LMFDB - Embedded newform 177.8.a.d.1.18

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:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + \cdots + 51\!\cdots\!48 $ x^18 - 6x^17 - 1798x^16 + 11087x^15 + 1326765x^14 - 8403720x^13 - 518334228x^12 + 3375594921x^11 + 115310342333x^10 - 774932111214x^9 - 14600047830166x^8 + 102185027148481x^7 + 988557475638619x^6 - 7379206238519716x^5 - 30152342836849520x^4 + 260578770749067175x^3 + 182609347488069978x^2 - 3481290425710753600*x + 5164646074739714048
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$20.8794$ 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)

$f(q)$	$=$	$q+21.8794 q^{2} +27.0000 q^{3} +350.709 q^{4} +336.305 q^{5} +590.745 q^{6} -653.674 q^{7} +4872.75 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+21.8794 q^{2} +27.0000 q^{3} +350.709 q^{4} +336.305 q^{5} +590.745 q^{6} -653.674 q^{7} +4872.75 q^{8} +729.000 q^{9} +7358.16 q^{10} +2824.81 q^{11} +9469.15 q^{12} -5709.07 q^{13} -14302.0 q^{14} +9080.24 q^{15} +61722.2 q^{16} +27447.6 q^{17} +15950.1 q^{18} -52668.8 q^{19} +117945. q^{20} -17649.2 q^{21} +61805.2 q^{22} -112683. q^{23} +131564. q^{24} +34976.1 q^{25} -124911. q^{26} +19683.0 q^{27} -229250. q^{28} +129138. q^{29} +198670. q^{30} +264379. q^{31} +726735. q^{32} +76269.8 q^{33} +600538. q^{34} -219834. q^{35} +255667. q^{36} -162719. q^{37} -1.15236e6 q^{38} -154145. q^{39} +1.63873e6 q^{40} -570791. q^{41} -386154. q^{42} +468651. q^{43} +990686. q^{44} +245166. q^{45} -2.46544e6 q^{46} -451069. q^{47} +1.66650e6 q^{48} -396253. q^{49} +765257. q^{50} +741086. q^{51} -2.00222e6 q^{52} -1.09397e6 q^{53} +430653. q^{54} +949997. q^{55} -3.18519e6 q^{56} -1.42206e6 q^{57} +2.82547e6 q^{58} +205379. q^{59} +3.18452e6 q^{60} -1.17187e6 q^{61} +5.78445e6 q^{62} -476529. q^{63} +8.00009e6 q^{64} -1.91999e6 q^{65} +1.66874e6 q^{66} -1.24840e6 q^{67} +9.62614e6 q^{68} -3.04245e6 q^{69} -4.80984e6 q^{70} +2.90922e6 q^{71} +3.55224e6 q^{72} +1.95003e6 q^{73} -3.56019e6 q^{74} +944354. q^{75} -1.84714e7 q^{76} -1.84650e6 q^{77} -3.37260e6 q^{78} +529861. q^{79} +2.07575e7 q^{80} +531441. q^{81} -1.24886e7 q^{82} +7.53928e6 q^{83} -6.18974e6 q^{84} +9.23078e6 q^{85} +1.02538e7 q^{86} +3.48673e6 q^{87} +1.37646e7 q^{88} +6.07893e6 q^{89} +5.36410e6 q^{90} +3.73187e6 q^{91} -3.95190e7 q^{92} +7.13822e6 q^{93} -9.86913e6 q^{94} -1.77128e7 q^{95} +1.96218e7 q^{96} -2.74881e6 q^{97} -8.66979e6 q^{98} +2.05928e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9}+O(q^{10}) $ 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9} + 3609 q^{10} + 15070 q^{11} + 36666 q^{12} + 13662 q^{13} + 20861 q^{14} + 18306 q^{15} + 60482 q^{16} + 71919 q^{17} + 17496 q^{18} + 56231 q^{19} + 143053 q^{20} + 83187 q^{21} + 274198 q^{22} + 150029 q^{23} + 110889 q^{24} + 399672 q^{25} + 182846 q^{26} + 354294 q^{27} + 434150 q^{28} + 591285 q^{29} + 97443 q^{30} + 426733 q^{31} + 1205630 q^{32} + 406890 q^{33} + 403548 q^{34} + 912879 q^{35} + 989982 q^{36} + 7703 q^{37} - 417859 q^{38} + 368874 q^{39} + 618020 q^{40} + 770959 q^{41} + 563247 q^{42} + 793050 q^{43} + 2591274 q^{44} + 494262 q^{45} - 4068019 q^{46} + 1410373 q^{47} + 1633014 q^{48} + 1637427 q^{49} + 1021549 q^{50} + 1941813 q^{51} - 3749190 q^{52} + 1037934 q^{53} + 472392 q^{54} + 331974 q^{55} - 391748 q^{56} + 1518237 q^{57} + 653724 q^{58} + 3696822 q^{59} + 3862431 q^{60} - 1374623 q^{61} + 5251718 q^{62} + 2246049 q^{63} + 5077197 q^{64} + 3257170 q^{65} + 7403346 q^{66} - 2436904 q^{67} + 14119909 q^{68} + 4050783 q^{69} + 5185580 q^{70} + 14289172 q^{71} + 2994003 q^{72} + 5482515 q^{73} + 14934154 q^{74} + 10791144 q^{75} + 3822912 q^{76} + 23157109 q^{77} + 4936842 q^{78} + 19786414 q^{79} + 31978143 q^{80} + 9565938 q^{81} + 9749509 q^{82} + 30227337 q^{83} + 11722050 q^{84} + 9946981 q^{85} + 44295864 q^{86} + 15964695 q^{87} + 39970897 q^{88} + 31061677 q^{89} + 2630961 q^{90} + 26377785 q^{91} + 4719698 q^{92} + 11521791 q^{93} + 44488296 q^{94} + 15534599 q^{95} + 32552010 q^{96} + 12084118 q^{97} + 42274744 q^{98} + 10986030 q^{99}+O(q^{100}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9 + 3609 * q^10 + 15070 * q^11 + 36666 * q^12 + 13662 * q^13 + 20861 * q^14 + 18306 * q^15 + 60482 * q^16 + 71919 * q^17 + 17496 * q^18 + 56231 * q^19 + 143053 * q^20 + 83187 * q^21 + 274198 * q^22 + 150029 * q^23 + 110889 * q^24 + 399672 * q^25 + 182846 * q^26 + 354294 * q^27 + 434150 * q^28 + 591285 * q^29 + 97443 * q^30 + 426733 * q^31 + 1205630 * q^32 + 406890 * q^33 + 403548 * q^34 + 912879 * q^35 + 989982 * q^36 + 7703 * q^37 - 417859 * q^38 + 368874 * q^39 + 618020 * q^40 + 770959 * q^41 + 563247 * q^42 + 793050 * q^43 + 2591274 * q^44 + 494262 * q^45 - 4068019 * q^46 + 1410373 * q^47 + 1633014 * q^48 + 1637427 * q^49 + 1021549 * q^50 + 1941813 * q^51 - 3749190 * q^52 + 1037934 * q^53 + 472392 * q^54 + 331974 * q^55 - 391748 * q^56 + 1518237 * q^57 + 653724 * q^58 + 3696822 * q^59 + 3862431 * q^60 - 1374623 * q^61 + 5251718 * q^62 + 2246049 * q^63 + 5077197 * q^64 + 3257170 * q^65 + 7403346 * q^66 - 2436904 * q^67 + 14119909 * q^68 + 4050783 * q^69 + 5185580 * q^70 + 14289172 * q^71 + 2994003 * q^72 + 5482515 * q^73 + 14934154 * q^74 + 10791144 * q^75 + 3822912 * q^76 + 23157109 * q^77 + 4936842 * q^78 + 19786414 * q^79 + 31978143 * q^80 + 9565938 * q^81 + 9749509 * q^82 + 30227337 * q^83 + 11722050 * q^84 + 9946981 * q^85 + 44295864 * q^86 + 15964695 * q^87 + 39970897 * q^88 + 31061677 * q^89 + 2630961 * q^90 + 26377785 * q^91 + 4719698 * q^92 + 11521791 * q^93 + 44488296 * q^94 + 15534599 * q^95 + 32552010 * q^96 + 12084118 * q^97 + 42274744 * q^98 + 10986030 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	21.8794	1.93389	0.966943	−	0.254992i	$-0.0820728\pi$
$2$	21.8794	1.93389	0.966943	+	0.254992i	$0.0820728\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	350.709	2.73992
$5$	336.305	1.20320	0.601601	−	0.798797i	$-0.294530\pi$
$5$	336.305	1.20320	0.601601	+	0.798797i	$0.294530\pi$
$6$	590.745	1.11653
$7$	−653.674	−0.720308	−0.360154	−	0.932893i	$-0.617276\pi$
$7$	−653.674	−0.720308	−0.360154	+	0.932893i	$0.617276\pi$
$8$	4872.75	3.36480
$9$	729.000	0.333333
$10$	7358.16	2.32685
$11$	2824.81	0.639903	0.319952	−	0.947434i	$-0.396333\pi$
$11$	2824.81	0.639903	0.319952	+	0.947434i	$0.396333\pi$
$12$	9469.15	1.58189
$13$	−5709.07	−0.720715	−0.360358	−	0.932814i	$-0.617345\pi$
$13$	−5709.07	−0.720715	−0.360358	+	0.932814i	$0.617345\pi$
$14$	−14302.0	−1.39299
$15$	9080.24	0.694669
$16$	61722.2	3.76723
$17$	27447.6	1.35498	0.677491	−	0.735531i	$-0.263067\pi$
$17$	27447.6	1.35498	0.677491	+	0.735531i	$0.263067\pi$
$18$	15950.1	0.644629
