LMFDB - Embedded newform 117.9.j.a.73.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [117,9,Mod(73,117)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(117, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 9, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("117.73"); S:= CuspForms(chi, 9); N := Newforms(S);

Level:	$ N $	$=$	$ 117 = 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	117.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [18,2,0,0,-166] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.6632973772$
Analytic rank:	$0$
Dimension:	$18$
Relative dimension:	$9$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 2 x^{17} + 13 x^{16} + 10976 x^{15} + 1201625 x^{14} + 122002 x^{13} + 46813351 x^{12} + \cdots + 12\!\cdots\!50 $ x^18 - 2x^17 + 13x^16 + 10976x^15 + 1201625x^14 + 122002x^13 + 46813351x^12 + 7862652956x^11 + 334844194775x^10 + 1307282975658x^9 + 11682756415941x^8 + 633414417845224x^7 + 18032980689164179x^6 + 45002614591373022x^5 + 390994490142779965x^4 + 14232520046144862044x^3 + 162776004109684584700x^2 + 701394949089602778920*x + 1264129294984265466250
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{16}\cdot 3^{8}\cdot 13^{4} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.7
Root			$9.74776 + 10.7478i$ of defining polynomial
Character	$\chi$	$=$	117.73
Dual form			117.9.j.a.109.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(9.74776 + 9.74776i) q^{2} -65.9624i q^{4} +(198.672 + 198.672i) q^{5} +(-1520.85 + 1520.85i) q^{7} +(3138.41 - 3138.41i) q^{8} +3873.22i q^{10} +(2962.81 - 2962.81i) q^{11} +(1826.58 - 28502.5i) q^{13} -29649.7 q^{14} +44298.6 q^{16} +32359.8i q^{17} +(-10661.6 - 10661.6i) q^{19} +(13104.9 - 13104.9i) q^{20} +57761.5 q^{22} +60462.2i q^{23} -311684. i q^{25} +(295641. - 260031. i) q^{26} +(100319. + 100319. i) q^{28} +549285. q^{29} +(-19449.9 - 19449.9i) q^{31} +(-371622. - 371622. i) q^{32} +(-315436. + 315436. i) q^{34} -604300. q^{35} +(1.93961e6 - 1.93961e6i) q^{37} -207854. i q^{38} +1.24703e6 q^{40} +(-253299. - 253299. i) q^{41} -5.09242e6i q^{43} +(-195434. - 195434. i) q^{44} +(-589371. + 589371. i) q^{46} +(6.02802e6 - 6.02802e6i) q^{47} +1.13885e6i q^{49} +(3.03822e6 - 3.03822e6i) q^{50} +(-1.88010e6 - 120486. i) q^{52} +1.10541e7 q^{53} +1.17726e6 q^{55} +9.54609e6i q^{56} +(5.35430e6 + 5.35430e6i) q^{58} +(-1.69750e6 + 1.69750e6i) q^{59} +1.11241e7 q^{61} -379185. i q^{62} -1.85854e7i q^{64} +(6.02555e6 - 5.29977e6i) q^{65} +(1.60890e7 + 1.60890e7i) q^{67} +2.13453e6 q^{68} +(-5.89057e6 - 5.89057e6i) q^{70} +(-2.04642e7 - 2.04642e7i) q^{71} +(-2.81599e7 + 2.81599e7i) q^{73} +3.78138e7 q^{74} +(-703266. + 703266. i) q^{76} +9.01196e6i q^{77} +6.53546e6 q^{79} +(8.80090e6 + 8.80090e6i) q^{80} -4.93819e6i q^{82} +(3.82312e7 + 3.82312e7i) q^{83} +(-6.42900e6 + 6.42900e6i) q^{85} +(4.96396e7 - 4.96396e7i) q^{86} -1.85970e7i q^{88} +(3.40081e7 - 3.40081e7i) q^{89} +(4.05700e7 + 4.61259e7i) q^{91} +3.98823e6 q^{92} +1.17519e8 q^{94} -4.23634e6i q^{95} +(-1.29141e7 - 1.29141e7i) q^{97} +(-1.11012e7 + 1.11012e7i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8} + 31556 q^{11} + 71300 q^{13} + 110260 q^{14} - 522860 q^{16} + 100288 q^{19} - 736268 q^{20} - 977312 q^{22} - 2952238 q^{26} + 4497084 q^{28} + 2479024 q^{29}+ \cdots - 588677614 q^{98}+O(q^{100}) $ 18 * q + 2 * q^2 - 166 * q^5 + 5308 * q^7 - 10464 * q^8 + 31556 * q^11 + 71300 * q^13 + 110260 * q^14 - 522860 * q^16 + 100288 * q^19 - 736268 * q^20 - 977312 * q^22 - 2952238 * q^26 + 4497084 * q^28 + 2479024 * q^29 - 1892664 * q^31 - 947212 * q^32 - 531576 * q^34 + 2918284 * q^35 - 8343978 * q^37 - 12691908 * q^40 - 1140178 * q^41 + 3867188 * q^44 + 2006148 * q^46 + 13368572 * q^47 - 37369598 * q^50 - 14821220 * q^52 - 50561348 * q^53 + 76994128 * q^55 + 22505716 * q^58 - 2127976 * q^59 - 52016516 * q^61 - 10413082 * q^65 + 960292 * q^67 - 283187508 * q^68 - 166635032 * q^70 - 67412140 * q^71 - 145213226 * q^73 + 233620024 * q^74 - 150533640 * q^76 - 76829120 * q^79 - 524889520 * q^80 + 241951556 * q^83 + 260737764 * q^85 + 579480384 * q^86 + 89187110 * q^89 + 232660948 * q^91 + 122690376 * q^92 + 1069637380 * q^94 + 331183146 * q^97 - 588677614 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/117\mathbb{Z}\right)^\times$.

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$

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 $

$2$	9.74776	+	9.74776i	0.609235	+	0.609235i	0.942746	−	0.333511i	$-0.108234\pi$
$2$	9.74776	+	9.74776i	0.609235	+	0.609235i	−0.333511	+	0.942746i	$0.608234\pi$
$3$		0			0
$3$		0			0
$4$		−	65.9624i		−	0.257666i
$5$	198.672	+	198.672i	0.317876	+	0.317876i	0.847951	−	0.530075i	$-0.177836\pi$
$5$	198.672	+	198.672i	0.317876	+	0.317876i	−0.530075	+	0.847951i	$0.677836\pi$
$6$		0			0
$7$	−1520.85	+	1520.85i	−0.633422	+	0.633422i	−0.948925	−	0.315502i	$-0.897827\pi$
$7$	−1520.85	+	1520.85i	−0.633422	+	0.633422i	0.315502	+	0.948925i	$0.397827\pi$
$8$	3138.41	−	3138.41i	0.766214	−	0.766214i
$9$		0			0
$10$			3873.22i			0.387322i
$11$	2962.81	−	2962.81i	0.202364	−	0.202364i	−0.598648	−	0.801012i	$-0.704295\pi$
$11$	2962.81	−	2962.81i	0.202364	−	0.202364i	0.801012	+	0.598648i	$0.204295\pi$
$12$		0			0
$13$	1826.58	−	28502.5i	0.0639537	−	0.997953i
$13$	1826.58	−	28502.5i	0.0639537	−	0.997953i
$14$	−29649.7			−0.771806
$15$		0			0
$16$	44298.6			0.675943
$17$			32359.8i			0.387445i	0.981056	+	0.193723i	$0.0620562\pi$
$17$			32359.8i			0.387445i	−0.981056	+	0.193723i	$0.937944\pi$
$18$		0			0
$19$	−10661.6	−	10661.6i	−0.0818105	−	0.0818105i	0.665017	−	0.746828i	$-0.268424\pi$
$19$	−10661.6	−	10661.6i	−0.0818105	−	0.0818105i	−0.746828	+	0.665017i	$0.768424\pi$
$20$	13104.9	−	13104.9i	0.0819056	−	0.0819056i
$21$		0			0
$22$	57761.5			0.246574
$23$			60462.2i			0.216059i	0.994148	+	0.108030i	$0.0344541\pi$
$23$			60462.2i			0.216059i	−0.994148	+	0.108030i	$0.965546\pi$
$24$		0			0
$25$		−	311684.i		−	0.797910i
$26$	295641.	−	260031.i	0.646951	−	0.569025i
$27$		0			0
$28$	100319.	+	100319.i	0.163211	+	0.163211i
$29$	549285.			0.776615			0.388308	−	0.921530i	$-0.373060\pi$
$29$	549285.			0.776615			0.388308	+	0.921530i	$0.373060\pi$
$30$		0			0
$31$	−19449.9	−	19449.9i	−0.0210605	−	0.0210605i	0.696498	−	0.717559i	$-0.254740\pi$
$31$	−19449.9	−	19449.9i	−0.0210605	−	0.0210605i	−0.717559	+	0.696498i	$0.754740\pi$
$32$	−371622.	−	371622.i	−0.354406	−	0.354406i
$33$		0			0
$34$	−315436.	+	315436.i	−0.236045	+	0.236045i
$35$	−604300.			−0.402699
$36$		0			0
$37$	1.93961e6	−	1.93961e6i	1.03492	−	1.03492i	0.0355564	−	0.999368i	$-0.488680\pi$
$37$	1.93961e6	−	1.93961e6i	1.03492	−	1.03492i	0.999368	−	0.0355564i	$-0.0113203\pi$
$38$		−	207854.i		−	0.0996836i
$39$		0			0
$40$	1.24703e6			0.487121
$41$	−253299.	−	253299.i	−0.0896392	−	0.0896392i	0.660865	−	0.750505i	$-0.270189\pi$
$41$	−253299.	−	253299.i	−0.0896392	−	0.0896392i	−0.750505	+	0.660865i	$0.770189\pi$
$42$		0			0
$43$		−	5.09242e6i		−	1.48953i	−0.667325	−	0.744766i	$-0.732561\pi$
$43$		−	5.09242e6i		−	1.48953i	0.667325	−	0.744766i	$-0.267439\pi$
$44$	−195434.	−	195434.i	−0.0521422	−	0.0521422i
$45$		0			0
$46$	−589371.	+	589371.i	−0.131631	+	0.131631i
$47$	6.02802e6	−	6.02802e6i	1.23533	−	1.23533i	0.273442	−	0.961888i	$-0.411838\pi$
$47$	6.02802e6	−	6.02802e6i	1.23533	−	1.23533i	0.961888	−	0.273442i	$-0.0881623\pi$
$48$		0			0
$49$			1.13885e6i			0.197552i
$50$	3.03822e6	−	3.03822e6i	0.486115	−	0.486115i
$51$		0			0
