LMFDB - Embedded newform 177.8.a.a.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [177,8,Mod(1,177)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(177, 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("177.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 177 = 3 \cdot 59 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	177.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:	$55.2921495107$
Analytic rank:	$1$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + \cdots - 23\!\cdots\!32 $ x^16 - 6x^15 - 1493x^14 + 8791x^13 + 890490x^12 - 5107725x^11 - 269092298x^10 + 1488374176x^9 + 42885295136x^8 - 226132003872x^7 - 3353576629440x^6 + 16796366777600x^5 + 99470801612800x^4 - 494039551757568x^3 - 493048066650624x^2 + 3193975642099712*x - 2385018853548032
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{9}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$-13.0039$ of defining polynomial
Character	$\chi$	$=$	177.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+13.0039 q^{2} -27.0000 q^{3} +41.1009 q^{4} +167.303 q^{5} -351.105 q^{6} +887.373 q^{7} -1130.03 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+13.0039 q^{2} -27.0000 q^{3} +41.1009 q^{4} +167.303 q^{5} -351.105 q^{6} +887.373 q^{7} -1130.03 q^{8} +729.000 q^{9} +2175.59 q^{10} -7784.37 q^{11} -1109.72 q^{12} +4609.99 q^{13} +11539.3 q^{14} -4517.19 q^{15} -19955.6 q^{16} -6055.74 q^{17} +9479.83 q^{18} +13552.7 q^{19} +6876.32 q^{20} -23959.1 q^{21} -101227. q^{22} +35274.4 q^{23} +30510.7 q^{24} -50134.6 q^{25} +59947.7 q^{26} -19683.0 q^{27} +36471.8 q^{28} +78335.1 q^{29} -58741.0 q^{30} -170575. q^{31} -114857. q^{32} +210178. q^{33} -78748.2 q^{34} +148460. q^{35} +29962.6 q^{36} -263573. q^{37} +176238. q^{38} -124470. q^{39} -189057. q^{40} -198905. q^{41} -311561. q^{42} -444435. q^{43} -319945. q^{44} +121964. q^{45} +458704. q^{46} -531622. q^{47} +538802. q^{48} -36112.2 q^{49} -651945. q^{50} +163505. q^{51} +189475. q^{52} -1.00107e6 q^{53} -255955. q^{54} -1.30235e6 q^{55} -1.00275e6 q^{56} -365924. q^{57} +1.01866e6 q^{58} +205379. q^{59} -185661. q^{60} -1.77312e6 q^{61} -2.21813e6 q^{62} +646895. q^{63} +1.06073e6 q^{64} +771266. q^{65} +2.73313e6 q^{66} -2.37225e6 q^{67} -248897. q^{68} -952409. q^{69} +1.93056e6 q^{70} -5.87275e6 q^{71} -823789. q^{72} -2.90906e6 q^{73} -3.42748e6 q^{74} +1.35363e6 q^{75} +557030. q^{76} -6.90764e6 q^{77} -1.61859e6 q^{78} -1.66441e6 q^{79} -3.33864e6 q^{80} +531441. q^{81} -2.58653e6 q^{82} +8.18425e6 q^{83} -984739. q^{84} -1.01315e6 q^{85} -5.77938e6 q^{86} -2.11505e6 q^{87} +8.79653e6 q^{88} +1.13356e7 q^{89} +1.58601e6 q^{90} +4.09078e6 q^{91} +1.44981e6 q^{92} +4.60552e6 q^{93} -6.91315e6 q^{94} +2.26742e6 q^{95} +3.10115e6 q^{96} +243947. q^{97} -469599. q^{98} -5.67480e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) $ 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9} - 3479 q^{10} + 898 q^{11} - 26298 q^{12} - 8172 q^{13} - 13315 q^{14} + 1836 q^{15} + 3138 q^{16} - 44985 q^{17} - 4374 q^{18} - 40137 q^{19} + 130657 q^{20} + 63261 q^{21} + 109394 q^{22} - 2833 q^{23} - 22113 q^{24} + 285746 q^{25} - 129420 q^{26} - 314928 q^{27} + 112890 q^{28} + 144375 q^{29} + 93933 q^{30} - 141759 q^{31} - 36224 q^{32} - 24246 q^{33} - 341332 q^{34} - 78859 q^{35} + 710046 q^{36} - 297971 q^{37} + 329075 q^{38} + 220644 q^{39} - 203048 q^{40} + 659077 q^{41} + 359505 q^{42} - 1431608 q^{43} + 254916 q^{44} - 49572 q^{45} + 873113 q^{46} - 1574073 q^{47} - 84726 q^{48} + 1893545 q^{49} + 302533 q^{50} + 1214595 q^{51} - 4972548 q^{52} + 587736 q^{53} + 118098 q^{54} - 4624036 q^{55} - 5798506 q^{56} + 1083699 q^{57} - 6991380 q^{58} + 3286064 q^{59} - 3527739 q^{60} - 6117131 q^{61} - 11570258 q^{62} - 1708047 q^{63} - 19063011 q^{64} - 5335514 q^{65} - 2953638 q^{66} - 16518710 q^{67} - 17284669 q^{68} + 76491 q^{69} - 39189486 q^{70} - 10882582 q^{71} + 597051 q^{72} - 21097441 q^{73} - 16717030 q^{74} - 7715142 q^{75} - 40864952 q^{76} - 3404601 q^{77} + 3494340 q^{78} - 3784458 q^{79} - 27466195 q^{80} + 8503056 q^{81} - 24990117 q^{82} - 1951425 q^{83} - 3048030 q^{84} - 23238675 q^{85} - 35910572 q^{86} - 3898125 q^{87} - 27843055 q^{88} + 10499443 q^{89} - 2536191 q^{90} + 699217 q^{91} - 20062766 q^{92} + 3827493 q^{93} - 59358988 q^{94} - 29236333 q^{95} + 978048 q^{96} - 25158976 q^{97} + 2120460 q^{98} + 654642 q^{99}+O(q^{100}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9 - 3479 * q^10 + 898 * q^11 - 26298 * q^12 - 8172 * q^13 - 13315 * q^14 + 1836 * q^15 + 3138 * q^16 - 44985 * q^17 - 4374 * q^18 - 40137 * q^19 + 130657 * q^20 + 63261 * q^21 + 109394 * q^22 - 2833 * q^23 - 22113 * q^24 + 285746 * q^25 - 129420 * q^26 - 314928 * q^27 + 112890 * q^28 + 144375 * q^29 + 93933 * q^30 - 141759 * q^31 - 36224 * q^32 - 24246 * q^33 - 341332 * q^34 - 78859 * q^35 + 710046 * q^36 - 297971 * q^37 + 329075 * q^38 + 220644 * q^39 - 203048 * q^40 + 659077 * q^41 + 359505 * q^42 - 1431608 * q^43 + 254916 * q^44 - 49572 * q^45 + 873113 * q^46 - 1574073 * q^47 - 84726 * q^48 + 1893545 * q^49 + 302533 * q^50 + 1214595 * q^51 - 4972548 * q^52 + 587736 * q^53 + 118098 * q^54 - 4624036 * q^55 - 5798506 * q^56 + 1083699 * q^57 - 6991380 * q^58 + 3286064 * q^59 - 3527739 * q^60 - 6117131 * q^61 - 11570258 * q^62 - 1708047 * q^63 - 19063011 * q^64 - 5335514 * q^65 - 2953638 * q^66 - 16518710 * q^67 - 17284669 * q^68 + 76491 * q^69 - 39189486 * q^70 - 10882582 * q^71 + 597051 * q^72 - 21097441 * q^73 - 16717030 * q^74 - 7715142 * q^75 - 40864952 * q^76 - 3404601 * q^77 + 3494340 * q^78 - 3784458 * q^79 - 27466195 * q^80 + 8503056 * q^81 - 24990117 * q^82 - 1951425 * q^83 - 3048030 * q^84 - 23238675 * q^85 - 35910572 * q^86 - 3898125 * q^87 - 27843055 * q^88 + 10499443 * q^89 - 2536191 * q^90 + 699217 * q^91 - 20062766 * q^92 + 3827493 * q^93 - 59358988 * q^94 - 29236333 * q^95 + 978048 * q^96 - 25158976 * q^97 + 2120460 * q^98 + 654642 * 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 $

$2$	13.0039	1.14939	0.574696	−	0.818367i	$-0.305120\pi$
$2$	13.0039	1.14939	0.574696	+	0.818367i	$0.305120\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	41.1009	0.321101
$5$	167.303	0.598562	0.299281	−	0.954165i	$-0.403253\pi$
$5$	167.303	0.598562	0.299281	+	0.954165i	$0.403253\pi$
$6$	−351.105	−0.663601
$7$	887.373	0.977829	0.488915	−	0.872332i	$-0.337393\pi$
$7$	887.373	0.977829	0.488915	+	0.872332i	$0.337393\pi$
$8$	−1130.03	−0.780321
$9$	729.000	0.333333
$10$	2175.59	0.687982
$11$	−7784.37	−1.76339	−0.881696	−	0.471818i	$-0.843598\pi$
$11$	−7784.37	−1.76339	−0.881696	+	0.471818i	$0.843598\pi$
$12$	−1109.72	−0.185388
$13$	4609.99	0.581967	0.290983	−	0.956728i	$-0.406018\pi$
$13$	4609.99	0.581967	0.290983	+	0.956728i	$0.406018\pi$
