LMFDB - Embedded newform 363.2.a.j.1.3

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			1.3
Root			$0.792287$ 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)

$f(q)$	$=$	\(q+0.792287 q^{2} +1.00000 q^{3} -1.37228 q^{4} +3.37228 q^{5} +0.792287 q^{6} +2.52434 q^{7} -2.67181 q^{8} +1.00000 q^{9} +2.67181 q^{10} -1.37228 q^{12} -5.84096 q^{13} +2.00000 q^{14} +3.37228 q^{15} +0.627719 q^{16} +2.67181 q^{17} +0.792287 q^{18} -0.939764 q^{19} -4.62772 q^{20} +2.52434 q^{21} +2.00000 q^{23} -2.67181 q^{24} +6.37228 q^{25} -4.62772 q^{26} +1.00000 q^{27} -3.46410 q^{28} -0.792287 q^{29} +2.67181 q^{30} +1.62772 q^{31} +5.84096 q^{32} +2.11684 q^{34} +8.51278 q^{35} -1.37228 q^{36} +5.00000 q^{37} -0.744563 q^{38} -5.84096 q^{39} -9.01011 q^{40} -10.8896 q^{41} +2.00000 q^{42} -6.63325 q^{43} +3.37228 q^{45} +1.58457 q^{46} -12.7446 q^{47} +0.627719 q^{48} -0.627719 q^{49} +5.04868 q^{50} +2.67181 q^{51} +8.01544 q^{52} -4.11684 q^{53} +0.792287 q^{54} -6.74456 q^{56} -0.939764 q^{57} -0.627719 q^{58} -6.00000 q^{59} -4.62772 q^{60} +5.98844 q^{61} +1.28962 q^{62} +2.52434 q^{63} +3.37228 q^{64} -19.6974 q^{65} -1.11684 q^{67} -3.66648 q^{68} +2.00000 q^{69} +6.74456 q^{70} -10.7446 q^{71} -2.67181 q^{72} +9.15759 q^{73} +3.96143 q^{74} +6.37228 q^{75} +1.28962 q^{76} -4.62772 q^{78} +4.10891 q^{79} +2.11684 q^{80} +1.00000 q^{81} -8.62772 q^{82} -1.87953 q^{83} -3.46410 q^{84} +9.01011 q^{85} -5.25544 q^{86} -0.792287 q^{87} -0.627719 q^{89} +2.67181 q^{90} -14.7446 q^{91} -2.74456 q^{92} +1.62772 q^{93} -10.0974 q^{94} -3.16915 q^{95} +5.84096 q^{96} +10.4891 q^{97} -0.497333 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.792287	0.560232	0.280116	−	0.959966i	$-0.409627\pi$
$2$	0.792287	0.560232	0.280116	+	0.959966i	$0.409627\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.37228	−0.686141
$5$	3.37228	1.50813	0.754065	−	0.656800i	$-0.228090\pi$
$5$	3.37228	1.50813	0.754065	+	0.656800i	$0.228090\pi$
$6$	0.792287	0.323450
$7$	2.52434	0.954110	0.477055	−	0.878873i	$-0.341704\pi$
$7$	2.52434	0.954110	0.477055	+	0.878873i	$0.341704\pi$
$8$	−2.67181	−0.944629
$9$	1.00000	0.333333
$10$	2.67181	0.844902
$11$	0	0
$11$	0	0
$12$	−1.37228	−0.396143
$13$	−5.84096	−1.61999	−0.809996	−	0.586436i	$-0.800531\pi$
$13$	−5.84096	−1.61999	−0.809996	+	0.586436i	$0.800531\pi$
$14$	2.00000	0.534522
$15$	3.37228	0.870719
$16$	0.627719	0.156930
$17$	2.67181	0.648010	0.324005	−	0.946055i	$-0.394970\pi$
$17$	2.67181	0.648010	0.324005	+	0.946055i	$0.394970\pi$
$18$	0.792287	0.186744
$19$	−0.939764	−0.215597	−0.107798	−	0.994173i	$-0.534380\pi$
$19$	−0.939764	−0.215597	−0.107798	+	0.994173i	$0.534380\pi$
$20$	−4.62772	−1.03479
$21$	2.52434	0.550856
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	−2.67181	−0.545382
$25$	6.37228	1.27446
$26$	−4.62772	−0.907570
$27$	1.00000	0.192450
$28$	−3.46410	−0.654654
$29$	−0.792287	−0.147124	−0.0735620	−	0.997291i	$-0.523437\pi$
$29$	−0.792287	−0.147124	−0.0735620	+	0.997291i	$0.523437\pi$
$30$	2.67181	0.487804
$31$	1.62772	0.292347	0.146173	−	0.989259i	$-0.453304\pi$
$31$	1.62772	0.292347	0.146173	+	0.989259i	$0.453304\pi$
$32$	5.84096	1.03255
$33$	0	0
$34$	2.11684	0.363036
$35$	8.51278	1.43892
$36$	−1.37228	−0.228714
$37$	5.00000	0.821995	0.410997	−	0.911636i	$-0.365181\pi$
$37$	5.00000	0.821995	0.410997	+	0.911636i	$0.365181\pi$
$38$	−0.744563	−0.120784
$39$	−5.84096	−0.935303
$40$	−9.01011	−1.42462
$41$	−10.8896	−1.70068	−0.850338	−	0.526237i	$-0.823602\pi$
$41$	−10.8896	−1.70068	−0.850338	+	0.526237i	$0.823602\pi$
$42$	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
$45$	3.37228	0.502710
$46$	1.58457	0.233633
$47$	−12.7446	−1.85899	−0.929493	−	0.368840i	$-0.879755\pi$
$47$	−12.7446	−1.85899	−0.929493	+	0.368840i	$0.879755\pi$
$48$	0.627719	0.0906034
$49$	−0.627719	−0.0896741
$50$	5.04868	0.713991
$51$	2.67181	0.374129
$52$	8.01544	1.11154
$53$	−4.11684	−0.565492	−0.282746	−	0.959195i	$-0.591245\pi$
$53$	−4.11684	−0.565492	−0.282746	+	0.959195i	$0.591245\pi$
$54$	0.792287	0.107817
$55$	0	0
$56$	−6.74456	−0.901280
$57$	−0.939764	−0.124475
$58$	−0.627719	−0.0824235
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−4.62772	−0.597436
$61$	5.98844	0.766741	0.383371	−	0.923595i	$-0.374763\pi$
$61$	5.98844	0.766741	0.383371	+	0.923595i	$0.374763\pi$
$62$	1.28962	0.163782
$63$	2.52434	0.318037
$64$	3.37228	0.421535
$65$	−19.6974	−2.44316
$66$	0	0
$67$	−1.11684	−0.136444	−0.0682221	−	0.997670i	$-0.521733\pi$
$67$	−1.11684	−0.136444	−0.0682221	+	0.997670i	$0.521733\pi$
$68$	−3.66648	−0.444626
$69$	2.00000	0.240772
$70$	6.74456	0.806129
$71$	−10.7446	−1.27514	−0.637572	−	0.770390i	$-0.720061\pi$
$71$	−10.7446	−1.27514	−0.637572	+	0.770390i	$0.720061\pi$
$72$	−2.67181	−0.314876
$73$	9.15759	1.07181	0.535907	−	0.844277i	$-0.319970\pi$
$73$	9.15759	1.07181	0.535907	+	0.844277i	$0.319970\pi$
$74$	3.96143	0.460507
$75$	6.37228	0.735808
$76$	1.28962	0.147930
$77$	0	0
$78$	−4.62772	−0.523986
$79$	4.10891	0.462289	0.231144	−	0.972919i	$-0.425753\pi$
$79$	4.10891	0.462289	0.231144	+	0.972919i	$0.425753\pi$
$80$	2.11684	0.236670
$81$	1.00000	0.111111
$82$	−8.62772	−0.952772
$83$	−1.87953	−0.206305	−0.103152	−	0.994666i	$-0.532893\pi$
$83$	−1.87953	−0.206305	−0.103152	+	0.994666i	$0.532893\pi$
$84$	−3.46410	−0.377964
$85$	9.01011	0.977284
$86$	−5.25544	−0.566708
