LMFDB - Embedded newform 69.6.a.a.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{29}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 7 $ x^2 - x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.19258$ 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)

$f(q)$	$=$	$q+10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +52.3852 q^{5} -96.9330 q^{6} -118.237 q^{7} +560.057 q^{8} +81.0000 q^{9} +O(q^{10})$	\(q+10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +52.3852 q^{5} -96.9330 q^{6} -118.237 q^{7} +560.057 q^{8} +81.0000 q^{9} +564.205 q^{10} +741.598 q^{11} -756.000 q^{12} -542.502 q^{13} -1273.45 q^{14} -471.466 q^{15} +3344.00 q^{16} -834.993 q^{17} +872.397 q^{18} -1309.79 q^{19} +4400.35 q^{20} +1064.13 q^{21} +7987.25 q^{22} +529.000 q^{23} -5040.51 q^{24} -380.795 q^{25} -5842.93 q^{26} -729.000 q^{27} -9931.89 q^{28} +5536.45 q^{29} -5077.85 q^{30} -7267.14 q^{31} +18094.2 q^{32} -6674.38 q^{33} -8993.15 q^{34} -6193.85 q^{35} +6804.00 q^{36} -10446.7 q^{37} -14106.8 q^{38} +4882.52 q^{39} +29338.7 q^{40} -4630.45 q^{41} +11461.0 q^{42} -9201.23 q^{43} +62294.2 q^{44} +4243.20 q^{45} +5697.50 q^{46} +15272.3 q^{47} -30096.0 q^{48} -2827.06 q^{49} -4101.28 q^{50} +7514.94 q^{51} -45570.2 q^{52} -36202.8 q^{53} -7851.57 q^{54} +38848.7 q^{55} -66219.4 q^{56} +11788.1 q^{57} +59629.4 q^{58} +6436.34 q^{59} -39603.2 q^{60} -2637.86 q^{61} -78269.5 q^{62} -9577.18 q^{63} +87872.0 q^{64} -28419.1 q^{65} -71885.3 q^{66} +21068.5 q^{67} -70139.4 q^{68} -4761.00 q^{69} -66709.9 q^{70} +74668.2 q^{71} +45364.6 q^{72} +80742.0 q^{73} -112514. q^{74} +3427.15 q^{75} -110022. q^{76} -87684.2 q^{77} +52586.3 q^{78} +39107.7 q^{79} +175176. q^{80} +6561.00 q^{81} -49871.4 q^{82} +101465. q^{83} +89387.0 q^{84} -43741.2 q^{85} -99100.3 q^{86} -49828.1 q^{87} +415337. q^{88} +22407.6 q^{89} +45700.6 q^{90} +64143.7 q^{91} +44436.0 q^{92} +65404.3 q^{93} +164488. q^{94} -68613.4 q^{95} -162847. q^{96} -29724.8 q^{97} -30448.3 q^{98} +60069.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9}+O(q^{10}) $ 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9} + 116 q^{10} + 320 q^{11} - 1512 q^{12} - 288 q^{13} - 1276 q^{14} - 846 q^{15} + 6688 q^{16} - 1810 q^{17} + 730 q^{19} + 7896 q^{20} + 1062 q^{21} + 12528 q^{22} + 1058 q^{23} - 1774 q^{25} - 8584 q^{26} - 1458 q^{27} - 9912 q^{28} + 8208 q^{29} - 1044 q^{30} + 1772 q^{31} - 2880 q^{33} + 1508 q^{34} - 6184 q^{35} + 13608 q^{36} - 23112 q^{37} - 36076 q^{38} + 2592 q^{39} + 6032 q^{40} + 5516 q^{41} + 11484 q^{42} + 10322 q^{43} + 26880 q^{44} + 7614 q^{45} + 42952 q^{47} - 60192 q^{48} - 19634 q^{49} + 10904 q^{50} + 16290 q^{51} - 24192 q^{52} - 25350 q^{53} + 21304 q^{55} - 66352 q^{56} - 6570 q^{57} + 30856 q^{58} + 18344 q^{59} - 71064 q^{60} + 37224 q^{61} - 175624 q^{62} - 9558 q^{63} + 175744 q^{64} - 17828 q^{65} - 112752 q^{66} - 7482 q^{67} - 152040 q^{68} - 9522 q^{69} - 66816 q^{70} + 126848 q^{71} + 137660 q^{73} + 23896 q^{74} + 15966 q^{75} + 61320 q^{76} - 87784 q^{77} + 77256 q^{78} + 62286 q^{79} + 314336 q^{80} + 13122 q^{81} - 159152 q^{82} + 83120 q^{83} + 89208 q^{84} - 84316 q^{85} - 309372 q^{86} - 73872 q^{87} + 651456 q^{88} + 69770 q^{89} + 9396 q^{90} + 64204 q^{91} + 88872 q^{92} - 15948 q^{93} - 133632 q^{94} + 16272 q^{95} - 170104 q^{97} + 150568 q^{98} + 25920 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9 + 116 * q^10 + 320 * q^11 - 1512 * q^12 - 288 * q^13 - 1276 * q^14 - 846 * q^15 + 6688 * q^16 - 1810 * q^17 + 730 * q^19 + 7896 * q^20 + 1062 * q^21 + 12528 * q^22 + 1058 * q^23 - 1774 * q^25 - 8584 * q^26 - 1458 * q^27 - 9912 * q^28 + 8208 * q^29 - 1044 * q^30 + 1772 * q^31 - 2880 * q^33 + 1508 * q^34 - 6184 * q^35 + 13608 * q^36 - 23112 * q^37 - 36076 * q^38 + 2592 * q^39 + 6032 * q^40 + 5516 * q^41 + 11484 * q^42 + 10322 * q^43 + 26880 * q^44 + 7614 * q^45 + 42952 * q^47 - 60192 * q^48 - 19634 * q^49 + 10904 * q^50 + 16290 * q^51 - 24192 * q^52 - 25350 * q^53 + 21304 * q^55 - 66352 * q^56 - 6570 * q^57 + 30856 * q^58 + 18344 * q^59 - 71064 * q^60 + 37224 * q^61 - 175624 * q^62 - 9558 * q^63 + 175744 * q^64 - 17828 * q^65 - 112752 * q^66 - 7482 * q^67 - 152040 * q^68 - 9522 * q^69 - 66816 * q^70 + 126848 * q^71 + 137660 * q^73 + 23896 * q^74 + 15966 * q^75 + 61320 * q^76 - 87784 * q^77 + 77256 * q^78 + 62286 * q^79 + 314336 * q^80 + 13122 * q^81 - 159152 * q^82 + 83120 * q^83 + 89208 * q^84 - 84316 * q^85 - 309372 * q^86 - 73872 * q^87 + 651456 * q^88 + 69770 * q^89 + 9396 * q^90 + 64204 * q^91 + 88872 * q^92 - 15948 * q^93 - 133632 * q^94 + 16272 * q^95 - 170104 * q^97 + 150568 * q^98 + 25920 * 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 $

$2$	10.7703	1.90394	0.951972	−	0.306186i	$-0.0990530\pi$
$2$	10.7703	1.90394	0.951972	+	0.306186i	$0.0990530\pi$
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	84.0000	2.62500
$5$	52.3852	0.937094	0.468547	−	0.883438i	$-0.344778\pi$
$5$	52.3852	0.937094	0.468547	+	0.883438i	$0.344778\pi$
$6$	−96.9330	−1.09924
$7$	−118.237	−0.912027	−0.456013	−	0.889973i	$-0.650723\pi$
$7$	−118.237	−0.912027	−0.456013	+	0.889973i	$0.650723\pi$
$8$	560.057	3.09391
$9$	81.0000	0.333333
$10$	564.205	1.78417
$11$	741.598	1.84794	0.923968	−	0.382471i	$-0.124927\pi$
$11$	741.598	1.84794	0.923968	+	0.382471i	$0.124927\pi$
$12$	−756.000	−1.51554
$13$	−542.502	−0.890314	−0.445157	−	0.895453i	$-0.646852\pi$
$13$	−542.502	−0.890314	−0.445157	+	0.895453i	$0.646852\pi$
$14$	−1273.45	−1.73645
$15$	−471.466	−0.541032
$16$	3344.00	3.26562
$17$	−834.993	−0.700746	−0.350373	−	0.936610i	$-0.613945\pi$
$17$	−834.993	−0.700746	−0.350373	+	0.936610i	$0.613945\pi$
$18$	872.397	0.634648
$19$	−1309.79	−0.832370	−0.416185	−	0.909280i	$-0.636633\pi$
$19$	−1309.79	−0.832370	−0.416185	+	0.909280i	$0.636633\pi$
$20$	4400.35	2.45987
$21$	1064.13	0.526559
$22$	7987.25	3.51836
$23$	529.000	0.208514
$23$	529.000	0.208514
$24$	−5040.51	−1.78627
$25$	−380.795	−0.121854
$26$	−5842.93	−1.69511
$27$	−729.000	−0.192450
$28$	−9931.89	−2.39407
$29$	5536.45	1.22247	0.611233	−	0.791451i	$-0.290674\pi$
$29$	5536.45	1.22247	0.611233	+	0.791451i	$0.290674\pi$
$30$	−5077.85	−1.03009
$31$	−7267.14	−1.35819	−0.679093	−	0.734052i	$-0.737627\pi$
$31$	−7267.14	−1.35819	−0.679093	+	0.734052i	$0.737627\pi$
$32$	18094.2	3.12366
$33$	−6674.38	−1.06691
$34$	−8993.15	−1.33418
