LMFDB - Embedded newform 363.2.a.j.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [363,2,Mod(1,363)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("363.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(363, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	363.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$2.89856959337$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 4 $ x^4 - 7*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.52434$ of defining polynomial
Character	$\chi$	$=$	363.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.52434 q^{2} +1.00000 q^{3} +4.37228 q^{4} -2.37228 q^{5} +2.52434 q^{6} +0.792287 q^{7} +5.98844 q^{8} +1.00000 q^{9} -5.98844 q^{10} +4.37228 q^{12} -4.10891 q^{13} +2.00000 q^{14} -2.37228 q^{15} +6.37228 q^{16} -5.98844 q^{17} +2.52434 q^{18} +4.25639 q^{19} -10.3723 q^{20} +0.792287 q^{21} +2.00000 q^{23} +5.98844 q^{24} +0.627719 q^{25} -10.3723 q^{26} +1.00000 q^{27} +3.46410 q^{28} -2.52434 q^{29} -5.98844 q^{30} +7.37228 q^{31} +4.10891 q^{32} -15.1168 q^{34} -1.87953 q^{35} +4.37228 q^{36} +5.00000 q^{37} +10.7446 q^{38} -4.10891 q^{39} -14.2063 q^{40} -5.69349 q^{41} +2.00000 q^{42} -6.63325 q^{43} -2.37228 q^{45} +5.04868 q^{46} -1.25544 q^{47} +6.37228 q^{48} -6.37228 q^{49} +1.58457 q^{50} -5.98844 q^{51} -17.9653 q^{52} +13.1168 q^{53} +2.52434 q^{54} +4.74456 q^{56} +4.25639 q^{57} -6.37228 q^{58} -6.00000 q^{59} -10.3723 q^{60} -2.67181 q^{61} +18.6101 q^{62} +0.792287 q^{63} -2.37228 q^{64} +9.74749 q^{65} +16.1168 q^{67} -26.1831 q^{68} +2.00000 q^{69} -4.74456 q^{70} +0.744563 q^{71} +5.98844 q^{72} +7.42554 q^{73} +12.6217 q^{74} +0.627719 q^{75} +18.6101 q^{76} -10.3723 q^{78} +5.84096 q^{79} -15.1168 q^{80} +1.00000 q^{81} -14.3723 q^{82} +8.51278 q^{83} +3.46410 q^{84} +14.2063 q^{85} -16.7446 q^{86} -2.52434 q^{87} -6.37228 q^{89} -5.98844 q^{90} -3.25544 q^{91} +8.74456 q^{92} +7.37228 q^{93} -3.16915 q^{94} -10.0974 q^{95} +4.10891 q^{96} -12.4891 q^{97} -16.0858 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 6 q^{4} + 2 q^{5} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 2 q^{15} + 14 q^{16} - 30 q^{20} + 8 q^{23} + 14 q^{25} - 30 q^{26} + 4 q^{27} + 18 q^{31} - 26 q^{34} + 6 q^{36} + 20 q^{37} + 20 q^{38}+ \cdots - 4 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 + 6 * q^4 + 2 * q^5 + 4 * q^9 + 6 * q^12 + 8 * q^14 + 2 * q^15 + 14 * q^16 - 30 * q^20 + 8 * q^23 + 14 * q^25 - 30 * q^26 + 4 * q^27 + 18 * q^31 - 26 * q^34 + 6 * q^36 + 20 * q^37 + 20 * q^38 + 8 * q^42 + 2 * q^45 - 28 * q^47 + 14 * q^48 - 14 * q^49 + 18 * q^53 - 4 * q^56 - 14 * q^58 - 24 * q^59 - 30 * q^60 + 2 * q^64 + 30 * q^67 + 8 * q^69 + 4 * q^70 - 20 * q^71 + 14 * q^75 - 30 * q^78 - 26 * q^80 + 4 * q^81 - 46 * q^82 - 44 * q^86 - 14 * q^89 - 36 * q^91 + 12 * q^92 + 18 * q^93 - 4 * q^97

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 $

$2$	2.52434	1.78498	0.892488	−	0.451071i	$-0.148958\pi$
$2$	2.52434	1.78498	0.892488	+	0.451071i	$0.148958\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.37228	2.18614
$5$	−2.37228	−1.06092	−0.530458	−	0.847711i	$-0.677980\pi$
$5$	−2.37228	−1.06092	−0.530458	+	0.847711i	$0.677980\pi$
$6$	2.52434	1.03056
$7$	0.792287	0.299456	0.149728	−	0.988727i	$-0.452160\pi$
$7$	0.792287	0.299456	0.149728	+	0.988727i	$0.452160\pi$
$8$	5.98844	2.11723
$9$	1.00000	0.333333
$10$	−5.98844	−1.89371
$11$	0	0
$11$	0	0
$12$	4.37228	1.26217
$13$	−4.10891	−1.13961	−0.569804	−	0.821781i	$-0.692981\pi$
$13$	−4.10891	−1.13961	−0.569804	+	0.821781i	$0.692981\pi$
$14$	2.00000	0.534522
$15$	−2.37228	−0.612520
$16$	6.37228	1.59307
$17$	−5.98844	−1.45241	−0.726205	−	0.687478i	$-0.758718\pi$
$17$	−5.98844	−1.45241	−0.726205	+	0.687478i	$0.758718\pi$
$18$	2.52434	0.594992
$19$	4.25639	0.976483	0.488241	−	0.872709i	$-0.337639\pi$
$19$	4.25639	0.976483	0.488241	+	0.872709i	$0.337639\pi$
$20$	−10.3723	−2.31931
$21$	0.792287	0.172891
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	5.98844	1.22239
$25$	0.627719	0.125544
$26$	−10.3723	−2.03417
$27$	1.00000	0.192450
$28$	3.46410	0.654654
$29$	−2.52434	−0.468758	−0.234379	−	0.972145i	$-0.575306\pi$
$29$	−2.52434	−0.468758	−0.234379	+	0.972145i	$0.575306\pi$
$30$	−5.98844	−1.09333
$31$	7.37228	1.32410	0.662050	−	0.749459i	$-0.269686\pi$
$31$	7.37228	1.32410	0.662050	+	0.749459i	$0.269686\pi$
$32$	4.10891	0.726360
$33$	0	0
$34$	−15.1168	−2.59252
$35$	−1.87953	−0.317698
$36$	4.37228	0.728714
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	10.7446	1.74300
$39$	−4.10891	−0.657952
$40$	−14.2063	−2.24621
$41$	−5.69349	−0.889173	−0.444587	−	0.895736i	$-0.646649\pi$
$41$	−5.69349	−0.889173	−0.444587	+	0.895736i	$0.646649\pi$
$42$	2.00000	0.308607
$43$	−6.63325	−1.01156	−0.505781	−	0.862662i	$-0.668795\pi$
$43$	−6.63325	−1.01156	−0.505781	+	0.862662i	$0.668795\pi$
$44$	0	0
$45$	−2.37228	−0.353639
$46$	5.04868	0.744387
$47$	−1.25544	−0.183124	−0.0915622	−	0.995799i	$-0.529186\pi$
$47$	−1.25544	−0.183124	−0.0915622	+	0.995799i	$0.529186\pi$
$48$	6.37228	0.919760
$49$	−6.37228	−0.910326
$50$	1.58457	0.224093
$51$	−5.98844	−0.838549
$52$	−17.9653	−2.49134
$53$	13.1168	1.80174	0.900869	−	0.434092i	$-0.142931\pi$
$53$	13.1168	1.80174	0.900869	+	0.434092i	$0.142931\pi$
$54$	2.52434	0.343519
$55$	0	0
$56$	4.74456	0.634019
$57$	4.25639	0.563772
$58$	−6.37228	−0.836722
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−10.3723	−1.33906
$61$	−2.67181	−0.342091	−0.171045	−	0.985263i	$-0.554714\pi$
$61$	−2.67181	−0.342091	−0.171045	+	0.985263i	$0.554714\pi$
$62$	18.6101	2.36349
