LMFDB - Embedded newform 378.2.k.d.269.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [378,2,Mod(215,378)]

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, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("378.215");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 378 = 2 \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	378.k (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:	$3.01834519640$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.4
Root			$-0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	378.269
Dual form			378.2.k.d.215.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +(2.12132 + 1.22474i) q^{10} +(3.67423 + 2.12132i) q^{11} -0.717439i q^{13} +(0.358719 + 2.62132i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.22474 - 2.12132i) q^{17} +(-4.24264 + 2.44949i) q^{19} +2.44949 q^{20} +4.24264 q^{22} +(5.19615 - 3.00000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.358719 - 0.621320i) q^{26} +(1.62132 + 2.09077i) q^{28} -1.75736i q^{29} +(-7.86396 - 4.54026i) q^{31} +(-0.866025 - 0.500000i) q^{32} -2.44949i q^{34} +(-6.42090 + 0.878680i) q^{35} +(-2.62132 - 4.54026i) q^{37} +(-2.44949 + 4.24264i) q^{38} +(2.12132 - 1.22474i) q^{40} +2.44949 q^{41} +7.00000 q^{43} +(3.67423 - 2.12132i) q^{44} +(3.00000 - 5.19615i) q^{46} +(6.42090 + 11.1213i) q^{47} +(-5.00000 - 4.89898i) q^{49} +1.00000i q^{50} +(-0.621320 - 0.358719i) q^{52} +(-12.5446 - 7.24264i) q^{53} +10.3923i q^{55} +(2.44949 + 1.00000i) q^{56} +(-0.878680 - 1.52192i) q^{58} +(-1.22474 + 2.12132i) q^{59} +(3.62132 - 2.09077i) q^{61} -9.08052 q^{62} -1.00000 q^{64} +(1.52192 - 0.878680i) q^{65} +(-6.74264 + 11.6786i) q^{67} +(-1.22474 - 2.12132i) q^{68} +(-5.12132 + 3.97141i) q^{70} -12.7279i q^{71} +(-4.75736 - 2.74666i) q^{73} +(-4.54026 - 2.62132i) q^{74} +4.89898i q^{76} +(-8.87039 + 6.87868i) q^{77} +(-0.378680 - 0.655892i) q^{79} +(1.22474 - 2.12132i) q^{80} +(2.12132 - 1.22474i) q^{82} -15.2913 q^{83} +6.00000 q^{85} +(6.06218 - 3.50000i) q^{86} +(2.12132 - 3.67423i) q^{88} +(1.52192 + 2.63604i) q^{89} +(1.75736 + 0.717439i) q^{91} -6.00000i q^{92} +(11.1213 + 6.42090i) q^{94} +(-10.3923 - 6.00000i) q^{95} +3.16693i q^{97} +(-6.77962 - 1.74264i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q + 4 * q^4 - 8 * q^7	$ 8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 4 q^{25} - 4 q^{28} - 12 q^{31} - 4 q^{37} + 56 q^{43} + 24 q^{46} - 40 q^{49} + 12 q^{52} - 24 q^{58} + 12 q^{61} - 8 q^{64} - 20 q^{67} - 24 q^{70} - 72 q^{73} - 20 q^{79} + 48 q^{85} + 48 q^{91} + 72 q^{94}+O(q^{100}) $ 8 * q + 4 * q^4 - 8 * q^7 - 4 * q^16 - 4 * q^25 - 4 * q^28 - 12 * q^31 - 4 * q^37 + 56 * q^43 + 24 * q^46 - 40 * q^49 + 12 * q^52 - 24 * q^58 + 12 * q^61 - 8 * q^64 - 20 * q^67 - 24 * q^70 - 72 * q^73 - 20 * q^79 + 48 * q^85 + 48 * q^91 + 72 * q^94

Display 100 coefficients 10 coefficients

Character values

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

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

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 $

$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$2$	0.866025	−	0.500000i	0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	−	0.866025i	0.250000	−	0.433013i
$5$	1.22474	+	2.12132i	0.547723	+	0.948683i	0.998430	+	0.0560116i	$0.0178384\pi$
$5$	1.22474	+	2.12132i	0.547723	+	0.948683i	−0.450708	+	0.892672i	$0.648828\pi$
$6$		0			0
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$7$	−1.00000	+	2.44949i	−0.377964	+	0.925820i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	2.12132	+	1.22474i	0.670820	+	0.387298i
$11$	3.67423	+	2.12132i	1.10782	+	0.639602i	0.938265	−	0.345918i	$-0.112432\pi$
$11$	3.67423	+	2.12132i	1.10782	+	0.639602i	0.169559	+	0.985520i	$0.445766\pi$
$12$		0			0
$13$		−	0.717439i		−	0.198982i	−0.995038	−	0.0994909i	$-0.968279\pi$
$13$		−	0.717439i		−	0.198982i	0.995038	−	0.0994909i	$-0.0317214\pi$
$14$	0.358719	+	2.62132i	0.0958718	+	0.700577i
$15$		0			0
$16$	−0.500000	−	0.866025i	−0.125000	−	0.216506i
$17$	1.22474	−	2.12132i	0.297044	−	0.514496i	−0.678414	−	0.734680i	$-0.737332\pi$
$17$	1.22474	−	2.12132i	0.297044	−	0.514496i	0.975458	+	0.220184i	$0.0706658\pi$
$18$		0			0
$19$	−4.24264	+	2.44949i	−0.973329	+	0.561951i	−0.900249	−	0.435375i	$-0.856616\pi$
$19$	−4.24264	+	2.44949i	−0.973329	+	0.561951i	−0.0730792	+	0.997326i	$0.523283\pi$
$20$	2.44949			0.547723
$21$		0			0
$22$	4.24264			0.904534
$23$	5.19615	−	3.00000i	1.08347	−	0.625543i	0.151642	−	0.988436i	$-0.451544\pi$
$23$	5.19615	−	3.00000i	1.08347	−	0.625543i	0.931831	+	0.362892i	$0.118211\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	−0.358719	−	0.621320i	−0.0703507	−	0.121851i
$27$		0			0
$28$	1.62132	+	2.09077i	0.306401	+	0.395118i
$29$		−	1.75736i		−	0.326333i	−0.986599	−	0.163167i	$-0.947829\pi$
$29$		−	1.75736i		−	0.326333i	0.986599	−	0.163167i	$-0.0521708\pi$
$30$		0			0
$31$	−7.86396	−	4.54026i	−1.41241	−	0.815455i	−0.416794	−	0.909001i	$-0.636846\pi$
$31$	−7.86396	−	4.54026i	−1.41241	−	0.815455i	−0.995615	+	0.0935461i	$0.970180\pi$
$32$	−0.866025	−	0.500000i	−0.153093	−	0.0883883i
$33$		0			0
$34$		−	2.44949i		−	0.420084i
$35$	−6.42090	+	0.878680i	−1.08533	+	0.148524i
$36$		0			0
$37$	−2.62132	−	4.54026i	−0.430942	−	0.746414i	0.566012	−	0.824397i	$-0.308485\pi$
$37$	−2.62132	−	4.54026i	−0.430942	−	0.746414i	−0.996955	+	0.0779826i	$0.975152\pi$
$38$	−2.44949	+	4.24264i	−0.397360	+	0.688247i
$39$		0			0
$40$	2.12132	−	1.22474i	0.335410	−	0.193649i
$41$	2.44949			0.382546			0.191273	−	0.981537i	$-0.438738\pi$
$41$	2.44949			0.382546			0.191273	+	0.981537i	$0.438738\pi$
$42$		0			0
$43$	7.00000			1.06749			0.533745	−	0.845645i	$-0.320784\pi$
$43$	7.00000			1.06749			0.533745	+	0.845645i	$0.320784\pi$
$44$	3.67423	−	2.12132i	0.553912	−	0.319801i
$45$		0			0
$46$	3.00000	−	5.19615i	0.442326	−	0.766131i
$47$	6.42090	+	11.1213i	0.936584	+	1.62221i	0.771784	+	0.635884i	$0.219364\pi$
$47$	6.42090	+	11.1213i	0.936584	+	1.62221i	0.164800	+	0.986327i	$0.447302\pi$
$48$		0			0
$49$	−5.00000	−	4.89898i	−0.714286	−	0.699854i
$50$			1.00000i			0.141421i
$51$		0			0
$52$	−0.621320	−	0.358719i	−0.0861616	−	0.0497454i
$53$	−12.5446	−	7.24264i	−1.72314	−	0.994853i	−0.912231	−	0.409675i	$-0.865642\pi$
$53$	−12.5446	−	7.24264i	−1.72314	−	0.994853i	−0.810905	−	0.585178i	$-0.801025\pi$
$54$		0			0
$55$			10.3923i			1.40130i
$56$	2.44949	+	1.00000i	0.327327	+	0.133631i
$57$		0			0
$58$	−0.878680	−	1.52192i	−0.115376	−	0.199838i
$59$	−1.22474	+	2.12132i	−0.159448	+	0.276172i	−0.934670	−	0.355517i	$-0.884305\pi$
$59$	−1.22474	+	2.12132i	−0.159448	+	0.276172i	0.775222	+	0.631689i	$0.217638\pi$
$60$		0			0
$61$	3.62132	−	2.09077i	0.463663	−	0.267696i	−0.249920	−	0.968266i	$-0.580404\pi$
$61$	3.62132	−	2.09077i	0.463663	−	0.267696i	0.713583	+	0.700571i	$0.247071\pi$
$62$	−9.08052			−1.15323
$63$		0			0
$64$	−1.00000			−0.125000
$65$	1.52192	−	0.878680i	0.188771	−	0.108987i
$66$		0			0
$67$	−6.74264	+	11.6786i	−0.823745	+	1.42677i	0.0791303	+	0.996864i	$0.474786\pi$
$67$	−6.74264	+	11.6786i	−0.823745	+	1.42677i	−0.902875	+	0.429903i	$0.858548\pi$
$68$	−1.22474	−	2.12132i	−0.148522	−	0.257248i
$69$		0			0
$70$	−5.12132	+	3.97141i	−0.612115	+	0.474674i
