LMFDB - Embedded newform 8281.2.a.cw.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q-2.43473 q^{2} -1.95805 q^{3} +3.92789 q^{4} +4.14976 q^{5} +4.76731 q^{6} -4.69388 q^{8} +0.833959 q^{9} +O(q^{10})$	\(q-2.43473 q^{2} -1.95805 q^{3} +3.92789 q^{4} +4.14976 q^{5} +4.76731 q^{6} -4.69388 q^{8} +0.833959 q^{9} -10.1035 q^{10} -0.673168 q^{11} -7.69100 q^{12} -8.12544 q^{15} +3.57253 q^{16} +0.100930 q^{17} -2.03046 q^{18} -5.44395 q^{19} +16.2998 q^{20} +1.63898 q^{22} +0.462300 q^{23} +9.19084 q^{24} +12.2205 q^{25} +4.24122 q^{27} +1.30939 q^{29} +19.7832 q^{30} +7.23091 q^{31} +0.689636 q^{32} +1.31810 q^{33} -0.245738 q^{34} +3.27570 q^{36} -5.24519 q^{37} +13.2545 q^{38} -19.4785 q^{40} -4.44603 q^{41} +4.83284 q^{43} -2.64413 q^{44} +3.46073 q^{45} -1.12557 q^{46} -2.83381 q^{47} -6.99518 q^{48} -29.7536 q^{50} -0.197627 q^{51} -14.3434 q^{53} -10.3262 q^{54} -2.79348 q^{55} +10.6595 q^{57} -3.18799 q^{58} +0.417258 q^{59} -31.9158 q^{60} -1.53617 q^{61} -17.6053 q^{62} -8.82412 q^{64} -3.20920 q^{66} +5.88527 q^{67} +0.396443 q^{68} -0.905207 q^{69} +10.0629 q^{71} -3.91450 q^{72} -6.39720 q^{73} +12.7706 q^{74} -23.9284 q^{75} -21.3832 q^{76} +0.287850 q^{79} +14.8251 q^{80} -10.8064 q^{81} +10.8249 q^{82} +9.43279 q^{83} +0.418837 q^{85} -11.7666 q^{86} -2.56384 q^{87} +3.15977 q^{88} +10.0571 q^{89} -8.42593 q^{90} +1.81586 q^{92} -14.1585 q^{93} +6.89954 q^{94} -22.5911 q^{95} -1.35034 q^{96} -3.84295 q^{97} -0.561394 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9}+O(q^{10}) $ 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9	$ 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}+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

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.43473	−1.72161	−0.860805	−	0.508934i	$-0.830040\pi$
$2$	−2.43473	−1.72161	−0.860805	+	0.508934i	$0.830040\pi$
$3$	−1.95805	−1.13048	−0.565240	−	0.824926i	$-0.691217\pi$
$3$	−1.95805	−1.13048	−0.565240	+	0.824926i	$0.691217\pi$
$4$	3.92789	1.96394
$5$	4.14976	1.85583	0.927915	−	0.372792i	$-0.121600\pi$
$5$	4.14976	1.85583	0.927915	+	0.372792i	$0.121600\pi$
$6$	4.76731	1.94625
$7$	0	0
$7$	0	0
$8$	−4.69388	−1.65954
$9$	0.833959	0.277986
$10$	−10.1035	−3.19502
$11$	−0.673168	−0.202968	−0.101484	−	0.994837i	$-0.532359\pi$
$11$	−0.673168	−0.202968	−0.101484	+	0.994837i	$0.532359\pi$
$12$	−7.69100	−2.22020
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−8.12544	−2.09798
$16$	3.57253	0.893131
$17$	0.100930	0.0244792	0.0122396	−	0.999925i	$-0.496104\pi$
$17$	0.100930	0.0244792	0.0122396	+	0.999925i	$0.496104\pi$
$18$	−2.03046	−0.478585
$19$	−5.44395	−1.24893	−0.624464	−	0.781054i	$-0.714682\pi$
$19$	−5.44395	−1.24893	−0.624464	+	0.781054i	$0.714682\pi$
$20$	16.2998	3.64475
$21$	0	0
$22$	1.63898	0.349431
$23$	0.462300	0.0963962	0.0481981	−	0.998838i	$-0.484652\pi$
$23$	0.462300	0.0963962	0.0481981	+	0.998838i	$0.484652\pi$
$24$	9.19084	1.87607
$25$	12.2205	2.44410
$26$	0	0
$27$	4.24122	0.816222
$28$	0	0
$29$	1.30939	0.243147	0.121573	−	0.992582i	$-0.461206\pi$
$29$	1.30939	0.243147	0.121573	+	0.992582i	$0.461206\pi$
$30$	19.7832	3.61190
$31$	7.23091	1.29871	0.649355	−	0.760485i	$-0.275039\pi$
$31$	7.23091	1.29871	0.649355	+	0.760485i	$0.275039\pi$
$32$	0.689636	0.121911
$33$	1.31810	0.229451
$34$	−0.245738	−0.0421437
$35$	0	0
$36$	3.27570	0.545950
$37$	−5.24519	−0.862303	−0.431152	−	0.902280i	$-0.641893\pi$
$37$	−5.24519	−0.862303	−0.431152	+	0.902280i	$0.641893\pi$
$38$	13.2545	2.15017
$39$	0	0
$40$	−19.4785	−3.07982
$41$	−4.44603	−0.694353	−0.347177	−	0.937800i	$-0.612860\pi$
$41$	−4.44603	−0.694353	−0.347177	+	0.937800i	$0.612860\pi$
$42$	0	0
$43$	4.83284	0.737002	0.368501	−	0.929627i	$-0.379871\pi$
$43$	4.83284	0.737002	0.368501	+	0.929627i	$0.379871\pi$
$44$	−2.64413	−0.398617
$45$	3.46073	0.515896
$46$	−1.12557	−0.165957
$47$	−2.83381	−0.413353	−0.206677	−	0.978409i	$-0.566265\pi$
$47$	−2.83381	−0.413353	−0.206677	+	0.978409i	$0.566265\pi$
$48$	−6.99518	−1.00967
$49$	0	0
$50$	−29.7536	−4.20780
$51$	−0.197627	−0.0276733
$52$	0	0
$53$	−14.3434	−1.97022	−0.985108	−	0.171937i	$-0.944998\pi$
$53$	−14.3434	−1.97022	−0.985108	+	0.171937i	$0.944998\pi$
$54$	−10.3262	−1.40522
$55$	−2.79348	−0.376673
$56$	0	0
$57$	10.6595	1.41189
$58$	−3.18799	−0.418604
$59$	0.417258	0.0543224	0.0271612	−	0.999631i	$-0.491353\pi$
$59$	0.417258	0.0543224	0.0271612	+	0.999631i	$0.491353\pi$
$60$	−31.9158	−4.12031
$61$	−1.53617	−0.196687	−0.0983433	−	0.995153i	$-0.531354\pi$
$61$	−1.53617	−0.196687	−0.0983433	+	0.995153i	$0.531354\pi$
$62$	−17.6053	−2.23587
$63$	0	0
$64$	−8.82412	−1.10302
$65$	0	0
$66$	−3.20920	−0.395025
$67$	5.88527	0.719000	0.359500	−	0.933145i	$-0.382947\pi$
$67$	5.88527	0.719000	0.359500	+	0.933145i	$0.382947\pi$
$68$	0.396443	0.0480758
$69$	−0.905207	−0.108974
$70$	0	0
$71$	10.0629	1.19425	0.597123	−	0.802150i	$-0.296310\pi$
$71$	10.0629	1.19425	0.597123	+	0.802150i	$0.296310\pi$
$72$	−3.91450	−0.461329
$73$	−6.39720	−0.748735	−0.374368	−	0.927280i	$-0.622140\pi$
$73$	−6.39720	−0.748735	−0.374368	+	0.927280i	$0.622140\pi$
$74$	12.7706	1.48455
$75$	−23.9284	−2.76301
$76$	−21.3832	−2.45282
$77$	0	0
$78$	0	0
$79$	0.287850	0.0323857	0.0161928	−	0.999869i	$-0.494845\pi$
