LMFDB - Embedded newform 43.8.a.b.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [43,8,Mod(1,43)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(43, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("43.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 43 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	43.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:	$13.4325560958$
Analytic rank:	$0$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + \cdots + 1551032970660 $ x^13 - 3x^12 - 1279x^11 + 3765x^10 + 598742x^9 - 1518614x^8 - 124677082x^7 + 193428526x^6 + 11160446785x^5 + 1754605765x^4 - 349352939351x^3 - 481872751923x^2 + 2098464001560x + 1551032970660
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$21.1822$ of defining polynomial
Character	$\chi$	$=$	43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+22.1822 q^{2} +2.16379 q^{3} +364.050 q^{4} +122.553 q^{5} +47.9976 q^{6} -90.8173 q^{7} +5236.10 q^{8} -2182.32 q^{9} +O(q^{10})$	\(q+22.1822 q^{2} +2.16379 q^{3} +364.050 q^{4} +122.553 q^{5} +47.9976 q^{6} -90.8173 q^{7} +5236.10 q^{8} -2182.32 q^{9} +2718.48 q^{10} +3410.31 q^{11} +787.727 q^{12} -11275.0 q^{13} -2014.53 q^{14} +265.178 q^{15} +69549.8 q^{16} +9720.37 q^{17} -48408.6 q^{18} +25351.7 q^{19} +44615.2 q^{20} -196.510 q^{21} +75648.2 q^{22} -52795.4 q^{23} +11329.8 q^{24} -63105.9 q^{25} -250104. q^{26} -9454.29 q^{27} -33062.0 q^{28} -216763. q^{29} +5882.23 q^{30} +142113. q^{31} +872545. q^{32} +7379.20 q^{33} +215619. q^{34} -11129.9 q^{35} -794472. q^{36} -74290.8 q^{37} +562357. q^{38} -24396.7 q^{39} +641697. q^{40} -273026. q^{41} -4359.01 q^{42} -79507.0 q^{43} +1.24152e6 q^{44} -267449. q^{45} -1.17112e6 q^{46} -112553. q^{47} +150491. q^{48} -815295. q^{49} -1.39983e6 q^{50} +21032.8 q^{51} -4.10466e6 q^{52} +1.41102e6 q^{53} -209717. q^{54} +417942. q^{55} -475528. q^{56} +54855.8 q^{57} -4.80827e6 q^{58} -1.44713e6 q^{59} +96537.9 q^{60} +3.19474e6 q^{61} +3.15239e6 q^{62} +198192. q^{63} +1.04526e7 q^{64} -1.38178e6 q^{65} +163687. q^{66} -3.52359e6 q^{67} +3.53870e6 q^{68} -114238. q^{69} -246885. q^{70} -327344. q^{71} -1.14268e7 q^{72} +2.80064e6 q^{73} -1.64793e6 q^{74} -136548. q^{75} +9.22928e6 q^{76} -309715. q^{77} -541173. q^{78} +3.35723e6 q^{79} +8.52350e6 q^{80} +4.75227e6 q^{81} -6.05633e6 q^{82} +7.10311e6 q^{83} -71539.2 q^{84} +1.19126e6 q^{85} -1.76364e6 q^{86} -469029. q^{87} +1.78567e7 q^{88} -1.32585e6 q^{89} -5.93259e6 q^{90} +1.02396e6 q^{91} -1.92201e7 q^{92} +307504. q^{93} -2.49668e6 q^{94} +3.10692e6 q^{95} +1.88801e6 q^{96} +1.29568e7 q^{97} -1.80850e7 q^{98} -7.44239e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9}+O(q^{10}) $ 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9} + 4667 q^{10} + 1620 q^{11} - 19681 q^{12} + 13550 q^{13} + 44160 q^{14} + 31412 q^{15} + 114026 q^{16} + 110880 q^{17} + 159267 q^{18} + 105058 q^{19} + 167251 q^{20} + 129840 q^{21} + 201504 q^{22} + 160184 q^{23} + 161289 q^{24} + 270149 q^{25} + 272104 q^{26} + 252544 q^{27} + 208172 q^{28} + 285546 q^{29} + 107580 q^{30} - 99616 q^{31} + 200126 q^{32} + 531468 q^{33} - 80941 q^{34} - 187104 q^{35} - 608975 q^{36} + 176038 q^{37} + 652165 q^{38} - 794680 q^{39} - 895387 q^{40} - 410260 q^{41} - 3413218 q^{42} - 1033591 q^{43} - 2177076 q^{44} - 1051178 q^{45} - 3975765 q^{46} - 424556 q^{47} - 2360477 q^{48} - 1561359 q^{49} - 4063801 q^{50} - 2375738 q^{51} - 4172312 q^{52} + 3992458 q^{53} - 10438626 q^{54} + 406960 q^{55} + 1559556 q^{56} - 3116152 q^{57} - 4052005 q^{58} + 2248836 q^{59} - 2911436 q^{60} + 6210394 q^{61} + 885317 q^{62} + 11622368 q^{63} - 3096318 q^{64} + 5600420 q^{65} - 2174604 q^{66} - 1993648 q^{67} + 9327135 q^{68} + 13366240 q^{69} - 1105098 q^{70} + 4978064 q^{71} + 11370663 q^{72} + 8224814 q^{73} - 3613563 q^{74} + 27115592 q^{75} + 10687121 q^{76} + 17261892 q^{77} - 15226630 q^{78} + 6945708 q^{79} + 15822799 q^{80} + 35113185 q^{81} - 508449 q^{82} + 22937328 q^{83} - 14010106 q^{84} - 575532 q^{85} - 1272112 q^{86} + 9081380 q^{87} + 11202656 q^{88} + 9291302 q^{89} + 2841402 q^{90} + 25581108 q^{91} - 14388137 q^{92} + 25930480 q^{93} - 24645805 q^{94} + 30750464 q^{95} - 22461255 q^{96} + 10001852 q^{97} - 32304856 q^{98} + 5055452 q^{99}+O(q^{100}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9 + 4667 * q^10 + 1620 * q^11 - 19681 * q^12 + 13550 * q^13 + 44160 * q^14 + 31412 * q^15 + 114026 * q^16 + 110880 * q^17 + 159267 * q^18 + 105058 * q^19 + 167251 * q^20 + 129840 * q^21 + 201504 * q^22 + 160184 * q^23 + 161289 * q^24 + 270149 * q^25 + 272104 * q^26 + 252544 * q^27 + 208172 * q^28 + 285546 * q^29 + 107580 * q^30 - 99616 * q^31 + 200126 * q^32 + 531468 * q^33 - 80941 * q^34 - 187104 * q^35 - 608975 * q^36 + 176038 * q^37 + 652165 * q^38 - 794680 * q^39 - 895387 * q^40 - 410260 * q^41 - 3413218 * q^42 - 1033591 * q^43 - 2177076 * q^44 - 1051178 * q^45 - 3975765 * q^46 - 424556 * q^47 - 2360477 * q^48 - 1561359 * q^49 - 4063801 * q^50 - 2375738 * q^51 - 4172312 * q^52 + 3992458 * q^53 - 10438626 * q^54 + 406960 * q^55 + 1559556 * q^56 - 3116152 * q^57 - 4052005 * q^58 + 2248836 * q^59 - 2911436 * q^60 + 6210394 * q^61 + 885317 * q^62 + 11622368 * q^63 - 3096318 * q^64 + 5600420 * q^65 - 2174604 * q^66 - 1993648 * q^67 + 9327135 * q^68 + 13366240 * q^69 - 1105098 * q^70 + 4978064 * q^71 + 11370663 * q^72 + 8224814 * q^73 - 3613563 * q^74 + 27115592 * q^75 + 10687121 * q^76 + 17261892 * q^77 - 15226630 * q^78 + 6945708 * q^79 + 15822799 * q^80 + 35113185 * q^81 - 508449 * q^82 + 22937328 * q^83 - 14010106 * q^84 - 575532 * q^85 - 1272112 * q^86 + 9081380 * q^87 + 11202656 * q^88 + 9291302 * q^89 + 2841402 * q^90 + 25581108 * q^91 - 14388137 * q^92 + 25930480 * q^93 - 24645805 * q^94 + 30750464 * q^95 - 22461255 * q^96 + 10001852 * q^97 - 32304856 * q^98 + 5055452 * 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 $

$2$	22.1822	1.96065	0.980324	−	0.197397i	$-0.0632489\pi$
$2$	22.1822	1.96065	0.980324	+	0.197397i	$0.0632489\pi$
$3$	2.16379	0.0462691	0.0231345	−	0.999732i	$-0.492635\pi$
$3$	2.16379	0.0462691	0.0231345	+	0.999732i	$0.492635\pi$
$4$	364.050	2.84414
$5$	122.553	0.438457	0.219229	−	0.975674i	$-0.429646\pi$
$5$	122.553	0.438457	0.219229	+	0.975674i	$0.429646\pi$
$6$	47.9976	0.0907173
$7$	−90.8173	−0.100075	−0.0500375	−	0.998747i	$-0.515934\pi$
$7$	−90.8173	−0.100075	−0.0500375	+	0.998747i	$0.515934\pi$
$8$	5236.10	3.61570
$9$	−2182.32	−0.997859
$10$	2718.48	0.859660
$11$	3410.31	0.772538	0.386269	−	0.922386i	$-0.373764\pi$
$11$	3410.31	0.772538	0.386269	+	0.922386i	$0.373764\pi$
$12$	787.727	0.131596
$13$	−11275.0	−1.42336	−0.711680	−	0.702504i	$-0.752065\pi$
$13$	−11275.0	−1.42336	−0.711680	+	0.702504i	$0.752065\pi$
$14$	−2014.53	−0.196212