$71$		−	12.7279i		−	1.51053i	−0.655422	−	0.755263i	$-0.727509\pi$
$71$		−	12.7279i		−	1.51053i	0.655422	−	0.755263i	$-0.272491\pi$
$72$		0			0
$73$	−4.75736	−	2.74666i	−0.556807	−	0.321473i	0.195056	−	0.980792i	$-0.437511\pi$
$73$	−4.75736	−	2.74666i	−0.556807	−	0.321473i	−0.751863	+	0.659320i	$0.770844\pi$
$74$	−4.54026	−	2.62132i	−0.527795	−	0.304722i
$75$		0			0
$76$			4.89898i			0.561951i
$77$	−8.87039	+	6.87868i	−1.01087	+	0.783898i
$78$		0			0
$79$	−0.378680	−	0.655892i	−0.0426048	−	0.0737937i	0.843937	−	0.536443i	$-0.180232\pi$
$79$	−0.378680	−	0.655892i	−0.0426048	−	0.0737937i	−0.886541	+	0.462649i	$0.846899\pi$
$80$	1.22474	−	2.12132i	0.136931	−	0.237171i
$81$		0			0
$82$	2.12132	−	1.22474i	0.234261	−	0.135250i
$83$	−15.2913			−1.67844			−0.839218	−	0.543795i	$-0.816987\pi$
$83$	−15.2913			−1.67844			−0.839218	+	0.543795i	$0.816987\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$	6.06218	−	3.50000i	0.653701	−	0.377415i
$87$		0			0
$88$	2.12132	−	3.67423i	0.226134	−	0.391675i
$89$	1.52192	+	2.63604i	0.161323	+	0.279420i	0.935343	−	0.353741i	$-0.115091\pi$
$89$	1.52192	+	2.63604i	0.161323	+	0.279420i	−0.774020	+	0.633161i	$0.781757\pi$
$90$		0			0
$91$	1.75736	+	0.717439i	0.184221	+	0.0752080i
$92$		−	6.00000i		−	0.625543i
$93$		0			0
$94$	11.1213	+	6.42090i	1.14708	+	0.662265i
$95$	−10.3923	−	6.00000i	−1.06623	−	0.615587i
$96$		0			0
$97$			3.16693i			0.321553i	0.986991	+	0.160776i	$0.0513998\pi$
$97$			3.16693i			0.321553i	−0.986991	+	0.160776i	$0.948600\pi$
$98$	−6.77962	−	1.74264i	−0.684845	−	0.176033i
$99$		0			0
$100$	0.500000	+	0.866025i	0.0500000	+	0.0866025i
$101$	3.67423	−	6.36396i	0.365600	−	0.633238i	−0.623272	−	0.782005i	$-0.714197\pi$
$101$	3.67423	−	6.36396i	0.365600	−	0.633238i	0.988872	+	0.148767i	$0.0475305\pi$
$102$		0			0
$103$	9.62132	−	5.55487i	0.948017	−	0.547338i	0.0555525	−	0.998456i	$-0.482308\pi$
$103$	9.62132	−	5.55487i	0.948017	−	0.547338i	0.892464	+	0.451118i	$0.148975\pi$
$104$	−0.717439			−0.0703507
$105$		0			0
$106$	−14.4853			−1.40693
$107$	−2.15232	+	1.24264i	−0.208072	+	0.120131i	−0.600415	−	0.799688i	$-0.704998\pi$
$107$	−2.15232	+	1.24264i	−0.208072	+	0.120131i	0.392343	+	0.919819i	$0.371665\pi$
$108$		0			0
$109$	8.86396	−	15.3528i	0.849013	−	1.47053i	−0.0330761	−	0.999453i	$-0.510530\pi$
$109$	8.86396	−	15.3528i	0.849013	−	1.47053i	0.882090	−	0.471082i	$-0.156136\pi$
$110$	5.19615	+	9.00000i	0.495434	+	0.858116i
$111$		0			0
$112$	2.62132	−	0.358719i	0.247691	−	0.0338958i
$113$			10.2426i			0.963547i	0.876296	+	0.481773i	$0.160007\pi$
$113$			10.2426i			0.963547i	−0.876296	+	0.481773i	$0.839993\pi$
$114$		0			0
$115$	12.7279	+	7.34847i	1.18688	+	0.685248i
$116$	−1.52192	−	0.878680i	−0.141307	−	0.0815834i
$117$		0			0
$118$			2.44949i			0.225494i
$119$	3.97141	+	5.12132i	0.364058	+	0.469471i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$	2.09077	−	3.62132i	0.189289	−	0.327859i
$123$		0			0
$124$	−7.86396	+	4.54026i	−0.706205	+	0.407727i
$125$	9.79796			0.876356
$126$		0			0
$127$	−7.72792			−0.685742			−0.342871	−	0.939382i	$-0.611399\pi$
$127$	−7.72792			−0.685742			−0.342871	+	0.939382i	$0.611399\pi$
$128$	−0.866025	+	0.500000i	−0.0765466	+	0.0441942i
$129$		0			0
$130$	0.878680	−	1.52192i	0.0770653	−	0.133481i
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−1.75736	−	12.8418i	−0.152382	−	1.11352i
$134$			13.4853i			1.16495i
$135$		0			0
$136$	−2.12132	−	1.22474i	−0.181902	−	0.105021i
$137$	−5.19615	−	3.00000i	−0.443937	−	0.256307i	0.261329	−	0.965250i	$-0.415839\pi$
$137$	−5.19615	−	3.00000i	−0.443937	−	0.256307i	−0.705266	+	0.708942i	$0.749173\pi$
$138$		0			0
$139$			8.06591i			0.684141i	0.939674	+	0.342071i	$0.111128\pi$
$139$			8.06591i			0.684141i	−0.939674	+	0.342071i	$0.888872\pi$
$140$	−2.44949	+	6.00000i	−0.207020	+	0.507093i
$141$		0			0
$142$	−6.36396	−	11.0227i	−0.534052	−	0.925005i
$143$	1.52192	−	2.63604i	0.127269	−	0.220437i
$144$		0			0
$145$	3.72792	−	2.15232i	0.309587	−	0.178740i
$146$	−5.49333			−0.454631
$147$		0			0
$148$	−5.24264			−0.430942
$149$	14.0665	−	8.12132i	1.15238	−	0.665324i	0.202911	−	0.979197i	$-0.434960\pi$
$149$	14.0665	−	8.12132i	1.15238	−	0.665324i	0.949465	+	0.313873i	$0.101627\pi$
$150$		0			0
$151$	−4.37868	+	7.58410i	−0.356332	+	0.617185i	−0.987345	−	0.158587i	$-0.949306\pi$
$151$	−4.37868	+	7.58410i	−0.356332	+	0.617185i	0.631013	+	0.775772i	$0.282639\pi$
$152$	2.44949	+	4.24264i	0.198680	+	0.344124i
$153$		0			0
$154$	−4.24264	+	10.3923i	−0.341882	+	0.837436i
$155$		−	22.2426i		−	1.78657i
$156$		0			0
$157$	−9.00000	−	5.19615i	−0.718278	−	0.414698i	0.0958404	−	0.995397i	$-0.469446\pi$
$157$	−9.00000	−	5.19615i	−0.718278	−	0.414698i	−0.814119	+	0.580699i	$0.802779\pi$
$158$	−0.655892	−	0.378680i	−0.0521800	−	0.0301261i
$159$		0			0
$160$		−	2.44949i		−	0.193649i
$161$	2.15232	+	15.7279i	0.169626	+	1.23953i
$162$		0			0
$163$	4.74264	+	8.21449i	0.371472	+	0.643409i	0.989792	−	0.142518i	$-0.0455197\pi$
$163$	4.74264	+	8.21449i	0.371472	+	0.643409i	−0.618320	+	0.785926i	$0.712186\pi$
$164$	1.22474	−	2.12132i	0.0956365	−	0.165647i
$165$		0			0
$166$	−13.2426	+	7.64564i	−1.02783	+	0.593417i
$167$	0.594346			0.0459919			0.0229959	−	0.999736i	$-0.492680\pi$
$167$	0.594346			0.0459919			0.0229959	+	0.999736i	$0.492680\pi$
$168$		0			0
$169$	12.4853			0.960406
$170$	5.19615	−	3.00000i	0.398527	−	0.230089i
$171$		0			0
$172$	3.50000	−	6.06218i	0.266872	−	0.462237i
$173$	10.3923	+	18.0000i	0.790112	+	1.36851i	0.925897	+	0.377776i	$0.123311\pi$
$173$	10.3923	+	18.0000i	0.790112	+	1.36851i	−0.135785	+	0.990738i	$0.543356\pi$
$174$		0			0
$175$	−1.62132	−	2.09077i	−0.122560	−	0.158047i
$176$		−	4.24264i		−	0.319801i
$177$		0			0
$178$	2.63604	+	1.52192i	0.197579	+	0.114073i
$179$	5.82655	+	3.36396i	0.435497	+	0.251434i	0.701686	−	0.712487i	$-0.252431\pi$
$179$	5.82655	+	3.36396i	0.435497	+	0.251434i	−0.266189	+	0.963921i	$0.585764\pi$
$180$		0			0
$181$		−	9.79796i		−	0.728277i	−0.931345	−	0.364138i	$-0.881364\pi$
$181$		−	9.79796i		−	0.728277i	0.931345	−	0.364138i	$-0.118636\pi$
$182$	1.88064	−	0.257359i	0.139402	−	0.0190767i
$183$		0			0
$184$	−3.00000	−	5.19615i	−0.221163	−	0.383065i
$185$	6.42090	−	11.1213i	0.472074	−	0.817656i
$186$		0			0
$187$	9.00000	−	5.19615i	0.658145	−	0.379980i
$188$	12.8418			0.936584
$189$		0			0
$190$	−12.0000			−0.870572
$191$	18.3712	−	10.6066i	1.32929	−	0.767467i	0.344101	−	0.938933i	$-0.388184\pi$
$191$	18.3712	−	10.6066i	1.32929	−	0.767467i	0.985190	+	0.171466i	$0.0548503\pi$
$192$		0			0
$193$	0.742641	−	1.28629i	0.0534564	−	0.0925893i	−0.838059	−	0.545580i	$-0.816310\pi$
$193$	0.742641	−	1.28629i	0.0534564	−	0.0925893i	0.891515	+	0.452990i	$0.149643\pi$
$194$	1.58346	+	2.74264i	0.113686	+	0.196910i
$195$		0			0
$196$	−6.74264	+	1.88064i	−0.481617	+	0.134331i
$197$			16.9706i			1.20910i	0.796566	+	0.604551i	$0.206648\pi$
$197$			16.9706i			1.20910i	−0.796566	+	0.604551i	$0.793352\pi$
$198$		0			0
$199$	−18.1066	−	10.4539i	−1.28354	−	0.741054i	−0.306049	−	0.952016i	$-0.599007\pi$
$199$	−18.1066	−	10.4539i	−1.28354	−	0.741054i	−0.977494	+	0.210962i	$0.932340\pi$
$200$	0.866025	+	0.500000i	0.0612372	+	0.0353553i
$201$		0			0
$202$		−	7.34847i		−	0.517036i
