LMFDB - Embedded newform 1369.2.a.n.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1369,2,Mod(1,1369)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1369, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("1369.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1369 = 37^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1369.a (trivial)

Newform invariants

comment:select newform

sage:traces = [27,-9,-1,25,-17,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.9315200367$
Analytic rank:	$1$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Character	$\chi$	$=$	1369.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.581223 q^{2} -1.17520 q^{3} -1.66218 q^{4} +0.135564 q^{5} +0.683054 q^{6} -3.40406 q^{7} +2.12854 q^{8} -1.61890 q^{9} -0.0787931 q^{10} +4.51469 q^{11} +1.95340 q^{12} +1.75123 q^{13} +1.97852 q^{14} -0.159316 q^{15} +2.08720 q^{16} -0.238914 q^{17} +0.940940 q^{18} +3.02781 q^{19} -0.225332 q^{20} +4.00046 q^{21} -2.62404 q^{22} +7.46588 q^{23} -2.50147 q^{24} -4.98162 q^{25} -1.01785 q^{26} +5.42814 q^{27} +5.65816 q^{28} -1.30151 q^{29} +0.0925978 q^{30} -8.68910 q^{31} -5.47021 q^{32} -5.30568 q^{33} +0.138862 q^{34} -0.461469 q^{35} +2.69090 q^{36} -1.75983 q^{38} -2.05805 q^{39} +0.288554 q^{40} +7.75460 q^{41} -2.32516 q^{42} -8.68570 q^{43} -7.50423 q^{44} -0.219465 q^{45} -4.33934 q^{46} -8.13636 q^{47} -2.45289 q^{48} +4.58762 q^{49} +2.89543 q^{50} +0.280773 q^{51} -2.91086 q^{52} +11.3476 q^{53} -3.15496 q^{54} +0.612031 q^{55} -7.24568 q^{56} -3.55829 q^{57} +0.756470 q^{58} -6.24883 q^{59} +0.264811 q^{60} -13.3101 q^{61} +5.05030 q^{62} +5.51083 q^{63} -0.994996 q^{64} +0.237404 q^{65} +3.08378 q^{66} +2.59709 q^{67} +0.397119 q^{68} -8.77393 q^{69} +0.268216 q^{70} +3.60176 q^{71} -3.44589 q^{72} -6.65769 q^{73} +5.85442 q^{75} -5.03277 q^{76} -15.3683 q^{77} +1.19618 q^{78} -4.55092 q^{79} +0.282950 q^{80} -1.52247 q^{81} -4.50715 q^{82} -4.73268 q^{83} -6.64949 q^{84} -0.0323883 q^{85} +5.04833 q^{86} +1.52954 q^{87} +9.60971 q^{88} +16.7206 q^{89} +0.127558 q^{90} -5.96128 q^{91} -12.4096 q^{92} +10.2115 q^{93} +4.72903 q^{94} +0.410463 q^{95} +6.42861 q^{96} -2.99941 q^{97} -2.66643 q^{98} -7.30883 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q - 9 q^{2} - q^{3} + 25 q^{4} - 17 q^{5} - 10 q^{6} - 3 q^{7} - 21 q^{8} + 20 q^{9} - 11 q^{10} - 5 q^{11} - 10 q^{12} - 15 q^{13} - 25 q^{14} - 23 q^{15} + 13 q^{16} - 26 q^{17} - 15 q^{18} - 27 q^{19}+ \cdots + 35 q^{99}+O(q^{100}) $ 27 * q - 9 * q^2 - q^3 + 25 * q^4 - 17 * q^5 - 10 * q^6 - 3 * q^7 - 21 * q^8 + 20 * q^9 - 11 * q^10 - 5 * q^11 - 10 * q^12 - 15 * q^13 - 25 * q^14 - 23 * q^15 + 13 * q^16 - 26 * q^17 - 15 * q^18 - 27 * q^19 - 38 * q^20 - 17 * q^21 + 4 * q^22 - 41 * q^23 - 27 * q^24 + 16 * q^25 - 23 * q^26 - 13 * q^27 + 48 * q^28 - 42 * q^29 + 46 * q^30 - 45 * q^31 - 31 * q^32 - 33 * q^33 + 49 * q^34 - 51 * q^35 - 25 * q^36 - 11 * q^38 - 47 * q^39 + 31 * q^40 + 7 * q^41 + 7 * q^42 - 11 * q^43 - 34 * q^44 - 24 * q^45 - 15 * q^46 - 21 * q^47 + 6 * q^48 + 6 * q^49 - 19 * q^50 - 37 * q^51 - 6 * q^52 - 7 * q^53 - 7 * q^54 - 35 * q^55 - 54 * q^56 + 3 * q^57 + 37 * q^58 - 74 * q^59 + 14 * q^60 - 26 * q^61 - 33 * q^62 - 50 * q^63 - 27 * q^64 - 6 * q^65 - 19 * q^66 + 2 * q^67 - 57 * q^68 - 28 * q^69 + 6 * q^70 + 5 * q^71 + 12 * q^72 - 23 * q^73 + 34 * q^75 - 27 * q^76 - 11 * q^77 + 17 * q^78 - 58 * q^79 - 36 * q^80 - 37 * q^81 + 34 * q^82 + 56 * q^83 - 40 * q^84 - 16 * q^85 - 45 * q^86 - 66 * q^87 + 33 * q^88 - 64 * q^89 - 34 * q^90 - 17 * q^91 - 34 * q^92 + 30 * q^93 + 5 * q^94 - 22 * q^95 - 19 * q^96 + 17 * q^97 + 6 * q^98 + 35 * 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 $

$2$	−0.581223	−0.410986	−0.205493	−	0.978659i	$-0.565880\pi$
$2$	−0.581223	−0.410986	−0.205493	+	0.978659i	$0.565880\pi$
$3$	−1.17520	−0.678504	−0.339252	−	0.940696i	$-0.610174\pi$
$3$	−1.17520	−0.678504	−0.339252	+	0.940696i	$0.610174\pi$
$4$	−1.66218	−0.831090
$5$	0.135564	0.0606262	0.0303131	−	0.999540i	$-0.490350\pi$
$5$	0.135564	0.0606262	0.0303131	+	0.999540i	$0.490350\pi$
$6$	0.683054	0.278856
$7$	−3.40406	−1.28661	−0.643307	−	0.765608i	$-0.722438\pi$
$7$	−3.40406	−1.28661	−0.643307	+	0.765608i	$0.722438\pi$
$8$	2.12854	0.752553
$9$	−1.61890	−0.539633
$10$	−0.0787931	−0.0249166
$11$	4.51469	1.36123	0.680615	−	0.732641i	$-0.261713\pi$
$11$	4.51469	1.36123	0.680615	+	0.732641i	$0.261713\pi$
$12$	1.95340	0.563898
$13$	1.75123	0.485703	0.242852	−	0.970063i	$-0.421917\pi$
$13$	1.75123	0.485703	0.242852	+	0.970063i	$0.421917\pi$
$14$	1.97852	0.528781
$15$	−0.159316	−0.0411351
$16$	2.08720	0.521801
$17$	−0.238914	−0.0579452	−0.0289726	−	0.999580i	$-0.509224\pi$
$17$	−0.238914	−0.0579452	−0.0289726	+	0.999580i	$0.509224\pi$
$18$	0.940940	0.221782
$19$	3.02781	0.694628	0.347314	−	0.937749i	$-0.387094\pi$
$19$	3.02781	0.694628	0.347314	+	0.937749i	$0.387094\pi$
$20$	−0.225332	−0.0503859
$21$	4.00046	0.872972
$22$	−2.62404	−0.559447
$23$	7.46588	1.55674	0.778372	−	0.627803i	$-0.216046\pi$
$23$	7.46588	1.55674	0.778372	+	0.627803i	$0.216046\pi$
$24$	−2.50147	−0.510610
$25$	−4.98162	−0.996324
$26$	−1.01785	−0.199617
$27$	5.42814	1.04465
$28$	5.65816	1.06929
$29$	−1.30151	−0.241685	−0.120843	−	0.992672i	$-0.538560\pi$
$29$	−1.30151	−0.241685	−0.120843	+	0.992672i	$0.538560\pi$
$30$	0.0925978	0.0169060
$31$	−8.68910	−1.56061	−0.780304	−	0.625400i	$-0.784936\pi$
$31$	−8.68910	−1.56061	−0.780304	+	0.625400i	$0.784936\pi$
$32$	−5.47021	−0.967006
$33$	−5.30568	−0.923600
$34$	0.138862	0.0238147
$35$	−0.461469	−0.0780025
$36$	2.69090	0.448483
$37$	0	0
$37$	0	0
$38$	−1.75983	−0.285482
$39$	−2.05805	−0.329551
$40$	0.288554	0.0456245
$41$	7.75460	1.21107	0.605533	−	0.795820i	$-0.292960\pi$
$41$	7.75460	1.21107	0.605533	+	0.795820i	$0.292960\pi$
$42$	−2.32516	−0.358780
$43$	−8.68570	−1.32456	−0.662279	−	0.749258i	$-0.730411\pi$
$43$	−8.68570	−1.32456	−0.662279	+	0.749258i	$0.730411\pi$
