LMFDB - Embedded newform 69.8.a.a.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,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, 8, names="a")

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
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:	$21.5545667584$
Analytic rank:	$1$
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} - 455x^{3} - 474x^{2} + 42284x + 127016 $ x^5 - 455x^3 - 474x^2 + 42284*x + 127016
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.24502$ 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)

$f(q)$	$=$	$q-3.24502 q^{2} -27.0000 q^{3} -117.470 q^{4} +168.083 q^{5} +87.6156 q^{6} +149.817 q^{7} +796.555 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-3.24502 q^{2} -27.0000 q^{3} -117.470 q^{4} +168.083 q^{5} +87.6156 q^{6} +149.817 q^{7} +796.555 q^{8} +729.000 q^{9} -545.434 q^{10} -18.5452 q^{11} +3171.69 q^{12} +10200.4 q^{13} -486.159 q^{14} -4538.25 q^{15} +12451.3 q^{16} -25924.5 q^{17} -2365.62 q^{18} -16166.5 q^{19} -19744.7 q^{20} -4045.05 q^{21} +60.1797 q^{22} +12167.0 q^{23} -21507.0 q^{24} -49873.0 q^{25} -33100.5 q^{26} -19683.0 q^{27} -17599.0 q^{28} -92885.5 q^{29} +14726.7 q^{30} -52229.9 q^{31} -142364. q^{32} +500.721 q^{33} +84125.5 q^{34} +25181.7 q^{35} -85635.5 q^{36} -76601.8 q^{37} +52460.7 q^{38} -275411. q^{39} +133888. q^{40} +108892. q^{41} +13126.3 q^{42} -791546. q^{43} +2178.50 q^{44} +122533. q^{45} -39482.2 q^{46} -992576. q^{47} -336185. q^{48} -801098. q^{49} +161839. q^{50} +699961. q^{51} -1.19824e6 q^{52} +1.60859e6 q^{53} +63871.8 q^{54} -3117.14 q^{55} +119337. q^{56} +436496. q^{57} +301416. q^{58} +733660. q^{59} +533107. q^{60} +114090. q^{61} +169487. q^{62} +109216. q^{63} -1.13179e6 q^{64} +1.71452e6 q^{65} -1624.85 q^{66} +447189. q^{67} +3.04534e6 q^{68} -328509. q^{69} -81715.2 q^{70} -671180. q^{71} +580689. q^{72} -406192. q^{73} +248575. q^{74} +1.34657e6 q^{75} +1.89908e6 q^{76} -2778.39 q^{77} +893714. q^{78} +883156. q^{79} +2.09286e6 q^{80} +531441. q^{81} -353358. q^{82} -4.07716e6 q^{83} +475172. q^{84} -4.35747e6 q^{85} +2.56858e6 q^{86} +2.50791e6 q^{87} -14772.3 q^{88} +5.07507e6 q^{89} -397621. q^{90} +1.52819e6 q^{91} -1.42926e6 q^{92} +1.41021e6 q^{93} +3.22093e6 q^{94} -2.71732e6 q^{95} +3.84382e6 q^{96} -1.23228e6 q^{97} +2.59958e6 q^{98} -13519.5 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) $ 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * 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 $

$2$	−3.24502	−0.286822	−0.143411	−	0.989663i	$-0.545807\pi$
$2$	−3.24502	−0.286822	−0.143411	+	0.989663i	$0.545807\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−117.470	−0.917733
$5$	168.083	0.601353	0.300677	−	0.953726i	$-0.402788\pi$
$5$	168.083	0.601353	0.300677	+	0.953726i	$0.402788\pi$
$6$	87.6156	0.165597
$7$	149.817	0.165089	0.0825444	−	0.996587i	$-0.473695\pi$
$7$	149.817	0.165089	0.0825444	+	0.996587i	$0.473695\pi$
$8$	796.555	0.550048
$9$	729.000	0.333333
$10$	−545.434	−0.172481
$11$	−18.5452	−0.00420105	−0.00210052	−	0.999998i	$-0.500669\pi$
$11$	−18.5452	−0.00420105	−0.00210052	+	0.999998i	$0.500669\pi$
$12$	3171.69	0.529853
$13$	10200.4	1.28770	0.643851	−	0.765151i	$-0.277336\pi$
$13$	10200.4	1.28770	0.643851	+	0.765151i	$0.277336\pi$
$14$	−486.159	−0.0473511
$15$	−4538.25	−0.347191
$16$	12451.3	0.759967
$17$	−25924.5	−1.27979	−0.639895	−	0.768462i	$-0.721022\pi$
$17$	−25924.5	−1.27979	−0.639895	+	0.768462i	$0.721022\pi$
$18$	−2365.62	−0.0956074
$19$	−16166.5	−0.540728	−0.270364	−	0.962758i	$-0.587144\pi$
$19$	−16166.5	−0.540728	−0.270364	+	0.962758i	$0.587144\pi$
$20$	−19744.7	−0.551882
$21$	−4045.05	−0.0953140
$22$	60.1797	0.00120495
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	−21507.0	−0.317571
$25$	−49873.0	−0.638374
$26$	−33100.5	−0.369341
$27$	−19683.0	−0.192450
$28$	−17599.0	−0.151507
$29$	−92885.5	−0.707221	−0.353610	−	0.935393i	$-0.615046\pi$
$29$	−92885.5	−0.707221	−0.353610	+	0.935393i	$0.615046\pi$
$30$	14726.7	0.0995822
$31$	−52229.9	−0.314886	−0.157443	−	0.987528i	$-0.550325\pi$
$31$	−52229.9	−0.314886	−0.157443	+	0.987528i	$0.550325\pi$
$32$	−142364.	−0.768024
$33$	500.721	0.00242548
$34$	84125.5	0.367072
$35$	25181.7	0.0992766
$36$	−85635.5	−0.305911
$37$	−76601.8	−0.248618	−0.124309	−	0.992244i	$-0.539671\pi$
$37$	−76601.8	−0.248618	−0.124309	+	0.992244i	$0.539671\pi$
$38$	52460.7	0.155093
$39$	−275411.	−0.743455
$40$	133888.	0.330773
$41$	108892.	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$41$	108892.	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$42$	13126.3	0.0273382
$43$	−791546.	−1.51823	−0.759113	−	0.650958i	$-0.774367\pi$
$43$	−791546.	−1.51823	−0.759113	+	0.650958i	$0.774367\pi$
$44$	2178.50	0.00385544
$45$	122533.	0.200451
$46$	−39482.2	−0.0598066
$47$	−992576.	−1.39451	−0.697254	−	0.716824i	$-0.745595\pi$
$47$	−992576.	−1.39451	−0.697254	+	0.716824i	$0.745595\pi$
$48$	−336185.	−0.438767
$49$	−801098.	−0.972746
$50$	161839.	0.183100
$51$	699961.	0.738887
$52$	−1.19824e6	−1.18177
$53$	1.60859e6	1.48416	0.742078	−	0.670313i	$-0.233840\pi$
$53$	1.60859e6	1.48416	0.742078	+	0.670313i	$0.233840\pi$
$54$	63871.8	0.0551989
$55$	−3117.14	−0.00252631
$56$	119337.	0.0908068
$57$	436496.	0.312190
$58$	301416.	0.202847
$59$	733660.	0.465064	0.232532	−	0.972589i	$-0.425299\pi$
$59$	733660.	0.465064	0.232532	+	0.972589i	$0.425299\pi$
$60$	533107.	0.318629
$61$	114090.	0.0643569	0.0321784	−	0.999482i	$-0.489756\pi$
$61$	114090.	0.0643569	0.0321784	+	0.999482i	$0.489756\pi$
$62$	169487.	0.0903163
$63$	109216.	0.0550296
$64$	−1.13179e6	−0.539681
$65$	1.71452e6	0.774364
$66$	−1624.85	−0.000695681 0
$67$	447189.	0.181647	0.0908237	−	0.995867i	$-0.471050\pi$
$67$	447189.	0.181647	0.0908237	+	0.995867i	$0.471050\pi$
$68$	3.04534e6	1.17451
$69$	−328509.	−0.120386
$70$	−81715.2	−0.0284747
$71$	−671180.	−0.222554	−0.111277	−	0.993789i	$-0.535494\pi$
$71$	−671180.	−0.222554	−0.111277	+	0.993789i	$0.535494\pi$
$72$	580689.	0.183349
$73$	−406192.	−0.122209	−0.0611043	−	0.998131i	$-0.519462\pi$
$73$	−406192.	−0.122209	−0.0611043	+	0.998131i	$0.519462\pi$
$74$	248575.	0.0713092