$87$	−0.792287	−0.0849421
$88$	0	0
$89$	−0.627719	−0.0665380	−0.0332690	−	0.999446i	$-0.510592\pi$
$89$	−0.627719	−0.0665380	−0.0332690	+	0.999446i	$0.510592\pi$
$90$	2.67181	0.281634
$91$	−14.7446	−1.54565
$92$	−2.74456	−0.286140
$93$	1.62772	0.168787
$94$	−10.0974	−1.04146
$95$	−3.16915	−0.325148
$96$	5.84096	0.596141
$97$	10.4891	1.06501	0.532505	−	0.846427i	$-0.321251\pi$
$97$	10.4891	1.06501	0.532505	+	0.846427i	$0.321251\pi$
$98$	−0.497333	−0.0502383
$99$	0	0
$100$	−8.74456	−0.874456
$101$	11.9769	1.19174	0.595872	−	0.803079i	$-0.296807\pi$
$101$	11.9769	1.19174	0.595872	+	0.803079i	$0.296807\pi$
$102$	2.11684	0.209599
$103$	−9.11684	−0.898309	−0.449155	−	0.893454i	$-0.648275\pi$
$103$	−9.11684	−0.898309	−0.449155	+	0.893454i	$0.648275\pi$
$104$	15.6060	1.53029
$105$	8.51278	0.830762
$106$	−3.26172	−0.316806
$107$	13.5615	1.31104	0.655518	−	0.755180i	$-0.272451\pi$
$107$	13.5615	1.31104	0.655518	+	0.755180i	$0.272451\pi$
$108$	−1.37228	−0.132048
$109$	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
$112$	1.58457	0.149728
$113$	1.37228	0.129093	0.0645467	−	0.997915i	$-0.479440\pi$
$113$	1.37228	0.129093	0.0645467	+	0.997915i	$0.479440\pi$
$114$	−0.744563	−0.0697347
$115$	6.74456	0.628934
$116$	1.08724	0.100948
$117$	−5.84096	−0.539997
$118$	−4.75372	−0.437616
$119$	6.74456	0.618273
$120$	−9.01011	−0.822507
$121$	0	0
$122$	4.74456	0.429553
$123$	−10.8896	−0.981886
$124$	−2.23369	−0.200591
$125$	4.62772	0.413916
$126$	2.00000	0.178174
$127$	−9.74749	−0.864950	−0.432475	−	0.901646i	$-0.642360\pi$
$127$	−9.74749	−0.864950	−0.432475	+	0.901646i	$0.642360\pi$
$128$	−9.01011	−0.796389
$129$	−6.63325	−0.584025
$130$	−15.6060	−1.36873
$131$	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
$133$	−2.37228	−0.205703
$134$	−0.884861	−0.0764403
$135$	3.37228	0.290240
$136$	−7.13859	−0.612129
$137$	−10.7446	−0.917970	−0.458985	−	0.888444i	$-0.651787\pi$
$137$	−10.7446	−0.917970	−0.458985	+	0.888444i	$0.651787\pi$
$138$	1.58457	0.134888
$139$	10.3923	0.881464	0.440732	−	0.897639i	$-0.354719\pi$
$139$	10.3923	0.881464	0.440732	+	0.897639i	$0.354719\pi$
$140$	−11.6819	−0.987303
$141$	−12.7446	−1.07329
$142$	−8.51278	−0.714376
$143$	0	0
$144$	0.627719	0.0523099
$145$	−2.67181	−0.221882
$146$	7.25544	0.600464
$147$	−0.627719	−0.0517734
$148$	−6.86141	−0.564004
$149$	2.67181	0.218884	0.109442	−	0.993993i	$-0.465094\pi$
$149$	2.67181	0.218884	0.109442	+	0.993993i	$0.465094\pi$
$150$	5.04868	0.412223
$151$	−22.0742	−1.79638	−0.898188	−	0.439612i	$-0.855116\pi$
$151$	−22.0742	−1.79638	−0.898188	+	0.439612i	$0.855116\pi$
$152$	2.51087	0.203659
$153$	2.67181	0.216003
$154$	0	0
$155$	5.48913	0.440897
$156$	8.01544	0.641749
$157$	12.3723	0.987416	0.493708	−	0.869628i	$-0.335641\pi$
$157$	12.3723	0.987416	0.493708	+	0.869628i	$0.335641\pi$
$158$	3.25544	0.258989
$159$	−4.11684	−0.326487
$160$	19.6974	1.55721
$161$	5.04868	0.397891
$162$	0.792287	0.0622479
$163$	−3.62772	−0.284145	−0.142072	−	0.989856i	$-0.545377\pi$
$163$	−3.62772	−0.284145	−0.142072	+	0.989856i	$0.545377\pi$
$164$	14.9436	1.16690
$165$	0	0
$166$	−1.48913	−0.115579
$167$	−23.6588	−1.83077	−0.915387	−	0.402576i	$-0.868115\pi$
$167$	−23.6588	−1.83077	−0.915387	+	0.402576i	$0.868115\pi$
$168$	−6.74456	−0.520354
$169$	21.1168	1.62437
$170$	7.13859	0.547505
$171$	−0.939764	−0.0718655
$172$	9.10268	0.694073
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	−0.627719	−0.0475872
$175$	16.0858	1.21597
$176$	0	0
$177$	−6.00000	−0.450988
$178$	−0.497333	−0.0372767
$179$	10.2337	0.764902	0.382451	−	0.923976i	$-0.375080\pi$
$179$	10.2337	0.764902	0.382451	+	0.923976i	$0.375080\pi$
$180$	−4.62772	−0.344930
$181$	17.0000	1.26360	0.631800	−	0.775131i	$-0.282316\pi$
$181$	17.0000	1.26360	0.631800	+	0.775131i	$0.282316\pi$
$182$	−11.6819	−0.865922
$183$	5.98844	0.442678
$184$	−5.34363	−0.393938
$185$	16.8614	1.23968
$186$	1.28962	0.0945596
$187$	0	0
$188$	17.4891	1.27553
$189$	2.52434	0.183619
$190$	−2.51087	−0.182158
$191$	5.48913	0.397179	0.198590	−	0.980083i	$-0.436364\pi$
$191$	5.48913	0.397179	0.198590	+	0.980083i	$0.436364\pi$
$192$	3.37228	0.243373
$193$	10.5398	0.758670	0.379335	−	0.925259i	$-0.376153\pi$
$193$	10.5398	0.758670	0.379335	+	0.925259i	$0.376153\pi$
$194$	8.31040	0.596652
$195$	−19.6974	−1.41056
$196$	0.861407	0.0615290
$197$	2.08191	0.148330	0.0741649	−	0.997246i	$-0.476371\pi$
$197$	2.08191	0.148330	0.0741649	+	0.997246i	$0.476371\pi$
$198$	0	0
$199$	21.8614	1.54971	0.774857	−	0.632137i	$-0.217822\pi$
$199$	21.8614	1.54971	0.774857	+	0.632137i	$0.217822\pi$
$200$	−17.0256	−1.20389
$201$	−1.11684	−0.0787761
$202$	9.48913	0.667653
$203$	−2.00000	−0.140372
$204$	−3.66648	−0.256705
$205$	−36.7229	−2.56484
$206$	−7.22316	−0.503261
$207$	2.00000	0.139010
$208$	−3.66648	−0.254225
$209$	0	0
$210$	6.74456	0.465419
$211$	10.7422	0.739521	0.369760	−	0.929127i	$-0.379440\pi$
$211$	10.7422	0.739521	0.369760	+	0.929127i	$0.379440\pi$
$212$	5.64947	0.388007
$213$	−10.7446	−0.736205
$214$	10.7446	0.734483
$215$	−22.3692	−1.52557
$216$	−2.67181	−0.181794
$217$	4.10891	0.278931
$218$	7.88316	0.533915
$219$	9.15759	0.618812
$220$	0	0
$221$	−15.6060	−1.04977
$222$	3.96143	0.265874
$223$	23.8614	1.59788	0.798939	−	0.601412i	$-0.205395\pi$
$223$	23.8614	1.59788	0.798939	+	0.601412i	$0.205395\pi$
$224$	14.7446	0.985163
$225$	6.37228	0.424819
$226$	1.08724	0.0723222
