$	9.80990	0.583138	0.291569	+	0.956550i	$0.405823\pi$
$284$	0	0
$285$	−8.45779	−0.500996
$286$	0	0
$287$	−2.91386	−0.172000
$288$	0	0
$289$	−2.28969	−0.134688
$290$	0	0
$291$	52.6182	3.08453
$292$	0	0
$293$	−18.3639	−1.07283	−0.536414	−	0.843955i	$-0.680221\pi$
$293$	−18.3639	−1.07283	−0.536414	+	0.843955i	$0.680221\pi$
$294$	0	0
$295$	14.9293	0.869215
$296$	0	0
$297$	−2.99226	−0.173628
$298$	0	0
$299$	−54.1316	−3.13051
$300$	0	0
$301$	16.7943	0.968006
$302$	0	0
$303$	10.3350	0.593732
$304$	0	0
$305$	46.6805	2.67292
$306$	0	0
$307$	15.6084	0.890817	0.445408	−	0.895327i	$-0.353058\pi$
$307$	15.6084	0.890817	0.445408	+	0.895327i	$0.353058\pi$
$308$	0	0
$309$	11.2776	0.641558
$310$	0	0
$311$	32.1901	1.82534	0.912668	−	0.408702i	$-0.134019\pi$
$311$	32.1901	1.82534	0.912668	+	0.408702i	$0.134019\pi$
$312$	0	0
$313$	7.86850	0.444754	0.222377	−	0.974961i	$-0.428618\pi$
$313$	7.86850	0.444754	0.222377	+	0.974961i	$0.428618\pi$
$314$	0	0
$315$	−57.5769	−3.24409
$316$	0	0
$317$	−17.1801	−0.964928	−0.482464	−	0.875916i	$-0.660258\pi$
$317$	−17.1801	−0.964928	−0.482464	+	0.875916i	$0.660258\pi$
$318$	0	0
$319$	−2.93699	−0.164440
$320$	0	0
$321$	−2.85030	−0.159088
$322$	0	0
$323$	3.83540	0.213407
$324$	0	0
$325$	−29.5857	−1.64112
$326$	0	0
$327$	−28.8643	−1.59620
$328$	0	0
$329$	40.0971	2.21062
$330$	0	0
$331$	6.19504	0.340510	0.170255	−	0.985400i	$-0.445541\pi$
$331$	6.19504	0.340510	0.170255	+	0.985400i	$0.445541\pi$
$332$	0	0
$333$	−1.10376	−0.0604855
$334$	0	0
$335$	5.82766	0.318399
$336$	0	0
$337$	19.8818	1.08303	0.541514	−	0.840691i	$-0.317851\pi$
$337$	19.8818	1.08303	0.541514	+	0.840691i	$0.317851\pi$
$338$	0	0
$339$	−50.6591	−2.75143
$340$	0	0
$341$	−0.499646	−0.0270573
$342$	0	0
$343$	11.3417	0.612396
$344$	0	0
$345$	68.7531	3.70154
$346$	0	0
$347$	−12.6134	−0.677121	−0.338561	−	0.940945i	$-0.609940\pi$
$347$	−12.6134	−0.677121	−0.338561	+	0.940945i	$0.609940\pi$
$348$	0	0
$349$	32.1149	1.71907	0.859536	−	0.511075i	$-0.170752\pi$
$349$	32.1149	1.71907	0.859536	+	0.511075i	$0.170752\pi$
$350$	0	0
$351$	28.8752	1.54124
$352$	0	0
$353$	−29.5964	−1.57526	−0.787630	−	0.616149i	$-0.788692\pi$
$353$	−29.5964	−1.57526	−0.787630	+	0.616149i	$0.788692\pi$
$354$	0	0
$355$	19.9704	1.05992
$356$	0	0
$357$	43.2291	2.28793
$358$	0	0
$359$	−3.45990	−0.182606	−0.0913032	−	0.995823i	$-0.529103\pi$
$359$	−3.45990	−0.182606	−0.0913032	+	0.995823i	$0.529103\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	28.9653	1.52028
$364$	0	0
$365$	2.65593	0.139018
$366$	0	0
$367$	−24.1357	−1.25987	−0.629937	−	0.776646i	$-0.716919\pi$
$367$	−24.1357	−1.25987	−0.629937	+	0.776646i	$0.716919\pi$
$368$	0	0
$369$	−3.25568	−0.169484
$370$	0	0
$371$	−5.93518	−0.308139
$372$	0	0
$373$	38.4644	1.99161	0.995806	−	0.0914933i	$-0.0291640\pi$
$373$	38.4644	1.99161	0.995806	+	0.0914933i	$0.0291640\pi$
$374$	0	0
$375$	−4.71185	−0.243319
$376$	0	0
$377$	28.3419	1.45968
$378$	0	0
$379$	23.0560	1.18431	0.592153	−	0.805825i	$-0.298278\pi$
$379$	23.0560	1.18431	0.592153	+	0.805825i	$0.298278\pi$
$380$	0	0
$381$	38.5342	1.97417
$382$	0	0
$383$	−1.07052	−0.0547009	−0.0273504	−	0.999626i	$-0.508707\pi$