$203$	4.30463	+	1.75736i	0.302126	+	0.123342i
$204$		0			0
$205$	3.00000	+	5.19615i	0.209529	+	0.362915i
$206$	5.55487	−	9.62132i	0.387026	−	0.670349i
$207$		0			0
$208$	−0.621320	+	0.358719i	−0.0430808	+	0.0248727i
$209$	−20.7846			−1.43770
$210$		0			0
$211$	3.48528			0.239937			0.119968	−	0.992778i	$-0.461721\pi$
$211$	3.48528			0.239937			0.119968	+	0.992778i	$0.461721\pi$
$212$	−12.5446	+	7.24264i	−0.861568	+	0.497427i
$213$		0			0
$214$	−1.24264	+	2.15232i	−0.0849452	+	0.147129i
$215$	8.57321	+	14.8492i	0.584688	+	1.01271i
$216$		0			0
$217$	18.9853	−	14.7224i	1.28880	−	0.999424i
$218$		−	17.7279i		−	1.20069i
$219$		0			0
$220$	9.00000	+	5.19615i	0.606780	+	0.350325i
$221$	−1.52192	−	0.878680i	−0.102375	−	0.0591064i
$222$		0			0
$223$		−	10.3923i		−	0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$		−	10.3923i		−	0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	2.09077	−	1.62132i	0.139695	−	0.108329i
$225$		0			0
$226$	5.12132	+	8.87039i	0.340665	+	0.590049i
$227$	−12.5446	+	21.7279i	−0.832616	+	1.44213i	0.0633412	+	0.997992i	$0.479824\pi$
$227$	−12.5446	+	21.7279i	−0.832616	+	1.44213i	−0.895957	+	0.444141i	$0.853509\pi$
$228$		0			0
$229$	−4.86396	+	2.80821i	−0.321420	+	0.185572i	−0.652025	−	0.758197i	$-0.726080\pi$
$229$	−4.86396	+	2.80821i	−0.321420	+	0.185572i	0.330606	+	0.943769i	$0.392747\pi$
$230$	14.6969			0.969087
$231$		0			0
$232$	−1.75736			−0.115376
$233$	17.7408	−	10.2426i	1.16224	−	0.671018i	0.210398	−	0.977616i	$-0.432524\pi$
$233$	17.7408	−	10.2426i	1.16224	−	0.671018i	0.951839	+	0.306598i	$0.0991908\pi$
$234$		0			0
$235$	−15.7279	+	27.2416i	−1.02598	+	1.77704i
$236$	1.22474	+	2.12132i	0.0797241	+	0.138086i
$237$		0			0
$238$	6.00000	+	2.44949i	0.388922	+	0.158777i
$239$			16.2426i			1.05065i	0.850902	+	0.525325i	$0.176056\pi$
$239$			16.2426i			1.05065i	−0.850902	+	0.525325i	$0.823944\pi$
$240$		0			0
$241$	−0.985281	−	0.568852i	−0.0634676	−	0.0366430i	0.467930	−	0.883765i	$-0.345000\pi$
$241$	−0.985281	−	0.568852i	−0.0634676	−	0.0366430i	−0.531398	+	0.847122i	$0.678333\pi$
$242$	6.06218	+	3.50000i	0.389692	+	0.224989i
$243$		0			0
$244$		−	4.18154i		−	0.267696i
$245$	4.26858	−	16.6066i	0.272710	−	1.06096i
$246$		0			0
$247$	1.75736	+	3.04384i	0.111818	+	0.193675i
$248$	−4.54026	+	7.86396i	−0.288307	+	0.499362i
$249$		0			0
$250$	8.48528	−	4.89898i	0.536656	−	0.309839i
$251$	15.2913			0.965177			0.482589	−	0.875847i	$-0.339697\pi$
$251$	15.2913			0.965177			0.482589	+	0.875847i	$0.339697\pi$
$252$		0			0
$253$	25.4558			1.60040
$254$	−6.69258	+	3.86396i	−0.419930	+	0.242446i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	−12.5446	−	21.7279i	−0.782512	−	1.35535i	−0.930474	−	0.366358i	$-0.880605\pi$
$257$	−12.5446	−	21.7279i	−0.782512	−	1.35535i	0.147962	−	0.988993i	$-0.452729\pi$
$258$		0			0
$259$	13.7426	−	1.88064i	0.853926	−	0.116857i
$260$		−	1.75736i		−	0.108987i
$261$		0			0
$262$		0			0
$263$	−5.82655	−	3.36396i	−0.359281	−	0.207431i	0.309485	−	0.950904i	$-0.399843\pi$
$263$	−5.82655	−	3.36396i	−0.359281	−	0.207431i	−0.668765	+	0.743474i	$0.733177\pi$
$264$		0			0
$265$		−	35.4815i		−	2.17961i
$266$	−7.94282	−	10.2426i	−0.487005	−	0.628017i
$267$		0			0
$268$	6.74264	+	11.6786i	0.411872	+	0.713384i
$269$	−4.89898	+	8.48528i	−0.298696	+	0.517357i	−0.975838	−	0.218496i	$-0.929885\pi$
$269$	−4.89898	+	8.48528i	−0.298696	+	0.517357i	0.677142	+	0.735853i	$0.263218\pi$
$270$		0			0
$271$	−16.3492	+	9.43924i	−0.993146	+	0.573393i	−0.906213	−	0.422821i	$-0.861040\pi$
$271$	−16.3492	+	9.43924i	−0.993146	+	0.573393i	−0.0869326	+	0.996214i	$0.527706\pi$
$272$	−2.44949			−0.148522
$273$		0			0
$274$	−6.00000			−0.362473
$275$	−3.67423	+	2.12132i	−0.221565	+	0.127920i
$276$		0			0
$277$	−11.8640	+	20.5490i	−0.712836	+	1.23467i	0.250952	+	0.968000i	$0.419256\pi$
$277$	−11.8640	+	20.5490i	−0.712836	+	1.23467i	−0.963788	+	0.266669i	$0.914077\pi$
$278$	4.03295	+	6.98528i	0.241881	+	0.418949i
$279$		0			0
$280$	0.878680	+	6.42090i	0.0525112	+	0.383722i
$281$		−	13.7574i		−	0.820695i	−0.911929	−	0.410348i	$-0.865407\pi$
$281$		−	13.7574i		−	0.820695i	0.911929	−	0.410348i	$-0.134593\pi$
$282$		0			0
$283$	5.22792	+	3.01834i	0.310768	+	0.179422i	0.647270	−	0.762261i	$-0.275911\pi$
$283$	5.22792	+	3.01834i	0.310768	+	0.179422i	−0.336502	+	0.941683i	$0.609244\pi$
$284$	−11.0227	−	6.36396i	−0.654077	−	0.377632i
$285$		0			0
$286$		−	3.04384i		−	0.179986i
$287$	−2.44949	+	6.00000i	−0.144589	+	0.354169i
$288$		0			0
$289$	5.50000	+	9.52628i	0.323529	+	0.560369i
$290$	2.15232	−	3.72792i	0.126388	−	0.218911i
$291$		0			0
$292$	−4.75736	+	2.74666i	−0.278403	+	0.160736i
$293$	12.8418			0.750226			0.375113	−	0.926979i	$-0.377604\pi$
$293$	12.8418			0.750226			0.375113	+	0.926979i	$0.377604\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$	−4.54026	+	2.62132i	−0.263897	+	0.152361i
$297$		0			0
$298$	8.12132	−	14.0665i	0.470455	−	0.814853i
$299$	−2.15232	−	3.72792i	−0.124472	−	0.215591i
$300$		0			0
$301$	−7.00000	+	17.1464i	−0.403473	+	0.988304i
$302$			8.75736i			0.503929i
$303$		0			0
$304$	4.24264	+	2.44949i	0.243332	+	0.140488i
$305$	8.87039	+	5.12132i	0.507917	+	0.293246i
$306$		0			0
$307$			26.8213i			1.53077i	0.643571	+	0.765386i	$0.277452\pi$
$307$			26.8213i			1.53077i	−0.643571	+	0.765386i	$0.722548\pi$
$308$	1.52192	+	11.1213i	0.0867193	+	0.633696i
$309$		0			0
$310$	−11.1213	−	19.2627i	−0.631649	−	1.09405i
$311$	8.57321	−	14.8492i	0.486142	−	0.842023i	−0.513731	−	0.857951i	$-0.671737\pi$
$311$	8.57321	−	14.8492i	0.486142	−	0.842023i	0.999873	+	0.0159282i	$0.00507031\pi$
$312$		0			0
$313$	−17.4853	+	10.0951i	−0.988327	+	0.570611i	−0.904774	−	0.425893i	$-0.859960\pi$
$313$	−17.4853	+	10.0951i	−0.988327	+	0.570611i	−0.0835529	+	0.996503i	$0.526627\pi$
$314$	−10.3923			−0.586472
$315$		0			0
$316$	−0.757359			−0.0426048
$317$	−19.2627	+	11.1213i	−1.08190	+	0.624636i	−0.931408	−	0.363976i	$-0.881419\pi$
$317$	−19.2627	+	11.1213i	−1.08190	+	0.624636i	−0.150492	+	0.988611i	$0.548086\pi$
$318$		0			0
$319$	3.72792	−	6.45695i	0.208724	−	0.361520i
$320$	−1.22474	−	2.12132i	−0.0684653	−	0.118585i
$321$		0			0
$322$	9.72792	+	12.5446i	0.542116	+	0.699084i
$323$			12.0000i			0.667698i
$324$		0			0
$325$	0.621320	+	0.358719i	0.0344647	+	0.0198982i
$326$	8.21449	+	4.74264i	0.454959	+	0.262671i
$327$		0			0
$328$		−	2.44949i		−	0.135250i
$329$	−33.6625	+	4.60660i	−1.85587	+	0.253970i
$330$		0			0
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	0.695266	−	0.718752i	$-0.255287\pi$
$331$	−5.00000	−	8.66025i	−0.274825	−	0.476011i	−0.970091	+	0.242742i	$0.921953\pi$
$332$	−7.64564	+	13.2426i	−0.419609	+	0.726784i
$333$		0			0
$334$	0.514719	−	0.297173i	0.0281642	−	0.0162606i
$335$	−33.0321			−1.80473
$336$		0			0
$337$	−21.4558			−1.16877			−0.584387	−	0.811475i	$-0.698665\pi$
$337$	−21.4558			−1.16877			−0.584387	+	0.811475i	$0.698665\pi$
$338$	10.8126	−	6.24264i	0.588126	−	0.339555i
$339$		0			0
$340$	3.00000	−	5.19615i	0.162698	−	0.281801i
$341$	−19.2627	−	33.3640i	−1.04313	−	1.80676i
$342$		0			0
$343$	17.0000	−	7.34847i	0.917914	−	0.396780i
$344$		−	7.00000i		−	0.377415i
$345$		0			0
$346$	18.0000	+	10.3923i	0.967686	+	0.558694i
$347$	−4.30463	−	2.48528i	−0.231085	−	0.133417i	0.379988	−	0.924992i	$-0.375928\pi$