$63$	0.792287	0.0998188
$64$	−2.37228	−0.296535
$65$	9.74749	1.20903
$66$	0	0
$67$	16.1168	1.96899	0.984493	−	0.175424i	$-0.0561297\pi$
$67$	16.1168	1.96899	0.984493	+	0.175424i	$0.0561297\pi$
$68$	−26.1831	−3.17517
$69$	2.00000	0.240772
$70$	−4.74456	−0.567084
$71$	0.744563	0.0883633	0.0441817	−	0.999024i	$-0.485932\pi$
$71$	0.744563	0.0883633	0.0441817	+	0.999024i	$0.485932\pi$
$72$	5.98844	0.705744
$73$	7.42554	0.869093	0.434547	−	0.900649i	$-0.356909\pi$
$73$	7.42554	0.869093	0.434547	+	0.900649i	$0.356909\pi$
$74$	12.6217	1.46724
$75$	0.627719	0.0724827
$76$	18.6101	2.13473
$77$	0	0
$78$	−10.3723	−1.17443
$79$	5.84096	0.657160	0.328580	−	0.944476i	$-0.393430\pi$
$79$	5.84096	0.657160	0.328580	+	0.944476i	$0.393430\pi$
$80$	−15.1168	−1.69011
$81$	1.00000	0.111111
$82$	−14.3723	−1.58715
$83$	8.51278	0.934399	0.467199	−	0.884152i	$-0.345263\pi$
$83$	8.51278	0.934399	0.467199	+	0.884152i	$0.345263\pi$
$84$	3.46410	0.377964
$85$	14.2063	1.54089
$86$	−16.7446	−1.80561
$87$	−2.52434	−0.270637
$88$	0	0
$89$	−6.37228	−0.675460	−0.337730	−	0.941243i	$-0.609659\pi$
$89$	−6.37228	−0.675460	−0.337730	+	0.941243i	$0.609659\pi$
$90$	−5.98844	−0.631237
$91$	−3.25544	−0.341263
$92$	8.74456	0.911684
$93$	7.37228	0.764470
$94$	−3.16915	−0.326873
$95$	−10.0974	−1.03597
$96$	4.10891	0.419364
$97$	−12.4891	−1.26808	−0.634039	−	0.773301i	$-0.718604\pi$
$97$	−12.4891	−1.26808	−0.634039	+	0.773301i	$0.718604\pi$
$98$	−16.0858	−1.62491
$99$	0	0
$100$	2.74456	0.274456
$101$	−5.34363	−0.531711	−0.265855	−	0.964013i	$-0.585654\pi$
$101$	−5.34363	−0.531711	−0.265855	+	0.964013i	$0.585654\pi$
$102$	−15.1168	−1.49679
$103$	8.11684	0.799776	0.399888	−	0.916564i	$-0.369049\pi$
$103$	8.11684	0.799776	0.399888	+	0.916564i	$0.369049\pi$
$104$	−24.6060	−2.41281
$105$	−1.87953	−0.183423
$106$	33.1113	3.21606
$107$	−0.294954	−0.0285142	−0.0142571	−	0.999898i	$-0.504538\pi$
$107$	−0.294954	−0.0285142	−0.0142571	+	0.999898i	$0.504538\pi$
$108$	4.37228	0.420723
$109$	9.94987	0.953025	0.476513	−	0.879168i	$-0.341901\pi$
$109$	9.94987	0.953025	0.476513	+	0.879168i	$0.341901\pi$
$110$	0	0
$111$	5.00000	0.474579
$112$	5.04868	0.477055
$113$	−4.37228	−0.411310	−0.205655	−	0.978625i	$-0.565932\pi$
$113$	−4.37228	−0.411310	−0.205655	+	0.978625i	$0.565932\pi$
$114$	10.7446	1.00632
$115$	−4.74456	−0.442433
$116$	−11.0371	−1.02477
$117$	−4.10891	−0.379869
$118$	−15.1460	−1.39430
$119$	−4.74456	−0.434933
$120$	−14.2063	−1.29685
$121$	0	0
$122$	−6.74456	−0.610624
$123$	−5.69349	−0.513364
$124$	32.2337	2.89467
$125$	10.3723	0.927725
$126$	2.00000	0.178174
$127$	19.6974	1.74786	0.873929	−	0.486053i	$-0.161564\pi$
$127$	19.6974	1.74786	0.873929	+	0.486053i	$0.161564\pi$
$128$	−14.2063	−1.25567
$129$	−6.63325	−0.584025
$130$	24.6060	2.15809
$131$	6.63325	0.579550	0.289775	−	0.957095i	$-0.406420\pi$
$131$	6.63325	0.579550	0.289775	+	0.957095i	$0.406420\pi$
$132$	0	0
$133$	3.37228	0.292414
$134$	40.6844	3.51459
$135$	−2.37228	−0.204173
$136$	−35.8614	−3.07509
$137$	0.744563	0.0636123	0.0318061	−	0.999494i	$-0.489874\pi$
$137$	0.744563	0.0636123	0.0318061	+	0.999494i	$0.489874\pi$
$138$	5.04868	0.429772
$139$	−10.3923	−0.881464	−0.440732	−	0.897639i	$-0.645281\pi$
$139$	−10.3923	−0.881464	−0.440732	+	0.897639i	$0.645281\pi$
$140$	−8.21782	−0.694533
$141$	−1.25544	−0.105727
$142$	1.87953	0.157726
$143$	0	0
$144$	6.37228	0.531023
$145$	5.98844	0.497313
$146$	18.7446	1.55131
$147$	−6.37228	−0.525577
$148$	21.8614	1.79700
$149$	−5.98844	−0.490592	−0.245296	−	0.969448i	$-0.578885\pi$
$149$	−5.98844	−0.490592	−0.245296	+	0.969448i	$0.578885\pi$
$150$	1.58457	0.129380
$151$	2.17448	0.176957	0.0884784	−	0.996078i	$-0.471800\pi$
$151$	2.17448	0.176957	0.0884784	+	0.996078i	$0.471800\pi$
$152$	25.4891	2.06744
$153$	−5.98844	−0.484137
$154$	0	0
$155$	−17.4891	−1.40476
$156$	−17.9653	−1.43838
$157$	6.62772	0.528950	0.264475	−	0.964393i	$-0.414801\pi$
$157$	6.62772	0.528950	0.264475	+	0.964393i	$0.414801\pi$
$158$	14.7446	1.17301
$159$	13.1168	1.04023
$160$	−9.74749	−0.770607
$161$	1.58457	0.124882
$162$	2.52434	0.198331
$163$	−9.37228	−0.734094	−0.367047	−	0.930202i	$-0.619631\pi$
$163$	−9.37228	−0.734094	−0.367047	+	0.930202i	$0.619631\pi$
$164$	−24.8935	−1.94386
$165$	0	0
$166$	21.4891	1.66788
$167$	−2.87419	−0.222412	−0.111206	−	0.993797i	$-0.535471\pi$
$167$	−2.87419	−0.222412	−0.111206	+	0.993797i	$0.535471\pi$
$168$	4.74456	0.366051
$169$	3.88316	0.298704
$170$	35.8614	2.75044
$171$	4.25639	0.325494
$172$	−29.0024	−2.21141
$173$	−6.92820	−0.526742	−0.263371	−	0.964695i	$-0.584834\pi$
$173$	−6.92820	−0.526742	−0.263371	+	0.964695i	$0.584834\pi$
$174$	−6.37228	−0.483081
$175$	0.497333	0.0375949
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−16.0858	−1.20568
$179$	−24.2337	−1.81131	−0.905655	−	0.424014i	$-0.860621\pi$
$179$	−24.2337	−1.81131	−0.905655	+	0.424014i	$0.860621\pi$
$180$	−10.3723	−0.773104
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	−8.21782	−0.609146
$183$	−2.67181	−0.197506
$184$	11.9769	0.882947
$185$	−11.8614	−0.872068
$186$	18.6101	1.36456
$187$	0	0
$188$	−5.48913	−0.400336
$189$	0.792287	0.0576304
$190$	−25.4891	−1.84918
$191$	−17.4891	−1.26547	−0.632734	−	0.774369i	$-0.718067\pi$
$191$	−17.4891	−1.26547	−0.632734	+	0.774369i	$0.718067\pi$
$192$	−2.37228	−0.171205
$193$	−17.1730	−1.23614	−0.618071	−	0.786122i	$-0.712086\pi$
$193$	−17.1730	−1.23614	−0.618071	+	0.786122i	$0.712086\pi$
$194$	−31.5268	−2.26349
$195$	9.74749	0.698033
$196$	−27.8614	−1.99010
$197$	21.1345	1.50577	0.752884	−	0.658153i	$-0.228662\pi$
