LMFDB - Embedded newform 117.9.j.a.109.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [117,9,Mod(73,117)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			109.9
Root			$21.6276 - 22.6276i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.9.j.a.73.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(21.6276 - 21.6276i) q^{2} -679.508i q^{4} +(706.219 - 706.219i) q^{5} +(966.123 + 966.123i) q^{7} +(-9159.48 - 9159.48i) q^{8} +O(q^{10})$	\(q+(21.6276 - 21.6276i) q^{2} -679.508i q^{4} +(706.219 - 706.219i) q^{5} +(966.123 + 966.123i) q^{7} +(-9159.48 - 9159.48i) q^{8} -30547.7i q^{10} +(431.183 + 431.183i) q^{11} +(13650.7 - 25087.6i) q^{13} +41789.9 q^{14} -222241. q^{16} +5328.69i q^{17} +(-7342.35 + 7342.35i) q^{19} +(-479882. - 479882. i) q^{20} +18650.9 q^{22} +87712.3i q^{23} -606866. i q^{25} +(-247354. - 837818. i) q^{26} +(656489. - 656489. i) q^{28} +787498. q^{29} +(-398923. + 398923. i) q^{31} +(-2.46173e6 + 2.46173e6i) q^{32} +(115247. + 115247. i) q^{34} +1.36459e6 q^{35} +(1.04786e6 + 1.04786e6i) q^{37} +317595. i q^{38} -1.29372e7 q^{40} +(-2.42006e6 + 2.42006e6i) q^{41} -1.12816e6i q^{43} +(292992. - 292992. i) q^{44} +(1.89701e6 + 1.89701e6i) q^{46} +(5.15032e6 + 5.15032e6i) q^{47} -3.89801e6i q^{49} +(-1.31251e7 - 1.31251e7i) q^{50} +(-1.70473e7 - 9.27576e6i) q^{52} -1.30921e7 q^{53} +609019. q^{55} -1.76984e7i q^{56} +(1.70317e7 - 1.70317e7i) q^{58} +(6.77137e6 + 6.77137e6i) q^{59} +3.48674e6 q^{61} +1.72555e7i q^{62} +4.95888e7i q^{64} +(-8.07699e6 - 2.73577e7i) q^{65} +(-1.71216e7 + 1.71216e7i) q^{67} +3.62089e6 q^{68} +(2.95128e7 - 2.95128e7i) q^{70} +(-1.55636e7 + 1.55636e7i) q^{71} +(3.33302e6 + 3.33302e6i) q^{73} +4.53253e7 q^{74} +(4.98919e6 + 4.98919e6i) q^{76} +833151. i q^{77} -4.95167e6 q^{79} +(-1.56951e8 + 1.56951e8i) q^{80} +1.04680e8i q^{82} +(3.26970e7 - 3.26970e7i) q^{83} +(3.76322e6 + 3.76322e6i) q^{85} +(-2.43995e7 - 2.43995e7i) q^{86} -7.89882e6i q^{88} +(8.25743e7 + 8.25743e7i) q^{89} +(3.74260e7 - 1.10495e7i) q^{91} +5.96012e7 q^{92} +2.22779e8 q^{94} +1.03706e7i q^{95} +(7.02251e7 - 7.02251e7i) q^{97} +(-8.43048e7 - 8.43048e7i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8}+O(q^{10}) $ 18 * q + 2 * q^2 - 166 * q^5 + 5308 * q^7 - 10464 * q^8	$ 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}+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

Display 100 coefficients 10 coefficients

Character values

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

$n$	$28$	$92$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$1$

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 $

$2$	21.6276	−	21.6276i	1.35173	−	1.35173i	0.467996	−	0.883731i	$-0.344976\pi$
$2$	21.6276	−	21.6276i	1.35173	−	1.35173i	0.883731	−	0.467996i	$-0.155024\pi$
$3$		0			0
$3$		0			0
$4$		−	679.508i		−	2.65433i
$5$	706.219	−	706.219i	1.12995	−	1.12995i	0.139766	−	0.990185i	$-0.455365\pi$
$5$	706.219	−	706.219i	1.12995	−	1.12995i	0.990185	−	0.139766i	$-0.0446351\pi$
$6$		0			0
$7$	966.123	+	966.123i	0.402384	+	0.402384i	0.879072	−	0.476689i	$-0.158163\pi$
$7$	966.123	+	966.123i	0.402384	+	0.402384i	−0.476689	+	0.879072i	$0.658163\pi$
$8$	−9159.48	−	9159.48i	−2.23620	−	2.23620i
$9$		0			0
$10$		−	30547.7i		−	3.05477i
$11$	431.183	+	431.183i	0.0294504	+	0.0294504i	0.721679	−	0.692228i	$-0.243371\pi$
$11$	431.183	+	431.183i	0.0294504	+	0.0294504i	−0.692228	+	0.721679i	$0.743371\pi$
$12$		0			0
$13$	13650.7	−	25087.6i	0.477949	−	0.878388i
$13$	13650.7	−	25087.6i	0.477949	−	0.878388i
$14$	41789.9			1.08783
$15$		0			0
$16$	−222241.			−3.39114
$17$			5328.69i			0.0638006i	0.999491	+	0.0319003i	$0.0101559\pi$
$17$			5328.69i			0.0638006i	−0.999491	+	0.0319003i	$0.989844\pi$
$18$		0			0
$19$	−7342.35	+	7342.35i	−0.0563405	+	0.0563405i	−0.734716	−	0.678375i	$-0.762684\pi$
$19$	−7342.35	+	7342.35i	−0.0563405	+	0.0563405i	0.678375	+	0.734716i	$0.262684\pi$
$20$	−479882.	−	479882.i	−2.99926	−	2.99926i
$21$		0			0
$22$	18650.9			0.0796177
$23$			87712.3i			0.313436i	0.987643	+	0.156718i	$0.0500914\pi$
$23$			87712.3i			0.313436i	−0.987643	+	0.156718i	$0.949909\pi$
$24$		0			0
$25$		−	606866.i		−	1.55358i
$26$	−247354.	−	837818.i	−0.541284	−	1.83340i
$27$		0			0
$28$	656489.	−	656489.i	1.06806	−	1.06806i
$29$	787498.			1.11342			0.556708	−	0.830708i	$-0.312064\pi$
$29$	787498.			1.11342			0.556708	+	0.830708i	$0.312064\pi$
$30$		0			0
$31$	−398923.	+	398923.i	−0.431959	+	0.431959i	−0.889294	−	0.457335i	$-0.848804\pi$
$31$	−398923.	+	398923.i	−0.431959	+	0.431959i	0.457335	+	0.889294i	$0.348804\pi$
$32$	−2.46173e6	+	2.46173e6i	−2.34769	+	2.34769i
$33$		0			0
$34$	115247.	+	115247.i	0.0862410	+	0.0862410i
$35$	1.36459e6			0.909347
$36$		0			0
$37$	1.04786e6	+	1.04786e6i	0.559107	+	0.559107i	0.929053	−	0.369946i	$-0.120624\pi$
$37$	1.04786e6	+	1.04786e6i	0.559107	+	0.559107i	−0.369946	+	0.929053i	$0.620624\pi$
$38$			317595.i			0.152314i
$39$		0			0
$40$	−1.29372e7			−5.05359
$41$	−2.42006e6	+	2.42006e6i	−0.856428	+	0.856428i	−0.990915	−	0.134487i	$-0.957061\pi$
$41$	−2.42006e6	+	2.42006e6i	−0.856428	+	0.856428i	0.134487	+	0.990915i	$0.457061\pi$
$42$		0			0
$43$		−	1.12816e6i		−	0.329988i	−0.986295	−	0.164994i	$-0.947240\pi$
$43$		−	1.12816e6i		−	0.329988i	0.986295	−	0.164994i	$-0.0527604\pi$
$44$	292992.	−	292992.i	0.0781709	−	0.0781709i
$45$		0			0
$46$	1.89701e6	+	1.89701e6i	0.423680	+	0.423680i
$47$	5.15032e6	+	5.15032e6i	1.05546	+	1.05546i	0.998369	+	0.0570943i	$0.0181836\pi$
$47$	5.15032e6	+	5.15032e6i	1.05546	+	1.05546i	0.0570943	+	0.998369i	$0.481816\pi$
$48$		0			0
$49$		−	3.89801e6i		−	0.676175i
$50$	−1.31251e7	−	1.31251e7i	−2.10001	−	2.10001i
$51$		0			0
$52$	−1.70473e7	−	9.27576e6i	−2.33153	−	1.26863i
$53$	−1.30921e7			−1.65922			−0.829611	−	0.558342i	$-0.811438\pi$
$53$	−1.30921e7			−1.65922			−0.829611	+	0.558342i	$0.811438\pi$
$54$		0			0
$55$	609019.			0.0665549
$56$		−	1.76984e7i		−	1.79962i
$57$		0			0
$58$	1.70317e7	−	1.70317e7i	1.50503	−	1.50503i
$59$	6.77137e6	+	6.77137e6i	0.558816	+	0.558816i	0.928970	−	0.370154i	$-0.120695\pi$
$59$	6.77137e6	+	6.77137e6i	0.558816	+	0.558816i	−0.370154	+	0.928970i	$0.620695\pi$
$60$		0			0
$61$	3.48674e6			0.251826			0.125913	−	0.992041i	$-0.459814\pi$
$61$	3.48674e6			0.251826			0.125913	+	0.992041i	$0.459814\pi$
$62$			1.72555e7i			1.16778i
$63$		0			0
$64$			4.95888e7i			2.95573i
$65$	−8.07699e6	−	2.73577e7i	−0.452477	−	1.53259i
$66$		0			0
$67$	−1.71216e7	+	1.71216e7i	−0.849660	+	0.849660i	−0.990091	−	0.140431i	$-0.955151\pi$
$67$	−1.71216e7	+	1.71216e7i	−0.849660	+	0.849660i	0.140431	+	0.990091i	$0.455151\pi$
$68$	3.62089e6			0.169348
$69$		0			0
$70$	2.95128e7	−	2.95128e7i	1.22919	−	1.22919i
$71$	−1.55636e7	+	1.55636e7i	−0.612457	+	0.612457i	−0.943586	−	0.331129i	$-0.892571\pi$
$71$	−1.55636e7	+	1.55636e7i	−0.612457	+	0.612457i	0.331129	+	0.943586i	$0.392571\pi$
$72$		0			0
$73$	3.33302e6	+	3.33302e6i	0.117367	+	0.117367i	0.763351	−	0.645984i	$-0.223553\pi$
$73$	3.33302e6	+	3.33302e6i	0.117367	+	0.117367i	−0.645984	+	0.763351i	$0.723553\pi$
$74$	4.53253e7			1.51152
$75$		0			0
$76$	4.98919e6	+	4.98919e6i	0.149546	+	0.149546i
$77$			833151.i			0.0237007i
$78$		0			0
$79$	−4.95167e6			−0.127129			−0.0635643	−	0.997978i	$-0.520247\pi$