$383$	−1.07052	−0.0547009	−0.0273504	+	0.999626i	$0.508707\pi$
$384$	0	0
$385$	−8.68366	−0.442560
$386$	0	0
$387$	18.7644	0.953847
$388$	0	0
$389$	25.0006	1.26758	0.633791	−	0.773504i	$-0.281498\pi$
$389$	25.0006	1.26758	0.633791	+	0.773504i	$0.281498\pi$
$390$	0	0
$391$	−31.1779	−1.57673
$392$	0	0
$393$	20.2972	1.02386
$394$	0	0
$395$	3.07293	0.154616
$396$	0	0
$397$	5.26580	0.264283	0.132142	−	0.991231i	$-0.457815\pi$
$397$	5.26580	0.264283	0.132142	+	0.991231i	$0.457815\pi$
$398$	0	0
$399$	11.2711	0.564260
$400$	0	0
$401$	22.0347	1.10036	0.550181	−	0.835045i	$-0.314559\pi$
$401$	22.0347	1.10036	0.550181	+	0.835045i	$0.314559\pi$
$402$	0	0
$403$	4.82157	0.240180
$404$	0	0
$405$	5.50542	0.273566
$406$	0	0
$407$	−0.166467	−0.00825147
$408$	0	0
$409$	10.7944	0.533751	0.266875	−	0.963731i	$-0.414009\pi$
$409$	10.7944	0.533751	0.266875	+	0.963731i	$0.414009\pi$
$410$	0	0
$411$	10.2674	0.506454
$412$	0	0
$413$	−19.8951	−0.978976
$414$	0	0
$415$	−23.9347	−1.17491
$416$	0	0
$417$	−43.2263	−2.11680
$418$	0	0
$419$	35.8295	1.75039	0.875193	−	0.483773i	$-0.160734\pi$
$419$	35.8295	1.75039	0.875193	+	0.483773i	$0.160734\pi$
$420$	0	0
$421$	−6.24853	−0.304535	−0.152267	−	0.988339i	$-0.548657\pi$
$421$	−6.24853	−0.304535	−0.152267	+	0.988339i	$0.548657\pi$
$422$	0	0
$423$	44.8008	2.17829
$424$	0	0
$425$	−17.0403	−0.826576
$426$	0	0
$427$	−62.2077	−3.01044
$428$	0	0
$429$	12.6476	0.610632
$430$	0	0
$431$	−7.55746	−0.364030	−0.182015	−	0.983296i	$-0.558262\pi$
$431$	−7.55746	−0.364030	−0.182015	+	0.983296i	$0.558262\pi$
$432$	0	0
$433$	−4.36395	−0.209718	−0.104859	−	0.994487i	$-0.533439\pi$
$433$	−4.36395	−0.209718	−0.104859	+	0.994487i	$0.533439\pi$
$434$	0	0
$435$	−35.9973	−1.72594
$436$	0	0
$437$	−8.12897	−0.388861
$438$	0	0
$439$	25.8412	1.23333	0.616667	−	0.787224i	$-0.288482\pi$
$439$	25.8412	1.23333	0.616667	+	0.787224i	$0.288482\pi$
$440$	0	0
$441$	44.7004	2.12859
$442$	0	0
$443$	13.6220	0.647198	0.323599	−	0.946194i	$-0.395107\pi$
$443$	13.6220	0.647198	0.323599	+	0.946194i	$0.395107\pi$
$444$	0	0
$445$	44.7295	2.12038
$446$	0	0
$447$	−51.6750	−2.44414
$448$	0	0
$449$	−30.5414	−1.44134	−0.720668	−	0.693280i	$-0.756165\pi$
$449$	−30.5414	−1.44134	−0.720668	+	0.693280i	$0.756165\pi$
$450$	0	0
$451$	−0.491017	−0.0231211
$452$	0	0
$453$	−5.75530	−0.270408
$454$	0	0
$455$	83.7971	3.92847
$456$	0	0
$457$	−2.53759	−0.118703	−0.0593517	−	0.998237i	$-0.518903\pi$
$457$	−2.53759	−0.118703	−0.0593517	+	0.998237i	$0.518903\pi$
$458$	0	0
$459$	16.6311	0.776273
$460$	0	0
$461$	19.4653	0.906588	0.453294	−	0.891361i	$-0.350249\pi$
$461$	19.4653	0.906588	0.453294	+	0.891361i	$0.350249\pi$
$462$	0	0
$463$	−15.7432	−0.731651	−0.365825	−	0.930684i	$-0.619213\pi$
$463$	−15.7432	−0.731651	−0.365825	+	0.930684i	$0.619213\pi$
$464$	0	0
$465$	−6.12393	−0.283991
$466$	0	0
$467$	4.12677	0.190964	0.0954820	−	0.995431i	$-0.469561\pi$
$467$	4.12677	0.190964	0.0954820	+	0.995431i	$0.469561\pi$
$468$	0	0
$469$	−7.76610	−0.358605
$470$	0	0
$471$	54.3338	2.50357
$472$	0	0
$473$	2.83002	0.130124
$474$	0	0
$475$	−4.44290	−0.203854
$476$	0	0
$477$	−6.63142	−0.303632
$478$	0	0
$479$	33.5207	1.53160	0.765801	−	0.643078i	$-0.222343\pi$
