LMFDB - Embedded newform 8281.2.a.bq.1.4

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			1.4
Root			$-0.456850$ 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)

$f(q)$	$=$	$q+2.18890 q^{2} +1.79129 q^{3} +2.79129 q^{4} -2.18890 q^{5} +3.92095 q^{6} +1.73205 q^{8} +0.208712 q^{9} +O(q^{10})$	\(q+2.18890 q^{2} +1.79129 q^{3} +2.79129 q^{4} -2.18890 q^{5} +3.92095 q^{6} +1.73205 q^{8} +0.208712 q^{9} -4.79129 q^{10} +1.27520 q^{11} +5.00000 q^{12} -3.92095 q^{15} -1.79129 q^{16} +3.00000 q^{17} +0.456850 q^{18} -6.56670 q^{19} -6.10985 q^{20} +2.79129 q^{22} -7.58258 q^{23} +3.10260 q^{24} -0.208712 q^{25} -5.00000 q^{27} -2.20871 q^{29} -8.58258 q^{30} +8.66025 q^{31} -7.38505 q^{32} +2.28425 q^{33} +6.56670 q^{34} +0.582576 q^{36} -6.92820 q^{37} -14.3739 q^{38} -3.79129 q^{40} -2.55040 q^{41} +4.37386 q^{43} +3.55945 q^{44} -0.456850 q^{45} -16.5975 q^{46} +4.28245 q^{47} -3.20871 q^{48} -0.456850 q^{50} +5.37386 q^{51} -12.1652 q^{53} -10.9445 q^{54} -2.79129 q^{55} -11.7629 q^{57} -4.83465 q^{58} +8.85095 q^{59} -10.9445 q^{60} -12.7477 q^{61} +18.9564 q^{62} -12.5826 q^{64} +5.00000 q^{66} +11.4014 q^{67} +8.37386 q^{68} -13.5826 q^{69} +0.913701 q^{71} +0.361500 q^{72} +3.46410 q^{73} -15.1652 q^{74} -0.373864 q^{75} -18.3296 q^{76} -6.00000 q^{79} +3.92095 q^{80} -9.58258 q^{81} -5.58258 q^{82} +3.55945 q^{83} -6.56670 q^{85} +9.57395 q^{86} -3.95644 q^{87} +2.20871 q^{88} -2.91190 q^{89} -1.00000 q^{90} -21.1652 q^{92} +15.5130 q^{93} +9.37386 q^{94} +14.3739 q^{95} -13.2288 q^{96} -15.2270 q^{97} +0.266150 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{4} + 10 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 2 * q^4 + 10 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.18890	1.54779	0.773893	−	0.633316i	$-0.218307\pi$
$2$	2.18890	1.54779	0.773893	+	0.633316i	$0.218307\pi$
$3$	1.79129	1.03420	0.517100	−	0.855925i	$-0.327011\pi$
$3$	1.79129	1.03420	0.517100	+	0.855925i	$0.327011\pi$
$4$	2.79129	1.39564
$5$	−2.18890	−0.978906	−0.489453	−	0.872030i	$-0.662804\pi$
$5$	−2.18890	−0.978906	−0.489453	+	0.872030i	$0.662804\pi$
$6$	3.92095	1.60072
$7$	0	0
$7$	0	0
$8$	1.73205	0.612372
$9$	0.208712	0.0695707
$10$	−4.79129	−1.51514
$11$	1.27520	0.384487	0.192244	−	0.981347i	$-0.438424\pi$
$11$	1.27520	0.384487	0.192244	+	0.981347i	$0.438424\pi$
$12$	5.00000	1.44338
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.92095	−1.01239
$16$	−1.79129	−0.447822
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0.456850	0.107681
$19$	−6.56670	−1.50651	−0.753253	−	0.657731i	$-0.771516\pi$
$19$	−6.56670	−1.50651	−0.753253	+	0.657731i	$0.771516\pi$
$20$	−6.10985	−1.36620
$21$	0	0
$22$	2.79129	0.595105
$23$	−7.58258	−1.58108	−0.790538	−	0.612413i	$-0.790199\pi$
$23$	−7.58258	−1.58108	−0.790538	+	0.612413i	$0.790199\pi$
$24$	3.10260	0.633316
$25$	−0.208712	−0.0417424
$26$	0	0
$27$	−5.00000	−0.962250
$28$	0	0
$29$	−2.20871	−0.410148	−0.205074	−	0.978747i	$-0.565743\pi$
$29$	−2.20871	−0.410148	−0.205074	+	0.978747i	$0.565743\pi$
$30$	−8.58258	−1.56696
$31$	8.66025	1.55543	0.777714	−	0.628619i	$-0.216379\pi$
$31$	8.66025	1.55543	0.777714	+	0.628619i	$0.216379\pi$
$32$	−7.38505	−1.30551
$33$	2.28425	0.397637
$34$	6.56670	1.12618
$35$	0	0
$36$	0.582576	0.0970959
$37$	−6.92820	−1.13899	−0.569495	−	0.821995i	$-0.692861\pi$
$37$	−6.92820	−1.13899	−0.569495	+	0.821995i	$0.692861\pi$
$38$	−14.3739	−2.33175
$39$	0	0
$40$	−3.79129	−0.599455
$41$	−2.55040	−0.398306	−0.199153	−	0.979968i	$-0.563819\pi$
$41$	−2.55040	−0.398306	−0.199153	+	0.979968i	$0.563819\pi$
$42$	0	0
$43$	4.37386	0.667008	0.333504	−	0.942749i	$-0.391769\pi$
$43$	4.37386	0.667008	0.333504	+	0.942749i	$0.391769\pi$
$44$	3.55945	0.536608
$45$	−0.456850	−0.0681032
$46$	−16.5975	−2.44717
$47$	4.28245	0.624660	0.312330	−	0.949974i	$-0.398891\pi$
$47$	4.28245	0.624660	0.312330	+	0.949974i	$0.398891\pi$
$48$	−3.20871	−0.463138
$49$	0	0
$50$	−0.456850	−0.0646084
$51$	5.37386	0.752491
$52$	0	0
$53$	−12.1652	−1.67101	−0.835506	−	0.549481i	$-0.814825\pi$
$53$	−12.1652	−1.67101	−0.835506	+	0.549481i	$0.814825\pi$
$54$	−10.9445	−1.48936
$55$	−2.79129	−0.376377
$56$	0	0
$57$	−11.7629	−1.55803
$58$	−4.83465	−0.634821
$59$	8.85095	1.15230	0.576148	−	0.817345i	$-0.304555\pi$
$59$	8.85095	1.15230	0.576148	+	0.817345i	$0.304555\pi$
$60$	−10.9445	−1.41293
$61$	−12.7477	−1.63218	−0.816090	−	0.577925i	$-0.803862\pi$
$61$	−12.7477	−1.63218	−0.816090	+	0.577925i	$0.803862\pi$
$62$	18.9564	2.40747
$63$	0	0
$64$	−12.5826	−1.57282
$65$	0	0
$66$	5.00000	0.615457
$67$	11.4014	1.39290	0.696449	−	0.717607i	$-0.254762\pi$
$67$	11.4014	1.39290	0.696449	+	0.717607i	$0.254762\pi$
$68$	8.37386	1.01548
$69$	−13.5826	−1.63515
$70$	0	0
$71$	0.913701	0.108436	0.0542181	−	0.998529i	$-0.482733\pi$
$71$	0.913701	0.108436	0.0542181	+	0.998529i	$0.482733\pi$
$72$	0.361500	0.0426032
$73$	3.46410	0.405442	0.202721	−	0.979236i	$-0.435021\pi$
$73$	3.46410	0.405442	0.202721	+	0.979236i	$0.435021\pi$
$74$	−15.1652	−1.76291
$75$	−0.373864	−0.0431700
$76$	−18.3296	−2.10254
$77$	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	$0.609590\pi$
$80$	3.92095	0.438376
$81$	−9.58258	−1.06473
$82$	−5.58258	−0.616492
$83$	3.55945	0.390701	0.195350	−	0.980734i	$-0.437416\pi$
