LMFDB - Embedded newform 5239.2.a.t.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5239,2,Mod(1,5239)] 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("5239.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5239, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5239 = 13^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5239.a (trivial)

Newform invariants

comment:select newform

sage:traces = [36,2,-5,28,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.8336256189$
Analytic rank:	$1$
Dimension:	$36$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Character	$\chi$	$=$	5239.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.92927 q^{2} +1.67028 q^{3} +1.72208 q^{4} +2.56491 q^{5} -3.22241 q^{6} +0.681322 q^{7} +0.536185 q^{8} -0.210175 q^{9} -4.94841 q^{10} -2.28267 q^{11} +2.87635 q^{12} -1.31445 q^{14} +4.28412 q^{15} -4.47860 q^{16} -5.55908 q^{17} +0.405484 q^{18} -7.89524 q^{19} +4.41698 q^{20} +1.13800 q^{21} +4.40389 q^{22} +8.68233 q^{23} +0.895578 q^{24} +1.57879 q^{25} -5.36188 q^{27} +1.17329 q^{28} -4.32363 q^{29} -8.26521 q^{30} -1.00000 q^{31} +7.56806 q^{32} -3.81269 q^{33} +10.7250 q^{34} +1.74753 q^{35} -0.361938 q^{36} +11.2267 q^{37} +15.2321 q^{38} +1.37527 q^{40} +6.80962 q^{41} -2.19550 q^{42} -7.36042 q^{43} -3.93094 q^{44} -0.539080 q^{45} -16.7505 q^{46} +10.8788 q^{47} -7.48051 q^{48} -6.53580 q^{49} -3.04590 q^{50} -9.28520 q^{51} -9.57178 q^{53} +10.3445 q^{54} -5.85486 q^{55} +0.365315 q^{56} -13.1872 q^{57} +8.34144 q^{58} +9.15512 q^{59} +7.37759 q^{60} +7.00621 q^{61} +1.92927 q^{62} -0.143197 q^{63} -5.64361 q^{64} +7.35571 q^{66} -4.18226 q^{67} -9.57317 q^{68} +14.5019 q^{69} -3.37146 q^{70} -9.67090 q^{71} -0.112693 q^{72} -7.54664 q^{73} -21.6594 q^{74} +2.63701 q^{75} -13.5962 q^{76} -1.55523 q^{77} +9.38808 q^{79} -11.4872 q^{80} -8.32530 q^{81} -13.1376 q^{82} -10.8075 q^{83} +1.95972 q^{84} -14.2586 q^{85} +14.2002 q^{86} -7.22165 q^{87} -1.22393 q^{88} +7.94388 q^{89} +1.04003 q^{90} +14.9516 q^{92} -1.67028 q^{93} -20.9880 q^{94} -20.2506 q^{95} +12.6408 q^{96} +0.383647 q^{97} +12.6093 q^{98} +0.479760 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 2 q^{2} - 5 q^{3} + 28 q^{4} + 5 q^{5} - 3 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9} - 15 q^{10} - q^{11} - 13 q^{12} - 19 q^{14} - 10 q^{15} + 4 q^{16} - 46 q^{17} - 9 q^{18} + 8 q^{19} - 5 q^{20} - 16 q^{21}+ \cdots - 47 q^{99}+O(q^{100}) $ 36 * q + 2 * q^2 - 5 * q^3 + 28 * q^4 + 5 * q^5 - 3 * q^6 + 5 * q^7 + 3 * q^8 + 5 * q^9 - 15 * q^10 - q^11 - 13 * q^12 - 19 * q^14 - 10 * q^15 + 4 * q^16 - 46 * q^17 - 9 * q^18 + 8 * q^19 - 5 * q^20 - 16 * q^21 - 21 * q^22 - 24 * q^23 + 57 * q^24 + 5 * q^25 - 11 * q^27 - 32 * q^28 - 73 * q^29 - 31 * q^30 - 36 * q^31 + 3 * q^32 - 8 * q^33 - 37 * q^34 - 31 * q^35 - 31 * q^36 + 6 * q^37 - 25 * q^38 - 22 * q^40 + 6 * q^41 - 37 * q^42 - 37 * q^43 - 16 * q^45 - 45 * q^46 + 13 * q^47 - 46 * q^48 - 5 * q^49 + 24 * q^50 - 46 * q^51 - 42 * q^53 - 26 * q^54 - 47 * q^55 - 95 * q^56 + 33 * q^57 + 9 * q^58 + 20 * q^59 - 40 * q^60 - 48 * q^61 - 2 * q^62 + 52 * q^63 + 21 * q^64 - 4 * q^66 - 50 * q^67 - 76 * q^68 - 33 * q^69 + 93 * q^70 + 16 * q^71 - 52 * q^72 - 43 * q^73 - 76 * q^74 - 40 * q^75 + 62 * q^76 - 47 * q^77 - 43 * q^79 - 46 * q^80 - 96 * q^81 + 7 * q^82 + 9 * q^83 - 77 * q^84 - 5 * q^85 + 64 * q^86 - 8 * q^87 - 54 * q^88 + 6 * q^89 + 27 * q^90 - 35 * q^92 + 5 * q^93 - 76 * q^94 - 63 * q^95 + 72 * q^96 - 4 * q^97 - 41 * q^98 - 47 * q^99

Coefficient data

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

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

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

$2$	−1.92927	−1.36420	−0.682100	−	0.731259i	$-0.738933\pi$
$2$	−1.92927	−1.36420	−0.682100	+	0.731259i	$0.738933\pi$
$3$	1.67028	0.964335	0.482167	−	0.876079i	$-0.339850\pi$
$3$	1.67028	0.964335	0.482167	+	0.876079i	$0.339850\pi$
$4$	1.72208	0.861039
$5$	2.56491	1.14706	0.573532	−	0.819183i	$-0.305573\pi$
$5$	2.56491	1.14706	0.573532	+	0.819183i	$0.305573\pi$
$6$	−3.22241	−1.31554
$7$	0.681322	0.257515	0.128758	−	0.991676i	$-0.458901\pi$
$7$	0.681322	0.257515	0.128758	+	0.991676i	$0.458901\pi$
$8$	0.536185	0.189570
$9$	−0.210175	−0.0700583
$10$	−4.94841	−1.56482
$11$	−2.28267	−0.688251	−0.344126	−	0.938924i	$-0.611825\pi$
$11$	−2.28267	−0.688251	−0.344126	+	0.938924i	$0.611825\pi$
$12$	2.87635	0.830330
$13$	0	0
$13$	0	0
$14$	−1.31445	−0.351302
$15$	4.28412	1.10615
$16$	−4.47860	−1.11965
$17$	−5.55908	−1.34827	−0.674137	−	0.738606i	$-0.735484\pi$
$17$	−5.55908	−1.34827	−0.674137	+	0.738606i	$0.735484\pi$
$18$	0.405484	0.0955734
$19$	−7.89524	−1.81129	−0.905647	−	0.424033i	$-0.860614\pi$
$19$	−7.89524	−1.81129	−0.905647	+	0.424033i	$0.860614\pi$
$20$	4.41698	0.987668
$21$	1.13800	0.248331
$22$	4.40389	0.938912
$23$	8.68233	1.81039	0.905195	−	0.424996i	$-0.139725\pi$
$23$	8.68233	1.81039	0.905195	+	0.424996i	$0.139725\pi$
$24$	0.895578	0.182809
$25$	1.57879	0.315757
$26$	0	0
$27$	−5.36188	−1.03189
$28$	1.17329	0.221731
$29$	−4.32363	−0.802877	−0.401439	−	0.915886i	$-0.631490\pi$
$29$	−4.32363	−0.802877	−0.401439	+	0.915886i	$0.631490\pi$
$30$	−8.26521	−1.50901
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	7.56806	1.33786
$33$	−3.81269	−0.663705
$34$	10.7250	1.83931
$35$	1.74753	0.295387
$36$	−0.361938	−0.0603229
$37$	11.2267	1.84566	0.922832	−	0.385202i	$-0.125868\pi$
$37$	11.2267	1.84566	0.922832	+	0.385202i	$0.125868\pi$
$38$	15.2321	2.47096
$39$	0	0
$40$	1.37527	0.217449
$41$	6.80962	1.06348	0.531742	−	0.846906i	$-0.321538\pi$
$41$	6.80962	1.06348	0.531742	+	0.846906i	$0.321538\pi$
$42$	−2.19550	−0.338773
$43$	−7.36042	−1.12245	−0.561227	−	0.827662i	$-0.689670\pi$
