LMFDB - Embedded newform 8281.2.a.be.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] 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("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2,0,2,3,2,0,-6,-1,0,-8,-10,0,0,8,4,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.688892 q^{2} +2.21432 q^{3} -1.52543 q^{4} +3.21432 q^{5} -1.52543 q^{6} +2.42864 q^{8} +1.90321 q^{9} -2.21432 q^{10} -2.68889 q^{11} -3.37778 q^{12} +7.11753 q^{15} +1.37778 q^{16} -3.59210 q^{17} -1.31111 q^{18} -8.54617 q^{19} -4.90321 q^{20} +1.85236 q^{22} -3.28100 q^{23} +5.37778 q^{24} +5.33185 q^{25} -2.42864 q^{27} +2.05086 q^{29} -4.90321 q^{30} +5.83654 q^{31} -5.80642 q^{32} -5.95407 q^{33} +2.47457 q^{34} -2.90321 q^{36} -3.93332 q^{37} +5.88739 q^{38} +7.80642 q^{40} +0.755569 q^{41} +8.80642 q^{43} +4.10171 q^{44} +6.11753 q^{45} +2.26025 q^{46} +1.88247 q^{47} +3.05086 q^{48} -3.67307 q^{50} -7.95407 q^{51} +2.52543 q^{53} +1.67307 q^{54} -8.64296 q^{55} -18.9240 q^{57} -1.41282 q^{58} +7.33185 q^{59} -10.8573 q^{60} -9.05086 q^{61} -4.02074 q^{62} +1.24443 q^{64} +4.10171 q^{66} -0.428639 q^{67} +5.47949 q^{68} -7.26517 q^{69} -8.98418 q^{71} +4.62222 q^{72} +5.79060 q^{73} +2.70964 q^{74} +11.8064 q^{75} +13.0366 q^{76} -4.47949 q^{79} +4.42864 q^{80} -11.0874 q^{81} -0.520505 q^{82} -10.8272 q^{83} -11.5462 q^{85} -6.06668 q^{86} +4.54125 q^{87} -6.53035 q^{88} +5.36196 q^{89} -4.21432 q^{90} +5.00492 q^{92} +12.9240 q^{93} -1.29682 q^{94} -27.4701 q^{95} -12.8573 q^{96} +9.62867 q^{97} -5.11753 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 6 q^{8} - q^{9} - 8 q^{11} - 10 q^{12} + 8 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{18} + q^{19} - 8 q^{20} + 12 q^{22} - 3 q^{23} + 16 q^{24} - 4 q^{25} + 6 q^{27}+ \cdots - 2 q^{99}+O(q^{100}) $ 3 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 6 * q^8 - q^9 - 8 * q^11 - 10 * q^12 + 8 * q^15 + 4 * q^16 - 4 * q^17 - 4 * q^18 + q^19 - 8 * q^20 + 12 * q^22 - 3 * q^23 + 16 * q^24 - 4 * q^25 + 6 * q^27 - 7 * q^29 - 8 * q^30 + 11 * q^31 - 4 * q^32 + 2 * q^33 + 14 * q^34 - 2 * q^36 - 12 * q^37 - 2 * q^38 + 10 * q^40 + 2 * q^41 + 13 * q^43 - 14 * q^44 + 5 * q^45 + 20 * q^46 + 19 * q^47 - 4 * q^48 + 2 * q^50 - 4 * q^51 + q^53 - 8 * q^54 - 6 * q^55 - 30 * q^57 + 22 * q^58 + 2 * q^59 - 6 * q^60 - 14 * q^61 + 8 * q^62 + 4 * q^64 - 14 * q^66 + 12 * q^67 - 10 * q^68 - 2 * q^69 - 14 * q^71 + 14 * q^72 - 9 * q^73 - 12 * q^74 + 22 * q^75 + 32 * q^76 + 13 * q^79 - 13 * q^81 - 28 * q^82 + q^83 - 8 * q^85 - 18 * q^86 + 20 * q^87 + 20 * q^88 + 3 * q^89 - 6 * q^90 - 18 * q^92 + 12 * q^93 - 10 * q^94 - 29 * q^95 - 12 * q^96 + 15 * q^97 - 2 * q^99

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 $

$2$	−0.688892	−0.487120	−0.243560	−	0.969886i	$-0.578315\pi$
$2$	−0.688892	−0.487120	−0.243560	+	0.969886i	$0.578315\pi$
$3$	2.21432	1.27844	0.639219	−	0.769025i	$-0.279258\pi$
$3$	2.21432	1.27844	0.639219	+	0.769025i	$0.279258\pi$
$4$	−1.52543	−0.762714
$5$	3.21432	1.43749	0.718744	−	0.695275i	$-0.244717\pi$
$5$	3.21432	1.43749	0.718744	+	0.695275i	$0.244717\pi$
$6$	−1.52543	−0.622753
$7$	0	0
$7$	0	0
$8$	2.42864	0.858654
$9$	1.90321	0.634404
$10$	−2.21432	−0.700229
$11$	−2.68889	−0.810731	−0.405366	−	0.914155i	$-0.632856\pi$
$11$	−2.68889	−0.810731	−0.405366	+	0.914155i	$0.632856\pi$
$12$	−3.37778	−0.975082
$13$	0	0
$13$	0	0
$14$	0	0
$15$	7.11753	1.83774
$16$	1.37778	0.344446
$17$	−3.59210	−0.871213	−0.435607	−	0.900137i	$-0.643466\pi$
$17$	−3.59210	−0.871213	−0.435607	+	0.900137i	$0.643466\pi$
$18$	−1.31111	−0.309031
$19$	−8.54617	−1.96063	−0.980313	−	0.197449i	$-0.936734\pi$
$19$	−8.54617	−1.96063	−0.980313	+	0.197449i	$0.936734\pi$
$20$	−4.90321	−1.09639
$21$	0	0
$22$	1.85236	0.394924
$23$	−3.28100	−0.684135	−0.342068	−	0.939675i	$-0.611127\pi$
$23$	−3.28100	−0.684135	−0.342068	+	0.939675i	$0.611127\pi$
$24$	5.37778	1.09774
$25$	5.33185	1.06637
$26$	0	0
$27$	−2.42864	−0.467392
$28$	0	0
$29$	2.05086	0.380834	0.190417	−	0.981703i	$-0.439016\pi$
$29$	2.05086	0.380834	0.190417	+	0.981703i	$0.439016\pi$
$30$	−4.90321	−0.895200
$31$	5.83654	1.04827	0.524136	−	0.851634i	$-0.324388\pi$
$31$	5.83654	1.04827	0.524136	+	0.851634i	$0.324388\pi$
$32$	−5.80642	−1.02644
$33$	−5.95407	−1.03647
$34$	2.47457	0.424386
$35$	0	0
$36$	−2.90321	−0.483869
$37$	−3.93332	−0.646634	−0.323317	−	0.946291i	$-0.604798\pi$
$37$	−3.93332	−0.646634	−0.323317	+	0.946291i	$0.604798\pi$
$38$	5.88739	0.955061
$39$	0	0
$40$	7.80642	1.23430
$41$	0.755569	0.118000	0.0590000	−	0.998258i	$-0.481209\pi$
$41$	0.755569	0.118000	0.0590000	+	0.998258i	$0.481209\pi$
$42$	0	0
$43$	8.80642	1.34297	0.671484	−	0.741019i	$-0.265657\pi$
$43$	8.80642	1.34297	0.671484	+	0.741019i	$0.265657\pi$
$44$	4.10171	0.618356
$45$	6.11753	0.911948
$46$	2.26025	0.333256
$47$	1.88247	0.274586	0.137293	−	0.990530i	$-0.456160\pi$
$47$	1.88247	0.274586	0.137293	+	0.990530i	$0.456160\pi$
$48$	3.05086	0.440353
$49$	0	0
$50$	−3.67307	−0.519451
$51$	−7.95407	−1.11379
$52$	0	0
$53$	2.52543	0.346894	0.173447	−	0.984843i	$-0.444509\pi$
$53$	2.52543	0.346894	0.173447	+	0.984843i	$0.444509\pi$
$54$	1.67307	0.227676
$55$	−8.64296	−1.16542
$56$	0	0
$57$	−18.9240	−2.50654
$58$	−1.41282	−0.185512
$59$	7.33185	0.954526	0.477263	−	0.878761i	$-0.341629\pi$
$59$	7.33185	0.954526	0.477263	+	0.878761i	$0.341629\pi$
$60$	−10.8573	−1.40167
$61$	−9.05086	−1.15884	−0.579422	−	0.815028i	$-0.696722\pi$
$61$	−9.05086	−1.15884	−0.579422	+	0.815028i	$0.696722\pi$