$75$	1.34657e6	0.368566
$76$	1.89908e6	0.496244
$77$	−2778.39	−0.000693546 0
$78$	893714.	0.213239
$79$	883156.	0.201531	0.100766	−	0.994910i	$-0.467871\pi$
$79$	883156.	0.201531	0.100766	+	0.994910i	$0.467871\pi$
$80$	2.09286e6	0.457009
$81$	531441.	0.111111
$82$	−353358.	−0.0707728
$83$	−4.07716e6	−0.782680	−0.391340	−	0.920246i	$-0.627988\pi$
$83$	−4.07716e6	−0.782680	−0.391340	+	0.920246i	$0.627988\pi$
$84$	475172.	0.0874728
$85$	−4.35747e6	−0.769606
$86$	2.56858e6	0.435461
$87$	2.50791e6	0.408314
$88$	−14772.3	−0.00231078
$89$	5.07507e6	0.763092	0.381546	−	0.924350i	$-0.375392\pi$
$89$	5.07507e6	0.763092	0.381546	+	0.924350i	$0.375392\pi$
$90$	−397621.	−0.0574938
$91$	1.52819e6	0.212585
$92$	−1.42926e6	−0.191361
$93$	1.41021e6	0.181800
$94$	3.22093e6	0.399976
$95$	−2.71732e6	−0.325169
$96$	3.84382e6	0.443419
$97$	−1.23228e6	−0.137091	−0.0685454	−	0.997648i	$-0.521836\pi$
$97$	−1.23228e6	−0.137091	−0.0685454	+	0.997648i	$0.521836\pi$
$98$	2.59958e6	0.279005
$99$	−13519.5	−0.00140035
$100$	5.85857e6	0.585857
$101$	4.65370e6	0.449442	0.224721	−	0.974423i	$-0.427853\pi$
$101$	4.65370e6	0.449442	0.224721	+	0.974423i	$0.427853\pi$
$102$	−2.27139e6	−0.211929
$103$	−1.60768e7	−1.44967	−0.724834	−	0.688923i	$-0.758084\pi$
$103$	−1.60768e7	−1.44967	−0.724834	+	0.688923i	$0.758084\pi$
$104$	8.12518e6	0.708298
$105$	−679906.	−0.0573174
$106$	−5.21991e6	−0.425689
$107$	−5.90678e6	−0.466131	−0.233065	−	0.972461i	$-0.574876\pi$
$107$	−5.90678e6	−0.466131	−0.233065	+	0.972461i	$0.574876\pi$
$108$	2.31216e6	0.176618
$109$	−2.42125e7	−1.79080	−0.895399	−	0.445264i	$-0.853110\pi$
$109$	−2.42125e7	−1.79080	−0.895399	+	0.445264i	$0.853110\pi$
$110$	10115.2	0.000724603 0
$111$	2.06825e6	0.143540
$112$	1.86541e6	0.125462
$113$	−1.27303e7	−0.829972	−0.414986	−	0.909828i	$-0.636214\pi$
$113$	−1.27303e7	−0.829972	−0.414986	+	0.909828i	$0.636214\pi$
$114$	−1.41644e6	−0.0895429
$115$	2.04507e6	0.125391
$116$	1.09112e7	0.649040
$117$	7.43609e6	0.429234
$118$	−2.38074e6	−0.133391
$119$	−3.88392e6	−0.211279
$120$	−3.61497e6	−0.190972
$121$	−1.94868e7	−0.999982
$122$	−370226.	−0.0184590
$123$	−2.94009e6	−0.142460
$124$	6.13544e6	0.288981
$125$	−2.15143e7	−0.985242
$126$	−354410.	−0.0157837
$127$	−4.28814e7	−1.85762	−0.928808	−	0.370562i	$-0.879165\pi$
$127$	−4.28814e7	−1.85762	−0.928808	+	0.370562i	$0.879165\pi$
$128$	2.18953e7	0.922816
$129$	2.13717e7	0.876549
$130$	−5.56364e6	−0.222105
$131$	−1.56206e7	−0.607084	−0.303542	−	0.952818i	$-0.598169\pi$
$131$	−1.56206e7	−0.607084	−0.303542	+	0.952818i	$0.598169\pi$
$132$	−58819.6	−0.00222594
$133$	−2.42202e6	−0.0892682
$134$	−1.45114e6	−0.0521005
$135$	−3.30838e6	−0.115730
$136$	−2.06503e7	−0.703946
$137$	2.94587e7	0.978793	0.489397	−	0.872061i	$-0.337217\pi$
$137$	2.94587e7	0.978793	0.489397	+	0.872061i	$0.337217\pi$
$138$	1.06602e6	0.0345293
$139$	2.71105e7	0.856219	0.428110	−	0.903727i	$-0.359180\pi$
$139$	2.71105e7	0.856219	0.428110	+	0.903727i	$0.359180\pi$
$140$	−2.95809e6	−0.0911095
$141$	2.67995e7	0.805120
$142$	2.17799e6	0.0638333
$143$	−189169.	−0.00540970
$144$	9.07700e6	0.253322
$145$	−1.56125e7	−0.425289
$146$	1.31810e6	0.0350521
$147$	2.16296e7	0.561615
$148$	8.99840e6	0.228165
$149$	2.30596e7	0.571085	0.285542	−	0.958366i	$-0.407826\pi$
$149$	2.30596e7	0.571085	0.285542	+	0.958366i	$0.407826\pi$
$150$	−4.36965e6	−0.105713
$151$	−5.65968e7	−1.33774	−0.668871	−	0.743378i	$-0.733222\pi$
$151$	−5.65968e7	−1.33774	−0.668871	+	0.743378i	$0.733222\pi$
$152$	−1.28775e7	−0.297427
$153$	−1.88989e7	−0.426597
$154$	9015.93	0.000198924 0
$155$	−8.77898e6	−0.189358
$156$	3.23524e7	0.682293
$157$	1.82778e7	0.376943	0.188472	−	0.982079i	$-0.439647\pi$
$157$	1.82778e7	0.376943	0.188472	+	0.982079i	$0.439647\pi$
$158$	−2.86586e6	−0.0578036
$159$	−4.34319e7	−0.856878
$160$	−2.39290e7	−0.461854
$161$	1.82282e6	0.0344234
$162$	−1.72454e6	−0.0318691
$163$	4.46297e7	0.807174	0.403587	−	0.914941i	$-0.367763\pi$
$163$	4.46297e7	0.807174	0.403587	+	0.914941i	$0.367763\pi$
$164$	−1.27916e7	−0.226449
$165$	84162.9	0.00145857
$166$	1.32305e7	0.224490
$167$	−9.93656e7	−1.65093	−0.825465	−	0.564453i	$-0.809087\pi$
$167$	−9.93656e7	−1.65093	−0.825465	+	0.564453i	$0.809087\pi$
$168$	−3.22211e6	−0.0524273
$169$	4.12995e7	0.658175
$170$	1.41401e7	0.220740
$171$	−1.17854e7	−0.180243
$172$	9.29828e7	1.39333
$173$	7.06790e7	1.03784	0.518919	−	0.854824i	$-0.326335\pi$
$173$	7.06790e7	1.03784	0.518919	+	0.854824i	$0.326335\pi$
$174$	−8.13822e6	−0.117114
$175$	−7.47181e6	−0.105388
$176$	−230912.	−0.00319266
$177$	−1.98088e7	−0.268505
$178$	−1.64687e7	−0.218872
$179$	5.86892e7	0.764844	0.382422	−	0.923988i	$-0.375090\pi$
$179$	5.86892e7	0.764844	0.382422	+	0.923988i	$0.375090\pi$
$180$	−1.43939e7	−0.183961
$181$	−7.29280e7	−0.914154	−0.457077	−	0.889427i	$-0.651104\pi$
$181$	−7.29280e7	−0.914154	−0.457077	+	0.889427i	$0.651104\pi$
$182$	−4.95901e6	−0.0609741
$183$	−3.08044e6	−0.0371564
$184$	9.69168e6	0.114693
$185$	−1.28755e7	−0.149507
$186$	−4.57616e6	−0.0521441
$187$	480775.	0.00537646
$188$	1.16598e8	1.27979
$189$	−2.94884e6	−0.0317713
$190$	8.81778e6	0.0932656
$191$	1.28471e8	1.33410	0.667049	−	0.745014i	$-0.267557\pi$
$191$	1.28471e8	1.33410	0.667049	+	0.745014i	$0.267557\pi$
$192$	3.05584e7	0.311585
$193$	1.81759e8	1.81989	0.909945	−	0.414730i	$-0.136124\pi$
$193$	1.81759e8	1.81989	0.909945	+	0.414730i	$0.136124\pi$
$194$	3.99878e6	0.0393207
$195$	−4.62919e7	−0.447079
$196$	9.41048e7	0.892721
$197$	5.33642e7	0.497300	0.248650	−	0.968593i	$-0.420013\pi$
$197$	5.33642e7	0.497300	0.248650	+	0.968593i	$0.420013\pi$
$198$	43871.0	0.000401651 0
$199$	4.64692e7	0.418003	0.209002	−	0.977915i	$-0.432979\pi$
$199$	4.64692e7	0.418003	0.209002	+	0.977915i	$0.432979\pi$
$200$	−3.97266e7	−0.351137
$201$	−1.20741e7	−0.104874
$202$	−1.51014e7	−0.128910
$203$	−1.39158e7	−0.116754
$204$	−8.22243e7	−0.678101
$205$	1.83030e7	0.148383
$206$	5.21695e7	0.415797
$207$	8.86974e6	0.0695048
$208$	1.27008e8	0.978611
$209$	299812.	0.00227163
$210$	2.20631e6	0.0164399
$211$	−3.75261e7	−0.275008	−0.137504	−	0.990501i	$-0.543908\pi$
$211$	−3.75261e7	−0.275008	−0.137504	+	0.990501i	$0.543908\pi$
