LMFDB - Embedded newform 8281.2.a.cj.1.7

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 = [8,0,-4,6,0,0,0,0,12,6,0,-18,0,0,0,-2,-8] 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:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 3 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 3
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			1.7
Root			$2.12549$ 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)

$f(q)$	$=$	\(q+2.12549 q^{2} +0.178854 q^{3} +2.51771 q^{4} +3.60603 q^{5} +0.380153 q^{6} +1.10038 q^{8} -2.96801 q^{9} +7.66457 q^{10} -3.99236 q^{11} +0.450303 q^{12} +0.644954 q^{15} -2.69656 q^{16} -4.78916 q^{17} -6.30848 q^{18} -3.15060 q^{19} +9.07892 q^{20} -8.48572 q^{22} -2.17885 q^{23} +0.196808 q^{24} +8.00343 q^{25} -1.06740 q^{27} -6.57198 q^{29} +1.37084 q^{30} +1.48672 q^{31} -7.93228 q^{32} -0.714051 q^{33} -10.1793 q^{34} -7.47259 q^{36} -4.96503 q^{37} -6.69656 q^{38} +3.96801 q^{40} -2.11931 q^{41} +1.43145 q^{43} -10.0516 q^{44} -10.7027 q^{45} -4.63113 q^{46} +1.01893 q^{47} -0.482292 q^{48} +17.0112 q^{50} -0.856562 q^{51} +6.03542 q^{53} -2.26876 q^{54} -14.3966 q^{55} -0.563498 q^{57} -13.9687 q^{58} -4.90322 q^{59} +1.62380 q^{60} +2.03542 q^{61} +3.16000 q^{62} -11.4669 q^{64} -1.51771 q^{66} -3.91090 q^{67} -12.0577 q^{68} -0.389698 q^{69} +8.80684 q^{71} -3.26595 q^{72} +3.08878 q^{73} -10.5531 q^{74} +1.43145 q^{75} -7.93228 q^{76} +1.96801 q^{79} -9.72388 q^{80} +8.71312 q^{81} -4.50457 q^{82} +7.66020 q^{83} -17.2698 q^{85} +3.04253 q^{86} -1.17543 q^{87} -4.39312 q^{88} -12.7992 q^{89} -22.7485 q^{90} -5.48572 q^{92} +0.265906 q^{93} +2.16572 q^{94} -11.3611 q^{95} -1.41872 q^{96} +1.35900 q^{97} +11.8494 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} + 6 q^{4} + 12 q^{9} + 6 q^{10} - 18 q^{12} - 2 q^{16} - 8 q^{17} - 18 q^{22} - 12 q^{23} - 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} - 34 q^{38} - 4 q^{40} - 8 q^{43} - 18 q^{48} + 16 q^{51}+ \cdots - 16 q^{95}+O(q^{100}) $ 8 * q - 4 * q^3 + 6 * q^4 + 12 * q^9 + 6 * q^10 - 18 * q^12 - 2 * q^16 - 8 * q^17 - 18 * q^22 - 12 * q^23 - 16 * q^27 - 8 * q^29 + 38 * q^30 + 28 * q^36 - 34 * q^38 - 4 * q^40 - 8 * q^43 - 18 * q^48 + 16 * q^51 + 20 * q^53 - 12 * q^55 - 12 * q^61 + 22 * q^62 - 44 * q^64 + 2 * q^66 - 2 * q^68 - 28 * q^69 - 42 * q^74 - 8 * q^75 - 20 * q^79 + 24 * q^81 + 16 * q^82 - 68 * q^87 + 4 * q^88 - 108 * q^90 + 6 * q^92 - 26 * q^94 - 16 * 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 $

$2$	2.12549	1.50295	0.751474	−	0.659762i	$-0.229343\pi$
$2$	2.12549	1.50295	0.751474	+	0.659762i	$0.229343\pi$
$3$	0.178854	0.103262	0.0516308	−	0.998666i	$-0.483558\pi$
$3$	0.178854	0.103262	0.0516308	+	0.998666i	$0.483558\pi$
$4$	2.51771	1.25885
$5$	3.60603	1.61266	0.806332	−	0.591463i	$-0.201449\pi$
$5$	3.60603	1.61266	0.806332	+	0.591463i	$0.201449\pi$
$6$	0.380153	0.155197
$7$	0	0
$7$	0	0
$8$	1.10038	0.389044
$9$	−2.96801	−0.989337
$10$	7.66457	2.42375
$11$	−3.99236	−1.20374	−0.601871	−	0.798594i	$-0.705578\pi$
$11$	−3.99236	−1.20374	−0.601871	+	0.798594i	$0.705578\pi$
$12$	0.450303	0.129991
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.644954	0.166526
$16$	−2.69656	−0.674141
$17$	−4.78916	−1.16154	−0.580771	−	0.814067i	$-0.697249\pi$
$17$	−4.78916	−1.16154	−0.580771	+	0.814067i	$0.697249\pi$
$18$	−6.30848	−1.48692
$19$	−3.15060	−0.722796	−0.361398	−	0.932412i	$-0.617701\pi$
$19$	−3.15060	−0.722796	−0.361398	+	0.932412i	$0.617701\pi$
$20$	9.07892	2.03011
$21$	0	0
$22$	−8.48572	−1.80916
$23$	−2.17885	−0.454323	−0.227161	−	0.973857i	$-0.572944\pi$
$23$	−2.17885	−0.454323	−0.227161	+	0.973857i	$0.572944\pi$
$24$	0.196808	0.0401733
$25$	8.00343	1.60069
$26$	0	0
$27$	−1.06740	−0.205422
$28$	0	0
$29$	−6.57198	−1.22039	−0.610193	−	0.792253i	$-0.708908\pi$
$29$	−6.57198	−1.22039	−0.610193	+	0.792253i	$0.708908\pi$
$30$	1.37084	0.250280
$31$	1.48672	0.267022	0.133511	−	0.991047i	$-0.457375\pi$
$31$	1.48672	0.267022	0.133511	+	0.991047i	$0.457375\pi$
$32$	−7.93228	−1.40224
$33$	−0.714051	−0.124300
$34$	−10.1793	−1.74574
$35$	0	0
$36$	−7.47259	−1.24543
$37$	−4.96503	−0.816246	−0.408123	−	0.912927i	$-0.633817\pi$
$37$	−4.96503	−0.816246	−0.408123	+	0.912927i	$0.633817\pi$
$38$	−6.69656	−1.08633
$39$	0	0
$40$	3.96801	0.627398
$41$	−2.11931	−0.330981	−0.165490	−	0.986211i	$-0.552921\pi$
$41$	−2.11931	−0.330981	−0.165490	+	0.986211i	$0.552921\pi$
$42$	0	0
$43$	1.43145	0.218294	0.109147	−	0.994026i	$-0.465188\pi$
$43$	1.43145	0.218294	0.109147	+	0.994026i	$0.465188\pi$
$44$	−10.0516	−1.51533
$45$	−10.7027	−1.59547
$46$	−4.63113	−0.682823
$47$	1.01893	0.148626	0.0743129	−	0.997235i	$-0.476324\pi$
$47$	1.01893	0.148626	0.0743129	+	0.997235i	$0.476324\pi$
$48$	−0.482292	−0.0696129
$49$	0	0
$50$	17.0112	2.40575
$51$	−0.856562	−0.119943
$52$	0	0
$53$	6.03542	0.829028	0.414514	−	0.910043i	$-0.363952\pi$
$53$	6.03542	0.829028	0.414514	+	0.910043i	$0.363952\pi$
$54$	−2.26876	−0.308739
$55$	−14.3966	−1.94123
$56$	0	0
$57$	−0.563498	−0.0746371
$58$	−13.9687	−1.83418
$59$	−4.90322	−0.638345	−0.319172	−	0.947697i	$-0.603405\pi$
$59$	−4.90322	−0.638345	−0.319172	+	0.947697i	$0.603405\pi$
$60$	1.62380	0.209632
$61$	2.03542	0.260608	0.130304	−	0.991474i	$-0.458405\pi$
$61$	2.03542	0.260608	0.130304	+	0.991474i	$0.458405\pi$
$62$	3.16000	0.401320
$63$	0	0
$64$	−11.4669	−1.43336
$65$	0	0
$66$	−1.51771	−0.186817
$67$	−3.91090	−0.477793	−0.238896	−	0.971045i	$-0.576786\pi$