$79$	0.287850	0.0323857	0.0161928	+	0.999869i	$0.494845\pi$
$80$	14.8251	1.65750
$81$	−10.8064	−1.20071
$82$	10.8249	1.19541
$83$	9.43279	1.03538	0.517692	−	0.855567i	$-0.326791\pi$
$83$	9.43279	1.03538	0.517692	+	0.855567i	$0.326791\pi$
$84$	0	0
$85$	0.418837	0.0454292
$86$	−11.7666	−1.26883
$87$	−2.56384	−0.274873
$88$	3.15977	0.336832
$89$	10.0571	1.06605	0.533026	−	0.846099i	$-0.321055\pi$
$89$	10.0571	1.06605	0.533026	+	0.846099i	$0.321055\pi$
$90$	−8.42593	−0.888171
$91$	0	0
$92$	1.81586	0.189317
$93$	−14.1585	−1.46817
$94$	6.89954	0.711633
$95$	−22.5911	−2.31780
$96$	−1.35034	−0.137819
$97$	−3.84295	−0.390192	−0.195096	−	0.980784i	$-0.562502\pi$
$97$	−3.84295	−0.390192	−0.195096	+	0.980784i	$0.562502\pi$
$98$	0	0
$99$	−0.561394	−0.0564223
$100$	48.0008	4.80008
$101$	0.0853667	0.00849430	0.00424715	−	0.999991i	$-0.498648\pi$
$101$	0.0853667	0.00849430	0.00424715	+	0.999991i	$0.498648\pi$
$102$	0.481167	0.0476426
$103$	−11.5770	−1.14072	−0.570360	−	0.821395i	$-0.693196\pi$
$103$	−11.5770	−1.14072	−0.570360	+	0.821395i	$0.693196\pi$
$104$	0	0
$105$	0	0
$106$	34.9222	3.39195
$107$	4.34606	0.420149	0.210075	−	0.977685i	$-0.432629\pi$
$107$	4.34606	0.420149	0.210075	+	0.977685i	$0.432629\pi$
$108$	16.6590	1.60301
$109$	7.85513	0.752385	0.376192	−	0.926542i	$-0.377233\pi$
$109$	7.85513	0.752385	0.376192	+	0.926542i	$0.377233\pi$
$110$	6.80137	0.648485
$111$	10.2703	0.974817
$112$	0	0
$113$	−0.189171	−0.0177957	−0.00889785	−	0.999960i	$-0.502832\pi$
$113$	−0.189171	−0.0177957	−0.00889785	+	0.999960i	$0.502832\pi$
$114$	−25.9530	−2.43072
$115$	1.91844	0.178895
$116$	5.14312	0.477527
$117$	0	0
$118$	−1.01591	−0.0935220
$119$	0	0
$120$	38.1398	3.48167
$121$	−10.5468	−0.958804
$122$	3.74016	0.338618
$123$	8.70555	0.784953
$124$	28.4022	2.55059
$125$	29.9634	2.68001
$126$	0	0
$127$	15.8811	1.40922	0.704609	−	0.709595i	$-0.251122\pi$
$127$	15.8811	1.40922	0.704609	+	0.709595i	$0.251122\pi$
$128$	20.1050	1.77705
$129$	−9.46295	−0.833166
$130$	0	0
$131$	−2.24654	−0.196281	−0.0981406	−	0.995173i	$-0.531289\pi$
$131$	−2.24654	−0.196281	−0.0981406	+	0.995173i	$0.531289\pi$
$132$	5.17733	0.450629
$133$	0	0
$134$	−14.3290	−1.23784
$135$	17.6000	1.51477
$136$	−0.473755	−0.0406241
$137$	−12.3624	−1.05619	−0.528097	−	0.849184i	$-0.677094\pi$
$137$	−12.3624	−1.05619	−0.528097	+	0.849184i	$0.677094\pi$
$138$	2.20393	0.187611
$139$	13.6036	1.15385	0.576923	−	0.816798i	$-0.304253\pi$
$139$	13.6036	1.15385	0.576923	+	0.816798i	$0.304253\pi$
$140$	0	0
$141$	5.54873	0.467288
$142$	−24.5004	−2.05603
$143$	0	0
$144$	2.97934	0.248278
$145$	5.43364	0.451239
$146$	15.5754	1.28903
$147$	0	0
$148$	−20.6025	−1.69351
$149$	11.7417	0.961920	0.480960	−	0.876743i	$-0.340288\pi$
$149$	11.7417	0.961920	0.480960	+	0.876743i	$0.340288\pi$
$150$	58.2591	4.75683
$151$	11.6323	0.946620	0.473310	−	0.880896i	$-0.343059\pi$
$151$	11.6323	0.946620	0.473310	+	0.880896i	$0.343059\pi$
$152$	25.5532	2.07264
$153$	0.0841718	0.00680489
$154$	0	0
$155$	30.0066	2.41018
$156$	0	0
$157$	−18.1348	−1.44731	−0.723656	−	0.690161i	$-0.757540\pi$
$157$	−18.1348	−1.44731	−0.723656	+	0.690161i	$0.757540\pi$
$158$	−0.700837	−0.0557556
$159$	28.0851	2.22729
$160$	2.86182	0.226247
$161$	0	0
$162$	26.3106	2.06716
$163$	8.63771	0.676558	0.338279	−	0.941046i	$-0.390155\pi$
$163$	8.63771	0.676558	0.338279	+	0.941046i	$0.390155\pi$
$164$	−17.4635	−1.36367
$165$	5.46978	0.425822
$166$	−22.9663	−1.78253
$167$	5.12092	0.396269	0.198134	−	0.980175i	$-0.436512\pi$
$167$	5.12092	0.396269	0.198134	+	0.980175i	$0.436512\pi$
$168$	0	0
$169$	0	0
$170$	−1.01975	−0.0782115
$171$	−4.54003	−0.347185
$172$	18.9829	1.44743
$173$	−15.4050	−1.17122	−0.585612	−	0.810592i	$-0.699146\pi$
$173$	−15.4050	−1.17122	−0.585612	+	0.810592i	$0.699146\pi$
$174$	6.24225	0.473224
$175$	0	0
$176$	−2.40491	−0.181277
$177$	−0.817012	−0.0614104
$178$	−24.4863	−1.83533
$179$	3.45101	0.257940	0.128970	−	0.991648i	$-0.458833\pi$
$179$	3.45101	0.257940	0.128970	+	0.991648i	$0.458833\pi$
$180$	13.5934	1.01319
$181$	6.36457	0.473074	0.236537	−	0.971622i	$-0.423987\pi$
$181$	6.36457	0.473074	0.236537	+	0.971622i	$0.423987\pi$
$182$	0	0
$183$	3.00790	0.222350
$184$	−2.16998	−0.159973
$185$	−21.7663	−1.60029
$186$	34.4720	2.52761
$187$	−0.0679430	−0.00496849
$188$	−11.1309	−0.811802
$189$	0	0
$190$	55.0031	3.99034
$191$	−22.7123	−1.64340	−0.821702	−	0.569917i	$-0.806975\pi$
$191$	−22.7123	−1.64340	−0.821702	+	0.569917i	$0.806975\pi$
$192$	17.2781	1.24694
$193$	24.9724	1.79756	0.898778	−	0.438404i	$-0.144456\pi$
$193$	24.9724	1.79756	0.898778	+	0.438404i	$0.144456\pi$
$194$	9.35652	0.671759
$195$	0	0
$196$	0	0
$197$	−11.4085	−0.812818	−0.406409	−	0.913691i	$-0.633219\pi$
$197$	−11.4085	−0.812818	−0.406409	+	0.913691i	$0.633219\pi$
$198$	1.36684	0.0971372
$199$	21.3686	1.51478	0.757390	−	0.652963i	$-0.226474\pi$
$199$	21.3686	1.51478	0.757390	+	0.652963i	$0.226474\pi$
$200$	−57.3616	−4.05608
$201$	−11.5236	−0.812816
$202$	−0.207844	−0.0146239
$203$	0	0
$204$	−0.776255	−0.0543487
$205$	−18.4500	−1.28860
$206$	28.1869	1.96388
$207$	0.385540	0.0267969
$208$	0	0
$209$	3.66469	0.253492
$210$	0	0
$211$	20.2000	1.39062	0.695312	−	0.718708i	$-0.255266\pi$
$211$	20.2000	1.39062	0.695312	+	0.718708i	$0.255266\pi$
$212$	−56.3392	−3.86939
$213$	−19.7037	−1.35007
