LMFDB - Embedded newform 405.2.e.f.136.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.e (of order $3$, degree $2$, 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:	$3.23394128186$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			136.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	405.136
Dual form			405.2.e.f.271.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +3.00000 q^{8} +O(q^{10})$	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +3.00000 q^{8} -1.00000 q^{10} +(2.00000 - 3.46410i) q^{11} +(1.00000 + 1.73205i) q^{13} +(0.500000 - 0.866025i) q^{16} +2.00000 q^{17} +4.00000 q^{19} +(0.500000 - 0.866025i) q^{20} +(-2.00000 - 3.46410i) q^{22} +(-0.500000 + 0.866025i) q^{25} +2.00000 q^{26} +(1.00000 - 1.73205i) q^{29} +(2.50000 + 4.33013i) q^{32} +(1.00000 - 1.73205i) q^{34} -10.0000 q^{37} +(2.00000 - 3.46410i) q^{38} +(-1.50000 - 2.59808i) q^{40} +(-5.00000 - 8.66025i) q^{41} +(-2.00000 + 3.46410i) q^{43} +4.00000 q^{44} +(-4.00000 + 6.92820i) q^{47} +(3.50000 + 6.06218i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-1.00000 + 1.73205i) q^{52} -10.0000 q^{53} -4.00000 q^{55} +(-1.00000 - 1.73205i) q^{58} +(2.00000 + 3.46410i) q^{59} +(1.00000 - 1.73205i) q^{61} +7.00000 q^{64} +(1.00000 - 1.73205i) q^{65} +(-6.00000 - 10.3923i) q^{67} +(1.00000 + 1.73205i) q^{68} -8.00000 q^{71} +10.0000 q^{73} +(-5.00000 + 8.66025i) q^{74} +(2.00000 + 3.46410i) q^{76} -1.00000 q^{80} -10.0000 q^{82} +(-6.00000 + 10.3923i) q^{83} +(-1.00000 - 1.73205i) q^{85} +(2.00000 + 3.46410i) q^{86} +(6.00000 - 10.3923i) q^{88} -6.00000 q^{89} +(4.00000 + 6.92820i) q^{94} +(-2.00000 - 3.46410i) q^{95} +(-1.00000 + 1.73205i) q^{97} +7.00000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + q^{4} - q^{5} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + q^4 - q^5 + 6 * q^8	$ 2 q + q^{2} + q^{4} - q^{5} + 6 q^{8} - 2 q^{10} + 4 q^{11} + 2 q^{13} + q^{16} + 4 q^{17} + 8 q^{19} + q^{20} - 4 q^{22} - q^{25} + 4 q^{26} + 2 q^{29} + 5 q^{32} + 2 q^{34} - 20 q^{37} + 4 q^{38} - 3 q^{40} - 10 q^{41} - 4 q^{43} + 8 q^{44} - 8 q^{47} + 7 q^{49} + q^{50} - 2 q^{52} - 20 q^{53} - 8 q^{55} - 2 q^{58} + 4 q^{59} + 2 q^{61} + 14 q^{64} + 2 q^{65} - 12 q^{67} + 2 q^{68} - 16 q^{71} + 20 q^{73} - 10 q^{74} + 4 q^{76} - 2 q^{80} - 20 q^{82} - 12 q^{83} - 2 q^{85} + 4 q^{86} + 12 q^{88} - 12 q^{89} + 8 q^{94} - 4 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q + q^2 + q^4 - q^5 + 6 * q^8 - 2 * q^10 + 4 * q^11 + 2 * q^13 + q^16 + 4 * q^17 + 8 * q^19 + q^20 - 4 * q^22 - q^25 + 4 * q^26 + 2 * q^29 + 5 * q^32 + 2 * q^34 - 20 * q^37 + 4 * q^38 - 3 * q^40 - 10 * q^41 - 4 * q^43 + 8 * q^44 - 8 * q^47 + 7 * q^49 + q^50 - 2 * q^52 - 20 * q^53 - 8 * q^55 - 2 * q^58 + 4 * q^59 + 2 * q^61 + 14 * q^64 + 2 * q^65 - 12 * q^67 + 2 * q^68 - 16 * q^71 + 20 * q^73 - 10 * q^74 + 4 * q^76 - 2 * q^80 - 20 * q^82 - 12 * q^83 - 2 * q^85 + 4 * q^86 + 12 * q^88 - 12 * q^89 + 8 * q^94 - 4 * q^95 - 2 * q^97 + 14 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

$n$	$82$	$326$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.500000	−	0.866025i	0.353553	−	0.612372i	−0.633316	−	0.773893i	$-0.718307\pi$
$2$	0.500000	−	0.866025i	0.353553	−	0.612372i	0.986869	+	0.161521i	$0.0516399\pi$
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$5$	−0.500000	−	0.866025i	−0.223607	−	0.387298i
$6$		0			0
$7$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$7$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$8$	3.00000			1.06066
$9$		0			0
$10$	−1.00000			−0.316228
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	−0.389338	−	0.921095i	$-0.627296\pi$
$11$	2.00000	−	3.46410i	0.603023	−	1.04447i	0.992361	−	0.123371i	$-0.0393705\pi$
$12$		0			0
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	0.970725	−	0.240192i	$-0.0772105\pi$
$13$	1.00000	+	1.73205i	0.277350	+	0.480384i	−0.693375	+	0.720577i	$0.743877\pi$
$14$		0			0
$15$		0			0
$16$	0.500000	−	0.866025i	0.125000	−	0.216506i
$17$	2.00000			0.485071			0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$	0.500000	−	0.866025i	0.111803	−	0.193649i
$21$		0			0
$22$	−2.00000	−	3.46410i	−0.426401	−	0.738549i
$23$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$23$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	2.00000			0.392232
$27$		0			0
$28$		0			0
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	−0.758115	−	0.652121i	$-0.773880\pi$
$29$	1.00000	−	1.73205i	0.185695	−	0.321634i	0.943811	+	0.330487i	$0.107213\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	2.50000	+	4.33013i	0.441942	+	0.765466i
$33$		0			0
$34$	1.00000	−	1.73205i	0.171499	−	0.297044i
$35$		0			0
$36$		0			0
$37$	−10.0000			−1.64399			−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000			−1.64399			−0.821995	+	0.569495i	$0.807139\pi$
$38$	2.00000	−	3.46410i	0.324443	−	0.561951i
$39$		0			0
$40$	−1.50000	−	2.59808i	−0.237171	−	0.410792i
$41$	−5.00000	−	8.66025i	−0.780869	−	1.35250i	−0.931436	−	0.363905i	$-0.881443\pi$
$41$	−5.00000	−	8.66025i	−0.780869	−	1.35250i	0.150567	−	0.988600i	$-0.451890\pi$
$42$		0			0
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	−0.977261	−	0.212041i	$-0.931989\pi$
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	0.672264	+	0.740312i	$0.265322\pi$
$44$	4.00000			0.603023
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$	0.500000	+	0.866025i	0.0707107	+	0.122474i
$51$		0			0
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$	−10.0000			−1.37361			−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000			−1.37361			−0.686803	+	0.726844i	$0.740986\pi$
$54$		0			0
$55$	−4.00000			−0.539360
$56$		0			0
$57$		0			0
$58$	−1.00000	−	1.73205i	−0.131306	−	0.227429i
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	0.966342	−	0.257260i	$-0.0828195\pi$
$59$	2.00000	+	3.46410i	0.260378	+	0.450988i	−0.705965	+	0.708247i	$0.749486\pi$
$60$		0			0
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	−0.794879	−	0.606768i	$-0.792466\pi$
$61$	1.00000	−	1.73205i	0.128037	−	0.221766i	0.922916	+	0.385002i	$0.125799\pi$
$62$		0			0
$63$		0			0
$64$	7.00000			0.875000
$65$	1.00000	−	1.73205i	0.124035	−	0.214834i
$66$		0			0
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	−0.955588	−	0.294706i	$-0.904778\pi$
$67$	−6.00000	−	10.3923i	−0.733017	−	1.26962i	0.222571	−	0.974916i	$-0.428555\pi$
$68$	1.00000	+	1.73205i	0.121268	+	0.210042i
$69$		0			0
$70$		0			0
$71$	−8.00000			−0.949425			−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000			−0.949425			−0.474713	+	0.880141i	$0.657448\pi$
$72$		0			0
