LMFDB - Embedded newform 420.2.q.b.121.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [420,2,Mod(121,420)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(420, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("420.121"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	420.q (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$3.35371688489$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			121.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	420.121
Dual form			420.2.q.b.361.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(2.00000 + 3.46410i) q^{11} +7.00000 q^{13} +1.00000 q^{15} +(3.00000 + 5.19615i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(-2.00000 - 1.73205i) q^{21} +(1.00000 - 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -2.00000 q^{29} +(-3.50000 - 6.06218i) q^{31} +(-2.00000 + 3.46410i) q^{33} +(-0.500000 + 2.59808i) q^{35} +(3.50000 - 6.06218i) q^{37} +(3.50000 + 6.06218i) q^{39} -8.00000 q^{41} +5.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +(-5.00000 + 8.66025i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-3.00000 + 5.19615i) q^{51} +(4.00000 + 6.92820i) q^{53} +4.00000 q^{55} -3.00000 q^{57} +(-5.00000 - 8.66025i) q^{59} +(3.00000 - 5.19615i) q^{61} +(0.500000 - 2.59808i) q^{63} +(3.50000 - 6.06218i) q^{65} +(-1.50000 - 2.59808i) q^{67} +2.00000 q^{69} +(-7.50000 - 12.9904i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-8.00000 - 6.92820i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +8.00000 q^{83} +6.00000 q^{85} +(-1.00000 - 1.73205i) q^{87} +(-1.00000 + 1.73205i) q^{89} +(-17.5000 + 6.06218i) q^{91} +(3.50000 - 6.06218i) q^{93} +(1.50000 + 2.59808i) q^{95} -10.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + q^{5} - 5 q^{7} - q^{9} + 4 q^{11} + 14 q^{13} + 2 q^{15} + 6 q^{17} - 3 q^{19} - 4 q^{21} + 2 q^{23} - q^{25} - 2 q^{27} - 4 q^{29} - 7 q^{31} - 4 q^{33} - q^{35} + 7 q^{37} + 7 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) $ 2 * q + q^3 + q^5 - 5 * q^7 - q^9 + 4 * q^11 + 14 * q^13 + 2 * q^15 + 6 * q^17 - 3 * q^19 - 4 * q^21 + 2 * q^23 - q^25 - 2 * q^27 - 4 * q^29 - 7 * q^31 - 4 * q^33 - q^35 + 7 * q^37 + 7 * q^39 - 16 * q^41 + 10 * q^43 + q^45 - 10 * q^47 + 11 * q^49 - 6 * q^51 + 8 * q^53 + 8 * q^55 - 6 * q^57 - 10 * q^59 + 6 * q^61 + q^63 + 7 * q^65 - 3 * q^67 + 4 * q^69 - 15 * q^73 + q^75 - 16 * q^77 - q^79 - q^81 + 16 * q^83 + 12 * q^85 - 2 * q^87 - 2 * q^89 - 35 * q^91 + 7 * q^93 + 3 * q^95 - 20 * q^97 - 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/420\mathbb{Z}\right)^\times$.

$n$	$211$	$241$	$281$	$337$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$	$1$

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			0
$2$		0			0
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$3$	0.500000	+	0.866025i	0.288675	+	0.500000i
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$7$	−2.50000	+	0.866025i	−0.944911	+	0.327327i
$8$		0			0
$9$	−0.500000	+	0.866025i	−0.166667	+	0.288675i
$10$		0			0
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	0.992361	+	0.123371i	$0.0393705\pi$
$11$	2.00000	+	3.46410i	0.603023	+	1.04447i	−0.389338	+	0.921095i	$0.627296\pi$
$12$		0			0
$13$	7.00000			1.94145			0.970725	−	0.240192i	$-0.0772105\pi$
$13$	7.00000			1.94145			0.970725	+	0.240192i	$0.0772105\pi$
$14$		0			0
$15$	1.00000			0.258199
$16$		0			0
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$		0			0
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	−0.985194	−	0.171442i	$-0.945157\pi$
$19$	−1.50000	+	2.59808i	−0.344124	+	0.596040i	0.641071	+	0.767482i	$0.278491\pi$
$20$		0			0
$21$	−2.00000	−	1.73205i	−0.436436	−	0.377964i
$22$		0			0
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	−0.742732	−	0.669588i	$-0.766471\pi$
$23$	1.00000	−	1.73205i	0.208514	−	0.361158i	0.951247	+	0.308431i	$0.0998038\pi$
$24$		0			0
$25$	−0.500000	−	0.866025i	−0.100000	−	0.173205i
$26$		0			0
$27$	−1.00000			−0.192450
$28$		0			0
$29$	−2.00000			−0.371391			−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000			−0.371391			−0.185695	+	0.982607i	$0.559454\pi$
$30$		0			0
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−3.50000	−	6.06218i	−0.628619	−	1.08880i	0.359211	−	0.933257i	$-0.383046\pi$
$32$		0			0
$33$	−2.00000	+	3.46410i	−0.348155	+	0.603023i
$34$		0			0
$35$	−0.500000	+	2.59808i	−0.0845154	+	0.439155i
$36$		0			0
$37$	3.50000	−	6.06218i	0.575396	−	0.996616i	−0.420602	−	0.907245i	$-0.638181\pi$
$37$	3.50000	−	6.06218i	0.575396	−	0.996616i	0.995998	−	0.0893706i	$-0.0284856\pi$
$38$		0			0
$39$	3.50000	+	6.06218i	0.560449	+	0.970725i
$40$		0			0
$41$	−8.00000			−1.24939			−0.624695	−	0.780869i	$-0.714777\pi$
$41$	−8.00000			−1.24939			−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$	5.00000			0.762493			0.381246	−	0.924473i	$-0.375495\pi$
$43$	5.00000			0.762493			0.381246	+	0.924473i	$0.375495\pi$
$44$		0			0
$45$	0.500000	+	0.866025i	0.0745356	+	0.129099i
$46$		0			0
$47$	−5.00000	+	8.66025i	−0.729325	+	1.26323i	0.227844	+	0.973698i	$0.426832\pi$
$47$	−5.00000	+	8.66025i	−0.729325	+	1.26323i	−0.957169	+	0.289530i	$0.906501\pi$
$48$		0			0
$49$	5.50000	−	4.33013i	0.785714	−	0.618590i
$50$		0			0
$51$	−3.00000	+	5.19615i	−0.420084	+	0.727607i
$52$		0			0
$53$	4.00000	+	6.92820i	0.549442	+	0.951662i	0.998313	+	0.0580651i	$0.0184931\pi$
$53$	4.00000	+	6.92820i	0.549442	+	0.951662i	−0.448871	+	0.893597i	$0.648174\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$	−3.00000			−0.397360
$58$		0			0
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	−0.982894	−	0.184172i	$-0.941040\pi$
$59$	−5.00000	−	8.66025i	−0.650945	−	1.12747i	0.331949	−	0.943297i	$-0.392294\pi$
$60$		0			0
$61$	3.00000	−	5.19615i	0.384111	−	0.665299i	−0.607535	−	0.794293i	$-0.707841\pi$
$61$	3.00000	−	5.19615i	0.384111	−	0.665299i	0.991645	+	0.128994i	$0.0411748\pi$
$62$		0			0
$63$	0.500000	−	2.59808i	0.0629941	−	0.327327i
$64$		0			0
$65$	3.50000	−	6.06218i	0.434122	−	0.751921i
$66$		0			0
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	0.759733	−	0.650236i	$-0.225330\pi$
$67$	−1.50000	−	2.59808i	−0.183254	−	0.317406i	−0.942987	+	0.332830i	$0.891996\pi$
$68$		0			0
$69$	2.00000			0.240772
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$		0			0
$73$	−7.50000	−	12.9904i	−0.877809	−	1.52041i	−0.853740	−	0.520699i	$-0.825671\pi$