$197$	21.1345	1.50577	0.752884	+	0.658153i	$0.228662\pi$
$198$	0	0
$199$	−6.86141	−0.486392	−0.243196	−	0.969977i	$-0.578196\pi$
$199$	−6.86141	−0.486392	−0.243196	+	0.969977i	$0.578196\pi$
$200$	3.75906	0.265805
$201$	16.1168	1.13679
$202$	−13.4891	−0.949092
$203$	−2.00000	−0.140372
$204$	−26.1831	−1.83319
$205$	13.5065	0.943338
$206$	20.4897	1.42758
$207$	2.00000	0.139010
$208$	−26.1831	−1.81547
$209$	0	0
$210$	−4.74456	−0.327406
$211$	12.4742	0.858760	0.429380	−	0.903124i	$-0.358732\pi$
$211$	12.4742	0.858760	0.429380	+	0.903124i	$0.358732\pi$
$212$	57.3505	3.93885
$213$	0.744563	0.0510166
$214$	−0.744563	−0.0508973
$215$	15.7359	1.07318
$216$	5.98844	0.407462
$217$	5.84096	0.396510
$218$	25.1168	1.70113
$219$	7.42554	0.501771
$220$	0	0
$221$	24.6060	1.65518
$222$	12.6217	0.847112
$223$	−4.86141	−0.325544	−0.162772	−	0.986664i	$-0.552043\pi$
$223$	−4.86141	−0.325544	−0.162772	+	0.986664i	$0.552043\pi$
$224$	3.25544	0.217513
$225$	0.627719	0.0418479
$226$	−11.0371	−0.734178
$227$	25.2434	1.67546	0.837731	−	0.546083i	$-0.183882\pi$
$227$	25.2434	1.67546	0.837731	+	0.546083i	$0.183882\pi$
$228$	18.6101	1.23249
$229$	−3.62772	−0.239726	−0.119863	−	0.992790i	$-0.538246\pi$
$229$	−3.62772	−0.239726	−0.119863	+	0.992790i	$0.538246\pi$
$230$	−11.9769	−0.789732
$231$	0	0
$232$	−15.1168	−0.992469
$233$	2.52434	0.165375	0.0826874	−	0.996576i	$-0.473650\pi$
$233$	2.52434	0.165375	0.0826874	+	0.996576i	$0.473650\pi$
$234$	−10.3723	−0.678057
$235$	2.97825	0.194280
$236$	−26.2337	−1.70767
$237$	5.84096	0.379411
$238$	−11.9769	−0.776346
$239$	−29.2974	−1.89509	−0.947545	−	0.319622i	$-0.896444\pi$
$239$	−29.2974	−1.89509	−0.947545	+	0.319622i	$0.896444\pi$
$240$	−15.1168	−0.975788
$241$	−13.2665	−0.854570	−0.427285	−	0.904117i	$-0.640530\pi$
$241$	−13.2665	−0.854570	−0.427285	+	0.904117i	$0.640530\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	−11.6819	−0.747859
$245$	15.1168	0.965780
$246$	−14.3723	−0.916343
$247$	−17.4891	−1.11281
$248$	44.1485	2.80343
$249$	8.51278	0.539475
$250$	26.1831	1.65597
$251$	−0.510875	−0.0322461	−0.0161231	−	0.999870i	$-0.505132\pi$
$251$	−0.510875	−0.0322461	−0.0161231	+	0.999870i	$0.505132\pi$
$252$	3.46410	0.218218
$253$	0	0
$254$	49.7228	3.11989
$255$	14.2063	0.889631
$256$	−31.1168	−1.94480
$257$	24.3723	1.52030	0.760151	−	0.649747i	$-0.225125\pi$
$257$	24.3723	1.52030	0.760151	+	0.649747i	$0.225125\pi$
$258$	−16.7446	−1.04247
$259$	3.96143	0.246152
$260$	42.6188	2.64311
$261$	−2.52434	−0.156253
$262$	16.7446	1.03448
$263$	−22.3692	−1.37934	−0.689671	−	0.724122i	$-0.742245\pi$
$263$	−22.3692	−1.37934	−0.689671	+	0.724122i	$0.742245\pi$
$264$	0	0
$265$	−31.1168	−1.91149
$266$	8.51278	0.521952
$267$	−6.37228	−0.389977
$268$	70.4674	4.30448
$269$	9.11684	0.555864	0.277932	−	0.960601i	$-0.410351\pi$
$269$	9.11684	0.555864	0.277932	+	0.960601i	$0.410351\pi$
$270$	−5.98844	−0.364445
$271$	−10.3923	−0.631288	−0.315644	−	0.948878i	$-0.602220\pi$
$271$	−10.3923	−0.631288	−0.315644	+	0.948878i	$0.602220\pi$
$272$	−38.1600	−2.31379
$273$	−3.25544	−0.197028
$274$	1.87953	0.113546
$275$	0	0
$276$	8.74456	0.526361
$277$	−8.66025	−0.520344	−0.260172	−	0.965562i	$-0.583779\pi$
$277$	−8.66025	−0.520344	−0.260172	+	0.965562i	$0.583779\pi$
$278$	−26.2337	−1.57339
$279$	7.37228	0.441367
$280$	−11.2554	−0.672641
$281$	−0.589907	−0.0351909	−0.0175955	−	0.999845i	$-0.505601\pi$
$281$	−0.589907	−0.0351909	−0.0175955	+	0.999845i	$0.505601\pi$
$282$	−3.16915	−0.188720
$283$	−2.37686	−0.141290	−0.0706449	−	0.997502i	$-0.522506\pi$
$283$	−2.37686	−0.141290	−0.0706449	+	0.997502i	$0.522506\pi$
$284$	3.25544	0.193175
$285$	−10.0974	−0.598115
$286$	0	0
$287$	−4.51087	−0.266269
$288$	4.10891	0.242120
$289$	18.8614	1.10949
$290$	15.1168	0.887692
$291$	−12.4891	−0.732125
$292$	32.4665	1.89996
$293$	6.57835	0.384311	0.192156	−	0.981364i	$-0.438452\pi$
$293$	6.57835	0.384311	0.192156	+	0.981364i	$0.438452\pi$
$294$	−16.0858	−0.938142
$295$	14.2337	0.828717
$296$	29.9422	1.74035
$297$	0	0
$298$	−15.1168	−0.875695
$299$	−8.21782	−0.475249
$300$	2.74456	0.158457
$301$	−5.25544	−0.302918
$302$	5.48913	0.315864
$303$	−5.34363	−0.306983
$304$	27.1229	1.55561
$305$	6.33830	0.362930
$306$	−15.1168	−0.864172
$307$	−17.8178	−1.01692	−0.508459	−	0.861086i	$-0.669785\pi$
$307$	−17.8178	−1.01692	−0.508459	+	0.861086i	$0.669785\pi$
$308$	0	0
$309$	8.11684	0.461751
$310$	−44.1485	−2.50746
$311$	−6.51087	−0.369198	−0.184599	−	0.982814i	$-0.559099\pi$
$311$	−6.51087	−0.369198	−0.184599	+	0.982814i	$0.559099\pi$
$312$	−24.6060	−1.39304
$313$	9.11684	0.515314	0.257657	−	0.966236i	$-0.417049\pi$
$313$	9.11684	0.515314	0.257657	+	0.966236i	$0.417049\pi$
$314$	16.7306	0.944162
$315$	−1.87953	−0.105899
$316$	25.5383	1.43664
$317$	15.2554	0.856831	0.428415	−	0.903582i	$-0.359072\pi$
$317$	15.2554	0.856831	0.428415	+	0.903582i	$0.359072\pi$
$318$	33.1113	1.85679
$319$	0	0
$320$	5.62772	0.314599
$321$	−0.294954	−0.0164627
$322$	4.00000	0.222911
$323$	−25.4891	−1.41825
$324$	4.37228	0.242905
$325$	−2.57924	−0.143071
$326$	−23.6588	−1.31034
$327$	9.94987	0.550229
$328$	−34.0951	−1.88259
$329$	−0.994667	−0.0548377
$330$	0	0
$331$	−0.627719	−0.0345025	−0.0172513	−	0.999851i	$-0.505492\pi$
$331$	−0.627719	−0.0345025	−0.0172513	+	0.999851i	$0.505492\pi$
$332$	37.2203	2.04273
$333$	5.00000	0.273998
$334$	−7.25544	−0.397000
$335$	−38.2337	−2.08893
$336$	5.04868	0.275428
$337$	15.2935	0.833090	0.416545	−	0.909115i	$-0.363241\pi$
$337$	15.2935	0.833090	0.416545	+	0.909115i	$0.363241\pi$
$338$	9.80240	0.533180
$339$	−4.37228	−0.237470
