LMFDB - Embedded newform 162.9.d.e.53.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,9,Mod(53,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.53"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	162.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,512,0,0,-308] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$65.9953348299$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{12} $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.2
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	162.53
Dual form			162.9.d.e.107.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-9.79796 + 5.65685i) q^{2} +(64.0000 - 110.851i) q^{4} +(7.36159 + 4.25022i) q^{5} +(496.073 + 859.223i) q^{7} +1448.15i q^{8} -96.1714 q^{10} +(5301.41 - 3060.77i) q^{11} +(-2509.96 + 4347.38i) q^{13} +(-9721.00 - 5612.42i) q^{14} +(-8192.00 - 14189.0i) q^{16} +81813.2i q^{17} +130359. q^{19} +(942.284 - 544.028i) q^{20} +(-34628.6 + 59978.6i) q^{22} +(-161887. - 93465.4i) q^{23} +(-195276. - 338229. i) q^{25} -56793.9i q^{26} +126995. q^{28} +(-420614. + 242842. i) q^{29} +(-260936. + 451955. i) q^{31} +(160530. + 92681.9i) q^{32} +(-462805. - 801602. i) q^{34} +8433.67i q^{35} +3.53201e6 q^{37} +(-1.27726e6 + 737424. i) q^{38} +(-6154.97 + 10660.7i) q^{40} +(387124. + 223506. i) q^{41} +(-1.79681e6 - 3.11216e6i) q^{43} -783557. i q^{44} +2.11488e6 q^{46} +(3.11345e6 - 1.79755e6i) q^{47} +(2.39022e6 - 4.13999e6i) q^{49} +(3.82662e6 + 2.20930e6i) q^{50} +(321275. + 556464. i) q^{52} +1.22010e7i q^{53} +52035.7 q^{55} +(-1.24429e6 + 718390. i) q^{56} +(2.74744e6 - 4.75871e6i) q^{58} +(-5.67074e6 - 3.27400e6i) q^{59} +(2.66550e6 + 4.61678e6i) q^{61} -5.90432e6i q^{62} -2.09715e6 q^{64} +(-36954.6 + 21335.7i) q^{65} +(1.11982e7 - 1.93958e7i) q^{67} +(9.06909e6 + 5.23604e6i) q^{68} +(-47708.0 - 82632.7i) q^{70} +3.53919e7i q^{71} -1.66778e7 q^{73} +(-3.46065e7 + 1.99801e7i) q^{74} +(8.34300e6 - 1.44505e7i) q^{76} +(5.25977e6 + 3.03673e6i) q^{77} +(1.69904e7 + 2.94282e7i) q^{79} -139271. i q^{80} -5.05737e6 q^{82} +(-4.39658e7 + 2.53837e7i) q^{83} +(-347724. + 602275. i) q^{85} +(3.52101e7 + 2.03285e7i) q^{86} +(4.43247e6 + 7.67726e6i) q^{88} +1.03941e8i q^{89} -4.98049e6 q^{91} +(-2.07215e7 + 1.19636e7i) q^{92} +(-2.03370e7 + 3.52247e7i) q^{94} +(959652. + 554056. i) q^{95} +(-3.54725e6 - 6.14401e6i) q^{97} +5.40846e7i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 512 q^{4} - 308 q^{7} + 26112 q^{10} + 44680 q^{13} - 65536 q^{16} - 113024 q^{19} + 222720 q^{22} - 876736 q^{25} - 78848 q^{28} + 2375428 q^{31} + 82944 q^{34} + 10394656 q^{37} + 1671168 q^{40}+ \cdots + 269355220 q^{97}+O(q^{100}) $ 8 * q + 512 * q^4 - 308 * q^7 + 26112 * q^10 + 44680 * q^13 - 65536 * q^16 - 113024 * q^19 + 222720 * q^22 - 876736 * q^25 - 78848 * q^28 + 2375428 * q^31 + 82944 * q^34 + 10394656 * q^37 + 1671168 * q^40 + 2755096 * q^43 + 13937664 * q^46 + 18463200 * q^49 - 5719040 * q^52 + 37596888 * q^55 + 21977088 * q^58 - 36842384 * q^61 - 16777216 * q^64 + 52170376 * q^67 - 15375360 * q^70 + 44881864 * q^73 - 7233536 * q^76 + 51146440 * q^79 + 28093440 * q^82 - 101486520 * q^85 - 28508160 * q^88 - 145355728 * q^91 + 4420608 * q^94 + 269355220 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−9.79796	+	5.65685i	−0.612372	+	0.353553i
$2$	−9.79796	+	5.65685i	−0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	64.0000	−	110.851i	0.250000	−	0.433013i
$5$	7.36159	+	4.25022i	0.0117785	+	0.00680035i	0.505878	−	0.862605i	$-0.331169\pi$
$5$	7.36159	+	4.25022i	0.0117785	+	0.00680035i	−0.494099	+	0.869406i	$0.664502\pi$
$6$		0			0
$7$	496.073	+	859.223i	0.206611	+	0.357861i	0.950645	−	0.310281i	$-0.100423\pi$
$7$	496.073	+	859.223i	0.206611	+	0.357861i	−0.744034	+	0.668142i	$0.767090\pi$
$8$			1448.15i			0.353553i
$9$		0			0
$10$	−96.1714			−0.00961714
$11$	5301.41	−	3060.77i	0.362093	−	0.209055i	−0.307905	−	0.951417i	$-0.599628\pi$
$11$	5301.41	−	3060.77i	0.362093	−	0.209055i	0.669999	+	0.742362i	$0.266295\pi$
$12$		0			0
$13$	−2509.96	+	4347.38i	−0.0878806	+	0.152214i	−0.906615	−	0.421959i	$-0.861343\pi$
$13$	−2509.96	+	4347.38i	−0.0878806	+	0.152214i	0.818734	+	0.574172i	$0.194676\pi$
$14$	−9721.00	−	5612.42i	−0.253046	−	0.146096i
$15$		0			0
$16$	−8192.00	−	14189.0i	−0.125000	−	0.216506i
$17$			81813.2i			0.979552i	0.871848	+	0.489776i	$0.162921\pi$
$17$			81813.2i			0.979552i	−0.871848	+	0.489776i	$0.837079\pi$
$18$		0			0
$19$	130359.			1.00029			0.500147	−	0.865940i	$-0.333279\pi$
$19$	130359.			1.00029			0.500147	+	0.865940i	$0.333279\pi$
$20$	942.284	−	544.028i	0.00588927	−	0.00340017i
$21$		0			0
$22$	−34628.6	+	59978.6i	−0.147824	+	0.256039i
$23$	−161887.	−	93465.4i	−0.578496	−	0.333995i	0.182039	−	0.983291i	$-0.441730\pi$
$23$	−161887.	−	93465.4i	−0.578496	−	0.333995i	−0.760535	+	0.649296i	$0.775063\pi$
$24$		0			0
$25$	−195276.	−	338229.i	−0.499908	−	0.865865i
$26$		−	56793.9i		−	0.124282i
$27$		0			0
$28$	126995.			0.206611
$29$	−420614.	+	242842.i	−0.594692	+	0.343346i	−0.766951	−	0.641706i	$-0.778227\pi$
$29$	−420614.	+	242842.i	−0.594692	+	0.343346i	0.172259	+	0.985052i	$0.444894\pi$
$30$		0			0
$31$	−260936.	+	451955.i	−0.282545	+	0.489382i	−0.972011	−	0.234936i	$-0.924512\pi$
$31$	−260936.	+	451955.i	−0.282545	+	0.489382i	0.689466	+	0.724318i	$0.257845\pi$
$32$	160530.	+	92681.9i	0.153093	+	0.0883883i
$33$		0			0
$34$	−462805.	−	801602.i	−0.346324	−	0.599851i
$35$			8433.67i			0.00562010i
$36$		0			0
$37$	3.53201e6			1.88458			0.942292	−	0.334793i	$-0.108666\pi$
$37$	3.53201e6			1.88458			0.942292	+	0.334793i	$0.108666\pi$
$38$	−1.27726e6	+	737424.i	−0.612553	+	0.353657i
$39$		0			0
$40$	−6154.97	+	10660.7i	−0.00240429	+	0.00416434i
$41$	387124.	+	223506.i	0.136998	+	0.0790959i	0.566933	−	0.823764i	$-0.308130\pi$
$41$	387124.	+	223506.i	0.136998	+	0.0790959i	−0.429934	+	0.902860i	$0.641463\pi$
$42$		0			0
$43$	−1.79681e6	−	3.11216e6i	−0.525566	−	0.910307i	−0.999557	−	0.0297772i	$-0.990520\pi$
$43$	−1.79681e6	−	3.11216e6i	−0.525566	−	0.910307i	0.473990	−	0.880530i	$-0.342813\pi$
$44$		−	783557.i		−	0.209055i
$45$		0			0
$46$	2.11488e6			0.472340
$47$	3.11345e6	−	1.79755e6i	0.638045	−	0.368375i	−0.145816	−	0.989312i	$-0.546581\pi$
$47$	3.11345e6	−	1.79755e6i	0.638045	−	0.368375i	0.783861	+	0.620936i	$0.213248\pi$
$48$		0			0
$49$	2.39022e6	−	4.13999e6i	0.414624	−	0.718150i
$50$	3.82662e6	+	2.20930e6i	0.612259	+	0.353488i
$51$		0			0
$52$	321275.	+	556464.i	0.0439403	+	0.0761069i
$53$			1.22010e7i			1.54630i	0.634226	+	0.773148i	$0.281319\pi$
$53$			1.22010e7i			1.54630i	−0.634226	+	0.773148i	$0.718681\pi$
$54$		0			0
$55$	52035.7			0.00568657
$56$	−1.24429e6	+	718390.i	−0.126523	+	0.0730480i
$57$		0			0
$58$	2.74744e6	−	4.75871e6i	0.242782	−	0.420511i
$59$	−5.67074e6	−	3.27400e6i	−0.467985	−	0.270191i	0.247411	−	0.968911i	$-0.420420\pi$
$59$	−5.67074e6	−	3.27400e6i	−0.467985	−	0.270191i	−0.715396	+	0.698720i	$0.753754\pi$
$60$		0			0
$61$	2.66550e6	+	4.61678e6i	0.192513	+	0.333442i	0.946082	−	0.323926i	$-0.105003\pi$
$61$	2.66550e6	+	4.61678e6i	0.192513	+	0.333442i	−0.753570	+	0.657368i	$0.771670\pi$
$62$		−	5.90432e6i		−	0.399579i
$63$		0			0
$64$	−2.09715e6			−0.125000
$65$	−36954.6	+	21335.7i	−0.00207021	+	0.00119524i
$66$		0			0
$67$	1.11982e7	−	1.93958e7i	0.555711	−	0.962519i	−0.442137	−	0.896948i	$-0.645779\pi$
$67$	1.11982e7	−	1.93958e7i	0.555711	−	0.962519i	0.997848	−	0.0655719i	$-0.0208872\pi$
$68$	9.06909e6	+	5.23604e6i	0.424159	+	0.244888i
$69$		0			0
$70$	−47708.0	−	82632.7i	−0.00198701	−	0.00344160i
$71$			3.53919e7i			1.39274i	0.717683	+	0.696370i	$0.245203\pi$
$71$			3.53919e7i			1.39274i	−0.717683	+	0.696370i	$0.754797\pi$