$73$	10.0000			1.17041			0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000			1.17041			0.585206	+	0.810885i	$0.301014\pi$
$74$	−5.00000	+	8.66025i	−0.581238	+	1.00673i
$75$		0			0
$76$	2.00000	+	3.46410i	0.229416	+	0.397360i
$77$		0			0
$78$		0			0
$79$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$79$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$80$	−1.00000			−0.111803
$81$		0			0
$82$	−10.0000			−1.10432
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	0.322396	+	0.946605i	$0.395512\pi$
$83$	−6.00000	+	10.3923i	−0.658586	+	1.14070i	−0.980982	+	0.194099i	$0.937822\pi$
$84$		0			0
$85$	−1.00000	−	1.73205i	−0.108465	−	0.187867i
$86$	2.00000	+	3.46410i	0.215666	+	0.373544i
$87$		0			0
$88$	6.00000	−	10.3923i	0.639602	−	1.10782i
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$	4.00000	+	6.92820i	0.412568	+	0.714590i
$95$	−2.00000	−	3.46410i	−0.205196	−	0.355409i
$96$		0			0
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	−0.912317	−	0.409484i	$-0.865709\pi$
$97$	−1.00000	+	1.73205i	−0.101535	+	0.175863i	0.810782	+	0.585348i	$0.199042\pi$
$98$	7.00000			0.707107
$99$		0			0
$100$	−1.00000			−0.100000
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	−0.975796	−	0.218685i	$-0.929823\pi$
$101$	−3.00000	+	5.19615i	−0.298511	+	0.517036i	0.677284	+	0.735721i	$0.263157\pi$
$102$		0			0
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	0.927030	+	0.374987i	$0.122353\pi$
$103$	8.00000	+	13.8564i	0.788263	+	1.36531i	−0.138767	+	0.990325i	$0.544314\pi$
$104$	3.00000	+	5.19615i	0.294174	+	0.509525i
$105$		0			0
$106$	−5.00000	+	8.66025i	−0.485643	+	0.841158i
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$	−2.00000	+	3.46410i	−0.190693	+	0.330289i
$111$		0			0
$112$		0			0
$113$	−1.00000	−	1.73205i	−0.0940721	−	0.162938i	0.815149	−	0.579252i	$-0.196655\pi$
$113$	−1.00000	−	1.73205i	−0.0940721	−	0.162938i	−0.909221	+	0.416314i	$0.863322\pi$
$114$		0			0
$115$		0			0
$116$	2.00000			0.185695
$117$		0			0
$118$	4.00000			0.368230
$119$		0			0
$120$		0			0
$121$	−2.50000	−	4.33013i	−0.227273	−	0.393648i
$122$	−1.00000	−	1.73205i	−0.0905357	−	0.156813i
$123$		0			0
$124$		0			0
$125$	1.00000			0.0894427
$126$		0			0
$127$	−8.00000			−0.709885			−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000			−0.709885			−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.50000	+	2.59808i	−0.132583	+	0.229640i
$129$		0			0
$130$	−1.00000	−	1.73205i	−0.0877058	−	0.151911i
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	0.999602	+	0.0281993i	$0.00897729\pi$
$131$	6.00000	+	10.3923i	0.524222	+	0.907980i	−0.475380	+	0.879781i	$0.657689\pi$
$132$		0			0
$133$		0			0
$134$	−12.0000			−1.03664
$135$		0			0
$136$	6.00000			0.514496
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	−0.708942	−	0.705266i	$-0.750827\pi$
$137$	3.00000	−	5.19615i	0.256307	−	0.443937i	0.965250	+	0.261329i	$0.0841608\pi$
$138$		0			0
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	0.938293	−	0.345843i	$-0.112407\pi$
$139$	2.00000	+	3.46410i	0.169638	+	0.293821i	−0.768655	+	0.639664i	$0.779074\pi$
$140$		0			0
$141$		0			0
$142$	−4.00000	+	6.92820i	−0.335673	+	0.581402i
$143$	8.00000			0.668994
$144$		0			0
$145$	−2.00000			−0.166091
$146$	5.00000	−	8.66025i	0.413803	−	0.716728i
$147$		0			0
$148$	−5.00000	−	8.66025i	−0.410997	−	0.711868i
$149$	−11.0000	−	19.0526i	−0.901155	−	1.56085i	−0.825997	−	0.563675i	$-0.809387\pi$
$149$	−11.0000	−	19.0526i	−0.901155	−	1.56085i	−0.0751583	−	0.997172i	$-0.523946\pi$
$150$		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	$0.0611289\pi$
$152$	12.0000			0.973329
$153$		0			0
$154$		0			0
$155$		0			0
$156$		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
$159$		0			0
$160$	2.50000	−	4.33013i	0.197642	−	0.342327i
$161$		0			0
$162$		0			0
$163$	−4.00000			−0.313304			−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000			−0.313304			−0.156652	+	0.987654i	$0.550070\pi$
$164$	5.00000	−	8.66025i	0.390434	−	0.676252i
$165$		0			0
$166$	6.00000	+	10.3923i	0.465690	+	0.806599i
$167$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$167$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$168$		0			0
$169$	4.50000	−	7.79423i	0.346154	−	0.599556i
$170$	−2.00000			−0.153393
$171$		0			0
$172$	−4.00000			−0.304997
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	−0.289412	−	0.957205i	$-0.593460\pi$
$173$	9.00000	−	15.5885i	0.684257	−	1.18517i	0.973670	−	0.227964i	$-0.0732068\pi$
$174$		0			0
$175$		0			0
$176$	−2.00000	−	3.46410i	−0.150756	−	0.261116i
$177$		0			0
$178$	−3.00000	+	5.19615i	−0.224860	+	0.389468i
$179$	20.0000			1.49487			0.747435	−	0.664335i	$-0.231285\pi$
$179$	20.0000			1.49487			0.747435	+	0.664335i	$0.231285\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.00000	+	8.66025i	0.367607	+	0.636715i
$186$		0			0
$187$	4.00000	−	6.92820i	0.292509	−	0.506640i
$188$	−8.00000			−0.583460
$189$		0			0
$190$	−4.00000			−0.290191
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	0.416751	+	0.909021i	$0.363169\pi$
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	−0.995610	+	0.0935936i	$0.970165\pi$
$192$		0			0
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	0.827788	−	0.561041i	$-0.189599\pi$
$193$	−1.00000	−	1.73205i	−0.0719816	−	0.124676i	−0.899770	+	0.436365i	$0.856266\pi$
$194$	1.00000	+	1.73205i	0.0717958	+	0.124354i
$195$		0			0
$196$	−3.50000	+	6.06218i	−0.250000	+	0.433013i
$197$	6.00000			0.427482			0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000			0.427482			0.213741	+	0.976890i	$0.431435\pi$
$198$		0			0
$199$	−8.00000			−0.567105			−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000			−0.567105			−0.283552	+	0.958957i	$0.591513\pi$
$200$	−1.50000	+	2.59808i	−0.106066	+	0.183712i
$201$		0			0
$202$	3.00000	+	5.19615i	0.211079	+	0.365600i
$203$		0			0
$204$		0			0
$205$	−5.00000	+	8.66025i	−0.349215	+	0.604858i
$206$	16.0000			1.11477
$207$		0			0
$208$	2.00000			0.138675
$209$	8.00000	−	13.8564i	0.553372	−	0.958468i
$210$		0			0
$211$	−10.0000	−	17.3205i	−0.688428	−	1.19239i	−0.972346	−	0.233544i	$-0.924968\pi$
$211$	−10.0000	−	17.3205i	−0.688428	−	1.19239i	0.283918	−	0.958849i	$-0.408366\pi$
$212$	−5.00000	−	8.66025i	−0.343401	−	0.594789i
$213$		0			0
$214$	−6.00000	+	10.3923i	−0.410152	+	0.710403i
$215$	4.00000			0.272798
$216$		0			0
$217$		0			0
$218$	7.00000	−	12.1244i	0.474100	−	0.821165i
$219$		0			0
$220$	−2.00000	−	3.46410i	−0.134840	−	0.233550i
$221$	2.00000	+	3.46410i	0.134535	+	0.233021i
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$		0			0
$225$		0			0
$226$	−2.00000			−0.133038
$227$	10.0000	−	17.3205i	0.663723	−	1.14960i	−0.315906	−	0.948790i	$-0.602309\pi$