$19$	−52668.8	−1.76163	−0.880817	−	0.473458i	$-0.843006\pi$
$19$	−52668.8	−1.76163	−0.880817	+	0.473458i	$0.843006\pi$
$20$	117945.	3.29667
$21$	−17649.2	−0.415870
$22$	61805.2	1.23750
$23$	−112683.	−1.93113	−0.965565	−	0.260161i	$-0.916224\pi$
$23$	−112683.	−1.93113	−0.965565	+	0.260161i	$0.916224\pi$
$24$	131564.	1.94267
$25$	34976.1	0.447694
$26$	−124911.	−1.39378
$27$	19683.0	0.192450
$28$	−229250.	−1.97358
$29$	129138.	0.983246	0.491623	−	0.870808i	$-0.336404\pi$
$29$	129138.	0.983246	0.491623	+	0.870808i	$0.336404\pi$
$30$	198670.	1.34341
$31$	264379.	1.59390	0.796949	−	0.604047i	$-0.206446\pi$
$31$	264379.	1.59390	0.796949	+	0.604047i	$0.206446\pi$
$32$	726735.	3.92059
$33$	76269.8	0.369448
$34$	600538.	2.62038
$35$	−219834.	−0.866676
$36$	255667.	0.913305
$37$	−162719.	−0.528118	−0.264059	−	0.964506i	$-0.585061\pi$
$37$	−162719.	−0.528118	−0.264059	+	0.964506i	$0.585061\pi$
$38$	−1.15236e6	−3.40680
$39$	−154145.	−0.416105
$40$	1.63873e6	4.04853
$41$	−570791.	−1.29340	−0.646701	−	0.762743i	$-0.723852\pi$
$41$	−570791.	−1.29340	−0.646701	+	0.762743i	$0.723852\pi$
$42$	−386154.	−0.804245
$43$	468651.	0.898898	0.449449	−	0.893306i	$-0.351620\pi$
$43$	468651.	0.898898	0.449449	+	0.893306i	$0.351620\pi$
$44$	990686.	1.75328
$45$	245166.	0.401067
$46$	−2.46544e6	−3.73459
$47$	−451069.	−0.633724	−0.316862	−	0.948472i	$-0.602629\pi$
$47$	−451069.	−0.633724	−0.316862	+	0.948472i	$0.602629\pi$
$48$	1.66650e6	2.17501
$49$	−396253.	−0.481156
$50$	765257.	0.865789
$51$	741086.	0.782299
$52$	−2.00222e6	−1.97470
$53$	−1.09397e6	−1.00935	−0.504674	−	0.863310i	$-0.668387\pi$
$53$	−1.09397e6	−1.00935	−0.504674	+	0.863310i	$0.668387\pi$
$54$	430653.	0.372177
$55$	949997.	0.769933
$56$	−3.18519e6	−2.42369
$57$	−1.42206e6	−1.01708
$58$	2.82547e6	1.90149
$59$	205379.	0.130189
$59$	205379.	0.130189
$60$	3.18452e6	1.90333
$61$	−1.17187e6	−0.661038	−0.330519	−	0.943799i	$-0.607224\pi$
$61$	−1.17187e6	−0.661038	−0.330519	+	0.943799i	$0.607224\pi$
$62$	5.78445e6	3.08242
$63$	−476529.	−0.240103
$64$	8.00009e6	3.81474
$65$	−1.91999e6	−0.867166
$66$	1.66874e6	0.714471
$67$	−1.24840e6	−0.507096	−0.253548	−	0.967323i	$-0.581598\pi$
$67$	−1.24840e6	−0.507096	−0.253548	+	0.967323i	$0.581598\pi$
$68$	9.62614e6	3.71254
$69$	−3.04245e6	−1.11494
$70$	−4.80984e6	−1.67605
$71$	2.90922e6	0.964655	0.482327	−	0.875991i	$-0.339792\pi$
$71$	2.90922e6	0.964655	0.482327	+	0.875991i	$0.339792\pi$
$72$	3.55224e6	1.12160
$73$	1.95003e6	0.586694	0.293347	−	0.956006i	$-0.405231\pi$
$73$	1.95003e6	0.586694	0.293347	+	0.956006i	$0.405231\pi$
$74$	−3.56019e6	−1.02132
$75$	944354.	0.258476
$76$	−1.84714e7	−4.82673
$77$	−1.84650e6	−0.460928
$78$	−3.37260e6	−0.804700
$79$	529861.	0.120911	0.0604557	−	0.998171i	$-0.480745\pi$
$79$	529861.	0.120911	0.0604557	+	0.998171i	$0.480745\pi$
$80$	2.07575e7	4.53273
$81$	531441.	0.111111
$82$	−1.24886e7	−2.50129
$83$	7.53928e6	1.44729	0.723646	−	0.690171i	$-0.242465\pi$
$83$	7.53928e6	1.44729	0.723646	+	0.690171i	$0.242465\pi$
$84$	−6.18974e6	−1.13945
$85$	9.23078e6	1.63032
$86$	1.02538e7	1.73837
$87$	3.48673e6	0.567677
$88$	1.37646e7	2.15315
$89$	6.07893e6	0.914034	0.457017	−	0.889458i	$-0.348918\pi$
$89$	6.07893e6	0.914034	0.457017	+	0.889458i	$0.348918\pi$
$90$	5.36410e6	0.775618
$91$	3.73187e6	0.519137
$92$	−3.95190e7	−5.29114
$93$	7.13822e6	0.920237
$94$	−9.86913e6	−1.22555
$95$	−1.77128e7	−2.11960
$96$	1.96218e7	2.26355
$97$	−2.74881e6	−0.305804	−0.152902	−	0.988241i	$-0.548862\pi$
$97$	−2.74881e6	−0.305804	−0.152902	+	0.988241i	$0.548862\pi$
$98$	−8.66979e6	−0.930502
$99$	2.05928e6	0.213301
$100$	1.22664e7	1.22664
$101$	−1.90082e7	−1.83576	−0.917880	−	0.396858i	$-0.870100\pi$
$101$	−1.90082e7	−1.83576	−0.917880	+	0.396858i	$0.870100\pi$
$102$	1.62145e7	1.51288
$103$	1.79174e7	1.61564	0.807818	−	0.589432i	$-0.200648\pi$
$103$	1.79174e7	1.61564	0.807818	+	0.589432i	$0.200648\pi$
$104$	−2.78189e7	−2.42506
$105$	−5.93552e6	−0.500375
$106$	−2.39355e7	−1.95196
$107$	85176.8	0.00672168	0.00336084	−	0.999994i	$-0.498930\pi$
$107$	85176.8	0.00672168	0.00336084	+	0.999994i	$0.498930\pi$
$108$	6.90301e6	0.527297
$109$	1.58698e7	1.17376	0.586881	−	0.809673i	$-0.300356\pi$
$109$	1.58698e7	1.17376	0.586881	+	0.809673i	$0.300356\pi$
$110$	2.07854e7	1.48896
$111$	−4.39340e6	−0.304909
$112$	−4.03462e7	−2.71356
$113$	−2.00702e7	−1.30851	−0.654257	−	0.756273i	$-0.727018\pi$
$113$	−2.00702e7	−1.30851	−0.654257	+	0.756273i	$0.727018\pi$
$114$	−3.11138e7	−1.96692
$115$	−3.78959e7	−2.32354
$116$	4.52900e7	2.69401
$117$	−4.16191e6	−0.240238
$118$	4.49357e6	0.251771
$119$	−1.79418e7	−0.976005
$120$	4.42457e7	2.33742
$121$	−1.15076e7	−0.590524
$122$	−2.56399e7	−1.27837
$123$	−1.54114e7	−0.746747
$124$	9.27200e7	4.36715
$125$	−1.45112e7	−0.664536
$126$	−1.04262e7	−0.464331
$127$	1.19818e7	0.519048	0.259524	−	0.965737i	$-0.416434\pi$
$127$	1.19818e7	0.519048	0.259524	+	0.965737i	$0.416434\pi$
$128$	8.20154e7	3.45669
$129$	1.26536e7	0.518979
$130$	−4.20083e7	−1.67700
$131$	8.95447e6	0.348009	0.174004	−	0.984745i	$-0.444329\pi$
$131$	8.95447e6	0.348009	0.174004	+	0.984745i	$0.444329\pi$
$132$	2.67485e7	1.01226
$133$	3.44282e7	1.26892
$134$	−2.73142e7	−0.980666
$135$	6.61949e6	0.231556
$136$	1.33745e8	4.55924
$137$	−1.70175e7	−0.565424	−0.282712	−	0.959205i	$-0.591234\pi$
$137$	−1.70175e7	−0.565424	−0.282712	+	0.959205i	$0.591234\pi$
$138$	−6.65670e7	−2.15616
$139$	−1.49569e7	−0.472379	−0.236189	−	0.971707i	$-0.575899\pi$
$139$	−1.49569e7	−0.472379	−0.236189	+	0.971707i	$0.575899\pi$
$140$	−7.70978e7	−2.37462
$141$	−1.21789e7	−0.365881
$142$	6.36520e7	1.86553
$143$	−1.61270e7	−0.461188
$144$	4.49955e7	1.25574
$145$	4.34299e7	1.18304
$146$	4.26655e7	1.13460
$147$	−1.06988e7	−0.277796
$148$	−5.70669e7	−1.44700
$149$	−5.19387e7	−1.28629	−0.643145	−	0.765744i	$-0.722371\pi$
$149$	−5.19387e7	−1.28629	−0.643145	+	0.765744i	$0.722371\pi$
$150$	2.06619e7	0.499863
$151$	−4.79328e6	−0.113296	−0.0566478	−	0.998394i	$-0.518041\pi$
$151$	−4.79328e6	−0.113296	−0.0566478	+	0.998394i	$0.518041\pi$
$152$	−2.56642e8	−5.92754
$153$	2.00093e7	0.451661
$154$	−4.04004e7	−0.891381
$155$	8.89119e7	1.91778
$156$	−5.40601e7	−1.14009
$157$	5.35304e7	1.10396	0.551978	−	0.833859i	$-0.313873\pi$
$157$	5.35304e7	1.10396	0.551978	+	0.833859i	$0.313873\pi$
$158$	1.15931e7	0.233829
$159$	−2.95372e7	−0.582747
$160$	2.44404e8	4.71725
$161$	7.36581e7	1.39101
$162$	1.16276e7	0.214876
$163$	−8.41640e7	−1.52219	−0.761096	−	0.648639i	$-0.775339\pi$
$163$	−8.41640e7	−1.52219	−0.761096	+	0.648639i	$0.775339\pi$
$164$	−2.00182e8	−3.54382
$165$	2.56499e7	0.444521
$166$	1.64955e8	2.79890
$167$	6.28714e6	0.104459	0.0522295	−	0.998635i	$-0.483367\pi$
