LMFDB - Embedded newform 8281.2.a.cw.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:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
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)

$f(q)$	$=$	\(q-2.06809 q^{2} +2.54401 q^{3} +2.27701 q^{4} +0.855789 q^{5} -5.26125 q^{6} -0.572885 q^{8} +3.47198 q^{9} -1.76985 q^{10} +6.04521 q^{11} +5.79274 q^{12} +2.17714 q^{15} -3.36924 q^{16} -4.70145 q^{17} -7.18038 q^{18} +2.96859 q^{19} +1.94864 q^{20} -12.5021 q^{22} +3.25452 q^{23} -1.45742 q^{24} -4.26762 q^{25} +1.20072 q^{27} +4.50008 q^{29} -4.50252 q^{30} +1.94060 q^{31} +8.11368 q^{32} +15.3791 q^{33} +9.72304 q^{34} +7.90574 q^{36} +8.47961 q^{37} -6.13933 q^{38} -0.490269 q^{40} -8.47502 q^{41} +3.13983 q^{43} +13.7650 q^{44} +2.97128 q^{45} -6.73066 q^{46} -3.04442 q^{47} -8.57138 q^{48} +8.82585 q^{50} -11.9605 q^{51} -7.96541 q^{53} -2.48321 q^{54} +5.17343 q^{55} +7.55212 q^{57} -9.30658 q^{58} +6.65182 q^{59} +4.95736 q^{60} +11.2195 q^{61} -4.01334 q^{62} -10.0414 q^{64} -31.8054 q^{66} -9.74400 q^{67} -10.7053 q^{68} +8.27953 q^{69} -5.35491 q^{71} -1.98905 q^{72} +3.57019 q^{73} -17.5366 q^{74} -10.8569 q^{75} +6.75952 q^{76} -0.811080 q^{79} -2.88336 q^{80} -7.36129 q^{81} +17.5271 q^{82} +17.0860 q^{83} -4.02345 q^{85} -6.49346 q^{86} +11.4482 q^{87} -3.46321 q^{88} +15.2000 q^{89} -6.14489 q^{90} +7.41058 q^{92} +4.93690 q^{93} +6.29614 q^{94} +2.54049 q^{95} +20.6413 q^{96} +15.2597 q^{97} +20.9889 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) $ 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

Coefficient data

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

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

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

$2$	−2.06809	−1.46236	−0.731182	−	0.682183i	$-0.761031\pi$
$2$	−2.06809	−1.46236	−0.731182	+	0.682183i	$0.761031\pi$
$3$	2.54401	1.46878	0.734392	−	0.678725i	$-0.237467\pi$
$3$	2.54401	1.46878	0.734392	+	0.678725i	$0.237467\pi$
$4$	2.27701	1.13851
$5$	0.855789	0.382721	0.191360	−	0.981520i	$-0.438710\pi$
$5$	0.855789	0.382721	0.191360	+	0.981520i	$0.438710\pi$
$6$	−5.26125	−2.14790
$7$	0	0
$7$	0	0
$8$	−0.572885	−0.202546
$9$	3.47198	1.15733
$10$	−1.76985	−0.559677
$11$	6.04521	1.82270	0.911350	−	0.411631i	$-0.135041\pi$
$11$	6.04521	1.82270	0.911350	+	0.411631i	$0.135041\pi$
$12$	5.79274	1.67222
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.17714	0.562134
$16$	−3.36924	−0.842311
$17$	−4.70145	−1.14027	−0.570135	−	0.821551i	$-0.693109\pi$
$17$	−4.70145	−1.14027	−0.570135	+	0.821551i	$0.693109\pi$
$18$	−7.18038	−1.69243
$19$	2.96859	0.681042	0.340521	−	0.940237i	$-0.389397\pi$
$19$	2.96859	0.681042	0.340521	+	0.940237i	$0.389397\pi$
$20$	1.94864	0.435730
$21$	0	0
$22$	−12.5021	−2.66545
$23$	3.25452	0.678615	0.339307	−	0.940676i	$-0.389807\pi$
$23$	3.25452	0.678615	0.339307	+	0.940676i	$0.389807\pi$
$24$	−1.45742	−0.297496
$25$	−4.26762	−0.853525
$26$	0	0
$27$	1.20072	0.231079
$28$	0	0
$29$	4.50008	0.835644	0.417822	−	0.908529i	$-0.362794\pi$
$29$	4.50008	0.835644	0.417822	+	0.908529i	$0.362794\pi$
$30$	−4.50252	−0.822044
$31$	1.94060	0.348542	0.174271	−	0.984698i	$-0.444243\pi$
$31$	1.94060	0.348542	0.174271	+	0.984698i	$0.444243\pi$
$32$	8.11368	1.43431
$33$	15.3791	2.67715
$34$	9.72304	1.66749
$35$	0	0
$36$	7.90574	1.31762
$37$	8.47961	1.39404	0.697020	−	0.717052i	$-0.254509\pi$
$37$	8.47961	1.39404	0.697020	+	0.717052i	$0.254509\pi$
$38$	−6.13933	−0.995930
$39$	0	0
$40$	−0.490269	−0.0775184
$41$	−8.47502	−1.32358	−0.661788	−	0.749691i	$-0.730202\pi$
$41$	−8.47502	−1.32358	−0.661788	+	0.749691i	$0.730202\pi$
$42$	0	0
$43$	3.13983	0.478819	0.239410	−	0.970919i	$-0.423046\pi$
$43$	3.13983	0.478819	0.239410	+	0.970919i	$0.423046\pi$
$44$	13.7650	2.07516
$45$	2.97128	0.442933
$46$	−6.73066	−0.992381
$47$	−3.04442	−0.444074	−0.222037	−	0.975038i	$-0.571271\pi$
$47$	−3.04442	−0.444074	−0.222037	+	0.975038i	$0.571271\pi$
$48$	−8.57138	−1.23717
$49$	0	0
$50$	8.82585	1.24816
$51$	−11.9605	−1.67481
$52$	0	0
$53$	−7.96541	−1.09413	−0.547067	−	0.837089i	$-0.684256\pi$
$53$	−7.96541	−1.09413	−0.547067	+	0.837089i	$0.684256\pi$
$54$	−2.48321	−0.337922
$55$	5.17343	0.697585
$56$	0	0
$57$	7.55212	1.00030
$58$	−9.30658	−1.22201
$59$	6.65182	0.865993	0.432996	−	0.901396i	$-0.357456\pi$
$59$	6.65182	0.865993	0.432996	+	0.901396i	$0.357456\pi$
$60$	4.95736	0.639993
$61$	11.2195	1.43651	0.718256	−	0.695779i	$-0.244941\pi$
$61$	11.2195	1.43651	0.718256	+	0.695779i	$0.244941\pi$
$62$	−4.01334	−0.509695
$63$	0	0
$64$	−10.0414	−1.25517
$65$	0	0
$66$	−31.8054	−3.91497
$67$	−9.74400	−1.19042	−0.595209	−	0.803571i	$-0.702931\pi$
$67$	−9.74400	−1.19042	−0.595209	+	0.803571i	$0.702931\pi$
$68$	−10.7053	−1.29820
$69$	8.27953	0.996739
$70$	0	0
$71$	−5.35491	−0.635510	−0.317755	−	0.948173i	$-0.602929\pi$
$71$	−5.35491	−0.635510	−0.317755	+	0.948173i	$0.602929\pi$
$72$	−1.98905	−0.234411
$73$	3.57019	0.417860	0.208930	−	0.977931i	$-0.433002\pi$
$73$	3.57019	0.417860	0.208930	+	0.977931i	$0.433002\pi$
$74$	−17.5366	−2.03859
$75$	−10.8569	−1.25364
$76$	6.75952	0.775370
$77$	0	0
$78$	0	0
$79$	−0.811080	−0.0912536	−0.0456268	−	0.998959i	$-0.514529\pi$
