LMFDB - Embedded newform 69.6.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,6,Mod(1,69)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	69.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$11.0664835671$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 113x^{3} - 257x^{2} + 1404x + 2197 $ x^5 - 113x^3 - 257x^2 + 1404*x + 2197
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.90234$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.24792 q^{2} +9.00000 q^{3} -26.9469 q^{4} +53.3906 q^{5} +20.2313 q^{6} +89.8688 q^{7} -132.508 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+2.24792 q^{2} +9.00000 q^{3} -26.9469 q^{4} +53.3906 q^{5} +20.2313 q^{6} +89.8688 q^{7} -132.508 q^{8} +81.0000 q^{9} +120.018 q^{10} +225.013 q^{11} -242.522 q^{12} +725.747 q^{13} +202.018 q^{14} +480.515 q^{15} +564.433 q^{16} +44.9540 q^{17} +182.081 q^{18} +1212.50 q^{19} -1438.71 q^{20} +808.819 q^{21} +505.810 q^{22} -529.000 q^{23} -1192.57 q^{24} -274.446 q^{25} +1631.42 q^{26} +729.000 q^{27} -2421.68 q^{28} +2592.49 q^{29} +1080.16 q^{30} +585.421 q^{31} +5509.05 q^{32} +2025.11 q^{33} +101.053 q^{34} +4798.15 q^{35} -2182.70 q^{36} -3849.04 q^{37} +2725.59 q^{38} +6531.73 q^{39} -7074.66 q^{40} -4299.92 q^{41} +1818.16 q^{42} -20567.4 q^{43} -6063.39 q^{44} +4324.64 q^{45} -1189.15 q^{46} -5221.50 q^{47} +5079.90 q^{48} -8730.61 q^{49} -616.931 q^{50} +404.586 q^{51} -19556.6 q^{52} -15753.7 q^{53} +1638.73 q^{54} +12013.6 q^{55} -11908.3 q^{56} +10912.5 q^{57} +5827.70 q^{58} -13780.6 q^{59} -12948.4 q^{60} -18988.5 q^{61} +1315.98 q^{62} +7279.37 q^{63} -5677.98 q^{64} +38748.1 q^{65} +4552.29 q^{66} +1951.43 q^{67} -1211.37 q^{68} -4761.00 q^{69} +10785.8 q^{70} +75281.4 q^{71} -10733.1 q^{72} -4177.11 q^{73} -8652.32 q^{74} -2470.01 q^{75} -32673.0 q^{76} +20221.6 q^{77} +14682.8 q^{78} -95896.7 q^{79} +30135.4 q^{80} +6561.00 q^{81} -9665.87 q^{82} +13366.7 q^{83} -21795.1 q^{84} +2400.12 q^{85} -46233.9 q^{86} +23332.4 q^{87} -29815.9 q^{88} +50932.1 q^{89} +9721.43 q^{90} +65222.0 q^{91} +14254.9 q^{92} +5268.79 q^{93} -11737.5 q^{94} +64735.9 q^{95} +49581.4 q^{96} +44280.7 q^{97} -19625.7 q^{98} +18226.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9}+O(q^{10}) $ 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99}+O(q^{100}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9 - 172 * q^10 + 1100 * q^11 + 1062 * q^12 - 978 * q^13 - 344 * q^14 + 846 * q^15 + 1218 * q^16 + 2522 * q^17 + 648 * q^18 + 2060 * q^19 + 7720 * q^20 + 2448 * q^21 - 2572 * q^22 - 2645 * q^23 + 2322 * q^24 + 12035 * q^25 + 9280 * q^26 + 3645 * q^27 + 8072 * q^28 + 1526 * q^29 - 1548 * q^30 - 7392 * q^31 - 5086 * q^32 + 9900 * q^33 - 15608 * q^34 + 6056 * q^35 + 9558 * q^36 - 8210 * q^37 - 14276 * q^38 - 8802 * q^39 - 37472 * q^40 + 21250 * q^41 - 3096 * q^42 - 4548 * q^43 - 4260 * q^44 + 7614 * q^45 - 4232 * q^46 + 536 * q^47 + 10962 * q^48 - 27979 * q^49 - 81872 * q^50 + 22698 * q^51 - 76380 * q^52 - 11482 * q^53 + 5832 * q^54 - 77064 * q^55 - 28624 * q^56 + 18540 * q^57 - 79680 * q^58 + 74676 * q^59 + 69480 * q^60 - 44618 * q^61 + 64880 * q^62 + 22032 * q^63 - 137382 * q^64 - 24388 * q^65 - 23148 * q^66 - 1412 * q^67 + 80196 * q^68 - 23805 * q^69 - 222304 * q^70 + 37912 * q^71 + 20898 * q^72 + 46546 * q^73 + 111604 * q^74 + 108315 * q^75 - 79548 * q^76 + 157008 * q^77 + 83520 * q^78 + 50544 * q^79 + 69424 * q^80 + 32805 * q^81 - 233720 * q^82 + 89588 * q^83 + 72648 * q^84 + 147892 * q^85 + 77428 * q^86 + 13734 * q^87 + 54484 * q^88 + 280410 * q^89 - 13932 * q^90 - 27416 * q^91 - 62422 * q^92 - 66528 * q^93 + 113632 * q^94 + 203120 * q^95 - 45774 * q^96 + 90074 * q^97 + 32976 * q^98 + 89100 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.24792	0.397380	0.198690	−	0.980062i	$-0.436331\pi$
$2$	2.24792	0.397380	0.198690	+	0.980062i	$0.436331\pi$
$3$	9.00000	0.577350
$3$	9.00000	0.577350
$4$	−26.9469	−0.842090
$5$	53.3906	0.955080	0.477540	−	0.878610i	$-0.341529\pi$
$5$	53.3906	0.955080	0.477540	+	0.878610i	$0.341529\pi$
$6$	20.2313	0.229427
$7$	89.8688	0.693208	0.346604	−	0.938012i	$-0.387335\pi$
$7$	89.8688	0.693208	0.346604	+	0.938012i	$0.387335\pi$
$8$	−132.508	−0.732009
$9$	81.0000	0.333333
$10$	120.018	0.379529
$11$	225.013	0.560693	0.280347	−	0.959899i	$-0.409551\pi$
$11$	225.013	0.560693	0.280347	+	0.959899i	$0.409551\pi$
$12$	−242.522	−0.486181
$13$	725.747	1.19104	0.595521	−	0.803340i	$-0.296946\pi$
$13$	725.747	1.19104	0.595521	+	0.803340i	$0.296946\pi$
$14$	202.018	0.275467
$15$	480.515	0.551416
$16$	564.433	0.551204
$17$	44.9540	0.0377265	0.0188632	−	0.999822i	$-0.493995\pi$
$17$	44.9540	0.0377265	0.0188632	+	0.999822i	$0.493995\pi$
$18$	182.081	0.132460
$19$	1212.50	0.770542	0.385271	−	0.922803i	$-0.374108\pi$
$19$	1212.50	0.770542	0.385271	+	0.922803i	$0.374108\pi$
$20$	−1438.71	−0.804263
$21$	808.819	0.400224
$22$	505.810	0.222808
$23$	−529.000	−0.208514
$23$	−529.000	−0.208514
$24$	−1192.57	−0.422625
$25$	−274.446	−0.0878226
$26$	1631.42	0.473296
$27$	729.000	0.192450
$28$	−2421.68	−0.583743
$29$	2592.49	0.572430	0.286215	−	0.958165i	$-0.407603\pi$
$29$	2592.49	0.572430	0.286215	+	0.958165i	$0.407603\pi$
$30$	1080.16	0.219121
$31$	585.421	0.109412	0.0547059	−	0.998503i	$-0.482578\pi$
$31$	585.421	0.109412	0.0547059	+	0.998503i	$0.482578\pi$
$32$	5509.05	0.951046
$33$	2025.11	0.323716
$34$	101.053	0.0149917
$35$	4798.15	0.662069
$36$	−2182.70	−0.280697
$37$	−3849.04	−0.462219	−0.231109	−	0.972928i	$-0.574236\pi$
$37$	−3849.04	−0.462219	−0.231109	+	0.972928i	$0.574236\pi$
$38$	2725.59	0.306198
$39$	6531.73	0.687648
$40$	−7074.66	−0.699127
$41$	−4299.92	−0.399485	−0.199743	−	0.979848i	$-0.564011\pi$
$41$	−4299.92	−0.399485	−0.199743	+	0.979848i	$0.564011\pi$
$42$	1818.16	0.159041
$43$	−20567.4	−1.69632	−0.848162	−	0.529737i	$-0.822291\pi$
$43$	−20567.4	−1.69632	−0.848162	+	0.529737i	$0.822291\pi$
$44$	−6063.39	−0.472154
$45$	4324.64	0.318360
$46$	−1189.15	−0.0828594
$47$	−5221.50	−0.344787	−0.172393	−	0.985028i	$-0.555150\pi$
$47$	−5221.50	−0.344787	−0.172393	+	0.985028i	$0.555150\pi$
$48$	5079.90	0.318238
$49$	−8730.61	−0.519462
$50$	−616.931	−0.0348989
$51$	404.586	0.0217814
$52$	−19556.6	−1.00296
$53$	−15753.7	−0.770360	−0.385180	−	0.922841i	$-0.625861\pi$
