LMFDB - Embedded newform 117.9.j.a.109.2

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			109.2
Root			$-13.3036 + 12.3036i$ of defining polynomial
Character	$\chi$	$=$	117.109
Dual form			117.9.j.a.73.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-13.3036 + 13.3036i) q^{2} -97.9733i q^{4} +(454.715 - 454.715i) q^{5} +(-1427.64 - 1427.64i) q^{7} +(-2102.33 - 2102.33i) q^{8} +O(q^{10})$	\(q+(-13.3036 + 13.3036i) q^{2} -97.9733i q^{4} +(454.715 - 454.715i) q^{5} +(-1427.64 - 1427.64i) q^{7} +(-2102.33 - 2102.33i) q^{8} +12098.7i q^{10} +(16692.4 + 16692.4i) q^{11} +(-24173.6 + 15210.8i) q^{13} +37985.5 q^{14} +81018.4 q^{16} +12812.2i q^{17} +(65286.1 - 65286.1i) q^{19} +(-44549.9 - 44549.9i) q^{20} -444139. q^{22} -107769. i q^{23} -22905.6i q^{25} +(119238. - 523955. i) q^{26} +(-139870. + 139870. i) q^{28} +106806. q^{29} +(-1.20080e6 + 1.20080e6i) q^{31} +(-539643. + 539643. i) q^{32} +(-170448. - 170448. i) q^{34} -1.29833e6 q^{35} +(-1.27244e6 - 1.27244e6i) q^{37} +1.73709e6i q^{38} -1.91192e6 q^{40} +(2.57699e6 - 2.57699e6i) q^{41} -5.33128e6i q^{43} +(1.63541e6 - 1.63541e6i) q^{44} +(1.43372e6 + 1.43372e6i) q^{46} +(-5.14845e6 - 5.14845e6i) q^{47} -1.68851e6i q^{49} +(304728. + 304728. i) q^{50} +(1.49025e6 + 2.36837e6i) q^{52} +2.13711e6 q^{53} +1.51805e7 q^{55} +6.00273e6i q^{56} +(-1.42091e6 + 1.42091e6i) q^{58} +(1.34172e7 + 1.34172e7i) q^{59} -9.15548e6 q^{61} -3.19500e7i q^{62} +6.38229e6i q^{64} +(-4.07553e6 + 1.79086e7i) q^{65} +(2.27853e7 - 2.27853e7i) q^{67} +1.25525e6 q^{68} +(1.72726e7 - 1.72726e7i) q^{70} +(1.70237e7 - 1.70237e7i) q^{71} +(5.23825e6 + 5.23825e6i) q^{73} +3.38562e7 q^{74} +(-6.39630e6 - 6.39630e6i) q^{76} -4.76613e7i q^{77} -1.73854e7 q^{79} +(3.68402e7 - 3.68402e7i) q^{80} +6.85667e7i q^{82} +(3.43716e7 - 3.43716e7i) q^{83} +(5.82587e6 + 5.82587e6i) q^{85} +(7.09254e7 + 7.09254e7i) q^{86} -7.01858e7i q^{88} +(-2.40569e7 - 2.40569e7i) q^{89} +(5.62266e7 + 1.27957e7i) q^{91} -1.05585e7 q^{92} +1.36986e8 q^{94} -5.93731e7i q^{95} +(1.99560e7 - 1.99560e7i) q^{97} +(2.24633e7 + 2.24633e7i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−13.3036	+	13.3036i	−0.831477	+	0.831477i	−0.987719	−	0.156242i	$-0.950062\pi$
$2$	−13.3036	+	13.3036i	−0.831477	+	0.831477i	0.156242	+	0.987719i	$0.450062\pi$
$3$		0			0
$3$		0			0
$4$		−	97.9733i		−	0.382708i
$5$	454.715	−	454.715i	0.727543	−	0.727543i	−0.242586	−	0.970130i	$-0.577996\pi$
$5$	454.715	−	454.715i	0.727543	−	0.727543i	0.970130	+	0.242586i	$0.0779958\pi$
$6$		0			0
$7$	−1427.64	−	1427.64i	−0.594601	−	0.594601i	0.344270	−	0.938871i	$-0.388127\pi$
$7$	−1427.64	−	1427.64i	−0.594601	−	0.594601i	−0.938871	+	0.344270i	$0.888127\pi$
$8$	−2102.33	−	2102.33i	−0.513264	−	0.513264i
$9$		0			0
$10$			12098.7i			1.20987i
$11$	16692.4	+	16692.4i	1.14011	+	1.14011i	0.988429	+	0.151683i	$0.0484692\pi$
$11$	16692.4	+	16692.4i	1.14011	+	1.14011i	0.151683	+	0.988429i	$0.451531\pi$
$12$		0			0
$13$	−24173.6	+	15210.8i	−0.846385	+	0.532571i
$13$	−24173.6	+	15210.8i	−0.846385	+	0.532571i
$14$	37985.5			0.988794
$15$		0			0
$16$	81018.4			1.23624
$17$			12812.2i			0.153400i	0.997054	+	0.0767002i	$0.0244384\pi$
$17$			12812.2i			0.153400i	−0.997054	+	0.0767002i	$0.975562\pi$
$18$		0			0
$19$	65286.1	−	65286.1i	0.500964	−	0.500964i	−0.410773	−	0.911737i	$-0.634741\pi$
$19$	65286.1	−	65286.1i	0.500964	−	0.500964i	0.911737	+	0.410773i	$0.134741\pi$
$20$	−44549.9	−	44549.9i	−0.278437	−	0.278437i
$21$		0			0
$22$	−444139.			−1.89595
$23$		−	107769.i		−	0.385108i	−0.981286	−	0.192554i	$-0.938323\pi$
$23$		−	107769.i		−	0.385108i	0.981286	−	0.192554i	$-0.0616770\pi$
$24$		0			0
$25$		−	22905.6i		−	0.0586385i
$26$	119238.	−	523955.i	0.260929	−	1.14657i
$27$		0			0
$28$	−139870.	+	139870.i	−0.227559	+	0.227559i
$29$	106806.			0.151009			0.0755047	−	0.997145i	$-0.475943\pi$
$29$	106806.			0.151009			0.0755047	+	0.997145i	$0.475943\pi$
$30$		0			0
$31$	−1.20080e6	+	1.20080e6i	−1.30024	+	1.30024i	−0.372015	+	0.928227i	$0.621333\pi$
$31$	−1.20080e6	+	1.20080e6i	−1.30024	+	1.30024i	−0.928227	+	0.372015i	$0.878667\pi$
$32$	−539643.	+	539643.i	−0.514643	+	0.514643i
$33$		0			0
$34$	−170448.	−	170448.i	−0.127549	−	0.127549i
$35$	−1.29833e6			−0.865196
$36$		0			0
$37$	−1.27244e6	−	1.27244e6i	−0.678940	−	0.678940i	0.280820	−	0.959760i	$-0.409394\pi$
$37$	−1.27244e6	−	1.27244e6i	−0.678940	−	0.678940i	−0.959760	+	0.280820i	$0.909394\pi$
$38$			1.73709e6i			0.833080i
$39$		0			0
$40$	−1.91192e6			−0.746844
$41$	2.57699e6	−	2.57699e6i	0.911964	−	0.911964i	−0.0844626	−	0.996427i	$-0.526917\pi$
$41$	2.57699e6	−	2.57699e6i	0.911964	−	0.911964i	0.996427	+	0.0844626i	$0.0269173\pi$
$42$		0			0
$43$		−	5.33128e6i		−	1.55940i	−0.626153	−	0.779700i	$-0.715372\pi$
$43$		−	5.33128e6i		−	1.55940i	0.626153	−	0.779700i	$-0.284628\pi$
$44$	1.63541e6	−	1.63541e6i	0.436330	−	0.436330i
$45$		0			0
$46$	1.43372e6	+	1.43372e6i	0.320208	+	0.320208i
$47$	−5.14845e6	−	5.14845e6i	−1.05508	−	1.05508i	−0.998392	−	0.0566865i	$-0.981946\pi$
$47$	−5.14845e6	−	5.14845e6i	−1.05508	−	1.05508i	−0.0566865	−	0.998392i	$-0.518054\pi$
$48$		0			0
$49$		−	1.68851e6i		−	0.292899i
$50$	304728.	+	304728.i	0.0487565	+	0.0487565i
$51$		0			0
$52$	1.49025e6	+	2.36837e6i	0.203819	+	0.323918i
$53$	2.13711e6			0.270847			0.135423	−	0.990788i	$-0.456761\pi$
$53$	2.13711e6			0.270847			0.135423	+	0.990788i	$0.456761\pi$
$54$		0			0
$55$	1.51805e7			1.65896
$56$			6.00273e6i			0.610375i
$57$		0			0
$58$	−1.42091e6	+	1.42091e6i	−0.125561	+	0.125561i
$59$	1.34172e7	+	1.34172e7i	1.10727	+	1.10727i	0.993508	+	0.113765i	$0.0362913\pi$
$59$	1.34172e7	+	1.34172e7i	1.10727	+	1.10727i	0.113765	+	0.993508i	$0.463709\pi$
$60$		0			0
$61$	−9.15548e6			−0.661244			−0.330622	−	0.943763i	$-0.607259\pi$
$61$	−9.15548e6			−0.661244			−0.330622	+	0.943763i	$0.607259\pi$
$62$		−	3.19500e7i		−	2.16224i
$63$		0			0
$64$			6.38229e6i			0.380414i
$65$	−4.07553e6	+	1.79086e7i	−0.228313	+	1.00325i
$66$		0			0
$67$	2.27853e7	−	2.27853e7i	1.13072	−	1.13072i	0.140664	−	0.990057i	$-0.455076\pi$
$67$	2.27853e7	−	2.27853e7i	1.13072	−	1.13072i	0.990057	−	0.140664i	$-0.0449239\pi$
$68$	1.25525e6			0.0587076
$69$		0			0
$70$	1.72726e7	−	1.72726e7i	0.719390	−	0.719390i
$71$	1.70237e7	−	1.70237e7i	0.669918	−	0.669918i	−0.287779	−	0.957697i	$-0.592917\pi$
$71$	1.70237e7	−	1.70237e7i	0.669918	−	0.669918i	0.957697	+	0.287779i	$0.0929168\pi$
$72$		0			0
$73$	5.23825e6	+	5.23825e6i	0.184457	+	0.184457i	0.793295	−	0.608838i	$-0.208364\pi$
$73$	5.23825e6	+	5.23825e6i	0.184457	+	0.184457i	−0.608838	+	0.793295i	$0.708364\pi$
$74$	3.38562e7			1.12905
$75$		0			0
$76$	−6.39630e6	−	6.39630e6i	−0.191723	−	0.191723i
$77$		−	4.76613e7i		−	1.35582i
$78$		0			0
$79$	−1.73854e7			−0.446351			−0.223176	−	0.974778i	$-0.571642\pi$
$79$	−1.73854e7			−0.446351			−0.223176	+	0.974778i	$0.571642\pi$
$80$	3.68402e7	−	3.68402e7i	0.899420	−	0.899420i
$81$		0			0
$82$			6.85667e7i			1.51655i
$83$	3.43716e7	−	3.43716e7i	0.724248	−	0.724248i	−0.245219	−	0.969468i	$-0.578860\pi$
$83$	3.43716e7	−	3.43716e7i	0.724248	−	0.724248i	0.969468	+	0.245219i	$0.0788600\pi$
