LMFDB - Embedded newform 8281.2.a.ck.1.3

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:	$0$
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			1.3
Root			$-1.07305$ 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)

$f(q)$	$=$	$q-1.07305 q^{2} -2.43140 q^{3} -0.848553 q^{4} +0.625432 q^{5} +2.60903 q^{6} +3.05665 q^{8} +2.91173 q^{9} +O(q^{10})$	\(q-1.07305 q^{2} -2.43140 q^{3} -0.848553 q^{4} +0.625432 q^{5} +2.60903 q^{6} +3.05665 q^{8} +2.91173 q^{9} -0.671123 q^{10} -0.708521 q^{11} +2.06318 q^{12} -1.52068 q^{15} -1.58285 q^{16} -3.34313 q^{17} -3.12445 q^{18} -5.20276 q^{19} -0.530712 q^{20} +0.760282 q^{22} -4.43140 q^{23} -7.43196 q^{24} -4.60883 q^{25} +0.214623 q^{27} -6.59711 q^{29} +1.63177 q^{30} -4.39061 q^{31} -4.41482 q^{32} +1.72270 q^{33} +3.58737 q^{34} -2.47076 q^{36} +0.423409 q^{37} +5.58285 q^{38} +1.91173 q^{40} -5.01604 q^{41} -11.2059 q^{43} +0.601218 q^{44} +1.82109 q^{45} +4.75514 q^{46} +8.07269 q^{47} +3.84855 q^{48} +4.94553 q^{50} +8.12851 q^{51} -0.697106 q^{53} -0.230302 q^{54} -0.443132 q^{55} +12.6500 q^{57} +7.07906 q^{58} +9.86319 q^{59} +1.29038 q^{60} +4.69711 q^{61} +4.71136 q^{62} +7.90305 q^{64} -1.84855 q^{66} +10.4208 q^{67} +2.83683 q^{68} +10.7745 q^{69} -14.0876 q^{71} +8.90015 q^{72} -5.08383 q^{73} -0.454341 q^{74} +11.2059 q^{75} +4.41482 q^{76} -3.91173 q^{79} -0.989966 q^{80} -9.25702 q^{81} +5.38249 q^{82} +10.2035 q^{83} -2.09090 q^{85} +12.0246 q^{86} +16.0402 q^{87} -2.16570 q^{88} -13.3791 q^{89} -1.95413 q^{90} +3.76028 q^{92} +10.6753 q^{93} -8.66244 q^{94} -3.25397 q^{95} +10.7342 q^{96} -0.202023 q^{97} -2.06302 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) $ 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9	$ 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}+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

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 $

$2$	−1.07305	−0.758764	−0.379382	−	0.925240i	$-0.623863\pi$
$2$	−1.07305	−0.758764	−0.379382	+	0.925240i	$0.623863\pi$
$3$	−2.43140	−1.40377	−0.701886	−	0.712289i	$-0.747658\pi$
$3$	−2.43140	−1.40377	−0.701886	+	0.712289i	$0.747658\pi$
$4$	−0.848553	−0.424277
$5$	0.625432	0.279702	0.139851	−	0.990173i	$-0.455338\pi$
$5$	0.625432	0.279702	0.139851	+	0.990173i	$0.455338\pi$
$6$	2.60903	1.06513
$7$	0	0
$7$	0	0
$8$	3.05665	1.08069
$9$	2.91173	0.970576
$10$	−0.671123	−0.212228
$11$	−0.708521	−0.213627	−0.106814	−	0.994279i	$-0.534065\pi$
$11$	−0.708521	−0.213627	−0.106814	+	0.994279i	$0.534065\pi$
$12$	2.06318	0.595588
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.52068	−0.392637
$16$	−1.58285	−0.395713
$17$	−3.34313	−0.810829	−0.405414	−	0.914133i	$-0.632873\pi$
$17$	−3.34313	−0.810829	−0.405414	+	0.914133i	$0.632873\pi$
$18$	−3.12445	−0.736439
$19$	−5.20276	−1.19360	−0.596798	−	0.802392i	$-0.703561\pi$
$19$	−5.20276	−1.19360	−0.596798	+	0.802392i	$0.703561\pi$
$20$	−0.530712	−0.118671
$21$	0	0
$22$	0.760282	0.162093
$23$	−4.43140	−0.924012	−0.462006	−	0.886877i	$-0.652870\pi$
$23$	−4.43140	−0.924012	−0.462006	+	0.886877i	$0.652870\pi$
$24$	−7.43196	−1.51704
$25$	−4.60883	−0.921767
$26$	0	0
$27$	0.214623	0.0413042
$28$	0	0
$29$	−6.59711	−1.22505	−0.612526	−	0.790450i	$-0.709847\pi$
$29$	−6.59711	−1.22505	−0.612526	+	0.790450i	$0.709847\pi$
$30$	1.63177	0.297919
$31$	−4.39061	−0.788576	−0.394288	−	0.918987i	$-0.629009\pi$
$31$	−4.39061	−0.788576	−0.394288	+	0.918987i	$0.629009\pi$
$32$	−4.41482	−0.780437
$33$	1.72270	0.299884
$34$	3.58737	0.615228
$35$	0	0
$36$	−2.47076	−0.411793
$37$	0.423409	0.0696080	0.0348040	−	0.999394i	$-0.488919\pi$
$37$	0.423409	0.0696080	0.0348040	+	0.999394i	$0.488919\pi$
$38$	5.58285	0.905658
$39$	0	0
$40$	1.91173	0.302271
$41$	−5.01604	−0.783374	−0.391687	−	0.920099i	$-0.628108\pi$
$41$	−5.01604	−0.783374	−0.391687	+	0.920099i	$0.628108\pi$
$42$	0	0
$43$	−11.2059	−1.70889	−0.854445	−	0.519542i	$-0.826103\pi$
$43$	−11.2059	−1.70889	−0.854445	+	0.519542i	$0.826103\pi$
$44$	0.601218	0.0906370
$45$	1.82109	0.271472
$46$	4.75514	0.701107
$47$	8.07269	1.17752	0.588762	−	0.808307i	$-0.299616\pi$
$47$	8.07269	1.17752	0.588762	+	0.808307i	$0.299616\pi$
$48$	3.84855	0.555491
$49$	0	0
$50$	4.94553	0.699404
$51$	8.12851	1.13822
$52$	0	0
$53$	−0.697106	−0.0957549	−0.0478774	−	0.998853i	$-0.515246\pi$
$53$	−0.697106	−0.0957549	−0.0478774	+	0.998853i	$0.515246\pi$
$54$	−0.230302	−0.0313401
$55$	−0.443132	−0.0597519
$56$	0	0
$57$	12.6500	1.67554
$58$	7.07906	0.929526
$59$	9.86319	1.28408	0.642039	−	0.766672i	$-0.278089\pi$
$59$	9.86319	1.28408	0.642039	+	0.766672i	$0.278089\pi$
$60$	1.29038	0.166587
$61$	4.69711	0.601403	0.300701	−	0.953718i	$-0.402779\pi$
$61$	4.69711	0.601403	0.300701	+	0.953718i	$0.402779\pi$
$62$	4.71136	0.598344
$63$	0	0
$64$	7.90305	0.987881
$65$	0	0
$66$	−1.84855	−0.227541
$67$	10.4208	1.27311	0.636553	−	0.771233i	$-0.280360\pi$
$67$	10.4208	1.27311	0.636553	+	0.771233i	$0.280360\pi$
$68$	2.83683	0.344016
$69$	10.7745	1.29710
$70$	0	0
$71$	−14.0876	−1.67189	−0.835946	−	0.548812i	$-0.815080\pi$
$71$	−14.0876	−1.67189	−0.835946	+	0.548812i	$0.815080\pi$
$72$	8.90015	1.04889
$73$	−5.08383	−0.595017	−0.297509	−	0.954719i	$-0.596156\pi$
$73$	−5.08383	−0.595017	−0.297509	+	0.954719i	$0.596156\pi$
$74$	−0.454341	−0.0528161
$75$	11.2059	1.29395
$76$	4.41482	0.506415
$77$	0	0
$78$	0	0
$79$	−3.91173	−0.440104	−0.220052	−	0.975488i	$-0.570623\pi$