$212$	−1.88961e8	−1.36206
$213$	1.81219e7	0.128491
$214$	1.91676e7	0.133697
$215$	−1.33046e8	−0.912991
$216$	−1.56786e7	−0.105857
$217$	−7.82492e6	−0.0519841
$218$	7.85701e7	0.513641
$219$	1.09672e7	0.0705571
$220$	366170.	0.00231848
$221$	−2.64440e8	−1.64799
$222$	−6.71151e6	−0.0411704
$223$	−1.04665e8	−0.632028	−0.316014	−	0.948755i	$-0.602345\pi$
$223$	−1.04665e8	−0.632028	−0.316014	+	0.948755i	$0.602345\pi$
$224$	−2.13285e7	−0.126792
$225$	−3.63574e7	−0.212791
$226$	4.13100e7	0.238054
$227$	−2.30736e7	−0.130926	−0.0654629	−	0.997855i	$-0.520852\pi$
$227$	−2.30736e7	−0.130926	−0.0654629	+	0.997855i	$0.520852\pi$
$228$	−5.12751e7	−0.286507
$229$	2.26806e8	1.24804	0.624022	−	0.781407i	$-0.285497\pi$
$229$	2.26806e8	1.24804	0.624022	+	0.781407i	$0.285497\pi$
$230$	−6.63630e6	−0.0359649
$231$	75016.5	0.000400419 0
$232$	−7.39884e7	−0.389006
$233$	−2.91144e8	−1.50787	−0.753933	−	0.656952i	$-0.771846\pi$
$233$	−2.91144e8	−1.50787	−0.753933	+	0.656952i	$0.771846\pi$
$234$	−2.41303e7	−0.123114
$235$	−1.66835e8	−0.838592
$236$	−8.61829e7	−0.426804
$237$	−2.38452e7	−0.116354
$238$	1.26034e7	0.0605995
$239$	1.45226e8	0.688098	0.344049	−	0.938952i	$-0.388201\pi$
$239$	1.45226e8	0.688098	0.344049	+	0.938952i	$0.388201\pi$
$240$	−5.65071e7	−0.263854
$241$	−2.91763e8	−1.34267	−0.671336	−	0.741153i	$-0.734279\pi$
$241$	−2.91763e8	−1.34267	−0.671336	+	0.741153i	$0.734279\pi$
$242$	6.32352e7	0.286817
$243$	−1.43489e7	−0.0641500
$244$	−1.34022e7	−0.0590624
$245$	−1.34651e8	−0.584964
$246$	9.54067e6	0.0408607
$247$	−1.64905e8	−0.696297
$248$	−4.16040e7	−0.173203
$249$	1.10083e8	0.451880
$250$	6.98145e7	0.282589
$251$	−3.16146e7	−0.126191	−0.0630957	−	0.998007i	$-0.520097\pi$
$251$	−3.16146e7	−0.126191	−0.0630957	+	0.998007i	$0.520097\pi$
$252$	−1.28296e7	−0.0505025
$253$	−225640.	−0.000875979 0
$254$	1.39151e8	0.532805
$255$	1.17652e8	0.444332
$256$	7.38189e7	0.274997
$257$	4.73473e6	0.0173992	0.00869960	−	0.999962i	$-0.497231\pi$
$257$	4.73473e6	0.0173992	0.00869960	+	0.999962i	$0.497231\pi$
$258$	−6.93518e7	−0.251414
$259$	−1.14762e7	−0.0410441
$260$	−2.01404e8	−0.710659
$261$	−6.77135e7	−0.235740
$262$	5.06893e7	0.174125
$263$	6.07617e7	0.205961	0.102980	−	0.994683i	$-0.467162\pi$
$263$	6.07617e7	0.205961	0.102980	+	0.994683i	$0.467162\pi$
$264$	398852.	0.00133413
$265$	2.70377e8	0.892502
$266$	7.85950e6	0.0256041
$267$	−1.37027e8	−0.440571
$268$	−5.25312e7	−0.166704
$269$	−8.38261e7	−0.262571	−0.131285	−	0.991345i	$-0.541910\pi$
$269$	−8.38261e7	−0.262571	−0.131285	+	0.991345i	$0.541910\pi$
$270$	1.07358e7	0.0331941
$271$	2.81295e8	0.858558	0.429279	−	0.903172i	$-0.358768\pi$
$271$	2.81295e8	0.858558	0.429279	+	0.903172i	$0.358768\pi$
$272$	−3.22793e8	−0.972598
$273$	−4.12611e7	−0.122736
$274$	−9.55940e7	−0.280740
$275$	924906.	0.00268184
$276$	3.85899e7	0.110482
$277$	3.90136e8	1.10290	0.551451	−	0.834207i	$-0.314074\pi$
$277$	3.90136e8	1.10290	0.551451	+	0.834207i	$0.314074\pi$
$278$	−8.79740e7	−0.245583
$279$	−3.80756e7	−0.104962
$280$	2.00586e7	0.0546069
$281$	2.37822e8	0.639410	0.319705	−	0.947517i	$-0.396416\pi$
$281$	2.37822e8	0.639410	0.319705	+	0.947517i	$0.396416\pi$
$282$	−8.69651e7	−0.230926
$283$	1.71342e8	0.449378	0.224689	−	0.974431i	$-0.427863\pi$
$283$	1.71342e8	0.449378	0.224689	+	0.974431i	$0.427863\pi$
$284$	7.88434e7	0.204245
$285$	7.33678e7	0.187736
$286$	613856.	0.00155162
$287$	1.63139e7	0.0407353
$288$	−1.03783e8	−0.256008
$289$	2.61739e8	0.637862
$290$	5.06629e7	0.121982
$291$	3.32716e7	0.0791495
$292$	4.77153e7	0.112155
$293$	1.93284e8	0.448909	0.224455	−	0.974485i	$-0.427940\pi$
$293$	1.93284e8	0.448909	0.224455	+	0.974485i	$0.427940\pi$
$294$	−7.01887e7	−0.161084
$295$	1.23316e8	0.279668
$296$	−6.10176e7	−0.136752
$297$	365026.	0.000808492 0
$298$	−7.48291e7	−0.163800
$299$	1.24108e8	0.268504
$300$	−1.58181e8	−0.338245
$301$	−1.18587e8	−0.250642
$302$	1.83658e8	0.383694
$303$	−1.25650e8	−0.259485
$304$	−2.01294e8	−0.410936
$305$	1.91767e7	0.0387012
$306$	6.13275e7	0.122357
$307$	−4.20067e8	−0.828579	−0.414290	−	0.910145i	$-0.635970\pi$
$307$	−4.20067e8	−0.828579	−0.414290	+	0.910145i	$0.635970\pi$
$308$	326377.	0.000636490 0
$309$	4.34073e8	0.836966
$310$	2.84880e7	0.0543120
$311$	−4.75218e8	−0.895842	−0.447921	−	0.894073i	$-0.647835\pi$
$311$	−4.75218e8	−0.895842	−0.447921	+	0.894073i	$0.647835\pi$
$312$	−2.19380e8	−0.408936
$313$	8.20300e8	1.51205	0.756027	−	0.654540i	$-0.227138\pi$
$313$	8.20300e8	1.51205	0.756027	+	0.654540i	$0.227138\pi$
$314$	−5.93120e7	−0.108116
$315$	1.83575e7	0.0330922
$316$	−1.03744e8	−0.184952
$317$	9.09355e8	1.60334	0.801671	−	0.597765i	$-0.203945\pi$
$317$	9.09355e8	1.60334	0.801671	+	0.597765i	$0.203945\pi$
$318$	1.40938e8	0.245772
$319$	1.72258e6	0.00297107
$320$	−1.90236e8	−0.324539
$321$	1.59483e8	0.269121
$322$	−5.91510e6	−0.00987339
$323$	4.19109e8	0.692019
$324$	−6.24283e7	−0.101970
$325$	−5.08724e8	−0.822036
$326$	−1.44824e8	−0.231515
$327$	6.53737e8	1.03392
$328$	8.67387e7	0.135723
$329$	−1.48705e8	−0.230218
$330$	−273110.	−0.000418350 0
$331$	−2.32731e8	−0.352741	−0.176370	−	0.984324i	$-0.556436\pi$
$331$	−2.32731e8	−0.352741	−0.176370	+	0.984324i	$0.556436\pi$
$332$	4.78943e8	0.718291
$333$	−5.58427e7	−0.0828728
$334$	3.22444e8	0.473523
$335$	7.51650e7	0.109234
$336$	−5.03662e7	−0.0724355
$337$	2.05391e8	0.292332	0.146166	−	0.989260i	$-0.453307\pi$
$337$	2.05391e8	0.292332	0.146166	+	0.989260i	$0.453307\pi$
$338$	−1.34018e8	−0.188779
$339$	3.43718e8	0.479185
$340$	5.11871e8	0.706293
$341$	968616.	0.00132285
$342$	3.82439e7	0.0516976
$343$	−2.43399e8	−0.325678
$344$	−6.30510e8	−0.835098
$345$	−5.52169e7	−0.0723944
$346$	−2.29355e8	−0.297675
$347$	4.46363e8	0.573502	0.286751	−	0.958005i	$-0.407425\pi$
$347$	4.46363e8	0.573502	0.286751	+	0.958005i	$0.407425\pi$
$348$	−2.94604e8	−0.374723
$349$	−7.98585e8	−1.00562	−0.502808	−	0.864398i	$-0.667700\pi$
$349$	−7.98585e8	−1.00562	−0.502808	+	0.864398i	$0.667700\pi$
$350$	2.42462e7	0.0302277
$351$	−2.00774e8	−0.247818
$352$	2.64017e6	0.00322651
$353$	1.49174e9	1.80502	0.902510	−	0.430670i	$-0.141723\pi$
$353$	1.49174e9	1.80502	0.902510	+	0.430670i	$0.141723\pi$