$340$	62.1138	3.36859
$341$	0	0
$342$	10.7446	0.580999
$343$	−10.5947	−0.572059
$344$	−39.7228	−2.14171
$345$	−4.74456	−0.255439
$346$	−17.4891	−0.940221
$347$	−18.0202	−0.967376	−0.483688	−	0.875240i	$-0.660703\pi$
$347$	−18.0202	−0.967376	−0.483688	+	0.875240i	$0.660703\pi$
$348$	−11.0371	−0.591651
$349$	−25.9808	−1.39072	−0.695359	−	0.718662i	$-0.744755\pi$
$349$	−25.9808	−1.39072	−0.695359	+	0.718662i	$0.744755\pi$
$350$	1.25544	0.0671059
$351$	−4.10891	−0.219317
$352$	0	0
$353$	−29.3505	−1.56217	−0.781086	−	0.624424i	$-0.785334\pi$
$353$	−29.3505	−1.56217	−0.781086	+	0.624424i	$0.785334\pi$
$354$	−15.1460	−0.805002
$355$	−1.76631	−0.0937461
$356$	−27.8614	−1.47665
$357$	−4.74456	−0.251109
$358$	−61.1740	−3.23315
$359$	−1.28962	−0.0680636	−0.0340318	−	0.999421i	$-0.510835\pi$
$359$	−1.28962	−0.0680636	−0.0340318	+	0.999421i	$0.510835\pi$
$360$	−14.2063	−0.748736
$361$	−0.883156	−0.0464819
$362$	42.9137	2.25550
$363$	0	0
$364$	−14.2337	−0.746048
$365$	−17.6155	−0.922035
$366$	−6.74456	−0.352544
$367$	−23.7228	−1.23832	−0.619160	−	0.785265i	$-0.712527\pi$
$367$	−23.7228	−1.23832	−0.619160	+	0.785265i	$0.712527\pi$
$368$	12.7446	0.664356
$369$	−5.69349	−0.296391
$370$	−29.9422	−1.55662
$371$	10.3923	0.539542
$372$	32.2337	1.67124
$373$	−24.4511	−1.26603	−0.633015	−	0.774140i	$-0.718183\pi$
$373$	−24.4511	−1.26603	−0.633015	+	0.774140i	$0.718183\pi$
$374$	0	0
$375$	10.3723	0.535622
$376$	−7.51811	−0.387717
$377$	10.3723	0.534200
$378$	2.00000	0.102869
$379$	23.7228	1.21856	0.609280	−	0.792956i	$-0.291459\pi$
$379$	23.7228	1.21856	0.609280	+	0.792956i	$0.291459\pi$
$380$	−44.1485	−2.26477
$381$	19.6974	1.00913
$382$	−44.1485	−2.25883
$383$	−23.4891	−1.20024	−0.600119	−	0.799911i	$-0.704880\pi$
$383$	−23.4891	−1.20024	−0.600119	+	0.799911i	$0.704880\pi$
$384$	−14.2063	−0.724960
$385$	0	0
$386$	−43.3505	−2.20648
$387$	−6.63325	−0.337187
$388$	−54.6060	−2.77220
$389$	−9.11684	−0.462242	−0.231121	−	0.972925i	$-0.574239\pi$
$389$	−9.11684	−0.462242	−0.231121	+	0.972925i	$0.574239\pi$
$390$	24.6060	1.24597
$391$	−11.9769	−0.605697
$392$	−38.1600	−1.92737
$393$	6.63325	0.334603
$394$	53.3505	2.68776
$395$	−13.8564	−0.697191
$396$	0	0
$397$	24.4891	1.22907	0.614537	−	0.788888i	$-0.289343\pi$
$397$	24.4891	1.22907	0.614537	+	0.788888i	$0.289343\pi$
$398$	−17.3205	−0.868199
$399$	3.37228	0.168825
$400$	4.00000	0.200000
$401$	−19.1168	−0.954650	−0.477325	−	0.878727i	$-0.658393\pi$
$401$	−19.1168	−0.954650	−0.477325	+	0.878727i	$0.658393\pi$
$402$	40.6844	2.02915
$403$	−30.2921	−1.50895
$404$	−23.3639	−1.16240
$405$	−2.37228	−0.117880
$406$	−5.04868	−0.250562
$407$	0	0
$408$	−35.8614	−1.77540
$409$	0.442430	0.0218768	0.0109384	−	0.999940i	$-0.496518\pi$
$409$	0.442430	0.0218768	0.0109384	+	0.999940i	$0.496518\pi$
$410$	34.0951	1.68384
$411$	0.744563	0.0367266
$412$	35.4891	1.74842
$413$	−4.75372	−0.233915
$414$	5.04868	0.248129
$415$	−20.1947	−0.991319
$416$	−16.8832	−0.827765
$417$	−10.3923	−0.508913
$418$	0	0
$419$	−8.51087	−0.415783	−0.207892	−	0.978152i	$-0.566660\pi$
$419$	−8.51087	−0.415783	−0.207892	+	0.978152i	$0.566660\pi$
$420$	−8.21782	−0.400989
$421$	6.13859	0.299177	0.149588	−	0.988748i	$-0.452205\pi$
$421$	6.13859	0.299177	0.149588	+	0.988748i	$0.452205\pi$
$422$	31.4891	1.53287
$423$	−1.25544	−0.0610415
$424$	78.5494	3.81470
$425$	−3.75906	−0.182341
$426$	1.87953	0.0910634
$427$	−2.11684	−0.102441
$428$	−1.28962	−0.0623362
$429$	0	0
$430$	39.7228	1.91560
$431$	23.6588	1.13960	0.569802	−	0.821782i	$-0.307020\pi$
$431$	23.6588	1.13960	0.569802	+	0.821782i	$0.307020\pi$
$432$	6.37228	0.306587
$433$	−23.2337	−1.11654	−0.558270	−	0.829659i	$-0.688535\pi$
$433$	−23.2337	−1.11654	−0.558270	+	0.829659i	$0.688535\pi$
$434$	14.7446	0.707762
$435$	5.98844	0.287124
$436$	43.5036	2.08345
$437$	8.51278	0.407221
$438$	18.7446	0.895650
$439$	24.4511	1.16699	0.583493	−	0.812118i	$-0.301685\pi$
$439$	24.4511	1.16699	0.583493	+	0.812118i	$0.301685\pi$
$440$	0	0
$441$	−6.37228	−0.303442
$442$	62.1138	2.95445
$443$	20.7446	0.985604	0.492802	−	0.870142i	$-0.335973\pi$
$443$	20.7446	0.985604	0.492802	+	0.870142i	$0.335973\pi$
$444$	21.8614	1.03750
$445$	15.1168	0.716607
$446$	−12.2718	−0.581088
$447$	−5.98844	−0.283243
$448$	−1.87953	−0.0887993
$449$	24.6060	1.16123	0.580614	−	0.814179i	$-0.302813\pi$
$449$	24.6060	1.16123	0.580614	+	0.814179i	$0.302813\pi$
$450$	1.58457	0.0746975
$451$	0	0
$452$	−19.1168	−0.899181
$453$	2.17448	0.102166
$454$	63.7228	2.99066
$455$	7.72281	0.362051
$456$	25.4891	1.19364
$457$	−8.45787	−0.395643	−0.197821	−	0.980238i	$-0.563387\pi$
$457$	−8.45787	−0.395643	−0.197821	+	0.980238i	$0.563387\pi$
$458$	−9.15759	−0.427906
$459$	−5.98844	−0.279516
$460$	−20.7446	−0.967220
$461$	14.5012	0.675389	0.337694	−	0.941256i	$-0.390353\pi$
$461$	14.5012	0.675389	0.337694	+	0.941256i	$0.390353\pi$
$462$	0	0
$463$	−3.25544	−0.151293	−0.0756465	−	0.997135i	$-0.524102\pi$
$463$	−3.25544	−0.151293	−0.0756465	+	0.997135i	$0.524102\pi$
$464$	−16.0858	−0.746764
$465$	−17.4891	−0.811039
$466$	6.37228	0.295190
$467$	−22.2337	−1.02885	−0.514426	−	0.857535i	$-0.671995\pi$
$467$	−22.2337	−1.02885	−0.514426	+	0.857535i	$0.671995\pi$
$468$	−17.9653	−0.830447
$469$	12.7692	0.589625
$470$	7.51811	0.346785
$471$	6.62772	0.305389
$472$	−35.9306	−1.65384
$473$	0	0
$474$	14.7446	0.677240
$475$	2.67181	0.122591
$476$	−20.7446	−0.950825
$477$	13.1168	0.600579
$478$	−73.9565	−3.38269
$479$	37.5152	1.71411	0.857057	−	0.515222i	$-0.172290\pi$
$479$	37.5152	1.71411	0.857057	+	0.515222i	$0.172290\pi$
$480$	−9.74749	−0.444910
