LMFDB - Embedded newform 405.4.e.w.136.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 2 x^{10} + 32 x^{9} + 583 x^{8} - 624 x^{7} + 594 x^{6} + 9450 x^{5} + 90513 x^{4} + \cdots + 746496 $ x^12 - 2x^11 + 2x^10 + 32x^9 + 583x^8 - 624x^7 + 594x^6 + 9450x^5 + 90513x^4 - 20304x^3 + 10368x^2 + 124416*x + 746496
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			136.3
Root			$-2.82176 - 2.82176i$ of defining polynomial
Character	$\chi$	$=$	405.136
Dual form			405.4.e.w.271.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.03663 + 1.79550i) q^{2} +(1.85079 + 3.20567i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(2.33056 - 4.03665i) q^{7} -24.2604 q^{8} +O(q^{10})$	\(q+(-1.03663 + 1.79550i) q^{2} +(1.85079 + 3.20567i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(2.33056 - 4.03665i) q^{7} -24.2604 q^{8} +10.3663 q^{10} +(-4.44705 + 7.70252i) q^{11} +(-17.3086 - 29.9793i) q^{13} +(4.83186 + 8.36903i) q^{14} +(10.3428 - 17.9142i) q^{16} +2.66659 q^{17} +125.599 q^{19} +(9.25397 - 16.0283i) q^{20} +(-9.21990 - 15.9693i) q^{22} +(-65.9874 - 114.294i) q^{23} +(-12.5000 + 21.6506i) q^{25} +71.7704 q^{26} +17.2535 q^{28} +(35.6053 - 61.6702i) q^{29} +(-6.71252 - 11.6264i) q^{31} +(-75.5985 - 130.940i) q^{32} +(-2.76427 + 4.78785i) q^{34} -23.3056 q^{35} +283.849 q^{37} +(-130.200 + 225.513i) q^{38} +(60.6511 + 105.051i) q^{40} +(-191.726 - 332.079i) q^{41} +(169.588 - 293.734i) q^{43} -32.9223 q^{44} +273.618 q^{46} +(-39.1434 + 67.7983i) q^{47} +(160.637 + 278.231i) q^{49} +(-25.9158 - 44.8874i) q^{50} +(64.0692 - 110.971i) q^{52} +254.626 q^{53} +44.4705 q^{55} +(-56.5404 + 97.9309i) q^{56} +(73.8191 + 127.858i) q^{58} +(16.4098 + 28.4226i) q^{59} +(-93.4914 + 161.932i) q^{61} +27.8336 q^{62} +478.955 q^{64} +(-86.5429 + 149.897i) q^{65} +(-203.307 - 352.138i) q^{67} +(4.93530 + 8.54819i) q^{68} +(24.1593 - 41.8451i) q^{70} +966.124 q^{71} +276.177 q^{73} +(-294.246 + 509.649i) q^{74} +(232.458 + 402.629i) q^{76} +(20.7282 + 35.9024i) q^{77} +(573.426 - 993.204i) q^{79} -103.428 q^{80} +794.996 q^{82} +(-89.0225 + 154.191i) q^{83} +(-6.66647 - 11.5467i) q^{85} +(351.599 + 608.988i) q^{86} +(107.887 - 186.867i) q^{88} +806.486 q^{89} -161.355 q^{91} +(244.258 - 423.068i) q^{92} +(-81.1544 - 140.564i) q^{94} +(-313.998 - 543.860i) q^{95} +(619.023 - 1072.18i) q^{97} -666.085 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8}+O(q^{10}) $ 12 * q - 4 * q^2 - 34 * q^4 - 30 * q^5 - 40 * q^7 + 132 * q^8	$ 12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8} + 40 q^{10} - 88 q^{11} - 20 q^{13} - 180 q^{14} - 58 q^{16} + 248 q^{17} - 92 q^{19} - 170 q^{20} + 74 q^{22} - 210 q^{23} - 150 q^{25} + 8 q^{26} + 704 q^{28} - 296 q^{29} + 104 q^{31} - 722 q^{32} + 428 q^{34} + 400 q^{35} - 408 q^{37} + 20 q^{38} - 330 q^{40} - 344 q^{41} - 512 q^{43} + 1432 q^{44} - 372 q^{46} - 238 q^{47} - 68 q^{49} - 100 q^{50} + 468 q^{52} + 1700 q^{53} + 880 q^{55} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} + 364 q^{61} + 2076 q^{62} - 1980 q^{64} - 100 q^{65} - 88 q^{67} - 236 q^{68} - 900 q^{70} + 2728 q^{71} + 1672 q^{73} - 1316 q^{74} + 2106 q^{76} - 840 q^{77} + 680 q^{79} + 580 q^{80} + 3484 q^{82} - 2148 q^{83} - 620 q^{85} - 2872 q^{86} - 1296 q^{88} + 6000 q^{89} - 6116 q^{91} - 1002 q^{92} + 3662 q^{94} + 230 q^{95} + 612 q^{97} + 3964 q^{98}+O(q^{100}) $ 12 * q - 4 * q^2 - 34 * q^4 - 30 * q^5 - 40 * q^7 + 132 * q^8 + 40 * q^10 - 88 * q^11 - 20 * q^13 - 180 * q^14 - 58 * q^16 + 248 * q^17 - 92 * q^19 - 170 * q^20 + 74 * q^22 - 210 * q^23 - 150 * q^25 + 8 * q^26 + 704 * q^28 - 296 * q^29 + 104 * q^31 - 722 * q^32 + 428 * q^34 + 400 * q^35 - 408 * q^37 + 20 * q^38 - 330 * q^40 - 344 * q^41 - 512 * q^43 + 1432 * q^44 - 372 * q^46 - 238 * q^47 - 68 * q^49 - 100 * q^50 + 468 * q^52 + 1700 * q^53 + 880 * q^55 - 2316 * q^56 - 890 * q^58 - 1840 * q^59 + 364 * q^61 + 2076 * q^62 - 1980 * q^64 - 100 * q^65 - 88 * q^67 - 236 * q^68 - 900 * q^70 + 2728 * q^71 + 1672 * q^73 - 1316 * q^74 + 2106 * q^76 - 840 * q^77 + 680 * q^79 + 580 * q^80 + 3484 * q^82 - 2148 * q^83 - 620 * q^85 - 2872 * q^86 - 1296 * q^88 + 6000 * q^89 - 6116 * q^91 - 1002 * q^92 + 3662 * q^94 + 230 * q^95 + 612 * q^97 + 3964 * q^98

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$	−1.03663	+	1.79550i	−0.366504	+	0.634804i	−0.989016	−	0.147806i	$-0.952779\pi$
$2$	−1.03663	+	1.79550i	−0.366504	+	0.634804i	0.622512	+	0.782610i	$0.286112\pi$
$3$		0			0
$3$		0			0
$4$	1.85079	+	3.20567i	0.231349	+	0.400709i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$6$		0			0
$7$	2.33056	−	4.03665i	0.125838	−	0.217959i	−0.796222	−	0.605005i	$-0.793171\pi$
$7$	2.33056	−	4.03665i	0.125838	−	0.217959i	0.922060	+	0.387046i	$0.126505\pi$
$8$	−24.2604			−1.07217
$9$		0			0
$10$	10.3663			0.327811
$11$	−4.44705	+	7.70252i	−0.121894	+	0.211127i	−0.920515	−	0.390708i	$-0.872230\pi$
$11$	−4.44705	+	7.70252i	−0.121894	+	0.211127i	0.798620	+	0.601835i	$0.205564\pi$
$12$		0			0
$13$	−17.3086	−	29.9793i	−0.369272	−	0.639598i	0.620180	−	0.784460i	$-0.287060\pi$
$13$	−17.3086	−	29.9793i	−0.369272	−	0.639598i	−0.989452	+	0.144862i	$0.953726\pi$
$14$	4.83186	+	8.36903i	0.0922407	+	0.159766i
$15$		0			0
$16$	10.3428	−	17.9142i	0.161606	−	0.279910i
$17$	2.66659			0.0380437			0.0190218	−	0.999819i	$-0.493945\pi$
$17$	2.66659			0.0380437			0.0190218	+	0.999819i	$0.493945\pi$
$18$		0			0
$19$	125.599			1.51655			0.758273	−	0.651937i	$-0.226043\pi$
$19$	125.599			1.51655			0.758273	+	0.651937i	$0.226043\pi$
$20$	9.25397	−	16.0283i	0.103463	−	0.179202i
$21$		0			0
$22$	−9.21990	−	15.9693i	−0.0893495	−	0.154758i
$23$	−65.9874	−	114.294i	−0.598231	−	1.03617i	−0.993082	−	0.117422i	$-0.962537\pi$
$23$	−65.9874	−	114.294i	−0.598231	−	1.03617i	0.394851	−	0.918745i	$-0.370796\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	71.7704			0.541359
$27$		0			0
$28$	17.2535			0.116451
$29$	35.6053	−	61.6702i	0.227991	−	0.394892i	−0.729222	−	0.684278i	$-0.760118\pi$
$29$	35.6053	−	61.6702i	0.227991	−	0.394892i	0.957213	+	0.289386i	$0.0934510\pi$
$30$		0			0
$31$	−6.71252	−	11.6264i	−0.0388905	−	0.0673603i	0.845925	−	0.533302i	$-0.179049\pi$
$31$	−6.71252	−	11.6264i	−0.0388905	−	0.0673603i	−0.884815	+	0.465942i	$0.845716\pi$
$32$	−75.5985	−	130.940i	−0.417627	−	0.723351i
$33$		0			0
$34$	−2.76427	+	4.78785i	−0.0139432	+	0.0241503i
$35$	−23.3056			−0.112553
$36$		0			0
$37$	283.849			1.26120			0.630600	−	0.776108i	$-0.282809\pi$
$37$	283.849			1.26120			0.630600	+	0.776108i	$0.282809\pi$
$38$	−130.200	+	225.513i	−0.555821	+	0.962710i
$39$		0			0
$40$	60.6511	+	105.051i	0.239745	+	0.415250i
$41$	−191.726	−	332.079i	−0.730307	−	1.26493i	−0.956752	−	0.290904i	$-0.906044\pi$
$41$	−191.726	−	332.079i	−0.730307	−	1.26493i	0.226445	−	0.974024i	$-0.427289\pi$
$42$		0			0
$43$	169.588	−	293.734i	0.601438	−	1.04172i	−0.391165	−	0.920321i	$-0.627928\pi$
$43$	169.588	−	293.734i	0.601438	−	1.04172i	0.992604	−	0.121401i	$-0.0387388\pi$
