LMFDB - Embedded newform 8649.2.a.be.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8649,2,Mod(1,8649)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8649, 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("8649.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8649 = 3^{2} \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8649.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,-2,0,8,-3,0,-2,9,0,-13,-18,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$69.0626127082$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.2051578125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 $ x^8 - 2x^7 - 9x^6 + 19x^5 + 14x^4 - 28x^3 - 11x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11562$ of defining polynomial
Character	$\chi$	$=$	8649.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.30753 q^{2} +3.32468 q^{4} +2.49846 q^{5} +1.60188 q^{7} -3.05673 q^{8} -5.76526 q^{10} +0.733657 q^{11} +1.89598 q^{13} -3.69638 q^{14} +0.404135 q^{16} -4.37924 q^{17} -4.63803 q^{19} +8.30658 q^{20} -1.69293 q^{22} -7.11846 q^{23} +1.24230 q^{25} -4.37501 q^{26} +5.32573 q^{28} -0.128939 q^{29} +5.18091 q^{32} +10.1052 q^{34} +4.00223 q^{35} -8.42948 q^{37} +10.7024 q^{38} -7.63712 q^{40} +7.37398 q^{41} -0.230492 q^{43} +2.43917 q^{44} +16.4260 q^{46} +8.03652 q^{47} -4.43399 q^{49} -2.86664 q^{50} +6.30351 q^{52} +5.73424 q^{53} +1.83301 q^{55} -4.89652 q^{56} +0.297530 q^{58} -9.50372 q^{59} +7.84044 q^{61} -12.7634 q^{64} +4.73702 q^{65} +4.82658 q^{67} -14.5596 q^{68} -9.23525 q^{70} +3.40502 q^{71} -2.69205 q^{73} +19.4512 q^{74} -15.4200 q^{76} +1.17523 q^{77} -4.52794 q^{79} +1.00972 q^{80} -17.0157 q^{82} +2.67035 q^{83} -10.9414 q^{85} +0.531866 q^{86} -2.24259 q^{88} -2.20459 q^{89} +3.03712 q^{91} -23.6666 q^{92} -18.5445 q^{94} -11.5879 q^{95} +12.2899 q^{97} +10.2315 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} - 18 q^{11} + 8 q^{13} + 9 q^{14} + 4 q^{16} - 14 q^{17} - 6 q^{19} + 7 q^{20} + 4 q^{22} - 22 q^{23} + 13 q^{25} - 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) $ 8 * q - 2 * q^2 + 8 * q^4 - 3 * q^5 - 2 * q^7 + 9 * q^8 - 13 * q^10 - 18 * q^11 + 8 * q^13 + 9 * q^14 + 4 * q^16 - 14 * q^17 - 6 * q^19 + 7 * q^20 + 4 * q^22 - 22 * q^23 + 13 * q^25 - 9 * q^26 - 5 * q^28 - 12 * q^29 + 21 * q^32 - 7 * q^34 + 12 * q^35 - 8 * q^37 - 7 * q^38 + 11 * q^40 + 22 * q^41 - 2 * q^43 - 4 * q^44 + 18 * q^46 + 18 * q^47 - 2 * q^49 - 27 * q^50 - q^52 - 6 * q^53 + 28 * q^55 - 30 * q^56 - 5 * q^58 + 4 * q^59 + 30 * q^61 + 9 * q^64 - 3 * q^65 - 13 * q^67 - 30 * q^68 - 39 * q^70 + q^71 + 2 * q^73 - 8 * q^74 - 33 * q^76 + 24 * q^77 + 8 * q^79 - 24 * q^80 - 19 * q^82 - 39 * q^83 - 26 * q^85 + 16 * q^86 - 17 * q^88 - 27 * q^89 + 16 * q^91 - 32 * q^92 + 22 * q^94 + 6 * q^95 + 34 * q^97 - 10 * q^98

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 $

$2$	−2.30753	−1.63167	−0.815834	−	0.578286i	$-0.803722\pi$
$2$	−2.30753	−1.63167	−0.815834	+	0.578286i	$0.803722\pi$
$3$	0	0
$3$	0	0
$4$	3.32468	1.66234
$5$	2.49846	1.11735	0.558673	−	0.829388i	$-0.311311\pi$
$5$	2.49846	1.11735	0.558673	+	0.829388i	$0.311311\pi$
$6$	0	0
$7$	1.60188	0.605453	0.302727	−	0.953077i	$-0.402103\pi$
$7$	1.60188	0.605453	0.302727	+	0.953077i	$0.402103\pi$
$8$	−3.05673	−1.08072
$9$	0	0
$10$	−5.76526	−1.82314
$11$	0.733657	0.221206	0.110603	−	0.993865i	$-0.464722\pi$
$11$	0.733657	0.221206	0.110603	+	0.993865i	$0.464722\pi$
$12$	0	0
$13$	1.89598	0.525849	0.262925	−	0.964816i	$-0.415313\pi$
$13$	1.89598	0.525849	0.262925	+	0.964816i	$0.415313\pi$
$14$	−3.69638	−0.987898
$15$	0	0
$16$	0.404135	0.101034
$17$	−4.37924	−1.06212	−0.531061	−	0.847334i	$-0.678206\pi$
$17$	−4.37924	−1.06212	−0.531061	+	0.847334i	$0.678206\pi$
$18$	0	0
$19$	−4.63803	−1.06404	−0.532018	−	0.846733i	$-0.678566\pi$
$19$	−4.63803	−1.06404	−0.532018	+	0.846733i	$0.678566\pi$
$20$	8.30658	1.85741
$21$	0	0
$22$	−1.69293	−0.360934
$23$	−7.11846	−1.48430	−0.742151	−	0.670233i	$-0.766194\pi$
$23$	−7.11846	−1.48430	−0.742151	+	0.670233i	$0.766194\pi$
$24$	0	0
$25$	1.24230	0.248460
$26$	−4.37501	−0.858011
$27$	0	0
$28$	5.32573	1.00647
$29$	−0.128939	−0.0239434	−0.0119717	−	0.999928i	$-0.503811\pi$
$29$	−0.128939	−0.0239434	−0.0119717	+	0.999928i	$0.503811\pi$
$30$	0	0
$31$	0	0
$31$	0	0
$32$	5.18091	0.915865
$33$	0	0
$34$	10.1052	1.73303
$35$	4.00223	0.676500
$36$	0	0
$37$	−8.42948	−1.38580	−0.692899	−	0.721035i	$-0.743667\pi$
$37$	−8.42948	−1.38580	−0.692899	+	0.721035i	$0.743667\pi$
$38$	10.7024	1.73615
$39$	0	0
$40$	−7.63712	−1.20754
$41$	7.37398	1.15162	0.575811	−	0.817583i	$-0.304686\pi$
$41$	7.37398	1.15162	0.575811	+	0.817583i	$0.304686\pi$
$42$	0	0
$43$	−0.230492	−0.0351497	−0.0175748	−	0.999846i	$-0.505595\pi$
$43$	−0.230492	−0.0351497	−0.0175748	+	0.999846i	$0.505595\pi$
$44$	2.43917	0.367719
$45$	0	0
$46$	16.4260	2.42189
$47$	8.03652	1.17225	0.586124	−	0.810222i	$-0.300653\pi$
$47$	8.03652	1.17225	0.586124	+	0.810222i	$0.300653\pi$
$48$	0	0
$49$	−4.43399	−0.633427
$50$	−2.86664	−0.405404
$51$	0	0
$52$	6.30351	0.874140
$53$	5.73424	0.787658	0.393829	−	0.919184i	$-0.371150\pi$
$53$	5.73424	0.787658	0.393829	+	0.919184i	$0.371150\pi$
$54$	0	0
$55$	1.83301	0.247163
$56$	−4.89652	−0.654324
$57$	0	0
$58$	0.297530	0.0390676
$59$	−9.50372	−1.23728	−0.618639	−	0.785675i	$-0.712316\pi$
$59$	−9.50372	−1.23728	−0.618639	+	0.785675i	$0.712316\pi$
$60$	0	0