$72$		0			0
$73$	−1.66778e7			−0.587282			−0.293641	−	0.955916i	$-0.594867\pi$
$73$	−1.66778e7			−0.587282			−0.293641	+	0.955916i	$0.594867\pi$
$74$	−3.46065e7	+	1.99801e7i	−1.15407	+	0.666301i
$75$		0			0
$76$	8.34300e6	−	1.44505e7i	0.250074	−	0.433140i
$77$	5.25977e6	+	3.03673e6i	0.149625	+	0.0863859i
$78$		0			0
$79$	1.69904e7	+	2.94282e7i	0.436209	+	0.755536i	0.997393	−	0.0721548i	$-0.0229876\pi$
$79$	1.69904e7	+	2.94282e7i	0.436209	+	0.755536i	−0.561185	+	0.827691i	$0.689654\pi$
$80$		−	139271.i		−	0.00340017i
$81$		0			0
$82$	−5.05737e6			−0.111859
$83$	−4.39658e7	+	2.53837e7i	−0.926408	+	0.534862i	−0.885674	−	0.464308i	$-0.846303\pi$
$83$	−4.39658e7	+	2.53837e7i	−0.926408	+	0.534862i	−0.0407343	+	0.999170i	$0.512970\pi$
$84$		0			0
$85$	−347724.	+	602275.i	−0.00666129	+	0.0115377i
$86$	3.52101e7	+	2.03285e7i	0.643684	+	0.371631i
$87$		0			0
$88$	4.43247e6	+	7.67726e6i	0.0739120	+	0.128019i
$89$			1.03941e8i			1.65664i	0.560255	+	0.828320i	$0.310703\pi$
$89$			1.03941e8i			1.65664i	−0.560255	+	0.828320i	$0.689297\pi$
$90$		0			0
$91$	−4.98049e6			−0.0726284
$92$	−2.07215e7	+	1.19636e7i	−0.289248	+	0.166997i
$93$		0			0
$94$	−2.03370e7	+	3.52247e7i	−0.260481	+	0.451166i
$95$	959652.	+	554056.i	0.0117820	+	0.00680235i
$96$		0			0
$97$	−3.54725e6	−	6.14401e6i	−0.0400686	−	0.0694009i	0.845296	−	0.534299i	$-0.179424\pi$
$97$	−3.54725e6	−	6.14401e6i	−0.0400686	−	0.0694009i	−0.885364	+	0.464898i	$0.846091\pi$
$98$			5.40846e7i			0.586367i
$99$		0			0
$100$	−4.99908e7			−0.499908
$101$	−5.57886e7	+	3.22095e7i	−0.536117	+	0.309527i	−0.743504	−	0.668732i	$-0.766837\pi$
$101$	−5.57886e7	+	3.22095e7i	−0.536117	+	0.309527i	0.207387	+	0.978259i	$0.433504\pi$
$102$		0			0
$103$	−1.05285e8	+	1.82359e8i	−0.935444	+	1.62024i	−0.161602	+	0.986856i	$0.551666\pi$
$103$	−1.05285e8	+	1.82359e8i	−0.935444	+	1.62024i	−0.773841	+	0.633380i	$0.781667\pi$
$104$	−6.29567e6	−	3.63481e6i	−0.0538157	−	0.0310705i
$105$		0			0
$106$	−6.90194e7	−	1.19545e8i	−0.546698	−	0.946909i
$107$			2.27173e8i			1.73309i	0.499095	+	0.866547i	$0.333666\pi$
$107$			2.27173e8i			1.73309i	−0.499095	+	0.866547i	$0.666334\pi$
$108$		0			0
$109$	8.71964e7			0.617722			0.308861	−	0.951107i	$-0.400052\pi$
$109$	8.71964e7			0.617722			0.308861	+	0.951107i	$0.400052\pi$
$110$	−509844.	+	294358.i	−0.00348230	+	0.00201051i
$111$		0			0
$112$	8.12766e6	−	1.40775e7i	0.0516527	−	0.0894651i
$113$	9.83564e7	+	5.67861e7i	0.603238	+	0.348280i	0.770314	−	0.637664i	$-0.220099\pi$
$113$	9.83564e7	+	5.67861e7i	0.603238	+	0.348280i	−0.167076	+	0.985944i	$0.553433\pi$
$114$		0			0
$115$	−794497.	−	1.37611e6i	−0.00454256	−	0.00786795i
$116$			6.21675e7i			0.343346i
$117$		0			0
$118$	7.40822e7			0.382108
$119$	−7.02958e7	+	4.05853e7i	−0.350543	+	0.202386i
$120$		0			0
$121$	−8.84428e7	+	1.53187e8i	−0.412592	+	0.714631i
$122$	−5.22330e7	−	3.01567e7i	−0.235779	−	0.136127i
$123$		0			0
$124$	3.33998e7	+	5.78502e7i	0.141273	+	0.244691i
$125$		−	6.64035e6i		−	0.0271989i
$126$		0			0
$127$	−1.91241e7			−0.0735134			−0.0367567	−	0.999324i	$-0.511703\pi$
$127$	−1.91241e7			−0.0735134			−0.0367567	+	0.999324i	$0.511703\pi$
$128$	2.05478e7	−	1.18633e7i	0.0765466	−	0.0441942i
$129$		0			0
$130$	241386.	−	418093.i	0.000845160	−	0.00146386i
$131$	2.53255e8	+	1.46217e8i	0.859949	+	0.496492i	0.863995	−	0.503500i	$-0.167955\pi$
$131$	2.53255e8	+	1.46217e8i	0.859949	+	0.496492i	−0.00404589	+	0.999992i	$0.501288\pi$
$132$		0			0
$133$	6.46677e7	+	1.12008e8i	0.206672	+	0.357966i
$134$			2.53386e8i			0.785894i
$135$		0			0
$136$	−1.18478e8			−0.346324
$137$	−4.88782e8	+	2.82198e8i	−1.38750	+	0.801073i	−0.993033	−	0.117837i	$-0.962404\pi$
$137$	−4.88782e8	+	2.82198e8i	−1.38750	+	0.801073i	−0.394466	+	0.918910i	$0.629070\pi$
$138$		0			0
$139$	−4.58767e6	+	7.94608e6i	−0.0122895	+	0.0212860i	−0.872105	−	0.489319i	$-0.837245\pi$
$139$	−4.58767e6	+	7.94608e6i	−0.0122895	+	0.0212860i	0.859815	+	0.510605i	$0.170579\pi$
$140$	934882.	+	539755.i	0.00243358	+	0.00140503i
$141$		0			0
$142$	−2.00207e8	−	3.46768e8i	−0.492408	−	0.852875i
$143$			3.07296e7i			0.0734874i
$144$		0			0
$145$	−4.12852e6			−0.00933948
$146$	1.63408e8	−	9.43438e7i	0.359635	−	0.207636i
$147$		0			0
$148$	2.26049e8	−	3.91528e8i	0.471146	−	0.816049i
$149$	2.67428e8	+	1.54400e8i	0.542578	+	0.313258i	0.746123	−	0.665808i	$-0.231913\pi$
$149$	2.67428e8	+	1.54400e8i	0.542578	+	0.313258i	−0.203545	+	0.979066i	$0.565246\pi$
$150$		0			0
$151$	2.62712e8	+	4.55031e8i	0.505327	+	0.875251i	0.999981	+	0.00616159i	$0.00196131\pi$
$151$	2.62712e8	+	4.55031e8i	0.505327	+	0.875251i	−0.494654	+	0.869090i	$0.664705\pi$
$152$			1.88781e8i			0.353657i
$153$		0			0
$154$	−6.87133e7			−0.122168
$155$	−3.84181e6	+	2.21807e6i	−0.00665594	+	0.00384281i
$156$		0			0
$157$	3.06098e8	−	5.30178e8i	0.503805	−	0.872616i	−0.496186	−	0.868217i	$-0.665266\pi$
$157$	3.06098e8	−	5.30178e8i	0.503805	−	0.872616i	0.999990	−	0.00439901i	$-0.00140025\pi$
$158$	−3.32942e8	−	1.92224e8i	−0.534245	−	0.308446i
$159$		0			0
$160$	787836.	+	1.36457e6i	0.00120214	+	0.00208217i
$161$		−	1.85463e8i		−	0.276028i
$162$		0			0
$163$	−6.85520e7			−0.0971113			−0.0485556	−	0.998820i	$-0.515462\pi$
$163$	−6.85520e7			−0.0971113			−0.0485556	+	0.998820i	$0.515462\pi$
$164$	4.95519e7	−	2.86088e7i	0.0684991	−	0.0395480i
$165$		0			0
$166$	2.87183e8	−	4.97416e8i	0.378205	−	0.655069i
$167$	−7.01101e8	−	4.04781e8i	−0.901395	−	0.520420i	−0.0237421	−	0.999718i	$-0.507558\pi$
$167$	−7.01101e8	−	4.04781e8i	−0.901395	−	0.520420i	−0.877652	+	0.479298i	$0.840891\pi$
$168$		0			0
$169$	3.95266e8	+	6.84620e8i	0.484554	+	0.839272i
$170$		−	7.86809e6i		−	0.00942049i
$171$		0			0
$172$	−4.59982e8			−0.525566
$173$	1.90694e8	−	1.10097e8i	0.212889	−	0.122911i	−0.389764	−	0.920915i	$-0.627444\pi$
$173$	1.90694e8	−	1.10097e8i	0.212889	−	0.122911i	0.602653	+	0.798003i	$0.294110\pi$
$174$		0			0
$175$	1.93743e8	−	3.35572e8i	0.206573	−	0.357794i
$176$	−8.68582e7	−	5.01476e7i	−0.0905233	−	0.0522637i
$177$		0			0
$178$	−5.87981e8	−	1.01841e9i	−0.585711	−	1.01448i
$179$		−	3.75120e8i		−	0.365391i	−0.983170	−	0.182695i	$-0.941518\pi$
$179$		−	3.75120e8i		−	0.365391i	0.983170	−	0.182695i	$-0.0584822\pi$
$180$		0			0
$181$	7.19277e8			0.670165			0.335083	−	0.942189i	$-0.391236\pi$
$181$	7.19277e8			0.670165			0.335083	+	0.942189i	$0.391236\pi$
$182$	4.87986e7	−	2.81739e7i	0.0444756	−	0.0256780i
$183$		0			0
$184$	1.35352e8	−	2.34437e8i	0.118085	−	0.204529i
$185$	2.60012e7	+	1.50118e7i	0.0221977	+	0.0128158i
$186$		0			0
$187$	2.50411e8	+	4.33725e8i	0.204780	+	0.354689i
$188$		−	4.60174e8i		−	0.368375i
$189$		0			0
$190$	−1.25368e7			−0.00961997
$191$	1.34194e9	−	7.74769e8i	1.00832	−	0.582155i	0.0976223	−	0.995224i	$-0.468876\pi$
$191$	1.34194e9	−	7.74769e8i	1.00832	−	0.582155i	0.910700	+	0.413068i	$0.135543\pi$
$192$		0			0
$193$	−6.61406e8	+	1.14559e9i	−0.476693	+	0.825656i	−0.999643	−	0.0267068i	$-0.991498\pi$
$193$	−6.61406e8	+	1.14559e9i	−0.476693	+	0.825656i	0.522950	+	0.852363i	$0.324831\pi$
$194$	6.95115e7	+	4.01325e7i	0.0490738	+	0.0283328i
$195$		0			0
$196$	−3.05949e8	−	5.29919e8i	−0.207312	−	0.359075i
$197$		−	2.27722e9i		−	1.51196i	−0.654596	−	0.755978i	$-0.727161\pi$
$197$		−	2.27722e9i		−	1.51196i	0.654596	−	0.755978i	$-0.272839\pi$
$198$		0			0
$199$	1.81429e9			1.15690			0.578449	−	0.815719i	$-0.303658\pi$
$199$	1.81429e9			1.15690			0.578449	+	0.815719i	$0.303658\pi$
$200$	4.89807e8	−	2.82790e8i	0.306130	−	0.176744i
$201$		0			0
$202$	3.64409e8	−	6.31176e8i	0.218869	−	0.379092i
$203$	−4.17311e8	−	2.40934e8i	−0.245740	−	0.141878i
$204$		0			0
$205$	1.89990e6	+	3.29072e6i	0.00107576	+	0.00186327i