$354$	6.42800e7	0.0770131
$355$	−1.12814e8	−0.133833
$356$	−5.96167e8	−0.700314
$357$	1.04866e8	0.121982
$358$	−1.90448e8	−0.219374
$359$	9.03992e8	1.03118	0.515590	−	0.856836i	$-0.327573\pi$
$359$	9.03992e8	1.03118	0.515590	+	0.856836i	$0.327573\pi$
$360$	9.76041e7	0.110258
$361$	−6.32515e8	−0.707613
$362$	2.36653e8	0.262199
$363$	5.26144e8	0.577340
$364$	−1.79516e8	−0.195096
$365$	−6.82741e7	−0.0734905
$366$	9.99610e6	0.0106573
$367$	−5.78376e8	−0.610772	−0.305386	−	0.952229i	$-0.598786\pi$
$367$	−5.78376e8	−0.610772	−0.305386	+	0.952229i	$0.598786\pi$
$368$	1.51495e8	0.158464
$369$	7.93825e7	0.0822494
$370$	4.17813e7	0.0428820
$371$	2.40994e8	0.245018
$372$	−1.65657e8	−0.166843
$373$	−9.27782e8	−0.925689	−0.462844	−	0.886440i	$-0.653171\pi$
$373$	−9.27782e8	−0.925689	−0.462844	+	0.886440i	$0.653171\pi$
$374$	−1.56013e6	−0.00154209
$375$	5.80887e8	0.568830
$376$	−7.90641e8	−0.767047
$377$	−9.47469e8	−0.910689
$378$	9.56907e6	0.00911272
$379$	−1.54337e9	−1.45624	−0.728119	−	0.685451i	$-0.759605\pi$
$379$	−1.54337e9	−1.45624	−0.728119	+	0.685451i	$0.759605\pi$
$380$	3.19204e8	0.298418
$381$	1.15780e9	1.07250
$382$	−4.16890e8	−0.382649
$383$	1.01378e8	0.0922040	0.0461020	−	0.998937i	$-0.485320\pi$
$383$	1.01378e8	0.0922040	0.0461020	+	0.998937i	$0.485320\pi$
$384$	−5.91172e8	−0.532788
$385$	−467001.	−0.000417066 0
$386$	−5.89811e8	−0.521985
$387$	−5.77037e8	−0.506076
$388$	1.44756e8	0.125813
$389$	−6.88033e8	−0.592633	−0.296316	−	0.955090i	$-0.595758\pi$
$389$	−6.88033e8	−0.592633	−0.296316	+	0.955090i	$0.595758\pi$
$390$	1.50218e8	0.128232
$391$	−3.15423e8	−0.266855
$392$	−6.38119e8	−0.535057
$393$	4.21757e8	0.350500
$394$	−1.73168e8	−0.142637
$395$	1.48444e8	0.121191
$396$	1.58813e6	0.00128515
$397$	5.50236e7	0.0441349	0.0220674	−	0.999756i	$-0.492975\pi$
$397$	5.50236e7	0.0441349	0.0220674	+	0.999756i	$0.492975\pi$
$398$	−1.50794e8	−0.119893
$399$	6.53945e7	0.0515390
$400$	−6.20984e8	−0.485143
$401$	1.90663e9	1.47660	0.738299	−	0.674474i	$-0.235630\pi$
$401$	1.90663e9	1.47660	0.738299	+	0.674474i	$0.235630\pi$
$402$	3.91807e7	0.0300802
$403$	−5.32766e8	−0.405479
$404$	−5.46669e8	−0.412468
$405$	8.93264e7	0.0668170
$406$	4.51571e7	0.0334877
$407$	1.42060e6	0.00104446
$408$	5.57557e8	0.406424
$409$	−1.37032e9	−0.990351	−0.495175	−	0.868793i	$-0.664896\pi$
$409$	−1.37032e9	−0.990351	−0.495175	+	0.868793i	$0.664896\pi$
$410$	−5.93936e7	−0.0425595
$411$	−7.95384e8	−0.565106
$412$	1.88854e9	1.33041
$413$	1.09915e8	0.0767768
$414$	−2.87825e7	−0.0199355
$415$	−6.85302e8	−0.470667
$416$	−1.45217e9	−0.988985
$417$	−7.31982e8	−0.494339
$418$	−972897.	−0.000651553 0
$419$	−2.11477e9	−1.40447	−0.702237	−	0.711943i	$-0.747815\pi$
$419$	−2.11477e9	−1.40447	−0.702237	+	0.711943i	$0.747815\pi$
$420$	7.98685e7	0.0526021
$421$	−1.59467e9	−1.04156	−0.520780	−	0.853691i	$-0.674359\pi$
$421$	−1.59467e9	−1.04156	−0.520780	+	0.853691i	$0.674359\pi$
$422$	1.21773e8	0.0788783
$423$	−7.23588e8	−0.464836
$424$	1.28133e9	0.816358
$425$	1.29293e9	0.816985
$426$	−5.88059e7	−0.0368542
$427$	1.70927e7	0.0106246
$428$	6.93869e8	0.427784
$429$	5.10755e6	0.00312329
$430$	4.31736e8	0.261866
$431$	−7.74220e6	−0.00465794	−0.00232897	−	0.999997i	$-0.500741\pi$
$431$	−7.74220e6	−0.00465794	−0.00232897	+	0.999997i	$0.500741\pi$
$432$	−2.45079e8	−0.146256
$433$	−7.28017e8	−0.430957	−0.215478	−	0.976509i	$-0.569131\pi$
$433$	−7.28017e8	−0.430957	−0.215478	+	0.976509i	$0.569131\pi$
$434$	2.53920e7	0.0149102
$435$	4.21538e8	0.245541
$436$	2.84424e9	1.64347
$437$	−1.96698e8	−0.112750
$438$	−3.55888e7	−0.0202373
$439$	−5.29464e8	−0.298683	−0.149341	−	0.988786i	$-0.547715\pi$
$439$	−5.29464e8	−0.298683	−0.149341	+	0.988786i	$0.547715\pi$
$440$	−2.48298e6	−0.00138960
$441$	−5.84000e8	−0.324249
$442$	8.58113e8	0.472679
$443$	2.17075e9	1.18631	0.593154	−	0.805089i	$-0.297882\pi$
$443$	2.17075e9	1.18631	0.593154	+	0.805089i	$0.297882\pi$
$444$	−2.42957e8	−0.131731
$445$	8.53034e8	0.458888
$446$	3.39642e8	0.181280
$447$	−6.22610e8	−0.329716
$448$	−1.69562e8	−0.0890952
$449$	2.08538e9	1.08723	0.543616	−	0.839334i	$-0.317055\pi$
$449$	2.08538e9	1.08723	0.543616	+	0.839334i	$0.317055\pi$
$450$	1.17981e8	0.0610333
$451$	−2.01943e6	−0.00103660
$452$	1.49542e9	0.761693
$453$	1.52811e9	0.772346
$454$	7.48744e7	0.0375524
$455$	2.56863e8	0.127839
$456$	3.47693e8	0.171719
$457$	2.95584e9	1.44868	0.724342	−	0.689440i	$-0.242143\pi$
$457$	2.95584e9	1.44868	0.724342	+	0.689440i	$0.242143\pi$
$458$	−7.35990e8	−0.357967
$459$	5.10271e8	0.246296
$460$	−2.40234e8	−0.115075
$461$	−2.49440e9	−1.18581	−0.592903	−	0.805274i	$-0.702018\pi$
$461$	−2.49440e9	−1.18581	−0.592903	+	0.805274i	$0.702018\pi$
$462$	−243430.	−0.000114849 0
$463$	−3.46592e9	−1.62288	−0.811438	−	0.584439i	$-0.801315\pi$
$463$	−3.46592e9	−1.62288	−0.811438	+	0.584439i	$0.801315\pi$
$464$	−1.15655e9	−0.537464
$465$	2.37032e8	0.109326
$466$	9.44770e8	0.432489
$467$	−3.83153e9	−1.74086	−0.870428	−	0.492296i	$-0.836158\pi$
$467$	−3.83153e9	−1.74086	−0.870428	+	0.492296i	$0.836158\pi$
$468$	−8.73516e8	−0.393922
$469$	6.69964e7	0.0299879
$470$	5.41385e8	0.240527
$471$	−4.93502e8	−0.217628
$472$	5.84400e8	0.255808
$473$	1.46794e7	0.00637815
$474$	7.73782e7	0.0333729
$475$	8.06273e8	0.345187
$476$	4.56244e8	0.193898
$477$	1.17266e9	0.494719
$478$	−4.71260e8	−0.197362
$479$	−2.73741e9	−1.13806	−0.569030	−	0.822317i	$-0.692681\pi$
$479$	−2.73741e9	−1.13806	−0.569030	+	0.822317i	$0.692681\pi$
$480$	6.46082e8	0.266651
$481$	−7.81369e8	−0.320146
$482$	9.46777e8	0.385108
$483$	−4.92162e7	−0.0198743
$484$	2.28911e9	0.917717
$485$	−2.07126e8	−0.0824400
$486$	4.65625e7	0.0183996
$487$	−9.56818e8	−0.375386	−0.187693	−	0.982228i	$-0.560101\pi$
$487$	−9.56818e8	−0.375386	−0.187693	+	0.982228i	$0.560101\pi$
$488$	9.08793e7	0.0353994
$489$	−1.20500e9	−0.466022
$490$	4.36946e8	0.167781
$491$	2.40556e9	0.917132	0.458566	−	0.888660i	$-0.348363\pi$
$491$	2.40556e9	0.917132	0.458566	+	0.888660i	$0.348363\pi$
$492$	3.45372e8	0.130740
$493$	2.40801e9	0.905094
$494$	5.35120e8	0.199713
$495$	−2.27240e6	−0.000842105 0
$496$	−6.50330e8	−0.239303
$497$	−1.00554e8	−0.0367411
