LMFDB - Embedded newform 61.8.b.a.60.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [61,8,Mod(60,61)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(61, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("61.60");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 61 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	61.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$19.0554865545$
Analytic rank:	$0$
Dimension:	$34$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			60.6
Character	$\chi$	$=$	61.60
Dual form			61.8.b.a.60.29

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-15.0747i q^{2} -81.9260 q^{3} -99.2480 q^{4} +263.133 q^{5} +1235.01i q^{6} +570.938i q^{7} -433.430i q^{8} +4524.86 q^{9} +O(q^{10})$	\(q-15.0747i q^{2} -81.9260 q^{3} -99.2480 q^{4} +263.133 q^{5} +1235.01i q^{6} +570.938i q^{7} -433.430i q^{8} +4524.86 q^{9} -3966.67i q^{10} -3865.41i q^{11} +8130.99 q^{12} +15091.2 q^{13} +8606.74 q^{14} -21557.5 q^{15} -19237.6 q^{16} +29113.1i q^{17} -68211.2i q^{18} +1309.28 q^{19} -26115.5 q^{20} -46774.6i q^{21} -58270.1 q^{22} -43506.9i q^{23} +35509.1i q^{24} -8885.81 q^{25} -227496. i q^{26} -191532. q^{27} -56664.4i q^{28} -49172.8i q^{29} +324973. i q^{30} -90314.1i q^{31} +234523. i q^{32} +316677. i q^{33} +438872. q^{34} +150233. i q^{35} -449084. q^{36} -167974. i q^{37} -19737.0i q^{38} -1.23636e6 q^{39} -114050. i q^{40} +460834. q^{41} -705116. q^{42} -966054. i q^{43} +383634. i q^{44} +1.19064e6 q^{45} -655856. q^{46} +65314.5 q^{47} +1.57606e6 q^{48} +497573. q^{49} +133951. i q^{50} -2.38512e6i q^{51} -1.49777e6 q^{52} -1.25201e6i q^{53} +2.88729e6i q^{54} -1.01712e6i q^{55} +247461. q^{56} -107264. q^{57} -741267. q^{58} +584945. i q^{59} +2.13953e6 q^{60} +(-1.45336e6 + 1.01513e6i) q^{61} -1.36146e6 q^{62} +2.58342e6i q^{63} +1.07296e6 q^{64} +3.97100e6 q^{65} +4.77383e6 q^{66} -4.09370e6i q^{67} -2.88941e6i q^{68} +3.56435e6i q^{69} +2.26472e6 q^{70} +4.91426e6i q^{71} -1.96121e6i q^{72} -3.29257e6 q^{73} -2.53216e6 q^{74} +727978. q^{75} -129943. q^{76} +2.20691e6 q^{77} +1.86378e7i q^{78} -7.17414e6i q^{79} -5.06205e6 q^{80} +5.79555e6 q^{81} -6.94696e6i q^{82} +6.00945e6 q^{83} +4.64229e6i q^{84} +7.66062e6i q^{85} -1.45630e7 q^{86} +4.02853e6i q^{87} -1.67538e6 q^{88} +4.07355e6i q^{89} -1.79486e7i q^{90} +8.61613e6i q^{91} +4.31798e6i q^{92} +7.39907e6i q^{93} -984600. i q^{94} +344515. q^{95} -1.92135e7i q^{96} +7.06088e6 q^{97} -7.50079e6i q^{98} -1.74905e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9}+O(q^{10}) $ 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9	$ 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9} - 8382 q^{12} + 19860 q^{13} + 3744 q^{14} - 24988 q^{15} + 110642 q^{16} + 48660 q^{19} - 148866 q^{20} - 147578 q^{22} + 220494 q^{25} + 371248 q^{27} + 382432 q^{34} + 241132 q^{36} - 908620 q^{39} + 569544 q^{41} + 421016 q^{42} - 306560 q^{45} - 342232 q^{46} - 1179240 q^{47} + 3464534 q^{48} - 1069538 q^{49} - 2494330 q^{52} + 3794484 q^{56} + 3106472 q^{57} - 6540834 q^{58} - 2370908 q^{60} + 2828770 q^{61} + 5869278 q^{62} + 3854362 q^{64} - 502500 q^{65} - 4330852 q^{66} - 16861390 q^{70} - 6245708 q^{73} + 7083030 q^{74} - 13741952 q^{75} - 2405382 q^{76} - 14437656 q^{77} - 5019522 q^{80} + 31255922 q^{81} + 1991136 q^{83} - 4399944 q^{86} + 25423434 q^{88} - 22196868 q^{95} - 6652408 q^{97}+O(q^{100}) $ 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9 - 8382 * q^12 + 19860 * q^13 + 3744 * q^14 - 24988 * q^15 + 110642 * q^16 + 48660 * q^19 - 148866 * q^20 - 147578 * q^22 + 220494 * q^25 + 371248 * q^27 + 382432 * q^34 + 241132 * q^36 - 908620 * q^39 + 569544 * q^41 + 421016 * q^42 - 306560 * q^45 - 342232 * q^46 - 1179240 * q^47 + 3464534 * q^48 - 1069538 * q^49 - 2494330 * q^52 + 3794484 * q^56 + 3106472 * q^57 - 6540834 * q^58 - 2370908 * q^60 + 2828770 * q^61 + 5869278 * q^62 + 3854362 * q^64 - 502500 * q^65 - 4330852 * q^66 - 16861390 * q^70 - 6245708 * q^73 + 7083030 * q^74 - 13741952 * q^75 - 2405382 * q^76 - 14437656 * q^77 - 5019522 * q^80 + 31255922 * q^81 + 1991136 * q^83 - 4399944 * q^86 + 25423434 * q^88 - 22196868 * q^95 - 6652408 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/61\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	15.0747i		−	1.33243i	−0.745759	−	0.666216i	$-0.767913\pi$
$2$		−	15.0747i		−	1.33243i	0.745759	−	0.666216i	$-0.232087\pi$
$3$	−81.9260			−1.75185			−0.875926	−	0.482446i	$-0.839748\pi$
$3$	−81.9260			−1.75185			−0.875926	+	0.482446i	$0.839748\pi$
$4$	−99.2480			−0.775375
$5$	263.133			0.941415			0.470707	−	0.882289i	$-0.343999\pi$
$5$	263.133			0.941415			0.470707	+	0.882289i	$0.343999\pi$
$6$			1235.01i			2.33422i
$7$			570.938i			0.629137i	0.949235	+	0.314569i	$0.101860\pi$
$7$			570.938i			0.629137i	−0.949235	+	0.314569i	$0.898140\pi$
$8$		−	433.430i		−	0.299298i
$9$	4524.86			2.06898
$10$		−	3966.67i		−	1.25437i
$11$		−	3865.41i		−	0.875631i	−0.899065	−	0.437816i	$-0.855752\pi$
$11$		−	3865.41i		−	0.875631i	0.899065	−	0.437816i	$-0.144248\pi$
$12$	8130.99			1.35834
$13$	15091.2			1.90512			0.952559	−	0.304354i	$-0.0984407\pi$
$13$	15091.2			1.90512			0.952559	+	0.304354i	$0.0984407\pi$
$14$	8606.74			0.838283
$15$	−21557.5			−1.64922
$16$	−19237.6			−1.17417
$17$			29113.1i			1.43720i	0.695424	+	0.718600i	$0.255217\pi$
$17$			29113.1i			1.43720i	−0.695424	+	0.718600i	$0.744783\pi$
$18$		−	68211.2i		−	2.75678i
$19$	1309.28			0.0437920			0.0218960	−	0.999760i	$-0.493030\pi$
$19$	1309.28			0.0437920			0.0218960	+	0.999760i	$0.493030\pi$
$20$	−26115.5			−0.729949
$21$		−	46774.6i		−	1.10216i
$22$	−58270.1			−1.16672
$23$		−	43506.9i		−	0.745609i	−0.927910	−	0.372805i	$-0.878396\pi$
$23$		−	43506.9i		−	0.745609i	0.927910	−	0.372805i	$-0.121604\pi$
$24$			35509.1i			0.524325i
$25$	−8885.81			−0.113738
$26$		−	227496.i		−	2.53844i
$27$	−191532.			−1.87270
$28$		−	56664.4i		−	0.487817i
$29$		−	49172.8i		−	0.374396i	−0.982322	−	0.187198i	$-0.940059\pi$
$29$		−	49172.8i		−	0.374396i	0.982322	−	0.187198i	$-0.0599407\pi$
$30$			324973.i			2.19747i
$31$		−	90314.1i		−	0.544490i	−0.962228	−	0.272245i	$-0.912234\pi$
$31$		−	90314.1i		−	0.544490i	0.962228	−	0.272245i	$-0.0877661\pi$
$32$			234523.i			1.26520i
$33$			316677.i			1.53398i
$34$	438872.			1.91497
$35$			150233.i			0.592279i
$36$	−449084.			−1.60424
$37$		−	167974.i		−	0.545174i	−0.962131	−	0.272587i	$-0.912121\pi$
$37$		−	167974.i		−	0.545174i	0.962131	−	0.272587i	$-0.0878793\pi$
$38$		−	19737.0i		−	0.0583498i
$39$	−1.23636e6			−3.33748
$40$		−	114050.i		−	0.281763i
$41$	460834.			1.04424			0.522121	−	0.852871i	$-0.325141\pi$
$41$	460834.			1.04424			0.522121	+	0.852871i	$0.325141\pi$
$42$	−705116.			−1.46855
$43$		−	966054.i		−	1.85294i	−0.376367	−	0.926471i	$-0.622827\pi$
$43$		−	966054.i		−	1.85294i	0.376367	−	0.926471i	$-0.377173\pi$
$44$			383634.i			0.678942i
$45$	1.19064e6			1.94777
$46$	−655856.			−0.993473
$47$	65314.5			0.0917629			0.0458815	−	0.998947i	$-0.485390\pi$