$73$	−7.50000	−	12.9904i	−0.877809	−	1.52041i	−0.0240681	−	0.999710i	$-0.507662\pi$
$74$		0			0
$75$	0.500000	−	0.866025i	0.0577350	−	0.100000i
$76$		0			0
$77$	−8.00000	−	6.92820i	−0.911685	−	0.789542i
$78$		0			0
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	−0.892781	−	0.450490i	$-0.851249\pi$
$79$	−0.500000	+	0.866025i	−0.0562544	+	0.0974355i	0.836527	+	0.547926i	$0.184582\pi$
$80$		0			0
$81$	−0.500000	−	0.866025i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	8.00000			0.878114			0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000			0.878114			0.439057	+	0.898459i	$0.355313\pi$
$84$		0			0
$85$	6.00000			0.650791
$86$		0			0
$87$	−1.00000	−	1.73205i	−0.107211	−	0.185695i
$88$		0			0
$89$	−1.00000	+	1.73205i	−0.106000	+	0.183597i	−0.914146	−	0.405385i	$-0.867138\pi$
$89$	−1.00000	+	1.73205i	−0.106000	+	0.183597i	0.808146	+	0.588982i	$0.200471\pi$
$90$		0			0
$91$	−17.5000	+	6.06218i	−1.83450	+	0.635489i
$92$		0			0
$93$	3.50000	−	6.06218i	0.362933	−	0.628619i
$94$		0			0
$95$	1.50000	+	2.59808i	0.153897	+	0.266557i
$96$		0			0
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	−0.993258	−	0.115924i	$-0.963017\pi$
$101$	−6.00000	−	10.3923i	−0.597022	−	1.03407i	0.396236	−	0.918149i	$-0.370316\pi$
$102$		0			0
$103$	3.50000	−	6.06218i	0.344865	−	0.597324i	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	3.50000	−	6.06218i	0.344865	−	0.597324i	0.985329	+	0.170664i	$0.0545913\pi$
$104$		0			0
$105$	−2.50000	+	0.866025i	−0.243975	+	0.0845154i
$106$		0			0
$107$	−1.00000	+	1.73205i	−0.0966736	+	0.167444i	−0.910306	−	0.413936i	$-0.864154\pi$
$107$	−1.00000	+	1.73205i	−0.0966736	+	0.167444i	0.813632	+	0.581380i	$0.197487\pi$
$108$		0			0
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	0.928877	−	0.370387i	$-0.120775\pi$
$109$	1.50000	+	2.59808i	0.143674	+	0.248851i	−0.785203	+	0.619238i	$0.787442\pi$
$110$		0			0
$111$	7.00000			0.664411
$112$		0			0
$113$	18.0000			1.69330			0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000			1.69330			0.846649	+	0.532152i	$0.178617\pi$
$114$		0			0
$115$	−1.00000	−	1.73205i	−0.0932505	−	0.161515i
$116$		0			0
$117$	−3.50000	+	6.06218i	−0.323575	+	0.560449i
$118$		0			0
$119$	−12.0000	−	10.3923i	−1.10004	−	0.952661i
$120$		0			0
$121$	−2.50000	+	4.33013i	−0.227273	+	0.393648i
$122$		0			0
$123$	−4.00000	−	6.92820i	−0.360668	−	0.624695i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	13.0000			1.15356			0.576782	−	0.816898i	$-0.304308\pi$
$127$	13.0000			1.15356			0.576782	+	0.816898i	$0.304308\pi$
$128$		0			0
$129$	2.50000	+	4.33013i	0.220113	+	0.381246i
$130$		0			0
$131$	−11.0000	+	19.0526i	−0.961074	+	1.66463i	−0.241264	+	0.970460i	$0.577562\pi$
$131$	−11.0000	+	19.0526i	−0.961074	+	1.66463i	−0.719811	+	0.694170i	$0.755772\pi$
$132$		0			0
$133$	1.50000	−	7.79423i	0.130066	−	0.675845i
$134$		0			0
$135$	−0.500000	+	0.866025i	−0.0430331	+	0.0745356i
$136$		0			0
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	−0.938148	−	0.346235i	$-0.887460\pi$
$137$	−9.00000	−	15.5885i	−0.768922	−	1.33181i	0.169226	−	0.985577i	$-0.445873\pi$
$138$		0			0
$139$	−13.0000			−1.10265			−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000			−1.10265			−0.551323	+	0.834292i	$0.685877\pi$
$140$		0			0
$141$	−10.0000			−0.842152
$142$		0			0
$143$	14.0000	+	24.2487i	1.17074	+	2.02778i
$144$		0			0
$145$	−1.00000	+	1.73205i	−0.0830455	+	0.143839i
$146$		0			0
$147$	6.50000	+	2.59808i	0.536111	+	0.214286i
$148$		0			0
$149$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$149$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$150$		0			0
$151$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$151$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$152$		0			0
$153$	−6.00000			−0.485071
$154$		0			0
$155$	−7.00000			−0.562254
$156$		0			0
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	−0.997609	−	0.0691164i	$-0.977982\pi$
$157$	−7.00000	−	12.1244i	−0.558661	−	0.967629i	0.438948	−	0.898513i	$-0.355351\pi$
$158$		0			0
$159$	−4.00000	+	6.92820i	−0.317221	+	0.549442i
$160$		0			0
$161$	−1.00000	+	5.19615i	−0.0788110	+	0.409514i
$162$		0			0
$163$	−2.00000	+	3.46410i	−0.156652	+	0.271329i	−0.933659	−	0.358162i	$-0.883403\pi$
$163$	−2.00000	+	3.46410i	−0.156652	+	0.271329i	0.777007	+	0.629492i	$0.216737\pi$
$164$		0			0
$165$	2.00000	+	3.46410i	0.155700	+	0.269680i
$166$		0			0
$167$	24.0000			1.85718			0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000			1.85718			0.928588	+	0.371113i	$0.121024\pi$
$168$		0			0
$169$	36.0000			2.76923
$170$		0			0
$171$	−1.50000	−	2.59808i	−0.114708	−	0.198680i
$172$		0			0
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	2.00000	+	1.73205i	0.151186	+	0.130931i
$176$		0			0
$177$	5.00000	−	8.66025i	0.375823	−	0.650945i
$178$		0			0
$179$	−7.00000	−	12.1244i	−0.523205	−	0.906217i	−0.999635	−	0.0270049i	$-0.991403\pi$
$179$	−7.00000	−	12.1244i	−0.523205	−	0.906217i	0.476431	−	0.879212i	$-0.341930\pi$
$180$		0			0
$181$	−3.00000			−0.222988			−0.111494	−	0.993765i	$-0.535564\pi$
$181$	−3.00000			−0.222988			−0.111494	+	0.993765i	$0.535564\pi$
$182$		0			0
$183$	6.00000			0.443533
$184$		0			0
$185$	−3.50000	−	6.06218i	−0.257325	−	0.445700i
$186$		0			0
$187$	−12.0000	+	20.7846i	−0.877527	+	1.51992i
$188$		0			0
$189$	2.50000	−	0.866025i	0.181848	−	0.0629941i
$190$		0			0
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	−0.331611	−	0.943416i	$-0.607592\pi$
$191$	9.00000	−	15.5885i	0.651217	−	1.12794i	0.982828	−	0.184525i	$-0.0590746\pi$
$192$		0			0
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	0.899770	+	0.436365i	$0.143734\pi$
$193$	11.5000	+	19.9186i	0.827788	+	1.43377i	−0.0719816	+	0.997406i	$0.522932\pi$
$194$		0			0
$195$	7.00000			0.501280
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$199$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$200$		0			0
$201$	1.50000	−	2.59808i	0.105802	−	0.183254i
$202$		0			0
$203$	5.00000	−	1.73205i	0.350931	−	0.121566i
$204$		0			0
$205$	−4.00000	+	6.92820i	−0.279372	+	0.483887i
$206$		0			0
$207$	1.00000	+	1.73205i	0.0695048	+	0.120386i
$208$		0			0
$209$	−12.0000			−0.830057
$210$		0			0
$211$	−4.00000			−0.275371			−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000			−0.275371			−0.137686	+	0.990476i	$0.543966\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2.50000	−	4.33013i	0.170499	−	0.295312i
