LMFDB - Embedded newform 1573.2.a.o.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1573,2,Mod(1,1573)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1573.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1573, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1573 = 11^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1573.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,1,-5,1,-5,3] 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:	$12.5604682379$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.661518125.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 7x^{6} + 5x^{5} + 15x^{4} - 7x^{3} - 10x^{2} + 2x + 1 $ x^8 - x^7 - 7x^6 + 5x^5 + 15x^4 - 7x^3 - 10x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 143)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-1.50656$ of defining polynomial
Character	$\chi$	$=$	1573.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.46083 q^{2} -0.833297 q^{3} +4.05570 q^{4} -0.985684 q^{5} -2.05061 q^{6} -3.88617 q^{7} +5.05874 q^{8} -2.30562 q^{9} -2.42560 q^{10} -3.37961 q^{12} -1.00000 q^{13} -9.56322 q^{14} +0.821367 q^{15} +4.33732 q^{16} +2.18287 q^{17} -5.67374 q^{18} -5.28179 q^{19} -3.99764 q^{20} +3.23833 q^{21} -2.41996 q^{23} -4.21544 q^{24} -4.02843 q^{25} -2.46083 q^{26} +4.42115 q^{27} -15.7611 q^{28} +6.81225 q^{29} +2.02125 q^{30} -3.29134 q^{31} +0.555943 q^{32} +5.37168 q^{34} +3.83053 q^{35} -9.35089 q^{36} -9.55991 q^{37} -12.9976 q^{38} +0.833297 q^{39} -4.98632 q^{40} +3.43173 q^{41} +7.96900 q^{42} -7.11470 q^{43} +2.27261 q^{45} -5.95513 q^{46} +1.92804 q^{47} -3.61428 q^{48} +8.10231 q^{49} -9.91329 q^{50} -1.81898 q^{51} -4.05570 q^{52} +4.61767 q^{53} +10.8797 q^{54} -19.6591 q^{56} +4.40130 q^{57} +16.7638 q^{58} +0.0436376 q^{59} +3.33122 q^{60} -10.6196 q^{61} -8.09944 q^{62} +8.96001 q^{63} -7.30656 q^{64} +0.985684 q^{65} -2.77390 q^{67} +8.85307 q^{68} +2.01655 q^{69} +9.42631 q^{70} +10.5199 q^{71} -11.6635 q^{72} +16.0318 q^{73} -23.5253 q^{74} +3.35688 q^{75} -21.4214 q^{76} +2.05061 q^{78} -9.29309 q^{79} -4.27523 q^{80} +3.23271 q^{81} +8.44492 q^{82} +12.2782 q^{83} +13.1337 q^{84} -2.15162 q^{85} -17.5081 q^{86} -5.67663 q^{87} +2.97373 q^{89} +5.59251 q^{90} +3.88617 q^{91} -9.81465 q^{92} +2.74266 q^{93} +4.74458 q^{94} +5.20618 q^{95} -0.463266 q^{96} -5.60356 q^{97} +19.9384 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 5 q^{3} + q^{4} - 5 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} - q^{9} - 6 q^{10} - 7 q^{12} - 8 q^{13} + 2 q^{14} + 5 q^{15} - 5 q^{16} - 13 q^{17} - 5 q^{18} + 6 q^{19} - 3 q^{20} - 4 q^{21}+ \cdots + 12 q^{98}+O(q^{100}) $ 8 * q + q^2 - 5 * q^3 + q^4 - 5 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 - q^9 - 6 * q^10 - 7 * q^12 - 8 * q^13 + 2 * q^14 + 5 * q^15 - 5 * q^16 - 13 * q^17 - 5 * q^18 + 6 * q^19 - 3 * q^20 - 4 * q^21 - 23 * q^23 - q^24 - q^25 - q^26 + q^27 - 32 * q^28 + 15 * q^29 - 4 * q^30 - 11 * q^31 + 12 * q^32 + 2 * q^34 + 8 * q^35 - 6 * q^36 - 11 * q^37 - 25 * q^38 + 5 * q^39 + 15 * q^40 + 11 * q^41 - 11 * q^42 - 8 * q^43 - 23 * q^45 + 12 * q^46 - 21 * q^47 + 8 * q^48 + 2 * q^49 - 12 * q^50 + 26 * q^51 - q^52 - 7 * q^53 + 3 * q^54 - 34 * q^56 - 51 * q^57 - 20 * q^58 - 22 * q^59 - 3 * q^60 + 6 * q^61 - q^62 + 14 * q^63 + 5 * q^64 + 5 * q^65 - 31 * q^67 - q^68 + 17 * q^69 - 26 * q^70 - 23 * q^71 - 27 * q^72 + 34 * q^73 + 6 * q^74 - 27 * q^75 - 2 * q^76 - 3 * q^78 - 5 * q^79 - 7 * q^80 - 32 * q^81 - 11 * q^82 + 11 * q^83 + 45 * q^84 - 4 * q^85 - 21 * q^86 + 2 * q^87 - 15 * q^89 + 3 * q^90 - 2 * q^91 - 6 * q^92 - 8 * q^93 - 8 * q^94 - 11 * q^95 - 41 * q^96 - 6 * q^97 + 12 * q^98

Coefficient data

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

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

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

$2$	2.46083	1.74007	0.870036	−	0.492988i	$-0.164095\pi$
$2$	2.46083	1.74007	0.870036	+	0.492988i	$0.164095\pi$
$3$	−0.833297	−0.481104	−0.240552	−	0.970636i	$-0.577329\pi$
$3$	−0.833297	−0.481104	−0.240552	+	0.970636i	$0.577329\pi$
$4$	4.05570	2.02785
$5$	−0.985684	−0.440811	−0.220406	−	0.975408i	$-0.570738\pi$
$5$	−0.985684	−0.440811	−0.220406	+	0.975408i	$0.570738\pi$
$6$	−2.05061	−0.837156
$7$	−3.88617	−1.46883	−0.734417	−	0.678699i	$-0.762544\pi$
$7$	−3.88617	−1.46883	−0.734417	+	0.678699i	$0.762544\pi$
$8$	5.05874	1.78854
$9$	−2.30562	−0.768539
$10$	−2.42560	−0.767043
$11$	0	0
$11$	0	0
$12$	−3.37961	−0.975608
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−9.56322	−2.55588
$15$	0.821367	0.212076
$16$	4.33732	1.08433
$17$	2.18287	0.529424	0.264712	−	0.964328i	$-0.414723\pi$
$17$	2.18287	0.529424	0.264712	+	0.964328i	$0.414723\pi$
$18$	−5.67374	−1.33731
$19$	−5.28179	−1.21173	−0.605863	−	0.795569i	$-0.707172\pi$
$19$	−5.28179	−1.21173	−0.605863	+	0.795569i	$0.707172\pi$
$20$	−3.99764	−0.893900
$21$	3.23833	0.706662
$22$	0	0
$23$	−2.41996	−0.504597	−0.252299	−	0.967649i	$-0.581186\pi$
$23$	−2.41996	−0.504597	−0.252299	+	0.967649i	$0.581186\pi$
$24$	−4.21544	−0.860473
$25$	−4.02843	−0.805686
$26$	−2.46083	−0.482609
$27$	4.42115	0.850852
$28$	−15.7611	−2.97858
$29$	6.81225	1.26500	0.632501	−	0.774559i	$-0.282028\pi$
$29$	6.81225	1.26500	0.632501	+	0.774559i	$0.282028\pi$
$30$	2.02125	0.369028
$31$	−3.29134	−0.591142	−0.295571	−	0.955321i	$-0.595510\pi$
$31$	−3.29134	−0.591142	−0.295571	+	0.955321i	$0.595510\pi$
$32$	0.555943	0.0982778
$33$	0	0
$34$	5.37168	0.921235
$35$	3.83053	0.647478
$36$	−9.35089	−1.55848
$37$	−9.55991	−1.57164	−0.785819	−	0.618456i	$-0.787758\pi$
$37$	−9.55991	−1.57164	−0.785819	+	0.618456i	$0.787758\pi$
$38$	−12.9976	−2.10849
$39$	0.833297	0.133434
$40$	−4.98632	−0.788407
$41$	3.43173	0.535946	0.267973	−	0.963426i	$-0.413646\pi$