$44$	−32.9223			−0.112801
$45$		0			0
$46$	273.618			0.877018
$47$	−39.1434	+	67.7983i	−0.121482	+	0.210413i	−0.920352	−	0.391090i	$-0.872098\pi$
$47$	−39.1434	+	67.7983i	−0.121482	+	0.210413i	0.798870	+	0.601503i	$0.205431\pi$
$48$		0			0
$49$	160.637	+	278.231i	0.468329	+	0.811170i
$50$	−25.9158	−	44.8874i	−0.0733009	−	0.126961i
$51$		0			0
$52$	64.0692	−	110.971i	0.170862	−	0.295941i
$53$	254.626			0.659915			0.329958	−	0.943996i	$-0.392966\pi$
$53$	254.626			0.659915			0.329958	+	0.943996i	$0.392966\pi$
$54$		0			0
$55$	44.4705			0.109026
$56$	−56.5404	+	97.9309i	−0.134920	+	0.233689i
$57$		0			0
$58$	73.8191	+	127.858i	0.167119	+	0.289459i
$59$	16.4098	+	28.4226i	0.0362097	+	0.0627171i	0.883562	−	0.468314i	$-0.155138\pi$
$59$	16.4098	+	28.4226i	0.0362097	+	0.0627171i	−0.847353	+	0.531031i	$0.821805\pi$
$60$		0			0
$61$	−93.4914	+	161.932i	−0.196235	+	0.339889i	−0.947305	−	0.320334i	$-0.896205\pi$
$61$	−93.4914	+	161.932i	−0.196235	+	0.339889i	0.751070	+	0.660223i	$0.229538\pi$
$62$	27.8336			0.0570141
$63$		0			0
$64$	478.955			0.935460
$65$	−86.5429	+	149.897i	−0.165143	+	0.286037i
$66$		0			0
$67$	−203.307	−	352.138i	−0.370715	−	0.642097i	0.618961	−	0.785422i	$-0.287554\pi$
$67$	−203.307	−	352.138i	−0.370715	−	0.642097i	−0.989676	+	0.143325i	$0.954221\pi$
$68$	4.93530	+	8.54819i	0.00880137	+	0.0152444i
$69$		0			0
$70$	24.1593	−	41.8451i	0.0412513	−	0.0714493i
$71$	966.124			1.61490			0.807450	−	0.589936i	$-0.200847\pi$
$71$	966.124			1.61490			0.807450	+	0.589936i	$0.200847\pi$
$72$		0			0
$73$	276.177			0.442796			0.221398	−	0.975184i	$-0.428938\pi$
$73$	276.177			0.442796			0.221398	+	0.975184i	$0.428938\pi$
$74$	−294.246	+	509.649i	−0.462235	+	0.800615i
$75$		0			0
$76$	232.458	+	402.629i	0.350852	+	0.607693i
$77$	20.7282	+	35.9024i	0.0306780	+	0.0531358i
$78$		0			0
$79$	573.426	−	993.204i	0.816652	−	1.41448i	−0.0914836	−	0.995807i	$-0.529161\pi$
$79$	573.426	−	993.204i	0.816652	−	1.41448i	0.908136	−	0.418676i	$-0.137506\pi$
$80$	−103.428			−0.144545
$81$		0			0
$82$	794.996			1.07064
$83$	−89.0225	+	154.191i	−0.117729	+	0.203912i	−0.918867	−	0.394567i	$-0.870895\pi$
$83$	−89.0225	+	154.191i	−0.117729	+	0.203912i	0.801139	+	0.598479i	$0.204228\pi$
$84$		0			0
$85$	−6.66647	−	11.5467i	−0.00850682	−	0.0147342i
$86$	351.599	+	608.988i	0.440860	+	0.763591i
$87$		0			0
$88$	107.887	−	186.867i	0.130691	−	0.226364i
$89$	806.486			0.960532			0.480266	−	0.877123i	$-0.340540\pi$
$89$	806.486			0.960532			0.480266	+	0.877123i	$0.340540\pi$
$90$		0			0
$91$	−161.355			−0.185874
$92$	244.258	−	423.068i	0.276801	−	0.479433i
$93$		0			0
$94$	−81.1544	−	140.564i	−0.0890472	−	0.154234i
$95$	−313.998	−	543.860i	−0.339110	−	0.587356i
$96$		0			0
$97$	619.023	−	1072.18i	0.647961	−	1.12230i	−0.335648	−	0.941988i	$-0.608955\pi$
$97$	619.023	−	1072.18i	0.647961	−	1.12230i	0.983609	−	0.180314i	$-0.0577115\pi$
$98$	−666.085			−0.686579
$99$		0			0
$100$	−92.5397			−0.0925397
$101$	−283.428	+	490.912i	−0.279229	+	0.483639i	−0.971193	−	0.238293i	$-0.923412\pi$
$101$	−283.428	+	490.912i	−0.279229	+	0.483639i	0.691964	+	0.721932i	$0.256746\pi$
$102$		0			0
$103$	409.509	+	709.290i	0.391748	+	0.678528i	0.992680	−	0.120772i	$-0.0385371\pi$
$103$	409.509	+	709.290i	0.391748	+	0.678528i	−0.600932	+	0.799300i	$0.705204\pi$
$104$	419.914	+	727.312i	0.395923	+	0.685758i
$105$		0			0
$106$	−263.953	+	457.179i	−0.241862	+	0.418917i
$107$	−543.772			−0.491293			−0.245647	−	0.969359i	$-0.579000\pi$
$107$	−543.772			−0.491293			−0.245647	+	0.969359i	$0.579000\pi$
$108$		0			0
$109$	−1636.54			−1.43809			−0.719046	−	0.694963i	$-0.755421\pi$
$109$	−1636.54			−1.43809			−0.719046	+	0.694963i	$0.755421\pi$
$110$	−46.0995	+	79.8467i	−0.0399583	+	0.0692098i
$111$		0			0
$112$	−48.2089	−	83.5003i	−0.0406725	−	0.0704468i
$113$	272.899	+	472.675i	0.227188	+	0.393500i	0.956974	−	0.290175i	$-0.0937136\pi$
$113$	272.899	+	472.675i	0.227188	+	0.393500i	−0.729786	+	0.683676i	$0.760380\pi$
$114$		0			0
$115$	−329.937	+	571.468i	−0.267537	+	0.463388i
$116$	263.592			0.210982
$117$		0			0
$118$	−68.0436			−0.0530841
$119$	6.21464	−	10.7641i	0.00478736	−	0.00829194i
$120$		0			0
$121$	625.947	+	1084.17i	0.470284	+	0.814555i
$122$	−193.832	−	335.727i	−0.143842	−	0.249142i
$123$		0			0
$124$	24.8470	−	43.0363i	0.0179946	−	0.0311675i
$125$	125.000			0.0894427
$126$		0			0
$127$	36.0480			0.0251869			0.0125935	−	0.999921i	$-0.495991\pi$
$127$	36.0480			0.0251869			0.0125935	+	0.999921i	$0.495991\pi$
$128$	108.288	−	187.561i	0.0747768	−	0.129517i
$129$		0			0
$130$	−179.426	−	310.775i	−0.121052	−	0.209667i
$131$	−655.075	−	1134.62i	−0.436902	−	0.756737i	0.560547	−	0.828123i	$-0.310591\pi$
$131$	−655.075	−	1134.62i	−0.436902	−	0.756737i	−0.997449	+	0.0713860i	$0.977258\pi$
$132$		0			0
$133$	292.716	−	506.999i	0.190840	−	0.330544i
$134$	843.016			0.543474
$135$		0			0
$136$	−64.6926			−0.0407893
$137$	1098.60	−	1902.83i	0.685109	−	1.18664i	−0.288294	−	0.957542i	$-0.593088\pi$
$137$	1098.60	−	1902.83i	0.685109	−	1.18664i	0.973403	−	0.229101i	$-0.0735786\pi$
$138$		0			0
$139$	−863.005	−	1494.77i	−0.526612	−	0.912119i	−0.999519	−	0.0310067i	$-0.990129\pi$
$139$	−863.005	−	1494.77i	−0.526612	−	0.912119i	0.472907	−	0.881112i	$-0.343205\pi$
$140$	−43.1339	−	74.7101i	−0.0260391	−	0.0451011i
$141$		0			0
$142$	−1001.51	+	1734.67i	−0.591868	+	1.02514i
$143$	307.889			0.180049
$144$		0			0
$145$	−356.053			−0.203921
$146$	−286.294	+	495.875i	−0.162286	+	0.281088i
$147$		0			0
$148$	525.345	+	909.925i	0.291778	+	0.505374i
$149$	900.699	+	1560.06i	0.495222	+	0.857750i	0.999985	−	0.00550809i	$-0.00175329\pi$
$149$	900.699	+	1560.06i	0.495222	+	0.857750i	−0.504763	+	0.863258i	$0.668420\pi$
$150$		0			0
$151$	−1664.09	+	2882.30i	−0.896835	+	1.55336i	−0.0653185	+	0.997864i	$0.520806\pi$
$151$	−1664.09	+	2882.30i	−0.896835	+	1.55336i	−0.831517	+	0.555500i	$0.812527\pi$
$152$	−3047.09			−1.62600
$153$		0			0
$154$	−85.9501			−0.0449744
$155$	−33.5626	+	58.1322i	−0.0173924	+	0.0301244i
$156$		0			0
$157$	−1915.09	−	3317.04i	−0.973510	−	1.68617i	−0.684766	−	0.728763i	$-0.740096\pi$
$157$	−1915.09	−	3317.04i	−0.973510	−	1.68617i	−0.288745	−	0.957406i	$-0.593238\pi$
$158$	1188.86	+	2059.17i	0.598613	+	1.03683i
$159$		0			0
$160$	−377.993	+	654.702i	−0.186768	+	0.323492i
$161$	−615.151			−0.301122
$162$		0			0
$163$	−2404.51			−1.15543			−0.577717	−	0.816237i	$-0.696056\pi$
$163$	−2404.51			−1.15543			−0.577717	+	0.816237i	$0.696056\pi$
$164$	709.691	−	1229.22i	0.337912	−	0.585280i
$165$		0			0
$166$	−184.567	−	319.679i	−0.0862962	−	0.149469i
$167$	−1785.48	−	3092.53i	−0.827331	−	1.43298i	−0.900125	−	0.435632i	$-0.856525\pi$
$167$	−1785.48	−	3092.53i	−0.827331	−	1.43298i	0.0727941	−	0.997347i	$-0.476808\pi$
$168$		0			0
$169$	499.326	−	864.859i	0.227276	−	0.393654i
$170$	27.6427			0.0124711
$171$		0			0
$172$	1255.49			0.556569
$173$	79.9681	−	138.509i	0.0351437	−	0.0608707i	−0.847919	−	0.530126i	$-0.822144\pi$