$44$	−7.50423	−1.13131
$45$	−0.219465	−0.0327159
$46$	−4.33934	−0.639801
$47$	−8.13636	−1.18681	−0.593405	−	0.804904i	$-0.702217\pi$
$47$	−8.13636	−1.18681	−0.593405	+	0.804904i	$0.702217\pi$
$48$	−2.45289	−0.354044
$49$	4.58762	0.655375
$50$	2.89543	0.409476
$51$	0.280773	0.0393161
$52$	−2.91086	−0.403663
$53$	11.3476	1.55871	0.779353	−	0.626585i	$-0.215548\pi$
$53$	11.3476	1.55871	0.779353	+	0.626585i	$0.215548\pi$
$54$	−3.15496	−0.429336
$55$	0.612031	0.0825263
$56$	−7.24568	−0.968245
$57$	−3.55829	−0.471307
$58$	0.756470	0.0993293
$59$	−6.24883	−0.813528	−0.406764	−	0.913533i	$-0.633343\pi$
$59$	−6.24883	−0.813528	−0.406764	+	0.913533i	$0.633343\pi$
$60$	0.264811	0.0341870
$61$	−13.3101	−1.70418	−0.852090	−	0.523395i	$-0.824665\pi$
$61$	−13.3101	−1.70418	−0.852090	+	0.523395i	$0.824665\pi$
$62$	5.05030	0.641389
$63$	5.51083	0.694299
$64$	−0.994996	−0.124375
$65$	0.237404	0.0294464
$66$	3.08378	0.379587
$67$	2.59709	0.317285	0.158642	−	0.987336i	$-0.449288\pi$
$67$	2.59709	0.317285	0.158642	+	0.987336i	$0.449288\pi$
$68$	0.397119	0.0481577
$69$	−8.77393	−1.05626
$70$	0.268216	0.0320580
$71$	3.60176	0.427451	0.213725	−	0.976894i	$-0.431440\pi$
$71$	3.60176	0.427451	0.213725	+	0.976894i	$0.431440\pi$
$72$	−3.44589	−0.406102
$73$	−6.65769	−0.779223	−0.389612	−	0.920979i	$-0.627391\pi$
$73$	−6.65769	−0.779223	−0.389612	+	0.920979i	$0.627391\pi$
$74$	0	0
$75$	5.85442	0.676010
$76$	−5.03277	−0.577298
$77$	−15.3683	−1.75138
$78$	1.19618	0.135441
$79$	−4.55092	−0.512018	−0.256009	−	0.966674i	$-0.582408\pi$
$79$	−4.55092	−0.512018	−0.256009	+	0.966674i	$0.582408\pi$
$80$	0.282950	0.0316348
$81$	−1.52247	−0.169164
$82$	−4.50715	−0.497732
$83$	−4.73268	−0.519480	−0.259740	−	0.965679i	$-0.583637\pi$
$83$	−4.73268	−0.519480	−0.259740	+	0.965679i	$0.583637\pi$
$84$	−6.64949	−0.725519
$85$	−0.0323883	−0.00351300
$86$	5.04833	0.544375
$87$	1.52954	0.163984
$88$	9.60971	1.02440
$89$	16.7206	1.77238	0.886192	−	0.463318i	$-0.153341\pi$
$89$	16.7206	1.77238	0.886192	+	0.463318i	$0.153341\pi$
$90$	0.127558	0.0134458
$91$	−5.96128	−0.624912
$92$	−12.4096	−1.29380
$93$	10.2115	1.05888
$94$	4.72903	0.487763
$95$	0.410463	0.0421126
$96$	6.42861	0.656117
$97$	−2.99941	−0.304544	−0.152272	−	0.988339i	$-0.548659\pi$
$97$	−2.99941	−0.304544	−0.152272	+	0.988339i	$0.548659\pi$
$98$	−2.66643	−0.269350
$99$	−7.30883	−0.734565
$100$	8.28035	0.828035
$101$	−11.7971	−1.17386	−0.586929	−	0.809638i	$-0.699663\pi$
$101$	−11.7971	−1.17386	−0.586929	+	0.809638i	$0.699663\pi$
$102$	−0.163191	−0.0161584
$103$	1.81129	0.178472	0.0892358	−	0.996011i	$-0.471558\pi$
$103$	1.81129	0.178472	0.0892358	+	0.996011i	$0.471558\pi$
$104$	3.72756	0.365518
$105$	0.542320	0.0529250
$106$	−6.59545	−0.640607
$107$	0.0937009	0.00905841	0.00452920	−	0.999990i	$-0.498558\pi$
$107$	0.0937009	0.00905841	0.00452920	+	0.999990i	$0.498558\pi$
$108$	−9.02255	−0.868195
$109$	−14.8590	−1.42323	−0.711616	−	0.702569i	$-0.752036\pi$
$109$	−14.8590	−1.42323	−0.711616	+	0.702569i	$0.752036\pi$
$110$	−0.355726	−0.0339172
$111$	0	0
$112$	−7.10497	−0.671356
$113$	4.47061	0.420559	0.210280	−	0.977641i	$-0.432563\pi$
$113$	4.47061	0.420559	0.210280	+	0.977641i	$0.432563\pi$
$114$	2.06816	0.193701
$115$	1.01211	0.0943795
$116$	2.16335	0.200862
$117$	−2.83506	−0.262101
$118$	3.63196	0.334349
$119$	0.813279	0.0745531
$120$	−0.339110	−0.0309564
$121$	9.38243	0.852948
$122$	7.73612	0.700395
$123$	−9.11323	−0.821713
$124$	14.4428	1.29701
$125$	−1.35315	−0.121030
$126$	−3.20302	−0.285347
$127$	−1.18304	−0.104978	−0.0524888	−	0.998622i	$-0.516715\pi$
$127$	−1.18304	−0.104978	−0.0524888	+	0.998622i	$0.516715\pi$
$128$	11.5187	1.01812
$129$	10.2075	0.898717
$130$	−0.137985	−0.0121021
$131$	−11.6279	−1.01593	−0.507967	−	0.861377i	$-0.669603\pi$
$131$	−11.6279	−1.01593	−0.507967	+	0.861377i	$0.669603\pi$
$132$	8.81899	0.767595
$133$	−10.3069	−0.893717
$134$	−1.50949	−0.130400
$135$	0.735863	0.0633330
$136$	−0.508539	−0.0436069
$137$	−9.18682	−0.784883	−0.392442	−	0.919777i	$-0.628369\pi$
$137$	−9.18682	−0.784883	−0.392442	+	0.919777i	$0.628369\pi$
$138$	5.09961	0.434107
$139$	−5.22192	−0.442917	−0.221459	−	0.975170i	$-0.571082\pi$
$139$	−5.22192	−0.442917	−0.221459	+	0.975170i	$0.571082\pi$
$140$	0.767045	0.0648271
$141$	9.56187	0.805255
$142$	−2.09343	−0.175676
$143$	7.90625	0.661154
$144$	−3.37897	−0.281581
$145$	−0.176439	−0.0146525
$146$	3.86960	0.320250
$147$	−5.39139	−0.444674
$148$	0	0
$149$	−3.97015	−0.325247	−0.162624	−	0.986688i	$-0.551996\pi$
$149$	−3.97015	−0.325247	−0.162624	+	0.986688i	$0.551996\pi$
$150$	−3.40272	−0.277831
$151$	−3.51533	−0.286074	−0.143037	−	0.989717i	$-0.545687\pi$
$151$	−3.51533	−0.286074	−0.143037	+	0.989717i	$0.545687\pi$
$152$	6.44482	0.522744
$153$	0.386778	0.0312691
$154$	8.93239	0.719793
$155$	−1.17793	−0.0946138
$156$	3.42085	0.273887
$157$	4.18187	0.333750	0.166875	−	0.985978i	$-0.446632\pi$
$157$	4.18187	0.333750	0.166875	+	0.985978i	$0.446632\pi$
$158$	2.64509	0.210432
$159$	−13.3357	−1.05759
$160$	−0.741566	−0.0586259
$161$	−25.4143	−2.00293
$162$	0.884896	0.0695240
$163$	−10.7692	−0.843513	−0.421756	−	0.906709i	$-0.638586\pi$
$163$	−10.7692	−0.843513	−0.421756	+	0.906709i	$0.638586\pi$
$164$	−12.8896	−1.00650
$165$	−0.719261	−0.0559944
$166$	2.75074	0.213499
$167$	−11.5114	−0.890783	−0.445391	−	0.895336i	$-0.646935\pi$
$167$	−11.5114	−0.890783	−0.445391	+	0.895336i	$0.646935\pi$
$168$	8.51515	0.656958
$169$	−9.93320	−0.764092
$170$	0.0188248	0.00144380
$171$	−4.90172	−0.374844
$172$	14.4372	1.10083
$173$	11.3357	0.861834	0.430917	−	0.902392i	$-0.358190\pi$
$173$	11.3357	0.861834	0.430917	+	0.902392i	$0.358190\pi$
$174$	−0.889005	−0.0673953
$175$	16.9577	1.28188
$176$	9.42308	0.710291
$177$	7.34364	0.551982
$178$	−9.71841	−0.728426
$179$	−14.6224	−1.09293	−0.546466	−	0.837481i	$-0.684027\pi$
$179$	−14.6224	−1.09293	−0.546466	+	0.837481i	$0.684027\pi$
$180$	0.364790	0.0271899
$181$	0.178071	0.0132359	0.00661796	−	0.999978i	$-0.497893\pi$
