LMFDB - Embedded newform 69.8.a.d.1.8

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 757x^{6} - 1170x^{5} + 170343x^{4} + 424132x^{3} - 9973075x^{2} - 5161010x + 130545120 $ x^8 - 757x^6 - 1170x^5 + 170343x^4 + 424132x^3 - 9973075x^2 - 5161010x + 130545120
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-18.5379$ 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)

$f(q)$	$=$	$q+21.5379 q^{2} +27.0000 q^{3} +335.881 q^{4} +54.7305 q^{5} +581.523 q^{6} -565.198 q^{7} +4477.33 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+21.5379 q^{2} +27.0000 q^{3} +335.881 q^{4} +54.7305 q^{5} +581.523 q^{6} -565.198 q^{7} +4477.33 q^{8} +729.000 q^{9} +1178.78 q^{10} -890.150 q^{11} +9068.80 q^{12} +12435.5 q^{13} -12173.2 q^{14} +1477.72 q^{15} +53439.5 q^{16} -2692.57 q^{17} +15701.1 q^{18} -56073.6 q^{19} +18383.0 q^{20} -15260.3 q^{21} -19172.0 q^{22} +12167.0 q^{23} +120888. q^{24} -75129.6 q^{25} +267836. q^{26} +19683.0 q^{27} -189839. q^{28} -72626.2 q^{29} +31827.1 q^{30} +207276. q^{31} +577876. q^{32} -24034.0 q^{33} -57992.3 q^{34} -30933.6 q^{35} +244857. q^{36} -160422. q^{37} -1.20771e6 q^{38} +335760. q^{39} +245047. q^{40} -562941. q^{41} -328676. q^{42} +7553.70 q^{43} -298985. q^{44} +39898.6 q^{45} +262052. q^{46} +374615. q^{47} +1.44287e6 q^{48} -504095. q^{49} -1.61813e6 q^{50} -72699.4 q^{51} +4.17687e6 q^{52} +96348.3 q^{53} +423931. q^{54} -48718.4 q^{55} -2.53058e6 q^{56} -1.51399e6 q^{57} -1.56422e6 q^{58} -1.29931e6 q^{59} +496340. q^{60} +1.20681e6 q^{61} +4.46429e6 q^{62} -412029. q^{63} +5.60599e6 q^{64} +680604. q^{65} -517643. q^{66} +3.81931e6 q^{67} -904384. q^{68} +328509. q^{69} -666244. q^{70} -649906. q^{71} +3.26397e6 q^{72} -2.81315e6 q^{73} -3.45515e6 q^{74} -2.02850e6 q^{75} -1.88341e7 q^{76} +503111. q^{77} +7.23156e6 q^{78} -807658. q^{79} +2.92477e6 q^{80} +531441. q^{81} -1.21246e7 q^{82} -4.40700e6 q^{83} -5.12566e6 q^{84} -147366. q^{85} +162691. q^{86} -1.96091e6 q^{87} -3.98549e6 q^{88} -1.28241e7 q^{89} +859331. q^{90} -7.02854e6 q^{91} +4.08667e6 q^{92} +5.59645e6 q^{93} +8.06842e6 q^{94} -3.06894e6 q^{95} +1.56027e7 q^{96} -2.57777e6 q^{97} -1.08571e7 q^{98} -648919. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) $ 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9} + 11720 q^{10} + 6932 q^{11} + 15174 q^{12} + 12404 q^{13} + 30222 q^{14} + 10206 q^{15} + 27058 q^{16} + 24434 q^{17} + 17496 q^{18} - 14682 q^{19} - 3760 q^{20} + 3402 q^{21} + 36294 q^{22} + 97336 q^{23} + 113076 q^{24} + 144644 q^{25} + 325840 q^{26} + 157464 q^{27} - 21566 q^{28} + 255356 q^{29} + 316440 q^{30} + 450764 q^{31} + 647588 q^{32} + 187164 q^{33} + 191822 q^{34} + 1022616 q^{35} + 409698 q^{36} + 206240 q^{37} + 737372 q^{38} + 334908 q^{39} + 590028 q^{40} + 1053344 q^{41} + 815994 q^{42} + 1587806 q^{43} + 589366 q^{44} + 275562 q^{45} + 292008 q^{46} + 443336 q^{47} + 730566 q^{48} + 1944828 q^{49} - 1556112 q^{50} + 659718 q^{51} - 614236 q^{52} - 375530 q^{53} + 472392 q^{54} + 407792 q^{55} - 1316922 q^{56} - 396414 q^{57} - 1413384 q^{58} + 624008 q^{59} - 101520 q^{60} - 2005568 q^{61} - 3908272 q^{62} + 91854 q^{63} - 5082310 q^{64} + 646124 q^{65} + 979938 q^{66} - 2712286 q^{67} - 2289698 q^{68} + 2628072 q^{69} - 16499468 q^{70} - 6287176 q^{71} + 3053052 q^{72} - 10358312 q^{73} - 2000150 q^{74} + 3905388 q^{75} - 25107464 q^{76} - 2156840 q^{77} + 8797680 q^{78} - 8800574 q^{79} + 2384344 q^{80} + 4251528 q^{81} - 31799800 q^{82} + 384948 q^{83} - 582282 q^{84} - 17826684 q^{85} - 11563928 q^{86} + 6894612 q^{87} - 25202782 q^{88} - 3445530 q^{89} + 8543880 q^{90} - 16316740 q^{91} + 6837854 q^{92} + 12170628 q^{93} - 24237616 q^{94} + 26164288 q^{95} + 17484876 q^{96} - 28043764 q^{97} - 9998012 q^{98} + 5053428 q^{99}+O(q^{100}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9 + 11720 * q^10 + 6932 * q^11 + 15174 * q^12 + 12404 * q^13 + 30222 * q^14 + 10206 * q^15 + 27058 * q^16 + 24434 * q^17 + 17496 * q^18 - 14682 * q^19 - 3760 * q^20 + 3402 * q^21 + 36294 * q^22 + 97336 * q^23 + 113076 * q^24 + 144644 * q^25 + 325840 * q^26 + 157464 * q^27 - 21566 * q^28 + 255356 * q^29 + 316440 * q^30 + 450764 * q^31 + 647588 * q^32 + 187164 * q^33 + 191822 * q^34 + 1022616 * q^35 + 409698 * q^36 + 206240 * q^37 + 737372 * q^38 + 334908 * q^39 + 590028 * q^40 + 1053344 * q^41 + 815994 * q^42 + 1587806 * q^43 + 589366 * q^44 + 275562 * q^45 + 292008 * q^46 + 443336 * q^47 + 730566 * q^48 + 1944828 * q^49 - 1556112 * q^50 + 659718 * q^51 - 614236 * q^52 - 375530 * q^53 + 472392 * q^54 + 407792 * q^55 - 1316922 * q^56 - 396414 * q^57 - 1413384 * q^58 + 624008 * q^59 - 101520 * q^60 - 2005568 * q^61 - 3908272 * q^62 + 91854 * q^63 - 5082310 * q^64 + 646124 * q^65 + 979938 * q^66 - 2712286 * q^67 - 2289698 * q^68 + 2628072 * q^69 - 16499468 * q^70 - 6287176 * q^71 + 3053052 * q^72 - 10358312 * q^73 - 2000150 * q^74 + 3905388 * q^75 - 25107464 * q^76 - 2156840 * q^77 + 8797680 * q^78 - 8800574 * q^79 + 2384344 * q^80 + 4251528 * q^81 - 31799800 * q^82 + 384948 * q^83 - 582282 * q^84 - 17826684 * q^85 - 11563928 * q^86 + 6894612 * q^87 - 25202782 * q^88 - 3445530 * q^89 + 8543880 * q^90 - 16316740 * q^91 + 6837854 * q^92 + 12170628 * q^93 - 24237616 * q^94 + 26164288 * q^95 + 17484876 * q^96 - 28043764 * q^97 - 9998012 * q^98 + 5053428 * 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 $

$2$	21.5379	1.90370	0.951850	−	0.306564i	$-0.0991795\pi$
$2$	21.5379	1.90370	0.951850	+	0.306564i	$0.0991795\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	335.881	2.62407
$5$	54.7305	0.195810	0.0979050	−	0.995196i	$-0.468786\pi$
$5$	54.7305	0.195810	0.0979050	+	0.995196i	$0.468786\pi$
$6$	581.523	1.09910
$7$	−565.198	−0.622812	−0.311406	−	0.950277i	$-0.600800\pi$
$7$	−565.198	−0.622812	−0.311406	+	0.950277i	$0.600800\pi$
$8$	4477.33	3.09175
$9$	729.000	0.333333
$10$	1178.78	0.372763
$11$	−890.150	−0.201646	−0.100823	−	0.994904i	$-0.532148\pi$
$11$	−890.150	−0.201646	−0.100823	+	0.994904i	$0.532148\pi$
$12$	9068.80	1.51501
$13$	12435.5	1.56987	0.784934	−	0.619579i	$-0.212697\pi$
$13$	12435.5	1.56987	0.784934	+	0.619579i	$0.212697\pi$
$14$	−12173.2	−1.18565
$15$	1477.72	0.113051
$16$	53439.5	3.26169
$17$	−2692.57	−0.132922	−0.0664608	−	0.997789i	$-0.521171\pi$
$17$	−2692.57	−0.132922	−0.0664608	+	0.997789i	$0.521171\pi$
$18$	15701.1	0.634567
$19$	−56073.6	−1.87551	−0.937757	−	0.347291i	$-0.887102\pi$
$19$	−56073.6	−1.87551	−0.937757	+	0.347291i	$0.887102\pi$
$20$	18383.0	0.513819
$21$	−15260.3	−0.359581
$22$	−19172.0	−0.383873
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	120888.	1.78502
$25$	−75129.6	−0.961658