$62$	−4.02074	−0.510635
$63$	0	0
$64$	1.24443	0.155554
$65$	0	0
$66$	4.10171	0.504886
$67$	−0.428639	−0.0523666	−0.0261833	−	0.999657i	$-0.508335\pi$
$67$	−0.428639	−0.0523666	−0.0261833	+	0.999657i	$0.508335\pi$
$68$	5.47949	0.664486
$69$	−7.26517	−0.874624
$70$	0	0
$71$	−8.98418	−1.06623	−0.533113	−	0.846044i	$-0.678978\pi$
$71$	−8.98418	−1.06623	−0.533113	+	0.846044i	$0.678978\pi$
$72$	4.62222	0.544733
$73$	5.79060	0.677739	0.338869	−	0.940833i	$-0.389956\pi$
$73$	5.79060	0.677739	0.338869	+	0.940833i	$0.389956\pi$
$74$	2.70964	0.314989
$75$	11.8064	1.36329
$76$	13.0366	1.49540
$77$	0	0
$78$	0	0
$79$	−4.47949	−0.503983	−0.251991	−	0.967730i	$-0.581085\pi$
$79$	−4.47949	−0.503983	−0.251991	+	0.967730i	$0.581085\pi$
$80$	4.42864	0.495137
$81$	−11.0874	−1.23194
$82$	−0.520505	−0.0574802
$83$	−10.8272	−1.18844	−0.594218	−	0.804304i	$-0.702538\pi$
$83$	−10.8272	−1.18844	−0.594218	+	0.804304i	$0.702538\pi$
$84$	0	0
$85$	−11.5462	−1.25236
$86$	−6.06668	−0.654187
$87$	4.54125	0.486873
$88$	−6.53035	−0.696138
$89$	5.36196	0.568367	0.284183	−	0.958770i	$-0.408278\pi$
$89$	5.36196	0.568367	0.284183	+	0.958770i	$0.408278\pi$
$90$	−4.21432	−0.444228
$91$	0	0
$92$	5.00492	0.521799
$93$	12.9240	1.34015
$94$	−1.29682	−0.133757
$95$	−27.4701	−2.81838
$96$	−12.8573	−1.31224
$97$	9.62867	0.977643	0.488822	−	0.872384i	$-0.337427\pi$
$97$	9.62867	0.977643	0.488822	+	0.872384i	$0.337427\pi$
$98$	0	0
$99$	−5.11753	−0.514331
$100$	−8.13335	−0.813335
$101$	−13.6938	−1.36259	−0.681293	−	0.732011i	$-0.738582\pi$
$101$	−13.6938	−1.36259	−0.681293	+	0.732011i	$0.738582\pi$
$102$	5.47949	0.542551
$103$	−12.2953	−1.21149	−0.605745	−	0.795659i	$-0.707125\pi$
$103$	−12.2953	−1.21149	−0.605745	+	0.795659i	$0.707125\pi$
$104$	0	0
$105$	0	0
$106$	−1.73975	−0.168979
$107$	−18.1891	−1.75841	−0.879205	−	0.476444i	$-0.841925\pi$
$107$	−18.1891	−1.75841	−0.879205	+	0.476444i	$0.841925\pi$
$108$	3.70471	0.356486
$109$	−8.36196	−0.800931	−0.400465	−	0.916312i	$-0.631152\pi$
$109$	−8.36196	−0.800931	−0.400465	+	0.916312i	$0.631152\pi$
$110$	5.95407	0.567698
$111$	−8.70964	−0.826682
$112$	0	0
$113$	−8.46520	−0.796339	−0.398170	−	0.917312i	$-0.630354\pi$
$113$	−8.46520	−0.796339	−0.398170	+	0.917312i	$0.630354\pi$
$114$	13.0366	1.22099
$115$	−10.5462	−0.983436
$116$	−3.12843	−0.290468
$117$	0	0
$118$	−5.05086	−0.464969
$119$	0	0
$120$	17.2859	1.57798
$121$	−3.76986	−0.342714
$122$	6.23506	0.564496
$123$	1.67307	0.150856
$124$	−8.90321	−0.799532
$125$	1.06668	0.0954065
$126$	0	0
$127$	4.08742	0.362700	0.181350	−	0.983419i	$-0.441953\pi$
$127$	4.08742	0.362700	0.181350	+	0.983419i	$0.441953\pi$
$128$	10.7556	0.950667
$129$	19.5002	1.71690
$130$	0	0
$131$	−5.93978	−0.518961	−0.259480	−	0.965748i	$-0.583551\pi$
$131$	−5.93978	−0.518961	−0.259480	+	0.965748i	$0.583551\pi$
$132$	9.08250	0.790530
$133$	0	0
$134$	0.295286	0.0255089
$135$	−7.80642	−0.671870
$136$	−8.72393	−0.748070
$137$	−16.3620	−1.39790	−0.698948	−	0.715172i	$-0.746348\pi$
$137$	−16.3620	−1.39790	−0.698948	+	0.715172i	$0.746348\pi$
$138$	5.00492	0.426047
$139$	−3.03011	−0.257011	−0.128505	−	0.991709i	$-0.541018\pi$
$139$	−3.03011	−0.257011	−0.128505	+	0.991709i	$0.541018\pi$
$140$	0	0
$141$	4.16839	0.351041
$142$	6.18913	0.519380
$143$	0	0
$144$	2.62222	0.218518
$145$	6.59210	0.547444
$146$	−3.98910	−0.330140
$147$	0	0
$148$	6.00000	0.493197
$149$	14.5368	1.19090	0.595451	−	0.803392i	$-0.296974\pi$
$149$	14.5368	1.19090	0.595451	+	0.803392i	$0.296974\pi$
$150$	−8.13335	−0.664085
$151$	19.9748	1.62553	0.812764	−	0.582594i	$-0.197962\pi$
$151$	19.9748	1.62553	0.812764	+	0.582594i	$0.197962\pi$
$152$	−20.7556	−1.68350
$153$	−6.83654	−0.552701
$154$	0	0
$155$	18.7605	1.50688
$156$	0	0
$157$	−7.39853	−0.590467	−0.295233	−	0.955425i	$-0.595397\pi$
$157$	−7.39853	−0.590467	−0.295233	+	0.955425i	$0.595397\pi$
$158$	3.08589	0.245500
$159$	5.59210	0.443483
$160$	−18.6637	−1.47550
$161$	0	0
$162$	7.63804	0.600101
$163$	−2.32693	−0.182259	−0.0911296	−	0.995839i	$-0.529048\pi$
$163$	−2.32693	−0.182259	−0.0911296	+	0.995839i	$0.529048\pi$
$164$	−1.15257	−0.0900002
$165$	−19.1383	−1.48991
$166$	7.45875	0.578911
$167$	−3.42219	−0.264817	−0.132408	−	0.991195i	$-0.542271\pi$
$167$	−3.42219	−0.264817	−0.132408	+	0.991195i	$0.542271\pi$
$168$	0	0
$169$	0	0
$170$	7.95407	0.610049
$171$	−16.2652	−1.24383
$172$	−13.4336	−1.02430
$173$	−8.27454	−0.629102	−0.314551	−	0.949241i	$-0.601854\pi$
$173$	−8.27454	−0.629102	−0.314551	+	0.949241i	$0.601854\pi$
$174$	−3.12843	−0.237166
$175$	0	0
$176$	−3.70471	−0.279253
$177$	16.2351	1.22030
$178$	−3.69381	−0.276863
$179$	−16.2257	−1.21277	−0.606383	−	0.795173i	$-0.707380\pi$
$179$	−16.2257	−1.21277	−0.606383	+	0.795173i	$0.707380\pi$
$180$	−9.33185	−0.695555
$181$	9.20495	0.684199	0.342099	−	0.939664i	$-0.388862\pi$
$181$	9.20495	0.684199	0.342099	+	0.939664i	$0.388862\pi$
$182$	0	0
$183$	−20.0415	−1.48151
$184$	−7.96836	−0.587435
$185$	−12.6430	−0.929529
$186$	−8.90321	−0.652815
$187$	9.65878	0.706320
$188$	−2.87157	−0.209431
$189$	0	0
$190$	18.9240	1.37289
$191$	−6.66815	−0.482490	−0.241245	−	0.970464i	$-0.577556\pi$
$191$	−6.66815	−0.482490	−0.241245	+	0.970464i	$0.577556\pi$
$192$	2.75557	0.198866
$193$	−20.6035	−1.48307	−0.741535	−	0.670914i	$-0.765902\pi$
$193$	−20.6035	−1.48307	−0.741535	+	0.670914i	$0.765902\pi$
$194$	−6.63311	−0.476230
$195$	0	0
$196$	0	0
$197$	8.36842	0.596225	0.298112	−	0.954531i	$-0.403643\pi$
$197$	8.36842	0.596225	0.298112	+	0.954531i	$0.403643\pi$