$83$	3.55945	0.390701	0.195350	+	0.980734i	$0.437416\pi$
$84$	0	0
$85$	−6.56670	−0.712259
$86$	9.57395	1.03239
$87$	−3.95644	−0.424175
$88$	2.20871	0.235450
$89$	−2.91190	−0.308661	−0.154330	−	0.988019i	$-0.549322\pi$
$89$	−2.91190	−0.308661	−0.154330	+	0.988019i	$0.549322\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	−21.1652	−2.20662
$93$	15.5130	1.60862
$94$	9.37386	0.966840
$95$	14.3739	1.47473
$96$	−13.2288	−1.35015
$97$	−15.2270	−1.54606	−0.773032	−	0.634367i	$-0.781261\pi$
$97$	−15.2270	−1.54606	−0.773032	+	0.634367i	$0.781261\pi$
$98$	0	0
$99$	0.266150	0.0267491
$100$	−0.582576	−0.0582576
$101$	9.79129	0.974270	0.487135	−	0.873327i	$-0.338042\pi$
$101$	9.79129	0.974270	0.487135	+	0.873327i	$0.338042\pi$
$102$	11.7629	1.16470
$103$	−4.58258	−0.451535	−0.225767	−	0.974181i	$-0.572489\pi$
$103$	−4.58258	−0.451535	−0.225767	+	0.974181i	$0.572489\pi$
$104$	0	0
$105$	0	0
$106$	−26.6283	−2.58637
$107$	−9.79129	−0.946560	−0.473280	−	0.880912i	$-0.656930\pi$
$107$	−9.79129	−0.946560	−0.473280	+	0.880912i	$0.656930\pi$
$108$	−13.9564	−1.34296
$109$	−7.93725	−0.760251	−0.380126	−	0.924935i	$-0.624119\pi$
$109$	−7.93725	−0.760251	−0.380126	+	0.924935i	$0.624119\pi$
$110$	−6.10985	−0.582552
$111$	−12.4104	−1.17794
$112$	0	0
$113$	1.41742	0.133340	0.0666700	−	0.997775i	$-0.478763\pi$
$113$	1.41742	0.133340	0.0666700	+	0.997775i	$0.478763\pi$
$114$	−25.7477	−2.41150
$115$	16.5975	1.54773
$116$	−6.16515	−0.572420
$117$	0	0
$118$	19.3739	1.78351
$119$	0	0
$120$	−6.79129	−0.619957
$121$	−9.37386	−0.852169
$122$	−27.9035	−2.52627
$123$	−4.56850	−0.411928
$124$	24.1733	2.17082
$125$	11.4014	1.01977
$126$	0	0
$127$	15.9564	1.41591	0.707953	−	0.706260i	$-0.249619\pi$
$127$	15.9564	1.41591	0.707953	+	0.706260i	$0.249619\pi$
$128$	−12.7719	−1.12889
$129$	7.83485	0.689820
$130$	0	0
$131$	3.62614	0.316817	0.158409	−	0.987374i	$-0.449364\pi$
$131$	3.62614	0.316817	0.158409	+	0.987374i	$0.449364\pi$
$132$	6.37600	0.554960
$133$	0	0
$134$	24.9564	2.15591
$135$	10.9445	0.941953
$136$	5.19615	0.445566
$137$	−17.1497	−1.46520	−0.732599	−	0.680660i	$-0.761693\pi$
$137$	−17.1497	−1.46520	−0.732599	+	0.680660i	$0.761693\pi$
$138$	−29.7309	−2.53086
$139$	0.791288	0.0671162	0.0335581	−	0.999437i	$-0.489316\pi$
$139$	0.791288	0.0671162	0.0335581	+	0.999437i	$0.489316\pi$
$140$	0	0
$141$	7.67110	0.646024
$142$	2.00000	0.167836
$143$	0	0
$144$	−0.373864	−0.0311553
$145$	4.83465	0.401496
$146$	7.58258	0.627538
$147$	0	0
$148$	−19.3386	−1.58962
$149$	−2.18890	−0.179322	−0.0896609	−	0.995972i	$-0.528578\pi$
$149$	−2.18890	−0.179322	−0.0896609	+	0.995972i	$0.528578\pi$
$150$	−0.818350	−0.0668180
$151$	−12.1244	−0.986666	−0.493333	−	0.869841i	$-0.664222\pi$
$151$	−12.1244	−0.986666	−0.493333	+	0.869841i	$0.664222\pi$
$152$	−11.3739	−0.922542
$153$	0.626136	0.0506201
$154$	0	0
$155$	−18.9564	−1.52262
$156$	0	0
$157$	−21.9564	−1.75231	−0.876157	−	0.482025i	$-0.839901\pi$
$157$	−21.9564	−1.75231	−0.876157	+	0.482025i	$0.839901\pi$
$158$	−13.1334	−1.04484
$159$	−21.7913	−1.72816
$160$	16.1652	1.27797
$161$	0	0
$162$	−20.9753	−1.64798
$163$	6.92820	0.542659	0.271329	−	0.962487i	$-0.412537\pi$
$163$	6.92820	0.542659	0.271329	+	0.962487i	$0.412537\pi$
$164$	−7.11890	−0.555893
$165$	−5.00000	−0.389249
$166$	7.79129	0.604721
$167$	19.9663	1.54504	0.772518	−	0.634993i	$-0.218997\pi$
$167$	19.9663	1.54504	0.772518	+	0.634993i	$0.218997\pi$
$168$	0	0
$169$	0	0
$170$	−14.3739	−1.10243
$171$	−1.37055	−0.104809
$172$	12.2087	0.930906
$173$	−7.74773	−0.589049	−0.294524	−	0.955644i	$-0.595161\pi$
$173$	−7.74773	−0.589049	−0.294524	+	0.955644i	$0.595161\pi$
$174$	−8.66025	−0.656532
$175$	0	0
$176$	−2.28425	−0.172182
$177$	15.8546	1.19171
$178$	−6.37386	−0.477741
$179$	9.00000	0.672692	0.336346	−	0.941739i	$-0.390809\pi$
$179$	9.00000	0.672692	0.336346	+	0.941739i	$0.390809\pi$
$180$	−1.27520	−0.0950478
$181$	9.16515	0.681240	0.340620	−	0.940201i	$-0.389363\pi$
$181$	9.16515	0.681240	0.340620	+	0.940201i	$0.389363\pi$
$182$	0	0
$183$	−22.8348	−1.68800
$184$	−13.1334	−0.968208
$185$	15.1652	1.11496
$186$	33.9564	2.48981
$187$	3.82560	0.279756
$188$	11.9536	0.871803
$189$	0	0
$190$	31.4630	2.28256
$191$	−0.626136	−0.0453056	−0.0226528	−	0.999743i	$-0.507211\pi$
$191$	−0.626136	−0.0453056	−0.0226528	+	0.999743i	$0.507211\pi$
$192$	−22.5390	−1.62661
$193$	12.4104	0.893321	0.446660	−	0.894704i	$-0.352613\pi$
$193$	12.4104	0.893321	0.446660	+	0.894704i	$0.352613\pi$
$194$	−33.3303	−2.39298
$195$	0	0
$196$	0	0
$197$	−10.9445	−0.779764	−0.389882	−	0.920865i	$-0.627484\pi$
$197$	−10.9445	−0.779764	−0.389882	+	0.920865i	$0.627484\pi$
$198$	0.582576	0.0414019
$199$	−11.0000	−0.779769	−0.389885	−	0.920864i	$-0.627485\pi$
$199$	−11.0000	−0.779769	−0.389885	+	0.920864i	$0.627485\pi$
$200$	−0.361500	−0.0255619
$201$	20.4231	1.44054
$202$	21.4322	1.50796
$203$	0	0
$204$	15.0000	1.05021
$205$	5.58258	0.389904
$206$	−10.0308	−0.698879
$207$	−1.58258	−0.109997
$208$	0	0
$209$	−8.37386	−0.579232
$210$	0	0
$211$	1.41742	0.0975795	0.0487898	−	0.998809i	$-0.484464\pi$
$211$	1.41742	0.0975795	0.0487898	+	0.998809i	$0.484464\pi$
$212$	−33.9564	−2.33214
$213$	1.63670	0.112145
$214$	−21.4322	−1.46507