$481$	−20.5446	−0.936751
$482$	−33.4891	−1.52539
$483$	1.58457	0.0721006
$484$	0	0
$485$	29.6277	1.34533
$486$	2.52434	0.114506
$487$	−12.7446	−0.577511	−0.288756	−	0.957403i	$-0.593242\pi$
$487$	−12.7446	−0.577511	−0.288756	+	0.957403i	$0.593242\pi$
$488$	−16.0000	−0.724286
$489$	−9.37228	−0.423829
$490$	38.1600	1.72389
$491$	18.6101	0.839863	0.419932	−	0.907556i	$-0.362054\pi$
$491$	18.6101	0.839863	0.419932	+	0.907556i	$0.362054\pi$
$492$	−24.8935	−1.12229
$493$	15.1168	0.680828
$494$	−44.1485	−1.98633
$495$	0	0
$496$	46.9783	2.10939
$497$	0.589907	0.0264610
$498$	21.4891	0.962951
$499$	−37.8397	−1.69394	−0.846968	−	0.531644i	$-0.821574\pi$
$499$	−37.8397	−1.69394	−0.846968	+	0.531644i	$0.821574\pi$
$500$	45.3505	2.02814
$501$	−2.87419	−0.128410
$502$	−1.28962	−0.0575586
$503$	22.0742	0.984241	0.492121	−	0.870527i	$-0.336222\pi$
$503$	22.0742	0.984241	0.492121	+	0.870527i	$0.336222\pi$
$504$	4.74456	0.211340
$505$	12.6766	0.564101
$506$	0	0
$507$	3.88316	0.172457
$508$	86.1224	3.82107
$509$	−40.9783	−1.81633	−0.908165	−	0.418613i	$-0.862516\pi$
$509$	−40.9783	−1.81633	−0.908165	+	0.418613i	$0.862516\pi$
$510$	35.8614	1.58797
$511$	5.88316	0.260256
$512$	−50.1369	−2.21576
$513$	4.25639	0.187924
$514$	61.5239	2.71370
$515$	−19.2554	−0.848496
$516$	−29.0024	−1.27676
$517$	0	0
$518$	10.0000	0.439375
$519$	−6.92820	−0.304114
$520$	58.3723	2.55979
$521$	16.9783	0.743831	0.371915	−	0.928267i	$-0.378701\pi$
$521$	16.9783	0.743831	0.371915	+	0.928267i	$0.378701\pi$
$522$	−6.37228	−0.278907
$523$	−6.83563	−0.298901	−0.149451	−	0.988769i	$-0.547750\pi$
$523$	−6.83563	−0.298901	−0.149451	+	0.988769i	$0.547750\pi$
$524$	29.0024	1.26698
$525$	0.497333	0.0217054
$526$	−56.4674	−2.46209
$527$	−44.1485	−1.92314
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−78.5494	−3.41197
$531$	−6.00000	−0.260378
$532$	14.7446	0.639258
$533$	23.3940	1.01331
$534$	−16.0858	−0.696100
$535$	0.699713	0.0302512
$536$	96.5147	4.16880
$537$	−24.2337	−1.04576
$538$	23.0140	0.992204
$539$	0	0
$540$	−10.3723	−0.446352
$541$	−27.7128	−1.19147	−0.595733	−	0.803182i	$-0.703139\pi$
$541$	−27.7128	−1.19147	−0.595733	+	0.803182i	$0.703139\pi$
$542$	−26.2337	−1.12683
$543$	17.0000	0.729540
$544$	−24.6060	−1.05497
$545$	−23.6039	−1.01108
$546$	−8.21782	−0.351690
$547$	41.9740	1.79468	0.897339	−	0.441342i	$-0.145498\pi$
$547$	41.9740	1.79468	0.897339	+	0.441342i	$0.145498\pi$
$548$	3.25544	0.139065
$549$	−2.67181	−0.114030
$550$	0	0
$551$	−10.7446	−0.457734
$552$	11.9769	0.509770
$553$	4.62772	0.196791
$554$	−21.8614	−0.928802
$555$	−11.8614	−0.503489
$556$	−45.4381	−1.92700
$557$	9.50744	0.402843	0.201422	−	0.979505i	$-0.435444\pi$
$557$	9.50744	0.402843	0.201422	+	0.979505i	$0.435444\pi$
$558$	18.6101	0.787830
$559$	27.2554	1.15278
$560$	−11.9769	−0.506116
$561$	0	0
$562$	−1.48913	−0.0628150
$563$	14.5561	0.613467	0.306734	−	0.951795i	$-0.400764\pi$
$563$	14.5561	0.613467	0.306734	+	0.951795i	$0.400764\pi$
$564$	−5.48913	−0.231134
$565$	10.3723	0.436365
$566$	−6.00000	−0.252199
$567$	0.792287	0.0332729
$568$	4.45877	0.187086
$569$	24.5437	1.02892	0.514462	−	0.857513i	$-0.327992\pi$
$569$	24.5437	1.02892	0.514462	+	0.857513i	$0.327992\pi$
$570$	−25.4891	−1.06762
$571$	4.55134	0.190468	0.0952339	−	0.995455i	$-0.469640\pi$
$571$	4.55134	0.190468	0.0952339	+	0.995455i	$0.469640\pi$
$572$	0	0
$573$	−17.4891	−0.730619
$574$	−11.3870	−0.475283
$575$	1.25544	0.0523554
$576$	−2.37228	−0.0988451
$577$	−16.4891	−0.686451	−0.343226	−	0.939253i	$-0.611520\pi$
$577$	−16.4891	−0.686451	−0.343226	+	0.939253i	$0.611520\pi$
$578$	47.6126	1.98042
$579$	−17.1730	−0.713687
$580$	26.1831	1.08720
$581$	6.74456	0.279812
$582$	−31.5268	−1.30683
$583$	0	0
$584$	44.4674	1.84007
$585$	9.74749	0.403009
$586$	16.6060	0.685986
$587$	38.9783	1.60880	0.804402	−	0.594085i	$-0.202486\pi$
$587$	38.9783	1.60880	0.804402	+	0.594085i	$0.202486\pi$
$588$	−27.8614	−1.14899
$589$	31.3793	1.29296
$590$	35.9306	1.47924
$591$	21.1345	0.869356
$592$	31.8614	1.30950
$593$	27.7677	1.14028	0.570142	−	0.821546i	$-0.306888\pi$
$593$	27.7677	1.14028	0.570142	+	0.821546i	$0.306888\pi$
$594$	0	0
$595$	11.2554	0.461428
$596$	−26.1831	−1.07250
$597$	−6.86141	−0.280819
$598$	−20.7446	−0.848308
$599$	40.4674	1.65345	0.826726	−	0.562605i	$-0.190201\pi$
$599$	40.4674	1.65345	0.826726	+	0.562605i	$0.190201\pi$
$600$	3.75906	0.153463
$601$	−40.2419	−1.64150	−0.820751	−	0.571286i	$-0.806445\pi$
$601$	−40.2419	−1.64150	−0.820751	+	0.571286i	$0.806445\pi$
$602$	−13.2665	−0.540702
$603$	16.1168	0.656329
$604$	9.50744	0.386852
$605$	0	0
$606$	−13.4891	−0.547958
$607$	26.1282	1.06051	0.530256	−	0.847837i	$-0.322096\pi$
$607$	26.1282	1.06051	0.530256	+	0.847837i	$0.322096\pi$
$608$	17.4891	0.709278
$609$	−2.00000	−0.0810441
$610$	16.0000	0.647821
$611$	5.15848	0.208690
$612$	−26.1831	−1.05839
$613$	−43.4111	−1.75336	−0.876678	−	0.481077i	$-0.840246\pi$
$613$	−43.4111	−1.75336	−0.876678	+	0.481077i	$0.840246\pi$
$614$	−44.9783	−1.81517
$615$	13.5065	0.544637
$616$	0	0
$617$	43.8614	1.76579	0.882897	−	0.469567i	$-0.155590\pi$
$617$	43.8614	1.76579	0.882897	+	0.469567i	$0.155590\pi$
$618$	20.4897	0.824215
$619$	10.2337	0.411327	0.205663	−	0.978623i	$-0.434065\pi$
$619$	10.2337	0.411327	0.205663	+	0.978623i	$0.434065\pi$
$620$	−76.4674	−3.07100
$621$	2.00000	0.0802572
$622$	−16.4356	−0.659009
$623$	−5.04868	−0.202271
$624$	−26.1831	−1.04816
$625$	−27.7446	−1.10978
$626$	23.0140	0.919824
$627$	0	0
$628$	28.9783	1.15636
$629$	−29.9422	−1.19387
$630$	−4.74456	−0.189028
$631$	22.2337	0.885109	0.442555	−	0.896742i	$-0.354072\pi$