$198$	3.52543	0.250541
$199$	0.601472	0.0426372	0.0213186	−	0.999773i	$-0.493214\pi$
$199$	0.601472	0.0426372	0.0213186	+	0.999773i	$0.493214\pi$
$200$	12.9491	0.915643
$201$	−0.949145	−0.0669475
$202$	9.43356	0.663743
$203$	0	0
$204$	12.1334	0.849505
$205$	2.42864	0.169624
$206$	8.47013	0.590142
$207$	−6.24443	−0.434018
$208$	0	0
$209$	22.9797	1.58954
$210$	0	0
$211$	−1.90321	−0.131023	−0.0655113	−	0.997852i	$-0.520868\pi$
$211$	−1.90321	−0.131023	−0.0655113	+	0.997852i	$0.520868\pi$
$212$	−3.85236	−0.264581
$213$	−19.8938	−1.36310
$214$	12.5303	0.856557
$215$	28.3067	1.93050
$216$	−5.89829	−0.401328
$217$	0	0
$218$	5.76049	0.390150
$219$	12.8222	0.866447
$220$	13.1842	0.888879
$221$	0	0
$222$	6.00000	0.402694
$223$	10.6336	0.712078	0.356039	−	0.934471i	$-0.384127\pi$
$223$	10.6336	0.712078	0.356039	+	0.934471i	$0.384127\pi$
$224$	0	0
$225$	10.1476	0.676510
$226$	5.83161	0.387913
$227$	15.0509	0.998960	0.499480	−	0.866325i	$-0.333524\pi$
$227$	15.0509	0.998960	0.499480	+	0.866325i	$0.333524\pi$
$228$	28.8671	1.91177
$229$	−6.53480	−0.431831	−0.215916	−	0.976412i	$-0.569274\pi$
$229$	−6.53480	−0.431831	−0.215916	+	0.976412i	$0.569274\pi$
$230$	7.26517	0.479051
$231$	0	0
$232$	4.98079	0.327005
$233$	−27.9590	−1.83165	−0.915827	−	0.401573i	$-0.868464\pi$
$233$	−27.9590	−1.83165	−0.915827	+	0.401573i	$0.868464\pi$
$234$	0	0
$235$	6.05086	0.394714
$236$	−11.1842	−0.728030
$237$	−9.91903	−0.644310
$238$	0	0
$239$	19.5812	1.26660	0.633301	−	0.773905i	$-0.281699\pi$
$239$	19.5812	1.26660	0.633301	+	0.773905i	$0.281699\pi$
$240$	9.80642	0.633002
$241$	−13.3575	−0.860433	−0.430217	−	0.902726i	$-0.641563\pi$
$241$	−13.3575	−0.860433	−0.430217	+	0.902726i	$0.641563\pi$
$242$	2.59703	0.166943
$243$	−17.2652	−1.10756
$244$	13.8064	0.883866
$245$	0	0
$246$	−1.15257	−0.0734849
$247$	0	0
$248$	14.1748	0.900103
$249$	−23.9748	−1.51934
$250$	−0.734825	−0.0464744
$251$	−3.29682	−0.208093	−0.104047	−	0.994572i	$-0.533179\pi$
$251$	−3.29682	−0.208093	−0.104047	+	0.994572i	$0.533179\pi$
$252$	0	0
$253$	8.82225	0.554650
$254$	−2.81579	−0.176678
$255$	−25.5669	−1.60106
$256$	−9.89829	−0.618643
$257$	23.6938	1.47798	0.738990	−	0.673717i	$-0.235303\pi$
$257$	23.6938	1.47798	0.738990	+	0.673717i	$0.235303\pi$
$258$	−13.4336	−0.836337
$259$	0	0
$260$	0	0
$261$	3.90321	0.241603
$262$	4.09187	0.252796
$263$	9.99063	0.616049	0.308024	−	0.951378i	$-0.400332\pi$
$263$	9.99063	0.616049	0.308024	+	0.951378i	$0.400332\pi$
$264$	−14.4603	−0.889969
$265$	8.11753	0.498656
$266$	0	0
$267$	11.8731	0.726622
$268$	0.653858	0.0399408
$269$	18.2034	1.10988	0.554941	−	0.831890i	$-0.312741\pi$
$269$	18.2034	1.10988	0.554941	+	0.831890i	$0.312741\pi$
$270$	5.37778	0.327282
$271$	−24.1748	−1.46852	−0.734258	−	0.678870i	$-0.762470\pi$
$271$	−24.1748	−1.46852	−0.734258	+	0.678870i	$0.762470\pi$
$272$	−4.94914	−0.300086
$273$	0	0
$274$	11.2716	0.680944
$275$	−14.3368	−0.864540
$276$	11.0825	0.667088
$277$	1.69535	0.101863	0.0509317	−	0.998702i	$-0.483781\pi$
$277$	1.69535	0.101863	0.0509317	+	0.998702i	$0.483781\pi$
$278$	2.08742	0.125195
$279$	11.1082	0.665028
$280$	0	0
$281$	−11.6479	−0.694854	−0.347427	−	0.937707i	$-0.612945\pi$
$281$	−11.6479	−0.694854	−0.347427	+	0.937707i	$0.612945\pi$
$282$	−2.87157	−0.170999
$283$	−12.1334	−0.721253	−0.360626	−	0.932710i	$-0.617437\pi$
$283$	−12.1334	−0.721253	−0.360626	+	0.932710i	$0.617437\pi$
$284$	13.7047	0.813225
$285$	−60.8276	−3.60312
$286$	0	0
$287$	0	0
$288$	−11.0509	−0.651178
$289$	−4.09679	−0.240988
$290$	−4.54125	−0.266671
$291$	21.3210	1.24986
$292$	−8.83314	−0.516921
$293$	11.4538	0.669140	0.334570	−	0.942371i	$-0.391409\pi$
$293$	11.4538	0.669140	0.334570	+	0.942371i	$0.391409\pi$
$294$	0	0
$295$	23.5669	1.37212
$296$	−9.55262	−0.555235
$297$	6.53035	0.378929
$298$	−10.0143	−0.580112
$299$	0	0
$300$	−18.0098	−1.03980
$301$	0	0
$302$	−13.7605	−0.791827
$303$	−30.3225	−1.74198
$304$	−11.7748	−0.675330
$305$	−29.0923	−1.66582
$306$	4.70964	0.269232
$307$	3.96989	0.226574	0.113287	−	0.993562i	$-0.463862\pi$
$307$	3.96989	0.226574	0.113287	+	0.993562i	$0.463862\pi$
$308$	0	0
$309$	−27.2257	−1.54882
$310$	−12.9240	−0.734031
$311$	−27.3481	−1.55077	−0.775386	−	0.631488i	$-0.782444\pi$
$311$	−27.3481	−1.55077	−0.775386	+	0.631488i	$0.782444\pi$
$312$	0	0
$313$	19.1032	1.07978	0.539890	−	0.841736i	$-0.318466\pi$
$313$	19.1032	1.07978	0.539890	+	0.841736i	$0.318466\pi$
$314$	5.09679	0.287628
$315$	0	0
$316$	6.83314	0.384394
$317$	−9.90813	−0.556496	−0.278248	−	0.960509i	$-0.589754\pi$
$317$	−9.90813	−0.556496	−0.278248	+	0.960509i	$0.589754\pi$
$318$	−3.85236	−0.216029
$319$	−5.51453	−0.308754
$320$	4.00000	0.223607
$321$	−40.2766	−2.24802
$322$	0	0
$323$	30.6987	1.70812
$324$	16.9131	0.939614
$325$	0	0
$326$	1.60300	0.0887821
$327$	−18.5161	−1.02394
$328$	1.83500	0.101321
$329$	0	0
$330$	13.1842	0.725767
$331$	−23.4193	−1.28724	−0.643620	−	0.765345i	$-0.722568\pi$
$331$	−23.4193	−1.28724	−0.643620	+	0.765345i	$0.722568\pi$
$332$	16.5161	0.906437
$333$	−7.48595	−0.410227
$334$	2.35752	0.128998
$335$	−1.37778	−0.0752764
$336$	0	0
$337$	−7.51606	−0.409426	−0.204713	−	0.978822i	$-0.565626\pi$
$337$	−7.51606	−0.409426	−0.204713	+	0.978822i	$0.565626\pi$
$338$	0	0
$339$	−18.7447	−1.01807
$340$	17.6128	0.955191
$341$	−15.6938	−0.849868
$342$	11.2050	0.605894
$343$	0	0
$344$	21.3876	1.15314
$345$	−23.3526	−1.25726
$346$	5.70027	0.306448
$347$	5.64449	0.303012	0.151506	−	0.988456i	$-0.451588\pi$