$53$	−15753.7	−0.770360	−0.385180	+	0.922841i	$0.625861\pi$
$54$	1638.73	0.0764757
$55$	12013.6	0.535507
$56$	−11908.3	−0.507434
$57$	10912.5	0.444873
$58$	5827.70	0.227472
$59$	−13780.6	−0.515392	−0.257696	−	0.966226i	$-0.582963\pi$
$59$	−13780.6	−0.515392	−0.257696	+	0.966226i	$0.582963\pi$
$60$	−12948.4	−0.464341
$61$	−18988.5	−0.653380	−0.326690	−	0.945131i	$-0.605933\pi$
$61$	−18988.5	−0.653380	−0.326690	+	0.945131i	$0.605933\pi$
$62$	1315.98	0.0434780
$63$	7279.37	0.231069
$64$	−5677.98	−0.173278
$65$	38748.1	1.13754
$66$	4552.29	0.128638
$67$	1951.43	0.0531088	0.0265544	−	0.999647i	$-0.491546\pi$
$67$	1951.43	0.0531088	0.0265544	+	0.999647i	$0.491546\pi$
$68$	−1211.37	−0.0317691
$69$	−4761.00	−0.120386
$70$	10785.8	0.263093
$71$	75281.4	1.77232	0.886160	−	0.463380i	$-0.153363\pi$
$71$	75281.4	1.77232	0.886160	+	0.463380i	$0.153363\pi$
$72$	−10733.1	−0.244003
$73$	−4177.11	−0.0917422	−0.0458711	−	0.998947i	$-0.514606\pi$
$73$	−4177.11	−0.0917422	−0.0458711	+	0.998947i	$0.514606\pi$
$74$	−8652.32	−0.183676
$75$	−2470.01	−0.0507044
$76$	−32673.0	−0.648865
$77$	20221.6	0.388677
$78$	14682.8	0.273257
$79$	−95896.7	−1.72877	−0.864383	−	0.502835i	$-0.832290\pi$
$79$	−95896.7	−1.72877	−0.864383	+	0.502835i	$0.832290\pi$
$80$	30135.4	0.526444
$81$	6561.00	0.111111
$82$	−9665.87	−0.158747
$83$	13366.7	0.212976	0.106488	−	0.994314i	$-0.466039\pi$
$83$	13366.7	0.212976	0.106488	+	0.994314i	$0.466039\pi$
$84$	−21795.1	−0.337024
$85$	2400.12	0.0360318
$86$	−46233.9	−0.674084
$87$	23332.4	0.330492
$88$	−29815.9	−0.410432
$89$	50932.1	0.681580	0.340790	−	0.940139i	$-0.389305\pi$
$89$	50932.1	0.681580	0.340790	+	0.940139i	$0.389305\pi$
$90$	9721.43	0.126510
$91$	65222.0	0.825640
$92$	14254.9	0.175588
$93$	5268.79	0.0631690
$94$	−11737.5	−0.137011
$95$	64735.9	0.735929
$96$	49581.4	0.549087
$97$	44280.7	0.477843	0.238921	−	0.971039i	$-0.423206\pi$
$97$	44280.7	0.477843	0.238921	+	0.971039i	$0.423206\pi$
$98$	−19625.7	−0.206424
$99$	18226.0	0.186898
$100$	7395.45	0.0739545
$101$	198402.	1.93527	0.967637	−	0.252346i	$-0.0812021\pi$
$101$	198402.	1.93527	0.967637	+	0.252346i	$0.0812021\pi$
$102$	909.476	0.00865548
$103$	201915.	1.87532	0.937658	−	0.347559i	$-0.112989\pi$
$103$	201915.	1.87532	0.937658	+	0.347559i	$0.112989\pi$
$104$	−96167.1	−0.871853
$105$	43183.3	0.382246
$106$	−35413.1	−0.306125
$107$	37688.4	0.318235	0.159117	−	0.987260i	$-0.449135\pi$
$107$	37688.4	0.318235	0.159117	+	0.987260i	$0.449135\pi$
$108$	−19644.3	−0.162060
$109$	−86493.8	−0.697298	−0.348649	−	0.937253i	$-0.613360\pi$
$109$	−86493.8	−0.697298	−0.348649	+	0.937253i	$0.613360\pi$
$110$	27005.5	0.212799
$111$	−34641.3	−0.266862
$112$	50724.9	0.382099
$113$	−150822.	−1.11114	−0.555571	−	0.831469i	$-0.687500\pi$
$113$	−150822.	−1.11114	−0.555571	+	0.831469i	$0.687500\pi$
$114$	24530.3	0.176783
$115$	−28243.6	−0.199148
$116$	−69859.5	−0.482037
$117$	58785.5	0.397014
$118$	−30977.6	−0.204806
$119$	4039.96	0.0261523
$120$	−63672.0	−0.403641
$121$	−110420.	−0.685623
$122$	−42684.6	−0.259640
$123$	−38699.3	−0.230643
$124$	−15775.3	−0.0921346
$125$	−181498.	−1.03896
$126$	16363.4	0.0918222
$127$	−47605.4	−0.261907	−0.130953	−	0.991389i	$-0.541804\pi$
$127$	−47605.4	−0.261907	−0.130953	+	0.991389i	$0.541804\pi$
$128$	−189053.	−1.01990
$129$	−185107.	−0.979373
$130$	87102.5	0.452035
$131$	−260443.	−1.32597	−0.662985	−	0.748632i	$-0.730711\pi$
$131$	−260443.	−1.32597	−0.662985	+	0.748632i	$0.730711\pi$
$132$	−54570.5	−0.272598
$133$	108966.	0.534146
$134$	4386.66	0.0211043
$135$	38921.7	0.183805
$136$	−5956.75	−0.0276161
$137$	121613.	0.553580	0.276790	−	0.960930i	$-0.410729\pi$
$137$	121613.	0.553580	0.276790	+	0.960930i	$0.410729\pi$
$138$	−10702.3	−0.0478389
$139$	61341.2	0.269287	0.134643	−	0.990894i	$-0.457011\pi$
$139$	61341.2	0.269287	0.134643	+	0.990894i	$0.457011\pi$
$140$	−129295.	−0.557521
$141$	−46993.5	−0.199063
$142$	169227.	0.704284
$143$	163302.	0.667809
$144$	45719.1	0.183735
$145$	138415.	0.546716
$146$	−9389.81	−0.0364565
$147$	−78575.4	−0.299912
$148$	103719.	0.389230
$149$	−171257.	−0.631950	−0.315975	−	0.948767i	$-0.602332\pi$
$149$	−171257.	−0.631950	−0.315975	+	0.948767i	$0.602332\pi$
$150$	−5552.38	−0.0201489
$151$	−459607.	−1.64038	−0.820189	−	0.572092i	$-0.806132\pi$
$151$	−459607.	−1.64038	−0.820189	+	0.572092i	$0.806132\pi$
$152$	−160665.	−0.564043
$153$	3641.27	0.0125755
$154$	45456.5	0.154452
$155$	31256.0	0.104497
$156$	−176010.	−0.579061
$157$	387631.	1.25508	0.627538	−	0.778586i	$-0.284063\pi$
$157$	387631.	1.25508	0.627538	+	0.778586i	$0.284063\pi$
$158$	−215568.	−0.686976
$159$	−141784.	−0.444767
$160$	294131.	0.908325
$161$	−47540.6	−0.144544
$162$	14748.6	0.0441533
$163$	−286971.	−0.845998	−0.422999	−	0.906130i	$-0.639023\pi$
$163$	−286971.	−0.845998	−0.422999	+	0.906130i	$0.639023\pi$
$164$	115869.	0.336402
$165$	108122.	0.309175
$166$	30047.3	0.0846322
$167$	580711.	1.61127	0.805636	−	0.592411i	$-0.201824\pi$
$167$	580711.	1.61127	0.805636	+	0.592411i	$0.201824\pi$
$168$	−107175.	−0.292967
$169$	155416.	0.418581
$170$	5395.27	0.0143183
$171$	98212.2	0.256847
$172$	554228.	1.42846
$173$	−191878.	−0.487427	−0.243714	−	0.969847i	$-0.578366\pi$
$173$	−191878.	−0.487427	−0.243714	+	0.969847i	$0.578366\pi$
$174$	52449.3	0.131331
$175$	−24664.1	−0.0608793
$176$	127005.	0.309056
$177$	−124025.	−0.297562
$178$	114491.	0.270846
$179$	18757.8	0.0437571	0.0218785	−	0.999761i	$-0.493035\pi$
$179$	18757.8	0.0437571	0.0218785	+	0.999761i	$0.493035\pi$
$180$	−116535.	−0.268088
$181$	47967.6	0.108831	0.0544154	−	0.998518i	$-0.482670\pi$
$181$	47967.6	0.108831	0.0544154	+	0.998518i	$0.482670\pi$
$182$	146614.	0.328092
$183$	−170897.	−0.377229
$184$	70096.6	0.152634
$185$	−205502.	−0.441456
$186$	11843.8	0.0251021
$187$	10115.2	0.0211530
$188$	140703.	0.290341
$189$	65514.3	0.133408
$190$	145521.	0.292443
$191$	−14294.3	−0.0283518	−0.0141759	−	0.999900i	$-0.504512\pi$
$191$	−14294.3	−0.0283518	−0.0141759	+	0.999900i	$0.504512\pi$
$192$	−51101.8	−0.100042
$193$	−422764.	−0.816968	−0.408484	−	0.912765i	$-0.633942\pi$
$193$	−422764.	−0.816968	−0.408484	+	0.912765i	$0.633942\pi$
$194$	99539.3	0.189885
$195$	348733.	0.656759
$196$	235262.	0.437434
$197$	−311005.	−0.570955	−0.285477	−	0.958385i	$-0.592152\pi$
$197$	−311005.	−0.570955	−0.285477	+	0.958385i	$0.592152\pi$