$79$	−4.95167e6			−0.127129			−0.0635643	+	0.997978i	$0.520247\pi$
$80$	−1.56951e8	+	1.56951e8i	−3.83182	+	3.83182i
$81$		0			0
$82$			1.04680e8i			2.31531i
$83$	3.26970e7	−	3.26970e7i	0.688961	−	0.688961i	−0.273041	−	0.962002i	$-0.588029\pi$
$83$	3.26970e7	−	3.26970e7i	0.688961	−	0.688961i	0.962002	+	0.273041i	$0.0880295\pi$
$84$		0			0
$85$	3.76322e6	+	3.76322e6i	0.0720915	+	0.0720915i
$86$	−2.43995e7	−	2.43995e7i	−0.446053	−	0.446053i
$87$		0			0
$88$		−	7.89882e6i		−	0.131714i
$89$	8.25743e7	+	8.25743e7i	1.31609	+	1.31609i	0.916846	+	0.399242i	$0.130727\pi$
$89$	8.25743e7	+	8.25743e7i	1.31609	+	1.31609i	0.399242	+	0.916846i	$0.369273\pi$
$90$		0			0
$91$	3.74260e7	−	1.10495e7i	0.545768	−	0.161130i
$92$	5.96012e7			0.831963
$93$		0			0
$94$	2.22779e8			2.85339
$95$			1.03706e7i			0.127324i
$96$		0			0
$97$	7.02251e7	−	7.02251e7i	0.793241	−	0.793241i	−0.188778	−	0.982020i	$-0.560453\pi$
$97$	7.02251e7	−	7.02251e7i	0.793241	−	0.793241i	0.982020	+	0.188778i	$0.0604528\pi$
$98$	−8.43048e7	−	8.43048e7i	−0.914003	−	0.914003i
$99$		0			0
$100$	−4.12371e8			−4.12371
$101$		−	1.07171e8i		−	1.02989i	−0.857222	−	0.514947i	$-0.827812\pi$
$101$		−	1.07171e8i		−	1.02989i	0.857222	−	0.514947i	$-0.172188\pi$
$102$		0			0
$103$		−	1.56166e8i		−	1.38751i	−0.720210	−	0.693756i	$-0.755955\pi$
$103$		−	1.56166e8i		−	1.38751i	0.720210	−	0.693756i	$-0.244045\pi$
$104$	−3.54823e8	+	1.04756e8i	−3.03304	+	0.895463i
$105$		0			0
$106$	−2.83150e8	+	2.83150e8i	−2.24281	+	2.24281i
$107$	1.89540e8			1.44599			0.722996	−	0.690852i	$-0.242764\pi$
$107$	1.89540e8			1.44599			0.722996	+	0.690852i	$0.242764\pi$
$108$		0			0
$109$	3.48339e7	−	3.48339e7i	0.246772	−	0.246772i	−0.572872	−	0.819645i	$-0.694171\pi$
$109$	3.48339e7	−	3.48339e7i	0.246772	−	0.246772i	0.819645	+	0.572872i	$0.194171\pi$
$110$	1.31716e7	−	1.31716e7i	0.0899640	−	0.0899640i
$111$		0			0
$112$	−2.14713e8	−	2.14713e8i	−1.36454	−	1.36454i
$113$	−9.85497e7			−0.604424			−0.302212	−	0.953241i	$-0.597725\pi$
$113$	−9.85497e7			−0.604424			−0.302212	+	0.953241i	$0.597725\pi$
$114$		0			0
$115$	6.19441e7	+	6.19441e7i	0.354167	+	0.354167i
$116$		−	5.35112e8i		−	2.95537i
$117$		0			0
$118$	2.92897e8			1.51073
$119$	−5.14817e6	+	5.14817e6i	−0.0256723	+	0.0256723i
$120$		0			0
$121$		−	2.13987e8i		−	0.998265i
$122$	7.54100e7	−	7.54100e7i	0.340400	−	0.340400i
$123$		0			0
$124$	2.71072e8	+	2.71072e8i	1.14656	+	1.14656i
$125$	−1.52714e8	−	1.52714e8i	−0.625515	−	0.625515i
$126$		0			0
$127$		−	2.44685e8i		−	0.940575i	−0.882513	−	0.470287i	$-0.844150\pi$
$127$		−	2.44685e8i		−	0.940575i	0.882513	−	0.470287i	$-0.155850\pi$
$128$	4.42287e8	+	4.42287e8i	1.64765	+	1.64765i
$129$		0			0
$130$	−7.66369e8	−	4.16997e8i	−2.68327	−	1.46002i
$131$	3.05747e8			1.03819			0.519095	−	0.854716i	$-0.326269\pi$
$131$	3.05747e8			1.03819			0.519095	+	0.854716i	$0.326269\pi$
$132$		0			0
$133$	−1.41872e7			−0.0453410
$134$			7.40599e8i			2.29702i
$135$		0			0
$136$	4.88080e7	−	4.88080e7i	0.142671	−	0.142671i
$137$	−1.87463e8	−	1.87463e8i	−0.532149	−	0.532149i	0.389062	−	0.921211i	$-0.372799\pi$
$137$	−1.87463e8	−	1.87463e8i	−0.532149	−	0.532149i	−0.921211	+	0.389062i	$0.872799\pi$
$138$		0			0
$139$	−4.76588e8			−1.27669			−0.638343	−	0.769752i	$-0.720380\pi$
$139$	−4.76588e8			−1.27669			−0.638343	+	0.769752i	$0.720380\pi$
$140$		−	9.27250e8i		−	2.41371i
$141$		0			0
$142$			6.73206e8i			1.65575i
$143$	1.67033e7	−	4.93141e6i	0.0399446	−	0.0117931i
$144$		0			0
$145$	5.56147e8	−	5.56147e8i	1.25811	−	1.25811i
$146$	1.44171e8			0.317297
$147$		0			0
$148$	7.12027e8	−	7.12027e8i	1.48405	−	1.48405i
$149$	−1.83994e8	+	1.83994e8i	−0.373301	+	0.373301i	−0.868678	−	0.495377i	$-0.835030\pi$
$149$	−1.83994e8	+	1.83994e8i	−0.373301	+	0.373301i	0.495377	+	0.868678i	$0.335030\pi$
$150$		0			0
$151$	2.92294e8	+	2.92294e8i	0.562228	+	0.562228i	0.929940	−	0.367712i	$-0.119859\pi$
$151$	2.92294e8	+	2.92294e8i	0.562228	+	0.562228i	−0.367712	+	0.929940i	$0.619859\pi$
$152$	1.34504e8			0.251977
$153$		0			0
$154$	1.80191e7	+	1.80191e7i	0.0320368	+	0.0320368i
$155$			5.63454e8i			0.976185i
$156$		0			0
$157$	1.69240e8			0.278551			0.139276	−	0.990254i	$-0.455523\pi$
$157$	1.69240e8			0.278551			0.139276	+	0.990254i	$0.455523\pi$
$158$	−1.07093e8	+	1.07093e8i	−0.171843	+	0.171843i
$159$		0			0
$160$			3.47704e9i			5.30554i
$161$	−8.47409e7	+	8.47409e7i	−0.126122	+	0.126122i
$162$		0			0
$163$	8.70441e8	+	8.70441e8i	1.23307	+	1.23307i	0.962777	+	0.270296i	$0.0871217\pi$
$163$	8.70441e8	+	8.70441e8i	1.23307	+	1.23307i	0.270296	+	0.962777i	$0.412878\pi$
$164$	1.64445e9	+	1.64445e9i	2.27324	+	2.27324i
$165$		0			0
$166$		−	1.41431e9i		−	1.86257i
$167$	−1.55595e8	−	1.55595e8i	−0.200046	−	0.200046i	0.599974	−	0.800020i	$-0.295178\pi$
$167$	−1.55595e8	−	1.55595e8i	−0.200046	−	0.200046i	−0.800020	+	0.599974i	$0.795178\pi$
$168$		0			0
$169$	−4.43048e8	−	6.84927e8i	−0.543130	−	0.839648i
$170$	1.62779e8			0.194896
$171$		0			0
$172$	−7.66595e8			−0.875896
$173$		−	6.17461e7i		−	0.0689327i	−0.999406	−	0.0344663i	$-0.989027\pi$
$173$		−	6.17461e7i		−	0.0689327i	0.999406	−	0.0344663i	$-0.0109731\pi$
$174$		0			0
$175$	5.86308e8	−	5.86308e8i	0.625134	−	0.625134i
$176$	−9.58267e7	−	9.58267e7i	−0.0998701	−	0.0998701i
$177$		0			0
$178$	3.57177e9			3.55798
$179$		−	6.89754e8i		−	0.671866i	−0.941886	−	0.335933i	$-0.890948\pi$
$179$		−	6.89754e8i		−	0.671866i	0.941886	−	0.335933i	$-0.109052\pi$
$180$		0			0
$181$		−	8.50963e8i		−	0.792860i	−0.918065	−	0.396430i	$-0.870249\pi$
$181$		−	8.50963e8i		−	0.792860i	0.918065	−	0.396430i	$-0.129751\pi$
$182$	5.70461e8	−	1.04841e9i	0.519925	−	0.955533i
$183$		0			0
$184$	8.03399e8	−	8.03399e8i	0.700906	−	0.700906i
$185$	1.48003e9			1.26353
$186$		0			0
$187$	−2.29764e6	+	2.29764e6i	−0.00187895	+	0.00187895i
$188$	3.49969e9	−	3.49969e9i	2.80155	−	2.80155i
$189$		0			0
$190$	2.24292e8	+	2.24292e8i	0.172107	+	0.172107i
$191$	6.91369e7			0.0519489			0.0259744	−	0.999663i	$-0.491731\pi$
$191$	6.91369e7			0.0519489			0.0259744	+	0.999663i	$0.491731\pi$
$192$		0			0
$193$	−2.46969e8	−	2.46969e8i	−0.177997	−	0.177997i	0.612485	−	0.790482i	$-0.290170\pi$
$193$	−2.46969e8	−	2.46969e8i	−0.177997	−	0.177997i	−0.790482	+	0.612485i	$0.790170\pi$
$194$		−	3.03760e9i		−	2.14449i
$195$		0			0
$196$	−2.64873e9			−1.79479
$197$	2.79924e8	−	2.79924e8i	0.185855	−	0.185855i	−0.608046	−	0.793902i	$-0.708046\pi$
$197$	2.79924e8	−	2.79924e8i	0.185855	−	0.185855i	0.793902	+	0.608046i	$0.208046\pi$
$198$		0			0
$199$		−	2.07496e8i		−	0.132311i	−0.997809	−	0.0661557i	$-0.978927\pi$
$199$		−	2.07496e8i		−	0.132311i	0.997809	−	0.0661557i	$-0.0210734\pi$
$200$	−5.55858e9	+	5.55858e9i	−3.47411	+	3.47411i
$201$		0			0
$202$	−2.31786e9	−	2.31786e9i	−1.39213	−	1.39213i
$203$	7.60821e8	+	7.60821e8i	0.448021	+	0.448021i
$204$		0			0
$205$			3.41819e9i			1.93544i
$206$	−3.37749e9	−	3.37749e9i	−1.87554	−	1.87554i
$207$		0			0
$208$	−3.03375e9	+	5.57551e9i	−1.62079	+	2.97873i
$209$	−6.33178e6			−0.00331849
$210$		0			0
$211$	−2.54699e6			−0.00128498			−0.000642491	−	1.00000i	$-0.500205\pi$
$211$	−2.54699e6			−0.00128498			−0.000642491		1.00000i	$0.500205\pi$
$212$			8.89616e9i			4.40412i
$213$		0			0
