LMFDB - Embedded newform 8281.2.a.ca.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [6,-2,1,4,1,-9,0,-3,-3,4,-4,5,0,0,2,-8,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6995813.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1 $ x^6 - x^5 - 6x^4 + 4x^3 + 7*x^2 - x - 1
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			1.4
Root			$-0.874884$ 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)

$f(q)$	$=$	\(q-0.268125 q^{2} +1.14301 q^{3} -1.92811 q^{4} +2.56175 q^{5} -0.306470 q^{6} +1.05323 q^{8} -1.69353 q^{9} -0.686871 q^{10} +3.94600 q^{11} -2.20385 q^{12} +2.92811 q^{15} +3.57382 q^{16} -0.785100 q^{17} +0.454078 q^{18} +7.49527 q^{19} -4.93934 q^{20} -1.05802 q^{22} -7.95518 q^{23} +1.20385 q^{24} +1.56259 q^{25} -5.36475 q^{27} +2.35173 q^{29} -0.785100 q^{30} +2.55437 q^{31} -3.06468 q^{32} +4.51032 q^{33} +0.210505 q^{34} +3.26531 q^{36} +6.75716 q^{37} -2.00967 q^{38} +2.69810 q^{40} +2.43747 q^{41} -2.24946 q^{43} -7.60832 q^{44} -4.33841 q^{45} +2.13298 q^{46} -1.31655 q^{47} +4.08491 q^{48} -0.418969 q^{50} -0.897376 q^{51} +9.27954 q^{53} +1.43842 q^{54} +10.1087 q^{55} +8.56716 q^{57} -0.630558 q^{58} +8.96671 q^{59} -5.64571 q^{60} -9.44547 q^{61} -0.684890 q^{62} -6.32592 q^{64} -1.20933 q^{66} -1.35256 q^{67} +1.51376 q^{68} -9.09284 q^{69} +12.3162 q^{71} -1.78367 q^{72} -0.768590 q^{73} -1.81176 q^{74} +1.78605 q^{75} -14.4517 q^{76} +6.19284 q^{79} +9.15525 q^{80} -1.05136 q^{81} -0.653548 q^{82} +1.07292 q^{83} -2.01123 q^{85} +0.603137 q^{86} +2.68805 q^{87} +4.15603 q^{88} -7.66299 q^{89} +1.16324 q^{90} +15.3384 q^{92} +2.91966 q^{93} +0.353001 q^{94} +19.2010 q^{95} -3.50296 q^{96} +2.37202 q^{97} -6.68267 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} - 9 q^{6} - 3 q^{8} - 3 q^{9} + 4 q^{10} - 4 q^{11} + 5 q^{12} + 2 q^{15} - 8 q^{16} + 5 q^{17} - 3 q^{18} - q^{19} - q^{20} + 5 q^{22} + q^{23} - 11 q^{24}+ \cdots - 10 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 + q^3 + 4 * q^4 + q^5 - 9 * q^6 - 3 * q^8 - 3 * q^9 + 4 * q^10 - 4 * q^11 + 5 * q^12 + 2 * q^15 - 8 * q^16 + 5 * q^17 - 3 * q^18 - q^19 - q^20 + 5 * q^22 + q^23 - 11 * q^24 - 7 * q^25 + 4 * q^27 - 3 * q^29 + 5 * q^30 + 16 * q^31 - 8 * q^32 + 16 * q^33 - 16 * q^34 + 21 * q^36 + 13 * q^37 - 17 * q^38 - 5 * q^40 - 8 * q^41 + 11 * q^43 - 21 * q^44 - 7 * q^45 - 16 * q^46 - q^47 + 21 * q^48 - 6 * q^50 + 20 * q^51 + 2 * q^53 - 18 * q^54 + 9 * q^55 + 21 * q^57 + 8 * q^58 + 13 * q^59 - 20 * q^60 - 5 * q^61 + 5 * q^62 - 15 * q^64 + 18 * q^66 + 11 * q^67 + 29 * q^68 + 23 * q^69 - 6 * q^71 - 25 * q^72 - 30 * q^73 + 3 * q^74 - 3 * q^75 - 9 * q^76 - 7 * q^79 - 7 * q^80 + 6 * q^81 + q^82 + 27 * q^83 + q^85 + 7 * q^86 + 16 * q^87 + 4 * q^89 + 8 * q^90 + 27 * q^92 + 7 * q^93 + 45 * q^94 + 6 * q^95 + 19 * q^96 - 35 * q^97 - 10 * 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 $

$2$	−0.268125	−0.189593	−0.0947966	−	0.995497i	$-0.530220\pi$
$2$	−0.268125	−0.189593	−0.0947966	+	0.995497i	$0.530220\pi$
$3$	1.14301	0.659917	0.329958	−	0.943996i	$-0.392965\pi$
$3$	1.14301	0.659917	0.329958	+	0.943996i	$0.392965\pi$
$4$	−1.92811	−0.964054
$5$	2.56175	1.14565	0.572826	−	0.819677i	$-0.305847\pi$
$5$	2.56175	1.14565	0.572826	+	0.819677i	$0.305847\pi$
$6$	−0.306470	−0.125116
$7$	0	0
$7$	0	0
$8$	1.05323	0.372371
$9$	−1.69353	−0.564510
$10$	−0.686871	−0.217208
$11$	3.94600	1.18976	0.594882	−	0.803813i	$-0.297199\pi$
$11$	3.94600	1.18976	0.594882	+	0.803813i	$0.297199\pi$
$12$	−2.20385	−0.636195
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.92811	0.756034
$16$	3.57382	0.893455
$17$	−0.785100	−0.190415	−0.0952073	−	0.995457i	$-0.530351\pi$
$17$	−0.785100	−0.190415	−0.0952073	+	0.995457i	$0.530351\pi$
$18$	0.454078	0.107027
$19$	7.49527	1.71953	0.859767	−	0.510687i	$-0.170609\pi$
$19$	7.49527	1.71953	0.859767	+	0.510687i	$0.170609\pi$
$20$	−4.93934	−1.10447
$21$	0	0
$22$	−1.05802	−0.225571
$23$	−7.95518	−1.65877	−0.829384	−	0.558678i	$-0.811309\pi$
$23$	−7.95518	−1.65877	−0.829384	+	0.558678i	$0.811309\pi$
$24$	1.20385	0.245734
$25$	1.56259	0.312518
$26$	0	0
$27$	−5.36475	−1.03245
$28$	0	0
$29$	2.35173	0.436705	0.218353	−	0.975870i	$-0.429932\pi$
$29$	2.35173	0.436705	0.218353	+	0.975870i	$0.429932\pi$
$30$	−0.785100	−0.143339
$31$	2.55437	0.458778	0.229389	−	0.973335i	$-0.426327\pi$
$31$	2.55437	0.458778	0.229389	+	0.973335i	$0.426327\pi$
$32$	−3.06468	−0.541764
$33$	4.51032	0.785145
$34$	0.210505	0.0361013
$35$	0	0
$36$	3.26531	0.544219
$37$	6.75716	1.11087	0.555435	−	0.831560i	$-0.312552\pi$
$37$	6.75716	1.11087	0.555435	+	0.831560i	$0.312552\pi$
$38$	−2.00967	−0.326012
$39$	0	0
$40$	2.69810	0.426608
$41$	2.43747	0.380669	0.190335	−	0.981719i	$-0.439043\pi$
$41$	2.43747	0.380669	0.190335	+	0.981719i	$0.439043\pi$
$42$	0	0
$43$	−2.24946	−0.343039	−0.171520	−	0.985181i	$-0.554868\pi$
$43$	−2.24946	−0.343039	−0.171520	+	0.985181i	$0.554868\pi$
$44$	−7.60832	−1.14700
$45$	−4.33841	−0.646732
$46$	2.13298	0.314491
$47$	−1.31655	−0.192039	−0.0960195	−	0.995379i	$-0.530611\pi$
$47$	−1.31655	−0.192039	−0.0960195	+	0.995379i	$0.530611\pi$
$48$	4.08491	0.589606
$49$	0	0
$50$	−0.418969	−0.0592512
$51$	−0.897376	−0.125658
$52$	0	0
$53$	9.27954	1.27464	0.637321	−	0.770598i	$-0.280042\pi$
$53$	9.27954	1.27464	0.637321	+	0.770598i	$0.280042\pi$
$54$	1.43842	0.195745
$55$	10.1087	1.36306
$56$	0	0
$57$	8.56716	1.13475
$58$	−0.630558	−0.0827963
$59$	8.96671	1.16737	0.583683	−	0.811982i	$-0.301611\pi$
$59$	8.96671	1.16737	0.583683	+	0.811982i	$0.301611\pi$
$60$	−5.64571	−0.728858