$52$	−1.88010e6	−	120486.i	−0.257138	−	0.0164787i
$53$	1.10541e7			1.40094			0.700470	−	0.713682i	$-0.252974\pi$
$53$	1.10541e7			1.40094			0.700470	+	0.713682i	$0.252974\pi$
$54$		0			0
$55$	1.17726e6			0.128653
$56$			9.54609e6i			0.970674i
$57$		0			0
$58$	5.35430e6	+	5.35430e6i	0.473141	+	0.473141i
$59$	−1.69750e6	+	1.69750e6i	−0.140088	+	0.140088i	−0.773673	−	0.633585i	$-0.781583\pi$
$59$	−1.69750e6	+	1.69750e6i	−0.140088	+	0.140088i	0.633585	+	0.773673i	$0.281583\pi$
$60$		0			0
$61$	1.11241e7			0.803423			0.401712	−	0.915766i	$-0.368415\pi$
$61$	1.11241e7			0.803423			0.401712	+	0.915766i	$0.368415\pi$
$62$		−	379185.i		−	0.0256616i
$63$		0			0
$64$		−	1.85854e7i		−	1.10778i
$65$	6.02555e6	−	5.29977e6i	0.337554	−	0.296896i
$66$		0			0
$67$	1.60890e7	+	1.60890e7i	0.798420	+	0.798420i	0.982846	−	0.184427i	$-0.0590428\pi$
$67$	1.60890e7	+	1.60890e7i	0.798420	+	0.798420i	−0.184427	+	0.982846i	$0.559043\pi$
$68$	2.13453e6			0.0998313
$69$		0			0
$70$	−5.89057e6	−	5.89057e6i	−0.245338	−	0.245338i
$71$	−2.04642e7	−	2.04642e7i	−0.805308	−	0.805308i	0.178612	−	0.983920i	$-0.442839\pi$
$71$	−2.04642e7	−	2.04642e7i	−0.805308	−	0.805308i	−0.983920	+	0.178612i	$0.942839\pi$
$72$		0			0
$73$	−2.81599e7	+	2.81599e7i	−0.991605	+	0.991605i	−0.999965	−	0.00835960i	$-0.997339\pi$
$73$	−2.81599e7	+	2.81599e7i	−0.991605	+	0.991605i	0.00835960	+	0.999965i	$0.497339\pi$
$74$	3.78138e7			1.26102
$75$		0			0
$76$	−703266.	+	703266.i	−0.0210798	+	0.0210798i
$77$			9.01196e6i			0.256364i
$78$		0			0
$79$	6.53546e6			0.167791			0.0838953	−	0.996475i	$-0.473264\pi$
$79$	6.53546e6			0.167791			0.0838953	+	0.996475i	$0.473264\pi$
$80$	8.80090e6	+	8.80090e6i	0.214866	+	0.214866i
$81$		0			0
$82$		−	4.93819e6i		−	0.109223i
$83$	3.82312e7	+	3.82312e7i	0.805574	+	0.805574i	0.983960	−	0.178387i	$-0.0570878\pi$
$83$	3.82312e7	+	3.82312e7i	0.805574	+	0.805574i	−0.178387	+	0.983960i	$0.557088\pi$
$84$		0			0
$85$	−6.42900e6	+	6.42900e6i	−0.123159	+	0.123159i
$86$	4.96396e7	−	4.96396e7i	0.907475	−	0.907475i
$87$		0			0
$88$		−	1.85970e7i		−	0.310108i
$89$	3.40081e7	−	3.40081e7i	0.542028	−	0.542028i	−0.382095	−	0.924123i	$-0.624797\pi$
$89$	3.40081e7	−	3.40081e7i	0.542028	−	0.542028i	0.924123	+	0.382095i	$0.124797\pi$
$90$		0			0
$91$	4.05700e7	+	4.61259e7i	0.591616	+	0.672635i
$92$	3.98823e6			0.0556710
$93$		0			0
$94$	1.17519e8			1.50521
$95$		−	4.23634e6i		−	0.0520111i
$96$		0			0
$97$	−1.29141e7	−	1.29141e7i	−0.145873	−	0.145873i	0.630398	−	0.776272i	$-0.282892\pi$
$97$	−1.29141e7	−	1.29141e7i	−0.145873	−	0.145873i	−0.776272	+	0.630398i	$0.782892\pi$
$98$	−1.11012e7	+	1.11012e7i	−0.120356	+	0.120356i
$99$		0			0
$100$	−2.05594e7			−0.205594
$101$			6.17366e7i			0.593276i	0.954990	+	0.296638i	$0.0958655\pi$
$101$			6.17366e7i			0.593276i	−0.954990	+	0.296638i	$0.904134\pi$
$102$		0			0
$103$		−	1.73240e8i		−	1.53922i	−0.638515	−	0.769609i	$-0.720451\pi$
$103$		−	1.73240e8i		−	1.53922i	0.638515	−	0.769609i	$-0.279549\pi$
$104$	−8.37201e7	−	9.51852e7i	−0.715643	−	0.813647i
$105$		0			0
$106$	1.07753e8	+	1.07753e8i	0.853501	+	0.853501i
$107$	−1.44603e8			−1.10317			−0.551583	−	0.834120i	$-0.685976\pi$
$107$	−1.44603e8			−1.10317			−0.551583	+	0.834120i	$0.685976\pi$
$108$		0			0
$109$	−1.61023e8	−	1.61023e8i	−1.14073	−	1.14073i	−0.988317	−	0.152411i	$-0.951296\pi$
$109$	−1.61023e8	−	1.61023e8i	−1.14073	−	1.14073i	−0.152411	−	0.988317i	$-0.548704\pi$
$110$	1.14756e7	+	1.14756e7i	0.0783799	+	0.0783799i
$111$		0			0
$112$	−6.73714e7	+	6.73714e7i	−0.428157	+	0.428157i
$113$	3.08494e8			1.89205			0.946025	−	0.324094i	$-0.105059\pi$
$113$	3.08494e8			1.89205			0.946025	+	0.324094i	$0.105059\pi$
$114$		0			0
$115$	−1.20122e7	+	1.20122e7i	−0.0686799	+	0.0686799i
$116$		−	3.62322e7i		−	0.200107i
$117$		0			0
$118$	−3.30937e7			−0.170694
$119$	−4.92143e7	−	4.92143e7i	−0.245416	−	0.245416i
$120$		0			0
$121$			1.96802e8i			0.918098i
$122$	1.08435e8	+	1.08435e8i	0.489473	+	0.489473i
$123$		0			0
$124$	−1.28296e6	+	1.28296e6i	−0.00542658	+	0.00542658i
$125$	1.39529e8	−	1.39529e8i	0.571512	−	0.571512i
$126$		0			0
$127$		−	2.77756e8i		−	1.06770i	−0.845580	−	0.533849i	$-0.820745\pi$
$127$		−	2.77756e8i		−	1.06770i	0.845580	−	0.533849i	$-0.179255\pi$
$128$	8.60308e7	−	8.60308e7i	0.320490	−	0.320490i
$129$		0			0
$130$	1.10397e8	+	7.07475e6i	0.386529	+	0.0247707i
$131$	−4.98211e8			−1.69172			−0.845860	−	0.533405i	$-0.820912\pi$
$131$	−4.98211e8			−1.69172			−0.845860	+	0.533405i	$0.820912\pi$
$132$		0			0
$133$	3.24294e7			0.103641
$134$			3.13664e8i			0.972850i
$135$		0			0
$136$	1.01558e8	+	1.01558e8i	0.296866	+	0.296866i
$137$	−1.87596e8	+	1.87596e8i	−0.532526	+	0.532526i	−0.921323	−	0.388797i	$-0.872891\pi$
$137$	−1.87596e8	+	1.87596e8i	−0.532526	+	0.532526i	0.388797	+	0.921323i	$0.372891\pi$
$138$		0			0
$139$	1.29769e8			0.347625			0.173812	−	0.984779i	$-0.444391\pi$
$139$	1.29769e8			0.347625			0.173812	+	0.984779i	$0.444391\pi$
$140$			3.98611e7i			0.103762i
$141$		0			0
$142$		−	3.98961e8i		−	0.981243i
$143$	−7.90357e7	−	8.98594e7i	−0.189008	−	0.214891i
$144$		0			0
$145$	1.09128e8	+	1.09128e8i	0.246867	+	0.246867i
$146$	−5.48991e8			−1.20824
$147$		0			0
$148$	−1.27942e8	−	1.27942e8i	−0.266664	−	0.266664i
$149$	−2.22073e8	−	2.22073e8i	−0.450559	−	0.450559i	0.444981	−	0.895540i	$-0.353210\pi$
$149$	−2.22073e8	−	2.22073e8i	−0.450559	−	0.450559i	−0.895540	+	0.444981i	$0.853210\pi$
$150$		0			0
$151$	−3.57612e8	+	3.57612e8i	−0.687868	+	0.687868i	−0.961760	−	0.273893i	$-0.911689\pi$
$151$	−3.57612e8	+	3.57612e8i	−0.687868	+	0.687868i	0.273893	+	0.961760i	$0.411689\pi$
$152$	−6.69211e7			−0.125369
$153$		0			0
$154$	−8.78464e7	+	8.78464e7i	−0.156186	+	0.156186i
$155$		−	7.72829e6i		−	0.0133893i
$156$		0			0
$157$	−8.70825e8			−1.43328			−0.716642	−	0.697441i	$-0.754322\pi$
$157$	−8.70825e8			−1.43328			−0.716642	+	0.697441i	$0.754322\pi$
$158$	6.37061e7	+	6.37061e7i	0.102224	+	0.102224i
$159$		0			0
$160$		−	1.47662e8i		−	0.225314i
$161$	−9.19538e7	−	9.19538e7i	−0.136857	−	0.136857i
$162$		0			0
$163$	−6.38819e7	+	6.38819e7i	−0.0904955	+	0.0904955i	−0.750905	−	0.660410i	$-0.770383\pi$
$163$	−6.38819e7	+	6.38819e7i	−0.0904955	+	0.0904955i	0.660410	+	0.750905i	$0.270383\pi$
$164$	−1.67082e7	+	1.67082e7i	−0.0230969	+	0.0230969i
$165$		0			0
$166$			7.45336e8i			0.981567i
$167$	−7.63495e8	+	7.63495e8i	−0.981613	+	0.981613i	−0.999834	−	0.0182207i	$-0.994200\pi$
$167$	−7.63495e8	+	7.63495e8i	−0.981613	+	0.981613i	0.0182207	+	0.999834i	$0.494200\pi$
$168$		0			0
$169$	−8.09058e8	−	1.04124e8i	−0.991820	−	0.127646i
$170$	−1.25337e8			−0.150066
$171$		0			0
$172$	−3.35908e8			−0.383801
$173$			3.08494e8i			0.344399i	0.985062	+	0.172200i	$0.0550875\pi$
$173$			3.08494e8i			0.344399i	−0.985062	+	0.172200i	$0.944913\pi$
$174$		0			0
$175$	4.74023e8	+	4.74023e8i	0.505414	+	0.505414i
$176$	1.31248e8	−	1.31248e8i	0.136786	−	0.136786i
$177$		0			0
$178$	6.63005e8			0.660445
$179$			3.95895e8i			0.385627i	0.981235	+	0.192814i	$0.0617612\pi$
$179$			3.95895e8i			0.385627i	−0.981235	+	0.192814i	$0.938239\pi$
$180$		0			0
$181$			1.38347e9i			1.28901i	0.764601	+	0.644504i	$0.222936\pi$
$181$			1.38347e9i			1.28901i	−0.764601	+	0.644504i	$0.777064\pi$
$182$	−5.41576e7	+	8.45092e8i	−0.0493598	+	0.770226i
$183$		0			0
$184$	1.89755e8	+	1.89755e8i	0.165547	+	0.165547i
$185$	7.70695e8			0.657954
$186$		0			0
$187$	9.58759e7	+	9.58759e7i	0.0784049	+	0.0784049i
$188$	−3.97623e8	−	3.97623e8i	−0.318302	−	0.318302i