$84$		0			0
$85$	5.82587e6	+	5.82587e6i	0.111605	+	0.111605i
$86$	7.09254e7	+	7.09254e7i	1.29661	+	1.29661i
$87$		0			0
$88$		−	7.01858e7i		−	1.17036i
$89$	−2.40569e7	−	2.40569e7i	−0.383425	−	0.383425i	0.488910	−	0.872334i	$-0.337395\pi$
$89$	−2.40569e7	−	2.40569e7i	−0.383425	−	0.383425i	−0.872334	+	0.488910i	$0.837395\pi$
$90$		0			0
$91$	5.62266e7	+	1.27957e7i	0.819929	+	0.186594i
$92$	−1.05585e7			−0.147384
$93$		0			0
$94$	1.36986e8			1.75455
$95$		−	5.93731e7i		−	0.728946i
$96$		0			0
$97$	1.99560e7	−	1.99560e7i	0.225417	−	0.225417i	−0.585358	−	0.810775i	$-0.699046\pi$
$97$	1.99560e7	−	1.99560e7i	0.225417	−	0.225417i	0.810775	+	0.585358i	$0.199046\pi$
$98$	2.24633e7	+	2.24633e7i	0.243539	+	0.243539i
$99$		0			0
$100$	−2.24414e6			−0.0224414
$101$		−	6.81167e7i		−	0.654588i	−0.944923	−	0.327294i	$-0.893863\pi$
$101$		−	6.81167e7i		−	0.654588i	0.944923	−	0.327294i	$-0.106137\pi$
$102$		0			0
$103$		−	7.00269e7i		−	0.622180i	−0.950380	−	0.311090i	$-0.899306\pi$
$103$		−	7.00269e7i		−	0.622180i	0.950380	−	0.311090i	$-0.100694\pi$
$104$	8.27989e7	+	1.88428e7i	0.707769	+	0.161069i
$105$		0			0
$106$	−2.84313e7	+	2.84313e7i	−0.225203	+	0.225203i
$107$	−1.48571e8			−1.13344			−0.566722	−	0.823909i	$-0.691789\pi$
$107$	−1.48571e8			−1.13344			−0.566722	+	0.823909i	$0.691789\pi$
$108$		0			0
$109$	−3.24334e7	+	3.24334e7i	−0.229767	+	0.229767i	−0.812595	−	0.582828i	$-0.801946\pi$
$109$	−3.24334e7	+	3.24334e7i	−0.229767	+	0.229767i	0.582828	+	0.812595i	$0.301946\pi$
$110$	−2.01956e8	+	2.01956e8i	−1.37939	+	1.37939i
$111$		0			0
$112$	−1.15665e8	−	1.15665e8i	−0.735071	−	0.735071i
$113$	−8.18150e6			−0.0501787			−0.0250893	−	0.999685i	$-0.507987\pi$
$113$	−8.18150e6			−0.0501787			−0.0250893	+	0.999685i	$0.507987\pi$
$114$		0			0
$115$	−4.90041e7	−	4.90041e7i	−0.280183	−	0.280183i
$116$		−	1.04641e7i		−	0.0577925i
$117$		0			0
$118$	−3.56996e8			−1.84134
$119$	1.82911e7	−	1.82911e7i	0.0912120	−	0.0912120i
$120$		0			0
$121$			3.42912e8i			1.59971i
$122$	1.21801e8	−	1.21801e8i	0.549809	−	0.549809i
$123$		0			0
$124$	1.17646e8	+	1.17646e8i	0.497613	+	0.497613i
$125$	1.67207e8	+	1.67207e8i	0.684881	+	0.684881i
$126$		0			0
$127$		−	2.30499e8i		−	0.886042i	−0.896511	−	0.443021i	$-0.853907\pi$
$127$		−	2.30499e8i		−	0.886042i	0.896511	−	0.443021i	$-0.146093\pi$
$128$	−2.23056e8	−	2.23056e8i	−0.830949	−	0.830949i
$129$		0			0
$130$	−1.84031e8	−	2.92469e8i	−0.644343	−	1.02402i
$131$	−1.67860e8			−0.569984			−0.284992	−	0.958530i	$-0.591991\pi$
$131$	−1.67860e8			−0.569984			−0.284992	+	0.958530i	$0.591991\pi$
$132$		0			0
$133$	−1.86410e8			−0.595747
$134$			6.06255e8i			1.88034i
$135$		0			0
$136$	2.69354e7	−	2.69354e7i	0.0787349	−	0.0787349i
$137$	−1.38902e8	−	1.38902e8i	−0.394299	−	0.394299i	0.481918	−	0.876217i	$-0.339940\pi$
$137$	−1.38902e8	−	1.38902e8i	−0.394299	−	0.394299i	−0.876217	+	0.481918i	$0.839940\pi$
$138$		0			0
$139$	1.62858e8			0.436264			0.218132	−	0.975919i	$-0.430004\pi$
$139$	1.62858e8			0.436264			0.218132	+	0.975919i	$0.430004\pi$
$140$			1.27202e8i			0.331117i
$141$		0			0
$142$			4.52955e8i			1.11404i
$143$	−6.57419e8	−	1.49611e8i	−1.57216	−	0.357783i
$144$		0			0
$145$	4.85663e7	−	4.85663e7i	0.109866	−	0.109866i
$146$	−1.39376e8			−0.306744
$147$		0			0
$148$	−1.24665e8	+	1.24665e8i	−0.259836	+	0.259836i
$149$	4.85659e8	−	4.85659e8i	0.985341	−	0.985341i	−0.0145536	−	0.999894i	$-0.504633\pi$
$149$	4.85659e8	−	4.85659e8i	0.985341	−	0.985341i	0.999894	+	0.0145536i	$0.00463271\pi$
$150$		0			0
$151$	−2.27459e8	−	2.27459e8i	−0.437518	−	0.437518i	0.453658	−	0.891176i	$-0.350119\pi$
$151$	−2.27459e8	−	2.27459e8i	−0.437518	−	0.437518i	−0.891176	+	0.453658i	$0.850119\pi$
$152$	−2.74506e8			−0.514254
$153$		0			0
$154$	6.34069e8	+	6.34069e8i	1.12734	+	1.12734i
$155$			1.09204e9i			1.89196i
$156$		0			0
$157$	5.02066e7			0.0826347			0.0413173	−	0.999146i	$-0.486845\pi$
$157$	5.02066e7			0.0826347			0.0413173	+	0.999146i	$0.486845\pi$
$158$	2.31289e8	−	2.31289e8i	0.371131	−	0.371131i
$159$		0			0
$160$			4.90767e8i			0.748851i
$161$	−1.53855e8	+	1.53855e8i	−0.228985	+	0.228985i
$162$		0			0
$163$	−4.26362e8	−	4.26362e8i	−0.603988	−	0.603988i	0.337381	−	0.941368i	$-0.390459\pi$
$163$	−4.26362e8	−	4.26362e8i	−0.603988	−	0.603988i	−0.941368	+	0.337381i	$0.890459\pi$
$164$	−2.52476e8	−	2.52476e8i	−0.349016	−	0.349016i
$165$		0			0
$166$			9.14534e8i			1.20439i
$167$	7.17804e8	+	7.17804e8i	0.922868	+	0.922868i	0.997231	−	0.0743630i	$-0.0236923\pi$
$167$	7.17804e8	+	7.17804e8i	0.922868	+	0.922868i	−0.0743630	+	0.997231i	$0.523692\pi$
$168$		0			0
$169$	3.52995e8	−	7.35398e8i	0.432735	−	0.901521i
$170$	−1.55011e8			−0.185595
$171$		0			0
$172$	−5.22323e8			−0.596795
$173$		−	1.10658e9i		−	1.23538i	−0.786423	−	0.617689i	$-0.788069\pi$
$173$		−	1.10658e9i		−	1.23538i	0.786423	−	0.617689i	$-0.211931\pi$
$174$		0			0
$175$	−3.27009e7	+	3.27009e7i	−0.0348665	+	0.0348665i
$176$	1.35239e9	+	1.35239e9i	1.40945	+	1.40945i
$177$		0			0
$178$	6.40089e8			0.637618
$179$		−	1.08843e9i		−	1.06020i	−0.847935	−	0.530100i	$-0.822155\pi$
$179$		−	1.08843e9i		−	1.06020i	0.847935	−	0.530100i	$-0.177845\pi$
$180$		0			0
$181$		−	1.33202e9i		−	1.24107i	−0.784179	−	0.620535i	$-0.786915\pi$
$181$		−	1.33202e9i		−	1.24107i	0.784179	−	0.620535i	$-0.213085\pi$
$182$	−9.18247e8	+	5.77789e8i	−0.836901	+	0.526603i
$183$		0			0
$184$	−2.26566e8	+	2.26566e8i	−0.197662	+	0.197662i
$185$	−1.15720e9			−0.987917
$186$		0			0
$187$	−2.13865e8	+	2.13865e8i	−0.174894	+	0.174894i
$188$	−5.04410e8	+	5.04410e8i	−0.403787	+	0.403787i
$189$		0			0
$190$	7.89878e8	+	7.89878e8i	0.606102	+	0.606102i
$191$	−6.31320e7			−0.0474369			−0.0237184	−	0.999719i	$-0.507551\pi$
$191$	−6.31320e7			−0.0474369			−0.0237184	+	0.999719i	$0.507551\pi$
$192$		0			0
$193$	6.15182e8	+	6.15182e8i	0.443378	+	0.443378i	0.893146	−	0.449768i	$-0.148493\pi$
$193$	6.15182e8	+	6.15182e8i	0.443378	+	0.443378i	−0.449768	+	0.893146i	$0.648493\pi$
$194$			5.30976e8i			0.374859i
$195$		0			0
$196$	−1.65429e8			−0.112095
$197$	−3.31842e8	+	3.31842e8i	−0.220326	+	0.220326i	−0.808636	−	0.588309i	$-0.799794\pi$
$197$	−3.31842e8	+	3.31842e8i	−0.220326	+	0.220326i	0.588309	+	0.808636i	$0.299794\pi$
$198$		0			0
$199$		−	5.42447e8i		−	0.345895i	−0.984931	−	0.172948i	$-0.944671\pi$
$199$		−	5.42447e8i		−	0.345895i	0.984931	−	0.172948i	$-0.0553292\pi$
$200$	−4.81552e7	+	4.81552e7i	−0.0300970	+	0.0300970i
$201$		0			0
$202$	9.06199e8	+	9.06199e8i	0.544275	+	0.544275i
$203$	−1.52480e8	−	1.52480e8i	−0.0897903	−	0.0897903i
$204$		0			0
$205$		−	2.34359e9i		−	1.32699i
$206$	9.31612e8	+	9.31612e8i	0.517328	+	0.517328i
$207$		0			0
$208$	−1.95851e9	+	1.23235e9i	−1.04634	+	0.658388i
$209$	2.17956e9			1.14231
$210$		0			0
$211$	−1.13712e8			−0.0573688			−0.0286844	−	0.999589i	$-0.509132\pi$
$211$	−1.13712e8			−0.0573688			−0.0286844	+	0.999589i	$0.509132\pi$
$212$		−	2.09380e8i		−	0.103655i
$213$		0			0
$214$	1.97654e9	−	1.97654e9i	0.942433	−	0.942433i
$215$	−2.42421e9	−	2.42421e9i	−1.13453	−	1.13453i
$216$		0			0