$26$	267836.	2.98856
$27$	19683.0	0.192450
$28$	−189839.	−1.63430
$29$	−72626.2	−0.552969	−0.276484	−	0.961018i	$-0.589169\pi$
$29$	−72626.2	−0.552969	−0.276484	+	0.961018i	$0.589169\pi$
$30$	31827.1	0.215215
$31$	207276.	1.24963	0.624817	−	0.780771i	$-0.285173\pi$
$31$	207276.	1.24963	0.624817	+	0.780771i	$0.285173\pi$
$32$	577876.	3.11752
$33$	−24034.0	−0.116420
$34$	−57992.3	−0.253043
$35$	−30933.6	−0.121953
$36$	244857.	0.874691
$37$	−160422.	−0.520664	−0.260332	−	0.965519i	$-0.583832\pi$
$37$	−160422.	−0.520664	−0.260332	+	0.965519i	$0.583832\pi$
$38$	−1.20771e6	−3.57042
$39$	335760.	0.906364
$40$	245047.	0.605395
$41$	−562941.	−1.27561	−0.637807	−	0.770196i	$-0.720158\pi$
$41$	−562941.	−1.27561	−0.637807	+	0.770196i	$0.720158\pi$
$42$	−328676.	−0.684534
$43$	7553.70	0.0144884	0.00724419	−	0.999974i	$-0.497694\pi$
$43$	7553.70	0.0144884	0.00724419	+	0.999974i	$0.497694\pi$
$44$	−298985.	−0.529133
$45$	39898.6	0.0652700
$46$	262052.	0.396949
$47$	374615.	0.526311	0.263156	−	0.964753i	$-0.415237\pi$
$47$	374615.	0.526311	0.263156	+	0.964753i	$0.415237\pi$
$48$	1.44287e6	1.88314
$49$	−504095.	−0.612105
$50$	−1.61813e6	−1.83071
$51$	−72699.4	−0.0767424
$52$	4.17687e6	4.11945
$53$	96348.3	0.0888952	0.0444476	−	0.999012i	$-0.485847\pi$
$53$	96348.3	0.0888952	0.0444476	+	0.999012i	$0.485847\pi$
$54$	423931.	0.366367
$55$	−48718.4	−0.0394842
$56$	−2.53058e6	−1.92558
$57$	−1.51399e6	−1.08283
$58$	−1.56422e6	−1.05269
$59$	−1.29931e6	−0.823625	−0.411812	−	0.911269i	$-0.635104\pi$
$59$	−1.29931e6	−0.823625	−0.411812	+	0.911269i	$0.635104\pi$
$60$	496340.	0.296654
$61$	1.20681e6	0.680747	0.340374	−	0.940290i	$-0.389446\pi$
$61$	1.20681e6	0.680747	0.340374	+	0.940290i	$0.389446\pi$
$62$	4.46429e6	2.37893
$63$	−412029.	−0.207604
$64$	5.60599e6	2.67314
$65$	680604.	0.307396
$66$	−517643.	−0.221629
$67$	3.81931e6	1.55140	0.775699	−	0.631103i	$-0.217398\pi$
$67$	3.81931e6	1.55140	0.775699	+	0.631103i	$0.217398\pi$
$68$	−904384.	−0.348796
$69$	328509.	0.120386
$70$	−666244.	−0.232162
$71$	−649906.	−0.215499	−0.107750	−	0.994178i	$-0.534365\pi$
$71$	−649906.	−0.215499	−0.107750	+	0.994178i	$0.534365\pi$
$72$	3.26397e6	1.03058
$73$	−2.81315e6	−0.846374	−0.423187	−	0.906042i	$-0.639089\pi$
$73$	−2.81315e6	−0.846374	−0.423187	+	0.906042i	$0.639089\pi$
$74$	−3.45515e6	−0.991187
$75$	−2.02850e6	−0.555214
$76$	−1.88341e7	−4.92149
$77$	503111.	0.125587
$78$	7.23156e6	1.72544
$79$	−807658.	−0.184303	−0.0921516	−	0.995745i	$-0.529374\pi$
$79$	−807658.	−0.184303	−0.0921516	+	0.995745i	$0.529374\pi$
$80$	2.92477e6	0.638670
$81$	531441.	0.111111
$82$	−1.21246e7	−2.42839
$83$	−4.40700e6	−0.845999	−0.422999	−	0.906130i	$-0.639023\pi$
$83$	−4.40700e6	−0.845999	−0.422999	+	0.906130i	$0.639023\pi$
$84$	−5.12566e6	−0.943566
$85$	−147366.	−0.0260274
$86$	162691.	0.0275815
$87$	−1.96091e6	−0.319257
$88$	−3.98549e6	−0.623437
$89$	−1.28241e7	−1.92824	−0.964121	−	0.265464i	$-0.914475\pi$
$89$	−1.28241e7	−1.92824	−0.964121	+	0.265464i	$0.914475\pi$
$90$	859331.	0.124254
$91$	−7.02854e6	−0.977733
$92$	4.08667e6	0.547157
$93$	5.59645e6	0.721477
$94$	8.06842e6	1.00194
$95$	−3.06894e6	−0.367244
$96$	1.56027e7	1.79990
$97$	−2.57777e6	−0.286776	−0.143388	−	0.989667i	$-0.545800\pi$
$97$	−2.57777e6	−0.286776	−0.143388	+	0.989667i	$0.545800\pi$
$98$	−1.08571e7	−1.16526
$99$	−648919.	−0.0672152
$100$	−2.52346e7	−2.52346
$101$	1.36859e7	1.32174	0.660872	−	0.750499i	$-0.270187\pi$
$101$	1.36859e7	1.32174	0.660872	+	0.750499i	$0.270187\pi$
$102$	−1.56579e6	−0.146094
$103$	1.78977e7	1.61386	0.806932	−	0.590645i	$-0.201127\pi$
$103$	1.78977e7	1.61386	0.806932	+	0.590645i	$0.201127\pi$
$104$	5.56780e7	4.85364
$105$	−835206.	−0.0704095
$106$	2.07514e6	0.169230
$107$	2.05198e7	1.61931	0.809655	−	0.586907i	$-0.199654\pi$
$107$	2.05198e7	1.61931	0.809655	+	0.586907i	$0.199654\pi$
$108$	6.61115e6	0.505003
$109$	−1.99672e7	−1.47681	−0.738405	−	0.674358i	$-0.764421\pi$
$109$	−1.99672e7	−1.47681	−0.738405	+	0.674358i	$0.764421\pi$
$110$	−1.04929e6	−0.0751661
$111$	−4.33139e6	−0.300605
$112$	−3.02039e7	−2.03142
$113$	3.03308e7	1.97747	0.988734	−	0.149685i	$-0.0478259\pi$
$113$	3.03308e7	1.97747	0.988734	+	0.149685i	$0.0478259\pi$
$114$	−3.26081e7	−2.06138
$115$	665906.	0.0408292
$116$	−2.43938e7	−1.45103
$117$	9.06551e6	0.523289
$118$	−2.79843e7	−1.56793
$119$	1.52183e6	0.0827853
$120$	6.61626e6	0.349525
$121$	−1.86948e7	−0.959339
$122$	2.59922e7	1.29594
$123$	−1.51994e7	−0.736476
$124$	6.96201e7	3.27913
$125$	−8.38770e6	−0.384112
$126$	−8.87424e6	−0.395216
$127$	1.95967e7	0.848926	0.424463	−	0.905445i	$-0.360463\pi$
$127$	1.95967e7	0.848926	0.424463	+	0.905445i	$0.360463\pi$
$128$	4.67731e7	1.97134
$129$	203950.	0.00836487
$130$	1.46588e7	0.585189
$131$	−1.49391e7	−0.580597	−0.290299	−	0.956936i	$-0.593755\pi$
$131$	−1.49391e7	−0.580597	−0.290299	+	0.956936i	$0.593755\pi$
$132$	−8.07259e6	−0.305495
$133$	3.16926e7	1.16809
$134$	8.22599e7	2.95339
$135$	1.07726e6	0.0376836
$136$	−1.20555e7	−0.410960
$137$	3.18774e7	1.05916	0.529578	−	0.848261i	$-0.322350\pi$
$137$	3.18774e7	1.05916	0.529578	+	0.848261i	$0.322350\pi$
$138$	7.07540e6	0.229179
$139$	4.39284e7	1.38737	0.693687	−	0.720277i	$-0.255985\pi$
$139$	4.39284e7	1.38737	0.693687	+	0.720277i	$0.255985\pi$
$140$	−1.03900e7	−0.320013
$141$	1.01146e7	0.303866
$142$	−1.39976e7	−0.410246
$143$	−1.10695e7	−0.316557
$144$	3.89574e7	1.08723
$145$	−3.97487e6	−0.108277
$146$	−6.05893e7	−1.61124
$147$	−1.36106e7	−0.353399
$148$	−5.38827e7	−1.36626
$149$	6.05704e7	1.50006	0.750030	−	0.661404i	$-0.230039\pi$
$149$	6.05704e7	1.50006	0.750030	+	0.661404i	$0.230039\pi$
$150$	−4.36896e7	−1.05696
$151$	1.59550e7	0.377117	0.188559	−	0.982062i	$-0.439618\pi$
$151$	1.59550e7	0.377117	0.188559	+	0.982062i	$0.439618\pi$
$152$	−2.51060e8	−5.79862
$153$	−1.96288e6	−0.0443072
$154$	1.08359e7	0.239081
$155$	1.13443e7	0.244691
$156$	1.12775e8	2.37836
$157$	−3.40669e7	−0.702562	−0.351281	−	0.936270i	$-0.614254\pi$
$157$	−3.40669e7	−0.702562	−0.351281	+	0.936270i	$0.614254\pi$
$158$	−1.73953e7	−0.350858
$159$	2.60140e6	0.0513237
$160$	3.16275e7	0.610442
$161$	−6.87676e6	−0.129865
$162$	1.14461e7	0.211522
$163$	−7.41885e7	−1.34178	−0.670888	−	0.741559i	$-0.734087\pi$
$163$	−7.41885e7	−1.34178	−0.670888	+	0.741559i	$0.734087\pi$
$164$	−1.89081e8	−3.34731
$165$	−1.31540e6	−0.0227962
$166$	−9.49175e7	−1.61053
$167$	1.05052e8	1.74540	0.872700	−	0.488258i	$-0.162367\pi$
$167$	1.05052e8	1.74540	0.872700	+	0.488258i	$0.162367\pi$
$168$	−6.83255e7	−1.11173
$169$	9.18943e7	1.46449
$170$	−3.17395e6	−0.0495483
$171$	−4.08776e7	−0.625172
$172$	2.53715e6	0.0380186
$173$	8.92240e7	1.31015	0.655074	−	0.755565i	$-0.272638\pi$