$79$	−3.91173	−0.440104	−0.220052	+	0.975488i	$0.570623\pi$
$80$	−0.989966	−0.110682
$81$	−9.25702	−1.02856
$82$	5.38249	0.594396
$83$	10.2035	1.11998	0.559990	−	0.828499i	$-0.310805\pi$
$83$	10.2035	1.11998	0.559990	+	0.828499i	$0.310805\pi$
$84$	0	0
$85$	−2.09090	−0.226790
$86$	12.0246	1.29665
$87$	16.0402	1.71969
$88$	−2.16570	−0.230865
$89$	−13.3791	−1.41818	−0.709090	−	0.705117i	$-0.750894\pi$
$89$	−13.3791	−1.41818	−0.709090	+	0.705117i	$0.750894\pi$
$90$	−1.95413	−0.205983
$91$	0	0
$92$	3.76028	0.392036
$93$	10.6753	1.10698
$94$	−8.66244	−0.893463
$95$	−3.25397	−0.333851
$96$	10.7342	1.09556
$97$	−0.202023	−0.0205123	−0.0102562	−	0.999947i	$-0.503265\pi$
$97$	−0.202023	−0.0205123	−0.0102562	+	0.999947i	$0.503265\pi$
$98$	0	0
$99$	−2.06302	−0.207341
$100$	3.91084	0.391084
$101$	−17.3345	−1.72484	−0.862421	−	0.506191i	$-0.831053\pi$
$101$	−17.3345	−1.72484	−0.862421	+	0.506191i	$0.831053\pi$
$102$	−8.72234	−0.863640
$103$	−10.8148	−1.06561	−0.532806	−	0.846237i	$-0.678862\pi$
$103$	−10.8148	−1.06561	−0.532806	+	0.846237i	$0.678862\pi$
$104$	0	0
$105$	0	0
$106$	0.748033	0.0726554
$107$	−6.11678	−0.591332	−0.295666	−	0.955291i	$-0.595542\pi$
$107$	−6.11678	−0.591332	−0.295666	+	0.955291i	$0.595542\pi$
$108$	−0.182119	−0.0175244
$109$	11.3992	1.09184	0.545921	−	0.837837i	$-0.316180\pi$
$109$	11.3992	1.09184	0.545921	+	0.837837i	$0.316180\pi$
$110$	0.475505	0.0453376
$111$	−1.02948	−0.0977137
$112$	0	0
$113$	−0.923456	−0.0868714	−0.0434357	−	0.999056i	$-0.513830\pi$
$113$	−0.923456	−0.0868714	−0.0434357	+	0.999056i	$0.513830\pi$
$114$	−13.5742	−1.27134
$115$	−2.77154	−0.258448
$116$	5.59800	0.519761
$117$	0	0
$118$	−10.5837	−0.974312
$119$	0	0
$120$	−4.64819	−0.424319
$121$	−10.4980	−0.954363
$122$	−5.04025	−0.456323
$123$	12.1960	1.09968
$124$	3.72566	0.334574
$125$	−6.00967	−0.537521
$126$	0	0
$127$	−8.50972	−0.755116	−0.377558	−	0.925986i	$-0.623236\pi$
$127$	−8.50972	−0.755116	−0.377558	+	0.925986i	$0.623236\pi$
$128$	0.349236	0.0308684
$129$	27.2462	2.39889
$130$	0	0
$131$	−7.00305	−0.611859	−0.305930	−	0.952054i	$-0.598967\pi$
$131$	−7.00305	−0.611859	−0.305930	+	0.952054i	$0.598967\pi$
$132$	−1.46180	−0.127234
$133$	0	0
$134$	−11.1821	−0.965988
$135$	0.134232	0.0115528
$136$	−10.2188	−0.876255
$137$	−6.21694	−0.531149	−0.265575	−	0.964090i	$-0.585562\pi$
$137$	−6.21694	−0.531149	−0.265575	+	0.964090i	$0.585562\pi$
$138$	−11.5617	−0.984195
$139$	6.53140	0.553986	0.276993	−	0.960872i	$-0.410662\pi$
$139$	6.53140	0.553986	0.276993	+	0.960872i	$0.410662\pi$
$140$	0	0
$141$	−19.6280	−1.65297
$142$	15.1168	1.26857
$143$	0	0
$144$	−4.60883	−0.384070
$145$	−4.12604	−0.342649
$146$	5.45523	0.451478
$147$	0	0
$148$	−0.359285	−0.0295330
$149$	−3.69738	−0.302901	−0.151451	−	0.988465i	$-0.548394\pi$
$149$	−3.69738	−0.302901	−0.151451	+	0.988465i	$0.548394\pi$
$150$	−12.0246	−0.981804
$151$	−4.87774	−0.396945	−0.198473	−	0.980106i	$-0.563598\pi$
$151$	−4.87774	−0.396945	−0.198473	+	0.980106i	$0.563598\pi$
$152$	−15.9030	−1.28991
$153$	−9.73430	−0.786971
$154$	0	0
$155$	−2.74603	−0.220566
$156$	0	0
$157$	−9.51968	−0.759753	−0.379876	−	0.925037i	$-0.624033\pi$
$157$	−9.51968	−0.759753	−0.379876	+	0.925037i	$0.624033\pi$
$158$	4.19750	0.333935
$159$	1.69495	0.134418
$160$	−2.76117	−0.218290
$161$	0	0
$162$	9.93329	0.780433
$163$	−23.7089	−1.85702	−0.928511	−	0.371305i	$-0.878910\pi$
$163$	−23.7089	−1.85702	−0.928511	+	0.371305i	$0.878910\pi$
$164$	4.25637	0.332367
$165$	1.07743	0.0838780
$166$	−10.9489	−0.849801
$167$	1.13193	0.0875914	0.0437957	−	0.999041i	$-0.486055\pi$
$167$	1.13193	0.0875914	0.0437957	+	0.999041i	$0.486055\pi$
$168$	0	0
$169$	0	0
$170$	2.24365	0.172080
$171$	−15.1490	−1.15848
$172$	9.50884	0.725042
$173$	11.9892	0.911519	0.455760	−	0.890103i	$-0.349368\pi$
$173$	11.9892	0.911519	0.455760	+	0.890103i	$0.349368\pi$
$174$	−17.2121	−1.30484
$175$	0	0
$176$	1.12148	0.0845350
$177$	−23.9814	−1.80255
$178$	14.3565	1.07607
$179$	9.47076	0.707878	0.353939	−	0.935269i	$-0.384842\pi$
$179$	9.47076	0.707878	0.353939	+	0.935269i	$0.384842\pi$
$180$	−1.54529	−0.115179
$181$	11.4314	0.849690	0.424845	−	0.905266i	$-0.360329\pi$
$181$	11.4314	0.849690	0.424845	+	0.905266i	$0.360329\pi$
$182$	0	0
$183$	−11.4206	−0.844233
$184$	−13.5453	−0.998571
$185$	0.264813	0.0194695
$186$	−11.4552	−0.839938
$187$	2.36868	0.173215
$188$	−6.85011	−0.499595
$189$	0	0
$190$	3.49169	0.253314
$191$	15.6875	1.13511	0.567555	−	0.823335i	$-0.307889\pi$
$191$	15.6875	1.13511	0.567555	+	0.823335i	$0.307889\pi$
$192$	−19.2155	−1.38676
$193$	23.0071	1.65609	0.828045	−	0.560662i	$-0.189453\pi$
$193$	23.0071	1.65609	0.828045	+	0.560662i	$0.189453\pi$
$194$	0.216782	0.0155640
$195$	0	0
$196$	0	0
$197$	−10.2035	−0.726970	−0.363485	−	0.931600i	$-0.618413\pi$
$197$	−10.2035	−0.726970	−0.363485	+	0.931600i	$0.618413\pi$
$198$	2.21373	0.157323
$199$	11.9235	0.845231	0.422616	−	0.906309i	$-0.361112\pi$
$199$	11.9235	0.845231	0.422616	+	0.906309i	$0.361112\pi$
$200$	−14.0876	−0.996145
$201$	−25.3372	−1.78715
$202$	18.6008	1.30875
$203$	0	0
$204$	−6.89747	−0.482920
$205$	−3.13719	−0.219111
$206$	11.6049	0.808548
$207$	−12.9030	−0.896824
$208$	0	0
$209$	3.68627	0.254984
$210$	0	0
$211$	−15.5893	−1.07321	−0.536606	−	0.843833i	$-0.680294\pi$
$211$	−15.5893	−1.07321	−0.536606	+	0.843833i	$0.680294\pi$
$212$	0.591531	0.0406265
$213$	34.2527	2.34696
