LMFDB - Embedded newform 405.2.e.g.136.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,2,Mod(136,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [2,2,0,-2,1,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
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.g.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +2.00000 q^{10} +(2.50000 - 4.33013i) q^{11} +(-2.00000 - 3.46410i) q^{13} +(2.00000 - 3.46410i) q^{16} +4.00000 q^{17} -5.00000 q^{19} +(1.00000 - 1.73205i) q^{20} +(-5.00000 - 8.66025i) q^{22} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} -8.00000 q^{26} +(-2.50000 + 4.33013i) q^{29} +(4.50000 + 7.79423i) q^{31} +(-4.00000 - 6.92820i) q^{32} +(4.00000 - 6.92820i) q^{34} -10.0000 q^{37} +(-5.00000 + 8.66025i) q^{38} +(3.50000 + 6.06218i) q^{41} +(1.00000 - 1.73205i) q^{43} -10.0000 q^{44} +12.0000 q^{46} +(1.00000 - 1.73205i) q^{47} +(3.50000 + 6.06218i) q^{49} +(1.00000 + 1.73205i) q^{50} +(-4.00000 + 6.92820i) q^{52} -8.00000 q^{53} +5.00000 q^{55} +(5.00000 + 8.66025i) q^{58} +(-0.500000 - 0.866025i) q^{59} +(1.00000 - 1.73205i) q^{61} +18.0000 q^{62} -8.00000 q^{64} +(2.00000 - 3.46410i) q^{65} +(-3.00000 - 5.19615i) q^{67} +(-4.00000 - 6.92820i) q^{68} -1.00000 q^{71} -8.00000 q^{73} +(-10.0000 + 17.3205i) q^{74} +(5.00000 + 8.66025i) q^{76} +(-6.00000 + 10.3923i) q^{79} +4.00000 q^{80} +14.0000 q^{82} +(3.00000 - 5.19615i) q^{83} +(2.00000 + 3.46410i) q^{85} +(-2.00000 - 3.46410i) q^{86} +9.00000 q^{89} +(6.00000 - 10.3923i) q^{92} +(-2.00000 - 3.46410i) q^{94} +(-2.50000 - 4.33013i) q^{95} +(-7.00000 + 12.1244i) q^{97} +14.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{10} + 5 q^{11} - 4 q^{13} + 4 q^{16} + 8 q^{17} - 10 q^{19} + 2 q^{20} - 10 q^{22} + 6 q^{23} - q^{25} - 16 q^{26} - 5 q^{29} + 9 q^{31} - 8 q^{32} + 8 q^{34} - 20 q^{37}+ \cdots + 28 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^4 + q^5 + 4 * q^10 + 5 * q^11 - 4 * q^13 + 4 * q^16 + 8 * q^17 - 10 * q^19 + 2 * q^20 - 10 * q^22 + 6 * q^23 - q^25 - 16 * q^26 - 5 * q^29 + 9 * q^31 - 8 * q^32 + 8 * q^34 - 20 * q^37 - 10 * q^38 + 7 * q^41 + 2 * q^43 - 20 * q^44 + 24 * q^46 + 2 * q^47 + 7 * q^49 + 2 * q^50 - 8 * q^52 - 16 * q^53 + 10 * q^55 + 10 * q^58 - q^59 + 2 * q^61 + 36 * q^62 - 16 * q^64 + 4 * q^65 - 6 * q^67 - 8 * q^68 - 2 * q^71 - 16 * q^73 - 20 * q^74 + 10 * q^76 - 12 * q^79 + 8 * q^80 + 28 * q^82 + 6 * q^83 + 4 * q^85 - 4 * q^86 + 18 * q^89 + 12 * q^92 - 4 * q^94 - 5 * q^95 - 14 * q^97 + 28 * q^98

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 $

$2$	1.00000	−	1.73205i	0.707107	−	1.22474i	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	1.00000	−	1.73205i	0.707107	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$3$		0			0
$3$		0			0
$4$	−1.00000	−	1.73205i	−0.500000	−	0.866025i
$5$	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	$0.166667\pi$
$8$		0			0
$9$		0			0
$10$	2.00000			0.632456
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	−0.192201	−	0.981356i	$-0.561563\pi$
$11$	2.50000	−	4.33013i	0.753778	−	1.30558i	0.945979	−	0.324227i	$-0.105104\pi$
$12$		0			0
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	−0.997927	−	0.0643593i	$-0.979500\pi$
$13$	−2.00000	−	3.46410i	−0.554700	−	0.960769i	0.443227	−	0.896410i	$-0.353834\pi$
$14$		0			0
$15$		0			0
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	4.00000			0.970143			0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000			0.970143			0.485071	+	0.874475i	$0.338794\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$	1.00000	−	1.73205i	0.223607	−	0.387298i
$21$		0			0
$22$	−5.00000	−	8.66025i	−1.06600	−	1.84637i
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	0.988436	+	0.151642i	$0.0484560\pi$
$23$	3.00000	+	5.19615i	0.625543	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	−8.00000			−1.56893
$27$		0			0
$28$		0			0
$29$	−2.50000	+	4.33013i	−0.464238	+	0.804084i	−0.999167	−	0.0408130i	$-0.987005\pi$
$29$	−2.50000	+	4.33013i	−0.464238	+	0.804084i	0.534928	+	0.844897i	$0.320339\pi$
$30$		0			0
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	0.914093	+	0.405505i	$0.132904\pi$
$31$	4.50000	+	7.79423i	0.808224	+	1.39988i	−0.105869	+	0.994380i	$0.533762\pi$
$32$	−4.00000	−	6.92820i	−0.707107	−	1.22474i
$33$		0			0
$34$	4.00000	−	6.92820i	0.685994	−	1.18818i
$35$		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	$0.807139\pi$
$38$	−5.00000	+	8.66025i	−0.811107	+	1.40488i
$39$		0			0
$40$		0			0
$41$	3.50000	+	6.06218i	0.546608	+	0.946753i	0.998504	+	0.0546823i	$0.0174146\pi$
$41$	3.50000	+	6.06218i	0.546608	+	0.946753i	−0.451896	+	0.892071i	$0.649252\pi$
$42$		0			0
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	−0.779647	−	0.626219i	$-0.784601\pi$
$43$	1.00000	−	1.73205i	0.152499	−	0.264135i	0.932145	+	0.362084i	$0.117935\pi$
$44$	−10.0000			−1.50756
$45$		0			0
$46$	12.0000			1.76930
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	−0.783830	−	0.620975i	$-0.786737\pi$
$47$	1.00000	−	1.73205i	0.145865	−	0.252646i	0.929695	+	0.368329i	$0.120070\pi$
$48$		0			0
$49$	3.50000	+	6.06218i	0.500000	+	0.866025i
$50$	1.00000	+	1.73205i	0.141421	+	0.244949i
$51$		0			0
$52$	−4.00000	+	6.92820i	−0.554700	+	0.960769i
$53$	−8.00000			−1.09888			−0.549442	−	0.835532i	$-0.685160\pi$
$53$	−8.00000			−1.09888			−0.549442	+	0.835532i	$0.685160\pi$
$54$		0			0
$55$	5.00000			0.674200
$56$		0			0
$57$		0			0
$58$	5.00000	+	8.66025i	0.656532	+	1.13715i
$59$	−0.500000	−	0.866025i	−0.0650945	−	0.112747i	0.831641	−	0.555313i	$-0.187402\pi$
$59$	−0.500000	−	0.866025i	−0.0650945	−	0.112747i	−0.896736	+	0.442566i	$0.854068\pi$
$60$		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	$0.125799\pi$
$62$	18.0000			2.28600
$63$		0			0
$64$	−8.00000			−1.00000
$65$	2.00000	−	3.46410i	0.248069	−	0.429669i
$66$		0			0
$67$	−3.00000	−	5.19615i	−0.366508	−	0.634811i	0.622509	−	0.782613i	$-0.286114\pi$
$67$	−3.00000	−	5.19615i	−0.366508	−	0.634811i	−0.989017	+	0.147802i	$0.952780\pi$
$68$	−4.00000	−	6.92820i	−0.485071	−	0.840168i
$69$		0			0
$70$		0			0
$71$	−1.00000			−0.118678			−0.0593391	−	0.998238i	$-0.518899\pi$
$71$	−1.00000			−0.118678			−0.0593391	+	0.998238i	$0.518899\pi$
$72$		0			0
$73$	−8.00000			−0.936329			−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000			−0.936329			−0.468165	+	0.883641i	$0.655085\pi$
$74$	−10.0000	+	17.3205i	−1.16248	+	2.01347i
$75$		0			0