$189$		0			0
$190$	4.12948e7	−	4.12948e7i	0.0316870	−	0.0316870i
$191$	−1.26344e9			−0.949338			−0.474669	−	0.880164i	$-0.657432\pi$
$191$	−1.26344e9			−0.949338			−0.474669	+	0.880164i	$0.657432\pi$
$192$		0			0
$193$	8.05420e8	−	8.05420e8i	0.580488	−	0.580488i	−0.354549	−	0.935037i	$-0.615366\pi$
$193$	8.05420e8	−	8.05420e8i	0.580488	−	0.580488i	0.935037	+	0.354549i	$0.115366\pi$
$194$		−	2.51766e8i		−	0.177742i
$195$		0			0
$196$	7.51212e7			0.0509024
$197$	1.32900e9	+	1.32900e9i	0.882386	+	0.882386i	0.993777	−	0.111391i	$-0.0355305\pi$
$197$	1.32900e9	+	1.32900e9i	0.882386	+	0.882386i	−0.111391	+	0.993777i	$0.535530\pi$
$198$		0			0
$199$		−	8.97449e8i		−	0.572265i	−0.958190	−	0.286133i	$-0.907630\pi$
$199$		−	8.97449e8i		−	0.572265i	0.958190	−	0.286133i	$-0.0923698\pi$
$200$	−9.78192e8	−	9.78192e8i	−0.611370	−	0.611370i
$201$		0			0
$202$	−6.01793e8	+	6.01793e8i	−0.361445	+	0.361445i
$203$	−8.35379e8	+	8.35379e8i	−0.491926	+	0.491926i
$204$		0			0
$205$		−	1.00647e8i		−	0.0569882i
$206$	1.68871e9	−	1.68871e9i	0.937746	−	0.937746i
$207$		0			0
$208$	8.09150e7	−	1.26262e9i	0.0432290	−	0.674559i
$209$	−6.31767e7			−0.0331110
$210$		0			0
$211$	−2.58040e9			−1.30184			−0.650919	−	0.759147i	$-0.725616\pi$
$211$	−2.58040e9			−1.30184			−0.650919	+	0.759147i	$0.725616\pi$
$212$		−	7.29154e8i		−	0.360974i
$213$		0			0
$214$	−1.40955e9	−	1.40955e9i	−0.672087	−	0.672087i
$215$	1.01172e9	−	1.01172e9i	0.473486	−	0.473486i
$216$		0			0
$217$	5.91605e7			0.0266804
$218$		−	3.13923e9i		−	1.38994i
$219$		0			0
$220$		−	7.76546e7i		−	0.0331495i
$221$	9.22336e8	+	5.91078e7i	0.386652	+	0.0247785i
$222$		0			0
$223$	2.48566e8	+	2.48566e8i	0.100513	+	0.100513i	0.755575	−	0.655062i	$-0.227358\pi$
$223$	2.48566e8	+	2.48566e8i	0.100513	+	0.100513i	−0.655062	+	0.755575i	$0.727358\pi$
$224$	1.13036e9			0.448977
$225$		0			0
$226$	3.00712e9	+	3.00712e9i	1.15270	+	1.15270i
$227$	3.09771e9	+	3.09771e9i	1.16664	+	1.16664i	0.982991	+	0.183651i	$0.0587917\pi$
$227$	3.09771e9	+	3.09771e9i	1.16664	+	1.16664i	0.183651	+	0.982991i	$0.441208\pi$
$228$		0			0
$229$	3.49571e9	−	3.49571e9i	1.27114	−	1.27114i	0.325650	−	0.945490i	$-0.394417\pi$
$229$	3.49571e9	−	3.49571e9i	1.27114	−	1.27114i	0.945490	−	0.325650i	$-0.105583\pi$
$230$	−2.34183e8			−0.0836844
$231$		0			0
$232$	1.72388e9	−	1.72388e9i	0.595053	−	0.595053i
$233$		−	2.91410e9i		−	0.988736i	−0.869253	−	0.494368i	$-0.835399\pi$
$233$		−	2.91410e9i		−	0.988736i	0.869253	−	0.494368i	$-0.164601\pi$
$234$		0			0
$235$	2.39520e9			0.785363
$236$	1.11971e8	+	1.11971e8i	0.0360960	+	0.0360960i
$237$		0			0
$238$		−	9.59459e8i		−	0.299033i
$239$	2.22243e9	+	2.22243e9i	0.681140	+	0.681140i	0.960257	−	0.279117i	$-0.0900417\pi$
$239$	2.22243e9	+	2.22243e9i	0.681140	+	0.681140i	−0.279117	+	0.960257i	$0.590042\pi$
$240$		0			0
$241$	1.19161e9	−	1.19161e9i	0.353237	−	0.353237i	−0.508076	−	0.861312i	$-0.669643\pi$
$241$	1.19161e9	−	1.19161e9i	0.353237	−	0.353237i	0.861312	+	0.508076i	$0.169643\pi$
$242$	−1.91838e9	+	1.91838e9i	−0.559337	+	0.559337i
$243$		0			0
$244$		−	7.33770e8i		−	0.207015i
$245$	−2.26258e8	+	2.26258e8i	−0.0627970	+	0.0627970i
$246$		0			0
$247$	−3.23358e8	+	2.84409e8i	−0.0868751	+	0.0764109i
$248$	−1.22083e8			−0.0322738
$249$		0			0
$250$	2.72020e9			0.696370
$251$		−	6.53718e9i		−	1.64701i	−0.567312	−	0.823503i	$-0.692016\pi$
$251$		−	6.53718e9i		−	1.64701i	0.567312	−	0.823503i	$-0.307984\pi$
$252$		0			0
$253$	1.79138e8	+	1.79138e8i	0.0437225	+	0.0437225i
$254$	2.70750e9	−	2.70750e9i	0.650479	−	0.650479i
$255$		0			0
$256$	−3.08065e9			−0.717269
$257$			2.65350e9i			0.608255i	0.952631	+	0.304128i	$0.0983649\pi$
$257$			2.65350e9i			0.608255i	−0.952631	+	0.304128i	$0.901635\pi$
$258$		0			0
$259$			5.89971e9i			1.31109i
$260$	−3.49586e8	−	3.97460e8i	−0.0764998	−	0.0869761i
$261$		0			0
$262$	−4.85644e9	−	4.85644e9i	−1.03065	−	1.03065i
$263$	5.80149e9			1.21260			0.606299	−	0.795237i	$-0.292653\pi$
$263$	5.80149e9			1.21260			0.606299	+	0.795237i	$0.292653\pi$
$264$		0			0
$265$	2.19614e9	+	2.19614e9i	0.445325	+	0.445325i
$266$	3.16114e8	+	3.16114e8i	0.0631418	+	0.0631418i
$267$		0			0
$268$	1.06127e9	−	1.06127e9i	0.205725	−	0.205725i
$269$	8.58856e8			0.164026			0.0820128	−	0.996631i	$-0.473865\pi$
$269$	8.58856e8			0.164026			0.0820128	+	0.996631i	$0.473865\pi$
$270$		0			0
$271$	−2.15727e9	+	2.15727e9i	−0.399969	+	0.399969i	−0.878222	−	0.478253i	$-0.841270\pi$
$271$	−2.15727e9	+	2.15727e9i	−0.399969	+	0.399969i	0.478253	+	0.878222i	$0.341270\pi$
$272$			1.43349e9i			0.261891i
$273$		0			0
$274$	−3.65728e9			−0.648867
$275$	−9.23459e8	−	9.23459e8i	−0.161468	−	0.161468i
$276$		0			0
$277$			6.75310e8i			0.114705i	0.998354	+	0.0573527i	$0.0182660\pi$
$277$			6.75310e8i			0.114705i	−0.998354	+	0.0573527i	$0.981734\pi$
$278$	1.26495e9	+	1.26495e9i	0.211785	+	0.211785i
$279$		0			0
$280$	−1.89654e9	+	1.89654e9i	−0.308554	+	0.308554i
$281$	−1.11588e9	+	1.11588e9i	−0.178975	+	0.178975i	−0.790909	−	0.611934i	$-0.790392\pi$
$281$	−1.11588e9	+	1.11588e9i	−0.178975	+	0.178975i	0.611934	+	0.790909i	$0.290392\pi$
$282$		0			0
$283$		−	2.25404e9i		−	0.351411i	−0.984443	−	0.175706i	$-0.943779\pi$
$283$		−	2.25404e9i		−	0.351411i	0.984443	−	0.175706i	$-0.0562207\pi$
$284$	−1.34987e9	+	1.34987e9i	−0.207500	+	0.207500i
$285$		0			0
$286$	1.05506e8	−	1.64635e9i	0.0157693	−	0.246069i
$287$	7.70458e8			0.113559
$288$		0			0
$289$	5.92860e9			0.849886
$290$			2.12750e9i			0.300800i
$291$		0			0
$292$	1.85749e9	+	1.85749e9i	0.255503	+	0.255503i
$293$	−6.60712e9	+	6.60712e9i	−0.896482	+	0.896482i	−0.995123	−	0.0986406i	$-0.968551\pi$
$293$	−6.60712e9	+	6.60712e9i	−0.896482	+	0.896482i	0.0986406	+	0.995123i	$0.468551\pi$
$294$		0			0
$295$	−6.74494e8			−0.0890614
$296$		−	1.21746e10i		−	1.58595i
$297$		0			0
$298$		−	4.32944e9i		−	0.548992i
$299$	1.72333e9	+	1.10439e8i	0.215617	+	0.0138178i
$300$		0			0
$301$	7.74478e9	+	7.74478e9i	0.943503	+	0.943503i
$302$	−6.97184e9			−0.838146
$303$		0			0
$304$	−4.72295e8	−	4.72295e8i	−0.0552992	−	0.0552992i
$305$	2.21004e9	+	2.21004e9i	0.255389	+	0.255389i
$306$		0			0
$307$	−8.84782e9	+	8.84782e9i	−0.996054	+	0.996054i	−0.999992	−	0.00393808i	$-0.998746\pi$
$307$	−8.84782e9	+	8.84782e9i	−0.996054	+	0.996054i	0.00393808	+	0.999992i	$0.498746\pi$
$308$	5.94450e8			0.0660561
$309$		0			0
$310$	7.53335e7	−	7.53335e7i	0.00815721	−	0.00815721i
$311$			1.33001e10i			1.42172i	0.703336	+	0.710858i	$0.251693\pi$
$311$			1.33001e10i			1.42172i	−0.703336	+	0.710858i	$0.748307\pi$
$312$		0			0
$313$	−2.77550e9			−0.289177			−0.144589	−	0.989492i	$-0.546186\pi$
$313$	−2.77550e9			−0.289177			−0.144589	+	0.989492i	$0.546186\pi$
$314$	−8.48859e9	−	8.48859e9i	−0.873206	−	0.873206i
$315$		0			0
$316$		−	4.31095e8i		−	0.0432339i
$317$	1.67710e9	+	1.67710e9i	0.166081	+	0.166081i	0.785254	−	0.619173i	$-0.212532\pi$
$317$	1.67710e9	+	1.67710e9i	0.166081	+	0.166081i	−0.619173	+	0.785254i	$0.712532\pi$
$318$		0			0
$319$	1.62743e9	−	1.62743e9i	0.157159	−	0.157159i
$320$	3.69240e9	−	3.69240e9i	0.352135	−	0.352135i
$321$		0			0
$322$		−	1.79269e9i		−	0.166756i
$323$	3.45008e8	−	3.45008e8i	0.0316971	−	0.0316971i
$324$		0			0
$325$	−8.88377e9	−	5.69316e8i	−0.796277	−	0.0510293i
$326$	−1.24541e9			−0.110266
$327$		0			0
$328$	−1.58991e9			−0.137366
$329$			1.83354e10i			1.56497i
$330$		0			0
$331$	5.86196e9	+	5.86196e9i	0.488349	+	0.488349i	0.907785	−	0.419436i	$-0.137772\pi$
$331$	5.86196e9	+	5.86196e9i	0.488349	+	0.488349i	−0.419436	+	0.907785i	$0.637772\pi$