$631$	22.2337	0.885109	0.442555	+	0.896742i	$0.354072\pi$
$632$	34.9783	1.39136
$633$	12.4742	0.495805
$634$	38.5099	1.52942
$635$	−46.7277	−1.85433
$636$	57.3505	2.27410
$637$	26.1831	1.03741
$638$	0	0
$639$	0.744563	0.0294544
$640$	33.7013	1.33216
$641$	7.11684	0.281098	0.140549	−	0.990074i	$-0.455113\pi$
$641$	7.11684	0.281098	0.140549	+	0.990074i	$0.455113\pi$
$642$	−0.744563	−0.0293855
$643$	21.0951	0.831909	0.415955	−	0.909385i	$-0.363447\pi$
$643$	21.0951	0.831909	0.415955	+	0.909385i	$0.363447\pi$
$644$	6.92820	0.273009
$645$	15.7359	0.619602
$646$	−64.3432	−2.53155
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	5.98844	0.235248
$649$	0	0
$650$	−6.51087	−0.255378
$651$	5.84096	0.228925
$652$	−40.9783	−1.60483
$653$	4.51087	0.176524	0.0882621	−	0.996097i	$-0.471869\pi$
$653$	4.51087	0.176524	0.0882621	+	0.996097i	$0.471869\pi$
$654$	25.1168	0.982146
$655$	−15.7359	−0.614854
$656$	−36.2805	−1.41652
$657$	7.42554	0.289698
$658$	−2.51087	−0.0978841
$659$	22.3692	0.871380	0.435690	−	0.900097i	$-0.356504\pi$
$659$	22.3692	0.871380	0.435690	+	0.900097i	$0.356504\pi$
$660$	0	0
$661$	−26.7228	−1.03940	−0.519698	−	0.854350i	$-0.673956\pi$
$661$	−26.7228	−1.03940	−0.519698	+	0.854350i	$0.673956\pi$
$662$	−1.58457	−0.0615862
$663$	24.6060	0.955617
$664$	50.9783	1.97834
$665$	−8.00000	−0.310227
$666$	12.6217	0.489081
$667$	−5.04868	−0.195485
$668$	−12.5668	−0.486224
$669$	−4.86141	−0.187953
$670$	−96.5147	−3.72869
$671$	0	0
$672$	3.25544	0.125581
$673$	41.4766	1.59881	0.799404	−	0.600794i	$-0.205149\pi$
$673$	41.4766	1.59881	0.799404	+	0.600794i	$0.205149\pi$
$674$	38.6060	1.48705
$675$	0.627719	0.0241609
$676$	16.9783	0.653010
$677$	36.2805	1.39437	0.697186	−	0.716890i	$-0.254435\pi$
$677$	36.2805	1.39437	0.697186	+	0.716890i	$0.254435\pi$
$678$	−11.0371	−0.423878
$679$	−9.89497	−0.379734
$680$	85.0733	3.26241
$681$	25.2434	0.967328
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	18.6101	0.711576
$685$	−1.76631	−0.0674873
$686$	−26.7446	−1.02111
$687$	−3.62772	−0.138406
$688$	−42.2689	−1.61149
$689$	−53.8960	−2.05327
$690$	−11.9769	−0.455952
$691$	−4.11684	−0.156612	−0.0783061	−	0.996929i	$-0.524951\pi$
$691$	−4.11684	−0.156612	−0.0783061	+	0.996929i	$0.524951\pi$
$692$	−30.2921	−1.15153
$693$	0	0
$694$	−45.4891	−1.72674
$695$	24.6535	0.935159
$696$	−15.1168	−0.573002
$697$	34.0951	1.29144
$698$	−65.5842	−2.48240
$699$	2.52434	0.0954792
$700$	2.17448	0.0821877
$701$	10.0424	0.379298	0.189649	−	0.981852i	$-0.439265\pi$
$701$	10.0424	0.379298	0.189649	+	0.981852i	$0.439265\pi$
$702$	−10.3723	−0.391477
$703$	21.2819	0.802664
$704$	0	0
$705$	2.97825	0.112167
$706$	−74.0907	−2.78844
$707$	−4.23369	−0.159224
$708$	−26.2337	−0.985922
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−4.45877	−0.167335
$711$	5.84096	0.219053
$712$	−38.1600	−1.43011
$713$	14.7446	0.552188
$714$	−11.9769	−0.448223
$715$	0	0
$716$	−105.957	−3.95978
$717$	−29.2974	−1.09413
$718$	−3.25544	−0.121492
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	−15.1168	−0.563372
$721$	6.43087	0.239498
$722$	−2.22938	−0.0829691
$723$	−13.2665	−0.493386
$724$	74.3288	2.76241
$725$	−1.58457	−0.0588496
$726$	0	0
$727$	7.25544	0.269089	0.134545	−	0.990908i	$-0.457043\pi$
$727$	7.25544	0.269089	0.134545	+	0.990908i	$0.457043\pi$
$728$	−19.4950	−0.722532
$729$	1.00000	0.0370370
$730$	−44.4674	−1.64581
$731$	39.7228	1.46920
$732$	−11.6819	−0.431776
$733$	−9.15759	−0.338243	−0.169122	−	0.985595i	$-0.554093\pi$
$733$	−9.15759	−0.338243	−0.169122	+	0.985595i	$0.554093\pi$
$734$	−59.8844	−2.21037
$735$	15.1168	0.557593
$736$	8.21782	0.302913
$737$	0	0
$738$	−14.3723	−0.529051
$739$	−38.8974	−1.43086	−0.715432	−	0.698682i	$-0.753770\pi$
$739$	−38.8974	−1.43086	−0.715432	+	0.698682i	$0.753770\pi$
$740$	−51.8614	−1.90646
$741$	−17.4891	−0.642479
$742$	26.2337	0.963069
$743$	−24.6535	−0.904448	−0.452224	−	0.891904i	$-0.649369\pi$
$743$	−24.6535	−0.904448	−0.452224	+	0.891904i	$0.649369\pi$
$744$	44.1485	1.61856
$745$	14.2063	0.520477
$746$	−61.7228	−2.25983
$747$	8.51278	0.311466
$748$	0	0
$749$	−0.233688	−0.00853877
$750$	26.1831	0.956073
$751$	−16.3505	−0.596639	−0.298320	−	0.954466i	$-0.596426\pi$
$751$	−16.3505	−0.596639	−0.298320	+	0.954466i	$0.596426\pi$
$752$	−8.00000	−0.291730
$753$	−0.510875	−0.0186173
$754$	26.1831	0.953534
$755$	−5.15848	−0.187736
$756$	3.46410	0.125988
$757$	9.51087	0.345679	0.172839	−	0.984950i	$-0.444706\pi$
$757$	9.51087	0.345679	0.172839	+	0.984950i	$0.444706\pi$
$758$	59.8844	2.17510
$759$	0	0
$760$	−60.4674	−2.19338
$761$	40.6295	1.47282	0.736408	−	0.676537i	$-0.236520\pi$
$761$	40.6295	1.47282	0.736408	+	0.676537i	$0.236520\pi$
$762$	49.7228	1.80127
$763$	7.88316	0.285389
$764$	−76.4674	−2.76649
$765$	14.2063	0.513629
$766$	−59.2945	−2.14240
$767$	24.6535	0.890185
$768$	−31.1168	−1.12283
$769$	34.7885	1.25451	0.627253	−	0.778816i	$-0.284179\pi$
$769$	34.7885	1.25451	0.627253	+	0.778816i	$0.284179\pi$
$770$	0	0
$771$	24.3723	0.877746
$772$	−75.0853	−2.70238
$773$	33.2119	1.19455	0.597275	−	0.802036i	$-0.296250\pi$
$773$	33.2119	1.19455	0.597275	+	0.802036i	$0.296250\pi$
$774$	−16.7446	−0.601871
$775$	4.62772	0.166233
$776$	−74.7904	−2.68482
$777$	3.96143	0.142116
$778$	−23.0140	−0.825092
$779$	−24.2337	−0.868262
$780$	42.6188	1.52600
$781$	0	0
$782$	−30.2337	−1.08115
$783$	−2.52434	−0.0902125
$784$	−40.6060	−1.45021
$785$	−15.7228	−0.561171
$786$	16.7446	0.597259
$787$	−4.75372	−0.169452	−0.0847259	−	0.996404i	$-0.527001\pi$
$787$	−4.75372	−0.169452	−0.0847259	+	0.996404i	$0.527001\pi$