$206$		−	2.38233e9i		−	1.32292i
$207$		0			0
$208$	8.22463e7			0.0439403
$209$	6.91088e8	−	3.99000e8i	0.362200	−	0.209116i
$210$		0			0
$211$	−1.26901e9	+	2.19799e9i	−0.640230	+	1.10891i	0.345151	+	0.938547i	$0.387828\pi$
$211$	−1.26901e9	+	2.19799e9i	−0.640230	+	1.10891i	−0.985381	+	0.170364i	$0.945506\pi$
$212$	1.35250e9	+	7.80865e8i	0.669565	+	0.386574i
$213$		0			0
$214$	−1.28509e9	−	2.22584e9i	−0.612741	−	1.06130i
$215$		−	3.05473e7i		−	0.0142961i
$216$		0			0
$217$	−5.17774e8			−0.233508
$218$	−8.54347e8	+	4.93258e8i	−0.378276	+	0.218398i
$219$		0			0
$220$	3.33029e6	−	5.76822e6i	0.00142164	−	0.00246236i
$221$	−3.55673e8	−	2.05348e8i	−0.149101	−	0.0860837i
$222$		0			0
$223$	5.67316e8	+	9.82620e8i	0.229406	+	0.397344i	0.957632	−	0.287994i	$-0.0929882\pi$
$223$	5.67316e8	+	9.82620e8i	0.229406	+	0.397344i	−0.728226	+	0.685337i	$0.759655\pi$
$224$			1.83908e8i			0.0730480i
$225$		0			0
$226$	−1.28492e9			−0.492542
$227$	4.03916e9	−	2.33201e9i	1.52120	−	0.878268i	0.521518	−	0.853240i	$-0.325366\pi$
$227$	4.03916e9	−	2.33201e9i	1.52120	−	0.878268i	0.999687	−	0.0250277i	$-0.00796740\pi$
$228$		0			0
$229$	−1.83905e9	+	3.18534e9i	−0.668733	+	1.15828i	0.309526	+	0.950891i	$0.399830\pi$
$229$	−1.83905e9	+	3.18534e9i	−0.668733	+	1.15828i	−0.978259	+	0.207388i	$0.933504\pi$
$230$	1.55689e7	+	8.98870e6i	0.00556348	+	0.00321208i
$231$		0			0
$232$	−3.51673e8	−	6.09115e8i	−0.121391	−	0.210255i
$233$		−	3.64703e9i		−	1.23742i	−0.785621	−	0.618708i	$-0.787656\pi$
$233$		−	3.64703e9i		−	1.23742i	0.785621	−	0.618708i	$-0.212344\pi$
$234$		0			0
$235$	3.05600e7			0.0100203
$236$	−7.25855e8	+	4.19072e8i	−0.233992	+	0.135096i
$237$		0			0
$238$	4.59170e8	−	7.95306e8i	0.143109	−	0.247871i
$239$	−4.95613e9	−	2.86142e9i	−1.51898	−	0.876982i	−0.999750	−	0.0223458i	$-0.992887\pi$
$239$	−4.95613e9	−	2.86142e9i	−1.51898	−	0.876982i	−0.519227	−	0.854636i	$-0.673780\pi$
$240$		0			0
$241$	1.17145e9	+	2.02901e9i	0.347260	+	0.601473i	0.985762	−	0.168148i	$-0.0537787\pi$
$241$	1.17145e9	+	2.02901e9i	0.347260	+	0.601473i	−0.638501	+	0.769621i	$0.720445\pi$
$242$		−	2.00123e9i		−	0.583494i
$243$		0			0
$244$	6.82369e8			0.192513
$245$	3.51917e7	−	2.03179e7i	0.00976733	−	0.00563917i
$246$		0			0
$247$	−3.27197e8	+	5.66721e8i	−0.0879065	+	0.152259i
$248$	−6.54501e8	−	3.77876e8i	−0.173023	−	0.0998948i
$249$		0			0
$250$	3.75635e7	+	6.50619e7i	0.00961625	+	0.0166558i
$251$			3.31445e9i			0.835058i	0.908664	+	0.417529i	$0.137104\pi$
$251$			3.31445e9i			0.835058i	−0.908664	+	0.417529i	$0.862896\pi$
$252$		0			0
$253$	−1.14430e9			−0.279293
$254$	1.87377e8	−	1.08182e8i	0.0450176	−	0.0259909i
$255$		0			0
$256$	−1.34218e8	+	2.32472e8i	−0.0312500	+	0.0541266i
$257$	1.73121e9	+	9.99513e8i	0.396841	+	0.229116i	0.685120	−	0.728430i	$-0.259750\pi$
$257$	1.73121e9	+	9.99513e8i	0.396841	+	0.229116i	−0.288279	+	0.957546i	$0.593083\pi$
$258$		0			0
$259$	1.75214e9	+	3.03479e9i	0.389375	+	0.674418i
$260$			5.46195e6i			0.00119524i
$261$		0			0
$262$	−3.30851e9			−0.702146
$263$	−7.09747e7	+	4.09773e7i	−0.0148348	+	0.00856486i	−0.507399	−	0.861711i	$-0.669393\pi$
$263$	−7.09747e7	+	4.09773e7i	−0.0148348	+	0.00856486i	0.492564	+	0.870276i	$0.336060\pi$
$264$		0			0
$265$	−5.18569e7	+	8.98189e7i	−0.0105153	+	0.0182131i
$266$	−1.26722e9	−	7.31632e8i	−0.253120	−	0.146139i
$267$		0			0
$268$	−1.43337e9	−	2.48267e9i	−0.277855	−	0.481260i
$269$		−	7.24861e8i		−	0.138435i	−0.997602	−	0.0692174i	$-0.977950\pi$
$269$		−	7.24861e8i		−	0.138435i	0.997602	−	0.0692174i	$-0.0220502\pi$
$270$		0			0
$271$	−3.93920e9			−0.730349			−0.365174	−	0.930939i	$-0.618991\pi$
$271$	−3.93920e9			−0.730349			−0.365174	+	0.930939i	$0.618991\pi$
$272$	1.16084e9	−	6.70214e8i	0.212079	−	0.122444i
$273$		0			0
$274$	3.19271e9	−	5.52993e9i	0.566444	−	0.981110i
$275$	−2.07048e9	−	1.19539e9i	−0.362026	−	0.209016i
$276$		0			0
$277$	3.86522e9	+	6.69476e9i	0.656532	+	1.13715i	0.981508	+	0.191424i	$0.0613104\pi$
$277$	3.86522e9	+	6.69476e9i	0.656532	+	1.13715i	−0.324976	+	0.945722i	$0.605356\pi$
$278$		−	1.03807e8i		−	0.0173799i
$279$		0			0
$280$	−1.22133e7			−0.00198701
$281$	−6.45705e9	+	3.72798e9i	−1.03564	+	0.597927i	−0.918595	−	0.395199i	$-0.870676\pi$
$281$	−6.45705e9	+	3.72798e9i	−1.03564	+	0.597927i	−0.117045	+	0.993127i	$0.537342\pi$
$282$		0			0
$283$	2.41445e9	−	4.18194e9i	0.376419	−	0.651977i	−0.614119	−	0.789213i	$-0.710489\pi$
$283$	2.41445e9	−	4.18194e9i	0.376419	−	0.651977i	0.990538	+	0.137236i	$0.0438219\pi$
$284$	3.92323e9	+	2.26508e9i	0.603074	+	0.348185i
$285$		0			0
$286$	−1.73833e8	−	3.01087e8i	−0.0259817	−	0.0450017i
$287$			4.43501e8i			0.0653683i
$288$		0			0
$289$	2.82361e8			0.0404775
$290$	4.04511e7	−	2.33544e7i	0.00571924	−	0.00330200i
$291$		0			0
$292$	−1.06738e9	+	1.84875e9i	−0.146821	+	0.254301i
$293$	1.08228e10	+	6.24857e9i	1.46849	+	0.847833i	0.999377	−	0.0353044i	$-0.0112401\pi$
$293$	1.08228e10	+	6.24857e9i	1.46849	+	0.847833i	0.469114	+	0.883138i	$0.344573\pi$
$294$		0			0
$295$	−2.78304e7	−	4.82037e7i	−0.00367479	−	0.00636492i
$296$			5.11490e9i			0.666301i
$297$		0			0
$298$	−3.49367e9			−0.443013
$299$	8.12659e8	−	4.69189e8i	0.101677	−	0.0587033i
$300$		0			0
$301$	1.78269e9	−	3.08771e9i	0.217175	−	0.376159i
$302$	−5.14808e9	−	2.97225e9i	−0.618896	−	0.357320i
$303$		0			0
$304$	−1.06790e9	−	1.84966e9i	−0.125037	−	0.216570i
$305$			4.53158e7i			0.00523662i
$306$		0			0
$307$	9.73759e9			1.09622			0.548110	−	0.836406i	$-0.315347\pi$
$307$	9.73759e9			1.09622			0.548110	+	0.836406i	$0.315347\pi$
$308$	6.73250e8	−	3.88701e8i	0.0748124	−	0.0431930i
$309$		0			0
$310$	2.50946e7	−	4.34652e7i	0.00271728	−	0.00470646i
$311$	3.93490e9	+	2.27182e9i	0.420622	+	0.242846i	0.695344	−	0.718678i	$-0.255252\pi$
$311$	3.93490e9	+	2.27182e9i	0.420622	+	0.242846i	−0.274721	+	0.961524i	$0.588586\pi$
$312$		0			0
$313$	−5.94496e9	−	1.02970e10i	−0.619401	−	1.07283i	−0.989595	−	0.143879i	$-0.954042\pi$
$313$	−5.94496e9	−	1.02970e10i	−0.619401	−	1.07283i	0.370194	−	0.928954i	$-0.379291\pi$
$314$			6.92621e9i			0.712488i
$315$		0			0
$316$	4.34953e9			0.436209
$317$	−8.96114e9	+	5.17372e9i	−0.887414	+	0.512349i	−0.873096	−	0.487549i	$-0.837891\pi$
$317$	−8.96114e9	+	5.17372e9i	−0.887414	+	0.512349i	−0.0143180	+	0.999897i	$0.504558\pi$
$318$		0			0
$319$	−1.48657e9	+	2.57481e9i	−0.143556	+	0.248646i
$320$	−1.54384e7	−	8.91335e6i	−0.00147232	−	0.000850043i
$321$		0			0
$322$	1.04914e9	+	1.81716e9i	0.0975906	+	0.169032i
$323$			1.06651e10i			0.979841i
$324$		0			0
$325$	1.96054e9			0.175729
$326$	6.71670e8	−	3.87789e8i	0.0594683	−	0.0343340i
$327$		0			0
$328$	−3.23672e8	+	5.60616e8i	−0.0279646	+	0.0484362i
$329$	3.08900e9	+	1.78343e9i	0.263654	+	0.152221i
$330$		0			0
$331$	−9.84250e8	−	1.70477e9i	−0.0819962	−	0.142022i	0.822111	−	0.569327i	$-0.192796\pi$
$331$	−9.84250e8	−	1.70477e9i	−0.0819962	−	0.142022i	−0.904107	+	0.427306i	$0.859463\pi$
$332$			6.49821e9i			0.534862i
$333$		0			0
$334$	9.15915e9			0.735986
$335$	1.64873e8	−	9.51895e7i	0.0130909	−	0.00755805i
$336$		0			0
$337$	2.44186e9	−	4.22943e9i	0.189322	−	0.327916i	−0.755702	−	0.654915i	$-0.772704\pi$
$337$	2.44186e9	−	4.22943e9i	0.189322	−	0.327916i	0.945025	+	0.327000i	$0.106038\pi$
$338$	−7.74559e9	−	4.47192e9i	−0.593455	−	0.342631i
$339$		0			0
$340$	4.45086e7	+	7.70912e7i	0.00333065	+	0.00576885i
$341$			3.19466e9i			0.236269i
$342$		0			0
$343$	1.04624e10			0.755885
$344$	4.50689e9	−	2.60205e9i	0.321842	−	0.185816i
$345$		0			0
$346$	−1.24561e9	+	2.15746e9i	−0.0869115	+	0.150535i
$347$	−2.49444e9	−	1.44016e9i	−0.172050	−	0.0993332i	0.411502	−	0.911409i	$-0.365004\pi$