$227$	7.92287	0.525859	0.262930	−	0.964815i	$-0.415311\pi$
$227$	7.92287	0.525859	0.262930	+	0.964815i	$0.415311\pi$
$228$	1.28962	0.0854072
$229$	−9.37228	−0.619338	−0.309669	−	0.950844i	$-0.600218\pi$
$229$	−9.37228	−0.619338	−0.309669	+	0.950844i	$0.600218\pi$
$230$	5.34363	0.352348
$231$	0	0
$232$	2.11684	0.138978
$233$	0.792287	0.0519044	0.0259522	−	0.999663i	$-0.491738\pi$
$233$	0.792287	0.0519044	0.0259522	+	0.999663i	$0.491738\pi$
$234$	−4.62772	−0.302523
$235$	−42.9783	−2.80359
$236$	8.23369	0.535967
$237$	4.10891	0.266903
$238$	5.34363	0.346376
$239$	22.6641	1.46602	0.733011	−	0.680217i	$-0.238115\pi$
$239$	22.6641	1.46602	0.733011	+	0.680217i	$0.238115\pi$
$240$	2.11684	0.136642
$241$	−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
$244$	−8.21782	−0.526092
$245$	−2.11684	−0.135240
$246$	−8.62772	−0.550083
$247$	5.48913	0.349265
$248$	−4.34896	−0.276159
$249$	−1.87953	−0.119110
$250$	3.66648	0.231889
$251$	−23.4891	−1.48262	−0.741310	−	0.671163i	$-0.765795\pi$
$251$	−23.4891	−1.48262	−0.741310	+	0.671163i	$0.765795\pi$
$252$	−3.46410	−0.218218
$253$	0	0
$254$	−7.72281	−0.484572
$255$	9.01011	0.564235
$256$	−13.8832	−0.867697
$257$	18.6277	1.16197	0.580983	−	0.813916i	$-0.302668\pi$
$257$	18.6277	1.16197	0.580983	+	0.813916i	$0.302668\pi$
$258$	−5.25544	−0.327189
$259$	12.6217	0.784274
$260$	27.0303	1.67635
$261$	−0.792287	−0.0490413
$262$	5.25544	0.324682
$263$	15.7359	0.970319	0.485160	−	0.874426i	$-0.338761\pi$
$263$	15.7359	0.970319	0.485160	+	0.874426i	$0.338761\pi$
$264$	0	0
$265$	−13.8832	−0.852835
$266$	−1.87953	−0.115241
$267$	−0.627719	−0.0384158
$268$	1.53262	0.0936199
$269$	−8.11684	−0.494893	−0.247446	−	0.968902i	$-0.579591\pi$
$269$	−8.11684	−0.494893	−0.247446	+	0.968902i	$0.579591\pi$
$270$	2.67181	0.162601
$271$	10.3923	0.631288	0.315644	−	0.948878i	$-0.397780\pi$
$271$	10.3923	0.631288	0.315644	+	0.948878i	$0.397780\pi$
$272$	1.67715	0.101692
$273$	−14.7446	−0.892382
$274$	−8.51278	−0.514276
$275$	0	0
$276$	−2.74456	−0.165203
$277$	8.66025	0.520344	0.260172	−	0.965562i	$-0.416221\pi$
$277$	8.66025	0.520344	0.260172	+	0.965562i	$0.416221\pi$
$278$	8.23369	0.493824
$279$	1.62772	0.0974490
$280$	−22.7446	−1.35925
$281$	27.1229	1.61802	0.809008	−	0.587797i	$-0.200005\pi$
$281$	27.1229	1.61802	0.809008	+	0.587797i	$0.200005\pi$
$282$	−10.0974	−0.601289
$283$	−7.57301	−0.450169	−0.225084	−	0.974339i	$-0.572266\pi$
$283$	−7.57301	−0.450169	−0.225084	+	0.974339i	$0.572266\pi$
$284$	14.7446	0.874929
$285$	−3.16915	−0.187724
$286$	0	0
$287$	−27.4891	−1.62263
$288$	5.84096	0.344182
$289$	−9.86141	−0.580083
$290$	−2.11684	−0.124305
$291$	10.4891	0.614883
$292$	−12.5668	−0.735416
$293$	−29.7947	−1.74063	−0.870313	−	0.492499i	$-0.836084\pi$
$293$	−29.7947	−1.74063	−0.870313	+	0.492499i	$0.836084\pi$
$294$	−0.497333	−0.0290051
$295$	−20.2337	−1.17805
$296$	−13.3591	−0.776480
$297$	0	0
$298$	2.11684	0.122625
$299$	−11.6819	−0.675583
$300$	−8.74456	−0.504868
$301$	−16.7446	−0.965141
$302$	−17.4891	−1.00639
$303$	11.9769	0.688054
$304$	−0.589907	−0.0338335
$305$	20.1947	1.15635
$306$	2.11684	0.121012
$307$	1.23472	0.0704690	0.0352345	−	0.999379i	$-0.488782\pi$
$307$	1.23472	0.0704690	0.0352345	+	0.999379i	$0.488782\pi$
$308$	0	0
$309$	−9.11684	−0.518639
$310$	4.34896	0.247004
$311$	−29.4891	−1.67217	−0.836087	−	0.548596i	$-0.815162\pi$
$311$	−29.4891	−1.67217	−0.836087	+	0.548596i	$0.815162\pi$
$312$	15.6060	0.883514
$313$	−8.11684	−0.458791	−0.229396	−	0.973333i	$-0.573675\pi$
$313$	−8.11684	−0.458791	−0.229396	+	0.973333i	$0.573675\pi$
$314$	9.80240	0.553181
$315$	8.51278	0.479641
$316$	−5.63858	−0.317195
$317$	26.7446	1.50212	0.751062	−	0.660232i	$-0.229542\pi$
$317$	26.7446	1.50212	0.751062	+	0.660232i	$0.229542\pi$
$318$	−3.26172	−0.182908
$319$	0	0
$320$	11.3723	0.635730
$321$	13.5615	0.756926
$322$	4.00000	0.222911
$323$	−2.51087	−0.139709
$324$	−1.37228	−0.0762379
$325$	−37.2203	−2.06461
$326$	−2.87419	−0.159187
$327$	9.94987	0.550229
$328$	29.0951	1.60651
$329$	−32.1716	−1.77368
$330$	0	0
$331$	−6.37228	−0.350252	−0.175126	−	0.984546i	$-0.556033\pi$
$331$	−6.37228	−0.350252	−0.175126	+	0.984546i	$0.556033\pi$
$332$	2.57924	0.141554
$333$	5.00000	0.273998
$334$	−18.7446	−1.02566
$335$	−3.76631	−0.205776
$336$	1.58457	0.0864456
$337$	−2.02700	−0.110418	−0.0552090	−	0.998475i	$-0.517583\pi$
$337$	−2.02700	−0.110418	−0.0552090	+	0.998475i	$0.517583\pi$
$338$	16.7306	0.910025
$339$	1.37228	0.0745321
$340$	−12.3644	−0.670554
$341$	0	0
$342$	−0.744563	−0.0402613
$343$	−19.2549	−1.03967
$344$	17.7228	0.955550
$345$	6.74456	0.363115
$346$	5.48913	0.295097
$347$	−28.4125	−1.52526	−0.762632	−	0.646832i	$-0.776093\pi$
$347$	−28.4125	−1.52526	−0.762632	+	0.646832i	$0.776093\pi$
$348$	1.08724	0.0582822
$349$	25.9808	1.39072	0.695359	−	0.718662i	$-0.255245\pi$
$349$	25.9808	1.39072	0.695359	+	0.718662i	$0.255245\pi$
$350$	12.7446	0.681226
$351$	−5.84096	−0.311768
$352$	0	0
$353$	22.3505	1.18960	0.594799	−	0.803874i	$-0.297231\pi$
$353$	22.3505	1.18960	0.594799	+	0.803874i	$0.297231\pi$
$354$	−4.75372	−0.252657
$355$	−36.2337	−1.92308
$356$	0.861407	0.0456545
$357$	6.74456	0.356960
$358$	8.10802	0.428522
$359$	−18.6101	−0.982205	−0.491103	−	0.871102i	$-0.663406\pi$
$359$	−18.6101	−0.982205	−0.491103	+	0.871102i	$0.663406\pi$
$360$	−9.01011	−0.474875
$361$	−18.1168	−0.953518
$362$	13.4689	0.707909
