LMFDB - Embedded newform 378.3.s.d.107.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,3,Mod(53,378)]

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([3, 4]))

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

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

chi := DirichletCharacter("378.53");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	378.s (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.2997539928$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.621801639936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 34x^{6} + 116x^{5} + 413x^{4} - 1024x^{3} - 1664x^{2} + 2196x + 4467 $ x^8 - 4x^7 - 34x^6 + 116x^5 + 413x^4 - 1024x^3 - 1664x^2 + 2196*x + 4467
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.3
Root			$-3.76613 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	378.107
Dual form			378.3.s.d.53.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.11254 - 1.79703i) q^{5} +(-6.04138 - 3.53578i) q^{7} +2.82843i q^{8} +O(q^{10})$	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.11254 - 1.79703i) q^{5} +(-6.04138 - 3.53578i) q^{7} +2.82843i q^{8} +(-2.54138 - 4.40180i) q^{10} +(13.5736 - 7.83670i) q^{11} +13.0828 q^{13} +(-4.89898 - 8.60233i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(22.9112 - 13.2278i) q^{17} +(6.50000 - 11.2583i) q^{19} -7.18811i q^{20} +22.1655 q^{22} +(17.8095 + 10.2823i) q^{23} +(-6.04138 - 10.4640i) q^{25} +(16.0230 + 9.25091i) q^{26} +(0.0827625 - 13.9998i) q^{28} -9.48500i q^{29} +(-15.6655 - 27.1335i) q^{31} +(-4.89898 + 2.82843i) q^{32} +37.4138 q^{34} +(12.4502 + 21.8618i) q^{35} +(-26.8724 + 46.5444i) q^{37} +(15.9217 - 9.19239i) q^{38} +(5.08276 - 8.80360i) q^{40} -19.2674i q^{41} -76.7449 q^{43} +(27.1471 + 15.6734i) q^{44} +(14.5414 + 25.1864i) q^{46} +(-20.6644 - 11.9306i) q^{47} +(23.9966 + 42.7219i) q^{49} -17.0876i q^{50} +(13.0828 + 22.6600i) q^{52} +(39.3397 - 22.7128i) q^{53} -56.3311 q^{55} +(10.0007 - 17.0876i) q^{56} +(6.70691 - 11.6167i) q^{58} +(71.2379 - 41.1292i) q^{59} +(-2.28967 + 3.96582i) q^{61} -44.3088i q^{62} -8.00000 q^{64} +(-40.7207 - 23.5101i) q^{65} +(43.7897 + 75.8459i) q^{67} +(45.8224 + 26.4556i) q^{68} +(-0.210331 + 35.5787i) q^{70} +100.526i q^{71} +(-29.9586 - 51.8898i) q^{73} +(-65.8237 + 38.0034i) q^{74} +26.0000 q^{76} +(-109.712 - 0.648585i) q^{77} +(36.8345 - 63.7992i) q^{79} +(12.4502 - 7.18811i) q^{80} +(13.6241 - 23.5977i) q^{82} -90.0415i q^{83} -95.0828 q^{85} +(-93.9929 - 54.2668i) q^{86} +(22.1655 + 38.3918i) q^{88} +(109.454 + 63.1934i) q^{89} +(-79.0380 - 46.2577i) q^{91} +41.1292i q^{92} +(-16.8724 - 29.2239i) q^{94} +(-40.4631 + 23.3614i) q^{95} -0.0827625 q^{97} +(-0.819293 + 69.2916i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} - 24 q^{7}+O(q^{10}) $ 8 * q + 8 * q^4 - 24 * q^7	$ 8 q + 8 q^{4} - 24 q^{7} + 4 q^{10} + 56 q^{13} - 16 q^{16} + 52 q^{19} + 80 q^{22} - 24 q^{25} - 48 q^{28} - 28 q^{31} + 56 q^{34} + 4 q^{37} - 8 q^{40} - 176 q^{43} + 92 q^{46} - 100 q^{49} + 56 q^{52} - 256 q^{55} - 68 q^{58} + 152 q^{61} - 64 q^{64} + 180 q^{67} - 172 q^{70} - 264 q^{73} + 208 q^{76} + 392 q^{79} + 36 q^{82} - 712 q^{85} + 80 q^{88} - 316 q^{91} + 84 q^{94} + 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^4 - 24 * q^7 + 4 * q^10 + 56 * q^13 - 16 * q^16 + 52 * q^19 + 80 * q^22 - 24 * q^25 - 48 * q^28 - 28 * q^31 + 56 * q^34 + 4 * q^37 - 8 * q^40 - 176 * q^43 + 92 * q^46 - 100 * q^49 + 56 * q^52 - 256 * q^55 - 68 * q^58 + 152 * q^61 - 64 * q^64 + 180 * q^67 - 172 * q^70 - 264 * q^73 + 208 * q^76 + 392 * q^79 + 36 * q^82 - 712 * q^85 + 80 * q^88 - 316 * q^91 + 84 * q^94 + 48 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	1.22474	+	0.707107i	0.612372	+	0.353553i
$2$	1.22474	+	0.707107i	0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	1.00000	+	1.73205i	0.250000	+	0.433013i
$5$	−3.11254	−	1.79703i	−0.622509	−	0.359406i	0.155336	−	0.987862i	$-0.450354\pi$
$5$	−3.11254	−	1.79703i	−0.622509	−	0.359406i	−0.777845	+	0.628456i	$0.783687\pi$
$6$		0			0
$7$	−6.04138	−	3.53578i	−0.863054	−	0.505111i
$7$	−6.04138	−	3.53578i	−0.863054	−	0.505111i
$8$			2.82843i			0.353553i
$9$		0			0
$10$	−2.54138	−	4.40180i	−0.254138	−	0.440180i
$11$	13.5736	−	7.83670i	1.23396	−	0.712427i	0.266107	−	0.963944i	$-0.414263\pi$
$11$	13.5736	−	7.83670i	1.23396	−	0.712427i	0.967853	+	0.251517i	$0.0809293\pi$
$12$		0			0
$13$	13.0828			1.00637			0.503183	−	0.864180i	$-0.332162\pi$
$13$	13.0828			1.00637			0.503183	+	0.864180i	$0.332162\pi$
$14$	−4.89898	−	8.60233i	−0.349927	−	0.614452i
$15$		0			0
$16$	−2.00000	+	3.46410i	−0.125000	+	0.216506i
$17$	22.9112	−	13.2278i	1.34772	−	0.778105i	0.359791	−	0.933033i	$-0.382848\pi$
$17$	22.9112	−	13.2278i	1.34772	−	0.778105i	0.987926	+	0.154928i	$0.0495146\pi$
$18$		0			0
$19$	6.50000	−	11.2583i	0.342105	−	0.592544i	−0.642718	−	0.766103i	$-0.722193\pi$
$19$	6.50000	−	11.2583i	0.342105	−	0.592544i	0.984823	+	0.173559i	$0.0555267\pi$
$20$		−	7.18811i		−	0.359406i
$21$		0			0
$22$	22.1655			1.00752
$23$	17.8095	+	10.2823i	0.774325	+	0.447057i	0.834415	−	0.551136i	$-0.185805\pi$
$23$	17.8095	+	10.2823i	0.774325	+	0.447057i	−0.0600901	+	0.998193i	$0.519139\pi$
$24$		0			0
$25$	−6.04138	−	10.4640i	−0.241655	−	0.418559i
$26$	16.0230	+	9.25091i	0.616271	+	0.355804i
$27$		0			0
$28$	0.0827625	−	13.9998i	0.00295580	−	0.499991i
$29$		−	9.48500i		−	0.327069i	−0.986538	−	0.163534i	$-0.947711\pi$
$29$		−	9.48500i		−	0.327069i	0.986538	−	0.163534i	$-0.0522895\pi$
$30$		0			0
$31$	−15.6655	−	27.1335i	−0.505340	−	0.875274i	−0.999981	−	0.00617656i	$-0.998034\pi$
$31$	−15.6655	−	27.1335i	−0.505340	−	0.875274i	0.494641	−	0.869097i	$-0.335299\pi$
$32$	−4.89898	+	2.82843i	−0.153093	+	0.0883883i
$33$		0			0
$34$	37.4138			1.10041
$35$	12.4502	+	21.8618i	0.355719	+	0.624623i
$36$		0			0
$37$	−26.8724	+	46.5444i	−0.726282	+	1.25796i	0.232162	+	0.972677i	$0.425420\pi$
$37$	−26.8724	+	46.5444i	−0.726282	+	1.25796i	−0.958444	+	0.285280i	$0.907913\pi$
$38$	15.9217	−	9.19239i	0.418992	−	0.241905i
$39$		0			0
$40$	5.08276	−	8.80360i	0.127069	−	0.220090i
$41$		−	19.2674i		−	0.469938i	−0.972003	−	0.234969i	$-0.924501\pi$
$41$		−	19.2674i		−	0.469938i	0.972003	−	0.234969i	$-0.0754988\pi$
$42$		0			0
$43$	−76.7449			−1.78476			−0.892382	−	0.451281i	$-0.850967\pi$
$43$	−76.7449			−1.78476			−0.892382	+	0.451281i	$0.850967\pi$
$44$	27.1471	+	15.6734i	0.616980	+	0.356213i
$45$		0			0
$46$	14.5414	+	25.1864i	0.316117	+	0.547531i
$47$	−20.6644	−	11.9306i	−0.439669	−	0.253843i	0.263788	−	0.964581i	$-0.415028\pi$
$47$	−20.6644	−	11.9306i	−0.439669	−	0.253843i	−0.703457	+	0.710738i	$0.748361\pi$
$48$		0			0
$49$	23.9966	+	42.7219i	0.489726	+	0.871876i
$50$		−	17.0876i		−	0.341752i
$51$		0			0
$52$	13.0828	+	22.6600i	0.251592	+	0.435769i
$53$	39.3397	−	22.7128i	0.742258	−	0.428543i	−0.0806316	−	0.996744i	$-0.525694\pi$
$53$	39.3397	−	22.7128i	0.742258	−	0.428543i	0.822890	+	0.568201i	$0.192360\pi$
$54$		0			0
$55$	−56.3311			−1.02420
$56$	10.0007	−	17.0876i	0.178584	−	0.305136i
$57$		0			0
$58$	6.70691	−	11.6167i	0.115636	−	0.200288i
$59$	71.2379	−	41.1292i	1.20742	−	0.697106i	0.245227	−	0.969466i	$-0.421137\pi$
$59$	71.2379	−	41.1292i	1.20742	−	0.697106i	0.962195	+	0.272360i	$0.0878041\pi$
$60$		0			0
