LMFDB - Embedded newform 8281.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 1 $ x^4 - 5*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.18890$ 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)

$f(q)$	$=$	\(q-0.456850 q^{2} -2.79129 q^{3} -1.79129 q^{4} +0.456850 q^{5} +1.27520 q^{6} +1.73205 q^{8} +4.79129 q^{9} -0.208712 q^{10} +3.92095 q^{11} +5.00000 q^{12} -1.27520 q^{15} +2.79129 q^{16} +3.00000 q^{17} -2.18890 q^{18} +1.37055 q^{19} -0.818350 q^{20} -1.79129 q^{22} +1.58258 q^{23} -4.83465 q^{24} -4.79129 q^{25} -5.00000 q^{27} -6.79129 q^{29} +0.582576 q^{30} +8.66025 q^{31} -4.73930 q^{32} -10.9445 q^{33} -1.37055 q^{34} -8.58258 q^{36} -6.92820 q^{37} -0.626136 q^{38} +0.791288 q^{40} -7.84190 q^{41} -9.37386 q^{43} -7.02355 q^{44} +2.18890 q^{45} -0.723000 q^{46} +9.57395 q^{47} -7.79129 q^{48} +2.18890 q^{50} -8.37386 q^{51} +6.16515 q^{53} +2.28425 q^{54} +1.79129 q^{55} -3.82560 q^{57} +3.10260 q^{58} -12.3151 q^{59} +2.28425 q^{60} +14.7477 q^{61} -3.95644 q^{62} -3.41742 q^{64} +5.00000 q^{66} -4.47315 q^{67} -5.37386 q^{68} -4.41742 q^{69} -4.37780 q^{71} +8.29875 q^{72} +3.46410 q^{73} +3.16515 q^{74} +13.3739 q^{75} -2.45505 q^{76} -6.00000 q^{79} +1.27520 q^{80} -0.417424 q^{81} +3.58258 q^{82} -7.02355 q^{83} +1.37055 q^{85} +4.28245 q^{86} +18.9564 q^{87} +6.79129 q^{88} -16.1407 q^{89} -1.00000 q^{90} -2.83485 q^{92} -24.1733 q^{93} -4.37386 q^{94} +0.626136 q^{95} +13.2288 q^{96} -7.28970 q^{97} +18.7864 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{4} + 10 q^{9} - 10 q^{10} + 20 q^{12} + 2 q^{16} + 12 q^{17} + 2 q^{22} - 12 q^{23} - 10 q^{25} - 20 q^{27} - 18 q^{29} - 16 q^{30} - 16 q^{36} - 30 q^{38} - 6 q^{40} - 10 q^{43} - 22 q^{48}+ \cdots + 30 q^{95}+O(q^{100}) $ 4 * q - 2 * q^3 + 2 * q^4 + 10 * q^9 - 10 * q^10 + 20 * q^12 + 2 * q^16 + 12 * q^17 + 2 * q^22 - 12 * q^23 - 10 * q^25 - 20 * q^27 - 18 * q^29 - 16 * q^30 - 16 * q^36 - 30 * q^38 - 6 * q^40 - 10 * q^43 - 22 * q^48 - 6 * q^51 - 12 * q^53 - 2 * q^55 + 4 * q^61 + 30 * q^62 - 32 * q^64 + 20 * q^66 + 6 * q^68 - 36 * q^69 - 24 * q^74 + 26 * q^75 - 24 * q^79 - 20 * q^81 - 4 * q^82 + 30 * q^87 + 18 * q^88 - 4 * q^90 - 48 * q^92 + 10 * q^94 + 30 * q^95

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 $

$2$	−0.456850	−0.323042	−0.161521	−	0.986869i	$-0.551640\pi$
$2$	−0.456850	−0.323042	−0.161521	+	0.986869i	$0.551640\pi$
$3$	−2.79129	−1.61155	−0.805775	−	0.592221i	$-0.798251\pi$
$3$	−2.79129	−1.61155	−0.805775	+	0.592221i	$0.798251\pi$
$4$	−1.79129	−0.895644
$5$	0.456850	0.204310	0.102155	−	0.994769i	$-0.467426\pi$
$5$	0.456850	0.204310	0.102155	+	0.994769i	$0.467426\pi$
$6$	1.27520	0.520598
$7$	0	0
$7$	0	0
$8$	1.73205	0.612372
$9$	4.79129	1.59710
$10$	−0.208712	−0.0660006
$11$	3.92095	1.18221	0.591106	−	0.806594i	$-0.298692\pi$
$11$	3.92095	1.18221	0.591106	+	0.806594i	$0.298692\pi$
$12$	5.00000	1.44338
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.27520	−0.329255
$16$	2.79129	0.697822
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	−2.18890	−0.515929
$19$	1.37055	0.314426	0.157213	−	0.987565i	$-0.449749\pi$
$19$	1.37055	0.314426	0.157213	+	0.987565i	$0.449749\pi$
$20$	−0.818350	−0.182989
$21$	0	0
$22$	−1.79129	−0.381904
$23$	1.58258	0.329990	0.164995	−	0.986294i	$-0.447239\pi$
$23$	1.58258	0.329990	0.164995	+	0.986294i	$0.447239\pi$
$24$	−4.83465	−0.986869
$25$	−4.79129	−0.958258
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−6.79129	−1.26111	−0.630555	−	0.776144i	$-0.717173\pi$
$29$	−6.79129	−1.26111	−0.630555	+	0.776144i	$0.717173\pi$
$30$	0.582576	0.106363
$31$	8.66025	1.55543	0.777714	−	0.628619i	$-0.216379\pi$
$31$	8.66025	1.55543	0.777714	+	0.628619i	$0.216379\pi$
$32$	−4.73930	−0.837798
$33$	−10.9445	−1.90519
$34$	−1.37055	−0.235048
$35$	0	0
$36$	−8.58258	−1.43043
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	−0.626136	−0.101573
$39$	0	0
$40$	0.791288	0.125114
$41$	−7.84190	−1.22470	−0.612350	−	0.790587i	$-0.709776\pi$
$41$	−7.84190	−1.22470	−0.612350	+	0.790587i	$0.709776\pi$
$42$	0	0
$43$	−9.37386	−1.42950	−0.714750	−	0.699380i	$-0.753460\pi$
$43$	−9.37386	−1.42950	−0.714750	+	0.699380i	$0.753460\pi$
$44$	−7.02355	−1.05884
$45$	2.18890	0.326302
$46$	−0.723000	−0.106601
$47$	9.57395	1.39650	0.698252	−	0.715852i	$-0.253961\pi$
$47$	9.57395	1.39650	0.698252	+	0.715852i	$0.253961\pi$
$48$	−7.79129	−1.12458
$49$	0	0
$50$	2.18890	0.309557
$51$	−8.37386	−1.17258
$52$	0	0
$53$	6.16515	0.846849	0.423424	−	0.905931i	$-0.360828\pi$
$53$	6.16515	0.846849	0.423424	+	0.905931i	$0.360828\pi$
$54$	2.28425	0.310847
$55$	1.79129	0.241537
$56$	0	0
$57$	−3.82560	−0.506713
$58$	3.10260	0.407392
$59$	−12.3151	−1.60328	−0.801642	−	0.597805i	$-0.796040\pi$
$59$	−12.3151	−1.60328	−0.801642	+	0.597805i	$0.796040\pi$
$60$	2.28425	0.294896
$61$	14.7477	1.88825	0.944126	−	0.329583i	$-0.106908\pi$
$61$	14.7477	1.88825	0.944126	+	0.329583i	$0.106908\pi$
$62$	−3.95644	−0.502468
$63$	0	0
$64$	−3.41742	−0.427178
$65$	0	0
$66$	5.00000	0.615457
$67$	−4.47315	−0.546483	−0.273241	−	0.961946i	$-0.588096\pi$
$67$	−4.47315	−0.546483	−0.273241	+	0.961946i	$0.588096\pi$
$68$	−5.37386	−0.651677
$69$	−4.41742	−0.531795
$70$	0	0
$71$	−4.37780	−0.519550	−0.259775	−	0.965669i	$-0.583648\pi$