$67$	−3.91090	−0.477793	−0.238896	+	0.971045i	$0.576786\pi$
$68$	−12.0577	−1.46221
$69$	−0.389698	−0.0469141
$70$	0	0
$71$	8.80684	1.04518	0.522590	−	0.852584i	$-0.324966\pi$
$71$	8.80684	1.04518	0.522590	+	0.852584i	$0.324966\pi$
$72$	−3.26595	−0.384896
$73$	3.08878	0.361515	0.180757	−	0.983528i	$-0.442145\pi$
$73$	3.08878	0.361515	0.180757	+	0.983528i	$0.442145\pi$
$74$	−10.5531	−1.22678
$75$	1.43145	0.165289
$76$	−7.93228	−0.909895
$77$	0	0
$78$	0	0
$79$	1.96801	0.221419	0.110709	−	0.993853i	$-0.464688\pi$
$79$	1.96801	0.221419	0.110709	+	0.993853i	$0.464688\pi$
$80$	−9.72388	−1.08716
$81$	8.71312	0.968125
$82$	−4.50457	−0.497447
$83$	7.66020	0.840816	0.420408	−	0.907335i	$-0.361887\pi$
$83$	7.66020	0.840816	0.420408	+	0.907335i	$0.361887\pi$
$84$	0	0
$85$	−17.2698	−1.87318
$86$	3.04253	0.328084
$87$	−1.17543	−0.126019
$88$	−4.39312	−0.468309
$89$	−12.7992	−1.35671	−0.678356	−	0.734733i	$-0.737307\pi$
$89$	−12.7992	−1.35671	−0.678356	+	0.734733i	$0.737307\pi$
$90$	−22.7485	−2.39791
$91$	0	0
$92$	−5.48572	−0.571926
$93$	0.265906	0.0275731
$94$	2.16572	0.223377
$95$	−11.3611	−1.16563
$96$	−1.41872	−0.144798
$97$	1.35900	0.137986	0.0689930	−	0.997617i	$-0.478021\pi$
$97$	1.35900	0.137986	0.0689930	+	0.997617i	$0.478021\pi$
$98$	0	0
$99$	11.8494	1.19091
$100$	20.1503	2.01503
$101$	−4.28801	−0.426673	−0.213336	−	0.976979i	$-0.568433\pi$
$101$	−4.28801	−0.426673	−0.213336	+	0.976979i	$0.568433\pi$
$102$	−1.82061	−0.180268
$103$	−14.4349	−1.42231	−0.711155	−	0.703035i	$-0.751828\pi$
$103$	−14.4349	−1.42231	−0.711155	+	0.703035i	$0.751828\pi$
$104$	0	0
$105$	0	0
$106$	12.8282	1.24599
$107$	−9.71884	−0.939556	−0.469778	−	0.882785i	$-0.655666\pi$
$107$	−9.71884	−0.939556	−0.469778	+	0.882785i	$0.655666\pi$
$108$	−2.68741	−0.258597
$109$	6.64855	0.636816	0.318408	−	0.947954i	$-0.396852\pi$
$109$	6.64855	0.636816	0.318408	+	0.947954i	$0.396852\pi$
$110$	−30.5997	−2.91757
$111$	−0.888018	−0.0842869
$112$	0	0
$113$	17.5434	1.65035	0.825173	−	0.564880i	$-0.191078\pi$
$113$	17.5434	1.65035	0.825173	+	0.564880i	$0.191078\pi$
$114$	−1.19771	−0.112176
$115$	−7.85701	−0.732670
$116$	−16.5463	−1.53629
$117$	0	0
$118$	−10.4217	−0.959399
$119$	0	0
$120$	0.709696	0.0647861
$121$	4.93893	0.448994
$122$	4.32626	0.391681
$123$	−0.379048	−0.0341776
$124$	3.74312	0.336142
$125$	10.8304	0.968704
$126$	0	0
$127$	19.5143	1.73162	0.865809	−	0.500375i	$-0.166805\pi$
$127$	19.5143	1.73162	0.865809	+	0.500375i	$0.166805\pi$
$128$	−8.50814	−0.752020
$129$	0.256021	0.0225414
$130$	0	0
$131$	−19.0743	−1.66653	−0.833263	−	0.552877i	$-0.813530\pi$
$131$	−19.0743	−1.66653	−0.833263	+	0.552877i	$0.813530\pi$
$132$	−1.79777	−0.156476
$133$	0	0
$134$	−8.31259	−0.718098
$135$	−3.84909	−0.331277
$136$	−5.26991	−0.451891
$137$	6.42890	0.549258	0.274629	−	0.961550i	$-0.411445\pi$
$137$	6.42890	0.549258	0.274629	+	0.961550i	$0.411445\pi$
$138$	−0.828298	−0.0705094
$139$	2.42854	0.205986	0.102993	−	0.994682i	$-0.467158\pi$
$139$	2.42854	0.205986	0.102993	+	0.994682i	$0.467158\pi$
$140$	0	0
$141$	0.182240	0.0153473
$142$	18.7188	1.57085
$143$	0	0
$144$	8.00343	0.666952
$145$	−23.6987	−1.96807
$146$	6.56518	0.543338
$147$	0	0
$148$	−12.5005	−1.02753
$149$	0.115353	0.00945007	0.00472503	−	0.999989i	$-0.498496\pi$
$149$	0.115353	0.00945007	0.00472503	+	0.999989i	$0.498496\pi$
$150$	3.04253	0.248421
$151$	−11.8031	−0.960523	−0.480262	−	0.877125i	$-0.659458\pi$
$151$	−11.8031	−0.960523	−0.480262	+	0.877125i	$0.659458\pi$
$152$	−3.46686	−0.281200
$153$	14.2143	1.14916
$154$	0	0
$155$	5.36114	0.430617
$156$	0	0
$157$	13.1469	1.04923	0.524617	−	0.851338i	$-0.324209\pi$
$157$	13.1469	1.04923	0.524617	+	0.851338i	$0.324209\pi$
$158$	4.18299	0.332781
$159$	1.07946	0.0856068
$160$	−28.6040	−2.26135
$161$	0	0
$162$	18.5197	1.45504
$163$	−18.6485	−1.46066	−0.730331	−	0.683094i	$-0.760634\pi$
$163$	−18.6485	−1.46066	−0.730331	+	0.683094i	$0.760634\pi$
$164$	−5.33581	−0.416656
$165$	−2.57489	−0.200455
$166$	16.2817	1.26370
$167$	0.972672	0.0752676	0.0376338	−	0.999292i	$-0.488018\pi$
$167$	0.972672	0.0752676	0.0376338	+	0.999292i	$0.488018\pi$
$168$	0	0
$169$	0	0
$170$	−36.7068	−2.81529
$171$	9.35101	0.715089
$172$	3.60397	0.274800
$173$	−2.45710	−0.186810	−0.0934050	−	0.995628i	$-0.529775\pi$
$173$	−2.45710	−0.186810	−0.0934050	+	0.995628i	$0.529775\pi$
$174$	−2.49836	−0.189400
$175$	0	0
$176$	10.7656	0.811491
$177$	−0.876962	−0.0659165
$178$	−27.2046	−2.03907
$179$	14.4726	1.08173	0.540866	−	0.841109i	$-0.318097\pi$
$179$	14.4726	1.08173	0.540866	+	0.841109i	$0.318097\pi$
$180$	−26.9463	−2.00846
$181$	−9.17885	−0.682259	−0.341129	−	0.940016i	$-0.610809\pi$
$181$	−9.17885	−0.682259	−0.341129	+	0.940016i	$0.610809\pi$
$182$	0	0
$183$	0.364043	0.0269108
$184$	−2.39757	−0.176752
$185$	−17.9040	−1.31633
$186$	0.565180	0.0414410
$187$	19.1200	1.39820
$188$	2.56536	0.187098
$189$	0	0
$190$	−24.1480	−1.75188
$191$	17.5840	1.27234	0.636168	−	0.771551i	$-0.280518\pi$
$191$	17.5840	1.27234	0.636168	+	0.771551i	$0.280518\pi$
$192$	−2.05090	−0.148011
$193$	−19.7558	−1.42206	−0.711028	−	0.703164i	$-0.751770\pi$
$193$	−19.7558	−1.42206	−0.711028	+	0.703164i	$0.751770\pi$
$194$	2.88855	0.207386
$195$	0	0
$196$	0	0
$197$	7.66020	0.545767	0.272883	−	0.962047i	$-0.412023\pi$
$197$	7.66020	0.545767	0.272883	+	0.962047i	$0.412023\pi$
$198$	25.1857	1.78987
$199$	6.54342	0.463851	0.231925	−	0.972734i	$-0.425498\pi$
$199$	6.54342	0.463851	0.231925	+	0.972734i	$0.425498\pi$