$227$	10.0000	−	17.3205i	0.663723	−	1.14960i	0.979630	−	0.200812i	$-0.0643581\pi$
$228$		0			0
$229$	−3.00000	−	5.19615i	−0.198246	−	0.343371i	0.749714	−	0.661762i	$-0.230191\pi$
$229$	−3.00000	−	5.19615i	−0.198246	−	0.343371i	−0.947960	+	0.318390i	$0.896858\pi$
$230$		0			0
$231$		0			0
$232$	3.00000	−	5.19615i	0.196960	−	0.341144i
$233$	−6.00000			−0.393073			−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000			−0.393073			−0.196537	+	0.980497i	$0.562969\pi$
$234$		0			0
$235$	8.00000			0.521862
$236$	−2.00000	+	3.46410i	−0.130189	+	0.225494i
$237$		0			0
$238$		0			0
$239$	8.00000	+	13.8564i	0.517477	+	0.896296i	0.999794	+	0.0202996i	$0.00646202\pi$
$239$	8.00000	+	13.8564i	0.517477	+	0.896296i	−0.482317	+	0.875997i	$0.660205\pi$
$240$		0			0
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	7.00000	−	12.1244i	0.450910	−	0.780998i	0.998443	+	0.0557856i	$0.0177663\pi$
$242$	−5.00000			−0.321412
$243$		0			0
$244$	2.00000			0.128037
$245$	3.50000	−	6.06218i	0.223607	−	0.387298i
$246$		0			0
$247$	4.00000	+	6.92820i	0.254514	+	0.440831i
$248$		0			0
$249$		0			0
$250$	0.500000	−	0.866025i	0.0316228	−	0.0547723i
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		0			0
$254$	−4.00000	+	6.92820i	−0.250982	+	0.434714i
$255$		0			0
$256$	8.50000	+	14.7224i	0.531250	+	0.920152i
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	−0.997374	−	0.0724199i	$-0.976928\pi$
$257$	−9.00000	−	15.5885i	−0.561405	−	0.972381i	0.435970	−	0.899961i	$-0.356405\pi$
$258$		0			0
$259$		0			0
$260$	2.00000			0.124035
$261$		0			0
$262$	12.0000			0.741362
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	−0.999970	−	0.00771799i	$-0.997543\pi$
$263$	−8.00000	+	13.8564i	−0.493301	+	0.854423i	0.506669	+	0.862141i	$0.330877\pi$
$264$		0			0
$265$	5.00000	+	8.66025i	0.307148	+	0.531995i
$266$		0			0
$267$		0			0
$268$	6.00000	−	10.3923i	0.366508	−	0.634811i
$269$	14.0000			0.853595			0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000			0.853595			0.426798	+	0.904347i	$0.359642\pi$
$270$		0			0
$271$	16.0000			0.971931			0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000			0.971931			0.485965	+	0.873978i	$0.338468\pi$
$272$	1.00000	−	1.73205i	0.0606339	−	0.105021i
$273$		0			0
$274$	−3.00000	−	5.19615i	−0.181237	−	0.313911i
$275$	2.00000	+	3.46410i	0.120605	+	0.208893i
$276$		0			0
$277$	−3.00000	+	5.19615i	−0.180253	+	0.312207i	−0.941966	−	0.335707i	$-0.891025\pi$
$277$	−3.00000	+	5.19615i	−0.180253	+	0.312207i	0.761714	+	0.647913i	$0.224358\pi$
$278$	4.00000			0.239904
$279$		0			0
$280$		0			0
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	−0.762561	−	0.646916i	$-0.776058\pi$
$281$	3.00000	−	5.19615i	0.178965	−	0.309976i	0.941526	+	0.336939i	$0.109392\pi$
$282$		0			0
$283$	6.00000	+	10.3923i	0.356663	+	0.617758i	0.987401	−	0.158237i	$-0.0505811\pi$
$283$	6.00000	+	10.3923i	0.356663	+	0.617758i	−0.630738	+	0.775996i	$0.717248\pi$
$284$	−4.00000	−	6.92820i	−0.237356	−	0.411113i
$285$		0			0
$286$	4.00000	−	6.92820i	0.236525	−	0.409673i
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$	−1.00000	+	1.73205i	−0.0587220	+	0.101710i
$291$		0			0
$292$	5.00000	+	8.66025i	0.292603	+	0.506803i
$293$	−3.00000	−	5.19615i	−0.175262	−	0.303562i	0.764990	−	0.644042i	$-0.222744\pi$
$293$	−3.00000	−	5.19615i	−0.175262	−	0.303562i	−0.940252	+	0.340480i	$0.889411\pi$
$294$		0			0
$295$	2.00000	−	3.46410i	0.116445	−	0.201688i
$296$	−30.0000			−1.74371
$297$		0			0
$298$	−22.0000			−1.27443
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−4.00000	−	6.92820i	−0.230174	−	0.398673i
$303$		0			0
$304$	2.00000	−	3.46410i	0.114708	−	0.198680i
$305$	−2.00000			−0.114520
$306$		0			0
$307$	28.0000			1.59804			0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000			1.59804			0.799022	+	0.601302i	$0.205351\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	0.974841	+	0.222900i	$0.0715523\pi$
$311$	12.0000	+	20.7846i	0.680458	+	1.17859i	−0.294384	+	0.955687i	$0.595114\pi$
$312$		0			0
$313$	−13.0000	+	22.5167i	−0.734803	+	1.27272i	0.220006	+	0.975499i	$0.429392\pi$
$313$	−13.0000	+	22.5167i	−0.734803	+	1.27272i	−0.954810	+	0.297218i	$0.903941\pi$
$314$	−14.0000			−0.790066
$315$		0			0
$316$		0			0
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	−0.836576	−	0.547852i	$-0.815446\pi$
$317$	1.00000	−	1.73205i	0.0561656	−	0.0972817i	0.892741	+	0.450570i	$0.148779\pi$
$318$		0			0
$319$	−4.00000	−	6.92820i	−0.223957	−	0.387905i
$320$	−3.50000	−	6.06218i	−0.195656	−	0.338886i
$321$		0			0
$322$		0			0
$323$	8.00000			0.445132
$324$		0			0
$325$	−2.00000			−0.110940
$326$	−2.00000	+	3.46410i	−0.110770	+	0.191859i
$327$		0			0
$328$	−15.0000	−	25.9808i	−0.828236	−	1.43455i
$329$		0			0
$330$		0			0
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	−0.982470	−	0.186421i	$-0.940311\pi$
$331$	−6.00000	+	10.3923i	−0.329790	+	0.571213i	0.652680	+	0.757634i	$0.273645\pi$
$332$	−12.0000			−0.658586
$333$		0			0
$334$		0			0
$335$	−6.00000	+	10.3923i	−0.327815	+	0.567792i
$336$		0			0
$337$	7.00000	+	12.1244i	0.381314	+	0.660456i	0.991250	−	0.131995i	$-0.0421382\pi$
$337$	7.00000	+	12.1244i	0.381314	+	0.660456i	−0.609936	+	0.792451i	$0.708805\pi$
$338$	−4.50000	−	7.79423i	−0.244768	−	0.423950i
$339$		0			0
$340$	1.00000	−	1.73205i	0.0542326	−	0.0939336i
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−6.00000	+	10.3923i	−0.323498	+	0.560316i
$345$		0			0
$346$	−9.00000	−	15.5885i	−0.483843	−	0.838041i
$347$	14.0000	+	24.2487i	0.751559	+	1.30174i	0.947067	+	0.321037i	$0.104031\pi$
$347$	14.0000	+	24.2487i	0.751559	+	1.30174i	−0.195507	+	0.980702i	$0.562635\pi$
$348$		0			0
$349$	1.00000	−	1.73205i	0.0535288	−	0.0927146i	−0.838019	−	0.545640i	$-0.816286\pi$
$349$	1.00000	−	1.73205i	0.0535288	−	0.0927146i	0.891548	+	0.452926i	$0.149620\pi$
$350$		0			0
$351$		0			0
$352$	20.0000			1.06600
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	−0.999711	−	0.0240566i	$-0.992342\pi$
$353$	−9.00000	+	15.5885i	−0.479022	+	0.829690i	0.520689	+	0.853746i	$0.325675\pi$
$354$		0			0
$355$	4.00000	+	6.92820i	0.212298	+	0.367711i
$356$	−3.00000	−	5.19615i	−0.159000	−	0.275396i
$357$		0			0
$358$	10.0000	−	17.3205i	0.528516	−	0.915417i
$359$	−24.0000			−1.26667			−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$	−5.00000	+	8.66025i	−0.262794	+	0.455173i
$363$		0			0
$364$		0			0
$365$	−5.00000	−	8.66025i	−0.261712	−	0.453298i
$366$		0			0
$367$	12.0000	−	20.7846i	0.626395	−	1.08495i	−0.361874	−	0.932227i	$-0.617863\pi$
$367$	12.0000	−	20.7846i	0.626395	−	1.08495i	0.988269	−	0.152721i	$-0.0488036\pi$