$498$	−3.57223e8	−0.129609
$499$	1.17148e9	0.422068	0.211034	−	0.977479i	$-0.432317\pi$
$499$	1.17148e9	0.422068	0.211034	+	0.977479i	$0.432317\pi$
$500$	2.52728e9	0.904189
$501$	2.68287e9	0.953165
$502$	1.02590e8	0.0361945
$503$	−1.89152e9	−0.662710	−0.331355	−	0.943506i	$-0.607506\pi$
$503$	−1.89152e9	−0.662710	−0.331355	+	0.943506i	$0.607506\pi$
$504$	8.69969e7	0.0302689
$505$	7.82209e8	0.270273
$506$	732206.	0.000251250 0
$507$	−1.11509e9	−0.379998
$508$	5.03727e9	1.70480
$509$	2.84363e9	0.955786	0.477893	−	0.878418i	$-0.341401\pi$
$509$	2.84363e9	0.955786	0.477893	+	0.878418i	$0.341401\pi$
$510$	−3.81782e8	−0.127444
$511$	−6.08544e7	−0.0201752
$512$	−3.04214e9	−1.00169
$513$	3.18206e8	0.104063
$514$	−1.53643e7	−0.00499047
$515$	−2.70224e9	−0.871763
$516$	−2.51054e9	−0.804438
$517$	1.84075e7	0.00585840
$518$	3.72407e7	0.0117724
$519$	−1.90833e9	−0.599196
$520$	1.36571e9	0.425937
$521$	1.32906e9	0.411731	0.205865	−	0.978580i	$-0.433999\pi$
$521$	1.32906e9	0.411731	0.205865	+	0.978580i	$0.433999\pi$
$522$	2.19732e8	0.0676155
$523$	2.97968e9	0.910779	0.455390	−	0.890292i	$-0.349500\pi$
$523$	2.97968e9	0.910779	0.455390	+	0.890292i	$0.349500\pi$
$524$	1.83495e9	0.557141
$525$	2.01739e8	0.0608460
$526$	−1.97173e8	−0.0590741
$527$	1.35403e9	0.402988
$528$	6.23463e6	0.00184328
$529$	1.48036e8	0.0434783
$530$	−8.77380e8	−0.255989
$531$	5.34838e8	0.155021
$532$	2.84514e8	0.0819244
$533$	1.11074e9	0.317738
$534$	4.44655e8	0.126366
$535$	−9.92832e8	−0.280309
$536$	3.56210e8	0.0999148
$537$	−1.58461e9	−0.441583
$538$	2.72017e8	0.0753111
$539$	1.48565e7	0.00408655
$540$	3.88635e8	0.106210
$541$	−2.27719e9	−0.618313	−0.309156	−	0.951011i	$-0.600047\pi$
$541$	−2.27719e9	−0.618313	−0.309156	+	0.951011i	$0.600047\pi$
$542$	−9.12809e8	−0.246253
$543$	1.96906e9	0.527787
$544$	3.69071e9	0.982909
$545$	−4.06972e9	−1.07690
$546$	1.33893e8	0.0352034
$547$	2.31024e9	0.603534	0.301767	−	0.953382i	$-0.402424\pi$
$547$	2.31024e9	0.603534	0.301767	+	0.953382i	$0.402424\pi$
$548$	−3.46050e9	−0.898271
$549$	8.31719e7	0.0214523
$550$	−3.00134e6	−0.000769212 0
$551$	1.50164e9	0.382414
$552$	−2.61675e8	−0.0662180
$553$	1.32312e8	0.0332705
$554$	−1.26600e9	−0.316337
$555$	3.47638e8	0.0863181
$556$	−3.18466e9	−0.785781
$557$	−2.07806e9	−0.509523	−0.254762	−	0.967004i	$-0.581997\pi$
$557$	−2.07806e9	−0.509523	−0.254762	+	0.967004i	$0.581997\pi$
$558$	1.23556e8	0.0301054
$559$	−8.07408e9	−1.95502
$560$	3.13545e8	0.0754470
$561$	−1.29809e7	−0.00310410
$562$	−7.71737e8	−0.183397
$563$	8.30273e9	1.96084	0.980418	−	0.196925i	$-0.0630957\pi$
$563$	8.30273e9	1.96084	0.980418	+	0.196925i	$0.0630957\pi$
$564$	−3.14814e9	−0.738885
$565$	−2.13975e9	−0.499106
$566$	−5.56009e8	−0.128892
$567$	7.96188e7	0.0183432
$568$	−5.34632e8	−0.122415
$569$	−2.36999e9	−0.539329	−0.269664	−	0.962954i	$-0.586913\pi$
$569$	−2.36999e9	−0.539329	−0.269664	+	0.962954i	$0.586913\pi$
$570$	−2.38080e8	−0.0538469
$571$	−2.85722e9	−0.642270	−0.321135	−	0.947033i	$-0.604064\pi$
$571$	−2.85722e9	−0.642270	−0.321135	+	0.947033i	$0.604064\pi$
$572$	2.22216e7	0.00496466
$573$	−3.46871e9	−0.770241
$574$	−5.29390e7	−0.0116838
$575$	−6.06805e8	−0.133110
$576$	−8.25077e8	−0.179894
$577$	5.93632e9	1.28648	0.643238	−	0.765666i	$-0.277590\pi$
$577$	5.93632e9	1.28648	0.643238	+	0.765666i	$0.277590\pi$
$578$	−8.49350e8	−0.182953
$579$	−4.90749e9	−1.05071
$580$	1.83400e9	0.390302
$581$	−6.10827e8	−0.129212
$582$	−1.07967e8	−0.0227018
$583$	−2.98317e7	−0.00623501
$584$	−3.23554e8	−0.0672206
$585$	1.24988e9	0.258121
$586$	−6.27210e8	−0.128757
$587$	3.99412e8	0.0815057	0.0407529	−	0.999169i	$-0.487024\pi$
$587$	3.99412e8	0.0815057	0.0407529	+	0.999169i	$0.487024\pi$
$588$	−2.54083e9	−0.515413
$589$	8.44376e8	0.170268
$590$	−4.00163e8	−0.0802149
$591$	−1.44083e9	−0.287116
$592$	−9.53792e8	−0.188942
$593$	4.93660e9	0.972157	0.486078	−	0.873915i	$-0.338427\pi$
$593$	4.93660e9	0.972157	0.486078	+	0.873915i	$0.338427\pi$
$594$	−1.18452e6	−0.000231894 0
$595$	−6.52823e8	−0.127053
$596$	−2.70881e9	−0.524103
$597$	−1.25467e9	−0.241334
$598$	−4.02734e8	−0.0770130
$599$	−6.69136e9	−1.27210	−0.636049	−	0.771649i	$-0.719432\pi$
$599$	−6.69136e9	−1.27210	−0.636049	+	0.771649i	$0.719432\pi$
$600$	1.07262e9	0.202729
$601$	−2.24773e9	−0.422361	−0.211181	−	0.977447i	$-0.567731\pi$
$601$	−2.24773e9	−0.422361	−0.211181	+	0.977447i	$0.567731\pi$
$602$	3.84817e8	0.0718897
$603$	3.26001e8	0.0605491
$604$	6.64841e9	1.22769
$605$	−3.27541e9	−0.601343
$606$	4.07737e8	0.0744262
$607$	8.07058e9	1.46468	0.732342	−	0.680937i	$-0.238427\pi$
$607$	8.07058e9	1.46468	0.732342	+	0.680937i	$0.238427\pi$
$608$	2.30153e9	0.415292
$609$	3.75727e8	0.0674080
$610$	−6.22288e7	−0.0111004
$611$	−1.01247e10	−1.79571
$612$	2.22006e9	0.391502
$613$	3.17148e9	0.556097	0.278048	−	0.960567i	$-0.410312\pi$
$613$	3.17148e9	0.556097	0.278048	+	0.960567i	$0.410312\pi$
$614$	1.36313e9	0.237655
$615$	−4.94181e8	−0.0856688
$616$	−2.21314e6	−0.000381484 0
$617$	−3.21472e9	−0.550993	−0.275496	−	0.961302i	$-0.588842\pi$
$617$	−3.21472e9	−0.550993	−0.275496	+	0.961302i	$0.588842\pi$
$618$	−1.40858e9	−0.240060
$619$	−3.88507e9	−0.658388	−0.329194	−	0.944262i	$-0.606777\pi$
$619$	−3.88507e9	−0.658388	−0.329194	+	0.944262i	$0.606777\pi$
$620$	1.03127e9	0.173780
$621$	−2.39483e8	−0.0401286
$622$	1.54209e9	0.256947
$623$	7.60330e8	0.125978
$624$	−3.42922e9	−0.565001
$625$	2.80127e8	0.0458960
$626$	−2.66189e9	−0.433691
$627$	−8.09492e6	−0.00131152
$628$	−2.14709e9	−0.345933
$629$	1.98586e9	0.318179
$630$	−5.95704e7	−0.00949158
$631$	−3.88896e9	−0.616213	−0.308107	−	0.951352i	$-0.599695\pi$
$631$	−3.88896e9	−0.616213	−0.308107	+	0.951352i	$0.599695\pi$
$632$	7.03482e8	0.110852
$633$	1.01321e9	0.158776
$634$	−2.95088e9	−0.459874
$635$	−7.20765e9	−1.11708
$636$	5.10194e9	0.786385
$637$	−8.17151e9	−1.25261
$638$	−5.58982e6	−0.000852168 0
$639$	−4.89290e8	−0.0741846
$640$	3.68023e9	0.554938
$641$	1.29473e10	1.94168	0.970839	−	0.239731i	$-0.0770593\pi$
$641$	1.29473e10	1.94168	0.970839	+	0.239731i	$0.0770593\pi$
$642$	−5.17526e8	−0.0771898
$643$	1.18526e10	1.75822	0.879112	−	0.476615i	$-0.158136\pi$