$15$	265.178	0.0202870
$16$	69549.8	4.24498
$17$	9720.37	0.479857	0.239928	−	0.970791i	$-0.422876\pi$
$17$	9720.37	0.479857	0.239928	+	0.970791i	$0.422876\pi$
$18$	−48408.6	−1.95645
$19$	25351.7	0.847949	0.423975	−	0.905674i	$-0.360635\pi$
$19$	25351.7	0.847949	0.423975	+	0.905674i	$0.360635\pi$
$20$	44615.2	1.24703
$21$	−196.510	−0.00463038
$22$	75648.2	1.51467
$23$	−52795.4	−0.904792	−0.452396	−	0.891817i	$-0.649431\pi$
$23$	−52795.4	−0.904792	−0.452396	+	0.891817i	$0.649431\pi$
$24$	11329.8	0.167295
$25$	−63105.9	−0.807755
$26$	−250104.	−2.79071
$27$	−9454.29	−0.0924391
$28$	−33062.0	−0.284627
$29$	−216763.	−1.65041	−0.825204	−	0.564835i	$-0.808940\pi$
$29$	−216763.	−1.65041	−0.825204	+	0.564835i	$0.808940\pi$
$30$	5882.23	0.0397757
$31$	142113.	0.856780	0.428390	−	0.903594i	$-0.359081\pi$
$31$	142113.	0.856780	0.428390	+	0.903594i	$0.359081\pi$
$32$	872545.	4.70721
$33$	7379.20	0.0357446
$34$	215619.	0.940830
$35$	−11129.9	−0.0438786
$36$	−794472.	−2.83805
$37$	−74290.8	−0.241118	−0.120559	−	0.992706i	$-0.538469\pi$
$37$	−74290.8	−0.241118	−0.120559	+	0.992706i	$0.538469\pi$
$38$	562357.	1.66253
$39$	−24396.7	−0.0658576
$40$	641697.	1.58533
$41$	−273026.	−0.618673	−0.309337	−	0.950953i	$-0.600107\pi$
$41$	−273026.	−0.618673	−0.309337	+	0.950953i	$0.600107\pi$
$42$	−4359.01	−0.00907853
$43$	−79507.0	−0.152499
$43$	−79507.0	−0.152499
$44$	1.24152e6	2.19720
$45$	−267449.	−0.437519
$46$	−1.17112e6	−1.77398
$47$	−112553.	−0.158130	−0.0790652	−	0.996869i	$-0.525194\pi$
$47$	−112553.	−0.158130	−0.0790652	+	0.996869i	$0.525194\pi$
$48$	150491.	0.196411
$49$	−815295.	−0.989985
$50$	−1.39983e6	−1.58372
$51$	21032.8	0.0222025
$52$	−4.10466e6	−4.04823
$53$	1.41102e6	1.30187	0.650936	−	0.759132i	$-0.274376\pi$
$53$	1.41102e6	1.30187	0.650936	+	0.759132i	$0.274376\pi$
$54$	−209717.	−0.181240
$55$	417942.	0.338725
$56$	−475528.	−0.361841
$57$	54855.8	0.0392338
$58$	−4.80827e6	−3.23587
$59$	−1.44713e6	−0.917329	−0.458665	−	0.888609i	$-0.651672\pi$
$59$	−1.44713e6	−0.917329	−0.458665	+	0.888609i	$0.651672\pi$
$60$	96537.9	0.0576990
$61$	3.19474e6	1.80211	0.901054	−	0.433708i	$-0.142795\pi$
$61$	3.19474e6	1.80211	0.901054	+	0.433708i	$0.142795\pi$
$62$	3.15239e6	1.67984
$63$	198192.	0.0998607
$64$	1.04526e7	4.98419
$65$	−1.38178e6	−0.624083
$66$	163687.	0.0700826
$67$	−3.52359e6	−1.43128	−0.715638	−	0.698472i	$-0.753864\pi$
$67$	−3.52359e6	−1.43128	−0.715638	+	0.698472i	$0.753864\pi$
$68$	3.53870e6	1.36478
$69$	−114238.	−0.0418639
$70$	−246885.	−0.0860304
$71$	−327344.	−0.108542	−0.0542712	−	0.998526i	$-0.517284\pi$
$71$	−327344.	−0.108542	−0.0542712	+	0.998526i	$0.517284\pi$
$72$	−1.14268e7	−3.60796
$73$	2.80064e6	0.842613	0.421306	−	0.906918i	$-0.361572\pi$
$73$	2.80064e6	0.842613	0.421306	+	0.906918i	$0.361572\pi$
$74$	−1.64793e6	−0.472746
$75$	−136548.	−0.0373741
$76$	9.22928e6	2.41168
$77$	−309715.	−0.0773117
$78$	−541173.	−0.129123
$79$	3.35723e6	0.766101	0.383051	−	0.923727i	$-0.374874\pi$
$79$	3.35723e6	0.766101	0.383051	+	0.923727i	$0.374874\pi$
$80$	8.52350e6	1.86124
$81$	4.75227e6	0.993582
$82$	−6.05633e6	−1.21300
$83$	7.10311e6	1.36356	0.681781	−	0.731556i	$-0.261206\pi$
$83$	7.10311e6	1.36356	0.681781	+	0.731556i	$0.261206\pi$
$84$	−71539.2	−0.0131694
$85$	1.19126e6	0.210397
$86$	−1.76364e6	−0.298996
$87$	−469029.	−0.0763628
$88$	1.78567e7	2.79327
$89$	−1.32585e6	−0.199356	−0.0996778	−	0.995020i	$-0.531781\pi$
$89$	−1.32585e6	−0.199356	−0.0996778	+	0.995020i	$0.531781\pi$
$90$	−5.93259e6	−0.857819
$91$	1.02396e6	0.142443
$92$	−1.92201e7	−2.57335
$93$	307504.	0.0396424
$94$	−2.49668e6	−0.310038
$95$	3.10692e6	0.371789
$96$	1.88801e6	0.217798
$97$	1.29568e7	1.44144	0.720718	−	0.693228i	$-0.243812\pi$
$97$	1.29568e7	1.44144	0.720718	+	0.693228i	$0.243812\pi$
$98$	−1.80850e7	−1.94101
$99$	−7.44239e6	−0.770884
$100$	−2.29737e7	−2.29737
$101$	309915.	0.0299308	0.0149654	−	0.999888i	$-0.495236\pi$
$101$	309915.	0.0299308	0.0149654	+	0.999888i	$0.495236\pi$
$102$	466555.	0.0435313
$103$	1.99810e7	1.80172	0.900858	−	0.434113i	$-0.142938\pi$
$103$	1.99810e7	1.80172	0.900858	+	0.434113i	$0.142938\pi$
$104$	−5.90370e7	−5.14645
$105$	−24082.7	−0.00203022
$106$	3.12996e7	2.55251
$107$	1.77323e6	0.139934	0.0699669	−	0.997549i	$-0.477711\pi$
$107$	1.77323e6	0.139934	0.0699669	+	0.997549i	$0.477711\pi$
$108$	−3.44183e6	−0.262910
$109$	7.12031e6	0.526631	0.263315	−	0.964710i	$-0.415184\pi$
$109$	7.12031e6	0.526631	0.263315	+	0.964710i	$0.415184\pi$
$110$	9.27088e6	0.664120
$111$	−160750.	−0.0111563
$112$	−6.31632e6	−0.424816
$113$	−9.47406e6	−0.617677	−0.308839	−	0.951114i	$-0.599940\pi$
$113$	−9.47406e6	−0.617677	−0.308839	+	0.951114i	$0.599940\pi$
$114$	1.21682e6	0.0769237
$115$	−6.47021e6	−0.396713
$116$	−7.89123e7	−4.69399
$117$	2.46056e7	1.42031
$118$	−3.21005e7	−1.79856
$119$	−882778.	−0.0480216
$120$	1.38850e6	0.0733518
$121$	−7.85694e6	−0.403185
$122$	7.08663e7	3.53330
$123$	−590772.	−0.0286254
$124$	5.17363e7	2.43680
$125$	−1.73082e7	−0.792623
$126$	4.39634e6	0.195792
$127$	2.94089e7	1.27399	0.636994	−	0.770868i	$-0.280177\pi$
$127$	2.94089e7	1.27399	0.636994	+	0.770868i	$0.280177\pi$
$128$	1.20176e8	5.06503
$129$	−172037.	−0.00705597
$130$	−3.06509e7	−1.22361
$131$	4.80770e7	1.86848	0.934238	−	0.356650i	$-0.116081\pi$
$131$	4.80770e7	1.86848	0.934238	+	0.356650i	$0.116081\pi$
$132$	2.68640e6	0.101663
$133$	−2.30237e6	−0.0848585
$134$	−7.81609e7	−2.80623
$135$	−1.15865e6	−0.0405306
$136$	5.08968e7	1.73502
$137$	5.87744e6	0.195284	0.0976419	−	0.995222i	$-0.468870\pi$
$137$	5.87744e6	0.195284	0.0976419	+	0.995222i	$0.468870\pi$
$138$	−2.53405e6	−0.0820803
$139$	−5.12513e7	−1.61865	−0.809326	−	0.587360i	$-0.800167\pi$
$139$	−5.12513e7	−1.61865	−0.809326	+	0.587360i	$0.800167\pi$
$140$	−4.05183e6	−0.124797
$141$	−243541.	−0.00731654
$142$	−7.26120e6	−0.212813
$143$	−3.84513e7	−1.09960
$144$	−1.51780e8	−4.23589
$145$	−2.65648e7	−0.723633
$146$	6.21244e7	1.65207
$147$	−1.76413e6	−0.0458057
$148$	−2.70455e7	−0.685771
$149$	−6.91179e6	−0.171174	−0.0855871	−	0.996331i	$-0.527277\pi$
$149$	−6.91179e6	−0.171174	−0.0855871	+	0.996331i	$0.527277\pi$
$150$	−3.02893e6	−0.0732774
$151$	−6.97335e7	−1.64825	−0.824124	−	0.566410i	$-0.808332\pi$
$151$	−6.97335e7	−1.64825	−0.824124	+	0.566410i	$0.808332\pi$
$152$	1.32744e8	3.06593
$153$	−2.12129e7	−0.478829
$154$	−6.87016e6	−0.151581
$155$	1.74164e7	0.375661
$156$	−8.88162e6	−0.187308
$157$	−6.10206e7	−1.25843	−0.629213	−	0.777233i	$-0.716623\pi$
$157$	−6.10206e7	−1.25843	−0.629213	+	0.777233i	$0.716623\pi$
$158$	7.44707e7	1.50205
$159$	3.05316e6	0.0602365
$160$	1.06933e8	2.06391
$161$	4.79474e6	0.0905470
$162$	1.05416e8	1.94806
$163$	−1.28985e6	−0.0233282	−0.0116641	−	0.999932i	$-0.503713\pi$
$163$	−1.28985e6	−0.0233282	−0.0116641	+	0.999932i	$0.503713\pi$