$61$	7.84044	1.00387	0.501933	−	0.864907i	$-0.332623\pi$
$61$	7.84044	1.00387	0.501933	+	0.864907i	$0.332623\pi$
$62$	0	0
$63$	0	0
$64$	−12.7634	−1.59542
$65$	4.73702	0.587555
$66$	0	0
$67$	4.82658	0.589660	0.294830	−	0.955550i	$-0.404737\pi$
$67$	4.82658	0.589660	0.294830	+	0.955550i	$0.404737\pi$
$68$	−14.5596	−1.76561
$69$	0	0
$70$	−9.23525	−1.10382
$71$	3.40502	0.404101	0.202051	−	0.979375i	$-0.435239\pi$
$71$	3.40502	0.404101	0.202051	+	0.979375i	$0.435239\pi$
$72$	0	0
$73$	−2.69205	−0.315080	−0.157540	−	0.987513i	$-0.550356\pi$
$73$	−2.69205	−0.315080	−0.157540	+	0.987513i	$0.550356\pi$
$74$	19.4512	2.26116
$75$	0	0
$76$	−15.4200	−1.76879
$77$	1.17523	0.133930
$78$	0	0
$79$	−4.52794	−0.509434	−0.254717	−	0.967016i	$-0.581982\pi$
$79$	−4.52794	−0.509434	−0.254717	+	0.967016i	$0.581982\pi$
$80$	1.00972	0.112890
$81$	0	0
$82$	−17.0157	−1.87906
$83$	2.67035	0.293109	0.146554	−	0.989203i	$-0.453182\pi$
$83$	2.67035	0.293109	0.146554	+	0.989203i	$0.453182\pi$
$84$	0	0
$85$	−10.9414	−1.18676
$86$	0.531866	0.0573526
$87$	0	0
$88$	−2.24259	−0.239061
$89$	−2.20459	−0.233686	−0.116843	−	0.993150i	$-0.537277\pi$
$89$	−2.20459	−0.233686	−0.116843	+	0.993150i	$0.537277\pi$
$90$	0	0
$91$	3.03712	0.318377
$92$	−23.6666	−2.46741
$93$	0	0
$94$	−18.5445	−1.91272
$95$	−11.5879	−1.18890
$96$	0	0
$97$	12.2899	1.24785	0.623923	−	0.781485i	$-0.285538\pi$
$97$	12.2899	1.24785	0.623923	+	0.781485i	$0.285538\pi$
$98$	10.2315	1.03354
$99$	0	0
$100$	4.13025	0.413025
$101$	−7.34432	−0.730787	−0.365394	−	0.930853i	$-0.619066\pi$
$101$	−7.34432	−0.730787	−0.365394	+	0.930853i	$0.619066\pi$
$102$	0	0
$103$	5.58875	0.550676	0.275338	−	0.961347i	$-0.411210\pi$
$103$	5.58875	0.550676	0.275338	+	0.961347i	$0.411210\pi$
$104$	−5.79549	−0.568295
$105$	0	0
$106$	−13.2319	−1.28520
$107$	−1.82416	−0.176348	−0.0881741	−	0.996105i	$-0.528103\pi$
$107$	−1.82416	−0.176348	−0.0881741	+	0.996105i	$0.528103\pi$
$108$	0	0
$109$	11.7653	1.12691	0.563455	−	0.826147i	$-0.309472\pi$
$109$	11.7653	1.12691	0.563455	+	0.826147i	$0.309472\pi$
$110$	−4.22972	−0.403288
$111$	0	0
$112$	0.647376	0.0611712
$113$	−11.0545	−1.03992	−0.519962	−	0.854190i	$-0.674054\pi$
$113$	−11.0545	−1.03992	−0.519962	+	0.854190i	$0.674054\pi$
$114$	0	0
$115$	−17.7852	−1.65848
$116$	−0.428681	−0.0398020
$117$	0	0
$118$	21.9301	2.01883
$119$	−7.01501	−0.643065
$120$	0	0
$121$	−10.4617	−0.951068
$122$	−18.0920	−1.63797
$123$	0	0
$124$	0	0
$125$	−9.38846	−0.839730
$126$	0	0
$127$	1.28703	0.114206	0.0571028	−	0.998368i	$-0.481814\pi$
$127$	1.28703	0.114206	0.0571028	+	0.998368i	$0.481814\pi$
$128$	19.0900	1.68733
$129$	0	0
$130$	−10.9308	−0.958694
$131$	8.02035	0.700742	0.350371	−	0.936611i	$-0.386056\pi$
$131$	8.02035	0.700742	0.350371	+	0.936611i	$0.386056\pi$
$132$	0	0
$133$	−7.42955	−0.644224
$134$	−11.1375	−0.962130
$135$	0	0
$136$	13.3862	1.14785
$137$	−16.4768	−1.40770	−0.703852	−	0.710347i	$-0.748538\pi$
$137$	−16.4768	−1.40770	−0.703852	+	0.710347i	$0.748538\pi$
$138$	0	0
$139$	−7.33982	−0.622555	−0.311278	−	0.950319i	$-0.600757\pi$
$139$	−7.33982	−0.622555	−0.311278	+	0.950319i	$0.600757\pi$
$140$	13.3061	1.12457
$141$	0	0
$142$	−7.85717	−0.659359
$143$	1.39100	0.116321
$144$	0	0
$145$	−0.322149	−0.0267530
$146$	6.21197	0.514106
$147$	0	0
$148$	−28.0253	−2.30367
$149$	6.36193	0.521189	0.260595	−	0.965448i	$-0.416081\pi$
$149$	6.36193	0.521189	0.260595	+	0.965448i	$0.416081\pi$
$150$	0	0
$151$	5.52473	0.449596	0.224798	−	0.974405i	$-0.427828\pi$
$151$	5.52473	0.449596	0.224798	+	0.974405i	$0.427828\pi$
$152$	14.1772	1.14992
$153$	0	0
$154$	−2.71187	−0.218529
$155$	0	0
$156$	0	0
$157$	7.99448	0.638029	0.319014	−	0.947750i	$-0.396648\pi$
$157$	7.99448	0.638029	0.319014	+	0.947750i	$0.396648\pi$
$158$	10.4484	0.831226
$159$	0	0
$160$	12.9443	1.02334
$161$	−11.4029	−0.898675
$162$	0	0
$163$	−17.0654	−1.33667	−0.668333	−	0.743863i	$-0.732992\pi$
$163$	−17.0654	−1.33667	−0.668333	+	0.743863i	$0.732992\pi$
$164$	24.5161	1.91439
$165$	0	0
$166$	−6.16190	−0.478256
$167$	−24.3722	−1.88598	−0.942989	−	0.332824i	$-0.891999\pi$
$167$	−24.3722	−1.88598	−0.942989	+	0.332824i	$0.891999\pi$
$168$	0	0
$169$	−9.40528	−0.723483
$170$	25.2475	1.93639
$171$	0	0
$172$	−0.766311	−0.0584307
$173$	−8.97360	−0.682250	−0.341125	−	0.940018i	$-0.610808\pi$
$173$	−8.97360	−0.682250	−0.341125	+	0.940018i	$0.610808\pi$
$174$	0	0
$175$	1.99001	0.150431
$176$	0.296497	0.0223493
$177$	0	0
$178$	5.08714	0.381297
$179$	−11.2905	−0.843893	−0.421946	−	0.906621i	$-0.638653\pi$
$179$	−11.2905	−0.843893	−0.421946	+	0.906621i	$0.638653\pi$
$180$	0	0
$181$	14.9410	1.11056	0.555279	−	0.831664i	$-0.312611\pi$
$181$	14.9410	1.11056	0.555279	+	0.831664i	$0.312611\pi$
$182$	−7.00824	−0.519485
$183$	0	0
$184$	21.7592	1.60411
$185$	−21.0607	−1.54841
$186$	0	0
$187$	−3.21286	−0.234947
$188$	26.7189	1.94867
$189$	0	0
$190$	26.7394	1.93988
$191$	−2.26093	−0.163595	−0.0817975	−	0.996649i	$-0.526066\pi$
$191$	−2.26093	−0.163595	−0.0817975	+	0.996649i	$0.526066\pi$
$192$	0	0
$193$	25.6643	1.84735	0.923677	−	0.383173i	$-0.125169\pi$
$193$	25.6643	1.84735	0.923677	+	0.383173i	$0.125169\pi$
$194$	−28.3592	−2.03607
$195$	0	0
$196$	−14.7416	−1.05297
$197$	−6.19571	−0.441426	−0.220713	−	0.975339i	$-0.570838\pi$
$197$	−6.19571	−0.441426	−0.220713	+	0.975339i	$0.570838\pi$
$198$	0	0
$199$	4.56933	0.323911	0.161956	−	0.986798i	$-0.448220\pi$
$199$	4.56933	0.323911	0.161956	+	0.986798i	$0.448220\pi$