$167$	6.28714e6	0.104459	0.0522295	+	0.998635i	$0.483367\pi$
$168$	−8.60002e7	−1.39932
$169$	−3.01550e7	−0.480569
$170$	2.01964e8	3.15285
$171$	−3.83955e7	−0.587211
$172$	1.64360e8	2.46290
$173$	−4.64860e7	−0.682591	−0.341296	−	0.939956i	$-0.610866\pi$
$173$	−4.64860e7	−0.682591	−0.341296	+	0.939956i	$0.610866\pi$
$174$	7.62877e7	1.09782
$175$	−2.28630e7	−0.322477
$176$	1.74353e8	2.41066
$177$	5.54523e6	0.0751646
$178$	1.33004e8	1.76764
$179$	1.12901e8	1.47134	0.735671	−	0.677339i	$-0.236867\pi$
$179$	1.12901e8	1.47134	0.735671	+	0.677339i	$0.236867\pi$
$180$	8.59821e7	1.09889
$181$	−1.04425e8	−1.30897	−0.654487	−	0.756073i	$-0.727115\pi$
$181$	−1.04425e8	−1.30897	−0.654487	+	0.756073i	$0.727115\pi$
$182$	8.16513e7	1.00395
$183$	−3.16406e7	−0.381651
$184$	−5.49077e8	−6.49787
$185$	−5.47231e7	−0.635433
$186$	1.56180e8	1.77963
$187$	7.75343e7	0.867058
$188$	−1.58194e8	−1.73635
$189$	−1.28663e7	−0.138623
$190$	−3.87545e8	−4.09906
$191$	−1.38829e8	−1.44167	−0.720833	−	0.693108i	$-0.756241\pi$
$191$	−1.38829e8	−1.44167	−0.720833	+	0.693108i	$0.756241\pi$
$192$	2.16002e8	2.20244
$193$	3.06880e7	0.307268	0.153634	−	0.988128i	$-0.450902\pi$
$193$	3.06880e7	0.307268	0.153634	+	0.988128i	$0.450902\pi$
$194$	−6.01424e7	−0.591391
$195$	−5.18397e7	−0.500658
$196$	−1.38970e8	−1.31833
$197$	−2.78823e7	−0.259834	−0.129917	−	0.991525i	$-0.541471\pi$
$197$	−2.78823e7	−0.259834	−0.129917	+	0.991525i	$0.541471\pi$
$198$	4.50560e7	0.412500
$199$	1.31062e8	1.17894	0.589469	−	0.807791i	$-0.299337\pi$
$199$	1.31062e8	1.17894	0.589469	+	0.807791i	$0.299337\pi$
$200$	1.70430e8	1.50640
$201$	−3.37067e7	−0.292772
$202$	−4.15888e8	−3.55015
$203$	−8.44144e7	−0.708240
$204$	2.59906e8	2.14343
$205$	−1.91960e8	−1.55622
$206$	3.92021e8	3.12446
$207$	−8.21460e7	−0.643710
$208$	−3.52377e8	−2.71510
$209$	−1.48779e8	−1.12727
$210$	−1.29866e8	−0.967669
$211$	7.00592e7	0.513424	0.256712	−	0.966488i	$-0.417361\pi$
$211$	7.00592e7	0.513424	0.256712	+	0.966488i	$0.417361\pi$
$212$	−3.83666e8	−2.76553
$213$	7.85489e7	0.556944
$214$	1.86362e6	0.0129990
$215$	1.57610e8	1.08155
$216$	9.59104e7	0.647556
$217$	−1.72817e8	−1.14810
$218$	3.47223e8	2.26992
$219$	5.26508e7	0.338728
$220$	3.33173e8	2.10955
$221$	−1.56700e8	−0.976557
$222$	−9.61252e7	−0.589660
$223$	1.42923e8	0.863049	0.431525	−	0.902101i	$-0.357976\pi$
$223$	1.42923e8	0.863049	0.431525	+	0.902101i	$0.357976\pi$
$224$	−4.75048e8	−2.82403
$225$	2.54976e7	0.149231
$226$	−4.39126e8	−2.53052
$227$	−1.16361e8	−0.660262	−0.330131	−	0.943935i	$-0.607093\pi$
$227$	−1.16361e8	−0.660262	−0.330131	+	0.943935i	$0.607093\pi$
$228$	−4.98728e8	−2.78671
$229$	2.48550e8	1.36770	0.683849	−	0.729624i	$-0.260305\pi$
$229$	2.48550e8	1.36770	0.683849	+	0.729624i	$0.260305\pi$
$230$	−8.29141e8	−4.49346
$231$	−4.98556e7	−0.266117
$232$	6.29259e8	3.30842
$233$	−1.28332e8	−0.664645	−0.332323	−	0.943166i	$-0.607832\pi$
$233$	−1.28332e8	−0.664645	−0.332323	+	0.943166i	$0.607832\pi$
$234$	−9.10603e7	−0.464594
$235$	−1.51697e8	−0.762498
$236$	7.20283e7	0.356707
$237$	1.43063e7	0.0698082
$238$	−3.92556e8	−1.88748
$239$	1.52392e8	0.722056	0.361028	−	0.932555i	$-0.382426\pi$
$239$	1.52392e8	0.722056	0.361028	+	0.932555i	$0.382426\pi$
$240$	5.60452e8	2.61697
$241$	−2.46545e8	−1.13458	−0.567292	−	0.823517i	$-0.692009\pi$
$241$	−2.46545e8	−1.13458	−0.567292	+	0.823517i	$0.692009\pi$
$242$	−2.51780e8	−1.14201
$243$	1.43489e7	0.0641500
$244$	−4.10987e8	−1.81119
$245$	−1.33262e8	−0.578928
$246$	−3.37192e8	−1.44412
$247$	3.00690e8	1.26964
$248$	1.28825e9	5.36315
$249$	2.03560e8	0.835595
$250$	−3.17497e8	−1.28514
$251$	−1.53947e7	−0.0614486	−0.0307243	−	0.999528i	$-0.509781\pi$
$251$	−1.53947e7	−0.0614486	−0.0307243	+	0.999528i	$0.509781\pi$
$252$	−1.67123e8	−0.657861
$253$	−3.18308e8	−1.23574
$254$	2.62154e8	1.00378
$255$	2.49231e8	0.941264
$256$	7.70438e8	2.87010
$257$	1.45269e7	0.0533836	0.0266918	−	0.999644i	$-0.491503\pi$
$257$	1.45269e7	0.0533836	0.0266918	+	0.999644i	$0.491503\pi$
$258$	2.76853e8	1.00365
$259$	1.06365e8	0.380408
$260$	−6.73358e8	−2.37596
$261$	9.41418e7	0.327749
$262$	1.95919e8	0.673010
$263$	4.08752e8	1.38553	0.692763	−	0.721166i	$-0.256393\pi$
$263$	4.08752e8	1.38553	0.692763	+	0.721166i	$0.256393\pi$
$264$	3.71644e8	1.24312
$265$	−3.67908e8	−1.21445
$266$	7.53270e8	2.45394
$267$	1.64131e8	0.527718
$268$	−4.37824e8	−1.38940
$269$	2.03154e6	0.00636345	0.00318173	−	0.999995i	$-0.498987\pi$
$269$	2.03154e6	0.00636345	0.00318173	+	0.999995i	$0.498987\pi$
$270$	1.44831e8	0.447803
$271$	1.02015e8	0.311366	0.155683	−	0.987807i	$-0.450242\pi$
$271$	1.02015e8	0.311366	0.155683	+	0.987807i	$0.450242\pi$
$272$	1.69413e9	5.10452
$273$	1.00761e8	0.299724
$274$	−3.72333e8	−1.09347
$275$	9.88007e7	0.286481
$276$	−1.06701e9	−3.05484
$277$	−1.80373e8	−0.509907	−0.254954	−	0.966953i	$-0.582060\pi$
$277$	−1.80373e8	−0.509907	−0.254954	+	0.966953i	$0.582060\pi$
$278$	−3.27249e8	−0.913526
$279$	1.92732e8	0.531299
$280$	−1.07120e9	−2.91619
$281$	3.39131e8	0.911791	0.455896	−	0.890033i	$-0.349319\pi$
$281$	3.39131e8	0.911791	0.455896	+	0.890033i	$0.349319\pi$
$282$	−2.66467e8	−0.707572
$283$	−1.22871e8	−0.322252	−0.161126	−	0.986934i	$-0.551513\pi$
$283$	−1.22871e8	−0.322252	−0.161126	+	0.986934i	$0.551513\pi$
$284$	1.02029e9	2.64307
$285$	−4.78245e8	−1.22375
$286$	−3.52850e8	−0.891886
$287$	3.73112e8	0.931649
$288$	5.29789e8	1.30686
$289$	3.43034e8	0.835977
$290$	9.50220e8	2.28787
$291$	−7.42179e7	−0.176556
$292$	6.83894e8	1.60749
$293$	−2.73920e8	−0.636190	−0.318095	−	0.948059i	$-0.603043\pi$
$293$	−2.73920e8	−0.636190	−0.318095	+	0.948059i	$0.603043\pi$
$294$	−2.34084e8	−0.537225
$295$	6.90700e7	0.156643
$296$	−7.92888e8	−1.77701
$297$	5.56007e7	0.123149
$298$	−1.13639e9	−2.48754
$299$	6.43316e8	1.39180
$300$	3.31194e8	0.708203
$301$	−3.06345e8	−0.647483
$302$	−1.04874e8	−0.219101
$303$	−5.13221e8	−1.05988
$304$	−3.25083e9	−6.63647
$305$	−3.94107e8	−0.795362
$306$	4.37792e8	0.873461
$307$	−2.51384e8	−0.495853	−0.247927	−	0.968779i	$-0.579749\pi$
$307$	−2.51384e8	−0.495853	−0.247927	+	0.968779i	$0.579749\pi$
$308$	−6.47586e8	−1.26290
$309$	4.83769e8	0.932788
$310$	1.94534e9	3.70877
$311$	3.46715e8	0.653599	0.326799	−	0.945094i	$-0.394030\pi$
$311$	3.46715e8	0.653599	0.326799	+	0.945094i	$0.394030\pi$
$312$	−7.51110e8	−1.40011
$313$	8.59091e8	1.58356	0.791779	−	0.610807i	$-0.209155\pi$
$313$	8.59091e8	1.58356	0.791779	+	0.610807i	$0.209155\pi$
$314$	1.17122e9	2.13493
$315$	−1.60259e8	−0.288892
$316$	1.85827e8	0.331287
$317$	6.36316e8	1.12193	0.560964	−	0.827840i	$-0.310431\pi$
$317$	6.36316e8	1.12193	0.560964	+	0.827840i	$0.310431\pi$
$318$	−6.46258e8	−1.12697
$319$	3.64791e8	0.629182
$320$	2.69047e9	4.58990