$71$	−4.37780	−0.519550	−0.259775	+	0.965669i	$0.583648\pi$
$72$	8.29875	0.978018
$73$	3.46410	0.405442	0.202721	−	0.979236i	$-0.435021\pi$
$73$	3.46410	0.405442	0.202721	+	0.979236i	$0.435021\pi$
$74$	3.16515	0.367941
$75$	13.3739	1.54428
$76$	−2.45505	−0.281614
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	1.27520	0.142572
$81$	−0.417424	−0.0463805
$82$	3.58258	0.395629
$83$	−7.02355	−0.770935	−0.385468	−	0.922721i	$-0.625960\pi$
$83$	−7.02355	−0.770935	−0.385468	+	0.922721i	$0.625960\pi$
$84$	0	0
$85$	1.37055	0.148657
$86$	4.28245	0.461789
$87$	18.9564	2.03234
$88$	6.79129	0.723954
$89$	−16.1407	−1.71091	−0.855453	−	0.517880i	$-0.826721\pi$
$89$	−16.1407	−1.71091	−0.855453	+	0.517880i	$0.826721\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−2.83485	−0.295553
$93$	−24.1733	−2.50665
$94$	−4.37386	−0.451130
$95$	0.626136	0.0642402
$96$	13.2288	1.35015
$97$	−7.28970	−0.740157	−0.370079	−	0.929000i	$-0.620669\pi$
$97$	−7.28970	−0.740157	−0.370079	+	0.929000i	$0.620669\pi$
$98$	0	0
$99$	18.7864	1.88811
$100$	8.58258	0.858258
$101$	5.20871	0.518286	0.259143	−	0.965839i	$-0.416560\pi$
$101$	5.20871	0.518286	0.259143	+	0.965839i	$0.416560\pi$
$102$	3.82560	0.378791
$103$	4.58258	0.451535	0.225767	−	0.974181i	$-0.427511\pi$
$103$	4.58258	0.451535	0.225767	+	0.974181i	$0.427511\pi$
$104$	0	0
$105$	0	0
$106$	−2.81655	−0.273568
$107$	−5.20871	−0.503545	−0.251773	−	0.967786i	$-0.581014\pi$
$107$	−5.20871	−0.503545	−0.251773	+	0.967786i	$0.581014\pi$
$108$	8.95644	0.861834
$109$	7.93725	0.760251	0.380126	−	0.924935i	$-0.375881\pi$
$109$	7.93725	0.760251	0.380126	+	0.924935i	$0.375881\pi$
$110$	−0.818350	−0.0780266
$111$	19.3386	1.83554
$112$	0	0
$113$	10.5826	0.995525	0.497762	−	0.867313i	$-0.334155\pi$
$113$	10.5826	0.995525	0.497762	+	0.867313i	$0.334155\pi$
$114$	1.74773	0.163690
$115$	0.723000	0.0674201
$116$	12.1652	1.12951
$117$	0	0
$118$	5.62614	0.517928
$119$	0	0
$120$	−2.20871	−0.201627
$121$	4.37386	0.397624
$122$	−6.73750	−0.609985
$123$	21.8890	1.97367
$124$	−15.5130	−1.39311
$125$	−4.47315	−0.400091
$126$	0	0
$127$	−6.95644	−0.617284	−0.308642	−	0.951178i	$-0.599875\pi$
$127$	−6.95644	−0.617284	−0.308642	+	0.951178i	$0.599875\pi$
$128$	11.0399	0.975795
$129$	26.1652	2.30371
$130$	0	0
$131$	17.3739	1.51796	0.758981	−	0.651113i	$-0.225698\pi$
$131$	17.3739	1.51796	0.758981	+	0.651113i	$0.225698\pi$
$132$	19.6048	1.70638
$133$	0	0
$134$	2.04356	0.176537
$135$	−2.28425	−0.196597
$136$	5.19615	0.445566
$137$	11.9536	1.02126	0.510631	−	0.859800i	$-0.329412\pi$
$137$	11.9536	1.02126	0.510631	+	0.859800i	$0.329412\pi$
$138$	2.01810	0.171792
$139$	−3.79129	−0.321573	−0.160786	−	0.986989i	$-0.551403\pi$
$139$	−3.79129	−0.321573	−0.160786	+	0.986989i	$0.551403\pi$
$140$	0	0
$141$	−26.7237	−2.25054
$142$	2.00000	0.167836
$143$	0	0
$144$	13.3739	1.11449
$145$	−3.10260	−0.257657
$146$	−1.58258	−0.130975
$147$	0	0
$148$	12.4104	1.02013
$149$	0.456850	0.0374266	0.0187133	−	0.999825i	$-0.494043\pi$
$149$	0.456850	0.0374266	0.0187133	+	0.999825i	$0.494043\pi$
$150$	−6.10985	−0.498867
$151$	−12.1244	−0.986666	−0.493333	−	0.869841i	$-0.664222\pi$
$151$	−12.1244	−0.986666	−0.493333	+	0.869841i	$0.664222\pi$
$152$	2.37386	0.192546
$153$	14.3739	1.16206
$154$	0	0
$155$	3.95644	0.317789
$156$	0	0
$157$	0.956439	0.0763322	0.0381661	−	0.999271i	$-0.487848\pi$
$157$	0.956439	0.0763322	0.0381661	+	0.999271i	$0.487848\pi$
$158$	2.74110	0.218070
$159$	−17.2087	−1.36474
$160$	−2.16515	−0.171170
$161$	0	0
$162$	0.190700	0.0149828
$163$	6.92820	0.542659	0.271329	−	0.962487i	$-0.412537\pi$
$163$	6.92820	0.542659	0.271329	+	0.962487i	$0.412537\pi$
$164$	14.0471	1.09689
$165$	−5.00000	−0.389249
$166$	3.20871	0.249044
$167$	14.6748	1.13557	0.567783	−	0.823178i	$-0.307801\pi$
$167$	14.6748	1.13557	0.567783	+	0.823178i	$0.307801\pi$
$168$	0	0
$169$	0	0
$170$	−0.626136	−0.0480225
$171$	6.56670	0.502168
$172$	16.7913	1.28032
$173$	19.7477	1.50139	0.750696	−	0.660648i	$-0.229718\pi$
$173$	19.7477	1.50139	0.750696	+	0.660648i	$0.229718\pi$
$174$	−8.66025	−0.656532
$175$	0	0
$176$	10.9445	0.824973
$177$	34.3749	2.58377
$178$	7.37386	0.552694
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	−3.92095	−0.292250
$181$	−9.16515	−0.681240	−0.340620	−	0.940201i	$-0.610637\pi$
$181$	−9.16515	−0.681240	−0.340620	+	0.940201i	$0.610637\pi$
$182$	0	0
$183$	−41.1652	−3.04302
$184$	2.74110	0.202077
$185$	−3.16515	−0.232707
$186$	11.0436	0.809753
$187$	11.7629	0.860185
$188$	−17.1497	−1.25077
$189$	0	0
$190$	−0.286051	−0.0207523
$191$	−14.3739	−1.04006	−0.520028	−	0.854149i	$-0.674079\pi$
$191$	−14.3739	−1.04006	−0.520028	+	0.854149i	$0.674079\pi$
$192$	9.53901	0.688419
$193$	−19.3386	−1.39202	−0.696012	−	0.718030i	$-0.745044\pi$
$193$	−19.3386	−1.39202	−0.696012	+	0.718030i	$0.745044\pi$
$194$	3.33030	0.239102
$195$	0	0
$196$	0	0
$197$	2.28425	0.162746	0.0813731	−	0.996684i	$-0.474069\pi$
$197$	2.28425	0.162746	0.0813731	+	0.996684i	$0.474069\pi$
$198$	−8.58258	−0.609937
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	−8.29875	−0.586811
$201$	12.4859	0.880684
$202$	−2.37960	−0.167428
$203$	0	0
$204$	15.0000	1.05021
$205$	−3.58258	−0.250218
$206$	−2.09355	−0.145865
$207$	7.58258	0.527025
$208$	0	0
$209$	5.37386	0.371718