$47$	65314.5			0.0917629			0.0458815	+	0.998947i	$0.485390\pi$
$48$	1.57606e6			2.05697
$49$	497573.			0.604186
$50$			133951.i			0.151549i
$51$		−	2.38512e6i		−	2.51776i
$52$	−1.49777e6			−1.47718
$53$		−	1.25201e6i		−	1.15516i	−0.816333	−	0.577581i	$-0.803997\pi$
$53$		−	1.25201e6i		−	1.15516i	0.816333	−	0.577581i	$-0.196003\pi$
$54$			2.88729e6i			2.49524i
$55$		−	1.01712e6i		−	0.824332i
$56$	247461.			0.188300
$57$	−107264.			−0.0767170
$58$	−741267.			−0.498858
$59$			584945.i			0.370795i	0.982664	+	0.185397i	$0.0593572\pi$
$59$			584945.i			0.370795i	−0.982664	+	0.185397i	$0.940643\pi$
$60$	2.13953e6			1.27876
$61$	−1.45336e6	+	1.01513e6i	−0.819819	+	0.572623i
$61$	−1.45336e6	+	1.01513e6i	−0.819819	+	0.572623i
$62$	−1.36146e6			−0.725496
$63$			2.58342e6i			1.30167i
$64$	1.07296e6			0.511627
$65$	3.97100e6			1.79351
$66$	4.77383e6			2.04392
$67$		−	4.09370e6i		−	1.66285i	−0.555635	−	0.831426i	$-0.687525\pi$
$67$		−	4.09370e6i		−	1.66285i	0.555635	−	0.831426i	$-0.312475\pi$
$68$		−	2.88941e6i		−	1.11437i
$69$			3.56435e6i			1.30620i
$70$	2.26472e6			0.789172
$71$			4.91426e6i			1.62950i	0.579813	+	0.814750i	$0.303126\pi$
$71$			4.91426e6i			1.62950i	−0.579813	+	0.814750i	$0.696874\pi$
$72$		−	1.96121e6i		−	0.619242i
$73$	−3.29257e6			−0.990616			−0.495308	−	0.868717i	$-0.664945\pi$
$73$	−3.29257e6			−0.990616			−0.495308	+	0.868717i	$0.664945\pi$
$74$	−2.53216e6			−0.726407
$75$	727978.			0.199253
$76$	−129943.			−0.0339552
$77$	2.20691e6			0.550892
$78$			1.86378e7i			4.44697i
$79$		−	7.17414e6i		−	1.63710i	−0.574435	−	0.818550i	$-0.694778\pi$
$79$		−	7.17414e6i		−	1.63710i	0.574435	−	0.818550i	$-0.305222\pi$
$80$	−5.06205e6			−1.10538
$81$	5.79555e6			1.21171
$82$		−	6.94696e6i		−	1.39138i
$83$	6.00945e6			1.15362			0.576808	−	0.816880i	$-0.304298\pi$
$83$	6.00945e6			1.15362			0.576808	+	0.816880i	$0.304298\pi$
$84$			4.64229e6i			0.854583i
$85$			7.66062e6i			1.35300i
$86$	−1.45630e7			−2.46892
$87$			4.02853e6i			0.655887i
$88$	−1.67538e6			−0.262075
$89$			4.07355e6i			0.612502i	0.951951	+	0.306251i	$0.0990748\pi$
$89$			4.07355e6i			0.612502i	−0.951951	+	0.306251i	$0.900925\pi$
$90$		−	1.79486e7i		−	2.59527i
$91$			8.61613e6i			1.19858i
$92$			4.31798e6i			0.578126i
$93$			7.39907e6i			0.953865i
$94$		−	984600.i		−	0.122268i
$95$	344515.			0.0412264
$96$		−	1.92135e7i		−	2.21645i
$97$	7.06088e6			0.785521			0.392760	−	0.919641i	$-0.371520\pi$
$97$	7.06088e6			0.785521			0.392760	+	0.919641i	$0.371520\pi$
$98$		−	7.50079e6i		−	0.805037i
$99$		−	1.74905e7i		−	1.81167i
$100$	881898.			0.0881898
$101$		−	8.16613e6i		−	0.788663i	−0.918968	−	0.394332i	$-0.870976\pi$
$101$		−	8.16613e6i		−	0.788663i	0.918968	−	0.394332i	$-0.129024\pi$
$102$	−3.59550e7			−3.35474
$103$	6.12071e6			0.551914			0.275957	−	0.961170i	$-0.411005\pi$
$103$	6.12071e6			0.551914			0.275957	+	0.961170i	$0.411005\pi$
$104$		−	6.54097e6i		−	0.570198i
$105$		−	1.23080e7i		−	1.03759i
$106$	−1.88738e7			−1.53918
$107$	1.36853e7			1.07997			0.539984	−	0.841675i	$-0.318430\pi$
$107$	1.36853e7			1.07997			0.539984	+	0.841675i	$0.318430\pi$
$108$	1.90091e7			1.45204
$109$	−6.97293e6			−0.515730			−0.257865	−	0.966181i	$-0.583019\pi$
$109$	−6.97293e6			−0.515730			−0.257865	+	0.966181i	$0.583019\pi$
$110$	−1.53328e7			−1.09837
$111$			1.37614e7i			0.955064i
$112$		−	1.09835e7i		−	0.738714i
$113$	−6.13760e6			−0.400151			−0.200075	−	0.979780i	$-0.564119\pi$
$113$	−6.13760e6			−0.400151			−0.200075	+	0.979780i	$0.564119\pi$
$114$			1.61698e6i			0.102220i
$115$		−	1.14481e7i		−	0.701927i
$116$			4.88030e6i			0.290298i
$117$	6.82856e7			3.94166
$118$	8.81790e6			0.494058
$119$	−1.66218e7			−0.904196
$120$			9.34364e6i			0.493608i
$121$	4.54577e6			0.233270
$122$	1.53029e7	+	2.19090e7i	0.762981	+	1.09235i
$123$	−3.77543e7			−1.82936
$124$			8.96349e6i			0.422184i
$125$	−2.28955e7			−1.04849
$126$	3.89443e7			1.73439
$127$	1.49635e6			0.0648215			0.0324108	−	0.999475i	$-0.489682\pi$
$127$	1.49635e6			0.0648215			0.0324108	+	0.999475i	$0.489682\pi$
$128$			1.38443e7i			0.583494i
$129$			7.91449e7i			3.24608i
$130$		−	5.98618e7i		−	2.38972i
$131$	−2.28813e7			−0.889266			−0.444633	−	0.895713i	$-0.646666\pi$
$131$	−2.28813e7			−0.889266			−0.444633	+	0.895713i	$0.646666\pi$
$132$		−	3.14296e7i		−	1.18941i
$133$			747516.i			0.0275512i
$134$	−6.17114e7			−2.21564
$135$	−5.03984e7			−1.76299
$136$	1.26185e7			0.430151
$137$	1.54183e7			0.512289			0.256145	−	0.966638i	$-0.417548\pi$
$137$	1.54183e7			0.512289			0.256145	+	0.966638i	$0.417548\pi$
$138$	5.37317e7			1.74042
$139$			7.07964e6i			0.223594i	0.993731	+	0.111797i	$0.0356606\pi$
$139$			7.07964e6i			0.223594i	−0.993731	+	0.111797i	$0.964339\pi$
$140$		−	1.49103e7i		−	0.459238i
$141$	−5.35095e6			−0.160755
$142$	7.40813e7			2.17120
$143$		−	5.83337e7i		−	1.66818i
$144$	−8.70474e7			−2.42933
$145$		−	1.29390e7i		−	0.352462i
$146$			4.96347e7i			1.31993i
$147$	−4.07642e7			−1.05844
$148$			1.66710e7i			0.422714i
$149$	−1.24928e7			−0.309390			−0.154695	−	0.987962i	$-0.549440\pi$
$149$	−1.24928e7			−0.309390			−0.154695	+	0.987962i	$0.549440\pi$
$150$		−	1.09741e7i		−	0.265491i
$151$			3.45282e7i			0.816120i	0.912955	+	0.408060i	$0.133795\pi$
$151$			3.45282e7i			0.816120i	−0.912955	+	0.408060i	$0.866205\pi$
$152$		−	567480.i		−	0.0131068i
$153$			1.31733e8i			2.97354i
$154$		−	3.32686e7i		−	0.734027i
$155$		−	2.37647e7i		−	0.512591i
$156$	1.22706e8			2.58780
$157$		−	8.66535e7i		−	1.78705i	−0.449012	−	0.893526i	$-0.648224\pi$
$157$		−	8.66535e7i		−	1.78705i	0.449012	−	0.893526i	$-0.351776\pi$
$158$	−1.08148e8			−2.18132
$159$			1.02572e8i			2.02367i
$160$			6.17107e7i			1.19108i
$161$	2.48398e7			0.469091
$162$		−	8.73665e7i		−	1.61452i
$163$	8.26608e7			1.49501			0.747503	−	0.664259i	$-0.231253\pi$
$163$	8.26608e7			1.49501			0.747503	+	0.664259i	$0.231253\pi$
$164$	−4.57369e7			−0.809679
$165$			8.33284e7i			1.44411i
$166$		−	9.05909e7i		−	1.53711i
$167$	1.16381e7			0.193364			0.0966818	−	0.995315i	$-0.469177\pi$
$167$	1.16381e7			0.193364			0.0966818	+	0.995315i	$0.469177\pi$
$168$	−2.02735e7			−0.329873
$169$	1.64996e8			2.62947
$170$	1.15482e8			1.80278
$171$	5.92431e6			0.0906048
$172$			9.58789e7i			1.43672i
$173$		−	1.08005e7i		−	0.158593i	−0.996851	−	0.0792963i	$-0.974733\pi$
$173$		−	1.08005e7i		−	0.158593i	0.996851	−	0.0792963i	$-0.0252673\pi$
$174$	6.07290e7			0.873924
$175$		−	5.07324e6i		−	0.0715571i
$176$			7.43611e7i			1.02814i
$177$		−	4.79222e7i		−	0.649577i
$178$	6.14077e7			0.816118
$179$	−7.37774e7			−0.961474			−0.480737	−	0.876865i	$-0.659631\pi$
$179$	−7.37774e7			−0.961474			−0.480737	+	0.876865i	$0.659631\pi$
$180$	−1.18169e8			−1.51025
$181$		−	7.37821e7i		−	0.924860i	−0.886656	−	0.462430i	$-0.846978\pi$
$181$		−	7.37821e7i		−	0.924860i	0.886656	−	0.462430i	$-0.153022\pi$
$182$	1.29886e8			1.59703
$183$	1.19068e8	−	8.31658e7i	1.43620	−	1.00315i
$184$	−1.88572e7			−0.223159
$185$		−	4.41995e7i		−	0.513235i
$186$	1.11539e8			1.27096
$187$	1.12534e8			1.25846
$188$	−6.48233e6			−0.0711506
$189$		−	1.09353e8i		−	1.17818i