$173$	8.92240e7	1.31015	0.655074	+	0.755565i	$0.272638\pi$
$174$	−4.22339e7	−0.607769
$175$	4.24631e7	0.598933
$176$	−4.75691e7	−0.657705
$177$	−3.50813e7	−0.475520
$178$	−2.76204e8	−3.67079
$179$	4.09274e7	0.533369	0.266685	−	0.963784i	$-0.414072\pi$
$179$	4.09274e7	0.533369	0.266685	+	0.963784i	$0.414072\pi$
$180$	1.34012e7	0.171273
$181$	4.48333e7	0.561986	0.280993	−	0.959710i	$-0.409336\pi$
$181$	4.48333e7	0.561986	0.280993	+	0.959710i	$0.409336\pi$
$182$	−1.51380e8	−1.86131
$183$	3.25840e7	0.393029
$184$	5.44757e7	0.644674
$185$	−8.77997e6	−0.101951
$186$	1.20536e8	1.37348
$187$	2.39679e6	0.0268031
$188$	1.25826e8	1.38108
$189$	−1.11248e7	−0.119860
$190$	−6.60984e7	−0.699123
$191$	1.08477e8	1.12648	0.563239	−	0.826294i	$-0.309555\pi$
$191$	1.08477e8	1.12648	0.563239	+	0.826294i	$0.309555\pi$
$192$	1.51362e8	1.54334
$193$	8.81302e6	0.0882417	0.0441209	−	0.999026i	$-0.485951\pi$
$193$	8.81302e6	0.0882417	0.0441209	+	0.999026i	$0.485951\pi$
$194$	−5.55197e7	−0.545935
$195$	1.83763e7	0.177475
$196$	−1.69316e8	−1.60621
$197$	1.04354e7	0.0972475	0.0486238	−	0.998817i	$-0.484516\pi$
$197$	1.04354e7	0.0972475	0.0486238	+	0.998817i	$0.484516\pi$
$198$	−1.39764e7	−0.127958
$199$	−1.41236e8	−1.27046	−0.635230	−	0.772323i	$-0.719095\pi$
$199$	−1.41236e8	−1.27046	−0.635230	+	0.772323i	$0.719095\pi$
$200$	−3.36380e8	−2.97320
$201$	1.03121e8	0.895700
$202$	2.94765e8	2.51620
$203$	4.10482e7	0.344396
$204$	−2.44184e7	−0.201378
$205$	−3.08101e7	−0.249778
$206$	3.85479e8	3.07231
$207$	8.86974e6	0.0695048
$208$	6.64549e8	5.12042
$209$	4.99139e7	0.378189
$210$	−1.79886e7	−0.134039
$211$	5.31949e7	0.389835	0.194918	−	0.980820i	$-0.437556\pi$
$211$	5.31949e7	0.389835	0.194918	+	0.980820i	$0.437556\pi$
$212$	3.23616e7	0.233267
$213$	−1.75475e7	−0.124419
$214$	4.41953e8	3.08268
$215$	413418.	0.00283697
$216$	8.81273e7	0.595007
$217$	−1.17152e8	−0.778288
$218$	−4.30052e8	−2.81140
$219$	−7.59550e7	−0.488654
$220$	−1.63636e7	−0.103609
$221$	−3.34836e7	−0.208670
$222$	−9.32890e7	−0.572262
$223$	−1.70709e8	−1.03084	−0.515419	−	0.856938i	$-0.672364\pi$
$223$	−1.70709e8	−1.03084	−0.515419	+	0.856938i	$0.672364\pi$
$224$	−3.26614e8	−1.94163
$225$	−5.47695e7	−0.320553
$226$	6.53262e8	3.76450
$227$	−1.95923e8	−1.11172	−0.555860	−	0.831276i	$-0.687611\pi$
$227$	−1.95923e8	−1.11172	−0.555860	+	0.831276i	$0.687611\pi$
$228$	−5.08520e8	−2.84142
$229$	−2.03625e8	−1.12049	−0.560244	−	0.828328i	$-0.689293\pi$
$229$	−2.03625e8	−1.12049	−0.560244	+	0.828328i	$0.689293\pi$
$230$	1.43422e7	0.0777265
$231$	1.35840e7	0.0725079
$232$	−3.25172e8	−1.70964
$233$	3.21437e7	0.166475	0.0832376	−	0.996530i	$-0.473474\pi$
$233$	3.21437e7	0.166475	0.0832376	+	0.996530i	$0.473474\pi$
$234$	1.95252e8	0.996186
$235$	2.05029e7	0.103057
$236$	−4.36413e8	−2.16125
$237$	−2.18068e7	−0.106407
$238$	3.27771e7	0.157598
$239$	4.43634e7	0.210200	0.105100	−	0.994462i	$-0.466484\pi$
$239$	4.43634e7	0.210200	0.105100	+	0.994462i	$0.466484\pi$
$240$	7.89688e7	0.368737
$241$	1.09153e8	0.502316	0.251158	−	0.967946i	$-0.419189\pi$
$241$	1.09153e8	0.502316	0.251158	+	0.967946i	$0.419189\pi$
$242$	−4.02647e8	−1.82629
$243$	1.43489e7	0.0641500
$244$	4.05346e8	1.78633
$245$	−2.75894e7	−0.119856
$246$	−3.27363e8	−1.40203
$247$	−6.97305e8	−2.94431
$248$	9.28043e8	3.86356
$249$	−1.18989e8	−0.488438
$250$	−1.80654e8	−0.731234
$251$	−1.36741e8	−0.545810	−0.272905	−	0.962041i	$-0.587984\pi$
$251$	−1.36741e8	−0.545810	−0.272905	+	0.962041i	$0.587984\pi$
$252$	−1.38393e8	−0.544768
$253$	−1.08305e7	−0.0420460
$254$	4.22072e8	1.61610
$255$	−3.97888e6	−0.0150269
$256$	2.89828e8	1.07969
$257$	4.31493e8	1.58565	0.792825	−	0.609449i	$-0.208609\pi$
$257$	4.31493e8	1.58565	0.792825	+	0.609449i	$0.208609\pi$
$258$	4.39265e6	0.0159242
$259$	9.06700e7	0.324276
$260$	2.28602e8	0.806629
$261$	−5.29445e7	−0.184323
$262$	−3.21757e8	−1.10528
$263$	2.44909e8	0.830157	0.415078	−	0.909786i	$-0.363754\pi$
$263$	2.44909e8	0.830157	0.415078	+	0.909786i	$0.363754\pi$
$264$	−1.07608e8	−0.359942
$265$	5.27319e6	0.0174066
$266$	6.82593e8	2.22370
$267$	−3.46250e8	−1.11327
$268$	1.28283e9	4.07098
$269$	−6.20750e8	−1.94439	−0.972195	−	0.234171i	$-0.924762\pi$
$269$	−6.20750e8	−1.94439	−0.972195	+	0.234171i	$0.924762\pi$
$270$	2.32019e7	0.0717383
$271$	8.27788e7	0.252654	0.126327	−	0.991989i	$-0.459681\pi$
$271$	8.27788e7	0.252654	0.126327	+	0.991989i	$0.459681\pi$
$272$	−1.43889e8	−0.433549
$273$	−1.89771e8	−0.564495
$274$	6.86571e8	2.01632
$275$	6.68766e7	0.193914
$276$	1.10340e8	0.315901
$277$	−5.28652e8	−1.49448	−0.747242	−	0.664552i	$-0.768622\pi$
$277$	−5.28652e8	−1.49448	−0.747242	+	0.664552i	$0.768622\pi$
$278$	9.46125e8	2.64114
$279$	1.51104e8	0.416545
$280$	−1.38500e8	−0.377047
$281$	−9.61867e7	−0.258608	−0.129304	−	0.991605i	$-0.541274\pi$
$281$	−9.61867e7	−0.258608	−0.129304	+	0.991605i	$0.541274\pi$
$282$	2.17847e8	0.578469
$283$	−3.04630e8	−0.798950	−0.399475	−	0.916744i	$-0.630808\pi$
$283$	−3.04630e8	−0.798950	−0.399475	+	0.916744i	$0.630808\pi$
$284$	−2.18291e8	−0.565486
$285$	−8.28613e7	−0.212029
$286$	−2.38414e8	−0.602630
$287$	3.18173e8	0.794468
$288$	4.21272e8	1.03917
$289$	−4.03089e8	−0.982332
$290$	−8.56104e7	−0.206126
$291$	−6.95997e7	−0.165570
$292$	−9.44883e8	−2.22095
$293$	5.54091e8	1.28690	0.643450	−	0.765488i	$-0.277502\pi$
$293$	5.54091e8	1.28690	0.643450	+	0.765488i	$0.277502\pi$
$294$	−2.93143e8	−0.672765
$295$	−7.11117e7	−0.161274
$296$	−7.18261e8	−1.60976
$297$	−1.75208e7	−0.0388067
$298$	1.30456e9	2.85566
$299$	1.51303e8	0.327340
$300$	−6.81335e8	−1.45692
$301$	−4.26933e6	−0.00902354
$302$	3.43636e8	0.717918
$303$	3.69518e8	0.763109
$304$	−2.99654e9	−6.11734
$305$	6.60495e7	0.133297
$306$	−4.22764e7	−0.0843477
$307$	6.73332e8	1.32814	0.664071	−	0.747669i	$-0.268827\pi$
$307$	6.73332e8	1.32814	0.664071	+	0.747669i	$0.268827\pi$
$308$	1.68985e8	0.329550
$309$	4.83238e8	0.931764
$310$	2.44333e8	0.465818
$311$	−4.87910e8	−0.919767	−0.459884	−	0.887979i	$-0.652109\pi$
$311$	−4.87910e8	−0.919767	−0.459884	+	0.887979i	$0.652109\pi$
$312$	1.50331e9	2.80225
$313$	−4.65266e8	−0.857622	−0.428811	−	0.903394i	$-0.641067\pi$
$313$	−4.65266e8	−0.857622	−0.428811	+	0.903394i	$0.641067\pi$
$314$	−7.33731e8	−1.33747
$315$	−2.25506e7	−0.0406509
$316$	−2.71277e8	−0.483625
$317$	−4.54827e7	−0.0801934	−0.0400967	−	0.999196i	$-0.512767\pi$
$317$	−4.54827e7	−0.0801934	−0.0400967	+	0.999196i	$0.512767\pi$
$318$	5.60288e7	0.0977049
$319$	6.46482e7	0.111504
$320$	3.06819e8	0.523428
$321$	5.54034e8	0.934909
$322$	−1.48111e8	−0.247225
$323$	1.50982e8	0.249297
$324$	1.78501e8	0.291564
$325$	−9.34277e8	−1.50968
$326$	−1.59786e9	−2.55434
$327$	−5.39115e8	−0.852636
$328$	−2.52047e9	−3.94388