$61$	−9.44547	−1.20937	−0.604684	−	0.796465i	$-0.706701\pi$
$61$	−9.44547	−1.20937	−0.604684	+	0.796465i	$0.706701\pi$
$62$	−0.684890	−0.0869811
$63$	0	0
$64$	−6.32592	−0.790741
$65$	0	0
$66$	−1.20933	−0.148858
$67$	−1.35256	−0.165242	−0.0826209	−	0.996581i	$-0.526329\pi$
$67$	−1.35256	−0.165242	−0.0826209	+	0.996581i	$0.526329\pi$
$68$	1.51376	0.183570
$69$	−9.09284	−1.09465
$70$	0	0
$71$	12.3162	1.46166	0.730829	−	0.682560i	$-0.239133\pi$
$71$	12.3162	1.46166	0.730829	+	0.682560i	$0.239133\pi$
$72$	−1.78367	−0.210207
$73$	−0.768590	−0.0899567	−0.0449783	−	0.998988i	$-0.514322\pi$
$73$	−0.768590	−0.0899567	−0.0449783	+	0.998988i	$0.514322\pi$
$74$	−1.81176	−0.210613
$75$	1.78605	0.206236
$76$	−14.4517	−1.65772
$77$	0	0
$78$	0	0
$79$	6.19284	0.696749	0.348375	−	0.937355i	$-0.386734\pi$
$79$	6.19284	0.696749	0.348375	+	0.937355i	$0.386734\pi$
$80$	9.15525	1.02359
$81$	−1.05136	−0.116818
$82$	−0.653548	−0.0721723
$83$	1.07292	0.117768	0.0588841	−	0.998265i	$-0.481246\pi$
$83$	1.07292	0.117768	0.0588841	+	0.998265i	$0.481246\pi$
$84$	0	0
$85$	−2.01123	−0.218149
$86$	0.603137	0.0650379
$87$	2.68805	0.288189
$88$	4.15603	0.443034
$89$	−7.66299	−0.812275	−0.406138	−	0.913812i	$-0.633125\pi$
$89$	−7.66299	−0.812275	−0.406138	+	0.913812i	$0.633125\pi$
$90$	1.16324	0.122616
$91$	0	0
$92$	15.3384	1.59914
$93$	2.91966	0.302755
$94$	0.353001	0.0364093
$95$	19.2010	1.96999
$96$	−3.50296	−0.357519
$97$	2.37202	0.240842	0.120421	−	0.992723i	$-0.461576\pi$
$97$	2.37202	0.240842	0.120421	+	0.992723i	$0.461576\pi$
$98$	0	0
$99$	−6.68267	−0.671634
$100$	−3.01284	−0.301284
$101$	0.797330	0.0793373	0.0396686	−	0.999213i	$-0.487370\pi$
$101$	0.797330	0.0793373	0.0396686	+	0.999213i	$0.487370\pi$
$102$	0.240609	0.0238239
$103$	−2.16618	−0.213440	−0.106720	−	0.994289i	$-0.534035\pi$
$103$	−2.16618	−0.213440	−0.106720	+	0.994289i	$0.534035\pi$
$104$	0	0
$105$	0	0
$106$	−2.48808	−0.241664
$107$	−11.5262	−1.11428	−0.557141	−	0.830418i	$-0.688102\pi$
$107$	−11.5262	−1.11428	−0.557141	+	0.830418i	$0.688102\pi$
$108$	10.3438	0.995334
$109$	8.07823	0.773754	0.386877	−	0.922131i	$-0.373554\pi$
$109$	8.07823	0.773754	0.386877	+	0.922131i	$0.373554\pi$
$110$	−2.71039	−0.258426
$111$	7.72349	0.733081
$112$	0	0
$113$	8.04135	0.756467	0.378233	−	0.925710i	$-0.376532\pi$
$113$	8.04135	0.756467	0.378233	+	0.925710i	$0.376532\pi$
$114$	−2.29707	−0.215141
$115$	−20.3792	−1.90037
$116$	−4.53439	−0.421007
$117$	0	0
$118$	−2.40420	−0.221325
$119$	0	0
$120$	3.08396	0.281526
$121$	4.57093	0.415539
$122$	2.53257	0.229288
$123$	2.78605	0.251210
$124$	−4.92510	−0.442287
$125$	−8.80581	−0.787615
$126$	0	0
$127$	1.78805	0.158663	0.0793317	−	0.996848i	$-0.474721\pi$
$127$	1.78805	0.158663	0.0793317	+	0.996848i	$0.474721\pi$
$128$	7.82550	0.691683
$129$	−2.57115	−0.226377
$130$	0	0
$131$	6.39091	0.558376	0.279188	−	0.960236i	$-0.409935\pi$
$131$	6.39091	0.558376	0.279188	+	0.960236i	$0.409935\pi$
$132$	−8.69638	−0.756923
$133$	0	0
$134$	0.362656	0.0313287
$135$	−13.7432	−1.18282
$136$	−0.826887	−0.0709050
$137$	10.0365	0.857480	0.428740	−	0.903428i	$-0.358958\pi$
$137$	10.0365	0.857480	0.428740	+	0.903428i	$0.358958\pi$
$138$	2.43802	0.207538
$139$	5.54557	0.470369	0.235184	−	0.971951i	$-0.424431\pi$
$139$	5.54557	0.470369	0.235184	+	0.971951i	$0.424431\pi$
$140$	0	0
$141$	−1.50483	−0.126730
$142$	−3.30227	−0.277121
$143$	0	0
$144$	−6.05238	−0.504365
$145$	6.02455	0.500312
$146$	0.206078	0.0170552
$147$	0	0
$148$	−13.0285	−1.07094
$149$	18.4651	1.51272	0.756359	−	0.654157i	$-0.226976\pi$
$149$	18.4651	1.51272	0.756359	+	0.654157i	$0.226976\pi$
$150$	−0.478886	−0.0391009
$151$	1.60736	0.130805	0.0654024	−	0.997859i	$-0.479167\pi$
$151$	1.60736	0.130805	0.0654024	+	0.997859i	$0.479167\pi$
$152$	7.89421	0.640305
$153$	1.32959	0.107491
$154$	0	0
$155$	6.54366	0.525600
$156$	0	0
$157$	1.64593	0.131360	0.0656799	−	0.997841i	$-0.479078\pi$
$157$	1.64593	0.131360	0.0656799	+	0.997841i	$0.479078\pi$
$158$	−1.66046	−0.132099
$159$	10.6066	0.841158
$160$	−7.85096	−0.620673
$161$	0	0
$162$	0.281897	0.0221479
$163$	−6.54819	−0.512894	−0.256447	−	0.966558i	$-0.582552\pi$
$163$	−6.54819	−0.512894	−0.256447	+	0.966558i	$0.582552\pi$
$164$	−4.69971	−0.366986
$165$	11.5543	0.899503
$166$	−0.287677	−0.0223281
$167$	−9.54880	−0.738908	−0.369454	−	0.929249i	$-0.620455\pi$
$167$	−9.54880	−0.738908	−0.369454	+	0.929249i	$0.620455\pi$
$168$	0	0
$169$	0	0
$170$	0.539262	0.0413595
$171$	−12.6935	−0.970694
$172$	4.33720	0.330709
$173$	−11.1316	−0.846322	−0.423161	−	0.906054i	$-0.639080\pi$
$173$	−11.1316	−0.846322	−0.423161	+	0.906054i	$0.639080\pi$
$174$	−0.720733	−0.0546386
$175$	0	0
$176$	14.1023	1.06300
$177$	10.2490	0.770364
$178$	2.05464	0.154002
$179$	−12.6435	−0.945017	−0.472508	−	0.881326i	$-0.656651\pi$
$179$	−12.6435	−0.945017	−0.472508	+	0.881326i	$0.656651\pi$
$180$	8.36493	0.623485
$181$	−14.9158	−1.10868	−0.554341	−	0.832289i	$-0.687030\pi$
$181$	−14.9158	−1.10868	−0.554341	+	0.832289i	$0.687030\pi$
$182$	0	0
$183$	−10.7963	−0.798082
$184$	−8.37859	−0.617678
$185$	17.3102	1.27267
$186$	−0.782836	−0.0574003
$187$	−3.09801	−0.226549
$188$	2.53846	0.185136
$189$	0	0
$190$	−5.14829	−0.373496
$191$	−14.1306	−1.02245	−0.511226	−	0.859447i	$-0.670808\pi$
$191$	−14.1306	−1.02245	−0.511226	+	0.859447i	$0.670808\pi$
$192$	−7.23059	−0.521823
$193$	3.89454	0.280335	0.140167	−	0.990128i	$-0.455236\pi$
$193$	3.89454	0.280335	0.140167	+	0.990128i	$0.455236\pi$
$194$	−0.635998	−0.0456620
$195$	0	0
$196$	0	0
$197$	−11.7089	−0.834224	−0.417112	−	0.908855i	$-0.636958\pi$
