LMFDB - Embedded newform 13.8.a.c.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [13,8,Mod(1,13)] 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("13.1"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(13, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	13.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.06100533129$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 354x^{2} - 640x + 20912 $ x^4 - x^3 - 354x^2 - 640x + 20912
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-12.6802$ of defining polynomial
Character	$\chi$	$=$	13.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+16.6802 q^{2} -6.69280 q^{3} +150.230 q^{4} +406.835 q^{5} -111.637 q^{6} -315.324 q^{7} +370.802 q^{8} -2142.21 q^{9} +6786.11 q^{10} -491.853 q^{11} -1005.46 q^{12} -2197.00 q^{13} -5259.68 q^{14} -2722.87 q^{15} -13044.4 q^{16} -17769.4 q^{17} -35732.5 q^{18} +32385.0 q^{19} +61118.9 q^{20} +2110.40 q^{21} -8204.23 q^{22} +70673.5 q^{23} -2481.70 q^{24} +87390.0 q^{25} -36646.5 q^{26} +28974.5 q^{27} -47371.1 q^{28} -192134. q^{29} -45418.1 q^{30} +247519. q^{31} -265046. q^{32} +3291.88 q^{33} -296397. q^{34} -128285. q^{35} -321824. q^{36} +593086. q^{37} +540189. q^{38} +14704.1 q^{39} +150855. q^{40} +212827. q^{41} +35202.0 q^{42} -822000. q^{43} -73891.2 q^{44} -871525. q^{45} +1.17885e6 q^{46} +49543.1 q^{47} +87303.4 q^{48} -724114. q^{49} +1.45769e6 q^{50} +118927. q^{51} -330055. q^{52} +793607. q^{53} +483301. q^{54} -200103. q^{55} -116923. q^{56} -216746. q^{57} -3.20484e6 q^{58} +126042. q^{59} -409056. q^{60} -1.77997e6 q^{61} +4.12868e6 q^{62} +675489. q^{63} -2.75135e6 q^{64} -893817. q^{65} +54909.3 q^{66} +37499.0 q^{67} -2.66949e6 q^{68} -473004. q^{69} -2.13982e6 q^{70} +4.96741e6 q^{71} -794334. q^{72} -1.85961e6 q^{73} +9.89281e6 q^{74} -584884. q^{75} +4.86520e6 q^{76} +155093. q^{77} +245267. q^{78} +6.07684e6 q^{79} -5.30692e6 q^{80} +4.49108e6 q^{81} +3.55001e6 q^{82} -5.68466e6 q^{83} +317046. q^{84} -7.22921e6 q^{85} -1.37112e7 q^{86} +1.28592e6 q^{87} -182380. q^{88} -2.92821e6 q^{89} -1.45372e7 q^{90} +692767. q^{91} +1.06173e7 q^{92} -1.65660e6 q^{93} +826390. q^{94} +1.31754e7 q^{95} +1.77390e6 q^{96} -4.72954e6 q^{97} -1.20784e7 q^{98} +1.05365e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9} - 4495 q^{10} + 1836 q^{11} - 3655 q^{12} - 8788 q^{13} - 18285 q^{14} - 29736 q^{15} - 36159 q^{16}+ \cdots + 6200852 q^{99}+O(q^{100}) $ 4 * q + 15 * q^2 + 80 * q^3 + 253 * q^4 + 258 * q^5 + 1579 * q^6 + 1692 * q^7 + 1893 * q^8 + 3494 * q^9 - 4495 * q^10 + 1836 * q^11 - 3655 * q^12 - 8788 * q^13 - 18285 * q^14 - 29736 * q^15 - 36159 * q^16 + 11814 * q^17 - 2738 * q^18 + 27660 * q^19 - 30369 * q^20 + 69930 * q^21 + 59930 * q^22 + 172920 * q^23 + 26553 * q^24 + 219414 * q^25 - 32955 * q^26 + 174800 * q^27 - 170903 * q^28 + 133344 * q^29 - 624695 * q^30 - 231748 * q^31 - 459291 * q^32 - 33868 * q^33 - 480249 * q^34 + 322296 * q^35 - 157930 * q^36 + 248026 * q^37 - 85326 * q^38 - 175760 * q^39 + 379895 * q^40 + 588108 * q^41 + 132275 * q^42 + 309304 * q^43 + 1270806 * q^44 - 1773144 * q^45 + 574948 * q^46 + 557916 * q^47 + 837429 * q^48 - 767378 * q^49 + 5737920 * q^50 - 1816496 * q^51 - 555841 * q^52 + 2022348 * q^53 - 479 * q^54 - 3770208 * q^55 + 1391625 * q^56 - 5609844 * q^57 - 3027194 * q^58 + 1162668 * q^59 - 8100077 * q^60 - 1340572 * q^61 + 2715564 * q^62 + 4213476 * q^63 - 6734935 * q^64 - 566826 * q^65 + 8124226 * q^66 - 598484 * q^67 + 4515993 * q^68 - 108848 * q^69 - 4329415 * q^70 + 697860 * q^71 + 1607430 * q^72 - 13725816 * q^73 + 14884053 * q^74 + 16604592 * q^75 + 12055070 * q^76 - 5326932 * q^77 - 3469063 * q^78 + 20079576 * q^79 + 3710667 * q^80 - 8368396 * q^81 + 857160 * q^82 - 2024724 * q^83 - 13603483 * q^84 - 16146514 * q^85 - 22922499 * q^86 + 22170952 * q^87 - 2100618 * q^88 + 17646240 * q^89 - 42372890 * q^90 - 3717324 * q^91 + 14504892 * q^92 - 17910680 * q^93 - 3729785 * q^94 + 12508152 * q^95 - 37637751 * q^96 - 6329096 * q^97 - 22166262 * q^98 + 6200852 * 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 $

$2$	16.6802	1.47434	0.737169	−	0.675708i	$-0.236162\pi$
$2$	16.6802	1.47434	0.737169	+	0.675708i	$0.236162\pi$
$3$	−6.69280	−0.143114	−0.0715572	−	0.997436i	$-0.522797\pi$
$3$	−6.69280	−0.143114	−0.0715572	+	0.997436i	$0.522797\pi$
$4$	150.230	1.17367
$5$	406.835	1.45554	0.727769	−	0.685822i	$-0.240557\pi$
$5$	406.835	1.45554	0.727769	+	0.685822i	$0.240557\pi$
$6$	−111.637	−0.210999
$7$	−315.324	−0.347467	−0.173734	−	0.984793i	$-0.555583\pi$
$7$	−315.324	−0.347467	−0.173734	+	0.984793i	$0.555583\pi$
$8$	370.802	0.256051
$9$	−2142.21	−0.979518
$10$	6786.11	2.14596
$11$	−491.853	−0.111420	−0.0557098	−	0.998447i	$-0.517742\pi$
$11$	−491.853	−0.111420	−0.0557098	+	0.998447i	$0.517742\pi$
$12$	−1005.46	−0.167969
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	−5259.68	−0.512284
$15$	−2722.87	−0.208309
$16$	−13044.4	−0.796166
$17$	−17769.4	−0.877205	−0.438602	−	0.898681i	$-0.644526\pi$
$17$	−17769.4	−0.877205	−0.438602	+	0.898681i	$0.644526\pi$
$18$	−35732.5	−1.44414
$19$	32385.0	1.08319	0.541597	−	0.840638i	$-0.317820\pi$
$19$	32385.0	1.08319	0.541597	+	0.840638i	$0.317820\pi$
$20$	61118.9	1.70832
$21$	2110.40	0.0497276
$22$	−8204.23	−0.164270
$23$	70673.5	1.21118	0.605591	−	0.795776i	$-0.292937\pi$
$23$	70673.5	1.21118	0.605591	+	0.795776i	$0.292937\pi$
$24$	−2481.70	−0.0366447
$25$	87390.0	1.11859
$26$	−36646.5	−0.408908
$27$	28974.5	0.283298
$28$	−47371.1	−0.407813
$29$	−192134.	−1.46289	−0.731446	−	0.681900i	$-0.761154\pi$
$29$	−192134.	−1.46289	−0.731446	+	0.681900i	$0.761154\pi$
$30$	−45418.1	−0.307117
$31$	247519.	1.49225	0.746127	−	0.665803i	$-0.231911\pi$
$31$	247519.	1.49225	0.746127	+	0.665803i	$0.231911\pi$
$32$	−265046.	−1.42987
$33$	3291.88	0.0159457
$34$	−296397.	−1.29330
$35$	−128285.	−0.505752
$36$	−321824.	−1.14963
$37$	593086.	1.92491	0.962457	−	0.271433i	$-0.0874977\pi$
$37$	593086.	1.92491	0.962457	+	0.271433i	$0.0874977\pi$
$38$	540189.	1.59699
$39$	14704.1	0.0396928
$40$	150855.	0.372693
$41$	212827.	0.482263	0.241132	−	0.970492i	$-0.422481\pi$
$41$	212827.	0.482263	0.241132	+	0.970492i	$0.422481\pi$
$42$	35202.0	0.0733153
$43$	−822000.	−1.57664	−0.788320	−	0.615266i	$-0.789049\pi$
$43$	−822000.	−1.57664	−0.788320	+	0.615266i	$0.789049\pi$
$44$	−73891.2	−0.130770
$45$	−871525.	−1.42573