$215$	−9.57395	−0.652938
$216$	−8.66025	−0.589256
$217$	0	0
$218$	−17.3739	−1.17671
$219$	6.20520	0.419309
$220$	−7.79129	−0.525289
$221$	0	0
$222$	−27.1652	−1.82321
$223$	20.7092	1.38679	0.693394	−	0.720559i	$-0.256115\pi$
$223$	20.7092	1.38679	0.693394	+	0.720559i	$0.256115\pi$
$224$	0	0
$225$	−0.0435608	−0.00290405
$226$	3.10260	0.206382
$227$	12.3151	0.817379	0.408689	−	0.912673i	$-0.365986\pi$
$227$	12.3151	0.817379	0.408689	+	0.912673i	$0.365986\pi$
$228$	−32.8335	−2.17445
$229$	6.92820	0.457829	0.228914	−	0.973447i	$-0.426482\pi$
$229$	6.92820	0.457829	0.228914	+	0.973447i	$0.426482\pi$
$230$	36.3303	2.39555
$231$	0	0
$232$	−3.82560	−0.251163
$233$	6.95644	0.455731	0.227866	−	0.973693i	$-0.426825\pi$
$233$	6.95644	0.455731	0.227866	+	0.973693i	$0.426825\pi$
$234$	0	0
$235$	−9.37386	−0.611483
$236$	24.7056	1.60820
$237$	−10.7477	−0.698140
$238$	0	0
$239$	13.2288	0.855697	0.427849	−	0.903850i	$-0.359272\pi$
$239$	13.2288	0.855697	0.427849	+	0.903850i	$0.359272\pi$
$240$	7.02355	0.453368
$241$	4.11165	0.264855	0.132427	−	0.991193i	$-0.457723\pi$
$241$	4.11165	0.264855	0.132427	+	0.991193i	$0.457723\pi$
$242$	−20.5185	−1.31898
$243$	−2.16515	−0.138895
$244$	−35.5826	−2.27794
$245$	0	0
$246$	−10.0000	−0.637577
$247$	0	0
$248$	15.0000	0.952501
$249$	6.37600	0.404063
$250$	24.9564	1.57838
$251$	21.1652	1.33593	0.667966	−	0.744192i	$-0.267165\pi$
$251$	21.1652	1.33593	0.667966	+	0.744192i	$0.267165\pi$
$252$	0	0
$253$	−9.66930	−0.607904
$254$	34.9271	2.19152
$255$	−11.7629	−0.736619
$256$	−2.79129	−0.174455
$257$	−27.9564	−1.74387	−0.871937	−	0.489617i	$-0.837136\pi$
$257$	−27.9564	−1.74387	−0.871937	+	0.489617i	$0.837136\pi$
$258$	17.1497	1.06769
$259$	0	0
$260$	0	0
$261$	−0.460985	−0.0285343
$262$	7.93725	0.490365
$263$	−27.3303	−1.68526	−0.842629	−	0.538494i	$-0.818993\pi$
$263$	−27.3303	−1.68526	−0.842629	+	0.538494i	$0.818993\pi$
$264$	3.95644	0.243502
$265$	26.6283	1.63576
$266$	0	0
$267$	−5.21605	−0.319217
$268$	31.8245	1.94399
$269$	−11.2087	−0.683407	−0.341704	−	0.939808i	$-0.611004\pi$
$269$	−11.2087	−0.683407	−0.341704	+	0.939808i	$0.611004\pi$
$270$	23.9564	1.45794
$271$	28.7219	1.74473	0.872364	−	0.488856i	$-0.162586\pi$
$271$	28.7219	1.74473	0.872364	+	0.488856i	$0.162586\pi$
$272$	−5.37386	−0.325838
$273$	0	0
$274$	−37.5390	−2.26781
$275$	−0.266150	−0.0160494
$276$	−37.9129	−2.28209
$277$	15.7477	0.946189	0.473095	−	0.881012i	$-0.343137\pi$
$277$	15.7477	0.946189	0.473095	+	0.881012i	$0.343137\pi$
$278$	1.73205	0.103882
$279$	1.80750	0.108212
$280$	0	0
$281$	−6.39590	−0.381548	−0.190774	−	0.981634i	$-0.561100\pi$
$281$	−6.39590	−0.381548	−0.190774	+	0.981634i	$0.561100\pi$
$282$	16.7913	0.999907
$283$	24.7477	1.47110	0.735550	−	0.677471i	$-0.236924\pi$
$283$	24.7477	1.47110	0.735550	+	0.677471i	$0.236924\pi$
$284$	2.55040	0.151338
$285$	25.7477	1.52516
$286$	0	0
$287$	0	0
$288$	−1.54135	−0.0908249
$289$	−8.00000	−0.470588
$290$	10.5826	0.621430
$291$	−27.2759	−1.59894
$292$	9.66930	0.565853
$293$	7.84190	0.458129	0.229064	−	0.973411i	$-0.426433\pi$
$293$	7.84190	0.458129	0.229064	+	0.973411i	$0.426433\pi$
$294$	0	0
$295$	−19.3739	−1.12799
$296$	−12.0000	−0.697486
$297$	−6.37600	−0.369973
$298$	−4.79129	−0.277552
$299$	0	0
$300$	−1.04356	−0.0602500
$301$	0	0
$302$	−26.5390	−1.52715
$303$	17.5390	1.00759
$304$	11.7629	0.674646
$305$	27.9035	1.59775
$306$	1.37055	0.0783492
$307$	−24.1733	−1.37964	−0.689820	−	0.723980i	$-0.742311\pi$
$307$	−24.1733	−1.37964	−0.689820	+	0.723980i	$0.742311\pi$
$308$	0	0
$309$	−8.20871	−0.466977
$310$	−41.4938	−2.35669
$311$	5.53901	0.314089	0.157044	−	0.987592i	$-0.449803\pi$
$311$	5.53901	0.314089	0.157044	+	0.987592i	$0.449803\pi$
$312$	0	0
$313$	20.7477	1.17273	0.586365	−	0.810047i	$-0.300558\pi$
$313$	20.7477	1.17273	0.586365	+	0.810047i	$0.300558\pi$
$314$	−48.0605	−2.71221
$315$	0	0
$316$	−16.7477	−0.942133
$317$	−18.5203	−1.04020	−0.520101	−	0.854105i	$-0.674106\pi$
$317$	−18.5203	−1.04020	−0.520101	+	0.854105i	$0.674106\pi$
$318$	−47.6990	−2.67483
$319$	−2.81655	−0.157697
$320$	27.5420	1.53965
$321$	−17.5390	−0.978932
$322$	0	0
$323$	−19.7001	−1.09614
$324$	−26.7477	−1.48598
$325$	0	0
$326$	15.1652	0.839920
$327$	−14.2179	−0.786252
$328$	−4.41742	−0.243911
$329$	0	0
$330$	−10.9445	−0.602475
$331$	−1.08450	−0.0596095	−0.0298048	−	0.999556i	$-0.509489\pi$
$331$	−1.08450	−0.0596095	−0.0298048	+	0.999556i	$0.509489\pi$
$332$	9.93545	0.545279
$333$	−1.44600	−0.0792403
$334$	43.7042	2.39139
$335$	−24.9564	−1.36352
$336$	0	0
$337$	12.9564	0.705782	0.352891	−	0.935664i	$-0.385199\pi$
$337$	12.9564	0.705782	0.352891	+	0.935664i	$0.385199\pi$
$338$	0	0
$339$	2.53901	0.137900
$340$	−18.3296	−0.994060
$341$	11.0436	0.598042
$342$	−3.00000	−0.162221
$343$	0	0
$344$	7.57575	0.408457
$345$	29.7309	1.60066
$346$	−16.9590	−0.911722
$347$	−4.41742	−0.237140	−0.118570	−	0.992946i	$-0.537831\pi$
$347$	−4.41742	−0.237140	−0.118570	+	0.992946i	$0.537831\pi$
$348$	−11.0436	−0.591997
$349$	−10.6784	−0.571599	−0.285800	−	0.958289i	$-0.592259\pi$
$349$	−10.6784	−0.571599	−0.285800	+	0.958289i	$0.592259\pi$
$350$	0	0
$351$	0	0
$352$	−9.41742	−0.501950
$353$	−26.8190	−1.42743	−0.713716	−	0.700435i	$-0.752989\pi$
$353$	−26.8190	−1.42743	−0.713716	+	0.700435i	$0.752989\pi$