$173$	79.9681	−	138.509i	0.0351437	−	0.0608707i	0.883062	+	0.469256i	$0.155478\pi$
$174$		0			0
$175$	58.2640	+	100.916i	0.0251677	+	0.0435917i
$176$	91.9897	+	159.331i	0.0393976	+	0.0682387i
$177$		0			0
$178$	−836.028	+	1448.04i	−0.352039	+	0.609750i
$179$	−1120.87			−0.468030			−0.234015	−	0.972233i	$-0.575187\pi$
$179$	−1120.87			−0.468030			−0.234015	+	0.972233i	$0.575187\pi$
$180$		0			0
$181$	−3856.65			−1.58377			−0.791886	−	0.610669i	$-0.790901\pi$
$181$	−3856.65			−1.58377			−0.791886	+	0.610669i	$0.790901\pi$
$182$	167.265	−	289.712i	0.0681238	−	0.117994i
$183$		0			0
$184$	1600.88	+	2772.81i	0.641406	+	1.11095i
$185$	−709.621	−	1229.10i	−0.282013	−	0.488461i
$186$		0			0
$187$	−11.8584	+	20.5394i	−0.00463730	+	0.00803204i
$188$	−289.785			−0.112419
$189$		0			0
$190$	1302.00			0.497141
$191$	39.5050	−	68.4246i	0.0149659	−	0.0259216i	−0.858445	−	0.512905i	$-0.828569\pi$
$191$	39.5050	−	68.4246i	0.0149659	−	0.0259216i	0.873411	+	0.486983i	$0.161903\pi$
$192$		0			0
$193$	−2285.75	−	3959.03i	−0.852496	−	1.47657i	−0.878949	−	0.476916i	$-0.841755\pi$
$193$	−2285.75	−	3959.03i	−0.852496	−	1.47657i	0.0264535	−	0.999650i	$-0.491579\pi$
$194$	1283.40	+	2222.91i	0.474961	+	0.822657i
$195$		0			0
$196$	−594.612	+	1029.90i	−0.216695	+	0.375327i
$197$	−4081.61			−1.47615			−0.738077	−	0.674717i	$-0.764266\pi$
$197$	−4081.61			−1.47615			−0.738077	+	0.674717i	$0.764266\pi$
$198$		0			0
$199$	−2518.98			−0.897316			−0.448658	−	0.893703i	$-0.648098\pi$
$199$	−2518.98			−0.897316			−0.448658	+	0.893703i	$0.648098\pi$
$200$	303.256	−	525.254i	0.107217	−	0.185705i
$201$		0			0
$202$	−587.620	−	1017.79i	−0.204677	−	0.354512i
$203$	−165.961	−	287.452i	−0.0573801	−	0.0993852i
$204$		0			0
$205$	−958.630	+	1660.40i	−0.326603	+	0.565693i
$206$	−1698.04			−0.574310
$207$		0			0
$208$	−716.075			−0.238706
$209$	−558.545	+	967.429i	−0.184858	+	0.320184i
$210$		0			0
$211$	−113.622	−	196.799i	−0.0370714	−	0.0642095i	0.846894	−	0.531761i	$-0.178470\pi$
$211$	−113.622	−	196.799i	−0.0370714	−	0.0642095i	−0.883966	+	0.467552i	$0.845136\pi$
$212$	471.259	+	816.245i	0.152671	+	0.264434i
$213$		0			0
$214$	563.690	−	976.340i	0.180061	−	0.311875i
$215$	−1695.88			−0.537943
$216$		0			0
$217$	−62.5758			−0.0195757
$218$	1696.49	−	2938.40i	0.527067	−	0.912906i
$219$		0			0
$220$	82.3058	+	142.558i	0.0252230	+	0.0436875i
$221$	−46.1548	−	79.9425i	−0.0140485	−	0.0243326i
$222$		0			0
$223$	257.612	−	446.196i	0.0773585	−	0.133989i	−0.824751	−	0.565496i	$-0.808685\pi$
$223$	257.612	−	446.196i	0.0773585	−	0.133989i	0.902109	+	0.431507i	$0.142018\pi$
$224$	−704.748			−0.210214
$225$		0			0
$226$	−1131.58			−0.333061
$227$	1978.02	−	3426.03i	0.578352	−	1.00173i	−0.417317	−	0.908761i	$-0.637029\pi$
$227$	1978.02	−	3426.03i	0.578352	−	1.00173i	0.995669	−	0.0929736i	$-0.0296372\pi$
$228$		0			0
$229$	838.285	+	1451.95i	0.241901	+	0.418985i	0.961256	−	0.275658i	$-0.0888956\pi$
$229$	838.285	+	1451.95i	0.241901	+	0.418985i	−0.719355	+	0.694643i	$0.755562\pi$
$230$	−684.046	−	1184.80i	−0.196107	−	0.339667i
$231$		0			0
$232$	−863.801	+	1496.15i	−0.244445	+	0.423392i
$233$	4610.01			1.29619			0.648095	−	0.761560i	$-0.275566\pi$
$233$	4610.01			1.29619			0.648095	+	0.761560i	$0.275566\pi$
$234$		0			0
$235$	391.434			0.108657
$236$	−60.7423	+	105.209i	−0.0167542	+	0.0290191i
$237$		0			0
$238$	12.8846	+	22.3167i	0.00350917	+	0.00607806i
$239$	3454.12	+	5982.71i	0.934846	+	1.61920i	0.774907	+	0.632075i	$0.217796\pi$
$239$	3454.12	+	5982.71i	0.934846	+	1.61920i	0.159939	+	0.987127i	$0.448870\pi$
$240$		0			0
$241$	1490.48	−	2581.59i	0.398383	−	0.690019i	−0.595144	−	0.803619i	$-0.702905\pi$
$241$	1490.48	−	2581.59i	0.398383	−	0.690019i	0.993527	+	0.113600i	$0.0362383\pi$
$242$	−2595.51			−0.689444
$243$		0			0
$244$	−692.133			−0.181595
$245$	803.185	−	1391.16i	0.209443	−	0.362766i
$246$		0			0
$247$	−2173.94	−	3765.37i	−0.560018	−	0.969980i
$248$	162.849	+	282.062i	0.0416972	+	0.0722217i
$249$		0			0
$250$	−129.579	+	224.437i	−0.0327811	+	0.0567786i
$251$	7132.90			1.79372			0.896862	−	0.442310i	$-0.145841\pi$
$251$	7132.90			1.79372			0.896862	+	0.442310i	$0.145841\pi$
$252$		0			0
$253$	1173.80			0.291684
$254$	−37.3684	+	64.7240i	−0.00923112	+	0.0159888i
$255$		0			0
$256$	2140.33	+	3707.16i	0.522542	+	0.905069i
$257$	−3281.28	−	5683.35i	−0.796423	−	1.37945i	−0.921932	−	0.387353i	$-0.873390\pi$
$257$	−3281.28	−	5683.35i	−0.796423	−	1.37945i	0.125509	−	0.992093i	$-0.459944\pi$
$258$		0			0
$259$	661.526	−	1145.80i	0.158708	−	0.274889i
$260$	−640.692			−0.152823
$261$		0			0
$262$	2716.28			0.640506
$263$	−2831.75	+	4904.73i	−0.663928	+	1.14996i	0.315647	+	0.948877i	$0.397779\pi$
$263$	−2831.75	+	4904.73i	−0.663928	+	1.14996i	−0.979575	+	0.201080i	$0.935555\pi$
$264$		0			0
$265$	−636.564	−	1102.56i	−0.147562	−	0.255584i
$266$	606.877	+	1051.14i	0.139887	+	0.242292i
$267$		0			0
$268$	752.558	−	1303.47i	0.171529	−	0.297097i
$269$	4018.43			0.910811			0.455406	−	0.890284i	$-0.349494\pi$
$269$	4018.43			0.910811			0.455406	+	0.890284i	$0.349494\pi$
$270$		0			0
$271$	−1518.33			−0.340340			−0.170170	−	0.985415i	$-0.554432\pi$
$271$	−1518.33			−0.340340			−0.170170	+	0.985415i	$0.554432\pi$
$272$	27.5799	−	47.7698i	0.00614808	−	0.0106488i
$273$		0			0
$274$	2277.69	+	3945.07i	0.502190	+	0.869819i
$275$	−111.176	−	192.563i	−0.0243788	−	0.0422254i
$276$		0			0
$277$	−1091.22	+	1890.05i	−0.236697	+	0.409971i	−0.959764	−	0.280806i	$-0.909398\pi$
$277$	−1091.22	+	1890.05i	−0.236697	+	0.409971i	0.723068	+	0.690777i	$0.242731\pi$
$278$	3578.47			0.772022
$279$		0			0
$280$	565.404			0.120676
$281$	−2551.28	+	4418.95i	−0.541626	+	0.938123i	0.457185	+	0.889372i	$0.348858\pi$
$281$	−2551.28	+	4418.95i	−0.541626	+	0.938123i	−0.998811	+	0.0487518i	$0.984476\pi$
$282$		0			0
$283$	1288.97	+	2232.56i	0.270747	+	0.468947i	0.969053	−	0.246852i	$-0.0793961\pi$
$283$	1288.97	+	2232.56i	0.270747	+	0.468947i	−0.698307	+	0.715799i	$0.746063\pi$
$284$	1788.10	+	3097.08i	0.373606	+	0.647104i
$285$		0			0
$286$	−319.167	+	552.813i	−0.0659885	+	0.114296i
$287$	−1787.32			−0.367603
$288$		0			0
$289$	−4905.89			−0.998553
$290$	369.095	−	639.292i	0.0747380	−	0.129450i
$291$		0			0
$292$	511.147	+	885.332i	0.102440	+	0.177432i
$293$	4169.93	+	7222.52i	0.831432	+	1.44008i	0.896902	+	0.442229i	$0.145812\pi$
$293$	4169.93	+	7222.52i	0.831432	+	1.44008i	−0.0654699	+	0.997855i	$0.520855\pi$
$294$		0			0
$295$	82.0490	−	142.113i	0.0161935	−	0.0280479i
$296$	−6886.29			−1.35222
$297$		0			0
$298$	−3734.77			−0.726004
$299$	−2284.30	+	3956.52i	−0.441820	+	0.765255i
$300$		0			0
$301$	−790.468	−	1369.13i	−0.151368	−	0.262177i
$302$	−3450.10	−	5975.75i	−0.657388	−	1.13863i
$303$		0			0
$304$	1299.04	−	2250.01i	0.245083	−	0.424496i
$305$	934.914			0.175518
$306$		0			0
$307$	7042.31			1.30921			0.654603	−	0.755973i	$-0.272836\pi$
$307$	7042.31			1.30921			0.654603	+	0.755973i	$0.272836\pi$
$308$	−76.7274	+	132.896i	−0.0141946	+	0.0245858i
$309$		0			0