$46$	1.17885e6	1.78569
$47$	49543.1	0.0696051	0.0348025	−	0.999394i	$-0.488920\pi$
$47$	49543.1	0.0696051	0.0348025	+	0.999394i	$0.488920\pi$
$48$	87303.4	0.113943
$49$	−724114.	−0.879266
$50$	1.45769e6	1.64918
$51$	118927.	0.125541
$52$	−330055.	−0.325518
$53$	793607.	0.732217	0.366109	−	0.930572i	$-0.380690\pi$
$53$	793607.	0.732217	0.366109	+	0.930572i	$0.380690\pi$
$54$	483301.	0.417676
$55$	−200103.	−0.162175
$56$	−116923.	−0.0889695
$57$	−216746.	−0.155021
$58$	−3.20484e6	−2.15680
$59$	126042.	0.0798976	0.0399488	−	0.999202i	$-0.487281\pi$
$59$	126042.	0.0798976	0.0399488	+	0.999202i	$0.487281\pi$
$60$	−409056.	−0.244486
$61$	−1.77997e6	−1.00406	−0.502028	−	0.864851i	$-0.667413\pi$
$61$	−1.77997e6	−1.00406	−0.502028	+	0.864851i	$0.667413\pi$
$62$	4.12868e6	2.20009
$63$	675489.	0.340351
$64$	−2.75135e6	−1.31194
$65$	−893817.	−0.403694
$66$	54909.3	0.0235094
$67$	37499.0	0.0152320	0.00761602	−	0.999971i	$-0.497576\pi$
$67$	37499.0	0.0152320	0.00761602	+	0.999971i	$0.497576\pi$
$68$	−2.66949e6	−1.02955
$69$	−473004.	−0.173338
$70$	−2.13982e6	−0.745649
$71$	4.96741e6	1.64712	0.823561	−	0.567228i	$-0.191984\pi$
$71$	4.96741e6	1.64712	0.823561	+	0.567228i	$0.191984\pi$
$72$	−794334.	−0.250807
$73$	−1.85961e6	−0.559491	−0.279745	−	0.960074i	$-0.590250\pi$
$73$	−1.85961e6	−0.559491	−0.279745	+	0.960074i	$0.590250\pi$
$74$	9.89281e6	2.83797
$75$	−584884.	−0.160087
$76$	4.86520e6	1.27132
$77$	155093.	0.0387146
$78$	245267.	0.0585206
$79$	6.07684e6	1.38670	0.693351	−	0.720600i	$-0.256134\pi$
$79$	6.07684e6	1.38670	0.693351	+	0.720600i	$0.256134\pi$
$80$	−5.30692e6	−1.15885
$81$	4.49108e6	0.938974
$82$	3.55001e6	0.711019
$83$	−5.68466e6	−1.09127	−0.545634	−	0.838024i	$-0.683711\pi$
$83$	−5.68466e6	−1.09127	−0.545634	+	0.838024i	$0.683711\pi$
$84$	317046.	0.0583639
$85$	−7.22921e6	−1.27681
$86$	−1.37112e7	−2.32450
$87$	1.28592e6	0.209361
$88$	−182380.	−0.0285291
$89$	−2.92821e6	−0.440288	−0.220144	−	0.975467i	$-0.570653\pi$
$89$	−2.92821e6	−0.440288	−0.220144	+	0.975467i	$0.570653\pi$
$90$	−1.45372e7	−2.10200
$91$	692767.	0.0963701
$92$	1.06173e7	1.42153
$93$	−1.65660e6	−0.213563
$94$	826390.	0.102621
$95$	1.31754e7	1.57663
$96$	1.77390e6	0.204635
$97$	−4.72954e6	−0.526160	−0.263080	−	0.964774i	$-0.584738\pi$
$97$	−4.72954e6	−0.526160	−0.263080	+	0.964774i	$0.584738\pi$
$98$	−1.20784e7	−1.29634
$99$	1.05365e6	0.109137
$100$	1.31286e7	1.31286
$101$	6.65679e6	0.642895	0.321447	−	0.946927i	$-0.395831\pi$
$101$	6.65679e6	0.642895	0.321447	+	0.946927i	$0.395831\pi$
$102$	1.98373e6	0.185089
$103$	−1.26910e7	−1.14437	−0.572183	−	0.820126i	$-0.693903\pi$
$103$	−1.26910e7	−1.14437	−0.572183	+	0.820126i	$0.693903\pi$
$104$	−814652.	−0.0710159
$105$	858586.	0.0723804
$106$	1.32375e7	1.07954
$107$	1.01337e7	0.799695	0.399847	−	0.916582i	$-0.369063\pi$
$107$	1.01337e7	0.799695	0.399847	+	0.916582i	$0.369063\pi$
$108$	4.35284e6	0.332499
$109$	−5.71864e6	−0.422961	−0.211480	−	0.977382i	$-0.567828\pi$
$109$	−5.71864e6	−0.422961	−0.211480	+	0.977382i	$0.567828\pi$
$110$	−3.33777e6	−0.239101
$111$	−3.96940e6	−0.275483
$112$	4.11321e6	0.276642
$113$	−4.59610e6	−0.299651	−0.149825	−	0.988712i	$-0.547871\pi$
$113$	−4.59610e6	−0.299651	−0.149825	+	0.988712i	$0.547871\pi$
$114$	−3.61538e6	−0.228553
$115$	2.87525e7	1.76292
$116$	−2.88643e7	−1.71695
$117$	4.70643e6	0.271669
$118$	2.10241e6	0.117796
$119$	5.60311e6	0.304800
$120$	−1.00964e6	−0.0533377
$121$	−1.92453e7	−0.987586
$122$	−2.96903e7	−1.48032
$123$	−1.42441e6	−0.0690188
$124$	3.71848e7	1.75142
$125$	3.76933e6	0.172615
$126$	1.12673e7	0.501792
$127$	1.36102e7	0.589594	0.294797	−	0.955560i	$-0.404748\pi$
$127$	1.36102e7	0.589594	0.294797	+	0.955560i	$0.404748\pi$
$128$	−1.19672e7	−0.504380
$129$	5.50148e6	0.225640
$130$	−1.49091e7	−0.595181
$131$	−1.49831e7	−0.582307	−0.291154	−	0.956676i	$-0.594039\pi$
$131$	−1.49831e7	−0.582307	−0.291154	+	0.956676i	$0.594039\pi$
$132$	494539.	0.0187151
$133$	−1.02118e7	−0.376375
$134$	625492.	0.0224572
$135$	1.17879e7	0.412351
$136$	−6.58892e6	−0.224610
$137$	−7.70284e6	−0.255934	−0.127967	−	0.991778i	$-0.540845\pi$
$137$	−7.70284e6	−0.255934	−0.127967	+	0.991778i	$0.540845\pi$
$138$	−7.88981e6	−0.255558
$139$	−4.65260e7	−1.46941	−0.734706	−	0.678386i	$-0.762680\pi$
$139$	−4.65260e7	−1.46941	−0.734706	+	0.678386i	$0.762680\pi$
$140$	−1.92723e7	−0.593587
$141$	−331582.	−0.00996149
$142$	8.28575e7	2.42841
$143$	1.08060e6	0.0309022
$144$	2.79438e7	0.779859
$145$	−7.81670e7	−2.12929
$146$	−3.10188e7	−0.824879
$147$	4.84635e6	0.125836
$148$	8.90993e7	2.25922
$149$	−4.26700e7	−1.05675	−0.528373	−	0.849012i	$-0.677198\pi$
$149$	−4.26700e7	−1.05675	−0.528373	+	0.849012i	$0.677198\pi$
$150$	−9.75600e6	−0.236022
$151$	−2.39173e7	−0.565318	−0.282659	−	0.959220i	$-0.591216\pi$
$151$	−2.39173e7	−0.565318	−0.282659	+	0.959220i	$0.591216\pi$
$152$	1.20084e7	0.277353
$153$	3.80657e7	0.859238
$154$	2.58699e6	0.0570785
$155$	1.00700e8	2.17203
$156$	2.20899e6	0.0465863
$157$	−2.54577e6	−0.0525013	−0.0262507	−	0.999655i	$-0.508357\pi$
$157$	−2.54577e6	−0.0525013	−0.0262507	+	0.999655i	$0.508357\pi$
$158$	1.01363e8	2.04447
$159$	−5.31145e6	−0.104791
$160$	−1.07830e8	−2.08123
$161$	−2.22851e7	−0.420846
$162$	7.49123e7	1.38437
$163$	7.24004e7	1.30944	0.654718	−	0.755873i	$-0.272787\pi$
$163$	7.24004e7	1.30944	0.654718	+	0.755873i	$0.272787\pi$
$164$	3.19731e7	0.566019
$165$	1.33925e6	0.0232096
$166$	−9.48215e7	−1.60890
$167$	1.72100e7	0.285939	0.142970	−	0.989727i	$-0.454335\pi$
$167$	1.72100e7	0.285939	0.142970	+	0.989727i	$0.454335\pi$
$168$	782541.	0.0127328
$169$	4.82681e6	0.0769231
$170$	−1.20585e8	−1.88244
$171$	−6.93754e7	−1.06101
$172$	−1.23489e8	−1.85046
$173$	1.71191e7	0.251374	0.125687	−	0.992070i	$-0.459887\pi$
$173$	1.71191e7	0.251374	0.125687	+	0.992070i	$0.459887\pi$
$174$	2.14494e7	0.308669
$175$	−2.75562e7	−0.388674
$176$	6.41592e6	0.0887084
$177$	−843575.	−0.0114345
$178$	−4.88432e7	−0.649134
$179$	−2.87284e7	−0.374392	−0.187196	−	0.982323i	$-0.559940\pi$
$179$	−2.87284e7	−0.374392	−0.187196	+	0.982323i	$0.559940\pi$
$180$	−1.30929e8	−1.67334
$181$	3.96381e7	0.496864	0.248432	−	0.968649i	$-0.420085\pi$
$181$	3.96381e7	0.496864	0.248432	+	0.968649i	$0.420085\pi$
$182$	1.15555e7	0.142082
$183$	1.19130e7	0.143695
$184$	2.62059e7	0.310125
$185$	2.41288e8	2.80179
$186$	−2.76324e7	−0.314864
$187$	8.73993e6	0.0977377