$214$	−10.5815	−0.723333
$215$	20.0551	1.36775
$216$	−19.9077	−1.35455
$217$	0	0
$218$	−19.1251	−1.29531
$219$	12.5260	0.846431
$220$	−10.9725	−0.739765
$221$	0	0
$222$	−25.0054	−1.67826
$223$	11.2430	0.752884	0.376442	−	0.926440i	$-0.377147\pi$
$223$	11.2430	0.752884	0.376442	+	0.926440i	$0.377147\pi$
$224$	0	0
$225$	10.1914	0.679428
$226$	0.460579	0.0306373
$227$	10.6289	0.705466	0.352733	−	0.935724i	$-0.385252\pi$
$227$	10.6289	0.705466	0.352733	+	0.935724i	$0.385252\pi$
$228$	41.8694	2.77287
$229$	−10.8139	−0.714600	−0.357300	−	0.933990i	$-0.616303\pi$
$229$	−10.8139	−0.714600	−0.357300	+	0.933990i	$0.616303\pi$
$230$	−4.67086	−0.307988
$231$	0	0
$232$	−6.14609	−0.403511
$233$	24.0724	1.57704	0.788519	−	0.615010i	$-0.210848\pi$
$233$	24.0724	1.57704	0.788519	+	0.615010i	$0.210848\pi$
$234$	0	0
$235$	−11.7596	−0.767113
$236$	1.63894	0.106686
$237$	−0.563625	−0.0366114
$238$	0	0
$239$	−8.54995	−0.553050	−0.276525	−	0.961007i	$-0.589183\pi$
$239$	−8.54995	−0.553050	−0.276525	+	0.961007i	$0.589183\pi$
$240$	−29.0283	−1.87377
$241$	−14.2733	−0.919426	−0.459713	−	0.888068i	$-0.652048\pi$
$241$	−14.2733	−0.919426	−0.459713	+	0.888068i	$0.652048\pi$
$242$	25.6787	1.65069
$243$	8.43580	0.541157
$244$	−6.03391	−0.386282
$245$	0	0
$246$	−21.1956	−1.35138
$247$	0	0
$248$	−33.9410	−2.15526
$249$	−18.4699	−1.17048
$250$	−72.9527	−4.61394
$251$	13.0948	0.826537	0.413269	−	0.910609i	$-0.364387\pi$
$251$	13.0948	0.826537	0.413269	+	0.910609i	$0.364387\pi$
$252$	0	0
$253$	−0.311205	−0.0195653
$254$	−38.6661	−2.42613
$255$	−0.820104	−0.0513569
$256$	−31.3020	−1.95638
$257$	−1.56599	−0.0976837	−0.0488418	−	0.998807i	$-0.515553\pi$
$257$	−1.56599	−0.0976837	−0.0488418	+	0.998807i	$0.515553\pi$
$258$	23.0397	1.43439
$259$	0	0
$260$	0	0
$261$	1.09197	0.0675915
$262$	5.46971	0.337920
$263$	−21.5104	−1.32639	−0.663194	−	0.748447i	$-0.730800\pi$
$263$	−21.5104	−1.32639	−0.663194	+	0.748447i	$0.730800\pi$
$264$	−6.18698	−0.380782
$265$	−59.5216	−3.65639
$266$	0	0
$267$	−19.6923	−1.20515
$268$	23.1167	1.41208
$269$	10.6973	0.652225	0.326113	−	0.945331i	$-0.394261\pi$
$269$	10.6973	0.652225	0.326113	+	0.945331i	$0.394261\pi$
$270$	−42.8512	−2.60784
$271$	14.5801	0.885678	0.442839	−	0.896601i	$-0.353971\pi$
$271$	14.5801	0.885678	0.442839	+	0.896601i	$0.353971\pi$
$272$	0.360576	0.0218631
$273$	0	0
$274$	30.0992	1.81836
$275$	−8.22646	−0.496074
$276$	−3.55555	−0.214019
$277$	−15.1270	−0.908891	−0.454445	−	0.890775i	$-0.650163\pi$
$277$	−15.1270	−0.908891	−0.454445	+	0.890775i	$0.650163\pi$
$278$	−33.1211	−1.98647
$279$	6.03029	0.361024
$280$	0	0
$281$	15.4331	0.920660	0.460330	−	0.887748i	$-0.347731\pi$
$281$	15.4331	0.920660	0.460330	+	0.887748i	$0.347731\pi$
$282$	−13.5096	−0.804487
$283$	−17.8852	−1.06316	−0.531582	−	0.847007i	$-0.678402\pi$
$283$	−17.8852	−1.06316	−0.531582	+	0.847007i	$0.678402\pi$
$284$	39.5259	2.34543
$285$	44.2345	2.62022
$286$	0	0
$287$	0	0
$288$	0.575128	0.0338897
$289$	−16.9898	−0.999401
$290$	−13.2294	−0.776858
$291$	7.52468	0.441105
$292$	−25.1275	−1.47047
$293$	33.9848	1.98541	0.992707	−	0.120549i	$-0.0384654\pi$
$293$	33.9848	1.98541	0.992707	+	0.120549i	$0.0384654\pi$
$294$	0	0
$295$	1.73152	0.100813
$296$	24.6203	1.43102
$297$	−2.85505	−0.165667
$298$	−28.5879	−1.65605
$299$	0	0
$300$	−93.9880	−5.42640
$301$	0	0
$302$	−28.3214	−1.62971
$303$	−0.167152	−0.00960265
$304$	−19.4486	−1.11546
$305$	−6.37475	−0.365017
$306$	−0.204935	−0.0117154
$307$	−7.14276	−0.407659	−0.203829	−	0.979006i	$-0.565339\pi$
$307$	−7.14276	−0.407659	−0.203829	+	0.979006i	$0.565339\pi$
$308$	0	0
$309$	22.6684	1.28956
$310$	−73.0577	−4.14940
$311$	14.8167	0.840180	0.420090	−	0.907483i	$-0.361999\pi$
$311$	14.8167	0.840180	0.420090	+	0.907483i	$0.361999\pi$
$312$	0	0
$313$	33.4914	1.89305	0.946523	−	0.322637i	$-0.104569\pi$
$313$	33.4914	1.89305	0.946523	+	0.322637i	$0.104569\pi$
$314$	44.1531	2.49171
$315$	0	0
$316$	1.13064	0.0636037
$317$	−8.77872	−0.493062	−0.246531	−	0.969135i	$-0.579291\pi$
$317$	−8.77872	−0.493062	−0.246531	+	0.969135i	$0.579291\pi$
$318$	−68.3794	−3.83453
$319$	−0.881436	−0.0493509
$320$	−36.6180	−2.04701
$321$	−8.50979	−0.474970
$322$	0	0
$323$	−0.549460	−0.0305728
$324$	−42.4463	−2.35813
$325$	0	0
$326$	−21.0305	−1.16477
$327$	−15.3807	−0.850557
$328$	20.8691	1.15230
$329$	0	0
$330$	−13.3174	−0.733100
$331$	−6.48851	−0.356641	−0.178320	−	0.983972i	$-0.557066\pi$
$331$	−6.48851	−0.356641	−0.178320	+	0.983972i	$0.557066\pi$
$332$	37.0509	2.03343
$333$	−4.37427	−0.239709
$334$	−12.4680	−0.682221
$335$	24.4225	1.33434
$336$	0	0
$337$	−25.4871	−1.38837	−0.694187	−	0.719795i	$-0.744236\pi$
$337$	−25.4871	−1.38837	−0.694187	+	0.719795i	$0.744236\pi$
$338$	0	0
$339$	0.370406	0.0201177
$340$	1.64514	0.0892205
$341$	−4.86762	−0.263596
$342$	11.0537	0.597717
$343$	0	0
$344$	−22.6848	−1.22308
$345$	−3.75639	−0.202237
$346$	37.5070	2.01639
$347$	−14.2034	−0.762480	−0.381240	−	0.924476i	$-0.624503\pi$
$347$	−14.2034	−0.762480	−0.381240	+	0.924476i	$0.624503\pi$
$348$	−10.0705	−0.539835
$349$	25.7839	1.38018	0.690089	−	0.723725i	$-0.257571\pi$
$349$	25.7839	1.38018	0.690089	+	0.723725i	$0.257571\pi$
$350$	0	0
$351$	0	0
$352$	−0.464240	−0.0247441
$353$	14.8854	0.792269	0.396134	−	0.918193i	$-0.370351\pi$