$210$	0	0
$211$	10.5826	0.728535	0.364267	−	0.931294i	$-0.381319\pi$
$211$	10.5826	0.728535	0.364267	+	0.931294i	$0.381319\pi$
$212$	−11.0436	−0.758475
$213$	12.2197	0.837280
$214$	2.37960	0.162666
$215$	−4.28245	−0.292061
$216$	−8.66025	−0.589256
$217$	0	0
$218$	−3.62614	−0.245593
$219$	−9.66930	−0.653391
$220$	−3.20871	−0.216331
$221$	0	0
$222$	−8.83485	−0.592956
$223$	−18.9771	−1.27080	−0.635401	−	0.772183i	$-0.719165\pi$
$223$	−18.9771	−1.27080	−0.635401	+	0.772183i	$0.719165\pi$
$224$	0	0
$225$	−22.9564	−1.53043
$226$	−4.83465	−0.321596
$227$	−8.85095	−0.587458	−0.293729	−	0.955889i	$-0.594896\pi$
$227$	−8.85095	−0.587458	−0.293729	+	0.955889i	$0.594896\pi$
$228$	6.85275	0.453835
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	−0.330303	−0.0217795
$231$	0	0
$232$	−11.7629	−0.772269
$233$	−15.9564	−1.04534	−0.522671	−	0.852535i	$-0.675064\pi$
$233$	−15.9564	−1.04534	−0.522671	+	0.852535i	$0.675064\pi$
$234$	0	0
$235$	4.37386	0.285319
$236$	22.0598	1.43597
$237$	16.7477	1.08788
$238$	0	0
$239$	−13.2288	−0.855697	−0.427849	−	0.903850i	$-0.640728\pi$
$239$	−13.2288	−0.855697	−0.427849	+	0.903850i	$0.640728\pi$
$240$	−3.55945	−0.229762
$241$	−19.7001	−1.26900	−0.634498	−	0.772925i	$-0.718793\pi$
$241$	−19.7001	−1.26900	−0.634498	+	0.772925i	$0.718793\pi$
$242$	−1.99820	−0.128449
$243$	16.1652	1.03699
$244$	−26.4174	−1.69120
$245$	0	0
$246$	−10.0000	−0.637577
$247$	0	0
$248$	15.0000	0.952501
$249$	19.6048	1.24240
$250$	2.04356	0.129246
$251$	2.83485	0.178934	0.0894670	−	0.995990i	$-0.471484\pi$
$251$	2.83485	0.178934	0.0894670	+	0.995990i	$0.471484\pi$
$252$	0	0
$253$	6.20520	0.390118
$254$	3.17805	0.199409
$255$	−3.82560	−0.239568
$256$	1.79129	0.111955
$257$	−5.04356	−0.314609	−0.157304	−	0.987550i	$-0.550280\pi$
$257$	−5.04356	−0.314609	−0.157304	+	0.987550i	$0.550280\pi$
$258$	−11.9536	−0.744196
$259$	0	0
$260$	0	0
$261$	−32.5390	−2.01411
$262$	−7.93725	−0.490365
$263$	9.33030	0.575331	0.287666	−	0.957731i	$-0.407121\pi$
$263$	9.33030	0.575331	0.287666	+	0.957731i	$0.407121\pi$
$264$	−18.9564	−1.16669
$265$	2.81655	0.173019
$266$	0	0
$267$	45.0532	2.75721
$268$	8.01270	0.489454
$269$	−15.7913	−0.962812	−0.481406	−	0.876498i	$-0.659874\pi$
$269$	−15.7913	−0.962812	−0.481406	+	0.876498i	$0.659874\pi$
$270$	1.04356	0.0635091
$271$	12.8474	0.780421	0.390211	−	0.920726i	$-0.372402\pi$
$271$	12.8474	0.780421	0.390211	+	0.920726i	$0.372402\pi$
$272$	8.37386	0.507740
$273$	0	0
$274$	−5.46099	−0.329910
$275$	−18.7864	−1.13286
$276$	7.91288	0.476299
$277$	−11.7477	−0.705853	−0.352926	−	0.935651i	$-0.614813\pi$
$277$	−11.7477	−0.705853	−0.352926	+	0.935651i	$0.614813\pi$
$278$	1.73205	0.103882
$279$	41.4938	2.48417
$280$	0	0
$281$	30.6446	1.82810	0.914052	−	0.405597i	$-0.132936\pi$
$281$	30.6446	1.82810	0.914052	+	0.405597i	$0.132936\pi$
$282$	12.2087	0.727018
$283$	−2.74773	−0.163335	−0.0816677	−	0.996660i	$-0.526025\pi$
$283$	−2.74773	−0.163335	−0.0816677	+	0.996660i	$0.526025\pi$
$284$	7.84190	0.465331
$285$	−1.74773	−0.103526
$286$	0	0
$287$	0	0
$288$	−22.7074	−1.33804
$289$	−8.00000	−0.470588
$290$	1.41742	0.0832340
$291$	20.3477	1.19280
$292$	−6.20520	−0.363132
$293$	2.55040	0.148996	0.0744980	−	0.997221i	$-0.476265\pi$
$293$	2.55040	0.148996	0.0744980	+	0.997221i	$0.476265\pi$
$294$	0	0
$295$	−5.62614	−0.327566
$296$	−12.0000	−0.697486
$297$	−19.6048	−1.13758
$298$	−0.208712	−0.0120904
$299$	0	0
$300$	−23.9564	−1.38313
$301$	0	0
$302$	5.53901	0.318734
$303$	−14.5390	−0.835245
$304$	3.82560	0.219413
$305$	6.73750	0.385788
$306$	−6.56670	−0.375393
$307$	15.5130	0.885374	0.442687	−	0.896676i	$-0.354025\pi$
$307$	15.5130	0.885374	0.442687	+	0.896676i	$0.354025\pi$
$308$	0	0
$309$	−12.7913	−0.727671
$310$	−1.80750	−0.102659
$311$	−26.5390	−1.50489	−0.752445	−	0.658655i	$-0.771126\pi$
$311$	−26.5390	−1.50489	−0.752445	+	0.658655i	$0.771126\pi$
$312$	0	0
$313$	−6.74773	−0.381404	−0.190702	−	0.981648i	$-0.561076\pi$
$313$	−6.74773	−0.381404	−0.190702	+	0.981648i	$0.561076\pi$
$314$	−0.436950	−0.0246585
$315$	0	0
$316$	10.7477	0.604607
$317$	18.5203	1.04020	0.520101	−	0.854105i	$-0.325894\pi$
$317$	18.5203	1.04020	0.520101	+	0.854105i	$0.325894\pi$
$318$	7.86180	0.440868
$319$	−26.6283	−1.49090
$320$	−1.56125	−0.0872766
$321$	14.5390	0.811489
$322$	0	0
$323$	4.11165	0.228778
$324$	0.747727	0.0415404
$325$	0	0
$326$	−3.16515	−0.175302
$327$	−22.1552	−1.22518
$328$	−13.5826	−0.749972
$329$	0	0
$330$	2.28425	0.125744
$331$	−24.8963	−1.36842	−0.684211	−	0.729284i	$-0.739853\pi$
$331$	−24.8963	−1.36842	−0.684211	+	0.729284i	$0.739853\pi$
$332$	12.5812	0.690483
$333$	−33.1950	−1.81908
$334$	−6.70417	−0.366836
$335$	−2.04356	−0.111652
$336$	0	0
$337$	−9.95644	−0.542362	−0.271181	−	0.962528i	$-0.587414\pi$
$337$	−9.95644	−0.542362	−0.271181	+	0.962528i	$0.587414\pi$
$338$	0	0
$339$	−29.5390	−1.60434
$340$	−2.45505	−0.133144
$341$	33.9564	1.83884
$342$	−3.00000	−0.162221
$343$	0	0
$344$	−16.2360	−0.875387
$345$	−2.01810	−0.108651
$346$	−9.02175	−0.485013
$347$	−13.5826	−0.729151	−0.364575	−	0.931174i	$-0.618786\pi$
$347$	−13.5826	−0.729151	−0.364575	+	0.931174i	$0.618786\pi$
$348$	−33.9564	−1.82026
$349$	21.0707	1.12789	0.563943	−	0.825814i	$-0.309284\pi$
$349$	21.0707	1.12789	0.563943	+	0.825814i	$0.309284\pi$
