LMFDB - Embedded newform 378.3.p.a.145.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,3,Mod(73,378)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(378, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("378.73");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.p (of order $6$, degree $2$, not 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:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 126)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.7
Character	$\chi$	$=$	378.145
Dual form			378.3.p.a.73.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +(6.01208 - 3.47108i) q^{5} +(-6.97949 + 0.535409i) q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +(6.01208 - 3.47108i) q^{5} +(-6.97949 + 0.535409i) q^{7} -2.82843 q^{8} +(-8.50237 + 4.90885i) q^{10} +(8.72855 - 15.1183i) q^{11} +(-13.3362 - 7.69965i) q^{13} +(9.87050 - 0.757183i) q^{14} +4.00000 q^{16} +(-12.3517 + 7.13128i) q^{17} +(1.25450 + 0.724286i) q^{19} +(12.0242 - 6.94216i) q^{20} +(-12.3440 + 21.3805i) q^{22} +(-6.43077 - 11.1384i) q^{23} +(11.5968 - 20.0862i) q^{25} +(18.8602 + 10.8890i) q^{26} +(-13.9590 + 1.07082i) q^{28} +(2.98967 + 5.17826i) q^{29} +16.4679i q^{31} -5.65685 q^{32} +(17.4680 - 10.0852i) q^{34} +(-40.1029 + 27.4453i) q^{35} +(-0.732115 + 1.26806i) q^{37} +(-1.77413 - 1.02430i) q^{38} +(-17.0047 + 9.81769i) q^{40} +(-10.5856 - 6.11162i) q^{41} +(-40.1499 - 69.5416i) q^{43} +(17.4571 - 30.2366i) q^{44} +(9.09448 + 15.7521i) q^{46} -83.6474i q^{47} +(48.4267 - 7.47377i) q^{49} +(-16.4003 + 28.4062i) q^{50} +(-26.6724 - 15.3993i) q^{52} +(-6.51752 - 11.2887i) q^{53} -121.190i q^{55} +(19.7410 - 1.51437i) q^{56} +(-4.22803 - 7.32317i) q^{58} -101.747i q^{59} +67.1191i q^{61} -23.2891i q^{62} +8.00000 q^{64} -106.904 q^{65} -39.4835 q^{67} +(-24.7035 + 14.2626i) q^{68} +(56.7140 - 38.8135i) q^{70} +93.8955 q^{71} +(17.2189 - 9.94134i) q^{73} +(1.03537 - 1.79331i) q^{74} +(2.50900 + 1.44857i) q^{76} +(-52.8264 + 110.191i) q^{77} -122.786 q^{79} +(24.0483 - 13.8843i) q^{80} +(14.9703 + 8.64313i) q^{82} +(-7.01741 + 4.05150i) q^{83} +(-49.5065 + 85.7477i) q^{85} +(56.7805 + 98.3467i) q^{86} +(-24.6881 + 42.7610i) q^{88} +(43.7047 + 25.2329i) q^{89} +(97.2023 + 46.5994i) q^{91} +(-12.8615 - 22.2768i) q^{92} +118.295i q^{94} +10.0562 q^{95} +(80.8837 - 46.6982i) q^{97} +(-68.4857 + 10.5695i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q + 64 q^{4} - 2 q^{7}+O(q^{10}) $ 32 * q + 64 * q^4 - 2 * q^7	$ 32 q + 64 q^{4} - 2 q^{7} + 12 q^{11} + 30 q^{13} + 12 q^{14} + 128 q^{16} - 54 q^{17} + 42 q^{23} + 80 q^{25} + 72 q^{26} - 4 q^{28} + 84 q^{29} + 66 q^{35} - 22 q^{37} + 396 q^{41} - 16 q^{43} + 24 q^{44} + 12 q^{46} + 50 q^{49} + 96 q^{50} + 60 q^{52} + 252 q^{53} + 24 q^{56} - 24 q^{58} + 256 q^{64} - 12 q^{65} - 140 q^{67} - 108 q^{68} + 72 q^{70} - 300 q^{71} + 72 q^{74} - 570 q^{77} - 212 q^{79} - 756 q^{83} - 60 q^{85} + 120 q^{86} - 414 q^{89} - 186 q^{91} + 84 q^{92} - 1104 q^{95} - 114 q^{97} + 96 q^{98}+O(q^{100}) $ 32 * q + 64 * q^4 - 2 * q^7 + 12 * q^11 + 30 * q^13 + 12 * q^14 + 128 * q^16 - 54 * q^17 + 42 * q^23 + 80 * q^25 + 72 * q^26 - 4 * q^28 + 84 * q^29 + 66 * q^35 - 22 * q^37 + 396 * q^41 - 16 * q^43 + 24 * q^44 + 12 * q^46 + 50 * q^49 + 96 * q^50 + 60 * q^52 + 252 * q^53 + 24 * q^56 - 24 * q^58 + 256 * q^64 - 12 * q^65 - 140 * q^67 - 108 * q^68 + 72 * q^70 - 300 * q^71 + 72 * q^74 - 570 * q^77 - 212 * q^79 - 756 * q^83 - 60 * q^85 + 120 * q^86 - 414 * q^89 - 186 * q^91 + 84 * q^92 - 1104 * q^95 - 114 * q^97 + 96 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/378\mathbb{Z}\right)^\times$.

$n$	$29$	$325$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$e\left(\frac{5}{6}\right)$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	6.01208	−	3.47108i	1.20242	−	0.694216i	0.241325	−	0.970444i	$-0.422418\pi$
$5$	6.01208	−	3.47108i	1.20242	−	0.694216i	0.961092	+	0.276229i	$0.0890847\pi$
$6$		0			0
$7$	−6.97949	+	0.535409i	−0.997071	+	0.0764870i
$7$	−6.97949	+	0.535409i	−0.997071	+	0.0764870i
$8$	−2.82843			−0.353553
$9$		0			0
$10$	−8.50237	+	4.90885i	−0.850237	+	0.490885i
$11$	8.72855	−	15.1183i	0.793504	−	1.37439i	−0.130280	−	0.991477i	$-0.541588\pi$
$11$	8.72855	−	15.1183i	0.793504	−	1.37439i	0.923785	−	0.382913i	$-0.125079\pi$
$12$		0			0
$13$	−13.3362	−	7.69965i	−1.02586	−	0.592281i	−0.110065	−	0.993924i	$-0.535106\pi$
$13$	−13.3362	−	7.69965i	−1.02586	−	0.592281i	−0.915796	+	0.401643i	$0.868439\pi$
$14$	9.87050	−	0.757183i	0.705035	−	0.0540845i
$15$		0			0
$16$	4.00000			0.250000
$17$	−12.3517	+	7.13128i	−0.726573	+	0.419487i	−0.817167	−	0.576401i	$-0.804457\pi$
$17$	−12.3517	+	7.13128i	−0.726573	+	0.419487i	0.0905943	+	0.995888i	$0.471123\pi$
$18$		0			0
$19$	1.25450	+	0.724286i	0.0660264	+	0.0381203i	0.532650	−	0.846336i	$-0.321196\pi$
$19$	1.25450	+	0.724286i	0.0660264	+	0.0381203i	−0.466623	+	0.884456i	$0.654530\pi$
$20$	12.0242	−	6.94216i	0.601208	−	0.347108i
$21$		0			0
$22$	−12.3440	+	21.3805i	−0.561092	+	0.971840i
$23$	−6.43077	−	11.1384i	−0.279599	−	0.484279i	0.691686	−	0.722198i	$-0.256868\pi$
$23$	−6.43077	−	11.1384i	−0.279599	−	0.484279i	−0.971285	+	0.237919i	$0.923535\pi$
$24$		0			0
$25$	11.5968	−	20.0862i	0.463871	−	0.803448i
$26$	18.8602	+	10.8890i	0.725393	+	0.418806i
$27$		0			0
$28$	−13.9590	+	1.07082i	−0.498535	+	0.0382435i
$29$	2.98967	+	5.17826i	0.103092	+	0.178561i	0.912957	−	0.408056i	$-0.133793\pi$
$29$	2.98967	+	5.17826i	0.103092	+	0.178561i	−0.809865	+	0.586616i	$0.800460\pi$
$30$		0			0
$31$			16.4679i			0.531223i	0.964080	+	0.265611i	$0.0855738\pi$
$31$			16.4679i			0.531223i	−0.964080	+	0.265611i	$0.914426\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$	17.4680	−	10.0852i	0.513765	−	0.296622i
$35$	−40.1029	+	27.4453i	−1.14580	+	0.784151i
$36$		0			0
$37$	−0.732115	+	1.26806i	−0.0197869	+	0.0342719i	−0.875749	−	0.482766i	$-0.839632\pi$
$37$	−0.732115	+	1.26806i	−0.0197869	+	0.0342719i	0.855962	+	0.517038i	$0.172965\pi$
$38$	−1.77413	−	1.02430i	−0.0466877	−	0.0269551i
$39$		0			0
$40$	−17.0047	+	9.81769i	−0.425119	+	0.245442i
$41$	−10.5856	−	6.11162i	−0.258186	−	0.149064i	0.365321	−	0.930882i	$-0.380959\pi$
$41$	−10.5856	−	6.11162i	−0.258186	−	0.149064i	−0.623507	+	0.781818i	$0.714293\pi$
$42$		0			0
$43$	−40.1499	−	69.5416i	−0.933718	−	1.61725i	−0.776904	−	0.629619i	$-0.783211\pi$
$43$	−40.1499	−	69.5416i	−0.933718	−	1.61725i	−0.156814	−	0.987628i	$-0.550122\pi$
$44$	17.4571	−	30.2366i	0.396752	−	0.687195i
$45$		0			0
$46$	9.09448	+	15.7521i	0.197706	+	0.342437i
$47$		−	83.6474i		−	1.77973i	−0.456223	−	0.889866i	$-0.650798\pi$
$47$		−	83.6474i		−	1.77973i	0.456223	−	0.889866i	$-0.349202\pi$
$48$		0			0
$49$	48.4267	−	7.47377i	0.988299	−	0.152526i
$50$	−16.4003	+	28.4062i	−0.328006	+	0.568123i
$51$		0			0
$52$	−26.6724	−	15.3993i	−0.512930	−	0.296140i
$53$	−6.51752	−	11.2887i	−0.122972	−	0.212994i	0.797966	−	0.602702i	$-0.205909\pi$
$53$	−6.51752	−	11.2887i	−0.122972	−	0.212994i	−0.920938	+	0.389708i	$0.872576\pi$
$54$		0			0
$55$		−	121.190i		−	2.20345i
$56$	19.7410	−	1.51437i	0.352518	−	0.0270423i
$57$		0			0
$58$	−4.22803	−	7.32317i	−0.0728971	−	0.126262i
$59$		−	101.747i		−	1.72452i	−0.506462	−	0.862262i	$-0.669047\pi$
$59$		−	101.747i		−	1.72452i	0.506462	−	0.862262i	$-0.330953\pi$
$60$		0			0
$61$			67.1191i			1.10031i	0.835062	+	0.550156i	$0.185432\pi$