$214$	6.56365	0.448682
$215$	−7.00855	−0.477979
$216$	0.656028	0.0446370
$217$	0	0
$218$	−12.2319	−0.828451
$219$	12.3608	0.835269
$220$	0.376021	0.0253513
$221$	0	0
$222$	1.10469	0.0741417
$223$	6.76662	0.453126	0.226563	−	0.973996i	$-0.427251\pi$
$223$	6.76662	0.453126	0.226563	+	0.973996i	$0.427251\pi$
$224$	0	0
$225$	−13.4197	−0.894645
$226$	0.990919	0.0659149
$227$	−16.8245	−1.11668	−0.558340	−	0.829612i	$-0.688562\pi$
$227$	−16.8245	−1.11668	−0.558340	+	0.829612i	$0.688562\pi$
$228$	−10.7342	−0.710891
$229$	−11.0257	−0.728599	−0.364300	−	0.931282i	$-0.618692\pi$
$229$	−11.0257	−0.728599	−0.364300	+	0.931282i	$0.618692\pi$
$230$	2.97402	0.196101
$231$	0	0
$232$	−20.1651	−1.32390
$233$	17.3549	1.13695	0.568477	−	0.822699i	$-0.307533\pi$
$233$	17.3549	1.13695	0.568477	+	0.822699i	$0.307533\pi$
$234$	0	0
$235$	5.04892	0.329355
$236$	−8.36944	−0.544804
$237$	9.51100	0.617806
$238$	0	0
$239$	−19.7223	−1.27573	−0.637865	−	0.770148i	$-0.720182\pi$
$239$	−19.7223	−1.27573	−0.637865	+	0.770148i	$0.720182\pi$
$240$	2.40701	0.155372
$241$	−2.78413	−0.179341	−0.0896706	−	0.995971i	$-0.528581\pi$
$241$	−2.78413	−0.179341	−0.0896706	+	0.995971i	$0.528581\pi$
$242$	11.2649	0.724137
$243$	21.8637	1.40256
$244$	−3.98574	−0.255161
$245$	0	0
$246$	−13.0870	−0.834397
$247$	0	0
$248$	−13.4206	−0.852207
$249$	−24.8088	−1.57220
$250$	6.44871	0.407852
$251$	23.5608	1.48714	0.743572	−	0.668655i	$-0.233130\pi$
$251$	23.5608	1.48714	0.743572	+	0.668655i	$0.233130\pi$
$252$	0	0
$253$	3.13974	0.197394
$254$	9.13140	0.572955
$255$	5.08383	0.318362
$256$	−16.1808	−1.01130
$257$	−3.43229	−0.214101	−0.107050	−	0.994254i	$-0.534141\pi$
$257$	−3.43229	−0.214101	−0.107050	+	0.994254i	$0.534141\pi$
$258$	−29.2367	−1.82019
$259$	0	0
$260$	0	0
$261$	−19.2090	−1.18901
$262$	7.51465	0.464257
$263$	−21.4491	−1.32261	−0.661303	−	0.750119i	$-0.729996\pi$
$263$	−21.4491	−1.32261	−0.661303	+	0.750119i	$0.729996\pi$
$264$	5.26570	0.324082
$265$	−0.435992	−0.0267828
$266$	0	0
$267$	32.5300	1.99080
$268$	−8.84262	−0.540149
$269$	14.6569	0.893645	0.446822	−	0.894623i	$-0.352556\pi$
$269$	14.6569	0.893645	0.446822	+	0.894623i	$0.352556\pi$
$270$	−0.144038	−0.00876589
$271$	2.04366	0.124143	0.0620717	−	0.998072i	$-0.480229\pi$
$271$	2.04366	0.124143	0.0620717	+	0.998072i	$0.480229\pi$
$272$	5.29168	0.320856
$273$	0	0
$274$	6.67112	0.403017
$275$	3.26546	0.196914
$276$	−9.14277	−0.550330
$277$	−5.43356	−0.326471	−0.163236	−	0.986587i	$-0.552193\pi$
$277$	−5.43356	−0.326471	−0.163236	+	0.986587i	$0.552193\pi$
$278$	−7.00855	−0.420345
$279$	−12.7843	−0.765373
$280$	0	0
$281$	−20.2356	−1.20715	−0.603577	−	0.797305i	$-0.706258\pi$
$281$	−20.2356	−1.20715	−0.603577	+	0.797305i	$0.706258\pi$
$282$	21.0619	1.25422
$283$	1.73519	0.103146	0.0515731	−	0.998669i	$-0.483576\pi$
$283$	1.73519	0.103146	0.0515731	+	0.998669i	$0.483576\pi$
$284$	11.9541	0.709345
$285$	7.91173	0.468650
$286$	0	0
$287$	0	0
$288$	−12.8548	−0.757474
$289$	−5.82346	−0.342556
$290$	4.42747	0.259990
$291$	0.491200	0.0287947
$292$	4.31390	0.252452
$293$	−27.2441	−1.59162	−0.795810	−	0.605547i	$-0.792954\pi$
$293$	−27.2441	−1.59162	−0.795810	+	0.605547i	$0.792954\pi$
$294$	0	0
$295$	6.16875	0.359159
$296$	1.29421	0.0752247
$297$	−0.152065	−0.00882369
$298$	3.96750	0.229831
$299$	0	0
$300$	−9.50884	−0.548993
$301$	0	0
$302$	5.23409	0.301188
$303$	42.1471	2.42129
$304$	8.23520	0.472321
$305$	2.93772	0.168213
$306$	10.4454	0.597126
$307$	−12.7138	−0.725612	−0.362806	−	0.931865i	$-0.618181\pi$
$307$	−12.7138	−0.725612	−0.362806	+	0.931865i	$0.618181\pi$
$308$	0	0
$309$	26.2951	1.49588
$310$	2.94664	0.167358
$311$	9.61879	0.545431	0.272716	−	0.962095i	$-0.412078\pi$
$311$	9.61879	0.545431	0.272716	+	0.962095i	$0.412078\pi$
$312$	0	0
$313$	−9.02547	−0.510149	−0.255075	−	0.966921i	$-0.582100\pi$
$313$	−9.02547	−0.510149	−0.255075	+	0.966921i	$0.582100\pi$
$314$	10.2151	0.576473
$315$	0	0
$316$	3.31931	0.186726
$317$	−24.6262	−1.38314	−0.691572	−	0.722307i	$-0.743082\pi$
$317$	−24.6262	−1.38314	−0.691572	+	0.722307i	$0.743082\pi$
$318$	−1.81877	−0.101992
$319$	4.67419	0.261704
$320$	4.94282	0.276312
$321$	14.8724	0.830095
$322$	0	0
$323$	17.3935	0.967802
$324$	7.85507	0.436393
$325$	0	0
$326$	25.4409	1.40904
$327$	−27.7160	−1.53270
$328$	−15.3323	−0.846584
$329$	0	0
$330$	−1.15614	−0.0636436
$331$	−13.1718	−0.723989	−0.361994	−	0.932180i	$-0.617904\pi$
$331$	−13.1718	−0.723989	−0.361994	+	0.932180i	$0.617904\pi$
$332$	−8.65821	−0.475181
$333$	1.23285	0.0675599
$334$	−1.21462	−0.0664612
$335$	6.51752	0.356090
$336$	0	0
$337$	−17.0307	−0.927720	−0.463860	−	0.885909i	$-0.653536\pi$
$337$	−17.0307	−0.927720	−0.463860	+	0.885909i	$0.653536\pi$
$338$	0	0
$339$	2.24529	0.121948
$340$	1.77424	0.0962218
$341$	3.11084	0.168461
$342$	16.2558	0.879010
$343$	0	0
$344$	−34.2527	−1.84678
$345$	6.73874	0.362802
$346$	−12.8650	−0.691628
$347$	0.459917	0.0246897	0.0123448	−	0.999924i	$-0.496070\pi$
$347$	0.459917	0.0246897	0.0123448	+	0.999924i	$0.496070\pi$
$348$	−13.6110	−0.729626
$349$	6.87822	0.368183	0.184091	−	0.982909i	$-0.441066\pi$
$349$	6.87822	0.368183	0.184091	+	0.982909i	$0.441066\pi$
$350$	0	0
$351$	0	0
$352$	3.12799	0.166723
$353$	−1.53326	−0.0816073	−0.0408036	−	0.999167i	$-0.512992\pi$
$353$	−1.53326	−0.0816073	−0.0408036	+	0.999167i	$0.512992\pi$
$354$	25.7334	1.36771
$355$	−8.81084	−0.467631
$356$	11.3529	0.601701
$357$	0	0