$14$	11539.3	1.12391
$15$	−4517.19	−0.345580
$16$	−19955.6	−1.21800
$17$	−6055.74	−0.298948	−0.149474	−	0.988766i	$-0.547758\pi$
$17$	−6055.74	−0.298948	−0.149474	+	0.988766i	$0.547758\pi$
$18$	9479.83	0.383131
$19$	13552.7	0.453304	0.226652	−	0.973976i	$-0.427222\pi$
$19$	13552.7	0.453304	0.226652	+	0.973976i	$0.427222\pi$
$20$	6876.32	0.192199
$21$	−23959.1	−0.564550
$22$	−101227.	−2.02683
$23$	35274.4	0.604523	0.302261	−	0.953225i	$-0.402258\pi$
$23$	35274.4	0.604523	0.302261	+	0.953225i	$0.402258\pi$
$24$	30510.7	0.450519
$25$	−50134.6	−0.641723
$26$	59947.7	0.668907
$27$	−19683.0	−0.192450
$28$	36471.8	0.313982
$29$	78335.1	0.596435	0.298218	−	0.954498i	$-0.403608\pi$
$29$	78335.1	0.596435	0.298218	+	0.954498i	$0.403608\pi$
$30$	−58741.0	−0.397207
$31$	−170575.	−1.02837	−0.514184	−	0.857680i	$-0.671905\pi$
$31$	−170575.	−1.02837	−0.514184	+	0.857680i	$0.671905\pi$
$32$	−114857.	−0.619632
$33$	210178.	1.01809
$34$	−78748.2	−0.343609
$35$	148460.	0.585292
$36$	29962.6	0.107034
$37$	−263573.	−0.855451	−0.427726	−	0.903909i	$-0.640685\pi$
$37$	−263573.	−0.855451	−0.427726	+	0.903909i	$0.640685\pi$
$38$	176238.	0.521024
$39$	−124470.	−0.335999
$40$	−189057.	−0.467071
$41$	−198905.	−0.450714	−0.225357	−	0.974276i	$-0.572355\pi$
$41$	−198905.	−0.450714	−0.225357	+	0.974276i	$0.572355\pi$
$42$	−311561.	−0.648889
$43$	−444435.	−0.852449	−0.426225	−	0.904617i	$-0.640157\pi$
$43$	−444435.	−0.852449	−0.426225	+	0.904617i	$0.640157\pi$
$44$	−319945.	−0.566227
$45$	121964.	0.199521
$46$	458704.	0.694833
$47$	−531622.	−0.746896	−0.373448	−	0.927651i	$-0.621825\pi$
$47$	−531622.	−0.746896	−0.373448	+	0.927651i	$0.621825\pi$
$48$	538802.	0.703210
$49$	−36112.2	−0.0438499
$50$	−651945.	−0.737591
$51$	163505.	0.172598
$52$	189475.	0.186870
$53$	−1.00107e6	−0.923631	−0.461816	−	0.886976i	$-0.652802\pi$
$53$	−1.00107e6	−0.923631	−0.461816	+	0.886976i	$0.652802\pi$
$54$	−255955.	−0.221200
$55$	−1.30235e6	−1.05550
$56$	−1.00275e6	−0.763021
$57$	−365924.	−0.261715
$58$	1.01866e6	0.685538
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	−185661.	−0.110966
$61$	−1.77312e6	−1.00019	−0.500097	−	0.865969i	$-0.666702\pi$
$61$	−1.77312e6	−1.00019	−0.500097	+	0.865969i	$0.666702\pi$
$62$	−2.21813e6	−1.18200
$63$	646895.	0.325943
$64$	1.06073e6	0.505795
$65$	771266.	0.348343
$66$	2.73313e6	1.17019
$67$	−2.37225e6	−0.963606	−0.481803	−	0.876280i	$-0.660018\pi$
$67$	−2.37225e6	−0.963606	−0.481803	+	0.876280i	$0.660018\pi$
$68$	−248897.	−0.0959926
$69$	−952409.	−0.349021
$70$	1.93056e6	0.672729
$71$	−5.87275e6	−1.94732	−0.973659	−	0.228008i	$-0.926779\pi$
$71$	−5.87275e6	−1.94732	−0.973659	+	0.228008i	$0.926779\pi$
$72$	−823789.	−0.260107
$73$	−2.90906e6	−0.875232	−0.437616	−	0.899162i	$-0.644177\pi$
$73$	−2.90906e6	−0.875232	−0.437616	+	0.899162i	$0.644177\pi$
$74$	−3.42748e6	−0.983249
$75$	1.35363e6	0.370499
$76$	557030.	0.145556
$77$	−6.90764e6	−1.72430
$78$	−1.61859e6	−0.386194
$79$	−1.66441e6	−0.379808	−0.189904	−	0.981803i	$-0.560818\pi$
$79$	−1.66441e6	−0.379808	−0.189904	+	0.981803i	$0.560818\pi$
$80$	−3.33864e6	−0.729046
$81$	531441.	0.111111
$82$	−2.58653e6	−0.518047
$83$	8.18425e6	1.57111	0.785553	−	0.618794i	$-0.212379\pi$
$83$	8.18425e6	1.57111	0.785553	+	0.618794i	$0.212379\pi$
$84$	−984739.	−0.181277
$85$	−1.01315e6	−0.178939
$86$	−5.77938e6	−0.979798
$87$	−2.11505e6	−0.344352
$88$	8.79653e6	1.37601
$89$	1.13356e7	1.70444	0.852219	−	0.523185i	$-0.175256\pi$
$89$	1.13356e7	1.70444	0.852219	+	0.523185i	$0.175256\pi$
$90$	1.58601e6	0.229327
$91$	4.09078e6	0.569064
$92$	1.44981e6	0.194113
$93$	4.60552e6	0.593729
$94$	−6.91315e6	−0.858476
$95$	2.26742e6	0.271331
$96$	3.10115e6	0.357745
$97$	243947.	0.0271390	0.0135695	−	0.999908i	$-0.495681\pi$
$97$	243947.	0.0271390	0.0135695	+	0.999908i	$0.495681\pi$
$98$	−469599.	−0.0504006
$99$	−5.67480e6	−0.587797
$100$	−2.06058e6	−0.206058
$101$	1.91745e7	1.85183	0.925913	−	0.377736i	$-0.123297\pi$
$101$	1.91745e7	1.85183	0.925913	+	0.377736i	$0.123297\pi$
$102$	2.12620e6	0.198383
$103$	8.63345e6	0.778492	0.389246	−	0.921134i	$-0.372736\pi$
$103$	8.63345e6	0.778492	0.389246	+	0.921134i	$0.372736\pi$
$104$	−5.20940e6	−0.454121
$105$	−4.00843e6	−0.337918
$106$	−1.30178e7	−1.06161
$107$	−1.22715e7	−0.968397	−0.484199	−	0.874958i	$-0.660889\pi$
$107$	−1.22715e7	−0.968397	−0.484199	+	0.874958i	$0.660889\pi$
$108$	−808989.	−0.0617959
$109$	−6.90858e6	−0.510971	−0.255485	−	0.966813i	$-0.582235\pi$
$109$	−6.90858e6	−0.510971	−0.255485	+	0.966813i	$0.582235\pi$
$110$	−1.69356e7	−1.21318
$111$	7.11648e6	0.493895
$112$	−1.77081e7	−1.19099
$113$	9.23590e6	0.602150	0.301075	−	0.953601i	$-0.402655\pi$
$113$	9.23590e6	0.602150	0.301075	+	0.953601i	$0.402655\pi$
$114$	−4.75843e6	−0.300813
$115$	5.90153e6	0.361844
$116$	3.21964e6	0.191516
$117$	3.36068e6	0.193989
$118$	2.67072e6	0.149638
$119$	−5.37370e6	−0.292320
$120$	5.10454e6	0.269663
$121$	4.11092e7	2.10955
$122$	−2.30575e7	−1.14961
$123$	5.37042e6	0.260220
$124$	−7.01078e6	−0.330210
$125$	−2.14583e7	−0.982674
$126$	8.41214e6	0.374636
$127$	3.43987e6	0.149015	0.0745073	−	0.997220i	$-0.476262\pi$
$127$	3.43987e6	0.149015	0.0745073	+	0.997220i	$0.476262\pi$
$128$	2.84953e7	1.20099
$129$	1.19997e7	0.492162
$130$	1.00295e7	0.400383
$131$	−2.36186e6	−0.0917919	−0.0458960	−	0.998946i	$-0.514614\pi$
$131$	−2.36186e6	−0.0917919	−0.0458960	+	0.998946i	$0.514614\pi$
$132$	8.63850e6	0.326911
$133$	1.20263e7	0.443254
$134$	−3.08485e7	−1.10756
$135$	−3.29303e6	−0.115193
$136$	6.84314e6	0.233276
$137$	3.14805e6	0.104597	0.0522985	−	0.998631i	$-0.483345\pi$
$137$	3.14805e6	0.104597	0.0522985	+	0.998631i	$0.483345\pi$
$138$	−1.23850e7	−0.401162
$139$	1.60628e7	0.507305	0.253652	−	0.967295i	$-0.418368\pi$
$139$	1.60628e7	0.507305	0.253652	+	0.967295i	$0.418368\pi$
$140$	6.10186e6	0.187938
$141$	1.43538e7	0.431221
$142$	−7.63685e7	−2.23823
$143$	−3.58858e7	−1.02624
$144$	−1.45477e7	−0.405998
$145$	1.31057e7	0.357004
$146$	−3.78291e7	−1.00598
$147$	975030.	0.0253167
$148$	−1.08331e7	−0.274686
$149$	−3.30906e7	−0.819506	−0.409753	−	0.912196i	$-0.634385\pi$
$149$	−3.30906e7	−0.819506	−0.409753	+	0.912196i	$0.634385\pi$
$150$	1.76025e7	0.425848
$151$	−8.23311e7	−1.94601	−0.973003	−	0.230791i	$-0.925869\pi$
$151$	−8.23311e7	−1.94601	−0.973003	+	0.230791i	$0.925869\pi$
$152$	−1.53149e7	−0.353722
$153$	−4.41464e6	−0.0996495
$154$	−8.98261e7	−1.98189
$155$	−2.85377e7	−0.615543
$156$	−5.11582e6	−0.107889
$157$	7.06660e7	1.45734	0.728671	−	0.684863i	$-0.240138\pi$
$157$	7.06660e7	1.45734	0.728671	+	0.684863i	$0.240138\pi$
$158$	−2.16437e7	−0.436548
$159$	2.70289e7	0.533259
$160$	−1.92160e7	−0.370889
$161$	3.13016e7	0.591120
$162$	6.91080e6	0.127710
$163$	−5.37840e7	−0.972740	−0.486370	−	0.873753i	$-0.661679\pi$
