LMFDB - Embedded newform 38.9.b.a.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [38,9,Mod(37,38)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(38, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("38.37");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 38 = 2 \cdot 19 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	38.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:	$15.4803871823$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 46118 x^{10} + 738386961 x^{8} + 5214446299656 x^{6} + \cdots + 92\!\cdots\!64 $ x^12 + 46118x^10 + 738386961x^8 + 5214446299656x^6 + 17370647163698184x^4 + 24830681474333400768*x^2 + 9218779084612644462864
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{21} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.2
Root			$-64.6816i$ of defining polynomial
Character	$\chi$	$=$	38.37
Dual form			38.9.b.a.37.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-11.3137i q^{2} -51.9537i q^{3} -128.000 q^{4} +629.221 q^{5} -587.789 q^{6} +2565.52 q^{7} +1448.15i q^{8} +3861.81 q^{9} +O(q^{10})$	\(q-11.3137i q^{2} -51.9537i q^{3} -128.000 q^{4} +629.221 q^{5} -587.789 q^{6} +2565.52 q^{7} +1448.15i q^{8} +3861.81 q^{9} -7118.82i q^{10} +9430.05 q^{11} +6650.08i q^{12} -7159.53i q^{13} -29025.6i q^{14} -32690.4i q^{15} +16384.0 q^{16} +10353.8 q^{17} -43691.4i q^{18} +(-129250. - 16674.4i) q^{19} -80540.3 q^{20} -133289. i q^{21} -106689. i q^{22} +84015.6 q^{23} +75237.0 q^{24} +5293.93 q^{25} -81000.9 q^{26} -541504. i q^{27} -328387. q^{28} -668157. i q^{29} -369849. q^{30} -442850. i q^{31} -185364. i q^{32} -489926. i q^{33} -117140. i q^{34} +1.61428e6 q^{35} -494312. q^{36} +593205. i q^{37} +(-188649. + 1.46230e6i) q^{38} -371964. q^{39} +911209. i q^{40} +3.30133e6i q^{41} -1.50799e6 q^{42} +4.37474e6 q^{43} -1.20705e6 q^{44} +2.42993e6 q^{45} -950528. i q^{46} -888620. q^{47} -851210. i q^{48} +817114. q^{49} -59894.0i q^{50} -537920. i q^{51} +916420. i q^{52} +6.29819e6i q^{53} -6.12642e6 q^{54} +5.93358e6 q^{55} +3.71528e6i q^{56} +(-866297. + 6.71501e6i) q^{57} -7.55933e6 q^{58} +1.98429e7i q^{59} +4.18437e6i q^{60} -4.10665e6 q^{61} -5.01027e6 q^{62} +9.90757e6 q^{63} -2.09715e6 q^{64} -4.50493e6i q^{65} -5.54288e6 q^{66} -3.39031e7i q^{67} -1.32529e6 q^{68} -4.36492e6i q^{69} -1.82635e7i q^{70} -3.72573e7i q^{71} +5.59250e6i q^{72} -65965.6 q^{73} +6.71135e6 q^{74} -275040. i q^{75} +(1.65440e7 + 2.13432e6i) q^{76} +2.41930e7 q^{77} +4.20830e6i q^{78} +3.84925e6i q^{79} +1.03092e7 q^{80} -2.79578e6 q^{81} +3.73503e7 q^{82} -3.76526e7 q^{83} +1.70609e7i q^{84} +6.51484e6 q^{85} -4.94946e7i q^{86} -3.47132e7 q^{87} +1.36562e7i q^{88} -7.00026e7i q^{89} -2.74916e7i q^{90} -1.83680e7i q^{91} -1.07540e7 q^{92} -2.30077e7 q^{93} +1.00536e7i q^{94} +(-8.13267e7 - 1.04919e7i) q^{95} -9.63034e6 q^{96} +1.21711e8i q^{97} -9.24459e6i q^{98} +3.64171e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	$ 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) $ 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/38\mathbb{Z}\right)^\times$.

$n$	$21$
$\chi(n)$	$-1$

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 $

$2$		−	11.3137i		−	0.707107i
$2$		−	11.3137i		−	0.707107i
$3$		−	51.9537i		−	0.641404i	−0.947180	−	0.320702i	$-0.896081\pi$
$3$		−	51.9537i		−	0.641404i	0.947180	−	0.320702i	$-0.103919\pi$
$4$	−128.000			−0.500000
$5$	629.221			1.00675			0.503377	−	0.864067i	$-0.332091\pi$
$5$	629.221			1.00675			0.503377	+	0.864067i	$0.332091\pi$
$6$	−587.789			−0.453541
$7$	2565.52			1.06852			0.534262	−	0.845319i	$-0.320590\pi$
$7$	2565.52			1.06852			0.534262	+	0.845319i	$0.320590\pi$
$8$			1448.15i			0.353553i
$9$	3861.81			0.588601
$10$		−	7118.82i		−	0.711882i
$11$	9430.05			0.644085			0.322042	−	0.946725i	$-0.395631\pi$
$11$	9430.05			0.644085			0.322042	+	0.946725i	$0.395631\pi$
$12$			6650.08i			0.320702i
$13$		−	7159.53i		−	0.250675i	−0.992114	−	0.125338i	$-0.959999\pi$
$13$		−	7159.53i		−	0.250675i	0.992114	−	0.125338i	$-0.0400014\pi$
$14$		−	29025.6i		−	0.755560i
$15$		−	32690.4i		−	0.645736i
$16$	16384.0			0.250000
$17$	10353.8			0.123967			0.0619834	−	0.998077i	$-0.480257\pi$
$17$	10353.8			0.123967			0.0619834	+	0.998077i	$0.480257\pi$
$18$		−	43691.4i		−	0.416204i
$19$	−129250.	−	16674.4i	−0.991781	−	0.127949i
$19$	−129250.	−	16674.4i	−0.991781	−	0.127949i
$20$	−80540.3			−0.503377
$21$		−	133289.i		−	0.685355i
$22$		−	106689.i		−	0.455437i
$23$	84015.6			0.300226			0.150113	−	0.988669i	$-0.452036\pi$
$23$	84015.6			0.300226			0.150113	+	0.988669i	$0.452036\pi$
$24$	75237.0			0.226771
$25$	5293.93			0.0135525
$26$	−81000.9			−0.177254
$27$		−	541504.i		−	1.01893i
$28$	−328387.			−0.534262
$29$		−	668157.i		−	0.944683i	−0.881416	−	0.472342i	$-0.843409\pi$
$29$		−	668157.i		−	0.944683i	0.881416	−	0.472342i	$-0.156591\pi$
$30$	−369849.			−0.456604
$31$		−	442850.i		−	0.479523i	−0.970832	−	0.239762i	$-0.922931\pi$
$31$		−	442850.i		−	0.479523i	0.970832	−	0.239762i	$-0.0770693\pi$
$32$		−	185364.i		−	0.176777i
$33$		−	489926.i		−	0.413119i
$34$		−	117140.i		−	0.0876577i
$35$	1.61428e6			1.07574
$36$	−494312.			−0.294301
$37$			593205.i			0.316518i	0.987398	+	0.158259i	$0.0505880\pi$
$37$			593205.i			0.316518i	−0.987398	+	0.158259i	$0.949412\pi$
$38$	−188649.	+	1.46230e6i	−0.0904734	+	0.701295i
$39$	−371964.			−0.160784
$40$			911209.i			0.355941i
$41$			3.30133e6i			1.16830i	0.811646	+	0.584149i	$0.198572\pi$
$41$			3.30133e6i			1.16830i	−0.811646	+	0.584149i	$0.801428\pi$
$42$	−1.50799e6			−0.484619
$43$	4.37474e6			1.27961			0.639807	−	0.768536i	$-0.279014\pi$
$43$	4.37474e6			1.27961			0.639807	+	0.768536i	$0.279014\pi$
$44$	−1.20705e6			−0.322042
$45$	2.42993e6			0.592576
$46$		−	950528.i		−	0.212292i
$47$	−888620.			−0.182106			−0.0910531	−	0.995846i	$-0.529023\pi$
$47$	−888620.			−0.182106			−0.0910531	+	0.995846i	$0.529023\pi$
$48$		−	851210.i		−	0.160351i
$49$	817114.			0.141742
$50$		−	59894.0i		−	0.00958304i
$51$		−	537920.i		−	0.0795128i
$52$			916420.i			0.125338i
$53$			6.29819e6i			0.798200i	0.916907	+	0.399100i	$0.130677\pi$
$53$			6.29819e6i			0.798200i	−0.916907	+	0.399100i	$0.869323\pi$
$54$	−6.12642e6			−0.720496
$55$	5.93358e6			0.648435
$56$			3.71528e6i			0.377780i
$57$	−866297.	+	6.71501e6i	−0.0820668	+	0.636132i
$58$	−7.55933e6			−0.667992
$59$			1.98429e7i			1.63756i	0.574108	+	0.818779i	$0.305349\pi$
$59$			1.98429e7i			1.63756i	−0.574108	+	0.818779i	$0.694651\pi$
$60$			4.18437e6i			0.322868i
$61$	−4.10665e6			−0.296598			−0.148299	−	0.988943i	$-0.547380\pi$
$61$	−4.10665e6			−0.296598			−0.148299	+	0.988943i	$0.547380\pi$
$62$	−5.01027e6			−0.339074
$63$	9.90757e6			0.628934
$64$	−2.09715e6			−0.125000
$65$		−	4.50493e6i		−	0.252368i
$66$	−5.54288e6			−0.292119
$67$		−	3.39031e7i		−	1.68244i	−0.540690	−	0.841222i	$-0.681837\pi$
$67$		−	3.39031e7i		−	1.68244i	0.540690	−	0.841222i	$-0.318163\pi$
$68$	−1.32529e6			−0.0619834
$69$		−	4.36492e6i		−	0.192566i
$70$		−	1.82635e7i		−	0.760663i
$71$		−	3.72573e7i		−	1.46615i	−0.680149	−	0.733074i	$-0.738085\pi$
$71$		−	3.72573e7i		−	1.46615i	0.680149	−	0.733074i	$-0.261915\pi$
$72$			5.59250e6i			0.208102i