$181$	0.178071	0.0132359	0.00661796	+	0.999978i	$0.497893\pi$
$182$	3.46483	0.256831
$183$	15.6420	1.15629
$184$	15.8914	1.17153
$185$	0	0
$186$	−5.93513	−0.435185
$187$	−1.07862	−0.0788768
$188$	13.5241	0.986346
$189$	−18.4777	−1.34406
$190$	−0.238571	−0.0173077
$191$	5.99804	0.434003	0.217001	−	0.976171i	$-0.430372\pi$
$191$	5.99804	0.434003	0.217001	+	0.976171i	$0.430372\pi$
$192$	1.16932	0.0843886
$193$	17.1747	1.23626	0.618131	−	0.786075i	$-0.287890\pi$
$193$	17.1747	1.23626	0.618131	+	0.786075i	$0.287890\pi$
$194$	1.74332	0.125163
$195$	−0.278998	−0.0199795
$196$	−7.62546	−0.544676
$197$	−9.78679	−0.697280	−0.348640	−	0.937257i	$-0.613356\pi$
$197$	−9.78679	−0.697280	−0.348640	+	0.937257i	$0.613356\pi$
$198$	4.24805	0.301896
$199$	−23.8805	−1.69285	−0.846423	−	0.532512i	$-0.821248\pi$
$199$	−23.8805	−1.69285	−0.846423	+	0.532512i	$0.821248\pi$
$200$	−10.6036	−0.749787
$201$	−3.05210	−0.215279
$202$	6.85676	0.482440
$203$	4.43043	0.310956
$204$	−0.466695	−0.0326752
$205$	1.05125	0.0734223
$206$	−1.05276	−0.0733494
$207$	−12.0865	−0.840070
$208$	3.65517	0.253440
$209$	13.6696	0.945548
$210$	−0.315209	−0.0217515
$211$	0.762065	0.0524627	0.0262314	−	0.999656i	$-0.491649\pi$
$211$	0.762065	0.0524627	0.0262314	+	0.999656i	$0.491649\pi$
$212$	−18.8617	−1.29543
$213$	−4.23280	−0.290027
$214$	−0.0544611	−0.00372288
$215$	−1.17747	−0.0803029
$216$	11.5540	0.786152
$217$	29.5782	2.00790
$218$	8.63637	0.584929
$219$	7.82413	0.528706
$220$	−1.01731	−0.0685868
$221$	−0.418393	−0.0281442
$222$	0	0
$223$	10.8657	0.727619	0.363810	−	0.931473i	$-0.381476\pi$
$223$	10.8657	0.727619	0.363810	+	0.931473i	$0.381476\pi$
$224$	18.6209	1.24416
$225$	8.06474	0.537649
$226$	−2.59842	−0.172844
$227$	2.62128	0.173981	0.0869904	−	0.996209i	$-0.472275\pi$
$227$	2.62128	0.173981	0.0869904	+	0.996209i	$0.472275\pi$
$228$	5.91452	0.391699
$229$	15.6829	1.03636	0.518178	−	0.855273i	$-0.326610\pi$
$229$	15.6829	1.03636	0.518178	+	0.855273i	$0.326610\pi$
$230$	−0.588260	−0.0387887
$231$	18.0608	1.18832
$232$	−2.77033	−0.181881
$233$	19.7685	1.29507	0.647537	−	0.762034i	$-0.275799\pi$
$233$	19.7685	1.29507	0.647537	+	0.762034i	$0.275799\pi$
$234$	1.64780	0.107720
$235$	−1.10300	−0.0719518
$236$	10.3867	0.676115
$237$	5.34825	0.347406
$238$	−0.472696	−0.0306403
$239$	−15.1048	−0.977046	−0.488523	−	0.872551i	$-0.662464\pi$
$239$	−15.1048	−0.977046	−0.488523	+	0.872551i	$0.662464\pi$
$240$	−0.332524	−0.0214643
$241$	12.3019	0.792432	0.396216	−	0.918157i	$-0.370323\pi$
$241$	12.3019	0.792432	0.396216	+	0.918157i	$0.370323\pi$
$242$	−5.45328	−0.350550
$243$	−14.4952	−0.929868
$244$	22.1237	1.41633
$245$	0.621918	0.0397329
$246$	5.29682	0.337713
$247$	5.30239	0.337383
$248$	−18.4951	−1.17444
$249$	5.56186	0.352469
$250$	0.786483	0.0497415
$251$	−29.0194	−1.83169	−0.915844	−	0.401534i	$-0.868477\pi$
$251$	−29.0194	−1.83169	−0.915844	+	0.401534i	$0.868477\pi$
$252$	−9.15999	−0.577025
$253$	33.7062	2.11909
$254$	0.687607	0.0431443
$255$	0.0380628	0.00238358
$256$	−4.70496	−0.294060
$257$	16.4204	1.02428	0.512139	−	0.858903i	$-0.328853\pi$
$257$	16.4204	1.02428	0.512139	+	0.858903i	$0.328853\pi$
$258$	−5.93281	−0.369361
$259$	0	0
$260$	−0.394608	−0.0244726
$261$	2.10702	0.130421
$262$	6.75839	0.417535
$263$	−1.41775	−0.0874222	−0.0437111	−	0.999044i	$-0.513918\pi$
$263$	−1.41775	−0.0874222	−0.0437111	+	0.999044i	$0.513918\pi$
$264$	−11.2934	−0.695058
$265$	1.53832	0.0944985
$266$	5.99057	0.367306
$267$	−19.6501	−1.20257
$268$	−4.31683	−0.263692
$269$	20.3605	1.24140	0.620700	−	0.784048i	$-0.286849\pi$
$269$	20.3605	1.24140	0.620700	+	0.784048i	$0.286849\pi$
$270$	−0.427700	−0.0260290
$271$	−16.6609	−1.01208	−0.506040	−	0.862510i	$-0.668891\pi$
$271$	−16.6609	−1.01208	−0.506040	+	0.862510i	$0.668891\pi$
$272$	−0.498663	−0.0302359
$273$	7.00572	0.424005
$274$	5.33959	0.322576
$275$	−22.4905	−1.35623
$276$	14.5839	0.877845
$277$	−10.7457	−0.645649	−0.322825	−	0.946459i	$-0.604632\pi$
$277$	−10.7457	−0.645649	−0.322825	+	0.946459i	$0.604632\pi$
$278$	3.03510	0.182033
$279$	14.0668	0.842155
$280$	−0.982257	−0.0587011
$281$	29.4052	1.75417	0.877084	−	0.480338i	$-0.159486\pi$
$281$	29.4052	1.75417	0.877084	+	0.480338i	$0.159486\pi$
$282$	−5.55758	−0.330949
$283$	9.89331	0.588096	0.294048	−	0.955791i	$-0.404997\pi$
$283$	9.89331	0.588096	0.294048	+	0.955791i	$0.404997\pi$
$284$	−5.98678	−0.355250
$285$	−0.482378	−0.0285736
$286$	−4.59529	−0.271725
$287$	−26.3971	−1.55817
$288$	8.85572	0.521828
$289$	−16.9429	−0.996642
$290$	0.102550	0.00602196
$291$	3.52491	0.206634
$292$	11.0663	0.647605
$293$	−17.1879	−1.00413	−0.502064	−	0.864831i	$-0.667426\pi$
$293$	−17.1879	−1.00413	−0.502064	+	0.864831i	$0.667426\pi$
$294$	3.13360	0.182755
$295$	−0.847118	−0.0493211
$296$	0	0
$297$	24.5064	1.42200
$298$	2.30754	0.133672
$299$	13.0745	0.756116
$300$	−9.73110	−0.561825
$301$	29.5667	1.70419
$302$	2.04319	0.117572
$303$	13.8640	0.796467
$304$	6.31966	0.362457
$305$	−1.80437	−0.103318
$306$	−0.224804	−0.0128512
$307$	−9.76967	−0.557585	−0.278792	−	0.960351i	$-0.589934\pi$
$307$	−9.76967	−0.557585	−0.278792	+	0.960351i	$0.589934\pi$
$308$	25.5449	1.45555
$309$	−2.12863	−0.121094
$310$	0.684641	0.0388850
$311$	−1.58619	−0.0899447	−0.0449723	−	0.998988i	$-0.514320\pi$
$311$	−1.58619	−0.0899447	−0.0449723	+	0.998988i	$0.514320\pi$
$312$	−4.38064	−0.248005
$313$	−11.9721	−0.676701	−0.338350	−	0.941020i	$-0.609869\pi$
$313$	−11.9721	−0.676701	−0.338350	+	0.941020i	$0.609869\pi$
$314$	−2.43060	−0.137167
$315$	0.747072	0.0420927
$316$	7.56444	0.425533
$317$	−4.30654	−0.241879	−0.120940	−	0.992660i	$-0.538591\pi$
$317$	−4.30654	−0.241879	−0.120940	+	0.992660i	$0.538591\pi$
$318$	7.75100	0.434654
$319$	−5.87594	−0.328989
$320$	−0.134886	−0.00754036
$321$	−0.110118	−0.00614616
$322$	14.7714	0.823177
$323$	−0.723387	−0.0402504
$324$	2.53063	0.140590
$325$	−8.72396	−0.483918
$326$	6.25933	0.346672
$327$	17.4623	0.965668
$328$	16.5060	0.911391