$197$	−11.7089	−0.834224	−0.417112	+	0.908855i	$0.636958\pi$
$198$	1.79179	0.127337
$199$	−3.49684	−0.247884	−0.123942	−	0.992289i	$-0.539554\pi$
$199$	−3.49684	−0.247884	−0.123942	+	0.992289i	$0.539554\pi$
$200$	1.64576	0.116373
$201$	−1.54599	−0.109046
$202$	−0.213784	−0.0150418
$203$	0	0
$204$	1.73024	0.121141
$205$	6.24421	0.436114
$206$	0.580807	0.0404667
$207$	13.4723	0.936392
$208$	0	0
$209$	29.5764	2.04584
$210$	0	0
$211$	19.0052	1.30837	0.654184	−	0.756335i	$-0.273012\pi$
$211$	19.0052	1.30837	0.654184	+	0.756335i	$0.273012\pi$
$212$	−17.8920	−1.22882
$213$	14.0775	0.964573
$214$	3.09047	0.211260
$215$	−5.76256	−0.393004
$216$	−5.65029	−0.384453
$217$	0	0
$218$	−2.16598	−0.146698
$219$	−0.878506	−0.0593639
$220$	−19.4907	−1.31406
$221$	0	0
$222$	−2.07086	−0.138987
$223$	11.9662	0.801317	0.400658	−	0.916228i	$-0.368781\pi$
$223$	11.9662	0.801317	0.400658	+	0.916228i	$0.368781\pi$
$224$	0	0
$225$	−2.64629	−0.176419
$226$	−2.15609	−0.143421
$227$	−15.3842	−1.02108	−0.510542	−	0.859853i	$-0.670555\pi$
$227$	−15.3842	−1.02108	−0.510542	+	0.859853i	$0.670555\pi$
$228$	−16.5184	−1.09396
$229$	−8.66168	−0.572380	−0.286190	−	0.958173i	$-0.592389\pi$
$229$	−8.66168	−0.572380	−0.286190	+	0.958173i	$0.592389\pi$
$230$	5.46418	0.360297
$231$	0	0
$232$	2.47690	0.162616
$233$	20.2507	1.32667	0.663333	−	0.748325i	$-0.269141\pi$
$233$	20.2507	1.32667	0.663333	+	0.748325i	$0.269141\pi$
$234$	0	0
$235$	−3.37269	−0.220010
$236$	−17.2888	−1.12540
$237$	7.07847	0.459796
$238$	0	0
$239$	16.5526	1.07070	0.535350	−	0.844630i	$-0.320180\pi$
$239$	16.5526	1.07070	0.535350	+	0.844630i	$0.320180\pi$
$240$	10.4645	0.675483
$241$	16.4008	1.05647	0.528233	−	0.849100i	$-0.322855\pi$
$241$	16.4008	1.05647	0.528233	+	0.849100i	$0.322855\pi$
$242$	−1.22558	−0.0787834
$243$	14.8925	0.955356
$244$	18.2119	1.16590
$245$	0	0
$246$	−0.747011	−0.0476277
$247$	0	0
$248$	2.69032	0.170836
$249$	1.22636	0.0777172
$250$	2.36106	0.149326
$251$	20.4307	1.28958	0.644788	−	0.764362i	$-0.276946\pi$
$251$	20.4307	1.28958	0.644788	+	0.764362i	$0.276946\pi$
$252$	0	0
$253$	−31.3911	−1.97354
$254$	−0.479420	−0.0300815
$255$	−2.29886	−0.143960
$256$	10.5536	0.659602
$257$	−13.7779	−0.859442	−0.429721	−	0.902962i	$-0.641388\pi$
$257$	−13.7779	−0.859442	−0.429721	+	0.902962i	$0.641388\pi$
$258$	0.689391	0.0429196
$259$	0	0
$260$	0	0
$261$	−3.98272	−0.246524
$262$	−1.71356	−0.105864
$263$	25.9173	1.59813	0.799065	−	0.601244i	$-0.205328\pi$
$263$	25.9173	1.59813	0.799065	+	0.601244i	$0.205328\pi$
$264$	4.75038	0.292366
$265$	23.7719	1.46030
$266$	0	0
$267$	−8.75887	−0.536034
$268$	2.60789	0.159302
$269$	30.0666	1.83319	0.916596	−	0.399814i	$-0.130925\pi$
$269$	30.0666	1.83319	0.916596	+	0.399814i	$0.130925\pi$
$270$	3.68489	0.224255
$271$	−14.4505	−0.877808	−0.438904	−	0.898534i	$-0.644633\pi$
$271$	−14.4505	−0.877808	−0.438904	+	0.898534i	$0.644633\pi$
$272$	−2.80581	−0.170127
$273$	0	0
$274$	−2.69105	−0.162572
$275$	6.16598	0.371822
$276$	17.5320	1.05530
$277$	−15.3255	−0.920819	−0.460409	−	0.887707i	$-0.652297\pi$
$277$	−15.3255	−0.920819	−0.460409	+	0.887707i	$0.652297\pi$
$278$	−1.48691	−0.0891787
$279$	−4.32590	−0.258985
$280$	0	0
$281$	5.29279	0.315741	0.157871	−	0.987460i	$-0.449537\pi$
$281$	5.29279	0.315741	0.157871	+	0.987460i	$0.449537\pi$
$282$	0.403483	0.0240271
$283$	30.7845	1.82995	0.914975	−	0.403511i	$-0.132210\pi$
$283$	30.7845	1.82995	0.914975	+	0.403511i	$0.132210\pi$
$284$	−23.7469	−1.40912
$285$	21.9470	1.30003
$286$	0	0
$287$	0	0
$288$	5.19013	0.305831
$289$	−16.3836	−0.963742
$290$	−1.61533	−0.0948557
$291$	2.71124	0.158935
$292$	1.48193	0.0867231
$293$	17.5173	1.02337	0.511685	−	0.859173i	$-0.329021\pi$
$293$	17.5173	1.02337	0.511685	+	0.859173i	$0.329021\pi$
$294$	0	0
$295$	22.9705	1.33739
$296$	7.11681	0.413656
$297$	−21.1693	−1.22837
$298$	−4.95095	−0.286801
$299$	0	0
$300$	−3.44370	−0.198822
$301$	0	0
$302$	−0.430973	−0.0247997
$303$	0.911355	0.0523560
$304$	26.7868	1.53633
$305$	−24.1970	−1.38551
$306$	−0.356497	−0.0203796
$307$	8.63573	0.492867	0.246434	−	0.969160i	$-0.420741\pi$
$307$	8.63573	0.492867	0.246434	+	0.969160i	$0.420741\pi$
$308$	0	0
$309$	−2.47596	−0.140852
$310$	−1.75452	−0.0996501
$311$	16.4226	0.931240	0.465620	−	0.884985i	$-0.345831\pi$
$311$	16.4226	0.931240	0.465620	+	0.884985i	$0.345831\pi$
$312$	0	0
$313$	−10.0462	−0.567843	−0.283921	−	0.958848i	$-0.591636\pi$
$313$	−10.0462	−0.567843	−0.283921	+	0.958848i	$0.591636\pi$
$314$	−0.441316	−0.0249049
$315$	0	0
$316$	−11.9405	−0.671704
$317$	10.1450	0.569799	0.284899	−	0.958557i	$-0.408040\pi$
$317$	10.1450	0.569799	0.284899	+	0.958557i	$0.408040\pi$
$318$	−2.84390	−0.159478
$319$	9.27992	0.519576
$320$	−16.2055	−0.905913
$321$	−13.1746	−0.735333
$322$	0	0
$323$	−5.88454	−0.327424
$324$	2.02714	0.112619
$325$	0	0
$326$	1.75573	0.0972411
$327$	9.23349	0.510613
$328$	2.56721	0.141750
$329$	0	0
$330$	−3.09801	−0.170540
$331$	2.31916	0.127473	0.0637363	−	0.997967i	$-0.479698\pi$
$331$	2.31916	0.127473	0.0637363	+	0.997967i	$0.479698\pi$
$332$	−2.06871	−0.113535
$333$	−11.4434	−0.627097
$334$	2.56027	0.140092
$335$	−3.46493	−0.189309
$336$	0	0
$337$	−15.9998	−0.871565	−0.435783	−	0.900052i	$-0.643528\pi$
$337$	−15.9998	−0.871565	−0.435783	+	0.900052i	$0.643528\pi$
$338$	0	0
$339$	9.19133	0.499205
$340$	3.87788	0.210307
$341$	10.0795	0.545837
$342$	3.40344	0.184037
$343$	0	0
$344$	−2.36919	−0.127738
$345$	−23.2936	−1.25409
$346$	2.98467	0.160457
$347$	−22.8208	−1.22509	−0.612543	−	0.790437i	$-0.709853\pi$
$347$	−22.8208	−1.22509	−0.612543	+	0.790437i	$0.709853\pi$