$79$	−0.811080	−0.0912536	−0.0456268	+	0.998959i	$0.514529\pi$
$80$	−2.88336	−0.322370
$81$	−7.36129	−0.817921
$82$	17.5271	1.93555
$83$	17.0860	1.87543	0.937717	−	0.347399i	$-0.112935\pi$
$83$	17.0860	1.87543	0.937717	+	0.347399i	$0.112935\pi$
$84$	0	0
$85$	−4.02345	−0.436405
$86$	−6.49346	−0.700207
$87$	11.4482	1.22738
$88$	−3.46321	−0.369180
$89$	15.2000	1.61119	0.805597	−	0.592464i	$-0.201845\pi$
$89$	15.2000	1.61119	0.805597	+	0.592464i	$0.201845\pi$
$90$	−6.14489	−0.647729
$91$	0	0
$92$	7.41058	0.772607
$93$	4.93690	0.511933
$94$	6.29614	0.649397
$95$	2.54049	0.260649
$96$	20.6413	2.10669
$97$	15.2597	1.54939	0.774694	−	0.632336i	$-0.217904\pi$
$97$	15.2597	1.54939	0.774694	+	0.632336i	$0.217904\pi$
$98$	0	0
$99$	20.9889	2.10946
$100$	−9.71743	−0.971743
$101$	−7.54035	−0.750293	−0.375146	−	0.926966i	$-0.622408\pi$
$101$	−7.54035	−0.750293	−0.375146	+	0.926966i	$0.622408\pi$
$102$	24.7355	2.44918
$103$	6.68286	0.658481	0.329241	−	0.944246i	$-0.393207\pi$
$103$	6.68286	0.658481	0.329241	+	0.944246i	$0.393207\pi$
$104$	0	0
$105$	0	0
$106$	16.4732	1.60002
$107$	2.10758	0.203748	0.101874	−	0.994797i	$-0.467516\pi$
$107$	2.10758	0.203748	0.101874	+	0.994797i	$0.467516\pi$
$108$	2.73406	0.263085
$109$	−6.77293	−0.648729	−0.324364	−	0.945932i	$-0.605150\pi$
$109$	−6.77293	−0.648729	−0.324364	+	0.945932i	$0.605150\pi$
$110$	−10.6991	−1.02012
$111$	21.5722	2.04754
$112$	0	0
$113$	−4.76130	−0.447905	−0.223953	−	0.974600i	$-0.571896\pi$
$113$	−4.76130	−0.447905	−0.223953	+	0.974600i	$0.571896\pi$
$114$	−15.6185	−1.46281
$115$	2.78519	0.259720
$116$	10.2467	0.951385
$117$	0	0
$118$	−13.7566	−1.26640
$119$	0	0
$120$	−1.24725	−0.113858
$121$	25.5446	2.32224
$122$	−23.2030	−2.10070
$123$	−21.5605	−1.94405
$124$	4.41877	0.396817
$125$	−7.93114	−0.709382
$126$	0	0
$127$	9.73157	0.863537	0.431769	−	0.901984i	$-0.357890\pi$
$127$	9.73157	0.863537	0.431769	+	0.901984i	$0.357890\pi$
$128$	4.53912	0.401205
$129$	7.98775	0.703282
$130$	0	0
$131$	6.87156	0.600371	0.300186	−	0.953881i	$-0.402951\pi$
$131$	6.87156	0.600371	0.300186	+	0.953881i	$0.402951\pi$
$132$	35.0183	3.04795
$133$	0	0
$134$	20.1515	1.74082
$135$	1.02757	0.0884388
$136$	2.69339	0.230956
$137$	8.45255	0.722150	0.361075	−	0.932537i	$-0.382410\pi$
$137$	8.45255	0.722150	0.361075	+	0.932537i	$0.382410\pi$
$138$	−17.1229	−1.45759
$139$	−3.83558	−0.325330	−0.162665	−	0.986681i	$-0.552009\pi$
$139$	−3.83558	−0.325330	−0.162665	+	0.986681i	$0.552009\pi$
$140$	0	0
$141$	−7.74502	−0.652249
$142$	11.0744	0.929347
$143$	0	0
$144$	−11.6979	−0.974829
$145$	3.85112	0.319818
$146$	−7.38349	−0.611062
$147$	0	0
$148$	19.3082	1.58712
$149$	−7.65108	−0.626801	−0.313400	−	0.949621i	$-0.601468\pi$
$149$	−7.65108	−0.626801	−0.313400	+	0.949621i	$0.601468\pi$
$150$	22.4530	1.83328
$151$	−2.84108	−0.231204	−0.115602	−	0.993296i	$-0.536880\pi$
$151$	−2.84108	−0.231204	−0.115602	+	0.993296i	$0.536880\pi$
$152$	−1.70066	−0.137942
$153$	−16.3234	−1.31966
$154$	0	0
$155$	1.66074	0.133394
$156$	0	0
$157$	−1.17859	−0.0940619	−0.0470309	−	0.998893i	$-0.514976\pi$
$157$	−1.17859	−0.0940619	−0.0470309	+	0.998893i	$0.514976\pi$
$158$	1.67739	0.133446
$159$	−20.2641	−1.60705
$160$	6.94360	0.548940
$161$	0	0
$162$	15.2238	1.19610
$163$	−1.66795	−0.130644	−0.0653220	−	0.997864i	$-0.520807\pi$
$163$	−1.66795	−0.130644	−0.0653220	+	0.997864i	$0.520807\pi$
$164$	−19.2977	−1.50690
$165$	13.1613	1.02460
$166$	−35.3355	−2.74257
$167$	7.52717	0.582470	0.291235	−	0.956652i	$-0.405934\pi$
$167$	7.52717	0.582470	0.291235	+	0.956652i	$0.405934\pi$
$168$	0	0
$169$	0	0
$170$	8.32088	0.638182
$171$	10.3069	0.788188
$172$	7.14942	0.545138
$173$	−13.2390	−1.00654	−0.503271	−	0.864129i	$-0.667870\pi$
$173$	−13.2390	−1.00654	−0.503271	+	0.864129i	$0.667870\pi$
$174$	−23.6760	−1.79488
$175$	0	0
$176$	−20.3678	−1.53528
$177$	16.9223	1.27196
$178$	−31.4350	−2.35615
$179$	2.59714	0.194119	0.0970596	−	0.995279i	$-0.469056\pi$
$179$	2.59714	0.194119	0.0970596	+	0.995279i	$0.469056\pi$
$180$	6.76565	0.504282
$181$	−10.1531	−0.754677	−0.377338	−	0.926075i	$-0.623161\pi$
$181$	−10.1531	−0.754677	−0.377338	+	0.926075i	$0.623161\pi$
$182$	0	0
$183$	28.5425	2.10992
$184$	−1.86447	−0.137450
$185$	7.25676	0.533528
$186$	−10.2100	−0.748631
$187$	−28.4213	−2.07837
$188$	−6.93217	−0.505581
$189$	0	0
$190$	−5.25397	−0.381163
$191$	14.9774	1.08373	0.541863	−	0.840467i	$-0.317719\pi$
$191$	14.9774	1.08373	0.541863	+	0.840467i	$0.317719\pi$
$192$	−25.5453	−1.84357
$193$	−19.2531	−1.38587	−0.692934	−	0.721001i	$-0.743682\pi$
$193$	−19.2531	−1.38587	−0.692934	+	0.721001i	$0.743682\pi$
$194$	−31.5585	−2.26577
$195$	0	0
$196$	0	0
$197$	8.92091	0.635589	0.317794	−	0.948160i	$-0.397058\pi$
$197$	8.92091	0.635589	0.317794	+	0.948160i	$0.397058\pi$
$198$	−43.4069	−3.08480
$199$	16.0004	1.13424	0.567118	−	0.823636i	$-0.308058\pi$
$199$	16.0004	1.13424	0.567118	+	0.823636i	$0.308058\pi$
$200$	2.44486	0.172878
$201$	−24.7888	−1.74847
$202$	15.5941	1.09720
$203$	0	0
$204$	−27.2343	−1.90678
$205$	−7.25284	−0.506560
$206$	−13.8208	−0.962939
$207$	11.2996	0.785379
$208$	0	0
$209$	17.9458	1.24134
$210$	0	0
$211$	−6.97420	−0.480124	−0.240062	−	0.970758i	$-0.577168\pi$
$211$	−6.97420	−0.480124	−0.240062	+	0.970758i	$0.577168\pi$