$347$	5.64449	0.303012	0.151506	+	0.988456i	$0.451588\pi$
$348$	−6.92735	−0.371345
$349$	24.1590	1.29320	0.646601	−	0.762828i	$-0.276190\pi$
$349$	24.1590	1.29320	0.646601	+	0.762828i	$0.276190\pi$
$350$	0	0
$351$	0	0
$352$	15.6128	0.832168
$353$	36.1289	1.92295	0.961474	−	0.274896i	$-0.0886436\pi$
$353$	36.1289	1.92295	0.961474	+	0.274896i	$0.0886436\pi$
$354$	−11.1842	−0.594434
$355$	−28.8780	−1.53269
$356$	−8.17929	−0.433501
$357$	0	0
$358$	11.1778	0.590763
$359$	−17.4128	−0.919013	−0.459507	−	0.888174i	$-0.651974\pi$
$359$	−17.4128	−0.919013	−0.459507	+	0.888174i	$0.651974\pi$
$360$	14.8573	0.783047
$361$	54.0370	2.84405
$362$	−6.34122	−0.333287
$363$	−8.34767	−0.438139
$364$	0	0
$365$	18.6128	0.974241
$366$	13.8064	0.721673
$367$	−2.93825	−0.153375	−0.0766876	−	0.997055i	$-0.524434\pi$
$367$	−2.93825	−0.153375	−0.0766876	+	0.997055i	$0.524434\pi$
$368$	−4.52051	−0.235648
$369$	1.43801	0.0748597
$370$	8.70964	0.452792
$371$	0	0
$372$	−19.7146	−1.02215
$373$	−18.7699	−0.971866	−0.485933	−	0.873996i	$-0.661520\pi$
$373$	−18.7699	−0.971866	−0.485933	+	0.873996i	$0.661520\pi$
$374$	−6.65386	−0.344063
$375$	2.36196	0.121971
$376$	4.57184	0.235774
$377$	0	0
$378$	0	0
$379$	−23.6894	−1.21684	−0.608421	−	0.793615i	$-0.708197\pi$
$379$	−23.6894	−1.21684	−0.608421	+	0.793615i	$0.708197\pi$
$380$	41.9037	2.14961
$381$	9.05086	0.463689
$382$	4.59364	0.235031
$383$	31.6128	1.61534	0.807671	−	0.589634i	$-0.200728\pi$
$383$	31.6128	1.61534	0.807671	+	0.589634i	$0.200728\pi$
$384$	23.8163	1.21537
$385$	0	0
$386$	14.1936	0.722434
$387$	16.7605	0.851984
$388$	−14.6878	−0.745662
$389$	−20.0558	−1.01687	−0.508434	−	0.861101i	$-0.669775\pi$
$389$	−20.0558	−1.01687	−0.508434	+	0.861101i	$0.669775\pi$
$390$	0	0
$391$	11.7857	0.596027
$392$	0	0
$393$	−13.1526	−0.663459
$394$	−5.76494	−0.290433
$395$	−14.3985	−0.724469
$396$	7.80642	0.392288
$397$	22.8731	1.14797	0.573984	−	0.818867i	$-0.305397\pi$
$397$	22.8731	1.14797	0.573984	+	0.818867i	$0.305397\pi$
$398$	−0.414349	−0.0207695
$399$	0	0
$400$	7.34614	0.367307
$401$	5.61285	0.280292	0.140146	−	0.990131i	$-0.455243\pi$
$401$	5.61285	0.280292	0.140146	+	0.990131i	$0.455243\pi$
$402$	0.653858	0.0326115
$403$	0	0
$404$	20.8889	1.03926
$405$	−35.6385	−1.77089
$406$	0	0
$407$	10.5763	0.524247
$408$	−19.3176	−0.956362
$409$	−26.1175	−1.29143	−0.645714	−	0.763579i	$-0.723440\pi$
$409$	−26.1175	−1.29143	−0.645714	+	0.763579i	$0.723440\pi$
$410$	−1.67307	−0.0826271
$411$	−36.2306	−1.78712
$412$	18.7556	0.924021
$413$	0	0
$414$	4.30174	0.211419
$415$	−34.8020	−1.70836
$416$	0	0
$417$	−6.70964	−0.328572
$418$	−15.8306	−0.774298
$419$	25.2464	1.23337	0.616685	−	0.787210i	$-0.288475\pi$
$419$	25.2464	1.23337	0.616685	+	0.787210i	$0.288475\pi$
$420$	0	0
$421$	8.22861	0.401038	0.200519	−	0.979690i	$-0.435737\pi$
$421$	8.22861	0.401038	0.200519	+	0.979690i	$0.435737\pi$
$422$	1.31111	0.0638237
$423$	3.58274	0.174199
$424$	6.13335	0.297862
$425$	−19.1526	−0.929036
$426$	13.7047	0.663996
$427$	0	0
$428$	27.7462	1.34116
$429$	0	0
$430$	−19.5002	−0.940385
$431$	1.43801	0.0692664	0.0346332	−	0.999400i	$-0.488974\pi$
$431$	1.43801	0.0692664	0.0346332	+	0.999400i	$0.488974\pi$
$432$	−3.34614	−0.160991
$433$	16.5018	0.793024	0.396512	−	0.918029i	$-0.370220\pi$
$433$	16.5018	0.793024	0.396512	+	0.918029i	$0.370220\pi$
$434$	0	0
$435$	14.5970	0.699874
$436$	12.7556	0.610881
$437$	28.0400	1.34133
$438$	−8.83314	−0.422064
$439$	10.1619	0.485003	0.242501	−	0.970151i	$-0.422032\pi$
$439$	10.1619	0.485003	0.242501	+	0.970151i	$0.422032\pi$
$440$	−20.9906	−1.00069
$441$	0	0
$442$	0	0
$443$	3.20787	0.152410	0.0762052	−	0.997092i	$-0.475720\pi$
$443$	3.20787	0.152410	0.0762052	+	0.997092i	$0.475720\pi$
$444$	13.2859	0.630522
$445$	17.2351	0.817020
$446$	−7.32540	−0.346868
$447$	32.1891	1.52249
$448$	0	0
$449$	18.4099	0.868817	0.434409	−	0.900716i	$-0.356957\pi$
$449$	18.4099	0.868817	0.434409	+	0.900716i	$0.356957\pi$
$450$	−6.99063	−0.329542
$451$	−2.03164	−0.0956663
$452$	12.9131	0.607379
$453$	44.2306	2.07814
$454$	−10.3684	−0.486614
$455$	0	0
$456$	−45.9595	−2.15225
$457$	−3.40297	−0.159184	−0.0795922	−	0.996828i	$-0.525362\pi$
$457$	−3.40297	−0.159184	−0.0795922	+	0.996828i	$0.525362\pi$
$458$	4.50177	0.210354
$459$	8.72393	0.407198
$460$	16.0874	0.750080
$461$	17.5714	0.818380	0.409190	−	0.912449i	$-0.365811\pi$
$461$	17.5714	0.818380	0.409190	+	0.912449i	$0.365811\pi$
$462$	0	0
$463$	−15.7714	−0.732959	−0.366479	−	0.930426i	$-0.619437\pi$
$463$	−15.7714	−0.732959	−0.366479	+	0.930426i	$0.619437\pi$
$464$	2.82564	0.131177
$465$	41.5417	1.92645
$466$	19.2607	0.892236
$467$	3.76694	0.174313	0.0871567	−	0.996195i	$-0.472222\pi$
$467$	3.76694	0.174313	0.0871567	+	0.996195i	$0.472222\pi$
$468$	0	0
$469$	0	0
$470$	−4.16839	−0.192273
$471$	−16.3827	−0.754875
$472$	17.8064	0.819607
$473$	−23.6795	−1.08879
$474$	6.83314	0.313857
$475$	−45.5669	−2.09075
$476$	0	0
$477$	4.80642	0.220071
$478$	−13.4893	−0.616988
$479$	−12.1032	−0.553011	−0.276506	−	0.961012i	$-0.589176\pi$
$479$	−12.1032	−0.553011	−0.276506	+	0.961012i	$0.589176\pi$
$480$	−41.3274	−1.88633
$481$	0	0
$482$	9.20189	0.419135
$483$	0	0
$484$	5.75065	0.261393
$485$	30.9496	1.40535
$486$	11.8938	0.539516
$487$	17.3778	0.787463	0.393731	−	0.919226i	$-0.371184\pi$
$487$	17.3778	0.787463	0.393731	+	0.919226i	$0.371184\pi$
$488$	−21.9813	−0.995045
$489$	−5.15257	−0.233007
$490$	0	0
$491$	−19.3921	−0.875152	−0.437576	−	0.899181i	$-0.644163\pi$
$491$	−19.3921	−0.875152	−0.437576	+	0.899181i	$0.644163\pi$