$41$	3.43173	0.535946	0.267973	+	0.963426i	$0.413646\pi$
$42$	7.96900	1.22964
$43$	−7.11470	−1.08498	−0.542491	−	0.840062i	$-0.682519\pi$
$43$	−7.11470	−1.08498	−0.542491	+	0.840062i	$0.682519\pi$
$44$	0	0
$45$	2.27261	0.338780
$46$	−5.95513	−0.878035
$47$	1.92804	0.281233	0.140616	−	0.990064i	$-0.455092\pi$
$47$	1.92804	0.281233	0.140616	+	0.990064i	$0.455092\pi$
$48$	−3.61428	−0.521676
$49$	8.10231	1.15747
$50$	−9.91329	−1.40195
$51$	−1.81898	−0.254708
$52$	−4.05570	−0.562425
$53$	4.61767	0.634285	0.317143	−	0.948378i	$-0.397277\pi$
$53$	4.61767	0.634285	0.317143	+	0.948378i	$0.397277\pi$
$54$	10.8797	1.48054
$55$	0	0
$56$	−19.6591	−2.62706
$57$	4.40130	0.582967
$58$	16.7638	2.20120
$59$	0.0436376	0.00568113	0.00284056	−	0.999996i	$-0.499096\pi$
$59$	0.0436376	0.00568113	0.00284056	+	0.999996i	$0.499096\pi$
$60$	3.33122	0.430059
$61$	−10.6196	−1.35970	−0.679852	−	0.733349i	$-0.737956\pi$
$61$	−10.6196	−1.35970	−0.679852	+	0.733349i	$0.737956\pi$
$62$	−8.09944	−1.02863
$63$	8.96001	1.12886
$64$	−7.30656	−0.913320
$65$	0.985684	0.122259
$66$	0	0
$67$	−2.77390	−0.338885	−0.169443	−	0.985540i	$-0.554197\pi$
$67$	−2.77390	−0.338885	−0.169443	+	0.985540i	$0.554197\pi$
$68$	8.85307	1.07359
$69$	2.01655	0.242764
$70$	9.42631	1.12666
$71$	10.5199	1.24848	0.624241	−	0.781232i	$-0.285408\pi$
$71$	10.5199	1.24848	0.624241	+	0.781232i	$0.285408\pi$
$72$	−11.6635	−1.37456
$73$	16.0318	1.87638	0.938189	−	0.346123i	$-0.112502\pi$
$73$	16.0318	1.87638	0.938189	+	0.346123i	$0.112502\pi$
$74$	−23.5253	−2.73477
$75$	3.35688	0.387619
$76$	−21.4214	−2.45720
$77$	0	0
$78$	2.05061	0.232185
$79$	−9.29309	−1.04555	−0.522777	−	0.852469i	$-0.675104\pi$
$79$	−9.29309	−1.04555	−0.522777	+	0.852469i	$0.675104\pi$
$80$	−4.27523	−0.477985
$81$	3.23271	0.359190
$82$	8.44492	0.932585
$83$	12.2782	1.34771	0.673853	−	0.738865i	$-0.264638\pi$
$83$	12.2782	1.34771	0.673853	+	0.738865i	$0.264638\pi$
$84$	13.1337	1.43301
$85$	−2.15162	−0.233376
$86$	−17.5081	−1.88795
$87$	−5.67663	−0.608598
$88$	0	0
$89$	2.97373	0.315215	0.157607	−	0.987502i	$-0.449622\pi$
$89$	2.97373	0.315215	0.157607	+	0.987502i	$0.449622\pi$
$90$	5.59251	0.589502
$91$	3.88617	0.407381
$92$	−9.81465	−1.02325
$93$	2.74266	0.284401
$94$	4.74458	0.489366
$95$	5.20618	0.534143
$96$	−0.463266	−0.0472819
$97$	−5.60356	−0.568955	−0.284478	−	0.958683i	$-0.591820\pi$
$97$	−5.60356	−0.568955	−0.284478	+	0.958683i	$0.591820\pi$
$98$	19.9384	2.01409
$99$	0	0
$100$	−16.3381	−1.63381
$101$	13.5577	1.34905	0.674523	−	0.738254i	$-0.264350\pi$
$101$	13.5577	1.34905	0.674523	+	0.738254i	$0.264350\pi$
$102$	−4.47620	−0.443210
$103$	1.93196	0.190362	0.0951809	−	0.995460i	$-0.469657\pi$
$103$	1.93196	0.190362	0.0951809	+	0.995460i	$0.469657\pi$
$104$	−5.05874	−0.496051
$105$	−3.19197	−0.311505
$106$	11.3633	1.10370
$107$	−14.4878	−1.40059	−0.700293	−	0.713856i	$-0.746947\pi$
$107$	−14.4878	−1.40059	−0.700293	+	0.713856i	$0.746947\pi$
$108$	17.9309	1.72540
$109$	11.8121	1.13139	0.565697	−	0.824613i	$-0.308607\pi$
$109$	11.8121	1.13139	0.565697	+	0.824613i	$0.308607\pi$
$110$	0	0
$111$	7.96624	0.756122
$112$	−16.8556	−1.59270
$113$	−17.0164	−1.60077	−0.800384	−	0.599487i	$-0.795371\pi$
$113$	−17.0164	−1.60077	−0.800384	+	0.599487i	$0.795371\pi$
$114$	10.8309	1.01440
$115$	2.38532	0.222432
$116$	27.6285	2.56524
$117$	2.30562	0.213154
$118$	0.107385	0.00988557
$119$	−8.48300	−0.777635
$120$	4.15509	0.379306
$121$	0	0
$122$	−26.1331	−2.36598
$123$	−2.85965	−0.257846
$124$	−13.3487	−1.19875
$125$	8.89917	0.795966
$126$	22.0491	1.96429
$127$	−2.96884	−0.263442	−0.131721	−	0.991287i	$-0.542050\pi$
$127$	−2.96884	−0.263442	−0.131721	+	0.991287i	$0.542050\pi$
$128$	−19.0921	−1.68752
$129$	5.92866	0.521989
$130$	2.42560	0.212740
$131$	−15.8031	−1.38073	−0.690363	−	0.723463i	$-0.742549\pi$
$131$	−15.8031	−1.38073	−0.690363	+	0.723463i	$0.742549\pi$
$132$	0	0
$133$	20.5259	1.77983
$134$	−6.82610	−0.589685
$135$	−4.35786	−0.375065
$136$	11.0426	0.946893
$137$	−10.4105	−0.889428	−0.444714	−	0.895673i	$-0.646695\pi$
$137$	−10.4105	−0.889428	−0.444714	+	0.895673i	$0.646695\pi$
$138$	4.96239	0.422427
$139$	17.4174	1.47733	0.738663	−	0.674075i	$-0.235457\pi$
$139$	17.4174	1.47733	0.738663	+	0.674075i	$0.235457\pi$
$140$	15.5355	1.31299
$141$	−1.60663	−0.135302
$142$	25.8877	2.17245
$143$	0	0
$144$	−10.0002	−0.833350
$145$	−6.71472	−0.557627
$146$	39.4516	3.26503
$147$	−6.75163	−0.556865
$148$	−38.7721	−3.18705
$149$	7.45391	0.610648	0.305324	−	0.952248i	$-0.401235\pi$
$149$	7.45391	0.610648	0.305324	+	0.952248i	$0.401235\pi$
$150$	8.26072	0.674485
$151$	−2.62280	−0.213440	−0.106720	−	0.994289i	$-0.534035\pi$
$151$	−2.62280	−0.213440	−0.106720	+	0.994289i	$0.534035\pi$
$152$	−26.7193	−2.16722
$153$	−5.03286	−0.406882
$154$	0	0
$155$	3.24422	0.260582
$156$	3.37961	0.270585
$157$	−14.1508	−1.12936	−0.564679	−	0.825310i	$-0.691000\pi$
$157$	−14.1508	−1.12936	−0.564679	+	0.825310i	$0.691000\pi$
$158$	−22.8688	−1.81934
$159$	−3.84789	−0.305157
$160$	−0.547984	−0.0433220
$161$	9.40438	0.741169
$162$	7.95517	0.625017
$163$	−17.8224	−1.39596	−0.697979	−	0.716119i	$-0.745917\pi$
$163$	−17.8224	−1.39596	−0.697979	+	0.716119i	$0.745917\pi$
$164$	13.9181	1.08682
$165$	0	0
$166$	30.2146	2.34511
$167$	19.9057	1.54035	0.770173	−	0.637834i	$-0.220170\pi$
$167$	19.9057	1.54035	0.770173	+	0.637834i	$0.220170\pi$
$168$	16.3819	1.26389
$169$	1.00000	0.0769231
$170$	−5.29478	−0.406091
$171$	12.1778	0.931259
$172$	−28.8551	−2.20018
$173$	2.06598	0.157073	0.0785366	−	0.996911i	$-0.474975\pi$
$173$	2.06598	0.157073	0.0785366	+	0.996911i	$0.474975\pi$
$174$	−13.9692	−1.05901
$175$	15.6552	1.18342
$176$	0	0
$177$	−0.0363631	−0.00273322
$178$	7.31786	0.548497