$348$	−5.18285	−0.277830
$349$	22.7022	1.21522	0.607612	−	0.794234i	$-0.292128\pi$
$349$	22.7022	1.21522	0.607612	+	0.794234i	$0.292128\pi$
$350$	0	0
$351$	0	0
$352$	−12.0932	−0.644572
$353$	27.2644	1.45114	0.725568	−	0.688150i	$-0.241577\pi$
$353$	27.2644	1.45114	0.725568	+	0.688150i	$0.241577\pi$
$354$	−2.74802	−0.146056
$355$	31.5510	1.67455
$356$	14.7751	0.783077
$357$	0	0
$358$	3.39003	0.179169
$359$	−14.4262	−0.761385	−0.380692	−	0.924702i	$-0.624314\pi$
$359$	−14.4262	−0.761385	−0.380692	+	0.924702i	$0.624314\pi$
$360$	−4.56932	−0.240824
$361$	37.1791	1.95679
$362$	3.99930	0.210199
$363$	5.22461	0.274221
$364$	0	0
$365$	−1.96894	−0.103059
$366$	2.89475	0.151311
$367$	11.3917	0.594643	0.297322	−	0.954777i	$-0.403907\pi$
$367$	11.3917	0.594643	0.297322	+	0.954777i	$0.403907\pi$
$368$	−28.4304	−1.48204
$369$	−4.12793	−0.214892
$370$	−4.64130	−0.241289
$371$	0	0
$372$	−5.62943	−0.291872
$373$	−30.9629	−1.60320	−0.801599	−	0.597862i	$-0.796017\pi$
$373$	−30.9629	−1.60320	−0.801599	+	0.597862i	$0.796017\pi$
$374$	0.830653	0.0429521
$375$	−10.0651	−0.519760
$376$	−1.38663	−0.0715098
$377$	0	0
$378$	0	0
$379$	10.5866	0.543797	0.271898	−	0.962326i	$-0.412349\pi$
$379$	10.5866	0.543797	0.271898	+	0.962326i	$0.412349\pi$
$380$	−37.0217	−1.89917
$381$	2.04375	0.104705
$382$	3.78876	0.193850
$383$	−30.7517	−1.57134	−0.785668	−	0.618648i	$-0.787681\pi$
$383$	−30.7517	−1.57134	−0.785668	+	0.618648i	$0.787681\pi$
$384$	8.94462	0.456453
$385$	0	0
$386$	−1.04422	−0.0531496
$387$	3.80953	0.193649
$388$	−4.57351	−0.232185
$389$	−16.3796	−0.830477	−0.415239	−	0.909713i	$-0.636302\pi$
$389$	−16.3796	−0.830477	−0.415239	+	0.909713i	$0.636302\pi$
$390$	0	0
$391$	6.24561	0.315854
$392$	0	0
$393$	7.30486	0.368482
$394$	3.13945	0.158163
$395$	15.8645	0.798232
$396$	12.8849	0.647492
$397$	15.8827	0.797127	0.398564	−	0.917141i	$-0.369509\pi$
$397$	15.8827	0.797127	0.398564	+	0.917141i	$0.369509\pi$
$398$	0.937591	0.0469972
$399$	0	0
$400$	5.58441	0.279221
$401$	6.63573	0.331373	0.165686	−	0.986178i	$-0.447016\pi$
$401$	6.63573	0.331373	0.165686	+	0.986178i	$0.447016\pi$
$402$	0.414519	0.0206743
$403$	0	0
$404$	−1.53734	−0.0764855
$405$	−2.69334	−0.133833
$406$	0	0
$407$	26.6637	1.32167
$408$	−0.945139	−0.0467914
$409$	−5.87235	−0.290369	−0.145184	−	0.989405i	$-0.546378\pi$
$409$	−5.87235	−0.290369	−0.145184	+	0.989405i	$0.546378\pi$
$410$	−1.67423	−0.0826843
$411$	11.4719	0.565865
$412$	4.17663	0.205768
$413$	0	0
$414$	−3.61227	−0.177533
$415$	2.74856	0.134921
$416$	0	0
$417$	6.33863	0.310404
$418$	−7.93017	−0.387877
$419$	30.1423	1.47255	0.736274	−	0.676683i	$-0.236583\pi$
$419$	30.1423	1.47255	0.736274	+	0.676683i	$0.236583\pi$
$420$	0	0
$421$	40.0580	1.95231	0.976153	−	0.217083i	$-0.0696543\pi$
$421$	40.0580	1.95231	0.976153	+	0.217083i	$0.0696543\pi$
$422$	−5.09576	−0.248058
$423$	2.22962	0.108408
$424$	9.77344	0.474640
$425$	−1.22679	−0.0595079
$426$	−3.77453	−0.182876
$427$	0	0
$428$	22.2238	1.07423
$429$	0	0
$430$	1.54509	0.0745108
$431$	−3.91587	−0.188621	−0.0943104	−	0.995543i	$-0.530065\pi$
$431$	−3.91587	−0.188621	−0.0943104	+	0.995543i	$0.530065\pi$
$432$	−19.1726	−0.922445
$433$	40.7925	1.96036	0.980182	−	0.198100i	$-0.0634770\pi$
$433$	40.7925	1.96036	0.980182	+	0.198100i	$0.0634770\pi$
$434$	0	0
$435$	6.88612	0.330164
$436$	−15.5757	−0.745941
$437$	−59.6262	−2.85231
$438$	0.235549	0.0112550
$439$	25.5623	1.22002	0.610010	−	0.792394i	$-0.291165\pi$
$439$	25.5623	1.22002	0.610010	+	0.792394i	$0.291165\pi$
$440$	10.6467	0.507563
$441$	0	0
$442$	0	0
$443$	−27.4564	−1.30449	−0.652247	−	0.758007i	$-0.726173\pi$
$443$	−27.4564	−1.30449	−0.652247	+	0.758007i	$0.726173\pi$
$444$	−14.8917	−0.706730
$445$	−19.6307	−0.930584
$446$	−3.20844	−0.151924
$447$	21.1057	0.998267
$448$	0	0
$449$	−14.8036	−0.698626	−0.349313	−	0.937006i	$-0.613585\pi$
$449$	−14.8036	−0.698626	−0.349313	+	0.937006i	$0.613585\pi$
$450$	0.709537	0.0334479
$451$	9.61827	0.452907
$452$	−15.5046	−0.729275
$453$	1.83722	0.0863203
$454$	4.12489	0.193590
$455$	0	0
$456$	9.02315	0.422548
$457$	−0.651951	−0.0304970	−0.0152485	−	0.999884i	$-0.504854\pi$
$457$	−0.651951	−0.0304970	−0.0152485	+	0.999884i	$0.504854\pi$
$458$	2.32241	0.108519
$459$	4.21186	0.196593
$460$	39.2933	1.83206
$461$	−12.4955	−0.581972	−0.290986	−	0.956727i	$-0.593983\pi$
$461$	−12.4955	−0.581972	−0.290986	+	0.956727i	$0.593983\pi$
$462$	0	0
$463$	−0.309503	−0.0143838	−0.00719190	−	0.999974i	$-0.502289\pi$
$463$	−0.309503	−0.0143838	−0.00719190	+	0.999974i	$0.502289\pi$
$464$	8.40466	0.390176
$465$	7.47946	0.346852
$466$	−5.42972	−0.251527
$467$	−24.4773	−1.13268	−0.566338	−	0.824173i	$-0.691640\pi$
$467$	−24.4773	−1.13268	−0.566338	+	0.824173i	$0.691640\pi$
$468$	0	0
$469$	0	0
$470$	0.904302	0.0417123
$471$	1.88132	0.0866865
$472$	9.44396	0.434694
$473$	−8.87637	−0.408136
$474$	−1.89792	−0.0871742
$475$	11.7120	0.537384
$476$	0	0
$477$	−15.7152	−0.719549
$478$	−4.43817	−0.202997
$479$	−8.13850	−0.371858	−0.185929	−	0.982563i	$-0.559529\pi$
$479$	−8.13850	−0.371858	−0.185929	+	0.982563i	$0.559529\pi$
$480$	−8.97372	−0.409593
$481$	0	0
$482$	−4.39746	−0.200299
$483$	0	0
$484$	−8.81325	−0.400602
$485$	6.07653	0.275921
$486$	−3.99306	−0.181129
$487$	4.60960	0.208881	0.104440	−	0.994531i	$-0.466695\pi$
$487$	4.60960	0.208881	0.104440	+	0.994531i	$0.466695\pi$
$488$	−9.94821	−0.450334
$489$	−7.48464	−0.338467
$490$	0	0
$491$	13.0189	0.587537	0.293768	−	0.955877i	$-0.405091\pi$
$491$	13.0189	0.587537	0.293768	+	0.955877i	$0.405091\pi$
$492$	−5.37181	−0.242180
$493$	−1.84634	−0.0831551