$61$	−2.28967	+	3.96582i	−0.0375356	+	0.0650135i	−0.884183	−	0.467141i	$-0.845284\pi$
$61$	−2.28967	+	3.96582i	−0.0375356	+	0.0650135i	0.846647	+	0.532154i	$0.178617\pi$
$62$		−	44.3088i		−	0.714658i
$63$		0			0
$64$	−8.00000			−0.125000
$65$	−40.7207	−	23.5101i	−0.626472	−	0.361694i
$66$		0			0
$67$	43.7897	+	75.8459i	0.653577	+	1.13203i	0.982248	+	0.187584i	$0.0600658\pi$
$67$	43.7897	+	75.8459i	0.653577	+	1.13203i	−0.328671	+	0.944444i	$0.606601\pi$
$68$	45.8224	+	26.4556i	0.673858	+	0.389052i
$69$		0			0
$70$	−0.210331	+	35.5787i	−0.00300473	+	0.508267i
$71$			100.526i			1.41586i	0.706282	+	0.707931i	$0.250371\pi$
$71$			100.526i			1.41586i	−0.706282	+	0.707931i	$0.749629\pi$
$72$		0			0
$73$	−29.9586	−	51.8898i	−0.410392	−	0.710820i	0.584541	−	0.811365i	$-0.301275\pi$
$73$	−29.9586	−	51.8898i	−0.410392	−	0.710820i	−0.994933	+	0.100545i	$0.967941\pi$
$74$	−65.8237	+	38.0034i	−0.889510	+	0.513559i
$75$		0			0
$76$	26.0000			0.342105
$77$	−109.712	−	0.648585i	−1.42483	−	0.00842318i
$78$		0			0
$79$	36.8345	−	63.7992i	0.466259	−	0.807585i	−0.532998	−	0.846116i	$-0.678935\pi$
$79$	36.8345	−	63.7992i	0.466259	−	0.807585i	0.999257	+	0.0385318i	$0.0122681\pi$
$80$	12.4502	−	7.18811i	0.155627	−	0.0898514i
$81$		0			0
$82$	13.6241	−	23.5977i	0.166148	−	0.287777i
$83$		−	90.0415i		−	1.08484i	−0.840108	−	0.542419i	$-0.817509\pi$
$83$		−	90.0415i		−	1.08484i	0.840108	−	0.542419i	$-0.182491\pi$
$84$		0			0
$85$	−95.0828			−1.11862
$86$	−93.9929	−	54.2668i	−1.09294	−	0.631009i
$87$		0			0
$88$	22.1655	+	38.3918i	0.251881	+	0.436271i
$89$	109.454	+	63.1934i	1.22982	+	0.710039i	0.966993	−	0.254804i	$-0.0820108\pi$
$89$	109.454	+	63.1934i	1.22982	+	0.710039i	0.262830	+	0.964842i	$0.415344\pi$
$90$		0			0
$91$	−79.0380	−	46.2577i	−0.868549	−	0.508327i
$92$			41.1292i			0.447057i
$93$		0			0
$94$	−16.8724	−	29.2239i	−0.179494	−	0.310893i
$95$	−40.4631	+	23.3614i	−0.425927	+	0.245909i
$96$		0			0
$97$	−0.0827625			−0.000853222			−0.000426611	−	1.00000i	$-0.500136\pi$
$97$	−0.0827625			−0.000853222			−0.000426611		1.00000i	$0.500136\pi$
$98$	−0.819293	+	69.2916i	−0.00836013	+	0.707057i
$99$		0			0
$100$	12.0828	−	20.9280i	0.120828	−	0.209280i
$101$	−125.625	+	72.5297i	−1.24381	+	0.718116i	−0.969868	−	0.243630i	$-0.921662\pi$
$101$	−125.625	+	72.5297i	−1.24381	+	0.718116i	−0.273945	+	0.961745i	$0.588329\pi$
$102$		0			0
$103$	3.12757	−	5.41711i	0.0303647	−	0.0525933i	−0.850444	−	0.526066i	$-0.823666\pi$
$103$	3.12757	−	5.41711i	0.0303647	−	0.0525933i	0.880808	+	0.473473i	$0.157000\pi$
$104$			37.0036i			0.355804i
$105$		0			0
$106$	64.2414			0.606051
$107$	30.6102	+	17.6728i	0.286077	+	0.165167i	0.636171	−	0.771548i	$-0.280517\pi$
$107$	30.6102	+	17.6728i	0.286077	+	0.165167i	−0.350094	+	0.936714i	$0.613850\pi$
$108$		0			0
$109$	72.0414	+	124.779i	0.660930	+	1.14476i	0.980372	+	0.197159i	$0.0631714\pi$
$109$	72.0414	+	124.779i	0.660930	+	1.14476i	−0.319442	+	0.947606i	$0.603495\pi$
$110$	−68.9912	−	39.8321i	−0.627192	−	0.362110i
$111$		0			0
$112$	24.3311	−	13.8564i	0.217242	−	0.123718i
$113$			78.3670i			0.693513i	0.937955	+	0.346756i	$0.112717\pi$
$113$			78.3670i			0.693513i	−0.937955	+	0.346756i	$0.887283\pi$
$114$		0			0
$115$	−36.9552	−	64.0083i	−0.321350	−	0.556594i
$116$	16.4285	−	9.48500i	0.141625	−	0.0817672i
$117$		0			0
$118$	116.331			0.985856
$119$	−185.186	−	1.09476i	−1.55618	−	0.00919970i
$120$		0			0
$121$	62.3276	−	107.955i	0.515104	−	0.892187i
$122$	−5.60852	+	3.23808i	−0.0459715	+	0.0265416i
$123$		0			0
$124$	31.3311	−	54.2670i	0.252670	−	0.437637i
$125$			133.278i			1.06622i
$126$		0			0
$127$	76.5793			0.602987			0.301493	−	0.953468i	$-0.402515\pi$
$127$	76.5793			0.602987			0.301493	+	0.953468i	$0.402515\pi$
$128$	−9.79796	−	5.65685i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	−33.2483	−	57.5877i	−0.255756	−	0.442982i
$131$	−169.974	−	98.1343i	−1.29751	−	0.749117i	−0.317535	−	0.948247i	$-0.602855\pi$
$131$	−169.974	−	98.1343i	−1.29751	−	0.749117i	−0.979973	+	0.199130i	$0.936188\pi$
$132$		0			0
$133$	−79.0759	+	45.0333i	−0.594556	+	0.338596i
$134$			123.856i			0.924298i
$135$		0			0
$136$	37.4138	+	64.8026i	0.275102	+	0.476490i
$137$	21.1796	−	12.2281i	0.154596	−	0.0892559i	−0.420706	−	0.907197i	$-0.638218\pi$
$137$	21.1796	−	12.2281i	0.154596	−	0.0892559i	0.575302	+	0.817941i	$0.304884\pi$
$138$		0			0
$139$	−64.5725			−0.464550			−0.232275	−	0.972650i	$-0.574617\pi$
$139$	−64.5725			−0.464550			−0.232275	+	0.972650i	$0.574617\pi$
$140$	−25.4156	+	43.4261i	−0.181540	+	0.310187i
$141$		0			0
$142$	−71.0828	+	123.119i	−0.500583	+	0.867035i
$143$	177.580	−	102.526i	1.24182	−	0.716963i
$144$		0			0
$145$	−17.0448	+	29.5225i	−0.117550	+	0.203603i
$146$		−	84.7358i		−	0.580382i
$147$		0			0
$148$	−107.490			−0.726282
$149$	−174.374	−	100.675i	−1.17030	−	0.675671i	−0.216546	−	0.976272i	$-0.569479\pi$
$149$	−174.374	−	100.675i	−1.17030	−	0.675671i	−0.953750	+	0.300602i	$0.902813\pi$
$150$		0			0
$151$	−56.8242	−	98.4224i	−0.376319	−	0.651804i	0.614204	−	0.789147i	$-0.289477\pi$
$151$	−56.8242	−	98.4224i	−0.376319	−	0.651804i	−0.990524	+	0.137343i	$0.956144\pi$
$152$	31.8434	+	18.3848i	0.209496	+	0.120952i
$153$		0			0
$154$	−133.910	−	78.3723i	−0.869548	−	0.508911i
$155$			112.606i			0.726487i
$156$		0			0
$157$	64.6173	+	111.920i	0.411575	+	0.712869i	0.995062	−	0.0992534i	$-0.0316454\pi$
$157$	64.6173	+	111.920i	0.411575	+	0.712869i	−0.583487	+	0.812122i	$0.698312\pi$
$158$	90.2257	−	52.0918i	0.571049	−	0.329695i
$159$		0			0
$160$	20.3311			0.127069
$161$	−71.2379	−	125.090i	−0.442472	−	0.776955i
$162$		0			0
$163$	48.5000	−	84.0045i	0.297546	−	0.515365i	−0.678028	−	0.735036i	$-0.737165\pi$
$163$	48.5000	−	84.0045i	0.297546	−	0.515365i	0.975574	+	0.219671i	$0.0704985\pi$
$164$	33.3722	−	19.2674i	0.203489	−	0.117484i
$165$		0			0
$166$	63.6689	−	110.278i	0.383548	−	0.664324i
$167$			159.328i			0.954061i	0.878887	+	0.477031i	$0.158287\pi$
$167$			159.328i			0.954061i	−0.878887	+	0.477031i	$0.841713\pi$
$168$		0			0
$169$	2.15868			0.0127732
$170$	−116.452	−	67.2337i	−0.685013	−	0.395492i
$171$		0			0
$172$	−76.7449	−	132.926i	−0.446191	−	0.772826i
$173$	173.251	+	100.026i	1.00145	+	0.578187i	0.908677	−	0.417500i	$-0.137094\pi$
$173$	173.251	+	100.026i	1.00145	+	0.578187i	0.0927725	+	0.995687i	$0.470427\pi$
$174$		0			0
$175$	−0.500000	+	84.5779i	−0.00285714	+	0.483302i
$176$			62.6936i			0.356213i
$177$		0			0
$178$	89.3690	+	154.792i	0.502073	+	0.869616i
$179$	−190.896	+	110.214i	−1.06646	+	0.615718i	−0.927211	−	0.374540i	$-0.877801\pi$
$179$	−190.896	+	110.214i	−1.06646	+	0.615718i	−0.139245	+	0.990258i	$0.544467\pi$
$180$		0			0
$181$	274.311			1.51553			0.757764	−	0.652529i	$-0.226292\pi$
$181$	274.311			1.51553			0.757764	+	0.652529i	$0.226292\pi$
$182$	−64.0922	−	112.542i	−0.352155	−	0.618364i
$183$		0			0
$184$	−29.0828	+	50.3728i	−0.158058	+	0.273765i
$185$	167.283	−	96.5810i	0.904234	−	0.522060i
$186$		0			0
$187$	207.324	−	359.096i	1.10869	−	1.92030i
$188$		−	47.7224i		−	0.253843i
$189$		0			0
$190$	−66.0759			−0.347768
$191$	−150.175	−	86.7035i	−0.786256	−	0.453945i	0.0523868	−	0.998627i	$-0.483317\pi$
$191$	−150.175	−	86.7035i	−0.786256	−	0.453945i	−0.838643	+	0.544682i	$0.816650\pi$
$192$		0			0
$193$	94.6173	+	163.882i	0.490245	+	0.849129i	0.999937	−	0.0112277i	$-0.00357395\pi$