$179$	5.92882	0.443141	0.221570	−	0.975144i	$-0.428882\pi$
$179$	5.92882	0.443141	0.221570	+	0.975144i	$0.428882\pi$
$180$	9.21702	0.686996
$181$	−21.0596	−1.56535	−0.782674	−	0.622432i	$-0.786145\pi$
$181$	−21.0596	−1.56535	−0.782674	+	0.622432i	$0.786145\pi$
$182$	9.56322	0.708873
$183$	8.84931	0.654160
$184$	−12.2420	−0.902490
$185$	9.42304	0.692796
$186$	6.74924	0.494878
$187$	0	0
$188$	7.81954	0.570299
$189$	−17.1814	−1.24976
$190$	12.8115	0.929447
$191$	−9.68456	−0.700750	−0.350375	−	0.936610i	$-0.613946\pi$
$191$	−9.68456	−0.700750	−0.350375	+	0.936610i	$0.613946\pi$
$192$	6.08854	0.439402
$193$	1.73040	0.124557	0.0622786	−	0.998059i	$-0.480163\pi$
$193$	1.73040	0.124557	0.0622786	+	0.998059i	$0.480163\pi$
$194$	−13.7894	−0.990023
$195$	−0.821367	−0.0588193
$196$	32.8606	2.34718
$197$	24.7399	1.76265	0.881323	−	0.472514i	$-0.156653\pi$
$197$	24.7399	1.76265	0.881323	+	0.472514i	$0.156653\pi$
$198$	0	0
$199$	−2.12759	−0.150821	−0.0754103	−	0.997153i	$-0.524027\pi$
$199$	−2.12759	−0.150821	−0.0754103	+	0.997153i	$0.524027\pi$
$200$	−20.3788	−1.44100
$201$	2.31148	0.163039
$202$	33.3633	2.34744
$203$	−26.4735	−1.85808
$204$	−7.37724	−0.516510
$205$	−3.38260	−0.236251
$206$	4.75423	0.331243
$207$	5.57950	0.387802
$208$	−4.33732	−0.300739
$209$	0	0
$210$	−7.85491	−0.542041
$211$	−3.55487	−0.244727	−0.122364	−	0.992485i	$-0.539047\pi$
$211$	−3.55487	−0.244727	−0.122364	+	0.992485i	$0.539047\pi$
$212$	18.7279	1.28624
$213$	−8.76620	−0.600650
$214$	−35.6520	−2.43712
$215$	7.01284	0.478272
$216$	22.3655	1.52178
$217$	12.7907	0.868289
$218$	29.0676	1.96871
$219$	−13.3592	−0.902734
$220$	0	0
$221$	−2.18287	−0.146836
$222$	19.6036	1.31571
$223$	−18.6891	−1.25152	−0.625758	−	0.780017i	$-0.715210\pi$
$223$	−18.6891	−1.25152	−0.625758	+	0.780017i	$0.715210\pi$
$224$	−2.16049	−0.144354
$225$	9.28801	0.619200
$226$	−41.8746	−2.78545
$227$	11.7004	0.776580	0.388290	−	0.921537i	$-0.373066\pi$
$227$	11.7004	0.776580	0.388290	+	0.921537i	$0.373066\pi$
$228$	17.8504	1.18217
$229$	5.43048	0.358856	0.179428	−	0.983771i	$-0.442575\pi$
$229$	5.43048	0.358856	0.179428	+	0.983771i	$0.442575\pi$
$230$	5.86987	0.387048
$231$	0	0
$232$	34.4614	2.26250
$233$	−13.1355	−0.860536	−0.430268	−	0.902701i	$-0.641581\pi$
$233$	−13.1355	−0.860536	−0.430268	+	0.902701i	$0.641581\pi$
$234$	5.67374	0.370904
$235$	−1.90043	−0.123971
$236$	0.176981	0.0115205
$237$	7.74391	0.503021
$238$	−20.8753	−1.35314
$239$	−8.86074	−0.573154	−0.286577	−	0.958057i	$-0.592517\pi$
$239$	−8.86074	−0.573154	−0.286577	+	0.958057i	$0.592517\pi$
$240$	3.56254	0.229961
$241$	−19.9825	−1.28719	−0.643593	−	0.765368i	$-0.722557\pi$
$241$	−19.9825	−1.28719	−0.643593	+	0.765368i	$0.722557\pi$
$242$	0	0
$243$	−15.9573	−1.02366
$244$	−43.0701	−2.75728
$245$	−7.98631	−0.510227
$246$	−7.03713	−0.448671
$247$	5.28179	0.336073
$248$	−16.6500	−1.05728
$249$	−10.2314	−0.648388
$250$	21.8994	1.38504
$251$	−29.5375	−1.86439	−0.932195	−	0.361956i	$-0.882109\pi$
$251$	−29.5375	−1.86439	−0.932195	+	0.361956i	$0.882109\pi$
$252$	36.3392	2.28915
$253$	0	0
$254$	−7.30583	−0.458409
$255$	1.79294	0.112278
$256$	−32.3694	−2.02309
$257$	−8.58239	−0.535355	−0.267677	−	0.963509i	$-0.586256\pi$
$257$	−8.58239	−0.535355	−0.267677	+	0.963509i	$0.586256\pi$
$258$	14.5894	0.908299
$259$	37.1514	2.30848
$260$	3.99764	0.247923
$261$	−15.7064	−0.972204
$262$	−38.8889	−2.40256
$263$	−16.7442	−1.03249	−0.516245	−	0.856441i	$-0.672671\pi$
$263$	−16.7442	−1.03249	−0.516245	+	0.856441i	$0.672671\pi$
$264$	0	0
$265$	−4.55156	−0.279600
$266$	50.5109	3.09702
$267$	−2.47800	−0.151651
$268$	−11.2501	−0.687209
$269$	7.33341	0.447126	0.223563	−	0.974689i	$-0.428231\pi$
$269$	7.33341	0.447126	0.223563	+	0.974689i	$0.428231\pi$
$270$	−10.7240	−0.652640
$271$	−21.9098	−1.33093	−0.665463	−	0.746431i	$-0.731766\pi$
$271$	−21.9098	−1.33093	−0.665463	+	0.746431i	$0.731766\pi$
$272$	9.46781	0.574070
$273$	−3.23833	−0.195993
$274$	−25.6185	−1.54767
$275$	0	0
$276$	8.17852	0.492289
$277$	−9.61513	−0.577717	−0.288859	−	0.957372i	$-0.593276\pi$
$277$	−9.61513	−0.577717	−0.288859	+	0.957372i	$0.593276\pi$
$278$	42.8614	2.57066
$279$	7.58857	0.454316
$280$	19.3777	1.15804
$281$	1.65128	0.0985071	0.0492535	−	0.998786i	$-0.484316\pi$
$281$	1.65128	0.0985071	0.0492535	+	0.998786i	$0.484316\pi$
$282$	−3.95364	−0.235436
$283$	14.6389	0.870189	0.435095	−	0.900385i	$-0.356715\pi$
$283$	14.6389	0.870189	0.435095	+	0.900385i	$0.356715\pi$
$284$	42.6656	2.53174
$285$	−4.33829	−0.256978
$286$	0	0
$287$	−13.3363	−0.787216
$288$	−1.28179	−0.0755303
$289$	−12.2351	−0.719711
$290$	−16.5238	−0.970312
$291$	4.66943	0.273727
$292$	65.0202	3.80502
$293$	−0.405770	−0.0237054	−0.0118527	−	0.999930i	$-0.503773\pi$
$293$	−0.405770	−0.0237054	−0.0118527	+	0.999930i	$0.503773\pi$
$294$	−16.6146	−0.968986
$295$	−0.0430128	−0.00250430
$296$	−48.3611	−2.81093
$297$	0	0
$298$	18.3428	1.06257
$299$	2.41996	0.139950
$300$	13.6145	0.786034
$301$	27.6489	1.59366
$302$	−6.45427	−0.371402
$303$	−11.2976	−0.649031
$304$	−22.9089	−1.31391
$305$	10.4676	0.599373
$306$	−12.3850	−0.708005
$307$	9.00182	0.513761	0.256880	−	0.966443i	$-0.417305\pi$
$307$	9.00182	0.513761	0.256880	+	0.966443i	$0.417305\pi$
$308$	0	0
$309$	−1.60990	−0.0915839
$310$	7.98349	0.453432
$311$	6.37201	0.361324	0.180662	−	0.983545i	$-0.442176\pi$
$311$	6.37201	0.361324	0.180662	+	0.983545i	$0.442176\pi$
$312$	4.21544	0.238652
$313$	18.3812	1.03897	0.519484	−	0.854480i	$-0.326124\pi$
$313$	18.3812	1.03897	0.519484	+	0.854480i	$0.326124\pi$
$314$	−34.8228	−1.96517
$315$	−8.83174	−0.497612
$316$	−37.6900	−2.12023
$317$	−6.18506	−0.347387	−0.173694	−	0.984800i	$-0.555570\pi$
$317$	−6.18506	−0.347387	−0.173694	+	0.984800i	$0.555570\pi$
$318$	−9.46902	−0.530996
$319$	0	0
$320$	7.20196	0.402602