$368$		0			0
$369$		0			0
$370$	10.0000			0.519875
$371$		0			0
$372$		0			0
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	0.977016	+	0.213165i	$0.0683772\pi$
$373$	13.0000	+	22.5167i	0.673114	+	1.16587i	−0.303902	+	0.952703i	$0.598289\pi$
$374$	−4.00000	−	6.92820i	−0.206835	−	0.358249i
$375$		0			0
$376$	−12.0000	+	20.7846i	−0.618853	+	1.07188i
$377$	4.00000			0.206010
$378$		0			0
$379$	−20.0000			−1.02733			−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000			−1.02733			−0.513665	+	0.857991i	$0.671713\pi$
$380$	2.00000	−	3.46410i	0.102598	−	0.177705i
$381$		0			0
$382$	8.00000	+	13.8564i	0.409316	+	0.708955i
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	0.990702	+	0.136047i	$0.0434398\pi$
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	−0.377531	+	0.925997i	$0.623227\pi$
$384$		0			0
$385$		0			0
$386$	−2.00000			−0.101797
$387$		0			0
$388$	−2.00000			−0.101535
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	−0.932002	−	0.362454i	$-0.881939\pi$
$389$	−3.00000	+	5.19615i	−0.152106	+	0.263455i	0.779895	+	0.625910i	$0.215272\pi$
$390$		0			0
$391$		0			0
$392$	10.5000	+	18.1865i	0.530330	+	0.918559i
$393$		0			0
$394$	3.00000	−	5.19615i	0.151138	−	0.261778i
$395$		0			0
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$	−4.00000	+	6.92820i	−0.200502	+	0.347279i
$399$		0			0
$400$	0.500000	+	0.866025i	0.0250000	+	0.0433013i
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$		0			0
$404$	−6.00000			−0.298511
$405$		0			0
$406$		0			0
$407$	−20.0000	+	34.6410i	−0.991363	+	1.71709i
$408$		0			0
$409$	−13.0000	−	22.5167i	−0.642809	−	1.11338i	−0.984803	−	0.173675i	$-0.944436\pi$
$409$	−13.0000	−	22.5167i	−0.642809	−	1.11338i	0.341994	−	0.939702i	$-0.388898\pi$
$410$	5.00000	+	8.66025i	0.246932	+	0.427699i
$411$		0			0
$412$	−8.00000	+	13.8564i	−0.394132	+	0.682656i
$413$		0			0
$414$		0			0
$415$	12.0000			0.589057
$416$	−5.00000	+	8.66025i	−0.245145	+	0.424604i
$417$		0			0
$418$	−8.00000	−	13.8564i	−0.391293	−	0.677739i
$419$	−2.00000	−	3.46410i	−0.0977064	−	0.169232i	0.813029	−	0.582224i	$-0.197817\pi$
$419$	−2.00000	−	3.46410i	−0.0977064	−	0.169232i	−0.910735	+	0.412991i	$0.864484\pi$
$420$		0			0
$421$	13.0000	−	22.5167i	0.633581	−	1.09739i	−0.353233	−	0.935536i	$-0.614918\pi$
$421$	13.0000	−	22.5167i	0.633581	−	1.09739i	0.986814	−	0.161859i	$-0.0517491\pi$
$422$	−20.0000			−0.973585
$423$		0			0
$424$	−30.0000			−1.45693
$425$	−1.00000	+	1.73205i	−0.0485071	+	0.0840168i
$426$		0			0
$427$		0			0
$428$	−6.00000	−	10.3923i	−0.290021	−	0.502331i
$429$		0			0
$430$	2.00000	−	3.46410i	0.0964486	−	0.167054i
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	−14.0000			−0.672797			−0.336399	−	0.941720i	$-0.609209\pi$
$433$	−14.0000			−0.672797			−0.336399	+	0.941720i	$0.609209\pi$
$434$		0			0
$435$		0			0
$436$	7.00000	+	12.1244i	0.335239	+	0.580651i
$437$		0			0
$438$		0			0
$439$	−20.0000	+	34.6410i	−0.954548	+	1.65333i	−0.219149	+	0.975691i	$0.570328\pi$
$439$	−20.0000	+	34.6410i	−0.954548	+	1.65333i	−0.735399	+	0.677634i	$0.763005\pi$
$440$	−12.0000			−0.572078
$441$		0			0
$442$	4.00000			0.190261
$443$	6.00000	−	10.3923i	0.285069	−	0.493753i	−0.687557	−	0.726130i	$-0.741317\pi$
$443$	6.00000	−	10.3923i	0.285069	−	0.493753i	0.972626	+	0.232377i	$0.0746503\pi$
$444$		0			0
$445$	3.00000	+	5.19615i	0.142214	+	0.246321i
$446$	4.00000	+	6.92820i	0.189405	+	0.328060i
$447$		0			0
$448$		0			0
$449$	2.00000			0.0943858			0.0471929	−	0.998886i	$-0.484972\pi$
$449$	2.00000			0.0943858			0.0471929	+	0.998886i	$0.484972\pi$
$450$		0			0
$451$	−40.0000			−1.88353
$452$	1.00000	−	1.73205i	0.0470360	−	0.0814688i
$453$		0			0
$454$	−10.0000	−	17.3205i	−0.469323	−	0.812892i
$455$		0			0
$456$		0			0
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	−0.958950	−	0.283577i	$-0.908479\pi$
$457$	−5.00000	+	8.66025i	−0.233890	+	0.405110i	0.725059	+	0.688686i	$0.241812\pi$
$458$	−6.00000			−0.280362
$459$		0			0
$460$		0			0
$461$	9.00000	−	15.5885i	0.419172	−	0.726027i	−0.576685	−	0.816967i	$-0.695654\pi$
$461$	9.00000	−	15.5885i	0.419172	−	0.726027i	0.995856	+	0.0909401i	$0.0289872\pi$
$462$		0			0
$463$	−12.0000	−	20.7846i	−0.557687	−	0.965943i	−0.997689	−	0.0679458i	$-0.978356\pi$
$463$	−12.0000	−	20.7846i	−0.557687	−	0.965943i	0.440002	−	0.897997i	$-0.354978\pi$
$464$	−1.00000	−	1.73205i	−0.0464238	−	0.0804084i
$465$		0			0
$466$	−3.00000	+	5.19615i	−0.138972	+	0.240707i
$467$	28.0000			1.29569			0.647843	−	0.761774i	$-0.275671\pi$
$467$	28.0000			1.29569			0.647843	+	0.761774i	$0.275671\pi$
$468$		0			0
$469$		0			0
$470$	4.00000	−	6.92820i	0.184506	−	0.319574i
$471$		0			0
$472$	6.00000	+	10.3923i	0.276172	+	0.478345i
$473$	8.00000	+	13.8564i	0.367840	+	0.637118i
$474$		0			0
$475$	−2.00000	+	3.46410i	−0.0917663	+	0.158944i
$476$		0			0
$477$		0			0
$478$	16.0000			0.731823
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	−10.0000	−	17.3205i	−0.455961	−	0.789747i
$482$	−7.00000	−	12.1244i	−0.318841	−	0.552249i
$483$		0			0
$484$	2.50000	−	4.33013i	0.113636	−	0.196824i
$485$	2.00000			0.0908153
$486$		0			0
$487$	32.0000			1.45006			0.725029	−	0.688718i	$-0.241826\pi$
$487$	32.0000			1.45006			0.725029	+	0.688718i	$0.241826\pi$
$488$	3.00000	−	5.19615i	0.135804	−	0.235219i
$489$		0			0
$490$	−3.50000	−	6.06218i	−0.158114	−	0.273861i
$491$	−14.0000	−	24.2487i	−0.631811	−	1.09433i	−0.987181	−	0.159603i	$-0.948978\pi$
$491$	−14.0000	−	24.2487i	−0.631811	−	1.09433i	0.355370	−	0.934726i	$-0.384355\pi$
$492$		0			0
$493$	2.00000	−	3.46410i	0.0900755	−	0.156015i
$494$	8.00000			0.359937
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	0.817781	−	0.575529i	$-0.195204\pi$
$499$	−2.00000	−	3.46410i	−0.0895323	−	0.155074i	−0.907314	+	0.420455i	$0.861871\pi$
$500$	0.500000	+	0.866025i	0.0223607	+	0.0387298i
$501$		0			0
$502$	6.00000	−	10.3923i	0.267793	−	0.463831i
$503$	−32.0000			−1.42681			−0.713405	−	0.700752i	$-0.752848\pi$
$503$	−32.0000			−1.42681			−0.713405	+	0.700752i	$0.752848\pi$
$504$		0			0
$505$	6.00000			0.266996
$506$		0			0
$507$		0			0
$508$	−4.00000	−	6.92820i	−0.177471	−	0.307389i
$509$	17.0000	+	29.4449i	0.753512	+	1.30512i	0.946111	+	0.323843i	$0.104975\pi$
$509$	17.0000	+	29.4449i	0.753512	+	1.30512i	−0.192599	+	0.981278i	$0.561692\pi$
$510$		0			0
$511$		0			0
$512$	11.0000			0.486136
$513$		0			0
$514$	−18.0000			−0.793946
$515$	8.00000	−	13.8564i	0.352522	−	0.610586i
$516$		0			0
$517$	16.0000	+	27.7128i	0.703679	+	1.21881i
$518$		0			0
$519$		0			0
$520$	3.00000	−	5.19615i	0.131559	−	0.227866i