$200$	−3.79738	−0.268515
$201$	0	0
$202$	16.9472	1.19240
$203$	−0.206545	−0.0144966
$204$	0	0
$205$	18.4236	1.28676
$206$	−12.8962	−0.898520
$207$	0	0
$208$	0.766231	0.0531285
$209$	−3.40272	−0.235371
$210$	0	0
$211$	−18.6168	−1.28163	−0.640816	−	0.767695i	$-0.721404\pi$
$211$	−18.6168	−1.28163	−0.640816	+	0.767695i	$0.721404\pi$
$212$	19.0645	1.30936
$213$	0	0
$214$	4.20930	0.287742
$215$	−0.575874	−0.0392743
$216$	0	0
$217$	0	0
$218$	−27.1487	−1.83874
$219$	0	0
$220$	6.09418	0.410869
$221$	−8.30293	−0.558516
$222$	0	0
$223$	−6.21495	−0.416184	−0.208092	−	0.978109i	$-0.566725\pi$
$223$	−6.21495	−0.416184	−0.208092	+	0.978109i	$0.566725\pi$
$224$	8.29919	0.554513
$225$	0	0
$226$	25.5086	1.69681
$227$	13.0478	0.866013	0.433006	−	0.901391i	$-0.357453\pi$
$227$	13.0478	0.866013	0.433006	+	0.901391i	$0.357453\pi$
$228$	0	0
$229$	−15.2779	−1.00959	−0.504797	−	0.863238i	$-0.668433\pi$
$229$	−15.2779	−1.00959	−0.504797	+	0.863238i	$0.668433\pi$
$230$	41.0398	2.70608
$231$	0	0
$232$	0.394132	0.0258760
$233$	−13.0613	−0.855671	−0.427836	−	0.903857i	$-0.640724\pi$
$233$	−13.0613	−0.855671	−0.427836	+	0.903857i	$0.640724\pi$
$234$	0	0
$235$	20.0789	1.30980
$236$	−31.5968	−2.05678
$237$	0	0
$238$	16.1873	1.04927
$239$	−23.4422	−1.51635	−0.758176	−	0.652050i	$-0.773909\pi$
$239$	−23.4422	−1.51635	−0.758176	+	0.652050i	$0.773909\pi$
$240$	0	0
$241$	−24.7121	−1.59185	−0.795923	−	0.605398i	$-0.793014\pi$
$241$	−24.7121	−1.59185	−0.795923	+	0.605398i	$0.793014\pi$
$242$	24.1408	1.55183
$243$	0	0
$244$	26.0669	1.66876
$245$	−11.0781	−0.707756
$246$	0	0
$247$	−8.79358	−0.559522
$248$	0	0
$249$	0	0
$250$	21.6641	1.37016
$251$	−23.1941	−1.46400	−0.732001	−	0.681304i	$-0.761413\pi$
$251$	−23.1941	−1.46400	−0.732001	+	0.681304i	$0.761413\pi$
$252$	0	0
$253$	−5.22251	−0.328336
$254$	−2.96986	−0.186346
$255$	0	0
$256$	−18.5239	−1.15774
$257$	−15.3461	−0.957264	−0.478632	−	0.878016i	$-0.658867\pi$
$257$	−15.3461	−0.957264	−0.478632	+	0.878016i	$0.658867\pi$
$258$	0	0
$259$	−13.5030	−0.839035
$260$	15.7491	0.976716
$261$	0	0
$262$	−18.5072	−1.14338
$263$	−6.80588	−0.419668	−0.209834	−	0.977737i	$-0.567292\pi$
$263$	−6.80588	−0.419668	−0.209834	+	0.977737i	$0.567292\pi$
$264$	0	0
$265$	14.3268	0.880086
$266$	17.1439	1.05116
$267$	0	0
$268$	16.0468	0.980216
$269$	10.5196	0.641394	0.320697	−	0.947182i	$-0.396083\pi$
$269$	10.5196	0.641394	0.320697	+	0.947182i	$0.396083\pi$
$270$	0	0
$271$	1.57714	0.0958044	0.0479022	−	0.998852i	$-0.484746\pi$
$271$	1.57714	0.0958044	0.0479022	+	0.998852i	$0.484746\pi$
$272$	−1.76981	−0.107310
$273$	0	0
$274$	38.0206	2.29691
$275$	0.911421	0.0549608
$276$	0	0
$277$	−12.1014	−0.727100	−0.363550	−	0.931575i	$-0.618435\pi$
$277$	−12.1014	−0.727100	−0.363550	+	0.931575i	$0.618435\pi$
$278$	16.9368	1.01580
$279$	0	0
$280$	−12.2337	−0.731106
$281$	21.4727	1.28096	0.640478	−	0.767976i	$-0.278736\pi$
$281$	21.4727	1.28096	0.640478	+	0.767976i	$0.278736\pi$
$282$	0	0
$283$	−0.688807	−0.0409453	−0.0204726	−	0.999790i	$-0.506517\pi$
$283$	−0.688807	−0.0409453	−0.0204726	+	0.999790i	$0.506517\pi$
$284$	11.3206	0.671753
$285$	0	0
$286$	−3.20976	−0.189797
$287$	11.8122	0.697253
$288$	0	0
$289$	2.17774	0.128102
$290$	0.743367	0.0436520
$291$	0	0
$292$	−8.95019	−0.523770
$293$	−9.01188	−0.526480	−0.263240	−	0.964730i	$-0.584791\pi$
$293$	−9.01188	−0.526480	−0.263240	+	0.964730i	$0.584791\pi$
$294$	0	0
$295$	−23.7447	−1.38247
$296$	25.7667	1.49766
$297$	0	0
$298$	−14.6803	−0.850408
$299$	−13.4964	−0.780519
$300$	0	0
$301$	−0.369220	−0.0212815
$302$	−12.7485	−0.733592
$303$	0	0
$304$	−1.87439	−0.107504
$305$	19.5890	1.12166
$306$	0	0
$307$	−31.3326	−1.78825	−0.894124	−	0.447819i	$-0.852201\pi$
$307$	−31.3326	−1.78825	−0.894124	+	0.447819i	$0.852201\pi$
$308$	3.90726	0.222637
$309$	0	0
$310$	0	0
$311$	17.7139	1.00447	0.502233	−	0.864732i	$-0.332512\pi$
$311$	17.7139	1.00447	0.502233	+	0.864732i	$0.332512\pi$
$312$	0	0
$313$	21.6687	1.22479	0.612394	−	0.790553i	$-0.290207\pi$
$313$	21.6687	1.22479	0.612394	+	0.790553i	$0.290207\pi$
$314$	−18.4475	−1.04105
$315$	0	0
$316$	−15.0540	−0.846852
$317$	−1.01605	−0.0570669	−0.0285334	−	0.999593i	$-0.509084\pi$
$317$	−1.01605	−0.0570669	−0.0285334	+	0.999593i	$0.509084\pi$
$318$	0	0
$319$	−0.0945970	−0.00529641
$320$	−31.8888	−1.78264
$321$	0	0
$322$	26.3125	1.46634
$323$	20.3110	1.13014
$324$	0	0
$325$	2.35537	0.130652
$326$	39.3789	2.18099
$327$	0	0
$328$	−22.5403	−1.24458
$329$	12.8735	0.709741
$330$	0	0
$331$	−26.6894	−1.46698	−0.733492	−	0.679698i	$-0.762111\pi$
$331$	−26.6894	−1.46698	−0.733492	+	0.679698i	$0.762111\pi$
$332$	8.87806	0.487247
$333$	0	0
$334$	56.2395	3.07729
$335$	12.0590	0.658854
$336$	0	0
$337$	−31.9415	−1.73997	−0.869983	−	0.493081i	$-0.835871\pi$
$337$	−31.9415	−1.73997	−0.869983	+	0.493081i	$0.835871\pi$
$338$	21.7029	1.18048
$339$	0	0
$340$	−36.3765	−1.97279
$341$	0	0
$342$	0	0
$343$	−18.3159	−0.988963
$344$	0.704552	0.0379869
$345$	0	0
$346$	20.7068	1.11321
$347$	3.12131	0.167561	0.0837804	−	0.996484i	$-0.473301\pi$
$347$	3.12131	0.167561	0.0837804	+	0.996484i	$0.473301\pi$
$348$	0	0
$349$	−22.0265	−1.17905	−0.589525	−	0.807750i	$-0.700685\pi$
$349$	−22.0265	−1.17905	−0.589525	+	0.807750i	$0.700685\pi$
$350$	−4.59201	−0.245453
$351$	0	0
$352$	3.80101	0.202595
$353$	4.93239	0.262525	0.131262	−	0.991348i	$-0.458097\pi$
$353$	4.93239	0.262525	0.131262	+	0.991348i	$0.458097\pi$