$217$	3.42861e9			1.54625
$218$		−	8.62965e8i		−	0.382091i
$219$		0			0
$220$		−	1.48729e9i		−	0.634898i
$221$	−1.94883e8	−	3.09716e8i	−0.0816967	−	0.129836i
$222$		0			0
$223$	−2.61690e9	+	2.61690e9i	−1.05820	+	1.05820i	−0.0600027	+	0.998198i	$0.519111\pi$
$223$	−2.61690e9	+	2.61690e9i	−1.05820	+	1.05820i	−0.998198	+	0.0600027i	$0.980889\pi$
$224$	1.54083e9			0.612015
$225$		0			0
$226$	1.08844e8	−	1.08844e8i	0.0417224	−	0.0417224i
$227$	1.07202e9	−	1.07202e9i	0.403739	−	0.403739i	−0.475809	−	0.879549i	$-0.657845\pi$
$227$	1.07202e9	−	1.07202e9i	0.403739	−	0.403739i	0.879549	+	0.475809i	$0.157845\pi$
$228$		0			0
$229$	−1.19273e9	−	1.19273e9i	−0.433710	−	0.433710i	0.456178	−	0.889888i	$-0.349218\pi$
$229$	−1.19273e9	−	1.19273e9i	−0.433710	−	0.433710i	−0.889888	+	0.456178i	$0.849218\pi$
$230$	1.30386e9			0.465931
$231$		0			0
$232$	−2.24542e8	−	2.24542e8i	−0.0775077	−	0.0775077i
$233$		−	4.33962e8i		−	0.147241i	−0.997286	−	0.0736205i	$-0.976545\pi$
$233$		−	4.33962e8i		−	0.147241i	0.997286	−	0.0736205i	$-0.0234553\pi$
$234$		0			0
$235$	−4.68215e9			−1.53523
$236$	1.31453e9	−	1.31453e9i	0.423762	−	0.423762i
$237$		0			0
$238$			4.86676e8i			0.151681i
$239$	−1.62550e9	+	1.62550e9i	−0.498189	+	0.498189i	−0.910874	−	0.412685i	$-0.864591\pi$
$239$	−1.62550e9	+	1.62550e9i	−0.498189	+	0.498189i	0.412685	+	0.910874i	$0.364591\pi$
$240$		0			0
$241$	−2.47661e9	−	2.47661e9i	−0.734156	−	0.734156i	0.237284	−	0.971440i	$-0.423743\pi$
$241$	−2.47661e9	−	2.47661e9i	−0.734156	−	0.734156i	−0.971440	+	0.237284i	$0.923743\pi$
$242$	−4.56198e9	−	4.56198e9i	−1.33012	−	1.33012i
$243$		0			0
$244$			8.96993e8i			0.253064i
$245$	−7.67789e8	−	7.67789e8i	−0.213097	−	0.213097i
$246$		0			0
$247$	−5.85148e8	+	2.57125e9i	−0.157209	+	0.690807i
$248$	5.04896e9			1.33473
$249$		0			0
$250$	−4.44893e9			−1.13893
$251$			5.04851e8i			0.127194i	0.997976	+	0.0635972i	$0.0202573\pi$
$251$			5.04851e8i			0.127194i	−0.997976	+	0.0635972i	$0.979743\pi$
$252$		0			0
$253$	1.79892e9	−	1.79892e9i	0.439066	−	0.439066i
$254$	3.06647e9	+	3.06647e9i	0.736723	+	0.736723i
$255$		0			0
$256$	4.30105e9			1.00142
$257$		−	5.03823e9i		−	1.15490i	−0.816425	−	0.577452i	$-0.804047\pi$
$257$		−	5.03823e9i		−	1.15490i	0.816425	−	0.577452i	$-0.195953\pi$
$258$		0			0
$259$			3.63317e9i			0.807397i
$260$	1.75457e9	+	3.99293e8i	0.383952	+	0.0873772i
$261$		0			0
$262$	2.23315e9	−	2.23315e9i	0.473928	−	0.473928i
$263$	5.21789e9			1.09062			0.545308	−	0.838236i	$-0.316413\pi$
$263$	5.21789e9			1.09062			0.545308	+	0.838236i	$0.316413\pi$
$264$		0			0
$265$	9.71776e8	−	9.71776e8i	0.197053	−	0.197053i
$266$	2.47993e9	−	2.47993e9i	0.495350	−	0.495350i
$267$		0			0
$268$	−2.23235e9	−	2.23235e9i	−0.432736	−	0.432736i
$269$	−8.85848e8			−0.169180			−0.0845902	−	0.996416i	$-0.526958\pi$
$269$	−8.85848e8			−0.169180			−0.0845902	+	0.996416i	$0.526958\pi$
$270$		0			0
$271$	−2.98082e9	−	2.98082e9i	−0.552660	−	0.552660i	0.374548	−	0.927208i	$-0.377798\pi$
$271$	−2.98082e9	−	2.98082e9i	−0.552660	−	0.552660i	−0.927208	+	0.374548i	$0.877798\pi$
$272$			1.03802e9i			0.189640i
$273$		0			0
$274$	3.69580e9			0.655701
$275$	3.82350e8	−	3.82350e8i	0.0668544	−	0.0668544i
$276$		0			0
$277$			6.84417e9i			1.16252i	0.813717	+	0.581262i	$0.197441\pi$
$277$			6.84417e9i			1.16252i	−0.813717	+	0.581262i	$0.802559\pi$
$278$	−2.16660e9	+	2.16660e9i	−0.362744	+	0.362744i
$279$		0			0
$280$	2.72953e9	+	2.72953e9i	0.444074	+	0.444074i
$281$	2.06205e9	+	2.06205e9i	0.330731	+	0.330731i	0.852864	−	0.522133i	$-0.174864\pi$
$281$	2.06205e9	+	2.06205e9i	0.330731	+	0.330731i	−0.522133	+	0.852864i	$0.674864\pi$
$282$		0			0
$283$		−	3.89823e9i		−	0.607745i	−0.952713	−	0.303873i	$-0.901720\pi$
$283$		−	3.89823e9i		−	0.607745i	0.952713	−	0.303873i	$-0.0982797\pi$
$284$	−1.66787e9	−	1.66787e9i	−0.256383	−	0.256383i
$285$		0			0
$286$	1.07364e10	−	6.75569e9i	1.60471	−	1.00973i
$287$	−7.35802e9			−1.08451
$288$		0			0
$289$	6.81161e9			0.976468
$290$			1.29222e9i			0.182702i
$291$		0			0
$292$	5.13209e8	−	5.13209e8i	0.0705932	−	0.0705932i
$293$	7.11985e8	+	7.11985e8i	0.0966052	+	0.0966052i	0.753758	−	0.657152i	$-0.228239\pi$
$293$	7.11985e8	+	7.11985e8i	0.0966052	+	0.0966052i	−0.657152	+	0.753758i	$0.728239\pi$
$294$		0			0
$295$	1.22020e10			1.61118
$296$			5.35019e9i			0.696951i
$297$		0			0
$298$			1.29221e10i			1.63858i
$299$	1.63925e9	+	2.60516e9i	0.205097	+	0.325949i
$300$		0			0
$301$	−7.61113e9	+	7.61113e9i	−0.927221	+	0.927221i
$302$	6.05207e9			0.727573
$303$		0			0
$304$	5.28938e9	−	5.28938e9i	0.619313	−	0.619313i
$305$	−4.16313e9	+	4.16313e9i	−0.481084	+	0.481084i
$306$		0			0
$307$	−3.68508e9	−	3.68508e9i	−0.414852	−	0.414852i	0.468573	−	0.883425i	$-0.344768\pi$
$307$	−3.68508e9	−	3.68508e9i	−0.414852	−	0.414852i	−0.883425	+	0.468573i	$0.844768\pi$
$308$	−4.66953e9			−0.518885
$309$		0			0
$310$	−1.45281e10	−	1.45281e10i	−1.57313	−	1.57313i
$311$		−	1.48054e10i		−	1.58263i	−0.611407	−	0.791316i	$-0.709396\pi$
$311$		−	1.48054e10i		−	1.58263i	0.611407	−	0.791316i	$-0.290604\pi$
$312$		0			0
$313$	1.71740e10			1.78934			0.894672	−	0.446724i	$-0.147409\pi$
$313$	1.71740e10			1.78934			0.894672	+	0.446724i	$0.147409\pi$
$314$	−6.67930e8	+	6.67930e8i	−0.0687088	+	0.0687088i
$315$		0			0
$316$			1.70331e9i			0.170822i
$317$	−3.79991e9	+	3.79991e9i	−0.376301	+	0.376301i	−0.869766	−	0.493464i	$-0.835730\pi$
$317$	−3.79991e9	+	3.79991e9i	−0.376301	+	0.376301i	0.493464	+	0.869766i	$0.335730\pi$
$318$		0			0
$319$	1.78285e9	+	1.78285e9i	0.172168	+	0.172168i
$320$	2.90212e9	+	2.90212e9i	0.276768	+	0.276768i
$321$		0			0
$322$		−	4.09366e9i		−	0.380792i
$323$	8.36456e8	+	8.36456e8i	0.0768481	+	0.0768481i
$324$		0			0
$325$	3.48413e8	+	5.53712e8i	0.0312292	+	0.0496307i
$326$	1.13443e10			1.00440
$327$		0			0
$328$	−1.08354e10			−0.936157
$329$			1.47002e10i			1.25470i
$330$		0			0
$331$	−1.49259e10	+	1.49259e10i	−1.24345	+	1.24345i	−0.284885	+	0.958562i	$0.591955\pi$
$331$	−1.49259e10	+	1.49259e10i	−1.24345	+	1.24345i	−0.958562	+	0.284885i	$0.908045\pi$
$332$	−3.36750e9	−	3.36750e9i	−0.277176	−	0.277176i
$333$		0			0
$334$	−1.90988e10			−1.53469
$335$		−	2.07216e10i		−	1.64530i
$336$		0			0
$337$		−	4.25747e9i		−	0.330089i	−0.986286	−	0.165045i	$-0.947223\pi$
$337$		−	4.25747e9i		−	0.330089i	0.986286	−	0.165045i	$-0.0527768\pi$
$338$	5.08735e9	+	1.44796e10i	0.389785	+	1.10940i
$339$		0			0
$340$	5.70780e8	−	5.70780e8i	0.0427123	−	0.0427123i
$341$	−4.00884e10			−2.96484
$342$		0			0
$343$	−1.06406e10	+	1.06406e10i	−0.768759	+	0.768759i
$344$	−1.12081e10	+	1.12081e10i	−0.800384	+	0.800384i
$345$		0			0
$346$	1.47216e10	+	1.47216e10i	1.02719	+	1.02719i
$347$	3.21964e9			0.222070			0.111035	−	0.993817i	$-0.464583\pi$
$347$	3.21964e9			0.222070			0.111035	+	0.993817i	$0.464583\pi$
$348$		0			0
$349$	−6.49502e9	−	6.49502e9i	−0.437803	−	0.437803i	0.453469	−	0.891272i	$-0.350186\pi$
$349$	−6.49502e9	−	6.49502e9i	−0.437803	−	0.437803i	−0.891272	+	0.453469i	$0.850186\pi$
$350$		−	8.70083e8i		−	0.0579814i
$351$		0			0
$352$	−1.80158e10			−1.17350