$214$	4.09930e9	−	4.09930e9i	1.95459	−	1.95459i
$215$	−7.96729e8	−	7.96729e8i	−0.372870	−	0.372870i
$216$		0			0
$217$	−7.70818e8			−0.347626
$218$		−	1.50675e9i		−	0.667137i
$219$		0			0
$220$		−	4.13833e8i		−	0.176659i
$221$	1.33684e8	+	7.27403e7i	0.0560417	+	0.0304934i
$222$		0			0
$223$	−2.15016e9	+	2.15016e9i	−0.869462	+	0.869462i	−0.992413	−	0.122951i	$-0.960764\pi$
$223$	−2.15016e9	+	2.15016e9i	−0.869462	+	0.869462i	0.122951	+	0.992413i	$0.460764\pi$
$224$	−4.75666e9			−1.88934
$225$		0			0
$226$	−2.13140e9	+	2.13140e9i	−0.817015	+	0.817015i
$227$	−1.19673e9	+	1.19673e9i	−0.450707	+	0.450707i	−0.895589	−	0.444882i	$-0.853246\pi$
$227$	−1.19673e9	+	1.19673e9i	−0.450707	+	0.450707i	0.444882	+	0.895589i	$0.353246\pi$
$228$		0			0
$229$	−1.41483e9	−	1.41483e9i	−0.514473	−	0.514473i	0.401421	−	0.915894i	$-0.368516\pi$
$229$	−1.41483e9	−	1.41483e9i	−0.514473	−	0.514473i	−0.915894	+	0.401421i	$0.868516\pi$
$230$	2.67941e9			0.957475
$231$		0			0
$232$	−7.21308e9	−	7.21308e9i	−2.48982	−	2.48982i
$233$			4.15641e9i			1.41024i	0.709086	+	0.705122i	$0.249108\pi$
$233$			4.15641e9i			1.41024i	−0.709086	+	0.705122i	$0.750892\pi$
$234$		0			0
$235$	7.27451e9			2.38524
$236$	4.60120e9	−	4.60120e9i	1.48328	−	1.48328i
$237$		0			0
$238$			2.22685e8i			0.0694039i
$239$	−1.82691e9	+	1.82691e9i	−0.559919	+	0.559919i	−0.929284	−	0.369365i	$-0.879575\pi$
$239$	−1.82691e9	+	1.82691e9i	−0.559919	+	0.559919i	0.369365	+	0.929284i	$0.379575\pi$
$240$		0			0
$241$	−2.91957e9	−	2.91957e9i	−0.865469	−	0.865469i	0.126498	−	0.991967i	$-0.459626\pi$
$241$	−2.91957e9	−	2.91957e9i	−0.865469	−	0.865469i	−0.991967	+	0.126498i	$0.959626\pi$
$242$	−4.62803e9	−	4.62803e9i	−1.34938	−	1.34938i
$243$		0			0
$244$		−	2.36927e9i		−	0.668430i
$245$	−2.75285e9	−	2.75285e9i	−0.764044	−	0.764044i
$246$		0			0
$247$	8.39740e7	+	2.84430e8i	0.0225609	+	0.0764166i
$248$	7.30786e9			1.93189
$249$		0			0
$250$	−6.60567e9			−1.69105
$251$			4.34046e9i			1.09356i	0.837278	+	0.546778i	$0.184146\pi$
$251$			4.34046e9i			1.09356i	−0.837278	+	0.546778i	$0.815854\pi$
$252$		0			0
$253$	−3.78200e7	+	3.78200e7i	−0.00923081	+	0.00923081i
$254$	−5.29197e9	−	5.29197e9i	−1.27140	−	1.27140i
$255$		0			0
$256$	6.43647e9			1.49861
$257$		−	5.42704e9i		−	1.24403i	−0.783005	−	0.622015i	$-0.786314\pi$
$257$		−	5.42704e9i		−	1.24403i	0.783005	−	0.622015i	$-0.213686\pi$
$258$		0			0
$259$			2.02472e9i			0.449951i
$260$	−1.85898e10	+	5.48838e9i	−4.06801	+	1.20102i
$261$		0			0
$262$	6.61258e9	−	6.61258e9i	1.40335	−	1.40335i
$263$	8.97912e9			1.87677			0.938385	−	0.345593i	$-0.112322\pi$
$263$	8.97912e9			1.87677			0.938385	+	0.345593i	$0.112322\pi$
$264$		0			0
$265$	−9.24586e9	+	9.24586e9i	−1.87484	+	1.87484i
$266$	−3.06836e8	+	3.06836e8i	−0.0612886	+	0.0612886i
$267$		0			0
$268$	1.16343e10	+	1.16343e10i	2.25528	+	2.25528i
$269$	2.61168e9			0.498783			0.249391	−	0.968403i	$-0.419769\pi$
$269$	2.61168e9			0.498783			0.249391	+	0.968403i	$0.419769\pi$
$270$		0			0
$271$	5.13079e9	+	5.13079e9i	0.951277	+	0.951277i	0.998867	−	0.0475900i	$-0.0151541\pi$
$271$	5.13079e9	+	5.13079e9i	0.951277	+	0.951277i	−0.0475900	+	0.998867i	$0.515154\pi$
$272$		−	1.18426e9i		−	0.216356i
$273$		0			0
$274$	−8.10876e9			−1.43864
$275$	2.61670e8	−	2.61670e8i	0.0457534	−	0.0457534i
$276$		0			0
$277$		−	2.85168e9i		−	0.484375i	−0.970229	−	0.242188i	$-0.922135\pi$
$277$		−	2.85168e9i		−	0.484375i	0.970229	−	0.242188i	$-0.0778649\pi$
$278$	−1.03075e10	+	1.03075e10i	−1.72573	+	1.72573i
$279$		0			0
$280$	−1.24989e10	−	1.24989e10i	−2.03348	−	2.03348i
$281$	9.40208e8	+	9.40208e8i	0.150799	+	0.150799i	0.778475	−	0.627676i	$-0.215994\pi$
$281$	9.40208e8	+	9.40208e8i	0.150799	+	0.150799i	−0.627676	+	0.778475i	$0.715994\pi$
$282$		0			0
$283$			4.24340e8i			0.0661559i	0.999453	+	0.0330779i	$0.0105310\pi$
$283$			4.24340e8i			0.0661559i	−0.999453	+	0.0330779i	$0.989469\pi$
$284$	1.05756e10	+	1.05756e10i	1.62566	+	1.62566i
$285$		0			0
$286$	2.54598e8	−	4.67907e8i	0.0380531	−	0.0699352i
$287$	−4.67616e9			−0.689226
$288$		0			0
$289$	6.94736e9			0.995929
$290$		−	2.40563e10i		−	3.40123i
$291$		0			0
$292$	2.26482e9	−	2.26482e9i	0.311531	−	0.311531i
$293$	4.76690e9	+	4.76690e9i	0.646794	+	0.646794i	0.952217	−	0.305423i	$-0.0987979\pi$
$293$	4.76690e9	+	4.76690e9i	0.646794	+	0.646794i	−0.305423	+	0.952217i	$0.598798\pi$
$294$		0			0
$295$	9.56415e9			1.26287
$296$		−	1.91956e10i		−	2.50055i
$297$		0			0
$298$			7.95872e9i			1.00920i
$299$	2.20049e9	+	1.19733e9i	0.275319	+	0.149806i
$300$		0			0
$301$	1.08994e9	−	1.08994e9i	0.132782	−	0.132782i
$302$	1.26433e10			1.51996
$303$		0			0
$304$	1.63177e9	−	1.63177e9i	0.191058	−	0.191058i
$305$	2.46241e9	−	2.46241e9i	0.284551	−	0.284551i
$306$		0			0
$307$	−2.28644e9	−	2.28644e9i	−0.257399	−	0.257399i	0.566596	−	0.823995i	$-0.308260\pi$
$307$	−2.28644e9	−	2.28644e9i	−0.257399	−	0.257399i	−0.823995	+	0.566596i	$0.808260\pi$
$308$	5.66133e8			0.0629094
$309$		0			0
$310$	1.21862e10	+	1.21862e10i	1.31953	+	1.31953i
$311$			1.65154e10i			1.76542i	0.469918	+	0.882710i	$0.344283\pi$
$311$			1.65154e10i			1.76542i	−0.469918	+	0.882710i	$0.655717\pi$
$312$		0			0
$313$	−7.14367e9			−0.744293			−0.372146	−	0.928174i	$-0.621378\pi$
$313$	−7.14367e9			−0.744293			−0.372146	+	0.928174i	$0.621378\pi$
$314$	3.66027e9	−	3.66027e9i	0.376525	−	0.376525i
$315$		0			0
$316$			3.36470e9i			0.337441i
$317$	−1.16773e10	+	1.16773e10i	−1.15639	+	1.15639i	−0.171142	+	0.985246i	$0.554746\pi$
$317$	−1.16773e10	+	1.16773e10i	−1.15639	+	1.15639i	−0.985246	+	0.171142i	$0.945254\pi$
$318$		0			0
$319$	3.39556e8	+	3.39556e8i	0.0327905	+	0.0327905i
$320$	3.50206e10	+	3.50206e10i	3.33982	+	3.33982i
$321$		0			0
$322$			3.66549e9i			0.340964i
$323$	−3.91251e7	−	3.91251e7i	−0.00359456	−	0.00359456i
$324$		0			0
$325$	−1.52248e10	−	8.28414e9i	−1.36464	−	0.742530i
$326$	3.76511e10			3.33356
$327$		0			0
$328$	4.43330e10			3.83029
$329$			9.95169e9i			0.849402i
$330$		0			0
$331$	−9.01779e9	+	9.01779e9i	−0.751256	+	0.751256i	−0.974714	−	0.223458i	$-0.928266\pi$
$331$	−9.01779e9	+	9.01779e9i	−0.751256	+	0.751256i	0.223458	+	0.974714i	$0.428266\pi$
$332$	−2.22179e10	−	2.22179e10i	−1.82873	−	1.82873i
$333$		0			0
$334$	−6.73031e9			−0.540816
$335$			2.41832e10i			1.92015i
$336$		0			0
$337$			1.63467e10i			1.26739i	0.773584	+	0.633694i	$0.218462\pi$
$337$			1.63467e10i			1.26739i	−0.773584	+	0.633694i	$0.781538\pi$
$338$	−2.43954e10	−	5.23126e9i	−1.86914	−	0.400811i
$339$		0			0
$340$	2.55714e9	−	2.55714e9i	0.191355	−	0.191355i
$341$	−3.44017e8			−0.0254427
$342$		0			0
$343$	9.33547e9	−	9.33547e9i	0.674465	−	0.674465i
$344$	−1.03334e10	+	1.03334e10i	−0.737918	+	0.737918i
$345$		0			0
$346$	−1.33542e9	−	1.33542e9i	−0.0931781	−	0.0931781i
$347$	−2.48764e10			−1.71581			−0.857906	−	0.513807i	$-0.828235\pi$
$347$	−2.48764e10			−1.71581			−0.857906	+	0.513807i	$0.828235\pi$
$348$		0			0
$349$	−9.43914e9	−	9.43914e9i	−0.636254	−	0.636254i	0.313375	−	0.949629i	$-0.398540\pi$
$349$	−9.43914e9	−	9.43914e9i	−0.636254	−	0.636254i	−0.949629	+	0.313375i	$0.898540\pi$
$350$		−	2.53609e10i		−	1.69002i
$351$		0			0
$352$	−2.12291e9			−0.138280
$353$	−4.31297e9	+	4.31297e9i	−0.277765	+	0.277765i	−0.832216	−	0.554451i	$-0.812928\pi$