$198$	40970.6	0.0742693
$199$	60066.4	0.107522	0.0537612	−	0.998554i	$-0.482879\pi$
$199$	60066.4	0.107522	0.0537612	+	0.998554i	$0.482879\pi$
$200$	36366.2	0.0642869
$201$	17562.9	0.0306624
$202$	445991.	0.769038
$203$	232984.	0.396813
$204$	−10902.3	−0.0183419
$205$	−229575.	−0.381540
$206$	453887.	0.745212
$207$	−42849.0	−0.0695048
$208$	409636.	0.656507
$209$	272827.	0.432038
$210$	97072.5	0.151897
$211$	−1.08896e6	−1.68385	−0.841926	−	0.539593i	$-0.818578\pi$
$211$	−1.08896e6	−1.68385	−0.841926	+	0.539593i	$0.818578\pi$
$212$	424514.	0.648712
$213$	677533.	1.02325
$214$	84720.3	0.126460
$215$	−1.09811e6	−1.62012
$216$	−96598.1	−0.140875
$217$	52611.1	0.0758452
$218$	−194431.	−0.277092
$219$	−37594.0	−0.0529674
$220$	−323728.	−0.450945
$221$	32625.2	0.0449338
$222$	−77870.9	−0.106046
$223$	365915.	0.492741	0.246370	−	0.969176i	$-0.420762\pi$
$223$	365915.	0.492741	0.246370	+	0.969176i	$0.420762\pi$
$224$	495091.	0.659273
$225$	−22230.1	−0.0292742
$226$	−339036.	−0.441545
$227$	1.08830e6	1.40179	0.700894	−	0.713265i	$-0.252784\pi$
$227$	1.08830e6	1.40179	0.700894	+	0.713265i	$0.252784\pi$
$228$	−294057.	−0.374623
$229$	−1.45107e6	−1.82853	−0.914263	−	0.405121i	$-0.867229\pi$
$229$	−1.45107e6	−1.82853	−0.914263	+	0.405121i	$0.867229\pi$
$230$	−63489.3	−0.0791373
$231$	181994.	0.224403
$232$	−343525.	−0.419023
$233$	−1.27334e6	−1.53657	−0.768286	−	0.640106i	$-0.778890\pi$
$233$	−1.27334e6	−1.53657	−0.768286	+	0.640106i	$0.778890\pi$
$234$	132145.	0.157765
$235$	−278779.	−0.329299
$236$	371344.	0.434006
$237$	−863070.	−0.998103
$238$	9081.50	0.0103924
$239$	799348.	0.905193	0.452596	−	0.891715i	$-0.350498\pi$
$239$	799348.	0.905193	0.452596	+	0.891715i	$0.350498\pi$
$240$	271219.	0.303943
$241$	727560.	0.806913	0.403456	−	0.914999i	$-0.367809\pi$
$241$	727560.	0.806913	0.403456	+	0.914999i	$0.367809\pi$
$242$	−248216.	−0.272453
$243$	59049.0	0.0641500
$244$	511681.	0.550205
$245$	−466132.	−0.496128
$246$	−86992.8	−0.0916528
$247$	879966.	0.917748
$248$	−77572.9	−0.0800904
$249$	120301.	0.122962
$250$	−407993.	−0.412860
$251$	960924.	0.962730	0.481365	−	0.876520i	$-0.340141\pi$
$251$	960924.	0.962730	0.481365	+	0.876520i	$0.340141\pi$
$252$	−196156.	−0.194581
$253$	−119032.	−0.116913
$254$	−107013.	−0.104076
$255$	21601.1	0.0208030
$256$	−243281.	−0.232010
$257$	−1.57691e6	−1.48927	−0.744637	−	0.667470i	$-0.767377\pi$
$257$	−1.57691e6	−1.48927	−0.744637	+	0.667470i	$0.767377\pi$
$258$	−416105.	−0.389183
$259$	−345908.	−0.320414
$260$	−1.04414e6	−0.957911
$261$	209992.	0.190810
$262$	−585454.	−0.526914
$263$	1.42682e6	1.27198	0.635991	−	0.771696i	$-0.280591\pi$
$263$	1.42682e6	1.27198	0.635991	+	0.771696i	$0.280591\pi$
$264$	−268343.	−0.236963
$265$	−841101.	−0.735755
$266$	244946.	0.212259
$267$	458389.	0.393510
$268$	−52585.0	−0.0447224
$269$	−341906.	−0.288089	−0.144044	−	0.989571i	$-0.546011\pi$
$269$	−341906.	−0.288089	−0.144044	+	0.989571i	$0.546011\pi$
$270$	87492.9	0.0730404
$271$	723709.	0.598605	0.299303	−	0.954158i	$-0.403246\pi$
$271$	723709.	0.598605	0.299303	+	0.954158i	$0.403246\pi$
$272$	25373.5	0.0207950
$273$	586998.	0.476683
$274$	273377.	0.219981
$275$	−61753.7	−0.0492415
$276$	128294.	0.101376
$277$	−1.82928e6	−1.43245	−0.716226	−	0.697868i	$-0.754132\pi$
$277$	−1.82928e6	−1.43245	−0.716226	+	0.697868i	$0.754132\pi$
$278$	137890.	0.107009
$279$	47419.1	0.0364706
$280$	−635791.	−0.484640
$281$	1.14935e6	0.868337	0.434169	−	0.900832i	$-0.357042\pi$
$281$	1.14935e6	0.868337	0.434169	+	0.900832i	$0.357042\pi$
$282$	−105638.	−0.0791035
$283$	−1.83262e6	−1.36021	−0.680105	−	0.733115i	$-0.738066\pi$
$283$	−1.83262e6	−1.36021	−0.680105	+	0.733115i	$0.738066\pi$
$284$	−2.02860e6	−1.49245
$285$	582623.	0.424889
$286$	367090.	0.265374
$287$	−386429.	−0.276926
$288$	446233.	0.317015
$289$	−1.41784e6	−0.998577
$290$	311145.	0.217254
$291$	398526.	0.275883
$292$	112560.	0.0772552
$293$	1.33169e6	0.906221	0.453111	−	0.891454i	$-0.350314\pi$
$293$	1.33169e6	0.906221	0.453111	+	0.891454i	$0.350314\pi$
$294$	−176631.	−0.119179
$295$	−735754.	−0.492241
$296$	510027.	0.338348
$297$	164034.	0.107905
$298$	−384972.	−0.251124
$299$	−383920.	−0.248349
$300$	66559.0	0.0426976
$301$	−1.84837e6	−1.17591
$302$	−1.03316e6	−0.651853
$303$	1.78562e6	1.11733
$304$	684373.	0.424726
$305$	−1.01381e6	−0.624031
$306$	8185.29	0.00499724
$307$	3.00299e6	1.81848	0.909239	−	0.416275i	$-0.136665\pi$
$307$	3.00299e6	1.81848	0.909239	+	0.416275i	$0.136665\pi$
$308$	−544909.	−0.327301
$309$	1.81723e6	1.08271
$310$	70260.9	0.0415250
$311$	2.75093e6	1.61279	0.806397	−	0.591374i	$-0.201414\pi$
$311$	2.75093e6	1.61279	0.806397	+	0.591374i	$0.201414\pi$
$312$	−865504.	−0.503364
$313$	−119894.	−0.0691727	−0.0345864	−	0.999402i	$-0.511011\pi$
$313$	−119894.	−0.0691727	−0.0345864	+	0.999402i	$0.511011\pi$
$314$	871364.	0.498741
$315$	388650.	0.220690
$316$	2.58412e6	1.45578
$317$	514254.	0.287428	0.143714	−	0.989619i	$-0.454095\pi$
$317$	514254.	0.287428	0.143714	+	0.989619i	$0.454095\pi$
$318$	−318718.	−0.176741
$319$	583343.	0.320957
$320$	−303151.	−0.165494
$321$	339195.	0.183733
$322$	−106867.	−0.0574388
$323$	54506.6	0.0290698
$324$	−176798.	−0.0935655
$325$	−199178.	−0.104600
$326$	−645088.	−0.336182
$327$	−778444.	−0.402585
$328$	569773.	0.292427
$329$	−469250.	−0.239009
$330$	243049.	0.122860
$331$	2.46021e6	1.23425	0.617124	−	0.786866i	$-0.288298\pi$
$331$	2.46021e6	1.23425	0.617124	+	0.786866i	$0.288298\pi$
$332$	−360192.	−0.179345
$333$	−311772.	−0.154073
$334$	1.30539e6	0.640286
$335$	104188.	0.0507231
$336$	456524.	0.220605
$337$	−1.42694e6	−0.684432	−0.342216	−	0.939621i	$-0.611177\pi$
$337$	−1.42694e6	−0.684432	−0.342216	+	0.939621i	$0.611177\pi$
$338$	349363.	0.166335
$339$	−1.35740e6	−0.641518
$340$	−64675.7	−0.0303420
$341$	131727.	0.0613465
$342$	220773.	0.102066
$343$	−2.29503e6	−1.05330
$344$	2.72534e6	1.24172
$345$	−254193.	−0.114978
$346$	−431326.	−0.193694
$347$	1.63440e6	0.728676	0.364338	−	0.931267i	$-0.381295\pi$
$347$	1.63440e6	0.728676	0.364338	+	0.931267i	$0.381295\pi$
$348$	−628735.	−0.278304
$349$	−2.65498e6	−1.16680	−0.583402	−	0.812184i	$-0.698279\pi$
$349$	−2.65498e6	−1.16680	−0.583402	+	0.812184i	$0.698279\pi$
$350$	−55442.8	−0.0241922
$351$	529070.	0.229216
$352$	1.23961e6	0.533245