$329$	−2.11731e8	−0.327793
$330$	−2.83309e7	−0.0433972
$331$	5.74770e8	0.871157	0.435578	−	0.900151i	$-0.356544\pi$
$331$	5.74770e8	0.871157	0.435578	+	0.900151i	$0.356544\pi$
$332$	−1.48023e9	−2.21996
$333$	−1.16947e8	−0.173555
$334$	2.26259e9	3.32272
$335$	2.09033e8	0.303779
$336$	−8.15504e8	−1.17284
$337$	6.94857e8	0.988987	0.494494	−	0.869181i	$-0.335354\pi$
$337$	6.94857e8	0.988987	0.494494	+	0.869181i	$0.335354\pi$
$338$	1.97921e9	2.78794
$339$	8.18932e8	1.14169
$340$	−4.94974e7	−0.0682977
$341$	−1.84507e8	−0.251983
$342$	−8.80418e8	−1.19014
$343$	7.50378e8	1.00404
$344$	3.38204e7	0.0447944
$345$	1.79795e7	0.0235727
$346$	1.92170e9	2.49413
$347$	1.21676e9	1.56334	0.781669	−	0.623694i	$-0.214369\pi$
$347$	1.21676e9	1.56334	0.781669	+	0.623694i	$0.214369\pi$
$348$	−6.58633e8	−0.837753
$349$	−1.31017e9	−1.64982	−0.824912	−	0.565261i	$-0.808775\pi$
$349$	−1.31017e9	−1.64982	−0.824912	+	0.565261i	$0.808775\pi$
$350$	9.14565e8	1.14019
$351$	2.44769e8	0.302121
$352$	−5.14396e8	−0.628635
$353$	−8.00528e8	−0.968646	−0.484323	−	0.874889i	$-0.660934\pi$
$353$	−8.00528e8	−0.968646	−0.484323	+	0.874889i	$0.660934\pi$
$354$	−7.55577e8	−0.905247
$355$	−3.55697e7	−0.0421969
$356$	−4.30737e9	−5.05985
$357$	4.10895e7	0.0477961
$358$	8.81490e8	1.01538
$359$	2.84701e7	0.0324757	0.0162378	−	0.999868i	$-0.494831\pi$
$359$	2.84701e7	0.0324757	0.0162378	+	0.999868i	$0.494831\pi$
$360$	1.78639e8	0.201798
$361$	2.25037e9	2.51756
$362$	9.65615e8	1.06985
$363$	−5.04760e8	−0.553875
$364$	−2.36076e9	−2.56564
$365$	−1.53965e8	−0.165728
$366$	7.01790e8	0.748210
$367$	−1.47636e8	−0.155906	−0.0779529	−	0.996957i	$-0.524838\pi$
$367$	−1.47636e8	−0.155906	−0.0779529	+	0.996957i	$0.524838\pi$
$368$	6.50198e8	0.680108
$369$	−4.10384e8	−0.425205
$370$	−1.89102e8	−0.194084
$371$	−5.44558e7	−0.0553650
$372$	1.87974e9	1.89321
$373$	−1.17802e9	−1.17537	−0.587683	−	0.809091i	$-0.699960\pi$
$373$	−1.17802e9	−1.17537	−0.587683	+	0.809091i	$0.699960\pi$
$374$	5.16219e7	0.0510250
$375$	−2.26468e8	−0.221767
$376$	1.67727e9	1.62722
$377$	−9.03147e8	−0.868088
$378$	−2.39605e8	−0.228178
$379$	−1.50793e9	−1.42280	−0.711399	−	0.702788i	$-0.751938\pi$
$379$	−1.50793e9	−1.42280	−0.711399	+	0.702788i	$0.751938\pi$
$380$	−1.03080e9	−0.963676
$381$	5.29111e8	0.490128
$382$	2.33638e9	2.14448
$383$	−7.42514e7	−0.0675319	−0.0337659	−	0.999430i	$-0.510750\pi$
$383$	−7.42514e7	−0.0675319	−0.0337659	+	0.999430i	$0.510750\pi$
$384$	1.26287e9	1.13815
$385$	2.75355e7	0.0245912
$386$	1.89814e8	0.167986
$387$	5.50665e6	0.00482946
$388$	−8.65824e8	−0.752521
$389$	1.56330e9	1.34654	0.673271	−	0.739396i	$-0.264889\pi$
$389$	1.56330e9	1.34654	0.673271	+	0.739396i	$0.264889\pi$
$390$	3.95787e8	0.337859
$391$	−3.27605e7	−0.0277161
$392$	−2.25700e9	−1.89247
$393$	−4.03356e8	−0.335208
$394$	2.24757e8	0.185130
$395$	−4.42036e7	−0.0360884
$396$	−2.17960e8	−0.176378
$397$	−3.60628e8	−0.289263	−0.144631	−	0.989486i	$-0.546200\pi$
$397$	−3.60628e8	−0.289263	−0.144631	+	0.989486i	$0.546200\pi$
$398$	−3.04194e9	−2.41857
$399$	8.55701e8	0.674399
$400$	−4.01488e9	−3.13663
$401$	−1.19634e9	−0.926507	−0.463253	−	0.886226i	$-0.653318\pi$
$401$	−1.19634e9	−0.926507	−0.463253	+	0.886226i	$0.653318\pi$
$402$	2.22102e9	1.70514
$403$	2.57759e9	1.96176
$404$	4.59682e9	3.46835
$405$	2.90860e7	0.0217567
$406$	8.84092e8	0.655626
$407$	1.42799e8	0.104990
$408$	−3.25499e8	−0.237268
$409$	6.44935e8	0.466106	0.233053	−	0.972464i	$-0.425129\pi$
$409$	6.44935e8	0.466106	0.233053	+	0.972464i	$0.425129\pi$
$410$	−6.63584e8	−0.475502
$411$	8.60688e8	0.611504
$412$	6.01150e9	4.23490
$413$	7.34365e8	0.512964
$414$	1.91036e8	0.132316
$415$	−2.41197e8	−0.165655
$416$	7.18620e9	4.89410
$417$	1.18607e9	0.801000
$418$	1.07504e9	0.719959
$419$	−1.87982e9	−1.24844	−0.624218	−	0.781250i	$-0.714582\pi$
$419$	−1.87982e9	−1.24844	−0.624218	+	0.781250i	$0.714582\pi$
$420$	−2.80530e8	−0.184760
$421$	1.34590e9	0.879076	0.439538	−	0.898224i	$-0.355142\pi$
$421$	1.34590e9	0.879076	0.439538	+	0.898224i	$0.355142\pi$
$422$	1.14571e9	0.742130
$423$	2.73094e8	0.175437
$424$	4.31383e8	0.274841
$425$	2.02292e8	0.127825
$426$	−3.77935e8	−0.236856
$427$	−6.82088e8	−0.423978
$428$	6.89222e9	4.24919
$429$	−2.98877e8	−0.182764
$430$	8.90415e6	0.00540074
$431$	−3.61149e8	−0.217278	−0.108639	−	0.994081i	$-0.534649\pi$
$431$	−3.61149e8	−0.217278	−0.108639	+	0.994081i	$0.534649\pi$
$432$	1.05185e9	0.627712
$433$	−2.03487e9	−1.20456	−0.602281	−	0.798284i	$-0.705741\pi$
$433$	−2.03487e9	−1.20456	−0.602281	+	0.798284i	$0.705741\pi$
$434$	−2.52321e9	−1.48163
$435$	−1.07322e8	−0.0625136
$436$	−6.70661e9	−3.87526
$437$	−6.82247e8	−0.391072
$438$	−1.63591e9	−0.930251
$439$	−2.51793e9	−1.42043	−0.710213	−	0.703987i	$-0.751401\pi$
$439$	−2.51793e9	−1.42043	−0.710213	+	0.703987i	$0.751401\pi$
$440$	−2.18128e8	−0.122075
$441$	−3.67485e8	−0.204035
$442$	−7.21166e8	−0.397244
$443$	8.85785e8	0.484078	0.242039	−	0.970267i	$-0.422184\pi$
$443$	8.85785e8	0.484078	0.242039	+	0.970267i	$0.422184\pi$
$444$	−1.45483e9	−0.788810
$445$	−7.01869e8	−0.377569
$446$	−3.67672e9	−1.96241
$447$	1.63540e9	0.866060
$448$	−3.16849e9	−1.66487
$449$	−8.25873e8	−0.430577	−0.215289	−	0.976550i	$-0.569069\pi$
$449$	−8.25873e8	−0.430577	−0.215289	+	0.976550i	$0.569069\pi$
$450$	−1.17962e9	−0.610236
$451$	5.01102e8	0.257222
$452$	1.01876e10	5.18902
$453$	4.30784e8	0.217729
$454$	−4.21978e9	−2.11638
$455$	−3.84676e8	−0.191450
$456$	−6.77861e9	−3.34783
$457$	−1.26103e9	−0.618041	−0.309021	−	0.951055i	$-0.600001\pi$
$457$	−1.26103e9	−0.618041	−0.309021	+	0.951055i	$0.600001\pi$
$458$	−4.38566e9	−2.13307
$459$	−5.29979e7	−0.0255808
$460$	2.23666e8	0.107139
$461$	−2.91819e9	−1.38727	−0.693634	−	0.720328i	$-0.743991\pi$
$461$	−2.91819e9	−1.38727	−0.693634	+	0.720328i	$0.743991\pi$
$462$	2.92571e8	0.138033
$463$	8.59303e8	0.402358	0.201179	−	0.979554i	$-0.435523\pi$
$463$	8.59303e8	0.402358	0.201179	+	0.979554i	$0.435523\pi$
$464$	−3.88111e9	−1.80361
$465$	3.06297e8	0.141272
$466$	6.92307e8	0.316919
$467$	3.45880e9	1.57151	0.785754	−	0.618539i	$-0.212275\pi$
$467$	3.45880e9	1.57151	0.785754	+	0.618539i	$0.212275\pi$
$468$	3.04494e9	1.37315
$469$	−2.15866e9	−0.966229
$470$	4.41589e8	0.196189
$471$	−9.19808e8	−0.405624
$472$	−5.81742e9	−2.54644
$473$	−6.72392e6	−0.00292152
$474$	−4.69672e8	−0.202568
$475$	4.21278e9	1.80360
$476$	5.11156e8	0.217235
$477$	7.02379e7	0.0296317
$478$	9.55495e8	0.400157
$479$	2.91208e9	1.21068	0.605339	−	0.795968i	$-0.293038\pi$
$479$	2.91208e9	1.21068	0.605339	+	0.795968i	$0.293038\pi$
$480$	8.53941e8	0.352439
$481$	−1.99493e9	−0.817373
$482$	2.35093e9	0.956259
$483$	−1.85673e8	−0.0749778
$484$	−6.27924e9	−2.51738
$485$	−1.41083e8	−0.0561536