$329$	27.6966	1.52697
$330$	0.418051	0.0230129
$331$	−8.08288	−0.444275	−0.222138	−	0.975015i	$-0.571303\pi$
$331$	−8.08288	−0.444275	−0.222138	+	0.975015i	$0.571303\pi$
$332$	7.86657	0.431734
$333$	0	0
$334$	6.69071	0.366100
$335$	0.352072	0.0192358
$336$	8.34978	0.455518
$337$	−2.16065	−0.117698	−0.0588491	−	0.998267i	$-0.518743\pi$
$337$	−2.16065	−0.117698	−0.0588491	+	0.998267i	$0.518743\pi$
$338$	5.77340	0.314032
$339$	−5.25387	−0.285351
$340$	0.0538351	0.00291962
$341$	−39.2286	−2.12435
$342$	2.84899	0.154056
$343$	8.21187	0.443399
$344$	−18.4879	−0.996800
$345$	−1.18943	−0.0640369
$346$	−6.58854	−0.354202
$347$	−31.7682	−1.70541	−0.852703	−	0.522396i	$-0.825038\pi$
$347$	−31.7682	−1.70541	−0.852703	+	0.522396i	$0.825038\pi$
$348$	−2.54238	−0.136286
$349$	17.6444	0.944481	0.472241	−	0.881470i	$-0.343445\pi$
$349$	17.6444	0.944481	0.472241	+	0.881470i	$0.343445\pi$
$350$	−9.85622	−0.526837
$351$	9.50591	0.507388
$352$	−24.6963	−1.31632
$353$	−24.8036	−1.32016	−0.660081	−	0.751195i	$-0.729478\pi$
$353$	−24.8036	−1.32016	−0.660081	+	0.751195i	$0.729478\pi$
$354$	−4.26829	−0.226857
$355$	0.488271	0.0259147
$356$	−27.7927	−1.47301
$357$	−0.955767	−0.0505846
$358$	8.49889	0.449180
$359$	−11.0967	−0.585664	−0.292832	−	0.956164i	$-0.594598\pi$
$359$	−11.0967	−0.585664	−0.292832	+	0.956164i	$0.594598\pi$
$360$	−0.467140	−0.0246205
$361$	−9.83236	−0.517493
$362$	−0.103499	−0.00543978
$363$	−11.0263	−0.578729
$364$	9.90873	0.519359
$365$	−0.902545	−0.0472414
$366$	−9.09151	−0.475221
$367$	22.4927	1.17411	0.587054	−	0.809548i	$-0.300288\pi$
$367$	22.4927	1.17411	0.587054	+	0.809548i	$0.300288\pi$
$368$	15.5828	0.812311
$369$	−12.5539	−0.653531
$370$	0	0
$371$	−38.6278	−2.00545
$372$	−16.9733	−0.880023
$373$	8.15249	0.422120	0.211060	−	0.977473i	$-0.432308\pi$
$373$	8.15249	0.422120	0.211060	+	0.977473i	$0.432308\pi$
$374$	0.626921	0.0324173
$375$	1.59023	0.0821190
$376$	−17.3186	−0.893137
$377$	−2.27925	−0.117387
$378$	10.7397	0.552389
$379$	27.6915	1.42242	0.711208	−	0.702981i	$-0.248148\pi$
$379$	27.6915	1.42242	0.711208	+	0.702981i	$0.248148\pi$
$380$	−0.682264	−0.0349994
$381$	1.39031	0.0712276
$382$	−3.48620	−0.178369
$383$	−30.0351	−1.53472	−0.767362	−	0.641214i	$-0.778431\pi$
$383$	−30.0351	−1.53472	−0.767362	+	0.641214i	$0.778431\pi$
$384$	−13.5369	−0.690800
$385$	−2.08339	−0.106179
$386$	−9.98233	−0.508087
$387$	14.0613	0.714775
$388$	4.98556	0.253103
$389$	−8.47319	−0.429608	−0.214804	−	0.976657i	$-0.568911\pi$
$389$	−8.47319	−0.429608	−0.214804	+	0.976657i	$0.568911\pi$
$390$	0.162160	0.00821129
$391$	−1.78371	−0.0902059
$392$	9.76495	0.493205
$393$	13.6651	0.689314
$394$	5.68830	0.286572
$395$	−0.616942	−0.0310417
$396$	12.1486	0.610489
$397$	5.93703	0.297971	0.148986	−	0.988839i	$-0.452399\pi$
$397$	5.93703	0.297971	0.148986	+	0.988839i	$0.452399\pi$
$398$	13.8799	0.695736
$399$	12.1126	0.606390
$400$	−10.3977	−0.519883
$401$	−8.27102	−0.413035	−0.206518	−	0.978443i	$-0.566213\pi$
$401$	−8.27102	−0.413035	−0.206518	+	0.978443i	$0.566213\pi$
$402$	1.77395	0.0884766
$403$	−15.2166	−0.757992
$404$	19.6090	0.975582
$405$	−0.206393	−0.0102558
$406$	−2.57507	−0.127798
$407$	0	0
$408$	0.597637	0.0295874
$409$	3.85500	0.190617	0.0953087	−	0.995448i	$-0.469616\pi$
$409$	3.85500	0.190617	0.0953087	+	0.995448i	$0.469616\pi$
$410$	−0.611009	−0.0301756
$411$	10.7964	0.532546
$412$	−3.01069	−0.148326
$413$	21.2714	1.04670
$414$	7.02495	0.345258
$415$	−0.641583	−0.0314941
$416$	−9.57959	−0.469678
$417$	6.13681	0.300521
$418$	−7.94510	−0.388607
$419$	−7.84489	−0.383248	−0.191624	−	0.981468i	$-0.561375\pi$
$419$	−7.84489	−0.383248	−0.191624	+	0.981468i	$0.561375\pi$
$420$	−0.901433	−0.0439855
$421$	8.68389	0.423227	0.211613	−	0.977353i	$-0.432128\pi$
$421$	8.68389	0.423227	0.211613	+	0.977353i	$0.432128\pi$
$422$	−0.442929	−0.0215615
$423$	13.1719	0.640441
$424$	24.1537	1.17301
$425$	1.19018	0.0577323
$426$	2.46020	0.119197
$427$	45.3083	2.19262
$428$	−0.155748	−0.00752835
$429$	−9.29145	−0.448595
$430$	0.684373	0.0330034
$431$	−10.7078	−0.515774	−0.257887	−	0.966175i	$-0.583026\pi$
$431$	−10.7078	−0.515774	−0.257887	+	0.966175i	$0.583026\pi$
$432$	11.3296	0.545098
$433$	−20.6300	−0.991415	−0.495707	−	0.868490i	$-0.665091\pi$
$433$	−20.6300	−0.991415	−0.495707	+	0.868490i	$0.665091\pi$
$434$	−17.1915	−0.825220
$435$	0.207352	0.00994175
$436$	24.6983	1.18283
$437$	22.6053	1.08136
$438$	−4.54756	−0.217291
$439$	−31.3169	−1.49467	−0.747337	−	0.664445i	$-0.768668\pi$
$439$	−31.3169	−1.49467	−0.747337	+	0.664445i	$0.768668\pi$
$440$	1.30273	0.0621054
$441$	−7.42690	−0.353662
$442$	0.243180	0.0115669
$443$	36.0868	1.71454	0.857268	−	0.514870i	$-0.172160\pi$
$443$	36.0868	1.71454	0.857268	+	0.514870i	$0.172160\pi$
$444$	0	0
$445$	2.26672	0.107453
$446$	−6.31537	−0.299042
$447$	4.66573	0.220681
$448$	3.38703	0.160022
$449$	−8.06309	−0.380521	−0.190260	−	0.981734i	$-0.560933\pi$
$449$	−8.06309	−0.380521	−0.190260	+	0.981734i	$0.560933\pi$
$450$	−4.68741	−0.220967
$451$	35.0096	1.64854
$452$	−7.43096	−0.349523
$453$	4.13123	0.194102
$454$	−1.52355	−0.0715037
$455$	−0.808138	−0.0378861
$456$	−7.57397	−0.354684
$457$	23.1181	1.08142	0.540709	−	0.841209i	$-0.318156\pi$
$457$	23.1181	1.08142	0.540709	+	0.841209i	$0.318156\pi$
$458$	−9.11527	−0.425928
$459$	−1.29686	−0.0605323
$460$	−1.68231	−0.0784379
$461$	1.36014	0.0633480	0.0316740	−	0.999498i	$-0.489916\pi$
$461$	1.36014	0.0633480	0.0316740	+	0.999498i	$0.489916\pi$
$462$	−10.4974	−0.488382
$463$	9.63998	0.448008	0.224004	−	0.974588i	$-0.428087\pi$
$463$	9.63998	0.448008	0.224004	+	0.974588i	$0.428087\pi$
$464$	−2.71653	−0.126112
$465$	1.38431	0.0641958
$466$	−11.4899	−0.532258
$467$	11.4646	0.530519	0.265260	−	0.964177i	$-0.414542\pi$
$467$	11.4646	0.530519	0.265260	+	0.964177i	$0.414542\pi$
$468$	4.71238	0.217830
$469$	−8.84064	−0.408223
$470$	0.641089	0.0295712
$471$	−4.91455	−0.226450
$472$	−13.3009	−0.612223
$473$	−39.2133	−1.80303