$354$	0	0
$355$	8.50730	0.451520
$356$	−7.32954	−0.388465
$357$	0	0
$358$	26.0532	1.37695
$359$	0.360860	0.0190455	0.00952273	−	0.999955i	$-0.496969\pi$
$359$	0.360860	0.0190455	0.00952273	+	0.999955i	$0.496969\pi$
$360$	0	0
$361$	2.51129	0.132173
$362$	−34.4769	−1.81206
$363$	0	0
$364$	10.0975	0.529251
$365$	−6.72597	−0.352053
$366$	0	0
$367$	29.3869	1.53398	0.766992	−	0.641657i	$-0.221753\pi$
$367$	29.3869	1.53398	0.766992	+	0.641657i	$0.221753\pi$
$368$	−2.87682	−0.149965
$369$	0	0
$370$	48.5981	2.52650
$371$	9.18555	0.476890
$372$	0	0
$373$	17.7284	0.917941	0.458971	−	0.888451i	$-0.348218\pi$
$373$	17.7284	0.917941	0.458971	+	0.888451i	$0.348218\pi$
$374$	7.41376	0.383356
$375$	0	0
$376$	−24.5655	−1.26687
$377$	−0.244465	−0.0125906
$378$	0	0
$379$	−2.46042	−0.126383	−0.0631917	−	0.998001i	$-0.520128\pi$
$379$	−2.46042	−0.126383	−0.0631917	+	0.998001i	$0.520128\pi$
$380$	−38.5261	−1.97635
$381$	0	0
$382$	5.21715	0.266933
$383$	−19.1179	−0.976880	−0.488440	−	0.872598i	$-0.662434\pi$
$383$	−19.1179	−0.976880	−0.488440	+	0.872598i	$0.662434\pi$
$384$	0	0
$385$	2.93626	0.149646
$386$	−59.2209	−3.01427
$387$	0	0
$388$	40.8599	2.07435
$389$	5.27106	0.267253	0.133627	−	0.991032i	$-0.457338\pi$
$389$	5.27106	0.267253	0.133627	+	0.991032i	$0.457338\pi$
$390$	0	0
$391$	31.1735	1.57651
$392$	13.5535	0.684556
$393$	0	0
$394$	14.2968	0.720261
$395$	−11.3129	−0.569213
$396$	0	0
$397$	16.9255	0.849467	0.424733	−	0.905319i	$-0.360368\pi$
$397$	16.9255	0.849467	0.424733	+	0.905319i	$0.360368\pi$
$398$	−10.5439	−0.528516
$399$	0	0
$400$	0.502057	0.0251029
$401$	38.0397	1.89961	0.949805	−	0.312843i	$-0.101281\pi$
$401$	38.0397	1.89961	0.949805	+	0.312843i	$0.101281\pi$
$402$	0	0
$403$	0	0
$404$	−24.4175	−1.21482
$405$	0	0
$406$	0.476607	0.0236536
$407$	−6.18434	−0.306546
$408$	0	0
$409$	−0.498684	−0.0246583	−0.0123292	−	0.999924i	$-0.503925\pi$
$409$	−0.498684	−0.0246583	−0.0123292	+	0.999924i	$0.503925\pi$
$410$	−42.5129	−2.09956
$411$	0	0
$412$	18.5808	0.915411
$413$	−15.2238	−0.749114
$414$	0	0
$415$	6.67176	0.327504
$416$	9.82289	0.481607
$417$	0	0
$418$	7.85186	0.384047
$419$	−13.5100	−0.660007	−0.330003	−	0.943980i	$-0.607050\pi$
$419$	−13.5100	−0.660007	−0.330003	+	0.943980i	$0.607050\pi$
$420$	0	0
$421$	−30.9151	−1.50671	−0.753355	−	0.657614i	$-0.771566\pi$
$421$	−30.9151	−1.50671	−0.753355	+	0.657614i	$0.771566\pi$
$422$	42.9587	2.09120
$423$	0	0
$424$	−17.5280	−0.851237
$425$	−5.44033	−0.263895
$426$	0	0
$427$	12.5594	0.607793
$428$	−6.06475	−0.293151
$429$	0	0
$430$	1.32885	0.0640826
$431$	12.0548	0.580659	0.290330	−	0.956927i	$-0.406235\pi$
$431$	12.0548	0.580659	0.290330	+	0.956927i	$0.406235\pi$
$432$	0	0
$433$	32.3919	1.55665	0.778327	−	0.627860i	$-0.216069\pi$
$433$	32.3919	1.55665	0.778327	+	0.627860i	$0.216069\pi$
$434$	0	0
$435$	0	0
$436$	39.1158	1.87331
$437$	33.0156	1.57935
$438$	0	0
$439$	12.3682	0.590300	0.295150	−	0.955451i	$-0.404630\pi$
$439$	12.3682	0.590300	0.295150	+	0.955451i	$0.404630\pi$
$440$	−5.60303	−0.267114
$441$	0	0
$442$	19.1592	0.911312
$443$	4.72341	0.224416	0.112208	−	0.993685i	$-0.464208\pi$
$443$	4.72341	0.224416	0.112208	+	0.993685i	$0.464208\pi$
$444$	0	0
$445$	−5.50807	−0.261108
$446$	14.3412	0.679074
$447$	0	0
$448$	−20.4454	−0.965953
$449$	−15.9842	−0.754341	−0.377170	−	0.926144i	$-0.623103\pi$
$449$	−15.9842	−0.754341	−0.377170	+	0.926144i	$0.623103\pi$
$450$	0	0
$451$	5.40997	0.254746
$452$	−36.7528	−1.72871
$453$	0	0
$454$	−30.1081	−1.41305
$455$	7.58813	0.355737
$456$	0	0
$457$	−4.55775	−0.213203	−0.106601	−	0.994302i	$-0.533997\pi$
$457$	−4.55775	−0.213203	−0.106601	+	0.994302i	$0.533997\pi$
$458$	35.2542	1.64732
$459$	0	0
$460$	−59.1301	−2.75695
$461$	−16.9910	−0.791349	−0.395675	−	0.918391i	$-0.629489\pi$
$461$	−16.9910	−0.791349	−0.395675	+	0.918391i	$0.629489\pi$
$462$	0	0
$463$	−27.4279	−1.27468	−0.637340	−	0.770582i	$-0.719965\pi$
$463$	−27.4279	−1.27468	−0.637340	+	0.770582i	$0.719965\pi$
$464$	−0.0521088	−0.00241909
$465$	0	0
$466$	30.1392	1.39617
$467$	22.9105	1.06017	0.530086	−	0.847944i	$-0.322160\pi$
$467$	22.9105	1.06017	0.530086	+	0.847944i	$0.322160\pi$
$468$	0	0
$469$	7.73159	0.357012
$470$	−46.3327	−2.13717
$471$	0	0
$472$	29.0503	1.33715
$473$	−0.169102	−0.00777531
$474$	0	0
$475$	−5.76182	−0.264370
$476$	−23.3227	−1.06899
$477$	0	0
$478$	54.0936	2.47418
$479$	−9.04705	−0.413370	−0.206685	−	0.978407i	$-0.566268\pi$
$479$	−9.04705	−0.413370	−0.206685	+	0.978407i	$0.566268\pi$
$480$	0	0
$481$	−15.9821	−0.728720
$482$	57.0238	2.59736
$483$	0	0
$484$	−34.7820	−1.58100
$485$	30.7057	1.39428
$486$	0	0
$487$	23.6227	1.07045	0.535223	−	0.844711i	$-0.320228\pi$
$487$	23.6227	1.07045	0.535223	+	0.844711i	$0.320228\pi$
$488$	−23.9661	−1.08490
$489$	0	0
$490$	25.5631	1.15482
$491$	9.22692	0.416405	0.208202	−	0.978086i	$-0.433239\pi$
$491$	9.22692	0.416405	0.208202	+	0.978086i	$0.433239\pi$
$492$	0	0
$493$	0.564655	0.0254308
$494$	20.2914	0.912955
$495$	0	0
$496$	0	0
$497$	5.45442	0.244664
$498$	0	0
$499$	−28.4139	−1.27198	−0.635992	−	0.771696i	$-0.719409\pi$
$499$	−28.4139	−1.27198	−0.635992	+	0.771696i	$0.719409\pi$
$500$	−31.2136	−1.39592
$501$	0	0
$502$	53.5211	2.38876
$503$	−18.2781	−0.814979	−0.407489	−	0.913210i	$-0.633596\pi$
$503$	−18.2781	−0.814979	−0.407489	+	0.913210i	$0.633596\pi$
$504$	0	0
$505$	−18.3495	−0.816541
$506$	12.0511	0.535736
$507$	0	0
$508$	4.27897	0.189849