$188$	7.44286e6	0.0816935
$189$	−9.13636e6	−0.0984367
$190$	2.19768e8	2.32449
$191$	−1.71987e8	−1.78599	−0.892995	−	0.450066i	$-0.851400\pi$
$191$	−1.71987e8	−1.78599	−0.892995	+	0.450066i	$0.851400\pi$
$192$	1.84142e7	0.187758
$193$	−4.94121e6	−0.0494746	−0.0247373	−	0.999694i	$-0.507875\pi$
$193$	−4.94121e6	−0.0494746	−0.0247373	+	0.999694i	$0.507875\pi$
$194$	−7.88898e7	−0.775737
$195$	5.98214e6	0.0577744
$196$	−1.08784e8	−1.03197
$197$	1.27289e8	1.18620	0.593101	−	0.805128i	$-0.297903\pi$
$197$	1.27289e8	1.18620	0.593101	+	0.805128i	$0.297903\pi$
$198$	1.75752e7	0.160905
$199$	1.87477e8	1.68641	0.843204	−	0.537594i	$-0.180667\pi$
$199$	1.87477e8	1.68641	0.843204	+	0.537594i	$0.180667\pi$
$200$	3.24044e7	0.286417
$201$	−250973.	−0.00217992
$202$	1.11037e8	0.947844
$203$	6.05846e7	0.508307
$204$	1.78664e7	0.147344
$205$	8.65857e7	0.701952
$206$	−2.11689e8	−1.68718
$207$	−1.51397e8	−1.18637
$208$	2.86585e7	0.220817
$209$	−1.59287e7	−0.120689
$210$	1.43214e7	0.106713
$211$	−5.11608e7	−0.374929	−0.187464	−	0.982271i	$-0.560027\pi$
$211$	−5.11608e7	−0.374929	−0.187464	+	0.982271i	$0.560027\pi$
$212$	1.19224e8	0.859383
$213$	−3.32459e7	−0.235727
$214$	1.69032e8	1.17902
$215$	−3.34419e8	−2.29486
$216$	1.07438e7	0.0725388
$217$	−7.80487e7	−0.518510
$218$	−9.53882e7	−0.623587
$219$	1.24460e7	0.0800712
$220$	−3.00615e7	−0.190341
$221$	3.90393e7	0.243293
$222$	−6.62106e7	−0.406155
$223$	−2.24752e8	−1.35717	−0.678587	−	0.734520i	$-0.737408\pi$
$223$	−2.24752e8	−1.35717	−0.678587	+	0.734520i	$0.737408\pi$
$224$	8.35753e7	0.496833
$225$	−1.87207e8	−1.09568
$226$	−7.66640e7	−0.441786
$227$	2.82010e8	1.60020	0.800100	−	0.599867i	$-0.204780\pi$
$227$	2.82010e8	1.60020	0.800100	+	0.599867i	$0.204780\pi$
$228$	−3.25618e7	−0.181944
$229$	−4.40878e7	−0.242602	−0.121301	−	0.992616i	$-0.538707\pi$
$229$	−4.40878e7	−0.242602	−0.121301	+	0.992616i	$0.538707\pi$
$230$	4.79598e8	2.59914
$231$	−1.03801e6	−0.00554062
$232$	−7.12438e7	−0.374575
$233$	1.91177e8	0.990123	0.495061	−	0.868858i	$-0.335146\pi$
$233$	1.91177e8	0.990123	0.495061	+	0.868858i	$0.335146\pi$
$234$	7.85043e7	0.400533
$235$	2.01559e7	0.101313
$236$	1.89353e7	0.0937736
$237$	−4.06711e7	−0.198457
$238$	9.34612e7	0.449378
$239$	1.02948e7	0.0487781	0.0243891	−	0.999703i	$-0.492236\pi$
$239$	1.02948e7	0.0487781	0.0243891	+	0.999703i	$0.492236\pi$
$240$	3.55181e7	0.165848
$241$	−1.99542e8	−0.918281	−0.459140	−	0.888364i	$-0.651842\pi$
$241$	−1.99542e8	−0.918281	−0.459140	+	0.888364i	$0.651842\pi$
$242$	−3.21015e8	−1.45603
$243$	−9.34252e7	−0.417678
$244$	−2.67405e8	−1.17843
$245$	−2.94595e8	−1.27981
$246$	−2.37595e7	−0.101757
$247$	−7.11499e7	−0.300424
$248$	9.17806e7	0.382094
$249$	3.80463e7	0.156176
$250$	6.28733e7	0.254493
$251$	1.68161e8	0.671223	0.335612	−	0.942000i	$-0.391057\pi$
$251$	1.68161e8	0.671223	0.335612	+	0.942000i	$0.391057\pi$
$252$	1.01479e8	0.399460
$253$	−3.47610e7	−0.134949
$254$	2.27022e8	0.869260
$255$	4.83837e7	0.182729
$256$	1.52557e8	0.568318
$257$	5.15465e8	1.89423	0.947116	−	0.320891i	$-0.103982\pi$
$257$	5.15465e8	1.89423	0.947116	+	0.320891i	$0.103982\pi$
$258$	9.17660e7	0.332669
$259$	−1.87014e8	−0.668845
$260$	−1.34278e8	−0.473804
$261$	4.11591e8	1.43293
$262$	−2.49922e8	−0.858517
$263$	−9.83463e7	−0.333360	−0.166680	−	0.986011i	$-0.553305\pi$
$263$	−9.83463e7	−0.333360	−0.166680	+	0.986011i	$0.553305\pi$
$264$	1.22063e6	0.00408293
$265$	3.22867e8	1.06577
$266$	−1.70335e8	−0.554903
$267$	1.95979e7	0.0630116
$268$	5.63348e6	0.0178774
$269$	−2.94959e8	−0.923909	−0.461955	−	0.886904i	$-0.652852\pi$
$269$	−2.94959e8	−0.923909	−0.461955	+	0.886904i	$0.652852\pi$
$270$	1.96624e8	0.607944
$271$	4.73501e8	1.44520	0.722601	−	0.691265i	$-0.242946\pi$
$271$	4.73501e8	1.44520	0.722601	+	0.691265i	$0.242946\pi$
$272$	2.31791e8	0.698401
$273$	−4.63655e6	−0.0137920
$274$	−1.28485e8	−0.377334
$275$	−4.29831e7	−0.124633
$276$	−7.10594e7	−0.203441
$277$	1.44246e8	0.407779	0.203890	−	0.978994i	$-0.434642\pi$
$277$	1.44246e8	0.407779	0.203890	+	0.978994i	$0.434642\pi$
$278$	−7.76064e8	−2.16641
$279$	−5.30237e8	−1.46169
$280$	−4.75683e7	−0.129498
$281$	−1.05385e8	−0.283339	−0.141669	−	0.989914i	$-0.545247\pi$
$281$	−1.05385e8	−0.283339	−0.141669	+	0.989914i	$0.545247\pi$
$282$	−5.53087e6	−0.0146866
$283$	−4.03722e8	−1.05884	−0.529419	−	0.848360i	$-0.677590\pi$
$283$	−4.03722e8	−1.05884	−0.529419	+	0.848360i	$0.677590\pi$
$284$	7.46254e8	1.93318
$285$	−8.81801e7	−0.225639
$286$	1.80247e7	0.0455603
$287$	−6.71096e7	−0.167571
$288$	5.67783e8	1.40058
$289$	−9.45879e7	−0.230512
$290$	−1.30384e9	−3.13930
$291$	3.16538e7	0.0753010
$292$	−2.79370e8	−0.656659
$293$	5.45177e8	1.26620	0.633098	−	0.774071i	$-0.281783\pi$
$293$	5.45177e8	1.26620	0.633098	+	0.774071i	$0.281783\pi$
$294$	8.08382e7	0.185524
$295$	5.12784e7	0.116294
$296$	2.19917e8	0.492877
$297$	−1.42512e7	−0.0315649
$298$	−7.11745e8	−1.55800
$299$	−1.55270e8	−0.335921
$300$	−8.78671e7	−0.187889
$301$	2.59196e8	0.547831
$302$	−3.98946e8	−0.833470
$303$	−4.45525e7	−0.0920075
$304$	−4.22442e8	−0.862402
$305$	−7.24155e8	−1.46144
$306$	6.34944e8	1.26681
$307$	4.34423e8	0.856898	0.428449	−	0.903566i	$-0.359060\pi$
$307$	4.34423e8	0.856898	0.428449	+	0.903566i	$0.359060\pi$
$308$	2.32997e7	0.0454383
$309$	8.49382e7	0.163775
$310$	1.67969e9	3.20231
$311$	3.64638e8	0.687386	0.343693	−	0.939082i	$-0.388322\pi$
$311$	3.64638e8	0.687386	0.343693	+	0.939082i	$0.388322\pi$
$312$	5.45230e6	0.0101634
$313$	3.46305e8	0.638342	0.319171	−	0.947697i	$-0.396595\pi$
$313$	3.46305e8	0.638342	0.319171	+	0.947697i	$0.396595\pi$
$314$	−4.24640e7	−0.0774047
$315$	2.74813e8	0.495393
$316$	9.12924e8	1.62753
$317$	7.18389e8	1.26664	0.633319	−	0.773891i	$-0.281692\pi$
$317$	7.18389e8	1.26664	0.633319	+	0.773891i	$0.281692\pi$
$318$	−8.85962e7	−0.154497
$319$	9.45019e7	0.162995
$320$	−1.11934e9	−1.90958
$321$	−6.78227e7	−0.114448
$322$	−3.71720e8	−0.620469
$323$	−5.75461e8	−0.950183
$324$	6.74696e8	1.10205
$325$	−1.91996e8	−0.310242
$326$	1.20766e9	1.93055
$327$	3.82737e7	0.0605318
$328$	7.89168e7	0.123484
$329$	−1.56221e7	−0.0241855
$330$	2.23390e7	0.0342189
$331$	−1.18927e8	−0.180253	−0.0901267	−	0.995930i	$-0.528727\pi$
$331$	−1.18927e8	−0.180253	−0.0901267	+	0.995930i	$0.528727\pi$
$332$	−8.54007e8	−1.28079
$333$	−1.27051e9	−1.88549
$334$	2.87067e8	0.421571
$335$	1.52559e7	0.0221708
$336$	−2.75289e7	−0.0395914