$492$	−2.55215	−0.115060
$493$	−7.36689	−0.331788
$494$	0	0
$495$	−16.4494	−0.739345
$496$	8.04149	0.361073
$497$	0	0
$498$	16.5161	0.740102
$499$	10.6702	0.477662	0.238831	−	0.971061i	$-0.423236\pi$
$499$	10.6702	0.477662	0.238831	+	0.971061i	$0.423236\pi$
$500$	−1.62714	−0.0727678
$501$	−7.57781	−0.338552
$502$	2.27115	0.101366
$503$	4.27655	0.190682	0.0953410	−	0.995445i	$-0.469606\pi$
$503$	4.27655	0.190682	0.0953410	+	0.995445i	$0.469606\pi$
$504$	0	0
$505$	−44.0163	−1.95870
$506$	−6.07758	−0.270181
$507$	0	0
$508$	−6.23506	−0.276636
$509$	33.0765	1.46609	0.733046	−	0.680180i	$-0.238098\pi$
$509$	33.0765	1.46609	0.733046	+	0.680180i	$0.238098\pi$
$510$	17.6128	0.779910
$511$	0	0
$512$	−14.6923	−0.649313
$513$	20.7556	0.916381
$514$	−16.3225	−0.719954
$515$	−39.5210	−1.74150
$516$	−29.7462	−1.30950
$517$	−5.06175	−0.222616
$518$	0	0
$519$	−18.3225	−0.804268
$520$	0	0
$521$	−24.3783	−1.06803	−0.534015	−	0.845475i	$-0.679318\pi$
$521$	−24.3783	−1.06803	−0.534015	+	0.845475i	$0.679318\pi$
$522$	−2.68889	−0.117690
$523$	−34.9403	−1.52783	−0.763915	−	0.645317i	$-0.776725\pi$
$523$	−34.9403	−1.52783	−0.763915	+	0.645317i	$0.776725\pi$
$524$	9.06070	0.395818
$525$	0	0
$526$	−6.88247	−0.300090
$527$	−20.9654	−0.913269
$528$	−8.20342	−0.357008
$529$	−12.2351	−0.531959
$530$	−5.59210	−0.242905
$531$	13.9541	0.605555
$532$	0	0
$533$	0	0
$534$	−8.17929	−0.353952
$535$	−58.4657	−2.52769
$536$	−1.04101	−0.0449648
$537$	−35.9289	−1.55045
$538$	−12.5402	−0.540646
$539$	0	0
$540$	11.9081	0.512445
$541$	30.4953	1.31110	0.655548	−	0.755153i	$-0.272438\pi$
$541$	30.4953	1.31110	0.655548	+	0.755153i	$0.272438\pi$
$542$	16.6539	0.715344
$543$	20.3827	0.874706
$544$	20.8573	0.894248
$545$	−26.8780	−1.15133
$546$	0	0
$547$	−10.0049	−0.427780	−0.213890	−	0.976858i	$-0.568613\pi$
$547$	−10.0049	−0.427780	−0.213890	+	0.976858i	$0.568613\pi$
$548$	24.9590	1.06620
$549$	−17.2257	−0.735175
$550$	9.87649	0.421135
$551$	−17.5270	−0.746674
$552$	−17.6445	−0.750999
$553$	0	0
$554$	−1.16791	−0.0496198
$555$	−27.9956	−1.18835
$556$	4.62222	0.196026
$557$	−14.1936	−0.601401	−0.300701	−	0.953719i	$-0.597220\pi$
$557$	−14.1936	−0.601401	−0.300701	+	0.953719i	$0.597220\pi$
$558$	−7.65233	−0.323949
$559$	0	0
$560$	0	0
$561$	21.3876	0.902986
$562$	8.02413	0.338478
$563$	21.3590	0.900177	0.450088	−	0.892984i	$-0.351393\pi$
$563$	21.3590	0.900177	0.450088	+	0.892984i	$0.351393\pi$
$564$	−6.35857	−0.267744
$565$	−27.2099	−1.14473
$566$	8.35857	0.351337
$567$	0	0
$568$	−21.8193	−0.915519
$569$	34.2672	1.43656	0.718278	−	0.695757i	$-0.244931\pi$
$569$	34.2672	1.43656	0.718278	+	0.695757i	$0.244931\pi$
$570$	41.9037	1.75515
$571$	−37.6494	−1.57558	−0.787789	−	0.615945i	$-0.788774\pi$
$571$	−37.6494	−1.57558	−0.787789	+	0.615945i	$0.788774\pi$
$572$	0	0
$573$	−14.7654	−0.616834
$574$	0	0
$575$	−17.4938	−0.729541
$576$	2.36842	0.0986840
$577$	28.3970	1.18218	0.591091	−	0.806605i	$-0.298697\pi$
$577$	28.3970	1.18218	0.591091	+	0.806605i	$0.298697\pi$
$578$	2.82225	0.117390
$579$	−45.6227	−1.89601
$580$	−10.0558	−0.417543
$581$	0	0
$582$	−14.6878	−0.608830
$583$	−6.79060	−0.281238
$584$	14.0633	0.581943
$585$	0	0
$586$	−7.89045	−0.325952
$587$	−6.23659	−0.257412	−0.128706	−	0.991683i	$-0.541082\pi$
$587$	−6.23659	−0.257412	−0.128706	+	0.991683i	$0.541082\pi$
$588$	0	0
$589$	−49.8800	−2.05527
$590$	−16.2351	−0.668387
$591$	18.5303	0.762237
$592$	−5.41927	−0.222731
$593$	17.5698	0.721506	0.360753	−	0.932661i	$-0.382520\pi$
$593$	17.5698	0.721506	0.360753	+	0.932661i	$0.382520\pi$
$594$	−4.49871	−0.184584
$595$	0	0
$596$	−22.1748	−0.908317
$597$	1.33185	0.0545090
$598$	0	0
$599$	16.9813	0.693836	0.346918	−	0.937896i	$-0.387228\pi$
$599$	16.9813	0.693836	0.346918	+	0.937896i	$0.387228\pi$
$600$	28.6735	1.17059
$601$	18.7052	0.763001	0.381500	−	0.924369i	$-0.375408\pi$
$601$	18.7052	0.763001	0.381500	+	0.924369i	$0.375408\pi$
$602$	0	0
$603$	−0.815792	−0.0332216
$604$	−30.4701	−1.23981
$605$	−12.1175	−0.492648
$606$	20.8889	0.848554
$607$	10.3575	0.420399	0.210199	−	0.977659i	$-0.432589\pi$
$607$	10.3575	0.420399	0.210199	+	0.977659i	$0.432589\pi$
$608$	49.6227	2.01247
$609$	0	0
$610$	20.0415	0.811456
$611$	0	0
$612$	10.4286	0.421553
$613$	7.02227	0.283627	0.141814	−	0.989893i	$-0.454707\pi$
$613$	7.02227	0.283627	0.141814	+	0.989893i	$0.454707\pi$
$614$	−2.73483	−0.110369
$615$	5.37778	0.216853
$616$	0	0
$617$	−29.9813	−1.20700	−0.603500	−	0.797363i	$-0.706228\pi$
$617$	−29.9813	−1.20700	−0.603500	+	0.797363i	$0.706228\pi$
$618$	18.7556	0.754460
$619$	0.0285802	0.00114874	0.000574368	−	1.00000i	$-0.499817\pi$
$619$	0.0285802	0.00114874	0.000574368		1.00000i	$0.499817\pi$
$620$	−28.6178	−1.14932
$621$	7.96836	0.319759
$622$	18.8399	0.755412
$623$	0	0
$624$	0	0
$625$	−23.2306	−0.929225
$626$	−13.1601	−0.525982
$627$	50.8845	2.03213
$628$	11.2859	0.450357
$629$	14.1289	0.563356
$630$	0	0
$631$	12.1936	0.485419	0.242709	−	0.970099i	$-0.421964\pi$
$631$	12.1936	0.485419	0.242709	+	0.970099i	$0.421964\pi$
$632$	−10.8791	−0.432746
$633$	−4.21432	−0.167504
$634$	6.82564	0.271081
$635$	13.1383	0.521377
$636$	−8.53035	−0.338250
$637$	0	0
$638$	3.79892	0.150401
$639$	−17.0988	−0.676418
$640$	34.5718	1.36657
$641$	−2.82516	−0.111587	−0.0557935	−	0.998442i	$-0.517769\pi$
$641$	−2.82516	−0.111587	−0.0557935	+	0.998442i	$0.517769\pi$
$642$	27.7462	1.09506
$643$	−37.7275	−1.48783	−0.743913	−	0.668276i	$-0.767032\pi$
$643$	−37.7275	−1.48783	−0.743913	+	0.668276i	$0.767032\pi$