$363$	0	0
$364$	20.2337	1.06053
$365$	30.8820	1.61644
$366$	4.74456	0.248002
$367$	33.7228	1.76032	0.880158	−	0.474680i	$-0.157436\pi$
$367$	33.7228	1.76032	0.880158	+	0.474680i	$0.157436\pi$
$368$	1.25544	0.0654442
$369$	−10.8896	−0.566892
$370$	13.3591	0.694505
$371$	−10.3923	−0.539542
$372$	−2.23369	−0.115811
$373$	−5.39853	−0.279525	−0.139763	−	0.990185i	$-0.544634\pi$
$373$	−5.39853	−0.279525	−0.139763	+	0.990185i	$0.544634\pi$
$374$	0	0
$375$	4.62772	0.238974
$376$	34.0511	1.75605
$377$	4.62772	0.238340
$378$	2.00000	0.102869
$379$	−33.7228	−1.73222	−0.866112	−	0.499849i	$-0.833389\pi$
$379$	−33.7228	−1.73222	−0.866112	+	0.499849i	$0.833389\pi$
$380$	4.34896	0.223097
$381$	−9.74749	−0.499379
$382$	4.34896	0.222512
$383$	−0.510875	−0.0261045	−0.0130522	−	0.999915i	$-0.504155\pi$
$383$	−0.510875	−0.0261045	−0.0130522	+	0.999915i	$0.504155\pi$
$384$	−9.01011	−0.459795
$385$	0	0
$386$	8.35053	0.425031
$387$	−6.63325	−0.337187
$388$	−14.3940	−0.730746
$389$	8.11684	0.411540	0.205770	−	0.978600i	$-0.434030\pi$
$389$	8.11684	0.411540	0.205770	+	0.978600i	$0.434030\pi$
$390$	−15.6060	−0.790239
$391$	5.34363	0.270239
$392$	1.67715	0.0847088
$393$	6.63325	0.334603
$394$	1.64947	0.0830990
$395$	13.8564	0.697191
$396$	0	0
$397$	1.51087	0.0758286	0.0379143	−	0.999281i	$-0.487929\pi$
$397$	1.51087	0.0758286	0.0379143	+	0.999281i	$0.487929\pi$
$398$	17.3205	0.868199
$399$	−2.37228	−0.118763
$400$	4.00000	0.200000
$401$	−1.88316	−0.0940403	−0.0470202	−	0.998894i	$-0.514973\pi$
$401$	−1.88316	−0.0940403	−0.0470202	+	0.998894i	$0.514973\pi$
$402$	−0.884861	−0.0441329
$403$	−9.50744	−0.473600
$404$	−16.4356	−0.817704
$405$	3.37228	0.167570
$406$	−1.58457	−0.0786411
$407$	0	0
$408$	−7.13859	−0.353413
$409$	−20.3422	−1.00586	−0.502928	−	0.864328i	$-0.667744\pi$
$409$	−20.3422	−1.00586	−0.502928	+	0.864328i	$0.667744\pi$
$410$	−29.0951	−1.43690
$411$	−10.7446	−0.529990
$412$	12.5109	0.616367
$413$	−15.1460	−0.745287
$414$	1.58457	0.0778776
$415$	−6.33830	−0.311135
$416$	−34.1168	−1.67272
$417$	10.3923	0.508913
$418$	0	0
$419$	−31.4891	−1.53834	−0.769172	−	0.639042i	$-0.779331\pi$
$419$	−31.4891	−1.53834	−0.769172	+	0.639042i	$0.779331\pi$
$420$	−11.6819	−0.570020
$421$	34.8614	1.69904	0.849520	−	0.527556i	$-0.176891\pi$
$421$	34.8614	1.69904	0.849520	+	0.527556i	$0.176891\pi$
$422$	8.51087	0.414303
$423$	−12.7446	−0.619662
$424$	10.9994	0.534180
$425$	17.0256	0.825861
$426$	−8.51278	−0.412445
$427$	15.1168	0.731555
$428$	−18.6101	−0.899555
$429$	0	0
$430$	−17.7228	−0.854670
$431$	2.87419	0.138445	0.0692225	−	0.997601i	$-0.477948\pi$
$431$	2.87419	0.138445	0.0692225	+	0.997601i	$0.477948\pi$
$432$	0.627719	0.0302011
$433$	11.2337	0.539857	0.269928	−	0.962880i	$-0.413000\pi$
$433$	11.2337	0.539857	0.269928	+	0.962880i	$0.413000\pi$
$434$	3.25544	0.156266
$435$	−2.67181	−0.128104
$436$	−13.6540	−0.653909
$437$	−1.87953	−0.0899100
$438$	7.25544	0.346678
$439$	5.39853	0.257658	0.128829	−	0.991667i	$-0.458878\pi$
$439$	5.39853	0.257658	0.128829	+	0.991667i	$0.458878\pi$
$440$	0	0
$441$	−0.627719	−0.0298914
$442$	−12.3644	−0.588115
$443$	9.25544	0.439739	0.219870	−	0.975529i	$-0.429437\pi$
$443$	9.25544	0.439739	0.219870	+	0.975529i	$0.429437\pi$
$444$	−6.86141	−0.325628
$445$	−2.11684	−0.100348
$446$	18.9051	0.895182
$447$	2.67181	0.126372
$448$	8.51278	0.402191
$449$	−15.6060	−0.736491	−0.368246	−	0.929729i	$-0.620041\pi$
$449$	−15.6060	−0.736491	−0.368246	+	0.929729i	$0.620041\pi$
$450$	5.04868	0.237997
$451$	0	0
$452$	−1.88316	−0.0885762
$453$	−22.0742	−1.03714
$454$	6.27719	0.294603
$455$	−49.7228	−2.33104
$456$	2.51087	0.117582
$457$	38.3075	1.79195	0.895975	−	0.444105i	$-0.146478\pi$
$457$	38.3075	1.79195	0.895975	+	0.444105i	$0.146478\pi$
$458$	−7.42554	−0.346973
$459$	2.67181	0.124710
$460$	−9.25544	−0.431537
$461$	−4.55134	−0.211977	−0.105989	−	0.994367i	$-0.533801\pi$
$461$	−4.55134	−0.211977	−0.105989	+	0.994367i	$0.533801\pi$
$462$	0	0
$463$	−14.7446	−0.685238	−0.342619	−	0.939474i	$-0.611314\pi$
$463$	−14.7446	−0.685238	−0.342619	+	0.939474i	$0.611314\pi$
$464$	−0.497333	−0.0230881
$465$	5.48913	0.254552
$466$	0.627719	0.0290785
$467$	12.2337	0.566108	0.283054	−	0.959104i	$-0.408653\pi$
$467$	12.2337	0.566108	0.283054	+	0.959104i	$0.408653\pi$
$468$	8.01544	0.370514
$469$	−2.81929	−0.130183
$470$	−34.0511	−1.57066
$471$	12.3723	0.570085
$472$	16.0309	0.737881
$473$	0	0
$474$	3.25544	0.149527
$475$	−5.98844	−0.274768
$476$	−9.25544	−0.424222
$477$	−4.11684	−0.188497
$478$	17.9565	0.821311
$479$	−10.9822	−0.501790	−0.250895	−	0.968014i	$-0.580725\pi$
$479$	−10.9822	−0.501790	−0.250895	+	0.968014i	$0.580725\pi$
$480$	19.6974	0.899058
$481$	−29.2048	−1.33162
$482$	−10.5109	−0.478757
$483$	5.04868	0.229723
$484$	0	0
$485$	35.3723	1.60617
$486$	0.792287	0.0359389
$487$	−1.25544	−0.0568893	−0.0284446	−	0.999595i	$-0.509055\pi$
$487$	−1.25544	−0.0568893	−0.0284446	+	0.999595i	$0.509055\pi$
$488$	−16.0000	−0.724286
$489$	−3.62772	−0.164051
$490$	−1.67715	−0.0757658
$491$	1.28962	0.0581998	0.0290999	−	0.999577i	$-0.490736\pi$
$491$	1.28962	0.0581998	0.0290999	+	0.999577i	$0.490736\pi$
$492$	14.9436	0.673712
$493$	−2.11684	−0.0953379
$494$	4.34896	0.195669
$495$	0	0
$496$	1.02175	0.0458779
$497$	−27.1229	−1.21663
$498$	−1.48913	−0.0667293
$499$	36.8397	1.64917	0.824585	−	0.565738i	$-0.191409\pi$
