LMFDB - Embedded newform 69.8.a.a.1.2

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			1.2
Root			$-9.64907$ 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)

$f(q)$	$=$	$q-9.64907 q^{2} -27.0000 q^{3} -34.8955 q^{4} -306.939 q^{5} +260.525 q^{6} +733.069 q^{7} +1571.79 q^{8} +729.000 q^{9} +O(q^{10})$	\(q-9.64907 q^{2} -27.0000 q^{3} -34.8955 q^{4} -306.939 q^{5} +260.525 q^{6} +733.069 q^{7} +1571.79 q^{8} +729.000 q^{9} +2961.68 q^{10} +3882.47 q^{11} +942.178 q^{12} -5973.30 q^{13} -7073.43 q^{14} +8287.36 q^{15} -10699.7 q^{16} +14030.7 q^{17} -7034.17 q^{18} +11012.4 q^{19} +10710.8 q^{20} -19792.9 q^{21} -37462.2 q^{22} +12167.0 q^{23} -42438.3 q^{24} +16086.7 q^{25} +57636.8 q^{26} -19683.0 q^{27} -25580.8 q^{28} +116.864 q^{29} -79965.3 q^{30} -65552.7 q^{31} -97947.0 q^{32} -104827. q^{33} -135383. q^{34} -225008. q^{35} -25438.8 q^{36} -382473. q^{37} -106260. q^{38} +161279. q^{39} -482444. q^{40} -370414. q^{41} +190983. q^{42} +368875. q^{43} -135480. q^{44} -223759. q^{45} -117400. q^{46} +830724. q^{47} +288892. q^{48} -286153. q^{49} -155222. q^{50} -378828. q^{51} +208441. q^{52} -545061. q^{53} +189923. q^{54} -1.19168e6 q^{55} +1.15223e6 q^{56} -297336. q^{57} -1127.63 q^{58} +1.87374e6 q^{59} -289191. q^{60} -1.20373e6 q^{61} +632523. q^{62} +534407. q^{63} +2.31466e6 q^{64} +1.83344e6 q^{65} +1.01148e6 q^{66} -4.04596e6 q^{67} -489607. q^{68} -328509. q^{69} +2.17111e6 q^{70} +2.77968e6 q^{71} +1.14583e6 q^{72} -2.58877e6 q^{73} +3.69051e6 q^{74} -434342. q^{75} -384284. q^{76} +2.84611e6 q^{77} -1.55619e6 q^{78} +1.28761e6 q^{79} +3.28415e6 q^{80} +531441. q^{81} +3.57415e6 q^{82} +5.20417e6 q^{83} +690681. q^{84} -4.30657e6 q^{85} -3.55930e6 q^{86} -3155.33 q^{87} +6.10242e6 q^{88} -2.72740e6 q^{89} +2.15906e6 q^{90} -4.37884e6 q^{91} -424573. q^{92} +1.76992e6 q^{93} -8.01571e6 q^{94} -3.38015e6 q^{95} +2.64457e6 q^{96} -7.55774e6 q^{97} +2.76111e6 q^{98} +2.83032e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−9.64907	−0.852865	−0.426433	−	0.904519i	$-0.640230\pi$
$2$	−9.64907	−0.852865	−0.426433	+	0.904519i	$0.640230\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−34.8955	−0.272621
$5$	−306.939	−1.09814	−0.549070	−	0.835777i	$-0.685018\pi$
$5$	−306.939	−1.09814	−0.549070	+	0.835777i	$0.685018\pi$
$6$	260.525	0.492402
$7$	733.069	0.807796	0.403898	−	0.914804i	$-0.367655\pi$
$7$	733.069	0.807796	0.403898	+	0.914804i	$0.367655\pi$
$8$	1571.79	1.08537
$9$	729.000	0.333333
$10$	2961.68	0.936565
$11$	3882.47	0.879495	0.439747	−	0.898122i	$-0.355068\pi$
$11$	3882.47	0.879495	0.439747	+	0.898122i	$0.355068\pi$
$12$	942.178	0.157398
$13$	−5973.30	−0.754071	−0.377036	−	0.926199i	$-0.623057\pi$
$13$	−5973.30	−0.754071	−0.377036	+	0.926199i	$0.623057\pi$
$14$	−7073.43	−0.688941
$15$	8287.36	0.634011
$16$	−10699.7	−0.653057
$17$	14030.7	0.692640	0.346320	−	0.938117i	$-0.387431\pi$
$17$	14030.7	0.692640	0.346320	+	0.938117i	$0.387431\pi$
$18$	−7034.17	−0.284288
$19$	11012.4	0.368337	0.184169	−	0.982895i	$-0.441041\pi$
$19$	11012.4	0.368337	0.184169	+	0.982895i	$0.441041\pi$
$20$	10710.8	0.299376
$21$	−19792.9	−0.466381
$22$	−37462.2	−0.750091
$23$	12167.0	0.208514
$23$	12167.0	0.208514
$24$	−42438.3	−0.626641
$25$	16086.7	0.205910
$26$	57636.8	0.643121
$27$	−19683.0	−0.192450
$28$	−25580.8	−0.220222
$29$	116.864	0.000889790 0	0.000444895	−	1.00000i	$-0.499858\pi$
$29$	116.864	0.000889790 0	0.000444895		1.00000i	$0.499858\pi$
$30$	−79965.3	−0.540726
$31$	−65552.7	−0.395207	−0.197604	−	0.980282i	$-0.563316\pi$
$31$	−65552.7	−0.395207	−0.197604	+	0.980282i	$0.563316\pi$
$32$	−97947.0	−0.528404
$33$	−104827.	−0.507777
$34$	−135383.	−0.590728
$35$	−225008.	−0.887072
$36$	−25438.8	−0.0908736
$37$	−382473.	−1.24135	−0.620676	−	0.784067i	$-0.713142\pi$
$37$	−382473.	−1.24135	−0.620676	+	0.784067i	$0.713142\pi$
$38$	−106260.	−0.314142
$39$	161279.	0.435363
$40$	−482444.	−1.19189
$41$	−370414.	−0.839352	−0.419676	−	0.907674i	$-0.637856\pi$
$41$	−370414.	−0.839352	−0.419676	+	0.907674i	$0.637856\pi$
$42$	190983.	0.397760
$43$	368875.	0.707522	0.353761	−	0.935336i	$-0.384903\pi$
$43$	368875.	0.707522	0.353761	+	0.935336i	$0.384903\pi$
$44$	−135480.	−0.239769
$45$	−223759.	−0.366046
$46$	−117400.	−0.177835
$47$	830724.	1.16712	0.583558	−	0.812071i	$-0.301660\pi$
$47$	830724.	1.16712	0.583558	+	0.812071i	$0.301660\pi$
$48$	288892.	0.377043
$49$	−286153.	−0.347466
$50$	−155222.	−0.175614
$51$	−378828.	−0.399896
$52$	208441.	0.205576
$53$	−545061.	−0.502897	−0.251449	−	0.967871i	$-0.580907\pi$
$53$	−545061.	−0.502897	−0.251449	+	0.967871i	$0.580907\pi$
$54$	189923.	0.164134
$55$	−1.19168e6	−0.965808
$56$	1.15223e6	0.876760
$57$	−297336.	−0.212660
$58$	−1127.63	−0.000758871 0
$59$	1.87374e6	1.18776	0.593880	−	0.804554i	$-0.297596\pi$
$59$	1.87374e6	1.18776	0.593880	+	0.804554i	$0.297596\pi$
$60$	−289191.	−0.172845
$61$	−1.20373e6	−0.679010	−0.339505	−	0.940604i	$-0.610260\pi$
$61$	−1.20373e6	−0.679010	−0.339505	+	0.940604i	$0.610260\pi$
$62$	632523.	0.337058
$63$	534407.	0.269265
$64$	2.31466e6	1.10371
$65$	1.83344e6	0.828076
$66$	1.01148e6	0.433065
$67$	−4.04596e6	−1.64346	−0.821731	−	0.569876i	$-0.806991\pi$
$67$	−4.04596e6	−1.64346	−0.821731	+	0.569876i	$0.806991\pi$
$68$	−489607.	−0.188828
$69$	−328509.	−0.120386
$70$	2.17111e6	0.756553
$71$	2.77968e6	0.921702	0.460851	−	0.887478i	$-0.347544\pi$
$71$	2.77968e6	0.921702	0.460851	+	0.887478i	$0.347544\pi$
$72$	1.14583e6	0.361791
$73$	−2.58877e6	−0.778867	−0.389434	−	0.921055i	$-0.627329\pi$
$73$	−2.58877e6	−0.778867	−0.389434	+	0.921055i	$0.627329\pi$
$74$	3.69051e6	1.05871
$75$	−434342.	−0.118882
$76$	−384284.	−0.100416
$77$	2.84611e6	0.710452
$78$	−1.55619e6	−0.371306
$79$	1.28761e6	0.293825	0.146913	−	0.989149i	$-0.453066\pi$
$79$	1.28761e6	0.293825	0.146913	+	0.989149i	$0.453066\pi$
$80$	3.28415e6	0.717148
