LMFDB - Embedded newform 1350.2.q.d.143.2

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			143.2
Root			$-0.258819 - 0.965926i$ of defining polynomial
Character	$\chi$	$=$	1350.143
Dual form			1350.2.q.d.557.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.965926 - 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(1.22474 + 4.57081i) q^{7} +(0.707107 - 0.707107i) q^{8} +O(q^{10})$	\(q+(0.965926 - 0.258819i) q^{2} +(0.866025 - 0.500000i) q^{4} +(1.22474 + 4.57081i) q^{7} +(0.707107 - 0.707107i) q^{8} +(-3.00000 - 1.73205i) q^{11} +(-1.22474 + 4.57081i) q^{13} +(2.36603 + 4.09808i) q^{14} +(0.500000 - 0.866025i) q^{16} +3.19615i q^{19} +(-3.34607 - 0.896575i) q^{22} +(2.12132 + 0.568406i) q^{23} +4.73205i q^{26} +(3.34607 + 3.34607i) q^{28} +(-5.36603 + 9.29423i) q^{29} +(-0.0980762 - 0.169873i) q^{31} +(0.258819 - 0.965926i) q^{32} +(5.79555 - 5.79555i) q^{37} +(0.827225 + 3.08725i) q^{38} +(1.50000 - 0.866025i) q^{41} +(0.448288 - 0.120118i) q^{43} -3.46410 q^{44} +2.19615 q^{46} +(-5.79555 + 1.55291i) q^{47} +(-13.3301 + 7.69615i) q^{49} +(1.22474 + 4.57081i) q^{52} +(5.79555 - 5.79555i) q^{53} +(4.09808 + 2.36603i) q^{56} +(-2.77766 + 10.3664i) q^{58} +(2.76795 + 4.79423i) q^{59} +(2.00000 - 3.46410i) q^{61} +(-0.138701 - 0.138701i) q^{62} -1.00000i q^{64} +(5.34727 + 1.43280i) q^{67} -7.26795i q^{71} +(-3.67423 - 3.67423i) q^{73} +(4.09808 - 7.09808i) q^{74} +(1.59808 + 2.76795i) q^{76} +(4.24264 - 15.8338i) q^{77} +(8.66025 + 5.00000i) q^{79} +(1.22474 - 1.22474i) q^{82} +(4.45069 + 16.6102i) q^{83} +(0.401924 - 0.232051i) q^{86} +(-3.34607 + 0.896575i) q^{88} +8.66025 q^{89} -22.3923 q^{91} +(2.12132 - 0.568406i) q^{92} +(-5.19615 + 3.00000i) q^{94} +(0.688524 + 2.56961i) q^{97} +(-10.8840 + 10.8840i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$e\left(\frac{3}{4}\right)$

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	0.663727	+	0.747975i	$0.268974\pi$
$7$	1.22474	+	4.57081i	0.462910	+	1.72760i	−0.200817	+	0.979629i	$0.564360\pi$
$8$	0.707107	−	0.707107i	0.250000	−	0.250000i
$9$		0			0
$10$		0			0
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.0258656	−	0.999665i	$-0.508234\pi$
$11$	−3.00000	−	1.73205i	−0.904534	−	0.522233i	−0.878668	+	0.477432i	$0.841568\pi$
$12$		0			0
$13$	−1.22474	+	4.57081i	−0.339683	+	1.26771i	0.559019	+	0.829155i	$0.311178\pi$
$13$	−1.22474	+	4.57081i	−0.339683	+	1.26771i	−0.898702	+	0.438560i	$0.855489\pi$
$14$	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	$-0.250000\pi$
$17$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$18$		0			0
$19$			3.19615i			0.733248i	0.930369	+	0.366624i	$0.119486\pi$
$19$			3.19615i			0.733248i	−0.930369	+	0.366624i	$0.880514\pi$
$20$		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	$-0.343133\pi$
$23$	2.12132	+	0.568406i	0.442326	+	0.118521i	−0.0307805	+	0.999526i	$0.509799\pi$
$24$		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	−0.425273	+	0.905065i	$0.639822\pi$
$29$	−5.36603	+	9.29423i	−0.996446	+	1.72589i	−0.571173	+	0.820830i	$0.693511\pi$
$30$		0			0
$31$	−0.0980762	−	0.169873i	−0.0176150	−	0.0305101i	0.857084	−	0.515177i	$-0.172274\pi$
$31$	−0.0980762	−	0.169873i	−0.0176150	−	0.0305101i	−0.874699	+	0.484667i	$0.838941\pi$
$32$	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	$-0.514696\pi$
$37$	5.79555	−	5.79555i	0.952783	−	0.952783i	0.998934	+	0.0461511i	$0.0146956\pi$
$38$	0.827225	+	3.08725i	0.134194	+	0.500817i
$39$		0			0
$40$		0			0
$41$	1.50000	−	0.866025i	0.234261	−	0.135250i	−0.378275	−	0.925693i	$-0.623483\pi$
$41$	1.50000	−	0.866025i	0.234261	−	0.135250i	0.612536	+	0.790443i	$0.290149\pi$
$42$		0			0
$43$	0.448288	−	0.120118i	0.0683632	−	0.0183179i	−0.224475	−	0.974480i	$-0.572067\pi$