$190$		−	5.19347e6i		−	0.0549314i
$191$		−	8.47020e6i		−	0.0879583i	−0.999032	−	0.0439791i	$-0.985996\pi$
$191$		−	8.47020e6i		−	0.0879583i	0.999032	−	0.0439791i	$-0.0140035\pi$
$192$	−8.79032e7			−0.896294
$193$			3.62512e7i			0.362971i	0.983394	+	0.181486i	$0.0580906\pi$
$193$			3.62512e7i			0.362971i	−0.983394	+	0.181486i	$0.941909\pi$
$194$		−	1.06441e8i		−	1.04665i
$195$	−3.25328e8			−3.14196
$196$	−4.93831e7			−0.468471
$197$	5.66844e7			0.528241			0.264120	−	0.964490i	$-0.414918\pi$
$197$	5.66844e7			0.528241			0.264120	+	0.964490i	$0.414918\pi$
$198$	−2.63664e8			−2.41392
$199$	−2.20225e8			−1.98098			−0.990490	−	0.137585i	$-0.956066\pi$
$199$	−2.20225e8			−1.98098			−0.990490	+	0.137585i	$0.956066\pi$
$200$			3.85137e6i			0.0340416i
$201$			3.35380e8i			2.91307i
$202$	−1.23102e8			−1.05084
$203$	2.80746e7			0.235547
$204$			2.36718e8i			1.95221i
$205$	1.21261e8			0.983065
$206$		−	9.22681e7i		−	0.735388i
$207$		−	1.96863e8i		−	1.54265i
$208$	−2.90318e8			−2.23693
$209$		−	5.06090e6i		−	0.0383456i
$210$	−1.85539e8			−1.38251
$211$		−	3.97247e7i		−	0.291120i	−0.989349	−	0.145560i	$-0.953502\pi$
$211$		−	3.97247e7i		−	0.291120i	0.989349	−	0.145560i	$-0.0464984\pi$
$212$			1.24260e8i			0.895684i
$213$		−	4.02606e8i		−	2.85464i
$214$		−	2.06302e8i		−	1.43898i
$215$		−	2.54201e8i		−	1.74439i
$216$			8.30156e7i			0.560495i
$217$	5.15637e7			0.342559
$218$			1.05115e8i			0.687175i
$219$	2.69747e8			1.73541
$220$			1.00947e8i			0.639166i
$221$			4.39351e8i			2.73803i
$222$	2.07450e8			1.27256
$223$			2.31298e8i			1.39670i	0.715755	+	0.698351i	$0.246083\pi$
$223$			2.31298e8i			1.39670i	−0.715755	+	0.698351i	$0.753917\pi$
$224$	−1.33898e8			−0.795986
$225$	−4.02071e7			−0.235323
$226$			9.25227e7i			0.533174i
$227$			3.25777e7i			0.184854i	0.995719	+	0.0924272i	$0.0294625\pi$
$227$			3.25777e7i			0.184854i	−0.995719	+	0.0924272i	$0.970537\pi$
$228$	1.06457e7			0.0594844
$229$	2.76726e8			1.52274			0.761369	−	0.648319i	$-0.224528\pi$
$229$	2.76726e8			1.52274			0.761369	+	0.648319i	$0.224528\pi$
$230$	−1.72578e8			−0.935270
$231$	−1.80803e8			−0.965081
$232$	−2.13129e7			−0.112056
$233$		−	2.81697e8i		−	1.45894i	−0.684015	−	0.729468i	$-0.739768\pi$
$233$		−	2.81697e8i		−	1.45894i	0.684015	−	0.729468i	$-0.260232\pi$
$234$		−	1.02939e9i		−	5.25199i
$235$	1.71864e7			0.0863870
$236$		−	5.80546e7i		−	0.287505i
$237$			5.87749e8i			2.86796i
$238$			2.50569e8i			1.20478i
$239$	−2.23596e8			−1.05943			−0.529715	−	0.848176i	$-0.677701\pi$
$239$	−2.23596e8			−1.05943			−0.529715	+	0.848176i	$0.677701\pi$
$240$	4.14713e8			1.93646
$241$	6.61396e7			0.304370			0.152185	−	0.988352i	$-0.451369\pi$
$241$	6.61396e7			0.304370			0.152185	+	0.988352i	$0.451369\pi$
$242$		−	6.85264e7i		−	0.310816i
$243$	−5.59260e7			−0.250030
$244$	1.44243e8	−	1.00750e8i	0.635667	−	0.443997i
$245$	1.30928e8			0.568790
$246$			5.69136e8i			2.43749i
$247$	1.97586e7			0.0834289
$248$	−3.91448e7			−0.162965
$249$	−4.92330e8			−2.02096
$250$			3.45143e8i			1.39704i
$251$			2.93483e8i			1.17145i	0.810509	+	0.585726i	$0.199191\pi$
$251$			2.93483e8i			1.17145i	−0.810509	+	0.585726i	$0.800809\pi$
$252$		−	2.56399e8i		−	1.00929i
$253$	−1.68172e8			−0.652879
$254$		−	2.25571e7i		−	0.0863703i
$255$		−	6.27604e8i		−	2.37026i
$256$	3.46038e8			1.28909
$257$	−4.34477e8			−1.59662			−0.798310	−	0.602247i	$-0.794272\pi$
$257$	−4.34477e8			−1.59662			−0.798310	+	0.602247i	$0.794272\pi$
$258$	1.19309e9			4.32518
$259$	9.59025e7			0.342989
$260$	−3.94113e8			−1.39064
$261$		−	2.22500e8i		−	0.774620i
$262$			3.44930e8i			1.18489i
$263$	2.05661e8			0.697120			0.348560	−	0.937287i	$-0.386671\pi$
$263$	2.05661e8			0.697120			0.348560	+	0.937287i	$0.386671\pi$
$264$	1.37257e8			0.459116
$265$		−	3.29446e8i		−	1.08749i
$266$	1.12686e7			0.0367101
$267$		−	3.33729e8i		−	1.07301i
$268$			4.06291e8i			1.28933i
$269$	1.28259e8			0.401750			0.200875	−	0.979617i	$-0.435622\pi$
$269$	1.28259e8			0.401750			0.200875	+	0.979617i	$0.435622\pi$
$270$			7.59743e8i			2.34906i
$271$	4.49745e7			0.137269			0.0686347	−	0.997642i	$-0.478136\pi$
$271$	4.49745e7			0.137269			0.0686347	+	0.997642i	$0.478136\pi$
$272$		−	5.60065e8i		−	1.68751i
$273$		−	7.05885e8i		−	2.09974i
$274$		−	2.32427e8i		−	0.682591i
$275$			3.43473e7i			0.0995928i
$276$		−	3.53754e8i		−	1.01279i
$277$			1.18584e8i			0.335232i	0.985852	+	0.167616i	$0.0536068\pi$
$277$			1.18584e8i			0.335232i	−0.985852	+	0.167616i	$0.946393\pi$
$278$	1.06724e8			0.297923
$279$		−	4.08659e8i		−	1.12654i
$280$	6.51153e7			0.177268
$281$			1.88864e8i			0.507783i	0.967233	+	0.253891i	$0.0817105\pi$
$281$			1.88864e8i			0.507783i	−0.967233	+	0.253891i	$0.918289\pi$
$282$			8.06643e7i			0.214195i
$283$	1.95058e8			0.511578			0.255789	−	0.966733i	$-0.417665\pi$
$283$	1.95058e8			0.511578			0.255789	+	0.966733i	$0.417665\pi$
$284$		−	4.87731e8i		−	1.26347i
$285$	−2.82247e7			−0.0722225
$286$	−8.79365e8			−2.22274
$287$			2.63108e8i			0.656972i
$288$			1.06118e9i			2.61768i
$289$	−4.37233e8			−1.06554
$290$	−1.95052e8			−0.469632
$291$	−5.78469e8			−1.37612
$292$	3.26781e8			0.768099
$293$	6.63917e8			1.54197			0.770987	−	0.636851i	$-0.219763\pi$
$293$	6.63917e8			1.54197			0.770987	+	0.636851i	$0.219763\pi$
$294$			6.14509e8i			1.41030i
$295$			1.53919e8i			0.349071i
$296$	−7.28048e7			−0.163169
$297$			7.40349e8i			1.63979i
$298$			1.88325e8i			0.412241i
$299$		−	6.56572e8i		−	1.42047i
$300$	−7.22504e7			−0.154495
$301$	5.51556e8			1.16575
$302$	5.20503e8			1.08742
$303$			6.69018e8i			1.38162i
$304$	−2.51874e7			−0.0514192
$305$	−3.82427e8	+	2.67115e8i	−0.771790	+	0.539075i
$306$	1.98584e9			3.96204
$307$			4.68699e8i			0.924505i	0.886748	+	0.462253i	$0.152959\pi$
$307$			4.68699e8i			0.924505i	−0.886748	+	0.462253i	$0.847041\pi$
$308$	−2.19031e8			−0.427148
$309$	−5.01445e8			−0.966871
$310$	−3.58246e8			−0.682992
$311$		−	2.29669e8i		−	0.432954i	−0.976288	−	0.216477i	$-0.930543\pi$
$311$		−	2.29669e8i		−	0.432954i	0.976288	−	0.216477i	$-0.0694567\pi$
$312$			5.35875e8i			0.998901i
$313$		−	2.55875e8i		−	0.471652i	−0.971795	−	0.235826i	$-0.924220\pi$
$313$		−	2.55875e8i		−	0.471652i	0.971795	−	0.235826i	$-0.0757796\pi$
$314$	−1.30628e9			−2.38112
$315$			6.79783e8i			1.22542i
$316$			7.12019e8i			1.26937i
$317$	−2.11883e8			−0.373584			−0.186792	−	0.982400i	$-0.559809\pi$
$317$	−2.11883e8			−0.373584			−0.186792	+	0.982400i	$0.559809\pi$
$318$	1.54625e9			2.69641
$319$	−1.90073e8			−0.327833
$320$	2.82331e8			0.481653
$321$	−1.12118e9			−1.89194
$322$		−	3.74453e8i		−	0.625031i
$323$			3.81171e7i			0.0629378i
$324$	−5.75197e8			−0.939526
$325$	−1.34097e8			−0.216685
$326$		−	1.24609e9i		−	1.99199i
$327$	5.71264e8			0.903482
$328$		−	1.99739e8i		−	0.312539i
$329$			3.72905e7i			0.0577315i
$330$	1.25615e9			1.92417
$331$			9.95780e8i			1.50926i	0.656148	+	0.754632i	$0.272185\pi$
$331$			9.95780e8i			1.50926i	−0.656148	+	0.754632i	$0.727815\pi$
$332$	−5.96425e8			−0.894484
$333$		−	7.60058e8i		−	1.12796i
$334$		−	1.75442e8i		−	0.257644i