$61$			67.1191i			1.10031i	−0.835062	+	0.550156i	$0.814568\pi$
$62$		−	23.2891i		−	0.375631i
$63$		0			0
$64$	8.00000			0.125000
$65$	−106.904			−1.64468
$66$		0			0
$67$	−39.4835			−0.589306			−0.294653	−	0.955604i	$-0.595204\pi$
$67$	−39.4835			−0.589306			−0.294653	+	0.955604i	$0.595204\pi$
$68$	−24.7035	+	14.2626i	−0.363286	+	0.209743i
$69$		0			0
$70$	56.7140	−	38.8135i	0.810200	−	0.554479i
$71$	93.8955			1.32247			0.661236	−	0.750178i	$-0.270032\pi$
$71$	93.8955			1.32247			0.661236	+	0.750178i	$0.270032\pi$
$72$		0			0
$73$	17.2189	−	9.94134i	0.235875	−	0.136183i	−0.377404	−	0.926049i	$-0.623183\pi$
$73$	17.2189	−	9.94134i	0.235875	−	0.136183i	0.613280	+	0.789866i	$0.289850\pi$
$74$	1.03537	−	1.79331i	0.0139914	−	0.0242339i
$75$		0			0
$76$	2.50900	+	1.44857i	0.0330132	+	0.0190602i
$77$	−52.8264	+	110.191i	−0.686057	+	1.43106i
$78$		0			0
$79$	−122.786			−1.55425			−0.777126	−	0.629345i	$-0.783323\pi$
$79$	−122.786			−1.55425			−0.777126	+	0.629345i	$0.783323\pi$
$80$	24.0483	−	13.8843i	0.300604	−	0.173554i
$81$		0			0
$82$	14.9703	+	8.64313i	0.182565	+	0.105404i
$83$	−7.01741	+	4.05150i	−0.0845471	+	0.0488133i	−0.541678	−	0.840586i	$-0.682211\pi$
$83$	−7.01741	+	4.05150i	−0.0845471	+	0.0488133i	0.457130	+	0.889400i	$0.348877\pi$
$84$		0			0
$85$	−49.5065	+	85.7477i	−0.582429	+	1.00880i
$86$	56.7805	+	98.3467i	0.660238	+	1.14357i
$87$		0			0
$88$	−24.6881	+	42.7610i	−0.280546	+	0.485920i
$89$	43.7047	+	25.2329i	0.491064	+	0.283516i	0.725016	−	0.688732i	$-0.241832\pi$
$89$	43.7047	+	25.2329i	0.491064	+	0.283516i	−0.233952	+	0.972248i	$0.575166\pi$
$90$		0			0
$91$	97.2023	+	46.5994i	1.06816	+	0.512081i
$92$	−12.8615	−	22.2768i	−0.139799	−	0.242139i
$93$		0			0
$94$			118.295i			1.25846i
$95$	10.0562			0.105855
$96$		0			0
$97$	80.8837	−	46.6982i	0.833853	−	0.481425i	−0.0213173	−	0.999773i	$-0.506786\pi$
$97$	80.8837	−	46.6982i	0.833853	−	0.481425i	0.855170	+	0.518348i	$0.173453\pi$
$98$	−68.4857	+	10.5695i	−0.698833	+	0.107852i
$99$		0			0
$100$	23.1935	−	40.1724i	0.231935	−	0.401724i
$101$	121.717	+	70.2731i	1.20511	+	0.695773i	0.961688	−	0.274146i	$-0.0883953\pi$
$101$	121.717	+	70.2731i	1.20511	+	0.695773i	0.243426	+	0.969919i	$0.421729\pi$
$102$		0			0
$103$	−31.2691	+	18.0532i	−0.303584	+	0.175274i	−0.644052	−	0.764982i	$-0.722748\pi$
$103$	−31.2691	+	18.0532i	−0.303584	+	0.175274i	0.340468	+	0.940256i	$0.389414\pi$
$104$	37.7204	+	21.7779i	0.362697	+	0.209403i
$105$		0			0
$106$	9.21716	+	15.9646i	0.0869544	+	0.150609i
$107$	−11.3624	+	19.6803i	−0.106191	+	0.183928i	−0.914224	−	0.405209i	$-0.867199\pi$
$107$	−11.3624	+	19.6803i	−0.106191	+	0.183928i	0.808033	+	0.589137i	$0.200532\pi$
$108$		0			0
$109$	78.3438	+	135.695i	0.718751	+	1.24491i	0.961495	+	0.274823i	$0.0886191\pi$
$109$	78.3438	+	135.695i	0.718751	+	1.24491i	−0.242744	+	0.970090i	$0.578048\pi$
$110$			171.388i			1.55808i
$111$		0			0
$112$	−27.9180	+	2.14164i	−0.249268	+	0.0191218i
$113$	66.4505	−	115.096i	0.588058	−	1.01855i	−0.406429	−	0.913682i	$-0.633226\pi$
$113$	66.4505	−	115.096i	0.588058	−	1.01855i	0.994487	−	0.104863i	$-0.0334405\pi$
$114$		0			0
$115$	−77.3246	−	44.6434i	−0.672388	−	0.388203i
$116$	5.97934	+	10.3565i	0.0515460	+	0.0892804i
$117$		0			0
$118$			143.892i			1.21942i
$119$	82.3907	−	56.3860i	0.692359	−	0.473832i
$120$		0			0
$121$	−91.8751	−	159.132i	−0.759298	−	1.31514i
$122$		−	94.9207i		−	0.778039i
$123$		0			0
$124$			32.9358i			0.265611i
$125$			12.5408i			0.100326i
$126$		0			0
$127$	245.690			1.93457			0.967284	−	0.253694i	$-0.0816458\pi$
$127$	245.690			1.93457			0.967284	+	0.253694i	$0.0816458\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	151.186			1.16297
$131$	−187.332	+	108.156i	−1.43002	+	0.825620i	−0.997121	−	0.0758237i	$-0.975841\pi$
$131$	−187.332	+	108.156i	−1.43002	+	0.825620i	−0.432895	+	0.901444i	$0.642508\pi$
$132$		0			0
$133$	−9.14357	−	4.38348i	−0.0687486	−	0.0329585i
$134$	55.8381			0.416702
$135$		0			0
$136$	34.9360	−	20.1703i	0.256882	−	0.148311i
$137$	−50.3719	+	87.2467i	−0.367678	+	0.636837i	−0.989202	−	0.146558i	$-0.953181\pi$
$137$	−50.3719	+	87.2467i	−0.367678	+	0.636837i	0.621524	+	0.783395i	$0.286514\pi$
$138$		0			0
$139$	152.411	+	87.9944i	1.09648	+	0.633054i	0.935294	−	0.353870i	$-0.115135\pi$
$139$	152.411	+	87.9944i	1.09648	+	0.633054i	0.161187	+	0.986924i	$0.448468\pi$
$140$	−80.2057	+	54.8906i	−0.572898	+	0.392076i
$141$		0			0
$142$	−132.788			−0.935128
$143$	−232.811	+	134.414i	−1.62805	+	0.939955i
$144$		0			0
$145$	35.9483	+	20.7548i	0.247919	+	0.143136i
$146$	−24.3512	+	14.0592i	−0.166789	+	0.0962957i
$147$		0			0
$148$	−1.46423	+	2.53612i	−0.00989344	+	0.0171359i
$149$	−100.465	−	174.010i	−0.674258	−	1.16785i	−0.976685	−	0.214677i	$-0.931130\pi$
$149$	−100.465	−	174.010i	−0.674258	−	1.16785i	0.302427	−	0.953173i	$-0.402203\pi$
$150$		0			0
$151$	118.477	−	205.209i	0.784619	−	1.35900i	−0.144608	−	0.989489i	$-0.546192\pi$
$151$	118.477	−	205.209i	0.784619	−	1.35900i	0.929227	−	0.369510i	$-0.120474\pi$
$152$	−3.54826	−	2.04859i	−0.0233438	−	0.0134776i
$153$		0			0
$154$	74.7078	−	155.834i	0.485115	−	1.01191i
$155$	57.1614	+	99.0064i	0.368783	+	0.638751i
$156$		0			0
$157$		−	13.1553i		−	0.0837919i	−0.999122	−	0.0418960i	$-0.986660\pi$
$157$		−	13.1553i		−	0.0837919i	0.999122	−	0.0418960i	$-0.0133398\pi$
$158$	173.646			1.09902
$159$		0			0
$160$	−34.0095	+	19.6354i	−0.212559	+	0.122721i
$161$	50.8471	+	74.2974i	0.315821	+	0.461475i
$162$		0			0
$163$	−91.5446	+	158.560i	−0.561623	+	0.972760i	0.435732	+	0.900077i	$0.356490\pi$
$163$	−91.5446	+	158.560i	−0.561623	+	0.972760i	−0.997355	+	0.0726836i	$0.976844\pi$
$164$	−21.1713	−	12.2232i	−0.129093	−	0.0745319i
$165$		0			0
$166$	9.92412	−	5.72969i	0.0597838	−	0.0345162i
$167$	178.551	+	103.087i	1.06917	+	0.617285i	0.927954	−	0.372694i	$-0.121566\pi$
$167$	178.551	+	103.087i	1.06917	+	0.617285i	0.141215	+	0.989979i	$0.454899\pi$
$168$		0			0
$169$	34.0693	+	59.0098i	0.201593	+	0.349170i
$170$	70.0127	−	121.266i	0.411839	−	0.713327i
$171$		0			0
$172$	−80.2998	−	139.083i	−0.466859	−	0.808624i
$173$			183.449i			1.06040i	0.847873	+	0.530199i	$0.177883\pi$
$173$			183.449i			1.06040i	−0.847873	+	0.530199i	$0.822117\pi$
$174$		0			0
$175$	−70.1852	+	146.400i	−0.401059	+	0.836574i
$176$	34.9142	−	60.4731i	0.198376	−	0.343597i
$177$		0			0
$178$	−61.8078	−	35.6847i	−0.347235	−	0.200476i
$179$	−19.7815	−	34.2626i	−0.110511	−	0.191411i	0.805465	−	0.592643i	$-0.201915\pi$
$179$	−19.7815	−	34.2626i	−0.110511	−	0.191411i	−0.915976	+	0.401232i	$0.868582\pi$
$180$		0			0
$181$		−	3.67614i		−	0.0203102i	−0.999948	−	0.0101551i	$-0.996767\pi$
$181$		−	3.67614i		−	0.0203102i	0.999948	−	0.0101551i	$-0.00323252\pi$
$182$	−137.465	−	65.9014i	−0.755301	−	0.362096i
$183$		0			0
$184$	18.1890	+	31.5042i	0.0988530	+	0.171218i
$185$			10.1649i			0.0549454i
$186$		0			0
$187$			248.983i			1.33146i
$188$		−	167.295i		−	0.889866i
$189$		0			0
$190$	−14.2216			−0.0748507
$191$	47.8780			0.250670			0.125335	−	0.992114i	$-0.459999\pi$
$191$	47.8780			0.250670			0.125335	+	0.992114i	$0.459999\pi$
$192$		0			0
$193$	58.4870			0.303041			0.151521	−	0.988454i	$-0.451583\pi$
$193$	58.4870			0.303041			0.151521	+	0.988454i	$0.451583\pi$
$194$	−114.387	+	66.0413i	−0.589623	+	0.340419i
$195$		0			0
$196$	96.8533	−	14.9475i	0.494150	−	0.0762630i
$197$	203.981			1.03544			0.517718	−	0.855552i	$-0.326782\pi$