$494$	0	0
$495$	−17.1194	−0.769459
$496$	9.12885	0.409897
$497$	0	0
$498$	−0.328817	−0.0147347
$499$	−32.3207	−1.44687	−0.723436	−	0.690391i	$-0.757438\pi$
$499$	−32.3207	−1.44687	−0.723436	+	0.690391i	$0.757438\pi$
$500$	16.9786	0.759304
$501$	−10.9144	−0.487618
$502$	−5.47799	−0.244495
$503$	31.8253	1.41902	0.709509	−	0.704696i	$-0.248917\pi$
$503$	31.8253	1.41902	0.709509	+	0.704696i	$0.248917\pi$
$504$	0	0
$505$	2.04256	0.0908929
$506$	8.41676	0.374170
$507$	0	0
$508$	−3.44755	−0.152960
$509$	−2.25575	−0.0999845	−0.0499922	−	0.998750i	$-0.515920\pi$
$509$	−2.25575	−0.0999845	−0.0499922	+	0.998750i	$0.515920\pi$
$510$	0.616382	0.0272938
$511$	0	0
$512$	−18.4807	−0.816739
$513$	−40.2102	−1.77533
$514$	3.69420	0.162944
$515$	−5.54922	−0.244528
$516$	4.95746	0.218240
$517$	−5.19512	−0.228481
$518$	0	0
$519$	−12.7236	−0.558502
$520$	0	0
$521$	−10.7712	−0.471897	−0.235948	−	0.971766i	$-0.575820\pi$
$521$	−10.7712	−0.471897	−0.235948	+	0.971766i	$0.575820\pi$
$522$	1.06787	0.0467393
$523$	−7.40793	−0.323926	−0.161963	−	0.986797i	$-0.551783\pi$
$523$	−7.40793	−0.323926	−0.161963	+	0.986797i	$0.551783\pi$
$524$	−12.3224	−0.538305
$525$	0	0
$526$	−6.94909	−0.302995
$527$	−2.00543	−0.0873580
$528$	16.1191	0.701492
$529$	40.2848	1.75151
$530$	−6.37385	−0.276862
$531$	−15.1854	−0.658990
$532$	0	0
$533$	0	0
$534$	2.34847	0.101628
$535$	−29.5274	−1.27658
$536$	−1.42455	−0.0615313
$537$	−14.4516	−0.623632
$538$	−8.06161	−0.347561
$539$	0	0
$540$	26.4983	1.14031
$541$	32.5481	1.39935	0.699676	−	0.714460i	$-0.253327\pi$
$541$	32.5481	1.39935	0.699676	+	0.714460i	$0.253327\pi$
$542$	3.87455	0.166426
$543$	−17.0489	−0.731638
$544$	2.40608	0.103160
$545$	20.6944	0.886453
$546$	0	0
$547$	−13.4997	−0.577206	−0.288603	−	0.957449i	$-0.593191\pi$
$547$	−13.4997	−0.577206	−0.288603	+	0.957449i	$0.593191\pi$
$548$	−19.3515	−0.826657
$549$	15.9962	0.682701
$550$	−1.65325	−0.0704950
$551$	17.6268	0.750929
$552$	−9.57680	−0.407616
$553$	0	0
$554$	4.10915	0.174581
$555$	19.7857	0.839856
$556$	−10.6925	−0.453461
$557$	−29.7703	−1.26141	−0.630703	−	0.776024i	$-0.717233\pi$
$557$	−29.7703	−1.26141	−0.630703	+	0.776024i	$0.717233\pi$
$558$	1.15988	0.0491017
$559$	0	0
$560$	0	0
$561$	−3.54105	−0.149503
$562$	−1.41913	−0.0598624
$563$	−14.1326	−0.595617	−0.297809	−	0.954626i	$-0.596256\pi$
$563$	−14.1326	−0.595617	−0.297809	+	0.954626i	$0.596256\pi$
$564$	2.90148	0.122174
$565$	20.6000	0.866647
$566$	−8.25410	−0.346946
$567$	0	0
$568$	12.9717	0.544280
$569$	−24.2540	−1.01678	−0.508391	−	0.861127i	$-0.669759\pi$
$569$	−24.2540	−1.01678	−0.508391	+	0.861127i	$0.669759\pi$
$570$	−5.88454	−0.246476
$571$	1.20832	0.0505665	0.0252832	−	0.999680i	$-0.491951\pi$
$571$	1.20832	0.0505665	0.0252832	+	0.999680i	$0.491951\pi$
$572$	0	0
$573$	−16.1514	−0.674732
$574$	0	0
$575$	−12.4307	−0.518394
$576$	10.7131	0.446381
$577$	14.6104	0.608237	0.304119	−	0.952634i	$-0.401638\pi$
$577$	14.6104	0.608237	0.304119	+	0.952634i	$0.401638\pi$
$578$	4.39286	0.182719
$579$	4.45149	0.184998
$580$	−11.6160	−0.482328
$581$	0	0
$582$	−0.726951	−0.0301331
$583$	36.6171	1.51652
$584$	−0.809498	−0.0334973
$585$	0	0
$586$	−4.69683	−0.194024
$587$	−21.5095	−0.887793	−0.443897	−	0.896078i	$-0.646404\pi$
$587$	−21.5095	−0.887793	−0.443897	+	0.896078i	$0.646404\pi$
$588$	0	0
$589$	19.1457	0.788884
$590$	−6.15897	−0.253561
$591$	−13.3834	−0.550518
$592$	24.1489	0.992512
$593$	2.64857	0.108764	0.0543820	−	0.998520i	$-0.482681\pi$
$593$	2.64857	0.108764	0.0543820	+	0.998520i	$0.482681\pi$
$594$	5.67602	0.232890
$595$	0	0
$596$	−35.6027	−1.45834
$597$	−3.99692	−0.163583
$598$	0	0
$599$	−40.2501	−1.64457	−0.822287	−	0.569074i	$-0.807302\pi$
$599$	−40.2501	−1.64457	−0.822287	+	0.569074i	$0.807302\pi$
$600$	1.88112	0.0767962
$601$	−38.3449	−1.56412	−0.782061	−	0.623202i	$-0.785831\pi$
$601$	−38.3449	−1.56412	−0.782061	+	0.623202i	$0.785831\pi$
$602$	0	0
$603$	2.29060	0.0932806
$604$	−3.09916	−0.126103
$605$	11.7096	0.476063
$606$	−0.244357	−0.00992634
$607$	−42.5547	−1.72724	−0.863620	−	0.504143i	$-0.831808\pi$
$607$	−42.5547	−1.72724	−0.863620	+	0.504143i	$0.831808\pi$
$608$	−22.9706	−0.931582
$609$	0	0
$610$	6.48782	0.262684
$611$	0	0
$612$	−2.56360	−0.103627
$613$	15.2652	0.616556	0.308278	−	0.951296i	$-0.400247\pi$
$613$	15.2652	0.616556	0.308278	+	0.951296i	$0.400247\pi$
$614$	−2.31546	−0.0934442
$615$	7.13718	0.287799
$616$	0	0
$617$	13.9812	0.562863	0.281431	−	0.959581i	$-0.409191\pi$
$617$	13.9812	0.562863	0.281431	+	0.959581i	$0.409191\pi$
$618$	0.663868	0.0267047
$619$	8.51585	0.342281	0.171140	−	0.985247i	$-0.445255\pi$
$619$	8.51585	0.342281	0.171140	+	0.985247i	$0.445255\pi$
$620$	−12.6169	−0.506707
$621$	42.6775	1.71259
$622$	−4.40331	−0.176557
$623$	0	0
$624$	0	0
$625$	−30.3713	−1.21485
$626$	2.69363	0.107659
$627$	33.8060	1.35008
$628$	−3.17354	−0.126638
$629$	−5.30504	−0.211526
$630$	0	0
$631$	36.8292	1.46615	0.733074	−	0.680149i	$-0.238085\pi$
$631$	36.8292	1.46615	0.733074	+	0.680149i	$0.238085\pi$
$632$	6.52246	0.259449
$633$	21.7231	0.863414
$634$	−2.72013	−0.108030
$635$	4.58053	0.181773
$636$	−20.4507	−0.810922
$637$	0	0
$638$	−2.48818	−0.0985081
$639$	−20.8578	−0.825121
$640$	20.0470	0.792428
$641$	25.8747	1.02199	0.510996	−	0.859583i	$-0.329277\pi$
$641$	25.8747	1.02199	0.510996	+	0.859583i	$0.329277\pi$
$642$	3.53244	0.139414
$643$	−40.5252	−1.59816	−0.799078	−	0.601227i	$-0.794679\pi$
$643$	−40.5252	−1.59816	−0.799078	+	0.601227i	$0.794679\pi$
$644$	0	0
$645$	−6.58666	−0.259350
$646$	1.57779	0.0620774
$647$	−1.78400	−0.0701364	−0.0350682	−	0.999385i	$-0.511165\pi$