$200$	8.80684	0.622737
$201$	−0.699482	−0.0493377
$202$	−9.11412	−0.641267
$203$	0	0
$204$	−2.15657	−0.150990
$205$	−7.64229	−0.533761
$206$	−30.6812	−2.13766
$207$	6.46686	0.449478
$208$	0	0
$209$	12.5783	0.870060
$210$	0	0
$211$	20.0452	1.37997	0.689983	−	0.723825i	$-0.257618\pi$
$211$	20.0452	1.37997	0.689983	+	0.723825i	$0.257618\pi$
$212$	15.1954	1.04363
$213$	1.57514	0.107927
$214$	−20.6573	−1.41210
$215$	5.16184	0.352035
$216$	−1.17455	−0.0799183
$217$	0	0
$218$	14.1314	0.957102
$219$	0.552443	0.0373306
$220$	−36.2463	−2.44373
$221$	0	0
$222$	−1.88747	−0.126679
$223$	−27.7139	−1.85586	−0.927931	−	0.372752i	$-0.878414\pi$
$223$	−27.7139	−1.85586	−0.927931	+	0.372752i	$0.878414\pi$
$224$	0	0
$225$	−23.7543	−1.58362
$226$	37.2884	2.48038
$227$	−11.3711	−0.754726	−0.377363	−	0.926065i	$-0.623169\pi$
$227$	−11.3711	−0.754726	−0.377363	+	0.926065i	$0.623169\pi$
$228$	−1.41872	−0.0939573
$229$	8.71113	0.575647	0.287824	−	0.957683i	$-0.407068\pi$
$229$	8.71113	0.575647	0.287824	+	0.957683i	$0.407068\pi$
$230$	−16.7000	−1.10116
$231$	0	0
$232$	−7.23170	−0.474784
$233$	−3.36456	−0.220420	−0.110210	−	0.993908i	$-0.535152\pi$
$233$	−3.36456	−0.220420	−0.110210	+	0.993908i	$0.535152\pi$
$234$	0	0
$235$	3.67428	0.239684
$236$	−12.3449	−0.803583
$237$	0.351987	0.0228640
$238$	0	0
$239$	−19.8798	−1.28592	−0.642958	−	0.765902i	$-0.722293\pi$
$239$	−19.8798	−1.28592	−0.642958	+	0.765902i	$0.722293\pi$
$240$	−1.73916	−0.112262
$241$	18.8719	1.21565	0.607823	−	0.794073i	$-0.292043\pi$
$241$	18.8719	1.21565	0.607823	+	0.794073i	$0.292043\pi$
$242$	10.4976	0.674814
$243$	4.76059	0.305392
$244$	5.12458	0.328068
$245$	0	0
$246$	−0.805663	−0.0513672
$247$	0	0
$248$	1.63596	0.103883
$249$	1.37006	0.0868240
$250$	23.0200	1.45591
$251$	9.79601	0.618319	0.309159	−	0.951010i	$-0.399952\pi$
$251$	9.79601	0.618319	0.309159	+	0.951010i	$0.399952\pi$
$252$	0	0
$253$	8.69877	0.546887
$254$	41.4775	2.60253
$255$	−3.08878	−0.193427
$256$	4.84976	0.303110
$257$	−20.9394	−1.30617	−0.653083	−	0.757286i	$-0.726525\pi$
$257$	−20.9394	−1.30617	−0.653083	+	0.757286i	$0.726525\pi$
$258$	0.544170	0.0338785
$259$	0	0
$260$	0	0
$261$	19.5057	1.20737
$262$	−40.5421	−2.50470
$263$	−7.38679	−0.455489	−0.227745	−	0.973721i	$-0.573135\pi$
$263$	−7.38679	−0.455489	−0.227745	+	0.973721i	$0.573135\pi$
$264$	−0.785730	−0.0483583
$265$	21.7639	1.33694
$266$	0	0
$267$	−2.28919	−0.140096
$268$	−9.84651	−0.601471
$269$	−22.7892	−1.38948	−0.694740	−	0.719261i	$-0.744480\pi$
$269$	−22.7892	−1.38948	−0.694740	+	0.719261i	$0.744480\pi$
$270$	−8.18120	−0.497892
$271$	4.16633	0.253086	0.126543	−	0.991961i	$-0.459612\pi$
$271$	4.16633	0.253086	0.126543	+	0.991961i	$0.459612\pi$
$272$	12.9143	0.783042
$273$	0	0
$274$	13.6646	0.825507
$275$	−31.9526	−1.92681
$276$	−0.981145	−0.0590580
$277$	0.777101	0.0466915	0.0233457	−	0.999727i	$-0.492568\pi$
$277$	0.777101	0.0466915	0.0233457	+	0.999727i	$0.492568\pi$
$278$	5.16184	0.309587
$279$	−4.41259	−0.264175
$280$	0	0
$281$	11.8988	0.709824	0.354912	−	0.934900i	$-0.384511\pi$
$281$	11.8988	0.709824	0.354912	+	0.934900i	$0.384511\pi$
$282$	0.387349	0.0230663
$283$	15.9040	0.945397	0.472698	−	0.881224i	$-0.343280\pi$
$283$	15.9040	0.945397	0.472698	+	0.881224i	$0.343280\pi$
$284$	22.1730	1.31573
$285$	−2.03199	−0.120365
$286$	0	0
$287$	0	0
$288$	23.5431	1.38729
$289$	5.93602	0.349178
$290$	−50.3714	−2.95791
$291$	0.243064	0.0142487
$292$	7.77666	0.455094
$293$	−6.73698	−0.393579	−0.196789	−	0.980446i	$-0.563051\pi$
$293$	−6.73698	−0.393579	−0.196789	+	0.980446i	$0.563051\pi$
$294$	0	0
$295$	−17.6811	−1.02944
$296$	−5.46344	−0.317556
$297$	4.26146	0.247275
$298$	0.245181	0.0142030
$299$	0	0
$300$	3.60397	0.208075
$301$	0	0
$302$	−25.0874	−1.44362
$303$	−0.766929	−0.0440589
$304$	8.49578	0.487266
$305$	7.33976	0.420274
$306$	30.2123	1.72712
$307$	14.7179	0.839996	0.419998	−	0.907525i	$-0.362031\pi$
$307$	14.7179	0.839996	0.419998	+	0.907525i	$0.362031\pi$
$308$	0	0
$309$	−2.58174	−0.146870
$310$	11.3950	0.647195
$311$	−28.6578	−1.62503	−0.812517	−	0.582938i	$-0.801903\pi$
$311$	−28.6578	−1.62503	−0.812517	+	0.582938i	$0.801903\pi$
$312$	0	0
$313$	32.8251	1.85538	0.927692	−	0.373347i	$-0.121790\pi$
$313$	32.8251	1.85538	0.927692	+	0.373347i	$0.121790\pi$
$314$	27.9435	1.57694
$315$	0	0
$316$	4.95488	0.278734
$317$	10.4121	0.584802	0.292401	−	0.956296i	$-0.405546\pi$
$317$	10.4121	0.584802	0.292401	+	0.956296i	$0.405546\pi$
$318$	2.29438	0.128663
$319$	26.2377	1.46903
$320$	−41.3498	−2.31152
$321$	−1.73826	−0.0970201
$322$	0	0
$323$	15.0887	0.839558
$324$	21.9371	1.21873
$325$	0	0
$326$	−39.6371	−2.19530
$327$	1.18912	0.0657587
$328$	−2.33205	−0.128766
$329$	0	0
$330$	−5.47290	−0.301273
$331$	−4.46015	−0.245152	−0.122576	−	0.992459i	$-0.539115\pi$
$331$	−4.46015	−0.245152	−0.122576	+	0.992459i	$0.539115\pi$
$332$	19.2861	1.05846
$333$	14.7363	0.807542
$334$	2.06740	0.113123
$335$	−14.1028	−0.770519
$336$	0	0
$337$	−10.7949	−0.588034	−0.294017	−	0.955800i	$-0.594992\pi$
$337$	−10.7949	−0.588034	−0.294017	+	0.955800i	$0.594992\pi$
$338$	0	0
$339$	3.13772	0.170417
$340$	−43.4804	−2.35805
$341$	−5.93550	−0.321425
$342$	19.8755	1.07474
$343$	0	0
$344$	1.57514	0.0849259
$345$	−1.40526	−0.0756567
$346$	−5.22255	−0.280766
$347$	−4.07031	−0.218506	−0.109253	−	0.994014i	$-0.534846\pi$
$347$	−4.07031	−0.218506	−0.109253	+	0.994014i	$0.534846\pi$
$348$	−2.95938	−0.158640
$349$	23.8727	1.27788	0.638938	−	0.769258i	$-0.279374\pi$