$193$	94.6173	+	163.882i	0.490245	+	0.849129i	−0.509692	+	0.860357i	$0.670241\pi$
$194$	−0.101363	−	0.0585219i	−0.000522490	−	0.000301660i
$195$		0			0
$196$	−50.0000	+	84.2852i	−0.255102	+	0.430027i
$197$			180.083i			0.914127i	0.889434	+	0.457063i	$0.151099\pi$
$197$			180.083i			0.914127i	−0.889434	+	0.457063i	$0.848901\pi$
$198$		0			0
$199$	2.86901	+	4.96927i	0.0144171	+	0.0249712i	0.873144	−	0.487462i	$-0.162077\pi$
$199$	2.86901	+	4.96927i	0.0144171	+	0.0249712i	−0.858727	+	0.512434i	$0.828744\pi$
$200$	29.5966	−	17.0876i	0.147983	−	0.0854380i
$201$		0			0
$202$	−205.145			−1.01557
$203$	−33.5368	+	57.3025i	−0.165206	+	0.282278i
$204$		0			0
$205$	−34.6241	+	59.9708i	−0.168898	+	0.292540i
$206$	7.66095	−	4.42305i	0.0371891	−	0.0214711i
$207$		0			0
$208$	−26.1655	+	45.3200i	−0.125796	+	0.217885i
$209$		−	203.754i		−	0.974900i
$210$		0			0
$211$	−75.7517			−0.359013			−0.179506	−	0.983757i	$-0.557450\pi$
$211$	−75.7517			−0.359013			−0.179506	+	0.983757i	$0.557450\pi$
$212$	78.6794	+	45.4256i	0.371129	+	0.214271i
$213$		0			0
$214$	24.9932	+	43.2894i	0.116790	+	0.202287i
$215$	238.872	+	137.913i	1.11103	+	0.641454i
$216$		0			0
$217$	−1.29652	+	219.314i	−0.00597474	+	1.01066i
$218$			203.764i			0.934696i
$219$		0			0
$220$	−56.3311	−	97.5682i	−0.256050	−	0.443492i
$221$	299.742	−	173.056i	1.35630	−	0.783058i
$222$		0			0
$223$	−249.235			−1.11764			−0.558822	−	0.829288i	$-0.688746\pi$
$223$	−249.235			−1.11764			−0.558822	+	0.829288i	$0.688746\pi$
$224$	39.5973	+	0.234088i	0.176774	+	0.00104503i
$225$		0			0
$226$	−55.4138	+	95.9795i	−0.245194	+	0.424688i
$227$	248.138	−	143.262i	1.09312	−	0.631112i	0.158713	−	0.987325i	$-0.449266\pi$
$227$	248.138	−	143.262i	1.09312	−	0.631112i	0.934405	+	0.356213i	$0.115932\pi$
$228$		0			0
$229$	−109.831	+	190.233i	−0.479612	+	0.830712i	−0.999727	−	0.0233846i	$-0.992556\pi$
$229$	−109.831	+	190.233i	−0.479612	+	0.830712i	0.520115	+	0.854096i	$0.325889\pi$
$230$		−	104.525i		−	0.454457i
$231$		0			0
$232$	26.8276			0.115636
$233$	−221.320	−	127.779i	−0.949871	−	0.548408i	−0.0568299	−	0.998384i	$-0.518099\pi$
$233$	−221.320	−	127.779i	−0.949871	−	0.548408i	−0.893041	+	0.449976i	$0.851433\pi$
$234$		0			0
$235$	42.8793	+	74.2691i	0.182465	+	0.316039i
$236$	142.476	+	82.2585i	0.603711	+	0.348553i
$237$		0			0
$238$	−226.031	−	132.287i	−0.949711	−	0.555827i
$239$			162.113i			0.678296i	0.940733	+	0.339148i	$0.110139\pi$
$239$			162.113i			0.678296i	−0.940733	+	0.339148i	$0.889861\pi$
$240$		0			0
$241$	38.3724	+	66.4630i	0.159222	+	0.275780i	0.934588	−	0.355731i	$-0.115768\pi$
$241$	38.3724	+	66.4630i	0.159222	+	0.275780i	−0.775367	+	0.631511i	$0.782435\pi$
$242$	152.671	−	88.1446i	0.630871	−	0.364234i
$243$		0			0
$244$	−9.15868			−0.0375356
$245$	2.08214	−	176.096i	0.00849851	−	0.718761i
$246$		0			0
$247$	85.0380	−	147.290i	0.344283	−	0.596316i
$248$	76.7451	−	44.3088i	0.309456	−	0.178665i
$249$		0			0
$250$	−94.2414	+	163.231i	−0.376966	+	0.652924i
$251$		−	343.815i		−	1.36978i	−0.728646	−	0.684890i	$-0.759850\pi$
$251$		−	343.815i		−	1.36978i	0.728646	−	0.684890i	$-0.240150\pi$
$252$		0			0
$253$	322.317			1.27398
$254$	93.7902	+	54.1498i	0.369253	+	0.213188i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.0312500	−	0.0541266i
$257$	192.277	+	111.011i	0.748158	+	0.431949i	0.825028	−	0.565092i	$-0.191159\pi$
$257$	192.277	+	111.011i	0.748158	+	0.431949i	−0.0768701	+	0.997041i	$0.524493\pi$
$258$		0			0
$259$	326.917	−	186.178i	1.26223	−	0.718833i
$260$		−	94.0404i		−	0.361694i
$261$		0			0
$262$	−138.783	−	240.379i	−0.529705	−	0.917477i
$263$	273.718	−	158.031i	1.04075	−	0.600879i	0.120706	−	0.992688i	$-0.461484\pi$
$263$	273.718	−	158.031i	1.04075	−	0.600879i	0.920046	+	0.391810i	$0.128151\pi$
$264$		0			0
$265$	−163.262			−0.616083
$266$	−128.691	−	0.760785i	−0.483801	−	0.00286010i
$267$		0			0
$268$	−87.5793	+	151.692i	−0.326789	+	0.566014i
$269$	239.645	−	138.359i	0.890872	−	0.514345i	0.0166443	−	0.999861i	$-0.494702\pi$
$269$	239.645	−	138.359i	0.890872	−	0.514345i	0.874228	+	0.485516i	$0.161368\pi$
$270$		0			0
$271$	−224.786	+	389.341i	−0.829470	+	1.43668i	0.0689854	+	0.997618i	$0.478024\pi$
$271$	−224.786	+	389.341i	−0.829470	+	1.43668i	−0.898455	+	0.439066i	$0.855310\pi$
$272$			105.822i			0.389052i
$273$		0			0
$274$	34.5862			0.126227
$275$	−164.006	−	94.6889i	−0.596386	−	0.344323i
$276$		0			0
$277$	135.366	+	234.460i	0.488684	+	0.846426i	0.999915	−	0.0130172i	$-0.00414362\pi$
$277$	135.366	+	234.460i	0.488684	+	0.846426i	−0.511231	+	0.859443i	$0.670810\pi$
$278$	−79.0848	−	45.6596i	−0.284478	−	0.164243i
$279$		0			0
$280$	−61.8345	+	35.2144i	−0.220837	+	0.125766i
$281$			550.948i			1.96067i	0.197340	+	0.980335i	$0.436770\pi$
$281$			550.948i			1.96067i	−0.197340	+	0.980335i	$0.563230\pi$
$282$		0			0
$283$	−91.9449	−	159.253i	−0.324894	−	0.562732i	0.656597	−	0.754241i	$-0.271995\pi$
$283$	−91.9449	−	159.253i	−0.324894	−	0.562732i	−0.981491	+	0.191509i	$0.938662\pi$
$284$	−174.116	+	100.526i	−0.613086	+	0.353966i
$285$		0			0
$286$	289.986			1.01394
$287$	−68.1254	+	116.402i	−0.237371	+	0.405582i
$288$		0			0
$289$	205.448	−	355.847i	0.710894	−	1.23130i
$290$	−41.7511	+	24.1050i	−0.143969	+	0.0831207i
$291$		0			0
$292$	59.9172	−	103.780i	0.205196	−	0.355410i
$293$		−	66.7998i		−	0.227986i	−0.993482	−	0.113993i	$-0.963636\pi$
$293$		−	66.7998i		−	0.227986i	0.993482	−	0.113993i	$-0.0363641\pi$
$294$		0			0
$295$	−295.642			−1.00217
$296$	−131.647	−	76.0067i	−0.444755	−	0.256779i
$297$		0			0
$298$	−142.376	−	246.602i	−0.477771	−	0.827524i
$299$	232.997	+	134.521i	0.779255	+	0.449903i
$300$		0			0
$301$	463.645	+	271.353i	1.54035	+	0.901504i
$302$		−	160.723i		−	0.532196i
$303$		0			0
$304$	26.0000	+	45.0333i	0.0855263	+	0.148136i
$305$	14.2534	−	8.22920i	0.0467324	−	0.0269810i
$306$		0			0
$307$	−165.331			−0.538538			−0.269269	−	0.963065i	$-0.586782\pi$
$307$	−165.331			−0.538538			−0.269269	+	0.963065i	$0.586782\pi$
$308$	−108.588	−	190.675i	−0.352560	−	0.619075i
$309$		0			0
$310$	−79.6241	+	137.913i	−0.256852	+	0.444881i
$311$	316.499	−	182.731i	1.01768	−	0.587559i	0.104251	−	0.994551i	$-0.466755\pi$
$311$	316.499	−	182.731i	1.01768	−	0.587559i	0.913432	+	0.406992i	$0.133422\pi$
$312$		0			0
$313$	28.1689	−	48.7901i	0.0899966	−	0.155879i	−0.817513	−	0.575910i	$-0.804648\pi$
$313$	28.1689	−	48.7901i	0.0899966	−	0.155879i	0.907509	+	0.420032i	$0.137981\pi$
$314$			182.765i			0.582055i
$315$		0			0
$316$	147.338			0.466259
$317$	363.116	+	209.645i	1.14548	+	0.661341i	0.947781	−	0.318923i	$-0.103321\pi$
$317$	363.116	+	209.645i	1.14548	+	0.661341i	0.197695	+	0.980264i	$0.436654\pi$
$318$		0			0
$319$	−74.3311	−	128.745i	−0.233013	−	0.403590i
$320$	24.9003	+	14.3762i	0.0778136	+	0.0449257i
$321$		0			0
$322$	1.20348	−	203.576i	0.00373752	−	0.632223i
$323$		−	343.922i		−	1.06477i
$324$		0			0
$325$	−79.0380	−	136.898i	−0.243194	−	0.421224i
$326$	118.800	−	68.5894i	0.364418	−	0.210397i
$327$		0			0
$328$	54.4966			0.166148
$329$	82.6577	+	145.142i	0.251239	+	0.441162i
$330$		0			0
$331$	237.366	−	411.129i	0.717117	−	1.24208i	−0.245021	−	0.969518i	$-0.578795\pi$
$331$	237.366	−	411.129i	0.717117	−	1.24208i	0.962138	−	0.272565i	$-0.0878718\pi$
$332$	155.956	−	90.0415i	0.469748	−	0.271209i
$333$		0			0
$334$	−112.662	+	195.136i	−0.337312	+	0.584241i