$521$	10.0000			0.438108			0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000			0.438108			0.219054	+	0.975713i	$0.429703\pi$
$522$		0			0
$523$	4.00000			0.174908			0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000			0.174908			0.0874539	+	0.996169i	$0.472127\pi$
$524$	−6.00000	+	10.3923i	−0.262111	+	0.453990i
$525$		0			0
$526$	8.00000	+	13.8564i	0.348817	+	0.604168i
$527$		0			0
$528$		0			0
$529$	11.5000	−	19.9186i	0.500000	−	0.866025i
$530$	10.0000			0.434372
$531$		0			0
$532$		0			0
$533$	10.0000	−	17.3205i	0.433148	−	0.750234i
$534$		0			0
$535$	6.00000	+	10.3923i	0.259403	+	0.449299i
$536$	−18.0000	−	31.1769i	−0.777482	−	1.34664i
$537$		0			0
$538$	7.00000	−	12.1244i	0.301791	−	0.522718i
$539$	28.0000			1.20605
$540$		0			0
$541$	30.0000			1.28980			0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000			1.28980			0.644900	+	0.764267i	$0.276899\pi$
$542$	8.00000	−	13.8564i	0.343629	−	0.595184i
$543$		0			0
$544$	5.00000	+	8.66025i	0.214373	+	0.371305i
$545$	−7.00000	−	12.1244i	−0.299847	−	0.519350i
$546$		0			0
$547$	10.0000	−	17.3205i	0.427569	−	0.740571i	−0.569087	−	0.822277i	$-0.692703\pi$
$547$	10.0000	−	17.3205i	0.427569	−	0.740571i	0.996657	+	0.0817056i	$0.0260367\pi$
$548$	6.00000			0.256307
$549$		0			0
$550$	4.00000			0.170561
$551$	4.00000	−	6.92820i	0.170406	−	0.295151i
$552$		0			0
$553$		0			0
$554$	3.00000	+	5.19615i	0.127458	+	0.220763i
$555$		0			0
$556$	−2.00000	+	3.46410i	−0.0848189	+	0.146911i
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$	−3.00000	−	5.19615i	−0.126547	−	0.219186i
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	0.711445	−	0.702742i	$-0.248041\pi$
$563$	−6.00000	−	10.3923i	−0.252870	−	0.437983i	−0.964315	+	0.264758i	$0.914708\pi$
$564$		0			0
$565$	−1.00000	+	1.73205i	−0.0420703	+	0.0728679i
$566$	12.0000			0.504398
$567$		0			0
$568$	−24.0000			−1.00702
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	−0.796266	−	0.604947i	$-0.793194\pi$
$569$	3.00000	−	5.19615i	0.125767	−	0.217834i	0.922032	+	0.387113i	$0.126528\pi$
$570$		0			0
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	0.904835	−	0.425762i	$-0.139994\pi$
$571$	2.00000	+	3.46410i	0.0836974	+	0.144968i	−0.821138	+	0.570730i	$0.806660\pi$
$572$	4.00000	+	6.92820i	0.167248	+	0.289683i
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	2.00000			0.0832611			0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000			0.0832611			0.0416305	+	0.999133i	$0.486745\pi$
$578$	−6.50000	+	11.2583i	−0.270364	+	0.468285i
$579$		0			0
$580$	−1.00000	−	1.73205i	−0.0415227	−	0.0719195i
$581$		0			0
$582$		0			0
$583$	−20.0000	+	34.6410i	−0.828315	+	1.43468i
$584$	30.0000			1.24141
$585$		0			0
$586$	−6.00000			−0.247858
$587$	6.00000	−	10.3923i	0.247647	−	0.428936i	−0.715226	−	0.698893i	$-0.753676\pi$
$587$	6.00000	−	10.3923i	0.247647	−	0.428936i	0.962872	+	0.269957i	$0.0870095\pi$
$588$		0			0
$589$		0			0
$590$	−2.00000	−	3.46410i	−0.0823387	−	0.142615i
$591$		0			0
$592$	−5.00000	+	8.66025i	−0.205499	+	0.355934i
$593$	34.0000			1.39621			0.698106	−	0.715994i	$-0.254026\pi$
$593$	34.0000			1.39621			0.698106	+	0.715994i	$0.254026\pi$
$594$		0			0
$595$		0			0
$596$	11.0000	−	19.0526i	0.450578	−	0.780423i
$597$		0			0
$598$		0			0
$599$	4.00000	+	6.92820i	0.163436	+	0.283079i	0.936099	−	0.351738i	$-0.114409\pi$
$599$	4.00000	+	6.92820i	0.163436	+	0.283079i	−0.772663	+	0.634816i	$0.781076\pi$
$600$		0			0
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	0.469095	+	0.883148i	$0.344580\pi$
$601$	−13.0000	+	22.5167i	−0.530281	+	0.918474i	−0.999376	+	0.0353259i	$0.988753\pi$
$602$		0			0
$603$		0			0
$604$	8.00000			0.325515
$605$	−2.50000	+	4.33013i	−0.101639	+	0.176045i
$606$		0			0
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	0.935713	−	0.352763i	$-0.114758\pi$
$607$	4.00000	+	6.92820i	0.162355	+	0.281207i	−0.773358	+	0.633970i	$0.781424\pi$
$608$	10.0000	+	17.3205i	0.405554	+	0.702439i
$609$		0			0
$610$	−1.00000	+	1.73205i	−0.0404888	+	0.0701287i
$611$	−16.0000			−0.647291
$612$		0			0
$613$	22.0000			0.888572			0.444286	−	0.895885i	$-0.353457\pi$
$613$	22.0000			0.888572			0.444286	+	0.895885i	$0.353457\pi$
$614$	14.0000	−	24.2487i	0.564994	−	0.978598i
$615$		0			0
$616$		0			0
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	0.920074	−	0.391745i	$-0.128129\pi$
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	−0.799298	+	0.600935i	$0.794795\pi$
$618$		0			0
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	−0.823029	−	0.567999i	$-0.807718\pi$
$619$	2.00000	−	3.46410i	0.0803868	−	0.139234i	0.903416	+	0.428765i	$0.141051\pi$
$620$		0			0
$621$		0			0
$622$	24.0000			0.962312
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	13.0000	+	22.5167i	0.519584	+	0.899947i
$627$		0			0
$628$	7.00000	−	12.1244i	0.279330	−	0.483814i
$629$	−20.0000			−0.797452
$630$		0			0
$631$	−8.00000			−0.318475			−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000			−0.318475			−0.159237	+	0.987240i	$0.550904\pi$
$632$		0			0
$633$		0			0
$634$	−1.00000	−	1.73205i	−0.0397151	−	0.0687885i
$635$	4.00000	+	6.92820i	0.158735	+	0.274937i
$636$		0			0
$637$	−7.00000	+	12.1244i	−0.277350	+	0.480384i
$638$	−8.00000			−0.316723
$639$		0			0
$640$	3.00000			0.118585
$641$	15.0000	−	25.9808i	0.592464	−	1.02618i	−0.401435	−	0.915888i	$-0.631488\pi$
$641$	15.0000	−	25.9808i	0.592464	−	1.02618i	0.993899	−	0.110291i	$-0.0351782\pi$
$642$		0			0
$643$	18.0000	+	31.1769i	0.709851	+	1.22950i	0.964912	+	0.262573i	$0.0845709\pi$
$643$	18.0000	+	31.1769i	0.709851	+	1.22950i	−0.255062	+	0.966925i	$0.582096\pi$
$644$		0			0
$645$		0			0
$646$	4.00000	−	6.92820i	0.157378	−	0.272587i
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		0			0
$649$	16.0000			0.628055
$650$	−1.00000	+	1.73205i	−0.0392232	+	0.0679366i
$651$		0			0
$652$	−2.00000	−	3.46410i	−0.0783260	−	0.135665i
$653$	−23.0000	−	39.8372i	−0.900060	−	1.55895i	−0.827415	−	0.561591i	$-0.810189\pi$
$653$	−23.0000	−	39.8372i	−0.900060	−	1.55895i	−0.0726446	−	0.997358i	$-0.523144\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$	−10.0000			−0.390434
$657$		0			0
$658$		0			0
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	−0.992388	−	0.123148i	$-0.960701\pi$
$659$	−10.0000	+	17.3205i	−0.389545	+	0.674711i	0.602844	+	0.797859i	$0.294034\pi$
$660$		0			0
$661$	−11.0000	−	19.0526i	−0.427850	−	0.741059i	0.568831	−	0.822454i	$-0.307396\pi$
$661$	−11.0000	−	19.0526i	−0.427850	−	0.741059i	−0.996682	+	0.0813955i	$0.974062\pi$
$662$	6.00000	+	10.3923i	0.233197	+	0.403908i
$663$		0			0
$664$	−18.0000	+	31.1769i	−0.698535	+	1.20990i
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$	6.00000	+	10.3923i	0.231800	+	0.401490i