$81$	531441.	0.111111
$82$	3.57415e6	0.715854
$83$	5.20417e6	0.999029	0.499514	−	0.866306i	$-0.333512\pi$
$83$	5.20417e6	0.999029	0.499514	+	0.866306i	$0.333512\pi$
$84$	690681.	0.127145
$85$	−4.30657e6	−0.760615
$86$	−3.55930e6	−0.603421
$87$	−3155.33	−0.000513720 0
$88$	6.10242e6	0.954581
$89$	−2.72740e6	−0.410094	−0.205047	−	0.978752i	$-0.565735\pi$
$89$	−2.72740e6	−0.410094	−0.205047	+	0.978752i	$0.565735\pi$
$90$	2.15906e6	0.312188
$91$	−4.37884e6	−0.609136
$92$	−424573.	−0.0568454
$93$	1.76992e6	0.228173
$94$	−8.01571e6	−0.995393
$95$	−3.38015e6	−0.404486
$96$	2.64457e6	0.305074
$97$	−7.55774e6	−0.840797	−0.420398	−	0.907340i	$-0.638110\pi$
$97$	−7.55774e6	−0.840797	−0.420398	+	0.907340i	$0.638110\pi$
$98$	2.76111e6	0.296342
$99$	2.83032e6	0.293165
$100$	−561354.	−0.0561354
$101$	−1.41037e7	−1.36209	−0.681047	−	0.732240i	$-0.738475\pi$
$101$	−1.41037e7	−1.36209	−0.681047	+	0.732240i	$0.738475\pi$
$102$	3.65534e6	0.341057
$103$	−228114.	−0.0205694	−0.0102847	−	0.999947i	$-0.503274\pi$
$103$	−228114.	−0.0205694	−0.0102847	+	0.999947i	$0.503274\pi$
$104$	−9.38877e6	−0.818450
$105$	6.07520e6	0.512151
$106$	5.25933e6	0.428904
$107$	−1.72470e7	−1.36104	−0.680519	−	0.732730i	$-0.738246\pi$
$107$	−1.72470e7	−1.36104	−0.680519	+	0.732730i	$0.738246\pi$
$108$	686847.	0.0524659
$109$	−3.22193e6	−0.238299	−0.119150	−	0.992876i	$-0.538017\pi$
$109$	−3.22193e6	−0.238299	−0.119150	+	0.992876i	$0.538017\pi$
$110$	1.14986e7	0.823704
$111$	1.03268e7	0.716695
$112$	−7.84361e6	−0.527537
$113$	1.74509e6	0.113774	0.0568869	−	0.998381i	$-0.481883\pi$
$113$	1.74509e6	0.113774	0.0568869	+	0.998381i	$0.481883\pi$
$114$	2.86901e6	0.181370
$115$	−3.73453e6	−0.228978
$116$	−4078.02	−0.000242575 0
$117$	−4.35453e6	−0.251357
$118$	−1.80799e7	−1.01300
$119$	1.02854e7	0.559511
$120$	1.30260e7	0.688139
$121$	−4.41363e6	−0.226489
$122$	1.16149e7	0.579104
$123$	1.00012e7	0.484600
$124$	2.28749e6	0.107742
$125$	1.90420e7	0.872021
$126$	−5.15653e6	−0.229647
$127$	−1.91418e7	−0.829222	−0.414611	−	0.909999i	$-0.636082\pi$
$127$	−1.91418e7	−0.829222	−0.414611	+	0.909999i	$0.636082\pi$
$128$	−9.79707e6	−0.412916
$129$	−9.95963e6	−0.408488
$130$	−1.76910e7	−0.706237
$131$	−3.17455e7	−1.23376	−0.616882	−	0.787056i	$-0.711604\pi$
$131$	−3.17455e7	−1.23376	−0.616882	+	0.787056i	$0.711604\pi$
$132$	3.65797e6	0.138430
$133$	8.07287e6	0.297541
$134$	3.90397e7	1.40165
$135$	6.04149e6	0.211337
$136$	2.20533e7	0.751773
$137$	−3.41056e7	−1.13319	−0.566597	−	0.823995i	$-0.691740\pi$
$137$	−3.41056e7	−1.13319	−0.566597	+	0.823995i	$0.691740\pi$
$138$	3.16981e6	0.102673
$139$	−3.40795e7	−1.07632	−0.538160	−	0.842843i	$-0.680880\pi$
$139$	−3.40795e7	−1.07632	−0.538160	+	0.842843i	$0.680880\pi$
$140$	7.85174e6	0.241834
$141$	−2.24295e7	−0.673835
$142$	−2.68213e7	−0.786087
$143$	−2.31911e7	−0.663202
$144$	−7.80007e6	−0.217686
$145$	−35870.1	−0.000977113 0
$146$	2.49792e7	0.664269
$147$	7.72614e6	0.200610
$148$	1.33466e7	0.338419
$149$	−5.32875e7	−1.31970	−0.659848	−	0.751399i	$-0.729379\pi$
$149$	−5.32875e7	−1.31970	−0.659848	+	0.751399i	$0.729379\pi$
$150$	4.19100e6	0.101391
$151$	6.20237e7	1.46601	0.733007	−	0.680221i	$-0.238116\pi$
$151$	6.20237e7	1.46601	0.733007	+	0.680221i	$0.238116\pi$
$152$	1.73092e7	0.399784
$153$	1.02284e7	0.230880
$154$	−2.74624e7	−0.605920
$155$	2.01207e7	0.433992
$156$	−5.62791e6	−0.118689
$157$	1.23372e7	0.254429	0.127214	−	0.991875i	$-0.459396\pi$
$157$	1.23372e7	0.254429	0.127214	+	0.991875i	$0.459396\pi$
$158$	−1.24242e7	−0.250593
$159$	1.47166e7	0.290348
$160$	3.00638e7	0.580262
$161$	8.91925e6	0.168437
$162$	−5.12791e6	−0.0947628
$163$	−9.91990e7	−1.79412	−0.897058	−	0.441912i	$-0.854300\pi$
$163$	−9.91990e7	−1.79412	−0.897058	+	0.441912i	$0.854300\pi$
$164$	1.29258e7	0.228825
$165$	3.21754e7	0.557609
$166$	−5.02154e7	−0.852037
$167$	4.36229e7	0.724782	0.362391	−	0.932026i	$-0.381961\pi$
$167$	4.36229e7	0.724782	0.362391	+	0.932026i	$0.381961\pi$
$168$	−3.11102e7	−0.506198
$169$	−2.70682e7	−0.431376
$170$	4.15543e7	0.648702
$171$	8.02806e6	0.122779
$172$	−1.28721e7	−0.192885
$173$	3.86357e7	0.567318	0.283659	−	0.958925i	$-0.408452\pi$
$173$	3.86357e7	0.567318	0.283659	+	0.958925i	$0.408452\pi$
$174$	30446.0	0.000438134 0
$175$	1.17927e7	0.166333
$176$	−4.15412e7	−0.574360
$177$	−5.05911e7	−0.685753
$178$	2.63169e7	0.349755
$179$	5.49433e7	0.716027	0.358014	−	0.933716i	$-0.383454\pi$
$179$	5.49433e7	0.716027	0.358014	+	0.933716i	$0.383454\pi$
$180$	7.80817e6	0.0997919
$181$	7.42706e7	0.930983	0.465492	−	0.885052i	$-0.345878\pi$
$181$	7.42706e7	0.930983	0.465492	+	0.885052i	$0.345878\pi$
$182$	4.22517e7	0.519511
$183$	3.25008e7	0.392027
$184$	1.91240e7	0.226316
$185$	1.17396e8	1.36318
$186$	−1.70781e7	−0.194601
$187$	5.44736e7	0.609173
$188$	−2.89885e7	−0.318180
$189$	−1.44290e7	−0.155460
$190$	3.26153e7	0.344972
$191$	−1.76294e8	−1.83072	−0.915358	−	0.402642i	$-0.868092\pi$
$191$	−1.76294e8	−1.83072	−0.915358	+	0.402642i	$0.868092\pi$
$192$	−6.24958e7	−0.637230
$193$	5.76947e7	0.577677	0.288839	−	0.957378i	$-0.406731\pi$
$193$	5.76947e7	0.577677	0.288839	+	0.957378i	$0.406731\pi$
$194$	7.29252e7	0.717086
$195$	−4.95029e7	−0.478090
$196$	9.98545e6	0.0947265
$197$	3.71726e7	0.346411	0.173205	−	0.984886i	$-0.444588\pi$
$197$	3.71726e7	0.346411	0.173205	+	0.984886i	$0.444588\pi$
$198$	−2.73099e7	−0.250030
$199$	9.70573e7	0.873057	0.436528	−	0.899690i	$-0.356208\pi$
$199$	9.70573e7	0.873057	0.436528	+	0.899690i	$0.356208\pi$
$200$	2.52850e7	0.223490
$201$	1.09241e8	0.948853
$202$	1.36087e8	1.16168
$203$	85669.3	0.000718768 0
$204$	1.32194e7	0.109020
$205$	1.13695e8	0.921725
$206$	2.20109e6	0.0175429
$207$	8.86974e6	0.0695048
$208$	6.39124e7	0.492452
$209$	4.27554e7	0.323951
$210$	−5.86201e7	−0.436796
$211$	−1.26885e8	−0.929870	−0.464935	−	0.885345i	$-0.653922\pi$
$211$	−1.26885e8	−0.929870	−0.464935	+	0.885345i	$0.653922\pi$