$321$	2.29977e6	0.00388076
$322$	1.61160e9	2.69005
$323$	−1.44563e9	−2.38698
$324$	1.86381e8	0.304435
$325$	−1.99681e8	−0.322660
$326$	−1.84146e9	−2.94375
$327$	4.28486e8	0.677672
$328$	−2.78132e9	−4.35204
$329$	2.94852e8	0.456477
$330$	5.61205e8	0.859653
$331$	4.08883e8	0.619728	0.309864	−	0.950781i	$-0.399716\pi$
$331$	4.08883e8	0.619728	0.309864	+	0.950781i	$0.399716\pi$
$332$	2.64409e9	3.96546
$333$	−1.18622e8	−0.176039
$334$	1.37559e8	0.202012
$335$	−4.19842e8	−0.610139
$336$	−1.08935e9	−1.56668
$337$	6.00455e8	0.854626	0.427313	−	0.904104i	$-0.359460\pi$
$337$	6.00455e8	0.854626	0.427313	+	0.904104i	$0.359460\pi$
$338$	−6.59774e8	−0.929366
$339$	−5.41897e8	−0.755471
$340$	3.23732e9	4.46693
$341$	7.46818e8	1.01994
$342$	−8.40072e8	−1.13560
$343$	7.97349e8	1.06689
$344$	2.28362e9	3.02461
$345$	−1.02319e9	−1.34150
$346$	−1.01709e9	−1.32005
$347$	1.81629e8	0.233363	0.116681	−	0.993169i	$-0.462774\pi$
$347$	1.81629e8	0.233363	0.116681	+	0.993169i	$0.462774\pi$
$348$	1.22283e9	1.55539
$349$	−7.06742e8	−0.889963	−0.444981	−	0.895540i	$-0.646790\pi$
$349$	−7.06742e8	−0.889963	−0.444981	+	0.895540i	$0.646790\pi$
$350$	−5.00229e8	−0.623635
$351$	−1.12372e8	−0.138702
$352$	2.05288e9	2.50880
$353$	2.31567e8	0.280199	0.140099	−	0.990137i	$-0.455258\pi$
$353$	2.31567e8	0.280199	0.140099	+	0.990137i	$0.455258\pi$
$354$	1.21327e8	0.145360
$355$	9.78384e8	1.16067
$356$	2.13194e9	2.50438
$357$	−4.84429e8	−0.563497
$358$	2.47022e9	2.84541
$359$	1.13982e9	1.30019	0.650093	−	0.759854i	$-0.274730\pi$
$359$	1.13982e9	1.30019	0.650093	+	0.759854i	$0.274730\pi$
$360$	1.19463e9	1.34951
$361$	1.88013e9	2.10335
$362$	−2.28477e9	−2.53141
$363$	−3.10706e8	−0.340939
$364$	1.30880e9	1.42239
$365$	6.55805e8	0.705911
$366$	−6.92278e8	−0.738069
$367$	6.44750e8	0.680864	0.340432	−	0.940269i	$-0.389427\pi$
$367$	6.44750e8	0.680864	0.340432	+	0.940269i	$0.389427\pi$
$368$	−6.95506e9	−7.27500
$369$	−4.16107e8	−0.431134
$370$	−1.19731e9	−1.22885
$371$	7.15101e8	0.727041
$372$	2.50344e9	2.52137
$373$	1.69263e8	0.168881	0.0844404	−	0.996429i	$-0.473090\pi$
$373$	1.69263e8	0.168881	0.0844404	+	0.996429i	$0.473090\pi$
$374$	1.69640e9	1.67679
$375$	−3.91802e8	−0.383670
$376$	−2.19795e9	−2.13236
$377$	−7.37260e8	−0.708640
$378$	−2.81507e8	−0.268082
$379$	−1.01373e9	−0.956503	−0.478252	−	0.878223i	$-0.658729\pi$
$379$	−1.01373e9	−0.956503	−0.478252	+	0.878223i	$0.658729\pi$
$380$	−6.21203e9	−5.80753
$381$	3.23508e8	0.299673
$382$	−3.03751e9	−2.78802
$383$	5.54096e8	0.503953	0.251976	−	0.967733i	$-0.418920\pi$
$383$	5.54096e8	0.503953	0.251976	+	0.967733i	$0.418920\pi$
$384$	2.21442e9	1.99572
$385$	−6.20988e8	−0.554589
$386$	6.71435e8	0.594221
$387$	3.41647e8	0.299633
$388$	−9.64033e8	−0.837879
$389$	1.17131e9	1.00890	0.504450	−	0.863441i	$-0.331695\pi$
$389$	1.17131e9	1.00890	0.504450	+	0.863441i	$0.331695\pi$
$390$	−1.13422e9	−0.968217
$391$	−3.09289e9	−2.61665
$392$	−1.93084e9	−1.61900
$393$	2.41771e8	0.200923
$394$	−6.10048e8	−0.502490
$395$	1.78195e8	0.145481
$396$	7.22210e8	0.584427
$397$	1.86752e9	1.49796	0.748978	−	0.662594i	$-0.230545\pi$
$397$	1.86752e9	1.49796	0.748978	+	0.662594i	$0.230545\pi$
$398$	2.86756e9	2.27993
$399$	9.29562e8	0.732610
$400$	2.15880e9	1.68656
$401$	2.23006e9	1.72708	0.863539	−	0.504282i	$-0.168243\pi$
$401$	2.23006e9	1.72708	0.863539	+	0.504282i	$0.168243\pi$
$402$	−7.37483e8	−0.566188
$403$	−1.50936e9	−1.14875
$404$	−6.66635e9	−5.02983
$405$	1.78726e8	0.133689
$406$	−1.84694e9	−1.36965
$407$	−4.59649e8	−0.337945
$408$	3.61113e9	2.63228
$409$	−1.03440e9	−0.747575	−0.373788	−	0.927514i	$-0.621941\pi$
$409$	−1.03440e9	−0.747575	−0.373788	+	0.927514i	$0.621941\pi$
$410$	−4.19997e9	−3.00956
$411$	−4.59473e8	−0.326448
$412$	6.28378e9	4.42671
$413$	−1.34251e8	−0.0937761
$414$	−1.79731e9	−1.24486
$415$	2.53550e9	1.74138
$416$	−4.14898e9	−2.82563
$417$	−4.03837e8	−0.272728
$418$	−3.25520e9	−2.18002
$419$	−1.07822e9	−0.716075	−0.358038	−	0.933707i	$-0.616554\pi$
$419$	−1.07822e9	−0.716075	−0.358038	+	0.933707i	$0.616554\pi$
$420$	−2.08164e9	−1.37099
$421$	−1.88974e9	−1.23428	−0.617140	−	0.786853i	$-0.711709\pi$
$421$	−1.88974e9	−1.23428	−0.617140	+	0.786853i	$0.711709\pi$
$422$	1.53285e9	0.992904
$423$	−3.28829e8	−0.211241
$424$	−5.33065e9	−3.39625
$425$	9.60010e8	0.606617
$426$	1.71860e9	1.07707
$427$	7.66024e8	0.476151
$428$	2.98723e7	0.0184168
$429$	−4.35430e8	−0.266267
$430$	3.44841e9	2.09160
$431$	−2.24349e9	−1.34975	−0.674876	−	0.737931i	$-0.735803\pi$
$431$	−2.24349e9	−1.34975	−0.674876	+	0.737931i	$0.735803\pi$
$432$	1.21488e9	0.725003
$433$	1.57772e9	0.933948	0.466974	−	0.884271i	$-0.345344\pi$
$433$	1.57772e9	0.933948	0.466974	+	0.884271i	$0.345344\pi$
$434$	−3.78115e9	−2.22029
$435$	1.17261e9	0.683030
$436$	5.56570e9	3.21601
$437$	5.93488e9	3.40194
$438$	1.15197e9	0.655061
$439$	2.36005e9	1.33136	0.665681	−	0.746236i	$-0.268141\pi$
$439$	2.36005e9	1.33136	0.665681	+	0.746236i	$0.268141\pi$
$440$	4.62910e9	2.59067
$441$	−2.88868e8	−0.160385
$442$	−3.42852e9	−1.88855
$443$	−4.93538e8	−0.269717	−0.134858	−	0.990865i	$-0.543058\pi$
$443$	−4.93538e8	−0.269717	−0.134858	+	0.990865i	$0.543058\pi$
$444$	−1.54081e9	−0.835426
$445$	2.04438e9	1.09977
$446$	3.12708e9	1.66904
$447$	−1.40234e9	−0.742640
$448$	−5.22945e9	−2.74779
$449$	−2.07688e9	−1.08280	−0.541401	−	0.840764i	$-0.682106\pi$
$449$	−2.07688e9	−1.08280	−0.541401	+	0.840764i	$0.682106\pi$
$450$	5.57872e8	0.288596
$451$	−1.61238e9	−0.827653
$452$	−7.03882e9	−3.58522
$453$	−1.29418e8	−0.0654112
$454$	−2.54591e9	−1.27687
$455$	1.25505e9	0.624627
$456$	−6.92933e9	−3.42227
$457$	−1.48289e9	−0.726777	−0.363389	−	0.931638i	$-0.618380\pi$
$457$	−1.48289e9	−0.726777	−0.363389	+	0.931638i	$0.618380\pi$
$458$	5.43814e9	2.64497
$459$	5.40252e8	0.260766
$460$	−1.32905e10	−6.36630
$461$	2.57673e9	1.22494	0.612471	−	0.790493i	$-0.290175\pi$
$461$	2.57673e9	1.22494	0.612471	+	0.790493i	$0.290175\pi$
$462$	−1.09081e9	−0.514639
$463$	−4.54013e8	−0.212586	−0.106293	−	0.994335i	$-0.533898\pi$
$463$	−4.54013e8	−0.212586	−0.106293	+	0.994335i	$0.533898\pi$
$464$	7.97070e9	3.70411
$465$	2.40062e9	1.10723
$466$	−2.80784e9	−1.28535
$467$	3.63008e9	1.64933	0.824663	−	0.565624i	$-0.191365\pi$
$467$	3.63008e9	1.64933	0.824663	+	0.565624i	$0.191365\pi$
$468$	−1.45962e9	−0.658233
$469$	8.16044e8	0.365265
$470$	−3.31904e9	−1.47458
$471$	1.44532e9	0.637370
$472$	1.00076e9	0.438060
$473$	1.32385e9	0.575208
$474$	3.13013e8	0.135001
$475$	−1.84215e9	−0.788672
$476$	−6.29236e9	−2.67417
$477$	−7.97506e8	−0.336449
$478$	3.33426e9	1.39637
$479$	−9.05516e7	−0.0376463	−0.0188231	−	0.999823i	$-0.505992\pi$
$479$	−9.05516e7	−0.0376463	−0.0188231	+	0.999823i	$0.505992\pi$
$480$	6.59892e9	2.72351
