LMFDB - Embedded newform 177.8.a.b.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:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} + \cdots - 58\!\cdots\!76 $ x^17 - 2x^16 - 1639x^15 + 1625x^14 + 1070274x^13 - 274939x^12 - 357079564x^11 - 89298188x^10 + 64650816672x^9 + 33122051904x^8 - 6210397064704x^7 - 2735256748800x^6 + 288860762071040x^5 - 34502173230080x^4 - 5633463408885760x^3 + 4719471961341952x^2 + 37636623107620864x - 58321181718347776
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	multiple of $ 2^{10}\cdot 3^{5} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$7.01814$ 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)

$f(q)$	$=$	$q+5.01814 q^{2} +27.0000 q^{3} -102.818 q^{4} -77.5687 q^{5} +135.490 q^{6} -216.829 q^{7} -1158.28 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+5.01814 q^{2} +27.0000 q^{3} -102.818 q^{4} -77.5687 q^{5} +135.490 q^{6} -216.829 q^{7} -1158.28 q^{8} +729.000 q^{9} -389.251 q^{10} +3376.21 q^{11} -2776.09 q^{12} +12448.6 q^{13} -1088.08 q^{14} -2094.36 q^{15} +7348.33 q^{16} -26362.1 q^{17} +3658.23 q^{18} +41401.4 q^{19} +7975.48 q^{20} -5854.37 q^{21} +16942.3 q^{22} -42118.5 q^{23} -31273.5 q^{24} -72108.1 q^{25} +62468.7 q^{26} +19683.0 q^{27} +22293.9 q^{28} -151717. q^{29} -10509.8 q^{30} -239434. q^{31} +185135. q^{32} +91157.8 q^{33} -132289. q^{34} +16819.1 q^{35} -74954.5 q^{36} -411153. q^{37} +207758. q^{38} +336111. q^{39} +89846.2 q^{40} -536752. q^{41} -29378.1 q^{42} +583229. q^{43} -347136. q^{44} -56547.6 q^{45} -211357. q^{46} -587465. q^{47} +198405. q^{48} -776528. q^{49} -361849. q^{50} -711776. q^{51} -1.27994e6 q^{52} -267710. q^{53} +98772.1 q^{54} -261889. q^{55} +251148. q^{56} +1.11784e6 q^{57} -761339. q^{58} -205379. q^{59} +215338. q^{60} -498824. q^{61} -1.20151e6 q^{62} -158068. q^{63} -11553.8 q^{64} -965620. q^{65} +457443. q^{66} +2.87144e6 q^{67} +2.71050e6 q^{68} -1.13720e6 q^{69} +84400.8 q^{70} +1.08176e6 q^{71} -844385. q^{72} -4746.50 q^{73} -2.06322e6 q^{74} -1.94692e6 q^{75} -4.25682e6 q^{76} -732060. q^{77} +1.68666e6 q^{78} -1.41454e6 q^{79} -570000. q^{80} +531441. q^{81} -2.69350e6 q^{82} +3.72243e6 q^{83} +601936. q^{84} +2.04487e6 q^{85} +2.92672e6 q^{86} -4.09637e6 q^{87} -3.91060e6 q^{88} +1.04746e7 q^{89} -283764. q^{90} -2.69921e6 q^{91} +4.33055e6 q^{92} -6.46471e6 q^{93} -2.94798e6 q^{94} -3.21145e6 q^{95} +4.99864e6 q^{96} -4.51283e6 q^{97} -3.89673e6 q^{98} +2.46126e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 32 q^{2} + 459 q^{3} + 1166 q^{4} - 1072 q^{5} - 864 q^{6} - 2407 q^{7} - 6645 q^{8} + 12393 q^{9}+O(q^{10}) $ 17 * q - 32 * q^2 + 459 * q^3 + 1166 * q^4 - 1072 * q^5 - 864 * q^6 - 2407 * q^7 - 6645 * q^8 + 12393 * q^9	\( 17 q - 32 q^{2} + 459 q^{3} + 1166 q^{4} - 1072 q^{5} - 864 q^{6} - 2407 q^{7} - 6645 q^{8} + 12393 q^{9} - 6391 q^{10} - 8888 q^{11} + 31482 q^{12} - 12702 q^{13} - 17555 q^{14} - 28944 q^{15} + 139226 q^{16} - 36167 q^{17} - 23328 q^{18} - 71037 q^{19} - 274883 q^{20} - 64989 q^{21} - 325182 q^{22} - 269995 q^{23} - 179415 q^{24} + 97329 q^{25} - 336906 q^{26} + 334611 q^{27} - 901362 q^{28} - 543825 q^{29} - 172557 q^{30} - 633109 q^{31} - 837062 q^{32} - 239976 q^{33} - 529288 q^{34} - 287621 q^{35} + 850014 q^{36} - 867607 q^{37} - 1727169 q^{38} - 342954 q^{39} - 815662 q^{40} - 1428939 q^{41} - 473985 q^{42} - 477060 q^{43} - 1667926 q^{44} - 781488 q^{45} + 5305549 q^{46} - 1217849 q^{47} + 3759102 q^{48} + 4350738 q^{49} + 4561369 q^{50} - 976509 q^{51} + 4175994 q^{52} - 3487068 q^{53} - 629856 q^{54} - 960484 q^{55} - 5363196 q^{56} - 1917999 q^{57} - 3082906 q^{58} - 3491443 q^{59} - 7421841 q^{60} + 998917 q^{61} - 5742614 q^{62} - 1754703 q^{63} + 17531621 q^{64} - 6075816 q^{65} - 8779914 q^{66} - 356026 q^{67} - 16149231 q^{68} - 7289865 q^{69} - 548798 q^{70} - 12879428 q^{71} - 4844205 q^{72} - 6176157 q^{73} - 5971906 q^{74} + 2627883 q^{75} - 17624580 q^{76} + 239687 q^{77} - 9096462 q^{78} - 18886490 q^{79} - 70463349 q^{80} + 9034497 q^{81} - 19351611 q^{82} - 22824893 q^{83} - 24336774 q^{84} - 7973079 q^{85} - 27502196 q^{86} - 14683275 q^{87} - 62527651 q^{88} - 30609647 q^{89} - 4659039 q^{90} - 36301521 q^{91} - 41388548 q^{92} - 17093943 q^{93} + 1010176 q^{94} - 29303629 q^{95} - 22600674 q^{96} - 26249806 q^{97} - 93110852 q^{98} - 6479352 q^{99}+O(q^{100}) \) 17 * q - 32 * q^2 + 459 * q^3 + 1166 * q^4 - 1072 * q^5 - 864 * q^6 - 2407 * q^7 - 6645 * q^8 + 12393 * q^9 - 6391 * q^10 - 8888 * q^11 + 31482 * q^12 - 12702 * q^13 - 17555 * q^14 - 28944 * q^15 + 139226 * q^16 - 36167 * q^17 - 23328 * q^18 - 71037 * q^19 - 274883 * q^20 - 64989 * q^21 - 325182 * q^22 - 269995 * q^23 - 179415 * q^24 + 97329 * q^25 - 336906 * q^26 + 334611 * q^27 - 901362 * q^28 - 543825 * q^29 - 172557 * q^30 - 633109 * q^31 - 837062 * q^32 - 239976 * q^33 - 529288 * q^34 - 287621 * q^35 + 850014 * q^36 - 867607 * q^37 - 1727169 * q^38 - 342954 * q^39 - 815662 * q^40 - 1428939 * q^41 - 473985 * q^42 - 477060 * q^43 - 1667926 * q^44 - 781488 * q^45 + 5305549 * q^46 - 1217849 * q^47 + 3759102 * q^48 + 4350738 * q^49 + 4561369 * q^50 - 976509 * q^51 + 4175994 * q^52 - 3487068 * q^53 - 629856 * q^54 - 960484 * q^55 - 5363196 * q^56 - 1917999 * q^57 - 3082906 * q^58 - 3491443 * q^59 - 7421841 * q^60 + 998917 * q^61 - 5742614 * q^62 - 1754703 * q^63 + 17531621 * q^64 - 6075816 * q^65 - 8779914 * q^66 - 356026 * q^67 - 16149231 * q^68 - 7289865 * q^69 - 548798 * q^70 - 12879428 * q^71 - 4844205 * q^72 - 6176157 * q^73 - 5971906 * q^74 + 2627883 * q^75 - 17624580 * q^76 + 239687 * q^77 - 9096462 * q^78 - 18886490 * q^79 - 70463349 * q^80 + 9034497 * q^81 - 19351611 * q^82 - 22824893 * q^83 - 24336774 * q^84 - 7973079 * q^85 - 27502196 * q^86 - 14683275 * q^87 - 62527651 * q^88 - 30609647 * q^89 - 4659039 * q^90 - 36301521 * q^91 - 41388548 * q^92 - 17093943 * q^93 + 1010176 * q^94 - 29303629 * q^95 - 22600674 * q^96 - 26249806 * q^97 - 93110852 * q^98 - 6479352 * 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 $

$2$	5.01814	0.443545	0.221773	−	0.975098i	$-0.428816\pi$
$2$	5.01814	0.443545	0.221773	+	0.975098i	$0.428816\pi$
$3$	27.0000	0.577350
$3$	27.0000	0.577350
$4$	−102.818	−0.803268
$5$	−77.5687	−0.277518	−0.138759	−	0.990326i	$-0.544311\pi$
$5$	−77.5687	−0.277518	−0.138759	+	0.990326i	$0.544311\pi$
$6$	135.490	0.256081
$7$	−216.829	−0.238932	−0.119466	−	0.992838i	$-0.538118\pi$
$7$	−216.829	−0.238932	−0.119466	+	0.992838i	$0.538118\pi$
$8$	−1158.28	−0.799831
$9$	729.000	0.333333
$10$	−389.251	−0.123092
$11$	3376.21	0.764813	0.382407	−	0.923994i	$-0.375095\pi$
$11$	3376.21	0.764813	0.382407	+	0.923994i	$0.375095\pi$
$12$	−2776.09	−0.463767
$13$	12448.6	1.57151	0.785756	−	0.618536i	$-0.212274\pi$
$13$	12448.6	1.57151	0.785756	+	0.618536i	$0.212274\pi$