$643$	1.18526e10	1.75822	0.879112	+	0.476615i	$0.158136\pi$
$644$	−2.14127e8	−0.0315915
$645$	3.59223e9	0.527115
$646$	−1.36002e9	−0.198486
$647$	−8.17682e9	−1.18691	−0.593457	−	0.804866i	$-0.702237\pi$
$647$	−8.17682e9	−1.18691	−0.593457	+	0.804866i	$0.702237\pi$
$648$	4.23322e8	0.0611165
$649$	−1.36059e7	−0.00195376
$650$	1.65082e9	0.235778
$651$	2.11273e8	0.0300131
$652$	−5.24264e9	−0.740770
$653$	1.45664e9	0.204718	0.102359	−	0.994748i	$-0.467361\pi$
$653$	1.45664e9	0.204718	0.102359	+	0.994748i	$0.467361\pi$
$654$	−2.12139e9	−0.296551
$655$	−2.62557e9	−0.365072
$656$	1.35585e9	0.187520
$657$	−2.96114e8	−0.0407362
$658$	4.82550e8	0.0660315
$659$	−7.97853e8	−0.108598	−0.0542992	−	0.998525i	$-0.517292\pi$
$659$	−7.97853e8	−0.108598	−0.0542992	+	0.998525i	$0.517292\pi$
$660$	−9.88660e6	−0.00133858
$661$	1.56979e8	0.0211416	0.0105708	−	0.999944i	$-0.496635\pi$
$661$	1.56979e8	0.0211416	0.0105708	+	0.999944i	$0.496635\pi$
$662$	7.55216e8	0.101174
$663$	7.13988e9	0.951466
$664$	−3.24768e9	−0.430512
$665$	−4.07101e8	−0.0536817
$666$	1.81211e8	0.0237697
$667$	−1.13014e9	−0.147466
$668$	1.16725e10	1.51511
$669$	2.82597e9	0.364901
$670$	−2.43912e8	−0.0313308
$671$	−2.11583e6	−0.000270366 0
$672$	5.75869e8	0.0732034
$673$	1.16317e10	1.47093	0.735463	−	0.677565i	$-0.236965\pi$
$673$	1.16317e10	1.47093	0.735463	+	0.677565i	$0.236965\pi$
$674$	−6.66498e8	−0.0838473
$675$	9.81650e8	0.122855
$676$	−4.85145e9	−0.604029
$677$	1.12531e10	1.39383	0.696917	−	0.717152i	$-0.254555\pi$
$677$	1.12531e10	1.39383	0.696917	+	0.717152i	$0.254555\pi$
$678$	−1.11537e9	−0.137441
$679$	−1.84616e8	−0.0226322
$680$	−3.47097e9	−0.423320
$681$	6.22988e8	0.0755900
$682$	−3.14318e6	−0.000379423 0
$683$	−2.86773e9	−0.344403	−0.172201	−	0.985062i	$-0.555088\pi$
$683$	−2.86773e9	−0.344403	−0.172201	+	0.985062i	$0.555088\pi$
$684$	1.38443e9	0.165415
$685$	4.95151e9	0.588600
$686$	7.89834e8	0.0934117
$687$	−6.12376e9	−0.720559
$688$	−9.85578e9	−1.15380
$689$	1.64082e10	1.91115
$690$	1.79180e8	0.0207643
$691$	−6.87429e9	−0.792602	−0.396301	−	0.918121i	$-0.629706\pi$
$691$	−6.87429e9	−0.792602	−0.396301	+	0.918121i	$0.629706\pi$
$692$	−8.30265e9	−0.952458
$693$	−2.02544e6	−0.000231182 0
$694$	−1.44846e9	−0.164493
$695$	4.55682e9	0.514890
$696$	1.99769e9	0.224592
$697$	−2.82298e9	−0.315786
$698$	2.59142e9	0.288433
$699$	7.86090e9	0.870566
$700$	8.77713e8	0.0967184
$701$	1.65668e10	1.81646	0.908231	−	0.418468i	$-0.137433\pi$
$701$	1.65668e10	1.81646	0.908231	+	0.418468i	$0.137433\pi$
$702$	6.51517e8	0.0710798
$703$	1.23839e9	0.134435
$704$	2.09894e7	0.00226723
$705$	4.50456e9	0.484161
$706$	−4.84073e9	−0.517719
$707$	6.97203e8	0.0741978
$708$	2.32694e9	0.246416
$709$	8.01915e9	0.845019	0.422510	−	0.906358i	$-0.361149\pi$
$709$	8.01915e9	0.845019	0.422510	+	0.906358i	$0.361149\pi$
$710$	3.66085e8	0.0383864
$711$	6.43820e8	0.0671771
$712$	4.04257e9	0.419737
$713$	−6.35481e8	−0.0656583
$714$	−3.40292e8	−0.0349871
$715$	−3.17961e7	−0.00325314
$716$	−6.89421e9	−0.701922
$717$	−3.92109e9	−0.397274
$718$	−2.93348e9	−0.295765
$719$	1.97264e8	0.0197923	0.00989617	−	0.999951i	$-0.496850\pi$
$719$	1.97264e8	0.0197923	0.00989617	+	0.999951i	$0.496850\pi$
$720$	1.52569e9	0.152336
$721$	−2.40857e9	−0.239324
$722$	2.05253e9	0.202959
$723$	7.87759e9	0.775193
$724$	8.56684e9	0.838949
$725$	4.63248e9	0.451471
$726$	−1.70735e9	−0.165594
$727$	−4.86101e9	−0.469198	−0.234599	−	0.972092i	$-0.575378\pi$
$727$	−4.86101e9	−0.469198	−0.234599	+	0.972092i	$0.575378\pi$
$728$	1.21729e9	0.116932
$729$	3.87420e8	0.0370370
$730$	2.21551e8	0.0210787
$731$	2.05204e10	1.94301
$732$	3.61859e8	0.0340997
$733$	7.01036e9	0.657471	0.328735	−	0.944422i	$-0.393378\pi$
$733$	7.01036e9	0.657471	0.328735	+	0.944422i	$0.393378\pi$
$734$	1.87684e9	0.175183
$735$	3.63558e9	0.337729
$736$	−1.73214e9	−0.160144
$737$	−8.29322e6	−0.000763109 0
$738$	−2.57598e8	−0.0235909
$739$	−2.09461e10	−1.90918	−0.954590	−	0.297923i	$-0.903706\pi$
$739$	−2.09461e10	−1.90918	−0.954590	+	0.297923i	$0.903706\pi$
$740$	1.51248e9	0.137208
$741$	4.45243e9	0.402007
$742$	−7.82030e8	−0.0702764
$743$	−1.60185e10	−1.43271	−0.716357	−	0.697734i	$-0.754192\pi$
$743$	−1.60185e10	−1.43271	−0.716357	+	0.697734i	$0.754192\pi$
$744$	1.12331e9	0.0999985
$745$	3.87594e9	0.343424
$746$	3.01067e9	0.265508
$747$	−2.97225e9	−0.260893
$748$	−5.64766e7	−0.00493416
$749$	−8.84935e8	−0.0769529
$750$	−1.88499e9	−0.163153
$751$	−9.22632e9	−0.794856	−0.397428	−	0.917633i	$-0.630097\pi$
$751$	−9.22632e9	−0.794856	−0.397428	+	0.917633i	$0.630097\pi$
$752$	−1.23589e10	−1.05978
$753$	8.53595e8	0.0728567
$754$	3.07456e9	0.261206
$755$	−9.51298e9	−0.804456
$756$	3.46400e8	0.0291576
$757$	−1.99486e10	−1.67139	−0.835694	−	0.549195i	$-0.814934\pi$
$757$	−1.99486e10	−1.67139	−0.835694	+	0.549195i	$0.814934\pi$
$758$	5.00826e9	0.417681
$759$	6.09227e6	0.000505747 0
$760$	−2.16450e9	−0.178859
$761$	−7.42197e9	−0.610482	−0.305241	−	0.952275i	$-0.598737\pi$
$761$	−7.42197e9	−0.610482	−0.305241	+	0.952275i	$0.598737\pi$
$762$	−3.75708e9	−0.307615
$763$	−3.62744e9	−0.295641
$764$	−1.50914e10	−1.22435
$765$	−3.17660e9	−0.256535
$766$	−3.28975e8	−0.0264461
$767$	7.48362e9	0.598863
$768$	−1.99311e9	−0.158769
$769$	7.97515e9	0.632407	0.316204	−	0.948691i	$-0.397592\pi$
$769$	7.97515e9	0.632407	0.316204	+	0.948691i	$0.397592\pi$
$770$	1.51543e6	0.000119624 0
$771$	−1.27838e8	−0.0100454
$772$	−2.13512e10	−1.67017
$773$	−6.46602e9	−0.503511	−0.251756	−	0.967791i	$-0.581008\pi$
$773$	−6.46602e9	−0.503511	−0.251756	+	0.967791i	$0.581008\pi$
$774$	1.87250e9	0.145154
$775$	2.60486e9	0.201015
$776$	−9.81579e8	−0.0754066
$777$	3.09859e8	0.0236968
$778$	2.23268e9	0.169980
$779$	−1.76041e9	−0.133424
$780$	5.43791e9	0.410299
$781$	1.24472e7	0.000934959 0
$782$	1.02355e9	0.0765398
$783$	1.82827e9	0.136105
$784$	−9.97471e9	−0.739255
$785$	3.07220e9	0.226676
$786$	−1.36861e9	−0.100531
$787$	−3.77628e9	−0.276155	−0.138077	−	0.990421i	$-0.544092\pi$
$787$	−3.77628e9	−0.276155	−0.138077	+	0.990421i	$0.544092\pi$
$788$	−6.26868e9	−0.456388
$789$	−1.64057e9	−0.118912
$790$	−4.81703e8	−0.0347604
$791$	−1.90721e9	−0.137019