$164$	−9.93952e7	−1.75959
$165$	904340.	0.0156725
$166$	1.57563e8	2.67347
$167$	−8.25906e7	−1.37222	−0.686109	−	0.727499i	$-0.740683\pi$
$167$	−8.25906e7	−1.37222	−0.686109	+	0.727499i	$0.740683\pi$
$168$	−1.02894e6	−0.0167421
$169$	6.43771e7	1.02595
$170$	2.64247e7	0.412514
$171$	−5.53255e7	−0.846134
$172$	−2.89445e7	−0.433727
$173$	−6.06382e6	−0.0890400	−0.0445200	−	0.999008i	$-0.514176\pi$
$173$	−6.06382e6	−0.0890400	−0.0445200	+	0.999008i	$0.514176\pi$
$174$	−1.04041e7	−0.149721
$175$	5.73110e6	0.0808361
$176$	2.37186e8	3.27941
$177$	−3.13128e6	−0.0424440
$178$	−2.94102e7	−0.390866
$179$	−5.06661e7	−0.660286	−0.330143	−	0.943931i	$-0.607097\pi$
$179$	−5.06661e7	−0.660286	−0.330143	+	0.943931i	$0.607097\pi$
$180$	−9.73645e7	−1.24436
$181$	−2.08989e6	−0.0261968	−0.0130984	−	0.999914i	$-0.504169\pi$
$181$	−2.08989e6	−0.0261968	−0.0130984	+	0.999914i	$0.504169\pi$
$182$	2.27138e7	0.279280
$183$	6.91274e6	0.0833818
$184$	−2.76442e8	−3.27146
$185$	−9.10452e6	−0.105720
$186$	6.82110e6	0.0777248
$187$	3.31495e7	0.370707
$188$	−4.09749e7	−0.449744
$189$	858613.	0.00925084
$190$	6.89182e7	0.728948
$191$	1.78113e8	1.84960	0.924800	−	0.380454i	$-0.124232\pi$
$191$	1.78113e8	1.84960	0.924800	+	0.380454i	$0.124232\pi$
$192$	2.26172e7	0.230614
$193$	1.26757e8	1.26917	0.634586	−	0.772852i	$-0.281170\pi$
$193$	1.26757e8	1.26917	0.634586	+	0.772852i	$0.281170\pi$
$194$	2.87410e8	2.82615
$195$	−2.98988e6	−0.0288757
$196$	−2.96808e8	−2.81565
$197$	−1.27815e8	−1.19110	−0.595550	−	0.803318i	$-0.703066\pi$
$197$	−1.27815e8	−1.19110	−0.595550	+	0.803318i	$0.703066\pi$
$198$	−1.65088e8	−1.51143
$199$	−1.29981e8	−1.16921	−0.584606	−	0.811318i	$-0.698751\pi$
$199$	−1.29981e8	−1.16921	−0.584606	+	0.811318i	$0.698751\pi$
$200$	−3.30429e8	−2.92060
$201$	−7.62431e6	−0.0662238
$202$	6.87459e6	0.0586837
$203$	1.96858e7	0.165164
$204$	7.65700e6	0.0631470
$205$	−3.34601e7	−0.271262
$206$	4.43222e8	3.53253
$207$	1.15216e8	0.902855
$208$	−7.84173e8	−6.04214
$209$	8.64573e7	0.655073
$210$	−534208.	−0.00398055
$211$	−1.85689e8	−1.36081	−0.680406	−	0.732836i	$-0.738196\pi$
$211$	−1.85689e8	−1.36081	−0.680406	+	0.732836i	$0.738196\pi$
$212$	5.13682e8	3.70270
$213$	−708303.	−0.00502216
$214$	3.93342e7	0.274361
$215$	−9.74378e6	−0.0668641
$216$	−4.95036e7	−0.334232
$217$	−1.29064e7	−0.0857422
$218$	1.57944e8	1.03254
$219$	6.06001e6	0.0389869
$220$	1.52152e8	0.963380
$221$	−1.09597e8	−0.683009
$222$	−3.56578e6	−0.0218735
$223$	1.57910e7	0.0953545	0.0476773	−	0.998863i	$-0.484818\pi$
$223$	1.57910e7	0.0953545	0.0476773	+	0.998863i	$0.484818\pi$
$224$	−7.92422e7	−0.471073
$225$	1.37717e8	0.806026
$226$	−2.10155e8	−1.21105
$227$	3.37705e7	0.191623	0.0958113	−	0.995400i	$-0.469455\pi$
$227$	3.37705e7	0.191623	0.0958113	+	0.995400i	$0.469455\pi$
$228$	1.99702e7	0.111586
$229$	2.60087e8	1.43118	0.715590	−	0.698520i	$-0.246158\pi$
$229$	2.60087e8	1.43118	0.715590	+	0.698520i	$0.246158\pi$
$230$	−1.43523e8	−0.777813
$231$	−670159.	−0.00357714
$232$	−1.13499e9	−5.96738
$233$	1.45083e8	0.751398	0.375699	−	0.926742i	$-0.377403\pi$
$233$	1.45083e8	0.751398	0.375699	+	0.926742i	$0.377403\pi$
$234$	5.45807e8	2.78473
$235$	−1.37937e7	−0.0693334
$236$	−5.26827e8	−2.60901
$237$	7.26434e6	0.0354468
$238$	−1.95819e7	−0.0941535
$239$	−1.55295e8	−0.735808	−0.367904	−	0.929864i	$-0.619924\pi$
$239$	−1.55295e8	−0.735808	−0.367904	+	0.929864i	$0.619924\pi$
$240$	1.84431e7	0.0861180
$241$	3.22999e8	1.48642	0.743211	−	0.669057i	$-0.233302\pi$
$241$	3.22999e8	1.48642	0.743211	+	0.669057i	$0.233302\pi$
$242$	−1.74284e8	−0.790504
$243$	3.09595e7	0.138411
$244$	1.16304e9	5.12544
$245$	−9.99165e7	−0.434066
$246$	−1.31046e7	−0.0561244
$247$	−2.85841e8	−1.20694
$248$	7.44119e8	3.09786
$249$	1.53696e7	0.0630908
$250$	−3.83934e8	−1.55405
$251$	−2.35414e8	−0.939668	−0.469834	−	0.882755i	$-0.655686\pi$
$251$	−2.35414e8	−0.939668	−0.469834	+	0.882755i	$0.655686\pi$
$252$	7.21518e7	0.284018
$253$	−1.80049e8	−0.698986
$254$	6.52354e8	2.49784
$255$	2.57763e6	0.00973486
$256$	1.32783e9	4.94655
$257$	−4.25190e7	−0.156249	−0.0781245	−	0.996944i	$-0.524893\pi$
$257$	−4.25190e7	−0.156249	−0.0781245	+	0.996944i	$0.524893\pi$
$258$	−3.81615e6	−0.0138343
$259$	6.74689e6	0.0241298
$260$	−5.03036e8	−1.77498
$261$	4.73045e8	1.64687
$262$	1.06645e9	3.66342
$263$	1.04522e8	0.354292	0.177146	−	0.984185i	$-0.443314\pi$
$263$	1.04522e8	0.354292	0.177146	+	0.984185i	$0.443314\pi$
$264$	3.86382e7	0.129242
$265$	1.72924e8	0.570815
$266$	−5.10717e7	−0.166378
$267$	−2.86885e6	−0.00922400
$268$	−1.28276e9	−4.07074
$269$	−1.86374e8	−0.583784	−0.291892	−	0.956451i	$-0.594285\pi$
$269$	−1.86374e8	−0.583784	−0.291892	+	0.956451i	$0.594285\pi$
$270$	−2.57013e7	−0.0794662
$271$	3.37243e8	1.02932	0.514660	−	0.857394i	$-0.327918\pi$
$271$	3.37243e8	1.02932	0.514660	+	0.857394i	$0.327918\pi$
$272$	6.76049e8	2.03698
$273$	2.21565e6	0.00659069
$274$	1.30375e8	0.382883
$275$	−2.15211e8	−0.624022
$276$	−4.15884e7	−0.119067
$277$	−4.22893e8	−1.19550	−0.597752	−	0.801681i	$-0.703939\pi$
$277$	−4.22893e8	−1.19550	−0.597752	+	0.801681i	$0.703939\pi$
$278$	−1.13687e9	−3.17360
$279$	−3.10137e8	−0.854945
$280$	−5.82772e7	−0.158652
$281$	3.41588e8	0.918396	0.459198	−	0.888334i	$-0.348137\pi$
$281$	3.41588e8	0.918396	0.459198	+	0.888334i	$0.348137\pi$
$282$	−5.40228e6	−0.0143452
$283$	−1.93785e8	−0.508238	−0.254119	−	0.967173i	$-0.581785\pi$
$283$	−1.93785e8	−0.508238	−0.254119	+	0.967173i	$0.581785\pi$
$284$	−1.19169e8	−0.308710
$285$	6.72272e6	0.0172024
$286$	−8.52934e8	−2.15593
$287$	2.47955e7	0.0619137
$288$	−1.90417e9	−4.69713
$289$	−3.15853e8	−0.769738
$290$	−5.89265e8	−1.41879
$291$	2.80357e7	0.0666940
$292$	1.01957e9	2.39651
$293$	4.63942e8	1.07752	0.538762	−	0.842458i	$-0.318892\pi$
$293$	4.63942e8	1.07752	0.538762	+	0.842458i	$0.318892\pi$
$294$	−3.91322e7	−0.0898088
$295$	−1.77349e8	−0.402210
$296$	−3.88994e8	−0.871809
$297$	−3.22421e7	−0.0714127
$298$	−1.53319e8	−0.335612
$299$	5.95268e8	1.28785
$300$	−4.97102e7	−0.106297
$301$	7.22061e6	0.0152613
$302$	−1.54684e9	−3.23163
$303$	670591.	0.00138487
$304$	1.76321e9	3.59953
$305$	3.91523e8	0.790147
$306$	−4.70549e8	−0.938816
$307$	1.10340e8	0.217644	0.108822	−	0.994061i	$-0.465292\pi$
$307$	1.10340e8	0.217644	0.108822	+	0.994061i	$0.465292\pi$
$308$	−1.12752e8	−0.219885
$309$	4.32347e7	0.0833638
$310$	3.86333e8	0.736539
$311$	−1.65769e8	−0.312494	−0.156247	−	0.987718i	$-0.549940\pi$
$311$	−1.65769e8	−0.312494	−0.156247	+	0.987718i	$0.549940\pi$
$312$	−1.27744e8	−0.238121
$313$	−1.02978e9	−1.89818	−0.949091	−	0.315003i	$-0.897994\pi$
$313$	−1.02978e9	−1.89818	−0.949091	+	0.315003i	$0.897994\pi$
$314$	−1.35357e9	−2.46733
$315$	2.42890e7	0.0437846
$316$	1.22220e9	2.17890
$317$	6.40081e8	1.12857	0.564284	−	0.825581i	$-0.309152\pi$