$14$	−1088.08	−0.105977
$15$	−2094.36	−0.160225
$16$	7348.33	0.448506
$17$	−26362.1	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$17$	−26362.1	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$18$	3658.23	0.147848
$19$	41401.4	1.38477	0.692384	−	0.721529i	$-0.256560\pi$
$19$	41401.4	1.38477	0.692384	+	0.721529i	$0.256560\pi$
$20$	7975.48	0.222921
$21$	−5854.37	−0.137947
$22$	16942.3	0.339229
$23$	−42118.5	−0.721815	−0.360907	−	0.932602i	$-0.617533\pi$
$23$	−42118.5	−0.721815	−0.360907	+	0.932602i	$0.617533\pi$
$24$	−31273.5	−0.461783
$25$	−72108.1	−0.922984
$26$	62468.7	0.697037
$27$	19683.0	0.192450
$28$	22293.9	0.191926
$29$	−151717.	−1.15516	−0.577580	−	0.816334i	$-0.696003\pi$
$29$	−151717.	−1.15516	−0.577580	+	0.816334i	$0.696003\pi$
$30$	−10509.8	−0.0710672
$31$	−239434.	−1.44351	−0.721754	−	0.692149i	$-0.756664\pi$
$31$	−239434.	−1.44351	−0.721754	+	0.692149i	$0.756664\pi$
$32$	185135.	0.998764
$33$	91157.8	0.441565
$34$	−132289.	−0.577226
$35$	16819.1	0.0663079
$36$	−74954.5	−0.267756
$37$	−411153.	−1.33444	−0.667218	−	0.744863i	$-0.732515\pi$
$37$	−411153.	−1.33444	−0.667218	+	0.744863i	$0.732515\pi$
$38$	207758.	0.614208
$39$	336111.	0.907313
$40$	89846.2	0.221968
$41$	−536752.	−1.21627	−0.608135	−	0.793834i	$-0.708082\pi$
$41$	−536752.	−1.21627	−0.608135	+	0.793834i	$0.708082\pi$
$42$	−29378.1	−0.0611858
$43$	583229.	1.11866	0.559331	−	0.828944i	$-0.311058\pi$
$43$	583229.	1.11866	0.559331	+	0.828944i	$0.311058\pi$
$44$	−347136.	−0.614350
$45$	−56547.6	−0.0925061
$46$	−211357.	−0.320158
$47$	−587465.	−0.825353	−0.412677	−	0.910878i	$-0.635406\pi$
$47$	−587465.	−0.825353	−0.412677	+	0.910878i	$0.635406\pi$
$48$	198405.	0.258945
$49$	−776528.	−0.942912
$50$	−361849.	−0.409385
$51$	−711776.	−0.751359
$52$	−1.27994e6	−1.26234
$53$	−267710.	−0.247001	−0.123501	−	0.992344i	$-0.539412\pi$
$53$	−267710.	−0.247001	−0.123501	+	0.992344i	$0.539412\pi$
$54$	98772.1	0.0853603
$55$	−261889.	−0.212250
$56$	251148.	0.191105
$57$	1.11784e6	0.799496
$58$	−761339.	−0.512366
$59$	−205379.	−0.130189
$59$	−205379.	−0.130189
$60$	215338.	0.128704
$61$	−498824.	−0.281380	−0.140690	−	0.990054i	$-0.544932\pi$
$61$	−498824.	−0.281380	−0.140690	+	0.990054i	$0.544932\pi$
$62$	−1.20151e6	−0.640261
$63$	−158068.	−0.0796439
$64$	−11553.8	−0.00550929
$65$	−965620.	−0.436124
$66$	457443.	0.195854
$67$	2.87144e6	1.16637	0.583187	−	0.812338i	$-0.301805\pi$
$67$	2.87144e6	1.16637	0.583187	+	0.812338i	$0.301805\pi$
$68$	2.71050e6	1.04537
$69$	−1.13720e6	−0.416740
$70$	84400.8	0.0294106
$71$	1.08176e6	0.358697	0.179349	−	0.983786i	$-0.442601\pi$
$71$	1.08176e6	0.358697	0.179349	+	0.983786i	$0.442601\pi$
$72$	−844385.	−0.266610
$73$	−4746.50	−0.00142805	−0.000714025	−	1.00000i	$-0.500227\pi$
$73$	−4746.50	−0.00142805	−0.000714025		1.00000i	$0.500227\pi$
$74$	−2.06322e6	−0.591883
$75$	−1.94692e6	−0.532885
$76$	−4.25682e6	−1.11234
$77$	−732060.	−0.182738
$78$	1.68666e6	0.402435
$79$	−1.41454e6	−0.322791	−0.161396	−	0.986890i	$-0.551600\pi$
$79$	−1.41454e6	−0.322791	−0.161396	+	0.986890i	$0.551600\pi$
$80$	−570000.	−0.124469
$81$	531441.	0.111111
$82$	−2.69350e6	−0.539471
$83$	3.72243e6	0.714583	0.357292	−	0.933993i	$-0.383700\pi$
$83$	3.72243e6	0.714583	0.357292	+	0.933993i	$0.383700\pi$
$84$	601936.	0.110809
$85$	2.04487e6	0.361160
$86$	2.92672e6	0.496178
$87$	−4.09637e6	−0.666932
$88$	−3.91060e6	−0.611721
$89$	1.04746e7	1.57497	0.787484	−	0.616336i	$-0.211384\pi$
$89$	1.04746e7	1.57497	0.787484	+	0.616336i	$0.211384\pi$
$90$	−283764.	−0.0410307
$91$	−2.69921e6	−0.375484
$92$	4.33055e6	0.579810
$93$	−6.46471e6	−0.833410
$94$	−2.94798e6	−0.366081
$95$	−3.21145e6	−0.384299
$96$	4.99864e6	0.576637
$97$	−4.51283e6	−0.502051	−0.251025	−	0.967980i	$-0.580768\pi$
$97$	−4.51283e6	−0.502051	−0.251025	+	0.967980i	$0.580768\pi$
$98$	−3.89673e6	−0.418224
$99$	2.46126e6	0.254938
$100$	7.41403e6	0.741403
$101$	−1.70633e7	−1.64793	−0.823963	−	0.566644i	$-0.808242\pi$
$101$	−1.70633e7	−1.64793	−0.823963	+	0.566644i	$0.808242\pi$
$102$	−3.57179e6	−0.333262
$103$	−4.18892e6	−0.377721	−0.188861	−	0.982004i	$-0.560479\pi$
$103$	−4.18892e6	−0.377721	−0.188861	+	0.982004i	$0.560479\pi$
$104$	−1.44189e7	−1.25694
$105$	454116.	0.0382829
$106$	−1.34341e6	−0.109556
$107$	−1.20271e7	−0.949116	−0.474558	−	0.880224i	$-0.657392\pi$
$107$	−1.20271e7	−0.949116	−0.474558	+	0.880224i	$0.657392\pi$
$108$	−2.02377e6	−0.154589
$109$	−1.96024e7	−1.44983	−0.724913	−	0.688840i	$-0.758120\pi$
$109$	−1.96024e7	−1.44983	−0.724913	+	0.688840i	$0.758120\pi$
$110$	−1.31419e6	−0.0941424
$111$	−1.11011e7	−0.770437
$112$	−1.59333e6	−0.107162
$113$	−7.54916e6	−0.492180	−0.246090	−	0.969247i	$-0.579146\pi$
$113$	−7.54916e6	−0.492180	−0.246090	+	0.969247i	$0.579146\pi$
$114$	5.60947e6	0.354613
$115$	3.26708e6	0.200317
$116$	1.55993e7	0.927903
$117$	9.07501e6	0.523837
$118$	−1.03062e6	−0.0577447
$119$	5.71605e6	0.310944
$120$	2.42585e6	0.128153
$121$	−8.08835e6	−0.415060
$122$	−2.50317e6	−0.124805
$123$	−1.44923e7	−0.702214
$124$	2.46182e7	1.15952
$125$	1.16534e7	0.533663
$126$	−793208.	−0.0353257
$127$	−1.36062e7	−0.589417	−0.294709	−	0.955587i	$-0.595223\pi$
$127$	−1.36062e7	−0.589417	−0.294709	+	0.955587i	$0.595223\pi$
$128$	−2.37552e7	−1.00121
$129$	1.57472e7	0.645860
$130$	−4.84562e6	−0.193441
$131$	−1.29419e7	−0.502978	−0.251489	−	0.967860i	$-0.580920\pi$
$131$	−1.29419e7	−0.502978	−0.251489	+	0.967860i	$0.580920\pi$
$132$	−9.37268e6	−0.354695
$133$	−8.97700e6	−0.330865
$134$	1.44093e7	0.517340
$135$	−1.52679e6	−0.0534084
$136$	3.05346e7	1.04089
$137$	−7.65865e6	−0.254466	−0.127233	−	0.991873i	$-0.540610\pi$
$137$	−7.65865e6	−0.254466	−0.127233	+	0.991873i	$0.540610\pi$
$138$	−5.70663e6	−0.184843
$139$	4.87660e6	0.154016	0.0770080	−	0.997030i	$-0.475463\pi$
$139$	4.87660e6	0.154016	0.0770080	+	0.997030i	$0.475463\pi$
$140$	−1.72931e6	−0.0532630
$141$	−1.58616e7	−0.476518
$142$	5.42844e6	0.159098
$143$	4.20290e7	1.20191
$144$	5.35693e6	0.149502
$145$	1.17685e7	0.320578
$146$	−23818.6	−0.000633405 0
$147$	−2.09663e7	−0.544390
$148$	4.22740e7	1.07191
$149$	1.05059e6	0.0260186	0.0130093	−	0.999915i	$-0.495859\pi$
$149$	1.05059e6	0.0260186	0.0130093	+	0.999915i	$0.495859\pi$
$150$	−9.76991e6	−0.236359
$151$	5.81661e7	1.37484	0.687418	−	0.726262i	$-0.258744\pi$
$151$	5.81661e7	1.37484	0.687418	+	0.726262i	$0.258744\pi$
$152$	−4.79543e7	−1.10758
$153$	−1.92179e7	−0.433797
$154$	−3.67358e6	−0.0810526
$155$	1.85726e7	0.400600
$156$	−3.45584e7	−0.728815
$157$	4.08109e7	0.841642	0.420821	−	0.907144i	$-0.361742\pi$
$157$	4.08109e7	0.841642	0.420821	+	0.907144i	$0.361742\pi$
$158$	−7.09838e6	−0.143173
$159$	−7.22818e6	−0.142606
$160$	−1.43607e7	−0.277175
$161$	9.13250e6	0.172464
$162$	2.66685e6	0.0492828
$163$	−7.46918e7	−1.35088	−0.675439	−	0.737416i	$-0.736046\pi$