$212$	−18.1373	−1.24568
$213$	−13.6229	−0.933428
$214$	−4.35868	−0.297953
$215$	2.68703	0.183254
$216$	−0.687876	−0.0468040
$217$	0	0
$218$	14.0070	0.948677
$219$	9.08260	0.613746
$220$	11.7800	0.794205
$221$	0	0
$222$	−44.6134	−2.99425
$223$	−2.51316	−0.168294	−0.0841470	−	0.996453i	$-0.526817\pi$
$223$	−2.51316	−0.168294	−0.0841470	+	0.996453i	$0.526817\pi$
$224$	0	0
$225$	−14.8171	−0.987807
$226$	9.84681	0.655000
$227$	−2.74119	−0.181939	−0.0909697	−	0.995854i	$-0.528997\pi$
$227$	−2.74119	−0.181939	−0.0909697	+	0.995854i	$0.528997\pi$
$228$	17.1963	1.13885
$229$	5.78487	0.382275	0.191137	−	0.981563i	$-0.438782\pi$
$229$	5.78487	0.382275	0.191137	+	0.981563i	$0.438782\pi$
$230$	−5.76003	−0.379805
$231$	0	0
$232$	−2.57803	−0.169256
$233$	−0.569277	−0.0372946	−0.0186473	−	0.999826i	$-0.505936\pi$
$233$	−0.569277	−0.0372946	−0.0186473	+	0.999826i	$0.505936\pi$
$234$	0	0
$235$	−2.60538	−0.169956
$236$	15.1463	0.985938
$237$	−2.06339	−0.134032
$238$	0	0
$239$	−3.52059	−0.227728	−0.113864	−	0.993496i	$-0.536323\pi$
$239$	−3.52059	−0.227728	−0.113864	+	0.993496i	$0.536323\pi$
$240$	−7.33530	−0.473491
$241$	22.9920	1.48104	0.740522	−	0.672032i	$-0.234578\pi$
$241$	22.9920	1.48104	0.740522	+	0.672032i	$0.234578\pi$
$242$	−52.8287	−3.39596
$243$	−22.3294	−1.43243
$244$	25.5469	1.63548
$245$	0	0
$246$	44.5892	2.84290
$247$	0	0
$248$	−1.11174	−0.0705956
$249$	43.4670	2.75461
$250$	16.4023	1.03737
$251$	−1.64123	−0.103594	−0.0517968	−	0.998658i	$-0.516495\pi$
$251$	−1.64123	−0.103594	−0.0517968	+	0.998658i	$0.516495\pi$
$252$	0	0
$253$	19.6743	1.23691
$254$	−20.1258	−1.26280
$255$	−10.2357	−0.640984
$256$	10.6954	0.668462
$257$	6.79574	0.423907	0.211953	−	0.977280i	$-0.432018\pi$
$257$	6.79574	0.423907	0.211953	+	0.977280i	$0.432018\pi$
$258$	−16.5194	−1.02845
$259$	0	0
$260$	0	0
$261$	15.6242	0.967113
$262$	−14.2110	−0.877961
$263$	3.30365	0.203712	0.101856	−	0.994799i	$-0.467522\pi$
$263$	3.30365	0.203712	0.101856	+	0.994799i	$0.467522\pi$
$264$	−8.81045	−0.542246
$265$	−6.81671	−0.418747
$266$	0	0
$267$	38.6689	2.36650
$268$	−22.1872	−1.35530
$269$	28.7189	1.75102	0.875512	−	0.483196i	$-0.160524\pi$
$269$	28.7189	1.75102	0.875512	+	0.483196i	$0.160524\pi$
$270$	−2.12510	−0.129330
$271$	18.5177	1.12487	0.562434	−	0.826842i	$-0.309865\pi$
$271$	18.5177	1.12487	0.562434	+	0.826842i	$0.309865\pi$
$272$	15.8403	0.960461
$273$	0	0
$274$	−17.4807	−1.05605
$275$	−25.7987	−1.55572
$276$	18.8526	1.13479
$277$	8.02277	0.482042	0.241021	−	0.970520i	$-0.422518\pi$
$277$	8.02277	0.482042	0.241021	+	0.970520i	$0.422518\pi$
$278$	7.93234	0.475750
$279$	6.73772	0.403377
$280$	0	0
$281$	−3.32260	−0.198210	−0.0991049	−	0.995077i	$-0.531598\pi$
$281$	−3.32260	−0.198210	−0.0991049	+	0.995077i	$0.531598\pi$
$282$	16.0174	0.953824
$283$	−14.9999	−0.891652	−0.445826	−	0.895120i	$-0.647090\pi$
$283$	−14.9999	−0.891652	−0.445826	+	0.895120i	$0.647090\pi$
$284$	−12.1932	−0.723532
$285$	6.46303	0.382837
$286$	0	0
$287$	0	0
$288$	28.1705	1.65996
$289$	5.10366	0.300215
$290$	−7.96448	−0.467690
$291$	38.8208	2.27572
$292$	8.12937	0.475735
$293$	−18.8858	−1.10332	−0.551659	−	0.834070i	$-0.686005\pi$
$293$	−18.8858	−1.10332	−0.551659	+	0.834070i	$0.686005\pi$
$294$	0	0
$295$	5.69255	0.331433
$296$	−4.85785	−0.282357
$297$	7.25863	0.421188
$298$	15.8231	0.916610
$299$	0	0
$300$	−24.7212	−1.42728
$301$	0	0
$302$	5.87561	0.338104
$303$	−19.1827	−1.10202
$304$	−10.0019	−0.573649
$305$	9.60154	0.549782
$306$	33.7582	1.92983
$307$	−2.18025	−0.124433	−0.0622166	−	0.998063i	$-0.519817\pi$
$307$	−2.18025	−0.124433	−0.0622166	+	0.998063i	$0.519817\pi$
$308$	0	0
$309$	17.0012	0.967167
$310$	−3.43457	−0.195071
$311$	3.39506	0.192516	0.0962582	−	0.995356i	$-0.469313\pi$
$311$	3.39506	0.192516	0.0962582	+	0.995356i	$0.469313\pi$
$312$	0	0
$313$	−23.4537	−1.32568	−0.662841	−	0.748760i	$-0.730649\pi$
$313$	−23.4537	−1.32568	−0.662841	+	0.748760i	$0.730649\pi$
$314$	2.43744	0.137553
$315$	0	0
$316$	−1.84684	−0.103893
$317$	−23.0761	−1.29608	−0.648041	−	0.761606i	$-0.724411\pi$
$317$	−23.0761	−1.29608	−0.648041	+	0.761606i	$0.724411\pi$
$318$	41.9080	2.35008
$319$	27.2039	1.52313
$320$	−8.59329	−0.480380
$321$	5.36171	0.299261
$322$	0	0
$323$	−13.9567	−0.776571
$324$	−16.7617	−0.931208
$325$	0	0
$326$	3.44948	0.191049
$327$	−17.2304	−0.952843
$328$	4.85522	0.268084
$329$	0	0
$330$	−27.2187	−1.49834
$331$	−16.9563	−0.932002	−0.466001	−	0.884784i	$-0.654306\pi$
$331$	−16.9563	−0.932002	−0.466001	+	0.884784i	$0.654306\pi$
$332$	38.9051	2.13519
$333$	29.4411	1.61336
$334$	−15.5669	−0.851783
$335$	−8.33881	−0.455598
$336$	0	0
$337$	−2.51749	−0.137136	−0.0685682	−	0.997646i	$-0.521843\pi$
$337$	−2.51749	−0.137136	−0.0685682	+	0.997646i	$0.521843\pi$
$338$	0	0
$339$	−12.1128	−0.657876
$340$	−9.16145	−0.496849
$341$	11.7313	0.635287
$342$	−21.3156	−1.15262
$343$	0	0
$344$	−1.79876	−0.0969826
$345$	7.08554	0.381473
$346$	27.3795	1.47193
$347$	13.5624	0.728068	0.364034	−	0.931386i	$-0.381399\pi$
$347$	13.5624	0.728068	0.364034	+	0.931386i	$0.381399\pi$
$348$	26.0678	1.39738
$349$	1.15064	0.0615923	0.0307962	−	0.999526i	$-0.490196\pi$
$349$	1.15064	0.0615923	0.0307962	+	0.999526i	$0.490196\pi$
$350$	0	0
$351$	0	0
$352$	49.0489	2.61432