$197$	203.981			1.03544			0.517718	+	0.855552i	$0.326782\pi$
$198$		0			0
$199$	−146.130	+	84.3682i	−0.734321	+	0.423961i	−0.820001	−	0.572362i	$-0.806027\pi$
$199$	−146.130	+	84.3682i	−0.734321	+	0.423961i	0.0856795	+	0.996323i	$0.472694\pi$
$200$	−32.8006	+	56.8123i	−0.164003	+	0.284062i
$201$		0			0
$202$	−172.133	−	99.3811i	−0.852145	−	0.491986i
$203$	−23.6389	−	34.5409i	−0.116448	−	0.170152i
$204$		0			0
$205$	−84.8556			−0.413930
$206$	44.2212	−	25.5311i	0.214666	−	0.123937i
$207$		0			0
$208$	−53.3448	−	30.7986i	−0.256465	−	0.148070i
$209$	21.8999	−	12.6439i	0.104784	−	0.0604973i
$210$		0			0
$211$	24.3937	−	42.2511i	0.115610	−	0.200242i	−0.802413	−	0.596768i	$-0.796451\pi$
$211$	24.3937	−	42.2511i	0.115610	−	0.200242i	0.918023	+	0.396526i	$0.129784\pi$
$212$	−13.0350	−	22.5773i	−0.0614860	−	0.106497i
$213$		0			0
$214$	16.0689	−	27.8322i	0.0750884	−	0.130057i
$215$	−482.769	−	278.727i	−2.24544	−	1.29640i
$216$		0			0
$217$	−8.81707	−	114.938i	−0.0406316	−	0.529666i
$218$	−110.795	−	191.902i	−0.508234	−	0.880286i
$219$		0			0
$220$		−	242.380i		−	1.10173i
$221$	219.633			0.993817
$222$		0			0
$223$	158.969	−	91.7807i	0.712865	−	0.411573i	−0.0992560	−	0.995062i	$-0.531646\pi$
$223$	158.969	−	91.7807i	0.712865	−	0.411573i	0.812121	+	0.583489i	$0.198313\pi$
$224$	39.4820	−	3.02873i	0.176259	−	0.0135211i
$225$		0			0
$226$	−93.9752	+	162.770i	−0.415820	+	0.720221i
$227$	70.3035	+	40.5897i	0.309707	+	0.178809i	0.646795	−	0.762664i	$-0.276109\pi$
$227$	70.3035	+	40.5897i	0.309707	+	0.178809i	−0.337088	+	0.941473i	$0.609442\pi$
$228$		0			0
$229$	−176.364	+	101.824i	−0.770149	+	0.444646i	−0.832928	−	0.553382i	$-0.813337\pi$
$229$	−176.364	+	101.824i	−0.770149	+	0.444646i	0.0627785	+	0.998027i	$0.480004\pi$
$230$	109.354	+	63.1353i	0.475450	+	0.274501i
$231$		0			0
$232$	−8.45607	−	14.6463i	−0.0364486	−	0.0631308i
$233$	−11.3573	+	19.6715i	−0.0487439	+	0.0844270i	−0.889368	−	0.457192i	$-0.848855\pi$
$233$	−11.3573	+	19.6715i	−0.0487439	+	0.0844270i	0.840624	+	0.541619i	$0.182188\pi$
$234$		0			0
$235$	−290.347	−	502.895i	−1.23552	−	2.13998i
$236$		−	203.494i		−	0.862262i
$237$		0			0
$238$	−116.518	+	79.7418i	−0.489572	+	0.335049i
$239$	17.2294	−	29.8421i	0.0720894	−	0.124863i	−0.827727	−	0.561130i	$-0.810367\pi$
$239$	17.2294	−	29.8421i	0.0720894	−	0.124863i	0.899817	+	0.436268i	$0.143700\pi$
$240$		0			0
$241$	106.441	+	61.4535i	0.441662	+	0.254994i	0.704302	−	0.709900i	$-0.251260\pi$
$241$	106.441	+	61.4535i	0.441662	+	0.254994i	−0.262640	+	0.964894i	$0.584593\pi$
$242$	129.931	+	225.047i	0.536905	+	0.929947i
$243$		0			0
$244$			134.238i			0.550156i
$245$	265.203	−	213.026i	1.08246	−	0.869493i
$246$		0			0
$247$	−11.1535	−	19.3184i	−0.0451559	−	0.0782123i
$248$		−	46.5783i		−	0.187816i
$249$		0			0
$250$		−	17.7353i		−	0.0709413i
$251$		−	63.3749i		−	0.252490i	−0.991999	−	0.126245i	$-0.959708\pi$
$251$		−	63.3749i		−	0.252490i	0.991999	−	0.126245i	$-0.0402925\pi$
$252$		0			0
$253$	−224.525			−0.887451
$254$	−347.458			−1.36795
$255$		0			0
$256$	16.0000			0.0625000
$257$	113.664	−	65.6238i	0.442272	−	0.255346i	−0.262289	−	0.964989i	$-0.584477\pi$
$257$	113.664	−	65.6238i	0.442272	−	0.255346i	0.704561	+	0.709644i	$0.251144\pi$
$258$		0			0
$259$	4.43086	−	9.24239i	0.0171076	−	0.0356849i
$260$	−213.809			−0.822341
$261$		0			0
$262$	264.928	−	152.956i	1.01117	−	0.583802i
$263$	136.997	−	237.285i	0.520899	−	0.902224i	−0.478805	−	0.877921i	$-0.658930\pi$
$263$	136.997	−	237.285i	0.520899	−	0.902224i	0.999705	−	0.0243030i	$-0.00773665\pi$
$264$		0			0
$265$	−78.3677	−	45.2456i	−0.295727	−	0.170738i
$266$	12.9310	+	6.19918i	0.0486126	+	0.0233052i
$267$		0			0
$268$	−78.9669			−0.294653
$269$	142.473	−	82.2566i	0.529638	−	0.305787i	−0.211231	−	0.977436i	$-0.567747\pi$
$269$	142.473	−	82.2566i	0.529638	−	0.305787i	0.740869	+	0.671649i	$0.234414\pi$
$270$		0			0
$271$	−62.3327	−	35.9878i	−0.230010	−	0.132796i	0.380567	−	0.924753i	$-0.375729\pi$
$271$	−62.3327	−	35.9878i	−0.230010	−	0.132796i	−0.610577	+	0.791957i	$0.709062\pi$
$272$	−49.4069	+	28.5251i	−0.181643	+	0.104872i
$273$		0			0
$274$	71.2366	−	123.385i	0.259988	−	0.450312i
$275$	−202.446	−	350.647i	−0.736167	−	1.27508i
$276$		0			0
$277$	−37.5087	+	64.9670i	−0.135411	+	0.234538i	−0.925754	−	0.378126i	$-0.876569\pi$
$277$	−37.5087	+	64.9670i	−0.135411	+	0.234538i	0.790344	+	0.612664i	$0.209902\pi$
$278$	−215.542	−	124.443i	−0.775329	−	0.447636i
$279$		0			0
$280$	113.428	−	77.6270i	0.405100	−	0.277239i
$281$	−185.301	−	320.950i	−0.659433	−	1.14217i	−0.980763	−	0.195205i	$-0.937463\pi$
$281$	−185.301	−	320.950i	−0.659433	−	1.14217i	0.321329	−	0.946968i	$-0.395870\pi$
$282$		0			0
$283$		−	139.136i		−	0.491648i	−0.969315	−	0.245824i	$-0.920942\pi$
$283$		−	139.136i		−	0.491648i	0.969315	−	0.245824i	$-0.0790585\pi$
$284$	187.791			0.661236
$285$		0			0
$286$	329.245	−	190.089i	1.15121	−	0.664649i
$287$	77.1546	+	36.9884i	0.268831	+	0.128879i
$288$		0			0
$289$	−42.7897	+	74.1140i	−0.148061	+	0.256450i
$290$	−50.8386	−	29.3517i	−0.175305	−	0.101213i
$291$		0			0
$292$	34.4378	−	19.8827i	0.117938	−	0.0680913i
$293$	−89.4372	−	51.6366i	−0.305247	−	0.176234i	0.339551	−	0.940588i	$-0.389725\pi$
$293$	−89.4372	−	51.6366i	−0.305247	−	0.176234i	−0.644797	+	0.764354i	$0.723058\pi$
$294$		0			0
$295$	−353.172	−	611.711i	−1.19719	−	2.07360i
$296$	2.07073	−	3.58661i	0.00699572	−	0.0121169i
$297$		0			0
$298$	142.078	+	246.087i	0.476773	+	0.825795i
$299$			198.059i			0.662404i
$300$		0			0
$301$	317.459	+	463.869i	1.05468	+	1.54109i
$302$	−167.552	+	290.209i	−0.554809	+	0.960957i
$303$		0			0
$304$	5.01800	+	2.89715i	0.0165066	+	0.00953008i
$305$	232.976	+	403.526i	0.763854	+	1.32303i
$306$		0			0
$307$			190.081i			0.619158i	0.950874	+	0.309579i	$0.100188\pi$
$307$			190.081i			0.619158i	−0.950874	+	0.309579i	$0.899812\pi$
$308$	−105.653	+	220.383i	−0.343028	+	0.715528i
$309$		0			0
$310$	−80.8384	−	140.016i	−0.260769	−	0.451665i
$311$			155.398i			0.499671i	0.968288	+	0.249836i	$0.0803766\pi$
$311$			155.398i			0.499671i	−0.968288	+	0.249836i	$0.919623\pi$
$312$		0			0
$313$			566.895i			1.81117i	0.424170	+	0.905583i	$0.360566\pi$
$313$			566.895i			1.81117i	−0.424170	+	0.905583i	$0.639434\pi$
$314$			18.6044i			0.0592498i
$315$		0			0
$316$	−245.572			−0.777126
$317$	191.408			0.603809			0.301905	−	0.953338i	$-0.402378\pi$
$317$	191.408			0.603809			0.301905	+	0.953338i	$0.402378\pi$
$318$		0			0
$319$	104.382			0.327216
$320$	48.0967	−	27.7686i	0.150302	−	0.0867770i
$321$		0			0
$322$	−71.9087	−	105.072i	−0.223319	−	0.326312i
$323$	−20.6604			−0.0639639
$324$		0			0
$325$	−309.313	+	178.582i	−0.951733	+	0.549484i
$326$	129.464	−	224.238i	0.397128	−	0.687845i
$327$		0			0
$328$	29.9407	+	17.2863i	0.0912826	+	0.0527020i
$329$	44.7856	+	583.816i	0.136126	+	1.77452i
$330$		0			0
$331$	526.847			1.59168			0.795842	−	0.605504i	$-0.207029\pi$
$331$	526.847			1.59168			0.795842	+	0.605504i	$0.207029\pi$
$332$	−14.0348	+	8.10301i	−0.0422736	+	0.0244067i
$333$		0			0
$334$	−252.510	−	145.786i	−0.756017	−	0.436486i
$335$	−237.378	+	137.050i	−0.708591	+	0.409105i
$336$		0			0
$337$	−168.251	+	291.419i	−0.499261	+	0.864746i	−1.00000		0.000852644i	$-0.999729\pi$
$337$	−168.251	+	291.419i	−0.499261	+	0.864746i	0.500738	+	0.865599i	$0.333062\pi$
$338$	−48.1813	−	83.4524i	−0.142548	−	0.246901i
$339$		0			0