$73$	−65965.6			−0.00232288			−0.00116144	−	0.999999i	$-0.500370\pi$
$73$	−65965.6			−0.00232288			−0.00116144	+	0.999999i	$0.500370\pi$
$74$	6.71135e6			0.223812
$75$		−	275040.i		−	0.00869261i
$76$	1.65440e7	+	2.13432e6i	0.495890	+	0.0639744i
$77$	2.41930e7			0.688220
$78$			4.20830e6i			0.113691i
$79$			3.84925e6i			0.0988251i	0.998778	+	0.0494126i	$0.0157349\pi$
$79$			3.84925e6i			0.0988251i	−0.998778	+	0.0494126i	$0.984265\pi$
$80$	1.03092e7			0.251688
$81$	−2.79578e6			−0.0649476
$82$	3.73503e7			0.826112
$83$	−3.76526e7			−0.793381			−0.396691	−	0.917952i	$-0.629841\pi$
$83$	−3.76526e7			−0.793381			−0.396691	+	0.917952i	$0.629841\pi$
$84$			1.70609e7i			0.342677i
$85$	6.51484e6			0.124804
$86$		−	4.94946e7i		−	0.904823i
$87$	−3.47132e7			−0.605923
$88$			1.36562e7i			0.227718i
$89$		−	7.00026e7i		−	1.11572i	−0.829936	−	0.557859i	$-0.811623\pi$
$89$		−	7.00026e7i		−	1.11572i	0.829936	−	0.557859i	$-0.188377\pi$
$90$		−	2.74916e7i		−	0.419015i
$91$		−	1.83680e7i		−	0.267852i
$92$	−1.07540e7			−0.150113
$93$	−2.30077e7			−0.307568
$94$			1.00536e7i			0.128769i
$95$	−8.13267e7	−	1.04919e7i	−0.998479	−	0.128813i
$96$	−9.63034e6			−0.113385
$97$			1.21711e8i			1.37481i	0.726274	+	0.687406i	$0.241250\pi$
$97$			1.21711e8i			1.37481i	−0.726274	+	0.687406i	$0.758750\pi$
$98$		−	9.24459e6i		−	0.100227i
$99$	3.64171e7			0.379109
$100$	−677624.			−0.00677624
$101$	−9.09160e7			−0.873685			−0.436842	−	0.899538i	$-0.643903\pi$
$101$	−9.09160e7			−0.873685			−0.436842	+	0.899538i	$0.643903\pi$
$102$	−6.08587e6			−0.0562240
$103$			1.97485e8i			1.75463i	0.479919	+	0.877313i	$0.340666\pi$
$103$			1.97485e8i			1.75463i	−0.479919	+	0.877313i	$0.659334\pi$
$104$	1.03681e7			0.0886271
$105$		−	8.38679e7i		−	0.689983i
$106$	7.12558e7			0.564413
$107$		−	1.36332e7i		−	0.104007i	−0.998647	−	0.0520035i	$-0.983439\pi$
$107$		−	1.36332e7i		−	0.104007i	0.998647	−	0.0520035i	$-0.0165607\pi$
$108$			6.93125e7i			0.509467i
$109$			6.37632e7i			0.451714i	0.974160	+	0.225857i	$0.0725183\pi$
$109$			6.37632e7i			0.451714i	−0.974160	+	0.225857i	$0.927482\pi$
$110$		−	6.71308e7i		−	0.458513i
$111$	3.08192e7			0.203016
$112$	4.20336e7			0.267131
$113$			1.98478e8i			1.21730i	0.793438	+	0.608652i	$0.208289\pi$
$113$			1.98478e8i			1.21730i	−0.793438	+	0.608652i	$0.791711\pi$
$114$	7.59717e7	+	9.80103e6i	0.449813	+	0.0580300i
$115$	5.28643e7			0.302254
$116$			8.55240e7i			0.472342i
$117$		−	2.76488e7i		−	0.147548i
$118$	2.24497e8			1.15793
$119$	2.65630e7			0.132461
$120$	4.73407e7			0.228302
$121$	−1.25433e8			−0.585155
$122$			4.64614e7i			0.209726i
$123$	1.71516e8			0.749351
$124$			5.66848e7i			0.239762i
$125$	−2.42458e8			−0.993109
$126$		−	1.12091e8i		−	0.444723i
$127$			1.85702e8i			0.713840i	0.934135	+	0.356920i	$0.116173\pi$
$127$			1.85702e8i			0.713840i	−0.934135	+	0.356920i	$0.883827\pi$
$128$			2.37266e7i			0.0883883i
$129$		−	2.27284e8i		−	0.820749i
$130$	−5.09675e7			−0.178451
$131$	4.67984e8			1.58908			0.794539	−	0.607213i	$-0.207712\pi$
$131$	4.67984e8			1.58908			0.794539	+	0.607213i	$0.207712\pi$
$132$			6.27105e7i			0.206559i
$133$	−3.31594e8	−	4.27786e7i	−1.05974	−	0.136716i
$134$	−3.83570e8			−1.18967
$135$		−	3.40725e8i		−	1.02582i
$136$			1.49939e7i			0.0438289i
$137$	−1.16856e8			−0.331718			−0.165859	−	0.986149i	$-0.553040\pi$
$137$	−1.16856e8			−0.331718			−0.165859	+	0.986149i	$0.553040\pi$
$138$	−4.93834e7			−0.136165
$139$	4.58364e8			1.22787			0.613933	−	0.789358i	$-0.289586\pi$
$139$	4.58364e8			1.22787			0.613933	+	0.789358i	$0.289586\pi$
$140$	−2.06628e8			−0.537870
$141$			4.61671e7i			0.116804i
$142$	−4.21518e8			−1.03672
$143$		−	6.75148e7i		−	0.161456i
$144$	6.32719e7			0.147150
$145$		−	4.20418e8i		−	0.951063i
$146$			746316.i			0.00164252i
$147$		−	4.24521e7i		−	0.0909138i
$148$		−	7.59302e7i		−	0.158259i
$149$	−4.29377e8			−0.871152			−0.435576	−	0.900152i	$-0.643455\pi$
$149$	−4.29377e8			−0.871152			−0.435576	+	0.900152i	$0.643455\pi$
$150$	−3.11172e6			−0.00614660
$151$			1.07932e8i			0.207608i	0.994598	+	0.103804i	$0.0331015\pi$
$151$			1.07932e8i			0.207608i	−0.994598	+	0.103804i	$0.966899\pi$
$152$	2.41471e7	−	1.87174e8i	0.0452367	−	0.350647i
$153$	3.99845e7			0.0729670
$154$		−	2.73713e8i		−	0.486645i
$155$		−	2.78650e8i		−	0.482762i
$156$	4.76114e7			0.0803920
$157$	−4.87378e7			−0.0802171			−0.0401086	−	0.999195i	$-0.512770\pi$
$157$	−4.87378e7			−0.0802171			−0.0401086	+	0.999195i	$0.512770\pi$
$158$	4.35492e7			0.0698799
$159$	3.27214e8			0.511969
$160$		−	1.16635e8i		−	0.177971i
$161$	2.15544e8			0.320799
$162$			3.16307e7i			0.0459249i
$163$	3.59771e8			0.509654			0.254827	−	0.966987i	$-0.417982\pi$
$163$	3.59771e8			0.509654			0.254827	+	0.966987i	$0.417982\pi$
$164$		−	4.22570e8i		−	0.584149i
$165$		−	3.08272e8i		−	0.415909i
$166$			4.25990e8i			0.561005i
$167$			3.42855e8i			0.440803i	0.975409	+	0.220402i	$0.0707368\pi$
$167$			3.42855e8i			0.440803i	−0.975409	+	0.220402i	$0.929263\pi$
$168$	1.93022e8			0.242310
$169$	7.64472e8			0.937162
$170$		−	7.37071e7i		−	0.0882497i
$171$	−4.99139e8	−	6.43934e7i	−0.583763	−	0.0753108i
$172$	−5.59967e8			−0.639807
$173$		−	6.11142e8i		−	0.682272i	−0.940014	−	0.341136i	$-0.889188\pi$
$173$		−	6.11142e8i		−	0.682272i	0.940014	−	0.341136i	$-0.110812\pi$
$174$			3.92735e8i			0.428453i
$175$	1.35817e7			0.0144811
$176$	1.54502e8			0.161021
$177$	1.03091e9			1.05034
$178$	−7.91989e8			−0.788931
$179$			3.06608e7i			0.0298656i	0.999888	+	0.0149328i	$0.00475344\pi$
$179$			3.06608e7i			0.0298656i	−0.999888	+	0.0149328i	$0.995247\pi$
$180$	−3.11031e8			−0.296288
$181$			2.14022e9i			1.99409i	0.0768112	+	0.997046i	$0.475526\pi$
$181$			2.14022e9i			1.99409i	−0.0768112	+	0.997046i	$0.524474\pi$
$182$	−2.07810e8			−0.189400
$183$			2.13356e8i			0.190239i
$184$			1.21668e8i			0.106146i
$185$			3.73257e8i			0.318655i
$186$			2.60302e8i			0.217483i
$187$	9.76371e7			0.0798451
$188$	1.13743e8			0.0910531
$189$		−	1.38924e9i		−	1.08876i
$190$	−1.18702e8	+	9.20107e8i	−0.0910844	+	0.706031i
$191$	8.42747e8			0.633233			0.316617	−	0.948554i	$-0.397453\pi$
$191$	8.42747e8			0.633233			0.316617	+	0.948554i	$0.397453\pi$
$192$			1.08955e8i			0.0801755i
$193$			1.64916e9i			1.18859i	0.804247	+	0.594296i	$0.202569\pi$
$193$			1.64916e9i			1.18859i	−0.804247	+	0.594296i	$0.797431\pi$
$194$	1.37700e9			0.972138
$195$	−2.34048e8			−0.161870
$196$	−1.04591e8			−0.0708710
$197$	−2.77435e9			−1.84203			−0.921014	−	0.389529i	$-0.872638\pi$
$197$	−2.77435e9			−1.84203			−0.921014	+	0.389529i	$0.872638\pi$
$198$		−	4.12012e8i		−	0.268071i
$199$	1.12521e9			0.717502			0.358751	−	0.933433i	$-0.383203\pi$
$199$	1.12521e9			0.717502			0.358751	+	0.933433i	$0.383203\pi$
$200$			7.66644e6i			0.00479152i
$201$	−1.76139e9			−1.07913
$202$			1.02860e9i			0.617788i
$203$		−	1.71417e9i		−	1.00942i
$204$			6.88537e7i			0.0397564i
$205$			2.07727e9i			1.17619i
$206$	2.23428e9			1.24071
$207$	3.24452e8			0.176713
$208$		−	1.17302e8i		−	0.0626688i
$209$	−1.21883e9	−	1.57240e8i	−0.638791	−	0.0824098i
$210$	−9.48857e8			−0.487892
$211$			1.96541e9i			0.991570i	0.868445	+	0.495785i	$0.165120\pi$
$211$			1.96541e9i			0.991570i	−0.868445	+	0.495785i	$0.834880\pi$