$347$	−4.30463	−	2.48528i	−0.231085	−	0.133417i	−0.611072	+	0.791575i	$0.709262\pi$
$348$		0			0
$349$		−	0.123093i		−	0.00658902i	−0.999995	−	0.00329451i	$-0.998951\pi$
$349$		−	0.123093i		−	0.00658902i	0.999995	−	0.00329451i	$-0.00104868\pi$
$350$	−2.44949	−	1.00000i	−0.130931	−	0.0534522i
$351$		0			0
$352$	−2.12132	−	3.67423i	−0.113067	−	0.195837i
$353$	−5.49333	+	9.51472i	−0.292380	+	0.506417i	−0.974372	−	0.224942i	$-0.927781\pi$
$353$	−5.49333	+	9.51472i	−0.292380	+	0.506417i	0.681992	+	0.731360i	$0.261114\pi$
$354$		0			0
$355$	27.0000	−	15.5885i	1.43301	−	0.827349i
$356$	3.04384			0.161323
$357$		0			0
$358$	6.72792			0.355582
$359$	−12.5446	+	7.24264i	−0.662080	+	0.382252i	−0.793069	−	0.609132i	$-0.791518\pi$
$359$	−12.5446	+	7.24264i	−0.662080	+	0.382252i	0.130989	+	0.991384i	$0.458185\pi$
$360$		0			0
$361$	2.50000	−	4.33013i	0.131579	−	0.227901i
$362$	−4.89898	−	8.48528i	−0.257485	−	0.445976i
$363$		0			0
$364$	1.50000	−	1.16320i	0.0786214	−	0.0609682i
$365$		−	13.4558i		−	0.704311i
$366$		0			0
$367$	−25.9706	−	14.9941i	−1.35565	−	0.782686i	−0.366618	−	0.930372i	$-0.619484\pi$
$367$	−25.9706	−	14.9941i	−1.35565	−	0.782686i	−0.989034	+	0.147685i	$0.952818\pi$
$368$	−5.19615	−	3.00000i	−0.270868	−	0.156386i
$369$		0			0
$370$		−	12.8418i		−	0.667613i
$371$	30.2854	−	23.4853i	1.57234	−	1.21930i
$372$		0			0
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	0.996610	+	0.0822766i	$0.0262191\pi$
$373$	11.0000	+	19.0526i	0.569558	+	0.986504i	−0.427051	+	0.904227i	$0.640448\pi$
$374$	5.19615	−	9.00000i	0.268687	−	0.465379i
$375$		0			0
$376$	11.1213	−	6.42090i	0.573538	−	0.331132i
$377$	−1.26080			−0.0649344
$378$		0			0
$379$	−7.48528			−0.384493			−0.192247	−	0.981347i	$-0.561577\pi$
$379$	−7.48528			−0.384493			−0.192247	+	0.981347i	$0.561577\pi$
$380$	−10.3923	+	6.00000i	−0.533114	+	0.307794i
$381$		0			0
$382$	10.6066	−	18.3712i	0.542681	−	0.939951i
$383$	−2.74666	−	4.75736i	−0.140348	−	0.243090i	0.787280	−	0.616596i	$-0.211489\pi$
$383$	−2.74666	−	4.75736i	−0.140348	−	0.243090i	−0.927628	+	0.373506i	$0.878155\pi$
$384$		0			0
$385$	−25.4558	−	10.3923i	−1.29735	−	0.529641i
$386$		−	1.48528i		−	0.0755988i
$387$		0			0
$388$	2.74264	+	1.58346i	0.139236	+	0.0803882i
$389$	13.4361	+	7.75736i	0.681239	+	0.393314i	0.800322	−	0.599571i	$-0.204662\pi$
$389$	13.4361	+	7.75736i	0.681239	+	0.393314i	−0.119082	+	0.992884i	$0.537995\pi$
$390$		0			0
$391$		−	14.6969i		−	0.743256i
$392$	−4.89898	+	5.00000i	−0.247436	+	0.252538i
$393$		0			0
$394$	8.48528	+	14.6969i	0.427482	+	0.740421i
$395$	0.927572	−	1.60660i	0.0466712	−	0.0808369i
$396$		0			0
$397$	−13.1360	+	7.58410i	−0.659279	+	0.380635i	−0.792002	−	0.610518i	$-0.790961\pi$
$397$	−13.1360	+	7.58410i	−0.659279	+	0.380635i	0.132723	+	0.991153i	$0.457628\pi$
$398$	−20.9077			−1.04801
$399$		0			0
$400$	1.00000			0.0500000
$401$	−17.1104	+	9.87868i	−0.854451	+	0.493318i	−0.862150	−	0.506653i	$-0.830883\pi$
$401$	−17.1104	+	9.87868i	−0.854451	+	0.493318i	0.00769892	+	0.999970i	$0.497549\pi$
$402$		0			0
$403$	−3.25736	+	5.64191i	−0.162261	+	0.281044i
$404$	−3.67423	−	6.36396i	−0.182800	−	0.316619i
$405$		0			0
$406$	4.60660	−	0.630399i	0.228622	−	0.0312862i
$407$		−	22.2426i		−	1.10253i
$408$		0			0
$409$	−3.25736	−	1.88064i	−0.161066	−	0.0929915i	0.417300	−	0.908769i	$-0.362976\pi$
$409$	−3.25736	−	1.88064i	−0.161066	−	0.0929915i	−0.578366	+	0.815777i	$0.696310\pi$
$410$	5.19615	+	3.00000i	0.256620	+	0.148159i
$411$		0			0
$412$		−	11.1097i		−	0.547338i
$413$	−3.97141	−	5.12132i	−0.195420	−	0.252004i
$414$		0			0
$415$	−18.7279	−	32.4377i	−0.919318	−	1.59230i
$416$	−0.358719	+	0.621320i	−0.0175877	+	0.0304627i
$417$		0			0
$418$	−18.0000	+	10.3923i	−0.880409	+	0.508304i
$419$	7.94282			0.388032			0.194016	−	0.980998i	$-0.437849\pi$
$419$	7.94282			0.388032			0.194016	+	0.980998i	$0.437849\pi$
$420$		0			0
$421$	−23.4558			−1.14317			−0.571584	−	0.820544i	$-0.693671\pi$
$421$	−23.4558			−1.14317			−0.571584	+	0.820544i	$0.693671\pi$
$422$	3.01834	−	1.74264i	0.146931	−	0.0848304i
$423$		0			0
$424$	−7.24264	+	12.5446i	−0.351734	+	0.609221i
$425$	1.22474	+	2.12132i	0.0594089	+	0.102899i
$426$		0			0
$427$	1.50000	+	10.9612i	0.0725901	+	0.530448i
$428$			2.48528i			0.120131i
$429$		0			0
$430$	14.8492	+	8.57321i	0.716094	+	0.413437i
$431$	1.52192	+	0.878680i	0.0733082	+	0.0423245i	0.536206	−	0.844087i	$-0.319857\pi$
$431$	1.52192	+	0.878680i	0.0733082	+	0.0423245i	−0.462898	+	0.886412i	$0.653190\pi$
$432$		0			0
$433$			2.57258i			0.123630i	0.998088	+	0.0618152i	$0.0196889\pi$
$433$			2.57258i			0.123630i	−0.998088	+	0.0618152i	$0.980311\pi$
$434$	9.08052	−	22.2426i	0.435879	−	1.06768i
$435$		0			0
$436$	−8.86396	−	15.3528i	−0.424507	−	0.735267i
$437$	−14.6969	+	25.4558i	−0.703050	+	1.21772i
$438$		0			0
$439$	3.72792	−	2.15232i	0.177924	−	0.102724i	−0.408393	−	0.912806i	$-0.633911\pi$
$439$	3.72792	−	2.15232i	0.177924	−	0.102724i	0.586317	+	0.810082i	$0.300577\pi$
$440$	10.3923			0.495434
$441$		0			0
$442$	−1.75736			−0.0835891
$443$	−22.0454	+	12.7279i	−1.04741	+	0.604722i	−0.921923	−	0.387374i	$-0.873382\pi$
$443$	−22.0454	+	12.7279i	−1.04741	+	0.604722i	−0.125486	+	0.992095i	$0.540049\pi$
$444$		0			0
$445$	−3.72792	+	6.45695i	−0.176720	+	0.306089i
$446$	−5.19615	−	9.00000i	−0.246045	−	0.426162i
$447$		0			0
$448$	1.00000	−	2.44949i	0.0472456	−	0.115728i
$449$			5.27208i			0.248805i	0.992232	+	0.124402i	$0.0397014\pi$
$449$			5.27208i			0.248805i	−0.992232	+	0.124402i	$0.960299\pi$
$450$		0			0
$451$	9.00000	+	5.19615i	0.423793	+	0.244677i
$452$	8.87039	+	5.12132i	0.417228	+	0.240887i
$453$		0			0
$454$			25.0892i			1.17750i
$455$	0.630399	+	4.60660i	0.0295536	+	0.215961i
$456$		0			0
$457$	11.5000	+	19.9186i	0.537947	+	0.931752i	0.999014	+	0.0443868i	$0.0141334\pi$
$457$	11.5000	+	19.9186i	0.537947	+	0.931752i	−0.461067	+	0.887365i	$0.652533\pi$
$458$	−2.80821	+	4.86396i	−0.131219	+	0.227278i
$459$		0			0
$460$	12.7279	−	7.34847i	0.593442	−	0.342624i
$461$	21.4511			0.999076			0.499538	−	0.866292i	$-0.333503\pi$
$461$	21.4511			0.999076			0.499538	+	0.866292i	$0.333503\pi$
$462$		0			0
$463$	−22.0000			−1.02243			−0.511213	−	0.859454i	$-0.670804\pi$
$463$	−22.0000			−1.02243			−0.511213	+	0.859454i	$0.670804\pi$
$464$	−1.52192	+	0.878680i	−0.0706533	+	0.0407917i
$465$		0			0
$466$	10.2426	−	17.7408i	0.474481	−	0.821825i
$467$	−8.87039	−	15.3640i	−0.410473	−	0.710959i	0.584469	−	0.811416i	$-0.301303\pi$
$467$	−8.87039	−	15.3640i	−0.410473	−	0.710959i	−0.994941	+	0.100457i	$0.967970\pi$
$468$		0			0
$469$	−21.8640	−	28.1946i	−1.00958	−	1.30191i
$470$			31.4558i			1.45095i
$471$		0			0
$472$	2.12132	+	1.22474i	0.0976417	+	0.0563735i
$473$	25.7196	+	14.8492i	1.18259	+	0.682769i
$474$		0			0
$475$		−	4.89898i		−	0.224781i
$476$	6.42090	−	0.878680i	0.294301	−	0.0402742i
$477$		0			0
$478$	8.12132	+	14.0665i	0.371461	+	0.643389i
$479$	−1.22474	+	2.12132i	−0.0559600	+	0.0969256i	−0.892648	−	0.450754i	$-0.851155\pi$
$479$	−1.22474	+	2.12132i	−0.0559600	+	0.0969256i	0.836688	+	0.547679i	$0.184489\pi$
$480$		0			0
$481$	−3.25736	+	1.88064i	−0.148523	+	0.0857497i
$482$	−1.13770			−0.0518210
$483$		0			0
$484$	7.00000			0.318182
$485$	−6.71807	+	3.87868i	−0.305052	+	0.176122i
