LMFDB - Embedded newform 1369.2.a.o.1.15

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:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
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	$-0.434120\pi$
$2$	0.581223	0.410986	0.205493	+	0.978659i	$0.434120\pi$
$3$	−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	$-0.509650\pi$
$5$	−0.135564	−0.0606262	−0.0303131	+	0.999540i	$0.509650\pi$
$6$	−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	$-0.578083\pi$
$13$	−1.75123	−0.485703	−0.242852	+	0.970063i	$0.578083\pi$
$14$	−1.97852	−0.528781
$15$	0.159316	0.0411351
$16$	2.08720	0.521801
$17$	0.238914	0.0579452	0.0289726	−	0.999580i	$-0.490776\pi$
$17$	0.238914	0.0579452	0.0289726	+	0.999580i	$0.490776\pi$
$18$	−0.940940	−0.221782
$19$	−3.02781	−0.694628	−0.347314	−	0.937749i	$-0.612906\pi$
$19$	−3.02781	−0.694628	−0.347314	+	0.937749i	$0.612906\pi$
$20$	0.225332	0.0503859
$21$	4.00046	0.872972
$22$	2.62404	0.559447
$23$	−7.46588	−1.55674	−0.778372	−	0.627803i	$-0.783954\pi$
$23$	−7.46588	−1.55674	−0.778372	+	0.627803i	$0.783954\pi$
$24$	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	$-0.461440\pi$
$29$	1.30151	0.241685	0.120843	+	0.992672i	$0.461440\pi$
$30$	0.0925978	0.0169060
$31$	8.68910	1.56061	0.780304	−	0.625400i	$-0.215064\pi$
$31$	8.68910	1.56061	0.780304	+	0.625400i	$0.215064\pi$
$32$	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	$-0.269589\pi$
$43$	8.68570	1.32456	0.662279	+	0.749258i	$0.269589\pi$
$44$	−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	$-0.366657\pi$
$59$	6.24883	0.813528	0.406764	+	0.913533i	$0.366657\pi$
$60$	−0.264811	−0.0341870
$61$	13.3101	1.70418	0.852090	−	0.523395i	$-0.175335\pi$
$61$	13.3101	1.70418	0.852090	+	0.523395i	$0.175335\pi$
$62$	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	$-0.417592\pi$
$79$	4.55092	0.512018	0.256009	+	0.966674i	$0.417592\pi$
$80$	−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	$-0.846659\pi$
$89$	−16.7206	−1.77238	−0.886192	+	0.463318i	$0.846659\pi$
$90$	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	$-0.451341\pi$
$97$	2.99941	0.304544	0.152272	+	0.988339i	$0.451341\pi$
$98$	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	$-0.528442\pi$
$103$	−1.81129	−0.178472	−0.0892358	+	0.996011i	$0.528442\pi$
$104$	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	$-0.247964\pi$
$109$	14.8590	1.42323	0.711616	+	0.702569i	$0.247964\pi$
$110$	−0.355726	−0.0339172
$111$	0	0
$112$	−7.10497	−0.671356
$113$	−4.47061	−0.420559	−0.210280	−	0.977641i	$-0.567437\pi$
$113$	−4.47061	−0.420559	−0.210280	+	0.977641i	$0.567437\pi$
$114$	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	$-0.330397\pi$
$131$	11.6279	1.01593	0.507967	+	0.861377i	$0.330397\pi$
$132$	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	$-0.361414\pi$
$163$	10.7692	0.843513	0.421756	+	0.906709i	$0.361414\pi$
$164$	−12.8896	−1.00650
$165$	0.719261	0.0559944
$166$	−2.75074	−0.213499
$167$	11.5114	0.890783	0.445391	−	0.895336i	$-0.353065\pi$
$167$	11.5114	0.890783	0.445391	+	0.895336i	$0.353065\pi$
$168$	−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	$-0.315973\pi$
$179$	14.6224	1.09293	0.546466	+	0.837481i	$0.315973\pi$
$180$	−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	$-0.569628\pi$
$191$	−5.99804	−0.434003	−0.217001	+	0.976171i	$0.569628\pi$
