LMFDB - Embedded newform 1350.2.q.d.557.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,2,Mod(143,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([2, 9]))

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

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

chi := DirichletCharacter("1350.143");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1350.q (of order $12$, degree $4$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$10.7798042729$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$2$ over $\Q(\zeta_{12})$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 450)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			557.1
Root			$0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1350.557
Dual form			1350.2.q.d.143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.22474 + 4.57081i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})$	\(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.22474 + 4.57081i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-3.00000 + 1.73205i) q^{11} +(1.22474 + 4.57081i) q^{13} +(2.36603 - 4.09808i) q^{14} +(0.500000 + 0.866025i) q^{16} -3.19615i q^{19} +(3.34607 - 0.896575i) q^{22} +(-2.12132 + 0.568406i) q^{23} -4.73205i q^{26} +(-3.34607 + 3.34607i) q^{28} +(-5.36603 - 9.29423i) q^{29} +(-0.0980762 + 0.169873i) q^{31} +(-0.258819 - 0.965926i) q^{32} +(-5.79555 - 5.79555i) q^{37} +(-0.827225 + 3.08725i) q^{38} +(1.50000 + 0.866025i) q^{41} +(-0.448288 - 0.120118i) q^{43} -3.46410 q^{44} +2.19615 q^{46} +(5.79555 + 1.55291i) q^{47} +(-13.3301 - 7.69615i) q^{49} +(-1.22474 + 4.57081i) q^{52} +(-5.79555 - 5.79555i) q^{53} +(4.09808 - 2.36603i) q^{56} +(2.77766 + 10.3664i) q^{58} +(2.76795 - 4.79423i) q^{59} +(2.00000 + 3.46410i) q^{61} +(0.138701 - 0.138701i) q^{62} +1.00000i q^{64} +(-5.34727 + 1.43280i) q^{67} +7.26795i q^{71} +(3.67423 - 3.67423i) q^{73} +(4.09808 + 7.09808i) q^{74} +(1.59808 - 2.76795i) q^{76} +(-4.24264 - 15.8338i) q^{77} +(8.66025 - 5.00000i) q^{79} +(-1.22474 - 1.22474i) q^{82} +(-4.45069 + 16.6102i) q^{83} +(0.401924 + 0.232051i) q^{86} +(3.34607 + 0.896575i) q^{88} +8.66025 q^{89} -22.3923 q^{91} +(-2.12132 - 0.568406i) q^{92} +(-5.19615 - 3.00000i) q^{94} +(-0.688524 + 2.56961i) q^{97} +(10.8840 + 10.8840i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q+O(q^{10}) $ 8 * q	$ 8 q - 24 q^{11} + 12 q^{14} + 4 q^{16} - 36 q^{29} + 20 q^{31} + 12 q^{41} - 24 q^{46} - 72 q^{49} + 12 q^{56} + 36 q^{59} + 16 q^{61} + 12 q^{74} - 8 q^{76} + 24 q^{86} - 96 q^{91}+O(q^{100}) $ 8 * q - 24 * q^11 + 12 * q^14 + 4 * q^16 - 36 * q^29 + 20 * q^31 + 12 * q^41 - 24 * q^46 - 72 * q^49 + 12 * q^56 + 36 * q^59 + 16 * q^61 + 12 * q^74 - 8 * q^76 + 24 * q^86 - 96 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{4}\right)$

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 $

$2$	−0.965926	−	0.258819i	−0.683013	−	0.183013i
$2$	−0.965926	−	0.258819i	−0.683013	−	0.183013i
$3$		0			0
$3$		0			0
$4$	0.866025	+	0.500000i	0.433013	+	0.250000i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−1.22474	+	4.57081i	−0.462910	+	1.72760i	0.200817	+	0.979629i	$0.435640\pi$
$7$	−1.22474	+	4.57081i	−0.462910	+	1.72760i	−0.663727	+	0.747975i	$0.731026\pi$
$8$	−0.707107	−	0.707107i	−0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	−3.00000	+	1.73205i	−0.904534	+	0.522233i	−0.878668	−	0.477432i	$-0.841568\pi$
$11$	−3.00000	+	1.73205i	−0.904534	+	0.522233i	−0.0258656	+	0.999665i	$0.508234\pi$
$12$		0			0
$13$	1.22474	+	4.57081i	0.339683	+	1.26771i	0.898702	+	0.438560i	$0.144511\pi$
$13$	1.22474	+	4.57081i	0.339683	+	1.26771i	−0.559019	+	0.829155i	$0.688822\pi$
$14$	2.36603	−	4.09808i	0.632347	−	1.09526i
$15$		0			0
$16$	0.500000	+	0.866025i	0.125000	+	0.216506i
$17$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$17$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$18$		0			0
$19$		−	3.19615i		−	0.733248i	−0.930369	−	0.366624i	$-0.880514\pi$
$19$		−	3.19615i		−	0.733248i	0.930369	−	0.366624i	$-0.119486\pi$
$20$		0			0
$21$		0			0
$22$	3.34607	−	0.896575i	0.713384	−	0.191151i
$23$	−2.12132	+	0.568406i	−0.442326	+	0.118521i	−0.473106	−	0.881005i	$-0.656867\pi$
$23$	−2.12132	+	0.568406i	−0.442326	+	0.118521i	0.0307805	+	0.999526i	$0.490201\pi$
$24$		0			0
$25$		0			0
$26$		−	4.73205i		−	0.928032i
$27$		0			0
$28$	−3.34607	+	3.34607i	−0.632347	+	0.632347i
$29$	−5.36603	−	9.29423i	−0.996446	−	1.72589i	−0.571173	−	0.820830i	$-0.693511\pi$
$29$	−5.36603	−	9.29423i	−0.996446	−	1.72589i	−0.425273	−	0.905065i	$-0.639822\pi$
$30$		0			0
$31$	−0.0980762	+	0.169873i	−0.0176150	+	0.0305101i	−0.874699	−	0.484667i	$-0.838941\pi$
$31$	−0.0980762	+	0.169873i	−0.0176150	+	0.0305101i	0.857084	+	0.515177i	$0.172274\pi$
$32$	−0.258819	−	0.965926i	−0.0457532	−	0.170753i
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−5.79555	−	5.79555i	−0.952783	−	0.952783i	0.0461511	−	0.998934i	$-0.485304\pi$
$37$	−5.79555	−	5.79555i	−0.952783	−	0.952783i	−0.998934	+	0.0461511i	$0.985304\pi$
$38$	−0.827225	+	3.08725i	−0.134194	+	0.500817i
$39$		0			0
$40$		0			0
$41$	1.50000	+	0.866025i	0.234261	+	0.135250i	0.612536	−	0.790443i	$-0.290149\pi$
$41$	1.50000	+	0.866025i	0.234261	+	0.135250i	−0.378275	+	0.925693i	$0.623483\pi$
$42$		0			0
$43$	−0.448288	−	0.120118i	−0.0683632	−	0.0183179i	0.224475	−	0.974480i	$-0.427933\pi$
$43$	−0.448288	−	0.120118i	−0.0683632	−	0.0183179i	−0.292839	+	0.956162i	$0.594600\pi$
$44$	−3.46410			−0.522233
$45$		0			0
$46$	2.19615			0.323805
$47$	5.79555	+	1.55291i	0.845369	+	0.226516i	0.655407	−	0.755276i	$-0.272497\pi$
$47$	5.79555	+	1.55291i	0.845369	+	0.226516i	0.189961	+	0.981792i	$0.439164\pi$
$48$		0			0
$49$	−13.3301	−	7.69615i	−1.90430	−	1.09945i
$50$		0			0
$51$		0			0
$52$	−1.22474	+	4.57081i	−0.169842	+	0.633857i
$53$	−5.79555	−	5.79555i	−0.796081	−	0.796081i	0.186394	−	0.982475i	$-0.440320\pi$
$53$	−5.79555	−	5.79555i	−0.796081	−	0.796081i	−0.982475	+	0.186394i	$0.940320\pi$
$54$		0			0
$55$		0			0
$56$	4.09808	−	2.36603i	0.547628	−	0.316173i
$57$		0			0
$58$	2.77766	+	10.3664i	0.364725	+	1.36117i
$59$	2.76795	−	4.79423i	0.360356	−	0.624155i	−0.627663	−	0.778485i	$-0.715988\pi$
$59$	2.76795	−	4.79423i	0.360356	−	0.624155i	0.988019	+	0.154330i	$0.0493218\pi$
$60$		0			0
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	0.965187	−	0.261562i	$-0.0842377\pi$
$61$	2.00000	+	3.46410i	0.256074	+	0.443533i	−0.709113	+	0.705095i	$0.750904\pi$
$62$	0.138701	−	0.138701i	0.0176150	−	0.0176150i
$63$		0			0
$64$			1.00000i			0.125000i
$65$		0			0
$66$		0			0
$67$	−5.34727	+	1.43280i	−0.653273	+	0.175044i	−0.570208	−	0.821500i	$-0.693137\pi$
$67$	−5.34727	+	1.43280i	−0.653273	+	0.175044i	−0.0830646	+	0.996544i	$0.526471\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			7.26795i			0.862547i	0.902221	+	0.431273i	$0.141936\pi$