$163$	−5.37840e7	−0.972740	−0.486370	+	0.873753i	$0.661679\pi$
$164$	−8.17516e6	−0.144725
$165$	3.51634e7	0.609393
$166$	1.06427e8	1.80582
$167$	4.12989e7	0.686169	0.343084	−	0.939305i	$-0.388528\pi$
$167$	4.12989e7	0.686169	0.343084	+	0.939305i	$0.388528\pi$
$168$	2.70744e7	0.440530
$169$	−4.14965e7	−0.661315
$170$	−1.31748e7	−0.205671
$171$	9.87994e6	0.151101
$172$	−1.82667e7	−0.273722
$173$	5.36462e7	0.787731	0.393865	−	0.919168i	$-0.371138\pi$
$173$	5.36462e7	0.787731	0.393865	+	0.919168i	$0.371138\pi$
$174$	−2.75038e7	−0.395795
$175$	−4.44881e7	−0.627496
$176$	1.55342e8	2.14780
$177$	−5.54523e6	−0.0751646
$178$	1.47407e8	1.95907
$179$	−3.62629e7	−0.472581	−0.236291	−	0.971682i	$-0.575932\pi$
$179$	−3.62629e7	−0.472581	−0.236291	+	0.971682i	$0.575932\pi$
$180$	5.01283e6	0.0640663
$181$	−1.64625e7	−0.206358	−0.103179	−	0.994663i	$-0.532901\pi$
$181$	−1.64625e7	−0.206358	−0.103179	+	0.994663i	$0.532901\pi$
$182$	5.31960e7	0.654077
$183$	4.78743e7	0.577462
$184$	−3.98610e7	−0.471722
$185$	−4.40967e7	−0.512041
$186$	5.98896e7	0.682427
$187$	4.71401e7	0.527163
$188$	−2.18501e7	−0.239829
$189$	−1.74662e7	−0.188183
$190$	2.94852e7	0.311865
$191$	−7.01561e7	−0.728532	−0.364266	−	0.931295i	$-0.618680\pi$
$191$	−7.01561e7	−0.728532	−0.364266	+	0.931295i	$0.618680\pi$
$192$	−2.86397e7	−0.292021
$193$	9.06871e7	0.908019	0.454010	−	0.890997i	$-0.349993\pi$
$193$	9.06871e7	0.908019	0.454010	+	0.890997i	$0.349993\pi$
$194$	3.17225e6	0.0311933
$195$	−2.08242e7	−0.201116
$196$	−1.48425e6	−0.0140802
$197$	−3.02563e7	−0.281958	−0.140979	−	0.990013i	$-0.545025\pi$
$197$	−3.02563e7	−0.281958	−0.140979	+	0.990013i	$0.545025\pi$
$198$	−7.37945e7	−0.675609
$199$	1.78612e8	1.60666	0.803332	−	0.595532i	$-0.203059\pi$
$199$	1.78612e8	1.60666	0.803332	+	0.595532i	$0.203059\pi$
$200$	5.66534e7	0.500750
$201$	6.40509e7	0.556338
$202$	2.49343e8	2.12847
$203$	6.95125e7	0.583212
$204$	6.72021e6	0.0554213
$205$	−3.32774e7	−0.269781
$206$	1.12268e8	0.894792
$207$	2.57151e7	0.201508
$208$	−9.19952e7	−0.708832
$209$	−1.05499e8	−0.799352
$210$	−5.21252e7	−0.388401
$211$	−1.01493e8	−0.743786	−0.371893	−	0.928276i	$-0.621291\pi$
$211$	−1.01493e8	−0.743786	−0.371893	+	0.928276i	$0.621291\pi$
$212$	−4.11449e7	−0.296579
$213$	1.58564e8	1.12428
$214$	−1.59577e8	−1.11307
$215$	−7.43554e7	−0.510244
$216$	2.22423e7	0.150173
$217$	−1.51363e8	−1.00557
$218$	−8.98383e7	−0.587305
$219$	7.85447e7	0.505315
$220$	−5.35278e7	−0.338922
$221$	−2.79169e7	−0.173978
$222$	9.25418e7	0.567679
$223$	1.17677e8	0.710600	0.355300	−	0.934752i	$-0.384379\pi$
$223$	1.17677e8	0.710600	0.355300	+	0.934752i	$0.384379\pi$
$224$	−1.01921e8	−0.605895
$225$	−3.65481e7	−0.213908
$226$	1.20102e8	0.692106
$227$	2.17121e8	1.23200	0.616000	−	0.787746i	$-0.288752\pi$
$227$	2.17121e8	1.23200	0.616000	+	0.787746i	$0.288752\pi$
$228$	−1.50398e7	−0.0840369
$229$	6.89428e7	0.379372	0.189686	−	0.981845i	$-0.439253\pi$
$229$	6.89428e7	0.379372	0.189686	+	0.981845i	$0.439253\pi$
$230$	7.67427e7	0.415901
$231$	1.86506e8	0.995523
$232$	−8.85207e7	−0.465411
$233$	2.42127e8	1.25400	0.627000	−	0.779019i	$-0.284282\pi$
$233$	2.42127e8	1.25400	0.627000	+	0.779019i	$0.284282\pi$
$234$	4.37019e7	0.222969
$235$	−8.89421e7	−0.447064
$236$	8.44126e6	0.0418038
$237$	4.49389e7	0.219282
$238$	−6.98790e7	−0.335991
$239$	1.66165e8	0.787312	0.393656	−	0.919258i	$-0.371210\pi$
$239$	1.66165e8	0.787312	0.393656	+	0.919258i	$0.371210\pi$
$240$	9.01433e7	0.420915
$241$	2.91718e7	0.134247	0.0671233	−	0.997745i	$-0.478618\pi$
$241$	2.91718e7	0.134247	0.0671233	+	0.997745i	$0.478618\pi$
$242$	5.34579e8	2.42470
$243$	−1.43489e7	−0.0641500
$244$	−7.28769e7	−0.321163
$245$	−6.04170e6	−0.0262469
$246$	6.98363e7	0.299095
$247$	6.24779e7	0.263808
$248$	1.92754e8	0.802458
$249$	−2.20975e8	−0.907078
$250$	−2.79041e8	−1.12948
$251$	2.96094e8	1.18188	0.590938	−	0.806717i	$-0.298758\pi$
$251$	2.96094e8	1.18188	0.590938	+	0.806717i	$0.298758\pi$
$252$	2.65880e7	0.104661
$253$	−2.74589e8	−1.06601
$254$	4.47317e7	0.171276
$255$	2.73549e7	0.103311
$256$	2.34777e8	0.874611
$257$	2.67505e8	0.983027	0.491514	−	0.870870i	$-0.336444\pi$
$257$	2.67505e8	0.983027	0.491514	+	0.870870i	$0.336444\pi$
$258$	1.56043e8	0.565686
$259$	−2.33888e8	−0.836485
$260$	3.16997e7	0.111853
$261$	5.71063e7	0.198812
$262$	−3.07133e7	−0.105505
$263$	2.10348e8	0.713006	0.356503	−	0.934294i	$-0.383969\pi$
$263$	2.10348e8	0.713006	0.356503	+	0.934294i	$0.383969\pi$
$264$	−2.37506e8	−0.794441
$265$	−1.67482e8	−0.552851
$266$	1.56389e8	0.509472
$267$	−3.06062e8	−0.984058
$268$	−9.75018e7	−0.309415
$269$	−4.76854e8	−1.49366	−0.746831	−	0.665014i	$-0.768426\pi$
$269$	−4.76854e8	−1.49366	−0.746831	+	0.665014i	$0.768426\pi$
$270$	−4.28222e7	−0.132402
$271$	−2.49889e8	−0.762701	−0.381350	−	0.924431i	$-0.624541\pi$
$271$	−2.49889e8	−0.762701	−0.381350	+	0.924431i	$0.624541\pi$
$272$	1.20846e8	0.364118
$273$	−1.10451e8	−0.328549
$274$	4.09368e7	0.120223
$275$	3.90266e8	1.13161
$276$	−3.91449e7	−0.112071
$277$	−2.02351e8	−0.572039	−0.286020	−	0.958224i	$-0.592332\pi$
$277$	−2.02351e8	−0.572039	−0.286020	+	0.958224i	$0.592332\pi$
$278$	2.08879e8	0.583092
$279$	−1.24349e8	−0.342790
$280$	−1.67764e8	−0.456715
$281$	−2.77424e8	−0.745885	−0.372942	−	0.927855i	$-0.621651\pi$
$281$	−2.77424e8	−0.745885	−0.372942	+	0.927855i	$0.621651\pi$
$282$	1.86655e8	0.495641
$283$	−3.44360e8	−0.903151	−0.451575	−	0.892233i	$-0.649138\pi$
$283$	−3.44360e8	−0.903151	−0.451575	+	0.892233i	$0.649138\pi$
$284$	−2.41375e8	−0.625286
$285$	−6.12203e7	−0.156653
$286$	−4.66655e8	−1.17955
$287$	−1.76503e8	−0.440722
$288$	−8.37310e7	−0.206544
$289$	−3.73667e8	−0.910630
$290$	1.70425e8	0.410337
$291$	−6.58656e6	−0.0156687
$292$	−1.19565e8	−0.281038
$293$	−3.56077e8	−0.827004	−0.413502	−	0.910503i	$-0.635694\pi$
$293$	−3.56077e8	−0.827004	−0.413502	+	0.910503i	$0.635694\pi$
$294$	1.26792e7	0.0290988
$295$	3.43606e7	0.0779262
$296$	2.97845e8	0.667527
$297$	1.53220e8	0.339365
$298$	−4.30306e8	−0.941934
$299$	1.62615e8	0.351812
$300$	5.56356e7	0.118968
$301$	−3.94379e8	−0.833550
$302$	−1.07062e9	−2.23672
$303$	−5.17713e8	−1.06915
$304$	−2.70453e8	−0.552122
$305$	−2.96649e8	−0.598679
$306$	−5.74074e7	−0.114536
$307$	−1.96195e8	−0.386993	−0.193496	−	0.981101i	$-0.561983\pi$
$307$	−1.96195e8	−0.386993	−0.193496	+	0.981101i	$0.561983\pi$
$308$	−2.83910e8	−0.553673
$309$	−2.33103e8	−0.449463
$310$	−3.71101e8	−0.707500
$311$	5.23320e7	0.0986519	0.0493260	−	0.998783i	$-0.484293\pi$
$311$	5.23320e7	0.0986519	0.0493260	+	0.998783i	$0.484293\pi$
$312$	1.40654e8	0.262187
$313$	−3.12015e8	−0.575136	−0.287568	−	0.957760i	$-0.592847\pi$
$313$	−3.12015e8	−0.575136	−0.287568	+	0.957760i	$0.592847\pi$
$314$	9.18932e8	1.67506
$315$	1.08228e8	0.195097
$316$	−6.84086e7	−0.121957