$192$	1.16932	0.0843886
$193$	−17.1747	−1.23626	−0.618131	−	0.786075i	$-0.712110\pi$
$193$	−17.1747	−1.23626	−0.618131	+	0.786075i	$0.712110\pi$
$194$	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	$-0.178752\pi$
$199$	23.8805	1.69285	0.846423	+	0.532512i	$0.178752\pi$
$200$	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	$-0.527725\pi$
$227$	−2.62128	−0.173981	−0.0869904	+	0.996209i	$0.527725\pi$
$228$	−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	$-0.337536\pi$
$239$	15.1048	0.977046	0.488523	+	0.872551i	$0.337536\pi$
$240$	0.332524	0.0214643
$241$	−12.3019	−0.792432	−0.396216	−	0.918157i	$-0.629677\pi$
$241$	−12.3019	−0.792432	−0.396216	+	0.918157i	$0.629677\pi$
$242$	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	$-0.131523\pi$
$251$	29.0194	1.83169	0.915844	+	0.401534i	$0.131523\pi$
$252$	−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	$-0.671147\pi$
$257$	−16.4204	−1.02428	−0.512139	+	0.858903i	$0.671147\pi$
$258$	−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	$-0.395368\pi$
$277$	10.7457	0.645649	0.322825	+	0.946459i	$0.395368\pi$
$278$	−3.03510	−0.182033
$279$	−14.0668	−0.842155
$280$	−0.982257	−0.0587011
$281$	−29.4052	−1.75417	−0.877084	−	0.480338i	$-0.840514\pi$
$281$	−29.4052	−1.75417	−0.877084	+	0.480338i	$0.840514\pi$
$282$	5.55758	0.330949
$283$	−9.89331	−0.588096	−0.294048	−	0.955791i	$-0.595003\pi$
$283$	−9.89331	−0.588096	−0.294048	+	0.955791i	$0.595003\pi$
$284$	−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	$-0.485680\pi$
$311$	1.58619	0.0899447	0.0449723	+	0.998988i	$0.485680\pi$
$312$	−4.38064	−0.248005
$313$	11.9721	0.676701	0.338350	−	0.941020i	$-0.390131\pi$
$313$	11.9721	0.676701	0.338350	+	0.941020i	$0.390131\pi$
$314$	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	$-0.428697\pi$
$331$	8.08288	0.444275	0.222138	+	0.975015i	$0.428697\pi$
$332$	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	$-0.174962\pi$
$347$	31.7682	1.70541	0.852703	+	0.522396i	$0.174962\pi$
$348$	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	$-0.270522\pi$
$353$	24.8036	1.32016	0.660081	+	0.751195i	$0.270522\pi$
$354$	−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	$-0.221569\pi$
$383$	30.0351	1.53472	0.767362	+	0.641214i	$0.221569\pi$
$384$	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	$-0.431089\pi$
$389$	8.47319	0.429608	0.214804	+	0.976657i	$0.431089\pi$
$390$	−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	$-0.433787\pi$
$401$	8.27102	0.413035	0.206518	+	0.978443i	$0.433787\pi$
$402$	−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	$-0.530384\pi$
$409$	−3.85500	−0.190617	−0.0953087	+	0.995448i	$0.530384\pi$
$410$	−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	$-0.567872\pi$
$421$	−8.68389	−0.423227	−0.211613	+	0.977353i	$0.567872\pi$
$422$	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	$-0.416974\pi$
$431$	10.7078	0.515774	0.257887	+	0.966175i	$0.416974\pi$
$432$	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	$-0.231332\pi$
$439$	31.3169	1.49467	0.747337	+	0.664445i	$0.231332\pi$
$440$	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	$-0.439067\pi$
$449$	8.06309	0.380521	0.190260	+	0.981734i	$0.439067\pi$
$450$	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	$-0.681844\pi$