$212$		−	8.06168e8i		−	0.399100i
$213$	−1.93565e9			−0.940393
$214$	−1.54242e8			−0.0735441
$215$	2.75268e9			1.28826
$216$	7.84181e8			0.360248
$217$		−	1.13614e9i		−	0.512382i
$218$	7.21398e8			0.319410
$219$			3.42716e6i			0.00148990i
$220$	−7.59499e8			−0.324217
$221$		−	7.41286e7i		−	0.0310754i
$222$		−	3.48679e8i		−	0.143554i
$223$			1.10202e9i			0.445627i	0.974861	+	0.222813i	$0.0715240\pi$
$223$			1.10202e9i			0.445627i	−0.974861	+	0.222813i	$0.928476\pi$
$224$		−	4.75555e8i		−	0.188890i
$225$	2.04442e7			0.00797700
$226$	2.24552e9			0.860763
$227$		−	3.82995e9i		−	1.44241i	−0.692719	−	0.721207i	$-0.743588\pi$
$227$		−	3.82995e9i		−	1.44241i	0.692719	−	0.721207i	$-0.256412\pi$
$228$	1.10886e8	−	8.59521e8i	0.0410334	−	0.318066i
$229$	−2.63235e9			−0.957199			−0.478599	−	0.878033i	$-0.658855\pi$
$229$	−2.63235e9			−0.957199			−0.478599	+	0.878033i	$0.658855\pi$
$230$		−	5.98092e8i		−	0.213726i
$231$		−	1.25692e9i		−	0.441427i
$232$	9.67594e8			0.333996
$233$	−4.48933e9			−1.52320			−0.761602	−	0.648045i	$-0.775587\pi$
$233$	−4.48933e9			−1.52320			−0.761602	+	0.648045i	$0.775587\pi$
$234$	−3.12810e8			−0.104332
$235$	−5.59138e8			−0.183336
$236$		−	2.53989e9i		−	0.818779i
$237$	1.99983e8			0.0633868
$238$		−	3.00526e8i		−	0.0936643i
$239$	3.53347e9			1.08295			0.541477	−	0.840716i	$-0.317865\pi$
$239$	3.53347e9			1.08295			0.541477	+	0.840716i	$0.317865\pi$
$240$		−	5.35599e8i		−	0.161434i
$241$		−	1.88534e9i		−	0.558882i	−0.960163	−	0.279441i	$-0.909851\pi$
$241$		−	1.88534e9i		−	0.558882i	0.960163	−	0.279441i	$-0.0901492\pi$
$242$			1.41911e9i			0.413767i
$243$		−	3.40755e9i		−	0.977277i
$244$	5.25651e8			0.148299
$245$	5.14145e8			0.142699
$246$		−	1.94049e9i		−	0.529871i
$247$	−1.19381e8	+	9.25369e8i	−0.0320736	+	0.248615i
$248$	6.41315e8			0.169537
$249$			1.95619e9i			0.508878i
$250$			2.74310e9i			0.702234i
$251$	1.74114e9			0.438670			0.219335	−	0.975650i	$-0.429611\pi$
$251$	1.74114e9			0.438670			0.219335	+	0.975650i	$0.429611\pi$
$252$	−1.26817e9			−0.314467
$253$	7.92271e8			0.193371
$254$	2.10097e9			0.504761
$255$		−	3.38470e8i		−	0.0800497i
$256$	2.68435e8			0.0625000
$257$			8.79555e8i			0.201619i	0.994906	+	0.100809i	$0.0321432\pi$
$257$			8.79555e8i			0.201619i	−0.994906	+	0.100809i	$0.967857\pi$
$258$	−2.57143e9			−0.580357
$259$			1.52188e9i			0.338206i
$260$			5.76631e8i			0.126184i
$261$		−	2.58029e9i		−	0.556042i
$262$		−	5.29463e9i		−	1.12365i
$263$	1.68818e9			0.352855			0.176428	−	0.984314i	$-0.443546\pi$
$263$	1.68818e9			0.352855			0.176428	+	0.984314i	$0.443546\pi$
$264$	7.09489e8			0.146059
$265$			3.96295e9i			0.803591i
$266$	−4.83984e8	+	3.75155e9i	−0.0966729	+	0.749350i
$267$	−3.63690e9			−0.715625
$268$			4.33960e9i			0.841222i
$269$		−	6.32329e9i		−	1.20763i	−0.797124	−	0.603815i	$-0.793646\pi$
$269$		−	6.32329e9i		−	1.20763i	0.797124	−	0.603815i	$-0.206354\pi$
$270$	−3.85487e9			−0.725362
$271$	−2.42690e9			−0.449961			−0.224981	−	0.974363i	$-0.572232\pi$
$271$	−2.42690e9			−0.449961			−0.224981	+	0.974363i	$0.572232\pi$
$272$	1.69637e8			0.0309917
$273$	−9.54284e8			−0.171801
$274$			1.32207e9i			0.234560i
$275$	4.99221e7			0.00872894
$276$			5.58710e8i			0.0962831i
$277$	6.96079e9			1.18233			0.591166	−	0.806550i	$-0.298668\pi$
$277$	6.96079e9			1.18233			0.591166	+	0.806550i	$0.298668\pi$
$278$		−	5.18579e9i		−	0.868232i
$279$		−	1.71020e9i		−	0.282248i
$280$			2.33773e9i			0.380331i
$281$		−	2.84558e9i		−	0.456399i	−0.973614	−	0.228200i	$-0.926716\pi$
$281$		−	2.84558e9i		−	0.456399i	0.973614	−	0.228200i	$-0.0732839\pi$
$282$	5.22321e8			0.0825926
$283$	−2.23602e9			−0.348603			−0.174301	−	0.984692i	$-0.555767\pi$
$283$	−2.23602e9			−0.348603			−0.174301	+	0.984692i	$0.555767\pi$
$284$			4.76893e9i			0.733074i
$285$	−5.45092e8	+	4.22522e9i	−0.0826210	+	0.640428i
$286$	−7.63842e8			−0.114167
$287$			8.46965e9i			1.24835i
$288$		−	7.15840e8i		−	0.104051i
$289$	−6.86856e9			−0.984632
$290$	−4.75649e9			−0.672503
$291$	6.32334e9			0.881809
$292$	8.44360e6			0.00116144
$293$			9.45960e9i			1.28352i	0.766906	+	0.641759i	$0.221795\pi$
$293$			9.45960e9i			1.28352i	−0.766906	+	0.641759i	$0.778205\pi$
$294$	−4.80291e8			−0.0642858
$295$			1.24856e10i			1.64862i
$296$	−8.59052e8			−0.111906
$297$		−	5.10641e9i		−	0.656281i
$298$			4.85785e9i			0.615997i
$299$		−	6.01512e8i		−	0.0752592i
$300$			3.52051e7i			0.00434630i
$301$	1.12235e10			1.36730
$302$	1.22112e9			0.146801
$303$			4.72342e9i			0.560385i
$304$	−2.11763e9	−	2.73193e8i	−0.247945	−	0.0319872i
$305$	−2.58399e9			−0.298601
$306$		−	4.52373e8i		−	0.0515954i
$307$		−	4.65687e9i		−	0.524252i	−0.965034	−	0.262126i	$-0.915576\pi$
$307$		−	4.65687e9i		−	0.524252i	0.965034	−	0.262126i	$-0.0844236\pi$
$308$	−3.09671e9			−0.344110
$309$	1.02601e10			1.12542
$310$	−3.15257e9			−0.341364
$311$	2.07150e9			0.221434			0.110717	−	0.993852i	$-0.464685\pi$
$311$	2.07150e9			0.221434			0.110717	+	0.993852i	$0.464685\pi$
$312$		−	5.38662e8i		−	0.0568457i
$313$	5.92446e9			0.617265			0.308632	−	0.951181i	$-0.400129\pi$
$313$	5.92446e9			0.617265			0.308632	+	0.951181i	$0.400129\pi$
$314$			5.51405e8i			0.0567221i
$315$	6.23405e9			0.633181
$316$		−	4.92703e8i		−	0.0494126i
$317$		−	8.43378e9i		−	0.835190i	−0.908633	−	0.417595i	$-0.862873\pi$
$317$		−	8.43378e9i		−	0.835190i	0.908633	−	0.417595i	$-0.137127\pi$
$318$		−	3.70200e9i		−	0.362017i
$319$		−	6.30075e9i		−	0.608456i
$320$	−1.31957e9			−0.125844
$321$	−7.08295e8			−0.0667105
$322$		−	2.43860e9i		−	0.226839i
$323$	−1.33823e9	−	1.72644e8i	−0.122948	−	0.0158614i
$324$	3.57860e8			0.0324738
$325$		−	3.79021e7i		−	0.00339727i
$326$		−	4.07034e9i		−	0.360380i
$327$	3.31273e9			0.289731
$328$	−4.78084e9			−0.413056
$329$	−2.27978e9			−0.194585
$330$	−3.48770e9			−0.294092
$331$			1.35397e10i			1.12797i	0.825786	+	0.563983i	$0.190732\pi$
$331$			1.35397e10i			1.12797i	−0.825786	+	0.563983i	$0.809268\pi$
$332$	4.81953e9			0.396691
$333$			2.29085e9i			0.186303i
$334$	3.87896e9			0.311695
$335$		−	2.13326e10i		−	1.69381i
$336$		−	2.18380e9i		−	0.171339i
$337$		−	3.50354e9i		−	0.271636i	−0.990734	−	0.135818i	$-0.956634\pi$
$337$		−	3.50354e9i		−	0.271636i	0.990734	−	0.135818i	$-0.0433662\pi$
$338$		−	8.64901e9i		−	0.662674i
$339$	1.03117e10			0.780783
$340$	−8.33900e8			−0.0624020
$341$		−	4.17609e9i		−	0.308854i
$342$	−7.28528e8	+	5.64711e9i	−0.0532527	+	0.412783i
$343$	−1.26934e10			−0.917069
$344$			6.33530e9i			0.452412i
$345$		−	2.74650e9i		−	0.193867i
$346$	−6.91428e9			−0.482439
$347$	−2.20519e10			−1.52100			−0.760498	−	0.649340i	$-0.775045\pi$
$347$	−2.20519e10			−1.52100			−0.760498	+	0.649340i	$0.775045\pi$
$348$	4.44329e9			0.302962
$349$	6.61994e9			0.446224			0.223112	−	0.974793i	$-0.428378\pi$
$349$	6.61994e9			0.446224			0.223112	+	0.974793i	$0.428378\pi$
$350$		−	1.53660e8i		−	0.0102397i
$351$	−3.87691e9			−0.255422
$352$		−	1.74799e9i		−	0.113859i
$353$	−2.55413e10			−1.64492			−0.822460	−	0.568824i	$-0.807399\pi$
$353$	−2.55413e10			−1.64492			−0.822460	+	0.568824i	$0.807399\pi$
$354$		−	1.16634e10i		−	0.742700i
$355$		−	2.34431e10i		−	1.47605i
$356$			8.96033e9i			0.557859i