$163$	−7.46918e7	−1.35088	−0.675439	+	0.737416i	$0.736046\pi$
$164$	5.51879e7	0.976990
$165$	−7.07099e6	−0.122542
$166$	1.86797e7	0.316950
$167$	−5.71426e7	−0.949407	−0.474704	−	0.880146i	$-0.657445\pi$
$167$	−5.71426e7	−0.949407	−0.474704	+	0.880146i	$0.657445\pi$
$168$	6.78100e6	0.110334
$169$	9.22184e7	1.46965
$170$	1.02615e7	0.160191
$171$	3.01816e7	0.461589
$172$	−5.99665e7	−0.898585
$173$	6.74021e7	0.989720	0.494860	−	0.868973i	$-0.335219\pi$
$173$	6.74021e7	0.989720	0.494860	+	0.868973i	$0.335219\pi$
$174$	−2.05562e7	−0.295815
$175$	1.56351e7	0.220530
$176$	2.48095e7	0.343024
$177$	−5.54523e6	−0.0751646
$178$	5.25629e7	0.698569
$179$	5.40298e7	0.704122	0.352061	−	0.935977i	$-0.385481\pi$
$179$	5.40298e7	0.704122	0.352061	+	0.935977i	$0.385481\pi$
$180$	5.81413e6	0.0743072
$181$	−2.98190e7	−0.373782	−0.186891	−	0.982381i	$-0.559841\pi$
$181$	−2.98190e7	−0.373782	−0.186891	+	0.982381i	$0.559841\pi$
$182$	−1.35450e7	−0.166544
$183$	−1.34683e7	−0.162455
$184$	4.87850e7	0.577330
$185$	3.18926e7	0.370330
$186$	−3.24408e7	−0.369655
$187$	−8.90040e7	−0.995322
$188$	6.04022e7	0.662979
$189$	−4.26784e6	−0.0459824
$190$	−1.61155e7	−0.170454
$191$	−1.03129e8	−1.07093	−0.535466	−	0.844557i	$-0.679864\pi$
$191$	−1.03129e8	−1.07093	−0.535466	+	0.844557i	$0.679864\pi$
$192$	−311953.	−0.00318079
$193$	1.72283e8	1.72501	0.862506	−	0.506047i	$-0.168894\pi$
$193$	1.72283e8	1.72501	0.862506	+	0.506047i	$0.168894\pi$
$194$	−2.26460e7	−0.222682
$195$	−2.60717e7	−0.251796
$196$	7.98413e7	0.757410
$197$	−1.27602e8	−1.18912	−0.594562	−	0.804050i	$-0.702674\pi$
$197$	−1.27602e8	−1.18912	−0.594562	+	0.804050i	$0.702674\pi$
$198$	1.23510e7	0.113076
$199$	−6.47395e7	−0.582349	−0.291175	−	0.956670i	$-0.594046\pi$
$199$	−6.47395e7	−0.582349	−0.291175	+	0.956670i	$0.594046\pi$
$200$	8.35213e7	0.738231
$201$	7.75289e7	0.673406
$202$	−8.56260e7	−0.730930
$203$	3.28967e7	0.276004
$204$	7.31835e7	0.603542
$205$	4.16351e7	0.337537
$206$	−2.10206e7	−0.167536
$207$	−3.07044e7	−0.240605
$208$	9.14762e7	0.704833
$209$	1.39780e8	1.05909
$210$	2.27882e6	0.0169802
$211$	1.42567e8	1.04479	0.522395	−	0.852704i	$-0.325039\pi$
$211$	1.42567e8	1.04479	0.522395	+	0.852704i	$0.325039\pi$
$212$	2.75255e7	0.198408
$213$	2.92076e7	0.207094
$214$	−6.03539e7	−0.420976
$215$	−4.52403e7	−0.310449
$216$	−2.27984e7	−0.153928
$217$	5.19161e7	0.344900
$218$	−9.83675e7	−0.643064
$219$	−128155.	−0.000824485 0
$220$	2.69269e7	0.170493
$221$	−3.28170e8	−2.04515
$222$	−5.57071e7	−0.341724
$223$	2.35480e8	1.42196	0.710979	−	0.703213i	$-0.248252\pi$
$223$	2.35480e8	1.42196	0.710979	+	0.703213i	$0.248252\pi$
$224$	−4.01425e7	−0.238636
$225$	−5.25668e7	−0.307661
$226$	−3.78828e7	−0.218304
$227$	3.35151e8	1.90174	0.950868	−	0.309598i	$-0.100194\pi$
$227$	3.35151e8	1.90174	0.950868	+	0.309598i	$0.100194\pi$
$228$	−1.14934e8	−0.642209
$229$	−1.68231e8	−0.925727	−0.462864	−	0.886430i	$-0.653178\pi$
$229$	−1.68231e8	−0.925727	−0.462864	+	0.886430i	$0.653178\pi$
$230$	1.63947e7	0.0888496
$231$	−1.97656e7	−0.105504
$232$	1.75731e8	0.923933
$233$	−2.76156e8	−1.43024	−0.715120	−	0.699002i	$-0.753628\pi$
$233$	−2.76156e8	−1.43024	−0.715120	+	0.699002i	$0.753628\pi$
$234$	4.55397e7	0.232346
$235$	4.55689e7	0.229051
$236$	2.11167e7	0.104577
$237$	−3.81927e7	−0.186364
$238$	2.86840e7	0.137918
$239$	1.64969e8	0.781647	0.390824	−	0.920466i	$-0.372190\pi$
$239$	1.64969e8	0.781647	0.390824	+	0.920466i	$0.372190\pi$
$240$	−1.53900e7	−0.0718621
$241$	2.19804e8	1.01152	0.505761	−	0.862674i	$-0.331212\pi$
$241$	2.19804e8	1.01152	0.505761	+	0.862674i	$0.331212\pi$
$242$	−4.05885e7	−0.184098
$243$	1.43489e7	0.0641500
$244$	5.12882e7	0.226023
$245$	6.02343e7	0.261675
$246$	−7.27244e7	−0.311464
$247$	5.15388e8	2.17618
$248$	2.77331e8	1.15456
$249$	1.00506e8	0.412565
$250$	5.84784e7	0.236704
$251$	−2.64722e8	−1.05665	−0.528327	−	0.849041i	$-0.677180\pi$
$251$	−2.64722e8	−1.05665	−0.528327	+	0.849041i	$0.677180\pi$
$252$	1.62523e7	0.0639753
$253$	−1.42201e8	−0.552054
$254$	−6.82777e7	−0.261433
$255$	5.52116e7	0.208516
$256$	−1.17728e8	−0.438572
$257$	−2.39775e8	−0.881125	−0.440563	−	0.897722i	$-0.645221\pi$
$257$	−2.39775e8	−0.881125	−0.440563	+	0.897722i	$0.645221\pi$
$258$	7.90215e7	0.286468
$259$	8.91498e7	0.318839
$260$	9.92834e7	0.350324
$261$	−1.10602e8	−0.385054
$262$	−6.49444e7	−0.223094
$263$	1.27661e8	0.432727	0.216364	−	0.976313i	$-0.430580\pi$
$263$	1.27661e8	0.432727	0.216364	+	0.976313i	$0.430580\pi$
$264$	−1.05586e8	−0.353178
$265$	2.07659e7	0.0685474
$266$	−4.50479e7	−0.146754
$267$	2.82814e8	0.909308
$268$	−2.95236e8	−0.936910
$269$	1.98210e8	0.620858	0.310429	−	0.950597i	$-0.399527\pi$
$269$	1.98210e8	0.620858	0.310429	+	0.950597i	$0.399527\pi$
$270$	−7.66163e6	−0.0236891
$271$	1.17159e8	0.357587	0.178794	−	0.983887i	$-0.442781\pi$
$271$	1.17159e8	0.357587	0.178794	+	0.983887i	$0.442781\pi$
$272$	−1.93717e8	−0.583683
$273$	−7.28786e7	−0.216786
$274$	−3.84322e7	−0.112867
$275$	−2.43452e8	−0.705910
$276$	1.16925e8	0.334754
$277$	2.80002e8	0.791555	0.395778	−	0.918346i	$-0.370475\pi$
$277$	2.80002e8	0.791555	0.395778	+	0.918346i	$0.370475\pi$
$278$	2.44715e7	0.0683131
$279$	−1.74547e8	−0.481169
$280$	−1.94812e7	−0.0530351
$281$	−6.16699e7	−0.165806	−0.0829032	−	0.996558i	$-0.526419\pi$
$281$	−6.16699e7	−0.165806	−0.0829032	+	0.996558i	$0.526419\pi$
$282$	−7.95956e7	−0.211357
$283$	−6.20410e8	−1.62715	−0.813573	−	0.581463i	$-0.802481\pi$
$283$	−6.20410e8	−1.62715	−0.813573	+	0.581463i	$0.802481\pi$
$284$	−1.11225e8	−0.288130
$285$	−8.67092e7	−0.221875
$286$	2.10908e8	0.533103
$287$	1.16383e8	0.290605
$288$	1.34963e8	0.332921
$289$	2.84620e8	0.693622
$290$	5.90561e7	0.142191
$291$	−1.21846e8	−0.289859
$292$	488027.	0.00114711
$293$	−7.47133e8	−1.73525	−0.867623	−	0.497222i	$-0.834353\pi$
$293$	−7.47133e8	−1.73525	−0.867623	+	0.497222i	$0.834353\pi$
$294$	−1.05212e8	−0.241462
$295$	1.59310e7	0.0361298
$296$	4.76230e8	1.06732
$297$	6.64540e7	0.147188
$298$	5.27203e6	0.0115404
$299$	−5.24316e8	−1.13434
$300$	2.00179e8	0.428049
$301$	−1.26461e8	−0.267284
$302$	2.91886e8	0.609802
$303$	−4.60708e8	−0.951430
$304$	3.04231e8	0.621077
$305$	3.86932e7	0.0780881
$306$	−9.64384e7	−0.192409
$307$	6.94608e8	1.37011	0.685056	−	0.728491i	$-0.259778\pi$
$307$	6.94608e8	1.37011	0.685056	+	0.728491i	$0.259778\pi$
$308$	7.52691e7	0.146788
$309$	−1.13101e8	−0.218077
$310$	9.31998e7	0.177684
$311$	1.27745e8	0.240815	0.120407	−	0.992725i	$-0.461580\pi$
$311$	1.27745e8	0.240815	0.120407	+	0.992725i	$0.461580\pi$
$312$	−3.89311e8	−0.725697
$313$	−9.74255e8	−1.79584	−0.897920	−	0.440159i	$-0.854922\pi$
$313$	−9.74255e8	−1.79584	−0.897920	+	0.440159i	$0.854922\pi$
$314$	2.04795e8	0.373306
$315$	1.22611e7	0.0221026
$316$	1.45441e8	0.259288
$317$	−8.99019e8	−1.58512	−0.792559	−	0.609795i	$-0.791252\pi$