$647$	−1.78400	−0.0701364	−0.0350682	+	0.999385i	$0.511165\pi$
$648$	−1.10732	−0.0434997
$649$	35.3826	1.38889
$650$	0	0
$651$	0	0
$652$	12.6256	0.494457
$653$	12.4042	0.485414	0.242707	−	0.970100i	$-0.421965\pi$
$653$	12.4042	0.485414	0.242707	+	0.970100i	$0.421965\pi$
$654$	−2.47573	−0.0968088
$655$	16.3719	0.639704
$656$	8.71109	0.340111
$657$	1.30163	0.0507815
$658$	0	0
$659$	−1.12867	−0.0439668	−0.0219834	−	0.999758i	$-0.506998\pi$
$659$	−1.12867	−0.0439668	−0.0219834	+	0.999758i	$0.506998\pi$
$660$	−22.2780	−0.867170
$661$	28.9254	1.12507	0.562534	−	0.826774i	$-0.309826\pi$
$661$	28.9254	1.12507	0.562534	+	0.826774i	$0.309826\pi$
$662$	−0.621826	−0.0241680
$663$	0	0
$664$	1.13003	0.0438535
$665$	0	0
$666$	3.06828	0.118893
$667$	−18.7084	−0.724393
$668$	18.4111	0.712348
$669$	13.6775	0.528802
$670$	0.929036	0.0358918
$671$	−37.2718	−1.43886
$672$	0	0
$673$	−7.09960	−0.273669	−0.136835	−	0.990594i	$-0.543693\pi$
$673$	−7.09960	−0.273669	−0.136835	+	0.990594i	$0.543693\pi$
$674$	4.28995	0.165243
$675$	−8.38289	−0.322658
$676$	0	0
$677$	50.4020	1.93710	0.968552	−	0.248811i	$-0.0800397\pi$
$677$	50.4020	1.93710	0.968552	+	0.248811i	$0.0800397\pi$
$678$	−2.46443	−0.0946458
$679$	0	0
$680$	−2.11828	−0.0812324
$681$	−17.5843	−0.673830
$682$	−2.70258	−0.103487
$683$	27.5282	1.05334	0.526669	−	0.850070i	$-0.323441\pi$
$683$	27.5282	1.05334	0.526669	+	0.850070i	$0.323441\pi$
$684$	24.4744	0.935802
$685$	25.7112	0.982373
$686$	0	0
$687$	−9.90037	−0.377723
$688$	−8.03917	−0.306490
$689$	0	0
$690$	6.24561	0.237766
$691$	24.3338	0.925702	0.462851	−	0.886436i	$-0.346826\pi$
$691$	24.3338	0.925702	0.462851	+	0.886436i	$0.346826\pi$
$692$	21.4630	0.815901
$693$	0	0
$694$	6.11884	0.232268
$695$	14.2064	0.538879
$696$	2.83112	0.107313
$697$	−1.91366	−0.0724850
$698$	−6.08704	−0.230398
$699$	23.1467	0.875488
$700$	0	0
$701$	−20.5588	−0.776495	−0.388248	−	0.921555i	$-0.626919\pi$
$701$	−20.5588	−0.776495	−0.388248	+	0.921555i	$0.626919\pi$
$702$	0	0
$703$	50.6467	1.91018
$704$	−24.9621	−0.940795
$705$	−3.85501	−0.145188
$706$	−7.31027	−0.275126
$707$	0	0
$708$	−19.7612	−0.742673
$709$	40.9089	1.53637	0.768183	−	0.640230i	$-0.221161\pi$
$709$	40.9089	1.53637	0.768183	+	0.640230i	$0.221161\pi$
$710$	−8.45961	−0.317484
$711$	−10.4878	−0.393322
$712$	−8.07085	−0.302468
$713$	−20.3204	−0.761006
$714$	0	0
$715$	0	0
$716$	24.3780	0.911048
$717$	18.9198	0.706572
$718$	3.86802	0.144353
$719$	1.19947	0.0447326	0.0223663	−	0.999750i	$-0.492880\pi$
$719$	1.19947	0.0447326	0.0223663	+	0.999750i	$0.492880\pi$
$720$	−15.5047	−0.577826
$721$	0	0
$722$	−9.96865	−0.370995
$723$	18.7462	0.697179
$724$	28.7593	1.06883
$725$	3.67478	0.136478
$726$	−1.40085	−0.0519904
$727$	2.06230	0.0764865	0.0382433	−	0.999268i	$-0.487824\pi$
$727$	2.06230	0.0764865	0.0382433	+	0.999268i	$0.487824\pi$
$728$	0	0
$729$	20.1764	0.747273
$730$	0.527922	0.0195393
$731$	1.76605	0.0653197
$732$	20.8164	0.769395
$733$	−30.0621	−1.11037	−0.555184	−	0.831728i	$-0.687352\pi$
$733$	−30.0621	−1.11037	−0.555184	+	0.831728i	$0.687352\pi$
$734$	−3.05441	−0.112740
$735$	0	0
$736$	24.3801	0.898662
$737$	−5.33721	−0.196599
$738$	1.10680	0.0407420
$739$	44.2548	1.62794	0.813969	−	0.580908i	$-0.197303\pi$
$739$	44.2548	1.62794	0.813969	+	0.580908i	$0.197303\pi$
$740$	−33.3759	−1.22692
$741$	0	0
$742$	0	0
$743$	−8.62651	−0.316476	−0.158238	−	0.987401i	$-0.550581\pi$
$743$	−8.62651	−0.316476	−0.158238	+	0.987401i	$0.550581\pi$
$744$	3.07506	0.112737
$745$	47.3030	1.73305
$746$	8.30194	0.303955
$747$	−1.81702	−0.0664814
$748$	5.97329	0.218405
$749$	0	0
$750$	2.69871	0.0985430
$751$	5.72211	0.208803	0.104401	−	0.994535i	$-0.466707\pi$
$751$	5.72211	0.208803	0.104401	+	0.994535i	$0.466707\pi$
$752$	−4.70512	−0.171578
$753$	23.3525	0.851012
$754$	0	0
$755$	4.11765	0.149857
$756$	0	0
$757$	−34.7222	−1.26200	−0.631000	−	0.775783i	$-0.717355\pi$
$757$	−34.7222	−1.26200	−0.631000	+	0.775783i	$0.717355\pi$
$758$	−2.83853	−0.103100
$759$	−35.8803	−1.30237
$760$	20.2230	0.733566
$761$	53.1735	1.92754	0.963768	−	0.266741i	$-0.0859467\pi$
$761$	53.1735	1.92754	0.963768	+	0.266741i	$0.0859467\pi$
$762$	−0.547981	−0.0198513
$763$	0	0
$764$	27.2452	0.985699
$765$	3.40609	0.123147
$766$	8.24529	0.297915
$767$	0	0
$768$	12.0629	0.435282
$769$	−4.91157	−0.177116	−0.0885578	−	0.996071i	$-0.528226\pi$
$769$	−4.91157	−0.177116	−0.0885578	+	0.996071i	$0.528226\pi$
$770$	0	0
$771$	−15.7483	−0.567160
$772$	−7.50909	−0.270258
$773$	22.9807	0.826557	0.413279	−	0.910605i	$-0.364384\pi$
$773$	22.9807	0.826557	0.413279	+	0.910605i	$0.364384\pi$
$774$	−1.02143	−0.0367146
$775$	3.99142	0.143376
$776$	2.49827	0.0896826
$777$	0	0
$778$	4.39178	0.157453
$779$	18.2695	0.654573
$780$	0	0
$781$	48.5996	1.73903
$782$	−1.67460	−0.0598837
$783$	−12.6164	−0.450874
$784$	0	0
$785$	4.21648	0.150493
$786$	−1.95862	−0.0698616
$787$	3.18774	0.113631	0.0568154	−	0.998385i	$-0.481905\pi$
$787$	3.18774	0.113631	0.0568154	+	0.998385i	$0.481905\pi$
$788$	22.5760	0.804237
$789$	29.6237	1.05463
$790$	−4.25368	−0.151339
$791$	0	0
$792$	−7.03836	−0.250097
$793$	0	0
$794$	−4.25854	−0.151130
$795$	27.1715	0.963674
$796$	6.74229	0.238974
$797$	−54.6509	−1.93584	−0.967918	−	0.251267i	$-0.919153\pi$
$797$	−54.6509	−1.93584	−0.967918	+	0.251267i	$0.919153\pi$
$798$	0	0
$799$	1.03363	0.0365670
$800$	−4.78884	−0.169311
$801$	12.9775	0.458538
$802$	−1.77921	−0.0628260
$803$	−3.03286	−0.107027
$804$	2.98084	0.105126
$805$	0	0
$806$	0	0
$807$	34.3664	1.20975
$808$	0.839768	0.0295429