$358$	−10.1626	−0.537112
$359$	27.2068	1.43592	0.717959	−	0.696085i	$-0.245077\pi$
$359$	27.2068	1.43592	0.717959	+	0.696085i	$0.245077\pi$
$360$	5.56644	0.293377
$361$	8.06875	0.424671
$362$	−12.2665	−0.644714
$363$	25.5249	1.33971
$364$	0	0
$365$	−3.17959	−0.166427
$366$	12.2549	0.640574
$367$	26.9814	1.40842	0.704208	−	0.709994i	$-0.251302\pi$
$367$	26.9814	1.40842	0.704208	+	0.709994i	$0.251302\pi$
$368$	7.01426	0.365643
$369$	−14.6053	−0.760324
$370$	−0.284159	−0.0147727
$371$	0	0
$372$	−9.05859	−0.469666
$373$	3.97238	0.205682	0.102841	−	0.994698i	$-0.467207\pi$
$373$	3.97238	0.205682	0.102841	+	0.994698i	$0.467207\pi$
$374$	−2.54172	−0.131429
$375$	14.6119	0.754558
$376$	24.6754	1.27254
$377$	0	0
$378$	0	0
$379$	11.4059	0.585884	0.292942	−	0.956130i	$-0.405366\pi$
$379$	11.4059	0.585884	0.292942	+	0.956130i	$0.405366\pi$
$380$	2.76117	0.141645
$381$	20.6906	1.06001
$382$	−16.8336	−0.861281
$383$	23.7920	1.21571	0.607856	−	0.794047i	$-0.292030\pi$
$383$	23.7920	1.21571	0.607856	+	0.794047i	$0.292030\pi$
$384$	−0.849134	−0.0433322
$385$	0	0
$386$	−24.6879	−1.25658
$387$	−32.6287	−1.65861
$388$	0.171427	0.00870290
$389$	−28.4110	−1.44049	−0.720247	−	0.693717i	$-0.755972\pi$
$389$	−28.4110	−1.44049	−0.720247	+	0.693717i	$0.755972\pi$
$390$	0	0
$391$	14.8148	0.749216
$392$	0	0
$393$	17.0272	0.858911
$394$	10.9489	0.551599
$395$	−2.44652	−0.123098
$396$	1.75058	0.0879701
$397$	9.85912	0.494815	0.247408	−	0.968912i	$-0.420421\pi$
$397$	9.85912	0.494815	0.247408	+	0.968912i	$0.420421\pi$
$398$	−12.7945	−0.641332
$399$	0	0
$400$	7.29510	0.364755
$401$	−12.6194	−0.630184	−0.315092	−	0.949061i	$-0.602035\pi$
$401$	−12.6194	−0.630184	−0.315092	+	0.949061i	$0.602035\pi$
$402$	27.1883	1.35603
$403$	0	0
$404$	14.7092	0.731810
$405$	−5.78964	−0.287689
$406$	0	0
$407$	−0.299994	−0.0148701
$408$	24.8460	1.23006
$409$	18.0573	0.892878	0.446439	−	0.894814i	$-0.352692\pi$
$409$	18.0573	0.892878	0.446439	+	0.894814i	$0.352692\pi$
$410$	3.36638	0.166254
$411$	15.1159	0.745613
$412$	9.17691	0.452114
$413$	0	0
$414$	13.8457	0.680478
$415$	6.38160	0.313260
$416$	0	0
$417$	−15.8805	−0.777671
$418$	−3.95557	−0.193473
$419$	14.2805	0.697647	0.348823	−	0.937188i	$-0.386581\pi$
$419$	14.2805	0.697647	0.348823	+	0.937188i	$0.386581\pi$
$420$	0	0
$421$	−4.27439	−0.208321	−0.104160	−	0.994561i	$-0.533216\pi$
$421$	−4.27439	−0.208321	−0.104160	+	0.994561i	$0.533216\pi$
$422$	16.7282	0.814316
$423$	23.5055	1.14288
$424$	−2.13081	−0.103481
$425$	15.4080	0.747395
$426$	−36.7550	−1.78079
$427$	0	0
$428$	5.19042	0.250888
$429$	0	0
$430$	7.52056	0.362674
$431$	−14.6067	−0.703580	−0.351790	−	0.936079i	$-0.614427\pi$
$431$	−14.6067	−0.703580	−0.351790	+	0.936079i	$0.614427\pi$
$432$	−0.339716	−0.0163446
$433$	−28.0099	−1.34607	−0.673035	−	0.739611i	$-0.735009\pi$
$433$	−28.0099	−1.34607	−0.673035	+	0.739611i	$0.735009\pi$
$434$	0	0
$435$	10.0321	0.481001
$436$	−9.67279	−0.463243
$437$	23.0556	1.10290
$438$	−13.2639	−0.633772
$439$	17.0774	0.815061	0.407531	−	0.913192i	$-0.366390\pi$
$439$	17.0774	0.815061	0.407531	+	0.913192i	$0.366390\pi$
$440$	−1.35450	−0.0645733
$441$	0	0
$442$	0	0
$443$	13.8157	0.656402	0.328201	−	0.944608i	$-0.393558\pi$
$443$	13.8157	0.656402	0.328201	+	0.944608i	$0.393558\pi$
$444$	0.873567	0.0414576
$445$	−8.36771	−0.396668
$446$	−7.26096	−0.343816
$447$	8.98984	0.425205
$448$	0	0
$449$	−32.6410	−1.54042	−0.770211	−	0.637789i	$-0.779849\pi$
$449$	−32.6410	−1.54042	−0.770211	+	0.637789i	$0.779849\pi$
$450$	14.4001	0.678825
$451$	3.55397	0.167350
$452$	0.783601	0.0368575
$453$	11.8598	0.557220
$454$	18.0536	0.847298
$455$	0	0
$456$	38.6667	1.81074
$457$	3.17034	0.148302	0.0741511	−	0.997247i	$-0.476375\pi$
$457$	3.17034	0.148302	0.0741511	+	0.997247i	$0.476375\pi$
$458$	11.8312	0.552835
$459$	−0.717513	−0.0334906
$460$	2.35180	0.109653
$461$	0.202023	0.00940915	0.00470458	−	0.999989i	$-0.498502\pi$
$461$	0.202023	0.00940915	0.00470458	+	0.999989i	$0.498502\pi$
$462$	0	0
$463$	17.2121	0.799912	0.399956	−	0.916534i	$-0.369025\pi$
$463$	17.2121	0.799912	0.399956	+	0.916534i	$0.369025\pi$
$464$	10.4422	0.484769
$465$	6.67670	0.309625
$466$	−18.6227	−0.862681
$467$	−0.191169	−0.00884625	−0.00442312	−	0.999990i	$-0.501408\pi$
$467$	−0.191169	−0.00884625	−0.00442312	+	0.999990i	$0.501408\pi$
$468$	0	0
$469$	0	0
$470$	−5.41777	−0.249903
$471$	23.1462	1.06652
$472$	30.1483	1.38769
$473$	7.93965	0.365065
$474$	−10.2058	−0.468769
$475$	23.9787	1.10022
$476$	0	0
$477$	−2.02978	−0.0929374
$478$	21.1631	0.967978
$479$	−21.4785	−0.981377	−0.490688	−	0.871335i	$-0.663255\pi$
$479$	−21.4785	−0.981377	−0.490688	+	0.871335i	$0.663255\pi$
$480$	6.71352	0.306429
$481$	0	0
$482$	2.98752	0.136078
$483$	0	0
$484$	8.90811	0.404914
$485$	−0.126352	−0.00573733
$486$	−23.4609	−1.06421
$487$	−19.0484	−0.863167	−0.431584	−	0.902073i	$-0.642045\pi$
$487$	−19.0484	−0.863167	−0.431584	+	0.902073i	$0.642045\pi$
$488$	14.3574	0.649930
$489$	57.6458	2.60684
$490$	0	0
$491$	35.7559	1.61364	0.806821	−	0.590796i	$-0.201186\pi$
$491$	35.7559	1.61364	0.806821	+	0.590796i	$0.201186\pi$
$492$	−10.3490	−0.466568
$493$	22.0550	0.993308
$494$	0	0
$495$	−1.29028	−0.0579937
$496$	6.94968	0.312050
$497$	0	0
$498$	26.6213	1.19293
$499$	−17.6891	−0.791875	−0.395937	−	0.918277i	$-0.629580\pi$
$499$	−17.6891	−0.791875	−0.395937	+	0.918277i	$0.629580\pi$
$500$	5.09953	0.228058
$501$	−2.75218	−0.122958