$486$	3.09045e8	0.122122
$487$	1.75754e9	0.689530	0.344765	−	0.938689i	$-0.387959\pi$
$487$	1.75754e9	0.689530	0.344765	+	0.938689i	$0.387959\pi$
$488$	5.40330e9	2.10470
$489$	−2.00309e9	−0.774674
$490$	−5.94217e8	−0.228170
$491$	−9.74766e8	−0.371634	−0.185817	−	0.982584i	$-0.559493\pi$
$491$	−9.74766e8	−0.371634	−0.185817	+	0.982584i	$0.559493\pi$
$492$	−5.10520e9	−1.93257
$493$	1.95551e8	0.0735015
$494$	−1.50185e10	−5.60508
$495$	−3.55157e7	−0.0131614
$496$	1.10767e10	4.07592
$497$	3.67325e8	0.134216
$498$	−2.56277e9	−0.929839
$499$	−4.71158e7	−0.0169752	−0.00848760	−	0.999964i	$-0.502702\pi$
$499$	−4.71158e7	−0.0169752	−0.00848760	+	0.999964i	$0.502702\pi$
$500$	−2.81727e9	−1.00794
$501$	2.83639e9	1.00771
$502$	−2.94512e9	−1.03906
$503$	5.67881e7	0.0198962	0.00994809	−	0.999951i	$-0.496833\pi$
$503$	5.67881e7	0.0198962	0.00994809	+	0.999951i	$0.496833\pi$
$504$	−1.84479e9	−0.641859
$505$	7.49034e8	0.258810
$506$	−2.33265e8	−0.0800430
$507$	2.48115e9	0.845522
$508$	6.58217e9	2.22764
$509$	−4.54481e9	−1.52758	−0.763789	−	0.645466i	$-0.776663\pi$
$509$	−4.54481e9	−1.52758	−0.763789	+	0.645466i	$0.776663\pi$
$510$	−8.56967e7	−0.0286067
$511$	1.58998e9	0.527132
$512$	2.55330e8	0.0840730
$513$	−1.10370e9	−0.360943
$514$	9.29345e9	3.01860
$515$	9.79550e8	0.316010
$516$	6.85029e7	0.0219500
$517$	−3.33463e8	−0.106128
$518$	1.95284e9	0.617324
$519$	2.40905e9	0.756414
$520$	3.04729e9	0.950390
$521$	−1.70629e7	−0.00528593	−0.00264297	−	0.999997i	$-0.500841\pi$
$521$	−1.70629e7	−0.00528593	−0.00264297	+	0.999997i	$0.500841\pi$
$522$	−1.14031e9	−0.350895
$523$	−1.82834e8	−0.0558856	−0.0279428	−	0.999610i	$-0.508896\pi$
$523$	−1.82834e8	−0.0558856	−0.0279428	+	0.999610i	$0.508896\pi$
$524$	−5.01776e9	−1.52353
$525$	1.14650e9	0.345794
$526$	5.27483e9	1.58037
$527$	−5.58105e8	−0.166104
$528$	−1.28437e9	−0.379726
$529$	1.48036e8	0.0434783
$530$	1.13573e8	0.0331369
$531$	−9.47194e8	−0.274542
$532$	1.06450e10	3.06516
$533$	−7.00048e9	−2.00255
$534$	−7.45751e9	−2.11933
$535$	1.12306e9	0.317077
$536$	1.71003e10	4.79653
$537$	1.10504e9	0.307941
$538$	−1.33696e10	−3.70154
$539$	4.48720e8	0.123428
$540$	3.61832e8	0.0988846
$541$	−1.96073e9	−0.532387	−0.266194	−	0.963920i	$-0.585766\pi$
$541$	−1.96073e9	−0.532387	−0.266194	+	0.963920i	$0.585766\pi$
$542$	1.78288e9	0.480978
$543$	1.21050e9	0.324463
$544$	−1.55597e9	−0.414386
$545$	−1.09282e9	−0.289174
$546$	−4.08726e9	−1.07463
$547$	−2.88969e9	−0.754910	−0.377455	−	0.926028i	$-0.623201\pi$
$547$	−2.88969e9	−0.754910	−0.377455	+	0.926028i	$0.623201\pi$
$548$	1.07070e10	2.77930
$549$	8.79767e8	0.226916
$550$	1.44038e9	0.369154
$551$	4.07241e9	1.03710
$552$	1.47084e9	0.372203
$553$	4.56486e8	0.114786
$554$	−1.13861e10	−2.84505
$555$	−2.37059e8	−0.0588615
$556$	1.47547e10	3.64057
$557$	4.50918e9	1.10562	0.552808	−	0.833309i	$-0.313556\pi$
$557$	4.50918e9	1.10562	0.552808	+	0.833309i	$0.313556\pi$
$558$	3.25447e9	0.792977
$559$	9.39344e7	0.0227449
$560$	−1.65307e9	−0.397772
$561$	6.47134e7	0.0154748
$562$	−2.07166e9	−0.492313
$563$	−2.34598e9	−0.554046	−0.277023	−	0.960863i	$-0.589348\pi$
$563$	−2.34598e9	−0.554046	−0.277023	+	0.960863i	$0.589348\pi$
$564$	3.39731e9	0.797366
$565$	1.66002e9	0.387208
$566$	−6.56108e9	−1.52096
$567$	−3.00369e8	−0.0692014
$568$	−2.90984e9	−0.666270
$569$	5.31334e9	1.20913	0.604567	−	0.796554i	$-0.293346\pi$
$569$	5.31334e9	1.20913	0.604567	+	0.796554i	$0.293346\pi$
$570$	−1.78466e9	−0.403639
$571$	−6.82398e9	−1.53395	−0.766975	−	0.641677i	$-0.778239\pi$
$571$	−6.82398e9	−1.53395	−0.766975	+	0.641677i	$0.778239\pi$
$572$	−3.71804e9	−0.830669
$573$	2.92889e9	0.650373
$574$	6.85278e9	1.51243
$575$	−9.14101e8	−0.200520
$576$	4.08676e9	0.891048
$577$	2.32589e9	0.504050	0.252025	−	0.967721i	$-0.418903\pi$
$577$	2.32589e9	0.504050	0.252025	+	0.967721i	$0.418903\pi$
$578$	−8.68169e9	−1.87006
$579$	2.37951e8	0.0509464
$580$	−1.33509e9	−0.284126
$581$	2.49083e9	0.526898
$582$	−1.49903e9	−0.315196
$583$	−8.57644e7	−0.0179253
$584$	−1.25954e10	−2.61677
$585$	4.96160e8	0.102465
$586$	1.19340e10	2.44987
$587$	−2.10001e8	−0.0428536	−0.0214268	−	0.999770i	$-0.506821\pi$
$587$	−2.10001e8	−0.0428536	−0.0214268	+	0.999770i	$0.506821\pi$
$588$	−4.57153e9	−0.927344
$589$	−1.16227e10	−2.34371
$590$	−1.53160e9	−0.307017
$591$	2.81757e8	0.0561459
$592$	−8.57285e9	−1.69824
$593$	−6.51352e7	−0.0128270	−0.00641348	−	0.999979i	$-0.502041\pi$
$593$	−6.51352e7	−0.0128270	−0.00641348	+	0.999979i	$0.502041\pi$
$594$	−3.77362e8	−0.0738763
$595$	8.32908e7	0.0162102
$596$	2.03445e10	3.93627
$597$	−3.81338e9	−0.733500
$598$	3.25876e9	0.623157
$599$	−2.77059e7	−0.00526718	−0.00263359	−	0.999997i	$-0.500838\pi$
$599$	−2.77059e7	−0.00526718	−0.00263359	+	0.999997i	$0.500838\pi$
$600$	−9.08225e9	−1.71658
$601$	6.25558e8	0.117546	0.0587729	−	0.998271i	$-0.481281\pi$
$601$	6.25558e8	0.117546	0.0587729	+	0.998271i	$0.481281\pi$
$602$	−9.19525e7	−0.0171781
$603$	2.78428e9	0.517132
$604$	5.35897e9	0.989583
$605$	−1.02318e9	−0.187848
$606$	7.95865e9	1.45273
$607$	5.77739e9	1.04851	0.524253	−	0.851562i	$-0.324344\pi$
$607$	5.77739e9	1.04851	0.524253	+	0.851562i	$0.324344\pi$
$608$	−3.24036e10	−5.84696
$609$	1.10830e9	0.198837
$610$	1.42257e9	0.253757
$611$	4.65854e9	0.826239
$612$	−6.59296e8	−0.116265
$613$	−3.43198e8	−0.0601774	−0.0300887	−	0.999547i	$-0.509579\pi$
$613$	−3.43198e8	−0.0601774	−0.0300887	+	0.999547i	$0.509579\pi$
$614$	1.45022e10	2.52838
$615$	−8.31872e8	−0.144209
$616$	2.25259e9	0.388284
$617$	6.69767e9	1.14796	0.573978	−	0.818871i	$-0.305399\pi$
$617$	6.69767e9	1.14796	0.573978	+	0.818871i	$0.305399\pi$
$618$	1.04079e10	1.77380
$619$	−9.34901e9	−1.58434	−0.792170	−	0.610301i	$-0.791049\pi$
$619$	−9.34901e9	−1.58434	−0.792170	+	0.610301i	$0.791049\pi$
$620$	3.81035e9	0.642087
$621$	2.39483e8	0.0401286
$622$	−1.05085e10	−1.75096
$623$	7.24814e9	1.20093
$624$	1.79428e10	2.95627
$625$	5.41043e9	0.886446
$626$	−1.00209e10	−1.63266
$627$	1.34767e9	0.218348
$628$	−1.14425e10	−1.84357
$629$	4.31947e8	0.0692075
$630$	−4.85692e8	−0.0773872
$631$	1.13983e10	1.80608	0.903038	−	0.429561i	$-0.141332\pi$
$631$	1.13983e10	1.80608	0.903038	+	0.429561i	$0.141332\pi$
$632$	−3.61615e9	−0.569819
$633$	1.43626e9	0.225072
$634$	−9.79601e8	−0.152664
$635$	1.07254e9	0.166228
$636$	8.73763e8	0.134677
$637$	−6.26869e9	−0.960924
$638$	1.39239e9	0.212270
$639$	−4.73781e8	−0.0718331
$640$	2.55992e9	0.386007
$641$	−1.05373e10	−1.58025	−0.790124	−	0.612947i	$-0.789984\pi$
$641$	−1.05373e10	−1.58025	−0.790124	+	0.612947i	$0.789984\pi$
$642$	1.19327e10	1.77979
$643$	9.34452e9	1.38618	0.693089	−	0.720852i	$-0.256249\pi$
$643$	9.34452e9	1.38618	0.693089	+	0.720852i	$0.256249\pi$