$317$	−8.99019e8	−1.58512	−0.792559	+	0.609795i	$0.791252\pi$
$318$	−3.62720e7	−0.0632524
$319$	−5.12230e8	−0.883482
$320$	896215.	0.00152893
$321$	−3.24733e8	−0.547972
$322$	4.58282e7	0.0764957
$323$	−1.09143e9	−1.80213
$324$	−5.46418e7	−0.0892519
$325$	−8.97643e8	−1.45048
$326$	−3.74814e8	−0.599176
$327$	−5.29264e8	−0.837058
$328$	6.21708e8	0.972810
$329$	1.27379e8	0.197203
$330$	−3.54833e7	−0.0543531
$331$	−1.16605e9	−1.76733	−0.883666	−	0.468119i	$-0.844932\pi$
$331$	−1.16605e9	−1.76733	−0.883666	+	0.468119i	$0.844932\pi$
$332$	−3.82733e8	−0.574002
$333$	−2.99731e8	−0.444812
$334$	−2.86750e8	−0.421105
$335$	−2.22734e8	−0.323690
$336$	−4.30198e7	−0.0618702
$337$	1.23311e9	1.75509	0.877544	−	0.479496i	$-0.159181\pi$
$337$	1.23311e9	1.75509	0.877544	+	0.479496i	$0.159181\pi$
$338$	4.62765e8	0.651857
$339$	−2.03827e8	−0.284160
$340$	−2.10250e8	−0.290108
$341$	−8.08379e8	−1.10401
$342$	1.51456e8	0.204736
$343$	3.46941e8	0.464223
$344$	−6.75541e8	−0.894741
$345$	8.82112e7	0.115653
$346$	3.38233e8	0.438986
$347$	−5.89436e8	−0.757327	−0.378664	−	0.925534i	$-0.623616\pi$
$347$	−5.89436e8	−0.757327	−0.378664	+	0.925534i	$0.623616\pi$
$348$	4.21182e8	0.535725
$349$	1.15878e9	1.45919	0.729596	−	0.683878i	$-0.239708\pi$
$349$	1.15878e9	1.45919	0.729596	+	0.683878i	$0.239708\pi$
$350$	7.84592e7	0.0978150
$351$	2.45025e8	0.302438
$352$	6.25054e8	0.763868
$353$	−5.26080e8	−0.636561	−0.318281	−	0.947997i	$-0.603105\pi$
$353$	−5.26080e8	−0.636561	−0.318281	+	0.947997i	$0.603105\pi$
$354$	−2.78268e7	−0.0333389
$355$	−8.39110e7	−0.0995450
$356$	−1.07698e9	−1.26512
$357$	1.54333e8	0.179523
$358$	2.71129e8	0.312310
$359$	1.09995e9	1.25470	0.627351	−	0.778737i	$-0.284139\pi$
$359$	1.09995e9	1.25470	0.627351	+	0.778737i	$0.284139\pi$
$360$	6.54979e7	0.0739893
$361$	8.20202e8	0.917583
$362$	−1.49636e8	−0.165789
$363$	−2.18385e8	−0.239635
$364$	2.77528e8	0.301614
$365$	368180.	0.000396310 0
$366$	−6.75856e7	−0.0720561
$367$	1.21755e9	1.28575	0.642874	−	0.765972i	$-0.277742\pi$
$367$	1.21755e9	1.28575	0.642874	+	0.765972i	$0.277742\pi$
$368$	−3.09501e8	−0.323738
$369$	−3.91292e8	−0.405423
$370$	1.60042e8	0.164258
$371$	5.80473e7	0.0590164
$372$	6.64690e8	0.669451
$373$	5.53595e8	0.552346	0.276173	−	0.961108i	$-0.410934\pi$
$373$	5.53595e8	0.552346	0.276173	+	0.961108i	$0.410934\pi$
$374$	−4.46635e8	−0.441471
$375$	3.14642e8	0.308111
$376$	6.80449e8	0.660143
$377$	−1.88866e9	−1.81535
$378$	−2.14166e7	−0.0203953
$379$	−2.47132e8	−0.233180	−0.116590	−	0.993180i	$-0.537196\pi$
$379$	−2.47132e8	−0.233180	−0.116590	+	0.993180i	$0.537196\pi$
$380$	3.30196e8	0.308695
$381$	−3.67367e8	−0.340300
$382$	−5.17514e8	−0.475007
$383$	−599846.	−0.000545562 0	−0.000272781	−	1.00000i	$-0.500087\pi$
$383$	−599846.	−0.000545562 0	−0.000272781		1.00000i	$0.500087\pi$
$384$	−6.41391e8	−0.578047
$385$	5.67850e7	0.0507132
$386$	8.64541e8	0.765121
$387$	4.25174e8	0.372888
$388$	4.64001e8	0.403281
$389$	−8.92101e8	−0.768406	−0.384203	−	0.923249i	$-0.625524\pi$
$389$	−8.92101e8	−0.768406	−0.384203	+	0.923249i	$0.625524\pi$
$390$	−1.30832e8	−0.111683
$391$	1.11033e9	0.939364
$392$	8.99436e8	0.754170
$393$	−3.49432e8	−0.290395
$394$	−6.40327e8	−0.527430
$395$	1.09724e8	0.0895805
$396$	−2.53062e8	−0.204783
$397$	−9.58791e8	−0.769055	−0.384527	−	0.923114i	$-0.625636\pi$
$397$	−9.58791e8	−0.769055	−0.384527	+	0.923114i	$0.625636\pi$
$398$	−3.24872e8	−0.258298
$399$	−2.42379e8	−0.191025
$400$	−5.29874e8	−0.413964
$401$	1.82830e9	1.41593	0.707966	−	0.706247i	$-0.249613\pi$
$401$	1.82830e9	1.41593	0.707966	+	0.706247i	$0.249613\pi$
$402$	3.89051e8	0.298686
$403$	−2.98061e9	−2.26849
$404$	1.75442e9	1.32373
$405$	−4.12232e7	−0.0308354
$406$	1.65080e8	0.122420
$407$	−1.38814e9	−1.02059
$408$	8.24435e8	0.600960
$409$	−7.15047e8	−0.516776	−0.258388	−	0.966041i	$-0.583191\pi$
$409$	−7.15047e8	−0.516776	−0.258388	+	0.966041i	$0.583191\pi$
$410$	2.08931e8	0.149713
$411$	−2.06784e8	−0.146916
$412$	4.30697e8	0.303411
$413$	4.45320e7	0.0311062
$414$	−1.54079e8	−0.106719
$415$	−2.88744e8	−0.198310
$416$	2.30466e9	1.56957
$417$	1.31668e8	0.0889212
$418$	7.01435e8	0.469754
$419$	2.22875e9	1.48017	0.740086	−	0.672513i	$-0.234785\pi$
$419$	2.22875e9	1.48017	0.740086	+	0.672513i	$0.234785\pi$
$420$	−4.66914e7	−0.0307514
$421$	1.49109e9	0.973906	0.486953	−	0.873428i	$-0.338108\pi$
$421$	1.49109e9	0.973906	0.486953	+	0.873428i	$0.338108\pi$
$422$	7.15419e8	0.463412
$423$	−4.28262e8	−0.275118
$424$	3.10083e8	0.197559
$425$	1.90092e9	1.20116
$426$	1.46568e8	0.0918555
$427$	1.08159e8	0.0672306
$428$	1.23661e9	0.762394
$429$	1.13478e9	0.693925
$430$	−2.27022e8	−0.137698
$431$	2.66273e9	1.60198	0.800988	−	0.598680i	$-0.204308\pi$
$431$	2.66273e9	1.60198	0.800988	+	0.598680i	$0.204308\pi$
$432$	1.44637e8	0.0863151
$433$	7.09435e7	0.0419957	0.0209979	−	0.999780i	$-0.493316\pi$
$433$	7.09435e7	0.0419957	0.0209979	+	0.999780i	$0.493316\pi$
$434$	2.60522e8	0.152979
$435$	3.17750e8	0.185086
$436$	2.01548e9	1.16460
$437$	−1.74376e9	−0.999546
$438$	−643103.	−0.000365697 0
$439$	6.26733e7	0.0353555	0.0176777	−	0.999844i	$-0.494373\pi$
$439$	6.26733e7	0.0353555	0.0176777	+	0.999844i	$0.494373\pi$
$440$	3.03340e8	0.169764
$441$	−5.66089e8	−0.314304
$442$	−1.64680e9	−0.907119
$443$	−6.41984e8	−0.350842	−0.175421	−	0.984494i	$-0.556129\pi$
$443$	−6.41984e8	−0.350842	−0.175421	+	0.984494i	$0.556129\pi$
$444$	1.14140e9	0.618867
$445$	−8.12500e8	−0.437082
$446$	1.18167e9	0.630703
$447$	2.83661e7	0.0150218
$448$	2.50520e6	0.00131634
$449$	−2.77507e9	−1.44681	−0.723406	−	0.690423i	$-0.757425\pi$
$449$	−2.77507e9	−1.44681	−0.723406	+	0.690423i	$0.757425\pi$
$450$	−2.63788e8	−0.136462
$451$	−1.81219e9	−0.930219
$452$	7.76191e8	0.395352
$453$	1.57048e9	0.793762
$454$	1.68184e9	0.843506
$455$	2.09374e8	0.104204
$456$	−1.29477e9	−0.639462
$457$	3.79393e9	1.85944	0.929721	−	0.368266i	$-0.120048\pi$
$457$	3.79393e9	1.85944	0.929721	+	0.368266i	$0.120048\pi$
$458$	−8.44209e8	−0.410602
$459$	−5.18885e8	−0.250453
$460$	−3.35916e8	−0.160908
$461$	−3.74187e9	−1.77884	−0.889418	−	0.457095i	$-0.848890\pi$
$461$	−3.74187e9	−1.77884	−0.889418	+	0.457095i	$0.848890\pi$
$462$	−9.91867e7	−0.0467958
$463$	1.68652e9	0.789691	0.394846	−	0.918748i	$-0.370798\pi$
$463$	1.68652e9	0.789691	0.394846	+	0.918748i	$0.370798\pi$
$464$	−1.11487e9	−0.518097
$465$	5.01459e8	0.231287
$466$	−1.38579e9	−0.634376
$467$	−1.90310e8	−0.0864675	−0.0432338	−	0.999065i	$-0.513766\pi$
$467$	−1.90310e8	−0.0864675	−0.0432338	+	0.999065i	$0.513766\pi$
$468$	−9.33077e8	−0.420782
$469$	−6.22610e8	−0.278684
$470$	2.28671e8	0.101594
$471$	1.10189e9	0.485922
$472$	2.37886e8	0.104129
$473$	1.96910e9	0.855568
$474$	−1.91656e8	−0.0826607
$475$	−2.98537e9	−1.27812
$476$	−5.87714e8	−0.249771
$477$	−1.95161e8	−0.0823338
$478$	8.27840e8	0.346696