$481$	9.28973e8	0.380623
$482$	−5.39427e9	−2.19416
$483$	1.98877e9	0.803099
$484$	−4.03584e9	−1.61799
$485$	−9.24439e8	−0.367944
$486$	3.13946e8	0.124059
$487$	2.49820e8	0.0980112	0.0490056	−	0.998799i	$-0.484395\pi$
$487$	2.49820e8	0.0980112	0.0490056	+	0.998799i	$0.484395\pi$
$488$	−5.71025e9	−2.22426
$489$	−2.27243e9	−0.878839
$490$	−2.91569e9	−1.11958
$491$	4.25867e8	0.162364	0.0811818	−	0.996699i	$-0.474131\pi$
$491$	4.25867e8	0.162364	0.0811818	+	0.996699i	$0.474131\pi$
$492$	−5.40491e9	−2.04602
$493$	3.54454e9	1.33228
$494$	6.57892e9	2.45533
$495$	6.92548e8	0.256644
$496$	1.63180e10	6.00457
$497$	−1.90168e9	−0.694849
$498$	4.45379e9	1.61595
$499$	−4.17328e9	−1.50358	−0.751789	−	0.659404i	$-0.770809\pi$
$499$	−4.17328e9	−1.50358	−0.751789	+	0.659404i	$0.770809\pi$
$500$	−5.08921e9	−1.82077
$501$	1.69753e8	0.0603094
$502$	−3.36826e8	−0.118835
$503$	4.99769e9	1.75098	0.875491	−	0.483234i	$-0.160538\pi$
$503$	4.99769e9	1.75098	0.875491	+	0.483234i	$0.160538\pi$
$504$	−2.32201e9	−0.807898
$505$	−6.39255e9	−2.20879
$506$	−6.96440e9	−2.38977
$507$	−8.14185e8	−0.277457
$508$	4.20212e9	1.42215
$509$	−3.79733e9	−1.27634	−0.638170	−	0.769895i	$-0.720308\pi$
$509$	−3.79733e9	−1.27634	−0.638170	+	0.769895i	$0.720308\pi$
$510$	5.45303e9	1.82030
$511$	−1.27468e9	−0.422600
$512$	6.35877e9	2.09377
$513$	−1.03668e9	−0.339026
$514$	3.17841e8	0.103238
$515$	6.02570e9	1.94394
$516$	4.43773e9	1.42196
$517$	−1.27418e9	−0.405522
$518$	2.32721e9	0.735666
$519$	−1.25512e9	−0.394094
$520$	−9.35563e9	−2.91784
$521$	4.83903e9	1.49909	0.749543	−	0.661956i	$-0.230273\pi$
$521$	4.83903e9	1.49909	0.749543	+	0.661956i	$0.230273\pi$
$522$	2.05977e9	0.633828
$523$	−1.38035e9	−0.421922	−0.210961	−	0.977494i	$-0.567659\pi$
$523$	−1.38035e9	−0.421922	−0.210961	+	0.977494i	$0.567659\pi$
$524$	3.14042e9	0.953515
$525$	−6.17300e8	−0.186182
$526$	8.94325e9	2.67945
$527$	7.25657e9	2.15970
$528$	4.70754e9	1.39180
$529$	9.29267e9	2.72926
$530$	−8.04962e9	−2.34860
$531$	1.49721e8	0.0433963
$532$	1.20743e10	3.47673
$533$	3.25869e9	0.932175
$534$	3.59110e9	1.02055
$535$	2.86454e7	0.00808754
$536$	−6.08312e9	−1.70628
$537$	3.04834e9	0.849480
$538$	4.44490e7	0.0123062
$539$	−1.11934e9	−0.307894
$540$	2.32152e9	0.634445
$541$	3.09741e9	0.841023	0.420512	−	0.907287i	$-0.361851\pi$
$541$	3.09741e9	0.841023	0.420512	+	0.907287i	$0.361851\pi$
$542$	2.23203e9	0.602146
$543$	−2.81949e9	−0.755736
$544$	1.99471e10	5.31232
$545$	5.33711e9	1.41227
$546$	2.20458e9	0.579632
$547$	4.36175e9	1.13948	0.569738	−	0.821826i	$-0.307045\pi$
$547$	4.36175e9	1.13948	0.569738	+	0.821826i	$0.307045\pi$
$548$	−5.96820e9	−1.54921
$549$	−8.54296e8	−0.220346
$550$	2.16170e9	0.554021
$551$	−6.80155e9	−1.73212
$552$	−1.48251e10	−3.75155
$553$	−3.46357e8	−0.0870935
$554$	−3.94645e9	−0.986103
$555$	−1.47752e9	−0.366867
$556$	−5.24553e9	−1.29428
$557$	7.54573e9	1.85016	0.925078	−	0.379778i	$-0.124000\pi$
$557$	7.54573e9	1.85016	0.925078	+	0.379778i	$0.124000\pi$
$558$	4.21687e9	1.02747
$559$	−2.67556e9	−0.647849
$560$	−1.35686e10	−3.26496
$561$	2.09342e9	0.500596
$562$	7.41999e9	1.76330
$563$	5.42932e9	1.28223	0.641115	−	0.767445i	$-0.278472\pi$
$563$	5.42932e9	1.28223	0.641115	+	0.767445i	$0.278472\pi$
$564$	−4.27124e9	−1.00248
$565$	−6.74973e9	−1.57441
$566$	−2.68834e9	−0.623199
$567$	−3.47389e8	−0.0800342
$568$	1.41759e10	3.24587
$569$	−7.43477e9	−1.69190	−0.845950	−	0.533262i	$-0.820966\pi$
$569$	−7.43477e9	−1.69190	−0.845950	+	0.533262i	$0.820966\pi$
$570$	−1.04637e10	−2.36660
$571$	−6.57051e9	−1.47697	−0.738487	−	0.674268i	$-0.764459\pi$
$571$	−6.57051e9	−1.47697	−0.738487	+	0.674268i	$0.764459\pi$
$572$	−5.65590e9	−1.26362
$573$	−3.74840e9	−0.832347
$574$	8.16347e9	1.80170
$575$	−3.94122e9	−0.864555
$576$	5.83207e9	1.27158
$577$	3.17169e9	0.687347	0.343673	−	0.939089i	$-0.388329\pi$
$577$	3.17169e9	0.687347	0.343673	+	0.939089i	$0.388329\pi$
$578$	7.50538e9	1.61668
$579$	8.28575e8	0.177401
$580$	1.52313e10	3.24144
$581$	−4.92823e9	−1.04250
$582$	−1.62384e9	−0.341440
$583$	−3.09026e9	−0.645885
$584$	9.50201e9	1.97411
$585$	−1.39967e9	−0.289055
$586$	−5.99321e9	−1.23032
$587$	−4.96828e9	−1.01385	−0.506923	−	0.861991i	$-0.669217\pi$
$587$	−4.96828e9	−1.01385	−0.506923	+	0.861991i	$0.669217\pi$
$588$	−3.75218e9	−0.761137
$589$	−1.39245e10	−2.80786
$590$	1.51121e9	0.302931
$591$	−7.52821e8	−0.150015
$592$	−1.00434e10	−1.98954
$593$	−1.11991e9	−0.220542	−0.110271	−	0.993902i	$-0.535172\pi$
$593$	−1.11991e9	−0.220542	−0.110271	+	0.993902i	$0.535172\pi$
$594$	1.21651e9	0.238157
$595$	−6.03392e9	−1.17433
$596$	−1.82154e10	−3.52433
$597$	3.53867e9	0.680660
$598$	1.40754e10	2.69157
$599$	1.35680e9	0.257942	0.128971	−	0.991648i	$-0.458833\pi$
$599$	1.35680e9	0.257942	0.128971	+	0.991648i	$0.458833\pi$
$600$	4.60160e9	0.869721
$601$	−8.43823e9	−1.58559	−0.792795	−	0.609489i	$-0.791375\pi$
$601$	−8.43823e9	−1.58559	−0.792795	+	0.609489i	$0.791375\pi$
$602$	−6.70266e9	−1.25216
$603$	−9.10080e8	−0.169032
$604$	−1.68105e9	−0.310420
$605$	−3.87008e9	−0.710519
$606$	−1.12290e10	−2.04968
$607$	2.84233e9	0.515839	0.257920	−	0.966166i	$-0.416963\pi$
$607$	2.84233e9	0.515839	0.257920	+	0.966166i	$0.416963\pi$
$608$	−3.82762e10	−6.90663
$609$	−2.27919e9	−0.408902
$610$	−8.62284e9	−1.53814
$611$	2.57519e9	0.456735
$612$	7.01746e9	1.23751
$613$	−5.43477e8	−0.0952949	−0.0476475	−	0.998864i	$-0.515172\pi$
$613$	−5.43477e8	−0.0952949	−0.0476475	+	0.998864i	$0.515172\pi$
$614$	−5.50014e9	−0.958924
$615$	−5.18292e9	−0.898487
$616$	−8.99755e9	−1.55093
$617$	5.89919e8	0.101110	0.0505550	−	0.998721i	$-0.483901\pi$
$617$	5.89919e8	0.101110	0.0505550	+	0.998721i	$0.483901\pi$
$618$	1.05846e10	1.80391
$619$	−8.29034e9	−1.40493	−0.702466	−	0.711717i	$-0.747918\pi$
$619$	−8.29034e9	−1.40493	−0.702466	+	0.711717i	$0.747918\pi$
$620$	3.11822e10	5.25456
$621$	−2.21794e9	−0.371646
$622$	7.58592e9	1.26399
$623$	−3.97364e9	−0.658386
$624$	−9.51417e9	−1.56756
$625$	−7.61270e9	−1.24726
$626$	1.87964e10	3.06242
$627$	−4.01703e9	−0.650832
$628$	1.87736e10	3.02475
$629$	−4.46624e9	−0.715591
$630$	−3.50637e9	−0.558684
$631$	3.36985e9	0.533959	0.266980	−	0.963702i	$-0.413974\pi$
$631$	3.36985e9	0.533959	0.266980	+	0.963702i	$0.413974\pi$
$632$	2.58188e9	0.406843
$633$	1.89160e9	0.296426
$634$	1.39222e10	2.16968
$635$	4.02953e9	0.624520
$636$	−1.03590e10	−1.59668
$637$	2.26224e9	0.346777
$638$	7.98141e9	1.21677
$639$	2.12082e9	0.321552
$640$	2.75822e10	4.15909
$641$	−6.77698e9	−1.01633	−0.508163	−	0.861261i	$-0.669675\pi$
$641$	−6.77698e9	−1.01633	−0.508163	+	0.861261i	$0.669675\pi$
$642$	5.03177e7	0.00750496
$643$	5.27093e9	0.781895	0.390948	−	0.920413i	$-0.372147\pi$
$643$	5.27093e9	0.781895	0.390948	+	0.920413i	$0.372147\pi$