$349$	23.8727	1.27788	0.638938	+	0.769258i	$0.279374\pi$
$350$	0	0
$351$	0	0
$352$	31.6685	1.68794
$353$	26.0839	1.38831	0.694154	−	0.719826i	$-0.255779\pi$
$353$	26.0839	1.38831	0.694154	+	0.719826i	$0.255779\pi$
$354$	−1.86397	−0.0990691
$355$	31.7577	1.68552
$356$	−32.2246	−1.70790
$357$	0	0
$358$	30.7613	1.62579
$359$	22.8944	1.20832	0.604160	−	0.796863i	$-0.293509\pi$
$359$	22.8944	1.20832	0.604160	+	0.796863i	$0.293509\pi$
$360$	−11.7771	−0.620708
$361$	−9.07374	−0.477565
$362$	−19.5096	−1.02540
$363$	0.883349	0.0463638
$364$	0	0
$365$	11.1382	0.583002
$366$	0.773770	0.0404456
$367$	18.1601	0.947947	0.473974	−	0.880539i	$-0.342819\pi$
$367$	18.1601	0.947947	0.473974	+	0.880539i	$0.342819\pi$
$368$	5.87542	0.306277
$369$	6.29014	0.327452
$370$	−38.0548	−1.97838
$371$	0	0
$372$	0.669473	0.0347105
$373$	−15.8691	−0.821673	−0.410836	−	0.911709i	$-0.634763\pi$
$373$	−15.8691	−0.821673	−0.410836	+	0.911709i	$0.634763\pi$
$374$	40.6394	2.10142
$375$	1.93707	0.100030
$376$	1.12121	0.0578220
$377$	0	0
$378$	0	0
$379$	−27.7634	−1.42611	−0.713055	−	0.701108i	$-0.752689\pi$
$379$	−27.7634	−1.42611	−0.713055	+	0.701108i	$0.752689\pi$
$380$	−28.6040	−1.46736
$381$	3.49022	0.178810
$382$	37.3747	1.91226
$383$	−26.2469	−1.34115	−0.670576	−	0.741841i	$-0.733953\pi$
$383$	−26.2469	−1.34115	−0.670576	+	0.741841i	$0.733953\pi$
$384$	−1.52172	−0.0776548
$385$	0	0
$386$	−41.9908	−2.13728
$387$	−4.24855	−0.215966
$388$	3.42158	0.173704
$389$	−25.2554	−1.28050	−0.640250	−	0.768167i	$-0.721169\pi$
$389$	−25.2554	−1.28050	−0.640250	+	0.768167i	$0.721169\pi$
$390$	0	0
$391$	10.4349	0.527714
$392$	0	0
$393$	−3.41151	−0.172088
$394$	16.2817	0.820259
$395$	7.09670	0.357074
$396$	29.8332	1.49918
$397$	−14.9765	−0.751651	−0.375826	−	0.926690i	$-0.622641\pi$
$397$	−14.9765	−0.751651	−0.375826	+	0.926690i	$0.622641\pi$
$398$	13.9080	0.697143
$399$	0	0
$400$	−21.5817	−1.07909
$401$	5.34408	0.266871	0.133435	−	0.991058i	$-0.457399\pi$
$401$	5.34408	0.266871	0.133435	+	0.991058i	$0.457399\pi$
$402$	−1.48674	−0.0741520
$403$	0	0
$404$	−10.7960	−0.537119
$405$	31.4198	1.56126
$406$	0	0
$407$	19.8222	0.982549
$408$	−0.942546	−0.0466630
$409$	−3.36517	−0.166397	−0.0831985	−	0.996533i	$-0.526514\pi$
$409$	−3.36517	−0.166397	−0.0831985	+	0.996533i	$0.526514\pi$
$410$	−16.2436	−0.802215
$411$	1.14984	0.0567173
$412$	−36.3428	−1.79048
$413$	0	0
$414$	13.7453	0.675542
$415$	27.6229	1.35595
$416$	0	0
$417$	0.434355	0.0212705
$418$	26.7351	1.30766
$419$	28.8639	1.41010	0.705048	−	0.709160i	$-0.250926\pi$
$419$	28.8639	1.41010	0.705048	+	0.709160i	$0.250926\pi$
$420$	0	0
$421$	16.6125	0.809644	0.404822	−	0.914395i	$-0.367333\pi$
$421$	16.6125	0.809644	0.404822	+	0.914395i	$0.367333\pi$
$422$	42.6058	2.07402
$423$	−3.02419	−0.147041
$424$	6.64127	0.322529
$425$	−38.3297	−1.85926
$426$	3.34795	0.162209
$427$	0	0
$428$	−24.4692	−1.18276
$429$	0	0
$430$	10.9714	0.529090
$431$	−20.5554	−0.990120	−0.495060	−	0.868859i	$-0.664854\pi$
$431$	−20.5554	−0.990120	−0.495060	+	0.868859i	$0.664854\pi$
$432$	2.87832	0.138483
$433$	−19.4092	−0.932748	−0.466374	−	0.884588i	$-0.654440\pi$
$433$	−19.4092	−0.932748	−0.466374	+	0.884588i	$0.654440\pi$
$434$	0	0
$435$	−4.23862	−0.203226
$436$	16.7391	0.801658
$437$	6.86469	0.328383
$438$	1.17421	0.0561060
$439$	−13.4251	−0.640746	−0.320373	−	0.947292i	$-0.603808\pi$
$439$	−13.4251	−0.640746	−0.320373	+	0.947292i	$0.603808\pi$
$440$	−15.8417	−0.755225
$441$	0	0
$442$	0	0
$443$	−33.5532	−1.59416	−0.797080	−	0.603874i	$-0.793623\pi$
$443$	−33.5532	−1.59416	−0.797080	+	0.603874i	$0.793623\pi$
$444$	−2.23577	−0.106105
$445$	−46.1542	−2.18792
$446$	−58.9057	−2.78927
$447$	0.0206313	0.000975829 0
$448$	0	0
$449$	−34.4284	−1.62478	−0.812388	−	0.583117i	$-0.801833\pi$
$449$	−34.4284	−1.62478	−0.812388	+	0.583117i	$0.801833\pi$
$450$	−50.4894	−2.38010
$451$	8.46105	0.398415
$452$	44.1692	2.07754
$453$	−2.11104	−0.0991852
$454$	−24.1691	−1.13431
$455$	0	0
$456$	−0.620064	−0.0290371
$457$	13.4793	0.630537	0.315269	−	0.949002i	$-0.397905\pi$
$457$	13.4793	0.630537	0.315269	+	0.949002i	$0.397905\pi$
$458$	18.5154	0.865168
$459$	5.11197	0.238606
$460$	−19.7816	−0.922324
$461$	−1.35900	−0.0632951	−0.0316476	−	0.999499i	$-0.510075\pi$
$461$	−1.35900	−0.0632951	−0.0316476	+	0.999499i	$0.510075\pi$
$462$	0	0
$463$	−2.49836	−0.116109	−0.0580543	−	0.998313i	$-0.518490\pi$
$463$	−2.49836	−0.116109	−0.0580543	+	0.998313i	$0.518490\pi$
$464$	17.7218	0.822712
$465$	0.958863	0.0444662
$466$	−7.15135	−0.331280
$467$	26.2182	1.21323	0.606617	−	0.794994i	$-0.292526\pi$
$467$	26.2182	1.21323	0.606617	+	0.794994i	$0.292526\pi$
$468$	0	0
$469$	0	0
$470$	7.80965	0.360232
$471$	2.35137	0.108346
$472$	−5.39542	−0.248344
$473$	−5.71485	−0.262769
$474$	0.748146	0.0343635
$475$	−25.2156	−1.15697
$476$	0	0
$477$	−17.9132	−0.820188
$478$	−42.2542	−1.93266
$479$	23.8461	1.08956	0.544778	−	0.838580i	$-0.316614\pi$
$479$	23.8461	1.08956	0.544778	+	0.838580i	$0.316614\pi$
$480$	−5.11595	−0.233510
$481$	0	0
$482$	40.1120	1.82705
$483$	0	0
$484$	12.4348	0.565217
$485$	4.90061	0.222525
$486$	10.1186	0.458989
$487$	−10.5947	−0.480090	−0.240045	−	0.970762i	$-0.577162\pi$
$487$	−10.5947	−0.480090	−0.240045	+	0.970762i	$0.577162\pi$
$488$	2.23974	0.101388
$489$	−3.33536	−0.150830
$490$	0	0
$491$	−19.7704	−0.892224	−0.446112	−	0.894977i	$-0.647192\pi$
$491$	−19.7704	−0.892224	−0.446112	+	0.894977i	$0.647192\pi$