$350$	0	0
$351$	0	0
$352$	−18.5826	−0.990455
$353$	18.1588	0.966493	0.483247	−	0.875484i	$-0.339457\pi$
$353$	18.1588	0.966493	0.483247	+	0.875484i	$0.339457\pi$
$354$	−15.7042	−0.834667
$355$	−2.00000	−0.106149
$356$	28.9126	1.53236
$357$	0	0
$358$	−4.11165	−0.217308
$359$	−0.552200	−0.0291440	−0.0145720	−	0.999894i	$-0.504639\pi$
$359$	−0.552200	−0.0291440	−0.0145720	+	0.999894i	$0.504639\pi$
$360$	3.79129	0.199818
$361$	−17.1216	−0.901136
$362$	4.18710	0.220069
$363$	−12.2087	−0.640791
$364$	0	0
$365$	1.58258	0.0828358
$366$	18.8063	0.983022
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	4.41742	0.230274
$369$	−37.5728	−1.95596
$370$	1.44600	0.0751740
$371$	0	0
$372$	43.3013	2.24507
$373$	32.2087	1.66770	0.833852	−	0.551988i	$-0.186131\pi$
$373$	32.2087	1.66770	0.833852	+	0.551988i	$0.186131\pi$
$374$	−5.37386	−0.277876
$375$	12.4859	0.644767
$376$	16.5826	0.855181
$377$	0	0
$378$	0	0
$379$	−28.3604	−1.45677	−0.728387	−	0.685166i	$-0.759730\pi$
$379$	−28.3604	−1.45677	−0.728387	+	0.685166i	$0.759730\pi$
$380$	−1.12159	−0.0575364
$381$	19.4174	0.994785
$382$	6.56670	0.335982
$383$	−1.27520	−0.0651597	−0.0325799	−	0.999469i	$-0.510372\pi$
$383$	−1.27520	−0.0651597	−0.0325799	+	0.999469i	$0.510372\pi$
$384$	−30.8154	−1.57254
$385$	0	0
$386$	8.83485	0.449682
$387$	−44.9129	−2.28305
$388$	13.0580	0.662917
$389$	0.330303	0.0167470	0.00837351	−	0.999965i	$-0.497335\pi$
$389$	0.330303	0.0167470	0.00837351	+	0.999965i	$0.497335\pi$
$390$	0	0
$391$	4.74773	0.240103
$392$	0	0
$393$	−48.4955	−2.44627
$394$	−1.04356	−0.0525738
$395$	−2.74110	−0.137920
$396$	−33.6519	−1.69107
$397$	32.4720	1.62972	0.814862	−	0.579655i	$-0.196813\pi$
$397$	32.4720	1.62972	0.814862	+	0.579655i	$0.196813\pi$
$398$	5.02535	0.251898
$399$	0	0
$400$	−13.3739	−0.668693
$401$	−31.2922	−1.56266	−0.781328	−	0.624121i	$-0.785457\pi$
$401$	−31.2922	−1.56266	−0.781328	+	0.624121i	$0.785457\pi$
$402$	−5.70417	−0.284498
$403$	0	0
$404$	−9.33030	−0.464200
$405$	−0.190700	−0.00947598
$406$	0	0
$407$	−27.1652	−1.34653
$408$	−14.5040	−0.718053
$409$	−8.29875	−0.410347	−0.205173	−	0.978726i	$-0.565776\pi$
$409$	−8.29875	−0.410347	−0.205173	+	0.978726i	$0.565776\pi$
$410$	1.63670	0.0808309
$411$	−33.3658	−1.64581
$412$	−8.20871	−0.404414
$413$	0	0
$414$	−3.46410	−0.170251
$415$	−3.20871	−0.157509
$416$	0	0
$417$	10.5826	0.518231
$418$	−2.45505	−0.120080
$419$	−1.74773	−0.0853821	−0.0426910	−	0.999088i	$-0.513593\pi$
$419$	−1.74773	−0.0853821	−0.0426910	+	0.999088i	$0.513593\pi$
$420$	0	0
$421$	−4.18710	−0.204067	−0.102033	−	0.994781i	$-0.532535\pi$
$421$	−4.18710	−0.204067	−0.102033	+	0.994781i	$0.532535\pi$
$422$	−4.83465	−0.235347
$423$	45.8716	2.23035
$424$	10.6784	0.518587
$425$	−14.3739	−0.697235
$426$	−5.58258	−0.270477
$427$	0	0
$428$	9.33030	0.450997
$429$	0	0
$430$	1.95644	0.0943479
$431$	34.6609	1.66956	0.834779	−	0.550585i	$-0.185595\pi$
$431$	34.6609	1.66956	0.834779	+	0.550585i	$0.185595\pi$
$432$	−13.9564	−0.671479
$433$	32.4955	1.56163	0.780816	−	0.624761i	$-0.214804\pi$
$433$	32.4955	1.56163	0.780816	+	0.624761i	$0.214804\pi$
$434$	0	0
$435$	8.66025	0.415227
$436$	−14.2179	−0.680914
$437$	2.16900	0.103757
$438$	4.41742	0.211073
$439$	20.5390	0.980274	0.490137	−	0.871645i	$-0.336947\pi$
$439$	20.5390	0.980274	0.490137	+	0.871645i	$0.336947\pi$
$440$	3.10260	0.147911
$441$	0	0
$442$	0	0
$443$	−15.1652	−0.720518	−0.360259	−	0.932852i	$-0.617312\pi$
$443$	−15.1652	−0.720518	−0.360259	+	0.932852i	$0.617312\pi$
$444$	−34.6410	−1.64399
$445$	−7.37386	−0.349555
$446$	8.66970	0.410522
$447$	−1.27520	−0.0603149
$448$	0	0
$449$	−25.1624	−1.18749	−0.593744	−	0.804654i	$-0.702351\pi$
$449$	−25.1624	−1.18749	−0.593744	+	0.804654i	$0.702351\pi$
$450$	10.4877	0.494393
$451$	−30.7477	−1.44785
$452$	−18.9564	−0.891636
$453$	33.8426	1.59006
$454$	4.04356	0.189774
$455$	0	0
$456$	−6.62614	−0.310297
$457$	22.8027	1.06667	0.533333	−	0.845905i	$-0.320939\pi$
$457$	22.8027	1.06667	0.533333	+	0.845905i	$0.320939\pi$
$458$	−3.16515	−0.147898
$459$	−15.0000	−0.700140
$460$	−1.29510	−0.0603844
$461$	−4.64395	−0.216290	−0.108145	−	0.994135i	$-0.534491\pi$
$461$	−4.64395	−0.216290	−0.108145	+	0.994135i	$0.534491\pi$
$462$	0	0
$463$	7.93725	0.368875	0.184438	−	0.982844i	$-0.440954\pi$
$463$	7.93725	0.368875	0.184438	+	0.982844i	$0.440954\pi$
$464$	−18.9564	−0.880031
$465$	−11.0436	−0.512133
$466$	7.28970	0.337689
$467$	−30.1652	−1.39588	−0.697938	−	0.716158i	$-0.745899\pi$
$467$	−30.1652	−1.39588	−0.697938	+	0.716158i	$0.745899\pi$
$468$	0	0
$469$	0	0
$470$	−1.99820	−0.0921701
$471$	−2.66970	−0.123013
$472$	−21.3303	−0.981807
$473$	−36.7545	−1.68997
$474$	−7.65120	−0.351431
$475$	−6.56670	−0.301301
$476$	0	0
$477$	29.5390	1.35250
$478$	6.04356	0.276426
$479$	18.8818	0.862730	0.431365	−	0.902178i	$-0.358032\pi$
$479$	18.8818	0.862730	0.431365	+	0.902178i	$0.358032\pi$
$480$	6.04356	0.275850
$481$	0	0
$482$	9.00000	0.409939
$483$	0	0
$484$	−7.83485	−0.356129
$485$	−3.33030	−0.151221
$486$	−7.38505	−0.334993
$487$	−29.3694	−1.33086	−0.665428	−	0.746462i	$-0.731751\pi$
$487$	−29.3694	−1.33086	−0.665428	+	0.746462i	$0.731751\pi$
$488$	25.5438	1.15631
$489$	−19.3386	−0.874522
$490$	0	0
$491$	4.12159	0.186005	0.0930024	−	0.995666i	$-0.470354\pi$