$332$	2.52182e9	−	2.52182e9i	0.207569	−	0.207569i
$333$		0			0
$334$	−1.48847e10			−1.19607
$335$			6.39290e9i			0.507596i
$336$		0			0
$337$			7.94739e9i			0.616176i	0.951358	+	0.308088i	$0.0996891\pi$
$337$			7.94739e9i			0.616176i	−0.951358	+	0.308088i	$0.900311\pi$
$338$	−6.87152e9	−	8.90148e9i	−0.526485	−	0.682017i
$339$		0			0
$340$	4.24072e8	+	4.24072e8i	0.0317339	+	0.0317339i
$341$	−1.15252e8			−0.00852378
$342$		0			0
$343$	−1.04994e10	−	1.04994e10i	−0.758556	−	0.758556i
$344$	−1.59821e10	−	1.59821e10i	−1.14130	−	1.14130i
$345$		0			0
$346$	−3.00713e9	+	3.00713e9i	−0.209820	+	0.209820i
$347$	−1.23874e10			−0.854401			−0.427201	−	0.904157i	$-0.640500\pi$
$347$	−1.23874e10			−0.854401			−0.427201	+	0.904157i	$0.640500\pi$
$348$		0			0
$349$	−2.78130e8	+	2.78130e8i	−0.0187476	+	0.0187476i	−0.716418	−	0.697671i	$-0.754220\pi$
$349$	−2.78130e8	+	2.78130e8i	−0.0187476	+	0.0187476i	0.697671	+	0.716418i	$0.254220\pi$
$350$			9.24133e9i			0.615832i
$351$		0			0
$352$	−2.20209e9			−0.143438
$353$	−1.59300e9	−	1.59300e9i	−0.102593	−	0.102593i	0.653947	−	0.756540i	$-0.273112\pi$
$353$	−1.59300e9	−	1.59300e9i	−0.102593	−	0.102593i	−0.756540	+	0.653947i	$0.773112\pi$
$354$		0			0
$355$		−	8.13135e9i		−	0.511975i
$356$	−2.24325e9	−	2.24325e9i	−0.139662	−	0.139662i
$357$		0			0
$358$	−3.85908e9	+	3.85908e9i	−0.234937	+	0.234937i
$359$	1.19822e10	−	1.19822e10i	0.721370	−	0.721370i	−0.247514	−	0.968884i	$-0.579614\pi$
$359$	1.19822e10	−	1.19822e10i	0.721370	−	0.721370i	0.968884	+	0.247514i	$0.0796136\pi$
$360$		0			0
$361$		−	1.67562e10i		−	0.986614i
$362$	−1.34857e10	+	1.34857e10i	−0.785309	+	0.785309i
$363$		0			0
$364$	3.04258e9	−	2.67610e9i	0.173315	−	0.152439i
$365$	−1.11892e10			−0.630415
$366$		0			0
$367$	3.46746e10			1.91138			0.955691	−	0.294372i	$-0.0951106\pi$
$367$	3.46746e10			1.91138			0.955691	+	0.294372i	$0.0951106\pi$
$368$			2.67839e9i			0.146044i
$369$		0			0
$370$	7.51255e9	+	7.51255e9i	0.400849	+	0.400849i
$371$	−1.68116e10	+	1.68116e10i	−0.887387	+	0.887387i
$372$		0			0
$373$	1.23140e9			0.0636157			0.0318078	−	0.999494i	$-0.489874\pi$
$373$	1.23140e9			0.0636157			0.0318078	+	0.999494i	$0.489874\pi$
$374$			1.86915e9i			0.0955340i
$375$		0			0
$376$		−	3.78368e10i		−	1.89305i
$377$	1.00331e9	−	1.56560e10i	0.0496674	−	0.775025i
$378$		0			0
$379$	2.31134e10	+	2.31134e10i	1.12023	+	1.12023i	0.991707	+	0.128523i	$0.0410236\pi$
$379$	2.31134e10	+	2.31134e10i	1.12023	+	1.12023i	0.128523	+	0.991707i	$0.458976\pi$
$380$	−2.79439e8			−0.0134015
$381$		0			0
$382$	−1.23157e10	−	1.23157e10i	−0.578370	−	0.578370i
$383$	2.23472e10	+	2.23472e10i	1.03855	+	1.03855i	0.999227	+	0.0393232i	$0.0125202\pi$
$383$	2.23472e10	+	2.23472e10i	1.03855	+	1.03855i	0.0393232	+	0.999227i	$0.487480\pi$
$384$		0			0
$385$	−1.79043e9	+	1.79043e9i	−0.0814917	+	0.0814917i
$386$	1.57021e10			0.707307
$387$		0			0
$388$	−8.51843e8	+	8.51843e8i	−0.0375866	+	0.0375866i
$389$		−	1.17421e10i		−	0.512801i	−0.966571	−	0.256401i	$-0.917463\pi$
$389$		−	1.17421e10i		−	0.512801i	0.966571	−	0.256401i	$-0.0825366\pi$
$390$		0			0
$391$	−1.95655e9			−0.0837111
$392$	3.57418e9	+	3.57418e9i	0.151367	+	0.151367i
$393$		0			0
$394$			2.59095e10i			1.07516i
$395$	1.29841e9	+	1.29841e9i	0.0533366	+	0.0533366i
$396$		0			0
$397$	−1.81552e10	+	1.81552e10i	−0.730868	+	0.730868i	−0.970792	−	0.239924i	$-0.922878\pi$
$397$	−1.81552e10	+	1.81552e10i	−0.730868	+	0.730868i	0.239924	+	0.970792i	$0.422878\pi$
$398$	8.74812e9	−	8.74812e9i	0.348644	−	0.348644i
$399$		0			0
$400$		−	1.38071e10i		−	0.539342i
$401$	1.78247e10	−	1.78247e10i	0.689358	−	0.689358i	−0.272732	−	0.962090i	$-0.587927\pi$
$401$	1.78247e10	−	1.78247e10i	0.689358	−	0.689358i	0.962090	+	0.272732i	$0.0879272\pi$
$402$		0			0
$403$	−5.89897e8	+	5.18843e8i	−0.0223643	+	0.0196705i
$404$	4.07229e9			0.152867
$405$		0			0
$406$	−1.62861e10			−0.599396
$407$		−	1.14934e10i		−	0.418862i
$408$		0			0
$409$	−2.43214e10	−	2.43214e10i	−0.869153	−	0.869153i	0.123226	−	0.992379i	$-0.460676\pi$
$409$	−2.43214e10	−	2.43214e10i	−0.869153	−	0.869153i	−0.992379	+	0.123226i	$0.960676\pi$
$410$	9.81082e8	−	9.81082e8i	0.0347192	−	0.0347192i
$411$		0			0
$412$	−1.14274e10			−0.396604
$413$		−	5.16328e9i		−	0.177470i
$414$		0			0
$415$			1.51910e10i			0.512145i
$416$	−1.12710e10	+	9.91336e9i	−0.376346	+	0.331015i
$417$		0			0
$418$	−6.15831e8	−	6.15831e8i	−0.0201724	−	0.0201724i
$419$	−2.20449e10			−0.715241			−0.357620	−	0.933867i	$-0.616412\pi$
$419$	−2.20449e10			−0.715241			−0.357620	+	0.933867i	$0.616412\pi$
$420$		0			0
$421$	4.70295e9	+	4.70295e9i	0.149707	+	0.149707i	0.777987	−	0.628280i	$-0.216241\pi$
$421$	4.70295e9	+	4.70295e9i	0.149707	+	0.149707i	−0.628280	+	0.777987i	$0.716241\pi$
$422$	−2.51531e10	−	2.51531e10i	−0.793125	−	0.793125i
$423$		0			0
$424$	3.46923e10	−	3.46923e10i	1.07342	−	1.07342i
$425$	1.00860e10			0.309146
$426$		0			0
$427$	−1.69180e10	+	1.69180e10i	−0.508906	+	0.508906i
$428$			9.53833e9i			0.284248i
$429$		0			0
$430$	1.97240e10			0.576929
$431$	3.01796e10	+	3.01796e10i	0.874590	+	0.874590i	0.992969	−	0.118379i	$-0.0377697\pi$
$431$	3.01796e10	+	3.01796e10i	0.874590	+	0.874590i	−0.118379	+	0.992969i	$0.537770\pi$
$432$		0			0
$433$			2.98973e10i			0.850510i	0.905073	+	0.425255i	$0.139816\pi$
$433$			2.98973e10i			0.850510i	−0.905073	+	0.425255i	$0.860184\pi$
$434$	5.76682e8	+	5.76682e8i	0.0162547	+	0.0162547i
$435$		0			0
$436$	−1.06215e10	+	1.06215e10i	−0.293927	+	0.293927i
$437$	6.44625e8	−	6.44625e8i	0.0176759	−	0.0176759i
$438$		0			0
$439$		−	5.11771e10i		−	1.37790i	−0.724809	−	0.688950i	$-0.758072\pi$
$439$		−	5.11771e10i		−	1.37790i	0.724809	−	0.688950i	$-0.241928\pi$
$440$	3.69471e9	−	3.69471e9i	0.0985758	−	0.0985758i
$441$		0			0
$442$	8.41454e9	+	9.56688e9i	0.220466	+	0.250658i
$443$	−4.25669e10			−1.10524			−0.552621	−	0.833433i	$-0.686372\pi$
$443$	−4.25669e10			−1.10524			−0.552621	+	0.833433i	$0.686372\pi$
$444$		0			0
$445$	1.35129e10			0.344595
$446$			4.84593e9i			0.122472i
$447$		0			0
$448$	2.82655e10	+	2.82655e10i	0.701690	+	0.701690i
$449$	1.54120e10	−	1.54120e10i	0.379204	−	0.379204i	−0.491611	−	0.870815i	$-0.663592\pi$
$449$	1.54120e10	−	1.54120e10i	0.379204	−	0.379204i	0.870815	+	0.491611i	$0.163592\pi$
$450$		0			0
$451$	−1.50095e9			−0.0362794
$452$		−	2.03490e10i		−	0.487516i
$453$		0			0
$454$			6.03915e10i			1.42152i
$455$	−1.10380e9	+	1.72241e10i	−0.0257541	+	0.401875i
$456$		0			0
$457$	2.33572e10	+	2.33572e10i	0.535495	+	0.535495i	0.922203	−	0.386707i	$-0.126388\pi$
$457$	2.33572e10	+	2.33572e10i	0.535495	+	0.535495i	−0.386707	+	0.922203i	$0.626388\pi$
$458$	6.81507e10			1.54885
$459$		0			0
$460$	7.92351e8	+	7.92351e8i	0.0176965	+	0.0176965i
$461$	5.81609e10	+	5.81609e10i	1.28774	+	1.28774i	0.936158	+	0.351579i	$0.114355\pi$
$461$	5.81609e10	+	5.81609e10i	1.28774	+	1.28774i	0.351579	+	0.936158i	$0.385645\pi$
$462$		0			0
$463$	2.79753e10	−	2.79753e10i	0.608766	−	0.608766i	−0.333857	−	0.942624i	$-0.608350\pi$
$463$	2.79753e10	−	2.79753e10i	0.608766	−	0.608766i	0.942624	+	0.333857i	$0.108350\pi$
$464$	2.43326e10			0.524948
$465$		0			0
$466$	2.84059e10	−	2.84059e10i	0.602373	−	0.602373i
$467$		−	4.29922e10i		−	0.903903i	−0.892042	−	0.451952i	$-0.850728\pi$
$467$		−	4.29922e10i		−	0.903903i	0.892042	−	0.451952i	$-0.149272\pi$
$468$		0			0
$469$	−4.89380e10			−1.01147
$470$	2.33478e10	+	2.33478e10i	0.478471	+	0.478471i
$471$		0			0