$353$	−4.31297e9	+	4.31297e9i	−0.277765	+	0.277765i	0.554451	+	0.832216i	$0.312928\pi$
$354$		0			0
$355$			2.19826e10i			1.38409i
$356$	5.61099e10	−	5.61099e10i	3.49333	−	3.49333i
$357$		0			0
$358$	−1.49178e10	−	1.49178e10i	−0.908179	−	0.908179i
$359$	−5.34220e9	−	5.34220e9i	−0.321619	−	0.321619i	0.527769	−	0.849388i	$-0.323029\pi$
$359$	−5.34220e9	−	5.34220e9i	−0.321619	−	0.321619i	−0.849388	+	0.527769i	$0.823029\pi$
$360$		0			0
$361$			1.68757e10i			0.993652i
$362$	−1.84043e10	−	1.84043e10i	−1.07173	−	1.07173i
$363$		0			0
$364$	−7.50823e9	−	2.54313e10i	−0.427693	−	1.44865i
$365$	4.70769e9			0.265238
$366$		0			0
$367$	6.12347e9			0.337546			0.168773	−	0.985655i	$-0.446019\pi$
$367$	6.12347e9			0.337546			0.168773	+	0.985655i	$0.446019\pi$
$368$		−	1.94933e10i		−	1.06290i
$369$		0			0
$370$	3.20096e10	−	3.20096e10i	1.70794	−	1.70794i
$371$	−1.26485e10	−	1.26485e10i	−0.667644	−	0.667644i
$372$		0			0
$373$	−3.24303e10			−1.67539			−0.837694	−	0.546139i	$-0.816097\pi$
$373$	−3.24303e10			−1.67539			−0.837694	+	0.546139i	$0.816097\pi$
$374$			9.93849e7i			0.00507965i
$375$		0			0
$376$		−	9.43486e10i		−	4.72046i
$377$	1.07499e10	−	1.97565e10i	0.532156	−	0.978012i
$378$		0			0
$379$	−1.35377e10	+	1.35377e10i	−0.656127	+	0.656127i	−0.954461	−	0.298335i	$-0.903569\pi$
$379$	−1.35377e10	+	1.35377e10i	−0.656127	+	0.656127i	0.298335	+	0.954461i	$0.403569\pi$
$380$	7.04692e9			0.337960
$381$		0			0
$382$	1.49527e9	−	1.49527e9i	0.0702207	−	0.0702207i
$383$	2.42793e10	−	2.42793e10i	1.12834	−	1.12834i	0.137894	−	0.990447i	$-0.455967\pi$
$383$	2.42793e10	−	2.42793e10i	1.12834	−	1.12834i	0.990447	−	0.137894i	$-0.0440332\pi$
$384$		0			0
$385$	5.88387e8	+	5.88387e8i	0.0267806	+	0.0267806i
$386$	−1.06827e10			−0.481206
$387$		0			0
$388$	−4.77185e10	−	4.77185e10i	−2.10552	−	2.10552i
$389$			3.75267e10i			1.63886i	0.573178	+	0.819431i	$0.305710\pi$
$389$			3.75267e10i			1.63886i	−0.573178	+	0.819431i	$0.694290\pi$
$390$		0			0
$391$	−4.67392e8			−0.0199974
$392$	−3.57038e10	+	3.57038e10i	−1.51206	+	1.51206i
$393$		0			0
$394$		−	1.21082e10i		−	0.502451i
$395$	−3.49696e9	+	3.49696e9i	−0.143649	+	0.143649i
$396$		0			0
$397$	2.65082e10	+	2.65082e10i	1.06713	+	1.06713i	0.997578	+	0.0695533i	$0.0221574\pi$
$397$	2.65082e10	+	2.65082e10i	1.06713	+	1.06713i	0.0695533	+	0.997578i	$0.477843\pi$
$398$	−4.48764e9	−	4.48764e9i	−0.178849	−	0.178849i
$399$		0			0
$400$			1.34871e11i			5.26839i
$401$	3.53870e9	+	3.53870e9i	0.136857	+	0.136857i	0.772216	−	0.635360i	$-0.219148\pi$
$401$	3.53870e9	+	3.53870e9i	0.136857	+	0.136857i	−0.635360	+	0.772216i	$0.719148\pi$
$402$		0			0
$403$	4.56246e9	+	1.54536e10i	0.172973	+	0.585882i
$404$	−7.28237e10			−2.73368
$405$		0			0
$406$	3.29095e10			1.21120
$407$			9.03635e8i			0.0329318i
$408$		0			0
$409$	−7.94399e9	+	7.94399e9i	−0.283887	+	0.283887i	−0.834657	−	0.550770i	$-0.814334\pi$
$409$	−7.94399e9	+	7.94399e9i	−0.283887	+	0.283887i	0.550770	+	0.834657i	$0.314334\pi$
$410$	7.39273e10	+	7.39273e10i	2.61619	+	2.61619i
$411$		0			0
$412$	−1.06116e11			−3.68292
$413$			1.30840e10i			0.449717i
$414$		0			0
$415$		−	4.61824e10i		−	1.55698i
$416$	2.81547e10	+	9.53632e10i	0.940106	+	3.18425i
$417$		0			0
$418$	−1.36941e8	+	1.36941e8i	−0.00448570	+	0.00448570i
$419$	2.26899e10			0.736167			0.368084	−	0.929793i	$-0.380014\pi$
$419$	2.26899e10			0.736167			0.368084	+	0.929793i	$0.380014\pi$
$420$		0			0
$421$	1.46275e10	−	1.46275e10i	0.465630	−	0.465630i	−0.434865	−	0.900496i	$-0.643204\pi$
$421$	1.46275e10	−	1.46275e10i	0.465630	−	0.465630i	0.900496	+	0.434865i	$0.143204\pi$
$422$	−5.50853e7	+	5.50853e7i	−0.00173694	+	0.00173694i
$423$		0			0
$424$	1.19916e11	+	1.19916e11i	3.71035	+	3.71035i
$425$	3.23380e9			0.0991192
$426$		0			0
$427$	3.36863e9	+	3.36863e9i	0.101331	+	0.101331i
$428$		−	1.28794e11i		−	3.83814i
$429$		0			0
$430$	−3.44627e10			−1.00804
$431$	3.07608e10	−	3.07608e10i	0.891433	−	0.891433i	−0.103225	−	0.994658i	$-0.532916\pi$
$431$	3.07608e10	−	3.07608e10i	0.891433	−	0.891433i	0.994658	+	0.103225i	$0.0329161\pi$
$432$		0			0
$433$			2.27162e10i			0.646227i	0.946360	+	0.323113i	$0.104730\pi$
$433$			2.27162e10i			0.646227i	−0.946360	+	0.323113i	$0.895270\pi$
$434$	−1.66710e10	+	1.66710e10i	−0.469896	+	0.469896i
$435$		0			0
$436$	−2.36699e10	−	2.36699e10i	−0.655015	−	0.655015i
$437$	−6.44014e8	−	6.44014e8i	−0.0176591	−	0.0176591i
$438$		0			0
$439$		−	3.91283e10i		−	1.05350i	−0.850022	−	0.526748i	$-0.823411\pi$
$439$		−	3.91283e10i		−	1.05350i	0.850022	−	0.526748i	$-0.176589\pi$
$440$	−5.57830e9	−	5.57830e9i	−0.148830	−	0.148830i
$441$		0			0
$442$	4.46447e9	−	1.31807e9i	0.116972	−	0.0345343i
$443$	−2.20125e10			−0.571550			−0.285775	−	0.958297i	$-0.592251\pi$
$443$	−2.20125e10			−0.571550			−0.285775	+	0.958297i	$0.592251\pi$
$444$		0			0
$445$	1.16631e11			2.97423
$446$			9.30056e10i			2.35055i
$447$		0			0
$448$	−4.79089e10	+	4.79089e10i	−1.18934	+	1.18934i
$449$	−1.45102e9	−	1.45102e9i	−0.0357015	−	0.0357015i	0.689031	−	0.724732i	$-0.258037\pi$
$449$	−1.45102e9	−	1.45102e9i	−0.0357015	−	0.0357015i	−0.724732	+	0.689031i	$0.758037\pi$
$450$		0			0
$451$	−2.08698e9			−0.0504442
$452$			6.69653e10i			1.60434i
$453$		0			0
$454$			5.17650e10i			1.21846i
$455$	1.86276e10	−	3.42343e10i	0.434621	−	0.798760i
$456$		0			0
$457$	4.42354e10	−	4.42354e10i	1.01416	−	1.01416i	0.0142587	−	0.999898i	$-0.495461\pi$
$457$	4.42354e10	−	4.42354e10i	1.01416	−	1.01416i	0.999898	−	0.0142587i	$-0.00453885\pi$
$458$	−6.11988e10			−1.39085
$459$		0			0
$460$	4.20915e10	−	4.20915e10i	0.940077	−	0.940077i
$461$	7.59452e9	−	7.59452e9i	0.168150	−	0.168150i	−0.618016	−	0.786166i	$-0.712063\pi$
$461$	7.59452e9	−	7.59452e9i	0.168150	−	0.168150i	0.786166	+	0.618016i	$0.212063\pi$
$462$		0			0
$463$	−5.43374e10	−	5.43374e10i	−1.18243	−	1.18243i	−0.979113	−	0.203316i	$-0.934828\pi$
$463$	−5.43374e10	−	5.43374e10i	−1.18243	−	1.18243i	−0.203316	−	0.979113i	$-0.565172\pi$
$464$	−1.75015e11			−3.77575
$465$		0			0
$466$	8.98932e10	+	8.98932e10i	1.90626	+	1.90626i
$467$			1.32796e10i			0.279201i	0.990208	+	0.139601i	$0.0445819\pi$
$467$			1.32796e10i			0.279201i	−0.990208	+	0.139601i	$0.955418\pi$
$468$		0			0
$469$	−3.30831e10			−0.683778
$470$	1.57330e11	−	1.57330e11i	3.22420	−	3.22420i
$471$		0			0
$472$		−	1.24045e11i		−	2.49925i
$473$	4.86444e8	−	4.86444e8i	0.00971825	−	0.00971825i
$474$		0			0
$475$	4.45582e9	+	4.45582e9i	0.0875293	+	0.0875293i
$476$	3.49822e9	+	3.49822e9i	0.0681428	+	0.0681428i
$477$		0			0
$478$			7.90233e10i			1.51371i
$479$	6.72197e10	+	6.72197e10i	1.27689	+	1.27689i	0.942397	+	0.334495i	$0.108566\pi$
$479$	6.72197e10	+	6.72197e10i	1.27689	+	1.27689i	0.334495	+	0.942397i	$0.391434\pi$
$480$		0			0
$481$	4.05922e10	−	1.19843e10i	0.758337	−	0.223888i
$482$	−1.26287e11			−2.33975
$483$		0			0
$484$	−1.45406e11			−2.64973
$485$		−	9.91886e10i		−	1.79265i
$486$		0			0
$487$	−2.47767e10	+	2.47767e10i	−0.440481	+	0.440481i	−0.892174	−	0.451692i	$-0.850820\pi$
$487$	−2.47767e10	+	2.47767e10i	−0.440481	+	0.440481i	0.451692	+	0.892174i	$0.350820\pi$
$488$	−3.19368e10	−	3.19368e10i	−0.563134	−	0.563134i
$489$		0			0
$490$	−1.19075e11			−2.06556