$479$	33.5207	1.53160	0.765801	+	0.643078i	$0.222343\pi$
$480$	0	0
$481$	1.60640	0.0732457
$482$	0	0
$483$	−91.6223	−4.16896
$484$	0	0
$485$	58.7468	2.66756
$486$	0	0
$487$	3.39572	0.153875	0.0769373	−	0.997036i	$-0.475486\pi$
$487$	3.39572	0.153875	0.0769373	+	0.997036i	$0.475486\pi$
$488$	0	0
$489$	0.992296	0.0448732
$490$	0	0
$491$	19.2967	0.870850	0.435425	−	0.900225i	$-0.356598\pi$
$491$	19.2967	0.870850	0.435425	+	0.900225i	$0.356598\pi$
$492$	0	0
$493$	16.3239	0.735192
$494$	0	0
$495$	−9.70232	−0.436087
$496$	0	0
$497$	−26.6131	−1.19376
$498$	0	0
$499$	−10.1453	−0.454167	−0.227083	−	0.973875i	$-0.572919\pi$
$499$	−10.1453	−0.454167	−0.227083	+	0.973875i	$0.572919\pi$
$500$	0	0
$501$	−3.89724	−0.174116
$502$	0	0
$503$	−40.1297	−1.78930	−0.894648	−	0.446771i	$-0.852574\pi$
$503$	−40.1297	−1.78930	−0.894648	+	0.446771i	$0.852574\pi$
$504$	0	0
$505$	11.5388	0.513469
$506$	0	0
$507$	−86.2684	−3.83132
$508$	0	0
$509$	44.1841	1.95843	0.979213	−	0.202833i	$-0.0650148\pi$
$509$	44.1841	1.95843	0.979213	+	0.202833i	$0.0650148\pi$
$510$	0	0
$511$	−3.53937	−0.156572
$512$	0	0
$513$	4.33621	0.191448
$514$	0	0
$515$	12.5911	0.554830
$516$	0	0
$517$	6.75678	0.297163
$518$	0	0
$519$	23.7549	1.04272
$520$	0	0
$521$	10.0166	0.438834	0.219417	−	0.975631i	$-0.429585\pi$
$521$	10.0166	0.438834	0.219417	+	0.975631i	$0.429585\pi$
$522$	0	0
$523$	3.90801	0.170885	0.0854426	−	0.996343i	$-0.472770\pi$
$523$	3.90801	0.170885	0.0854426	+	0.996343i	$0.472770\pi$
$524$	0	0
$525$	−50.0763	−2.18551
$526$	0	0
$527$	2.77705	0.120970
$528$	0	0
$529$	43.0801	1.87305
$530$	0	0
$531$	−22.2290	−0.964656
$532$	0	0
$533$	4.73830	0.205239
$534$	0	0
$535$	−3.18228	−0.137582
$536$	0	0
$537$	−30.7387	−1.32647
$538$	0	0
$539$	6.74164	0.290383
$540$	0	0
$541$	17.2207	0.740377	0.370188	−	0.928957i	$-0.379293\pi$
$541$	17.2207	0.740377	0.370188	+	0.928957i	$0.379293\pi$
$542$	0	0
$543$	−51.3197	−2.20234
$544$	0	0
$545$	−32.2263	−1.38042
$546$	0	0
$547$	−21.9443	−0.938270	−0.469135	−	0.883126i	$-0.655434\pi$
$547$	−21.9443	−0.938270	−0.469135	+	0.883126i	$0.655434\pi$
$548$	0	0
$549$	−69.5051	−2.96641
$550$	0	0
$551$	4.25611	0.181317
$552$	0	0
$553$	−4.09507	−0.174140
$554$	0	0
$555$	−2.04031	−0.0866064
$556$	0	0
$557$	2.59692	0.110035	0.0550176	−	0.998485i	$-0.482479\pi$
$557$	2.59692	0.110035	0.0550176	+	0.998485i	$0.482479\pi$
$558$	0	0
$559$	−27.3096	−1.15507
$560$	0	0
$561$	7.28457	0.307555
$562$	0	0
$563$	32.0173	1.34937	0.674684	−	0.738107i	$-0.264280\pi$
$563$	32.0173	1.34937	0.674684	+	0.738107i	$0.264280\pi$
$564$	0	0
$565$	−56.5596	−2.37948
$566$	0	0
$567$	−7.33667	−0.308111
$568$	0	0
$569$	−22.1351	−0.927952	−0.463976	−	0.885848i	$-0.653578\pi$
$569$	−22.1351	−0.927952	−0.463976	+	0.885848i	$0.653578\pi$
$570$	0	0
$571$	1.61670	0.0676569	0.0338285	−	0.999428i	$-0.489230\pi$
$571$	1.61670	0.0676569	0.0338285	+	0.999428i	$0.489230\pi$
$572$	0	0
$573$	−12.9372	−0.540461
$574$	0	0
$575$	36.1162	1.50615
$576$	0	0
$577$	−32.2690	−1.34337	−0.671687	−	0.740835i	$-0.734430\pi$
$577$	−32.2690	−1.34337	−0.671687	+	0.740835i	$0.734430\pi$
$578$	0	0
$579$	54.8072	2.27771
$580$	0	0