$492$	−0.954332	−0.0430246
$493$	31.4742	1.41753
$494$	0	0
$495$	42.7291	1.92053
$496$	−4.00902	−0.180010
$497$	0	0
$498$	2.91205	0.130492
$499$	13.1906	0.590492	0.295246	−	0.955421i	$-0.404598\pi$
$499$	13.1906	0.590492	0.295246	+	0.955421i	$0.404598\pi$
$500$	27.2679	1.21946
$501$	0.173967	0.00777226
$502$	20.8213	0.929301
$503$	−37.9046	−1.69008	−0.845040	−	0.534703i	$-0.820424\pi$
$503$	−37.9046	−1.69008	−0.845040	+	0.534703i	$0.820424\pi$
$504$	0	0
$505$	−15.4627	−0.688080
$506$	18.4891	0.821943
$507$	0	0
$508$	49.1314	2.17985
$509$	−27.6626	−1.22612	−0.613062	−	0.790035i	$-0.710062\pi$
$509$	−27.6626	−1.22612	−0.613062	+	0.790035i	$0.710062\pi$
$510$	−6.56518	−0.290711
$511$	0	0
$512$	27.3244	1.20758
$513$	3.36296	0.148478
$514$	−44.5066	−1.96310
$515$	−52.0525	−2.29371
$516$	0.644585	0.0283763
$517$	−4.06792	−0.178907
$518$	0	0
$519$	−0.439464	−0.0192903
$520$	0	0
$521$	−15.5668	−0.681993	−0.340996	−	0.940065i	$-0.610764\pi$
$521$	−15.5668	−0.681993	−0.340996	+	0.940065i	$0.610764\pi$
$522$	41.4592	1.81462
$523$	−27.2338	−1.19085	−0.595425	−	0.803411i	$-0.703016\pi$
$523$	−27.2338	−1.19085	−0.595425	+	0.803411i	$0.703016\pi$
$524$	−48.0234	−2.09791
$525$	0	0
$526$	−15.7005	−0.684577
$527$	−7.12011	−0.310157
$528$	1.92548	0.0837959
$529$	−18.2526	−0.793591
$530$	46.2589	2.00936
$531$	14.5528	0.631538
$532$	0	0
$533$	0	0
$534$	−4.86566	−0.210557
$535$	−35.0464	−1.51519
$536$	−4.30349	−0.185883
$537$	2.58849	0.111701
$538$	−48.4381	−2.08832
$539$	0	0
$540$	−9.69089	−0.417029
$541$	24.2055	1.04068	0.520338	−	0.853961i	$-0.325806\pi$
$541$	24.2055	1.04068	0.520338	+	0.853961i	$0.325806\pi$
$542$	8.85548	0.380376
$543$	−1.64168	−0.0704512
$544$	37.9889	1.62876
$545$	23.9749	1.02697
$546$	0	0
$547$	−22.2177	−0.949960	−0.474980	−	0.879997i	$-0.657545\pi$
$547$	−22.2177	−0.949960	−0.474980	+	0.879997i	$0.657545\pi$
$548$	16.1861	0.691436
$549$	−6.04114	−0.257829
$550$	−67.9148	−2.89590
$551$	20.7057	0.882091
$552$	−0.428817	−0.0182517
$553$	0	0
$554$	1.65172	0.0701749
$555$	−3.20221	−0.135926
$556$	6.11436	0.259306
$557$	−22.3204	−0.945746	−0.472873	−	0.881131i	$-0.656783\pi$
$557$	−22.3204	−0.945746	−0.472873	+	0.881131i	$0.656783\pi$
$558$	−9.37891	−0.397041
$559$	0	0
$560$	0	0
$561$	3.41970	0.144380
$562$	25.2908	1.06683
$563$	−26.7039	−1.12543	−0.562717	−	0.826649i	$-0.690244\pi$
$563$	−26.7039	−1.12543	−0.562717	+	0.826649i	$0.690244\pi$
$564$	0.458826	0.0193201
$565$	63.2620	2.66145
$566$	33.8039	1.42088
$567$	0	0
$568$	9.69090	0.406621
$569$	6.61021	0.277114	0.138557	−	0.990354i	$-0.455754\pi$
$569$	6.61021	0.277114	0.138557	+	0.990354i	$0.455754\pi$
$570$	−4.31897	−0.180902
$571$	−42.1286	−1.76303	−0.881513	−	0.472159i	$-0.843475\pi$
$571$	−42.1286	−1.76303	−0.881513	+	0.472159i	$0.843475\pi$
$572$	0	0
$573$	3.14498	0.131383
$574$	0	0
$575$	−17.4383	−0.727227
$576$	34.0338	1.41807
$577$	−15.8839	−0.661255	−0.330628	−	0.943761i	$-0.607260\pi$
$577$	−15.8839	−0.661255	−0.330628	+	0.943761i	$0.607260\pi$
$578$	12.6170	0.524796
$579$	−3.53342	−0.146844
$580$	−59.6665	−2.47752
$581$	0	0
$582$	0.516630	0.0214150
$583$	−24.0955	−0.997936
$584$	3.39885	0.140645
$585$	0	0
$586$	−14.3194	−0.591528
$587$	−18.5676	−0.766366	−0.383183	−	0.923672i	$-0.625172\pi$
$587$	−18.5676	−0.766366	−0.383183	+	0.923672i	$0.625172\pi$
$588$	0	0
$589$	−4.68404	−0.193003
$590$	−37.5811	−1.54719
$591$	1.37006	0.0563567
$592$	13.3885	0.550265
$593$	20.6099	0.846349	0.423175	−	0.906048i	$-0.360916\pi$
$593$	20.6099	0.846349	0.423175	+	0.906048i	$0.360916\pi$
$594$	9.05770	0.371642
$595$	0	0
$596$	0.290425	0.0118963
$597$	1.17032	0.0478980
$598$	0	0
$599$	−13.6045	−0.555864	−0.277932	−	0.960601i	$-0.589649\pi$
$599$	−13.6045	−0.555864	−0.277932	+	0.960601i	$0.589649\pi$
$600$	1.57514	0.0643049
$601$	12.1503	0.495621	0.247810	−	0.968809i	$-0.420289\pi$
$601$	12.1503	0.495621	0.247810	+	0.968809i	$0.420289\pi$
$602$	0	0
$603$	11.6076	0.472698
$604$	−29.7168	−1.20916
$605$	17.8099	0.724076
$606$	−1.63010	−0.0662183
$607$	−35.2332	−1.43007	−0.715035	−	0.699088i	$-0.753589\pi$
$607$	−35.2332	−1.43007	−0.715035	+	0.699088i	$0.753589\pi$
$608$	24.9914	1.01354
$609$	0	0
$610$	15.6006	0.631650
$611$	0	0
$612$	35.7874	1.44662
$613$	30.0621	1.21419	0.607097	−	0.794627i	$-0.292334\pi$
$613$	30.0621	1.21419	0.607097	+	0.794627i	$0.292334\pi$
$614$	31.2828	1.26247
$615$	−1.36686	−0.0551170
$616$	0	0
$617$	−7.01712	−0.282499	−0.141249	−	0.989974i	$-0.545112\pi$
$617$	−7.01712	−0.282499	−0.141249	+	0.989974i	$0.545112\pi$
$618$	−5.48746	−0.220738
$619$	−43.8482	−1.76241	−0.881203	−	0.472737i	$-0.843266\pi$
$619$	−43.8482	−1.76241	−0.881203	+	0.472737i	$0.843266\pi$
$620$	13.4978	0.542084
$621$	2.32572	0.0933279
$622$	−60.9118	−2.44234
$623$	0	0
$624$	0	0
$625$	−0.962290	−0.0384916
$626$	69.7694	2.78855
$627$	2.24969	0.0898438
$628$	33.1000	1.32083
$629$	23.7783	0.948103
$630$	0	0
$631$	23.4936	0.935267	0.467634	−	0.883922i	$-0.345107\pi$
$631$	23.4936	0.935267	0.467634	+	0.883922i	$0.345107\pi$
$632$	2.16557	0.0861416
$633$	3.58517	0.142498
$634$	22.1308	0.878927
$635$	70.3692	2.79252
$636$	2.71777	0.107766
$637$	0	0
$638$	55.7680	2.20788
$639$	−26.1388	−1.03403
$640$	−30.6806	−1.21276
$641$	7.40466	0.292467	0.146233	−	0.989250i	$-0.453285\pi$
$641$	7.40466	0.292467	0.146233	+	0.989250i	$0.453285\pi$
$642$	−3.69465	−0.145816
$643$	39.9607	1.57590	0.787948	−	0.615742i	$-0.211144\pi$
$643$	39.9607	1.57590	0.787948	+	0.615742i	$0.211144\pi$