$353$	934019.	0.398950	0.199475	−	0.979903i	$-0.436076\pi$
$353$	934019.	0.398950	0.199475	+	0.979903i	$0.436076\pi$
$354$	−278799.	−0.118245
$355$	4.01932e6	1.69271
$356$	−1.37246e6	−0.573952
$357$	36359.7	0.0150990
$358$	42165.9	0.0173882
$359$	2.98227e6	1.22127	0.610634	−	0.791913i	$-0.290915\pi$
$359$	2.98227e6	1.22127	0.610634	+	0.791913i	$0.290915\pi$
$360$	−573048.	−0.233042
$361$	−1.00595e6	−0.406265
$362$	107827.	0.0432471
$363$	−993783.	−0.395845
$364$	−1.75753e6	−0.695263
$365$	−223019.	−0.0876211
$366$	−384161.	−0.149903
$367$	4.17728e6	1.61893	0.809466	−	0.587167i	$-0.199757\pi$
$367$	4.17728e6	1.61893	0.809466	+	0.587167i	$0.199757\pi$
$368$	−298585.	−0.114934
$369$	−348294.	−0.133162
$370$	−461952.	−0.175426
$371$	−1.41577e6	−0.534020
$372$	−141977.	−0.0531939
$373$	−1.12351e6	−0.418124	−0.209062	−	0.977902i	$-0.567041\pi$
$373$	−1.12351e6	−0.418124	−0.209062	+	0.977902i	$0.567041\pi$
$374$	22738.2	0.00840576
$375$	−1.63349e6	−0.599842
$376$	691889.	0.252387
$377$	1.88149e6	0.681788
$378$	147271.	0.0530136
$379$	3.45021e6	1.23381	0.616904	−	0.787039i	$-0.288387\pi$
$379$	3.45021e6	1.23381	0.616904	+	0.787039i	$0.288387\pi$
$380$	−1.74443e6	−0.619718
$381$	−428448.	−0.151212
$382$	−32132.5	−0.0112664
$383$	−573250.	−0.199686	−0.0998430	−	0.995003i	$-0.531834\pi$
$383$	−573250.	−0.199686	−0.0998430	+	0.995003i	$0.531834\pi$
$384$	−1.70148e6	−0.588841
$385$	1.07964e6	0.371218
$386$	−950339.	−0.324646
$387$	−1.66596e6	−0.565441
$388$	−1.19323e6	−0.402386
$389$	−5.10842e6	−1.71164	−0.855820	−	0.517274i	$-0.826947\pi$
$389$	−5.10842e6	−1.71164	−0.855820	+	0.517274i	$0.826947\pi$
$390$	783922.	0.260983
$391$	−23780.7	−0.00786651
$392$	1.15687e6	0.380251
$393$	−2.34398e6	−0.765550
$394$	−699114.	−0.226886
$395$	−5.11998e6	−1.65111
$396$	−491134.	−0.157385
$397$	−417826.	−0.133051	−0.0665256	−	0.997785i	$-0.521191\pi$
$397$	−417826.	−0.133051	−0.0665256	+	0.997785i	$0.521191\pi$
$398$	135024.	0.0427272
$399$	980690.	0.308389
$400$	−154906.	−0.0484082
$401$	3.79619e6	1.17893	0.589463	−	0.807795i	$-0.299339\pi$
$401$	3.79619e6	1.17893	0.589463	+	0.807795i	$0.299339\pi$
$402$	39479.9	0.0121846
$403$	424868.	0.130314
$404$	−5.34631e6	−1.62967
$405$	350296.	0.106120
$406$	523729.	0.157685
$407$	−866082.	−0.259163
$408$	−53610.8	−0.0159442
$409$	4.95389e6	1.46433	0.732164	−	0.681129i	$-0.238511\pi$
$409$	4.95389e6	1.46433	0.732164	+	0.681129i	$0.238511\pi$
$410$	−516066.	−0.151616
$411$	1.09452e6	0.319609
$412$	−5.44096e6	−1.57918
$413$	−1.23844e6	−0.357274
$414$	−96321.0	−0.0276198
$415$	713658.	0.203409
$416$	3.99818e6	1.13274
$417$	552071.	0.155473
$418$	613293.	0.171683
$419$	−2.38845e6	−0.664631	−0.332316	−	0.943168i	$-0.607830\pi$
$419$	−2.38845e6	−0.664631	−0.332316	+	0.943168i	$0.607830\pi$
$420$	−1.16365e6	−0.321885
$421$	−582907.	−0.160285	−0.0801427	−	0.996783i	$-0.525538\pi$
$421$	−582907.	−0.160285	−0.0801427	+	0.996783i	$0.525538\pi$
$422$	−2.44788e6	−0.669128
$423$	−422942.	−0.114929
$424$	2.08749e6	0.563910
$425$	−12337.4	−0.00331324
$426$	1.52304e6	0.406618
$427$	−1.70647e6	−0.452929
$428$	−1.01558e6	−0.267982
$429$	1.46972e6	0.385560
$430$	−2.46845e6	−0.643804
$431$	−1.17212e6	−0.303935	−0.151967	−	0.988386i	$-0.548561\pi$
$431$	−1.17212e6	−0.303935	−0.151967	+	0.988386i	$0.548561\pi$
$432$	411472.	0.106079
$433$	−5.34133e6	−1.36908	−0.684542	−	0.728974i	$-0.739998\pi$
$433$	−5.34133e6	−1.36908	−0.684542	+	0.728974i	$0.739998\pi$
$434$	118265.	0.0301393
$435$	1.24573e6	0.315647
$436$	2.33074e6	0.587188
$437$	−641410.	−0.160669
$438$	−84508.3	−0.0210482
$439$	3.68652e6	0.912967	0.456483	−	0.889732i	$-0.349109\pi$
$439$	3.68652e6	0.912967	0.456483	+	0.889732i	$0.349109\pi$
$440$	−1.59189e6	−0.391996
$441$	−707179.	−0.173154
$442$	73338.9	0.0178558
$443$	−705619.	−0.170829	−0.0854143	−	0.996346i	$-0.527221\pi$
$443$	−705619.	−0.170829	−0.0854143	+	0.996346i	$0.527221\pi$
$444$	933475.	0.224722
$445$	2.71930e6	0.650963
$446$	822548.	0.195805
$447$	−1.54131e6	−0.364857
$448$	−510273.	−0.120118
$449$	442688.	0.103629	0.0518146	−	0.998657i	$-0.483500\pi$
$449$	442688.	0.103629	0.0518146	+	0.998657i	$0.483500\pi$
$450$	−49971.4	−0.0116330
$451$	−967537.	−0.223989
$452$	4.06419e6	0.935682
$453$	−4.13646e6	−0.947073
$454$	2.44640e6	0.557042
$455$	3.48224e6	0.788552
$456$	−1.44599e6	−0.325651
$457$	4.85276e6	1.08692	0.543461	−	0.839434i	$-0.317113\pi$
$457$	4.85276e6	1.08692	0.543461	+	0.839434i	$0.317113\pi$
$458$	−3.26190e6	−0.726619
$459$	32771.5	0.00726046
$460$	761077.	0.167700
$461$	8.87833e6	1.94571	0.972857	−	0.231408i	$-0.0743331\pi$
$461$	8.87833e6	1.94571	0.972857	+	0.231408i	$0.0743331\pi$
$462$	409109.	0.0891731
$463$	−3.58364e6	−0.776911	−0.388456	−	0.921467i	$-0.626991\pi$
$463$	−3.58364e6	−0.776911	−0.388456	+	0.921467i	$0.626991\pi$
$464$	1.46329e6	0.315526
$465$	281304.	0.0603314
$466$	−2.86235e6	−0.610603
$467$	−6.13857e6	−1.30249	−0.651246	−	0.758867i	$-0.725753\pi$
$467$	−6.13857e6	−1.30249	−0.651246	+	0.758867i	$0.725753\pi$
$468$	−1.58409e6	−0.334321
$469$	175373.	0.0368155
$470$	−626672.	−0.130857
$471$	3.48868e6	0.724618
$472$	1.82603e6	0.377271
$473$	−4.62793e6	−0.951117
$474$	−1.94011e6	−0.396626
$475$	−332764.	−0.0676710
$476$	−108864.	−0.0220226
$477$	−1.27605e6	−0.256787
$478$	1.79687e6	0.359705
$479$	505577.	0.100681	0.0503406	−	0.998732i	$-0.483969\pi$
$479$	505577.	0.100681	0.0503406	+	0.998732i	$0.483969\pi$
$480$	2.64718e6	0.524422
$481$	−2.79343e6	−0.550522
$482$	1.63550e6	0.320651
$483$	−427865.	−0.0834525
$484$	2.97548e6	0.577356
$485$	2.36417e6	0.456378
$486$	132737.	0.0254919
$487$	−751626.	−0.143608	−0.0718041	−	0.997419i	$-0.522876\pi$
$487$	−751626.	−0.143608	−0.0718041	+	0.997419i	$0.522876\pi$
$488$	2.51612e6	0.478280
$489$	−2.58274e6	−0.488437
$490$	−1.04783e6	−0.197151
$491$	−2.19083e6	−0.410115	−0.205057	−	0.978750i	$-0.565738\pi$
$491$	−2.19083e6	−0.410115	−0.205057	+	0.978750i	$0.565738\pi$
$492$	1.04282e6	0.194222
$493$	116543.	0.0215957
$494$	1.97809e6	0.364694
$495$	973098.	0.178502
$496$	330431.	0.0603083
$497$	6.76545e6	1.22859
$498$	270426.	0.0488624
$499$	2.21183e6	0.397649	0.198824	−	0.980035i	$-0.436288\pi$
$499$	2.21183e6	0.397649	0.198824	+	0.980035i	$0.436288\pi$
$500$	4.89081e6	0.874895
$501$	5.22640e6	0.930268
$502$	2.16008e6	0.382569
$503$	−1.05756e7	−1.86373	−0.931866	−	0.362803i	$-0.881820\pi$