$581$	31.8961	1.32327
$582$	0	0
$583$	−1.00014	−0.0414216
$584$	0	0
$585$	93.6272	3.87101
$586$	0	0
$587$	12.0872	0.498891	0.249445	−	0.968389i	$-0.419752\pi$
$587$	12.0872	0.498891	0.249445	+	0.968389i	$0.419752\pi$
$588$	0	0
$589$	0.724058	0.0298343
$590$	0	0
$591$	−14.4047	−0.592532
$592$	0	0
$593$	17.8568	0.733291	0.366645	−	0.930361i	$-0.380506\pi$
$593$	17.8568	0.733291	0.366645	+	0.930361i	$0.380506\pi$
$594$	0	0
$595$	48.2642	1.97864
$596$	0	0
$597$	13.1158	0.536792
$598$	0	0
$599$	34.3282	1.40261	0.701307	−	0.712859i	$-0.252600\pi$
$599$	34.3282	1.40261	0.701307	+	0.712859i	$0.252600\pi$
$600$	0	0
$601$	−34.1561	−1.39326	−0.696629	−	0.717432i	$-0.745317\pi$
$601$	−34.1561	−1.39326	−0.696629	+	0.717432i	$0.745317\pi$
$602$	0	0
$603$	−8.67712	−0.353360
$604$	0	0
$605$	32.3389	1.31476
$606$	0	0
$607$	28.1406	1.14219	0.571095	−	0.820884i	$-0.306519\pi$
$607$	28.1406	1.14219	0.571095	+	0.820884i	$0.306519\pi$
$608$	0	0
$609$	47.9710	1.94388
$610$	0	0
$611$	−65.2028	−2.63782
$612$	0	0
$613$	−8.15476	−0.329368	−0.164684	−	0.986346i	$-0.552660\pi$
$613$	−8.15476	−0.329368	−0.164684	+	0.986346i	$0.552660\pi$
$614$	0	0
$615$	−6.01817	−0.242676
$616$	0	0
$617$	3.37109	0.135715	0.0678575	−	0.997695i	$-0.478384\pi$
$617$	3.37109	0.135715	0.0678575	+	0.997695i	$0.478384\pi$
$618$	0	0
$619$	−26.0969	−1.04892	−0.524462	−	0.851434i	$-0.675734\pi$
$619$	−26.0969	−1.04892	−0.524462	+	0.851434i	$0.675734\pi$
$620$	0	0
$621$	−35.2489	−1.41449
$622$	0	0
$623$	−59.6078	−2.38814
$624$	0	0
$625$	−27.4751	−1.09901
$626$	0	0
$627$	1.89930	0.0758507
$628$	0	0
$629$	0.925232	0.0368914
$630$	0	0
$631$	−11.9433	−0.475454	−0.237727	−	0.971332i	$-0.576402\pi$
$631$	−11.9433	−0.475454	−0.237727	+	0.971332i	$0.576402\pi$
$632$	0	0
$633$	37.2920	1.48222
$634$	0	0
$635$	43.0224	1.70729
$636$	0	0
$637$	−65.0567	−2.57764
$638$	0	0
$639$	−29.7350	−1.17630
$640$	0	0
$641$	−30.3034	−1.19691	−0.598457	−	0.801155i	$-0.704219\pi$
$641$	−30.3034	−1.19691	−0.598457	+	0.801155i	$0.704219\pi$
$642$	0	0
$643$	49.0747	1.93532	0.967658	−	0.252266i	$-0.0811757\pi$
$643$	49.0747	1.93532	0.967658	+	0.252266i	$0.0811757\pi$
$644$	0	0
$645$	34.6862	1.36577
$646$	0	0
$647$	12.9865	0.510551	0.255276	−	0.966868i	$-0.417834\pi$
$647$	12.9865	0.510551	0.255276	+	0.966868i	$0.417834\pi$
$648$	0	0
$649$	−3.35254	−0.131599
$650$	0	0
$651$	8.16092	0.319852
$652$	0	0
$653$	−6.90905	−0.270372	−0.135186	−	0.990820i	$-0.543163\pi$
$653$	−6.90905	−0.270372	−0.135186	+	0.990820i	$0.543163\pi$
$654$	0	0
$655$	22.6613	0.885449
$656$	0	0
$657$	−3.95456	−0.154282
$658$	0	0
$659$	−40.4899	−1.57726	−0.788632	−	0.614865i	$-0.789210\pi$
$659$	−40.4899	−1.57726	−0.788632	+	0.614865i	$0.789210\pi$
$660$	0	0
$661$	2.50357	0.0973776	0.0486888	−	0.998814i	$-0.484496\pi$
$661$	2.50357	0.0973776	0.0486888	+	0.998814i	$0.484496\pi$
$662$	0	0
$663$	−70.2959	−2.73007
$664$	0	0
$665$	12.5839	0.487981
$666$	0	0
$667$	−34.5978	−1.33963
$668$	0	0
$669$	−53.5378	−2.06989
$670$	0	0
$671$	−10.4827	−0.404679
$672$	0	0
$673$	−13.0513	−0.503092	−0.251546	−	0.967845i	$-0.580939\pi$
$673$	−13.0513	−0.503092	−0.251546	+	0.967845i	$0.580939\pi$
$674$	0	0