$357$		−	1.38005e9i		−	0.0849612i
$358$	3.46888e8			0.0211182
$359$	−1.95409e10			−1.17643			−0.588216	−	0.808704i	$-0.700170\pi$
$359$	−1.95409e10			−1.17643			−0.588216	+	0.808704i	$0.700170\pi$
$360$			3.51892e9i			0.209507i
$361$	1.64275e10	+	4.31033e9i	0.967258	+	0.253794i
$362$	2.42139e10			1.41004
$363$			6.51671e9i			0.375320i
$364$			2.35110e9i			0.133926i
$365$	−4.15070e7			−0.00233857
$366$	2.41384e9			0.134519
$367$	−1.96784e10			−1.08474			−0.542370	−	0.840140i	$-0.682473\pi$
$367$	−1.96784e10			−1.08474			−0.542370	+	0.840140i	$0.682473\pi$
$368$	1.37651e9			0.0750565
$369$			1.27491e10i			0.687662i
$370$	4.22292e9			0.225323
$371$			1.61581e10i			0.852896i
$372$	2.94498e9			0.153784
$373$		−	2.75980e10i		−	1.42575i	−0.701292	−	0.712874i	$-0.747393\pi$
$373$		−	2.75980e10i		−	1.42575i	0.701292	−	0.712874i	$-0.252607\pi$
$374$		−	1.10464e9i		−	0.0564590i
$375$			1.25966e10i			0.636984i
$376$		−	1.28686e9i		−	0.0643843i
$377$	−4.78369e9			−0.236809
$378$	−1.57175e10			−0.769866
$379$		−	3.06260e10i		−	1.48434i	−0.670212	−	0.742170i	$-0.733797\pi$
$379$		−	3.06260e10i		−	1.48434i	0.670212	−	0.742170i	$-0.266203\pi$
$380$	1.04098e10	+	1.34296e9i	0.499239	+	0.0644064i
$381$	9.64789e9			0.457860
$382$		−	9.53459e9i		−	0.447763i
$383$			1.04982e10i			0.487888i	0.969789	+	0.243944i	$0.0784413\pi$
$383$			1.04982e10i			0.487888i	−0.969789	+	0.243944i	$0.921559\pi$
$384$	1.23268e9			0.0566926
$385$	1.52228e10			0.692868
$386$	1.86581e10			0.840461
$387$	1.68944e10			0.753182
$388$		−	1.55790e10i		−	0.687406i
$389$	−1.88651e10			−0.823875			−0.411937	−	0.911212i	$-0.635148\pi$
$389$	−1.88651e10			−0.823875			−0.411937	+	0.911212i	$0.635148\pi$
$390$			2.64795e9i			0.114459i
$391$	8.69883e8			0.0372181
$392$			1.18331e9i			0.0501134i
$393$		−	2.43135e10i		−	1.01924i
$394$			3.13882e10i			1.30251i
$395$			2.42203e9i			0.0994925i
$396$	−4.66138e9			−0.189555
$397$	4.56248e10			1.83670			0.918352	−	0.395765i	$-0.129521\pi$
$397$	4.56248e10			1.83670			0.918352	+	0.395765i	$0.129521\pi$
$398$		−	1.27303e10i		−	0.507350i
$399$	−2.22251e9	+	1.72275e10i	−0.0876903	+	0.679722i
$400$	8.67358e7			0.00338812
$401$			4.78635e10i			1.85109i	0.378639	+	0.925545i	$0.376392\pi$
$401$			4.78635e10i			1.85109i	−0.378639	+	0.925545i	$0.623608\pi$
$402$			1.99279e10i			0.763057i
$403$	−3.17060e9			−0.120205
$404$	1.16372e10			0.436842
$405$	−1.75916e9			−0.0653862
$406$	−1.93936e10			−0.713765
$407$			5.59395e9i			0.203864i
$408$	7.78991e8			0.0281120
$409$			3.65614e10i			1.30656i	0.757116	+	0.653280i	$0.226608\pi$
$409$			3.65614e10i			1.30656i	−0.757116	+	0.653280i	$0.773392\pi$
$410$	2.35016e10			0.831691
$411$			6.07110e9i			0.212765i
$412$		−	2.52780e10i		−	0.877313i
$413$			5.09074e10i			1.74977i
$414$		−	3.67076e9i		−	0.124955i
$415$	−2.36918e10			−0.798740
$416$	−1.32712e9			−0.0443135
$417$		−	2.38137e10i		−	0.787558i
$418$	−1.77897e9	+	1.37895e10i	−0.0582726	+	0.451693i
$419$	−1.76316e10			−0.572051			−0.286025	−	0.958222i	$-0.592334\pi$
$419$	−1.76316e10			−0.572051			−0.286025	+	0.958222i	$0.592334\pi$
$420$			1.07351e10i			0.344992i
$421$			3.77588e10i			1.20196i	0.799264	+	0.600980i	$0.205223\pi$
$421$			3.77588e10i			1.20196i	−0.799264	+	0.600980i	$0.794777\pi$
$422$	2.22361e10			0.701146
$423$	−3.43168e9			−0.107188
$424$	−9.12075e9			−0.282206
$425$	5.48125e7			0.00168006
$426$			2.18994e10i			0.664958i
$427$	−1.05357e10			−0.316922
$428$			1.74505e9i			0.0520035i
$429$	−3.50764e9			−0.103559
$430$		−	3.11430e10i		−	0.910934i
$431$		−	2.15502e10i		−	0.624514i	−0.949998	−	0.312257i	$-0.898915\pi$
$431$		−	2.15502e10i		−	0.624514i	0.949998	−	0.312257i	$-0.101085\pi$
$432$		−	8.87200e9i		−	0.254734i
$433$		−	3.21072e10i		−	0.913379i	−0.889626	−	0.456690i	$-0.849035\pi$
$433$		−	3.21072e10i		−	0.913379i	0.889626	−	0.456690i	$-0.150965\pi$
$434$	−1.28540e10			−0.362309
$435$	−2.18423e10			−0.610016
$436$		−	8.16169e9i		−	0.225857i
$437$	−1.08590e10	−	1.40091e9i	−0.297758	−	0.0384135i
$438$	3.87739e7			0.00105352
$439$		−	1.90093e10i		−	0.511808i	−0.966702	−	0.255904i	$-0.917627\pi$
$439$		−	1.90093e10i		−	0.511808i	0.966702	−	0.255904i	$-0.0823731\pi$
$440$			8.59275e9i			0.229256i
$441$	3.15554e9			0.0834295
$442$	−8.38669e8			−0.0219736
$443$	−4.72912e10			−1.22791			−0.613953	−	0.789342i	$-0.710422\pi$
$443$	−4.72912e10			−1.22791			−0.613953	+	0.789342i	$0.710422\pi$
$444$	−3.94486e9			−0.101508
$445$		−	4.40471e10i		−	1.12325i
$446$	1.24680e10			0.315106
$447$			2.23077e10i			0.558760i
$448$	−5.38029e9			−0.133565
$449$		−	3.40864e10i		−	0.838679i	−0.907830	−	0.419339i	$-0.862262\pi$
$449$		−	3.40864e10i		−	0.838679i	0.907830	−	0.419339i	$-0.137738\pi$
$450$		−	2.31299e8i		−	0.00564059i
$451$			3.11317e10i			0.752483i
$452$		−	2.54052e10i		−	0.608652i
$453$	5.60749e9			0.133161
$454$	−4.33310e10			−1.01994
$455$		−	1.15575e10i		−	0.269661i
$456$	−9.72437e9	−	1.25453e9i	−0.224907	−	0.0290150i
$457$	4.13362e10			0.947689			0.473845	−	0.880608i	$-0.342866\pi$
$457$	4.13362e10			0.947689			0.473845	+	0.880608i	$0.342866\pi$
$458$			2.97817e10i			0.676842i
$459$		−	5.60664e9i		−	0.126314i
$460$	−6.76664e9			−0.151127
$461$	8.72135e10			1.93099			0.965495	−	0.260423i	$-0.0838620\pi$
$461$	8.72135e10			1.93099			0.965495	+	0.260423i	$0.0838620\pi$
$462$	−1.42204e10			−0.312136
$463$	3.84840e10			0.837446			0.418723	−	0.908114i	$-0.362478\pi$
$463$	3.84840e10			0.837446			0.418723	+	0.908114i	$0.362478\pi$
$464$		−	1.09471e10i		−	0.236171i
$465$	−1.44769e10			−0.309645
$466$			5.07910e10i			1.07707i
$467$	3.07954e10			0.647468			0.323734	−	0.946148i	$-0.395062\pi$
$467$	3.07954e10			0.647468			0.323734	+	0.946148i	$0.395062\pi$
$468$			3.53904e9i			0.0737739i
$469$		−	8.69793e10i		−	1.79773i
$470$			6.32593e9i			0.129638i
$471$			2.53211e9i			0.0514516i
$472$	−2.87356e10			−0.578964
$473$	4.12540e10			0.824180
$474$		−	2.26254e9i		−	0.0448212i
$475$	−6.84240e8	−	8.82732e7i	−0.0134411	−	0.00173402i
$476$	−3.40006e9			−0.0662307
$477$			2.43224e10i			0.469822i
$478$		−	3.99767e10i		−	0.765764i
$479$	3.46197e10			0.657630			0.328815	−	0.944394i	$-0.393351\pi$
$479$	3.46197e10			0.657630			0.328815	+	0.944394i	$0.393351\pi$
$480$	−6.05961e9			−0.114151
$481$	4.24707e9			0.0793431
$482$	−2.13301e10			−0.395190
$483$		−	1.11983e10i		−	0.205761i
$484$	1.60554e10			0.292577
$485$			7.65831e10i			1.38410i
$486$	−3.85521e10			−0.691039
$487$		−	1.04862e11i		−	1.86425i	−0.362138	−	0.932125i	$-0.617953\pi$
$487$		−	1.04862e11i		−	1.86425i	0.362138	−	0.932125i	$-0.382047\pi$
$488$		−	5.94706e9i		−	0.104863i
$489$		−	1.86914e10i		−	0.326894i
$490$		−	5.81689e9i		−	0.100904i
$491$	−7.40091e10			−1.27338			−0.636692	−	0.771118i	$-0.719698\pi$
$491$	−7.40091e10			−1.27338			−0.636692	+	0.771118i	$0.719698\pi$
$492$	−2.19541e10			−0.374675
$493$		−	6.91798e9i		−	0.117109i
$494$	1.04694e10	+	1.35064e9i	0.175797	+	0.0226794i
$495$	2.29144e10			0.381669
$496$		−	7.25565e9i		−	0.119881i
$497$		−	9.55845e10i		−	1.56661i
$498$	2.21318e10			0.359831
$499$	5.35198e10			0.863202			0.431601	−	0.902065i	$-0.357949\pi$
$499$	5.35198e10			0.863202			0.431601	+	0.902065i	$0.357949\pi$