$491$			1.23003e10i			0.211636i	0.994385	+	0.105818i	$0.0337461\pi$
$491$			1.23003e10i			0.211636i	−0.994385	+	0.105818i	$0.966254\pi$
$492$		0			0
$493$			4.19633e9i			0.0710366i
$494$	7.96771e9	+	4.33539e9i	0.133791	+	0.0727982i
$495$		0			0
$496$	8.86573e10	−	8.86573e10i	1.46483	−	1.46483i
$497$	−3.00726e10			−0.492885
$498$		0			0
$499$	3.51105e10	−	3.51105e10i	0.566284	−	0.566284i	−0.364801	−	0.931085i	$-0.618863\pi$
$499$	3.51105e10	−	3.51105e10i	0.566284	−	0.566284i	0.931085	+	0.364801i	$0.118863\pi$
$500$	−1.03770e11	+	1.03770e11i	−1.66032	+	1.66032i
$501$		0			0
$502$	9.38738e10	+	9.38738e10i	1.47819	+	1.47819i
$503$	−9.14102e10			−1.42798			−0.713991	−	0.700155i	$-0.753114\pi$
$503$	−9.14102e10			−1.42798			−0.713991	+	0.700155i	$0.753114\pi$
$504$		0			0
$505$	−7.56863e10	−	7.56863e10i	−1.16373	−	1.16373i
$506$			1.63591e9i			0.0249551i
$507$		0			0
$508$	−1.66266e11			−2.49660
$509$	2.06174e10	−	2.06174e10i	0.307159	−	0.307159i	−0.536648	−	0.843806i	$-0.680310\pi$
$509$	2.06174e10	−	2.06174e10i	0.307159	−	0.307159i	0.843806	+	0.536648i	$0.180310\pi$
$510$		0			0
$511$			6.44022e9i			0.0944533i
$512$	2.59803e10	−	2.59803e10i	0.378063	−	0.378063i
$513$		0			0
$514$	−1.17374e11	−	1.17374e11i	−1.68159	−	1.68159i
$515$	−1.10287e11	−	1.10287e11i	−1.56782	−	1.56782i
$516$		0			0
$517$			4.44146e9i			0.0621675i
$518$	4.37898e10	+	4.37898e10i	0.608211	+	0.608211i
$519$		0			0
$520$	−1.76602e11	+	3.24564e11i	−2.41536	+	4.43902i
$521$	7.52460e10			1.02125			0.510626	−	0.859803i	$-0.329414\pi$
$521$	7.52460e10			1.02125			0.510626	+	0.859803i	$0.329414\pi$
$522$		0			0
$523$	−2.50012e10			−0.334160			−0.167080	−	0.985943i	$-0.553434\pi$
$523$	−2.50012e10			−0.334160			−0.167080	+	0.985943i	$0.553434\pi$
$524$		−	2.07758e11i		−	2.75570i
$525$		0			0
$526$	1.94197e11	−	1.94197e11i	2.53688	−	2.53688i
$527$	−2.12574e9	−	2.12574e9i	−0.0275592	−	0.0275592i
$528$		0			0
$529$	7.06175e10			0.901758
$530$			3.99932e11i			5.06854i
$531$		0			0
$532$			9.64033e9i			0.120350i
$533$	2.76781e10	+	9.37492e10i	0.342948	+	1.16161i
$534$		0			0
$535$	1.33857e11	−	1.33857e11i	1.63390	−	1.63390i
$536$	3.13650e11			3.80002
$537$		0			0
$538$	5.64845e10	−	5.64845e10i	0.674218	−	0.674218i
$539$	1.68076e9	−	1.68076e9i	0.0199136	−	0.0199136i
$540$		0			0
$541$	−1.46837e9	−	1.46837e9i	−0.0171415	−	0.0171415i	0.698484	−	0.715626i	$-0.253858\pi$
$541$	−1.46837e9	−	1.46837e9i	−0.0171415	−	0.0171415i	−0.715626	+	0.698484i	$0.753858\pi$
$542$	2.21934e11			2.57173
$543$		0			0
$544$	−1.31178e10	−	1.31178e10i	−0.149784	−	0.149784i
$545$		−	4.92008e10i		−	0.557681i
$546$		0			0
$547$	−2.13596e10			−0.238585			−0.119293	−	0.992859i	$-0.538063\pi$
$547$	−2.13596e10			−0.238585			−0.119293	+	0.992859i	$0.538063\pi$
$548$	−1.27383e11	+	1.27383e11i	−1.41250	+	1.41250i
$549$		0			0
$550$		−	1.13186e10i		−	0.123692i
$551$	−5.78209e9	+	5.78209e9i	−0.0627304	+	0.0627304i
$552$		0			0
$553$	−4.78392e9	−	4.78392e9i	−0.0511545	−	0.0511545i
$554$	−6.16751e10	−	6.16751e10i	−0.654743	−	0.654743i
$555$		0			0
$556$			3.23846e11i			3.38874i
$557$	−8.24194e10	−	8.24194e10i	−0.856266	−	0.856266i	0.134630	−	0.990896i	$-0.457015\pi$
$557$	−8.24194e10	−	8.24194e10i	−0.856266	−	0.856266i	−0.990896	+	0.134630i	$0.957015\pi$
$558$		0			0
$559$	−2.83029e10	−	1.54002e10i	−0.289857	−	0.157717i
$560$	−3.03268e11			−3.08372
$561$		0			0
$562$	4.06689e10			0.407678
$563$		−	2.19548e10i		−	0.218522i	−0.994013	−	0.109261i	$-0.965151\pi$
$563$		−	2.19548e10i		−	0.218522i	0.994013	−	0.109261i	$-0.0348485\pi$
$564$		0			0
$565$	−6.95977e10	+	6.95977e10i	−0.682969	+	0.682969i
$566$	9.17747e9	+	9.17747e9i	0.0894247	+	0.0894247i
$567$		0			0
$568$	2.85108e11			2.73915
$569$		−	4.42507e9i		−	0.0422154i	−0.999777	−	0.0211077i	$-0.993281\pi$
$569$		−	4.42507e9i		−	0.0422154i	0.999777	−	0.0211077i	$-0.00671929\pi$
$570$		0			0
$571$			5.68177e10i			0.534490i	0.963629	+	0.267245i	$0.0861133\pi$
$571$			5.68177e10i			0.534490i	−0.963629	+	0.267245i	$0.913887\pi$
$572$	−3.35094e9	−	1.13500e10i	−0.0313027	−	0.106026i
$573$		0			0
$574$	−1.01134e11	+	1.01134e11i	−0.931645	+	0.931645i
$575$	5.32296e10			0.486947
$576$		0			0
$577$	−7.62085e10	+	7.62085e10i	−0.687543	+	0.687543i	−0.961688	−	0.274145i	$-0.911605\pi$
$577$	−7.62085e10	+	7.62085e10i	−0.687543	+	0.687543i	0.274145	+	0.961688i	$0.411605\pi$
$578$	1.50255e11	−	1.50255e11i	1.34622	−	1.34622i
$579$		0			0
$580$	−3.77906e11	−	3.77906e11i	−3.33943	−	3.33943i
$581$	6.31786e10			0.554454
$582$		0			0
$583$	−5.64507e9	−	5.64507e9i	−0.0488647	−	0.0488647i
$584$		−	6.10575e10i		−	0.524913i
$585$		0			0
$586$	2.06194e11			1.74858
$587$	6.09649e9	−	6.09649e9i	0.0513484	−	0.0513484i	−0.680966	−	0.732315i	$-0.738440\pi$
$587$	6.09649e9	−	6.09649e9i	0.0513484	−	0.0513484i	0.732315	+	0.680966i	$0.238440\pi$
$588$		0			0
$589$		−	5.85806e9i		−	0.0486735i
$590$	2.06850e11	−	2.06850e11i	1.70705	−	1.70705i
$591$		0			0
$592$	−2.32877e11	−	2.32877e11i	−1.89601	−	1.89601i
$593$	8.50831e10	+	8.50831e10i	0.688057	+	0.688057i	0.961802	−	0.273745i	$-0.0882625\pi$
$593$	8.50831e10	+	8.50831e10i	0.688057	+	0.688057i	−0.273745	+	0.961802i	$0.588263\pi$
$594$		0			0
$595$			7.27147e9i			0.0580169i
$596$	1.25026e11	+	1.25026e11i	0.990865	+	0.990865i
$597$		0			0
$598$	7.34869e10	−	2.16960e10i	0.574653	−	0.169658i
$599$	−9.14223e10			−0.710142			−0.355071	−	0.934839i	$-0.615543\pi$
$599$	−9.14223e10			−0.710142			−0.355071	+	0.934839i	$0.615543\pi$
$600$		0			0
$601$	−3.88100e8			−0.00297472			−0.00148736	−	0.999999i	$-0.500473\pi$
$601$	−3.88100e8			−0.00297472			−0.00148736	+	0.999999i	$0.500473\pi$
$602$		−	4.71458e10i		−	0.358969i
$603$		0			0
$604$	1.98616e11	−	1.98616e11i	1.49234	−	1.49234i
$605$	−1.51122e11	−	1.51122e11i	−1.12799	−	1.12799i
$606$		0			0
$607$	−9.47209e10			−0.697736			−0.348868	−	0.937172i	$-0.613434\pi$
$607$	−9.47209e10			−0.697736			−0.348868	+	0.937172i	$0.613434\pi$
$608$		−	3.61497e10i		−	0.264540i
$609$		0			0
$610$		−	1.06512e11i		−	0.769271i
$611$	1.99515e11	−	5.89040e10i	1.43156	−	0.422649i
$612$		0			0
$613$	−6.37233e10	+	6.37233e10i	−0.451290	+	0.451290i	−0.895783	−	0.444492i	$-0.853384\pi$
$613$	−6.37233e10	+	6.37233e10i	−0.451290	+	0.451290i	0.444492	+	0.895783i	$0.353384\pi$
$614$	−9.89006e10			−0.695866
$615$		0			0
$616$	7.63123e9	−	7.63123e9i	0.0529995	−	0.0529995i
$617$	2.65061e10	−	2.65061e10i	0.182896	−	0.182896i	−0.609720	−	0.792617i	$-0.708718\pi$
$617$	2.65061e10	−	2.65061e10i	0.182896	−	0.182896i	0.792617	+	0.609720i	$0.208718\pi$
$618$		0			0
$619$	1.76381e11	+	1.76381e11i	1.20141	+	1.20141i	0.973739	+	0.227668i	$0.0731102\pi$
$619$	1.76381e11	+	1.76381e11i	1.20141	+	1.20141i	0.227668	+	0.973739i	$0.426890\pi$
$620$	3.82872e11			2.59112
$621$		0			0
$622$	3.57189e11	+	3.57189e11i	2.38636	+	2.38636i
$623$			1.59554e11i			1.05914i
$624$		0			0
$625$	2.13584e10			0.139975
$626$	−1.54501e11	+	1.54501e11i	−1.00608	+	1.00608i
$627$		0			0
$628$		−	1.15000e11i		−	0.739367i
$629$	−5.58370e9	+	5.58370e9i	−0.0356713	+	0.0356713i
$630$		0			0
$631$	−9.29590e10	−	9.29590e10i	−0.586373	−	0.586373i	0.350274	−	0.936647i	$-0.386088\pi$
$631$	−9.29590e10	−	9.29590e10i	−0.586373	−	0.586373i	−0.936647	+	0.350274i	$0.886088\pi$