$354$	34.7042	1.84451
$355$	−2.00000	−0.106149
$356$	−8.12795	−0.430781
$357$	0	0
$358$	19.7001	1.04118
$359$	12.6766	0.669043	0.334522	−	0.942388i	$-0.391425\pi$
$359$	12.6766	0.669043	0.334522	+	0.942388i	$0.391425\pi$
$360$	−0.791288	−0.0417045
$361$	24.1216	1.26956
$362$	20.0616	1.05441
$363$	−16.7913	−0.881314
$364$	0	0
$365$	−7.58258	−0.396890
$366$	−49.9832	−2.61266
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	13.5826	0.708041
$369$	−0.532300	−0.0277104
$370$	33.1950	1.72573
$371$	0	0
$372$	43.3013	2.24507
$373$	36.7913	1.90498	0.952490	−	0.304569i	$-0.0985124\pi$
$373$	36.7913	1.90498	0.952490	+	0.304569i	$0.0985124\pi$
$374$	8.37386	0.433002
$375$	20.4231	1.05464
$376$	7.41742	0.382524
$377$	0	0
$378$	0	0
$379$	−4.54860	−0.233646	−0.116823	−	0.993153i	$-0.537271\pi$
$379$	−4.54860	−0.233646	−0.116823	+	0.993153i	$0.537271\pi$
$380$	40.1216	2.05819
$381$	28.5826	1.46433
$382$	−1.37055	−0.0701235
$383$	−3.92095	−0.200351	−0.100176	−	0.994970i	$-0.531940\pi$
$383$	−3.92095	−0.200351	−0.100176	+	0.994970i	$0.531940\pi$
$384$	−22.8782	−1.16750
$385$	0	0
$386$	27.1652	1.38267
$387$	0.912878	0.0464042
$388$	−42.5028	−2.15775
$389$	−36.3303	−1.84202	−0.921010	−	0.389540i	$-0.872634\pi$
$389$	−36.3303	−1.84202	−0.921010	+	0.389540i	$0.872634\pi$
$390$	0	0
$391$	−22.7477	−1.15040
$392$	0	0
$393$	6.49545	0.327652
$394$	−23.9564	−1.20691
$395$	13.1334	0.660813
$396$	0.742901	0.0373322
$397$	−15.1515	−0.760432	−0.380216	−	0.924898i	$-0.624150\pi$
$397$	−15.1515	−0.760432	−0.380216	+	0.924898i	$0.624150\pi$
$398$	−24.0779	−1.20692
$399$	0	0
$400$	0.373864	0.0186932
$401$	29.5601	1.47616	0.738081	−	0.674712i	$-0.235732\pi$
$401$	29.5601	1.47616	0.738081	+	0.674712i	$0.235732\pi$
$402$	44.7042	2.22964
$403$	0	0
$404$	27.3303	1.35973
$405$	20.9753	1.04227
$406$	0	0
$407$	−8.83485	−0.437927
$408$	9.30780	0.460805
$409$	−0.361500	−0.0178750	−0.00893751	−	0.999960i	$-0.502845\pi$
$409$	−0.361500	−0.0178750	−0.00893751	+	0.999960i	$0.502845\pi$
$410$	12.2197	0.603488
$411$	−30.7201	−1.51531
$412$	−12.7913	−0.630182
$413$	0	0
$414$	−3.46410	−0.170251
$415$	−7.79129	−0.382459
$416$	0	0
$417$	1.41742	0.0694116
$418$	−18.3296	−0.896528
$419$	25.7477	1.25786	0.628929	−	0.777462i	$-0.283493\pi$
$419$	25.7477	1.25786	0.628929	+	0.777462i	$0.283493\pi$
$420$	0	0
$421$	−20.0616	−0.977743	−0.488872	−	0.872356i	$-0.662591\pi$
$421$	−20.0616	−0.977743	−0.488872	+	0.872356i	$0.662591\pi$
$422$	3.10260	0.151032
$423$	0.893800	0.0434580
$424$	−21.0707	−1.02328
$425$	−0.626136	−0.0303721
$426$	3.58258	0.173576
$427$	0	0
$428$	−27.3303	−1.32106
$429$	0	0
$430$	−20.9564	−1.01061
$431$	−15.6084	−0.751828	−0.375914	−	0.926655i	$-0.622671\pi$
$431$	−15.6084	−0.751828	−0.375914	+	0.926655i	$0.622671\pi$
$432$	8.95644	0.430917
$433$	−22.4955	−1.08106	−0.540531	−	0.841324i	$-0.681777\pi$
$433$	−22.4955	−1.08106	−0.540531	+	0.841324i	$0.681777\pi$
$434$	0	0
$435$	8.66025	0.415227
$436$	−22.1552	−1.06104
$437$	49.7925	2.38190
$438$	13.5826	0.649001
$439$	−11.5390	−0.550727	−0.275364	−	0.961340i	$-0.588798\pi$
$439$	−11.5390	−0.550727	−0.275364	+	0.961340i	$0.588798\pi$
$440$	−4.83465	−0.230483
$441$	0	0
$442$	0	0
$443$	3.16515	0.150381	0.0751904	−	0.997169i	$-0.476044\pi$
$443$	3.16515	0.150381	0.0751904	+	0.997169i	$0.476044\pi$
$444$	−34.6410	−1.64399
$445$	6.37386	0.302150
$446$	45.3303	2.14645
$447$	−3.92095	−0.185455
$448$	0	0
$449$	−19.8709	−0.937766	−0.468883	−	0.883260i	$-0.655343\pi$
$449$	−19.8709	−0.937766	−0.468883	+	0.883260i	$0.655343\pi$
$450$	−0.0953502	−0.00449485
$451$	−3.25227	−0.153144
$452$	3.95644	0.186095
$453$	−21.7182	−1.02041
$454$	26.9564	1.26513
$455$	0	0
$456$	−20.3739	−0.954094
$457$	−8.94630	−0.418490	−0.209245	−	0.977863i	$-0.567101\pi$
$457$	−8.94630	−0.418490	−0.209245	+	0.977863i	$0.567101\pi$
$458$	15.1652	0.708621
$459$	−15.0000	−0.700140
$460$	46.3284	2.16007
$461$	−17.8727	−0.832415	−0.416208	−	0.909270i	$-0.636641\pi$
$461$	−17.8727	−0.832415	−0.416208	+	0.909270i	$0.636641\pi$
$462$	0	0
$463$	−7.93725	−0.368875	−0.184438	−	0.982844i	$-0.559046\pi$
$463$	−7.93725	−0.368875	−0.184438	+	0.982844i	$0.559046\pi$
$464$	3.95644	0.183673
$465$	−33.9564	−1.57469
$466$	15.2270	0.705375
$467$	−11.8348	−0.547651	−0.273826	−	0.961779i	$-0.588289\pi$
$467$	−11.8348	−0.547651	−0.273826	+	0.961779i	$0.588289\pi$
$468$	0	0
$469$	0	0
$470$	−20.5185	−0.946446
$471$	−39.3303	−1.81224
$472$	15.3303	0.705634
$473$	5.57755	0.256456
$474$	−23.5257	−1.08057
$475$	1.37055	0.0628852
$476$	0	0
$477$	−2.53901	−0.116254
$478$	28.9564	1.32444
$479$	−10.2215	−0.467032	−0.233516	−	0.972353i	$-0.575023\pi$
$479$	−10.2215	−0.467032	−0.233516	+	0.972353i	$0.575023\pi$
$480$	28.9564	1.32167
$481$	0	0
$482$	9.00000	0.409939
$483$	0	0
$484$	−26.1652	−1.18933
$485$	33.3303	1.51345
$486$	−4.73930	−0.214979
$487$	10.3169	0.467501	0.233751	−	0.972297i	$-0.424900\pi$
$487$	10.3169	0.467501	0.233751	+	0.972297i	$0.424900\pi$
$488$	−22.0797	−0.999502
$489$	12.4104	0.561218
$490$	0	0
$491$	−37.1216	−1.67527	−0.837637	−	0.546227i	$-0.816063\pi$
$491$	−37.1216	−1.67527	−0.837637	+	0.546227i	$0.816063\pi$
$492$	−12.7520	−0.574905
$493$	−6.62614	−0.298426
$494$	0	0
$495$	−0.582576	−0.0261848