$310$	−69.5841	−	120.523i	−0.0127487	−	0.0220815i
$311$	−1171.46	−	2029.03i	−0.213593	−	0.369954i	0.739243	−	0.673438i	$-0.235183\pi$
$311$	−1171.46	−	2029.03i	−0.213593	−	0.369954i	−0.952836	+	0.303485i	$0.901850\pi$
$312$		0			0
$313$	1916.73	−	3319.88i	0.346135	−	0.599523i	−0.639425	−	0.768854i	$-0.720827\pi$
$313$	1916.73	−	3319.88i	0.346135	−	0.599523i	0.985559	+	0.169331i	$0.0541607\pi$
$314$	7940.98			1.42718
$315$		0			0
$316$	4245.18			0.755727
$317$	−2944.99	+	5100.87i	−0.521789	+	0.903765i	0.477890	+	0.878420i	$0.341402\pi$
$317$	−2944.99	+	5100.87i	−0.521789	+	0.903765i	−0.999679	+	0.0253451i	$0.991932\pi$
$318$		0			0
$319$	316.677	+	548.501i	0.0555816	+	0.0962701i
$320$	−1197.39	−	2073.94i	−0.209175	−	0.362302i
$321$		0			0
$322$	637.684	−	1104.50i	0.110363	−	0.191154i
$323$	334.921			0.0576950
$324$		0			0
$325$	865.429			0.147709
$326$	2492.59	−	4317.29i	0.423472	−	0.733474i
$327$		0			0
$328$	4651.36	+	8056.39i	0.783013	+	1.35622i
$329$	182.452	+	316.016i	0.0305742	+	0.0529560i
$330$		0			0
$331$	3169.48	−	5489.69i	0.526315	−	0.911604i	−0.473215	−	0.880947i	$-0.656907\pi$
$331$	3169.48	−	5489.69i	0.526315	−	0.911604i	0.999530	−	0.0306570i	$-0.00975997\pi$
$332$	−659.049			−0.108946
$333$		0			0
$334$	7403.51			1.21288
$335$	−1016.53	+	1760.69i	−0.165789	+	0.287154i
$336$		0			0
$337$	−484.171	−	838.609i	−0.0782625	−	0.135555i	0.824238	−	0.566244i	$-0.191604\pi$
$337$	−484.171	−	838.609i	−0.0782625	−	0.135555i	−0.902500	+	0.430689i	$0.858271\pi$
$338$	1035.23	+	1793.08i	0.166596	+	0.288552i
$339$		0			0
$340$	24.6765	−	42.7410i	0.00393609	−	0.00681751i
$341$	119.404			0.0189621
$342$		0			0
$343$	3096.26			0.487412
$344$	−4114.27	+	7126.12i	−0.644845	+	1.11690i
$345$		0			0
$346$	165.795	+	287.165i	0.0257606	+	0.0446188i
$347$	−3228.85	−	5592.54i	−0.499521	−	0.865196i	0.500479	−	0.865749i	$-0.333157\pi$
$347$	−3228.85	−	5592.54i	−0.499521	−	0.865196i	−1.00000		0.000552683i	$0.999824\pi$
$348$		0			0
$349$	3077.59	−	5330.54i	0.472033	−	0.817586i	−0.527455	−	0.849583i	$-0.676854\pi$
$349$	3077.59	−	5330.54i	0.472033	−	0.817586i	0.999488	+	0.0319976i	$0.0101869\pi$
$350$	−241.593			−0.0368963
$351$		0			0
$352$	1344.76			0.203625
$353$	−1831.77	+	3172.71i	−0.276190	+	0.478375i	−0.970435	−	0.241364i	$-0.922405\pi$
$353$	−1831.77	+	3172.71i	−0.276190	+	0.478375i	0.694245	+	0.719739i	$0.255739\pi$
$354$		0			0
$355$	−2415.31	−	4183.44i	−0.361103	−	0.625448i
$356$	1492.64	+	2585.33i	0.222218	+	0.384894i
$357$		0			0
$358$	1161.92	−	2012.51i	0.171535	−	0.297107i
$359$	12112.6			1.78073			0.890364	−	0.455250i	$-0.150450\pi$
$359$	12112.6			1.78073			0.890364	+	0.455250i	$0.150450\pi$
$360$		0			0
$361$	8916.11			1.29991
$362$	3997.92	−	6924.61i	0.580459	−	1.00538i
$363$		0			0
$364$	−298.634	−	517.250i	−0.0430019	−	0.0744815i
$365$	−690.443	−	1195.88i	−0.0990121	−	0.171494i
$366$		0			0
$367$	−5808.08	+	10059.9i	−0.826101	+	1.43085i	0.0749736	+	0.997186i	$0.476113\pi$
$367$	−5808.08	+	10059.9i	−0.826101	+	1.43085i	−0.901075	+	0.433664i	$0.857221\pi$
$368$	−2729.97			−0.386711
$369$		0			0
$370$	2942.46			0.413436
$371$	593.420	−	1027.83i	0.0830427	−	0.143834i
$372$		0			0
$373$	1107.86	+	1918.87i	0.153788	+	0.266369i	0.932617	−	0.360868i	$-0.117519\pi$
$373$	1107.86	+	1918.87i	0.153788	+	0.266369i	−0.778829	+	0.627236i	$0.784186\pi$
$374$	−24.5857	−	42.5836i	−0.00339918	−	0.00588756i
$375$		0			0
$376$	949.636	−	1644.82i	0.130249	−	0.225598i
$377$	−2465.11			−0.336763
$378$		0			0
$379$	−6539.83			−0.886354			−0.443177	−	0.896434i	$-0.646149\pi$
$379$	−6539.83			−0.886354			−0.443177	+	0.896434i	$0.646149\pi$
$380$	1162.29	−	2013.14i	0.156906	−	0.271769i
$381$		0			0
$382$	81.9041	+	141.862i	0.0109701	+	0.0190008i
$383$	−4247.63	−	7357.12i	−0.566694	−	0.981543i	−0.996890	−	0.0788075i	$-0.974889\pi$
$383$	−4247.63	−	7357.12i	−0.566694	−	0.981543i	0.430196	−	0.902736i	$-0.358445\pi$
$384$		0			0
$385$	103.641	−	179.512i	0.0137196	−	0.0237630i
$386$	9477.90			1.24977
$387$		0			0
$388$	4582.74			0.599621
$389$	1768.42	−	3062.99i	0.230494	−	0.399228i	−0.727459	−	0.686151i	$-0.759299\pi$
$389$	1768.42	−	3062.99i	0.230494	−	0.399228i	0.957954	+	0.286923i	$0.0926323\pi$
$390$		0			0
$391$	−175.961	−	304.774i	−0.0227589	−	0.0394196i
$392$	−3897.13	−	6750.02i	−0.502129	−	0.869713i
$393$		0			0
$394$	4231.12	−	7328.51i	0.541017	−	0.937068i
$395$	−5734.26			−0.730436
$396$		0			0
$397$	8586.22			1.08547			0.542733	−	0.839905i	$-0.317390\pi$
$397$	8586.22			1.08547			0.542733	+	0.839905i	$0.317390\pi$
$398$	2611.25	−	4522.82i	0.328870	−	0.569620i
$399$		0			0
$400$	258.569	+	447.855i	0.0323212	+	0.0559819i
$401$	3616.39	+	6263.78i	0.450359	+	0.780045i	0.998408	−	0.0564010i	$-0.0179625\pi$
$401$	3616.39	+	6263.78i	0.450359	+	0.780045i	−0.548049	+	0.836446i	$0.684629\pi$
$402$		0			0
$403$	−232.369	+	402.474i	−0.0287223	+	0.0497485i
$404$	−2098.27			−0.258398
$405$		0			0
$406$	688.160			0.0841202
$407$	−1262.29	+	2186.35i	−0.153733	+	0.266274i
$408$		0			0
$409$	−615.480	−	1066.04i	−0.0744096	−	0.128881i	0.826420	−	0.563054i	$-0.190374\pi$
$409$	−615.480	−	1066.04i	−0.0744096	−	0.128881i	−0.900829	+	0.434173i	$0.857041\pi$
$410$	−1987.49	−	3442.43i	−0.239403	−	0.414658i
$411$		0			0
$412$	−1515.83	+	2625.50i	−0.181261	+	0.313954i
$413$	152.976			0.0182263
$414$		0			0
$415$	890.225			0.105300
$416$	−2617.01	+	4532.79i	−0.308436	+	0.534226i
$417$		0			0
$418$	−1158.01	−	2005.73i	−0.135503	−	0.234698i
$419$	291.219	+	504.407i	0.0339547	+	0.0588112i	0.882503	−	0.470306i	$-0.155856\pi$
$419$	291.219	+	504.407i	0.0339547	+	0.0588112i	−0.848549	+	0.529117i	$0.822523\pi$
$420$		0			0
$421$	6281.14	−	10879.3i	0.727135	−	1.25944i	−0.230954	−	0.972965i	$-0.574185\pi$
$421$	6281.14	−	10879.3i	0.727135	−	1.25944i	0.958089	−	0.286471i	$-0.0924821\pi$
$422$	471.136			0.0543473
$423$		0			0
$424$	−6177.33			−0.707542
$425$	−33.3323	+	57.7333i	−0.00380437	+	0.00658936i
$426$		0			0
$427$	435.775	+	754.784i	0.0493878	+	0.0855422i
$428$	−1006.41	−	1743.15i	−0.113660	−	0.196865i
$429$		0			0
$430$	1758.00	−	3044.94i	0.197158	−	0.341488i
$431$	−9612.85			−1.07433			−0.537163	−	0.843478i	$-0.680504\pi$
$431$	−9612.85			−1.07433			−0.537163	+	0.843478i	$0.680504\pi$
$432$		0			0
$433$	8285.20			0.919541			0.459771	−	0.888038i	$-0.347932\pi$
$433$	8285.20			0.919541			0.459771	+	0.888038i	$0.347932\pi$
$434$	64.8680	−	112.355i	0.00717457	−	0.0124267i
$435$		0			0
$436$	−3028.90	−	5246.20i	−0.332701	−	0.576256i
$437$	−8287.95	−	14355.2i	−0.907246	−	1.57140i
$438$		0			0
$439$	−5384.48	+	9326.20i	−0.585393	+	1.01393i	0.409434	+	0.912340i	$0.365726\pi$
$439$	−5384.48	+	9326.20i	−0.585393	+	1.01393i	−0.994826	+	0.101590i	$0.967607\pi$
$440$	−1078.87			−0.116894
$441$		0			0
$442$	191.382			0.0205953
$443$	−4404.96	+	7629.62i	−0.472429	+	0.818271i	−0.999502	−	0.0315487i	$-0.989956\pi$
$443$	−4404.96	+	7629.62i	−0.472429	+	0.818271i	0.527073	+	0.849820i	$0.323289\pi$
$444$		0			0