$502$	−25.2820	−1.12839
$503$	11.3305	0.505203	0.252601	−	0.967570i	$-0.418714\pi$
$503$	11.3305	0.505203	0.252601	+	0.967570i	$0.418714\pi$
$504$	0	0
$505$	−10.8415	−0.482441
$506$	−3.36912	−0.149776
$507$	0	0
$508$	7.22095	0.320378
$509$	19.3514	0.857735	0.428868	−	0.903367i	$-0.358913\pi$
$509$	19.3514	0.857735	0.428868	+	0.903367i	$0.358913\pi$
$510$	−5.45523	−0.241562
$511$	0	0
$512$	16.6645	0.736472
$513$	−1.11663	−0.0493005
$514$	3.68304	0.162452
$515$	−6.76391	−0.298053
$516$	−23.1198	−1.01779
$517$	−5.71967	−0.251551
$518$	0	0
$519$	−29.1505	−1.27957
$520$	0	0
$521$	−7.71099	−0.337825	−0.168912	−	0.985631i	$-0.554025\pi$
$521$	−7.71099	−0.337825	−0.168912	+	0.985631i	$0.554025\pi$
$522$	20.6123	0.902176
$523$	−35.0501	−1.53263	−0.766317	−	0.642462i	$-0.777913\pi$
$523$	−35.0501	−1.53263	−0.766317	+	0.642462i	$0.777913\pi$
$524$	5.94246	0.259597
$525$	0	0
$526$	23.0160	1.00355
$527$	14.6784	0.639401
$528$	−2.72678	−0.118668
$529$	−3.36265	−0.146202
$530$	0.467844	0.0203218
$531$	28.7189	1.24630
$532$	0	0
$533$	0	0
$534$	−34.9065	−1.51055
$535$	−3.82563	−0.165396
$536$	31.8529	1.37583
$537$	−23.0272	−0.993699
$538$	−15.7276	−0.678066
$539$	0	0
$540$	−0.113903	−0.00490160
$541$	−12.1335	−0.521659	−0.260829	−	0.965385i	$-0.583996\pi$
$541$	−12.1335	−0.521659	−0.260829	+	0.965385i	$0.583996\pi$
$542$	−2.19296	−0.0941956
$543$	−27.7944	−1.19277
$544$	14.7593	0.632801
$545$	7.12940	0.305390
$546$	0	0
$547$	−5.12546	−0.219149	−0.109575	−	0.993979i	$-0.534949\pi$
$547$	−5.12546	−0.219149	−0.109575	+	0.993979i	$0.534949\pi$
$548$	5.27541	0.225354
$549$	13.6767	0.583707
$550$	−3.50401	−0.149412
$551$	34.3232	1.46222
$552$	32.9340	1.40177
$553$	0	0
$554$	5.83051	0.247715
$555$	−0.643868	−0.0273307
$556$	−5.54224	−0.235043
$557$	−37.5586	−1.59141	−0.795705	−	0.605685i	$-0.792899\pi$
$557$	−37.5586	−1.59141	−0.795705	+	0.605685i	$0.792899\pi$
$558$	13.7182	0.580738
$559$	0	0
$560$	0	0
$561$	−5.75922	−0.243154
$562$	21.7139	0.915945
$563$	28.7009	1.20960	0.604799	−	0.796378i	$-0.293253\pi$
$563$	28.7009	1.20960	0.604799	+	0.796378i	$0.293253\pi$
$564$	16.6554	0.701318
$565$	−0.577559	−0.0242981
$566$	−1.86195	−0.0782636
$567$	0	0
$568$	−43.0610	−1.80680
$569$	17.9483	0.752434	0.376217	−	0.926532i	$-0.377225\pi$
$569$	17.9483	0.752434	0.376217	+	0.926532i	$0.377225\pi$
$570$	−8.48972	−0.355595
$571$	17.8274	0.746053	0.373027	−	0.927821i	$-0.378320\pi$
$571$	17.8274	0.746053	0.373027	+	0.927821i	$0.378320\pi$
$572$	0	0
$573$	−38.1428	−1.59344
$574$	0	0
$575$	20.4236	0.851724
$576$	23.0115	0.958814
$577$	−33.0570	−1.37618	−0.688091	−	0.725624i	$-0.741551\pi$
$577$	−33.0570	−1.37618	−0.688091	+	0.725624i	$0.741551\pi$
$578$	6.24889	0.259920
$579$	−55.9396	−2.32477
$580$	3.50117	0.145378
$581$	0	0
$582$	−0.527085	−0.0218484
$583$	0.493914	0.0204558
$584$	−15.5395	−0.643029
$585$	0	0
$586$	29.2345	1.20766
$587$	−14.7295	−0.607953	−0.303976	−	0.952680i	$-0.598314\pi$
$587$	−14.7295	−0.607953	−0.303976	+	0.952680i	$0.598314\pi$
$588$	0	0
$589$	22.8433	0.941241
$590$	−6.61941	−0.272517
$591$	24.8088	1.02050
$592$	−0.670193	−0.0275448
$593$	9.20987	0.378204	0.189102	−	0.981957i	$-0.439442\pi$
$593$	9.20987	0.378204	0.189102	+	0.981957i	$0.439442\pi$
$594$	0.163174	0.00669510
$595$	0	0
$596$	3.13743	0.128514
$597$	−28.9907	−1.18651
$598$	0	0
$599$	−10.5745	−0.432064	−0.216032	−	0.976386i	$-0.569312\pi$
$599$	−10.5745	−0.432064	−0.216032	+	0.976386i	$0.569312\pi$
$600$	34.2527	1.39836
$601$	4.08916	0.166800	0.0834001	−	0.996516i	$-0.473422\pi$
$601$	4.08916	0.166800	0.0834001	+	0.996516i	$0.473422\pi$
$602$	0	0
$603$	30.3426	1.23565
$604$	4.13902	0.168414
$605$	−6.56578	−0.266937
$606$	−45.2261	−1.83719
$607$	3.60706	0.146406	0.0732030	−	0.997317i	$-0.476678\pi$
$607$	3.60706	0.146406	0.0732030	+	0.997317i	$0.476678\pi$
$608$	22.9693	0.931527
$609$	0	0
$610$	−3.15233	−0.127634
$611$	0	0
$612$	8.26007	0.333893
$613$	38.4845	1.55437	0.777186	−	0.629271i	$-0.216646\pi$
$613$	38.4845	1.55437	0.777186	+	0.629271i	$0.216646\pi$
$614$	13.6426	0.550569
$615$	7.62778	0.307582
$616$	0	0
$617$	3.09503	0.124601	0.0623007	−	0.998057i	$-0.480156\pi$
$617$	3.09503	0.124601	0.0623007	+	0.998057i	$0.480156\pi$
$618$	−28.2161	−1.13502
$619$	−12.2692	−0.493142	−0.246571	−	0.969125i	$-0.579304\pi$
$619$	−12.2692	−0.493142	−0.246571	+	0.969125i	$0.579304\pi$
$620$	2.33015	0.0935810
$621$	−0.951081	−0.0381655
$622$	−10.3215	−0.413854
$623$	0	0
$624$	0	0
$625$	19.2855	0.771421
$626$	9.68482	0.387083
$627$	−8.96281	−0.357940
$628$	8.07795	0.322345
$629$	−1.41551	−0.0564402
$630$	0	0
$631$	5.31780	0.211698	0.105849	−	0.994382i	$-0.466244\pi$
$631$	5.31780	0.211698	0.105849	+	0.994382i	$0.466244\pi$
$632$	−11.9568	−0.475616
$633$	37.9039	1.50655
$634$	26.4253	1.04948
$635$	−5.32225	−0.211207
$636$	−1.43825	−0.0570304
$637$	0	0
$638$	−5.01566	−0.198572
$639$	−41.0193	−1.62270
$640$	0.218423	0.00863394
$641$	12.1904	0.481493	0.240746	−	0.970588i	$-0.422608\pi$
$641$	12.1904	0.481493	0.240746	+	0.970588i	$0.422608\pi$
$642$	−15.9589	−0.629847
$643$	18.9733	0.748235	0.374117	−	0.927381i	$-0.377946\pi$
$643$	18.9733	0.748235	0.374117	+	0.927381i	$0.377946\pi$
$644$	0	0
$645$	17.0406	0.670974
$646$	−18.6642	−0.734334
$647$	19.7117	0.774948	0.387474	−	0.921881i	$-0.373348\pi$
$647$	19.7117	0.774948	0.387474	+	0.921881i	$0.373348\pi$
$648$	−28.2955	−1.11155
$649$	−6.98827	−0.274314