$347$	−2.49444e9	−	1.44016e9i	−0.172050	−	0.0993332i	−0.583552	+	0.812076i	$0.698338\pi$
$348$		0			0
$349$	−7.75259e9	−	1.34279e10i	−0.522571	−	0.905119i	−0.999655	−	0.0262617i	$-0.991640\pi$
$349$	−7.75259e9	−	1.34279e10i	−0.522571	−	0.905119i	0.477084	−	0.878858i	$-0.341694\pi$
$350$			4.38389e9i			0.292138i
$351$		0			0
$352$	1.13471e9			0.0739120
$353$	−1.70487e10	+	9.84308e9i	−1.09798	+	0.633916i	−0.935689	−	0.352827i	$-0.885220\pi$
$353$	−1.70487e10	+	9.84308e9i	−1.09798	+	0.633916i	−0.162287	+	0.986744i	$0.551887\pi$
$354$		0			0
$355$	−1.50423e8	+	2.60540e8i	−0.00947111	+	0.0164044i
$356$	1.15220e10	+	6.65225e9i	0.717346	+	0.414160i
$357$		0			0
$358$	2.12200e9	+	3.67541e9i	0.129185	+	0.223755i
$359$			2.70309e10i			1.62735i	0.581317	+	0.813677i	$0.302538\pi$
$359$			2.70309e10i			1.62735i	−0.581317	+	0.813677i	$0.697462\pi$
$360$		0			0
$361$	1.00026e7			0.000588959
$362$	−7.04745e9	+	4.06885e9i	−0.410391	+	0.236939i
$363$		0			0
$364$	−3.18751e8	+	5.52093e8i	−0.0181571	+	0.0314490i
$365$	−1.22775e8	−	7.08842e7i	−0.00691733	−	0.00399372i
$366$		0			0
$367$	−2.97841e9	−	5.15875e9i	−0.164180	−	0.284368i	0.772184	−	0.635399i	$-0.219164\pi$
$367$	−2.97841e9	−	5.15875e9i	−0.164180	−	0.284368i	−0.936364	+	0.351031i	$0.885831\pi$
$368$			3.06268e9i			0.166997i
$369$		0			0
$370$	−3.39679e8			−0.0181243
$371$	−1.04834e10	+	6.05259e9i	−0.553358	+	0.319481i
$372$		0			0
$373$	−7.70280e9	+	1.33416e10i	−0.397936	+	0.689245i	−0.993471	−	0.114084i	$-0.963607\pi$
$373$	−7.70280e9	+	1.33416e10i	−0.397936	+	0.689245i	0.595535	+	0.803329i	$0.296940\pi$
$374$	−4.90704e9	−	2.83308e9i	−0.250803	−	0.144801i
$375$		0			0
$376$	2.60314e9	+	4.50876e9i	0.130240	+	0.225583i
$377$		−	2.43809e9i		−	0.120694i
$378$		0			0
$379$	−2.29401e10			−1.11183			−0.555914	−	0.831240i	$-0.687632\pi$
$379$	−2.29401e10			−1.11183			−0.555914	+	0.831240i	$0.687632\pi$
$380$	1.22836e8	−	7.09191e7i	0.00589101	−	0.00340117i
$381$		0			0
$382$	−8.76551e9	+	1.51823e10i	−0.411646	+	0.712992i
$383$	1.12172e10	+	6.47626e9i	0.521302	+	0.300974i	0.737467	−	0.675383i	$-0.236022\pi$
$383$	1.12172e10	+	6.47626e9i	0.521302	+	0.300974i	−0.216165	+	0.976357i	$0.569355\pi$
$384$		0			0
$385$	2.58135e7	+	4.47103e7i	0.00117491	+	0.00203500i
$386$		−	1.49659e10i		−	0.674146i
$387$		0			0
$388$	−9.08095e8			−0.0400686
$389$	2.70953e9	−	1.56435e9i	0.118330	−	0.0683180i	−0.439667	−	0.898161i	$-0.644903\pi$
$389$	2.70953e9	−	1.56435e9i	0.118330	−	0.0683180i	0.557997	+	0.829843i	$0.311570\pi$
$390$		0			0
$391$	7.64671e9	−	1.32445e10i	0.327165	−	0.566667i
$392$	5.99535e9	+	3.46141e9i	0.253904	+	0.146592i
$393$		0			0
$394$	1.28819e10	+	2.23121e10i	0.534558	+	0.925881i
$395$			2.88851e8i			0.0118655i
$396$		0			0
$397$	−2.56318e10			−1.03185			−0.515926	−	0.856633i	$-0.672552\pi$
$397$	−2.56318e10			−1.03185			−0.515926	+	0.856633i	$0.672552\pi$
$398$	−1.77764e10	+	1.02632e10i	−0.708452	+	0.409025i
$399$		0			0
$400$	−3.19941e9	+	5.54154e9i	−0.124977	+	0.216466i
$401$	7.38149e9	+	4.26171e9i	0.285474	+	0.164819i	0.635899	−	0.771772i	$-0.280629\pi$
$401$	7.38149e9	+	4.26171e9i	0.285474	+	0.164819i	−0.350425	+	0.936591i	$0.613963\pi$
$402$		0			0
$403$	−1.30988e9	−	2.26878e9i	−0.0496605	−	0.0860145i
$404$			8.24564e9i			0.309527i
$405$		0			0
$406$	5.45172e9			0.200646
$407$	1.87246e10	−	1.08107e10i	0.682395	−	0.393981i
$408$		0			0
$409$	1.61262e10	−	2.79314e10i	0.576287	−	0.998158i	−0.419614	−	0.907703i	$-0.637835\pi$
$409$	1.61262e10	−	2.79314e10i	0.576287	−	0.998158i	0.995901	−	0.0904551i	$-0.0288322\pi$
$410$	−3.72303e7	−	2.14949e7i	−0.00131753	−	0.000760677i
$411$		0			0
$412$	1.34765e10	+	2.33419e10i	0.467722	+	0.810118i
$413$		−	6.49657e9i		−	0.223298i
$414$		0			0
$415$	−4.31544e8			−0.0145490
$416$	−8.05846e8	+	4.65255e8i	−0.0269078	+	0.0155352i
$417$		0			0
$418$	−4.51417e9	+	7.81877e9i	−0.147867	+	0.256114i
$419$	−4.10509e10	−	2.37007e10i	−1.33188	−	0.768964i	−0.346296	−	0.938125i	$-0.612561\pi$
$419$	−4.10509e10	−	2.37007e10i	−1.33188	−	0.768964i	−0.985588	+	0.169162i	$0.945894\pi$
$420$		0			0
$421$	7.04824e9	+	1.22079e10i	0.224364	+	0.388609i	0.956128	−	0.292948i	$-0.0946363\pi$
$421$	7.04824e9	+	1.22079e10i	0.224364	+	0.388609i	−0.731765	+	0.681557i	$0.761303\pi$
$422$		−	2.87145e10i		−	0.905422i
$423$		0			0
$424$	−1.76690e10			−0.546698
$425$	2.76716e10	−	1.59762e10i	0.848160	−	0.489686i
$426$		0			0
$427$	−2.64457e9	+	4.58052e9i	−0.0795505	+	0.137785i
$428$	2.51824e10	+	1.45391e10i	0.750452	+	0.433274i
$429$		0			0
$430$	1.72801e8	+	2.99301e8i	0.00505444	+	0.00875455i
$431$		−	1.05886e10i		−	0.306852i	−0.988160	−	0.153426i	$-0.950969\pi$
$431$		−	1.05886e10i		−	0.306852i	0.988160	−	0.153426i	$-0.0490306\pi$
$432$		0			0
$433$	−3.17408e10			−0.902955			−0.451477	−	0.892283i	$-0.649103\pi$
$433$	−3.17408e10			−0.902955			−0.451477	+	0.892283i	$0.649103\pi$
$434$	5.07312e9	−	2.92897e9i	0.142994	−	0.0825574i
$435$		0			0
$436$	5.58057e9	−	9.66583e9i	0.154430	−	0.267481i
$437$	−2.11035e10	−	1.21841e10i	−0.578666	−	0.334093i
$438$		0			0
$439$	8.88113e9	+	1.53826e10i	0.239117	+	0.414162i	0.960461	−	0.278414i	$-0.0898088\pi$
$439$	8.88113e9	+	1.53826e10i	0.239117	+	0.414162i	−0.721344	+	0.692577i	$0.756475\pi$
$440$			7.53558e7i			0.00201051i
$441$		0			0
$442$	4.64649e9			0.121741
$443$	1.75054e10	−	1.01067e10i	0.454523	−	0.262419i	−0.255215	−	0.966884i	$-0.582146\pi$
$443$	1.75054e10	−	1.01067e10i	0.454523	−	0.262419i	0.709739	+	0.704465i	$0.248813\pi$
$444$		0			0
$445$	−4.41773e8	+	7.65174e8i	−0.0112657	+	0.0195128i
$446$	−1.11171e10	−	6.41845e9i	−0.280964	−	0.162215i
$447$		0			0
$448$	−1.04034e9	−	1.80192e9i	−0.0258264	−	0.0447326i
$449$		−	4.74282e8i		−	0.0116695i	−0.999983	−	0.00583474i	$-0.998143\pi$
$449$		−	4.74282e8i		−	0.0116695i	0.999983	−	0.00583474i	$-0.00185727\pi$
$450$		0			0
$451$	2.73640e9			0.0661415
$452$	1.25896e10	−	7.26862e9i	0.301619	−	0.174140i
$453$		0			0
$454$	−2.63837e10	+	4.56979e10i	−0.621029	+	1.07565i
$455$	−3.66643e7	−	2.11682e7i	−0.000855457	−	0.000493898i
$456$		0			0
$457$	−3.55019e10	−	6.14910e10i	−0.813928	−	1.40977i	−0.910095	−	0.414400i	$-0.863991\pi$
$457$	−3.55019e10	−	6.14910e10i	−0.813928	−	1.40977i	0.0961664	−	0.995365i	$-0.469342\pi$
$458$		−	4.16130e10i		−	0.945731i
$459$		0			0
$460$	−2.03391e8			−0.00454256
$461$	5.74429e10	−	3.31647e10i	1.27184	−	0.734298i	0.296506	−	0.955031i	$-0.404178\pi$
$461$	5.74429e10	−	3.31647e10i	1.27184	−	0.734298i	0.975334	+	0.220733i	$0.0708451\pi$
$462$		0			0
$463$	7.92359e9	−	1.37241e10i	0.172424	−	0.298647i	−0.766843	−	0.641835i	$-0.778173\pi$
$463$	7.92359e9	−	1.37241e10i	0.172424	−	0.298647i	0.939267	+	0.343188i	$0.111507\pi$
$464$	6.89135e9	+	3.97872e9i	0.148673	+	0.0858364i
$465$		0			0
$466$	2.06307e10	+	3.57335e10i	0.437493	+	0.757760i
$467$		−	6.51992e10i		−	1.37080i	−0.728166	−	0.685401i	$-0.759627\pi$
$467$		−	6.51992e10i		−	1.37080i	0.728166	−	0.685401i	$-0.240373\pi$
$468$		0			0
$469$	2.22205e10			0.459264
$470$	−2.99425e8	+	1.72873e8i	−0.00613617	+	0.00354272i
$471$		0			0
$472$	4.74126e9	−	8.21211e9i	0.0955270	−	0.165458i
$473$	−1.90512e10	−	1.09992e10i	−0.380608	−	0.219744i
$474$		0			0
$475$	−2.54561e10	−	4.40913e10i	−0.500055	−	0.866120i
$476$			1.03898e10i			0.202386i
$477$		0			0
$478$	6.47466e10			1.24024
$479$	−8.30989e9	+	4.79772e9i	−0.157853	+	0.0911366i	−0.576846	−	0.816853i	$-0.695717\pi$
$479$	−8.30989e9	+	4.79772e9i	−0.157853	+	0.0911366i	0.418993	+	0.907990i	$0.362383\pi$
$480$		0			0
$481$	−8.86521e9	+	1.53550e10i	−0.165618	+	0.286859i
$482$	−2.29556e10	−	1.32534e10i	−0.425305	−	0.245550i
$483$		0			0
$484$	1.13207e10	+	1.96080e10i	0.206296	+	0.357315i
$485$		−	6.03062e7i		−	0.00108992i