$76$	5.00000	+	8.66025i	0.573539	+	0.993399i
$77$		0			0
$78$		0			0
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	0.301401	+	0.953498i	$0.402546\pi$
$79$	−6.00000	+	10.3923i	−0.675053	+	1.16923i	−0.976453	+	0.215728i	$0.930788\pi$
$80$	4.00000			0.447214
$81$		0			0
$82$	14.0000			1.54604
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	−0.653079	−	0.757290i	$-0.726523\pi$
$83$	3.00000	−	5.19615i	0.329293	−	0.570352i	0.982372	+	0.186938i	$0.0598564\pi$
$84$		0			0
$85$	2.00000	+	3.46410i	0.216930	+	0.375735i
$86$	−2.00000	−	3.46410i	−0.215666	−	0.373544i
$87$		0			0
$88$		0			0
$89$	9.00000			0.953998			0.476999	−	0.878904i	$-0.341725\pi$
$89$	9.00000			0.953998			0.476999	+	0.878904i	$0.341725\pi$
$90$		0			0
$91$		0			0
$92$	6.00000	−	10.3923i	0.625543	−	1.08347i
$93$		0			0
$94$	−2.00000	−	3.46410i	−0.206284	−	0.357295i
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	−7.00000	+	12.1244i	−0.710742	+	1.23104i	0.253837	+	0.967247i	$0.418307\pi$
$97$	−7.00000	+	12.1244i	−0.710742	+	1.23104i	−0.964579	+	0.263795i	$0.915026\pi$
$98$	14.0000			1.41421
$99$		0			0
$100$	2.00000			0.200000
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	−0.930953	−	0.365140i	$-0.881021\pi$
$101$	−1.50000	+	2.59808i	−0.149256	+	0.258518i	0.781697	+	0.623658i	$0.214354\pi$
$102$		0			0
$103$	−1.00000	−	1.73205i	−0.0985329	−	0.170664i	0.812545	−	0.582899i	$-0.198082\pi$
$103$	−1.00000	−	1.73205i	−0.0985329	−	0.170664i	−0.911078	+	0.412235i	$0.864748\pi$
$104$		0			0
$105$		0			0
$106$	−8.00000	+	13.8564i	−0.777029	+	1.34585i
$107$	6.00000			0.580042			0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000			0.580042			0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$	5.00000	−	8.66025i	0.476731	−	0.825723i
$111$		0			0
$112$		0			0
$113$	−8.00000	−	13.8564i	−0.752577	−	1.30350i	−0.946570	−	0.322498i	$-0.895477\pi$
$113$	−8.00000	−	13.8564i	−0.752577	−	1.30350i	0.193993	−	0.981003i	$-0.437856\pi$
$114$		0			0
$115$	−3.00000	+	5.19615i	−0.279751	+	0.484544i
$116$	10.0000			0.928477
$117$		0			0
$118$	−2.00000			−0.184115
$119$		0			0
$120$		0			0
$121$	−7.00000	−	12.1244i	−0.636364	−	1.10221i
$122$	−2.00000	−	3.46410i	−0.181071	−	0.313625i
$123$		0			0
$124$	9.00000	−	15.5885i	0.808224	−	1.39988i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	16.0000			1.41977			0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000			1.41977			0.709885	+	0.704317i	$0.248747\pi$
$128$		0			0
$129$		0			0
$130$	−4.00000	−	6.92820i	−0.350823	−	0.607644i
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	0.981824	+	0.189794i	$0.0607819\pi$
$131$	7.50000	+	12.9904i	0.655278	+	1.13497i	−0.326546	+	0.945181i	$0.605885\pi$
$132$		0			0
$133$		0			0
$134$	−12.0000			−1.03664
$135$		0			0
$136$		0			0
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	−0.487278	−	0.873247i	$-0.662010\pi$
$137$	6.00000	−	10.3923i	0.512615	−	0.887875i	0.999893	−	0.0146279i	$-0.00465636\pi$
$138$		0			0
$139$	9.50000	+	16.4545i	0.805779	+	1.39565i	0.915764	+	0.401718i	$0.131587\pi$
$139$	9.50000	+	16.4545i	0.805779	+	1.39565i	−0.109984	+	0.993933i	$0.535080\pi$
$140$		0			0
$141$		0			0
$142$	−1.00000	+	1.73205i	−0.0839181	+	0.145350i
$143$	−20.0000			−1.67248
$144$		0			0
$145$	−5.00000			−0.415227
$146$	−8.00000	+	13.8564i	−0.662085	+	1.14676i
$147$		0			0
$148$	10.0000	+	17.3205i	0.821995	+	1.42374i
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	0.822153	−	0.569267i	$-0.192773\pi$
$149$	−1.00000	−	1.73205i	−0.0819232	−	0.141895i	−0.904076	+	0.427372i	$0.859440\pi$
$150$		0			0
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	−0.746190	−	0.665733i	$-0.768119\pi$
$151$	2.50000	−	4.33013i	0.203447	−	0.352381i	0.949637	+	0.313353i	$0.101452\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.50000	+	7.79423i	−0.361449	+	0.626048i
$156$		0			0
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	0.823359	−	0.567521i	$-0.192098\pi$
$157$	−1.00000	−	1.73205i	−0.0798087	−	0.138233i	−0.903167	+	0.429289i	$0.858764\pi$
$158$	12.0000	+	20.7846i	0.954669	+	1.65353i
$159$		0			0
$160$	4.00000	−	6.92820i	0.316228	−	0.547723i
$161$		0			0
$162$		0			0
$163$	8.00000			0.626608			0.313304	−	0.949653i	$-0.398564\pi$
$163$	8.00000			0.626608			0.313304	+	0.949653i	$0.398564\pi$
$164$	7.00000	−	12.1244i	0.546608	−	0.946753i
$165$		0			0
$166$	−6.00000	−	10.3923i	−0.465690	−	0.806599i
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	0.999169	−	0.0407502i	$-0.0129748\pi$
$167$	6.00000	+	10.3923i	0.464294	+	0.804181i	−0.534875	+	0.844931i	$0.679641\pi$
$168$		0			0
$169$	−1.50000	+	2.59808i	−0.115385	+	0.199852i
$170$	8.00000			0.613572
$171$		0			0
$172$	−4.00000			−0.304997
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	−0.957241	−	0.289292i	$-0.906580\pi$
$173$	−3.00000	+	5.19615i	−0.228086	+	0.395056i	0.729155	+	0.684349i	$0.239913\pi$
$174$		0			0
$175$		0			0
$176$	−10.0000	−	17.3205i	−0.753778	−	1.30558i
$177$		0			0
$178$	9.00000	−	15.5885i	0.674579	−	1.16840i
$179$	−23.0000			−1.71910			−0.859550	−	0.511051i	$-0.829256\pi$
$179$	−23.0000			−1.71910			−0.859550	+	0.511051i	$0.829256\pi$
$180$		0			0
$181$	−25.0000			−1.85824			−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000			−1.85824			−0.929118	+	0.369784i	$0.879432\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.00000	−	8.66025i	−0.367607	−	0.636715i
$186$		0			0
$187$	10.0000	−	17.3205i	0.731272	−	1.26660i
$188$	−4.00000			−0.291730
$189$		0			0
$190$	−10.0000			−0.725476
$191$	0.500000	−	0.866025i	0.0361787	−	0.0626634i	−0.847369	−	0.531004i	$-0.821815\pi$
$191$	0.500000	−	0.866025i	0.0361787	−	0.0626634i	0.883548	+	0.468341i	$0.155148\pi$
$192$		0			0
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.773272	−	0.634074i	$-0.781381\pi$
$193$	−13.0000	−	22.5167i	−0.935760	−	1.62078i	−0.162488	−	0.986710i	$-0.551952\pi$
$194$	14.0000	+	24.2487i	1.00514	+	1.74096i
$195$		0			0
$196$	7.00000	−	12.1244i	0.500000	−	0.866025i
$197$	−12.0000			−0.854965			−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000			−0.854965			−0.427482	+	0.904024i	$0.640599\pi$
$198$		0			0
$199$	16.0000			1.13421			0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000			1.13421			0.567105	+	0.823646i	$0.308063\pi$
$200$		0			0
$201$		0			0
$202$	3.00000	+	5.19615i	0.211079	+	0.365600i
$203$		0			0
$204$		0			0
$205$	−3.50000	+	6.06218i	−0.244451	+	0.423401i
