LMFDB - Embedded newform 135.2.e.b.46.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,2,Mod(46,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, 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("135.46");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	135.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:	$1.07798042729$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.954288.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 $ x^6 - x^5 - 2x^4 + 3x^3 - 6x^2 - 9x + 27
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			46.2
Root			$-1.62241 + 0.606458i$ of defining polynomial
Character	$\chi$	$=$	135.46
Dual form			135.2.e.b.91.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.285997 - 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.500000 + 0.866025i) q^{5} +(-0.714003 + 1.23669i) q^{7} +2.10083 q^{8} +O(q^{10})$	\(q+(0.285997 - 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.500000 + 0.866025i) q^{5} +(-0.714003 + 1.23669i) q^{7} +2.10083 q^{8} +0.571993 q^{10} +(1.33641 - 2.31473i) q^{11} +(-2.33641 - 4.04678i) q^{13} +(0.408405 + 0.707378i) q^{14} +(-1.07199 + 1.85675i) q^{16} -2.67282 q^{17} +4.67282 q^{19} +(-0.836412 + 1.44871i) q^{20} +(-0.764419 - 1.32401i) q^{22} +(-2.95882 - 5.12483i) q^{23} +(-0.500000 + 0.866025i) q^{25} -2.67282 q^{26} -2.38880 q^{28} +(-4.74482 + 8.21826i) q^{29} +(-3.48040 - 6.02823i) q^{31} +(2.71400 + 4.70079i) q^{32} +(-0.764419 + 1.32401i) q^{34} -1.42801 q^{35} -1.81681 q^{37} +(1.33641 - 2.31473i) q^{38} +(1.05042 + 1.81937i) q^{40} +(-0.735581 - 1.27406i) q^{41} +(-0.235581 + 0.408039i) q^{43} +4.47116 q^{44} -3.38485 q^{46} +(3.47842 - 6.02480i) q^{47} +(2.48040 + 4.29618i) q^{49} +(0.285997 + 0.495361i) q^{50} +(3.90841 - 6.76956i) q^{52} +1.14399 q^{53} +2.67282 q^{55} +(-1.50000 + 2.59808i) q^{56} +(2.71400 + 4.70079i) q^{58} +(-0.571993 - 0.990721i) q^{59} +(1.26442 - 2.19004i) q^{61} -3.98153 q^{62} -1.18319 q^{64} +(2.33641 - 4.04678i) q^{65} +(3.29523 + 5.70751i) q^{67} +(-2.23558 - 3.87214i) q^{68} +(-0.408405 + 0.707378i) q^{70} +12.8745 q^{71} -1.71203 q^{73} +(-0.519602 + 0.899976i) q^{74} +(3.90841 + 6.76956i) q^{76} +(1.90841 + 3.30545i) q^{77} +(0.143987 - 0.249392i) q^{79} -2.14399 q^{80} -0.841495 q^{82} +(-2.14201 + 3.71007i) q^{83} +(-1.33641 - 2.31473i) q^{85} +(0.134751 + 0.233396i) q^{86} +(2.80757 - 4.86286i) q^{88} +3.00000 q^{89} +6.67282 q^{91} +(4.94958 - 8.57293i) q^{92} +(-1.98963 - 3.44615i) q^{94} +(2.33641 + 4.04678i) q^{95} +(-3.91764 + 6.78555i) q^{97} +2.83754 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} - 5 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) $ 6 * q + q^2 - 5 * q^4 + 3 * q^5 - 5 * q^7 - 6 * q^8	$ 6 q + q^{2} - 5 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{13} - 9 q^{14} - 5 q^{16} + 4 q^{17} + 8 q^{19} + 5 q^{20} + 4 q^{22} + 3 q^{23} - 3 q^{25} + 4 q^{26} + 10 q^{28} - 7 q^{29} - 8 q^{31} + 17 q^{32} + 4 q^{34} - 10 q^{35} + 12 q^{37} - 2 q^{38} - 3 q^{40} - 13 q^{41} - 10 q^{43} + 44 q^{44} - 6 q^{46} + 13 q^{47} + 2 q^{49} + q^{50} + 12 q^{52} + 4 q^{53} - 4 q^{55} - 9 q^{56} + 17 q^{58} - 2 q^{59} - q^{61} - 84 q^{62} - 30 q^{64} + 4 q^{65} - 11 q^{67} - 22 q^{68} + 9 q^{70} + 20 q^{71} - 16 q^{73} - 16 q^{74} + 12 q^{76} - 2 q^{79} - 10 q^{80} - 58 q^{82} - 15 q^{83} + 2 q^{85} + 28 q^{86} + 24 q^{88} + 18 q^{89} + 20 q^{91} + 39 q^{92} + 31 q^{94} + 4 q^{95} + 18 q^{97} + 80 q^{98}+O(q^{100}) $ 6 * q + q^2 - 5 * q^4 + 3 * q^5 - 5 * q^7 - 6 * q^8 + 2 * q^10 - 2 * q^11 - 4 * q^13 - 9 * q^14 - 5 * q^16 + 4 * q^17 + 8 * q^19 + 5 * q^20 + 4 * q^22 + 3 * q^23 - 3 * q^25 + 4 * q^26 + 10 * q^28 - 7 * q^29 - 8 * q^31 + 17 * q^32 + 4 * q^34 - 10 * q^35 + 12 * q^37 - 2 * q^38 - 3 * q^40 - 13 * q^41 - 10 * q^43 + 44 * q^44 - 6 * q^46 + 13 * q^47 + 2 * q^49 + q^50 + 12 * q^52 + 4 * q^53 - 4 * q^55 - 9 * q^56 + 17 * q^58 - 2 * q^59 - q^61 - 84 * q^62 - 30 * q^64 + 4 * q^65 - 11 * q^67 - 22 * q^68 + 9 * q^70 + 20 * q^71 - 16 * q^73 - 16 * q^74 + 12 * q^76 - 2 * q^79 - 10 * q^80 - 58 * q^82 - 15 * q^83 + 2 * q^85 + 28 * q^86 + 24 * q^88 + 18 * q^89 + 20 * q^91 + 39 * q^92 + 31 * q^94 + 4 * q^95 + 18 * q^97 + 80 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$	0.285997	−	0.495361i	0.202230	−	0.350273i	−0.747017	−	0.664805i	$-0.768514\pi$
$2$	0.285997	−	0.495361i	0.202230	−	0.350273i	0.949247	+	0.314533i	$0.101848\pi$
$3$		0			0
$3$		0			0
$4$	0.836412	+	1.44871i	0.418206	+	0.724354i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	−0.714003	+	1.23669i	−0.269868	+	0.467425i	−0.968828	−	0.247736i	$-0.920313\pi$
$7$	−0.714003	+	1.23669i	−0.269868	+	0.467425i	0.698960	+	0.715161i	$0.253647\pi$
$8$	2.10083			0.742756
$9$		0			0
$10$	0.571993			0.180880
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	−0.591136	−	0.806572i	$-0.701321\pi$
$11$	1.33641	−	2.31473i	0.402943	−	0.697918i	0.994080	+	0.108653i	$0.0346538\pi$
$12$		0			0
$13$	−2.33641	−	4.04678i	−0.648004	−	1.12238i	−0.983599	−	0.180370i	$-0.942271\pi$
$13$	−2.33641	−	4.04678i	−0.648004	−	1.12238i	0.335595	−	0.942006i	$-0.391063\pi$
$14$	0.408405	+	0.707378i	0.109151	+	0.189055i
$15$		0			0
$16$	−1.07199	+	1.85675i	−0.267998	+	0.464187i
$17$	−2.67282			−0.648255			−0.324127	−	0.946013i	$-0.605071\pi$
$17$	−2.67282			−0.648255			−0.324127	+	0.946013i	$0.605071\pi$
$18$		0			0
$19$	4.67282			1.07202			0.536010	−	0.844212i	$-0.319931\pi$
$19$	4.67282			1.07202			0.536010	+	0.844212i	$0.319931\pi$
$20$	−0.836412	+	1.44871i	−0.187027	+	0.323941i
$21$		0			0
$22$	−0.764419	−	1.32401i	−0.162975	−	0.282280i
$23$	−2.95882	−	5.12483i	−0.616957	−	1.06860i	−0.990038	−	0.140802i	$-0.955032\pi$
$23$	−2.95882	−	5.12483i	−0.616957	−	1.06860i	0.373081	−	0.927799i	$-0.378301\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$	−2.67282			−0.524184
$27$		0			0
$28$	−2.38880			−0.451441
$29$	−4.74482	+	8.21826i	−0.881090	+	1.52609i	−0.0309603	+	0.999521i	$0.509857\pi$
$29$	−4.74482	+	8.21826i	−0.881090	+	1.52609i	−0.850130	+	0.526573i	$0.823477\pi$
$30$		0			0
$31$	−3.48040	−	6.02823i	−0.625098	−	1.08270i	−0.988522	−	0.151078i	$-0.951726\pi$
$31$	−3.48040	−	6.02823i	−0.625098	−	1.08270i	0.363424	−	0.931624i	$-0.381608\pi$
$32$	2.71400	+	4.70079i	0.479773	+	0.830990i
$33$		0			0
$34$	−0.764419	+	1.32401i	−0.131097	+	0.227066i
$35$	−1.42801			−0.241377
$36$		0			0
$37$	−1.81681			−0.298682			−0.149341	−	0.988786i	$-0.547715\pi$
$37$	−1.81681			−0.298682			−0.149341	+	0.988786i	$0.547715\pi$
$38$	1.33641	−	2.31473i	0.216795	−	0.375499i
$39$		0			0
$40$	1.05042	+	1.81937i	0.166085	+	0.287668i
$41$	−0.735581	−	1.27406i	−0.114879	−	0.198975i	0.802853	−	0.596177i	$-0.203315\pi$
$41$	−0.735581	−	1.27406i	−0.114879	−	0.198975i	−0.917731	+	0.397202i	$0.869981\pi$
$42$		0			0
$43$	−0.235581	+	0.408039i	−0.0359258	+	0.0622254i	−0.883429	−	0.468565i	$-0.844771\pi$
$43$	−0.235581	+	0.408039i	−0.0359258	+	0.0622254i	0.847503	+	0.530790i	$0.178105\pi$
$44$	4.47116			0.674053
$45$		0			0
$46$	−3.38485			−0.499069
$47$	3.47842	−	6.02480i	0.507380	−	0.878808i	−0.492584	−	0.870265i	$-0.663947\pi$
$47$	3.47842	−	6.02480i	0.507380	−	0.878808i	0.999964	−	0.00854274i	$-0.00271927\pi$
$48$		0			0
$49$	2.48040	+	4.29618i	0.354343	+	0.613739i
$50$	0.285997	+	0.495361i	0.0404460	+	0.0700546i
$51$		0			0
$52$	3.90841	−	6.76956i	0.541998	−	0.938769i
$53$	1.14399			0.157139			0.0785693	−	0.996909i	$-0.474965\pi$
$53$	1.14399			0.157139			0.0785693	+	0.996909i	$0.474965\pi$
$54$		0			0
$55$	2.67282			0.360403
$56$	−1.50000	+	2.59808i	−0.200446	+	0.347183i