$445$	−2016.22	−	3492.19i	−0.214782	−	0.372013i
$446$	534.096	+	925.082i	0.0567045	+	0.0982150i
$447$		0			0
$448$	1116.23	−	1933.37i	0.117717	−	0.203891i
$449$	−9060.10			−0.952277			−0.476139	−	0.879370i	$-0.657964\pi$
$449$	−9060.10			−0.952277			−0.476139	+	0.879370i	$0.657964\pi$
$450$		0			0
$451$	3410.46			0.356081
$452$	−1010.16	+	1749.65i	−0.105119	+	0.182072i
$453$		0			0
$454$	4100.95	+	7103.06i	0.423937	+	0.734280i
$455$	403.387	+	698.687i	0.0415628	+	0.0719889i
$456$		0			0
$457$	−1238.78	+	2145.62i	−0.126800	+	0.219624i	−0.922435	−	0.386152i	$-0.873804\pi$
$457$	−1238.78	+	2145.62i	−0.126800	+	0.219624i	0.795635	+	0.605776i	$0.207137\pi$
$458$	−3475.97			−0.354631
$459$		0			0
$460$	−2442.58			−0.247578
$461$	7650.77	−	13251.5i	0.772954	−	1.33880i	−0.162983	−	0.986629i	$-0.552112\pi$
$461$	7650.77	−	13251.5i	0.772954	−	1.33880i	0.935937	−	0.352167i	$-0.114555\pi$
$462$		0			0
$463$	341.506	+	591.506i	0.0342789	+	0.0593728i	0.882656	−	0.470020i	$-0.155753\pi$
$463$	341.506	+	591.506i	0.0342789	+	0.0593728i	−0.848377	+	0.529393i	$0.822420\pi$
$464$	−736.515	−	1275.68i	−0.0736894	−	0.127634i
$465$		0			0
$466$	−4778.88	+	8277.27i	−0.475059	+	0.822826i
$467$	6569.32			0.650946			0.325473	−	0.945551i	$-0.394477\pi$
$467$	6569.32			0.650946			0.325473	+	0.945551i	$0.394477\pi$
$468$		0			0
$469$	−1895.28			−0.186601
$470$	−405.772	+	702.818i	−0.0398231	+	0.0689757i
$471$		0			0
$472$	−398.109	−	689.545i	−0.0388230	−	0.0672434i
$473$	1508.33	+	2612.50i	0.146624	+	0.253960i
$474$		0			0
$475$	−1569.99	+	2719.30i	−0.151655	+	0.262674i
$476$	46.0081			0.00443020
$477$		0			0
$478$	−14322.6			−1.37050
$479$	−5644.33	+	9776.26i	−0.538405	+	0.932544i	0.460585	+	0.887615i	$0.347639\pi$
$479$	−5644.33	+	9776.26i	−0.538405	+	0.932544i	−0.998990	+	0.0449289i	$0.985694\pi$
$480$		0			0
$481$	−4913.02	−	8509.59i	−0.465726	−	0.806661i
$482$	3090.15	+	5352.30i	0.292018	+	0.505790i
$483$		0			0
$484$	−2317.00	+	4013.16i	−0.217599	+	0.376893i
$485$	−6190.23			−0.579554
$486$		0			0
$487$	−8937.53			−0.831618			−0.415809	−	0.909452i	$-0.636502\pi$
$487$	−8937.53			−0.831618			−0.415809	+	0.909452i	$0.636502\pi$
$488$	2268.14	−	3928.54i	0.210397	−	0.364419i
$489$		0			0
$490$	1665.21	+	2884.23i	0.153524	+	0.265911i
$491$	−1846.57	−	3198.35i	−0.169724	−	0.293971i	0.768599	−	0.639731i	$-0.220954\pi$
$491$	−1846.57	−	3198.35i	−0.169724	−	0.293971i	−0.938323	+	0.345760i	$0.887621\pi$
$492$		0			0
$493$	94.9446	−	164.449i	0.00867361	−	0.0150231i
$494$	9014.29			0.820996
$495$		0			0
$496$	−277.705			−0.0251397
$497$	2251.61	−	3899.91i	0.203216	−	0.351981i
$498$		0			0
$499$	5593.22	+	9687.74i	0.501777	+	0.869104i	0.999998	+	0.00205364i	$0.000653694\pi$
$499$	5593.22	+	9687.74i	0.501777	+	0.869104i	−0.498220	+	0.867050i	$0.666013\pi$
$500$	231.349	+	400.709i	0.0206925	+	0.0358405i
$501$		0			0
$502$	−7394.19	+	12807.1i	−0.657408	+	1.13866i
$503$	6553.81			0.580954			0.290477	−	0.956882i	$-0.406186\pi$
$503$	6553.81			0.580954			0.290477	+	0.956882i	$0.406186\pi$
$504$		0			0
$505$	2834.28			0.249750
$506$	−1216.79	+	2107.55i	−0.106903	+	0.185162i
$507$		0			0
$508$	66.7174	+	115.558i	0.00582698	+	0.0100926i
$509$	7513.86	+	13014.4i	0.654314	+	1.13331i	0.982065	+	0.188542i	$0.0603760\pi$
$509$	7513.86	+	13014.4i	0.654314	+	1.13331i	−0.327751	+	0.944764i	$0.606291\pi$
$510$		0			0
$511$	643.647	−	1114.83i	0.0557207	−	0.0965111i
$512$	−7142.32			−0.616502
$513$		0			0
$514$	13605.9			1.16757
$515$	2047.54	−	3546.45i	0.175195	−	0.303447i
$516$		0			0
$517$	−348.145	−	603.005i	−0.0296159	−	0.0512962i
$518$	1371.52	+	2375.54i	0.116334	+	0.201496i
$519$		0			0
$520$	2099.57	−	3636.56i	0.177062	−	0.306680i
$521$	−835.969			−0.0702965			−0.0351483	−	0.999382i	$-0.511190\pi$
$521$	−835.969			−0.0702965			−0.0351483	+	0.999382i	$0.511190\pi$
$522$		0			0
$523$	−13032.4			−1.08961			−0.544805	−	0.838563i	$-0.683396\pi$
$523$	−13032.4			−1.08961			−0.544805	+	0.838563i	$0.683396\pi$
$524$	2424.82	−	4199.91i	0.202154	−	0.350141i
$525$		0			0
$526$	−5870.95	−	10168.8i	−0.486665	−	0.842928i
$527$	−17.8995	−	31.0029i	−0.00147954	−	0.00256263i
$528$		0			0
$529$	−2625.17	+	4546.93i	−0.215762	+	0.373710i
$530$	2639.53			0.216328
$531$		0			0
$532$	2167.03			0.176603
$533$	−6637.01	+	11495.6i	−0.539364	+	0.934205i
$534$		0			0
$535$	1359.43	+	2354.60i	0.109857	+	0.190277i
$536$	4932.32	+	8543.02i	0.397469	+	0.688437i
$537$		0			0
$538$	−4165.63	+	7215.09i	−0.333816	+	0.578187i
$539$	−2857.44			−0.228347
$540$		0			0
$541$	−4182.33			−0.332370			−0.166185	−	0.986095i	$-0.553145\pi$
$541$	−4182.33			−0.332370			−0.166185	+	0.986095i	$0.553145\pi$
$542$	1573.95	−	2726.16i	0.124736	−	0.216049i
$543$		0			0
$544$	−201.590	−	349.164i	−0.0158881	−	0.0275189i
$545$	4091.35	+	7086.42i	0.321567	+	0.556970i
$546$		0			0
$547$	−2506.95	+	4342.17i	−0.195959	+	0.339411i	−0.947214	−	0.320601i	$-0.896115\pi$
$547$	−2506.95	+	4342.17i	−0.195959	+	0.339411i	0.751256	+	0.660011i	$0.229449\pi$
$548$	8133.14			0.633997
$549$		0			0
$550$	460.995			0.0357398
$551$	4471.99	−	7745.72i	0.345759	−	0.598872i
$552$		0			0
$553$	−2672.81	−	4629.44i	−0.205532	−	0.355993i
$554$	−2262.38	−	3918.56i	−0.173501	−	0.300512i
$555$		0			0
$556$	3194.49	−	5533.01i	0.243663	−	0.422036i
$557$	−2611.86			−0.198686			−0.0993430	−	0.995053i	$-0.531674\pi$
$557$	−2611.86			−0.198686			−0.0993430	+	0.995053i	$0.531674\pi$
$558$		0			0
$559$	−11741.3			−0.888377
$560$	−241.045	+	417.502i	−0.0181893	+	0.0315048i
$561$		0			0
$562$	−5289.48	−	9161.64i	−0.397016	−	0.687652i
$563$	9337.00	+	16172.2i	0.698948	+	1.21061i	0.968832	+	0.247720i	$0.0796813\pi$
$563$	9337.00	+	16172.2i	0.698948	+	1.21061i	−0.269884	+	0.962893i	$0.586985\pi$
$564$		0			0
$565$	1364.50	−	2363.38i	0.101601	−	0.175979i
$566$	−5344.74			−0.396919
$567$		0			0
$568$	−23438.6			−1.73145
$569$	11835.5	−	20499.7i	0.872004	−	1.51036i	0.0120843	−	0.999927i	$-0.496153\pi$
$569$	11835.5	−	20499.7i	0.872004	−	1.51036i	0.859920	−	0.510429i	$-0.170513\pi$
$570$		0			0
$571$	4677.77	+	8102.14i	0.342835	+	0.593807i	0.984958	−	0.172794i	$-0.0552796\pi$
$571$	4677.77	+	8102.14i	0.342835	+	0.593807i	−0.642123	+	0.766601i	$0.721946\pi$
$572$	569.838	+	986.989i	0.0416541	+	0.0721470i
$573$		0			0
$574$	1852.79	−	3209.12i	0.134728	−	0.233356i
$575$	3299.37			0.239293
$576$		0			0
$577$	−21695.2			−1.56531			−0.782653	−	0.622459i	$-0.786134\pi$
$577$	−21695.2			−1.56531			−0.782653	+	0.622459i	$0.786134\pi$
$578$	5085.60	−	8808.51i	0.365974	−	0.633885i
$579$		0			0
$580$	−658.981	−	1141.39i	−0.0471770	−	0.0817130i
$581$	414.944	+	718.705i	0.0296296	+	0.0513200i
$582$		0			0
$583$	−1132.33	+	1961.26i	−0.0804399	+	0.139326i
$584$	−6700.18			−0.474752
$585$		0			0
$586$	−17290.7			−1.21889
$587$	−368.574	+	638.389i	−0.0259160	+	0.0448878i	−0.878693	−	0.477388i	$-0.841584\pi$
$587$	−368.574	+	638.389i	−0.0259160	+	0.0448878i	0.852777	+	0.522276i	$0.174917\pi$
$588$		0			0
$589$	−843.086	−	1460.27i	−0.0589792	−	0.102155i