$486$		0			0
$487$	2.39483e10			0.425755			0.212877	−	0.977079i	$-0.431716\pi$
$487$	2.39483e10			0.425755			0.212877	+	0.977079i	$0.431716\pi$
$488$	−6.68582e9	+	3.86006e9i	−0.117890	+	0.0680636i
$489$		0			0
$490$	−2.29871e8	+	3.98149e8i	−0.00398750	+	0.00690655i
$491$	5.22717e10	+	3.01791e10i	0.899374	+	0.519254i	0.876997	−	0.480496i	$-0.159543\pi$
$491$	5.22717e10	+	3.01791e10i	0.899374	+	0.519254i	0.0223770	+	0.999750i	$0.492877\pi$
$492$		0			0
$493$	−1.98677e10	−	3.44118e10i	−0.336325	−	0.582532i
$494$		−	7.40361e9i		−	0.124319i
$495$		0			0
$496$	8.55036e9			0.141273
$497$	−3.04095e10	+	1.75569e10i	−0.498407	+	0.287755i
$498$		0			0
$499$	2.10288e10	−	3.64229e10i	0.339165	−	0.587452i	−0.645111	−	0.764089i	$-0.723189\pi$
$499$	2.10288e10	−	3.64229e10i	0.339165	−	0.587452i	0.984276	+	0.176638i	$0.0565221\pi$
$500$	−7.36091e8	−	4.24982e8i	−0.0117775	−	0.00679972i
$501$		0			0
$502$	−1.87494e10	−	3.24748e10i	−0.295238	−	0.511366i
$503$		−	1.03916e11i		−	1.62335i	−0.584113	−	0.811673i	$-0.698557\pi$
$503$		−	1.03916e11i		−	1.62335i	0.584113	−	0.811673i	$-0.301443\pi$
$504$		0			0
$505$	−5.47590e8			−0.00841957
$506$	1.12118e10	−	6.47316e9i	0.171031	−	0.0987448i
$507$		0			0
$508$	−1.22394e9	+	2.11993e9i	−0.0183784	+	0.0318322i
$509$	−3.10782e10	−	1.79430e10i	−0.463004	−	0.267315i	0.250303	−	0.968168i	$-0.419470\pi$
$509$	−3.10782e10	−	1.79430e10i	−0.463004	−	0.267315i	−0.713306	+	0.700852i	$0.752803\pi$
$510$		0			0
$511$	−8.27339e9	−	1.43299e10i	−0.121339	−	0.210165i
$512$		−	3.03700e9i		−	0.0441942i
$513$		0			0
$514$	−2.26164e10			−0.324019
$515$	−1.55013e9	+	8.94968e8i	−0.0220363	+	0.0127227i
$516$		0			0
$517$	1.10038e10	−	1.90591e10i	0.154021	−	0.266772i
$518$	−3.43347e10	−	1.98231e10i	−0.476886	−	0.275330i
$519$		0			0
$520$	−3.08974e7	−	5.35159e7i	−0.000422580	−	0.000731930i
$521$		−	1.79939e10i		−	0.244217i	−0.992517	−	0.122108i	$-0.961034\pi$
$521$		−	1.79939e10i		−	0.244217i	0.992517	−	0.122108i	$-0.0389655\pi$
$522$		0			0
$523$	1.40716e11			1.88078			0.940390	−	0.340098i	$-0.110460\pi$
$523$	1.40716e11			1.88078			0.940390	+	0.340098i	$0.110460\pi$
$524$	3.24166e10	−	1.87158e10i	0.429975	−	0.248246i
$525$		0			0
$526$	4.63605e8	−	8.02987e8i	0.00605627	−	0.0104898i
$527$	−3.69759e10	−	2.13480e10i	−0.479376	−	0.276768i
$528$		0			0
$529$	−2.16839e10	−	3.75576e10i	−0.276895	−	0.479596i
$530$		−	1.17339e9i		−	0.0148709i
$531$		0			0
$532$	1.65549e10			0.206672
$533$	−1.94333e9	+	1.12198e9i	−0.0240790	+	0.0139020i
$534$		0			0
$535$	−9.65536e8	+	1.67236e9i	−0.0117856	+	0.0204133i
$536$	2.80882e10	+	1.62167e10i	0.340302	+	0.196473i
$537$		0			0
$538$	4.10043e9	+	7.10216e9i	0.0489441	+	0.0847737i
$539$		−	2.92637e10i		−	0.346716i
$540$		0			0
$541$	−1.09373e11			−1.27680			−0.638399	−	0.769706i	$-0.720403\pi$
$541$	−1.09373e11			−1.27680			−0.638399	+	0.769706i	$0.720403\pi$
$542$	3.85961e10	−	2.22835e10i	0.447246	−	0.258217i
$543$		0			0
$544$	−7.58260e9	+	1.31335e10i	−0.0865810	+	0.149963i
$545$	6.41905e8	+	3.70604e8i	0.00727586	+	0.00420072i
$546$		0			0
$547$	5.96676e10	+	1.03347e11i	0.666483	+	1.15438i	0.978881	+	0.204432i	$0.0655346\pi$
$547$	5.96676e10	+	1.03347e11i	0.666483	+	1.15438i	−0.312397	+	0.949951i	$0.601132\pi$
$548$			7.22428e10i			0.801073i
$549$		0			0
$550$	2.70486e10			0.295593
$551$	−5.48310e10	+	3.16567e10i	−0.594867	+	0.343447i
$552$		0			0
$553$	−1.68569e10	+	2.91970e10i	−0.180251	+	0.312204i
$554$	−7.57426e10	−	4.37300e10i	−0.804084	−	0.464238i
$555$		0			0
$556$	5.87222e8	+	1.01710e9i	0.00614474	+	0.0106430i
$557$			1.58000e10i			0.164148i	0.996626	+	0.0820740i	$0.0261544\pi$
$557$			1.58000e10i			0.164148i	−0.996626	+	0.0820740i	$0.973846\pi$
$558$		0			0
$559$	1.80396e10			0.184748
$560$	1.19665e8	−	6.90886e7i	0.00121679	−	0.000702513i
$561$		0			0
$562$	4.21773e10	−	7.30532e10i	0.422798	−	0.732308i
$563$	1.17697e11	+	6.79522e10i	1.17147	+	0.676348i	0.954026	−	0.299725i	$-0.0968949\pi$
$563$	1.17697e11	+	6.79522e10i	1.17147	+	0.676348i	0.217444	+	0.976073i	$0.430228\pi$
$564$		0			0
$565$	4.82706e8	+	8.36072e8i	0.00473685	+	0.00820446i
$566$			5.46327e10i			0.532337i
$567$		0			0
$568$	−5.12529e10			−0.492408
$569$	1.07502e11	−	6.20665e10i	1.02558	−	0.592118i	0.109864	−	0.993947i	$-0.464958\pi$
$569$	1.07502e11	−	6.20665e10i	1.02558	−	0.592118i	0.915715	+	0.401828i	$0.131625\pi$
$570$		0			0
$571$	6.47455e10	−	1.12142e11i	0.609067	−	1.05493i	−0.382328	−	0.924027i	$-0.624877\pi$
$571$	6.47455e10	−	1.12142e11i	0.609067	−	1.05493i	0.991395	−	0.130908i	$-0.0417892\pi$
$572$	3.40642e9	+	1.96669e9i	0.0318210	+	0.0183718i
$573$		0			0
$574$	−2.50882e9	−	4.34541e9i	−0.0231112	−	0.0400298i
$575$			7.30064e10i			0.667866i
$576$		0			0
$577$	−1.61077e11			−1.45321			−0.726607	−	0.687053i	$-0.758904\pi$
$577$	−1.61077e11			−1.45321			−0.726607	+	0.687053i	$0.758904\pi$
$578$	−2.76656e9	+	1.59728e9i	−0.0247873	+	0.0143110i
$579$		0			0
$580$	−2.64225e8	+	4.57652e8i	−0.00233487	+	0.00404411i
$581$	−4.36204e10	−	2.51843e10i	−0.382812	−	0.221017i
$582$		0			0
$583$	3.73445e10	+	6.46825e10i	0.323260	+	0.559903i
$584$		−	2.41520e10i		−	0.207636i
$585$		0			0
$586$	−1.41389e11			−1.19902
$587$	−1.16832e11	+	6.74532e10i	−0.984036	+	0.568133i	−0.903486	−	0.428617i	$-0.859001\pi$
$587$	−1.16832e11	+	6.74532e10i	−0.984036	+	0.568133i	−0.0805496	+	0.996751i	$0.525668\pi$
$588$		0			0
$589$	−3.40155e10	+	5.89166e10i	−0.282628	+	0.489527i
$590$	5.45363e8	+	3.14865e8i	0.00450067	+	0.00259847i
$591$		0			0
$592$	−2.89343e10	−	5.01156e10i	−0.235573	−	0.408024i
$593$			3.92983e10i			0.317800i	0.987295	+	0.158900i	$0.0507948\pi$
$593$			3.92983e10i			0.317800i	−0.987295	+	0.158900i	$0.949205\pi$
$594$		0			0
$595$	−6.89985e8			−0.00550518
$596$	3.42308e10	−	1.97632e10i	0.271289	−	0.156629i
$597$		0			0
$598$	−5.30826e9	+	9.19418e9i	−0.0415095	+	0.0718966i
$599$	1.07635e11	+	6.21432e10i	0.836078	+	0.482710i	0.855929	−	0.517093i	$-0.172986\pi$
$599$	1.07635e11	+	6.21432e10i	0.836078	+	0.482710i	−0.0198514	+	0.999803i	$0.506319\pi$
$600$		0			0
$601$	7.24590e10	+	1.25503e11i	0.555385	+	0.961956i	0.997873	+	0.0651814i	$0.0207626\pi$
$601$	7.24590e10	+	1.25503e11i	0.555385	+	0.961956i	−0.442488	+	0.896774i	$0.645904\pi$
$602$			4.03377e10i			0.307132i
$603$		0			0
$604$	6.72543e10			0.505327
$605$	−1.30216e9	+	7.51802e8i	−0.00971948	+	0.00561154i
$606$		0			0
$607$	−1.07280e10	+	1.85814e10i	−0.0790246	+	0.136875i	−0.902829	−	0.429999i	$-0.858514\pi$
$607$	−1.07280e10	+	1.85814e10i	−0.0790246	+	0.136875i	0.823805	+	0.566874i	$0.191847\pi$
$608$	2.09266e10	+	1.20820e10i	0.153138	+	0.0884144i
$609$		0			0
$610$	−2.56345e8	−	4.44003e8i	−0.00185142	−	0.00320676i
$611$			1.80471e10i			0.129492i
$612$		0			0
$613$	4.66213e10			0.330174			0.165087	−	0.986279i	$-0.447210\pi$
$613$	4.66213e10			0.330174			0.165087	+	0.986279i	$0.447210\pi$
$614$	−9.54085e10	+	5.50841e10i	−0.671295	+	0.387572i
$615$		0			0
$616$	−4.39765e9	+	7.61696e9i	−0.0305420	+	0.0529004i
$617$	6.84308e10	+	3.95086e10i	0.472184	+	0.272616i	0.717154	−	0.696915i	$-0.245445\pi$
$617$	6.84308e10	+	3.95086e10i	0.472184	+	0.272616i	−0.244970	+	0.969531i	$0.578778\pi$
$618$		0			0
$619$	5.55033e10	+	9.61346e10i	0.378056	+	0.654812i	0.990779	−	0.135485i	$-0.0432593\pi$
$619$	5.55033e10	+	9.61346e10i	0.378056	+	0.654812i	−0.612723	+	0.790298i	$0.709926\pi$
$620$			5.67826e8i			0.00384281i
$621$		0			0
$622$	−5.14053e10			−0.343437
$623$	−8.93088e10	+	5.15625e10i	−0.592846	+	0.342280i
$624$		0			0
$625$	−7.62516e10	+	1.32072e11i	−0.499723	+	0.865545i
$626$	1.16497e11	+	6.72596e10i	0.758608	+	0.437983i
$627$		0			0
$628$	−3.91806e10	−	6.78628e10i	−0.251902	−	0.436308i