$71$			7.26795i			0.862547i	−0.902221	+	0.431273i	$0.858064\pi$
$72$		0			0
$73$	3.67423	−	3.67423i	0.430037	−	0.430037i	−0.458604	−	0.888641i	$-0.651650\pi$
$73$	3.67423	−	3.67423i	0.430037	−	0.430037i	0.888641	+	0.458604i	$0.151650\pi$
$74$	4.09808	+	7.09808i	0.476392	+	0.825135i
$75$		0			0
$76$	1.59808	−	2.76795i	0.183312	−	0.317506i
$77$	−4.24264	−	15.8338i	−0.483494	−	1.80442i
$78$		0			0
$79$	8.66025	−	5.00000i	0.974355	−	0.562544i	0.0737937	−	0.997274i	$-0.476489\pi$
$79$	8.66025	−	5.00000i	0.974355	−	0.562544i	0.900561	+	0.434730i	$0.143156\pi$
$80$		0			0
$81$		0			0
$82$	−1.22474	−	1.22474i	−0.135250	−	0.135250i
$83$	−4.45069	+	16.6102i	−0.488527	+	1.82321i	0.0750978	+	0.997176i	$0.476073\pi$
$83$	−4.45069	+	16.6102i	−0.488527	+	1.82321i	−0.563625	+	0.826031i	$0.690594\pi$
$84$		0			0
$85$		0			0
$86$	0.401924	+	0.232051i	0.0433406	+	0.0250227i
$87$		0			0
$88$	3.34607	+	0.896575i	0.356692	+	0.0955753i
$89$	8.66025			0.917985			0.458993	−	0.888440i	$-0.348210\pi$
$89$	8.66025			0.917985			0.458993	+	0.888440i	$0.348210\pi$
$90$		0			0
$91$	−22.3923			−2.34735
$92$	−2.12132	−	0.568406i	−0.221163	−	0.0592604i
$93$		0			0
$94$	−5.19615	−	3.00000i	−0.535942	−	0.309426i
$95$		0			0
$96$		0			0
$97$	−0.688524	+	2.56961i	−0.0699091	+	0.260904i	−0.992031	−	0.125996i	$-0.959787\pi$
$97$	−0.688524	+	2.56961i	−0.0699091	+	0.260904i	0.922122	+	0.386900i	$0.126454\pi$
$98$	10.8840	+	10.8840i	1.09945	+	1.09945i
$99$		0			0
$100$		0			0
$101$	−6.29423	+	3.63397i	−0.626299	+	0.361594i	−0.779317	−	0.626629i	$-0.784434\pi$
$101$	−6.29423	+	3.63397i	−0.626299	+	0.361594i	0.153018	+	0.988223i	$0.451101\pi$
$102$		0			0
$103$	−2.68973	−	10.0382i	−0.265027	−	0.989093i	−0.962234	−	0.272223i	$-0.912241\pi$
$103$	−2.68973	−	10.0382i	−0.265027	−	0.989093i	0.697207	−	0.716869i	$-0.254426\pi$
$104$	2.36603	−	4.09808i	0.232008	−	0.401849i
$105$		0			0
$106$	4.09808	+	7.09808i	0.398040	+	0.689426i
$107$	−7.91688	+	7.91688i	−0.765353	+	0.765353i	−0.977285	−	0.211931i	$-0.932025\pi$
$107$	−7.91688	+	7.91688i	−0.765353	+	0.765353i	0.211931	+	0.977285i	$0.432025\pi$
$108$		0			0
$109$			1.80385i			0.172777i	0.996262	+	0.0863886i	$0.0275327\pi$
$109$			1.80385i			0.172777i	−0.996262	+	0.0863886i	$0.972467\pi$
$110$		0			0
$111$		0			0
$112$	−4.57081	+	1.22474i	−0.431901	+	0.115728i
$113$	−12.9360	+	3.46618i	−1.21691	+	0.326071i	−0.809471	−	0.587160i	$-0.800246\pi$
$113$	−12.9360	+	3.46618i	−1.21691	+	0.326071i	−0.407443	+	0.913231i	$0.633579\pi$
$114$		0			0
$115$		0			0
$116$		−	10.7321i		−	0.996446i
$117$		0			0
$118$	−3.91447	+	3.91447i	−0.360356	+	0.360356i
$119$		0			0
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$	−1.03528	−	3.86370i	−0.0937295	−	0.349803i
$123$		0			0
$124$	−0.169873	+	0.0980762i	−0.0152550	+	0.00880750i
$125$		0			0
$126$		0			0
$127$	−5.79555	−	5.79555i	−0.514272	−	0.514272i	0.401560	−	0.915833i	$-0.368468\pi$
$127$	−5.79555	−	5.79555i	−0.514272	−	0.514272i	−0.915833	+	0.401560i	$0.868468\pi$
$128$	0.258819	−	0.965926i	0.0228766	−	0.0853766i
$129$		0			0
$130$		0			0
$131$	6.00000	+	3.46410i	0.524222	+	0.302660i	0.738661	−	0.674078i	$-0.235459\pi$
$131$	6.00000	+	3.46410i	0.524222	+	0.302660i	−0.214438	+	0.976738i	$0.568792\pi$
$132$		0			0
$133$	14.6090	+	3.91447i	1.26676	+	0.339428i
$134$	5.53590			0.478229
$135$		0			0
$136$		0			0
$137$	−12.9360	−	3.46618i	−1.10519	−	0.296136i	−0.340317	−	0.940311i	$-0.610534\pi$
$137$	−12.9360	−	3.46618i	−1.10519	−	0.296136i	−0.764878	+	0.644175i	$0.777201\pi$
$138$		0			0
$139$	−6.92820	−	4.00000i	−0.587643	−	0.339276i	0.176522	−	0.984297i	$-0.443515\pi$
$139$	−6.92820	−	4.00000i	−0.587643	−	0.339276i	−0.764165	+	0.645021i	$0.776849\pi$
$140$		0			0
$141$		0			0
$142$	1.88108	−	7.02030i	0.157857	−	0.589130i
$143$	−11.5911	−	11.5911i	−0.969297	−	0.969297i
$144$		0			0
$145$		0			0
$146$	−4.50000	+	2.59808i	−0.372423	+	0.215018i
$147$		0			0
$148$	−2.12132	−	7.91688i	−0.174371	−	0.650763i
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	0.216394	+	0.976306i	$0.430570\pi$
$149$	−9.00000	+	15.5885i	−0.737309	+	1.27706i	−0.953703	+	0.300750i	$0.902763\pi$
$150$		0			0
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	0.981617	−	0.190864i	$-0.0611289\pi$
$151$	4.00000	+	6.92820i	0.325515	+	0.563809i	−0.656101	+	0.754673i	$0.727796\pi$
$152$	−2.26002	+	2.26002i	−0.183312	+	0.183312i
$153$		0			0
$154$			16.3923i			1.32093i
$155$		0			0
$156$		0			0
$157$	19.1798	−	5.13922i	1.53072	−	0.410154i	0.607463	−	0.794348i	$-0.292187\pi$
$157$	19.1798	−	5.13922i	1.53072	−	0.410154i	0.923253	+	0.384194i	$0.125520\pi$
$158$	−9.65926	+	2.58819i	−0.768449	+	0.205905i
$159$		0			0
$160$		0			0
$161$		−	10.3923i		−	0.819028i
$162$		0			0
$163$	−10.1261	+	10.1261i	−0.793140	+	0.793140i	−0.982003	−	0.188864i	$-0.939520\pi$
$163$	−10.1261	+	10.1261i	−0.793140	+	0.793140i	0.188864	+	0.982003i	$0.439520\pi$
$164$	0.866025	+	1.50000i	0.0676252	+	0.117130i
$165$		0			0
$166$	8.59808	−	14.8923i	0.667340	−	1.15587i
$167$	4.24264	+	15.8338i	0.328305	+	1.22525i	0.910947	+	0.412523i	$0.135352\pi$
$167$	4.24264	+	15.8338i	0.328305	+	1.22525i	−0.582642	+	0.812729i	$0.697981\pi$
$168$		0			0
$169$	−8.13397	+	4.69615i	−0.625690	+	0.361242i
$170$		0			0
$171$		0			0
$172$	−0.328169	−	0.328169i	−0.0250227	−	0.0250227i
$173$	4.24264	−	15.8338i	0.322562	−	1.20382i	−0.594178	−	0.804334i	$-0.702523\pi$
$173$	4.24264	−	15.8338i	0.322562	−	1.20382i	0.916740	−	0.399484i	$-0.130811\pi$
$174$		0			0
$175$		0			0
$176$	−3.00000	−	1.73205i	−0.226134	−	0.130558i
$177$		0			0
$178$	−8.36516	−	2.24144i	−0.626995	−	0.168003i
$179$	−22.8564			−1.70837			−0.854184	−	0.519971i	$-0.825943\pi$
$179$	−22.8564			−1.70837			−0.854184	+	0.519971i	$0.825943\pi$
$180$		0			0
$181$	−12.3923			−0.921113			−0.460556	−	0.887630i	$-0.652350\pi$
$181$	−12.3923			−0.921113			−0.460556	+	0.887630i	$0.652350\pi$
$182$	21.6293	+	5.79555i	1.60327	+	0.429595i
$183$		0			0
$184$	1.90192	+	1.09808i	0.140212	+	0.0809513i
$185$		0			0
$186$		0			0
$187$		0			0
$188$	4.24264	+	4.24264i	0.309426	+	0.309426i
$189$		0			0
$190$		0			0
$191$	−3.00000	+	1.73205i	−0.217072	+	0.125327i	−0.604594	−	0.796534i	$-0.706665\pi$
$191$	−3.00000	+	1.73205i	−0.217072	+	0.125327i	0.387522	+	0.921861i	$0.373331\pi$
$192$		0			0
$193$	−2.86559	−	10.6945i	−0.206270	−	0.769809i	−0.989059	−	0.147522i	$-0.952870\pi$
$193$	−2.86559	−	10.6945i	−0.206270	−	0.769809i	0.782789	−	0.622287i	$-0.213796\pi$
$194$	1.33013	−	2.30385i	0.0954976	−	0.165407i
$195$		0			0
$196$	−7.69615	−	13.3301i	−0.549725	−	0.952152i
$197$	10.0382	−	10.0382i	0.715192	−	0.715192i	−0.252425	−	0.967617i	$-0.581228\pi$
$197$	10.0382	−	10.0382i	0.715192	−	0.715192i	0.967617	+	0.252425i	$0.0812280\pi$
$198$		0			0
$199$			20.3923i			1.44557i	0.691072	+	0.722786i	$0.257139\pi$