$335$		−	314.765i		−	0.939597i
$336$		0			0
$337$	467.304			1.38666			0.693329	−	0.720621i	$-0.256143\pi$
$337$	467.304			1.38666			0.693329	+	0.720621i	$0.256143\pi$
$338$	2.64383	+	1.52641i	0.00782197	+	0.00451602i
$339$		0			0
$340$	−95.0828	−	164.688i	−0.279655	−	0.484377i
$341$	−425.274	−	245.532i	−1.24714	−	0.720035i
$342$		0			0
$343$	6.08276	−	342.946i	0.0177340	−	0.999843i
$344$		−	217.067i		−	0.631009i
$345$		0			0
$346$	141.459	+	245.014i	0.408840	+	0.708132i
$347$	41.9370	−	24.2124i	0.120856	−	0.0697762i	−0.438353	−	0.898803i	$-0.644438\pi$
$347$	41.9370	−	24.2124i	0.120856	−	0.0697762i	0.559209	+	0.829026i	$0.311105\pi$
$348$		0			0
$349$	−199.345			−0.571188			−0.285594	−	0.958351i	$-0.592191\pi$
$349$	−199.345			−0.571188			−0.285594	+	0.958351i	$0.592191\pi$
$350$	−60.4180	+	103.233i	−0.172623	+	0.294951i
$351$		0			0
$352$	−44.3311	+	76.7836i	−0.125940	+	0.218135i
$353$	−502.129	+	289.904i	−1.42246	+	0.821258i	−0.996509	−	0.0834878i	$-0.973394\pi$
$353$	−502.129	+	289.904i	−1.42246	+	0.821258i	−0.425952	+	0.904746i	$0.640061\pi$
$354$		0			0
$355$	180.648	−	312.892i	0.508869	−	0.881387i
$356$			252.774i			0.710039i
$357$		0			0
$358$	−311.731			−0.870757
$359$	−493.142	−	284.715i	−1.37365	−	0.793079i	−0.382267	−	0.924052i	$-0.624857\pi$
$359$	−493.142	−	284.715i	−1.37365	−	0.793079i	−0.991386	+	0.130973i	$0.958190\pi$
$360$		0			0
$361$	96.0000	+	166.277i	0.265928	+	0.460601i
$362$	335.960	+	193.967i	0.928067	+	0.535820i
$363$		0			0
$364$	1.08276	−	183.155i	0.00297462	−	0.503174i
$365$			215.346i			0.589989i
$366$		0			0
$367$	−114.110	−	197.645i	−0.310928	−	0.538542i	0.667636	−	0.744488i	$-0.267306\pi$
$367$	−114.110	−	197.645i	−0.310928	−	0.538542i	−0.978563	+	0.205946i	$0.933973\pi$
$368$	−71.2379	+	41.1292i	−0.193581	+	0.111764i
$369$		0			0
$370$	273.172			0.738304
$371$	−317.973	−	1.87977i	−0.857071	−	0.00506676i
$372$		0			0
$373$	−208.811	+	361.670i	−0.559814	+	0.969626i	0.437698	+	0.899122i	$0.355794\pi$
$373$	−208.811	+	361.670i	−0.559814	+	0.969626i	−0.997512	+	0.0705036i	$0.977539\pi$
$374$	507.839	−	293.201i	1.35786	−	0.783959i
$375$		0			0
$376$	33.7449	−	58.4478i	0.0897470	−	0.155446i
$377$		−	124.090i		−	0.329151i
$378$		0			0
$379$	−235.552			−0.621509			−0.310755	−	0.950490i	$-0.600582\pi$
$379$	−235.552			−0.621509			−0.310755	+	0.950490i	$0.600582\pi$
$380$	−80.9261	−	46.7227i	−0.212964	−	0.122955i
$381$		0			0
$382$	−122.617	−	212.379i	−0.320988	−	0.555967i
$383$	224.010	+	129.332i	0.584883	+	0.337682i	0.763072	−	0.646314i	$-0.223690\pi$
$383$	224.010	+	129.332i	0.584883	+	0.337682i	−0.178189	+	0.983996i	$0.557024\pi$
$384$		0			0
$385$	340.317	+	199.174i	0.883941	+	0.517335i
$386$			267.618i			0.693311i
$387$		0			0
$388$	−0.0827625	−	0.143349i	−0.000213305	−	0.000369456i
$389$	153.123	−	88.4055i	0.393632	−	0.227263i	−0.290101	−	0.956996i	$-0.593689\pi$
$389$	153.123	−	88.4055i	0.393632	−	0.227263i	0.683733	+	0.729733i	$0.260355\pi$
$390$		0			0
$391$	544.049			1.39143
$392$	−120.836	+	67.8726i	−0.308255	+	0.173144i
$393$		0			0
$394$	−127.338	+	220.556i	−0.323193	+	0.559786i
$395$	−229.298	+	132.385i	−0.580501	+	0.335152i
$396$		0			0
$397$	−54.7999	+	94.9163i	−0.138035	+	0.239084i	−0.926753	−	0.375672i	$-0.877412\pi$
$397$	−54.7999	+	94.9163i	−0.138035	+	0.239084i	0.788718	+	0.614756i	$0.210745\pi$
$398$			8.11478i			0.0203889i
$399$		0			0
$400$	48.3311			0.120828
$401$	−204.541	−	118.092i	−0.510077	−	0.294493i	0.222788	−	0.974867i	$-0.428484\pi$
$401$	−204.541	−	118.092i	−0.510077	−	0.294493i	−0.732865	+	0.680374i	$0.761817\pi$
$402$		0			0
$403$	−204.948	−	354.981i	−0.508557	−	0.880846i
$404$	−251.250	−	145.059i	−0.621907	−	0.359058i
$405$		0			0
$406$	−81.5930	+	46.4668i	−0.200968	+	0.114450i
$407$			842.364i			2.06969i
$408$		0			0
$409$	209.114	+	362.196i	0.511281	+	0.885564i	0.999915	+	0.0130754i	$0.00416214\pi$
$409$	209.114	+	362.196i	0.511281	+	0.885564i	−0.488634	+	0.872489i	$0.662505\pi$
$410$	−84.8115	+	48.9659i	−0.206857	+	0.119429i
$411$		0			0
$412$	12.5103			0.0303647
$413$	−575.799	−	3.40396i	−1.39419	−	0.00824203i
$414$		0			0
$415$	−161.807	+	280.258i	−0.389897	+	0.675321i
$416$	−64.0922	+	37.0036i	−0.154068	+	0.0889511i
$417$		0			0
$418$	144.076	−	249.547i	0.344679	−	0.597002i
$419$			572.537i			1.36644i	0.730214	+	0.683219i	$0.239420\pi$
$419$			572.537i			1.36644i	−0.730214	+	0.683219i	$0.760580\pi$
$420$		0			0
$421$	−237.407			−0.563912			−0.281956	−	0.959427i	$-0.590983\pi$
$421$	−237.407			−0.563912			−0.281956	+	0.959427i	$0.590983\pi$
$422$	−92.7765	−	53.5645i	−0.219850	−	0.126930i
$423$		0			0
$424$	64.2414	+	111.269i	0.151513	+	0.262428i
$425$	−276.830	−	159.828i	−0.651366	−	0.376066i
$426$		0			0
$427$	27.8550	−	15.8633i	0.0652342	−	0.0371506i
$428$			70.6913i			0.165167i
$429$		0			0
$430$	195.038	+	337.816i	0.453577	+	0.785618i
$431$	−492.297	+	284.228i	−1.14222	+	0.659461i	−0.946980	−	0.321294i	$-0.895882\pi$
$431$	−492.297	+	284.228i	−1.14222	+	0.659461i	−0.195241	+	0.980755i	$0.562549\pi$
$432$		0			0
$433$	−739.724			−1.70837			−0.854185	−	0.519969i	$-0.825943\pi$
$433$	−739.724			−1.70837			−0.854185	+	0.519969i	$0.825943\pi$
$434$	−156.666	+	267.686i	−0.360982	+	0.616789i
$435$		0			0
$436$	−144.083	+	249.559i	−0.330465	+	0.572382i
$437$	231.523	−	133.670i	0.529802	−	0.305881i
$438$		0			0
$439$	−363.521	+	629.637i	−0.828066	+	1.43425i	0.0714879	+	0.997441i	$0.477225\pi$
$439$	−363.521	+	629.637i	−0.828066	+	1.43425i	−0.899554	+	0.436810i	$0.856108\pi$
$440$		−	159.328i		−	0.362110i
$441$		0			0
$442$	489.476			1.10741
$443$	687.128	+	396.714i	1.55108	+	0.895516i	0.998054	+	0.0623521i	$0.0198602\pi$
$443$	687.128	+	396.714i	1.55108	+	0.895516i	0.553026	+	0.833164i	$0.313473\pi$
$444$		0			0
$445$	−227.121	−	393.385i	−0.510384	−	0.884010i
$446$	−305.249	−	176.235i	−0.684414	−	0.395147i
$447$		0			0
$448$	48.3311	+	28.2862i	0.107882	+	0.0631389i
$449$			81.6882i			0.181934i	0.995854	+	0.0909668i	$0.0289957\pi$
$449$			81.6882i			0.181934i	−0.995854	+	0.0909668i	$0.971004\pi$
$450$		0			0
$451$	−150.993	−	261.528i	−0.334796	−	0.579884i
$452$	−135.736	+	78.3670i	−0.300300	+	0.173378i
$453$		0			0
$454$	405.207			0.892527
$455$	162.883	+	286.013i	0.357984	+	0.628599i
$456$		0			0
$457$	−314.441	+	544.629i	−0.688056	+	1.19175i	0.284410	+	0.958703i	$0.408202\pi$
$457$	−314.441	+	544.629i	−0.688056	+	1.19175i	−0.972466	+	0.233045i	$0.925131\pi$
$458$	−269.030	+	155.325i	−0.587402	+	0.339137i
$459$		0			0
$460$	73.9104	−	128.017i	0.160675	−	0.278297i
$461$		−	518.090i		−	1.12384i	−0.827192	−	0.561919i	$-0.810063\pi$
$461$		−	518.090i		−	1.12384i	0.827192	−	0.561919i	$-0.189937\pi$
$462$		0			0
$463$	−591.062			−1.27659			−0.638296	−	0.769791i	$-0.720360\pi$
$463$	−591.062			−1.27659			−0.638296	+	0.769791i	$0.720360\pi$
$464$	32.8570	+	18.9700i	0.0708125	+	0.0408836i
$465$		0			0
$466$	−180.707	−	312.994i	−0.387783	−	0.671660i
$467$	−84.7185	−	48.9123i	−0.181410	−	0.104737i	0.406545	−	0.913631i	$-0.366733\pi$
$467$	−84.7185	−	48.9123i	−0.181410	−	0.104737i	−0.587955	+	0.808894i	$0.700067\pi$
$468$		0			0
$469$	3.62414	−	613.045i	0.00772739	−	1.30713i
$470$			121.281i			0.258045i
$471$		0			0
$472$	116.331	+	201.491i	0.246464	+	0.426888i
$473$	−1041.70	+	601.426i	−2.20233	+	1.27151i
$474$		0			0