$335$		−	1.07719e9i		−	1.56543i
$336$			8.99831e8i			1.29412i
$337$		−	1.11690e9i		−	1.58968i	−0.606819	−	0.794840i	$-0.707555\pi$
$337$		−	1.11690e9i		−	1.58968i	0.606819	−	0.794840i	$-0.292445\pi$
$338$		−	2.48727e9i		−	3.50359i
$339$	5.02829e8			0.701005
$340$		−	7.60301e8i		−	1.04908i
$341$	−3.49101e8			−0.476772
$342$		−	8.93074e7i		−	0.120725i
$343$			7.54275e8i			1.00925i
$344$	−4.18716e8			−0.554581
$345$			9.37899e8i			1.22967i
$346$	−1.62815e8			−0.211314
$347$	−7.64828e8			−0.982677			−0.491338	−	0.870969i	$-0.663492\pi$
$347$	−7.64828e8			−0.982677			−0.491338	+	0.870969i	$0.663492\pi$
$348$		−	3.99823e8i		−	0.508558i
$349$		−	1.04697e9i		−	1.31839i	−0.751970	−	0.659197i	$-0.770896\pi$
$349$		−	1.04697e9i		−	1.31839i	0.751970	−	0.659197i	$-0.229104\pi$
$350$	−7.64778e7			−0.0953449
$351$	−2.89044e9			−3.56771
$352$	9.06526e8			1.10785
$353$	1.13235e9			1.37015			0.685074	−	0.728473i	$-0.259770\pi$
$353$	1.13235e9			1.37015			0.685074	+	0.728473i	$0.259770\pi$
$354$	−7.22415e8			−0.865517
$355$			1.29311e9i			1.53403i
$356$		−	4.04291e8i		−	0.474919i
$357$	1.36175e9			1.58402
$358$			1.11218e9i			1.28110i
$359$			3.46360e8i			0.395091i	0.980294	+	0.197546i	$0.0632970\pi$
$359$			3.46360e8i			0.395091i	−0.980294	+	0.197546i	$0.936703\pi$
$360$		−	5.16060e8i		−	0.582964i
$361$	−8.92158e8			−0.998082
$362$	−1.11225e9			−1.23231
$363$	−3.72417e8			−0.408654
$364$		−	8.55133e8i		−	0.929349i
$365$	−8.66386e8			−0.932580
$366$	−1.25370e9	−	1.79492e9i	−1.33663	−	1.91364i
$367$	−1.50266e9			−1.58683			−0.793415	−	0.608681i	$-0.791699\pi$
$367$	−1.50266e9			−1.58683			−0.793415	+	0.608681i	$0.791699\pi$
$368$			8.36968e8i			0.875471i
$369$	2.08521e9			2.16052
$370$	−6.66296e8			−0.683851
$371$	7.14821e8			0.726756
$372$		−	7.34343e8i		−	0.739603i
$373$			1.59677e9i			1.59317i	0.604526	+	0.796585i	$0.293362\pi$
$373$			1.59677e9i			1.59317i	−0.604526	+	0.796585i	$0.706638\pi$
$374$		−	1.69642e9i		−	1.67681i
$375$	1.87573e9			1.83680
$376$		−	2.83092e7i		−	0.0274644i
$377$		−	7.42076e8i		−	0.713269i
$378$	−1.64846e9			−1.56985
$379$	1.42276e9			1.34244			0.671220	−	0.741258i	$-0.265771\pi$
$379$	1.42276e9			1.34244			0.671220	+	0.741258i	$0.265771\pi$
$380$	−3.41924e7			−0.0319659
$381$	−1.22590e8			−0.113558
$382$	−1.27686e8			−0.117198
$383$			6.85554e8i			0.623514i	0.950162	+	0.311757i	$0.100917\pi$
$383$			6.85554e8i			0.623514i	−0.950162	+	0.311757i	$0.899083\pi$
$384$		−	1.13421e9i		−	1.02219i
$385$	5.80711e8			0.518618
$386$	5.46478e8			0.483635
$387$		−	4.37126e9i		−	3.83370i
$388$	−7.00778e8			−0.609073
$389$			5.86921e8i			0.505541i	0.967526	+	0.252770i	$0.0813417\pi$
$389$			5.86921e8i			0.505541i	−0.967526	+	0.252770i	$0.918658\pi$
$390$			4.90423e9i			4.18644i
$391$	1.26662e9			1.07159
$392$		−	2.15663e8i		−	0.180832i
$393$	1.87458e9			1.55786
$394$		−	8.54503e8i		−	0.703845i
$395$		−	1.88776e9i		−	1.54119i
$396$			1.73589e9i			1.40472i
$397$			1.93470e9i			1.55184i	0.630830	+	0.775921i	$0.282715\pi$
$397$			1.93470e9i			1.55184i	−0.630830	+	0.775921i	$0.717285\pi$
$398$			3.31983e9i			2.63952i
$399$		−	6.12410e7i		−	0.0482655i
$400$	1.70941e8			0.133548
$401$			2.13836e9i			1.65606i	0.560683	+	0.828031i	$0.310539\pi$
$401$			2.13836e9i			1.65606i	−0.560683	+	0.828031i	$0.689461\pi$
$402$	5.05577e9			3.88147
$403$		−	1.36295e9i		−	1.03732i
$404$			8.10472e8i			0.611510i
$405$	1.52500e9			1.14072
$406$		−	4.23217e8i		−	0.313850i
$407$	−6.49287e8			−0.477371
$408$	−1.03378e9			−0.753560
$409$			8.12527e8i			0.587228i	0.955924	+	0.293614i	$0.0948580\pi$
$409$			8.12527e8i			0.587228i	−0.955924	+	0.293614i	$0.905142\pi$
$410$		−	1.82798e9i		−	1.30987i
$411$	−1.26316e9			−0.897455
$412$	−6.07468e8			−0.427940
$413$	−3.33967e8			−0.233281
$414$	−2.96766e9			−2.05548
$415$	1.58129e9			1.08603
$416$			3.53923e9i			2.41036i
$417$		−	5.80007e8i		−	0.391703i
$418$	−7.62918e7			−0.0510929
$419$			2.19692e9i			1.45903i	0.683963	+	0.729516i	$0.260255\pi$
$419$			2.19692e9i			1.45903i	−0.683963	+	0.729516i	$0.739745\pi$
$420$			1.22154e9i			0.804517i
$421$		−	6.08151e8i		−	0.397213i	−0.980079	−	0.198607i	$-0.936358\pi$
$421$		−	6.08151e8i		−	0.397213i	0.980079	−	0.198607i	$-0.0636416\pi$
$422$	−5.98840e8			−0.387898
$423$	2.95539e8			0.189856
$424$	−5.42659e8			−0.345738
$425$		−	2.58693e8i		−	0.163465i
$426$	−6.06918e9			−3.80361
$427$	−5.79578e8	−	8.29777e8i	−0.360258	−	0.515779i
$428$	−1.35824e9			−0.837380
$429$			4.77904e9i			2.92240i
$430$	−3.83202e9			−2.32428
$431$	1.24399e9			0.748420			0.374210	−	0.927344i	$-0.377914\pi$
$431$	1.24399e9			0.748420			0.374210	+	0.927344i	$0.377914\pi$
$432$	3.68461e9			2.19886
$433$		−	2.52366e9i		−	1.49391i	−0.664877	−	0.746953i	$-0.731516\pi$
$433$		−	2.52366e9i		−	1.49391i	0.664877	−	0.746953i	$-0.268484\pi$
$434$		−	7.77310e8i		−	0.456436i
$435$			1.06004e9i			0.617461i
$436$	6.92049e8			0.399884
$437$		−	5.69627e7i		−	0.0326517i
$438$		−	4.06637e9i		−	2.31232i
$439$	−1.14038e9			−0.643313			−0.321657	−	0.946856i	$-0.604240\pi$
$439$	−1.14038e9			−0.643313			−0.321657	+	0.946856i	$0.604240\pi$
$440$	−4.40849e8			−0.246721
$441$	2.25145e9			1.25005
$442$	6.62311e9			3.64824
$443$	−1.14282e9			−0.624548			−0.312274	−	0.949992i	$-0.601091\pi$
$443$	−1.14282e9			−0.624548			−0.312274	+	0.949992i	$0.601091\pi$
$444$		−	1.36579e9i		−	0.740532i
$445$			1.07189e9i			0.576619i
$446$	3.48675e9			1.86101
$447$	1.02348e9			0.542005
$448$			6.12593e8i			0.321884i
$449$	−1.32619e9			−0.691423			−0.345711	−	0.938341i	$-0.612362\pi$
$449$	−1.32619e9			−0.691423			−0.345711	+	0.938341i	$0.612362\pi$
$450$			6.06112e8i			0.313551i
$451$		−	1.78131e9i		−	0.914371i
$452$	6.09144e8			0.310267
$453$		−	2.82875e9i		−	1.42972i
$454$	4.91100e8			0.246306
$455$			2.26719e9i			1.12836i
$456$			4.64914e7i			0.0229612i
$457$		−	1.60514e9i		−	0.786695i	−0.919390	−	0.393348i	$-0.871317\pi$
$457$		−	1.60514e9i		−	0.786695i	0.919390	−	0.393348i	$-0.128683\pi$
$458$		−	4.17157e9i		−	2.02894i
$459$		−	5.57608e9i		−	2.69144i
$460$			1.13620e9i			0.544257i
$461$	3.32704e9			1.58163			0.790816	−	0.612054i	$-0.209657\pi$
$461$	3.32704e9			1.58163			0.790816	+	0.612054i	$0.209657\pi$
$462$			2.72556e9i			1.28591i
$463$	1.27197e9			0.595584			0.297792	−	0.954631i	$-0.403750\pi$
$463$	1.27197e9			0.595584			0.297792	+	0.954631i	$0.403750\pi$
$464$			9.45965e8i			0.439605i
$465$			1.94694e9i			0.897983i
$466$	−4.24651e9			−1.94393
$467$			1.69921e9i			0.772037i	0.922491	+	0.386018i	$0.126150\pi$
$467$			1.69921e9i			0.772037i	−0.922491	+	0.386018i	$0.873850\pi$
$468$	−6.77721e9			−3.05626
$469$	2.33724e9			1.04616
$470$		−	2.59081e8i		−	0.115105i
$471$			7.09917e9i			3.13065i
$472$	2.53533e8			0.110978
$473$	−3.73419e9			−1.62249
$474$	8.86016e9			3.82136
$475$	−1.16340e7			−0.00498083
$476$	1.64968e9			0.701091
$477$		−	5.66519e9i		−	2.39001i
$478$			3.37066e9i			1.41162i
$479$	−2.45765e8			−0.102175			−0.0510876	−	0.998694i	$-0.516269\pi$
$479$	−2.45765e8			−0.102175			−0.0510876	+	0.998694i	$0.516269\pi$
$480$		−	5.05571e9i		−	2.08659i