$199$			20.3923i			1.44557i	−0.691072	+	0.722786i	$0.742861\pi$
$200$		0			0
$201$		0			0
$202$	7.02030	−	1.88108i	0.493947	−	0.132353i
$203$	49.0542	−	13.1440i	3.44293	−	0.922530i
$204$		0			0
$205$		0			0
$206$			10.3923i			0.724066i
$207$		0			0
$208$	−3.34607	+	3.34607i	−0.232008	+	0.232008i
$209$	5.53590	+	9.58846i	0.382926	+	0.663247i
$210$		0			0
$211$	−0.598076	+	1.03590i	−0.0411733	+	0.0713142i	−0.885878	−	0.463919i	$-0.846443\pi$
$211$	−0.598076	+	1.03590i	−0.0411733	+	0.0713142i	0.844704	+	0.535233i	$0.179776\pi$
$212$	−2.12132	−	7.91688i	−0.145693	−	0.543733i
$213$		0			0
$214$	9.69615	−	5.59808i	0.662815	−	0.382677i
$215$		0			0
$216$		0			0
$217$	−0.656339	−	0.656339i	−0.0445552	−	0.0445552i
$218$	0.466870	−	1.74238i	0.0316204	−	0.118009i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−2.44949	−	0.656339i	−0.164030	−	0.0439517i	0.175869	−	0.984414i	$-0.443726\pi$
$223$	−2.44949	−	0.656339i	−0.164030	−	0.0439517i	−0.339899	+	0.940462i	$0.610393\pi$
$224$	4.73205			0.316173
$225$		0			0
$226$	13.3923			0.890843
$227$	−12.3676	−	3.31388i	−0.820864	−	0.219950i	−0.176140	−	0.984365i	$-0.556361\pi$
$227$	−12.3676	−	3.31388i	−0.820864	−	0.219950i	−0.644724	+	0.764415i	$0.723028\pi$
$228$		0			0
$229$	−5.02628	−	2.90192i	−0.332146	−	0.191765i	0.324648	−	0.945835i	$-0.394754\pi$
$229$	−5.02628	−	2.90192i	−0.332146	−	0.191765i	−0.656793	+	0.754071i	$0.728088\pi$
$230$		0			0
$231$		0			0
$232$	−2.77766	+	10.3664i	−0.182362	+	0.680585i
$233$	−3.25813	−	3.25813i	−0.213447	−	0.213447i	0.592283	−	0.805730i	$-0.298227\pi$
$233$	−3.25813	−	3.25813i	−0.213447	−	0.213447i	−0.805730	+	0.592283i	$0.798227\pi$
$234$		0			0
$235$		0			0
$236$	4.79423	−	2.76795i	0.312078	−	0.180178i
$237$		0			0
$238$		0			0
$239$	1.90192	−	3.29423i	0.123025	−	0.213086i	−0.797934	−	0.602745i	$-0.794074\pi$
$239$	1.90192	−	3.29423i	0.123025	−	0.213086i	0.920959	+	0.389659i	$0.127407\pi$
$240$		0			0
$241$	−4.69615	−	8.13397i	−0.302506	−	0.523955i	0.674197	−	0.738551i	$-0.264490\pi$
$241$	−4.69615	−	8.13397i	−0.302506	−	0.523955i	−0.976703	+	0.214596i	$0.931156\pi$
$242$	−0.707107	+	0.707107i	−0.0454545	+	0.0454545i
$243$		0			0
$244$			4.00000i			0.256074i
$245$		0			0
$246$		0			0
$247$	14.6090	−	3.91447i	0.929549	−	0.249072i
$248$	0.189469	−	0.0507680i	0.0120313	−	0.00322377i
$249$		0			0
$250$		0			0
$251$			9.00000i			0.568075i	0.958813	+	0.284037i	$0.0916740\pi$
$251$			9.00000i			0.568075i	−0.958813	+	0.284037i	$0.908326\pi$
$252$		0			0
$253$	5.37945	−	5.37945i	0.338203	−	0.338203i
$254$	4.09808	+	7.09808i	0.257136	+	0.445373i
$255$		0			0
$256$	−0.500000	+	0.866025i	−0.0312500	+	0.0541266i
$257$	6.57201	+	24.5271i	0.409951	+	1.52996i	0.794738	+	0.606952i	$0.207608\pi$
$257$	6.57201	+	24.5271i	0.409951	+	1.52996i	−0.384787	+	0.923005i	$0.625725\pi$
$258$		0			0
$259$	33.5885	−	19.3923i	2.08709	−	1.20498i
$260$		0			0
$261$		0			0
$262$	−4.89898	−	4.89898i	−0.302660	−	0.302660i
$263$	3.25813	−	12.1595i	0.200905	−	0.749788i	−0.789754	−	0.613424i	$-0.789792\pi$
$263$	3.25813	−	12.1595i	0.200905	−	0.749788i	0.990659	−	0.136364i	$-0.0435416\pi$
$264$		0			0
$265$		0			0
$266$	−13.0981	−	7.56218i	−0.803095	−	0.463667i
$267$		0			0
$268$	−5.34727	−	1.43280i	−0.326636	−	0.0875219i
$269$	10.0526			0.612915			0.306458	−	0.951884i	$-0.400856\pi$
$269$	10.0526			0.612915			0.306458	+	0.951884i	$0.400856\pi$
$270$		0			0
$271$	26.5885			1.61513			0.807567	−	0.589776i	$-0.200784\pi$
$271$	26.5885			1.61513			0.807567	+	0.589776i	$0.200784\pi$
$272$		0			0
$273$		0			0
$274$	11.5981	+	6.69615i	0.700665	+	0.404529i
$275$		0			0
$276$		0			0
$277$	−5.05128	+	18.8516i	−0.303502	+	1.13269i	0.630725	+	0.776007i	$0.282758\pi$
$277$	−5.05128	+	18.8516i	−0.303502	+	1.13269i	−0.934227	+	0.356679i	$0.883909\pi$
$278$	5.65685	+	5.65685i	0.339276	+	0.339276i
$279$		0			0
$280$		0			0
$281$	−9.00000	+	5.19615i	−0.536895	+	0.309976i	−0.743820	−	0.668380i	$-0.766988\pi$
$281$	−9.00000	+	5.19615i	−0.536895	+	0.309976i	0.206925	+	0.978357i	$0.433655\pi$
$282$		0			0
$283$	1.01669	+	3.79435i	0.0604362	+	0.225551i	0.989538	−	0.144274i	$-0.0460845\pi$
$283$	1.01669	+	3.79435i	0.0604362	+	0.225551i	−0.929102	+	0.369824i	$0.879418\pi$
$284$	−3.63397	+	6.29423i	−0.215637	+	0.373494i
$285$		0			0
$286$	8.19615	+	14.1962i	0.484649	+	0.839436i
$287$	−5.79555	+	5.79555i	−0.342101	+	0.342101i
$288$		0			0
$289$			17.0000i			1.00000i
$290$		0			0
$291$		0			0
$292$	5.01910	−	1.34486i	0.293720	−	0.0787022i
$293$	−3.67423	+	0.984508i	−0.214651	+	0.0575156i	−0.364542	−	0.931187i	$-0.618774\pi$
$293$	−3.67423	+	0.984508i	−0.214651	+	0.0575156i	0.149891	+	0.988703i	$0.452108\pi$
$294$		0			0
$295$		0			0
$296$			8.19615i			0.476392i
$297$		0			0
$298$	12.7279	−	12.7279i	0.737309	−	0.737309i
$299$	−5.19615	−	9.00000i	−0.300501	−	0.520483i
$300$		0			0
$301$	1.09808	−	1.90192i	0.0632921	−	0.109625i
$302$	−2.07055	−	7.72741i	−0.119147	−	0.444662i
$303$		0			0
$304$	2.76795	−	1.59808i	0.158753	−	0.0916560i
$305$		0			0
$306$		0			0
$307$	−2.44949	−	2.44949i	−0.139800	−	0.139800i	0.633743	−	0.773543i	$-0.281517\pi$
$307$	−2.44949	−	2.44949i	−0.139800	−	0.139800i	−0.773543	+	0.633743i	$0.781517\pi$
$308$	4.24264	−	15.8338i	0.241747	−	0.902212i
$309$		0			0
$310$		0			0
$311$	9.58846	+	5.53590i	0.543712	+	0.313912i	0.746582	−	0.665294i	$-0.231694\pi$
$311$	9.58846	+	5.53590i	0.543712	+	0.313912i	−0.202870	+	0.979206i	$0.565027\pi$
$312$		0			0
$313$	−19.9563	−	5.34727i	−1.12800	−	0.302245i	−0.353880	−	0.935291i	$-0.615138\pi$
$313$	−19.9563	−	5.34727i	−1.12800	−	0.302245i	−0.774115	+	0.633045i	$0.781805\pi$
$314$	−19.8564			−1.12056
$315$		0			0
$316$	10.0000			0.562544
$317$	6.36396	+	1.70522i	0.357436	+	0.0957746i	0.433068	−	0.901361i	$-0.357431\pi$
$317$	6.36396	+	1.70522i	0.357436	+	0.0957746i	−0.0756325	+	0.997136i	$0.524098\pi$
$318$		0			0
$319$	32.1962	+	18.5885i	1.80264	+	1.04075i
$320$		0			0
$321$		0			0
$322$	−2.68973	+	10.0382i	−0.149893	+	0.559407i
$323$		0			0
$324$		0			0
$325$		0			0
$326$	12.4019	−	7.16025i	0.686879	−	0.396570i
$327$		0			0
$328$	−0.448288	−	1.67303i	−0.0247525	−	0.0923778i
$329$	−14.1962	+	24.5885i	−0.782659	+	1.35561i
$330$		0			0
$331$	6.79423	+	11.7679i	0.373445	+	0.646825i	0.990093	−	0.140414i	$-0.0448433\pi$
$331$	6.79423	+	11.7679i	0.373445	+	0.646825i	−0.616648	+	0.787239i	$0.711510\pi$
$332$	−12.1595	+	12.1595i	−0.667340	+	0.667340i
$333$		0			0
$334$		−	16.3923i		−	0.896947i
$335$		0			0
$336$		0			0
$337$	19.8362	−	5.31508i	1.08054	−	0.289531i	0.325725	−	0.945464i	$-0.394391\pi$
$337$	19.8362	−	5.31508i	1.08054	−	0.289531i	0.754819	+	0.655933i	$0.227725\pi$
$338$	9.07227	−	2.43091i	0.493466	−	0.132224i
$339$		0			0
$340$		0			0
$341$		−	0.679492i		−	0.0367966i
$342$		0			0
$343$	28.0812	−	28.0812i	1.51624	−	1.51624i
$344$	0.232051	+	0.401924i	0.0125113	+	0.0216703i