$212$	1.90202e7	0.137100
$213$	−7.50513e7	−0.532145
$214$	1.66418e8	1.16078
$215$	−1.13222e8	−0.776957
$216$	−3.09375e7	−0.208880
$217$	−4.80546e7	−0.319247
$218$	3.10886e7	0.203237
$219$	6.98968e7	0.449679
$220$	4.15843e7	0.263299
$221$	−8.38094e7	−0.522300
$222$	−9.96438e7	−0.611245
$223$	2.83332e7	0.171091	0.0855457	−	0.996334i	$-0.472737\pi$
$223$	2.83332e7	0.171091	0.0855457	+	0.996334i	$0.472737\pi$
$224$	−7.18019e7	−0.426843
$225$	1.17272e7	0.0686368
$226$	−1.68385e7	−0.0970338
$227$	−2.73863e8	−1.55397	−0.776987	−	0.629517i	$-0.783253\pi$
$227$	−2.73863e8	−1.55397	−0.776987	+	0.629517i	$0.783253\pi$
$228$	1.03757e7	0.0579754
$229$	−1.97587e8	−1.08726	−0.543632	−	0.839324i	$-0.682951\pi$
$229$	−1.97587e8	−1.08726	−0.543632	+	0.839324i	$0.682951\pi$
$230$	3.60347e7	0.195287
$231$	−7.68451e7	−0.410180
$232$	183685.	0.000965755 0
$233$	1.14212e8	0.591517	0.295758	−	0.955263i	$-0.404428\pi$
$233$	1.14212e8	0.591517	0.295758	+	0.955263i	$0.404428\pi$
$234$	4.20172e7	0.214374
$235$	−2.54982e8	−1.28166
$236$	−6.53852e7	−0.323808
$237$	−3.47654e7	−0.169640
$238$	−9.92450e7	−0.477188
$239$	−3.98583e8	−1.88854	−0.944270	−	0.329173i	$-0.893230\pi$
$239$	−3.98583e8	−1.88854	−0.944270	+	0.329173i	$0.893230\pi$
$240$	−8.86722e7	−0.414045
$241$	1.02788e8	0.473024	0.236512	−	0.971629i	$-0.423996\pi$
$241$	1.02788e8	0.473024	0.236512	+	0.971629i	$0.423996\pi$
$242$	4.25874e7	0.193165
$243$	−1.43489e7	−0.0641500
$244$	4.20049e7	0.185112
$245$	8.78317e7	0.381566
$246$	−9.65021e7	−0.413299
$247$	−6.57805e7	−0.277753
$248$	−1.03035e8	−0.428947
$249$	−1.40513e8	−0.576790
$250$	−1.83737e8	−0.743717
$251$	−4.18803e8	−1.67168	−0.835838	−	0.548977i	$-0.815018\pi$
$251$	−4.18803e8	−1.67168	−0.835838	+	0.548977i	$0.815018\pi$
$252$	−1.86484e7	−0.0734073
$253$	4.72380e7	0.183387
$254$	1.84701e8	0.707214
$255$	1.16277e8	0.439141
$256$	−2.01744e8	−0.751553
$257$	3.90111e7	0.143358	0.0716791	−	0.997428i	$-0.477164\pi$
$257$	3.90111e7	0.143358	0.0716791	+	0.997428i	$0.477164\pi$
$258$	9.61011e7	0.348385
$259$	−2.80379e8	−1.00276
$260$	−6.39787e7	−0.225751
$261$	85193.8	0.000296597 0
$262$	3.06314e8	1.05223
$263$	3.22526e8	1.09325	0.546625	−	0.837377i	$-0.315912\pi$
$263$	3.22526e8	1.09325	0.546625	+	0.837377i	$0.315912\pi$
$264$	−1.64765e8	−0.551127
$265$	1.67301e8	0.552252
$266$	−7.78956e7	−0.253763
$267$	7.36398e7	0.236768
$268$	1.41186e8	0.448042
$269$	9.39578e7	0.294306	0.147153	−	0.989114i	$-0.452989\pi$
$269$	9.39578e7	0.294306	0.147153	+	0.989114i	$0.452989\pi$
$270$	−5.82947e7	−0.180242
$271$	3.27712e8	1.00023	0.500115	−	0.865959i	$-0.333291\pi$
$271$	3.27712e8	1.00023	0.500115	+	0.865959i	$0.333291\pi$
$272$	−1.50124e8	−0.452333
$273$	1.18229e8	0.351685
$274$	3.29088e8	0.966462
$275$	6.24562e7	0.181097
$276$	1.14635e7	0.0328197
$277$	4.33983e8	1.22685	0.613427	−	0.789751i	$-0.289790\pi$
$277$	4.33983e8	1.22685	0.613427	+	0.789751i	$0.289790\pi$
$278$	3.28836e8	0.917956
$279$	−4.77879e7	−0.131736
$280$	−3.53665e8	−0.962805
$281$	−4.53055e8	−1.21809	−0.609044	−	0.793136i	$-0.708447\pi$
$281$	−4.53055e8	−1.21809	−0.609044	+	0.793136i	$0.708447\pi$
$282$	2.16424e8	0.574691
$283$	5.47208e8	1.43516	0.717580	−	0.696477i	$-0.245250\pi$
$283$	5.47208e8	1.43516	0.717580	+	0.696477i	$0.245250\pi$
$284$	−9.69982e7	−0.251275
$285$	9.12640e7	0.233530
$286$	2.23773e8	0.565622
$287$	−2.71539e8	−0.678025
$288$	−7.14034e7	−0.176135
$289$	−2.13479e8	−0.520250
$290$	346113.	0.000833346 0
$291$	2.04059e8	0.485434
$292$	9.03363e7	0.212335
$293$	7.13317e6	0.0165671	0.00828354	−	0.999966i	$-0.497363\pi$
$293$	7.13317e6	0.0165671	0.00828354	+	0.999966i	$0.497363\pi$
$294$	−7.45501e7	−0.171093
$295$	−5.75126e8	−1.30433
$296$	−6.01168e8	−1.34733
$297$	−7.64186e7	−0.169259
$298$	5.14175e8	1.12552
$299$	−7.26771e7	−0.157235
$300$	1.51566e7	0.0324098
$301$	2.70411e8	0.571533
$302$	−5.98471e8	−1.25031
$303$	3.80799e8	0.786405
$304$	−1.17830e8	−0.240545
$305$	3.69473e8	0.745648
$306$	−9.86942e7	−0.196909
$307$	9.59422e8	1.89245	0.946227	−	0.323503i	$-0.104860\pi$
$307$	9.59422e8	1.89245	0.946227	+	0.323503i	$0.104860\pi$
$308$	−9.93165e7	−0.193684
$309$	6.15907e6	0.0118757
$310$	−1.94146e8	−0.370137
$311$	8.24708e8	1.55467	0.777336	−	0.629085i	$-0.216570\pi$
$311$	8.24708e8	1.55467	0.777336	+	0.629085i	$0.216570\pi$
$312$	2.53497e8	0.472532
$313$	−5.01561e7	−0.0924526	−0.0462263	−	0.998931i	$-0.514720\pi$
$313$	−5.01561e7	−0.0924526	−0.0462263	+	0.998931i	$0.514720\pi$
$314$	−1.19042e8	−0.216994
$315$	−1.64031e8	−0.295691
$316$	−4.49317e7	−0.0801029
$317$	3.58019e8	0.631247	0.315623	−	0.948885i	$-0.397786\pi$
$317$	3.58019e8	0.631247	0.315623	+	0.948885i	$0.397786\pi$
$318$	−1.42002e8	−0.247628
$319$	453720.	0.000782565 0
$320$	−7.10459e8	−1.21203
$321$	4.65669e8	0.785796
$322$	−8.60624e7	−0.143654
$323$	1.54512e8	0.255125
$324$	−1.85449e7	−0.0302912
$325$	−9.60909e7	−0.155271
$326$	9.57178e8	1.53014
$327$	8.69920e7	0.137582
$328$	−5.82213e8	−0.911011
$329$	6.08978e8	0.942792
$330$	−3.10463e8	−0.475566
$331$	−4.42027e8	−0.669963	−0.334981	−	0.942225i	$-0.608730\pi$
$331$	−4.42027e8	−0.669963	−0.334981	+	0.942225i	$0.608730\pi$
$332$	−1.81602e8	−0.272356
$333$	−2.78823e8	−0.413784
$334$	−4.20921e8	−0.618141
$335$	1.24186e9	1.80475
$336$	2.11777e8	0.304573
$337$	9.16771e8	1.30484	0.652418	−	0.757859i	$-0.273755\pi$
$337$	9.16771e8	1.30484	0.652418	+	0.757859i	$0.273755\pi$
$338$	2.61183e8	0.367906
$339$	−4.71174e7	−0.0656874
$340$	1.50280e8	0.207359
$341$	−2.54506e8	−0.347582
$342$	−7.74633e7	−0.104714
$343$	−8.13484e8	−1.08848
$344$	5.79794e8	0.767926
$345$	1.00832e8	0.132200
$346$	−3.72798e8	−0.483846
$347$	7.77286e8	0.998683	0.499341	−	0.866405i	$-0.333575\pi$
$347$	7.77286e8	0.998683	0.499341	+	0.866405i	$0.333575\pi$
$348$	110107.	0.000140051 0
$349$	−1.30337e9	−1.64126	−0.820631	−	0.571458i	$-0.806378\pi$
$349$	−1.30337e9	−1.64126	−0.820631	+	0.571458i	$0.806378\pi$