$644$	0	0
$645$	0.923218	0.0363517
$646$	32.0709	1.26181
$647$	−27.2468	−1.07118	−0.535591	−	0.844478i	$-0.679911\pi$
$647$	−27.2468	−1.07118	−0.535591	+	0.844478i	$0.679911\pi$
$648$	9.58777	0.376643
$649$	19.5754	0.768402
$650$	0	0
$651$	0	0
$652$	−46.9514	−1.83876
$653$	19.1451	0.749205	0.374603	−	0.927185i	$-0.377779\pi$
$653$	19.1451	0.749205	0.374603	+	0.927185i	$0.377779\pi$
$654$	2.52747	0.0988319
$655$	−68.7823	−2.68755
$656$	5.71485	0.223128
$657$	−9.16755	−0.357660
$658$	0	0
$659$	41.5725	1.61943	0.809717	−	0.586820i	$-0.199620\pi$
$659$	41.5725	1.61943	0.809717	+	0.586820i	$0.199620\pi$
$660$	−6.48281	−0.252343
$661$	34.2046	1.33041	0.665203	−	0.746662i	$-0.268345\pi$
$661$	34.2046	1.33041	0.665203	+	0.746662i	$0.268345\pi$
$662$	−9.48000	−0.368451
$663$	0	0
$664$	8.42915	0.327115
$665$	0	0
$666$	31.3218	1.21369
$667$	14.3194	0.554449
$668$	2.44890	0.0947510
$669$	−4.95676	−0.191639
$670$	−29.9754	−1.15805
$671$	−8.12611	−0.313705
$672$	0	0
$673$	21.4308	0.826098	0.413049	−	0.910709i	$-0.364464\pi$
$673$	21.4308	0.826098	0.413049	+	0.910709i	$0.364464\pi$
$674$	−22.9444	−0.883786
$675$	−8.54290	−0.328816
$676$	0	0
$677$	−9.78166	−0.375940	−0.187970	−	0.982175i	$-0.560191\pi$
$677$	−9.78166	−0.375940	−0.187970	+	0.982175i	$0.560191\pi$
$678$	6.66919	0.256129
$679$	0	0
$680$	−19.0034	−0.728748
$681$	−2.03377	−0.0779342
$682$	−12.6159	−0.483086
$683$	−15.2764	−0.584533	−0.292267	−	0.956337i	$-0.594409\pi$
$683$	−15.2764	−0.584533	−0.292267	+	0.956337i	$0.594409\pi$
$684$	23.5431	0.900193
$685$	23.1828	0.885769
$686$	0	0
$687$	1.55802	0.0594423
$688$	−3.85999	−0.147161
$689$	0	0
$690$	−2.98687	−0.113708
$691$	42.4572	1.61515	0.807573	−	0.589767i	$-0.200780\pi$
$691$	42.4572	1.61515	0.807573	+	0.589767i	$0.200780\pi$
$692$	−6.18627	−0.235167
$693$	0	0
$694$	−8.65141	−0.328403
$695$	8.75738	0.332186
$696$	−1.29342	−0.0490270
$697$	10.1497	0.384448
$698$	50.7412	1.92058
$699$	−0.601767	−0.0227609
$700$	0	0
$701$	−2.79985	−0.105749	−0.0528744	−	0.998601i	$-0.516838\pi$
$701$	−2.79985	−0.105749	−0.0528744	+	0.998601i	$0.516838\pi$
$702$	0	0
$703$	15.6428	0.589980
$704$	45.7798	1.72539
$705$	0.657161	0.0247501
$706$	55.4412	2.08656
$707$	0	0
$708$	−2.20793	−0.0829793
$709$	14.5664	0.547052	0.273526	−	0.961865i	$-0.411810\pi$
$709$	14.5664	0.547052	0.273526	+	0.961865i	$0.411810\pi$
$710$	67.5006	2.53325
$711$	−5.84108	−0.219058
$712$	−14.0840	−0.527821
$713$	−3.23934	−0.121314
$714$	0	0
$715$	0	0
$716$	36.4377	1.36174
$717$	−3.55558	−0.132786
$718$	48.6618	1.81604
$719$	−34.5057	−1.28685	−0.643423	−	0.765511i	$-0.722486\pi$
$719$	−34.5057	−1.28685	−0.643423	+	0.765511i	$0.722486\pi$
$720$	28.8606	1.07557
$721$	0	0
$722$	−19.2861	−0.717756
$723$	3.37532	0.125530
$724$	−23.1097	−0.858864
$725$	−52.5984	−1.95345
$726$	1.87755	0.0696824
$727$	−35.7571	−1.32616	−0.663078	−	0.748550i	$-0.730750\pi$
$727$	−35.7571	−1.32616	−0.663078	+	0.748550i	$0.730750\pi$
$728$	0	0
$729$	−25.2879	−0.936590
$730$	23.6742	0.876222
$731$	−6.85543	−0.253557
$732$	0.916554	0.0338768
$733$	−41.0500	−1.51622	−0.758108	−	0.652129i	$-0.773876\pi$
$733$	−41.0500	−1.51622	−0.758108	+	0.652129i	$0.773876\pi$
$734$	38.5990	1.42472
$735$	0	0
$736$	17.2833	0.637071
$737$	15.6137	0.575139
$738$	13.3696	0.492143
$739$	0.726409	0.0267214	0.0133607	−	0.999911i	$-0.495747\pi$
$739$	0.726409	0.0267214	0.0133607	+	0.999911i	$0.495747\pi$
$740$	−45.0771	−1.65707
$741$	0	0
$742$	0	0
$743$	−16.4547	−0.603664	−0.301832	−	0.953361i	$-0.597598\pi$
$743$	−16.4547	−0.603664	−0.301832	+	0.953361i	$0.597598\pi$
$744$	0.292598	0.0107272
$745$	0.415965	0.0152398
$746$	−33.7297	−1.23493
$747$	−22.7356	−0.831850
$748$	48.1387	1.76012
$749$	0	0
$750$	4.11723	0.150340
$751$	25.1707	0.918494	0.459247	−	0.888309i	$-0.348119\pi$
$751$	25.1707	0.918494	0.459247	+	0.888309i	$0.348119\pi$
$752$	−2.74760	−0.100195
$753$	1.75206	0.0638486
$754$	0	0
$755$	−42.5623	−1.54900
$756$	0	0
$757$	44.0743	1.60191	0.800953	−	0.598727i	$-0.204327\pi$
$757$	44.0743	1.60191	0.800953	+	0.598727i	$0.204327\pi$
$758$	−59.0108	−2.14337
$759$	1.55581	0.0564724
$760$	−12.5016	−0.453481
$761$	−38.7022	−1.40295	−0.701477	−	0.712692i	$-0.747476\pi$
$761$	−38.7022	−1.40295	−0.701477	+	0.712692i	$0.747476\pi$
$762$	7.41844	0.268742
$763$	0	0
$764$	44.2715	1.60169
$765$	51.2570	1.85320
$766$	−55.7874	−2.01568
$767$	0	0
$768$	0.867401	0.0312996
$769$	−36.1506	−1.30362	−0.651811	−	0.758381i	$-0.725991\pi$
$769$	−36.1506	−1.30362	−0.651811	+	0.758381i	$0.725991\pi$
$770$	0	0
$771$	−3.74511	−0.134877
$772$	−49.7394	−1.79016
$773$	30.0731	1.08165	0.540827	−	0.841134i	$-0.318111\pi$
$773$	30.0731	1.08165	0.540827	+	0.841134i	$0.318111\pi$
$774$	−9.03026	−0.324586
$775$	11.8988	0.427418
$776$	1.49543	0.0536827
$777$	0	0
$778$	−53.6801	−1.92453
$779$	6.67709	0.239232
$780$	0	0
$781$	−35.1601	−1.25813
$782$	22.1792	0.793127
$783$	7.01496	0.250694
$784$	0	0
$785$	47.4079	1.69206
$786$	−7.25114	−0.258640
$787$	−36.8472	−1.31346	−0.656730	−	0.754126i	$-0.728061\pi$
$787$	−36.8472	−1.31346	−0.656730	+	0.754126i	$0.728061\pi$
$788$	19.2861	0.687040
$789$	−1.32116	−0.0470345
$790$	15.0840	0.536663
$791$	0	0
$792$	13.0388	0.463315
$793$	0	0
$794$	−31.8325	−1.12969
$795$	3.89256	0.138055
$796$	16.4744	0.583920
$797$	−27.5910	−0.977323	−0.488661	−	0.872474i	$-0.662515\pi$
$797$	−27.5910	−0.977323	−0.488661	+	0.872474i	$0.662515\pi$
$798$	0	0
$799$	−4.87980	−0.172635
$800$	−63.4854	−2.24455