$337$	−3.19783e8	−0.455145	−0.227573	−	0.973761i	$-0.573079\pi$
$337$	−3.19783e8	−0.455145	−0.227573	+	0.973761i	$0.573079\pi$
$338$	8.05123e7	0.113411
$339$	3.07608e7	0.0428843
$340$	−1.08604e9	−1.49855
$341$	−1.21743e8	−0.166266
$342$	−1.15720e9	−1.56429
$343$	4.88013e8	0.652984
$344$	−3.04799e8	−0.403701
$345$	−1.92435e8	−0.252299
$346$	2.85551e8	0.370610
$347$	8.75021e8	1.12426	0.562128	−	0.827050i	$-0.309983\pi$
$347$	8.75021e8	1.12426	0.562128	+	0.827050i	$0.309983\pi$
$348$	1.93183e8	0.245721
$349$	−1.01412e9	−1.27703	−0.638513	−	0.769611i	$-0.720450\pi$
$349$	−1.01412e9	−1.27703	−0.638513	+	0.769611i	$0.720450\pi$
$350$	−4.59643e8	−0.573037
$351$	−6.36570e7	−0.0785726
$352$	1.30364e8	0.159315
$353$	2.10542e8	0.254757	0.127379	−	0.991854i	$-0.459344\pi$
$353$	2.10542e8	0.254757	0.127379	+	0.991854i	$0.459344\pi$
$354$	−1.40710e7	−0.0168583
$355$	2.02092e9	2.39745
$356$	−4.39905e8	−0.516754
$357$	−3.75005e7	−0.0436213
$358$	−4.79197e8	−0.551980
$359$	3.91966e8	0.447113	0.223557	−	0.974691i	$-0.428233\pi$
$359$	3.91966e8	0.447113	0.223557	+	0.974691i	$0.428233\pi$
$360$	−3.23163e8	−0.365059
$361$	1.54917e8	0.173310
$362$	6.61173e8	0.732546
$363$	1.28805e8	0.141338
$364$	1.04074e8	0.113107
$365$	−7.56557e8	−0.814360
$366$	1.98711e8	0.211855
$367$	−1.45296e9	−1.53434	−0.767170	−	0.641444i	$-0.778336\pi$
$367$	−1.45296e9	−1.53434	−0.767170	+	0.641444i	$0.778336\pi$
$368$	−9.21892e8	−0.964301
$369$	−4.55920e8	−0.472385
$370$	4.02474e9	4.13078
$371$	−2.50243e8	−0.254421
$372$	−2.48871e8	−0.250653
$373$	8.55462e8	0.853532	0.426766	−	0.904362i	$-0.359653\pi$
$373$	8.55462e8	0.853532	0.426766	+	0.904362i	$0.359653\pi$
$374$	1.45784e8	0.144098
$375$	−2.52274e7	−0.0247038
$376$	1.83707e7	0.0178225
$377$	4.22119e8	0.405733
$378$	−1.52397e8	−0.145129
$379$	−7.02370e8	−0.662718	−0.331359	−	0.943505i	$-0.607507\pi$
$379$	−7.02370e8	−0.662718	−0.331359	+	0.943505i	$0.607507\pi$
$380$	1.97934e9	1.85045
$381$	−9.10906e7	−0.0843794
$382$	−2.86878e9	−2.63315
$383$	−7.29638e8	−0.663608	−0.331804	−	0.943348i	$-0.607657\pi$
$383$	−7.29638e8	−0.663608	−0.331804	+	0.943348i	$0.607657\pi$
$384$	8.00941e7	0.0721840
$385$	6.30974e7	0.0563506
$386$	−8.24204e7	−0.0729423
$387$	1.76089e9	1.54435
$388$	−7.10518e8	−0.617539
$389$	−1.67929e9	−1.44645	−0.723223	−	0.690615i	$-0.757340\pi$
$389$	−1.67929e9	−1.44645	−0.723223	+	0.690615i	$0.757340\pi$
$390$	9.97835e7	0.0851790
$391$	−1.25582e9	−1.06245
$392$	−2.68503e8	−0.225137
$393$	1.00279e8	0.0833366
$394$	2.12321e9	1.74886
$395$	2.47227e9	2.01840
$396$	1.58290e8	0.128092
$397$	−1.61054e9	−1.29183	−0.645914	−	0.763410i	$-0.723524\pi$
$397$	−1.61054e9	−1.29183	−0.645914	+	0.763410i	$0.723524\pi$
$398$	3.12716e9	2.48634
$399$	6.83453e7	0.0538646
$400$	−1.13995e9	−0.890585
$401$	−2.03617e9	−1.57692	−0.788458	−	0.615089i	$-0.789120\pi$
$401$	−2.03617e9	−1.57692	−0.788458	+	0.615089i	$0.789120\pi$
$402$	−4.18629e6	−0.00321395
$403$	−5.43800e8	−0.413877
$404$	1.00005e9	0.754548
$405$	1.82713e9	1.36671
$406$	1.01056e9	0.749416
$407$	−2.91711e8	−0.214473
$408$	4.40983e7	0.0321449
$409$	−3.92496e8	−0.283664	−0.141832	−	0.989891i	$-0.545299\pi$
$409$	−3.92496e8	−0.283664	−0.141832	+	0.989891i	$0.545299\pi$
$410$	1.44427e9	1.03491
$411$	5.15535e7	0.0366279
$412$	−1.90657e9	−1.34311
$413$	−3.97441e7	−0.0277618
$414$	−2.52534e9	−1.74912
$415$	−2.31272e9	−1.58838
$416$	5.82306e8	0.396574
$417$	3.11389e8	0.210294
$418$	−2.65694e8	−0.177936
$419$	1.55932e9	1.03559	0.517794	−	0.855505i	$-0.326753\pi$
$419$	1.55932e9	1.03559	0.517794	+	0.855505i	$0.326753\pi$
$420$	1.28985e8	0.0849509
$421$	1.61013e9	1.05166	0.525829	−	0.850591i	$-0.323755\pi$
$421$	1.61013e9	1.05166	0.525829	+	0.850591i	$0.323755\pi$
$422$	−8.53373e8	−0.552771
$423$	−1.06132e8	−0.0681794
$424$	2.94271e8	0.187485
$425$	−1.55287e9	−0.981234
$426$	−5.54549e8	−0.347541
$427$	5.61267e8	0.348877
$428$	1.52238e9	0.938579
$429$	−7.23225e6	−0.00442255
$430$	−5.57818e9	−3.38340
$431$	4.33581e8	0.260855	0.130428	−	0.991458i	$-0.458365\pi$
$431$	4.33581e8	0.260855	0.130428	+	0.991458i	$0.458365\pi$
$432$	−3.77955e8	−0.225552
$433$	1.27596e9	0.755315	0.377658	−	0.925945i	$-0.376730\pi$
$433$	1.27596e9	0.755315	0.377658	+	0.925945i	$0.376730\pi$
$434$	−1.30187e9	−0.764458
$435$	5.23156e8	0.304733
$436$	−8.59111e8	−0.496417
$437$	2.28876e9	1.31195
$438$	2.07603e8	0.118052
$439$	−2.91733e9	−1.64573	−0.822867	−	0.568233i	$-0.807627\pi$
$439$	−2.91733e9	−1.64573	−0.822867	+	0.568233i	$0.807627\pi$
$440$	−7.41987e7	−0.0415252
$441$	1.55120e9	0.861258
$442$	6.51185e8	0.358696
$443$	2.93905e9	1.60618	0.803089	−	0.595859i	$-0.203188\pi$
$443$	2.93905e9	1.60618	0.803089	+	0.595859i	$0.203188\pi$
$444$	−5.96324e8	−0.323327
$445$	−1.19130e9	−0.640857
$446$	−3.74891e9	−2.00093
$447$	2.85582e8	0.151236
$448$	8.67565e8	0.455858
$449$	1.11724e9	0.582483	0.291242	−	0.956650i	$-0.405932\pi$
$449$	1.11724e9	0.582483	0.291242	+	0.956650i	$0.405932\pi$
$450$	−3.12266e9	−1.61540
$451$	−1.04680e8	−0.0537335
$452$	−6.90472e8	−0.351691
$453$	1.60074e8	0.0809052
$454$	4.70399e9	2.35923
$455$	2.81842e8	0.140270
$456$	−8.03700e7	−0.0396933
$457$	7.26762e8	0.356193	0.178096	−	0.984013i	$-0.443006\pi$
$457$	7.26762e8	0.356193	0.178096	+	0.984013i	$0.443006\pi$
$458$	−7.35395e8	−0.357678
$459$	−5.14859e8	−0.248510
$460$	4.31949e9	2.06909
$461$	7.53566e8	0.358235	0.179118	−	0.983828i	$-0.442676\pi$
$461$	7.53566e8	0.358235	0.179118	+	0.983828i	$0.442676\pi$
$462$	−1.73142e7	−0.00816875
$463$	5.71307e8	0.267508	0.133754	−	0.991015i	$-0.457297\pi$
$463$	5.71307e8	0.267508	0.133754	+	0.991015i	$0.457297\pi$
$464$	2.50627e9	1.16470
$465$	−6.73962e8	−0.310849
$466$	3.18887e9	1.45978
$467$	2.04209e9	0.927825	0.463912	−	0.885881i	$-0.346445\pi$
$467$	2.04209e9	0.927825	0.463912	+	0.885881i	$0.346445\pi$
$468$	7.07047e8	0.318851
$469$	−1.18243e7	−0.00529263
$470$	3.36205e8	0.149369
$471$	1.70383e7	0.00751370
$472$	4.67367e7	0.0204579
$473$	4.04304e8	0.175668
$474$	−6.78403e8	−0.292593
$475$	2.83013e9	1.21165
$476$	8.41756e8	0.357735
$477$	−1.70007e9	−0.717220
$478$	1.71720e8	0.0719155
$479$	−3.76312e9	−1.56449	−0.782247	−	0.622968i	$-0.785927\pi$
$479$	−3.76312e9	−1.56449	−0.782247	+	0.622968i	$0.785927\pi$
$480$	7.21685e8	0.297854
$481$	−1.30301e9	−0.533875
$482$	−3.32841e9	−1.35386
$483$	1.49149e8	0.0602291
$484$	−2.89121e9	−1.15910
$485$	−1.92414e9	−0.765846