$479$	1.04486e9	0.434395	0.217197	−	0.976128i	$-0.430308\pi$
$479$	1.04486e9	0.434395	0.217197	+	0.976128i	$0.430308\pi$
$480$	−3.87738e8	−0.160027
$481$	−5.11827e9	−2.09708
$482$	1.10301e9	0.448656
$483$	2.46578e8	0.0995723
$484$	8.31630e8	0.333404
$485$	3.50054e8	0.139328
$486$	7.20049e7	0.0284534
$487$	−4.52457e9	−1.77511	−0.887557	−	0.460697i	$-0.847599\pi$
$487$	−4.52457e9	−1.77511	−0.887557	+	0.460697i	$0.847599\pi$
$488$	5.77778e8	0.225056
$489$	−2.01668e9	−0.779930
$490$	3.02264e8	0.116065
$491$	1.28164e9	0.488630	0.244315	−	0.969696i	$-0.421437\pi$
$491$	1.28164e9	0.488630	0.244315	+	0.969696i	$0.421437\pi$
$492$	1.49007e9	0.564065
$493$	3.99958e9	1.50332
$494$	2.58629e9	0.965235
$495$	−1.90917e8	−0.0707499
$496$	−1.75944e9	−0.647423
$497$	−2.34557e8	−0.0857041
$498$	5.04351e8	0.182991
$499$	−3.26234e9	−1.17538	−0.587689	−	0.809087i	$-0.699962\pi$
$499$	−3.26234e9	−1.17538	−0.587689	+	0.809087i	$0.699962\pi$
$500$	−1.19818e9	−0.428674
$501$	−1.54285e9	−0.548140
$502$	−1.32841e9	−0.468674
$503$	5.11806e8	0.179315	0.0896577	−	0.995973i	$-0.471423\pi$
$503$	5.11806e8	0.179315	0.0896577	+	0.995973i	$0.471423\pi$
$504$	1.83087e8	0.0637016
$505$	1.32358e9	0.457330
$506$	−7.13586e8	−0.244861
$507$	2.48990e9	0.848504
$508$	1.39896e9	0.473460
$509$	5.38317e9	1.80936	0.904682	−	0.426088i	$-0.140108\pi$
$509$	5.38317e9	1.80936	0.904682	+	0.426088i	$0.140108\pi$
$510$	2.77059e8	0.0924863
$511$	1.02918e6	0.000341206 0
$512$	2.44989e9	0.806681
$513$	8.14903e8	0.266499
$514$	−1.20322e9	−0.390819
$515$	3.24929e8	0.104825
$516$	−1.61910e9	−0.518799
$517$	−1.98341e9	−0.631241
$518$	4.47366e8	0.141419
$519$	1.81986e9	0.571415
$520$	1.11846e9	0.348825
$521$	1.92620e9	0.596720	0.298360	−	0.954453i	$-0.403560\pi$
$521$	1.92620e9	0.596720	0.298360	+	0.954453i	$0.403560\pi$
$522$	−5.55016e8	−0.170789
$523$	3.47164e9	1.06115	0.530577	−	0.847637i	$-0.321975\pi$
$523$	3.47164e9	1.06115	0.530577	+	0.847637i	$0.321975\pi$
$524$	1.33067e9	0.404026
$525$	4.22148e8	0.127323
$526$	6.40623e8	0.191934
$527$	6.31197e9	1.87857
$528$	6.69857e8	0.198045
$529$	−1.63086e9	−0.478984
$530$	1.04206e8	0.0304039
$531$	−1.49721e8	−0.0433963
$532$	9.23000e8	0.265773
$533$	−6.68179e9	−1.91138
$534$	1.41920e9	0.403319
$535$	9.32930e8	0.263397
$536$	−3.32593e9	−0.932902
$537$	1.45880e9	0.406525
$538$	9.94645e8	0.275379
$539$	−2.62173e9	−0.721152
$540$	1.56981e8	0.0429013
$541$	−1.13960e9	−0.309430	−0.154715	−	0.987959i	$-0.549446\pi$
$541$	−1.13960e9	−0.309430	−0.154715	+	0.987959i	$0.549446\pi$
$542$	5.87919e8	0.158606
$543$	−8.05114e8	−0.215803
$544$	−4.88053e9	−1.29978
$545$	1.52053e9	0.402353
$546$	−3.65715e8	−0.0961543
$547$	−3.40030e9	−0.888303	−0.444152	−	0.895952i	$-0.646495\pi$
$547$	−3.40030e9	−0.888303	−0.444152	+	0.895952i	$0.646495\pi$
$548$	7.87449e8	0.204405
$549$	−3.63643e8	−0.0937933
$550$	−1.22168e9	−0.313103
$551$	−6.28131e9	−1.59963
$552$	1.31719e9	0.333321
$553$	3.06714e8	0.0771250
$554$	1.40509e9	0.351091
$555$	8.61101e8	0.213810
$556$	−5.01404e8	−0.123716
$557$	1.94268e9	0.476331	0.238165	−	0.971225i	$-0.423454\pi$
$557$	1.94268e9	0.476331	0.238165	+	0.971225i	$0.423454\pi$
$558$	−8.75903e8	−0.213420
$559$	7.26036e9	1.75799
$560$	1.23592e8	0.0297395
$561$	−2.40311e9	−0.574650
$562$	−3.09468e8	−0.0735427
$563$	6.16858e9	1.45682	0.728410	−	0.685142i	$-0.240260\pi$
$563$	6.16858e9	1.45682	0.728410	+	0.685142i	$0.240260\pi$
$564$	1.63086e9	0.382771
$565$	5.85579e8	0.136589
$566$	−3.11331e9	−0.721713
$567$	−1.15232e8	−0.0265480
$568$	−1.25298e9	−0.286897
$569$	−5.43476e9	−1.23677	−0.618383	−	0.785877i	$-0.712212\pi$
$569$	−5.43476e9	−1.23677	−0.618383	+	0.785877i	$0.712212\pi$
$570$	−4.35119e8	−0.0984116
$571$	−5.20959e9	−1.17105	−0.585527	−	0.810653i	$-0.699112\pi$
$571$	−5.20959e9	−1.17105	−0.585527	+	0.810653i	$0.699112\pi$
$572$	−4.32135e9	−0.965458
$573$	−2.78447e9	−0.618303
$574$	5.84027e8	0.128897
$575$	3.03709e9	0.666223
$576$	−8.42274e6	−0.00183643
$577$	2.67350e9	0.579381	0.289691	−	0.957120i	$-0.406448\pi$
$577$	2.67350e9	0.579381	0.289691	+	0.957120i	$0.406448\pi$
$578$	1.42826e9	0.307653
$579$	4.65164e9	0.995936
$580$	−1.21002e9	−0.257510
$581$	−8.07129e8	−0.170737
$582$	−6.11443e8	−0.128566
$583$	−9.03847e8	−0.188910
$584$	5.49777e6	0.00114220
$585$	−7.03937e8	−0.145375
$586$	−3.74922e9	−0.769660
$587$	4.23836e9	0.864898	0.432449	−	0.901658i	$-0.357650\pi$
$587$	4.23836e9	0.864898	0.432449	+	0.901658i	$0.357650\pi$
$588$	2.15571e9	0.437291
$589$	−9.91288e9	−1.99892
$590$	7.99440e7	0.0160252
$591$	−3.44526e9	−0.686541
$592$	−3.02129e9	−0.598503
$593$	−5.59276e9	−1.10137	−0.550687	−	0.834712i	$-0.685634\pi$
$593$	−5.59276e9	−1.10137	−0.550687	+	0.834712i	$0.685634\pi$
$594$	3.33476e8	0.0652847
$595$	−4.43387e8	−0.0862926
$596$	−1.08020e8	−0.0208999
$597$	−1.74797e9	−0.336220
$598$	−2.63109e9	−0.503132
$599$	−2.76072e9	−0.524842	−0.262421	−	0.964953i	$-0.584521\pi$
$599$	−2.76072e9	−0.524842	−0.262421	+	0.964953i	$0.584521\pi$
$600$	2.25507e9	0.426218
$601$	−3.38464e9	−0.635992	−0.317996	−	0.948092i	$-0.603010\pi$
$601$	−3.38464e9	−0.635992	−0.317996	+	0.948092i	$0.603010\pi$
$602$	−6.34598e8	−0.118552
$603$	2.09328e9	0.388791
$604$	−5.98054e9	−1.10436
$605$	6.27403e8	0.115187
$606$	−2.31190e9	−0.422003
$607$	3.57949e9	0.649622	0.324811	−	0.945779i	$-0.394699\pi$
$607$	3.57949e9	0.649622	0.324811	+	0.945779i	$0.394699\pi$
$608$	7.66483e9	1.38306
$609$	8.88210e8	0.159351
$610$	1.94168e8	0.0346356
$611$	−7.31310e9	−1.29705
$612$	1.97596e9	0.348455
$613$	9.92114e9	1.73960	0.869801	−	0.493403i	$-0.164247\pi$
$613$	9.92114e9	1.73960	0.869801	+	0.493403i	$0.164247\pi$
$614$	3.48564e9	0.607706
$615$	1.12415e9	0.194877
$616$	8.47929e8	0.146160
$617$	−2.67237e9	−0.458034	−0.229017	−	0.973422i	$-0.573551\pi$
$617$	−2.67237e9	−0.458034	−0.229017	+	0.973422i	$0.573551\pi$
$618$	−5.67556e8	−0.0967272
$619$	5.77104e9	0.977996	0.488998	−	0.872285i	$-0.337363\pi$
$619$	5.77104e9	0.977996	0.488998	+	0.872285i	$0.337363\pi$
$620$	−1.90960e9	−0.321789
$621$	−8.29019e8	−0.138913
$622$	6.41044e8	0.106812
$623$	−2.27119e9	−0.376309
$624$	2.46986e9	0.406936
$625$	4.72951e9	0.774882
$626$	−4.88895e9	−0.796536
$627$	3.77406e9	0.611466
$628$	−4.19610e9	−0.676064
$629$	1.08388e10	1.73662
$630$	6.15282e7	0.00980352
$631$	−1.33017e9	−0.210768	−0.105384	−	0.994432i	$-0.533607\pi$
$631$	−1.33017e9	−0.210768	−0.105384	+	0.994432i	$0.533607\pi$
$632$	1.63844e9	0.258178
$633$	3.84930e9	0.603210
$634$	−4.51141e9	−0.703072
$635$	1.05541e9	0.163574
$636$	7.43188e8	0.114551
$637$	−9.66667e9	−1.48180
$638$	−2.57044e9	−0.391864
$639$	7.88605e8	0.119566
$640$	1.84266e9	0.277853
$641$	8.97295e8	0.134565	0.0672825	−	0.997734i	$-0.478567\pi$
$641$	8.97295e8	0.134565	0.0672825	+	0.997734i	$0.478567\pi$
$642$	−1.62956e9	−0.243051