$43$	0.448288	−	0.120118i	0.0683632	−	0.0183179i	0.292839	+	0.956162i	$0.405400\pi$
$44$	−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	$-0.727503\pi$
$47$	−5.79555	+	1.55291i	−0.845369	+	0.226516i	−0.189961	+	0.981792i	$0.560836\pi$
$48$		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	$-0.559680\pi$
$53$	5.79555	−	5.79555i	0.796081	−	0.796081i	0.982475	+	0.186394i	$0.0596802\pi$
$54$		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	0.988019	−	0.154330i	$-0.0493218\pi$
$59$	2.76795	+	4.79423i	0.360356	+	0.624155i	−0.627663	+	0.778485i	$0.715988\pi$
$60$		0			0
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	−0.709113	−	0.705095i	$-0.750904\pi$
$61$	2.00000	−	3.46410i	0.256074	−	0.443533i	0.965187	+	0.261562i	$0.0842377\pi$
$62$	−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	$-0.306863\pi$
$67$	5.34727	+	1.43280i	0.653273	+	0.175044i	0.0830646	+	0.996544i	$0.473529\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.26795i		−	0.862547i	−0.902221	−	0.431273i	$-0.858064\pi$
$71$		−	7.26795i		−	0.862547i	0.902221	−	0.431273i	$-0.141936\pi$
$72$		0			0
$73$	−3.67423	−	3.67423i	−0.430037	−	0.430037i	0.458604	−	0.888641i	$-0.348350\pi$
$73$	−3.67423	−	3.67423i	−0.430037	−	0.430037i	−0.888641	+	0.458604i	$0.848350\pi$
$74$	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	0.900561	−	0.434730i	$-0.143156\pi$
$79$	8.66025	+	5.00000i	0.974355	+	0.562544i	0.0737937	+	0.997274i	$0.476489\pi$
$80$		0			0
$81$		0			0
$82$	1.22474	−	1.22474i	0.135250	−	0.135250i
$83$	4.45069	+	16.6102i	0.488527	+	1.82321i	0.563625	+	0.826031i	$0.309406\pi$
$83$	4.45069	+	16.6102i	0.488527	+	1.82321i	−0.0750978	+	0.997176i	$0.523927\pi$
$84$		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	$-0.0402126\pi$
$97$	0.688524	+	2.56961i	0.0699091	+	0.260904i	−0.922122	+	0.386900i	$0.873546\pi$
$98$	−10.8840	+	10.8840i	−1.09945	+	1.09945i
$99$		0			0
$100$		0			0
$101$	−6.29423	−	3.63397i	−0.626299	−	0.361594i	0.153018	−	0.988223i	$-0.451101\pi$
$101$	−6.29423	−	3.63397i	−0.626299	−	0.361594i	−0.779317	+	0.626629i	$0.784434\pi$
$102$		0			0
$103$	2.68973	−	10.0382i	0.265027	−	0.989093i	−0.697207	−	0.716869i	$-0.745574\pi$
$103$	2.68973	−	10.0382i	0.265027	−	0.989093i	0.962234	−	0.272223i	$-0.0877590\pi$
$104$	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	$-0.0679753\pi$
$107$	7.91688	+	7.91688i	0.765353	+	0.765353i	−0.211931	+	0.977285i	$0.567975\pi$
$108$		0			0
$109$		−	1.80385i		−	0.172777i	−0.996262	−	0.0863886i	$-0.972467\pi$
$109$		−	1.80385i		−	0.172777i	0.996262	−	0.0863886i	$-0.0275327\pi$
$110$		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	$-0.199754\pi$
$113$	12.9360	+	3.46618i	1.21691	+	0.326071i	0.407443	+	0.913231i	$0.366421\pi$
$114$		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	$-0.631532\pi$
$127$	5.79555	−	5.79555i	0.514272	−	0.514272i	0.915833	+	0.401560i	$0.131532\pi$
$128$	−0.258819	−	0.965926i	−0.0228766	−	0.0853766i
$129$		0			0
$130$		0			0
$131$	6.00000	−	3.46410i	0.524222	−	0.302660i	−0.214438	−	0.976738i	$-0.568792\pi$
$131$	6.00000	−	3.46410i	0.524222	−	0.302660i	0.738661	+	0.674078i	$0.235459\pi$
$132$		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	$-0.389466\pi$
$137$	12.9360	−	3.46618i	1.10519	−	0.296136i	0.764878	+	0.644175i	$0.222799\pi$
$138$		0			0
$139$	−6.92820	+	4.00000i	−0.587643	+	0.339276i	−0.764165	−	0.645021i	$-0.776849\pi$
$139$	−6.92820	+	4.00000i	−0.587643	+	0.339276i	0.176522	+	0.984297i	$0.443515\pi$
$140$		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	−0.953703	−	0.300750i	$-0.902763\pi$
$149$	−9.00000	−	15.5885i	−0.737309	−	1.27706i	0.216394	−	0.976306i	$-0.430570\pi$
$150$		0			0
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	−0.656101	−	0.754673i	$-0.727796\pi$