$675$	−19.2653	−0.741523
$676$	0	0
$677$	−10.8672	−0.417659	−0.208830	−	0.977952i	$-0.566965\pi$
$677$	−10.8672	−0.417659	−0.208830	+	0.977952i	$0.566965\pi$
$678$	0	0
$679$	−78.2876	−3.00440
$680$	0	0
$681$	−38.7129	−1.48348
$682$	0	0
$683$	−21.9949	−0.841611	−0.420805	−	0.907151i	$-0.638252\pi$
$683$	−21.9949	−0.841611	−0.420805	+	0.907151i	$0.638252\pi$
$684$	0	0
$685$	11.4633	0.437990
$686$	0	0
$687$	30.3919	1.15952
$688$	0	0
$689$	9.65134	0.367687
$690$	0	0
$691$	18.4999	0.703770	0.351885	−	0.936043i	$-0.385541\pi$
$691$	18.4999	0.703770	0.351885	+	0.936043i	$0.385541\pi$
$692$	0	0
$693$	12.9296	0.491154
$694$	0	0
$695$	−48.2610	−1.83065
$696$	0	0
$697$	2.72909	0.103372
$698$	0	0
$699$	2.53002	0.0956941
$700$	0	0
$701$	32.2728	1.21893	0.609464	−	0.792814i	$-0.291385\pi$
$701$	32.2728	1.21893	0.609464	+	0.792814i	$0.291385\pi$
$702$	0	0
$703$	0.241235	0.00909833
$704$	0	0
$705$	82.8148	3.11899
$706$	0	0
$707$	−15.3769	−0.578308
$708$	0	0
$709$	−47.9623	−1.80126	−0.900630	−	0.434586i	$-0.856895\pi$
$709$	−47.9623	−1.80126	−0.900630	+	0.434586i	$0.856895\pi$
$710$	0	0
$711$	−4.57545	−0.171593
$712$	0	0
$713$	−5.88585	−0.220427
$714$	0	0
$715$	14.1207	0.528085
$716$	0	0
$717$	−1.83473	−0.0685194
$718$	0	0
$719$	39.1801	1.46117	0.730586	−	0.682820i	$-0.239247\pi$
$719$	39.1801	1.46117	0.730586	+	0.682820i	$0.239247\pi$
$720$	0	0
$721$	−16.7792	−0.624891
$722$	0	0
$723$	7.30325	0.271611
$724$	0	0
$725$	−18.9095	−0.702281
$726$	0	0
$727$	8.71352	0.323167	0.161583	−	0.986859i	$-0.448340\pi$
$727$	8.71352	0.323167	0.161583	+	0.986859i	$0.448340\pi$
$728$	0	0
$729$	−44.0016	−1.62969
$730$	0	0
$731$	−15.7294	−0.581771
$732$	0	0
$733$	−23.8072	−0.879341	−0.439670	−	0.898159i	$-0.644905\pi$
$733$	−23.8072	−0.879341	−0.439670	+	0.898159i	$0.644905\pi$
$734$	0	0
$735$	82.6293	3.04783
$736$	0	0
$737$	−1.30867	−0.0482055
$738$	0	0
$739$	19.8429	0.729934	0.364967	−	0.931021i	$-0.381080\pi$
$739$	19.8429	0.729934	0.364967	+	0.931021i	$0.381080\pi$
$740$	0	0
$741$	−18.3282	−0.673303
$742$	0	0
$743$	9.54940	0.350333	0.175167	−	0.984539i	$-0.443954\pi$
$743$	9.54940	0.350333	0.175167	+	0.984539i	$0.443954\pi$
$744$	0	0
$745$	−57.6937	−2.11373
$746$	0	0
$747$	35.6377	1.30392
$748$	0	0
$749$	4.24080	0.154955
$750$	0	0
$751$	22.0785	0.805655	0.402828	−	0.915276i	$-0.368027\pi$
$751$	22.0785	0.805655	0.402828	+	0.915276i	$0.368027\pi$
$752$	0	0
$753$	−64.9978	−2.36865
$754$	0	0
$755$	−6.42564	−0.233853
$756$	0	0
$757$	−18.4016	−0.668817	−0.334409	−	0.942428i	$-0.608537\pi$
$757$	−18.4016	−0.668817	−0.334409	+	0.942428i	$0.608537\pi$
$758$	0	0
$759$	−15.4393	−0.560412
$760$	0	0
$761$	29.0714	1.05384	0.526919	−	0.849916i	$-0.323347\pi$
$761$	29.0714	1.05384	0.526919	+	0.849916i	$0.323347\pi$
$762$	0	0
$763$	42.9456	1.55473
$764$	0	0
$765$	53.9259	1.94970
$766$	0	0
$767$	32.3520	1.16816
$768$	0	0
$769$	−5.69248	−0.205276	−0.102638	−	0.994719i	$-0.532728\pi$
$769$	−5.69248	−0.205276	−0.102638	+	0.994719i	$0.532728\pi$
$770$	0	0
$771$	−32.9960	−1.18832
$772$	0	0
$773$	32.9332	1.18453	0.592263	−	0.805745i	$-0.298235\pi$
$773$	32.9332	1.18453	0.592263	+	0.805745i	$0.298235\pi$
$774$	0	0