$671$	−4.00000	−	6.92820i	−0.154418	−	0.267460i
$672$		0			0
$673$	15.0000	−	25.9808i	0.578208	−	1.00148i	−0.417477	−	0.908687i	$-0.637086\pi$
$673$	15.0000	−	25.9808i	0.578208	−	1.00148i	0.995685	−	0.0927975i	$-0.0295809\pi$
$674$	14.0000			0.539260
$675$		0			0
$676$	9.00000			0.346154
$677$	−3.00000	+	5.19615i	−0.115299	+	0.199704i	−0.917899	−	0.396813i	$-0.870116\pi$
$677$	−3.00000	+	5.19615i	−0.115299	+	0.199704i	0.802600	+	0.596518i	$0.203449\pi$
$678$		0			0
$679$		0			0
$680$	−3.00000	−	5.19615i	−0.115045	−	0.199263i
$681$		0			0
$682$		0			0
$683$	36.0000			1.37750			0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000			1.37750			0.688751	+	0.724998i	$0.258159\pi$
$684$		0			0
$685$	−6.00000			−0.229248
$686$		0			0
$687$		0			0
$688$	2.00000	+	3.46410i	0.0762493	+	0.132068i
$689$	−10.0000	−	17.3205i	−0.380970	−	0.659859i
$690$		0			0
$691$	22.0000	−	38.1051i	0.836919	−	1.44959i	−0.0555386	−	0.998457i	$-0.517688\pi$
$691$	22.0000	−	38.1051i	0.836919	−	1.44959i	0.892458	−	0.451130i	$-0.148979\pi$
$692$	18.0000			0.684257
$693$		0			0
$694$	28.0000			1.06287
$695$	2.00000	−	3.46410i	0.0758643	−	0.131401i
$696$		0			0
$697$	−10.0000	−	17.3205i	−0.378777	−	0.656061i
$698$	−1.00000	−	1.73205i	−0.0378506	−	0.0655591i
$699$		0			0
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$	−40.0000			−1.50863
$704$	14.0000	−	24.2487i	0.527645	−	0.913908i
$705$		0			0
$706$	9.00000	+	15.5885i	0.338719	+	0.586679i
$707$		0			0
$708$		0			0
$709$	13.0000	−	22.5167i	0.488225	−	0.845631i	−0.511683	−	0.859174i	$-0.670978\pi$
$709$	13.0000	−	22.5167i	0.488225	−	0.845631i	0.999908	+	0.0135434i	$0.00431112\pi$
$710$	8.00000			0.300235
$711$		0			0
$712$	−18.0000			−0.674579
$713$		0			0
$714$		0			0
$715$	−4.00000	−	6.92820i	−0.149592	−	0.259100i
$716$	10.0000	+	17.3205i	0.373718	+	0.647298i
$717$		0			0
$718$	−12.0000	+	20.7846i	−0.447836	+	0.775675i
$719$	−48.0000			−1.79010			−0.895049	−	0.445968i	$-0.852860\pi$
$719$	−48.0000			−1.79010			−0.895049	+	0.445968i	$0.852860\pi$
$720$		0			0
$721$		0			0
$722$	−1.50000	+	2.59808i	−0.0558242	+	0.0966904i
$723$		0			0
$724$	−5.00000	−	8.66025i	−0.185824	−	0.321856i
$725$	1.00000	+	1.73205i	0.0371391	+	0.0643268i
$726$		0			0
$727$	8.00000	−	13.8564i	0.296704	−	0.513906i	−0.678676	−	0.734438i	$-0.737446\pi$
$727$	8.00000	−	13.8564i	0.296704	−	0.513906i	0.975380	+	0.220532i	$0.0707793\pi$
$728$		0			0
$729$		0			0
$730$	−10.0000			−0.370117
$731$	−4.00000	+	6.92820i	−0.147945	+	0.256249i
$732$		0			0
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	0.707303	−	0.706910i	$-0.249912\pi$
$733$	−7.00000	−	12.1244i	−0.258551	−	0.447823i	−0.965854	+	0.259087i	$0.916578\pi$
$734$	−12.0000	−	20.7846i	−0.442928	−	0.767174i
$735$		0			0
$736$		0			0
$737$	−48.0000			−1.76810
$738$		0			0
$739$	−44.0000			−1.61857			−0.809283	−	0.587419i	$-0.800144\pi$
$739$	−44.0000			−1.61857			−0.809283	+	0.587419i	$0.800144\pi$
$740$	−5.00000	+	8.66025i	−0.183804	+	0.318357i
$741$		0			0
$742$		0			0
$743$	8.00000	+	13.8564i	0.293492	+	0.508342i	0.974633	−	0.223810i	$-0.0718494\pi$
$743$	8.00000	+	13.8564i	0.293492	+	0.508342i	−0.681141	+	0.732152i	$0.738516\pi$
$744$		0			0
$745$	−11.0000	+	19.0526i	−0.403009	+	0.698032i
$746$	26.0000			0.951928
$747$		0			0
$748$	8.00000			0.292509
$749$		0			0
$750$		0			0
$751$	−8.00000	−	13.8564i	−0.291924	−	0.505627i	0.682341	−	0.731034i	$-0.260962\pi$
$751$	−8.00000	−	13.8564i	−0.291924	−	0.505627i	−0.974265	+	0.225407i	$0.927629\pi$
$752$	4.00000	+	6.92820i	0.145865	+	0.252646i
$753$		0			0
$754$	2.00000	−	3.46410i	0.0728357	−	0.126155i
$755$	−8.00000			−0.291150
$756$		0			0
$757$	−26.0000			−0.944986			−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000			−0.944986			−0.472493	+	0.881334i	$0.656646\pi$
$758$	−10.0000	+	17.3205i	−0.363216	+	0.629109i
$759$		0			0
$760$	−6.00000	−	10.3923i	−0.217643	−	0.376969i
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	0.915264	−	0.402854i	$-0.131982\pi$
$761$	3.00000	+	5.19615i	0.108750	+	0.188360i	−0.806514	+	0.591215i	$0.798649\pi$
$762$		0			0
$763$		0			0
$764$	−16.0000			−0.578860
$765$		0			0
$766$	24.0000			0.867155
$767$	−4.00000	+	6.92820i	−0.144432	+	0.250163i
$768$		0			0
$769$	−1.00000	−	1.73205i	−0.0360609	−	0.0624593i	0.847432	−	0.530904i	$-0.178148\pi$
$769$	−1.00000	−	1.73205i	−0.0360609	−	0.0624593i	−0.883493	+	0.468445i	$0.844814\pi$
$770$		0			0
$771$		0			0
$772$	1.00000	−	1.73205i	0.0359908	−	0.0623379i
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$		0			0
$776$	−3.00000	+	5.19615i	−0.107694	+	0.186531i
$777$		0			0
$778$	3.00000	+	5.19615i	0.107555	+	0.186291i
$779$	−20.0000	−	34.6410i	−0.716574	−	1.24114i
$780$		0			0
$781$	−16.0000	+	27.7128i	−0.572525	+	0.991642i
$782$		0			0
$783$		0			0
$784$	7.00000			0.250000
$785$	−7.00000	+	12.1244i	−0.249841	+	0.432737i
$786$		0			0
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	0.500953	−	0.865474i	$-0.332983\pi$
$787$	−14.0000	−	24.2487i	−0.499046	−	0.864373i	−0.999999	+	0.00110111i	$0.999650\pi$
$788$	3.00000	+	5.19615i	0.106871	+	0.185105i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	4.00000			0.142044
$794$	−1.00000	+	1.73205i	−0.0354887	+	0.0614682i
$795$		0			0
$796$	−4.00000	−	6.92820i	−0.141776	−	0.245564i
$797$	1.00000	+	1.73205i	0.0354218	+	0.0613524i	0.883193	−	0.469010i	$-0.155389\pi$
$797$	1.00000	+	1.73205i	0.0354218	+	0.0613524i	−0.847771	+	0.530362i	$0.822056\pi$
$798$		0			0
$799$	−8.00000	+	13.8564i	−0.283020	+	0.490204i
$800$	−5.00000			−0.176777
$801$		0			0
$802$	−18.0000			−0.635602
$803$	20.0000	−	34.6410i	0.705785	−	1.22245i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$	−9.00000	+	15.5885i	−0.316619	+	0.548400i
$809$	10.0000			0.351581			0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000			0.351581			0.175791	+	0.984428i	$0.443752\pi$
$810$		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
$813$		0			0
$814$	20.0000	+	34.6410i	0.701000	+	1.21417i
$815$	2.00000	+	3.46410i	0.0700569	+	0.121342i
$816$		0			0
$817$	−8.00000	+	13.8564i	−0.279885	+	0.484774i
$818$	−26.0000			−0.909069
$819$		0			0
$820$	−10.0000			−0.349215
$821$	−27.0000	+	46.7654i	−0.942306	+	1.63212i	−0.181250	+	0.983437i	$0.558014\pi$
$821$	−27.0000	+	46.7654i	−0.942306	+	1.63212i	−0.761056	+	0.648686i	$0.775319\pi$
$822$		0			0
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	−0.997686	−	0.0679910i	$-0.978341\pi$
$823$	−16.0000	−	27.7128i	−0.557725	−	0.966008i	0.439961	−	0.898017i	$-0.354992\pi$
$824$	24.0000	+	41.5692i	0.836080	+	1.44813i
$825$		0			0
$826$		0			0
