LMFDB - Embedded newform 4284.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4284,2,Mod(1,4284)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4284, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4284.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4284.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:	$34.2079122259$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.65544$ of defining polynomial
Character	$\chi$	$=$	4284.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.39593 q^{5} -1.00000 q^{7} +O(q^{10})$	$q+1.39593 q^{5} -1.00000 q^{7} +6.31088 q^{11} +3.65544 q^{13} -1.00000 q^{17} +1.39593 q^{19} -8.15412 q^{23} -3.05137 q^{25} -0.259511 q^{29} +8.23917 q^{31} -1.39593 q^{35} +11.0177 q^{37} +3.65544 q^{41} -0.208136 q^{43} +10.7582 q^{47} +1.00000 q^{49} -2.39593 q^{53} +8.80957 q^{55} +9.44731 q^{59} -10.7582 q^{61} +5.10275 q^{65} -15.7759 q^{67} -0.362259 q^{71} -4.51902 q^{73} -6.31088 q^{77} -3.01770 q^{79} +10.0000 q^{83} -1.39593 q^{85} -7.41363 q^{89} -3.65544 q^{91} +1.94863 q^{95} -2.79186 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{5} - 3 q^{7}+O(q^{10}) $ 3 * q + q^5 - 3 * q^7	$ 3 q + q^{5} - 3 q^{7} + 7 q^{11} + 5 q^{13} - 3 q^{17} + q^{19} + 3 q^{23} + 2 q^{29} + 6 q^{31} - q^{35} + 6 q^{37} + 5 q^{41} - 7 q^{43} + 8 q^{47} + 3 q^{49} - 4 q^{53} - 7 q^{55} + 16 q^{59} - 8 q^{61} - 3 q^{65} + 4 q^{67} + 20 q^{71} - 8 q^{73} - 7 q^{77} + 18 q^{79} + 30 q^{83} - q^{85} + 8 q^{89} - 5 q^{91} + 15 q^{95} - 2 q^{97}+O(q^{100}) $ 3 * q + q^5 - 3 * q^7 + 7 * q^11 + 5 * q^13 - 3 * q^17 + q^19 + 3 * q^23 + 2 * q^29 + 6 * q^31 - q^35 + 6 * q^37 + 5 * q^41 - 7 * q^43 + 8 * q^47 + 3 * q^49 - 4 * q^53 - 7 * q^55 + 16 * q^59 - 8 * q^61 - 3 * q^65 + 4 * q^67 + 20 * q^71 - 8 * q^73 - 7 * q^77 + 18 * q^79 + 30 * q^83 - q^85 + 8 * q^89 - 5 * q^91 + 15 * q^95 - 2 * 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
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.39593	0.624280	0.312140	−	0.950036i	$-0.398954\pi$
$5$	1.39593	0.624280	0.312140	+	0.950036i	$0.398954\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	6.31088	1.90280	0.951402	−	0.307953i	$-0.0996438\pi$
$11$	6.31088	1.90280	0.951402	+	0.307953i	$0.0996438\pi$
$12$	0	0
$13$	3.65544	1.01384	0.506919	−	0.861994i	$-0.330784\pi$
$13$	3.65544	1.01384	0.506919	+	0.861994i	$0.330784\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	1.39593	0.320249	0.160124	−	0.987097i	$-0.448810\pi$
$19$	1.39593	0.320249	0.160124	+	0.987097i	$0.448810\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.15412	−1.70025	−0.850126	−	0.526579i	$-0.823474\pi$
$23$	−8.15412	−1.70025	−0.850126	+	0.526579i	$0.823474\pi$
$24$	0	0
$25$	−3.05137	−0.610275
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−0.259511	−0.0481899	−0.0240949	−	0.999710i	$-0.507670\pi$
$29$	−0.259511	−0.0481899	−0.0240949	+	0.999710i	$0.507670\pi$
$30$	0	0
$31$	8.23917	1.47980	0.739899	−	0.672718i	$-0.234873\pi$
$31$	8.23917	1.47980	0.739899	+	0.672718i	$0.234873\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−1.39593	−0.235956
$36$	0	0
$37$	11.0177	1.81130	0.905649	−	0.424027i	$-0.139384\pi$
$37$	11.0177	1.81130	0.905649	+	0.424027i	$0.139384\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.65544	0.570884	0.285442	−	0.958396i	$-0.407860\pi$
$41$	3.65544	0.570884	0.285442	+	0.958396i	$0.407860\pi$
$42$	0	0
$43$	−0.208136	−0.0317405	−0.0158702	−	0.999874i	$-0.505052\pi$
$43$	−0.208136	−0.0317405	−0.0158702	+	0.999874i	$0.505052\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.7582	1.56924	0.784622	−	0.619975i	$-0.212857\pi$
$47$	10.7582	1.56924	0.784622	+	0.619975i	$0.212857\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−2.39593	−0.329107	−0.164553	−	0.986368i	$-0.552618\pi$
$53$	−2.39593	−0.329107	−0.164553	+	0.986368i	$0.552618\pi$
$54$	0	0
$55$	8.80957	1.18788
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.44731	1.22993	0.614967	−	0.788553i	$-0.289169\pi$
$59$	9.44731	1.22993	0.614967	+	0.788553i	$0.289169\pi$
$60$	0	0
$61$	−10.7582	−1.37745	−0.688723	−	0.725025i	$-0.741828\pi$
$61$	−10.7582	−1.37745	−0.688723	+	0.725025i	$0.741828\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.10275	0.632918
$66$	0	0
$67$	−15.7759	−1.92733	−0.963666	−	0.267110i	$-0.913931\pi$
$67$	−15.7759	−1.92733	−0.963666	+	0.267110i	$0.913931\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−0.362259	−0.0429922	−0.0214961	−	0.999769i	$-0.506843\pi$
$71$	−0.362259	−0.0429922	−0.0214961	+	0.999769i	$0.506843\pi$
$72$	0	0