$353$	14.8854	0.792269	0.396134	+	0.918193i	$0.370351\pi$
$354$	1.98920	0.105725
$355$	41.7586	2.21632
$356$	39.5032	2.09367
$357$	0	0
$358$	−8.40225	−0.444073
$359$	−2.43200	−0.128356	−0.0641780	−	0.997938i	$-0.520443\pi$
$359$	−2.43200	−0.128356	−0.0641780	+	0.997938i	$0.520443\pi$
$360$	−16.2443	−0.856147
$361$	10.6366	0.559820
$362$	−15.4960	−0.814450
$363$	20.6512	1.08391
$364$	0	0
$365$	−26.5468	−1.38953
$366$	−7.32341	−0.382801
$367$	15.9888	0.834610	0.417305	−	0.908767i	$-0.362975\pi$
$367$	15.9888	0.834610	0.417305	+	0.908767i	$0.362975\pi$
$368$	1.65158	0.0860945
$369$	−3.70781	−0.193021
$370$	52.9949	2.75507
$371$	0	0
$372$	−55.6129	−2.88340
$373$	−7.03416	−0.364215	−0.182107	−	0.983279i	$-0.558292\pi$
$373$	−7.03416	−0.364215	−0.182107	+	0.983279i	$0.558292\pi$
$374$	0.165423	0.00855380
$375$	−58.6699	−3.02970
$376$	13.3015	0.685974
$377$	0	0
$378$	0	0
$379$	−23.4139	−1.20269	−0.601346	−	0.798988i	$-0.705369\pi$
$379$	−23.4139	−1.20269	−0.601346	+	0.798988i	$0.705369\pi$
$380$	−88.7352	−4.55202
$381$	−31.0960	−1.59309
$382$	55.2982	2.82930
$383$	35.9429	1.83660	0.918299	−	0.395888i	$-0.129563\pi$
$383$	35.9429	1.83660	0.918299	+	0.395888i	$0.129563\pi$
$384$	−39.3667	−2.00892
$385$	0	0
$386$	−60.8010	−3.09469
$387$	4.03039	0.204876
$388$	−15.0947	−0.766315
$389$	−14.0560	−0.712670	−0.356335	−	0.934358i	$-0.615974\pi$
$389$	−14.0560	−0.712670	−0.356335	+	0.934358i	$0.615974\pi$
$390$	0	0
$391$	0.0466601	0.00235970
$392$	0	0
$393$	4.39884	0.221892
$394$	27.7764	1.39936
$395$	1.19451	0.0601023
$396$	−2.20509	−0.110810
$397$	3.17888	0.159543	0.0797716	−	0.996813i	$-0.474581\pi$
$397$	3.17888	0.159543	0.0797716	+	0.996813i	$0.474581\pi$
$398$	−52.0267	−2.60786
$399$	0	0
$400$	43.6581	2.18291
$401$	21.6892	1.08311	0.541553	−	0.840667i	$-0.317837\pi$
$401$	21.6892	1.08311	0.541553	+	0.840667i	$0.317837\pi$
$402$	28.0569	1.39935
$403$	0	0
$404$	0.335311	0.0166823
$405$	−44.8439	−2.22831
$406$	0	0
$407$	3.53089	0.175020
$408$	0.927635	0.0459248
$409$	25.4642	1.25912	0.629562	−	0.776951i	$-0.283235\pi$
$409$	25.4642	1.25912	0.629562	+	0.776951i	$0.283235\pi$
$410$	44.9206	2.21847
$411$	24.2063	1.19401
$412$	−45.4733	−2.24031
$413$	0	0
$414$	−0.938683	−0.0461337
$415$	39.1438	1.92150
$416$	0	0
$417$	−26.6366	−1.30440
$418$	−8.92251	−0.436414
$419$	−7.01689	−0.342798	−0.171399	−	0.985202i	$-0.554829\pi$
$419$	−7.01689	−0.342798	−0.171399	+	0.985202i	$0.554829\pi$
$420$	0	0
$421$	−1.00282	−0.0488744	−0.0244372	−	0.999701i	$-0.507779\pi$
$421$	−1.00282	−0.0488744	−0.0244372	+	0.999701i	$0.507779\pi$
$422$	−49.1814	−2.39411
$423$	−2.36328	−0.114907
$424$	67.3261	3.26964
$425$	1.23342	0.0598297
$426$	47.9730	2.32430
$427$	0	0
$428$	17.0708	0.825149
$429$	0	0
$430$	−48.8288	−2.35473
$431$	26.2281	1.26336	0.631682	−	0.775228i	$-0.282365\pi$
$431$	26.2281	1.26336	0.631682	+	0.775228i	$0.282365\pi$
$432$	15.1518	0.728994
$433$	−14.8546	−0.713864	−0.356932	−	0.934130i	$-0.616177\pi$
$433$	−14.8546	−0.713864	−0.356932	+	0.934130i	$0.616177\pi$
$434$	0	0
$435$	−10.6393	−0.510117
$436$	30.8541	1.47764
$437$	−2.51674	−0.120392
$438$	−30.4974	−1.45722
$439$	−14.6741	−0.700355	−0.350178	−	0.936683i	$-0.613879\pi$
$439$	−14.6741	−0.700355	−0.350178	+	0.936683i	$0.613879\pi$
$440$	13.1123	0.625103
$441$	0	0
$442$	0	0
$443$	−5.56688	−0.264491	−0.132245	−	0.991217i	$-0.542219\pi$
$443$	−5.56688	−0.264491	−0.132245	+	0.991217i	$0.542219\pi$
$444$	40.3407	1.91449
$445$	41.7346	1.97841
$446$	−27.3735	−1.29617
$447$	−22.9909	−1.08743
$448$	0	0
$449$	5.99261	0.282809	0.141404	−	0.989952i	$-0.454838\pi$
$449$	5.99261	0.282809	0.141404	+	0.989952i	$0.454838\pi$
$450$	−24.8133	−1.16971
$451$	2.99292	0.140931
$452$	−0.743042	−0.0349498
$453$	−22.7766	−1.07014
$454$	−25.8785	−1.21454
$455$	0	0
$456$	−50.0345	−2.34308
$457$	−18.1998	−0.851352	−0.425676	−	0.904876i	$-0.639964\pi$
$457$	−18.1998	−0.851352	−0.425676	+	0.904876i	$0.639964\pi$
$458$	26.3288	1.23026
$459$	0.428067	0.0199805
$460$	7.53540	0.351340
$461$	−6.54384	−0.304777	−0.152389	−	0.988321i	$-0.548697\pi$
$461$	−6.54384	−0.304777	−0.152389	+	0.988321i	$0.548697\pi$
$462$	0	0
$463$	−35.9986	−1.67300	−0.836499	−	0.547968i	$-0.815401\pi$
$463$	−35.9986	−1.67300	−0.836499	+	0.547968i	$0.815401\pi$
$464$	4.67781	0.217162
$465$	−58.7543	−2.72467
$466$	−58.6098	−2.71505
$467$	−6.59368	−0.305119	−0.152560	−	0.988294i	$-0.548752\pi$
$467$	−6.59368	−0.305119	−0.152560	+	0.988294i	$0.548752\pi$
$468$	0	0
$469$	0	0
$470$	28.6314	1.32067
$471$	35.5088	1.63616
$472$	−1.95856	−0.0901500
$473$	−3.25331	−0.149587
$474$	1.37227	0.0630306
$475$	−66.5279	−3.05251
$476$	0	0
$477$	−11.9618	−0.547693
$478$	20.8168	0.952137
$479$	27.9239	1.27588	0.637939	−	0.770087i	$-0.279787\pi$
$479$	27.9239	1.27588	0.637939	+	0.770087i	$0.279787\pi$
$480$	−5.60359	−0.255768
$481$	0	0
$482$	34.7516	1.58289
$483$	0	0
$484$	−41.4268	−1.88304
$485$	−15.9473	−0.724130
$486$	−20.5389	−0.931662
$487$	−16.6528	−0.754611	−0.377305	−	0.926089i	$-0.623149\pi$
$487$	−16.6528	−0.754611	−0.377305	+	0.926089i	$0.623149\pi$
$488$	7.21060	0.326409
$489$	−16.9131	−0.764835
$490$	0	0
$491$	22.2229	1.00291	0.501454	−	0.865184i	$-0.332799\pi$
$491$	22.2229	1.00291	0.501454	+	0.865184i	$0.332799\pi$
$492$	34.1944	1.54160