$475$	−157.076			−0.330686
$476$	−183.290	−	321.846i	−0.385062	−	0.676147i
$477$		0			0
$478$	−114.631	+	198.547i	−0.239814	+	0.415370i
$479$	−301.967	+	174.341i	−0.630411	+	0.363968i	−0.780911	−	0.624642i	$-0.785245\pi$
$479$	−301.967	+	174.341i	−0.630411	+	0.363968i	0.150500	+	0.988610i	$0.451912\pi$
$480$		0			0
$481$	−351.566	+	608.930i	−0.730906	+	1.26597i
$482$			108.534i			0.225173i
$483$		0			0
$484$	249.311			0.515104
$485$	0.257602	+	0.148727i	0.000531138	+	0.000306653i
$486$		0			0
$487$	280.969	+	486.653i	0.576939	+	0.999287i	0.995828	+	0.0912499i	$0.0290862\pi$
$487$	280.969	+	486.653i	0.576939	+	0.999287i	−0.418889	+	0.908037i	$0.637580\pi$
$488$	−11.2170	−	6.47616i	−0.0229857	−	0.0132708i
$489$		0			0
$490$	127.069	−	214.201i	0.259325	−	0.437145i
$491$		−	784.859i		−	1.59849i	−0.601004	−	0.799246i	$-0.705232\pi$
$491$		−	784.859i		−	1.59849i	0.601004	−	0.799246i	$-0.294768\pi$
$492$		0			0
$493$	−125.465	−	217.313i	−0.254494	−	0.440796i
$494$	208.300	−	120.262i	0.421659	−	0.243445i
$495$		0			0
$496$	125.324			0.252670
$497$	355.438	−	607.317i	0.715167	−	1.22197i
$498$		0			0
$499$	383.659	−	664.516i	0.768855	−	1.33170i	−0.169329	−	0.985560i	$-0.554160\pi$
$499$	383.659	−	664.516i	0.768855	−	1.33170i	0.938184	−	0.346136i	$-0.112507\pi$
$500$	−230.843	+	133.278i	−0.461687	+	0.266555i
$501$		0			0
$502$	243.114	−	421.086i	0.484291	−	0.838816i
$503$			187.891i			0.373540i	0.982404	+	0.186770i	$0.0598019\pi$
$503$			187.891i			0.373540i	−0.982404	+	0.186770i	$0.940198\pi$
$504$		0			0
$505$	521.352			1.03238
$506$	394.757	+	227.913i	0.780151	+	0.450421i
$507$		0			0
$508$	76.5793	+	132.639i	0.150747	+	0.261101i
$509$	100.725	+	58.1535i	0.197888	+	0.114250i	0.595670	−	0.803229i	$-0.296887\pi$
$509$	100.725	+	58.1535i	0.197888	+	0.114250i	−0.397782	+	0.917480i	$0.630220\pi$
$510$		0			0
$511$	−2.47945	+	419.413i	−0.00485215	+	0.820770i
$512$		−	22.6274i		−	0.0441942i
$513$		0			0
$514$	156.993	+	271.920i	0.305434	+	0.529027i
$515$	−19.4694	+	11.2407i	−0.0378046	+	0.0218265i
$516$		0			0
$517$	−373.986			−0.723378
$518$	532.038	+	3.14525i	1.02710	+	0.00607192i
$519$		0			0
$520$	66.4966	−	115.175i	0.127878	−	0.221491i
$521$	300.679	−	173.597i	0.577119	−	0.333200i	−0.182869	−	0.983137i	$-0.558538\pi$
$521$	300.679	−	173.597i	0.577119	−	0.333200i	0.759988	+	0.649937i	$0.225205\pi$
$522$		0			0
$523$	−464.221	+	804.054i	−0.887612	+	1.53739i	−0.0449209	+	0.998991i	$0.514304\pi$
$523$	−464.221	+	804.054i	−0.887612	+	1.53739i	−0.842691	+	0.538398i	$0.819030\pi$
$524$		−	392.537i		−	0.749117i
$525$		0			0
$526$	446.979			0.849771
$527$	−717.832	−	414.440i	−1.36211	−	0.786414i
$528$		0			0
$529$	−53.0482	−	91.8822i	−0.100280	−	0.173690i
$530$	−199.954	−	115.444i	−0.377272	−	0.217818i
$531$		0			0
$532$	−157.076	−	91.9302i	−0.295255	−	0.172801i
$533$		−	252.071i		−	0.472930i
$534$		0			0
$535$	−63.5171	−	110.015i	−0.118724	−	0.205635i
$536$	−214.525	+	123.856i	−0.400233	+	0.231074i
$537$		0			0
$538$	391.338			0.727394
$539$	660.518	+	391.835i	1.22545	+	0.726966i
$540$		0			0
$541$	453.173	−	784.918i	0.837657	−	1.45087i	−0.0541909	−	0.998531i	$-0.517258\pi$
$541$	453.173	−	784.918i	0.837657	−	1.45087i	0.891848	−	0.452335i	$-0.149409\pi$
$542$	−550.612	+	317.896i	−1.01589	+	0.586524i
$543$		0			0
$544$	−74.8276	+	129.605i	−0.137551	+	0.238245i
$545$		−	517.841i		−	0.950168i
$546$		0			0
$547$	833.263			1.52333			0.761666	−	0.647970i	$-0.224382\pi$
$547$	833.263			1.52333			0.761666	+	0.647970i	$0.224382\pi$
$548$	42.3593	+	24.4561i	0.0772979	+	0.0446280i
$549$		0			0
$550$	−133.910	−	231.940i	−0.243473	−	0.421708i
$551$	−106.785	−	61.6525i	−0.193803	−	0.111892i
$552$		0			0
$553$	−448.111	+	255.197i	−0.810327	+	0.461477i
$554$			382.872i			0.691104i
$555$		0			0
$556$	−64.5725	−	111.843i	−0.116138	−	0.201156i
$557$	188.320	−	108.726i	0.338096	−	0.195200i	−0.321334	−	0.946966i	$-0.604131\pi$
$557$	188.320	−	108.726i	0.338096	−	0.195200i	0.659430	+	0.751766i	$0.270798\pi$
$558$		0			0
$559$	−1004.03			−1.79613
$560$	−100.632	−	0.594906i	−0.179700	−	0.00106233i
$561$		0			0
$562$	−389.579	+	674.771i	−0.693202	+	1.20066i
$563$	70.9803	−	40.9805i	0.126075	−	0.0727895i	−0.435636	−	0.900123i	$-0.643477\pi$
$563$	70.9803	−	40.9805i	0.126075	−	0.0727895i	0.561711	+	0.827333i	$0.310143\pi$
$564$		0			0
$565$	140.828	−	243.921i	0.249252	−	0.431718i
$566$		−	260.060i		−	0.459469i
$567$		0			0
$568$	−284.331			−0.500583
$569$	732.693	+	423.021i	1.28769	+	0.743446i	0.978241	−	0.207472i	$-0.0665235\pi$
$569$	732.693	+	423.021i	1.28769	+	0.743446i	0.309445	+	0.950917i	$0.399857\pi$
$570$		0			0
$571$	−312.759	−	541.714i	−0.547739	−	0.948711i	−0.998429	−	0.0560310i	$-0.982155\pi$
$571$	−312.759	−	541.714i	−0.547739	−	0.948711i	0.450690	−	0.892680i	$-0.351178\pi$
$572$	355.159	+	205.051i	0.620908	+	0.358481i
$573$		0			0
$574$	−165.745	+	94.3908i	−0.288754	+	0.164444i
$575$		−	248.477i		−	0.432135i
$576$		0			0
$577$	−149.203	−	258.428i	−0.258585	−	0.447882i	0.707278	−	0.706935i	$-0.249923\pi$
$577$	−149.203	−	258.428i	−0.258585	−	0.447882i	−0.965863	+	0.259053i	$0.916590\pi$
$578$	503.244	−	290.548i	0.870664	−	0.502678i
$579$		0			0
$580$	−68.1792			−0.117550
$581$	−318.367	+	543.975i	−0.547963	+	0.936274i
$582$		0			0
$583$	355.986	−	616.586i	0.610611	−	1.05761i
$584$	146.767	−	84.7358i	0.251313	−	0.145095i
$585$		0			0
$586$	47.2346	−	81.8127i	0.0806051	−	0.139612i
$587$			476.498i			0.811751i	0.913928	+	0.405875i	$0.133033\pi$
$587$			476.498i			0.811751i	−0.913928	+	0.405875i	$0.866967\pi$
$588$		0			0
$589$	−407.304			−0.691517
$590$	−362.085	−	209.050i	−0.613704	−	0.354322i
$591$		0			0
$592$	−107.490	−	186.178i	−0.181570	−	0.314489i
$593$	244.324	+	141.061i	0.412014	+	0.237876i	0.691655	−	0.722229i	$-0.256882\pi$
$593$	244.324	+	141.061i	0.412014	+	0.237876i	−0.279641	+	0.960105i	$0.590215\pi$
$594$		0			0
$595$	574.431	+	336.191i	0.965431	+	0.565027i
$596$		−	402.700i		−	0.675671i
$597$		0			0
$598$	190.241	+	329.508i	0.318129	+	0.551016i
$599$	547.550	−	316.128i	0.914107	−	0.527760i	0.0323567	−	0.999476i	$-0.489699\pi$
$599$	547.550	−	316.128i	0.914107	−	0.527760i	0.881750	+	0.471716i	$0.156365\pi$
$600$		0			0
$601$	−629.738			−1.04782			−0.523908	−	0.851775i	$-0.675527\pi$
$601$	−629.738			−1.04782			−0.523908	+	0.851775i	$0.675527\pi$
$602$	375.972	+	660.184i	0.624537	+	1.09665i
$603$		0			0
$604$	113.648	−	196.845i	0.188160	−	0.325902i
$605$	−387.995	+	224.009i	−0.641314	+	0.370263i
$606$		0			0
$607$	−68.6724	+	118.944i	−0.113134	+	0.195954i	−0.917032	−	0.398813i	$-0.869422\pi$
$607$	−68.6724	+	118.944i	−0.113134	+	0.195954i	0.803898	+	0.594767i	$0.202756\pi$
$608$			73.5391i			0.120952i
$609$		0			0
$610$	23.2757			0.0381569
$611$	−270.348	−	156.085i	−0.442468	−	0.255459i
$612$		0			0
$613$	266.717	+	461.968i	0.435101	+	0.753618i	0.997304	−	0.0733818i	$-0.0233792\pi$
$613$	266.717	+	461.968i	0.435101	+	0.753618i	−0.562202	+	0.827000i	$0.690046\pi$
$614$	−202.488	−	116.907i	−0.329786	−	0.190402i
$615$		0			0
$616$	1.83447	−	310.312i	0.00297804	−	0.503753i
$617$		−	172.407i		−	0.279428i	−0.990192	−	0.139714i	$-0.955382\pi$
$617$		−	172.407i		−	0.279428i	0.990192	−	0.139714i	$-0.0446184\pi$
$618$		0			0
$619$	361.293	+	625.778i	0.583672	+	1.01095i	0.995040	+	0.0994803i	$0.0317180\pi$