$317$	6.40081e8	1.12857	0.564284	+	0.825581i	$0.309152\pi$
$318$	6.77257e7	0.118102
$319$	−7.39228e8	−1.27500
$320$	1.28099e9	2.18535
$321$	3.83691e6	0.00647461
$322$	1.06358e8	0.177531
$323$	2.46428e8	0.406894
$324$	1.73006e9	2.82588
$325$	7.11519e8	1.14973
$326$	−2.86116e7	−0.0457383
$327$	1.54069e7	0.0243667
$328$	−1.42959e9	−2.23694
$329$	1.02218e7	0.0158249
$330$	2.00602e7	0.0307282
$331$	−4.29838e8	−0.651489	−0.325744	−	0.945458i	$-0.605615\pi$
$331$	−4.29838e8	−0.651489	−0.325744	+	0.945458i	$0.605615\pi$
$332$	2.58588e9	3.87816
$333$	1.62126e8	0.240601
$334$	−1.83204e9	−2.69044
$335$	−4.31824e8	−0.627553
$336$	−1.36672e7	−0.0196559
$337$	−3.78144e8	−0.538211	−0.269105	−	0.963111i	$-0.586728\pi$
$337$	−3.78144e8	−0.538211	−0.269105	+	0.963111i	$0.586728\pi$
$338$	1.42803e9	2.01153
$339$	−2.04999e7	−0.0285794
$340$	4.33676e8	0.598397
$341$	4.84651e8	0.661895
$342$	−1.22724e9	−1.65897
$343$	1.48835e8	0.199148
$344$	−4.16306e8	−0.551389
$345$	−1.40002e7	−0.0183555
$346$	−1.34509e8	−0.174576
$347$	−3.18524e8	−0.409250	−0.204625	−	0.978840i	$-0.565597\pi$
$347$	−3.18524e8	−0.409250	−0.204625	+	0.978840i	$0.565597\pi$
$348$	−1.70750e8	−0.217186
$349$	9.98844e8	1.25779	0.628896	−	0.777489i	$-0.283507\pi$
$349$	9.98844e8	1.25779	0.628896	+	0.777489i	$0.283507\pi$
$350$	1.27128e8	0.158491
$351$	1.06597e8	0.131574
$352$	2.97565e9	3.63649
$353$	1.22295e8	0.147978	0.0739890	−	0.997259i	$-0.476427\pi$
$353$	1.22295e8	0.147978	0.0739890	+	0.997259i	$0.476427\pi$
$354$	−6.94588e7	−0.0832177
$355$	−4.01168e7	−0.0475912
$356$	−4.82674e8	−0.566994
$357$	−1.91015e6	−0.00222192
$358$	−1.12389e9	−1.29459
$359$	−1.04011e9	−1.18645	−0.593225	−	0.805037i	$-0.702146\pi$
$359$	−1.04011e9	−1.18645	−0.593225	+	0.805037i	$0.702146\pi$
$360$	−1.40039e9	−1.58194
$361$	−2.51162e8	−0.280982
$362$	−4.63583e7	−0.0513627
$363$	−1.70008e7	−0.0186550
$364$	3.72774e8	0.405127
$365$	3.43226e8	0.369450
$366$	1.53340e8	0.163482
$367$	3.08148e8	0.325408	0.162704	−	0.986675i	$-0.447978\pi$
$367$	3.08148e8	0.325408	0.162704	+	0.986675i	$0.447978\pi$
$368$	−3.67191e9	−3.84082
$369$	5.95831e8	0.617349
$370$	−2.01958e8	−0.207279
$371$	−1.28145e8	−0.130285
$372$	1.11947e8	0.112748
$373$	−1.52740e9	−1.52395	−0.761977	−	0.647604i	$-0.775771\pi$
$373$	−1.52740e9	−1.52395	−0.761977	+	0.647604i	$0.775771\pi$
$374$	7.35329e8	0.726827
$375$	−3.74513e7	−0.0366740
$376$	−5.89339e8	−0.571752
$377$	2.44400e9	2.34912
$378$	1.90459e7	0.0181376
$379$	−6.94059e8	−0.654876	−0.327438	−	0.944873i	$-0.606185\pi$
$379$	−6.94059e8	−0.654876	−0.327438	+	0.944873i	$0.606185\pi$
$380$	1.13107e9	1.05742
$381$	6.36347e7	0.0589463
$382$	3.95093e9	3.62641
$383$	4.76633e8	0.433500	0.216750	−	0.976227i	$-0.430454\pi$
$383$	4.76633e8	0.433500	0.216750	+	0.976227i	$0.430454\pi$
$384$	2.60035e8	0.234354
$385$	−3.79564e7	−0.0338979
$386$	2.81174e9	2.48840
$387$	1.73510e8	0.152172
$388$	4.71691e9	4.09964
$389$	7.93506e8	0.683482	0.341741	−	0.939794i	$-0.388984\pi$
$389$	7.93506e8	0.683482	0.341741	+	0.939794i	$0.388984\pi$
$390$	−6.63221e7	−0.0566151
$391$	−5.13191e8	−0.434171
$392$	−4.26896e9	−3.57949
$393$	1.04028e8	0.0864527
$394$	−2.83521e9	−2.33533
$395$	4.11437e8	0.335903
$396$	−2.70940e9	−2.19250
$397$	1.68006e9	1.34759	0.673795	−	0.738918i	$-0.264663\pi$
$397$	1.68006e9	1.34759	0.673795	+	0.738918i	$0.264663\pi$
$398$	−2.88326e9	−2.29241
$399$	−4.98186e6	−0.00392632
$400$	−4.38900e9	−3.42891
$401$	−7.52355e8	−0.582663	−0.291331	−	0.956622i	$-0.594098\pi$
$401$	−7.52355e8	−0.582663	−0.291331	+	0.956622i	$0.594098\pi$
$402$	−1.69124e8	−0.129841
$403$	−1.60233e9	−1.21951
$404$	1.12824e8	0.0851272
$405$	5.82403e8	0.435643
$406$	4.36674e8	0.323829
$407$	−2.53355e8	−0.186272
$408$	1.10130e8	0.0802777
$409$	−4.62613e8	−0.334338	−0.167169	−	0.985928i	$-0.553463\pi$
$409$	−4.62613e8	−0.334338	−0.167169	+	0.985928i	$0.553463\pi$
$410$	−7.42218e8	−0.531848
$411$	1.27176e7	0.00903560
$412$	7.27407e9	5.12433
$413$	1.31424e8	0.0918017
$414$	2.55575e9	1.77018
$415$	8.70504e8	0.597864
$416$	−9.83795e9	−6.70005
$417$	−1.10897e8	−0.0748935
$418$	1.91781e9	1.28437
$419$	1.30611e9	0.867420	0.433710	−	0.901053i	$-0.357204\pi$
$419$	1.30611e9	0.867420	0.433710	+	0.901053i	$0.357204\pi$
$420$	−8.76731e6	−0.00577423
$421$	−2.16516e8	−0.141417	−0.0707087	−	0.997497i	$-0.522526\pi$
$421$	−2.16516e8	−0.141417	−0.0707087	+	0.997497i	$0.522526\pi$
$422$	−4.11899e9	−2.66807
$423$	2.45627e8	0.157792
$424$	7.38825e9	4.70718
$425$	−6.13413e8	−0.387607
$426$	−1.57117e7	−0.00984668
$427$	−2.90137e8	−0.180346
$428$	6.45545e8	0.397991
$429$	−8.32005e7	−0.0508775
$430$	−2.16138e8	−0.131097
$431$	2.46067e9	1.48041	0.740207	−	0.672379i	$-0.234727\pi$
$431$	2.46067e9	1.48041	0.740207	+	0.672379i	$0.234727\pi$
$432$	−6.57544e8	−0.392402
$433$	−1.42408e9	−0.843001	−0.421500	−	0.906828i	$-0.638496\pi$
$433$	−1.42408e9	−0.843001	−0.421500	+	0.906828i	$0.638496\pi$
$434$	−2.86291e8	−0.168110
$435$	−5.74807e7	−0.0334818
$436$	2.59215e9	1.49781
$437$	−1.33845e9	−0.767218
$438$	1.34424e8	0.0764396
$439$	−1.86431e9	−1.05170	−0.525851	−	0.850577i	$-0.676253\pi$
$439$	−1.86431e9	−1.05170	−0.525851	+	0.850577i	$0.676253\pi$
$440$	2.18839e9	1.22473
$441$	1.77923e9	0.987866
$442$	−2.43111e9	−1.33914
$443$	−4.59781e8	−0.251269	−0.125634	−	0.992077i	$-0.540097\pi$
$443$	−4.59781e8	−0.251269	−0.125634	+	0.992077i	$0.540097\pi$
$444$	−5.85209e7	−0.0317300
$445$	−1.62486e8	−0.0874089
$446$	3.50278e8	0.186957
$447$	−1.49557e7	−0.00792007
$448$	−9.49277e8	−0.498792
$449$	1.81677e9	0.947192	0.473596	−	0.880742i	$-0.342956\pi$
$449$	1.81677e9	0.947192	0.473596	+	0.880742i	$0.342956\pi$
$450$	3.05487e9	1.58033
$451$	−9.31106e8	−0.477948
$452$	−3.44903e9	−1.75676
$453$	−1.50889e8	−0.0762629
$454$	7.49103e8	0.375704
$455$	1.25489e8	0.0624550
$456$	2.87230e8	0.141858
$457$	−1.28773e9	−0.631131	−0.315566	−	0.948904i	$-0.602194\pi$
$457$	−1.28773e9	−0.631131	−0.315566	+	0.948904i	$0.602194\pi$
$458$	5.76930e9	2.80604
$459$	−9.18992e7	−0.0443575
$460$	−2.35548e9	−1.12831
$461$	−1.09265e8	−0.0519430	−0.0259715	−	0.999663i	$-0.508268\pi$
$461$	−1.09265e8	−0.0519430	−0.0259715	+	0.999663i	$0.508268\pi$
$462$	−1.48656e7	−0.00701351
$463$	−1.73669e9	−0.813185	−0.406592	−	0.913610i	$-0.633283\pi$
$463$	−1.73669e9	−0.813185	−0.406592	+	0.913610i	$0.633283\pi$
$464$	−1.50758e10	−7.00595
$465$	3.76853e7	0.0173815
$466$	3.21825e9	1.47323
$467$	−2.48860e9	−1.13070	−0.565349	−	0.824852i	$-0.691259\pi$
$467$	−2.48860e9	−1.13070	−0.565349	+	0.824852i	$0.691259\pi$
$468$	8.95767e9	4.03957
$469$	3.20003e8	0.143235
$470$	−3.05974e8	−0.135938
$471$	−1.32036e8	−0.0582262
$472$	−7.57731e9	−3.31679
$473$	−2.71144e8	−0.117811
$474$	1.61139e8	0.0694986
$475$	−1.59984e9	−0.684935
$476$	−3.21375e8	−0.136580
$477$	−3.07930e9	−1.29909
$478$	−3.44478e9	−1.44266