$216$		0			0
$217$	14.0000	+	12.1244i	0.950382	+	0.823055i
$218$		0			0
$219$	7.50000	−	12.9904i	0.506803	−	0.877809i
$220$		0			0
$221$	21.0000	+	36.3731i	1.41261	+	2.44672i
$222$		0			0
$223$	−12.0000			−0.803579			−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000			−0.803579			−0.401790	+	0.915732i	$0.631612\pi$
$224$		0			0
$225$	1.00000			0.0666667
$226$		0			0
$227$	−4.00000	−	6.92820i	−0.265489	−	0.459841i	0.702202	−	0.711977i	$-0.252200\pi$
$227$	−4.00000	−	6.92820i	−0.265489	−	0.459841i	−0.967692	+	0.252136i	$0.918867\pi$
$228$		0			0
$229$	−8.50000	+	14.7224i	−0.561696	+	0.972886i	0.435653	+	0.900115i	$0.356518\pi$
$229$	−8.50000	+	14.7224i	−0.561696	+	0.972886i	−0.997349	+	0.0727709i	$0.976816\pi$
$230$		0			0
$231$	2.00000	−	10.3923i	0.131590	−	0.683763i
$232$		0			0
$233$	−3.00000	+	5.19615i	−0.196537	+	0.340411i	−0.947403	−	0.320043i	$-0.896303\pi$
$233$	−3.00000	+	5.19615i	−0.196537	+	0.340411i	0.750867	+	0.660454i	$0.229636\pi$
$234$		0			0
$235$	5.00000	+	8.66025i	0.326164	+	0.564933i
$236$		0			0
$237$	−1.00000			−0.0649570
$238$		0			0
$239$	6.00000			0.388108			0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000			0.388108			0.194054	+	0.980991i	$0.437836\pi$
$240$		0			0
$241$	7.00000	+	12.1244i	0.450910	+	0.780998i	0.998443	−	0.0557856i	$-0.0177663\pi$
$241$	7.00000	+	12.1244i	0.450910	+	0.780998i	−0.547533	+	0.836784i	$0.684433\pi$
$242$		0			0
$243$	0.500000	−	0.866025i	0.0320750	−	0.0555556i
$244$		0			0
$245$	−1.00000	−	6.92820i	−0.0638877	−	0.442627i
$246$		0			0
$247$	−10.5000	+	18.1865i	−0.668099	+	1.15718i
$248$		0			0
$249$	4.00000	+	6.92820i	0.253490	+	0.439057i
$250$		0			0
$251$	22.0000			1.38863			0.694314	−	0.719672i	$-0.255708\pi$
$251$	22.0000			1.38863			0.694314	+	0.719672i	$0.255708\pi$
$252$		0			0
$253$	8.00000			0.502956
$254$		0			0
$255$	3.00000	+	5.19615i	0.187867	+	0.325396i
$256$		0			0
$257$	−8.00000	+	13.8564i	−0.499026	+	0.864339i	−0.999999	−	0.00112398i	$-0.999642\pi$
$257$	−8.00000	+	13.8564i	−0.499026	+	0.864339i	0.500973	+	0.865463i	$0.332976\pi$
$258$		0			0
$259$	−3.50000	+	18.1865i	−0.217479	+	1.13006i
$260$		0			0
$261$	1.00000	−	1.73205i	0.0618984	−	0.107211i
$262$		0			0
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	0.619586	−	0.784929i	$-0.287301\pi$
$263$	−6.00000	−	10.3923i	−0.369976	−	0.640817i	−0.989561	+	0.144112i	$0.953967\pi$
$264$		0			0
$265$	8.00000			0.491436
$266$		0			0
$267$	−2.00000			−0.122398
$268$		0			0
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	0.977229	−	0.212187i	$-0.0680585\pi$
$269$	5.00000	+	8.66025i	0.304855	+	0.528025i	−0.672374	+	0.740212i	$0.734725\pi$
$270$		0			0
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	−0.513905	−	0.857847i	$-0.671801\pi$
$271$	8.00000	−	13.8564i	0.485965	−	0.841717i	0.999870	+	0.0161307i	$0.00513477\pi$
$272$		0			0
$273$	−14.0000	−	12.1244i	−0.847319	−	0.733799i
$274$		0			0
$275$	2.00000	−	3.46410i	0.120605	−	0.208893i
$276$		0			0
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	−0.971521	−	0.236953i	$-0.923851\pi$
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	0.280553	−	0.959839i	$-0.409482\pi$
$278$		0			0
$279$	7.00000			0.419079
$280$		0			0
$281$	−16.0000			−0.954480			−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000			−0.954480			−0.477240	+	0.878773i	$0.658363\pi$
$282$		0			0
$283$	−12.5000	−	21.6506i	−0.743048	−	1.28700i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	−12.5000	−	21.6506i	−0.743048	−	1.28700i	0.208053	−	0.978117i	$-0.433287\pi$
$284$		0			0
$285$	−1.50000	+	2.59808i	−0.0888523	+	0.153897i
$286$		0			0
$287$	20.0000	−	6.92820i	1.18056	−	0.408959i
$288$		0			0
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$		0			0
$291$	−5.00000	−	8.66025i	−0.293105	−	0.507673i
$292$		0			0
$293$	−12.0000			−0.701047			−0.350524	−	0.936554i	$-0.613996\pi$
$293$	−12.0000			−0.701047			−0.350524	+	0.936554i	$0.613996\pi$
$294$		0			0
$295$	−10.0000			−0.582223
$296$		0			0
$297$	−2.00000	−	3.46410i	−0.116052	−	0.201008i
$298$		0			0
$299$	7.00000	−	12.1244i	0.404820	−	0.701170i
$300$		0			0
$301$	−12.5000	+	4.33013i	−0.720488	+	0.249584i
$302$		0			0
$303$	6.00000	−	10.3923i	0.344691	−	0.597022i
$304$		0			0
$305$	−3.00000	−	5.19615i	−0.171780	−	0.297531i
$306$		0			0
$307$	11.0000			0.627803			0.313902	−	0.949456i	$-0.398364\pi$
$307$	11.0000			0.627803			0.313902	+	0.949456i	$0.398364\pi$
$308$		0			0
$309$	7.00000			0.398216
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	−10.5000	+	18.1865i	−0.593495	+	1.02796i	0.400262	+	0.916401i	$0.368919\pi$
$313$	−10.5000	+	18.1865i	−0.593495	+	1.02796i	−0.993757	+	0.111563i	$0.964414\pi$
$314$		0			0
$315$	−2.00000	−	1.73205i	−0.112687	−	0.0975900i
$316$		0			0
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	0.494489	+	0.869184i	$0.335355\pi$
$317$	−9.00000	+	15.5885i	−0.505490	+	0.875535i	−0.999980	+	0.00635137i	$0.997978\pi$
$318$		0			0
$319$	−4.00000	−	6.92820i	−0.223957	−	0.387905i
$320$		0			0
$321$	−2.00000			−0.111629
$322$		0			0
$323$	−18.0000			−1.00155
$324$		0			0
$325$	−3.50000	−	6.06218i	−0.194145	−	0.336269i
$326$		0			0
$327$	−1.50000	+	2.59808i	−0.0829502	+	0.143674i
$328$		0			0
$329$	5.00000	−	25.9808i	0.275659	−	1.43237i
$330$		0			0
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	−0.753660	−	0.657264i	$-0.771714\pi$
$331$	3.50000	−	6.06218i	0.192377	−	0.333207i	0.946038	+	0.324057i	$0.105047\pi$
$332$		0			0
$333$	3.50000	+	6.06218i	0.191799	+	0.332205i
$334$		0			0
$335$	−3.00000			−0.163908
$336$		0			0
$337$	15.0000			0.817102			0.408551	−	0.912735i	$-0.366034\pi$
$337$	15.0000			0.817102			0.408551	+	0.912735i	$0.366034\pi$
$338$		0			0
$339$	9.00000	+	15.5885i	0.488813	+	0.846649i
$340$		0			0
$341$	14.0000	−	24.2487i	0.758143	−	1.31314i
$342$		0			0
$343$	−10.0000	+	15.5885i	−0.539949	+	0.841698i
$344$		0			0
$345$	1.00000	−	1.73205i	0.0538382	−	0.0932505i
$346$		0			0
$347$	−4.00000	−	6.92820i	−0.214731	−	0.371925i	0.738458	−	0.674299i	$-0.235554\pi$
$347$	−4.00000	−	6.92820i	−0.214731	−	0.371925i	−0.953189	+	0.302374i	$0.902221\pi$
$348$		0			0
$349$	−6.00000			−0.321173			−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000			−0.321173			−0.160586	+	0.987022i	$0.551338\pi$
$350$		0			0