$629$			2.88965e11i			1.84605i
$630$		0			0
$631$	5.01528e10			0.316357			0.158179	−	0.987410i	$-0.449438\pi$
$631$	5.01528e10			0.316357			0.158179	+	0.987410i	$0.449438\pi$
$632$	−4.26166e10	+	2.46047e10i	−0.267122	+	0.154223i
$633$		0			0
$634$	5.85339e10	−	1.01384e11i	0.362285	−	0.627496i
$635$	−1.40784e8	−	8.12817e7i	−0.000865881	−	0.000499917i
$636$		0			0
$637$	1.19987e10	+	2.07824e10i	0.0728748	+	0.126223i
$638$		−	3.36371e10i		−	0.203019i
$639$		0			0
$640$	2.01686e8			0.00120214
$641$	−1.73422e11	+	1.00125e11i	−1.02724	+	0.593076i	−0.916192	−	0.400739i	$-0.868753\pi$
$641$	−1.73422e11	+	1.00125e11i	−1.02724	+	0.593076i	−0.111046	+	0.993815i	$0.535420\pi$
$642$		0			0
$643$	−1.44951e11	+	2.51062e11i	−0.847962	+	1.46871i	0.0350628	+	0.999385i	$0.488837\pi$
$643$	−1.44951e11	+	2.51062e11i	−0.847962	+	1.46871i	−0.883024	+	0.469327i	$0.844496\pi$
$644$	−2.05588e10	−	1.18696e10i	−0.119524	−	0.0690070i
$645$		0			0
$646$	−6.03310e10	−	1.04496e11i	−0.346426	−	0.600027i
$647$			9.25383e10i			0.528085i	0.964511	+	0.264043i	$0.0850560\pi$
$647$			9.25383e10i			0.528085i	−0.964511	+	0.264043i	$0.914944\pi$
$648$		0			0
$649$	−4.00839e10			−0.225939
$650$	−1.92093e10	+	1.10905e10i	−0.107611	+	0.0621295i
$651$		0			0
$652$	−4.38733e9	+	7.59907e9i	−0.0242778	+	0.0420504i
$653$	−2.00779e11	−	1.15920e11i	−1.10424	−	0.637536i	−0.166912	−	0.985972i	$-0.553379\pi$
$653$	−2.00779e11	−	1.15920e11i	−1.10424	−	0.637536i	−0.937332	+	0.348436i	$0.886713\pi$
$654$		0			0
$655$	1.24291e9	+	2.15278e9i	0.00675264	+	0.0116959i
$656$		−	7.32385e9i		−	0.0395480i
$657$		0			0
$658$	−4.03545e10			−0.215273
$659$	−3.28529e10	+	1.89677e10i	−0.174194	+	0.100571i	−0.584562	−	0.811349i	$-0.698734\pi$
$659$	−3.28529e10	+	1.89677e10i	−0.174194	+	0.100571i	0.410368	+	0.911920i	$0.365400\pi$
$660$		0			0
$661$	−2.62985e10	+	4.55503e10i	−0.137761	+	0.238608i	−0.926649	−	0.375929i	$-0.877324\pi$
$661$	−2.62985e10	+	4.55503e10i	−0.137761	+	0.238608i	0.788888	+	0.614537i	$0.210657\pi$
$662$	1.92873e10	+	1.11355e10i	0.100424	+	0.0579800i
$663$		0			0
$664$	−3.67595e10	−	6.36692e10i	−0.189102	−	0.327535i
$665$			1.09941e9i			0.00562176i
$666$		0			0
$667$	9.07893e10			0.458703
$668$	−8.97410e10	+	5.18120e10i	−0.450697	+	0.260210i
$669$		0			0
$670$	−1.07695e9	+	1.86533e9i	−0.00534435	+	0.00925669i
$671$	2.82618e10	+	1.63170e10i	0.139415	+	0.0804914i
$672$		0			0
$673$	−1.19388e11	−	2.06786e11i	−0.581968	−	1.00800i	−0.995246	−	0.0973938i	$-0.968949\pi$
$673$	−1.19388e11	−	2.06786e11i	−0.581968	−	1.00800i	0.413277	−	0.910605i	$-0.364384\pi$
$674$			5.52531e10i			0.267742i
$675$		0			0
$676$	1.01188e11			0.484554
$677$	1.45655e11	−	8.40942e10i	0.693381	−	0.400324i	−0.111496	−	0.993765i	$-0.535564\pi$
$677$	1.45655e11	−	8.40942e10i	0.693381	−	0.400324i	0.804877	+	0.593441i	$0.202231\pi$
$678$		0			0
$679$	3.51938e9	−	6.09575e9i	0.0165572	−	0.0286779i
$680$	−8.72188e8	−	5.03558e8i	−0.00407919	−	0.00235512i
$681$		0			0
$682$	−1.80717e10	−	3.13012e10i	−0.0835338	−	0.144685i
$683$			2.35683e11i			1.08304i	0.840688	+	0.541520i	$0.182151\pi$
$683$			2.35683e11i			1.08304i	−0.840688	+	0.541520i	$0.817849\pi$
$684$		0			0
$685$	−4.79762e9			−0.0217903
$686$	−1.02510e11	+	5.91844e10i	−0.462883	+	0.267246i
$687$		0			0
$688$	−2.94389e10	+	5.09896e10i	−0.131392	+	0.227577i
$689$	−5.30424e10	−	3.06240e10i	−0.235367	−	0.135889i
$690$		0			0
$691$	1.15486e11	+	2.00028e11i	0.506545	+	0.877361i	0.999971	+	0.00757388i	$0.00241087\pi$
$691$	1.15486e11	+	2.00028e11i	0.506545	+	0.877361i	−0.493426	+	0.869788i	$0.664256\pi$
$692$		−	2.81849e10i		−	0.122911i
$693$		0			0
$694$	3.25872e10			0.140478
$695$	−6.75452e7	+	3.89972e7i	−0.000289504	+	0.000167145i
$696$		0			0
$697$	−1.82858e10	+	3.16719e10i	−0.0774786	+	0.134197i
$698$	1.51919e11	+	8.77106e10i	0.640016	+	0.369513i
$699$		0			0
$700$	−2.47990e10	−	4.29532e10i	−0.103286	−	0.178897i
$701$			4.43245e11i			1.83557i	0.397076	+	0.917786i	$0.370025\pi$
$701$			4.43245e11i			1.83557i	−0.397076	+	0.917786i	$0.629975\pi$
$702$		0			0
$703$	4.60431e11			1.88514
$704$	−1.11179e10	+	6.41890e9i	−0.0452616	+	0.0261318i
$705$		0			0
$706$	1.11362e11	−	1.92884e11i	0.448247	−	0.776386i
$707$	−5.53504e10	−	3.19565e10i	−0.221535	−	0.127903i
$708$		0			0
$709$	−7.25421e10	−	1.25647e11i	−0.287081	−	0.497240i	0.686030	−	0.727573i	$-0.259352\pi$
$709$	−7.25421e10	−	1.25647e11i	−0.287081	−	0.497240i	−0.973112	+	0.230333i	$0.926018\pi$
$710$		−	3.40369e9i		−	0.0133942i
$711$		0			0
$712$	−1.50523e11			−0.585711
$713$	8.44843e10	−	4.87771e10i	0.326902	−	0.188737i
$714$		0			0
$715$	−1.30607e8	+	2.26219e8i	−0.000499740	+	0.000865575i
$716$	−4.15825e10	−	2.40077e10i	−0.158219	−	0.0913477i
$717$		0			0
$718$	−1.52910e11	−	2.64847e11i	−0.575357	−	0.996547i
$719$		−	4.24829e11i		−	1.58964i	−0.606845	−	0.794820i	$-0.707565\pi$
$719$		−	4.24829e11i		−	1.58964i	0.606845	−	0.794820i	$-0.292435\pi$
$720$		0			0
$721$	−2.08916e11			−0.773091
$722$	−9.80054e7	+	5.65834e7i	−0.000360663	+	0.000208229i
$723$		0			0
$724$	4.60337e10	−	7.97328e10i	0.167541	−	0.290190i
$725$	1.64272e11	+	9.48425e10i	0.594582	+	0.343282i
$726$		0			0
$727$	1.80472e11	+	3.12586e11i	0.646058	+	1.11901i	0.984056	+	0.177859i	$0.0569170\pi$
$727$	1.80472e11	+	3.12586e11i	0.646058	+	1.11901i	−0.337998	+	0.941147i	$0.609750\pi$
$728$		−	7.21252e9i		−	0.0256780i
$729$		0			0
$730$	1.60393e9			0.00564798
$731$	2.54616e11	−	1.47002e11i	0.891693	−	0.514819i
$732$		0			0
$733$	−8.94545e10	+	1.54940e11i	−0.309875	+	0.536719i	−0.978335	−	0.207029i	$-0.933621\pi$
$733$	−8.94545e10	+	1.54940e11i	−0.309875	+	0.536719i	0.668460	+	0.743748i	$0.266954\pi$
$734$	5.83646e10	+	3.36968e10i	0.201078	+	0.116093i
$735$		0			0
$736$	−1.73251e10	−	3.00080e10i	−0.0590425	−	0.102265i
$737$		−	1.37100e11i		−	0.464696i
$738$		0			0
$739$	3.36137e11			1.12704			0.563518	−	0.826103i	$-0.309447\pi$
$739$	3.36137e11			1.12704			0.563518	+	0.826103i	$0.309447\pi$
$740$	3.32816e9	−	1.92151e9i	0.0110988	−	0.00640791i
$741$		0			0
$742$	6.84772e10	−	1.18606e11i	0.225907	−	0.391283i
$743$	−2.28344e11	−	1.31835e11i	−0.749265	−	0.432588i	0.0761636	−	0.997095i	$-0.475733\pi$
$743$	−2.28344e11	−	1.31835e11i	−0.749265	−	0.432588i	−0.825428	+	0.564507i	$0.809066\pi$
$744$		0			0
$745$	1.31247e9	+	2.27326e9i	0.00426052	+	0.00737944i
$746$		−	1.74294e11i		−	0.562767i
$747$		0			0
$748$	6.41053e10			0.204780
$749$	−1.95193e11	+	1.12695e11i	−0.620206	+	0.358076i
$750$		0			0
$751$	2.60589e11	−	4.51353e11i	0.819211	−	1.41891i	−0.0870541	−	0.996204i	$-0.527745\pi$
$751$	2.60589e11	−	4.51353e11i	0.819211	−	1.41891i	0.906265	−	0.422711i	$-0.138921\pi$
$752$	−5.10108e10	−	2.94511e10i	−0.159511	−	0.0920938i
$753$		0			0
$754$	1.37919e10	+	2.38883e10i	0.0426717	+	0.0739095i
$755$			4.46633e9i			0.0137456i
$756$		0			0
$757$	−5.02584e11			−1.53047			−0.765235	−	0.643751i	$-0.777377\pi$
$757$	−5.02584e11			−1.53047			−0.765235	+	0.643751i	$0.777377\pi$
$758$	2.24766e11	−	1.29769e11i	0.680853	−	0.393091i
$759$		0			0
$760$	−8.02358e8	+	1.38973e9i	−0.00240499	+	0.00416557i
$761$	3.61474e11	+	2.08697e11i	1.07780	+	0.622269i	0.930303	−	0.366793i	$-0.119544\pi$
$761$	3.61474e11	+	2.08697e11i	1.07780	+	0.622269i	0.147499	+	0.989062i	$0.452878\pi$
$762$		0			0
$763$	4.32558e10	+	7.49212e10i	0.127628	+	0.221058i
$764$		−	1.98341e11i		−	0.582155i
$765$		0			0
$766$	−1.46541e11			−0.425642
$767$	2.84666e10	−	1.64352e10i	0.0822536	−	0.0474891i
$768$		0			0
$769$	1.41880e11	−	2.45743e11i	0.405709	−	0.702708i	−0.588695	−	0.808355i	$-0.700358\pi$
$769$	1.41880e11	−	2.45743e11i	0.405709	−	0.702708i	0.994404	+	0.105647i	$0.0336914\pi$