$206$	−4.00000			−0.278693
$207$		0			0
$208$	−16.0000			−1.10940
$209$	−12.5000	+	21.6506i	−0.864643	+	1.49761i
$210$		0			0
$211$	−5.50000	−	9.52628i	−0.378636	−	0.655816i	0.612228	−	0.790681i	$-0.290273\pi$
$211$	−5.50000	−	9.52628i	−0.378636	−	0.655816i	−0.990864	+	0.134865i	$0.956940\pi$
$212$	8.00000	+	13.8564i	0.549442	+	0.951662i
$213$		0			0
$214$	6.00000	−	10.3923i	0.410152	−	0.710403i
$215$	2.00000			0.136399
$216$		0			0
$217$		0			0
$218$	−1.00000	+	1.73205i	−0.0677285	+	0.117309i
$219$		0			0
$220$	−5.00000	−	8.66025i	−0.337100	−	0.583874i
$221$	−8.00000	−	13.8564i	−0.538138	−	0.932083i
$222$		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
$226$	−32.0000			−2.12861
$227$	2.00000	−	3.46410i	0.132745	−	0.229920i	−0.791989	−	0.610535i	$-0.790954\pi$
$227$	2.00000	−	3.46410i	0.132745	−	0.229920i	0.924734	+	0.380615i	$0.124288\pi$
$228$		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	$0.896858\pi$
$230$	6.00000	+	10.3923i	0.395628	+	0.685248i
$231$		0			0
$232$		0			0
$233$	−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
$235$	2.00000			0.130466
$236$	−1.00000	+	1.73205i	−0.0650945	+	0.112747i
$237$		0			0
$238$		0			0
$239$	−8.00000	−	13.8564i	−0.517477	−	0.896296i	−0.999794	−	0.0202996i	$-0.993538\pi$
$239$	−8.00000	−	13.8564i	−0.517477	−	0.896296i	0.482317	−	0.875997i	$-0.339795\pi$
$240$		0			0
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	−0.632709	−	0.774389i	$-0.718057\pi$
$241$	5.50000	−	9.52628i	0.354286	−	0.613642i	0.986996	+	0.160748i	$0.0513906\pi$
$242$	−28.0000			−1.79991
$243$		0			0
$244$	−4.00000			−0.256074
$245$	−3.50000	+	6.06218i	−0.223607	+	0.387298i
$246$		0			0
$247$	10.0000	+	17.3205i	0.636285	+	1.10208i
$248$		0			0
$249$		0			0
$250$	−1.00000	+	1.73205i	−0.0632456	+	0.109545i
$251$	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
$253$	30.0000			1.88608
$254$	16.0000	−	27.7128i	1.00393	−	1.73886i
$255$		0			0
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	0.757159	−	0.653231i	$-0.226587\pi$
$257$	−3.00000	−	5.19615i	−0.187135	−	0.324127i	−0.944294	+	0.329104i	$0.893253\pi$
$258$		0			0
$259$		0			0
$260$	−8.00000			−0.496139
$261$		0			0
$262$	30.0000			1.85341
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	−0.669680	−	0.742650i	$-0.733569\pi$
$263$	5.00000	−	8.66025i	0.308313	−	0.534014i	0.977993	+	0.208635i	$0.0669022\pi$
$264$		0			0
$265$	−4.00000	−	6.92820i	−0.245718	−	0.425596i
$266$		0			0
$267$		0			0
$268$	−6.00000	+	10.3923i	−0.366508	+	0.634811i
$269$	31.0000			1.89010			0.945052	−	0.326921i	$-0.106011\pi$
$269$	31.0000			1.89010			0.945052	+	0.326921i	$0.106011\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	8.00000	−	13.8564i	0.485071	−	0.840168i
$273$		0			0
$274$	−12.0000	−	20.7846i	−0.724947	−	1.25564i
$275$	2.50000	+	4.33013i	0.150756	+	0.261116i
$276$		0			0
$277$	9.00000	−	15.5885i	0.540758	−	0.936620i	−0.458103	−	0.888899i	$-0.651471\pi$
$277$	9.00000	−	15.5885i	0.540758	−	0.936620i	0.998861	−	0.0477206i	$-0.0151957\pi$
$278$	38.0000			2.27909
$279$		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
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	0.762978	−	0.646425i	$-0.223737\pi$
$283$	−3.00000	−	5.19615i	−0.178331	−	0.308879i	−0.941309	+	0.337546i	$0.890403\pi$
$284$	1.00000	+	1.73205i	0.0593391	+	0.102778i
$285$		0			0
$286$	−20.0000	+	34.6410i	−1.18262	+	2.04837i
$287$		0			0
$288$		0			0
$289$	−1.00000			−0.0588235
$290$	−5.00000	+	8.66025i	−0.293610	+	0.508548i
$291$		0			0
$292$	8.00000	+	13.8564i	0.468165	+	0.810885i
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	0.999549	+	0.0300351i	$0.00956192\pi$
$293$	9.00000	+	15.5885i	0.525786	+	0.910687i	−0.473763	+	0.880652i	$0.657105\pi$
$294$		0			0
$295$	0.500000	−	0.866025i	0.0291111	−	0.0504219i
$296$		0			0
$297$		0			0
$298$	−4.00000			−0.231714
$299$	12.0000	−	20.7846i	0.693978	−	1.20201i
$300$		0			0
$301$		0			0
$302$	−5.00000	−	8.66025i	−0.287718	−	0.498342i
$303$		0			0
$304$	−10.0000	+	17.3205i	−0.573539	+	0.993399i
$305$	2.00000			0.114520
$306$		0			0
$307$	10.0000			0.570730			0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000			0.570730			0.285365	+	0.958419i	$0.407885\pi$
$308$		0			0
$309$		0			0
$310$	9.00000	+	15.5885i	0.511166	+	0.885365i
$311$	−4.50000	−	7.79423i	−0.255172	−	0.441970i	0.709771	−	0.704433i	$-0.248799\pi$
$311$	−4.50000	−	7.79423i	−0.255172	−	0.441970i	−0.964942	+	0.262463i	$0.915465\pi$
$312$		0			0
$313$	2.00000	−	3.46410i	0.113047	−	0.195803i	−0.803951	−	0.594696i	$-0.797272\pi$
$313$	2.00000	−	3.46410i	0.113047	−	0.195803i	0.916997	+	0.398894i	$0.130606\pi$
$314$	−4.00000			−0.225733
$315$		0			0
$316$	24.0000			1.35011
$317$	−1.00000	+	1.73205i	−0.0561656	+	0.0972817i	−0.892741	−	0.450570i	$-0.851221\pi$
$317$	−1.00000	+	1.73205i	−0.0561656	+	0.0972817i	0.836576	+	0.547852i	$0.184554\pi$
$318$		0			0
$319$	12.5000	+	21.6506i	0.699866	+	1.21220i
$320$	−4.00000	−	6.92820i	−0.223607	−	0.387298i
$321$		0			0
$322$		0			0
$323$	−20.0000			−1.11283
$324$		0			0
$325$	4.00000			0.221880
$326$	8.00000	−	13.8564i	0.443079	−	0.767435i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	10.5000	−	18.1865i	0.577132	−	0.999622i	−0.418674	−	0.908137i	$-0.637505\pi$
$331$	10.5000	−	18.1865i	0.577132	−	0.999622i	0.995806	−	0.0914858i	$-0.0291616\pi$
$332$	−12.0000			−0.658586
$333$		0			0
$334$	24.0000			1.31322
$335$	3.00000	−	5.19615i	0.163908	−	0.283896i
$336$		0			0
$337$	4.00000	+	6.92820i	0.217894	+	0.377403i	0.954164	−	0.299285i	$-0.0967480\pi$
$337$	4.00000	+	6.92820i	0.217894	+	0.377403i	−0.736270	+	0.676688i	$0.763415\pi$
$338$	3.00000	+	5.19615i	0.163178	+	0.282633i
$339$		0			0
$340$	4.00000	−	6.92820i	0.216930	−	0.375735i
$341$	45.0000			2.43689
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	6.00000	+	10.3923i	0.322562	+	0.558694i
$347$	10.0000	+	17.3205i	0.536828	+	0.929814i	0.999072	+	0.0430610i	$0.0137110\pi$
$347$	10.0000	+	17.3205i	0.536828	+	0.929814i	−0.462244	+	0.886753i	$0.652956\pi$
$348$		0			0
$349$	−6.50000	+	11.2583i	−0.347937	+	0.602645i	−0.985883	−	0.167437i	$-0.946451\pi$
$349$	−6.50000	+	11.2583i	−0.347937	+	0.602645i	0.637946	+	0.770081i	$0.279784\pi$
$350$		0			0
$351$		0			0
$352$	−40.0000			−2.13201
$353$	−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
$355$	−0.500000	−	0.866025i	−0.0265372	−	0.0459639i
$356$	−9.00000	−	15.5885i	−0.476999	−	0.826187i