$775$	−3.21692	−0.115555
$776$	0	0
$777$	2.71898	0.0975427
$778$	0	0
$779$	0.711554	0.0254941
$780$	0	0
$781$	−4.48460	−0.160471
$782$	0	0
$783$	18.4554	0.659542
$784$	0	0
$785$	60.6622	2.16513
$786$	0	0
$787$	−8.68073	−0.309435	−0.154717	−	0.987959i	$-0.549447\pi$
$787$	−8.68073	−0.309435	−0.154717	+	0.987959i	$0.549447\pi$
$788$	0	0
$789$	86.7347	3.08784
$790$	0	0
$791$	75.3728	2.67995
$792$	0	0
$793$	101.157	3.59221
$794$	0	0
$795$	−12.2583	−0.434756
$796$	0	0
$797$	−37.3098	−1.32158	−0.660790	−	0.750571i	$-0.729778\pi$
$797$	−37.3098	−1.32158	−0.660790	+	0.750571i	$0.729778\pi$
$798$	0	0
$799$	−37.5545	−1.32858
$800$	0	0
$801$	−66.6002	−2.35320
$802$	0	0
$803$	−0.596421	−0.0210472
$804$	0	0
$805$	−102.294	−3.60538
$806$	0	0
$807$	−65.9859	−2.32281
$808$	0	0
$809$	14.3819	0.505642	0.252821	−	0.967513i	$-0.418642\pi$
$809$	14.3819	0.505642	0.252821	+	0.967513i	$0.418642\pi$
$810$	0	0
$811$	−8.86429	−0.311267	−0.155634	−	0.987815i	$-0.549742\pi$
$811$	−8.86429	−0.311267	−0.155634	+	0.987815i	$0.549742\pi$
$812$	0	0
$813$	−42.8224	−1.50185
$814$	0	0
$815$	1.10787	0.0388070
$816$	0	0
$817$	−4.10110	−0.143479
$818$	0	0
$819$	−124.770	−4.35982
$820$	0	0
$821$	37.8312	1.32032	0.660159	−	0.751126i	$-0.270489\pi$
$821$	37.8312	1.32032	0.660159	+	0.751126i	$0.270489\pi$
$822$	0	0
$823$	20.1001	0.700646	0.350323	−	0.936629i	$-0.386072\pi$
$823$	20.1001	0.700646	0.350323	+	0.936629i	$0.386072\pi$
$824$	0	0
$825$	−8.43839	−0.293787
$826$	0	0
$827$	−25.4631	−0.885440	−0.442720	−	0.896660i	$-0.645986\pi$
$827$	−25.4631	−0.885440	−0.442720	+	0.896660i	$0.645986\pi$
$828$	0	0
$829$	−48.3880	−1.68059	−0.840293	−	0.542132i	$-0.817617\pi$
$829$	−48.3880	−1.68059	−0.840293	+	0.542132i	$0.817617\pi$
$830$	0	0
$831$	24.4431	0.847922
$832$	0	0
$833$	−37.4704	−1.29827
$834$	0	0
$835$	−4.35117	−0.150578
$836$	0	0
$837$	3.13967	0.108523
$838$	0	0
$839$	−24.5393	−0.847189	−0.423595	−	0.905852i	$-0.639232\pi$
$839$	−24.5393	−0.847189	−0.423595	+	0.905852i	$0.639232\pi$
$840$	0	0
$841$	−10.8855	−0.375362
$842$	0	0
$843$	−66.6383	−2.29514
$844$	0	0
$845$	−96.3164	−3.31339
$846$	0	0
$847$	−43.0958	−1.48079
$848$	0	0
$849$	−27.0003	−0.926649
$850$	0	0
$851$	−1.96099	−0.0672218
$852$	0	0
$853$	−50.6573	−1.73447	−0.867236	−	0.497897i	$-0.834106\pi$
$853$	−50.6573	−1.73447	−0.867236	+	0.497897i	$0.834106\pi$
$854$	0	0
$855$	14.0600	0.480843
$856$	0	0
$857$	49.2765	1.68325	0.841627	−	0.540059i	$-0.181598\pi$
$857$	49.2765	1.68325	0.841627	+	0.540059i	$0.181598\pi$
$858$	0	0
$859$	−53.5723	−1.82786	−0.913932	−	0.405867i	$-0.866970\pi$
$859$	−53.5723	−1.82786	−0.913932	+	0.405867i	$0.866970\pi$
$860$	0	0
$861$	8.01998	0.273320
$862$	0	0
$863$	41.3812	1.40863	0.704316	−	0.709886i	$-0.251254\pi$
$863$	41.3812	1.40863	0.704316	+	0.709886i	$0.251254\pi$
$864$	0	0
$865$	26.5217	0.901764
$866$	0	0
$867$	6.30204	0.214029
$868$	0	0
$869$	−0.690063	−0.0234088
$870$	0	0
$871$	12.6286	0.427905
$872$	0	0
$873$	−87.4714	−2.96046
$874$	0	0
$875$	7.01050	0.236998
$876$	0	0
$877$	41.9024	1.41494	0.707472	−	0.706741i	$-0.249836\pi$
$877$	41.9024	1.41494	0.707472	+	0.706741i	$0.249836\pi$
$878$	0	0
$879$	50.5438	1.70480