$486$	−1.55835e9	−0.615799
$487$	−2.02785e9	−0.795582	−0.397791	−	0.917476i	$-0.630223\pi$
$487$	−2.02785e9	−0.795582	−0.397791	+	0.917476i	$0.630223\pi$
$488$	−6.60017e8	−0.257090
$489$	−4.84561e8	−0.187399
$490$	−4.91391e9	−1.88687
$491$	2.51831e9	0.960117	0.480059	−	0.877236i	$-0.340615\pi$
$491$	2.51831e9	0.960117	0.480059	+	0.877236i	$0.340615\pi$
$492$	−2.13989e8	−0.0810054
$493$	3.41411e9	1.28326
$494$	−1.18680e9	−0.442927
$495$	4.28663e8	0.158854
$496$	−3.22873e9	−1.18808
$497$	−1.56634e9	−0.572321
$498$	6.34621e8	0.230256
$499$	−1.20416e9	−0.433842	−0.216921	−	0.976189i	$-0.569601\pi$
$499$	−1.20416e9	−0.433842	−0.216921	+	0.976189i	$0.569601\pi$
$500$	5.66267e8	0.202594
$501$	−1.15183e8	−0.0409221
$502$	2.80496e9	0.989610
$503$	−3.14109e9	−1.10051	−0.550253	−	0.834998i	$-0.685469\pi$
$503$	−3.14109e9	−1.10051	−0.550253	+	0.834998i	$0.685469\pi$
$504$	2.50473e8	0.0871472
$505$	2.70822e9	0.935758
$506$	−5.79822e8	−0.198961
$507$	−3.23049e7	−0.0110088
$508$	2.04467e9	0.691990
$509$	−5.23496e9	−1.75955	−0.879773	−	0.475393i	$-0.842306\pi$
$509$	−5.23496e9	−1.75955	−0.879773	+	0.475393i	$0.842306\pi$
$510$	8.07051e8	0.269405
$511$	5.86381e8	0.194405
$512$	4.07648e9	1.34227
$513$	9.38340e8	0.306866
$514$	8.59807e9	2.79274
$515$	−5.16314e9	−1.66567
$516$	8.26488e8	0.264827
$517$	−2.43679e7	−0.00775536
$518$	−3.11944e9	−0.986103
$519$	−1.14575e8	−0.0359752
$520$	−3.31429e8	−0.103366
$521$	−2.82652e8	−0.0875628	−0.0437814	−	0.999041i	$-0.513941\pi$
$521$	−2.82652e8	−0.0875628	−0.0437814	+	0.999041i	$0.513941\pi$
$522$	6.86544e9	2.11262
$523$	−1.39625e9	−0.426784	−0.213392	−	0.976967i	$-0.568451\pi$
$523$	−1.39625e9	−0.426784	−0.213392	+	0.976967i	$0.568451\pi$
$524$	−2.25091e9	−0.683438
$525$	1.84428e8	0.0556249
$526$	−1.64044e9	−0.491485
$527$	−4.39826e9	−1.30901
$528$	−4.29405e7	−0.0126955
$529$	1.58992e9	0.466961
$530$	5.38550e9	1.57131
$531$	−2.70008e8	−0.0782611
$532$	−1.53411e9	−0.441740
$533$	−4.67582e8	−0.133756
$534$	3.26898e8	0.0929004
$535$	4.12274e9	1.16399
$536$	1.39047e7	0.00390018
$537$	1.92274e8	0.0535808
$538$	−4.91999e9	−1.36215
$539$	3.56158e8	0.0979675
$540$	1.77089e9	0.483964
$541$	4.38257e9	1.18998	0.594988	−	0.803735i	$-0.297157\pi$
$541$	4.38257e9	1.18998	0.594988	+	0.803735i	$0.297157\pi$
$542$	7.89811e9	2.13072
$543$	−2.65290e8	−0.0711085
$544$	4.70970e9	1.25429
$545$	−2.32654e9	−0.615635
$546$	−7.73387e7	−0.0203340
$547$	−1.61792e9	−0.422670	−0.211335	−	0.977414i	$-0.567781\pi$
$547$	−1.61792e9	−0.422670	−0.211335	+	0.977414i	$0.567781\pi$
$548$	−1.15720e9	−0.300383
$549$	3.81306e9	0.983492
$550$	−7.16968e8	−0.183751
$551$	−6.22227e9	−1.58460
$552$	−1.75391e8	−0.0443833
$553$	−1.91617e9	−0.481834
$554$	2.40606e9	0.601205
$555$	−1.61489e9	−0.400976
$556$	−6.98960e9	−1.72461
$557$	7.81037e9	1.91504	0.957521	−	0.288362i	$-0.0931106\pi$
$557$	7.81037e9	1.91504	0.957521	+	0.288362i	$0.0931106\pi$
$558$	−8.84448e9	−2.15503
$559$	1.80593e9	0.437281
$560$	1.67340e9	0.402662
$561$	−5.84946e7	−0.0139877
$562$	−1.75784e9	−0.417737
$563$	−2.17176e9	−0.512899	−0.256449	−	0.966558i	$-0.582553\pi$
$563$	−2.17176e9	−0.512899	−0.256449	+	0.966558i	$0.582553\pi$
$564$	−4.98136e7	−0.0116915
$565$	−1.86986e9	−0.436153
$566$	−6.73417e9	−1.56109
$567$	−1.41615e9	−0.326263
$568$	1.84192e9	0.421748
$569$	1.67837e9	0.381940	0.190970	−	0.981596i	$-0.438837\pi$
$569$	1.67837e9	0.381940	0.190970	+	0.981596i	$0.438837\pi$
$570$	−1.47086e9	−0.332668
$571$	−1.15705e9	−0.260091	−0.130045	−	0.991508i	$-0.541512\pi$
$571$	−1.15705e9	−0.260091	−0.130045	+	0.991508i	$0.541512\pi$
$572$	1.62339e8	0.0362691
$573$	1.15108e9	0.255601
$574$	−1.11940e9	−0.247056
$575$	6.17616e9	1.35482
$576$	5.89395e9	1.28507
$577$	4.37749e9	0.948659	0.474330	−	0.880347i	$-0.342691\pi$
$577$	4.37749e9	0.948659	0.474330	+	0.880347i	$0.342691\pi$
$578$	−1.57775e9	−0.339852
$579$	3.30705e7	0.00708053
$580$	−1.17430e10	−2.49909
$581$	1.79251e9	0.379180
$582$	5.27993e8	0.111019
$583$	−3.90338e8	−0.0815833
$584$	−6.89549e8	−0.143258
$585$	1.91474e9	0.395425
$586$	9.09368e9	1.86680
$587$	−1.48897e9	−0.303846	−0.151923	−	0.988392i	$-0.548547\pi$
$587$	−1.48897e9	−0.303846	−0.151923	+	0.988392i	$0.548547\pi$
$588$	7.28067e8	0.147690
$589$	8.01591e9	1.61640
$590$	8.55335e8	0.171457
$591$	−8.51919e8	−0.169763
$592$	−7.73644e9	−1.53255
$593$	5.19852e8	0.102374	0.0511869	−	0.998689i	$-0.483700\pi$
$593$	5.19852e8	0.102374	0.0511869	+	0.998689i	$0.483700\pi$
$594$	−2.37714e8	−0.0465373
$595$	2.27954e9	0.443648
$596$	−6.41032e9	−1.24027
$597$	−1.25475e9	−0.241349
$598$	−2.58993e9	−0.495262
$599$	2.61581e9	0.497294	0.248647	−	0.968594i	$-0.420014\pi$
$599$	2.61581e9	0.497294	0.248647	+	0.968594i	$0.420014\pi$
$600$	−2.16876e8	−0.0409904
$601$	−1.22782e9	−0.230713	−0.115357	−	0.993324i	$-0.536801\pi$
$601$	−1.22782e9	−0.230713	−0.115357	+	0.993324i	$0.536801\pi$
$602$	4.32346e9	0.807687
$603$	−8.03306e7	−0.0149201
$604$	−3.59310e9	−0.663498
$605$	−7.82965e9	−1.43747
$606$	−7.43147e8	−0.135650
$607$	1.23808e9	0.224692	0.112346	−	0.993669i	$-0.464163\pi$
$607$	1.23808e9	0.224692	0.112346	+	0.993669i	$0.464163\pi$
$608$	−8.58351e9	−1.54883
$609$	−4.05480e8	−0.0727460
$610$	−1.20791e10	−2.15466
$611$	−1.08846e8	−0.0193050
$612$	5.71861e9	1.00846
$613$	−4.61781e9	−0.809700	−0.404850	−	0.914383i	$-0.632676\pi$
$613$	−4.61781e9	−0.809700	−0.404850	+	0.914383i	$0.632676\pi$
$614$	7.24628e9	1.26336
$615$	−5.79501e8	−0.100460
$616$	5.75089e7	0.00991294
$617$	2.74596e9	0.470649	0.235324	−	0.971917i	$-0.424385\pi$
$617$	2.74596e9	0.470649	0.235324	+	0.971917i	$0.424385\pi$
$618$	1.41679e9	0.241460
$619$	−5.84319e9	−0.990223	−0.495112	−	0.868829i	$-0.664873\pi$
$619$	−5.84319e9	−0.990223	−0.495112	+	0.868829i	$0.664873\pi$
$620$	1.51281e10	2.54926
$621$	2.04773e9	0.343125
$622$	6.08225e9	1.01344
$623$	9.23335e8	0.152986
$624$	−1.91806e8	−0.0316021
$625$	−5.29385e9	−0.867344
$626$	5.77645e9	0.941132
$627$	1.06607e8	0.0172723
$628$	−3.82451e8	−0.0616194
$629$	−1.05388e10	−1.68854
$630$	4.58394e9	0.730377
$631$	9.22601e9	1.46188	0.730939	−	0.682443i	$-0.239082\pi$
$631$	9.22601e9	1.46188	0.730939	+	0.682443i	$0.239082\pi$
$632$	2.25331e9	0.355067
$633$	3.42409e8	0.0536577
$634$	1.19829e10	1.86745
$635$	5.53713e9	0.858176
$636$	−7.97940e8	−0.122990
$637$	1.59088e9	0.243865
$638$	1.57631e9	0.240309
$639$	−1.06412e10	−1.61339
