LMFDB - Embedded newform 1573.2.a.n.1.1

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			1.1
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	$-0.835905\pi$
$2$	−2.46083	−1.74007	−0.870036	+	0.492988i	$0.835905\pi$
$3$	−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	$-0.237456\pi$
$7$	3.88617	1.46883	0.734417	+	0.678699i	$0.237456\pi$
$8$	−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	$-0.585277\pi$
$17$	−2.18287	−0.529424	−0.264712	+	0.964328i	$0.585277\pi$
$18$	5.67374	1.33731
$19$	5.28179	1.21173	0.605863	−	0.795569i	$-0.292828\pi$
$19$	5.28179	1.21173	0.605863	+	0.795569i	$0.292828\pi$
$20$	−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	$-0.717972\pi$
$29$	−6.81225	−1.26500	−0.632501	+	0.774559i	$0.717972\pi$
$30$	−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	$-0.586354\pi$
$41$	−3.43173	−0.535946	−0.267973	+	0.963426i	$0.586354\pi$
$42$	7.96900	1.22964
$43$	7.11470	1.08498	0.542491	−	0.840062i	$-0.317481\pi$
$43$	7.11470	1.08498	0.542491	+	0.840062i	$0.317481\pi$
$44$	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	$-0.262044\pi$
$61$	10.6196	1.35970	0.679852	+	0.733349i	$0.262044\pi$
$62$	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	$-0.887498\pi$
$73$	−16.0318	−1.87638	−0.938189	+	0.346123i	$0.887498\pi$
$74$	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	$-0.324896\pi$
$79$	9.29309	1.04555	0.522777	+	0.852469i	$0.324896\pi$
$80$	−4.27523	−0.477985
$81$	3.23271	0.359190
$82$	8.44492	0.932585
$83$	−12.2782	−1.34771	−0.673853	−	0.738865i	$-0.735362\pi$
$83$	−12.2782	−1.34771	−0.673853	+	0.738865i	$0.735362\pi$
$84$	−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	$-0.735650\pi$
$101$	−13.5577	−1.34905	−0.674523	+	0.738254i	$0.735650\pi$
$102$	−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	$-0.253053\pi$
$107$	14.4878	1.40059	0.700293	+	0.713856i	$0.253053\pi$
$108$	17.9309	1.72540
$109$	−11.8121	−1.13139	−0.565697	−	0.824613i	$-0.691393\pi$
$109$	−11.8121	−1.13139	−0.565697	+	0.824613i	$0.691393\pi$
$110$	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	$-0.457950\pi$
$127$	2.96884	0.263442	0.131721	+	0.991287i	$0.457950\pi$
$128$	19.0921	1.68752
$129$	−5.92866	−0.521989
$130$	2.42560	0.212740
$131$	15.8031	1.38073	0.690363	−	0.723463i	$-0.257451\pi$
$131$	15.8031	1.38073	0.690363	+	0.723463i	$0.257451\pi$
$132$	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	$-0.764543\pi$
$139$	−17.4174	−1.47733	−0.738663	+	0.674075i	$0.764543\pi$
$140$	−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	$-0.598765\pi$
$149$	−7.45391	−0.610648	−0.305324	+	0.952248i	$0.598765\pi$
$150$	−8.26072	−0.674485
$151$	2.62280	0.213440	0.106720	−	0.994289i	$-0.465965\pi$
$151$	2.62280	0.213440	0.106720	+	0.994289i	$0.465965\pi$
$152$	−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	$-0.779830\pi$
$167$	−19.9057	−1.54035	−0.770173	+	0.637834i	$0.779830\pi$
$168$	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	$-0.525025\pi$
$173$	−2.06598	−0.157073	−0.0785366	+	0.996911i	$0.525025\pi$
$174$	−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	$-0.519837\pi$
$193$	−1.73040	−0.124557	−0.0622786	+	0.998059i	$0.519837\pi$
$194$	13.7894	0.990023
$195$	0.821367	0.0588193
$196$	32.8606	2.34718
$197$	−24.7399	−1.76265	−0.881323	−	0.472514i	$-0.843347\pi$
$197$	−24.7399	−1.76265	−0.881323	+	0.472514i	$0.843347\pi$
$198$	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	$-0.460953\pi$
$211$	3.55487	0.244727	0.122364	+	0.992485i	$0.460953\pi$
$212$	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	$-0.626934\pi$
$227$	−11.7004	−0.776580	−0.388290	+	0.921537i	$0.626934\pi$
$228$	−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	$-0.358419\pi$
$233$	13.1355	0.860536	0.430268	+	0.902701i	$0.358419\pi$
$234$	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	$-0.407483\pi$
$239$	8.86074	0.573154	0.286577	+	0.958057i	$0.407483\pi$
$240$	3.56254	0.229961
$241$	19.9825	1.28719	0.643593	−	0.765368i	$-0.277443\pi$
$241$	19.9825	1.28719	0.643593	+	0.765368i	$0.277443\pi$
$242$	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	$-0.327329\pi$