$500$	3.10347e10			0.496555
$501$	1.78126e10			0.282733
$502$		−	1.96987e10i		−	0.310187i
$503$	−2.44696e10			−0.382256			−0.191128	−	0.981565i	$-0.561215\pi$
$503$	−2.44696e10			−0.382256			−0.191128	+	0.981565i	$0.561215\pi$
$504$			1.43477e10i			0.222362i
$505$	−5.72062e10			−0.879585
$506$		−	8.96352e9i		−	0.136734i
$507$		−	3.97171e10i		−	0.601099i
$508$		−	2.37698e10i		−	0.356920i
$509$		−	9.78265e10i		−	1.45742i	−0.684821	−	0.728711i	$-0.740120\pi$
$509$		−	9.78265e10i		−	1.45742i	0.684821	−	0.728711i	$-0.259880\pi$
$510$	−3.82936e9			−0.0566037
$511$	−1.69236e8			−0.00248205
$512$		−	3.03700e9i		−	0.0441942i
$513$	−9.02925e9	+	6.99893e10i	−0.130371	+	1.01056i
$514$	9.95103e9			0.142566
$515$			1.24261e11i			1.76648i
$516$			2.90924e10i			0.410374i
$517$	−8.37973e9			−0.117292
$518$	1.72181e10			0.239148
$519$	−3.17511e10			−0.437612
$520$	6.52383e9			0.0892256
$521$			1.19739e11i			1.62512i	0.582878	+	0.812560i	$0.301927\pi$
$521$			1.19739e11i			1.62512i	−0.582878	+	0.812560i	$0.698073\pi$
$522$	−2.91927e10			−0.393181
$523$			1.40096e11i			1.87248i	0.351355	+	0.936242i	$0.385721\pi$
$523$			1.40096e11i			1.87248i	−0.351355	+	0.936242i	$0.614279\pi$
$524$	−5.99019e10			−0.794539
$525$		−	7.05621e8i		−	0.00928825i
$526$		−	1.90996e10i		−	0.249506i
$527$		−	4.58519e9i		−	0.0594449i
$528$		−	8.02695e9i		−	0.103280i
$529$	−7.12524e10			−0.909864
$530$	4.48357e10			0.568225
$531$			7.66295e10i			0.963869i
$532$	4.24440e10	+	5.47566e9i	0.529870	+	0.0683581i
$533$	2.36360e10			0.292863
$534$			4.11468e10i			0.506024i
$535$		−	8.57829e9i		−	0.104709i
$536$	4.90970e10			0.594834
$537$	1.59294e9			0.0191559
$538$	−7.15399e10			−0.853924
$539$	7.70543e9			0.0912939
$540$			4.36129e10i			0.512908i
$541$	−5.50648e10			−0.642814			−0.321407	−	0.946941i	$-0.604156\pi$
$541$	−5.50648e10			−0.642814			−0.321407	+	0.946941i	$0.604156\pi$
$542$			2.74573e10i			0.318171i
$543$	1.11193e11			1.27902
$544$		−	1.91922e9i		−	0.0219144i
$545$			4.01211e10i			0.454765i
$546$			1.07965e10i			0.121482i
$547$		−	1.23981e11i		−	1.38486i	−0.721484	−	0.692431i	$-0.756540\pi$
$547$		−	1.23981e11i		−	1.38486i	0.721484	−	0.692431i	$-0.243460\pi$
$548$	1.49576e10			0.165859
$549$	−1.58591e10			−0.174578
$550$		−	5.64804e8i		−	0.00617229i
$551$	−1.11411e10	+	8.63591e10i	−0.120871	+	0.936919i
$552$	6.32108e9			0.0680824
$553$			9.87533e9i			0.105597i
$554$		−	7.87523e10i		−	0.836035i
$555$	1.93921e10			0.204387
$556$	−5.86706e10			−0.613933
$557$	7.17677e10			0.745604			0.372802	−	0.927911i	$-0.378397\pi$
$557$	7.17677e10			0.745604			0.372802	+	0.927911i	$0.378397\pi$
$558$	−1.93487e10			−0.199579
$559$		−	3.13211e10i		−	0.320767i
$560$	2.64484e10			0.268935
$561$		−	5.07261e9i		−	0.0512130i
$562$	−3.21940e10			−0.322723
$563$		−	5.66128e10i		−	0.563483i	−0.959490	−	0.281742i	$-0.909088\pi$
$563$		−	5.66128e10i		−	0.563483i	0.959490	−	0.281742i	$-0.0909121\pi$
$564$		−	5.90939e9i		−	0.0584018i
$565$			1.24887e11i			1.22552i
$566$			2.52977e10i			0.246499i
$567$	−7.17265e9			−0.0693980
$568$	5.39543e10			0.518362
$569$		−	5.37834e10i		−	0.513096i	−0.966531	−	0.256548i	$-0.917415\pi$
$569$		−	5.37834e10i		−	0.513096i	0.966531	−	0.256548i	$-0.0825852\pi$
$570$	4.78030e10	+	6.16702e9i	0.452851	+	0.0584219i
$571$	−9.28075e9			−0.0873049			−0.0436524	−	0.999047i	$-0.513899\pi$
$571$	−9.28075e9			−0.0873049			−0.0436524	+	0.999047i	$0.513899\pi$
$572$			8.64189e9i			0.0807281i
$573$		−	4.37838e10i		−	0.406158i
$574$	9.58231e10			0.882719
$575$	4.44773e8			0.00406881
$576$	−8.09881e9			−0.0735751
$577$	−1.78113e11			−1.60691			−0.803457	−	0.595363i	$-0.797008\pi$
$577$	−1.78113e11			−1.60691			−0.803457	+	0.595363i	$0.797008\pi$
$578$			7.77088e10i			0.696240i
$579$	8.56798e10			0.762367
$580$			5.38135e10i			0.475532i
$581$	−9.65985e10			−0.847747
$582$		−	7.15404e10i		−	0.623533i
$583$			5.93922e10i			0.514109i
$584$		−	9.55285e7i		−	0.000821261i
$585$		−	1.73972e10i		−	0.148544i
$586$	1.07023e11			0.907585
$587$	−5.63747e10			−0.474823			−0.237412	−	0.971409i	$-0.576299\pi$
$587$	−5.63747e10			−0.474823			−0.237412	+	0.971409i	$0.576299\pi$
$588$			5.43387e9i			0.0454569i
$589$	−7.38425e9	+	5.72383e10i	−0.0613544	+	0.475582i
$590$	1.41258e11			1.16575
$591$			1.44138e11i			1.18148i
$592$			9.71907e9i			0.0791294i
$593$	−1.13204e11			−0.915469			−0.457734	−	0.889089i	$-0.651339\pi$
$593$	−1.13204e11			−0.915469			−0.457734	+	0.889089i	$0.651339\pi$
$594$	−5.77724e10			−0.464060
$595$	1.67140e10			0.133356
$596$	5.49603e10			0.435576
$597$		−	5.84591e10i		−	0.460208i
$598$	−6.80534e9			−0.0532163
$599$		−	1.34269e11i		−	1.04297i	−0.853262	−	0.521483i	$-0.825379\pi$
$599$		−	1.34269e11i		−	1.04297i	0.853262	−	0.521483i	$-0.174621\pi$
$600$	3.98300e8			0.00307330
$601$		−	7.09025e10i		−	0.543455i	−0.962374	−	0.271727i	$-0.912405\pi$
$601$		−	7.09025e10i		−	0.543455i	0.962374	−	0.271727i	$-0.0875949\pi$
$602$		−	1.26980e11i		−	0.966825i
$603$		−	1.30928e11i		−	0.990288i
$604$		−	1.38154e10i		−	0.103804i
$605$	−7.89251e10			−0.589106
$606$	5.34394e10			0.396252
$607$		−	1.25913e11i		−	0.927506i	−0.885965	−	0.463753i	$-0.846503\pi$
$607$		−	1.25913e11i		−	0.927506i	0.885965	−	0.463753i	$-0.153497\pi$
$608$	−3.09083e9	+	2.39582e10i	−0.0226184	+	0.175324i
$609$	−8.90576e10			−0.647443
$610$			2.92345e10i			0.211143i
$611$			6.36211e9i			0.0456495i
$612$	−5.11802e9			−0.0364835
$613$	2.12847e11			1.50739			0.753696	−	0.657224i	$-0.228269\pi$
$613$	2.12847e11			1.50739			0.753696	+	0.657224i	$0.228269\pi$
$614$	−5.26864e10			−0.370702
$615$	1.07922e11			0.754412
$616$			3.50352e10i			0.243322i
$617$	−1.70248e11			−1.17474			−0.587369	−	0.809320i	$-0.699836\pi$
$617$	−1.70248e11			−1.17474			−0.587369	+	0.809320i	$0.699836\pi$
$618$		−	1.16079e11i		−	0.795795i
$619$	2.10445e11			1.43343			0.716714	−	0.697368i	$-0.245645\pi$
$619$	2.10445e11			1.43343			0.716714	+	0.697368i	$0.245645\pi$
$620$			3.56672e10i			0.241381i
$621$		−	4.54947e10i		−	0.305911i
$622$		−	2.34364e10i		−	0.156577i
$623$		−	1.79593e11i		−	1.19217i
$624$	−6.09426e9			−0.0401960
$625$	−1.54628e11			−1.01337
$626$		−	6.70276e10i		−	0.436472i
$627$	−8.16922e9	+	6.33229e10i	−0.0528580	+	0.409723i
$628$	6.23844e9			0.0401086
$629$			6.14194e9i			0.0392377i
$630$		−	7.05302e10i		−	0.447727i
$631$	−1.85646e11			−1.17103			−0.585515	−	0.810661i	$-0.699108\pi$
$631$	−1.85646e11			−1.17103			−0.585515	+	0.810661i	$0.699108\pi$
$632$	−5.57430e9			−0.0349399
$633$	1.02110e11			0.635997
$634$	−9.54173e10			−0.590568
$635$			1.16847e11i			0.718661i
$636$	−4.18834e10			−0.255984
$637$		−	5.85016e9i		−	0.0355312i
$638$	−7.12848e10			−0.430244
$639$		−	1.43881e11i		−	0.862977i
$640$			1.49293e10i			0.0889853i
$641$			1.32645e11i			0.785702i	0.919602	+	0.392851i	$0.128511\pi$
$641$			1.32645e11i			0.785702i	−0.919602	+	0.392851i	$0.871489\pi$
$642$			8.01345e9i			0.0471715i
$643$	1.56526e11			0.915680			0.457840	−	0.889035i	$-0.348623\pi$
$643$	1.56526e11			0.915680			0.457840	+	0.889035i	$0.348623\pi$
$644$	−2.75896e10			−0.160399
$645$		−	1.43012e11i		−	0.826292i
$646$	−1.95324e9	+	1.51404e10i	−0.0112157	+	0.0869373i
$647$	−2.26765e11			−1.29407			−0.647036	−	0.762460i	$-0.723992\pi$