$827$	−28.0000			−0.973655			−0.486828	−	0.873498i	$-0.661846\pi$
$827$	−28.0000			−0.973655			−0.486828	+	0.873498i	$0.661846\pi$
$828$		0			0
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$	6.00000	−	10.3923i	0.208263	−	0.360722i
$831$		0			0
$832$	7.00000	+	12.1244i	0.242681	+	0.420336i
$833$	7.00000	+	12.1244i	0.242536	+	0.420084i
$834$		0			0
$835$		0			0
$836$	16.0000			0.553372
$837$		0			0
$838$	−4.00000			−0.138178
$839$	−20.0000	+	34.6410i	−0.690477	+	1.19594i	0.281205	+	0.959648i	$0.409266\pi$
$839$	−20.0000	+	34.6410i	−0.690477	+	1.19594i	−0.971682	+	0.236293i	$0.924067\pi$
$840$		0			0
$841$	12.5000	+	21.6506i	0.431034	+	0.746574i
$842$	−13.0000	−	22.5167i	−0.448010	−	0.775975i
$843$		0			0
$844$	10.0000	−	17.3205i	0.344214	−	0.596196i
$845$	−9.00000			−0.309609
$846$		0			0
$847$		0			0
$848$	−5.00000	+	8.66025i	−0.171701	+	0.297394i
$849$		0			0
$850$	1.00000	+	1.73205i	0.0342997	+	0.0594089i
$851$		0			0
$852$		0			0
$853$	−3.00000	+	5.19615i	−0.102718	+	0.177913i	−0.912804	−	0.408399i	$-0.866087\pi$
$853$	−3.00000	+	5.19615i	−0.102718	+	0.177913i	0.810086	+	0.586312i	$0.199421\pi$
$854$		0			0
$855$		0			0
$856$	−36.0000			−1.23045
$857$	11.0000	−	19.0526i	0.375753	−	0.650823i	−0.614687	−	0.788771i	$-0.710717\pi$
$857$	11.0000	−	19.0526i	0.375753	−	0.650823i	0.990439	+	0.137948i	$0.0440508\pi$
$858$		0			0
$859$	10.0000	+	17.3205i	0.341196	+	0.590968i	0.984655	−	0.174512i	$-0.0558348\pi$
$859$	10.0000	+	17.3205i	0.341196	+	0.590968i	−0.643459	+	0.765480i	$0.722501\pi$
$860$	2.00000	+	3.46410i	0.0681994	+	0.118125i
$861$		0			0
$862$		0			0
$863$	−56.0000			−1.90626			−0.953131	−	0.302558i	$-0.902160\pi$
$863$	−56.0000			−1.90626			−0.953131	+	0.302558i	$0.902160\pi$
$864$		0			0
$865$	−18.0000			−0.612018
$866$	−7.00000	+	12.1244i	−0.237870	+	0.412002i
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$	12.0000	−	20.7846i	0.406604	−	0.704260i
$872$	42.0000			1.42230
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−15.0000	−	25.9808i	−0.506514	−	0.877308i	−0.999972	−	0.00753813i	$-0.997601\pi$
$877$	−15.0000	−	25.9808i	−0.506514	−	0.877308i	0.493458	−	0.869770i	$-0.335733\pi$
$878$	20.0000	+	34.6410i	0.674967	+	1.16908i
$879$		0			0
$880$	−2.00000	+	3.46410i	−0.0674200	+	0.116775i
$881$	−46.0000			−1.54978			−0.774890	−	0.632096i	$-0.782195\pi$
$881$	−46.0000			−1.54978			−0.774890	+	0.632096i	$0.782195\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$	−2.00000	+	3.46410i	−0.0672673	+	0.116510i
$885$		0			0
$886$	−6.00000	−	10.3923i	−0.201574	−	0.349136i
$887$	−24.0000	−	41.5692i	−0.805841	−	1.39576i	−0.915722	−	0.401813i	$-0.868380\pi$
$887$	−24.0000	−	41.5692i	−0.805841	−	1.39576i	0.109881	−	0.993945i	$-0.464953\pi$
$888$		0			0
$889$		0			0
$890$	6.00000			0.201120
$891$		0			0
$892$	−8.00000			−0.267860
$893$	−16.0000	+	27.7128i	−0.535420	+	0.927374i
$894$		0			0
$895$	−10.0000	−	17.3205i	−0.334263	−	0.578961i
$896$		0			0
$897$		0			0
$898$	1.00000	−	1.73205i	0.0333704	−	0.0577993i
$899$		0			0
$900$		0			0
$901$	−20.0000			−0.666297
$902$	−20.0000	+	34.6410i	−0.665927	+	1.15342i
$903$		0			0
$904$	−3.00000	−	5.19615i	−0.0997785	−	0.172821i
$905$	5.00000	+	8.66025i	0.166206	+	0.287877i
$906$		0			0
$907$	6.00000	−	10.3923i	0.199227	−	0.345071i	−0.749051	−	0.662512i	$-0.769490\pi$
$907$	6.00000	−	10.3923i	0.199227	−	0.345071i	0.948278	+	0.317441i	$0.102824\pi$
$908$	20.0000			0.663723
$909$		0			0
$910$		0			0
$911$	−16.0000	+	27.7128i	−0.530104	+	0.918166i	0.469280	+	0.883050i	$0.344514\pi$
$911$	−16.0000	+	27.7128i	−0.530104	+	0.918166i	−0.999383	+	0.0351168i	$0.988820\pi$
$912$		0			0
$913$	24.0000	+	41.5692i	0.794284	+	1.37574i
$914$	5.00000	+	8.66025i	0.165385	+	0.286456i
$915$		0			0
$916$	3.00000	−	5.19615i	0.0991228	−	0.171686i
$917$		0			0
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$	−9.00000	−	15.5885i	−0.296399	−	0.513378i
$923$	−8.00000	−	13.8564i	−0.263323	−	0.456089i
$924$		0			0
$925$	5.00000	−	8.66025i	0.164399	−	0.284747i
$926$	−24.0000			−0.788689
$927$		0			0
$928$	10.0000			0.328266
$929$	−17.0000	+	29.4449i	−0.557752	+	0.966055i	0.439932	+	0.898031i	$0.355003\pi$
$929$	−17.0000	+	29.4449i	−0.557752	+	0.966055i	−0.997684	+	0.0680235i	$0.978331\pi$
$930$		0			0
$931$	14.0000	+	24.2487i	0.458831	+	0.794719i
$932$	−3.00000	−	5.19615i	−0.0982683	−	0.170206i
$933$		0			0
$934$	14.0000	−	24.2487i	0.458094	−	0.793442i
$935$	−8.00000			−0.261628
$936$		0			0
$937$	−54.0000			−1.76410			−0.882052	−	0.471153i	$-0.843838\pi$
$937$	−54.0000			−1.76410			−0.882052	+	0.471153i	$0.843838\pi$
$938$		0			0
$939$		0			0
$940$	4.00000	+	6.92820i	0.130466	+	0.225973i
$941$	25.0000	+	43.3013i	0.814977	+	1.41158i	0.909345	+	0.416044i	$0.136584\pi$
$941$	25.0000	+	43.3013i	0.814977	+	1.41158i	−0.0943679	+	0.995537i	$0.530083\pi$
$942$		0			0
$943$		0			0
$944$	4.00000			0.130189
$945$		0			0
$946$	16.0000			0.520205
$947$	18.0000	−	31.1769i	0.584921	−	1.01311i	−0.409964	−	0.912102i	$-0.634459\pi$
$947$	18.0000	−	31.1769i	0.584921	−	1.01311i	0.994885	−	0.101012i	$-0.0322080\pi$
$948$		0			0
$949$	10.0000	+	17.3205i	0.324614	+	0.562247i
$950$	2.00000	+	3.46410i	0.0648886	+	0.112390i
$951$		0			0
$952$		0			0
$953$	−22.0000			−0.712650			−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000			−0.712650			−0.356325	+	0.934362i	$0.615970\pi$
$954$		0			0
$955$	16.0000			0.517748
$956$	−8.00000	+	13.8564i	−0.258738	+	0.448148i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	15.5000	−	26.8468i	0.500000	−	0.866025i
$962$	−20.0000			−0.644826
$963$		0			0
$964$	14.0000			0.450910
$965$	−1.00000	+	1.73205i	−0.0321911	+	0.0557567i
$966$		0			0
$967$	−16.0000	−	27.7128i	−0.514525	−	0.891184i	−0.999858	−	0.0168544i	$-0.994635\pi$
$967$	−16.0000	−	27.7128i	−0.514525	−	0.891184i	0.485333	−	0.874330i	$-0.338699\pi$
$968$	−7.50000	−	12.9904i	−0.241059	−	0.417527i
$969$		0			0
$970$	1.00000	−	1.73205i	0.0321081	−	0.0556128i
$971$	60.0000			1.92549			0.962746	−	0.270408i	$-0.0871586\pi$
$971$	60.0000			1.92549			0.962746	+	0.270408i	$0.0871586\pi$
$972$		0			0
$973$		0			0
$974$	16.0000	−	27.7128i	0.512673	−	0.887976i
$975$		0			0
$976$	−1.00000	−	1.73205i	−0.0320092	−	0.0554416i
$977$	−1.00000	−	1.73205i	−0.0319928	−	0.0554132i	0.849586	−	0.527451i	$-0.176852\pi$
$977$	−1.00000	−	1.73205i	−0.0319928	−	0.0554132i	−0.881579	+	0.472037i	$0.843519\pi$
$978$		0			0
$979$	−12.0000	+	20.7846i	−0.383522	+	0.664279i
$980$	7.00000			0.223607
$981$		0			0
$982$	−28.0000			−0.893516
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$	−3.00000	−	5.19615i	−0.0955879	−	0.165563i
$986$	−2.00000	−	3.46410i	−0.0636930	−	0.110319i