$503$	−1.05756e7	−1.86373	−0.931866	+	0.362803i	$0.881820\pi$
$504$	−964573.	−0.169145
$505$	1.05928e7	1.84834
$506$	−267573.	−0.0464587
$507$	1.39874e6	0.241668
$508$	1.28282e6	0.220549
$509$	−1.00676e7	−1.72239	−0.861195	−	0.508275i	$-0.830283\pi$
$509$	−1.00676e7	−1.72239	−0.861195	+	0.508275i	$0.830283\pi$
$510$	48557.5	0.00826667
$511$	−375392.	−0.0635964
$512$	5.50282e6	0.927707
$513$	883910.	0.148291
$514$	−3.54477e6	−0.591807
$515$	1.07803e7	1.79108
$516$	4.98805e6	0.824720
$517$	−1.17490e6	−0.193320
$518$	−777573.	−0.127326
$519$	−1.72690e6	−0.281416
$520$	−5.13442e6	−0.832689
$521$	7.71005e6	1.24441	0.622204	−	0.782855i	$-0.286237\pi$
$521$	7.71005e6	1.24441	0.622204	+	0.782855i	$0.286237\pi$
$522$	472044.	0.0758239
$523$	153540.	0.0245453	0.0122726	−	0.999925i	$-0.496093\pi$
$523$	153540.	0.0245453	0.0122726	+	0.999925i	$0.496093\pi$
$524$	7.01811e6	1.11659
$525$	−221977.	−0.0351487
$526$	3.20738e6	0.505460
$527$	26317.0	0.00412772
$528$	1.14304e6	0.178434
$529$	279841.	0.0434783
$530$	−1.89073e6	−0.292374
$531$	−1.11623e6	−0.171797
$532$	−2.93628e6	−0.449799
$533$	−3.12066e6	−0.475804
$534$	1.03042e6	0.156373
$535$	2.01220e6	0.303940
$536$	−258580.	−0.0388761
$537$	168820.	0.0252632
$538$	−768577.	−0.114481
$539$	−1.96450e6	−0.291259
$540$	−1.04882e6	−0.154780
$541$	267983.	0.0393653	0.0196827	−	0.999806i	$-0.493734\pi$
$541$	267983.	0.0393653	0.0196827	+	0.999806i	$0.493734\pi$
$542$	1.62684e6	0.237873
$543$	431708.	0.0628334
$544$	247654.	0.0358796
$545$	−4.61795e6	−0.665976
$546$	1.31952e6	0.189424
$547$	6.44942e6	0.921621	0.460810	−	0.887499i	$-0.347559\pi$
$547$	6.44942e6	0.921621	0.460810	+	0.887499i	$0.347559\pi$
$548$	−3.27710e6	−0.466164
$549$	−1.53807e6	−0.217793
$550$	−138817.	−0.0195676
$551$	3.14338e6	0.441081
$552$	630869.	0.0881235
$553$	−8.61812e6	−1.19839
$554$	−4.11207e6	−0.569227
$555$	−1.84952e6	−0.254875
$556$	−1.65295e6	−0.226764
$557$	−1.36650e7	−1.86626	−0.933132	−	0.359534i	$-0.882936\pi$
$557$	−1.36650e7	−1.86626	−0.933132	+	0.359534i	$0.882936\pi$
$558$	106594.	0.0144927
$559$	−1.49268e7	−2.02039
$560$	2.70823e6	0.364935
$561$	91037.0	0.0122127
$562$	2.58366e6	0.345059
$563$	564712.	0.0750855	0.0375427	−	0.999295i	$-0.488047\pi$
$563$	564712.	0.0750855	0.0375427	+	0.999295i	$0.488047\pi$
$564$	1.26633e6	0.167629
$565$	−8.05250e6	−1.06123
$566$	−4.11957e6	−0.540519
$567$	589629.	0.0770231
$568$	−9.97537e6	−1.29735
$569$	5.68519e6	0.736147	0.368074	−	0.929797i	$-0.380017\pi$
$569$	5.68519e6	0.736147	0.368074	+	0.929797i	$0.380017\pi$
$570$	1.30969e6	0.168842
$571$	5.94914e6	0.763597	0.381798	−	0.924246i	$-0.375305\pi$
$571$	5.94914e6	0.763597	0.381798	+	0.924246i	$0.375305\pi$
$572$	−4.40049e6	−0.562355
$573$	−128649.	−0.0163689
$574$	−868660.	−0.110045
$575$	145182.	0.0183123
$576$	−459916.	−0.0577594
$577$	−7.28162e6	−0.910518	−0.455259	−	0.890359i	$-0.650453\pi$
$577$	−7.28162e6	−0.910518	−0.455259	+	0.890359i	$0.650453\pi$
$578$	−3.18718e6	−0.396814
$579$	−3.80488e6	−0.471677
$580$	−3.72984e6	−0.460384
$581$	1.20125e6	0.147637
$582$	895854.	0.109630
$583$	−3.54479e6	−0.431935
$584$	553500.	0.0671561
$585$	3.13859e6	0.379180
$586$	2.99353e6	0.360114
$587$	9.14151e6	1.09502	0.547511	−	0.836799i	$-0.315575\pi$
$587$	9.14151e6	1.09502	0.547511	+	0.836799i	$0.315575\pi$
$588$	2.11736e6	0.252553
$589$	709821.	0.0843064
$590$	−1.65391e6	−0.195606
$591$	−2.79904e6	−0.329641
$592$	−2.17252e6	−0.254777
$593$	6.14133e6	0.717176	0.358588	−	0.933496i	$-0.383258\pi$
$593$	6.14133e6	0.717176	0.358588	+	0.933496i	$0.383258\pi$
$594$	368735.	0.0428794
$595$	215696.	0.0249775
$596$	4.61484e6	0.532159
$597$	540598.	0.0620781
$598$	−863021.	−0.0986890
$599$	−6.53791e6	−0.744512	−0.372256	−	0.928130i	$-0.621416\pi$
$599$	−6.53791e6	−0.744512	−0.372256	+	0.928130i	$0.621416\pi$
$600$	327295.	0.0371161
$601$	−9.92849e6	−1.12124	−0.560618	−	0.828075i	$-0.689436\pi$
$601$	−9.92849e6	−1.12124	−0.560618	+	0.828075i	$0.689436\pi$
$602$	−4.15498e6	−0.467281
$603$	158066.	0.0177029
$604$	1.23850e7	1.38135
$605$	−5.89540e6	−0.654825
$606$	4.01392e6	0.444004
$607$	−2.59619e6	−0.285999	−0.143000	−	0.989723i	$-0.545675\pi$
$607$	−2.59619e6	−0.285999	−0.143000	+	0.989723i	$0.545675\pi$
$608$	6.67970e6	0.732821
$609$	2.09685e6	0.229100
$610$	−2.27896e6	−0.247977
$611$	−3.78949e6	−0.410656
$612$	−98120.9	−0.0105897
$613$	−4.12434e6	−0.443306	−0.221653	−	0.975126i	$-0.571145\pi$
$613$	−4.12434e6	−0.443306	−0.221653	+	0.975126i	$0.571145\pi$
$614$	6.75047e6	0.722626
$615$	−2.06618e6	−0.220282
$616$	−2.67952e6	−0.284515
$617$	1.17375e7	1.24126	0.620631	−	0.784103i	$-0.286876\pi$
$617$	1.17375e7	1.24126	0.620631	+	0.784103i	$0.286876\pi$
$618$	4.08499e6	0.430249
$619$	6.90815e6	0.724662	0.362331	−	0.932050i	$-0.381981\pi$
$619$	6.90815e6	0.724662	0.362331	+	0.932050i	$0.381981\pi$
$620$	−842251.	−0.0879959
$621$	−385641.	−0.0401286
$622$	6.18387e6	0.640892
$623$	4.57721e6	0.472477
$624$	3.68672e6	0.379035
$625$	−8.83266e6	−0.904465
$626$	−269511.	−0.0274878
$627$	2.45544e6	0.249437
$628$	−1.04455e7	−1.05689
$629$	−173030.	−0.0174379
$630$	873653.	0.0876976
$631$	793455.	0.0793321	0.0396660	−	0.999213i	$-0.487371\pi$
$631$	793455.	0.0793321	0.0396660	+	0.999213i	$0.487371\pi$
$632$	1.27071e7	1.26547
$633$	−9.80060e6	−0.972173
$634$	1.15600e6	0.114218
$635$	−2.54168e6	−0.250142
$636$	3.82062e6	0.374534
$637$	−6.33621e6	−0.618702
$638$	1.31131e6	0.127542
$639$	6.09780e6	0.590773
$640$	−1.00937e7	−0.974089
$641$	3.78009e6	0.363377	0.181689	−	0.983356i	$-0.441844\pi$
$641$	3.78009e6	0.363377	0.181689	+	0.983356i	$0.441844\pi$
$642$	762483.	0.0730117
$643$	1.21266e7	1.15667	0.578336	−	0.815799i	$-0.303702\pi$
$643$	1.21266e7	1.15667	0.578336	+	0.815799i	$0.303702\pi$
$644$	1.28107e6	0.121719
$645$	−9.88296e6	−0.935380
$646$	122526.	0.0115518
$647$	−6.56812e6	−0.616851	−0.308425	−	0.951249i	$-0.599802\pi$
$647$	−6.56812e6	−0.616851	−0.308425	+	0.951249i	$0.599802\pi$
$648$	−869383.	−0.0813343
$649$	−3.10081e6	−0.288977
$650$	−447736.	−0.0415660
$651$	473500.	0.0437892
$652$	7.73297e6	0.712406
$653$	−9.97904e6	−0.915811	−0.457905	−	0.889001i	$-0.651400\pi$
$653$	−9.97904e6	−0.915811	−0.457905	+	0.889001i	$0.651400\pi$
$654$	−1.74988e6	−0.159979
$655$	−1.39052e7	−1.26641
$656$	−2.42702e6	−0.220198