$801$	37.9882	1.34225
$802$	11.3588	0.401093
$803$	−12.3315	−0.435170
$804$	−1.76109	−0.0621089
$805$	0	0
$806$	0	0
$807$	−4.07594	−0.143480
$808$	−4.71845	−0.165995
$809$	35.7102	1.25550	0.627752	−	0.778413i	$-0.283975\pi$
$809$	35.7102	1.25550	0.627752	+	0.778413i	$0.283975\pi$
$810$	66.7824	2.34649
$811$	2.22418	0.0781015	0.0390508	−	0.999237i	$-0.487567\pi$
$811$	2.22418	0.0781015	0.0390508	+	0.999237i	$0.487567\pi$
$812$	0	0
$813$	0.745166	0.0261341
$814$	42.1319	1.47672
$815$	−67.2469	−2.35556
$816$	2.30977	0.0808582
$817$	−4.50992	−0.157782
$818$	−7.15264	−0.250086
$819$	0	0
$820$	−19.2411	−0.671927
$821$	53.3694	1.86260	0.931302	−	0.364248i	$-0.118674\pi$
$821$	53.3694	1.86260	0.931302	+	0.364248i	$0.118674\pi$
$822$	2.44397	0.0852432
$823$	51.2086	1.78502	0.892509	−	0.451029i	$-0.148943\pi$
$823$	51.2086	1.78502	0.892509	+	0.451029i	$0.148943\pi$
$824$	−15.8839	−0.553342
$825$	−5.71485	−0.198966
$826$	0	0
$827$	−8.97196	−0.311986	−0.155993	−	0.987758i	$-0.549858\pi$
$827$	−8.97196	−0.311986	−0.155993	+	0.987758i	$0.549858\pi$
$828$	16.2817	0.565827
$829$	−40.5716	−1.40911	−0.704554	−	0.709650i	$-0.748853\pi$
$829$	−40.5716	−1.40911	−0.704554	+	0.709650i	$0.748853\pi$
$830$	58.7122	2.03793
$831$	0.138988	0.00482144
$832$	0	0
$833$	0	0
$834$	0.923218	0.0319684
$835$	3.50748	0.121381
$836$	31.6685	1.09528
$837$	−1.58693	−0.0548522
$838$	61.3500	2.11930
$839$	−32.3005	−1.11514	−0.557568	−	0.830131i	$-0.688266\pi$
$839$	−32.3005	−1.11514	−0.557568	+	0.830131i	$0.688266\pi$
$840$	0	0
$841$	14.1909	0.489342
$842$	35.3097	1.21685
$843$	2.12816	0.0732976
$844$	50.4679	1.73718
$845$	0	0
$846$	−6.42788	−0.220995
$847$	0	0
$848$	−16.2749	−0.558882
$849$	2.84451	0.0976232
$850$	−81.4693	−2.79437
$851$	10.8181	0.370839
$852$	3.96575	0.135864
$853$	35.5887	1.21853	0.609267	−	0.792965i	$-0.291464\pi$
$853$	35.5887	1.21853	0.609267	+	0.792965i	$0.291464\pi$
$854$	0	0
$855$	33.7200	1.15320
$856$	−10.6945	−0.365529
$857$	−46.0229	−1.57211	−0.786055	−	0.618156i	$-0.787880\pi$
$857$	−46.0229	−1.57211	−0.786055	+	0.618156i	$0.787880\pi$
$858$	0	0
$859$	−25.2458	−0.861377	−0.430689	−	0.902501i	$-0.641729\pi$
$859$	−25.2458	−0.861377	−0.430689	+	0.902501i	$0.641729\pi$
$860$	12.9960	0.443160
$861$	0	0
$862$	−43.6903	−1.48810
$863$	−13.8337	−0.470904	−0.235452	−	0.971886i	$-0.575657\pi$
$863$	−13.8337	−0.470904	−0.235452	+	0.971886i	$0.575657\pi$
$864$	8.46696	0.288052
$865$	−8.86038	−0.301262
$866$	−41.2541	−1.40187
$867$	1.06168	0.0360567
$868$	0	0
$869$	−7.85701	−0.266531
$870$	−9.00915	−0.305439
$871$	0	0
$872$	7.31596	0.247750
$873$	−4.03354	−0.136515
$874$	14.5908	0.493542
$875$	0	0
$876$	1.39089	0.0469938
$877$	3.65416	0.123392	0.0616961	−	0.998095i	$-0.480349\pi$
$877$	3.65416	0.123392	0.0616961	+	0.998095i	$0.480349\pi$
$878$	−28.5349	−0.963008
$879$	−1.20494	−0.0406416
$880$	38.8212	1.30866
$881$	−36.6320	−1.23416	−0.617082	−	0.786899i	$-0.711685\pi$
$881$	−36.6320	−1.23416	−0.617082	+	0.786899i	$0.711685\pi$
$882$	0	0
$883$	−7.11145	−0.239319	−0.119660	−	0.992815i	$-0.538180\pi$
$883$	−7.11145	−0.239319	−0.119660	+	0.992815i	$0.538180\pi$
$884$	0	0
$885$	−3.16235	−0.106301
$886$	−71.3169	−2.39594
$887$	6.73546	0.226155	0.113077	−	0.993586i	$-0.463929\pi$
$887$	6.73546	0.226155	0.113077	+	0.993586i	$0.463929\pi$
$888$	−0.977160	−0.0327913
$889$	0	0
$890$	−98.1004	−3.28833
$891$	−34.7859	−1.16537
$892$	−69.7756	−2.33626
$893$	−3.21023	−0.107426
$894$	0.0438517	0.00146662
$895$	52.1885	1.74447
$896$	0	0
$897$	0	0
$898$	−73.1772	−2.44195
$899$	−9.77066	−0.325870
$900$	−59.8063	−1.99354
$901$	−28.9046	−0.962950
$902$	17.9839	0.598798
$903$	0	0
$904$	19.3045	0.642058
$905$	−33.0992	−1.10025
$906$	−4.48699	−0.149070
$907$	−4.93260	−0.163784	−0.0818921	−	0.996641i	$-0.526096\pi$
$907$	−4.93260	−0.163784	−0.0818921	+	0.996641i	$0.526096\pi$
$908$	−28.6291	−0.950090
$909$	12.7269	0.422123
$910$	0	0
$911$	−26.6258	−0.882152	−0.441076	−	0.897470i	$-0.645403\pi$
$911$	−26.6258	−0.882152	−0.441076	+	0.897470i	$0.645403\pi$
$912$	1.51951	0.0503159
$913$	−30.5823	−1.01213
$914$	28.6502	0.947665
$915$	1.31275	0.0433981
$916$	21.9321	0.724656
$917$	0	0
$918$	10.8654	0.358613
$919$	−32.3836	−1.06824	−0.534118	−	0.845410i	$-0.679356\pi$
$919$	−32.3836	−1.06824	−0.534118	+	0.845410i	$0.679356\pi$
$920$	−8.64572	−0.285041
$921$	2.63237	0.0867394
$922$	−2.88855	−0.0951293
$923$	0	0
$924$	0	0
$925$	−39.7373	−1.30655
$926$	−5.31024	−0.174505
$927$	42.8429	1.40714
$928$	52.1308	1.71128
$929$	28.0271	0.919539	0.459769	−	0.888038i	$-0.347932\pi$
$929$	28.0271	0.919539	0.459769	+	0.888038i	$0.347932\pi$
$930$	2.03805	0.0668304
$931$	0	0
$932$	−8.47099	−0.277476
$933$	−5.12557	−0.167804
$934$	55.7266	1.82343
$935$	68.9473	2.25482
$936$	0	0
$937$	−14.1324	−0.461686	−0.230843	−	0.972991i	$-0.574148\pi$
$937$	−14.1324	−0.461686	−0.230843	+	0.972991i	$0.574148\pi$
$938$	0	0
$939$	5.87091	0.191590
$940$	9.25076	0.301727
$941$	8.97295	0.292510	0.146255	−	0.989247i	$-0.453278\pi$
$941$	8.97295	0.292510	0.146255	+	0.989247i	$0.453278\pi$
$942$	4.99782	0.162838
$943$	4.61767	0.150372
$944$	13.2218	0.430334
$945$	0	0
$946$	−12.1469	−0.394929
$947$	46.2958	1.50441	0.752205	−	0.658929i	$-0.228990\pi$
$947$	46.2958	1.50441	0.752205	+	0.658929i	$0.228990\pi$
$948$	0.886202	0.0287825
$949$	0	0
$950$	−53.5954	−1.73887
$951$	1.86225	0.0603876
$952$	0	0
$953$	19.1097	0.619023	0.309512	−	0.950896i	$-0.399834\pi$