$650$	0	0
$651$	0	0
$652$	20.1182	0.787891
$653$	−20.3973	−0.798206	−0.399103	−	0.916906i	$-0.630678\pi$
$653$	−20.3973	−0.798206	−0.399103	+	0.916906i	$0.630678\pi$
$654$	29.7408	1.16296
$655$	−4.37993	−0.171138
$656$	7.93965	0.309991
$657$	−14.8027	−0.577510
$658$	0	0
$659$	32.6628	1.27236	0.636181	−	0.771540i	$-0.280513\pi$
$659$	32.6628	1.27236	0.636181	+	0.771540i	$0.280513\pi$
$660$	−0.914258	−0.0355875
$661$	9.73692	0.378722	0.189361	−	0.981908i	$-0.439358\pi$
$661$	9.73692	0.378722	0.189361	+	0.981908i	$0.439358\pi$
$662$	14.1341	0.549337
$663$	0	0
$664$	31.1886	1.21035
$665$	0	0
$666$	−1.32292	−0.0512620
$667$	29.2345	1.13196
$668$	−0.960502	−0.0371630
$669$	−16.4524	−0.636086
$670$	−6.99365	−0.270188
$671$	−3.32800	−0.128476
$672$	0	0
$673$	39.4512	1.52073	0.760367	−	0.649494i	$-0.225019\pi$
$673$	39.4512	1.52073	0.760367	+	0.649494i	$0.225019\pi$
$674$	18.2748	0.703921
$675$	−0.989161	−0.0380728
$676$	0	0
$677$	−48.6339	−1.86915	−0.934576	−	0.355764i	$-0.884221\pi$
$677$	−48.6339	−1.86915	−0.934576	+	0.355764i	$0.884221\pi$
$678$	−2.40932	−0.0925296
$679$	0	0
$680$	−6.39117	−0.245090
$681$	40.9072	1.56757
$682$	−3.33810	−0.127822
$683$	−5.70773	−0.218400	−0.109200	−	0.994020i	$-0.534829\pi$
$683$	−5.70773	−0.218400	−0.109200	+	0.994020i	$0.534829\pi$
$684$	12.8548	0.491514
$685$	−3.88828	−0.148563
$686$	0	0
$687$	26.8080	1.02279
$688$	17.7373	0.676230
$689$	0	0
$690$	−7.23104	−0.275281
$691$	41.7732	1.58913	0.794563	−	0.607182i	$-0.207700\pi$
$691$	41.7732	1.58913	0.794563	+	0.607182i	$0.207700\pi$
$692$	−10.1734	−0.386736
$693$	0	0
$694$	−0.493517	−0.0187336
$695$	4.08495	0.154951
$696$	49.0295	1.85846
$697$	16.7693	0.635182
$698$	−7.38071	−0.279364
$699$	−42.1967	−1.59603
$700$	0	0
$701$	−22.4361	−0.847399	−0.423700	−	0.905803i	$-0.639269\pi$
$701$	−22.4361	−0.847399	−0.423700	+	0.905803i	$0.639269\pi$
$702$	0	0
$703$	−2.20290	−0.0830838
$704$	−5.59948	−0.211038
$705$	−12.2760	−0.462340
$706$	1.64527	0.0619207
$707$	0	0
$708$	20.3495	0.764780
$709$	28.7468	1.07961	0.539804	−	0.841790i	$-0.318498\pi$
$709$	28.7468	1.07961	0.539804	+	0.841790i	$0.318498\pi$
$710$	9.45452	0.354822
$711$	−11.3899	−0.427154
$712$	−40.8953	−1.53261
$713$	19.4566	0.728654
$714$	0	0
$715$	0	0
$716$	−8.03644	−0.300336
$717$	47.9529	1.79083
$718$	−29.1944	−1.08952
$719$	−4.20899	−0.156969	−0.0784844	−	0.996915i	$-0.525008\pi$
$719$	−4.20899	−0.156969	−0.0784844	+	0.996915i	$0.525008\pi$
$720$	−2.88251	−0.107425
$721$	0	0
$722$	−8.65821	−0.322225
$723$	6.76934	0.251754
$724$	−9.70015	−0.360503
$725$	30.4050	1.12921
$726$	−27.3896	−1.01652
$727$	43.4680	1.61214	0.806070	−	0.591820i	$-0.201591\pi$
$727$	43.4680	1.61214	0.806070	+	0.591820i	$0.201591\pi$
$728$	0	0
$729$	−25.3884	−0.940312
$730$	3.41187	0.126279
$731$	37.4630	1.38562
$732$	9.69096	0.358188
$733$	9.09421	0.335902	0.167951	−	0.985795i	$-0.446285\pi$
$733$	9.09421	0.335902	0.167951	+	0.985795i	$0.446285\pi$
$734$	−28.9525	−1.06866
$735$	0	0
$736$	19.5639	0.721133
$737$	−7.38337	−0.271970
$738$	15.6723	0.576907
$739$	9.60867	0.353461	0.176730	−	0.984259i	$-0.443448\pi$
$739$	9.60867	0.353461	0.176730	+	0.984259i	$0.443448\pi$
$740$	−0.224708	−0.00826044
$741$	0	0
$742$	0	0
$743$	32.1771	1.18046	0.590231	−	0.807234i	$-0.299036\pi$
$743$	32.1771	1.18046	0.590231	+	0.807234i	$0.299036\pi$
$744$	32.6308	1.19630
$745$	−2.31246	−0.0847220
$746$	−4.26258	−0.156064
$747$	29.7098	1.08703
$748$	−2.00995	−0.0734911
$749$	0	0
$750$	−15.6794	−0.572531
$751$	7.79784	0.284547	0.142274	−	0.989827i	$-0.454559\pi$
$751$	7.79784	0.284547	0.142274	+	0.989827i	$0.454559\pi$
$752$	−12.7779	−0.465961
$753$	−57.2858	−2.08761
$754$	0	0
$755$	−3.05070	−0.111026
$756$	0	0
$757$	17.9970	0.654110	0.327055	−	0.945005i	$-0.393944\pi$
$757$	17.9970	0.654110	0.327055	+	0.945005i	$0.393944\pi$
$758$	−12.2392	−0.444548
$759$	−7.63399	−0.277096
$760$	−9.94627	−0.360789
$761$	40.6790	1.47461	0.737306	−	0.675559i	$-0.236098\pi$
$761$	40.6790	1.47461	0.737306	+	0.675559i	$0.236098\pi$
$762$	−22.2021	−0.804299
$763$	0	0
$764$	−13.3117	−0.481601
$765$	−6.08814	−0.220117
$766$	−25.5301	−0.922439
$767$	0	0
$768$	39.3422	1.41964
$769$	−39.3098	−1.41755	−0.708774	−	0.705435i	$-0.750752\pi$
$769$	−39.3098	−1.41755	−0.708774	+	0.705435i	$0.750752\pi$
$770$	0	0
$771$	8.34529	0.300548
$772$	−19.5228	−0.702640
$773$	−13.4736	−0.484611	−0.242306	−	0.970200i	$-0.577904\pi$
$773$	−13.4736	−0.484611	−0.242306	+	0.970200i	$0.577904\pi$
$774$	35.0124	1.25849
$775$	20.2356	0.726884
$776$	−0.617515	−0.0221675
$777$	0	0
$778$	30.4866	1.09300
$779$	26.0973	0.935032
$780$	0	0
$781$	9.98137	0.357161
$782$	−15.8971	−0.568478
$783$	−1.41589	−0.0505998
$784$	0	0
$785$	−5.95391	−0.212504
$786$	−18.2712	−0.651711
$787$	−39.8291	−1.41975	−0.709877	−	0.704326i	$-0.751249\pi$
$787$	−39.8291	−1.41975	−0.709877	+	0.704326i	$0.751249\pi$
$788$	8.65821	0.308436
$789$	52.1514	1.85664
$790$	2.62525	0.0934022
$791$	0	0
$792$	−6.30594	−0.224072
$793$	0	0
$794$	−10.5794	−0.375448
$795$	1.06007	0.0375969
$796$	−10.1177	−0.358612
$797$	−21.2530	−0.752821	−0.376410	−	0.926453i	$-0.622842\pi$
$797$	−21.2530	−0.752821	−0.376410	+	0.926453i	$0.622842\pi$
$798$	0	0
$799$	−26.9881	−0.954770
$800$	20.3472	0.719382
$801$	−38.9563	−1.37645
$802$	13.5413	0.478161
$803$	3.60200	0.127112
$804$	21.5000	0.758246
$805$	0	0
$806$	0	0
$807$	−35.6368	−1.25447