$640$	−4.86868e9	−0.734144
$641$	7.09598e8	0.106417	0.0532083	−	0.998583i	$-0.483055\pi$
$641$	7.09598e8	0.106417	0.0532083	+	0.998583i	$0.483055\pi$
$642$	−1.13130e9	−0.168735
$643$	1.08531e10	1.60996	0.804981	−	0.593300i	$-0.202175\pi$
$643$	1.08531e10	1.60996	0.804981	+	0.593300i	$0.202175\pi$
$644$	−3.34788e9	−0.493935
$645$	2.23820e9	0.328428
$646$	−9.59883e9	−1.40089
$647$	4.80470e9	0.697431	0.348715	−	0.937229i	$-0.386618\pi$
$647$	4.80470e9	0.697431	0.348715	+	0.937229i	$0.386618\pi$
$648$	1.66530e9	0.240426
$649$	−6.19942e7	−0.00890215
$650$	−3.20253e9	−0.457401
$651$	5.22365e8	0.0742062
$652$	1.08767e10	1.53685
$653$	1.44168e9	0.202615	0.101307	−	0.994855i	$-0.467697\pi$
$653$	1.44168e9	0.202615	0.101307	+	0.994855i	$0.467697\pi$
$654$	6.38414e8	0.0892443
$655$	−6.09565e9	−0.847570
$656$	−2.77620e9	−0.383961
$657$	3.98368e9	0.548032
$658$	−2.60581e8	−0.0356576
$659$	7.21670e9	0.982289	0.491145	−	0.871078i	$-0.336579\pi$
$659$	7.21670e9	0.982289	0.491145	+	0.871078i	$0.336579\pi$
$660$	2.01196e8	0.0272405
$661$	−2.21635e9	−0.298493	−0.149246	−	0.988800i	$-0.547685\pi$
$661$	−2.21635e9	−0.298493	−0.149246	+	0.988800i	$0.547685\pi$
$662$	−1.98373e9	−0.265754
$663$	−2.61282e8	−0.0348187
$664$	−2.10788e9	−0.279421
$665$	−4.15451e9	−0.547828
$666$	−2.11924e10	−2.77985
$667$	−1.35788e10	−1.77183
$668$	2.58546e9	0.335599
$669$	1.50422e9	0.194231
$670$	2.54472e8	0.0326873
$671$	8.75485e8	0.111872
$672$	−5.59353e8	−0.0711039
$673$	−2.41738e9	−0.305697	−0.152849	−	0.988250i	$-0.548845\pi$
$673$	−2.41738e9	−0.305697	−0.152849	+	0.988250i	$0.548845\pi$
$674$	−5.33405e9	−0.671038
$675$	2.53208e9	0.316895
$676$	7.25132e8	0.0902825
$677$	−9.39011e9	−1.16308	−0.581541	−	0.813517i	$-0.697550\pi$
$677$	−9.39011e9	−1.16308	−0.581541	+	0.813517i	$0.697550\pi$
$678$	5.13097e8	0.0632260
$679$	1.49134e9	0.182823
$680$	−2.68061e9	−0.326928
$681$	−1.88744e9	−0.229012
$682$	−2.03070e9	−0.245133
$683$	4.92749e9	0.591770	0.295885	−	0.955223i	$-0.404385\pi$
$683$	4.92749e9	0.591770	0.295885	+	0.955223i	$0.404385\pi$
$684$	−1.04223e10	−1.24528
$685$	−3.13379e9	−0.372522
$686$	8.14017e9	0.962718
$687$	2.95071e8	0.0347199
$688$	1.07225e10	1.25527
$689$	−1.74355e9	−0.203080
$690$	−3.20985e9	−0.371975
$691$	3.13469e9	0.361428	0.180714	−	0.983536i	$-0.442159\pi$
$691$	3.13469e9	0.361428	0.180714	+	0.983536i	$0.442159\pi$
$692$	2.57180e9	0.295030
$693$	−3.32242e8	−0.0379217
$694$	1.45956e10	1.65753
$695$	−1.89284e10	−2.13879
$696$	4.76821e8	0.0536071
$697$	−3.78181e9	−0.423043
$698$	−1.69157e10	−1.88277
$699$	−1.27951e9	−0.141701
$700$	−4.13976e9	−0.456176
$701$	6.09252e9	0.668012	0.334006	−	0.942571i	$-0.391599\pi$
$701$	6.09252e9	0.668012	0.334006	+	0.942571i	$0.391599\pi$
$702$	−1.06181e9	−0.115843
$703$	1.92071e10	2.08506
$704$	1.35326e9	0.146176
$705$	−1.34899e8	−0.0144993
$706$	3.51188e9	0.375598
$707$	−2.09904e9	−0.223385
$708$	−1.26730e8	−0.0134203
$709$	1.05836e10	1.11525	0.557626	−	0.830092i	$-0.311712\pi$
$709$	1.05836e10	1.11525	0.557626	+	0.830092i	$0.311712\pi$
$710$	3.37094e10	3.53465
$711$	−1.30179e10	−1.35830
$712$	−1.08579e9	−0.112736
$713$	1.74931e10	1.80739
$714$	−6.25517e8	−0.0643125
$715$	4.39627e8	0.0449794
$716$	−4.31587e9	−0.439413
$717$	−6.89010e7	−0.00698086
$718$	6.53808e9	0.659196
$719$	1.67642e9	0.168202	0.0841011	−	0.996457i	$-0.473198\pi$
$719$	1.67642e9	0.168202	0.0841011	+	0.996457i	$0.473198\pi$
$720$	1.13685e10	1.13511
$721$	4.00177e9	0.397630
$722$	2.58405e9	0.255518
$723$	1.33550e9	0.131419
$724$	5.95483e9	0.583156
$725$	−1.67906e10	−1.63638
$726$	2.14849e9	0.208380
$727$	5.67428e9	0.547697	0.273848	−	0.961773i	$-0.411703\pi$
$727$	5.67428e9	0.547697	0.273848	+	0.961773i	$0.411703\pi$
$728$	2.56879e8	0.0246757
$729$	−9.19673e9	−0.879198
$730$	−1.26195e10	−1.20064
$731$	1.46064e10	1.38304
$732$	1.78969e9	0.168651
$733$	−1.70540e9	−0.159942	−0.0799712	−	0.996797i	$-0.525483\pi$
$733$	−1.70540e9	−0.159942	−0.0799712	+	0.996797i	$0.525483\pi$
$734$	−2.42357e10	−2.26214
$735$	1.97167e9	0.183159
$736$	−1.87317e10	−1.73183
$737$	−1.84440e7	−0.00169715
$738$	−7.60485e9	−0.696456
$739$	−1.54475e10	−1.40800	−0.703998	−	0.710202i	$-0.748604\pi$
$739$	−1.54475e10	−1.40800	−0.703998	+	0.710202i	$0.748604\pi$
$740$	3.62487e10	3.28838
$741$	4.76192e8	0.0429950
$742$	−4.17412e9	−0.375103
$743$	1.73310e10	1.55011	0.775056	−	0.631892i	$-0.217721\pi$
$743$	1.73310e10	1.55011	0.775056	+	0.631892i	$0.217721\pi$
$744$	−6.14269e8	−0.0546832
$745$	−1.73597e10	−1.53813
$746$	1.42693e10	1.25839
$747$	1.21777e10	1.06892
$748$	1.31300e9	0.114712
$749$	−3.19539e9	−0.277868
$750$	−4.20799e8	−0.0364217
$751$	−1.43783e10	−1.23871	−0.619354	−	0.785112i	$-0.712605\pi$
$751$	−1.43783e10	−1.23871	−0.619354	+	0.785112i	$0.712605\pi$
$752$	−6.46259e8	−0.0554172
$753$	−1.12547e9	−0.0960618
$754$	7.04104e9	0.598188
$755$	−9.73040e9	−0.822842
$756$	−1.37256e9	−0.115532
$757$	−2.36759e10	−1.98368	−0.991840	−	0.127485i	$-0.959309\pi$
$757$	−2.36759e10	−1.98368	−0.991840	+	0.127485i	$0.959309\pi$
$758$	−1.17157e10	−0.977071
$759$	2.32648e8	0.0193132
$760$	4.88545e9	0.403699
$761$	−1.37187e10	−1.12841	−0.564205	−	0.825635i	$-0.690817\pi$
$761$	−1.37187e10	−1.12841	−0.564205	+	0.825635i	$0.690817\pi$
$762$	−1.51941e9	−0.124404
$763$	1.80322e9	0.146965
$764$	−2.58376e10	−2.09617
$765$	1.54865e10	1.25065
$766$	−1.21705e10	−0.978382
$767$	−2.76914e8	−0.0221596
$768$	−1.02103e9	−0.0813345
$769$	1.56104e10	1.23786	0.618932	−	0.785444i	$-0.287566\pi$
$769$	1.56104e10	1.23786	0.618932	+	0.785444i	$0.287566\pi$
$770$	1.05248e9	0.0830799
$771$	−3.44990e9	−0.271092
$772$	−7.42318e8	−0.0580670
$773$	1.25044e10	0.973725	0.486862	−	0.873479i	$-0.338141\pi$
$773$	1.25044e10	0.973725	0.486862	+	0.873479i	$0.338141\pi$
$774$	2.93721e10	2.27689
$775$	2.16307e10	1.66922
$776$	−1.75372e9	−0.134724
$777$	1.25165e9	0.0957213
$778$	−2.80109e10	−2.13255
$779$	6.89242e9	0.522385
$780$	8.98697e8	0.0678082
$781$	−2.44324e9	−0.183521
$782$	−2.09474e10	−1.56642
$783$	−5.56700e9	−0.414434
$784$	9.44562e9	0.700042
$785$	−1.03571e9	−0.0764177
$786$	1.67267e9	0.122866
$787$	−1.35499e10	−0.990890	−0.495445	−	0.868639i	$-0.664995\pi$
$787$	−1.35499e10	−0.990890	−0.495445	+	0.868639i	$0.664995\pi$
$788$	1.91226e10	1.39221
$789$	6.58212e8	0.0477086
$790$	4.12381e10	2.97580
$791$	1.44926e9	0.104119
$792$	3.90696e8	0.0279448
$793$	3.91059e9	0.278475