$151$	4.00000	−	6.92820i	0.325515	−	0.563809i	0.981617	+	0.190864i	$0.0611289\pi$
$152$	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	$-0.707813\pi$
$157$	−19.1798	−	5.13922i	−1.53072	−	0.410154i	−0.923253	+	0.384194i	$0.874480\pi$
$158$	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	$-0.0604804\pi$
$163$	10.1261	+	10.1261i	0.793140	+	0.793140i	−0.188864	+	0.982003i	$0.560480\pi$
$164$	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	0.582642	+	0.812729i	$0.302019\pi$
$167$	−4.24264	+	15.8338i	−0.328305	+	1.22525i	−0.910947	+	0.412523i	$0.864648\pi$
$168$		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	−0.916740	−	0.399484i	$-0.869189\pi$
$173$	−4.24264	−	15.8338i	−0.322562	−	1.20382i	0.594178	−	0.804334i	$-0.297477\pi$
$174$		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	0.387522	−	0.921861i	$-0.373331\pi$
$191$	−3.00000	−	1.73205i	−0.217072	−	0.125327i	−0.604594	+	0.796534i	$0.706665\pi$
$192$		0			0
$193$	2.86559	−	10.6945i	0.206270	−	0.769809i	−0.782789	−	0.622287i	$-0.786204\pi$
$193$	2.86559	−	10.6945i	0.206270	−	0.769809i	0.989059	−	0.147522i	$-0.0471298\pi$
$194$	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	$-0.418772\pi$
$197$	−10.0382	−	10.0382i	−0.715192	−	0.715192i	−0.967617	+	0.252425i	$0.918772\pi$
$198$		0			0
$199$		−	20.3923i		−	1.44557i	−0.691072	−	0.722786i	$-0.742861\pi$
$199$		−	20.3923i		−	1.44557i	0.691072	−	0.722786i	$-0.257139\pi$
$200$		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	0.844704	−	0.535233i	$-0.179776\pi$
$211$	−0.598076	−	1.03590i	−0.0411733	−	0.0713142i	−0.885878	+	0.463919i	$0.846443\pi$
$212$	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	$-0.556274\pi$
$223$	2.44949	−	0.656339i	0.164030	−	0.0439517i	0.339899	+	0.940462i	$0.389607\pi$
$224$	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	$-0.443639\pi$
$227$	12.3676	−	3.31388i	0.820864	−	0.219950i	0.644724	+	0.764415i	$0.276972\pi$
$228$		0			0
$229$	−5.02628	+	2.90192i	−0.332146	+	0.191765i	−0.656793	−	0.754071i	$-0.728088\pi$
$229$	−5.02628	+	2.90192i	−0.332146	+	0.191765i	0.324648	+	0.945835i	$0.394754\pi$
$230$		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	$-0.701773\pi$
$233$	3.25813	−	3.25813i	0.213447	−	0.213447i	0.805730	+	0.592283i	$0.201773\pi$
$234$		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	0.920959	−	0.389659i	$-0.127407\pi$
$239$	1.90192	+	3.29423i	0.123025	+	0.213086i	−0.797934	+	0.602745i	$0.794074\pi$
$240$		0			0
$241$	−4.69615	+	8.13397i	−0.302506	+	0.523955i	−0.976703	−	0.214596i	$-0.931156\pi$
$241$	−4.69615	+	8.13397i	−0.302506	+	0.523955i	0.674197	+	0.738551i	$0.264490\pi$
$242$	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	$-0.908326\pi$
$251$		−	9.00000i		−	0.568075i	0.958813	−	0.284037i	$-0.0916740\pi$
$252$		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	0.384787	+	0.923005i	$0.374275\pi$
$257$	−6.57201	+	24.5271i	−0.409951	+	1.52996i	−0.794738	+	0.606952i	$0.792392\pi$
$258$		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	−0.990659	−	0.136364i	$-0.956458\pi$
$263$	−3.25813	−	12.1595i	−0.200905	−	0.749788i	0.789754	−	0.613424i	$-0.210208\pi$
$264$		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	0.934227	+	0.356679i	$0.116091\pi$
$277$	5.05128	+	18.8516i	0.303502	+	1.13269i	−0.630725	+	0.776007i	$0.717242\pi$
$278$	−5.65685	+	5.65685i	−0.339276	+	0.339276i
$279$		0			0
$280$		0			0
$281$	−9.00000	−	5.19615i	−0.536895	−	0.309976i	0.206925	−	0.978357i	$-0.433655\pi$
$281$	−9.00000	−	5.19615i	−0.536895	−	0.309976i	−0.743820	+	0.668380i	$0.766988\pi$
$282$		0			0
$283$	−1.01669	+	3.79435i	−0.0604362	+	0.225551i	−0.989538	−	0.144274i	$-0.953915\pi$
$283$	−1.01669	+	3.79435i	−0.0604362	+	0.225551i	0.929102	+	0.369824i	$0.120582\pi$
$284$	−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	$-0.381226\pi$