$792$	−1.07690e7	−0.000770260 0
$793$	1.16377e9	0.0828724
$794$	−1.78553e8	−0.0126589
$795$	−7.30018e9	−0.515286
$796$	−5.45873e9	−0.383615
$797$	1.79598e9	0.125660	0.0628300	−	0.998024i	$-0.479987\pi$
$797$	1.79598e9	0.125660	0.0628300	+	0.998024i	$0.479987\pi$
$798$	−2.12207e8	−0.0147825
$799$	2.57320e10	1.78468
$800$	7.10011e9	0.490287
$801$	3.69972e9	0.254364
$802$	−6.18707e9	−0.423521
$803$	7.53292e6	0.000513404 0
$804$	1.41834e9	0.0962465
$805$	3.06386e8	0.0207006
$806$	1.72884e9	0.116300
$807$	2.26330e9	0.151595
$808$	3.70693e9	0.247215
$809$	−1.21567e10	−0.807226	−0.403613	−	0.914930i	$-0.632246\pi$
$809$	−1.21567e10	−0.807226	−0.403613	+	0.914930i	$0.632246\pi$
$810$	−2.89866e8	−0.0191646
$811$	−1.56560e9	−0.103064	−0.0515320	−	0.998671i	$-0.516410\pi$
$811$	−1.56560e9	−0.103064	−0.0515320	+	0.998671i	$0.516410\pi$
$812$	1.63469e9	0.107149
$813$	−7.59497e9	−0.495689
$814$	−4.60987e6	−0.000299574 0
$815$	7.50151e9	0.485397
$816$	8.71542e9	0.561530
$817$	1.27966e10	0.820948
$818$	4.44670e9	0.284055
$819$	1.11405e9	0.0708617
$820$	−2.15005e9	−0.136176
$821$	−2.53876e10	−1.60110	−0.800552	−	0.599263i	$-0.795460\pi$
$821$	−2.53876e10	−1.60110	−0.800552	+	0.599263i	$0.795460\pi$
$822$	2.58104e9	0.162085
$823$	2.74439e10	1.71611	0.858056	−	0.513556i	$-0.171672\pi$
$823$	2.74439e10	1.71611	0.858056	+	0.513556i	$0.171672\pi$
$824$	−1.28060e10	−0.797388
$825$	−2.49725e7	−0.00154836
$826$	−3.56675e8	−0.0220213
$827$	−3.05050e10	−1.87543	−0.937716	−	0.347402i	$-0.887064\pi$
$827$	−3.05050e10	−1.87543	−0.937716	+	0.347402i	$0.887064\pi$
$828$	−1.04193e9	−0.0637869
$829$	1.27483e10	0.777162	0.388581	−	0.921414i	$-0.372965\pi$
$829$	1.27483e10	0.777162	0.388581	+	0.921414i	$0.372965\pi$
$830$	2.22382e9	0.134998
$831$	−1.05337e10	−0.636761
$832$	−1.15447e10	−0.694948
$833$	2.07680e10	1.24491
$834$	2.37530e9	0.141787
$835$	−1.67017e10	−0.992792
$836$	−3.52189e7	−0.00208475
$837$	1.02804e9	0.0605998
$838$	6.86247e9	0.402834
$839$	1.26629e10	0.740230	0.370115	−	0.928986i	$-0.379318\pi$
$839$	1.26629e10	0.740230	0.370115	+	0.928986i	$0.379318\pi$
$840$	−5.41583e8	−0.0315273
$841$	−8.62216e9	−0.499839
$842$	5.17474e9	0.298742
$843$	−6.42119e9	−0.369164
$844$	4.40819e9	0.252384
$845$	6.94176e9	0.395796
$846$	2.34806e9	0.133325
$847$	−2.91945e9	−0.165086
$848$	2.00290e10	1.12791
$849$	−4.62624e9	−0.259449
$850$	−4.19559e9	−0.234329
$851$	−9.32014e8	−0.0518405
$852$	−2.12877e9	−0.117921
$853$	5.68247e9	0.313484	0.156742	−	0.987640i	$-0.449901\pi$
$853$	5.68247e9	0.313484	0.156742	+	0.987640i	$0.449901\pi$
$854$	−5.54661e7	−0.00304737
$855$	−1.98093e9	−0.108390
$856$	−4.70508e9	−0.256394
$857$	1.26235e10	0.685088	0.342544	−	0.939502i	$-0.388711\pi$
$857$	1.26235e10	0.685088	0.342544	+	0.939502i	$0.388711\pi$
$858$	−1.65741e7	−0.000895829 0
$859$	−1.84247e10	−0.991803	−0.495902	−	0.868379i	$-0.665162\pi$
$859$	−1.84247e10	−0.991803	−0.495902	+	0.868379i	$0.665162\pi$
$860$	1.56289e10	0.837882
$861$	−4.40475e8	−0.0235186
$862$	2.51236e7	0.00133600
$863$	−1.37297e10	−0.727151	−0.363576	−	0.931565i	$-0.618444\pi$
$863$	−1.37297e10	−0.727151	−0.363576	+	0.931565i	$0.618444\pi$
$864$	2.80215e9	0.147806
$865$	1.18800e10	0.624107
$866$	2.36243e9	0.123608
$867$	−7.06697e9	−0.368270
$868$	9.19192e8	0.0477076
$869$	−1.63783e7	−0.000846643 0
$870$	−1.36790e9	−0.0704266
$871$	4.56150e9	0.233908
$872$	−1.92866e10	−0.985026
$873$	−8.98333e8	−0.0456970
$874$	6.38290e8	0.0323391
$875$	−3.22321e9	−0.162652
$876$	−1.28831e9	−0.0647526
$877$	−2.79796e10	−1.40069	−0.700346	−	0.713803i	$-0.746971\pi$
$877$	−2.79796e10	−1.40069	−0.700346	+	0.713803i	$0.746971\pi$
$878$	1.71812e9	0.0856689
$879$	−5.21866e9	−0.259178
$880$	−3.88125e7	−0.00191992
$881$	−2.85364e10	−1.40600	−0.702998	−	0.711192i	$-0.748156\pi$
$881$	−2.85364e10	−1.40600	−0.702998	+	0.711192i	$0.748156\pi$
$882$	1.89509e9	0.0930017
$883$	2.80960e10	1.37335	0.686676	−	0.726964i	$-0.259069\pi$
$883$	2.80960e10	1.37335	0.686676	+	0.726964i	$0.259069\pi$
$884$	3.10637e10	1.51241
$885$	−3.32953e9	−0.161466
$886$	−7.04414e9	−0.340259
$887$	−3.35283e10	−1.61317	−0.806583	−	0.591121i	$-0.798686\pi$
$887$	−3.35283e10	−1.61317	−0.806583	+	0.591121i	$0.798686\pi$
$888$	1.64747e9	0.0789538
$889$	−6.42435e9	−0.306671
$890$	−2.76811e9	−0.131619
$891$	−9.85569e6	−0.000466783 0
$892$	1.22950e10	0.580033
$893$	1.60465e10	0.754050
$894$	2.02038e9	0.0945699
$895$	9.86468e9	0.459941
$896$	3.28028e9	0.152347
$897$	−3.35092e9	−0.155021
$898$	−6.76710e9	−0.311842
$899$	4.85140e9	0.222694
$900$	4.27090e9	0.195286
$901$	−4.17018e10	−1.89941
$902$	6.55311e6	0.000297320 0
$903$	3.20185e9	0.144708
$904$	−1.01404e10	−0.456525
$905$	−1.22580e10	−0.549729
$906$	−4.95876e9	−0.221526
$907$	3.90397e10	1.73733	0.868663	−	0.495404i	$-0.164980\pi$
$907$	3.90397e10	1.73733	0.868663	+	0.495404i	$0.164980\pi$
$908$	2.71045e9	0.120155
$909$	3.39255e9	0.149814
$910$	−8.33527e8	−0.0366670
$911$	3.74924e10	1.64297	0.821483	−	0.570233i	$-0.193147\pi$
$911$	3.74924e10	1.64297	0.821483	+	0.570233i	$0.193147\pi$
$912$	5.43495e9	0.237254
$913$	7.56118e7	0.00328808
$914$	−9.59176e9	−0.415515
$915$	−5.17771e8	−0.0223441
$916$	−2.66428e10	−1.14537
$917$	−2.34023e9	−0.100223
$918$	−1.65584e9	−0.0706431
$919$	2.71337e10	1.15320	0.576600	−	0.817026i	$-0.304379\pi$
$919$	2.71337e10	1.15320	0.576600	+	0.817026i	$0.304379\pi$
$920$	1.62901e9	0.0689710
$921$	1.13418e10	0.478380
$922$	8.09440e9	0.340115
$923$	−6.84630e9	−0.286583
$924$	−8.81217e6	−0.000367478 0
$925$	3.82036e9	0.158712
$926$	1.12470e10	0.465477
$927$	−1.17200e10	−0.483223
$928$	1.32235e10	0.543162
$929$	1.17269e10	0.479875	0.239938	−	0.970788i	$-0.422873\pi$
$929$	1.17269e10	0.479875	0.239938	+	0.970788i	$0.422873\pi$
$930$	−7.69175e8	−0.0313570
$931$	1.29510e10	0.525991
$932$	3.42007e10	1.38382
$933$	1.28309e10	0.517214
$934$	1.24334e10	0.499316
$935$	8.08103e7	0.00323315
$936$	5.92325e9	0.236099
$937$	2.65223e10	1.05323	0.526615	−	0.850104i	$-0.323461\pi$
$937$	2.65223e10	1.05323	0.526615	+	0.850104i	$0.323461\pi$
$938$	−2.17405e8	−0.00860120
$939$	−2.21481e10	−0.872985
$940$	1.95981e10	0.769604
$941$	−4.09678e10	−1.60280	−0.801399	−	0.598130i	$-0.795911\pi$