$457$	−23.1181	−1.08142	−0.540709	+	0.841209i	$0.681844\pi$
$458$	9.11527	0.425928
$459$	1.29686	0.0605323
$460$	−1.68231	−0.0784379
$461$	−1.36014	−0.0633480	−0.0316740	−	0.999498i	$-0.510084\pi$
$461$	−1.36014	−0.0633480	−0.0316740	+	0.999498i	$0.510084\pi$
$462$	10.4974	0.488382
$463$	−9.63998	−0.448008	−0.224004	−	0.974588i	$-0.571913\pi$
$463$	−9.63998	−0.448008	−0.224004	+	0.974588i	$0.571913\pi$
$464$	2.71653	0.126112
$465$	1.38431	0.0641958
$466$	11.4899	0.532258
$467$	−11.4646	−0.530519	−0.265260	−	0.964177i	$-0.585458\pi$
$467$	−11.4646	−0.530519	−0.265260	+	0.964177i	$0.585458\pi$
$468$	−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	$-0.349157\pi$
$479$	19.9753	0.912695	0.456348	+	0.889802i	$0.349157\pi$
$480$	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	$-0.476970\pi$
$487$	3.19056	0.144578	0.0722890	+	0.997384i	$0.476970\pi$
$488$	−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	$-0.507765\pi$
$499$	−1.08975	−0.0487838	−0.0243919	+	0.999702i	$0.507765\pi$
$500$	−2.24918	−0.100587
$501$	−13.5283	−0.604399
$502$	16.8667	0.752799
$503$	0.148635	0.00662731	0.00331366	−	0.999995i	$-0.498945\pi$
$503$	0.148635	0.00662731	0.00331366	+	0.999995i	$0.498945\pi$
$504$	−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	$-0.343769\pi$
$523$	21.5585	0.942688	0.471344	+	0.881949i	$0.343769\pi$
$524$	−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	$-0.271566\pi$
$541$	30.5914	1.31523	0.657614	+	0.753355i	$0.271566\pi$
$542$	−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	$-0.696653\pi$
$547$	−27.0949	−1.15849	−0.579246	+	0.815153i	$0.696653\pi$
$548$	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	$-0.230649\pi$
$557$	35.3428	1.49752	0.748762	+	0.662839i	$0.230649\pi$
$558$	−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	$-0.646597\pi$
$563$	−21.0910	−0.888881	−0.444441	+	0.895808i	$0.646597\pi$
$564$	−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	$-0.855248\pi$
$569$	−42.8589	−1.79674	−0.898370	+	0.439239i	$0.855248\pi$
$570$	−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	$-0.312790\pi$
$577$	26.6541	1.10962	0.554812	+	0.831976i	$0.312790\pi$
$578$	−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	$-0.241133\pi$
$587$	35.2047	1.45305	0.726527	+	0.687138i	$0.241133\pi$
$588$	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	$-0.476269\pi$
$607$	3.67023	0.148970	0.0744849	+	0.997222i	$0.476269\pi$
$608$	−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	$-0.449513\pi$
$631$	7.93504	0.315889	0.157945	+	0.987448i	$0.449513\pi$
$632$	−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	$-0.586914\pi$
$643$	−13.6761	−0.539334	−0.269667	+	0.962954i	$0.586914\pi$
$644$	−42.2432	−1.66461
$645$	1.38377	0.0544858
$646$	−0.420449	−0.0165423
$647$	−4.72308	−0.185683	−0.0928417	−	0.995681i	$-0.529595\pi$
$647$	−4.72308	−0.185683	−0.0928417	+	0.995681i	$0.529595\pi$
$648$	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	$-0.629648\pi$
$653$	−20.2454	−0.792263	−0.396132	+	0.918194i	$0.629648\pi$
$654$	−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	$-0.447398\pi$
$661$	8.45867	0.329004	0.164502	+	0.986377i	$0.447398\pi$
$662$	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	$-0.225448\pi$
$683$	39.6975	1.51898	0.759492	+	0.650516i	$0.225448\pi$