$43$	−7.36042	−1.12245	−0.561227	+	0.827662i	$0.689670\pi$
$44$	−3.93094	−0.592612
$45$	−0.539080	−0.0803614
$46$	−16.7505	−2.46973
$47$	10.8788	1.58683	0.793415	−	0.608681i	$-0.208301\pi$
$47$	10.8788	1.58683	0.793415	+	0.608681i	$0.208301\pi$
$48$	−7.48051	−1.07972
$49$	−6.53580	−0.933686
$50$	−3.04590	−0.430756
$51$	−9.28520	−1.30019
$52$	0	0
$53$	−9.57178	−1.31479	−0.657393	−	0.753548i	$-0.728341\pi$
$53$	−9.57178	−1.31479	−0.657393	+	0.753548i	$0.728341\pi$
$54$	10.3445	1.40771
$55$	−5.85486	−0.789469
$56$	0.365315	0.0488172
$57$	−13.1872	−1.74669
$58$	8.34144	1.09528
$59$	9.15512	1.19190	0.595948	−	0.803023i	$-0.296777\pi$
$59$	9.15512	1.19190	0.595948	+	0.803023i	$0.296777\pi$
$60$	7.37759	0.952442
$61$	7.00621	0.897053	0.448527	−	0.893769i	$-0.351949\pi$
$61$	7.00621	0.897053	0.448527	+	0.893769i	$0.351949\pi$
$62$	1.92927	0.245017
$63$	−0.143197	−0.0180411
$64$	−5.64361	−0.705452
$65$	0	0
$66$	7.35571	0.905426
$67$	−4.18226	−0.510945	−0.255472	−	0.966816i	$-0.582231\pi$
$67$	−4.18226	−0.510945	−0.255472	+	0.966816i	$0.582231\pi$
$68$	−9.57317	−1.16092
$69$	14.5019	1.74582
$70$	−3.37146	−0.402966
$71$	−9.67090	−1.14772	−0.573862	−	0.818952i	$-0.694556\pi$
$71$	−9.67090	−1.14772	−0.573862	+	0.818952i	$0.694556\pi$
$72$	−0.112693	−0.0132810
$73$	−7.54664	−0.883267	−0.441633	−	0.897196i	$-0.645601\pi$
$73$	−7.54664	−0.883267	−0.441633	+	0.897196i	$0.645601\pi$
$74$	−21.6594	−2.51785
$75$	2.63701	0.304496
$76$	−13.5962	−1.55959
$77$	−1.55523	−0.177235
$78$	0	0
$79$	9.38808	1.05624	0.528121	−	0.849169i	$-0.322897\pi$
$79$	9.38808	1.05624	0.528121	+	0.849169i	$0.322897\pi$
$80$	−11.4872	−1.28431
$81$	−8.32530	−0.925034
$82$	−13.1376	−1.45080
$83$	−10.8075	−1.18628	−0.593139	−	0.805100i	$-0.702111\pi$
$83$	−10.8075	−1.18628	−0.593139	+	0.805100i	$0.702111\pi$
$84$	1.95972	0.213823
$85$	−14.2586	−1.54656
$86$	14.2002	1.53125
$87$	−7.22165	−0.774243
$88$	−1.22393	−0.130472
$89$	7.94388	0.842050	0.421025	−	0.907049i	$-0.361670\pi$
$89$	7.94388	0.842050	0.421025	+	0.907049i	$0.361670\pi$
$90$	1.04003	0.109629
$91$	0	0
$92$	14.9516	1.55882
$93$	−1.67028	−0.173200
$94$	−20.9880	−2.16475
$95$	−20.2506	−2.07767
$96$	12.6408	1.29014
$97$	0.383647	0.0389535	0.0194767	−	0.999810i	$-0.493800\pi$
$97$	0.383647	0.0389535	0.0194767	+	0.999810i	$0.493800\pi$
$98$	12.6093	1.27373
$99$	0.479760	0.0482177
$100$	2.71879	0.271879
$101$	−12.8886	−1.28246	−0.641230	−	0.767348i	$-0.721576\pi$
$101$	−12.8886	−1.28246	−0.641230	+	0.767348i	$0.721576\pi$
$102$	17.9136	1.77372
$103$	−8.36750	−0.824475	−0.412237	−	0.911076i	$-0.635253\pi$
$103$	−8.36750	−0.824475	−0.412237	+	0.911076i	$0.635253\pi$
$104$	0	0
$105$	2.91886	0.284852
$106$	18.4665	1.79363
$107$	−11.9738	−1.15755	−0.578774	−	0.815488i	$-0.696469\pi$
$107$	−11.9738	−1.15755	−0.578774	+	0.815488i	$0.696469\pi$
$108$	−9.23358	−0.888502
$109$	−5.55713	−0.532277	−0.266138	−	0.963935i	$-0.585748\pi$
$109$	−5.55713	−0.532277	−0.266138	+	0.963935i	$0.585748\pi$
$110$	11.2956	1.07699
$111$	18.7518	1.77984
$112$	−3.05137	−0.288327
$113$	1.98514	0.186746	0.0933731	−	0.995631i	$-0.470235\pi$
$113$	1.98514	0.186746	0.0933731	+	0.995631i	$0.470235\pi$
$114$	25.4417	2.38284
$115$	22.2694	2.07663
$116$	−7.44563	−0.691309
$117$	0	0
$118$	−17.6627	−1.62598
$119$	−3.78752	−0.347201
$120$	2.29708	0.209694
$121$	−5.78941	−0.526310
$122$	−13.5169	−1.22376
$123$	11.3740	1.02556
$124$	−1.72208	−0.154647
$125$	−8.77512	−0.784871
$126$	0.276265	0.0246116
$127$	−8.35141	−0.741068	−0.370534	−	0.928819i	$-0.620825\pi$
$127$	−8.35141	−0.741068	−0.370534	+	0.928819i	$0.620825\pi$
$128$	−4.24807	−0.375480
$129$	−12.2939	−1.08242
$130$	0	0
$131$	−0.186139	−0.0162631	−0.00813154	−	0.999967i	$-0.502588\pi$
$131$	−0.186139	−0.0162631	−0.00813154	+	0.999967i	$0.502588\pi$
$132$	−6.56576	−0.571476
$133$	−5.37920	−0.466436
$134$	8.06871	0.697031
$135$	−13.7528	−1.18365
$136$	−2.98070	−0.255592
$137$	9.97675	0.852371	0.426186	−	0.904636i	$-0.359857\pi$
$137$	9.97675	0.852371	0.426186	+	0.904636i	$0.359857\pi$
$138$	−27.9780	−2.38165
$139$	1.91146	0.162128	0.0810639	−	0.996709i	$-0.474168\pi$
$139$	1.91146	0.162128	0.0810639	+	0.996709i	$0.474168\pi$
$140$	3.00939	0.254340
$141$	18.1705	1.53024
$142$	18.6578	1.56572
$143$	0	0
$144$	0.941289	0.0784408
$145$	−11.0897	−0.920952
$146$	14.5595	1.20495
$147$	−10.9166	−0.900386
$148$	19.3333	1.58919
$149$	−19.2851	−1.57990	−0.789949	−	0.613173i	$-0.789893\pi$
$149$	−19.2851	−1.57990	−0.789949	+	0.613173i	$0.789893\pi$
$150$	−5.08750	−0.415393
$151$	9.96896	0.811262	0.405631	−	0.914037i	$-0.367052\pi$
$151$	9.96896	0.811262	0.405631	+	0.914037i	$0.367052\pi$
$152$	−4.23331	−0.343367
$153$	1.16838	0.0944578
$154$	3.00046	0.241784
$155$	−2.56491	−0.206019
$156$	0	0
$157$	−14.0155	−1.11856	−0.559278	−	0.828980i	$-0.688922\pi$
$157$	−14.0155	−1.11856	−0.559278	+	0.828980i	$0.688922\pi$
$158$	−18.1121	−1.44092
$159$	−15.9875	−1.26789
$160$	19.4114	1.53461
$161$	5.91546	0.466203
$162$	16.0617	1.26193
$163$	−12.3709	−0.968962	−0.484481	−	0.874802i	$-0.660992\pi$
$163$	−12.3709	−0.968962	−0.484481	+	0.874802i	$0.660992\pi$
$164$	11.7267	0.915702
$165$	−9.77924	−0.761312
$166$	20.8506	1.61832
$167$	−21.4372	−1.65886	−0.829430	−	0.558611i	$-0.811334\pi$
$167$	−21.4372	−1.65886	−0.829430	+	0.558611i	$0.811334\pi$
$168$	0.610177	0.0470761
$169$	0	0
$170$	27.5086	2.10981
$171$	1.65938	0.126896
$172$	−12.6752	−0.966477
$173$	1.18863	0.0903698	0.0451849	−	0.998979i	$-0.485612\pi$
$173$	1.18863	0.0903698	0.0451849	+	0.998979i	$0.485612\pi$
$174$	13.9325	1.05622
$175$	1.07566	0.0813123
$176$	10.2232	0.770601
$177$	15.2916	1.14939
$178$	−15.3259	−1.14872
$179$	−6.84998	−0.511991	−0.255996	−	0.966678i	$-0.582403\pi$
$179$	−6.84998	−0.511991	−0.255996	+	0.966678i	$0.582403\pi$