$353$	−5.61498e9	+	5.61498e9i	−0.361617	+	0.361617i	−0.864408	−	0.502791i	$-0.832307\pi$
$353$	−5.61498e9	+	5.61498e9i	−0.361617	+	0.361617i	0.502791	+	0.864408i	$0.332307\pi$
$354$		0			0
$355$		−	1.54819e10i		−	0.974789i
$356$	−2.35694e9	+	2.35694e9i	−0.146740	+	0.146740i
$357$		0			0
$358$	1.44800e10	+	1.44800e10i	0.881531	+	0.881531i
$359$	−5.17602e9	−	5.17602e9i	−0.311615	−	0.311615i	0.533920	−	0.845535i	$-0.320718\pi$
$359$	−5.17602e9	−	5.17602e9i	−0.311615	−	0.311615i	−0.845535	+	0.533920i	$0.820718\pi$
$360$		0			0
$361$			8.45901e9i			0.498070i
$362$	1.77207e10	+	1.77207e10i	1.03192	+	1.03192i
$363$		0			0
$364$	1.25363e9	−	5.50870e9i	0.0714110	−	0.313793i
$365$	4.76382e9			0.268401
$366$		0			0
$367$	−1.49404e10			−0.823566			−0.411783	−	0.911282i	$-0.635094\pi$
$367$	−1.49404e10			−0.823566			−0.411783	+	0.911282i	$0.635094\pi$
$368$		−	8.73126e9i		−	0.476087i
$369$		0			0
$370$	1.53949e10	−	1.53949e10i	0.821430	−	0.821430i
$371$	−3.05102e9	−	3.05102e9i	−0.161046	−	0.161046i
$372$		0			0
$373$	−3.25719e9			−0.168270			−0.0841351	−	0.996454i	$-0.526813\pi$
$373$	−3.25719e9			−0.168270			−0.0841351	+	0.996454i	$0.526813\pi$
$374$		−	5.69037e9i		−	0.290840i
$375$		0			0
$376$			2.16475e10i			1.08307i
$377$	−2.58189e9	+	1.62460e9i	−0.127812	+	0.0804233i
$378$		0			0
$379$	−2.12322e10	+	2.12322e10i	−1.02905	+	1.02905i	−0.0294868	+	0.999565i	$0.509387\pi$
$379$	−2.12322e10	+	2.12322e10i	−1.02905	+	1.02905i	−0.999565	+	0.0294868i	$0.990613\pi$
$380$	−5.81698e9			−0.278974
$381$		0			0
$382$	8.39885e8	−	8.39885e8i	0.0394427	−	0.0394427i
$383$	−8.26210e9	+	8.26210e9i	−0.383968	+	0.383968i	−0.872530	−	0.488561i	$-0.837522\pi$
$383$	−8.26210e9	+	8.26210e9i	−0.383968	+	0.383968i	0.488561	+	0.872530i	$0.337522\pi$
$384$		0			0
$385$	−2.16723e10	−	2.16723e10i	−0.986420	−	0.986420i
$386$	−1.63683e10			−0.737317
$387$		0			0
$388$	−1.95516e9	−	1.95516e9i	−0.0862691	−	0.0862691i
$389$		−	1.49728e10i		−	0.653890i	−0.945043	−	0.326945i	$-0.893981\pi$
$389$		−	1.49728e10i		−	0.653890i	0.945043	−	0.326945i	$-0.106019\pi$
$390$		0			0
$391$	1.38075e9			0.0590757
$392$	−3.54980e9	+	3.54980e9i	−0.150335	+	0.150335i
$393$		0			0
$394$		−	8.82941e9i		−	0.366393i
$395$	−7.90540e9	+	7.90540e9i	−0.324740	+	0.324740i
$396$		0			0
$397$	3.01605e10	+	3.01605e10i	1.21416	+	1.21416i	0.969645	+	0.244516i	$0.0786292\pi$
$397$	3.01605e10	+	3.01605e10i	1.21416	+	1.21416i	0.244516	+	0.969645i	$0.421371\pi$
$398$	7.21651e9	+	7.21651e9i	0.287604	+	0.287604i
$399$		0			0
$400$		−	1.85578e9i		−	0.0724914i
$401$	2.03772e10	+	2.03772e10i	0.788074	+	0.788074i	0.981178	−	0.193105i	$-0.0618557\pi$
$401$	2.03772e10	+	2.03772e10i	0.788074	+	0.788074i	−0.193105	+	0.981178i	$0.561856\pi$
$402$		0			0
$403$	1.07626e10	−	4.72928e10i	0.408034	−	1.79298i
$404$	−6.67361e9			−0.250516
$405$		0			0
$406$	4.05708e9			0.149317
$407$		−	4.24802e10i		−	1.54814i
$408$		0			0
$409$	−1.78691e9	+	1.78691e9i	−0.0638572	+	0.0638572i	−0.738314	−	0.674457i	$-0.764378\pi$
$409$	−1.78691e9	+	1.78691e9i	−0.0638572	+	0.0638572i	0.674457	+	0.738314i	$0.264378\pi$
$410$	3.11783e10	+	3.11783e10i	1.10336	+	1.10336i
$411$		0			0
$412$	−6.86077e9			−0.238113
$413$		−	3.83099e10i		−	1.31677i
$414$		0			0
$415$		−	3.12585e10i		−	1.05384i
$416$	4.83673e9	−	2.12535e10i	0.161502	−	0.709671i
$417$		0			0
$418$	−2.89961e10	+	2.89961e10i	−0.949804	+	0.949804i
$419$	5.68142e10			1.84332			0.921659	−	0.388000i	$-0.126834\pi$
$419$	5.68142e10			1.84332			0.921659	+	0.388000i	$0.126834\pi$
$420$		0			0
$421$	−1.81664e9	+	1.81664e9i	−0.0578283	+	0.0578283i	−0.735430	−	0.677601i	$-0.763020\pi$
$421$	−1.81664e9	+	1.81664e9i	−0.0578283	+	0.0578283i	0.677601	+	0.735430i	$0.263020\pi$
$422$	1.51278e9	−	1.51278e9i	0.0477008	−	0.0477008i
$423$		0			0
$424$	−4.49291e9	−	4.49291e9i	−0.139016	−	0.139016i
$425$	2.93471e8			0.00899516
$426$		0			0
$427$	1.30707e10	+	1.30707e10i	0.393176	+	0.393176i
$428$			1.45560e10i			0.433778i
$429$		0			0
$430$	6.45016e10			1.88667
$431$	−3.27031e10	+	3.27031e10i	−0.947720	+	0.947720i	−0.998700	−	0.0509798i	$-0.983766\pi$
$431$	−3.27031e10	+	3.27031e10i	−0.947720	+	0.947720i	0.0509798	+	0.998700i	$0.483766\pi$
$432$		0			0
$433$		−	1.71901e10i		−	0.489021i	−0.969647	−	0.244511i	$-0.921373\pi$
$433$		−	1.71901e10i		−	0.489021i	0.969647	−	0.244511i	$-0.0786273\pi$
$434$	−4.56130e10	+	4.56130e10i	−1.28567	+	1.28567i
$435$		0			0
$436$	3.17761e9	+	3.17761e9i	0.0879336	+	0.0879336i
$437$	−7.03581e9	−	7.03581e9i	−0.192925	−	0.192925i
$438$		0			0
$439$		−	2.23742e10i		−	0.602406i	−0.953560	−	0.301203i	$-0.902612\pi$
$439$		−	2.23742e10i		−	0.602406i	0.953560	−	0.301203i	$-0.0973882\pi$
$440$	−3.19145e10	−	3.19145e10i	−0.851485	−	0.851485i
$441$		0			0
$442$	6.71299e9	+	1.52770e9i	0.175884	+	0.0400266i
$443$	−2.37096e10			−0.615615			−0.307807	−	0.951449i	$-0.599595\pi$
$443$	−2.37096e10			−0.615615			−0.307807	+	0.951449i	$0.599595\pi$
$444$		0			0
$445$	−2.18781e10			−0.557917
$446$		−	6.96286e10i		−	1.75974i
$447$		0			0
$448$	9.11160e9	−	9.11160e9i	0.226195	−	0.226195i
$449$	3.81458e10	+	3.81458e10i	0.938560	+	0.938560i	0.998219	−	0.0596591i	$-0.0190014\pi$
$449$	3.81458e10	+	3.81458e10i	0.938560	+	0.938560i	−0.0596591	+	0.998219i	$0.519001\pi$
$450$		0			0
$451$	8.60323e10			2.07948
$452$			8.01568e8i			0.0192038i
$453$		0			0
$454$			2.85236e10i			0.671400i
$455$	3.13854e10	−	1.97487e10i	0.732289	−	0.460779i
$456$		0			0
$457$	3.02656e10	−	3.02656e10i	0.693881	−	0.693881i	−0.269202	−	0.963084i	$-0.586760\pi$
$457$	3.02656e10	−	3.02656e10i	0.693881	−	0.693881i	0.963084	+	0.269202i	$0.0867601\pi$
$458$	3.17352e10			0.721240
$459$		0			0
$460$	−4.80109e9	+	4.80109e9i	−0.107228	+	0.107228i
$461$	7.11579e9	−	7.11579e9i	0.157550	−	0.157550i	−0.623930	−	0.781480i	$-0.714465\pi$
$461$	7.11579e9	−	7.11579e9i	0.157550	−	0.157550i	0.781480	+	0.623930i	$0.214465\pi$
$462$		0			0
$463$	4.58822e10	+	4.58822e10i	0.998437	+	0.998437i	0.999999	−	0.00156224i	$-0.000497278\pi$
$463$	4.58822e10	+	4.58822e10i	0.998437	+	0.998437i	−0.00156224	+	0.999999i	$0.500497\pi$
$464$	8.65326e9			0.186684
$465$		0			0
$466$	5.77328e9	+	5.77328e9i	0.122427	+	0.122427i
$467$			4.41131e10i			0.927470i	0.885974	+	0.463735i	$0.153491\pi$
$467$			4.41131e10i			0.927470i	−0.885974	+	0.463735i	$0.846509\pi$
$468$		0			0
$469$	−6.50583e10			−1.34466
$470$	6.22896e10	−	6.22896e10i	1.27651	−	1.27651i
$471$		0			0
$472$		−	5.64149e10i		−	1.13665i
$473$	8.89917e10	−	8.89917e10i	1.77789	−	1.77789i
$474$		0			0
$475$	−1.49542e9	−	1.49542e9i	−0.0293758	−	0.0293758i
$476$	−1.79204e9	−	1.79204e9i	−0.0349076	−	0.0349076i
$477$		0			0
$478$		−	4.32500e10i		−	0.828466i
$479$	2.27074e10	+	2.27074e10i	0.431345	+	0.431345i	0.889086	−	0.457740i	$-0.151341\pi$
$479$	2.27074e10	+	2.27074e10i	0.431345	+	0.431345i	−0.457740	+	0.889086i	$0.651341\pi$
$480$		0			0
$481$	5.01144e10	+	1.14047e10i	0.936229	+	0.213061i
$482$	6.58957e10			1.22087
$483$		0			0
$484$	3.35962e10			0.612222
$485$		−	1.81486e10i		−	0.328002i
$486$		0			0
$487$	−3.41454e10	+	3.41454e10i	−0.607039	+	0.607039i	−0.942171	−	0.335132i	$-0.891219\pi$