$644$	2.58326e10	3.81125
$645$	4.25546e9	0.624436
$646$	−3.16296e10	−4.61615
$647$	−1.36234e10	−1.97752	−0.988762	−	0.149498i	$-0.952234\pi$
$647$	−1.36234e10	−1.97752	−0.988762	+	0.149498i	$0.952234\pi$
$648$	2.58958e9	0.373867
$649$	5.80156e8	0.0833083
$650$	−4.36890e9	−0.623987
$651$	−4.66607e9	−0.662854
$652$	−2.95171e10	−4.17068
$653$	7.55574e9	1.06189	0.530947	−	0.847405i	$-0.321836\pi$
$653$	7.55574e9	1.06189	0.530947	+	0.847405i	$0.321836\pi$
$654$	9.37502e9	1.31054
$655$	3.01143e9	0.418725
$656$	−3.52305e10	−4.87254
$657$	1.42157e9	0.195565
$658$	6.45120e9	0.882774
$659$	6.65668e9	0.906063	0.453032	−	0.891494i	$-0.350342\pi$
$659$	6.65668e9	0.906063	0.453032	+	0.891494i	$0.350342\pi$
$660$	8.99566e9	1.21795
$661$	−9.57345e9	−1.28933	−0.644664	−	0.764466i	$-0.723003\pi$
$661$	−9.57345e9	−1.28933	−0.644664	+	0.764466i	$0.723003\pi$
$662$	8.94613e9	1.19848
$663$	−4.23091e9	−0.563815
$664$	3.67370e10	4.86985
$665$	1.15784e10	1.52676
$666$	−2.59538e9	−0.340440
$667$	−1.45517e10	−1.89878
$668$	2.20496e9	0.286209
$669$	3.85892e9	0.498282
$670$	−9.18589e9	−1.17994
$671$	−3.31032e9	−0.423001
$672$	−1.28263e10	−1.63045
$673$	8.00895e9	1.01280	0.506399	−	0.862299i	$-0.330976\pi$
$673$	8.00895e9	1.01280	0.506399	+	0.862299i	$0.330976\pi$
$674$	1.31376e10	1.65275
$675$	6.88434e8	0.0861587
$676$	−1.05756e10	−1.31672
$677$	−3.66833e8	−0.0454368	−0.0227184	−	0.999742i	$-0.507232\pi$
$677$	−3.66833e8	−0.0454368	−0.0227184	+	0.999742i	$0.507232\pi$
$678$	−1.18564e10	−1.46099
$679$	1.79683e9	0.220273
$680$	4.49793e10	5.48569
$681$	−3.14174e9	−0.381202
$682$	1.63400e10	1.97245
$683$	−9.58167e9	−1.15072	−0.575359	−	0.817901i	$-0.695137\pi$
$683$	−9.58167e9	−1.15072	−0.575359	+	0.817901i	$0.695137\pi$
$684$	−1.34657e10	−1.60891
$685$	−5.72307e9	−0.680319
$686$	1.74455e10	2.06324
$687$	6.71086e9	0.789641
$688$	2.89262e10	3.38635
$689$	6.24557e9	0.727452
$690$	−2.23868e10	−2.59430
$691$	1.25570e10	1.44781	0.723907	−	0.689898i	$-0.242345\pi$
$691$	1.25570e10	1.44781	0.723907	+	0.689898i	$0.242345\pi$
$692$	−1.63031e10	−1.87024
$693$	−1.34610e9	−0.153643
$694$	3.97393e9	0.451297
$695$	−5.03008e9	−0.568367
$696$	1.69900e10	1.91012
$697$	−1.56669e10	−1.75254
$698$	−1.54631e10	−1.72109
$699$	−3.46497e9	−0.383733
$700$	−8.01825e9	−0.883561
$701$	−2.90472e9	−0.318487	−0.159243	−	0.987239i	$-0.550905\pi$
$701$	−2.90472e9	−0.318487	−0.159243	+	0.987239i	$0.550905\pi$
$702$	−2.45863e9	−0.268233
$703$	8.57019e9	0.930351
$704$	2.25987e10	2.44107
$705$	−4.09581e9	−0.440229
$706$	5.06656e9	0.541872
$707$	1.24252e10	1.32231
$708$	1.94476e9	0.205945
$709$	4.07802e9	0.429721	0.214861	−	0.976645i	$-0.431070\pi$
$709$	4.07802e9	0.429721	0.214861	+	0.976645i	$0.431070\pi$
$710$	2.14065e10	2.24461
$711$	3.86269e8	0.0403038
$712$	2.96211e10	3.07554
$713$	−2.97910e10	−3.07802
$714$	−1.05990e10	−1.08974
$715$	−5.42360e9	−0.554902
$716$	3.95956e10	4.03136
$717$	4.11460e9	0.416879
$718$	2.49386e10	2.51441
$719$	−6.67220e8	−0.0669450	−0.0334725	−	0.999440i	$-0.510657\pi$
$719$	−6.67220e8	−0.0669450	−0.0334725	+	0.999440i	$0.510657\pi$
$720$	1.51322e10	1.51091
$721$	−1.17121e10	−1.16376
$722$	4.11361e10	4.06764
$723$	−6.65672e9	−0.655052
$724$	−3.66229e10	−3.58648
$725$	4.51675e9	0.440193
$726$	−6.79807e9	−0.659337
$727$	−1.54917e10	−1.49530	−0.747651	−	0.664092i	$-0.768819\pi$
$727$	−1.54917e10	−1.49530	−0.747651	+	0.664092i	$0.768819\pi$
$728$	1.81845e10	1.74679
$729$	3.87420e8	0.0370370
$730$	1.43486e10	1.36515
$731$	1.28634e10	1.21799
$732$	−1.10967e10	−1.04569
$733$	1.32907e10	1.24647	0.623237	−	0.782033i	$-0.285817\pi$
$733$	1.32907e10	1.24647	0.623237	+	0.782033i	$0.285817\pi$
$734$	1.41068e10	1.31671
$735$	−3.59807e9	−0.334244
$736$	−8.18908e10	−7.57116
$737$	−3.52648e9	−0.324493
$738$	−9.10418e9	−0.833765
$739$	1.38709e10	1.26430	0.632148	−	0.774847i	$-0.282173\pi$
$739$	1.38709e10	1.26430	0.632148	+	0.774847i	$0.282173\pi$
$740$	−1.91919e10	−1.74103
$741$	8.11862e9	0.733025
$742$	1.56460e10	1.40601
$743$	−9.01349e9	−0.806181	−0.403090	−	0.915160i	$-0.632064\pi$
$743$	−9.01349e9	−0.806181	−0.403090	+	0.915160i	$0.632064\pi$
$744$	3.47828e10	3.09641
$745$	−1.74672e10	−1.54767
$746$	3.70337e9	0.326596
$747$	5.49613e9	0.482431
$748$	2.71920e10	2.37567
$749$	−5.56779e7	−0.00484168
$750$	−8.57241e9	−0.741974
$751$	−6.25646e9	−0.539000	−0.269500	−	0.963000i	$-0.586858\pi$
$751$	−6.25646e9	−0.539000	−0.269500	+	0.963000i	$0.586858\pi$
$752$	−2.78410e10	−2.38738
$753$	−4.15656e8	−0.0354774
$754$	−1.61308e10	−1.37043
$755$	−1.61200e9	−0.136317
$756$	−4.51232e9	−0.379816
$757$	−2.50426e9	−0.209819	−0.104909	−	0.994482i	$-0.533455\pi$
$757$	−2.50426e9	−0.209819	−0.104909	+	0.994482i	$0.533455\pi$
$758$	−2.21799e10	−1.84977
$759$	−8.59432e9	−0.713453
$760$	−8.63099e10	−7.13203
$761$	6.57794e9	0.541057	0.270529	−	0.962712i	$-0.412802\pi$
$761$	6.57794e9	0.541057	0.270529	+	0.962712i	$0.412802\pi$
$762$	7.07816e9	0.579533
$763$	−1.03737e10	−0.845470
$764$	−4.86888e10	−3.95005
$765$	6.72924e9	0.543439
$766$	1.21233e10	0.974587
$767$	−1.17252e9	−0.0938292
$768$	2.08018e10	1.65706
$769$	5.82071e9	0.461566	0.230783	−	0.973005i	$-0.425871\pi$
$769$	5.82071e9	0.461566	0.230783	+	0.973005i	$0.425871\pi$
$770$	−1.35869e10	−1.07251
$771$	3.92227e8	0.0308210
$772$	1.07626e10	0.841889
$773$	5.98583e9	0.466118	0.233059	−	0.972463i	$-0.425126\pi$
$773$	5.98583e9	0.466118	0.233059	+	0.972463i	$0.425126\pi$
$774$	7.47504e9	0.579455
$775$	9.24693e9	0.713578
$776$	−1.33943e10	−1.02897
$777$	2.87186e9	0.219629
$778$	2.56276e10	1.95110
$779$	3.00629e10	2.27850
$780$	−1.81807e10	−1.37176
$781$	8.21798e9	0.617286
$782$	−6.76706e10	−5.06030
$783$	2.54183e9	0.189226
$784$	−2.44576e10	−1.81262
$785$	1.80026e10	1.32828
$786$	5.28980e9	0.388562
$787$	−4.85827e9	−0.355279	−0.177640	−	0.984096i	$-0.556846\pi$
$787$	−4.85827e9	−0.355279	−0.177640	+	0.984096i	$0.556846\pi$
$788$	−9.77857e9	−0.711924
$789$	1.10363e10	0.799933
$790$	3.89880e9	0.281343
$791$	1.31194e10	0.942533
$792$	1.00344e10	0.717716
$793$	6.69032e9	0.476421
$794$	4.08603e10	2.89688
$795$	−9.93352e9	−0.701162
$796$	4.59647e10	3.23019
$797$	−7.28406e9	−0.509647	−0.254823	−	0.966988i	$-0.582017\pi$
$797$	−7.28406e9	−0.509647	−0.254823	+	0.966988i	$0.582017\pi$
$798$	2.03383e10	1.41679
$799$	−1.23808e10	−0.858685
$800$	2.54183e10	1.75522
$801$	4.43154e9	0.304678
$802$	4.87925e10	3.33997
$803$	5.50846e9	0.375427
$804$	−1.18212e10	−0.802171
$805$	2.47716e10	1.67366
$806$	−3.30239e10	−2.22155
$807$	5.48516e7	0.00367394
$808$	−9.26222e10	−6.17697
$809$	3.79381e9	0.251916	0.125958	−	0.992036i	$-0.459799\pi$
$809$	3.79381e9	0.251916	0.125958	+	0.992036i	$0.459799\pi$