$496$	−15.5130	−0.696555
$497$	0	0
$498$	13.9564	0.625403
$499$	−42.2168	−1.88988	−0.944941	−	0.327240i	$-0.893881\pi$
$499$	−42.2168	−1.88988	−0.944941	+	0.327240i	$0.893881\pi$
$500$	31.8245	1.42323
$501$	35.7653	1.59788
$502$	46.3284	2.06774
$503$	−22.1216	−0.986353	−0.493176	−	0.869929i	$-0.664164\pi$
$503$	−22.1216	−0.986353	−0.493176	+	0.869929i	$0.664164\pi$
$504$	0	0
$505$	−21.4322	−0.953719
$506$	−21.1652	−0.940906
$507$	0	0
$508$	44.5390	1.97610
$509$	21.9844	0.974440	0.487220	−	0.873279i	$-0.338011\pi$
$509$	21.9844	0.974440	0.487220	+	0.873279i	$0.338011\pi$
$510$	−25.7477	−1.14013
$511$	0	0
$512$	19.4340	0.858868
$513$	32.8335	1.44964
$514$	−61.1939	−2.69915
$515$	10.0308	0.442010
$516$	21.8693	0.962743
$517$	5.46099	0.240174
$518$	0	0
$519$	−13.8784	−0.609195
$520$	0	0
$521$	25.5826	1.12079	0.560396	−	0.828224i	$-0.310649\pi$
$521$	25.5826	1.12079	0.560396	+	0.828224i	$0.310649\pi$
$522$	−1.00905	−0.0441649
$523$	−12.3303	−0.539166	−0.269583	−	0.962977i	$-0.586886\pi$
$523$	−12.3303	−0.539166	−0.269583	+	0.962977i	$0.586886\pi$
$524$	10.1216	0.442164
$525$	0	0
$526$	−59.8233	−2.60842
$527$	25.9808	1.13174
$528$	−4.09175	−0.178071
$529$	34.4955	1.49980
$530$	58.2867	2.53181
$531$	1.84730	0.0801661
$532$	0	0
$533$	0	0
$534$	−11.4174	−0.494080
$535$	21.4322	0.926593
$536$	19.7477	0.852972
$537$	16.1216	0.695698
$538$	−24.5348	−1.05777
$539$	0	0
$540$	30.5493	1.31463
$541$	30.0924	1.29377	0.646887	−	0.762586i	$-0.276071\pi$
$541$	30.0924	1.29377	0.646887	+	0.762586i	$0.276071\pi$
$542$	62.8693	2.70047
$543$	16.4174	0.704539
$544$	−22.1552	−0.949895
$545$	17.3739	0.744215
$546$	0	0
$547$	15.7477	0.673324	0.336662	−	0.941626i	$-0.390702\pi$
$547$	15.7477	0.673324	0.336662	+	0.941626i	$0.390702\pi$
$548$	−47.8698	−2.04490
$549$	−2.66061	−0.113552
$550$	−0.582576	−0.0248411
$551$	14.5040	0.617889
$552$	−23.5257	−1.00132
$553$	0	0
$554$	34.4702	1.46450
$555$	27.1652	1.15310
$556$	2.20871	0.0936703
$557$	−27.8281	−1.17911	−0.589556	−	0.807727i	$-0.700697\pi$
$557$	−27.8281	−1.17911	−0.589556	+	0.807727i	$0.700697\pi$
$558$	3.95644	0.167489
$559$	0	0
$560$	0	0
$561$	6.85275	0.289323
$562$	−14.0000	−0.590554
$563$	0.330303	0.0139206	0.00696030	−	0.999976i	$-0.497784\pi$
$563$	0.330303	0.0139206	0.00696030	+	0.999976i	$0.497784\pi$
$564$	21.4123	0.901619
$565$	−3.10260	−0.130527
$566$	54.1703	2.27695
$567$	0	0
$568$	1.58258	0.0664034
$569$	10.7477	0.450568	0.225284	−	0.974293i	$-0.427669\pi$
$569$	10.7477	0.450568	0.225284	+	0.974293i	$0.427669\pi$
$570$	56.3592	2.36063
$571$	24.9564	1.04439	0.522197	−	0.852825i	$-0.325112\pi$
$571$	24.9564	1.04439	0.522197	+	0.852825i	$0.325112\pi$
$572$	0	0
$573$	−1.12159	−0.0468551
$574$	0	0
$575$	1.58258	0.0659980
$576$	−2.62614	−0.109422
$577$	−19.7756	−0.823267	−0.411634	−	0.911349i	$-0.635042\pi$
$577$	−19.7756	−0.823267	−0.411634	+	0.911349i	$0.635042\pi$
$578$	−17.5112	−0.728370
$579$	22.2306	0.923873
$580$	13.4949	0.560345
$581$	0	0
$582$	−59.7042	−2.47482
$583$	−15.5130	−0.642483
$584$	6.00000	0.248282
$585$	0	0
$586$	17.1652	0.709086
$587$	35.4793	1.46439	0.732193	−	0.681097i	$-0.238497\pi$
$587$	35.4793	1.46439	0.732193	+	0.681097i	$0.238497\pi$
$588$	0	0
$589$	−56.8693	−2.34326
$590$	−42.4075	−1.74589
$591$	−19.6048	−0.806432
$592$	12.4104	0.510065
$593$	19.6048	0.805071	0.402535	−	0.915404i	$-0.368129\pi$
$593$	19.6048	0.805071	0.402535	+	0.915404i	$0.368129\pi$
$594$	−13.9564	−0.572640
$595$	0	0
$596$	−6.10985	−0.250269
$597$	−19.7042	−0.806438
$598$	0	0
$599$	−20.3739	−0.832453	−0.416227	−	0.909261i	$-0.636648\pi$
$599$	−20.3739	−0.832453	−0.416227	+	0.909261i	$0.636648\pi$
$600$	−0.647551	−0.0264361
$601$	1.37386	0.0560411	0.0280205	−	0.999607i	$-0.491080\pi$
$601$	1.37386	0.0560411	0.0280205	+	0.999607i	$0.491080\pi$
$602$	0	0
$603$	2.37960	0.0969049
$604$	−33.8426	−1.37703
$605$	20.5185	0.834194
$606$	38.3912	1.55953
$607$	7.74773	0.314471	0.157235	−	0.987561i	$-0.449742\pi$
$607$	7.74773	0.314471	0.157235	+	0.987561i	$0.449742\pi$
$608$	48.4955	1.96675
$609$	0	0
$610$	61.0780	2.47298
$611$	0	0
$612$	1.74773	0.0706477
$613$	37.3067	1.50680	0.753401	−	0.657561i	$-0.228412\pi$
$613$	37.3067	1.50680	0.753401	+	0.657561i	$0.228412\pi$
$614$	−52.9129	−2.13539
$615$	10.0000	0.403239
$616$	0	0
$617$	27.8082	1.11951	0.559757	−	0.828657i	$-0.310894\pi$
$617$	27.8082	1.11951	0.559757	+	0.828657i	$0.310894\pi$
$618$	−17.9681	−0.722781
$619$	−12.4104	−0.498816	−0.249408	−	0.968398i	$-0.580236\pi$
$619$	−12.4104	−0.498816	−0.249408	+	0.968398i	$0.580236\pi$
$620$	−52.9129	−2.12503
$621$	37.9129	1.52139
$622$	12.1244	0.486142
$623$	0	0
$624$	0	0
$625$	−23.9129	−0.956515
$626$	45.4147	1.81514
$627$	−15.0000	−0.599042
$628$	−61.2867	−2.44561
$629$	−20.7846	−0.828737
$630$	0	0
$631$	−12.0489	−0.479659	−0.239830	−	0.970815i	$-0.577092\pi$
$631$	−12.0489	−0.479659	−0.239830	+	0.970815i	$0.577092\pi$
$632$	−10.3923	−0.413384
$633$	2.53901	0.100917
$634$	−40.5390	−1.61001
$635$	−34.9271	−1.38604
$636$	−60.8258	−2.41190
$637$	0	0
$638$	−6.16515	−0.244081
$639$	0.190700	0.00754399
$640$	27.9564	1.10508
$641$	−15.6261	−0.617195	−0.308598	−	0.951193i	$-0.599860\pi$
$641$	−15.6261	−0.617195	−0.308598	+	0.951193i	$0.599860\pi$