$73$	−4.51902	−0.528911	−0.264456	−	0.964398i	$-0.585192\pi$
$73$	−4.51902	−0.528911	−0.264456	+	0.964398i	$0.585192\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−6.31088	−0.719192
$78$	0	0
$79$	−3.01770	−0.339518	−0.169759	−	0.985486i	$-0.554299\pi$
$79$	−3.01770	−0.339518	−0.169759	+	0.985486i	$0.554299\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	10.0000	1.09764	0.548821	−	0.835940i	$-0.315077\pi$
$83$	10.0000	1.09764	0.548821	+	0.835940i	$0.315077\pi$
$84$	0	0
$85$	−1.39593	−0.151410
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.41363	−0.785844	−0.392922	−	0.919572i	$-0.628536\pi$
$89$	−7.41363	−0.785844	−0.392922	+	0.919572i	$0.628536\pi$
$90$	0	0
$91$	−3.65544	−0.383194
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.94863	0.199925
$96$	0	0
$97$	−2.79186	−0.283471	−0.141735	−	0.989905i	$-0.545268\pi$
$97$	−2.79186	−0.283471	−0.141735	+	0.989905i	$0.545268\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.06908	−0.205881	−0.102940	−	0.994688i	$-0.532825\pi$
$101$	−2.06908	−0.205881	−0.102940	+	0.994688i	$0.532825\pi$
$102$	0	0
$103$	−14.9663	−1.47468	−0.737338	−	0.675524i	$-0.763917\pi$
$103$	−14.9663	−1.47468	−0.737338	+	0.675524i	$0.763917\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.84324	0.468213	0.234107	−	0.972211i	$-0.424783\pi$
$107$	4.84324	0.468213	0.234107	+	0.972211i	$0.424783\pi$
$108$	0	0
$109$	−11.2905	−1.08144	−0.540719	−	0.841203i	$-0.681848\pi$
$109$	−11.2905	−1.08144	−0.540719	+	0.841203i	$0.681848\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	12.9327	1.21660	0.608301	−	0.793707i	$-0.291851\pi$
$113$	12.9327	1.21660	0.608301	+	0.793707i	$0.291851\pi$
$114$	0	0
$115$	−11.3826	−1.06143
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.00000	0.0916698
$120$	0	0
$121$	28.8273	2.62066
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−11.2392	−1.00526
$126$	0	0
$127$	8.29755	0.736289	0.368144	−	0.929769i	$-0.379993\pi$
$127$	8.29755	0.736289	0.368144	+	0.929769i	$0.379993\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0.187796	0.0164078	0.00820389	−	0.999966i	$-0.497389\pi$
$131$	0.187796	0.0164078	0.00820389	+	0.999966i	$0.497389\pi$
$132$	0	0
$133$	−1.39593	−0.121043
$134$	0	0
$135$	0	0
$136$	0	0
$137$	13.6041	1.16227	0.581137	−	0.813806i	$-0.302608\pi$
$137$	13.6041	1.16227	0.581137	+	0.813806i	$0.302608\pi$
$138$	0	0
$139$	−8.20550	−0.695981	−0.347991	−	0.937498i	$-0.613136\pi$
$139$	−8.20550	−0.695981	−0.347991	+	0.937498i	$0.613136\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	23.0691	1.92913
$144$	0	0
$145$	−0.362259	−0.0300840
$146$	0	0
$147$	0	0
$148$	0	0
$149$	11.3109	0.926624	0.463312	−	0.886195i	$-0.346661\pi$
$149$	11.3109	0.926624	0.463312	+	0.886195i	$0.346661\pi$
$150$	0	0
$151$	17.0868	1.39050	0.695251	−	0.718767i	$-0.255293\pi$
$151$	17.0868	1.39050	0.695251	+	0.718767i	$0.255293\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11.5013	0.923808
$156$	0	0
$157$	−7.49868	−0.598460	−0.299230	−	0.954181i	$-0.596730\pi$
$157$	−7.49868	−0.598460	−0.299230	+	0.954181i	$0.596730\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.15412	0.642635
$162$	0	0
$163$	0.519021	0.0406529	0.0203264	−	0.999793i	$-0.493529\pi$
$163$	0.519021	0.0406529	0.0203264	+	0.999793i	$0.493529\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	19.7582	1.52893	0.764467	−	0.644662i	$-0.223002\pi$
$167$	19.7582	1.52893	0.764467	+	0.644662i	$0.223002\pi$
$168$	0	0
$169$	0.362259	0.0278661
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−7.39593	−0.562302	−0.281151	−	0.959664i	$-0.590716\pi$
$173$	−7.39593	−0.562302	−0.281151	+	0.959664i	$0.590716\pi$
$174$	0	0
$175$	3.05137	0.230662
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.17009	0.311687	0.155844	−	0.987782i	$-0.450190\pi$
$179$	4.17009	0.311687	0.155844	+	0.987782i	$0.450190\pi$
$180$	0	0
$181$	20.5544	1.52780	0.763899	−	0.645336i	$-0.223282\pi$
$181$	20.5544	1.52780	0.763899	+	0.645336i	$0.223282\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	15.3800	1.13076
$186$	0	0
$187$	−6.31088	−0.461498
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−4.62177	−0.334420	−0.167210	−	0.985921i	$-0.553476\pi$
$191$	−4.62177	−0.334420	−0.167210	+	0.985921i	$0.553476\pi$
$192$	0	0
$193$	−3.60407	−0.259427	−0.129713	−	0.991552i	$-0.541406\pi$
$193$	−3.60407	−0.259427	−0.129713	+	0.991552i	$0.541406\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2.41363	−0.171964	−0.0859821	−	0.996297i	$-0.527403\pi$
$197$	−2.41363	−0.171964	−0.0859821	+	0.996297i	$0.527403\pi$