$479$	8.40928e8	0.349611	0.174805	−	0.984603i	$-0.444070\pi$
$479$	8.40928e8	0.349611	0.174805	+	0.984603i	$0.444070\pi$
$480$	2.31380e8	0.0954951
$481$	8.37628e8	0.343197
$482$	7.16483e9	2.91435
$483$	1.03748e7	0.00418953
$484$	−2.86032e9	−1.14671
$485$	1.58788e9	0.632008
$486$	6.86749e8	0.271376
$487$	−1.41392e9	−0.554718	−0.277359	−	0.960766i	$-0.589459\pi$
$487$	−1.41392e9	−0.554718	−0.277359	+	0.960766i	$0.589459\pi$
$488$	1.67279e10	6.51588
$489$	−2.79096e6	−0.00107937
$490$	−2.21637e9	−0.851050
$491$	−2.99438e9	−1.14162	−0.570810	−	0.821082i	$-0.693371\pi$
$491$	−2.99438e9	−1.14162	−0.570810	+	0.821082i	$0.693371\pi$
$492$	−2.15070e8	−0.0814147
$493$	−2.10701e9	−0.791959
$494$	−6.34057e9	−2.36638
$495$	−9.12083e8	−0.338000
$496$	9.88395e9	3.63701
$497$	2.97285e7	0.0108624
$498$	3.40932e8	0.123699
$499$	−3.88719e9	−1.40050	−0.700251	−	0.713897i	$-0.746929\pi$
$499$	−3.88719e9	−1.40050	−0.700251	+	0.713897i	$0.746929\pi$
$500$	−6.30104e9	−2.25433
$501$	−1.78709e8	−0.0634913
$502$	−5.22200e9	−1.84236
$503$	−2.25318e9	−0.789420	−0.394710	−	0.918806i	$-0.629155\pi$
$503$	−2.25318e9	−0.789420	−0.394710	+	0.918806i	$0.629155\pi$
$504$	1.03775e9	0.361067
$505$	3.79809e7	0.0131234
$506$	−3.99388e9	−1.37047
$507$	1.39299e8	0.0474700
$508$	1.07063e10	3.62340
$509$	2.23850e9	0.752393	0.376196	−	0.926540i	$-0.377232\pi$
$509$	2.23850e9	0.752393	0.376196	+	0.926540i	$0.377232\pi$
$510$	5.71774e7	0.0190866
$511$	−2.54347e8	−0.0843244
$512$	1.40717e10	4.63341
$513$	−2.39682e8	−0.0783837
$514$	−9.43165e8	−0.306349
$515$	2.44872e9	0.789976
$516$	−6.26298e7	−0.0200681
$517$	−3.83841e8	−0.122162
$518$	1.49661e8	0.0473101
$519$	−1.31208e7	−0.00411980
$520$	−7.23513e9	−2.25650
$521$	3.53193e7	0.0109416	0.00547079	−	0.999985i	$-0.498259\pi$
$521$	3.53193e7	0.0109416	0.00547079	+	0.999985i	$0.498259\pi$
$522$	1.04932e10	3.22894
$523$	1.93242e9	0.590671	0.295335	−	0.955394i	$-0.404569\pi$
$523$	1.93242e9	0.590671	0.295335	+	0.955394i	$0.404569\pi$
$524$	1.75024e10	5.31420
$525$	1.24009e7	0.00374021
$526$	2.31852e9	0.694642
$527$	1.38139e9	0.411131
$528$	5.13222e8	0.151735
$529$	−6.17469e8	−0.181351
$530$	3.83584e9	1.11917
$531$	3.15810e9	0.915365
$532$	−8.38178e8	−0.241349
$533$	3.07837e9	0.880595
$534$	−6.36375e7	−0.0180850
$535$	2.17314e8	0.0613550
$536$	−1.84498e10	−5.17507
$537$	−1.09631e8	−0.0305508
$538$	−4.13419e9	−1.14460
$539$	−2.78041e9	−0.764801
$540$	−4.21805e8	−0.115275
$541$	4.16649e9	1.13130	0.565652	−	0.824644i	$-0.308625\pi$
$541$	4.16649e9	1.13130	0.565652	+	0.824644i	$0.308625\pi$
$542$	7.48079e9	2.01813
$543$	−4.52208e6	−0.00121210
$544$	8.48147e9	2.25878
$545$	8.72612e8	0.230905
$546$	4.91479e7	0.0129220
$547$	−2.80094e9	−0.731725	−0.365862	−	0.930669i	$-0.619226\pi$
$547$	−2.80094e9	−0.731725	−0.365862	+	0.930669i	$0.619226\pi$
$548$	2.13968e9	0.555414
$549$	−6.97193e9	−1.79825
$550$	−4.77385e9	−1.22349
$551$	−5.49530e9	−1.39946
$552$	−5.98162e8	−0.151367
$553$	−3.04894e8	−0.0766675
$554$	−9.38068e9	−2.34396
$555$	−1.97003e7	−0.00489155
$556$	−1.86580e10	−4.60367
$557$	4.87182e9	1.19453	0.597266	−	0.802043i	$-0.296254\pi$
$557$	4.87182e9	1.19453	0.597266	+	0.802043i	$0.296254\pi$
$558$	−6.87951e9	−1.67625
$559$	8.96441e8	0.217060
$560$	−7.74081e8	−0.186264
$561$	7.17286e7	0.0171523
$562$	7.57716e9	1.80065
$563$	6.20948e9	1.46648	0.733240	−	0.679970i	$-0.238007\pi$
$563$	6.20948e9	1.46648	0.733240	+	0.679970i	$0.238007\pi$
$564$	−8.86612e7	−0.0208093
$565$	−1.16107e9	−0.270825
$566$	−4.29857e9	−0.996475
$567$	−4.31588e8	−0.0994327
$568$	−1.71400e9	−0.392457
$569$	−5.71470e9	−1.30047	−0.650235	−	0.759733i	$-0.725330\pi$
$569$	−5.71470e9	−1.30047	−0.650235	+	0.759733i	$0.725330\pi$
$570$	1.49125e8	0.0337277
$571$	2.96607e9	0.666737	0.333368	−	0.942797i	$-0.391815\pi$
$571$	2.96607e9	0.666737	0.333368	+	0.942797i	$0.391815\pi$
$572$	−1.39982e10	−3.12741
$573$	3.85398e8	0.0855793
$574$	5.50019e8	0.121391
$575$	3.33170e9	0.730851
$576$	−2.28109e10	−4.97352
$577$	5.24837e9	1.13739	0.568694	−	0.822549i	$-0.307449\pi$
$577$	5.24837e9	1.13739	0.568694	+	0.822549i	$0.307449\pi$
$578$	−7.00631e9	−1.50918
$579$	2.74275e8	0.0587235
$580$	−9.67090e9	−2.05811
$581$	−6.45085e8	−0.136458
$582$	6.21894e8	0.130763
$583$	4.81203e9	1.00575
$584$	1.46644e10	3.04664
$585$	3.01548e9	0.622747
$586$	1.02912e10	2.11264
$587$	1.22772e9	0.250534	0.125267	−	0.992123i	$-0.460021\pi$
$587$	1.22772e9	0.250534	0.125267	+	0.992123i	$0.460021\pi$
$588$	−6.42230e8	−0.130278
$589$	3.60282e9	0.726505
$590$	−3.93400e9	−0.788591
$591$	−2.76564e8	−0.0551111
$592$	−5.16690e9	−1.02354
$593$	6.05113e9	1.19164	0.595820	−	0.803118i	$-0.296827\pi$
$593$	6.05113e9	1.19164	0.595820	+	0.803118i	$0.296827\pi$
$594$	−7.15200e8	−0.140015
$595$	−1.08187e8	−0.0210554
$596$	−2.51623e9	−0.486843
$597$	−2.81251e8	−0.0540983
$598$	1.32044e10	2.52501
$599$	5.84902e9	1.11196	0.555980	−	0.831196i	$-0.312343\pi$
$599$	5.84902e9	1.11196	0.555980	+	0.831196i	$0.312343\pi$
$600$	−7.14978e8	−0.135134
$601$	−8.45352e9	−1.58846	−0.794231	−	0.607616i	$-0.792126\pi$
$601$	−8.45352e9	−1.58846	−0.794231	+	0.607616i	$0.792126\pi$
$602$	1.60169e8	0.0299220
$603$	7.68959e9	1.42821
$604$	−2.53865e10	−4.68784
$605$	−9.62888e8	−0.176779
$606$	1.48752e7	0.00271524
$607$	−9.03739e8	−0.164015	−0.0820074	−	0.996632i	$-0.526133\pi$
$607$	−9.03739e8	−0.164015	−0.0820074	+	0.996632i	$0.526133\pi$
$608$	2.21205e10	3.99147
$609$	4.25959e7	0.00764201
$610$	8.68484e9	1.54920
$611$	1.26904e9	0.225076
$612$	−7.72256e9	−1.36186
$613$	−3.64313e9	−0.638797	−0.319399	−	0.947620i	$-0.603481\pi$
$613$	−3.64313e9	−0.638797	−0.319399	+	0.947620i	$0.603481\pi$
$614$	2.44757e9	0.426723
$615$	−7.24006e7	−0.0125510
$616$	−1.62170e9	−0.279536
$617$	8.10529e9	1.38922	0.694609	−	0.719387i	$-0.255577\pi$
$617$	8.10529e9	1.38922	0.694609	+	0.719387i	$0.255577\pi$
$618$	9.59039e8	0.163447
$619$	−7.22020e9	−1.22358	−0.611789	−	0.791021i	$-0.709550\pi$
$619$	−7.22020e9	−1.22358	−0.611789	+	0.791021i	$0.709550\pi$
$620$	6.34042e9	1.06843
$621$	4.99143e8	0.0836382
$622$	−3.67712e9	−0.612691
$623$	1.20410e8	0.0199505
$624$	−1.69679e9	−0.279564
$625$	2.80898e9	0.460224
$626$	−2.28427e10	−3.72166
$627$	1.87075e8	0.0303096
$628$	−2.22145e10	−3.57914
$629$	−7.22134e8	−0.115702
$630$	5.38782e8	0.0858462
$631$	4.23517e8	0.0671070	0.0335535	−	0.999437i	$-0.489318\pi$
$631$	4.23517e8	0.0671070	0.0335535	+	0.999437i	$0.489318\pi$
$632$	1.75788e10	2.76999
$633$	−4.01793e8	−0.0629635
$634$	1.41984e10	2.21272
$635$	3.60413e9	0.558590
$636$	1.11150e9	0.171321
$637$	9.19245e9	1.40911
$638$	−1.63977e10	−2.49983
$639$	7.14368e8	0.108310
$640$	1.47278e10	2.22080
$641$	3.03264e9	0.454797	0.227398	−	0.973802i	$-0.426978\pi$
$641$	3.03264e9	0.454797	0.227398	+	0.973802i	$0.426978\pi$
$642$	8.51110e7	0.0126944