$642$	−38.3912	−1.51518
$643$	−13.5704	−0.535163	−0.267581	−	0.963535i	$-0.586224\pi$
$643$	−13.5704	−0.535163	−0.267581	+	0.963535i	$0.586224\pi$
$644$	0	0
$645$	−17.1497	−0.675269
$646$	−43.1216	−1.69660
$647$	29.0780	1.14318	0.571588	−	0.820541i	$-0.306328\pi$
$647$	29.0780	1.14318	0.571588	+	0.820541i	$0.306328\pi$
$648$	−16.5975	−0.652012
$649$	11.2867	0.443043
$650$	0	0
$651$	0	0
$652$	19.3386	0.757358
$653$	11.2087	0.438631	0.219315	−	0.975654i	$-0.429618\pi$
$653$	11.2087	0.438631	0.219315	+	0.975654i	$0.429618\pi$
$654$	−31.1216	−1.21695
$655$	−7.93725	−0.310134
$656$	4.56850	0.178370
$657$	0.723000	0.0282069
$658$	0	0
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	−13.9564	−0.543254
$661$	18.7665	0.729933	0.364966	−	0.931021i	$-0.381081\pi$
$661$	18.7665	0.729933	0.364966	+	0.931021i	$0.381081\pi$
$662$	−2.37386	−0.0922628
$663$	0	0
$664$	6.16515	0.239254
$665$	0	0
$666$	−3.16515	−0.122647
$667$	16.7477	0.648475
$668$	55.7316	2.15632
$669$	37.0961	1.43422
$670$	−54.6272	−2.11043
$671$	−16.2559	−0.627552
$672$	0	0
$673$	28.4955	1.09842	0.549210	−	0.835685i	$-0.314929\pi$
$673$	28.4955	1.09842	0.549210	+	0.835685i	$0.314929\pi$
$674$	28.3604	1.09240
$675$	1.04356	0.0401667
$676$	0	0
$677$	−33.7913	−1.29870	−0.649352	−	0.760488i	$-0.724960\pi$
$677$	−33.7913	−1.29870	−0.649352	+	0.760488i	$0.724960\pi$
$678$	5.55765	0.213440
$679$	0	0
$680$	−11.3739	−0.436168
$681$	22.0598	0.845334
$682$	24.1733	0.925642
$683$	25.0671	0.959164	0.479582	−	0.877497i	$-0.340788\pi$
$683$	25.0671	0.959164	0.479582	+	0.877497i	$0.340788\pi$
$684$	−3.82560	−0.146276
$685$	37.5390	1.43429
$686$	0	0
$687$	12.4104	0.473487
$688$	−7.83485	−0.298701
$689$	0	0
$690$	65.0780	2.47748
$691$	29.3694	1.11727	0.558633	−	0.829415i	$-0.311326\pi$
$691$	29.3694	1.11727	0.558633	+	0.829415i	$0.311326\pi$
$692$	−21.6261	−0.822102
$693$	0	0
$694$	−9.66930	−0.367042
$695$	−1.73205	−0.0657004
$696$	−6.85275	−0.259753
$697$	−7.65120	−0.289810
$698$	−23.3739	−0.884714
$699$	12.4610	0.471318
$700$	0	0
$701$	−31.9129	−1.20533	−0.602666	−	0.797993i	$-0.705895\pi$
$701$	−31.9129	−1.20533	−0.602666	+	0.797993i	$0.705895\pi$
$702$	0	0
$703$	45.4955	1.71589
$704$	−16.0453	−0.604730
$705$	−16.7913	−0.632396
$706$	−58.7042	−2.20936
$707$	0	0
$708$	44.2548	1.66320
$709$	−7.28970	−0.273771	−0.136885	−	0.990587i	$-0.543709\pi$
$709$	−7.28970	−0.273771	−0.136885	+	0.990587i	$0.543709\pi$
$710$	−4.37780	−0.164296
$711$	−1.25227	−0.0469639
$712$	−5.04356	−0.189015
$713$	−65.6670	−2.45925
$714$	0	0
$715$	0	0
$716$	25.1216	0.938838
$717$	23.6965	0.884962
$718$	27.7477	1.03554
$719$	−5.83485	−0.217603	−0.108802	−	0.994063i	$-0.534701\pi$
$719$	−5.83485	−0.217603	−0.108802	+	0.994063i	$0.534701\pi$
$720$	0.818350	0.0304981
$721$	0	0
$722$	52.7998	1.96500
$723$	7.36515	0.273913
$724$	25.5826	0.950769
$725$	0.460985	0.0171206
$726$	−36.7545	−1.36409
$727$	27.7477	1.02911	0.514553	−	0.857459i	$-0.327958\pi$
$727$	27.7477	1.02911	0.514553	+	0.857459i	$0.327958\pi$
$728$	0	0
$729$	24.8693	0.921086
$730$	−16.5975	−0.614301
$731$	13.1216	0.485320
$732$	−63.7386	−2.35585
$733$	−9.02175	−0.333226	−0.166613	−	0.986022i	$-0.553283\pi$
$733$	−9.02175	−0.333226	−0.166613	+	0.986022i	$0.553283\pi$
$734$	−39.4002	−1.45429
$735$	0	0
$736$	55.9977	2.06410
$737$	14.5390	0.535551
$738$	−1.16515	−0.0428898
$739$	12.4104	0.456524	0.228262	−	0.973600i	$-0.426696\pi$
$739$	12.4104	0.456524	0.228262	+	0.973600i	$0.426696\pi$
$740$	42.3303	1.55609
$741$	0	0
$742$	0	0
$743$	5.36695	0.196894	0.0984472	−	0.995142i	$-0.468612\pi$
$743$	5.36695	0.196894	0.0984472	+	0.995142i	$0.468612\pi$
$744$	26.8693	0.985077
$745$	4.79129	0.175539
$746$	80.5325	2.94850
$747$	0.742901	0.0271813
$748$	10.6784	0.390439
$749$	0	0
$750$	44.7042	1.63237
$751$	3.74773	0.136757	0.0683783	−	0.997659i	$-0.478218\pi$
$751$	3.74773	0.136757	0.0683783	+	0.997659i	$0.478218\pi$
$752$	−7.67110	−0.279736
$753$	37.9129	1.38162
$754$	0	0
$755$	26.5390	0.965854
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	−9.95644	−0.361634
$759$	−17.3205	−0.628695
$760$	24.8963	0.903082
$761$	−32.0152	−1.16055	−0.580274	−	0.814421i	$-0.697055\pi$
$761$	−32.0152	−1.16055	−0.580274	+	0.814421i	$0.697055\pi$
$762$	62.5644	2.26647
$763$	0	0
$764$	−1.74773	−0.0632305
$765$	−1.37055	−0.0495524
$766$	−8.58258	−0.310101
$767$	0	0
$768$	−5.00000	−0.180422
$769$	−25.2578	−0.910818	−0.455409	−	0.890282i	$-0.650507\pi$
$769$	−25.2578	−0.910818	−0.455409	+	0.890282i	$0.650507\pi$
$770$	0	0
$771$	−50.0780	−1.80352
$772$	34.6410	1.24676
$773$	22.8981	0.823586	0.411793	−	0.911277i	$-0.364903\pi$
$773$	22.8981	0.823586	0.411793	+	0.911277i	$0.364903\pi$
$774$	1.99820	0.0718238
$775$	−1.80750	−0.0649273
$776$	−26.3739	−0.946767
$777$	0	0
$778$	−79.5234	−2.85105
$779$	16.7477	0.600050
$780$	0	0
$781$	1.16515	0.0416924
$782$	−49.7925	−1.78058
$783$	11.0436	0.394665
$784$	0	0
$785$	48.0605	1.71535
$786$	14.2179	0.507136
$787$	−5.48220	−0.195419	−0.0977097	−	0.995215i	$-0.531152\pi$
$787$	−5.48220	−0.195419	−0.0977097	+	0.995215i	$0.531152\pi$
$788$	−30.5493	−1.08827
$789$	−48.9564	−1.74290
$790$	28.7477	1.02280
$791$	0	0
$792$	0.460985	0.0163804
$793$	0	0
$794$	−33.1652	−1.17699
$795$	47.6990	1.69171