$57$		0			0
$58$	2.71400	+	4.70079i	0.356366	+	0.617244i
$59$	−0.571993	−	0.990721i	−0.0744672	−	0.128981i	0.826387	−	0.563102i	$-0.190392\pi$
$59$	−0.571993	−	0.990721i	−0.0744672	−	0.128981i	−0.900854	+	0.434121i	$0.857059\pi$
$60$		0			0
$61$	1.26442	−	2.19004i	0.161892	−	0.280406i	−0.773655	−	0.633607i	$-0.781574\pi$
$61$	1.26442	−	2.19004i	0.161892	−	0.280406i	0.935547	+	0.353201i	$0.114907\pi$
$62$	−3.98153			−0.505655
$63$		0			0
$64$	−1.18319			−0.147899
$65$	2.33641	−	4.04678i	0.289796	−	0.501942i
$66$		0			0
$67$	3.29523	+	5.70751i	0.402577	+	0.697283i	0.994036	−	0.109051i	$-0.0347813\pi$
$67$	3.29523	+	5.70751i	0.402577	+	0.697283i	−0.591459	+	0.806335i	$0.701448\pi$
$68$	−2.23558	−	3.87214i	−0.271104	−	0.469566i
$69$		0			0
$70$	−0.408405	+	0.707378i	−0.0488137	+	0.0845479i
$71$	12.8745			1.52792			0.763960	−	0.645263i	$-0.223252\pi$
$71$	12.8745			1.52792			0.763960	+	0.645263i	$0.223252\pi$
$72$		0			0
$73$	−1.71203			−0.200378			−0.100189	−	0.994968i	$-0.531945\pi$
$73$	−1.71203			−0.200378			−0.100189	+	0.994968i	$0.531945\pi$
$74$	−0.519602	+	0.899976i	−0.0604025	+	0.104620i
$75$		0			0
$76$	3.90841	+	6.76956i	0.448325	+	0.776521i
$77$	1.90841	+	3.30545i	0.217483	+	0.376692i
$78$		0			0
$79$	0.143987	−	0.249392i	0.0161998	−	0.0280588i	−0.857812	−	0.513964i	$-0.828177\pi$
$79$	0.143987	−	0.249392i	0.0161998	−	0.0280588i	0.874012	+	0.485905i	$0.161510\pi$
$80$	−2.14399			−0.239705
$81$		0			0
$82$	−0.841495			−0.0929276
$83$	−2.14201	+	3.71007i	−0.235116	+	0.407233i	−0.959306	−	0.282367i	$-0.908880\pi$
$83$	−2.14201	+	3.71007i	−0.235116	+	0.407233i	0.724190	+	0.689600i	$0.242214\pi$
$84$		0			0
$85$	−1.33641	−	2.31473i	−0.144954	−	0.251068i
$86$	0.134751	+	0.233396i	0.0145306	+	0.0251677i
$87$		0			0
$88$	2.80757	−	4.86286i	0.299288	−	0.518383i
$89$	3.00000			0.317999			0.159000	−	0.987279i	$-0.449173\pi$
$89$	3.00000			0.317999			0.159000	+	0.987279i	$0.449173\pi$
$90$		0			0
$91$	6.67282			0.699502
$92$	4.94958	−	8.57293i	0.516030	−	0.893790i
$93$		0			0
$94$	−1.98963	−	3.44615i	−0.205215	−	0.355443i
$95$	2.33641	+	4.04678i	0.239711	+	0.415191i
$96$		0			0
$97$	−3.91764	+	6.78555i	−0.397776	+	0.688968i	−0.993451	−	0.114257i	$-0.963551\pi$
$97$	−3.91764	+	6.78555i	−0.397776	+	0.688968i	0.595675	+	0.803225i	$0.296885\pi$
$98$	2.83754			0.286635
$99$		0			0
$100$	−1.67282			−0.167282
$101$	−2.10083	+	3.63875i	−0.209040	+	0.362069i	−0.951413	−	0.307919i	$-0.900367\pi$
$101$	−2.10083	+	3.63875i	−0.209040	+	0.362069i	0.742372	+	0.669988i	$0.233701\pi$
$102$		0			0
$103$	0.908405	+	1.57340i	0.0895078	+	0.155032i	0.907303	−	0.420477i	$-0.138137\pi$
$103$	0.908405	+	1.57340i	0.0895078	+	0.155032i	−0.817795	+	0.575509i	$0.804804\pi$
$104$	−4.90841	−	8.50161i	−0.481309	−	0.833651i
$105$		0			0
$106$	0.327176	−	0.566686i	0.0317782	−	0.0550414i
$107$	−11.9176			−1.15212			−0.576061	−	0.817407i	$-0.695411\pi$
$107$	−11.9176			−1.15212			−0.576061	+	0.817407i	$0.695411\pi$
$108$		0			0
$109$	−16.6521			−1.59498			−0.797491	−	0.603331i	$-0.793840\pi$
$109$	−16.6521			−1.59498			−0.797491	+	0.603331i	$0.793840\pi$
$110$	0.764419	−	1.32401i	0.0728845	−	0.126240i
$111$		0			0
$112$	−1.53081	−	2.65145i	−0.144648	−	0.250538i
$113$	10.0616	+	17.4272i	0.946518	+	1.63942i	0.752682	+	0.658384i	$0.228760\pi$
$113$	10.0616	+	17.4272i	0.946518	+	1.63942i	0.193836	+	0.981034i	$0.437907\pi$
$114$		0			0
$115$	2.95882	−	5.12483i	0.275911	−	0.477893i
$116$	−15.8745			−1.47391
$117$		0			0
$118$	−0.654353			−0.0602380
$119$	1.90841	−	3.30545i	0.174943	−	0.303011i
$120$		0			0
$121$	1.92801	+	3.33941i	0.175273	+	0.303582i
$122$	−0.723239	−	1.25269i	−0.0654790	−	0.113413i
$123$		0			0
$124$	5.82209	−	10.0842i	0.522839	−	0.905584i
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	2.18714			0.194078			0.0970388	−	0.995281i	$-0.469063\pi$
$127$	2.18714			0.194078			0.0970388	+	0.995281i	$0.469063\pi$
$128$	−5.76640	+	9.98769i	−0.509682	+	0.882795i
$129$		0			0
$130$	−1.33641	−	2.31473i	−0.117211	−	0.203016i
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	0.704692	−	0.709514i	$-0.251085\pi$
$131$	−3.00000	−	5.19615i	−0.262111	−	0.453990i	−0.966803	+	0.255524i	$0.917752\pi$
$132$		0			0
$133$	−3.33641	+	5.77883i	−0.289304	+	0.501089i
$134$	3.76970			0.325653
$135$		0			0
$136$	−5.61515			−0.481495
$137$	−5.10083	+	8.83490i	−0.435793	+	0.754816i	−0.997360	−	0.0726153i	$-0.976865\pi$
$137$	−5.10083	+	8.83490i	−0.435793	+	0.754816i	0.561567	+	0.827431i	$0.310199\pi$
$138$		0			0
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	0.645021	−	0.764165i	$-0.276849\pi$
$139$	−4.00000	−	6.92820i	−0.339276	−	0.587643i	−0.984297	+	0.176522i	$0.943515\pi$
$140$	−1.19440	−	2.06876i	−0.100945	−	0.174843i
$141$		0			0
$142$	3.68206	−	6.37751i	0.308992	−	0.535189i
$143$	−12.4896			−1.04444
$144$		0			0
$145$	−9.48963			−0.788071
$146$	−0.489634	+	0.848071i	−0.0405224	+	0.0701868i
$147$		0			0
$148$	−1.51960	−	2.63203i	−0.124910	−	0.216351i
$149$	−10.0381	−	17.3865i	−0.822351	−	1.42435i	−0.903927	−	0.427687i	$-0.859329\pi$
$149$	−10.0381	−	17.3865i	−0.822351	−	1.42435i	0.0815762	−	0.996667i	$-0.474005\pi$
$150$		0			0
$151$	−1.51960	+	2.63203i	−0.123663	+	0.214191i	−0.921210	−	0.389066i	$-0.872798\pi$
$151$	−1.51960	+	2.63203i	−0.123663	+	0.214191i	0.797546	+	0.603258i	$0.206131\pi$
$152$	9.81681			0.796248
$153$		0			0
$154$	2.18319			0.175926
$155$	3.48040	−	6.02823i	0.279552	−	0.484199i
$156$		0			0
$157$	−0.100830	−	0.174643i	−0.00804714	−	0.0139381i	0.861974	−	0.506953i	$-0.169228\pi$
$157$	−0.100830	−	0.174643i	−0.00804714	−	0.0139381i	−0.870021	+	0.493015i	$0.835895\pi$
$158$	−0.0823593	−	0.142651i	−0.00655216	−	0.0113487i
$159$		0			0
$160$	−2.71400	+	4.70079i	−0.214561	+	0.371630i
$161$	8.45043			0.665987
$162$		0			0
$163$	17.8168			1.39552			0.697760	−	0.716331i	$-0.254180\pi$
$163$	17.8168			1.39552			0.697760	+	0.716331i	$0.254180\pi$
$164$	1.23050	−	2.13129i	0.0960858	−	0.166425i
$165$		0			0
$166$	1.22522	+	2.12214i	0.0950952	+	0.164710i
$167$	7.05042	+	12.2117i	0.545578	+	0.944968i	0.998570	+	0.0534538i	$0.0170230\pi$
$167$	7.05042	+	12.2117i	0.545578	+	0.944968i	−0.452993	+	0.891514i	$0.649644\pi$
$168$		0			0
$169$	−4.41764	+	7.65158i	−0.339819	+	0.588583i
$170$	−1.52884			−0.117256
$171$		0			0
$172$	−0.788172			−0.0600976
$173$	2.18319	−	3.78140i	0.165985	−	0.287494i	−0.771020	−	0.636811i	$-0.780253\pi$
$173$	2.18319	−	3.78140i	0.165985	−	0.287494i	0.937005	+	0.349317i	$0.113586\pi$
$174$		0			0
$175$	−0.714003	−	1.23669i	−0.0539736	−	0.0934850i
$176$	2.86525	+	4.96276i	0.215976	+	0.374082i
$177$		0			0
$178$	0.857990	−	1.48608i	0.0643091	−	0.111387i
$179$	15.1625			1.13330			0.566648	−	0.823960i	$-0.308240\pi$
$179$	15.1625			1.13330			0.566648	+	0.823960i	$0.308240\pi$
$180$		0			0
$181$	3.20166			0.237978			0.118989	−	0.992896i	$-0.462035\pi$
$181$	3.20166			0.237978			0.118989	+	0.992896i	$0.462035\pi$
$182$	1.90841	−	3.30545i	0.141460	−	0.245017i
$183$		0			0
$184$	−6.21598	−	10.7664i	−0.458248	−	0.793709i
$185$	−0.908405	−	1.57340i	−0.0667873	−	0.115679i
$186$		0			0
$187$	−3.57199	+	6.18687i	−0.261210	+	0.452429i
$188$	11.6376			0.848757
$189$		0			0
$190$	2.67282			0.193907
$191$	−1.41877	+	2.45738i	−0.102659	+	0.177810i	−0.912779	−	0.408453i	$-0.866068\pi$
$191$	−1.41877	+	2.45738i	−0.102659	+	0.177810i	0.810121	+	0.586263i	$0.199402\pi$
$192$		0			0
$193$	9.39409	+	16.2710i	0.676201	+	1.17121i	0.976116	+	0.217249i	$0.0697083\pi$
$193$	9.39409	+	16.2710i	0.676201	+	1.17121i	−0.299915	+	0.953966i	$0.596958\pi$