$353$	19.4859	1.03713	0.518565	−	0.855038i	$-0.326467\pi$
$353$	19.4859	1.03713	0.518565	+	0.855038i	$0.326467\pi$
$354$	−34.9969	−1.86006
$355$	−4.58267	−0.243223
$356$	34.6105	1.83435
$357$	0	0
$358$	−5.37112	−0.283873
$359$	−23.2966	−1.22955	−0.614775	−	0.788703i	$-0.710753\pi$
$359$	−23.2966	−1.22955	−0.614775	+	0.788703i	$0.710753\pi$
$360$	−1.70220	−0.0897141
$361$	−10.1875	−0.536182
$362$	20.9976	1.10361
$363$	64.9857	3.41087
$364$	0	0
$365$	3.05533	0.159924
$366$	−59.0286	−3.08548
$367$	−36.0668	−1.88267	−0.941337	−	0.337469i	$-0.890429\pi$
$367$	−36.0668	−1.88267	−0.941337	+	0.337469i	$0.890429\pi$
$368$	−10.9653	−0.571604
$369$	−29.4251	−1.53181
$370$	−15.0077	−0.780212
$371$	0	0
$372$	11.2414	0.582838
$373$	29.5939	1.53231	0.766157	−	0.642653i	$-0.222166\pi$
$373$	29.5939	1.53231	0.766157	+	0.642653i	$0.222166\pi$
$374$	58.7779	3.03933
$375$	−20.1769	−1.04193
$376$	1.74410	0.0899452
$377$	0	0
$378$	0	0
$379$	−12.0481	−0.618869	−0.309435	−	0.950921i	$-0.600140\pi$
$379$	−12.0481	−0.618869	−0.309435	+	0.950921i	$0.600140\pi$
$380$	5.78472	0.296750
$381$	24.7572	1.26835
$382$	−30.9746	−1.58480
$383$	15.7567	0.805128	0.402564	−	0.915392i	$-0.368119\pi$
$383$	15.7567	0.805128	0.402564	+	0.915392i	$0.368119\pi$
$384$	11.5476	0.589284
$385$	0	0
$386$	39.8172	2.02664
$387$	10.9014	0.554150
$388$	34.7465	1.76399
$389$	−37.3109	−1.89174	−0.945869	−	0.324549i	$-0.894787\pi$
$389$	−37.3109	−1.89174	−0.945869	+	0.324549i	$0.894787\pi$
$390$	0	0
$391$	−15.3010	−0.773804
$392$	0	0
$393$	17.4813	0.881816
$394$	−18.4493	−0.929461
$395$	−0.694114	−0.0349247
$396$	47.7919	2.40163
$397$	−24.3492	−1.22205	−0.611027	−	0.791610i	$-0.709243\pi$
$397$	−24.3492	−1.22205	−0.611027	+	0.791610i	$0.709243\pi$
$398$	−33.0903	−1.65867
$399$	0	0
$400$	14.3787	0.718933
$401$	32.4065	1.61830	0.809151	−	0.587600i	$-0.199927\pi$
$401$	32.4065	1.61830	0.809151	+	0.587600i	$0.199927\pi$
$402$	51.2656	2.55690
$403$	0	0
$404$	−17.1695	−0.854212
$405$	−6.29972	−0.313035
$406$	0	0
$407$	51.2611	2.54092
$408$	6.85201	0.339225
$409$	−8.39823	−0.415266	−0.207633	−	0.978207i	$-0.566576\pi$
$409$	−8.39823	−0.415266	−0.207633	+	0.978207i	$0.566576\pi$
$410$	14.9995	0.740775
$411$	21.5034	1.06068
$412$	15.2169	0.749685
$413$	0	0
$414$	−23.3687	−1.14851
$415$	14.6220	0.717768
$416$	0	0
$417$	−9.75775	−0.477839
$418$	−37.1135	−1.81528
$419$	13.8684	0.677518	0.338759	−	0.940873i	$-0.389993\pi$
$419$	13.8684	0.677518	0.338759	+	0.940873i	$0.389993\pi$
$420$	0	0
$421$	10.1628	0.495305	0.247653	−	0.968849i	$-0.420341\pi$
$421$	10.1628	0.495305	0.247653	+	0.968849i	$0.420341\pi$
$422$	14.4233	0.702115
$423$	−10.5702	−0.513939
$424$	4.56327	0.221612
$425$	20.0640	0.973249
$426$	28.1735	1.36501
$427$	0	0
$428$	4.79899	0.231968
$429$	0	0
$430$	−5.55703	−0.267984
$431$	14.8560	0.715590	0.357795	−	0.933800i	$-0.383529\pi$
$431$	14.8560	0.715590	0.357795	+	0.933800i	$0.383529\pi$
$432$	−4.04553	−0.194640
$433$	24.5778	1.18113	0.590566	−	0.806989i	$-0.298904\pi$
$433$	24.5778	1.18113	0.590566	+	0.806989i	$0.298904\pi$
$434$	0	0
$435$	9.79728	0.469744
$436$	−15.4220	−0.738581
$437$	9.66135	0.462165
$438$	−18.7837	−0.897519
$439$	−4.58169	−0.218672	−0.109336	−	0.994005i	$-0.534872\pi$
$439$	−4.58169	−0.218672	−0.109336	+	0.994005i	$0.534872\pi$
$440$	−2.96378	−0.141293
$441$	0	0
$442$	0	0
$443$	−19.1552	−0.910092	−0.455046	−	0.890468i	$-0.650377\pi$
$443$	−19.1552	−0.910092	−0.455046	+	0.890468i	$0.650377\pi$
$444$	49.1202	2.33114
$445$	13.0080	0.616637
$446$	5.19746	0.246107
$447$	−19.4644	−0.920635
$448$	0	0
$449$	14.4360	0.681276	0.340638	−	0.940195i	$-0.389357\pi$
$449$	14.4360	0.681276	0.340638	+	0.940195i	$0.389357\pi$
$450$	30.6432	1.44453
$451$	−51.2333	−2.41248
$452$	−10.8415	−0.509943
$453$	−7.22772	−0.339588
$454$	5.66905	0.266062
$455$	0	0
$456$	−4.32650	−0.202607
$457$	28.9906	1.35612	0.678062	−	0.735005i	$-0.262820\pi$
$457$	28.9906	1.35612	0.678062	+	0.735005i	$0.262820\pi$
$458$	−11.9636	−0.559024
$459$	−5.64514	−0.263493
$460$	6.34190	0.295693
$461$	31.7874	1.48049	0.740243	−	0.672339i	$-0.234710\pi$
$461$	31.7874	1.48049	0.740243	+	0.672339i	$0.234710\pi$
$462$	0	0
$463$	−35.1655	−1.63428	−0.817139	−	0.576441i	$-0.804441\pi$
$463$	−35.1655	−1.63428	−0.817139	+	0.576441i	$0.804441\pi$
$464$	−15.1619	−0.703871
$465$	4.22495	0.195927
$466$	1.17732	0.0545382
$467$	27.3442	1.26534	0.632668	−	0.774423i	$-0.281960\pi$
$467$	27.3442	1.26534	0.632668	+	0.774423i	$0.281960\pi$
$468$	0	0
$469$	0	0
$470$	5.38817	0.248538
$471$	−2.99835	−0.138157
$472$	−3.81073	−0.175403
$473$	18.9809	0.872744
$474$	4.26729	0.196003
$475$	−12.6688	−0.581286
$476$	0	0
$477$	−27.6558	−1.26627
$478$	7.28092	0.333021
$479$	9.54597	0.436166	0.218083	−	0.975930i	$-0.430020\pi$
$479$	9.54597	0.436166	0.218083	+	0.975930i	$0.430020\pi$
$480$	17.6646	0.806274
$481$	0	0
$482$	−47.5496	−2.16582
$483$	0	0
$484$	58.1654	2.64388
$485$	13.0591	0.592983
$486$	46.1792	2.09473
$487$	−16.0354	−0.726633	−0.363316	−	0.931666i	$-0.618356\pi$
$487$	−16.0354	−0.726633	−0.363316	+	0.931666i	$0.618356\pi$
$488$	−6.42749	−0.290959
$489$	−4.24328	−0.191888
$490$	0	0
$491$	−14.4662	−0.652849	−0.326425	−	0.945223i	$-0.605844\pi$