$481$		−	2.53492e9i		−	1.03862i
$482$		−	9.97037e8i		−	0.405552i
$483$	−2.03502e9			−0.821777
$484$	−4.51159e8			−0.180872
$485$	1.85795e9			0.739501
$486$			8.43071e8i			0.333148i
$487$	1.40649e9			0.551805			0.275903	−	0.961186i	$-0.411023\pi$
$487$	1.40649e9			0.551805			0.275903	+	0.961186i	$0.411023\pi$
$488$	4.39989e8	+	6.29928e8i	0.171385	+	0.245370i
$489$	−6.77206e9			−2.61903
$490$		−	1.97371e9i		−	0.757873i
$491$	−1.30739e9			−0.498448			−0.249224	−	0.968446i	$-0.580175\pi$
$491$	−1.30739e9			−0.498448			−0.249224	+	0.968446i	$0.580175\pi$
$492$	3.74704e9			1.41844
$493$	1.43157e9			0.538082
$494$		−	2.97855e8i		−	0.111163i
$495$		−	4.60232e9i		−	1.70553i
$496$			1.73743e9i			0.639323i
$497$	−2.80574e9			−1.02518
$498$			7.42174e9i			2.69279i
$499$			2.03340e9i			0.732607i	0.930495	+	0.366304i	$0.119377\pi$
$499$			2.03340e9i			0.732607i	−0.930495	+	0.366304i	$0.880623\pi$
$500$	2.27233e9			0.812972
$501$	−9.53464e8			−0.338744
$502$	4.42418e9			1.56088
$503$	4.94050e9			1.73095			0.865473	−	0.500956i	$-0.167018\pi$
$503$	4.94050e9			1.73095			0.865473	+	0.500956i	$0.167018\pi$
$504$	1.11973e9			0.389588
$505$		−	2.14878e9i		−	0.742459i
$506$			2.53515e9i			0.869916i
$507$	−1.35174e10			−4.60645
$508$	−1.48509e8			−0.0502610
$509$			4.62846e9i			1.55570i	0.628453	+	0.777848i	$0.283688\pi$
$509$			4.62846e9i			1.55570i	−0.628453	+	0.777848i	$0.716312\pi$
$510$	−9.46097e9			−3.15820
$511$		−	1.87985e9i		−	0.623234i
$512$		−	3.44437e9i		−	1.13413i
$513$	−2.50768e8			−0.0820091
$514$			6.54964e9i			2.12739i
$515$	1.61056e9			0.519580
$516$		−	7.85497e9i		−	2.51693i
$517$		−	2.52467e8i		−	0.0803505i
$518$		−	1.44571e9i		−	0.457010i
$519$			8.84842e8i			0.277831i
$520$		−	1.72115e9i		−	0.536792i
$521$			5.14104e9i			1.59265i	0.604872	+	0.796323i	$0.293224\pi$
$521$			5.14104e9i			1.59265i	−0.604872	+	0.796323i	$0.706776\pi$
$522$	−3.35413e9			−1.03213
$523$		−	3.56494e9i		−	1.08967i	−0.838543	−	0.544836i	$-0.816592\pi$
$523$		−	3.56494e9i		−	1.08967i	0.838543	−	0.544836i	$-0.183408\pi$
$524$	2.27093e9			0.689515
$525$			4.15630e8i			0.125357i
$526$		−	3.10029e9i		−	0.928864i
$527$	2.62932e9			0.782540
$528$		−	6.09211e9i		−	1.80115i
$529$	1.51197e9			0.444067
$530$	−4.96632e9			−1.44900
$531$			2.64680e9i			0.767167i
$532$		−	7.41895e7i		−	0.0213625i
$533$	6.95454e9			1.98940
$534$	−5.03089e9			−1.42972
$535$	3.60106e9			1.01670
$536$	−1.77433e9			−0.497688
$537$	6.04428e9			1.68436
$538$		−	1.93348e9i		−	0.535304i
$539$		−	1.92332e9i		−	0.529044i
$540$	5.00194e9			1.36697
$541$			4.66946e9i			1.26787i	0.773385	+	0.633937i	$0.218562\pi$
$541$			4.66946e9i			1.26787i	−0.773385	+	0.633937i	$0.781438\pi$
$542$		−	6.77979e8i		−	0.182902i
$543$			6.04467e9i			1.62022i
$544$	−6.82768e9			−1.81835
$545$	−1.83481e9			−0.485516
$546$	−1.06410e10			−2.79775
$547$			4.59252e8i			0.119976i	0.998199	+	0.0599882i	$0.0191063\pi$
$547$			4.59252e8i			0.119976i	−0.998199	+	0.0599882i	$0.980894\pi$
$548$	−1.53024e9			−0.397216
$549$	−6.57625e9	+	4.59334e9i	−1.69619	+	1.18475i
$550$	5.17777e8			0.132701
$551$		−	6.43808e7i		−	0.0163956i
$552$	1.54489e9			0.390942
$553$	4.09599e9			1.02996
$554$	1.78762e9			0.446673
$555$			3.62109e9i			0.899111i
$556$		−	7.02640e8i		−	0.173369i
$557$			2.40414e9i			0.589477i	0.955578	+	0.294739i	$0.0952326\pi$
$557$			2.40414e9i			0.589477i	−0.955578	+	0.294739i	$0.904767\pi$
$558$	−6.16043e9			−1.50104
$559$		−	1.45789e10i		−	3.53007i
$560$		−	2.89012e9i		−	0.695436i
$561$	−9.21946e9			−2.20463
$562$	2.84708e9			0.676586
$563$	−5.71625e9			−1.34999			−0.674997	−	0.737820i	$-0.735855\pi$
$563$	−5.71625e9			−1.34999			−0.674997	+	0.737820i	$0.735855\pi$
$564$	5.31071e8			0.124645
$565$	−1.61501e9			−0.376708
$566$		−	2.94045e9i		−	0.681642i
$567$			3.30890e9i			0.762329i
$568$	2.12999e9			0.487706
$569$	5.36287e9			1.22041			0.610203	−	0.792245i	$-0.291088\pi$
$569$	5.36287e9			1.22041			0.610203	+	0.792245i	$0.291088\pi$
$570$			4.25480e8i			0.0962316i
$571$	−1.78252e9			−0.400689			−0.200344	−	0.979726i	$-0.564206\pi$
$571$	−1.78252e9			−0.400689			−0.200344	+	0.979726i	$0.564206\pi$
$572$			5.78950e9i			1.29346i
$573$			6.93929e8i			0.154090i
$574$	3.96628e9			0.875370
$575$			3.86594e8i			0.0848043i
$576$	4.85499e9			1.05855
$577$		−	1.77758e9i		−	0.385225i	−0.981275	−	0.192613i	$-0.938304\pi$
$577$		−	1.77758e9i		−	0.385225i	0.981275	−	0.192613i	$-0.0616960\pi$
$578$			6.59117e9i			1.41976i
$579$		−	2.96992e9i		−	0.635872i
$580$			1.28417e9i			0.273290i
$581$			3.43102e9i			0.725783i
$582$			8.72028e9i			1.83358i
$583$	−4.83954e9			−1.01150
$584$			1.42710e9i			0.296489i
$585$	1.79682e10			3.71073
$586$		−	1.00084e10i		−	2.05458i
$587$			2.59186e9i			0.528905i	0.964399	+	0.264452i	$0.0851912\pi$
$587$			2.59186e9i			0.528905i	−0.964399	+	0.264452i	$0.914809\pi$
$588$	4.04576e9			0.820691
$589$		−	1.18246e8i		−	0.0238443i
$590$	2.32029e9			0.465114
$591$	−4.64392e9			−0.925399
$592$			3.23141e9i			0.640126i
$593$			5.65169e8i			0.111298i	0.998450	+	0.0556490i	$0.0177228\pi$
$593$			5.65169e8i			0.111298i	−0.998450	+	0.0556490i	$0.982277\pi$
$594$	1.11606e10			2.18491
$595$	−4.37374e9			−0.851223
$596$	1.23988e9			0.239893
$597$	1.80421e10			3.47038
$598$	−9.89765e9			−1.89268
$599$		−	6.43604e9i		−	1.22356i	−0.791029	−	0.611779i	$-0.790454\pi$
$599$		−	6.43604e9i		−	1.22356i	0.791029	−	0.611779i	$-0.209546\pi$
$600$		−	3.15527e8i		−	0.0596359i
$601$	−2.62363e9			−0.492995			−0.246498	−	0.969143i	$-0.579280\pi$
$601$	−2.62363e9			−0.492995			−0.246498	+	0.969143i	$0.579280\pi$
$602$		−	8.31457e9i		−	1.55329i
$603$		−	1.85234e10i		−	3.44041i
$604$		−	3.42685e9i		−	0.632799i
$605$	1.19614e9			0.219604
$606$	1.00853e10			1.84092
$607$	4.55957e8			0.0827492			0.0413746	−	0.999144i	$-0.486826\pi$
$607$	4.55957e8			0.0827492			0.0413746	+	0.999144i	$0.486826\pi$
$608$			3.07055e8i			0.0554057i
$609$	−2.30004e9			−0.412643
$610$	4.02670e9	+	5.76499e9i	0.718281	+	1.02836i
$611$	9.85674e8			0.174819
$612$		−	1.30742e10i		−	2.30561i
$613$	1.10881e9			0.194422			0.0972109	−	0.995264i	$-0.469008\pi$
$613$	1.10881e9			0.194422			0.0972109	+	0.995264i	$0.469008\pi$
$614$	7.06551e9			1.23184
$615$	−9.93441e9			−1.72218
$616$		−	9.56539e8i		−	0.164881i
$617$			4.43122e9i			0.759496i	0.925090	+	0.379748i	$0.123989\pi$
$617$			4.43122e9i			0.759496i	−0.925090	+	0.379748i	$0.876011\pi$
$618$			7.55916e9i			1.28829i
$619$	−7.83651e9			−1.32802			−0.664011	−	0.747723i	$-0.731147\pi$
$619$	−7.83651e9			−1.32802			−0.664011	+	0.747723i	$0.731147\pi$
$620$			2.35859e9i			0.397450i
$621$			8.33296e9i			1.39630i
$622$	−3.46221e9			−0.576882
$623$	−2.32574e9			−0.385348
$624$	2.37846e10			3.91877
$625$	−5.33035e9			−0.873325
$626$	−3.85724e9			−0.628445
$627$			4.14619e8i			0.0671758i
$628$			8.60018e9i			1.38563i
$629$	4.89023e9			0.783524
$630$	1.02476e10			1.63278
$631$			3.88114e9i			0.614974i	0.951552	+	0.307487i	$0.0994881\pi$
$631$			3.88114e9i			0.614974i	−0.951552	+	0.307487i	$0.900512\pi$