$619$	361.293	+	625.778i	0.583672	+	1.01095i	−0.411367	+	0.911470i	$0.634949\pi$
$620$	−195.039	+	112.606i	−0.314578	+	0.181622i
$621$		0			0
$622$	516.841			0.830935
$623$	−437.817	−	768.781i	−0.702756	−	1.23400i
$624$		0			0
$625$	88.4689	−	153.233i	0.141550	−	0.245172i
$626$	68.9996	−	39.8369i	0.110223	−	0.0636372i
$627$		0			0
$628$	−129.235	+	223.841i	−0.205788	+	0.356435i
$629$			1421.85i			2.26049i
$630$		0			0
$631$	−204.772			−0.324520			−0.162260	−	0.986748i	$-0.551878\pi$
$631$	−204.772			−0.324520			−0.162260	+	0.986748i	$0.551878\pi$
$632$	180.451	+	104.184i	0.285524	+	0.164848i
$633$		0			0
$634$	296.483	+	513.523i	0.467639	+	0.809974i
$635$	−238.357	−	137.615i	−0.375365	−	0.216717i
$636$		0			0
$637$	313.941	+	558.921i	0.492844	+	0.877427i
$638$		−	210.240i		−	0.329530i
$639$		0			0
$640$	20.3311	+	35.2144i	0.0317673	+	0.0550225i
$641$	−44.1121	+	25.4682i	−0.0688177	+	0.0397319i	−0.534014	−	0.845476i	$-0.679317\pi$
$641$	−44.1121	+	25.4682i	−0.0688177	+	0.0397319i	0.465196	+	0.885208i	$0.345984\pi$
$642$		0			0
$643$	114.034			0.177347			0.0886736	−	0.996061i	$-0.471737\pi$
$643$	114.034			0.177347			0.0886736	+	0.996061i	$0.471737\pi$
$644$	145.424	−	248.477i	0.225813	−	0.385834i
$645$		0			0
$646$	243.190	−	421.217i	0.376455	−	0.652039i
$647$	810.385	−	467.876i	1.25253	−	0.723147i	0.280916	−	0.959732i	$-0.409362\pi$
$647$	810.385	−	467.876i	1.25253	−	0.723147i	0.971611	+	0.236586i	$0.0760284\pi$
$648$		0			0
$649$	644.635	−	1116.54i	0.993274	−	1.72040i
$650$		−	223.553i		−	0.343928i
$651$		0			0
$652$	194.000			0.297546
$653$	728.829	+	420.790i	1.11612	+	0.644395i	0.940409	−	0.340046i	$-0.110443\pi$
$653$	728.829	+	420.790i	1.11612	+	0.644395i	0.175716	+	0.984441i	$0.443776\pi$
$654$		0			0
$655$	352.700	+	610.894i	0.538473	+	0.932663i
$656$	66.7444	+	38.5349i	0.101745	+	0.0587422i
$657$		0			0
$658$	−1.39641	+	236.210i	−0.00212220	+	0.358982i
$659$		−	744.564i		−	1.12984i	−0.825146	−	0.564920i	$-0.808907\pi$
$659$		−	744.564i		−	1.12984i	0.825146	−	0.564920i	$-0.191093\pi$
$660$		0			0
$661$	−591.817	−	1025.06i	−0.895336	−	1.55077i	−0.833388	−	0.552689i	$-0.813602\pi$
$661$	−591.817	−	1025.06i	−0.895336	−	1.55077i	−0.0619486	−	0.998079i	$-0.519732\pi$
$662$	581.425	−	335.686i	0.878285	−	0.507078i
$663$		0			0
$664$	254.676			0.383548
$665$	327.053	+	1.93345i	0.491810	+	0.00290744i
$666$		0			0
$667$	97.5277	−	168.923i	0.146218	−	0.253258i
$668$	−275.965	+	159.328i	−0.413121	+	0.238515i
$669$		0			0
$670$	222.572	−	385.507i	0.332198	−	0.575383i
$671$			71.7738i			0.106965i
$672$		0			0
$673$	−247.724			−0.368090			−0.184045	−	0.982918i	$-0.558919\pi$
$673$	−247.724			−0.368090			−0.184045	+	0.982918i	$0.558919\pi$
$674$	572.328	+	330.434i	0.849151	+	0.490258i
$675$		0			0
$676$	2.15868	+	3.73894i	0.00319331	+	0.00553097i
$677$	294.218	+	169.867i	0.434590	+	0.250911i	0.701300	−	0.712866i	$-0.252603\pi$
$677$	294.218	+	169.867i	0.434590	+	0.250911i	−0.266710	+	0.963777i	$0.585937\pi$
$678$		0			0
$679$	0.500000	+	0.292630i	0.000736377	+	0.000430972i
$680$		−	268.935i		−	0.395492i
$681$		0			0
$682$	−347.235	−	601.428i	−0.509142	−	0.881859i
$683$	527.494	−	304.549i	0.772319	−	0.445899i	−0.0613823	−	0.998114i	$-0.519551\pi$
$683$	527.494	−	304.549i	0.772319	−	0.445899i	0.833701	+	0.552216i	$0.186218\pi$
$684$		0			0
$685$	−87.8967			−0.128316
$686$	249.949	−	415.720i	0.364358	−	0.606006i
$687$		0			0
$688$	153.490	−	265.852i	0.223096	−	0.386413i
$689$	514.672	−	297.146i	0.746984	−	0.431271i
$690$		0			0
$691$	384.842	−	666.565i	0.556934	−	0.964639i	−0.440816	−	0.897598i	$-0.645311\pi$
$691$	384.842	−	666.565i	0.556934	−	0.964639i	0.997750	−	0.0670410i	$-0.0213558\pi$
$692$			400.105i			0.578187i
$693$		0			0
$694$	68.4829			0.0986785
$695$	200.985	+	116.039i	0.289187	+	0.166962i
$696$		0			0
$697$	−254.866	−	441.440i	−0.365661	−	0.633343i
$698$	−244.146	−	140.958i	−0.349780	−	0.201946i
$699$		0			0
$700$	−146.993	+	83.7118i	−0.209990	+	0.119588i
$701$		−	1309.78i		−	1.86845i	−0.356687	−	0.934224i	$-0.616094\pi$
$701$		−	1309.78i		−	1.86845i	0.356687	−	0.934224i	$-0.383906\pi$
$702$		0			0
$703$	349.342	+	605.077i	0.496930	+	0.860708i
$704$	−108.588	+	62.6936i	−0.154245	+	0.0890534i
$705$		0			0
$706$	−819.973			−1.16143
$707$	1015.40	+	6.00274i	1.43621	+	0.00849044i
$708$		0			0
$709$	−195.628	+	338.837i	−0.275920	+	0.477908i	−0.970367	−	0.241636i	$-0.922316\pi$
$709$	−195.628	+	338.837i	−0.275920	+	0.477908i	0.694447	+	0.719544i	$0.255649\pi$
$710$	442.496	−	255.475i	0.623234	−	0.359825i
$711$		0			0
$712$	−178.738	+	309.583i	−0.251037	+	0.434808i
$713$		−	644.311i		−	0.903662i
$714$		0			0
$715$	−736.966			−1.03072
$716$	−381.791	−	220.427i	−0.533228	−	0.307859i
$717$		0			0
$718$	−402.648	−	697.407i	−0.560792	−	0.971320i
$719$	165.315	+	95.4449i	0.229924	+	0.132747i	0.610537	−	0.791988i	$-0.290954\pi$
$719$	165.315	+	95.4449i	0.229924	+	0.132747i	−0.380613	+	0.924734i	$0.624287\pi$
$720$		0			0
$721$	−38.0485	+	21.6684i	−0.0527719	+	0.0300533i
$722$			271.529i			0.376079i
$723$		0			0
$724$	274.311	+	475.120i	0.378882	+	0.656243i
$725$	−99.2508	+	57.3025i	−0.136898	+	0.0790379i
$726$		0			0
$727$	−125.131			−0.172119			−0.0860596	−	0.996290i	$-0.527428\pi$
$727$	−125.131			−0.172119			−0.0860596	+	0.996290i	$0.527428\pi$
$728$	130.837	−	223.553i	0.179721	−	0.307078i
$729$		0			0
$730$	−152.273	+	263.744i	−0.208593	+	0.361293i
$731$	−1758.32	+	1015.16i	−2.40536	+	1.38873i
$732$		0			0
$733$	207.214	−	358.905i	0.282693	−	0.489638i	−0.689354	−	0.724424i	$-0.742106\pi$
$733$	207.214	−	358.905i	0.282693	−	0.489638i	0.972047	+	0.234786i	$0.0754390\pi$
$734$		−	322.753i		−	0.439718i
$735$		0			0
$736$	−116.331			−0.158058
$737$	1188.76	+	686.333i	1.61298	+	0.931252i
$738$		0			0
$739$	−211.638	−	366.568i	−0.286384	−	0.496032i	0.686560	−	0.727074i	$-0.259120\pi$
$739$	−211.638	−	366.568i	−0.286384	−	0.496032i	−0.972944	+	0.231041i	$0.925787\pi$
$740$	334.566	+	193.162i	0.452117	+	0.261030i
$741$		0			0
$742$	−388.107	−	227.143i	−0.523055	−	0.306123i
$743$		−	454.884i		−	0.612226i	−0.951995	−	0.306113i	$-0.900971\pi$
$743$		−	454.884i		−	0.612226i	0.951995	−	0.306113i	$-0.0990286\pi$
$744$		0			0
$745$	361.831	+	626.710i	0.485680	+	0.841222i
$746$	−511.479	+	295.303i	−0.685629	+	0.395848i
$747$		0			0
$748$	829.297			1.10869
$749$	−122.441	−	214.999i	−0.163473	−	0.287048i
$750$		0			0
$751$	−125.766	+	217.833i	−0.167464	+	0.290057i	−0.937528	−	0.347911i	$-0.886891\pi$
$751$	−125.766	+	217.833i	−0.167464	+	0.290057i	0.770063	+	0.637967i	$0.220225\pi$
$752$	82.6577	−	47.7224i	0.109917	−	0.0634607i
$753$		0			0
$754$	87.7449	−	151.979i	0.116372	−	0.201563i
$755$			408.459i			0.541005i
$756$		0			0
$757$	485.572			0.641443			0.320722	−	0.947174i	$-0.396075\pi$
$757$	485.572			0.641443			0.320722	+	0.947174i	$0.396075\pi$
$758$	−288.491	−	166.560i	−0.380595	−	0.219737i
$759$		0			0
$760$	−66.0759	−	114.447i	−0.0869420	−	0.150588i
$761$	−748.041	−	431.882i	−0.982971	−	0.567519i	−0.0798052	−	0.996810i	$-0.525430\pi$
$761$	−748.041	−	431.882i	−0.982971	−	0.567519i	−0.903166	+	0.429292i	$0.858763\pi$
$762$		0			0
$763$	5.96233	−	1008.56i	0.00781432	−	1.32184i
$764$		−	346.814i		−	0.453945i
$765$		0			0
$766$	182.904	+	316.798i	0.238777	+	0.413575i
$767$	931.989	−	538.084i	1.21511	−	0.701544i