$351$	−7.00000			−0.373632
$352$		0			0
$353$	6.00000	+	10.3923i	0.319348	+	0.553127i	0.980352	−	0.197256i	$-0.0632029\pi$
$353$	6.00000	+	10.3923i	0.319348	+	0.553127i	−0.661004	+	0.750382i	$0.729870\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	3.00000	−	15.5885i	0.158777	−	0.825029i
$358$		0			0
$359$	7.00000	−	12.1244i	0.369446	−	0.639899i	−0.620033	−	0.784576i	$-0.712881\pi$
$359$	7.00000	−	12.1244i	0.369446	−	0.639899i	0.989479	+	0.144677i	$0.0462142\pi$
$360$		0			0
$361$	5.00000	+	8.66025i	0.263158	+	0.455803i
$362$		0			0
$363$	−5.00000			−0.262432
$364$		0			0
$365$	−15.0000			−0.785136
$366$		0			0
$367$	2.50000	+	4.33013i	0.130499	+	0.226031i	0.923869	−	0.382709i	$-0.125009\pi$
$367$	2.50000	+	4.33013i	0.130499	+	0.226031i	−0.793370	+	0.608740i	$0.791675\pi$
$368$		0			0
$369$	4.00000	−	6.92820i	0.208232	−	0.360668i
$370$		0			0
$371$	−16.0000	−	13.8564i	−0.830679	−	0.719389i
$372$		0			0
$373$	−4.50000	+	7.79423i	−0.233001	+	0.403570i	−0.958690	−	0.284453i	$-0.908188\pi$
$373$	−4.50000	+	7.79423i	−0.233001	+	0.403570i	0.725689	+	0.688023i	$0.241521\pi$
$374$		0			0
$375$	−0.500000	−	0.866025i	−0.0258199	−	0.0447214i
$376$		0			0
$377$	−14.0000			−0.721037
$378$		0			0
$379$	−7.00000			−0.359566			−0.179783	−	0.983706i	$-0.557540\pi$
$379$	−7.00000			−0.359566			−0.179783	+	0.983706i	$0.557540\pi$
$380$		0			0
$381$	6.50000	+	11.2583i	0.333005	+	0.576782i
$382$		0			0
$383$	−4.00000	+	6.92820i	−0.204390	+	0.354015i	−0.949938	−	0.312437i	$-0.898855\pi$
$383$	−4.00000	+	6.92820i	−0.204390	+	0.354015i	0.745548	+	0.666452i	$0.232188\pi$
$384$		0			0
$385$	−10.0000	+	3.46410i	−0.509647	+	0.176547i
$386$		0			0
$387$	−2.50000	+	4.33013i	−0.127082	+	0.220113i
$388$		0			0
$389$	−8.00000	−	13.8564i	−0.405616	−	0.702548i	0.588777	−	0.808296i	$-0.299610\pi$
$389$	−8.00000	−	13.8564i	−0.405616	−	0.702548i	−0.994393	+	0.105748i	$0.966276\pi$
$390$		0			0
$391$	12.0000			0.606866
$392$		0			0
$393$	−22.0000			−1.10975
$394$		0			0
$395$	0.500000	+	0.866025i	0.0251577	+	0.0435745i
$396$		0			0
$397$	9.50000	−	16.4545i	0.476791	−	0.825827i	−0.522855	−	0.852422i	$-0.675133\pi$
$397$	9.50000	−	16.4545i	0.476791	−	0.825827i	0.999646	+	0.0265948i	$0.00846640\pi$
$398$		0			0
$399$	7.50000	−	2.59808i	0.375470	−	0.130066i
$400$		0			0
$401$	16.0000	−	27.7128i	0.799002	−	1.38391i	−0.121265	−	0.992620i	$-0.538695\pi$
$401$	16.0000	−	27.7128i	0.799002	−	1.38391i	0.920267	−	0.391292i	$-0.127972\pi$
$402$		0			0
$403$	−24.5000	−	42.4352i	−1.22043	−	2.11385i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$	28.0000			1.38791
$408$		0			0
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	0.999402	−	0.0345902i	$-0.0110126\pi$
$409$	9.50000	+	16.4545i	0.469745	+	0.813622i	−0.529657	+	0.848212i	$0.677679\pi$
$410$		0			0
$411$	9.00000	−	15.5885i	0.443937	−	0.768922i
$412$		0			0
$413$	20.0000	+	17.3205i	0.984136	+	0.852286i
$414$		0			0
$415$	4.00000	−	6.92820i	0.196352	−	0.340092i
$416$		0			0
$417$	−6.50000	−	11.2583i	−0.318306	−	0.551323i
$418$		0			0
$419$	−26.0000			−1.27018			−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000			−1.27018			−0.635092	+	0.772437i	$0.719038\pi$
$420$		0			0
$421$	13.0000			0.633581			0.316791	−	0.948495i	$-0.397395\pi$
$421$	13.0000			0.633581			0.316791	+	0.948495i	$0.397395\pi$
$422$		0			0
$423$	−5.00000	−	8.66025i	−0.243108	−	0.421076i
$424$		0			0
$425$	3.00000	−	5.19615i	0.145521	−	0.252050i
$426$		0			0
$427$	−3.00000	+	15.5885i	−0.145180	+	0.754378i
$428$		0			0
$429$	−14.0000	+	24.2487i	−0.675926	+	1.17074i
$430$		0			0
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	0.959985	+	0.280052i	$0.0903517\pi$
$431$	15.0000	+	25.9808i	0.722525	+	1.25145i	−0.237460	+	0.971397i	$0.576315\pi$
$432$		0			0
$433$	−23.0000			−1.10531			−0.552655	−	0.833410i	$-0.686385\pi$
$433$	−23.0000			−1.10531			−0.552655	+	0.833410i	$0.686385\pi$
$434$		0			0
$435$	−2.00000			−0.0958927
$436$		0			0
$437$	3.00000	+	5.19615i	0.143509	+	0.248566i
$438$		0			0
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	−0.310228	−	0.950662i	$-0.600405\pi$
$439$	14.0000	−	24.2487i	0.668184	−	1.15733i	0.978412	−	0.206666i	$-0.0662612\pi$
$440$		0			0
$441$	1.00000	+	6.92820i	0.0476190	+	0.329914i
$442$		0			0
$443$	−2.00000	+	3.46410i	−0.0950229	+	0.164584i	−0.909618	−	0.415445i	$-0.863626\pi$
$443$	−2.00000	+	3.46410i	−0.0950229	+	0.164584i	0.814595	+	0.580030i	$0.196959\pi$
$444$		0			0
$445$	1.00000	+	1.73205i	0.0474045	+	0.0821071i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−34.0000			−1.60456			−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000			−1.60456			−0.802280	+	0.596948i	$0.796380\pi$
$450$		0			0
$451$	−16.0000	−	27.7128i	−0.753411	−	1.30495i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.50000	+	18.1865i	−0.164083	+	0.852598i
$456$		0			0
$457$	4.50000	−	7.79423i	0.210501	−	0.364599i	−0.741370	−	0.671096i	$-0.765824\pi$
$457$	4.50000	−	7.79423i	0.210501	−	0.364599i	0.951871	+	0.306497i	$0.0991571\pi$
$458$		0			0
$459$	−3.00000	−	5.19615i	−0.140028	−	0.242536i
$460$		0			0
$461$	30.0000			1.39724			0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000			1.39724			0.698620	+	0.715493i	$0.253798\pi$
$462$		0			0
$463$	−1.00000			−0.0464739			−0.0232370	−	0.999730i	$-0.507397\pi$
$463$	−1.00000			−0.0464739			−0.0232370	+	0.999730i	$0.507397\pi$
$464$		0			0
$465$	−3.50000	−	6.06218i	−0.162309	−	0.281127i
$466$		0			0
$467$	−9.00000	+	15.5885i	−0.416470	+	0.721348i	−0.995582	−	0.0939008i	$-0.970066\pi$
$467$	−9.00000	+	15.5885i	−0.416470	+	0.721348i	0.579111	+	0.815249i	$0.303400\pi$
$468$		0			0
$469$	6.00000	+	5.19615i	0.277054	+	0.239936i
$470$		0			0
$471$	7.00000	−	12.1244i	0.322543	−	0.558661i
$472$		0			0
$473$	10.0000	+	17.3205i	0.459800	+	0.796398i
$474$		0			0
$475$	3.00000			0.137649
$476$		0			0
$477$	−8.00000			−0.366295
$478$		0			0
$479$	−20.0000	−	34.6410i	−0.913823	−	1.58279i	−0.808615	−	0.588338i	$-0.799782\pi$
$479$	−20.0000	−	34.6410i	−0.913823	−	1.58279i	−0.105208	−	0.994450i	$-0.533551\pi$
$480$		0			0
$481$	24.5000	−	42.4352i	1.11710	−	1.93488i
$482$		0			0
$483$	−5.00000	+	1.73205i	−0.227508	+	0.0788110i
$484$		0			0
$485$	−5.00000	+	8.66025i	−0.227038	+	0.393242i
$486$		0			0
$487$	−11.5000	−	19.9186i	−0.521115	−	0.902597i	−0.999698	−	0.0245553i	$-0.992183\pi$