$644$	−2.30978e9	−0.340776
$645$	1.11623e7	0.00163793
$646$	3.25184e9	0.474586
$647$	−6.73299e9	−0.977334	−0.488667	−	0.872471i	$-0.662517\pi$
$647$	−6.73299e9	−0.977334	−0.488667	+	0.872471i	$0.662517\pi$
$648$	2.37944e9	0.343527
$649$	1.15658e9	0.166080
$650$	−2.01224e10	−2.87397
$651$	−3.16310e9	−0.449345
$652$	−2.49185e10	−3.52092
$653$	2.89741e9	0.407205	0.203603	−	0.979054i	$-0.434735\pi$
$653$	2.89741e9	0.407205	0.203603	+	0.979054i	$0.434735\pi$
$654$	−1.16114e10	−1.62316
$655$	−8.17625e8	−0.113687
$656$	−3.00833e10	−4.16065
$657$	−2.05078e9	−0.282125
$658$	−4.56025e9	−0.624020
$659$	4.09714e9	0.557675	0.278838	−	0.960338i	$-0.410051\pi$
$659$	4.09714e9	0.557675	0.278838	+	0.960338i	$0.410051\pi$
$660$	−4.41817e8	−0.0598189
$661$	1.19350e10	1.60738	0.803691	−	0.595047i	$-0.202867\pi$
$661$	1.19350e10	1.60738	0.803691	+	0.595047i	$0.202867\pi$
$662$	1.23793e10	1.65842
$663$	−9.04057e8	−0.120475
$664$	−1.97316e10	−2.61561
$665$	1.73456e9	0.228724
$666$	−2.51880e9	−0.330396
$667$	−8.83643e8	−0.115302
$668$	3.52849e10	4.58005
$669$	−4.60915e9	−0.595155
$670$	4.50213e9	0.578304
$671$	−1.07424e9	−0.137270
$672$	−8.81858e9	−1.12100
$673$	−9.75181e9	−1.23320	−0.616599	−	0.787278i	$-0.711490\pi$
$673$	−9.75181e9	−1.23320	−0.616599	+	0.787278i	$0.711490\pi$
$674$	1.49658e10	1.88273
$675$	−1.47878e9	−0.185071
$676$	3.08656e10	3.84292
$677$	−1.09587e10	−1.35737	−0.678684	−	0.734430i	$-0.737449\pi$
$677$	−1.09587e10	−1.35737	−0.678684	+	0.734430i	$0.737449\pi$
$678$	1.76381e10	2.17344
$679$	1.45695e9	0.178608
$680$	−6.59805e8	−0.0804701
$681$	−5.28993e9	−0.641852
$682$	−3.97389e9	−0.479701
$683$	−5.39763e9	−0.648233	−0.324116	−	0.946017i	$-0.605067\pi$
$683$	−5.39763e9	−0.648233	−0.324116	+	0.946017i	$0.605067\pi$
$684$	−1.37300e10	−1.64050
$685$	1.74466e9	0.207393
$686$	1.61616e10	1.91139
$687$	−5.49788e9	−0.646914
$688$	4.03665e8	0.0472566
$689$	1.19814e9	0.139554
$690$	3.87240e8	0.0448754
$691$	−1.91440e9	−0.220729	−0.110364	−	0.993891i	$-0.535202\pi$
$691$	−1.91440e9	−0.220729	−0.110364	+	0.993891i	$0.535202\pi$
$692$	2.99687e10	3.43792
$693$	3.66768e8	0.0418625
$694$	2.62065e10	2.97612
$695$	2.40422e9	0.271661
$696$	−8.77963e9	−0.987061
$697$	1.51576e9	0.169557
$698$	−2.82183e10	−3.14077
$699$	8.67879e8	0.0961145
$700$	1.42625e10	1.57164
$701$	1.00810e10	1.10532	0.552661	−	0.833406i	$-0.313612\pi$
$701$	1.00810e10	1.10532	0.552661	+	0.833406i	$0.313612\pi$
$702$	5.27181e9	0.575148
$703$	8.99542e9	0.976512
$704$	−4.99017e9	−0.539027
$705$	5.53578e8	0.0594999
$706$	−1.72417e10	−1.84401
$707$	−7.73521e9	−0.823198
$708$	−1.17831e10	−1.24780
$709$	−1.08851e9	−0.114702	−0.0573509	−	0.998354i	$-0.518265\pi$
$709$	−1.08851e9	−0.114702	−0.0573509	+	0.998354i	$0.518265\pi$
$710$	−7.66097e8	−0.0803303
$711$	−5.88783e8	−0.0614344
$712$	−5.74177e10	−5.96164
$713$	2.52193e9	0.260567
$714$	8.84982e8	0.0909894
$715$	−6.05840e8	−0.0619850
$716$	1.37467e10	1.39960
$717$	1.19781e9	0.121359
$718$	6.13186e8	0.0618240
$719$	1.83367e10	1.83980	0.919899	−	0.392156i	$-0.128271\pi$
$719$	1.83367e10	1.83980	0.919899	+	0.392156i	$0.128271\pi$
$720$	2.13216e9	0.212890
$721$	−1.01157e10	−1.00513
$722$	4.84683e10	4.79267
$723$	2.94714e9	0.290012
$724$	1.50587e10	1.47469
$725$	5.45638e9	0.531767
$726$	−1.08715e10	−1.05441
$727$	1.27508e9	0.123074	0.0615369	−	0.998105i	$-0.480400\pi$
$727$	1.27508e9	0.123074	0.0615369	+	0.998105i	$0.480400\pi$
$728$	−3.14691e10	−3.02290
$729$	3.87420e8	0.0370370
$730$	−3.31608e9	−0.315497
$731$	−2.03389e7	−0.00192582
$732$	1.09443e10	1.03134
$733$	−1.44780e10	−1.35782	−0.678912	−	0.734220i	$-0.737548\pi$
$733$	−1.44780e10	−1.35782	−0.678912	+	0.734220i	$0.737548\pi$
$734$	−3.17978e9	−0.296798
$735$	−7.44913e8	−0.0691990
$736$	7.03102e9	0.650049
$737$	−3.39976e9	−0.312832
$738$	−8.83881e9	−0.809462
$739$	−1.82585e9	−0.166422	−0.0832109	−	0.996532i	$-0.526518\pi$
$739$	−1.82585e9	−0.166422	−0.0832109	+	0.996532i	$0.526518\pi$
$740$	−2.94903e9	−0.267527
$741$	−1.88272e10	−1.69990
$742$	−1.17286e9	−0.105398
$743$	−1.29792e9	−0.116088	−0.0580439	−	0.998314i	$-0.518486\pi$
$743$	−1.29792e9	−0.116088	−0.0580439	+	0.998314i	$0.518486\pi$
$744$	2.50572e10	2.23062
$745$	3.31505e9	0.293727
$746$	−2.53722e10	−2.23754
$747$	−3.21270e9	−0.282000
$748$	8.05037e8	0.0703332
$749$	−1.15977e10	−1.00853
$750$	−4.87765e9	−0.422178
$751$	−1.52504e10	−1.31383	−0.656917	−	0.753963i	$-0.728140\pi$
$751$	−1.52504e10	−1.31383	−0.656917	+	0.753963i	$0.728140\pi$
$752$	2.00192e10	1.71666
$753$	−3.69201e9	−0.315123
$754$	−1.94519e10	−1.65258
$755$	8.73224e8	0.0738433
$756$	−3.73661e9	−0.314522
$757$	−1.60955e9	−0.134856	−0.0674280	−	0.997724i	$-0.521479\pi$
$757$	−1.60955e9	−0.134856	−0.0674280	+	0.997724i	$0.521479\pi$
$758$	−3.24776e10	−2.70858
$759$	−2.92422e8	−0.0242753
$760$	−1.37406e10	−1.13543
$761$	−1.60947e9	−0.132384	−0.0661920	−	0.997807i	$-0.521085\pi$
$761$	−1.60947e9	−0.132384	−0.0661920	+	0.997807i	$0.521085\pi$
$762$	1.13959e10	0.933056
$763$	1.12854e10	0.919775
$764$	3.64356e10	2.95596
$765$	−1.07430e8	−0.00867579
$766$	−1.59922e9	−0.128560
$767$	−1.61576e10	−1.29298
$768$	7.82535e9	0.623361
$769$	−1.07804e10	−0.854855	−0.427428	−	0.904050i	$-0.640580\pi$
$769$	−1.07804e10	−0.854855	−0.427428	+	0.904050i	$0.640580\pi$
$770$	5.93057e8	0.0468144
$771$	1.16503e10	0.915476
$772$	2.96013e9	0.231553
$773$	−2.94937e9	−0.229668	−0.114834	−	0.993385i	$-0.536634\pi$
$773$	−2.94937e9	−0.229668	−0.114834	+	0.993385i	$0.536634\pi$
$774$	1.18602e8	0.00919385
$775$	−1.55726e10	−1.20172
$776$	−1.15415e10	−0.886638
$777$	2.44809e9	0.187221
$778$	3.36703e10	2.56341
$779$	3.15661e10	2.39243
$780$	6.17226e9	0.465707
$781$	5.78514e8	0.0434545
$782$	−7.05593e8	−0.0527631
$783$	−1.42950e9	−0.106419
$784$	−2.69385e10	−1.99649
$785$	−1.86450e9	−0.137569
$786$	−8.68743e9	−0.638135
$787$	−1.46253e10	−1.06953	−0.534765	−	0.845001i	$-0.679600\pi$
$787$	−1.46253e10	−1.06953	−0.534765	+	0.845001i	$0.679600\pi$
$788$	3.50507e9	0.255185
$789$	6.61255e9	0.479291
$790$	−9.52052e8	−0.0687014
$791$	−1.71429e10	−1.23159
$792$	−2.90542e9	−0.207812
$793$	1.50074e10	1.06868
$794$	−7.76717e9	−0.550669
$795$	1.42376e8	0.0100497
$796$	−4.74387e10	−3.33378
$797$	4.11710e9	0.288063	0.144031	−	0.989573i	$-0.453993\pi$
$797$	4.11710e9	0.288063	0.144031	+	0.989573i	$0.453993\pi$
$798$	1.84300e10	1.28385
$799$	−1.00868e9	−0.0699582
$800$	−4.34156e10	−2.99799
$801$	−9.34876e9	−0.642747
$802$	−2.57666e10	−1.76379
$803$	2.50412e9	0.170668
$804$	3.46365e10	2.35038
$805$	−3.76369e8	−0.0254289
$806$	5.55159e10	3.73461
$807$	−1.67602e10	−1.12259
$808$	6.12761e10	4.08650
$809$	−9.51685e9	−0.631936	−0.315968	−	0.948770i	$-0.602329\pi$