$491$	4.12159	0.186005	0.0930024	+	0.995666i	$0.470354\pi$
$492$	−39.2095	−1.76770
$493$	−20.3739	−0.917593
$494$	0	0
$495$	8.58258	0.385758
$496$	24.1733	1.08541
$497$	0	0
$498$	−8.95644	−0.401348
$499$	−18.4050	−0.823921	−0.411961	−	0.911202i	$-0.635156\pi$
$499$	−18.4050	−0.823921	−0.411961	+	0.911202i	$0.635156\pi$
$500$	8.01270	0.358339
$501$	−40.9615	−1.83002
$502$	−1.29510	−0.0578032
$503$	19.1216	0.852590	0.426295	−	0.904584i	$-0.359819\pi$
$503$	19.1216	0.852590	0.426295	+	0.904584i	$0.359819\pi$
$504$	0	0
$505$	2.37960	0.105891
$506$	−2.83485	−0.126024
$507$	0	0
$508$	12.4610	0.552867
$509$	−15.0562	−0.667352	−0.333676	−	0.942688i	$-0.608289\pi$
$509$	−15.0562	−0.667352	−0.333676	+	0.942688i	$0.608289\pi$
$510$	1.74773	0.0773907
$511$	0	0
$512$	−22.8981	−1.01196
$513$	−6.85275	−0.302556
$514$	2.30415	0.101632
$515$	2.09355	0.0922529
$516$	−46.8693	−2.06331
$517$	37.5390	1.65096
$518$	0	0
$519$	−55.1216	−2.41957
$520$	0	0
$521$	16.4174	0.719260	0.359630	−	0.933095i	$-0.382903\pi$
$521$	16.4174	0.719260	0.359630	+	0.933095i	$0.382903\pi$
$522$	14.8655	0.650643
$523$	24.3303	1.06389	0.531945	−	0.846779i	$-0.321461\pi$
$523$	24.3303	1.06389	0.531945	+	0.846779i	$0.321461\pi$
$524$	−31.1216	−1.35955
$525$	0	0
$526$	−4.26255	−0.185856
$527$	25.9808	1.13174
$528$	−30.5493	−1.32949
$529$	−20.4955	−0.891107
$530$	−1.28674	−0.0558925
$531$	−59.0050	−2.56060
$532$	0	0
$533$	0	0
$534$	−20.5826	−0.890695
$535$	−2.37960	−0.102879
$536$	−7.74773	−0.334651
$537$	−25.1216	−1.08408
$538$	7.21425	0.311029
$539$	0	0
$540$	4.09175	0.176081
$541$	6.28065	0.270026	0.135013	−	0.990844i	$-0.456892\pi$
$541$	6.28065	0.270026	0.135013	+	0.990844i	$0.456892\pi$
$542$	−5.86932	−0.252109
$543$	25.5826	1.09785
$544$	−14.2179	−0.609588
$545$	3.62614	0.155327
$546$	0	0
$547$	−11.7477	−0.502297	−0.251148	−	0.967949i	$-0.580808\pi$
$547$	−11.7477	−0.502297	−0.251148	+	0.967949i	$0.580808\pi$
$548$	−21.4123	−0.914686
$549$	70.6606	3.01572
$550$	8.58258	0.365962
$551$	−9.30780	−0.396526
$552$	−7.65120	−0.325657
$553$	0	0
$554$	5.36695	0.228020
$555$	8.83485	0.375018
$556$	6.79129	0.288015
$557$	33.0242	1.39928	0.699640	−	0.714495i	$-0.253344\pi$
$557$	33.0242	1.39928	0.699640	+	0.714495i	$0.253344\pi$
$558$	−18.9564	−0.802490
$559$	0	0
$560$	0	0
$561$	−32.8335	−1.38623
$562$	−14.0000	−0.590554
$563$	−36.3303	−1.53114	−0.765570	−	0.643353i	$-0.777543\pi$
$563$	−36.3303	−1.53114	−0.765570	+	0.643353i	$0.777543\pi$
$564$	47.8698	2.01568
$565$	4.83465	0.203395
$566$	1.25530	0.0527642
$567$	0	0
$568$	−7.58258	−0.318158
$569$	−16.7477	−0.702101	−0.351051	−	0.936356i	$-0.614175\pi$
$569$	−16.7477	−0.702101	−0.351051	+	0.936356i	$0.614175\pi$
$570$	0.798450	0.0334434
$571$	2.04356	0.0855204	0.0427602	−	0.999085i	$-0.486385\pi$
$571$	2.04356	0.0855204	0.0427602	+	0.999085i	$0.486385\pi$
$572$	0	0
$573$	40.1216	1.67610
$574$	0	0
$575$	−7.58258	−0.316215
$576$	−16.3739	−0.682244
$577$	−35.6501	−1.48413	−0.742066	−	0.670327i	$-0.766154\pi$
$577$	−35.6501	−1.48413	−0.742066	+	0.670327i	$0.766154\pi$
$578$	3.65480	0.152020
$579$	53.9796	2.24332
$580$	5.55765	0.230769
$581$	0	0
$582$	−9.29583	−0.385325
$583$	24.1733	1.00115
$584$	6.00000	0.248282
$585$	0	0
$586$	−1.16515	−0.0481320
$587$	−9.49851	−0.392045	−0.196023	−	0.980599i	$-0.562803\pi$
$587$	−9.49851	−0.392045	−0.196023	+	0.980599i	$0.562803\pi$
$588$	0	0
$589$	11.8693	0.489067
$590$	2.57030	0.105818
$591$	−6.37600	−0.262274
$592$	−19.3386	−0.794812
$593$	6.37600	0.261831	0.130916	−	0.991394i	$-0.458208\pi$
$593$	6.37600	0.261831	0.130916	+	0.991394i	$0.458208\pi$
$594$	8.95644	0.367487
$595$	0	0
$596$	−0.818350	−0.0335209
$597$	30.7042	1.25664
$598$	0	0
$599$	−6.62614	−0.270737	−0.135368	−	0.990795i	$-0.543222\pi$
$599$	−6.62614	−0.270737	−0.135368	+	0.990795i	$0.543222\pi$
$600$	23.1642	0.945675
$601$	−12.3739	−0.504740	−0.252370	−	0.967631i	$-0.581210\pi$
$601$	−12.3739	−0.504740	−0.252370	+	0.967631i	$0.581210\pi$
$602$	0	0
$603$	−21.4322	−0.872785
$604$	21.7182	0.883701
$605$	1.99820	0.0812384
$606$	6.64215	0.269819
$607$	−19.7477	−0.801536	−0.400768	−	0.916180i	$-0.631257\pi$
$607$	−19.7477	−0.801536	−0.400768	+	0.916180i	$0.631257\pi$
$608$	−6.49545	−0.263425
$609$	0	0
$610$	−3.07803	−0.124626
$611$	0	0
$612$	−25.7477	−1.04079
$613$	−18.2541	−0.737277	−0.368638	−	0.929573i	$-0.620176\pi$
$613$	−18.2541	−0.737277	−0.368638	+	0.929573i	$0.620176\pi$
$614$	−7.08712	−0.286013
$615$	10.0000	0.403239
$616$	0	0
$617$	17.2252	0.693459	0.346729	−	0.937965i	$-0.387292\pi$
$617$	17.2252	0.693459	0.346729	+	0.937965i	$0.387292\pi$
$618$	5.84370	0.235068
$619$	19.3386	0.777284	0.388642	−	0.921389i	$-0.372944\pi$
$619$	19.3386	0.777284	0.388642	+	0.921389i	$0.372944\pi$
$620$	−7.08712	−0.284626
$621$	−7.91288	−0.317533
$622$	12.1244	0.486142
$623$	0	0
$624$	0	0
$625$	21.9129	0.876515
$626$	3.08270	0.123210
$627$	−15.0000	−0.599042
$628$	−1.71326	−0.0683664
$629$	−20.7846	−0.828737
$630$	0	0
$631$	27.6374	1.10023	0.550113	−	0.835090i	$-0.314585\pi$
$631$	27.6374	1.10023	0.550113	+	0.835090i	$0.314585\pi$
$632$	−10.3923	−0.413384
$633$	−29.5390	−1.17407
$634$	−8.46099	−0.336029
$635$	−3.17805	−0.126117
$636$	30.8258	1.22232
$637$	0	0
$638$	12.1652	0.481623
$639$	−20.9753	−0.829770