$194$	2.24086	+	3.88129i	0.160885	+	0.278660i
$195$		0			0
$196$	−4.14927	+	7.18675i	−0.296376	+	0.513339i
$197$	−5.83528			−0.415747			−0.207873	−	0.978156i	$-0.566654\pi$
$197$	−5.83528			−0.415747			−0.207873	+	0.978156i	$0.566654\pi$
$198$		0			0
$199$	13.0761			0.926943			0.463472	−	0.886112i	$-0.346604\pi$
$199$	13.0761			0.926943			0.463472	+	0.886112i	$0.346604\pi$
$200$	−1.05042	+	1.81937i	−0.0742756	+	0.128649i
$201$		0			0
$202$	1.20166	+	2.08134i	0.0845486	+	0.146442i
$203$	−6.77563	−	11.7357i	−0.475556	−	0.823687i
$204$		0			0
$205$	0.735581	−	1.27406i	0.0513752	−	0.0889845i
$206$	1.03920			0.0724047
$207$		0			0
$208$	10.0185			0.694656
$209$	6.24482	−	10.8163i	0.431963	−	0.748182i
$210$		0			0
$211$	−4.19243	−	7.26149i	−0.288618	−	0.499902i	0.684862	−	0.728673i	$-0.259863\pi$
$211$	−4.19243	−	7.26149i	−0.288618	−	0.499902i	−0.973480	+	0.228771i	$0.926529\pi$
$212$	0.956844	+	1.65730i	0.0657163	+	0.113824i
$213$		0			0
$214$	−3.40841	+	5.90353i	−0.232994	+	0.403557i
$215$	−0.471163			−0.0321330
$216$		0			0
$217$	9.94006			0.674776
$218$	−4.76244	+	8.24879i	−0.322553	+	0.558679i
$219$		0			0
$220$	2.23558	+	3.87214i	0.150723	+	0.261060i
$221$	6.24482	+	10.8163i	0.420072	+	0.727586i
$222$		0			0
$223$	−4.58321	+	7.93834i	−0.306914	+	0.531591i	−0.977686	−	0.210073i	$-0.932630\pi$
$223$	−4.58321	+	7.93834i	−0.306914	+	0.531591i	0.670772	+	0.741664i	$0.265963\pi$
$224$	−7.75123			−0.517901
$225$		0			0
$226$	11.5104			0.765658
$227$	−1.33641	+	2.31473i	−0.0887008	+	0.153634i	−0.906962	−	0.421212i	$-0.861605\pi$
$227$	−1.33641	+	2.31473i	−0.0887008	+	0.153634i	0.818261	+	0.574846i	$0.194938\pi$
$228$		0			0
$229$	−1.27365	−	2.20603i	−0.0841654	−	0.145779i	0.820870	−	0.571115i	$-0.193489\pi$
$229$	−1.27365	−	2.20603i	−0.0841654	−	0.145779i	−0.905035	+	0.425336i	$0.860156\pi$
$230$	−1.69243	−	2.93137i	−0.111595	−	0.193289i
$231$		0			0
$232$	−9.96806	+	17.2652i	−0.654435	+	1.13351i
$233$	6.22013			0.407494			0.203747	−	0.979024i	$-0.434688\pi$
$233$	6.22013			0.407494			0.203747	+	0.979024i	$0.434688\pi$
$234$		0			0
$235$	6.95684			0.453814
$236$	0.956844	−	1.65730i	0.0622852	−	0.107881i
$237$		0			0
$238$	−1.09159	−	1.89070i	−0.0707576	−	0.122556i
$239$	4.06163	+	7.03494i	0.262725	+	0.455053i	0.966965	−	0.254909i	$-0.0820455\pi$
$239$	4.06163	+	7.03494i	0.262725	+	0.455053i	−0.704240	+	0.709962i	$0.748712\pi$
$240$		0			0
$241$	13.1821	−	22.8320i	0.849131	−	1.47074i	−0.0328536	−	0.999460i	$-0.510460\pi$
$241$	13.1821	−	22.8320i	0.849131	−	1.47074i	0.881985	−	0.471278i	$-0.156207\pi$
$242$	2.20561			0.141782
$243$		0			0
$244$	4.23030			0.270817
$245$	−2.48040	+	4.29618i	−0.158467	+	0.274473i
$246$		0			0
$247$	−10.9176	−	18.9099i	−0.694673	−	1.20321i
$248$	−7.31173	−	12.6643i	−0.464295	−	0.804183i
$249$		0			0
$250$	−0.285997	+	0.495361i	−0.0180880	+	0.0313294i
$251$	0.549569			0.0346885			0.0173443	−	0.999850i	$-0.494479\pi$
$251$	0.549569			0.0346885			0.0173443	+	0.999850i	$0.494479\pi$
$252$		0			0
$253$	−15.8168			−0.994394
$254$	0.625515	−	1.08342i	0.0392483	−	0.0679801i
$255$		0			0
$256$	2.11515	+	3.66355i	0.132197	+	0.228972i
$257$	−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
$259$	1.29721	−	2.24683i	0.0806046	−	0.139611i
$260$	7.81681			0.484778
$261$		0			0
$262$	−3.43196			−0.212027
$263$	5.94761	−	10.3016i	0.366745	−	0.635221i	−0.622309	−	0.782771i	$-0.713805\pi$
$263$	5.94761	−	10.3016i	0.366745	−	0.635221i	0.989055	+	0.147550i	$0.0471387\pi$
$264$		0			0
$265$	0.571993	+	0.990721i	0.0351373	+	0.0608595i
$266$	1.90841	+	3.30545i	0.117012	+	0.202670i
$267$		0			0
$268$	−5.51234	+	9.54766i	−0.336720	+	0.583216i
$269$	−28.5737			−1.74217			−0.871084	−	0.491134i	$-0.836583\pi$
$269$	−28.5737			−1.74217			−0.871084	+	0.491134i	$0.836583\pi$
$270$		0			0
$271$	−23.3641			−1.41927			−0.709635	−	0.704570i	$-0.751140\pi$
$271$	−23.3641			−1.41927			−0.709635	+	0.704570i	$0.751140\pi$
$272$	2.86525	−	4.96276i	0.173731	−	0.300911i
$273$		0			0
$274$	2.91764	+	5.05350i	0.176261	+	0.305293i
$275$	1.33641	+	2.31473i	0.0805887	+	0.139584i
$276$		0			0
$277$	7.53807	−	13.0563i	0.452919	−	0.784479i	−0.545647	−	0.838015i	$-0.683716\pi$
$277$	7.53807	−	13.0563i	0.452919	−	0.784479i	0.998566	+	0.0535366i	$0.0170494\pi$
$278$	−4.57595			−0.274447
$279$		0			0
$280$	−3.00000			−0.179284
$281$	3.32605	−	5.76088i	0.198415	−	0.343665i	−0.749599	−	0.661892i	$-0.769754\pi$
$281$	3.32605	−	5.76088i	0.198415	−	0.343665i	0.948015	+	0.318226i	$0.103087\pi$
$282$		0			0
$283$	−13.4485	−	23.2934i	−0.799428	−	1.38465i	−0.919989	−	0.391943i	$-0.871803\pi$
$283$	−13.4485	−	23.2934i	−0.799428	−	1.38465i	0.120562	−	0.992706i	$-0.461530\pi$
$284$	10.7684	+	18.6514i	0.638985	+	1.10675i
$285$		0			0
$286$	−3.57199	+	6.18687i	−0.211216	+	0.365838i
$287$	2.10083			0.124008
$288$		0			0
$289$	−9.85601			−0.579765
$290$	−2.71400	+	4.70079i	−0.159372	+	0.276040i
$291$		0			0
$292$	−1.43196	−	2.48023i	−0.0837991	−	0.145144i
$293$	6.19243	+	10.7256i	0.361765	+	0.626596i	0.988251	−	0.152837i	$-0.0488408\pi$
$293$	6.19243	+	10.7256i	0.361765	+	0.626596i	−0.626486	+	0.779433i	$0.715507\pi$
$294$		0			0
$295$	0.571993	−	0.990721i	0.0333027	−	0.0576820i
$296$	−3.81681			−0.221848
$297$		0			0
$298$	−11.4834			−0.665217
$299$	−13.8260	+	23.9474i	−0.799581	+	1.38491i
$300$		0			0
$301$	−0.336412	−	0.582682i	−0.0193905	−	0.0335853i
$302$	0.869202	+	1.50550i	0.0500169	+	0.0866319i
$303$		0			0
$304$	−5.00924	+	8.67625i	−0.287299	+	0.497617i
$305$	2.52884			0.144801
$306$		0			0
$307$	−2.49359			−0.142317			−0.0711583	−	0.997465i	$-0.522670\pi$
$307$	−2.49359			−0.142317			−0.0711583	+	0.997465i	$0.522670\pi$
$308$	−3.19243	+	5.52944i	−0.181905	+	0.315069i
$309$		0			0
$310$	−1.99076	−	3.44811i	−0.113068	−	0.195839i
$311$	−12.1101	−	20.9752i	−0.686699	−	1.18940i	−0.972900	−	0.231228i	$-0.925726\pi$
$311$	−12.1101	−	20.9752i	−0.686699	−	1.18940i	0.286201	−	0.958170i	$-0.407608\pi$
$312$		0			0
$313$	17.5420	−	30.3837i	0.991534	−	1.71739i	0.383315	−	0.923618i	$-0.374782\pi$
$313$	17.5420	−	30.3837i	0.991534	−	1.71739i	0.608219	−	0.793770i	$-0.291884\pi$
$314$	−0.115349			−0.00650950
$315$		0			0
$316$	0.481728			0.0270993
$317$	−5.23558	+	9.06829i	−0.294060	+	0.509326i	−0.974766	−	0.223231i	$-0.928340\pi$
$317$	−5.23558	+	9.06829i	−0.294060	+	0.509326i	0.680706	+	0.732557i	$0.261673\pi$
$318$		0			0
$319$	12.6821	+	21.9660i	0.710059	+	1.22986i
$320$	−0.591595	−	1.02467i	−0.0330712	−	0.0572809i
$321$		0			0
$322$	2.41679	−	4.18601i	0.134683	−	0.233277i
$323$	−12.4896			−0.694942
$324$		0			0
$325$	4.67282			0.259202
$326$	5.09555	−	8.82575i	0.282216	−	0.488813i
$327$		0			0
$328$	−1.54533	−	2.67659i	−0.0853267	−	0.147790i
$329$	4.96721	+	8.60346i	0.273851	+	0.474324i
$330$		0			0
$331$	−8.38880	+	14.5298i	−0.461090	+	0.798632i	−0.999016	−	0.0443606i	$-0.985875\pi$
$331$	−8.38880	+	14.5298i	−0.461090	+	0.798632i	0.537925	+	0.842993i	$0.319208\pi$
$332$	−7.16641			−0.393308
$333$		0			0
$334$	8.06558			0.441329
$335$	−3.29523	+	5.70751i	−0.180038	+	0.311835i
$336$		0			0
$337$	13.5905	+	23.5394i	0.740320	+	1.28227i	0.952350	+	0.305008i	$0.0986592\pi$
$337$	13.5905	+	23.5394i	0.740320	+	1.28227i	−0.212030	+	0.977263i	$0.568007\pi$
$338$	2.52686	+	4.37665i	0.137443	+	0.238058i
$339$		0			0
$340$	2.23558	−	3.87214i	0.121241	−	0.209996i
$341$	−18.6050			−1.00752
$342$		0			0
$343$	−17.0801			−0.922239
$344$	−0.494917	+	0.857221i	−0.0266841	+	0.0462182i
$345$		0			0
$346$	−1.24877	−	2.16293i	−0.0671343	−	0.116280i