$941$	−4.09678e10	−1.60280	−0.801399	+	0.598130i	$0.795911\pi$
$942$	1.60142e9	0.0624206
$943$	1.32489e9	0.0514505
$944$	9.13502e9	0.353433
$945$	−4.95652e8	−0.0191058
$946$	−4.76350e7	−0.00182939
$947$	9.37947e8	0.0358883	0.0179442	−	0.999839i	$-0.494288\pi$
$947$	9.37947e8	0.0358883	0.0179442	+	0.999839i	$0.494288\pi$
$948$	2.80109e9	0.106782
$949$	−4.14332e9	−0.157368
$950$	−2.61637e9	−0.0990073
$951$	−2.45526e10	−0.925690
$952$	−3.09376e9	−0.116214
$953$	3.41919e10	1.27967	0.639836	−	0.768512i	$-0.279002\pi$
$953$	3.41919e10	1.27967	0.639836	+	0.768512i	$0.279002\pi$
$954$	−3.80531e9	−0.141896
$955$	2.15938e10	0.802264
$956$	−1.70596e10	−0.631490
$957$	−4.65097e7	−0.00171535
$958$	8.88294e9	0.326421
$959$	4.41340e9	0.161588
$960$	5.13636e9	0.187373
$961$	−2.47847e10	−0.900847
$962$	2.53556e9	0.0918250
$963$	−4.30604e9	−0.155377
$964$	3.42733e10	1.23222
$965$	3.05506e10	1.09440
$966$	1.59708e8	0.00570040
$967$	3.92498e10	1.39587	0.697934	−	0.716162i	$-0.254103\pi$
$967$	3.92498e10	1.39587	0.697934	+	0.716162i	$0.254103\pi$
$968$	−1.55223e10	−0.550039
$969$	−1.13159e10	−0.399537
$970$	6.72128e8	0.0236456
$971$	−4.68207e10	−1.64123	−0.820617	−	0.571478i	$-0.806370\pi$
$971$	−4.68207e10	−1.64123	−0.820617	+	0.571478i	$0.806370\pi$
$972$	1.68556e9	0.0588726
$973$	4.06160e9	0.141352
$974$	3.10490e9	0.107669
$975$	1.37356e10	0.474602
$976$	1.42057e9	0.0489091
$977$	−4.91556e10	−1.68633	−0.843164	−	0.537657i	$-0.819310\pi$
$977$	−4.91556e10	−1.68633	−0.843164	+	0.537657i	$0.819310\pi$
$978$	3.91026e9	0.133665
$979$	−9.41183e7	−0.00320579
$980$	1.58175e10	0.536841
$981$	−1.76509e10	−0.596933
$982$	−7.80611e9	−0.263054
$983$	5.36541e10	1.80163	0.900814	−	0.434205i	$-0.142971\pi$
$983$	5.36541e10	1.80163	0.900814	+	0.434205i	$0.142971\pi$
$984$	−2.34195e9	−0.0783599
$985$	8.96963e9	0.299053
$986$	−7.81404e9	−0.259601
$987$	4.01502e9	0.132916
$988$	1.93714e10	0.639015
$989$	−9.63074e9	−0.316572
$990$	7.37398e6	0.000241534 0
$991$	3.66380e9	0.119584	0.0597921	−	0.998211i	$-0.480956\pi$
$991$	3.66380e9	0.119584	0.0597921	+	0.998211i	$0.480956\pi$
$992$	7.43565e9	0.241840
$993$	6.28373e9	0.203655
$994$	3.26300e8	0.0105382
$995$	7.81070e9	0.251368
$996$	−1.29315e10	−0.414706
$997$	5.69179e10	1.81893	0.909464	−	0.415783i	$-0.136492\pi$
$997$	5.69179e10	1.81893	0.909464	+	0.415783i	$0.136492\pi$
$998$	−3.80147e9	−0.121058
$999$	1.50775e9	0.0478466

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.a.1.3	✓	5
3.2	odd	2		207.8.a.a.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.a.1.3	✓	5	1.1	even	1	trivial
207.8.a.a.1.3		5	3.2	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q-3.24502 q^{2} -27.0000 q^{3} -117.470 q^{4} +168.083 q^{5} +87.6156 q^{6} +149.817 q^{7} +796.555 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-3.24502 q^{2} -27.0000 q^{3} -117.470 q^{4} +168.083 q^{5} +87.6156 q^{6} +149.817 q^{7} +796.555 q^{8} +729.000 q^{9} -545.434 q^{10} -18.5452 q^{11} +3171.69 q^{12} +10200.4 q^{13} -486.159 q^{14} -4538.25 q^{15} +12451.3 q^{16} -25924.5 q^{17} -2365.62 q^{18} -16166.5 q^{19} -19744.7 q^{20} -4045.05 q^{21} +60.1797 q^{22} +12167.0 q^{23} -21507.0 q^{24} -49873.0 q^{25} -33100.5 q^{26} -19683.0 q^{27} -17599.0 q^{28} -92885.5 q^{29} +14726.7 q^{30} -52229.9 q^{31} -142364. q^{32} +500.721 q^{33} +84125.5 q^{34} +25181.7 q^{35} -85635.5 q^{36} -76601.8 q^{37} +52460.7 q^{38} -275411. q^{39} +133888. q^{40} +108892. q^{41} +13126.3 q^{42} -791546. q^{43} +2178.50 q^{44} +122533. q^{45} -39482.2 q^{46} -992576. q^{47} -336185. q^{48} -801098. q^{49} +161839. q^{50} +699961. q^{51} -1.19824e6 q^{52} +1.60859e6 q^{53} +63871.8 q^{54} -3117.14 q^{55} +119337. q^{56} +436496. q^{57} +301416. q^{58} +733660. q^{59} +533107. q^{60} +114090. q^{61} +169487. q^{62} +109216. q^{63} -1.13179e6 q^{64} +1.71452e6 q^{65} -1624.85 q^{66} +447189. q^{67} +3.04534e6 q^{68} -328509. q^{69} -81715.2 q^{70} -671180. q^{71} +580689. q^{72} -406192. q^{73} +248575. q^{74} +1.34657e6 q^{75} +1.89908e6 q^{76} -2778.39 q^{77} +893714. q^{78} +883156. q^{79} +2.09286e6 q^{80} +531441. q^{81} -353358. q^{82} -4.07716e6 q^{83} +475172. q^{84} -4.35747e6 q^{85} +2.56858e6 q^{86} +2.50791e6 q^{87} -14772.3 q^{88} +5.07507e6 q^{89} -397621. q^{90} +1.52819e6 q^{91} -1.42926e6 q^{92} +1.41021e6 q^{93} +3.22093e6 q^{94} -2.71732e6 q^{95} +3.84382e6 q^{96} -1.23228e6 q^{97} +2.59958e6 q^{98} -13519.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99}+O(q^{100}) \) 5 * q - 135 * q^3 + 270 * q^4 - 266 * q^5 - 496 * q^7 + 1422 * q^8 + 3645 * q^9 + 1452 * q^10 - 1148 * q^11 - 7290 * q^12 - 642 * q^13 + 5756 * q^14 + 7182 * q^15 - 22606 * q^16 - 5798 * q^17 - 6036 * q^19 - 27376 * q^20 + 13392 * q^21 - 97896 * q^22 + 60835 * q^23 - 38394 * q^24 - 262477 * q^25 - 355992 * q^26 - 98415 * q^27 - 507124 * q^28 - 169162 * q^29 - 39204 * q^30 - 199640 * q^31 - 284794 * q^32 + 30996 * q^33 - 1027740 * q^34 - 137680 * q^35 + 196830 * q^36 - 202002 * q^37 - 554924 * q^38 + 17334 * q^39 - 340904 * q^40 + 541282 * q^41 - 155412 * q^42 - 909596 * q^43 - 1236032 * q^44 - 193914 * q^45 + 80208 * q^47 + 610362 * q^48 + 850589 * q^49 - 941416 * q^50 + 156546 * q^51 + 146940 * q^52 - 278138 * q^53 - 933560 * q^55 - 539932 * q^56 + 162972 * q^57 - 3522712 * q^58 - 3177380 * q^59 + 739152 * q^60 + 147782 * q^61 + 4606456 * q^62 - 361584 * q^63 - 4142622 * q^64 + 3877332 * q^65 + 2643192 * q^66 - 464916 * q^67 + 7513072 * q^68 - 1642545 * q^69 + 2093200 * q^70 + 1576792 * q^71 + 1036638 * q^72 - 38190 * q^73 + 12164864 * q^74 + 7086879 * q^75 + 6889436 * q^76 + 10332384 * q^77 + 9611784 * q^78 - 3913336 * q^79 + 6334776 * q^80 + 2657205 * q^81 + 6799360 * q^82 + 15774716 * q^83 + 13692348 * q^84 - 8520740 * q^85 + 24874084 * q^86 + 4567374 * q^87 + 53216 * q^88 + 1116482 * q^89 + 1058508 * q^90 - 27369552 * q^91 + 3285090 * q^92 + 5390280 * q^93 - 7153744 * q^94 - 6067832 * q^95 + 7689438 * q^96 - 15738566 * q^97 + 11730488 * q^98 - 836892 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

Properties

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