$644$	0	0
$645$	62.6800	2.46802
$646$	−21.1481	−0.832062
$647$	−12.3664	−0.486174	−0.243087	−	0.970005i	$-0.578160\pi$
$647$	−12.3664	−0.486174	−0.243087	+	0.970005i	$0.578160\pi$
$648$	−26.9273	−1.05781
$649$	−19.7146	−0.773864
$650$	0	0
$651$	0	0
$652$	3.54956	0.139012
$653$	−33.0005	−1.29141	−0.645704	−	0.763588i	$-0.723436\pi$
$653$	−33.0005	−1.29141	−0.645704	+	0.763588i	$0.723436\pi$
$654$	12.7556	0.498782
$655$	−19.0923	−0.746000
$656$	1.04101	0.0406446
$657$	11.0207	0.429960
$658$	0	0
$659$	32.8118	1.27817	0.639084	−	0.769137i	$-0.279314\pi$
$659$	32.8118	1.27817	0.639084	+	0.769137i	$0.279314\pi$
$660$	29.1941	1.13638
$661$	−14.7067	−0.572025	−0.286013	−	0.958226i	$-0.592330\pi$
$661$	−14.7067	−0.572025	−0.286013	+	0.958226i	$0.592330\pi$
$662$	16.1334	0.627041
$663$	0	0
$664$	−26.2953	−1.02046
$665$	0	0
$666$	5.15701	0.199830
$667$	−6.72885	−0.260542
$668$	5.22030	0.201979
$669$	23.5462	0.910348
$670$	0.949145	0.0366687
$671$	24.3368	0.939511
$672$	0	0
$673$	21.2908	0.820702	0.410351	−	0.911928i	$-0.365406\pi$
$673$	21.2908	0.820702	0.410351	+	0.911928i	$0.365406\pi$
$674$	5.17775	0.199440
$675$	−12.9491	−0.498413
$676$	0	0
$677$	3.07160	0.118051	0.0590256	−	0.998256i	$-0.481201\pi$
$677$	3.07160	0.118051	0.0590256	+	0.998256i	$0.481201\pi$
$678$	12.9131	0.495923
$679$	0	0
$680$	−28.0415	−1.07534
$681$	33.3274	1.27711
$682$	10.8113	0.413988
$683$	24.7971	0.948833	0.474416	−	0.880301i	$-0.342659\pi$
$683$	24.7971	0.948833	0.474416	+	0.880301i	$0.342659\pi$
$684$	24.8113	0.948686
$685$	−52.5926	−2.00946
$686$	0	0
$687$	−14.4701	−0.552070
$688$	12.1334	0.462580
$689$	0	0
$690$	16.0874	0.612438
$691$	−27.0953	−1.03075	−0.515376	−	0.856964i	$-0.672348\pi$
$691$	−27.0953	−1.03075	−0.515376	+	0.856964i	$0.672348\pi$
$692$	12.6222	0.479825
$693$	0	0
$694$	−3.88845	−0.147603
$695$	−9.73975	−0.369450
$696$	11.0291	0.418055
$697$	−2.71408	−0.102803
$698$	−16.6430	−0.629945
$699$	−61.9101	−2.34166
$700$	0	0
$701$	13.5205	0.510662	0.255331	−	0.966854i	$-0.417815\pi$
$701$	13.5205	0.510662	0.255331	+	0.966854i	$0.417815\pi$
$702$	0	0
$703$	33.6149	1.26781
$704$	−3.34614	−0.126112
$705$	13.3985	0.504618
$706$	−24.8889	−0.936707
$707$	0	0
$708$	−24.7654	−0.930741
$709$	50.9753	1.91442	0.957209	−	0.289399i	$-0.0934554\pi$
$709$	50.9753	1.91442	0.957209	+	0.289399i	$0.0934554\pi$
$710$	19.8938	0.746603
$711$	−8.52543	−0.319729
$712$	13.0223	0.488030
$713$	−19.1497	−0.717160
$714$	0	0
$715$	0	0
$716$	24.7511	0.924993
$717$	43.3590	1.61927
$718$	11.9956	0.447670
$719$	−41.5417	−1.54924	−0.774622	−	0.632424i	$-0.782060\pi$
$719$	−41.5417	−1.54924	−0.774622	+	0.632424i	$0.782060\pi$
$720$	8.42864	0.314117
$721$	0	0
$722$	−37.2257	−1.38540
$723$	−29.5778	−1.10001
$724$	−14.0415	−0.521848
$725$	10.9349	0.406110
$726$	5.75065	0.213427
$727$	20.8988	0.775092	0.387546	−	0.921850i	$-0.373323\pi$
$727$	20.8988	0.775092	0.387546	+	0.921850i	$0.373323\pi$
$728$	0	0
$729$	−4.96836	−0.184013
$730$	−12.8222	−0.474573
$731$	−31.6336	−1.17001
$732$	30.5718	1.12997
$733$	19.3575	0.714986	0.357493	−	0.933916i	$-0.383632\pi$
$733$	19.3575	0.714986	0.357493	+	0.933916i	$0.383632\pi$
$734$	2.02413	0.0747122
$735$	0	0
$736$	19.0509	0.702224
$737$	1.15257	0.0424553
$738$	−0.990632	−0.0364657
$739$	1.31111	0.0482299	0.0241149	−	0.999709i	$-0.492323\pi$
$739$	1.31111	0.0482299	0.0241149	+	0.999709i	$0.492323\pi$
$740$	19.2859	0.708964
$741$	0	0
$742$	0	0
$743$	21.5210	0.789528	0.394764	−	0.918783i	$-0.370826\pi$
$743$	21.5210	0.789528	0.394764	+	0.918783i	$0.370826\pi$
$744$	31.3876	1.15073
$745$	46.7259	1.71191
$746$	12.9304	0.473416
$747$	−20.6064	−0.753949
$748$	−14.7338	−0.538720
$749$	0	0
$750$	−1.62714	−0.0594147
$751$	−16.3176	−0.595436	−0.297718	−	0.954654i	$-0.596226\pi$
$751$	−16.3176	−0.595436	−0.297718	+	0.954654i	$0.596226\pi$
$752$	2.59364	0.0945802
$753$	−7.30021	−0.266034
$754$	0	0
$755$	64.2054	2.33667
$756$	0	0
$757$	−18.7462	−0.681342	−0.340671	−	0.940183i	$-0.610654\pi$
$757$	−18.7462	−0.681342	−0.340671	+	0.940183i	$0.610654\pi$
$758$	16.3194	0.592748
$759$	19.5353	0.709085
$760$	−66.7150	−2.42001
$761$	−10.2968	−0.373259	−0.186630	−	0.982430i	$-0.559756\pi$
$761$	−10.2968	−0.373259	−0.186630	+	0.982430i	$0.559756\pi$
$762$	−6.23506	−0.225873
$763$	0	0
$764$	10.1718	0.368002
$765$	−21.9748	−0.794501
$766$	−21.7778	−0.786865
$767$	0	0
$768$	−21.9180	−0.790897
$769$	9.36641	0.337761	0.168881	−	0.985637i	$-0.445985\pi$
$769$	9.36641	0.337761	0.168881	+	0.985637i	$0.445985\pi$
$770$	0	0
$771$	52.4657	1.88951
$772$	31.4291	1.13116
$773$	−3.71456	−0.133603	−0.0668017	−	0.997766i	$-0.521279\pi$
$773$	−3.71456	−0.133603	−0.0668017	+	0.997766i	$0.521279\pi$
$774$	−11.5462	−0.415019
$775$	31.1195	1.11785
$776$	23.3846	0.839457
$777$	0	0
$778$	13.8163	0.495337
$779$	−6.45722	−0.231354
$780$	0	0
$781$	24.1575	0.864423
$782$	−8.11906	−0.290337
$783$	−4.98079	−0.177999
$784$	0	0
$785$	−23.7812	−0.848789
$786$	9.06070	0.323184
$787$	6.23659	0.222311	0.111155	−	0.993803i	$-0.464545\pi$
$787$	6.23659	0.222311	0.111155	+	0.993803i	$0.464545\pi$
$788$	−12.7654	−0.454749
$789$	22.1225	0.787580
$790$	9.91903	0.352903
$791$	0	0
$792$	−12.4286	−0.441632
$793$	0	0
$794$	−15.7571	−0.559199
$795$	17.9748	0.637501
$796$	−0.917502	−0.0325200
$797$	40.0701	1.41935	0.709677	−	0.704527i	$-0.248841\pi$
$797$	40.0701	1.41935	0.709677	+	0.704527i	$0.248841\pi$
$798$	0	0
$799$	−6.76202	−0.239223
$800$	−30.9590	−1.09457
$801$	10.2050	0.360574