$347$	−11.7829	−	20.4086i	−0.632539	−	1.09559i	−0.987031	−	0.160530i	$-0.948680\pi$
$347$	−11.7829	−	20.4086i	−0.632539	−	1.09559i	0.354492	−	0.935059i	$-0.384654\pi$
$348$		0			0
$349$	5.35601	−	9.27689i	0.286701	−	0.496580i	−0.686319	−	0.727300i	$-0.740775\pi$
$349$	5.35601	−	9.27689i	0.286701	−	0.496580i	0.973020	+	0.230720i	$0.0741081\pi$
$350$	−0.816810			−0.0436603
$351$		0			0
$352$	14.5081			0.773285
$353$	13.6336	−	23.6141i	0.725644	−	1.25685i	−0.233064	−	0.972461i	$-0.574875\pi$
$353$	13.6336	−	23.6141i	0.725644	−	1.25685i	0.958708	−	0.284392i	$-0.0917916\pi$
$354$		0			0
$355$	6.43724	+	11.1496i	0.341653	+	0.591761i
$356$	2.50924	+	4.34612i	0.132989	+	0.230344i
$357$		0			0
$358$	4.33641	−	7.51089i	0.229186	−	0.396963i
$359$	10.6807			0.563707			0.281854	−	0.959457i	$-0.409051\pi$
$359$	10.6807			0.563707			0.281854	+	0.959457i	$0.409051\pi$
$360$		0			0
$361$	2.83528			0.149225
$362$	0.915664	−	1.58598i	0.0481262	−	0.0833571i
$363$		0			0
$364$	5.58123	+	9.66697i	0.292536	+	0.506687i
$365$	−0.856013	−	1.48266i	−0.0448058	−	0.0776059i
$366$		0			0
$367$	4.23558	−	7.33624i	0.221096	−	0.382949i	−0.734045	−	0.679100i	$-0.762370\pi$
$367$	4.23558	−	7.33624i	0.221096	−	0.382949i	0.955141	+	0.296152i	$0.0957034\pi$
$368$	12.6873			0.661373
$369$		0			0
$370$	−1.03920			−0.0540256
$371$	−0.816810	+	1.41476i	−0.0424067	+	0.0734505i
$372$		0			0
$373$	−5.06163	−	8.76700i	−0.262081	−	0.453938i	0.704714	−	0.709492i	$-0.251075\pi$
$373$	−5.06163	−	8.76700i	−0.262081	−	0.453938i	−0.966795	+	0.255554i	$0.917742\pi$
$374$	2.04316	+	3.53885i	0.105649	+	0.182990i
$375$		0			0
$376$	7.30757	−	12.6571i	0.376859	−	0.652740i
$377$	44.3434			2.28380
$378$		0			0
$379$	11.9216			0.612371			0.306186	−	0.951972i	$-0.400947\pi$
$379$	11.9216			0.612371			0.306186	+	0.951972i	$0.400947\pi$
$380$	−3.90841	+	6.76956i	−0.200497	+	0.347271i
$381$		0			0
$382$	0.811528	+	1.40561i	0.0415214	+	0.0719171i
$383$	4.90841	+	8.50161i	0.250808	+	0.434412i	0.963748	−	0.266813i	$-0.0859705\pi$
$383$	4.90841	+	8.50161i	0.250808	+	0.434412i	−0.712941	+	0.701224i	$0.752637\pi$
$384$		0			0
$385$	−1.90841	+	3.30545i	−0.0972613	+	0.168462i
$386$	10.7467			0.546993
$387$		0			0
$388$	−13.1070			−0.665409
$389$	4.61007	−	7.98487i	0.233740	−	0.404849i	−0.725166	−	0.688574i	$-0.758237\pi$
$389$	4.61007	−	7.98487i	0.233740	−	0.404849i	0.958906	+	0.283725i	$0.0915703\pi$
$390$		0			0
$391$	7.90841	+	13.6978i	0.399945	+	0.692725i
$392$	5.21090	+	9.02554i	0.263190	+	0.455858i
$393$		0			0
$394$	−1.66887	+	2.89057i	−0.0840765	+	0.145625i
$395$	0.287973			0.0144895
$396$		0			0
$397$	−22.9793			−1.15330			−0.576648	−	0.816993i	$-0.695640\pi$
$397$	−22.9793			−1.15330			−0.576648	+	0.816993i	$0.695640\pi$
$398$	3.73973	−	6.47741i	0.187456	−	0.324683i
$399$		0			0
$400$	−1.07199	−	1.85675i	−0.0535997	−	0.0928373i
$401$	−5.53279	−	9.58307i	−0.276294	−	0.478556i	0.694167	−	0.719814i	$-0.255773\pi$
$401$	−5.53279	−	9.58307i	−0.276294	−	0.478556i	−0.970461	+	0.241259i	$0.922440\pi$
$402$		0			0
$403$	−16.2633	+	28.1688i	−0.810132	+	1.40319i
$404$	−7.02864			−0.349688
$405$		0			0
$406$	−7.75123			−0.384687
$407$	−2.42801	+	4.20543i	−0.120352	+	0.208455i
$408$		0			0
$409$	8.81681	+	15.2712i	0.435963	+	0.755110i	0.997374	−	0.0724270i	$-0.0230744\pi$
$409$	8.81681	+	15.2712i	0.435963	+	0.755110i	−0.561411	+	0.827537i	$0.689741\pi$
$410$	−0.420748	−	0.728756i	−0.0207792	−	0.0359907i
$411$		0			0
$412$	−1.51960	+	2.63203i	−0.0748654	+	0.129671i
$413$	1.63362			0.0803852
$414$		0			0
$415$	−4.28402			−0.210294
$416$	12.6821	−	21.9660i	0.621789	−	1.07697i
$417$		0			0
$418$	−3.57199	−	6.18687i	−0.174712	−	0.302610i
$419$	18.5173	+	32.0730i	0.904631	+	1.56687i	0.821411	+	0.570336i	$0.193187\pi$
$419$	18.5173	+	32.0730i	0.904631	+	1.56687i	0.0832199	+	0.996531i	$0.473480\pi$
$420$		0			0
$421$	−2.52884	+	4.38007i	−0.123248	+	0.213472i	−0.921047	−	0.389452i	$-0.872664\pi$
$421$	−2.52884	+	4.38007i	−0.123248	+	0.213472i	0.797799	+	0.602924i	$0.205998\pi$
$422$	−4.79608			−0.233469
$423$		0			0
$424$	2.40332			0.116716
$425$	1.33641	−	2.31473i	0.0648255	−	0.112281i
$426$		0			0
$427$	1.80560	+	3.12739i	0.0873790	+	0.151345i
$428$	−9.96806	−	17.2652i	−0.481824	−	0.834544i
$429$		0			0
$430$	−0.134751	+	0.233396i	−0.00649827	+	0.0112553i
$431$	5.23030			0.251935			0.125967	−	0.992034i	$-0.459797\pi$
$431$	5.23030			0.251935			0.125967	+	0.992034i	$0.459797\pi$
$432$		0			0
$433$	34.3434			1.65044			0.825219	−	0.564813i	$-0.191052\pi$
$433$	34.3434			1.65044			0.825219	+	0.564813i	$0.191052\pi$
$434$	2.84283	−	4.92392i	0.136460	−	0.236356i
$435$		0			0
$436$	−13.9280	−	24.1240i	−0.667031	−	1.15533i
$437$	−13.8260	−	23.9474i	−0.661389	−	1.14556i
$438$		0			0
$439$	9.77365	−	16.9285i	0.466471	−	0.807952i	−0.532796	−	0.846244i	$-0.678859\pi$
$439$	9.77365	−	16.9285i	0.466471	−	0.807952i	0.999267	+	0.0382924i	$0.0121918\pi$
$440$	5.61515			0.267692
$441$		0			0
$442$	7.14399			0.339805
$443$	5.25208	−	9.09686i	0.249534	−	0.432205i	−0.713863	−	0.700286i	$-0.753056\pi$
$443$	5.25208	−	9.09686i	0.249534	−	0.432205i	0.963396	+	0.268081i	$0.0863893\pi$
$444$		0			0
$445$	1.50000	+	2.59808i	0.0711068	+	0.123161i
$446$	2.62156	+	4.54068i	0.124135	+	0.215007i
$447$		0			0
$448$	0.844801	−	1.46324i	0.0399131	−	0.0691315i
$449$	−22.8560			−1.07864			−0.539321	−	0.842100i	$-0.681319\pi$
$449$	−22.8560			−1.07864			−0.539321	+	0.842100i	$0.681319\pi$
$450$		0			0
$451$	−3.93216			−0.185158
$452$	−16.8313	+	29.1527i	−0.791679	+	1.37123i
$453$		0			0
$454$	0.764419	+	1.32401i	0.0358759	+	0.0621390i
$455$	3.33641	+	5.77883i	0.156413	+	0.270916i
$456$		0			0
$457$	7.01319	−	12.1472i	0.328063	−	0.568222i	−0.654064	−	0.756439i	$-0.726937\pi$
$457$	7.01319	−	12.1472i	0.328063	−	0.568222i	0.982127	+	0.188217i	$0.0602708\pi$
$458$	−1.45704			−0.0680832
$459$		0			0
$460$	9.89917			0.461551
$461$	−12.0513	+	20.8734i	−0.561283	+	0.972171i	0.436102	+	0.899897i	$0.356359\pi$
$461$	−12.0513	+	20.8734i	−0.561283	+	0.972171i	−0.997385	+	0.0722736i	$0.976975\pi$
$462$		0			0
$463$	16.9700	+	29.3930i	0.788664	+	1.36601i	0.926785	+	0.375591i	$0.122560\pi$
$463$	16.9700	+	29.3930i	0.788664	+	1.36601i	−0.138121	+	0.990415i	$0.544106\pi$
$464$	−10.1728	−	17.6198i	−0.472261	−	0.817981i
$465$		0			0
$466$	1.77894	−	3.08121i	0.0824077	−	0.142734i
$467$	27.3720			1.26663			0.633313	−	0.773896i	$-0.281695\pi$
$467$	27.3720			1.26663			0.633313	+	0.773896i	$0.281695\pi$
$468$		0			0
$469$	−9.41123			−0.434570
$470$	1.98963	−	3.44615i	0.0917750	−	0.158959i
$471$		0			0
$472$	−1.20166	−	2.08134i	−0.0553109	−	0.0958013i
$473$	0.629668	+	1.09062i	0.0289521	+	0.0501466i
$474$		0			0
$475$	−2.33641	+	4.04678i	−0.107202	+	0.185679i
$476$	6.38485			0.292649
$477$		0			0
$478$	4.64645			0.212524
$479$	2.61515	−	4.52957i	0.119489	−	0.206961i	−0.800076	−	0.599898i	$-0.795208\pi$
$479$	2.61515	−	4.52957i	0.119489	−	0.206961i	0.919565	+	0.392937i	$0.128541\pi$
$480$		0			0
$481$	4.24482	+	7.35224i	0.193547	+	0.335233i
$482$	−7.54005	−	13.0597i	−0.343440	−	0.594855i
$483$		0			0
$484$	−3.22522	+	5.58624i	−0.146601	+	0.253920i
$485$	−7.83528			−0.355782
$486$		0			0
$487$	−24.0185			−1.08838			−0.544190	−	0.838962i	$-0.683163\pi$
$487$	−24.0185			−1.08838			−0.544190	+	0.838962i	$0.683163\pi$
$488$	2.65633	−	4.60090i	0.120246	−	0.208273i
$489$		0			0
$490$	1.41877	+	2.45738i	0.0640935	+	0.111013i
$491$	7.38880	+	12.7978i	0.333452	+	0.577556i	0.983186	−	0.182606i	$-0.0584531\pi$
$491$	7.38880	+	12.7978i	0.333452	+	0.577556i	−0.649734	+	0.760161i	$0.725120\pi$