$809$	−20.2995	−0.713694	−0.356847	−	0.934163i	$-0.616148\pi$
$809$	−20.2995	−0.713694	−0.356847	+	0.934163i	$0.616148\pi$
$810$	0.722151	0.0253738
$811$	−2.43587	−0.0855350	−0.0427675	−	0.999085i	$-0.513617\pi$
$811$	−2.43587	−0.0855350	−0.0427675	+	0.999085i	$0.513617\pi$
$812$	0	0
$813$	−16.5171	−0.579280
$814$	−7.14922	−0.250580
$815$	−16.7748	−0.587597
$816$	−3.20706	−0.112270
$817$	−16.8603	−0.589867
$818$	1.57452	0.0550519
$819$	0	0
$820$	−12.0395	−0.420438
$821$	45.3524	1.58281	0.791405	−	0.611292i	$-0.209350\pi$
$821$	45.3524	1.58281	0.791405	+	0.611292i	$0.209350\pi$
$822$	−3.07589	−0.107284
$823$	−2.75742	−0.0961177	−0.0480588	−	0.998845i	$-0.515303\pi$
$823$	−2.75742	−0.0961177	−0.0480588	+	0.998845i	$0.515303\pi$
$824$	−2.28147	−0.0794789
$825$	7.04776	0.245372
$826$	0	0
$827$	−8.64504	−0.300618	−0.150309	−	0.988639i	$-0.548027\pi$
$827$	−8.64504	−0.300618	−0.150309	+	0.988639i	$0.548027\pi$
$828$	−25.9761	−0.902733
$829$	29.5741	1.02715	0.513576	−	0.858044i	$-0.328320\pi$
$829$	29.5741	1.02715	0.513576	+	0.858044i	$0.328320\pi$
$830$	−0.736958	−0.0255802
$831$	−17.5172	−0.607663
$832$	0	0
$833$	0	0
$834$	−1.69955	−0.0588505
$835$	−24.4617	−0.846531
$836$	−57.0264	−1.97230
$837$	−13.7035	−0.473663
$838$	−8.08191	−0.279185
$839$	−25.2473	−0.871633	−0.435817	−	0.900035i	$-0.643540\pi$
$839$	−25.2473	−0.871633	−0.435817	+	0.900035i	$0.643540\pi$
$840$	0	0
$841$	−23.4694	−0.809289
$842$	−10.7406	−0.370144
$843$	6.04971	0.208363
$844$	−36.6440	−1.26134
$845$	0	0
$846$	−0.597818	−0.0205534
$847$	0	0
$848$	33.1634	1.13884
$849$	35.1870	1.20761
$850$	0.328933	0.0112823
$851$	−53.7544	−1.84268
$852$	−27.1429	−0.929901
$853$	35.1368	1.20306	0.601531	−	0.798850i	$-0.294558\pi$
$853$	35.1368	1.20306	0.601531	+	0.798850i	$0.294558\pi$
$854$	0	0
$855$	−32.5176	−1.11208
$856$	−12.1397	−0.414927
$857$	1.34269	0.0458654	0.0229327	−	0.999737i	$-0.492700\pi$
$857$	1.34269	0.0458654	0.0229327	+	0.999737i	$0.492700\pi$
$858$	0	0
$859$	−4.76772	−0.162673	−0.0813363	−	0.996687i	$-0.525919\pi$
$859$	−4.76772	−0.162673	−0.0813363	+	0.996687i	$0.525919\pi$
$860$	11.1108	0.378877
$861$	0	0
$862$	1.04994	0.0357612
$863$	−26.6105	−0.905830	−0.452915	−	0.891554i	$-0.649616\pi$
$863$	−26.6105	−0.905830	−0.452915	+	0.891554i	$0.649616\pi$
$864$	16.4412	0.559343
$865$	−28.5165	−0.969591
$866$	−10.9375	−0.371672
$867$	−18.7266	−0.635989
$868$	0	0
$869$	24.4370	0.828967
$870$	−1.84634	−0.0625968
$871$	0	0
$872$	8.50819	0.288124
$873$	−4.01708	−0.135958
$874$	15.9873	0.540778
$875$	0	0
$876$	1.69385	0.0572300
$877$	8.03696	0.271389	0.135695	−	0.990751i	$-0.456673\pi$
$877$	8.03696	0.271389	0.135695	+	0.990751i	$0.456673\pi$
$878$	−6.85389	−0.231307
$879$	20.0224	0.675339
$880$	36.1266	1.21783
$881$	54.6697	1.84187	0.920935	−	0.389716i	$-0.127427\pi$
$881$	54.6697	1.84187	0.920935	+	0.389716i	$0.127427\pi$
$882$	0	0
$883$	−8.45085	−0.284394	−0.142197	−	0.989838i	$-0.545417\pi$
$883$	−8.45085	−0.284394	−0.142197	+	0.989838i	$0.545417\pi$
$884$	0	0
$885$	26.2555	0.882569
$886$	7.36176	0.247323
$887$	10.3557	0.347710	0.173855	−	0.984771i	$-0.444378\pi$
$887$	10.3557	0.347710	0.173855	+	0.984771i	$0.444378\pi$
$888$	8.13457	0.272978
$889$	0	0
$890$	5.26349	0.176432
$891$	−4.14868	−0.138986
$892$	−23.0722	−0.772513
$893$	−9.86792	−0.330217
$894$	−5.65898	−0.189265
$895$	−32.3895	−1.08266
$896$	0	0
$897$	0	0
$898$	3.96922	0.132455
$899$	6.00718	0.200351
$900$	5.10234	0.170078
$901$	−7.28536	−0.242711
$902$	−2.57890	−0.0858680
$903$	0	0
$904$	8.46935	0.281686
$905$	−38.2106	−1.27016
$906$	−0.492606	−0.0163657
$907$	18.4804	0.613631	0.306815	−	0.951769i	$-0.400737\pi$
$907$	18.4804	0.613631	0.306815	+	0.951769i	$0.400737\pi$
$908$	29.6624	0.984380
$909$	−1.35030	−0.0447867
$910$	0	0
$911$	−26.6282	−0.882230	−0.441115	−	0.897451i	$-0.645417\pi$
$911$	−26.6282	−0.882230	−0.441115	+	0.897451i	$0.645417\pi$
$912$	30.6175	1.01385
$913$	4.23374	0.140116
$914$	0.174804	0.00578202
$915$	−27.6574	−0.914324
$916$	16.7007	0.551805
$917$	0	0
$918$	−1.12931	−0.0372727
$919$	−11.1493	−0.367783	−0.183891	−	0.982947i	$-0.558869\pi$
$919$	−11.1493	−0.367783	−0.183891	+	0.982947i	$0.558869\pi$
$920$	−21.4639	−0.707644
$921$	9.87072	0.325251
$922$	3.35035	0.110338
$923$	0	0
$924$	0	0
$925$	10.5587	0.347166
$926$	0.0829855	0.00272707
$927$	3.66849	0.120489
$928$	−7.20730	−0.236591
$929$	−7.74510	−0.254108	−0.127054	−	0.991896i	$-0.540552\pi$
$929$	−7.74510	−0.254108	−0.127054	+	0.991896i	$0.540552\pi$
$930$	−2.00543	−0.0657607
$931$	0	0
$932$	−39.0455	−1.27898
$933$	18.7712	0.614541
$934$	6.56299	0.214748
$935$	−7.93633	−0.259546
$936$	0	0
$937$	−36.4239	−1.18992	−0.594959	−	0.803756i	$-0.702832\pi$
$937$	−36.4239	−1.18992	−0.594959	+	0.803756i	$0.702832\pi$
$938$	0	0
$939$	−11.4829	−0.374729
$940$	6.50291	0.212101
$941$	19.7893	0.645114	0.322557	−	0.946550i	$-0.395458\pi$
$941$	19.7893	0.645114	0.322557	+	0.946550i	$0.395458\pi$
$942$	−0.504429	−0.0164352
$943$	−19.3905	−0.631442
$944$	32.0454	1.04299
$945$	0	0
$946$	2.37998	0.0773798
$947$	9.94796	0.323265	0.161633	−	0.986851i	$-0.448324\pi$
$947$	9.94796	0.323265	0.161633	+	0.986851i	$0.448324\pi$
$948$	−13.6481	−0.443269
$949$	0	0
$950$	−3.14029	−0.101884
$951$	11.5958	0.376020
$952$	0	0
$953$	0.0211134	0.000683929 0	0.000341965	−	1.00000i	$-0.499891\pi$
$953$	0.0211134	0.000683929 0	0.000341965		1.00000i	$0.499891\pi$
$954$	4.21364	0.136422
$955$	−36.1990	−1.17137
$956$	−31.9152	−1.03221
$957$	10.6070	0.342877
$958$	2.18214	0.0705017
$959$	0	0
$960$	−18.5230	−0.597827
$961$	−24.4752	−0.789523
$962$	0	0