$293$	3.67423	+	0.984508i	0.214651	+	0.0575156i	−0.149891	+	0.988703i	$0.547892\pi$
$294$		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	$-0.718483\pi$
$307$	2.44949	−	2.44949i	0.139800	−	0.139800i	0.773543	+	0.633743i	$0.218483\pi$
$308$	−4.24264	−	15.8338i	−0.241747	−	0.902212i
$309$		0			0
$310$		0			0
$311$	9.58846	−	5.53590i	0.543712	−	0.313912i	−0.202870	−	0.979206i	$-0.565027\pi$
$311$	9.58846	−	5.53590i	0.543712	−	0.313912i	0.746582	+	0.665294i	$0.231694\pi$
$312$		0			0
$313$	19.9563	−	5.34727i	1.12800	−	0.302245i	0.353880	−	0.935291i	$-0.384862\pi$
$313$	19.9563	−	5.34727i	1.12800	−	0.302245i	0.774115	+	0.633045i	$0.218195\pi$
$314$	−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	$-0.642569\pi$
$317$	−6.36396	+	1.70522i	−0.357436	+	0.0957746i	0.0756325	+	0.997136i	$0.475902\pi$
$318$		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	−0.616648	−	0.787239i	$-0.711510\pi$
$331$	6.79423	−	11.7679i	0.373445	−	0.646825i	0.990093	+	0.140414i	$0.0448433\pi$
$332$	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	$-0.605609\pi$
$337$	−19.8362	−	5.31508i	−1.08054	−	0.289531i	−0.754819	+	0.655933i	$0.772275\pi$
$338$	−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	$-0.992435\pi$
$347$	−2.27362	+	8.48528i	−0.122055	+	0.455514i	0.877663	+	0.479278i	$0.159102\pi$
$348$		0			0
$349$	1.73205	+	1.00000i	0.0927146	+	0.0535288i	0.545640	−	0.838019i	$-0.316286\pi$
$349$	1.73205	+	1.00000i	0.0927146	+	0.0535288i	−0.452926	+	0.891548i	$0.649620\pi$
$350$		0			0
$351$		0			0
$352$	−2.44949	+	2.44949i	−0.130558	+	0.130558i
$353$	3.88229	+	14.4889i	0.206633	+	0.771166i	0.988946	+	0.148279i	$0.0473735\pi$
$353$	3.88229	+	14.4889i	0.206633	+	0.771166i	−0.782312	+	0.622886i	$0.785960\pi$
$354$		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	$-0.0488416\pi$
$367$	1.07244	+	4.00240i	0.0559810	+	0.208924i	−0.932270	+	0.361763i	$0.882175\pi$
$368$	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	−0.613754	−	0.789498i	$-0.710341\pi$
$373$	6.03579	−	22.5259i	0.312521	−	1.16635i	0.926275	−	0.376848i	$-0.122992\pi$
$374$		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	$-0.996793\pi$
$379$		−	0.392305i		−	0.0201513i	0.999949	−	0.0100757i	$-0.00320724\pi$
$380$		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	$-0.372516\pi$
$383$	5.22715	+	1.40061i	0.267095	+	0.0715678i	−0.122786	+	0.992433i	$0.539183\pi$
$384$		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	−0.967158	−	0.254177i	$-0.918196\pi$
$389$	−5.19615	+	9.00000i	−0.263455	+	0.456318i	0.703702	+	0.710495i	$0.251529\pi$
$390$		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	$-0.876855\pi$
$397$	−10.9348	+	10.9348i	−0.548800	+	0.548800i	0.377293	+	0.926094i	$0.376855\pi$
$398$	−5.27792	−	19.6975i	−0.264558	−	0.987344i
$399$		0			0
$400$		0			0
$401$	−9.00000	+	5.19615i	−0.449439	+	0.259483i	−0.707593	−	0.706620i	$-0.750219\pi$
$401$	−9.00000	+	5.19615i	−0.449439	+	0.259483i	0.258154	+	0.966104i	$0.416886\pi$
$402$		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	−0.882378	−	0.470542i	$-0.844059\pi$
$409$	−18.5263	+	10.6962i	−0.916066	+	0.528891i	−0.0336878	+	0.999432i	$0.510725\pi$
$410$		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	0.954759	−	0.297382i	$-0.0961133\pi$
$419$	4.50000	+	7.79423i	0.219839	+	0.380773i	−0.734919	+	0.678155i	$0.762780\pi$
$420$		0			0
$421$	−11.2942	+	19.5622i	−0.550447	+	0.953402i	0.447795	+	0.894136i	$0.352209\pi$
$421$	−11.2942	+	19.5622i	−0.550447	+	0.953402i	−0.998242	+	0.0592661i	$0.981124\pi$
$422$	−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	$0.169897\pi$
$431$			21.1244i			1.01752i	−0.860907	+	0.508762i	$0.830103\pi$
$432$		0			0
$433$	12.0716	+	12.0716i	0.580123	+	0.580123i	0.934937	−	0.354814i	$-0.115456\pi$