$474$	−3.10852	−0.142779
$475$	−15.0834	−0.692074
$476$	−1.35182	−0.0619604
$477$	−18.3705	−0.841129
$478$	8.77923	0.401552
$479$	−19.9753	−0.912695	−0.456348	−	0.889802i	$-0.650843\pi$
$479$	−19.9753	−0.912695	−0.456348	+	0.889802i	$0.650843\pi$
$480$	0.871490	0.0397779
$481$	0	0
$482$	−7.15012	−0.325679
$483$	29.8670	1.35899
$484$	−15.5953	−0.708877
$485$	−0.406613	−0.0184633
$486$	8.42494	0.382163
$487$	−3.19056	−0.144578	−0.0722890	−	0.997384i	$-0.523030\pi$
$487$	−3.19056	−0.144578	−0.0722890	+	0.997384i	$0.523030\pi$
$488$	−28.3311	−1.28249
$489$	12.6561	0.572327
$490$	−0.361473	−0.0163297
$491$	5.42371	0.244769	0.122384	−	0.992483i	$-0.460946\pi$
$491$	5.42371	0.244769	0.122384	+	0.992483i	$0.460946\pi$
$492$	15.1478	0.682917
$493$	0.310950	0.0140045
$494$	−3.08187	−0.138660
$495$	−0.990816	−0.0445339
$496$	−18.1359	−0.814327
$497$	−12.2606	−0.549964
$498$	−3.23268	−0.144860
$499$	1.08975	0.0487838	0.0243919	−	0.999702i	$-0.492235\pi$
$499$	1.08975	0.0487838	0.0243919	+	0.999702i	$0.492235\pi$
$500$	2.24918	0.100587
$501$	13.5283	0.604399
$502$	16.8667	0.752799
$503$	−0.148635	−0.00662731	−0.00331366	−	0.999995i	$-0.501055\pi$
$503$	−0.148635	−0.00662731	−0.00331366	+	0.999995i	$0.501055\pi$
$504$	11.7300	0.522497
$505$	−1.59927	−0.0711666
$506$	−19.5908	−0.870916
$507$	11.6735	0.518439
$508$	1.96642	0.0872458
$509$	−12.8435	−0.569279	−0.284639	−	0.958635i	$-0.591874\pi$
$509$	−12.8435	−0.569279	−0.284639	+	0.958635i	$0.591874\pi$
$510$	−0.0221229	−0.000979621 0
$511$	22.6632	1.00256
$512$	−20.3029	−0.897268
$513$	16.4354	0.725640
$514$	−9.54392	−0.420964
$515$	0.245546	0.0108201
$516$	−16.9666	−0.746915
$517$	−36.7331	−1.61552
$518$	0	0
$519$	−13.3217	−0.584758
$520$	0.505325	0.0221599
$521$	−33.9439	−1.48711	−0.743555	−	0.668675i	$-0.766862\pi$
$521$	−33.9439	−1.48711	−0.743555	+	0.668675i	$0.766862\pi$
$522$	−1.22465	−0.0536014
$523$	−21.5585	−0.942688	−0.471344	−	0.881949i	$-0.656231\pi$
$523$	−21.5585	−0.942688	−0.471344	+	0.881949i	$0.656231\pi$
$524$	19.3276	0.844332
$525$	−19.9288	−0.869764
$526$	0.824028	0.0359293
$527$	2.07595	0.0904298
$528$	−11.0740	−0.481935
$529$	32.7394	1.42345
$530$	−0.894109	−0.0388376
$531$	10.1162	0.439006
$532$	17.1318	0.742760
$533$	13.5801	0.588219
$534$	11.4211	0.494240
$535$	0.0127025	0.000549177 0
$536$	5.52801	0.238774
$537$	17.1843	0.741559
$538$	−11.8340	−0.510198
$539$	20.7117	0.892116
$540$	−1.22314	−0.0526354
$541$	−30.5914	−1.31523	−0.657614	−	0.753355i	$-0.728434\pi$
$541$	−30.5914	−1.31523	−0.657614	+	0.753355i	$0.728434\pi$
$542$	9.68371	0.415951
$543$	−0.209270	−0.00898062
$544$	1.30691	0.0560334
$545$	−2.01435	−0.0862852
$546$	−4.07188	−0.174260
$547$	27.0949	1.15849	0.579246	−	0.815153i	$-0.303347\pi$
$547$	27.0949	1.15849	0.579246	+	0.815153i	$0.303347\pi$
$548$	15.2702	0.652309
$549$	21.5477	0.919632
$550$	13.0720	0.557391
$551$	−3.94074	−0.167881
$552$	−18.6757	−0.794890
$553$	15.4916	0.658769
$554$	6.24567	0.265353
$555$	0	0
$556$	8.67977	0.368104
$557$	−35.3428	−1.49752	−0.748762	−	0.662839i	$-0.769351\pi$
$557$	−35.3428	−1.49752	−0.748762	+	0.662839i	$0.769351\pi$
$558$	−8.17592	−0.346114
$559$	−15.2106	−0.643342
$560$	−0.963180	−0.0407018
$561$	1.26760	0.0535182
$562$	−17.0910	−0.720939
$563$	21.0910	0.888881	0.444441	−	0.895808i	$-0.353403\pi$
$563$	21.0910	0.888881	0.444441	+	0.895808i	$0.353403\pi$
$564$	−15.8936	−0.669239
$565$	0.606055	0.0254969
$566$	−5.75021	−0.241700
$567$	5.18259	0.217648
$568$	7.66650	0.321679
$569$	42.8589	1.79674	0.898370	−	0.439239i	$-0.144752\pi$
$569$	42.8589	1.79674	0.898370	+	0.439239i	$0.144752\pi$
$570$	0.280369	0.0117434
$571$	18.5662	0.776969	0.388485	−	0.921455i	$-0.372999\pi$
$571$	18.5662	0.776969	0.388485	+	0.921455i	$0.372999\pi$
$572$	−13.1416	−0.549479
$573$	−7.04891	−0.294473
$574$	15.3426	0.640388
$575$	−37.1922	−1.55102
$576$	1.61080	0.0671166
$577$	−26.6541	−1.10962	−0.554812	−	0.831976i	$-0.687210\pi$
$577$	−26.6541	−1.10962	−0.554812	+	0.831976i	$0.687210\pi$
$578$	9.84761	0.409606
$579$	−20.1838	−0.838809
$580$	0.293273	0.0121775
$581$	16.1103	0.668369
$582$	−2.04876	−0.0849238
$583$	51.2307	2.12176
$584$	−14.1712	−0.586407
$585$	−0.384333	−0.0158902
$586$	9.98999	0.412683
$587$	−35.2047	−1.45305	−0.726527	−	0.687138i	$-0.758867\pi$
$587$	−35.2047	−1.45305	−0.726527	+	0.687138i	$0.758867\pi$
$588$	8.96146	0.369564
$589$	−26.3089	−1.08404
$590$	0.492364	0.0202703
$591$	11.5015	0.473107
$592$	0	0
$593$	35.8341	1.47153	0.735764	−	0.677238i	$-0.236823\pi$
$593$	35.8341	1.47153	0.735764	+	0.677238i	$0.236823\pi$
$594$	−14.2437	−0.584425
$595$	0.110252	0.00451987
$596$	6.59910	0.270310
$597$	28.0645	1.14860
$598$	−7.59917	−0.310753
$599$	−22.8314	−0.932865	−0.466432	−	0.884557i	$-0.654461\pi$
$599$	−22.8314	−0.932865	−0.466432	+	0.884557i	$0.654461\pi$
$600$	12.4614	0.508733
$601$	−44.8434	−1.82920	−0.914600	−	0.404359i	$-0.867495\pi$
$601$	−44.8434	−1.82920	−0.914600	+	0.404359i	$0.867495\pi$
$602$	−17.1848	−0.700401
$603$	−4.20442	−0.171217
$604$	5.84312	0.237753
$605$	1.27192	0.0517110
$606$	−8.05808	−0.327337
$607$	−3.67023	−0.148970	−0.0744849	−	0.997222i	$-0.523731\pi$
$607$	−3.67023	−0.148970	−0.0744849	+	0.997222i	$0.523731\pi$
$608$	−16.5628	−0.671709
$609$	−5.20666	−0.210984
$610$	1.04874	0.0424623
$611$	−14.2486	−0.576437
$612$	−0.642895	−0.0259875
$613$	−11.5735	−0.467448	−0.233724	−	0.972303i	$-0.575091\pi$
$613$	−11.5735	−0.467448	−0.233724	+	0.972303i	$0.575091\pi$
$614$	5.67836	0.229160
$615$	−1.23543	−0.0498173
$616$	−32.7120	−1.31801
$617$	17.3327	0.697790	0.348895	−	0.937162i	$-0.386557\pi$
$617$	17.3327	0.697790	0.348895	+	0.937162i	$0.386557\pi$
$618$	1.23721	0.0497678
$619$	20.0946	0.807670	0.403835	−	0.914832i	$-0.367677\pi$
$619$	20.0946	0.807670	0.403835	+	0.914832i	$0.367677\pi$
$620$	1.95794	0.0786326
$621$	40.5259	1.62625
$622$	0.921930	0.0369660
$623$	−56.9181	−2.28037
$624$	−4.29557	−0.171960
$625$	24.7247	0.988987
$626$	6.95843	0.278115