$487$	−11.5000	−	19.9186i	−0.521115	−	0.902597i	0.478584	−	0.878042i	$-0.341150\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$	−4.00000			−0.180517			−0.0902587	−	0.995918i	$-0.528769\pi$
$491$	−4.00000			−0.180517			−0.0902587	+	0.995918i	$0.528769\pi$
$492$		0			0
$493$	−6.00000	−	10.3923i	−0.270226	−	0.468046i
$494$		0			0
$495$	−2.00000	+	3.46410i	−0.0898933	+	0.155700i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	−0.933670	−	0.358134i	$-0.883413\pi$
$499$	−3.50000	+	6.06218i	−0.156682	+	0.271380i	0.776989	+	0.629515i	$0.216746\pi$
$500$		0			0
$501$	12.0000	+	20.7846i	0.536120	+	0.928588i
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$	−12.0000			−0.533993
$506$		0			0
$507$	18.0000	+	31.1769i	0.799408	+	1.38462i
$508$		0			0
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	−0.978421	−	0.206623i	$-0.933753\pi$
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	0.668151	+	0.744026i	$0.267086\pi$
$510$		0			0
$511$	30.0000	+	25.9808i	1.32712	+	1.14932i
$512$		0			0
$513$	1.50000	−	2.59808i	0.0662266	−	0.114708i
$514$		0			0
$515$	−3.50000	−	6.06218i	−0.154228	−	0.267131i
$516$		0			0
$517$	−40.0000			−1.75920
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	−0.854887	−	0.518815i	$-0.826373\pi$
$523$	0.500000	−	0.866025i	0.0218635	−	0.0378686i	0.876750	+	0.480946i	$0.159707\pi$
$524$		0			0
$525$	−0.500000	+	2.59808i	−0.0218218	+	0.113389i
$526$		0			0
$527$	21.0000	−	36.3731i	0.914774	−	1.58444i
$528$		0			0
$529$	9.50000	+	16.4545i	0.413043	+	0.715412i
$530$		0			0
$531$	10.0000			0.433963
$532$		0			0
$533$	−56.0000			−2.42563
$534$		0			0
$535$	1.00000	+	1.73205i	0.0432338	+	0.0748831i
$536$		0			0
$537$	7.00000	−	12.1244i	0.302072	−	0.523205i
$538$		0			0
$539$	26.0000	+	10.3923i	1.11990	+	0.447628i
$540$		0			0
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	−0.988847	−	0.148933i	$-0.952416\pi$
$541$	−8.50000	+	14.7224i	−0.365444	+	0.632967i	0.623404	+	0.781900i	$0.285749\pi$
$542$		0			0
$543$	−1.50000	−	2.59808i	−0.0643712	−	0.111494i
$544$		0			0
$545$	3.00000			0.128506
$546$		0			0
$547$	16.0000			0.684111			0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000			0.684111			0.342055	+	0.939680i	$0.388877\pi$
$548$		0			0
$549$	3.00000	+	5.19615i	0.128037	+	0.221766i
$550$		0			0
$551$	3.00000	−	5.19615i	0.127804	−	0.221364i
$552$		0			0
$553$	0.500000	−	2.59808i	0.0212622	−	0.110481i
$554$		0			0
$555$	3.50000	−	6.06218i	0.148567	−	0.257325i
$556$		0			0
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	0.986394	+	0.164399i	$0.0525683\pi$
$557$	15.0000	+	25.9808i	0.635570	+	1.10084i	−0.350824	+	0.936442i	$0.614098\pi$
$558$		0			0
$559$	35.0000			1.48034
$560$		0			0
$561$	−24.0000			−1.01328
$562$		0			0
$563$	7.00000	+	12.1244i	0.295015	+	0.510981i	0.974988	−	0.222256i	$-0.0713421\pi$
$563$	7.00000	+	12.1244i	0.295015	+	0.510981i	−0.679974	+	0.733237i	$0.738009\pi$
$564$		0			0
$565$	9.00000	−	15.5885i	0.378633	−	0.655811i
$566$		0			0
$567$	2.00000	+	1.73205i	0.0839921	+	0.0727393i
$568$		0			0
$569$	−6.00000	+	10.3923i	−0.251533	+	0.435668i	−0.963948	−	0.266090i	$-0.914268\pi$
$569$	−6.00000	+	10.3923i	−0.251533	+	0.435668i	0.712415	+	0.701758i	$0.247601\pi$
$570$		0			0
$571$	20.5000	+	35.5070i	0.857898	+	1.48592i	0.873930	+	0.486052i	$0.161563\pi$
$571$	20.5000	+	35.5070i	0.857898	+	1.48592i	−0.0160316	+	0.999871i	$0.505103\pi$
$572$		0			0
$573$	18.0000			0.751961
$574$		0			0
$575$	−2.00000			−0.0834058
$576$		0			0
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	0.957503	+	0.288425i	$0.0931316\pi$
$577$	17.5000	+	30.3109i	0.728535	+	1.26186i	−0.228968	+	0.973434i	$0.573535\pi$
$578$		0			0
$579$	−11.5000	+	19.9186i	−0.477924	+	0.827788i
$580$		0			0
$581$	−20.0000	+	6.92820i	−0.829740	+	0.287430i
$582$		0			0
$583$	−16.0000	+	27.7128i	−0.662652	+	1.14775i
$584$		0			0
$585$	3.50000	+	6.06218i	0.144707	+	0.250640i
$586$		0			0
$587$	−18.0000			−0.742940			−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000			−0.742940			−0.371470	+	0.928445i	$0.621146\pi$
$588$		0			0
$589$	21.0000			0.865290
$590$		0			0
$591$	1.00000	+	1.73205i	0.0411345	+	0.0712470i
$592$		0			0
$593$	16.0000	−	27.7128i	0.657041	−	1.13803i	−0.324337	−	0.945942i	$-0.605141\pi$
$593$	16.0000	−	27.7128i	0.657041	−	1.13803i	0.981378	−	0.192087i	$-0.0615256\pi$
$594$		0			0
$595$	−15.0000	+	5.19615i	−0.614940	+	0.213021i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.0000	−	17.3205i	−0.408589	−	0.707697i	0.586143	−	0.810208i	$-0.300646\pi$
$599$	−10.0000	−	17.3205i	−0.408589	−	0.707697i	−0.994732	+	0.102511i	$0.967312\pi$
$600$		0			0
$601$	−17.0000			−0.693444			−0.346722	−	0.937968i	$-0.612705\pi$
$601$	−17.0000			−0.693444			−0.346722	+	0.937968i	$0.612705\pi$
$602$		0			0
$603$	3.00000			0.122169
$604$		0			0
$605$	2.50000	+	4.33013i	0.101639	+	0.176045i
$606$		0			0
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	−0.703429	−	0.710766i	$-0.748349\pi$
$607$	6.50000	−	11.2583i	0.263827	−	0.456962i	0.967256	+	0.253804i	$0.0816819\pi$
$608$		0			0
$609$	4.00000	+	3.46410i	0.162088	+	0.140372i
$610$		0			0
$611$	−35.0000	+	60.6218i	−1.41595	+	2.45249i
$612$		0			0
$613$	−11.0000	−	19.0526i	−0.444286	−	0.769526i	0.553716	−	0.832705i	$-0.313209\pi$
$613$	−11.0000	−	19.0526i	−0.444286	−	0.769526i	−0.998002	+	0.0631797i	$0.979876\pi$
$614$		0			0
$615$	−8.00000			−0.322591
$616$		0			0
$617$	−2.00000			−0.0805170			−0.0402585	−	0.999189i	$-0.512818\pi$
$617$	−2.00000			−0.0805170			−0.0402585	+	0.999189i	$0.512818\pi$
$618$		0			0
$619$	−14.5000	−	25.1147i	−0.582804	−	1.00945i	−0.995145	−	0.0984169i	$-0.968622\pi$
$619$	−14.5000	−	25.1147i	−0.582804	−	1.00945i	0.412341	−	0.911030i	$-0.364711\pi$
$620$		0			0
$621$	−1.00000	+	1.73205i	−0.0401286	+	0.0695048i
$622$		0			0
$623$	1.00000	−	5.19615i	0.0400642	−	0.208179i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$	−6.00000	−	10.3923i	−0.239617	−	0.415029i
$628$		0			0
$629$	42.0000			1.67465
$630$		0			0
$631$	−32.0000			−1.27390			−0.636950	−	0.770905i	$-0.719804\pi$
$631$	−32.0000			−1.27390			−0.636950	+	0.770905i	$0.719804\pi$
$632$		0			0
$633$	−2.00000	−	3.46410i	−0.0794929	−	0.137686i
$634$		0			0
$635$	6.50000	−	11.2583i	0.257945	−	0.446773i
$636$		0			0
$637$	38.5000	−	30.3109i	1.52543	−	1.20096i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−3.00000	−	5.19615i	−0.118493	−	0.205236i	0.800678	−	0.599095i	$-0.204473\pi$