$770$	−5.05839e8	−	2.92046e8i	−0.00143896	−	0.000830786i
$771$		0			0
$772$	8.46599e10	+	1.46635e11i	0.238346	+	0.412828i
$773$		−	6.99303e11i		−	1.95861i	−0.202398	−	0.979303i	$-0.564873\pi$
$773$		−	6.99303e11i		−	1.95861i	0.202398	−	0.979303i	$-0.435127\pi$
$774$		0			0
$775$	2.03819e11			0.564986
$776$	8.89748e9	−	5.13696e9i	0.0245369	−	0.0141664i
$777$		0			0
$778$	−1.76986e10	+	3.06549e10i	−0.0483082	+	0.0836722i
$779$	5.04653e10	+	2.91361e10i	0.137039	+	0.0791192i
$780$		0			0
$781$	1.08326e11	+	1.87627e11i	0.291159	+	0.504302i
$782$			1.73025e11i			0.462682i
$783$		0			0
$784$	−7.83229e10			−0.207312
$785$	4.50674e9	−	2.60197e9i	0.0118682	−	0.00685209i
$786$		0			0
$787$	1.59246e11	−	2.75822e11i	0.415117	−	0.719003i	−0.580324	−	0.814386i	$-0.697074\pi$
$787$	1.59246e11	−	2.75822e11i	0.415117	−	0.719003i	0.995441	+	0.0953827i	$0.0304075\pi$
$788$	−2.52432e11	−	1.45742e11i	−0.654697	−	0.377989i
$789$		0			0
$790$	−1.63399e9	−	2.83015e9i	−0.00419508	−	0.00726610i
$791$			1.12680e11i			0.287834i
$792$		0			0
$793$	−2.67612e10			−0.0676726
$794$	2.51140e11	−	1.44995e11i	0.631878	−	0.364815i
$795$		0			0
$796$	1.16115e11	−	2.01117e11i	0.289224	−	0.500951i
$797$	−4.78481e11	−	2.76251e11i	−1.18585	−	0.684654i	−0.228493	−	0.973546i	$-0.573380\pi$
$797$	−4.78481e11	−	2.76251e11i	−1.18585	−	0.684654i	−0.957362	+	0.288892i	$0.906713\pi$
$798$		0			0
$799$	1.47064e11	+	2.54722e11i	0.360843	+	0.624998i
$800$		−	7.23943e10i		−	0.176744i
$801$		0			0
$802$	−9.64314e10			−0.233089
$803$	−8.84157e10	+	5.10468e10i	−0.212651	+	0.122774i
$804$		0			0
$805$	7.88256e8	−	1.36530e9i	0.00187709	−	0.00325121i
$806$	2.56683e10	+	1.48196e10i	0.0608214	+	0.0351153i
$807$		0			0
$808$	−4.66444e10	−	8.07905e10i	−0.109434	−	0.189546i
$809$		−	7.14496e11i		−	1.66804i	−0.551736	−	0.834019i	$-0.686034\pi$
$809$		−	7.14496e11i		−	1.66804i	0.551736	−	0.834019i	$-0.313966\pi$
$810$		0			0
$811$	−1.57160e11			−0.363294			−0.181647	−	0.983364i	$-0.558143\pi$
$811$	−1.57160e11			−0.363294			−0.181647	+	0.983364i	$0.558143\pi$
$812$	−5.34158e10	+	3.08396e10i	−0.122870	+	0.0709389i
$813$		0			0
$814$	−1.22309e11	+	2.11845e11i	−0.278587	+	0.482526i
$815$	−5.04652e8	−	2.91361e8i	−0.00114383	−	0.000660390i
$816$		0			0
$817$	−2.34230e11	−	4.05699e11i	−0.525721	−	0.910575i
$818$			3.64894e11i			0.814992i
$819$		0			0
$820$	4.86374e8			0.00107576
$821$	5.64977e11	−	3.26189e11i	1.24353	−	0.717955i	0.273723	−	0.961809i	$-0.411745\pi$
$821$	5.64977e11	−	3.26189e11i	1.24353	−	0.717955i	0.969812	+	0.243853i	$0.0784116\pi$
$822$		0			0
$823$	−3.42736e11	+	5.93636e11i	−0.747068	+	1.29396i	0.202155	+	0.979354i	$0.435206\pi$
$823$	−3.42736e11	+	5.93636e11i	−0.747068	+	1.29396i	−0.949223	+	0.314605i	$0.898128\pi$
$824$	−2.64084e11	−	1.52469e11i	−0.572840	−	0.330729i
$825$		0			0
$826$	3.67502e10	+	6.36532e10i	0.0789476	+	0.136741i
$827$		−	4.89963e11i		−	1.04747i	−0.851881	−	0.523735i	$-0.824538\pi$
$827$		−	4.89963e11i		−	1.04747i	0.851881	−	0.523735i	$-0.175462\pi$
$828$		0			0
$829$	−2.91865e11			−0.617964			−0.308982	−	0.951068i	$-0.599988\pi$
$829$	−2.91865e11			−0.617964			−0.308982	+	0.951068i	$0.599988\pi$
$830$	4.22825e9	−	2.44118e9i	0.00890940	−	0.00514384i
$831$		0			0
$832$	5.26376e9	−	9.11711e9i	0.0109851	−	0.0190267i
$833$	3.38706e11	+	1.95552e11i	0.703465	+	0.406146i
$834$		0			0
$835$	−3.44081e9	−	5.95967e9i	−0.00707808	−	0.0122596i
$836$		−	1.02144e11i		−	0.209116i
$837$		0			0
$838$	5.36287e11			1.08748
$839$	4.66181e11	−	2.69150e11i	0.940821	−	0.543183i	0.0506037	−	0.998719i	$-0.483885\pi$
$839$	4.66181e11	−	2.69150e11i	0.940821	−	0.543183i	0.890218	+	0.455535i	$0.150552\pi$
$840$		0			0
$841$	−1.32179e11	+	2.28941e11i	−0.264228	+	0.457656i
$842$	−1.38117e11	−	7.97418e10i	−0.274788	−	0.158649i
$843$		0			0
$844$	1.62434e11	+	2.81343e11i	0.320115	+	0.554456i
$845$			6.71986e9i			0.0131805i
$846$		0			0
$847$	−1.75496e11			−0.340984
$848$	1.73120e11	−	9.99507e10i	0.334783	−	0.193287i
$849$		0			0
$850$	−1.80750e11	+	3.13068e11i	−0.346260	+	0.599740i
$851$	−5.71787e11	−	3.30121e11i	−1.09022	−	0.629441i
$852$		0			0
$853$	−3.03741e11	−	5.26096e11i	−0.573731	−	0.993730i	−0.996178	−	0.0873432i	$-0.972162\pi$
$853$	−3.03741e11	−	5.26096e11i	−0.573731	−	0.993730i	0.422448	−	0.906387i	$-0.361171\pi$
$854$		−	5.98397e10i		−	0.112501i
$855$		0			0
$856$	−3.28982e11			−0.612741
$857$	−9.10093e11	+	5.25442e11i	−1.68718	+	0.974096i	−0.730525	+	0.682886i	$0.760725\pi$
$857$	−9.10093e11	+	5.25442e11i	−1.68718	+	0.974096i	−0.956659	+	0.291211i	$0.905942\pi$
$858$		0			0
$859$	5.84786e10	−	1.01288e11i	0.107405	−	0.186031i	−0.807313	−	0.590123i	$-0.799079\pi$
$859$	5.84786e10	−	1.01288e11i	0.107405	−	0.186031i	0.914718	+	0.404092i	$0.132413\pi$
$860$	−3.38620e9	−	1.95502e9i	−0.00619040	−	0.00357403i
$861$		0			0
$862$	5.98981e10	+	1.03746e11i	0.108489	+	0.187908i
$863$			5.62940e11i			1.01489i	0.861684	+	0.507445i	$0.169410\pi$
$863$			5.62940e11i			1.01489i	−0.861684	+	0.507445i	$0.830590\pi$
$864$		0			0
$865$	1.87175e9			0.00334336
$866$	3.10995e11	−	1.79553e11i	0.552945	−	0.319243i
$867$		0			0
$868$	−3.31375e10	+	5.73958e10i	−0.0583769	+	0.101112i
$869$	1.80146e11	+	1.04007e11i	0.315896	+	0.182383i
$870$		0			0
$871$	5.62140e10	+	9.73655e10i	0.0976724	+	0.169174i
$872$			1.26274e11i			0.218398i
$873$		0			0
$874$	2.75695e11			0.472479
$875$	5.70554e9	−	3.29410e9i	0.00973340	−	0.00561958i
$876$		0			0
$877$	−2.47007e11	+	4.27829e11i	−0.417553	+	0.723222i	−0.995693	−	0.0927152i	$-0.970445\pi$
$877$	−2.47007e11	+	4.27829e11i	−0.417553	+	0.723222i	0.578140	+	0.815938i	$0.303779\pi$
$878$	−1.74034e11	−	1.00478e11i	−0.292857	−	0.169081i
$879$		0			0
$880$	−4.26277e8	−	7.38333e8i	−0.000710822	−	0.00123118i
$881$		−	8.97673e11i		−	1.49010i	−0.667010	−	0.745049i	$-0.732426\pi$
$881$		−	8.97673e11i		−	1.49010i	0.667010	−	0.745049i	$-0.267574\pi$
$882$		0			0
$883$	1.00275e12			1.64949			0.824747	−	0.565502i	$-0.191318\pi$
$883$	1.00275e12			1.64949			0.824747	+	0.565502i	$0.191318\pi$
$884$	−4.55261e10	+	2.62845e10i	−0.0745506	+	0.0430418i
$885$		0			0
$886$	−1.14345e11	+	1.98051e11i	−0.185558	+	0.321396i
$887$	3.45989e11	+	1.99757e11i	0.558944	+	0.322706i	0.752721	−	0.658339i	$-0.228741\pi$
$887$	3.45989e11	+	1.99757e11i	0.558944	+	0.322706i	−0.193778	+	0.981045i	$0.562074\pi$
$888$		0			0
$889$	−9.48696e9	−	1.64319e10i	−0.0151887	−	0.0263076i
$890$		−	9.99619e9i		−	0.0159321i
$891$		0			0
$892$	1.45233e11			0.229406
$893$	4.05868e11	−	2.34328e11i	0.638233	−	0.368484i
$894$		0			0
$895$	1.59434e9	−	2.76148e9i	0.00248479	−	0.00430377i
$896$	2.03864e10	+	1.17701e10i	0.0316307	+	0.0182620i
$897$		0			0
$898$	2.68295e9	+	4.64700e9i	0.00412578	+	0.00714607i
$899$		−	2.53465e11i		−	0.388042i
$900$		0			0
$901$	−9.98204e11			−1.51468
$902$	−2.68112e10	+	1.54794e10i	−0.0405032	+	0.0233845i
$903$		0			0
$904$	−8.22350e10	+	1.42435e11i	−0.123135	+	0.213277i
$905$	5.29502e9	+	3.05708e9i	0.00789357	+	0.00455736i
$906$		0			0
$907$	−2.23457e11	−	3.87039e11i	−0.330191	−	0.571907i	0.652358	−	0.757911i	$-0.273780\pi$
$907$	−2.23457e11	−	3.87039e11i	−0.330191	−	0.571907i	−0.982549	+	0.186004i	$0.940446\pi$
$908$		−	5.96995e11i		−	0.878268i
$909$		0			0
$910$	4.78981e8			0.000698477
$911$	7.48252e10	−	4.32004e10i	0.108636	−	0.0627212i	−0.444697	−	0.895681i	$-0.646689\pi$
$911$	7.48252e10	−	4.32004e10i	0.108636	−	0.0627212i	0.553334	+	0.832960i	$0.313355\pi$
$912$		0			0
$913$	−1.55387e11	+	2.69138e11i	−0.223631	+	0.387340i
$914$	6.95691e11	+	4.01658e11i	0.996855	+	0.575534i
$915$		0			0
$916$	2.35399e11	+	4.07723e11i	0.334366	+	0.579140i
$917$			2.90137e11i			0.410323i