$35$	−6193.85	−0.854655
$36$	6804.00	0.875000
$37$	−10446.7	−1.25451	−0.627253	−	0.778815i	$-0.715821\pi$
$37$	−10446.7	−1.25451	−0.627253	+	0.778815i	$0.715821\pi$
$38$	−14106.8	−1.58479
$39$	4882.52	0.514023
$40$	29338.7	2.89928
$41$	−4630.45	−0.430193	−0.215096	−	0.976593i	$-0.569007\pi$
$41$	−4630.45	−0.430193	−0.215096	+	0.976593i	$0.569007\pi$
$42$	11461.0	1.00254
$43$	−9201.23	−0.758883	−0.379442	−	0.925216i	$-0.623884\pi$
$43$	−9201.23	−0.758883	−0.379442	+	0.925216i	$0.623884\pi$
$44$	62294.2	4.85083
$45$	4243.20	0.312365
$46$	5697.50	0.397000
$47$	15272.3	1.00846	0.504231	−	0.863569i	$-0.331776\pi$
$47$	15272.3	1.00846	0.504231	+	0.863569i	$0.331776\pi$
$48$	−30096.0	−1.88541
$49$	−2827.06	−0.168207
$50$	−4101.28	−0.232004
$51$	7514.94	0.404576
$52$	−45570.2	−2.33707
$53$	−36202.8	−1.77032	−0.885161	−	0.465285i	$-0.845952\pi$
$53$	−36202.8	−1.77032	−0.885161	+	0.465285i	$0.845952\pi$
$54$	−7851.57	−0.366414
$55$	38848.7	1.73169
$56$	−66219.4	−2.82173
$57$	11788.1	0.480569
$58$	59629.4	2.32751
$59$	6436.34	0.240718	0.120359	−	0.992730i	$-0.461595\pi$
$59$	6436.34	0.240718	0.120359	+	0.992730i	$0.461595\pi$
$60$	−39603.2	−1.42021
$61$	−2637.86	−0.0907668	−0.0453834	−	0.998970i	$-0.514451\pi$
$61$	−2637.86	−0.0907668	−0.0453834	+	0.998970i	$0.514451\pi$
$62$	−78269.5	−2.58591
$63$	−9577.18	−0.304009
$64$	87872.0	2.68164
$65$	−28419.1	−0.834308
$66$	−71885.3	−2.03133
$67$	21068.5	0.573384	0.286692	−	0.958023i	$-0.407444\pi$
$67$	21068.5	0.573384	0.286692	+	0.958023i	$0.407444\pi$
$68$	−70139.4	−1.83946
$69$	−4761.00	−0.120386
$70$	−66709.9	−1.62721
$71$	74668.2	1.75788	0.878941	−	0.476930i	$-0.158250\pi$
$71$	74668.2	1.75788	0.878941	+	0.476930i	$0.158250\pi$
$72$	45364.6	1.03130
$73$	80742.0	1.77334	0.886671	−	0.462402i	$-0.153012\pi$
$73$	80742.0	1.77334	0.886671	+	0.462402i	$0.153012\pi$
$74$	−112514.	−2.38851
$75$	3427.15	0.0703526
$76$	−110022.	−2.18497
$77$	−87684.2	−1.68537
$78$	52586.3	0.978671
$79$	39107.7	0.705008	0.352504	−	0.935810i	$-0.385330\pi$
$79$	39107.7	0.705008	0.352504	+	0.935810i	$0.385330\pi$
$80$	175176.	3.06020
$81$	6561.00	0.111111
$82$	−49871.4	−0.819063
$83$	101465.	1.61666	0.808331	−	0.588728i	$-0.200371\pi$
$83$	101465.	1.61666	0.808331	+	0.588728i	$0.200371\pi$
$84$	89387.0	1.38222
$85$	−43741.2	−0.656665
$86$	−99100.3	−1.44487
$87$	−49828.1	−0.705791
$88$	415337.	5.71734
$89$	22407.6	0.299861	0.149930	−	0.988697i	$-0.452095\pi$
$89$	22407.6	0.299861	0.149930	+	0.988697i	$0.452095\pi$
$90$	45700.6	0.594725
$91$	64143.7	0.811990
$92$	44436.0	0.547350
$93$	65404.3	0.784149
$94$	164488.	1.92005
$95$	−68613.4	−0.780009
$96$	−162847.	−1.80344
$97$	−29724.8	−0.320767	−0.160384	−	0.987055i	$-0.551273\pi$
$97$	−29724.8	−0.320767	−0.160384	+	0.987055i	$0.551273\pi$
$98$	−30448.3	−0.320257
$99$	60069.4	0.615978
$100$	−31986.7	−0.319867
$101$	96859.7	0.944800	0.472400	−	0.881384i	$-0.343388\pi$
$101$	96859.7	0.944800	0.472400	+	0.881384i	$0.343388\pi$
$102$	80938.3	0.770289
$103$	2081.11	0.0193287	0.00966434	−	0.999953i	$-0.496924\pi$
$103$	2081.11	0.0193287	0.00966434	+	0.999953i	$0.496924\pi$
$104$	−303832.	−2.75455
$105$	55744.7	0.493435
$106$	−389916.	−3.37059
$107$	−76907.3	−0.649393	−0.324697	−	0.945818i	$-0.605262\pi$
$107$	−76907.3	−0.649393	−0.324697	+	0.945818i	$0.605262\pi$
$108$	−61236.0	−0.505181
$109$	−27250.0	−0.219685	−0.109842	−	0.993949i	$-0.535035\pi$
$109$	−27250.0	−0.219685	−0.109842	+	0.993949i	$0.535035\pi$
$110$	418414.	3.29704
$111$	94019.9	0.724290
$112$	−395384.	−2.97834
$113$	−120738.	−0.889505	−0.444753	−	0.895653i	$-0.646708\pi$
$113$	−120738.	−0.889505	−0.444753	+	0.895653i	$0.646708\pi$
$114$	126961.	0.914976
$115$	27711.8	0.195398
$116$	465062.	3.20897
$117$	−43942.7	−0.296771
$118$	69321.5	0.458314
$119$	98726.9	0.639099
$120$	−264048.	−1.67390
$121$	388916.	2.41486
$122$	−28410.6	−0.172815
$123$	41674.0	0.248372
$124$	−610440.	−3.56524
$125$	−183652.	−1.05128
$126$	−103149.	−0.578816
$127$	−140823.	−0.774755	−0.387377	−	0.921921i	$-0.626619\pi$
$127$	−140823.	−0.774755	−0.387377	+	0.921921i	$0.626619\pi$
$128$	367397.	1.98203
$129$	82811.1	0.438141
$130$	−306083.	−1.58848
$131$	−137026.	−0.697630	−0.348815	−	0.937192i	$-0.613416\pi$
$131$	−137026.	−0.697630	−0.348815	+	0.937192i	$0.613416\pi$
$132$	−560648.	−2.80063
$133$	154865.	0.759144
$134$	226914.	1.09169
$135$	−38188.8	−0.180344
$136$	−467644.	−2.16804
$137$	73627.5	0.335149	0.167575	−	0.985859i	$-0.446406\pi$
$137$	73627.5	0.335149	0.167575	+	0.985859i	$0.446406\pi$
$138$	−51277.5	−0.229208
$139$	214252.	0.940562	0.470281	−	0.882517i	$-0.344152\pi$
$139$	214252.	0.940562	0.470281	+	0.882517i	$0.344152\pi$
$140$	−520284.	−2.24347
$141$	−137451.	−0.582236
$142$	804201.	3.34691
$143$	−402318.	−1.64524
$144$	270864.	1.08854
$145$	290028.	1.14557
$146$	869618.	3.37634
$147$	25443.5	0.0971144
$148$	−877519.	−3.29308
$149$	132470.	0.488823	0.244412	−	0.969672i	$-0.421405\pi$
$149$	132470.	0.488823	0.244412	+	0.969672i	$0.421405\pi$
$150$	36911.5	0.133947
$151$	171660.	0.612669	0.306334	−	0.951924i	$-0.400897\pi$
$151$	171660.	0.612669	0.306334	+	0.951924i	$0.400897\pi$
$152$	−733555.	−2.57528
$153$	−67634.4	−0.233582
$154$	−944387.	−3.20884
$155$	−380690.	−1.27275
$156$	410132.	1.34931
$157$	−359852.	−1.16513	−0.582565	−	0.812784i	$-0.697951\pi$
$157$	−359852.	−1.16513	−0.582565	+	0.812784i	$0.697951\pi$
$158$	421202.	1.34230
$159$	325825.	1.02210
$160$	947865.	2.92716
$161$	−62547.3	−0.190171
$162$	70664.1	0.211549
$163$	454656.	1.34034	0.670168	−	0.742210i	$-0.266222\pi$
$163$	454656.	1.34034	0.670168	+	0.742210i	$0.266222\pi$
$164$	−388957.	−1.12926
$165$	−349639.	−0.999791
$166$	1.09281e6	3.07803
$167$	−359837.	−0.998424	−0.499212	−	0.866480i	$-0.666377\pi$
$167$	−359837.	−0.998424	−0.499212	+	0.866480i	$0.666377\pi$
$168$	595974.	1.62912
$169$	−76984.4	−0.207341
$170$	−471108.	−1.25025
$171$	−106093.	−0.277457
$172$	−772904.	−1.99207
$173$	−308025.	−0.782476	−0.391238	−	0.920289i	$-0.627953\pi$
$173$	−308025.	−0.782476	−0.391238	+	0.920289i	$0.627953\pi$
$174$	−536665.	−1.34379
$175$	45023.9	0.111134
$176$	2.47990e6	6.03466
$177$	−57927.0	−0.138979
$178$	241337.	0.570918
$179$	628893.	1.46705	0.733524	−	0.679663i	$-0.237874\pi$
$179$	628893.	1.46705	0.733524	+	0.679663i	$0.237874\pi$
$180$	356429.	0.819958
$181$	−667015.	−1.51335	−0.756675	−	0.653791i	$-0.773177\pi$