$810$	3.91043e9	0.258539
$811$	−4.79601e9	−0.315724	−0.157862	−	0.987461i	$-0.550460\pi$
$811$	−4.79601e9	−0.315724	−0.157862	+	0.987461i	$0.550460\pi$
$812$	−2.96049e10	−1.94052
$813$	2.75440e9	0.179767
$814$	−1.00569e10	−0.653547
$815$	−2.83048e10	−1.83150
$816$	4.57415e10	2.94710
$817$	−2.46833e10	−1.58353
$818$	−2.26320e10	−1.44573
$819$	2.72054e9	0.173046
$820$	−6.73222e10	−4.26392
$821$	1.01365e10	0.639272	0.319636	−	0.947540i	$-0.396439\pi$
$821$	1.01365e10	0.639272	0.319636	+	0.947540i	$0.396439\pi$
$822$	−1.00530e10	−0.631313
$823$	−9.27062e9	−0.579708	−0.289854	−	0.957071i	$-0.593607\pi$
$823$	−9.27062e9	−0.579708	−0.289854	+	0.957071i	$0.593607\pi$
$824$	8.73068e10	5.43629
$825$	2.66762e9	0.165400
$826$	−2.93733e9	−0.181352
$827$	−1.15153e10	−0.707953	−0.353977	−	0.935254i	$-0.615171\pi$
$827$	−1.15153e10	−0.707953	−0.353977	+	0.935254i	$0.615171\pi$
$828$	−2.88094e10	−1.76371
$829$	−7.01951e9	−0.427923	−0.213962	−	0.976842i	$-0.568637\pi$
$829$	−7.01951e9	−0.427923	−0.213962	+	0.976842i	$0.568637\pi$
$830$	5.54752e10	3.36764
$831$	−4.87006e9	−0.294395
$832$	−4.56731e10	−2.74934
$833$	−1.08762e10	−0.651958
$834$	−8.83571e9	−0.527425
$835$	2.11440e9	0.125685
$836$	−5.21782e10	−3.08864
$837$	5.20376e9	0.306746
$838$	−2.35908e10	−1.38481
$839$	2.55234e10	1.49201	0.746004	−	0.665942i	$-0.231970\pi$
$839$	2.55234e10	1.49201	0.746004	+	0.665942i	$0.231970\pi$
$840$	−2.89223e10	−1.68366
$841$	−5.73184e8	−0.0332283
$842$	−4.13463e10	−2.38696
$843$	9.15654e9	0.526423
$844$	2.45704e10	1.40674
$845$	−1.01413e10	−0.578222
$846$	−7.19460e9	−0.408517
$847$	7.52225e9	0.425359
$848$	−6.75224e10	−3.80244
$849$	−3.31751e9	−0.186052
$850$	2.10045e10	1.17313
$851$	1.83357e10	1.01987
$852$	2.75478e10	1.52598
$853$	1.30151e10	0.718004	0.359002	−	0.933337i	$-0.383117\pi$
$853$	1.30151e10	0.718004	0.359002	+	0.933337i	$0.383117\pi$
$854$	1.67602e10	0.920822
$855$	−1.29126e10	−0.706533
$856$	4.15045e8	0.0226171
$857$	−7.46263e9	−0.405004	−0.202502	−	0.979282i	$-0.564907\pi$
$857$	−7.46263e9	−0.405004	−0.202502	+	0.979282i	$0.564907\pi$
$858$	−9.52695e9	−0.514930
$859$	2.06753e9	0.111295	0.0556476	−	0.998450i	$-0.482278\pi$
$859$	2.06753e9	0.111295	0.0556476	+	0.998450i	$0.482278\pi$
$860$	5.52752e10	2.96337
$861$	1.00740e10	0.537888
$862$	−4.90863e10	−2.61027
$863$	−2.78352e10	−1.47420	−0.737099	−	0.675785i	$-0.763805\pi$
$863$	−2.78352e10	−1.47420	−0.737099	+	0.675785i	$0.763805\pi$
$864$	1.43043e10	0.754517
$865$	−1.56335e10	−0.821295
$866$	3.45196e10	1.80615
$867$	9.26191e9	0.482651
$868$	−6.06087e10	−3.14569
$869$	1.49676e9	0.0773716
$870$	2.56559e10	1.32090
$871$	7.12718e9	0.365472
$872$	7.73298e10	3.94947
$873$	−2.00388e9	−0.101935
$874$	1.29852e11	6.57897
$875$	9.48560e9	0.478670
$876$	1.84651e10	0.928086
$877$	−1.33724e10	−0.669436	−0.334718	−	0.942318i	$-0.608641\pi$
$877$	−1.33724e10	−0.669436	−0.334718	+	0.942318i	$0.608641\pi$
$878$	5.16366e10	2.57470
$879$	−7.39584e9	−0.367305
$880$	5.86359e10	2.90051
$881$	−3.66543e10	−1.80596	−0.902982	−	0.429678i	$-0.858627\pi$
$881$	−3.66543e10	−1.80596	−0.902982	+	0.429678i	$0.858627\pi$
$882$	−6.32027e9	−0.310167
$883$	3.23252e10	1.58008	0.790040	−	0.613055i	$-0.210060\pi$
$883$	3.23252e10	1.58008	0.790040	+	0.613055i	$0.210060\pi$
$884$	−5.49563e10	−2.67568
$885$	1.86489e9	0.0904382
$886$	−1.07983e10	−0.521602
$887$	2.43296e10	1.17058	0.585292	−	0.810823i	$-0.300980\pi$
$887$	2.43296e10	1.17058	0.585292	+	0.810823i	$0.300980\pi$
$888$	−2.14080e10	−1.02596
$889$	−7.83217e9	−0.373875
$890$	4.47298e10	2.12682
$891$	1.50122e9	0.0711004
$892$	5.01245e10	2.36468
$893$	2.37572e10	1.11639
$894$	−3.06825e10	−1.43618
$895$	3.79693e10	1.77032
$896$	−5.36113e10	−2.48988
$897$	1.73695e10	0.803554
$898$	−4.54410e10	−2.09402
$899$	3.41414e10	1.56719
$900$	8.94223e9	0.408881
$901$	−3.00269e10	−1.36765
$902$	−3.52778e10	−1.60059
$903$	−8.27132e9	−0.373825
$904$	−9.77973e10	−4.40289
$905$	−3.51188e10	−1.57496
$906$	−2.83160e9	−0.126498
$907$	−1.00143e10	−0.445653	−0.222827	−	0.974858i	$-0.571528\pi$
$907$	−1.00143e10	−0.445653	−0.222827	+	0.974858i	$0.571528\pi$
$908$	−4.08088e10	−1.80906
$909$	−1.38570e10	−0.611920
$910$	2.74597e10	1.20796
$911$	−1.43309e10	−0.627998	−0.313999	−	0.949423i	$-0.601669\pi$
$911$	−1.43309e10	−0.627998	−0.313999	+	0.949423i	$0.601669\pi$
$912$	−8.77725e10	−3.83157
$913$	2.12970e10	0.926127
$914$	−3.24447e10	−1.40550
$915$	−1.06409e10	−0.459203
$916$	8.71689e10	3.74738
$917$	−5.85331e9	−0.250674
$918$	1.18204e10	0.504293
$919$	5.16709e9	0.219605	0.109802	−	0.993953i	$-0.464978\pi$
$919$	5.16709e9	0.219605	0.109802	+	0.993953i	$0.464978\pi$
$920$	−1.84657e11	−7.81825
$921$	−6.78737e9	−0.286281
$922$	5.63774e10	2.36890
$923$	−1.66089e10	−0.695242
$924$	−1.74848e10	−0.729137
$925$	−5.69126e9	−0.236435
$926$	−9.93354e9	−0.411117
$927$	1.30617e10	0.538545
$928$	9.38492e10	3.85490
$929$	−3.79974e10	−1.55489	−0.777444	−	0.628953i	$-0.783484\pi$
$929$	−3.79974e10	−1.55489	−0.777444	+	0.628953i	$0.783484\pi$
$930$	5.25242e10	2.14126
$931$	2.08701e10	0.847621
$932$	−4.50073e10	−1.82107
$933$	9.36130e9	0.377355
$934$	7.94240e10	3.18961
$935$	2.60752e10	1.04325
$936$	−2.02800e10	−0.808355
$937$	−3.56133e10	−1.41424	−0.707121	−	0.707093i	$-0.750006\pi$
$937$	−3.56133e10	−1.41424	−0.707121	+	0.707093i	$0.750006\pi$
$938$	1.78546e10	0.706382
$939$	2.31955e10	0.914268
$940$	−5.32015e10	−2.08918
$941$	1.98729e10	0.777495	0.388748	−	0.921344i	$-0.372908\pi$
$941$	1.98729e10	0.777495	0.388748	+	0.921344i	$0.372908\pi$
$942$	3.16228e10	1.23260
$943$	6.43186e10	2.49773
$944$	1.26764e10	0.490451
$945$	−4.32699e9	−0.166792
$946$	2.89651e10	1.11239
$947$	1.94063e10	0.742537	0.371269	−	0.928525i	$-0.378923\pi$
$947$	1.94063e10	0.742537	0.371269	+	0.928525i	$0.378923\pi$
$948$	5.01734e9	0.191269
$949$	−1.11329e10	−0.422839
$950$	−4.03051e10	−1.52520
$951$	1.71805e10	0.647746
$952$	−8.74260e10	−3.28406
$953$	−2.57497e10	−0.963711	−0.481855	−	0.876251i	$-0.660037\pi$
$953$	−2.57497e10	−0.963711	−0.481855	+	0.876251i	$0.660037\pi$
$954$	−1.74490e10	−0.650654
$955$	−4.66891e10	−1.73462
$956$	5.34454e10	1.97837
$957$	9.84935e9	0.363258
$958$	−1.98122e9	−0.0728036
$959$	1.11239e10	0.407279
$960$	7.26427e10	2.64998
$961$	4.23834e10	1.54051
$962$	2.03254e10	0.736082
$963$	6.20939e7	0.00224056
$964$	−8.64657e10	−3.10867
$965$	1.03205e10	0.369705
$966$	4.35131e10	1.55310
$967$	−4.00078e10	−1.42283	−0.711413	−	0.702774i	$-0.751945\pi$
$967$	−4.00078e10	−1.42283	−0.711413	+	0.702774i	$0.751945\pi$
$968$	−5.60739e10	−1.98699
$969$	−3.90321e10	−1.37812
$970$	−2.02262e10	−0.711562
$971$	1.34901e10	0.472878	0.236439	−	0.971646i	$-0.424020\pi$
$971$	1.34901e10	0.472878	0.236439	+	0.971646i	$0.424020\pi$