$491$	−14.4662	−0.652849	−0.326425	+	0.945223i	$0.605844\pi$
$492$	−49.0936	−2.21331
$493$	−21.1569	−0.952859
$494$	0	0
$495$	17.9621	0.807334
$496$	−6.53835	−0.293580
$497$	0	0
$498$	−89.8938	−4.02824
$499$	10.7450	0.481014	0.240507	−	0.970647i	$-0.422686\pi$
$499$	10.7450	0.481014	0.240507	+	0.970647i	$0.422686\pi$
$500$	−18.0593	−0.807636
$501$	19.1492	0.855523
$502$	3.39422	0.151492
$503$	−37.0876	−1.65366	−0.826828	−	0.562455i	$-0.809857\pi$
$503$	−37.0876	−1.65366	−0.826828	+	0.562455i	$0.809857\pi$
$504$	0	0
$505$	−6.45295	−0.287153
$506$	−40.6883	−1.80881
$507$	0	0
$508$	22.1589	0.983142
$509$	22.0815	0.978747	0.489373	−	0.872074i	$-0.337226\pi$
$509$	22.0815	0.978747	0.489373	+	0.872074i	$0.337226\pi$
$510$	21.1684	0.937352
$511$	0	0
$512$	−31.1973	−1.37874
$513$	3.56446	0.157375
$514$	−14.0542	−0.619906
$515$	5.71912	0.252014
$516$	18.1882	0.800690
$517$	−18.4042	−0.809414
$518$	0	0
$519$	−33.6801	−1.47839
$520$	0	0
$521$	−10.7999	−0.473153	−0.236576	−	0.971613i	$-0.576025\pi$
$521$	−10.7999	−0.473153	−0.236576	+	0.971613i	$0.576025\pi$
$522$	−32.3123	−1.41427
$523$	−9.71457	−0.424789	−0.212394	−	0.977184i	$-0.568126\pi$
$523$	−9.71457	−0.424789	−0.212394	+	0.977184i	$0.568126\pi$
$524$	15.6466	0.683526
$525$	0	0
$526$	−6.83226	−0.297901
$527$	−9.12363	−0.397432
$528$	−51.8158	−2.25500
$529$	−12.4081	−0.539482
$530$	14.0976	0.612361
$531$	23.0950	1.00224
$532$	0	0
$533$	0	0
$534$	−79.9708	−3.46068
$535$	1.80365	0.0779785
$536$	5.58219	0.241114
$537$	6.60714	0.285119
$538$	−59.3935	−2.56063
$539$	0	0
$540$	2.33978	0.100688
$541$	3.28022	0.141028	0.0705139	−	0.997511i	$-0.477536\pi$
$541$	3.28022	0.141028	0.0705139	+	0.997511i	$0.477536\pi$
$542$	−38.2962	−1.64496
$543$	−25.8297	−1.10846
$544$	−38.1461	−1.63550
$545$	−5.79620	−0.248282
$546$	0	0
$547$	−24.8189	−1.06118	−0.530589	−	0.847629i	$-0.678029\pi$
$547$	−24.8189	−1.06118	−0.530589	+	0.847629i	$0.678029\pi$
$548$	19.2466	0.822172
$549$	38.9539	1.66251
$550$	53.3541	2.27503
$551$	13.3589	0.569108
$552$	−4.74322	−0.201885
$553$	0	0
$554$	−16.5918	−0.704920
$555$	18.4613	0.783637
$556$	−8.73366	−0.370390
$557$	11.6933	0.495461	0.247730	−	0.968829i	$-0.420315\pi$
$557$	11.6933	0.495461	0.247730	+	0.968829i	$0.420315\pi$
$558$	−13.9342	−0.589883
$559$	0	0
$560$	0	0
$561$	−72.3040	−3.05268
$562$	6.87145	0.289855
$563$	−28.5085	−1.20149	−0.600745	−	0.799441i	$-0.705129\pi$
$563$	−28.5085	−1.20149	−0.600745	+	0.799441i	$0.705129\pi$
$564$	−17.6355	−0.742589
$565$	−4.07467	−0.171423
$566$	31.0212	1.30392
$567$	0	0
$568$	3.06775	0.128720
$569$	3.73705	0.156665	0.0783326	−	0.996927i	$-0.475040\pi$
$569$	3.73705	0.156665	0.0783326	+	0.996927i	$0.475040\pi$
$570$	−13.3661	−0.559846
$571$	−34.0436	−1.42468	−0.712341	−	0.701833i	$-0.752365\pi$
$571$	−34.0436	−1.42468	−0.712341	+	0.701833i	$0.752365\pi$
$572$	0	0
$573$	38.1026	1.59176
$574$	0	0
$575$	−13.8891	−0.579215
$576$	−34.8634	−1.45264
$577$	−17.6755	−0.735840	−0.367920	−	0.929857i	$-0.619930\pi$
$577$	−17.6755	−0.735840	−0.367920	+	0.929857i	$0.619930\pi$
$578$	−10.5548	−0.439023
$579$	−48.9800	−2.03554
$580$	8.76904	0.364115
$581$	0	0
$582$	−80.2851	−3.32792
$583$	−48.1526	−1.99428
$584$	−2.04531	−0.0846356
$585$	0	0
$586$	39.0575	1.61345
$587$	24.4054	1.00732	0.503660	−	0.863902i	$-0.331987\pi$
$587$	24.4054	1.00732	0.503660	+	0.863902i	$0.331987\pi$
$588$	0	0
$589$	5.76085	0.237371
$590$	−11.7727	−0.484676
$591$	22.6949	0.933542
$592$	−28.5699	−1.17421
$593$	−17.9229	−0.736004	−0.368002	−	0.929825i	$-0.619958\pi$
$593$	−17.9229	−0.736004	−0.368002	+	0.929825i	$0.619958\pi$
$594$	−15.0115	−0.615930
$595$	0	0
$596$	−17.4216	−0.713616
$597$	40.7051	1.66595
$598$	0	0
$599$	12.5797	0.513992	0.256996	−	0.966413i	$-0.417267\pi$
$599$	12.5797	0.513992	0.256996	+	0.966413i	$0.417267\pi$
$600$	6.21974	0.253920
$601$	14.3960	0.587227	0.293614	−	0.955924i	$-0.405142\pi$
$601$	14.3960	0.587227	0.293614	+	0.955924i	$0.405142\pi$
$602$	0	0
$603$	−33.8310	−1.37770
$604$	−6.46916	−0.263227
$605$	21.8608	0.888769
$606$	39.6716	1.61155
$607$	8.14443	0.330572	0.165286	−	0.986246i	$-0.447145\pi$
$607$	8.14443	0.330572	0.165286	+	0.986246i	$0.447145\pi$
$608$	24.0862	0.976824
$609$	0	0
$610$	−19.8569	−0.803982
$611$	0	0
$612$	−37.1685	−1.50245
$613$	15.6422	0.631781	0.315890	−	0.948796i	$-0.397697\pi$
$613$	15.6422	0.631781	0.315890	+	0.948796i	$0.397697\pi$
$614$	4.50895	0.181967
$615$	−18.4513	−0.744027
$616$	0	0
$617$	−31.9466	−1.28612	−0.643062	−	0.765814i	$-0.722336\pi$
$617$	−31.9466	−1.28612	−0.643062	+	0.765814i	$0.722336\pi$
$618$	−35.1602	−1.41435
$619$	28.1268	1.13051	0.565255	−	0.824916i	$-0.308778\pi$
$619$	28.1268	1.13051	0.565255	+	0.824916i	$0.308778\pi$
$620$	3.78153	0.151870
$621$	3.90778	0.156814
$622$	−7.02131	−0.281529
$623$	0	0
$624$	0	0
$625$	14.5507	0.582030
$626$	48.5044	1.93863
$627$	45.6542	1.82325
$628$	−2.68367	−0.107090
$629$	−39.8665	−1.58958
$630$	0	0
$631$	−43.5970	−1.73557	−0.867784	−	0.496941i	$-0.834456\pi$
$631$	−43.5970	−1.73557	−0.867784	+	0.496941i	$0.834456\pi$
$632$	0.464656	0.0184830
$633$	−17.7424	−0.705198
$634$	47.7235	1.89534
$635$	8.32818	0.330494
$636$	−46.1415	−1.82963
$637$	0	0
$638$	−56.2603	−2.22737
$639$	−18.5921	−0.735493
$640$	3.88453	0.153550