$643$	7.92605e9	1.17576	0.587879	−	0.808949i	$-0.299963\pi$
$643$	7.92605e9	1.17576	0.587879	+	0.808949i	$0.299963\pi$
$644$	−9.38988e8	−0.138535
$645$	−1.22149e9	−0.179238
$646$	−5.47693e9	−0.799325
$647$	−9.72853e9	−1.41215	−0.706077	−	0.708135i	$-0.749537\pi$
$647$	−9.72853e9	−1.41215	−0.706077	+	0.708135i	$0.749537\pi$
$648$	−6.15557e8	−0.0888701
$649$	−6.93403e8	−0.0995702
$650$	−4.50450e9	−0.643354
$651$	1.40173e9	0.199128
$652$	7.67968e9	1.08512
$653$	−4.58349e9	−0.644170	−0.322085	−	0.946711i	$-0.604384\pi$
$653$	−4.58349e9	−0.644170	−0.322085	+	0.946711i	$0.604384\pi$
$654$	−2.65592e9	−0.371273
$655$	1.00389e9	0.139586
$656$	−3.94423e9	−0.545505
$657$	−3.46020e6	−0.000476017 0
$658$	6.39208e8	0.0874684
$659$	−1.15331e9	−0.156982	−0.0784908	−	0.996915i	$-0.525010\pi$
$659$	−1.15331e9	−0.156982	−0.0784908	+	0.996915i	$0.525010\pi$
$660$	7.27027e8	0.0984344
$661$	−1.03408e8	−0.0139267	−0.00696334	−	0.999976i	$-0.502217\pi$
$661$	−1.03408e8	−0.0139267	−0.00696334	+	0.999976i	$0.502217\pi$
$662$	−5.85139e9	−0.783892
$663$	−8.86059e9	−1.18077
$664$	−4.31161e9	−0.571546
$665$	6.96335e8	0.0918211
$666$	−1.50409e9	−0.197294
$667$	6.39011e9	0.833812
$668$	5.87530e9	0.762628
$669$	6.35796e9	0.820968
$670$	−1.11771e9	−0.143571
$671$	−1.68414e9	−0.215203
$672$	−1.08385e9	−0.137777
$673$	−8.65901e9	−1.09500	−0.547502	−	0.836804i	$-0.684421\pi$
$673$	−8.65901e9	−1.09500	−0.547502	+	0.836804i	$0.684421\pi$
$674$	6.18795e9	0.778461
$675$	−1.41930e9	−0.177628
$676$	−9.48174e9	−1.18052
$677$	1.41759e10	1.75586	0.877930	−	0.478789i	$-0.158924\pi$
$677$	1.41759e10	1.75586	0.877930	+	0.478789i	$0.158924\pi$
$678$	−1.02283e9	−0.126038
$679$	9.78511e8	0.119956
$680$	−2.36853e9	−0.288867
$681$	9.04908e9	1.09797
$682$	−4.05656e9	−0.489681
$683$	2.10450e9	0.252742	0.126371	−	0.991983i	$-0.459667\pi$
$683$	2.10450e9	0.252742	0.126371	+	0.991983i	$0.459667\pi$
$684$	−3.10322e9	−0.370780
$685$	5.94072e8	0.0706191
$686$	1.74100e9	0.205904
$687$	−4.54225e9	−0.534469
$688$	4.28575e9	0.501727
$689$	−3.33261e9	−0.388166
$690$	4.42656e8	0.0512973
$691$	8.19673e9	0.945078	0.472539	−	0.881310i	$-0.343338\pi$
$691$	8.19673e9	0.945078	0.472539	+	0.881310i	$0.343338\pi$
$692$	−6.93017e9	−0.795010
$693$	−5.33672e8	−0.0609127
$694$	−2.95787e9	−0.335909
$695$	−3.78272e8	−0.0427423
$696$	4.74474e9	0.533433
$697$	1.41499e10	1.58284
$698$	5.81493e9	0.647218
$699$	−7.45621e9	−0.825749
$700$	−1.60757e9	−0.177145
$701$	−1.58794e9	−0.174109	−0.0870547	−	0.996204i	$-0.527745\pi$
$701$	−1.58794e9	−0.174109	−0.0870547	+	0.996204i	$0.527745\pi$
$702$	1.22957e9	0.134145
$703$	−1.70223e10	−1.84788
$704$	−3.90082e7	−0.00421358
$705$	1.23036e9	0.132242
$706$	−2.63994e9	−0.282344
$707$	3.69981e9	0.393741
$708$	5.70151e8	0.0603773
$709$	−1.39994e10	−1.47519	−0.737596	−	0.675242i	$-0.764039\pi$
$709$	−1.39994e10	−1.47519	−0.737596	+	0.675242i	$0.764039\pi$
$710$	−4.21077e8	−0.0441527
$711$	−1.03120e9	−0.107597
$712$	−1.21325e10	−1.25971
$713$	1.00846e10	1.04195
$714$	7.74467e8	0.0796268
$715$	−3.26014e9	−0.333553
$716$	−5.55525e9	−0.565598
$717$	4.45417e9	0.451284
$718$	5.51968e9	0.556517
$719$	−3.63177e8	−0.0364391	−0.0182195	−	0.999834i	$-0.505800\pi$
$719$	−3.63177e8	−0.0364391	−0.0182195	+	0.999834i	$0.505800\pi$
$720$	−4.15530e8	−0.0414896
$721$	9.08277e8	0.0902495
$722$	4.11589e9	0.406990
$723$	5.93470e9	0.584002
$724$	3.06594e9	0.300247
$725$	1.09401e10	1.06619
$726$	−1.09589e9	−0.106289
$727$	−7.18128e9	−0.693157	−0.346578	−	0.938021i	$-0.612657\pi$
$727$	−7.18128e9	−0.693157	−0.346578	+	0.938021i	$0.612657\pi$
$728$	3.12643e9	0.300324
$729$	3.87420e8	0.0370370
$730$	1.84758e6	0.000175782 0
$731$	−1.53751e10	−1.45582
$732$	1.38478e9	0.130495
$733$	1.12100e10	1.05133	0.525667	−	0.850691i	$-0.323816\pi$
$733$	1.12100e10	1.05133	0.525667	+	0.850691i	$0.323816\pi$
$734$	6.10985e9	0.570288
$735$	1.62633e9	0.151078
$736$	−7.79760e9	−0.720922
$737$	9.69459e9	0.892059
$738$	−1.96356e9	−0.179824
$739$	−2.61736e9	−0.238566	−0.119283	−	0.992860i	$-0.538060\pi$
$739$	−2.61736e9	−0.238566	−0.119283	+	0.992860i	$0.538060\pi$
$740$	−3.27914e9	−0.297474
$741$	1.39155e10	1.25642
$742$	2.91289e8	0.0261765
$743$	−2.16513e10	−1.93653	−0.968264	−	0.249930i	$-0.919593\pi$
$743$	−2.16513e10	−1.93653	−0.968264	+	0.249930i	$0.919593\pi$
$744$	7.48794e9	0.666587
$745$	−8.14933e7	−0.00722063
$746$	2.77802e9	0.244991
$747$	2.71365e9	0.238194
$748$	9.15123e9	0.799510
$749$	2.60783e9	0.226774
$750$	1.57892e9	0.136661
$751$	−1.23735e10	−1.06599	−0.532995	−	0.846118i	$-0.678934\pi$
$751$	−1.23735e10	−1.06599	−0.532995	+	0.846118i	$0.678934\pi$
$752$	−4.31689e9	−0.370176
$753$	−7.14750e9	−0.610059
$754$	−9.47759e9	−0.805190
$755$	−4.51187e9	−0.381542
$756$	4.38812e8	0.0369362
$757$	−1.68097e9	−0.140840	−0.0704198	−	0.997517i	$-0.522434\pi$
$757$	−1.68097e9	−0.140840	−0.0704198	+	0.997517i	$0.522434\pi$
$758$	−1.24014e9	−0.103426
$759$	−3.83943e9	−0.318728
$760$	3.71976e9	0.307374
$761$	1.31161e10	1.07884	0.539421	−	0.842036i	$-0.318643\pi$
$761$	1.31161e10	1.07884	0.539421	+	0.842036i	$0.318643\pi$
$762$	−1.84350e9	−0.150939
$763$	4.25036e9	0.346409
$764$	1.06035e10	0.860246
$765$	1.49071e9	0.120387
$766$	−3.01011e6	−0.000241981 0
$767$	−2.55668e9	−0.204593
$768$	−3.17866e9	−0.253209
$769$	1.42811e10	1.13245	0.566224	−	0.824251i	$-0.308404\pi$
$769$	1.42811e10	1.13245	0.566224	+	0.824251i	$0.308404\pi$
$770$	2.84955e8	0.0224936
$771$	−6.47392e9	−0.508718
$772$	−1.77138e10	−1.38565
$773$	−8.34783e9	−0.650048	−0.325024	−	0.945706i	$-0.605372\pi$
$773$	−8.34783e9	−0.650048	−0.325024	+	0.945706i	$0.605372\pi$
$774$	2.13358e9	0.165393
$775$	1.72651e10	1.33233
$776$	5.22711e9	0.401556
$777$	2.40704e9	0.184082
$778$	−4.47669e9	−0.340823
$779$	−2.22223e10	−1.68425
$780$	2.68065e9	0.202260
$781$	3.65226e9	0.274336
$782$	5.57180e9	0.416651
$783$	−2.98625e9	−0.222311
$784$	−5.70618e9	−0.422902
$785$	−3.16565e9	−0.233571
$786$	−1.75350e9	−0.128803
$787$	1.47026e10	1.07518	0.537591	−	0.843206i	$-0.319334\pi$
$787$	1.47026e10	1.07518	0.537591	+	0.843206i	$0.319334\pi$
$788$	1.31199e10	0.955184
$789$	3.44685e9	0.249835
$790$	5.50613e8	0.0397330
$791$	1.63687e9	0.117597
$792$	−2.85083e9	−0.203907
$793$	−6.20965e9	−0.442192
$794$	−4.81135e9	−0.341111
$795$	5.60681e8	0.0395759
$796$	6.65640e9	0.467782
$797$	3.94070e9	0.275721	0.137860	−	0.990452i	$-0.455977\pi$
$797$	3.94070e9	0.275721	0.137860	+	0.990452i	$0.455977\pi$
$798$	−1.21629e9	−0.0847282
$799$	1.54868e10	1.07411
$800$	−1.33497e10	−0.921843
$801$	7.63597e9	0.524989
$802$	9.17467e9	0.628030
$803$	−1.60252e7	−0.00109219
$804$	−7.97138e9	−0.540925
$805$	−7.08397e8	−0.0478620
$806$	−1.49571e10	−1.00618
$807$	5.35167e9	0.358452
$808$	1.97640e10	1.31806
$809$	2.12121e10	1.40852	0.704260	−	0.709942i	$-0.251279\pi$
$809$	2.12121e10	1.40852	0.704260	+	0.709942i	$0.251279\pi$