$493$	0.132157	0.00595204
$494$	0	0
$495$	−2.32965	−0.104710
$496$	25.8326	1.15992
$497$	0	0
$498$	44.9691	2.01511
$499$	−23.9155	−1.07061	−0.535303	−	0.844660i	$-0.679803\pi$
$499$	−23.9155	−1.07061	−0.535303	+	0.844660i	$0.679803\pi$
$500$	117.693	5.26339
$501$	−10.0270	−0.447974
$502$	−31.8823	−1.42298
$503$	17.5269	0.781487	0.390744	−	0.920500i	$-0.372218\pi$
$503$	17.5269	0.781487	0.390744	+	0.920500i	$0.372218\pi$
$504$	0	0
$505$	0.354251	0.0157640
$506$	0.757700	0.0336839
$507$	0	0
$508$	62.3791	2.76763
$509$	−20.5731	−0.911888	−0.455944	−	0.890008i	$-0.650698\pi$
$509$	−20.5731	−0.911888	−0.455944	+	0.890008i	$0.650698\pi$
$510$	1.99673	0.0884166
$511$	0	0
$512$	36.0017	1.59107
$513$	−23.0890	−1.01940
$514$	3.81275	0.168173
$515$	−48.0420	−2.11698
$516$	−37.1694	−1.63629
$517$	1.90763	0.0838973
$518$	0	0
$519$	30.1638	1.32404
$520$	0	0
$521$	38.7372	1.69711	0.848555	−	0.529108i	$-0.177473\pi$
$521$	38.7372	1.69711	0.848555	+	0.529108i	$0.177473\pi$
$522$	−2.65866	−0.116366
$523$	−15.5881	−0.681619	−0.340810	−	0.940132i	$-0.610701\pi$
$523$	−15.5881	−0.681619	−0.340810	+	0.940132i	$0.610701\pi$
$524$	−8.82416	−0.385485
$525$	0	0
$526$	52.3719	2.28352
$527$	0.729819	0.0317914
$528$	4.70893	0.204930
$529$	−22.7863	−0.990708
$530$	144.919	6.29487
$531$	0.347976	0.0151009
$532$	0	0
$533$	0	0
$534$	47.9454	2.07480
$535$	18.0351	0.779725
$536$	−27.6247	−1.19321
$537$	−6.75724	−0.291597
$538$	−26.0450	−1.12288
$539$	0	0
$540$	69.1309	2.97492
$541$	7.31760	0.314608	0.157304	−	0.987550i	$-0.449720\pi$
$541$	7.31760	0.314608	0.157304	+	0.987550i	$0.449720\pi$
$542$	−35.4985	−1.52479
$543$	−12.4621	−0.534802
$544$	0.0696052	0.00298430
$545$	32.5969	1.39630
$546$	0	0
$547$	35.8280	1.53189	0.765946	−	0.642904i	$-0.222271\pi$
$547$	35.8280	1.53189	0.765946	+	0.642904i	$0.222271\pi$
$548$	−48.5583	−2.07431
$549$	−1.28111	−0.0546762
$550$	20.0292	0.854046
$551$	−7.12823	−0.303673
$552$	4.24893	0.180846
$553$	0	0
$554$	36.8300	1.56476
$555$	42.6194	1.80909
$556$	53.4336	2.26609
$557$	25.0691	1.06221	0.531105	−	0.847306i	$-0.321777\pi$
$557$	25.0691	1.06221	0.531105	+	0.847306i	$0.321777\pi$
$558$	−14.6821	−0.621543
$559$	0	0
$560$	0	0
$561$	0.133036	0.00561678
$562$	−37.5753	−1.58502
$563$	−33.9716	−1.43173	−0.715866	−	0.698237i	$-0.753968\pi$
$563$	−33.9716	−1.43173	−0.715866	+	0.698237i	$0.753968\pi$
$564$	21.7948	0.917726
$565$	−0.785014	−0.0330258
$566$	43.5455	1.83036
$567$	0	0
$568$	−47.2340	−1.98189
$569$	9.74692	0.408612	0.204306	−	0.978907i	$-0.434506\pi$
$569$	9.74692	0.408612	0.204306	+	0.978907i	$0.434506\pi$
$570$	−107.699	−4.51101
$571$	17.0974	0.715502	0.357751	−	0.933817i	$-0.383544\pi$
$571$	17.0974	0.715502	0.357751	+	0.933817i	$0.383544\pi$
$572$	0	0
$573$	44.4718	1.85784
$574$	0	0
$575$	5.64955	0.235602
$576$	−7.35896	−0.306623
$577$	−23.8507	−0.992916	−0.496458	−	0.868061i	$-0.665366\pi$
$577$	−23.8507	−0.992916	−0.496458	+	0.868061i	$0.665366\pi$
$578$	41.3655	1.72058
$579$	−48.8973	−2.03210
$580$	21.3427	0.886208
$581$	0	0
$582$	−18.3205	−0.759411
$583$	9.65550	0.399890
$584$	30.0277	1.24255
$585$	0	0
$586$	−82.7437	−3.41811
$587$	−32.5384	−1.34300	−0.671502	−	0.741003i	$-0.734350\pi$
$587$	−32.5384	−1.34300	−0.671502	+	0.741003i	$0.734350\pi$
$588$	0	0
$589$	−39.3647	−1.62199
$590$	−4.21578	−0.173561
$591$	22.3383	0.918876
$592$	−18.7386	−0.770150
$593$	31.4379	1.29100	0.645501	−	0.763760i	$-0.276649\pi$
$593$	31.4379	1.29100	0.645501	+	0.763760i	$0.276649\pi$
$594$	6.95126	0.285214
$595$	0	0
$596$	46.1202	1.88916
$597$	−41.8408	−1.71243
$598$	0	0
$599$	34.6860	1.41723	0.708615	−	0.705595i	$-0.249320\pi$
$599$	34.6860	1.41723	0.708615	+	0.705595i	$0.249320\pi$
$600$	112.317	4.58532
$601$	−10.4187	−0.424986	−0.212493	−	0.977163i	$-0.568158\pi$
$601$	−10.4187	−0.424986	−0.212493	+	0.977163i	$0.568158\pi$
$602$	0	0
$603$	4.90808	0.199872
$604$	45.6902	1.85911
$605$	−43.7669	−1.77938
$606$	0.406970	0.0165320
$607$	−14.3796	−0.583648	−0.291824	−	0.956472i	$-0.594262\pi$
$607$	−14.3796	−0.583648	−0.291824	+	0.956472i	$0.594262\pi$
$608$	−3.75434	−0.152259
$609$	0	0
$610$	15.5208	0.628417
$611$	0	0
$612$	0.330617	0.0133644
$613$	−23.3341	−0.942457	−0.471228	−	0.882011i	$-0.656189\pi$
$613$	−23.3341	−0.942457	−0.471228	+	0.882011i	$0.656189\pi$
$614$	17.3907	0.701830
$615$	36.1260	1.45674
$616$	0	0
$617$	−28.2233	−1.13623	−0.568115	−	0.822949i	$-0.692327\pi$
$617$	−28.2233	−1.13623	−0.568115	+	0.822949i	$0.692327\pi$
$618$	−55.1914	−2.22012
$619$	30.2898	1.21745	0.608725	−	0.793382i	$-0.291681\pi$
$619$	30.2898	1.21745	0.608725	+	0.793382i	$0.291681\pi$
$620$	117.862	4.73347
$621$	1.96071	0.0786808
$622$	−36.0747	−1.44646
$623$	0	0
$624$	0	0
$625$	63.2385	2.52954
$626$	−81.5423	−3.25909
$627$	−7.17564	−0.286568
$628$	−71.2313	−2.84244
$629$	−0.529398	−0.0211085
$630$	0	0
$631$	25.3506	1.00919	0.504596	−	0.863355i	$-0.331641\pi$
$631$	25.3506	1.00919	0.504596	+	0.863355i	$0.331641\pi$
$632$	−1.35113	−0.0537452
$633$	−39.5525	−1.57207
$634$	21.3738	0.848861
$635$	65.9027	2.61527
$636$	110.315	4.37427
$637$	0	0
$638$	2.14605	0.0849631
$639$	8.39205	0.331984
$640$	83.4311	3.29791
$641$	−10.2681	−0.405565	−0.202782	−	0.979224i	$-0.564998\pi$
$641$	−10.2681	−0.405565	−0.202782	+	0.979224i	$0.564998\pi$