$487$	−3.41454e10	+	3.41454e10i	−0.607039	+	0.607039i	0.335132	+	0.942171i	$0.391219\pi$
$488$	1.92478e10	+	1.92478e10i	0.339393	+	0.339393i
$489$		0			0
$490$	2.04288e10			0.354371
$491$			1.10281e10i			0.189747i	0.995489	+	0.0948736i	$0.0302447\pi$
$491$			1.10281e10i			0.189747i	−0.995489	+	0.0948736i	$0.969755\pi$
$492$		0			0
$493$			1.36842e9i			0.0231649i
$494$	−2.64224e10	−	4.19916e10i	−0.443675	−	0.705106i
$495$		0			0
$496$	−9.72870e10	+	9.72870e10i	−1.60741	+	1.60741i
$497$	−4.86075e10			−0.796668
$498$		0			0
$499$	−7.35316e9	+	7.35316e9i	−0.118596	+	0.118596i	−0.763914	−	0.645318i	$-0.776725\pi$
$499$	−7.35316e9	+	7.35316e9i	−0.118596	+	0.118596i	0.645318	+	0.763914i	$0.276725\pi$
$500$	1.63819e10	−	1.63819e10i	0.262110	−	0.262110i
$501$		0			0
$502$	−6.71635e9	−	6.71635e9i	−0.105759	−	0.105759i
$503$	2.91201e10			0.454905			0.227453	−	0.973789i	$-0.426960\pi$
$503$	2.91201e10			0.454905			0.227453	+	0.973789i	$0.426960\pi$
$504$		0			0
$505$	−3.09736e10	−	3.09736e10i	−0.476241	−	0.476241i
$506$			4.78643e10i			0.730146i
$507$		0			0
$508$	−2.25827e10			−0.339095
$509$	−5.00869e10	+	5.00869e10i	−0.746196	+	0.746196i	−0.973762	−	0.227567i	$-0.926923\pi$
$509$	−5.00869e10	+	5.00869e10i	−0.746196	+	0.746196i	0.227567	+	0.973762i	$0.426923\pi$
$510$		0			0
$511$		−	1.49567e10i		−	0.219357i
$512$	−1.17177e8	+	1.17177e8i	−0.00170514	+	0.00170514i
$513$		0			0
$514$	6.70268e10	+	6.70268e10i	0.960276	+	0.960276i
$515$	−3.18423e10	−	3.18423e10i	−0.452663	−	0.452663i
$516$		0			0
$517$		−	1.71880e11i		−	2.40582i
$518$	−4.83344e10	−	4.83344e10i	−0.671332	−	0.671332i
$519$		0			0
$520$	4.62180e10	−	2.90818e10i	0.632117	−	0.397748i
$521$	−5.55952e10			−0.754547			−0.377274	−	0.926102i	$-0.623138\pi$
$521$	−5.55952e10			−0.754547			−0.377274	+	0.926102i	$0.623138\pi$
$522$		0			0
$523$	−1.37190e11			−1.83365			−0.916825	−	0.399290i	$-0.869257\pi$
$523$	−1.37190e11			−1.83365			−0.916825	+	0.399290i	$0.869257\pi$
$524$			1.64458e10i			0.218137i
$525$		0			0
$526$	−6.94169e10	+	6.94169e10i	−0.906822	+	0.906822i
$527$	−1.53848e10	−	1.53848e10i	−0.199458	−	0.199458i
$528$		0			0
$529$	6.66968e10			0.851692
$530$			2.58563e10i			0.327690i
$531$		0			0
$532$			1.82632e10i			0.227997i
$533$	−2.30971e10	+	1.01493e11i	−0.286187	+	1.25756i
$534$		0			0
$535$	−6.75576e10	+	6.75576e10i	−0.824630	+	0.824630i
$536$	−9.58045e10			−1.16072
$537$		0			0
$538$	1.17850e10	−	1.17850e10i	0.140670	−	0.140670i
$539$	2.81852e10	−	2.81852e10i	0.333938	−	0.333938i
$540$		0			0
$541$	−1.19569e11	−	1.19569e11i	−1.39582	−	1.39582i	−0.811583	−	0.584237i	$-0.801394\pi$
$541$	−1.19569e11	−	1.19569e11i	−1.39582	−	1.39582i	−0.584237	−	0.811583i	$-0.698606\pi$
$542$	7.93114e10			0.919048
$543$		0			0
$544$	−6.91398e9	−	6.91398e9i	−0.0789465	−	0.0789465i
$545$			2.94959e10i			0.334330i
$546$		0			0
$547$	2.60344e10			0.290802			0.145401	−	0.989373i	$-0.453553\pi$
$547$	2.60344e10			0.290802			0.145401	+	0.989373i	$0.453553\pi$
$548$	−1.36087e10	+	1.36087e10i	−0.150901	+	0.150901i
$549$		0			0
$550$			1.01733e10i			0.111176i
$551$	6.97295e9	−	6.97295e9i	0.0756503	−	0.0756503i
$552$		0			0
$553$	2.48201e10	+	2.48201e10i	0.265401	+	0.265401i
$554$	−9.10523e10	−	9.10523e10i	−0.966612	−	0.966612i
$555$		0			0
$556$		−	1.59557e10i		−	0.166962i
$557$	−3.51980e10	−	3.51980e10i	−0.365677	−	0.365677i	0.500221	−	0.865898i	$-0.333252\pi$
$557$	−3.51980e10	−	3.51980e10i	−0.365677	−	0.365677i	−0.865898	+	0.500221i	$0.833252\pi$
$558$		0			0
$559$	8.10929e10	+	1.28876e11i	0.830492	+	1.31985i
$560$	−1.05189e11			−1.06959
$561$		0			0
$562$	−5.48656e10			−0.549990
$563$			1.77452e11i			1.76623i	0.469153	+	0.883117i	$0.344559\pi$
$563$			1.77452e11i			1.76623i	−0.469153	+	0.883117i	$0.655441\pi$
$564$		0			0
$565$	−3.72025e9	+	3.72025e9i	−0.0365072	+	0.0365072i
$566$	5.18606e10	+	5.18606e10i	0.505326	+	0.505326i
$567$		0			0
$568$	−7.15791e10			−0.687690
$569$		−	1.62042e11i		−	1.54589i	−0.634473	−	0.772945i	$-0.718783\pi$
$569$		−	1.62042e11i		−	1.54589i	0.634473	−	0.772945i	$-0.281217\pi$
$570$		0			0
$571$		−	4.68420e10i		−	0.440647i	−0.975427	−	0.220324i	$-0.929289\pi$
$571$		−	4.68420e10i		−	0.440647i	0.975427	−	0.220324i	$-0.0707113\pi$
$572$	−1.46579e10	+	6.44095e10i	−0.136926	+	0.601680i
$573$		0			0
$574$	9.78884e10	−	9.78884e10i	0.901745	−	0.901745i
$575$	−2.46852e9			−0.0225821
$576$		0			0
$577$	−4.37702e10	+	4.37702e10i	−0.394890	+	0.394890i	−0.876426	−	0.481537i	$-0.840079\pi$
$577$	−4.37702e10	+	4.37702e10i	−0.394890	+	0.394890i	0.481537	+	0.876426i	$0.340079\pi$
$578$	−9.06191e10	+	9.06191e10i	−0.811911	+	0.811911i
$579$		0			0
$580$	−4.75820e9	−	4.75820e9i	−0.0420466	−	0.0420466i
$581$	−9.81403e10			−0.861277
$582$		0			0
$583$	3.56735e10	+	3.56735e10i	0.308796	+	0.308796i
$584$		−	2.20251e10i		−	0.189350i
$585$		0			0
$586$	−1.89440e10			−0.160650
$587$	5.28064e10	−	5.28064e10i	0.444768	−	0.444768i	−0.448843	−	0.893611i	$-0.648164\pi$
$587$	5.28064e10	−	5.28064e10i	0.444768	−	0.444768i	0.893611	+	0.448843i	$0.148164\pi$
$588$		0			0
$589$			1.56791e11i			1.30275i
$590$	−1.62331e11	+	1.62331e11i	−1.33966	+	1.33966i
$591$		0			0
$592$	−1.03091e11	−	1.03091e11i	−0.839335	−	0.839335i
$593$	7.03776e10	+	7.03776e10i	0.569136	+	0.569136i	0.931886	−	0.362750i	$-0.118162\pi$
$593$	7.03776e10	+	7.03776e10i	0.569136	+	0.569136i	−0.362750	+	0.931886i	$0.618162\pi$
$594$		0			0
$595$		−	1.66345e10i		−	0.132721i
$596$	−4.75816e10	−	4.75816e10i	−0.377098	−	0.377098i
$597$		0			0
$598$	−5.64661e10	−	1.28502e10i	−0.441553	−	0.100486i
$599$	−1.05639e9			−0.00820575			−0.00410287	−	0.999992i	$-0.501306\pi$
$599$	−1.05639e9			−0.00820575			−0.00410287	+	0.999992i	$0.501306\pi$
$600$		0			0
$601$	1.91196e11			1.46549			0.732743	−	0.680506i	$-0.238240\pi$
$601$	1.91196e11			1.46549			0.732743	+	0.680506i	$0.238240\pi$
$602$		−	2.02511e11i		−	1.54193i
$603$		0			0
$604$	−2.22850e10	+	2.22850e10i	−0.167442	+	0.167442i
$605$	1.55927e11	+	1.55927e11i	1.16386	+	1.16386i
$606$		0			0
$607$	1.53315e11			1.12935			0.564676	−	0.825313i	$-0.309001\pi$
$607$	1.53315e11			1.12935			0.564676	+	0.825313i	$0.309001\pi$
$608$			7.04623e10i			0.515635i
$609$		0			0
$610$		−	1.10770e11i		−	0.800020i
$611$	2.02768e11	+	4.61447e10i	1.45491	+	0.331098i
$612$		0			0
$613$	−1.57802e9	+	1.57802e9i	−0.0111756	+	0.0111756i	−0.712673	−	0.701497i	$-0.752515\pi$
$613$	−1.57802e9	+	1.57802e9i	−0.0111756	+	0.0111756i	0.701497	+	0.712673i	$0.252515\pi$
$614$	9.80500e10			0.689881
$615$		0			0
$616$	−1.00200e11	+	1.00200e11i	−0.695895	+	0.695895i
$617$	9.21851e10	−	9.21851e10i	0.636092	−	0.636092i	−0.313497	−	0.949589i	$-0.601501\pi$
$617$	9.21851e10	−	9.21851e10i	0.636092	−	0.636092i	0.949589	+	0.313497i	$0.101501\pi$
$618$		0			0
$619$	−1.04144e11	−	1.04144e11i	−0.709371	−	0.709371i	0.257032	−	0.966403i	$-0.417255\pi$
$619$	−1.04144e11	−	1.04144e11i	−0.709371	−	0.709371i	−0.966403	+	0.257032i	$0.917255\pi$
$620$	1.06991e11			0.724070
$621$		0			0
$622$	1.96966e11	+	1.96966e11i	1.31592	+	1.31592i
$623$			6.86892e10i			0.455970i
$624$		0			0