$180$	−0.928339	−0.0691943
$181$	14.6857	1.09158	0.545790	−	0.837922i	$-0.316230\pi$
$181$	14.6857	1.09158	0.545790	+	0.837922i	$0.316230\pi$
$182$	0	0
$183$	11.7023	0.865060
$184$	4.65533	0.343196
$185$	28.7956	2.11710
$186$	3.22241	0.236279
$187$	12.6895	0.927952
$188$	18.7341	1.36632
$189$	−3.65317	−0.265729
$190$	39.0689	2.83436
$191$	−16.7694	−1.21339	−0.606697	−	0.794933i	$-0.707506\pi$
$191$	−16.7694	−1.21339	−0.606697	+	0.794933i	$0.707506\pi$
$192$	−9.42640	−0.680292
$193$	−16.7070	−1.20259	−0.601297	−	0.799026i	$-0.705349\pi$
$193$	−16.7070	−1.20259	−0.601297	+	0.799026i	$0.705349\pi$
$194$	−0.740159	−0.0531403
$195$	0	0
$196$	−11.2552	−0.803940
$197$	−6.67130	−0.475310	−0.237655	−	0.971350i	$-0.576379\pi$
$197$	−6.67130	−0.475310	−0.237655	+	0.971350i	$0.576379\pi$
$198$	−0.925586	−0.0657786
$199$	−9.25209	−0.655863	−0.327932	−	0.944701i	$-0.606352\pi$
$199$	−9.25209	−0.655863	−0.327932	+	0.944701i	$0.606352\pi$
$200$	0.846522	0.0598581
$201$	−6.98554	−0.492722
$202$	24.8655	1.74953
$203$	−2.94578	−0.206753
$204$	−15.9898	−1.11951
$205$	17.4661	1.21989
$206$	16.1432	1.12475
$207$	−1.82481	−0.126833
$208$	0	0
$209$	18.0223	1.24663
$210$	−5.63127	−0.388595
$211$	7.27534	0.500855	0.250428	−	0.968135i	$-0.419429\pi$
$211$	7.27534	0.500855	0.250428	+	0.968135i	$0.419429\pi$
$212$	−16.4834	−1.13208
$213$	−16.1531	−1.10679
$214$	23.1006	1.57913
$215$	−18.8788	−1.28753
$216$	−2.87496	−0.195616
$217$	−0.681322	−0.0462511
$218$	10.7212	0.726131
$219$	−12.6050	−0.851765
$220$	−10.0825	−0.679764
$221$	0	0
$222$	−36.1772	−2.42805
$223$	−12.1265	−0.812050	−0.406025	−	0.913862i	$-0.633085\pi$
$223$	−12.1265	−0.812050	−0.406025	+	0.913862i	$0.633085\pi$
$224$	5.15628	0.344519
$225$	−0.331821	−0.0221214
$226$	−3.82987	−0.254759
$227$	9.19394	0.610223	0.305112	−	0.952317i	$-0.401306\pi$
$227$	9.19394	0.610223	0.305112	+	0.952317i	$0.401306\pi$
$228$	−22.7095	−1.50397
$229$	−5.81860	−0.384504	−0.192252	−	0.981346i	$-0.561579\pi$
$229$	−5.81860	−0.384504	−0.192252	+	0.981346i	$0.561579\pi$
$230$	−42.9637	−2.83294
$231$	−2.59767	−0.170914
$232$	−2.31826	−0.152202
$233$	3.10580	0.203468	0.101734	−	0.994812i	$-0.467561\pi$
$233$	3.10580	0.203468	0.101734	+	0.994812i	$0.467561\pi$
$234$	0	0
$235$	27.9031	1.82020
$236$	15.7658	1.02627
$237$	15.6807	1.01857
$238$	7.30714	0.473652
$239$	14.2363	0.920868	0.460434	−	0.887694i	$-0.347694\pi$
$239$	14.2363	0.920868	0.460434	+	0.887694i	$0.347694\pi$
$240$	−19.1869	−1.23851
$241$	−20.0648	−1.29249	−0.646244	−	0.763131i	$-0.723661\pi$
$241$	−20.0648	−1.29249	−0.646244	+	0.763131i	$0.723661\pi$
$242$	11.1693	0.717991
$243$	2.18008	0.139852
$244$	12.0652	0.772398
$245$	−16.7638	−1.07100
$246$	−21.9434	−1.39906
$247$	0	0
$248$	−0.536185	−0.0340478
$249$	−18.0515	−1.14397
$250$	16.9296	1.07072
$251$	−0.936679	−0.0591226	−0.0295613	−	0.999563i	$-0.509411\pi$
$251$	−0.936679	−0.0591226	−0.0295613	+	0.999563i	$0.509411\pi$
$252$	−0.246596	−0.0155341
$253$	−19.8189	−1.24600
$254$	16.1121	1.01096
$255$	−23.8157	−1.49140
$256$	19.4829	1.21768
$257$	12.9018	0.804793	0.402396	−	0.915466i	$-0.368177\pi$
$257$	12.9018	0.804793	0.402396	+	0.915466i	$0.368177\pi$
$258$	23.7183	1.47664
$259$	7.64902	0.475287
$260$	0	0
$261$	0.908717	0.0562482
$262$	0.359113	0.0221861
$263$	−15.7166	−0.969129	−0.484565	−	0.874755i	$-0.661022\pi$
$263$	−15.7166	−0.969129	−0.484565	+	0.874755i	$0.661022\pi$
$264$	−2.04431	−0.125819
$265$	−24.5508	−1.50814
$266$	10.3779	0.636311
$267$	13.2685	0.812018
$268$	−7.20219	−0.439944
$269$	−27.8154	−1.69593	−0.847967	−	0.530049i	$-0.822174\pi$
$269$	−27.8154	−1.69593	−0.847967	+	0.530049i	$0.822174\pi$
$270$	26.5328	1.61473
$271$	6.34889	0.385668	0.192834	−	0.981231i	$-0.438232\pi$
$271$	6.34889	0.385668	0.192834	+	0.981231i	$0.438232\pi$
$272$	24.8969	1.50960
$273$	0	0
$274$	−19.2478	−1.16280
$275$	−3.60385	−0.217320
$276$	24.9734	1.50322
$277$	16.5006	0.991424	0.495712	−	0.868487i	$-0.334907\pi$
$277$	16.5006	0.991424	0.495712	+	0.868487i	$0.334907\pi$
$278$	−3.68772	−0.221175
$279$	0.210175	0.0125828
$280$	0.937001	0.0559965
$281$	4.73139	0.282251	0.141126	−	0.989992i	$-0.454928\pi$
$281$	4.73139	0.282251	0.141126	+	0.989992i	$0.454928\pi$
$282$	−35.0558	−2.08755
$283$	2.83565	0.168562	0.0842809	−	0.996442i	$-0.473141\pi$
$283$	2.83565	0.168562	0.0842809	+	0.996442i	$0.473141\pi$
$284$	−16.6540	−0.988236
$285$	−33.8242	−2.00357
$286$	0	0
$287$	4.63954	0.273864
$288$	−1.59062	−0.0937279
$289$	13.9033	0.817844
$290$	21.3951	1.25636
$291$	0.640797	0.0375642
$292$	−12.9959	−0.760528
$293$	−26.3914	−1.54180	−0.770900	−	0.636956i	$-0.780193\pi$
$293$	−26.3914	−1.54180	−0.770900	+	0.636956i	$0.780193\pi$
$294$	21.0611	1.22831
$295$	23.4821	1.36718
$296$	6.01961	0.349883
$297$	12.2394	0.710203
$298$	37.2062	2.15530
$299$	0	0
$300$	4.54114	0.262183
$301$	−5.01481	−0.289049
$302$	−19.2328	−1.10672
$303$	−21.5275	−1.23672
$304$	35.3597	2.02802
$305$	17.9703	1.02898
$306$	−2.25412	−0.128859
$307$	6.66645	0.380474	0.190237	−	0.981738i	$-0.439074\pi$
$307$	6.66645	0.380474	0.190237	+	0.981738i	$0.439074\pi$
$308$	−2.67823	−0.152607
$309$	−13.9760	−0.795070
$310$	4.94841	0.281051
$311$	33.6723	1.90938	0.954691	−	0.297598i	$-0.0961855\pi$
$311$	33.6723	1.90938	0.954691	+	0.297598i	$0.0961855\pi$
$312$	0	0
$313$	−3.61362	−0.204254	−0.102127	−	0.994771i	$-0.532565\pi$
$313$	−3.61362	−0.204254	−0.102127	+	0.994771i	$0.532565\pi$
$314$	27.0396	1.52593
$315$	−0.367287	−0.0206943
$316$	16.1670	0.909466
$317$	−4.11892	−0.231342	−0.115671	−	0.993288i	$-0.536902\pi$
$317$	−4.11892	−0.231342	−0.115671	+	0.993288i	$0.536902\pi$
$318$	30.8442	1.72966
$319$	9.86942	0.552582
$320$	−14.4754	−0.809199
$321$	−19.9995	−1.11626
$322$	−11.4125	−0.635994
$323$	43.8903	2.44212
$324$	−14.3368	−0.796490