$492$		0			0
$493$	12.6821	−	21.9660i	0.571171	−	0.989298i
$494$	−12.4896			−0.561935
$495$		0			0
$496$	14.9239			0.670101
$497$	−9.19243	+	15.9217i	−0.412337	+	0.714188i
$498$		0			0
$499$	−12.4280	−	21.5259i	−0.556354	−	0.963633i	−0.997797	−	0.0663440i	$-0.978867\pi$
$499$	−12.4280	−	21.5259i	−0.556354	−	0.963633i	0.441443	−	0.897289i	$-0.354467\pi$
$500$	−0.836412	−	1.44871i	−0.0374055	−	0.0647882i
$501$		0			0
$502$	0.157175	−	0.272235i	0.00701506	−	0.0121504i
$503$	−38.9154			−1.73515			−0.867576	−	0.497305i	$-0.834323\pi$
$503$	−38.9154			−1.73515			−0.867576	+	0.497305i	$0.834323\pi$
$504$		0			0
$505$	−4.20166			−0.186971
$506$	−4.52355	+	7.83503i	−0.201097	+	0.348309i
$507$		0			0
$508$	1.82935	+	3.16853i	0.0811644	+	0.140581i
$509$	1.01037	+	1.75001i	0.0447837	+	0.0775676i	0.887548	−	0.460715i	$-0.152407\pi$
$509$	1.01037	+	1.75001i	0.0447837	+	0.0775676i	−0.842765	+	0.538282i	$0.819073\pi$
$510$		0			0
$511$	1.22239	−	2.11725i	0.0540755	−	0.0936615i
$512$	−20.6459			−0.912428
$513$		0			0
$514$	−10.2959			−0.454132
$515$	−0.908405	+	1.57340i	−0.0400291	+	0.0693325i
$516$		0			0
$517$	−9.29721	−	16.1032i	−0.408891	−	0.708220i
$518$	−0.741995	−	1.28517i	−0.0326014	−	0.0564672i
$519$		0			0
$520$	4.90841	−	8.50161i	0.215248	−	0.372820i
$521$	23.0290			1.00892			0.504460	−	0.863435i	$-0.331692\pi$
$521$	23.0290			1.00892			0.504460	+	0.863435i	$0.331692\pi$
$522$		0			0
$523$	−41.1170			−1.79792			−0.898961	−	0.438028i	$-0.855677\pi$
$523$	−41.1170			−1.79792			−0.898961	+	0.438028i	$0.855677\pi$
$524$	5.01847	−	8.69225i	0.219233	−	0.379723i
$525$		0			0
$526$	−3.40199	−	5.89242i	−0.148334	−	0.256922i
$527$	9.30249	+	16.1124i	0.405223	+	0.701867i
$528$		0			0
$529$	−6.00924	+	10.4083i	−0.261271	+	0.452535i
$530$	0.654353			0.0284233
$531$		0			0
$532$	−11.1625			−0.483954
$533$	−3.43724	+	5.95348i	−0.148883	+	0.257874i
$534$		0			0
$535$	−5.95882	−	10.3210i	−0.257622	−	0.446215i
$536$	6.92272	+	11.9905i	0.299016	+	0.517911i
$537$		0			0
$538$	−8.17198	+	14.1543i	−0.352319	+	0.610234i
$539$	13.2593			0.571120
$540$		0			0
$541$	5.20957			0.223977			0.111988	−	0.993710i	$-0.464278\pi$
$541$	5.20957			0.223977			0.111988	+	0.993710i	$0.464278\pi$
$542$	−6.68206	+	11.5737i	−0.287019	+	0.497132i
$543$		0			0
$544$	−7.25405	−	12.5644i	−0.311015	−	0.538694i
$545$	−8.32605	−	14.4211i	−0.356649	−	0.617734i
$546$		0			0
$547$	−20.0204	+	34.6764i	−0.856013	+	1.48266i	0.0196900	+	0.999806i	$0.493732\pi$
$547$	−20.0204	+	34.6764i	−0.856013	+	1.48266i	−0.875702	+	0.482851i	$0.839601\pi$
$548$	−17.0656			−0.729005
$549$		0			0
$550$	1.52884			0.0651898
$551$	−22.1717	+	38.4025i	−0.944546	+	1.63600i
$552$		0			0
$553$	0.205614	+	0.356133i	0.00874359	+	0.0151443i
$554$	−4.31173	−	7.46813i	−0.183188	−	0.317290i
$555$		0			0
$556$	6.69129	−	11.5897i	0.283774	−	0.491511i
$557$	14.4033			0.610288			0.305144	−	0.952306i	$-0.401295\pi$
$557$	14.4033			0.610288			0.305144	+	0.952306i	$0.401295\pi$
$558$		0			0
$559$	2.20166			0.0931203
$560$	1.53081	−	2.65145i	0.0646887	−	0.112044i
$561$		0			0
$562$	−1.90248	−	3.29518i	−0.0802511	−	0.138999i
$563$	14.6840	+	25.4335i	0.618858	+	1.07189i	0.989694	+	0.143196i	$0.0457378\pi$
$563$	14.6840	+	25.4335i	0.618858	+	1.07189i	−0.370836	+	0.928698i	$0.620929\pi$
$564$		0			0
$565$	−10.0616	+	17.4272i	−0.423296	+	0.733170i
$566$	−15.3849			−0.646674
$567$		0			0
$568$	27.0471			1.13487
$569$	23.4033	−	40.5357i	0.981118	−	1.69935i	0.323062	−	0.946378i	$-0.395288\pi$
$569$	23.4033	−	40.5357i	0.981118	−	1.69935i	0.658056	−	0.752969i	$-0.271379\pi$
$570$		0			0
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	0.577301	−	0.816532i	$-0.304106\pi$
$571$	−10.0000	−	17.3205i	−0.418487	−	0.724841i	−0.995788	+	0.0916910i	$0.970773\pi$
$572$	−10.4465	−	18.0938i	−0.436789	−	0.756541i
$573$		0			0
$574$	0.600830	−	1.04067i	0.0250782	−	0.0434367i
$575$	5.91764			0.246783
$576$		0			0
$577$	28.2386			1.17559			0.587794	−	0.809010i	$-0.299996\pi$
$577$	28.2386			1.17559			0.587794	+	0.809010i	$0.299996\pi$
$578$	−2.81879	+	4.88228i	−0.117246	+	0.203076i
$579$		0			0
$580$	−7.93724	−	13.7477i	−0.329576	−	0.570842i
$581$	−3.05880	−	5.29801i	−0.126901	−	0.219798i
$582$		0			0
$583$	1.52884	−	2.64802i	0.0633180	−	0.109670i
$584$	−3.59668			−0.148832
$585$		0			0
$586$	7.08405			0.292639
$587$	9.04118	−	15.6598i	0.373169	−	0.646348i	−0.616882	−	0.787056i	$-0.711604\pi$
$587$	9.04118	−	15.6598i	0.373169	−	0.646348i	0.990051	+	0.140707i	$0.0449377\pi$
$588$		0			0
$589$	−16.2633	−	28.1688i	−0.670117	−	1.16068i
$590$	−0.327176	−	0.566686i	−0.0134696	−	0.0233301i
$591$		0			0
$592$	1.94761	−	3.37336i	0.0800462	−	0.138644i
$593$	7.73840			0.317778			0.158889	−	0.987296i	$-0.449209\pi$
$593$	7.73840			0.317778			0.158889	+	0.987296i	$0.449209\pi$
$594$		0			0
$595$	3.81681			0.156474
$596$	16.7919	−	29.0845i	0.687824	−	1.19135i
$597$		0			0
$598$	7.90841	+	13.6978i	0.323399	+	0.560143i
$599$	−13.9608	−	24.1808i	−0.570423	−	0.988001i	−0.996522	−	0.0833249i	$-0.973446\pi$
$599$	−13.9608	−	24.1808i	−0.570423	−	0.988001i	0.426100	−	0.904676i	$-0.359887\pi$
$600$		0			0
$601$	−19.2201	+	33.2902i	−0.784006	+	1.35794i	0.145586	+	0.989346i	$0.453493\pi$
$601$	−19.2201	+	33.2902i	−0.784006	+	1.35794i	−0.929591	+	0.368592i	$0.879840\pi$
$602$	−0.384851			−0.0156853
$603$		0			0
$604$	−5.08405			−0.206867
$605$	−1.92801	+	3.33941i	−0.0783846	+	0.135766i
$606$		0			0
$607$	0.319917	+	0.554113i	0.0129850	+	0.0224907i	0.872445	−	0.488712i	$-0.162533\pi$
$607$	0.319917	+	0.554113i	0.0129850	+	0.0224907i	−0.859460	+	0.511203i	$0.829200\pi$
$608$	12.6821	+	21.9660i	0.514325	+	0.890838i
$609$		0			0
$610$	0.723239	−	1.25269i	0.0292831	−	0.0507198i
$611$	−32.5081			−1.31514
$612$		0			0
$613$	42.7467			1.72652			0.863262	−	0.504757i	$-0.168418\pi$
$613$	42.7467			1.72652			0.863262	+	0.504757i	$0.168418\pi$
$614$	−0.713157	+	1.23522i	−0.0287807	+	0.0498496i
$615$		0			0
$616$	4.00924	+	6.94420i	0.161537	+	0.279790i
$617$	10.5513	+	18.2753i	0.424778	+	0.735737i	0.996400	−	0.0847805i	$-0.0270189\pi$
$617$	10.5513	+	18.2753i	0.424778	+	0.735737i	−0.571622	+	0.820517i	$0.693686\pi$
$618$		0			0
$619$	6.82605	−	11.8231i	0.274362	−	0.475209i	−0.695612	−	0.718418i	$-0.744867\pi$
$619$	6.82605	−	11.8231i	0.274362	−	0.475209i	0.969974	+	0.243209i	$0.0782000\pi$
$620$	11.6442			0.467642
$621$		0			0
$622$	−13.8538			−0.555485
$623$	−2.14201	+	3.71007i	−0.0858178	+	0.148641i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$	−10.0339	−	17.3793i	−0.401036	−	0.694615i
$627$		0			0
$628$	0.168672	−	0.292148i	0.00673073	−	0.0116580i
$629$	4.85601			0.193622
$630$		0			0
$631$	33.2593			1.32403			0.662017	−	0.749489i	$-0.269701\pi$
$631$	33.2593			1.32403			0.662017	+	0.749489i	$0.269701\pi$
$632$	0.302491	−	0.523930i	0.0120325	−	0.0208408i
$633$		0			0
$634$	2.99472	+	5.18700i	0.118935	+	0.206002i
$635$	1.09357	+	1.89412i	0.0433971	+	0.0751659i
$636$		0			0
$637$	11.5905	−	20.0753i	0.459231	−	0.795411i
$638$	14.5081			0.574381
$639$		0			0
$640$	−11.5328			−0.455874
$641$	13.1429	−	22.7641i	0.519112	−	0.899128i	−0.480642	−	0.876917i	$-0.659596\pi$
$641$	13.1429	−	22.7641i	0.519112	−	0.899128i	0.999753	−	0.0222106i	$-0.00707044\pi$
$642$		0			0
$643$	−10.2913	−	17.8250i	−0.405848	−	0.702950i	0.588571	−	0.808445i	$-0.299691\pi$
$643$	−10.2913	−	17.8250i	−0.405848	−	0.702950i	−0.994420	+	0.105495i	$0.966357\pi$
$644$	7.06804	+	12.2422i	0.278520	+	0.482410i
$645$		0			0
$646$	−3.57199	+	6.18687i	−0.140538	+	0.243419i