$357$		0			0
$358$	−23.0000	+	39.8372i	−1.21559	+	2.10546i
$359$	27.0000			1.42501			0.712503	−	0.701669i	$-0.247562\pi$
$359$	27.0000			1.42501			0.712503	+	0.701669i	$0.247562\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$	−25.0000	+	43.3013i	−1.31397	+	2.27586i
$363$		0			0
$364$		0			0
$365$	−4.00000	−	6.92820i	−0.209370	−	0.362639i
$366$		0			0
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	−0.999404	−	0.0345320i	$-0.989006\pi$
$367$	−9.00000	+	15.5885i	−0.469796	+	0.813711i	0.529607	+	0.848243i	$0.322339\pi$
$368$	24.0000			1.25109
$369$		0			0
$370$	−20.0000			−1.03975
$371$		0			0
$372$		0			0
$373$	−8.00000	−	13.8564i	−0.414224	−	0.717458i	0.581122	−	0.813816i	$-0.302614\pi$
$373$	−8.00000	−	13.8564i	−0.414224	−	0.717458i	−0.995347	+	0.0963587i	$0.969280\pi$
$374$	−20.0000	−	34.6410i	−1.03418	−	1.79124i
$375$		0			0
$376$		0			0
$377$	20.0000			1.03005
$378$		0			0
$379$	4.00000			0.205466			0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000			0.205466			0.102733	+	0.994709i	$0.467241\pi$
$380$	−5.00000	+	8.66025i	−0.256495	+	0.444262i
$381$		0			0
$382$	−1.00000	−	1.73205i	−0.0511645	−	0.0886194i
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.799783	−	0.600289i	$-0.795052\pi$
$383$	−18.0000	−	31.1769i	−0.919757	−	1.59307i	−0.119974	−	0.992777i	$-0.538281\pi$
$384$		0			0
$385$		0			0
$386$	−52.0000			−2.64673
$387$		0			0
$388$	28.0000			1.42148
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	−0.779895	−	0.625910i	$-0.784728\pi$
$389$	3.00000	−	5.19615i	0.152106	−	0.263455i	0.932002	+	0.362454i	$0.118061\pi$
$390$		0			0
$391$	12.0000	+	20.7846i	0.606866	+	1.05112i
$392$		0			0
$393$		0			0
$394$	−12.0000	+	20.7846i	−0.604551	+	1.04711i
$395$	−12.0000			−0.603786
$396$		0			0
$397$	−38.0000			−1.90717			−0.953583	−	0.301131i	$-0.902636\pi$
$397$	−38.0000			−1.90717			−0.953583	+	0.301131i	$0.902636\pi$
$398$	16.0000	−	27.7128i	0.802008	−	1.38912i
$399$		0			0
$400$	2.00000	+	3.46410i	0.100000	+	0.173205i
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	0.948272	+	0.317460i	$0.102830\pi$
$401$	15.0000	+	25.9808i	0.749064	+	1.29742i	−0.199207	+	0.979957i	$0.563837\pi$
$402$		0			0
$403$	18.0000	−	31.1769i	0.896644	−	1.55303i
$404$	6.00000			0.298511
$405$		0			0
$406$		0			0
$407$	−25.0000	+	43.3013i	−1.23920	+	2.14636i
$408$		0			0
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	0.639430	−	0.768849i	$-0.279170\pi$
$409$	−7.00000	−	12.1244i	−0.346128	−	0.599511i	−0.985558	+	0.169338i	$0.945837\pi$
$410$	7.00000	+	12.1244i	0.345705	+	0.598779i
$411$		0			0
$412$	−2.00000	+	3.46410i	−0.0985329	+	0.170664i
$413$		0			0
$414$		0			0
$415$	6.00000			0.294528
$416$	−16.0000	+	27.7128i	−0.784465	+	1.35873i
$417$		0			0
$418$	25.0000	+	43.3013i	1.22279	+	2.11793i
$419$	8.00000	+	13.8564i	0.390826	+	0.676930i	0.992559	−	0.121768i	$-0.0388562\pi$
$419$	8.00000	+	13.8564i	0.390826	+	0.676930i	−0.601733	+	0.798697i	$0.705523\pi$
$420$		0			0
$421$	−6.50000	+	11.2583i	−0.316791	+	0.548697i	−0.979817	−	0.199899i	$-0.935939\pi$
$421$	−6.50000	+	11.2583i	−0.316791	+	0.548697i	0.663026	+	0.748596i	$0.269272\pi$
$422$	−22.0000			−1.07094
$423$		0			0
$424$		0			0
$425$	−2.00000	+	3.46410i	−0.0970143	+	0.168034i
$426$		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
$431$	−3.00000			−0.144505			−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000			−0.144505			−0.0722525	+	0.997386i	$0.523019\pi$
$432$		0			0
$433$	−26.0000			−1.24948			−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000			−1.24948			−0.624740	+	0.780833i	$0.714795\pi$
$434$		0			0
$435$		0			0
$436$	1.00000	+	1.73205i	0.0478913	+	0.0829502i
$437$	−15.0000	−	25.9808i	−0.717547	−	1.24283i
$438$		0			0
$439$	14.5000	−	25.1147i	0.692047	−	1.19866i	−0.279119	−	0.960257i	$-0.590042\pi$
$439$	14.5000	−	25.1147i	0.692047	−	1.19866i	0.971166	−	0.238404i	$-0.0766244\pi$
$440$		0			0
$441$		0			0
$442$	−32.0000			−1.52208
$443$	3.00000	−	5.19615i	0.142534	−	0.246877i	−0.785916	−	0.618333i	$-0.787808\pi$
$443$	3.00000	−	5.19615i	0.142534	−	0.246877i	0.928450	+	0.371457i	$0.121142\pi$
$444$		0			0
$445$	4.50000	+	7.79423i	0.213320	+	0.369482i
$446$	8.00000	+	13.8564i	0.378811	+	0.656120i
$447$		0			0
$448$		0			0
$449$	−17.0000			−0.802280			−0.401140	−	0.916017i	$-0.631386\pi$
$449$	−17.0000			−0.802280			−0.401140	+	0.916017i	$0.631386\pi$
$450$		0			0
$451$	35.0000			1.64809
$452$	−16.0000	+	27.7128i	−0.752577	+	1.30350i
$453$		0			0
$454$	−4.00000	−	6.92820i	−0.187729	−	0.325157i
$455$		0			0
$456$		0			0
$457$	19.0000	−	32.9090i	0.888783	−	1.53942i	0.0474665	−	0.998873i	$-0.484885\pi$
$457$	19.0000	−	32.9090i	0.888783	−	1.53942i	0.841316	−	0.540544i	$-0.181781\pi$
$458$	−12.0000			−0.560723
$459$		0			0
$460$	12.0000			0.559503
$461$	−7.50000	+	12.9904i	−0.349310	+	0.605022i	−0.986127	−	0.165992i	$-0.946917\pi$
$461$	−7.50000	+	12.9904i	−0.349310	+	0.605022i	0.636817	+	0.771015i	$0.280251\pi$
$462$		0			0
$463$	3.00000	+	5.19615i	0.139422	+	0.241486i	0.927278	−	0.374374i	$-0.122142\pi$
$463$	3.00000	+	5.19615i	0.139422	+	0.241486i	−0.787856	+	0.615859i	$0.788809\pi$
$464$	10.0000	+	17.3205i	0.464238	+	0.804084i
$465$		0			0
$466$	−6.00000	+	10.3923i	−0.277945	+	0.481414i
$467$	−4.00000			−0.185098			−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000			−0.185098			−0.0925490	+	0.995708i	$0.529501\pi$
$468$		0			0
$469$		0			0
$470$	2.00000	−	3.46410i	0.0922531	−	0.159787i
$471$		0			0
$472$		0			0
$473$	−5.00000	−	8.66025i	−0.229900	−	0.398199i
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$	−32.0000			−1.46365
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	−0.984930	−	0.172953i	$-0.944669\pi$
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	0.642246	+	0.766498i	$0.278003\pi$
$480$		0			0
$481$	20.0000	+	34.6410i	0.911922	+	1.57949i
$482$	−11.0000	−	19.0526i	−0.501036	−	0.867820i
$483$		0			0
$484$	−14.0000	+	24.2487i	−0.636364	+	1.10221i
$485$	−14.0000			−0.635707
$486$		0			0
$487$	8.00000			0.362515			0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000			0.362515			0.181257	+	0.983436i	$0.441983\pi$
$488$		0			0
$489$		0			0
$490$	7.00000	+	12.1244i	0.316228	+	0.547723i
$491$	21.5000	+	37.2391i	0.970281	+	1.68058i	0.694701	+	0.719299i	$0.255537\pi$
$491$	21.5000	+	37.2391i	0.970281	+	1.68058i	0.275581	+	0.961278i	$0.411130\pi$
$492$		0			0