$918$		0			0
$919$	8.18319e11			1.14726			0.573628	−	0.819116i	$-0.305535\pi$
$919$	8.18319e11			1.14726			0.573628	+	0.819116i	$0.305535\pi$
$920$	1.99282e9	−	1.15055e9i	0.00278174	−	0.00160604i
$921$		0			0
$922$	−3.75215e11	+	6.49892e11i	−0.519227	+	0.899327i
$923$	−1.53862e11	−	8.88321e10i	−0.211994	−	0.122395i
$924$		0			0
$925$	−6.89719e11	−	1.19463e12i	−0.942118	−	1.63180i
$926$			1.79290e11i			0.243845i
$927$		0			0
$928$	−9.00282e10			−0.121391
$929$	−8.98785e11	+	5.18914e11i	−1.20668	+	0.696678i	−0.962033	−	0.272933i	$-0.912006\pi$
$929$	−8.98785e11	+	5.18914e11i	−1.20668	+	0.696678i	−0.244649	+	0.969612i	$0.578673\pi$
$930$		0			0
$931$	3.11588e11	−	5.39686e11i	0.414746	−	0.718361i
$932$	−4.04278e11	−	2.33410e11i	−0.535817	−	0.309354i
$933$		0			0
$934$	3.68822e11	+	6.38819e11i	0.484652	+	0.839442i
$935$			4.25721e9i			0.00557030i
$936$		0			0
$937$	−1.10403e12			−1.43226			−0.716130	−	0.697966i	$-0.754088\pi$
$937$	−1.10403e12			−1.43226			−0.716130	+	0.697966i	$0.754088\pi$
$938$	−2.17715e11	+	1.25698e11i	−0.281240	+	0.162374i
$939$		0			0
$940$	1.95584e9	−	3.38761e9i	0.00250508	−	0.00433892i
$941$	1.41006e11	+	8.14096e10i	0.179836	+	0.103829i	0.587216	−	0.809430i	$-0.300224\pi$
$941$	1.41006e11	+	8.14096e10i	0.179836	+	0.103829i	−0.407379	+	0.913259i	$0.633557\pi$
$942$		0			0
$943$	−4.17802e10	−	7.23655e10i	−0.0528353	−	0.0915134i
$944$			1.07283e11i			0.135096i
$945$		0			0
$946$	2.48884e11			0.310765
$947$	1.03032e12	−	5.94854e11i	1.28107	−	0.739624i	0.304023	−	0.952665i	$-0.401670\pi$
$947$	1.03032e12	−	5.94854e11i	1.28107	−	0.739624i	0.977043	+	0.213041i	$0.0683369\pi$
$948$		0			0
$949$	4.18605e10	−	7.25046e10i	0.0516107	−	0.0893924i
$950$	4.98836e11	+	2.88003e11i	0.612439	+	0.353592i
$951$		0			0
$952$	−5.87738e10	−	1.01799e11i	−0.0715543	−	0.123936i
$953$			8.10559e11i			0.982682i	0.870967	+	0.491341i	$0.163493\pi$
$953$			8.10559e11i			0.982682i	−0.870967	+	0.491341i	$0.836507\pi$
$954$		0			0
$955$	1.31717e10			0.0158354
$956$	−6.34385e11	+	3.66262e11i	−0.759489	+	0.438491i
$957$		0			0
$958$	5.42800e10	−	9.40157e10i	0.0644433	−	0.111619i
$959$	−4.84943e11	−	2.79982e11i	−0.573345	−	0.331021i
$960$		0			0
$961$	2.90270e11	+	5.02762e11i	0.340337	+	0.589480i
$962$		−	2.00597e11i		−	0.234220i
$963$		0			0
$964$	2.99891e11			0.347260
$965$	−9.73800e9	+	5.62223e9i	−0.0112295	+	0.00648335i
$966$		0			0
$967$	8.07043e11	−	1.39784e12i	0.922976	−	1.59864i	0.128192	−	0.991749i	$-0.459083\pi$
$967$	8.07043e11	−	1.39784e12i	0.922976	−	1.59864i	0.794784	−	0.606892i	$-0.207584\pi$
$968$	−2.21839e11	−	1.28079e11i	−0.252660	−	0.145873i
$969$		0			0
$970$	3.41144e8	+	5.90878e8i	0.000385346	+	0.000667438i
$971$			1.33036e12i			1.49656i	0.663385	+	0.748278i	$0.269119\pi$
$971$			1.33036e12i			1.49656i	−0.663385	+	0.748278i	$0.730881\pi$
$972$		0			0
$973$	−9.10328e9			−0.0101566
$974$	−2.34645e11	+	1.35472e11i	−0.260721	+	0.150527i
$975$		0			0
$976$	4.36716e10	−	7.56414e10i	0.0481282	−	0.0833605i
$977$	8.48522e11	+	4.89894e11i	0.931290	+	0.537680i	0.887219	−	0.461348i	$-0.152634\pi$
$977$	8.48522e11	+	4.89894e11i	0.931290	+	0.537680i	0.0440705	+	0.999028i	$0.485967\pi$
$978$		0			0
$979$	3.18140e11	+	5.51035e11i	0.346328	+	0.599858i
$980$		−	5.20139e9i		−	0.00563917i
$981$		0			0
$982$	−6.82874e11			−0.734336
$983$	−1.46873e10	+	8.47973e9i	−0.0157300	+	0.00908171i	−0.507844	−	0.861449i	$-0.669558\pi$
$983$	−1.46873e10	+	8.47973e9i	−0.0157300	+	0.00908171i	0.492114	+	0.870531i	$0.336224\pi$
$984$		0			0
$985$	9.67866e9	−	1.67639e10i	0.0102818	−	0.0178087i
$986$	3.89325e11	+	2.24777e11i	0.411912	+	0.237818i
$987$		0			0
$988$	4.18812e10	+	7.25403e10i	0.0439532	+	0.0761293i
$989$			6.71757e11i			0.702145i
$990$		0			0
$991$	4.31377e11			0.447262			0.223631	−	0.974674i	$-0.428209\pi$
$991$	4.31377e11			0.447262			0.223631	+	0.974674i	$0.428209\pi$
$992$	−8.37761e10	+	4.83681e10i	−0.0865114	+	0.0499474i
$993$		0			0
$994$	1.98634e11	−	3.44044e11i	0.203474	−	0.352427i
$995$	1.33561e10	+	7.71113e9i	0.0136266	+	0.00786730i
$996$		0			0
$997$	−2.89657e10	−	5.01701e10i	−0.0293159	−	0.0507767i	0.850995	−	0.525173i	$-0.176000\pi$
$997$	−2.89657e10	−	5.01701e10i	−0.0293159	−	0.0507767i	−0.880311	+	0.474397i	$0.842666\pi$
$998$			4.75826e11i			0.479652i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.9.d.e.53.2		8
3.2	odd	2	inner	162.9.d.e.53.3		8
9.2	odd	6	inner	162.9.d.e.107.2		8
9.4	even	3		54.9.b.c.53.1	✓	4
9.5	odd	6		54.9.b.c.53.4	yes	4
9.7	even	3	inner	162.9.d.e.107.3		8
36.23	even	6		432.9.e.h.161.3		4
36.31	odd	6		432.9.e.h.161.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.9.b.c.53.1	✓	4	9.4	even	3
54.9.b.c.53.4	yes	4	9.5	odd	6
162.9.d.e.53.2		8	1.1	even	1	trivial
162.9.d.e.53.3		8	3.2	odd	2	inner
162.9.d.e.107.2		8	9.2	odd	6	inner
162.9.d.e.107.3		8	9.7	even	3	inner
432.9.e.h.161.2		4	36.31	odd	6
432.9.e.h.161.3		4	36.23	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-9.79796 + 5.65685i) q^{2} +(64.0000 - 110.851i) q^{4} +(7.36159 + 4.25022i) q^{5} +(496.073 + 859.223i) q^{7} +1448.15i q^{8} -96.1714 q^{10} +(5301.41 - 3060.77i) q^{11} +(-2509.96 + 4347.38i) q^{13} +(-9721.00 - 5612.42i) q^{14} +(-8192.00 - 14189.0i) q^{16} +81813.2i q^{17} +130359. q^{19} +(942.284 - 544.028i) q^{20} +(-34628.6 + 59978.6i) q^{22} +(-161887. - 93465.4i) q^{23} +(-195276. - 338229. i) q^{25} -56793.9i q^{26} +126995. q^{28} +(-420614. + 242842. i) q^{29} +(-260936. + 451955. i) q^{31} +(160530. + 92681.9i) q^{32} +(-462805. - 801602. i) q^{34} +8433.67i q^{35} +3.53201e6 q^{37} +(-1.27726e6 + 737424. i) q^{38} +(-6154.97 + 10660.7i) q^{40} +(387124. + 223506. i) q^{41} +(-1.79681e6 - 3.11216e6i) q^{43} -783557. i q^{44} +2.11488e6 q^{46} +(3.11345e6 - 1.79755e6i) q^{47} +(2.39022e6 - 4.13999e6i) q^{49} +(3.82662e6 + 2.20930e6i) q^{50} +(321275. + 556464. i) q^{52} +1.22010e7i q^{53} +52035.7 q^{55} +(-1.24429e6 + 718390. i) q^{56} +(2.74744e6 - 4.75871e6i) q^{58} +(-5.67074e6 - 3.27400e6i) q^{59} +(2.66550e6 + 4.61678e6i) q^{61} -5.90432e6i q^{62} -2.09715e6 q^{64} +(-36954.6 + 21335.7i) q^{65} +(1.11982e7 - 1.93958e7i) q^{67} +(9.06909e6 + 5.23604e6i) q^{68} +(-47708.0 - 82632.7i) q^{70} +3.53919e7i q^{71} -1.66778e7 q^{73} +(-3.46065e7 + 1.99801e7i) q^{74} +(8.34300e6 - 1.44505e7i) q^{76} +(5.25977e6 + 3.03673e6i) q^{77} +(1.69904e7 + 2.94282e7i) q^{79} -139271. i q^{80} -5.05737e6 q^{82} +(-4.39658e7 + 2.53837e7i) q^{83} +(-347724. + 602275. i) q^{85} +(3.52101e7 + 2.03285e7i) q^{86} +(4.43247e6 + 7.67726e6i) q^{88} +1.03941e8i q^{89} -4.98049e6 q^{91} +(-2.07215e7 + 1.19636e7i) q^{92} +(-2.03370e7 + 3.52247e7i) q^{94} +(959652. + 554056. i) q^{95} +(-3.54725e6 - 6.14401e6i) q^{97} +5.40846e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 512 q^{4} - 308 q^{7} + 26112 q^{10} + 44680 q^{13} - 65536 q^{16} - 113024 q^{19} + 222720 q^{22} - 876736 q^{25} - 78848 q^{28} + 2375428 q^{31} + 82944 q^{34} + 10394656 q^{37} + 1671168 q^{40}+ \cdots + 269355220 q^{97}+O(q^{100}) \) 8 * q + 512 * q^4 - 308 * q^7 + 26112 * q^10 + 44680 * q^13 - 65536 * q^16 - 113024 * q^19 + 222720 * q^22 - 876736 * q^25 - 78848 * q^28 + 2375428 * q^31 + 82944 * q^34 + 10394656 * q^37 + 1671168 * q^40 + 2755096 * q^43 + 13937664 * q^46 + 18463200 * q^49 - 5719040 * q^52 + 37596888 * q^55 + 21977088 * q^58 - 36842384 * q^61 - 16777216 * q^64 + 52170376 * q^67 - 15375360 * q^70 + 44881864 * q^73 - 7233536 * q^76 + 51146440 * q^79 + 28093440 * q^82 - 101486520 * q^85 - 28508160 * q^88 - 145355728 * q^91 + 4420608 * q^94 + 269355220 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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