$181$	−667015.	−1.51335	−0.756675	+	0.653791i	$0.773177\pi$
$182$	690849.	1.54598
$183$	23740.7	0.0524043
$184$	296270.	0.645124
$185$	−547250.	−1.17559
$186$	704425.	1.49298
$187$	−619229.	−1.29493
$188$	1.28287e6	2.64721
$189$	86194.6	0.175520
$190$	−738989.	−1.48509
$191$	573327.	1.13715	0.568577	−	0.822630i	$-0.307494\pi$
$191$	573327.	1.13715	0.568577	+	0.822630i	$0.307494\pi$
$192$	−790848.	−1.54825
$193$	21198.3	0.0409646	0.0204823	−	0.999790i	$-0.493480\pi$
$193$	21198.3	0.0409646	0.0204823	+	0.999790i	$0.493480\pi$
$194$	−320146.	−0.610722
$195$	255772.	0.481688
$196$	−237473.	−0.441544
$197$	−342207.	−0.628237	−0.314118	−	0.949384i	$-0.601709\pi$
$197$	−342207.	−0.628237	−0.314118	+	0.949384i	$0.601709\pi$
$198$	646967.	1.17279
$199$	−32335.5	−0.0578825	−0.0289412	−	0.999581i	$-0.509214\pi$
$199$	−32335.5	−0.0578825	−0.0289412	+	0.999581i	$0.509214\pi$
$200$	−213267.	−0.377006
$201$	−189616.	−0.331043
$202$	1.04321e6	1.79885
$203$	−654613.	−1.11492
$204$	631255.	1.06201
$205$	−242567.	−0.403131
$206$	22414.3	0.0368007
$207$	42849.0	0.0695048
$208$	−1.81413e6	−2.90743
$209$	−971335.	−1.53817
$210$	600389.	0.939473
$211$	−1.03399e6	−1.59886	−0.799430	−	0.600759i	$-0.794865\pi$
$211$	−1.03399e6	−1.59886	−0.799430	+	0.600759i	$0.794865\pi$
$212$	−3.04103e6	−4.64710
$213$	−672014.	−1.01491
$214$	−828317.	−1.23641
$215$	−482008.	−0.711145
$216$	−408282.	−0.595423
$217$	859243.	1.23870
$218$	−293491.	−0.418267
$219$	−726678.	−1.02384
$220$	3.26329e6	4.54568
$221$	452985.	0.623884
$222$	1.01263e6	1.37901
$223$	990187.	1.33338	0.666692	−	0.745333i	$-0.267710\pi$
$223$	990187.	1.33338	0.666692	+	0.745333i	$0.267710\pi$
$224$	−2.13940e6	−2.84886
$225$	−30844.4	−0.0406181
$226$	−1.30039e6	−1.69357
$227$	−26685.3	−0.0343723	−0.0171861	−	0.999852i	$-0.505471\pi$
$227$	−26685.3	−0.0343723	−0.0171861	+	0.999852i	$0.505471\pi$
$228$	990198.	1.26149
$229$	150586.	0.189756	0.0948778	−	0.995489i	$-0.469754\pi$
$229$	150586.	0.189756	0.0948778	+	0.995489i	$0.469754\pi$
$230$	298465.	0.372026
$231$	789157.	0.973047
$232$	3.10073e6	3.78220
$233$	1.29421e6	1.56176	0.780881	−	0.624680i	$-0.214770\pi$
$233$	1.29421e6	1.56176	0.780881	+	0.624680i	$0.214770\pi$
$234$	−473277.	−0.565036
$235$	800041.	0.945024
$236$	540652.	0.631885
$237$	−351969.	−0.407037
$238$	1.06332e6	1.21681
$239$	−136898.	−0.155025	−0.0775123	−	0.996991i	$-0.524698\pi$
$239$	−136898.	−0.155025	−0.0775123	+	0.996991i	$0.524698\pi$
$240$	−1.57658e6	−1.76681
$241$	−284796.	−0.315858	−0.157929	−	0.987450i	$-0.550482\pi$
$241$	−284796.	−0.315858	−0.157929	+	0.987450i	$0.550482\pi$
$242$	4.18876e6	4.59776
$243$	−59049.0	−0.0641500
$244$	−221580.	−0.238263
$245$	−148096.	−0.157626
$246$	448843.	0.472886
$247$	710562.	0.741071
$248$	−4.07001e6	−4.20210
$249$	−913181.	−0.933380
$250$	−1.97799e6	−2.00158
$251$	1.26148e6	1.26385	0.631927	−	0.775028i	$-0.282264\pi$
$251$	1.26148e6	1.26385	0.631927	+	0.775028i	$0.282264\pi$
$252$	−804483.	−0.798023
$253$	392305.	0.385321
$254$	−1.51671e6	−1.47509
$255$	393671.	0.379126
$256$	1.14509e6	1.09204
$257$	552720.	0.522002	0.261001	−	0.965338i	$-0.415947\pi$
$257$	552720.	0.522002	0.261001	+	0.965338i	$0.415947\pi$
$258$	891903.	0.834196
$259$	1.23518e6	1.14414
$260$	−2.38720e6	−2.19006
$261$	448453.	0.407489
$262$	−1.47582e6	−1.32825
$263$	−651514.	−0.580811	−0.290405	−	0.956904i	$-0.593790\pi$
$263$	−651514.	−0.580811	−0.290405	+	0.956904i	$0.593790\pi$
$264$	−3.73803e6	−3.30091
$265$	−1.89649e6	−1.65896
$266$	1.66795e6	1.44537
$267$	−201668.	−0.173125
$268$	1.76975e6	1.50513
$269$	1.62659e6	1.37056	0.685280	−	0.728280i	$-0.259680\pi$
$269$	1.62659e6	1.37056	0.685280	+	0.728280i	$0.259680\pi$
$270$	−411306.	−0.343365
$271$	526548.	0.435527	0.217763	−	0.976002i	$-0.430124\pi$
$271$	526548.	0.435527	0.217763	+	0.976002i	$0.430124\pi$
$272$	−2.79222e6	−2.28837
$273$	−577294.	−0.468803
$274$	792992.	0.638106
$275$	−282396.	−0.225179
$276$	−399924.	−0.316013
$277$	102974.	0.0806358	0.0403179	−	0.999187i	$-0.487163\pi$
$277$	102974.	0.0806358	0.0403179	+	0.999187i	$0.487163\pi$
$278$	2.30756e6	1.79078
$279$	−588638.	−0.452729
$280$	−3.46891e6	−2.64422
$281$	−1.79209e6	−1.35392	−0.676960	−	0.736020i	$-0.736703\pi$
$281$	−1.79209e6	−1.35392	−0.676960	+	0.736020i	$0.736703\pi$
$282$	−1.48039e6	−1.10854
$283$	−1.06565e6	−0.790946	−0.395473	−	0.918478i	$-0.629419\pi$
$283$	−1.06565e6	−0.790946	−0.395473	+	0.918478i	$0.629419\pi$
$284$	6.27213e6	4.61444
$285$	617520.	0.450339
$286$	−4.33310e6	−3.13245
$287$	547489.	0.392347
$288$	1.46563e6	1.04122
$289$	−722644.	−0.508955
$290$	3.12370e6	2.18109
$291$	267523.	0.185195
$292$	6.78233e6	4.65502
$293$	1.85858e6	1.26477	0.632387	−	0.774653i	$-0.282075\pi$
$293$	1.85858e6	1.26477	0.632387	+	0.774653i	$0.282075\pi$
$294$	274035.	0.184900
$295$	337169.	0.225576
$296$	−5.85072e6	−3.88133
$297$	−540625.	−0.355635
$298$	1.42675e6	0.930692
$299$	−286984.	−0.185643
$300$	287881.	0.184676
$301$	1.08792e6	0.692122
$302$	1.84883e6	1.16649
$303$	−871738.	−0.545480
$304$	−4.37993e6	−2.71821
$305$	−138185.	−0.0850571
$306$	−728445.	−0.444727
$307$	−2.13805e6	−1.29471	−0.647355	−	0.762189i	$-0.724125\pi$
$307$	−2.13805e6	−1.29471	−0.647355	+	0.762189i	$0.724125\pi$
$308$	−7.36547e6	−4.42409
$309$	−18730.0	−0.0111594
$310$	−4.10016e6	−2.42324
$311$	1.35485e6	0.794311	0.397155	−	0.917751i	$-0.369997\pi$
$311$	1.35485e6	0.794311	0.397155	+	0.917751i	$0.369997\pi$
$312$	2.73449e6	1.59034
$313$	−523789.	−0.302201	−0.151100	−	0.988518i	$-0.548282\pi$
$313$	−523789.	−0.302201	−0.151100	+	0.988518i	$0.548282\pi$
$314$	−3.87572e6	−2.21834
$315$	−501702.	−0.284885
$316$	3.28504e6	1.85065
$317$	2.59918e6	1.45274	0.726371	−	0.687302i	$-0.241205\pi$
$317$	2.59918e6	1.45274	0.726371	+	0.687302i	$0.241205\pi$
$318$	3.50924e6	1.94601
$319$	4.10582e6	2.25904
$320$	4.60319e6	2.51295
$321$	692165.	0.374928
$322$	−673655.	−0.362074
$323$	1.09366e6	0.583280
$324$	551124.	0.291667
$325$	206582.	0.108489
$326$	4.89679e6	2.55192
$327$	245250.	0.126835
$328$	−2.59331e6	−1.33098
$329$	−1.80575e6	−0.919744
$330$	−3.76572e6	−1.90355
$331$	−1.02887e6	−0.516168	−0.258084	−	0.966122i	$-0.583091\pi$
$331$	−1.02887e6	−0.516168	−0.258084	+	0.966122i	$0.583091\pi$
$332$	8.52302e6	4.24374
$333$	−846179.	−0.418169
$334$	−3.87557e6	−1.90094
$335$	1.10367e6	0.537315
$336$	3.55846e6	1.71954
$337$	1.13501e6	0.544406	0.272203	−	0.962240i	$-0.412248\pi$