$809$	−9.51685e9	−0.631936	−0.315968	+	0.948770i	$0.602329\pi$
$810$	6.26453e8	0.0414181
$811$	−1.01219e10	−0.666327	−0.333164	−	0.942869i	$-0.608116\pi$
$811$	−1.01219e10	−0.666327	−0.333164	+	0.942869i	$0.608116\pi$
$812$	1.37873e10	0.903719
$813$	2.23503e9	0.145870
$814$	3.07560e9	0.199869
$815$	−4.06037e9	−0.262733
$816$	−3.88502e9	−0.250309
$817$	−4.23563e8	−0.0271732
$818$	1.38905e10	0.887325
$819$	−5.12381e9	−0.325911
$820$	−1.03485e10	−0.655436
$821$	2.18569e10	1.37844	0.689218	−	0.724554i	$-0.257954\pi$
$821$	2.18569e10	1.37844	0.689218	+	0.724554i	$0.257954\pi$
$822$	1.85374e10	1.16412
$823$	−3.86655e9	−0.241782	−0.120891	−	0.992666i	$-0.538575\pi$
$823$	−3.86655e9	−0.241782	−0.120891	+	0.992666i	$0.538575\pi$
$824$	8.01338e10	4.98966
$825$	1.80567e9	0.111956
$826$	1.58167e10	0.976529
$827$	1.12726e10	0.693035	0.346518	−	0.938044i	$-0.387364\pi$
$827$	1.12726e10	0.693035	0.346518	+	0.938044i	$0.387364\pi$
$828$	2.97918e9	0.182386
$829$	−1.59797e10	−0.974154	−0.487077	−	0.873359i	$-0.661937\pi$
$829$	−1.59797e10	−0.974154	−0.487077	+	0.873359i	$0.661937\pi$
$830$	−5.19489e9	−0.315357
$831$	−1.42736e10	−0.862840
$832$	6.97135e10	4.19648
$833$	1.35731e9	0.0813620
$834$	2.55454e10	1.52486
$835$	5.74953e9	0.341766
$836$	1.67651e10	0.992396
$837$	4.07981e9	0.240492
$838$	−4.04873e10	−2.37665
$839$	2.40926e9	0.140837	0.0704186	−	0.997518i	$-0.477567\pi$
$839$	2.40926e9	0.140837	0.0704186	+	0.997518i	$0.477567\pi$
$840$	−3.73949e9	−0.217688
$841$	−1.19753e10	−0.694226
$842$	2.89879e10	1.67350
$843$	−2.59704e9	−0.149308
$844$	1.78672e10	1.02296
$845$	5.02943e9	0.286761
$846$	5.88188e9	0.333979
$847$	1.05663e10	0.597488
$848$	5.14880e9	0.289948
$849$	−8.22500e9	−0.461274
$850$	4.35694e9	0.243341
$851$	−1.95185e9	−0.108566
$852$	−5.89386e9	−0.326484
$853$	1.71764e10	0.947570	0.473785	−	0.880640i	$-0.342887\pi$
$853$	1.71764e10	0.947570	0.473785	+	0.880640i	$0.342887\pi$
$854$	−1.46907e10	−0.807126
$855$	−2.23725e9	−0.122415
$856$	9.18738e10	5.00649
$857$	−4.41092e9	−0.239385	−0.119692	−	0.992811i	$-0.538191\pi$
$857$	−4.41092e9	−0.239385	−0.119692	+	0.992811i	$0.538191\pi$
$858$	−6.43717e9	−0.347928
$859$	2.39855e10	1.29114	0.645570	−	0.763701i	$-0.276620\pi$
$859$	2.39855e10	1.29114	0.645570	+	0.763701i	$0.276620\pi$
$860$	1.38859e8	0.00744441
$861$	8.59067e9	0.458687
$862$	−7.77839e9	−0.413632
$863$	−1.76729e10	−0.935985	−0.467993	−	0.883732i	$-0.655023\pi$
$863$	−1.76729e10	−0.935985	−0.467993	+	0.883732i	$0.655023\pi$
$864$	1.13743e10	0.599968
$865$	4.88327e9	0.256540
$866$	−4.38268e10	−2.29312
$867$	−1.08834e10	−0.567150
$868$	−3.93491e10	−2.04228
$869$	7.18937e8	0.0371639
$870$	−2.31148e9	−0.119007
$871$	4.74952e10	2.43549
$872$	−8.93997e10	−4.56592
$873$	−1.87919e9	−0.0955920
$874$	−1.46942e10	−0.744483
$875$	4.74071e9	0.239230
$876$	−2.55119e10	−1.28226
$877$	−1.28667e10	−0.644122	−0.322061	−	0.946719i	$-0.604376\pi$
$877$	−1.28667e10	−0.644122	−0.322061	+	0.946719i	$0.604376\pi$
$878$	−5.42310e10	−2.70407
$879$	1.49605e10	0.742992
$880$	−2.60348e9	−0.128785
$881$	−4.58163e8	−0.0225738	−0.0112869	−	0.999936i	$-0.503593\pi$
$881$	−4.58163e8	−0.0225738	−0.0112869	+	0.999936i	$0.503593\pi$
$882$	−7.91486e9	−0.388421
$883$	−3.51665e9	−0.171897	−0.0859483	−	0.996300i	$-0.527392\pi$
$883$	−3.51665e9	−0.171897	−0.0859483	+	0.996300i	$0.527392\pi$
$884$	−1.12465e10	−0.547564
$885$	−1.92002e9	−0.0931115
$886$	1.90780e10	0.921539
$887$	−8.81338e9	−0.424043	−0.212021	−	0.977265i	$-0.568005\pi$
$887$	−8.81338e9	−0.424043	−0.212021	+	0.977265i	$0.568005\pi$
$888$	−1.93930e10	−0.929396
$889$	−1.10760e10	−0.528722
$890$	−1.51168e10	−0.718778
$891$	−4.73062e8	−0.0224051
$892$	−5.73381e10	−2.70499
$893$	−2.10060e10	−0.987104
$894$	3.52231e10	1.64872
$895$	2.23998e9	0.104439
$896$	−2.64360e10	−1.22777
$897$	4.08519e9	0.188990
$898$	−1.77876e10	−0.819690
$899$	−1.50537e10	−0.691009
$900$	−1.83960e10	−0.841154
$901$	−2.59424e8	−0.0118161
$902$	1.07927e10	0.489674
$903$	−1.15272e8	−0.00520975
$904$	1.35801e11	6.11383
$905$	2.45375e9	0.110042
$906$	9.27818e9	0.414490
$907$	2.98196e10	1.32701	0.663507	−	0.748170i	$-0.269067\pi$
$907$	2.98196e10	1.32701	0.663507	+	0.748170i	$0.269067\pi$
$908$	−6.58070e10	−2.91724
$909$	9.97699e9	0.440581
$910$	−8.28511e9	−0.364463
$911$	−1.93274e10	−0.846951	−0.423475	−	0.905908i	$-0.639190\pi$
$911$	−1.93274e10	−0.846951	−0.423475	+	0.905908i	$0.639190\pi$
$912$	−8.09066e10	−3.53185
$913$	3.92289e9	0.170592
$914$	−2.71599e10	−1.17656
$915$	1.78334e9	0.0769591
$916$	−6.83939e10	−2.94024
$917$	8.44354e9	0.361603
$918$	−1.14146e9	−0.0486981
$919$	1.92553e9	0.0818361	0.0409180	−	0.999163i	$-0.486972\pi$
$919$	1.92553e9	0.0818361	0.0409180	+	0.999163i	$0.486972\pi$
$920$	2.98148e9	0.126234
$921$	1.81800e10	0.766803
$922$	−6.28517e10	−2.64094
$923$	−8.08193e9	−0.338306
$924$	4.56261e9	0.190266
$925$	1.20524e10	0.500701
$926$	1.85076e10	0.765969
$927$	1.30474e10	0.537954
$928$	−4.19690e10	−1.72389
$929$	−2.99271e10	−1.22464	−0.612321	−	0.790609i	$-0.709764\pi$
$929$	−2.99271e10	−1.22464	−0.612321	+	0.790609i	$0.709764\pi$
$930$	6.59699e9	0.268940
$931$	2.82664e10	1.14801
$932$	1.07965e10	0.436843
$933$	−1.31736e10	−0.531028
$934$	7.44954e10	2.99168
$935$	1.31178e8	0.00524831
$936$	4.05893e10	1.61788
$937$	−6.02696e9	−0.239337	−0.119668	−	0.992814i	$-0.538183\pi$
$937$	−6.02696e9	−0.239337	−0.119668	+	0.992814i	$0.538183\pi$
$938$	−4.64931e10	−1.83941
$939$	−1.25622e10	−0.495149
$940$	6.88653e9	0.270429
$941$	1.83486e10	0.717860	0.358930	−	0.933364i	$-0.383142\pi$
$941$	1.83486e10	0.717860	0.358930	+	0.933364i	$0.383142\pi$
$942$	−1.98107e10	−0.772187
$943$	−6.84930e9	−0.265984
$944$	−6.94342e10	−2.68641
$945$	−6.08865e8	−0.0234698
$946$	−1.44819e8	−0.00556170
$947$	1.62700e10	0.622533	0.311267	−	0.950323i	$-0.399247\pi$
$947$	1.62700e10	0.622533	0.311267	+	0.950323i	$0.399247\pi$
$948$	−7.32449e9	−0.279221
$949$	−3.49830e10	−1.32870
$950$	9.07345e10	3.43352
$951$	−1.22803e9	−0.0462997
$952$	6.81375e9	0.255951
$953$	4.87166e10	1.82327	0.911636	−	0.410998i	$-0.134820\pi$
$953$	4.87166e10	1.82327	0.911636	+	0.410998i	$0.134820\pi$
$954$	1.51278e9	0.0564099
$955$	5.93703e9	0.220576
$956$	1.49008e10	0.551580
$957$	1.74550e9	0.0643767
$958$	6.27200e10	2.30477
$959$	−1.80170e10	−0.659656
$960$	8.28410e9	0.302201
$961$	1.54507e10	0.561587
$962$	−4.29667e10	−1.55603
$963$	1.49589e10	0.539770
$964$	3.66625e10	1.31811
$965$	4.82341e8	0.0172786
$966$	−3.99900e9	−0.142735
$967$	3.96535e10	1.41023	0.705113	−	0.709095i	$-0.250896\pi$
$967$	3.96535e10	1.41023	0.705113	+	0.709095i	$0.250896\pi$
$968$	−8.37028e10	−2.96603
$969$	4.07651e9	0.143931
$970$	−3.03862e9	−0.106900