$340$	−99.0129	+	171.495i	−0.291214	+	0.504398i
$341$	248.966	+	143.741i	0.730107	+	0.421527i
$342$		0			0
$343$	−333.992	+	78.0912i	−0.973738	+	0.227671i
$344$	113.561	+	196.693i	0.330119	+	0.571783i
$345$		0			0
$346$		−	259.436i		−	0.749815i
$347$	−185.827			−0.535523			−0.267762	−	0.963485i	$-0.586284\pi$
$347$	−185.827			−0.535523			−0.267762	+	0.963485i	$0.586284\pi$
$348$		0			0
$349$	−217.753	+	125.720i	−0.623935	+	0.360229i	−0.778400	−	0.627769i	$-0.783968\pi$
$349$	−217.753	+	125.720i	−0.623935	+	0.360229i	0.154464	+	0.987998i	$0.450635\pi$
$350$	99.2569	−	207.042i	0.283591	−	0.591547i
$351$		0			0
$352$	−49.3761	+	85.5219i	−0.140273	+	0.242960i
$353$	66.3740	+	38.3210i	0.188028	+	0.108558i	0.591059	−	0.806628i	$-0.298710\pi$
$353$	66.3740	+	38.3210i	0.188028	+	0.108558i	−0.403031	+	0.915186i	$0.632043\pi$
$354$		0			0
$355$	564.507	−	325.919i	1.59016	−	0.918080i
$356$	87.4094	+	50.4658i	0.245532	+	0.141758i
$357$		0			0
$358$	27.9753	+	48.4546i	0.0781432	+	0.135348i
$359$	236.702	−	409.979i	0.659336	−	1.14200i	−0.321451	−	0.946926i	$-0.604171\pi$
$359$	236.702	−	409.979i	0.659336	−	1.14200i	0.980788	−	0.195078i	$-0.0624960\pi$
$360$		0			0
$361$	−179.451	−	310.818i	−0.497094	−	0.860992i
$362$			5.19885i			0.0143615i
$363$		0			0
$364$	194.405	+	93.1987i	0.534079	+	0.256040i
$365$	69.0143	−	119.536i	0.189080	−	0.327497i
$366$		0			0
$367$	−42.3520	−	24.4520i	−0.115401	−	0.0666266i	0.441189	−	0.897414i	$-0.354557\pi$
$367$	−42.3520	−	24.4520i	−0.115401	−	0.0666266i	−0.556589	+	0.830788i	$0.687890\pi$
$368$	−25.7231	−	44.5537i	−0.0698996	−	0.121070i
$369$		0			0
$370$		−	14.3753i		−	0.0388523i
$371$	51.5330	+	75.2997i	0.138903	+	0.202964i
$372$		0			0
$373$	−273.081	−	472.991i	−0.732121	−	1.26807i	−0.955975	−	0.293448i	$-0.905197\pi$
$373$	−273.081	−	472.991i	−0.732121	−	1.26807i	0.223854	−	0.974623i	$-0.428136\pi$
$374$		−	352.115i		−	0.941484i
$375$		0			0
$376$			236.590i			0.629230i
$377$		−	92.0777i		−	0.244238i
$378$		0			0
$379$	385.626			1.01748			0.508741	−	0.860920i	$-0.330111\pi$
$379$	385.626			1.01748			0.508741	+	0.860920i	$0.330111\pi$
$380$	20.1124			0.0529275
$381$		0			0
$382$	−67.7097			−0.177250
$383$	42.1074	−	24.3107i	0.109941	−	0.0634744i	−0.444021	−	0.896016i	$-0.646449\pi$
$383$	42.1074	−	24.3107i	0.109941	−	0.0634744i	0.553962	+	0.832542i	$0.313115\pi$
$384$		0			0
$385$	64.8862	+	845.844i	0.168536	+	2.19700i
$386$	−82.7131			−0.214283
$387$		0			0
$388$	161.767	−	93.3965i	0.416926	−	0.240713i
$389$	184.898	−	320.253i	0.475316	−	0.823272i	−0.524284	−	0.851543i	$-0.675667\pi$
$389$	184.898	−	320.253i	0.475316	−	0.823272i	0.999600	+	0.0282716i	$0.00900032\pi$
$390$		0			0
$391$	158.862	+	91.7192i	0.406297	+	0.234576i
$392$	−136.971	+	21.1390i	−0.349417	+	0.0539261i
$393$		0			0
$394$	−288.472			−0.732163
$395$	−738.199	+	426.199i	−1.86886	+	1.07899i
$396$		0			0
$397$	570.204	+	329.208i	1.43628	+	0.829238i	0.997589	−	0.0693984i	$-0.0221080\pi$
$397$	570.204	+	329.208i	1.43628	+	0.829238i	0.438694	+	0.898637i	$0.355441\pi$
$398$	206.659	−	119.315i	0.519244	−	0.299785i
$399$		0			0
$400$	46.3871	−	80.3448i	0.115968	−	0.200862i
$401$	10.9763	+	19.0114i	0.0273722	+	0.0474101i	0.879387	−	0.476108i	$-0.157953\pi$
$401$	10.9763	+	19.0114i	0.0273722	+	0.0474101i	−0.852015	+	0.523518i	$0.824619\pi$
$402$		0			0
$403$	126.797	−	219.619i	0.314633	−	0.544960i
$404$	243.433	+	140.546i	0.602557	+	0.347887i
$405$		0			0
$406$	33.4304	+	48.8483i	0.0823409	+	0.120316i
$407$	12.7806	+	22.1366i	0.0314019	+	0.0543898i
$408$		0			0
$409$		−	229.660i		−	0.561517i	−0.959778	−	0.280758i	$-0.909414\pi$
$409$		−	229.660i		−	0.561517i	0.959778	−	0.280758i	$-0.0905860\pi$
$410$	120.004			0.292693
$411$		0			0
$412$	−62.5382	+	36.1065i	−0.151792	+	0.0876370i
$413$	54.4763	+	710.142i	0.131904	+	1.71947i
$414$		0			0
$415$	−28.1262	+	48.7160i	−0.0677739	+	0.117388i
$416$	75.4409	+	43.5558i	0.181348	+	0.104701i
$417$		0			0
$418$	−30.9712	+	17.8812i	−0.0740938	+	0.0427780i
$419$	−504.537	−	291.295i	−1.20415	−	0.695214i	−0.242672	−	0.970108i	$-0.578024\pi$
$419$	−504.537	−	291.295i	−1.20415	−	0.695214i	−0.961474	+	0.274894i	$0.911357\pi$
$420$		0			0
$421$	31.2793	+	54.1774i	0.0742977	+	0.128687i	0.900781	−	0.434274i	$-0.142995\pi$
$421$	31.2793	+	54.1774i	0.0742977	+	0.128687i	−0.826483	+	0.562962i	$0.809662\pi$
$422$	−34.4979	+	59.7521i	−0.0817486	+	0.141593i
$423$		0			0
$424$	18.4343	+	31.9292i	0.0434772	+	0.0753047i
$425$			330.799i			0.778351i
$426$		0			0
$427$	−35.9362	−	468.457i	−0.0841597	−	1.09709i
$428$	−22.7249	+	39.3606i	−0.0530955	+	0.0919641i
$429$		0			0
$430$	682.738	+	394.179i	1.58776	+	0.916696i
$431$	162.412	+	281.305i	0.376826	+	0.652681i	0.990598	−	0.136802i	$-0.0436823\pi$
$431$	162.412	+	281.305i	0.376826	+	0.652681i	−0.613773	+	0.789483i	$0.710349\pi$
$432$		0			0
$433$		−	222.186i		−	0.513131i	−0.966527	−	0.256566i	$-0.917409\pi$
$433$		−	222.186i		−	0.513131i	0.966527	−	0.256566i	$-0.0825909\pi$
$434$	12.4692	+	162.546i	0.0287309	+	0.374531i
$435$		0			0
$436$	156.688	+	271.391i	0.359375	+	0.622456i
$437$		−	18.6309i		−	0.0426336i
$438$		0			0
$439$			382.234i			0.870693i	0.900263	+	0.435346i	$0.143374\pi$
$439$			382.234i			0.870693i	−0.900263	+	0.435346i	$0.856626\pi$
$440$			342.777i			0.779038i
$441$		0			0
$442$	−310.609			−0.702734
$443$	−18.4259			−0.0415934			−0.0207967	−	0.999784i	$-0.506620\pi$
$443$	−18.4259			−0.0415934			−0.0207967	+	0.999784i	$0.506620\pi$
$444$		0			0
$445$	350.342			0.787285
$446$	−224.816	+	129.798i	−0.504072	+	0.291026i
$447$		0			0
$448$	−55.8360	+	4.28327i	−0.124634	+	0.00956088i
$449$	−383.046			−0.853109			−0.426554	−	0.904462i	$-0.640273\pi$
$449$	−383.046			−0.853109			−0.426554	+	0.904462i	$0.640273\pi$
$450$		0			0
$451$	−184.794	+	106.691i	−0.409744	+	0.236566i
$452$	132.901	−	230.191i	0.294029	−	0.509273i
$453$		0			0
$454$	−99.4241	−	57.4025i	−0.218996	−	0.126437i
$455$	746.139	−	57.2376i	1.63986	−	0.125797i
$456$		0			0
$457$	120.265			0.263163			0.131581	−	0.991305i	$-0.457995\pi$
$457$	120.265			0.263163			0.131581	+	0.991305i	$0.457995\pi$
$458$	249.417	−	144.001i	0.544578	−	0.314412i
$459$		0			0
$460$	−154.649	−	89.2868i	−0.336194	−	0.194102i
$461$	−17.9439	+	10.3599i	−0.0389239	+	0.0224727i	−0.519336	−	0.854570i	$-0.673821\pi$
$461$	−17.9439	+	10.3599i	−0.0389239	+	0.0224727i	0.480412	+	0.877043i	$0.340487\pi$
$462$		0			0
$463$	181.356	−	314.119i	0.391699	−	0.678442i	−0.600975	−	0.799268i	$-0.705221\pi$
$463$	181.356	−	314.119i	0.391699	−	0.678442i	0.992674	+	0.120826i	$0.0385542\pi$
$464$	11.9587	+	20.7130i	0.0257730	+	0.0446402i
$465$		0			0
$466$	16.0617	−	27.8197i	0.0344672	−	0.0596989i
$467$	−359.265	−	207.422i	−0.769304	−	0.444158i	0.0633224	−	0.997993i	$-0.479830\pi$
$467$	−359.265	−	207.422i	−0.769304	−	0.444158i	−0.832626	+	0.553835i	$0.813164\pi$
$468$		0			0
$469$	275.575	−	21.1398i	0.587579	−	0.0450742i
$470$	410.612	+	711.201i	0.873643	+	1.51319i
$471$		0			0
$472$			287.784i			0.609712i
$473$	−1401.80			−2.96364
$474$		0			0
$475$	29.0963	−	16.7988i	0.0612554	−	0.0353658i
$476$	164.781	−	112.772i	0.346180	−	0.236916i
$477$		0			0
$478$	−24.3660	+	42.2032i	−0.0509749	+	0.0882911i
$479$	129.582	+	74.8144i	0.270527	+	0.156189i	0.629127	−	0.777303i	$-0.283413\pi$
$479$	129.582	+	74.8144i	0.270527	+	0.156189i	−0.358600	+	0.933491i	$0.616746\pi$
$480$		0			0