$345$		0			0
$346$	−8.19615	+	14.1962i	−0.440628	+	0.763190i
$347$	2.27362	+	8.48528i	0.122055	+	0.455514i	0.999718	−	0.0237644i	$-0.00756516\pi$
$347$	2.27362	+	8.48528i	0.122055	+	0.455514i	−0.877663	+	0.479278i	$0.840898\pi$
$348$		0			0
$349$	1.73205	−	1.00000i	0.0927146	−	0.0535288i	−0.452926	−	0.891548i	$-0.649620\pi$
$349$	1.73205	−	1.00000i	0.0927146	−	0.0535288i	0.545640	+	0.838019i	$0.316286\pi$
$350$		0			0
$351$		0			0
$352$	2.44949	+	2.44949i	0.130558	+	0.130558i
$353$	−3.88229	+	14.4889i	−0.206633	+	0.771166i	0.782312	+	0.622886i	$0.214040\pi$
$353$	−3.88229	+	14.4889i	−0.206633	+	0.771166i	−0.988946	+	0.148279i	$0.952627\pi$
$354$		0			0
$355$		0			0
$356$	7.50000	+	4.33013i	0.397499	+	0.229496i
$357$		0			0
$358$	22.0776	+	5.91567i	1.16684	+	0.312653i
$359$	35.3205			1.86415			0.932073	−	0.362272i	$-0.117999\pi$
$359$	35.3205			1.86415			0.932073	+	0.362272i	$0.117999\pi$
$360$		0			0
$361$	8.78461			0.462348
$362$	11.9700	+	3.20736i	0.629132	+	0.168575i
$363$		0			0
$364$	−19.3923	−	11.1962i	−1.01643	−	0.586838i
$365$		0			0
$366$		0			0
$367$	−1.07244	+	4.00240i	−0.0559810	+	0.208924i	−0.988251	−	0.152839i	$-0.951158\pi$
$367$	−1.07244	+	4.00240i	−0.0559810	+	0.208924i	0.932270	+	0.361763i	$0.117825\pi$
$368$	−1.55291	−	1.55291i	−0.0809513	−	0.0809513i
$369$		0			0
$370$		0			0
$371$	33.5885	−	19.3923i	1.74383	−	1.00680i
$372$		0			0
$373$	−6.03579	−	22.5259i	−0.312521	−	1.16635i	−0.926275	−	0.376848i	$-0.877008\pi$
$373$	−6.03579	−	22.5259i	−0.312521	−	1.16635i	0.613754	−	0.789498i	$-0.289659\pi$
$374$		0			0
$375$		0			0
$376$	−3.00000	−	5.19615i	−0.154713	−	0.267971i
$377$	35.9101	−	35.9101i	1.84947	−	1.84947i
$378$		0			0
$379$			0.392305i			0.0201513i	0.999949	+	0.0100757i	$0.00320724\pi$
$379$			0.392305i			0.0201513i	−0.999949	+	0.0100757i	$0.996793\pi$
$380$		0			0
$381$		0			0
$382$	3.34607	−	0.896575i	0.171200	−	0.0458728i
$383$	−5.22715	+	1.40061i	−0.267095	+	0.0715678i	−0.389881	−	0.920865i	$-0.627484\pi$
$383$	−5.22715	+	1.40061i	−0.267095	+	0.0715678i	0.122786	+	0.992433i	$0.460817\pi$
$384$		0			0
$385$		0			0
$386$			11.0718i			0.563540i
$387$		0			0
$388$	−1.88108	+	1.88108i	−0.0954976	+	0.0954976i
$389$	−5.19615	−	9.00000i	−0.263455	−	0.456318i	0.703702	−	0.710495i	$-0.251529\pi$
$389$	−5.19615	−	9.00000i	−0.263455	−	0.456318i	−0.967158	+	0.254177i	$0.918196\pi$
$390$		0			0
$391$		0			0
$392$	3.98382	+	14.8678i	0.201213	+	0.750939i
$393$		0			0
$394$	−12.2942	+	7.09808i	−0.619374	+	0.357596i
$395$		0			0
$396$		0			0
$397$	10.9348	+	10.9348i	0.548800	+	0.548800i	0.926094	−	0.377293i	$-0.123145\pi$
$397$	10.9348	+	10.9348i	0.548800	+	0.548800i	−0.377293	+	0.926094i	$0.623145\pi$
$398$	5.27792	−	19.6975i	0.264558	−	0.987344i
$399$		0			0
$400$		0			0
$401$	−9.00000	−	5.19615i	−0.449439	−	0.259483i	0.258154	−	0.966104i	$-0.416886\pi$
$401$	−9.00000	−	5.19615i	−0.449439	−	0.259483i	−0.707593	+	0.706620i	$0.750219\pi$
$402$		0			0
$403$	−0.896575	−	0.240237i	−0.0446616	−	0.0119670i
$404$	−7.26795			−0.361594
$405$		0			0
$406$	−50.7846			−2.52040
$407$	27.4249	+	7.34847i	1.35940	+	0.364250i
$408$		0			0
$409$	−18.5263	−	10.6962i	−0.916066	−	0.528891i	−0.0336878	−	0.999432i	$-0.510725\pi$
$409$	−18.5263	−	10.6962i	−0.916066	−	0.528891i	−0.882378	+	0.470542i	$0.844059\pi$
$410$		0			0
$411$		0			0
$412$	2.68973	−	10.0382i	0.132513	−	0.494546i
$413$	18.5235	+	18.5235i	0.911481	+	0.911481i
$414$		0			0
$415$		0			0
$416$	4.09808	−	2.36603i	0.200925	−	0.116004i
$417$		0			0
$418$	−2.86559	−	10.6945i	−0.140161	−	0.523087i
$419$	4.50000	−	7.79423i	0.219839	−	0.380773i	−0.734919	−	0.678155i	$-0.762780\pi$
$419$	4.50000	−	7.79423i	0.219839	−	0.380773i	0.954759	+	0.297382i	$0.0961133\pi$
$420$		0			0
$421$	−11.2942	−	19.5622i	−0.550447	−	0.953402i	−0.998242	−	0.0592661i	$-0.981124\pi$
$421$	−11.2942	−	19.5622i	−0.550447	−	0.953402i	0.447795	−	0.894136i	$-0.352209\pi$
$422$	0.845807	−	0.845807i	0.0411733	−	0.0411733i
$423$		0			0
$424$			8.19615i			0.398040i
$425$		0			0
$426$		0			0
$427$	−18.2832	+	4.89898i	−0.884788	+	0.237078i
$428$	−10.8147	+	2.89778i	−0.522746	+	0.140069i
$429$		0			0
$430$		0			0
$431$		−	21.1244i		−	1.01752i	−0.860907	−	0.508762i	$-0.830103\pi$
$431$		−	21.1244i		−	1.01752i	0.860907	−	0.508762i	$-0.169897\pi$
$432$		0			0
$433$	−12.0716	+	12.0716i	−0.580123	+	0.580123i	−0.934937	−	0.354814i	$-0.884544\pi$
$433$	−12.0716	+	12.0716i	−0.580123	+	0.580123i	0.354814	+	0.934937i	$0.384544\pi$
$434$	0.464102	+	0.803848i	0.0222776	+	0.0385859i
$435$		0			0
$436$	−0.901924	+	1.56218i	−0.0431943	+	0.0748147i
$437$	1.81671	+	6.78006i	0.0869051	+	0.324334i
$438$		0			0
$439$	−1.22243	+	0.705771i	−0.0583435	+	0.0336846i	−0.528888	−	0.848692i	$-0.677391\pi$
$439$	−1.22243	+	0.705771i	−0.0583435	+	0.0336846i	0.470545	+	0.882376i	$0.344058\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.68973	+	10.0382i	−0.127793	+	0.476929i	−0.999924	−	0.0123433i	$-0.996071\pi$
$443$	−2.68973	+	10.0382i	−0.127793	+	0.476929i	0.872131	+	0.489272i	$0.162738\pi$
$444$		0			0
$445$		0			0
$446$	2.19615	+	1.26795i	0.103991	+	0.0600391i
$447$		0			0
$448$	−4.57081	−	1.22474i	−0.215950	−	0.0578638i
$449$	−30.1244			−1.42166			−0.710828	−	0.703366i	$-0.751680\pi$
$449$	−30.1244			−1.42166			−0.710828	+	0.703366i	$0.751680\pi$
$450$		0			0
$451$	−6.00000			−0.282529
$452$	−12.9360	−	3.46618i	−0.608457	−	0.163036i
$453$		0			0
$454$	11.0885	+	6.40192i	0.520407	+	0.300457i
$455$		0			0
$456$		0			0
$457$	7.14042	−	26.6484i	0.334015	−	1.24656i	−0.570919	−	0.821007i	$-0.693413\pi$
$457$	7.14042	−	26.6484i	0.334015	−	1.24656i	0.904933	−	0.425553i	$-0.139920\pi$
$458$	4.10394	+	4.10394i	0.191765	+	0.191765i
$459$		0			0
$460$		0			0
$461$	8.70577	−	5.02628i	0.405468	−	0.234097i	−0.283373	−	0.959010i	$-0.591453\pi$
$461$	8.70577	−	5.02628i	0.405468	−	0.234097i	0.688841	+	0.724913i	$0.258120\pi$
$462$		0			0
$463$	−0.568406	−	2.12132i	−0.0264161	−	0.0985861i	0.951459	−	0.307775i	$-0.0995844\pi$
$463$	−0.568406	−	2.12132i	−0.0264161	−	0.0985861i	−0.977875	+	0.209189i	$0.932918\pi$
$464$	5.36603	−	9.29423i	0.249111	−	0.431474i
$465$		0			0
$466$	2.30385	+	3.99038i	0.106724	+	0.184851i
$467$	−6.78006	+	6.78006i	−0.313744	+	0.313744i	−0.846358	−	0.532614i	$-0.821210\pi$
$467$	−6.78006	+	6.78006i	−0.313744	+	0.313744i	0.532614	+	0.846358i	$0.321210\pi$
$468$		0			0
$469$		−	26.1962i		−	1.20963i
$470$		0			0
$471$		0			0
$472$	−5.34727	+	1.43280i	−0.246128	+	0.0659498i
$473$	1.55291	−	0.416102i	0.0714031	−	0.0191324i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$	−2.68973	+	2.68973i	−0.123025	+	0.123025i
$479$	1.56218	+	2.70577i	0.0713777	+	0.123630i	0.899505	−	0.436910i	$-0.143927\pi$
$479$	1.56218	+	2.70577i	0.0713777	+	0.123630i	−0.828128	+	0.560540i	$0.810594\pi$
$480$		0			0