$657$	−338346.	−0.0305807
$658$	−1.05484e6	−0.0949773
$659$	1.22411e7	1.09801	0.549006	−	0.835818i	$-0.315006\pi$
$659$	1.22411e7	1.09801	0.549006	+	0.835818i	$0.315006\pi$
$660$	−2.91355e6	−0.260353
$661$	−7.26433e6	−0.646684	−0.323342	−	0.946282i	$-0.604806\pi$
$661$	−7.26433e6	−0.646684	−0.323342	+	0.946282i	$0.604806\pi$
$662$	5.53035e6	0.490465
$663$	293627.	0.0259425
$664$	−1.77120e6	−0.155900
$665$	5.81773e6	0.510152
$666$	−700838.	−0.0612254
$667$	−1.37143e6	−0.119360
$668$	−1.56483e7	−1.35683
$669$	3.29324e6	0.284484
$670$	234206.	0.0201563
$671$	−4.27265e6	−0.366346
$672$	4.45582e6	0.380631
$673$	1.76906e7	1.50558	0.752791	−	0.658260i	$-0.228707\pi$
$673$	1.76906e7	1.50558	0.752791	+	0.658260i	$0.228707\pi$
$674$	−3.20764e6	−0.271979
$675$	−200071.	−0.0169015
$676$	−4.18798e6	−0.352482
$677$	8.88669e6	0.745193	0.372596	−	0.927994i	$-0.378468\pi$
$677$	8.88669e6	0.745193	0.372596	+	0.927994i	$0.378468\pi$
$678$	−3.05133e6	−0.254926
$679$	3.97945e6	0.331244
$680$	−318035.	−0.0263756
$681$	9.79467e6	0.809323
$682$	296112.	0.0243778
$683$	−3.93961e6	−0.323148	−0.161574	−	0.986861i	$-0.551657\pi$
$683$	−3.93961e6	−0.323148	−0.161574	+	0.986861i	$0.551657\pi$
$684$	−2.64651e6	−0.216288
$685$	6.49301e6	0.528713
$686$	−5.15905e6	−0.418561
$687$	−1.30597e7	−1.05570
$688$	−1.16089e7	−0.935021
$689$	−1.14332e7	−0.917531
$690$	−571404.	−0.0456899
$691$	2.19400e7	1.74800	0.873999	−	0.485927i	$-0.161518\pi$
$691$	2.19400e7	1.74800	0.873999	+	0.485927i	$0.161518\pi$
$692$	5.17051e6	0.410457
$693$	1.63795e6	0.129559
$694$	3.67399e6	0.289561
$695$	3.27504e6	0.257190
$696$	−3.09172e6	−0.241923
$697$	−193299.	−0.0150712
$698$	−5.96818e6	−0.463664
$699$	−1.14600e7	−0.887141
$700$	664620.	0.0512658
$701$	1.20994e7	0.929969	0.464984	−	0.885319i	$-0.346060\pi$
$701$	1.20994e7	0.929969	0.464984	+	0.885319i	$0.346060\pi$
$702$	1.18931e6	0.0910858
$703$	−4.66694e6	−0.356159
$704$	−1.27762e6	−0.0971559
$705$	−2.50901e6	−0.190121
$706$	2.09960e6	0.158535
$707$	1.78301e7	1.34155
$708$	3.34209e6	0.250574
$709$	1.95715e7	1.46221	0.731103	−	0.682267i	$-0.239006\pi$
$709$	1.95715e7	1.46221	0.731103	+	0.682267i	$0.239006\pi$
$710$	9.03510e6	0.672647
$711$	−7.76763e6	−0.576255
$712$	−6.74890e6	−0.498923
$713$	−309688.	−0.0228139
$714$	81733.5	0.00600005
$715$	8.71881e6	0.637811
$716$	−505463.	−0.0368474
$717$	7.19413e6	0.522613
$718$	6.70390e6	0.485307
$719$	−1.98434e7	−1.43151	−0.715753	−	0.698354i	$-0.753916\pi$
$719$	−1.98434e7	−1.43151	−0.715753	+	0.698354i	$0.753916\pi$
$720$	2.44097e6	0.175481
$721$	1.81458e7	1.29998
$722$	−2.26130e6	−0.161441
$723$	6.54804e6	0.465871
$724$	−1.29258e6	−0.0916452
$725$	−711497.	−0.0502722
$726$	−2.23394e6	−0.157301
$727$	2.50056e7	1.75469	0.877345	−	0.479860i	$-0.159312\pi$
$727$	2.50056e7	1.75469	0.877345	+	0.479860i	$0.159312\pi$
$728$	−8.64242e6	−0.604376
$729$	531441.	0.0370370
$730$	−501327.	−0.0348188
$731$	−924588.	−0.0639963
$732$	4.60513e6	0.317661
$733$	2.53798e7	1.74473	0.872365	−	0.488855i	$-0.162585\pi$
$733$	2.53798e7	1.74473	0.872365	+	0.488855i	$0.162585\pi$
$734$	9.39018e6	0.643330
$735$	−4.19519e6	−0.286440
$736$	−2.91429e6	−0.198307
$737$	439097.	0.0297777
$738$	−782935.	−0.0529158
$739$	1.50131e7	1.01125	0.505625	−	0.862753i	$-0.331262\pi$
$739$	1.50131e7	1.01125	0.505625	+	0.862753i	$0.331262\pi$
$740$	5.53764e6	0.371745
$741$	7.91969e6	0.529862
$742$	−3.18253e6	−0.212209
$743$	−2.53859e7	−1.68702	−0.843512	−	0.537111i	$-0.819516\pi$
$743$	−2.53859e7	−1.68702	−0.843512	+	0.537111i	$0.819516\pi$
$744$	−698156.	−0.0462402
$745$	−9.14351e6	−0.603563
$746$	−2.52556e6	−0.166154
$747$	1.08271e6	0.0709919
$748$	−272573.	−0.0178127
$749$	3.38701e6	0.220603
$750$	−3.67194e6	−0.238365
$751$	1.25539e7	0.812230	0.406115	−	0.913822i	$-0.366883\pi$
$751$	1.25539e7	0.812230	0.406115	+	0.913822i	$0.366883\pi$
$752$	−2.94719e6	−0.190048
$753$	8.64831e6	0.555833
$754$	4.22944e6	0.270928
$755$	−2.45387e7	−1.56669
$756$	−1.76541e6	−0.112341
$757$	6.15033e6	0.390085	0.195042	−	0.980795i	$-0.437516\pi$
$757$	6.15033e6	0.390085	0.195042	+	0.980795i	$0.437516\pi$
$758$	7.75579e6	0.490290
$759$	−1.07129e6	−0.0674995
$760$	−8.57800e6	−0.538706
$761$	−6.22830e6	−0.389859	−0.194930	−	0.980817i	$-0.562448\pi$
$761$	−6.22830e6	−0.389859	−0.194930	+	0.980817i	$0.562448\pi$
$762$	−963117.	−0.0600885
$763$	−7.77309e6	−0.483373
$764$	385187.	0.0238747
$765$	194410.	0.0120106
$766$	−1.28862e6	−0.0793511
$767$	−1.00012e7	−0.613853
$768$	−2.18953e6	−0.133951
$769$	8.72415e6	0.531995	0.265997	−	0.963974i	$-0.414299\pi$
$769$	8.72415e6	0.531995	0.265997	+	0.963974i	$0.414299\pi$
$770$	2.42695e6	0.147514
$771$	−1.41922e7	−0.859832
$772$	1.13922e7	0.687960
$773$	−7.23571e6	−0.435545	−0.217772	−	0.976000i	$-0.569879\pi$
$773$	−7.23571e6	−0.435545	−0.217772	+	0.976000i	$0.569879\pi$
$774$	−3.74494e6	−0.224695
$775$	−160666.	−0.00960883
$776$	−5.86753e6	−0.349785
$777$	−3.11317e6	−0.184991
$778$	−1.14833e7	−0.680171
$779$	−5.21364e6	−0.307820
$780$	−9.39725e6	−0.553050
$781$	1.69393e7	0.993728
$782$	−53457.0	−0.00312599
$783$	1.88992e6	0.110164
$784$	−4.92784e6	−0.286330
$785$	2.06959e7	1.19870
$786$	−5.26908e6	−0.304214
$787$	−1.35431e7	−0.779440	−0.389720	−	0.920933i	$-0.627428\pi$
$787$	−1.35431e7	−0.779440	−0.389720	+	0.920933i	$0.627428\pi$
$788$	8.38061e6	0.480795
$789$	1.28414e7	0.734379
$790$	−1.15093e7	−0.656117
$791$	−1.35542e7	−0.770253
$792$	−2.41509e6	−0.136811
$793$	−1.37809e7	−0.778203
$794$	−939238.	−0.0528718
$795$	−7.56991e6	−0.424788
$796$	−1.61860e6	−0.0905435
$797$	1.01044e7	0.563461	0.281731	−	0.959494i	$-0.409092\pi$
$797$	1.01044e7	0.563461	0.281731	+	0.959494i	$0.409092\pi$
$798$	2.20451e6	0.122548
$799$	−234727.	−0.0130076
$800$	−1.51193e6	−0.0835233
$801$	4.12550e6	0.227193
$802$	8.53352e6	0.468481
$803$	−939904.	−0.0514392
$804$	−473265.	−0.0258205
$805$	−2.53822e6	−0.138051
$806$	955068.	0.0517841
$807$	−3.07716e6	−0.166328
$808$	−2.62898e7	−1.41664
$809$	−3.23912e6	−0.174003	−0.0870013	−	0.996208i	$-0.527728\pi$
$809$	−3.23912e6	−0.174003	−0.0870013	+	0.996208i	$0.527728\pi$
$810$	787436.	0.0421699
$811$	6.20038e6	0.331029	0.165515	−	0.986207i	$-0.447072\pi$
$811$	6.20038e6	0.331029	0.165515	+	0.986207i	$0.447072\pi$
$812$	−6.27818e6	−0.334152
$813$	6.51338e6	0.345605
$814$	−1.94688e6	−0.102986