$481$	19.5272	−	11.2741i	0.0405972	−	0.0234388i
$482$	−150.530	−	86.9084i	−0.312302	−	0.180308i
$483$		0			0
$484$	−183.750	−	318.265i	−0.379649	−	0.657571i
$485$	324.186	−	561.507i	0.668426	−	1.15775i
$486$		0			0
$487$	−46.3139	−	80.2181i	−0.0951004	−	0.164719i	0.814550	−	0.580093i	$-0.196984\pi$
$487$	−46.3139	−	80.2181i	−0.0951004	−	0.164719i	−0.909651	+	0.415374i	$0.863651\pi$
$488$		−	189.841i		−	0.389019i
$489$		0			0
$490$	−375.054	+	301.264i	−0.765416	+	0.614824i
$491$	164.023	−	284.097i	0.334060	−	0.578609i	−0.649244	−	0.760580i	$-0.724915\pi$
$491$	164.023	−	284.097i	0.334060	−	0.578609i	0.983304	+	0.181972i	$0.0582479\pi$
$492$		0			0
$493$	−73.8553	−	42.6403i	−0.149808	−	0.0864916i
$494$	15.7734	+	27.3204i	0.0319300	+	0.0553045i
$495$		0			0
$496$			65.8716i			0.132806i
$497$	−655.343	+	50.2725i	−1.31860	+	0.101152i
$498$		0			0
$499$	−83.8764	−	145.278i	−0.168089	−	0.291139i	0.769659	−	0.638455i	$-0.220426\pi$
$499$	−83.8764	−	145.278i	−0.168089	−	0.291139i	−0.937748	+	0.347317i	$0.887093\pi$
$500$			25.0815i			0.0501630i
$501$		0			0
$502$			89.6256i			0.178537i
$503$			320.995i			0.638161i	0.947728	+	0.319080i	$0.103374\pi$
$503$			320.995i			0.638161i	−0.947728	+	0.319080i	$0.896626\pi$
$504$		0			0
$505$	975.693			1.93207
$506$	317.526			0.627522
$507$		0			0
$508$	491.380			0.967284
$509$	−347.866	+	200.840i	−0.683429	+	0.394578i	−0.801146	−	0.598469i	$-0.795776\pi$
$509$	−347.866	+	200.840i	−0.683429	+	0.394578i	0.117716	+	0.993047i	$0.462443\pi$
$510$		0			0
$511$	−114.857	+	78.6047i	−0.224768	+	0.153825i
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−160.745	+	92.8061i	−0.312733	+	0.180557i
$515$	−125.328	+	217.075i	−0.243356	+	0.421505i
$516$		0			0
$517$	−1264.60	−	730.120i	−2.44604	−	1.41222i
$518$	−6.26618	+	13.0707i	−0.0120969	+	0.0252331i
$519$		0			0
$520$	302.371			0.581483
$521$	807.871	−	466.425i	1.55062	−	0.895249i	0.552525	−	0.833496i	$-0.313664\pi$
$521$	807.871	−	466.425i	1.55062	−	0.895249i	0.998091	−	0.0617529i	$-0.0196691\pi$
$522$		0			0
$523$	−631.376	−	364.525i	−1.20722	−	0.696989i	−0.245069	−	0.969506i	$-0.578810\pi$
$523$	−631.376	−	364.525i	−1.20722	−	0.696989i	−0.962151	+	0.272517i	$0.912144\pi$
$524$	−374.664	+	216.313i	−0.715008	+	0.412810i
$525$		0			0
$526$	−193.742	+	335.572i	−0.368331	+	0.637969i
$527$	−117.437	−	203.407i	−0.222841	−	0.385972i
$528$		0			0
$529$	181.790	−	314.870i	0.343649	−	0.595218i
$530$	110.829	+	63.9870i	0.209111	+	0.120730i
$531$		0			0
$532$	−18.2871	−	8.76696i	−0.0343743	−	0.0164793i
$533$	94.1147	+	163.011i	0.176575	+	0.305838i
$534$		0			0
$535$			157.760i			0.294878i
$536$	111.676			0.208351
$537$		0			0
$538$	−201.487	+	116.328i	−0.374511	+	0.216224i
$539$	309.704	−	797.364i	0.574590	−	1.47934i
$540$		0			0
$541$	222.378	−	385.171i	0.411051	−	0.711960i	−0.583954	−	0.811786i	$-0.698495\pi$
$541$	222.378	−	385.171i	0.411051	−	0.711960i	0.995005	+	0.0998261i	$0.0318286\pi$
$542$	88.1517	+	50.8944i	0.162642	+	0.0939011i
$543$		0			0
$544$	69.8720	−	40.3406i	0.128441	−	0.0741555i
$545$	942.019	+	543.875i	1.72848	+	0.997936i
$546$		0			0
$547$	−32.0268	−	55.4721i	−0.0585500	−	0.101411i	0.835265	−	0.549848i	$-0.185314\pi$
$547$	−32.0268	−	55.4721i	−0.0585500	−	0.101411i	−0.893815	+	0.448436i	$0.851981\pi$
$548$	−100.744	+	174.493i	−0.183839	+	0.318419i
$549$		0			0
$550$	286.302	+	495.889i	0.520549	+	0.901617i
$551$			8.66151i			0.0157196i
$552$		0			0
$553$	856.984	−	65.7407i	1.54970	−	0.118880i
$554$	53.0453	−	91.8772i	0.0957497	−	0.165843i
$555$		0			0
$556$	304.822	+	175.989i	0.548240	+	0.316527i
$557$	−321.689	−	557.182i	−0.577539	−	1.00033i	−0.995761	−	0.0919818i	$-0.970680\pi$
$557$	−321.689	−	557.182i	−0.577539	−	1.00033i	0.418222	−	0.908345i	$-0.362654\pi$
$558$		0			0
$559$			1236.56i			2.21209i
$560$	−160.411	+	109.781i	−0.286449	+	0.196038i
$561$		0			0
$562$	262.055	+	453.892i	0.466290	+	0.807638i
$563$			563.957i			1.00170i	0.865534	+	0.500850i	$0.166979\pi$
$563$			563.957i			1.00170i	−0.865534	+	0.500850i	$0.833021\pi$
$564$		0			0
$565$		−	922.620i		−	1.63296i
$566$			196.768i			0.347647i
$567$		0			0
$568$	−265.576			−0.467564
$569$	−608.965			−1.07024			−0.535118	−	0.844777i	$-0.679733\pi$
$569$	−608.965			−1.07024			−0.535118	+	0.844777i	$0.679733\pi$
$570$		0			0
$571$	−706.133			−1.23666			−0.618330	−	0.785918i	$-0.712191\pi$
$571$	−706.133			−1.23666			−0.618330	+	0.785918i	$0.712191\pi$
$572$	−465.622	+	268.827i	−0.814025	+	0.469977i
$573$		0			0
$574$	−109.113	−	52.3094i	−0.190092	−	0.0911314i
$575$	−298.304			−0.518790
$576$		0			0
$577$	−774.773	+	447.315i	−1.34276	+	0.775243i	−0.987212	−	0.159415i	$-0.949039\pi$
$577$	−774.773	+	447.315i	−1.34276	+	0.775243i	−0.355548	+	0.934658i	$0.615706\pi$
$578$	60.5138	−	104.813i	0.104695	−	0.181337i
$579$		0			0
$580$	71.8966	+	41.5095i	0.123960	+	0.0715681i
$581$	46.8088	−	32.0346i	0.0805659	−	0.0551371i
$582$		0			0
$583$	−227.554			−0.390315
$584$	−48.7024	+	28.1183i	−0.0833945	+	0.0481479i
$585$		0			0
$586$	126.483	+	73.0252i	0.215842	+	0.124616i
$587$	−98.0159	+	56.5895i	−0.166978	+	0.0964046i	−0.581160	−	0.813789i	$-0.697401\pi$
$587$	−98.0159	+	56.5895i	−0.166978	+	0.0964046i	0.414182	+	0.910194i	$0.364068\pi$
$588$		0			0
$589$	−11.9275	+	20.6590i	−0.0202504	+	0.0350747i
$590$	499.460	+	865.090i	0.846543	+	1.46625i
$591$		0			0
$592$	−2.92846	+	5.07224i	−0.00494672	+	0.00856797i
$593$	416.988	+	240.748i	0.703184	+	0.405983i	0.808532	−	0.588452i	$-0.200262\pi$
$593$	416.988	+	240.748i	0.703184	+	0.405983i	−0.105348	+	0.994435i	$0.533596\pi$
$594$		0			0
$595$	299.620	−	624.982i	0.503563	−	1.05039i
$596$	−200.929	−	348.019i	−0.337129	−	0.583925i
$597$		0			0
$598$		−	280.097i		−	0.468390i
$599$	−181.833			−0.303561			−0.151781	−	0.988414i	$-0.548501\pi$
$599$	−181.833			−0.303561			−0.151781	+	0.988414i	$0.548501\pi$
$600$		0			0
$601$	873.672	−	504.415i	1.45370	−	0.839292i	0.455008	−	0.890487i	$-0.349636\pi$
$601$	873.672	−	504.415i	1.45370	−	0.839292i	0.998689	+	0.0511946i	$0.0163029\pi$
$602$	−448.955	−	656.010i	−0.745772	−	1.08972i
$603$		0			0
$604$	236.955	−	410.418i	0.392309	−	0.679500i
$605$	−1104.72	−	637.811i	−1.82599	−	1.05423i
$606$		0			0
$607$	434.243	−	250.710i	0.715392	−	0.413032i	−0.0976624	−	0.995220i	$-0.531137\pi$
$607$	434.243	−	250.710i	0.715392	−	0.413032i	0.813054	+	0.582188i	$0.197803\pi$
$608$	−7.09653	−	4.09718i	−0.0116719	−	0.00673879i
$609$		0			0
$610$	−329.477	−	570.671i	−0.540127	−	0.935527i
$611$	−644.056	+	1115.54i	−1.05410	+	1.82576i
$612$		0			0
$613$	212.902	+	368.757i	0.347311	+	0.601561i	0.985771	−	0.168095i	$-0.0537614\pi$
$613$	212.902	+	368.757i	0.347311	+	0.601561i	−0.638460	+	0.769655i	$0.720428\pi$
$614$		−	268.816i		−	0.437811i
$615$		0			0
$616$	149.416	−	311.668i	0.242558	−	0.505955i
$617$	227.440	−	393.937i	0.368622	−	0.638472i	−0.620729	−	0.784026i	$-0.713163\pi$
$617$	227.440	−	393.937i	0.368622	−	0.638472i	0.989350	+	0.145554i	$0.0464964\pi$
$618$		0			0
$619$	433.600	+	250.339i	0.700485	+	0.404425i	0.807528	−	0.589829i	$-0.200805\pi$
$619$	433.600	+	250.339i	0.700485	+	0.404425i	−0.107043	+	0.994254i	$0.534138\pi$
$620$	114.323	+	198.013i	0.184392	+	0.319375i
$621$		0			0
$622$		−	219.766i		−	0.353321i
$623$	−318.547	−	152.713i	−0.511311	−	0.245125i
$624$		0			0
$625$	333.449	+	577.551i	0.533519	+	0.924081i
$626$		−	801.710i		−	1.28069i