$321$	12.0726	0.673828
$322$	23.1426	1.28969
$323$	−11.5295	−0.641517
$324$	13.1109	0.728384
$325$	4.02843	0.223457
$326$	−43.8579	−2.42907
$327$	−9.84298	−0.544318
$328$	17.3603	0.958559
$329$	−7.49267	−0.413084
$330$	0	0
$331$	7.43488	0.408658	0.204329	−	0.978902i	$-0.434499\pi$
$331$	7.43488	0.408658	0.204329	+	0.978902i	$0.434499\pi$
$332$	49.7967	2.73295
$333$	22.0415	1.20787
$334$	48.9845	2.68032
$335$	2.73418	0.149384
$336$	14.0457	0.766256
$337$	36.0293	1.96264	0.981321	−	0.192380i	$-0.0616206\pi$
$337$	36.0293	1.96264	0.981321	+	0.192380i	$0.0616206\pi$
$338$	2.46083	0.133852
$339$	14.1797	0.770137
$340$	−8.72633	−0.473251
$341$	0	0
$342$	29.9675	1.62046
$343$	−4.28376	−0.231301
$344$	−35.9914	−1.94053
$345$	−1.98768	−0.107013
$346$	5.08402	0.273319
$347$	−22.6272	−1.21469	−0.607346	−	0.794437i	$-0.707766\pi$
$347$	−22.6272	−1.21469	−0.607346	+	0.794437i	$0.707766\pi$
$348$	−23.0227	−1.23415
$349$	−3.92582	−0.210145	−0.105072	−	0.994465i	$-0.533507\pi$
$349$	−3.92582	−0.210145	−0.105072	+	0.994465i	$0.533507\pi$
$350$	38.5247	2.05923
$351$	−4.42115	−0.235984
$352$	0	0
$353$	−15.8328	−0.842694	−0.421347	−	0.906900i	$-0.638443\pi$
$353$	−15.8328	−0.842694	−0.421347	+	0.906900i	$0.638443\pi$
$354$	−0.0894835	−0.00475599
$355$	−10.3693	−0.550345
$356$	12.0606	0.639209
$357$	7.06886	0.374124
$358$	14.5898	0.771097
$359$	−2.39931	−0.126631	−0.0633154	−	0.997994i	$-0.520167\pi$
$359$	−2.39931	−0.126631	−0.0633154	+	0.997994i	$0.520167\pi$
$360$	11.4965	0.605921
$361$	8.89735	0.468282
$362$	−51.8242	−2.72382
$363$	0	0
$364$	15.7611	0.826109
$365$	−15.8023	−0.827128
$366$	21.7767	1.13829
$367$	−18.2688	−0.953624	−0.476812	−	0.879005i	$-0.658208\pi$
$367$	−18.2688	−0.953624	−0.476812	+	0.879005i	$0.658208\pi$
$368$	−10.4962	−0.547150
$369$	−7.91225	−0.411895
$370$	23.1885	1.20551
$371$	−17.9450	−0.931660
$372$	11.1234	0.576723
$373$	9.31655	0.482393	0.241196	−	0.970476i	$-0.422460\pi$
$373$	9.31655	0.482393	0.241196	+	0.970476i	$0.422460\pi$
$374$	0	0
$375$	−7.41566	−0.382943
$376$	9.75344	0.502995
$377$	−6.81225	−0.350849
$378$	−42.2805	−2.17467
$379$	19.6966	1.01175	0.505874	−	0.862607i	$-0.331170\pi$
$379$	19.6966	1.01175	0.505874	+	0.862607i	$0.331170\pi$
$380$	21.1147	1.08316
$381$	2.47393	0.126743
$382$	−23.8321	−1.21936
$383$	4.90797	0.250786	0.125393	−	0.992107i	$-0.459981\pi$
$383$	4.90797	0.250786	0.125393	+	0.992107i	$0.459981\pi$
$384$	15.9094	0.811874
$385$	0	0
$386$	4.25823	0.216738
$387$	16.4038	0.833850
$388$	−22.7264	−1.15376
$389$	−16.9903	−0.861441	−0.430721	−	0.902485i	$-0.641741\pi$
$389$	−16.9903	−0.861441	−0.430721	+	0.902485i	$0.641741\pi$
$390$	−2.02125	−0.102350
$391$	−5.28246	−0.267146
$392$	40.9875	2.07018
$393$	13.1687	0.664273
$394$	60.8808	3.06713
$395$	9.16005	0.460892
$396$	0	0
$397$	16.8443	0.845391	0.422695	−	0.906272i	$-0.361084\pi$
$397$	16.8443	0.845391	0.422695	+	0.906272i	$0.361084\pi$
$398$	−5.23563	−0.262439
$399$	−17.1042	−0.856282
$400$	−17.4726	−0.873630
$401$	35.1136	1.75349	0.876745	−	0.480956i	$-0.159710\pi$
$401$	35.1136	1.75349	0.876745	+	0.480956i	$0.159710\pi$
$402$	5.68817	0.283700
$403$	3.29134	0.163953
$404$	54.9862	2.73566
$405$	−3.18643	−0.158335
$406$	−65.1470	−3.23319
$407$	0	0
$408$	−9.20175	−0.455554
$409$	−15.3136	−0.757209	−0.378605	−	0.925559i	$-0.623596\pi$
$409$	−15.3136	−0.757209	−0.378605	+	0.925559i	$0.623596\pi$
$410$	−8.32402	−0.411094
$411$	8.67503	0.427908
$412$	7.83546	0.386025
$413$	−0.169583	−0.00834463
$414$	13.7302	0.674804
$415$	−12.1024	−0.594084
$416$	−0.555943	−0.0272574
$417$	−14.5139	−0.710748
$418$	0	0
$419$	14.5369	0.710173	0.355086	−	0.934833i	$-0.384451\pi$
$419$	14.5369	0.710173	0.355086	+	0.934833i	$0.384451\pi$
$420$	−12.9457	−0.631685
$421$	−6.03720	−0.294235	−0.147118	−	0.989119i	$-0.547000\pi$
$421$	−6.03720	−0.294235	−0.147118	+	0.989119i	$0.547000\pi$
$422$	−8.74794	−0.425843
$423$	−4.44531	−0.216138
$424$	23.3596	1.13444
$425$	−8.79353	−0.426549
$426$	−21.5722	−1.04518
$427$	41.2697	1.99718
$428$	−58.7581	−2.84018
$429$	0	0
$430$	17.2574	0.832227
$431$	4.93130	0.237532	0.118766	−	0.992922i	$-0.462106\pi$
$431$	4.93130	0.237532	0.118766	+	0.992922i	$0.462106\pi$
$432$	19.1760	0.922605
$433$	−5.53167	−0.265835	−0.132918	−	0.991127i	$-0.542435\pi$
$433$	−5.53167	−0.265835	−0.132918	+	0.991127i	$0.542435\pi$
$434$	31.4758	1.51089
$435$	5.59536	0.268277
$436$	47.9063	2.29430
$437$	12.7817	0.611434
$438$	−32.8749	−1.57082
$439$	16.1221	0.769468	0.384734	−	0.923028i	$-0.374293\pi$
$439$	16.1221	0.769468	0.384734	+	0.923028i	$0.374293\pi$
$440$	0	0
$441$	−18.6808	−0.889562
$442$	−5.37168	−0.255505
$443$	2.21666	0.105317	0.0526584	−	0.998613i	$-0.483231\pi$
$443$	2.21666	0.105317	0.0526584	+	0.998613i	$0.483231\pi$
$444$	32.3087	1.53330
$445$	−2.93116	−0.138950
$446$	−45.9908	−2.17773
$447$	−6.21132	−0.293786
$448$	28.3945	1.34152
$449$	0.209393	0.00988186	0.00494093	−	0.999988i	$-0.498427\pi$
$449$	0.209393	0.00988186	0.00494093	+	0.999988i	$0.498427\pi$
$450$	22.8562	1.07745
$451$	0	0
$452$	−69.0135	−3.24612
$453$	2.18557	0.102687
$454$	28.7926	1.35131
$455$	−3.83053	−0.179578
$456$	22.2651	1.04266
$457$	−33.6647	−1.57477	−0.787383	−	0.616464i	$-0.788565\pi$
$457$	−33.6647	−1.57477	−0.787383	+	0.616464i	$0.788565\pi$
$458$	13.3635	0.624436
$459$	9.65080	0.450461
$460$	9.67414	0.451059
$461$	21.4300	0.998093	0.499046	−	0.866575i	$-0.333684\pi$
$461$	21.4300	0.998093	0.499046	+	0.866575i	$0.333684\pi$
$462$	0	0
$463$	−14.4598	−0.672005	−0.336002	−	0.941861i	$-0.609075\pi$
$463$	−14.4598	−0.672005	−0.336002	+	0.941861i	$0.609075\pi$
$464$	29.5469	1.37168
$465$	−2.70340	−0.125367
$466$	−32.3243	−1.49739
$467$	−11.7989	−0.545987	−0.272993	−	0.962016i	$-0.588014\pi$