$647$	−2.26765e11			−1.29407			−0.647036	+	0.762460i	$0.723992\pi$
$648$		−	4.04872e9i		−	0.0229624i
$649$			1.87119e11i			1.05473i
$650$	−4.28813e8			−0.00240223
$651$	−5.90268e10			−0.328644
$652$	−4.60506e10			−0.254827
$653$	1.87061e11			1.02880			0.514399	−	0.857551i	$-0.328015\pi$
$653$	1.87061e11			1.02880			0.514399	+	0.857551i	$0.328015\pi$
$654$		−	3.74793e10i		−	0.204871i
$655$	2.94465e11			1.59981
$656$			5.40890e10i			0.292075i
$657$	−2.54747e8			−0.00136725
$658$			2.57927e10i			0.137592i
$659$		−	1.64068e11i		−	0.869924i	−0.900449	−	0.434962i	$-0.856762\pi$
$659$		−	1.64068e11i		−	0.869924i	0.900449	−	0.434962i	$-0.143238\pi$
$660$			3.94588e10i			0.207954i
$661$			2.29350e11i			1.20142i	0.799468	+	0.600708i	$0.205115\pi$
$661$			2.29350e11i			1.20142i	−0.799468	+	0.600708i	$0.794885\pi$
$662$	1.53184e11			0.797592
$663$	−3.85126e9			−0.0199319
$664$		−	5.45267e10i		−	0.280503i
$665$	−2.08646e11	−	2.69172e10i	−1.06690	−	0.137639i
$666$	2.59180e10			0.131736
$667$		−	5.61355e10i		−	0.283619i
$668$		−	4.38854e10i		−	0.220402i
$669$	5.72542e10			0.285827
$670$	−2.41350e11			−1.19770
$671$	−3.87259e10			−0.191034
$672$	−2.47069e10			−0.121155
$673$		−	1.52714e11i		−	0.744419i	−0.928149	−	0.372210i	$-0.878600\pi$
$673$		−	1.52714e11i		−	0.744419i	0.928149	−	0.372210i	$-0.121400\pi$
$674$	−3.96380e10			−0.192075
$675$		−	2.86669e9i		−	0.0138091i
$676$	−9.78524e10			−0.468581
$677$			4.69398e10i			0.223453i	0.993739	+	0.111727i	$0.0356381\pi$
$677$			4.69398e10i			0.223453i	−0.993739	+	0.111727i	$0.964362\pi$
$678$		−	1.16663e11i		−	0.552097i
$679$			3.12253e11i			1.46902i
$680$			9.43450e9i			0.0441249i
$681$	−1.98980e11			−0.925170
$682$	−4.72471e10			−0.218392
$683$		−	3.13546e10i		−	0.144085i	−0.997402	−	0.0720424i	$-0.977048\pi$
$683$		−	3.13546e10i		−	0.144085i	0.997402	−	0.0720424i	$-0.0229517\pi$
$684$	6.38897e10	+	8.24236e9i	0.291882	+	0.0376554i
$685$	−7.35282e10			−0.333958
$686$			1.43610e11i			0.648465i
$687$			1.36760e11i			0.613951i
$688$	7.16758e10			0.319903
$689$	4.50921e10			0.200089
$690$	−3.10731e10			−0.137084
$691$	3.40031e11			1.49144			0.745720	−	0.666259i	$-0.232106\pi$
$691$	3.40031e11			1.49144			0.745720	+	0.666259i	$0.232106\pi$
$692$			7.82262e10i			0.341136i
$693$	9.34289e10			0.405087
$694$			2.49489e11i			1.07551i
$695$	2.88412e11			1.23616
$696$		−	5.02701e10i		−	0.214226i
$697$			3.41814e10i			0.144830i
$698$		−	7.48961e10i		−	0.315528i
$699$			2.33238e11i			0.976989i
$700$	−1.73846e9			−0.00724057
$701$	3.02363e11			1.25215			0.626076	−	0.779762i	$-0.284660\pi$
$701$	3.02363e11			1.25215			0.626076	+	0.779762i	$0.284660\pi$
$702$			4.38623e10i			0.180610i
$703$	9.89134e9	−	7.66716e10i	0.0404980	−	0.313916i
$704$	−1.97762e10			−0.0805106
$705$			2.90493e10i			0.117592i
$706$			2.88967e11i			1.16313i
$707$	−2.33247e11			−0.933552
$708$	−1.31957e11			−0.525168
$709$	3.28980e11			1.30192			0.650960	−	0.759112i	$-0.274367\pi$
$709$	3.28980e11			1.30192			0.650960	+	0.759112i	$0.274367\pi$
$710$	−2.65228e11			−1.04372
$711$			1.48651e10i			0.0581686i
$712$	1.01375e11			0.394466
$713$		−	3.72063e10i		−	0.143965i
$714$	−1.56134e10			−0.0600767
$715$		−	4.24817e10i		−	0.162547i
$716$		−	3.92459e9i		−	0.0149328i
$717$		−	1.83577e11i		−	0.694611i
$718$			2.21080e11i			0.831864i
$719$	3.55862e11			1.33157			0.665787	−	0.746142i	$-0.268096\pi$
$719$	3.55862e11			1.33157			0.665787	+	0.746142i	$0.268096\pi$
$720$	3.98120e10			0.148144
$721$			5.06652e11i			1.87486i
$722$	4.87658e10	−	1.85856e11i	0.179460	−	0.683955i
$723$	−9.79502e10			−0.358469
$724$		−	2.73949e11i		−	0.997046i
$725$		−	3.53718e9i		−	0.0128028i
$726$	7.37282e10			0.265392
$727$	−3.07617e11			−1.10122			−0.550609	−	0.834763i	$-0.685604\pi$
$727$	−3.07617e11			−1.10122			−0.550609	+	0.834763i	$0.685604\pi$
$728$	2.65996e10			0.0947001
$729$	−1.95378e11			−0.691777
$730$			4.69598e8i			0.00165362i
$731$	4.52953e10			0.158630
$732$		−	2.73095e10i		−	0.0951195i
$733$	2.79041e10			0.0966611			0.0483306	−	0.998831i	$-0.484610\pi$
$733$	2.79041e10			0.0966611			0.0483306	+	0.998831i	$0.484610\pi$
$734$			2.22636e11i			0.767027i
$735$		−	2.67118e10i		−	0.0915278i
$736$		−	1.55734e10i		−	0.0530730i
$737$		−	3.19708e11i		−	1.08364i
$738$	1.44240e11			0.486250
$739$	1.34245e11			0.450110			0.225055	−	0.974346i	$-0.427744\pi$
$739$	1.34245e11			0.450110			0.225055	+	0.974346i	$0.427744\pi$
$740$		−	4.77769e10i		−	0.159328i
$741$	4.80763e10	+	6.20228e9i	0.159463	+	0.0205721i
$742$	1.82809e11			0.603088
$743$		−	2.98682e11i		−	0.980062i	−0.871705	−	0.490031i	$-0.836985\pi$
$743$		−	2.98682e11i		−	0.980062i	0.871705	−	0.490031i	$-0.163015\pi$
$744$		−	3.33187e10i		−	0.108742i
$745$	−2.70173e11			−0.877035
$746$	−3.12236e11			−1.00816
$747$	−1.45407e11			−0.466985
$748$	−1.24975e10			−0.0399226
$749$		−	3.49763e10i		−	0.111134i
$750$	1.42514e11			0.450416
$751$			3.11095e11i			0.977987i	0.872287	+	0.488994i	$0.162636\pi$
$751$			3.11095e11i			0.977987i	−0.872287	+	0.488994i	$0.837364\pi$
$752$	−1.45592e10			−0.0455265
$753$		−	9.04585e10i		−	0.281365i
$754$			5.41213e10i			0.167449i
$755$			6.79134e10i			0.209010i
$756$			1.77823e11i			0.544378i
$757$	3.22982e11			0.983547			0.491774	−	0.870723i	$-0.336349\pi$
$757$	3.22982e11			0.983547			0.491774	+	0.870723i	$0.336349\pi$
$758$	−3.46494e11			−1.04959
$759$		−	4.11614e10i		−	0.124029i
$760$	1.51939e10	−	1.17774e11i	0.0455422	−	0.353016i
$761$	−3.99502e11			−1.19119			−0.595594	−	0.803286i	$-0.703083\pi$
$761$	−3.99502e11			−1.19119			−0.595594	+	0.803286i	$0.703083\pi$
$762$		−	1.09153e11i		−	0.323756i
$763$			1.63586e11i			0.482667i
$764$	−1.07872e11			−0.316617
$765$	2.51591e10			0.0734598
$766$	1.18774e11			0.344989
$767$	1.42066e11			0.410495
$768$		−	1.39462e10i		−	0.0400877i
$769$	−6.27345e11			−1.79391			−0.896955	−	0.442121i	$-0.854226\pi$
$769$	−6.27345e11			−1.79391			−0.896955	+	0.442121i	$0.854226\pi$
$770$		−	1.72226e11i		−	0.489931i
$771$	4.56961e10			0.129319
$772$		−	2.11092e11i		−	0.594296i
$773$		−	4.75374e10i		−	0.133143i	−0.997782	−	0.0665714i	$-0.978794\pi$
$773$		−	4.75374e10i		−	0.133143i	0.997782	−	0.0665714i	$-0.0212060\pi$
$774$		−	1.91139e11i		−	0.532580i
$775$		−	2.34442e9i		−	0.00649872i
$776$	−1.76256e11			−0.486069
$777$	7.90674e10			0.216927
$778$			2.13434e11i			0.582567i
$779$	5.50477e10	−	4.26697e11i	0.149482	−	1.15870i
$780$	2.99581e10			0.0809349
$781$		−	3.51338e11i		−	0.944324i
$782$		−	9.84160e9i		−	0.0263171i
$783$	−3.61809e11			−0.962571
$784$	1.33876e10			0.0354355
$785$	−3.06668e10			−0.0807589
$786$	−2.75076e11			−0.720712
$787$		−	5.68284e11i		−	1.48138i	−0.671847	−	0.740690i	$-0.734499\pi$
$787$		−	5.68284e11i		−	1.48138i	0.671847	−	0.740690i	$-0.265501\pi$
$788$	3.55117e11			0.921014
$789$		−	8.77074e10i		−	0.226323i
$790$	2.74021e10			0.0703518
$791$			5.09200e11i			1.30072i
$792$			5.27376e10i			0.134035i
$793$			2.94017e10i			0.0743497i
$794$		−	5.16186e11i		−	1.29875i
$795$	2.05890e11			0.515426
$796$	−1.44027e11			−0.358751
$797$			1.02216e10i			0.0253329i	0.999920	+	0.0126665i	$0.00403197\pi$
$797$			1.02216e10i			0.0253329i	−0.999920	+	0.0126665i	$0.995968\pi$