$350$	−1.13788e8	−0.141860
$351$	1.17572e8	0.145121
$352$	−3.80276e8	−0.464729
$353$	8.94399e8	1.08223	0.541115	−	0.840949i	$-0.318002\pi$
$353$	8.94399e8	1.08223	0.541115	+	0.840949i	$0.318002\pi$
$354$	4.88157e8	0.584855
$355$	−8.53193e8	−1.01216
$356$	9.51739e7	0.111800
$357$	−2.77707e8	−0.323034
$358$	−5.30152e8	−0.610675
$359$	−1.09890e8	−0.125351	−0.0626757	−	0.998034i	$-0.519963\pi$
$359$	−1.09890e8	−0.125351	−0.0626757	+	0.998034i	$0.519963\pi$
$360$	−3.51702e8	−0.397297
$361$	−7.72598e8	−0.864328
$362$	−7.16642e8	−0.794003
$363$	1.19168e8	0.130763
$364$	1.52802e8	0.166063
$365$	7.94595e8	0.855305
$366$	−3.13603e8	−0.334346
$367$	3.67335e8	0.387910	0.193955	−	0.981010i	$-0.437868\pi$
$367$	3.67335e8	0.387910	0.193955	+	0.981010i	$0.437868\pi$
$368$	−1.30183e8	−0.136172
$369$	−2.70032e8	−0.279784
$370$	−1.13276e9	−1.16261
$371$	−3.99567e8	−0.406238
$372$	−6.17623e7	−0.0622047
$373$	1.07675e7	0.0107432	0.00537160	−	0.999986i	$-0.498290\pi$
$373$	1.07675e7	0.0107432	0.00537160	+	0.999986i	$0.498290\pi$
$374$	−5.25620e8	−0.519542
$375$	−5.14133e8	−0.503462
$376$	1.30572e9	1.26676
$377$	−698063.	−0.000670965 0
$378$	1.39226e8	0.132587
$379$	−1.58481e9	−1.49534	−0.747670	−	0.664071i	$-0.768827\pi$
$379$	−1.58481e9	−1.49534	−0.747670	+	0.664071i	$0.768827\pi$
$380$	1.17952e8	0.110271
$381$	5.16830e8	0.478751
$382$	1.70107e9	1.56135
$383$	1.17134e9	1.06534	0.532669	−	0.846324i	$-0.321189\pi$
$383$	1.17134e9	1.06534	0.532669	+	0.846324i	$0.321189\pi$
$384$	2.64521e8	0.238397
$385$	−8.73584e8	−0.780175
$386$	−5.56700e8	−0.492681
$387$	2.68910e8	0.235841
$388$	2.63731e8	0.229219
$389$	−1.30652e8	−0.112536	−0.0562680	−	0.998416i	$-0.517920\pi$
$389$	−1.30652e8	−0.112536	−0.0562680	+	0.998416i	$0.517920\pi$
$390$	4.77657e8	0.407746
$391$	1.70711e8	0.144425
$392$	−4.49773e8	−0.377131
$393$	8.57127e8	0.712314
$394$	−3.58681e8	−0.295442
$395$	−3.95218e8	−0.322661
$396$	−9.87652e7	−0.0799229
$397$	−2.35598e9	−1.88975	−0.944874	−	0.327434i	$-0.893816\pi$
$397$	−2.35598e9	−1.88975	−0.944874	+	0.327434i	$0.893816\pi$
$398$	−9.36513e8	−0.744600
$399$	−2.17967e8	−0.171785
$400$	−1.72123e8	−0.134471
$401$	5.75742e8	0.445884	0.222942	−	0.974832i	$-0.428434\pi$
$401$	5.75742e8	0.445884	0.222942	+	0.974832i	$0.428434\pi$
$402$	−1.05407e9	−0.809244
$403$	3.91566e8	0.298014
$404$	4.92154e8	0.371335
$405$	−1.63120e8	−0.122015
$406$	−826629.	−0.000613012 0
$407$	−1.48494e9	−1.09176
$408$	−5.95438e8	−0.434036
$409$	2.19710e9	1.58788	0.793942	−	0.607994i	$-0.208025\pi$
$409$	2.19710e9	1.58788	0.793942	+	0.607994i	$0.208025\pi$
$410$	−1.09705e9	−0.786108
$411$	9.20852e8	0.654250
$412$	7.96014e6	0.00560764
$413$	1.37358e9	0.959467
$414$	−8.55848e7	−0.0592782
$415$	−1.59736e9	−1.09707
$416$	5.85067e8	0.398455
$417$	9.20147e8	0.621414
$418$	−4.12550e8	−0.276286
$419$	−1.88674e9	−1.25304	−0.626518	−	0.779407i	$-0.715521\pi$
$419$	−1.88674e9	−1.25304	−0.626518	+	0.779407i	$0.715521\pi$
$420$	−2.11997e8	−0.139623
$421$	−4.52763e8	−0.295722	−0.147861	−	0.989008i	$-0.547239\pi$
$421$	−4.52763e8	−0.295722	−0.147861	+	0.989008i	$0.547239\pi$
$422$	1.22432e9	0.793054
$423$	6.05598e8	0.389039
$424$	−8.56721e8	−0.545832
$425$	2.25708e8	0.142622
$426$	7.24175e8	0.453848
$427$	−8.82420e8	−0.548501
$428$	6.01842e8	0.371047
$429$	6.26160e8	0.382900
$430$	1.09249e9	0.662640
$431$	4.63280e8	0.278723	0.139361	−	0.990242i	$-0.455495\pi$
$431$	4.63280e8	0.278723	0.139361	+	0.990242i	$0.455495\pi$
$432$	2.10602e8	0.125681
$433$	−1.17035e9	−0.692802	−0.346401	−	0.938087i	$-0.612596\pi$
$433$	−1.17035e9	−0.692802	−0.346401	+	0.938087i	$0.612596\pi$
$434$	4.63682e8	0.272274
$435$	968493.	0.000564137 0
$436$	1.12431e8	0.0649654
$437$	1.33988e8	0.0768036
$438$	−6.74439e8	−0.383516
$439$	1.64913e9	0.930313	0.465157	−	0.885228i	$-0.345998\pi$
$439$	1.64913e9	0.930313	0.465157	+	0.885228i	$0.345998\pi$
$440$	−1.87307e9	−1.04826
$441$	−2.08606e8	−0.115822
$442$	8.08683e8	0.445451
$443$	−7.14982e8	−0.390735	−0.195367	−	0.980730i	$-0.562590\pi$
$443$	−7.14982e8	−0.390735	−0.195367	+	0.980730i	$0.562590\pi$
$444$	−3.60358e8	−0.195386
$445$	8.37146e8	0.450341
$446$	−2.73389e8	−0.145918
$447$	1.43876e9	0.761926
$448$	1.69680e9	0.891576
$449$	−2.68585e9	−1.40030	−0.700149	−	0.713997i	$-0.746883\pi$
$449$	−2.68585e9	−1.40030	−0.700149	+	0.713997i	$0.746883\pi$
$450$	−1.13157e8	−0.0585379
$451$	−1.43812e9	−0.738206
$452$	−6.08956e7	−0.0310171
$453$	−1.67464e9	−0.846404
$454$	2.64253e9	1.32533
$455$	1.34404e9	0.668916
$456$	−4.67349e8	−0.230815
$457$	−2.96284e9	−1.45212	−0.726059	−	0.687632i	$-0.758650\pi$
$457$	−2.96284e9	−1.45212	−0.726059	+	0.687632i	$0.758650\pi$
$458$	1.90653e9	0.927290
$459$	−2.76166e8	−0.133299
$460$	1.30318e8	0.0624241
$461$	7.80077e8	0.370838	0.185419	−	0.982660i	$-0.440636\pi$
$461$	7.80077e8	0.370838	0.185419	+	0.982660i	$0.440636\pi$
$462$	7.41483e8	0.349828
$463$	1.50379e9	0.704131	0.352066	−	0.935975i	$-0.385479\pi$
$463$	1.50379e9	0.704131	0.352066	+	0.935975i	$0.385479\pi$
$464$	−1.25041e6	−0.000581083 0
$465$	−5.43259e8	−0.250566
$466$	−1.10204e9	−0.504484
$467$	−2.49049e9	−1.13156	−0.565779	−	0.824557i	$-0.691424\pi$
$467$	−2.49049e9	−1.13156	−0.565779	+	0.824557i	$0.691424\pi$
$468$	1.51954e8	0.0685252
$469$	−2.96597e9	−1.32758
$470$	2.46034e9	1.09308
$471$	−3.33103e8	−0.146895
$472$	2.94513e9	1.28916
$473$	1.43214e9	0.622261
$474$	3.35454e8	0.144680
$475$	1.77154e8	0.0758444
$476$	−3.58915e8	−0.152534
$477$	−3.97349e8	−0.167632
$478$	3.84595e9	1.61067
$479$	−2.48477e9	−1.03303	−0.516514	−	0.856279i	$-0.672771\pi$
$479$	−2.48477e9	−1.03303	−0.516514	+	0.856279i	$0.672771\pi$
$480$	−8.11723e8	−0.335014
$481$	2.28463e9	0.936069
$482$	−9.91810e8	−0.403426
$483$	−2.40820e8	−0.0972472
$484$	1.54016e8	0.0617456
$485$	2.31977e9	0.923312
$486$	1.38454e8	0.0547113
$487$	−3.44290e9	−1.35075	−0.675373	−	0.737476i	$-0.736017\pi$