$802$	−3.86665	−0.136536
$803$	−15.5703	−0.549464
$804$	1.44785	0.0510618
$805$	0	0
$806$	0	0
$807$	40.3082	1.41892
$808$	−33.2573	−1.16999
$809$	−2.57136	−0.0904042	−0.0452021	−	0.998978i	$-0.514393\pi$
$809$	−2.57136	−0.0904042	−0.0452021	+	0.998978i	$0.514393\pi$
$810$	24.5511	0.862637
$811$	−28.3654	−0.996042	−0.498021	−	0.867165i	$-0.665940\pi$
$811$	−28.3654	−0.996042	−0.498021	+	0.867165i	$0.665940\pi$
$812$	0	0
$813$	−53.5308	−1.87741
$814$	−7.28592	−0.255371
$815$	−7.47949	−0.261995
$816$	−10.9590	−0.383641
$817$	−75.2612	−2.63306
$818$	17.9922	0.629081
$819$	0	0
$820$	−3.70471	−0.129374
$821$	31.7846	1.10929	0.554646	−	0.832087i	$-0.312854\pi$
$821$	31.7846	1.10929	0.554646	+	0.832087i	$0.312854\pi$
$822$	24.9590	0.870545
$823$	30.9131	1.07756	0.538781	−	0.842446i	$-0.318885\pi$
$823$	30.9131	1.07756	0.538781	+	0.842446i	$0.318885\pi$
$824$	−29.8608	−1.04025
$825$	−31.7462	−1.10526
$826$	0	0
$827$	−49.7560	−1.73019	−0.865094	−	0.501610i	$-0.832741\pi$
$827$	−49.7560	−1.73019	−0.865094	+	0.501610i	$0.832741\pi$
$828$	9.52543	0.331031
$829$	19.9190	0.691817	0.345908	−	0.938268i	$-0.387571\pi$
$829$	19.9190	0.691817	0.345908	+	0.938268i	$0.387571\pi$
$830$	23.9748	0.832178
$831$	3.75404	0.130226
$832$	0	0
$833$	0	0
$834$	4.62222	0.160054
$835$	−11.0000	−0.380671
$836$	−35.0539	−1.21237
$837$	−14.1748	−0.489954
$838$	−17.3921	−0.600799
$839$	11.5625	0.399181	0.199590	−	0.979879i	$-0.436039\pi$
$839$	11.5625	0.399181	0.199590	+	0.979879i	$0.436039\pi$
$840$	0	0
$841$	−24.7940	−0.854965
$842$	−5.66862	−0.195354
$843$	−25.7921	−0.888328
$844$	2.90321	0.0999327
$845$	0	0
$846$	−2.46812	−0.0848557
$847$	0	0
$848$	3.47949	0.119486
$849$	−26.8671	−0.922077
$850$	13.1941	0.452552
$851$	12.9052	0.442385
$852$	30.3466	1.03966
$853$	1.74467	0.0597363	0.0298682	−	0.999554i	$-0.490491\pi$
$853$	1.74467	0.0597363	0.0298682	+	0.999554i	$0.490491\pi$
$854$	0	0
$855$	−52.2815	−1.78799
$856$	−44.1748	−1.50986
$857$	42.3368	1.44620	0.723098	−	0.690745i	$-0.242717\pi$
$857$	42.3368	1.44620	0.723098	+	0.690745i	$0.242717\pi$
$858$	0	0
$859$	5.37778	0.183488	0.0917438	−	0.995783i	$-0.470756\pi$
$859$	5.37778	0.183488	0.0917438	+	0.995783i	$0.470756\pi$
$860$	−43.1798	−1.47242
$861$	0	0
$862$	−0.990632	−0.0337411
$863$	−15.4291	−0.525213	−0.262607	−	0.964903i	$-0.584582\pi$
$863$	−15.4291	−0.525213	−0.262607	+	0.964903i	$0.584582\pi$
$864$	14.1017	0.479750
$865$	−26.5970	−0.904326
$866$	−11.3679	−0.386298
$867$	−9.07160	−0.308088
$868$	0	0
$869$	12.0449	0.408595
$870$	−10.0558	−0.340923
$871$	0	0
$872$	−20.3082	−0.687722
$873$	18.3254	0.620221
$874$	−19.3165	−0.653391
$875$	0	0
$876$	−19.5594	−0.660851
$877$	−38.5181	−1.30066	−0.650331	−	0.759651i	$-0.725370\pi$
$877$	−38.5181	−1.30066	−0.650331	+	0.759651i	$0.725370\pi$
$878$	−7.00048	−0.236255
$879$	25.3624	0.855454
$880$	−11.9081	−0.401423
$881$	41.7373	1.40617	0.703083	−	0.711108i	$-0.251806\pi$
$881$	41.7373	1.40617	0.703083	+	0.711108i	$0.251806\pi$
$882$	0	0
$883$	52.8439	1.77834	0.889170	−	0.457577i	$-0.151283\pi$
$883$	52.8439	1.77834	0.889170	+	0.457577i	$0.151283\pi$
$884$	0	0
$885$	52.1847	1.75417
$886$	−2.20987	−0.0742422
$887$	−8.99063	−0.301876	−0.150938	−	0.988543i	$-0.548229\pi$
$887$	−8.99063	−0.301876	−0.150938	+	0.988543i	$0.548229\pi$
$888$	−21.1526	−0.709834
$889$	0	0
$890$	−11.8731	−0.397987
$891$	29.8129	0.998769
$892$	−16.2208	−0.543112
$893$	−16.0879	−0.538361
$894$	−22.1748	−0.741638
$895$	−52.1546	−1.74334
$896$	0	0
$897$	0	0
$898$	−12.6824	−0.423218
$899$	11.9699	0.399218
$900$	−15.4795	−0.515983
$901$	−9.07160	−0.302219
$902$	1.39958	0.0466010
$903$	0	0
$904$	−20.5589	−0.683780
$905$	29.5877	0.983527
$906$	−30.4701	−1.01230
$907$	−20.7003	−0.687341	−0.343671	−	0.939090i	$-0.611670\pi$
$907$	−20.7003	−0.687341	−0.343671	+	0.939090i	$0.611670\pi$
$908$	−22.9590	−0.761921
$909$	−26.0622	−0.864430
$910$	0	0
$911$	29.8988	0.990590	0.495295	−	0.868725i	$-0.335060\pi$
$911$	29.8988	0.990590	0.495295	+	0.868725i	$0.335060\pi$
$912$	−26.0731	−0.863368
$913$	29.1131	0.963503
$914$	2.34428	0.0775420
$915$	−64.4197	−2.12965
$916$	9.96836	0.329364
$917$	0	0
$918$	−6.00984	−0.198354
$919$	38.5847	1.27279	0.636397	−	0.771362i	$-0.280424\pi$
$919$	38.5847	1.27279	0.636397	+	0.771362i	$0.280424\pi$
$920$	−25.6128	−0.844431
$921$	8.79060	0.289660
$922$	−12.1048	−0.398649
$923$	0	0
$924$	0	0
$925$	−20.9719	−0.689552
$926$	10.8648	0.357039
$927$	−23.4005	−0.768574
$928$	−11.9081	−0.390904
$929$	3.25581	0.106820	0.0534098	−	0.998573i	$-0.482991\pi$
$929$	3.25581	0.106820	0.0534098	+	0.998573i	$0.482991\pi$
$930$	−28.6178	−0.938414
$931$	0	0
$932$	42.6494	1.39703
$933$	−60.5575	−1.98257
$934$	−2.59502	−0.0849116
$935$	31.0464	1.01533
$936$	0	0
$937$	11.6840	0.381699	0.190849	−	0.981619i	$-0.438876\pi$
$937$	11.6840	0.381699	0.190849	+	0.981619i	$0.438876\pi$
$938$	0	0
$939$	42.3007	1.38043
$940$	−9.23014	−0.301054
$941$	−41.9699	−1.36818	−0.684090	−	0.729398i	$-0.739800\pi$
$941$	−41.9699	−1.36818	−0.684090	+	0.729398i	$0.739800\pi$
$942$	11.2859	0.367715
$943$	−2.47902	−0.0807279
$944$	10.1017	0.328783
$945$	0	0
$946$	16.3126	0.530370
$947$	20.3555	0.661465	0.330733	−	0.943725i	$-0.392704\pi$
$947$	20.3555	0.661465	0.330733	+	0.943725i	$0.392704\pi$
$948$	15.1308	0.491424
$949$	0	0
$950$	31.3907	1.01845
$951$	−21.9398	−0.711446
$952$	0	0
$953$	−30.5496	−0.989597	−0.494799	−	0.869008i	$-0.664758\pi$
$953$	−30.5496	−0.989597	−0.494799	+	0.869008i	$0.664758\pi$