$953$	19.1097	0.619023	0.309512	+	0.950896i	$0.399834\pi$
$954$	−38.0743	−1.23270
$955$	63.4085	2.05185
$956$	−50.0514	−1.61878
$957$	4.69273	0.151694
$958$	50.6846	1.63755
$959$	0	0
$960$	−7.39560	−0.238692
$961$	−28.7897	−0.928699
$962$	0	0
$963$	28.8456	0.929538
$964$	47.5139	1.53032
$965$	−71.2400	−2.29330
$966$	0	0
$967$	−22.5432	−0.724942	−0.362471	−	0.931995i	$-0.618067\pi$
$967$	−22.5432	−0.724942	−0.362471	+	0.931995i	$0.618067\pi$
$968$	5.43472	0.174678
$969$	2.69868	0.0866941
$970$	10.4162	0.334444
$971$	−27.2857	−0.875640	−0.437820	−	0.899063i	$-0.644249\pi$
$971$	−27.2857	−0.875640	−0.437820	+	0.899063i	$0.644249\pi$
$972$	11.9858	0.384444
$973$	0	0
$974$	−22.5188	−0.721550
$975$	0	0
$976$	−5.48863	−0.175687
$977$	56.1841	1.79749	0.898744	−	0.438474i	$-0.144481\pi$
$977$	56.1841	1.79749	0.898744	+	0.438474i	$0.144481\pi$
$978$	−7.08928	−0.226690
$979$	51.0990	1.63313
$980$	0	0
$981$	−19.7330	−0.630026
$982$	−42.0217	−1.34097
$983$	26.4890	0.844869	0.422435	−	0.906393i	$-0.361176\pi$
$983$	26.4890	0.844869	0.422435	+	0.906393i	$0.361176\pi$
$984$	−0.417098	−0.0132966
$985$	27.6229	0.880138
$986$	66.8982	2.13047
$987$	0	0
$988$	0	0
$989$	−3.11892	−0.0991758
$990$	90.8203	2.88646
$991$	5.11258	0.162407	0.0812033	−	0.996698i	$-0.474124\pi$
$991$	5.11258	0.162407	0.0812033	+	0.996698i	$0.474124\pi$
$992$	−11.7930	−0.374430
$993$	−0.797717	−0.0253148
$994$	0	0
$995$	23.5957	0.748035
$996$	3.44941	0.109299
$997$	−2.03542	−0.0644623	−0.0322311	−	0.999480i	$-0.510261\pi$
$997$	−2.03542	−0.0644623	−0.0322311	+	0.999480i	$0.510261\pi$
$998$	28.0365	0.887480
$999$	5.29970	0.167675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cj.1.7		8
7.3	odd	6		1183.2.e.i.170.2		16
7.5	odd	6		1183.2.e.i.508.2		16
7.6	odd	2		8281.2.a.ck.1.7		8
13.5	odd	4		637.2.c.e.246.2		8
13.8	odd	4		637.2.c.e.246.7		8
13.12	even	2	inner	8281.2.a.cj.1.2		8
91.5	even	12		91.2.r.a.25.2	✓	16
91.12	odd	6		1183.2.e.i.508.7		16
91.18	odd	12		637.2.r.f.324.7		16
91.31	even	12		91.2.r.a.51.7	yes	16
91.34	even	4		637.2.c.f.246.7		8
91.38	odd	6		1183.2.e.i.170.7		16
91.44	odd	12		637.2.r.f.116.2		16
91.47	even	12		91.2.r.a.25.7	yes	16
91.60	odd	12		637.2.r.f.324.2		16
91.73	even	12		91.2.r.a.51.2	yes	16
91.83	even	4		637.2.c.f.246.2		8
91.86	odd	12		637.2.r.f.116.7		16
91.90	odd	2		8281.2.a.ck.1.2		8
273.5	odd	12		819.2.dl.e.298.7		16
273.47	odd	12		819.2.dl.e.298.2		16
273.122	odd	12		819.2.dl.e.415.2		16
273.164	odd	12		819.2.dl.e.415.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.r.a.25.2	✓	16	91.5	even	12
91.2.r.a.25.7	yes	16	91.47	even	12
91.2.r.a.51.2	yes	16	91.73	even	12
91.2.r.a.51.7	yes	16	91.31	even	12
637.2.c.e.246.2		8	13.5	odd	4
637.2.c.e.246.7		8	13.8	odd	4
637.2.c.f.246.2		8	91.83	even	4
637.2.c.f.246.7		8	91.34	even	4
637.2.r.f.116.2		16	91.44	odd	12
637.2.r.f.116.7		16	91.86	odd	12
637.2.r.f.324.2		16	91.60	odd	12
637.2.r.f.324.7		16	91.18	odd	12
819.2.dl.e.298.2		16	273.47	odd	12
819.2.dl.e.298.7		16	273.5	odd	12
819.2.dl.e.415.2		16	273.122	odd	12
819.2.dl.e.415.7		16	273.164	odd	12
1183.2.e.i.170.2		16	7.3	odd	6
1183.2.e.i.170.7		16	91.38	odd	6
1183.2.e.i.508.2		16	7.5	odd	6
1183.2.e.i.508.7		16	91.12	odd	6
8281.2.a.cj.1.2		8	13.12	even	2	inner
8281.2.a.cj.1.7		8	1.1	even	1	trivial
8281.2.a.ck.1.2		8	91.90	odd	2
8281.2.a.ck.1.7		8	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.12549 q^{2} +0.178854 q^{3} +2.51771 q^{4} +3.60603 q^{5} +0.380153 q^{6} +1.10038 q^{8} -2.96801 q^{9} +7.66457 q^{10} -3.99236 q^{11} +0.450303 q^{12} +0.644954 q^{15} -2.69656 q^{16} -4.78916 q^{17} -6.30848 q^{18} -3.15060 q^{19} +9.07892 q^{20} -8.48572 q^{22} -2.17885 q^{23} +0.196808 q^{24} +8.00343 q^{25} -1.06740 q^{27} -6.57198 q^{29} +1.37084 q^{30} +1.48672 q^{31} -7.93228 q^{32} -0.714051 q^{33} -10.1793 q^{34} -7.47259 q^{36} -4.96503 q^{37} -6.69656 q^{38} +3.96801 q^{40} -2.11931 q^{41} +1.43145 q^{43} -10.0516 q^{44} -10.7027 q^{45} -4.63113 q^{46} +1.01893 q^{47} -0.482292 q^{48} +17.0112 q^{50} -0.856562 q^{51} +6.03542 q^{53} -2.26876 q^{54} -14.3966 q^{55} -0.563498 q^{57} -13.9687 q^{58} -4.90322 q^{59} +1.62380 q^{60} +2.03542 q^{61} +3.16000 q^{62} -11.4669 q^{64} -1.51771 q^{66} -3.91090 q^{67} -12.0577 q^{68} -0.389698 q^{69} +8.80684 q^{71} -3.26595 q^{72} +3.08878 q^{73} -10.5531 q^{74} +1.43145 q^{75} -7.93228 q^{76} +1.96801 q^{79} -9.72388 q^{80} +8.71312 q^{81} -4.50457 q^{82} +7.66020 q^{83} -17.2698 q^{85} +3.04253 q^{86} -1.17543 q^{87} -4.39312 q^{88} -12.7992 q^{89} -22.7485 q^{90} -5.48572 q^{92} +0.265906 q^{93} +2.16572 q^{94} -11.3611 q^{95} -1.41872 q^{96} +1.35900 q^{97} +11.8494 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} + 6 q^{4} + 12 q^{9} + 6 q^{10} - 18 q^{12} - 2 q^{16} - 8 q^{17} - 18 q^{22} - 12 q^{23} - 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} - 34 q^{38} - 4 q^{40} - 8 q^{43} - 18 q^{48} + 16 q^{51}+ \cdots - 16 q^{95}+O(q^{100}) \) 8 * q - 4 * q^3 + 6 * q^4 + 12 * q^9 + 6 * q^10 - 18 * q^12 - 2 * q^16 - 8 * q^17 - 18 * q^22 - 12 * q^23 - 16 * q^27 - 8 * q^29 + 38 * q^30 + 28 * q^36 - 34 * q^38 - 4 * q^40 - 8 * q^43 - 18 * q^48 + 16 * q^51 + 20 * q^53 - 12 * q^55 - 12 * q^61 + 22 * q^62 - 44 * q^64 + 2 * q^66 - 2 * q^68 - 28 * q^69 - 42 * q^74 - 8 * q^75 - 20 * q^79 + 24 * q^81 + 16 * q^82 - 68 * q^87 + 4 * q^88 - 108 * q^90 + 6 * q^92 - 26 * q^94 - 16 * q^95

Introduction

L-functions

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