$627$	−16.0646	−0.641558
$628$	−6.95103	−0.277376
$629$	0	0
$630$	−0.434215	−0.0172995
$631$	−7.93504	−0.315889	−0.157945	−	0.987448i	$-0.550487\pi$
$631$	−7.93504	−0.315889	−0.157945	+	0.987448i	$0.550487\pi$
$632$	−9.68681	−0.385321
$633$	−0.895581	−0.0355961
$634$	2.50306	0.0994090
$635$	−0.160378	−0.00636439
$636$	22.1663	0.878951
$637$	8.03398	0.318318
$638$	3.41523	0.135210
$639$	−5.83089	−0.230666
$640$	1.56153	0.0617249
$641$	25.2280	0.996446	0.498223	−	0.867049i	$-0.333986\pi$
$641$	25.2280	0.996446	0.498223	+	0.867049i	$0.333986\pi$
$642$	0.0640028	0.00252599
$643$	13.6761	0.539334	0.269667	−	0.962954i	$-0.413086\pi$
$643$	13.6761	0.539334	0.269667	+	0.962954i	$0.413086\pi$
$644$	42.2432	1.66461
$645$	1.38377	0.0544858
$646$	0.420449	0.0165423
$647$	4.72308	0.185683	0.0928417	−	0.995681i	$-0.470405\pi$
$647$	4.72308	0.185683	0.0928417	+	0.995681i	$0.470405\pi$
$648$	−3.24065	−0.127305
$649$	−28.2115	−1.10740
$650$	5.07056	0.198884
$651$	−34.7604	−1.36237
$652$	17.9004	0.701035
$653$	20.2454	0.792263	0.396132	−	0.918194i	$-0.370352\pi$
$653$	20.2454	0.792263	0.396132	+	0.918194i	$0.370352\pi$
$654$	−10.1495	−0.396877
$655$	−1.57633	−0.0615922
$656$	16.1854	0.631935
$657$	10.7781	0.420494
$658$	−16.0979	−0.627562
$659$	−18.1715	−0.707862	−0.353931	−	0.935271i	$-0.615155\pi$
$659$	−18.1715	−0.707862	−0.353931	+	0.935271i	$0.615155\pi$
$660$	1.19554	0.0465364
$661$	−8.45867	−0.329004	−0.164502	−	0.986377i	$-0.552602\pi$
$661$	−8.45867	−0.329004	−0.164502	+	0.986377i	$0.552602\pi$
$662$	4.69795	0.182591
$663$	0.491697	0.0190959
$664$	−10.0737	−0.390936
$665$	−1.39724	−0.0541827
$666$	0	0
$667$	−9.71696	−0.376242
$668$	19.1341	0.740321
$669$	−12.7694	−0.493692
$670$	−0.204632	−0.00790564
$671$	−60.0909	−2.31978
$672$	−21.8834	−0.844170
$673$	−47.9374	−1.84785	−0.923925	−	0.382574i	$-0.875038\pi$
$673$	−47.9374	−1.84785	−0.923925	+	0.382574i	$0.875038\pi$
$674$	1.25582	0.0483723
$675$	−27.0410	−1.04081
$676$	16.5108	0.635030
$677$	7.63281	0.293353	0.146676	−	0.989185i	$-0.453142\pi$
$677$	7.63281	0.293353	0.146676	+	0.989185i	$0.453142\pi$
$678$	3.05367	0.117275
$679$	10.2102	0.391830
$680$	−0.0689398	−0.00264372
$681$	−3.08054	−0.118047
$682$	22.8005	0.873078
$683$	−39.6975	−1.51898	−0.759492	−	0.650516i	$-0.774552\pi$
$683$	−39.6975	−1.51898	−0.759492	+	0.650516i	$0.774552\pi$
$684$	8.14754	0.311529
$685$	−1.24541	−0.0475845
$686$	−4.77292	−0.182231
$687$	−18.4306	−0.703172
$688$	−18.1288	−0.691155
$689$	19.8722	0.757069
$690$	0.691325	0.0263183
$691$	−5.98406	−0.227644	−0.113822	−	0.993501i	$-0.536309\pi$
$691$	−5.98406	−0.227644	−0.113822	+	0.993501i	$0.536309\pi$
$692$	−18.8419	−0.716262
$693$	24.8797	0.945101
$694$	18.4644	0.700899
$695$	−0.707906	−0.0268524
$696$	3.25570	0.123407
$697$	−1.85269	−0.0701755
$698$	−10.2553	−0.388169
$699$	−23.2319	−0.878713
$700$	−28.1868	−1.06536
$701$	−21.2280	−0.801769	−0.400884	−	0.916129i	$-0.631297\pi$
$701$	−21.2280	−0.801769	−0.400884	+	0.916129i	$0.631297\pi$
$702$	−5.52505	−0.208530
$703$	0	0
$704$	−4.49210	−0.169302
$705$	1.29625	0.0488196
$706$	14.4164	0.542568
$707$	40.1581	1.51030
$708$	−12.2065	−0.458747
$709$	−24.9373	−0.936541	−0.468270	−	0.883585i	$-0.655123\pi$
$709$	−24.9373	−0.936541	−0.468270	+	0.883585i	$0.655123\pi$
$710$	−0.283794	−0.0106506
$711$	7.36747	0.276302
$712$	35.5906	1.33381
$713$	−64.8718	−2.42947
$714$	0.555514	0.0207896
$715$	1.07181	0.0400833
$716$	24.3051	0.908325
$717$	17.7512	0.662929
$718$	6.44968	0.240700
$719$	2.23576	0.0833797	0.0416898	−	0.999131i	$-0.486726\pi$
$719$	2.23576	0.0833797	0.0416898	+	0.999131i	$0.486726\pi$
$720$	−0.458068	−0.0170712
$721$	−6.16573	−0.229624
$722$	5.71479	0.212682
$723$	−14.4572	−0.537668
$724$	−0.295986	−0.0110002
$725$	6.48365	0.240797
$726$	6.40871	0.237850
$727$	−36.4649	−1.35241	−0.676204	−	0.736715i	$-0.736376\pi$
$727$	−36.4649	−1.35241	−0.676204	+	0.736715i	$0.736376\pi$
$728$	−12.6888	−0.470280
$729$	21.6022	0.800083
$730$	0.524579	0.0194156
$731$	2.07514	0.0767518
$732$	−25.9999	−0.960984
$733$	2.09671	0.0774438	0.0387219	−	0.999250i	$-0.487671\pi$
$733$	2.09671	0.0774438	0.0387219	+	0.999250i	$0.487671\pi$
$734$	−13.0733	−0.482543
$735$	−0.730880	−0.0269589
$736$	−40.8400	−1.50538
$737$	11.7250	0.431897
$738$	7.29662	0.268592
$739$	−28.5648	−1.05077	−0.525386	−	0.850864i	$-0.676079\pi$
$739$	−28.5648	−1.05077	−0.525386	+	0.850864i	$0.676079\pi$
$740$	0	0
$741$	−6.23138	−0.228915
$742$	22.4513	0.824214
$743$	14.6071	0.535882	0.267941	−	0.963435i	$-0.413657\pi$
$743$	14.6071	0.535882	0.267941	+	0.963435i	$0.413657\pi$
$744$	21.7355	0.796862
$745$	−0.538211	−0.0197185
$746$	−4.73841	−0.173485
$747$	7.66173	0.280328
$748$	1.79287	0.0655537
$749$	−0.318963	−0.0116547
$750$	−0.924277	−0.0337498
$751$	−54.7210	−1.99680	−0.998399	−	0.0565662i	$-0.981985\pi$
$751$	−54.7210	−1.99680	−0.998399	+	0.0565662i	$0.981985\pi$
$752$	−16.9822	−0.619279
$753$	34.1037	1.24281
$754$	1.32475	0.0482446
$755$	−0.476554	−0.0173436
$756$	30.7133	1.11703
$757$	19.2216	0.698622	0.349311	−	0.937007i	$-0.386416\pi$
$757$	19.2216	0.698622	0.349311	+	0.937007i	$0.386416\pi$
$758$	−16.0949	−0.584594
$759$	−39.6116	−1.43781
$760$	0.873688	0.0316920
$761$	40.4818	1.46747	0.733733	−	0.679438i	$-0.237777\pi$
$761$	40.4818	1.46747	0.733733	+	0.679438i	$0.237777\pi$
$762$	−0.808078	−0.0292736
$763$	50.5809	1.83115
$764$	−9.96982	−0.360696
$765$	0.0524333	0.00189573
$766$	17.4571	0.630751
$767$	−10.9431	−0.395133
$768$	5.52928	0.199521
$769$	12.6633	0.456650	0.228325	−	0.973585i	$-0.426675\pi$
$769$	12.6633	0.456650	0.228325	+	0.973585i	$0.426675\pi$
$770$	1.21091	0.0436383
$771$	−19.2973	−0.694976
$772$	−28.5475	−1.02745
$773$	28.5896	1.02830	0.514148	−	0.857701i	$-0.328108\pi$
$773$	28.5896	1.02830	0.514148	+	0.857701i	$0.328108\pi$
$774$	−8.17273	−0.293763
$775$	43.2858	1.55487
$776$	−6.38437	−0.229185
$777$	0	0
$778$	4.92481	0.176563
$779$	23.4795	0.841240
$780$	0.463745	0.0166047