$954$	−3.31111	−0.107201
$955$	−21.4336	−0.693574
$956$	−29.8697	−0.966055
$957$	−12.2109	−0.394723
$958$	8.33783	0.269383
$959$	0	0
$960$	8.85728	0.285867
$961$	3.06515	0.0988757
$962$	0	0
$963$	−34.6178	−1.11554
$964$	20.3759	0.656264
$965$	−66.2262	−2.13190
$966$	0	0
$967$	−54.4548	−1.75115	−0.875574	−	0.483084i	$-0.839516\pi$
$967$	−54.4548	−1.75115	−0.875574	+	0.483084i	$0.839516\pi$
$968$	−9.15563	−0.294273
$969$	67.9768	2.18373
$970$	−21.3210	−0.684575
$971$	54.6035	1.75231	0.876155	−	0.482030i	$-0.160101\pi$
$971$	54.6035	1.75231	0.876155	+	0.482030i	$0.160101\pi$
$972$	26.3368	0.844752
$973$	0	0
$974$	−11.9714	−0.383589
$975$	0	0
$976$	−12.4701	−0.399159
$977$	−32.6474	−1.04448	−0.522242	−	0.852798i	$-0.674904\pi$
$977$	−32.6474	−1.04448	−0.522242	+	0.852798i	$0.674904\pi$
$978$	3.54956	0.113502
$979$	−14.4177	−0.460793
$980$	0	0
$981$	−15.9146	−0.508114
$982$	13.3590	0.426304
$983$	−27.5052	−0.877278	−0.438639	−	0.898663i	$-0.644539\pi$
$983$	−27.5052	−0.877278	−0.438639	+	0.898663i	$0.644539\pi$
$984$	4.06329	0.129533
$985$	26.8988	0.857066
$986$	5.07499	0.161621
$987$	0	0
$988$	0	0
$989$	−28.8938	−0.918771
$990$	11.3319	0.360150
$991$	−6.29390	−0.199932	−0.0999662	−	0.994991i	$-0.531873\pi$
$991$	−6.29390	−0.199932	−0.0999662	+	0.994991i	$0.531873\pi$
$992$	−33.8894	−1.07599
$993$	−51.8578	−1.64566
$994$	0	0
$995$	1.93332	0.0612905
$996$	36.5718	1.15882
$997$	20.8702	0.660965	0.330483	−	0.943812i	$-0.392788\pi$
$997$	20.8702	0.660965	0.330483	+	0.943812i	$0.392788\pi$
$998$	−7.35059	−0.232679
$999$	9.55262	0.302232

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.be.1.2		3
7.6	odd	2		1183.2.a.h.1.2		3
13.5	odd	4		637.2.c.d.246.4		6
13.8	odd	4		637.2.c.d.246.3		6
13.12	even	2		8281.2.a.bi.1.2		3
91.5	even	12		637.2.r.e.116.4		12
91.18	odd	12		637.2.r.d.324.3		12
91.31	even	12		637.2.r.e.324.3		12
91.34	even	4		91.2.c.a.64.3	✓	6
91.44	odd	12		637.2.r.d.116.4		12
91.47	even	12		637.2.r.e.116.3		12
91.60	odd	12		637.2.r.d.324.4		12
91.73	even	12		637.2.r.e.324.4		12
91.83	even	4		91.2.c.a.64.4	yes	6
91.86	odd	12		637.2.r.d.116.3		12
91.90	odd	2		1183.2.a.j.1.2		3
273.83	odd	4		819.2.c.b.64.3		6
273.125	odd	4		819.2.c.b.64.4		6
364.83	odd	4		1456.2.k.c.337.6		6
364.307	odd	4		1456.2.k.c.337.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.c.a.64.3	✓	6	91.34	even	4
91.2.c.a.64.4	yes	6	91.83	even	4
637.2.c.d.246.3		6	13.8	odd	4
637.2.c.d.246.4		6	13.5	odd	4
637.2.r.d.116.3		12	91.86	odd	12
637.2.r.d.116.4		12	91.44	odd	12
637.2.r.d.324.3		12	91.18	odd	12
637.2.r.d.324.4		12	91.60	odd	12
637.2.r.e.116.3		12	91.47	even	12
637.2.r.e.116.4		12	91.5	even	12
637.2.r.e.324.3		12	91.31	even	12
637.2.r.e.324.4		12	91.73	even	12
819.2.c.b.64.3		6	273.83	odd	4
819.2.c.b.64.4		6	273.125	odd	4
1183.2.a.h.1.2		3	7.6	odd	2
1183.2.a.j.1.2		3	91.90	odd	2
1456.2.k.c.337.5		6	364.307	odd	4
1456.2.k.c.337.6		6	364.83	odd	4
8281.2.a.be.1.2		3	1.1	even	1	trivial
8281.2.a.bi.1.2		3	13.12	even	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-0.688892 q^{2} +2.21432 q^{3} -1.52543 q^{4} +3.21432 q^{5} -1.52543 q^{6} +2.42864 q^{8} +1.90321 q^{9} -2.21432 q^{10} -2.68889 q^{11} -3.37778 q^{12} +7.11753 q^{15} +1.37778 q^{16} -3.59210 q^{17} -1.31111 q^{18} -8.54617 q^{19} -4.90321 q^{20} +1.85236 q^{22} -3.28100 q^{23} +5.37778 q^{24} +5.33185 q^{25} -2.42864 q^{27} +2.05086 q^{29} -4.90321 q^{30} +5.83654 q^{31} -5.80642 q^{32} -5.95407 q^{33} +2.47457 q^{34} -2.90321 q^{36} -3.93332 q^{37} +5.88739 q^{38} +7.80642 q^{40} +0.755569 q^{41} +8.80642 q^{43} +4.10171 q^{44} +6.11753 q^{45} +2.26025 q^{46} +1.88247 q^{47} +3.05086 q^{48} -3.67307 q^{50} -7.95407 q^{51} +2.52543 q^{53} +1.67307 q^{54} -8.64296 q^{55} -18.9240 q^{57} -1.41282 q^{58} +7.33185 q^{59} -10.8573 q^{60} -9.05086 q^{61} -4.02074 q^{62} +1.24443 q^{64} +4.10171 q^{66} -0.428639 q^{67} +5.47949 q^{68} -7.26517 q^{69} -8.98418 q^{71} +4.62222 q^{72} +5.79060 q^{73} +2.70964 q^{74} +11.8064 q^{75} +13.0366 q^{76} -4.47949 q^{79} +4.42864 q^{80} -11.0874 q^{81} -0.520505 q^{82} -10.8272 q^{83} -11.5462 q^{85} -6.06668 q^{86} +4.54125 q^{87} -6.53035 q^{88} +5.36196 q^{89} -4.21432 q^{90} +5.00492 q^{92} +12.9240 q^{93} -1.29682 q^{94} -27.4701 q^{95} -12.8573 q^{96} +9.62867 q^{97} -5.11753 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 2 q^{4} + 3 q^{5} + 2 q^{6} - 6 q^{8} - q^{9} - 8 q^{11} - 10 q^{12} + 8 q^{15} + 4 q^{16} - 4 q^{17} - 4 q^{18} + q^{19} - 8 q^{20} + 12 q^{22} - 3 q^{23} + 16 q^{24} - 4 q^{25} + 6 q^{27}+ \cdots - 2 q^{99}+O(q^{100}) \) 3 * q - 2 * q^2 + 2 * q^4 + 3 * q^5 + 2 * q^6 - 6 * q^8 - q^9 - 8 * q^11 - 10 * q^12 + 8 * q^15 + 4 * q^16 - 4 * q^17 - 4 * q^18 + q^19 - 8 * q^20 + 12 * q^22 - 3 * q^23 + 16 * q^24 - 4 * q^25 + 6 * q^27 - 7 * q^29 - 8 * q^30 + 11 * q^31 - 4 * q^32 + 2 * q^33 + 14 * q^34 - 2 * q^36 - 12 * q^37 - 2 * q^38 + 10 * q^40 + 2 * q^41 + 13 * q^43 - 14 * q^44 + 5 * q^45 + 20 * q^46 + 19 * q^47 - 4 * q^48 + 2 * q^50 - 4 * q^51 + q^53 - 8 * q^54 - 6 * q^55 - 30 * q^57 + 22 * q^58 + 2 * q^59 - 6 * q^60 - 14 * q^61 + 8 * q^62 + 4 * q^64 - 14 * q^66 + 12 * q^67 - 10 * q^68 - 2 * q^69 - 14 * q^71 + 14 * q^72 - 9 * q^73 - 12 * q^74 + 22 * q^75 + 32 * q^76 + 13 * q^79 - 13 * q^81 - 28 * q^82 + q^83 - 8 * q^85 - 18 * q^86 + 20 * q^87 + 20 * q^88 + 3 * q^89 - 6 * q^90 - 18 * q^92 + 12 * q^93 - 10 * q^94 - 29 * q^95 - 12 * q^96 + 15 * q^97 - 2 * q^99

Introduction

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