$198$	0	0
$199$	11.3446	0.804194	0.402097	−	0.915597i	$-0.368281\pi$
$199$	11.3446	0.804194	0.402097	+	0.915597i	$0.368281\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0.259511	0.0182141
$204$	0	0
$205$	5.10275	0.356391
$206$	0	0
$207$	0	0
$208$	0	0
$209$	8.80957	0.609370
$210$	0	0
$211$	9.33123	0.642388	0.321194	−	0.947013i	$-0.395916\pi$
$211$	9.33123	0.642388	0.321194	+	0.947013i	$0.395916\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.290544	−0.0198149
$216$	0	0
$217$	−8.23917	−0.559311
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.65544	−0.245892
$222$	0	0
$223$	−9.17182	−0.614191	−0.307095	−	0.951679i	$-0.599357\pi$
$223$	−9.17182	−0.614191	−0.307095	+	0.951679i	$0.599357\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	0.760830	0.0504981	0.0252490	−	0.999681i	$-0.491962\pi$
$227$	0.760830	0.0504981	0.0252490	+	0.999681i	$0.491962\pi$
$228$	0	0
$229$	17.3596	1.14716	0.573578	−	0.819151i	$-0.305555\pi$
$229$	17.3596	1.14716	0.573578	+	0.819151i	$0.305555\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.08942	−0.595467	−0.297734	−	0.954649i	$-0.596231\pi$
$233$	−9.08942	−0.595467	−0.297734	+	0.954649i	$0.596231\pi$
$234$	0	0
$235$	15.0177	0.979647
$236$	0	0
$237$	0	0
$238$	0	0
$239$	26.3286	1.70305	0.851527	−	0.524310i	$-0.175677\pi$
$239$	26.3286	1.70305	0.851527	+	0.524310i	$0.175677\pi$
$240$	0	0
$241$	−6.62177	−0.426546	−0.213273	−	0.976993i	$-0.568412\pi$
$241$	−6.62177	−0.426546	−0.213273	+	0.976993i	$0.568412\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.39593	0.0891828
$246$	0	0
$247$	5.10275	0.324680
$248$	0	0
$249$	0	0
$250$	0	0
$251$	17.3800	1.09701	0.548507	−	0.836146i	$-0.315197\pi$
$251$	17.3800	1.09701	0.548507	+	0.836146i	$0.315197\pi$
$252$	0	0
$253$	−51.4597	−3.23525
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.2365	−1.69897	−0.849484	−	0.527614i	$-0.823087\pi$
$257$	−27.2365	−1.69897	−0.849484	+	0.527614i	$0.823087\pi$
$258$	0	0
$259$	−11.0177	−0.684607
$260$	0	0
$261$	0	0
$262$	0	0
$263$	16.9974	1.04810	0.524051	−	0.851687i	$-0.324420\pi$
$263$	16.9974	1.04810	0.524051	+	0.851687i	$0.324420\pi$
$264$	0	0
$265$	−3.34456	−0.205455
$266$	0	0
$267$	0	0
$268$	0	0
$269$	11.9283	0.727280	0.363640	−	0.931539i	$-0.381534\pi$
$269$	11.9283	0.727280	0.363640	+	0.931539i	$0.381534\pi$
$270$	0	0
$271$	30.0531	1.82560	0.912798	−	0.408411i	$-0.133917\pi$
$271$	30.0531	1.82560	0.912798	+	0.408411i	$0.133917\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−19.2569	−1.16123
$276$	0	0
$277$	−12.9974	−0.780936	−0.390468	−	0.920617i	$-0.627687\pi$
$277$	−12.9974	−0.780936	−0.390468	+	0.920617i	$0.627687\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	5.75382	0.343244	0.171622	−	0.985163i	$-0.445099\pi$
$281$	5.75382	0.343244	0.171622	+	0.985163i	$0.445099\pi$
$282$	0	0
$283$	7.82290	0.465023	0.232511	−	0.972594i	$-0.425306\pi$
$283$	7.82290	0.465023	0.232511	+	0.972594i	$0.425306\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.65544	−0.215774
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−25.4136	−1.48468	−0.742340	−	0.670023i	$-0.766284\pi$
$293$	−25.4136	−1.48468	−0.742340	+	0.670023i	$0.766284\pi$
$294$	0	0
$295$	13.1878	0.767823
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−29.8069	−1.72378
$300$	0	0
$301$	0.208136	0.0119968
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−15.0177	−0.859911
$306$	0	0
$307$	−28.4110	−1.62150	−0.810751	−	0.585392i	$-0.800941\pi$
$307$	−28.4110	−1.62150	−0.810751	+	0.585392i	$0.800941\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.1382	−1.14193	−0.570965	−	0.820974i	$-0.693431\pi$
$311$	−20.1382	−1.14193	−0.570965	+	0.820974i	$0.693431\pi$
$312$	0	0
$313$	1.51465	0.0856132	0.0428066	−	0.999083i	$-0.486370\pi$
$313$	1.51465	0.0856132	0.0428066	+	0.999083i	$0.486370\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	27.2435	1.53015	0.765075	−	0.643941i	$-0.222702\pi$
$317$	27.2435	1.53015	0.765075	+	0.643941i	$0.222702\pi$
$318$	0	0
$319$	−1.63774	−0.0916959
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−1.39593	−0.0776717
$324$	0	0
$325$	−11.1541	−0.618719
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−10.7582	−0.593118
$330$	0	0
$331$	−8.98667	−0.493952	−0.246976	−	0.969022i	$-0.579437\pi$
$331$	−8.98667	−0.493952	−0.246976	+	0.969022i	$0.579437\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−22.0221	−1.20319
$336$	0	0
$337$	11.2905	0.615035	0.307518	−	0.951542i	$-0.400502\pi$