$433$	12.0716	+	12.0716i	0.580123	+	0.580123i	−0.354814	+	0.934937i	$0.615456\pi$
$434$	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	0.470545	−	0.882376i	$-0.344058\pi$
$439$	−1.22243	−	0.705771i	−0.0583435	−	0.0336846i	−0.528888	+	0.848692i	$0.677391\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.68973	+	10.0382i	0.127793	+	0.476929i	0.999924	−	0.0123433i	$-0.00392908\pi$
$443$	2.68973	+	10.0382i	0.127793	+	0.476929i	−0.872131	+	0.489272i	$0.837262\pi$
$444$		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	−0.904933	−	0.425553i	$-0.860080\pi$
$457$	−7.14042	−	26.6484i	−0.334015	−	1.24656i	0.570919	−	0.821007i	$-0.306587\pi$
$458$	−4.10394	+	4.10394i	−0.191765	+	0.191765i
$459$		0			0
$460$		0			0
$461$	8.70577	+	5.02628i	0.405468	+	0.234097i	0.688841	−	0.724913i	$-0.258120\pi$
$461$	8.70577	+	5.02628i	0.405468	+	0.234097i	−0.283373	+	0.959010i	$0.591453\pi$
$462$		0			0
$463$	0.568406	−	2.12132i	0.0264161	−	0.0985861i	−0.951459	−	0.307775i	$-0.900416\pi$
$463$	0.568406	−	2.12132i	0.0264161	−	0.0985861i	0.977875	+	0.209189i	$0.0670823\pi$
$464$	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	$-0.178790\pi$
$467$	6.78006	+	6.78006i	0.313744	+	0.313744i	−0.532614	+	0.846358i	$0.678790\pi$
$468$		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	−0.828128	−	0.560540i	$-0.810594\pi$
$479$	1.56218	−	2.70577i	0.0713777	−	0.123630i	0.899505	+	0.436910i	$0.143927\pi$
$480$		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	$-0.631511\pi$
$487$	11.3509	−	11.3509i	0.514357	−	0.514357i	0.915858	+	0.401501i	$0.131511\pi$
$488$	−1.03528	−	3.86370i	−0.0468648	−	0.174902i
$489$		0			0
$490$		0			0
$491$	16.2058	−	9.35641i	0.731356	−	0.422249i	−0.0875619	−	0.996159i	$-0.527908\pi$
$491$	16.2058	−	9.35641i	0.731356	−	0.422249i	0.818918	+	0.573910i	$0.194574\pi$
$492$		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	−0.258253	−	0.966077i	$-0.583147\pi$
$499$	10.0359	−	5.79423i	0.449269	−	0.259385i	0.707521	+	0.706692i	$0.249813\pi$
$500$		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	$-0.711504\pi$
$503$	3.82654	−	3.82654i	0.170617	−	0.170617i	0.787250	+	0.616633i	$0.211504\pi$
$504$		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	0.538362	−	0.842714i	$-0.319043\pi$
$509$	−10.3923	−	18.0000i	−0.460631	−	0.797836i	−0.998992	+	0.0448779i	$0.985710\pi$
$510$		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	$-0.980572\pi$
$521$		−	2.78461i		−	0.121996i	0.998138	−	0.0609980i	$-0.0194283\pi$
$522$		0			0
$523$	23.9909	+	23.9909i	1.04905	+	1.04905i	0.998733	+	0.0503137i	$0.0160221\pi$
$523$	23.9909	+	23.9909i	1.04905	+	1.04905i	0.0503137	+	0.998733i	$0.483978\pi$
$524$	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	0.716964	+	0.697110i	$0.245531\pi$
$547$	10.3986	+	38.8079i	0.444610	+	1.65931i	−0.272354	+	0.962197i	$0.587802\pi$
$548$	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	$-0.220337\pi$
$557$	3.10583	+	3.10583i	0.131598	+	0.131598i	−0.638240	+	0.769838i	$0.720337\pi$
$558$		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	$-0.432849\pi$
$563$	−2.32937	−	0.624153i	−0.0981713	−	0.0263049i	−0.307570	+	0.951525i	$0.599516\pi$
$564$		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	−0.477134	−	0.878830i	$-0.658324\pi$
$569$	12.4641	−	21.5885i	0.522522	−	0.905035i	0.999657	−	0.0262048i	$-0.00834220\pi$
$570$		0			0
$571$	−9.59808	−	16.6244i	−0.401667	−	0.695708i	0.592260	−	0.805747i	$-0.298236\pi$
$571$	−9.59808	−	16.6244i	−0.401667	−	0.695708i	−0.993927	+	0.110039i	$0.964902\pi$
$572$	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	$-0.905274\pi$
$577$	−15.9217	+	15.9217i	−0.662828	+	0.662828i	0.293217	+	0.956046i	$0.405274\pi$
$578$	−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	$-0.554441\pi$