$781$	16.2608	0.581859
$782$	1.03673	0.0370734
$783$	−7.06481	−0.252476
$784$	9.57531	0.341975
$785$	0.566913	0.0202340
$786$	−7.94248	−0.283299
$787$	−46.2315	−1.64797	−0.823987	−	0.566608i	$-0.808255\pi$
$787$	−46.2315	−1.64797	−0.823987	+	0.566608i	$0.808255\pi$
$788$	16.2674	0.579502
$789$	1.66614	0.0593163
$790$	0.358581	0.0127577
$791$	−15.2182	−0.541098
$792$	−15.5571	−0.552799
$793$	−23.3090	−0.827726
$794$	−3.45074	−0.122462
$795$	−1.80784	−0.0641176
$796$	39.6937	1.40691
$797$	27.6489	0.979373	0.489687	−	0.871899i	$-0.337111\pi$
$797$	27.6489	0.979373	0.489687	+	0.871899i	$0.337111\pi$
$798$	−7.04014	−0.249218
$799$	1.94389	0.0687700
$800$	27.2505	0.963452
$801$	−27.0690	−0.956436
$802$	4.80731	0.169752
$803$	−30.0574	−1.06070
$804$	5.07315	0.178916
$805$	−3.44528	−0.121430
$806$	8.84423	0.311525
$807$	−23.9277	−0.842294
$808$	−25.1107	−0.883391
$809$	−17.0004	−0.597704	−0.298852	−	0.954299i	$-0.596604\pi$
$809$	−17.0004	−0.597704	−0.298852	+	0.954299i	$0.596604\pi$
$810$	0.119960	0.00421498
$811$	3.43649	0.120671	0.0603357	−	0.998178i	$-0.480783\pi$
$811$	3.43649	0.120671	0.0603357	+	0.998178i	$0.480783\pi$
$812$	−7.36418	−0.258432
$813$	19.5800	0.686699
$814$	0	0
$815$	−1.45993	−0.0511390
$816$	0.586030	0.0205152
$817$	−26.2987	−0.920074
$818$	−2.24061	−0.0783411
$819$	9.65071	0.337223
$820$	−1.74736	−0.0610206
$821$	−16.5396	−0.577236	−0.288618	−	0.957444i	$-0.593196\pi$
$821$	−16.5396	−0.577236	−0.288618	+	0.957444i	$0.593196\pi$
$822$	−6.27510	−0.218869
$823$	−30.0080	−1.04601	−0.523006	−	0.852329i	$-0.675189\pi$
$823$	−30.0080	−1.04601	−0.523006	+	0.852329i	$0.675189\pi$
$824$	3.85540	0.134309
$825$	26.4309	0.920205
$826$	−12.3634	−0.430178
$827$	−0.382375	−0.0132965	−0.00664824	−	0.999978i	$-0.502116\pi$
$827$	−0.382375	−0.0132965	−0.00664824	+	0.999978i	$0.502116\pi$
$828$	20.0900	0.698174
$829$	−0.605322	−0.0210237	−0.0105119	−	0.999945i	$-0.503346\pi$
$829$	−0.605322	−0.0210237	−0.0105119	+	0.999945i	$0.503346\pi$
$830$	0.372903	0.0129436
$831$	12.6284	0.438075
$832$	−1.74247	−0.0604091
$833$	−1.09605	−0.0379759
$834$	−3.56685	−0.123510
$835$	−1.56054	−0.0540048
$836$	−22.7214	−0.785836
$837$	−47.1657	−1.63028
$838$	4.55962	0.157510
$839$	29.0031	1.00130	0.500650	−	0.865650i	$-0.333095\pi$
$839$	29.0031	1.00130	0.500650	+	0.865650i	$0.333095\pi$
$840$	1.15435	0.0398289
$841$	−27.3061	−0.941588
$842$	−5.04727	−0.173940
$843$	−34.5571	−1.19021
$844$	−1.26669	−0.0436012
$845$	−1.34659	−0.0463240
$846$	−7.65583	−0.263213
$847$	−31.9384	−1.09742
$848$	23.6847	0.813335
$849$	−11.6266	−0.399025
$850$	−0.691760	−0.0237272
$851$	0	0
$852$	7.03568	0.241038
$853$	−56.5657	−1.93677	−0.968387	−	0.249453i	$-0.919749\pi$
$853$	−56.5657	−1.93677	−0.968387	+	0.249453i	$0.919749\pi$
$854$	−26.3342	−0.901138
$855$	−0.664498	−0.0227254
$856$	0.199446	0.00681693
$857$	−0.990424	−0.0338322	−0.0169161	−	0.999857i	$-0.505385\pi$
$857$	−0.990424	−0.0338322	−0.0169161	+	0.999857i	$0.505385\pi$
$858$	5.40040	0.184367
$859$	4.84248	0.165223	0.0826116	−	0.996582i	$-0.473674\pi$
$859$	4.84248	0.165223	0.0826116	+	0.996582i	$0.473674\pi$
$860$	1.95717	0.0667390
$861$	31.0220	1.05723
$862$	6.22359	0.211976
$863$	30.9388	1.05317	0.526585	−	0.850122i	$-0.323472\pi$
$863$	30.9388	1.05317	0.526585	+	0.850122i	$0.323472\pi$
$864$	−29.6931	−1.01018
$865$	1.53671	0.0522497
$866$	11.9906	0.407458
$867$	19.9114	0.676226
$868$	−49.1643	−1.66875
$869$	−20.5460	−0.696975
$870$	−0.120517	−0.00408592
$871$	4.54809	0.154106
$872$	−31.6280	−1.07106
$873$	4.85574	0.164342
$874$	−13.1387	−0.444423
$875$	4.60621	0.155718
$876$	−13.0051	−0.439402
$877$	54.8462	1.85203	0.926013	−	0.377492i	$-0.123213\pi$
$877$	54.8462	1.85203	0.926013	+	0.377492i	$0.123213\pi$
$878$	18.2021	0.614291
$879$	20.1993	0.681304
$880$	1.27743	0.0430623
$881$	39.1318	1.31838	0.659192	−	0.751974i	$-0.270898\pi$
$881$	39.1318	1.31838	0.659192	+	0.751974i	$0.270898\pi$
$882$	4.31668	0.145350
$883$	16.6988	0.561959	0.280979	−	0.959714i	$-0.409341\pi$
$883$	16.6988	0.561959	0.280979	+	0.959714i	$0.409341\pi$
$884$	0.695445	0.0233904
$885$	0.995536	0.0334646
$886$	−20.9745	−0.704651
$887$	22.4696	0.754457	0.377228	−	0.926120i	$-0.376877\pi$
$887$	22.4696	0.754457	0.377228	+	0.926120i	$0.376877\pi$
$888$	0	0
$889$	4.02713	0.135066
$890$	−1.31747	−0.0441617
$891$	−6.87350	−0.230271
$892$	−18.0607	−0.604717
$893$	−24.6354	−0.824391
$894$	−2.71183	−0.0906971
$895$	−1.98228	−0.0662604
$896$	−39.2105	−1.30993
$897$	−15.3651	−0.513027
$898$	4.68645	0.156389
$899$	11.3090	0.377176
$900$	−13.4051	−0.446835
$901$	−2.71109	−0.0903196
$902$	−20.3484	−0.677527
$903$	−34.7468	−1.15630
$904$	9.51588	0.316493
$905$	0.0241401	0.000802443 0
$906$	−2.40116	−0.0797733
$907$	1.57002	0.0521316	0.0260658	−	0.999660i	$-0.491702\pi$
$907$	1.57002	0.0521316	0.0260658	+	0.999660i	$0.491702\pi$
$908$	−4.35705	−0.144594
$909$	19.0984	0.633452
$910$	0.469708	0.0155707
$911$	−23.1298	−0.766324	−0.383162	−	0.923681i	$-0.625165\pi$
$911$	−23.1298	−0.766324	−0.383162	+	0.923681i	$0.625165\pi$
$912$	−7.42688	−0.245929
$913$	−21.3666	−0.707131
$914$	−13.4368	−0.444448
$915$	2.12050	0.0701017
$916$	−26.0678	−0.861306
$917$	39.5820	1.30711
$918$	0.753765	0.0248779
$919$	31.8709	1.05132	0.525662	−	0.850693i	$-0.323818\pi$
$919$	31.8709	1.05132	0.525662	+	0.850693i	$0.323818\pi$
$920$	2.15431	0.0710256
$921$	11.4813	0.378323
$922$	−0.790544	−0.0260352
$923$	6.30751	0.207614
$924$	−30.0204	−0.987598
$925$	0	0
$926$	−5.60297	−0.184125
$927$	−2.93229	−0.0963091
$928$	7.11956	0.233711
$929$	−28.0517	−0.920346	−0.460173	−	0.887829i	$-0.652213\pi$
$929$	−28.0517	−0.920346	−0.460173	+	0.887829i	$0.652213\pi$
$930$	−0.804592	−0.0263836
$931$	13.8905	0.455241
$932$	−32.8587	−1.07632
$933$	1.86410	0.0610278
$934$	−6.66350	−0.218036
$935$	−0.146223	−0.00478200
$936$	−6.03454	−0.197245
$937$	38.6097	1.26133	0.630663	−	0.776057i	$-0.282783\pi$
$937$	38.6097	1.26133	0.630663	+	0.776057i	$0.282783\pi$