$493$	−10.0000	+	17.3205i	−0.450377	+	0.780076i
$494$	40.0000			1.79969
$495$		0			0
$496$	36.0000			1.61645
$497$		0			0
$498$		0			0
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	0.776989	−	0.629515i	$-0.216746\pi$
$499$	−3.50000	−	6.06218i	−0.156682	−	0.271380i	−0.933670	+	0.358134i	$0.883413\pi$
$500$	1.00000	+	1.73205i	0.0447214	+	0.0774597i
$501$		0			0
$502$	12.0000	−	20.7846i	0.535586	−	0.927663i
$503$	20.0000			0.891756			0.445878	−	0.895094i	$-0.352892\pi$
$503$	20.0000			0.891756			0.445878	+	0.895094i	$0.352892\pi$
$504$		0			0
$505$	−3.00000			−0.133498
$506$	30.0000	−	51.9615i	1.33366	−	2.30997i
$507$		0			0
$508$	−16.0000	−	27.7128i	−0.709885	−	1.22956i
$509$	1.00000	+	1.73205i	0.0443242	+	0.0767718i	0.887336	−	0.461123i	$-0.152553\pi$
$509$	1.00000	+	1.73205i	0.0443242	+	0.0767718i	−0.843012	+	0.537895i	$0.819220\pi$
$510$		0			0
$511$		0			0
$512$	−32.0000			−1.41421
$513$		0			0
$514$	−12.0000			−0.529297
$515$	1.00000	−	1.73205i	0.0440653	−	0.0763233i
$516$		0			0
$517$	−5.00000	−	8.66025i	−0.219900	−	0.380878i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$	16.0000			0.699631			0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000			0.699631			0.349816	+	0.936819i	$0.386244\pi$
$524$	15.0000	−	25.9808i	0.655278	−	1.13497i
$525$		0			0
$526$	−10.0000	−	17.3205i	−0.436021	−	0.755210i
$527$	18.0000	+	31.1769i	0.784092	+	1.35809i
$528$		0			0
$529$	−6.50000	+	11.2583i	−0.282609	+	0.489493i
$530$	−16.0000			−0.694996
$531$		0			0
$532$		0			0
$533$	14.0000	−	24.2487i	0.606407	−	1.05033i
$534$		0			0
$535$	3.00000	+	5.19615i	0.129701	+	0.224649i
$536$		0			0
$537$		0			0
$538$	31.0000	−	53.6936i	1.33650	−	2.31489i
$539$	35.0000			1.50756
$540$		0			0
$541$	3.00000			0.128980			0.0644900	−	0.997918i	$-0.479458\pi$
$541$	3.00000			0.128980			0.0644900	+	0.997918i	$0.479458\pi$
$542$	−8.00000	+	13.8564i	−0.343629	+	0.595184i
$543$		0			0
$544$	−16.0000	−	27.7128i	−0.685994	−	1.18818i
$545$	−0.500000	−	0.866025i	−0.0214176	−	0.0370965i
$546$		0			0
$547$	−14.0000	+	24.2487i	−0.598597	+	1.03680i	0.394432	+	0.918925i	$0.370941\pi$
$547$	−14.0000	+	24.2487i	−0.598597	+	1.03680i	−0.993028	+	0.117875i	$0.962392\pi$
$548$	−24.0000			−1.02523
$549$		0			0
$550$	10.0000			0.426401
$551$	12.5000	−	21.6506i	0.532518	−	0.922348i
$552$		0			0
$553$		0			0
$554$	−18.0000	−	31.1769i	−0.764747	−	1.32458i
$555$		0			0
$556$	19.0000	−	32.9090i	0.805779	−	1.39565i
$557$	24.0000			1.01691			0.508456	−	0.861088i	$-0.330216\pi$
$557$	24.0000			1.01691			0.508456	+	0.861088i	$0.330216\pi$
$558$		0			0
$559$	−8.00000			−0.338364
$560$		0			0
$561$		0			0
$562$	−6.00000	−	10.3923i	−0.253095	−	0.438373i
$563$	−9.00000	−	15.5885i	−0.379305	−	0.656975i	0.611656	−	0.791123i	$-0.290503\pi$
$563$	−9.00000	−	15.5885i	−0.379305	−	0.656975i	−0.990961	+	0.134148i	$0.957170\pi$
$564$		0			0
$565$	8.00000	−	13.8564i	0.336563	−	0.582943i
$566$	−12.0000			−0.504398
$567$		0			0
$568$		0			0
$569$	−1.50000	+	2.59808i	−0.0628833	+	0.108917i	−0.895753	−	0.444552i	$-0.853363\pi$
$569$	−1.50000	+	2.59808i	−0.0628833	+	0.108917i	0.832870	+	0.553469i	$0.186696\pi$
$570$		0			0
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	0.808962	−	0.587861i	$-0.200030\pi$
$571$	−2.50000	−	4.33013i	−0.104622	−	0.181210i	−0.913584	+	0.406651i	$0.866697\pi$
$572$	20.0000	+	34.6410i	0.836242	+	1.44841i
$573$		0			0
$574$		0			0
$575$	−6.00000			−0.250217
$576$		0			0
$577$	−16.0000			−0.666089			−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000			−0.666089			−0.333044	+	0.942911i	$0.608076\pi$
$578$	−1.00000	+	1.73205i	−0.0415945	+	0.0720438i
$579$		0			0
$580$	5.00000	+	8.66025i	0.207614	+	0.359597i
$581$		0			0
$582$		0			0
$583$	−20.0000	+	34.6410i	−0.828315	+	1.43468i
$584$		0			0
$585$		0			0
$586$	36.0000			1.48715
$587$	9.00000	−	15.5885i	0.371470	−	0.643404i	−0.618322	−	0.785925i	$-0.712187\pi$
$587$	9.00000	−	15.5885i	0.371470	−	0.643404i	0.989792	+	0.142520i	$0.0455206\pi$
$588$		0			0
$589$	−22.5000	−	38.9711i	−0.927096	−	1.60578i
$590$	−1.00000	−	1.73205i	−0.0411693	−	0.0713074i
$591$		0			0
$592$	−20.0000	+	34.6410i	−0.821995	+	1.42374i
$593$	14.0000			0.574911			0.287456	−	0.957794i	$-0.407191\pi$
$593$	14.0000			0.574911			0.287456	+	0.957794i	$0.407191\pi$
$594$		0			0
$595$		0			0
$596$	−2.00000	+	3.46410i	−0.0819232	+	0.141895i
$597$		0			0
$598$	−24.0000	−	41.5692i	−0.981433	−	1.69989i
$599$	−8.50000	−	14.7224i	−0.347301	−	0.601542i	0.638468	−	0.769648i	$-0.279568\pi$
$599$	−8.50000	−	14.7224i	−0.347301	−	0.601542i	−0.985769	+	0.168106i	$0.946235\pi$
$600$		0			0
$601$	9.50000	−	16.4545i	0.387513	−	0.671192i	−0.604601	−	0.796528i	$-0.706668\pi$
$601$	9.50000	−	16.4545i	0.387513	−	0.671192i	0.992114	+	0.125336i	$0.0400009\pi$
$602$		0			0
$603$		0			0
$604$	−10.0000			−0.406894
$605$	7.00000	−	12.1244i	0.284590	−	0.492925i
$606$		0			0
$607$	−17.0000	−	29.4449i	−0.690009	−	1.19513i	−0.971834	−	0.235665i	$-0.924273\pi$
$607$	−17.0000	−	29.4449i	−0.690009	−	1.19513i	0.281826	−	0.959466i	$-0.409060\pi$
$608$	20.0000	+	34.6410i	0.811107	+	1.40488i
$609$		0			0
$610$	2.00000	−	3.46410i	0.0809776	−	0.140257i
$611$	−8.00000			−0.323645
$612$		0			0
$613$	4.00000			0.161558			0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000			0.161558			0.0807792	+	0.996732i	$0.474259\pi$
$614$	10.0000	−	17.3205i	0.403567	−	0.698999i
$615$		0			0
$616$		0			0
$617$	−12.0000	−	20.7846i	−0.483102	−	0.836757i	0.516710	−	0.856161i	$-0.327157\pi$
$617$	−12.0000	−	20.7846i	−0.483102	−	0.836757i	−0.999812	+	0.0194037i	$0.993823\pi$
$618$		0			0
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	−0.434551	−	0.900647i	$-0.643093\pi$
$619$	14.0000	−	24.2487i	0.562708	−	0.974638i	0.997259	−	0.0739910i	$-0.0235736\pi$
$620$	18.0000			0.722897
$621$		0			0
$622$	−18.0000			−0.721734
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	−4.00000	−	6.92820i	−0.159872	−	0.276907i
$627$		0			0
$628$	−2.00000	+	3.46410i	−0.0798087	+	0.138233i
$629$	−40.0000			−1.59490
$630$		0			0
$631$	−17.0000			−0.676759			−0.338380	−	0.941010i	$-0.609879\pi$
$631$	−17.0000			−0.676759			−0.338380	+	0.941010i	$0.609879\pi$
$632$		0			0
$633$		0			0
$634$	2.00000	+	3.46410i	0.0794301	+	0.137577i
$635$	8.00000	+	13.8564i	0.317470	+	0.549875i
$636$		0			0
$637$	14.0000	−	24.2487i	0.554700	−	0.960769i
$638$	50.0000			1.97952
$639$		0			0
$640$		0			0