$317$	5.38220e8	0.948970	0.474485	−	0.880264i	$-0.342634\pi$
$317$	5.38220e8	0.948970	0.474485	+	0.880264i	$0.342634\pi$
$318$	3.51480e8	0.612923
$319$	−6.09789e8	−1.05175
$320$	1.77463e8	0.302750
$321$	3.31330e8	0.559104
$322$	4.07042e8	0.679428
$323$	−8.20719e7	−0.135514
$324$	2.18427e7	0.0356779
$325$	−2.31120e8	−0.373461
$326$	−6.99401e8	−1.11806
$327$	1.86532e8	0.295009
$328$	2.24767e8	0.351702
$329$	−4.71747e8	−0.730337
$330$	4.57261e8	0.700431
$331$	−7.08848e8	−1.07437	−0.537187	−	0.843463i	$-0.680513\pi$
$331$	−7.08848e8	−1.07437	−0.537187	+	0.843463i	$0.680513\pi$
$332$	3.36380e8	0.504483
$333$	−1.92145e8	−0.285150
$334$	5.37046e8	0.788676
$335$	−3.96886e8	−0.576778
$336$	4.78118e8	0.687619
$337$	1.28631e8	0.183079	0.0915397	−	0.995801i	$-0.470821\pi$
$337$	1.28631e8	0.183079	0.0915397	+	0.995801i	$0.470821\pi$
$338$	−5.39616e8	−0.760110
$339$	−2.49369e8	−0.347651
$340$	−4.16412e7	−0.0574575
$341$	1.32782e9	1.81342
$342$	1.28478e8	0.173675
$343$	−7.62835e8	−1.02071
$344$	5.02223e8	0.665184
$345$	−1.59341e8	−0.208911
$346$	6.97609e8	0.905411
$347$	−6.00582e7	−0.0771648	−0.0385824	−	0.999255i	$-0.512284\pi$
$347$	−6.00582e7	−0.0771648	−0.0385824	+	0.999255i	$0.512284\pi$
$348$	−8.69304e7	−0.110572
$349$	1.11674e9	1.40625	0.703124	−	0.711067i	$-0.251788\pi$
$349$	1.11674e9	1.40625	0.703124	+	0.711067i	$0.251788\pi$
$350$	−5.78518e8	−0.721238
$351$	−9.07384e7	−0.112000
$352$	8.94092e8	1.09265
$353$	3.99147e8	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$353$	3.99147e8	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$354$	−7.21095e7	−0.0863936
$355$	−9.82530e8	−1.16559
$356$	4.65905e8	0.547296
$357$	1.45090e8	0.168771
$358$	−4.71558e8	−0.543181
$359$	−6.35935e8	−0.725408	−0.362704	−	0.931904i	$-0.618146\pi$
$359$	−6.35935e8	−0.725408	−0.362704	+	0.931904i	$0.618146\pi$
$360$	−1.37823e8	−0.155690
$361$	−7.10195e8	−0.794516
$362$	−2.14077e8	−0.237186
$363$	−1.10995e9	−1.21795
$364$	1.68135e8	0.182727
$365$	−4.86696e8	−0.523881
$366$	6.22552e8	0.663730
$367$	−2.61675e8	−0.276332	−0.138166	−	0.990409i	$-0.544121\pi$
$367$	−2.61675e8	−0.276332	−0.138166	+	0.990409i	$0.544121\pi$
$368$	−7.03923e8	−0.736305
$369$	−1.45001e8	−0.150238
$370$	−5.73428e8	−0.588536
$371$	−8.88322e8	−0.903154
$372$	1.89291e8	0.190647
$373$	1.69724e9	1.69341	0.846706	−	0.532060i	$-0.178582\pi$
$373$	1.69724e9	1.69341	0.846706	+	0.532060i	$0.178582\pi$
$374$	6.13004e8	0.605917
$375$	5.79373e8	0.567347
$376$	6.00746e8	0.582819
$377$	3.61124e8	0.347106
$378$	−2.27128e8	−0.216296
$379$	1.10612e9	1.04367	0.521837	−	0.853045i	$-0.325247\pi$
$379$	1.10612e9	1.04367	0.521837	+	0.853045i	$0.325247\pi$
$380$	9.31929e7	0.0871245
$381$	−9.28765e7	−0.0860337
$382$	−9.12301e8	−0.837368
$383$	−6.71833e8	−0.611034	−0.305517	−	0.952187i	$-0.598829\pi$
$383$	−6.71833e8	−0.611034	−0.305517	+	0.952187i	$0.598829\pi$
$384$	−7.69374e8	−0.693391
$385$	−1.15567e9	−1.03210
$386$	1.17928e9	1.04367
$387$	−3.23993e8	−0.284150
$388$	1.00264e7	0.00871435
$389$	1.16525e9	1.00368	0.501839	−	0.864961i	$-0.332657\pi$
$389$	1.16525e9	1.00368	0.501839	+	0.864961i	$0.332657\pi$
$390$	−2.70795e8	−0.231161
$391$	−2.13613e8	−0.180721
$392$	4.08077e7	0.0342170
$393$	6.37702e7	0.0529961
$394$	−3.93450e8	−0.324080
$395$	−2.78460e8	−0.227339
$396$	−2.33240e8	−0.188742
$397$	6.93011e8	0.555870	0.277935	−	0.960600i	$-0.410350\pi$
$397$	6.93011e8	0.555870	0.277935	+	0.960600i	$0.410350\pi$
$398$	2.32265e9	1.84669
$399$	−3.24711e8	−0.255913
$400$	1.00047e9	0.781616
$401$	−2.25957e9	−1.74993	−0.874965	−	0.484186i	$-0.839116\pi$
$401$	−2.25957e9	−1.74993	−0.874965	+	0.484186i	$0.839116\pi$
$402$	8.32910e8	0.639450
$403$	−7.86348e8	−0.598476
$404$	7.88091e8	0.594623
$405$	8.89118e7	0.0665069
$406$	9.03932e8	0.670339
$407$	2.05175e9	1.50850
$408$	−1.84765e8	−0.134682
$409$	8.72327e7	0.0630446	0.0315223	−	0.999503i	$-0.489964\pi$
$409$	8.72327e7	0.0630446	0.0315223	+	0.999503i	$0.489964\pi$
$410$	−4.32735e8	−0.310084
$411$	−8.49973e7	−0.0603891
$412$	3.54843e8	0.249974
$413$	1.82248e8	0.127303
$414$	3.34395e8	0.231611
$415$	1.36925e9	0.940405
$416$	−5.29491e8	−0.360605
$417$	−4.33695e8	−0.292893
$418$	−1.37190e9	−0.918769
$419$	−8.24265e8	−0.547416	−0.273708	−	0.961813i	$-0.588250\pi$
$419$	−8.24265e8	−0.547416	−0.273708	+	0.961813i	$0.588250\pi$
$420$	−1.64750e8	−0.108506
$421$	−1.82215e9	−1.19014	−0.595068	−	0.803675i	$-0.702875\pi$
$421$	−1.82215e9	−1.19014	−0.595068	+	0.803675i	$0.702875\pi$
$422$	−1.31980e9	−0.854901
$423$	−3.87552e8	−0.248965
$424$	1.13123e9	0.720729
$425$	3.03602e8	0.191842
$426$	2.06195e9	1.29224
$427$	−1.57342e9	−0.978019
$428$	−5.04369e8	−0.310953
$429$	9.68918e8	0.592497
$430$	−9.66909e8	−0.586470
$431$	3.24182e9	1.95038	0.975189	−	0.221375i	$-0.0710544\pi$
$431$	3.24182e9	1.95038	0.975189	+	0.221375i	$0.0710544\pi$
$432$	3.92787e8	0.234403
$433$	1.71326e9	1.01418	0.507090	−	0.861893i	$-0.330721\pi$
$433$	1.71326e9	1.01418	0.507090	+	0.861893i	$0.330721\pi$
$434$	−1.96831e9	−1.15579
$435$	−3.53854e8	−0.206116
$436$	−2.83949e8	−0.164073
$437$	4.78065e8	0.274032
$438$	1.02139e9	0.580805
$439$	−1.70403e9	−0.961283	−0.480642	−	0.876917i	$-0.659596\pi$
$439$	−1.70403e9	−0.961283	−0.480642	+	0.876917i	$0.659596\pi$
$440$	1.47169e9	0.823629
$441$	−2.63258e7	−0.0146166
$442$	−3.63028e8	−0.199969
$443$	8.96154e7	0.0489745	0.0244872	−	0.999700i	$-0.492205\pi$
$443$	8.96154e7	0.0489745	0.0244872	+	0.999700i	$0.492205\pi$
$444$	2.92494e8	0.158590
$445$	1.89649e9	1.02021
$446$	1.53026e9	0.816757
$447$	8.93445e8	0.473142
$448$	9.41262e8	0.494581
$449$	−2.38975e9	−1.24592	−0.622959	−	0.782255i	$-0.714070\pi$
$449$	−2.38975e9	−1.24592	−0.622959	+	0.782255i	$0.714070\pi$
$450$	−4.75268e8	−0.245864
$451$	1.54835e9	0.794786
$452$	3.79604e8	0.193351
$453$	2.22294e9	1.12353
$454$	2.82341e9	1.41605
$455$	6.84401e8	0.340620
$456$	4.13503e8	0.204222
$457$	3.10251e9	1.52057	0.760286	−	0.649588i	$-0.225059\pi$
$457$	3.10251e9	1.52057	0.760286	+	0.649588i	$0.225059\pi$
$458$	8.96525e8	0.436047
$459$	1.19195e8	0.0575326
$460$	2.42558e8	0.116189
$461$	−9.42015e8	−0.447821	−0.223911	−	0.974610i	$-0.571882\pi$
$461$	−9.42015e8	−0.447821	−0.223911	+	0.974610i	$0.571882\pi$
$462$	2.42530e9	1.14425
$463$	−2.76277e9	−1.29363	−0.646817	−	0.762645i	$-0.723900\pi$
$463$	−2.76277e9	−1.29363	−0.646817	+	0.762645i	$0.723900\pi$
$464$	−1.56323e9	−0.726455
$465$	7.70518e8	0.355384
$466$	3.14859e9	1.44134
$467$	602426.	0.000273712 0	0.000136856	−	1.00000i	$-0.499956\pi$
$467$	602426.	0.000273712 0	0.000136856		1.00000i	$0.499956\pi$
$468$	1.38127e8	0.0622900
$469$	−2.10507e9	−0.942242
$470$	−1.15659e9	−0.513852
$471$	−1.90798e9	−0.841397
$472$	−2.32083e8	−0.101589
$473$	3.45964e9	1.50320
$474$	5.84381e8	0.252041
$475$	−6.79461e8	−0.290896