$587$	4.24264	−	1.13681i	0.175113	−	0.0469213i	0.345310	+	0.938489i	$0.387774\pi$
$588$		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	$-0.891904\pi$
$593$	−14.8492	+	14.8492i	−0.609785	+	0.609785i	0.333105	+	0.942890i	$0.391904\pi$
$594$		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	0.930666	−	0.365870i	$-0.119229\pi$
$599$	3.63397	+	6.29423i	0.148480	+	0.257175i	−0.782186	+	0.623045i	$0.785895\pi$
$600$		0			0
$601$	14.3923	−	24.9282i	0.587074	−	1.01684i	−0.407539	−	0.913188i	$-0.633613\pi$
$601$	14.3923	−	24.9282i	0.587074	−	1.01684i	0.994613	−	0.103655i	$-0.0330537\pi$
$602$	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	0.995519	+	0.0945643i	$0.0301458\pi$
$607$	46.9328	+	12.5756i	1.90495	+	0.510429i	0.909427	+	0.415864i	$0.136521\pi$
$608$	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	$-0.157445\pi$
$613$	10.0382	+	10.0382i	0.405439	+	0.405439i	−0.474706	+	0.880145i	$0.657445\pi$
$614$	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	$-0.840432\pi$
$617$	3.05008	−	11.3831i	0.122792	−	0.458265i	0.999751	+	0.0222993i	$0.00709866\pi$
$618$		0			0
$619$	−41.2128	−	23.7942i	−1.65648	−	0.956371i	−0.974319	−	0.225171i	$-0.927706\pi$
$619$	−41.2128	−	23.7942i	−1.65648	−	0.956371i	−0.682164	−	0.731200i	$-0.738961\pi$
$620$		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	0.974499	−	0.224393i	$-0.0720400\pi$
$641$	32.0885	+	18.5263i	1.26742	+	0.731744i	0.292919	+	0.956137i	$0.405373\pi$
$642$		0			0
$643$	−9.67784	+	36.1182i	−0.381657	+	1.42436i	0.461713	+	0.887029i	$0.347235\pi$
$643$	−9.67784	+	36.1182i	−0.381657	+	1.42436i	−0.843370	+	0.537333i	$0.819432\pi$
$644$	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	$-0.159979\pi$
$647$	10.0382	+	10.0382i	0.394642	+	0.394642i	−0.481696	+	0.876338i	$0.659979\pi$
$648$		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	$-0.467935\pi$
$653$	−7.91688	−	2.12132i	−0.309811	−	0.0830137i	−0.410375	+	0.911917i	$0.634602\pi$
$654$		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	0.147925	+	0.988999i	$0.452740\pi$
$659$	−20.0885	+	34.7942i	−0.782535	+	1.35539i	−0.930461	+	0.366392i	$0.880593\pi$
$660$		0			0
$661$	19.5885	+	33.9282i	0.761903	+	1.31965i	0.941869	+	0.335981i	$0.109068\pi$
$661$	19.5885	+	33.9282i	0.761903	+	1.31965i	−0.179966	+	0.983673i	$0.557599\pi$
$662$	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	$-0.441290\pi$
$673$	21.6293	−	5.79555i	0.833748	−	0.223402i	0.650348	+	0.759636i	$0.274623\pi$
$674$	−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	$-0.426683\pi$
$677$	23.7506	−	6.36396i	0.912811	−	0.244587i	0.684510	+	0.729004i	$0.260016\pi$
$678$		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	$-0.698551\pi$
$683$	5.94786	−	5.94786i	0.227588	−	0.227588i	0.811685	+	0.584096i	$0.198551\pi$
$684$		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	0.0696602	+	0.997571i	$0.477808\pi$
$691$	−21.7942	+	37.7487i	−0.829092	+	1.43603i	−0.898752	+	0.438458i	$0.855525\pi$
$692$	−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	$-0.911485\pi$
$701$		−	14.5359i		−	0.549013i	0.961585	−	0.274507i	$-0.0885145\pi$
$702$		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	−0.221876	−	0.975075i	$-0.571218\pi$
$709$	−31.3468	−	18.0981i	−1.17725	−	0.679688i	−0.955378	+	0.295387i	$0.904551\pi$
$710$		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	$-0.124342\pi$
$727$	−1.79315	−	6.69213i	−0.0665043	−	0.248197i	−0.991173	+	0.132575i	$0.957675\pi$
$728$	−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	$-0.997527\pi$
$733$	−3.52193	+	13.1440i	−0.130085	+	0.485486i	0.869884	+	0.493256i	$0.164193\pi$
$734$	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	$0.117321\pi$
$739$			19.5885i			0.720573i	−0.932842	+	0.360287i	$0.882679\pi$