$641$	−14.5144	−0.573286	−0.286643	−	0.958037i	$-0.592539\pi$
$641$	−14.5144	−0.573286	−0.286643	+	0.958037i	$0.592539\pi$
$642$	−11.0885	−0.437629
$643$	19.0778	0.752355	0.376177	−	0.926548i	$-0.377238\pi$
$643$	19.0778	0.752355	0.376177	+	0.926548i	$0.377238\pi$
$644$	0	0
$645$	6.83583	0.269161
$646$	28.8637	1.13563
$647$	−15.2560	−0.599774	−0.299887	−	0.953975i	$-0.596949\pi$
$647$	−15.2560	−0.599774	−0.299887	+	0.953975i	$0.596949\pi$
$648$	4.21718	0.165666
$649$	40.2117	1.57845
$650$	0	0
$651$	0	0
$652$	−3.79794	−0.148739
$653$	8.16961	0.319701	0.159851	−	0.987141i	$-0.448899\pi$
$653$	8.16961	0.319701	0.159851	+	0.987141i	$0.448899\pi$
$654$	35.6341	1.39340
$655$	5.88061	0.229774
$656$	28.5544	1.11486
$657$	12.3956	0.483600
$658$	0	0
$659$	1.86780	0.0727590	0.0363795	−	0.999338i	$-0.488417\pi$
$659$	1.86780	0.0727590	0.0363795	+	0.999338i	$0.488417\pi$
$660$	29.9683	1.16652
$661$	−9.73187	−0.378526	−0.189263	−	0.981926i	$-0.560610\pi$
$661$	−9.73187	−0.378526	−0.189263	+	0.981926i	$0.560610\pi$
$662$	35.0672	1.36292
$663$	0	0
$664$	−9.78833	−0.379861
$665$	0	0
$666$	−60.8869	−2.35932
$667$	14.6456	0.567080
$668$	17.1395	0.663146
$669$	−6.39351	−0.247188
$670$	17.2454	0.666250
$671$	67.8243	2.61833
$672$	0	0
$673$	−20.5640	−0.792683	−0.396341	−	0.918103i	$-0.629720\pi$
$673$	−20.5640	−0.792683	−0.396341	+	0.918103i	$0.629720\pi$
$674$	5.20640	0.200543
$675$	−5.12423	−0.197232
$676$	0	0
$677$	38.9226	1.49592	0.747959	−	0.663745i	$-0.231034\pi$
$677$	38.9226	1.49592	0.747959	+	0.663745i	$0.231034\pi$
$678$	25.0504	0.962054
$679$	0	0
$680$	2.30498	0.0883918
$681$	−6.97362	−0.267230
$682$	−24.2615	−0.929021
$683$	−37.2679	−1.42602	−0.713008	−	0.701156i	$-0.752668\pi$
$683$	−37.2679	−1.42602	−0.713008	+	0.701156i	$0.752668\pi$
$684$	23.4689	0.897356
$685$	7.23361	0.276382
$686$	0	0
$687$	14.7167	0.561479
$688$	−10.5788	−0.403314
$689$	0	0
$690$	−14.6536	−0.557851
$691$	−15.0565	−0.572776	−0.286388	−	0.958114i	$-0.592455\pi$
$691$	−15.0565	−0.572776	−0.286388	+	0.958114i	$0.592455\pi$
$692$	−30.1453	−1.14595
$693$	0	0
$694$	−28.0483	−1.06470
$695$	−3.28245	−0.124510
$696$	−6.55853	−0.248600
$697$	39.8449	1.50923
$698$	−2.37963	−0.0900704
$699$	−1.44825	−0.0547777
$700$	0	0
$701$	27.1442	1.02522	0.512612	−	0.858621i	$-0.328678\pi$
$701$	27.1442	1.02522	0.512612	+	0.858621i	$0.328678\pi$
$702$	0	0
$703$	25.1725	0.949399
$704$	−60.7022	−2.28780
$705$	−6.62811	−0.249629
$706$	−40.2987	−1.51666
$707$	0	0
$708$	38.5322	1.44813
$709$	8.79731	0.330390	0.165195	−	0.986261i	$-0.447175\pi$
$709$	8.79731	0.330390	0.165195	+	0.986261i	$0.447175\pi$
$710$	9.47740	0.355680
$711$	−2.81605	−0.105610
$712$	−8.70784	−0.326340
$713$	6.31572	0.236526
$714$	0	0
$715$	0	0
$716$	5.91371	0.221006
$717$	−8.95642	−0.334484
$718$	48.1796	1.79805
$719$	−22.5057	−0.839320	−0.419660	−	0.907681i	$-0.637851\pi$
$719$	−22.5057	−0.839320	−0.419660	+	0.907681i	$0.637851\pi$
$720$	−10.0110	−0.373087
$721$	0	0
$722$	21.0686	0.784093
$723$	58.4918	2.17533
$724$	−23.1188	−0.859204
$725$	−19.2046	−0.713243
$726$	−134.397	−4.98793
$727$	−34.9184	−1.29505	−0.647525	−	0.762044i	$-0.724196\pi$
$727$	−34.9184	−1.29505	−0.647525	+	0.762044i	$0.724196\pi$
$728$	0	0
$729$	−34.7222	−1.28601
$730$	−6.31872	−0.233866
$731$	−14.7617	−0.545983
$732$	64.9917	2.40216
$733$	−46.9386	−1.73372	−0.866859	−	0.498553i	$-0.833865\pi$
$733$	−46.9386	−1.73372	−0.866859	+	0.498553i	$0.833865\pi$
$734$	74.5896	2.75315
$735$	0	0
$736$	26.4061	0.973344
$737$	−58.9046	−2.16978
$738$	60.8539	2.24006
$739$	36.1012	1.32801	0.664003	−	0.747730i	$-0.268856\pi$
$739$	36.1012	1.32801	0.664003	+	0.747730i	$0.268856\pi$
$740$	16.5237	0.607425
$741$	0	0
$742$	0	0
$743$	24.6750	0.905238	0.452619	−	0.891704i	$-0.350490\pi$
$743$	24.6750	0.905238	0.452619	+	0.891704i	$0.350490\pi$
$744$	−2.82828	−0.103690
$745$	−6.54771	−0.239890
$746$	−61.2030	−2.24080
$747$	59.3223	2.17049
$748$	−64.7156	−2.36624
$749$	0	0
$750$	41.7277	1.52368
$751$	−51.3369	−1.87331	−0.936656	−	0.350252i	$-0.886096\pi$
$751$	−51.3369	−1.87331	−0.936656	+	0.350252i	$0.886096\pi$
$752$	10.2574	0.374048
$753$	−4.17531	−0.152157
$754$	0	0
$755$	−2.43136	−0.0884864
$756$	0	0
$757$	43.1811	1.56944	0.784722	−	0.619848i	$-0.212806\pi$
$757$	43.1811	1.56944	0.784722	+	0.619848i	$0.212806\pi$
$758$	24.9166	0.905011
$759$	50.0516	1.81676
$760$	−1.45541	−0.0527932
$761$	29.2586	1.06062	0.530311	−	0.847803i	$-0.322075\pi$
$761$	29.2586	1.06062	0.530311	+	0.847803i	$0.322075\pi$
$762$	−51.2002	−1.85479
$763$	0	0
$764$	34.1037	1.23383
$765$	−13.9694	−0.505063
$766$	−32.5863	−1.17739
$767$	0	0
$768$	27.2092	0.981827
$769$	−32.3727	−1.16739	−0.583695	−	0.811973i	$-0.698394\pi$
$769$	−32.3727	−1.16739	−0.583695	+	0.811973i	$0.698394\pi$
$770$	0	0
$771$	17.2884	0.622628
$772$	−43.8395	−1.57782
$773$	7.69157	0.276647	0.138323	−	0.990387i	$-0.455829\pi$
$773$	7.69157	0.276647	0.138323	+	0.990387i	$0.455829\pi$
$774$	−22.5451	−0.810369
$775$	−8.28175	−0.297489
$776$	−8.74206	−0.313822
$777$	0	0
$778$	77.1624	2.76641
$779$	−25.1589	−0.901411
$780$	0	0
$781$	−32.3716	−1.15835
$782$	31.6439	1.13158
$783$	5.40335	0.193100
$784$	0	0
$785$	−1.00863	−0.0359994
$786$	−36.1530	−1.28953