$476$	−2.20864e8	−0.0938643
$477$	−7.29780e8	−0.307877
$478$	2.16079e9	0.904930
$479$	−2.45402e9	−1.02024	−0.510121	−	0.860103i	$-0.670399\pi$
$479$	−2.45402e9	−1.02024	−0.510121	+	0.860103i	$0.670399\pi$
$480$	5.18832e8	0.214133
$481$	−1.21507e9	−0.497844
$482$	3.79346e8	0.154302
$483$	−8.45142e8	−0.341283
$484$	1.68963e9	0.677379
$485$	4.08131e7	0.0162444
$486$	−1.86591e8	−0.0737335
$487$	−1.78601e9	−0.700701	−0.350350	−	0.936619i	$-0.613937\pi$
$487$	−1.78601e9	−0.700701	−0.350350	+	0.936619i	$0.613937\pi$
$488$	2.00367e9	0.780472
$489$	1.45217e9	0.561612
$490$	−7.85655e7	−0.0301679
$491$	−2.06965e9	−0.789063	−0.394532	−	0.918882i	$-0.629093\pi$
$491$	−2.06965e9	−0.789063	−0.394532	+	0.918882i	$0.629093\pi$
$492$	2.20729e8	0.0835569
$493$	−4.74377e8	−0.178303
$494$	8.12456e8	0.303218
$495$	−9.49413e8	−0.351833
$496$	3.40393e9	1.25255
$497$	−5.21132e9	−1.90415
$498$	−2.87353e9	−1.04259
$499$	−1.63444e9	−0.588866	−0.294433	−	0.955672i	$-0.595131\pi$
$499$	−1.63444e9	−0.588866	−0.294433	+	0.955672i	$0.595131\pi$
$500$	−8.81954e8	−0.315537
$501$	−1.11507e9	−0.396160
$502$	3.85037e9	1.35844
$503$	−1.96230e9	−0.687507	−0.343754	−	0.939060i	$-0.611699\pi$
$503$	−1.96230e9	−0.687507	−0.343754	+	0.939060i	$0.611699\pi$
$504$	−7.31008e8	−0.254340
$505$	3.20796e9	1.10843
$506$	−3.57072e9	−1.22526
$507$	1.12041e9	0.381810
$508$	1.41382e8	0.0478487
$509$	−1.74020e8	−0.0584906	−0.0292453	−	0.999572i	$-0.509310\pi$
$509$	−1.74020e8	−0.0584906	−0.0292453	+	0.999572i	$0.509310\pi$
$510$	3.55720e8	0.118744
$511$	−2.58142e9	−0.855827
$512$	−5.94396e8	−0.195718
$513$	−2.66759e8	−0.0872384
$514$	3.47860e9	1.12988
$515$	1.44440e9	0.465976
$516$	4.93200e8	0.158034
$517$	4.13834e9	1.31707
$518$	−3.04145e9	−0.961449
$519$	−1.44845e9	−0.454797
$520$	−8.71550e8	−0.271820
$521$	−1.04646e9	−0.324184	−0.162092	−	0.986776i	$-0.551824\pi$
$521$	−1.04646e9	−0.324184	−0.162092	+	0.986776i	$0.551824\pi$
$522$	7.42603e8	0.228513
$523$	−3.71160e9	−1.13450	−0.567252	−	0.823545i	$-0.691993\pi$
$523$	−3.71160e9	−1.13450	−0.567252	+	0.823545i	$0.691993\pi$
$524$	−9.70745e7	−0.0294745
$525$	1.20118e9	0.362285
$526$	2.73534e9	0.819523
$527$	1.03296e9	0.307429
$528$	−4.19423e9	−1.24003
$529$	−2.16054e9	−0.634553
$530$	−2.17792e9	−0.635442
$531$	1.49721e8	0.0433963
$532$	4.94293e8	0.142329
$533$	−9.16948e8	−0.262301
$534$	−3.98000e9	−1.13107
$535$	−2.05306e9	−0.579646
$536$	2.68071e9	0.751922
$537$	9.79097e8	0.272845
$538$	−6.20095e9	−1.71680
$539$	2.81111e8	0.0773245
$540$	−1.35347e8	−0.0369887
$541$	3.26372e9	0.886180	0.443090	−	0.896477i	$-0.353882\pi$
$541$	3.26372e9	0.886180	0.443090	+	0.896477i	$0.353882\pi$
$542$	−3.24952e9	−0.876642
$543$	4.44488e8	0.119141
$544$	6.95547e8	0.185238
$545$	−1.15583e9	−0.305848
$546$	−1.43629e9	−0.377632
$547$	7.46387e8	0.194988	0.0974941	−	0.995236i	$-0.468917\pi$
$547$	7.46387e8	0.194988	0.0974941	+	0.995236i	$0.468917\pi$
$548$	1.29388e8	0.0335862
$549$	−1.29261e9	−0.333398
$550$	5.07498e9	1.30066
$551$	1.06166e9	0.270366
$552$	1.07625e9	0.272349
$553$	−1.47695e9	−0.371388
$554$	−2.63135e9	−0.657497
$555$	1.19061e9	0.295627
$556$	6.60195e8	0.162896
$557$	5.33461e9	1.30801	0.654003	−	0.756492i	$-0.273088\pi$
$557$	5.33461e9	1.30801	0.654003	+	0.756492i	$0.273088\pi$
$558$	−1.61702e9	−0.393999
$559$	−2.04884e9	−0.496097
$560$	−2.96262e9	−0.712883
$561$	−1.27278e9	−0.304358
$562$	−3.60759e9	−0.857313
$563$	7.90248e9	1.86631	0.933156	−	0.359472i	$-0.117043\pi$
$563$	7.90248e9	1.86631	0.933156	+	0.359472i	$0.117043\pi$
$564$	5.89954e8	0.138465
$565$	1.54520e9	0.360424
$566$	−4.47802e9	−1.03807
$567$	4.71586e8	0.108648
$568$	6.63635e9	1.51953
$569$	5.54526e9	1.26191	0.630956	−	0.775819i	$-0.282663\pi$
$569$	5.54526e9	1.26191	0.630956	+	0.775819i	$0.282663\pi$
$570$	−7.96101e8	−0.180055
$571$	5.09241e9	1.14471	0.572357	−	0.820004i	$-0.306029\pi$
$571$	5.09241e9	1.14471	0.572357	+	0.820004i	$0.306029\pi$
$572$	−1.47494e9	−0.329525
$573$	1.89421e9	0.420618
$574$	−2.29522e9	−0.506562
$575$	−1.76847e9	−0.387936
$576$	7.73271e8	0.168598
$577$	1.69420e9	0.367154	0.183577	−	0.983005i	$-0.441232\pi$
$577$	1.69420e9	0.367154	0.183577	+	0.983005i	$0.441232\pi$
$578$	−4.85912e9	−1.04667
$579$	−2.44855e9	−0.524245
$580$	5.38657e8	0.114634
$581$	7.26248e9	1.53627
$582$	−8.56508e7	−0.0180095
$583$	7.79269e9	1.62872
$584$	3.28732e9	0.682962
$585$	5.62253e8	0.116114
$586$	−4.63039e9	−0.950551
$587$	−8.65701e9	−1.76659	−0.883293	−	0.468821i	$-0.844679\pi$
$587$	−8.65701e9	−1.76659	−0.883293	+	0.468821i	$0.844679\pi$
$588$	4.00746e7	0.00812922
$589$	−2.31175e9	−0.466163
$590$	4.46821e8	0.0895677
$591$	8.16921e8	0.162789
$592$	5.25977e9	1.04194
$593$	−7.39803e9	−1.45688	−0.728441	−	0.685108i	$-0.759755\pi$
$593$	−7.39803e9	−1.45688	−0.728441	+	0.685108i	$0.759755\pi$
$594$	1.99245e9	0.390063
$595$	−8.99038e8	−0.174972
$596$	−1.36005e9	−0.263144
$597$	−4.82252e9	−0.927607
$598$	2.11462e9	0.404370
$599$	4.39303e9	0.835162	0.417581	−	0.908640i	$-0.362878\pi$
$599$	4.39303e9	0.835162	0.417581	+	0.908640i	$0.362878\pi$
$600$	−1.52964e9	−0.289108
$601$	−1.08818e9	−0.204475	−0.102238	−	0.994760i	$-0.532600\pi$
$601$	−1.08818e9	−0.204475	−0.102238	+	0.994760i	$0.532600\pi$
$602$	−5.12846e9	−0.958075
$603$	−1.72937e9	−0.321202
$604$	−3.38388e9	−0.624864
$605$	6.87770e9	1.26270
$606$	−6.73227e9	−1.22887
$607$	−5.98074e9	−1.08541	−0.542706	−	0.839923i	$-0.682600\pi$
$607$	−5.98074e9	−1.08541	−0.542706	+	0.839923i	$0.682600\pi$
$608$	−1.55663e9	−0.280882
$609$	−1.87684e9	−0.336718
$610$	−3.85759e9	−0.688116
$611$	−2.45077e9	−0.434669
$612$	−1.81446e8	−0.0319975
$613$	−5.60656e9	−0.983070	−0.491535	−	0.870858i	$-0.663564\pi$
$613$	−5.60656e9	−0.983070	−0.491535	+	0.870858i	$0.663564\pi$
$614$	−2.55129e9	−0.444806
$615$	8.98489e8	0.155758
$616$	7.80580e9	1.34550
$617$	−9.07967e9	−1.55622	−0.778112	−	0.628126i	$-0.783822\pi$
$617$	−9.07967e9	−1.55622	−0.778112	+	0.628126i	$0.783822\pi$
$618$	−3.03125e9	−0.516609
$619$	−3.61181e9	−0.612079	−0.306040	−	0.952019i	$-0.599004\pi$
$619$	−3.61181e9	−0.612079	−0.306040	+	0.952019i	$0.599004\pi$
$620$	−1.17293e9	−0.197651
$621$	−6.94306e8	−0.116340
$622$	6.80518e8	0.113390
$623$	1.00589e10	1.66665
$624$	2.48387e9	0.409245
$625$	3.26731e8	0.0535316
$626$	−4.05741e9	−0.661056
$627$	2.84849e9	0.461506
$628$	2.90444e9	0.467954
$629$	1.59613e9	0.255736
$630$	1.40738e9	0.224243
$631$	2.73746e9	0.433756	0.216878	−	0.976199i	$-0.430413\pi$
$631$	2.73746e9	0.433756	0.216878	+	0.976199i	$0.430413\pi$
$632$	1.88082e9	0.296372
$633$	2.74031e9	0.429425
$634$	6.99895e9	1.09074
$635$	5.75502e8	0.0891946
$636$	1.11091e9	0.171230
$637$	−1.66477e8	−0.0255191
$638$	−7.92963e9	−1.20887
$639$	−4.28123e9	−0.649106
$640$	4.76736e9	0.718867
$641$	−7.16076e9	−1.07388	−0.536940	−	0.843620i	$-0.680420\pi$