$987$		0			0
$988$	−4.00000	+	6.92820i	−0.127257	+	0.220416i
$989$		0			0
$990$		0			0
$991$	32.0000			1.01651			0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000			1.01651			0.508257	+	0.861206i	$0.330290\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.00000	+	6.92820i	0.126809	+	0.219639i
$996$		0			0
$997$	−27.0000	+	46.7654i	−0.855099	+	1.48107i	0.0214550	+	0.999770i	$0.493170\pi$
$997$	−27.0000	+	46.7654i	−0.855099	+	1.48107i	−0.876554	+	0.481304i	$0.840163\pi$
$998$	−4.00000			−0.126618
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.e.f.136.1		2
3.2	odd	2		405.2.e.c.136.1		2
9.2	odd	6		45.2.a.a.1.1		1
9.4	even	3	inner	405.2.e.f.271.1		2
9.5	odd	6		405.2.e.c.271.1		2
9.7	even	3		15.2.a.a.1.1	✓	1
36.7	odd	6		240.2.a.d.1.1		1
36.11	even	6		720.2.a.c.1.1		1
45.2	even	12		225.2.b.b.199.2		2
45.7	odd	12		75.2.b.b.49.1		2
45.29	odd	6		225.2.a.b.1.1		1
45.34	even	6		75.2.a.b.1.1		1
45.38	even	12		225.2.b.b.199.1		2
45.43	odd	12		75.2.b.b.49.2		2
63.16	even	3		735.2.i.e.361.1		2
63.20	even	6		2205.2.a.i.1.1		1
63.25	even	3		735.2.i.e.226.1		2
63.34	odd	6		735.2.a.c.1.1		1
63.52	odd	6		735.2.i.d.226.1		2
63.61	odd	6		735.2.i.d.361.1		2
72.11	even	6		2880.2.a.bc.1.1		1
72.29	odd	6		2880.2.a.y.1.1		1
72.43	odd	6		960.2.a.a.1.1		1
72.61	even	6		960.2.a.l.1.1		1
99.43	odd	6		1815.2.a.d.1.1		1
99.65	even	6		5445.2.a.c.1.1		1
117.25	even	6		2535.2.a.j.1.1		1
117.38	odd	6		7605.2.a.g.1.1		1
144.43	odd	12		3840.2.k.r.1921.1		2
144.61	even	12		3840.2.k.m.1921.1		2
144.115	odd	12		3840.2.k.r.1921.2		2
144.133	even	12		3840.2.k.m.1921.2		2
153.16	even	6		4335.2.a.c.1.1		1
171.151	odd	6		5415.2.a.j.1.1		1
180.7	even	12		1200.2.f.h.49.1		2
180.43	even	12		1200.2.f.h.49.2		2
180.47	odd	12		3600.2.f.e.2449.2		2
180.79	odd	6		1200.2.a.e.1.1		1
180.83	odd	12		3600.2.f.e.2449.1		2
180.119	even	6		3600.2.a.u.1.1		1
207.160	odd	6		7935.2.a.d.1.1		1
315.34	odd	6		3675.2.a.j.1.1		1
360.43	even	12		4800.2.f.c.3649.1		2
360.133	odd	12		4800.2.f.bf.3649.2		2
360.187	even	12		4800.2.f.c.3649.2		2
360.259	odd	6		4800.2.a.bz.1.1		1
360.277	odd	12		4800.2.f.bf.3649.1		2
360.349	even	6		4800.2.a.t.1.1		1
495.439	odd	6		9075.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.2.a.a.1.1	✓	1	9.7	even	3
45.2.a.a.1.1		1	9.2	odd	6
75.2.a.b.1.1		1	45.34	even	6
75.2.b.b.49.1		2	45.7	odd	12
75.2.b.b.49.2		2	45.43	odd	12
225.2.a.b.1.1		1	45.29	odd	6
225.2.b.b.199.1		2	45.38	even	12
225.2.b.b.199.2		2	45.2	even	12
240.2.a.d.1.1		1	36.7	odd	6
405.2.e.c.136.1		2	3.2	odd	2
405.2.e.c.271.1		2	9.5	odd	6
405.2.e.f.136.1		2	1.1	even	1	trivial
405.2.e.f.271.1		2	9.4	even	3	inner
720.2.a.c.1.1		1	36.11	even	6
735.2.a.c.1.1		1	63.34	odd	6
735.2.i.d.226.1		2	63.52	odd	6
735.2.i.d.361.1		2	63.61	odd	6
735.2.i.e.226.1		2	63.25	even	3
735.2.i.e.361.1		2	63.16	even	3
960.2.a.a.1.1		1	72.43	odd	6
960.2.a.l.1.1		1	72.61	even	6
1200.2.a.e.1.1		1	180.79	odd	6
1200.2.f.h.49.1		2	180.7	even	12
1200.2.f.h.49.2		2	180.43	even	12
1815.2.a.d.1.1		1	99.43	odd	6
2205.2.a.i.1.1		1	63.20	even	6
2535.2.a.j.1.1		1	117.25	even	6
2880.2.a.y.1.1		1	72.29	odd	6
2880.2.a.bc.1.1		1	72.11	even	6
3600.2.a.u.1.1		1	180.119	even	6
3600.2.f.e.2449.1		2	180.83	odd	12
3600.2.f.e.2449.2		2	180.47	odd	12
3675.2.a.j.1.1		1	315.34	odd	6
3840.2.k.m.1921.1		2	144.61	even	12
3840.2.k.m.1921.2		2	144.133	even	12
3840.2.k.r.1921.1		2	144.43	odd	12
3840.2.k.r.1921.2		2	144.115	odd	12
4335.2.a.c.1.1		1	153.16	even	6
4800.2.a.t.1.1		1	360.349	even	6
4800.2.a.bz.1.1		1	360.259	odd	6
4800.2.f.c.3649.1		2	360.43	even	12
4800.2.f.c.3649.2		2	360.187	even	12
4800.2.f.bf.3649.1		2	360.277	odd	12
4800.2.f.bf.3649.2		2	360.133	odd	12
5415.2.a.j.1.1		1	171.151	odd	6
5445.2.a.c.1.1		1	99.65	even	6
7605.2.a.g.1.1		1	117.38	odd	6
7935.2.a.d.1.1		1	207.160	odd	6
9075.2.a.g.1.1		1	495.439	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +3.00000 q^{8} +O(q^{10})\)	\(q+(0.500000 - 0.866025i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.500000 - 0.866025i) q^{5} +3.00000 q^{8} -1.00000 q^{10} +(2.00000 - 3.46410i) q^{11} +(1.00000 + 1.73205i) q^{13} +(0.500000 - 0.866025i) q^{16} +2.00000 q^{17} +4.00000 q^{19} +(0.500000 - 0.866025i) q^{20} +(-2.00000 - 3.46410i) q^{22} +(-0.500000 + 0.866025i) q^{25} +2.00000 q^{26} +(1.00000 - 1.73205i) q^{29} +(2.50000 + 4.33013i) q^{32} +(1.00000 - 1.73205i) q^{34} -10.0000 q^{37} +(2.00000 - 3.46410i) q^{38} +(-1.50000 - 2.59808i) q^{40} +(-5.00000 - 8.66025i) q^{41} +(-2.00000 + 3.46410i) q^{43} +4.00000 q^{44} +(-4.00000 + 6.92820i) q^{47} +(3.50000 + 6.06218i) q^{49} +(0.500000 + 0.866025i) q^{50} +(-1.00000 + 1.73205i) q^{52} -10.0000 q^{53} -4.00000 q^{55} +(-1.00000 - 1.73205i) q^{58} +(2.00000 + 3.46410i) q^{59} +(1.00000 - 1.73205i) q^{61} +7.00000 q^{64} +(1.00000 - 1.73205i) q^{65} +(-6.00000 - 10.3923i) q^{67} +(1.00000 + 1.73205i) q^{68} -8.00000 q^{71} +10.0000 q^{73} +(-5.00000 + 8.66025i) q^{74} +(2.00000 + 3.46410i) q^{76} -1.00000 q^{80} -10.0000 q^{82} +(-6.00000 + 10.3923i) q^{83} +(-1.00000 - 1.73205i) q^{85} +(2.00000 + 3.46410i) q^{86} +(6.00000 - 10.3923i) q^{88} -6.00000 q^{89} +(4.00000 + 6.92820i) q^{94} +(-2.00000 - 3.46410i) q^{95} +(-1.00000 + 1.73205i) q^{97} +7.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + q^{4} - q^{5} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + q^4 - q^5 + 6 * q^8	\( 2 q + q^{2} + q^{4} - q^{5} + 6 q^{8} - 2 q^{10} + 4 q^{11} + 2 q^{13} + q^{16} + 4 q^{17} + 8 q^{19} + q^{20} - 4 q^{22} - q^{25} + 4 q^{26} + 2 q^{29} + 5 q^{32} + 2 q^{34} - 20 q^{37} + 4 q^{38} - 3 q^{40} - 10 q^{41} - 4 q^{43} + 8 q^{44} - 8 q^{47} + 7 q^{49} + q^{50} - 2 q^{52} - 20 q^{53} - 8 q^{55} - 2 q^{58} + 4 q^{59} + 2 q^{61} + 14 q^{64} + 2 q^{65} - 12 q^{67} + 2 q^{68} - 16 q^{71} + 20 q^{73} - 10 q^{74} + 4 q^{76} - 2 q^{80} - 20 q^{82} - 12 q^{83} - 2 q^{85} + 4 q^{86} + 12 q^{88} - 12 q^{89} + 8 q^{94} - 4 q^{95} - 2 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q + q^2 + q^4 - q^5 + 6 * q^8 - 2 * q^10 + 4 * q^11 + 2 * q^13 + q^16 + 4 * q^17 + 8 * q^19 + q^20 - 4 * q^22 - q^25 + 4 * q^26 + 2 * q^29 + 5 * q^32 + 2 * q^34 - 20 * q^37 + 4 * q^38 - 3 * q^40 - 10 * q^41 - 4 * q^43 + 8 * q^44 - 8 * q^47 + 7 * q^49 + q^50 - 2 * q^52 - 20 * q^53 - 8 * q^55 - 2 * q^58 + 4 * q^59 + 2 * q^61 + 14 * q^64 + 2 * q^65 - 12 * q^67 + 2 * q^68 - 16 * q^71 + 20 * q^73 - 10 * q^74 + 4 * q^76 - 2 * q^80 - 20 * q^82 - 12 * q^83 - 2 * q^85 + 4 * q^86 + 12 * q^88 - 12 * q^89 + 8 * q^94 - 4 * q^95 - 2 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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