$788$	92.4058	3.29182
$789$	−22.3692	−0.796364
$790$	−34.9783	−1.24447
$791$	−3.46410	−0.123169
$792$	0	0
$793$	10.9783	0.389849
$794$	61.8188	2.19387
$795$	−31.1168	−1.10360
$796$	−30.0000	−1.06332
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	8.51278	0.301349
$799$	7.51811	0.265972
$800$	2.57924	0.0911899
$801$	−6.37228	−0.225153
$802$	−48.2574	−1.70403
$803$	0	0
$804$	70.4674	2.48519
$805$	−3.75906	−0.132489
$806$	−76.4674	−2.69345
$807$	9.11684	0.320928
$808$	−32.0000	−1.12576
$809$	−53.6559	−1.88644	−0.943221	−	0.332167i	$-0.892220\pi$
$809$	−53.6559	−1.88644	−0.943221	+	0.332167i	$0.892220\pi$
$810$	−5.98844	−0.210412
$811$	44.0559	1.54701	0.773506	−	0.633789i	$-0.218501\pi$
$811$	44.0559	1.54701	0.773506	+	0.633789i	$0.218501\pi$
$812$	−8.74456	−0.306874
$813$	−10.3923	−0.364474
$814$	0	0
$815$	22.2337	0.778812
$816$	−38.1600	−1.33587
$817$	−28.2337	−0.987772
$818$	1.11684	0.0390495
$819$	−3.25544	−0.113754
$820$	59.0544	2.06227
$821$	−12.8617	−0.448878	−0.224439	−	0.974488i	$-0.572055\pi$
$821$	−12.8617	−0.448878	−0.224439	+	0.974488i	$0.572055\pi$
$822$	1.87953	0.0655561
$823$	−32.3505	−1.12767	−0.563834	−	0.825888i	$-0.690674\pi$
$823$	−32.3505	−1.12767	−0.563834	+	0.825888i	$0.690674\pi$
$824$	48.6072	1.69331
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−16.1407	−0.561267	−0.280633	−	0.959815i	$-0.590545\pi$
$827$	−16.1407	−0.561267	−0.280633	+	0.959815i	$0.590545\pi$
$828$	8.74456	0.303895
$829$	−22.2554	−0.772963	−0.386482	−	0.922297i	$-0.626310\pi$
$829$	−22.2554	−0.772963	−0.386482	+	0.922297i	$0.626310\pi$
$830$	−50.9783	−1.76948
$831$	−8.66025	−0.300421
$832$	9.74749	0.337934
$833$	38.1600	1.32217
$834$	−26.2337	−0.908398
$835$	6.81840	0.235960
$836$	0	0
$837$	7.37228	0.254823
$838$	−21.4843	−0.742164
$839$	26.9783	0.931393	0.465696	−	0.884945i	$-0.345804\pi$
$839$	26.9783	0.931393	0.465696	+	0.884945i	$0.345804\pi$
$840$	−11.2554	−0.388349
$841$	−22.6277	−0.780266
$842$	15.4959	0.534023
$843$	−0.589907	−0.0203175
$844$	54.5408	1.87737
$845$	−9.21194	−0.316900
$846$	−3.16915	−0.108958
$847$	0	0
$848$	83.5842	2.87029
$849$	−2.37686	−0.0815737
$850$	−9.48913	−0.325474
$851$	10.0000	0.342796
$852$	3.25544	0.111529
$853$	31.7292	1.08639	0.543193	−	0.839608i	$-0.317215\pi$
$853$	31.7292	1.08639	0.543193	+	0.839608i	$0.317215\pi$
$854$	−5.34363	−0.182855
$855$	−10.0974	−0.345322
$856$	−1.76631	−0.0603713
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	−12.1168	−0.413421	−0.206711	−	0.978402i	$-0.566276\pi$
$859$	−12.1168	−0.413421	−0.206711	+	0.978402i	$0.566276\pi$
$860$	68.8019	2.34613
$861$	−4.51087	−0.153730
$862$	59.7228	2.03417
$863$	16.5109	0.562037	0.281018	−	0.959702i	$-0.409328\pi$
$863$	16.5109	0.562037	0.281018	+	0.959702i	$0.409328\pi$
$864$	4.10891	0.139788
$865$	16.4356	0.558829
$866$	−58.6497	−1.99300
$867$	18.8614	0.640567
$868$	25.5383	0.866827
$869$	0	0
$870$	15.1168	0.512509
$871$	−66.2227	−2.24387
$872$	59.5842	2.01778
$873$	−12.4891	−0.422693
$874$	21.4891	0.726881
$875$	8.21782	0.277813
$876$	32.4665	1.09694
$877$	25.3909	0.857388	0.428694	−	0.903450i	$-0.358974\pi$
$877$	25.3909	0.857388	0.428694	+	0.903450i	$0.358974\pi$
$878$	61.7228	2.08304
$879$	6.57835	0.221882
$880$	0	0
$881$	−5.39403	−0.181730	−0.0908648	−	0.995863i	$-0.528963\pi$
$881$	−5.39403	−0.181730	−0.0908648	+	0.995863i	$0.528963\pi$
$882$	−16.0858	−0.541637
$883$	−4.86141	−0.163599	−0.0817997	−	0.996649i	$-0.526067\pi$
$883$	−4.86141	−0.163599	−0.0817997	+	0.996649i	$0.526067\pi$
$884$	107.584	3.61845
$885$	14.2337	0.478460
$886$	52.3663	1.75928
$887$	29.4072	0.987397	0.493698	−	0.869633i	$-0.335645\pi$
$887$	29.4072	0.987397	0.493698	+	0.869633i	$0.335645\pi$
$888$	29.9422	1.00479
$889$	15.6060	0.523407
$890$	38.1600	1.27913
$891$	0	0
$892$	−21.2554	−0.711685
$893$	−5.34363	−0.178818
$894$	−15.1168	−0.505583
$895$	57.4891	1.92165
$896$	−11.2554	−0.376018
$897$	−8.21782	−0.274385
$898$	62.1138	2.07276
$899$	−18.6101	−0.620683
$900$	2.74456	0.0914854
$901$	−78.5494	−2.61686
$902$	0	0
$903$	−5.25544	−0.174890
$904$	−26.1831	−0.870838
$905$	−40.3288	−1.34057
$906$	5.48913	0.182364
$907$	−36.3505	−1.20700	−0.603500	−	0.797363i	$-0.706228\pi$
$907$	−36.3505	−1.20700	−0.603500	+	0.797363i	$0.706228\pi$
$908$	110.371	3.66280
$909$	−5.34363	−0.177237
$910$	19.4950	0.646253
$911$	38.2337	1.26674	0.633369	−	0.773850i	$-0.281671\pi$
$911$	38.2337	1.26674	0.633369	+	0.773850i	$0.281671\pi$
$912$	27.1229	0.898129
$913$	0	0
$914$	−21.3505	−0.706213
$915$	6.33830	0.209538
$916$	−15.8614	−0.524076
$917$	5.25544	0.173550
$918$	−15.1168	−0.498930
$919$	−31.6742	−1.04484	−0.522419	−	0.852689i	$-0.674970\pi$
$919$	−31.6742	−1.04484	−0.522419	+	0.852689i	$0.674970\pi$
$920$	−28.4125	−0.936733
$921$	−17.8178	−0.587118
$922$	36.6060	1.20555
$923$	−3.05934	−0.100699
$924$	0	0
$925$	3.13859	0.103196
$926$	−8.21782	−0.270054
$927$	8.11684	0.266592
$928$	−10.3723	−0.340487
$929$	−34.8832	−1.14448	−0.572240	−	0.820086i	$-0.693925\pi$
$929$	−34.8832	−1.14448	−0.572240	+	0.820086i	$0.693925\pi$
$930$	−44.1485	−1.44769
$931$	−27.1229	−0.888917
$932$	11.0371	0.361533
$933$	−6.51087	−0.213156
$934$	−56.1253	−1.83648
$935$	0	0
$936$	−24.6060	−0.804271
$937$	40.2419	1.31465	0.657323	−	0.753609i	$-0.271689\pi$
$937$	40.2419	1.31465	0.657323	+	0.753609i	$0.271689\pi$
$938$	32.2337	1.05247
$939$	9.11684	0.297517
$940$	13.0217	0.424723
$941$	16.3807	0.533997	0.266999	−	0.963697i	$-0.413968\pi$
$941$	16.3807	0.533997	0.266999	+	0.963697i	$0.413968\pi$
$942$	16.7306	0.545112
$943$	−11.3870	−0.370811