$627$		0			0
$628$		−	26.3107i		−	0.0418960i
$629$		−	20.8837i		−	0.0332014i
$630$		0			0
$631$	790.195			1.25229			0.626145	−	0.779706i	$-0.284632\pi$
$631$	790.195			1.25229			0.626145	+	0.779706i	$0.284632\pi$
$632$	347.291			0.549511
$633$		0			0
$634$	−270.691			−0.426958
$635$	1477.11	−	852.810i	2.32616	−	1.34301i
$636$		0			0
$637$	−703.373	−	273.197i	−1.10420	−	0.428881i
$638$	−147.618			−0.231377
$639$		0			0
$640$	−68.0190	+	39.2708i	−0.106280	+	0.0613606i
$641$	186.447	−	322.936i	0.290869	−	0.503800i	−0.683146	−	0.730281i	$-0.739389\pi$
$641$	186.447	−	322.936i	0.290869	−	0.503800i	0.974015	+	0.226481i	$0.0727223\pi$
$642$		0			0
$643$	−356.344	−	205.735i	−0.554190	−	0.319962i	0.196620	−	0.980480i	$-0.437003\pi$
$643$	−356.344	−	205.735i	−0.554190	−	0.319962i	−0.750810	+	0.660518i	$0.770337\pi$
$644$	101.694	+	148.595i	0.157910	+	0.230737i
$645$		0			0
$646$	29.2181			0.0452293
$647$	508.421	−	293.537i	0.785813	−	0.453689i	−0.0526736	−	0.998612i	$-0.516774\pi$
$647$	508.421	−	293.537i	0.785813	−	0.453689i	0.838486	+	0.544923i	$0.183441\pi$
$648$		0			0
$649$	−1538.24	−	888.103i	−2.37017	−	1.36842i
$650$	437.435	−	252.553i	0.672977	−	0.388544i
$651$		0			0
$652$	−183.089	+	317.120i	−0.280812	+	0.486380i
$653$	283.053	+	490.263i	0.433466	+	0.750786i	0.997169	−	0.0751921i	$-0.0239570\pi$
$653$	283.053	+	490.263i	0.433466	+	0.750786i	−0.563703	+	0.825978i	$0.690624\pi$
$654$		0			0
$655$	−750.838	+	1300.49i	−1.14632	+	1.98548i
$656$	−42.3425	−	24.4465i	−0.0645466	−	0.0372660i
$657$		0			0
$658$	−63.3364	−	825.641i	−0.0962559	−	1.25477i
$659$	−368.775	−	638.738i	−0.559599	−	0.969253i	−0.997530	−	0.0702447i	$-0.977622\pi$
$659$	−368.775	−	638.738i	−0.559599	−	0.969253i	0.437931	−	0.899008i	$-0.355711\pi$
$660$		0			0
$661$			1010.40i			1.52859i	0.644865	+	0.764297i	$0.276914\pi$
$661$			1010.40i			1.52859i	−0.644865	+	0.764297i	$0.723086\pi$
$662$	−745.075			−1.12549
$663$		0			0
$664$	19.8482	−	11.4594i	0.0298919	−	0.0172581i
$665$	−70.1873	+	5.38419i	−0.105545	+	0.00809653i
$666$		0			0
$667$	38.4518	−	66.6004i	0.0576488	−	0.0998507i
$668$	357.102	+	206.173i	0.534584	+	0.308642i
$669$		0			0
$670$	335.703	−	193.818i	0.501049	−	0.289281i
$671$	1014.73	+	585.852i	1.51226	+	0.873103i
$672$		0			0
$673$	−78.3833	−	135.764i	−0.116468	−	0.201729i	0.801897	−	0.597462i	$-0.203824\pi$
$673$	−78.3833	−	135.764i	−0.116468	−	0.201729i	−0.918366	+	0.395733i	$0.870491\pi$
$674$	237.943	−	412.129i	0.353031	−	0.611468i
$675$		0			0
$676$	68.1386	+	118.020i	0.100797	+	0.174585i
$677$		−	238.466i		−	0.352240i	−0.984369	−	0.176120i	$-0.943645\pi$
$677$		−	238.466i		−	0.352240i	0.984369	−	0.176120i	$-0.0563547\pi$
$678$		0			0
$679$	−539.525	+	369.236i	−0.794587	+	0.543794i
$680$	140.025	−	242.531i	0.205920	−	0.356663i
$681$		0			0
$682$	−352.092	−	203.280i	−0.516264	−	0.298065i
$683$	−324.836	−	562.632i	−0.475602	−	0.823766i	0.524008	−	0.851713i	$-0.324436\pi$
$683$	−324.836	−	562.632i	−0.475602	−	0.823766i	−0.999609	+	0.0279474i	$0.991103\pi$
$684$		0			0
$685$			699.379i			1.02099i
$686$	472.336	−	110.438i	0.688537	−	0.160988i
$687$		0			0
$688$	−160.600	−	278.167i	−0.233430	−	0.404312i
$689$			200.731i			0.291336i
$690$		0			0
$691$		−	499.800i		−	0.723300i	−0.932314	−	0.361650i	$-0.882214\pi$
$691$		−	499.800i		−	0.723300i	0.932314	−	0.361650i	$-0.117786\pi$
$692$			366.898i			0.530199i
$693$		0			0
$694$	262.799			0.378672
$695$	1221.74			1.75790
$696$		0			0
$697$	174.335			0.250121
$698$	307.950	−	177.795i	0.441189	−	0.254721i
$699$		0			0
$700$	−140.370	+	292.801i	−0.200529	+	0.418287i
$701$	−142.184			−0.202830			−0.101415	−	0.994844i	$-0.532337\pi$
$701$	−142.184			−0.202830			−0.101415	+	0.994844i	$0.532337\pi$
$702$		0			0
$703$	−1.83688	+	1.06052i	−0.00261291	+	0.00150856i
$704$	69.8284	−	120.946i	0.0991880	−	0.171799i
$705$		0			0
$706$	−93.8670	−	54.1941i	−0.132956	−	0.0767622i
$707$	−887.145	−	425.302i	−1.25480	−	0.601559i
$708$		0			0
$709$	−573.786			−0.809289			−0.404645	−	0.914474i	$-0.632605\pi$
$709$	−573.786			−0.809289			−0.404645	+	0.914474i	$0.632605\pi$
$710$	−798.334	+	460.918i	−1.12441	+	0.649181i
$711$		0			0
$712$	−123.616	−	71.3694i	−0.173617	−	0.100238i
$713$	183.426	−	105.901i	0.257260	−	0.148529i
$714$		0			0
$715$	−933.120	+	1616.21i	−1.30506	+	2.26044i
$716$	−39.5630	−	68.5251i	−0.0552556	−	0.0957055i
$717$		0			0
$718$	−334.747	+	579.799i	−0.466221	+	0.807519i
$719$	−161.137	−	93.0327i	−0.224113	−	0.129392i	0.383740	−	0.923441i	$-0.374636\pi$
$719$	−161.137	−	93.0327i	−0.224113	−	0.129392i	−0.607853	+	0.794049i	$0.707969\pi$
$720$		0			0
$721$	208.577	−	142.744i	0.289288	−	0.197981i
$722$	253.782	+	439.563i	0.351498	+	0.608813i
$723$		0			0
$724$		−	7.35228i		−	0.0101551i
$725$	138.682			0.191286
$726$		0			0
$727$	−47.5157	+	27.4332i	−0.0653587	+	0.0377348i	−0.532323	−	0.846541i	$-0.678681\pi$
$727$	−47.5157	+	27.4332i	−0.0653587	+	0.0377348i	0.466965	+	0.884276i	$0.345348\pi$
$728$	−274.930	−	131.803i	−0.377651	−	0.181048i
$729$		0			0
$730$	−97.6010	+	169.050i	−0.133700	+	0.231575i
$731$	991.842	+	572.640i	1.35683	+	0.783365i
$732$		0			0
$733$	−319.740	+	184.602i	−0.436208	+	0.251845i	−0.701988	−	0.712189i	$-0.747704\pi$
$733$	−319.740	+	184.602i	−0.436208	+	0.251845i	0.265780	+	0.964034i	$0.414370\pi$
$734$	59.8948	+	34.5803i	0.0816006	+	0.0471121i
$735$		0			0
$736$	36.3779	+	63.0084i	0.0494265	+	0.0856092i
$737$	−344.633	+	596.923i	−0.467617	+	0.809936i
$738$		0			0
$739$	−552.319	−	956.645i	−0.747387	−	1.29451i	−0.949071	−	0.315062i	$-0.897975\pi$
$739$	−552.319	−	956.645i	−0.747387	−	1.29451i	0.201684	−	0.979451i	$-0.435359\pi$
$740$			20.3298i			0.0274727i
$741$		0			0
$742$	−72.8787	−	106.490i	−0.0982193	−	0.143517i
$743$	184.560	−	319.667i	0.248398	−	0.430238i	−0.714683	−	0.699448i	$-0.753429\pi$
$743$	184.560	−	319.667i	0.248398	−	0.430238i	0.963082	+	0.269210i	$0.0867625\pi$
$744$		0			0
$745$	−1208.00	−	697.440i	−1.62148	−	0.936161i
$746$	386.195	+	668.910i	0.517688	+	0.896662i
$747$		0			0
$748$			497.966i			0.665729i
$749$	68.7670	−	143.442i	0.0918118	−	0.191512i
$750$		0			0
$751$	327.659	+	567.522i	0.436297	+	0.755689i	0.997401	−	0.0720568i	$-0.0229563\pi$
$751$	327.659	+	567.522i	0.436297	+	0.755689i	−0.561103	+	0.827746i	$0.689623\pi$
$752$		−	334.589i		−	0.444933i
$753$		0			0
$754$			130.218i			0.172702i
$755$		−	1644.98i		−	2.17878i
$756$		0			0
$757$	251.264			0.331921			0.165961	−	0.986132i	$-0.446928\pi$
$757$	251.264			0.331921			0.165961	+	0.986132i	$0.446928\pi$
$758$	−545.357			−0.719468
$759$		0			0
$760$	−28.4433			−0.0374254
$761$	750.816	−	433.484i	0.986617	−	0.569624i	0.0823559	−	0.996603i	$-0.473756\pi$
$761$	750.816	−	433.484i	0.986617	−	0.569624i	0.904261	+	0.426979i	$0.140422\pi$
$762$		0			0
$763$	−619.453	−	905.140i	−0.811865	−	1.18629i
$764$	95.7559			0.125335
$765$		0			0
$766$	−59.5488	+	34.3805i	−0.0777400	+	0.0448832i
$767$	−783.416	+	1356.92i	−1.02140	+	1.76912i
$768$		0			0
$769$	40.8679	+	23.5951i	0.0531442	+	0.0306828i	0.526337	−	0.850276i	$-0.323565\pi$
$769$	40.8679	+	23.5951i	0.0531442	+	0.0306828i	−0.473193	+	0.880959i	$0.656898\pi$
$770$	−91.7629	−	1196.20i	−0.119173	−	1.55351i
$771$		0			0
$772$	116.974			0.151521
$773$	190.903	−	110.218i	0.246963	−	0.142584i	−0.371410	−	0.928469i	$-0.621125\pi$
$773$	190.903	−	110.218i	0.246963	−	0.142584i	0.618373	+	0.785885i	$0.287792\pi$