$815$	−1.53216e7	−0.807995
$816$	228362.	0.0120060
$817$	−2.49379e7	−1.30709
$818$	1.11359e7	0.581894
$819$	5.28298e6	0.275213
$820$	6.18633e6	0.321291
$821$	1.75836e7	0.910439	0.455219	−	0.890379i	$-0.349561\pi$
$821$	1.75836e7	0.910439	0.455219	+	0.890379i	$0.349561\pi$
$822$	2.46039e6	0.127006
$823$	−3.80649e7	−1.95896	−0.979478	−	0.201549i	$-0.935402\pi$
$823$	−3.80649e7	−1.95896	−0.979478	+	0.201549i	$0.935402\pi$
$824$	−2.67552e7	−1.37275
$825$	−555784.	−0.0284296
$826$	−2.78392e6	−0.141973
$827$	8.39081e6	0.426619	0.213310	−	0.976985i	$-0.431576\pi$
$827$	8.39081e6	0.426619	0.213310	+	0.976985i	$0.431576\pi$
$828$	1.15465e6	0.0585293
$829$	394283.	0.0199261	0.00996305	−	0.999950i	$-0.496829\pi$
$829$	394283.	0.0199261	0.00996305	+	0.999950i	$0.496829\pi$
$830$	1.60424e6	0.0808305
$831$	−1.64635e7	−0.827027
$832$	−4.12078e6	−0.206382
$833$	−392476.	−0.0195975
$834$	1.24101e6	0.0617817
$835$	3.10045e7	1.53889
$836$	−7.35183e6	−0.363814
$837$	426772.	0.0210563
$838$	−5.36904e6	−0.264111
$839$	1.63376e7	0.801276	0.400638	−	0.916236i	$-0.368788\pi$
$839$	1.63376e7	0.801276	0.400638	+	0.916236i	$0.368788\pi$
$840$	−5.72212e6	−0.279807
$841$	−1.37901e7	−0.672324
$842$	−1.31033e6	−0.0636941
$843$	1.03442e7	0.501335
$844$	2.93439e7	1.41795
$845$	8.29775e6	0.399778
$846$	−950738.	−0.0456704
$847$	−9.92334e6	−0.475280
$848$	−8.89193e6	−0.424626
$849$	−1.64936e7	−0.785317
$850$	−27733.5	−0.00131661
$851$	2.03614e6	0.0963793
$852$	−1.82574e7	−0.861668
$853$	2.38816e6	0.112380	0.0561901	−	0.998420i	$-0.482105\pi$
$853$	2.38816e6	0.112380	0.0561901	+	0.998420i	$0.482105\pi$
$854$	−3.83601e6	−0.179985
$855$	5.24361e6	0.245310
$856$	−4.99400e6	−0.232951
$857$	−1.41990e7	−0.660398	−0.330199	−	0.943911i	$-0.607116\pi$
$857$	−1.41990e7	−0.660398	−0.330199	+	0.943911i	$0.607116\pi$
$858$	3.30381e6	0.153214
$859$	1.13234e7	0.523595	0.261797	−	0.965123i	$-0.415685\pi$
$859$	1.13234e7	0.523595	0.261797	+	0.965123i	$0.415685\pi$
$860$	2.95905e7	1.36429
$861$	−3.47786e6	−0.159884
$862$	−2.63484e6	−0.120777
$863$	2.55589e7	1.16819	0.584097	−	0.811684i	$-0.301449\pi$
$863$	2.55589e7	1.16819	0.584097	+	0.811684i	$0.301449\pi$
$864$	4.01609e6	0.183029
$865$	−1.02445e7	−0.465532
$866$	−1.20069e7	−0.544046
$867$	−1.27605e7	−0.576529
$868$	−1.41770e6	−0.0638684
$869$	−2.15780e7	−0.969307
$870$	2.80030e6	0.125431
$871$	1.41625e6	0.0632548
$872$	1.14611e7	0.510428
$873$	3.58673e6	0.159281
$874$	−1.44184e6	−0.0638466
$875$	−1.63110e7	−0.720214
$876$	1.01304e6	0.0446033
$877$	1.94757e7	0.855056	0.427528	−	0.904002i	$-0.359385\pi$
$877$	1.94757e7	0.855056	0.427528	+	0.904002i	$0.359385\pi$
$878$	8.28699e6	0.362794
$879$	1.19852e7	0.523207
$880$	6.78085e6	0.295174
$881$	9.76547e6	0.423890	0.211945	−	0.977282i	$-0.432020\pi$
$881$	9.76547e6	0.423890	0.211945	+	0.977282i	$0.432020\pi$
$882$	−1.58968e6	−0.0688079
$883$	4.86226e6	0.209863	0.104932	−	0.994479i	$-0.466538\pi$
$883$	4.86226e6	0.209863	0.104932	+	0.994479i	$0.466538\pi$
$884$	−879148.	−0.0378383
$885$	−6.62178e6	−0.284195
$886$	−1.58617e6	−0.0678838
$887$	−3.00831e7	−1.28385	−0.641925	−	0.766767i	$-0.721864\pi$
$887$	−3.00831e7	−1.28385	−0.641925	+	0.766767i	$0.721864\pi$
$888$	4.59024e6	0.195345
$889$	−4.27824e6	−0.181556
$890$	6.11276e6	0.258680
$891$	1.47631e6	0.0622992
$892$	−9.86027e6	−0.414932
$893$	−6.33105e6	−0.265673
$894$	−3.46475e6	−0.144987
$895$	1.00149e6	0.0417915
$896$	−1.69900e7	−0.707005
$897$	−3.45528e6	−0.143385
$898$	995127.	0.0411801
$899$	1.51770e6	0.0626306
$900$	599031.	0.0246515
$901$	−708193.	−0.0290630
$902$	−2.17494e6	−0.0890085
$903$	−1.66353e7	−0.678909
$904$	1.99851e7	0.813366
$905$	2.56102e6	0.103942
$906$	−9.29842e6	−0.376347
$907$	−3.27277e7	−1.32098	−0.660491	−	0.750834i	$-0.729652\pi$
$907$	−3.27277e7	−1.32098	−0.660491	+	0.750834i	$0.729652\pi$
$908$	−2.93262e7	−1.18043
$909$	1.60706e7	0.645091
$910$	7.82779e6	0.313354
$911$	−1.44550e6	−0.0577062	−0.0288531	−	0.999584i	$-0.509186\pi$
$911$	−1.44550e6	−0.0577062	−0.0288531	+	0.999584i	$0.509186\pi$
$912$	6.15936e6	0.245216
$913$	3.00769e6	0.119414
$914$	1.09086e7	0.431921
$915$	−9.12427e6	−0.360284
$916$	3.91019e7	1.53978
$917$	−2.34057e7	−0.919174
$918$	73667.6	0.00288516
$919$	1.02478e7	0.400258	0.200129	−	0.979770i	$-0.435864\pi$
$919$	1.02478e7	0.400258	0.200129	+	0.979770i	$0.435864\pi$
$920$	3.74250e6	0.145778
$921$	2.70269e7	1.04990
$922$	1.99578e7	0.773187
$923$	5.46353e7	2.11091
$924$	−4.90418e6	−0.188967
$925$	1.05635e6	0.0405933
$926$	−8.05572e6	−0.308729
$927$	1.63551e7	0.625105
$928$	1.42821e7	0.544407
$929$	4.79655e7	1.82343	0.911716	−	0.410821i	$-0.134758\pi$
$929$	4.79655e7	1.82343	0.911716	+	0.410821i	$0.134758\pi$
$930$	632348.	0.0239745
$931$	−1.05858e7	−0.400268
$932$	3.43124e7	1.29393
$933$	2.47584e7	0.931147
$934$	−1.37990e7	−0.517583
$935$	540058.	0.0202028
$936$	−7.78954e6	−0.290618
$937$	−1.18919e7	−0.442489	−0.221244	−	0.975218i	$-0.571012\pi$
$937$	−1.18919e7	−0.442489	−0.221244	+	0.975218i	$0.571012\pi$
$938$	394224.	0.0146297
$939$	−1.07904e6	−0.0399369
$940$	7.51222e6	0.277299
$941$	3.72086e7	1.36984	0.684919	−	0.728619i	$-0.259838\pi$
$941$	3.72086e7	1.36984	0.684919	+	0.728619i	$0.259838\pi$
$942$	7.84227e6	0.287948
$943$	2.27466e6	0.0832985
$944$	−7.77822e6	−0.284086
$945$	3.49785e6	0.127415
$946$	−1.04032e7	−0.377954
$947$	2.24159e7	0.812235	0.406118	−	0.913821i	$-0.366882\pi$
$947$	2.24159e7	0.812235	0.406118	+	0.913821i	$0.366882\pi$
$948$	2.32570e7	0.840492
$949$	−3.03153e6	−0.109269
$950$	−748027.	−0.0268911
$951$	4.62829e6	0.165947
$952$	−535326.	−0.0191437
$953$	1.19242e7	0.425302	0.212651	−	0.977128i	$-0.431790\pi$
$953$	1.19242e7	0.425302	0.212651	+	0.977128i	$0.431790\pi$
$954$	−2.86846e6	−0.102042
$955$	−763182.	−0.0270782
$956$	−2.15399e7	−0.762253
$957$	5.25009e6	0.185305
$958$	1.13649e6	0.0400086
$959$	1.09292e7	0.383746
$960$	−2.72836e6	−0.0955483
$961$	−2.82864e7	−0.988029
$962$	−6.27940e6	−0.218766
$963$	3.05276e6	0.106078
$964$	−1.96055e7	−0.679493
$965$	−2.25716e7	−0.780270
$966$	−961806.	−0.0331623
$967$	1.40918e7	0.484619	0.242309	−	0.970199i	$-0.422095\pi$
$967$	1.40918e7	0.484619	0.242309	+	0.970199i	$0.422095\pi$
$968$	1.46315e7	0.501882
$969$	490559.	0.0167835
$970$	5.31446e6	0.181355
$971$	−2.06293e7	−0.702160	−0.351080	−	0.936346i	$-0.614185\pi$