$808$	−52.9854	−1.86402
$809$	21.4175	0.753000	0.376500	−	0.926417i	$-0.377128\pi$
$809$	21.4175	0.753000	0.376500	+	0.926417i	$0.377128\pi$
$810$	6.21260	0.218288
$811$	11.0116	0.386669	0.193335	−	0.981133i	$-0.438070\pi$
$811$	11.0116	0.386669	0.193335	+	0.981133i	$0.438070\pi$
$812$	0	0
$813$	−4.96896	−0.174269
$814$	0.321910	0.0112829
$815$	−14.8283	−0.519412
$816$	−12.8662	−0.450408
$817$	58.3019	2.03972
$818$	−19.3765	−0.677484
$819$	0	0
$820$	2.66207	0.0929636
$821$	38.6685	1.34954	0.674769	−	0.738029i	$-0.264243\pi$
$821$	38.6685	1.34954	0.674769	+	0.738029i	$0.264243\pi$
$822$	−16.2202	−0.565744
$823$	−20.4566	−0.713073	−0.356537	−	0.934281i	$-0.616043\pi$
$823$	−20.4566	−0.713073	−0.356537	+	0.934281i	$0.616043\pi$
$824$	−33.0570	−1.15160
$825$	−7.93965	−0.276423
$826$	0	0
$827$	−27.3451	−0.950881	−0.475440	−	0.879748i	$-0.657711\pi$
$827$	−27.3451	−0.950881	−0.475440	+	0.879748i	$0.657711\pi$
$828$	10.9489	0.380501
$829$	−25.0086	−0.868585	−0.434292	−	0.900772i	$-0.643002\pi$
$829$	−25.0086	−0.868585	−0.434292	+	0.900772i	$0.643002\pi$
$830$	−6.84780	−0.237691
$831$	13.2112	0.458291
$832$	0	0
$833$	0	0
$834$	17.0406	0.590069
$835$	0.707945	0.0244994
$836$	−3.12799	−0.108184
$837$	−0.942324	−0.0325715
$838$	−15.3237	−0.529350
$839$	−8.76981	−0.302768	−0.151384	−	0.988475i	$-0.548373\pi$
$839$	−8.76981	−0.302768	−0.151384	+	0.988475i	$0.548373\pi$
$840$	0	0
$841$	14.5218	0.500753
$842$	4.58665	0.158066
$843$	49.2009	1.69457
$844$	13.2284	0.455339
$845$	0	0
$846$	−25.2227	−0.867174
$847$	0	0
$848$	1.10342	0.0378914
$849$	−4.21894	−0.144794
$850$	−16.5336	−0.567097
$851$	−1.87630	−0.0643186
$852$	−29.0652	−0.995758
$853$	−19.8232	−0.678734	−0.339367	−	0.940654i	$-0.610213\pi$
$853$	−19.8232	−0.678734	−0.339367	+	0.940654i	$0.610213\pi$
$854$	0	0
$855$	−9.47469	−0.324028
$856$	−18.6969	−0.639047
$857$	−2.67037	−0.0912181	−0.0456090	−	0.998959i	$-0.514523\pi$
$857$	−2.67037	−0.0912181	−0.0456090	+	0.998959i	$0.514523\pi$
$858$	0	0
$859$	−38.9597	−1.32929	−0.664644	−	0.747160i	$-0.731417\pi$
$859$	−38.9597	−1.32929	−0.664644	+	0.747160i	$0.731417\pi$
$860$	5.94713	0.202795
$861$	0	0
$862$	15.6738	0.533851
$863$	24.7976	0.844121	0.422060	−	0.906568i	$-0.361307\pi$
$863$	24.7976	0.844121	0.422060	+	0.906568i	$0.361307\pi$
$864$	−0.947521	−0.0322353
$865$	7.49840	0.254953
$866$	30.0561	1.02135
$867$	14.1592	0.480871
$868$	0	0
$869$	2.77154	0.0940181
$870$	−10.7650	−0.364967
$871$	0	0
$872$	34.8433	1.17994
$873$	−0.588237	−0.0199088
$874$	−24.7399	−0.836839
$875$	0	0
$876$	−10.4888	−0.354385
$877$	−1.97840	−0.0668059	−0.0334029	−	0.999442i	$-0.510634\pi$
$877$	−1.97840	−0.0668059	−0.0334029	+	0.999442i	$0.510634\pi$
$878$	−18.3250	−0.618440
$879$	66.2415	2.23427
$880$	0.701412	0.0236446
$881$	17.1466	0.577683	0.288841	−	0.957377i	$-0.406730\pi$
$881$	17.1466	0.577683	0.288841	+	0.957377i	$0.406730\pi$
$882$	0	0
$883$	−10.2168	−0.343822	−0.171911	−	0.985112i	$-0.554994\pi$
$883$	−10.2168	−0.343822	−0.171911	+	0.985112i	$0.554994\pi$
$884$	0	0
$885$	−14.9987	−0.504177
$886$	−14.8250	−0.498055
$887$	50.9931	1.71218	0.856090	−	0.516826i	$-0.172887\pi$
$887$	50.9931	1.71218	0.856090	+	0.516826i	$0.172887\pi$
$888$	−3.14676	−0.105598
$889$	0	0
$890$	8.97901	0.300977
$891$	6.55879	0.219728
$892$	−5.74184	−0.192251
$893$	−42.0003	−1.40549
$894$	−9.64659	−0.322630
$895$	5.92331	0.197995
$896$	0	0
$897$	0	0
$898$	35.0255	1.16882
$899$	28.9653	0.966047
$900$	11.3873	0.379577
$901$	2.33052	0.0776408
$902$	−3.81360	−0.126979
$903$	0	0
$904$	−2.82268	−0.0938811
$905$	7.14957	0.237660
$906$	−12.7262	−0.422799
$907$	−5.78538	−0.192100	−0.0960501	−	0.995376i	$-0.530621\pi$
$907$	−5.78538	−0.192100	−0.0960501	+	0.995376i	$0.530621\pi$
$908$	14.2765	0.473782
$909$	−50.4732	−1.67409
$910$	0	0
$911$	−1.70706	−0.0565573	−0.0282787	−	0.999600i	$-0.509003\pi$
$911$	−1.70706	−0.0565573	−0.0282787	+	0.999600i	$0.509003\pi$
$912$	−20.0231	−0.663032
$913$	−7.22940	−0.239258
$914$	−3.40195	−0.112526
$915$	−7.14279	−0.236133
$916$	9.35590	0.309128
$917$	0	0
$918$	0.769931	0.0254115
$919$	37.2050	1.22728	0.613640	−	0.789586i	$-0.289705\pi$
$919$	37.2050	1.22728	0.613640	+	0.789586i	$0.289705\pi$
$920$	−8.47164	−0.279302
$921$	30.9123	1.01859
$922$	−0.216782	−0.00713933
$923$	0	0
$924$	0	0
$925$	−1.95142	−0.0641623
$926$	−18.4695	−0.606945
$927$	−31.4897	−1.03426
$928$	29.1251	0.956077
$929$	19.9046	0.653048	0.326524	−	0.945189i	$-0.394123\pi$
$929$	19.9046	0.653048	0.326524	+	0.945189i	$0.394123\pi$
$930$	−7.16447	−0.234932
$931$	0	0
$932$	−14.7265	−0.482383
$933$	−23.3872	−0.765661
$934$	0.205135	0.00671222
$935$	1.48145	0.0484485
$936$	0	0
$937$	7.16949	0.234217	0.117109	−	0.993119i	$-0.462637\pi$
$937$	7.16949	0.234217	0.117109	+	0.993119i	$0.462637\pi$
$938$	0	0
$939$	21.9446	0.716134
$940$	−4.28428	−0.139738
$941$	3.06072	0.0997766	0.0498883	−	0.998755i	$-0.484113\pi$
$941$	3.06072	0.0997766	0.0498883	+	0.998755i	$0.484113\pi$
$942$	−24.8371	−0.809237
$943$	22.2281	0.723846
$944$	−15.6120	−0.508126
$945$	0	0
$946$	−8.51968	−0.276999
$947$	−44.2056	−1.43649	−0.718244	−	0.695791i	$-0.755054\pi$
$947$	−44.2056	−1.43649	−0.718244	+	0.695791i	$0.755054\pi$
$948$	−8.07058	−0.262120
$949$	0	0
$950$	−25.7304	−0.834806
$951$	59.8762	1.94162
$952$	0	0
$953$	−13.7002	−0.443791	−0.221896	−	0.975070i	$-0.571224\pi$
$953$	−13.7002	−0.443791	−0.221896	+	0.975070i	$0.571224\pi$