$798$	1.94907e11	+	2.51448e10i	0.480636	+	0.0620064i
$799$	−9.20062e9			−0.0225751
$800$		−	9.81304e8i		−	0.00239576i
$801$		−	2.70337e11i		−	0.656712i
$802$	5.41514e11			1.30892
$803$	−6.22059e8			−0.00149613
$804$	2.25458e11			0.539563
$805$	1.35625e11			0.322965
$806$			3.58712e10i			0.0849975i
$807$	−3.28519e11			−0.774579
$808$		−	1.31660e11i		−	0.308894i
$809$	−1.08281e10			−0.0252788			−0.0126394	−	0.999920i	$-0.504023\pi$
$809$	−1.08281e10			−0.0252788			−0.0126394	+	0.999920i	$0.504023\pi$
$810$			1.99027e10i			0.0462350i
$811$			7.19179e11i			1.66247i	0.555922	+	0.831235i	$0.312365\pi$
$811$			7.19179e11i			1.66247i	−0.555922	+	0.831235i	$0.687635\pi$
$812$			2.19414e11i			0.504708i
$813$			1.26087e11i			0.288607i
$814$	6.32883e10			0.144154
$815$	2.26375e11			0.513096
$816$		−	8.81328e9i		−	0.0198782i
$817$	−5.65435e11	−	7.29462e10i	−1.26910	−	0.163725i
$818$	4.13645e11			0.923878
$819$		−	7.09336e10i		−	0.157658i
$820$		−	2.65890e11i		−	0.588094i
$821$	5.54646e11			1.22080			0.610399	−	0.792094i	$-0.291009\pi$
$821$	5.54646e11			1.22080			0.610399	+	0.792094i	$0.291009\pi$
$822$	6.86867e10			0.150448
$823$	−2.40806e11			−0.524890			−0.262445	−	0.964947i	$-0.584529\pi$
$823$	−2.40806e11			−0.524890			−0.262445	+	0.964947i	$0.584529\pi$
$824$	−2.85988e11			−0.620354
$825$		−	2.59364e9i		−	0.00559878i
$826$	5.75952e11			1.23727
$827$		−	5.12223e11i		−	1.09506i	−0.836787	−	0.547529i	$-0.815569\pi$
$827$		−	5.12223e11i		−	1.09506i	0.836787	−	0.547529i	$-0.184431\pi$
$828$	−4.15299e10			−0.0883567
$829$			1.09954e11i			0.232806i	0.993202	+	0.116403i	$0.0371364\pi$
$829$			1.09954e11i			0.232806i	−0.993202	+	0.116403i	$0.962864\pi$
$830$			2.68042e11i			0.564794i
$831$		−	3.61639e11i		−	0.758352i
$832$			1.50146e10i			0.0313344i
$833$	8.46026e9			0.0175713
$834$	−2.69421e11			−0.556888
$835$			2.15732e11i			0.443780i
$836$	1.56011e11	+	2.01268e10i	0.319396	+	0.0412049i
$837$	−2.39805e11			−0.488603
$838$			1.99478e11i			0.404501i
$839$		−	9.95906e10i		−	0.200988i	−0.994938	−	0.100494i	$-0.967958\pi$
$839$		−	9.95906e10i		−	0.200988i	0.994938	−	0.100494i	$-0.0320423\pi$
$840$	1.21454e11			0.243946
$841$	5.38133e10			0.107574
$842$	4.27193e11			0.849915
$843$	−1.47838e11			−0.292736
$844$		−	2.51572e11i		−	0.495785i
$845$	4.81022e11			0.943491
$846$			3.88251e10i			0.0757933i
$847$	−3.21802e11			−0.625251
$848$			1.03189e11i			0.199550i
$849$			1.16170e11i			0.223595i
$850$		−	6.20132e8i		−	0.00118798i
$851$			4.98384e10i			0.0950268i
$852$	2.47764e11			0.470197
$853$	5.09046e11			0.961526			0.480763	−	0.876850i	$-0.340360\pi$
$853$	5.09046e11			0.961526			0.480763	+	0.876850i	$0.340360\pi$
$854$			1.19198e11i			0.224097i
$855$	−3.14068e11	−	4.05177e10i	−0.587706	−	0.0758194i
$856$	1.97430e10			0.0367720
$857$		−	9.79208e10i		−	0.181531i	−0.995872	−	0.0907657i	$-0.971069\pi$
$857$		−	9.79208e10i		−	0.181531i	0.995872	−	0.0907657i	$-0.0289314\pi$
$858$			3.96844e10i			0.0732270i
$859$	−7.97870e11			−1.46541			−0.732705	−	0.680546i	$-0.761743\pi$
$859$	−7.97870e11			−1.46541			−0.732705	+	0.680546i	$0.761743\pi$
$860$	−3.52343e11			−0.644128
$861$	4.40030e11			0.800699
$862$	−2.43813e11			−0.441598
$863$			8.98087e11i			1.61911i	0.587047	+	0.809553i	$0.300290\pi$
$863$			8.98087e11i			1.61911i	−0.587047	+	0.809553i	$0.699710\pi$
$864$	−1.00375e11			−0.180124
$865$		−	3.84543e11i		−	0.686880i
$866$	−3.63252e11			−0.645857
$867$			3.56847e11i			0.631547i
$868$			1.45426e11i			0.256191i
$869$			3.62986e10i			0.0636518i
$870$			2.47117e11i			0.431346i
$871$	−2.42731e11			−0.421747
$872$	−9.23390e10			−0.159705
$873$			4.70025e11i			0.809215i
$874$	−1.58495e10	+	1.22856e11i	−0.0271625	+	0.210547i
$875$	−6.22033e11			−1.06116
$876$		−	4.38676e8i		−	0.000744951i
$877$			5.69218e9i			0.00962232i	0.999988	+	0.00481116i	$0.00153145\pi$
$877$			5.69218e9i			0.00962232i	−0.999988	+	0.00481116i	$0.998469\pi$
$878$	−2.15065e11			−0.361903
$879$	4.91461e11			0.823254
$880$	9.72158e10			0.162109
$881$	4.83103e11			0.801929			0.400964	−	0.916094i	$-0.368675\pi$
$881$	4.83103e11			0.801929			0.400964	+	0.916094i	$0.368675\pi$
$882$		−	3.57009e10i		−	0.0589936i
$883$	−8.36775e11			−1.37647			−0.688234	−	0.725489i	$-0.741614\pi$
$883$	−8.36775e11			−1.37647			−0.688234	+	0.725489i	$0.741614\pi$
$884$			9.48846e9i			0.0155377i
$885$	6.48671e11			1.05743
$886$			5.35039e11i			0.868261i
$887$			1.04262e12i			1.68435i	0.539201	+	0.842177i	$0.318726\pi$
$887$			1.04262e12i			1.68435i	−0.539201	+	0.842177i	$0.681274\pi$
$888$			4.46310e10i			0.0717768i
$889$			4.76422e11i			0.762755i
$890$	−4.98336e11			−0.794259
$891$	−2.63644e10			−0.0418318
$892$		−	1.41059e11i		−	0.222813i
$893$	1.14854e11	+	1.48172e10i	0.180609	+	0.0233003i
$894$	2.52383e11			0.395103
$895$			1.92924e10i			0.0300673i
$896$			6.08711e10i			0.0944450i
$897$	−3.12508e10			−0.0482716
$898$	−3.85643e11			−0.593035
$899$	−2.95893e11			−0.452997
$900$	−2.61685e9			−0.00398850
$901$			6.52103e10i			0.0989503i
$902$	3.52215e11			0.532086
$903$		−	5.83103e11i		−	0.876989i
$904$	−2.87427e11			−0.430382
$905$			1.34667e12i			2.00756i
$906$		−	6.34415e10i		−	0.0941588i
$907$		−	1.20473e12i		−	1.78017i	−0.455793	−	0.890086i	$-0.650644\pi$
$907$		−	1.20473e12i		−	1.78017i	0.455793	−	0.890086i	$-0.349356\pi$
$908$			4.90234e11i			0.721207i
$909$	−3.51100e11			−0.514252
$910$	−1.30758e11			−0.190679
$911$		−	8.37061e11i		−	1.21530i	−0.794205	−	0.607650i	$-0.792112\pi$
$911$		−	8.37061e11i		−	1.21530i	0.794205	−	0.607650i	$-0.207888\pi$
$912$	−1.41934e10	+	1.10019e11i	−0.0205167	+	0.159033i
$913$	−3.55065e11			−0.511005
$914$		−	4.67666e11i		−	0.670118i
$915$			1.34248e11i			0.191524i
$916$	3.36941e11			0.478599
$917$	1.20062e12			1.69797
$918$	−6.34319e10			−0.0893175
$919$	9.03481e11			1.26665			0.633326	−	0.773885i	$-0.281689\pi$
$919$	9.03481e11			1.26665			0.633326	+	0.773885i	$0.281689\pi$
$920$			7.65558e10i			0.106863i
$921$	−2.41942e11			−0.336257
$922$		−	9.86708e11i		−	1.36542i
$923$	−2.66745e11			−0.367527
$924$			1.60885e11i			0.220713i
$925$			3.14039e9i			0.00428959i
$926$		−	4.35397e11i		−	0.592164i
$927$			7.62649e11i			1.03277i
$928$	−1.23852e11			−0.166998
$929$	1.24286e12			1.66863			0.834314	−	0.551290i	$-0.185864\pi$
$929$	1.24286e12			1.66863			0.834314	+	0.551290i	$0.185864\pi$
$930$			1.63788e11i			0.218952i
$931$	−1.05612e11	−	1.36249e10i	−0.140577	−	0.0181357i
$932$	5.74635e11			0.761602
$933$		−	1.07622e11i		−	0.142028i
$934$		−	3.48410e11i		−	0.457829i
$935$	6.14353e10			0.0803844
$936$	4.00397e10			0.0521660
$937$	6.09647e10			0.0790897			0.0395449	−	0.999218i	$-0.487409\pi$
$937$	6.09647e10			0.0790897			0.0395449	+	0.999218i	$0.487409\pi$
$938$	−9.84058e11			−1.27119
$939$		−	3.07798e11i		−	0.395916i
$940$	7.15697e10			0.0916680
$941$		−	1.19986e12i		−	1.53028i	−0.643864	−	0.765140i	$-0.722670\pi$
$941$		−	1.19986e12i		−	1.53028i	0.643864	−	0.765140i	$-0.277330\pi$
$942$	2.86475e10			0.0363818
$943$			2.77363e11i			0.350754i
$944$			3.25106e11i			0.409390i
$945$		−	8.74140e11i		−	1.09611i
$946$		−	4.66736e11i		−	0.582783i
$947$	2.42458e11			0.301465			0.150733	−	0.988575i	$-0.451837\pi$