$263$	16.7442	1.03249	0.516245	+	0.856441i	$0.327329\pi$
$264$	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	$-0.268234\pi$
$271$	21.9098	1.33093	0.665463	+	0.746431i	$0.268234\pi$
$272$	−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	$-0.406724\pi$
$277$	9.61513	0.577717	0.288859	+	0.957372i	$0.406724\pi$
$278$	42.8614	2.57066
$279$	7.58857	0.454316
$280$	19.3777	1.15804
$281$	−1.65128	−0.0985071	−0.0492535	−	0.998786i	$-0.515684\pi$
$281$	−1.65128	−0.0985071	−0.0492535	+	0.998786i	$0.515684\pi$
$282$	3.95364	0.235436
$283$	−14.6389	−0.870189	−0.435095	−	0.900385i	$-0.643285\pi$
$283$	−14.6389	−0.870189	−0.435095	+	0.900385i	$0.643285\pi$
$284$	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	$-0.496227\pi$
$293$	0.405770	0.0237054	0.0118527	+	0.999930i	$0.496227\pi$
$294$	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	$-0.582695\pi$
$307$	−9.00182	−0.513761	−0.256880	+	0.966443i	$0.582695\pi$
$308$	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	$-0.938379\pi$
$337$	−36.0293	−1.96264	−0.981321	+	0.192380i	$0.938379\pi$
$338$	−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	$-0.292234\pi$
$347$	22.6272	1.21469	0.607346	+	0.794437i	$0.292234\pi$
$348$	23.0227	1.23415
$349$	3.92582	0.210145	0.105072	−	0.994465i	$-0.466493\pi$
$349$	3.92582	0.210145	0.105072	+	0.994465i	$0.466493\pi$
$350$	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	$-0.479833\pi$
$359$	2.39931	0.126631	0.0633154	+	0.997994i	$0.479833\pi$
$360$	−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	$-0.577540\pi$
$373$	−9.31655	−0.482393	−0.241196	+	0.970476i	$0.577540\pi$
$374$	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	$-0.376404\pi$
$409$	15.3136	0.757209	0.378605	+	0.925559i	$0.376404\pi$
$410$	−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	$-0.537894\pi$
$431$	−4.93130	−0.237532	−0.118766	+	0.992922i	$0.537894\pi$
$432$	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	$-0.625707\pi$
$439$	−16.1221	−0.769468	−0.384734	+	0.923028i	$0.625707\pi$
$440$	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	$-0.211435\pi$
$457$	33.6647	1.57477	0.787383	+	0.616464i	$0.211435\pi$
$458$	−13.3635	−0.624436
$459$	−9.65080	−0.450461
$460$	9.67414	0.451059
$461$	−21.4300	−0.998093	−0.499046	−	0.866575i	$-0.666316\pi$
$461$	−21.4300	−0.998093	−0.499046	+	0.866575i	$0.666316\pi$
$462$	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	$-0.734427\pi$
$479$	−29.4009	−1.34336	−0.671681	+	0.740841i	$0.734427\pi$
$480$	−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	$-0.263904\pi$
$491$	29.9386	1.35111	0.675555	+	0.737310i	$0.263904\pi$
$492$	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	$-0.451129\pi$
$503$	6.85971	0.305859	0.152930	+	0.988237i	$0.451129\pi$
$504$	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	$-0.320926\pi$
$523$	24.3954	1.06674	0.533369	+	0.845883i	$0.320926\pi$
$524$	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	$-0.605901\pi$
$541$	−15.1928	−0.653188	−0.326594	+	0.945165i	$0.605901\pi$
$542$	−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	$-0.468963\pi$
$547$	4.55377	0.194705	0.0973525	+	0.995250i	$0.468963\pi$
$548$	−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	$-0.580610\pi$
$557$	−11.8262	−0.501091	−0.250546	+	0.968105i	$0.580610\pi$
$558$	−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	$-0.540260\pi$
$563$	−5.98613	−0.252285	−0.126143	+	0.992012i	$0.540260\pi$
$564$	−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	$-0.617232\pi$
$569$	−17.1759	−0.720051	−0.360025	+	0.932942i	$0.617232\pi$
$570$	−10.6758	−0.447161
$571$	17.1160	0.716282	0.358141	−	0.933668i	$-0.383411\pi$
$571$	17.1160	0.716282	0.358141	+	0.933668i	$0.383411\pi$
$572$	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	$-0.494189\pi$
$593$	0.889024	0.0365078	0.0182539	+	0.999833i	$0.494189\pi$
$594$	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	$-0.309594\pi$
$601$	27.6110	1.12628	0.563138	+	0.826363i	$0.309594\pi$
$602$	−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	$-0.164238\pi$
$607$	42.8599	1.73963	0.869814	+	0.493379i	$0.164238\pi$