$467$	−11.7989	−0.545987	−0.272993	+	0.962016i	$0.588014\pi$
$468$	9.35089	0.432245
$469$	10.7798	0.497766
$470$	−4.67665	−0.215718
$471$	11.7918	0.543340
$472$	0.220751	0.0101609
$473$	0	0
$474$	19.0565	0.875293
$475$	21.2773	0.976271
$476$	−34.4045	−1.57693
$477$	−10.6466	−0.487473
$478$	−21.8048	−0.997329
$479$	29.4009	1.34336	0.671681	−	0.740841i	$-0.265573\pi$
$479$	29.4009	1.34336	0.671681	+	0.740841i	$0.265573\pi$
$480$	0.456634	0.0208424
$481$	9.55991	0.435894
$482$	−49.1736	−2.23980
$483$	−7.83665	−0.356580
$484$	0	0
$485$	5.52334	0.250802
$486$	−39.2682	−1.78124
$487$	0.0842815	0.00381916	0.00190958	−	0.999998i	$-0.499392\pi$
$487$	0.0842815	0.00381916	0.00190958	+	0.999998i	$0.499392\pi$
$488$	−53.7220	−2.43188
$489$	14.8513	0.671601
$490$	−19.6530	−0.887832
$491$	−29.9386	−1.35111	−0.675555	−	0.737310i	$-0.736096\pi$
$491$	−29.9386	−1.35111	−0.675555	+	0.737310i	$0.736096\pi$
$492$	−11.5979	−0.522874
$493$	14.8702	0.669722
$494$	12.9976	0.584791
$495$	0	0
$496$	−14.2756	−0.640994
$497$	−40.8821	−1.83381
$498$	−25.1777	−1.12824
$499$	−32.3127	−1.44652	−0.723258	−	0.690578i	$-0.757356\pi$
$499$	−32.3127	−1.44652	−0.723258	+	0.690578i	$0.757356\pi$
$500$	36.0924	1.61410
$501$	−16.5873	−0.741068
$502$	−72.6869	−3.24417
$503$	−6.85971	−0.305859	−0.152930	−	0.988237i	$-0.548871\pi$
$503$	−6.85971	−0.305859	−0.152930	+	0.988237i	$0.548871\pi$
$504$	45.3264	2.01900
$505$	−13.3636	−0.594674
$506$	0	0
$507$	−0.833297	−0.0370080
$508$	−12.0408	−0.534222
$509$	2.89107	0.128145	0.0640723	−	0.997945i	$-0.479591\pi$
$509$	2.89107	0.128145	0.0640723	+	0.997945i	$0.479591\pi$
$510$	4.41212	0.195372
$511$	−62.3022	−2.75609
$512$	−41.4715	−1.83280
$513$	−23.3516	−1.03100
$514$	−21.1198	−0.931556
$515$	−1.90430	−0.0839136
$516$	24.0449	1.05852
$517$	0	0
$518$	91.4234	4.01692
$519$	−1.72157	−0.0755686
$520$	4.98632	0.218665
$521$	22.5461	0.987763	0.493882	−	0.869529i	$-0.335578\pi$
$521$	22.5461	0.987763	0.493882	+	0.869529i	$0.335578\pi$
$522$	−38.6509	−1.69170
$523$	−24.3954	−1.06674	−0.533369	−	0.845883i	$-0.679074\pi$
$523$	−24.3954	−1.06674	−0.533369	+	0.845883i	$0.679074\pi$
$524$	−64.0928	−2.79991
$525$	−13.0454	−0.569348
$526$	−41.2046	−1.79661
$527$	−7.18456	−0.312965
$528$	0	0
$529$	−17.1438	−0.745382
$530$	−11.2006	−0.486524
$531$	−0.100611	−0.00436617
$532$	83.2472	3.60922
$533$	−3.43173	−0.148645
$534$	−6.09795	−0.263884
$535$	14.2804	0.617394
$536$	−14.0324	−0.606109
$537$	−4.94047	−0.213197
$538$	18.0463	0.778032
$539$	0	0
$540$	−17.6742	−0.760576
$541$	15.1928	0.653188	0.326594	−	0.945165i	$-0.394099\pi$
$541$	15.1928	0.653188	0.326594	+	0.945165i	$0.394099\pi$
$542$	−53.9164	−2.31591
$543$	17.5489	0.753096
$544$	1.21355	0.0520306
$545$	−11.6430	−0.498731
$546$	−7.96900	−0.341042
$547$	−4.55377	−0.194705	−0.0973525	−	0.995250i	$-0.531037\pi$
$547$	−4.55377	−0.194705	−0.0973525	+	0.995250i	$0.531037\pi$
$548$	−42.2218	−1.80363
$549$	24.4848	1.04499
$550$	0	0
$551$	−35.9809	−1.53284
$552$	10.2012	0.434192
$553$	36.1145	1.53575
$554$	−23.6612	−1.00527
$555$	−7.85219	−0.333307
$556$	70.6399	2.99580
$557$	11.8262	0.501091	0.250546	−	0.968105i	$-0.419390\pi$
$557$	11.8262	0.501091	0.250546	+	0.968105i	$0.419390\pi$
$558$	18.6742	0.790542
$559$	7.11470	0.300920
$560$	16.6143	0.702081
$561$	0	0
$562$	4.06352	0.171409
$563$	5.98613	0.252285	0.126143	−	0.992012i	$-0.459740\pi$
$563$	5.98613	0.252285	0.126143	+	0.992012i	$0.459740\pi$
$564$	−6.51600	−0.274373
$565$	16.7728	0.705637
$566$	36.0238	1.51419
$567$	−12.5629	−0.527591
$568$	53.2175	2.23296
$569$	17.1759	0.720051	0.360025	−	0.932942i	$-0.382768\pi$
$569$	17.1759	0.720051	0.360025	+	0.932942i	$0.382768\pi$
$570$	−10.6758	−0.447161
$571$	−17.1160	−0.716282	−0.358141	−	0.933668i	$-0.616589\pi$
$571$	−17.1160	−0.716282	−0.358141	+	0.933668i	$0.616589\pi$
$572$	0	0
$573$	8.07011	0.337134
$574$	−32.8184	−1.36981
$575$	9.74864	0.406547
$576$	16.8461	0.701922
$577$	16.0156	0.666738	0.333369	−	0.942796i	$-0.391814\pi$
$577$	16.0156	0.666738	0.333369	+	0.942796i	$0.391814\pi$
$578$	−30.1085	−1.25235
$579$	−1.44194	−0.0599250
$580$	−27.2329	−1.13079
$581$	−47.7151	−1.97956
$582$	11.4907	0.476304
$583$	0	0
$584$	81.1007	3.35597
$585$	−2.27261	−0.0939608
$586$	−0.998533	−0.0412490
$587$	−18.6575	−0.770079	−0.385039	−	0.922900i	$-0.625812\pi$
$587$	−18.6575	−0.770079	−0.385039	+	0.922900i	$0.625812\pi$
$588$	−27.3826	−1.12924
$589$	17.3842	0.716303
$590$	−0.105847	−0.00435767
$591$	−20.6157	−0.848017
$592$	−41.4644	−1.70418
$593$	−0.889024	−0.0365078	−0.0182539	−	0.999833i	$-0.505811\pi$
$593$	−0.889024	−0.0365078	−0.0182539	+	0.999833i	$0.505811\pi$
$594$	0	0
$595$	8.36155	0.342790
$596$	30.2309	1.23830
$597$	1.77291	0.0725604
$598$	5.95513	0.243523
$599$	−5.61187	−0.229295	−0.114647	−	0.993406i	$-0.536574\pi$
$599$	−5.61187	−0.229295	−0.114647	+	0.993406i	$0.536574\pi$
$600$	16.9816	0.693270
$601$	−27.6110	−1.12628	−0.563138	−	0.826363i	$-0.690406\pi$
$601$	−27.6110	−1.12628	−0.563138	+	0.826363i	$0.690406\pi$
$602$	68.0394	2.77308
$603$	6.39554	0.260446
$604$	−10.6373	−0.432825
$605$	0	0
$606$	−27.8016	−1.12936
$607$	−42.8599	−1.73963	−0.869814	−	0.493379i	$-0.835762\pi$
$607$	−42.8599	−1.73963	−0.869814	+	0.493379i	$0.835762\pi$
$608$	−2.93638	−0.119086
$609$	22.0603	0.893930
$610$	25.7590	1.04295
$611$	−1.92804	−0.0780000
$612$	−20.4118	−0.825097
$613$	18.1307	0.732291	0.366145	−	0.930558i	$-0.380677\pi$
$613$	18.1307	0.732291	0.366145	+	0.930558i	$0.380677\pi$
$614$	22.1520	0.893981
$615$	2.81871	0.113661
$616$	0	0
$617$	17.6030	0.708671	0.354336	−	0.935118i	$-0.384707\pi$
$617$	17.6030	0.708671	0.354336	+	0.935118i	$0.384707\pi$
$618$	−3.96169	−0.159363
$619$	13.2971	0.534454	0.267227	−	0.963634i	$-0.413893\pi$