$337$	1.13501e6	0.544406	0.272203	+	0.962240i	$0.412248\pi$
$338$	−829147.	−0.394766
$339$	1.08664e6	0.513556
$340$	−3.67426e6	−1.72375
$341$	−5.38929e6	−2.50984
$342$	−1.14265e6	−0.528262
$343$	2.32147e6	1.06544
$344$	−5.15322e6	−2.34791
$345$	−249406.	−0.112813
$346$	−3.31753e6	−1.48979
$347$	1.08736e6	0.484787	0.242393	−	0.970178i	$-0.422068\pi$
$347$	1.08736e6	0.484787	0.242393	+	0.970178i	$0.422068\pi$
$348$	−4.18556e6	−1.85270
$349$	147242.	0.0647097	0.0323548	−	0.999476i	$-0.489699\pi$
$349$	147242.	0.0647097	0.0323548	+	0.999476i	$0.489699\pi$
$350$	484923.	0.211593
$351$	395484.	0.171341
$352$	1.34186e7	5.77232
$353$	1.58645e6	0.677623	0.338812	−	0.940854i	$-0.389975\pi$
$353$	1.58645e6	0.677623	0.338812	+	0.940854i	$0.389975\pi$
$354$	−623893.	−0.264607
$355$	3.91151e6	1.64730
$356$	1.88224e6	0.787135
$357$	−888542.	−0.368984
$358$	6.77339e6	2.79318
$359$	−1.85640e6	−0.760211	−0.380106	−	0.924943i	$-0.624112\pi$
$359$	−1.85640e6	−0.760211	−0.380106	+	0.924943i	$0.624112\pi$
$360$	2.37643e6	0.966428
$361$	−760559.	−0.307160
$362$	−7.18397e6	−2.88133
$363$	−3.50025e6	−1.39422
$364$	5.38807e6	2.13147
$365$	4.22968e6	1.66179
$366$	255696.	0.0997747
$367$	−2.01535e6	−0.781061	−0.390531	−	0.920590i	$-0.627708\pi$
$367$	−2.01535e6	−0.781061	−0.390531	+	0.920590i	$0.627708\pi$
$368$	1.76898e6	0.680930
$369$	−375066.	−0.143398
$370$	−5.89406e6	−2.23826
$371$	4.28050e6	1.61458
$372$	5.49396e6	2.05839
$373$	−1.33526e6	−0.496928	−0.248464	−	0.968641i	$-0.579926\pi$
$373$	−1.33526e6	−0.496928	−0.248464	+	0.968641i	$0.579926\pi$
$374$	−6.66930e6	−2.46548
$375$	1.65286e6	0.606959
$376$	8.55336e6	3.12009
$377$	−3.00354e6	−1.08838
$378$	928345.	0.334179
$379$	−2.84308e6	−1.01669	−0.508347	−	0.861152i	$-0.669743\pi$
$379$	−2.84308e6	−1.01669	−0.508347	+	0.861152i	$0.669743\pi$
$380$	−5.76352e6	−2.04752
$381$	1.26741e6	0.447305
$382$	6.17492e6	2.16508
$383$	3.29481e6	1.14771	0.573857	−	0.818956i	$-0.305447\pi$
$383$	3.29481e6	1.14771	0.573857	+	0.818956i	$0.305447\pi$
$384$	−3.30658e6	−1.14433
$385$	−4.59335e6	−1.57935
$386$	228313.	0.0779942
$387$	−745300.	−0.252961
$388$	−2.49688e6	−0.842014
$389$	−4.92017e6	−1.64857	−0.824283	−	0.566177i	$-0.808422\pi$
$389$	−4.92017e6	−1.64857	−0.824283	+	0.566177i	$0.808422\pi$
$390$	2.75474e6	0.917107
$391$	−441711.	−0.146116
$392$	−1.58331e6	−0.520417
$393$	1.23323e6	0.402777
$394$	−3.68568e6	−1.19613
$395$	2.04866e6	0.660659
$396$	5.04583e6	1.61694
$397$	−4.13215e6	−1.31583	−0.657915	−	0.753092i	$-0.728561\pi$
$397$	−4.13215e6	−1.31583	−0.657915	+	0.753092i	$0.728561\pi$
$398$	−348264.	−0.110205
$399$	−1.39378e6	−0.438292
$400$	−1.27338e6	−0.397930
$401$	−3.07921e6	−0.956266	−0.478133	−	0.878287i	$-0.658686\pi$
$401$	−3.07921e6	−0.956266	−0.478133	+	0.878287i	$0.658686\pi$
$402$	−2.04223e6	−0.630288
$403$	3.94244e6	1.20921
$404$	8.13622e6	2.48010
$405$	343699.	0.104122
$406$	−7.05039e6	−2.12275
$407$	−7.74722e6	−2.31825
$408$	4.20879e6	1.25172
$409$	−483481.	−0.142913	−0.0714564	−	0.997444i	$-0.522765\pi$
$409$	−483481.	−0.142913	−0.0714564	+	0.997444i	$0.522765\pi$
$410$	−2.61252e6	−0.767539
$411$	−662647.	−0.193499
$412$	174813.	0.0507378
$413$	−761012.	−0.219541
$414$	461498.	0.132333
$415$	5.31524e6	1.51496
$416$	−9.81612e6	−2.78103
$417$	−1.92827e6	−0.543034
$418$	−1.04616e7	−2.92858
$419$	1.51752e6	0.422279	0.211140	−	0.977456i	$-0.432282\pi$
$419$	1.51752e6	0.422279	0.211140	+	0.977456i	$0.432282\pi$
$420$	4.68255e6	1.29527
$421$	3.76256e6	1.03461	0.517306	−	0.855800i	$-0.326935\pi$
$421$	3.76256e6	1.03461	0.517306	+	0.855800i	$0.326935\pi$
$422$	−1.11364e7	−3.04414
$423$	1.23706e6	0.336154
$424$	−2.02756e7	−5.47721
$425$	317961.	0.0853888
$426$	−7.23781e6	−1.93234
$427$	311892.	0.0827818
$428$	−6.46021e6	−1.70466
$429$	3.62087e6	0.949881
$430$	−5.19139e6	−1.35398
$431$	2.80235e6	0.726658	0.363329	−	0.931661i	$-0.381640\pi$
$431$	2.80235e6	0.726658	0.363329	+	0.931661i	$0.381640\pi$
$432$	−2.43778e6	−0.628470
$433$	3.06756e6	0.786272	0.393136	−	0.919480i	$-0.371390\pi$
$433$	3.06756e6	0.786272	0.393136	+	0.919480i	$0.371390\pi$
$434$	9.25433e6	2.35842
$435$	−2.61025e6	−0.661393
$436$	−2.28900e6	−0.576672
$437$	−692877.	−0.173561
$438$	−7.82656e6	−1.94933
$439$	−4.01436e6	−0.994158	−0.497079	−	0.867705i	$-0.665594\pi$
$439$	−4.01436e6	−0.994158	−0.497079	+	0.867705i	$0.665594\pi$
$440$	2.17575e7	5.35769
$441$	−228992.	−0.0560690
$442$	4.87880e6	1.18784
$443$	4.74625e6	1.14906	0.574528	−	0.818485i	$-0.305186\pi$
$443$	4.74625e6	1.14906	0.574528	+	0.818485i	$0.305186\pi$
$444$	7.89767e6	1.90126
$445$	1.17382e6	0.280998
$446$	1.06646e7	2.53869
$447$	−1.19223e6	−0.282222
$448$	−1.03897e7	−2.44573
$449$	1.57070e6	0.367687	0.183843	−	0.982956i	$-0.441146\pi$
$449$	1.57070e6	0.367687	0.183843	+	0.982956i	$0.441146\pi$
$450$	−332204.	−0.0773345
$451$	−3.43393e6	−0.794968
$452$	−1.01420e7	−2.33495
$453$	−1.54494e6	−0.353725
$454$	−287410.	−0.0654429
$455$	3.36018e6	0.760911
$456$	6.60200e6	1.48684
$457$	4.94573e6	1.10775	0.553873	−	0.832601i	$-0.313149\pi$
$457$	4.94573e6	1.10775	0.553873	+	0.832601i	$0.313149\pi$
$458$	1.62186e6	0.361284
$459$	608710.	0.134859
$460$	2.32779e6	0.512919
$461$	4.86491e6	1.06616	0.533080	−	0.846065i	$-0.321035\pi$
$461$	4.86491e6	1.06616	0.533080	+	0.846065i	$0.321035\pi$
$462$	8.49949e6	1.85263
$463$	−2.04379e6	−0.443082	−0.221541	−	0.975151i	$-0.571109\pi$
$463$	−2.04379e6	−0.443082	−0.221541	+	0.975151i	$0.571109\pi$
$464$	1.85139e7	3.99211
$465$	3.42621e6	0.734822
$466$	1.39391e7	2.97351
$467$	−6.33570e6	−1.34432	−0.672159	−	0.740407i	$-0.734633\pi$
$467$	−6.33570e6	−1.34432	−0.672159	+	0.740407i	$0.734633\pi$
$468$	−3.69118e6	−0.779025
$469$	−2.49107e6	−0.522942
$470$	8.61671e6	1.79927
$471$	3.23866e6	0.672688
$472$	3.60472e6	0.744760
$473$	−6.82362e6	−1.40237
$474$	−3.79082e6	−0.774975
$475$	498759.	0.101428
$476$	8.29306e6	1.67763
$477$	−2.93243e6	−0.590107
$478$	−1.47443e6	−0.295158
$479$	2.39044e6	0.476036	0.238018	−	0.971261i	$-0.423502\pi$
$479$	2.39044e6	0.476036	0.238018	+	0.971261i	$0.423502\pi$
$480$	−8.53079e6	−1.69000
$481$	5.66733e6	1.11690
$482$	−3.06735e6	−0.601375
$483$	562925.	0.109795
$484$	3.26690e7	6.33902
$485$	−1.55714e6	−0.300589
$486$	−635977.	−0.122138
$487$	−8.69235e6	−1.66079	−0.830395	−	0.557175i	$-0.811885\pi$
$487$	−8.69235e6	−1.66079	−0.830395	+	0.557175i	$0.811885\pi$
$488$	−1.47735e6	−0.280824