$787$	−51.1838	−1.82451	−0.912253	−	0.409628i	$-0.865659\pi$
$787$	−51.1838	−1.82451	−0.912253	+	0.409628i	$0.865659\pi$
$788$	20.3130	0.723621
$789$	8.40452	0.299209
$790$	1.43549	0.0510725
$791$	0	0
$792$	−12.0242	−0.427262
$793$	0	0
$794$	50.3565	1.78709
$795$	−17.3418	−0.615050
$796$	36.4330	1.29133
$797$	3.10414	0.109954	0.0549771	−	0.998488i	$-0.482491\pi$
$797$	3.10414	0.109954	0.0549771	+	0.998488i	$0.482491\pi$
$798$	0	0
$799$	14.3132	0.506364
$800$	−34.6261	−1.22422
$801$	52.7740	1.86468
$802$	−67.0196	−2.36655
$803$	21.5826	0.761633
$804$	−56.4444	−1.99064
$805$	0	0
$806$	0	0
$807$	73.0612	2.57188
$808$	4.31975	0.151968
$809$	37.6810	1.32479	0.662397	−	0.749153i	$-0.269539\pi$
$809$	37.6810	1.32479	0.662397	+	0.749153i	$0.269539\pi$
$810$	13.0284	0.457771
$811$	22.0872	0.775587	0.387793	−	0.921746i	$-0.373237\pi$
$811$	22.0872	0.775587	0.387793	+	0.921746i	$0.373237\pi$
$812$	0	0
$813$	47.1091	1.65219
$814$	−106.013	−3.71574
$815$	−1.42741	−0.0500001
$816$	40.2979	1.41071
$817$	9.32086	0.326096
$818$	17.3683	0.607269
$819$	0	0
$820$	−16.5148	−0.576721
$821$	23.3585	0.815216	0.407608	−	0.913157i	$-0.366363\pi$
$821$	23.3585	0.815216	0.407608	+	0.913157i	$0.366363\pi$
$822$	−44.4710	−1.55110
$823$	−46.3395	−1.61529	−0.807647	−	0.589666i	$-0.799259\pi$
$823$	−46.3395	−1.61529	−0.807647	+	0.589666i	$0.799259\pi$
$824$	−3.82851	−0.133372
$825$	−65.6321	−2.28502
$826$	0	0
$827$	32.2703	1.12215	0.561074	−	0.827766i	$-0.310388\pi$
$827$	32.2703	1.12215	0.561074	+	0.827766i	$0.310388\pi$
$828$	25.7294	0.894159
$829$	50.1803	1.74283	0.871417	−	0.490543i	$-0.163202\pi$
$829$	50.1803	1.74283	0.871417	+	0.490543i	$0.163202\pi$
$830$	−30.2398	−1.04964
$831$	20.4100	0.708015
$832$	0	0
$833$	0	0
$834$	20.1800	0.698775
$835$	6.44168	0.222923
$836$	40.8627	1.41327
$837$	2.33012	0.0805408
$838$	−28.6812	−0.990777
$839$	33.2701	1.14861	0.574306	−	0.818640i	$-0.305272\pi$
$839$	33.2701	1.14861	0.574306	+	0.818640i	$0.305272\pi$
$840$	0	0
$841$	−8.74929	−0.301700
$842$	−21.0176	−0.724316
$843$	−8.45272	−0.291127
$844$	−15.8803	−0.546623
$845$	0	0
$846$	21.8601	0.751565
$847$	0	0
$848$	26.8374	0.921600
$849$	−38.1599	−1.30964
$850$	−41.4943	−1.42324
$851$	27.5971	0.946016
$852$	−31.0196	−1.06271
$853$	50.9815	1.74557	0.872787	−	0.488102i	$-0.162310\pi$
$853$	50.9815	1.74557	0.872787	+	0.488102i	$0.162310\pi$
$854$	0	0
$855$	8.82053	0.301656
$856$	−1.20740	−0.0412682
$857$	4.56461	0.155924	0.0779621	−	0.996956i	$-0.475159\pi$
$857$	4.56461	0.155924	0.0779621	+	0.996956i	$0.475159\pi$
$858$	0	0
$859$	−11.2727	−0.384620	−0.192310	−	0.981334i	$-0.561598\pi$
$859$	−11.2727	−0.384620	−0.192310	+	0.981334i	$0.561598\pi$
$860$	6.11840	0.208636
$861$	0	0
$862$	−30.7237	−1.04645
$863$	14.3561	0.488689	0.244344	−	0.969689i	$-0.421427\pi$
$863$	14.3561	0.488689	0.244344	+	0.969689i	$0.421427\pi$
$864$	9.74228	0.331439
$865$	−11.3298	−0.385224
$866$	−50.8291	−1.72724
$867$	12.9837	0.440951
$868$	0	0
$869$	−4.90315	−0.166328
$870$	−20.2617	−0.686936
$871$	0	0
$872$	3.88011	0.131397
$873$	52.9814	1.79315
$874$	−19.9806	−0.675853
$875$	0	0
$876$	20.6812	0.698753
$877$	50.3066	1.69873	0.849367	−	0.527802i	$-0.176984\pi$
$877$	50.3066	1.69873	0.849367	+	0.527802i	$0.176984\pi$
$878$	9.47537	0.319778
$879$	−48.0456	−1.62054
$880$	−17.4305	−0.587584
$881$	−30.3462	−1.02239	−0.511194	−	0.859465i	$-0.670797\pi$
$881$	−30.3462	−1.02239	−0.511194	+	0.859465i	$0.670797\pi$
$882$	0	0
$883$	−34.6742	−1.16688	−0.583440	−	0.812156i	$-0.698294\pi$
$883$	−34.6742	−1.16688	−0.583440	+	0.812156i	$0.698294\pi$
$884$	0	0
$885$	14.4819	0.486804
$886$	39.6148	1.33088
$887$	56.7734	1.90626	0.953132	−	0.302555i	$-0.0978395\pi$
$887$	56.7734	1.90626	0.953132	+	0.302555i	$0.0978395\pi$
$888$	−12.3584	−0.414721
$889$	0	0
$890$	−26.9017	−0.901747
$891$	−44.5006	−1.49083
$892$	−5.72250	−0.191604
$893$	−9.03763	−0.302433
$894$	40.2542	1.34630
$895$	2.22260	0.0742934
$896$	0	0
$897$	0	0
$898$	−29.8549	−0.996273
$899$	8.73285	0.291257
$900$	−33.7387	−1.12462
$901$	37.4490	1.24761
$902$	105.955	3.52793
$903$	0	0
$904$	2.72768	0.0907212
$905$	−8.68895	−0.288830
$906$	14.9476	0.496601
$907$	9.37033	0.311137	0.155568	−	0.987825i	$-0.450279\pi$
$907$	9.37033	0.311137	0.155568	+	0.987825i	$0.450279\pi$
$908$	−6.24173	−0.207139
$909$	−26.1799	−0.868334
$910$	0	0
$911$	−10.0569	−0.333200	−0.166600	−	0.986025i	$-0.553279\pi$
$911$	−10.0569	−0.333200	−0.166600	+	0.986025i	$0.553279\pi$
$912$	−25.4449	−0.842566
$913$	103.289	3.41836
$914$	−59.9553	−1.98314
$915$	24.4264	0.807512
$916$	13.1722	0.435222
$917$	0	0
$918$	11.6747	0.385322
$919$	38.8620	1.28194	0.640969	−	0.767567i	$-0.278533\pi$
$919$	38.8620	1.28194	0.640969	+	0.767567i	$0.278533\pi$
$920$	−1.59559	−0.0526051
$921$	−5.54657	−0.182766
$922$	−65.7393	−2.16501
$923$	0	0
$924$	0	0
$925$	−36.1878	−1.18985
$926$	72.7255	2.38991
$927$	23.2027	0.762078
$928$	36.5122	1.19857
$929$	35.2899	1.15782	0.578912	−	0.815390i	$-0.303478\pi$
$929$	35.2899	1.15782	0.578912	+	0.815390i	$0.303478\pi$
$930$	−8.73759	−0.286517
$931$	0	0
$932$	−1.29625	−0.0424601
$933$	8.63707	0.282765
$934$	−56.5503	−1.85038
$935$	−24.3226	−0.795435
$936$	0	0
$937$	−12.0937	−0.395085	−0.197543	−	0.980294i	$-0.563296\pi$