$619$	13.2971	0.534454	0.267227	+	0.963634i	$0.413893\pi$
$620$	13.1576	0.528422
$621$	−10.6990	−0.429337
$622$	15.6805	0.628730
$623$	−11.5564	−0.462998
$624$	3.61428	0.144687
$625$	11.3704	0.454815
$626$	45.2331	1.80788
$627$	0	0
$628$	−57.3916	−2.29017
$629$	−20.8680	−0.832063
$630$	−21.7334	−0.865881
$631$	−39.1399	−1.55814	−0.779068	−	0.626939i	$-0.784307\pi$
$631$	−39.1399	−1.55814	−0.779068	+	0.626939i	$0.784307\pi$
$632$	−47.0114	−1.87001
$633$	2.96226	0.117739
$634$	−15.2204	−0.604479
$635$	2.92634	0.116128
$636$	−15.6059	−0.618814
$637$	−8.10231	−0.321025
$638$	0	0
$639$	−24.2548	−0.959507
$640$	18.8188	0.743878
$641$	4.17668	0.164969	0.0824846	−	0.996592i	$-0.473714\pi$
$641$	4.17668	0.164969	0.0824846	+	0.996592i	$0.473714\pi$
$642$	29.7087	1.17251
$643$	−43.8692	−1.73003	−0.865017	−	0.501743i	$-0.832692\pi$
$643$	−43.8692	−1.73003	−0.865017	+	0.501743i	$0.832692\pi$
$644$	38.1414	1.50298
$645$	−5.84378	−0.230099
$646$	−28.3721	−1.11629
$647$	−3.38093	−0.132918	−0.0664590	−	0.997789i	$-0.521170\pi$
$647$	−3.38093	−0.132918	−0.0664590	+	0.997789i	$0.521170\pi$
$648$	16.3535	0.642425
$649$	0	0
$650$	9.91329	0.388831
$651$	−10.6585	−0.417738
$652$	−72.2823	−2.83079
$653$	−19.0214	−0.744365	−0.372182	−	0.928160i	$-0.621390\pi$
$653$	−19.0214	−0.744365	−0.372182	+	0.928160i	$0.621390\pi$
$654$	−24.2219	−0.947153
$655$	15.5769	0.608639
$656$	14.8845	0.581143
$657$	−36.9631	−1.44207
$658$	−18.4382	−0.718797
$659$	2.06130	0.0802970	0.0401485	−	0.999194i	$-0.487217\pi$
$659$	2.06130	0.0802970	0.0401485	+	0.999194i	$0.487217\pi$
$660$	0	0
$661$	−42.9729	−1.67145	−0.835725	−	0.549148i	$-0.814952\pi$
$661$	−42.9729	−1.67145	−0.835725	+	0.549148i	$0.814952\pi$
$662$	18.2960	0.711094
$663$	1.81898	0.0706433
$664$	62.1122	2.41042
$665$	−20.2321	−0.784567
$666$	54.2404	2.10177
$667$	−16.4854	−0.638317
$668$	80.7315	3.12360
$669$	15.5736	0.602110
$670$	6.72837	0.259940
$671$	0	0
$672$	1.80033	0.0694492
$673$	9.57417	0.369057	0.184529	−	0.982827i	$-0.440924\pi$
$673$	9.57417	0.369057	0.184529	+	0.982827i	$0.440924\pi$
$674$	88.6621	3.41514
$675$	−17.8103	−0.685519
$676$	4.05570	0.155989
$677$	2.78059	0.106867	0.0534333	−	0.998571i	$-0.482984\pi$
$677$	2.78059	0.106867	0.0534333	+	0.998571i	$0.482984\pi$
$678$	34.8939	1.34009
$679$	21.7764	0.835701
$680$	−10.8845	−0.417401
$681$	−9.74988	−0.373616
$682$	0	0
$683$	−48.0798	−1.83972	−0.919861	−	0.392244i	$-0.871699\pi$
$683$	−48.0798	−1.83972	−0.919861	+	0.392244i	$0.871699\pi$
$684$	49.3895	1.88845
$685$	10.2614	0.392070
$686$	−10.5416	−0.402481
$687$	−4.52521	−0.172647
$688$	−30.8587	−1.17648
$689$	−4.61767	−0.175919
$690$	−4.89135	−0.186210
$691$	25.3178	0.963134	0.481567	−	0.876409i	$-0.340068\pi$
$691$	25.3178	0.963134	0.481567	+	0.876409i	$0.340068\pi$
$692$	8.37898	0.318521
$693$	0	0
$694$	−55.6818	−2.11365
$695$	−17.1681	−0.651222
$696$	−28.7166	−1.08850
$697$	7.49102	0.283743
$698$	−9.66080	−0.365667
$699$	10.9458	0.414008
$700$	63.4927	2.39980
$701$	−12.0776	−0.456164	−0.228082	−	0.973642i	$-0.573245\pi$
$701$	−12.0776	−0.456164	−0.228082	+	0.973642i	$0.573245\pi$
$702$	−10.8797	−0.410629
$703$	50.4935	1.90440
$704$	0	0
$705$	1.58363	0.0596428
$706$	−38.9618	−1.46635
$707$	−52.6877	−1.98152
$708$	−0.147478	−0.00554256
$709$	−11.6489	−0.437483	−0.218742	−	0.975783i	$-0.570195\pi$
$709$	−11.6489	−0.437483	−0.218742	+	0.975783i	$0.570195\pi$
$710$	−25.5171	−0.957640
$711$	21.4263	0.803549
$712$	15.0433	0.563773
$713$	7.96492	0.298289
$714$	17.3953	0.651002
$715$	0	0
$716$	24.0455	0.898624
$717$	7.38363	0.275747
$718$	−5.90431	−0.220347
$719$	30.5599	1.13969	0.569846	−	0.821751i	$-0.307003\pi$
$719$	30.5599	1.13969	0.569846	+	0.821751i	$0.307003\pi$
$720$	9.85703	0.367350
$721$	−7.50793	−0.279610
$722$	21.8949	0.814844
$723$	16.6514	0.619271
$724$	−85.4115	−3.17429
$725$	−27.4427	−1.01919
$726$	0	0
$727$	4.42686	0.164183	0.0820915	−	0.996625i	$-0.473840\pi$
$727$	4.42686	0.164183	0.0820915	+	0.996625i	$0.473840\pi$
$728$	19.6591	0.728616
$729$	3.59902	0.133297
$730$	−38.8868	−1.43926
$731$	−15.5305	−0.574414
$732$	35.8902	1.32654
$733$	47.1905	1.74302	0.871511	−	0.490376i	$-0.163140\pi$
$733$	47.1905	1.74302	0.871511	+	0.490376i	$0.163140\pi$
$734$	−44.9565	−1.65937
$735$	6.65497	0.245472
$736$	−1.34536	−0.0495907
$737$	0	0
$738$	−19.4707	−0.716728
$739$	−3.94349	−0.145064	−0.0725318	−	0.997366i	$-0.523108\pi$
$739$	−3.94349	−0.145064	−0.0725318	+	0.997366i	$0.523108\pi$
$740$	38.2171	1.40489
$741$	−4.40130	−0.161686
$742$	−44.1598	−1.62116
$743$	−12.7375	−0.467293	−0.233647	−	0.972322i	$-0.575066\pi$
$743$	−12.7375	−0.467293	−0.233647	+	0.972322i	$0.575066\pi$
$744$	13.8744	0.508662
$745$	−7.34720	−0.269181
$746$	22.9265	0.839398
$747$	−28.3088	−1.03576
$748$	0	0
$749$	56.3019	2.05723
$750$	−18.2487	−0.666348
$751$	50.7641	1.85241	0.926205	−	0.377020i	$-0.123051\pi$
$751$	50.7641	1.85241	0.926205	+	0.377020i	$0.123051\pi$
$752$	8.36252	0.304950
$753$	24.6135	0.896966
$754$	−16.7638	−0.610502
$755$	2.58525	0.0940869
$756$	−69.6825	−2.53433
$757$	−5.29781	−0.192552	−0.0962761	−	0.995355i	$-0.530693\pi$
$757$	−5.29781	−0.192552	−0.0962761	+	0.995355i	$0.530693\pi$
$758$	48.4701	1.76051
$759$	0	0
$760$	26.3367	0.955333
$761$	40.4037	1.46463	0.732317	−	0.680964i	$-0.238439\pi$
$761$	40.4037	1.46463	0.732317	+	0.680964i	$0.238439\pi$
$762$	6.08793	0.220542
$763$	−45.9038	−1.66183
$764$	−39.2777	−1.42102
$765$	4.96081	0.179358
$766$	12.0777	0.436385
$767$	−0.0436376	−0.00157566
$768$	26.9733	0.973317
$769$	3.32366	0.119854	0.0599270	−	0.998203i	$-0.480913\pi$
$769$	3.32366	0.119854	0.0599270	+	0.998203i	$0.480913\pi$
$770$	0	0
$771$	7.15168	0.257562
$772$	7.01800	0.252583
$773$	−27.9917	−1.00679	−0.503396	−	0.864056i	$-0.667916\pi$