$810$	−2.06864e8	−0.0136769
$811$	4.13209e9	0.272018	0.136009	−	0.990708i	$-0.456572\pi$
$811$	4.13209e9	0.272018	0.136009	+	0.990708i	$0.456572\pi$
$812$	−3.38238e9	−0.221705
$813$	3.16329e9	0.206453
$814$	−6.96589e9	−0.452680
$815$	5.79375e9	0.374893
$816$	−5.23036e9	−0.336989
$817$	2.41465e10	1.54909
$818$	−3.58821e9	−0.229214
$819$	−1.96772e9	−0.125161
$820$	−4.28085e9	−0.271133
$821$	4.99283e9	0.314880	0.157440	−	0.987529i	$-0.449676\pi$
$821$	4.99283e9	0.314880	0.157440	+	0.987529i	$0.449676\pi$
$822$	−1.03767e9	−0.0651640
$823$	8.56606e9	0.535650	0.267825	−	0.963468i	$-0.413695\pi$
$823$	8.56606e9	0.535650	0.267825	+	0.963468i	$0.413695\pi$
$824$	4.85193e9	0.302113
$825$	−6.57321e9	−0.407557
$826$	2.23468e8	0.0137970
$827$	2.19884e10	1.35184	0.675920	−	0.736975i	$-0.263747\pi$
$827$	2.19884e10	1.35184	0.675920	+	0.736975i	$0.263747\pi$
$828$	3.15697e9	0.193270
$829$	−1.74819e9	−0.106573	−0.0532867	−	0.998579i	$-0.516970\pi$
$829$	−1.74819e9	−0.106573	−0.0532867	+	0.998579i	$0.516970\pi$
$830$	−1.44896e9	−0.0879595
$831$	7.56004e9	0.457005
$832$	−1.43829e8	−0.00865792
$833$	2.04709e10	1.22710
$834$	6.60730e8	0.0394406
$835$	4.43248e9	0.263478
$836$	−1.43719e10	−0.850732
$837$	−4.71277e9	−0.277803
$838$	1.11842e10	0.656523
$839$	7.57544e8	0.0442834	0.0221417	−	0.999755i	$-0.492952\pi$
$839$	7.57544e8	0.0442834	0.0221417	+	0.999755i	$0.492952\pi$
$840$	−5.25993e8	−0.0306198
$841$	5.76829e9	0.334396
$842$	7.48251e9	0.431972
$843$	−1.66509e9	−0.0957284
$844$	−1.46584e10	−0.839246
$845$	−7.15327e9	−0.407855
$846$	−2.14908e9	−0.122027
$847$	1.75379e9	0.0991710
$848$	−1.96722e9	−0.110782
$849$	−1.67511e10	−0.939433
$850$	9.53908e9	0.532771
$851$	1.73172e10	0.963215
$852$	−3.00307e9	−0.166352
$853$	−1.72683e9	−0.0952636	−0.0476318	−	0.998865i	$-0.515167\pi$
$853$	−1.72683e9	−0.0952636	−0.0476318	+	0.998865i	$0.515167\pi$
$854$	5.42759e8	0.0298198
$855$	−2.34115e9	−0.128100
$856$	1.39308e10	0.759132
$857$	−3.43055e10	−1.86179	−0.930895	−	0.365286i	$-0.880971\pi$
$857$	−3.43055e10	−1.86179	−0.930895	+	0.365286i	$0.880971\pi$
$858$	5.69451e9	0.307787
$859$	−1.16013e10	−0.624499	−0.312249	−	0.950000i	$-0.601082\pi$
$859$	−1.16013e10	−0.624499	−0.312249	+	0.950000i	$0.601082\pi$
$860$	4.65153e9	0.249374
$861$	3.14234e9	0.167781
$862$	1.33619e10	0.710549
$863$	1.06656e10	0.564870	0.282435	−	0.959286i	$-0.408858\pi$
$863$	1.06656e10	0.564870	0.282435	+	0.959286i	$0.408858\pi$
$864$	3.64401e9	0.192212
$865$	−5.22830e9	−0.274665
$866$	3.56005e8	0.0186270
$867$	7.68474e9	0.400463
$868$	−5.33792e9	−0.277047
$869$	−4.77580e9	−0.246875
$870$	1.59452e9	0.0820940
$871$	3.57453e10	1.83297
$872$	2.27050e10	1.15962
$873$	−3.28985e9	−0.167350
$874$	−8.75046e9	−0.443344
$875$	−2.52679e9	−0.127509
$876$	1.31767e7	0.000662282 0
$877$	−5.25940e9	−0.263292	−0.131646	−	0.991297i	$-0.542026\pi$
$877$	−5.25940e9	−0.263292	−0.131646	+	0.991297i	$0.542026\pi$
$878$	3.14503e8	0.0156817
$879$	−2.01726e10	−1.00184
$880$	−1.92444e9	−0.0951954
$881$	−2.79482e10	−1.37701	−0.688507	−	0.725230i	$-0.741734\pi$
$881$	−2.79482e10	−1.37701	−0.688507	+	0.725230i	$0.741734\pi$
$882$	−2.84072e9	−0.139408
$883$	3.72423e9	0.182043	0.0910214	−	0.995849i	$-0.470987\pi$
$883$	3.72423e9	0.182043	0.0910214	+	0.995849i	$0.470987\pi$
$884$	3.37419e10	1.64281
$885$	4.30137e8	0.0208596
$886$	−3.22157e9	−0.155614
$887$	6.26596e9	0.301477	0.150739	−	0.988574i	$-0.451835\pi$
$887$	6.26596e9	0.301477	0.150739	+	0.988574i	$0.451835\pi$
$888$	1.28582e10	0.616219
$889$	2.95021e9	0.140830
$890$	−4.07724e9	−0.193866
$891$	1.79426e9	0.0849793
$892$	−2.42116e10	−1.14221
$893$	−2.43219e10	−1.14292
$894$	1.42345e8	0.00666286
$895$	−4.19102e9	−0.195407
$896$	5.15081e9	0.239220
$897$	−1.41565e10	−0.654912
$898$	−1.39257e10	−0.641727
$899$	3.63263e10	1.66748
$900$	5.40483e9	0.247134
$901$	7.05740e9	0.321446
$902$	−9.09382e9	−0.412594
$903$	−3.41444e9	−0.154316
$904$	8.74403e9	0.393661
$905$	2.31302e9	0.103731
$906$	7.88092e9	0.352069
$907$	8.21053e9	0.365381	0.182690	−	0.983170i	$-0.441519\pi$
$907$	8.21053e9	0.365381	0.182690	+	0.983170i	$0.441519\pi$
$908$	−3.44596e10	−1.52760
$909$	−1.24391e10	−0.549309
$910$	1.05067e9	0.0462191
$911$	−1.57505e10	−0.690207	−0.345103	−	0.938565i	$-0.612156\pi$
$911$	−1.57505e10	−0.690207	−0.345103	+	0.938565i	$0.612156\pi$
$912$	8.21423e9	0.358579
$913$	1.25677e10	0.546523
$914$	1.90385e10	0.824746
$915$	1.04472e9	0.0450842
$916$	1.72973e10	0.743607
$917$	2.80618e9	0.120177
$918$	−2.60384e9	−0.111087
$919$	−6.94140e9	−0.295014	−0.147507	−	0.989061i	$-0.547125\pi$
$919$	−6.94140e9	−0.295014	−0.147507	+	0.989061i	$0.547125\pi$
$920$	−3.78419e9	−0.160220
$921$	1.87544e10	0.791034
$922$	−1.87773e10	−0.788994
$923$	1.34664e10	0.563697
$924$	2.03227e9	0.0847478
$925$	2.96475e10	1.23166
$926$	8.46318e9	0.350264
$927$	−3.05372e9	−0.125907
$928$	−2.80881e10	−1.15373
$929$	−1.64084e9	−0.0671448	−0.0335724	−	0.999436i	$-0.510688\pi$
$929$	−1.64084e9	−0.0671448	−0.0335724	+	0.999436i	$0.510688\pi$
$930$	2.51639e9	0.102586
$931$	−3.21493e10	−1.30571
$932$	2.83939e10	1.14887
$933$	3.44912e9	0.139035
$934$	−9.55004e8	−0.0383523
$935$	6.90393e9	0.276220
$936$	−1.05114e10	−0.418981
$937$	3.84565e10	1.52715	0.763575	−	0.645719i	$-0.223442\pi$
$937$	3.84565e10	1.52715	0.763575	+	0.645719i	$0.223442\pi$
$938$	−3.12435e9	−0.123609
$939$	−2.63049e10	−1.03683
$940$	−4.68532e9	−0.183989
$941$	3.13483e10	1.22645	0.613226	−	0.789907i	$-0.289871\pi$
$941$	3.13483e10	1.22645	0.613226	+	0.789907i	$0.289871\pi$
$942$	5.52946e9	0.215529
$943$	2.26072e10	0.877921
$944$	−1.50919e9	−0.0583905
$945$	3.31051e8	0.0127610
$946$	9.88124e9	0.379483
$947$	−4.56816e10	−1.74790	−0.873950	−	0.486016i	$-0.838450\pi$
$947$	−4.56816e10	−1.74790	−0.873950	+	0.486016i	$0.838450\pi$
$948$	3.92691e9	0.149700
$949$	−5.90871e7	−0.00224420
$950$	−1.49810e10	−0.566903
$951$	−2.42735e10	−0.915168
$952$	−6.62078e9	−0.248702
$953$	−2.57557e10	−0.963937	−0.481968	−	0.876189i	$-0.660078\pi$
$953$	−2.57557e10	−0.963937	−0.481968	+	0.876189i	$0.660078\pi$
$954$	−9.79345e8	−0.0365188
$955$	7.99955e9	0.297204
$956$	−1.69619e10	−0.627872
$957$	−1.38302e10	−0.510079
$958$	5.24327e9	0.192674
$959$	1.66061e9	0.0608000
$960$	2.41978e7	0.000882728 0
$961$	2.98159e10	1.08372
$962$	−2.56842e10	−0.930151
$963$	−8.76779e9	−0.316372
$964$	−2.25998e10	−0.812522
$965$	−1.33638e10	−0.478723
$966$	1.23736e9	0.0441648
$967$	2.61784e10	0.931002	0.465501	−	0.885047i	$-0.345874\pi$
$967$	2.61784e10	0.931002	0.465501	+	0.885047i	$0.345874\pi$
$968$	9.36857e9	0.331978
$969$	−2.94685e10	−1.04046
$970$	1.75662e9	0.0617984
$971$	−7.63541e9	−0.267648	−0.133824	−	0.991005i	$-0.542726\pi$
$971$	−7.63541e9	−0.267648	−0.133824	+	0.991005i	$0.542726\pi$
$972$	−1.47533e9	−0.0515296
$973$	−1.05739e9	−0.0367993