$880$	0	0
$881$	−19.1277	−0.644428	−0.322214	−	0.946667i	$-0.604427\pi$
$881$	−19.1277	−0.644428	−0.322214	+	0.946667i	$0.604427\pi$
$882$	0	0
$883$	23.5181	0.791448	0.395724	−	0.918370i	$-0.370494\pi$
$883$	23.5181	0.791448	0.395724	+	0.918370i	$0.370494\pi$
$884$	0	0
$885$	−41.0906	−1.38125
$886$	0	0
$887$	−15.7267	−0.528052	−0.264026	−	0.964516i	$-0.585051\pi$
$887$	−15.7267	−0.528052	−0.264026	+	0.964516i	$0.585051\pi$
$888$	0	0
$889$	−57.3328	−1.92288
$890$	0	0
$891$	−1.23631	−0.0414178
$892$	0	0
$893$	−9.79154	−0.327662
$894$	0	0
$895$	−34.3189	−1.14715
$896$	0	0
$897$	148.989	4.97461
$898$	0	0
$899$	3.08167	0.102780
$900$	0	0
$901$	5.55883	0.185192
$902$	0	0
$903$	−46.2238	−1.53823
$904$	0	0
$905$	−57.2971	−1.90462
$906$	0	0
$907$	−11.3161	−0.375746	−0.187873	−	0.982193i	$-0.560159\pi$
$907$	−11.3161	−0.375746	−0.187873	+	0.982193i	$0.560159\pi$
$908$	0	0
$909$	−17.1807	−0.569849
$910$	0	0
$911$	36.0019	1.19280	0.596398	−	0.802689i	$-0.296598\pi$
$911$	36.0019	1.19280	0.596398	+	0.802689i	$0.296598\pi$
$912$	0	0
$913$	5.37483	0.177881
$914$	0	0
$915$	−128.481	−4.24746
$916$	0	0
$917$	−30.1990	−0.997260
$918$	0	0
$919$	−1.07307	−0.0353973	−0.0176986	−	0.999843i	$-0.505634\pi$
$919$	−1.07307	−0.0353973	−0.0176986	+	0.999843i	$0.505634\pi$
$920$	0	0
$921$	−42.9598	−1.41557
$922$	0	0
$923$	43.2762	1.42445
$924$	0	0
$925$	−1.07178	−0.0352399
$926$	0	0
$927$	−18.7476	−0.615751
$928$	0	0
$929$	−36.6879	−1.20369	−0.601846	−	0.798612i	$-0.705568\pi$
$929$	−36.6879	−1.20369	−0.601846	+	0.798612i	$0.705568\pi$
$930$	0	0
$931$	−9.76961	−0.320186
$932$	0	0
$933$	−88.5987	−2.90059
$934$	0	0
$935$	8.13303	0.265978
$936$	0	0
$937$	48.1991	1.57460	0.787298	−	0.616573i	$-0.211479\pi$
$937$	48.1991	1.57460	0.787298	+	0.616573i	$0.211479\pi$
$938$	0	0
$939$	−21.6569	−0.706746
$940$	0	0
$941$	1.31426	0.0428435	0.0214218	−	0.999771i	$-0.493181\pi$
$941$	1.31426	0.0428435	0.0214218	+	0.999771i	$0.493181\pi$
$942$	0	0
$943$	−5.78420	−0.188359
$944$	0	0
$945$	54.5662	1.77504
$946$	0	0
$947$	−3.29365	−0.107029	−0.0535147	−	0.998567i	$-0.517042\pi$
$947$	−3.29365	−0.107029	−0.0535147	+	0.998567i	$0.517042\pi$
$948$	0	0
$949$	5.75545	0.186830
$950$	0	0
$951$	47.2856	1.53334
$952$	0	0
$953$	5.97190	0.193449	0.0967244	−	0.995311i	$-0.469163\pi$
$953$	5.97190	0.193449	0.0967244	+	0.995311i	$0.469163\pi$
$954$	0	0
$955$	−14.4441	−0.467399
$956$	0	0
$957$	8.08363	0.261307
$958$	0	0
$959$	−15.2763	−0.493297
$960$	0	0
$961$	−30.4757	−0.983088
$962$	0	0
$963$	4.73828	0.152689
$964$	0	0
$965$	61.1908	1.96980
$966$	0	0
$967$	−27.1086	−0.871753	−0.435877	−	0.900006i	$-0.643562\pi$
$967$	−27.1086	−0.871753	−0.435877	+	0.900006i	$0.643562\pi$
$968$	0	0
$969$	−10.5564	−0.339120
$970$	0	0
$971$	43.8450	1.40705	0.703526	−	0.710670i	$-0.251608\pi$
$971$	43.8450	1.40705	0.703526	+	0.710670i	$0.251608\pi$
$972$	0	0
$973$	64.3140	2.06181
$974$	0	0
$975$	81.4302	2.60785
$976$	0	0
$977$	10.8144	0.345985	0.172992	−	0.984923i	$-0.444656\pi$
$977$	10.8144	0.345985	0.172992	+	0.984923i	$0.444656\pi$
$978$	0	0
$979$	−10.0445	−0.321025
$980$	0	0
$981$	47.9834	1.53199
$982$	0	0