$768$		0			0
$769$	696.910			0.906255			0.453128	−	0.891446i	$-0.350308\pi$
$769$	696.910			0.906255			0.453128	+	0.891446i	$0.350308\pi$
$770$	275.965	+	484.578i	0.358396	+	0.629322i
$771$		0			0
$772$	−189.235	+	327.764i	−0.245123	+	0.424565i
$773$	−916.941	+	529.396i	−1.18621	+	0.684859i	−0.957443	−	0.288621i	$-0.906803\pi$
$773$	−916.941	+	529.396i	−1.18621	+	0.684859i	−0.228768	+	0.973481i	$0.573470\pi$
$774$		0			0
$775$	−189.283	+	327.847i	−0.244236	+	0.423029i
$776$		−	0.234088i		−	0.000301660i
$777$		0			0
$778$	250.049			0.321399
$779$	−216.919	−	125.238i	−0.278459	−	0.160768i
$780$		0			0
$781$	787.793	+	1364.50i	1.00870	+	1.74712i
$782$	666.321	+	384.700i	0.852072	+	0.491944i
$783$		0			0
$784$	−195.986	−	2.31731i	−0.249983	−	0.00295575i
$785$		−	464.476i		−	0.591690i
$786$		0			0
$787$	240.845	+	417.155i	0.306029	+	0.530058i	0.977490	−	0.210982i	$-0.0676662\pi$
$787$	240.845	+	417.155i	0.306029	+	0.530058i	−0.671461	+	0.741040i	$0.734333\pi$
$788$	−311.913	+	180.083i	−0.395829	+	0.228532i
$789$		0			0
$790$	−374.442			−0.473977
$791$	277.088	−	473.445i	0.350301	−	0.598539i
$792$		0			0
$793$	−29.9552	+	51.8839i	−0.0377745	+	0.0654274i
$794$	−134.232	+	77.4988i	−0.169058	+	0.0976056i
$795$		0			0
$796$	−5.73801	+	9.93853i	−0.00720856	+	0.0124856i
$797$			257.367i			0.322920i	0.986879	+	0.161460i	$0.0516203\pi$
$797$			257.367i			0.322920i	−0.986879	+	0.161460i	$0.948380\pi$
$798$		0			0
$799$	−631.262			−0.790065
$800$	59.1932	+	34.1752i	0.0739915	+	0.0427190i
$801$		0			0
$802$	−167.007	−	289.264i	−0.208238	−	0.360679i
$803$	−813.290	−	469.553i	−1.01281	−	0.584749i
$804$		0			0
$805$	−3.05851	+	517.364i	−0.00379939	+	0.642688i
$806$		−	579.681i		−	0.719208i
$807$		0			0
$808$	−205.145	−	355.322i	−0.253892	−	0.439754i
$809$	955.107	−	551.431i	1.18060	−	0.681621i	0.224449	−	0.974486i	$-0.427942\pi$
$809$	955.107	−	551.431i	1.18060	−	0.681621i	0.956154	+	0.292865i	$0.0946086\pi$
$810$		0			0
$811$	874.325			1.07808			0.539041	−	0.842279i	$-0.318787\pi$
$811$	874.325			1.07808			0.539041	+	0.842279i	$0.318787\pi$
$812$	−132.788	−	0.785002i	−0.163532	−	0.000966752i
$813$		0			0
$814$	−595.642	+	1031.68i	−0.731746	+	1.26742i
$815$	−301.917	+	174.312i	−0.370450	+	0.213879i
$816$		0			0
$817$	−498.842	+	864.019i	−0.610577	+	1.05755i
$818$			591.463i			0.723060i
$819$		0			0
$820$	−138.497			−0.168898
$821$	−343.267	−	198.185i	−0.418108	−	0.241395i	0.276159	−	0.961112i	$-0.410938\pi$
$821$	−343.267	−	198.185i	−0.418108	−	0.241395i	−0.694268	+	0.719717i	$0.744272\pi$
$822$		0			0
$823$	540.000	+	935.308i	0.656136	+	1.13646i	0.981608	+	0.190910i	$0.0611438\pi$
$823$	540.000	+	935.308i	0.656136	+	1.13646i	−0.325471	+	0.945552i	$0.605523\pi$
$824$	15.3219	+	8.84610i	0.0185945	+	0.0107356i
$825$		0			0
$826$	−702.800	−	411.321i	−0.850848	−	0.497967i
$827$			994.446i			1.20247i	0.799071	+	0.601237i	$0.205325\pi$
$827$			994.446i			1.20247i	−0.799071	+	0.601237i	$0.794675\pi$
$828$		0			0
$829$	−712.273	−	1233.69i	−0.859195	−	1.48817i	−0.872698	−	0.488261i	$-0.837631\pi$
$829$	−712.273	−	1233.69i	−0.859195	−	1.48817i	0.0135025	−	0.999909i	$-0.495702\pi$
$830$	−396.345	+	228.830i	−0.477524	+	0.275699i
$831$		0			0
$832$	−104.662			−0.125796
$833$	1114.91	+	661.389i	1.33842	+	0.793984i
$834$		0			0
$835$	286.317	−	495.916i	0.342895	−	0.593912i
$836$	352.912	−	203.754i	0.422144	−	0.243725i
$837$		0			0
$838$	−404.845	+	701.212i	−0.483109	+	0.836769i
$839$		−	21.8372i		−	0.0260276i	−0.999915	−	0.0130138i	$-0.995857\pi$
$839$		−	21.8372i		−	0.0260276i	0.999915	−	0.0130138i	$-0.00414254\pi$
$840$		0			0
$841$	751.035			0.893026
$842$	−290.763	−	167.872i	−0.345324	−	0.199373i
$843$		0			0
$844$	−75.7517	−	131.206i	−0.0897532	−	0.155457i
$845$	−6.71897	−	3.87920i	−0.00795145	−	0.00459077i
$846$		0			0
$847$	−758.248	+	431.818i	−0.895216	+	0.509821i
$848$			181.702i			0.214271i
$849$		0			0
$850$	−226.031	−	391.497i	−0.265919	−	0.460585i
$851$	−957.168	+	552.621i	−1.12476	+	0.649379i
$852$		0			0
$853$	142.235			0.166746			0.0833731	−	0.996518i	$-0.473431\pi$
$853$	142.235			0.166746			0.0833731	+	0.996518i	$0.473431\pi$
$854$	45.3323	+	0.267992i	0.0530824	+	0.000313808i
$855$		0			0
$856$	−49.9863	+	86.5788i	−0.0583952	+	0.101143i
$857$	−1259.35	+	727.086i	−1.46949	+	0.848409i	−0.999414	−	0.0342192i	$-0.989106\pi$
$857$	−1259.35	+	727.086i	−1.46949	+	0.848409i	−0.470072	+	0.882628i	$0.655772\pi$
$858$		0			0
$859$	−185.003	+	320.435i	−0.215371	+	0.373033i	−0.953387	−	0.301750i	$-0.902429\pi$
$859$	−185.003	+	320.435i	−0.215371	+	0.373033i	0.738016	+	0.674783i	$0.235763\pi$
$860$			551.651i			0.641454i
$861$		0			0
$862$	−803.918			−0.932619
$863$	986.397	+	569.497i	1.14299	+	0.659904i	0.947168	−	0.320737i	$-0.103930\pi$
$863$	986.397	+	569.497i	1.14299	+	0.659904i	0.195818	+	0.980640i	$0.437264\pi$
$864$		0			0
$865$	−359.500	−	622.673i	−0.415607	−	0.719853i
$866$	−905.974	−	523.064i	−1.04616	−	0.604000i
$867$		0			0
$868$	−381.159	+	217.068i	−0.439123	+	0.250078i
$869$		−	1154.64i		−	1.32870i
$870$		0			0
$871$	572.890	+	992.274i	0.657738	+	1.13924i
$872$	−352.929	+	203.764i	−0.404735	+	0.233674i
$873$		0			0
$874$	378.076			0.432581
$875$	471.239	−	805.180i	0.538559	−	0.920206i
$876$		0			0
$877$	132.652	−	229.760i	0.151256	−	0.261984i	−0.780433	−	0.625239i	$-0.785001\pi$
$877$	132.652	−	229.760i	0.151256	−	0.261984i	0.931690	+	0.363255i	$0.118335\pi$
$878$	−890.441	+	514.096i	−1.01417	+	0.585531i
$879$		0			0
$880$	112.662	−	195.136i	0.128025	−	0.221746i
$881$			431.799i			0.490124i	0.969507	+	0.245062i	$0.0788083\pi$
$881$			431.799i			0.490124i	−0.969507	+	0.245062i	$0.921192\pi$
$882$		0			0
$883$	−608.034			−0.688601			−0.344300	−	0.938860i	$-0.611884\pi$
$883$	−608.034			−0.688601			−0.344300	+	0.938860i	$0.611884\pi$
$884$	599.483	+	346.112i	0.678148	+	0.391529i
$885$		0			0
$886$	561.038	+	971.746i	0.633226	+	1.09678i
$887$	−786.894	−	454.314i	−0.887141	−	0.512191i	−0.0141350	−	0.999900i	$-0.504499\pi$
$887$	−786.894	−	454.314i	−0.887141	−	0.512191i	−0.873006	+	0.487709i	$0.837833\pi$
$888$		0			0
$889$	−462.645	−	270.767i	−0.520411	−	0.304575i
$890$		−	642.394i		−	0.721791i
$891$		0			0
$892$	−249.235	−	431.687i	−0.279411	−	0.483954i
$893$	−268.638	+	155.098i	−0.300826	+	0.173682i
$894$		0			0
$895$	792.228			0.885171
$896$	39.1918	+	68.8186i	0.0437409	+	0.0768065i
$897$		0			0
$898$	−57.7623	+	100.047i	−0.0643232	+	0.111411i
$899$	−257.361	+	148.587i	−0.286275	+	0.165281i
$900$		0			0
$901$	600.879	−	1040.75i	0.666903	−	1.15511i
$902$		−	427.073i		−	0.473474i
$903$		0			0
$904$	−221.655			−0.245194
$905$	−853.803	−	492.944i	−0.943429	−	0.544689i
$906$		0			0
$907$	581.076	+	1006.45i	0.640657	+	1.10965i	0.985286	+	0.170912i	$0.0546713\pi$
$907$	581.076	+	1006.45i	0.640657	+	1.10965i	−0.344629	+	0.938739i	$0.611995\pi$
$908$	496.275	+	286.525i	0.546559	+	0.315556i
$909$		0			0
$910$	−2.75171	+	465.468i	−0.00302386	+	0.511503i
$911$			371.319i			0.407595i	0.979013	+	0.203798i	$0.0653285\pi$
$911$			371.319i			0.407595i	−0.979013	+	0.203798i	$0.934672\pi$
$912$		0			0
$913$	−705.628	−	1222.18i	−0.772867	−	1.33865i
$914$	−770.221	+	444.687i	−0.842693	+	0.486529i
$915$		0			0
$916$	−439.324			−0.479612
$917$	679.894	+	1193.85i	0.741433	+	1.30191i
$918$		0			0