$481$	19.3923	−	33.5885i	0.884213	−	1.53150i
$482$	2.43091	+	9.07227i	0.110725	+	0.413231i
$483$		0			0
$484$	0.866025	−	0.500000i	0.0393648	−	0.0227273i
$485$		0			0
$486$		0			0
$487$	−11.3509	−	11.3509i	−0.514357	−	0.514357i	0.401501	−	0.915858i	$-0.368489\pi$
$487$	−11.3509	−	11.3509i	−0.514357	−	0.514357i	−0.915858	+	0.401501i	$0.868489\pi$
$488$	1.03528	−	3.86370i	0.0468648	−	0.174902i
$489$		0			0
$490$		0			0
$491$	16.2058	+	9.35641i	0.731356	+	0.422249i	0.818918	−	0.573910i	$-0.194574\pi$
$491$	16.2058	+	9.35641i	0.731356	+	0.422249i	−0.0875619	+	0.996159i	$0.527908\pi$
$492$		0			0
$493$		0			0
$494$	−15.1244			−0.680477
$495$		0			0
$496$	−0.196152			−0.00880750
$497$	−33.2204	−	8.90138i	−1.49014	−	0.399282i
$498$		0			0
$499$	10.0359	+	5.79423i	0.449269	+	0.259385i	0.707521	−	0.706692i	$-0.249813\pi$
$499$	10.0359	+	5.79423i	0.449269	+	0.259385i	−0.258253	+	0.966077i	$0.583147\pi$
$500$		0			0
$501$		0			0
$502$	2.32937	−	8.69333i	0.103965	−	0.388002i
$503$	−3.82654	−	3.82654i	−0.170617	−	0.170617i	0.616633	−	0.787250i	$-0.288496\pi$
$503$	−3.82654	−	3.82654i	−0.170617	−	0.170617i	−0.787250	+	0.616633i	$0.788496\pi$
$504$		0			0
$505$		0			0
$506$	−6.58846	+	3.80385i	−0.292893	+	0.169102i
$507$		0			0
$508$	−2.12132	−	7.91688i	−0.0941184	−	0.351255i
$509$	−10.3923	+	18.0000i	−0.460631	+	0.797836i	−0.998992	−	0.0448779i	$-0.985710\pi$
$509$	−10.3923	+	18.0000i	−0.460631	+	0.797836i	0.538362	+	0.842714i	$0.319043\pi$
$510$		0			0
$511$	12.2942	+	21.2942i	0.543865	+	0.942001i
$512$	0.707107	−	0.707107i	0.0312500	−	0.0312500i
$513$		0			0
$514$		−	25.3923i		−	1.12001i
$515$		0			0
$516$		0			0
$517$	−20.0764	+	5.37945i	−0.882959	+	0.236588i
$518$	−37.4631	+	10.0382i	−1.64603	+	0.441053i
$519$		0			0
$520$		0			0
$521$			2.78461i			0.121996i	0.998138	+	0.0609980i	$0.0194283\pi$
$521$			2.78461i			0.121996i	−0.998138	+	0.0609980i	$0.980572\pi$
$522$		0			0
$523$	−23.9909	+	23.9909i	−1.04905	+	1.04905i	−0.0503137	+	0.998733i	$0.516022\pi$
$523$	−23.9909	+	23.9909i	−1.04905	+	1.04905i	−0.998733	+	0.0503137i	$0.983978\pi$
$524$	3.46410	+	6.00000i	0.151330	+	0.262111i
$525$		0			0
$526$	−6.29423	+	10.9019i	−0.274441	+	0.475346i
$527$		0			0
$528$		0			0
$529$	−15.7417	+	9.08846i	−0.684420	+	0.395150i
$530$		0			0
$531$		0			0
$532$	10.6945	+	10.6945i	0.463667	+	0.463667i
$533$	−2.12132	+	7.91688i	−0.0918846	+	0.342918i
$534$		0			0
$535$		0			0
$536$	4.79423	+	2.76795i	0.207079	+	0.119557i
$537$		0			0
$538$	−9.71003	−	2.60179i	−0.418629	−	0.112171i
$539$	53.3205			2.29668
$540$		0			0
$541$	−8.39230			−0.360813			−0.180407	−	0.983592i	$-0.557741\pi$
$541$	−8.39230			−0.360813			−0.180407	+	0.983592i	$0.557741\pi$
$542$	−25.6825	−	6.88160i	−1.10316	−	0.295590i
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−10.3986	+	38.8079i	−0.444610	+	1.65931i	0.272354	+	0.962197i	$0.412198\pi$
$547$	−10.3986	+	38.8079i	−0.444610	+	1.65931i	−0.716964	+	0.697110i	$0.754469\pi$
$548$	−9.46979	−	9.46979i	−0.404529	−	0.404529i
$549$		0			0
$550$		0			0
$551$	−29.7058	+	17.1506i	−1.26551	+	0.730642i
$552$		0			0
$553$	12.2474	+	45.7081i	0.520814	+	1.94371i
$554$	9.75833	−	16.9019i	0.414592	−	0.718094i
$555$		0			0
$556$	−4.00000	−	6.92820i	−0.169638	−	0.293821i
$557$	−3.10583	+	3.10583i	−0.131598	+	0.131598i	−0.769838	−	0.638240i	$-0.779663\pi$
$557$	−3.10583	+	3.10583i	−0.131598	+	0.131598i	0.638240	+	0.769838i	$0.279663\pi$
$558$		0			0
$559$		−	2.19615i		−	0.0928874i
$560$		0			0
$561$		0			0
$562$	10.0382	−	2.68973i	0.423436	−	0.113459i
$563$	2.32937	−	0.624153i	0.0981713	−	0.0263049i	−0.209399	−	0.977830i	$-0.567151\pi$
$563$	2.32937	−	0.624153i	0.0981713	−	0.0263049i	0.307570	+	0.951525i	$0.400484\pi$
$564$		0			0
$565$		0			0
$566$		−	3.92820i		−	0.165115i
$567$		0			0
$568$	5.13922	−	5.13922i	0.215637	−	0.215637i
$569$	12.4641	+	21.5885i	0.522522	+	0.905035i	0.999657	+	0.0262048i	$0.00834220\pi$
$569$	12.4641	+	21.5885i	0.522522	+	0.905035i	−0.477134	+	0.878830i	$0.658324\pi$
$570$		0			0
$571$	−9.59808	+	16.6244i	−0.401667	+	0.695708i	−0.993927	−	0.110039i	$-0.964902\pi$
$571$	−9.59808	+	16.6244i	−0.401667	+	0.695708i	0.592260	+	0.805747i	$0.298236\pi$
$572$	−4.24264	−	15.8338i	−0.177394	−	0.662042i
$573$		0			0
$574$	7.09808	−	4.09808i	0.296268	−	0.171050i
$575$		0			0
$576$		0			0
$577$	15.9217	+	15.9217i	0.662828	+	0.662828i	0.956046	−	0.293217i	$-0.0947260\pi$
$577$	15.9217	+	15.9217i	0.662828	+	0.662828i	−0.293217	+	0.956046i	$0.594726\pi$
$578$	4.39992	−	16.4207i	0.183013	−	0.683013i
$579$		0			0
$580$		0			0
$581$	−70.4711	−	40.6865i	−2.92364	−	1.68796i
$582$		0			0
$583$	27.4249	+	7.34847i	1.13582	+	0.304342i
$584$	−5.19615			−0.215018
$585$		0			0
$586$	3.80385			0.157135
$587$	−4.24264	−	1.13681i	−0.175113	−	0.0469213i	0.170197	−	0.985410i	$-0.445559\pi$
$587$	−4.24264	−	1.13681i	−0.175113	−	0.0469213i	−0.345310	+	0.938489i	$0.612226\pi$
$588$		0			0
$589$	0.542940	+	0.313467i	0.0223715	+	0.0129162i
$590$		0			0
$591$		0			0
$592$	2.12132	−	7.91688i	0.0871857	−	0.325382i
$593$	14.8492	+	14.8492i	0.609785	+	0.609785i	0.942890	−	0.333105i	$-0.108096\pi$
$593$	14.8492	+	14.8492i	0.609785	+	0.609785i	−0.333105	+	0.942890i	$0.608096\pi$
$594$		0			0
$595$		0			0
$596$	−15.5885	+	9.00000i	−0.638528	+	0.368654i
$597$		0			0
$598$	2.68973	+	10.0382i	0.109991	+	0.410492i
$599$	3.63397	−	6.29423i	0.148480	−	0.257175i	−0.782186	−	0.623045i	$-0.785895\pi$
$599$	3.63397	−	6.29423i	0.148480	−	0.257175i	0.930666	+	0.365870i	$0.119229\pi$
$600$		0			0
$601$	14.3923	+	24.9282i	0.587074	+	1.01684i	0.994613	+	0.103655i	$0.0330537\pi$
$601$	14.3923	+	24.9282i	0.587074	+	1.01684i	−0.407539	+	0.913188i	$0.633613\pi$
$602$	−1.55291	+	1.55291i	−0.0632921	+	0.0632921i
$603$		0			0
$604$			8.00000i			0.325515i
$605$		0			0
$606$		0			0
$607$	−46.9328	+	12.5756i	−1.90495	+	0.510429i	−0.909427	+	0.415864i	$0.863479\pi$
$607$	−46.9328	+	12.5756i	−1.90495	+	0.510429i	−0.995519	+	0.0945643i	$0.969854\pi$
$608$	−3.08725	+	0.827225i	−0.125204	+	0.0335484i
$609$		0			0
$610$		0			0
$611$			28.3923i			1.14863i
$612$		0			0
$613$	−10.0382	+	10.0382i	−0.405439	+	0.405439i	−0.880145	−	0.474706i	$-0.842555\pi$
$613$	−10.0382	+	10.0382i	−0.405439	+	0.405439i	0.474706	+	0.880145i	$0.342555\pi$
$614$	1.73205	+	3.00000i	0.0698999	+	0.121070i
$615$		0			0
$616$	−8.19615	+	14.1962i	−0.330232	+	0.571979i
$617$	−3.05008	−	11.3831i	−0.122792	−	0.458265i	0.876960	−	0.480564i	$-0.159568\pi$
$617$	−3.05008	−	11.3831i	−0.122792	−	0.458265i	−0.999751	+	0.0222993i	$0.992901\pi$
$618$		0			0
$619$	−41.2128	+	23.7942i	−1.65648	+	0.956371i	−0.682164	+	0.731200i	$0.738961\pi$
$619$	−41.2128	+	23.7942i	−1.65648	+	0.956371i	−0.974319	+	0.225171i	$0.927706\pi$
$620$		0			0
$621$		0			0
$622$	−7.82894	−	7.82894i	−0.313912	−	0.313912i
$623$	−10.6066	+	39.5844i	−0.424945	+	1.58591i