$641$	−3.00000	−	5.19615i	−0.118493	−	0.205236i	−0.919171	+	0.393860i	$0.871140\pi$
$642$		0			0
$643$	5.00000			0.197181			0.0985904	−	0.995128i	$-0.468567\pi$
$643$	5.00000			0.197181			0.0985904	+	0.995128i	$0.468567\pi$
$644$		0			0
$645$	5.00000			0.196875
$646$		0			0
$647$	22.0000	+	38.1051i	0.864909	+	1.49807i	0.867137	+	0.498069i	$0.165957\pi$
$647$	22.0000	+	38.1051i	0.864909	+	1.49807i	−0.00222801	+	0.999998i	$0.500709\pi$
$648$		0			0
$649$	20.0000	−	34.6410i	0.785069	−	1.35978i
$650$		0			0
$651$	−3.50000	+	18.1865i	−0.137176	+	0.712786i
$652$		0			0
$653$	−14.0000	+	24.2487i	−0.547862	+	0.948925i	0.450558	+	0.892747i	$0.351225\pi$
$653$	−14.0000	+	24.2487i	−0.547862	+	0.948925i	−0.998421	+	0.0561784i	$0.982108\pi$
$654$		0			0
$655$	11.0000	+	19.0526i	0.429806	+	0.744445i
$656$		0			0
$657$	15.0000			0.585206
$658$		0			0
$659$	12.0000			0.467454			0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000			0.467454			0.233727	+	0.972302i	$0.424908\pi$
$660$		0			0
$661$	−7.50000	−	12.9904i	−0.291716	−	0.505267i	0.682499	−	0.730886i	$-0.260893\pi$
$661$	−7.50000	−	12.9904i	−0.291716	−	0.505267i	−0.974216	+	0.225619i	$0.927560\pi$
$662$		0			0
$663$	−21.0000	+	36.3731i	−0.815572	+	1.41261i
$664$		0			0
$665$	−6.00000	−	5.19615i	−0.232670	−	0.201498i
$666$		0			0
$667$	−2.00000	+	3.46410i	−0.0774403	+	0.134131i
$668$		0			0
$669$	−6.00000	−	10.3923i	−0.231973	−	0.401790i
$670$		0			0
$671$	24.0000			0.926510
$672$		0			0
$673$	37.0000			1.42625			0.713123	−	0.701039i	$-0.247280\pi$
$673$	37.0000			1.42625			0.713123	+	0.701039i	$0.247280\pi$
$674$		0			0
$675$	0.500000	+	0.866025i	0.0192450	+	0.0333333i
$676$		0			0
$677$	−1.00000	+	1.73205i	−0.0384331	+	0.0665681i	−0.884602	−	0.466347i	$-0.845570\pi$
$677$	−1.00000	+	1.73205i	−0.0384331	+	0.0665681i	0.846169	+	0.532915i	$0.178903\pi$
$678$		0			0
$679$	25.0000	−	8.66025i	0.959412	−	0.332350i
$680$		0			0
$681$	4.00000	−	6.92820i	0.153280	−	0.265489i
$682$		0			0
$683$	−5.00000	−	8.66025i	−0.191320	−	0.331375i	0.754368	−	0.656452i	$-0.227943\pi$
$683$	−5.00000	−	8.66025i	−0.191320	−	0.331375i	−0.945688	+	0.325076i	$0.894610\pi$
$684$		0			0
$685$	−18.0000			−0.687745
$686$		0			0
$687$	−17.0000			−0.648590
$688$		0			0
$689$	28.0000	+	48.4974i	1.06672	+	1.84760i
$690$		0			0
$691$	25.5000	−	44.1673i	0.970066	−	1.68020i	0.274725	−	0.961523i	$-0.411413\pi$
$691$	25.5000	−	44.1673i	0.970066	−	1.68020i	0.695341	−	0.718680i	$-0.255253\pi$
$692$		0			0
$693$	10.0000	−	3.46410i	0.379869	−	0.131590i
$694$		0			0
$695$	−6.50000	+	11.2583i	−0.246559	+	0.427053i
$696$		0			0
$697$	−24.0000	−	41.5692i	−0.909065	−	1.57455i
$698$		0			0
$699$	−6.00000			−0.226941
$700$		0			0
$701$	42.0000			1.58632			0.793159	−	0.609015i	$-0.208435\pi$
$701$	42.0000			1.58632			0.793159	+	0.609015i	$0.208435\pi$
$702$		0			0
$703$	10.5000	+	18.1865i	0.396015	+	0.685918i
$704$		0			0
$705$	−5.00000	+	8.66025i	−0.188311	+	0.326164i
$706$		0			0
$707$	24.0000	+	20.7846i	0.902613	+	0.781686i
$708$		0			0
$709$	−7.00000	+	12.1244i	−0.262891	+	0.455340i	−0.967009	−	0.254743i	$-0.918009\pi$
$709$	−7.00000	+	12.1244i	−0.262891	+	0.455340i	0.704118	+	0.710083i	$0.251342\pi$
$710$		0			0
$711$	−0.500000	−	0.866025i	−0.0187515	−	0.0324785i
$712$		0			0
$713$	−14.0000			−0.524304
$714$		0			0
$715$	28.0000			1.04714
$716$		0			0
$717$	3.00000	+	5.19615i	0.112037	+	0.194054i
$718$		0			0
$719$	−25.0000	+	43.3013i	−0.932343	+	1.61486i	−0.153037	+	0.988220i	$0.548906\pi$
$719$	−25.0000	+	43.3013i	−0.932343	+	1.61486i	−0.779305	+	0.626644i	$0.784428\pi$
$720$		0			0
$721$	−3.50000	+	18.1865i	−0.130347	+	0.677302i
$722$		0			0
$723$	−7.00000	+	12.1244i	−0.260333	+	0.450910i
$724$		0			0
$725$	1.00000	+	1.73205i	0.0371391	+	0.0643268i
$726$		0			0
$727$	23.0000			0.853023			0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000			0.853023			0.426511	+	0.904482i	$0.359742\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	15.0000	+	25.9808i	0.554795	+	0.960933i
$732$		0			0
$733$	6.50000	−	11.2583i	0.240083	−	0.415836i	−0.720655	−	0.693294i	$-0.756159\pi$
$733$	6.50000	−	11.2583i	0.240083	−	0.415836i	0.960738	+	0.277458i	$0.0894920\pi$
$734$		0			0
$735$	5.50000	−	4.33013i	0.202871	−	0.159719i
$736$		0			0
$737$	6.00000	−	10.3923i	0.221013	−	0.382805i
$738$		0			0
$739$	0.500000	+	0.866025i	0.0183928	+	0.0318573i	0.875075	−	0.483987i	$-0.160812\pi$
$739$	0.500000	+	0.866025i	0.0183928	+	0.0318573i	−0.856683	+	0.515844i	$0.827478\pi$
$740$		0			0
$741$	−21.0000			−0.771454
$742$		0			0
$743$	−44.0000			−1.61420			−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000			−1.61420			−0.807102	+	0.590412i	$0.798965\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−4.00000	+	6.92820i	−0.146352	+	0.253490i
$748$		0			0
$749$	1.00000	−	5.19615i	0.0365392	−	0.189863i
$750$		0			0
$751$	12.5000	−	21.6506i	0.456131	−	0.790043i	−0.542621	−	0.839978i	$-0.682568\pi$
$751$	12.5000	−	21.6506i	0.456131	−	0.790043i	0.998752	+	0.0499348i	$0.0159013\pi$
$752$		0			0
$753$	11.0000	+	19.0526i	0.400862	+	0.694314i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$	4.00000	+	6.92820i	0.145191	+	0.251478i
$760$		0			0
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	−0.454935	−	0.890525i	$-0.650337\pi$
$761$	15.0000	−	25.9808i	0.543750	−	0.941802i	0.998684	−	0.0512772i	$-0.0163292\pi$
$762$		0			0
$763$	−6.00000	−	5.19615i	−0.217215	−	0.188113i
$764$		0			0
$765$	−3.00000	+	5.19615i	−0.108465	+	0.187867i
$766$		0			0
$767$	−35.0000	−	60.6218i	−1.26378	−	2.18893i
$768$		0			0
$769$	−29.0000			−1.04577			−0.522883	−	0.852404i	$-0.675144\pi$
$769$	−29.0000			−1.04577			−0.522883	+	0.852404i	$0.675144\pi$
$770$		0			0
$771$	−16.0000			−0.576226
$772$		0			0
$773$	6.00000	+	10.3923i	0.215805	+	0.373785i	0.953521	−	0.301326i	$-0.0974291\pi$
$773$	6.00000	+	10.3923i	0.215805	+	0.373785i	−0.737716	+	0.675111i	$0.764096\pi$
$774$		0			0
$775$	−3.50000	+	6.06218i	−0.125724	+	0.217760i
$776$		0			0
$777$	−17.5000	+	6.06218i	−0.627809	+	0.217479i
$778$		0			0
$779$	12.0000	−	20.7846i	0.429945	−	0.744686i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	2.00000			0.0714742
$784$		0			0
$785$	−14.0000			−0.499681
$786$		0			0