$625$	1.61011e11			1.05520
$626$	−2.28476e11	+	2.28476e11i	−1.48780	+	1.48780i
$627$		0			0
$628$		−	4.91891e9i		−	0.0316250i
$629$	1.63027e10	−	1.63027e10i	0.104150	−	0.104150i
$630$		0			0
$631$	1.21599e11	+	1.21599e11i	0.767032	+	0.767032i	0.977583	−	0.210551i	$-0.0675258\pi$
$631$	1.21599e11	+	1.21599e11i	0.767032	+	0.767032i	−0.210551	+	0.977583i	$0.567526\pi$
$632$	3.65499e10	+	3.65499e10i	0.229096	+	0.229096i
$633$		0			0
$634$		−	1.01105e11i		−	0.625772i
$635$	−1.04811e11	−	1.04811e11i	−0.644634	−	0.644634i
$636$		0			0
$637$	2.56835e10	+	4.08173e10i	0.155990	+	0.247906i
$638$	−4.74367e10			−0.286307
$639$		0			0
$640$	−2.02854e11			−1.20910
$641$		−	3.14879e10i		−	0.186514i	−0.995642	−	0.0932570i	$-0.970272\pi$
$641$		−	3.14879e10i		−	0.186514i	0.995642	−	0.0932570i	$-0.0297278\pi$
$642$		0			0
$643$	1.07133e10	−	1.07133e10i	0.0626731	−	0.0626731i	−0.675076	−	0.737749i	$-0.735889\pi$
$643$	1.07133e10	−	1.07133e10i	0.0626731	−	0.0626731i	0.737749	+	0.675076i	$0.235889\pi$
$644$	1.50737e10	+	1.50737e10i	0.0876346	+	0.0876346i
$645$		0			0
$646$	−2.22558e10			−0.127795
$647$			1.73313e11i			0.989040i	0.869166	+	0.494520i	$0.164656\pi$
$647$			1.73313e11i			0.989040i	−0.869166	+	0.494520i	$0.835344\pi$
$648$		0			0
$649$			4.47931e11i			2.52483i
$650$	−1.20015e10	−	2.73123e9i	−0.0672331	−	0.0153005i
$651$		0			0
$652$	−4.17721e10	+	4.17721e10i	−0.231151	+	0.231151i
$653$	−9.91395e10			−0.545248			−0.272624	−	0.962121i	$-0.587891\pi$
$653$	−9.91395e10			−0.545248			−0.272624	+	0.962121i	$0.587891\pi$
$654$		0			0
$655$	−7.63285e10	+	7.63285e10i	−0.414688	+	0.414688i
$656$	2.08784e11	−	2.08784e11i	1.12741	−	1.12741i
$657$		0			0
$658$	−1.95566e11	−	1.95566e11i	−1.04326	−	1.04326i
$659$	−8.82278e10			−0.467804			−0.233902	−	0.972260i	$-0.575149\pi$
$659$	−8.82278e10			−0.467804			−0.233902	+	0.972260i	$0.575149\pi$
$660$		0			0
$661$	−8.53976e10	−	8.53976e10i	−0.447342	−	0.447342i	0.447128	−	0.894470i	$-0.352447\pi$
$661$	−8.53976e10	−	8.53976e10i	−0.447342	−	0.447342i	−0.894470	+	0.447128i	$0.852447\pi$
$662$		−	3.97136e11i		−	2.06779i
$663$		0			0
$664$	−1.44521e11			−0.743461
$665$	−8.47632e10	+	8.47632e10i	−0.433432	+	0.433432i
$666$		0			0
$667$		−	1.15104e10i		−	0.0581549i
$668$	7.03256e10	−	7.03256e10i	0.353189	−	0.353189i
$669$		0			0
$670$	2.75673e11	+	2.75673e11i	1.36803	+	1.36803i
$671$	−1.52827e11	−	1.52827e11i	−0.753892	−	0.753892i
$672$		0			0
$673$		−	1.76664e11i		−	0.861166i	−0.902551	−	0.430583i	$-0.858308\pi$
$673$		−	1.76664e11i		−	0.861166i	0.902551	−	0.430583i	$-0.141692\pi$
$674$	5.66398e10	+	5.66398e10i	0.274462	+	0.274462i
$675$		0			0
$676$	−7.20494e10	−	3.45841e10i	−0.345019	−	0.165611i
$677$	−4.23853e10			−0.201772			−0.100886	−	0.994898i	$-0.532168\pi$
$677$	−4.23853e10			−0.201772			−0.100886	+	0.994898i	$0.532168\pi$
$678$		0			0
$679$	−5.69800e10			−0.268067
$680$		−	2.44958e10i		−	0.114566i
$681$		0			0
$682$	5.33322e11	−	5.33322e11i	2.46520	−	2.46520i
$683$	1.52947e11	+	1.52947e11i	0.702844	+	0.702844i	0.965020	−	0.262176i	$-0.0844402\pi$
$683$	1.52947e11	+	1.52947e11i	0.702844	+	0.702844i	−0.262176	+	0.965020i	$0.584440\pi$
$684$		0			0
$685$	−1.26321e11			−0.573739
$686$		−	2.83118e11i		−	1.27841i
$687$		0			0
$688$		−	4.31932e11i		−	1.92780i
$689$	−5.16617e10	+	3.25071e10i	−0.229241	+	0.144245i
$690$		0			0
$691$	1.46459e11	−	1.46459e11i	0.642397	−	0.642397i	−0.308747	−	0.951144i	$-0.599910\pi$
$691$	1.46459e11	−	1.46459e11i	0.642397	−	0.642397i	0.951144	+	0.308747i	$0.0999097\pi$
$692$	−1.08416e11			−0.472789
$693$		0			0
$694$	−4.28329e10	+	4.28329e10i	−0.184646	+	0.184646i
$695$	7.40538e10	−	7.40538e10i	0.317401	−	0.317401i
$696$		0			0
$697$	3.30168e10	+	3.30168e10i	0.139896	+	0.139896i
$698$	1.72815e11			0.728047
$699$		0			0
$700$	3.20382e9	+	3.20382e9i	0.0133437	+	0.0133437i
$701$		−	5.59274e10i		−	0.231608i	−0.993272	−	0.115804i	$-0.963056\pi$
$701$		−	5.59274e10i		−	0.231608i	0.993272	−	0.115804i	$-0.0369444\pi$
$702$		0			0
$703$	−1.66146e11			−0.680249
$704$	−1.06536e11	+	1.06536e11i	−0.433715	+	0.433715i
$705$		0			0
$706$		−	1.49399e11i		−	0.601353i
$707$	−9.72458e10	+	9.72458e10i	−0.389218	+	0.389218i
$708$		0			0
$709$	2.82885e11	+	2.82885e11i	1.11950	+	1.11950i	0.991814	+	0.127687i	$0.0407554\pi$
$709$	2.82885e11	+	2.82885e11i	1.11950	+	1.11950i	0.127687	+	0.991814i	$0.459245\pi$
$710$	2.05965e11	+	2.05965e11i	0.810515	+	0.810515i
$711$		0			0
$712$			1.01151e11i			0.393596i
$713$	1.29409e11	+	1.29409e11i	0.500733	+	0.500733i
$714$		0			0
$715$	−3.66968e11	+	2.30908e11i	−1.40412	+	0.883515i
$716$	−1.06637e11			−0.405747
$717$		0			0
$718$	1.37720e11			0.518201
$719$			2.59427e11i			0.970733i	0.874311	+	0.485367i	$0.161314\pi$
$719$			2.59427e11i			0.970733i	−0.874311	+	0.485367i	$0.838686\pi$
$720$		0			0
$721$	−9.99730e10	+	9.99730e10i	−0.369949	+	0.369949i
$722$	−1.12536e11	−	1.12536e11i	−0.414134	−	0.414134i
$723$		0			0
$724$	−1.30502e11			−0.474968
$725$		−	2.44646e9i		−	0.00885496i
$726$		0			0
$727$			1.73130e11i			0.619776i	0.950773	+	0.309888i	$0.100292\pi$
$727$			1.73130e11i			0.619776i	−0.950773	+	0.309888i	$0.899708\pi$
$728$	−9.13061e10	−	1.45108e11i	−0.325068	−	0.516612i
$729$		0			0
$730$	−6.33761e10	+	6.33761e10i	−0.223169	+	0.223169i
$731$	6.83052e10			0.239213
$732$		0			0
$733$	3.59911e10	−	3.59911e10i	0.124675	−	0.124675i	−0.642016	−	0.766691i	$-0.721902\pi$
$733$	3.59911e10	−	3.59911e10i	0.124675	−	0.124675i	0.766691	+	0.642016i	$0.221902\pi$
$734$	1.98762e11	−	1.98762e11i	0.684776	−	0.684776i
$735$		0			0
$736$	5.81567e10	+	5.81567e10i	0.198193	+	0.198193i
$737$	7.60682e11			2.57830
$738$		0			0
$739$	−8.20726e10	−	8.20726e10i	−0.275182	−	0.275182i	0.556000	−	0.831182i	$-0.312335\pi$
$739$	−8.20726e10	−	8.20726e10i	−0.275182	−	0.275182i	−0.831182	+	0.556000i	$0.812335\pi$
$740$			1.13374e11i			0.378084i
$741$		0			0
$742$	8.11793e10			0.267812
$743$	−2.30046e11	+	2.30046e11i	−0.754848	+	0.754848i	−0.975380	−	0.220532i	$-0.929221\pi$
$743$	−2.30046e11	+	2.30046e11i	−0.754848	+	0.754848i	0.220532	+	0.975380i	$0.429221\pi$
$744$		0			0
$745$		−	4.41672e11i		−	1.43376i
$746$	4.33324e10	−	4.33324e10i	0.139913	−	0.139913i
$747$		0			0
$748$	2.09531e10	+	2.09531e10i	0.0669332	+	0.0669332i
$749$	2.12106e11	+	2.12106e11i	0.673947	+	0.673947i
$750$		0			0
$751$		−	1.25891e11i		−	0.395763i	−0.980226	−	0.197881i	$-0.936594\pi$
$751$		−	1.25891e11i		−	0.395763i	0.980226	−	0.197881i	$-0.0634061\pi$
$752$	−4.17119e11	−	4.17119e11i	−1.30433	−	1.30433i
$753$		0			0
$754$	1.27354e10	−	5.59616e10i	0.0394027	−	0.173143i
$755$	−2.06858e11			−0.636627
$756$		0			0
$757$	−3.08759e10			−0.0940234			−0.0470117	−	0.998894i	$-0.514970\pi$
$757$	−3.08759e10			−0.0940234			−0.0470117	+	0.998894i	$0.514970\pi$
$758$		−	5.64930e11i		−	1.71127i
$759$		0			0
$760$	−1.24822e11	+	1.24822e11i	−0.374142	+	0.374142i
$761$	2.00557e10	+	2.00557e10i	0.0597997	+	0.0597997i	0.736374	−	0.676574i	$-0.236536\pi$
$761$	2.00557e10	+	2.00557e10i	0.0597997	+	0.0597997i	−0.676574	+	0.736374i	$0.736536\pi$
$762$		0			0
$763$	9.26064e10			0.273239