$684$	−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	$-0.368703\pi$
$701$	21.2280	0.801769	0.400884	+	0.916129i	$0.368703\pi$
$702$	−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	$-0.344877\pi$
$709$	24.9373	0.936541	0.468270	+	0.883585i	$0.344877\pi$
$710$	−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	$-0.263624\pi$
$727$	36.4649	1.35241	0.676204	+	0.736715i	$0.263624\pi$
$728$	−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	$-0.613584\pi$
$757$	−19.2216	−0.698622	−0.349311	+	0.937007i	$0.613584\pi$
$758$	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	$-0.573325\pi$
$769$	−12.6633	−0.456650	−0.228325	+	0.973585i	$0.573325\pi$
$770$	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	$-0.662889\pi$
$797$	−27.6489	−0.979373	−0.489687	+	0.871899i	$0.662889\pi$
$798$	−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	$-0.403396\pi$
$809$	17.0004	0.597704	0.298852	+	0.954299i	$0.403396\pi$
$810$	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	$-0.497884\pi$
$827$	0.382375	0.0132965	0.00664824	+	0.999978i	$0.497884\pi$
$828$	−20.0900	−0.698174
$829$	0.605322	0.0210237	0.0105119	−	0.999945i	$-0.496654\pi$
$829$	0.605322	0.0210237	0.0105119	+	0.999945i	$0.496654\pi$
$830$	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	$-0.0802508\pi$
$853$	56.5657	1.93677	0.968387	+	0.249453i	$0.0802508\pi$
$854$	−26.3342	−0.901138
$855$	−0.664498	−0.0227254
$856$	−0.199446	−0.00681693
$857$	0.990424	0.0338322	0.0169161	−	0.999857i	$-0.494615\pi$
$857$	0.990424	0.0338322	0.0169161	+	0.999857i	$0.494615\pi$
$858$	5.40040	0.184367
$859$	−4.84248	−0.165223	−0.0826116	−	0.996582i	$-0.526326\pi$
$859$	−4.84248	−0.165223	−0.0826116	+	0.996582i	$0.526326\pi$
$860$	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	$-0.590659\pi$
$883$	−16.6988	−0.561959	−0.280979	+	0.959714i	$0.590659\pi$
$884$	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	$-0.508298\pi$
$907$	−1.57002	−0.0521316	−0.0260658	+	0.999660i	$0.508298\pi$
$908$	4.35705	0.144594
$909$	19.0984	0.633452
$910$	−0.469708	−0.0155707
$911$	23.1298	0.766324	0.383162	−	0.923681i	$-0.374835\pi$
$911$	23.1298	0.766324	0.383162	+	0.923681i	$0.374835\pi$
$912$	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	$-0.676182\pi$
$919$	−31.8709	−1.05132	−0.525662	+	0.850693i	$0.676182\pi$
$920$	−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	$-0.682585\pi$
$947$	−33.3992	−1.08533	−0.542665	+	0.839949i	$0.682585\pi$
$948$	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	$-0.621169\pi$
$967$	−23.1071	−0.743076	−0.371538	+	0.928418i	$0.621169\pi$
$968$	−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	$-0.692945\pi$
$977$	−35.6149	−1.13942	−0.569711	+	0.821845i	$0.692945\pi$
$978$	−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	$-0.561500\pi$
$991$	−12.0888	−0.384013	−0.192007	+	0.981394i	$0.561500\pi$
$992$	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	$-0.240345\pi$
$997$	45.9879	1.45645	0.728226	+	0.685338i	$0.240345\pi$
$998$	−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.o.1.15	yes	27
37.6	odd	4		1369.2.b.h.1368.22		54
37.31	odd	4		1369.2.b.h.1368.33		54
37.36	even	2		1369.2.a.n.1.13	✓	27

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

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