$472$			1.06549e10i			0.214675i
$473$	−1.50879e10	−	1.50879e10i	−0.301427	−	0.301427i
$474$		0			0
$475$	−3.32305e9	+	3.32305e9i	−0.0652774	+	0.0652774i
$476$	−3.24630e9	+	3.24630e9i	−0.0632354	+	0.0632354i
$477$		0			0
$478$			4.33274e10i			0.829949i
$479$	−1.00892e10	+	1.00892e10i	−0.191653	+	0.191653i	−0.796410	−	0.604757i	$-0.793270\pi$
$479$	−1.00892e10	+	1.00892e10i	−0.191653	+	0.191653i	0.604757	+	0.796410i	$0.293270\pi$
$480$		0			0
$481$	−5.17411e10	−	5.88268e10i	−0.966618	−	1.09899i
$482$	2.32310e10			0.430408
$483$		0			0
$484$	1.29816e10			0.236562
$485$		−	5.13133e9i		−	0.0927392i
$486$		0			0
$487$	−4.63051e10	−	4.63051e10i	−0.823214	−	0.823214i	0.163353	−	0.986568i	$-0.447769\pi$
$487$	−4.63051e10	−	4.63051e10i	−0.823214	−	0.823214i	−0.986568	+	0.163353i	$0.947769\pi$
$488$	3.49119e10	−	3.49119e10i	0.615594	−	0.615594i
$489$		0			0
$490$	−4.41101e9			−0.0765163
$491$		−	3.75763e10i		−	0.646530i	−0.946309	−	0.323265i	$-0.895220\pi$
$491$		−	3.75763e10i		−	0.646530i	0.946309	−	0.323265i	$-0.104780\pi$
$492$		0			0
$493$			1.77748e10i			0.300896i
$494$	−5.92436e9	−	3.79662e8i	−0.0994795	−	0.00637514i
$495$		0			0
$496$	−8.61601e8	−	8.61601e8i	−0.0142357	−	0.0142357i
$497$	6.22459e10			1.02020
$498$		0			0
$499$	6.17259e10	+	6.17259e10i	0.995555	+	0.995555i	0.999990	−	0.00443528i	$-0.00141180\pi$
$499$	6.17259e10	+	6.17259e10i	0.995555	+	0.995555i	−0.00443528	+	0.999990i	$0.501412\pi$
$500$	−9.20369e9	−	9.20369e9i	−0.147259	−	0.147259i
$501$		0			0
$502$	6.37228e10	−	6.37228e10i	1.00341	−	1.00341i
$503$	−7.87518e10			−1.23024			−0.615118	−	0.788435i	$-0.710892\pi$
$503$	−7.87518e10			−1.23024			−0.615118	+	0.788435i	$0.710892\pi$
$504$		0			0
$505$	−1.22653e10	+	1.22653e10i	−0.188588	+	0.188588i
$506$			3.49239e9i			0.0532746i
$507$		0			0
$508$	−1.83215e10			−0.275109
$509$	4.37746e9	+	4.37746e9i	0.0652155	+	0.0652155i	0.738962	−	0.673747i	$-0.235316\pi$
$509$	4.37746e9	+	4.37746e9i	0.0652155	+	0.0652155i	−0.673747	+	0.738962i	$0.735316\pi$
$510$		0			0
$511$		−	8.56537e10i		−	1.25621i
$512$	−5.20533e10	−	5.20533e10i	−0.757475	−	0.757475i
$513$		0			0
$514$	−2.58656e10	+	2.58656e10i	−0.370570	+	0.370570i
$515$	3.44181e10	−	3.44181e10i	0.489280	−	0.489280i
$516$		0			0
$517$		−	3.57197e10i		−	0.499972i
$518$	−5.75090e10	+	5.75090e10i	−0.798761	+	0.798761i
$519$		0			0
$520$	2.27780e9	−	3.55435e10i	0.0311532	−	0.486124i
$521$	−5.45813e10			−0.740786			−0.370393	−	0.928875i	$-0.620777\pi$
$521$	−5.45813e10			−0.740786			−0.370393	+	0.928875i	$0.620777\pi$
$522$		0			0
$523$	2.48896e10			0.332668			0.166334	−	0.986069i	$-0.446807\pi$
$523$	2.48896e10			0.332668			0.166334	+	0.986069i	$0.446807\pi$
$524$			3.28632e10i			0.435898i
$525$		0			0
$526$	5.65516e10	+	5.65516e10i	0.738757	+	0.738757i
$527$	6.29393e8	−	6.29393e8i	0.00815980	−	0.00815980i
$528$		0			0
$529$	7.46553e10			0.953318
$530$			4.28149e10i			0.542615i
$531$		0			0
$532$		−	2.13912e9i		−	0.0267048i
$533$	−7.68233e9	+	6.75699e9i	−0.0951884	+	0.0837229i
$534$		0			0
$535$	−2.87285e10	−	2.87285e10i	−0.350670	−	0.350670i
$536$	1.00988e11			1.22352
$537$		0			0
$538$	8.37192e9	+	8.37192e9i	0.0999301	+	0.0999301i
$539$	3.37419e9	+	3.37419e9i	0.0399774	+	0.0399774i
$540$		0			0
$541$	−4.97594e9	+	4.97594e9i	−0.0580880	+	0.0580880i	−0.735554	−	0.677466i	$-0.763078\pi$
$541$	−4.97594e9	+	4.97594e9i	−0.0580880	+	0.0580880i	0.677466	+	0.735554i	$0.263078\pi$
$542$	−4.20570e10			−0.487350
$543$		0			0
$544$	1.20256e10	−	1.20256e10i	0.137313	−	0.137313i
$545$		−	6.39817e10i		−	0.725220i
$546$		0			0
$547$	−4.15465e10			−0.464072			−0.232036	−	0.972707i	$-0.574539\pi$
$547$	−4.15465e10			−0.464072			−0.232036	+	0.972707i	$0.574539\pi$
$548$	1.23743e10	+	1.23743e10i	0.137214	+	0.137214i
$549$		0			0
$550$		−	1.80033e10i		−	0.196744i
$551$	−5.85627e9	−	5.85627e9i	−0.0635353	−	0.0635353i
$552$		0			0
$553$	−9.93944e9	+	9.93944e9i	−0.106282	+	0.106282i
$554$	−6.58276e9	+	6.58276e9i	−0.0698825	+	0.0698825i
$555$		0			0
$556$		−	8.55985e9i		−	0.0895709i
$557$	1.92737e10	−	1.92737e10i	0.200237	−	0.200237i	−0.599865	−	0.800102i	$-0.704779\pi$
$557$	1.92737e10	−	1.92737e10i	0.200237	−	0.200237i	0.800102	+	0.599865i	$0.204779\pi$
$558$		0			0
$559$	−1.45147e11	−	9.30171e9i	−1.48648	−	0.0952611i
$560$	−2.67697e10			−0.272202
$561$		0			0
$562$	−2.17547e10			−0.218076
$563$		−	2.18965e10i		−	0.217942i	−0.994045	−	0.108971i	$-0.965244\pi$
$563$		−	2.18965e10i		−	0.217942i	0.994045	−	0.108971i	$-0.0347555\pi$
$564$		0			0
$565$	6.12892e10	+	6.12892e10i	0.601437	+	0.601437i
$566$	2.19718e10	−	2.19718e10i	0.214092	−	0.214092i
$567$		0			0
$568$	−1.28450e11			−1.23408
$569$		−	2.89805e10i		−	0.276475i	−0.990399	−	0.138238i	$-0.955856\pi$
$569$		−	2.89805e10i		−	0.276475i	0.990399	−	0.138238i	$-0.0441438\pi$
$570$		0			0
$571$			8.79447e10i			0.827304i	0.910435	+	0.413652i	$0.135747\pi$
$571$			8.79447e10i			0.827304i	−0.910435	+	0.413652i	$0.864253\pi$
$572$	−5.92734e9	+	5.21339e9i	−0.0553701	+	0.0487008i
$573$		0			0
$574$	7.51024e9	+	7.51024e9i	0.0691841	+	0.0691841i
$575$	1.88451e10			0.172396
$576$		0			0
$577$	−3.21868e10	−	3.21868e10i	−0.290385	−	0.290385i	0.546847	−	0.837232i	$-0.315828\pi$
$577$	−3.21868e10	−	3.21868e10i	−0.290385	−	0.290385i	−0.837232	+	0.546847i	$0.815828\pi$
$578$	5.77906e10	+	5.77906e10i	0.517780	+	0.517780i
$579$		0			0
$580$	7.19833e9	−	7.19833e9i	0.0636092	−	0.0636092i
$581$	−1.16288e11			−1.02054
$582$		0			0
$583$	3.27511e10	−	3.27511e10i	0.283499	−	0.283499i
$584$			1.76754e11i			1.51956i
$585$		0			0
$586$	−1.28809e11			−1.09234
$587$	−1.17900e11	−	1.17900e11i	−0.993028	−	0.993028i	0.00694814	−	0.999976i	$-0.497788\pi$
$587$	−1.17900e11	−	1.17900e11i	−0.993028	−	0.993028i	−0.999976	+	0.00694814i	$0.997788\pi$
$588$		0			0
$589$			4.14734e8i			0.00344595i
$590$	−6.57480e9	−	6.57480e9i	−0.0542593	−	0.0542593i
$591$		0			0
$592$	8.59222e10	−	8.59222e10i	0.699549	−	0.699549i
$593$	−6.24190e10	+	6.24190e10i	−0.504775	+	0.504775i	−0.912918	−	0.408143i	$-0.866177\pi$
$593$	−6.24190e10	+	6.24190e10i	−0.504775	+	0.504775i	0.408143	+	0.912918i	$0.366177\pi$
$594$		0			0
$595$		−	1.95550e10i		−	0.156024i
$596$	−1.46485e10	+	1.46485e10i	−0.116094	+	0.116094i
$597$		0			0
$598$	1.57220e10	+	1.78751e10i	0.122943	+	0.139780i
$599$	−6.50863e10			−0.505572			−0.252786	−	0.967522i	$-0.581347\pi$
$599$	−6.50863e10			−0.505572			−0.252786	+	0.967522i	$0.581347\pi$
$600$		0			0
$601$	2.21879e11			1.70067			0.850333	−	0.526245i	$-0.176401\pi$
$601$	2.21879e11			1.70067			0.850333	+	0.526245i	$0.176401\pi$
$602$			1.50989e11i			1.14963i
$603$		0			0
$604$	2.35890e10	+	2.35890e10i	0.177240	+	0.177240i
$605$	−3.90992e10	+	3.90992e10i	−0.291841	+	0.291841i
$606$		0			0
$607$	−1.43440e11			−1.05662			−0.528308	−	0.849053i	$-0.677173\pi$
$607$	−1.43440e11			−1.05662			−0.528308	+	0.849053i	$0.677173\pi$
$608$			7.92418e9i			0.0579882i
$609$		0			0
$610$			4.30860e10i			0.311183i
$611$	−1.60803e11	−	1.82824e11i	−1.15380	−	1.31181i
$612$		0			0
$613$	−1.14293e10	−	1.14293e10i	−0.0809424	−	0.0809424i	0.665476	−	0.746419i	$-0.268228\pi$
$613$	−1.14293e10	−	1.14293e10i	−0.0809424	−	0.0809424i	−0.746419	+	0.665476i	$0.768228\pi$
$614$	−1.72493e11			−1.21366
$615$		0			0
$616$	2.82832e10	+	2.82832e10i	0.196429	+	0.196429i
$617$	5.19557e10	+	5.19557e10i	0.358503	+	0.358503i	0.863261	−	0.504758i	$-0.168418\pi$
$617$	5.19557e10	+	5.19557e10i	0.358503	+	0.358503i	−0.504758	+	0.863261i	$0.668418\pi$
$618$		0			0