$325$	0	0
$326$	23.8668	1.32186
$327$	−9.28195	−0.513293
$328$	3.65122	0.201605
$329$	7.41193	0.408633
$330$	18.8668	1.03858
$331$	33.8238	1.85913	0.929563	−	0.368664i	$-0.120185\pi$
$331$	33.8238	1.85913	0.929563	+	0.368664i	$0.120185\pi$
$332$	−18.6114	−1.02143
$333$	−2.35958	−0.129304
$334$	41.3581	2.26301
$335$	−10.7272	−0.586087
$336$	−5.09663	−0.278044
$337$	−11.2868	−0.614832	−0.307416	−	0.951575i	$-0.599464\pi$
$337$	−11.2868	−0.614832	−0.307416	+	0.951575i	$0.599464\pi$
$338$	0	0
$339$	3.31573	0.180086
$340$	−24.5544	−1.33165
$341$	2.28267	0.123614
$342$	−3.20139	−0.173112
$343$	−9.22223	−0.497954
$344$	−3.94655	−0.212784
$345$	37.1961	2.00257
$346$	−2.29319	−0.123282
$347$	25.0553	1.34504	0.672520	−	0.740079i	$-0.265212\pi$
$347$	25.0553	1.34504	0.672520	+	0.740079i	$0.265212\pi$
$348$	−12.4363	−0.666653
$349$	16.5925	0.888174	0.444087	−	0.895984i	$-0.353528\pi$
$349$	16.5925	0.888174	0.444087	+	0.895984i	$0.353528\pi$
$350$	−2.07524	−0.110926
$351$	0	0
$352$	−17.2754	−0.920782
$353$	11.1089	0.591266	0.295633	−	0.955302i	$-0.404469\pi$
$353$	11.1089	0.591266	0.295633	+	0.955302i	$0.404469\pi$
$354$	−29.5016	−1.56799
$355$	−24.8050	−1.31651
$356$	13.6800	0.725038
$357$	−6.32621	−0.334818
$358$	13.2154	0.698458
$359$	23.0012	1.21396	0.606980	−	0.794717i	$-0.292381\pi$
$359$	23.0012	1.21396	0.606980	+	0.794717i	$0.292381\pi$
$360$	−0.289047	−0.0152341
$361$	43.3349	2.28078
$362$	−28.3327	−1.48913
$363$	−9.66992	−0.507539
$364$	0	0
$365$	−19.3565	−1.01316
$366$	−22.5769	−1.18011
$367$	−10.8570	−0.566733	−0.283366	−	0.959012i	$-0.591451\pi$
$367$	−10.8570	−0.566733	−0.283366	+	0.959012i	$0.591451\pi$
$368$	−38.8847	−2.02700
$369$	−1.43121	−0.0745059
$370$	−55.5545	−2.88814
$371$	−6.52146	−0.338577
$372$	−2.87635	−0.149132
$373$	12.9528	0.670671	0.335335	−	0.942099i	$-0.391150\pi$
$373$	12.9528	0.670671	0.335335	+	0.942099i	$0.391150\pi$
$374$	−24.4816	−1.26591
$375$	−14.6569	−0.756878
$376$	5.83303	0.300815
$377$	0	0
$378$	7.04794	0.362507
$379$	−30.4334	−1.56326	−0.781629	−	0.623743i	$-0.785611\pi$
$379$	−30.4334	−1.56326	−0.781629	+	0.623743i	$0.785611\pi$
$380$	−34.8732	−1.78896
$381$	−13.9492	−0.714638
$382$	32.3528	1.65531
$383$	−32.9654	−1.68445	−0.842227	−	0.539123i	$-0.818756\pi$
$383$	−32.9654	−1.68445	−0.842227	+	0.539123i	$0.818756\pi$
$384$	−7.09545	−0.362088
$385$	−3.98904	−0.203300
$386$	32.2322	1.64058
$387$	1.54698	0.0786372
$388$	0.660671	0.0335405
$389$	17.8666	0.905871	0.452936	−	0.891543i	$-0.350377\pi$
$389$	17.8666	0.905871	0.452936	+	0.891543i	$0.350377\pi$
$390$	0	0
$391$	−48.2657	−2.44090
$392$	−3.50440	−0.176999
$393$	−0.310904	−0.0156831
$394$	12.8707	0.648418
$395$	24.0796	1.21158
$396$	0.826185	0.0415173
$397$	28.5950	1.43514	0.717570	−	0.696486i	$-0.245254\pi$
$397$	28.5950	1.43514	0.717570	+	0.696486i	$0.245254\pi$
$398$	17.8498	0.894728
$399$	−8.98476	−0.449800
$400$	−7.07076	−0.353538
$401$	−7.85670	−0.392345	−0.196172	−	0.980569i	$-0.562851\pi$
$401$	−7.85670	−0.392345	−0.196172	+	0.980569i	$0.562851\pi$
$402$	13.4770	0.672171
$403$	0	0
$404$	−22.1951	−1.10425
$405$	−21.3537	−1.06107
$406$	5.68320	0.282053
$407$	−25.6270	−1.27028
$408$	−4.97859	−0.246477
$409$	−1.80403	−0.0892036	−0.0446018	−	0.999005i	$-0.514202\pi$
$409$	−1.80403	−0.0892036	−0.0446018	+	0.999005i	$0.514202\pi$
$410$	−33.6968	−1.66417
$411$	16.6639	0.821971
$412$	−14.4095	−0.709905
$413$	6.23758	0.306931
$414$	3.52054	0.173025
$415$	−27.7203	−1.36074
$416$	0	0
$417$	3.19267	0.156346
$418$	−34.7698	−1.70065
$419$	−27.4795	−1.34246	−0.671230	−	0.741249i	$-0.734234\pi$
$419$	−27.4795	−1.34246	−0.671230	+	0.741249i	$0.734234\pi$
$420$	5.02651	0.245269
$421$	14.4423	0.703875	0.351937	−	0.936024i	$-0.385523\pi$
$421$	14.4423	0.703875	0.351937	+	0.936024i	$0.385523\pi$
$422$	−14.0361	−0.683266
$423$	−2.28644	−0.111171
$424$	−5.13225	−0.249244
$425$	−8.77659	−0.425727
$426$	31.1636	1.50988
$427$	4.77348	0.231005
$428$	−20.6198	−0.996695
$429$	0	0
$430$	36.4224	1.75644
$431$	−28.2615	−1.36131	−0.680654	−	0.732605i	$-0.738304\pi$
$431$	−28.2615	−1.36131	−0.680654	+	0.732605i	$0.738304\pi$
$432$	24.0137	1.15536
$433$	−8.67081	−0.416692	−0.208346	−	0.978055i	$-0.566808\pi$
$433$	−8.67081	−0.416692	−0.208346	+	0.978055i	$0.566808\pi$
$434$	1.31445	0.0630957
$435$	−18.5229	−0.888106
$436$	−9.56982	−0.458311
$437$	−68.5491	−3.27915
$438$	24.3184	1.16198
$439$	−16.3451	−0.780109	−0.390055	−	0.920792i	$-0.627544\pi$
$439$	−16.3451	−0.780109	−0.390055	+	0.920792i	$0.627544\pi$
$440$	−3.13929	−0.149660
$441$	1.37366	0.0654124
$442$	0	0
$443$	−7.99653	−0.379926	−0.189963	−	0.981791i	$-0.560837\pi$
$443$	−7.99653	−0.379926	−0.189963	+	0.981791i	$0.560837\pi$
$444$	32.2920	1.53251
$445$	20.3754	0.965886
$446$	23.3953	1.10780
$447$	−32.2115	−1.52355
$448$	−3.84512	−0.181665
$449$	25.5091	1.20385	0.601924	−	0.798553i	$-0.294401\pi$
$449$	25.5091	1.20385	0.601924	+	0.798553i	$0.294401\pi$
$450$	0.640172	0.0301780
$451$	−15.5441	−0.731945
$452$	3.41857	0.160796
$453$	16.6509	0.782329
$454$	−17.7376	−0.832466
$455$	0	0
$456$	−7.07081	−0.331121
$457$	−21.0600	−0.985143	−0.492571	−	0.870272i	$-0.663943\pi$
$457$	−21.0600	−0.985143	−0.492571	+	0.870272i	$0.663943\pi$
$458$	11.2256	0.524540
$459$	29.8071	1.39128
$460$	38.3497	1.78806
$461$	7.96798	0.371106	0.185553	−	0.982634i	$-0.440592\pi$
$461$	7.96798	0.371106	0.185553	+	0.982634i	$0.440592\pi$
$462$	5.01161	0.233161
$463$	−0.760281	−0.0353333	−0.0176666	−	0.999844i	$-0.505624\pi$
$463$	−0.760281	−0.0353333	−0.0176666	+	0.999844i	$0.505624\pi$
$464$	19.3638	0.898942
$465$	−4.28412	−0.198671
$466$	−5.99192	−0.277570
$467$	−0.977457	−0.0452313	−0.0226157	−	0.999744i	$-0.507199\pi$
$467$	−0.977457	−0.0452313	−0.0226157	+	0.999744i	$0.507199\pi$