$972$	5.03230e9	0.175766
$973$	9.77695e9	0.340258
$974$	5.46592e9	0.189542
$975$	−5.39139e9	−0.186288
$976$	−7.23307e10	−2.49028
$977$	4.28309e9	0.146935	0.0734676	−	0.997298i	$-0.476593\pi$
$977$	4.28309e9	0.146935	0.0734676	+	0.997298i	$0.476593\pi$
$978$	−4.97194e10	−1.69957
$979$	1.71718e10	0.584893
$980$	−4.67362e10	−1.58621
$981$	1.15691e10	0.391254
$982$	9.31773e9	0.313993
$983$	−1.23005e10	−0.413034	−0.206517	−	0.978443i	$-0.566213\pi$
$983$	−1.23005e10	−0.413034	−0.206517	+	0.978443i	$0.566213\pi$
$984$	−7.50958e10	−2.51265
$985$	−9.37695e9	−0.312633
$986$	7.75525e10	2.57648
$987$	7.96101e9	0.263547
$988$	1.05455e11	3.47870
$989$	−5.28091e10	−1.73589
$990$	1.51525e10	0.496321
$991$	−2.97106e10	−0.969737	−0.484868	−	0.874587i	$-0.661133\pi$
$991$	−2.97106e10	−0.969737	−0.484868	+	0.874587i	$0.661133\pi$
$992$	1.92133e11	6.24901
$993$	1.10398e10	0.357800
$994$	−4.16077e10	−1.34376
$995$	4.40768e10	1.41850
$996$	7.13905e10	2.28946
$997$	1.27080e10	0.406112	0.203056	−	0.979167i	$-0.434913\pi$
$997$	1.27080e10	0.406112	0.203056	+	0.979167i	$0.434913\pi$
$998$	−9.13090e10	−2.90775
$999$	−3.20279e9	−0.101636

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.d.1.18	✓	18
3.2	odd	2		531.8.a.e.1.1		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.d.1.18	✓	18	1.1	even	1	trivial
531.8.a.e.1.1		18	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)

\(f(q)\)	\(=\)	\(q+21.8794 q^{2} +27.0000 q^{3} +350.709 q^{4} +336.305 q^{5} +590.745 q^{6} -653.674 q^{7} +4872.75 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+21.8794 q^{2} +27.0000 q^{3} +350.709 q^{4} +336.305 q^{5} +590.745 q^{6} -653.674 q^{7} +4872.75 q^{8} +729.000 q^{9} +7358.16 q^{10} +2824.81 q^{11} +9469.15 q^{12} -5709.07 q^{13} -14302.0 q^{14} +9080.24 q^{15} +61722.2 q^{16} +27447.6 q^{17} +15950.1 q^{18} -52668.8 q^{19} +117945. q^{20} -17649.2 q^{21} +61805.2 q^{22} -112683. q^{23} +131564. q^{24} +34976.1 q^{25} -124911. q^{26} +19683.0 q^{27} -229250. q^{28} +129138. q^{29} +198670. q^{30} +264379. q^{31} +726735. q^{32} +76269.8 q^{33} +600538. q^{34} -219834. q^{35} +255667. q^{36} -162719. q^{37} -1.15236e6 q^{38} -154145. q^{39} +1.63873e6 q^{40} -570791. q^{41} -386154. q^{42} +468651. q^{43} +990686. q^{44} +245166. q^{45} -2.46544e6 q^{46} -451069. q^{47} +1.66650e6 q^{48} -396253. q^{49} +765257. q^{50} +741086. q^{51} -2.00222e6 q^{52} -1.09397e6 q^{53} +430653. q^{54} +949997. q^{55} -3.18519e6 q^{56} -1.42206e6 q^{57} +2.82547e6 q^{58} +205379. q^{59} +3.18452e6 q^{60} -1.17187e6 q^{61} +5.78445e6 q^{62} -476529. q^{63} +8.00009e6 q^{64} -1.91999e6 q^{65} +1.66874e6 q^{66} -1.24840e6 q^{67} +9.62614e6 q^{68} -3.04245e6 q^{69} -4.80984e6 q^{70} +2.90922e6 q^{71} +3.55224e6 q^{72} +1.95003e6 q^{73} -3.56019e6 q^{74} +944354. q^{75} -1.84714e7 q^{76} -1.84650e6 q^{77} -3.37260e6 q^{78} +529861. q^{79} +2.07575e7 q^{80} +531441. q^{81} -1.24886e7 q^{82} +7.53928e6 q^{83} -6.18974e6 q^{84} +9.23078e6 q^{85} +1.02538e7 q^{86} +3.48673e6 q^{87} +1.37646e7 q^{88} +6.07893e6 q^{89} +5.36410e6 q^{90} +3.73187e6 q^{91} -3.95190e7 q^{92} +7.13822e6 q^{93} -9.86913e6 q^{94} -1.77128e7 q^{95} +1.96218e7 q^{96} -2.74881e6 q^{97} -8.66979e6 q^{98} +2.05928e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9}+O(q^{10}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9	\( 18 q + 24 q^{2} + 486 q^{3} + 1358 q^{4} + 678 q^{5} + 648 q^{6} + 3081 q^{7} + 4107 q^{8} + 13122 q^{9} + 3609 q^{10} + 15070 q^{11} + 36666 q^{12} + 13662 q^{13} + 20861 q^{14} + 18306 q^{15} + 60482 q^{16} + 71919 q^{17} + 17496 q^{18} + 56231 q^{19} + 143053 q^{20} + 83187 q^{21} + 274198 q^{22} + 150029 q^{23} + 110889 q^{24} + 399672 q^{25} + 182846 q^{26} + 354294 q^{27} + 434150 q^{28} + 591285 q^{29} + 97443 q^{30} + 426733 q^{31} + 1205630 q^{32} + 406890 q^{33} + 403548 q^{34} + 912879 q^{35} + 989982 q^{36} + 7703 q^{37} - 417859 q^{38} + 368874 q^{39} + 618020 q^{40} + 770959 q^{41} + 563247 q^{42} + 793050 q^{43} + 2591274 q^{44} + 494262 q^{45} - 4068019 q^{46} + 1410373 q^{47} + 1633014 q^{48} + 1637427 q^{49} + 1021549 q^{50} + 1941813 q^{51} - 3749190 q^{52} + 1037934 q^{53} + 472392 q^{54} + 331974 q^{55} - 391748 q^{56} + 1518237 q^{57} + 653724 q^{58} + 3696822 q^{59} + 3862431 q^{60} - 1374623 q^{61} + 5251718 q^{62} + 2246049 q^{63} + 5077197 q^{64} + 3257170 q^{65} + 7403346 q^{66} - 2436904 q^{67} + 14119909 q^{68} + 4050783 q^{69} + 5185580 q^{70} + 14289172 q^{71} + 2994003 q^{72} + 5482515 q^{73} + 14934154 q^{74} + 10791144 q^{75} + 3822912 q^{76} + 23157109 q^{77} + 4936842 q^{78} + 19786414 q^{79} + 31978143 q^{80} + 9565938 q^{81} + 9749509 q^{82} + 30227337 q^{83} + 11722050 q^{84} + 9946981 q^{85} + 44295864 q^{86} + 15964695 q^{87} + 39970897 q^{88} + 31061677 q^{89} + 2630961 q^{90} + 26377785 q^{91} + 4719698 q^{92} + 11521791 q^{93} + 44488296 q^{94} + 15534599 q^{95} + 32552010 q^{96} + 12084118 q^{97} + 42274744 q^{98} + 10986030 q^{99}+O(q^{100}) \) 18 * q + 24 * q^2 + 486 * q^3 + 1358 * q^4 + 678 * q^5 + 648 * q^6 + 3081 * q^7 + 4107 * q^8 + 13122 * q^9 + 3609 * q^10 + 15070 * q^11 + 36666 * q^12 + 13662 * q^13 + 20861 * q^14 + 18306 * q^15 + 60482 * q^16 + 71919 * q^17 + 17496 * q^18 + 56231 * q^19 + 143053 * q^20 + 83187 * q^21 + 274198 * q^22 + 150029 * q^23 + 110889 * q^24 + 399672 * q^25 + 182846 * q^26 + 354294 * q^27 + 434150 * q^28 + 591285 * q^29 + 97443 * q^30 + 426733 * q^31 + 1205630 * q^32 + 406890 * q^33 + 403548 * q^34 + 912879 * q^35 + 989982 * q^36 + 7703 * q^37 - 417859 * q^38 + 368874 * q^39 + 618020 * q^40 + 770959 * q^41 + 563247 * q^42 + 793050 * q^43 + 2591274 * q^44 + 494262 * q^45 - 4068019 * q^46 + 1410373 * q^47 + 1633014 * q^48 + 1637427 * q^49 + 1021549 * q^50 + 1941813 * q^51 - 3749190 * q^52 + 1037934 * q^53 + 472392 * q^54 + 331974 * q^55 - 391748 * q^56 + 1518237 * q^57 + 653724 * q^58 + 3696822 * q^59 + 3862431 * q^60 - 1374623 * q^61 + 5251718 * q^62 + 2246049 * q^63 + 5077197 * q^64 + 3257170 * q^65 + 7403346 * q^66 - 2436904 * q^67 + 14119909 * q^68 + 4050783 * q^69 + 5185580 * q^70 + 14289172 * q^71 + 2994003 * q^72 + 5482515 * q^73 + 14934154 * q^74 + 10791144 * q^75 + 3822912 * q^76 + 23157109 * q^77 + 4936842 * q^78 + 19786414 * q^79 + 31978143 * q^80 + 9565938 * q^81 + 9749509 * q^82 + 30227337 * q^83 + 11722050 * q^84 + 9946981 * q^85 + 44295864 * q^86 + 15964695 * q^87 + 39970897 * q^88 + 31061677 * q^89 + 2630961 * q^90 + 26377785 * q^91 + 4719698 * q^92 + 11521791 * q^93 + 44488296 * q^94 + 15534599 * q^95 + 32552010 * q^96 + 12084118 * q^97 + 42274744 * q^98 + 10986030 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more