$486$		0			0
$487$	−11.0000	+	19.0526i	−0.498458	+	0.863354i	−0.999998	−	0.00178012i	$-0.999433\pi$
$487$	−11.0000	+	19.0526i	−0.498458	+	0.863354i	0.501541	+	0.865134i	$0.332767\pi$
$488$	−2.09077	−	3.62132i	−0.0946447	−	0.163929i
$489$		0			0
$490$	−4.60660	−	16.5160i	−0.208105	−	0.746118i
$491$			34.9706i			1.57820i	0.614265	+	0.789100i	$0.289453\pi$
$491$			34.9706i			1.57820i	−0.614265	+	0.789100i	$0.710547\pi$
$492$		0			0
$493$	−3.72792	−	2.15232i	−0.167897	−	0.0969355i
$494$	3.04384	+	1.75736i	0.136949	+	0.0790673i
$495$		0			0
$496$			9.08052i			0.407727i
$497$	31.1769	+	12.7279i	1.39848	+	0.570925i
$498$		0			0
$499$	12.2279	+	21.1794i	0.547397	+	0.948119i	0.998452	+	0.0556231i	$0.0177145\pi$
$499$	12.2279	+	21.1794i	0.547397	+	0.948119i	−0.451055	+	0.892496i	$0.648952\pi$
$500$	4.89898	−	8.48528i	0.219089	−	0.379473i
$501$		0			0
$502$	13.2426	−	7.64564i	0.591048	−	0.341242i
$503$	−0.594346			−0.0265006			−0.0132503	−	0.999912i	$-0.504218\pi$
$503$	−0.594346			−0.0265006			−0.0132503	+	0.999912i	$0.504218\pi$
$504$		0			0
$505$	18.0000			0.800989
$506$	22.0454	−	12.7279i	0.980038	−	0.565825i
$507$		0			0
$508$	−3.86396	+	6.69258i	−0.171436	+	0.296935i
$509$	4.60181	+	7.97056i	0.203971	+	0.353289i	0.949805	−	0.312844i	$-0.101282\pi$
$509$	4.60181	+	7.97056i	0.203971	+	0.353289i	−0.745833	+	0.666133i	$0.767948\pi$
$510$		0			0
$511$	11.4853	−	8.90644i	0.508079	−	0.393998i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	−21.7279	−	12.5446i	−0.958378	−	0.553320i
$515$	23.5673	+	13.6066i	1.03850	+	0.599579i
$516$		0			0
$517$			54.4831i			2.39616i
$518$	10.9612	−	8.50000i	0.481606	−	0.373469i
$519$		0			0
$520$	−0.878680	−	1.52192i	−0.0385327	−	0.0667405i
$521$	14.9941	−	25.9706i	0.656904	−	1.13779i	−0.324509	−	0.945883i	$-0.605199\pi$
$521$	14.9941	−	25.9706i	0.656904	−	1.13779i	0.981413	−	0.191908i	$-0.0614676\pi$
$522$		0			0
$523$	23.7426	−	13.7078i	1.03819	−	0.599401i	0.118872	−	0.992910i	$-0.462072\pi$
$523$	23.7426	−	13.7078i	1.03819	−	0.599401i	0.919321	+	0.393508i	$0.128739\pi$
$524$		0			0
$525$		0			0
$526$	−6.72792			−0.293351
$527$	−19.2627	+	11.1213i	−0.839096	+	0.484452i
$528$		0			0
$529$	6.50000	−	11.2583i	0.282609	−	0.489493i
$530$	−17.7408	−	30.7279i	−0.770610	−	1.33474i
$531$		0			0
$532$	−12.0000	−	4.89898i	−0.520266	−	0.212398i
$533$		−	1.75736i		−	0.0761197i
$534$		0			0
$535$	−5.27208	−	3.04384i	−0.227932	−	0.131596i
$536$	11.6786	+	6.74264i	0.504439	+	0.291238i
$537$		0			0
$538$			9.79796i			0.422420i
$539$	−7.97887	−	28.6066i	−0.343674	−	1.23217i
$540$		0			0
$541$	2.72792	+	4.72490i	0.117283	+	0.203139i	0.918690	−	0.394980i	$-0.129248\pi$
$541$	2.72792	+	4.72490i	0.117283	+	0.203139i	−0.801407	+	0.598119i	$0.795915\pi$
$542$	−9.43924	+	16.3492i	−0.405450	+	0.702260i
$543$		0			0
$544$	−2.12132	+	1.22474i	−0.0909509	+	0.0525105i
$545$	43.4244			1.86010
$546$		0			0
$547$	39.9706			1.70902			0.854509	−	0.519437i	$-0.173858\pi$
$547$	39.9706			1.70902			0.854509	+	0.519437i	$0.173858\pi$
$548$	−5.19615	+	3.00000i	−0.221969	+	0.128154i
$549$		0			0
$550$	−2.12132	+	3.67423i	−0.0904534	+	0.156670i
$551$	4.30463	+	7.45584i	0.183384	+	0.317630i
$552$		0			0
$553$	1.98528	−	0.271680i	0.0844228	−	0.0115530i
$554$			23.7279i			1.00810i
$555$		0			0
$556$	6.98528	+	4.03295i	0.296242	+	0.171035i
$557$	−18.3712	−	10.6066i	−0.778412	−	0.449416i	0.0574555	−	0.998348i	$-0.481701\pi$
$557$	−18.3712	−	10.6066i	−0.778412	−	0.449416i	−0.835867	+	0.548932i	$0.815035\pi$
$558$		0			0
$559$		−	5.02207i		−	0.212411i
$560$	3.97141	+	5.12132i	0.167823	+	0.216415i
$561$		0			0
$562$	−6.87868	−	11.9142i	−0.290160	−	0.502571i
$563$	22.9369	−	39.7279i	0.966676	−	1.67433i	0.261634	−	0.965167i	$-0.415739\pi$
$563$	22.9369	−	39.7279i	0.966676	−	1.67433i	0.705043	−	0.709165i	$-0.250928\pi$
$564$		0			0
$565$	−21.7279	+	12.5446i	−0.914101	+	0.527756i
$566$	6.03668			0.253741
$567$		0			0
$568$	−12.7279			−0.534052
$569$	−8.87039	+	5.12132i	−0.371866	+	0.214697i	−0.674273	−	0.738482i	$-0.735543\pi$
$569$	−8.87039	+	5.12132i	−0.371866	+	0.214697i	0.302407	+	0.953179i	$0.402210\pi$
$570$		0			0
$571$	11.0000	−	19.0526i	0.460336	−	0.797325i	−0.538642	−	0.842535i	$-0.681062\pi$
$571$	11.0000	−	19.0526i	0.460336	−	0.797325i	0.998978	+	0.0452101i	$0.0143957\pi$
$572$	−1.52192	−	2.63604i	−0.0636346	−	0.110218i
$573$		0			0
$574$	0.878680	+	6.42090i	0.0366754	+	0.268003i
$575$			6.00000i			0.250217i
$576$		0			0
$577$	23.7426	+	13.7078i	0.988419	+	0.570664i	0.904801	−	0.425834i	$-0.140019\pi$
$577$	23.7426	+	13.7078i	0.988419	+	0.570664i	0.0836177	+	0.996498i	$0.473353\pi$
$578$	9.52628	+	5.50000i	0.396241	+	0.228770i
$579$		0			0
$580$		−	4.30463i		−	0.178740i
$581$	15.2913	−	37.4558i	0.634389	−	1.55393i
$582$		0			0
$583$	−30.7279	−	53.2223i	−1.27262	−	2.20424i
$584$	−2.74666	+	4.75736i	−0.113658	+	0.196861i
$585$		0			0
$586$	11.1213	−	6.42090i	0.459418	−	0.265245i
$587$	32.4377			1.33885			0.669424	−	0.742881i	$-0.266541\pi$
$587$	32.4377			1.33885			0.669424	+	0.742881i	$0.266541\pi$
$588$		0			0
$589$	44.4853			1.83298
$590$	−5.19615	+	3.00000i	−0.213922	+	0.123508i
$591$		0			0
$592$	−2.62132	+	4.54026i	−0.107736	+	0.186604i
$593$	0.927572	+	1.60660i	0.0380908	+	0.0659752i	0.884442	−	0.466650i	$-0.154539\pi$
$593$	0.927572	+	1.60660i	0.0380908	+	0.0659752i	−0.846352	+	0.532625i	$0.821206\pi$
$594$		0			0
$595$	−6.00000	+	14.6969i	−0.245976	+	0.602516i
$596$		−	16.2426i		−	0.665324i
$597$		0			0
$598$	−3.72792	−	2.15232i	−0.152446	−	0.0880148i
$599$	3.04384	+	1.75736i	0.124368	+	0.0718038i	0.560893	−	0.827888i	$-0.310458\pi$
$599$	3.04384	+	1.75736i	0.124368	+	0.0718038i	−0.436526	+	0.899692i	$0.643791\pi$
$600$		0			0
$601$			41.5182i			1.69356i	0.531940	+	0.846782i	$0.321463\pi$
$601$			41.5182i			1.69356i	−0.531940	+	0.846782i	$0.678537\pi$
$602$	2.51104	+	18.3492i	0.102342	+	0.747859i
$603$		0			0
$604$	4.37868	+	7.58410i	0.178166	+	0.308592i
$605$	−8.57321	+	14.8492i	−0.348551	+	0.603708i
$606$		0			0
$607$	31.2426	−	18.0379i	1.26810	−	0.732138i	0.293471	−	0.955968i	$-0.405189\pi$
$607$	31.2426	−	18.0379i	1.26810	−	0.732138i	0.974628	+	0.223830i	$0.0718561\pi$
$608$	4.89898			0.198680
$609$		0			0
$610$	10.2426			0.414712
$611$	7.97887	−	4.60660i	0.322790	−	0.186363i
$612$		0			0
$613$	14.1066	−	24.4334i	0.569760	−	0.986854i	−0.426829	−	0.904332i	$-0.640369\pi$
$613$	14.1066	−	24.4334i	0.569760	−	0.986854i	0.996589	−	0.0825214i	$-0.0262973\pi$
$614$	13.4106	+	23.2279i	0.541210	+	0.937403i
$615$		0			0
$616$	6.87868	+	8.87039i	0.277150	+	0.357398i
$617$			4.24264i			0.170802i	0.996347	+	0.0854011i	$0.0272172\pi$
$617$			4.24264i			0.170802i	−0.996347	+	0.0854011i	$0.972783\pi$
$618$		0			0
$619$	−11.0147	−	6.35935i	−0.442719	−	0.255604i	0.262031	−	0.965059i	$-0.415608\pi$
$619$	−11.0147	−	6.35935i	−0.442719	−	0.255604i	−0.704750	+	0.709455i	$0.748941\pi$
$620$	−19.2627	−	11.1213i	−0.773608	−	0.446643i
$621$		0			0
$622$		−	17.1464i		−	0.687509i
$623$	−7.97887	+	1.09188i	−0.319667	+	0.0437454i
$624$		0			0
$625$	14.5000	+	25.1147i	0.580000	+	1.00459i
$626$	−10.0951	+	17.4853i	−0.403483	+	0.698852i
$627$		0			0
$628$	−9.00000	+	5.19615i	−0.359139	+	0.207349i
$629$	−12.8418			−0.512036