$337$	11.2905	0.615035	0.307518	+	0.951542i	$0.400502\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	51.9965	2.81577
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	30.3977	1.63183	0.815916	−	0.578170i	$-0.196233\pi$
$347$	30.3977	1.63183	0.815916	+	0.578170i	$0.196233\pi$
$348$	0	0
$349$	−18.8096	−1.00685	−0.503426	−	0.864038i	$-0.667928\pi$
$349$	−18.8096	−1.00685	−0.503426	+	0.864038i	$0.667928\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−3.99299	−0.212525	−0.106263	−	0.994338i	$-0.533888\pi$
$353$	−3.99299	−0.212525	−0.106263	+	0.994338i	$0.533888\pi$
$354$	0	0
$355$	−0.505689	−0.0268392
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.0531	1.32225	0.661126	−	0.750275i	$-0.270079\pi$
$359$	25.0531	1.32225	0.661126	+	0.750275i	$0.270079\pi$
$360$	0	0
$361$	−17.0514	−0.897441
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.30825	−0.330189
$366$	0	0
$367$	31.0328	1.61990	0.809949	−	0.586501i	$-0.199495\pi$
$367$	31.0328	1.61990	0.809949	+	0.586501i	$0.199495\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.39593	0.124391
$372$	0	0
$373$	−12.1922	−0.631286	−0.315643	−	0.948878i	$-0.602220\pi$
$373$	−12.1922	−0.631286	−0.315643	+	0.948878i	$0.602220\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−0.948626	−0.0488567
$378$	0	0
$379$	−21.1675	−1.08730	−0.543650	−	0.839312i	$-0.682958\pi$
$379$	−21.1675	−1.08730	−0.543650	+	0.839312i	$0.682958\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−3.48799	−0.178228	−0.0891139	−	0.996021i	$-0.528404\pi$
$383$	−3.48799	−0.178228	−0.0891139	+	0.996021i	$0.528404\pi$
$384$	0	0
$385$	−8.80957	−0.448977
$386$	0	0
$387$	0	0
$388$	0	0
$389$	22.8946	1.16080	0.580401	−	0.814330i	$-0.302896\pi$
$389$	22.8946	1.16080	0.580401	+	0.814330i	$0.302896\pi$
$390$	0	0
$391$	8.15412	0.412372
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4.21251	−0.211954
$396$	0	0
$397$	−0.655442	−0.0328957	−0.0164479	−	0.999865i	$-0.505236\pi$
$397$	−0.655442	−0.0328957	−0.0164479	+	0.999865i	$0.505236\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.79450	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$401$	−4.79450	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$402$	0	0
$403$	30.1178	1.50028
$404$	0	0
$405$	0	0
$406$	0	0
$407$	69.5314	3.44655
$408$	0	0
$409$	18.4066	0.910149	0.455075	−	0.890453i	$-0.349613\pi$
$409$	18.4066	0.910149	0.455075	+	0.890453i	$0.349613\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.44731	−0.464872
$414$	0	0
$415$	13.9593	0.685236
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−24.9840	−1.22055	−0.610275	−	0.792190i	$-0.708941\pi$
$419$	−24.9840	−1.22055	−0.610275	+	0.792190i	$0.708941\pi$
$420$	0	0
$421$	−6.98667	−0.340509	−0.170255	−	0.985400i	$-0.554459\pi$
$421$	−6.98667	−0.340509	−0.170255	+	0.985400i	$0.554459\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.05137	0.148013
$426$	0	0
$427$	10.7582	0.520625
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−14.5811	−0.702346	−0.351173	−	0.936311i	$-0.614217\pi$
$431$	−14.5811	−0.702346	−0.351173	+	0.936311i	$0.614217\pi$
$432$	0	0
$433$	−19.9504	−0.958753	−0.479376	−	0.877609i	$-0.659137\pi$
$433$	−19.9504	−0.958753	−0.479376	+	0.877609i	$0.659137\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−11.3826	−0.544504
$438$	0	0
$439$	−9.13378	−0.435932	−0.217966	−	0.975956i	$-0.569942\pi$
$439$	−9.13378	−0.435932	−0.217966	+	0.975956i	$0.569942\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−20.2879	−0.963907	−0.481954	−	0.876197i	$-0.660073\pi$
$443$	−20.2879	−0.963907	−0.481954	+	0.876197i	$0.660073\pi$
$444$	0	0
$445$	−10.3489	−0.490586
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.67314	−0.173346	−0.0866732	−	0.996237i	$-0.527624\pi$
$449$	−3.67314	−0.173346	−0.0866732	+	0.996237i	$0.527624\pi$
$450$	0	0
$451$	23.0691	1.08628
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.10275	−0.239221
$456$	0	0
$457$	−8.46765	−0.396100	−0.198050	−	0.980192i	$-0.563461\pi$
$457$	−8.46765	−0.396100	−0.198050	+	0.980192i	$0.563461\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−23.1338	−1.07745	−0.538724	−	0.842482i	$-0.681093\pi$
$461$	−23.1338	−1.07745	−0.538724	+	0.842482i	$0.681093\pi$
$462$	0	0
$463$	−14.5324	−0.675376	−0.337688	−	0.941258i	$-0.609645\pi$
$463$	−14.5324	−0.675376	−0.337688	+	0.941258i	$0.609645\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.20814	0.0559059	0.0279529	−	0.999609i	$-0.491101\pi$
$467$	1.20814	0.0559059	0.0279529	+	0.999609i	$0.491101\pi$
$468$	0	0
$469$	15.7759	0.728463
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.31352	−0.0603959