$509$	10.3864	0.460367	0.230184	−	0.973147i	$-0.426067\pi$
$509$	10.3864	0.460367	0.230184	+	0.973147i	$0.426067\pi$
$510$	0	0
$511$	−4.31233	−0.190766
$512$	4.56446	0.201722
$513$	0	0
$514$	35.4116	1.56194
$515$	13.9633	0.615295
$516$	0	0
$517$	5.89605	0.259308
$518$	31.1585	1.36903
$519$	0	0
$520$	−14.4798	−0.634981
$521$	−9.25044	−0.405269	−0.202635	−	0.979254i	$-0.564950\pi$
$521$	−9.25044	−0.405269	−0.202635	+	0.979254i	$0.564950\pi$
$522$	0	0
$523$	6.57516	0.287512	0.143756	−	0.989613i	$-0.454082\pi$
$523$	6.57516	0.287512	0.143756	+	0.989613i	$0.454082\pi$
$524$	26.6651	1.16487
$525$	0	0
$526$	15.7047	0.684759
$527$	0	0
$528$	0	0
$529$	27.6725	1.20315
$530$	−33.0594	−1.43601
$531$	0	0
$532$	−24.7009	−1.07092
$533$	13.9809	0.605579
$534$	0	0
$535$	−4.55759	−0.197042
$536$	−14.7536	−0.637257
$537$	0	0
$538$	−24.2744	−1.04654
$539$	−3.25302	−0.140118
$540$	0	0
$541$	−2.20328	−0.0947262	−0.0473631	−	0.998878i	$-0.515082\pi$
$541$	−2.20328	−0.0947262	−0.0473631	+	0.998878i	$0.515082\pi$
$542$	−3.63929	−0.156321
$543$	0	0
$544$	−22.6885	−0.972760
$545$	29.3951	1.25915
$546$	0	0
$547$	−4.86344	−0.207946	−0.103973	−	0.994580i	$-0.533156\pi$
$547$	−4.86344	−0.207946	−0.103973	+	0.994580i	$0.533156\pi$
$548$	−54.7799	−2.34008
$549$	0	0
$550$	−2.10313	−0.0896777
$551$	0.598022	0.0254766
$552$	0	0
$553$	−7.25322	−0.308438
$554$	27.9242	1.18639
$555$	0	0
$556$	−24.4025	−1.03490
$557$	11.0363	0.467623	0.233811	−	0.972282i	$-0.424880\pi$
$557$	11.0363	0.467623	0.233811	+	0.972282i	$0.424880\pi$
$558$	0	0
$559$	−0.437007	−0.0184834
$560$	1.61744	0.0683494
$561$	0	0
$562$	−49.5489	−2.09010
$563$	11.1924	0.471704	0.235852	−	0.971789i	$-0.424212\pi$
$563$	11.1924	0.471704	0.235852	+	0.971789i	$0.424212\pi$
$564$	0	0
$565$	−27.6193	−1.16195
$566$	1.58944	0.0668091
$567$	0	0
$568$	−10.4082	−0.436720
$569$	−46.9220	−1.96707	−0.983536	−	0.180713i	$-0.942160\pi$
$569$	−46.9220	−1.96707	−0.983536	+	0.180713i	$0.942160\pi$
$570$	0	0
$571$	20.4597	0.856210	0.428105	−	0.903729i	$-0.359181\pi$
$571$	20.4597	0.856210	0.428105	+	0.903729i	$0.359181\pi$
$572$	4.62461	0.193365
$573$	0	0
$574$	−27.2570	−1.13769
$575$	−8.84326	−0.368789
$576$	0	0
$577$	3.64835	0.151883	0.0759413	−	0.997112i	$-0.475804\pi$
$577$	3.64835	0.151883	0.0759413	+	0.997112i	$0.475804\pi$
$578$	−5.02520	−0.209021
$579$	0	0
$580$	−1.07104	−0.0444726
$581$	4.27758	0.177464
$582$	0	0
$583$	4.20696	0.174235
$584$	8.22887	0.340513
$585$	0	0
$586$	20.7952	0.859040
$587$	−22.8726	−0.944053	−0.472027	−	0.881584i	$-0.656477\pi$
$587$	−22.8726	−0.944053	−0.472027	+	0.881584i	$0.656477\pi$
$588$	0	0
$589$	0	0
$590$	54.7914	2.25573
$591$	0	0
$592$	−3.40665	−0.140012
$593$	19.4598	0.799117	0.399558	−	0.916708i	$-0.369163\pi$
$593$	19.4598	0.799117	0.399558	+	0.916708i	$0.369163\pi$
$594$	0	0
$595$	−17.5267	−0.718525
$596$	21.1514	0.866394
$597$	0	0
$598$	31.1434	1.27355
$599$	−35.1718	−1.43708	−0.718541	−	0.695484i	$-0.755190\pi$
$599$	−35.1718	−1.43708	−0.718541	+	0.695484i	$0.755190\pi$
$600$	0	0
$601$	−5.88806	−0.240179	−0.120089	−	0.992763i	$-0.538318\pi$
$601$	−5.88806	−0.240179	−0.120089	+	0.992763i	$0.538318\pi$
$602$	0.851984	0.0347243
$603$	0	0
$604$	18.3680	0.747382
$605$	−26.1383	−1.06267
$606$	0	0
$607$	38.8848	1.57829	0.789143	−	0.614209i	$-0.210525\pi$
$607$	38.8848	1.57829	0.789143	+	0.614209i	$0.210525\pi$
$608$	−24.0292	−0.974513
$609$	0	0
$610$	−45.2022	−1.83018
$611$	15.2371	0.616425
$612$	0	0
$613$	45.6024	1.84186	0.920931	−	0.389725i	$-0.127430\pi$
$613$	45.6024	1.84186	0.920931	+	0.389725i	$0.127430\pi$
$614$	72.3009	2.91783
$615$	0	0
$616$	−3.59236	−0.144740
$617$	6.62648	0.266772	0.133386	−	0.991064i	$-0.457415\pi$
$617$	6.62648	0.266772	0.133386	+	0.991064i	$0.457415\pi$
$618$	0	0
$619$	−41.5360	−1.66947	−0.834736	−	0.550650i	$-0.814380\pi$
$619$	−41.5360	−1.66947	−0.834736	+	0.550650i	$0.814380\pi$
$620$	0	0
$621$	0	0
$622$	−40.8754	−1.63895
$623$	−3.53148	−0.141486
$624$	0	0
$625$	−29.6682	−1.18673
$626$	−50.0011	−1.99845
$627$	0	0
$628$	26.5791	1.06062
$629$	36.9147	1.47189
$630$	0	0
$631$	−11.1447	−0.443664	−0.221832	−	0.975085i	$-0.571204\pi$
$631$	−11.1447	−0.443664	−0.221832	+	0.975085i	$0.571204\pi$
$632$	13.8407	0.550554
$633$	0	0
$634$	2.34455	0.0931141
$635$	3.21560	0.127607
$636$	0	0
$637$	−8.40673	−0.333087
$638$	0.218285	0.00864199
$639$	0	0
$640$	47.6956	1.88533
$641$	17.5533	0.693314	0.346657	−	0.937992i	$-0.387317\pi$
$641$	17.5533	0.693314	0.346657	+	0.937992i	$0.387317\pi$
$642$	0	0
$643$	2.19605	0.0866038	0.0433019	−	0.999062i	$-0.486212\pi$
$643$	2.19605	0.0866038	0.0433019	+	0.999062i	$0.486212\pi$
$644$	−37.9110	−1.49390
$645$	0	0
$646$	−46.8682	−1.84401
$647$	−32.4446	−1.27553	−0.637764	−	0.770232i	$-0.720140\pi$
$647$	−32.4446	−1.27553	−0.637764	+	0.770232i	$0.720140\pi$
$648$	0	0
$649$	−6.97247	−0.273693
$650$	−5.43508	−0.213181
$651$	0	0
$652$	−56.7370	−2.22199
$653$	−16.4900	−0.645302	−0.322651	−	0.946518i	$-0.604574\pi$
$653$	−16.4900	−0.645302	−0.322651	+	0.946518i	$0.604574\pi$
$654$	0	0
$655$	20.0385	0.782970
$656$	2.98009	0.116353
$657$	0	0
$658$	−29.7060	−1.15806
$659$	19.7414	0.769015	0.384508	−	0.923122i	$-0.374371\pi$
$659$	19.7414	0.769015	0.384508	+	0.923122i	$0.374371\pi$
$660$	0	0
$661$	11.7808	0.458221	0.229110	−	0.973400i	$-0.426418\pi$
$661$	11.7808	0.458221	0.229110	+	0.973400i	$0.426418\pi$
$662$	61.5866	2.39363
$663$	0	0