$630$		0			0
$631$	−14.7574			−0.587481			−0.293741	−	0.955885i	$-0.594900\pi$
$631$	−14.7574			−0.587481			−0.293741	+	0.955885i	$0.594900\pi$
$632$	−0.655892	+	0.378680i	−0.0260900	+	0.0150631i
$633$		0			0
$634$	−11.1213	+	19.2627i	−0.441684	+	0.765019i
$635$	−9.46473	−	16.3934i	−0.375596	−	0.650552i
$636$		0			0
$637$	−3.51472	+	3.58719i	−0.139258	+	0.142130i
$638$		−	7.45584i		−	0.295180i
$639$		0			0
$640$	−2.12132	−	1.22474i	−0.0838525	−	0.0484123i
$641$	28.7635	+	16.6066i	1.13609	+	0.655921i	0.945459	−	0.325741i	$-0.105614\pi$
$641$	28.7635	+	16.6066i	1.13609	+	0.655921i	0.190630	+	0.981662i	$0.438947\pi$
$642$		0			0
$643$		−	1.73205i		−	0.0683054i	−0.999417	−	0.0341527i	$-0.989127\pi$
$643$		−	1.73205i		−	0.0683054i	0.999417	−	0.0341527i	$-0.0108733\pi$
$644$	14.6969	+	6.00000i	0.579141	+	0.236433i
$645$		0			0
$646$	6.00000	+	10.3923i	0.236067	+	0.408880i
$647$	−10.3923	+	18.0000i	−0.408564	+	0.707653i	−0.994729	−	0.102538i	$-0.967304\pi$
$647$	−10.3923	+	18.0000i	−0.408564	+	0.707653i	0.586165	+	0.810191i	$0.300637\pi$
$648$		0			0
$649$	−9.00000	+	5.19615i	−0.353281	+	0.203967i
$650$	0.717439			0.0281403
$651$		0			0
$652$	9.48528			0.371472
$653$	−2.15232	+	1.24264i	−0.0842267	+	0.0486283i	−0.541522	−	0.840687i	$-0.682152\pi$
$653$	−2.15232	+	1.24264i	−0.0842267	+	0.0486283i	0.457295	+	0.889315i	$0.348818\pi$
$654$		0			0
$655$		0			0
$656$	−1.22474	−	2.12132i	−0.0478183	−	0.0828236i
$657$		0			0
$658$	−26.8492	+	20.8207i	−1.04669	+	0.811674i
$659$			22.2426i			0.866450i	0.901286	+	0.433225i	$0.142624\pi$
$659$			22.2426i			0.866450i	−0.901286	+	0.433225i	$0.857376\pi$
$660$		0			0
$661$	−4.24264	−	2.44949i	−0.165020	−	0.0952741i	0.415216	−	0.909723i	$-0.363706\pi$
$661$	−4.24264	−	2.44949i	−0.165020	−	0.0952741i	−0.580235	+	0.814449i	$0.697039\pi$
$662$	−8.66025	−	5.00000i	−0.336590	−	0.194331i
$663$		0			0
$664$			15.2913i			0.593417i
$665$	25.0892	−	19.4558i	0.972919	−	0.754465i
$666$		0			0
$667$	−5.27208	−	9.13151i	−0.204136	−	0.353573i
$668$	0.297173	−	0.514719i	0.0114980	−	0.0199151i
$669$		0			0
$670$	−28.6066	+	16.5160i	−1.10517	+	0.638070i
$671$	17.7408			0.684875
$672$		0			0
$673$	−45.4558			−1.75219			−0.876097	−	0.482135i	$-0.839862\pi$
$673$	−45.4558			−1.75219			−0.876097	+	0.482135i	$0.839862\pi$
$674$	−18.5813	+	10.7279i	−0.715725	+	0.413224i
$675$		0			0
$676$	6.24264	−	10.8126i	0.240102	−	0.415868i
$677$	7.34847	+	12.7279i	0.282425	+	0.489174i	0.971981	−	0.235058i	$-0.0755280\pi$
$677$	7.34847	+	12.7279i	0.282425	+	0.489174i	−0.689557	+	0.724232i	$0.742195\pi$
$678$		0			0
$679$	−7.75736	−	3.16693i	−0.297700	−	0.121536i
$680$		−	6.00000i		−	0.230089i
$681$		0			0
$682$	−33.3640	−	19.2627i	−1.27757	−	0.737607i
$683$	8.87039	+	5.12132i	0.339416	+	0.195962i	0.660014	−	0.751254i	$-0.270550\pi$
$683$	8.87039	+	5.12132i	0.339416	+	0.195962i	−0.320598	+	0.947215i	$0.603884\pi$
$684$		0			0
$685$		−	14.6969i		−	0.561541i
$686$	11.0482	−	14.8640i	0.421822	−	0.567509i
$687$		0			0
$688$	−3.50000	−	6.06218i	−0.133436	−	0.231118i
$689$	−5.19615	+	9.00000i	−0.197958	+	0.342873i
$690$		0			0
$691$	2.22792	−	1.28629i	0.0847541	−	0.0489328i	−0.457024	−	0.889454i	$-0.651085\pi$
$691$	2.22792	−	1.28629i	0.0847541	−	0.0489328i	0.541778	+	0.840522i	$0.317751\pi$
$692$	20.7846			0.790112
$693$		0			0
$694$	−4.97056			−0.188680
$695$	−17.1104	+	9.87868i	−0.649034	+	0.374720i
$696$		0			0
$697$	3.00000	−	5.19615i	0.113633	−	0.196818i
$698$	−0.0615465	−	0.106602i	−0.00232957	−	0.00403493i
$699$		0			0
$700$	−2.62132	+	0.358719i	−0.0990766	+	0.0135583i
$701$			20.4853i			0.773718i	0.922139	+	0.386859i	$0.126440\pi$
$701$			20.4853i			0.773718i	−0.922139	+	0.386859i	$0.873560\pi$
$702$		0			0
$703$	22.2426	+	12.8418i	0.838897	+	0.484337i
$704$	−3.67423	−	2.12132i	−0.138478	−	0.0799503i
$705$		0			0
$706$			10.9867i			0.413488i
$707$	11.9142	+	15.3640i	0.448080	+	0.577821i
$708$		0			0
$709$	−8.10660	−	14.0410i	−0.304450	−	0.527323i	0.672689	−	0.739925i	$-0.265139\pi$
$709$	−8.10660	−	14.0410i	−0.304450	−	0.527323i	−0.977139	+	0.212603i	$0.931806\pi$
$710$	15.5885	−	27.0000i	0.585024	−	1.01329i
$711$		0			0
$712$	2.63604	−	1.52192i	0.0987897	−	0.0570363i
$713$	−54.4831			−2.04041
$714$		0			0
$715$	7.45584			0.278833
$716$	5.82655	−	3.36396i	0.217748	−	0.125717i
$717$		0			0
$718$	−7.24264	+	12.5446i	−0.270293	+	0.468161i
$719$	−26.3140	−	45.5772i	−0.981346	−	1.69974i	−0.657166	−	0.753746i	$-0.728245\pi$
$719$	−26.3140	−	45.5772i	−0.981346	−	1.69974i	−0.324181	−	0.945995i	$-0.605089\pi$
$720$		0			0
$721$	3.98528	+	29.1222i	0.148420	+	1.08457i
$722$		−	5.00000i		−	0.186081i
$723$		0			0
$724$	−8.48528	−	4.89898i	−0.315353	−	0.182069i
$725$	1.52192	+	0.878680i	0.0565226	+	0.0326333i
$726$		0			0
$727$		−	28.0821i		−	1.04151i	−0.853707	−	0.520754i	$-0.825651\pi$
$727$		−	28.0821i		−	1.04151i	0.853707	−	0.520754i	$-0.174349\pi$
$728$	0.717439	−	1.75736i	0.0265901	−	0.0651321i
$729$		0			0
$730$	−6.72792	−	11.6531i	−0.249012	−	0.431301i
$731$	8.57321	−	14.8492i	0.317092	−	0.549219i
$732$		0			0
$733$	1.13604	−	0.655892i	0.0419606	−	0.0242259i	−0.478873	−	0.877884i	$-0.658955\pi$
$733$	1.13604	−	0.655892i	0.0419606	−	0.0242259i	0.520834	+	0.853658i	$0.325621\pi$
$734$	−29.9882			−1.10689
$735$		0			0
$736$	−6.00000			−0.221163
$737$	−49.5481	+	28.6066i	−1.82513	+	1.05374i
$738$		0			0
$739$	4.22792	−	7.32298i	0.155527	−	0.269380i	−0.777724	−	0.628606i	$-0.783626\pi$
$739$	4.22792	−	7.32298i	0.155527	−	0.269380i	0.933251	+	0.359226i	$0.116959\pi$
$740$	−6.42090	−	11.1213i	−0.236037	−	0.408828i
$741$		0			0
$742$	14.4853	−	35.4815i	0.531771	−	1.30257i
$743$			18.7279i			0.687061i	0.939142	+	0.343530i	$0.111623\pi$
$743$			18.7279i			0.687061i	−0.939142	+	0.343530i	$0.888377\pi$
$744$		0			0
$745$	34.4558	+	19.8931i	1.26236	+	0.728826i
$746$	19.0526	+	11.0000i	0.697564	+	0.402739i
$747$		0			0
$748$		−	10.3923i		−	0.379980i
$749$	−0.891519	−	6.51472i	−0.0325754	−	0.238043i
$750$		0			0
$751$	−26.7279	−	46.2941i	−0.975316	−	1.68930i	−0.678889	−	0.734241i	$-0.737538\pi$
$751$	−26.7279	−	46.2941i	−0.975316	−	1.68930i	−0.296427	−	0.955056i	$-0.595795\pi$
$752$	6.42090	−	11.1213i	0.234146	−	0.405553i
$753$		0			0
$754$	−1.09188	+	0.630399i	−0.0397640	+	0.0229578i
$755$	−21.4511			−0.780684
$756$		0			0
$757$	32.7574			1.19059			0.595293	−	0.803509i	$-0.297036\pi$
$757$	32.7574			1.19059			0.595293	+	0.803509i	$0.297036\pi$
$758$	−6.48244	+	3.74264i	−0.235453	+	0.135939i
$759$		0			0
$760$	−6.00000	+	10.3923i	−0.217643	+	0.376969i
$761$	−12.2474	−	21.2132i	−0.443970	−	0.768978i	0.554010	−	0.832510i	$-0.313097\pi$
$761$	−12.2474	−	21.2132i	−0.443970	−	0.768978i	−0.997980	+	0.0635319i	$0.979764\pi$
$762$		0			0
$763$	28.7426	+	37.0650i	1.04055	+	1.34184i
$764$		−	21.2132i		−	0.767467i
$765$		0			0
$766$	−4.75736	−	2.74666i	−0.171890	−	0.0992410i
$767$	1.52192	+	0.878680i	0.0549533	+	0.0317273i
$768$		0			0
$769$		−	40.3805i		−	1.45616i	−0.685493	−	0.728080i	$-0.740413\pi$
$769$		−	40.3805i		−	1.45616i	0.685493	−	0.728080i	$-0.259587\pi$
$770$	−27.2416	+	3.72792i	−0.981718	+	0.134345i
$771$		0			0
$772$	−0.742641	−	1.28629i	−0.0267282	−	0.0462946i
$773$	4.89898	−	8.48528i	0.176204	−	0.305194i	−0.764373	−	0.644774i	$-0.776951\pi$