$487$	−3.44290e9	−1.35075	−0.675373	+	0.737476i	$0.736017\pi$
$488$	−1.89202e9	−0.736980
$489$	2.67837e9	1.03583
$490$	−8.47494e8	−0.325425
$491$	−2.47473e9	−0.943501	−0.471750	−	0.881732i	$-0.656378\pi$
$491$	−2.47473e9	−0.943501	−0.471750	+	0.881732i	$0.656378\pi$
$492$	−3.48996e8	−0.132112
$493$	1.63968e6	0.000616304 0
$494$	6.34721e8	0.236885
$495$	−8.68736e8	−0.321936
$496$	7.01393e8	0.258093
$497$	2.03770e9	0.744547
$498$	1.35582e9	0.491924
$499$	−1.61485e9	−0.581808	−0.290904	−	0.956752i	$-0.593956\pi$
$499$	−1.61485e9	−0.581808	−0.290904	+	0.956752i	$0.593956\pi$
$500$	−6.64479e8	−0.237731
$501$	−1.17782e9	−0.418453
$502$	4.04106e9	1.42571
$503$	8.25520e8	0.289228	0.144614	−	0.989488i	$-0.453806\pi$
$503$	8.25520e8	0.289228	0.144614	+	0.989488i	$0.453806\pi$
$504$	8.39975e8	0.292253
$505$	4.32897e9	1.49577
$506$	−4.55802e8	−0.156405
$507$	7.30842e8	0.249055
$508$	6.67963e8	0.226063
$509$	−3.14630e9	−1.05752	−0.528759	−	0.848772i	$-0.677343\pi$
$509$	−3.14630e9	−1.05752	−0.528759	+	0.848772i	$0.677343\pi$
$510$	−1.12197e9	−0.374528
$511$	−1.89775e9	−0.629165
$512$	3.20066e9	1.05389
$513$	−2.16758e8	−0.0708865
$514$	−3.76421e8	−0.122265
$515$	7.00171e7	0.0225880
$516$	3.47546e8	0.111362
$517$	3.22526e9	1.02647
$518$	2.70540e9	0.855219
$519$	−1.04316e9	−0.327541
$520$	2.88178e9	0.898772
$521$	2.98365e9	0.924307	0.462154	−	0.886800i	$-0.347077\pi$
$521$	2.98365e9	0.924307	0.462154	+	0.886800i	$0.347077\pi$
$522$	−822041.	−0.000252957 0
$523$	3.33480e9	1.01933	0.509663	−	0.860374i	$-0.329770\pi$
$523$	3.33480e9	1.01933	0.509663	+	0.860374i	$0.329770\pi$
$524$	1.10777e9	0.336350
$525$	−3.18403e8	−0.0960327
$526$	−3.11208e9	−0.932395
$527$	−9.19749e8	−0.273736
$528$	1.12161e9	0.331607
$529$	1.48036e8	0.0434783
$530$	−1.61430e9	−0.470996
$531$	1.36596e9	0.395920
$532$	−2.81706e8	−0.0811159
$533$	2.21259e9	0.632931
$534$	−7.10556e8	−0.201931
$535$	5.29378e9	1.49461
$536$	−6.35940e9	−1.78377
$537$	−1.48347e9	−0.413398
$538$	−9.06605e8	−0.251004
$539$	−1.11098e9	−0.305595
$540$	−2.10820e8	−0.0576149
$541$	−2.22555e9	−0.604292	−0.302146	−	0.953262i	$-0.597703\pi$
$541$	−2.22555e9	−0.604292	−0.302146	+	0.953262i	$0.597703\pi$
$542$	−3.16212e9	−0.853062
$543$	−2.00531e9	−0.537503
$544$	−1.37426e9	−0.365994
$545$	9.88936e8	0.261686
$546$	−1.14080e9	−0.299940
$547$	1.06154e9	0.277318	0.138659	−	0.990340i	$-0.455721\pi$
$547$	1.06154e9	0.277318	0.138659	+	0.990340i	$0.455721\pi$
$548$	1.19013e9	0.308932
$549$	−8.77522e8	−0.226337
$550$	−6.02644e8	−0.154451
$551$	1.28696e6	0.000327743 0
$552$	−5.16347e8	−0.130664
$553$	9.43906e8	0.237351
$554$	−4.18753e9	−1.04634
$555$	−3.16970e9	−0.787032
$556$	1.18922e9	0.293427
$557$	−5.05682e9	−1.23989	−0.619947	−	0.784644i	$-0.712846\pi$
$557$	−5.05682e9	−1.23989	−0.619947	+	0.784644i	$0.712846\pi$
$558$	4.61109e8	0.112353
$559$	−2.20340e9	−0.533522
$560$	2.40751e9	0.579309
$561$	−1.47079e9	−0.351706
$562$	4.37156e9	1.03887
$563$	−2.75862e9	−0.651498	−0.325749	−	0.945456i	$-0.605616\pi$
$563$	−2.75862e9	−0.651498	−0.325749	+	0.945456i	$0.605616\pi$
$564$	7.82690e8	0.183701
$565$	−5.35636e8	−0.124940
$566$	−5.28005e9	−1.22400
$567$	3.89583e8	0.0897551
$568$	4.36907e9	1.00039
$569$	5.48258e9	1.24765	0.623824	−	0.781565i	$-0.285578\pi$
$569$	5.48258e9	1.24765	0.623824	+	0.781565i	$0.285578\pi$
$570$	−8.80612e8	−0.199170
$571$	−3.26474e9	−0.733875	−0.366938	−	0.930246i	$-0.619594\pi$
$571$	−3.26474e9	−0.733875	−0.366938	+	0.930246i	$0.619594\pi$
$572$	8.09265e8	0.180803
$573$	4.75994e9	1.05696
$574$	2.62010e9	0.578264
$575$	1.95727e8	0.0429353
$576$	1.68739e9	0.367905
$577$	−1.20637e9	−0.261436	−0.130718	−	0.991420i	$-0.541728\pi$
$577$	−1.20637e9	−0.261436	−0.130718	+	0.991420i	$0.541728\pi$
$578$	2.05987e9	0.443703
$579$	−1.55776e9	−0.333522
$580$	1.25170e6	0.000266381 0
$581$	3.81501e9	0.807011
$582$	−1.96898e9	−0.414010
$583$	−2.11618e9	−0.442296
$584$	−4.06900e9	−0.845362
$585$	1.33658e9	0.276025
$586$	−6.88285e7	−0.0141295
$587$	−4.66286e9	−0.951523	−0.475761	−	0.879574i	$-0.657827\pi$
$587$	−4.66286e9	−0.951523	−0.475761	+	0.879574i	$0.657827\pi$
$588$	−2.69607e8	−0.0546904
$589$	−7.21894e8	−0.145569
$590$	5.54943e9	1.11241
$591$	−1.00366e9	−0.200000
$592$	4.09235e9	0.810674
$593$	4.02461e9	0.792560	0.396280	−	0.918130i	$-0.370301\pi$
$593$	4.02461e9	0.792560	0.396280	+	0.918130i	$0.370301\pi$
$594$	7.37368e8	0.144355
$595$	−3.15701e9	−0.614421
$596$	1.85949e9	0.359776
$597$	−2.62055e9	−0.504060
$598$	7.01267e8	0.134100
$599$	4.66630e9	0.887113	0.443557	−	0.896246i	$-0.353717\pi$
$599$	4.66630e9	0.887113	0.443557	+	0.896246i	$0.353717\pi$
$600$	−6.82694e8	−0.129032
$601$	−7.89419e9	−1.48336	−0.741681	−	0.670753i	$-0.765971\pi$
$601$	−7.89419e9	−1.48336	−0.741681	+	0.670753i	$0.765971\pi$
$602$	−2.60921e9	−0.487440
$603$	−2.94950e9	−0.547821
$604$	−2.16434e9	−0.399666
$605$	1.35472e9	0.248717
$606$	−3.67435e9	−0.670698
$607$	−7.35070e9	−1.33404	−0.667019	−	0.745041i	$-0.732430\pi$
$607$	−7.35070e9	−1.33404	−0.667019	+	0.745041i	$0.732430\pi$
$608$	−1.07863e9	−0.194631
$609$	−2.31307e6	−0.000414981 0
$610$	−3.56507e9	−0.635937
$611$	−4.96216e9	−0.880089
$612$	−3.56923e8	−0.0629427
$613$	3.32089e9	0.582294	0.291147	−	0.956678i	$-0.405963\pi$
$613$	3.32089e9	0.582294	0.291147	+	0.956678i	$0.405963\pi$
$614$	−9.25753e9	−1.61401
$615$	−3.06976e9	−0.532158
$616$	4.47349e9	0.771106
$617$	−6.89774e8	−0.118225	−0.0591125	−	0.998251i	$-0.518827\pi$
$617$	−6.89774e8	−0.118225	−0.0591125	+	0.998251i	$0.518827\pi$
$618$	−5.94293e7	−0.0101284
$619$	−1.04648e10	−1.77342	−0.886711	−	0.462325i	$-0.847016\pi$
$619$	−1.04648e10	−1.77342	−0.886711	+	0.462325i	$0.847016\pi$
$620$	−7.02121e8	−0.118315
$621$	−2.39483e8	−0.0401286
$622$	−7.95767e9	−1.32593
$623$	−1.99937e9	−0.331272
$624$	−1.72564e9	−0.284317
$625$	−7.10151e9	−1.16351
$626$	4.83960e8	0.0788496