$641$	−7.16076e9	−1.07388	−0.536940	+	0.843620i	$0.680420\pi$
$642$	4.30857e9	0.642630
$643$	−1.59933e9	−0.237247	−0.118623	−	0.992939i	$-0.537848\pi$
$643$	−1.59933e9	−0.237247	−0.118623	+	0.992939i	$0.537848\pi$
$644$	1.28652e9	0.189809
$645$	2.00760e9	0.294589
$646$	−1.06725e9	−0.155759
$647$	−1.56410e9	−0.227038	−0.113519	−	0.993536i	$-0.536212\pi$
$647$	−1.56410e9	−0.227038	−0.113519	+	0.993536i	$0.536212\pi$
$648$	−6.00542e8	−0.0867023
$649$	−1.59875e9	−0.229574
$650$	−3.00546e9	−0.429253
$651$	4.08681e9	0.580566
$652$	−2.21057e9	−0.312348
$653$	1.10948e10	1.55928	0.779639	−	0.626229i	$-0.215402\pi$
$653$	1.10948e10	1.55928	0.779639	+	0.626229i	$0.215402\pi$
$654$	2.42563e9	0.339081
$655$	−3.95147e8	−0.0549432
$656$	3.96927e9	0.548968
$657$	−2.12071e9	−0.291744
$658$	−6.13454e9	−0.839443
$659$	8.31935e9	1.13238	0.566188	−	0.824276i	$-0.308418\pi$
$659$	8.31935e9	1.13238	0.566188	+	0.824276i	$0.308418\pi$
$660$	1.44525e9	0.195677
$661$	5.78095e9	0.778563	0.389282	−	0.921119i	$-0.372723\pi$
$661$	5.78095e9	0.778563	0.389282	+	0.921119i	$0.372723\pi$
$662$	−9.21778e9	−1.23488
$663$	7.53756e8	0.100446
$664$	−9.24841e9	−1.22597
$665$	2.01204e9	0.265315
$666$	−2.49863e9	−0.327750
$667$	2.76323e9	0.360559
$668$	1.69742e9	0.220329
$669$	−3.17728e9	−0.410265
$670$	−5.16106e9	−0.662944
$671$	1.38026e10	1.76373
$672$	2.75188e9	0.349813
$673$	−6.94500e9	−0.878252	−0.439126	−	0.898425i	$-0.644712\pi$
$673$	−6.94500e9	−0.878252	−0.439126	+	0.898425i	$0.644712\pi$
$674$	1.67270e9	0.210430
$675$	9.86800e8	0.123500
$676$	−1.70554e9	−0.212349
$677$	2.90331e9	0.359611	0.179806	−	0.983702i	$-0.442453\pi$
$677$	2.90331e9	0.359611	0.179806	+	0.983702i	$0.442453\pi$
$678$	−3.24277e9	−0.399587
$679$	2.16472e8	0.0265373
$680$	1.14488e9	0.139630
$681$	−5.86226e9	−0.711296
$682$	1.72668e10	2.08433
$683$	−4.42751e9	−0.531726	−0.265863	−	0.964011i	$-0.585657\pi$
$683$	−4.42751e9	−0.531726	−0.265863	+	0.964011i	$0.585657\pi$
$684$	4.06075e8	0.0485187
$685$	5.26679e8	0.0626078
$686$	−9.91981e9	−1.17319
$687$	−1.86146e9	−0.219030
$688$	8.86898e9	1.03828
$689$	−4.61492e9	−0.537523
$690$	−2.07205e9	−0.240120
$691$	3.87979e9	0.447337	0.223669	−	0.974665i	$-0.428197\pi$
$691$	3.87979e9	0.447337	0.223669	+	0.974665i	$0.428197\pi$
$692$	2.20491e9	0.252941
$693$	−5.03567e9	−0.574765
$694$	−7.80989e8	−0.0886925
$695$	2.68736e9	0.303654
$696$	2.39006e9	0.268705
$697$	1.20451e9	0.134740
$698$	1.45219e10	1.61633
$699$	−6.53743e9	−0.723998
$700$	−1.82850e9	−0.201489
$701$	−1.56016e10	−1.71063	−0.855315	−	0.518108i	$-0.826636\pi$
$701$	−1.56016e10	−1.71063	−0.855315	+	0.518108i	$0.826636\pi$
$702$	−1.17995e9	−0.128731
$703$	−3.57214e9	−0.387779
$704$	−8.25710e9	−0.891915
$705$	2.40144e9	0.258113
$706$	5.19046e9	0.555123
$707$	1.70150e10	1.81077
$708$	−2.27914e8	−0.0241354
$709$	1.35775e10	1.43073	0.715364	−	0.698752i	$-0.246261\pi$
$709$	1.35775e10	1.43073	0.715364	+	0.698752i	$0.246261\pi$
$710$	−1.27767e10	−1.33972
$711$	−1.21335e9	−0.126603
$712$	−1.28096e10	−1.33001
$713$	−6.01693e9	−0.621672
$714$	1.88673e9	0.193984
$715$	−6.00382e9	−0.614266
$716$	−1.49044e9	−0.151746
$717$	−4.48646e9	−0.454555
$718$	−8.26963e9	−0.833778
$719$	1.51786e10	1.52293	0.761466	−	0.648205i	$-0.224480\pi$
$719$	1.51786e10	1.52293	0.761466	+	0.648205i	$0.224480\pi$
$720$	−2.43387e9	−0.243015
$721$	7.66109e9	0.761232
$722$	−9.23529e9	−0.913210
$723$	−7.87638e8	−0.0775073
$724$	−6.76625e8	−0.0662618
$725$	−3.92730e9	−0.382746
$726$	−1.44336e10	−1.39990
$727$	−1.49034e10	−1.43852	−0.719260	−	0.694741i	$-0.755519\pi$
$727$	−1.49034e10	−1.43852	−0.719260	+	0.694741i	$0.755519\pi$
$728$	−4.62268e9	−0.444053
$729$	3.87420e8	0.0370370
$730$	−6.32893e9	−0.602144
$731$	2.69138e9	0.254838
$732$	1.96768e9	0.185424
$733$	1.05230e10	0.986905	0.493453	−	0.869773i	$-0.335735\pi$
$733$	1.05230e10	0.986905	0.493453	+	0.869773i	$0.335735\pi$
$734$	−3.40279e9	−0.317614
$735$	1.63126e8	0.0151536
$736$	−4.05153e9	−0.374582
$737$	1.84665e10	1.69921
$738$	−1.88558e9	−0.172682
$739$	1.82290e10	1.66153	0.830764	−	0.556624i	$-0.187904\pi$
$739$	1.82290e10	1.66153	0.830764	+	0.556624i	$0.187904\pi$
$740$	−1.81241e9	−0.164417
$741$	−1.68690e9	−0.152309
$742$	−1.15516e10	−1.03808
$743$	−1.41694e9	−0.126733	−0.0633667	−	0.997990i	$-0.520184\pi$
$743$	−1.41694e9	−0.126733	−0.0633667	+	0.997990i	$0.520184\pi$
$744$	−5.20435e9	−0.463299
$745$	−5.53616e9	−0.490526
$746$	2.20707e10	1.94639
$747$	5.96632e9	0.523702
$748$	1.93750e9	0.169273
$749$	−1.08894e10	−0.946927
$750$	7.53409e9	0.652104
$751$	−1.51027e10	−1.30111	−0.650554	−	0.759460i	$-0.725463\pi$
$751$	−1.51027e10	−1.30111	−0.650554	+	0.759460i	$0.725463\pi$
$752$	1.06088e10	0.909716
$753$	−7.99454e9	−0.682356
$754$	4.69601e9	0.398960
$755$	−1.37743e10	−1.16481
$756$	−7.17875e8	−0.0604258
$757$	−8.01056e9	−0.671162	−0.335581	−	0.942011i	$-0.608933\pi$
$757$	−8.01056e9	−0.671162	−0.335581	+	0.942011i	$0.608933\pi$
$758$	1.43838e10	1.19959
$759$	7.41390e9	0.615461
$760$	−2.56224e9	−0.211725
$761$	9.09051e9	0.747725	0.373863	−	0.927484i	$-0.378033\pi$
$761$	9.09051e9	0.747725	0.373863	+	0.927484i	$0.378033\pi$
$762$	−1.20775e9	−0.0988864
$763$	−6.13049e9	−0.499642
$764$	−2.88348e9	−0.233932
$765$	−7.38583e8	−0.0596464
$766$	−8.73643e9	−0.702317
$767$	9.46795e8	0.0757656
$768$	−6.33897e9	−0.504957
$769$	−1.56881e10	−1.24402	−0.622009	−	0.783010i	$-0.713683\pi$
$769$	−1.56881e10	−1.24402	−0.622009	+	0.783010i	$0.713683\pi$
$770$	−1.50282e10	−1.18629
$771$	−7.22263e9	−0.567551
$772$	3.72732e9	0.291566
$773$	2.02623e10	1.57783	0.788916	−	0.614502i	$-0.210643\pi$
$773$	2.02623e10	1.57783	0.788916	+	0.614502i	$0.210643\pi$
$774$	−4.21317e9	−0.326599
$775$	8.55170e9	0.659928
$776$	−2.75666e8	−0.0211771
$777$	6.31497e9	0.482945
$778$	1.51527e10	1.15362
$779$	−2.69570e9	−0.204310
$780$	−8.55893e8	−0.0645785
$781$	4.57156e10	3.43389
$782$	−2.77780e9	−0.207719
$783$	−1.54187e9	−0.114784
$784$	7.20643e8	0.0534089
$785$	1.18227e10	0.872311
$786$	8.29260e8	0.0609133
$787$	2.40184e9	0.175644	0.0878219	−	0.996136i	$-0.472009\pi$
$787$	2.40184e9	0.175644	0.0878219	+	0.996136i	$0.472009\pi$
$788$	−1.24356e9	−0.0905370
$789$	−5.67939e9	−0.411654
$790$	−3.62107e9	−0.261301
$791$	8.19568e9	0.588800
$792$	6.41267e9	0.458671
$793$	−8.17407e9	−0.582080
$794$	9.01184e9	0.638913
$795$	4.52202e9	0.319189
$796$	7.34111e9	0.515901
$797$	2.37890e9	0.166446	0.0832228	−	0.996531i	$-0.473479\pi$
$797$	2.37890e9	0.166446	0.0832228	+	0.996531i	$0.473479\pi$
$798$	−4.22250e9	−0.294144
$799$	3.21936e9	0.223283
$800$	5.75833e9	0.397632
$801$	8.26368e9	0.568146
$802$	−2.93832e10	−2.01136
$803$	2.26452e10	1.54338
$804$	2.63255e9	0.178641
$805$	5.23685e9	0.353822
$806$	−1.02256e10	−0.687884
$807$	1.28751e10	0.862366
$808$	−2.16677e10	−1.44502