$632$	4.53547e10	+	4.53547e10i	0.284285	+	0.284285i
$633$		0			0
$634$			5.05103e11i			3.12624i
$635$	−1.72802e11	−	1.72802e11i	−1.06280	−	1.06280i
$636$		0			0
$637$	−9.77919e10	−	5.32106e10i	−0.593944	−	0.323177i
$638$	1.46876e10			0.0886476
$639$		0			0
$640$	6.24703e11			3.72352
$641$			1.24740e11i			0.738880i	0.929255	+	0.369440i	$0.120450\pi$
$641$			1.24740e11i			0.738880i	−0.929255	+	0.369440i	$0.879550\pi$
$642$		0			0
$643$	−1.21411e11	+	1.21411e11i	−0.710256	+	0.710256i	−0.966589	−	0.256333i	$-0.917486\pi$
$643$	−1.21411e11	+	1.21411e11i	−0.710256	+	0.710256i	0.256333	+	0.966589i	$0.417486\pi$
$644$	5.75821e10	+	5.75821e10i	0.334768	+	0.334768i
$645$		0			0
$646$	−1.69237e9			−0.00971771
$647$			3.31192e10i			0.189000i	0.995525	+	0.0945001i	$0.0301253\pi$
$647$			3.31192e10i			0.189000i	−0.995525	+	0.0945001i	$0.969875\pi$
$648$		0			0
$649$			5.83940e9i			0.0329147i
$650$	−5.08443e11	+	1.50111e11i	−2.84832	+	0.840927i
$651$		0			0
$652$	5.91472e11	−	5.91472e11i	3.27298	−	3.27298i
$653$	4.14146e10			0.227772			0.113886	−	0.993494i	$-0.463670\pi$
$653$	4.14146e10			0.227772			0.113886	+	0.993494i	$0.463670\pi$
$654$		0			0
$655$	2.15924e11	−	2.15924e11i	1.17310	−	1.17310i
$656$	5.37838e11	−	5.37838e11i	2.90426	−	2.90426i
$657$		0			0
$658$	2.15231e11	+	2.15231e11i	1.14816	+	1.14816i
$659$	2.91147e10			0.154373			0.0771865	−	0.997017i	$-0.475406\pi$
$659$	2.91147e10			0.154373			0.0771865	+	0.997017i	$0.475406\pi$
$660$		0			0
$661$	−1.10872e11	−	1.10872e11i	−0.580786	−	0.580786i	0.354333	−	0.935119i	$-0.384708\pi$
$661$	−1.10872e11	−	1.10872e11i	−0.580786	−	0.580786i	−0.935119	+	0.354333i	$0.884708\pi$
$662$			3.90067e11i			2.03099i
$663$		0			0
$664$	−5.98974e11			−3.08131
$665$	−1.00193e10	+	1.00193e10i	−0.0512331	+	0.0512331i
$666$		0			0
$667$			6.90733e10i			0.348985i
$668$	−1.05728e11	+	1.05728e11i	−0.530989	+	0.530989i
$669$		0			0
$670$	5.23025e11	+	5.23025e11i	2.59551	+	2.59551i
$671$	1.50342e9	+	1.50342e9i	0.00741637	+	0.00741637i
$672$		0			0
$673$		−	1.07689e11i		−	0.524943i	−0.964940	−	0.262472i	$-0.915462\pi$
$673$		−	1.07689e11i		−	0.524943i	0.964940	−	0.262472i	$-0.0845376\pi$
$674$	3.53540e11	+	3.53540e11i	1.71316	+	1.71316i
$675$		0			0
$676$	−4.65414e11	+	3.01055e11i	−2.22870	+	1.44165i
$677$	−1.69059e11			−0.804793			−0.402396	−	0.915466i	$-0.631823\pi$
$677$	−1.69059e11			−0.804793			−0.402396	+	0.915466i	$0.631823\pi$
$678$		0			0
$679$	1.35692e11			0.638375
$680$		−	6.89383e10i		−	0.322422i
$681$		0			0
$682$	−7.44028e9	+	7.44028e9i	−0.0343916	+	0.0343916i
$683$	5.19296e10	+	5.19296e10i	0.238634	+	0.238634i	0.816284	−	0.577650i	$-0.196030\pi$
$683$	5.19296e10	+	5.19296e10i	0.238634	+	0.238634i	−0.577650	+	0.816284i	$0.696030\pi$
$684$		0			0
$685$	−2.64780e11			−1.20260
$686$		−	4.03808e11i		−	1.82339i
$687$		0			0
$688$			2.50724e11i			1.11903i
$689$	−1.78716e11	+	3.28449e11i	−0.793023	+	1.45744i
$690$		0			0
$691$	−2.33370e11	+	2.33370e11i	−1.02361	+	1.02361i	−0.0238933	+	0.999715i	$0.507606\pi$
$691$	−2.33370e11	+	2.33370e11i	−1.02361	+	1.02361i	−0.999715	+	0.0238933i	$0.992394\pi$
$692$	−4.19570e10			−0.182970
$693$		0			0
$694$	−5.38017e11	+	5.38017e11i	−2.31931	+	2.31931i
$695$	−3.36576e11	+	3.36576e11i	−1.44259	+	1.44259i
$696$		0			0
$697$	−1.28958e10	−	1.28958e10i	−0.0546406	−	0.0546406i
$698$	−4.08292e11			−1.72008
$699$		0			0
$700$	−3.98401e11	−	3.98401e11i	−1.65931	−	1.65931i
$701$		−	1.62006e11i		−	0.670900i	−0.942058	−	0.335450i	$-0.891112\pi$
$701$		−	1.62006e11i		−	0.670900i	0.942058	−	0.335450i	$-0.108888\pi$
$702$		0			0
$703$	−1.53874e10			−0.0630007
$704$	−2.13818e10	+	2.13818e10i	−0.0870472	+	0.0870472i
$705$		0			0
$706$			1.86559e11i			0.750925i
$707$	1.03541e11	−	1.03541e11i	0.414413	−	0.414413i
$708$		0			0
$709$	7.20230e10	+	7.20230e10i	0.285027	+	0.285027i	0.835110	−	0.550083i	$-0.185404\pi$
$709$	7.20230e10	+	7.20230e10i	0.285027	+	0.285027i	−0.550083	+	0.835110i	$0.685404\pi$
$710$	4.75431e11	+	4.75431e11i	1.87091	+	1.87091i
$711$		0			0
$712$		−	1.51267e12i		−	5.88607i
$713$	−3.49905e10	−	3.49905e10i	−0.135392	−	0.135392i
$714$		0			0
$715$	8.31353e9	−	1.52788e10i	0.0318098	−	0.0584610i
$716$	−4.68694e11			−1.78335
$717$		0			0
$718$	−2.31078e11			−0.869482
$719$		−	2.25844e11i		−	0.845072i	−0.906346	−	0.422536i	$-0.861140\pi$
$719$		−	2.25844e11i		−	0.845072i	0.906346	−	0.422536i	$-0.138860\pi$
$720$		0			0
$721$	1.50875e11	−	1.50875e11i	0.558312	−	0.558312i
$722$	3.64982e11	+	3.64982e11i	1.34315	+	1.34315i
$723$		0			0
$724$	−5.78237e11			−2.10451
$725$		−	4.77906e11i		−	1.72978i
$726$		0			0
$727$		−	3.72838e11i		−	1.33470i	−0.744746	−	0.667348i	$-0.767429\pi$
$727$		−	3.72838e11i		−	1.33470i	0.744746	−	0.667348i	$-0.232571\pi$
$728$	−4.44010e11	−	2.41595e11i	−1.58077	−	0.860126i
$729$		0			0
$730$	1.01816e11	−	1.01816e11i	0.358530	−	0.358530i
$731$	6.01162e9			0.0210534
$732$		0			0
$733$	4.31522e10	−	4.31522e10i	0.149481	−	0.149481i	−0.628405	−	0.777886i	$-0.716292\pi$
$733$	4.31522e10	−	4.31522e10i	0.149481	−	0.149481i	0.777886	+	0.628405i	$0.216292\pi$
$734$	1.32436e11	−	1.32436e11i	0.456270	−	0.456270i
$735$		0			0
$736$	−2.15924e11	−	2.15924e11i	−0.735850	−	0.735850i
$737$	−1.47651e10			−0.0500456
$738$		0			0
$739$	2.19357e11	+	2.19357e11i	0.735485	+	0.735485i	0.971701	−	0.236216i	$-0.0759072\pi$
$739$	2.19357e11	+	2.19357e11i	0.735485	+	0.735485i	−0.236216	+	0.971701i	$0.575907\pi$
$740$		−	1.00569e12i		−	3.35381i
$741$		0			0
$742$	−5.47116e11			−1.80494
$743$	3.11348e11	−	3.11348e11i	1.02162	−	1.02162i	0.0218612	−	0.999761i	$-0.493041\pi$
$743$	3.11348e11	−	3.11348e11i	1.02162	−	1.02162i	0.999761	−	0.0218612i	$-0.00695919\pi$
$744$		0			0
$745$			2.59881e11i			0.843624i
$746$	−7.01390e11	+	7.01390e11i	−2.26467	+	2.26467i
$747$		0			0
$748$	1.56126e9	+	1.56126e9i	0.00498735	+	0.00498735i
$749$	1.83119e11	+	1.83119e11i	0.581844	+	0.581844i
$750$		0			0
$751$			4.29225e10i			0.134935i	0.997721	+	0.0674676i	$0.0214919\pi$
$751$			4.29225e10i			0.134935i	−0.997721	+	0.0674676i	$0.978508\pi$
$752$	−1.14462e12	−	1.14462e12i	−3.57922	−	3.57922i
$753$		0			0
$754$	−1.94791e11	−	6.59780e11i	−0.602675	−	2.04133i
$755$	4.12847e11			1.27058
$756$		0			0
$757$	4.54071e10			0.138274			0.0691370	−	0.997607i	$-0.477975\pi$
$757$	4.54071e10			0.138274			0.0691370	+	0.997607i	$0.477975\pi$
$758$			5.85576e11i			1.77381i
$759$		0			0
$760$	9.49894e10	−	9.49894e10i	0.284722	−	0.284722i
$761$	−3.88630e11	−	3.88630e11i	−1.15877	−	1.15877i	−0.984740	−	0.174032i	$-0.944321\pi$
$761$	−3.88630e11	−	3.88630e11i	−1.15877	−	1.15877i	−0.174032	−	0.984740i	$-0.555679\pi$
$762$		0			0
$763$	6.73077e10			0.198594
$764$		−	4.69791e10i		−	0.137889i
$765$		0			0
$766$		−	1.05021e12i		−	3.05042i
$767$	2.62312e11	−	7.74438e10i	0.757942	−	0.223772i
$768$		0			0
$769$	−3.38675e11	+	3.38675e11i	−0.968452	+	0.968452i	−0.999517	−	0.0310653i	$-0.990110\pi$
$769$	−3.38675e11	+	3.38675e11i	−0.968452	+	0.968452i	0.0310653	+	0.999517i	$0.490110\pi$
$770$	2.54508e10			0.0724001
$771$		0			0
$772$	−1.67817e11	+	1.67817e11i	−0.472462	+	0.472462i
$773$	−3.10254e11	+	3.10254e11i	−0.868958	+	0.868958i	−0.992357	−	0.123399i	$-0.960621\pi$