$947$	2.42458e11			0.301465			0.150733	+	0.988575i	$0.451837\pi$
$948$	−2.55978e10			−0.0316934
$949$			4.72283e8i			0.000582288i
$950$	−9.98697e8	+	7.74130e9i	−0.00122614	+	0.00950428i
$951$	−4.38166e11			−0.535694
$952$			3.84673e10i			0.0468322i
$953$		−	9.39609e11i		−	1.13914i	−0.821944	−	0.569568i	$-0.807110\pi$
$953$		−	9.39609e11i		−	1.13914i	0.821944	−	0.569568i	$-0.192890\pi$
$954$	2.75177e11			0.332214
$955$	5.30274e11			0.637510
$956$	−4.52284e11			−0.541477
$957$	−3.27347e11			−0.390266
$958$		−	3.91677e11i		−	0.465014i
$959$	−2.99797e11			−0.354448
$960$			6.85567e10i			0.0807169i
$961$	6.56775e11			0.770058
$962$		−	4.80501e10i		−	0.0561040i
$963$		−	5.26489e10i		−	0.0612187i
$964$			2.41323e11i			0.279441i
$965$			1.03768e12i			1.19662i
$966$	−1.26694e11			−0.145495
$967$	5.80683e10			0.0664099			0.0332050	−	0.999449i	$-0.489429\pi$
$967$	5.80683e10			0.0664099			0.0332050	+	0.999449i	$0.489429\pi$
$968$		−	1.81647e11i		−	0.206883i
$969$	−8.96949e9	+	6.95261e10i	−0.0101736	+	0.0788592i
$970$	8.66439e11			0.978704
$971$			1.52782e11i			0.171868i	0.996301	+	0.0859342i	$0.0273875\pi$
$971$			1.52782e11i			0.171868i	−0.996301	+	0.0859342i	$0.972613\pi$
$972$			4.36167e11i			0.488639i
$973$	1.17594e12			1.31200
$974$	−1.18638e12			−1.31822
$975$	−1.96916e9			−0.00217902
$976$	−6.72833e10			−0.0741495
$977$			1.39585e11i			0.153200i	0.997062	+	0.0766001i	$0.0244065\pi$
$977$			1.39585e11i			0.153200i	−0.997062	+	0.0766001i	$0.975594\pi$
$978$	−2.11469e11			−0.231149
$979$		−	6.60128e11i		−	0.718617i
$980$	−6.58106e10			−0.0713496
$981$			2.46241e11i			0.265880i
$982$			8.37318e11i			0.900418i
$983$		−	8.23010e11i		−	0.881437i	−0.897645	−	0.440718i	$-0.854724\pi$
$983$		−	8.23010e11i		−	0.881437i	0.897645	−	0.440718i	$-0.145276\pi$
$984$			2.48382e11i			0.264936i
$985$	−1.74568e12			−1.85447
$986$	−7.82680e10			−0.0828088
$987$			1.18443e11i			0.124807i
$988$	1.52808e10	−	1.18447e11i	0.0160368	−	0.124307i
$989$	3.67546e11			0.384173
$990$		−	2.59247e11i		−	0.269881i
$991$		−	3.88275e11i		−	0.402573i	−0.979532	−	0.201287i	$-0.935488\pi$
$991$		−	3.88275e11i		−	0.402573i	0.979532	−	0.201287i	$-0.0645123\pi$
$992$	−8.20883e10			−0.0847685
$993$	7.03436e11			0.723482
$994$	−1.08142e12			−1.10776
$995$	7.08009e11			0.722347
$996$		−	2.50392e11i		−	0.254439i
$997$	−2.15719e11			−0.218327			−0.109164	−	0.994024i	$-0.534817\pi$
$997$	−2.15719e11			−0.218327			−0.109164	+	0.994024i	$0.534817\pi$
$998$		−	6.05508e11i		−	0.610376i
$999$	3.21223e11			0.322511

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	38.9.b.a.37.2	✓	12
3.2	odd	2		342.9.d.a.37.8		12
4.3	odd	2		304.9.e.e.113.8		12
19.18	odd	2	inner	38.9.b.a.37.11	yes	12
57.56	even	2		342.9.d.a.37.2		12
76.75	even	2		304.9.e.e.113.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
38.9.b.a.37.2	✓	12	1.1	even	1	trivial
38.9.b.a.37.11	yes	12	19.18	odd	2	inner
304.9.e.e.113.5		12	76.75	even	2
304.9.e.e.113.8		12	4.3	odd	2
342.9.d.a.37.2		12	57.56	even	2
342.9.d.a.37.8		12	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:	\( N \)	\(=\)	\( 38 = 2 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	38.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-11.3137i q^{2} -51.9537i q^{3} -128.000 q^{4} +629.221 q^{5} -587.789 q^{6} +2565.52 q^{7} +1448.15i q^{8} +3861.81 q^{9} +O(q^{10})\)	\(q-11.3137i q^{2} -51.9537i q^{3} -128.000 q^{4} +629.221 q^{5} -587.789 q^{6} +2565.52 q^{7} +1448.15i q^{8} +3861.81 q^{9} -7118.82i q^{10} +9430.05 q^{11} +6650.08i q^{12} -7159.53i q^{13} -29025.6i q^{14} -32690.4i q^{15} +16384.0 q^{16} +10353.8 q^{17} -43691.4i q^{18} +(-129250. - 16674.4i) q^{19} -80540.3 q^{20} -133289. i q^{21} -106689. i q^{22} +84015.6 q^{23} +75237.0 q^{24} +5293.93 q^{25} -81000.9 q^{26} -541504. i q^{27} -328387. q^{28} -668157. i q^{29} -369849. q^{30} -442850. i q^{31} -185364. i q^{32} -489926. i q^{33} -117140. i q^{34} +1.61428e6 q^{35} -494312. q^{36} +593205. i q^{37} +(-188649. + 1.46230e6i) q^{38} -371964. q^{39} +911209. i q^{40} +3.30133e6i q^{41} -1.50799e6 q^{42} +4.37474e6 q^{43} -1.20705e6 q^{44} +2.42993e6 q^{45} -950528. i q^{46} -888620. q^{47} -851210. i q^{48} +817114. q^{49} -59894.0i q^{50} -537920. i q^{51} +916420. i q^{52} +6.29819e6i q^{53} -6.12642e6 q^{54} +5.93358e6 q^{55} +3.71528e6i q^{56} +(-866297. + 6.71501e6i) q^{57} -7.55933e6 q^{58} +1.98429e7i q^{59} +4.18437e6i q^{60} -4.10665e6 q^{61} -5.01027e6 q^{62} +9.90757e6 q^{63} -2.09715e6 q^{64} -4.50493e6i q^{65} -5.54288e6 q^{66} -3.39031e7i q^{67} -1.32529e6 q^{68} -4.36492e6i q^{69} -1.82635e7i q^{70} -3.72573e7i q^{71} +5.59250e6i q^{72} -65965.6 q^{73} +6.71135e6 q^{74} -275040. i q^{75} +(1.65440e7 + 2.13432e6i) q^{76} +2.41930e7 q^{77} +4.20830e6i q^{78} +3.84925e6i q^{79} +1.03092e7 q^{80} -2.79578e6 q^{81} +3.73503e7 q^{82} -3.76526e7 q^{83} +1.70609e7i q^{84} +6.51484e6 q^{85} -4.94946e7i q^{86} -3.47132e7 q^{87} +1.36562e7i q^{88} -7.00026e7i q^{89} -2.74916e7i q^{90} -1.83680e7i q^{91} -1.07540e7 q^{92} -2.30077e7 q^{93} +1.00536e7i q^{94} +(-8.13267e7 - 1.04919e7i) q^{95} -9.63034e6 q^{96} +1.21711e8i q^{97} -9.24459e6i q^{98} +3.64171e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9}+O(q^{10}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9	\( 12 q - 1536 q^{4} + 558 q^{5} + 1792 q^{6} - 5422 q^{7} - 15592 q^{9} - 12546 q^{11} + 196608 q^{16} + 270810 q^{17} + 41512 q^{19} - 71424 q^{20} - 823956 q^{23} - 229376 q^{24} + 865538 q^{25} - 431616 q^{26} + 694016 q^{28} + 71168 q^{30} - 1194378 q^{35} + 1995776 q^{36} + 998784 q^{38} + 5786100 q^{39} - 8383744 q^{42} + 7586646 q^{43} + 1605888 q^{44} + 2226046 q^{45} - 20260530 q^{47} - 19498842 q^{49} + 16933888 q^{54} - 14858554 q^{55} + 14430564 q^{57} - 5506560 q^{58} - 41363266 q^{61} + 32266752 q^{62} + 84235798 q^{63} - 25165824 q^{64} + 14371328 q^{66} - 34663680 q^{68} + 87906498 q^{73} - 2149632 q^{74} - 5313536 q^{76} - 78817962 q^{77} + 9142272 q^{80} - 100904812 q^{81} - 49609728 q^{82} - 55944960 q^{83} + 25440254 q^{85} + 119189604 q^{87} + 105466368 q^{92} + 105500856 q^{93} + 81396774 q^{95} + 29360128 q^{96} - 85554938 q^{99}+O(q^{100}) \) 12 * q - 1536 * q^4 + 558 * q^5 + 1792 * q^6 - 5422 * q^7 - 15592 * q^9 - 12546 * q^11 + 196608 * q^16 + 270810 * q^17 + 41512 * q^19 - 71424 * q^20 - 823956 * q^23 - 229376 * q^24 + 865538 * q^25 - 431616 * q^26 + 694016 * q^28 + 71168 * q^30 - 1194378 * q^35 + 1995776 * q^36 + 998784 * q^38 + 5786100 * q^39 - 8383744 * q^42 + 7586646 * q^43 + 1605888 * q^44 + 2226046 * q^45 - 20260530 * q^47 - 19498842 * q^49 + 16933888 * q^54 - 14858554 * q^55 + 14430564 * q^57 - 5506560 * q^58 - 41363266 * q^61 + 32266752 * q^62 + 84235798 * q^63 - 25165824 * q^64 + 14371328 * q^66 - 34663680 * q^68 + 87906498 * q^73 - 2149632 * q^74 - 5313536 * q^76 - 78817962 * q^77 + 9142272 * q^80 - 100904812 * q^81 - 49609728 * q^82 - 55944960 * q^83 + 25440254 * q^85 + 119189604 * q^87 + 105466368 * q^92 + 105500856 * q^93 + 81396774 * q^95 + 29360128 * q^96 - 85554938 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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