$643$	−2.78734e9	−0.413478	−0.206739	−	0.978396i	$-0.566285\pi$
$643$	−2.78734e9	−0.413478	−0.206739	+	0.978396i	$0.566285\pi$
$644$	1.74552e9	0.257528
$645$	−2.10835e7	−0.00309374
$646$	5.46631e9	0.797776
$647$	−1.73353e9	−0.251632	−0.125816	−	0.992054i	$-0.540155\pi$
$647$	−1.73353e9	−0.251632	−0.125816	+	0.992054i	$0.540155\pi$
$648$	2.48834e10	3.59250
$649$	−4.93516e9	−0.708672
$650$	1.57830e10	2.25421
$651$	−2.79266e7	−0.00396721
$652$	−4.69568e8	−0.0663486
$653$	−9.51546e9	−1.33731	−0.668657	−	0.743571i	$-0.733131\pi$
$653$	−9.51546e9	−1.33731	−0.668657	+	0.743571i	$0.733131\pi$
$654$	3.41758e8	0.0477745
$655$	5.89195e9	0.819247
$656$	−1.89889e10	−2.62626
$657$	−6.11190e9	−0.840809
$658$	2.26741e8	0.0310270
$659$	7.55803e9	1.02875	0.514375	−	0.857565i	$-0.328024\pi$
$659$	7.55803e9	1.02875	0.514375	+	0.857565i	$0.328024\pi$
$660$	3.29225e8	0.0445747
$661$	−1.39999e9	−0.188547	−0.0942733	−	0.995546i	$-0.530053\pi$
$661$	−1.39999e9	−0.188547	−0.0942733	+	0.995546i	$0.530053\pi$
$662$	−9.53475e9	−1.27734
$663$	−2.37145e8	−0.0316022
$664$	3.71926e10	4.93024
$665$	−2.82162e8	−0.0372068
$666$	3.59631e9	0.471734
$667$	1.14441e10	1.49328
$668$	−3.00671e10	−3.90278
$669$	3.41683e7	0.00441197
$670$	−9.57881e9	−1.23041
$671$	1.08950e10	1.39220
$672$	−1.71464e8	−0.0217961
$673$	−5.04985e8	−0.0638595	−0.0319298	−	0.999490i	$-0.510165\pi$
$673$	−5.04985e8	−0.0638595	−0.0319298	+	0.999490i	$0.510165\pi$
$674$	−8.38806e9	−1.05524
$675$	5.96621e8	0.0746682
$676$	2.34365e10	2.91796
$677$	−8.77580e9	−1.08699	−0.543496	−	0.839412i	$-0.682900\pi$
$677$	−8.77580e9	−1.08699	−0.543496	+	0.839412i	$0.682900\pi$
$678$	−4.54732e8	−0.0560340
$679$	−1.17670e9	−0.144252
$680$	6.23753e9	0.760732
$681$	7.30723e7	0.00886620
$682$	1.07506e10	1.29774
$683$	5.66666e9	0.680541	0.340271	−	0.940328i	$-0.389481\pi$
$683$	5.66666e9	0.680541	0.340271	+	0.940328i	$0.389481\pi$
$684$	−2.01412e10	−2.40652
$685$	7.20295e8	0.0856236
$686$	3.30148e9	0.390458
$687$	5.62774e8	0.0662194
$688$	−5.52969e9	−0.647353
$689$	−1.59093e10	−1.85303
$690$	−3.10555e8	−0.0359887
$691$	−7.72356e9	−0.890522	−0.445261	−	0.895401i	$-0.646889\pi$
$691$	−7.72356e9	−0.890522	−0.445261	+	0.895401i	$0.646889\pi$
$692$	−2.20753e9	−0.253242
$693$	6.75897e8	0.0771462
$694$	−7.06555e9	−0.802394
$695$	−6.28098e9	−0.709709
$696$	−2.45588e9	−0.276105
$697$	−2.65392e9	−0.296874
$698$	2.21566e10	2.46609
$699$	3.13929e8	0.0347665
$700$	2.08641e9	0.229909
$701$	−9.02472e9	−0.989511	−0.494755	−	0.869032i	$-0.664742\pi$
$701$	−9.02472e9	−0.989511	−0.494755	+	0.869032i	$0.664742\pi$
$702$	2.36456e9	0.257970
$703$	−1.88340e9	−0.204455
$704$	3.56466e10	3.85047
$705$	−2.98466e7	−0.00320799
$706$	2.71277e9	0.290133
$707$	−2.81456e7	−0.00299532
$708$	−1.13994e9	−0.120717
$709$	1.01828e10	1.07301	0.536505	−	0.843897i	$-0.319744\pi$
$709$	1.01828e10	1.07301	0.536505	+	0.843897i	$0.319744\pi$
$710$	−8.89878e8	−0.0933096
$711$	−7.32654e9	−0.764461
$712$	−6.94226e9	−0.720810
$713$	−7.50294e9	−0.775207
$714$	−4.23712e7	−0.00435640
$715$	−4.71230e9	−0.482127
$716$	−1.84450e10	−1.87794
$717$	−3.36026e8	−0.0340452
$718$	−2.30720e10	−2.32621
$719$	−3.88344e9	−0.389642	−0.194821	−	0.980839i	$-0.562413\pi$
$719$	−3.88344e9	−0.389642	−0.194821	+	0.980839i	$0.562413\pi$
$720$	−1.86010e10	−1.85726
$721$	−1.81462e9	−0.180307
$722$	−5.57133e9	−0.550907
$723$	6.98903e8	0.0687753
$724$	−7.60823e8	−0.0745073
$725$	1.36790e10	1.33313
$726$	−3.77114e8	−0.0365759
$727$	3.21178e9	0.310009	0.155005	−	0.987914i	$-0.450461\pi$
$727$	3.21178e9	0.310009	0.155005	+	0.987914i	$0.450461\pi$
$728$	5.36158e9	0.515030
$729$	−1.03262e10	−0.987178
$730$	7.61351e9	0.724360
$731$	−7.72837e8	−0.0731775
$732$	2.51658e9	0.237149
$733$	2.53300e9	0.237559	0.118779	−	0.992921i	$-0.462102\pi$
$733$	2.53300e9	0.237559	0.118779	+	0.992921i	$0.462102\pi$
$734$	6.83540e9	0.638010
$735$	−2.16198e8	−0.0200838
$736$	−4.60664e10	−4.25904
$737$	−1.20165e10	−1.10571
$738$	1.32168e10	1.21040
$739$	8.64031e9	0.787542	0.393771	−	0.919209i	$-0.371170\pi$
$739$	8.64031e9	0.787542	0.393771	+	0.919209i	$0.371170\pi$
$740$	−3.31450e9	−0.300681
$741$	−6.18499e8	−0.0558439
$742$	−2.84254e9	−0.255443
$743$	−7.96383e9	−0.712297	−0.356148	−	0.934429i	$-0.615910\pi$
$743$	−7.96383e9	−0.712297	−0.356148	+	0.934429i	$0.615910\pi$
$744$	1.61012e9	0.143335
$745$	−8.47057e8	−0.0750526
$746$	−3.38811e10	−2.98794
$747$	−1.55012e10	−1.36064
$748$	1.20681e10	1.05434
$749$	−1.61040e8	−0.0140039
$750$	−8.30752e8	−0.0719047
$751$	3.65497e9	0.314879	0.157439	−	0.987529i	$-0.449676\pi$
$751$	3.65497e9	0.314879	0.157439	+	0.987529i	$0.449676\pi$
$752$	−7.82804e9	−0.671260
$753$	−5.09387e8	−0.0434776
$754$	5.42132e10	4.60580
$755$	−8.54602e9	−0.722686
$756$	3.12578e8	0.0263107
$757$	1.20006e9	0.100546	0.0502732	−	0.998736i	$-0.483991\pi$
$757$	1.20006e9	0.100546	0.0502732	+	0.998736i	$0.483991\pi$
$758$	−1.53958e10	−1.28398
$759$	−3.89588e8	−0.0323414
$760$	1.62681e10	1.34428
$761$	2.32910e9	0.191576	0.0957880	−	0.995402i	$-0.469463\pi$
$761$	2.32910e9	0.191576	0.0957880	+	0.995402i	$0.469463\pi$
$762$	1.41156e9	0.115573
$763$	−6.46647e8	−0.0527025
$764$	6.48418e10	5.26052
$765$	−2.59970e9	−0.209946
$766$	1.05728e10	0.849940
$767$	1.63164e10	1.30569
$768$	2.87314e9	0.228872
$769$	−4.56990e9	−0.362381	−0.181190	−	0.983448i	$-0.557995\pi$
$769$	−4.56990e9	−0.362381	−0.181190	+	0.983448i	$0.557995\pi$
$770$	−8.41956e8	−0.0664618
$771$	−9.20022e7	−0.00722950
$772$	4.61458e10	3.60970
$773$	1.34361e10	1.04627	0.523135	−	0.852250i	$-0.324762\pi$
$773$	1.34361e10	1.04627	0.523135	+	0.852250i	$0.324762\pi$
$774$	3.84882e9	0.298356
$775$	−8.96819e9	−0.692068
$776$	6.78429e10	5.21181
$777$	1.45988e7	0.00111646
$778$	1.76017e10	1.34007
$779$	−6.92169e9	−0.524603
$780$	−1.08847e9	−0.0821265
$781$	−1.11634e9	−0.0838532
$782$	−1.13837e10	−0.851255
$783$	2.04934e9	0.152562
$784$	−5.67036e10	−4.20247
$785$	−7.47823e9	−0.551766
$786$	2.30758e9	0.169503
$787$	2.05668e10	1.50403	0.752014	−	0.659147i	$-0.229082\pi$
$787$	2.05668e10	1.50403	0.752014	+	0.659147i	$0.229082\pi$
$788$	−4.65308e10	−3.38765
$789$	2.26163e8	0.0163928
$790$	9.12657e9	0.658586
$791$	8.60408e8	0.0618140
$792$	−3.89691e10	−2.78729
$793$	−3.60207e10	−2.56505
$794$	3.72674e10	2.64215
$795$	3.74172e8	0.0264111
$796$	−4.73194e10	−3.32540
$797$	1.49135e10	1.04346	0.521729	−	0.853111i	$-0.325287\pi$
$797$	1.49135e10	1.04346	0.521729	+	0.853111i	$0.325287\pi$
$798$	−1.10508e8	−0.00769813
$799$	−1.09406e9	−0.0758799
$800$	−5.50628e10	−3.80227
$801$	2.89342e9	0.198929
$802$	−1.66889e10	−1.14240
$803$	9.55107e9	0.650950
$804$	−2.77563e9	−0.188350
$805$	5.87607e8	0.0397010
$806$	−3.55432e10	−2.39102
$807$	−4.03275e8	−0.0270112
$808$	1.62275e9	0.108221
$809$	−2.29446e10	−1.52356	−0.761781	−	0.647835i	$-0.775675\pi$