$468$	0	0
$469$	−2.84947	−0.131576
$470$	−53.8325	−2.48311
$471$	−23.4097	−1.07866
$472$	4.90884	0.225948
$473$	16.8014	0.772530
$474$	−30.2523	−1.38953
$475$	−12.4649	−0.571929
$476$	−6.52241	−0.298954
$477$	2.01175	0.0921116
$478$	−27.4656	−1.25625
$479$	−25.9857	−1.18732	−0.593659	−	0.804716i	$-0.702317\pi$
$479$	−25.9857	−1.18732	−0.593659	+	0.804716i	$0.702317\pi$
$480$	32.4225	1.47988
$481$	0	0
$482$	38.7104	1.76321
$483$	9.88045	0.449576
$484$	−9.96982	−0.453173
$485$	0.984022	0.0446822
$486$	−4.20597	−0.190786
$487$	17.2531	0.781812	0.390906	−	0.920431i	$-0.372162\pi$
$487$	17.2531	0.781812	0.390906	+	0.920431i	$0.372162\pi$
$488$	3.75663	0.170055
$489$	−20.6628	−0.934404
$490$	32.3418	1.46105
$491$	−7.26010	−0.327644	−0.163822	−	0.986490i	$-0.552382\pi$
$491$	−7.26010	−0.327644	−0.163822	+	0.986490i	$0.552382\pi$
$492$	19.5868	0.883043
$493$	24.0354	1.08250
$494$	0	0
$495$	1.23054	0.0553088
$496$	4.47860	0.201095
$497$	−6.58899	−0.295557
$498$	34.8262	1.56060
$499$	13.8502	0.620018	0.310009	−	0.950734i	$-0.399668\pi$
$499$	13.8502	0.620018	0.310009	+	0.950734i	$0.399668\pi$
$500$	−15.1114	−0.675804
$501$	−35.8060	−1.59970
$502$	1.80711	0.0806551
$503$	36.9033	1.64544	0.822719	−	0.568448i	$-0.192456\pi$
$503$	36.9033	1.64544	0.822719	+	0.568448i	$0.192456\pi$
$504$	−0.0767799	−0.00342005
$505$	−33.0581	−1.47107
$506$	38.2360	1.69980
$507$	0	0
$508$	−14.3818	−0.638088
$509$	41.9785	1.86067	0.930333	−	0.366716i	$-0.119518\pi$
$509$	41.9785	1.86067	0.930333	+	0.366716i	$0.119518\pi$
$510$	45.9470	2.03457
$511$	−5.14169	−0.227455
$512$	−29.0916	−1.28568
$513$	42.3334	1.86906
$514$	−24.8911	−1.09790
$515$	−21.4619	−0.945726
$516$	−21.1711	−0.932007
$517$	−24.8326	−1.09214
$518$	−14.7570	−0.648386
$519$	1.98534	0.0871468
$520$	0	0
$521$	−20.5597	−0.900736	−0.450368	−	0.892843i	$-0.648707\pi$
$521$	−20.5597	−0.900736	−0.450368	+	0.892843i	$0.648707\pi$
$522$	−1.75316	−0.0767337
$523$	7.42281	0.324577	0.162288	−	0.986743i	$-0.448112\pi$
$523$	7.42281	0.324577	0.162288	+	0.986743i	$0.448112\pi$
$524$	−0.320547	−0.0140031
$525$	1.79665	0.0784123
$526$	30.3216	1.32209
$527$	5.55908	0.242157
$528$	17.0755	0.743118
$529$	52.3828	2.27751
$530$	47.3651	2.05741
$531$	−1.92418	−0.0835021
$532$	−9.26341	−0.401620
$533$	0	0
$534$	−25.5985	−1.10775
$535$	−30.7117	−1.32778
$536$	−2.24247	−0.0968599
$537$	−11.4414	−0.493731
$538$	53.6634	2.31359
$539$	14.9191	0.642611
$540$	−23.6833	−1.01917
$541$	7.22124	0.310465	0.155233	−	0.987878i	$-0.450387\pi$
$541$	7.22124	0.310465	0.155233	+	0.987878i	$0.450387\pi$
$542$	−12.2487	−0.526127
$543$	24.5292	1.05265
$544$	−42.0714	−1.80380
$545$	−14.2536	−0.610556
$546$	0	0
$547$	34.1669	1.46087	0.730435	−	0.682983i	$-0.239318\pi$
$547$	34.1669	1.46087	0.730435	+	0.682983i	$0.239318\pi$
$548$	17.1807	0.733925
$549$	−1.47253	−0.0628460
$550$	6.95280	0.296468
$551$	34.1361	1.45425
$552$	7.77570	0.330956
$553$	6.39630	0.271999
$554$	−31.8341	−1.35250
$555$	48.0967	2.04159
$556$	3.29168	0.139598
$557$	−17.8338	−0.755643	−0.377822	−	0.925878i	$-0.623327\pi$
$557$	−17.8338	−0.755643	−0.377822	+	0.925878i	$0.623327\pi$
$558$	−0.405484	−0.0171655
$559$	0	0
$560$	−7.82650	−0.330730
$561$	21.1951	0.894856
$562$	−9.12813	−0.385047
$563$	−11.7075	−0.493412	−0.246706	−	0.969090i	$-0.579348\pi$
$563$	−11.7075	−0.493412	−0.246706	+	0.969090i	$0.579348\pi$
$564$	31.2911	1.31759
$565$	5.09172	0.214210
$566$	−5.47073	−0.229952
$567$	−5.67221	−0.238210
$568$	−5.18539	−0.217574
$569$	−14.3584	−0.601935	−0.300967	−	0.953634i	$-0.597310\pi$
$569$	−14.3584	−0.601935	−0.300967	+	0.953634i	$0.597310\pi$
$570$	65.2559	2.73327
$571$	22.4767	0.940622	0.470311	−	0.882501i	$-0.344142\pi$
$571$	22.4767	0.940622	0.470311	+	0.882501i	$0.344142\pi$
$572$	0	0
$573$	−28.0096	−1.17012
$574$	−8.95093	−0.373604
$575$	13.7075	0.571644
$576$	1.18615	0.0494227
$577$	19.4532	0.809846	0.404923	−	0.914351i	$-0.367298\pi$
$577$	19.4532	0.809846	0.404923	+	0.914351i	$0.367298\pi$
$578$	−26.8233	−1.11570
$579$	−27.9053	−1.15970
$580$	−19.0974	−0.792976
$581$	−7.36339	−0.305485
$582$	−1.23627	−0.0512450
$583$	21.8492	0.904903
$584$	−4.04639	−0.167441
$585$	0	0
$586$	50.9160	2.10332
$587$	32.5707	1.34434	0.672168	−	0.740398i	$-0.265363\pi$
$587$	32.5707	1.34434	0.672168	+	0.740398i	$0.265363\pi$
$588$	−18.7992	−0.775268
$589$	7.89524	0.325318
$590$	−45.3033	−1.86511
$591$	−11.1429	−0.458358
$592$	−50.2801	−2.06650
$593$	3.92792	0.161300	0.0806502	−	0.996742i	$-0.474300\pi$
$593$	3.92792	0.161300	0.0806502	+	0.996742i	$0.474300\pi$
$594$	−23.6131	−0.968858
$595$	−9.71466	−0.398262
$596$	−33.2105	−1.36035
$597$	−15.4536	−0.632472
$598$	0	0
$599$	−28.4471	−1.16232	−0.581159	−	0.813790i	$-0.697401\pi$
$599$	−28.4471	−1.16232	−0.581159	+	0.813790i	$0.697401\pi$
$600$	1.41393	0.0577233
$601$	−22.1654	−0.904147	−0.452073	−	0.891981i	$-0.649316\pi$
$601$	−22.1654	−0.904147	−0.452073	+	0.891981i	$0.649316\pi$
$602$	9.67492	0.394320
$603$	0.879007	0.0357959
$604$	17.1673	0.698529
$605$	−14.8493	−0.603711
$606$	41.5323	1.68713
$607$	27.6407	1.12190	0.560950	−	0.827850i	$-0.310436\pi$
$607$	27.6407	1.12190	0.560950	+	0.827850i	$0.310436\pi$
$608$	−59.7517	−2.42325
$609$	−4.92027	−0.199379
$610$	−34.6696	−1.40373
$611$	0	0
$612$	2.01204	0.0813318
$613$	15.6537	0.632248	0.316124	−	0.948718i	$-0.397618\pi$
$613$	15.6537	0.632248	0.316124	+	0.948718i	$0.397618\pi$
$614$	−12.8614	−0.519043
$615$	29.1732	1.17638
$616$	−0.833893	−0.0335985
$617$	9.55908	0.384834	0.192417	−	0.981313i	$-0.438367\pi$
$617$	9.55908	0.384834	0.192417	+	0.981313i	$0.438367\pi$
$618$	26.9636	1.08463
$619$	−36.5043	−1.46723	−0.733616	−	0.679565i	$-0.762169\pi$
$619$	−36.5043	−1.46723	−0.733616	+	0.679565i	$0.762169\pi$
$620$	−4.41698	−0.177390