$809$	1.39600e10	0.926967	0.463483	−	0.886106i	$-0.346599\pi$
$809$	1.39600e10	0.926967	0.463483	+	0.886106i	$0.346599\pi$
$810$	1.15620e9	0.0764425
$811$	−1.88244e10	−1.23922	−0.619611	−	0.784909i	$-0.712710\pi$
$811$	−1.88244e10	−1.23922	−0.619611	+	0.784909i	$0.712710\pi$
$812$	2.85703e9	0.187270
$813$	6.74700e9	0.440346
$814$	2.66807e10	1.73385
$815$	−8.99825e9	−0.582246
$816$	−3.26285e9	−0.210223
$817$	−6.02331e9	−0.386418
$818$	1.13436e9	0.0724629
$819$	2.98218e9	0.189688
$820$	−1.36773e9	−0.0866268
$821$	7.81215e9	0.492685	0.246343	−	0.969183i	$-0.420771\pi$
$821$	7.81215e9	0.492685	0.246343	+	0.969183i	$0.420771\pi$
$822$	−1.10529e9	−0.0694107
$823$	−1.87867e10	−1.17476	−0.587381	−	0.809310i	$-0.699841\pi$
$823$	−1.87867e10	−1.17476	−0.587381	+	0.809310i	$0.699841\pi$
$824$	−9.75602e9	−0.607474
$825$	−1.05372e10	−0.653335
$826$	2.36993e9	0.146320
$827$	−1.42506e10	−0.876119	−0.438059	−	0.898946i	$-0.644334\pi$
$827$	−1.42506e10	−0.876119	−0.438059	+	0.898946i	$0.644334\pi$
$828$	1.05691e9	0.0647042
$829$	7.49011e8	0.0456612	0.0228306	−	0.999739i	$-0.492732\pi$
$829$	7.49011e8	0.0456612	0.0228306	+	0.999739i	$0.492732\pi$
$830$	1.78056e10	1.08089
$831$	5.46347e9	0.330267
$832$	4.88995e9	0.294356
$833$	2.18686e8	0.0131088
$834$	−5.63972e9	−0.336648
$835$	6.90944e9	0.410715
$836$	−4.33612e9	−0.256673
$837$	3.35742e9	0.197910
$838$	−1.07186e10	−0.629196
$839$	−2.65131e10	−1.54987	−0.774933	−	0.632043i	$-0.782217\pi$
$839$	−2.65131e10	−1.54987	−0.774933	+	0.632043i	$0.782217\pi$
$840$	4.52963e9	0.263685
$841$	−1.11135e10	−0.644265
$842$	−2.36950e10	−1.36793
$843$	7.49044e9	0.430637
$844$	−4.17146e9	−0.238830
$845$	−6.94250e9	−0.395838
$846$	−5.03968e9	−0.286159
$847$	3.64792e10	2.06278
$848$	1.99770e10	1.12498
$849$	9.29772e9	0.521434
$850$	3.94801e9	0.220502
$851$	−9.29739e9	−0.517140
$852$	6.51713e9	0.361009
$853$	−1.34893e10	−0.744163	−0.372081	−	0.928200i	$-0.621356\pi$
$853$	−1.34893e10	−0.744163	−0.372081	+	0.928200i	$0.621356\pi$
$854$	−2.04606e10	−1.12413
$855$	1.65295e9	0.0904435
$856$	1.38671e10	0.755660
$857$	2.26740e10	1.23054	0.615269	−	0.788317i	$-0.289047\pi$
$857$	2.26740e10	1.23054	0.615269	+	0.788317i	$0.289047\pi$
$858$	1.25997e10	0.681011
$859$	−1.97914e10	−1.06537	−0.532685	−	0.846314i	$-0.678817\pi$
$859$	−1.97914e10	−1.06537	−0.532685	+	0.846314i	$0.678817\pi$
$860$	−3.05607e9	−0.163840
$861$	4.76557e9	0.254451
$862$	4.21563e10	2.24175
$863$	−1.08103e10	−0.572534	−0.286267	−	0.958150i	$-0.592415\pi$
$863$	−1.08103e10	−0.572534	−0.286267	+	0.958150i	$0.592415\pi$
$864$	2.26074e9	0.119248
$865$	8.97519e9	0.471506
$866$	2.22790e10	1.16569
$867$	1.00890e10	0.525752
$868$	−6.22117e9	−0.322889
$869$	1.29563e10	0.669751
$870$	−4.60148e9	−0.236908
$871$	−1.09361e10	−0.560786
$872$	7.80687e9	0.398721
$873$	1.77837e8	0.00904633
$874$	6.21670e9	0.314970
$875$	−1.90415e10	−0.960887
$876$	3.22826e9	0.162257
$877$	−8.46440e7	−0.00423738	−0.00211869	−	0.999998i	$-0.500674\pi$
$877$	−8.46440e7	−0.00423738	−0.00211869	+	0.999998i	$0.500674\pi$
$878$	−2.21590e10	−1.10489
$879$	9.61408e9	0.477471
$880$	2.59892e10	1.28559
$881$	−3.78092e10	−1.86286	−0.931432	−	0.363915i	$-0.881440\pi$
$881$	−3.78092e10	−1.86286	−0.931432	+	0.363915i	$0.881440\pi$
$882$	−3.42338e8	−0.0168002
$883$	3.77215e10	1.84385	0.921927	−	0.387364i	$-0.126614\pi$
$883$	3.77215e10	1.84385	0.921927	+	0.387364i	$0.126614\pi$
$884$	−1.14741e9	−0.0558645
$885$	−9.27736e8	−0.0449907
$886$	1.16535e9	0.0562908
$887$	1.24757e10	0.600250	0.300125	−	0.953900i	$-0.402972\pi$
$887$	1.24757e10	0.600250	0.300125	+	0.953900i	$0.402972\pi$
$888$	−8.04180e9	−0.385397
$889$	3.05245e9	0.145711
$890$	2.46617e10	1.17262
$891$	−4.13693e9	−0.195932
$892$	4.83664e9	0.228174
$893$	−7.20493e9	−0.338571
$894$	1.16183e10	0.543826
$895$	−6.06689e9	−0.282869
$896$	2.52860e10	1.17436
$897$	−4.39060e9	−0.203119
$898$	−3.10760e10	−1.43205
$899$	−1.33620e10	−0.613356
$900$	−1.50216e9	−0.0686859
$901$	6.06222e9	0.276118
$902$	2.01345e10	0.913520
$903$	1.06482e10	0.481250
$904$	−1.04368e10	−0.469870
$905$	−2.75424e9	−0.123518
$906$	2.89068e10	1.29137
$907$	1.59259e8	0.00708725	0.00354363	−	0.999994i	$-0.498872\pi$
$907$	1.59259e8	0.00708725	0.00354363	+	0.999994i	$0.498872\pi$
$908$	8.92386e9	0.395596
$909$	1.39782e10	0.617275
$910$	8.89986e9	0.391506
$911$	2.28968e10	1.00337	0.501683	−	0.865051i	$-0.332714\pi$
$911$	2.28968e10	1.00337	0.501683	+	0.865051i	$0.332714\pi$
$912$	7.30224e9	0.318768
$913$	−6.37092e10	−2.77048
$914$	4.03447e10	1.74773
$915$	8.00953e9	0.345647
$916$	2.83361e9	0.121817
$917$	−2.09585e9	−0.0897568
$918$	1.55000e9	0.0661275
$919$	−1.19936e10	−0.509734	−0.254867	−	0.966976i	$-0.582032\pi$
$919$	−1.19936e10	−0.509734	−0.254867	+	0.966976i	$0.582032\pi$
$920$	−6.66887e9	−0.282355
$921$	5.29726e9	0.223430
$922$	−1.22498e10	−0.514722
$923$	−2.70733e10	−1.13327
$924$	7.66557e9	0.319663
$925$	1.32141e10	0.548963
$926$	−3.59268e10	−1.48689
$927$	6.29379e9	0.259497
$928$	−8.99737e9	−0.369571
$929$	−5.52148e9	−0.225944	−0.112972	−	0.993598i	$-0.536037\pi$
$929$	−5.52148e9	−0.225944	−0.112972	+	0.993598i	$0.536037\pi$
$930$	1.00197e10	0.408475
$931$	−4.89420e8	−0.0198773
$932$	9.95165e9	0.402661
$933$	−1.41296e9	−0.0569567
$934$	7.83387e6	0.000314603 0
$935$	7.88670e9	0.315540
$936$	−3.79766e9	−0.151374
$937$	4.28153e10	1.70024	0.850120	−	0.526589i	$-0.176529\pi$
$937$	4.28153e10	1.70024	0.850120	+	0.526589i	$0.176529\pi$
$938$	−2.73741e10	−1.08300
$939$	8.42441e9	0.332055
$940$	−3.65560e9	−0.143553
$941$	4.62683e10	1.81017	0.905087	−	0.425227i	$-0.139806\pi$
$941$	4.62683e10	1.81017	0.905087	+	0.425227i	$0.139806\pi$
$942$	−2.48112e10	−0.967095
$943$	−7.01624e9	−0.272467
$944$	−4.09847e9	−0.158569
$945$	−2.92215e9	−0.112639
$946$	4.49888e10	1.72777
$947$	1.84946e10	0.707651	0.353826	−	0.935311i	$-0.384881\pi$
$947$	1.84946e10	0.707651	0.353826	+	0.935311i	$0.384881\pi$
$948$	1.84703e9	0.0704117
$949$	−1.34107e10	−0.509356
$950$	−8.83563e9	−0.334353
$951$	−1.45319e10	−0.547888
$952$	6.07242e9	0.228104
$953$	−1.08055e10	−0.404409	−0.202205	−	0.979343i	$-0.564811\pi$
$953$	−1.08055e10	−0.404409	−0.202205	+	0.979343i	$0.564811\pi$
$954$	−9.48997e9	−0.353871
$955$	−1.17373e10	−0.436072
$956$	6.82953e9	0.252807
$957$	1.64643e10	0.607228
$958$	−3.19117e10	−1.17266
$959$	2.79349e9	0.102278
$960$	−4.79151e9	−0.174793
$961$	1.58314e9	0.0575423
$962$	−1.58006e10	−0.572218
$963$	−8.94590e9	−0.322799
$964$	1.19899e9	0.0431067
$965$	1.51723e10	0.543506
$966$	−1.09901e10	−0.392268
$967$	2.41210e10	0.857834	0.428917	−	0.903344i	$-0.358895\pi$
$967$	2.41210e10	0.857834	0.428917	+	0.903344i	$0.358895\pi$
$968$	−4.64544e10	−1.64613
$969$	2.21594e9	0.0782393
$970$	5.30728e8	0.0186712
$971$	−3.65187e10	−1.28011	−0.640057	−	0.768328i	$-0.721089\pi$