$489$	−4.09190e6	−0.773843
$490$	−1.59504e6	−0.300111
$491$	3.02753e6	0.566742	0.283371	−	0.959010i	$-0.408547\pi$
$491$	3.02753e6	0.566742	0.283371	+	0.959010i	$0.408547\pi$
$492$	3.50062e6	0.651976
$493$	−4.62290e6	−0.856638
$494$	7.65299e6	1.41096
$495$	3.14675e6	0.577230
$496$	−2.43013e7	−4.43533
$497$	−8.82853e6	−1.60324
$498$	−9.83526e6	−1.77710
$499$	3.24808e6	0.583950	0.291975	−	0.956426i	$-0.405688\pi$
$499$	3.24808e6	0.583950	0.291975	+	0.956426i	$0.405688\pi$
$500$	−1.54267e7	−2.75962
$501$	3.23853e6	0.576440
$502$	1.35866e7	2.40631
$503$	−1.14587e6	−0.201937	−0.100969	−	0.994890i	$-0.532194\pi$
$503$	−1.14587e6	−0.201937	−0.100969	+	0.994890i	$0.532194\pi$
$504$	−5.36377e6	−0.940576
$505$	5.07401e6	0.885367
$506$	4.22526e6	0.733630
$507$	692859.	0.119709
$508$	−1.18291e7	−2.03373
$509$	−7.47756e6	−1.27928	−0.639640	−	0.768675i	$-0.720917\pi$
$509$	−7.47756e6	−1.27928	−0.639640	+	0.768675i	$0.720917\pi$
$510$	4.23997e6	0.721834
$511$	−9.54667e6	−1.61733
$512$	576256.	0.0971494
$513$	954834.	0.160190
$514$	5.95298e6	0.993863
$515$	109019.	0.0181128
$516$	6.95613e6	1.15012
$517$	1.13259e7	1.86357
$518$	1.33033e7	2.17838
$519$	2.77223e6	0.451763
$520$	−1.59163e7	−2.58127
$521$	−568216.	−0.0917106	−0.0458553	−	0.998948i	$-0.514601\pi$
$521$	−568216.	−0.0917106	−0.0458553	+	0.998948i	$0.514601\pi$
$522$	4.82998e6	0.775835
$523$	6.11824e6	0.978076	0.489038	−	0.872262i	$-0.337348\pi$
$523$	6.11824e6	0.978076	0.489038	+	0.872262i	$0.337348\pi$
$524$	−1.15102e7	−1.83128
$525$	−405215.	−0.0641634
$526$	−7.01703e6	−1.10583
$527$	6.06801e6	0.951743
$528$	−2.23191e7	−3.48411
$529$	279841.	0.0434783
$530$	−2.04258e7	−3.15856
$531$	521343.	0.0802394
$532$	1.30087e7	1.99275
$533$	2.51203e6	0.383007
$534$	−2.17203e6	−0.329620
$535$	−4.02880e6	−0.608543
$536$	1.17995e7	1.77400
$537$	−5.66004e6	−0.847001
$538$	1.75189e7	2.60947
$539$	−2.09654e6	−0.310836
$540$	−3.20786e6	−0.473403
$541$	−4.31434e6	−0.633755	−0.316878	−	0.948466i	$-0.602634\pi$
$541$	−4.31434e6	−0.633755	−0.316878	+	0.948466i	$0.602634\pi$
$542$	5.67110e6	0.829219
$543$	6.00314e6	0.873733
$544$	−1.51085e7	−2.18889
$545$	−1.42749e6	−0.205865
$546$	−6.21764e6	−0.892574
$547$	7.34052e6	1.04896	0.524480	−	0.851423i	$-0.324260\pi$
$547$	7.34052e6	1.04896	0.524480	+	0.851423i	$0.324260\pi$
$548$	6.18471e6	0.879767
$549$	−213667.	−0.0302556
$550$	−3.04150e6	−0.428728
$551$	−7.25157e6	−1.01754
$552$	−2.66643e6	−0.372463
$553$	−4.62396e6	−0.642986
$554$	1.10906e6	0.153526
$555$	4.92525e6	0.678728
$556$	1.79972e7	2.46898
$557$	8.10617e6	1.10708	0.553538	−	0.832824i	$-0.313277\pi$
$557$	8.10617e6	1.10708	0.553538	+	0.832824i	$0.313277\pi$
$558$	−6.33983e6	−0.861970
$559$	4.99169e6	0.675644
$560$	−2.07123e7	−2.79098
$561$	5.57306e6	0.747630
$562$	−1.93013e7	−2.57779
$563$	1.15177e7	1.53142	0.765709	−	0.643187i	$-0.222388\pi$
$563$	1.15177e7	1.53142	0.765709	+	0.643187i	$0.222388\pi$
$564$	−1.15459e7	−1.52837
$565$	−6.32489e6	−0.833550
$566$	−1.14774e7	−1.50592
$567$	−775752.	−0.101336
$568$	4.18185e7	5.43873
$569$	248635.	0.0321945	0.0160972	−	0.999870i	$-0.494876\pi$
$569$	248635.	0.0321945	0.0160972	+	0.999870i	$0.494876\pi$
$570$	6.65090e6	0.857419
$571$	2.55456e6	0.327888	0.163944	−	0.986470i	$-0.447578\pi$
$571$	2.55456e6	0.327888	0.163944	+	0.986470i	$0.447578\pi$
$572$	−3.37947e7	−4.31876
$573$	−5.15994e6	−0.656536
$574$	5.89664e6	0.747007
$575$	−201440.	−0.0254084
$576$	7.11763e6	0.893880
$577$	3.96849e6	0.496233	0.248116	−	0.968730i	$-0.420188\pi$
$577$	3.96849e6	0.496233	0.248116	+	0.968730i	$0.420188\pi$
$578$	−7.78311e6	−0.969022
$579$	−190785.	−0.0236509
$580$	2.43624e7	3.00711
$581$	−1.19968e7	−1.47444
$582$	2.88131e6	0.352601
$583$	−2.68479e7	−3.27144
$584$	4.52201e7	5.48655
$585$	−2.30194e6	−0.278103
$586$	2.00175e7	2.40806
$587$	7.38560e6	0.884689	0.442344	−	0.896845i	$-0.354147\pi$
$587$	7.38560e6	0.884689	0.442344	+	0.896845i	$0.354147\pi$
$588$	2.13725e6	0.254925
$589$	9.51840e6	1.13051
$590$	3.63142e6	0.429483
$591$	3.07986e6	0.362713
$592$	−3.49336e7	−4.09675
$593$	−1.34366e7	−1.56910	−0.784552	−	0.620063i	$-0.787107\pi$
$593$	−1.34366e7	−1.56910	−0.784552	+	0.620063i	$0.787107\pi$
$594$	−5.82271e6	−0.677109
$595$	5.17182e6	0.598896
$596$	1.11275e7	1.28316
$597$	291020.	0.0334185
$598$	−3.09091e6	−0.353454
$599$	1.65576e6	0.188552	0.0942760	−	0.995546i	$-0.469946\pi$
$599$	1.65576e6	0.188552	0.0942760	+	0.995546i	$0.469946\pi$
$600$	1.91940e6	0.217664
$601$	−8.29015e6	−0.936216	−0.468108	−	0.883671i	$-0.655064\pi$
$601$	−8.29015e6	−0.936216	−0.468108	+	0.883671i	$0.655064\pi$
$602$	1.17173e7	1.31776
$603$	1.70654e6	0.191128
$604$	1.44194e7	1.60826
$605$	2.03734e7	2.26296
$606$	−9.38890e6	−1.03856
$607$	−1.24477e7	−1.37125	−0.685624	−	0.727956i	$-0.740470\pi$
$607$	−1.24477e7	−1.37125	−0.685624	+	0.727956i	$0.740470\pi$
$608$	−2.36995e7	−2.60004
$609$	5.89151e6	0.643700
$610$	−1.48830e6	−0.161944
$611$	−8.28525e6	−0.897848
$612$	−5.68129e6	−0.613153
$613$	−9.98753e6	−1.07351	−0.536756	−	0.843737i	$-0.680351\pi$
$613$	−9.98753e6	−1.07351	−0.536756	+	0.843737i	$0.680351\pi$
$614$	−2.30275e7	−2.46505
$615$	2.18310e6	0.232748
$616$	−4.91081e7	−5.21437
$617$	−1.14994e7	−1.21608	−0.608042	−	0.793905i	$-0.708045\pi$
$617$	−1.14994e7	−1.21608	−0.608042	+	0.793905i	$0.708045\pi$
$618$	−201728.	−0.0212469
$619$	−5.70831e6	−0.598799	−0.299400	−	0.954128i	$-0.596786\pi$
$619$	−5.70831e6	−0.598799	−0.299400	+	0.954128i	$0.596786\pi$
$620$	−3.19780e7	−3.34096
$621$	−385641.	−0.0401286
$622$	1.45922e7	1.51232
$623$	−2.64940e6	−0.273481
$624$	1.63271e7	1.67861
$625$	−8.43064e6	−0.863297
$626$	−5.64138e6	−0.575373
$627$	8.74201e6	0.888060
$628$	−3.02275e7	−3.05847
$629$	8.72288e6	0.879090
$630$	−5.40350e6	−0.542405
$631$	6.75550e6	0.675436	0.337718	−	0.941247i	$-0.390345\pi$
$631$	6.75550e6	0.675436	0.337718	+	0.941247i	$0.390345\pi$
$632$	2.19025e7	2.18123
$633$	9.30592e6	0.923102
$634$	2.79941e7	2.76594
$635$	−7.37703e6	−0.726018
$636$	2.73693e7	2.68300
$637$	1.53368e6	0.149757
$638$	4.42211e7	4.30108
$639$	6.04813e6	0.585961
$640$	1.92462e7	1.85735
$641$	355815.	0.0342042	0.0171021	−	0.999854i	$-0.494556\pi$
$641$	355815.	0.0342042	0.0171021	+	0.999854i	$0.494556\pi$
$642$	7.45485e6	0.713841
$643$	1.12269e6	0.107086	0.0535432	−	0.998566i	$-0.482949\pi$
$643$	1.12269e6	0.107086	0.0535432	+	0.998566i	$0.482949\pi$