$640$	5.04356	0.199364
$641$	−29.3739	−1.16020	−0.580099	−	0.814546i	$-0.696986\pi$
$641$	−29.3739	−1.16020	−0.580099	+	0.814546i	$0.696986\pi$
$642$	−6.64215	−0.262145
$643$	−45.3194	−1.78722	−0.893611	−	0.448843i	$-0.851836\pi$
$643$	−45.3194	−1.78722	−0.893611	+	0.448843i	$0.851836\pi$
$644$	0	0
$645$	11.9536	0.470671
$646$	−1.87841	−0.0739050
$647$	−35.0780	−1.37906	−0.689530	−	0.724257i	$-0.742183\pi$
$647$	−35.0780	−1.37906	−0.689530	+	0.724257i	$0.742183\pi$
$648$	−0.723000	−0.0284021
$649$	−48.2867	−1.89542
$650$	0	0
$651$	0	0
$652$	−12.4104	−0.486029
$653$	15.7913	0.617961	0.308980	−	0.951068i	$-0.400012\pi$
$653$	15.7913	0.617961	0.308980	+	0.951068i	$0.400012\pi$
$654$	10.1216	0.395786
$655$	7.93725	0.310134
$656$	−21.8890	−0.854622
$657$	16.5975	0.647530
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	8.95644	0.348629
$661$	50.5155	1.96483	0.982413	−	0.186720i	$-0.0597859\pi$
$661$	50.5155	1.96483	0.982413	+	0.186720i	$0.0597859\pi$
$662$	11.3739	0.442058
$663$	0	0
$664$	−12.1652	−0.472099
$665$	0	0
$666$	15.1652	0.587638
$667$	−10.7477	−0.416154
$668$	−26.2867	−1.01706
$669$	52.9706	2.04796
$670$	0.933601	0.0360682
$671$	57.8251	2.23231
$672$	0	0
$673$	−26.4955	−1.02132	−0.510662	−	0.859781i	$-0.670600\pi$
$673$	−26.4955	−1.02132	−0.510662	+	0.859781i	$0.670600\pi$
$674$	4.54860	0.175206
$675$	23.9564	0.922084
$676$	0	0
$677$	−29.2087	−1.12258	−0.561291	−	0.827619i	$-0.689695\pi$
$677$	−29.2087	−1.12258	−0.561291	+	0.827619i	$0.689695\pi$
$678$	13.4949	0.518269
$679$	0	0
$680$	2.37386	0.0910335
$681$	24.7056	0.946719
$682$	−15.5130	−0.594024
$683$	30.3586	1.16164	0.580819	−	0.814033i	$-0.302732\pi$
$683$	30.3586	1.16164	0.580819	+	0.814033i	$0.302732\pi$
$684$	−11.7629	−0.449764
$685$	5.46099	0.208654
$686$	0	0
$687$	−19.3386	−0.737814
$688$	−26.1652	−0.997537
$689$	0	0
$690$	0.921970	0.0350988
$691$	−10.3169	−0.392472	−0.196236	−	0.980557i	$-0.562872\pi$
$691$	−10.3169	−0.392472	−0.196236	+	0.980557i	$0.562872\pi$
$692$	−35.3739	−1.34471
$693$	0	0
$694$	6.20520	0.235546
$695$	−1.73205	−0.0657004
$696$	32.8335	1.24455
$697$	−23.5257	−0.891100
$698$	−9.62614	−0.364355
$699$	44.5390	1.68462
$700$	0	0
$701$	13.9129	0.525482	0.262741	−	0.964866i	$-0.415373\pi$
$701$	13.9129	0.525482	0.262741	+	0.964866i	$0.415373\pi$
$702$	0	0
$703$	−9.49545	−0.358128
$704$	−13.3996	−0.505015
$705$	−12.2087	−0.459807
$706$	−8.29583	−0.312218
$707$	0	0
$708$	−61.5753	−2.31414
$709$	−15.2270	−0.571860	−0.285930	−	0.958250i	$-0.592303\pi$
$709$	−15.2270	−0.571860	−0.285930	+	0.958250i	$0.592303\pi$
$710$	0.913701	0.0342906
$711$	−28.7477	−1.07812
$712$	−27.9564	−1.04771
$713$	13.7055	0.513275
$714$	0	0
$715$	0	0
$716$	−16.1216	−0.602492
$717$	36.9253	1.37900
$718$	0.252273	0.00941474
$719$	−24.1652	−0.901208	−0.450604	−	0.892724i	$-0.648791\pi$
$719$	−24.1652	−0.901208	−0.450604	+	0.892724i	$0.648791\pi$
$720$	6.10985	0.227701
$721$	0	0
$722$	7.82200	0.291105
$723$	54.9887	2.04505
$724$	16.4174	0.610149
$725$	32.5390	1.20847
$726$	5.57755	0.207002
$727$	0.252273	0.00935628	0.00467814	−	0.999989i	$-0.498511\pi$
$727$	0.252273	0.00935628	0.00467814	+	0.999989i	$0.498511\pi$
$728$	0	0
$729$	−43.8693	−1.62479
$730$	−0.723000	−0.0267594
$731$	−28.1216	−1.04011
$732$	73.7386	2.72546
$733$	−16.9590	−0.626395	−0.313198	−	0.949688i	$-0.601400\pi$
$733$	−16.9590	−0.626395	−0.313198	+	0.949688i	$0.601400\pi$
$734$	8.22330	0.303528
$735$	0	0
$736$	−7.50030	−0.276465
$737$	−17.5390	−0.646058
$738$	17.1652	0.631858
$739$	−19.3386	−0.711382	−0.355691	−	0.934604i	$-0.615754\pi$
$739$	−19.3386	−0.711382	−0.355691	+	0.934604i	$0.615754\pi$
$740$	5.66970	0.208422
$741$	0	0
$742$	0	0
$743$	34.4702	1.26459	0.632295	−	0.774728i	$-0.282113\pi$
$743$	34.4702	1.26459	0.632295	+	0.774728i	$0.282113\pi$
$744$	−41.8693	−1.53500
$745$	0.208712	0.00764662
$746$	−14.7146	−0.538738
$747$	−33.6519	−1.23126
$748$	−21.0707	−0.770420
$749$	0	0
$750$	−5.70417	−0.208287
$751$	−23.7477	−0.866567	−0.433283	−	0.901258i	$-0.642645\pi$
$751$	−23.7477	−0.866567	−0.433283	+	0.901258i	$0.642645\pi$
$752$	26.7237	0.974512
$753$	−7.91288	−0.288361
$754$	0	0
$755$	−5.53901	−0.201585
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	12.9564	0.470599
$759$	−17.3205	−0.628695
$760$	1.08450	0.0393390
$761$	12.9626	0.469894	0.234947	−	0.972008i	$-0.424508\pi$
$761$	12.9626	0.469894	0.234947	+	0.972008i	$0.424508\pi$
$762$	−8.87086	−0.321357
$763$	0	0
$764$	25.7477	0.931520
$765$	6.56670	0.237420
$766$	0.582576	0.0210493
$767$	0	0
$768$	−5.00000	−0.180422
$769$	−9.38325	−0.338369	−0.169184	−	0.985584i	$-0.554113\pi$
$769$	−9.38325	−0.338369	−0.169184	+	0.985584i	$0.554113\pi$
$770$	0	0
$771$	14.0780	0.507008
$772$	34.6410	1.24676
$773$	−19.4340	−0.698991	−0.349495	−	0.936938i	$-0.613647\pi$
$773$	−19.4340	−0.698991	−0.349495	+	0.936938i	$0.613647\pi$
$774$	20.5185	0.737521
$775$	−41.4938	−1.49050
$776$	−12.6261	−0.453252
$777$	0	0
$778$	−0.150899	−0.00540999
$779$	−10.7477	−0.385077
$780$	0	0
$781$	−17.1652	−0.614217
$782$	−2.16900	−0.0775633
$783$	33.9564	1.21350
$784$	0	0
$785$	0.436950	0.0155954
$786$	22.1552	0.790248
$787$	26.2668	0.936311	0.468155	−	0.883646i	$-0.344919\pi$
$787$	26.2668	0.936311	0.468155	+	0.883646i	$0.344919\pi$