$794$	−2.68642e10	−1.90459
$795$	−2.16089e9	−0.152527
$796$	2.81647e10	1.97929
$797$	−3.06558e9	−0.214491	−0.107245	−	0.994233i	$-0.534203\pi$
$797$	−3.06558e9	−0.214491	−0.107245	+	0.994233i	$0.534203\pi$
$798$	1.14002e9	0.0794147
$799$	−8.80350e8	−0.0610579
$800$	−2.31624e10	−1.59944
$801$	6.27283e9	0.431271
$802$	−3.39637e10	−2.32491
$803$	9.14658e8	0.0623382
$804$	−3.77037e7	−0.00255852
$805$	−9.06635e9	−0.612557
$806$	−9.07070e9	−0.610195
$807$	1.97410e9	0.132225
$808$	2.46835e9	0.164614
$809$	−1.15847e10	−0.769243	−0.384622	−	0.923074i	$-0.625668\pi$
$809$	−1.15847e10	−0.769243	−0.384622	+	0.923074i	$0.625668\pi$
$810$	3.04770e10	2.01500
$811$	1.46919e9	0.0967173	0.0483587	−	0.998830i	$-0.484601\pi$
$811$	1.46919e9	0.0967173	0.0483587	+	0.998830i	$0.484601\pi$
$812$	9.10162e9	0.596586
$813$	−3.16905e9	−0.206829
$814$	−4.86581e9	−0.316206
$815$	2.94550e10	1.90593
$816$	−1.55133e9	−0.0999512
$817$	−2.66205e10	−1.70781
$818$	−6.54692e9	−0.418216
$819$	−1.48405e9	−0.0943963
$820$	1.30078e10	0.823862
$821$	9.59583e8	0.0605176	0.0302588	−	0.999542i	$-0.490367\pi$
$821$	9.59583e8	0.0605176	0.0302588	+	0.999542i	$0.490367\pi$
$822$	8.59925e8	0.0540019
$823$	6.57904e9	0.411399	0.205700	−	0.978615i	$-0.434053\pi$
$823$	6.57904e9	0.411399	0.205700	+	0.978615i	$0.434053\pi$
$824$	−4.70584e9	−0.293017
$825$	2.87677e8	0.0178368
$826$	−6.62941e8	−0.0409303
$827$	4.71138e8	0.0289654	0.0144827	−	0.999895i	$-0.495390\pi$
$827$	4.71138e8	0.0289654	0.0144827	+	0.999895i	$0.495390\pi$
$828$	−2.27444e10	−1.39241
$829$	2.35273e10	1.43427	0.717136	−	0.696933i	$-0.245452\pi$
$829$	2.35273e10	1.43427	0.717136	+	0.696933i	$0.245452\pi$
$830$	−3.85767e10	−2.34181
$831$	−9.65411e8	−0.0583591
$832$	6.04471e9	0.363868
$833$	1.28671e10	0.771297
$834$	5.19404e9	0.310044
$835$	7.00165e9	0.416196
$836$	−2.39297e9	−0.141649
$837$	7.17175e9	0.422752
$838$	2.60099e10	1.52681
$839$	2.07659e10	1.21390	0.606951	−	0.794739i	$-0.292392\pi$
$839$	2.07659e10	1.21390	0.606951	+	0.794739i	$0.292392\pi$
$840$	3.18365e8	0.0185331
$841$	1.96657e10	1.14005
$842$	2.68574e10	1.55050
$843$	7.05320e8	0.0405499
$844$	−7.68588e9	−0.440043
$845$	1.96372e9	0.111964
$846$	−1.77030e9	−0.100519
$847$	6.06849e9	0.343154
$848$	−1.03521e10	−0.582966
$849$	2.70203e9	0.151535
$850$	−2.59022e10	−1.44667
$851$	4.19154e10	2.33142
$852$	−4.99453e9	−0.276666
$853$	−1.82054e10	−1.00434	−0.502168	−	0.864770i	$-0.667464\pi$
$853$	−1.82054e10	−1.00434	−0.502168	+	0.864770i	$0.667464\pi$
$854$	9.36207e9	0.514362
$855$	−2.82244e10	−1.54434
$856$	3.75759e9	0.204763
$857$	−9.18289e9	−0.498364	−0.249182	−	0.968457i	$-0.580162\pi$
$857$	−9.18289e9	−0.498364	−0.249182	+	0.968457i	$0.580162\pi$
$858$	−1.20636e8	−0.00652034
$859$	−2.29543e10	−1.23563	−0.617813	−	0.786325i	$-0.711981\pi$
$859$	−2.29543e10	−1.23563	−0.617813	+	0.786325i	$0.711981\pi$
$860$	−5.02398e10	−2.69341
$861$	4.49151e8	0.0239818
$862$	7.23223e9	0.384589
$863$	−1.54093e10	−0.816101	−0.408051	−	0.912959i	$-0.633791\pi$
$863$	−1.54093e10	−0.816101	−0.408051	+	0.912959i	$0.633791\pi$
$864$	−7.67958e9	−0.405078
$865$	6.96466e9	0.365884
$866$	2.12832e10	1.11359
$867$	6.33058e8	0.0329896
$868$	−1.17253e10	−0.608560
$869$	−2.98892e9	−0.154506
$870$	8.72637e9	0.449279
$871$	−8.23853e7	−0.00422461
$872$	−2.12048e9	−0.108300
$873$	1.01316e10	0.515383
$874$	3.81771e10	1.93425
$875$	−1.18856e9	−0.0599782
$876$	1.86977e9	0.0939774
$877$	−5.07934e8	−0.0254278	−0.0127139	−	0.999919i	$-0.504047\pi$
$877$	−5.07934e8	−0.0254278	−0.0127139	+	0.999919i	$0.504047\pi$
$878$	−4.86617e10	−2.42637
$879$	−3.64876e9	−0.181211
$880$	2.61022e9	0.129119
$881$	−4.16725e9	−0.205321	−0.102661	−	0.994716i	$-0.532736\pi$
$881$	−4.16725e9	−0.205321	−0.102661	+	0.994716i	$0.532736\pi$
$882$	2.58744e10	1.26978
$883$	−3.46362e10	−1.69304	−0.846522	−	0.532354i	$-0.821308\pi$
$883$	−3.46362e10	−1.69304	−0.846522	+	0.532354i	$0.821308\pi$
$884$	5.86488e9	0.285546
$885$	−3.43196e8	−0.0166433
$886$	4.90240e10	2.36805
$887$	−1.48995e9	−0.0716868	−0.0358434	−	0.999357i	$-0.511412\pi$
$887$	−1.48995e9	−0.0716868	−0.0358434	+	0.999357i	$0.511412\pi$
$888$	−1.47186e9	−0.0705378
$889$	−4.29164e9	−0.204865
$890$	−1.98711e10	−0.944839
$891$	−2.20896e9	−0.104620
$892$	−3.37644e10	−1.59288
$893$	1.60445e9	0.0753958
$894$	4.76357e9	0.222972
$895$	−1.16877e10	−0.544941
$896$	3.77355e9	0.175255
$897$	1.03919e9	0.0480752
$898$	1.86358e10	0.858777
$899$	−4.75569e10	−2.18301
$900$	−2.81242e10	−1.28597
$901$	−1.41019e10	−0.642304
$902$	−1.74608e9	−0.0792214
$903$	−1.73475e9	−0.0784025
$904$	−1.70424e9	−0.0767259
$905$	1.61262e10	0.723205
$906$	2.67007e9	0.119282
$907$	−3.55548e10	−1.58224	−0.791121	−	0.611660i	$-0.790502\pi$
$907$	−3.55548e10	−1.58224	−0.791121	+	0.611660i	$0.790502\pi$
$908$	4.23664e10	1.87811
$909$	−1.42602e10	−0.629727
$910$	4.70119e9	0.206806
$911$	1.65880e10	0.726910	0.363455	−	0.931612i	$-0.381597\pi$
$911$	1.65880e10	0.726910	0.363455	+	0.931612i	$0.381597\pi$
$912$	2.82732e9	0.123422
$913$	2.79602e9	0.121589
$914$	1.21225e10	0.525149
$915$	4.84662e9	0.209154
$916$	−6.62332e9	−0.284735
$917$	4.72453e9	0.202333
$918$	−8.58797e9	−0.366388
$919$	2.77137e10	1.17785	0.588925	−	0.808188i	$-0.299552\pi$
$919$	2.77137e10	1.17785	0.588925	+	0.808188i	$0.299552\pi$
$920$	1.06615e10	0.451398
$921$	−2.90751e9	−0.122634
$922$	1.25697e10	0.528159
$923$	−1.09134e10	−0.456829
$924$	−1.55940e8	−0.00650288
$925$	5.18298e10	2.15319
$926$	9.52953e9	0.394396
$927$	2.71867e10	1.12093
$928$	5.09244e10	2.09174
$929$	−3.90142e10	−1.59650	−0.798248	−	0.602329i	$-0.794240\pi$
$929$	−3.90142e10	−1.59650	−0.798248	+	0.602329i	$0.794240\pi$
$930$	−1.12418e10	−0.458297
$931$	−2.34504e10	−0.952417
$932$	2.87205e10	1.16208
$933$	−2.44045e9	−0.0983749
$934$	3.40625e10	1.36793
$935$	3.55571e9	0.142261
$936$	1.74515e9	0.0695613
$937$	1.92322e10	0.763730	0.381865	−	0.924218i	$-0.375282\pi$
$937$	1.92322e10	0.763730	0.381865	+	0.924218i	$0.375282\pi$
$938$	−1.97233e8	−0.00780313
$939$	−2.31775e9	−0.0913560
$940$	3.02802e9	0.118908
$941$	3.97163e10	1.55383	0.776917	−	0.629603i	$-0.216782\pi$
$941$	3.97163e10	1.55383	0.776917	+	0.629603i	$0.216782\pi$
$942$	2.84203e8	0.0110777
$943$	1.50413e10	0.584108
$944$	−1.64414e9	−0.0636117
$945$	−3.71699e9	−0.143278
$946$	6.74388e9	0.258995
$947$	−5.90090e9	−0.225784	−0.112892	−	0.993607i	$-0.536011\pi$