$740$		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	$-0.206964\pi$
$743$	32.2359	+	8.63759i	1.18262	+	0.316882i	0.386655	+	0.922224i	$0.373630\pi$
$744$		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	0.753116	−	0.657887i	$-0.228550\pi$
$751$	−5.29423	−	9.16987i	−0.193189	−	0.334613i	−0.946305	+	0.323274i	$0.895216\pi$
$752$	−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	0.00552030	−	0.999985i	$-0.498243\pi$
$757$	27.6651	−	27.6651i	1.00551	−	1.00551i	0.999985	−	0.00552030i	$-0.00175717\pi$
$758$	−0.101536	−	0.378937i	−0.00368795	−	0.0137636i
$759$		0			0
$760$		0			0
$761$	41.0885	−	23.7224i	1.48946	−	0.859937i	0.489528	−	0.871988i	$-0.337169\pi$
$761$	41.0885	−	23.7224i	1.48946	−	0.859937i	0.999927	+	0.0120501i	$0.00383575\pi$
$762$		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	−0.290043	−	0.957014i	$-0.593669\pi$
$769$	10.9186	−	6.30385i	0.393734	−	0.227323i	0.683777	+	0.729691i	$0.260336\pi$
$770$		0			0
$771$		0			0
$772$	−2.86559	−	10.6945i	−0.103135	−	0.384905i
$773$	35.9101	−	35.9101i	1.29160	−	1.29160i	0.357800	−	0.933798i	$-0.383527\pi$
$773$	35.9101	−	35.9101i	1.29160	−	1.29160i	0.933798	−	0.357800i	$-0.116473\pi$
$774$		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	$-0.516562\pi$
$787$	−16.7303	−	4.48288i	−0.596372	−	0.159797i	−0.544364	+	0.838849i	$0.683229\pi$
$788$	−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	−0.476607	−	0.879117i	$-0.658133\pi$
$797$	10.6066	−	39.5844i	0.375705	−	1.40215i	0.852312	−	0.523034i	$-0.175200\pi$
$798$		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	0.263978	−	0.964529i	$-0.414966\pi$
$821$	−12.5885	−	7.26795i	−0.439340	−	0.253653i	−0.703318	+	0.710876i	$0.748299\pi$
$822$		0			0
$823$	−10.5187	+	39.2562i	−0.366658	+	1.36839i	0.498502	+	0.866889i	$0.333884\pi$
$823$	−10.5187	+	39.2562i	−0.366658	+	1.36839i	−0.865160	+	0.501497i	$0.832783\pi$
$824$	−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	$-0.245551\pi$
$827$	0.568406	+	0.568406i	0.0197654	+	0.0197654i	−0.697155	+	0.716920i	$0.745551\pi$
$828$		0			0
$829$			28.5885i			0.992918i	0.868060	+	0.496459i	$0.165367\pi$
$829$			28.5885i			0.992918i	−0.868060	+	0.496459i	$0.834633\pi$
$830$		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	−0.725461	−	0.688264i	$-0.758373\pi$
$839$	6.75833	−	11.7058i	0.233323	−	0.404128i	0.958784	+	0.284136i	$0.0917066\pi$
$840$		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	$-0.686945\pi$
$853$	−18.0430	+	4.83461i	−0.617781	+	0.165534i	−0.0636620	+	0.997972i	$0.520278\pi$
$854$	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	$-0.330366\pi$
$857$	40.3608	−	10.8147i	1.37870	−	0.369422i	0.870649	+	0.491905i	$0.163699\pi$
$858$		0			0
$859$	36.0167	−	20.7942i	1.22887	−	0.709490i	0.262079	−	0.965047i	$-0.415592\pi$
$859$	36.0167	−	20.7942i	1.22887	−	0.709490i	0.966794	+	0.255557i	$0.0822587\pi$
$860$		0			0
$861$		0			0
$862$	5.46739	+	20.4046i	0.186220	+	0.694982i
$863$	−33.2204	+	33.2204i	−1.13084	+	1.13084i	−0.140798	+	0.990038i	$0.544967\pi$
$863$	−33.2204	+	33.2204i	−1.13084	+	1.13084i	−0.990038	+	0.140798i	$0.955033\pi$
$864$		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	$-0.630952\pi$
$877$	−35.6699	−	9.55772i	−1.20449	−	0.322741i	−0.804597	+	0.593822i	$0.797618\pi$
$878$	−1.36345	−	0.365334i	−0.0460141	−	0.0123294i
$879$		0			0
$880$		0			0
$881$			18.6795i			0.629328i	0.949203	+	0.314664i	$0.101892\pi$
$881$			18.6795i			0.629328i	−0.949203	+	0.314664i	$0.898108\pi$
$882$		0			0
$883$	−18.2832	−	18.2832i	−0.615280	−	0.615280i	0.329037	−	0.944317i	$-0.393276\pi$
$883$	−18.2832	−	18.2832i	−0.615280	−	0.615280i	−0.944317	+	0.329037i	$0.893276\pi$
$884$		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	$-0.972488\pi$