$499$	36.8397	1.64917	0.824585	+	0.565738i	$0.191409\pi$
$500$	−6.35053	−0.284004
$501$	−23.6588	−1.05700
$502$	−18.6101	−0.830611
$503$	−2.17448	−0.0969553	−0.0484777	−	0.998824i	$-0.515437\pi$
$503$	−2.17448	−0.0969553	−0.0484777	+	0.998824i	$0.515437\pi$
$504$	−6.74456	−0.300427
$505$	40.3894	1.79730
$506$	0	0
$507$	21.1168	0.937832
$508$	13.3763	0.593478
$509$	4.97825	0.220657	0.110329	−	0.993895i	$-0.464810\pi$
$509$	4.97825	0.220657	0.110329	+	0.993895i	$0.464810\pi$
$510$	7.13859	0.316102
$511$	23.1168	1.02263
$512$	7.02078	0.310277
$513$	−0.939764	−0.0414916
$514$	14.7585	0.650969
$515$	−30.7446	−1.35477
$516$	9.10268	0.400723
$517$	0	0
$518$	10.0000	0.439375
$519$	6.92820	0.304114
$520$	52.6277	2.30788
$521$	−28.9783	−1.26956	−0.634780	−	0.772693i	$-0.718909\pi$
$521$	−28.9783	−1.26956	−0.634780	+	0.772693i	$0.718909\pi$
$522$	−0.627719	−0.0274745
$523$	−36.2805	−1.58644	−0.793218	−	0.608938i	$-0.791596\pi$
$523$	−36.2805	−1.58644	−0.793218	+	0.608938i	$0.791596\pi$
$524$	−9.10268	−0.397653
$525$	16.0858	0.702041
$526$	12.4674	0.543603
$527$	4.34896	0.189444
$528$	0	0
$529$	−19.0000	−0.826087
$530$	−10.9994	−0.477785
$531$	−6.00000	−0.260378
$532$	3.25544	0.141141
$533$	63.6060	2.75508
$534$	−0.497333	−0.0215217
$535$	45.7330	1.97721
$536$	2.98400	0.128889
$537$	10.2337	0.441616
$538$	−6.43087	−0.277254
$539$	0	0
$540$	−4.62772	−0.199145
$541$	27.7128	1.19147	0.595733	−	0.803182i	$-0.296861\pi$
$541$	27.7128	1.19147	0.595733	+	0.803182i	$0.296861\pi$
$542$	8.23369	0.353667
$543$	17.0000	0.729540
$544$	15.6060	0.669100
$545$	33.5538	1.43729
$546$	−11.6819	−0.499940
$547$	17.7253	0.757878	0.378939	−	0.925422i	$-0.376289\pi$
$547$	17.7253	0.757878	0.378939	+	0.925422i	$0.376289\pi$
$548$	14.7446	0.629857
$549$	5.98844	0.255580
$550$	0	0
$551$	0.744563	0.0317194
$552$	−5.34363	−0.227440
$553$	10.3723	0.441074
$554$	6.86141	0.291513
$555$	16.8614	0.715727
$556$	−14.2612	−0.604808
$557$	30.2921	1.28352	0.641758	−	0.766907i	$-0.278205\pi$
$557$	30.2921	1.28352	0.641758	+	0.766907i	$0.278205\pi$
$558$	1.28962	0.0545940
$559$	38.7446	1.63872
$560$	5.34363	0.225810
$561$	0	0
$562$	21.4891	0.906464
$563$	31.8766	1.34344	0.671720	−	0.740805i	$-0.265556\pi$
$563$	31.8766	1.34344	0.671720	+	0.740805i	$0.265556\pi$
$564$	17.4891	0.736425
$565$	4.62772	0.194690
$566$	−6.00000	−0.252199
$567$	2.52434	0.106012
$568$	28.7075	1.20454
$569$	−37.8102	−1.58508	−0.792542	−	0.609817i	$-0.791243\pi$
$569$	−37.8102	−1.58508	−0.792542	+	0.609817i	$0.791243\pi$
$570$	−2.51087	−0.105169
$571$	−14.5012	−0.606857	−0.303429	−	0.952854i	$-0.598131\pi$
$571$	−14.5012	−0.606857	−0.303429	+	0.952854i	$0.598131\pi$
$572$	0	0
$573$	5.48913	0.229311
$574$	−21.7793	−0.909049
$575$	12.7446	0.531485
$576$	3.37228	0.140512
$577$	6.48913	0.270146	0.135073	−	0.990836i	$-0.456873\pi$
$577$	6.48913	0.270146	0.135073	+	0.990836i	$0.456873\pi$
$578$	−7.81306	−0.324981
$579$	10.5398	0.438018
$580$	3.66648	0.152242
$581$	−4.74456	−0.196838
$582$	8.31040	0.344477
$583$	0	0
$584$	−24.4674	−1.01247
$585$	−19.6974	−0.814386
$586$	−23.6060	−0.975154
$587$	−6.97825	−0.288023	−0.144012	−	0.989576i	$-0.546000\pi$
$587$	−6.97825	−0.288023	−0.144012	+	0.989576i	$0.546000\pi$
$588$	0.861407	0.0355238
$589$	−1.52967	−0.0630290
$590$	−16.0309	−0.659981
$591$	2.08191	0.0856382
$592$	3.13859	0.128995
$593$	8.71516	0.357889	0.178944	−	0.983859i	$-0.442732\pi$
$593$	8.71516	0.357889	0.178944	+	0.983859i	$0.442732\pi$
$594$	0	0
$595$	22.7446	0.932436
$596$	−3.66648	−0.150185
$597$	21.8614	0.894728
$598$	−9.25544	−0.378483
$599$	−28.4674	−1.16315	−0.581573	−	0.813494i	$-0.697562\pi$
$599$	−28.4674	−1.16315	−0.581573	+	0.813494i	$0.697562\pi$
$600$	−17.0256	−0.695065
$601$	−19.4573	−0.793681	−0.396840	−	0.917888i	$-0.629893\pi$
$601$	−19.4573	−0.793681	−0.396840	+	0.917888i	$0.629893\pi$
$602$	−13.2665	−0.540702
$603$	−1.11684	−0.0454814
$604$	30.2921	1.23257
$605$	0	0
$606$	9.48913	0.385469
$607$	−32.7615	−1.32975	−0.664874	−	0.746956i	$-0.731515\pi$
$607$	−32.7615	−1.32975	−0.664874	+	0.746956i	$0.731515\pi$
$608$	−5.48913	−0.222613
$609$	−2.00000	−0.0810441
$610$	16.0000	0.647821
$611$	74.4405	3.01154
$612$	−3.66648	−0.148209
$613$	−29.5547	−1.19370	−0.596851	−	0.802352i	$-0.703582\pi$
$613$	−29.5547	−1.19370	−0.596851	+	0.802352i	$0.703582\pi$
$614$	0.978251	0.0394790
$615$	−36.7229	−1.48081
$616$	0	0
$617$	15.1386	0.609457	0.304728	−	0.952439i	$-0.401434\pi$
$617$	15.1386	0.609457	0.304728	+	0.952439i	$0.401434\pi$
$618$	−7.22316	−0.290558
$619$	−24.2337	−0.974034	−0.487017	−	0.873392i	$-0.661915\pi$
$619$	−24.2337	−0.974034	−0.487017	+	0.873392i	$0.661915\pi$
$620$	−7.53262	−0.302517
$621$	2.00000	0.0802572
$622$	−23.3639	−0.936805
$623$	−1.58457	−0.0634846
$624$	−3.66648	−0.146777
$625$	−16.2554	−0.650217
$626$	−6.43087	−0.257029
$627$	0	0
$628$	−16.9783	−0.677506
$629$	13.3591	0.532661
$630$	6.74456	0.268710
$631$	−12.2337	−0.487015	−0.243508	−	0.969899i	$-0.578298\pi$
$631$	−12.2337	−0.487015	−0.243508	+	0.969899i	$0.578298\pi$
$632$	−10.9783	−0.436691
$633$	10.7422	0.426963
$634$	21.1894	0.841537
$635$	−32.8713	−1.30446
$636$	5.64947	0.224016
$637$	3.66648	0.145271
$638$	0	0
$639$	−10.7446	−0.425048
$640$	−30.3846	−1.20106
$641$	−10.1168	−0.399591	−0.199796	−	0.979838i	$-0.564028\pi$
$641$	−10.1168	−0.399591	−0.199796	+	0.979838i	$0.564028\pi$
$642$	10.7446	0.424054