$764$			6.18525e9i			0.0181545i
$765$		0			0
$766$		−	2.19832e11i		−	0.638522i
$767$	−5.28429e11	−	1.20256e11i	−1.52688	−	0.347477i
$768$		0			0
$769$	−2.88028e11	+	2.88028e11i	−0.823625	+	0.823625i	−0.986626	−	0.163001i	$-0.947883\pi$
$769$	−2.88028e11	+	2.88028e11i	−0.823625	+	0.823625i	0.163001	+	0.986626i	$0.447883\pi$
$770$	5.76640e11			1.64037
$771$		0			0
$772$	6.02714e10	−	6.02714e10i	0.169684	−	0.169684i
$773$	1.88615e11	−	1.88615e11i	0.528272	−	0.528272i	−0.391785	−	0.920057i	$-0.628142\pi$
$773$	1.88615e11	−	1.88615e11i	0.528272	−	0.528272i	0.920057	+	0.391785i	$0.128142\pi$
$774$		0			0
$775$	2.75051e10	+	2.75051e10i	0.0762442	+	0.0762442i
$776$	−8.39083e10			−0.231397
$777$		0			0
$778$	1.99193e11	+	1.99193e11i	0.543695	+	0.543695i
$779$		−	3.36484e11i		−	0.913722i
$780$		0			0
$781$	5.68334e11			1.52756
$782$	−1.83690e10	+	1.83690e10i	−0.0491201	+	0.0491201i
$783$		0			0
$784$		−	1.36800e11i		−	0.362095i
$785$	2.28297e10	−	2.28297e10i	0.0601203	−	0.0601203i
$786$		0			0
$787$	−5.09222e11	−	5.09222e11i	−1.32742	−	1.32742i	−0.907613	−	0.419808i	$-0.862097\pi$
$787$	−5.09222e11	−	5.09222e11i	−1.32742	−	1.32742i	−0.419808	−	0.907613i	$-0.637903\pi$
$788$	3.25117e10	+	3.25117e10i	0.0843207	+	0.0843207i
$789$		0			0
$790$		−	2.10341e11i		−	0.540028i
$791$	1.16802e10	+	1.16802e10i	0.0298363	+	0.0298363i
$792$		0			0
$793$	2.21321e11	−	1.39262e11i	0.559667	−	0.352160i
$794$	−8.02488e11			−2.01909
$795$		0			0
$796$	−5.31453e10			−0.132377
$797$		−	2.31315e11i		−	0.573285i	−0.958038	−	0.286642i	$-0.907461\pi$
$797$		−	2.31315e11i		−	0.573285i	0.958038	−	0.286642i	$-0.0925391\pi$
$798$		0			0
$799$	6.59627e10	−	6.59627e10i	0.161849	−	0.161849i
$800$	1.23609e10	+	1.23609e10i	0.0301779	+	0.0301779i
$801$		0			0
$802$	−5.42181e11			−1.31053
$803$			1.74878e11i			0.420603i
$804$		0			0
$805$			1.39920e11i			0.333194i
$806$	4.85985e11	+	7.72347e11i	1.15155	+	1.83009i
$807$		0			0
$808$	−1.43204e11	+	1.43204e11i	−0.335976	+	0.335976i
$809$	8.27873e11			1.93272			0.966362	−	0.257187i	$-0.0827956\pi$
$809$	8.27873e11			1.93272			0.966362	+	0.257187i	$0.0827956\pi$
$810$		0			0
$811$	2.84489e11	−	2.84489e11i	0.657631	−	0.657631i	−0.297188	−	0.954819i	$-0.596049\pi$
$811$	2.84489e11	−	2.84489e11i	0.657631	−	0.657631i	0.954819	+	0.297188i	$0.0960489\pi$
$812$	−1.49390e10	+	1.49390e10i	−0.0343635	+	0.0343635i
$813$		0			0
$814$	5.65141e11	+	5.65141e11i	1.28724	+	1.28724i
$815$	−3.87746e11			−0.878854
$816$		0			0
$817$	−3.48059e11	−	3.48059e11i	−0.781203	−	0.781203i
$818$		−	4.75448e10i		−	0.106192i
$819$		0			0
$820$	−2.29609e11			−0.507849
$821$	3.55602e11	−	3.55602e11i	0.782693	−	0.782693i	−0.197592	−	0.980284i	$-0.563312\pi$
$821$	3.55602e11	−	3.55602e11i	0.782693	−	0.782693i	0.980284	+	0.197592i	$0.0633120\pi$
$822$		0			0
$823$			5.54344e11i			1.20831i	0.796865	+	0.604157i	$0.206490\pi$
$823$			5.54344e11i			1.20831i	−0.796865	+	0.604157i	$0.793510\pi$
$824$	−1.47220e11	+	1.47220e11i	−0.319343	+	0.319343i
$825$		0			0
$826$	5.09660e11	+	5.09660e11i	1.09487	+	1.09487i
$827$	−2.23010e11	−	2.23010e11i	−0.476763	−	0.476763i	0.427332	−	0.904095i	$-0.359454\pi$
$827$	−2.23010e11	−	2.23010e11i	−0.476763	−	0.476763i	−0.904095	+	0.427332i	$0.859454\pi$
$828$		0			0
$829$			4.96966e11i			1.05222i	0.850415	+	0.526112i	$0.176351\pi$
$829$			4.96966e11i			1.05222i	−0.850415	+	0.526112i	$0.823649\pi$
$830$	4.15852e11	+	4.15852e11i	0.876247	+	0.876247i
$831$		0			0
$832$	−9.70796e10	−	1.54283e11i	−0.202598	−	0.321977i
$833$	2.16334e10			0.0449309
$834$		0			0
$835$	6.52791e11			1.34285
$836$		−	2.13539e11i		−	0.437171i
$837$		0			0
$838$	−7.55835e11	+	7.55835e11i	−1.53268	+	1.53268i
$839$	4.81134e11	+	4.81134e11i	0.970997	+	0.970997i	0.999591	−	0.0285940i	$-0.00910301\pi$
$839$	4.81134e11	+	4.81134e11i	0.970997	+	0.970997i	−0.0285940	+	0.999591i	$0.509103\pi$
$840$		0			0
$841$	−4.88839e11			−0.977196
$842$		−	4.83358e10i		−	0.0961659i
$843$		0			0
$844$			1.11407e10i			0.0219555i
$845$	−1.73884e11	−	4.94909e11i	−0.341062	−	0.970729i
$846$		0			0
$847$	4.89554e11	−	4.89554e11i	0.951189	−	0.951189i
$848$	1.73145e11			0.334832
$849$		0			0
$850$	−3.90423e9	+	3.90423e9i	−0.00747927	+	0.00747927i
$851$	−1.37130e11	+	1.37130e11i	−0.261465	+	0.261465i
$852$		0			0
$853$	1.45038e11	+	1.45038e11i	0.273959	+	0.273959i	0.830692	−	0.556733i	$-0.187945\pi$
$853$	1.45038e11	+	1.45038e11i	0.273959	+	0.273959i	−0.556733	+	0.830692i	$0.687945\pi$
$854$	−3.47776e11			−0.653834
$855$		0			0
$856$	3.12346e11	+	3.12346e11i	0.581756	+	0.581756i
$857$			1.85724e11i			0.344307i	0.985070	+	0.172153i	$0.0550725\pi$
$857$			1.85724e11i			0.344307i	−0.985070	+	0.172153i	$0.944928\pi$
$858$		0			0
$859$	1.98445e10			0.0364474			0.0182237	−	0.999834i	$-0.494199\pi$
$859$	1.98445e10			0.0364474			0.0182237	+	0.999834i	$0.494199\pi$
$860$	−2.37508e11	+	2.37508e11i	−0.434194	+	0.434194i
$861$		0			0
$862$		−	8.70140e11i		−	1.57601i
$863$	−6.96281e11	+	6.96281e11i	−1.25528	+	1.25528i	−0.301963	+	0.953320i	$0.597642\pi$
$863$	−6.96281e11	+	6.96281e11i	−1.25528	+	1.25528i	−0.953320	+	0.301963i	$0.902358\pi$
$864$		0			0
$865$	−5.03179e11	−	5.03179e11i	−0.898790	−	0.898790i
$866$	2.28691e11	+	2.28691e11i	0.406610	+	0.406610i
$867$		0			0
$868$		−	3.35913e11i		−	0.591763i
$869$	−2.90204e11	−	2.90204e11i	−0.508890	−	0.508890i
$870$		0			0
$871$	−2.04221e11	+	8.97385e11i	−0.354836	+	1.55922i
$872$	1.36372e11			0.235862
$873$		0			0
$874$	1.87204e11			0.320826
$875$		−	4.77423e11i		−	0.814462i
$876$		0			0
$877$	6.64746e11	−	6.64746e11i	1.12372	−	1.12372i	0.132540	−	0.991178i	$-0.457687\pi$
$877$	6.64746e11	−	6.64746e11i	1.12372	−	1.12372i	0.991178	−	0.132540i	$-0.0423132\pi$
$878$	2.97658e11	+	2.97658e11i	0.500887	+	0.500887i
$879$		0			0
$880$	1.22990e12			2.05088
$881$		−	5.97328e11i		−	0.991538i	−0.868454	−	0.495769i	$-0.834886\pi$
$881$		−	5.97328e11i		−	0.991538i	0.868454	−	0.495769i	$-0.165114\pi$
$882$		0			0
$883$			6.33871e10i			0.104270i	0.998640	+	0.0521348i	$0.0166026\pi$
$883$			6.33871e10i			0.104270i	−0.998640	+	0.0521348i	$0.983397\pi$
$884$	−3.03439e10	+	1.90933e10i	−0.0496892	+	0.0312660i
$885$		0			0
$886$	3.15424e11	−	3.15424e11i	0.511870	−	0.511870i
$887$	9.33374e11			1.50786			0.753930	−	0.656955i	$-0.228156\pi$
$887$	9.33374e11			1.50786			0.753930	+	0.656955i	$0.228156\pi$
$888$		0			0
$889$	−3.29069e11	+	3.29069e11i	−0.526841	+	0.526841i
$890$	2.91058e11	−	2.91058e11i	0.463895	−	0.463895i
$891$		0			0
$892$	2.56387e11	+	2.56387e11i	0.404982	+	0.404982i
$893$	−6.72244e11			−1.05711
$894$		0			0
$895$	−4.94924e11	−	4.94924e11i	−0.771341	−	0.771341i
$896$			6.36887e11i			0.988166i
$897$		0			0
$898$	−1.01496e12			−1.56078
$899$	−1.28253e11	+	1.28253e11i	−0.196349	+	0.196349i
$900$		0			0
$901$			2.73810e10i			0.0415480i
$902$	−1.14454e12	+	1.14454e12i	−1.72904	+	1.72904i
$903$		0			0
$904$	1.72002e10	+	1.72002e10i	0.0257549	+	0.0257549i
$905$	−6.05689e11	−	6.05689e11i	−0.902932	−	0.902932i
$906$		0			0
$907$		−	4.10137e11i		−	0.606038i	−0.952985	−	0.303019i	$-0.902006\pi$