$641$	1.50000	−	2.59808i	0.0592464	−	0.102618i	−0.834881	−	0.550431i	$-0.814464\pi$
$641$	1.50000	−	2.59808i	0.0592464	−	0.102618i	0.894127	+	0.447813i	$0.147797\pi$
$642$		0			0
$643$	−3.00000	−	5.19615i	−0.118308	−	0.204916i	0.800789	−	0.598947i	$-0.204414\pi$
$643$	−3.00000	−	5.19615i	−0.118308	−	0.204916i	−0.919097	+	0.394030i	$0.871080\pi$
$644$		0			0
$645$		0			0
$646$	−20.0000	+	34.6410i	−0.786889	+	1.36293i
$647$	10.0000			0.393141			0.196570	−	0.980490i	$-0.437020\pi$
$647$	10.0000			0.393141			0.196570	+	0.980490i	$0.437020\pi$
$648$		0			0
$649$	−5.00000			−0.196267
$650$	4.00000	−	6.92820i	0.156893	−	0.271746i
$651$		0			0
$652$	−8.00000	−	13.8564i	−0.313304	−	0.542659i
$653$	−4.00000	−	6.92820i	−0.156532	−	0.271122i	0.777084	−	0.629397i	$-0.216698\pi$
$653$	−4.00000	−	6.92820i	−0.156532	−	0.271122i	−0.933616	+	0.358276i	$0.883365\pi$
$654$		0			0
$655$	−7.50000	+	12.9904i	−0.293049	+	0.507576i
$656$	28.0000			1.09322
$657$		0			0
$658$		0			0
$659$	22.0000	−	38.1051i	0.856998	−	1.48436i	−0.0177803	−	0.999842i	$-0.505660\pi$
$659$	22.0000	−	38.1051i	0.856998	−	1.48436i	0.874779	−	0.484523i	$-0.161007\pi$
$660$		0			0
$661$	−12.5000	−	21.6506i	−0.486194	−	0.842112i	0.513680	−	0.857982i	$-0.328282\pi$
$661$	−12.5000	−	21.6506i	−0.486194	−	0.842112i	−0.999874	+	0.0158695i	$0.994948\pi$
$662$	−21.0000	−	36.3731i	−0.816188	−	1.41368i
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−30.0000			−1.16160
$668$	12.0000	−	20.7846i	0.464294	−	0.804181i
$669$		0			0
$670$	−6.00000	−	10.3923i	−0.231800	−	0.401490i
$671$	−5.00000	−	8.66025i	−0.193023	−	0.334325i
$672$		0			0
$673$	−21.0000	+	36.3731i	−0.809491	+	1.40208i	0.103727	+	0.994606i	$0.466923\pi$
$673$	−21.0000	+	36.3731i	−0.809491	+	1.40208i	−0.913217	+	0.407473i	$0.866410\pi$
$674$	16.0000			0.616297
$675$		0			0
$676$	6.00000			0.230769
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	−0.802600	−	0.596518i	$-0.796551\pi$
$677$	3.00000	−	5.19615i	0.115299	−	0.199704i	0.917899	+	0.396813i	$0.129884\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$	45.0000	−	77.9423i	1.72314	−	2.98456i
$683$	−48.0000			−1.83667			−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000			−1.83667			−0.918334	+	0.395805i	$0.870466\pi$
$684$		0			0
$685$	12.0000			0.458496
$686$		0			0
$687$		0			0
$688$	−4.00000	−	6.92820i	−0.152499	−	0.264135i
$689$	16.0000	+	27.7128i	0.609551	+	1.05577i
$690$		0			0
$691$	−2.00000	+	3.46410i	−0.0760836	+	0.131781i	−0.901557	−	0.432660i	$-0.857575\pi$
$691$	−2.00000	+	3.46410i	−0.0760836	+	0.131781i	0.825473	+	0.564441i	$0.190908\pi$
$692$	12.0000			0.456172
$693$		0			0
$694$	40.0000			1.51838
$695$	−9.50000	+	16.4545i	−0.360356	+	0.624154i
$696$		0			0
$697$	14.0000	+	24.2487i	0.530288	+	0.918485i
$698$	13.0000	+	22.5167i	0.492057	+	0.852268i
$699$		0			0
$700$		0			0
$701$	−19.0000			−0.717620			−0.358810	−	0.933411i	$-0.616817\pi$
$701$	−19.0000			−0.717620			−0.358810	+	0.933411i	$0.616817\pi$
$702$		0			0
$703$	50.0000			1.88579
$704$	−20.0000	+	34.6410i	−0.753778	+	1.30558i
$705$		0			0
$706$	18.0000	+	31.1769i	0.677439	+	1.17336i
$707$		0			0
$708$		0			0
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	−0.995228	−	0.0975728i	$-0.968892\pi$
$709$	−11.0000	+	19.0526i	−0.413114	+	0.715534i	0.582115	+	0.813107i	$0.302225\pi$
$710$	−2.00000			−0.0750587
$711$		0			0
$712$		0			0
$713$	−27.0000	+	46.7654i	−1.01116	+	1.75138i
$714$		0			0
$715$	−10.0000	−	17.3205i	−0.373979	−	0.647750i
$716$	23.0000	+	39.8372i	0.859550	+	1.48878i
$717$		0			0
$718$	27.0000	−	46.7654i	1.00763	−	1.74527i
$719$	−27.0000			−1.00693			−0.503465	−	0.864016i	$-0.667942\pi$
$719$	−27.0000			−1.00693			−0.503465	+	0.864016i	$0.667942\pi$
$720$		0			0
$721$		0			0
$722$	6.00000	−	10.3923i	0.223297	−	0.386762i
$723$		0			0
$724$	25.0000	+	43.3013i	0.929118	+	1.60928i
$725$	−2.50000	−	4.33013i	−0.0928477	−	0.160817i
$726$		0			0
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	0.0927199	+	0.995692i	$0.470444\pi$
$727$	−22.0000	+	38.1051i	−0.815935	+	1.41324i	−0.908655	+	0.417548i	$0.862889\pi$
$728$		0			0
$729$		0			0
$730$	−16.0000			−0.592187
$731$	4.00000	−	6.92820i	0.147945	−	0.256249i
$732$		0			0
$733$	23.0000	+	39.8372i	0.849524	+	1.47142i	0.881633	+	0.471935i	$0.156444\pi$
$733$	23.0000	+	39.8372i	0.849524	+	1.47142i	−0.0321090	+	0.999484i	$0.510222\pi$
$734$	18.0000	+	31.1769i	0.664392	+	1.15076i
$735$		0			0
$736$	24.0000	−	41.5692i	0.884652	−	1.53226i
$737$	−30.0000			−1.10506
$738$		0			0
$739$	−35.0000			−1.28750			−0.643748	−	0.765238i	$-0.722621\pi$
$739$	−35.0000			−1.28750			−0.643748	+	0.765238i	$0.722621\pi$
$740$	−10.0000	+	17.3205i	−0.367607	+	0.636715i
$741$		0			0
$742$		0			0
$743$	−2.00000	−	3.46410i	−0.0733729	−	0.127086i	0.827005	−	0.562195i	$-0.190043\pi$
$743$	−2.00000	−	3.46410i	−0.0733729	−	0.127086i	−0.900378	+	0.435110i	$0.856710\pi$
$744$		0			0
$745$	1.00000	−	1.73205i	0.0366372	−	0.0634574i
$746$	−32.0000			−1.17160
$747$		0			0
$748$	−40.0000			−1.46254
$749$		0			0
$750$		0			0
$751$	16.0000	+	27.7128i	0.583848	+	1.01125i	0.995018	+	0.0996961i	$0.0317870\pi$
$751$	16.0000	+	27.7128i	0.583848	+	1.01125i	−0.411170	+	0.911559i	$0.634880\pi$
$752$	−4.00000	−	6.92820i	−0.145865	−	0.252646i
$753$		0			0
$754$	20.0000	−	34.6410i	0.728357	−	1.26155i
$755$	5.00000			0.181969
$756$		0			0
$757$	28.0000			1.01768			0.508839	−	0.860862i	$-0.330075\pi$
$757$	28.0000			1.01768			0.508839	+	0.860862i	$0.330075\pi$
$758$	4.00000	−	6.92820i	0.145287	−	0.251644i
$759$		0			0
$760$		0			0
$761$	−13.5000	−	23.3827i	−0.489375	−	0.847622i	0.510551	−	0.859848i	$-0.329442\pi$
$761$	−13.5000	−	23.3827i	−0.489375	−	0.847622i	−0.999925	+	0.0122260i	$0.996108\pi$
$762$		0			0
$763$		0			0
$764$	−2.00000			−0.0723575
$765$		0			0
$766$	−72.0000			−2.60147
$767$	−2.00000	+	3.46410i	−0.0722158	+	0.125081i
$768$		0			0
$769$	−2.50000	−	4.33013i	−0.0901523	−	0.156148i	0.817423	−	0.576038i	$-0.195402\pi$
$769$	−2.50000	−	4.33013i	−0.0901523	−	0.156148i	−0.907575	+	0.419890i	$0.862069\pi$
$770$		0			0
$771$		0			0
$772$	−26.0000	+	45.0333i	−0.935760	+	1.62078i
$773$	−18.0000			−0.647415			−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000			−0.647415			−0.323708	+	0.946157i	$0.604929\pi$
$774$		0			0
$775$	−9.00000			−0.323290
$776$		0			0
$777$		0			0
$778$	−6.00000	−	10.3923i	−0.215110	−	0.372582i
$779$	−17.5000	−	30.3109i	−0.627003	−	1.08600i