$947$	−5.90090e9	−0.225784	−0.112892	+	0.993607i	$0.536011\pi$
$948$	−6.11002e9	−0.232924
$949$	4.08557e9	0.155175
$950$	4.72072e10	1.78639
$951$	−4.80804e9	−0.181274
$952$	2.07765e9	0.0780445
$953$	−2.77255e10	−1.03766	−0.518830	−	0.854878i	$-0.673632\pi$
$953$	−2.77255e10	−1.03766	−0.518830	+	0.854878i	$0.673632\pi$
$954$	−2.83576e10	−1.05742
$955$	−6.99704e10	−2.59958
$956$	1.54659e9	0.0572495
$957$	−6.32483e8	−0.0233269
$958$	−6.27697e10	−2.30659
$959$	2.42889e9	0.0889288
$960$	7.49155e9	0.273289
$961$	3.37531e10	1.22682
$962$	−2.17345e10	−0.787112
$963$	−2.17084e10	−0.783315
$964$	−2.99773e10	−1.07776
$965$	−2.01026e9	−0.0720122
$966$	2.48785e9	0.0887981
$967$	−4.38801e10	−1.56054	−0.780270	−	0.625443i	$-0.784918\pi$
$967$	−4.38801e10	−1.56054	−0.780270	+	0.625443i	$0.784918\pi$
$968$	−7.13618e9	−0.252873
$969$	3.85145e9	0.135985
$970$	−3.20951e10	−1.12912
$971$	−1.92062e10	−0.673246	−0.336623	−	0.941640i	$-0.609285\pi$
$971$	−1.92062e10	−0.673246	−0.336623	+	0.941640i	$0.609285\pi$
$972$	−1.40353e10	−0.490218
$973$	1.46708e10	0.510573
$974$	−3.38250e10	−1.17296
$975$	1.28499e9	0.0444001
$976$	2.32186e10	0.799396
$977$	4.63629e9	0.159052	0.0795261	−	0.996833i	$-0.474659\pi$
$977$	4.63629e9	0.159052	0.0795261	+	0.996833i	$0.474659\pi$
$978$	−8.08259e9	−0.276290
$979$	1.44025e9	0.0490567
$980$	−4.42570e10	−1.50207
$981$	1.22505e10	0.414298
$982$	4.20060e10	1.41554
$983$	4.18641e10	1.40574	0.702869	−	0.711319i	$-0.251902\pi$
$983$	4.18641e10	1.40574	0.702869	+	0.711319i	$0.251902\pi$
$984$	−5.28174e8	−0.0176724
$985$	5.17856e10	1.72656
$986$	5.69481e10	1.89195
$987$	1.04556e8	0.00346129
$988$	−1.06888e10	−0.352599
$989$	−5.80937e10	−1.90960
$990$	7.15019e9	0.234204
$991$	−3.91684e10	−1.27843	−0.639217	−	0.769026i	$-0.720741\pi$
$991$	−3.91684e10	−1.27843	−0.639217	+	0.769026i	$0.720741\pi$
$992$	−6.56040e10	−2.13373
$993$	7.95956e8	0.0257969
$994$	−2.61270e10	−0.843794
$995$	7.62723e10	2.45463
$996$	5.71570e9	0.183300
$997$	3.98653e10	1.27398	0.636989	−	0.770873i	$-0.280180\pi$
$997$	3.98653e10	1.27398	0.636989	+	0.770873i	$0.280180\pi$
$998$	−2.00856e10	−0.639630
$999$	1.71844e10	0.545324

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	13.8.a.c.1.4	✓	4
3.2	odd	2		117.8.a.e.1.1		4
4.3	odd	2		208.8.a.k.1.3		4
5.4	even	2		325.8.a.c.1.1		4
13.5	odd	4		169.8.b.c.168.1		8
13.8	odd	4		169.8.b.c.168.8		8
13.12	even	2		169.8.a.c.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.a.c.1.4	✓	4	1.1	even	1	trivial
117.8.a.e.1.1		4	3.2	odd	2
169.8.a.c.1.1		4	13.12	even	2
169.8.b.c.168.1		8	13.5	odd	4
169.8.b.c.168.8		8	13.8	odd	4
208.8.a.k.1.3		4	4.3	odd	2
325.8.a.c.1.1		4	5.4	even	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:	\( N \)	\(=\)	\( 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	13.a (trivial)

\(f(q)\)	\(=\)	\(q+16.6802 q^{2} -6.69280 q^{3} +150.230 q^{4} +406.835 q^{5} -111.637 q^{6} -315.324 q^{7} +370.802 q^{8} -2142.21 q^{9} +6786.11 q^{10} -491.853 q^{11} -1005.46 q^{12} -2197.00 q^{13} -5259.68 q^{14} -2722.87 q^{15} -13044.4 q^{16} -17769.4 q^{17} -35732.5 q^{18} +32385.0 q^{19} +61118.9 q^{20} +2110.40 q^{21} -8204.23 q^{22} +70673.5 q^{23} -2481.70 q^{24} +87390.0 q^{25} -36646.5 q^{26} +28974.5 q^{27} -47371.1 q^{28} -192134. q^{29} -45418.1 q^{30} +247519. q^{31} -265046. q^{32} +3291.88 q^{33} -296397. q^{34} -128285. q^{35} -321824. q^{36} +593086. q^{37} +540189. q^{38} +14704.1 q^{39} +150855. q^{40} +212827. q^{41} +35202.0 q^{42} -822000. q^{43} -73891.2 q^{44} -871525. q^{45} +1.17885e6 q^{46} +49543.1 q^{47} +87303.4 q^{48} -724114. q^{49} +1.45769e6 q^{50} +118927. q^{51} -330055. q^{52} +793607. q^{53} +483301. q^{54} -200103. q^{55} -116923. q^{56} -216746. q^{57} -3.20484e6 q^{58} +126042. q^{59} -409056. q^{60} -1.77997e6 q^{61} +4.12868e6 q^{62} +675489. q^{63} -2.75135e6 q^{64} -893817. q^{65} +54909.3 q^{66} +37499.0 q^{67} -2.66949e6 q^{68} -473004. q^{69} -2.13982e6 q^{70} +4.96741e6 q^{71} -794334. q^{72} -1.85961e6 q^{73} +9.89281e6 q^{74} -584884. q^{75} +4.86520e6 q^{76} +155093. q^{77} +245267. q^{78} +6.07684e6 q^{79} -5.30692e6 q^{80} +4.49108e6 q^{81} +3.55001e6 q^{82} -5.68466e6 q^{83} +317046. q^{84} -7.22921e6 q^{85} -1.37112e7 q^{86} +1.28592e6 q^{87} -182380. q^{88} -2.92821e6 q^{89} -1.45372e7 q^{90} +692767. q^{91} +1.06173e7 q^{92} -1.65660e6 q^{93} +826390. q^{94} +1.31754e7 q^{95} +1.77390e6 q^{96} -4.72954e6 q^{97} -1.20784e7 q^{98} +1.05365e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 15 q^{2} + 80 q^{3} + 253 q^{4} + 258 q^{5} + 1579 q^{6} + 1692 q^{7} + 1893 q^{8} + 3494 q^{9} - 4495 q^{10} + 1836 q^{11} - 3655 q^{12} - 8788 q^{13} - 18285 q^{14} - 29736 q^{15} - 36159 q^{16}+ \cdots + 6200852 q^{99}+O(q^{100}) \) 4 * q + 15 * q^2 + 80 * q^3 + 253 * q^4 + 258 * q^5 + 1579 * q^6 + 1692 * q^7 + 1893 * q^8 + 3494 * q^9 - 4495 * q^10 + 1836 * q^11 - 3655 * q^12 - 8788 * q^13 - 18285 * q^14 - 29736 * q^15 - 36159 * q^16 + 11814 * q^17 - 2738 * q^18 + 27660 * q^19 - 30369 * q^20 + 69930 * q^21 + 59930 * q^22 + 172920 * q^23 + 26553 * q^24 + 219414 * q^25 - 32955 * q^26 + 174800 * q^27 - 170903 * q^28 + 133344 * q^29 - 624695 * q^30 - 231748 * q^31 - 459291 * q^32 - 33868 * q^33 - 480249 * q^34 + 322296 * q^35 - 157930 * q^36 + 248026 * q^37 - 85326 * q^38 - 175760 * q^39 + 379895 * q^40 + 588108 * q^41 + 132275 * q^42 + 309304 * q^43 + 1270806 * q^44 - 1773144 * q^45 + 574948 * q^46 + 557916 * q^47 + 837429 * q^48 - 767378 * q^49 + 5737920 * q^50 - 1816496 * q^51 - 555841 * q^52 + 2022348 * q^53 - 479 * q^54 - 3770208 * q^55 + 1391625 * q^56 - 5609844 * q^57 - 3027194 * q^58 + 1162668 * q^59 - 8100077 * q^60 - 1340572 * q^61 + 2715564 * q^62 + 4213476 * q^63 - 6734935 * q^64 - 566826 * q^65 + 8124226 * q^66 - 598484 * q^67 + 4515993 * q^68 - 108848 * q^69 - 4329415 * q^70 + 697860 * q^71 + 1607430 * q^72 - 13725816 * q^73 + 14884053 * q^74 + 16604592 * q^75 + 12055070 * q^76 - 5326932 * q^77 - 3469063 * q^78 + 20079576 * q^79 + 3710667 * q^80 - 8368396 * q^81 + 857160 * q^82 - 2024724 * q^83 - 13603483 * q^84 - 16146514 * q^85 - 22922499 * q^86 + 22170952 * q^87 - 2100618 * q^88 + 17646240 * q^89 - 42372890 * q^90 - 3717324 * q^91 + 14504892 * q^92 - 17910680 * q^93 - 3729785 * q^94 + 12508152 * q^95 - 37637751 * q^96 - 6329096 * q^97 - 22166262 * q^98 + 6200852 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more