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	0.828176	−	0.560469i	$-0.189379\pi$
$787$	−2.00000	−	3.46410i	−0.0712923	−	0.123482i	−0.899468	+	0.436987i	$0.856046\pi$
$788$		0			0
$789$	6.00000	−	10.3923i	0.213606	−	0.369976i
$790$		0			0
$791$	−45.0000	+	15.5885i	−1.60002	+	0.554262i
$792$		0			0
$793$	21.0000	−	36.3731i	0.745732	−	1.29165i
$794$		0			0
$795$	4.00000	+	6.92820i	0.141865	+	0.245718i
$796$		0			0
$797$	−42.0000			−1.48772			−0.743858	−	0.668338i	$-0.767006\pi$
$797$	−42.0000			−1.48772			−0.743858	+	0.668338i	$0.767006\pi$
$798$		0			0
$799$	−60.0000			−2.12265
$800$		0			0
$801$	−1.00000	−	1.73205i	−0.0353333	−	0.0611990i
$802$		0			0
$803$	30.0000	−	51.9615i	1.05868	−	1.83368i
$804$		0			0
$805$	4.00000	+	3.46410i	0.140981	+	0.122094i
$806$		0			0
$807$	−5.00000	+	8.66025i	−0.176008	+	0.304855i
$808$		0			0
$809$	3.00000	+	5.19615i	0.105474	+	0.182687i	0.913932	−	0.405868i	$-0.133031\pi$
$809$	3.00000	+	5.19615i	0.105474	+	0.182687i	−0.808458	+	0.588555i	$0.799697\pi$
$810$		0			0
$811$	12.0000			0.421377			0.210688	−	0.977553i	$-0.432429\pi$
$811$	12.0000			0.421377			0.210688	+	0.977553i	$0.432429\pi$
$812$		0			0
$813$	16.0000			0.561144
$814$		0			0
$815$	2.00000	+	3.46410i	0.0700569	+	0.121342i
$816$		0			0
$817$	−7.50000	+	12.9904i	−0.262392	+	0.454476i
$818$		0			0
$819$	3.50000	−	18.1865i	0.122300	−	0.635489i
$820$		0			0
$821$	6.00000	−	10.3923i	0.209401	−	0.362694i	−0.742125	−	0.670262i	$-0.766182\pi$
$821$	6.00000	−	10.3923i	0.209401	−	0.362694i	0.951526	+	0.307568i	$0.0995151\pi$
$822$		0			0
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	0.995768	−	0.0919029i	$-0.0292950\pi$
$823$	12.0000	+	20.7846i	0.418294	+	0.724506i	−0.577474	+	0.816409i	$0.695962\pi$
$824$		0			0
$825$	4.00000			0.139262
$826$		0			0
$827$	−10.0000			−0.347734			−0.173867	−	0.984769i	$-0.555626\pi$
$827$	−10.0000			−0.347734			−0.173867	+	0.984769i	$0.555626\pi$
$828$		0			0
$829$	−12.5000	−	21.6506i	−0.434143	−	0.751958i	0.563082	−	0.826401i	$-0.309615\pi$
$829$	−12.5000	−	21.6506i	−0.434143	−	0.751958i	−0.997225	+	0.0744432i	$0.976282\pi$
$830$		0			0
$831$	11.5000	−	19.9186i	0.398931	−	0.690968i
$832$		0			0
$833$	39.0000	+	15.5885i	1.35127	+	0.540108i
$834$		0			0
$835$	12.0000	−	20.7846i	0.415277	−	0.719281i
$836$		0			0
$837$	3.50000	+	6.06218i	0.120978	+	0.209540i
$838$		0			0
$839$	−22.0000			−0.759524			−0.379762	−	0.925084i	$-0.623994\pi$
$839$	−22.0000			−0.759524			−0.379762	+	0.925084i	$0.623994\pi$
$840$		0			0
$841$	−25.0000			−0.862069
$842$		0			0
$843$	−8.00000	−	13.8564i	−0.275535	−	0.477240i
$844$		0			0
$845$	18.0000	−	31.1769i	0.619219	−	1.07252i
$846$		0			0
$847$	2.50000	−	12.9904i	0.0859010	−	0.446355i
$848$		0			0
$849$	12.5000	−	21.6506i	0.428999	−	0.743048i
$850$		0			0
$851$	−7.00000	−	12.1244i	−0.239957	−	0.415618i
$852$		0			0
$853$	−23.0000			−0.787505			−0.393753	−	0.919216i	$-0.628823\pi$
$853$	−23.0000			−0.787505			−0.393753	+	0.919216i	$0.628823\pi$
$854$		0			0
$855$	−3.00000			−0.102598
$856$		0			0
$857$	−11.0000	−	19.0526i	−0.375753	−	0.650823i	0.614687	−	0.788771i	$-0.289283\pi$
$857$	−11.0000	−	19.0526i	−0.375753	−	0.650823i	−0.990439	+	0.137948i	$0.955949\pi$
$858$		0			0
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	0.376381	+	0.926465i	$0.377169\pi$
$859$	−18.0000	+	31.1769i	−0.614152	+	1.06374i	−0.990533	+	0.137277i	$0.956165\pi$
$860$		0			0
$861$	16.0000	+	13.8564i	0.545279	+	0.472225i
$862$		0			0
$863$	1.00000	−	1.73205i	0.0340404	−	0.0589597i	−0.848503	−	0.529190i	$-0.822496\pi$
$863$	1.00000	−	1.73205i	0.0340404	−	0.0589597i	0.882544	+	0.470230i	$0.155829\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−19.0000			−0.645274
$868$		0			0
$869$	−4.00000			−0.135691
$870$		0			0
$871$	−10.5000	−	18.1865i	−0.355779	−	0.616227i
$872$		0			0
$873$	5.00000	−	8.66025i	0.169224	−	0.293105i
$874$		0			0
$875$	2.50000	−	0.866025i	0.0845154	−	0.0292770i
$876$		0			0
$877$	−25.0000	+	43.3013i	−0.844190	+	1.46218i	0.0421327	+	0.999112i	$0.486585\pi$
$877$	−25.0000	+	43.3013i	−0.844190	+	1.46218i	−0.886323	+	0.463068i	$0.846749\pi$
$878$		0			0
$879$	−6.00000	−	10.3923i	−0.202375	−	0.350524i
$880$		0			0
$881$	4.00000			0.134763			0.0673817	−	0.997727i	$-0.478535\pi$
$881$	4.00000			0.134763			0.0673817	+	0.997727i	$0.478535\pi$
$882$		0			0
$883$	27.0000			0.908622			0.454311	−	0.890843i	$-0.349885\pi$
$883$	27.0000			0.908622			0.454311	+	0.890843i	$0.349885\pi$
$884$		0			0
$885$	−5.00000	−	8.66025i	−0.168073	−	0.291111i
$886$		0			0
$887$	−16.0000	+	27.7128i	−0.537227	+	0.930505i	0.461825	+	0.886971i	$0.347195\pi$
$887$	−16.0000	+	27.7128i	−0.537227	+	0.930505i	−0.999052	+	0.0435339i	$0.986138\pi$
$888$		0			0
$889$	−32.5000	+	11.2583i	−1.09002	+	0.377592i
$890$		0			0
$891$	2.00000	−	3.46410i	0.0670025	−	0.116052i
$892$		0			0
$893$	−15.0000	−	25.9808i	−0.501956	−	0.869413i
$894$		0			0
$895$	−14.0000			−0.467968
$896$		0			0
$897$	14.0000			0.467446
$898$		0			0
$899$	7.00000	+	12.1244i	0.233463	+	0.404370i
$900$		0			0
$901$	−24.0000	+	41.5692i	−0.799556	+	1.38487i
$902$		0			0
$903$	−10.0000	−	8.66025i	−0.332779	−	0.288195i
$904$		0			0
$905$	−1.50000	+	2.59808i	−0.0498617	+	0.0863630i
$906$		0			0
$907$	18.5000	+	32.0429i	0.614282	+	1.06397i	0.990510	+	0.137441i	$0.0438878\pi$
$907$	18.5000	+	32.0429i	0.614282	+	1.06397i	−0.376228	+	0.926527i	$0.622779\pi$
$908$		0			0
$909$	12.0000			0.398015
$910$		0			0
$911$	−50.0000			−1.65657			−0.828287	−	0.560304i	$-0.810684\pi$
$911$	−50.0000			−1.65657			−0.828287	+	0.560304i	$0.810684\pi$
$912$		0			0
$913$	16.0000	+	27.7128i	0.529523	+	0.917160i
$914$		0			0
$915$	3.00000	−	5.19615i	0.0991769	−	0.171780i
$916$		0			0
$917$	11.0000	−	57.1577i	0.363252	−	1.88751i
$918$		0			0
$919$	13.5000	−	23.3827i	0.445324	−	0.771324i	−0.552751	−	0.833347i	$-0.686422\pi$
$919$	13.5000	−	23.3827i	0.445324	−	0.771324i	0.998075	+	0.0620230i	$0.0197552\pi$
$920$		0			0
$921$	5.50000	+	9.52628i	0.181231	+	0.313902i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−7.00000			−0.230159
$926$		0			0
$927$	3.50000	+	6.06218i	0.114955	+	0.199108i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	3.00000	+	20.7846i	0.0983210	+	0.681188i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	+	20.7846i	0.392442	+	0.679729i
$936$		0			0
$937$	9.00000			0.294017			0.147009	−	0.989135i	$-0.453036\pi$