$621$	−46.5536	−1.86813
$622$	−64.9630	−2.60478
$623$	5.41234	0.216841
$624$	0	0
$625$	−30.4014	−1.21605
$626$	6.97164	0.278643
$627$	30.1022	1.20216
$628$	−24.1357	−0.963121
$629$	−62.4103	−2.48846
$630$	0.708596	0.0282311
$631$	20.3233	0.809057	0.404528	−	0.914525i	$-0.367436\pi$
$631$	20.3233	0.809057	0.404528	+	0.914525i	$0.367436\pi$
$632$	5.03375	0.200232
$633$	12.1518	0.482992
$634$	7.94651	0.315596
$635$	−21.4207	−0.850053
$636$	−27.5318	−1.09171
$637$	0	0
$638$	−19.0408	−0.753831
$639$	2.03258	0.0804076
$640$	−10.8959	−0.430699
$641$	−32.4780	−1.28280	−0.641402	−	0.767205i	$-0.721647\pi$
$641$	−32.4780	−1.28280	−0.641402	+	0.767205i	$0.721647\pi$
$642$	38.5845	1.52281
$643$	8.83568	0.348445	0.174223	−	0.984706i	$-0.444259\pi$
$643$	8.83568	0.348445	0.174223	+	0.984706i	$0.444259\pi$
$644$	10.1869	0.401419
$645$	−31.5329	−1.24161
$646$	−84.6761	−3.33154
$647$	7.40134	0.290977	0.145488	−	0.989360i	$-0.453525\pi$
$647$	7.40134	0.290977	0.145488	+	0.989360i	$0.453525\pi$
$648$	−4.46390	−0.175359
$649$	−20.8981	−0.820324
$650$	0	0
$651$	−1.13800	−0.0446016
$652$	−21.3036	−0.834315
$653$	13.4030	0.524500	0.262250	−	0.965000i	$-0.415536\pi$
$653$	13.4030	0.524500	0.262250	+	0.965000i	$0.415536\pi$
$654$	17.9074	0.700234
$655$	−0.477432	−0.0186548
$656$	−30.4976	−1.19073
$657$	1.58611	0.0618802
$658$	−14.2996	−0.557457
$659$	−10.6344	−0.414258	−0.207129	−	0.978314i	$-0.566412\pi$
$659$	−10.6344	−0.414258	−0.207129	+	0.978314i	$0.566412\pi$
$660$	−16.8406	−0.655520
$661$	−26.6564	−1.03681	−0.518407	−	0.855134i	$-0.673475\pi$
$661$	−26.6564	−1.03681	−0.518407	+	0.855134i	$0.673475\pi$
$662$	−65.2552	−2.53622
$663$	0	0
$664$	−5.79482	−0.224883
$665$	−13.7972	−0.535032
$666$	4.55226	0.176397
$667$	−37.5391	−1.45352
$668$	−36.9165	−1.42834
$669$	−20.2546	−0.783088
$670$	20.6956	0.799539
$671$	−15.9929	−0.617398
$672$	8.61242	0.332231
$673$	−34.8029	−1.34155	−0.670777	−	0.741659i	$-0.734039\pi$
$673$	−34.8029	−1.34155	−0.670777	+	0.741659i	$0.734039\pi$
$674$	21.7753	0.838754
$675$	−8.46526	−0.325828
$676$	0	0
$677$	17.3182	0.665592	0.332796	−	0.942999i	$-0.392008\pi$
$677$	17.3182	0.665592	0.332796	+	0.942999i	$0.392008\pi$
$678$	−6.39694	−0.245673
$679$	0.261387	0.0100311
$680$	−7.64523	−0.293181
$681$	15.3564	0.588460
$682$	−4.40389	−0.168634
$683$	11.5364	0.441429	0.220714	−	0.975338i	$-0.429161\pi$
$683$	11.5364	0.441429	0.220714	+	0.975338i	$0.429161\pi$
$684$	2.85759	0.109263
$685$	25.5895	0.977725
$686$	17.7922	0.679308
$687$	−9.71867	−0.370790
$688$	32.9644	1.25676
$689$	0	0
$690$	−71.7613	−2.73191
$691$	31.9993	1.21731	0.608655	−	0.793435i	$-0.291709\pi$
$691$	31.9993	1.21731	0.608655	+	0.793435i	$0.291709\pi$
$692$	2.04691	0.0778120
$693$	0.326871	0.0124168
$694$	−48.3384	−1.83490
$695$	4.90273	0.185971
$696$	−3.87214	−0.146773
$697$	−37.8552	−1.43387
$698$	−32.0113	−1.21165
$699$	5.18754	0.196211
$700$	1.85237	0.0700131
$701$	−35.4582	−1.33924	−0.669618	−	0.742706i	$-0.733542\pi$
$701$	−35.4582	−1.33924	−0.669618	+	0.742706i	$0.733542\pi$
$702$	0	0
$703$	−88.6379	−3.34304
$704$	12.8825	0.485528
$705$	46.6059	1.75528
$706$	−21.4320	−0.806605
$707$	−8.78126	−0.330253
$708$	26.3333	0.989667
$709$	11.9194	0.447643	0.223822	−	0.974630i	$-0.428147\pi$
$709$	11.9194	0.447643	0.223822	+	0.974630i	$0.428147\pi$
$710$	47.8556	1.79599
$711$	−1.97314	−0.0739985
$712$	4.25939	0.159627
$713$	−8.68233	−0.325156
$714$	12.2050	0.456759
$715$	0	0
$716$	−11.7962	−0.440845
$717$	23.7785	0.888025
$718$	−44.3756	−1.65608
$719$	−19.5383	−0.728655	−0.364328	−	0.931271i	$-0.618701\pi$
$719$	−19.5383	−0.728655	−0.364328	+	0.931271i	$0.618701\pi$
$720$	2.41433	0.0899767
$721$	−5.70096	−0.212315
$722$	−83.6047	−3.11144
$723$	−33.5138	−1.24639
$724$	25.2900	0.939894
$725$	−6.82608	−0.253514
$726$	18.6559	0.692384
$727$	22.0923	0.819360	0.409680	−	0.912229i	$-0.365640\pi$
$727$	22.0923	0.819360	0.409680	+	0.912229i	$0.365640\pi$
$728$	0	0
$729$	28.6172	1.05990
$730$	37.3438	1.38216
$731$	40.9171	1.51338
$732$	20.1523	0.744851
$733$	0.408574	0.0150910	0.00754551	−	0.999972i	$-0.497598\pi$
$733$	0.408574	0.0150910	0.00754551	+	0.999972i	$0.497598\pi$
$734$	20.9461	0.773136
$735$	−28.0001	−1.03280
$736$	65.7083	2.42204
$737$	9.54674	0.351659
$738$	2.76119	0.101641
$739$	50.4023	1.85408	0.927039	−	0.374964i	$-0.122345\pi$
$739$	50.4023	1.85408	0.927039	+	0.374964i	$0.122345\pi$
$740$	49.5883	1.82290
$741$	0	0
$742$	12.5817	0.461887
$743$	5.60492	0.205625	0.102812	−	0.994701i	$-0.467216\pi$
$743$	5.60492	0.205625	0.102812	+	0.994701i	$0.467216\pi$
$744$	−0.895578	−0.0328335
$745$	−49.4647	−1.81225
$746$	−24.9895	−0.914929
$747$	2.27146	0.0831086
$748$	21.8524	0.799003
$749$	−8.15799	−0.298087
$750$	28.2771	1.03253
$751$	−11.4272	−0.416983	−0.208492	−	0.978024i	$-0.566855\pi$
$751$	−11.4272	−0.416983	−0.208492	+	0.978024i	$0.566855\pi$
$752$	−48.7216	−1.77669
$753$	−1.56451	−0.0570140
$754$	0	0
$755$	25.5695	0.930570
$756$	−6.29104	−0.228803
$757$	9.82642	0.357147	0.178574	−	0.983927i	$-0.442852\pi$
$757$	9.82642	0.357147	0.178574	+	0.983927i	$0.442852\pi$
$758$	58.7142	2.13260
$759$	−33.1031	−1.20156
$760$	−10.8581	−0.393864
$761$	−24.5095	−0.888470	−0.444235	−	0.895910i	$-0.646525\pi$
$761$	−24.5095	−0.888470	−0.444235	+	0.895910i	$0.646525\pi$
$762$	26.9117	0.974908
$763$	−3.78619	−0.137069
$764$	−28.8783	−1.04478
$765$	2.99679	0.108349
$766$	63.5991	2.29793
$767$	0	0
$768$	32.5418	1.17425
$769$	9.64255	0.347719	0.173860	−	0.984770i	$-0.444376\pi$
$769$	9.64255	0.347719	0.173860	+	0.984770i	$0.444376\pi$
$770$	7.69593	0.277342
$771$	21.5496	0.776090
$772$	−28.7707	−1.03548
$773$	2.29144	0.0824172	0.0412086	−	0.999151i	$-0.486879\pi$
$773$	2.29144	0.0824172	0.0412086	+	0.999151i	$0.486879\pi$
$774$	−2.98453	−0.107277