$788$	−4.09175	−0.145763
$789$	−26.0436	−0.927175
$790$	1.25227	0.0445539
$791$	0	0
$792$	32.5390	1.15622
$793$	0	0
$794$	−14.8348	−0.526469
$795$	−7.86180	−0.278829
$796$	19.7042	0.698396
$797$	12.0000	0.425062	0.212531	−	0.977154i	$-0.431829\pi$
$797$	12.0000	0.425062	0.212531	+	0.977154i	$0.431829\pi$
$798$	0	0
$799$	28.7219	1.01611
$800$	22.7074	0.802826
$801$	−77.3345	−2.73248
$802$	14.2958	0.504803
$803$	13.5826	0.479319
$804$	−22.3658	−0.788780
$805$	0	0
$806$	0	0
$807$	44.0780	1.55162
$808$	9.02175	0.317384
$809$	53.2432	1.87193	0.935965	−	0.352092i	$-0.114530\pi$
$809$	53.2432	1.87193	0.935965	+	0.352092i	$0.114530\pi$
$810$	0.0871215	0.00306114
$811$	−27.6374	−0.970479	−0.485240	−	0.874381i	$-0.661268\pi$
$811$	−27.6374	−0.970479	−0.485240	+	0.874381i	$0.661268\pi$
$812$	0	0
$813$	−35.8607	−1.25769
$814$	12.4104	0.434985
$815$	3.16515	0.110870
$816$	−23.3739	−0.818249
$817$	−12.8474	−0.449472
$818$	3.79129	0.132559
$819$	0	0
$820$	6.41742	0.224106
$821$	12.6567	0.441720	0.220860	−	0.975305i	$-0.429114\pi$
$821$	12.6567	0.441720	0.220860	+	0.975305i	$0.429114\pi$
$822$	15.2432	0.531667
$823$	22.4955	0.784142	0.392071	−	0.919935i	$-0.371759\pi$
$823$	22.4955	0.784142	0.392071	+	0.919935i	$0.371759\pi$
$824$	7.93725	0.276507
$825$	52.4383	1.82567
$826$	0	0
$827$	−35.3839	−1.23042	−0.615210	−	0.788364i	$-0.710929\pi$
$827$	−35.3839	−1.23042	−0.615210	+	0.788364i	$0.710929\pi$
$828$	−13.5826	−0.472027
$829$	−34.3739	−1.19385	−0.596927	−	0.802296i	$-0.703612\pi$
$829$	−34.3739	−1.19385	−0.596927	+	0.802296i	$0.703612\pi$
$830$	1.46590	0.0508822
$831$	32.7913	1.13752
$832$	0	0
$833$	0	0
$834$	−4.83465	−0.167410
$835$	6.70417	0.232007
$836$	−9.62614	−0.332927
$837$	−43.3013	−1.49671
$838$	0.798450	0.0275820
$839$	−27.9790	−0.965941	−0.482971	−	0.875637i	$-0.660442\pi$
$839$	−27.9790	−0.965941	−0.482971	+	0.875637i	$0.660442\pi$
$840$	0	0
$841$	17.1216	0.590400
$842$	1.91288	0.0659221
$843$	−85.5379	−2.94608
$844$	−18.9564	−0.652508
$845$	0	0
$846$	−20.9564	−0.720497
$847$	0	0
$848$	17.2087	0.590950
$849$	7.66970	0.263223
$850$	6.56670	0.225236
$851$	−10.9644	−0.375855
$852$	−21.8890	−0.749905
$853$	−14.1425	−0.484229	−0.242114	−	0.970248i	$-0.577841\pi$
$853$	−14.1425	−0.484229	−0.242114	+	0.970248i	$0.577841\pi$
$854$	0	0
$855$	3.00000	0.102598
$856$	−9.02175	−0.308357
$857$	−15.4610	−0.528137	−0.264069	−	0.964504i	$-0.585065\pi$
$857$	−15.4610	−0.528137	−0.264069	+	0.964504i	$0.585065\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	7.67110	0.261582
$861$	0	0
$862$	−15.8348	−0.539337
$863$	−45.4147	−1.54594	−0.772968	−	0.634446i	$-0.781228\pi$
$863$	−45.4147	−1.54594	−0.772968	+	0.634446i	$0.781228\pi$
$864$	23.6965	0.806172
$865$	9.02175	0.306749
$866$	−14.8456	−0.504473
$867$	22.3303	0.758377
$868$	0	0
$869$	−23.5257	−0.798055
$870$	−3.95644	−0.134136
$871$	0	0
$872$	13.7477	0.465557
$873$	−34.9271	−1.18210
$874$	−0.990908	−0.0335180
$875$	0	0
$876$	17.3205	0.585206
$877$	−3.17805	−0.107315	−0.0536576	−	0.998559i	$-0.517088\pi$
$877$	−3.17805	−0.107315	−0.0536576	+	0.998559i	$0.517088\pi$
$878$	−9.38325	−0.316669
$879$	−7.11890	−0.240115
$880$	5.00000	0.168550
$881$	−16.5826	−0.558681	−0.279341	−	0.960192i	$-0.590116\pi$
$881$	−16.5826	−0.558681	−0.279341	+	0.960192i	$0.590116\pi$
$882$	0	0
$883$	−29.2432	−0.984111	−0.492056	−	0.870564i	$-0.663754\pi$
$883$	−29.2432	−0.984111	−0.492056	+	0.870564i	$0.663754\pi$
$884$	0	0
$885$	15.7042	0.527890
$886$	6.92820	0.232758
$887$	22.4174	0.752703	0.376352	−	0.926477i	$-0.377178\pi$
$887$	22.4174	0.752703	0.376352	+	0.926477i	$0.377178\pi$
$888$	33.4955	1.12403
$889$	0	0
$890$	3.36875	0.112921
$891$	−1.63670	−0.0548315
$892$	33.9935	1.13819
$893$	13.1216	0.439097
$894$	0.582576	0.0194842
$895$	4.11165	0.137437
$896$	0	0
$897$	0	0
$898$	11.4955	0.383608
$899$	−58.8143	−1.96157
$900$	41.1216	1.37072
$901$	18.4955	0.616173
$902$	14.0471	0.467717
$903$	0	0
$904$	18.3296	0.609632
$905$	−4.18710	−0.139184
$906$	−15.4610	−0.513657
$907$	41.0780	1.36397	0.681987	−	0.731364i	$-0.261116\pi$
$907$	41.0780	1.36397	0.681987	+	0.731364i	$0.261116\pi$
$908$	15.8546	0.526154
$909$	24.9564	0.827753
$910$	0	0
$911$	−43.1216	−1.42868	−0.714341	−	0.699798i	$-0.753273\pi$
$911$	−43.1216	−1.42868	−0.714341	+	0.699798i	$0.753273\pi$
$912$	−10.6784	−0.353596
$913$	−27.5390	−0.911408
$914$	−10.4174	−0.344578
$915$	−18.8063	−0.621717
$916$	−12.4104	−0.410051
$917$	0	0
$918$	6.85275	0.226175
$919$	25.9129	0.854787	0.427393	−	0.904066i	$-0.359432\pi$
$919$	25.9129	0.854787	0.427393	+	0.904066i	$0.359432\pi$
$920$	1.25227	0.0412862
$921$	−43.3013	−1.42683
$922$	2.12159	0.0698709
$923$	0	0
$924$	0	0
$925$	33.1950	1.09145
$926$	−3.62614	−0.119162
$927$	21.9564	0.721144
$928$	32.1860	1.05656
$929$	57.9205	1.90031	0.950155	−	0.311779i	$-0.100925\pi$
$929$	57.9205	1.90031	0.950155	+	0.311779i	$0.100925\pi$
$930$	5.04525	0.165440
$931$	0	0
$932$	28.5826	0.936253
$933$	74.0780	2.42521
$934$	13.7810	0.450927
$935$	5.37386	0.175744
$936$	0	0
$937$	20.4955	0.669557	0.334779	−	0.942297i	$-0.391338\pi$
$937$	20.4955	0.669557	0.334779	+	0.942297i	$0.391338\pi$
$938$	0	0
$939$	18.8348	0.614652
$940$	−7.83485	−0.255545
$941$	39.3049	1.28130	0.640651	−	0.767832i	$-0.278665\pi$