$619$	−2.55973e10	+	2.55973e10i	−0.174354	+	0.174354i	−0.788889	−	0.614535i	$-0.789344\pi$
$619$	−2.55973e10	+	2.55973e10i	−0.174354	+	0.174354i	0.614535	+	0.788889i	$0.289344\pi$
$620$	−5.09777e8			−0.00344995
$621$		0			0
$622$	−1.29646e11	+	1.29646e11i	−0.866159	+	0.866159i
$623$			1.03442e11i			0.686666i
$624$		0			0
$625$	−6.63102e10			−0.434571
$626$	−2.70549e10	−	2.70549e10i	−0.176177	−	0.176177i
$627$		0			0
$628$			5.74417e10i			0.369308i
$629$	6.27655e10	+	6.27655e10i	0.400976	+	0.400976i
$630$		0			0
$631$	−8.19590e10	+	8.19590e10i	−0.516986	+	0.516986i	−0.916658	−	0.399672i	$-0.869124\pi$
$631$	−8.19590e10	+	8.19590e10i	−0.516986	+	0.516986i	0.399672	+	0.916658i	$0.369124\pi$
$632$	2.05110e10	−	2.05110e10i	0.128564	−	0.128564i
$633$		0			0
$634$			3.26958e10i			0.202365i
$635$	5.51825e10	−	5.51825e10i	0.339396	−	0.339396i
$636$		0			0
$637$	3.24601e10	+	2.08020e9i	0.197148	+	0.0126342i
$638$	3.17275e10			0.191493
$639$		0			0
$640$	3.41839e10			0.203752
$641$		−	9.79403e10i		−	0.580135i	−0.957006	−	0.290068i	$-0.906322\pi$
$641$		−	9.79403e10i		−	0.580135i	0.957006	−	0.290068i	$-0.0936778\pi$
$642$		0			0
$643$	1.79267e10	+	1.79267e10i	0.104871	+	0.104871i	0.757596	−	0.652724i	$-0.226374\pi$
$643$	1.79267e10	+	1.79267e10i	0.104871	+	0.104871i	−0.652724	+	0.757596i	$0.726374\pi$
$644$	−6.06549e9	+	6.06549e9i	−0.0352633	+	0.0352633i
$645$		0			0
$646$	6.72611e9			0.0386219
$647$			2.17508e11i			1.24125i	0.784108	+	0.620624i	$0.213121\pi$
$647$			2.17508e11i			1.24125i	−0.784108	+	0.620624i	$0.786879\pi$
$648$		0			0
$649$			1.00588e10i			0.0566977i
$650$	−8.10473e10	−	9.21464e10i	−0.454031	−	0.516208i
$651$		0			0
$652$	4.21380e9	+	4.21380e9i	0.0233176	+	0.0233176i
$653$	5.11592e10			0.281366			0.140683	−	0.990055i	$-0.455070\pi$
$653$	5.11592e10			0.281366			0.140683	+	0.990055i	$0.455070\pi$
$654$		0			0
$655$	−9.89808e10	−	9.89808e10i	−0.537757	−	0.537757i
$656$	−1.12208e10	−	1.12208e10i	−0.0605910	−	0.0605910i
$657$		0			0
$658$	−1.78729e11	+	1.78729e11i	−0.953436	+	0.953436i
$659$	3.05213e11			1.61831			0.809153	−	0.587598i	$-0.199926\pi$
$659$	3.05213e11			1.61831			0.809153	+	0.587598i	$0.199926\pi$
$660$		0			0
$661$	1.86671e10	−	1.86671e10i	0.0977848	−	0.0977848i	−0.656522	−	0.754307i	$-0.727973\pi$
$661$	1.86671e10	−	1.86671e10i	0.0977848	−	0.0977848i	0.754307	+	0.656522i	$0.227973\pi$
$662$			1.14282e11i			0.595039i
$663$		0			0
$664$	2.39970e11			1.23448
$665$	6.44282e9	+	6.44282e9i	0.0329450	+	0.0329450i
$666$		0			0
$667$			3.32110e10i			0.167795i
$668$	5.03620e10	+	5.03620e10i	0.252928	+	0.252928i
$669$		0			0
$670$	−6.23164e10	+	6.23164e10i	−0.309245	+	0.309245i
$671$	3.29585e10	−	3.29585e10i	0.162584	−	0.162584i
$672$		0			0
$673$		−	2.79274e11i		−	1.36135i	−0.732585	−	0.680675i	$-0.761687\pi$
$673$		−	2.79274e11i		−	1.36135i	0.732585	−	0.680675i	$-0.238313\pi$
$674$	−7.74692e10	+	7.74692e10i	−0.375396	+	0.375396i
$675$		0			0
$676$	−6.86830e9	+	5.33674e10i	−0.0328899	+	0.255558i
$677$	−1.55893e11			−0.742118			−0.371059	−	0.928609i	$-0.621005\pi$
$677$	−1.55893e11			−0.742118			−0.371059	+	0.928609i	$0.621005\pi$
$678$		0			0
$679$	3.92806e10			0.184799
$680$			4.03537e10i			0.188733i
$681$		0			0
$682$	−1.12345e9	−	1.12345e9i	−0.00519299	−	0.00519299i
$683$	2.79459e10	−	2.79459e10i	0.128421	−	0.128421i	−0.639975	−	0.768396i	$-0.721055\pi$
$683$	2.79459e10	−	2.79459e10i	0.128421	−	0.128421i	0.768396	+	0.639975i	$0.221055\pi$
$684$		0			0
$685$	−7.45401e10			−0.338554
$686$		−	2.04691e11i		−	0.924278i
$687$		0			0
$688$		−	2.25587e11i		−	1.00684i
$689$	2.01912e10	−	3.15069e11i	0.0895953	−	1.39807i
$690$		0			0
$691$	2.61253e11	+	2.61253e11i	1.14591	+	1.14591i	0.987350	+	0.158557i	$0.0506840\pi$
$691$	2.61253e11	+	2.61253e11i	1.14591	+	1.14591i	0.158557	+	0.987350i	$0.449316\pi$
$692$	2.03490e10			0.0887399
$693$		0			0
$694$	−1.20749e11	−	1.20749e11i	−0.520531	−	0.520531i
$695$	2.57814e10	+	2.57814e10i	0.110501	+	0.110501i
$696$		0			0
$697$	8.19670e9	−	8.19670e9i	0.0347303	−	0.0347303i
$698$	−5.42228e9			−0.0228434
$699$		0			0
$700$	3.12677e10	−	3.12677e10i	0.130228	−	0.130228i
$701$			1.43737e11i			0.595245i	0.954684	+	0.297623i	$0.0961937\pi$
$701$			1.43737e11i			0.595245i	−0.954684	+	0.297623i	$0.903806\pi$
$702$		0			0
$703$	−4.13589e10			−0.169335
$704$	−5.50650e10	−	5.50650e10i	−0.224174	−	0.224174i
$705$		0			0
$706$		−	3.10564e10i		−	0.125006i
$707$	−9.38919e10	−	9.38919e10i	−0.375794	−	0.375794i
$708$		0			0
$709$	2.16872e11	−	2.16872e11i	0.858260	−	0.858260i	−0.132873	−	0.991133i	$-0.542420\pi$
$709$	2.16872e11	−	2.16872e11i	0.858260	−	0.858260i	0.991133	+	0.132873i	$0.0424201\pi$
$710$	7.92624e10	−	7.92624e10i	0.311913	−	0.311913i
$711$		0			0
$712$		−	2.13463e11i		−	0.830619i
$713$	1.17598e9	−	1.17598e9i	0.00455032	−	0.00455032i
$714$		0			0
$715$	2.15035e9	−	3.35548e10i	0.00822784	−	0.128390i
$716$	2.61142e10			0.0993628
$717$		0			0
$718$	2.33599e11			0.878968
$719$			7.13407e10i			0.266945i	0.991053	+	0.133472i	$0.0426128\pi$
$719$			7.13407e10i			0.266945i	−0.991053	+	0.133472i	$0.957387\pi$
$720$		0			0
$721$	2.63472e11	+	2.63472e11i	0.974975	+	0.974975i
$722$	1.63336e11	−	1.63336e11i	0.601080	−	0.601080i
$723$		0			0
$724$	9.12570e10			0.332133
$725$		−	1.71203e11i		−	0.619669i
$726$		0			0
$727$		−	3.07369e11i		−	1.10033i	−0.835056	−	0.550164i	$-0.814565\pi$
$727$		−	3.07369e11i		−	1.10033i	0.835056	−	0.550164i	$-0.185435\pi$
$728$	2.72088e11	+	1.74367e10i	0.968687	+	0.0620782i
$729$		0			0
$730$	−1.09069e11	−	1.09069e11i	−0.384071	−	0.384071i
$731$	1.64790e11			0.577112
$732$		0			0
$733$	1.09412e11	+	1.09412e11i	0.379009	+	0.379009i	0.870745	−	0.491736i	$-0.163637\pi$
$733$	1.09412e11	+	1.09412e11i	0.379009	+	0.379009i	−0.491736	+	0.870745i	$0.663637\pi$
$734$	3.38000e11	+	3.38000e11i	1.16448	+	1.16448i
$735$		0			0
$736$	2.24691e10	−	2.24691e10i	0.0765726	−	0.0765726i
$737$	9.53375e10			0.323142
$738$		0			0
$739$	−5.54700e10	+	5.54700e10i	−0.185986	+	0.185986i	−0.793958	−	0.607972i	$-0.791983\pi$
$739$	−5.54700e10	+	5.54700e10i	−0.185986	+	0.185986i	0.607972	+	0.793958i	$0.291983\pi$
$740$		−	5.08369e10i		−	0.169532i
$741$		0			0
$742$	−3.27750e11			−1.08125
$743$	−1.03488e11	−	1.03488e11i	−0.339573	−	0.339573i	0.516634	−	0.856207i	$-0.327185\pi$
$743$	−1.03488e11	−	1.03488e11i	−0.339573	−	0.339573i	−0.856207	+	0.516634i	$0.827185\pi$
$744$		0			0
$745$		−	8.82397e10i		−	0.286443i
$746$	1.20034e10	+	1.20034e10i	0.0387569	+	0.0387569i
$747$		0			0
$748$	6.32421e9	−	6.32421e9i	0.0202022	−	0.0202022i
$749$	2.19918e11	−	2.19918e11i	0.698770	−	0.698770i
$750$		0			0
$751$			1.33830e11i			0.420721i	0.977624	+	0.210360i	$0.0674637\pi$
$751$			1.33830e11i			0.420721i	−0.977624	+	0.210360i	$0.932536\pi$
$752$	2.67033e11	−	2.67033e11i	0.835013	−	0.835013i
$753$		0			0
$754$	1.62391e11	−	1.42831e11i	0.502432	−	0.441913i
$755$	−1.42095e11			−0.437313
$756$		0			0
$757$	−2.85512e11			−0.869442			−0.434721	−	0.900565i	$-0.643153\pi$
$757$	−2.85512e11			−0.869442			−0.434721	+	0.900565i	$0.643153\pi$
$758$			4.50608e11i			1.36497i
$759$		0			0
$760$	−1.32954e10	−	1.32954e10i	−0.0398516	−	0.0398516i
$761$	4.60437e11	−	4.60437e11i	1.37288	−	1.37288i	0.516728	−	0.856149i	$-0.327150\pi$
$761$	4.60437e11	−	4.60437e11i	1.37288	−	1.37288i	0.856149	−	0.516728i	$-0.172850\pi$
$762$		0			0
$763$	4.89783e11			1.44513
$764$			8.33395e10i			0.244612i
$765$		0			0
$766$			4.35669e11i			1.26544i