$887$	−2.68973	+	10.0382i	−0.0903122	+	0.337050i	0.905955	+	0.423374i	$0.139154\pi$
$888$		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	0.949856	+	0.312687i	$0.101229\pi$
$907$	8.54103	+	31.8756i	0.283600	+	1.05841i	−0.666256	+	0.745723i	$0.732104\pi$
$908$	9.05369	−	9.05369i	0.300457	−	0.300457i
$909$		0			0
$910$		0			0
$911$	−33.8827	−	19.5622i	−1.12258	−	0.648124i	−0.180524	−	0.983571i	$-0.557779\pi$
$911$	−33.8827	−	19.5622i	−1.12258	−	0.648124i	−0.942059	+	0.335447i	$0.891113\pi$
$912$		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	$0.417162\pi$
$919$			58.5885i			1.93265i	−0.257316	+	0.966327i	$0.582838\pi$
$920$		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	−0.477067	−	0.878867i	$-0.658300\pi$
$929$	15.9282	−	27.5885i	0.522587	−	0.905148i	0.999655	−	0.0262811i	$-0.00836650\pi$
$930$		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	$-0.884198\pi$
$937$	−17.7148	+	17.7148i	−0.578718	+	0.578718i	0.355832	+	0.934550i	$0.384198\pi$
$938$	6.78006	+	25.3035i	0.221377	+	0.826190i
$939$		0			0
$940$		0			0
$941$	−9.88269	+	5.70577i	−0.322166	+	0.186003i	−0.652358	−	0.757911i	$-0.726220\pi$
$941$	−9.88269	+	5.70577i	−0.322166	+	0.186003i	0.330191	+	0.943914i	$0.392887\pi$
$942$		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	$-0.825051\pi$
$947$	−40.9292	+	10.9670i	−1.33002	+	0.356378i	−0.477298	+	0.878741i	$0.658384\pi$
$948$		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	$-0.852869\pi$
$953$	−13.8647	+	13.8647i	−0.449123	+	0.449123i	0.445940	+	0.895063i	$0.352869\pi$
$954$		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	$-0.183651\pi$
$967$	40.1528	+	10.7589i	1.29123	+	0.345983i	0.453100	+	0.891460i	$0.350318\pi$
$968$	0.965926	+	0.258819i	0.0310460	+	0.00831876i
$969$		0			0
$970$		0			0
$971$			54.0333i			1.73401i	0.498298	+	0.867006i	$0.333959\pi$
$971$			54.0333i			1.73401i	−0.498298	+	0.867006i	$0.666041\pi$
$972$		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	0.459923	+	0.887959i	$0.347877\pi$
$977$	−11.9515	+	44.6035i	−0.382361	+	1.42699i	−0.842285	+	0.539033i	$0.818790\pi$
$978$		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	−0.878679	−	0.477414i	$-0.841574\pi$
$983$	−11.1750	−	41.7057i	−0.356427	−	1.33021i	0.522251	−	0.852792i	$-0.325092\pi$
$984$		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	0.995887	+	0.0905999i	$0.0288784\pi$
$997$	5.64325	+	21.0609i	0.178724	+	0.667005i	−0.817164	+	0.576405i	$0.804455\pi$
$998$	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.143.2		8
3.2	odd	2		450.2.p.c.443.1	yes	8
5.2	odd	4		1350.2.q.a.1007.2		8
5.3	odd	4		1350.2.q.a.1007.1		8
5.4	even	2	inner	1350.2.q.d.143.1		8
9.4	even	3		450.2.p.e.293.1	yes	8
9.5	odd	6		1350.2.q.a.1043.2		8
15.2	even	4		450.2.p.e.407.1	yes	8
15.8	even	4		450.2.p.e.407.2	yes	8
15.14	odd	2		450.2.p.c.443.2	yes	8
45.4	even	6		450.2.p.e.293.2	yes	8
45.13	odd	12		450.2.p.c.257.2	yes	8
45.14	odd	6		1350.2.q.a.1043.1		8
45.22	odd	12		450.2.p.c.257.1	✓	8
45.23	even	12	inner	1350.2.q.d.557.1		8
45.32	even	12	inner	1350.2.q.d.557.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
450.2.p.c.257.1	✓	8	45.22	odd	12
450.2.p.c.257.2	yes	8	45.13	odd	12
450.2.p.c.443.1	yes	8	3.2	odd	2
450.2.p.c.443.2	yes	8	15.14	odd	2
450.2.p.e.293.1	yes	8	9.4	even	3
450.2.p.e.293.2	yes	8	45.4	even	6
450.2.p.e.407.1	yes	8	15.2	even	4
450.2.p.e.407.2	yes	8	15.8	even	4
1350.2.q.a.1007.1		8	5.3	odd	4
1350.2.q.a.1007.2		8	5.2	odd	4
1350.2.q.a.1043.1		8	45.14	odd	6
1350.2.q.a.1043.2		8	9.5	odd	6
1350.2.q.d.143.1		8	5.4	even	2	inner
1350.2.q.d.143.2		8	1.1	even	1	trivial
1350.2.q.d.557.1		8	45.23	even	12	inner
1350.2.q.d.557.2		8	45.32	even	12	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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