$644$	−5.25397e6	−0.499198
$645$	4.33807e6	0.410580
$646$	1.17791e7	1.11053
$647$	−1.91791e7	−1.80123	−0.900613	−	0.434621i	$-0.856882\pi$
$647$	−1.91791e7	−1.80123	−0.900613	+	0.434621i	$0.856882\pi$
$648$	3.67453e6	0.343768
$649$	4.77317e6	0.444831
$650$	2.22495e6	0.206556
$651$	−7.73319e6	−0.715165
$652$	3.81911e7	3.51838
$653$	−9.62631e6	−0.883440	−0.441720	−	0.897153i	$-0.645632\pi$
$653$	−9.62631e6	−0.883440	−0.441720	+	0.897153i	$0.645632\pi$
$654$	2.64142e6	0.241487
$655$	−7.17813e6	−0.653745
$656$	−1.54842e7	−1.40485
$657$	6.54010e6	0.591114
$658$	−1.94485e7	−1.75114
$659$	−6.82930e6	−0.612579	−0.306290	−	0.951938i	$-0.599088\pi$
$659$	−6.82930e6	−0.612579	−0.306290	+	0.951938i	$0.599088\pi$
$660$	−2.93696e7	−2.62445
$661$	9.20374e6	0.819333	0.409667	−	0.912235i	$-0.365645\pi$
$661$	9.20374e6	0.819333	0.409667	+	0.912235i	$0.365645\pi$
$662$	−1.10813e7	−0.982755
$663$	−4.07687e6	−0.360199
$664$	5.68260e7	5.00180
$665$	8.11263e6	0.711389
$666$	−9.11363e6	−0.796170
$667$	2.92878e6	0.254902
$668$	−3.02263e7	−2.62086
$669$	−8.91169e6	−0.769830
$670$	1.18869e7	1.02302
$671$	−1.95623e6	−0.167731
$672$	1.92546e7	1.64479
$673$	−9.25649e6	−0.787787	−0.393893	−	0.919156i	$-0.628872\pi$
$673$	−9.25649e6	−0.787787	−0.393893	+	0.919156i	$0.628872\pi$
$674$	1.22244e7	1.03652
$675$	277599.	0.0234509
$676$	−6.46669e6	−0.544271
$677$	−4.89629e6	−0.410578	−0.205289	−	0.978701i	$-0.565813\pi$
$677$	−4.89629e6	−0.410578	−0.205289	+	0.978701i	$0.565813\pi$
$678$	1.17035e7	0.977782
$679$	3.51457e6	0.292548
$680$	−2.44976e7	−2.03166
$681$	240168.	0.0198448
$682$	−5.80445e7	−4.77859
$683$	−1.01443e6	−0.0832086	−0.0416043	−	0.999134i	$-0.513247\pi$
$683$	−1.01443e6	−0.0832086	−0.0416043	+	0.999134i	$0.513247\pi$
$684$	−8.91179e6	−0.728324
$685$	3.85699e6	0.314067
$686$	2.50030e7	2.02853
$687$	−1.35527e6	−0.109555
$688$	−3.07689e7	−2.47823
$689$	1.96401e7	1.57614
$690$	−2.68618e6	−0.214789
$691$	−9.64799e6	−0.768673	−0.384336	−	0.923193i	$-0.625570\pi$
$691$	−9.64799e6	−0.768673	−0.384336	+	0.923193i	$0.625570\pi$
$692$	−2.58741e7	−2.05400
$693$	−7.10242e6	−0.561789
$694$	1.17113e7	0.923007
$695$	1.12236e7	0.881396
$696$	−2.79066e7	−2.18365
$697$	3.86639e6	0.301456
$698$	1.58585e6	0.123204
$699$	−1.16479e7	−0.901684
$700$	3.78201e6	0.291728
$701$	−1.47825e7	−1.13620	−0.568098	−	0.822961i	$-0.692321\pi$
$701$	−1.47825e7	−1.13620	−0.568098	+	0.822961i	$0.692321\pi$
$702$	4.25949e6	0.326224
$703$	1.36829e7	1.04421
$704$	6.51657e7	4.95550
$705$	−7.20037e6	−0.545610
$706$	1.70865e7	1.29016
$707$	−1.14524e7	−0.861683
$708$	−4.86587e6	−0.364819
$709$	−585189.	−0.0437201	−0.0218600	−	0.999761i	$-0.506959\pi$
$709$	−585189.	−0.0437201	−0.0218600	+	0.999761i	$0.506959\pi$
$710$	4.21282e7	3.13637
$711$	3.16772e6	0.235003
$712$	1.25495e7	0.927742
$713$	−3.84432e6	−0.283201
$714$	−9.56989e6	−0.702524
$715$	−2.10755e7	−1.54175
$716$	5.28271e7	3.85100
$717$	1.23208e6	0.0895035
$718$	−1.99940e7	−1.44740
$719$	−3.42448e6	−0.247043	−0.123521	−	0.992342i	$-0.539419\pi$
$719$	−3.42448e6	−0.247043	−0.123521	+	0.992342i	$0.539419\pi$
$720$	1.41893e7	1.02007
$721$	−246064.	−0.0176283
$722$	−8.19147e6	−0.584816
$723$	2.56317e6	0.182361
$724$	−5.60293e7	−3.97254
$725$	−2.10825e6	−0.148963
$726$	−3.76988e7	−2.65452
$727$	−1.91984e7	−1.34719	−0.673595	−	0.739100i	$-0.735251\pi$
$727$	−1.91984e7	−1.34719	−0.673595	+	0.739100i	$0.735251\pi$
$728$	3.59242e7	2.51222
$729$	531441.	0.0370370
$730$	4.55551e7	3.16395
$731$	7.68297e6	0.531784
$732$	1.99422e6	0.137561
$733$	2.66362e7	1.83110	0.915552	−	0.402200i	$-0.131754\pi$
$733$	2.66362e7	1.83110	0.915552	+	0.402200i	$0.131754\pi$
$734$	−2.17060e7	−1.48710
$735$	1.33286e6	0.0910053
$736$	9.57181e6	0.651327
$737$	1.56243e7	1.05958
$738$	−4.03959e6	−0.273021
$739$	−6.81746e6	−0.459210	−0.229605	−	0.973284i	$-0.573743\pi$
$739$	−6.81746e6	−0.459210	−0.229605	+	0.973284i	$0.573743\pi$
$740$	−4.59690e7	−3.08593
$741$	−6.39506e6	−0.427857
$742$	4.61024e7	3.07407
$743$	−377932.	−0.0251155	−0.0125577	−	0.999921i	$-0.503997\pi$
$743$	−377932.	−0.0251155	−0.0125577	+	0.999921i	$0.503997\pi$
$744$	3.66301e7	2.42608
$745$	6.93946e6	0.458073
$746$	−1.43812e7	−0.946122
$747$	8.21863e6	0.538887
$748$	−5.20152e7	−3.39920
$749$	9.09327e6	0.592264
$750$	1.78019e7	1.15561
$751$	−8.02610e6	−0.519283	−0.259642	−	0.965705i	$-0.583604\pi$
$751$	−8.02610e6	−0.519283	−0.259642	+	0.965705i	$0.583604\pi$
$752$	5.10705e7	3.29326
$753$	−1.13533e7	−0.729686
$754$	−3.23491e7	−2.07221
$755$	8.99242e6	0.574129
$756$	7.24035e6	0.460739
$757$	−4.55894e6	−0.289150	−0.144575	−	0.989494i	$-0.546182\pi$
$757$	−4.55894e6	−0.289150	−0.144575	+	0.989494i	$0.546182\pi$
$758$	−3.06209e7	−1.93573
$759$	−3.53075e6	−0.222465
$760$	−3.84274e7	−2.41328
$761$	2.77860e7	1.73926	0.869629	−	0.493705i	$-0.164358\pi$
$761$	2.77860e7	1.73926	0.869629	+	0.493705i	$0.164358\pi$
$762$	1.36504e7	0.851643
$763$	3.22195e6	0.200358
$764$	4.81595e7	2.98503
$765$	−3.54304e6	−0.218888
$766$	3.54862e7	2.18518
$767$	−3.49173e6	−0.214315
$768$	−1.03058e7	−0.630490
$769$	1.47056e7	0.896742	0.448371	−	0.893847i	$-0.352004\pi$
$769$	1.47056e7	0.896742	0.448371	+	0.893847i	$0.352004\pi$
$770$	−4.94719e7	−3.00699
$771$	−4.97448e6	−0.301378
$772$	1.78066e6	0.107532
$773$	1.46941e7	0.884495	0.442248	−	0.896893i	$-0.354181\pi$
$773$	1.46941e7	0.884495	0.442248	+	0.896893i	$0.354181\pi$
$774$	−8.02713e6	−0.481624
$775$	2.76729e6	0.165501
$776$	−1.66476e7	−0.992424
$777$	−1.11166e7	−0.660572
$778$	−5.29919e7	−3.13878
$779$	6.06489e6	0.358080
$780$	2.14848e7	1.26443
$781$	5.53738e7	3.24845
$782$	−4.75738e6	−0.278196
$783$	−4.03607e6	−0.235264
$784$	−9.45368e6	−0.549301
$785$	−1.88509e7	−1.09184
$786$	1.32823e7	0.766864
$787$	1.02770e7	0.591465	0.295732	−	0.955271i	$-0.404436\pi$
$787$	1.02770e7	0.591465	0.295732	+	0.955271i	$0.404436\pi$
$788$	−2.87454e7	−1.64912
$789$	5.86363e6	0.335331
$790$	2.20648e7	1.25786
$791$	1.42757e7	0.811253
$792$	3.36423e7	1.90578
$793$	1.43105e6	0.0808110
$794$	−4.45046e7	−2.50527
$795$	1.70684e7	0.957800
$796$	−2.71618e6	−0.151942
$797$	2.13182e7	1.18879	0.594395	−	0.804173i	$-0.297392\pi$
$797$	2.13182e7	1.18879	0.594395	+	0.804173i	$0.297392\pi$
$798$	−1.50115e7	−0.834483
$799$	−1.27523e7	−0.706675
$800$	−6.89015e6	−0.380631
$801$	1.81501e6	0.0999536
$802$	−3.31641e7	−1.82068
$803$	5.98781e7	3.27702
$804$	−1.59278e7	−0.868989
$805$	−3.27655e6	−0.178208