$775$	−1.57879	−0.0567117
$776$	0.205706	0.00738441
$777$	12.7760	0.458336
$778$	−34.4694	−1.23579
$779$	−53.7636	−1.92628
$780$	0	0
$781$	22.0755	0.789923
$782$	93.1176	3.32988
$783$	23.1828	0.828485
$784$	29.2713	1.04540
$785$	−35.9485	−1.28306
$786$	0.599818	0.0213948
$787$	12.5303	0.446657	0.223328	−	0.974743i	$-0.428308\pi$
$787$	12.5303	0.446657	0.223328	+	0.974743i	$0.428308\pi$
$788$	−11.4885	−0.409261
$789$	−26.2511	−0.934565
$790$	−46.4561	−1.65283
$791$	1.35252	0.0480900
$792$	0.257240	0.00914064
$793$	0	0
$794$	−55.1674	−1.95782
$795$	−41.0066	−1.45436
$796$	−15.9328	−0.564724
$797$	−17.2101	−0.609612	−0.304806	−	0.952414i	$-0.598592\pi$
$797$	−17.2101	−0.609612	−0.304806	+	0.952414i	$0.598592\pi$
$798$	17.3340	0.613617
$799$	−60.4758	−2.13948
$800$	11.9483	0.422438
$801$	−1.66960	−0.0589926
$802$	15.1577	0.535236
$803$	17.2265	0.607910
$804$	−12.0296	−0.424253
$805$	15.1726	0.534765
$806$	0	0
$807$	−46.4594	−1.63545
$808$	−6.91066	−0.243116
$809$	−51.0461	−1.79469	−0.897343	−	0.441334i	$-0.854505\pi$
$809$	−51.0461	−1.79469	−0.897343	+	0.441334i	$0.854505\pi$
$810$	41.1970	1.44752
$811$	18.6496	0.654877	0.327439	−	0.944872i	$-0.393815\pi$
$811$	18.6496	0.654877	0.327439	+	0.944872i	$0.393815\pi$
$812$	−5.07287	−0.178023
$813$	10.6044	0.371913
$814$	49.4413	1.73292
$815$	−31.7303	−1.11146
$816$	41.5847	1.45576
$817$	58.1123	2.03309
$818$	3.48046	0.121691
$819$	0	0
$820$	30.0780	1.05037
$821$	−15.7296	−0.548968	−0.274484	−	0.961592i	$-0.588507\pi$
$821$	−15.7296	−0.548968	−0.274484	+	0.961592i	$0.588507\pi$
$822$	−32.1492	−1.12133
$823$	−15.0590	−0.524923	−0.262462	−	0.964942i	$-0.584534\pi$
$823$	−15.0590	−0.524923	−0.262462	+	0.964942i	$0.584534\pi$
$824$	−4.48653	−0.156296
$825$	−6.01943	−0.209570
$826$	−12.0340	−0.418716
$827$	27.6179	0.960368	0.480184	−	0.877168i	$-0.340570\pi$
$827$	27.6179	0.960368	0.480184	+	0.877168i	$0.340570\pi$
$828$	−3.14246	−0.109208
$829$	35.2019	1.22261	0.611307	−	0.791394i	$-0.290644\pi$
$829$	35.2019	1.22261	0.611307	+	0.791394i	$0.290644\pi$
$830$	53.4800	1.85632
$831$	27.5606	0.956065
$832$	0	0
$833$	36.3330	1.25886
$834$	−6.15951	−0.213286
$835$	−54.9845	−1.90282
$836$	31.0357	1.07339
$837$	5.36188	0.185334
$838$	53.0153	1.83138
$839$	10.0927	0.348439	0.174219	−	0.984707i	$-0.444260\pi$
$839$	10.0927	0.348439	0.174219	+	0.984707i	$0.444260\pi$
$840$	1.56505	0.0539994
$841$	−10.3063	−0.355388
$842$	−27.8631	−0.960226
$843$	7.90274	0.272185
$844$	12.5287	0.431256
$845$	0	0
$846$	4.41116	0.151659
$847$	−3.94445	−0.135533
$848$	42.8682	1.47210
$849$	4.73632	0.162550
$850$	16.9324	0.580777
$851$	97.4742	3.34137
$852$	−27.8169	−0.952990
$853$	−26.2755	−0.899657	−0.449828	−	0.893115i	$-0.648515\pi$
$853$	−26.2755	−0.899657	−0.449828	+	0.893115i	$0.648515\pi$
$854$	−9.20933	−0.315137
$855$	4.25617	0.145558
$856$	−6.42016	−0.219437
$857$	−13.4215	−0.458469	−0.229235	−	0.973371i	$-0.573622\pi$
$857$	−13.4215	−0.458469	−0.229235	+	0.973371i	$0.573622\pi$
$858$	0	0
$859$	19.3058	0.658705	0.329352	−	0.944207i	$-0.393170\pi$
$859$	19.3058	0.658705	0.329352	+	0.944207i	$0.393170\pi$
$860$	−32.5109	−1.10861
$861$	7.74932	0.264096
$862$	54.5240	1.85710
$863$	−49.3900	−1.68126	−0.840628	−	0.541613i	$-0.817814\pi$
$863$	−49.3900	−1.68126	−0.840628	+	0.541613i	$0.817814\pi$
$864$	−40.5790	−1.38053
$865$	3.04873	0.103660
$866$	16.7283	0.568451
$867$	23.2224	0.788675
$868$	−1.17329	−0.0398240
$869$	−21.4299	−0.726960
$870$	35.7357	1.21155
$871$	0	0
$872$	−2.97965	−0.100904
$873$	−0.0806330	−0.00272901
$874$	132.250	4.47341
$875$	−5.97868	−0.202116
$876$	−21.7068	−0.733403
$877$	22.5910	0.762844	0.381422	−	0.924401i	$-0.375434\pi$
$877$	22.5910	0.762844	0.381422	+	0.924401i	$0.375434\pi$
$878$	31.5341	1.06422
$879$	−44.0809	−1.48681
$880$	26.2216	0.883929
$881$	−34.6527	−1.16748	−0.583740	−	0.811941i	$-0.698411\pi$
$881$	−34.6527	−1.16748	−0.583740	+	0.811941i	$0.698411\pi$
$882$	−2.65016	−0.0892356
$883$	7.76871	0.261438	0.130719	−	0.991419i	$-0.458271\pi$
$883$	7.76871	0.261438	0.130719	+	0.991419i	$0.458271\pi$
$884$	0	0
$885$	39.2216	1.31842
$886$	15.4275	0.518295
$887$	26.9127	0.903641	0.451821	−	0.892109i	$-0.350775\pi$
$887$	26.9127	0.903641	0.451821	+	0.892109i	$0.350775\pi$
$888$	10.0544	0.337404
$889$	−5.69000	−0.190836
$890$	−39.3096	−1.31766
$891$	19.0039	0.636656
$892$	−20.8828	−0.699207
$893$	−85.8904	−2.87421
$894$	62.1446	2.07843
$895$	−17.5696	−0.587287
$896$	−2.89430	−0.0966918
$897$	0	0
$898$	−49.2139	−1.64229
$899$	4.32363	0.144201
$900$	−0.571422	−0.0190474
$901$	53.2103	1.77269
$902$	29.9888	0.998518
$903$	−8.37613	−0.278740
$904$	1.06440	0.0354015
$905$	37.6676	1.25211
$906$	−32.1241	−1.06725
$907$	3.77677	0.125406	0.0627028	−	0.998032i	$-0.480028\pi$
$907$	3.77677	0.125406	0.0627028	+	0.998032i	$0.480028\pi$
$908$	15.8327	0.525426
$909$	2.70885	0.0898470
$910$	0	0
$911$	−12.6436	−0.418900	−0.209450	−	0.977819i	$-0.567167\pi$
$911$	−12.6436	−0.418900	−0.209450	+	0.977819i	$0.567167\pi$
$912$	59.0604	1.95569
$913$	24.6700	0.816457
$914$	40.6303	1.34393
$915$	30.0154	0.992280
$916$	−10.0201	−0.331073
$917$	−0.126821	−0.00418799
$918$	−57.5059	−1.89798
$919$	26.4585	0.872787	0.436393	−	0.899756i	$-0.356256\pi$
$919$	26.4585	0.872787	0.436393	+	0.899756i	$0.356256\pi$
$920$	11.9405	0.393668
$921$	11.1348	0.366905
$922$	−15.3724	−0.506262
$923$	0	0
$924$	−4.47339	−0.147164
$925$	17.7246	0.582782
$926$	1.46679	0.0482016
$927$	1.75864	0.0577613
$928$	−32.7215	−1.07413
$929$	9.75416	0.320024	0.160012	−	0.987115i	$-0.448847\pi$
$929$	9.75416	0.320024	0.160012	+	0.987115i	$0.448847\pi$
$930$	8.26521	0.271027
$931$	51.6017	1.69118
$932$	5.34843	0.175194
$933$	56.2421	1.84128
$934$	1.88578	0.0617045
$935$	32.5476	1.06442
$936$	0	0