$474$	0	0
$475$	−4.25951	−0.195440
$476$	0	0
$477$	0	0
$478$	0	0
$479$	20.5907	0.940815	0.470407	−	0.882449i	$-0.344107\pi$
$479$	20.5907	0.940815	0.470407	+	0.882449i	$0.344107\pi$
$480$	0	0
$481$	40.2746	1.83636
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−3.89725	−0.176965
$486$	0	0
$487$	14.4163	0.653264	0.326632	−	0.945152i	$-0.394086\pi$
$487$	14.4163	0.653264	0.326632	+	0.945152i	$0.394086\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	17.0647	0.770119	0.385060	−	0.922892i	$-0.374181\pi$
$491$	17.0647	0.770119	0.385060	+	0.922892i	$0.374181\pi$
$492$	0	0
$493$	0.259511	0.0116878
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0.362259	0.0162495
$498$	0	0
$499$	−27.7422	−1.24191	−0.620956	−	0.783845i	$-0.713256\pi$
$499$	−27.7422	−1.24191	−0.620956	+	0.783845i	$0.713256\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−16.2905	−0.726359	−0.363180	−	0.931719i	$-0.618309\pi$
$503$	−16.2905	−0.726359	−0.363180	+	0.931719i	$0.618309\pi$
$504$	0	0
$505$	−2.88829	−0.128527
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.61913	0.426360	0.213180	−	0.977013i	$-0.431618\pi$
$509$	9.61913	0.426360	0.213180	+	0.977013i	$0.431618\pi$
$510$	0	0
$511$	4.51902	0.199910
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−20.8920	−0.920610
$516$	0	0
$517$	67.8937	2.98596
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−12.0310	−0.527089	−0.263545	−	0.964647i	$-0.584892\pi$
$521$	−12.0310	−0.527089	−0.263545	+	0.964647i	$0.584892\pi$
$522$	0	0
$523$	−39.3730	−1.72166	−0.860829	−	0.508893i	$-0.830055\pi$
$523$	−39.3730	−1.72166	−0.860829	+	0.508893i	$0.830055\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.23917	−0.358904
$528$	0	0
$529$	43.4897	1.89086
$530$	0	0
$531$	0	0
$532$	0	0
$533$	13.3623	0.578784
$534$	0	0
$535$	6.76083	0.292296
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.31088	0.271829
$540$	0	0
$541$	−39.3056	−1.68988	−0.844940	−	0.534861i	$-0.820364\pi$
$541$	−39.3056	−1.68988	−0.844940	+	0.534861i	$0.820364\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−15.7608	−0.675120
$546$	0	0
$547$	41.3259	1.76697	0.883485	−	0.468460i	$-0.155191\pi$
$547$	41.3259	1.76697	0.883485	+	0.468460i	$0.155191\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.362259	−0.0154328
$552$	0	0
$553$	3.01770	0.128326
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−34.4934	−1.46153	−0.730766	−	0.682628i	$-0.760837\pi$
$557$	−34.4934	−1.46153	−0.730766	+	0.682628i	$0.760837\pi$
$558$	0	0
$559$	−0.760830	−0.0321797
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.41363	0.228157	0.114079	−	0.993472i	$-0.463608\pi$
$563$	5.41363	0.228157	0.114079	+	0.993472i	$0.463608\pi$
$564$	0	0
$565$	18.0531	0.759500
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.75382	−0.241213	−0.120606	−	0.992700i	$-0.538484\pi$
$569$	−5.75382	−0.241213	−0.120606	+	0.992700i	$0.538484\pi$
$570$	0	0
$571$	−32.9300	−1.37808	−0.689039	−	0.724724i	$-0.741967\pi$
$571$	−32.9300	−1.37808	−0.689039	+	0.724724i	$0.741967\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	24.8813	1.03762
$576$	0	0
$577$	−16.3800	−0.681907	−0.340953	−	0.940080i	$-0.610750\pi$
$577$	−16.3800	−0.681907	−0.340953	+	0.940080i	$0.610750\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.0000	−0.414870
$582$	0	0
$583$	−15.1204	−0.626225
$584$	0	0
$585$	0	0
$586$	0	0
$587$	26.6502	1.09997	0.549985	−	0.835175i	$-0.314633\pi$
$587$	26.6502	1.09997	0.549985	+	0.835175i	$0.314633\pi$
$588$	0	0
$589$	11.5013	0.473904
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−11.8636	−0.487179	−0.243589	−	0.969878i	$-0.578325\pi$
$593$	−11.8636	−0.487179	−0.243589	+	0.969878i	$0.578325\pi$
$594$	0	0
$595$	1.39593	0.0572276
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.16745	0.211136	0.105568	−	0.994412i	$-0.466334\pi$
$599$	5.16745	0.211136	0.105568	+	0.994412i	$0.466334\pi$
$600$	0	0
$601$	−8.34192	−0.340274	−0.170137	−	0.985420i	$-0.554421\pi$
$601$	−8.34192	−0.340274	−0.170137	+	0.985420i	$0.554421\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	40.2409	1.63603
$606$	0	0
$607$	−30.6502	−1.24405	−0.622026	−	0.782997i	$-0.713690\pi$
$607$	−30.6502	−1.24405	−0.622026	+	0.782997i	$0.713690\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	39.3259	1.59096
$612$	0	0
$613$	−8.41363	−0.339823	−0.169912	−	0.985459i	$-0.554348\pi$
$613$	−8.41363	−0.339823	−0.169912	+	0.985459i	$0.554348\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.2188	−0.813979	−0.406990	−	0.913433i	$-0.633422\pi$