$919$	−560.287	+	970.445i	−0.609670	+	1.05598i	0.381625	+	0.924317i	$0.375365\pi$
$919$	−560.287	+	970.445i	−0.609670	+	1.05598i	−0.991295	+	0.131662i	$0.957969\pi$
$920$	181.043	−	104.525i	0.196786	−	0.113614i
$921$		0			0
$922$	366.345	−	634.528i	0.397337	−	0.688208i
$923$			1315.16i			1.42488i
$924$		0			0
$925$	649.386			0.702039
$926$	−723.900	−	417.944i	−0.781750	−	0.451344i
$927$		0			0
$928$	26.8276	+	46.4668i	0.0289091	+	0.0500720i
$929$	308.471	+	178.096i	0.332046	+	0.191707i	0.656749	−	0.754109i	$-0.271931\pi$
$929$	308.471	+	178.096i	0.332046	+	0.191707i	−0.324703	+	0.945816i	$0.605264\pi$
$930$		0			0
$931$	636.955	+	7.53126i	0.684163	+	0.00808943i
$932$		−	511.116i		−	0.548408i
$933$		0			0
$934$	−69.1724	−	119.810i	−0.0740604	−	0.128276i
$935$	−1290.61	+	745.135i	−1.38033	+	0.796936i
$936$		0			0
$937$	466.711			0.498090			0.249045	−	0.968492i	$-0.419883\pi$
$937$	466.711			0.498090			0.249045	+	0.968492i	$0.419883\pi$
$938$	437.927	−	748.261i	0.466873	−	0.797719i
$939$		0			0
$940$	−85.7586	+	148.538i	−0.0912325	+	0.158019i
$941$	−211.932	+	122.359i	−0.225220	+	0.130031i	−0.608365	−	0.793657i	$-0.708174\pi$
$941$	−211.932	+	122.359i	−0.225220	+	0.130031i	0.383145	+	0.923688i	$0.374841\pi$
$942$		0			0
$943$	198.114	−	343.143i	0.210089	−	0.363885i
$944$			329.034i			0.348553i
$945$		0			0
$946$	−1701.09			−1.79819
$947$	1089.61	+	629.084i	1.15059	+	0.664291i	0.949030	−	0.315186i	$-0.102067\pi$
$947$	1089.61	+	629.084i	1.15059	+	0.664291i	0.201556	+	0.979477i	$0.435400\pi$
$948$		0			0
$949$	−391.941	−	678.863i	−0.413005	−	0.715345i
$950$	−192.378	−	111.069i	−0.202503	−	0.116915i
$951$		0			0
$952$	3.09646	−	523.784i	0.00325259	−	0.550194i
$953$			111.953i			0.117474i	0.998273	+	0.0587369i	$0.0187073\pi$
$953$			111.953i			0.117474i	−0.998273	+	0.0587369i	$0.981293\pi$
$954$		0			0
$955$	311.617	+	539.737i	0.326301	+	0.565170i
$956$	−280.787	+	162.113i	−0.293711	+	0.169574i
$957$		0			0
$958$	−493.110			−0.514729
$959$	−171.190	−	1.01203i	−0.178509	−	0.00105529i
$960$		0			0
$961$	−10.3174	+	17.8702i	−0.0107361	+	0.0185954i
$962$	−861.156	+	497.189i	−0.895173	+	0.516828i
$963$		0			0
$964$	−76.7449	+	132.926i	−0.0796109	+	0.137890i
$965$		−	680.120i		−	0.704787i
$966$		0			0
$967$	−288.296			−0.298135			−0.149067	−	0.988827i	$-0.547627\pi$
$967$	−288.296			−0.298135			−0.149067	+	0.988827i	$0.547627\pi$
$968$	305.342	+	176.289i	0.315436	+	0.182117i
$969$		0			0
$970$	0.210331	+	0.364304i	0.000216836	+	0.000375571i
$971$	−1036.20	−	598.249i	−1.06715	−	0.616117i	−0.139745	−	0.990187i	$-0.544628\pi$
$971$	−1036.20	−	598.249i	−1.06715	−	0.616117i	−0.927400	+	0.374071i	$0.877962\pi$
$972$		0			0
$973$	390.107	+	228.314i	0.400932	+	0.234649i
$974$			794.701i			0.815915i
$975$		0			0
$976$	−9.15868	−	15.8633i	−0.00938389	−	0.0162534i
$977$	−496.605	+	286.715i	−0.508295	+	0.293465i	−0.732133	−	0.681162i	$-0.761475\pi$
$977$	−496.605	+	286.715i	−0.508295	+	0.293465i	0.223837	+	0.974627i	$0.428142\pi$
$978$		0			0
$979$	1980.91			2.02340
$980$	307.090	−	172.490i	0.313357	−	0.176010i
$981$		0			0
$982$	554.979	−	961.253i	0.565152	−	0.978872i
$983$	167.748	−	96.8494i	0.170649	−	0.0985243i	−0.412243	−	0.911074i	$-0.635254\pi$
$983$	167.748	−	96.8494i	0.170649	−	0.0985243i	0.582892	+	0.812550i	$0.301921\pi$
$984$		0			0
$985$	323.614	−	560.516i	0.328542	−	0.569052i
$986$		−	354.870i		−	0.359909i
$987$		0			0
$988$	340.152			0.344283
$989$	−1366.79	−	789.114i	−1.38199	−	0.797891i
$990$		0			0
$991$	967.059	+	1675.00i	0.975842	+	1.69021i	0.677129	+	0.735864i	$0.263224\pi$
$991$	967.059	+	1675.00i	0.975842	+	1.69021i	0.298712	+	0.954343i	$0.403443\pi$
$992$	153.490	+	88.6176i	0.154728	+	0.0893323i
$993$		0			0
$994$	864.759	−	492.476i	0.869979	−	0.495449i
$995$		−	20.6227i		−	0.0207264i
$996$		0			0
$997$	−950.097	−	1645.62i	−0.952956	−	1.65057i	−0.738979	−	0.673729i	$-0.764692\pi$
$997$	−950.097	−	1645.62i	−0.952956	−	1.65057i	−0.213977	−	0.976839i	$-0.568642\pi$
$998$	939.768	−	542.575i	0.941651	−	0.543663i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.3.s.d.107.3	yes	8
3.2	odd	2	inner	378.3.s.d.107.2	yes	8
7.4	even	3	inner	378.3.s.d.53.2	✓	8
21.11	odd	6	inner	378.3.s.d.53.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.3.s.d.53.2	✓	8	7.4	even	3	inner
378.3.s.d.53.3	yes	8	21.11	odd	6	inner
378.3.s.d.107.2	yes	8	3.2	odd	2	inner
378.3.s.d.107.3	yes	8	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.s (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.11254 - 1.79703i) q^{5} +(-6.04138 - 3.53578i) q^{7} +2.82843i q^{8} +O(q^{10})\)	\(q+(1.22474 + 0.707107i) q^{2} +(1.00000 + 1.73205i) q^{4} +(-3.11254 - 1.79703i) q^{5} +(-6.04138 - 3.53578i) q^{7} +2.82843i q^{8} +(-2.54138 - 4.40180i) q^{10} +(13.5736 - 7.83670i) q^{11} +13.0828 q^{13} +(-4.89898 - 8.60233i) q^{14} +(-2.00000 + 3.46410i) q^{16} +(22.9112 - 13.2278i) q^{17} +(6.50000 - 11.2583i) q^{19} -7.18811i q^{20} +22.1655 q^{22} +(17.8095 + 10.2823i) q^{23} +(-6.04138 - 10.4640i) q^{25} +(16.0230 + 9.25091i) q^{26} +(0.0827625 - 13.9998i) q^{28} -9.48500i q^{29} +(-15.6655 - 27.1335i) q^{31} +(-4.89898 + 2.82843i) q^{32} +37.4138 q^{34} +(12.4502 + 21.8618i) q^{35} +(-26.8724 + 46.5444i) q^{37} +(15.9217 - 9.19239i) q^{38} +(5.08276 - 8.80360i) q^{40} -19.2674i q^{41} -76.7449 q^{43} +(27.1471 + 15.6734i) q^{44} +(14.5414 + 25.1864i) q^{46} +(-20.6644 - 11.9306i) q^{47} +(23.9966 + 42.7219i) q^{49} -17.0876i q^{50} +(13.0828 + 22.6600i) q^{52} +(39.3397 - 22.7128i) q^{53} -56.3311 q^{55} +(10.0007 - 17.0876i) q^{56} +(6.70691 - 11.6167i) q^{58} +(71.2379 - 41.1292i) q^{59} +(-2.28967 + 3.96582i) q^{61} -44.3088i q^{62} -8.00000 q^{64} +(-40.7207 - 23.5101i) q^{65} +(43.7897 + 75.8459i) q^{67} +(45.8224 + 26.4556i) q^{68} +(-0.210331 + 35.5787i) q^{70} +100.526i q^{71} +(-29.9586 - 51.8898i) q^{73} +(-65.8237 + 38.0034i) q^{74} +26.0000 q^{76} +(-109.712 - 0.648585i) q^{77} +(36.8345 - 63.7992i) q^{79} +(12.4502 - 7.18811i) q^{80} +(13.6241 - 23.5977i) q^{82} -90.0415i q^{83} -95.0828 q^{85} +(-93.9929 - 54.2668i) q^{86} +(22.1655 + 38.3918i) q^{88} +(109.454 + 63.1934i) q^{89} +(-79.0380 - 46.2577i) q^{91} +41.1292i q^{92} +(-16.8724 - 29.2239i) q^{94} +(-40.4631 + 23.3614i) q^{95} -0.0827625 q^{97} +(-0.819293 + 69.2916i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} - 24 q^{7}+O(q^{10}) \) 8 * q + 8 * q^4 - 24 * q^7	\( 8 q + 8 q^{4} - 24 q^{7} + 4 q^{10} + 56 q^{13} - 16 q^{16} + 52 q^{19} + 80 q^{22} - 24 q^{25} - 48 q^{28} - 28 q^{31} + 56 q^{34} + 4 q^{37} - 8 q^{40} - 176 q^{43} + 92 q^{46} - 100 q^{49} + 56 q^{52} - 256 q^{55} - 68 q^{58} + 152 q^{61} - 64 q^{64} + 180 q^{67} - 172 q^{70} - 264 q^{73} + 208 q^{76} + 392 q^{79} + 36 q^{82} - 712 q^{85} + 80 q^{88} - 316 q^{91} + 84 q^{94} + 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^4 - 24 * q^7 + 4 * q^10 + 56 * q^13 - 16 * q^16 + 52 * q^19 + 80 * q^22 - 24 * q^25 - 48 * q^28 - 28 * q^31 + 56 * q^34 + 4 * q^37 - 8 * q^40 - 176 * q^43 + 92 * q^46 - 100 * q^49 + 56 * q^52 - 256 * q^55 - 68 * q^58 + 152 * q^61 - 64 * q^64 + 180 * q^67 - 172 * q^70 - 264 * q^73 + 208 * q^76 + 392 * q^79 + 36 * q^82 - 712 * q^85 + 80 * q^88 - 316 * q^91 + 84 * q^94 + 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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