$774$		0			0
$775$	330.777	+	190.974i	0.426810	+	0.246419i
$776$	−228.774	+	132.083i	−0.294811	+	0.170209i
$777$		0			0
$778$	−261.485	+	452.906i	−0.336099	+	0.582141i
$779$	−8.85312	−	15.3341i	−0.0113647	−	0.0196843i
$780$		0			0
$781$	819.571	−	1419.54i	1.04939	−	1.81759i
$782$	−224.665	−	129.711i	−0.287296	−	0.165870i
$783$		0			0
$784$	193.707	−	29.8951i	0.247075	−	0.0381315i
$785$	−45.6632	−	79.0910i	−0.0581697	−	0.100753i
$786$		0			0
$787$		−	975.179i		−	1.23911i	−0.784953	−	0.619555i	$-0.787313\pi$
$787$		−	975.179i		−	1.23911i	0.784953	−	0.619555i	$-0.212687\pi$
$788$	407.962			0.517718
$789$		0			0
$790$	1043.97	−	602.737i	1.32148	−	0.762958i
$791$	−402.168	+	838.888i	−0.508429	+	1.06054i
$792$		0			0
$793$	516.794	−	895.113i	0.651694	−	1.12877i
$794$	−806.391	−	465.570i	−1.01561	−	0.586360i
$795$		0			0
$796$	−292.260	+	168.736i	−0.367161	+	0.211980i
$797$	−1053.12	−	608.022i	−1.32136	−	0.762888i	−0.337415	−	0.941356i	$-0.609553\pi$
$797$	−1053.12	−	608.022i	−1.32136	−	0.762888i	−0.983946	+	0.178468i	$0.942886\pi$
$798$		0			0
$799$	596.513	+	1033.19i	0.746574	+	1.29310i
$800$	−65.6012	+	113.625i	−0.0820015	+	0.142031i
$801$		0			0
$802$	−15.5228	−	26.8862i	−0.0193551	−	0.0335240i
$803$		−	347.094i		−	0.432246i
$804$		0			0
$805$	563.589	+	270.188i	0.700111	+	0.335637i
$806$	−179.318	+	310.588i	−0.222479	+	0.385345i
$807$		0			0
$808$	−344.266	−	198.762i	−0.426072	−	0.245993i
$809$	−677.925	−	1174.20i	−0.837979	−	1.45142i	−0.891581	−	0.452861i	$-0.850404\pi$
$809$	−677.925	−	1174.20i	−0.837979	−	1.45142i	0.0536018	−	0.998562i	$-0.482930\pi$
$810$		0			0
$811$		−	838.233i		−	1.03358i	−0.856112	−	0.516790i	$-0.827127\pi$
$811$		−	838.233i		−	1.03358i	0.856112	−	0.516790i	$-0.172873\pi$
$812$	−47.2778	−	69.0819i	−0.0582238	−	0.0850762i
$813$		0			0
$814$	−18.0745	−	31.3059i	−0.0222045	−	0.0384594i
$815$			1271.03i			1.55955i
$816$		0			0
$817$		−	116.320i		−	0.142375i
$818$			324.789i			0.397052i
$819$		0			0
$820$	−169.711			−0.206965
$821$	−788.378			−0.960265			−0.480133	−	0.877196i	$-0.659411\pi$
$821$	−788.378			−0.960265			−0.480133	+	0.877196i	$0.659411\pi$
$822$		0			0
$823$	400.688			0.486863			0.243432	−	0.969918i	$-0.421727\pi$
$823$	400.688			0.486863			0.243432	+	0.969918i	$0.421727\pi$
$824$	88.4424	−	51.0622i	0.107333	−	0.0619687i
$825$		0			0
$826$	−77.0411	−	1004.29i	−0.0932701	−	1.21585i
$827$	1250.02			1.51151			0.755757	−	0.654852i	$-0.227269\pi$
$827$	1250.02			1.51151			0.755757	+	0.654852i	$0.227269\pi$
$828$		0			0
$829$	916.487	−	529.134i	1.10553	−	0.638280i	0.167864	−	0.985810i	$-0.446313\pi$
$829$	916.487	−	529.134i	1.10553	−	0.638280i	0.937669	+	0.347530i	$0.112980\pi$
$830$	39.7764	−	68.8948i	0.0479234	−	0.0830058i
$831$		0			0
$832$	−106.690	−	61.5972i	−0.128233	−	0.0740351i
$833$	−544.856	+	437.658i	−0.654089	+	0.525400i
$834$		0			0
$835$	1431.29			1.71412
$836$	43.7999	−	25.2879i	0.0523922	−	0.0302486i
$837$		0			0
$838$	713.523	+	411.953i	0.851460	+	0.491591i
$839$	−677.714	+	391.278i	−0.807764	+	0.466363i	−0.846179	−	0.532899i	$-0.821103\pi$
$839$	−677.714	+	391.278i	−0.807764	+	0.466363i	0.0384146	+	0.999262i	$0.487769\pi$
$840$		0			0
$841$	402.624	−	697.365i	0.478744	−	0.829209i
$842$	−44.2356	−	76.6184i	−0.0525364	−	0.0909957i
$843$		0			0
$844$	48.7874	−	84.5022i	0.0578050	−	0.100121i
$845$	409.655	+	236.514i	0.484799	+	0.279899i
$846$		0			0
$847$	726.442	+	1061.47i	0.857665	+	1.25321i
$848$	−26.0701	−	45.1547i	−0.0307430	−	0.0532485i
$849$		0			0
$850$		−	467.821i		−	0.550377i
$851$	18.8322			0.0221295
$852$		0			0
$853$	−72.1792	+	41.6727i	−0.0846180	+	0.0488542i	−0.541712	−	0.840564i	$-0.682224\pi$
$853$	−72.1792	+	41.6727i	−0.0846180	+	0.0488542i	0.457094	+	0.889418i	$0.348890\pi$
$854$	50.8214	+	662.499i	0.0595099	+	0.775759i
$855$		0			0
$856$	32.1378	−	55.6643i	0.0375442	−	0.0650284i
$857$	223.760	+	129.188i	0.261097	+	0.150744i	0.624835	−	0.780757i	$-0.285166\pi$
$857$	223.760	+	129.188i	0.261097	+	0.150744i	−0.363738	+	0.931501i	$0.618500\pi$
$858$		0			0
$859$	−343.478	+	198.307i	−0.399858	+	0.230858i	−0.686423	−	0.727203i	$-0.740820\pi$
$859$	−343.478	+	198.307i	−0.399858	+	0.230858i	0.286565	+	0.958061i	$0.407487\pi$
$860$	−965.538	−	557.454i	−1.12272	−	0.648202i
$861$		0			0
$862$	−229.685	−	397.826i	−0.266456	−	0.461515i
$863$	−195.031	+	337.804i	−0.225992	+	0.391430i	−0.956617	−	0.291349i	$-0.905896\pi$
$863$	−195.031	+	337.804i	−0.225992	+	0.391430i	0.730624	+	0.682780i	$0.239229\pi$
$864$		0			0
$865$	636.765	+	1102.91i	0.736145	+	1.27504i
$866$			314.218i			0.362838i
$867$		0			0
$868$	−17.6341	−	229.875i	−0.0203158	−	0.264833i
$869$	−1071.74	+	1856.31i	−1.23331	+	2.13615i
$870$		0			0
$871$	526.559	+	304.009i	0.604545	+	0.349034i
$872$	−221.590	−	383.805i	−0.254117	−	0.440143i
$873$		0			0
$874$			26.3480i			0.0301465i
$875$	−6.71444	−	87.5282i	−0.00767365	−	0.100032i
$876$		0			0
$877$	198.926	+	344.551i	0.226826	+	0.392874i	0.956866	−	0.290530i	$-0.0938318\pi$
$877$	198.926	+	344.551i	0.226826	+	0.392874i	−0.730040	+	0.683405i	$0.760498\pi$
$878$		−	540.561i		−	0.615673i
$879$		0			0
$880$		−	484.760i		−	0.550863i
$881$		−	845.162i		−	0.959321i	−0.877454	−	0.479660i	$-0.840760\pi$
$881$		−	845.162i		−	0.959321i	0.877454	−	0.479660i	$-0.159240\pi$
$882$		0			0
$883$	16.3649			0.0185333			0.00926666	−	0.999957i	$-0.497050\pi$
$883$	16.3649			0.0185333			0.00926666	+	0.999957i	$0.497050\pi$
$884$	439.267			0.496908
$885$		0			0
$886$	26.0581			0.0294110
$887$	558.172	−	322.261i	0.629281	−	0.363315i	−0.151193	−	0.988504i	$-0.548311\pi$
$887$	558.172	−	322.261i	0.629281	−	0.363315i	0.780473	+	0.625189i	$0.214978\pi$
$888$		0			0
$889$	−1714.79	+	131.545i	−1.92890	+	0.147969i
$890$	−495.458			−0.556694
$891$		0			0
$892$	317.938	−	183.561i	0.356432	−	0.205786i
$893$	60.5846	−	104.936i	0.0678439	−	0.117509i
$894$		0			0
$895$	−237.856	−	137.326i	−0.265761	−	0.153437i
$896$	78.9640	−	6.05746i	0.0881294	−	0.00676056i
$897$		0			0
$898$	541.709			0.603239
$899$	−85.2751	+	49.2336i	−0.0948555	+	0.0547649i
$900$		0			0
$901$	161.005	+	92.9565i	0.178696	+	0.103170i
$902$	261.339	−	150.884i	0.289733	−	0.167277i
$903$		0			0
$904$	−187.950	+	325.540i	−0.207910	+	0.360110i
$905$	−12.7602	−	22.1013i	−0.0140996	−	0.0244213i
$906$		0			0
$907$	−793.972	+	1375.20i	−0.875383	+	1.51621i	−0.0190283	+	0.999819i	$0.506057\pi$
$907$	−793.972	+	1375.20i	−0.875383	+	1.51621i	−0.856354	+	0.516389i	$0.827276\pi$
$908$	140.607	+	81.1795i	0.154853	+	0.0894047i
$909$		0			0
$910$	−1055.20	+	80.9462i	−1.15956	+	0.0889519i
$911$	752.842	+	1303.96i	0.826391	+	1.43135i	0.900852	+	0.434127i	$0.142943\pi$
$911$	752.842	+	1303.96i	0.826391	+	1.43135i	−0.0744605	+	0.997224i	$0.523723\pi$
$912$		0			0
$913$			141.455i			0.154934i
$914$	−170.081			−0.186084
$915$		0			0
$916$	−352.728	+	203.648i	−0.385075	+	0.222323i
$917$	1249.58	−	855.176i	1.36268	−	0.932580i
$918$		0			0
$919$	−583.949	+	1011.43i	−0.635418	+	1.10058i	0.351008	+	0.936372i	$0.385839\pi$
$919$	−583.949	+	1011.43i	−0.635418	+	1.10058i	−0.986426	+	0.164204i	$0.947494\pi$
$920$	218.707	+	126.271i	0.237725	+	0.137251i
$921$		0			0
$922$	25.3765	−	14.6512i	0.0275234	−	0.0158906i
$923$	−1252.21	−	722.962i	−1.35667	−	0.783275i
$924$		0			0
$925$	16.9803	+	29.4108i	0.0183571	+	0.0317954i
$926$	−256.477	+	444.231i	−0.276973	+	0.479731i