$937$	9.00000			0.294017			0.147009	+	0.989135i	$0.453036\pi$
$938$		0			0
$939$	−21.0000			−0.685309
$940$		0			0
$941$	−27.0000	−	46.7654i	−0.880175	−	1.52451i	−0.851146	−	0.524929i	$-0.824092\pi$
$941$	−27.0000	−	46.7654i	−0.880175	−	1.52451i	−0.0290288	−	0.999579i	$-0.509241\pi$
$942$		0			0
$943$	−8.00000	+	13.8564i	−0.260516	+	0.451227i
$944$		0			0
$945$	0.500000	−	2.59808i	0.0162650	−	0.0845154i
$946$		0			0
$947$	−2.00000	+	3.46410i	−0.0649913	+	0.112568i	−0.896690	−	0.442659i	$-0.854035\pi$
$947$	−2.00000	+	3.46410i	−0.0649913	+	0.112568i	0.831699	+	0.555227i	$0.187369\pi$
$948$		0			0
$949$	−52.5000	−	90.9327i	−1.70422	−	2.95180i
$950$		0			0
$951$	−18.0000			−0.583690
$952$		0			0
$953$	16.0000			0.518291			0.259145	−	0.965838i	$-0.416559\pi$
$953$	16.0000			0.518291			0.259145	+	0.965838i	$0.416559\pi$
$954$		0			0
$955$	−9.00000	−	15.5885i	−0.291233	−	0.504431i
$956$		0			0
$957$	4.00000	−	6.92820i	0.129302	−	0.223957i
$958$		0			0
$959$	36.0000	+	31.1769i	1.16250	+	1.00676i
$960$		0			0
$961$	−9.00000	+	15.5885i	−0.290323	+	0.502853i
$962$		0			0
$963$	−1.00000	−	1.73205i	−0.0322245	−	0.0558146i
$964$		0			0
$965$	23.0000			0.740396
$966$		0			0
$967$	−17.0000			−0.546683			−0.273342	−	0.961917i	$-0.588129\pi$
$967$	−17.0000			−0.546683			−0.273342	+	0.961917i	$0.588129\pi$
$968$		0			0
$969$	−9.00000	−	15.5885i	−0.289122	−	0.500773i
$970$		0			0
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	0.167258	+	0.985913i	$0.446509\pi$
$971$	−24.0000	+	41.5692i	−0.770197	+	1.33402i	−0.937455	+	0.348107i	$0.886825\pi$
$972$		0			0
$973$	32.5000	−	11.2583i	1.04190	−	0.360925i
$974$		0			0
$975$	3.50000	−	6.06218i	0.112090	−	0.194145i
$976$		0			0
$977$	12.0000	+	20.7846i	0.383914	+	0.664959i	0.991618	−	0.129205i	$-0.0412426\pi$
$977$	12.0000	+	20.7846i	0.383914	+	0.664959i	−0.607704	+	0.794164i	$0.707909\pi$
$978$		0			0
$979$	−8.00000			−0.255681
$980$		0			0
$981$	−3.00000			−0.0957826
$982$		0			0
$983$	7.00000	+	12.1244i	0.223265	+	0.386707i	0.955798	−	0.294025i	$-0.0949950\pi$
$983$	7.00000	+	12.1244i	0.223265	+	0.386707i	−0.732532	+	0.680732i	$0.761662\pi$
$984$		0			0
$985$	1.00000	−	1.73205i	0.0318626	−	0.0551877i
$986$		0			0
$987$	25.0000	−	8.66025i	0.795759	−	0.275659i
$988$		0			0
$989$	5.00000	−	8.66025i	0.158991	−	0.275380i
$990$		0			0
$991$	−15.5000	−	26.8468i	−0.492374	−	0.852816i	0.507588	−	0.861600i	$-0.330537\pi$
$991$	−15.5000	−	26.8468i	−0.492374	−	0.852816i	−0.999961	+	0.00878379i	$0.997204\pi$
$992$		0			0
$993$	7.00000			0.222138
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−8.50000	−	14.7224i	−0.269198	−	0.466264i	0.699457	−	0.714675i	$-0.253425\pi$
$997$	−8.50000	−	14.7224i	−0.269198	−	0.466264i	−0.968655	+	0.248410i	$0.920092\pi$
$998$		0			0
$999$	−3.50000	+	6.06218i	−0.110735	+	0.191799i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	420.2.q.b.121.1	✓	2
3.2	odd	2		1260.2.s.a.541.1		2
4.3	odd	2		1680.2.bg.j.961.1		2
5.2	odd	4		2100.2.bc.d.1549.2		4
5.3	odd	4		2100.2.bc.d.1549.1		4
5.4	even	2		2100.2.q.d.1801.1		2
7.2	even	3		2940.2.a.b.1.1		1
7.3	odd	6		2940.2.q.c.361.1		2
7.4	even	3	inner	420.2.q.b.361.1	yes	2
7.5	odd	6		2940.2.a.k.1.1		1
7.6	odd	2		2940.2.q.c.961.1		2
21.2	odd	6		8820.2.a.bb.1.1		1
21.5	even	6		8820.2.a.l.1.1		1
21.11	odd	6		1260.2.s.a.361.1		2
28.11	odd	6		1680.2.bg.j.1201.1		2
35.4	even	6		2100.2.q.d.1201.1		2
35.18	odd	12		2100.2.bc.d.949.2		4
35.32	odd	12		2100.2.bc.d.949.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.q.b.121.1	✓	2	1.1	even	1	trivial
420.2.q.b.361.1	yes	2	7.4	even	3	inner
1260.2.s.a.361.1		2	21.11	odd	6
1260.2.s.a.541.1		2	3.2	odd	2
1680.2.bg.j.961.1		2	4.3	odd	2
1680.2.bg.j.1201.1		2	28.11	odd	6
2100.2.q.d.1201.1		2	35.4	even	6
2100.2.q.d.1801.1		2	5.4	even	2
2100.2.bc.d.949.1		4	35.32	odd	12
2100.2.bc.d.949.2		4	35.18	odd	12
2100.2.bc.d.1549.1		4	5.3	odd	4
2100.2.bc.d.1549.2		4	5.2	odd	4
2940.2.a.b.1.1		1	7.2	even	3
2940.2.a.k.1.1		1	7.5	odd	6
2940.2.q.c.361.1		2	7.3	odd	6
2940.2.q.c.961.1		2	7.6	odd	2
8820.2.a.l.1.1		1	21.5	even	6
8820.2.a.bb.1.1		1	21.2	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 420 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	420.q (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{3} +(0.500000 - 0.866025i) q^{5} +(-2.50000 + 0.866025i) q^{7} +(-0.500000 + 0.866025i) q^{9} +(2.00000 + 3.46410i) q^{11} +7.00000 q^{13} +1.00000 q^{15} +(3.00000 + 5.19615i) q^{17} +(-1.50000 + 2.59808i) q^{19} +(-2.00000 - 1.73205i) q^{21} +(1.00000 - 1.73205i) q^{23} +(-0.500000 - 0.866025i) q^{25} -1.00000 q^{27} -2.00000 q^{29} +(-3.50000 - 6.06218i) q^{31} +(-2.00000 + 3.46410i) q^{33} +(-0.500000 + 2.59808i) q^{35} +(3.50000 - 6.06218i) q^{37} +(3.50000 + 6.06218i) q^{39} -8.00000 q^{41} +5.00000 q^{43} +(0.500000 + 0.866025i) q^{45} +(-5.00000 + 8.66025i) q^{47} +(5.50000 - 4.33013i) q^{49} +(-3.00000 + 5.19615i) q^{51} +(4.00000 + 6.92820i) q^{53} +4.00000 q^{55} -3.00000 q^{57} +(-5.00000 - 8.66025i) q^{59} +(3.00000 - 5.19615i) q^{61} +(0.500000 - 2.59808i) q^{63} +(3.50000 - 6.06218i) q^{65} +(-1.50000 - 2.59808i) q^{67} +2.00000 q^{69} +(-7.50000 - 12.9904i) q^{73} +(0.500000 - 0.866025i) q^{75} +(-8.00000 - 6.92820i) q^{77} +(-0.500000 + 0.866025i) q^{79} +(-0.500000 - 0.866025i) q^{81} +8.00000 q^{83} +6.00000 q^{85} +(-1.00000 - 1.73205i) q^{87} +(-1.00000 + 1.73205i) q^{89} +(-17.5000 + 6.06218i) q^{91} +(3.50000 - 6.06218i) q^{93} +(1.50000 + 2.59808i) q^{95} -10.0000 q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + q^{5} - 5 q^{7} - q^{9} + 4 q^{11} + 14 q^{13} + 2 q^{15} + 6 q^{17} - 3 q^{19} - 4 q^{21} + 2 q^{23} - q^{25} - 2 q^{27} - 4 q^{29} - 7 q^{31} - 4 q^{33} - q^{35} + 7 q^{37} + 7 q^{39}+ \cdots - 8 q^{99}+O(q^{100}) \) 2 * q + q^3 + q^5 - 5 * q^7 - q^9 + 4 * q^11 + 14 * q^13 + 2 * q^15 + 6 * q^17 - 3 * q^19 - 4 * q^21 + 2 * q^23 - q^25 - 2 * q^27 - 4 * q^29 - 7 * q^31 - 4 * q^33 - q^35 + 7 * q^37 + 7 * q^39 - 16 * q^41 + 10 * q^43 + q^45 - 10 * q^47 + 11 * q^49 - 6 * q^51 + 8 * q^53 + 8 * q^55 - 6 * q^57 - 10 * q^59 + 6 * q^61 + q^63 + 7 * q^65 - 3 * q^67 + 4 * q^69 - 15 * q^73 + q^75 - 16 * q^77 - q^79 - q^81 + 16 * q^83 + 12 * q^85 - 2 * q^87 - 2 * q^89 - 35 * q^91 + 7 * q^93 + 3 * q^95 - 20 * q^97 - 8 * q^99

Introduction

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