$617$	−20.2188	−0.813979	−0.406990	+	0.913433i	$0.633422\pi$
$618$	0	0
$619$	23.3800	0.939720	0.469860	−	0.882741i	$-0.344304\pi$
$619$	23.3800	0.939720	0.469860	+	0.882741i	$0.344304\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	7.41363	0.297021
$624$	0	0
$625$	−0.432244	−0.0172898
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−11.0177	−0.439305
$630$	0	0
$631$	−25.7866	−1.02655	−0.513274	−	0.858225i	$-0.671567\pi$
$631$	−25.7866	−1.02655	−0.513274	+	0.858225i	$0.671567\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	11.5828	0.459650
$636$	0	0
$637$	3.65544	0.144834
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−18.5030	−0.730827	−0.365413	−	0.930845i	$-0.619072\pi$
$641$	−18.5030	−0.730827	−0.365413	+	0.930845i	$0.619072\pi$
$642$	0	0
$643$	−14.4429	−0.569574	−0.284787	−	0.958591i	$-0.591923\pi$
$643$	−14.4429	−0.569574	−0.284787	+	0.958591i	$0.591923\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.85921	0.348291	0.174146	−	0.984720i	$-0.444284\pi$
$647$	8.85921	0.348291	0.174146	+	0.984720i	$0.444284\pi$
$648$	0	0
$649$	59.6209	2.34032
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−44.8520	−1.75519	−0.877597	−	0.479400i	$-0.840854\pi$
$653$	−44.8520	−1.75519	−0.877597	+	0.479400i	$0.840854\pi$
$654$	0	0
$655$	0.262150	0.0102430
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.9150	0.970549	0.485274	−	0.874362i	$-0.338720\pi$
$659$	24.9150	0.970549	0.485274	+	0.874362i	$0.338720\pi$
$660$	0	0
$661$	−0.774162	−0.0301114	−0.0150557	−	0.999887i	$-0.504793\pi$
$661$	−0.774162	−0.0301114	−0.0150557	+	0.999887i	$0.504793\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.94863	−0.0755645
$666$	0	0
$667$	2.11608	0.0819350
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−67.8937	−2.62101
$672$	0	0
$673$	−0.108027	−0.00416414	−0.00208207	−	0.999998i	$-0.500663\pi$
$673$	−0.108027	−0.00416414	−0.00208207	+	0.999998i	$0.500663\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−48.1425	−1.85027	−0.925134	−	0.379641i	$-0.876047\pi$
$677$	−48.1425	−1.85027	−0.925134	+	0.379641i	$0.876047\pi$
$678$	0	0
$679$	2.79186	0.107142
$680$	0	0
$681$	0	0
$682$	0	0
$683$	13.0354	0.498786	0.249393	−	0.968402i	$-0.419769\pi$
$683$	13.0354	0.498786	0.249393	+	0.968402i	$0.419769\pi$
$684$	0	0
$685$	18.9904	0.725584
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.75819	−0.333660
$690$	0	0
$691$	35.2099	1.33945	0.669723	−	0.742611i	$-0.266413\pi$
$691$	35.2099	1.33945	0.669723	+	0.742611i	$0.266413\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11.4543	−0.434487
$696$	0	0
$697$	−3.65544	−0.138460
$698$	0	0
$699$	0	0
$700$	0	0
$701$	35.9327	1.35716	0.678579	−	0.734528i	$-0.262596\pi$
$701$	35.9327	1.35716	0.678579	+	0.734528i	$0.262596\pi$
$702$	0	0
$703$	15.3800	0.580066
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.06908	0.0778156
$708$	0	0
$709$	14.8325	0.557048	0.278524	−	0.960429i	$-0.410155\pi$
$709$	14.8325	0.557048	0.278524	+	0.960429i	$0.410155\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−67.1832	−2.51603
$714$	0	0
$715$	32.2029	1.20432
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−11.8122	−0.440521	−0.220260	−	0.975441i	$-0.570691\pi$
$719$	−11.8122	−0.440521	−0.220260	+	0.975441i	$0.570691\pi$
$720$	0	0
$721$	14.9663	0.557375
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.791864	0.0294091
$726$	0	0
$727$	−49.5164	−1.83646	−0.918230	−	0.396046i	$-0.870382\pi$
$727$	−49.5164	−1.83646	−0.918230	+	0.396046i	$0.870382\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.208136	0.00769820
$732$	0	0
$733$	−18.3843	−0.679041	−0.339520	−	0.940599i	$-0.610265\pi$
$733$	−18.3843	−0.679041	−0.339520	+	0.940599i	$0.610265\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−99.5598	−3.66733
$738$	0	0
$739$	−23.5190	−0.865161	−0.432581	−	0.901595i	$-0.642397\pi$
$739$	−23.5190	−0.865161	−0.432581	+	0.901595i	$0.642397\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.6377	0.573693	0.286847	−	0.957977i	$-0.407393\pi$
$743$	15.6377	0.573693	0.286847	+	0.957977i	$0.407393\pi$
$744$	0	0
$745$	15.7892	0.578472
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−4.84324	−0.176968
$750$	0	0
$751$	−44.2206	−1.61363	−0.806816	−	0.590803i	$-0.798811\pi$
$751$	−44.2206	−1.61363	−0.806816	+	0.590803i	$0.798811\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	23.8520	0.868062
$756$	0	0
$757$	18.1541	0.659823	0.329911	−	0.944012i	$-0.392981\pi$
$757$	18.1541	0.659823	0.329911	+	0.944012i	$0.392981\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−45.6865	−1.65613	−0.828067	−	0.560630i	$-0.810559\pi$