$642$	20.7190	0.817714
$643$	−3.80764	−0.150159	−0.0750793	−	0.997178i	$-0.523921\pi$
$643$	−3.80764	−0.150159	−0.0750793	+	0.997178i	$0.523921\pi$
$644$	0	0
$645$	−39.2690	−1.54621
$646$	1.33778	0.0526344
$647$	27.7864	1.09240	0.546198	−	0.837656i	$-0.316075\pi$
$647$	27.7864	1.09240	0.546198	+	0.837656i	$0.316075\pi$
$648$	50.7239	1.99262
$649$	−0.280885	−0.0110257
$650$	0	0
$651$	0	0
$652$	33.9280	1.32872
$653$	−4.00325	−0.156659	−0.0783297	−	0.996928i	$-0.524959\pi$
$653$	−4.00325	−0.156659	−0.0783297	+	0.996928i	$0.524959\pi$
$654$	37.4479	1.46433
$655$	−9.32261	−0.364264
$656$	−15.8836	−0.620149
$657$	−5.33500	−0.208138
$658$	0	0
$659$	−0.740310	−0.0288384	−0.0144192	−	0.999896i	$-0.504590\pi$
$659$	−0.740310	−0.0288384	−0.0144192	+	0.999896i	$0.504590\pi$
$660$	21.4847	0.836290
$661$	28.3894	1.10422	0.552110	−	0.833771i	$-0.313823\pi$
$661$	28.3894	1.10422	0.552110	+	0.833771i	$0.313823\pi$
$662$	15.7977	0.613997
$663$	0	0
$664$	−44.2764	−1.71826
$665$	0	0
$666$	10.6501	0.412685
$667$	0.605329	0.0234384
$668$	20.1144	0.778250
$669$	−22.0143	−0.851121
$670$	−59.4620	−2.29722
$671$	1.03410	0.0399210
$672$	0	0
$673$	−39.3719	−1.51768	−0.758838	−	0.651279i	$-0.774233\pi$
$673$	−39.3719	−1.51768	−0.758838	+	0.651279i	$0.774233\pi$
$674$	62.0542	2.39024
$675$	51.8299	1.99493
$676$	0	0
$677$	26.1688	1.00575	0.502874	−	0.864359i	$-0.332276\pi$
$677$	26.1688	1.00575	0.502874	+	0.864359i	$0.332276\pi$
$678$	−0.901837	−0.0346349
$679$	0	0
$680$	−1.96597	−0.0753915
$681$	−20.8119	−0.797515
$682$	11.8513	0.453810
$683$	41.6669	1.59434	0.797170	−	0.603756i	$-0.206330\pi$
$683$	41.6669	1.59434	0.797170	+	0.603756i	$0.206330\pi$
$684$	−17.8327	−0.681852
$685$	−51.3012	−1.96012
$686$	0	0
$687$	21.1741	0.807841
$688$	17.2655	0.658239
$689$	0	0
$690$	9.14578	0.348174
$691$	−1.89238	−0.0719894	−0.0359947	−	0.999352i	$-0.511460\pi$
$691$	−1.89238	−0.0719894	−0.0359947	+	0.999352i	$0.511460\pi$
$692$	−60.5092	−2.30022
$693$	0	0
$694$	34.5815	1.31269
$695$	56.4519	2.14134
$696$	12.0344	0.456161
$697$	−0.448739	−0.0169972
$698$	−62.7766	−2.37613
$699$	−47.1350	−1.78281
$700$	0	0
$701$	28.5701	1.07908	0.539539	−	0.841961i	$-0.318599\pi$
$701$	28.5701	1.07908	0.539539	+	0.841961i	$0.318599\pi$
$702$	0	0
$703$	28.5545	1.07695
$704$	5.94011	0.223876
$705$	23.0259	0.867206
$706$	−36.2418	−1.36398
$707$	0	0
$708$	−3.20913	−0.120607
$709$	−37.2019	−1.39715	−0.698574	−	0.715538i	$-0.746182\pi$
$709$	−37.2019	−1.39715	−0.698574	+	0.715538i	$0.746182\pi$
$710$	−101.671	−3.81564
$711$	0.240056	0.00900279
$712$	−47.2069	−1.76915
$713$	3.34285	0.125191
$714$	0	0
$715$	0	0
$716$	13.5552	0.506580
$717$	16.7412	0.625212
$718$	5.92125	0.220979
$719$	5.71706	0.213210	0.106605	−	0.994301i	$-0.466002\pi$
$719$	5.71706	0.213210	0.106605	+	0.994301i	$0.466002\pi$
$720$	12.3636	0.460762
$721$	0	0
$722$	−25.8971	−0.963791
$723$	27.9479	1.03939
$724$	24.9993	0.929092
$725$	16.0014	0.594276
$726$	−50.2801	−1.86607
$727$	1.37012	0.0508149	0.0254074	−	0.999677i	$-0.491912\pi$
$727$	1.37012	0.0508149	0.0254074	+	0.999677i	$0.491912\pi$
$728$	0	0
$729$	15.9014	0.588942
$730$	64.6343	2.39222
$731$	0.487781	0.0180412
$732$	11.8147	0.436684
$733$	3.31302	0.122369	0.0611846	−	0.998126i	$-0.480512\pi$
$733$	3.31302	0.122369	0.0611846	+	0.998126i	$0.480512\pi$
$734$	−38.9284	−1.43687
$735$	0	0
$736$	0.318819	0.0117518
$737$	−3.96177	−0.145934
$738$	9.02750	0.332307
$739$	4.72593	0.173846	0.0869231	−	0.996215i	$-0.472297\pi$
$739$	4.72593	0.173846	0.0869231	+	0.996215i	$0.472297\pi$
$740$	−85.4954	−3.14288
$741$	0	0
$742$	0	0
$743$	27.9917	1.02692	0.513458	−	0.858115i	$-0.328364\pi$
$743$	27.9917	1.02692	0.513458	+	0.858115i	$0.328364\pi$
$744$	66.4582	2.43648
$745$	48.7254	1.78516
$746$	17.1262	0.627036
$747$	7.86656	0.287823
$748$	−0.266873	−0.00975783
$749$	0	0
$750$	142.845	5.21597
$751$	−18.1573	−0.662571	−0.331285	−	0.943531i	$-0.607482\pi$
$751$	−18.1573	−0.662571	−0.331285	+	0.943531i	$0.607482\pi$
$752$	−10.1238	−0.369178
$753$	−25.6403	−0.934384
$754$	0	0
$755$	48.2711	1.75677
$756$	0	0
$757$	−25.1014	−0.912327	−0.456164	−	0.889896i	$-0.650777\pi$
$757$	−25.1014	−0.912327	−0.456164	+	0.889896i	$0.650777\pi$
$758$	57.0065	2.07057
$759$	0.609356	0.0221182
$760$	106.040	3.84647
$761$	34.9690	1.26763	0.633814	−	0.773486i	$-0.281489\pi$
$761$	34.9690	1.26763	0.633814	+	0.773486i	$0.281489\pi$
$762$	75.7101	2.74269
$763$	0	0
$764$	−89.2114	−3.22755
$765$	0.349293	0.0126287
$766$	−87.5112	−3.16191
$767$	0	0
$768$	61.2909	2.21165
$769$	37.5705	1.35483	0.677413	−	0.735603i	$-0.263101\pi$
$769$	37.5705	1.35483	0.677413	+	0.735603i	$0.263101\pi$
$770$	0	0
$771$	3.06628	0.110430
$772$	98.0889	3.53030
$773$	−41.9589	−1.50916	−0.754578	−	0.656211i	$-0.772158\pi$
$773$	−41.9589	−1.50916	−0.754578	+	0.656211i	$0.772158\pi$
$774$	−9.81290	−0.352718
$775$	88.3655	3.17418
$776$	18.0383	0.647538
$777$	0	0
$778$	34.2226	1.22694
$779$	24.2040	0.867197
$780$	0	0
$781$	−6.77401	−0.242393
$782$	−0.113605	−0.00406249
$783$	5.55339	0.198462
$784$	0	0
$785$	−75.2549	−2.68596
$786$	−10.7100	−0.382012
$787$	33.8960	1.20826	0.604131	−	0.796885i	$-0.293520\pi$
$787$	33.8960	1.20826	0.604131	+	0.796885i	$0.293520\pi$
$788$	−44.8111	−1.59633
$789$	42.1184	1.49946