$809$	−2.29446e10	−1.52356	−0.761781	+	0.647835i	$0.775675\pi$
$810$	1.29190e10	0.854143
$811$	4.15707e9	0.273662	0.136831	−	0.990594i	$-0.456308\pi$
$811$	4.15707e9	0.273662	0.136831	+	0.990594i	$0.456308\pi$
$812$	7.16660e9	0.469750
$813$	7.29723e8	0.0476257
$814$	−5.61996e9	−0.365215
$815$	−1.58074e8	−0.0102284
$816$	1.46283e9	0.0942493
$817$	−2.01564e9	−0.129311
$818$	−1.02618e10	−0.655520
$819$	−2.23462e9	−0.142138
$820$	−1.21811e10	−0.771506
$821$	1.06838e10	0.673790	0.336895	−	0.941542i	$-0.390623\pi$
$821$	1.06838e10	0.673790	0.336895	+	0.941542i	$0.390623\pi$
$822$	2.82103e8	0.0177156
$823$	−1.14304e10	−0.714766	−0.357383	−	0.933958i	$-0.616331\pi$
$823$	−1.14304e10	−0.714766	−0.357383	+	0.933958i	$0.616331\pi$
$824$	1.04622e11	6.51447
$825$	−4.65671e8	−0.0288729
$826$	2.91528e9	0.179991
$827$	−1.17022e10	−0.719448	−0.359724	−	0.933059i	$-0.617129\pi$
$827$	−1.17022e10	−0.719448	−0.359724	+	0.933059i	$0.617129\pi$
$828$	4.19445e10	2.56784
$829$	−1.53161e10	−0.933699	−0.466850	−	0.884337i	$-0.654611\pi$
$829$	−1.53161e10	−0.933699	−0.466850	+	0.884337i	$0.654611\pi$
$830$	1.93097e10	1.17220
$831$	−9.15051e8	−0.0553148
$832$	−1.17853e11	−7.09430
$833$	−7.92497e9	−0.475051
$834$	−2.45994e9	−0.146840
$835$	−1.01217e10	−0.601659
$836$	3.14747e10	1.86312
$837$	−1.34358e9	−0.0791999
$838$	2.89723e10	1.70070
$839$	1.05214e10	0.615043	0.307521	−	0.951541i	$-0.400500\pi$
$839$	1.05214e10	0.615043	0.307521	+	0.951541i	$0.400500\pi$
$840$	−1.26100e8	−0.00734068
$841$	2.97361e10	1.72385
$842$	−4.80280e9	−0.277270
$843$	7.39124e8	0.0424933
$844$	−6.76001e10	−3.87034
$845$	7.88958e9	0.449837
$846$	5.44854e9	0.309374
$847$	7.13546e8	0.0403487
$848$	9.81363e10	5.52642
$849$	−4.19310e8	−0.0235157
$850$	−1.36068e10	−0.759960
$851$	3.92221e9	0.218161
$852$	−2.57857e8	−0.0142837
$853$	−2.95826e10	−1.63198	−0.815989	−	0.578068i	$-0.803807\pi$
$853$	−2.95826e10	−1.63198	−0.815989	+	0.578068i	$0.803807\pi$
$854$	−6.43588e9	−0.353594
$855$	−6.78028e9	−0.370993
$856$	9.28482e9	0.505959
$857$	6.19614e9	0.336270	0.168135	−	0.985764i	$-0.446225\pi$
$857$	6.19614e9	0.336270	0.168135	+	0.985764i	$0.446225\pi$
$858$	−1.84557e9	−0.0997528
$859$	2.78077e10	1.49689	0.748443	−	0.663199i	$-0.230802\pi$
$859$	2.78077e10	1.49689	0.748443	+	0.663199i	$0.230802\pi$
$860$	−3.54722e9	−0.190171
$861$	5.36523e7	0.00286469
$862$	5.45831e10	2.90257
$863$	3.15999e10	1.67359	0.836793	−	0.547519i	$-0.184428\pi$
$863$	3.15999e10	1.67359	0.836793	+	0.547519i	$0.184428\pi$
$864$	−8.24930e9	−0.435130
$865$	−7.43137e8	−0.0390402
$866$	−3.15893e10	−1.65283
$867$	−6.83440e8	−0.0356150
$868$	−4.69855e9	−0.243863
$869$	1.14492e10	0.591842
$870$	−1.27505e9	−0.0656461
$871$	3.97284e10	2.03722
$872$	3.72826e10	1.90414
$873$	−2.82758e10	−1.43835
$874$	−2.96899e10	−1.50424
$875$	1.57188e9	0.0793217
$876$	2.20614e9	0.110884
$877$	2.48099e10	1.24201	0.621006	−	0.783806i	$-0.286724\pi$
$877$	2.48099e10	1.24201	0.621006	+	0.783806i	$0.286724\pi$
$878$	−4.13545e10	−2.06201
$879$	1.00387e9	0.0498560
$880$	2.90678e10	1.43788
$881$	2.37579e9	0.117056	0.0585279	−	0.998286i	$-0.481359\pi$
$881$	2.37579e9	0.117056	0.0585279	+	0.998286i	$0.481359\pi$
$882$	3.94673e10	1.93686
$883$	−1.16100e10	−0.567505	−0.283752	−	0.958898i	$-0.591579\pi$
$883$	−1.16100e10	−0.567505	−0.283752	+	0.958898i	$0.591579\pi$
$884$	−3.98988e10	−1.94257
$885$	−3.83747e8	−0.0186099
$886$	−1.01990e10	−0.492649
$887$	1.53556e10	0.738813	0.369407	−	0.929268i	$-0.379561\pi$
$887$	1.53556e10	0.738813	0.369407	+	0.929268i	$0.379561\pi$
$888$	−8.41701e8	−0.0403378
$889$	−2.67084e9	−0.127494
$890$	−3.60429e9	−0.171378
$891$	1.62067e10	0.767580
$892$	5.74869e9	0.271201
$893$	−2.85342e9	−0.134086
$894$	−3.31749e8	−0.0155285
$895$	−6.20926e9	−0.289507
$896$	−1.09140e10	−0.506883
$897$	1.28804e9	0.0595874
$898$	4.03000e10	1.85711
$899$	−3.08049e10	−1.41404
$900$	5.01359e10	2.29245
$901$	1.37157e10	0.624712
$902$	−2.06540e10	−0.937088
$903$	1.56239e7	0.000706126 0
$904$	−4.96071e10	−2.23334
$905$	−2.56121e8	−0.0114862
$906$	−3.34704e9	−0.149525
$907$	3.31807e9	0.147659	0.0738295	−	0.997271i	$-0.476478\pi$
$907$	3.31807e9	0.147659	0.0738295	+	0.997271i	$0.476478\pi$
$908$	1.22941e10	0.545001
$909$	−6.76333e8	−0.0298667
$910$	2.78363e9	0.122452
$911$	−3.21961e9	−0.141087	−0.0705437	−	0.997509i	$-0.522473\pi$
$911$	−3.21961e9	−0.141087	−0.0705437	+	0.997509i	$0.522473\pi$
$912$	3.81521e9	0.166547
$913$	2.42238e10	1.05340
$914$	−2.85648e10	−1.23743
$915$	8.47174e8	0.0365594
$916$	9.46846e10	4.07047
$917$	−4.36622e9	−0.186988
$918$	−2.03853e9	−0.0869695
$919$	7.57125e9	0.321783	0.160892	−	0.986972i	$-0.448563\pi$
$919$	7.57125e9	0.321783	0.160892	+	0.986972i	$0.448563\pi$
$920$	−3.38787e10	−1.43439
$921$	2.38752e8	0.0100702
$922$	−2.42373e9	−0.101842
$923$	3.69080e9	0.154495
$924$	−2.43971e8	−0.0101739
$925$	4.68818e9	0.194764
$926$	−3.85236e10	−1.59437
$927$	−4.36049e10	−1.79786
$928$	−1.89135e11	−7.76881
$929$	−2.89017e10	−1.18268	−0.591342	−	0.806421i	$-0.701402\pi$
$929$	−2.89017e10	−1.18268	−0.591342	+	0.806421i	$0.701402\pi$
$930$	8.35943e8	0.0340790
$931$	−2.06691e10	−0.839457
$932$	5.28173e10	2.13708
$933$	−3.58690e8	−0.0144588
$934$	−5.52027e10	−2.21690
$935$	4.06255e9	0.162539
$936$	1.28837e11	5.13543
$937$	3.91577e10	1.55499	0.777496	−	0.628888i	$-0.216489\pi$
$937$	3.91577e10	1.55499	0.777496	+	0.628888i	$0.216489\pi$
$938$	7.09836e9	0.280833
$939$	−2.22822e9	−0.0878271
$940$	−5.02158e9	−0.197194
$941$	−2.41576e10	−0.945127	−0.472564	−	0.881297i	$-0.656671\pi$
$941$	−2.41576e10	−0.945127	−0.472564	+	0.881297i	$0.656671\pi$
$942$	−2.92884e9	−0.114161
$943$	1.44145e10	0.559771
$944$	−1.00647e11	−3.89404
$945$	1.05225e8	0.00405610
$946$	−6.01456e9	−0.230986
$947$	3.20511e10	1.22636	0.613179	−	0.789944i	$-0.289890\pi$
$947$	3.20511e10	1.22636	0.613179	+	0.789944i	$0.289890\pi$
$948$	2.64458e9	0.100816
$949$	−3.15773e10	−1.19934
$950$	−3.54880e10	−1.34292
$951$	1.38500e9	0.0522178
$952$	−4.62231e9	−0.173632
$953$	4.93663e9	0.184759	0.0923794	−	0.995724i	$-0.470553\pi$
$953$	4.93663e9	0.184759	0.0923794	+	0.995724i	$0.470553\pi$
$954$	−6.83056e10	−2.54705
$955$	2.18281e10	0.810970
$956$	−5.65350e10	−2.09274
$957$	−1.59953e9	−0.0589932
$958$	1.86536e10	0.685463
$959$	−5.33773e8	−0.0195430
$960$	2.77180e9	0.101114
$961$	−7.31640e9	−0.265929
$962$	1.85804e10	0.672888
$963$	−3.86976e9	−0.139634
$964$	1.17588e11	4.22759
$965$	1.55344e10	0.556478
$966$	2.30136e8	0.00821419
$967$	3.10919e10	1.10574	0.552872	−	0.833266i	$-0.313532\pi$
$967$	3.10919e10	1.10574	0.552872	+	0.833266i	$0.313532\pi$
$968$	−4.11397e10	−1.45780
$969$	5.33219e8	0.0188266
$970$	3.52228e10	1.23915
$971$	3.37297e10	1.18235	0.591174	−	0.806544i	$-0.298665\pi$
$971$	3.37297e10	1.18235	0.591174	+	0.806544i	$0.298665\pi$
$972$	1.12708e10	0.393661