$773$	−3.10254e11	+	3.10254e11i	−0.868958	+	0.868958i	0.123399	+	0.992357i	$0.460621\pi$
$774$		0			0
$775$	2.42093e11	+	2.42093e11i	0.671082	+	0.671082i
$776$	−1.28645e12			−3.54769
$777$		0			0
$778$	8.11614e11	+	8.11614e11i	2.21529	+	2.21529i
$779$		−	3.55379e10i		−	0.0965032i
$780$		0			0
$781$	−1.34215e10			−0.0360741
$782$	−1.01086e10	+	1.01086e10i	−0.0270310	+	0.0270310i
$783$		0			0
$784$			8.66300e11i			2.29300i
$785$	1.19521e11	−	1.19521e11i	0.314749	−	0.314749i
$786$		0			0
$787$	−9.04871e10	−	9.04871e10i	−0.235878	−	0.235878i	0.579263	−	0.815141i	$-0.303341\pi$
$787$	−9.04871e10	−	9.04871e10i	−0.235878	−	0.235878i	−0.815141	+	0.579263i	$0.803341\pi$
$788$	−1.90211e11	−	1.90211e11i	−0.493321	−	0.493321i
$789$		0			0
$790$			1.51262e11i			0.388348i
$791$	−9.52111e10	−	9.52111e10i	−0.243210	−	0.243210i
$792$		0			0
$793$	4.75965e10	−	8.74742e10i	0.120360	−	0.221201i
$794$	1.14662e12			2.88494
$795$		0			0
$796$	−1.40995e11			−0.351198
$797$			5.52731e11i			1.36987i	0.728602	+	0.684937i	$0.240170\pi$
$797$			5.52731e11i			1.36987i	−0.728602	+	0.684937i	$0.759830\pi$
$798$		0			0
$799$	−2.74445e10	+	2.74445e10i	−0.0673392	+	0.0673392i
$800$	1.49394e12	+	1.49394e12i	3.64731	+	3.64731i
$801$		0			0
$802$	1.53067e11			0.369985
$803$			2.87428e9i			0.00691301i
$804$		0			0
$805$			1.19691e11i			0.285022i
$806$	4.32900e11	+	2.35550e11i	1.02576	+	0.558139i
$807$		0			0
$808$	−9.81632e11	+	9.81632e11i	−2.30305	+	2.30305i
$809$	−5.92169e11			−1.38246			−0.691228	−	0.722637i	$-0.742930\pi$
$809$	−5.92169e11			−1.38246			−0.691228	+	0.722637i	$0.742930\pi$
$810$		0			0
$811$	−1.31991e10	+	1.31991e10i	−0.0305114	+	0.0305114i	−0.722198	−	0.691686i	$-0.756868\pi$
$811$	−1.31991e10	+	1.31991e10i	−0.0305114	+	0.0305114i	0.691686	+	0.722198i	$0.256868\pi$
$812$	5.16984e11	−	5.16984e11i	1.18919	−	1.18919i
$813$		0			0
$814$	1.95435e10	+	1.95435e10i	0.0445148	+	0.0445148i
$815$	1.22944e12			2.78662
$816$		0			0
$817$	8.28335e9	+	8.28335e9i	0.0185917	+	0.0185917i
$818$			3.43619e11i			0.767475i
$819$		0			0
$820$	2.32269e12			5.13731
$821$	−4.27080e11	+	4.27080e11i	−0.940020	+	0.940020i	−0.998300	−	0.0582807i	$-0.981438\pi$
$821$	−4.27080e11	+	4.27080e11i	−0.940020	+	0.940020i	0.0582807	+	0.998300i	$0.481438\pi$
$822$		0			0
$823$			2.42038e11i			0.527574i	0.964581	+	0.263787i	$0.0849716\pi$
$823$			2.42038e11i			0.527574i	−0.964581	+	0.263787i	$0.915028\pi$
$824$	−1.43040e12	+	1.43040e12i	−3.10276	+	3.10276i
$825$		0			0
$826$	2.82975e11	+	2.82975e11i	0.607894	+	0.607894i
$827$	3.93278e11	+	3.93278e11i	0.840771	+	0.840771i	0.988959	−	0.148188i	$-0.0473440\pi$
$827$	3.93278e11	+	3.93278e11i	0.840771	+	0.840771i	−0.148188	+	0.988959i	$0.547344\pi$
$828$		0			0
$829$		−	7.05049e11i		−	1.49280i	−0.665499	−	0.746399i	$-0.731781\pi$
$829$		−	7.05049e11i		−	1.49280i	0.665499	−	0.746399i	$-0.268219\pi$
$830$	−9.98816e11	−	9.98816e11i	−2.10462	−	2.10462i
$831$		0			0
$832$	1.24407e12	+	6.76922e11i	2.59627	+	1.41268i
$833$	2.07713e10			0.0431404
$834$		0			0
$835$	−2.19769e11			−0.452085
$836$			4.30250e9i			0.00880838i
$837$		0			0
$838$	4.90729e11	−	4.90729e11i	0.995097	−	0.995097i
$839$	−5.54376e10	−	5.54376e10i	−0.111881	−	0.111881i	0.648950	−	0.760831i	$-0.275208\pi$
$839$	−5.54376e10	−	5.54376e10i	−0.111881	−	0.111881i	−0.760831	+	0.648950i	$0.775208\pi$
$840$		0			0
$841$	1.19907e11			0.239697
$842$		−	6.32715e11i		−	1.25881i
$843$		0			0
$844$			1.73070e9i			0.00341076i
$845$	−7.96598e11	−	1.70819e11i	−1.56247	−	0.335051i
$846$		0			0
$847$	2.06738e11	−	2.06738e11i	0.401686	−	0.401686i
$848$	2.90960e12			5.62665
$849$		0			0
$850$	6.99394e10	−	6.99394e10i	0.133982	−	0.133982i
$851$	−9.19099e10	+	9.19099e10i	−0.175244	+	0.175244i
$852$		0			0
$853$	9.32541e10	+	9.32541e10i	0.176146	+	0.176146i	0.789673	−	0.613528i	$-0.210250\pi$
$853$	9.32541e10	+	9.32541e10i	0.176146	+	0.176146i	−0.613528	+	0.789673i	$0.710250\pi$
$854$	1.45711e11			0.273943
$855$		0			0
$856$	−1.73609e12	−	1.73609e12i	−3.23353	−	3.23353i
$857$		−	1.87150e11i		−	0.346949i	−0.984838	−	0.173474i	$-0.944501\pi$
$857$		−	1.87150e11i		−	0.346949i	0.984838	−	0.173474i	$-0.0554994\pi$
$858$		0			0
$859$	−3.60656e11			−0.662400			−0.331200	−	0.943561i	$-0.607454\pi$
$859$	−3.60656e11			−0.662400			−0.331200	+	0.943561i	$0.607454\pi$
$860$	−5.41384e11	+	5.41384e11i	−0.989719	+	0.989719i
$861$		0			0
$862$		−	1.33057e12i		−	2.40995i
$863$	4.53196e11	−	4.53196e11i	0.817039	−	0.817039i	−0.168639	−	0.985678i	$-0.553937\pi$
$863$	4.53196e11	−	4.53196e11i	0.817039	−	0.817039i	0.985678	+	0.168639i	$0.0539371\pi$
$864$		0			0
$865$	−4.36063e10	−	4.36063e10i	−0.0778905	−	0.0778905i
$866$	4.91299e11	+	4.91299e11i	0.873522	+	0.873522i
$867$		0			0
$868$			5.23777e11i			0.922715i
$869$	−2.13507e9	−	2.13507e9i	−0.00374398	−	0.00374398i
$870$		0			0
$871$	1.95819e11	+	6.63262e11i	0.340237	+	1.15242i
$872$	−6.38121e11			−1.10367
$873$		0			0
$874$	−2.78570e10			−0.0477407
$875$		−	2.95080e11i		−	0.503394i
$876$		0			0
$877$	−3.62151e11	+	3.62151e11i	−0.612197	+	0.612197i	−0.943518	−	0.331321i	$-0.892506\pi$
$877$	−3.62151e11	+	3.62151e11i	−0.612197	+	0.612197i	0.331321	+	0.943518i	$0.392506\pi$
$878$	−8.46252e11	−	8.46252e11i	−1.42404	−	1.42404i
$879$		0			0
$880$	−1.35349e11			−0.225697
$881$			5.54599e11i			0.920610i	0.887761	+	0.460305i	$0.152260\pi$
$881$			5.54599e11i			0.920610i	−0.887761	+	0.460305i	$0.847740\pi$
$882$		0			0
$883$			1.02832e12i			1.69155i	0.533539	+	0.845775i	$0.320862\pi$
$883$			1.02832e12i			1.69155i	−0.533539	+	0.845775i	$0.679138\pi$
$884$	4.94276e10	−	9.08395e10i	0.0809395	−	0.148753i
$885$		0			0
$886$	−4.76078e11	+	4.76078e11i	−0.772580	+	0.772580i
$887$	3.28797e11			0.531170			0.265585	−	0.964087i	$-0.414435\pi$
$887$	3.28797e11			0.531170			0.265585	+	0.964087i	$0.414435\pi$
$888$		0			0
$889$	2.36396e11	−	2.36396e11i	0.378472	−	0.378472i
$890$	2.52245e12	−	2.52245e12i	4.02034	−	4.02034i
$891$		0			0
$892$	1.46105e12	+	1.46105e12i	2.30784	+	2.30784i
$893$	−7.56309e10			−0.118931
$894$		0			0
$895$	−4.87118e11	−	4.87118e11i	−0.759175	−	0.759175i
$896$			8.54607e11i			1.32597i
$897$		0			0
$898$	−6.27641e10			−0.0965174
$899$	−3.14151e11	+	3.14151e11i	−0.480950	+	0.480950i
$900$		0			0
$901$		−	6.97635e10i		−	0.105859i
$902$	−4.51364e10	+	4.51364e10i	−0.0681868	+	0.0681868i
$903$		0			0
$904$	9.02664e11	+	9.02664e11i	1.35161	+	1.35161i
$905$	−6.00967e11	−	6.00967e11i	−0.895893	−	0.895893i
$906$		0			0
$907$		−	1.16707e12i		−	1.72452i	−0.506464	−	0.862261i	$-0.669048\pi$
$907$		−	1.16707e12i		−	1.72452i	0.506464	−	0.862261i	$-0.330952\pi$
$908$	8.13191e11	+	8.13191e11i	1.19632	+	1.19632i
$909$		0			0
$910$	−3.37537e11	−	1.14328e12i	−0.492216	−	1.66719i
$911$	−1.42075e11			−0.206273			−0.103137	−	0.994667i	$-0.532888\pi$
$911$	−1.42075e11			−0.206273			−0.103137	+	0.994667i	$0.532888\pi$
$912$		0			0
$913$	2.81967e10			0.0405803
$914$		−	1.91341e12i		−	2.74173i
$915$		0			0
$916$	−9.61389e11	+	9.61389e11i	−1.36558	+	1.36558i
$917$	2.95389e11	+	2.95389e11i	0.417751	+	0.417751i
$918$		0			0
$919$	6.85305e11			0.960775			0.480387	−	0.877056i	$-0.340496\pi$
$919$	6.85305e11			0.960775			0.480387	+	0.877056i	$0.340496\pi$
$920$		−	1.13475e12i		−	1.58398i