$647$	−23.2527			−0.914159			−0.457079	−	0.889426i	$-0.651104\pi$
$647$	−23.2527			−0.914159			−0.457079	+	0.889426i	$0.651104\pi$
$648$		0			0
$649$	−3.05767			−0.120024
$650$	1.33641	−	2.31473i	0.0524184	−	0.0907913i
$651$		0			0
$652$	14.9022	+	25.8114i	0.583615	+	1.01085i
$653$	9.37957	+	16.2459i	0.367051	+	0.635751i	0.989103	−	0.147225i	$-0.0470342\pi$
$653$	9.37957	+	16.2459i	0.367051	+	0.635751i	−0.622052	+	0.782976i	$0.713701\pi$
$654$		0			0
$655$	3.00000	−	5.19615i	0.117220	−	0.203030i
$656$	3.15415			0.123149
$657$		0			0
$658$	5.68242			0.221524
$659$	−0.140034	+	0.242545i	−0.00545494	+	0.00944823i	−0.868740	−	0.495268i	$-0.835070\pi$
$659$	−0.140034	+	0.242545i	−0.00545494	+	0.00944823i	0.863285	+	0.504717i	$0.168403\pi$
$660$		0			0
$661$	19.8930	+	34.4556i	0.773746	+	1.34017i	0.935496	+	0.353336i	$0.114953\pi$
$661$	19.8930	+	34.4556i	0.773746	+	1.34017i	−0.161750	+	0.986832i	$0.551714\pi$
$662$	4.79834	+	8.31097i	0.186493	+	0.323015i
$663$		0			0
$664$	−4.50000	+	7.79423i	−0.174634	+	0.302475i
$665$	−6.67282			−0.258761
$666$		0			0
$667$	56.1562			2.17438
$668$	−11.7941	+	20.4280i	−0.456327	+	0.790382i
$669$		0			0
$670$	1.88485	+	3.26466i	0.0728181	+	0.126125i
$671$	−3.37957	−	5.85358i	−0.130467	−	0.225975i
$672$		0			0
$673$	−16.7644	+	29.0368i	−0.646221	+	1.11929i	0.337797	+	0.941219i	$0.390318\pi$
$673$	−16.7644	+	29.0368i	−0.646221	+	1.11929i	−0.984018	+	0.178068i	$0.943015\pi$
$674$	15.5473			0.598860
$675$		0			0
$676$	−14.7799			−0.568456
$677$	13.7437	−	23.8048i	0.528213	−	0.914891i	−0.471246	−	0.882002i	$-0.656196\pi$
$677$	13.7437	−	23.8048i	0.528213	−	0.914891i	0.999459	−	0.0328897i	$-0.0104710\pi$
$678$		0			0
$679$	−5.59442	−	9.68981i	−0.214694	−	0.371861i
$680$	−2.80757	−	4.86286i	−0.107666	−	0.186482i
$681$		0			0
$682$	−5.32096	+	9.21618i	−0.203750	+	0.352906i
$683$	34.5865			1.32342			0.661708	−	0.749762i	$-0.269832\pi$
$683$	34.5865			1.32342			0.661708	+	0.749762i	$0.269832\pi$
$684$		0			0
$685$	−10.2017			−0.389785
$686$	−4.88485	+	8.46081i	−0.186504	+	0.323035i
$687$		0			0
$688$	−0.505083	−	0.874830i	−0.0192561	−	0.0333526i
$689$	−2.67282	−	4.62947i	−0.101826	−	0.176369i
$690$		0			0
$691$	−20.3641	+	35.2717i	−0.774688	+	1.34180i	0.160282	+	0.987071i	$0.448760\pi$
$691$	−20.3641	+	35.2717i	−0.774688	+	1.34180i	−0.934970	+	0.354727i	$0.884574\pi$
$692$	7.30418			0.277663
$693$		0			0
$694$	−13.4795			−0.511674
$695$	4.00000	−	6.92820i	0.151729	−	0.262802i
$696$		0			0
$697$	1.96608	+	3.40535i	0.0744706	+	0.128987i
$698$	−3.06360	−	5.30632i	−0.115959	−	0.200847i
$699$		0			0
$700$	1.19440	−	2.06876i	0.0451441	−	0.0781919i
$701$	19.4712			0.735416			0.367708	−	0.929941i	$-0.380143\pi$
$701$	19.4712			0.735416			0.367708	+	0.929941i	$0.380143\pi$
$702$		0			0
$703$	−8.48963			−0.320193
$704$	−1.58123	+	2.73877i	−0.0595948	+	0.103221i
$705$		0			0
$706$	−7.79834	−	13.5071i	−0.293494	−	0.508347i
$707$	−3.00000	−	5.19615i	−0.112827	−	0.195421i
$708$		0			0
$709$	−7.54316	+	13.0651i	−0.283289	+	0.490671i	−0.972193	−	0.234182i	$-0.924759\pi$
$709$	−7.54316	+	13.0651i	−0.283289	+	0.490671i	0.688904	+	0.724853i	$0.258092\pi$
$710$	7.36412			0.276370
$711$		0			0
$712$	6.30249			0.236196
$713$	−20.5957	+	35.6729i	−0.771317	+	1.33596i
$714$		0			0
$715$	−6.24482	−	10.8163i	−0.233543	−	0.404508i
$716$	12.6821	+	21.9660i	0.473951	+	0.820907i
$717$		0			0
$718$	3.05465	−	5.29081i	0.113999	−	0.197451i
$719$	3.43196			0.127990			0.0639952	−	0.997950i	$-0.479616\pi$
$719$	3.43196			0.127990			0.0639952	+	0.997950i	$0.479616\pi$
$720$		0			0
$721$	−2.59442			−0.0966211
$722$	0.810881	−	1.40449i	0.0301779	−	0.0522696i
$723$		0			0
$724$	2.67791	+	4.63827i	0.0995236	+	0.172380i
$725$	−4.74482	−	8.21826i	−0.176218	−	0.305219i
$726$		0			0
$727$	17.8857	−	30.9789i	0.663344	−	1.14895i	−0.316388	−	0.948630i	$-0.602470\pi$
$727$	17.8857	−	30.9789i	0.663344	−	1.14895i	0.979732	−	0.200315i	$-0.0641966\pi$
$728$	14.0185			0.519559
$729$		0			0
$730$	−0.979268			−0.0362443
$731$	0.629668	−	1.09062i	0.0232891	−	0.0403379i
$732$		0			0
$733$	11.0000	+	19.0526i	0.406294	+	0.703722i	0.994471	−	0.105010i	$-0.0334875\pi$
$733$	11.0000	+	19.0526i	0.406294	+	0.703722i	−0.588177	+	0.808732i	$0.700154\pi$
$734$	−2.42272	−	4.19628i	−0.0894244	−	0.154888i
$735$		0			0
$736$	16.0605	−	27.8176i	0.591998	−	1.02537i
$737$	17.6151			0.648862
$738$		0			0
$739$	6.08631			0.223889			0.111944	−	0.993714i	$-0.464292\pi$
$739$	6.08631			0.223889			0.111944	+	0.993714i	$0.464292\pi$
$740$	1.51960	−	2.63203i	0.0558617	−	0.0967552i
$741$		0			0
$742$	0.467210	+	0.809231i	0.0171518	+	0.0297078i
$743$	−12.7509	−	22.0853i	−0.467787	−	0.810231i	0.531536	−	0.847036i	$-0.321615\pi$
$743$	−12.7509	−	22.0853i	−0.467787	−	0.810231i	−0.999322	+	0.0368054i	$0.988282\pi$
$744$		0			0
$745$	10.0381	−	17.3865i	0.367767	−	0.636990i
$746$	−5.79043			−0.212003
$747$		0			0
$748$	−11.9506			−0.436958
$749$	8.50924	−	14.7384i	0.310921	−	0.538530i
$750$		0			0
$751$	9.19638	+	15.9286i	0.335581	+	0.581243i	0.983596	−	0.180384i	$-0.0577342\pi$
$751$	9.19638	+	15.9286i	0.335581	+	0.581243i	−0.648016	+	0.761627i	$0.724401\pi$
$752$	7.45769	+	12.9171i	0.271954	+	0.471038i
$753$		0			0
$754$	12.6821	−	21.9660i	0.461853	−	0.799953i
$755$	−3.03920			−0.110608
$756$		0			0
$757$	−41.8986			−1.52283			−0.761415	−	0.648264i	$-0.775495\pi$
$757$	−41.8986			−1.52283			−0.761415	+	0.648264i	$0.775495\pi$
$758$	3.40954	−	5.90549i	0.123840	−	0.214497i
$759$		0			0
$760$	4.90841	+	8.50161i	0.178047	+	0.308386i
$761$	−3.98568	−	6.90340i	−0.144481	−	0.250248i	0.784698	−	0.619878i	$-0.212818\pi$
$761$	−3.98568	−	6.90340i	−0.144481	−	0.250248i	−0.929179	+	0.369630i	$0.879485\pi$
$762$		0			0
$763$	11.8896	−	20.5935i	0.430434	−	0.745534i
$764$	−4.74671			−0.171730
$765$		0			0
$766$	5.61515			0.202884
$767$	−2.67282	+	4.62947i	−0.0965101	+	0.167160i
$768$		0			0
$769$	−3.01432	−	5.22095i	−0.108699	−	0.188272i	0.806544	−	0.591174i	$-0.201335\pi$
$769$	−3.01432	−	5.22095i	−0.108699	−	0.188272i	−0.915244	+	0.402901i	$0.868002\pi$
$770$	1.09159	+	1.89070i	0.0393383	+	0.0681360i
$771$		0			0
$772$	−15.7147	+	27.2186i	−0.565583	+	0.979618i
$773$	−44.4033			−1.59708			−0.798538	−	0.601944i	$-0.794393\pi$
$773$	−44.4033			−1.59708			−0.798538	+	0.601944i	$0.794393\pi$
$774$		0			0
$775$	6.96080			0.250039
$776$	−8.23030	+	14.2553i	−0.295451	+	0.511735i
$777$		0			0
$778$	−2.63693	−	4.56729i	−0.0945384	−	0.163745i
$779$	−3.43724	−	5.95348i	−0.123152	−	0.213305i
$780$		0			0
$781$	17.2056	−	29.8010i	0.615665	−	1.06636i
$782$	9.04711			0.323524
$783$		0			0
$784$	−10.6359			−0.379853
$785$	0.100830	−	0.174643i	0.00359879	−	0.00623329i
$786$		0			0
$787$	−8.41877	−	14.5817i	−0.300097	−	0.519783i	0.676061	−	0.736846i	$-0.263686\pi$
$787$	−8.41877	−	14.5817i	−0.300097	−	0.519783i	−0.976158	+	0.217063i	$0.930352\pi$
$788$	−4.88070	−	8.45362i	−0.173868	−	0.301148i
$789$		0			0
$790$	0.0823593	−	0.142651i	0.00293021	−	0.00507528i
$791$	−28.7361			−1.02174
$792$		0			0
$793$	−11.8168			−0.419627
$794$	−6.57199	+	11.3830i	−0.233231	+	0.403968i
$795$		0			0
$796$	10.9370	+	18.9435i	0.387653	+	0.671435i
$797$	−16.6860	−	28.9010i	−0.591049	−	1.02373i	−0.994091	−	0.108545i	$-0.965381\pi$
$797$	−16.6860	−	28.9010i	−0.591049	−	1.02373i	0.403043	−	0.915181i	$-0.367953\pi$
$798$		0			0
$799$	−9.29721	+	16.1032i	−0.328912	+	0.569692i
$800$	−5.42801			−0.191909
$801$		0			0
$802$	−6.32944			−0.223500
$803$	−2.28797	+	3.96289i	−0.0807408	+	0.139847i
$804$		0			0
$805$	4.22522	+	7.31829i	0.148919	+	0.257936i