$773$	−27.9917	−1.00679	−0.503396	+	0.864056i	$0.667916\pi$
$774$	40.3669	1.45096
$775$	13.2589	0.476275
$776$	−28.3470	−1.01760
$777$	−30.9582	−1.11062
$778$	−41.8102	−1.49897
$779$	−18.1257	−0.649421
$780$	−3.33122	−0.119277
$781$	0	0
$782$	−12.9993	−0.464853
$783$	30.1180	1.07633
$784$	35.1423	1.25508
$785$	13.9482	0.497834
$786$	32.4060	1.15588
$787$	−24.3657	−0.868542	−0.434271	−	0.900782i	$-0.642994\pi$
$787$	−24.3657	−0.868542	−0.434271	+	0.900782i	$0.642994\pi$
$788$	100.338	3.57439
$789$	13.9529	0.496735
$790$	22.5414	0.801986
$791$	66.1286	2.35126
$792$	0	0
$793$	10.6196	0.377114
$794$	41.4510	1.47104
$795$	3.79280	0.134517
$796$	−8.62885	−0.305842
$797$	51.7817	1.83420	0.917101	−	0.398654i	$-0.130523\pi$
$797$	51.7817	1.83420	0.917101	+	0.398654i	$0.130523\pi$
$798$	−42.0906	−1.48999
$799$	4.20865	0.148891
$800$	−2.23958	−0.0791810
$801$	−6.85628	−0.242255
$802$	86.4087	3.05120
$803$	0	0
$804$	9.37468	0.330619
$805$	−9.26975	−0.326716
$806$	8.09944	0.285291
$807$	−6.11091	−0.215114
$808$	68.5851	2.41282
$809$	22.5900	0.794222	0.397111	−	0.917770i	$-0.370013\pi$
$809$	22.5900	0.794222	0.397111	+	0.917770i	$0.370013\pi$
$810$	−7.84128	−0.275514
$811$	12.7361	0.447226	0.223613	−	0.974678i	$-0.428215\pi$
$811$	12.7361	0.447226	0.223613	+	0.974678i	$0.428215\pi$
$812$	−107.369	−3.76791
$813$	18.2574	0.640314
$814$	0	0
$815$	17.5672	0.615353
$816$	−7.88950	−0.276188
$817$	37.5784	1.31470
$818$	−37.6842	−1.31760
$819$	−8.96001	−0.313088
$820$	−13.7188	−0.479082
$821$	−35.6034	−1.24257	−0.621283	−	0.783586i	$-0.713388\pi$
$821$	−35.6034	−1.24257	−0.621283	+	0.783586i	$0.713388\pi$
$822$	21.3478	0.744590
$823$	22.5552	0.786225	0.393113	−	0.919490i	$-0.371398\pi$
$823$	22.5552	0.786225	0.393113	+	0.919490i	$0.371398\pi$
$824$	9.77330	0.340469
$825$	0	0
$826$	−0.417316	−0.0145203
$827$	−26.8873	−0.934963	−0.467482	−	0.884003i	$-0.654839\pi$
$827$	−26.8873	−0.934963	−0.467482	+	0.884003i	$0.654839\pi$
$828$	22.6288	0.786406
$829$	32.3304	1.12288	0.561441	−	0.827517i	$-0.310247\pi$
$829$	32.3304	1.12288	0.561441	+	0.827517i	$0.310247\pi$
$830$	−29.7820	−1.03375
$831$	8.01226	0.277942
$832$	7.30656	0.253309
$833$	17.6863	0.612793
$834$	−35.7163	−1.23675
$835$	−19.6207	−0.679002
$836$	0	0
$837$	−14.5515	−0.502974
$838$	35.7728	1.23575
$839$	39.6152	1.36767	0.683834	−	0.729637i	$-0.260311\pi$
$839$	39.6152	1.36767	0.683834	+	0.729637i	$0.260311\pi$
$840$	−16.1474	−0.557137
$841$	17.4067	0.600232
$842$	−14.8565	−0.511991
$843$	−1.37601	−0.0473922
$844$	−14.4175	−0.496270
$845$	−0.985684	−0.0339085
$846$	−10.9392	−0.376096
$847$	0	0
$848$	20.0283	0.687775
$849$	−12.1985	−0.418652
$850$	−21.6394	−0.742226
$851$	23.1346	0.793044
$852$	−35.5531	−1.21803
$853$	9.06728	0.310458	0.155229	−	0.987879i	$-0.450389\pi$
$853$	9.06728	0.310458	0.155229	+	0.987879i	$0.450389\pi$
$854$	101.558	3.47524
$855$	−12.0034	−0.410509
$856$	−73.2899	−2.50500
$857$	−31.9836	−1.09254	−0.546269	−	0.837610i	$-0.683952\pi$
$857$	−31.9836	−1.09254	−0.546269	+	0.837610i	$0.683952\pi$
$858$	0	0
$859$	−30.4821	−1.04004	−0.520018	−	0.854155i	$-0.674075\pi$
$859$	−30.4821	−1.04004	−0.520018	+	0.854155i	$0.674075\pi$
$860$	28.4420	0.969864
$861$	11.1131	0.378733
$862$	12.1351	0.413323
$863$	−11.2299	−0.382269	−0.191134	−	0.981564i	$-0.561217\pi$
$863$	−11.2299	−0.382269	−0.191134	+	0.981564i	$0.561217\pi$
$864$	2.45791	0.0836198
$865$	−2.03640	−0.0692396
$866$	−13.6125	−0.462573
$867$	10.1955	0.346256
$868$	51.8753	1.76076
$869$	0	0
$870$	13.7692	0.466821
$871$	2.77390	0.0939899
$872$	59.7544	2.02354
$873$	12.9197	0.437264
$874$	31.4538	1.06394
$875$	−34.5837	−1.16914
$876$	−54.1811	−1.83061
$877$	−17.4892	−0.590568	−0.295284	−	0.955410i	$-0.595414\pi$
$877$	−17.4892	−0.590568	−0.295284	+	0.955410i	$0.595414\pi$
$878$	39.6739	1.33893
$879$	0.338127	0.0114047
$880$	0	0
$881$	35.5824	1.19880	0.599400	−	0.800450i	$-0.295406\pi$
$881$	35.5824	1.19880	0.599400	+	0.800450i	$0.295406\pi$
$882$	−45.9704	−1.54790
$883$	−30.3668	−1.02192	−0.510962	−	0.859603i	$-0.670711\pi$
$883$	−30.3668	−1.02192	−0.510962	+	0.859603i	$0.670711\pi$
$884$	−8.85307	−0.297761
$885$	0.0358425	0.00120483
$886$	5.45484	0.183259
$887$	9.74009	0.327040	0.163520	−	0.986540i	$-0.447715\pi$
$887$	9.74009	0.327040	0.163520	+	0.986540i	$0.447715\pi$
$888$	40.2992	1.35235
$889$	11.5374	0.386953
$890$	−7.21309	−0.241783
$891$	0	0
$892$	−75.7975	−2.53789
$893$	−10.1835	−0.340778
$894$	−15.2850	−0.511208
$895$	−5.84394	−0.195341
$896$	74.1952	2.47869
$897$	−2.01655	−0.0673306
$898$	0.515281	0.0171952
$899$	−22.4214	−0.747796
$900$	37.6694	1.25565
$901$	10.0798	0.335806
$902$	0	0
$903$	−23.0398	−0.766715
$904$	−86.0817	−2.86303
$905$	20.7581	0.690023
$906$	5.37833	0.178683
$907$	−3.10515	−0.103105	−0.0515525	−	0.998670i	$-0.516417\pi$
$907$	−3.10515	−0.103105	−0.0515525	+	0.998670i	$0.516417\pi$
$908$	47.4532	1.57479
$909$	−31.2589	−1.03679
$910$	−9.42631	−0.312479
$911$	13.0287	0.431660	0.215830	−	0.976431i	$-0.430754\pi$
$911$	13.0287	0.431660	0.215830	+	0.976431i	$0.430754\pi$
$912$	19.0899	0.632129
$913$	0	0
$914$	−82.8431	−2.74021
$915$	−8.72262	−0.288361
$916$	22.0244	0.727708
$917$	61.4136	2.02806
$918$	23.7490	0.783835
$919$	34.3125	1.13187	0.565933	−	0.824451i	$-0.308516\pi$
$919$	34.3125	1.13187	0.565933	+	0.824451i	$0.308516\pi$
$920$	12.0667	0.397828
$921$	−7.50119	−0.247173
$922$	52.7356	1.73675
$923$	−10.5199	−0.346267
$924$	0	0
$925$	38.5114	1.26625
$926$	−35.5832	−1.16934
$927$	−4.45436	−0.146300
$928$	3.78722	0.124322
$929$	35.8697	1.17685	0.588424	−	0.808553i	$-0.299749\pi$
$929$	35.8697	1.17685	0.588424	+	0.808553i	$0.299749\pi$
$930$	−6.65262	−0.218148
$931$	−42.7947	−1.40254
$932$	−53.2737	−1.74504
$933$	−5.30978	−0.173834
$934$	−29.0351	−0.950057