$938$	5.13838	0.167774
$939$	14.0696	0.459144
$940$	1.83339	0.0597984
$941$	12.2716	0.400043	0.200021	−	0.979792i	$-0.435899\pi$
$941$	12.2716	0.400043	0.200021	+	0.979792i	$0.435899\pi$
$942$	2.85645	0.0930681
$943$	57.8950	1.88532
$944$	−13.0426	−0.424500
$945$	−2.50492	−0.0814851
$946$	22.7916	0.741020
$947$	33.3992	1.08533	0.542665	−	0.839949i	$-0.317415\pi$
$947$	33.3992	1.08533	0.542665	+	0.839949i	$0.317415\pi$
$948$	−8.88975	−0.288726
$949$	−11.6591	−0.378471
$950$	8.76682	0.284433
$951$	5.06105	0.164116
$952$	1.73110	0.0561052
$953$	−1.71781	−0.0556454	−0.0278227	−	0.999613i	$-0.508857\pi$
$953$	−1.71781	−0.0556454	−0.0278227	+	0.999613i	$0.508857\pi$
$954$	10.6774	0.345693
$955$	0.813120	0.0263120
$956$	25.1068	0.812013
$957$	6.90542	0.223220
$958$	11.6101	0.375105
$959$	31.2725	1.00984
$960$	0.158518	0.00511616
$961$	44.5004	1.43550
$962$	0	0
$963$	−0.151692	−0.00488821
$964$	−20.4479	−0.658583
$965$	2.32828	0.0749499
$966$	−17.3594	−0.558528
$967$	23.1071	0.743076	0.371538	−	0.928418i	$-0.378831\pi$
$967$	23.1071	0.743076	0.371538	+	0.928418i	$0.378831\pi$
$968$	19.9709	0.641889
$969$	0.850127	0.0273100
$970$	0.236333	0.00758818
$971$	37.9148	1.21674	0.608372	−	0.793652i	$-0.291823\pi$
$971$	37.9148	1.21674	0.608372	+	0.793652i	$0.291823\pi$
$972$	24.0937	0.772804
$973$	17.7757	0.569864
$974$	1.85442	0.0594196
$975$	10.2524	0.328340
$976$	−27.7808	−0.889243
$977$	35.6149	1.13942	0.569711	−	0.821845i	$-0.307055\pi$
$977$	35.6149	1.13942	0.569711	+	0.821845i	$0.307055\pi$
$978$	−7.35598	−0.235218
$979$	75.4885	2.41262
$980$	−1.03374	−0.0330216
$981$	24.0552	0.768023
$982$	−3.15238	−0.100597
$983$	12.1991	0.389092	0.194546	−	0.980893i	$-0.437677\pi$
$983$	12.1991	0.389092	0.194546	+	0.980893i	$0.437677\pi$
$984$	−19.3979	−0.618382
$985$	−1.32674	−0.0422734
$986$	−0.180731	−0.00575566
$987$	−32.5492	−1.03605
$988$	−8.81352	−0.280396
$989$	−64.8465	−2.06200
$990$	0.575885	0.0183028
$991$	12.0888	0.384013	0.192007	−	0.981394i	$-0.438500\pi$
$991$	12.0888	0.384013	0.192007	+	0.981394i	$0.438500\pi$
$992$	47.5312	1.50912
$993$	9.49902	0.301442
$994$	7.12615	0.226028
$995$	−3.23735	−0.102631
$996$	−9.24482	−0.292933
$997$	−45.9879	−1.45645	−0.728226	−	0.685338i	$-0.759655\pi$
$997$	−45.9879	−1.45645	−0.728226	+	0.685338i	$0.759655\pi$
$998$	−0.633386	−0.0200495
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1369.2.a.n.1.13	✓	27
37.6	odd	4		1369.2.b.h.1368.33		54
37.31	odd	4		1369.2.b.h.1368.22		54
37.36	even	2		1369.2.a.o.1.15	yes	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1369.2.a.n.1.13	✓	27	1.1	even	1	trivial
1369.2.a.o.1.15	yes	27	37.36	even	2
1369.2.b.h.1368.22		54	37.31	odd	4
1369.2.b.h.1368.33		54	37.6	odd	4

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:	\( N \)	\(=\)	\( 1369 = 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1369.a (trivial)

\(f(q)\)	\(=\)	\(q-0.581223 q^{2} -1.17520 q^{3} -1.66218 q^{4} +0.135564 q^{5} +0.683054 q^{6} -3.40406 q^{7} +2.12854 q^{8} -1.61890 q^{9} -0.0787931 q^{10} +4.51469 q^{11} +1.95340 q^{12} +1.75123 q^{13} +1.97852 q^{14} -0.159316 q^{15} +2.08720 q^{16} -0.238914 q^{17} +0.940940 q^{18} +3.02781 q^{19} -0.225332 q^{20} +4.00046 q^{21} -2.62404 q^{22} +7.46588 q^{23} -2.50147 q^{24} -4.98162 q^{25} -1.01785 q^{26} +5.42814 q^{27} +5.65816 q^{28} -1.30151 q^{29} +0.0925978 q^{30} -8.68910 q^{31} -5.47021 q^{32} -5.30568 q^{33} +0.138862 q^{34} -0.461469 q^{35} +2.69090 q^{36} -1.75983 q^{38} -2.05805 q^{39} +0.288554 q^{40} +7.75460 q^{41} -2.32516 q^{42} -8.68570 q^{43} -7.50423 q^{44} -0.219465 q^{45} -4.33934 q^{46} -8.13636 q^{47} -2.45289 q^{48} +4.58762 q^{49} +2.89543 q^{50} +0.280773 q^{51} -2.91086 q^{52} +11.3476 q^{53} -3.15496 q^{54} +0.612031 q^{55} -7.24568 q^{56} -3.55829 q^{57} +0.756470 q^{58} -6.24883 q^{59} +0.264811 q^{60} -13.3101 q^{61} +5.05030 q^{62} +5.51083 q^{63} -0.994996 q^{64} +0.237404 q^{65} +3.08378 q^{66} +2.59709 q^{67} +0.397119 q^{68} -8.77393 q^{69} +0.268216 q^{70} +3.60176 q^{71} -3.44589 q^{72} -6.65769 q^{73} +5.85442 q^{75} -5.03277 q^{76} -15.3683 q^{77} +1.19618 q^{78} -4.55092 q^{79} +0.282950 q^{80} -1.52247 q^{81} -4.50715 q^{82} -4.73268 q^{83} -6.64949 q^{84} -0.0323883 q^{85} +5.04833 q^{86} +1.52954 q^{87} +9.60971 q^{88} +16.7206 q^{89} +0.127558 q^{90} -5.96128 q^{91} -12.4096 q^{92} +10.2115 q^{93} +4.72903 q^{94} +0.410463 q^{95} +6.42861 q^{96} -2.99941 q^{97} -2.66643 q^{98} -7.30883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q - 9 q^{2} - q^{3} + 25 q^{4} - 17 q^{5} - 10 q^{6} - 3 q^{7} - 21 q^{8} + 20 q^{9} - 11 q^{10} - 5 q^{11} - 10 q^{12} - 15 q^{13} - 25 q^{14} - 23 q^{15} + 13 q^{16} - 26 q^{17} - 15 q^{18} - 27 q^{19}+ \cdots + 35 q^{99}+O(q^{100}) \) 27 * q - 9 * q^2 - q^3 + 25 * q^4 - 17 * q^5 - 10 * q^6 - 3 * q^7 - 21 * q^8 + 20 * q^9 - 11 * q^10 - 5 * q^11 - 10 * q^12 - 15 * q^13 - 25 * q^14 - 23 * q^15 + 13 * q^16 - 26 * q^17 - 15 * q^18 - 27 * q^19 - 38 * q^20 - 17 * q^21 + 4 * q^22 - 41 * q^23 - 27 * q^24 + 16 * q^25 - 23 * q^26 - 13 * q^27 + 48 * q^28 - 42 * q^29 + 46 * q^30 - 45 * q^31 - 31 * q^32 - 33 * q^33 + 49 * q^34 - 51 * q^35 - 25 * q^36 - 11 * q^38 - 47 * q^39 + 31 * q^40 + 7 * q^41 + 7 * q^42 - 11 * q^43 - 34 * q^44 - 24 * q^45 - 15 * q^46 - 21 * q^47 + 6 * q^48 + 6 * q^49 - 19 * q^50 - 37 * q^51 - 6 * q^52 - 7 * q^53 - 7 * q^54 - 35 * q^55 - 54 * q^56 + 3 * q^57 + 37 * q^58 - 74 * q^59 + 14 * q^60 - 26 * q^61 - 33 * q^62 - 50 * q^63 - 27 * q^64 - 6 * q^65 - 19 * q^66 + 2 * q^67 - 57 * q^68 - 28 * q^69 + 6 * q^70 + 5 * q^71 + 12 * q^72 - 23 * q^73 + 34 * q^75 - 27 * q^76 - 11 * q^77 + 17 * q^78 - 58 * q^79 - 36 * q^80 - 37 * q^81 + 34 * q^82 + 56 * q^83 - 40 * q^84 - 16 * q^85 - 45 * q^86 - 66 * q^87 + 33 * q^88 - 64 * q^89 - 34 * q^90 - 17 * q^91 - 34 * q^92 + 30 * q^93 + 5 * q^94 - 22 * q^95 - 19 * q^96 + 17 * q^97 + 6 * q^98 + 35 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more