$937$	−12.0937	−0.395085	−0.197543	+	0.980294i	$0.563296\pi$
$938$	0	0
$939$	−59.6664	−1.94714
$940$	−5.93248	−0.193496
$941$	−5.71098	−0.186173	−0.0930863	−	0.995658i	$-0.529673\pi$
$941$	−5.71098	−0.186173	−0.0930863	+	0.995658i	$0.529673\pi$
$942$	6.20087	0.202035
$943$	−27.5822	−0.898198
$944$	−22.4116	−0.729435
$945$	0	0
$946$	−39.2543	−1.27627
$947$	10.7318	0.348736	0.174368	−	0.984681i	$-0.444212\pi$
$947$	10.7318	0.348736	0.174368	+	0.984681i	$0.444212\pi$
$948$	−4.69837	−0.152596
$949$	0	0
$950$	26.2003	0.850051
$951$	−58.7057	−1.90366
$952$	0	0
$953$	39.1509	1.26822	0.634111	−	0.773242i	$-0.281366\pi$
$953$	39.1509	1.26822	0.634111	+	0.773242i	$0.281366\pi$
$954$	57.1947	1.85175
$955$	12.8175	0.414764
$956$	−8.01643	−0.259270
$957$	69.2071	2.23715
$958$	−19.7420	−0.637834
$959$	0	0
$960$	−21.8614	−0.705574
$961$	−27.2341	−0.878519
$962$	0	0
$963$	7.31749	0.235803
$964$	52.3530	1.68618
$965$	−16.4766	−0.530400
$966$	0	0
$967$	0.254316	0.00817824	0.00408912	−	0.999992i	$-0.498698\pi$
$967$	0.254316	0.00817824	0.00408912	+	0.999992i	$0.498698\pi$
$968$	−14.6341	−0.470359
$969$	−35.5059	−1.14062
$970$	−27.0074	−0.867156
$971$	−6.64898	−0.213376	−0.106688	−	0.994293i	$-0.534025\pi$
$971$	−6.64898	−0.213376	−0.106688	+	0.994293i	$0.534025\pi$
$972$	−50.8442	−1.63083
$973$	0	0
$974$	33.1627	1.06260
$975$	0	0
$976$	−37.8012	−1.20999
$977$	6.30276	0.201643	0.100822	−	0.994905i	$-0.467853\pi$
$977$	6.30276	0.201643	0.100822	+	0.994905i	$0.467853\pi$
$978$	8.77550	0.280609
$979$	91.8871	2.93672
$980$	0	0
$981$	−23.5155	−0.750791
$982$	29.9174	0.954703
$983$	−20.2273	−0.645151	−0.322575	−	0.946544i	$-0.604549\pi$
$983$	−20.2273	−0.645151	−0.322575	+	0.946544i	$0.604549\pi$
$984$	12.3517	0.393758
$985$	7.63442	0.243253
$986$	43.7545	1.39343
$987$	0	0
$988$	0	0
$989$	10.2186	0.324934
$990$	−37.1472	−1.18062
$991$	−48.3650	−1.53637	−0.768183	−	0.640230i	$-0.778839\pi$
$991$	−48.3650	−1.53637	−0.768183	+	0.640230i	$0.778839\pi$
$992$	15.7454	0.499917
$993$	−43.1369	−1.36891
$994$	0	0
$995$	13.6930	0.434096
$996$	98.9748	3.13614
$997$	−1.75489	−0.0555779	−0.0277890	−	0.999614i	$-0.508847\pi$
$997$	−1.75489	−0.0555779	−0.0277890	+	0.999614i	$0.508847\pi$
$998$	−22.2217	−0.703417
$999$	10.1817	0.322134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cw.1.4		24
7.3	odd	6		1183.2.e.k.170.21	✓	48
7.5	odd	6		1183.2.e.k.508.21	yes	48
7.6	odd	2		8281.2.a.cv.1.4		24
13.12	even	2		8281.2.a.ct.1.21		24
91.12	odd	6		1183.2.e.l.508.4	yes	48
91.38	odd	6		1183.2.e.l.170.4	yes	48
91.90	odd	2		8281.2.a.cu.1.21		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.21	✓	48	7.3	odd	6
1183.2.e.k.508.21	yes	48	7.5	odd	6
1183.2.e.l.170.4	yes	48	91.38	odd	6
1183.2.e.l.508.4	yes	48	91.12	odd	6
8281.2.a.ct.1.21		24	13.12	even	2
8281.2.a.cu.1.21		24	91.90	odd	2
8281.2.a.cv.1.4		24	7.6	odd	2
8281.2.a.cw.1.4		24	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-2.06809 q^{2} +2.54401 q^{3} +2.27701 q^{4} +0.855789 q^{5} -5.26125 q^{6} -0.572885 q^{8} +3.47198 q^{9} -1.76985 q^{10} +6.04521 q^{11} +5.79274 q^{12} +2.17714 q^{15} -3.36924 q^{16} -4.70145 q^{17} -7.18038 q^{18} +2.96859 q^{19} +1.94864 q^{20} -12.5021 q^{22} +3.25452 q^{23} -1.45742 q^{24} -4.26762 q^{25} +1.20072 q^{27} +4.50008 q^{29} -4.50252 q^{30} +1.94060 q^{31} +8.11368 q^{32} +15.3791 q^{33} +9.72304 q^{34} +7.90574 q^{36} +8.47961 q^{37} -6.13933 q^{38} -0.490269 q^{40} -8.47502 q^{41} +3.13983 q^{43} +13.7650 q^{44} +2.97128 q^{45} -6.73066 q^{46} -3.04442 q^{47} -8.57138 q^{48} +8.82585 q^{50} -11.9605 q^{51} -7.96541 q^{53} -2.48321 q^{54} +5.17343 q^{55} +7.55212 q^{57} -9.30658 q^{58} +6.65182 q^{59} +4.95736 q^{60} +11.2195 q^{61} -4.01334 q^{62} -10.0414 q^{64} -31.8054 q^{66} -9.74400 q^{67} -10.7053 q^{68} +8.27953 q^{69} -5.35491 q^{71} -1.98905 q^{72} +3.57019 q^{73} -17.5366 q^{74} -10.8569 q^{75} +6.75952 q^{76} -0.811080 q^{79} -2.88336 q^{80} -7.36129 q^{81} +17.5271 q^{82} +17.0860 q^{83} -4.02345 q^{85} -6.49346 q^{86} +11.4482 q^{87} -3.46321 q^{88} +15.2000 q^{89} -6.14489 q^{90} +7.41058 q^{92} +4.93690 q^{93} +6.29614 q^{94} +2.54049 q^{95} +20.6413 q^{96} +15.2597 q^{97} +20.9889 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27}+ \cdots + 39 q^{99}+O(q^{100}) \) 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 - 5 * q^10 + q^11 - 5 * q^12 - 5 * q^15 + 17 * q^16 + 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 + 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 + 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 + 4 * q^37 + 29 * q^38 + 4 * q^40 + 49 * q^41 + 13 * q^43 - 10 * q^44 + 58 * q^45 + 10 * q^46 + 62 * q^47 - 89 * q^48 + 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 + 14 * q^55 + 13 * q^57 - 56 * q^58 + 79 * q^59 - 22 * q^60 - 13 * q^61 - 12 * q^62 + 18 * q^64 + 38 * q^66 + 2 * q^67 + 12 * q^68 + 28 * q^69 + 19 * q^71 - 81 * q^72 + 17 * q^73 + 17 * q^74 - 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 + 22 * q^82 + 81 * q^83 + 34 * q^85 - 22 * q^86 - 70 * q^87 - 33 * q^88 + 72 * q^89 - q^90 - 4 * q^92 - 19 * q^93 + 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 + 39 * q^99

Introduction

L-functions

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