$790$	−2.90830	−0.103473
$791$	0	0
$792$	2.63512	0.0936348
$793$	0	0
$794$	−7.73969	−0.274671
$795$	116.546	4.13347
$796$	83.9335	2.97494
$797$	34.4758	1.22119	0.610597	−	0.791941i	$-0.290929\pi$
$797$	34.4758	1.22119	0.610597	+	0.791941i	$0.290929\pi$
$798$	0	0
$799$	−0.286017	−0.0101186
$800$	8.42770	0.297964
$801$	8.38723	0.296348
$802$	−52.8072	−1.86469
$803$	4.30639	0.151969
$804$	−45.2636	−1.59632
$805$	0	0
$806$	0	0
$807$	−20.9458	−0.737328
$808$	−0.400701	−0.0140966
$809$	39.4865	1.38827	0.694135	−	0.719845i	$-0.255787\pi$
$809$	39.4865	1.38827	0.694135	+	0.719845i	$0.255787\pi$
$810$	109.183	3.83629
$811$	18.5399	0.651023	0.325512	−	0.945538i	$-0.394463\pi$
$811$	18.5399	0.651023	0.325512	+	0.945538i	$0.394463\pi$
$812$	0	0
$813$	−28.5486	−1.00124
$814$	−8.59674	−0.301316
$815$	35.8444	1.25558
$816$	−0.706026	−0.0247159
$817$	−26.3097	−0.920461
$818$	−61.9983	−2.16772
$819$	0	0
$820$	−72.4694	−2.53074
$821$	44.7239	1.56087	0.780437	−	0.625234i	$-0.214997\pi$
$821$	44.7239	1.56087	0.780437	+	0.625234i	$0.214997\pi$
$822$	−58.9356	−2.05562
$823$	−25.4742	−0.887976	−0.443988	−	0.896033i	$-0.646437\pi$
$823$	−25.4742	−0.887976	−0.443988	+	0.896033i	$0.646437\pi$
$824$	54.3412	1.89307
$825$	16.1078	0.560802
$826$	0	0
$827$	−14.1027	−0.490399	−0.245200	−	0.969473i	$-0.578853\pi$
$827$	−14.1027	−0.490399	−0.245200	+	0.969473i	$0.578853\pi$
$828$	1.51436	0.0526275
$829$	34.8050	1.20883	0.604415	−	0.796670i	$-0.293407\pi$
$829$	34.8050	1.20883	0.604415	+	0.796670i	$0.293407\pi$
$830$	−95.3045	−3.30807
$831$	29.6193	1.02748
$832$	0	0
$833$	0	0
$834$	64.8528	2.24567
$835$	21.2506	0.735407
$836$	14.3945	0.497844
$837$	30.6679	1.06004
$838$	17.0842	0.590164
$839$	32.3178	1.11573	0.557867	−	0.829931i	$-0.311620\pi$
$839$	32.3178	1.11573	0.557867	+	0.829931i	$0.311620\pi$
$840$	0	0
$841$	−27.2855	−0.940880
$842$	2.44159	0.0841427
$843$	−30.2187	−1.04079
$844$	79.3432	2.73111
$845$	0	0
$846$	5.75393	0.197824
$847$	0	0
$848$	−51.2421	−1.75966
$849$	35.0201	1.20189
$850$	−3.00304	−0.103004
$851$	−2.42485	−0.0831228
$852$	−77.3937	−2.65147
$853$	−5.51656	−0.188883	−0.0944417	−	0.995530i	$-0.530107\pi$
$853$	−5.51656	−0.188883	−0.0944417	+	0.995530i	$0.530107\pi$
$854$	0	0
$855$	−18.8400	−0.644316
$856$	−20.3998	−0.697253
$857$	−41.1525	−1.40574	−0.702871	−	0.711317i	$-0.748099\pi$
$857$	−41.1525	−1.40574	−0.702871	+	0.711317i	$0.748099\pi$
$858$	0	0
$859$	−7.92936	−0.270546	−0.135273	−	0.990808i	$-0.543191\pi$
$859$	−7.92936	−0.270546	−0.135273	+	0.990808i	$0.543191\pi$
$860$	78.7743	2.68618
$861$	0	0
$862$	−63.8583	−2.17502
$863$	7.27113	0.247512	0.123756	−	0.992313i	$-0.460506\pi$
$863$	7.27113	0.247512	0.123756	+	0.992313i	$0.460506\pi$
$864$	2.92489	0.0995069
$865$	−63.9272	−2.17359
$866$	36.1668	1.22900
$867$	33.2669	1.12980
$868$	0	0
$869$	−0.193772	−0.00657325
$870$	25.9039	0.878223
$871$	0	0
$872$	−36.8710	−1.24861
$873$	−3.20486	−0.108468
$874$	6.12757	0.207268
$875$	0	0
$876$	49.2008	1.66234
$877$	−12.1073	−0.408834	−0.204417	−	0.978884i	$-0.565530\pi$
$877$	−12.1073	−0.408834	−0.204417	+	0.978884i	$0.565530\pi$
$878$	35.7273	1.20574
$879$	−66.5440	−2.24447
$880$	−9.97979	−0.336419
$881$	−3.32436	−0.112001	−0.0560003	−	0.998431i	$-0.517835\pi$
$881$	−3.32436	−0.112001	−0.0560003	+	0.998431i	$0.517835\pi$
$882$	0	0
$883$	−21.1635	−0.712208	−0.356104	−	0.934446i	$-0.615895\pi$
$883$	−21.1635	−0.712208	−0.356104	+	0.934446i	$0.615895\pi$
$884$	0	0
$885$	−3.39041	−0.113967
$886$	13.5538	0.455350
$887$	−26.7516	−0.898229	−0.449115	−	0.893474i	$-0.648261\pi$
$887$	−26.7516	−0.898229	−0.449115	+	0.893474i	$0.648261\pi$
$888$	−48.2077	−1.61774
$889$	0	0
$890$	−101.612	−3.40605
$891$	7.27451	0.243705
$892$	44.1611	1.47862
$893$	15.4271	0.516248
$894$	55.9765	1.87213
$895$	14.3209	0.478693
$896$	0	0
$897$	0	0
$898$	−14.5904	−0.486887
$899$	9.46805	0.315777
$900$	40.0307	1.33436
$901$	−1.44768	−0.0482293
$902$	−7.28695	−0.242629
$903$	0	0
$904$	0.887945	0.0295326
$905$	26.4114	0.877946
$906$	55.4547	1.84236
$907$	42.2576	1.40314	0.701570	−	0.712601i	$-0.252483\pi$
$907$	42.2576	1.40314	0.701570	+	0.712601i	$0.252483\pi$
$908$	41.7492	1.38549
$909$	0.0711924	0.00236130
$910$	0	0
$911$	−27.4466	−0.909347	−0.454673	−	0.890658i	$-0.650244\pi$
$911$	−27.4466	−0.909347	−0.454673	+	0.890658i	$0.650244\pi$
$912$	38.0814	1.26100
$913$	−6.34985	−0.210149
$914$	44.3116	1.46570
$915$	12.4821	0.412645
$916$	−42.4756	−1.40343
$917$	0	0
$918$	−1.04223	−0.0343986
$919$	0.918740	0.0303064	0.0151532	−	0.999885i	$-0.495176\pi$
$919$	0.918740	0.0303064	0.0151532	+	0.999885i	$0.495176\pi$
$920$	−9.00490	−0.296883
$921$	13.9859	0.460850
$922$	15.9325	0.524708
$923$	0	0
$924$	0	0
$925$	−64.0989	−2.10756
$926$	87.6468	2.88025
$927$	−9.65479	−0.317105
$928$	0.902999	0.0296424
$929$	47.2496	1.55021	0.775104	−	0.631834i	$-0.217697\pi$
$929$	47.2496	1.55021	0.775104	+	0.631834i	$0.217697\pi$
$930$	143.051	4.69082
$931$	0	0
$932$	94.5539	3.09721
$933$	−29.0119	−0.949807
$934$	16.0538	0.525297
$935$	−0.281947	−0.00922067
$936$	0	0
$937$	−11.3635	−0.371230	−0.185615	−	0.982623i	$-0.559428\pi$
$937$	−11.3635	−0.371230	−0.185615	+	0.982623i	$0.559428\pi$
$938$	0	0
$939$	−65.5778	−2.14005
$940$	−46.1904	−1.50657
$941$	20.3832	0.664474	0.332237	−	0.943196i	$-0.392197\pi$