$590$	170.109	+	294.637i	0.0118700	+	0.0205594i
$591$		0			0
$592$	2935.78	−	5084.92i	0.203817	−	0.353022i
$593$	−21908.1			−1.51713			−0.758565	−	0.651597i	$-0.774099\pi$
$593$	−21908.1			−1.51713			−0.758565	+	0.651597i	$0.774099\pi$
$594$		0			0
$595$	−62.1464			−0.00428194
$596$	−3334.02	+	5774.68i	−0.229139	+	0.396880i
$597$		0			0
$598$	−4735.94	−	8202.89i	−0.323858	−	0.560939i
$599$	3797.92	+	6578.19i	0.259063	+	0.448710i	0.965991	−	0.258575i	$-0.0832529\pi$
$599$	3797.92	+	6578.19i	0.259063	+	0.448710i	−0.706928	+	0.707285i	$0.749920\pi$
$600$		0			0
$601$	3251.26	−	5631.35i	0.220668	−	0.382209i	−0.734343	−	0.678779i	$-0.762509\pi$
$601$	3251.26	−	5631.35i	0.220668	−	0.382209i	0.955011	+	0.296570i	$0.0958428\pi$
$602$	3277.69			0.221908
$603$		0			0
$604$	−12319.6			−0.829929
$605$	3129.74	−	5420.86i	0.210317	−	0.364280i
$606$		0			0
$607$	13078.4	+	22652.5i	0.874524	+	1.51472i	0.857268	+	0.514870i	$0.172160\pi$
$607$	13078.4	+	22652.5i	0.874524	+	1.51472i	0.0172560	+	0.999851i	$0.494507\pi$
$608$	−9495.10	−	16446.0i	−0.633351	−	1.09700i
$609$		0			0
$610$	−969.160	+	1678.63i	−0.0643281	+	0.111420i
$611$	2710.06			0.179439
$612$		0			0
$613$	18172.8			1.19738			0.598690	−	0.800981i	$-0.295688\pi$
$613$	18172.8			1.19738			0.598690	+	0.800981i	$0.295688\pi$
$614$	−7300.28	+	12644.4i	−0.479829	+	0.831089i
$615$		0			0
$616$	−502.877	−	871.008i	−0.0328920	−	0.0569706i
$617$	13292.4	+	23023.1i	0.867312	+	1.50223i	0.864733	+	0.502231i	$0.167487\pi$
$617$	13292.4	+	23023.1i	0.867312	+	1.50223i	0.00257822	+	0.999997i	$0.499179\pi$
$618$		0			0
$619$	−10843.9	+	18782.2i	−0.704124	+	1.21958i	0.262882	+	0.964828i	$0.415327\pi$
$619$	−10843.9	+	18782.2i	−0.704124	+	1.21958i	−0.967007	+	0.254751i	$0.918006\pi$
$620$	−248.470			−0.0160948
$621$		0			0
$622$	4857.48			0.313131
$623$	1879.57	−	3255.50i	0.120872	−	0.209356i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	3973.89	+	6882.98i	0.253720	+	0.439455i
$627$		0			0
$628$	7088.89	−	12278.3i	0.450442	−	0.780188i
$629$	756.907			0.0479807
$630$		0			0
$631$	−7685.15			−0.484851			−0.242426	−	0.970170i	$-0.577943\pi$
$631$	−7685.15			−0.484851			−0.242426	+	0.970170i	$0.577943\pi$
$632$	−13911.6	+	24095.6i	−0.875590	+	1.51657i
$633$		0			0
$634$	−6105.73	−	10575.4i	−0.382476	−	0.662467i
$635$	−90.1199	−	156.092i	−0.00563197	−	0.00975486i
$636$		0			0
$637$	5560.79	−	9631.58i	0.345882	−	0.599085i
$638$	−1313.11			−0.0814835
$639$		0			0
$640$	−1082.88			−0.0668824
$641$	4027.84	−	6976.43i	0.248191	−	0.429879i	−0.714833	−	0.699295i	$-0.753497\pi$
$641$	4027.84	−	6976.43i	0.248191	−	0.429879i	0.963024	+	0.269416i	$0.0868307\pi$
$642$		0			0
$643$	3561.01	+	6167.85i	0.218402	+	0.378283i	0.954320	−	0.298788i	$-0.0965823\pi$
$643$	3561.01	+	6167.85i	0.218402	+	0.378283i	−0.735918	+	0.677071i	$0.763249\pi$
$644$	−1138.52	−	1971.97i	−0.0696644	−	0.120662i
$645$		0			0
$646$	−347.189	+	601.349i	−0.0211455	+	0.0366250i
$647$	10280.7			0.624692			0.312346	−	0.949968i	$-0.398885\pi$
$647$	10280.7			0.624692			0.312346	+	0.949968i	$0.398885\pi$
$648$		0			0
$649$	−291.901			−0.0176550
$650$	−897.130	+	1553.87i	−0.0541359	+	0.0937661i
$651$		0			0
$652$	−4450.25	−	7708.06i	−0.267309	−	0.462992i
$653$	−4492.52	−	7781.27i	−0.269228	−	0.466316i	0.699435	−	0.714696i	$-0.253435\pi$
$653$	−4492.52	−	7781.27i	−0.269228	−	0.466316i	−0.968663	+	0.248380i	$0.920102\pi$
$654$		0			0
$655$	−3275.38	+	5673.12i	−0.195389	+	0.338423i
$656$	−7931.92			−0.472087
$657$		0			0
$658$	−756.541			−0.0448223
$659$	6269.36	−	10858.9i	0.370591	−	0.641883i	−0.619065	−	0.785340i	$-0.712488\pi$
$659$	6269.36	−	10858.9i	0.370591	−	0.641883i	0.989657	+	0.143457i	$0.0458217\pi$
$660$		0			0
$661$	6239.95	+	10807.9i	0.367180	+	0.635974i	0.989123	−	0.147088i	$-0.0469900\pi$
$661$	6239.95	+	10807.9i	0.367180	+	0.635974i	−0.621944	+	0.783062i	$0.713657\pi$
$662$	6571.15	+	11381.6i	0.385793	+	0.668213i
$663$		0			0
$664$	2159.72	−	3740.75i	0.126225	−	0.218629i
$665$	−2927.16			−0.170692
$666$		0			0
$667$	−9398.01			−0.545566
$668$	6609.09	−	11447.3i	0.382805	−	0.663037i
$669$		0			0
$670$	−2107.54	−	3650.37i	−0.121524	−	0.210487i
$671$	−831.522	−	1440.24i	−0.0478399	−	0.0828611i
$672$		0			0
$673$	−6592.82	+	11419.1i	−0.377614	+	0.654047i	−0.990715	−	0.135958i	$-0.956589\pi$
$673$	−6592.82	+	11419.1i	−0.377614	+	0.654047i	0.613100	+	0.790005i	$0.289922\pi$
$674$	2007.63			0.114734
$675$		0			0
$676$	3696.60			0.210321
$677$	13284.1	−	23008.8i	0.754136	−	1.30620i	−0.191666	−	0.981460i	$-0.561389\pi$
$677$	13284.1	−	23008.8i	0.754136	−	1.30620i	0.945802	−	0.324742i	$-0.105278\pi$
$678$		0			0
$679$	−2885.34	−	4997.56i	−0.163077	−	0.282457i
$680$	161.731	+	280.127i	0.00912076	+	0.0157976i
$681$		0			0
$682$	−123.778	+	214.389i	−0.00694969	+	0.0120372i
$683$	7602.40			0.425912			0.212956	−	0.977062i	$-0.431691\pi$
$683$	7602.40			0.425912			0.212956	+	0.977062i	$0.431691\pi$
$684$		0			0
$685$	−10986.0			−0.612780
$686$	−3209.68	+	5559.33i	−0.178639	+	0.309411i
$687$		0			0
$688$	−3508.01	−	6076.05i	−0.194392	−	0.336697i
$689$	−4407.21	−	7633.51i	−0.243688	−	0.422080i
$690$		0			0
$691$	−3128.66	+	5418.99i	−0.172243	+	0.298333i	−0.939204	−	0.343361i	$-0.888435\pi$
$691$	−3128.66	+	5418.99i	−0.172243	+	0.298333i	0.766961	+	0.641694i	$0.221768\pi$
$692$	592.018			0.0325219
$693$		0			0
$694$	13388.5			0.732307
$695$	−4315.02	+	7473.84i	−0.235508	+	0.407912i
$696$		0			0
$697$	−511.254	−	885.518i	−0.0277835	−	0.0481225i
$698$	6380.65	+	11051.6i	0.346004	+	0.599297i
$699$		0			0
$700$	−215.669	+	373.550i	−0.0116451	+	0.0201698i
$701$	−769.271			−0.0414479			−0.0207239	−	0.999785i	$-0.506597\pi$
$701$	−769.271			−0.0414479			−0.0207239	+	0.999785i	$0.506597\pi$
$702$		0			0
$703$	35651.1			1.91267
$704$	−2129.94	+	3689.16i	−0.114027	+	0.197501i
$705$		0			0
$706$	−3797.73	−	6577.87i	−0.202450	−	0.350653i
$707$	1321.09	+	2288.20i	0.0702755	+	0.121721i
$708$		0			0
$709$	4225.16	−	7318.19i	0.223807	−	0.387645i	−0.732154	−	0.681139i	$-0.761485\pi$
$709$	4225.16	−	7318.19i	0.223807	−	0.387645i	0.955961	+	0.293494i	$0.0948181\pi$
$710$	10015.1			0.529383
$711$		0			0
$712$	−19565.7			−1.02985
$713$	−885.884	+	1534.40i	−0.0465310	+	0.0805941i
$714$		0			0
$715$	−769.721	−	1333.20i	−0.0402601	−	0.0697325i
$716$	−2074.49	−	3593.12i	−0.108278	−	0.187544i
$717$		0			0
$718$	−12556.3	+	21748.2i	−0.652644	+	1.13041i
$719$	−25350.0			−1.31488			−0.657438	−	0.753508i	$-0.728360\pi$
$719$	−25350.0			−1.31488			−0.657438	+	0.753508i	$0.728360\pi$
$720$		0			0
$721$	3817.54			0.197188
$722$	−9242.71	+	16008.9i	−0.476424	+	0.825191i
$723$		0			0
$724$	−7137.87	−	12363.2i	−0.366404	−	0.634631i
$725$	890.133	+	1541.75i	0.0455982	+	0.0789784i
$726$		0			0
$727$	1912.57	−	3312.68i	0.0975701	−	0.168996i	−0.813108	−	0.582113i	$-0.802226\pi$
$727$	1912.57	−	3312.68i	0.0975701	−	0.168996i	0.910678	+	0.413116i	$0.135560\pi$
$728$	3914.54			0.199289
$729$		0			0
$730$	2862.94			0.145153