$643$	−42.0951	−1.66007	−0.830034	−	0.557712i	$-0.811679\pi$
$643$	−42.0951	−1.66007	−0.830034	+	0.557712i	$0.811679\pi$
$644$	−6.92820	−0.273009
$645$	−22.3692	−0.880786
$646$	−1.98933	−0.0782693
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	−2.67181	−0.104959
$649$	0	0
$650$	−29.4891	−1.15666
$651$	4.10891	0.161041
$652$	4.97825	0.194963
$653$	27.4891	1.07573	0.537866	−	0.843030i	$-0.319231\pi$
$653$	27.4891	1.07573	0.537866	+	0.843030i	$0.319231\pi$
$654$	7.88316	0.308256
$655$	22.3692	0.874036
$656$	−6.83563	−0.266886
$657$	9.15759	0.357272
$658$	−25.4891	−0.993670
$659$	−15.7359	−0.612985	−0.306492	−	0.951873i	$-0.599155\pi$
$659$	−15.7359	−0.612985	−0.306492	+	0.951873i	$0.599155\pi$
$660$	0	0
$661$	30.7228	1.19498	0.597489	−	0.801877i	$-0.296165\pi$
$661$	30.7228	1.19498	0.597489	+	0.801877i	$0.296165\pi$
$662$	−5.04868	−0.196222
$663$	−15.6060	−0.606086
$664$	5.02175	0.194882
$665$	−8.00000	−0.310227
$666$	3.96143	0.153502
$667$	−1.58457	−0.0613549
$668$	32.4665	1.25617
$669$	23.8614	0.922535
$670$	−2.98400	−0.115282
$671$	0	0
$672$	14.7446	0.568784
$673$	1.63948	0.0631972	0.0315986	−	0.999501i	$-0.489940\pi$
$673$	1.63948	0.0631972	0.0315986	+	0.999501i	$0.489940\pi$
$674$	−1.60597	−0.0618596
$675$	6.37228	0.245269
$676$	−28.9783	−1.11455
$677$	6.83563	0.262715	0.131357	−	0.991335i	$-0.458066\pi$
$677$	6.83563	0.262715	0.131357	+	0.991335i	$0.458066\pi$
$678$	1.08724	0.0417552
$679$	26.4781	1.01614
$680$	−24.0733	−0.923171
$681$	7.92287	0.303605
$682$	0	0
$683$	2.00000	0.0765279	0.0382639	−	0.999268i	$-0.487817\pi$
$683$	2.00000	0.0765279	0.0382639	+	0.999268i	$0.487817\pi$
$684$	1.28962	0.0493099
$685$	−36.2337	−1.38442
$686$	−15.2554	−0.582455
$687$	−9.37228	−0.357575
$688$	−4.16381	−0.158744
$689$	24.0463	0.916092
$690$	5.34363	0.203428
$691$	13.1168	0.498988	0.249494	−	0.968376i	$-0.419736\pi$
$691$	13.1168	0.498988	0.249494	+	0.968376i	$0.419736\pi$
$692$	−9.50744	−0.361419
$693$	0	0
$694$	−22.5109	−0.854501
$695$	35.0458	1.32936
$696$	2.11684	0.0802388
$697$	−29.0951	−1.10206
$698$	20.5842	0.779124
$699$	0.792287	0.0299670
$700$	−22.0742	−0.834327
$701$	−33.2588	−1.25617	−0.628084	−	0.778145i	$-0.716161\pi$
$701$	−33.2588	−1.25617	−0.628084	+	0.778145i	$0.716161\pi$
$702$	−4.62772	−0.174662
$703$	−4.69882	−0.177219
$704$	0	0
$705$	−42.9783	−1.61865
$706$	17.7080	0.666451
$707$	30.2337	1.13705
$708$	8.23369	0.309441
$709$	6.00000	0.225335	0.112667	−	0.993633i	$-0.464061\pi$
$709$	6.00000	0.225335	0.112667	+	0.993633i	$0.464061\pi$
$710$	−28.7075	−1.07737
$711$	4.10891	0.154096
$712$	1.67715	0.0628538
$713$	3.25544	0.121917
$714$	5.34363	0.199980
$715$	0	0
$716$	−14.0435	−0.524830
$717$	22.6641	0.846408
$718$	−14.7446	−0.550262
$719$	−6.00000	−0.223762	−0.111881	−	0.993722i	$-0.535688\pi$
$719$	−6.00000	−0.223762	−0.111881	+	0.993722i	$0.535688\pi$
$720$	2.11684	0.0788901
$721$	−23.0140	−0.857086
$722$	−14.3537	−0.534191
$723$	−13.2665	−0.493386
$724$	−23.3288	−0.867007
$725$	−5.04868	−0.187503
$726$	0	0
$727$	18.7446	0.695197	0.347599	−	0.937643i	$-0.386997\pi$
$727$	18.7446	0.695197	0.347599	+	0.937643i	$0.386997\pi$
$728$	39.3947	1.46007
$729$	1.00000	0.0370370
$730$	24.4674	0.905578
$731$	−17.7228	−0.655502
$732$	−8.21782	−0.303739
$733$	−7.42554	−0.274268	−0.137134	−	0.990552i	$-0.543789\pi$
$733$	−7.42554	−0.274268	−0.137134	+	0.990552i	$0.543789\pi$
$734$	26.7181	0.986185
$735$	−2.11684	−0.0780810
$736$	11.6819	0.430601
$737$	0	0
$738$	−8.62772	−0.317591
$739$	35.5808	1.30886	0.654430	−	0.756123i	$-0.272909\pi$
$739$	35.5808	1.30886	0.654430	+	0.756123i	$0.272909\pi$
$740$	−23.1386	−0.850592
$741$	5.48913	0.201648
$742$	−8.23369	−0.302268
$743$	−35.0458	−1.28570	−0.642852	−	0.765990i	$-0.722249\pi$
$743$	−35.0458	−1.28570	−0.642852	+	0.765990i	$0.722249\pi$
$744$	−4.34896	−0.159441
$745$	9.01011	0.330105
$746$	−4.27719	−0.156599
$747$	−1.87953	−0.0687683
$748$	0	0
$749$	34.2337	1.25087
$750$	3.66648	0.133881
$751$	35.3505	1.28996	0.644980	−	0.764200i	$-0.276866\pi$
$751$	35.3505	1.28996	0.644980	+	0.764200i	$0.276866\pi$
$752$	−8.00000	−0.291730
$753$	−23.4891	−0.855991
$754$	3.66648	0.133525
$755$	−74.4405	−2.70917
$756$	−3.46410	−0.125988
$757$	32.4891	1.18084	0.590419	−	0.807097i	$-0.298963\pi$
$757$	32.4891	1.18084	0.590419	+	0.807097i	$0.298963\pi$
$758$	−26.7181	−0.970447
$759$	0	0
$760$	8.46738	0.307144
$761$	−37.3128	−1.35259	−0.676295	−	0.736631i	$-0.736415\pi$
$761$	−37.3128	−1.35259	−0.676295	+	0.736631i	$0.736415\pi$
$762$	−7.72281	−0.279768
$763$	25.1168	0.909291
$764$	−7.53262	−0.272521
$765$	9.01011	0.325761
$766$	−0.404759	−0.0146246
$767$	35.0458	1.26543
$768$	−13.8832	−0.500965
$769$	−41.4217	−1.49371	−0.746853	−	0.664989i	$-0.768436\pi$
$769$	−41.4217	−1.49371	−0.746853	+	0.664989i	$0.768436\pi$
$770$	0	0
$771$	18.6277	0.670861
$772$	−14.4635	−0.520554
$773$	−47.2119	−1.69810	−0.849048	−	0.528316i	$-0.822824\pi$
$773$	−47.2119	−1.69810	−0.849048	+	0.528316i	$0.822824\pi$
$774$	−5.25544	−0.188903
$775$	10.3723	0.372583
$776$	−28.0250	−1.00604
$777$	12.6217	0.452801
$778$	6.43087	0.230558
$779$	10.2337	0.366660
$780$	27.0303	0.967841
$781$	0	0
$782$	4.23369	0.151396
$783$	−0.792287	−0.0283140
$784$	−0.394031	−0.0140725
$785$	41.7228	1.48915
$786$	5.25544	0.187455
$787$	−15.1460	−0.539898	−0.269949	−	0.962875i	$-0.587007\pi$