$935$	0	0
$936$	11.6635	0.381234
$937$	−22.0230	−0.719461	−0.359730	−	0.933056i	$-0.617131\pi$
$937$	−22.0230	−0.719461	−0.359730	+	0.933056i	$0.617131\pi$
$938$	26.5274	0.866149
$939$	−15.3170	−0.499852
$940$	−7.70759	−0.251394
$941$	−1.81992	−0.0593278	−0.0296639	−	0.999560i	$-0.509444\pi$
$941$	−1.81992	−0.0593278	−0.0296639	+	0.999560i	$0.509444\pi$
$942$	29.0178	0.945450
$943$	−8.30466	−0.270437
$944$	0.189270	0.00616022
$945$	16.9354	0.550908
$946$	0	0
$947$	25.5160	0.829160	0.414580	−	0.910013i	$-0.363929\pi$
$947$	25.5160	0.829160	0.414580	+	0.910013i	$0.363929\pi$
$948$	31.4070	1.02005
$949$	−16.0318	−0.520414
$950$	52.3600	1.69878
$951$	5.15399	0.167130
$952$	−42.9133	−1.39083
$953$	20.0693	0.650109	0.325055	−	0.945695i	$-0.394617\pi$
$953$	20.0693	0.650109	0.325055	+	0.945695i	$0.394617\pi$
$954$	−26.1994	−0.848238
$955$	9.54591	0.308898
$956$	−35.9365	−1.16227
$957$	0	0
$958$	72.3507	2.33755
$959$	40.4569	1.30642
$960$	−6.00137	−0.193693
$961$	−20.1671	−0.650551
$962$	23.5253	0.758487
$963$	33.4032	1.07640
$964$	−81.0431	−2.61022
$965$	−1.70563	−0.0549062
$966$	−19.2847	−0.620475
$967$	−40.9837	−1.31795	−0.658973	−	0.752167i	$-0.729009\pi$
$967$	−40.9837	−1.31795	−0.658973	+	0.752167i	$0.729009\pi$
$968$	0	0
$969$	9.60747	0.308636
$970$	13.5920	0.436413
$971$	20.9281	0.671613	0.335807	−	0.941931i	$-0.390991\pi$
$971$	20.9281	0.671613	0.335807	+	0.941931i	$0.390991\pi$
$972$	−64.7180	−2.07583
$973$	−67.6870	−2.16995
$974$	0.207403	0.00664561
$975$	−3.35688	−0.107506
$976$	−46.0608	−1.47437
$977$	−15.7760	−0.504719	−0.252359	−	0.967634i	$-0.581207\pi$
$977$	−15.7760	−0.504719	−0.252359	+	0.967634i	$0.581207\pi$
$978$	36.5467	1.16863
$979$	0	0
$980$	−32.3901	−1.03466
$981$	−27.2341	−0.869519
$982$	−73.6739	−2.35103
$983$	44.7595	1.42761	0.713804	−	0.700346i	$-0.246971\pi$
$983$	44.7595	1.42761	0.713804	+	0.700346i	$0.246971\pi$
$984$	−14.4662	−0.461167
$985$	−24.3857	−0.776994
$986$	36.5932	1.16537
$987$	6.24362	0.198737
$988$	21.4214	0.681505
$989$	17.2173	0.547478
$990$	0	0
$991$	13.7020	0.435258	0.217629	−	0.976032i	$-0.430168\pi$
$991$	13.7020	0.435258	0.217629	+	0.976032i	$0.430168\pi$
$992$	−1.82980	−0.0580962
$993$	−6.19546	−0.196607
$994$	−100.604	−3.19097
$995$	2.09713	0.0664834
$996$	−41.4955	−1.31483
$997$	−11.0927	−0.351308	−0.175654	−	0.984452i	$-0.556204\pi$
$997$	−11.0927	−0.351308	−0.175654	+	0.984452i	$0.556204\pi$
$998$	−79.5162	−2.51704
$999$	−42.2658	−1.33723

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1573.2.a.o.1.8		8
11.3	even	5		143.2.h.b.53.4	yes	16
11.4	even	5		143.2.h.b.27.4	✓	16
11.10	odd	2		1573.2.a.n.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.h.b.27.4	✓	16	11.4	even	5
143.2.h.b.53.4	yes	16	11.3	even	5
1573.2.a.n.1.1		8	11.10	odd	2
1573.2.a.o.1.8		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1573 = 11^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1573.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46083 q^{2} -0.833297 q^{3} +4.05570 q^{4} -0.985684 q^{5} -2.05061 q^{6} -3.88617 q^{7} +5.05874 q^{8} -2.30562 q^{9} -2.42560 q^{10} -3.37961 q^{12} -1.00000 q^{13} -9.56322 q^{14} +0.821367 q^{15} +4.33732 q^{16} +2.18287 q^{17} -5.67374 q^{18} -5.28179 q^{19} -3.99764 q^{20} +3.23833 q^{21} -2.41996 q^{23} -4.21544 q^{24} -4.02843 q^{25} -2.46083 q^{26} +4.42115 q^{27} -15.7611 q^{28} +6.81225 q^{29} +2.02125 q^{30} -3.29134 q^{31} +0.555943 q^{32} +5.37168 q^{34} +3.83053 q^{35} -9.35089 q^{36} -9.55991 q^{37} -12.9976 q^{38} +0.833297 q^{39} -4.98632 q^{40} +3.43173 q^{41} +7.96900 q^{42} -7.11470 q^{43} +2.27261 q^{45} -5.95513 q^{46} +1.92804 q^{47} -3.61428 q^{48} +8.10231 q^{49} -9.91329 q^{50} -1.81898 q^{51} -4.05570 q^{52} +4.61767 q^{53} +10.8797 q^{54} -19.6591 q^{56} +4.40130 q^{57} +16.7638 q^{58} +0.0436376 q^{59} +3.33122 q^{60} -10.6196 q^{61} -8.09944 q^{62} +8.96001 q^{63} -7.30656 q^{64} +0.985684 q^{65} -2.77390 q^{67} +8.85307 q^{68} +2.01655 q^{69} +9.42631 q^{70} +10.5199 q^{71} -11.6635 q^{72} +16.0318 q^{73} -23.5253 q^{74} +3.35688 q^{75} -21.4214 q^{76} +2.05061 q^{78} -9.29309 q^{79} -4.27523 q^{80} +3.23271 q^{81} +8.44492 q^{82} +12.2782 q^{83} +13.1337 q^{84} -2.15162 q^{85} -17.5081 q^{86} -5.67663 q^{87} +2.97373 q^{89} +5.59251 q^{90} +3.88617 q^{91} -9.81465 q^{92} +2.74266 q^{93} +4.74458 q^{94} +5.20618 q^{95} -0.463266 q^{96} -5.60356 q^{97} +19.9384 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 5 q^{3} + q^{4} - 5 q^{5} + 3 q^{6} + 2 q^{7} + 3 q^{8} - q^{9} - 6 q^{10} - 7 q^{12} - 8 q^{13} + 2 q^{14} + 5 q^{15} - 5 q^{16} - 13 q^{17} - 5 q^{18} + 6 q^{19} - 3 q^{20} - 4 q^{21}+ \cdots + 12 q^{98}+O(q^{100}) \) 8 * q + q^2 - 5 * q^3 + q^4 - 5 * q^5 + 3 * q^6 + 2 * q^7 + 3 * q^8 - q^9 - 6 * q^10 - 7 * q^12 - 8 * q^13 + 2 * q^14 + 5 * q^15 - 5 * q^16 - 13 * q^17 - 5 * q^18 + 6 * q^19 - 3 * q^20 - 4 * q^21 - 23 * q^23 - q^24 - q^25 - q^26 + q^27 - 32 * q^28 + 15 * q^29 - 4 * q^30 - 11 * q^31 + 12 * q^32 + 2 * q^34 + 8 * q^35 - 6 * q^36 - 11 * q^37 - 25 * q^38 + 5 * q^39 + 15 * q^40 + 11 * q^41 - 11 * q^42 - 8 * q^43 - 23 * q^45 + 12 * q^46 - 21 * q^47 + 8 * q^48 + 2 * q^49 - 12 * q^50 + 26 * q^51 - q^52 - 7 * q^53 + 3 * q^54 - 34 * q^56 - 51 * q^57 - 20 * q^58 - 22 * q^59 - 3 * q^60 + 6 * q^61 - q^62 + 14 * q^63 + 5 * q^64 + 5 * q^65 - 31 * q^67 - q^68 + 17 * q^69 - 26 * q^70 - 23 * q^71 - 27 * q^72 + 34 * q^73 + 6 * q^74 - 27 * q^75 - 2 * q^76 - 3 * q^78 - 5 * q^79 - 7 * q^80 - 32 * q^81 - 11 * q^82 + 11 * q^83 + 45 * q^84 - 4 * q^85 - 21 * q^86 + 2 * q^87 - 15 * q^89 + 3 * q^90 - 2 * q^91 - 6 * q^92 - 8 * q^93 - 8 * q^94 - 11 * q^95 - 41 * q^96 - 6 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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