$608$	−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	$-0.619323\pi$
$613$	−18.1307	−0.732291	−0.366145	+	0.930558i	$0.619323\pi$
$614$	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	$-0.512783\pi$
$659$	−2.06130	−0.0802970	−0.0401485	+	0.999194i	$0.512783\pi$
$660$	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	$-0.559076\pi$
$673$	−9.57417	−0.369057	−0.184529	+	0.982827i	$0.559076\pi$
$674$	88.6621	3.41514
$675$	−17.8103	−0.685519
$676$	4.05570	0.155989
$677$	−2.78059	−0.106867	−0.0534333	−	0.998571i	$-0.517016\pi$
$677$	−2.78059	−0.106867	−0.0534333	+	0.998571i	$0.517016\pi$
$678$	−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	$-0.426755\pi$
$701$	12.0776	0.456164	0.228082	+	0.973642i	$0.426755\pi$
$702$	−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	$-0.836860\pi$
$733$	−47.1905	−1.74302	−0.871511	+	0.490376i	$0.836860\pi$
$734$	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	$-0.476892\pi$
$739$	3.94349	0.145064	0.0725318	+	0.997366i	$0.476892\pi$
$740$	38.2171	1.40489
$741$	−4.40130	−0.161686
$742$	−44.1598	−1.62116
$743$	12.7375	0.467293	0.233647	−	0.972322i	$-0.424934\pi$
$743$	12.7375	0.467293	0.233647	+	0.972322i	$0.424934\pi$
$744$	−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	$-0.761561\pi$
$761$	−40.4037	−1.46463	−0.732317	+	0.680964i	$0.761561\pi$
$762$	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	$-0.519087\pi$
$769$	−3.32366	−0.119854	−0.0599270	+	0.998203i	$0.519087\pi$
$770$	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	$-0.357006\pi$
$787$	24.3657	0.868542	0.434271	+	0.900782i	$0.357006\pi$
$788$	−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	$-0.629987\pi$
$809$	−22.5900	−0.794222	−0.397111	+	0.917770i	$0.629987\pi$
$810$	7.84128	0.275514
$811$	−12.7361	−0.447226	−0.223613	−	0.974678i	$-0.571785\pi$
$811$	−12.7361	−0.447226	−0.223613	+	0.974678i	$0.571785\pi$
$812$	−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	$-0.286612\pi$
$821$	35.6034	1.24257	0.621283	+	0.783586i	$0.286612\pi$
$822$	−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	$-0.345161\pi$
$827$	26.8873	0.934963	0.467482	+	0.884003i	$0.345161\pi$
$828$	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	$-0.549611\pi$
$853$	−9.06728	−0.310458	−0.155229	+	0.987879i	$0.549611\pi$
$854$	−101.558	−3.47524
$855$	12.0034	0.410509
$856$	−73.2899	−2.50500
$857$	31.9836	1.09254	0.546269	−	0.837610i	$-0.316048\pi$
$857$	31.9836	1.09254	0.546269	+	0.837610i	$0.316048\pi$
$858$	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	$-0.404586\pi$
$877$	17.4892	0.590568	0.295284	+	0.955410i	$0.404586\pi$
$878$	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	$-0.552285\pi$
$887$	−9.74009	−0.327040	−0.163520	+	0.986540i	$0.552285\pi$
$888$	−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	$-0.691484\pi$
$919$	−34.3125	−1.13187	−0.565933	+	0.824451i	$0.691484\pi$
$920$	−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	$-0.382869\pi$
$937$	22.0230	0.719461	0.359730	+	0.933056i	$0.382869\pi$
$938$	26.5274	0.866149
$939$	−15.3170	−0.499852
$940$	−7.70759	−0.251394
$941$	1.81992	0.0593278	0.0296639	−	0.999560i	$-0.490556\pi$
$941$	1.81992	0.0593278	0.0296639	+	0.999560i	$0.490556\pi$
$942$	−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	$-0.605383\pi$
$953$	−20.0693	−0.650109	−0.325055	+	0.945695i	$0.605383\pi$
$954$	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	$-0.270991\pi$
$967$	40.9837	1.31795	0.658973	+	0.752167i	$0.270991\pi$
$968$	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	$-0.443796\pi$
$997$	11.0927	0.351308	0.175654	+	0.984452i	$0.443796\pi$
$998$	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.n.1.1		8
11.7	odd	10		143.2.h.b.27.4	✓	16
11.8	odd	10		143.2.h.b.53.4	yes	16
11.10	odd	2		1573.2.a.o.1.8		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
143.2.h.b.27.4	✓	16	11.7	odd	10
143.2.h.b.53.4	yes	16	11.8	odd	10
1573.2.a.n.1.1		8	1.1	even	1	trivial
1573.2.a.o.1.8		8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1573 = 11^{2} \cdot 13 \)
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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