$664$	−8.16255	−0.316768
$665$	−18.5624	−0.719820
$666$	0	0
$667$	0.917847	0.0355392
$668$	−81.0298	−3.13514
$669$	0	0
$670$	−27.8265	−1.07503
$671$	5.75219	0.222061
$672$	0	0
$673$	−50.3821	−1.94209	−0.971044	−	0.238901i	$-0.923213\pi$
$673$	−50.3821	−1.94209	−0.971044	+	0.238901i	$0.923213\pi$
$674$	73.7060	2.83905
$675$	0	0
$676$	−31.2695	−1.20267
$677$	−3.97995	−0.152962	−0.0764810	−	0.997071i	$-0.524368\pi$
$677$	−3.97995	−0.152962	−0.0764810	+	0.997071i	$0.524368\pi$
$678$	0	0
$679$	19.6869	0.755513
$680$	33.4448	1.28255
$681$	0	0
$682$	0	0
$683$	−5.23244	−0.200214	−0.100107	−	0.994977i	$-0.531918\pi$
$683$	−5.23244	−0.200214	−0.100107	+	0.994977i	$0.531918\pi$
$684$	0	0
$685$	−41.1665	−1.57289
$686$	42.2643	1.61366
$687$	0	0
$688$	−0.0931499	−0.00355130
$689$	10.8720	0.414189
$690$	0	0
$691$	9.19553	0.349815	0.174907	−	0.984585i	$-0.444037\pi$
$691$	9.19553	0.349815	0.174907	+	0.984585i	$0.444037\pi$
$692$	−29.8344	−1.13413
$693$	0	0
$694$	−7.20251	−0.273404
$695$	−18.3382	−0.695609
$696$	0	0
$697$	−32.2924	−1.22316
$698$	50.8266	1.92382
$699$	0	0
$700$	6.61615	0.250067
$701$	40.5543	1.53172	0.765858	−	0.643010i	$-0.222315\pi$
$701$	40.5543	1.53172	0.765858	+	0.643010i	$0.222315\pi$
$702$	0	0
$703$	39.0961	1.47454
$704$	−9.36393	−0.352916
$705$	0	0
$706$	−11.3816	−0.428353
$707$	−11.7647	−0.442457
$708$	0	0
$709$	37.2861	1.40031	0.700154	−	0.713992i	$-0.253115\pi$
$709$	37.2861	1.40031	0.700154	+	0.713992i	$0.253115\pi$
$710$	−19.6308	−0.736731
$711$	0	0
$712$	6.73883	0.252548
$713$	0	0
$714$	0	0
$715$	3.47535	0.129971
$716$	−37.5374	−1.40284
$717$	0	0
$718$	−0.832694	−0.0310759
$719$	−18.5799	−0.692912	−0.346456	−	0.938066i	$-0.612615\pi$
$719$	−18.5799	−0.692912	−0.346456	+	0.938066i	$0.612615\pi$
$720$	0	0
$721$	8.95250	0.333409
$722$	−5.79486	−0.215662
$723$	0	0
$724$	49.6742	1.84613
$725$	−0.160181	−0.00594897
$726$	0	0
$727$	16.3772	0.607396	0.303698	−	0.952768i	$-0.401779\pi$
$727$	16.3772	0.607396	0.303698	+	0.952768i	$0.401779\pi$
$728$	−9.28367	−0.344076
$729$	0	0
$730$	15.5203	0.574434
$731$	1.00938	0.0373332
$732$	0	0
$733$	−1.15134	−0.0425256	−0.0212628	−	0.999774i	$-0.506769\pi$
$733$	−1.15134	−0.0425256	−0.0212628	+	0.999774i	$0.506769\pi$
$734$	−67.8111	−2.50295
$735$	0	0
$736$	−36.8801	−1.35942
$737$	3.54105	0.130436
$738$	0	0
$739$	20.7158	0.762043	0.381022	−	0.924566i	$-0.375572\pi$
$739$	20.7158	0.762043	0.381022	+	0.924566i	$0.375572\pi$
$740$	−70.0201	−2.57399
$741$	0	0
$742$	−21.1959	−0.778126
$743$	−35.2367	−1.29271	−0.646354	−	0.763038i	$-0.723707\pi$
$743$	−35.2367	−1.29271	−0.646354	+	0.763038i	$0.723707\pi$
$744$	0	0
$745$	15.8950	0.582348
$746$	−40.9087	−1.49778
$747$	0	0
$748$	−10.6817	−0.390563
$749$	−2.92208	−0.106771
$750$	0	0
$751$	−43.1459	−1.57441	−0.787207	−	0.616688i	$-0.788474\pi$
$751$	−43.1459	−1.57441	−0.787207	+	0.616688i	$0.788474\pi$
$752$	3.24784	0.118437
$753$	0	0
$754$	0.564110	0.0205437
$755$	13.8033	0.502354
$756$	0	0
$757$	27.3386	0.993637	0.496819	−	0.867854i	$-0.334501\pi$
$757$	27.3386	0.993637	0.496819	+	0.867854i	$0.334501\pi$
$758$	5.67749	0.206216
$759$	0	0
$760$	35.4212	1.28486
$761$	17.1920	0.623208	0.311604	−	0.950212i	$-0.399134\pi$
$761$	17.1920	0.623208	0.311604	+	0.950212i	$0.399134\pi$
$762$	0	0
$763$	18.8466	0.682291
$764$	−7.51686	−0.271951
$765$	0	0
$766$	44.1151	1.59394
$767$	−18.0188	−0.650622
$768$	0	0
$769$	14.0567	0.506899	0.253450	−	0.967349i	$-0.418435\pi$
$769$	14.0567	0.506899	0.253450	+	0.967349i	$0.418435\pi$
$770$	−6.77550	−0.244172
$771$	0	0
$772$	85.3254	3.07093
$773$	−22.2368	−0.799803	−0.399902	−	0.916558i	$-0.630956\pi$
$773$	−22.2368	−0.799803	−0.399902	+	0.916558i	$0.630956\pi$
$774$	0	0
$775$	0	0
$776$	−37.5668	−1.34857
$777$	0	0
$778$	−12.1631	−0.436069
$779$	−34.2007	−1.22537
$780$	0	0
$781$	2.49811	0.0893895
$782$	−71.9336	−2.57234
$783$	0	0
$784$	−1.79193	−0.0639975
$785$	19.9739	0.712898
$786$	0	0
$787$	−32.0338	−1.14188	−0.570941	−	0.820991i	$-0.693422\pi$
$787$	−32.0338	−1.14188	−0.570941	+	0.820991i	$0.693422\pi$
$788$	−20.5988	−0.733800
$789$	0	0
$790$	26.1048	0.928767
$791$	−17.7080	−0.629625
$792$	0	0
$793$	14.8653	0.527881
$794$	−39.0560	−1.38605
$795$	0	0
$796$	15.1916	0.538451
$797$	26.0493	0.922712	0.461356	−	0.887215i	$-0.347363\pi$
$797$	26.0493	0.922712	0.461356	+	0.887215i	$0.347363\pi$
$798$	0	0
$799$	−35.1939	−1.24507
$800$	6.43625	0.227556
$801$	0	0
$802$	−87.7775	−3.09953
$803$	−1.97504	−0.0696976
$804$	0	0
$805$	−28.4897	−1.00413
$806$	0	0
$807$	0	0
$808$	22.4496	0.789775
$809$	−8.72941	−0.306910	−0.153455	−	0.988156i	$-0.549040\pi$
$809$	−8.72941	−0.306910	−0.153455	+	0.988156i	$0.549040\pi$
$810$	0	0
$811$	−50.0784	−1.75849	−0.879245	−	0.476369i	$-0.841953\pi$
$811$	−50.0784	−1.75849	−0.879245	+	0.476369i	$0.841953\pi$
$812$	−0.686695	−0.0240983
$813$	0	0
$814$	14.2705	0.500182
$815$	−42.6372	−1.49352
$816$	0	0
$817$	1.06903	0.0374005
$818$	1.15073	0.0402342
$819$	0	0
$820$	61.2525	2.13903
$821$	−50.7831	−1.77234	−0.886171	−	0.463359i	$-0.846644\pi$
$821$	−50.7831	−1.77234	−0.886171	+	0.463359i	$0.846644\pi$
$822$	0	0
$823$	6.32920	0.220622	0.110311	−	0.993897i	$-0.464815\pi$
$823$	6.32920	0.220622	0.110311	+	0.993897i	$0.464815\pi$
$824$	−17.0833	−0.595126
$825$	0	0
$826$	35.1293	1.22231
$827$	3.69745	0.128573	0.0642865	−	0.997931i	$-0.479523\pi$
$827$	3.69745	0.128573	0.0642865	+	0.997931i	$0.479523\pi$
$828$	0	0