$806$	4.24614e7	2.30227
$807$	−1.46393e7	−0.791293
$808$	5.42470e7	2.92312
$809$	2.64965e7	1.42337	0.711685	−	0.702499i	$-0.247932\pi$
$809$	2.64965e7	1.42337	0.711685	+	0.702499i	$0.247932\pi$
$810$	3.70175e6	0.198242
$811$	1.59727e7	0.852758	0.426379	−	0.904545i	$-0.359789\pi$
$811$	1.59727e7	0.852758	0.426379	+	0.904545i	$0.359789\pi$
$812$	−5.49875e7	−2.92667
$813$	−4.73893e6	−0.251452
$814$	−8.34401e7	−4.41381
$815$	2.38172e7	1.25602
$816$	2.51299e7	1.32119
$817$	1.20517e7	0.631672
$818$	−5.20725e6	−0.272098
$819$	5.19564e6	0.270663
$820$	−2.03756e7	−1.05822
$821$	2.19797e7	1.13806	0.569029	−	0.822317i	$-0.307319\pi$
$821$	2.19797e7	1.13806	0.569029	+	0.822317i	$0.307319\pi$
$822$	−7.13693e6	−0.368410
$823$	−3.47354e7	−1.78761	−0.893806	−	0.448454i	$-0.851975\pi$
$823$	−3.47354e7	−1.78761	−0.893806	+	0.448454i	$0.851975\pi$
$824$	1.16554e6	0.0598012
$825$	2.54157e6	0.130007
$826$	−8.19635e6	−0.417994
$827$	−1.25679e7	−0.638997	−0.319499	−	0.947587i	$-0.603515\pi$
$827$	−1.25679e7	−0.638997	−0.319499	+	0.947587i	$0.603515\pi$
$828$	3.59932e6	0.182450
$829$	−3.35386e7	−1.69495	−0.847477	−	0.530831i	$-0.821880\pi$
$829$	−3.35386e7	−1.69495	−0.847477	+	0.530831i	$0.821880\pi$
$830$	5.72469e7	2.88441
$831$	−926766.	−0.0465551
$832$	−4.76708e7	−2.38750
$833$	2.36057e6	0.117870
$834$	−2.07681e7	−1.03391
$835$	−1.88501e7	−0.935617
$836$	−8.15921e7	−4.03769
$837$	5.29774e6	0.261383
$838$	1.63442e7	0.803996
$839$	1.15158e6	0.0564791	0.0282395	−	0.999601i	$-0.491010\pi$
$839$	1.15158e6	0.0564791	0.0282395	+	0.999601i	$0.491010\pi$
$840$	3.12202e7	1.52664
$841$	1.01412e7	0.494422
$842$	4.05240e7	1.96984
$843$	1.61288e7	0.781686
$844$	−8.68552e7	−4.19701
$845$	−4.03284e6	−0.194298
$846$	1.33235e7	0.640018
$847$	−4.59842e7	−2.20242
$848$	−1.21062e8	−5.78121
$849$	9.59081e6	0.456653
$850$	3.42454e6	0.162576
$851$	−5.52628e6	−0.261583
$852$	−5.64492e7	−2.66415
$853$	−1.19102e7	−0.560465	−0.280232	−	0.959932i	$-0.590411\pi$
$853$	−1.19102e7	−0.560465	−0.280232	+	0.959932i	$0.590411\pi$
$854$	3.35918e6	0.157612
$855$	−5.55768e6	−0.260003
$856$	−4.30725e7	−2.00916
$857$	−1.59627e7	−0.742430	−0.371215	−	0.928547i	$-0.621059\pi$
$857$	−1.59627e7	−0.742430	−0.371215	+	0.928547i	$0.621059\pi$
$858$	3.89979e7	1.80852
$859$	1.22935e7	0.568452	0.284226	−	0.958757i	$-0.408263\pi$
$859$	1.22935e7	0.568452	0.284226	+	0.958757i	$0.408263\pi$
$860$	−4.04887e7	−1.86676
$861$	−4.92740e6	−0.226522
$862$	3.01823e7	1.38351
$863$	−2.14243e7	−0.979219	−0.489609	−	0.871942i	$-0.662861\pi$
$863$	−2.14243e7	−0.979219	−0.489609	+	0.871942i	$0.662861\pi$
$864$	−1.31906e7	−0.601148
$865$	−1.61360e7	−0.733254
$866$	3.30386e7	1.49702
$867$	6.50380e6	0.293846
$868$	7.21764e7	3.25159
$869$	2.90022e7	1.30281
$870$	−2.81133e7	−1.25925
$871$	−1.14297e7	−0.510492
$872$	−1.52615e7	−0.679684
$873$	−2.40771e6	−0.106922
$874$	−7.46251e6	−0.330451
$875$	2.17144e7	0.958799
$876$	−6.10409e7	−2.68758
$877$	−3.70233e6	−0.162546	−0.0812730	−	0.996692i	$-0.525899\pi$
$877$	−3.70233e6	−0.162546	−0.0812730	+	0.996692i	$0.525899\pi$
$878$	−4.32360e7	−1.89282
$879$	−1.67272e7	−0.730217
$880$	1.29910e8	5.65505
$881$	5.90856e6	0.256473	0.128237	−	0.991744i	$-0.459068\pi$
$881$	5.90856e6	0.256473	0.128237	+	0.991744i	$0.459068\pi$
$882$	−2.46631e6	−0.106752
$883$	1.83087e7	0.790233	0.395117	−	0.918631i	$-0.370704\pi$
$883$	1.83087e7	0.790233	0.395117	+	0.918631i	$0.370704\pi$
$884$	3.80508e7	1.63769
$885$	−3.03452e6	−0.130236
$886$	5.11186e7	2.18774
$887$	2.01751e7	0.861005	0.430503	−	0.902589i	$-0.358336\pi$
$887$	2.01751e7	0.861005	0.430503	+	0.902589i	$0.358336\pi$
$888$	5.26565e7	2.24089
$889$	1.66505e7	0.706597
$890$	1.26425e7	0.535004
$891$	4.86562e6	0.205326
$892$	8.31757e7	3.50013
$893$	−2.00034e7	−0.839413
$894$	−1.28407e7	−0.537335
$895$	3.29447e7	1.37476
$896$	−4.34399e7	−1.80767
$897$	2.58285e6	0.107181
$898$	1.69170e7	0.700055
$899$	−4.02342e7	−1.66034
$900$	−2.59093e6	−0.106622
$901$	3.02291e7	1.24055
$902$	−3.69845e7	−1.51357
$903$	−9.79132e6	−0.399597
$904$	−6.76203e7	−2.75205
$905$	−3.49417e7	−1.41815
$906$	−1.66395e7	−0.673471
$907$	−3.53066e7	−1.42508	−0.712538	−	0.701634i	$-0.752454\pi$
$907$	−3.53066e7	−1.42508	−0.712538	+	0.701634i	$0.752454\pi$
$908$	−2.24157e6	−0.0902272
$909$	7.84564e6	0.314933
$910$	3.61902e7	1.44873
$911$	−1.91557e7	−0.764720	−0.382360	−	0.924014i	$-0.624889\pi$
$911$	−1.91557e7	−0.764720	−0.382360	+	0.924014i	$0.624889\pi$
$912$	3.94193e7	1.56936
$913$	7.52459e7	2.98749
$914$	5.32672e7	2.10909
$915$	1.24366e6	0.0491077
$916$	1.26492e7	0.498108
$917$	1.62015e7	0.636257
$918$	6.55601e6	0.256763
$919$	−3.36053e7	−1.31256	−0.656279	−	0.754518i	$-0.727871\pi$
$919$	−3.36053e7	−1.31256	−0.656279	+	0.754518i	$0.727871\pi$
$920$	1.55202e7	0.604542
$921$	1.92425e7	0.747501
$922$	5.23966e7	2.02991
$923$	−4.05077e7	−1.56507
$924$	6.62892e7	2.55425
$925$	3.97803e6	0.152867
$926$	−2.20123e7	−0.843602
$927$	168570.	0.00644289
$928$	1.00177e8	3.81856
$929$	−2.51698e7	−0.956842	−0.478421	−	0.878131i	$-0.658791\pi$
$929$	−2.51698e7	−0.956842	−0.478421	+	0.878131i	$0.658791\pi$
$930$	3.69014e7	1.39906
$931$	3.70284e6	0.140011
$932$	1.08714e8	4.09963
$933$	−1.21937e7	−0.458596
$934$	−6.82375e7	−2.55951
$935$	−3.24384e7	−1.21347
$936$	−2.46104e7	−0.918183
$937$	−2.32062e7	−0.863484	−0.431742	−	0.901997i	$-0.642101\pi$
$937$	−2.32062e7	−0.863484	−0.431742	+	0.901997i	$0.642101\pi$
$938$	−2.68296e7	−0.995651
$939$	4.71410e6	0.174476
$940$	6.72035e7	2.48069
$941$	−1.44758e7	−0.532930	−0.266465	−	0.963845i	$-0.585856\pi$
$941$	−1.44758e7	−0.532930	−0.266465	+	0.963845i	$0.585856\pi$
$942$	3.48815e7	1.28076
$943$	−2.44951e6	−0.0897014
$944$	2.15231e7	0.786095
$945$	4.51532e6	0.164478
$946$	−7.34926e7	−2.67003
$947$	−3.57094e7	−1.29392	−0.646960	−	0.762524i	$-0.723960\pi$
$947$	−3.57094e7	−1.29392	−0.646960	+	0.762524i	$0.723960\pi$
$948$	−2.95654e7	−1.06847
$949$	−4.38027e7	−1.57883
$950$	5.37180e6	0.193113
$951$	−2.33926e7	−0.838741
$952$	5.52927e7	1.97731
$953$	−6.09472e6	−0.217381	−0.108691	−	0.994076i	$-0.534666\pi$
$953$	−6.09472e6	−0.217381	−0.108691	+	0.994076i	$0.534666\pi$
$954$	−3.15832e7	−1.12353
$955$	3.00338e7	1.06562
$956$	−1.14994e7	−0.406940
$957$	−3.69524e7	−1.30426
$958$	2.57459e7	0.906345
$959$	−8.70548e6	−0.305665
$960$	−4.14287e7	−1.45085
$961$	2.41822e7	0.844669
$962$	6.10391e7	2.12652
$963$	−6.22949e6	−0.216464