$796$	−30.7042	−1.08828
$797$	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
$799$	12.8474	0.454507
$800$	1.54135	0.0544950
$801$	−0.607749	−0.0214738
$802$	64.7042	2.28478
$803$	4.41742	0.155888
$804$	57.0068	2.01047
$805$	0	0
$806$	0	0
$807$	−20.0780	−0.706780
$808$	16.9590	0.596616
$809$	−29.2432	−1.02814	−0.514068	−	0.857750i	$-0.671862\pi$
$809$	−29.2432	−1.02814	−0.514068	+	0.857750i	$0.671862\pi$
$810$	45.9129	1.61321
$811$	12.0489	0.423094	0.211547	−	0.977368i	$-0.432150\pi$
$811$	12.0489	0.423094	0.211547	+	0.977368i	$0.432150\pi$
$812$	0	0
$813$	51.4491	1.80440
$814$	−19.3386	−0.677818
$815$	−15.1652	−0.531212
$816$	−9.62614	−0.336982
$817$	−28.7219	−1.00485
$818$	−0.791288	−0.0276667
$819$	0	0
$820$	15.5826	0.544167
$821$	49.6972	1.73444	0.867222	−	0.497922i	$-0.165904\pi$
$821$	49.6972	1.73444	0.867222	+	0.497922i	$0.165904\pi$
$822$	−67.2432	−2.34538
$823$	−32.4955	−1.13272	−0.566360	−	0.824158i	$-0.691649\pi$
$823$	−32.4955	−1.13272	−0.566360	+	0.824158i	$0.691649\pi$
$824$	−7.93725	−0.276507
$825$	−0.476751	−0.0165983
$826$	0	0
$827$	−0.989150	−0.0343961	−0.0171981	−	0.999852i	$-0.505475\pi$
$827$	−0.989150	−0.0343961	−0.0171981	+	0.999852i	$0.505475\pi$
$828$	−4.41742	−0.153516
$829$	−20.6261	−0.716375	−0.358188	−	0.933650i	$-0.616605\pi$
$829$	−20.6261	−0.716375	−0.358188	+	0.933650i	$0.616605\pi$
$830$	−17.0544	−0.591965
$831$	28.2087	0.978549
$832$	0	0
$833$	0	0
$834$	3.10260	0.107434
$835$	−43.7042	−1.51245
$836$	−23.3739	−0.808402
$837$	−43.3013	−1.49671
$838$	56.3592	1.94690
$839$	−46.4992	−1.60533	−0.802666	−	0.596429i	$-0.796586\pi$
$839$	−46.4992	−1.60533	−0.802666	+	0.596429i	$0.796586\pi$
$840$	0	0
$841$	−24.1216	−0.831779
$842$	−43.9129	−1.51334
$843$	−11.4569	−0.394597
$844$	3.95644	0.136186
$845$	0	0
$846$	1.95644	0.0672638
$847$	0	0
$848$	21.7913	0.748316
$849$	44.3303	1.52141
$850$	−1.37055	−0.0470095
$851$	52.5336	1.80083
$852$	4.56850	0.156514
$853$	17.6066	0.602837	0.301419	−	0.953492i	$-0.402540\pi$
$853$	17.6066	0.602837	0.301419	+	0.953492i	$0.402540\pi$
$854$	0	0
$855$	3.00000	0.102598
$856$	−16.9590	−0.579647
$857$	−47.5390	−1.62390	−0.811951	−	0.583726i	$-0.801594\pi$
$857$	−47.5390	−1.62390	−0.811951	+	0.583726i	$0.801594\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	−26.7237	−0.911269
$861$	0	0
$862$	−34.1652	−1.16367
$863$	−3.08270	−0.104936	−0.0524682	−	0.998623i	$-0.516709\pi$
$863$	−3.08270	−0.104936	−0.0524682	+	0.998623i	$0.516709\pi$
$864$	36.9253	1.25622
$865$	16.9590	0.576624
$866$	−49.2403	−1.67325
$867$	−14.3303	−0.486683
$868$	0	0
$869$	−7.65120	−0.259549
$870$	18.9564	0.642683
$871$	0	0
$872$	−13.7477	−0.465557
$873$	−3.17805	−0.107561
$874$	108.991	3.68667
$875$	0	0
$876$	17.3205	0.585206
$877$	−34.9271	−1.17940	−0.589702	−	0.807621i	$-0.700755\pi$
$877$	−34.9271	−1.17940	−0.589702	+	0.807621i	$0.700755\pi$
$878$	−25.2578	−0.852408
$879$	14.0471	0.473797
$880$	5.00000	0.168550
$881$	−7.41742	−0.249899	−0.124950	−	0.992163i	$-0.539877\pi$
$881$	−7.41742	−0.249899	−0.124950	+	0.992163i	$0.539877\pi$
$882$	0	0
$883$	53.2432	1.79178	0.895888	−	0.444280i	$-0.146541\pi$
$883$	53.2432	1.79178	0.895888	+	0.444280i	$0.146541\pi$
$884$	0	0
$885$	−34.7042	−1.16657
$886$	6.92820	0.232758
$887$	31.5826	1.06044	0.530220	−	0.847860i	$-0.322109\pi$
$887$	31.5826	1.06044	0.530220	+	0.847860i	$0.322109\pi$
$888$	−21.4955	−0.721340
$889$	0	0
$890$	13.9518	0.467664
$891$	−12.2197	−0.409376
$892$	57.8052	1.93546
$893$	−28.1216	−0.941053
$894$	−8.58258	−0.287044
$895$	−19.7001	−0.658502
$896$	0	0
$897$	0	0
$898$	−43.4955	−1.45146
$899$	−19.1280	−0.637955
$900$	−0.121591	−0.00405302
$901$	−36.4955	−1.21584
$902$	−7.11890	−0.237034
$903$	0	0
$904$	2.45505	0.0816538
$905$	−20.0616	−0.666870
$906$	−47.5390	−1.57938
$907$	−23.0780	−0.766293	−0.383147	−	0.923688i	$-0.625160\pi$
$907$	−23.0780	−0.766293	−0.383147	+	0.923688i	$0.625160\pi$
$908$	34.3749	1.14077
$909$	2.04356	0.0677806
$910$	0	0
$911$	−1.87841	−0.0622345	−0.0311172	−	0.999516i	$-0.509907\pi$
$911$	−1.87841	−0.0622345	−0.0311172	+	0.999516i	$0.509907\pi$
$912$	21.0707	0.697719
$913$	4.53901	0.150219
$914$	−19.5826	−0.647734
$915$	49.9832	1.65239
$916$	19.3386	0.638966
$917$	0	0
$918$	−32.8335	−1.08367
$919$	−19.9129	−0.656865	−0.328433	−	0.944527i	$-0.606520\pi$
$919$	−19.9129	−0.656865	−0.328433	+	0.944527i	$0.606520\pi$
$920$	28.7477	0.947784
$921$	−43.3013	−1.42683
$922$	−39.1216	−1.28840
$923$	0	0
$924$	0	0
$925$	1.44600	0.0475442
$926$	−17.3739	−0.570941
$927$	−0.956439	−0.0314136
$928$	16.3115	0.535450
$929$	−26.7436	−0.877428	−0.438714	−	0.898627i	$-0.644566\pi$
$929$	−26.7436	−0.877428	−0.438714	+	0.898627i	$0.644566\pi$
$930$	−74.3273	−2.43729
$931$	0	0
$932$	19.4174	0.636039
$933$	9.92197	0.324831
$934$	−25.9053	−0.847648
$935$	−8.37386	−0.273855
$936$	0	0
$937$	−34.4955	−1.12692	−0.563459	−	0.826144i	$-0.690530\pi$
$937$	−34.4955	−1.12692	−0.563459	+	0.826144i	$0.690530\pi$
$938$	0	0
$939$	37.1652	1.21284
$940$	−26.1652	−0.853413
$941$	2.26435	0.0738157	0.0369079	−	0.999319i	$-0.488249\pi$
$941$	2.26435	0.0738157	0.0369079	+	0.999319i	$0.488249\pi$
$942$	−86.0901	−2.80497
$943$	19.3386	0.629752
$944$	−15.8546	−0.516024
$945$	0	0
$946$	12.2087	0.396939