$787$	−15.1460	−0.539898	−0.269949	+	0.962875i	$0.587007\pi$
$788$	−2.85696	−0.101775
$789$	15.7359	0.560214
$790$	10.9783	0.390589
$791$	3.46410	0.123169
$792$	0	0
$793$	−34.9783	−1.24211
$794$	1.19705	0.0424816
$795$	−13.8832	−0.492385
$796$	−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	$0.466111\pi$
$798$	−1.87953	−0.0665346
$799$	−34.0511	−1.20464
$800$	37.2203	1.31593
$801$	−0.627719	−0.0221793
$802$	−1.49200	−0.0526844
$803$	0	0
$804$	1.53262	0.0540515
$805$	17.0256	0.600072
$806$	−7.53262	−0.265325
$807$	−8.11684	−0.285726
$808$	−32.0000	−1.12576
$809$	−25.9431	−0.912111	−0.456055	−	0.889951i	$-0.650738\pi$
$809$	−25.9431	−0.912111	−0.456055	+	0.889951i	$0.650738\pi$
$810$	2.67181	0.0938780
$811$	38.8597	1.36455	0.682275	−	0.731096i	$-0.260991\pi$
$811$	38.8597	1.36455	0.682275	+	0.731096i	$0.260991\pi$
$812$	2.74456	0.0963153
$813$	10.3923	0.364474
$814$	0	0
$815$	−12.2337	−0.428527
$816$	1.67715	0.0587119
$817$	6.23369	0.218089
$818$	−16.1168	−0.563512
$819$	−14.7446	−0.515217
$820$	50.3942	1.75984
$821$	46.0280	1.60639	0.803194	−	0.595718i	$-0.203132\pi$
$821$	46.0280	1.60639	0.803194	+	0.595718i	$0.203132\pi$
$822$	−8.51278	−0.296917
$823$	19.3505	0.674517	0.337259	−	0.941412i	$-0.390500\pi$
$823$	19.3505	0.674517	0.337259	+	0.941412i	$0.390500\pi$
$824$	24.3585	0.848569
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−36.9253	−1.28402	−0.642009	−	0.766697i	$-0.721899\pi$
$827$	−36.9253	−1.28402	−0.642009	+	0.766697i	$0.721899\pi$
$828$	−2.74456	−0.0953801
$829$	−33.7446	−1.17200	−0.585999	−	0.810312i	$-0.699298\pi$
$829$	−33.7446	−1.17200	−0.585999	+	0.810312i	$0.699298\pi$
$830$	−5.02175	−0.174307
$831$	8.66025	0.300421
$832$	−19.6974	−0.682883
$833$	−1.67715	−0.0581097
$834$	8.23369	0.285109
$835$	−79.7841	−2.76104
$836$	0	0
$837$	1.62772	0.0562622
$838$	−24.9484	−0.861829
$839$	−18.9783	−0.655202	−0.327601	−	0.944816i	$-0.606240\pi$
$839$	−18.9783	−0.655202	−0.327601	+	0.944816i	$0.606240\pi$
$840$	−22.7446	−0.784762
$841$	−28.3723	−0.978355
$842$	27.6202	0.951856
$843$	27.1229	0.934162
$844$	−14.7413	−0.507415
$845$	71.2119	2.44977
$846$	−10.0974	−0.347154
$847$	0	0
$848$	−2.58422	−0.0887425
$849$	−7.57301	−0.259905
$850$	13.4891	0.462673
$851$	10.0000	0.342796
$852$	14.7446	0.505140
$853$	21.3368	0.730560	0.365280	−	0.930898i	$-0.380973\pi$
$853$	21.3368	0.730560	0.365280	+	0.930898i	$0.380973\pi$
$854$	11.9769	0.409840
$855$	−3.16915	−0.108383
$856$	−36.2337	−1.23844
$857$	0	0		−	1.00000i	$-0.5\pi$
$857$	0	0			1.00000i	$0.5\pi$
$858$	0	0
$859$	5.11684	0.174584	0.0872922	−	0.996183i	$-0.472179\pi$
$859$	5.11684	0.174584	0.0872922	+	0.996183i	$0.472179\pi$
$860$	30.6968	1.04675
$861$	−27.4891	−0.936827
$862$	2.27719	0.0775613
$863$	39.4891	1.34423	0.672113	−	0.740449i	$-0.265387\pi$
$863$	39.4891	1.34423	0.672113	+	0.740449i	$0.265387\pi$
$864$	5.84096	0.198714
$865$	23.3639	0.794395
$866$	8.90030	0.302445
$867$	−9.86141	−0.334911
$868$	−5.63858	−0.191386
$869$	0	0
$870$	−2.11684	−0.0717677
$871$	6.52344	0.221038
$872$	−26.5842	−0.900255
$873$	10.4891	0.355003
$874$	−1.48913	−0.0503704
$875$	11.6819	0.394921
$876$	−12.5668	−0.424592
$877$	1.14214	0.0385674	0.0192837	−	0.999814i	$-0.493861\pi$
$877$	1.14214	0.0385674	0.0192837	+	0.999814i	$0.493861\pi$
$878$	4.27719	0.144348
$879$	−29.7947	−1.00495
$880$	0	0
$881$	−45.6060	−1.53650	−0.768252	−	0.640147i	$-0.778873\pi$
$881$	−45.6060	−1.53650	−0.768252	+	0.640147i	$0.778873\pi$
$882$	−0.497333	−0.0167461
$883$	23.8614	0.803000	0.401500	−	0.915859i	$-0.368489\pi$
$883$	23.8614	0.803000	0.401500	+	0.915859i	$0.368489\pi$
$884$	21.4158	0.720291
$885$	−20.2337	−0.680148
$886$	7.33296	0.246356
$887$	50.1918	1.68528	0.842638	−	0.538481i	$-0.181001\pi$
$887$	50.1918	1.68528	0.842638	+	0.538481i	$0.181001\pi$
$888$	−13.3591	−0.448301
$889$	−24.6060	−0.825258
$890$	−1.67715	−0.0562181
$891$	0	0
$892$	−32.7446	−1.09637
$893$	11.9769	0.400791
$894$	2.11684	0.0707979
$895$	34.5109	1.15357
$896$	−22.7446	−0.759843
$897$	−11.6819	−0.390048
$898$	−12.3644	−0.412606
$899$	−1.28962	−0.0430112
$900$	−8.74456	−0.291485
$901$	−10.9994	−0.366445
$902$	0	0
$903$	−16.7446	−0.557224
$904$	−3.66648	−0.121945
$905$	57.3288	1.90567
$906$	−17.4891	−0.581037
$907$	15.3505	0.509706	0.254853	−	0.966980i	$-0.417973\pi$
$907$	15.3505	0.509706	0.254853	+	0.966980i	$0.417973\pi$
$908$	−10.8724	−0.360813
$909$	11.9769	0.397248
$910$	−39.3947	−1.30592
$911$	3.76631	0.124783	0.0623917	−	0.998052i	$-0.480127\pi$
$911$	3.76631	0.124783	0.0623917	+	0.998052i	$0.480127\pi$
$912$	−0.589907	−0.0195338
$913$	0	0
$914$	30.3505	1.00391
$915$	20.1947	0.667616
$916$	12.8614	0.424953
$917$	16.7446	0.552954
$918$	2.11684	0.0698663
$919$	15.0911	0.497810	0.248905	−	0.968528i	$-0.419929\pi$
$919$	15.0911	0.497810	0.248905	+	0.968528i	$0.419929\pi$
$920$	−18.0202	−0.594109
$921$	1.23472	0.0406853
$922$	−3.60597	−0.118756
$923$	62.7586	2.06572
$924$	0	0
$925$	31.8614	1.04760
$926$	−11.6819	−0.383892
$927$	−9.11684	−0.299436
$928$	−4.62772	−0.151912
$929$	−52.1168	−1.70990	−0.854949	−	0.518712i	$-0.826412\pi$
$929$	−52.1168	−1.70990	−0.854949	+	0.518712i	$0.826412\pi$
$930$	4.34896	0.142608
$931$	0.589907	0.0193334
$932$	−1.08724	−0.0356138
$933$	−29.4891	−0.965431
$934$	9.69259	0.317151
$935$	0	0
$936$	15.6060	0.510097
$937$	19.4573	0.635643	0.317821	−	0.948151i	$-0.397049\pi$