$954$	2.17807	0.0705176
$955$	9.81149	0.317492
$956$	16.7354	0.541262
$957$	−11.3648	−0.367373
$958$	23.0476	0.744634
$959$	0	0
$960$	−12.0180	−0.387879
$961$	−11.7226	−0.378147
$962$	0	0
$963$	−17.8104	−0.573933
$964$	2.36248	0.0760903
$965$	14.3894	0.463211
$966$	0	0
$967$	−43.9429	−1.41311	−0.706554	−	0.707659i	$-0.749751\pi$
$967$	−43.9429	−1.41311	−0.706554	+	0.707659i	$0.749751\pi$
$968$	−32.0887	−1.03137
$969$	−42.2907	−1.35857
$970$	0.135582	0.00435329
$971$	−21.3171	−0.684098	−0.342049	−	0.939682i	$-0.611121\pi$
$971$	−21.3171	−0.684098	−0.342049	+	0.939682i	$0.611121\pi$
$972$	−18.5525	−0.595072
$973$	0	0
$974$	20.4400	0.654941
$975$	0	0
$976$	−7.43482	−0.237983
$977$	−16.4466	−0.526173	−0.263087	−	0.964772i	$-0.584740\pi$
$977$	−16.4466	−0.526173	−0.263087	+	0.964772i	$0.584740\pi$
$978$	−61.8572	−1.97797
$979$	9.47937	0.302962
$980$	0	0
$981$	33.1913	1.05972
$982$	−38.3681	−1.22437
$983$	11.7104	0.373504	0.186752	−	0.982407i	$-0.440204\pi$
$983$	11.7104	0.373504	0.186752	+	0.982407i	$0.440204\pi$
$984$	37.2790	1.18841
$985$	−6.38160	−0.203335
$986$	−23.6662	−0.753687
$987$	0	0
$988$	0	0
$989$	49.6581	1.57903
$990$	1.38454	0.0440036
$991$	−31.3747	−0.996650	−0.498325	−	0.866990i	$-0.666051\pi$
$991$	−31.3747	−0.996650	−0.498325	+	0.866990i	$0.666051\pi$
$992$	19.3837	0.615434
$993$	32.0260	1.01632
$994$	0	0
$995$	7.45731	0.236413
$996$	21.0516	0.667046
$997$	−4.69711	−0.148759	−0.0743794	−	0.997230i	$-0.523698\pi$
$997$	−4.69711	−0.148759	−0.0743794	+	0.997230i	$0.523698\pi$
$998$	18.9814	0.600847
$999$	0.0908732	0.00287510

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ck.1.3		8
7.2	even	3		1183.2.e.i.508.6		16
7.4	even	3		1183.2.e.i.170.6		16
7.6	odd	2		8281.2.a.cj.1.3		8
13.5	odd	4		637.2.c.f.246.6		8
13.8	odd	4		637.2.c.f.246.3		8
13.12	even	2	inner	8281.2.a.ck.1.6		8
91.5	even	12		637.2.r.f.116.6		16
91.18	odd	12		91.2.r.a.51.3	yes	16
91.25	even	6		1183.2.e.i.170.3		16
91.31	even	12		637.2.r.f.324.3		16
91.34	even	4		637.2.c.e.246.3		8
91.44	odd	12		91.2.r.a.25.6	yes	16
91.47	even	12		637.2.r.f.116.3		16
91.51	even	6		1183.2.e.i.508.3		16
91.60	odd	12		91.2.r.a.51.6	yes	16
91.73	even	12		637.2.r.f.324.6		16
91.83	even	4		637.2.c.e.246.6		8
91.86	odd	12		91.2.r.a.25.3	✓	16
91.90	odd	2		8281.2.a.cj.1.6		8
273.44	even	12		819.2.dl.e.298.3		16
273.86	even	12		819.2.dl.e.298.6		16
273.200	even	12		819.2.dl.e.415.6		16
273.242	even	12		819.2.dl.e.415.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.r.a.25.3	✓	16	91.86	odd	12
91.2.r.a.25.6	yes	16	91.44	odd	12
91.2.r.a.51.3	yes	16	91.18	odd	12
91.2.r.a.51.6	yes	16	91.60	odd	12
637.2.c.e.246.3		8	91.34	even	4
637.2.c.e.246.6		8	91.83	even	4
637.2.c.f.246.3		8	13.8	odd	4
637.2.c.f.246.6		8	13.5	odd	4
637.2.r.f.116.3		16	91.47	even	12
637.2.r.f.116.6		16	91.5	even	12
637.2.r.f.324.3		16	91.31	even	12
637.2.r.f.324.6		16	91.73	even	12
819.2.dl.e.298.3		16	273.44	even	12
819.2.dl.e.298.6		16	273.86	even	12
819.2.dl.e.415.3		16	273.242	even	12
819.2.dl.e.415.6		16	273.200	even	12
1183.2.e.i.170.3		16	91.25	even	6
1183.2.e.i.170.6		16	7.4	even	3
1183.2.e.i.508.3		16	91.51	even	6
1183.2.e.i.508.6		16	7.2	even	3
8281.2.a.cj.1.3		8	7.6	odd	2
8281.2.a.cj.1.6		8	91.90	odd	2
8281.2.a.ck.1.3		8	1.1	even	1	trivial
8281.2.a.ck.1.6		8	13.12	even	2	inner

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)

\(f(q)\)	\(=\)	\(q-1.07305 q^{2} -2.43140 q^{3} -0.848553 q^{4} +0.625432 q^{5} +2.60903 q^{6} +3.05665 q^{8} +2.91173 q^{9} +O(q^{10})\)	\(q-1.07305 q^{2} -2.43140 q^{3} -0.848553 q^{4} +0.625432 q^{5} +2.60903 q^{6} +3.05665 q^{8} +2.91173 q^{9} -0.671123 q^{10} -0.708521 q^{11} +2.06318 q^{12} -1.52068 q^{15} -1.58285 q^{16} -3.34313 q^{17} -3.12445 q^{18} -5.20276 q^{19} -0.530712 q^{20} +0.760282 q^{22} -4.43140 q^{23} -7.43196 q^{24} -4.60883 q^{25} +0.214623 q^{27} -6.59711 q^{29} +1.63177 q^{30} -4.39061 q^{31} -4.41482 q^{32} +1.72270 q^{33} +3.58737 q^{34} -2.47076 q^{36} +0.423409 q^{37} +5.58285 q^{38} +1.91173 q^{40} -5.01604 q^{41} -11.2059 q^{43} +0.601218 q^{44} +1.82109 q^{45} +4.75514 q^{46} +8.07269 q^{47} +3.84855 q^{48} +4.94553 q^{50} +8.12851 q^{51} -0.697106 q^{53} -0.230302 q^{54} -0.443132 q^{55} +12.6500 q^{57} +7.07906 q^{58} +9.86319 q^{59} +1.29038 q^{60} +4.69711 q^{61} +4.71136 q^{62} +7.90305 q^{64} -1.84855 q^{66} +10.4208 q^{67} +2.83683 q^{68} +10.7745 q^{69} -14.0876 q^{71} +8.90015 q^{72} -5.08383 q^{73} -0.454341 q^{74} +11.2059 q^{75} +4.41482 q^{76} -3.91173 q^{79} -0.989966 q^{80} -9.25702 q^{81} +5.38249 q^{82} +10.2035 q^{83} -2.09090 q^{85} +12.0246 q^{86} +16.0402 q^{87} -2.16570 q^{88} -13.3791 q^{89} -1.95413 q^{90} +3.76028 q^{92} +10.6753 q^{93} -8.66244 q^{94} -3.25397 q^{95} +10.7342 q^{96} -0.202023 q^{97} -2.06302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9}+O(q^{10}) \) 8 * q + 4 * q^3 + 6 * q^4 + 12 * q^9	\( 8 q + 4 q^{3} + 6 q^{4} + 12 q^{9} - 6 q^{10} + 18 q^{12} - 2 q^{16} + 8 q^{17} - 18 q^{22} - 12 q^{23} + 16 q^{27} - 8 q^{29} + 38 q^{30} + 28 q^{36} + 34 q^{38} + 4 q^{40} - 8 q^{43} + 18 q^{48} + 16 q^{51} + 20 q^{53} + 12 q^{55} + 12 q^{61} - 22 q^{62} - 44 q^{64} - 2 q^{66} + 2 q^{68} + 28 q^{69} - 42 q^{74} + 8 q^{75} - 20 q^{79} + 24 q^{81} - 16 q^{82} + 68 q^{87} + 4 q^{88} + 108 q^{90} + 6 q^{92} + 26 q^{94} - 16 q^{95}+O(q^{100}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more