$927$		0			0
$928$	−16.9121	−	29.2927i	−0.0182243	−	0.0315654i
$929$			598.256i			0.643978i	0.946743	+	0.321989i	$0.104351\pi$
$929$			598.256i			0.643978i	−0.946743	+	0.321989i	$0.895649\pi$
$930$		0			0
$931$	66.1644	+	25.6989i	0.0710682	+	0.0276036i
$932$	−22.7147	+	39.3430i	−0.0243720	+	0.0422135i
$933$		0			0
$934$	508.077	+	293.339i	0.543980	+	0.314067i
$935$	864.239	+	1496.91i	0.924320	+	1.60097i
$936$		0			0
$937$			56.5072i			0.0603066i	0.999545	+	0.0301533i	$0.00959954\pi$
$937$			56.5072i			0.0603066i	−0.999545	+	0.0301533i	$0.990400\pi$
$938$	−389.721	+	29.8962i	−0.415481	+	0.0318723i
$939$		0			0
$940$	−580.693	−	1005.79i	−0.617759	−	1.06999i
$941$			1684.00i			1.78958i	0.446483	+	0.894792i	$0.352676\pi$
$941$			1684.00i			1.78958i	−0.446483	+	0.894792i	$0.647324\pi$
$942$		0			0
$943$			157.210i			0.166712i
$944$		−	406.988i		−	0.431131i
$945$		0			0
$946$	1982.45			2.09561
$947$	888.217			0.937927			0.468964	−	0.883218i	$-0.344628\pi$
$947$	888.217			0.937927			0.468964	+	0.883218i	$0.344628\pi$
$948$		0			0
$949$	−306.179			−0.322634
$950$	−41.1484	+	23.7570i	−0.0433141	+	0.0250074i
$951$		0			0
$952$	−233.036	+	159.484i	−0.244786	+	0.167525i
$953$	455.922			0.478407			0.239204	−	0.970969i	$-0.423114\pi$
$953$	455.922			0.478407			0.239204	+	0.970969i	$0.423114\pi$
$954$		0			0
$955$	287.846	−	166.188i	0.301410	−	0.174019i
$956$	34.4587	−	59.6843i	0.0360447	−	0.0624313i
$957$		0			0
$958$	−183.257	−	105.804i	−0.191291	−	0.110442i
$959$	304.858	−	635.907i	0.317891	−	0.663094i
$960$		0			0
$961$	689.808			0.717803
$962$	−27.6157	+	15.9439i	−0.0287065	+	0.0165737i
$963$		0			0
$964$	212.881	+	122.907i	0.220831	+	0.127497i
$965$	351.629	−	203.013i	0.364382	−	0.210376i
$966$		0			0
$967$	−761.975	+	1319.78i	−0.787979	+	1.36482i	0.139225	+	0.990261i	$0.455539\pi$
$967$	−761.975	+	1319.78i	−0.787979	+	1.36482i	−0.927203	+	0.374558i	$0.877794\pi$
$968$	259.862	+	450.094i	0.268452	+	0.464973i
$969$		0			0
$970$	−458.469	+	794.091i	−0.472648	+	0.818651i
$971$	294.389	+	169.966i	0.303182	+	0.175042i	0.643871	−	0.765134i	$-0.277327\pi$
$971$	294.389	+	169.966i	0.303182	+	0.175042i	−0.340690	+	0.940176i	$0.610661\pi$
$972$		0			0
$973$	−1110.86	−	532.555i	−1.14169	−	0.547333i
$974$	65.4978	+	113.445i	0.0672462	+	0.116474i
$975$		0			0
$976$			268.476i			0.275078i
$977$	452.332			0.462981			0.231490	−	0.972837i	$-0.425640\pi$
$977$	452.332			0.462981			0.231490	+	0.972837i	$0.425640\pi$
$978$		0			0
$979$	762.957	−	440.493i	0.779323	−	0.449942i
$980$	530.406	−	426.051i	0.541231	−	0.434746i
$981$		0			0
$982$	−231.964	+	401.774i	−0.236216	+	0.409138i
$983$	−770.670	−	444.947i	−0.783998	−	0.452642i	0.0538472	−	0.998549i	$-0.482852\pi$
$983$	−770.670	−	444.947i	−0.783998	−	0.452642i	−0.837845	+	0.545908i	$0.816185\pi$
$984$		0			0
$985$	1226.35	−	708.033i	1.24502	−	0.718815i
$986$	104.447	+	60.3026i	0.105930	+	0.0611588i
$987$		0			0
$988$	−22.3070	−	38.6369i	−0.0225779	−	0.0391062i
$989$	−516.389	+	894.412i	−0.522133	+	0.904360i
$990$		0			0
$991$	−152.141	−	263.517i	−0.153523	−	0.265910i	0.778997	−	0.627027i	$-0.215729\pi$
$991$	−152.141	−	263.517i	−0.153523	−	0.265910i	−0.932520	+	0.361118i	$0.882395\pi$
$992$		−	93.1565i		−	0.0939078i
$993$		0			0
$994$	926.795	−	71.0961i	0.932389	−	0.0715252i
$995$	−585.697	+	1014.46i	−0.588640	+	1.01955i
$996$		0			0
$997$	1582.27	+	913.526i	1.58704	+	0.916275i	0.993792	+	0.111252i	$0.0354860\pi$
$997$	1582.27	+	913.526i	1.58704	+	0.916275i	0.593243	+	0.805023i	$0.297847\pi$
$998$	118.619	+	205.454i	0.118857	+	0.205866i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.p.a.145.7		32
3.2	odd	2		126.3.p.a.103.9	yes	32
7.3	odd	6		378.3.j.a.199.10		32
9.2	odd	6		126.3.j.a.61.3	yes	32
9.7	even	3		378.3.j.a.19.15		32
21.17	even	6		126.3.j.a.31.3	✓	32
63.38	even	6		126.3.p.a.115.9	yes	32
63.52	odd	6	inner	378.3.p.a.73.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.3.j.a.31.3	✓	32	21.17	even	6
126.3.j.a.61.3	yes	32	9.2	odd	6
126.3.p.a.103.9	yes	32	3.2	odd	2
126.3.p.a.115.9	yes	32	63.38	even	6
378.3.j.a.19.15		32	9.7	even	3
378.3.j.a.199.10		32	7.3	odd	6
378.3.p.a.73.7		32	63.52	odd	6	inner
378.3.p.a.145.7		32	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:	\( N \)	\(=\)	\( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	378.p (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(6.01208 - 3.47108i) q^{5} +(-6.97949 + 0.535409i) q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(6.01208 - 3.47108i) q^{5} +(-6.97949 + 0.535409i) q^{7} -2.82843 q^{8} +(-8.50237 + 4.90885i) q^{10} +(8.72855 - 15.1183i) q^{11} +(-13.3362 - 7.69965i) q^{13} +(9.87050 - 0.757183i) q^{14} +4.00000 q^{16} +(-12.3517 + 7.13128i) q^{17} +(1.25450 + 0.724286i) q^{19} +(12.0242 - 6.94216i) q^{20} +(-12.3440 + 21.3805i) q^{22} +(-6.43077 - 11.1384i) q^{23} +(11.5968 - 20.0862i) q^{25} +(18.8602 + 10.8890i) q^{26} +(-13.9590 + 1.07082i) q^{28} +(2.98967 + 5.17826i) q^{29} +16.4679i q^{31} -5.65685 q^{32} +(17.4680 - 10.0852i) q^{34} +(-40.1029 + 27.4453i) q^{35} +(-0.732115 + 1.26806i) q^{37} +(-1.77413 - 1.02430i) q^{38} +(-17.0047 + 9.81769i) q^{40} +(-10.5856 - 6.11162i) q^{41} +(-40.1499 - 69.5416i) q^{43} +(17.4571 - 30.2366i) q^{44} +(9.09448 + 15.7521i) q^{46} -83.6474i q^{47} +(48.4267 - 7.47377i) q^{49} +(-16.4003 + 28.4062i) q^{50} +(-26.6724 - 15.3993i) q^{52} +(-6.51752 - 11.2887i) q^{53} -121.190i q^{55} +(19.7410 - 1.51437i) q^{56} +(-4.22803 - 7.32317i) q^{58} -101.747i q^{59} +67.1191i q^{61} -23.2891i q^{62} +8.00000 q^{64} -106.904 q^{65} -39.4835 q^{67} +(-24.7035 + 14.2626i) q^{68} +(56.7140 - 38.8135i) q^{70} +93.8955 q^{71} +(17.2189 - 9.94134i) q^{73} +(1.03537 - 1.79331i) q^{74} +(2.50900 + 1.44857i) q^{76} +(-52.8264 + 110.191i) q^{77} -122.786 q^{79} +(24.0483 - 13.8843i) q^{80} +(14.9703 + 8.64313i) q^{82} +(-7.01741 + 4.05150i) q^{83} +(-49.5065 + 85.7477i) q^{85} +(56.7805 + 98.3467i) q^{86} +(-24.6881 + 42.7610i) q^{88} +(43.7047 + 25.2329i) q^{89} +(97.2023 + 46.5994i) q^{91} +(-12.8615 - 22.2768i) q^{92} +118.295i q^{94} +10.0562 q^{95} +(80.8837 - 46.6982i) q^{97} +(-68.4857 + 10.5695i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q + 64 q^{4} - 2 q^{7}+O(q^{10}) \) 32 * q + 64 * q^4 - 2 * q^7	\( 32 q + 64 q^{4} - 2 q^{7} + 12 q^{11} + 30 q^{13} + 12 q^{14} + 128 q^{16} - 54 q^{17} + 42 q^{23} + 80 q^{25} + 72 q^{26} - 4 q^{28} + 84 q^{29} + 66 q^{35} - 22 q^{37} + 396 q^{41} - 16 q^{43} + 24 q^{44} + 12 q^{46} + 50 q^{49} + 96 q^{50} + 60 q^{52} + 252 q^{53} + 24 q^{56} - 24 q^{58} + 256 q^{64} - 12 q^{65} - 140 q^{67} - 108 q^{68} + 72 q^{70} - 300 q^{71} + 72 q^{74} - 570 q^{77} - 212 q^{79} - 756 q^{83} - 60 q^{85} + 120 q^{86} - 414 q^{89} - 186 q^{91} + 84 q^{92} - 1104 q^{95} - 114 q^{97} + 96 q^{98}+O(q^{100}) \) 32 * q + 64 * q^4 - 2 * q^7 + 12 * q^11 + 30 * q^13 + 12 * q^14 + 128 * q^16 - 54 * q^17 + 42 * q^23 + 80 * q^25 + 72 * q^26 - 4 * q^28 + 84 * q^29 + 66 * q^35 - 22 * q^37 + 396 * q^41 - 16 * q^43 + 24 * q^44 + 12 * q^46 + 50 * q^49 + 96 * q^50 + 60 * q^52 + 252 * q^53 + 24 * q^56 - 24 * q^58 + 256 * q^64 - 12 * q^65 - 140 * q^67 - 108 * q^68 + 72 * q^70 - 300 * q^71 + 72 * q^74 - 570 * q^77 - 212 * q^79 - 756 * q^83 - 60 * q^85 + 120 * q^86 - 414 * q^89 - 186 * q^91 + 84 * q^92 - 1104 * q^95 - 114 * q^97 + 96 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more