$773$	4.89898	−	8.48528i	0.176204	−	0.305194i	0.940577	+	0.339580i	$0.110285\pi$
$774$		0			0
$775$	7.86396	−	4.54026i	0.282482	−	0.163091i
$776$	3.16693			0.113686
$777$		0			0
$778$	15.5147			0.556230
$779$	−10.3923	+	6.00000i	−0.372343	+	0.214972i
$780$		0			0
$781$	27.0000	−	46.7654i	0.966136	−	1.67340i
$782$	−7.34847	−	12.7279i	−0.262781	−	0.455150i
$783$		0			0
$784$	−1.74264	+	6.77962i	−0.0622372	+	0.242129i
$785$		−	25.4558i		−	0.908558i
$786$		0			0
$787$	−24.4706	−	14.1281i	−0.872281	−	0.503612i	−0.00417567	−	0.999991i	$-0.501329\pi$
$787$	−24.4706	−	14.1281i	−0.872281	−	0.503612i	−0.868106	+	0.496379i	$0.834662\pi$
$788$	14.6969	+	8.48528i	0.523557	+	0.302276i
$789$		0			0
$790$		−	1.85514i		−	0.0660031i
$791$	−25.0892	−	10.2426i	−0.892071	−	0.364186i
$792$		0			0
$793$	−1.50000	−	2.59808i	−0.0532666	−	0.0922604i
$794$	−7.58410	+	13.1360i	−0.269149	+	0.466181i
$795$		0			0
$796$	−18.1066	+	10.4539i	−0.641771	+	0.370527i
$797$	−17.7408			−0.628410			−0.314205	−	0.949355i	$-0.601738\pi$
$797$	−17.7408			−0.628410			−0.314205	+	0.949355i	$0.601738\pi$
$798$		0			0
$799$	31.4558			1.11283
$800$	0.866025	−	0.500000i	0.0306186	−	0.0176777i
$801$		0			0
$802$	−9.87868	+	17.1104i	−0.348828	+	0.604188i
$803$	−11.6531	−	20.1838i	−0.411229	−	0.712270i
$804$		0			0
$805$	−30.7279	+	23.8284i	−1.08302	+	0.839842i
$806$			6.51472i			0.229471i
$807$		0			0
$808$	−6.36396	−	3.67423i	−0.223883	−	0.129259i
$809$	12.5446	+	7.24264i	0.441045	+	0.254638i	0.704041	−	0.710159i	$-0.251377\pi$
$809$	12.5446	+	7.24264i	0.441045	+	0.254638i	−0.262996	+	0.964797i	$0.584710\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$	3.67423	−	2.84924i	0.128940	−	0.0999888i
$813$		0			0
$814$	−11.1213	−	19.2627i	−0.389802	−	0.675157i
$815$	−11.6170	+	20.1213i	−0.406928	+	0.704819i
$816$		0			0
$817$	−29.6985	+	17.1464i	−1.03902	+	0.599878i
$818$	−3.76127			−0.131510
$819$		0			0
$820$	6.00000			0.209529
$821$	19.8931	−	11.4853i	0.694274	−	0.400839i	−0.110937	−	0.993827i	$-0.535385\pi$
$821$	19.8931	−	11.4853i	0.694274	−	0.400839i	0.805211	+	0.592988i	$0.202052\pi$
$822$		0			0
$823$	1.37868	−	2.38794i	0.0480578	−	0.0832385i	−0.840996	−	0.541042i	$-0.818030\pi$
$823$	1.37868	−	2.38794i	0.0480578	−	0.0832385i	0.889054	+	0.457803i	$0.151364\pi$
$824$	−5.55487	−	9.62132i	−0.193513	−	0.335175i
$825$		0			0
$826$	−6.00000	−	2.44949i	−0.208767	−	0.0852286i
$827$		−	3.51472i		−	0.122219i	−0.998131	−	0.0611094i	$-0.980536\pi$
$827$		−	3.51472i		−	0.122219i	0.998131	−	0.0611094i	$-0.0194638\pi$
$828$		0			0
$829$	8.48528	+	4.89898i	0.294706	+	0.170149i	0.640062	−	0.768323i	$-0.278909\pi$
$829$	8.48528	+	4.89898i	0.294706	+	0.170149i	−0.345356	+	0.938472i	$0.612242\pi$
$830$	−32.4377	−	18.7279i	−1.12593	−	0.650056i
$831$		0			0
$832$			0.717439i			0.0248727i
$833$	−16.5160	+	4.60660i	−0.572246	+	0.159609i
$834$		0			0
$835$	0.727922	+	1.26080i	0.0251908	+	0.0436317i
$836$	−10.3923	+	18.0000i	−0.359425	+	0.622543i
$837$		0			0
$838$	6.87868	−	3.97141i	0.237620	−	0.137190i
$839$	15.2913			0.527914			0.263957	−	0.964534i	$-0.414972\pi$
$839$	15.2913			0.527914			0.263957	+	0.964534i	$0.414972\pi$
$840$		0			0
$841$	25.9117			0.893506
$842$	−20.3134	+	11.7279i	−0.700045	+	0.404171i
$843$		0			0
$844$	1.74264	−	3.01834i	0.0599842	−	0.103896i
$845$	15.2913	+	26.4853i	0.526036	+	0.911121i
$846$		0			0
$847$	−18.3492	+	2.51104i	−0.630487	+	0.0862802i
$848$			14.4853i			0.497427i
$849$		0			0
$850$	2.12132	+	1.22474i	0.0727607	+	0.0420084i
$851$	−27.2416	−	15.7279i	−0.933829	−	0.539146i
$852$		0			0
$853$			31.7713i			1.08783i	0.839141	+	0.543914i	$0.183058\pi$
$853$			31.7713i			1.08783i	−0.839141	+	0.543914i	$0.816942\pi$
$854$	6.77962	+	8.74264i	0.231994	+	0.299167i
$855$		0			0
$856$	1.24264	+	2.15232i	0.0424726	+	0.0735647i
$857$	13.7694	−	23.8492i	0.470353	−	0.814675i	−0.529073	−	0.848577i	$-0.677460\pi$
$857$	13.7694	−	23.8492i	0.470353	−	0.814675i	0.999425	+	0.0339020i	$0.0107934\pi$
$858$		0			0
$859$	24.4706	−	14.1281i	0.834925	−	0.482044i	−0.0206111	−	0.999788i	$-0.506561\pi$
$859$	24.4706	−	14.1281i	0.834925	−	0.482044i	0.855536	+	0.517743i	$0.173228\pi$
$860$	17.1464			0.584688
$861$		0			0
$862$	1.75736			0.0598559
$863$	32.4377	−	18.7279i	1.10419	−	0.637506i	0.166874	−	0.985978i	$-0.446633\pi$
$863$	32.4377	−	18.7279i	1.10419	−	0.637506i	0.937319	+	0.348472i	$0.113300\pi$
$864$		0			0
$865$	−25.4558	+	44.0908i	−0.865525	+	1.49913i
$866$	1.28629	+	2.22792i	0.0437100	+	0.0757079i
$867$		0			0
$868$	−3.25736	−	23.8030i	−0.110562	−	0.807925i
$869$		−	3.21320i		−	0.109000i
$870$		0			0
$871$	8.37868	+	4.83743i	0.283901	+	0.163910i
$872$	−15.3528	−	8.86396i	−0.519912	−	0.300172i
$873$		0			0
$874$			29.3939i			0.994263i
$875$	−9.79796	+	24.0000i	−0.331231	+	0.811348i
$876$		0			0
$877$	23.1066	+	40.0218i	0.780254	+	1.35144i	0.931793	+	0.362989i	$0.118244\pi$
$877$	23.1066	+	40.0218i	0.780254	+	1.35144i	−0.151539	+	0.988451i	$0.548423\pi$
$878$	2.15232	−	3.72792i	0.0726372	−	0.125811i
$879$		0			0
$880$	9.00000	−	5.19615i	0.303390	−	0.175162i
$881$	−25.0892			−0.845278			−0.422639	−	0.906298i	$-0.638896\pi$
$881$	−25.0892			−0.845278			−0.422639	+	0.906298i	$0.638896\pi$
$882$		0			0
$883$	−2.00000			−0.0673054			−0.0336527	−	0.999434i	$-0.510714\pi$
$883$	−2.00000			−0.0673054			−0.0336527	+	0.999434i	$0.510714\pi$
$884$	−1.52192	+	0.878680i	−0.0511876	+	0.0295532i
$885$		0			0
$886$	−12.7279	+	22.0454i	−0.427603	+	0.740630i
$887$	−4.60181	−	7.97056i	−0.154514	−	0.267625i	0.778368	−	0.627808i	$-0.216048\pi$
$887$	−4.60181	−	7.97056i	−0.154514	−	0.267625i	−0.932882	+	0.360183i	$0.882714\pi$
$888$		0			0
$889$	7.72792	−	18.9295i	0.259186	−	0.634874i
$890$			7.45584i			0.249920i
$891$		0			0
$892$	−9.00000	−	5.19615i	−0.301342	−	0.173980i
$893$	−54.4831	−	31.4558i	−1.82321	−	1.05263i
$894$		0			0
$895$			16.4800i			0.550865i
$896$	−0.358719	−	2.62132i	−0.0119840	−	0.0875722i
$897$		0			0
$898$	2.63604	+	4.56575i	0.0879658	+	0.152361i
$899$	−7.97887	+	13.8198i	−0.266110	+	0.460916i
$900$		0			0
$901$	−30.7279	+	17.7408i	−1.02370	+	0.591031i
$902$	10.3923			0.346026
$903$		0			0
$904$	10.2426			0.340665
$905$	20.7846	−	12.0000i	0.690904	−	0.398893i
$906$		0			0
$907$	6.74264	−	11.6786i	0.223886	−	0.387781i	−0.732099	−	0.681198i	$-0.761459\pi$
$907$	6.74264	−	11.6786i	0.223886	−	0.387781i	0.955985	+	0.293417i	$0.0947924\pi$
$908$	12.5446	+	21.7279i	0.416308	+	0.721066i
$909$		0			0
$910$	2.84924	+	3.67423i	0.0944515	+	0.121800i
$911$		−	10.2426i		−	0.339354i	−0.985500	−	0.169677i	$-0.945728\pi$
$911$		−	10.2426i		−	0.339354i	0.985500	−	0.169677i	$-0.0542724\pi$
$912$		0			0
$913$	−56.1838	−	32.4377i	−1.85941	−	1.07353i
$914$	19.9186	+	11.5000i	0.658848	+	0.380386i
$915$		0			0
$916$			5.61642i			0.185572i
$917$		0			0
$918$		0			0
$919$	−12.8640	−	22.2810i	−0.424343	−	0.734983i	0.572016	−	0.820243i	$-0.306162\pi$
$919$	−12.8640	−	22.2810i	−0.424343	−	0.734983i	−0.996359	+	0.0852590i	$0.972828\pi$
$920$	7.34847	−	12.7279i	0.242272	−	0.419627i
$921$		0			0
$922$	18.5772	−	10.7255i	0.611806	−	0.353227i
$923$	−9.13151			−0.300567
$924$		0			0
$925$	5.24264			0.172377
$926$	−19.0526	+	11.0000i	−0.626106	+	0.361482i
$927$		0			0