$937$	19.4573	0.635643	0.317821	+	0.948151i	$0.397049\pi$
$938$	−2.23369	−0.0729325
$939$	−8.11684	−0.264883
$940$	58.9783	1.92366
$941$	−13.0641	−0.425878	−0.212939	−	0.977065i	$-0.568304\pi$
$941$	−13.0641	−0.425878	−0.212939	+	0.977065i	$0.568304\pi$
$942$	9.80240	0.319379
$943$	−21.7793	−0.709231
$944$	−3.76631	−0.122583
$945$	8.51278	0.276921
$946$	0	0
$947$	−8.51087	−0.276566	−0.138283	−	0.990393i	$-0.544158\pi$
$947$	−8.51087	−0.276566	−0.138283	+	0.990393i	$0.544158\pi$
$948$	−5.63858	−0.183133
$949$	−53.4891	−1.73633
$950$	−4.74456	−0.153934
$951$	26.7446	0.867252
$952$	−18.0202	−0.584039
$953$	30.6796	0.993809	0.496905	−	0.867805i	$-0.334470\pi$
$953$	30.6796	0.993809	0.496905	+	0.867805i	$0.334470\pi$
$954$	−3.26172	−0.105602
$955$	18.5109	0.598998
$956$	−31.1016	−1.00590
$957$	0	0
$958$	−8.70106	−0.281119
$959$	−27.1229	−0.875844
$960$	11.3723	0.367039
$961$	−28.3505	−0.914533
$962$	−23.1386	−0.746018
$963$	13.5615	0.437012
$964$	18.2054	0.586355
$965$	35.5431	1.14417
$966$	4.00000	0.128698
$967$	25.2983	0.813538	0.406769	−	0.913531i	$-0.366655\pi$
$967$	25.2983	0.813538	0.406769	+	0.913531i	$0.366655\pi$
$968$	0	0
$969$	−2.51087	−0.0806609
$970$	28.0250	0.899828
$971$	36.5109	1.17169	0.585845	−	0.810423i	$-0.300763\pi$
$971$	36.5109	1.17169	0.585845	+	0.810423i	$0.300763\pi$
$972$	−1.37228	−0.0440159
$973$	26.2337	0.841013
$974$	−0.994667	−0.0318712
$975$	−37.2203	−1.19200
$976$	3.75906	0.120324
$977$	−4.11684	−0.131710	−0.0658548	−	0.997829i	$-0.520977\pi$
$977$	−4.11684	−0.131710	−0.0658548	+	0.997829i	$0.520977\pi$
$978$	−2.87419	−0.0919066
$979$	0	0
$980$	2.90491	0.0927938
$981$	9.94987	0.317675
$982$	1.02175	0.0326053
$983$	−21.7663	−0.694238	−0.347119	−	0.937821i	$-0.612840\pi$
$983$	−21.7663	−0.694238	−0.347119	+	0.937821i	$0.612840\pi$
$984$	29.0951	0.927518
$985$	7.02078	0.223701
$986$	−1.67715	−0.0534113
$987$	−32.1716	−1.02403
$988$	−7.53262	−0.239645
$989$	−13.2665	−0.421850
$990$	0	0
$991$	57.4891	1.82620	0.913101	−	0.407733i	$-0.133681\pi$
$991$	57.4891	1.82620	0.913101	+	0.407733i	$0.133681\pi$
$992$	9.50744	0.301862
$993$	−6.37228	−0.202218
$994$	−21.4891	−0.681594
$995$	73.7228	2.33717
$996$	2.57924	0.0817264
$997$	31.1392	0.986190	0.493095	−	0.869976i	$-0.335866\pi$
$997$	31.1392	0.986190	0.493095	+	0.869976i	$0.335866\pi$
$998$	29.1876	0.923917
$999$	5.00000	0.158193

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.2.a.j.1.3	yes	4
3.2	odd	2		1089.2.a.u.1.2		4
4.3	odd	2		5808.2.a.ck.1.3		4
5.4	even	2		9075.2.a.cv.1.2		4
11.2	odd	10		363.2.e.n.202.3		16
11.3	even	5		363.2.e.n.130.3		16
11.4	even	5		363.2.e.n.148.3		16
11.5	even	5		363.2.e.n.124.2		16
11.6	odd	10		363.2.e.n.124.3		16
11.7	odd	10		363.2.e.n.148.2		16
11.8	odd	10		363.2.e.n.130.2		16
11.9	even	5		363.2.e.n.202.2		16
11.10	odd	2	inner	363.2.a.j.1.2	✓	4
33.32	even	2		1089.2.a.u.1.3		4
44.43	even	2		5808.2.a.ck.1.4		4
55.54	odd	2		9075.2.a.cv.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.2.a.j.1.2	✓	4	11.10	odd	2	inner
363.2.a.j.1.3	yes	4	1.1	even	1	trivial
363.2.e.n.124.2		16	11.5	even	5
363.2.e.n.124.3		16	11.6	odd	10
363.2.e.n.130.2		16	11.8	odd	10
363.2.e.n.130.3		16	11.3	even	5
363.2.e.n.148.2		16	11.7	odd	10
363.2.e.n.148.3		16	11.4	even	5
363.2.e.n.202.2		16	11.9	even	5
363.2.e.n.202.3		16	11.2	odd	10
1089.2.a.u.1.2		4	3.2	odd	2
1089.2.a.u.1.3		4	33.32	even	2
5808.2.a.ck.1.3		4	4.3	odd	2
5808.2.a.ck.1.4		4	44.43	even	2
9075.2.a.cv.1.2		4	5.4	even	2
9075.2.a.cv.1.3		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)

\(f(q)\)	\(=\)	\(q+0.792287 q^{2} +1.00000 q^{3} -1.37228 q^{4} +3.37228 q^{5} +0.792287 q^{6} +2.52434 q^{7} -2.67181 q^{8} +1.00000 q^{9} +2.67181 q^{10} -1.37228 q^{12} -5.84096 q^{13} +2.00000 q^{14} +3.37228 q^{15} +0.627719 q^{16} +2.67181 q^{17} +0.792287 q^{18} -0.939764 q^{19} -4.62772 q^{20} +2.52434 q^{21} +2.00000 q^{23} -2.67181 q^{24} +6.37228 q^{25} -4.62772 q^{26} +1.00000 q^{27} -3.46410 q^{28} -0.792287 q^{29} +2.67181 q^{30} +1.62772 q^{31} +5.84096 q^{32} +2.11684 q^{34} +8.51278 q^{35} -1.37228 q^{36} +5.00000 q^{37} -0.744563 q^{38} -5.84096 q^{39} -9.01011 q^{40} -10.8896 q^{41} +2.00000 q^{42} -6.63325 q^{43} +3.37228 q^{45} +1.58457 q^{46} -12.7446 q^{47} +0.627719 q^{48} -0.627719 q^{49} +5.04868 q^{50} +2.67181 q^{51} +8.01544 q^{52} -4.11684 q^{53} +0.792287 q^{54} -6.74456 q^{56} -0.939764 q^{57} -0.627719 q^{58} -6.00000 q^{59} -4.62772 q^{60} +5.98844 q^{61} +1.28962 q^{62} +2.52434 q^{63} +3.37228 q^{64} -19.6974 q^{65} -1.11684 q^{67} -3.66648 q^{68} +2.00000 q^{69} +6.74456 q^{70} -10.7446 q^{71} -2.67181 q^{72} +9.15759 q^{73} +3.96143 q^{74} +6.37228 q^{75} +1.28962 q^{76} -4.62772 q^{78} +4.10891 q^{79} +2.11684 q^{80} +1.00000 q^{81} -8.62772 q^{82} -1.87953 q^{83} -3.46410 q^{84} +9.01011 q^{85} -5.25544 q^{86} -0.792287 q^{87} -0.627719 q^{89} +2.67181 q^{90} -14.7446 q^{91} -2.74456 q^{92} +1.62772 q^{93} -10.0974 q^{94} -3.16915 q^{95} +5.84096 q^{96} +10.4891 q^{97} -0.497333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 6 q^{4} + 2 q^{5} + 4 q^{9} + 6 q^{12} + 8 q^{14} + 2 q^{15} + 14 q^{16} - 30 q^{20} + 8 q^{23} + 14 q^{25} - 30 q^{26} + 4 q^{27} + 18 q^{31} - 26 q^{34} + 6 q^{36} + 20 q^{37} + 20 q^{38}+ \cdots - 4 q^{97}+O(q^{100}) \) 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

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