$921$		0			0
$922$		−	3.28503e11i		−	0.454586i
$923$	1.78000e11	+	6.02906e11i	0.245252	+	0.830698i
$924$		0			0
$925$	6.35908e11	−	6.35908e11i	0.868616	−	0.868616i
$926$	−2.35038e12			−3.19664
$927$		0			0
$928$	−1.93861e12	+	1.93861e12i	−2.61395	+	2.61395i
$929$	−6.19559e11	+	6.19559e11i	−0.831802	+	0.831802i	−0.987763	−	0.155962i	$-0.950152\pi$
$929$	−6.19559e11	+	6.19559e11i	−0.831802	+	0.831802i	0.155962	+	0.987763i	$0.450152\pi$
$930$		0			0
$931$	2.86206e10	+	2.86206e10i	0.0380960	+	0.0380960i
$932$	2.82431e12			3.74325
$933$		0			0
$934$	2.87206e11	+	2.87206e11i	0.377404	+	0.377404i
$935$			3.24527e9i			0.00424624i
$936$		0			0
$937$	−1.22051e11			−0.158337			−0.0791686	−	0.996861i	$-0.525227\pi$
$937$	−1.22051e11			−0.158337			−0.0791686	+	0.996861i	$0.525227\pi$
$938$	−7.15510e11	+	7.15510e11i	−0.924282	+	0.924282i
$939$		0			0
$940$		−	4.94309e12i		−	6.33122i
$941$	6.18228e10	−	6.18228e10i	0.0788479	−	0.0788479i	−0.666583	−	0.745431i	$-0.732244\pi$
$941$	6.18228e10	−	6.18228e10i	0.0788479	−	0.0788479i	0.745431	+	0.666583i	$0.232244\pi$
$942$		0			0
$943$	−2.12269e11	−	2.12269e11i	−0.268436	−	0.268436i
$944$	−1.50488e12	−	1.50488e12i	−1.89502	−	1.89502i
$945$		0			0
$946$		−	2.10412e10i		−	0.0262728i
$947$	6.33654e11	+	6.33654e11i	0.787866	+	0.787866i	0.981144	−	0.193278i	$-0.0619119\pi$
$947$	6.33654e11	+	6.33654e11i	0.787866	+	0.787866i	−0.193278	+	0.981144i	$0.561912\pi$
$948$		0			0
$949$	1.29116e11	−	3.81196e10i	0.159189	−	0.0469984i
$950$	1.92738e11			0.236631
$951$		0			0
$952$	9.43091e10			0.114817
$953$		−	5.68408e11i		−	0.689110i	−0.938766	−	0.344555i	$-0.888030\pi$
$953$		−	5.68408e11i		−	0.689110i	0.938766	−	0.344555i	$-0.111970\pi$
$954$		0			0
$955$	4.88258e10	−	4.88258e10i	0.0586997	−	0.0586997i
$956$	1.24140e12	+	1.24140e12i	1.48621	+	1.48621i
$957$		0			0
$958$	2.90760e12			3.45202
$959$		−	3.62225e11i		−	0.428256i
$960$		0			0
$961$			5.34612e11i			0.626823i
$962$	6.18721e11	−	1.13710e12i	0.722428	−	1.32770i
$963$		0			0
$964$	−1.98387e12	+	1.98387e12i	−2.29724	+	2.29724i
$965$	−3.48828e11			−0.402255
$966$		0			0
$967$	7.72254e11	−	7.72254e11i	0.883190	−	0.883190i	−0.110667	−	0.993858i	$-0.535299\pi$
$967$	7.72254e11	−	7.72254e11i	0.883190	−	0.883190i	0.993858	+	0.110667i	$0.0352988\pi$
$968$	−1.96001e12	+	1.96001e12i	−2.23232	+	2.23232i
$969$		0			0
$970$	−2.14521e12	−	2.14521e12i	−2.42317	−	2.42317i
$971$	1.17024e11			0.131643			0.0658217	−	0.997831i	$-0.479033\pi$
$971$	1.17024e11			0.131643			0.0658217	+	0.997831i	$0.479033\pi$
$972$		0			0
$973$	−4.60443e11	−	4.60443e11i	−0.513717	−	0.513717i
$974$			1.07172e12i			1.19082i
$975$		0			0
$976$	−7.74899e11			−0.853977
$977$	1.12272e12	−	1.12272e12i	1.23224	−	1.23224i	0.269138	−	0.963102i	$-0.413261\pi$
$977$	1.12272e12	−	1.12272e12i	1.23224	−	1.23224i	0.963102	−	0.269138i	$-0.0867387\pi$
$978$		0			0
$979$			7.12092e10i			0.0775185i
$980$	−1.87059e12	+	1.87059e12i	−2.02802	+	2.02802i
$981$		0			0
$982$	2.66026e11	+	2.66026e11i	0.286074	+	0.286074i
$983$	6.78634e11	+	6.78634e11i	0.726811	+	0.726811i	0.969983	−	0.243172i	$-0.0781881\pi$
$983$	6.78634e11	+	6.78634e11i	0.726811	+	0.726811i	−0.243172	+	0.969983i	$0.578188\pi$
$984$		0			0
$985$		−	3.95375e11i		−	0.420015i
$986$	9.07568e10	+	9.07568e10i	0.0960221	+	0.0960221i
$987$		0			0
$988$	1.93273e11	−	5.70610e10i	0.202835	−	0.0598842i
$989$	9.89536e10			0.103430
$990$		0			0
$991$	1.48575e12			1.54046			0.770231	−	0.637765i	$-0.220141\pi$
$991$	1.48575e12			1.54046			0.770231	+	0.637765i	$0.220141\pi$
$992$		−	1.96408e12i		−	2.02821i
$993$		0			0
$994$	−6.50400e11	+	6.50400e11i	−0.666246	+	0.666246i
$995$	−1.46538e11	−	1.46538e11i	−0.149505	−	0.149505i
$996$		0			0
$997$	1.65986e11			0.167992			0.0839962	−	0.996466i	$-0.473232\pi$
$997$	1.65986e11			0.167992			0.0839962	+	0.996466i	$0.473232\pi$
$998$		−	1.51871e12i		−	1.53092i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.9.j.a.109.9		18
3.2	odd	2		13.9.d.a.5.1	✓	18
13.8	odd	4	inner	117.9.j.a.73.9		18
39.8	even	4		13.9.d.a.8.1	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.9.d.a.5.1	✓	18	3.2	odd	2
13.9.d.a.8.1	yes	18	39.8	even	4
117.9.j.a.73.9		18	13.8	odd	4	inner
117.9.j.a.109.9		18	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q+(21.6276 - 21.6276i) q^{2} -679.508i q^{4} +(706.219 - 706.219i) q^{5} +(966.123 + 966.123i) q^{7} +(-9159.48 - 9159.48i) q^{8} +O(q^{10})\)	\(q+(21.6276 - 21.6276i) q^{2} -679.508i q^{4} +(706.219 - 706.219i) q^{5} +(966.123 + 966.123i) q^{7} +(-9159.48 - 9159.48i) q^{8} -30547.7i q^{10} +(431.183 + 431.183i) q^{11} +(13650.7 - 25087.6i) q^{13} +41789.9 q^{14} -222241. q^{16} +5328.69i q^{17} +(-7342.35 + 7342.35i) q^{19} +(-479882. - 479882. i) q^{20} +18650.9 q^{22} +87712.3i q^{23} -606866. i q^{25} +(-247354. - 837818. i) q^{26} +(656489. - 656489. i) q^{28} +787498. q^{29} +(-398923. + 398923. i) q^{31} +(-2.46173e6 + 2.46173e6i) q^{32} +(115247. + 115247. i) q^{34} +1.36459e6 q^{35} +(1.04786e6 + 1.04786e6i) q^{37} +317595. i q^{38} -1.29372e7 q^{40} +(-2.42006e6 + 2.42006e6i) q^{41} -1.12816e6i q^{43} +(292992. - 292992. i) q^{44} +(1.89701e6 + 1.89701e6i) q^{46} +(5.15032e6 + 5.15032e6i) q^{47} -3.89801e6i q^{49} +(-1.31251e7 - 1.31251e7i) q^{50} +(-1.70473e7 - 9.27576e6i) q^{52} -1.30921e7 q^{53} +609019. q^{55} -1.76984e7i q^{56} +(1.70317e7 - 1.70317e7i) q^{58} +(6.77137e6 + 6.77137e6i) q^{59} +3.48674e6 q^{61} +1.72555e7i q^{62} +4.95888e7i q^{64} +(-8.07699e6 - 2.73577e7i) q^{65} +(-1.71216e7 + 1.71216e7i) q^{67} +3.62089e6 q^{68} +(2.95128e7 - 2.95128e7i) q^{70} +(-1.55636e7 + 1.55636e7i) q^{71} +(3.33302e6 + 3.33302e6i) q^{73} +4.53253e7 q^{74} +(4.98919e6 + 4.98919e6i) q^{76} +833151. i q^{77} -4.95167e6 q^{79} +(-1.56951e8 + 1.56951e8i) q^{80} +1.04680e8i q^{82} +(3.26970e7 - 3.26970e7i) q^{83} +(3.76322e6 + 3.76322e6i) q^{85} +(-2.43995e7 - 2.43995e7i) q^{86} -7.89882e6i q^{88} +(8.25743e7 + 8.25743e7i) q^{89} +(3.74260e7 - 1.10495e7i) q^{91} +5.96012e7 q^{92} +2.22779e8 q^{94} +1.03706e7i q^{95} +(7.02251e7 - 7.02251e7i) q^{97} +(-8.43048e7 - 8.43048e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8}+O(q^{10}) \) 18 * q + 2 * q^2 - 166 * q^5 + 5308 * q^7 - 10464 * q^8	\( 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8} + 31556 q^{11} + 71300 q^{13} + 110260 q^{14} - 522860 q^{16} + 100288 q^{19} - 736268 q^{20} - 977312 q^{22} - 2952238 q^{26} + 4497084 q^{28} + 2479024 q^{29} - 1892664 q^{31} - 947212 q^{32} - 531576 q^{34} + 2918284 q^{35} - 8343978 q^{37} - 12691908 q^{40} - 1140178 q^{41} + 3867188 q^{44} + 2006148 q^{46} + 13368572 q^{47} - 37369598 q^{50} - 14821220 q^{52} - 50561348 q^{53} + 76994128 q^{55} + 22505716 q^{58} - 2127976 q^{59} - 52016516 q^{61} - 10413082 q^{65} + 960292 q^{67} - 283187508 q^{68} - 166635032 q^{70} - 67412140 q^{71} - 145213226 q^{73} + 233620024 q^{74} - 150533640 q^{76} - 76829120 q^{79} - 524889520 q^{80} + 241951556 q^{83} + 260737764 q^{85} + 579480384 q^{86} + 89187110 q^{89} + 232660948 q^{91} + 122690376 q^{92} + 1069637380 q^{94} + 331183146 q^{97} - 588677614 q^{98}+O(q^{100}) \) 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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