$761$	−45.6865	−1.65613	−0.828067	+	0.560630i	$0.810559\pi$
$762$	0	0
$763$	11.2905	0.408745
$764$	0	0
$765$	0	0
$766$	0	0
$767$	34.5341	1.24695
$768$	0	0
$769$	−35.7936	−1.29075	−0.645375	−	0.763866i	$-0.723299\pi$
$769$	−35.7936	−1.29075	−0.645375	+	0.763866i	$0.723299\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23.1692	0.833338	0.416669	−	0.909058i	$-0.363197\pi$
$773$	23.1692	0.833338	0.416669	+	0.909058i	$0.363197\pi$
$774$	0	0
$775$	−25.1408	−0.903084
$776$	0	0
$777$	0	0
$778$	0	0
$779$	5.10275	0.182825
$780$	0	0
$781$	−2.28617	−0.0818058
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.4676	−0.373606
$786$	0	0
$787$	−15.6795	−0.558913	−0.279456	−	0.960158i	$-0.590154\pi$
$787$	−15.6795	−0.558913	−0.279456	+	0.960158i	$0.590154\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−12.9327	−0.459832
$792$	0	0
$793$	−39.3259	−1.39651
$794$	0	0
$795$	0	0
$796$	0	0
$797$	26.5491	0.940419	0.470209	−	0.882555i	$-0.344178\pi$
$797$	26.5491	0.940419	0.470209	+	0.882555i	$0.344178\pi$
$798$	0	0
$799$	−10.7582	−0.380597
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−28.5190	−1.00641
$804$	0	0
$805$	11.3826	0.401184
$806$	0	0
$807$	0	0
$808$	0	0
$809$	49.1329	1.72742	0.863710	−	0.503989i	$-0.168135\pi$
$809$	49.1329	1.72742	0.863710	+	0.503989i	$0.168135\pi$
$810$	0	0
$811$	12.7919	0.449183	0.224592	−	0.974453i	$-0.427895\pi$
$811$	12.7919	0.449183	0.224592	+	0.974453i	$0.427895\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0.724518	0.0253788
$816$	0	0
$817$	−0.290544	−0.0101649
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−17.6705	−0.616705	−0.308352	−	0.951272i	$-0.599778\pi$
$821$	−17.6705	−0.616705	−0.308352	+	0.951272i	$0.599778\pi$
$822$	0	0
$823$	−37.4020	−1.30375	−0.651876	−	0.758325i	$-0.726018\pi$
$823$	−37.4020	−1.30375	−0.651876	+	0.758325i	$0.726018\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−35.6519	−1.23974	−0.619869	−	0.784705i	$-0.712814\pi$
$827$	−35.6519	−1.23974	−0.619869	+	0.784705i	$0.712814\pi$
$828$	0	0
$829$	−6.34019	−0.220204	−0.110102	−	0.993920i	$-0.535118\pi$
$829$	−6.34019	−0.220204	−0.110102	+	0.993920i	$0.535118\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	27.5811	0.954483
$836$	0	0
$837$	0	0
$838$	0	0
$839$	41.0778	1.41816	0.709082	−	0.705126i	$-0.249110\pi$
$839$	41.0778	1.41816	0.709082	+	0.705126i	$0.249110\pi$
$840$	0	0
$841$	−28.9327	−0.997678
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.505689	0.0173962
$846$	0	0
$847$	−28.8273	−0.990517
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−89.8397	−3.07966
$852$	0	0
$853$	15.4206	0.527993	0.263996	−	0.964524i	$-0.414959\pi$
$853$	15.4206	0.527993	0.263996	+	0.964524i	$0.414959\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−9.45431	−0.322953	−0.161477	−	0.986877i	$-0.551626\pi$
$857$	−9.45431	−0.322953	−0.161477	+	0.986877i	$0.551626\pi$
$858$	0	0
$859$	−29.9947	−1.02341	−0.511703	−	0.859162i	$-0.670985\pi$
$859$	−29.9947	−1.02341	−0.511703	+	0.859162i	$0.670985\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−47.3666	−1.61238	−0.806189	−	0.591658i	$-0.798474\pi$
$863$	−47.3666	−1.61238	−0.806189	+	0.591658i	$0.798474\pi$
$864$	0	0
$865$	−10.3242	−0.351034
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−19.0444	−0.646036
$870$	0	0
$871$	−57.6679	−1.95400
$872$	0	0
$873$	0	0
$874$	0	0
$875$	11.2392	0.379953
$876$	0	0
$877$	−37.7068	−1.27327	−0.636634	−	0.771166i	$-0.719674\pi$
$877$	−37.7068	−1.27327	−0.636634	+	0.771166i	$0.719674\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−34.2728	−1.15468	−0.577341	−	0.816503i	$-0.695910\pi$
$881$	−34.2728	−1.15468	−0.577341	+	0.816503i	$0.695910\pi$
$882$	0	0
$883$	−14.0107	−0.471497	−0.235749	−	0.971814i	$-0.575754\pi$
$883$	−14.0107	−0.471497	−0.235749	+	0.971814i	$0.575754\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−54.8176	−1.84060	−0.920298	−	0.391219i	$-0.872054\pi$
$887$	−54.8176	−1.84060	−0.920298	+	0.391219i	$0.872054\pi$
$888$	0	0
$889$	−8.29755	−0.278291
$890$	0	0
$891$	0	0
$892$	0	0
$893$	15.0177	0.502548
$894$	0	0
$895$	5.82117	0.194580
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.13815	−0.0713113
$900$	0	0
$901$	2.39593	0.0798201
$902$	0	0
$903$	0	0
$904$	0	0
$905$	28.6926	0.953773
$906$	0	0
$907$	37.5164	1.24571	0.622856	−	0.782337i	$-0.285972\pi$
$907$	37.5164	1.24571	0.622856	+	0.782337i	$0.285972\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.52166	0.282335	0.141168	−	0.989986i	$-0.454914\pi$