$624$		0			0
$625$		0			0
$626$	17.8923	+	10.3301i	0.715120	+	0.412875i
$627$		0			0
$628$	19.1798	+	5.13922i	0.765358	+	0.205077i
$629$		0			0
$630$		0			0
$631$	−0.784610			−0.0312348			−0.0156174	−	0.999878i	$-0.504971\pi$
$631$	−0.784610			−0.0312348			−0.0156174	+	0.999878i	$0.504971\pi$
$632$	−9.65926	−	2.58819i	−0.384225	−	0.102953i
$633$		0			0
$634$	−5.70577	−	3.29423i	−0.226605	−	0.130831i
$635$		0			0
$636$		0			0
$637$	18.8516	−	70.3553i	0.746929	−	2.78758i
$638$	−26.2880	−	26.2880i	−1.04075	−	1.04075i
$639$		0			0
$640$		0			0
$641$	32.0885	−	18.5263i	1.26742	−	0.731744i	0.292919	−	0.956137i	$-0.405373\pi$
$641$	32.0885	−	18.5263i	1.26742	−	0.731744i	0.974499	+	0.224393i	$0.0720400\pi$
$642$		0			0
$643$	9.67784	+	36.1182i	0.381657	+	1.42436i	0.843370	+	0.537333i	$0.180568\pi$
$643$	9.67784	+	36.1182i	0.381657	+	1.42436i	−0.461713	+	0.887029i	$0.652765\pi$
$644$	5.19615	−	9.00000i	0.204757	−	0.354650i
$645$		0			0
$646$		0			0
$647$	−10.0382	+	10.0382i	−0.394642	+	0.394642i	−0.876338	−	0.481696i	$-0.840021\pi$
$647$	−10.0382	+	10.0382i	−0.394642	+	0.394642i	0.481696	+	0.876338i	$0.340021\pi$
$648$		0			0
$649$			19.1769i			0.752760i
$650$		0			0
$651$		0			0
$652$	−13.8325	+	3.70642i	−0.541724	+	0.145155i
$653$	7.91688	−	2.12132i	0.309811	−	0.0830137i	−0.100564	−	0.994931i	$-0.532065\pi$
$653$	7.91688	−	2.12132i	0.309811	−	0.0830137i	0.410375	+	0.911917i	$0.365398\pi$
$654$		0			0
$655$		0			0
$656$			1.73205i			0.0676252i
$657$		0			0
$658$	20.0764	−	20.0764i	0.782659	−	0.782659i
$659$	−20.0885	−	34.7942i	−0.782535	−	1.35539i	−0.930461	−	0.366392i	$-0.880593\pi$
$659$	−20.0885	−	34.7942i	−0.782535	−	1.35539i	0.147925	−	0.988999i	$-0.452740\pi$
$660$		0			0
$661$	19.5885	−	33.9282i	0.761903	−	1.31965i	−0.179966	−	0.983673i	$-0.557599\pi$
$661$	19.5885	−	33.9282i	0.761903	−	1.31965i	0.941869	−	0.335981i	$-0.109068\pi$
$662$	−3.51695	−	13.1254i	−0.136690	−	0.510135i
$663$		0			0
$664$	14.8923	−	8.59808i	0.577934	−	0.333670i
$665$		0			0
$666$		0			0
$667$	16.6660	+	16.6660i	0.645308	+	0.645308i
$668$	−4.24264	+	15.8338i	−0.164153	+	0.612626i
$669$		0			0
$670$		0			0
$671$	−12.0000	−	6.92820i	−0.463255	−	0.267460i
$672$		0			0
$673$	−21.6293	−	5.79555i	−0.833748	−	0.223402i	−0.183400	−	0.983038i	$-0.558710\pi$
$673$	−21.6293	−	5.79555i	−0.833748	−	0.223402i	−0.650348	+	0.759636i	$0.725377\pi$
$674$	−20.5359			−0.791013
$675$		0			0
$676$	−9.39230			−0.361242
$677$	−23.7506	−	6.36396i	−0.912811	−	0.244587i	−0.228301	−	0.973591i	$-0.573317\pi$
$677$	−23.7506	−	6.36396i	−0.912811	−	0.244587i	−0.684510	+	0.729004i	$0.739984\pi$
$678$		0			0
$679$	−10.9019	−	6.29423i	−0.418377	−	0.241550i
$680$		0			0
$681$		0			0
$682$	−0.175865	+	0.656339i	−0.00673424	+	0.0251325i
$683$	−5.94786	−	5.94786i	−0.227588	−	0.227588i	0.584096	−	0.811685i	$-0.301449\pi$
$683$	−5.94786	−	5.94786i	−0.227588	−	0.227588i	−0.811685	+	0.584096i	$0.801449\pi$
$684$		0			0
$685$		0			0
$686$	−34.3923	+	19.8564i	−1.31310	+	0.758121i
$687$		0			0
$688$	−0.120118	−	0.448288i	−0.00457947	−	0.0170908i
$689$	19.3923	−	33.5885i	0.738788	−	1.27962i
$690$		0			0
$691$	−21.7942	−	37.7487i	−0.829092	−	1.43603i	−0.898752	−	0.438458i	$-0.855525\pi$
$691$	−21.7942	−	37.7487i	−0.829092	−	1.43603i	0.0696602	−	0.997571i	$-0.477808\pi$
$692$	11.5911	−	11.5911i	0.440628	−	0.440628i
$693$		0			0
$694$		−	8.78461i		−	0.333459i
$695$		0			0
$696$		0			0
$697$		0			0
$698$	−1.93185	+	0.517638i	−0.0731217	+	0.0195929i
$699$		0			0
$700$		0			0
$701$			14.5359i			0.549013i	0.961585	+	0.274507i	$0.0885145\pi$
$701$			14.5359i			0.549013i	−0.961585	+	0.274507i	$0.911485\pi$
$702$		0			0
$703$	−18.5235	+	18.5235i	−0.698626	+	0.698626i
$704$	−1.73205	−	3.00000i	−0.0652791	−	0.113067i
$705$		0			0
$706$	7.50000	−	12.9904i	0.282266	−	0.488899i
$707$	−8.90138	−	33.2204i	−0.334771	−	1.24938i
$708$		0			0
$709$	−31.3468	+	18.0981i	−1.17725	+	0.679688i	−0.955378	−	0.295387i	$-0.904551\pi$
$709$	−31.3468	+	18.0981i	−1.17725	+	0.679688i	−0.221876	+	0.975075i	$0.571218\pi$
$710$		0			0
$711$		0			0
$712$	−6.12372	−	6.12372i	−0.229496	−	0.229496i
$713$	0.111494	−	0.416102i	0.00417549	−	0.0155831i
$714$		0			0
$715$		0			0
$716$	−19.7942	−	11.4282i	−0.739745	−	0.427092i
$717$		0			0
$718$	−34.1170	−	9.14162i	−1.27323	−	0.341162i
$719$	14.8756			0.554768			0.277384	−	0.960759i	$-0.410533\pi$
$719$	14.8756			0.554768			0.277384	+	0.960759i	$0.410533\pi$
$720$		0			0
$721$	49.1769			1.83144
$722$	−8.48528	−	2.27362i	−0.315789	−	0.0846155i
$723$		0			0
$724$	−10.7321	−	6.19615i	−0.398854	−	0.230278i
$725$		0			0
$726$		0			0
$727$	1.79315	−	6.69213i	0.0665043	−	0.248197i	−0.924669	−	0.380773i	$-0.875658\pi$
$727$	1.79315	−	6.69213i	0.0665043	−	0.248197i	0.991173	+	0.132575i	$0.0423247\pi$
$728$	15.8338	+	15.8338i	0.586838	+	0.586838i
$729$		0			0
$730$		0			0
$731$		0			0
$732$		0			0
$733$	3.52193	+	13.1440i	0.130085	+	0.485486i	0.999970	−	0.00777015i	$-0.00247334\pi$
$733$	3.52193	+	13.1440i	0.130085	+	0.485486i	−0.869884	+	0.493256i	$0.835807\pi$
$734$	2.07180	−	3.58846i	0.0764714	−	0.132452i
$735$		0			0
$736$	1.09808	+	1.90192i	0.0404756	+	0.0701058i
$737$	13.5601	−	13.5601i	0.499494	−	0.499494i
$738$		0			0
$739$		−	19.5885i		−	0.720573i	−0.932842	−	0.360287i	$-0.882679\pi$
$739$		−	19.5885i		−	0.720573i	0.932842	−	0.360287i	$-0.117321\pi$
$740$		0			0
$741$		0			0
$742$	−37.4631	+	10.0382i	−1.37531	+	0.368514i
$743$	−32.2359	+	8.63759i	−1.18262	+	0.316882i	−0.795965	−	0.605342i	$-0.793036\pi$
$743$	−32.2359	+	8.63759i	−1.18262	+	0.316882i	−0.386655	+	0.922224i	$0.626370\pi$
$744$		0			0
$745$		0			0
$746$			23.3205i			0.853824i
$747$		0			0
$748$		0			0
$749$	−26.4904	−	45.8827i	−0.967937	−	1.67652i
$750$		0			0
$751$	−5.29423	+	9.16987i	−0.193189	+	0.334613i	−0.946305	−	0.323274i	$-0.895216\pi$
$751$	−5.29423	+	9.16987i	−0.193189	+	0.334613i	0.753116	+	0.657887i	$0.228550\pi$
$752$	1.55291	+	5.79555i	0.0566290	+	0.211342i
$753$		0			0
$754$	−43.9808	+	25.3923i	−1.60168	+	0.924733i
$755$		0			0
$756$		0			0
$757$	−27.6651	−	27.6651i	−1.00551	−	1.00551i	−0.999985	−	0.00552030i	$-0.998243\pi$
$757$	−27.6651	−	27.6651i	−1.00551	−	1.00551i	−0.00552030	−	0.999985i	$-0.501757\pi$
$758$	0.101536	−	0.378937i	0.00368795	−	0.0137636i
$759$		0			0
$760$		0			0
$761$	41.0885	+	23.7224i	1.48946	+	0.859937i	0.999927	−	0.0120501i	$-0.00383575\pi$
$761$	41.0885	+	23.7224i	1.48946	+	0.859937i	0.489528	+	0.871988i	$0.337169\pi$
$762$		0			0
$763$	−8.24504	−	2.20925i	−0.298491	−	0.0799803i
$764$	−3.46410			−0.125327
$765$		0			0
$766$	5.41154			0.195527
$767$	25.3035	+	6.78006i	0.913658	+	0.244814i
$768$		0			0
$769$	10.9186	+	6.30385i	0.393734	+	0.227323i	0.683777	−	0.729691i	$-0.260336\pi$
$769$	10.9186	+	6.30385i	0.393734	+	0.227323i	−0.290043	+	0.957014i	$0.593669\pi$
$770$		0			0
$771$		0			0