$937$	45.1905	1.47631	0.738154	−	0.674632i	$-0.235698\pi$
$937$	45.1905	1.47631	0.738154	+	0.674632i	$0.235698\pi$
$938$	5.49739	0.179496
$939$	−6.03574	−0.196969
$940$	48.0513	1.56726
$941$	1.35919	0.0443085	0.0221542	−	0.999755i	$-0.492948\pi$
$941$	1.35919	0.0443085	0.0221542	+	0.999755i	$0.492948\pi$
$942$	45.1636	1.47151
$943$	59.1234	1.92532
$944$	−41.0022	−1.33451
$945$	−9.37006	−0.304808
$946$	−32.4145	−1.05389
$947$	−7.80919	−0.253765	−0.126882	−	0.991918i	$-0.540497\pi$
$947$	−7.80919	−0.253765	−0.126882	+	0.991918i	$0.540497\pi$
$948$	27.0034	0.877030
$949$	0	0
$950$	24.0482	0.780225
$951$	−6.87974	−0.223091
$952$	−2.03081	−0.0658190
$953$	−23.4731	−0.760368	−0.380184	−	0.924911i	$-0.624139\pi$
$953$	−23.4731	−0.760368	−0.380184	+	0.924911i	$0.624139\pi$
$954$	−3.88120	−0.125659
$955$	−43.0122	−1.39184
$956$	24.5160	0.792904
$957$	16.4847	0.532874
$958$	50.1335	1.61974
$959$	6.79737	0.219499
$960$	−24.1779	−0.780339
$961$	1.00000	0.0322581
$962$	0	0
$963$	2.51659	0.0810959
$964$	−34.5532	−1.11288
$965$	−42.8519	−1.37945
$966$	−19.0620	−0.613311
$967$	−27.1193	−0.872098	−0.436049	−	0.899923i	$-0.643623\pi$
$967$	−27.1193	−0.872098	−0.436049	+	0.899923i	$0.643623\pi$
$968$	−3.10420	−0.0997726
$969$	73.3089	2.35502
$970$	−1.89844	−0.0609554
$971$	−7.79697	−0.250217	−0.125108	−	0.992143i	$-0.539928\pi$
$971$	−7.79697	−0.250217	−0.125108	+	0.992143i	$0.539928\pi$
$972$	3.75427	0.120418
$973$	1.30232	0.0417504
$974$	−33.2858	−1.06655
$975$	0	0
$976$	−31.3780	−1.00439
$977$	34.6144	1.10741	0.553707	−	0.832712i	$-0.313213\pi$
$977$	34.6144	1.10741	0.553707	+	0.832712i	$0.313213\pi$
$978$	39.8641	1.27471
$979$	−18.1333	−0.579542
$980$	−28.8685	−0.922171
$981$	1.16797	0.0372904
$982$	14.0067	0.446971
$983$	−35.2990	−1.12586	−0.562932	−	0.826503i	$-0.690327\pi$
$983$	−35.2990	−1.12586	−0.562932	+	0.826503i	$0.690327\pi$
$984$	6.09855	0.194415
$985$	−17.1113	−0.545212
$986$	−46.3707	−1.47674
$987$	12.3800	0.394059
$988$	0	0
$989$	−63.9056	−2.03208
$990$	−2.37405	−0.0754523
$991$	43.9798	1.39706	0.698532	−	0.715579i	$-0.253837\pi$
$991$	43.9798	1.39706	0.698532	+	0.715579i	$0.253837\pi$
$992$	−7.56806	−0.240286
$993$	56.4951	1.79282
$994$	12.7119	0.403198
$995$	−23.7308	−0.752318
$996$	−31.0861	−0.985002
$997$	55.2614	1.75015	0.875073	−	0.483992i	$-0.160814\pi$
$997$	55.2614	1.75015	0.875073	+	0.483992i	$0.160814\pi$
$998$	−26.7207	−0.845828
$999$	−60.1965	−1.90453

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5239.2.a.t.1.6	yes	36
13.12	even	2		5239.2.a.s.1.31	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5239.2.a.s.1.31	✓	36	13.12	even	2
5239.2.a.t.1.6	yes	36	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:	\( N \)	\(=\)	\( 5239 = 13^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5239.a (trivial)

\(f(q)\)	\(=\)	\(q-1.92927 q^{2} +1.67028 q^{3} +1.72208 q^{4} +2.56491 q^{5} -3.22241 q^{6} +0.681322 q^{7} +0.536185 q^{8} -0.210175 q^{9} -4.94841 q^{10} -2.28267 q^{11} +2.87635 q^{12} -1.31445 q^{14} +4.28412 q^{15} -4.47860 q^{16} -5.55908 q^{17} +0.405484 q^{18} -7.89524 q^{19} +4.41698 q^{20} +1.13800 q^{21} +4.40389 q^{22} +8.68233 q^{23} +0.895578 q^{24} +1.57879 q^{25} -5.36188 q^{27} +1.17329 q^{28} -4.32363 q^{29} -8.26521 q^{30} -1.00000 q^{31} +7.56806 q^{32} -3.81269 q^{33} +10.7250 q^{34} +1.74753 q^{35} -0.361938 q^{36} +11.2267 q^{37} +15.2321 q^{38} +1.37527 q^{40} +6.80962 q^{41} -2.19550 q^{42} -7.36042 q^{43} -3.93094 q^{44} -0.539080 q^{45} -16.7505 q^{46} +10.8788 q^{47} -7.48051 q^{48} -6.53580 q^{49} -3.04590 q^{50} -9.28520 q^{51} -9.57178 q^{53} +10.3445 q^{54} -5.85486 q^{55} +0.365315 q^{56} -13.1872 q^{57} +8.34144 q^{58} +9.15512 q^{59} +7.37759 q^{60} +7.00621 q^{61} +1.92927 q^{62} -0.143197 q^{63} -5.64361 q^{64} +7.35571 q^{66} -4.18226 q^{67} -9.57317 q^{68} +14.5019 q^{69} -3.37146 q^{70} -9.67090 q^{71} -0.112693 q^{72} -7.54664 q^{73} -21.6594 q^{74} +2.63701 q^{75} -13.5962 q^{76} -1.55523 q^{77} +9.38808 q^{79} -11.4872 q^{80} -8.32530 q^{81} -13.1376 q^{82} -10.8075 q^{83} +1.95972 q^{84} -14.2586 q^{85} +14.2002 q^{86} -7.22165 q^{87} -1.22393 q^{88} +7.94388 q^{89} +1.04003 q^{90} +14.9516 q^{92} -1.67028 q^{93} -20.9880 q^{94} -20.2506 q^{95} +12.6408 q^{96} +0.383647 q^{97} +12.6093 q^{98} +0.479760 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 2 q^{2} - 5 q^{3} + 28 q^{4} + 5 q^{5} - 3 q^{6} + 5 q^{7} + 3 q^{8} + 5 q^{9} - 15 q^{10} - q^{11} - 13 q^{12} - 19 q^{14} - 10 q^{15} + 4 q^{16} - 46 q^{17} - 9 q^{18} + 8 q^{19} - 5 q^{20} - 16 q^{21}+ \cdots - 47 q^{99}+O(q^{100}) \) 36 * q + 2 * q^2 - 5 * q^3 + 28 * q^4 + 5 * q^5 - 3 * q^6 + 5 * q^7 + 3 * q^8 + 5 * q^9 - 15 * q^10 - q^11 - 13 * q^12 - 19 * q^14 - 10 * q^15 + 4 * q^16 - 46 * q^17 - 9 * q^18 + 8 * q^19 - 5 * q^20 - 16 * q^21 - 21 * q^22 - 24 * q^23 + 57 * q^24 + 5 * q^25 - 11 * q^27 - 32 * q^28 - 73 * q^29 - 31 * q^30 - 36 * q^31 + 3 * q^32 - 8 * q^33 - 37 * q^34 - 31 * q^35 - 31 * q^36 + 6 * q^37 - 25 * q^38 - 22 * q^40 + 6 * q^41 - 37 * q^42 - 37 * q^43 - 16 * q^45 - 45 * q^46 + 13 * q^47 - 46 * q^48 - 5 * q^49 + 24 * q^50 - 46 * q^51 - 42 * q^53 - 26 * q^54 - 47 * q^55 - 95 * q^56 + 33 * q^57 + 9 * q^58 + 20 * q^59 - 40 * q^60 - 48 * q^61 - 2 * q^62 + 52 * q^63 + 21 * q^64 - 4 * q^66 - 50 * q^67 - 76 * q^68 - 33 * q^69 + 93 * q^70 + 16 * q^71 - 52 * q^72 - 43 * q^73 - 76 * q^74 - 40 * q^75 + 62 * q^76 - 47 * q^77 - 43 * q^79 - 46 * q^80 - 96 * q^81 + 7 * q^82 + 9 * q^83 - 77 * q^84 - 5 * q^85 + 64 * q^86 - 8 * q^87 - 54 * q^88 + 6 * q^89 + 27 * q^90 - 35 * q^92 + 5 * q^93 - 76 * q^94 - 63 * q^95 + 72 * q^96 - 4 * q^97 - 41 * q^98 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

Properties

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