$632$	−3.10949e9			−0.489981
$633$			3.25449e9i			0.509999i
$634$			3.19408e9i			0.497775i
$635$	3.93739e8			0.0610240
$636$		−	1.01801e10i		−	1.56910i
$637$	7.50897e9			1.15105
$638$			2.86530e9i			0.436815i
$639$			2.22364e10i			3.37141i
$640$			3.64290e9i			0.549310i
$641$		−	5.14316e9i		−	0.771307i	−0.922644	−	0.385653i	$-0.873976\pi$
$641$		−	5.14316e9i		−	0.771307i	0.922644	−	0.385653i	$-0.126024\pi$
$642$			1.69015e10i			2.52088i
$643$		−	4.61160e9i		−	0.684090i	−0.939684	−	0.342045i	$-0.888880\pi$
$643$		−	4.61160e9i		−	0.684090i	0.939684	−	0.342045i	$-0.111120\pi$
$644$	−2.46530e9			−0.363721
$645$			2.08257e10i			3.05590i
$646$	5.74606e8			0.0838603
$647$		−	3.88068e9i		−	0.563304i	−0.959517	−	0.281652i	$-0.909118\pi$
$647$		−	3.88068e9i		−	0.563304i	0.959517	−	0.281652i	$-0.0908824\pi$
$648$		−	2.51196e9i		−	0.362661i
$649$	2.26105e9			0.324679
$650$			2.02149e9i			0.288718i
$651$	−4.22441e9			−0.600112
$652$	−8.20391e9			−1.15919
$653$			4.88319e9i			0.686290i	0.939282	+	0.343145i	$0.111492\pi$
$653$			4.88319e9i			0.686290i	−0.939282	+	0.343145i	$0.888508\pi$
$654$		−	8.61166e9i		−	1.20383i
$655$	−6.02084e9			−0.837168
$656$	−8.86534e9			−1.22612
$657$	−1.48984e10			−2.04957
$658$	5.62145e8			0.0769233
$659$	−6.23414e9			−0.848550			−0.424275	−	0.905533i	$-0.639471\pi$
$659$	−6.23414e9			−0.848550			−0.424275	+	0.905533i	$0.639471\pi$
$660$		−	8.27018e9i		−	1.11972i
$661$			5.54175e9i			0.746349i	0.927761	+	0.373174i	$0.121731\pi$
$661$			5.54175e9i			0.746349i	−0.927761	+	0.373174i	$0.878269\pi$
$662$	1.50111e10			2.01099
$663$		−	3.59943e10i		−	4.79663i
$664$		−	2.60467e9i		−	0.345275i
$665$			1.96697e8i			0.0259371i
$666$	−1.14577e10			−1.50292
$667$	−2.13936e9			−0.279153
$668$	−1.15506e9			−0.149929
$669$		−	1.89493e10i		−	2.44682i
$670$	−1.62383e10			−2.08583
$671$	3.92391e9	+	5.61782e9i	0.501406	+	0.717859i
$672$	1.09697e10			1.39445
$673$		−	1.10409e10i		−	1.39621i	−0.715994	−	0.698107i	$-0.754026\pi$
$673$		−	1.10409e10i		−	1.39621i	0.715994	−	0.698107i	$-0.245974\pi$
$674$	−1.68370e10			−2.11814
$675$	1.70192e9			0.212998
$676$	−1.63755e10			−2.03883
$677$		−	9.47361e9i		−	1.17343i	−0.809795	−	0.586713i	$-0.800422\pi$
$677$		−	9.47361e9i		−	1.17343i	0.809795	−	0.586713i	$-0.199578\pi$
$678$		−	7.58001e9i		−	0.934041i
$679$			4.03132e9i			0.494201i
$680$	3.32034e9			0.404950
$681$		−	2.66896e9i		−	0.323837i
$682$			5.26261e9i			0.635267i
$683$	−5.67045e9			−0.680997			−0.340499	−	0.940245i	$-0.610596\pi$
$683$	−5.67045e9			−0.680997			−0.340499	+	0.940245i	$0.610596\pi$
$684$	−5.87975e8			−0.0702527
$685$	4.05708e9			0.482277
$686$	1.13705e10			1.34476
$687$	−2.26710e10			−2.66761
$688$			1.85845e10i			2.17567i
$689$		−	1.88944e10i		−	2.20072i
$690$	1.41386e10			1.63845
$691$	8.00162e9			0.922582			0.461291	−	0.887249i	$-0.347386\pi$
$691$	8.00162e9			0.922582			0.461291	+	0.887249i	$0.347386\pi$
$692$			1.07193e9i			0.122969i
$693$	9.98596e9			1.13979
$694$			1.15296e10i			1.30935i
$695$			1.86289e9i			0.210494i
$696$	1.74608e9			0.196306
$697$			1.34163e10i			1.50078i
$698$	−1.57828e10			−1.75667
$699$			2.30783e10i			2.55584i
$700$			5.03509e8i			0.0554835i
$701$			1.90623e9i			0.209008i	0.994524	+	0.104504i	$0.0333255\pi$
$701$			1.90623e9i			0.209008i	−0.994524	+	0.104504i	$0.966674\pi$
$702$			4.35727e10i			4.75373i
$703$		−	2.19924e8i		−	0.0238742i
$704$		−	4.14743e9i		−	0.447996i
$705$	−1.40801e9			−0.151337
$706$		−	1.70698e10i		−	1.82563i
$707$	4.66235e9			0.496178
$708$			4.75618e9i			0.503665i
$709$		−	1.06413e10i		−	1.12133i	−0.828042	−	0.560667i	$-0.810545\pi$
$709$		−	1.06413e10i		−	1.12133i	0.828042	−	0.560667i	$-0.189455\pi$
$710$	1.94933e10			2.04400
$711$		−	3.24620e10i		−	3.38713i
$712$	1.76560e9			0.183321
$713$	−3.92929e9			−0.405977
$714$		−	2.05281e10i		−	2.11059i
$715$		−	1.53495e10i		−	1.57045i
$716$	7.32226e9			0.745503
$717$	1.83183e10			1.85596
$718$	5.22129e9			0.526432
$719$	−3.25505e9			−0.326592			−0.163296	−	0.986577i	$-0.552213\pi$
$719$	−3.25505e9			−0.326592			−0.163296	+	0.986577i	$0.552213\pi$
$720$	−2.29051e10			−2.28701
$721$			3.49454e9i			0.347230i
$722$			1.34490e10i			1.32988i
$723$	−5.41855e9			−0.533211
$724$			7.32273e9i			0.717113i
$725$			4.36940e8i			0.0425832i
$726$			5.61409e9i			0.544504i
$727$	−1.36754e10			−1.31998			−0.659992	−	0.751273i	$-0.729440\pi$
$727$	−1.36754e10			−1.31998			−0.659992	+	0.751273i	$0.729440\pi$
$728$	3.73449e9			0.358733
$729$	−8.09307e9			−0.773690
$730$			1.30605e10i			1.24260i
$731$	2.81248e10			2.66305
$732$	−1.18172e10	+	8.25403e9i	−1.11359	+	0.777817i
$733$	3.65132e9			0.342442			0.171221	−	0.985233i	$-0.445229\pi$
$733$	3.65132e9			0.342442			0.171221	+	0.985233i	$0.445229\pi$
$734$			2.26523e10i			2.11434i
$735$	−1.07264e10			−0.996435
$736$	1.02034e10			0.943346
$737$	−1.58238e10			−1.45605
$738$		−	3.14340e10i		−	2.87874i
$739$		−	1.27074e10i		−	1.15825i	−0.815240	−	0.579124i	$-0.803395\pi$
$739$		−	1.27074e10i		−	1.15825i	0.815240	−	0.579124i	$-0.196605\pi$
$740$			4.38671e9i			0.397949i
$741$	−1.61874e9			−0.146155
$742$		−	1.07757e10i		−	0.968353i
$743$		−	3.42182e9i		−	0.306053i	−0.988222	−	0.153026i	$-0.951098\pi$
$743$		−	3.42182e9i		−	0.306053i	0.988222	−	0.153026i	$-0.0489020\pi$
$744$	3.20698e9			0.285490
$745$	−3.28726e9			−0.291264
$746$	2.40709e10			2.12279
$747$	2.71919e10			2.38681
$748$	−1.11688e10			−0.975775
$749$			7.81345e9i			0.679448i
$750$		−	2.82762e10i		−	2.44741i
$751$	−8.55632e9			−0.737135			−0.368568	−	0.929601i	$-0.620152\pi$
$751$	−8.55632e9			−0.737135			−0.368568	+	0.929601i	$0.620152\pi$
$752$	−1.25649e9			−0.107745
$753$		−	2.40439e10i		−	2.05221i
$754$	−1.11866e10			−0.950383
$755$			9.08551e9i			0.768307i
$756$			1.08530e10i			0.913534i
$757$	1.54252e10			1.29239			0.646197	−	0.763170i	$-0.276358\pi$
$757$	1.54252e10			1.29239			0.646197	+	0.763170i	$0.276358\pi$
$758$		−	2.14478e10i		−	1.78871i
$759$	1.37777e10			1.14375
$760$		−	1.49323e8i		−	0.0123390i
$761$		−	2.74981e9i		−	0.226181i	−0.993585	−	0.113091i	$-0.963925\pi$
$761$		−	2.74981e9i		−	0.226181i	0.993585	−	0.113091i	$-0.0360750\pi$
$762$			1.84801e9i			0.151308i
$763$		−	3.98111e9i		−	0.324465i
$764$			8.40650e8i			0.0682006i
$765$			3.46633e10i			2.79933i
$766$	1.03346e10			0.830790
$767$			8.82752e9i			0.706407i
$768$	−2.83495e10			−2.25830
$769$		−	1.64685e10i		−	1.30591i	−0.757397	−	0.652954i	$-0.773529\pi$
$769$		−	1.64685e10i		−	1.30591i	0.757397	−	0.652954i	$-0.226471\pi$
$770$		−	8.75408e9i		−	0.691023i
$771$	3.55950e10			2.79704
$772$		−	3.59786e9i		−	0.281439i
$773$	−2.25048e10			−1.75246			−0.876228	−	0.481898i	$-0.839948\pi$
$773$	−2.25048e10			−1.75246			−0.876228	+	0.481898i	$0.839948\pi$
$774$	−6.58957e10			−5.10815
$775$			8.02514e8i			0.0619294i
$776$		−	3.06039e9i		−	0.235105i
$777$	−7.85691e9			−0.600867
$778$	8.84769e9			0.673598
$779$	6.03360e8			0.0457294
$780$	3.22881e10			2.43619
$781$	1.89956e10			1.42684
$782$		−	1.90940e10i		−	1.42782i
$783$			9.41815e9i			0.701131i
$784$	−9.57210e9			−0.709416