$964$	−2.39229e7	−0.829127
$965$	1.11048e6	0.0383877
$966$	6.06289e6	0.209044
$967$	1.48405e7	0.510366	0.255183	−	0.966893i	$-0.417864\pi$
$967$	1.48405e7	0.510366	0.255183	+	0.966893i	$0.417864\pi$
$968$	2.17815e8	7.47137
$969$	−9.84296e6	−0.336757
$970$	−1.67709e7	−0.572304
$971$	1.73779e7	0.591493	0.295746	−	0.955267i	$-0.404432\pi$
$971$	1.73779e7	0.591493	0.295746	+	0.955267i	$0.404432\pi$
$972$	−4.96012e6	−0.168394
$973$	−2.53325e7	−0.857818
$974$	−9.36194e7	−3.16205
$975$	−1.85924e6	−0.0626359
$976$	−8.82100e6	−0.296410
$977$	−1.30124e7	−0.436136	−0.218068	−	0.975934i	$-0.569976\pi$
$977$	−1.30124e7	−0.436136	−0.218068	+	0.975934i	$0.569976\pi$
$978$	−4.40711e7	−1.47335
$979$	1.66174e7	0.554123
$980$	−1.24400e7	−0.413768
$981$	−2.20725e6	−0.0732282
$982$	3.26075e7	1.07904
$983$	1.94435e7	0.641787	0.320893	−	0.947115i	$-0.396017\pi$
$983$	1.94435e7	0.641787	0.320893	+	0.947115i	$0.396017\pi$
$984$	2.33398e7	0.768440
$985$	−1.79266e7	−0.588717
$986$	−4.97902e7	−1.63099
$987$	1.62517e7	0.531015
$988$	5.96872e7	1.94531
$989$	−4.86745e6	−0.158238
$990$	3.38915e7	1.09901
$991$	−3.56706e6	−0.115379	−0.0576894	−	0.998335i	$-0.518373\pi$
$991$	−3.56706e6	−0.115379	−0.0576894	+	0.998335i	$0.518373\pi$
$992$	−1.31493e8	−4.24251
$993$	9.25984e6	0.298010
$994$	−9.50862e7	−3.05247
$995$	−1.69390e6	−0.0542414
$996$	−7.67072e7	−2.45012
$997$	3.36336e7	1.07161	0.535804	−	0.844342i	$-0.320009\pi$
$997$	3.36336e7	1.07161	0.535804	+	0.844342i	$0.320009\pi$
$998$	3.49829e7	1.11181
$999$	7.61561e6	0.241430

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.6.a.a.1.2	✓	2
3.2	odd	2		207.6.a.a.1.1		2
4.3	odd	2		1104.6.a.h.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.6.a.a.1.2	✓	2	1.1	even	1	trivial
207.6.a.a.1.1		2	3.2	odd	2
1104.6.a.h.1.2		2	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)

\(f(q)\)	\(=\)	\(q+10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +52.3852 q^{5} -96.9330 q^{6} -118.237 q^{7} +560.057 q^{8} +81.0000 q^{9} +O(q^{10})\)	\(q+10.7703 q^{2} -9.00000 q^{3} +84.0000 q^{4} +52.3852 q^{5} -96.9330 q^{6} -118.237 q^{7} +560.057 q^{8} +81.0000 q^{9} +564.205 q^{10} +741.598 q^{11} -756.000 q^{12} -542.502 q^{13} -1273.45 q^{14} -471.466 q^{15} +3344.00 q^{16} -834.993 q^{17} +872.397 q^{18} -1309.79 q^{19} +4400.35 q^{20} +1064.13 q^{21} +7987.25 q^{22} +529.000 q^{23} -5040.51 q^{24} -380.795 q^{25} -5842.93 q^{26} -729.000 q^{27} -9931.89 q^{28} +5536.45 q^{29} -5077.85 q^{30} -7267.14 q^{31} +18094.2 q^{32} -6674.38 q^{33} -8993.15 q^{34} -6193.85 q^{35} +6804.00 q^{36} -10446.7 q^{37} -14106.8 q^{38} +4882.52 q^{39} +29338.7 q^{40} -4630.45 q^{41} +11461.0 q^{42} -9201.23 q^{43} +62294.2 q^{44} +4243.20 q^{45} +5697.50 q^{46} +15272.3 q^{47} -30096.0 q^{48} -2827.06 q^{49} -4101.28 q^{50} +7514.94 q^{51} -45570.2 q^{52} -36202.8 q^{53} -7851.57 q^{54} +38848.7 q^{55} -66219.4 q^{56} +11788.1 q^{57} +59629.4 q^{58} +6436.34 q^{59} -39603.2 q^{60} -2637.86 q^{61} -78269.5 q^{62} -9577.18 q^{63} +87872.0 q^{64} -28419.1 q^{65} -71885.3 q^{66} +21068.5 q^{67} -70139.4 q^{68} -4761.00 q^{69} -66709.9 q^{70} +74668.2 q^{71} +45364.6 q^{72} +80742.0 q^{73} -112514. q^{74} +3427.15 q^{75} -110022. q^{76} -87684.2 q^{77} +52586.3 q^{78} +39107.7 q^{79} +175176. q^{80} +6561.00 q^{81} -49871.4 q^{82} +101465. q^{83} +89387.0 q^{84} -43741.2 q^{85} -99100.3 q^{86} -49828.1 q^{87} +415337. q^{88} +22407.6 q^{89} +45700.6 q^{90} +64143.7 q^{91} +44436.0 q^{92} +65404.3 q^{93} +164488. q^{94} -68613.4 q^{95} -162847. q^{96} -29724.8 q^{97} -30448.3 q^{98} +60069.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9}+O(q^{10}) \) 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9	\( 2 q - 18 q^{3} + 168 q^{4} + 94 q^{5} - 118 q^{7} + 162 q^{9} + 116 q^{10} + 320 q^{11} - 1512 q^{12} - 288 q^{13} - 1276 q^{14} - 846 q^{15} + 6688 q^{16} - 1810 q^{17} + 730 q^{19} + 7896 q^{20} + 1062 q^{21} + 12528 q^{22} + 1058 q^{23} - 1774 q^{25} - 8584 q^{26} - 1458 q^{27} - 9912 q^{28} + 8208 q^{29} - 1044 q^{30} + 1772 q^{31} - 2880 q^{33} + 1508 q^{34} - 6184 q^{35} + 13608 q^{36} - 23112 q^{37} - 36076 q^{38} + 2592 q^{39} + 6032 q^{40} + 5516 q^{41} + 11484 q^{42} + 10322 q^{43} + 26880 q^{44} + 7614 q^{45} + 42952 q^{47} - 60192 q^{48} - 19634 q^{49} + 10904 q^{50} + 16290 q^{51} - 24192 q^{52} - 25350 q^{53} + 21304 q^{55} - 66352 q^{56} - 6570 q^{57} + 30856 q^{58} + 18344 q^{59} - 71064 q^{60} + 37224 q^{61} - 175624 q^{62} - 9558 q^{63} + 175744 q^{64} - 17828 q^{65} - 112752 q^{66} - 7482 q^{67} - 152040 q^{68} - 9522 q^{69} - 66816 q^{70} + 126848 q^{71} + 137660 q^{73} + 23896 q^{74} + 15966 q^{75} + 61320 q^{76} - 87784 q^{77} + 77256 q^{78} + 62286 q^{79} + 314336 q^{80} + 13122 q^{81} - 159152 q^{82} + 83120 q^{83} + 89208 q^{84} - 84316 q^{85} - 309372 q^{86} - 73872 q^{87} + 651456 q^{88} + 69770 q^{89} + 9396 q^{90} + 64204 q^{91} + 88872 q^{92} - 15948 q^{93} - 133632 q^{94} + 16272 q^{95} - 170104 q^{97} + 150568 q^{98} + 25920 q^{99}+O(q^{100}) \) 2 * q - 18 * q^3 + 168 * q^4 + 94 * q^5 - 118 * q^7 + 162 * q^9 + 116 * q^10 + 320 * q^11 - 1512 * q^12 - 288 * q^13 - 1276 * q^14 - 846 * q^15 + 6688 * q^16 - 1810 * q^17 + 730 * q^19 + 7896 * q^20 + 1062 * q^21 + 12528 * q^22 + 1058 * q^23 - 1774 * q^25 - 8584 * q^26 - 1458 * q^27 - 9912 * q^28 + 8208 * q^29 - 1044 * q^30 + 1772 * q^31 - 2880 * q^33 + 1508 * q^34 - 6184 * q^35 + 13608 * q^36 - 23112 * q^37 - 36076 * q^38 + 2592 * q^39 + 6032 * q^40 + 5516 * q^41 + 11484 * q^42 + 10322 * q^43 + 26880 * q^44 + 7614 * q^45 + 42952 * q^47 - 60192 * q^48 - 19634 * q^49 + 10904 * q^50 + 16290 * q^51 - 24192 * q^52 - 25350 * q^53 + 21304 * q^55 - 66352 * q^56 - 6570 * q^57 + 30856 * q^58 + 18344 * q^59 - 71064 * q^60 + 37224 * q^61 - 175624 * q^62 - 9558 * q^63 + 175744 * q^64 - 17828 * q^65 - 112752 * q^66 - 7482 * q^67 - 152040 * q^68 - 9522 * q^69 - 66816 * q^70 + 126848 * q^71 + 137660 * q^73 + 23896 * q^74 + 15966 * q^75 + 61320 * q^76 - 87784 * q^77 + 77256 * q^78 + 62286 * q^79 + 314336 * q^80 + 13122 * q^81 - 159152 * q^82 + 83120 * q^83 + 89208 * q^84 - 84316 * q^85 - 309372 * q^86 - 73872 * q^87 + 651456 * q^88 + 69770 * q^89 + 9396 * q^90 + 64204 * q^91 + 88872 * q^92 - 15948 * q^93 - 133632 * q^94 + 16272 * q^95 - 170104 * q^97 + 150568 * q^98 + 25920 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more