$731$	452.220	−	783.268i	0.0228809	−	0.0396309i
$732$		0			0
$733$	1846.41	+	3198.08i	0.0930407	+	0.161151i	0.908789	−	0.417256i	$-0.137008\pi$
$733$	1846.41	+	3198.08i	0.0930407	+	0.161151i	−0.815749	+	0.578407i	$0.803675\pi$
$734$	−12041.7	−	20856.8i	−0.605539	−	1.04882i
$735$		0			0
$736$	−9977.10	+	17280.8i	−0.499675	+	0.865462i
$737$	3616.46			0.180752
$738$		0			0
$739$	4181.32			0.208136			0.104068	−	0.994570i	$-0.466814\pi$
$739$	4181.32			0.208136			0.104068	+	0.994570i	$0.466814\pi$
$740$	2626.73	−	4549.62i	0.130487	−	0.226010i
$741$		0			0
$742$	1230.32	+	2130.97i	0.0608710	+	0.105432i
$743$	463.991	+	803.656i	0.0229101	+	0.0396814i	0.877253	−	0.480028i	$-0.159374\pi$
$743$	463.991	+	803.656i	0.0229101	+	0.0396814i	−0.854343	+	0.519709i	$0.826040\pi$
$744$		0			0
$745$	4503.49	−	7800.28i	0.221470	−	0.383598i
$746$	−4593.78			−0.225456
$747$		0			0
$748$	−87.7902			−0.00429135
$749$	−1267.29	+	2195.02i	−0.0618236	+	0.107082i
$750$		0			0
$751$	11184.7	+	19372.5i	0.543458	+	0.941297i	0.998702	+	0.0509306i	$0.0162187\pi$
$751$	11184.7	+	19372.5i	0.543458	+	0.941297i	−0.455244	+	0.890367i	$0.650448\pi$
$752$	809.702	+	1402.45i	0.0392644	+	0.0680079i
$753$		0			0
$754$	2555.41	−	4426.09i	0.123425	−	0.213778i
$755$	16640.9			0.802154
$756$		0			0
$757$	35390.0			1.69917			0.849586	−	0.527451i	$-0.176852\pi$
$757$	35390.0			1.69917			0.849586	+	0.527451i	$0.176852\pi$
$758$	6779.38	−	11742.2i	0.324853	−	0.562661i
$759$		0			0
$760$	7617.72	+	13194.3i	0.363584	+	0.629746i
$761$	−16717.3	−	28955.2i	−0.796323	−	1.37927i	−0.921996	−	0.387200i	$-0.873442\pi$
$761$	−16717.3	−	28955.2i	−0.796323	−	1.37927i	0.125673	−	0.992072i	$-0.459891\pi$
$762$		0			0
$763$	−3814.05	+	6606.13i	−0.180967	+	0.313444i
$764$	292.462			0.0138494
$765$		0			0
$766$	17612.9			0.830783
$767$	568.060	−	983.909i	0.0267425	−	0.0463193i
$768$		0			0
$769$	−12510.7	−	21669.2i	−0.586668	−	1.01614i	−0.994665	−	0.103155i	$-0.967106\pi$
$769$	−12510.7	−	21669.2i	−0.586668	−	1.01614i	0.407998	−	0.912983i	$-0.366227\pi$
$770$	214.875	+	372.175i	0.0100566	+	0.0174185i
$771$		0			0
$772$	8460.89	−	14654.7i	0.394448	−	0.683205i
$773$	−3065.27			−0.142626			−0.0713132	−	0.997454i	$-0.522719\pi$
$773$	−3065.27			−0.142626			−0.0713132	+	0.997454i	$0.522719\pi$
$774$		0			0
$775$	335.626			0.0155562
$776$	−15017.8	+	26011.5i	−0.694725	+	1.20330i
$777$		0			0
$778$	3666.39	+	6350.38i	0.168954	+	0.292638i
$779$	−24080.6	−	41708.8i	−1.10754	−	1.91832i
$780$		0			0
$781$	−4296.41	+	7441.59i	−0.196847	+	0.340949i
$782$	729.627			0.0333650
$783$		0			0
$784$	6645.73			0.302739
$785$	−9575.47	+	16585.2i	−0.435367	+	0.754078i
$786$		0			0
$787$	17841.6	+	30902.5i	0.808111	+	1.39969i	0.914171	+	0.405329i	$0.132843\pi$
$787$	17841.6	+	30902.5i	0.808111	+	1.39969i	−0.106060	+	0.994360i	$0.533824\pi$
$788$	−7554.21	−	13084.3i	−0.341507	−	0.591508i
$789$		0			0
$790$	5944.31	−	10295.9i	0.267708	−	0.463684i
$791$	2544.03			0.114356
$792$		0			0
$793$	6472.81			0.289856
$794$	−8900.73	+	15416.5i	−0.397828	+	0.689058i
$795$		0			0
$796$	−4662.12	−	8075.02i	−0.207593	−	0.359562i
$797$	2262.54	+	3918.83i	0.100556	+	0.174168i	0.911914	−	0.410382i	$-0.134604\pi$
$797$	2262.54	+	3918.83i	0.100556	+	0.174168i	−0.811358	+	0.584550i	$0.801271\pi$
$798$		0			0
$799$	−104.379	+	180.790i	−0.00462162	+	0.00800487i
$800$	3779.93			0.167051
$801$		0			0
$802$	−14995.5			−0.660235
$803$	−1228.17	+	2127.26i	−0.0539742	+	0.0934861i
$804$		0			0
$805$	1537.88	+	2663.68i	0.0673329	+	0.116624i
$806$	−481.761	−	834.434i	−0.0210537	−	0.0364661i
$807$		0			0
$808$	6876.09	−	11909.7i	0.299381	−	0.518544i
$809$	−16234.6			−0.705535			−0.352767	−	0.935711i	$-0.614759\pi$
$809$	−16234.6			−0.705535			−0.352767	+	0.935711i	$0.614759\pi$
$810$		0			0
$811$	2197.06			0.0951286			0.0475643	−	0.998868i	$-0.484854\pi$
$811$	2197.06			0.0951286			0.0475643	+	0.998868i	$0.484854\pi$
$812$	614.318	−	1064.03i	0.0265497	−	0.0459854i
$813$		0			0
$814$	−2617.06	−	4532.87i	−0.112688	−	0.195181i
$815$	6011.28	+	10411.8i	0.258363	+	0.447498i
$816$		0			0
$817$	21300.0	−	36892.7i	0.912109	−	1.57982i
$818$	2552.10			0.109086
$819$		0			0
$820$	−7096.91			−0.302237
$821$	10070.8	−	17443.1i	0.428104	−	0.741498i	−0.568601	−	0.822614i	$-0.692515\pi$
$821$	10070.8	−	17443.1i	0.428104	−	0.741498i	0.996705	+	0.0811157i	$0.0258483\pi$
$822$		0			0
$823$	−7758.51	−	13438.1i	−0.328608	−	0.569166i	0.653628	−	0.756816i	$-0.273246\pi$
$823$	−7758.51	−	13438.1i	−0.328608	−	0.569166i	−0.982236	+	0.187650i	$0.939913\pi$
$824$	−9934.86	−	17207.7i	−0.420021	−	0.727498i
$825$		0			0
$826$	−158.580	+	274.668i	−0.00668002	+	0.0115701i
$827$	5582.51			0.234732			0.117366	−	0.993089i	$-0.462555\pi$
$827$	5582.51			0.234732			0.117366	+	0.993089i	$0.462555\pi$
$828$		0			0
$829$	26037.6			1.09086			0.545430	−	0.838157i	$-0.316367\pi$
$829$	26037.6			1.09086			0.545430	+	0.838157i	$0.316367\pi$
$830$	−922.834	+	1598.40i	−0.0385928	+	0.0668447i
$831$		0			0
$832$	−8290.04	−	14358.8i	−0.345439	−	0.598318i
$833$	428.352	+	741.928i	0.0178170	+	0.0308599i
$834$		0			0
$835$	−8927.38	+	15462.7i	−0.369994	+	0.640848i
$836$	−4135.01			−0.171067
$837$		0			0
$838$	−1207.55			−0.0497781
$839$	9043.99	−	15664.6i	0.372149	−	0.644581i	−0.617747	−	0.786377i	$-0.711954\pi$
$839$	9043.99	−	15664.6i	0.372149	−	0.644581i	0.989896	+	0.141796i	$0.0452877\pi$
$840$		0			0
$841$	9659.02	+	16729.9i	0.396040	+	0.685962i
$842$	13022.4	+	22555.5i	0.532996	+	0.923177i
$843$		0			0
$844$	420.582	−	728.469i	0.0171529	−	0.0297096i
$845$	−4993.26			−0.203282
$846$		0			0
$847$	5835.23			0.236719
$848$	2633.53	−	4561.42i	0.106646	−	0.184717i
$849$		0			0
$850$	−69.1066	−	119.696i	−0.00278863	−	0.00483005i
$851$	−18730.4	−	32442.1i	−0.754490	−	1.30681i
$852$		0			0
$853$	12674.5	−	21952.9i	0.508753	−	0.881187i	−0.491195	−	0.871049i	$-0.663440\pi$
$853$	12674.5	−	21952.9i	0.508753	−	0.881187i	0.999949	−	0.0101371i	$-0.00322680\pi$
$854$	−1806.95			−0.0724034
$855$		0			0
$856$	13192.1			0.526750
$857$	281.899	−	488.263i	0.0112363	−	0.0194618i	−0.860353	−	0.509699i	$-0.829757\pi$
$857$	281.899	−	488.263i	0.0112363	−	0.0194618i	0.871589	+	0.490238i	$0.163090\pi$
$858$		0			0
$859$	10279.4	+	17804.5i	0.408300	+	0.707197i	0.994699	−	0.102826i	$-0.0327884\pi$
$859$	10279.4	+	17804.5i	0.408300	+	0.707197i	−0.586399	+	0.810022i	$0.699455\pi$
$860$	−3138.71	−	5436.41i	−0.124453	−	0.215558i
$861$		0			0
$862$	9964.97	−	17259.8i	0.393745	−	0.681987i
$863$	−7056.31			−0.278331			−0.139166	−	0.990269i	$-0.544442\pi$
$863$	−7056.31			−0.278331			−0.139166	+	0.990269i	$0.544442\pi$
$864$		0			0
$865$	−799.681			−0.0314335
$866$	−8588.70	+	14876.1i	−0.337016	+	0.583729i
$867$		0			0
$868$	−115.815	−	200.597i	−0.00452882	−	0.00784414i
$869$	5100.11	+	8833.66i	0.199090	+	0.344835i
$870$		0			0
$871$	−7037.90	+	12190.0i	−0.273789	+	0.474217i
$872$	39703.2			1.54188
$873$		0			0
$874$	34366.2			1.33004
$875$	291.320	−	504.581i	0.0112553	−	0.0194948i