$806$	9.30249	+	16.1124i	0.327666	+	0.567535i
$807$		0			0
$808$	−4.41349	+	7.64439i	−0.155266	+	0.268929i
$809$	−29.1809			−1.02595			−0.512973	−	0.858404i	$-0.671456\pi$
$809$	−29.1809			−1.02595			−0.512973	+	0.858404i	$0.671456\pi$
$810$		0			0
$811$	−15.5552			−0.546217			−0.273109	−	0.961983i	$-0.588052\pi$
$811$	−15.5552			−0.546217			−0.273109	+	0.961983i	$0.588052\pi$
$812$	11.3344	−	19.6318i	0.397761	−	0.688942i
$813$		0			0
$814$	1.38880	+	2.40548i	0.0486775	+	0.0843120i
$815$	8.90841	+	15.4298i	0.312048	+	0.540483i
$816$		0			0
$817$	−1.10083	+	1.90669i	−0.0385132	+	0.0667068i
$818$	10.0863			0.352660
$819$		0			0
$820$	2.46100			0.0859417
$821$	−11.8588	+	20.5401i	−0.413876	+	0.716855i	−0.995310	−	0.0967393i	$-0.969159\pi$
$821$	−11.8588	+	20.5401i	−0.413876	+	0.716855i	0.581434	+	0.813594i	$0.302492\pi$
$822$		0			0
$823$	−9.68404	−	16.7732i	−0.337564	−	0.584678i	0.646410	−	0.762990i	$-0.276270\pi$
$823$	−9.68404	−	16.7732i	−0.337564	−	0.584678i	−0.983974	+	0.178312i	$0.942936\pi$
$824$	1.90841	+	3.30545i	0.0664824	+	0.115151i
$825$		0			0
$826$	0.467210	−	0.809231i	0.0162563	−	0.0281568i
$827$	−52.2241			−1.81601			−0.908005	−	0.418960i	$-0.862395\pi$
$827$	−52.2241			−1.81601			−0.908005	+	0.418960i	$0.862395\pi$
$828$		0			0
$829$	−26.6442			−0.925391			−0.462695	−	0.886517i	$-0.653118\pi$
$829$	−26.6442			−0.925391			−0.462695	+	0.886517i	$0.653118\pi$
$830$	−1.22522	+	2.12214i	−0.0425278	+	0.0736604i
$831$		0			0
$832$	2.76442	+	4.78811i	0.0958390	+	0.165998i
$833$	−6.62967	−	11.4829i	−0.229704	−	0.397860i
$834$		0			0
$835$	−7.05042	+	12.2117i	−0.243990	+	0.422603i
$836$	20.8930			0.722598
$837$		0			0
$838$	21.1836			0.731775
$839$	−12.5196	+	21.6846i	−0.432225	+	0.748635i	−0.997065	−	0.0765655i	$-0.975605\pi$
$839$	−12.5196	+	21.6846i	−0.432225	+	0.748635i	0.564840	+	0.825201i	$0.308938\pi$
$840$		0			0
$841$	−30.5266	−	52.8736i	−1.05264	−	1.82323i
$842$	1.44648	+	2.50537i	0.0498489	+	0.0863409i
$843$		0			0
$844$	7.01319	−	12.1472i	0.241404	−	0.418124i
$845$	−8.83528			−0.303943
$846$		0			0
$847$	−5.50641			−0.189203
$848$	−1.22635	+	2.12409i	−0.0421129	+	0.0729417i
$849$		0			0
$850$	−0.764419	−	1.32401i	−0.0262193	−	0.0454132i
$851$	5.37562	+	9.31084i	0.184274	+	0.319171i
$852$		0			0
$853$	5.43724	−	9.41758i	0.186168	−	0.322452i	−0.757802	−	0.652485i	$-0.773727\pi$
$853$	5.43724	−	9.41758i	0.186168	−	0.322452i	0.943969	+	0.330033i	$0.107060\pi$
$854$	2.06558			0.0706827
$855$		0			0
$856$	−25.0369			−0.855745
$857$	−9.29721	+	16.1032i	−0.317587	+	0.550076i	−0.979984	−	0.199077i	$-0.936206\pi$
$857$	−9.29721	+	16.1032i	−0.317587	+	0.550076i	0.662397	+	0.749153i	$0.269539\pi$
$858$		0			0
$859$	2.33246	+	4.03994i	0.0795825	+	0.137841i	0.903070	−	0.429494i	$-0.141308\pi$
$859$	2.33246	+	4.03994i	0.0795825	+	0.137841i	−0.823487	+	0.567335i	$0.807975\pi$
$860$	−0.394086	−	0.682577i	−0.0134382	−	0.0232757i
$861$		0			0
$862$	1.49585	−	2.59088i	0.0509488	−	0.0882459i
$863$	28.0594			0.955152			0.477576	−	0.878590i	$-0.341516\pi$
$863$	28.0594			0.955152			0.477576	+	0.878590i	$0.341516\pi$
$864$		0			0
$865$	4.36638			0.148461
$866$	9.82209	−	17.0124i	0.333768	−	0.578104i
$867$		0			0
$868$	8.31399	+	14.4002i	0.282195	+	0.488776i
$869$	−0.384851	−	0.666581i	−0.0130552	−	0.0226122i
$870$		0			0
$871$	15.3980	−	26.6702i	0.521743	−	0.903685i
$872$	−34.9832			−1.18468
$873$		0			0
$874$	−15.8168			−0.535012
$875$	0.714003	−	1.23669i	0.0241377	−	0.0418078i
$876$		0			0
$877$	−17.3342	−	30.0236i	−0.585333	−	1.01383i	−0.994834	−	0.101516i	$-0.967631\pi$
$877$	−17.3342	−	30.0236i	−0.585333	−	1.01383i	0.409501	−	0.912310i	$-0.365703\pi$
$878$	−5.59046	−	9.68297i	−0.188669	−	0.326784i
$879$		0			0
$880$	−2.86525	+	4.96276i	−0.0965875	+	0.167294i
$881$	5.29854			0.178512			0.0892561	−	0.996009i	$-0.471551\pi$
$881$	5.29854			0.178512			0.0892561	+	0.996009i	$0.471551\pi$
$882$		0			0
$883$	−10.3025			−0.346706			−0.173353	−	0.984860i	$-0.555460\pi$
$883$	−10.3025			−0.346706			−0.173353	+	0.984860i	$0.555460\pi$
$884$	−10.4465	+	18.0938i	−0.351353	+	0.608561i
$885$		0			0
$886$	−3.00415	−	5.20334i	−0.100926	−	0.174810i
$887$	20.7252	+	35.8971i	0.695885	+	1.20531i	0.969882	+	0.243577i	$0.0783208\pi$
$887$	20.7252	+	35.8971i	0.695885	+	1.20531i	−0.273997	+	0.961731i	$0.588346\pi$
$888$		0			0
$889$	−1.56163	+	2.70482i	−0.0523753	+	0.0907167i
$890$	1.71598			0.0575198
$891$		0			0
$892$	−15.3338			−0.513413
$893$	16.2541	−	28.1528i	0.543921	−	0.942099i
$894$		0			0
$895$	7.58123	+	13.1311i	0.253413	+	0.438923i
$896$	−8.23445	−	14.2625i	−0.275094	−	0.476476i
$897$		0			0
$898$	−6.53674	+	11.3220i	−0.218134	+	0.377819i
$899$	66.0554			2.20307
$900$		0			0
$901$	−3.05767			−0.101866
$902$	−1.12458	+	1.94784i	−0.0374446	+	0.0648559i
$903$		0			0
$904$	21.1378	+	36.6117i	0.703032	+	1.21769i
$905$	1.60083	+	2.77272i	0.0532134	+	0.0921683i
$906$		0			0
$907$	8.39606	−	14.5424i	0.278787	−	0.482873i	−0.692297	−	0.721613i	$-0.743401\pi$
$907$	8.39606	−	14.5424i	0.278787	−	0.482873i	0.971083	+	0.238740i	$0.0767344\pi$
$908$	−4.47116			−0.148381
$909$		0			0
$910$	3.81681			0.126526
$911$	18.6768	−	32.3491i	0.618789	−	1.07177i	−0.370918	−	0.928666i	$-0.620957\pi$
$911$	18.6768	−	32.3491i	0.618789	−	1.07177i	0.989707	−	0.143109i	$-0.0457098\pi$
$912$		0			0
$913$	5.72522	+	9.91636i	0.189477	+	0.328184i
$914$	−4.01150	−	6.94812i	−0.132689	−	0.229823i
$915$		0			0
$916$	2.13060	−	3.69031i	0.0703970	−	0.121931i
$917$	8.56804			0.282942
$918$		0			0
$919$	37.1316			1.22486			0.612429	−	0.790526i	$-0.290193\pi$
$919$	37.1316			1.22486			0.612429	+	0.790526i	$0.290193\pi$
$920$	6.21598	−	10.7664i	0.204935	−	0.354957i
$921$		0			0
$922$	6.89324	+	11.9394i	0.227017	+	0.393205i
$923$	−30.0801	−	52.1003i	−0.990098	−	1.71490i
$924$		0			0
$925$	0.908405	−	1.57340i	0.0298682	−	0.0517332i
$926$	19.4135			0.637967
$927$		0			0
$928$	−51.5098			−1.69089
$929$	−23.9977	+	41.5653i	−0.787340	+	1.36371i	0.140251	+	0.990116i	$0.455209\pi$
$929$	−23.9977	+	41.5653i	−0.787340	+	1.36371i	−0.927591	+	0.373597i	$0.878124\pi$
$930$		0			0
$931$	11.5905	+	20.0753i	0.379862	+	0.657941i
$932$	5.20259	+	9.01115i	0.170417	+	0.295170i
$933$		0			0
$934$	7.82831	−	13.5590i	0.256150	−	0.443665i
$935$	−7.14399			−0.233633
$936$		0			0
$937$	22.0079			0.718967			0.359483	−	0.933151i	$-0.382953\pi$
$937$	22.0079			0.718967			0.359483	+	0.933151i	$0.382953\pi$
$938$	−2.69158	+	4.66195i	−0.0878832	+	0.152218i
$939$		0			0
$940$	5.81879	+	10.0784i	0.189788	+	0.328722i
$941$	20.2921	+	35.1470i	0.661504	+	1.14576i	0.980220	+	0.197909i	$0.0634150\pi$
$941$	20.2921	+	35.1470i	0.661504	+	1.14576i	−0.318716	+	0.947850i	$0.603252\pi$
$942$		0			0
$943$	−4.35291	+	7.53946i	−0.141750	+	0.245518i
$944$	2.45269			0.0798283
$945$		0			0
$946$	0.720331			0.0234200
$947$	15.1160	−	26.1817i	0.491204	−	0.850790i	−0.508745	−	0.860917i	$-0.669890\pi$
$947$	15.1160	−	26.1817i	0.491204	−	0.850790i	0.999949	+	0.0101273i	$0.00322368\pi$
$948$		0			0
$949$	4.00000	+	6.92820i	0.129845	+	0.224899i
$950$	1.33641	+	2.31473i	0.0433589	+	0.0750999i
$951$		0			0
$952$	4.00924	−	6.94420i	0.129940	−	0.225063i
$953$	−2.50811			−0.0812455			−0.0406227	−	0.999175i	$-0.512934\pi$
$953$	−2.50811			−0.0812455			−0.0406227	+	0.999175i	$0.512934\pi$
$954$		0			0
$955$	−2.83754			−0.0918207
$956$	−6.79439	+	11.7682i	−0.219746	+	0.380612i
$957$		0			0
$958$	−1.49585	−	2.59088i	−0.0483286	−	0.0837077i
$959$	−7.28402	−	12.6163i	−0.235213	−	0.407401i
$960$		0			0