$944$	−38.2337	−1.24440
$945$	−1.87953	−0.0611410
$946$	0	0
$947$	−31.4891	−1.02326	−0.511630	−	0.859206i	$-0.670958\pi$
$947$	−31.4891	−1.02326	−0.511630	+	0.859206i	$0.670958\pi$
$948$	25.5383	0.829446
$949$	−30.5109	−0.990425
$950$	6.74456	0.218823
$951$	15.2554	0.494691
$952$	−28.4125	−0.920855
$953$	−47.2627	−1.53099	−0.765495	−	0.643442i	$-0.777506\pi$
$953$	−47.2627	−1.53099	−0.765495	+	0.643442i	$0.777506\pi$
$954$	33.1113	1.07202
$955$	41.4891	1.34256
$956$	−128.096	−4.14293
$957$	0	0
$958$	94.7011	3.05965
$959$	0.589907	0.0190491
$960$	5.62772	0.181634
$961$	23.3505	0.753243
$962$	−51.8614	−1.67208
$963$	−0.294954	−0.00950475
$964$	−58.0049	−1.86821
$965$	40.7393	1.31144
$966$	4.00000	0.128698
$967$	44.3508	1.42623	0.713113	−	0.701049i	$-0.247284\pi$
$967$	44.3508	1.42623	0.713113	+	0.701049i	$0.247284\pi$
$968$	0	0
$969$	−25.4891	−0.818829
$970$	74.7904	2.40137
$971$	59.4891	1.90910	0.954548	−	0.298056i	$-0.0963382\pi$
$971$	59.4891	1.90910	0.954548	+	0.298056i	$0.0963382\pi$
$972$	4.37228	0.140241
$973$	−8.23369	−0.263960
$974$	−32.1716	−1.03084
$975$	−2.57924	−0.0826018
$976$	−17.0256	−0.544975
$977$	13.1168	0.419645	0.209823	−	0.977739i	$-0.432711\pi$
$977$	13.1168	0.419645	0.209823	+	0.977739i	$0.432711\pi$
$978$	−23.6588	−0.756525
$979$	0	0
$980$	66.0951	2.11133
$981$	9.94987	0.317675
$982$	46.9783	1.49914
$983$	−56.2337	−1.79358	−0.896788	−	0.442460i	$-0.854106\pi$
$983$	−56.2337	−1.79358	−0.896788	+	0.442460i	$0.854106\pi$
$984$	−34.0951	−1.08691
$985$	−50.1369	−1.59749
$986$	38.1600	1.21526
$987$	−0.994667	−0.0316606
$988$	−76.4674	−2.43275
$989$	−13.2665	−0.421850
$990$	0	0
$991$	34.5109	1.09627	0.548137	−	0.836389i	$-0.315337\pi$
$991$	34.5109	1.09627	0.548137	+	0.836389i	$0.315337\pi$
$992$	30.2921	0.961774
$993$	−0.627719	−0.0199201
$994$	1.48913	0.0472322
$995$	16.2772	0.516022
$996$	37.2203	1.17937
$997$	48.4598	1.53474	0.767368	−	0.641207i	$-0.221566\pi$
$997$	48.4598	1.53474	0.767368	+	0.641207i	$0.221566\pi$
$998$	−95.5201	−3.02364
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.a.j.1.4	yes	4
3.2	odd	2		1089.2.a.u.1.1		4
4.3	odd	2		5808.2.a.ck.1.1		4
5.4	even	2		9075.2.a.cv.1.1		4
11.2	odd	10		363.2.e.n.202.4		16
11.3	even	5		363.2.e.n.130.4		16
11.4	even	5		363.2.e.n.148.4		16
11.5	even	5		363.2.e.n.124.1		16
11.6	odd	10		363.2.e.n.124.4		16
11.7	odd	10		363.2.e.n.148.1		16
11.8	odd	10		363.2.e.n.130.1		16
11.9	even	5		363.2.e.n.202.1		16
11.10	odd	2	inner	363.2.a.j.1.1	✓	4
33.32	even	2		1089.2.a.u.1.4		4
44.43	even	2		5808.2.a.ck.1.2		4
55.54	odd	2		9075.2.a.cv.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.j.1.1	✓	4	11.10	odd	2	inner
363.2.a.j.1.4	yes	4	1.1	even	1	trivial
363.2.e.n.124.1		16	11.5	even	5
363.2.e.n.124.4		16	11.6	odd	10
363.2.e.n.130.1		16	11.8	odd	10
363.2.e.n.130.4		16	11.3	even	5
363.2.e.n.148.1		16	11.7	odd	10
363.2.e.n.148.4		16	11.4	even	5
363.2.e.n.202.1		16	11.9	even	5
363.2.e.n.202.4		16	11.2	odd	10
1089.2.a.u.1.1		4	3.2	odd	2
1089.2.a.u.1.4		4	33.32	even	2
5808.2.a.ck.1.1		4	4.3	odd	2
5808.2.a.ck.1.2		4	44.43	even	2
9075.2.a.cv.1.1		4	5.4	even	2
9075.2.a.cv.1.4		4	55.54	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	363.a (trivial)

\(f(q)\)	\(=\)	\(q+2.52434 q^{2} +1.00000 q^{3} +4.37228 q^{4} -2.37228 q^{5} +2.52434 q^{6} +0.792287 q^{7} +5.98844 q^{8} +1.00000 q^{9} -5.98844 q^{10} +4.37228 q^{12} -4.10891 q^{13} +2.00000 q^{14} -2.37228 q^{15} +6.37228 q^{16} -5.98844 q^{17} +2.52434 q^{18} +4.25639 q^{19} -10.3723 q^{20} +0.792287 q^{21} +2.00000 q^{23} +5.98844 q^{24} +0.627719 q^{25} -10.3723 q^{26} +1.00000 q^{27} +3.46410 q^{28} -2.52434 q^{29} -5.98844 q^{30} +7.37228 q^{31} +4.10891 q^{32} -15.1168 q^{34} -1.87953 q^{35} +4.37228 q^{36} +5.00000 q^{37} +10.7446 q^{38} -4.10891 q^{39} -14.2063 q^{40} -5.69349 q^{41} +2.00000 q^{42} -6.63325 q^{43} -2.37228 q^{45} +5.04868 q^{46} -1.25544 q^{47} +6.37228 q^{48} -6.37228 q^{49} +1.58457 q^{50} -5.98844 q^{51} -17.9653 q^{52} +13.1168 q^{53} +2.52434 q^{54} +4.74456 q^{56} +4.25639 q^{57} -6.37228 q^{58} -6.00000 q^{59} -10.3723 q^{60} -2.67181 q^{61} +18.6101 q^{62} +0.792287 q^{63} -2.37228 q^{64} +9.74749 q^{65} +16.1168 q^{67} -26.1831 q^{68} +2.00000 q^{69} -4.74456 q^{70} +0.744563 q^{71} +5.98844 q^{72} +7.42554 q^{73} +12.6217 q^{74} +0.627719 q^{75} +18.6101 q^{76} -10.3723 q^{78} +5.84096 q^{79} -15.1168 q^{80} +1.00000 q^{81} -14.3723 q^{82} +8.51278 q^{83} +3.46410 q^{84} +14.2063 q^{85} -16.7446 q^{86} -2.52434 q^{87} -6.37228 q^{89} -5.98844 q^{90} -3.25544 q^{91} +8.74456 q^{92} +7.37228 q^{93} -3.16915 q^{94} -10.0974 q^{95} +4.10891 q^{96} -12.4891 q^{97} -16.0858 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 6 q^{4} + 2 q^{5} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 2 q^{15} + 14 q^{16} - 30 q^{20} + 8 q^{23} + 14 q^{25} - 30 q^{26} + 4 q^{27} + 18 q^{31} - 26 q^{34} + 6 q^{36} + 20 q^{37} + 20 q^{38}+ \cdots - 4 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 + 6 * q^4 + 2 * q^5 + 4 * q^9 + 6 * q^12 + 8 * q^14 + 2 * q^15 + 14 * q^16 - 30 * q^20 + 8 * q^23 + 14 * q^25 - 30 * q^26 + 4 * q^27 + 18 * q^31 - 26 * q^34 + 6 * q^36 + 20 * q^37 + 20 * q^38 + 8 * q^42 + 2 * q^45 - 28 * q^47 + 14 * q^48 - 14 * q^49 + 18 * q^53 - 4 * q^56 - 14 * q^58 - 24 * q^59 - 30 * q^60 + 2 * q^64 + 30 * q^67 + 8 * q^69 + 4 * q^70 - 20 * q^71 + 14 * q^75 - 30 * q^78 - 26 * q^80 + 4 * q^81 - 46 * q^82 - 44 * q^86 - 14 * q^89 - 36 * q^91 + 12 * q^92 + 18 * q^93 - 4 * q^97

Introduction

L-functions

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