$971$	4.46486e10	1.56509	0.782547	−	0.622591i	$-0.213920\pi$
$971$	4.46486e10	1.56509	0.782547	+	0.622591i	$0.213920\pi$
$972$	4.81953e9	0.168334
$973$	−2.48282e10	−0.864073
$974$	3.78536e10	1.31266
$975$	−2.52255e10	−0.871612
$976$	6.44915e10	2.22038
$977$	2.37557e10	0.814962	0.407481	−	0.913214i	$-0.366407\pi$
$977$	2.37557e10	0.814962	0.407481	+	0.913214i	$0.366407\pi$
$978$	−4.31423e10	−1.47475
$979$	1.14154e10	0.388821
$980$	−9.26675e9	−0.314511
$981$	−1.45561e10	−0.492270
$982$	−2.09944e10	−0.707479
$983$	2.14111e10	0.718955	0.359477	−	0.933154i	$-0.382955\pi$
$983$	2.14111e10	0.718955	0.359477	+	0.933154i	$0.382955\pi$
$984$	−6.80527e10	−2.27700
$985$	5.71137e8	0.0190420
$986$	4.21176e9	0.139925
$987$	−5.71675e9	−0.189251
$988$	−2.34212e11	−7.72609
$989$	9.19058e7	0.00302104
$990$	−7.64934e8	−0.0250554
$991$	1.26347e10	0.412387	0.206194	−	0.978511i	$-0.433892\pi$
$991$	1.26347e10	0.412387	0.206194	+	0.978511i	$0.433892\pi$
$992$	1.19780e11	3.89577
$993$	1.55188e10	0.502963
$994$	7.91141e9	0.255506
$995$	−7.72994e9	−0.248769
$996$	−3.99662e10	−1.28170
$997$	−6.85313e9	−0.219006	−0.109503	−	0.993986i	$-0.534926\pi$
$997$	−6.85313e9	−0.219006	−0.109503	+	0.993986i	$0.534926\pi$
$998$	−1.01478e9	−0.0323157
$999$	−3.15758e9	−0.100202

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.d.1.8	✓	8
3.2	odd	2		207.8.a.e.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.d.1.8	✓	8	1.1	even	1	trivial
207.8.a.e.1.1		8	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)

\(f(q)\)	\(=\)	\(q+21.5379 q^{2} +27.0000 q^{3} +335.881 q^{4} +54.7305 q^{5} +581.523 q^{6} -565.198 q^{7} +4477.33 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+21.5379 q^{2} +27.0000 q^{3} +335.881 q^{4} +54.7305 q^{5} +581.523 q^{6} -565.198 q^{7} +4477.33 q^{8} +729.000 q^{9} +1178.78 q^{10} -890.150 q^{11} +9068.80 q^{12} +12435.5 q^{13} -12173.2 q^{14} +1477.72 q^{15} +53439.5 q^{16} -2692.57 q^{17} +15701.1 q^{18} -56073.6 q^{19} +18383.0 q^{20} -15260.3 q^{21} -19172.0 q^{22} +12167.0 q^{23} +120888. q^{24} -75129.6 q^{25} +267836. q^{26} +19683.0 q^{27} -189839. q^{28} -72626.2 q^{29} +31827.1 q^{30} +207276. q^{31} +577876. q^{32} -24034.0 q^{33} -57992.3 q^{34} -30933.6 q^{35} +244857. q^{36} -160422. q^{37} -1.20771e6 q^{38} +335760. q^{39} +245047. q^{40} -562941. q^{41} -328676. q^{42} +7553.70 q^{43} -298985. q^{44} +39898.6 q^{45} +262052. q^{46} +374615. q^{47} +1.44287e6 q^{48} -504095. q^{49} -1.61813e6 q^{50} -72699.4 q^{51} +4.17687e6 q^{52} +96348.3 q^{53} +423931. q^{54} -48718.4 q^{55} -2.53058e6 q^{56} -1.51399e6 q^{57} -1.56422e6 q^{58} -1.29931e6 q^{59} +496340. q^{60} +1.20681e6 q^{61} +4.46429e6 q^{62} -412029. q^{63} +5.60599e6 q^{64} +680604. q^{65} -517643. q^{66} +3.81931e6 q^{67} -904384. q^{68} +328509. q^{69} -666244. q^{70} -649906. q^{71} +3.26397e6 q^{72} -2.81315e6 q^{73} -3.45515e6 q^{74} -2.02850e6 q^{75} -1.88341e7 q^{76} +503111. q^{77} +7.23156e6 q^{78} -807658. q^{79} +2.92477e6 q^{80} +531441. q^{81} -1.21246e7 q^{82} -4.40700e6 q^{83} -5.12566e6 q^{84} -147366. q^{85} +162691. q^{86} -1.96091e6 q^{87} -3.98549e6 q^{88} -1.28241e7 q^{89} +859331. q^{90} -7.02854e6 q^{91} +4.08667e6 q^{92} +5.59645e6 q^{93} +8.06842e6 q^{94} -3.06894e6 q^{95} +1.56027e7 q^{96} -2.57777e6 q^{97} -1.08571e7 q^{98} -648919. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9}+O(q^{10}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9	\( 8 q + 24 q^{2} + 216 q^{3} + 562 q^{4} + 378 q^{5} + 648 q^{6} + 126 q^{7} + 4188 q^{8} + 5832 q^{9} + 11720 q^{10} + 6932 q^{11} + 15174 q^{12} + 12404 q^{13} + 30222 q^{14} + 10206 q^{15} + 27058 q^{16} + 24434 q^{17} + 17496 q^{18} - 14682 q^{19} - 3760 q^{20} + 3402 q^{21} + 36294 q^{22} + 97336 q^{23} + 113076 q^{24} + 144644 q^{25} + 325840 q^{26} + 157464 q^{27} - 21566 q^{28} + 255356 q^{29} + 316440 q^{30} + 450764 q^{31} + 647588 q^{32} + 187164 q^{33} + 191822 q^{34} + 1022616 q^{35} + 409698 q^{36} + 206240 q^{37} + 737372 q^{38} + 334908 q^{39} + 590028 q^{40} + 1053344 q^{41} + 815994 q^{42} + 1587806 q^{43} + 589366 q^{44} + 275562 q^{45} + 292008 q^{46} + 443336 q^{47} + 730566 q^{48} + 1944828 q^{49} - 1556112 q^{50} + 659718 q^{51} - 614236 q^{52} - 375530 q^{53} + 472392 q^{54} + 407792 q^{55} - 1316922 q^{56} - 396414 q^{57} - 1413384 q^{58} + 624008 q^{59} - 101520 q^{60} - 2005568 q^{61} - 3908272 q^{62} + 91854 q^{63} - 5082310 q^{64} + 646124 q^{65} + 979938 q^{66} - 2712286 q^{67} - 2289698 q^{68} + 2628072 q^{69} - 16499468 q^{70} - 6287176 q^{71} + 3053052 q^{72} - 10358312 q^{73} - 2000150 q^{74} + 3905388 q^{75} - 25107464 q^{76} - 2156840 q^{77} + 8797680 q^{78} - 8800574 q^{79} + 2384344 q^{80} + 4251528 q^{81} - 31799800 q^{82} + 384948 q^{83} - 582282 q^{84} - 17826684 q^{85} - 11563928 q^{86} + 6894612 q^{87} - 25202782 q^{88} - 3445530 q^{89} + 8543880 q^{90} - 16316740 q^{91} + 6837854 q^{92} + 12170628 q^{93} - 24237616 q^{94} + 26164288 q^{95} + 17484876 q^{96} - 28043764 q^{97} - 9998012 q^{98} + 5053428 q^{99}+O(q^{100}) \) 8 * q + 24 * q^2 + 216 * q^3 + 562 * q^4 + 378 * q^5 + 648 * q^6 + 126 * q^7 + 4188 * q^8 + 5832 * q^9 + 11720 * q^10 + 6932 * q^11 + 15174 * q^12 + 12404 * q^13 + 30222 * q^14 + 10206 * q^15 + 27058 * q^16 + 24434 * q^17 + 17496 * q^18 - 14682 * q^19 - 3760 * q^20 + 3402 * q^21 + 36294 * q^22 + 97336 * q^23 + 113076 * q^24 + 144644 * q^25 + 325840 * q^26 + 157464 * q^27 - 21566 * q^28 + 255356 * q^29 + 316440 * q^30 + 450764 * q^31 + 647588 * q^32 + 187164 * q^33 + 191822 * q^34 + 1022616 * q^35 + 409698 * q^36 + 206240 * q^37 + 737372 * q^38 + 334908 * q^39 + 590028 * q^40 + 1053344 * q^41 + 815994 * q^42 + 1587806 * q^43 + 589366 * q^44 + 275562 * q^45 + 292008 * q^46 + 443336 * q^47 + 730566 * q^48 + 1944828 * q^49 - 1556112 * q^50 + 659718 * q^51 - 614236 * q^52 - 375530 * q^53 + 472392 * q^54 + 407792 * q^55 - 1316922 * q^56 - 396414 * q^57 - 1413384 * q^58 + 624008 * q^59 - 101520 * q^60 - 2005568 * q^61 - 3908272 * q^62 + 91854 * q^63 - 5082310 * q^64 + 646124 * q^65 + 979938 * q^66 - 2712286 * q^67 - 2289698 * q^68 + 2628072 * q^69 - 16499468 * q^70 - 6287176 * q^71 + 3053052 * q^72 - 10358312 * q^73 - 2000150 * q^74 + 3905388 * q^75 - 25107464 * q^76 - 2156840 * q^77 + 8797680 * q^78 - 8800574 * q^79 + 2384344 * q^80 + 4251528 * q^81 - 31799800 * q^82 + 384948 * q^83 - 582282 * q^84 - 17826684 * q^85 - 11563928 * q^86 + 6894612 * q^87 - 25202782 * q^88 - 3445530 * q^89 + 8543880 * q^90 - 16316740 * q^91 + 6837854 * q^92 + 12170628 * q^93 - 24237616 * q^94 + 26164288 * q^95 + 17484876 * q^96 - 28043764 * q^97 - 9998012 * q^98 + 5053428 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more