$772$	2.86559	−	10.6945i	0.103135	−	0.384905i
$773$	−35.9101	−	35.9101i	−1.29160	−	1.29160i	−0.933798	−	0.357800i	$-0.883527\pi$
$773$	−35.9101	−	35.9101i	−1.29160	−	1.29160i	−0.357800	−	0.933798i	$-0.616473\pi$
$774$		0			0
$775$		0			0
$776$	2.30385	−	1.33013i	0.0827033	−	0.0477488i
$777$		0			0
$778$	2.68973	+	10.0382i	0.0964314	+	0.359887i
$779$	2.76795	−	4.79423i	0.0991721	−	0.171771i
$780$		0			0
$781$	−12.5885	−	21.8038i	−0.450450	−	0.780203i
$782$		0			0
$783$		0			0
$784$		−	15.3923i		−	0.549725i
$785$		0			0
$786$		0			0
$787$	16.7303	−	4.48288i	0.596372	−	0.159797i	0.0520081	−	0.998647i	$-0.483438\pi$
$787$	16.7303	−	4.48288i	0.596372	−	0.159797i	0.544364	+	0.838849i	$0.316771\pi$
$788$	13.7124	−	3.67423i	0.488485	−	0.130889i
$789$		0			0
$790$		0			0
$791$		−	63.3731i		−	2.25329i
$792$		0			0
$793$	−13.3843	+	13.3843i	−0.475289	+	0.475289i
$794$	−7.73205	−	13.3923i	−0.274400	−	0.475275i
$795$		0			0
$796$	−10.1962	+	17.6603i	−0.361393	+	0.625951i
$797$	−10.6066	−	39.5844i	−0.375705	−	1.40215i	−0.852312	−	0.523034i	$-0.824800\pi$
$797$	−10.6066	−	39.5844i	−0.375705	−	1.40215i	0.476607	−	0.879117i	$-0.341867\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$	7.34847	+	7.34847i	0.259483	+	0.259483i
$803$	−4.65874	+	17.3867i	−0.164403	+	0.613562i
$804$		0			0
$805$		0			0
$806$	0.803848	+	0.464102i	0.0283143	+	0.0163473i
$807$		0			0
$808$	7.02030	+	1.88108i	0.246973	+	0.0661763i
$809$	1.73205			0.0608957			0.0304478	−	0.999536i	$-0.490307\pi$
$809$	1.73205			0.0608957			0.0304478	+	0.999536i	$0.490307\pi$
$810$		0			0
$811$	34.3731			1.20700			0.603501	−	0.797362i	$-0.293772\pi$
$811$	34.3731			1.20700			0.603501	+	0.797362i	$0.293772\pi$
$812$	49.0542	+	13.1440i	1.72146	+	0.461265i
$813$		0			0
$814$	−24.5885	−	14.1962i	−0.861825	−	0.497575i
$815$		0			0
$816$		0			0
$817$	−0.383917	+	1.43280i	−0.0134315	+	0.0501272i
$818$	15.1266	+	15.1266i	0.528891	+	0.528891i
$819$		0			0
$820$		0			0
$821$	−12.5885	+	7.26795i	−0.439340	+	0.253653i	−0.703318	−	0.710876i	$-0.748299\pi$
$821$	−12.5885	+	7.26795i	−0.439340	+	0.253653i	0.263978	+	0.964529i	$0.414966\pi$
$822$		0			0
$823$	10.5187	+	39.2562i	0.366658	+	1.36839i	0.865160	+	0.501497i	$0.167217\pi$
$823$	10.5187	+	39.2562i	0.366658	+	1.36839i	−0.498502	+	0.866889i	$0.666116\pi$
$824$	−5.19615	+	9.00000i	−0.181017	+	0.313530i
$825$		0			0
$826$	−13.0981	−	22.6865i	−0.455740	−	0.789365i
$827$	−0.568406	+	0.568406i	−0.0197654	+	0.0197654i	−0.716920	−	0.697155i	$-0.754449\pi$
$827$	−0.568406	+	0.568406i	−0.0197654	+	0.0197654i	0.697155	+	0.716920i	$0.254449\pi$
$828$		0			0
$829$		−	28.5885i		−	0.992918i	−0.868060	−	0.496459i	$-0.834633\pi$
$829$		−	28.5885i		−	0.992918i	0.868060	−	0.496459i	$-0.165367\pi$
$830$		0			0
$831$		0			0
$832$	−4.57081	+	1.22474i	−0.158464	+	0.0424604i
$833$		0			0
$834$		0			0
$835$		0			0
$836$			11.0718i			0.382926i
$837$		0			0
$838$	−6.36396	+	6.36396i	−0.219839	+	0.219839i
$839$	6.75833	+	11.7058i	0.233323	+	0.404128i	0.958784	−	0.284136i	$-0.0917066\pi$
$839$	6.75833	+	11.7058i	0.233323	+	0.404128i	−0.725461	+	0.688264i	$0.758373\pi$
$840$		0			0
$841$	−43.0885	+	74.6314i	−1.48581	+	2.57350i
$842$	5.84632	+	21.8188i	0.201478	+	0.751925i
$843$		0			0
$844$	−1.03590	+	0.598076i	−0.0356571	+	0.0205866i
$845$		0			0
$846$		0			0
$847$	3.34607	+	3.34607i	0.114972	+	0.114972i
$848$	2.12132	−	7.91688i	0.0728464	−	0.271867i
$849$		0			0
$850$		0			0
$851$	15.5885	+	9.00000i	0.534365	+	0.308516i
$852$		0			0
$853$	18.0430	+	4.83461i	0.617781	+	0.165534i	0.554119	−	0.832438i	$-0.313055\pi$
$853$	18.0430	+	4.83461i	0.617781	+	0.165534i	0.0636620	+	0.997972i	$0.479722\pi$
$854$	18.9282			0.647710
$855$		0			0
$856$	11.1962			0.382677
$857$	−40.3608	−	10.8147i	−1.37870	−	0.369422i	−0.508051	−	0.861327i	$-0.669634\pi$
$857$	−40.3608	−	10.8147i	−1.37870	−	0.369422i	−0.870649	+	0.491905i	$0.836301\pi$
$858$		0			0
$859$	36.0167	+	20.7942i	1.22887	+	0.709490i	0.966794	−	0.255557i	$-0.0822587\pi$
$859$	36.0167	+	20.7942i	1.22887	+	0.709490i	0.262079	+	0.965047i	$0.415592\pi$
$860$		0			0
$861$		0			0
$862$	−5.46739	+	20.4046i	−0.186220	+	0.694982i
$863$	33.2204	+	33.2204i	1.13084	+	1.13084i	0.990038	+	0.140798i	$0.0449667\pi$
$863$	33.2204	+	33.2204i	1.13084	+	1.13084i	0.140798	+	0.990038i	$0.455033\pi$
$864$		0			0
$865$		0			0
$866$	14.7846	−	8.53590i	0.502401	−	0.290062i
$867$		0			0
$868$	−0.240237	−	0.896575i	−0.00815416	−	0.0304318i
$869$	−17.3205	+	30.0000i	−0.587558	+	1.01768i
$870$		0			0
$871$	−13.0981	−	22.6865i	−0.443811	−	0.768704i
$872$	1.27551	−	1.27551i	0.0431943	−	0.0431943i
$873$		0			0
$874$		−	7.01924i		−	0.237429i
$875$		0			0
$876$		0			0
$877$	35.6699	−	9.55772i	1.20449	−	0.322741i	0.399890	−	0.916563i	$-0.369048\pi$
$877$	35.6699	−	9.55772i	1.20449	−	0.322741i	0.804597	+	0.593822i	$0.202382\pi$
$878$	1.36345	−	0.365334i	0.0460141	−	0.0123294i
$879$		0			0
$880$		0			0
$881$		−	18.6795i		−	0.629328i	−0.949203	−	0.314664i	$-0.898108\pi$
$881$		−	18.6795i		−	0.629328i	0.949203	−	0.314664i	$-0.101892\pi$
$882$		0			0
$883$	18.2832	−	18.2832i	0.615280	−	0.615280i	−0.329037	−	0.944317i	$-0.606724\pi$
$883$	18.2832	−	18.2832i	0.615280	−	0.615280i	0.944317	+	0.329037i	$0.106724\pi$
$884$		0			0
$885$		0			0
$886$	5.19615	−	9.00000i	0.174568	−	0.302361i
$887$	2.68973	+	10.0382i	0.0903122	+	0.337050i	0.996267	−	0.0863246i	$-0.0275122\pi$
$887$	2.68973	+	10.0382i	0.0903122	+	0.337050i	−0.905955	+	0.423374i	$0.860846\pi$
$888$		0			0
$889$	33.5885	−	19.3923i	1.12652	−	0.650397i
$890$		0			0
$891$		0			0
$892$	−1.79315	−	1.79315i	−0.0600391	−	0.0600391i
$893$	4.96335	−	18.5235i	0.166092	−	0.619865i
$894$		0			0
$895$		0			0
$896$	4.09808	+	2.36603i	0.136907	+	0.0790434i
$897$		0			0
$898$	29.0979	+	7.79676i	0.971009	+	0.260181i
$899$	2.10512			0.0702096
$900$		0			0
$901$		0			0
$902$	5.79555	+	1.55291i	0.192971	+	0.0517064i
$903$		0			0
$904$	11.5981	+	6.69615i	0.385746	+	0.222711i
$905$		0			0
$906$		0			0
$907$	−8.54103	+	31.8756i	−0.283600	+	1.05841i	0.666256	+	0.745723i	$0.267896\pi$
$907$	−8.54103	+	31.8756i	−0.283600	+	1.05841i	−0.949856	+	0.312687i	$0.898771\pi$
$908$	−9.05369	−	9.05369i	−0.300457	−	0.300457i
$909$		0			0
$910$		0			0
$911$	−33.8827	+	19.5622i	−1.12258	+	0.648124i	−0.942059	−	0.335447i	$-0.891113\pi$
$911$	−33.8827	+	19.5622i	−1.12258	+	0.648124i	−0.180524	+	0.983571i	$0.557779\pi$
$912$		0			0
$913$	−15.4176	−	57.5394i	−0.510250	−	1.90428i
$914$	−13.7942	+	23.8923i	−0.456273	+	0.790287i
$915$		0			0
$916$	−2.90192	−	5.02628i	−0.0958823	−	0.166073i
$917$	−23.1822	+	23.1822i	−0.765544	+	0.765544i
$918$		0			0
$919$		−	58.5885i		−	1.93265i	−0.257316	−	0.966327i	$-0.582838\pi$
$919$		−	58.5885i		−	1.93265i	0.257316	−	0.966327i	$-0.417162\pi$
$920$		0			0
$921$		0			0
$922$	−9.71003	+	2.60179i	−0.319783	+	0.0856855i