$974$	−2.27050e10	−0.787344
$975$	−2.42364e10	−0.837435
$976$	−3.66552e9	−0.126201
$977$	−1.18991e10	−0.408211	−0.204105	−	0.978949i	$-0.565429\pi$
$977$	−1.18991e10	−0.408211	−0.204105	+	0.978949i	$0.565429\pi$
$978$	−1.01200e10	−0.345934
$979$	3.53644e10	1.20456
$980$	−6.19319e9	−0.210195
$981$	−1.42901e10	−0.483275
$982$	6.43144e9	0.216729
$983$	−3.88296e10	−1.30384	−0.651921	−	0.758287i	$-0.726037\pi$
$983$	−3.88296e10	−1.30384	−0.651921	+	0.758287i	$0.726037\pi$
$984$	1.67861e10	0.561652
$985$	9.89796e9	0.330004
$986$	2.00705e10	0.666789
$987$	3.43924e9	0.113855
$988$	−5.29913e10	−1.74806
$989$	−2.45647e10	−0.807467
$990$	−9.58048e8	−0.0313808
$991$	−4.18861e10	−1.36714	−0.683569	−	0.729886i	$-0.739573\pi$
$991$	−4.18861e10	−1.36714	−0.683569	+	0.729886i	$0.739573\pi$
$992$	−4.43275e10	−1.44172
$993$	−3.14833e10	−1.02037
$994$	−1.17704e9	−0.0380136
$995$	5.02176e9	0.161613
$996$	−1.03338e10	−0.331400
$997$	−2.75354e10	−0.879950	−0.439975	−	0.898010i	$-0.645013\pi$
$997$	−2.75354e10	−0.879950	−0.439975	+	0.898010i	$0.645013\pi$
$998$	−1.63709e10	−0.521333
$999$	−8.09273e9	−0.256812

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	177.8.a.b.1.12	✓	17
3.2	odd	2		531.8.a.d.1.6		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.8.a.b.1.12	✓	17	1.1	even	1	trivial
531.8.a.d.1.6		17	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)

\(f(q)\)	\(=\)	\(q+5.01814 q^{2} +27.0000 q^{3} -102.818 q^{4} -77.5687 q^{5} +135.490 q^{6} -216.829 q^{7} -1158.28 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+5.01814 q^{2} +27.0000 q^{3} -102.818 q^{4} -77.5687 q^{5} +135.490 q^{6} -216.829 q^{7} -1158.28 q^{8} +729.000 q^{9} -389.251 q^{10} +3376.21 q^{11} -2776.09 q^{12} +12448.6 q^{13} -1088.08 q^{14} -2094.36 q^{15} +7348.33 q^{16} -26362.1 q^{17} +3658.23 q^{18} +41401.4 q^{19} +7975.48 q^{20} -5854.37 q^{21} +16942.3 q^{22} -42118.5 q^{23} -31273.5 q^{24} -72108.1 q^{25} +62468.7 q^{26} +19683.0 q^{27} +22293.9 q^{28} -151717. q^{29} -10509.8 q^{30} -239434. q^{31} +185135. q^{32} +91157.8 q^{33} -132289. q^{34} +16819.1 q^{35} -74954.5 q^{36} -411153. q^{37} +207758. q^{38} +336111. q^{39} +89846.2 q^{40} -536752. q^{41} -29378.1 q^{42} +583229. q^{43} -347136. q^{44} -56547.6 q^{45} -211357. q^{46} -587465. q^{47} +198405. q^{48} -776528. q^{49} -361849. q^{50} -711776. q^{51} -1.27994e6 q^{52} -267710. q^{53} +98772.1 q^{54} -261889. q^{55} +251148. q^{56} +1.11784e6 q^{57} -761339. q^{58} -205379. q^{59} +215338. q^{60} -498824. q^{61} -1.20151e6 q^{62} -158068. q^{63} -11553.8 q^{64} -965620. q^{65} +457443. q^{66} +2.87144e6 q^{67} +2.71050e6 q^{68} -1.13720e6 q^{69} +84400.8 q^{70} +1.08176e6 q^{71} -844385. q^{72} -4746.50 q^{73} -2.06322e6 q^{74} -1.94692e6 q^{75} -4.25682e6 q^{76} -732060. q^{77} +1.68666e6 q^{78} -1.41454e6 q^{79} -570000. q^{80} +531441. q^{81} -2.69350e6 q^{82} +3.72243e6 q^{83} +601936. q^{84} +2.04487e6 q^{85} +2.92672e6 q^{86} -4.09637e6 q^{87} -3.91060e6 q^{88} +1.04746e7 q^{89} -283764. q^{90} -2.69921e6 q^{91} +4.33055e6 q^{92} -6.46471e6 q^{93} -2.94798e6 q^{94} -3.21145e6 q^{95} +4.99864e6 q^{96} -4.51283e6 q^{97} -3.89673e6 q^{98} +2.46126e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 32 q^{2} + 459 q^{3} + 1166 q^{4} - 1072 q^{5} - 864 q^{6} - 2407 q^{7} - 6645 q^{8} + 12393 q^{9}+O(q^{10}) \) 17 * q - 32 * q^2 + 459 * q^3 + 1166 * q^4 - 1072 * q^5 - 864 * q^6 - 2407 * q^7 - 6645 * q^8 + 12393 * q^9	\( 17 q - 32 q^{2} + 459 q^{3} + 1166 q^{4} - 1072 q^{5} - 864 q^{6} - 2407 q^{7} - 6645 q^{8} + 12393 q^{9} - 6391 q^{10} - 8888 q^{11} + 31482 q^{12} - 12702 q^{13} - 17555 q^{14} - 28944 q^{15} + 139226 q^{16} - 36167 q^{17} - 23328 q^{18} - 71037 q^{19} - 274883 q^{20} - 64989 q^{21} - 325182 q^{22} - 269995 q^{23} - 179415 q^{24} + 97329 q^{25} - 336906 q^{26} + 334611 q^{27} - 901362 q^{28} - 543825 q^{29} - 172557 q^{30} - 633109 q^{31} - 837062 q^{32} - 239976 q^{33} - 529288 q^{34} - 287621 q^{35} + 850014 q^{36} - 867607 q^{37} - 1727169 q^{38} - 342954 q^{39} - 815662 q^{40} - 1428939 q^{41} - 473985 q^{42} - 477060 q^{43} - 1667926 q^{44} - 781488 q^{45} + 5305549 q^{46} - 1217849 q^{47} + 3759102 q^{48} + 4350738 q^{49} + 4561369 q^{50} - 976509 q^{51} + 4175994 q^{52} - 3487068 q^{53} - 629856 q^{54} - 960484 q^{55} - 5363196 q^{56} - 1917999 q^{57} - 3082906 q^{58} - 3491443 q^{59} - 7421841 q^{60} + 998917 q^{61} - 5742614 q^{62} - 1754703 q^{63} + 17531621 q^{64} - 6075816 q^{65} - 8779914 q^{66} - 356026 q^{67} - 16149231 q^{68} - 7289865 q^{69} - 548798 q^{70} - 12879428 q^{71} - 4844205 q^{72} - 6176157 q^{73} - 5971906 q^{74} + 2627883 q^{75} - 17624580 q^{76} + 239687 q^{77} - 9096462 q^{78} - 18886490 q^{79} - 70463349 q^{80} + 9034497 q^{81} - 19351611 q^{82} - 22824893 q^{83} - 24336774 q^{84} - 7973079 q^{85} - 27502196 q^{86} - 14683275 q^{87} - 62527651 q^{88} - 30609647 q^{89} - 4659039 q^{90} - 36301521 q^{91} - 41388548 q^{92} - 17093943 q^{93} + 1010176 q^{94} - 29303629 q^{95} - 22600674 q^{96} - 26249806 q^{97} - 93110852 q^{98} - 6479352 q^{99}+O(q^{100}) \) 17 * q - 32 * q^2 + 459 * q^3 + 1166 * q^4 - 1072 * q^5 - 864 * q^6 - 2407 * q^7 - 6645 * q^8 + 12393 * q^9 - 6391 * q^10 - 8888 * q^11 + 31482 * q^12 - 12702 * q^13 - 17555 * q^14 - 28944 * q^15 + 139226 * q^16 - 36167 * q^17 - 23328 * q^18 - 71037 * q^19 - 274883 * q^20 - 64989 * q^21 - 325182 * q^22 - 269995 * q^23 - 179415 * q^24 + 97329 * q^25 - 336906 * q^26 + 334611 * q^27 - 901362 * q^28 - 543825 * q^29 - 172557 * q^30 - 633109 * q^31 - 837062 * q^32 - 239976 * q^33 - 529288 * q^34 - 287621 * q^35 + 850014 * q^36 - 867607 * q^37 - 1727169 * q^38 - 342954 * q^39 - 815662 * q^40 - 1428939 * q^41 - 473985 * q^42 - 477060 * q^43 - 1667926 * q^44 - 781488 * q^45 + 5305549 * q^46 - 1217849 * q^47 + 3759102 * q^48 + 4350738 * q^49 + 4561369 * q^50 - 976509 * q^51 + 4175994 * q^52 - 3487068 * q^53 - 629856 * q^54 - 960484 * q^55 - 5363196 * q^56 - 1917999 * q^57 - 3082906 * q^58 - 3491443 * q^59 - 7421841 * q^60 + 998917 * q^61 - 5742614 * q^62 - 1754703 * q^63 + 17531621 * q^64 - 6075816 * q^65 - 8779914 * q^66 - 356026 * q^67 - 16149231 * q^68 - 7289865 * q^69 - 548798 * q^70 - 12879428 * q^71 - 4844205 * q^72 - 6176157 * q^73 - 5971906 * q^74 + 2627883 * q^75 - 17624580 * q^76 + 239687 * q^77 - 9096462 * q^78 - 18886490 * q^79 - 70463349 * q^80 + 9034497 * q^81 - 19351611 * q^82 - 22824893 * q^83 - 24336774 * q^84 - 7973079 * q^85 - 27502196 * q^86 - 14683275 * q^87 - 62527651 * q^88 - 30609647 * q^89 - 4659039 * q^90 - 36301521 * q^91 - 41388548 * q^92 - 17093943 * q^93 + 1010176 * q^94 - 29303629 * q^95 - 22600674 * q^96 - 26249806 * q^97 - 93110852 * q^98 - 6479352 * q^99

Introduction

L-functions

Modular forms

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