$941$	20.3832	0.664474	0.332237	+	0.943196i	$0.392197\pi$
$942$	−86.4541	−2.81683
$943$	−2.05540	−0.0669330
$944$	1.49067	0.0485170
$945$	0	0
$946$	7.92092	0.257531
$947$	−48.2179	−1.56687	−0.783435	−	0.621473i	$-0.786534\pi$
$947$	−48.2179	−1.56687	−0.783435	+	0.621473i	$0.786534\pi$
$948$	−2.21386	−0.0719027
$949$	0	0
$950$	161.977	5.25523
$951$	17.1892	0.557397
$952$	0	0
$953$	55.8697	1.80980	0.904899	−	0.425627i	$-0.139946\pi$
$953$	55.8697	1.80980	0.904899	+	0.425627i	$0.139946\pi$
$954$	29.1237	0.942915
$955$	−94.2506	−3.04988
$956$	−33.5832	−1.08616
$957$	1.72590	0.0557903
$958$	−67.9871	−2.19657
$959$	0	0
$960$	71.6999	2.31410
$961$	21.2861	0.686648
$962$	0	0
$963$	3.62443	0.116796
$964$	−56.0640	−1.80570
$965$	103.630	3.33596
$966$	0	0
$967$	32.7379	1.05278	0.526391	−	0.850243i	$-0.323545\pi$
$967$	32.7379	1.05278	0.526391	+	0.850243i	$0.323545\pi$
$968$	49.5056	1.59117
$969$	1.07587	0.0345619
$970$	38.8273	1.24667
$971$	3.30568	0.106084	0.0530422	−	0.998592i	$-0.483108\pi$
$971$	3.30568	0.106084	0.0530422	+	0.998592i	$0.483108\pi$
$972$	33.1349	1.06280
$973$	0	0
$974$	40.5450	1.29915
$975$	0	0
$976$	−5.48801	−0.175667
$977$	3.28424	0.105072	0.0525360	−	0.998619i	$-0.483270\pi$
$977$	3.28424	0.105072	0.0525360	+	0.998619i	$0.483270\pi$
$978$	41.1787	1.31675
$979$	−6.77013	−0.216374
$980$	0	0
$981$	6.55086	0.209153
$982$	−54.1068	−1.72662
$983$	−34.2240	−1.09157	−0.545787	−	0.837924i	$-0.683769\pi$
$983$	−34.2240	−1.09157	−0.545787	+	0.837924i	$0.683769\pi$
$984$	−40.8628	−1.30266
$985$	−47.3423	−1.50845
$986$	−0.321765	−0.0102471
$987$	0	0
$988$	0	0
$989$	2.23422	0.0710442
$990$	5.67207	0.180270
$991$	−44.7352	−1.42106	−0.710531	−	0.703666i	$-0.751545\pi$
$991$	−44.7352	−1.42106	−0.710531	+	0.703666i	$0.751545\pi$
$992$	4.98669	0.158328
$993$	12.7048	0.403176
$994$	0	0
$995$	88.6746	2.81117
$996$	−72.5476	−2.29876
$997$	9.47078	0.299943	0.149971	−	0.988690i	$-0.452082\pi$
$997$	9.47078	0.299943	0.149971	+	0.988690i	$0.452082\pi$
$998$	58.2278	1.84317
$999$	−22.2460	−0.703831

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cw.1.2		24
7.3	odd	6		1183.2.e.k.170.23	✓	48
7.5	odd	6		1183.2.e.k.508.23	yes	48
7.6	odd	2		8281.2.a.cv.1.2		24
13.12	even	2		8281.2.a.ct.1.23		24
91.12	odd	6		1183.2.e.l.508.2	yes	48
91.38	odd	6		1183.2.e.l.170.2	yes	48
91.90	odd	2		8281.2.a.cu.1.23		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.23	✓	48	7.3	odd	6
1183.2.e.k.508.23	yes	48	7.5	odd	6
1183.2.e.l.170.2	yes	48	91.38	odd	6
1183.2.e.l.508.2	yes	48	91.12	odd	6
8281.2.a.ct.1.23		24	13.12	even	2
8281.2.a.cu.1.23		24	91.90	odd	2
8281.2.a.cv.1.2		24	7.6	odd	2
8281.2.a.cw.1.2		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)

\(f(q)\)	\(=\)	\(q-2.43473 q^{2} -1.95805 q^{3} +3.92789 q^{4} +4.14976 q^{5} +4.76731 q^{6} -4.69388 q^{8} +0.833959 q^{9} +O(q^{10})\)	\(q-2.43473 q^{2} -1.95805 q^{3} +3.92789 q^{4} +4.14976 q^{5} +4.76731 q^{6} -4.69388 q^{8} +0.833959 q^{9} -10.1035 q^{10} -0.673168 q^{11} -7.69100 q^{12} -8.12544 q^{15} +3.57253 q^{16} +0.100930 q^{17} -2.03046 q^{18} -5.44395 q^{19} +16.2998 q^{20} +1.63898 q^{22} +0.462300 q^{23} +9.19084 q^{24} +12.2205 q^{25} +4.24122 q^{27} +1.30939 q^{29} +19.7832 q^{30} +7.23091 q^{31} +0.689636 q^{32} +1.31810 q^{33} -0.245738 q^{34} +3.27570 q^{36} -5.24519 q^{37} +13.2545 q^{38} -19.4785 q^{40} -4.44603 q^{41} +4.83284 q^{43} -2.64413 q^{44} +3.46073 q^{45} -1.12557 q^{46} -2.83381 q^{47} -6.99518 q^{48} -29.7536 q^{50} -0.197627 q^{51} -14.3434 q^{53} -10.3262 q^{54} -2.79348 q^{55} +10.6595 q^{57} -3.18799 q^{58} +0.417258 q^{59} -31.9158 q^{60} -1.53617 q^{61} -17.6053 q^{62} -8.82412 q^{64} -3.20920 q^{66} +5.88527 q^{67} +0.396443 q^{68} -0.905207 q^{69} +10.0629 q^{71} -3.91450 q^{72} -6.39720 q^{73} +12.7706 q^{74} -23.9284 q^{75} -21.3832 q^{76} +0.287850 q^{79} +14.8251 q^{80} -10.8064 q^{81} +10.8249 q^{82} +9.43279 q^{83} +0.418837 q^{85} -11.7666 q^{86} -2.56384 q^{87} +3.15977 q^{88} +10.0571 q^{89} -8.42593 q^{90} +1.81586 q^{92} -14.1585 q^{93} +6.89954 q^{94} -22.5911 q^{95} -1.35034 q^{96} -3.84295 q^{97} -0.561394 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9}+O(q^{10}) \) 24 * q + q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9	\( 24 q + q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} - 5 q^{10} + q^{11} - 5 q^{12} - 5 q^{15} + 17 q^{16} + 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} + 21 q^{27} + 4 q^{29} - 22 q^{30} + 40 q^{31} + 6 q^{32} + 24 q^{33} + 36 q^{34} - 15 q^{36} + 4 q^{37} + 29 q^{38} + 4 q^{40} + 49 q^{41} + 13 q^{43} - 10 q^{44} + 58 q^{45} + 10 q^{46} + 62 q^{47} - 89 q^{48} + 23 q^{50} - 21 q^{51} - 18 q^{53} + 12 q^{54} + 14 q^{55} + 13 q^{57} - 56 q^{58} + 79 q^{59} - 22 q^{60} - 13 q^{61} - 12 q^{62} + 18 q^{64} + 38 q^{66} + 2 q^{67} + 12 q^{68} + 28 q^{69} + 19 q^{71} - 81 q^{72} + 17 q^{73} + 17 q^{74} - 24 q^{75} + 58 q^{76} - 9 q^{79} + 63 q^{80} + 16 q^{81} + 22 q^{82} + 81 q^{83} + 34 q^{85} - 22 q^{86} - 70 q^{87} - 33 q^{88} + 72 q^{89} - q^{90} - 4 q^{92} - 19 q^{93} + 30 q^{94} + 13 q^{95} + 11 q^{96} + 45 q^{97} + 39 q^{99}+O(q^{100}) \) 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

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