$973$	4.65451e9	0.161986
$974$	−3.13637e10	−1.08761
$975$	1.53958e9	0.0531968
$976$	2.22193e11	7.64991
$977$	−2.60034e10	−0.892072	−0.446036	−	0.895015i	$-0.647165\pi$
$977$	−2.60034e10	−0.892072	−0.446036	+	0.895015i	$0.647165\pi$
$978$	−6.19095e7	−0.00211627
$979$	−4.52155e9	−0.154010
$980$	−3.63746e10	−1.23454
$981$	−1.55388e10	−0.525503
$982$	−6.64219e10	−2.23831
$983$	−4.35623e9	−0.146276	−0.0731381	−	0.997322i	$-0.523301\pi$
$983$	−4.35623e9	−0.146276	−0.0731381	+	0.997322i	$0.523301\pi$
$984$	−3.09334e9	−0.103501
$985$	−1.56640e10	−0.522247
$986$	−4.67381e10	−1.55275
$987$	2.21178e7	0.000732203 0
$988$	−1.04060e11	−3.43269
$989$	4.19761e9	0.137980
$990$	−2.02320e10	−0.662698
$991$	−3.99586e10	−1.30423	−0.652113	−	0.758122i	$-0.726117\pi$
$991$	−3.99586e10	−1.30423	−0.652113	+	0.758122i	$0.726117\pi$
$992$	1.24000e11	4.03304
$993$	−9.30080e8	−0.0301438
$994$	6.59442e8	0.0212973
$995$	−1.59295e10	−0.512649
$996$	5.59531e9	0.179439
$997$	1.79270e10	0.572894	0.286447	−	0.958096i	$-0.407526\pi$
$997$	1.79270e10	0.572894	0.286447	+	0.958096i	$0.407526\pi$
$998$	−8.62264e10	−2.74589
$999$	7.02366e8	0.0222887

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	43.8.a.b.1.13	✓	13
3.2	odd	2		387.8.a.d.1.1		13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.8.a.b.1.13	✓	13	1.1	even	1	trivial
387.8.a.d.1.1		13	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:	\( N \)	\(=\)	\( 43 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	43.a (trivial)

\(f(q)\)	\(=\)	\(q+22.1822 q^{2} +2.16379 q^{3} +364.050 q^{4} +122.553 q^{5} +47.9976 q^{6} -90.8173 q^{7} +5236.10 q^{8} -2182.32 q^{9} +O(q^{10})\)	\(q+22.1822 q^{2} +2.16379 q^{3} +364.050 q^{4} +122.553 q^{5} +47.9976 q^{6} -90.8173 q^{7} +5236.10 q^{8} -2182.32 q^{9} +2718.48 q^{10} +3410.31 q^{11} +787.727 q^{12} -11275.0 q^{13} -2014.53 q^{14} +265.178 q^{15} +69549.8 q^{16} +9720.37 q^{17} -48408.6 q^{18} +25351.7 q^{19} +44615.2 q^{20} -196.510 q^{21} +75648.2 q^{22} -52795.4 q^{23} +11329.8 q^{24} -63105.9 q^{25} -250104. q^{26} -9454.29 q^{27} -33062.0 q^{28} -216763. q^{29} +5882.23 q^{30} +142113. q^{31} +872545. q^{32} +7379.20 q^{33} +215619. q^{34} -11129.9 q^{35} -794472. q^{36} -74290.8 q^{37} +562357. q^{38} -24396.7 q^{39} +641697. q^{40} -273026. q^{41} -4359.01 q^{42} -79507.0 q^{43} +1.24152e6 q^{44} -267449. q^{45} -1.17112e6 q^{46} -112553. q^{47} +150491. q^{48} -815295. q^{49} -1.39983e6 q^{50} +21032.8 q^{51} -4.10466e6 q^{52} +1.41102e6 q^{53} -209717. q^{54} +417942. q^{55} -475528. q^{56} +54855.8 q^{57} -4.80827e6 q^{58} -1.44713e6 q^{59} +96537.9 q^{60} +3.19474e6 q^{61} +3.15239e6 q^{62} +198192. q^{63} +1.04526e7 q^{64} -1.38178e6 q^{65} +163687. q^{66} -3.52359e6 q^{67} +3.53870e6 q^{68} -114238. q^{69} -246885. q^{70} -327344. q^{71} -1.14268e7 q^{72} +2.80064e6 q^{73} -1.64793e6 q^{74} -136548. q^{75} +9.22928e6 q^{76} -309715. q^{77} -541173. q^{78} +3.35723e6 q^{79} +8.52350e6 q^{80} +4.75227e6 q^{81} -6.05633e6 q^{82} +7.10311e6 q^{83} -71539.2 q^{84} +1.19126e6 q^{85} -1.76364e6 q^{86} -469029. q^{87} +1.78567e7 q^{88} -1.32585e6 q^{89} -5.93259e6 q^{90} +1.02396e6 q^{91} -1.92201e7 q^{92} +307504. q^{93} -2.49668e6 q^{94} +3.10692e6 q^{95} +1.88801e6 q^{96} +1.29568e7 q^{97} -1.80850e7 q^{98} -7.44239e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9}+O(q^{10}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9	\( 13 q + 16 q^{2} + 94 q^{3} + 922 q^{4} + 998 q^{5} + 183 q^{6} + 1360 q^{7} + 3870 q^{8} + 10011 q^{9} + 4667 q^{10} + 1620 q^{11} - 19681 q^{12} + 13550 q^{13} + 44160 q^{14} + 31412 q^{15} + 114026 q^{16} + 110880 q^{17} + 159267 q^{18} + 105058 q^{19} + 167251 q^{20} + 129840 q^{21} + 201504 q^{22} + 160184 q^{23} + 161289 q^{24} + 270149 q^{25} + 272104 q^{26} + 252544 q^{27} + 208172 q^{28} + 285546 q^{29} + 107580 q^{30} - 99616 q^{31} + 200126 q^{32} + 531468 q^{33} - 80941 q^{34} - 187104 q^{35} - 608975 q^{36} + 176038 q^{37} + 652165 q^{38} - 794680 q^{39} - 895387 q^{40} - 410260 q^{41} - 3413218 q^{42} - 1033591 q^{43} - 2177076 q^{44} - 1051178 q^{45} - 3975765 q^{46} - 424556 q^{47} - 2360477 q^{48} - 1561359 q^{49} - 4063801 q^{50} - 2375738 q^{51} - 4172312 q^{52} + 3992458 q^{53} - 10438626 q^{54} + 406960 q^{55} + 1559556 q^{56} - 3116152 q^{57} - 4052005 q^{58} + 2248836 q^{59} - 2911436 q^{60} + 6210394 q^{61} + 885317 q^{62} + 11622368 q^{63} - 3096318 q^{64} + 5600420 q^{65} - 2174604 q^{66} - 1993648 q^{67} + 9327135 q^{68} + 13366240 q^{69} - 1105098 q^{70} + 4978064 q^{71} + 11370663 q^{72} + 8224814 q^{73} - 3613563 q^{74} + 27115592 q^{75} + 10687121 q^{76} + 17261892 q^{77} - 15226630 q^{78} + 6945708 q^{79} + 15822799 q^{80} + 35113185 q^{81} - 508449 q^{82} + 22937328 q^{83} - 14010106 q^{84} - 575532 q^{85} - 1272112 q^{86} + 9081380 q^{87} + 11202656 q^{88} + 9291302 q^{89} + 2841402 q^{90} + 25581108 q^{91} - 14388137 q^{92} + 25930480 q^{93} - 24645805 q^{94} + 30750464 q^{95} - 22461255 q^{96} + 10001852 q^{97} - 32304856 q^{98} + 5055452 q^{99}+O(q^{100}) \) 13 * q + 16 * q^2 + 94 * q^3 + 922 * q^4 + 998 * q^5 + 183 * q^6 + 1360 * q^7 + 3870 * q^8 + 10011 * q^9 + 4667 * q^10 + 1620 * q^11 - 19681 * q^12 + 13550 * q^13 + 44160 * q^14 + 31412 * q^15 + 114026 * q^16 + 110880 * q^17 + 159267 * q^18 + 105058 * q^19 + 167251 * q^20 + 129840 * q^21 + 201504 * q^22 + 160184 * q^23 + 161289 * q^24 + 270149 * q^25 + 272104 * q^26 + 252544 * q^27 + 208172 * q^28 + 285546 * q^29 + 107580 * q^30 - 99616 * q^31 + 200126 * q^32 + 531468 * q^33 - 80941 * q^34 - 187104 * q^35 - 608975 * q^36 + 176038 * q^37 + 652165 * q^38 - 794680 * q^39 - 895387 * q^40 - 410260 * q^41 - 3413218 * q^42 - 1033591 * q^43 - 2177076 * q^44 - 1051178 * q^45 - 3975765 * q^46 - 424556 * q^47 - 2360477 * q^48 - 1561359 * q^49 - 4063801 * q^50 - 2375738 * q^51 - 4172312 * q^52 + 3992458 * q^53 - 10438626 * q^54 + 406960 * q^55 + 1559556 * q^56 - 3116152 * q^57 - 4052005 * q^58 + 2248836 * q^59 - 2911436 * q^60 + 6210394 * q^61 + 885317 * q^62 + 11622368 * q^63 - 3096318 * q^64 + 5600420 * q^65 - 2174604 * q^66 - 1993648 * q^67 + 9327135 * q^68 + 13366240 * q^69 - 1105098 * q^70 + 4978064 * q^71 + 11370663 * q^72 + 8224814 * q^73 - 3613563 * q^74 + 27115592 * q^75 + 10687121 * q^76 + 17261892 * q^77 - 15226630 * q^78 + 6945708 * q^79 + 15822799 * q^80 + 35113185 * q^81 - 508449 * q^82 + 22937328 * q^83 - 14010106 * q^84 - 575532 * q^85 - 1272112 * q^86 + 9081380 * q^87 + 11202656 * q^88 + 9291302 * q^89 + 2841402 * q^90 + 25581108 * q^91 - 14388137 * q^92 + 25930480 * q^93 - 24645805 * q^94 + 30750464 * q^95 - 22461255 * q^96 + 10001852 * q^97 - 32304856 * q^98 + 5055452 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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