$627$	−1.15440e9	−0.187033
$628$	−4.30511e8	−0.0693626
$629$	−5.36636e9	−0.859810
$630$	1.58274e9	0.252184
$631$	3.22289e9	0.510673	0.255336	−	0.966852i	$-0.417814\pi$
$631$	3.22289e9	0.510673	0.255336	+	0.966852i	$0.417814\pi$
$632$	2.02385e9	0.318910
$633$	3.42590e9	0.536861
$634$	−3.45455e9	−0.538368
$635$	5.87538e9	0.910601
$636$	−5.13544e8	−0.0791549
$637$	1.70928e9	0.262014
$638$	−4.37798e6	−0.000667423 0
$639$	2.02639e9	0.307234
$640$	3.00711e9	0.453439
$641$	5.23897e9	0.785675	0.392837	−	0.919608i	$-0.371494\pi$
$641$	5.23897e9	0.785675	0.392837	+	0.919608i	$0.371494\pi$
$642$	−4.49327e9	−0.670178
$643$	6.78028e9	1.00579	0.502897	−	0.864346i	$-0.332268\pi$
$643$	6.78028e9	1.00579	0.502897	+	0.864346i	$0.332268\pi$
$644$	−3.11241e8	−0.0459194
$645$	3.05700e9	0.448577
$646$	−1.49090e9	−0.217587
$647$	−8.41553e8	−0.122156	−0.0610782	−	0.998133i	$-0.519454\pi$
$647$	−8.41553e8	−0.122156	−0.0610782	+	0.998133i	$0.519454\pi$
$648$	8.35313e8	0.120597
$649$	7.27475e9	1.04463
$650$	9.27188e8	0.132425
$651$	1.29747e9	0.184317
$652$	3.46160e9	0.489114
$653$	7.33932e9	1.03148	0.515739	−	0.856746i	$-0.327518\pi$
$653$	7.33932e9	1.03148	0.515739	+	0.856746i	$0.327518\pi$
$654$	−8.39392e8	−0.117339
$655$	9.74393e9	1.35485
$656$	3.96332e9	0.548145
$657$	−1.88721e9	−0.259622
$658$	−5.87607e9	−0.804074
$659$	1.14657e10	1.56064	0.780319	−	0.625382i	$-0.215057\pi$
$659$	1.14657e10	1.56064	0.780319	+	0.625382i	$0.215057\pi$
$660$	−1.12278e9	−0.152016
$661$	1.40403e10	1.89091	0.945453	−	0.325758i	$-0.105619\pi$
$661$	1.40403e10	1.89091	0.945453	+	0.325758i	$0.105619\pi$
$662$	4.26515e9	0.571388
$663$	2.26285e9	0.301550
$664$	8.17986e9	1.08432
$665$	−2.47788e9	−0.326742
$666$	2.69038e9	0.352902
$667$	1.42188e6	0.000185534 0
$668$	−1.52224e9	−0.197591
$669$	−7.64995e8	−0.0987796
$670$	−1.19828e10	−1.53921
$671$	−4.67346e9	−0.597186
$672$	1.93865e9	0.246438
$673$	1.07497e10	1.35939	0.679693	−	0.733496i	$-0.262113\pi$
$673$	1.07497e10	1.35939	0.679693	+	0.733496i	$0.262113\pi$
$674$	−8.84599e9	−1.11285
$675$	−3.16635e8	−0.0396275
$676$	9.44558e8	0.117602
$677$	1.29380e10	1.60254	0.801269	−	0.598304i	$-0.204159\pi$
$677$	1.29380e10	1.60254	0.801269	+	0.598304i	$0.204159\pi$
$678$	4.54639e8	0.0560225
$679$	−5.54034e9	−0.679192
$680$	−6.76902e9	−0.825552
$681$	7.39431e9	0.897187
$682$	2.45575e9	0.296441
$683$	−1.12095e10	−1.34621	−0.673104	−	0.739548i	$-0.735039\pi$
$683$	−1.12095e10	−1.34621	−0.673104	+	0.739548i	$0.735039\pi$
$684$	−2.80143e8	−0.0334721
$685$	1.04684e10	1.24440
$686$	7.84936e9	0.928325
$687$	5.33486e9	0.627732
$688$	−3.94685e9	−0.462052
$689$	3.25581e9	0.379221
$690$	−9.72938e8	−0.112749
$691$	2.40363e9	0.277137	0.138568	−	0.990353i	$-0.455750\pi$
$691$	2.40363e9	0.277137	0.138568	+	0.990353i	$0.455750\pi$
$692$	−1.34821e9	−0.154663
$693$	2.07482e9	0.236817
$694$	−7.50008e9	−0.851742
$695$	1.04603e10	1.18195
$696$	−4.95951e6	−0.000557579 0
$697$	−5.19716e9	−0.581368
$698$	1.25763e10	1.39978
$699$	−3.08373e9	−0.341512
$700$	−4.11511e8	−0.0453460
$701$	1.28427e9	0.140813	0.0704064	−	0.997518i	$-0.477570\pi$
$701$	1.28427e9	0.140813	0.0704064	+	0.997518i	$0.477570\pi$
$702$	−1.13446e9	−0.123769
$703$	−4.21196e9	−0.457236
$704$	8.98658e9	0.970711
$705$	6.88451e9	0.739965
$706$	−8.63011e9	−0.922997
$707$	−1.03389e10	−1.10029
$708$	1.76540e9	0.186951
$709$	−1.19734e10	−1.26170	−0.630851	−	0.775904i	$-0.717294\pi$
$709$	−1.19734e10	−1.26170	−0.630851	+	0.775904i	$0.717294\pi$
$710$	8.23252e9	0.863234
$711$	9.38667e8	0.0979418
$712$	−4.28690e9	−0.445106
$713$	−7.97580e8	−0.0824064
$714$	2.67961e9	0.275504
$715$	7.11827e9	0.728288
$716$	−1.91727e9	−0.195204
$717$	1.07617e10	1.09035
$718$	1.06034e9	0.106908
$719$	3.94160e8	0.0395477	0.0197738	−	0.999804i	$-0.493705\pi$
$719$	3.94160e8	0.0395477	0.0197738	+	0.999804i	$0.493705\pi$
$720$	2.39415e9	0.239049
$721$	−1.67223e8	−0.0166159
$722$	7.45485e9	0.737155
$723$	−2.77528e9	−0.273101
$724$	−2.59171e9	−0.253805
$725$	1.87996e6	0.000183217 0
$726$	−1.14986e9	−0.111524
$727$	−1.16145e10	−1.12106	−0.560530	−	0.828134i	$-0.689402\pi$
$727$	−1.16145e10	−1.12106	−0.560530	+	0.828134i	$0.689402\pi$
$728$	−6.88261e9	−0.661140
$729$	3.87420e8	0.0370370
$730$	−7.66710e9	−0.729460
$731$	5.17557e9	0.490057
$732$	−1.13413e9	−0.106875
$733$	7.63834e9	0.716366	0.358183	−	0.933651i	$-0.383396\pi$
$733$	7.63834e9	0.716366	0.358183	+	0.933651i	$0.383396\pi$
$734$	−3.54444e9	−0.330835
$735$	−2.37146e9	−0.220297
$736$	−1.19172e9	−0.110180
$737$	−1.57083e10	−1.44542
$738$	2.60556e9	0.238618
$739$	7.94049e9	0.723755	0.361878	−	0.932226i	$-0.382136\pi$
$739$	7.94049e9	0.723755	0.361878	+	0.932226i	$0.382136\pi$
$740$	−4.09659e9	−0.371631
$741$	1.77607e9	0.160361
$742$	3.85545e9	0.346467
$743$	5.34188e9	0.477786	0.238893	−	0.971046i	$-0.423216\pi$
$743$	5.34188e9	0.477786	0.238893	+	0.971046i	$0.423216\pi$
$744$	2.78195e9	0.247653
$745$	1.63560e10	1.44921
$746$	−1.03896e8	−0.00916250
$747$	3.79384e9	0.333010
$748$	−1.90088e9	−0.166073
$749$	−1.26432e10	−1.09944
$750$	4.96091e9	0.429385
$751$	−1.70844e10	−1.47184	−0.735921	−	0.677068i	$-0.763250\pi$
$751$	−1.70844e10	−1.47184	−0.735921	+	0.677068i	$0.763250\pi$
$752$	−8.88849e9	−0.762194
$753$	1.13077e10	0.965142
$754$	6.73566e6	0.000572243 0
$755$	−1.90375e10	−1.60989
$756$	5.03506e8	0.0423817
$757$	−1.18119e10	−0.989655	−0.494828	−	0.868991i	$-0.664769\pi$
$757$	−1.18119e10	−0.989655	−0.494828	+	0.868991i	$0.664769\pi$
$758$	1.52919e10	1.27532
$759$	−1.27542e9	−0.105879
$760$	−5.31288e9	−0.439018
$761$	−1.10997e10	−0.912987	−0.456494	−	0.889727i	$-0.650895\pi$
$761$	−1.10997e10	−0.912987	−0.456494	+	0.889727i	$0.650895\pi$
$762$	−4.98692e9	−0.408310
$763$	−2.36189e9	−0.192497
$764$	6.15186e9	0.499091
$765$	−3.13949e9	−0.253538
$766$	−1.13023e10	−0.908589
$767$	−1.11924e10	−0.895655
$768$	5.44708e9	0.433910
$769$	−2.76235e9	−0.219047	−0.109523	−	0.993984i	$-0.534932\pi$