$983$	43.2760	1.38029	0.690145	−	0.723671i	$-0.257547\pi$
$983$	43.2760	1.38029	0.690145	+	0.723671i	$0.257547\pi$
$984$	0	0
$985$	−16.0825	−0.512431
$986$	0	0
$987$	−110.361	−3.51284
$988$	0	0
$989$	33.3377	1.06008
$990$	0	0
$991$	17.8777	0.567903	0.283951	−	0.958839i	$-0.408355\pi$
$991$	17.8777	0.567903	0.283951	+	0.958839i	$0.408355\pi$
$992$	0	0
$993$	−17.0509	−0.541096
$994$	0	0
$995$	14.6434	0.464227
$996$	0	0
$997$	−5.00864	−0.158625	−0.0793126	−	0.996850i	$-0.525273\pi$
$997$	−5.00864	−0.158625	−0.0793126	+	0.996850i	$0.525273\pi$
$998$	0	0
$999$	1.04604	0.0330953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6004.2.a.h.1.6	✓	31

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6004.2.a.h.1.6	✓	31	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:	\( N \)	\(=\)	\( 6004 = 2^{2} \cdot 19 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6004.a (trivial)

\(f(q)\)	\(=\)	\(q-2.75235 q^{3} -3.07293 q^{5} +4.09507 q^{7} +4.57545 q^{9} +O(q^{10})\)	\(q-2.75235 q^{3} -3.07293 q^{5} +4.09507 q^{7} +4.57545 q^{9} +0.690063 q^{11} -6.65909 q^{13} +8.45779 q^{15} -3.83540 q^{17} -1.00000 q^{19} -11.2711 q^{21} +8.12897 q^{23} +4.44290 q^{25} -4.33621 q^{27} -4.25611 q^{29} -0.724058 q^{31} -1.89930 q^{33} -12.5839 q^{35} -0.241235 q^{37} +18.3282 q^{39} -0.711554 q^{41} +4.10110 q^{43} -14.0600 q^{45} +9.79154 q^{47} +9.76961 q^{49} +10.5564 q^{51} -1.44935 q^{53} -2.12052 q^{55} +2.75235 q^{57} -4.85831 q^{59} -15.1909 q^{61} +18.7368 q^{63} +20.4629 q^{65} -1.89645 q^{67} -22.3738 q^{69} -6.49882 q^{71} -0.864300 q^{73} -12.2284 q^{75} +2.82586 q^{77} -1.00000 q^{79} -1.79159 q^{81} +7.78889 q^{83} +11.7859 q^{85} +11.7143 q^{87} -14.5560 q^{89} -27.2695 q^{91} +1.99286 q^{93} +3.07293 q^{95} -19.1175 q^{97} +3.15735 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9}+O(q^{10}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9	\( 31 q - 4 q^{3} + 10 q^{5} + 11 q^{7} + 47 q^{9} - 4 q^{11} + 11 q^{13} + 5 q^{15} + 14 q^{17} - 31 q^{19} + 22 q^{21} + 15 q^{23} + 59 q^{25} + 5 q^{27} + 34 q^{29} - 12 q^{31} + 10 q^{33} + 8 q^{35} + 16 q^{37} + 18 q^{39} + 27 q^{41} + 2 q^{43} + 22 q^{45} + 30 q^{47} + 62 q^{49} - 14 q^{51} + 35 q^{53} + 8 q^{55} + 4 q^{57} - 16 q^{59} + 37 q^{61} + 31 q^{63} + 80 q^{65} + 16 q^{67} + q^{69} + 19 q^{71} + 38 q^{73} + 21 q^{75} + 44 q^{77} - 31 q^{79} + 55 q^{81} - 12 q^{83} + 66 q^{85} + 58 q^{87} + 16 q^{89} - 42 q^{91} + 10 q^{93} - 10 q^{95} - 12 q^{97} + 12 q^{99}+O(q^{100}) \) 31 * q - 4 * q^3 + 10 * q^5 + 11 * q^7 + 47 * q^9 - 4 * q^11 + 11 * q^13 + 5 * q^15 + 14 * q^17 - 31 * q^19 + 22 * q^21 + 15 * q^23 + 59 * q^25 + 5 * q^27 + 34 * q^29 - 12 * q^31 + 10 * q^33 + 8 * q^35 + 16 * q^37 + 18 * q^39 + 27 * q^41 + 2 * q^43 + 22 * q^45 + 30 * q^47 + 62 * q^49 - 14 * q^51 + 35 * q^53 + 8 * q^55 + 4 * q^57 - 16 * q^59 + 37 * q^61 + 31 * q^63 + 80 * q^65 + 16 * q^67 + q^69 + 19 * q^71 + 38 * q^73 + 21 * q^75 + 44 * q^77 - 31 * q^79 + 55 * q^81 - 12 * q^83 + 66 * q^85 + 58 * q^87 + 16 * q^89 - 42 * q^91 + 10 * q^93 - 10 * q^95 - 12 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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