$829$	−26.0416	−0.904461	−0.452230	−	0.891901i	$-0.649372\pi$
$829$	−26.0416	−0.904461	−0.452230	+	0.891901i	$0.649372\pi$
$830$	−15.3953	−0.534377
$831$	0	0
$832$	−24.1990	−0.838951
$833$	19.4175	0.672776
$834$	0	0
$835$	−60.8930	−2.10729
$836$	−11.3130	−0.391267
$837$	0	0
$838$	31.1747	1.07691
$839$	−7.49300	−0.258687	−0.129343	−	0.991600i	$-0.541287\pi$
$839$	−7.49300	−0.258687	−0.129343	+	0.991600i	$0.541287\pi$
$840$	0	0
$841$	−28.9834	−0.999427
$842$	71.3374	2.45845
$843$	0	0
$844$	−61.8948	−2.13051
$845$	−23.4987	−0.808380
$846$	0	0
$847$	−16.7584	−0.575827
$848$	2.31741	0.0795801
$849$	0	0
$850$	12.5537	0.430588
$851$	60.0049	2.05694
$852$	0	0
$853$	−43.5869	−1.49239	−0.746194	−	0.665728i	$-0.768121\pi$
$853$	−43.5869	−1.49239	−0.746194	+	0.665728i	$0.768121\pi$
$854$	−28.9812	−0.991717
$855$	0	0
$856$	5.57597	0.190583
$857$	−39.2228	−1.33982	−0.669912	−	0.742440i	$-0.733668\pi$
$857$	−39.2228	−1.33982	−0.669912	+	0.742440i	$0.733668\pi$
$858$	0	0
$859$	18.2013	0.621020	0.310510	−	0.950570i	$-0.399500\pi$
$859$	18.2013	0.621020	0.310510	+	0.950570i	$0.399500\pi$
$860$	−1.91460	−0.0652872
$861$	0	0
$862$	−27.8168	−0.947443
$863$	44.9477	1.53004	0.765018	−	0.644009i	$-0.222730\pi$
$863$	44.9477	1.53004	0.765018	+	0.644009i	$0.222730\pi$
$864$	0	0
$865$	−22.4202	−0.762309
$866$	−74.7451	−2.53994
$867$	0	0
$868$	0	0
$869$	−3.32196	−0.112690
$870$	0	0
$871$	9.15108	0.310072
$872$	−35.9633	−1.21787
$873$	0	0
$874$	−76.1844	−2.57698
$875$	−15.0392	−0.508417
$876$	0	0
$877$	−42.6511	−1.44022	−0.720112	−	0.693858i	$-0.755910\pi$
$877$	−42.6511	−1.44022	−0.720112	+	0.693858i	$0.755910\pi$
$878$	−28.5399	−0.963173
$879$	0	0
$880$	0.740785	0.0249718
$881$	−8.91652	−0.300405	−0.150203	−	0.988655i	$-0.547993\pi$
$881$	−8.91652	−0.300405	−0.150203	+	0.988655i	$0.547993\pi$
$882$	0	0
$883$	−47.0683	−1.58398	−0.791988	−	0.610537i	$-0.790954\pi$
$883$	−47.0683	−1.58398	−0.791988	+	0.610537i	$0.790954\pi$
$884$	−27.6046	−0.928443
$885$	0	0
$886$	−10.8994	−0.366172
$887$	−3.15657	−0.105987	−0.0529936	−	0.998595i	$-0.516876\pi$
$887$	−3.15657	−0.105987	−0.0529936	+	0.998595i	$0.516876\pi$
$888$	0	0
$889$	2.06167	0.0691462
$890$	12.7100	0.426041
$891$	0	0
$892$	−20.6627	−0.691839
$893$	−37.2736	−1.24731
$894$	0	0
$895$	−28.2089	−0.942920
$896$	30.5798	1.02160
$897$	0	0
$898$	36.8840	1.23083
$899$	0	0
$900$	0	0
$901$	−25.1116	−0.836589
$902$	−12.4836	−0.415660
$903$	0	0
$904$	33.7908	1.12386
$905$	37.3296	1.24088
$906$	0	0
$907$	26.9632	0.895297	0.447649	−	0.894210i	$-0.352262\pi$
$907$	26.9632	0.895297	0.447649	+	0.894210i	$0.352262\pi$
$908$	43.3797	1.43961
$909$	0	0
$910$	−17.5098	−0.580444
$911$	5.07802	0.168242	0.0841212	−	0.996456i	$-0.473192\pi$
$911$	5.07802	0.168242	0.0841212	+	0.996456i	$0.473192\pi$
$912$	0	0
$913$	1.95912	0.0648374
$914$	10.5171	0.347876
$915$	0	0
$916$	−50.7942	−1.67829
$917$	12.8476	0.424266
$918$	0	0
$919$	−49.2770	−1.62550	−0.812749	−	0.582615i	$-0.802030\pi$
$919$	−49.2770	−1.62550	−0.812749	+	0.582615i	$0.802030\pi$
$920$	54.3646	1.79235
$921$	0	0
$922$	39.2072	1.29122
$923$	6.45583	0.212496
$924$	0	0
$925$	−10.4719	−0.344315
$926$	63.2905	2.07986
$927$	0	0
$928$	−0.668022	−0.0219289
$929$	−39.9606	−1.31107	−0.655533	−	0.755167i	$-0.727556\pi$
$929$	−39.9606	−1.31107	−0.655533	+	0.755167i	$0.727556\pi$
$930$	0	0
$931$	20.5649	0.673989
$932$	−43.4245	−1.42242
$933$	0	0
$934$	−52.8666	−1.72985
$935$	−8.02720	−0.262517
$936$	0	0
$937$	45.0893	1.47300	0.736502	−	0.676436i	$-0.236476\pi$
$937$	45.0893	1.47300	0.736502	+	0.676436i	$0.236476\pi$
$938$	−17.8409	−0.582525
$939$	0	0
$940$	66.7560	2.17734
$941$	50.1923	1.63622	0.818112	−	0.575060i	$-0.195021\pi$
$941$	50.1923	1.63622	0.818112	+	0.575060i	$0.195021\pi$
$942$	0	0
$943$	−52.4914	−1.70936
$944$	−3.84079	−0.125007
$945$	0	0
$946$	0.390207	0.0126867
$947$	30.3259	0.985461	0.492730	−	0.870182i	$-0.335999\pi$
$947$	30.3259	0.985461	0.492730	+	0.870182i	$0.335999\pi$
$948$	0	0
$949$	−5.10405	−0.165685
$950$	13.2955	0.431365
$951$	0	0
$952$	21.4430	0.694972
$953$	−35.4775	−1.14923	−0.574614	−	0.818425i	$-0.694848\pi$
$953$	−35.4775	−1.14923	−0.574614	+	0.818425i	$0.694848\pi$
$954$	0	0
$955$	−5.64884	−0.182792
$956$	−77.9379	−2.52069
$957$	0	0
$958$	20.8763	0.674483
$959$	−26.3938	−0.852299
$960$	0	0
$961$	0	0
$962$	36.8791	1.18903
$963$	0	0
$964$	−82.1598	−2.64619
$965$	64.1211	2.06413
$966$	0	0
$967$	35.4443	1.13981	0.569906	−	0.821710i	$-0.306979\pi$
$967$	35.4443	1.13981	0.569906	+	0.821710i	$0.306979\pi$
$968$	31.9788	1.02784
$969$	0	0
$970$	−70.8543	−2.27499
$971$	40.0972	1.28678	0.643391	−	0.765538i	$-0.277527\pi$
$971$	40.0972	1.28678	0.643391	+	0.765538i	$0.277527\pi$
$972$	0	0
$973$	−11.7575	−0.376928
$974$	−54.5099	−1.74661
$975$	0	0
$976$	3.16860	0.101424
$977$	38.6708	1.23719	0.618595	−	0.785710i	$-0.287702\pi$
$977$	38.6708	1.23719	0.618595	+	0.785710i	$0.287702\pi$
$978$	0	0
$979$	−1.61741	−0.0516926
$980$	−36.8312	−1.17653
$981$	0	0
$982$	−21.2914	−0.679434
$983$	−21.5796	−0.688282	−0.344141	−	0.938918i	$-0.611830\pi$
$983$	−21.5796	−0.688282	−0.344141	+	0.938918i	$0.611830\pi$
$984$	0	0
$985$	−15.4797	−0.493225
$986$	−1.30296	−0.0414946
$987$	0	0
$988$	−29.2359	−0.930116
$989$	1.64075	0.0521727
$990$	0	0
$991$	−46.8764	−1.48908	−0.744538	−	0.667580i	$-0.767330\pi$
$991$	−46.8764	−1.48908	−0.744538	+	0.667580i	$0.767330\pi$
$992$	0	0
$993$	0	0
$994$	−12.5862	−0.399211