$907$		−	4.10137e11i		−	0.606038i	0.952985	−	0.303019i	$-0.0979945\pi$
$908$	−1.05030e11	−	1.05030e11i	−0.154514	−	0.154514i
$909$		0			0
$910$	−1.54811e11	+	6.80269e11i	−0.225755	+	0.992008i
$911$	−1.08000e12			−1.56802			−0.784008	−	0.620751i	$-0.786828\pi$
$911$	−1.08000e12			−1.56802			−0.784008	+	0.620751i	$0.786828\pi$
$912$		0			0
$913$	1.14749e12			1.65145
$914$			8.05286e11i			1.15389i
$915$		0			0
$916$	−1.16855e11	+	1.16855e11i	−0.165984	+	0.165984i
$917$	2.39643e11	+	2.39643e11i	0.338913	+	0.338913i
$918$		0			0
$919$	−4.49584e11			−0.630302			−0.315151	−	0.949042i	$-0.602055\pi$
$919$	−4.49584e11			−0.630302			−0.315151	+	0.949042i	$0.602055\pi$
$920$			2.06046e11i			0.287615i
$921$		0			0
$922$			1.89332e11i			0.261999i
$923$	−1.52581e11	+	6.70470e11i	−0.210229	+	0.923788i
$924$		0			0
$925$	−2.91461e10	+	2.91461e10i	−0.0398120	+	0.0398120i
$926$	−1.22080e12			−1.66035
$927$		0			0
$928$	−5.76371e10	+	5.76371e10i	−0.0777160	+	0.0777160i
$929$	−6.93108e11	+	6.93108e11i	−0.930547	+	0.930547i	−0.997740	−	0.0671929i	$-0.978596\pi$
$929$	−6.93108e11	+	6.93108e11i	−0.930547	+	0.930547i	0.0671929	+	0.997740i	$0.478596\pi$
$930$		0			0
$931$	−1.10236e11	−	1.10236e11i	−0.146732	−	0.146732i
$932$	−4.25167e10			−0.0563503
$933$		0			0
$934$	−5.86864e11	−	5.86864e11i	−0.771170	−	0.771170i
$935$			1.94495e11i			0.254485i
$936$		0			0
$937$	−1.43771e12			−1.86514			−0.932571	−	0.360986i	$-0.882440\pi$
$937$	−1.43771e12			−1.86514			−0.932571	+	0.360986i	$0.882440\pi$
$938$	8.65512e11	−	8.65512e11i	1.11805	−	1.11805i
$939$		0			0
$940$			4.58725e11i			0.587545i
$941$	7.92200e11	−	7.92200e11i	1.01036	−	1.01036i	0.0104147	−	0.999946i	$-0.496685\pi$
$941$	7.92200e11	−	7.92200e11i	1.01036	−	1.01036i	0.999946	−	0.0104147i	$-0.00331517\pi$
$942$		0			0
$943$	−2.77720e11	−	2.77720e11i	−0.351204	−	0.351204i
$944$	1.08704e12	+	1.08704e12i	1.36886	+	1.36886i
$945$		0			0
$946$			2.36783e12i			2.95655i
$947$	−9.98907e11	−	9.98907e11i	−1.24201	−	1.24201i	−0.959166	−	0.282844i	$-0.908722\pi$
$947$	−9.98907e11	−	9.98907e11i	−1.24201	−	1.24201i	−0.282844	−	0.959166i	$-0.591278\pi$
$948$		0			0
$949$	−2.06305e11	−	4.69496e10i	−0.254358	−	0.0578851i
$950$	3.97891e10			0.0488505
$951$		0			0
$952$	−7.69078e10			−0.0936317
$953$		−	8.26394e11i		−	1.00188i	−0.865482	−	0.500940i	$-0.832988\pi$
$953$		−	8.26394e11i		−	1.00188i	0.865482	−	0.500940i	$-0.167012\pi$
$954$		0			0
$955$	−2.87070e10	+	2.87070e10i	−0.0345124	+	0.0345124i
$956$	1.59255e11	+	1.59255e11i	0.190661	+	0.190661i
$957$		0			0
$958$	−6.04182e11			−0.717308
$959$			3.96603e11i			0.468901i
$960$		0			0
$961$		−	2.03095e12i		−	2.38126i
$962$	−8.18427e11	+	5.14979e11i	−0.955608	+	0.601298i
$963$		0			0
$964$	−2.42641e11	+	2.42641e11i	−0.280968	+	0.280968i
$965$	5.59464e11			0.645153
$966$		0			0
$967$	−7.94834e11	+	7.94834e11i	−0.909014	+	0.909014i	−0.996193	−	0.0871790i	$-0.972215\pi$
$967$	−7.94834e11	+	7.94834e11i	−0.909014	+	0.909014i	0.0871790	+	0.996193i	$0.472215\pi$
$968$	7.20914e11	−	7.20914e11i	0.821074	−	0.821074i
$969$		0			0
$970$	2.41442e11	+	2.41442e11i	0.272726	+	0.272726i
$971$	−1.04403e12			−1.17445			−0.587227	−	0.809422i	$-0.699780\pi$
$971$	−1.04403e12			−1.17445			−0.587227	+	0.809422i	$0.699780\pi$
$972$		0			0
$973$	−2.32502e11	−	2.32502e11i	−0.259403	−	0.259403i
$974$		−	9.08516e11i		−	1.00948i
$975$		0			0
$976$	−7.41762e11			−0.817458
$977$	5.03137e10	−	5.03137e10i	0.0552214	−	0.0552214i	−0.678957	−	0.734178i	$-0.737568\pi$
$977$	5.03137e10	−	5.03137e10i	0.0552214	−	0.0552214i	0.734178	+	0.678957i	$0.237568\pi$
$978$		0			0
$979$		−	8.03135e11i		−	0.874295i
$980$	−7.52228e10	+	7.52228e10i	−0.0815540	+	0.0815540i
$981$		0			0
$982$	−1.46714e11	−	1.46714e11i	−0.157771	−	0.157771i
$983$	1.05726e12	+	1.05726e12i	1.13232	+	1.13232i	0.989791	+	0.142525i	$0.0455223\pi$
$983$	1.05726e12	+	1.05726e12i	1.13232	+	1.13232i	0.142525	+	0.989791i	$0.454478\pi$
$984$		0			0
$985$			3.01787e11i			0.320594i
$986$	−1.82049e10	−	1.82049e10i	−0.0192611	−	0.0192611i
$987$		0			0
$988$	2.51914e11	+	5.73289e10i	0.264378	+	0.0601653i
$989$	−5.74546e11			−0.600537
$990$		0			0
$991$	2.21712e11			0.229877			0.114938	−	0.993373i	$-0.463333\pi$
$991$	2.21712e11			0.229877			0.114938	+	0.993373i	$0.463333\pi$
$992$		−	1.29601e12i		−	1.33832i
$993$		0			0
$994$	6.46656e11	−	6.46656e11i	0.662411	−	0.662411i
$995$	−2.46658e11	−	2.46658e11i	−0.251654	−	0.251654i
$996$		0			0
$997$	−2.65133e10			−0.0268339			−0.0134169	−	0.999910i	$-0.504271\pi$
$997$	−2.65133e10			−0.0268339			−0.0134169	+	0.999910i	$0.504271\pi$
$998$		−	1.95647e11i		−	0.197220i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	117.9.j.a.109.2		18
3.2	odd	2		13.9.d.a.5.8	✓	18
13.8	odd	4	inner	117.9.j.a.73.2		18
39.8	even	4		13.9.d.a.8.8	yes	18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.9.d.a.5.8	✓	18	3.2	odd	2
13.9.d.a.8.8	yes	18	39.8	even	4
117.9.j.a.73.2		18	13.8	odd	4	inner
117.9.j.a.109.2		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)

\(f(q)\)	\(=\)	\(q+(-13.3036 + 13.3036i) q^{2} -97.9733i q^{4} +(454.715 - 454.715i) q^{5} +(-1427.64 - 1427.64i) q^{7} +(-2102.33 - 2102.33i) q^{8} +O(q^{10})\)	\(q+(-13.3036 + 13.3036i) q^{2} -97.9733i q^{4} +(454.715 - 454.715i) q^{5} +(-1427.64 - 1427.64i) q^{7} +(-2102.33 - 2102.33i) q^{8} +12098.7i q^{10} +(16692.4 + 16692.4i) q^{11} +(-24173.6 + 15210.8i) q^{13} +37985.5 q^{14} +81018.4 q^{16} +12812.2i q^{17} +(65286.1 - 65286.1i) q^{19} +(-44549.9 - 44549.9i) q^{20} -444139. q^{22} -107769. i q^{23} -22905.6i q^{25} +(119238. - 523955. i) q^{26} +(-139870. + 139870. i) q^{28} +106806. q^{29} +(-1.20080e6 + 1.20080e6i) q^{31} +(-539643. + 539643. i) q^{32} +(-170448. - 170448. i) q^{34} -1.29833e6 q^{35} +(-1.27244e6 - 1.27244e6i) q^{37} +1.73709e6i q^{38} -1.91192e6 q^{40} +(2.57699e6 - 2.57699e6i) q^{41} -5.33128e6i q^{43} +(1.63541e6 - 1.63541e6i) q^{44} +(1.43372e6 + 1.43372e6i) q^{46} +(-5.14845e6 - 5.14845e6i) q^{47} -1.68851e6i q^{49} +(304728. + 304728. i) q^{50} +(1.49025e6 + 2.36837e6i) q^{52} +2.13711e6 q^{53} +1.51805e7 q^{55} +6.00273e6i q^{56} +(-1.42091e6 + 1.42091e6i) q^{58} +(1.34172e7 + 1.34172e7i) q^{59} -9.15548e6 q^{61} -3.19500e7i q^{62} +6.38229e6i q^{64} +(-4.07553e6 + 1.79086e7i) q^{65} +(2.27853e7 - 2.27853e7i) q^{67} +1.25525e6 q^{68} +(1.72726e7 - 1.72726e7i) q^{70} +(1.70237e7 - 1.70237e7i) q^{71} +(5.23825e6 + 5.23825e6i) q^{73} +3.38562e7 q^{74} +(-6.39630e6 - 6.39630e6i) q^{76} -4.76613e7i q^{77} -1.73854e7 q^{79} +(3.68402e7 - 3.68402e7i) q^{80} +6.85667e7i q^{82} +(3.43716e7 - 3.43716e7i) q^{83} +(5.82587e6 + 5.82587e6i) q^{85} +(7.09254e7 + 7.09254e7i) q^{86} -7.01858e7i q^{88} +(-2.40569e7 - 2.40569e7i) q^{89} +(5.62266e7 + 1.27957e7i) q^{91} -1.05585e7 q^{92} +1.36986e8 q^{94} -5.93731e7i q^{95} +(1.99560e7 - 1.99560e7i) q^{97} +(2.24633e7 + 2.24633e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 2 q^{2} - 166 q^{5} + 5308 q^{7} - 10464 q^{8}+O(q^{10}) \) 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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