$911$	8.52166	0.282335	0.141168	+	0.989986i	$0.454914\pi$
$912$	0	0
$913$	63.1088	2.08860
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−0.187796	−0.00620156
$918$	0	0
$919$	−13.1294	−0.433099	−0.216550	−	0.976272i	$-0.569480\pi$
$919$	−13.1294	−0.433099	−0.216550	+	0.976272i	$0.569480\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.32422	−0.0435871
$924$	0	0
$925$	−33.6191	−1.10539
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−36.6795	−1.20341	−0.601707	−	0.798717i	$-0.705513\pi$
$929$	−36.6795	−1.20341	−0.601707	+	0.798717i	$0.705513\pi$
$930$	0	0
$931$	1.39593	0.0457498
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−8.80957	−0.288104
$936$	0	0
$937$	−3.33823	−0.109055	−0.0545277	−	0.998512i	$-0.517365\pi$
$937$	−3.33823	−0.109055	−0.0545277	+	0.998512i	$0.517365\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.48098	0.0482785	0.0241393	−	0.999709i	$-0.492315\pi$
$941$	1.48098	0.0482785	0.0241393	+	0.999709i	$0.492315\pi$
$942$	0	0
$943$	−29.8069	−0.970647
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57.1842	1.85824	0.929119	−	0.369780i	$-0.120567\pi$
$947$	57.1842	1.85824	0.929119	+	0.369780i	$0.120567\pi$
$948$	0	0
$949$	−16.5190	−0.536230
$950$	0	0
$951$	0	0
$952$	0	0
$953$	48.4261	1.56867	0.784337	−	0.620335i	$-0.213003\pi$
$953$	48.4261	1.56867	0.784337	+	0.620335i	$0.213003\pi$
$954$	0	0
$955$	−6.45168	−0.208771
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−13.6041	−0.439298
$960$	0	0
$961$	36.8839	1.18980
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.03103	−0.161955
$966$	0	0
$967$	1.43834	0.0462540	0.0231270	−	0.999733i	$-0.492638\pi$
$967$	1.43834	0.0462540	0.0231270	+	0.999733i	$0.492638\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−38.2409	−1.22721	−0.613604	−	0.789614i	$-0.710281\pi$
$971$	−38.2409	−1.22721	−0.613604	+	0.789614i	$0.710281\pi$
$972$	0	0
$973$	8.20550	0.263056
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−1.07872	−0.0345114	−0.0172557	−	0.999851i	$-0.505493\pi$
$977$	−1.07872	−0.0345114	−0.0172557	+	0.999851i	$0.505493\pi$
$978$	0	0
$979$	−46.7866	−1.49531
$980$	0	0
$981$	0	0
$982$	0	0
$983$	23.8876	0.761896	0.380948	−	0.924596i	$-0.375598\pi$
$983$	23.8876	0.761896	0.380948	+	0.924596i	$0.375598\pi$
$984$	0	0
$985$	−3.36927	−0.107354
$986$	0	0
$987$	0	0
$988$	0	0
$989$	1.69717	0.0539668
$990$	0	0
$991$	41.0594	1.30430	0.652148	−	0.758092i	$-0.273868\pi$
$991$	41.0594	1.30430	0.652148	+	0.758092i	$0.273868\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.8362	0.502042
$996$	0	0
$997$	−0.478340	−0.0151492	−0.00757458	−	0.999971i	$-0.502411\pi$
$997$	−0.478340	−0.0151492	−0.00757458	+	0.999971i	$0.502411\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4284.2.a.u.1.2	yes	3
3.2	odd	2		4284.2.a.t.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4284.2.a.t.1.2	✓	3	3.2	odd	2
4284.2.a.u.1.2	yes	3	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:	\( N \)	\(=\)	\( 4284 = 2^{2} \cdot 3^{2} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4284.a (trivial)

\(f(q)\)	\(=\)	\(q+1.39593 q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q+1.39593 q^{5} -1.00000 q^{7} +6.31088 q^{11} +3.65544 q^{13} -1.00000 q^{17} +1.39593 q^{19} -8.15412 q^{23} -3.05137 q^{25} -0.259511 q^{29} +8.23917 q^{31} -1.39593 q^{35} +11.0177 q^{37} +3.65544 q^{41} -0.208136 q^{43} +10.7582 q^{47} +1.00000 q^{49} -2.39593 q^{53} +8.80957 q^{55} +9.44731 q^{59} -10.7582 q^{61} +5.10275 q^{65} -15.7759 q^{67} -0.362259 q^{71} -4.51902 q^{73} -6.31088 q^{77} -3.01770 q^{79} +10.0000 q^{83} -1.39593 q^{85} -7.41363 q^{89} -3.65544 q^{91} +1.94863 q^{95} -2.79186 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{5} - 3 q^{7}+O(q^{10}) \) 3 * q + q^5 - 3 * q^7	\( 3 q + q^{5} - 3 q^{7} + 7 q^{11} + 5 q^{13} - 3 q^{17} + q^{19} + 3 q^{23} + 2 q^{29} + 6 q^{31} - q^{35} + 6 q^{37} + 5 q^{41} - 7 q^{43} + 8 q^{47} + 3 q^{49} - 4 q^{53} - 7 q^{55} + 16 q^{59} - 8 q^{61} - 3 q^{65} + 4 q^{67} + 20 q^{71} - 8 q^{73} - 7 q^{77} + 18 q^{79} + 30 q^{83} - q^{85} + 8 q^{89} - 5 q^{91} + 15 q^{95} - 2 q^{97}+O(q^{100}) \) 3 * q + q^5 - 3 * q^7 + 7 * q^11 + 5 * q^13 - 3 * q^17 + q^19 + 3 * q^23 + 2 * q^29 + 6 * q^31 - q^35 + 6 * q^37 + 5 * q^41 - 7 * q^43 + 8 * q^47 + 3 * q^49 - 4 * q^53 - 7 * q^55 + 16 * q^59 - 8 * q^61 - 3 * q^65 + 4 * q^67 + 20 * q^71 - 8 * q^73 - 7 * q^77 + 18 * q^79 + 30 * q^83 - q^85 + 8 * q^89 - 5 * q^91 + 15 * q^95 - 2 * q^97

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