$995$	11.4163	0.361921
$996$	0	0
$997$	44.5389	1.41056	0.705281	−	0.708928i	$-0.250821\pi$
$997$	44.5389	1.41056	0.705281	+	0.708928i	$0.250821\pi$
$998$	65.5659	2.07545
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8649.2.a.be.1.1		8
3.2	odd	2		961.2.a.j.1.8		8
31.22	odd	30		279.2.y.c.19.2		16
31.24	odd	30		279.2.y.c.235.2		16
31.30	odd	2		8649.2.a.bf.1.1		8
93.2	odd	10		961.2.d.q.531.4		16
93.5	odd	6		961.2.c.i.521.8		16
93.8	odd	10		961.2.d.n.374.1		16
93.11	even	30		961.2.g.k.338.1		16
93.14	odd	30		961.2.g.j.816.1		16
93.17	even	30		961.2.g.k.816.1		16
93.20	odd	30		961.2.g.j.338.1		16
93.23	even	10		961.2.d.o.374.1		16
93.26	even	6		961.2.c.j.521.8		16
93.29	even	10		961.2.d.p.531.4		16
93.35	odd	10		961.2.d.n.388.1		16
93.38	odd	30		961.2.g.l.235.1		16
93.41	odd	30		961.2.g.m.844.2		16
93.44	even	30		961.2.g.t.448.2		16
93.47	odd	10		961.2.d.q.628.4		16
93.50	odd	30		961.2.g.n.547.2		16
93.53	even	30		31.2.g.a.19.1	yes	16
93.56	odd	6		961.2.c.i.439.8		16
93.59	odd	30		961.2.g.m.846.2		16
93.65	even	30		961.2.g.s.846.2		16
93.68	even	6		961.2.c.j.439.8		16
93.71	odd	30		961.2.g.l.732.1		16
93.74	even	30		961.2.g.t.547.2		16
93.77	even	10		961.2.d.p.628.4		16
93.80	odd	30		961.2.g.n.448.2		16
93.83	even	30		961.2.g.s.844.2		16
93.86	even	30		31.2.g.a.18.1	✓	16
93.89	even	10		961.2.d.o.388.1		16
93.92	even	2		961.2.a.i.1.8		8
372.179	odd	30		496.2.bg.c.49.2		16
372.239	odd	30		496.2.bg.c.81.2		16
465.53	odd	60		775.2.ck.a.174.1		32
465.179	even	30		775.2.bl.a.576.2		16
465.239	even	30		775.2.bl.a.701.2		16
465.272	odd	60		775.2.ck.a.49.1		32
465.332	odd	60		775.2.ck.a.174.4		32
465.458	odd	60		775.2.ck.a.49.4		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.2.g.a.18.1	✓	16	93.86	even	30
31.2.g.a.19.1	yes	16	93.53	even	30
279.2.y.c.19.2		16	31.22	odd	30
279.2.y.c.235.2		16	31.24	odd	30
496.2.bg.c.49.2		16	372.179	odd	30
496.2.bg.c.81.2		16	372.239	odd	30
775.2.bl.a.576.2		16	465.179	even	30
775.2.bl.a.701.2		16	465.239	even	30
775.2.ck.a.49.1		32	465.272	odd	60
775.2.ck.a.49.4		32	465.458	odd	60
775.2.ck.a.174.1		32	465.53	odd	60
775.2.ck.a.174.4		32	465.332	odd	60
961.2.a.i.1.8		8	93.92	even	2
961.2.a.j.1.8		8	3.2	odd	2
961.2.c.i.439.8		16	93.56	odd	6
961.2.c.i.521.8		16	93.5	odd	6
961.2.c.j.439.8		16	93.68	even	6
961.2.c.j.521.8		16	93.26	even	6
961.2.d.n.374.1		16	93.8	odd	10
961.2.d.n.388.1		16	93.35	odd	10
961.2.d.o.374.1		16	93.23	even	10
961.2.d.o.388.1		16	93.89	even	10
961.2.d.p.531.4		16	93.29	even	10
961.2.d.p.628.4		16	93.77	even	10
961.2.d.q.531.4		16	93.2	odd	10
961.2.d.q.628.4		16	93.47	odd	10
961.2.g.j.338.1		16	93.20	odd	30
961.2.g.j.816.1		16	93.14	odd	30
961.2.g.k.338.1		16	93.11	even	30
961.2.g.k.816.1		16	93.17	even	30
961.2.g.l.235.1		16	93.38	odd	30
961.2.g.l.732.1		16	93.71	odd	30
961.2.g.m.844.2		16	93.41	odd	30
961.2.g.m.846.2		16	93.59	odd	30
961.2.g.n.448.2		16	93.80	odd	30
961.2.g.n.547.2		16	93.50	odd	30
961.2.g.s.844.2		16	93.83	even	30
961.2.g.s.846.2		16	93.65	even	30
961.2.g.t.448.2		16	93.44	even	30
961.2.g.t.547.2		16	93.74	even	30
8649.2.a.be.1.1		8	1.1	even	1	trivial
8649.2.a.bf.1.1		8	31.30	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8649 = 3^{2} \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8649.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30753 q^{2} +3.32468 q^{4} +2.49846 q^{5} +1.60188 q^{7} -3.05673 q^{8} -5.76526 q^{10} +0.733657 q^{11} +1.89598 q^{13} -3.69638 q^{14} +0.404135 q^{16} -4.37924 q^{17} -4.63803 q^{19} +8.30658 q^{20} -1.69293 q^{22} -7.11846 q^{23} +1.24230 q^{25} -4.37501 q^{26} +5.32573 q^{28} -0.128939 q^{29} +5.18091 q^{32} +10.1052 q^{34} +4.00223 q^{35} -8.42948 q^{37} +10.7024 q^{38} -7.63712 q^{40} +7.37398 q^{41} -0.230492 q^{43} +2.43917 q^{44} +16.4260 q^{46} +8.03652 q^{47} -4.43399 q^{49} -2.86664 q^{50} +6.30351 q^{52} +5.73424 q^{53} +1.83301 q^{55} -4.89652 q^{56} +0.297530 q^{58} -9.50372 q^{59} +7.84044 q^{61} -12.7634 q^{64} +4.73702 q^{65} +4.82658 q^{67} -14.5596 q^{68} -9.23525 q^{70} +3.40502 q^{71} -2.69205 q^{73} +19.4512 q^{74} -15.4200 q^{76} +1.17523 q^{77} -4.52794 q^{79} +1.00972 q^{80} -17.0157 q^{82} +2.67035 q^{83} -10.9414 q^{85} +0.531866 q^{86} -2.24259 q^{88} -2.20459 q^{89} +3.03712 q^{91} -23.6666 q^{92} -18.5445 q^{94} -11.5879 q^{95} +12.2899 q^{97} +10.2315 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{2} + 8 q^{4} - 3 q^{5} - 2 q^{7} + 9 q^{8} - 13 q^{10} - 18 q^{11} + 8 q^{13} + 9 q^{14} + 4 q^{16} - 14 q^{17} - 6 q^{19} + 7 q^{20} + 4 q^{22} - 22 q^{23} + 13 q^{25} - 9 q^{26} - 5 q^{28}+ \cdots - 10 q^{98}+O(q^{100}) \) 8 * q - 2 * q^2 + 8 * q^4 - 3 * q^5 - 2 * q^7 + 9 * q^8 - 13 * q^10 - 18 * q^11 + 8 * q^13 + 9 * q^14 + 4 * q^16 - 14 * q^17 - 6 * q^19 + 7 * q^20 + 4 * q^22 - 22 * q^23 + 13 * q^25 - 9 * q^26 - 5 * q^28 - 12 * q^29 + 21 * q^32 - 7 * q^34 + 12 * q^35 - 8 * q^37 - 7 * q^38 + 11 * q^40 + 22 * q^41 - 2 * q^43 - 4 * q^44 + 18 * q^46 + 18 * q^47 - 2 * q^49 - 27 * q^50 - q^52 - 6 * q^53 + 28 * q^55 - 30 * q^56 - 5 * q^58 + 4 * q^59 + 30 * q^61 + 9 * q^64 - 3 * q^65 - 13 * q^67 - 30 * q^68 - 39 * q^70 + q^71 + 2 * q^73 - 8 * q^74 - 33 * q^76 + 24 * q^77 + 8 * q^79 - 24 * q^80 - 19 * q^82 - 39 * q^83 - 26 * q^85 + 16 * q^86 - 17 * q^88 - 27 * q^89 + 16 * q^91 - 32 * q^92 + 22 * q^94 + 6 * q^95 + 34 * q^97 - 10 * q^98

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