$971$	−2.06293e7	−0.702160	−0.351080	+	0.936346i	$0.614185\pi$
$972$	−1.59119e6	−0.0540201
$973$	5.51265e6	0.186672
$974$	−1.68959e6	−0.0570669
$975$	−1.79260e6	−0.0603911
$976$	−1.07177e7	−0.360146
$977$	2.77805e7	0.931116	0.465558	−	0.885017i	$-0.345854\pi$
$977$	2.77805e7	0.931116	0.465558	+	0.885017i	$0.345854\pi$
$978$	−5.80579e6	−0.194095
$979$	1.14604e7	0.382157
$980$	1.25608e7	0.417784
$981$	−7.00600e6	−0.232433
$982$	−4.92481e6	−0.162971
$983$	−8.68777e6	−0.286764	−0.143382	−	0.989667i	$-0.545798\pi$
$983$	−8.68777e6	−0.286764	−0.143382	+	0.989667i	$0.545798\pi$
$984$	5.12795e6	0.168833
$985$	−1.66047e7	−0.545307
$986$	261979.	0.00858171
$987$	−4.22325e6	−0.137992
$988$	−2.37123e7	−0.772826
$989$	1.08802e7	0.353708
$990$	2.18744e6	0.0709331
$991$	−5.74667e7	−1.85880	−0.929399	−	0.369077i	$-0.879674\pi$
$991$	−5.74667e7	−1.85880	−0.929399	+	0.369077i	$0.879674\pi$
$992$	3.22511e6	0.104056
$993$	2.21419e7	0.712594
$994$	1.52082e7	0.488215
$995$	3.20698e6	0.102692
$996$	−3.24173e6	−0.103545
$997$	−3.52375e7	−1.12271	−0.561354	−	0.827576i	$-0.689719\pi$
$997$	−3.52375e7	−1.12271	−0.561354	+	0.827576i	$0.689719\pi$
$998$	4.97200e6	0.158017
$999$	−2.80595e6	−0.0889541

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.e.1.3	✓	5
3.2	odd	2		207.6.a.f.1.3		5
4.3	odd	2		1104.6.a.r.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.e.1.3	✓	5	1.1	even	1	trivial
207.6.a.f.1.3		5	3.2	odd	2
1104.6.a.r.1.3		5	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+2.24792 q^{2} +9.00000 q^{3} -26.9469 q^{4} +53.3906 q^{5} +20.2313 q^{6} +89.8688 q^{7} -132.508 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+2.24792 q^{2} +9.00000 q^{3} -26.9469 q^{4} +53.3906 q^{5} +20.2313 q^{6} +89.8688 q^{7} -132.508 q^{8} +81.0000 q^{9} +120.018 q^{10} +225.013 q^{11} -242.522 q^{12} +725.747 q^{13} +202.018 q^{14} +480.515 q^{15} +564.433 q^{16} +44.9540 q^{17} +182.081 q^{18} +1212.50 q^{19} -1438.71 q^{20} +808.819 q^{21} +505.810 q^{22} -529.000 q^{23} -1192.57 q^{24} -274.446 q^{25} +1631.42 q^{26} +729.000 q^{27} -2421.68 q^{28} +2592.49 q^{29} +1080.16 q^{30} +585.421 q^{31} +5509.05 q^{32} +2025.11 q^{33} +101.053 q^{34} +4798.15 q^{35} -2182.70 q^{36} -3849.04 q^{37} +2725.59 q^{38} +6531.73 q^{39} -7074.66 q^{40} -4299.92 q^{41} +1818.16 q^{42} -20567.4 q^{43} -6063.39 q^{44} +4324.64 q^{45} -1189.15 q^{46} -5221.50 q^{47} +5079.90 q^{48} -8730.61 q^{49} -616.931 q^{50} +404.586 q^{51} -19556.6 q^{52} -15753.7 q^{53} +1638.73 q^{54} +12013.6 q^{55} -11908.3 q^{56} +10912.5 q^{57} +5827.70 q^{58} -13780.6 q^{59} -12948.4 q^{60} -18988.5 q^{61} +1315.98 q^{62} +7279.37 q^{63} -5677.98 q^{64} +38748.1 q^{65} +4552.29 q^{66} +1951.43 q^{67} -1211.37 q^{68} -4761.00 q^{69} +10785.8 q^{70} +75281.4 q^{71} -10733.1 q^{72} -4177.11 q^{73} -8652.32 q^{74} -2470.01 q^{75} -32673.0 q^{76} +20221.6 q^{77} +14682.8 q^{78} -95896.7 q^{79} +30135.4 q^{80} +6561.00 q^{81} -9665.87 q^{82} +13366.7 q^{83} -21795.1 q^{84} +2400.12 q^{85} -46233.9 q^{86} +23332.4 q^{87} -29815.9 q^{88} +50932.1 q^{89} +9721.43 q^{90} +65222.0 q^{91} +14254.9 q^{92} +5268.79 q^{93} -11737.5 q^{94} +64735.9 q^{95} +49581.4 q^{96} +44280.7 q^{97} -19625.7 q^{98} +18226.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9}+O(q^{10}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9	\( 5 q + 8 q^{2} + 45 q^{3} + 118 q^{4} + 94 q^{5} + 72 q^{6} + 272 q^{7} + 258 q^{8} + 405 q^{9} - 172 q^{10} + 1100 q^{11} + 1062 q^{12} - 978 q^{13} - 344 q^{14} + 846 q^{15} + 1218 q^{16} + 2522 q^{17} + 648 q^{18} + 2060 q^{19} + 7720 q^{20} + 2448 q^{21} - 2572 q^{22} - 2645 q^{23} + 2322 q^{24} + 12035 q^{25} + 9280 q^{26} + 3645 q^{27} + 8072 q^{28} + 1526 q^{29} - 1548 q^{30} - 7392 q^{31} - 5086 q^{32} + 9900 q^{33} - 15608 q^{34} + 6056 q^{35} + 9558 q^{36} - 8210 q^{37} - 14276 q^{38} - 8802 q^{39} - 37472 q^{40} + 21250 q^{41} - 3096 q^{42} - 4548 q^{43} - 4260 q^{44} + 7614 q^{45} - 4232 q^{46} + 536 q^{47} + 10962 q^{48} - 27979 q^{49} - 81872 q^{50} + 22698 q^{51} - 76380 q^{52} - 11482 q^{53} + 5832 q^{54} - 77064 q^{55} - 28624 q^{56} + 18540 q^{57} - 79680 q^{58} + 74676 q^{59} + 69480 q^{60} - 44618 q^{61} + 64880 q^{62} + 22032 q^{63} - 137382 q^{64} - 24388 q^{65} - 23148 q^{66} - 1412 q^{67} + 80196 q^{68} - 23805 q^{69} - 222304 q^{70} + 37912 q^{71} + 20898 q^{72} + 46546 q^{73} + 111604 q^{74} + 108315 q^{75} - 79548 q^{76} + 157008 q^{77} + 83520 q^{78} + 50544 q^{79} + 69424 q^{80} + 32805 q^{81} - 233720 q^{82} + 89588 q^{83} + 72648 q^{84} + 147892 q^{85} + 77428 q^{86} + 13734 q^{87} + 54484 q^{88} + 280410 q^{89} - 13932 q^{90} - 27416 q^{91} - 62422 q^{92} - 66528 q^{93} + 113632 q^{94} + 203120 q^{95} - 45774 q^{96} + 90074 q^{97} + 32976 q^{98} + 89100 q^{99}+O(q^{100}) \) 5 * q + 8 * q^2 + 45 * q^3 + 118 * q^4 + 94 * q^5 + 72 * q^6 + 272 * q^7 + 258 * q^8 + 405 * q^9 - 172 * q^10 + 1100 * q^11 + 1062 * q^12 - 978 * q^13 - 344 * q^14 + 846 * q^15 + 1218 * q^16 + 2522 * q^17 + 648 * q^18 + 2060 * q^19 + 7720 * q^20 + 2448 * q^21 - 2572 * q^22 - 2645 * q^23 + 2322 * q^24 + 12035 * q^25 + 9280 * q^26 + 3645 * q^27 + 8072 * q^28 + 1526 * q^29 - 1548 * q^30 - 7392 * q^31 - 5086 * q^32 + 9900 * q^33 - 15608 * q^34 + 6056 * q^35 + 9558 * q^36 - 8210 * q^37 - 14276 * q^38 - 8802 * q^39 - 37472 * q^40 + 21250 * q^41 - 3096 * q^42 - 4548 * q^43 - 4260 * q^44 + 7614 * q^45 - 4232 * q^46 + 536 * q^47 + 10962 * q^48 - 27979 * q^49 - 81872 * q^50 + 22698 * q^51 - 76380 * q^52 - 11482 * q^53 + 5832 * q^54 - 77064 * q^55 - 28624 * q^56 + 18540 * q^57 - 79680 * q^58 + 74676 * q^59 + 69480 * q^60 - 44618 * q^61 + 64880 * q^62 + 22032 * q^63 - 137382 * q^64 - 24388 * q^65 - 23148 * q^66 - 1412 * q^67 + 80196 * q^68 - 23805 * q^69 - 222304 * q^70 + 37912 * q^71 + 20898 * q^72 + 46546 * q^73 + 111604 * q^74 + 108315 * q^75 - 79548 * q^76 + 157008 * q^77 + 83520 * q^78 + 50544 * q^79 + 69424 * q^80 + 32805 * q^81 - 233720 * q^82 + 89588 * q^83 + 72648 * q^84 + 147892 * q^85 + 77428 * q^86 + 13734 * q^87 + 54484 * q^88 + 280410 * q^89 - 13932 * q^90 - 27416 * q^91 - 62422 * q^92 - 66528 * q^93 + 113632 * q^94 + 203120 * q^95 - 45774 * q^96 + 90074 * q^97 + 32976 * q^98 + 89100 * q^99

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