$785$		−	2.28014e10i		−	1.68236i
$786$		−	2.82587e10i		−	2.07575i
$787$			1.95625e10i			1.43058i	0.698826	+	0.715291i	$0.253706\pi$
$787$			1.95625e10i			1.43058i	−0.698826	+	0.715291i	$0.746294\pi$
$788$	−5.62581e9			−0.409584
$789$	−1.68490e10			−1.22125
$790$	−2.84575e10			−2.05353
$791$		−	3.50418e9i		−	0.251750i
$792$	−7.58088e9			−0.542228
$793$	−2.19329e10	+	1.53196e10i	−1.56185	+	1.09091i
$794$	2.91652e10			2.06772
$795$			2.69902e10i			1.90512i
$796$	2.18568e10			1.53600
$797$	−8.78323e9			−0.614540			−0.307270	−	0.951622i	$-0.599415\pi$
$797$	−8.78323e9			−0.614540			−0.307270	+	0.951622i	$0.599415\pi$
$798$	−9.23193e8			−0.0643105
$799$			1.90151e9i			0.131882i
$800$		−	2.08392e9i		−	0.143902i
$801$			1.84323e10i			1.26726i
$802$	3.22353e10			2.20659
$803$			1.27271e10i			0.867414i
$804$		−	3.32858e10i		−	2.25872i
$805$	6.53617e9			0.441609
$806$	−2.05461e10			−1.38215
$807$	−1.05078e10			−0.703806
$808$	−3.53944e9			−0.236045
$809$	−5.19577e9			−0.345009			−0.172504	−	0.985009i	$-0.555186\pi$
$809$	−5.19577e9			−0.345009			−0.172504	+	0.985009i	$0.555186\pi$
$810$		−	2.29890e10i		−	1.51993i
$811$			1.68432e10i			1.10880i	0.832252	+	0.554398i	$0.187051\pi$
$811$			1.68432e10i			1.10880i	−0.832252	+	0.554398i	$0.812949\pi$
$812$	−2.78635e9			−0.182637
$813$	−3.68458e9			−0.240475
$814$			9.78784e9i			0.636065i
$815$	2.17508e10			1.40742
$816$			4.58839e10i			2.95627i
$817$		−	1.26483e9i		−	0.0811439i
$818$	1.22486e10			0.782441
$819$			3.89868e10i			2.47984i
$820$	−1.20349e10			−0.762244
$821$		−	6.65304e9i		−	0.419584i	−0.977746	−	0.209792i	$-0.932721\pi$
$821$		−	6.65304e9i		−	0.419584i	0.977746	−	0.209792i	$-0.0672787\pi$
$822$			1.90418e10i			1.19580i
$823$			2.78630e10i			1.74232i	0.490995	+	0.871162i	$0.336633\pi$
$823$			2.78630e10i			1.74232i	−0.490995	+	0.871162i	$0.663367\pi$
$824$		−	2.65290e9i		−	0.165187i
$825$		−	2.81394e9i		−	0.174472i
$826$			5.03447e9i			0.310831i
$827$	2.06307e10			1.26837			0.634183	−	0.773183i	$-0.281336\pi$
$827$	2.06307e10			1.26837			0.634183	+	0.773183i	$0.281336\pi$
$828$			1.95383e10i			1.19613i
$829$	−1.32409e10			−0.807190			−0.403595	−	0.914938i	$-0.632240\pi$
$829$	−1.32409e10			−0.807190			−0.403595	+	0.914938i	$0.632240\pi$
$830$		−	2.38375e10i		−	1.44706i
$831$		−	9.71507e9i		−	0.587276i
$832$	1.61922e10			0.974709
$833$			1.44859e10i			0.868336i
$834$	−8.74345e9			−0.521917
$835$	3.06238e9			0.182035
$836$			5.02284e8i			0.0297322i
$837$			1.72980e10i			1.01966i
$838$	3.31180e10			1.94406
$839$	9.62441e9			0.562610			0.281305	−	0.959618i	$-0.409233\pi$
$839$	9.62441e9			0.562610			0.281305	+	0.959618i	$0.409233\pi$
$840$	−5.33464e9			−0.310547
$841$	1.48319e10			0.859827
$842$	−9.16772e9			−0.529260
$843$		−	1.54729e10i		−	0.889559i
$844$			3.94260e9i			0.225727i
$845$	4.34158e10			2.47543
$846$		−	4.45518e9i		−	0.252970i
$847$			2.59535e9i			0.146759i
$848$			2.40857e10i			1.35636i
$849$	−1.59803e10			−0.896208
$850$	−3.89974e9			−0.217806
$851$	−7.30802e9			−0.406487
$852$			3.99578e10i			2.21342i
$853$	−2.49736e10			−1.37771			−0.688857	−	0.724897i	$-0.741887\pi$
$853$	−2.49736e10			−1.37771			−0.688857	+	0.724897i	$0.741887\pi$
$854$	−1.25087e10	+	8.73699e9i	−0.687240	+	0.480020i
$855$	1.55888e9			0.0852967
$856$		−	5.93161e9i		−	0.323232i
$857$	2.38197e8			0.0129272			0.00646359	−	0.999979i	$-0.497943\pi$
$857$	2.38197e8			0.0129272			0.00646359	+	0.999979i	$0.497943\pi$
$858$	7.20428e10			3.89390
$859$	3.10481e10			1.67132			0.835658	−	0.549250i	$-0.185087\pi$
$859$	3.10481e10			1.67132			0.835658	+	0.549250i	$0.185087\pi$
$860$			2.52289e10i			1.35255i
$861$		−	2.15553e10i		−	1.15092i
$862$		−	1.87528e10i		−	0.997219i
$863$	2.62701e10			1.39131			0.695654	−	0.718377i	$-0.255114\pi$
$863$	2.62701e10			1.39131			0.695654	+	0.718377i	$0.255114\pi$
$864$		−	4.49185e10i		−	2.36934i
$865$		−	2.84197e9i		−	0.149301i
$866$	−3.80435e10			−1.99053
$867$	3.58207e10			1.86667
$868$	−5.11760e9			−0.265612
$869$	−2.77310e10			−1.43350
$870$	1.59798e10			0.822725
$871$		−	6.17787e10i		−	3.16793i
$872$			3.02227e9i			0.154357i
$873$	3.19495e10			1.62523
$874$	−8.58698e8			−0.0435061
$875$		−	1.30719e10i		−	0.659644i
$876$	−2.67719e10			−1.34559
$877$			3.52078e8i			0.0176255i	0.999961	+	0.00881274i	$0.00280522\pi$
$877$			3.52078e8i			0.0176255i	−0.999961	+	0.00881274i	$0.997195\pi$
$878$			1.71909e10i			0.857171i
$879$	−5.43920e10			−2.70131
$880$			1.95669e10i			0.967905i
$881$	−2.12783e10			−1.04839			−0.524194	−	0.851599i	$-0.675633\pi$
$881$	−2.12783e10			−1.04839			−0.524194	+	0.851599i	$0.675633\pi$
$882$		−	3.39401e10i		−	1.66561i
$883$			1.73110e10i			0.846175i	0.906089	+	0.423088i	$0.139054\pi$
$883$			1.73110e10i			0.846175i	−0.906089	+	0.423088i	$0.860946\pi$
$884$		−	4.36047e10i		−	2.12300i
$885$		−	1.26099e10i		−	0.611521i
$886$			1.72278e10i			0.832168i
$887$			1.78510e10i			0.858874i	0.903097	+	0.429437i	$0.141288\pi$
$887$			1.78510e10i			0.858874i	−0.903097	+	0.429437i	$0.858712\pi$
$888$	5.96460e9			0.285849
$889$			8.54321e8i			0.0407817i
$890$	1.61584e10			0.768305
$891$		−	2.24022e10i		−	1.06101i
$892$		−	2.29558e10i		−	1.08297i
$893$	8.55149e7			0.00401848
$894$		−	1.54287e10i		−	0.722185i
$895$	−1.94133e10			−0.905146
$896$	−7.90424e9			−0.367098
$897$			5.37903e10i			2.48846i
$898$			1.99920e10i			0.921274i
$899$	−4.44100e9			−0.203855
$900$	3.99047e9			0.182463
$901$	3.64499e10			1.66020
$902$	−2.68528e10			−1.21834
$903$	−4.51868e10			−2.04223
$904$			2.66022e9i			0.119764i
$905$		−	1.94145e10i		−	0.870677i
$906$	−4.26427e10			−1.90501
$907$			2.30401e10i			1.02532i	0.858592	+	0.512659i	$0.171339\pi$
$907$			2.30401e10i			1.02532i	−0.858592	+	0.512659i	$0.828661\pi$
$908$		−	3.23327e9i		−	0.143331i
$909$		−	3.69506e10i		−	1.63173i
$910$	3.41773e10			1.50347
$911$	−5.55766e9			−0.243544			−0.121772	−	0.992558i	$-0.538858\pi$
$911$	−5.55766e9			−0.243544			−0.121772	+	0.992558i	$0.538858\pi$
$912$	2.06350e9			0.0900787
$913$		−	2.32290e10i		−	1.01014i
$914$	−2.41971e10			−1.04822
$915$	3.13307e10	−	2.18837e10i	1.35206	−	0.944380i
$916$	−2.74645e10			−1.18069
$917$		−	1.30638e10i		−	0.559471i
$918$	−8.40580e10			−3.58616
$919$	−2.69169e10			−1.14398			−0.571992	−	0.820259i	$-0.693829\pi$
$919$	−2.69169e10			−1.14398			−0.571992	+	0.820259i	$0.693829\pi$
$920$	−4.96196e9			−0.210085
$921$		−	3.83986e10i		−	1.61960i
$922$		−	5.01543e10i		−	2.10742i
$923$			7.41621e10i			3.10439i
$924$	1.79443e10			0.748300
$925$			1.49258e9i			0.0620072i
$926$		−	1.91746e10i		−	0.793576i
$927$	2.76954e10			1.14190
$928$	1.15321e10			0.473687
$929$	−2.01199e10			−0.823324			−0.411662	−	0.911336i	$-0.635052\pi$
$929$	−2.01199e10			−0.823324			−0.411662	+	0.911336i	$0.635052\pi$
$930$	2.93497e10			1.19650
$931$	6.51462e8			0.0264585
$932$			2.79578e10i			1.13122i
$933$			1.88159e10i			0.758471i
$934$	2.56152e10			1.02869
$935$	2.96115e10			1.18473
$936$		−	2.95970e10i		−	1.17973i
$937$	−4.18094e10			−1.66030			−0.830148	−	0.557544i	$-0.811744\pi$