$941$	39.3049	1.28130	0.640651	+	0.767832i	$0.278665\pi$
$942$	1.21965	0.0397384
$943$	−12.4104	−0.404138
$944$	−34.3749	−1.11881
$945$	0	0
$946$	16.7913	0.545932
$947$	38.3713	1.24690	0.623449	−	0.781864i	$-0.285731\pi$
$947$	38.3713	1.24690	0.623449	+	0.781864i	$0.285731\pi$
$948$	−30.0000	−0.974355
$949$	0	0
$950$	3.00000	0.0973329
$951$	−51.6954	−1.67634
$952$	0	0
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	−13.4949	−0.436914
$955$	−6.56670	−0.212494
$956$	23.6965	0.766400
$957$	74.3273	2.40266
$958$	−8.62614	−0.278698
$959$	0	0
$960$	4.35790	0.140651
$961$	44.0000	1.41935
$962$	0	0
$963$	−24.9564	−0.804210
$964$	35.2886	1.13657
$965$	−8.83485	−0.284404
$966$	0	0
$967$	23.8118	0.765735	0.382867	−	0.923803i	$-0.374937\pi$
$967$	23.8118	0.765735	0.382867	+	0.923803i	$0.374937\pi$
$968$	7.57575	0.243494
$969$	−11.4768	−0.368688
$970$	1.52145	0.0488508
$971$	22.2523	0.714109	0.357055	−	0.934083i	$-0.383781\pi$
$971$	22.2523	0.714109	0.357055	+	0.934083i	$0.383781\pi$
$972$	−28.9564	−0.928778
$973$	0	0
$974$	13.4174	0.429922
$975$	0	0
$976$	41.1652	1.31766
$977$	−1.39045	−0.0444845	−0.0222422	−	0.999753i	$-0.507081\pi$
$977$	−1.39045	−0.0444845	−0.0222422	+	0.999753i	$0.507081\pi$
$978$	8.83485	0.282507
$979$	−63.2867	−2.02265
$980$	0	0
$981$	38.0297	1.21419
$982$	−1.88295	−0.0600873
$983$	19.8908	0.634418	0.317209	−	0.948356i	$-0.397254\pi$
$983$	19.8908	0.634418	0.317209	+	0.948356i	$0.397254\pi$
$984$	37.9129	1.20862
$985$	1.04356	0.0332506
$986$	9.30780	0.296421
$987$	0	0
$988$	0	0
$989$	−14.8348	−0.471721
$990$	−3.92095	−0.124616
$991$	−40.3739	−1.28252	−0.641259	−	0.767324i	$-0.721588\pi$
$991$	−40.3739	−1.28252	−0.641259	+	0.767324i	$0.721588\pi$
$992$	−41.0436	−1.30313
$993$	69.4926	2.20528
$994$	0	0
$995$	−5.02535	−0.159314
$996$	−35.1178	−1.11275
$997$	−14.0780	−0.445856	−0.222928	−	0.974835i	$-0.571561\pi$
$997$	−14.0780	−0.445856	−0.222928	+	0.974835i	$0.571561\pi$
$998$	8.40833	0.266161
$999$	34.6410	1.09599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bq.1.2		4
7.6	odd	2		8281.2.a.bs.1.2		4
13.2	odd	12		637.2.q.f.589.1	yes	4
13.7	odd	12		637.2.q.f.491.1	yes	4
13.12	even	2	inner	8281.2.a.bq.1.3		4
91.2	odd	12		637.2.k.f.459.2		4
91.20	even	12		637.2.q.e.491.1	✓	4
91.33	even	12		637.2.u.e.361.2		4
91.41	even	12		637.2.q.e.589.1	yes	4
91.46	odd	12		637.2.k.f.569.1		4
91.54	even	12		637.2.k.d.459.2		4
91.59	even	12		637.2.k.d.569.1		4
91.67	odd	12		637.2.u.d.30.2		4
91.72	odd	12		637.2.u.d.361.2		4
91.80	even	12		637.2.u.e.30.2		4
91.90	odd	2		8281.2.a.bs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.k.d.459.2		4	91.54	even	12
637.2.k.d.569.1		4	91.59	even	12
637.2.k.f.459.2		4	91.2	odd	12
637.2.k.f.569.1		4	91.46	odd	12
637.2.q.e.491.1	✓	4	91.20	even	12
637.2.q.e.589.1	yes	4	91.41	even	12
637.2.q.f.491.1	yes	4	13.7	odd	12
637.2.q.f.589.1	yes	4	13.2	odd	12
637.2.u.d.30.2		4	91.67	odd	12
637.2.u.d.361.2		4	91.72	odd	12
637.2.u.e.30.2		4	91.80	even	12
637.2.u.e.361.2		4	91.33	even	12
8281.2.a.bq.1.2		4	1.1	even	1	trivial
8281.2.a.bq.1.3		4	13.12	even	2	inner
8281.2.a.bs.1.2		4	7.6	odd	2
8281.2.a.bs.1.3		4	91.90	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.456850 q^{2} -2.79129 q^{3} -1.79129 q^{4} +0.456850 q^{5} +1.27520 q^{6} +1.73205 q^{8} +4.79129 q^{9} -0.208712 q^{10} +3.92095 q^{11} +5.00000 q^{12} -1.27520 q^{15} +2.79129 q^{16} +3.00000 q^{17} -2.18890 q^{18} +1.37055 q^{19} -0.818350 q^{20} -1.79129 q^{22} +1.58258 q^{23} -4.83465 q^{24} -4.79129 q^{25} -5.00000 q^{27} -6.79129 q^{29} +0.582576 q^{30} +8.66025 q^{31} -4.73930 q^{32} -10.9445 q^{33} -1.37055 q^{34} -8.58258 q^{36} -6.92820 q^{37} -0.626136 q^{38} +0.791288 q^{40} -7.84190 q^{41} -9.37386 q^{43} -7.02355 q^{44} +2.18890 q^{45} -0.723000 q^{46} +9.57395 q^{47} -7.79129 q^{48} +2.18890 q^{50} -8.37386 q^{51} +6.16515 q^{53} +2.28425 q^{54} +1.79129 q^{55} -3.82560 q^{57} +3.10260 q^{58} -12.3151 q^{59} +2.28425 q^{60} +14.7477 q^{61} -3.95644 q^{62} -3.41742 q^{64} +5.00000 q^{66} -4.47315 q^{67} -5.37386 q^{68} -4.41742 q^{69} -4.37780 q^{71} +8.29875 q^{72} +3.46410 q^{73} +3.16515 q^{74} +13.3739 q^{75} -2.45505 q^{76} -6.00000 q^{79} +1.27520 q^{80} -0.417424 q^{81} +3.58258 q^{82} -7.02355 q^{83} +1.37055 q^{85} +4.28245 q^{86} +18.9564 q^{87} +6.79129 q^{88} -16.1407 q^{89} -1.00000 q^{90} -2.83485 q^{92} -24.1733 q^{93} -4.37386 q^{94} +0.626136 q^{95} +13.2288 q^{96} -7.28970 q^{97} +18.7864 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{4} + 10 q^{9} - 10 q^{10} + 20 q^{12} + 2 q^{16} + 12 q^{17} + 2 q^{22} - 12 q^{23} - 10 q^{25} - 20 q^{27} - 18 q^{29} - 16 q^{30} - 16 q^{36} - 30 q^{38} - 6 q^{40} - 10 q^{43} - 22 q^{48}+ \cdots + 30 q^{95}+O(q^{100}) \) 4 * q - 2 * q^3 + 2 * q^4 + 10 * q^9 - 10 * q^10 + 20 * q^12 + 2 * q^16 + 12 * q^17 + 2 * q^22 - 12 * q^23 - 10 * q^25 - 20 * q^27 - 18 * q^29 - 16 * q^30 - 16 * q^36 - 30 * q^38 - 6 * q^40 - 10 * q^43 - 22 * q^48 - 6 * q^51 - 12 * q^53 - 2 * q^55 + 4 * q^61 + 30 * q^62 - 32 * q^64 + 20 * q^66 + 6 * q^68 - 36 * q^69 - 24 * q^74 + 26 * q^75 - 24 * q^79 - 20 * q^81 - 4 * q^82 + 30 * q^87 + 18 * q^88 - 4 * q^90 - 48 * q^92 + 10 * q^94 + 30 * q^95

Introduction

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