$971$	−3.65187e10	−1.28011	−0.640057	+	0.768328i	$0.721089\pi$
$972$	−5.89753e8	−0.0205986
$973$	1.42537e10	0.496058
$974$	−2.32251e10	−0.805379
$975$	6.24024e9	0.215618
$976$	3.53838e10	1.21823
$977$	1.82046e10	0.624524	0.312262	−	0.949996i	$-0.398913\pi$
$977$	1.82046e10	0.624524	0.312262	+	0.949996i	$0.398913\pi$
$978$	1.88838e10	0.645512
$979$	−8.82408e10	−3.00559
$980$	−2.48319e8	−0.00842789
$981$	−5.03635e9	−0.170324
$982$	−2.69135e10	−0.906942
$983$	−4.38690e10	−1.47306	−0.736531	−	0.676404i	$-0.763537\pi$
$983$	−4.38690e10	−1.47306	−0.736531	+	0.676404i	$0.763537\pi$
$984$	−6.06872e9	−0.203055
$985$	−5.06199e9	−0.168770
$986$	−6.16875e9	−0.204940
$987$	1.27372e10	0.421660
$988$	2.56790e9	0.0847089
$989$	−1.56772e10	−0.515325
$990$	−1.23461e10	−0.404394
$991$	−2.39803e10	−0.782702	−0.391351	−	0.920242i	$-0.627992\pi$
$991$	−2.39803e10	−0.782702	−0.391351	+	0.920242i	$0.627992\pi$
$992$	1.95918e10	0.637210
$993$	1.91389e10	0.620290
$994$	−6.77673e10	−2.18861
$995$	2.98824e10	0.961688
$996$	−9.08226e9	−0.291264
$997$	5.54581e10	1.77228	0.886139	−	0.463419i	$-0.153378\pi$
$997$	5.54581e10	1.77228	0.886139	+	0.463419i	$0.153378\pi$
$998$	−2.12540e10	−0.676837
$999$	5.18791e9	0.164632

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.a.1.12	✓	16
3.2	odd	2		531.8.a.b.1.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.a.1.12	✓	16	1.1	even	1	trivial
531.8.a.b.1.5		16	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:	\( N \)	\(=\)	\( 177 = 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	177.a (trivial)

\(f(q)\)	\(=\)	\(q+13.0039 q^{2} -27.0000 q^{3} +41.1009 q^{4} +167.303 q^{5} -351.105 q^{6} +887.373 q^{7} -1130.03 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+13.0039 q^{2} -27.0000 q^{3} +41.1009 q^{4} +167.303 q^{5} -351.105 q^{6} +887.373 q^{7} -1130.03 q^{8} +729.000 q^{9} +2175.59 q^{10} -7784.37 q^{11} -1109.72 q^{12} +4609.99 q^{13} +11539.3 q^{14} -4517.19 q^{15} -19955.6 q^{16} -6055.74 q^{17} +9479.83 q^{18} +13552.7 q^{19} +6876.32 q^{20} -23959.1 q^{21} -101227. q^{22} +35274.4 q^{23} +30510.7 q^{24} -50134.6 q^{25} +59947.7 q^{26} -19683.0 q^{27} +36471.8 q^{28} +78335.1 q^{29} -58741.0 q^{30} -170575. q^{31} -114857. q^{32} +210178. q^{33} -78748.2 q^{34} +148460. q^{35} +29962.6 q^{36} -263573. q^{37} +176238. q^{38} -124470. q^{39} -189057. q^{40} -198905. q^{41} -311561. q^{42} -444435. q^{43} -319945. q^{44} +121964. q^{45} +458704. q^{46} -531622. q^{47} +538802. q^{48} -36112.2 q^{49} -651945. q^{50} +163505. q^{51} +189475. q^{52} -1.00107e6 q^{53} -255955. q^{54} -1.30235e6 q^{55} -1.00275e6 q^{56} -365924. q^{57} +1.01866e6 q^{58} +205379. q^{59} -185661. q^{60} -1.77312e6 q^{61} -2.21813e6 q^{62} +646895. q^{63} +1.06073e6 q^{64} +771266. q^{65} +2.73313e6 q^{66} -2.37225e6 q^{67} -248897. q^{68} -952409. q^{69} +1.93056e6 q^{70} -5.87275e6 q^{71} -823789. q^{72} -2.90906e6 q^{73} -3.42748e6 q^{74} +1.35363e6 q^{75} +557030. q^{76} -6.90764e6 q^{77} -1.61859e6 q^{78} -1.66441e6 q^{79} -3.33864e6 q^{80} +531441. q^{81} -2.58653e6 q^{82} +8.18425e6 q^{83} -984739. q^{84} -1.01315e6 q^{85} -5.77938e6 q^{86} -2.11505e6 q^{87} +8.79653e6 q^{88} +1.13356e7 q^{89} +1.58601e6 q^{90} +4.09078e6 q^{91} +1.44981e6 q^{92} +4.60552e6 q^{93} -6.91315e6 q^{94} +2.26742e6 q^{95} +3.10115e6 q^{96} +243947. q^{97} -469599. q^{98} -5.67480e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9}+O(q^{10}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9	\( 16 q - 6 q^{2} - 432 q^{3} + 974 q^{4} - 68 q^{5} + 162 q^{6} - 2343 q^{7} + 819 q^{8} + 11664 q^{9} - 3479 q^{10} + 898 q^{11} - 26298 q^{12} - 8172 q^{13} - 13315 q^{14} + 1836 q^{15} + 3138 q^{16} - 44985 q^{17} - 4374 q^{18} - 40137 q^{19} + 130657 q^{20} + 63261 q^{21} + 109394 q^{22} - 2833 q^{23} - 22113 q^{24} + 285746 q^{25} - 129420 q^{26} - 314928 q^{27} + 112890 q^{28} + 144375 q^{29} + 93933 q^{30} - 141759 q^{31} - 36224 q^{32} - 24246 q^{33} - 341332 q^{34} - 78859 q^{35} + 710046 q^{36} - 297971 q^{37} + 329075 q^{38} + 220644 q^{39} - 203048 q^{40} + 659077 q^{41} + 359505 q^{42} - 1431608 q^{43} + 254916 q^{44} - 49572 q^{45} + 873113 q^{46} - 1574073 q^{47} - 84726 q^{48} + 1893545 q^{49} + 302533 q^{50} + 1214595 q^{51} - 4972548 q^{52} + 587736 q^{53} + 118098 q^{54} - 4624036 q^{55} - 5798506 q^{56} + 1083699 q^{57} - 6991380 q^{58} + 3286064 q^{59} - 3527739 q^{60} - 6117131 q^{61} - 11570258 q^{62} - 1708047 q^{63} - 19063011 q^{64} - 5335514 q^{65} - 2953638 q^{66} - 16518710 q^{67} - 17284669 q^{68} + 76491 q^{69} - 39189486 q^{70} - 10882582 q^{71} + 597051 q^{72} - 21097441 q^{73} - 16717030 q^{74} - 7715142 q^{75} - 40864952 q^{76} - 3404601 q^{77} + 3494340 q^{78} - 3784458 q^{79} - 27466195 q^{80} + 8503056 q^{81} - 24990117 q^{82} - 1951425 q^{83} - 3048030 q^{84} - 23238675 q^{85} - 35910572 q^{86} - 3898125 q^{87} - 27843055 q^{88} + 10499443 q^{89} - 2536191 q^{90} + 699217 q^{91} - 20062766 q^{92} + 3827493 q^{93} - 59358988 q^{94} - 29236333 q^{95} + 978048 q^{96} - 25158976 q^{97} + 2120460 q^{98} + 654642 q^{99}+O(q^{100}) \) 16 * q - 6 * q^2 - 432 * q^3 + 974 * q^4 - 68 * q^5 + 162 * q^6 - 2343 * q^7 + 819 * q^8 + 11664 * q^9 - 3479 * q^10 + 898 * q^11 - 26298 * q^12 - 8172 * q^13 - 13315 * q^14 + 1836 * q^15 + 3138 * q^16 - 44985 * q^17 - 4374 * q^18 - 40137 * q^19 + 130657 * q^20 + 63261 * q^21 + 109394 * q^22 - 2833 * q^23 - 22113 * q^24 + 285746 * q^25 - 129420 * q^26 - 314928 * q^27 + 112890 * q^28 + 144375 * q^29 + 93933 * q^30 - 141759 * q^31 - 36224 * q^32 - 24246 * q^33 - 341332 * q^34 - 78859 * q^35 + 710046 * q^36 - 297971 * q^37 + 329075 * q^38 + 220644 * q^39 - 203048 * q^40 + 659077 * q^41 + 359505 * q^42 - 1431608 * q^43 + 254916 * q^44 - 49572 * q^45 + 873113 * q^46 - 1574073 * q^47 - 84726 * q^48 + 1893545 * q^49 + 302533 * q^50 + 1214595 * q^51 - 4972548 * q^52 + 587736 * q^53 + 118098 * q^54 - 4624036 * q^55 - 5798506 * q^56 + 1083699 * q^57 - 6991380 * q^58 + 3286064 * q^59 - 3527739 * q^60 - 6117131 * q^61 - 11570258 * q^62 - 1708047 * q^63 - 19063011 * q^64 - 5335514 * q^65 - 2953638 * q^66 - 16518710 * q^67 - 17284669 * q^68 + 76491 * q^69 - 39189486 * q^70 - 10882582 * q^71 + 597051 * q^72 - 21097441 * q^73 - 16717030 * q^74 - 7715142 * q^75 - 40864952 * q^76 - 3404601 * q^77 + 3494340 * q^78 - 3784458 * q^79 - 27466195 * q^80 + 8503056 * q^81 - 24990117 * q^82 - 1951425 * q^83 - 3048030 * q^84 - 23238675 * q^85 - 35910572 * q^86 - 3898125 * q^87 - 27843055 * q^88 + 10499443 * q^89 - 2536191 * q^90 + 699217 * q^91 - 20062766 * q^92 + 3827493 * q^93 - 59358988 * q^94 - 29236333 * q^95 + 978048 * q^96 - 25158976 * q^97 + 2120460 * q^98 + 654642 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more