$963$	19.5200	0.629024
$964$	−31.6224	−1.01849
$965$	9.97685	0.321166
$966$	0	0
$967$	−19.8102	−0.637053	−0.318526	−	0.947914i	$-0.603188\pi$
$967$	−19.8102	−0.637053	−0.318526	+	0.947914i	$0.603188\pi$
$968$	4.81422	0.154735
$969$	−6.72608	−0.216073
$970$	−1.62927	−0.0523127
$971$	3.61774	0.116099	0.0580493	−	0.998314i	$-0.481512\pi$
$971$	3.61774	0.116099	0.0580493	+	0.998314i	$0.481512\pi$
$972$	−28.7144	−0.921015
$973$	0	0
$974$	−1.23595	−0.0396024
$975$	0	0
$976$	−33.7564	−1.08052
$977$	2.16169	0.0691586	0.0345793	−	0.999402i	$-0.488991\pi$
$977$	2.16169	0.0691586	0.0345793	+	0.999402i	$0.488991\pi$
$978$	2.00682	0.0641710
$979$	−30.2382	−0.966416
$980$	0	0
$981$	−13.6807	−0.436792
$982$	−3.49071	−0.111393
$983$	30.1091	0.960331	0.480165	−	0.877178i	$-0.340577\pi$
$983$	30.1091	0.960331	0.480165	+	0.877178i	$0.340577\pi$
$984$	2.93434	0.0935433
$985$	−29.9953	−0.955730
$986$	0.495051	0.0157656
$987$	0	0
$988$	0	0
$989$	17.8948	0.569023
$990$	4.59014	0.145884
$991$	−27.1460	−0.862323	−0.431161	−	0.902275i	$-0.641896\pi$
$991$	−27.1460	−0.862323	−0.431161	+	0.902275i	$0.641896\pi$
$992$	−7.82832	−0.248549
$993$	2.65082	0.0841213
$994$	0	0
$995$	−8.95805	−0.283989
$996$	−2.36455	−0.0749236
$997$	−50.8009	−1.60888	−0.804441	−	0.594033i	$-0.797535\pi$
$997$	−50.8009	−1.60888	−0.804441	+	0.594033i	$0.797535\pi$
$998$	8.66599	0.274317
$999$	−36.2504	−1.14691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.ca.1.4		6
7.3	odd	6		1183.2.e.h.170.3		12
7.5	odd	6		1183.2.e.h.508.3		12
7.6	odd	2		8281.2.a.bz.1.4		6
13.3	even	3		637.2.f.j.295.3		12
13.9	even	3		637.2.f.j.393.3		12
13.12	even	2		8281.2.a.cf.1.3		6
91.3	odd	6		91.2.g.b.9.3	✓	12
91.9	even	3		637.2.g.l.263.3		12
91.12	odd	6		1183.2.e.g.508.4		12
91.16	even	3		637.2.h.l.165.4		12
91.38	odd	6		1183.2.e.g.170.4		12
91.48	odd	6		637.2.f.k.393.3		12
91.55	odd	6		637.2.f.k.295.3		12
91.61	odd	6		91.2.g.b.81.3	yes	12
91.68	odd	6		91.2.h.b.74.4	yes	12
91.74	even	3		637.2.h.l.471.4		12
91.81	even	3		637.2.g.l.373.3		12
91.87	odd	6		91.2.h.b.16.4	yes	12
91.90	odd	2		8281.2.a.ce.1.3		6
273.68	even	6		819.2.s.d.802.3		12
273.152	even	6		819.2.n.d.172.4		12
273.185	even	6		819.2.n.d.100.4		12
273.269	even	6		819.2.s.d.289.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.b.9.3	✓	12	91.3	odd	6
91.2.g.b.81.3	yes	12	91.61	odd	6
91.2.h.b.16.4	yes	12	91.87	odd	6
91.2.h.b.74.4	yes	12	91.68	odd	6
637.2.f.j.295.3		12	13.3	even	3
637.2.f.j.393.3		12	13.9	even	3
637.2.f.k.295.3		12	91.55	odd	6
637.2.f.k.393.3		12	91.48	odd	6
637.2.g.l.263.3		12	91.9	even	3
637.2.g.l.373.3		12	91.81	even	3
637.2.h.l.165.4		12	91.16	even	3
637.2.h.l.471.4		12	91.74	even	3
819.2.n.d.100.4		12	273.185	even	6
819.2.n.d.172.4		12	273.152	even	6
819.2.s.d.289.3		12	273.269	even	6
819.2.s.d.802.3		12	273.68	even	6
1183.2.e.g.170.4		12	91.38	odd	6
1183.2.e.g.508.4		12	91.12	odd	6
1183.2.e.h.170.3		12	7.3	odd	6
1183.2.e.h.508.3		12	7.5	odd	6
8281.2.a.bz.1.4		6	7.6	odd	2
8281.2.a.ca.1.4		6	1.1	even	1	trivial
8281.2.a.ce.1.3		6	91.90	odd	2
8281.2.a.cf.1.3		6	13.12	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-0.268125 q^{2} +1.14301 q^{3} -1.92811 q^{4} +2.56175 q^{5} -0.306470 q^{6} +1.05323 q^{8} -1.69353 q^{9} -0.686871 q^{10} +3.94600 q^{11} -2.20385 q^{12} +2.92811 q^{15} +3.57382 q^{16} -0.785100 q^{17} +0.454078 q^{18} +7.49527 q^{19} -4.93934 q^{20} -1.05802 q^{22} -7.95518 q^{23} +1.20385 q^{24} +1.56259 q^{25} -5.36475 q^{27} +2.35173 q^{29} -0.785100 q^{30} +2.55437 q^{31} -3.06468 q^{32} +4.51032 q^{33} +0.210505 q^{34} +3.26531 q^{36} +6.75716 q^{37} -2.00967 q^{38} +2.69810 q^{40} +2.43747 q^{41} -2.24946 q^{43} -7.60832 q^{44} -4.33841 q^{45} +2.13298 q^{46} -1.31655 q^{47} +4.08491 q^{48} -0.418969 q^{50} -0.897376 q^{51} +9.27954 q^{53} +1.43842 q^{54} +10.1087 q^{55} +8.56716 q^{57} -0.630558 q^{58} +8.96671 q^{59} -5.64571 q^{60} -9.44547 q^{61} -0.684890 q^{62} -6.32592 q^{64} -1.20933 q^{66} -1.35256 q^{67} +1.51376 q^{68} -9.09284 q^{69} +12.3162 q^{71} -1.78367 q^{72} -0.768590 q^{73} -1.81176 q^{74} +1.78605 q^{75} -14.4517 q^{76} +6.19284 q^{79} +9.15525 q^{80} -1.05136 q^{81} -0.653548 q^{82} +1.07292 q^{83} -2.01123 q^{85} +0.603137 q^{86} +2.68805 q^{87} +4.15603 q^{88} -7.66299 q^{89} +1.16324 q^{90} +15.3384 q^{92} +2.91966 q^{93} +0.353001 q^{94} +19.2010 q^{95} -3.50296 q^{96} +2.37202 q^{97} -6.68267 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} - 9 q^{6} - 3 q^{8} - 3 q^{9} + 4 q^{10} - 4 q^{11} + 5 q^{12} + 2 q^{15} - 8 q^{16} + 5 q^{17} - 3 q^{18} - q^{19} - q^{20} + 5 q^{22} + q^{23} - 11 q^{24}+ \cdots - 10 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 + q^3 + 4 * q^4 + q^5 - 9 * q^6 - 3 * q^8 - 3 * q^9 + 4 * q^10 - 4 * q^11 + 5 * q^12 + 2 * q^15 - 8 * q^16 + 5 * q^17 - 3 * q^18 - q^19 - q^20 + 5 * q^22 + q^23 - 11 * q^24 - 7 * q^25 + 4 * q^27 - 3 * q^29 + 5 * q^30 + 16 * q^31 - 8 * q^32 + 16 * q^33 - 16 * q^34 + 21 * q^36 + 13 * q^37 - 17 * q^38 - 5 * q^40 - 8 * q^41 + 11 * q^43 - 21 * q^44 - 7 * q^45 - 16 * q^46 - q^47 + 21 * q^48 - 6 * q^50 + 20 * q^51 + 2 * q^53 - 18 * q^54 + 9 * q^55 + 21 * q^57 + 8 * q^58 + 13 * q^59 - 20 * q^60 - 5 * q^61 + 5 * q^62 - 15 * q^64 + 18 * q^66 + 11 * q^67 + 29 * q^68 + 23 * q^69 - 6 * q^71 - 25 * q^72 - 30 * q^73 + 3 * q^74 - 3 * q^75 - 9 * q^76 - 7 * q^79 - 7 * q^80 + 6 * q^81 + q^82 + 27 * q^83 + q^85 + 7 * q^86 + 16 * q^87 + 4 * q^89 + 8 * q^90 + 27 * q^92 + 7 * q^93 + 45 * q^94 + 6 * q^95 + 19 * q^96 - 35 * q^97 - 10 * q^99

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