$937$	−4.18094e10			−1.66030			−0.830148	+	0.557544i	$0.811744\pi$
$938$		−	3.52334e10i		−	1.39394i
$939$			2.09628e10i			0.826265i
$940$	−1.70572e9			−0.0669823
$941$			3.22036e10i			1.25992i	0.776630	+	0.629958i	$0.216928\pi$
$941$			3.22036e10i			1.25992i	−0.776630	+	0.629958i	$0.783072\pi$
$942$	1.07018e11			4.17138
$943$		−	2.00495e10i		−	0.778596i
$944$		−	1.12529e10i		−	0.435375i
$945$		−	2.87744e10i		−	1.10916i
$946$			5.62920e10i			2.16186i
$947$		−	3.07667e10i		−	1.17722i	−0.808418	−	0.588608i	$-0.799676\pi$
$947$		−	3.07667e10i		−	1.17722i	0.808418	−	0.588608i	$-0.200324\pi$
$948$		−	5.83329e10i		−	2.22374i
$949$	−4.96888e10			−1.88724
$950$			1.75380e8i			0.00663661i
$951$	1.73587e10			0.654463
$952$			7.20436e9i			0.270624i
$953$			4.70246e10i			1.75995i	0.475023	+	0.879973i	$0.342440\pi$
$953$			4.70246e10i			1.75995i	−0.475023	+	0.879973i	$0.657560\pi$
$954$	−8.54012e10			−3.18453
$955$		−	2.22879e9i		−	0.0828052i
$956$	2.21915e10			0.821455
$957$	1.55719e10			0.574315
$958$			3.70484e9i			0.136141i
$959$			8.80291e9i			0.322300i
$960$	−2.31303e10			−0.843784
$961$	1.93560e10			0.703531
$962$	−3.82133e10			−1.38389
$963$	6.19241e10			2.23443
$964$	−6.56422e9			−0.236001
$965$			9.53891e9i			0.341707i
$966$			3.06774e10i			1.09496i
$967$	1.36791e10			0.486481			0.243241	−	0.969966i	$-0.421790\pi$
$967$	1.36791e10			0.486481			0.243241	+	0.969966i	$0.421790\pi$
$968$		−	1.97027e9i		−	0.0698172i
$969$		−	3.12278e9i		−	0.110258i
$970$		−	2.80082e10i		−	0.985334i
$971$	−5.58533e10			−1.95786			−0.978930	−	0.204198i	$-0.934541\pi$
$971$	−5.58533e10			−1.95786			−0.978930	+	0.204198i	$0.934541\pi$
$972$	5.55055e9			0.193867
$973$	−4.04203e9			−0.140671
$974$		−	2.12025e10i		−	0.735243i
$975$	1.09861e10			0.379600
$976$	2.79591e10	−	1.95287e10i	0.962606	−	0.672356i
$977$	−2.44876e10			−0.840069			−0.420034	−	0.907508i	$-0.637982\pi$
$977$	−2.44876e10			−0.840069			−0.420034	+	0.907508i	$0.637982\pi$
$978$			1.02087e11i			3.48967i
$979$	1.57459e10			0.536326
$980$	−1.29944e10			−0.441025
$981$	−3.15515e10			−1.06704
$982$			1.97086e10i			0.664148i
$983$			1.34441e10i			0.451435i	0.974193	+	0.225718i	$0.0724727\pi$
$983$			1.34441e10i			0.451435i	−0.974193	+	0.225718i	$0.927527\pi$
$984$			1.63638e10i			0.547523i
$985$	1.49156e10			0.497294
$986$		−	2.15806e10i		−	0.716958i
$987$		−	3.05506e9i		−	0.101137i
$988$	−1.96100e9			−0.0646886
$989$	−4.20300e10			−1.38157
$990$	−6.93789e10			−2.27250
$991$	3.39993e10			1.10972			0.554859	−	0.831944i	$-0.312772\pi$
$991$	3.39993e10			1.10972			0.554859	+	0.831944i	$0.312772\pi$
$992$	2.11807e10			0.688890
$993$		−	8.15802e10i		−	2.64401i
$994$			4.22958e10i			1.36598i
$995$	−5.79485e10			−1.86492
$996$	4.88627e10			1.56700
$997$		−	3.86927e10i		−	1.23651i	−0.785979	−	0.618253i	$-0.787841\pi$
$997$		−	3.86927e10i		−	1.23651i	0.785979	−	0.618253i	$-0.212159\pi$
$998$	3.06530e10			0.976149
$999$			3.21723e10i			1.02095i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	61.8.b.a.60.6	✓	34
61.60	even	2	inner	61.8.b.a.60.29	yes	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.8.b.a.60.6	✓	34	1.1	even	1	trivial
61.8.b.a.60.29	yes	34	61.60	even	2	inner

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:	\( N \)	\(=\)	\( 61 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	61.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-15.0747i q^{2} -81.9260 q^{3} -99.2480 q^{4} +263.133 q^{5} +1235.01i q^{6} +570.938i q^{7} -433.430i q^{8} +4524.86 q^{9} +O(q^{10})\)	\(q-15.0747i q^{2} -81.9260 q^{3} -99.2480 q^{4} +263.133 q^{5} +1235.01i q^{6} +570.938i q^{7} -433.430i q^{8} +4524.86 q^{9} -3966.67i q^{10} -3865.41i q^{11} +8130.99 q^{12} +15091.2 q^{13} +8606.74 q^{14} -21557.5 q^{15} -19237.6 q^{16} +29113.1i q^{17} -68211.2i q^{18} +1309.28 q^{19} -26115.5 q^{20} -46774.6i q^{21} -58270.1 q^{22} -43506.9i q^{23} +35509.1i q^{24} -8885.81 q^{25} -227496. i q^{26} -191532. q^{27} -56664.4i q^{28} -49172.8i q^{29} +324973. i q^{30} -90314.1i q^{31} +234523. i q^{32} +316677. i q^{33} +438872. q^{34} +150233. i q^{35} -449084. q^{36} -167974. i q^{37} -19737.0i q^{38} -1.23636e6 q^{39} -114050. i q^{40} +460834. q^{41} -705116. q^{42} -966054. i q^{43} +383634. i q^{44} +1.19064e6 q^{45} -655856. q^{46} +65314.5 q^{47} +1.57606e6 q^{48} +497573. q^{49} +133951. i q^{50} -2.38512e6i q^{51} -1.49777e6 q^{52} -1.25201e6i q^{53} +2.88729e6i q^{54} -1.01712e6i q^{55} +247461. q^{56} -107264. q^{57} -741267. q^{58} +584945. i q^{59} +2.13953e6 q^{60} +(-1.45336e6 + 1.01513e6i) q^{61} -1.36146e6 q^{62} +2.58342e6i q^{63} +1.07296e6 q^{64} +3.97100e6 q^{65} +4.77383e6 q^{66} -4.09370e6i q^{67} -2.88941e6i q^{68} +3.56435e6i q^{69} +2.26472e6 q^{70} +4.91426e6i q^{71} -1.96121e6i q^{72} -3.29257e6 q^{73} -2.53216e6 q^{74} +727978. q^{75} -129943. q^{76} +2.20691e6 q^{77} +1.86378e7i q^{78} -7.17414e6i q^{79} -5.06205e6 q^{80} +5.79555e6 q^{81} -6.94696e6i q^{82} +6.00945e6 q^{83} +4.64229e6i q^{84} +7.66062e6i q^{85} -1.45630e7 q^{86} +4.02853e6i q^{87} -1.67538e6 q^{88} +4.07355e6i q^{89} -1.79486e7i q^{90} +8.61613e6i q^{91} +4.31798e6i q^{92} +7.39907e6i q^{93} -984600. i q^{94} +344515. q^{95} -1.92135e7i q^{96} +7.06088e6 q^{97} -7.50079e6i q^{98} -1.74905e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9}+O(q^{10}) \) 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9	\( 34 q + 52 q^{3} - 1974 q^{4} + 696 q^{5} + 18902 q^{9} - 8382 q^{12} + 19860 q^{13} + 3744 q^{14} - 24988 q^{15} + 110642 q^{16} + 48660 q^{19} - 148866 q^{20} - 147578 q^{22} + 220494 q^{25} + 371248 q^{27} + 382432 q^{34} + 241132 q^{36} - 908620 q^{39} + 569544 q^{41} + 421016 q^{42} - 306560 q^{45} - 342232 q^{46} - 1179240 q^{47} + 3464534 q^{48} - 1069538 q^{49} - 2494330 q^{52} + 3794484 q^{56} + 3106472 q^{57} - 6540834 q^{58} - 2370908 q^{60} + 2828770 q^{61} + 5869278 q^{62} + 3854362 q^{64} - 502500 q^{65} - 4330852 q^{66} - 16861390 q^{70} - 6245708 q^{73} + 7083030 q^{74} - 13741952 q^{75} - 2405382 q^{76} - 14437656 q^{77} - 5019522 q^{80} + 31255922 q^{81} + 1991136 q^{83} - 4399944 q^{86} + 25423434 q^{88} - 22196868 q^{95} - 6652408 q^{97}+O(q^{100}) \) 34 * q + 52 * q^3 - 1974 * q^4 + 696 * q^5 + 18902 * q^9 - 8382 * q^12 + 19860 * q^13 + 3744 * q^14 - 24988 * q^15 + 110642 * q^16 + 48660 * q^19 - 148866 * q^20 - 147578 * q^22 + 220494 * q^25 + 371248 * q^27 + 382432 * q^34 + 241132 * q^36 - 908620 * q^39 + 569544 * q^41 + 421016 * q^42 - 306560 * q^45 - 342232 * q^46 - 1179240 * q^47 + 3464534 * q^48 - 1069538 * q^49 - 2494330 * q^52 + 3794484 * q^56 + 3106472 * q^57 - 6540834 * q^58 - 2370908 * q^60 + 2828770 * q^61 + 5869278 * q^62 + 3854362 * q^64 - 502500 * q^65 - 4330852 * q^66 - 16861390 * q^70 - 6245708 * q^73 + 7083030 * q^74 - 13741952 * q^75 - 2405382 * q^76 - 14437656 * q^77 - 5019522 * q^80 + 31255922 * q^81 + 1991136 * q^83 - 4399944 * q^86 + 25423434 * q^88 - 22196868 * q^95 - 6652408 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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