$923$	−33.2204	+	8.90138i	−1.09346	+	0.292993i
$924$		0			0
$925$		0			0
$926$			2.19615i			0.0721700i
$927$		0			0
$928$	−7.58871	+	7.58871i	−0.249111	+	0.249111i
$929$	15.9282	+	27.5885i	0.522587	+	0.905148i	0.999655	+	0.0262811i	$0.00836650\pi$
$929$	15.9282	+	27.5885i	0.522587	+	0.905148i	−0.477067	+	0.878867i	$0.658300\pi$
$930$		0			0
$931$	−24.5981	+	42.6051i	−0.806169	+	1.39633i
$932$	−1.19256	−	4.45069i	−0.0390636	−	0.145787i
$933$		0			0
$934$	8.30385	−	4.79423i	0.271710	−	0.156872i
$935$		0			0
$936$		0			0
$937$	17.7148	+	17.7148i	0.578718	+	0.578718i	0.934550	−	0.355832i	$-0.115802\pi$
$937$	17.7148	+	17.7148i	0.578718	+	0.578718i	−0.355832	+	0.934550i	$0.615802\pi$
$938$	−6.78006	+	25.3035i	−0.221377	+	0.826190i
$939$		0			0
$940$		0			0
$941$	−9.88269	−	5.70577i	−0.322166	−	0.186003i	0.330191	−	0.943914i	$-0.392887\pi$
$941$	−9.88269	−	5.70577i	−0.322166	−	0.186003i	−0.652358	+	0.757911i	$0.726220\pi$
$942$		0			0
$943$	−3.67423	−	0.984508i	−0.119650	−	0.0320600i
$944$	5.53590			0.180178
$945$		0			0
$946$	−1.60770			−0.0522707
$947$	40.9292	+	10.9670i	1.33002	+	0.356378i	0.852723	−	0.522363i	$-0.174949\pi$
$947$	40.9292	+	10.9670i	1.33002	+	0.356378i	0.477298	+	0.878741i	$0.341616\pi$
$948$		0			0
$949$	21.2942	+	12.2942i	0.691240	+	0.399088i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	13.8647	+	13.8647i	0.449123	+	0.449123i	0.895063	−	0.445940i	$-0.147131\pi$
$953$	13.8647	+	13.8647i	0.449123	+	0.449123i	−0.445940	+	0.895063i	$0.647131\pi$
$954$		0			0
$955$		0			0
$956$	3.29423	−	1.90192i	0.106543	−	0.0615126i
$957$		0			0
$958$	−0.808643	−	3.01790i	−0.0261261	−	0.0975038i
$959$	31.6865	−	54.8827i	1.02321	−	1.77225i
$960$		0			0
$961$	15.4808	+	26.8135i	0.499379	+	0.864951i
$962$	−27.4249	+	27.4249i	−0.884213	+	0.884213i
$963$		0			0
$964$		−	9.39230i		−	0.302506i
$965$		0			0
$966$		0			0
$967$	−40.1528	+	10.7589i	−1.29123	+	0.345983i	−0.838126	−	0.545476i	$-0.816349\pi$
$967$	−40.1528	+	10.7589i	−1.29123	+	0.345983i	−0.453100	+	0.891460i	$0.649682\pi$
$968$	−0.965926	+	0.258819i	−0.0310460	+	0.00831876i
$969$		0			0
$970$		0			0
$971$		−	54.0333i		−	1.73401i	−0.498298	−	0.867006i	$-0.666041\pi$
$971$		−	54.0333i		−	1.73401i	0.498298	−	0.867006i	$-0.333959\pi$
$972$		0			0
$973$	26.7685	−	26.7685i	0.858159	−	0.858159i
$974$	8.02628	+	13.9019i	0.257179	+	0.445446i
$975$		0			0
$976$	−2.00000	+	3.46410i	−0.0640184	+	0.110883i
$977$	11.9515	+	44.6035i	0.382361	+	1.42699i	0.842285	+	0.539033i	$0.181210\pi$
$977$	11.9515	+	44.6035i	0.382361	+	1.42699i	−0.459923	+	0.887959i	$0.652123\pi$
$978$		0			0
$979$	−25.9808	+	15.0000i	−0.830349	+	0.479402i
$980$		0			0
$981$		0			0
$982$	−13.2320	−	13.2320i	−0.422249	−	0.422249i
$983$	11.1750	−	41.7057i	0.356427	−	1.33021i	−0.522251	−	0.852792i	$-0.674908\pi$
$983$	11.1750	−	41.7057i	0.356427	−	1.33021i	0.878679	−	0.477414i	$-0.158426\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	14.6090	+	3.91447i	0.464774	+	0.124536i
$989$	1.01924			0.0324099
$990$		0			0
$991$	−29.1769			−0.926835			−0.463418	−	0.886140i	$-0.653377\pi$
$991$	−29.1769			−0.926835			−0.463418	+	0.886140i	$0.653377\pi$
$992$	0.189469	+	0.0507680i	0.00601564	+	0.00161189i
$993$		0			0
$994$	29.7846	+	17.1962i	0.944710	+	0.545429i
$995$		0			0
$996$		0			0
$997$	−5.64325	+	21.0609i	−0.178724	+	0.667005i	0.817164	+	0.576405i	$0.195545\pi$
$997$	−5.64325	+	21.0609i	−0.178724	+	0.667005i	−0.995887	+	0.0905999i	$0.971122\pi$
$998$	−8.19428	−	8.19428i	−0.259385	−	0.259385i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.2.q.d.557.1		8
3.2	odd	2		450.2.p.c.257.2	yes	8
5.2	odd	4		1350.2.q.a.1043.2		8
5.3	odd	4		1350.2.q.a.1043.1		8
5.4	even	2	inner	1350.2.q.d.557.2		8
9.2	odd	6		1350.2.q.a.1007.1		8
9.7	even	3		450.2.p.e.407.2	yes	8
15.2	even	4		450.2.p.e.293.1	yes	8
15.8	even	4		450.2.p.e.293.2	yes	8
15.14	odd	2		450.2.p.c.257.1	✓	8
45.2	even	12	inner	1350.2.q.d.143.2		8
45.7	odd	12		450.2.p.c.443.1	yes	8
45.29	odd	6		1350.2.q.a.1007.2		8
45.34	even	6		450.2.p.e.407.1	yes	8
45.38	even	12	inner	1350.2.q.d.143.1		8
45.43	odd	12		450.2.p.c.443.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.p.c.257.1	✓	8	15.14	odd	2
450.2.p.c.257.2	yes	8	3.2	odd	2
450.2.p.c.443.1	yes	8	45.7	odd	12
450.2.p.c.443.2	yes	8	45.43	odd	12
450.2.p.e.293.1	yes	8	15.2	even	4
450.2.p.e.293.2	yes	8	15.8	even	4
450.2.p.e.407.1	yes	8	45.34	even	6
450.2.p.e.407.2	yes	8	9.7	even	3
1350.2.q.a.1007.1		8	9.2	odd	6
1350.2.q.a.1007.2		8	45.29	odd	6
1350.2.q.a.1043.1		8	5.3	odd	4
1350.2.q.a.1043.2		8	5.2	odd	4
1350.2.q.d.143.1		8	45.38	even	12	inner
1350.2.q.d.143.2		8	45.2	even	12	inner
1350.2.q.d.557.1		8	1.1	even	1	trivial
1350.2.q.d.557.2		8	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1350.q (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.22474 + 4.57081i) q^{7} +(-0.707107 - 0.707107i) q^{8} +O(q^{10})\)	\(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(-1.22474 + 4.57081i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-3.00000 + 1.73205i) q^{11} +(1.22474 + 4.57081i) q^{13} +(2.36603 - 4.09808i) q^{14} +(0.500000 + 0.866025i) q^{16} -3.19615i q^{19} +(3.34607 - 0.896575i) q^{22} +(-2.12132 + 0.568406i) q^{23} -4.73205i q^{26} +(-3.34607 + 3.34607i) q^{28} +(-5.36603 - 9.29423i) q^{29} +(-0.0980762 + 0.169873i) q^{31} +(-0.258819 - 0.965926i) q^{32} +(-5.79555 - 5.79555i) q^{37} +(-0.827225 + 3.08725i) q^{38} +(1.50000 + 0.866025i) q^{41} +(-0.448288 - 0.120118i) q^{43} -3.46410 q^{44} +2.19615 q^{46} +(5.79555 + 1.55291i) q^{47} +(-13.3301 - 7.69615i) q^{49} +(-1.22474 + 4.57081i) q^{52} +(-5.79555 - 5.79555i) q^{53} +(4.09808 - 2.36603i) q^{56} +(2.77766 + 10.3664i) q^{58} +(2.76795 - 4.79423i) q^{59} +(2.00000 + 3.46410i) q^{61} +(0.138701 - 0.138701i) q^{62} +1.00000i q^{64} +(-5.34727 + 1.43280i) q^{67} +7.26795i q^{71} +(3.67423 - 3.67423i) q^{73} +(4.09808 + 7.09808i) q^{74} +(1.59808 - 2.76795i) q^{76} +(-4.24264 - 15.8338i) q^{77} +(8.66025 - 5.00000i) q^{79} +(-1.22474 - 1.22474i) q^{82} +(-4.45069 + 16.6102i) q^{83} +(0.401924 + 0.232051i) q^{86} +(3.34607 + 0.896575i) q^{88} +8.66025 q^{89} -22.3923 q^{91} +(-2.12132 - 0.568406i) q^{92} +(-5.19615 - 3.00000i) q^{94} +(-0.688524 + 2.56961i) q^{97} +(10.8840 + 10.8840i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q+O(q^{10}) \) 8 * q	\( 8 q - 24 q^{11} + 12 q^{14} + 4 q^{16} - 36 q^{29} + 20 q^{31} + 12 q^{41} - 24 q^{46} - 72 q^{49} + 12 q^{56} + 36 q^{59} + 16 q^{61} + 12 q^{74} - 8 q^{76} + 24 q^{86} - 96 q^{91}+O(q^{100}) \) 8 * q - 24 * q^11 + 12 * q^14 + 4 * q^16 - 36 * q^29 + 20 * q^31 + 12 * q^41 - 24 * q^46 - 72 * q^49 + 12 * q^56 + 36 * q^59 + 16 * q^61 + 12 * q^74 - 8 * q^76 + 24 * q^86 - 96 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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