$769$	−2.76235e9	−0.219047	−0.109523	+	0.993984i	$0.534932\pi$
$770$	8.42928e9	0.665385
$771$	−1.05330e9	−0.0827679
$772$	−2.01328e9	−0.157487
$773$	1.66549e9	0.129692	0.0648460	−	0.997895i	$-0.479344\pi$
$773$	1.66549e9	0.129692	0.0648460	+	0.997895i	$0.479344\pi$
$774$	−2.59473e9	−0.201140
$775$	−1.05453e9	−0.0813772
$776$	−1.18792e10	−0.912579
$777$	7.57024e9	0.578943
$778$	1.26067e9	0.0959780
$779$	−4.07916e9	−0.309164
$780$	1.72743e9	0.130337
$781$	1.07920e10	0.810632
$782$	−1.64720e9	−0.123175
$783$	−2.30023e6	−0.000171240 0
$784$	3.06175e9	0.226915
$785$	−3.78676e9	−0.279398
$786$	−8.27048e9	−0.607508
$787$	3.50520e9	0.256331	0.128165	−	0.991753i	$-0.459091\pi$
$787$	3.50520e9	0.256331	0.128165	+	0.991753i	$0.459091\pi$
$788$	−1.29716e9	−0.0944388
$789$	−8.70820e9	−0.631188
$790$	3.81348e9	0.275186
$791$	1.27927e9	0.0919060
$792$	4.44866e9	0.318194
$793$	7.19027e9	0.512022
$794$	2.27330e10	1.61170
$795$	−4.51712e9	−0.318843
$796$	−3.38686e9	−0.238013
$797$	−2.46740e10	−1.72638	−0.863189	−	0.504881i	$-0.831536\pi$
$797$	−2.46740e10	−1.72638	−0.863189	+	0.504881i	$0.831536\pi$
$798$	2.10318e9	0.146510
$799$	1.16556e10	0.808391
$800$	−1.57565e9	−0.108804
$801$	−1.98827e9	−0.136698
$802$	−5.55537e9	−0.380279
$803$	−1.00508e10	−0.685009
$804$	−3.81201e9	−0.258677
$805$	−2.73767e9	−0.184967
$806$	−3.77825e9	−0.254166
$807$	−2.53686e9	−0.169918
$808$	−2.21680e10	−1.47838
$809$	2.43326e9	0.161573	0.0807865	−	0.996731i	$-0.474257\pi$
$809$	2.43326e9	0.161573	0.0807865	+	0.996731i	$0.474257\pi$
$810$	1.57396e9	0.104063
$811$	−4.26801e9	−0.280965	−0.140482	−	0.990083i	$-0.544865\pi$
$811$	−4.26801e9	−0.280965	−0.140482	+	0.990083i	$0.544865\pi$
$812$	−2.98947e6	−0.000195951 0
$813$	−8.84823e9	−0.577484
$814$	1.43283e10	0.931127
$815$	3.04481e10	1.97019
$816$	4.05334e9	0.261155
$817$	4.06221e9	0.260606
$818$	−2.12000e10	−1.35425
$819$	−3.19217e9	−0.203045
$820$	−3.96743e9	−0.251282
$821$	1.69378e10	1.06821	0.534103	−	0.845419i	$-0.320649\pi$
$821$	1.69378e10	1.06821	0.534103	+	0.845419i	$0.320649\pi$
$822$	−8.88537e9	−0.557987
$823$	−1.49405e10	−0.934258	−0.467129	−	0.884189i	$-0.654712\pi$
$823$	−1.49405e10	−0.934258	−0.467129	+	0.884189i	$0.654712\pi$
$824$	−3.58547e8	−0.0223255
$825$	−1.68632e9	−0.104556
$826$	−1.32538e10	−0.818296
$827$	2.62579e10	1.61432	0.807162	−	0.590330i	$-0.201002\pi$
$827$	2.62579e10	1.61432	0.807162	+	0.590330i	$0.201002\pi$
$828$	−3.09514e8	−0.0189485
$829$	1.26847e10	0.773282	0.386641	−	0.922230i	$-0.373635\pi$
$829$	1.26847e10	0.773282	0.386641	+	0.922230i	$0.373635\pi$
$830$	1.54131e10	0.935656
$831$	−1.17175e10	−0.708325
$832$	−1.38261e10	−0.832280
$833$	−4.01492e9	−0.240669
$834$	−8.87856e9	−0.529982
$835$	−1.33896e10	−0.795911
$836$	−1.49197e9	−0.0883157
$837$	1.29027e9	0.0760576
$838$	1.82053e10	1.06867
$839$	1.31687e10	0.769796	0.384898	−	0.922959i	$-0.374237\pi$
$839$	1.31687e10	0.769796	0.384898	+	0.922959i	$0.374237\pi$
$840$	9.54894e9	0.555876
$841$	−1.72499e10	−0.999999
$842$	4.36875e9	0.252211
$843$	1.22325e10	0.703264
$844$	4.42772e9	0.253502
$845$	8.30830e9	0.473711
$846$	−5.84346e9	−0.331798
$847$	−3.23549e9	−0.182957
$848$	5.83198e9	0.328421
$849$	−1.47746e10	−0.828589
$850$	−2.17787e9	−0.121637
$851$	−4.65355e9	−0.258840
$852$	2.61895e9	0.145074
$853$	−1.52335e8	−0.00840386	−0.00420193	−	0.999991i	$-0.501338\pi$
$853$	−1.52335e8	−0.00840386	−0.00420193	+	0.999991i	$0.501338\pi$
$854$	8.51453e9	0.467798
$855$	−2.46413e9	−0.134829
$856$	−2.71087e10	−1.47724
$857$	−8.43291e9	−0.457661	−0.228831	−	0.973466i	$-0.573490\pi$
$857$	−8.43291e9	−0.457661	−0.228831	+	0.973466i	$0.573490\pi$
$858$	−6.04187e9	−0.326562
$859$	3.11811e10	1.67847	0.839237	−	0.543766i	$-0.183002\pi$
$859$	3.11811e10	1.67847	0.839237	+	0.543766i	$0.183002\pi$
$860$	3.95094e9	0.211815
$861$	7.33155e9	0.391458
$862$	−4.47022e9	−0.237713
$863$	1.07163e10	0.567553	0.283776	−	0.958891i	$-0.408413\pi$
$863$	1.07163e10	0.567553	0.283776	+	0.958891i	$0.408413\pi$
$864$	1.92789e9	0.101691
$865$	−1.18588e10	−0.622995
$866$	1.12928e10	0.590866
$867$	5.76393e9	0.300367
$868$	1.67689e9	0.0870332
$869$	4.99910e9	0.258418
$870$	−9.34506e6	−0.000481133 0
$871$	2.41677e10	1.23929
$872$	−5.06419e9	−0.258644
$873$	−5.50959e9	−0.280266
$874$	−1.29286e9	−0.0655031
$875$	1.39591e10	0.704415
$876$	−2.43908e9	−0.122592
$877$	1.75959e9	0.0880874	0.0440437	−	0.999030i	$-0.485976\pi$
$877$	1.75959e9	0.0880874	0.0440437	+	0.999030i	$0.485976\pi$
$878$	−1.59126e10	−0.793432
$879$	−1.92596e8	−0.00956501
$880$	1.27506e10	0.630728
$881$	−2.66596e10	−1.31353	−0.656763	−	0.754097i	$-0.728075\pi$
$881$	−2.66596e10	−1.31353	−0.656763	+	0.754097i	$0.728075\pi$
$882$	2.01285e9	0.0987806
$883$	2.55224e10	1.24755	0.623776	−	0.781603i	$-0.285598\pi$
$883$	2.55224e10	1.24755	0.623776	+	0.781603i	$0.285598\pi$
$884$	2.92457e9	0.142390
$885$	1.55284e10	0.753053
$886$	6.89891e9	0.333244
$887$	−1.78494e10	−0.858800	−0.429400	−	0.903114i	$-0.641275\pi$
$887$	−1.78494e10	−0.858800	−0.429400	+	0.903114i	$0.641275\pi$
$888$	1.62315e10	0.777883
$889$	−1.40323e10	−0.669842
$890$	−8.07768e9	−0.384080
$891$	2.06330e9	0.0977216
$892$	−9.88699e8	−0.0466431
$893$	9.14829e9	0.429892
$894$	−1.38827e10	−0.649821
$895$	−1.68643e10	−0.786298
$896$	−7.18193e9	−0.333551
$897$	1.96228e9	0.0907795
$898$	2.59160e10	1.19427
$899$	−7.66074e6	−0.000351651 0
$900$	−4.09227e8	−0.0187118
$901$	−7.64757e9	−0.348327
$902$	1.38765e10	0.629590
$903$	−7.30109e9	−0.329975
$904$	2.74291e9	0.123487
$905$	−2.27966e10	−1.02235
$906$	1.61587e10	0.721868
$907$	−8.37198e9	−0.372566	−0.186283	−	0.982496i	$-0.559644\pi$
$907$	−8.37198e9	−0.372566	−0.186283	+	0.982496i	$0.559644\pi$
$908$	9.55659e9	0.423645
$909$	−1.02816e10	−0.454031
$910$	−1.29687e10	−0.570495
$911$	−2.02213e10	−0.886126	−0.443063	−	0.896490i	$-0.646108\pi$