$947$	56.8915	1.84873	0.924363	−	0.381514i	$-0.124597\pi$
$947$	56.8915	1.84873	0.924363	+	0.381514i	$0.124597\pi$
$948$	−30.0000	−0.974355
$949$	0	0
$950$	3.00000	0.0973329
$951$	−33.1751	−1.07578
$952$	0	0
$953$	15.0000	0.485898	0.242949	−	0.970039i	$-0.421885\pi$
$953$	15.0000	0.485898	0.242949	+	0.970039i	$0.421885\pi$
$954$	−5.55765	−0.179936
$955$	1.37055	0.0443500
$956$	36.9253	1.19425
$957$	−5.04525	−0.163090
$958$	−22.3739	−0.722867
$959$	0	0
$960$	49.3357	1.59230
$961$	44.0000	1.41935
$962$	0	0
$963$	−2.04356	−0.0658528
$964$	11.4768	0.369643
$965$	−27.1652	−0.874477
$966$	0	0
$967$	−23.8118	−0.765735	−0.382867	−	0.923803i	$-0.625063\pi$
$967$	−23.8118	−0.765735	−0.382867	+	0.923803i	$0.625063\pi$
$968$	−16.2360	−0.521845
$969$	−35.2886	−1.13363
$970$	72.9567	2.34250
$971$	49.7477	1.59648	0.798240	−	0.602339i	$-0.205765\pi$
$971$	49.7477	1.59648	0.798240	+	0.602339i	$0.205765\pi$
$972$	−6.04356	−0.193847
$973$	0	0
$974$	22.5826	0.723592
$975$	0	0
$976$	22.8348	0.730926
$977$	56.8161	1.81771	0.908854	−	0.417115i	$-0.136959\pi$
$977$	56.8161	1.81771	0.908854	+	0.417115i	$0.136959\pi$
$978$	27.1652	0.868646
$979$	−3.71326	−0.118676
$980$	0	0
$981$	−1.65660	−0.0528912
$982$	−81.2555	−2.59297
$983$	−25.0870	−0.800150	−0.400075	−	0.916482i	$-0.631016\pi$
$983$	−25.0870	−0.800150	−0.400075	+	0.916482i	$0.631016\pi$
$984$	−7.91288	−0.252253
$985$	23.9564	0.763316
$986$	−14.5040	−0.461900
$987$	0	0
$988$	0	0
$989$	−33.1652	−1.05459
$990$	−1.27520	−0.0405285
$991$	−26.6261	−0.845807	−0.422904	−	0.906175i	$-0.638989\pi$
$991$	−26.6261	−0.845807	−0.422904	+	0.906175i	$0.638989\pi$
$992$	−63.9564	−2.03062
$993$	−1.94265	−0.0616482
$994$	0	0
$995$	24.0779	0.763321
$996$	17.7973	0.563928
$997$	50.0780	1.58599	0.792994	−	0.609230i	$-0.208521\pi$
$997$	50.0780	1.58599	0.792994	+	0.609230i	$0.208521\pi$
$998$	−92.4083	−2.92513
$999$	34.6410	1.09599

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bq.1.4		4
7.6	odd	2		8281.2.a.bs.1.4		4
13.2	odd	12		637.2.q.f.589.2	yes	4
13.7	odd	12		637.2.q.f.491.2	yes	4
13.12	even	2	inner	8281.2.a.bq.1.1		4
91.2	odd	12		637.2.k.f.459.1		4
91.20	even	12		637.2.q.e.491.2	✓	4
91.33	even	12		637.2.u.e.361.1		4
91.41	even	12		637.2.q.e.589.2	yes	4
91.46	odd	12		637.2.k.f.569.2		4
91.54	even	12		637.2.k.d.459.1		4
91.59	even	12		637.2.k.d.569.2		4
91.67	odd	12		637.2.u.d.30.1		4
91.72	odd	12		637.2.u.d.361.1		4
91.80	even	12		637.2.u.e.30.1		4
91.90	odd	2		8281.2.a.bs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.k.d.459.1		4	91.54	even	12
637.2.k.d.569.2		4	91.59	even	12
637.2.k.f.459.1		4	91.2	odd	12
637.2.k.f.569.2		4	91.46	odd	12
637.2.q.e.491.2	✓	4	91.20	even	12
637.2.q.e.589.2	yes	4	91.41	even	12
637.2.q.f.491.2	yes	4	13.7	odd	12
637.2.q.f.589.2	yes	4	13.2	odd	12
637.2.u.d.30.1		4	91.67	odd	12
637.2.u.d.361.1		4	91.72	odd	12
637.2.u.e.30.1		4	91.80	even	12
637.2.u.e.361.1		4	91.33	even	12
8281.2.a.bq.1.1		4	13.12	even	2	inner
8281.2.a.bq.1.4		4	1.1	even	1	trivial
8281.2.a.bs.1.1		4	91.90	odd	2
8281.2.a.bs.1.4		4	7.6	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)

\(f(q)\)	\(=\)	\(q+2.18890 q^{2} +1.79129 q^{3} +2.79129 q^{4} -2.18890 q^{5} +3.92095 q^{6} +1.73205 q^{8} +0.208712 q^{9} +O(q^{10})\)	\(q+2.18890 q^{2} +1.79129 q^{3} +2.79129 q^{4} -2.18890 q^{5} +3.92095 q^{6} +1.73205 q^{8} +0.208712 q^{9} -4.79129 q^{10} +1.27520 q^{11} +5.00000 q^{12} -3.92095 q^{15} -1.79129 q^{16} +3.00000 q^{17} +0.456850 q^{18} -6.56670 q^{19} -6.10985 q^{20} +2.79129 q^{22} -7.58258 q^{23} +3.10260 q^{24} -0.208712 q^{25} -5.00000 q^{27} -2.20871 q^{29} -8.58258 q^{30} +8.66025 q^{31} -7.38505 q^{32} +2.28425 q^{33} +6.56670 q^{34} +0.582576 q^{36} -6.92820 q^{37} -14.3739 q^{38} -3.79129 q^{40} -2.55040 q^{41} +4.37386 q^{43} +3.55945 q^{44} -0.456850 q^{45} -16.5975 q^{46} +4.28245 q^{47} -3.20871 q^{48} -0.456850 q^{50} +5.37386 q^{51} -12.1652 q^{53} -10.9445 q^{54} -2.79129 q^{55} -11.7629 q^{57} -4.83465 q^{58} +8.85095 q^{59} -10.9445 q^{60} -12.7477 q^{61} +18.9564 q^{62} -12.5826 q^{64} +5.00000 q^{66} +11.4014 q^{67} +8.37386 q^{68} -13.5826 q^{69} +0.913701 q^{71} +0.361500 q^{72} +3.46410 q^{73} -15.1652 q^{74} -0.373864 q^{75} -18.3296 q^{76} -6.00000 q^{79} +3.92095 q^{80} -9.58258 q^{81} -5.58258 q^{82} +3.55945 q^{83} -6.56670 q^{85} +9.57395 q^{86} -3.95644 q^{87} +2.20871 q^{88} -2.91190 q^{89} -1.00000 q^{90} -21.1652 q^{92} +15.5130 q^{93} +9.37386 q^{94} +14.3739 q^{95} -13.2288 q^{96} -15.2270 q^{97} +0.266150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{4} + 10 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 2 * q^4 + 10 * q^9	\( 4 q - 2 q^{3} + 2 q^{4} + 10 q^{9} - 10 q^{10} + 20 q^{12} + 2 q^{16} + 12 q^{17} + 2 q^{22} - 12 q^{23} - 10 q^{25} - 20 q^{27} - 18 q^{29} - 16 q^{30} - 16 q^{36} - 30 q^{38} - 6 q^{40} - 10 q^{43} - 22 q^{48} - 6 q^{51} - 12 q^{53} - 2 q^{55} + 4 q^{61} + 30 q^{62} - 32 q^{64} + 20 q^{66} + 6 q^{68} - 36 q^{69} - 24 q^{74} + 26 q^{75} - 24 q^{79} - 20 q^{81} - 4 q^{82} + 30 q^{87} + 18 q^{88} - 4 q^{90} - 48 q^{92} + 10 q^{94} + 30 q^{95}+O(q^{100}) \) 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

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