$767$	4.52825e10	+	5.14837e10i	0.130843	+	0.148761i
$768$		0			0
$769$	−1.98615e11	−	1.98615e11i	−0.567946	−	0.567946i	0.363607	−	0.931552i	$-0.381545\pi$
$769$	−1.98615e11	−	1.98615e11i	−0.567946	−	0.567946i	−0.931552	+	0.363607i	$0.881545\pi$
$770$	−3.49053e10			−0.0992952
$771$		0			0
$772$	−5.31274e10	−	5.31274e10i	−0.149572	−	0.149572i
$773$	−1.36661e11	−	1.36661e11i	−0.382759	−	0.382759i	0.489336	−	0.872095i	$-0.337239\pi$
$773$	−1.36661e11	−	1.36661e11i	−0.382759	−	0.382759i	−0.872095	+	0.489336i	$0.837239\pi$
$774$		0			0
$775$	−6.06220e9	+	6.06220e9i	−0.0168044	+	0.0168044i
$776$	−8.10593e10			−0.223540
$777$		0			0
$778$	1.14460e11	−	1.14460e11i	0.312416	−	0.312416i
$779$			5.40116e9i			0.0146669i
$780$		0			0
$781$	−1.21263e11			−0.325930
$782$	−1.90719e10	−	1.90719e10i	−0.0509997	−	0.0509997i
$783$		0			0
$784$			5.04494e10i			0.133534i
$785$	−1.73009e11	−	1.73009e11i	−0.455606	−	0.455606i
$786$		0			0
$787$	1.21616e11	−	1.21616e11i	0.317023	−	0.317023i	−0.530600	−	0.847623i	$-0.678033\pi$
$787$	1.21616e11	−	1.21616e11i	0.317023	−	0.317023i	0.847623	+	0.530600i	$0.178033\pi$
$788$	8.76637e10	−	8.76637e10i	0.227361	−	0.227361i
$789$		0			0
$790$			2.53133e10i			0.0649890i
$791$	−4.69172e11	+	4.69172e11i	−1.19847	+	1.19847i
$792$		0			0
$793$	2.03190e10	−	3.17064e11i	0.0513819	−	0.801778i
$794$	−3.53945e11			−0.890541
$795$		0			0
$796$	−5.91979e10			−0.147453
$797$			3.79315e11i			0.940085i	0.882644	+	0.470043i	$0.155761\pi$
$797$			3.79315e11i			0.940085i	−0.882644	+	0.470043i	$0.844239\pi$
$798$		0			0
$799$	1.95066e11	+	1.95066e11i	0.478623	+	0.478623i
$800$	−1.15828e11	+	1.15828e11i	−0.282784	+	0.282784i
$801$		0			0
$802$	3.47502e11			0.839962
$803$			1.66864e11i			0.401330i
$804$		0			0
$805$		−	3.65373e10i		−	0.0870068i
$806$	−1.08077e10	−	6.92612e8i	−0.0256091	−	0.00164116i
$807$		0			0
$808$	1.93755e11	+	1.93755e11i	0.454577	+	0.454577i
$809$	6.12456e11			1.42982			0.714909	−	0.699217i	$-0.246468\pi$
$809$	6.12456e11			1.42982			0.714909	+	0.699217i	$0.246468\pi$
$810$		0			0
$811$	−2.94473e11	−	2.94473e11i	−0.680710	−	0.680710i	0.279450	−	0.960160i	$-0.409848\pi$
$811$	−2.94473e11	−	2.94473e11i	−0.680710	−	0.680710i	−0.960160	+	0.279450i	$0.909848\pi$
$812$	5.51036e10	+	5.51036e10i	0.126752	+	0.126752i
$813$		0			0
$814$	1.12035e11	−	1.12035e11i	0.255186	−	0.255186i
$815$	−2.53831e10			−0.0575327
$816$		0			0
$817$	−5.42934e10	+	5.42934e10i	−0.121859	+	0.121859i
$818$		−	4.74159e11i		−	1.05904i
$819$		0			0
$820$	−6.63891e9			−0.0146839
$821$	3.09941e11	+	3.09941e11i	0.682193	+	0.682193i	0.960494	−	0.278301i	$-0.0897714\pi$
$821$	3.09941e11	+	3.09941e11i	0.682193	+	0.682193i	−0.278301	+	0.960494i	$0.589771\pi$
$822$		0			0
$823$			7.50104e11i			1.63502i	0.575918	+	0.817508i	$0.304645\pi$
$823$			7.50104e11i			1.63502i	−0.575918	+	0.817508i	$0.695355\pi$
$824$	−5.43700e11	−	5.43700e11i	−1.17937	−	1.17937i
$825$		0			0
$826$	5.03304e10	−	5.03304e10i	0.108121	−	0.108121i
$827$	−4.09857e11	+	4.09857e11i	−0.876214	+	0.876214i	−0.993141	−	0.116926i	$-0.962696\pi$
$827$	−4.09857e11	+	4.09857e11i	−0.876214	+	0.876214i	0.116926	+	0.993141i	$0.462696\pi$
$828$		0			0
$829$			8.73555e11i			1.84958i	0.380483	+	0.924788i	$0.375758\pi$
$829$			8.73555e11i			1.84958i	−0.380483	+	0.924788i	$0.624242\pi$
$830$	−1.48078e11	+	1.48078e11i	−0.312016	+	0.312016i
$831$		0			0
$832$	−5.29731e11	−	3.39477e10i	−1.10551	−	0.0708463i
$833$	−3.68529e10			−0.0765406
$834$		0			0
$835$	−3.03371e11			−0.624062
$836$			4.16729e9i			0.00853156i
$837$		0			0
$838$	−2.14889e11	−	2.14889e11i	−0.435750	−	0.435750i
$839$	3.17828e11	−	3.17828e11i	0.641424	−	0.641424i	−0.309482	−	0.950905i	$-0.600156\pi$
$839$	3.17828e11	−	3.17828e11i	0.641424	−	0.641424i	0.950905	+	0.309482i	$0.100156\pi$
$840$		0			0
$841$	−1.98532e11			−0.396869
$842$			9.16864e10i			0.182413i
$843$		0			0
$844$			1.70209e11i			0.335439i
$845$	−1.40051e11	−	1.81424e11i	−0.274700	−	0.355851i
$846$		0			0
$847$	−2.99306e11	−	2.99306e11i	−0.581544	−	0.581544i
$848$	4.89680e11			0.946955
$849$		0			0
$850$	9.83161e10	+	9.83161e10i	0.188343	+	0.188343i
$851$	1.17273e11	+	1.17273e11i	0.223605	+	0.223605i
$852$		0			0
$853$	4.21575e11	−	4.21575e11i	0.796304	−	0.796304i	−0.186207	−	0.982511i	$-0.559619\pi$
$853$	4.21575e11	−	4.21575e11i	0.796304	−	0.796304i	0.982511	+	0.186207i	$0.0596195\pi$
$854$	−3.29825e11			−0.620087
$855$		0			0
$856$	−4.53822e11	+	4.53822e11i	−0.845261	+	0.845261i
$857$			6.32523e10i			0.117261i	0.998280	+	0.0586305i	$0.0186734\pi$
$857$			6.32523e10i			0.117261i	−0.998280	+	0.0586305i	$0.981327\pi$
$858$		0			0
$859$	−3.83127e11			−0.703671			−0.351836	−	0.936062i	$-0.614442\pi$
$859$	−3.83127e11			−0.703671			−0.351836	+	0.936062i	$0.614442\pi$
$860$	−6.67356e10	−	6.67356e10i	−0.122001	−	0.122001i
$861$		0			0
$862$			5.88367e11i			1.06566i
$863$	−1.74780e11	−	1.74780e11i	−0.315100	−	0.315100i	0.531781	−	0.846882i	$-0.321523\pi$
$863$	−1.74780e11	−	1.74780e11i	−0.315100	−	0.315100i	−0.846882	+	0.531781i	$0.821523\pi$
$864$		0			0
$865$	−6.12892e10	+	6.12892e10i	−0.109476	+	0.109476i
$866$	−2.91431e11	+	2.91431e11i	−0.518161	+	0.518161i
$867$		0			0
$868$		−	3.90237e9i		−	0.00687463i
$869$	1.93633e10	−	1.93633e10i	0.0339548	−	0.0339548i
$870$		0			0
$871$	4.87967e11	−	4.29191e11i	0.847847	−	0.745723i
$872$	−1.01071e12			−1.74808
$873$		0			0
$874$	1.25673e10			0.0215376
$875$			4.24405e11i			0.724017i
$876$		0			0
$877$	−4.24104e11	−	4.24104e11i	−0.716926	−	0.716926i	0.251049	−	0.967974i	$-0.419225\pi$
$877$	−4.24104e11	−	4.24104e11i	−0.716926	−	0.716926i	−0.967974	+	0.251049i	$0.919225\pi$
$878$	4.98862e11	−	4.98862e11i	0.839465	−	0.839465i
$879$		0			0
$880$	5.21508e10			0.0869621
$881$			7.73052e11i			1.28323i	0.767026	+	0.641616i	$0.221736\pi$
$881$			7.73052e11i			1.28323i	−0.767026	+	0.641616i	$0.778264\pi$
$882$		0			0
$883$			4.46632e10i			0.0734694i	0.999325	+	0.0367347i	$0.0116957\pi$
$883$			4.46632e10i			0.0734694i	−0.999325	+	0.0367347i	$0.988304\pi$
$884$	3.89889e9	−	6.08395e10i	0.00638458	−	0.0996269i
$885$		0			0
$886$	−4.14932e11	−	4.14932e11i	−0.673352	−	0.673352i
$887$	−5.99299e11			−0.968164			−0.484082	−	0.875023i	$-0.660846\pi$
$887$	−5.99299e11			−0.968164			−0.484082	+	0.875023i	$0.660846\pi$
$888$		0			0
$889$	4.22425e11	+	4.22425e11i	0.676304	+	0.676304i
$890$	1.31721e11	+	1.31721e11i	0.209939	+	0.209939i
$891$		0			0
$892$	1.63960e10	−	1.63960e10i	0.0258988	−	0.0258988i
$893$	−1.28537e11			−0.202126
$894$		0			0
$895$	−7.86533e10	+	7.86533e10i	−0.122581	+	0.122581i
$896$			2.61679e11i			0.406011i
$897$		0			0
$898$	3.00464e11			0.462048
$899$	−1.06835e10	−	1.06835e10i	−0.0163559	−	0.0163559i
$900$		0			0
$901$			3.57708e11i			0.542787i
$902$	−1.46309e10	−	1.46309e10i	−0.0221027	−	0.0221027i
$903$		0			0
$904$	9.68180e11	−	9.68180e11i	1.44971	−	1.44971i
$905$	−2.74857e11	+	2.74857e11i	−0.409744	+	0.409744i
$906$		0			0
$907$			7.40058e11i			1.09354i	0.837281	+	0.546772i	$0.184144\pi$
$907$			7.40058e11i			1.09354i	−0.837281	+	0.546772i	$0.815856\pi$
$908$	2.04333e11	−	2.04333e11i	0.300604	−	0.300604i
$909$		0			0
$910$	−1.78656e11	+	1.57137e11i	−0.260526	+	0.229146i
$911$	−1.00029e12			−1.45228			−0.726142	−	0.687545i	$-0.758688\pi$
$911$	−1.00029e12			−1.45228			−0.726142	+	0.687545i	$0.758688\pi$
$912$		0			0
$913$	2.26543e11			0.326038
$914$			4.55361e11i			0.652485i
$915$		0			0
$916$	−2.30585e11	−	2.30585e11i	−0.327529	−	0.327529i
$917$	7.57703e11	−	7.57703e11i	1.07157	−	1.07157i