$911$	−2.02213e10	−0.886126	−0.443063	+	0.896490i	$0.646108\pi$
$912$	3.18140e9	0.138879
$913$	2.02050e10	0.878641
$914$	2.85887e10	1.23846
$915$	−9.97578e9	−0.430500
$916$	6.89490e9	0.296411
$917$	−2.32716e10	−0.996629
$918$	2.66474e9	0.113686
$919$	1.96145e10	0.833629	0.416815	−	0.908992i	$-0.363146\pi$
$919$	1.96145e10	0.833629	0.416815	+	0.908992i	$0.363146\pi$
$920$	−5.86990e9	−0.248527
$921$	−2.59044e10	−1.09261
$922$	−7.52702e9	−0.316275
$923$	−1.66039e10	−0.695029
$924$	2.68154e9	0.111824
$925$	−6.15275e9	−0.255607
$926$	−1.45102e10	−0.600529
$927$	−1.66295e8	−0.00685646
$928$	−1.14465e7	−0.000470169 0
$929$	1.66505e10	0.681355	0.340677	−	0.940180i	$-0.389344\pi$
$929$	1.66505e10	0.681355	0.340677	+	0.940180i	$0.389344\pi$
$930$	5.24194e9	0.213699
$931$	−3.15124e9	−0.127985
$932$	−3.98549e9	−0.161260
$933$	−2.22671e10	−0.897591
$934$	2.40310e10	0.965066
$935$	−1.67201e10	−0.668957
$936$	−6.84441e9	−0.272817
$937$	4.92026e10	1.95389	0.976944	−	0.213496i	$-0.0684849\pi$
$937$	4.92026e10	1.95389	0.976944	+	0.213496i	$0.0684849\pi$
$938$	2.86188e10	1.13225
$939$	1.35422e9	0.0533775
$940$	8.89771e9	0.349406
$941$	−9.05047e9	−0.354085	−0.177042	−	0.984203i	$-0.556653\pi$
$941$	−9.05047e9	−0.354085	−0.177042	+	0.984203i	$0.556653\pi$
$942$	3.21414e9	0.125281
$943$	−4.50683e9	−0.175017
$944$	−2.00485e10	−0.775675
$945$	4.42882e9	0.170717
$946$	−1.38189e10	−0.530705
$947$	−4.80454e9	−0.183834	−0.0919171	−	0.995767i	$-0.529299\pi$
$947$	−4.80454e9	−0.183834	−0.0919171	+	0.995767i	$0.529299\pi$
$948$	1.21316e9	0.0462474
$949$	1.54635e10	0.587321
$950$	−1.70937e9	−0.0646851
$951$	−9.66653e9	−0.364451
$952$	1.61666e10	0.607279
$953$	3.86498e10	1.44651	0.723256	−	0.690580i	$-0.242645\pi$
$953$	3.86498e10	1.44651	0.723256	+	0.690580i	$0.242645\pi$
$954$	3.83405e9	0.142968
$955$	5.41116e10	2.01038
$956$	1.39087e10	0.514855
$957$	−1.22504e7	−0.000451814 0
$958$	2.39757e10	0.881034
$959$	−2.50018e10	−0.915389
$960$	1.91824e10	0.699767
$961$	−2.32155e10	−0.843811
$962$	−2.20445e10	−0.798341
$963$	−1.25731e10	−0.453680
$964$	−3.58684e9	−0.128956
$965$	−1.77088e10	−0.634370
$966$	2.32369e9	0.0829387
$967$	2.31025e10	0.821612	0.410806	−	0.911723i	$-0.365247\pi$
$967$	2.31025e10	0.821612	0.410806	+	0.911723i	$0.365247\pi$
$968$	−6.93730e9	−0.245825
$969$	−4.17182e9	−0.147296
$970$	−2.23836e10	−0.787461
$971$	3.14383e10	1.10203	0.551013	−	0.834497i	$-0.314242\pi$
$971$	3.14383e10	1.10203	0.551013	+	0.834497i	$0.314242\pi$
$972$	5.00712e8	0.0174886
$973$	−2.49826e10	−0.869447
$974$	3.32208e10	1.15200
$975$	2.59446e9	0.0896458
$976$	1.28796e10	0.443432
$977$	3.59756e10	1.23418	0.617089	−	0.786894i	$-0.288312\pi$
$977$	3.59756e10	1.23418	0.617089	+	0.786894i	$0.288312\pi$
$978$	−2.58438e10	−0.883427
$979$	−1.05890e10	−0.360676
$980$	−3.06493e9	−0.104023
$981$	−2.34878e9	−0.0794331
$982$	2.38788e10	0.804679
$983$	3.43584e10	1.15371	0.576853	−	0.816848i	$-0.304281\pi$
$983$	3.43584e10	1.15371	0.576853	+	0.816848i	$0.304281\pi$
$984$	1.57198e10	0.525972
$985$	−1.14097e10	−0.380407
$986$	−1.58214e7	−0.000525624 0
$987$	−1.64424e10	−0.544321
$988$	2.29544e9	0.0757211
$989$	4.48810e9	0.147528
$990$	8.38249e9	0.274568
$991$	−5.34838e9	−0.174568	−0.0872839	−	0.996183i	$-0.527819\pi$
$991$	−5.34838e9	−0.174568	−0.0872839	+	0.996183i	$0.527819\pi$
$992$	6.42069e9	0.208829
$993$	1.19347e10	0.386803
$994$	−1.96619e10	−0.634998
$995$	−2.97907e10	−0.958738
$996$	4.90325e9	0.157245
$997$	−5.37617e10	−1.71807	−0.859034	−	0.511919i	$-0.828935\pi$
$997$	−5.37617e10	−1.71807	−0.859034	+	0.511919i	$0.828935\pi$
$998$	1.55818e10	0.496204
$999$	7.52823e9	0.238898

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.a.1.2	✓	5
3.2	odd	2		207.8.a.a.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.a.1.2	✓	5	1.1	even	1	trivial
207.8.a.a.1.4		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)

\(f(q)\)	\(=\)	\(q-9.64907 q^{2} -27.0000 q^{3} -34.8955 q^{4} -306.939 q^{5} +260.525 q^{6} +733.069 q^{7} +1571.79 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q-9.64907 q^{2} -27.0000 q^{3} -34.8955 q^{4} -306.939 q^{5} +260.525 q^{6} +733.069 q^{7} +1571.79 q^{8} +729.000 q^{9} +2961.68 q^{10} +3882.47 q^{11} +942.178 q^{12} -5973.30 q^{13} -7073.43 q^{14} +8287.36 q^{15} -10699.7 q^{16} +14030.7 q^{17} -7034.17 q^{18} +11012.4 q^{19} +10710.8 q^{20} -19792.9 q^{21} -37462.2 q^{22} +12167.0 q^{23} -42438.3 q^{24} +16086.7 q^{25} +57636.8 q^{26} -19683.0 q^{27} -25580.8 q^{28} +116.864 q^{29} -79965.3 q^{30} -65552.7 q^{31} -97947.0 q^{32} -104827. q^{33} -135383. q^{34} -225008. q^{35} -25438.8 q^{36} -382473. q^{37} -106260. q^{38} +161279. q^{39} -482444. q^{40} -370414. q^{41} +190983. q^{42} +368875. q^{43} -135480. q^{44} -223759. q^{45} -117400. q^{46} +830724. q^{47} +288892. q^{48} -286153. q^{49} -155222. q^{50} -378828. q^{51} +208441. q^{52} -545061. q^{53} +189923. q^{54} -1.19168e6 q^{55} +1.15223e6 q^{56} -297336. q^{57} -1127.63 q^{58} +1.87374e6 q^{59} -289191. q^{60} -1.20373e6 q^{61} +632523. q^{62} +534407. q^{63} +2.31466e6 q^{64} +1.83344e6 q^{65} +1.01148e6 q^{66} -4.04596e6 q^{67} -489607. q^{68} -328509. q^{69} +2.17111e6 q^{70} +2.77968e6 q^{71} +1.14583e6 q^{72} -2.58877e6 q^{73} +3.69051e6 q^{74} -434342. q^{75} -384284. q^{76} +2.84611e6 q^{77} -1.55619e6 q^{78} +1.28761e6 q^{79} +3.28415e6 q^{80} +531441. q^{81} +3.57415e6 q^{82} +5.20417e6 q^{83} +690681. q^{84} -4.30657e6 q^{85} -3.55930e6 q^{86} -3155.33 q^{87} +6.10242e6 q^{88} -2.72740e6 q^{89} +2.15906e6 q^{90} -4.37884e6 q^{91} -424573. q^{92} +1.76992e6 q^{93} -8.01571e6 q^{94} -3.38015e6 q^{95} +2.64457e6 q^{96} -7.55774e6 q^{97} +2.76111e6 q^{98} +2.83032e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9}+O(q^{10}) \) 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

Groups

Database

Properties

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