$780$		0			0
$781$	−2.50000	+	4.33013i	−0.0894570	+	0.154944i
$782$	48.0000			1.71648
$783$		0			0
$784$	28.0000			1.00000
$785$	1.00000	−	1.73205i	0.0356915	−	0.0618195i
$786$		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	$0.856046\pi$
$788$	12.0000	+	20.7846i	0.427482	+	0.740421i
$789$		0			0
$790$	−12.0000	+	20.7846i	−0.426941	+	0.739483i
$791$		0			0
$792$		0			0
$793$	−8.00000			−0.284088
$794$	−38.0000	+	65.8179i	−1.34857	+	2.33579i
$795$		0			0
$796$	−16.0000	−	27.7128i	−0.567105	−	0.982255i
$797$	2.00000	+	3.46410i	0.0708436	+	0.122705i	0.899271	−	0.437391i	$-0.144098\pi$
$797$	2.00000	+	3.46410i	0.0708436	+	0.122705i	−0.828428	+	0.560096i	$0.810764\pi$
$798$		0			0
$799$	4.00000	−	6.92820i	0.141510	−	0.245102i
$800$	8.00000			0.282843
$801$		0			0
$802$	60.0000			2.11867
$803$	−20.0000	+	34.6410i	−0.705785	+	1.22245i
$804$		0			0
$805$		0			0
$806$	−36.0000	−	62.3538i	−1.26805	−	2.19632i
$807$		0			0
$808$		0			0
$809$	−7.00000			−0.246107			−0.123053	−	0.992400i	$-0.539269\pi$
$809$	−7.00000			−0.246107			−0.123053	+	0.992400i	$0.539269\pi$
$810$		0			0
$811$	−21.0000			−0.737410			−0.368705	−	0.929547i	$-0.620199\pi$
$811$	−21.0000			−0.737410			−0.368705	+	0.929547i	$0.620199\pi$
$812$		0			0
$813$		0			0
$814$	50.0000	+	86.6025i	1.75250	+	3.03542i
$815$	4.00000	+	6.92820i	0.140114	+	0.242684i
$816$		0			0
$817$	−5.00000	+	8.66025i	−0.174928	+	0.302984i
$818$	−28.0000			−0.978997
$819$		0			0
$820$	14.0000			0.488901
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	−0.838663	−	0.544651i	$-0.816662\pi$
$821$	1.50000	−	2.59808i	0.0523504	−	0.0906735i	0.891013	+	0.453978i	$0.149995\pi$
$822$		0			0
$823$	23.0000	+	39.8372i	0.801730	+	1.38864i	0.918477	+	0.395475i	$0.129420\pi$
$823$	23.0000	+	39.8372i	0.801730	+	1.38864i	−0.116747	+	0.993162i	$0.537247\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−44.0000			−1.53003			−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000			−1.53003			−0.765015	+	0.644013i	$0.777268\pi$
$828$		0			0
$829$	27.0000			0.937749			0.468874	−	0.883265i	$-0.344660\pi$
$829$	27.0000			0.937749			0.468874	+	0.883265i	$0.344660\pi$
$830$	6.00000	−	10.3923i	0.208263	−	0.360722i
$831$		0			0
$832$	16.0000	+	27.7128i	0.554700	+	0.960769i
$833$	14.0000	+	24.2487i	0.485071	+	0.840168i
$834$		0			0
$835$	−6.00000	+	10.3923i	−0.207639	+	0.359641i
$836$	50.0000			1.72929
$837$		0			0
$838$	32.0000			1.10542
$839$	6.50000	−	11.2583i	0.224405	−	0.388681i	−0.731736	−	0.681588i	$-0.761290\pi$
$839$	6.50000	−	11.2583i	0.224405	−	0.388681i	0.956141	+	0.292908i	$0.0946228\pi$
$840$		0			0
$841$	2.00000	+	3.46410i	0.0689655	+	0.119452i
$842$	13.0000	+	22.5167i	0.448010	+	0.775975i
$843$		0			0
$844$	−11.0000	+	19.0526i	−0.378636	+	0.655816i
$845$	−3.00000			−0.103203
$846$		0			0
$847$		0			0
$848$	−16.0000	+	27.7128i	−0.549442	+	0.951662i
$849$		0			0
$850$	4.00000	+	6.92820i	0.137199	+	0.237635i
$851$	−30.0000	−	51.9615i	−1.02839	−	1.78122i
$852$		0			0
$853$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$853$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	16.0000	−	27.7128i	0.546550	−	0.946652i	−0.451958	−	0.892039i	$-0.649274\pi$
$857$	16.0000	−	27.7128i	0.546550	−	0.946652i	0.998508	−	0.0546125i	$-0.0173923\pi$
$858$		0			0
$859$	−27.5000	−	47.6314i	−0.938288	−	1.62516i	−0.768663	−	0.639654i	$-0.779078\pi$
$859$	−27.5000	−	47.6314i	−0.938288	−	1.62516i	−0.169625	−	0.985509i	$-0.554255\pi$
$860$	−2.00000	−	3.46410i	−0.0681994	−	0.118125i
$861$		0			0
$862$	−3.00000	+	5.19615i	−0.102180	+	0.176982i
$863$	44.0000			1.49778			0.748889	−	0.662696i	$-0.230588\pi$
$863$	44.0000			1.49778			0.748889	+	0.662696i	$0.230588\pi$
$864$		0			0
$865$	−6.00000			−0.204006
$866$	−26.0000	+	45.0333i	−0.883516	+	1.53029i
$867$		0			0
$868$		0			0
$869$	30.0000	+	51.9615i	1.01768	+	1.76267i
$870$		0			0
$871$	−12.0000	+	20.7846i	−0.406604	+	0.704260i
$872$		0			0
$873$		0			0
$874$	−60.0000			−2.02953
$875$		0			0
$876$		0			0
$877$	12.0000	+	20.7846i	0.405211	+	0.701846i	0.994346	−	0.106188i	$-0.0338646\pi$
$877$	12.0000	+	20.7846i	0.405211	+	0.701846i	−0.589135	+	0.808035i	$0.700531\pi$
$878$	−29.0000	−	50.2295i	−0.978703	−	1.69516i
$879$		0			0
$880$	10.0000	−	17.3205i	0.337100	−	0.583874i
$881$	25.0000			0.842271			0.421136	−	0.906998i	$-0.361632\pi$
$881$	25.0000			0.842271			0.421136	+	0.906998i	$0.361632\pi$
$882$		0			0
$883$	−52.0000			−1.74994			−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000			−1.74994			−0.874970	+	0.484178i	$0.839119\pi$
$884$	−16.0000	+	27.7128i	−0.538138	+	0.932083i
$885$		0			0
$886$	−6.00000	−	10.3923i	−0.201574	−	0.349136i
$887$	3.00000	+	5.19615i	0.100730	+	0.174470i	0.911986	−	0.410222i	$-0.134549\pi$
$887$	3.00000	+	5.19615i	0.100730	+	0.174470i	−0.811256	+	0.584692i	$0.801215\pi$
$888$		0			0
$889$		0			0
$890$	18.0000			0.603361
$891$		0			0
$892$	16.0000			0.535720
$893$	−5.00000	+	8.66025i	−0.167319	+	0.289804i
$894$		0			0
$895$	−11.5000	−	19.9186i	−0.384403	−	0.665805i
$896$		0			0
$897$		0			0
$898$	−17.0000	+	29.4449i	−0.567297	+	0.982588i
$899$	−45.0000			−1.50083
$900$		0			0
$901$	−32.0000			−1.06607
$902$	35.0000	−	60.6218i	1.16537	−	2.01848i
$903$		0			0
$904$		0			0
$905$	−12.5000	−	21.6506i	−0.415514	−	0.719691i
$906$		0			0
$907$	9.00000	−	15.5885i	0.298840	−	0.517606i	−0.677031	−	0.735955i	$-0.736734\pi$
$907$	9.00000	−	15.5885i	0.298840	−	0.517606i	0.975871	+	0.218348i	$0.0700669\pi$
$908$	−8.00000			−0.265489
$909$		0			0
$910$		0			0
$911$	−24.5000	+	42.4352i	−0.811721	+	1.40594i	0.0999373	+	0.994994i	$0.468136\pi$
$911$	−24.5000	+	42.4352i	−0.811721	+	1.40594i	−0.911658	+	0.410949i	$0.865198\pi$
$912$		0			0
$913$	−15.0000	−	25.9808i	−0.496428	−	0.859838i
$914$	−38.0000	−	65.8179i	−1.25693	−	2.17706i
$915$		0			0
$916$	−6.00000	+	10.3923i	−0.198246	+	0.343371i
$917$		0			0
$918$		0			0
$919$	−11.0000			−0.362857			−0.181428	−	0.983404i	$-0.558072\pi$
$919$	−11.0000			−0.362857			−0.181428	+	0.983404i	$0.558072\pi$
$920$		0			0
$921$		0			0
$922$	15.0000	+	25.9808i	0.493999	+	0.855631i
$923$	2.00000	+	3.46410i	0.0658308	+	0.114022i
$924$		0			0
$925$	5.00000	−	8.66025i	0.164399	−	0.284747i
$926$	12.0000			0.394344
$927$		0			0
$928$	40.0000			1.31306
$929$	0.500000	−	0.866025i	0.0164045	−	0.0284134i	−0.857707	−	0.514139i	$-0.828111\pi$
$929$	0.500000	−	0.866025i	0.0164045	−	0.0284134i	0.874111	+	0.485726i	$0.161445\pi$