$918$		0			0
$919$	−8.17593e11			−1.14624			−0.573120	−	0.819472i	$-0.694267\pi$
$919$	−8.17593e11			−1.14624			−0.573120	+	0.819472i	$0.694267\pi$
$920$			7.53982e10i			0.105247i
$921$		0			0
$922$			1.13388e12i			1.56907i
$923$	−6.20662e11	+	5.45902e11i	−0.855161	+	0.752157i
$924$		0			0
$925$	−6.04546e11	−	6.04546e11i	−0.825776	−	0.825776i
$926$	5.45393e11			0.741763
$927$		0			0
$928$	−2.04126e11	−	2.04126e11i	−0.275237	−	0.275237i
$929$	2.63479e11	+	2.63479e11i	0.353739	+	0.353739i	0.861499	−	0.507760i	$-0.169526\pi$
$929$	2.63479e11	+	2.63479e11i	0.353739	+	0.353739i	−0.507760	+	0.861499i	$0.669526\pi$
$930$		0			0
$931$	1.21420e10	−	1.21420e10i	0.0161618	−	0.0161618i
$932$	−1.92221e11			−0.254763
$933$		0			0
$934$	4.19077e11	−	4.19077e11i	0.550689	−	0.550689i
$935$			3.80958e10i			0.0498460i
$936$		0			0
$937$	−3.72371e11			−0.483079			−0.241539	−	0.970391i	$-0.577652\pi$
$937$	−3.72371e11			−0.483079			−0.241539	+	0.970391i	$0.577652\pi$
$938$	−4.77035e11	−	4.77035e11i	−0.616225	−	0.616225i
$939$		0			0
$940$		−	1.57993e11i		−	0.202361i
$941$	−9.59317e10	−	9.59317e10i	−0.122350	−	0.122350i	0.643281	−	0.765630i	$-0.277573\pi$
$941$	−9.59317e10	−	9.59317e10i	−0.122350	−	0.122350i	−0.765630	+	0.643281i	$0.777573\pi$
$942$		0			0
$943$	1.53150e10	−	1.53150e10i	0.0193674	−	0.0193674i
$944$	−7.51970e10	+	7.51970e10i	−0.0946918	+	0.0946918i
$945$		0			0
$946$		−	2.94145e11i		−	0.367280i
$947$	−4.53656e11	+	4.53656e11i	−0.564062	+	0.564062i	−0.930459	−	0.366397i	$-0.880591\pi$
$947$	−4.53656e11	+	4.53656e11i	−0.564062	+	0.564062i	0.366397	+	0.930459i	$0.380591\pi$
$948$		0			0
$949$	7.51191e11	+	8.54063e11i	0.926159	+	1.05299i
$950$	−6.47847e10			−0.0795386
$951$		0			0
$952$	−3.08910e11			−0.376083
$953$			1.13945e11i			0.138141i	0.997612	+	0.0690706i	$0.0220034\pi$
$953$			1.13945e11i			0.138141i	−0.997612	+	0.0690706i	$0.977997\pi$
$954$		0			0
$955$	−2.51010e11	−	2.51010e11i	−0.301771	−	0.301771i
$956$	1.46597e11	−	1.46597e11i	0.175506	−	0.175506i
$957$		0			0
$958$	−1.96695e11			−0.233523
$959$		−	5.70609e11i		−	0.674627i
$960$		0			0
$961$		−	8.52134e11i		−	0.999113i
$962$	6.90700e10	−	1.07779e12i	0.0806471	−	1.25844i
$963$		0			0
$964$	−7.86014e10	−	7.86014e10i	−0.0910169	−	0.0910169i
$965$	3.20029e11			0.369046
$966$		0			0
$967$	−1.21100e11	−	1.21100e11i	−0.138497	−	0.138497i	0.634459	−	0.772956i	$-0.281223\pi$
$967$	−1.21100e11	−	1.21100e11i	−0.138497	−	0.138497i	−0.772956	+	0.634459i	$0.781223\pi$
$968$	6.17647e11	+	6.17647e11i	0.703459	+	0.703459i
$969$		0			0
$970$	5.00190e10	−	5.00190e10i	0.0565000	−	0.0565000i
$971$	7.06062e11			0.794266			0.397133	−	0.917761i	$-0.370005\pi$
$971$	7.06062e11			0.794266			0.397133	+	0.917761i	$0.370005\pi$
$972$		0			0
$973$	−1.97358e11	+	1.97358e11i	−0.220193	+	0.220193i
$974$		−	9.02742e11i		−	1.00306i
$975$		0			0
$976$	4.92781e11			0.543068
$977$	−6.78732e11	−	6.78732e11i	−0.744938	−	0.744938i	0.228585	−	0.973524i	$-0.426590\pi$
$977$	−6.78732e11	−	6.78732e11i	−0.744938	−	0.744938i	−0.973524	+	0.228585i	$0.926590\pi$
$978$		0			0
$979$		−	2.01519e11i		−	0.219374i
$980$	1.49245e10	+	1.49245e10i	0.0161806	+	0.0161806i
$981$		0			0
$982$	3.66285e11	−	3.66285e11i	0.393888	−	0.393888i
$983$	4.75495e11	−	4.75495e11i	0.509251	−	0.509251i	−0.405046	−	0.914296i	$-0.632744\pi$
$983$	4.75495e11	−	4.75495e11i	0.509251	−	0.509251i	0.914296	+	0.405046i	$0.132744\pi$
$984$		0			0
$985$			5.28069e11i			0.560978i
$986$	−1.73264e11	+	1.73264e11i	−0.183316	+	0.183316i
$987$		0			0
$988$	1.87603e10	+	2.13294e10i	0.0196885	+	0.0223847i
$989$	3.07899e11			0.321827
$990$		0			0
$991$	7.81292e10			0.0810063			0.0405031	−	0.999179i	$-0.487104\pi$
$991$	7.81292e10			0.0810063			0.0405031	+	0.999179i	$0.487104\pi$
$992$			1.44560e10i			0.0149280i
$993$		0			0
$994$	6.06758e11	+	6.06758e11i	0.621541	+	0.621541i
$995$	1.78298e11	−	1.78298e11i	0.181909	−	0.181909i
$996$		0			0
$997$	9.95206e11			1.00724			0.503619	−	0.863926i	$-0.332001\pi$
$997$	9.95206e11			1.00724			0.503619	+	0.863926i	$0.332001\pi$
$998$			1.20338e12i			1.21305i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.9.j.a.73.7		18
3.2	odd	2		13.9.d.a.8.3	yes	18
13.5	odd	4	inner	117.9.j.a.109.7		18
39.5	even	4		13.9.d.a.5.3	✓	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.9.d.a.5.3	✓	18	39.5	even	4
13.9.d.a.8.3	yes	18	3.2	odd	2
117.9.j.a.73.7		18	1.1	even	1	trivial
117.9.j.a.109.7		18	13.5	odd	4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 117 = 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	117.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(9.74776 + 9.74776i) q^{2} -65.9624i q^{4} +(198.672 + 198.672i) q^{5} +(-1520.85 + 1520.85i) q^{7} +(3138.41 - 3138.41i) q^{8} +3873.22i q^{10} +(2962.81 - 2962.81i) q^{11} +(1826.58 - 28502.5i) q^{13} -29649.7 q^{14} +44298.6 q^{16} +32359.8i q^{17} +(-10661.6 - 10661.6i) q^{19} +(13104.9 - 13104.9i) q^{20} +57761.5 q^{22} +60462.2i q^{23} -311684. i q^{25} +(295641. - 260031. i) q^{26} +(100319. + 100319. i) q^{28} +549285. q^{29} +(-19449.9 - 19449.9i) q^{31} +(-371622. - 371622. i) q^{32} +(-315436. + 315436. i) q^{34} -604300. q^{35} +(1.93961e6 - 1.93961e6i) q^{37} -207854. i q^{38} +1.24703e6 q^{40} +(-253299. - 253299. i) q^{41} -5.09242e6i q^{43} +(-195434. - 195434. i) q^{44} +(-589371. + 589371. i) q^{46} +(6.02802e6 - 6.02802e6i) q^{47} +1.13885e6i q^{49} +(3.03822e6 - 3.03822e6i) q^{50} +(-1.88010e6 - 120486. i) q^{52} +1.10541e7 q^{53} +1.17726e6 q^{55} +9.54609e6i q^{56} +(5.35430e6 + 5.35430e6i) q^{58} +(-1.69750e6 + 1.69750e6i) q^{59} +1.11241e7 q^{61} -379185. i q^{62} -1.85854e7i q^{64} +(6.02555e6 - 5.29977e6i) q^{65} +(1.60890e7 + 1.60890e7i) q^{67} +2.13453e6 q^{68} +(-5.89057e6 - 5.89057e6i) q^{70} +(-2.04642e7 - 2.04642e7i) q^{71} +(-2.81599e7 + 2.81599e7i) q^{73} +3.78138e7 q^{74} +(-703266. + 703266. i) q^{76} +9.01196e6i q^{77} +6.53546e6 q^{79} +(8.80090e6 + 8.80090e6i) q^{80} -4.93819e6i q^{82} +(3.82312e7 + 3.82312e7i) q^{83} +(-6.42900e6 + 6.42900e6i) q^{85} +(4.96396e7 - 4.96396e7i) q^{86} -1.85970e7i q^{88} +(3.40081e7 - 3.40081e7i) q^{89} +(4.05700e7 + 4.61259e7i) q^{91} +3.98823e6 q^{92} +1.17519e8 q^{94} -4.23634e6i q^{95} +(-1.29141e7 - 1.29141e7i) q^{97} +(-1.11012e7 + 1.11012e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8} + 31556 q^{11} + 71300 q^{13} + 110260 q^{14} - 522860 q^{16} + 100288 q^{19} - 736268 q^{20} - 977312 q^{22} - 2952238 q^{26} + 4497084 q^{28} + 2479024 q^{29}+ \cdots - 588677614 q^{98}+O(q^{100}) \) 18 * q + 2 * q^2 - 166 * q^5 + 5308 * q^7 - 10464 * q^8 + 31556 * q^11 + 71300 * q^13 + 110260 * q^14 - 522860 * q^16 + 100288 * q^19 - 736268 * q^20 - 977312 * q^22 - 2952238 * q^26 + 4497084 * q^28 + 2479024 * q^29 - 1892664 * q^31 - 947212 * q^32 - 531576 * q^34 + 2918284 * q^35 - 8343978 * q^37 - 12691908 * q^40 - 1140178 * q^41 + 3867188 * q^44 + 2006148 * q^46 + 13368572 * q^47 - 37369598 * q^50 - 14821220 * q^52 - 50561348 * q^53 + 76994128 * q^55 + 22505716 * q^58 - 2127976 * q^59 - 52016516 * q^61 - 10413082 * q^65 + 960292 * q^67 - 283187508 * q^68 - 166635032 * q^70 - 67412140 * q^71 - 145213226 * q^73 + 233620024 * q^74 - 150533640 * q^76 - 76829120 * q^79 - 524889520 * q^80 + 241951556 * q^83 + 260737764 * q^85 + 579480384 * q^86 + 89187110 * q^89 + 232660948 * q^91 + 122690376 * q^92 + 1069637380 * q^94 + 331183146 * q^97 - 588677614 * q^98

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