$961$	−8.72635	+	15.1145i	−0.281495	+	0.487564i
$962$	4.85601			0.156564
$963$		0			0
$964$	44.1025			1.42045
$965$	−9.39409	+	16.2710i	−0.302406	+	0.523783i
$966$		0			0
$967$	10.8765	+	18.8386i	0.349763	+	0.605808i	0.986207	−	0.165515i	$-0.0529286\pi$
$967$	10.8765	+	18.8386i	0.349763	+	0.605808i	−0.636444	+	0.771323i	$0.719595\pi$
$968$	4.05042	+	7.01552i	0.130185	+	0.225488i
$969$		0			0
$970$	−2.24086	+	3.88129i	−0.0719498	+	0.124621i
$971$	−22.7512			−0.730122			−0.365061	−	0.930984i	$-0.618952\pi$
$971$	−22.7512			−0.730122			−0.365061	+	0.930984i	$0.618952\pi$
$972$		0			0
$973$	11.4241			0.366238
$974$	−6.86920	+	11.8978i	−0.220103	+	0.381230i
$975$		0			0
$976$	2.71090	+	4.69541i	0.0867737	+	0.150296i
$977$	19.6389	+	34.0156i	0.628304	+	1.08825i	0.987892	+	0.155143i	$0.0495840\pi$
$977$	19.6389	+	34.0156i	0.628304	+	1.08825i	−0.359588	+	0.933111i	$0.617083\pi$
$978$		0			0
$979$	4.00924	−	6.94420i	0.128136	−	0.221938i
$980$	−8.29854			−0.265087
$981$		0			0
$982$	8.45269			0.269736
$983$	16.8949	−	29.2629i	0.538865	−	0.933341i	−0.460101	−	0.887867i	$-0.652187\pi$
$983$	16.8949	−	29.2629i	0.538865	−	0.933341i	0.998966	−	0.0454743i	$-0.0144799\pi$
$984$		0			0
$985$	−2.91764	−	5.05350i	−0.0929638	−	0.161018i
$986$	−7.25405	−	12.5644i	−0.231016	−	0.400132i
$987$		0			0
$988$	18.2633	−	31.6329i	0.581033	−	1.00638i
$989$	2.78817			0.0886587
$990$		0			0
$991$	−23.7983			−0.755979			−0.377990	−	0.925810i	$-0.623384\pi$
$991$	−23.7983			−0.755979			−0.377990	+	0.925810i	$0.623384\pi$
$992$	18.8916	−	32.7213i	0.599810	−	1.03890i
$993$		0			0
$994$	5.25801	+	9.10713i	0.166774	+	0.288861i
$995$	6.53807	+	11.3243i	0.207271	+	0.359004i
$996$		0			0
$997$	−25.8392	+	44.7549i	−0.818337	+	1.41740i	0.0885702	+	0.996070i	$0.471770\pi$
$997$	−25.8392	+	44.7549i	−0.818337	+	1.41740i	−0.906907	+	0.421331i	$0.861563\pi$
$998$	−14.2175			−0.450046
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.2.e.b.46.2		6
3.2	odd	2		45.2.e.b.16.2	✓	6
4.3	odd	2		2160.2.q.k.721.2		6
5.2	odd	4		675.2.k.b.424.4		12
5.3	odd	4		675.2.k.b.424.3		12
5.4	even	2		675.2.e.b.451.2		6
9.2	odd	6		405.2.a.j.1.2		3
9.4	even	3	inner	135.2.e.b.91.2		6
9.5	odd	6		45.2.e.b.31.2	yes	6
9.7	even	3		405.2.a.i.1.2		3
12.11	even	2		720.2.q.i.241.2		6
15.2	even	4		225.2.k.b.124.3		12
15.8	even	4		225.2.k.b.124.4		12
15.14	odd	2		225.2.e.b.151.2		6
36.7	odd	6		6480.2.a.bs.1.2		3
36.11	even	6		6480.2.a.bv.1.2		3
36.23	even	6		720.2.q.i.481.2		6
36.31	odd	6		2160.2.q.k.1441.2		6
45.2	even	12		2025.2.b.l.649.4		6
45.4	even	6		675.2.e.b.226.2		6
45.7	odd	12		2025.2.b.m.649.3		6
45.13	odd	12		675.2.k.b.199.4		12
45.14	odd	6		225.2.e.b.76.2		6
45.22	odd	12		675.2.k.b.199.3		12
45.23	even	12		225.2.k.b.49.3		12
45.29	odd	6		2025.2.a.n.1.2		3
45.32	even	12		225.2.k.b.49.4		12
45.34	even	6		2025.2.a.o.1.2		3
45.38	even	12		2025.2.b.l.649.3		6
45.43	odd	12		2025.2.b.m.649.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.2	✓	6	3.2	odd	2
45.2.e.b.31.2	yes	6	9.5	odd	6
135.2.e.b.46.2		6	1.1	even	1	trivial
135.2.e.b.91.2		6	9.4	even	3	inner
225.2.e.b.76.2		6	45.14	odd	6
225.2.e.b.151.2		6	15.14	odd	2
225.2.k.b.49.3		12	45.23	even	12
225.2.k.b.49.4		12	45.32	even	12
225.2.k.b.124.3		12	15.2	even	4
225.2.k.b.124.4		12	15.8	even	4
405.2.a.i.1.2		3	9.7	even	3
405.2.a.j.1.2		3	9.2	odd	6
675.2.e.b.226.2		6	45.4	even	6
675.2.e.b.451.2		6	5.4	even	2
675.2.k.b.199.3		12	45.22	odd	12
675.2.k.b.199.4		12	45.13	odd	12
675.2.k.b.424.3		12	5.3	odd	4
675.2.k.b.424.4		12	5.2	odd	4
720.2.q.i.241.2		6	12.11	even	2
720.2.q.i.481.2		6	36.23	even	6
2025.2.a.n.1.2		3	45.29	odd	6
2025.2.a.o.1.2		3	45.34	even	6
2025.2.b.l.649.3		6	45.38	even	12
2025.2.b.l.649.4		6	45.2	even	12
2025.2.b.m.649.3		6	45.7	odd	12
2025.2.b.m.649.4		6	45.43	odd	12
2160.2.q.k.721.2		6	4.3	odd	2
2160.2.q.k.1441.2		6	36.31	odd	6
6480.2.a.bs.1.2		3	36.7	odd	6
6480.2.a.bv.1.2		3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	135.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.285997 - 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.500000 + 0.866025i) q^{5} +(-0.714003 + 1.23669i) q^{7} +2.10083 q^{8} +O(q^{10})\)	\(q+(0.285997 - 0.495361i) q^{2} +(0.836412 + 1.44871i) q^{4} +(0.500000 + 0.866025i) q^{5} +(-0.714003 + 1.23669i) q^{7} +2.10083 q^{8} +0.571993 q^{10} +(1.33641 - 2.31473i) q^{11} +(-2.33641 - 4.04678i) q^{13} +(0.408405 + 0.707378i) q^{14} +(-1.07199 + 1.85675i) q^{16} -2.67282 q^{17} +4.67282 q^{19} +(-0.836412 + 1.44871i) q^{20} +(-0.764419 - 1.32401i) q^{22} +(-2.95882 - 5.12483i) q^{23} +(-0.500000 + 0.866025i) q^{25} -2.67282 q^{26} -2.38880 q^{28} +(-4.74482 + 8.21826i) q^{29} +(-3.48040 - 6.02823i) q^{31} +(2.71400 + 4.70079i) q^{32} +(-0.764419 + 1.32401i) q^{34} -1.42801 q^{35} -1.81681 q^{37} +(1.33641 - 2.31473i) q^{38} +(1.05042 + 1.81937i) q^{40} +(-0.735581 - 1.27406i) q^{41} +(-0.235581 + 0.408039i) q^{43} +4.47116 q^{44} -3.38485 q^{46} +(3.47842 - 6.02480i) q^{47} +(2.48040 + 4.29618i) q^{49} +(0.285997 + 0.495361i) q^{50} +(3.90841 - 6.76956i) q^{52} +1.14399 q^{53} +2.67282 q^{55} +(-1.50000 + 2.59808i) q^{56} +(2.71400 + 4.70079i) q^{58} +(-0.571993 - 0.990721i) q^{59} +(1.26442 - 2.19004i) q^{61} -3.98153 q^{62} -1.18319 q^{64} +(2.33641 - 4.04678i) q^{65} +(3.29523 + 5.70751i) q^{67} +(-2.23558 - 3.87214i) q^{68} +(-0.408405 + 0.707378i) q^{70} +12.8745 q^{71} -1.71203 q^{73} +(-0.519602 + 0.899976i) q^{74} +(3.90841 + 6.76956i) q^{76} +(1.90841 + 3.30545i) q^{77} +(0.143987 - 0.249392i) q^{79} -2.14399 q^{80} -0.841495 q^{82} +(-2.14201 + 3.71007i) q^{83} +(-1.33641 - 2.31473i) q^{85} +(0.134751 + 0.233396i) q^{86} +(2.80757 - 4.86286i) q^{88} +3.00000 q^{89} +6.67282 q^{91} +(4.94958 - 8.57293i) q^{92} +(-1.98963 - 3.44615i) q^{94} +(2.33641 + 4.04678i) q^{95} +(-3.91764 + 6.78555i) q^{97} +2.83754 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} - 5 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{8}+O(q^{10}) \) 6 * q + q^2 - 5 * q^4 + 3 * q^5 - 5 * q^7 - 6 * q^8	\( 6 q + q^{2} - 5 q^{4} + 3 q^{5} - 5 q^{7} - 6 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{13} - 9 q^{14} - 5 q^{16} + 4 q^{17} + 8 q^{19} + 5 q^{20} + 4 q^{22} + 3 q^{23} - 3 q^{25} + 4 q^{26} + 10 q^{28} - 7 q^{29} - 8 q^{31} + 17 q^{32} + 4 q^{34} - 10 q^{35} + 12 q^{37} - 2 q^{38} - 3 q^{40} - 13 q^{41} - 10 q^{43} + 44 q^{44} - 6 q^{46} + 13 q^{47} + 2 q^{49} + q^{50} + 12 q^{52} + 4 q^{53} - 4 q^{55} - 9 q^{56} + 17 q^{58} - 2 q^{59} - q^{61} - 84 q^{62} - 30 q^{64} + 4 q^{65} - 11 q^{67} - 22 q^{68} + 9 q^{70} + 20 q^{71} - 16 q^{73} - 16 q^{74} + 12 q^{76} - 2 q^{79} - 10 q^{80} - 58 q^{82} - 15 q^{83} + 2 q^{85} + 28 q^{86} + 24 q^{88} + 18 q^{89} + 20 q^{91} + 39 q^{92} + 31 q^{94} + 4 q^{95} + 18 q^{97} + 80 q^{98}+O(q^{100}) \) 6 * q + q^2 - 5 * q^4 + 3 * q^5 - 5 * q^7 - 6 * q^8 + 2 * q^10 - 2 * q^11 - 4 * q^13 - 9 * q^14 - 5 * q^16 + 4 * q^17 + 8 * q^19 + 5 * q^20 + 4 * q^22 + 3 * q^23 - 3 * q^25 + 4 * q^26 + 10 * q^28 - 7 * q^29 - 8 * q^31 + 17 * q^32 + 4 * q^34 - 10 * q^35 + 12 * q^37 - 2 * q^38 - 3 * q^40 - 13 * q^41 - 10 * q^43 + 44 * q^44 - 6 * q^46 + 13 * q^47 + 2 * q^49 + q^50 + 12 * q^52 + 4 * q^53 - 4 * q^55 - 9 * q^56 + 17 * q^58 - 2 * q^59 - q^61 - 84 * q^62 - 30 * q^64 + 4 * q^65 - 11 * q^67 - 22 * q^68 + 9 * q^70 + 20 * q^71 - 16 * q^73 - 16 * q^74 + 12 * q^76 - 2 * q^79 - 10 * q^80 - 58 * q^82 - 15 * q^83 + 2 * q^85 + 28 * q^86 + 24 * q^88 + 18 * q^89 + 20 * q^91 + 39 * q^92 + 31 * q^94 + 4 * q^95 + 18 * q^97 + 80 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more