$930$		0			0
$931$	−17.5000	−	30.3109i	−0.573539	−	0.993399i
$932$	6.00000	+	10.3923i	0.196537	+	0.340411i
$933$		0			0
$934$	−4.00000	+	6.92820i	−0.130884	+	0.226698i
$935$	20.0000			0.654070
$936$		0			0
$937$	42.0000			1.37208			0.686040	−	0.727564i	$-0.259347\pi$
$937$	42.0000			1.37208			0.686040	+	0.727564i	$0.259347\pi$
$938$		0			0
$939$		0			0
$940$	−2.00000	−	3.46410i	−0.0652328	−	0.112987i
$941$	23.0000	+	39.8372i	0.749779	+	1.29865i	0.947929	+	0.318483i	$0.103173\pi$
$941$	23.0000	+	39.8372i	0.749779	+	1.29865i	−0.198150	+	0.980172i	$0.563493\pi$
$942$		0			0
$943$	−21.0000	+	36.3731i	−0.683854	+	1.18447i
$944$	−4.00000			−0.130189
$945$		0			0
$946$	−20.0000			−0.650256
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	−0.681930	−	0.731417i	$-0.738859\pi$
$947$	9.00000	−	15.5885i	0.292461	−	0.506557i	0.974391	+	0.224860i	$0.0721926\pi$
$948$		0			0
$949$	16.0000	+	27.7128i	0.519382	+	0.899596i
$950$	−5.00000	−	8.66025i	−0.162221	−	0.280976i
$951$		0			0
$952$		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
$955$	1.00000			0.0323592
$956$	−16.0000	+	27.7128i	−0.517477	+	0.896296i
$957$		0			0
$958$	15.0000	+	25.9808i	0.484628	+	0.839400i
$959$		0			0
$960$		0			0
$961$	−25.0000	+	43.3013i	−0.806452	+	1.39682i
$962$	80.0000			2.57930
$963$		0			0
$964$	−22.0000			−0.708572
$965$	13.0000	−	22.5167i	0.418485	−	0.724837i
$966$		0			0
$967$	20.0000	+	34.6410i	0.643157	+	1.11398i	0.984724	+	0.174123i	$0.0557089\pi$
$967$	20.0000	+	34.6410i	0.643157	+	1.11398i	−0.341567	+	0.939857i	$0.610958\pi$
$968$		0			0
$969$		0			0
$970$	−14.0000	+	24.2487i	−0.449513	+	0.778579i
$971$	27.0000			0.866471			0.433236	−	0.901281i	$-0.357372\pi$
$971$	27.0000			0.866471			0.433236	+	0.901281i	$0.357372\pi$
$972$		0			0
$973$		0			0
$974$	8.00000	−	13.8564i	0.256337	−	0.443988i
$975$		0			0
$976$	−4.00000	−	6.92820i	−0.128037	−	0.221766i
$977$	−14.0000	−	24.2487i	−0.447900	−	0.775785i	0.550349	−	0.834934i	$-0.314494\pi$
$977$	−14.0000	−	24.2487i	−0.447900	−	0.775785i	−0.998249	+	0.0591494i	$0.981161\pi$
$978$		0			0
$979$	22.5000	−	38.9711i	0.719103	−	1.24552i
$980$	14.0000			0.447214
$981$		0			0
$982$	86.0000			2.74437
$983$	24.0000	−	41.5692i	0.765481	−	1.32585i	−0.174511	−	0.984655i	$-0.555834\pi$
$983$	24.0000	−	41.5692i	0.765481	−	1.32585i	0.939992	−	0.341197i	$-0.110832\pi$
$984$		0			0
$985$	−6.00000	−	10.3923i	−0.191176	−	0.331126i
$986$	20.0000	+	34.6410i	0.636930	+	1.10319i
$987$		0			0
$988$	20.0000	−	34.6410i	0.636285	−	1.10208i
$989$	12.0000			0.381578
$990$		0			0
$991$	−1.00000			−0.0317660			−0.0158830	−	0.999874i	$-0.505056\pi$
$991$	−1.00000			−0.0317660			−0.0158830	+	0.999874i	$0.505056\pi$
$992$	36.0000	−	62.3538i	1.14300	−	1.97974i
$993$		0			0
$994$		0			0
$995$	8.00000	+	13.8564i	0.253617	+	0.439278i
$996$		0			0
$997$	−24.0000	+	41.5692i	−0.760088	+	1.31651i	0.182717	+	0.983165i	$0.441511\pi$
$997$	−24.0000	+	41.5692i	−0.760088	+	1.31651i	−0.942805	+	0.333345i	$0.891823\pi$
$998$	−14.0000			−0.443162
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.e.g.136.1		2
3.2	odd	2		405.2.e.a.136.1		2
9.2	odd	6		405.2.a.f.1.1	yes	1
9.4	even	3	inner	405.2.e.g.271.1		2
9.5	odd	6		405.2.e.a.271.1		2
9.7	even	3		405.2.a.a.1.1	✓	1
36.7	odd	6		6480.2.a.f.1.1		1
36.11	even	6		6480.2.a.r.1.1		1
45.2	even	12		2025.2.b.b.649.2		2
45.7	odd	12		2025.2.b.a.649.1		2
45.29	odd	6		2025.2.a.a.1.1		1
45.34	even	6		2025.2.a.f.1.1		1
45.38	even	12		2025.2.b.b.649.1		2
45.43	odd	12		2025.2.b.a.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.a.1.1	✓	1	9.7	even	3
405.2.a.f.1.1	yes	1	9.2	odd	6
405.2.e.a.136.1		2	3.2	odd	2
405.2.e.a.271.1		2	9.5	odd	6
405.2.e.g.136.1		2	1.1	even	1	trivial
405.2.e.g.271.1		2	9.4	even	3	inner
2025.2.a.a.1.1		1	45.29	odd	6
2025.2.a.f.1.1		1	45.34	even	6
2025.2.b.a.649.1		2	45.7	odd	12
2025.2.b.a.649.2		2	45.43	odd	12
2025.2.b.b.649.1		2	45.38	even	12
2025.2.b.b.649.2		2	45.2	even	12
6480.2.a.f.1.1		1	36.7	odd	6
6480.2.a.r.1.1		1	36.11	even	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-1.00000 - 1.73205i) q^{4} +(0.500000 + 0.866025i) q^{5} +2.00000 q^{10} +(2.50000 - 4.33013i) q^{11} +(-2.00000 - 3.46410i) q^{13} +(2.00000 - 3.46410i) q^{16} +4.00000 q^{17} -5.00000 q^{19} +(1.00000 - 1.73205i) q^{20} +(-5.00000 - 8.66025i) q^{22} +(3.00000 + 5.19615i) q^{23} +(-0.500000 + 0.866025i) q^{25} -8.00000 q^{26} +(-2.50000 + 4.33013i) q^{29} +(4.50000 + 7.79423i) q^{31} +(-4.00000 - 6.92820i) q^{32} +(4.00000 - 6.92820i) q^{34} -10.0000 q^{37} +(-5.00000 + 8.66025i) q^{38} +(3.50000 + 6.06218i) q^{41} +(1.00000 - 1.73205i) q^{43} -10.0000 q^{44} +12.0000 q^{46} +(1.00000 - 1.73205i) q^{47} +(3.50000 + 6.06218i) q^{49} +(1.00000 + 1.73205i) q^{50} +(-4.00000 + 6.92820i) q^{52} -8.00000 q^{53} +5.00000 q^{55} +(5.00000 + 8.66025i) q^{58} +(-0.500000 - 0.866025i) q^{59} +(1.00000 - 1.73205i) q^{61} +18.0000 q^{62} -8.00000 q^{64} +(2.00000 - 3.46410i) q^{65} +(-3.00000 - 5.19615i) q^{67} +(-4.00000 - 6.92820i) q^{68} -1.00000 q^{71} -8.00000 q^{73} +(-10.0000 + 17.3205i) q^{74} +(5.00000 + 8.66025i) q^{76} +(-6.00000 + 10.3923i) q^{79} +4.00000 q^{80} +14.0000 q^{82} +(3.00000 - 5.19615i) q^{83} +(2.00000 + 3.46410i) q^{85} +(-2.00000 - 3.46410i) q^{86} +9.00000 q^{89} +(6.00000 - 10.3923i) q^{92} +(-2.00000 - 3.46410i) q^{94} +(-2.50000 - 4.33013i) q^{95} +(-7.00000 + 12.1244i) q^{97} +14.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{4} + q^{5} + 4 q^{10} + 5 q^{11} - 4 q^{13} + 4 q^{16} + 8 q^{17} - 10 q^{19} + 2 q^{20} - 10 q^{22} + 6 q^{23} - q^{25} - 16 q^{26} - 5 q^{29} + 9 q^{31} - 8 q^{32} + 8 q^{34} - 20 q^{37}+ \cdots + 28 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^4 + q^5 + 4 * q^10 + 5 * q^11 - 4 * q^13 + 4 * q^16 + 8 * q^17 - 10 * q^19 + 2 * q^20 - 10 * q^22 + 6 * q^23 - q^25 - 16 * q^26 - 5 * q^29 + 9 * q^31 - 8 * q^32 + 8 * q^34 - 20 * q^37 - 10 * q^38 + 7 * q^41 + 2 * q^43 - 20 * q^44 + 24 * q^46 + 2 * q^47 + 7 * q^49 + 2 * q^50 - 8 * q^52 - 16 * q^53 + 10 * q^55 + 10 * q^58 - q^59 + 2 * q^61 + 36 * q^62 - 16 * q^64 + 4 * q^65 - 6 * q^67 - 8 * q^68 - 2 * q^71 - 16 * q^73 - 20 * q^74 + 10 * q^76 - 12 * q^79 + 8 * q^80 + 28 * q^82 + 6 * q^83 + 4 * q^85 - 4 * q^86 + 18 * q^89 + 12 * q^92 - 4 * q^94 - 5 * q^95 - 14 * q^97 + 28 * q^98

Introduction

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