$928$	−0.878680	+	1.52192i	−0.0288441	+	0.0499594i
$929$	−14.3637	−	24.8787i	−0.471258	−	0.816243i	0.528201	−	0.849119i	$-0.322867\pi$
$929$	−14.3637	−	24.8787i	−0.471258	−	0.816243i	−0.999459	+	0.0328762i	$0.989533\pi$
$930$		0			0
$931$	33.2132	+	8.53716i	1.08852	+	0.279794i
$932$		−	20.4853i		−	0.671018i
$933$		0			0
$934$	−15.3640	−	8.87039i	−0.502724	−	0.290248i
$935$	22.0454	+	12.7279i	0.720962	+	0.416248i
$936$		0			0
$937$		−	33.5033i		−	1.09451i	−0.836967	−	0.547253i	$-0.815674\pi$
$937$		−	33.5033i		−	1.09451i	0.836967	−	0.547253i	$-0.184326\pi$
$938$	−33.0321	−	13.4853i	−1.07853	−	0.440310i
$939$		0			0
$940$	15.7279	+	27.2416i	0.512988	+	0.888522i
$941$	−21.1178	+	36.5772i	−0.688422	+	1.19238i	0.283927	+	0.958846i	$0.408363\pi$
$941$	−21.1178	+	36.5772i	−0.688422	+	1.19238i	−0.972348	+	0.233535i	$0.924971\pi$
$942$		0			0
$943$	12.7279	−	7.34847i	0.414478	−	0.239299i
$944$	2.44949			0.0797241
$945$		0			0
$946$	29.6985			0.965581
$947$	−14.0665	+	8.12132i	−0.457101	+	0.263907i	−0.710825	−	0.703369i	$-0.751678\pi$
$947$	−14.0665	+	8.12132i	−0.457101	+	0.263907i	0.253723	+	0.967277i	$0.418345\pi$
$948$		0			0
$949$	−1.97056	+	3.41311i	−0.0639672	+	0.110794i
$950$	−2.44949	−	4.24264i	−0.0794719	−	0.137649i
$951$		0			0
$952$	5.12132	−	3.97141i	0.165983	−	0.128714i
$953$		−	1.02944i		−	0.0333467i	−0.999861	−	0.0166734i	$-0.994692\pi$
$953$		−	1.02944i		−	0.0333467i	0.999861	−	0.0166734i	$-0.00530755\pi$
$954$		0			0
$955$	45.0000	+	25.9808i	1.45617	+	0.840718i
$956$	14.0665	+	8.12132i	0.454944	+	0.262662i
$957$		0			0
$958$			2.44949i			0.0791394i
$959$	12.5446	−	9.72792i	0.405087	−	0.314131i
$960$		0			0
$961$	25.7279	+	44.5621i	0.829933	+	1.43749i
$962$	−1.88064	+	3.25736i	−0.0606342	+	0.105021i
$963$		0			0
$964$	−0.985281	+	0.568852i	−0.0317338	+	0.0183215i
$965$	3.63818			0.117117
$966$		0			0
$967$	−42.6985			−1.37309			−0.686545	−	0.727087i	$-0.740874\pi$
$967$	−42.6985			−1.37309			−0.686545	+	0.727087i	$0.740874\pi$
$968$	6.06218	−	3.50000i	0.194846	−	0.112494i
$969$		0			0
$970$	−3.87868	+	6.71807i	−0.124537	+	0.215704i
$971$	20.1903	+	34.9706i	0.647936	+	1.12226i	0.983615	+	0.180282i	$0.0577011\pi$
$971$	20.1903	+	34.9706i	0.647936	+	1.12226i	−0.335679	+	0.941977i	$0.608966\pi$
$972$		0			0
$973$	−19.7574	−	8.06591i	−0.633392	−	0.258581i
$974$			22.0000i			0.704925i
$975$		0			0
$976$	−3.62132	−	2.09077i	−0.115916	−	0.0669239i
$977$	−5.19615	−	3.00000i	−0.166240	−	0.0959785i	0.414572	−	0.910017i	$-0.363931\pi$
$977$	−5.19615	−	3.00000i	−0.166240	−	0.0959785i	−0.580812	+	0.814038i	$0.697265\pi$
$978$		0			0
$979$			12.9139i			0.412730i
$980$	−12.2474	−	12.0000i	−0.391230	−	0.383326i
$981$		0			0
$982$	17.4853	+	30.2854i	0.557978	+	0.966446i
$983$	−11.6170	+	20.1213i	−0.370526	+	0.641770i	−0.989647	−	0.143526i	$-0.954156\pi$
$983$	−11.6170	+	20.1213i	−0.370526	+	0.641770i	0.619120	+	0.785296i	$0.287489\pi$
$984$		0			0
$985$	−36.0000	+	20.7846i	−1.14706	+	0.662253i
$986$	−4.30463			−0.137087
$987$		0			0
$988$	3.51472			0.111818
$989$	36.3731	−	21.0000i	1.15660	−	0.667761i
$990$		0			0
$991$	10.1066	−	17.5051i	0.321047	−	0.556069i	−0.659658	−	0.751566i	$-0.729299\pi$
$991$	10.1066	−	17.5051i	0.321047	−	0.556069i	0.980704	+	0.195497i	$0.0626320\pi$
$992$	4.54026	+	7.86396i	0.144153	+	0.249681i
$993$		0			0
$994$	33.3640	−	4.56575i	1.05824	−	0.144817i
$995$		−	51.2132i		−	1.62357i
$996$		0			0
$997$	16.8640	+	9.73641i	0.534087	+	0.308355i	0.742679	−	0.669647i	$-0.233555\pi$
$997$	16.8640	+	9.73641i	0.534087	+	0.308355i	−0.208592	+	0.978003i	$0.566888\pi$
$998$	21.1794	+	12.2279i	0.670422	+	0.387068i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	378.2.k.d.269.4	yes	8
3.2	odd	2	inner	378.2.k.d.269.1	yes	8
7.3	odd	6		2646.2.d.d.2645.7		8
7.4	even	3		2646.2.d.d.2645.5		8
7.5	odd	6	inner	378.2.k.d.215.1	✓	8
9.2	odd	6		1134.2.t.f.1025.4		8
9.4	even	3		1134.2.l.e.269.4		8
9.5	odd	6		1134.2.l.e.269.1		8
9.7	even	3		1134.2.t.f.1025.1		8
21.5	even	6	inner	378.2.k.d.215.4	yes	8
21.11	odd	6		2646.2.d.d.2645.4		8
21.17	even	6		2646.2.d.d.2645.2		8
63.5	even	6		1134.2.t.f.593.1		8
63.40	odd	6		1134.2.t.f.593.4		8
63.47	even	6		1134.2.l.e.215.2		8
63.61	odd	6		1134.2.l.e.215.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
378.2.k.d.215.1	✓	8	7.5	odd	6	inner
378.2.k.d.215.4	yes	8	21.5	even	6	inner
378.2.k.d.269.1	yes	8	3.2	odd	2	inner
378.2.k.d.269.4	yes	8	1.1	even	1	trivial
1134.2.l.e.215.2		8	63.47	even	6
1134.2.l.e.215.3		8	63.61	odd	6
1134.2.l.e.269.1		8	9.5	odd	6
1134.2.l.e.269.4		8	9.4	even	3
1134.2.t.f.593.1		8	63.5	even	6
1134.2.t.f.593.4		8	63.40	odd	6
1134.2.t.f.1025.1		8	9.7	even	3
1134.2.t.f.1025.4		8	9.2	odd	6
2646.2.d.d.2645.2		8	21.17	even	6
2646.2.d.d.2645.4		8	21.11	odd	6
2646.2.d.d.2645.5		8	7.4	even	3
2646.2.d.d.2645.7		8	7.3	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(0.866025 - 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(1.22474 + 2.12132i) q^{5} +(-1.00000 + 2.44949i) q^{7} -1.00000i q^{8} +(2.12132 + 1.22474i) q^{10} +(3.67423 + 2.12132i) q^{11} -0.717439i q^{13} +(0.358719 + 2.62132i) q^{14} +(-0.500000 - 0.866025i) q^{16} +(1.22474 - 2.12132i) q^{17} +(-4.24264 + 2.44949i) q^{19} +2.44949 q^{20} +4.24264 q^{22} +(5.19615 - 3.00000i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(-0.358719 - 0.621320i) q^{26} +(1.62132 + 2.09077i) q^{28} -1.75736i q^{29} +(-7.86396 - 4.54026i) q^{31} +(-0.866025 - 0.500000i) q^{32} -2.44949i q^{34} +(-6.42090 + 0.878680i) q^{35} +(-2.62132 - 4.54026i) q^{37} +(-2.44949 + 4.24264i) q^{38} +(2.12132 - 1.22474i) q^{40} +2.44949 q^{41} +7.00000 q^{43} +(3.67423 - 2.12132i) q^{44} +(3.00000 - 5.19615i) q^{46} +(6.42090 + 11.1213i) q^{47} +(-5.00000 - 4.89898i) q^{49} +1.00000i q^{50} +(-0.621320 - 0.358719i) q^{52} +(-12.5446 - 7.24264i) q^{53} +10.3923i q^{55} +(2.44949 + 1.00000i) q^{56} +(-0.878680 - 1.52192i) q^{58} +(-1.22474 + 2.12132i) q^{59} +(3.62132 - 2.09077i) q^{61} -9.08052 q^{62} -1.00000 q^{64} +(1.52192 - 0.878680i) q^{65} +(-6.74264 + 11.6786i) q^{67} +(-1.22474 - 2.12132i) q^{68} +(-5.12132 + 3.97141i) q^{70} -12.7279i q^{71} +(-4.75736 - 2.74666i) q^{73} +(-4.54026 - 2.62132i) q^{74} +4.89898i q^{76} +(-8.87039 + 6.87868i) q^{77} +(-0.378680 - 0.655892i) q^{79} +(1.22474 - 2.12132i) q^{80} +(2.12132 - 1.22474i) q^{82} -15.2913 q^{83} +6.00000 q^{85} +(6.06218 - 3.50000i) q^{86} +(2.12132 - 3.67423i) q^{88} +(1.52192 + 2.63604i) q^{89} +(1.75736 + 0.717439i) q^{91} -6.00000i q^{92} +(11.1213 + 6.42090i) q^{94} +(-10.3923 - 6.00000i) q^{95} +3.16693i q^{97} +(-6.77962 - 1.74264i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q + 4 * q^4 - 8 * q^7	\( 8 q + 4 q^{4} - 8 q^{7} - 4 q^{16} - 4 q^{25} - 4 q^{28} - 12 q^{31} - 4 q^{37} + 56 q^{43} + 24 q^{46} - 40 q^{49} + 12 q^{52} - 24 q^{58} + 12 q^{61} - 8 q^{64} - 20 q^{67} - 24 q^{70} - 72 q^{73} - 20 q^{79} + 48 q^{85} + 48 q^{91} + 72 q^{94}+O(q^{100}) \) 8 * q + 4 * q^4 - 8 * q^7 - 4 * q^16 - 4 * q^25 - 4 * q^28 - 12 * q^31 - 4 * q^37 + 56 * q^43 + 24 * q^46 - 40 * q^49 + 12 * q^52 - 24 * q^58 + 12 * q^61 - 8 * q^64 - 20 * q^67 - 24 * q^70 - 72 * q^73 - 20 * q^79 + 48 * q^85 + 48 * q^91 + 72 * q^94

Introduction

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