$876$		0			0
$877$	−1258.15	−	2179.18i	−0.0484432	−	0.0839061i	0.840787	−	0.541366i	$-0.182093\pi$
$877$	−1258.15	−	2179.18i	−0.0484432	−	0.0839061i	−0.889230	+	0.457460i	$0.848759\pi$
$878$	−11163.4	−	19335.6i	−0.429098	−	0.743219i
$879$		0			0
$880$	459.949	−	796.654i	0.0176192	−	0.0305173i
$881$	31571.2			1.20733			0.603666	−	0.797237i	$-0.293706\pi$
$881$	31571.2			1.20733			0.603666	+	0.797237i	$0.293706\pi$
$882$		0			0
$883$	47743.5			1.81959			0.909794	−	0.415059i	$-0.136239\pi$
$883$	47743.5			1.81959			0.909794	+	0.415059i	$0.136239\pi$
$884$	170.846	−	295.914i	0.00650020	−	0.0112587i
$885$		0			0
$886$	−9132.64	−	15818.2i	−0.346295	−	0.599800i
$887$	16514.4	+	28603.8i	0.625141	+	1.08278i	0.988514	+	0.151132i	$0.0482919\pi$
$887$	16514.4	+	28603.8i	0.625141	+	1.08278i	−0.363372	+	0.931644i	$0.618375\pi$
$888$		0			0
$889$	84.0120	−	145.513i	0.00316948	−	0.00548971i
$890$	8360.28			0.314873
$891$		0			0
$892$	1907.14			0.0715873
$893$	−4916.37	+	8515.40i	−0.184233	+	0.319101i
$894$		0			0
$895$	2802.16	+	4853.49i	0.104655	+	0.181267i
$896$	−504.745	−	874.244i	−0.0188196	−	0.0325965i
$897$		0			0
$898$	9391.98	−	16267.4i	0.349014	−	0.604510i
$899$	−956.006			−0.0354667
$900$		0			0
$901$	678.981			0.0251056
$902$	−3535.39	+	6123.47i	−0.130505	+	0.226041i
$903$		0			0
$904$	−6620.66	−	11467.3i	−0.243584	−	0.421900i
$905$	9641.63	+	16699.8i	0.354142	+	0.613392i
$906$		0			0
$907$	21665.3	−	37525.3i	0.793146	−	1.37377i	−0.130864	−	0.991400i	$-0.541775\pi$
$907$	21665.3	−	37525.3i	0.793146	−	1.37377i	0.924010	−	0.382369i	$-0.124892\pi$
$908$	14643.6			0.535205
$909$		0			0
$910$	−1672.65			−0.0609318
$911$	−11177.2	+	19359.5i	−0.406496	+	0.704071i	−0.994494	−	0.104791i	$-0.966583\pi$
$911$	−11177.2	+	19359.5i	−0.406496	+	0.704071i	0.587999	+	0.808862i	$0.299916\pi$
$912$		0			0
$913$	−791.775	−	1371.39i	−0.0287009	−	0.0497114i
$914$	−2568.31	−	4448.44i	−0.0929454	−	0.160986i
$915$		0			0
$916$	−3102.98	+	5374.53i	−0.111927	+	0.193864i
$917$	−6106.77			−0.219916
$918$		0			0
$919$	1306.08			0.0468811			0.0234406	−	0.999725i	$-0.492538\pi$
$919$	1306.08			0.0468811			0.0234406	+	0.999725i	$0.492538\pi$
$920$	8004.42	−	13864.1i	0.286846	−	0.496831i
$921$		0			0
$922$	15862.0	+	27473.9i	0.566582	+	0.981349i
$923$	−16722.2	−	28963.8i	−0.596337	−	1.03289i
$924$		0			0
$925$	−3548.11	+	6145.50i	−0.126120	+	0.218446i
$926$	−1416.06			−0.0502535
$927$		0			0
$928$	−10766.8			−0.380861
$929$	25331.9	−	43876.2i	0.894633	−	1.54955i	0.0603743	−	0.998176i	$-0.480771\pi$
$929$	25331.9	−	43876.2i	0.894633	−	1.54955i	0.834258	−	0.551374i	$-0.185896\pi$
$930$		0			0
$931$	20175.8	+	34945.6i	0.710243	+	1.23018i
$932$	8532.18	+	14778.2i	0.299872	+	0.519394i
$933$		0			0
$934$	−6809.95	+	11795.2i	−0.238574	+	0.413223i
$935$	118.584			0.00414773
$936$		0			0
$937$	−12652.4			−0.441126			−0.220563	−	0.975373i	$-0.570789\pi$
$937$	−12652.4			−0.441126			−0.220563	+	0.975373i	$0.570789\pi$
$938$	1964.70	−	3402.96i	0.0683899	−	0.118455i
$939$		0			0
$940$	724.463	+	1254.81i	0.0251376	+	0.0435397i
$941$	20338.7	+	35227.7i	0.704594	+	1.22039i	0.966838	+	0.255392i	$0.0822044\pi$
$941$	20338.7	+	35227.7i	0.704594	+	1.22039i	−0.262243	+	0.965002i	$0.584462\pi$
$942$		0			0
$943$	−25303.0	+	43826.1i	−0.873785	+	1.51344i
$944$	678.891			0.0234068
$945$		0			0
$946$	−6254.32			−0.214953
$947$	−13458.4	+	23310.6i	−0.461816	+	0.799888i	−0.999052	−	0.0435439i	$-0.986135\pi$
$947$	−13458.4	+	23310.6i	−0.461816	+	0.799888i	0.537236	+	0.843432i	$0.319468\pi$
$948$		0			0
$949$	−4780.23	−	8279.60i	−0.163512	−	0.283211i
$950$	−3254.99	−	5637.82i	−0.111164	−	0.192542i
$951$		0			0
$952$	−150.770	+	261.141i	−0.00513286	+	0.00889038i
$953$	−29644.9			−1.00765			−0.503827	−	0.863805i	$-0.668075\pi$
$953$	−29644.9			−1.00765			−0.503827	+	0.863805i	$0.668075\pi$
$954$		0			0
$955$	−395.050			−0.0133859
$956$	−12785.7	+	22145.5i	−0.432552	+	0.749202i
$957$		0			0
$958$	−11702.2	−	20268.7i	−0.394655	−	0.683563i
$959$	−5120.72	−	8869.34i	−0.172426	−	0.298651i
$960$		0			0
$961$	14805.4	−	25643.7i	0.496975	−	0.860786i
$962$	20371.9			0.682762
$963$		0			0
$964$	11034.3			0.368662
$965$	−11428.7	+	19795.1i	−0.381248	+	0.660340i
$966$		0			0
$967$	−10726.0	−	18578.1i	−0.356697	−	0.617818i	0.630710	−	0.776019i	$-0.282764\pi$
$967$	−10726.0	−	18578.1i	−0.356697	−	0.617818i	−0.987407	+	0.158201i	$0.949431\pi$
$968$	−15185.8	−	26302.5i	−0.504224	−	0.873342i
$969$		0			0
$970$	6416.98	−	11114.5i	0.212409	−	0.367903i
$971$	42684.6			1.41073			0.705363	−	0.708846i	$-0.250784\pi$
$971$	42684.6			1.41073			0.705363	+	0.708846i	$0.250784\pi$
$972$		0			0
$973$	−8045.14			−0.265072
$974$	9264.91	−	16047.3i	0.304792	−	0.527914i
$975$		0			0
$976$	1933.92	+	3349.65i	0.0634255	+	0.109856i
$977$	30193.1	+	52296.0i	0.988704	+	1.71248i	0.624155	+	0.781300i	$0.285443\pi$
$977$	30193.1	+	52296.0i	0.988704	+	1.71248i	0.364548	+	0.931184i	$0.381223\pi$
$978$		0			0
$979$	−3586.49	+	6211.98i	−0.117083	+	0.202794i
$980$	5946.12			0.193818
$981$		0			0
$982$	7656.85			0.248818
$983$	−29504.6	+	51103.4i	−0.957325	+	1.65814i	−0.228369	+	0.973575i	$0.573339\pi$
$983$	−29504.6	+	51103.4i	−0.957325	+	1.65814i	−0.728956	+	0.684560i	$0.759994\pi$
$984$		0			0
$985$	10204.0	+	17673.9i	0.330078	+	0.571712i
$986$	196.845	+	340.946i	0.00635783	+	0.0110121i
$987$		0			0
$988$	8047.03	−	13937.9i	0.259120	−	0.448808i
$989$	−44762.6			−1.43920
$990$		0			0
$991$	−30899.0			−0.990454			−0.495227	−	0.868764i	$-0.664915\pi$
$991$	−30899.0			−0.990454			−0.495227	+	0.868764i	$0.664915\pi$
$992$	−1014.91	+	1757.88i	−0.0324834	+	0.0562629i
$993$		0			0
$994$	4668.18	+	8085.52i	0.148959	+	0.258005i
$995$	6297.46	+	10907.5i	0.200646	+	0.347529i
$996$		0			0
$997$	16933.4	−	29329.6i	0.537901	−	0.931671i	−0.461116	−	0.887340i	$-0.652551\pi$
$997$	16933.4	−	29329.6i	0.537901	−	0.931671i	0.999017	−	0.0443315i	$-0.0141158\pi$
$998$	−23192.4			−0.735614
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.w.136.3		12
3.2	odd	2		405.4.e.x.136.4		12
9.2	odd	6		405.4.a.k.1.3	✓	6
9.4	even	3	inner	405.4.e.w.271.3		12
9.5	odd	6		405.4.e.x.271.4		12
9.7	even	3		405.4.a.l.1.4	yes	6
45.29	odd	6		2025.4.a.z.1.4		6
45.34	even	6		2025.4.a.y.1.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.3	✓	6	9.2	odd	6
405.4.a.l.1.4	yes	6	9.7	even	3
405.4.e.w.136.3		12	1.1	even	1	trivial
405.4.e.w.271.3		12	9.4	even	3	inner
405.4.e.x.136.4		12	3.2	odd	2
405.4.e.x.271.4		12	9.5	odd	6
2025.4.a.y.1.3		6	45.34	even	6
2025.4.a.z.1.4		6	45.29	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.03663 + 1.79550i) q^{2} +(1.85079 + 3.20567i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(2.33056 - 4.03665i) q^{7} -24.2604 q^{8} +O(q^{10})\)	\(q+(-1.03663 + 1.79550i) q^{2} +(1.85079 + 3.20567i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(2.33056 - 4.03665i) q^{7} -24.2604 q^{8} +10.3663 q^{10} +(-4.44705 + 7.70252i) q^{11} +(-17.3086 - 29.9793i) q^{13} +(4.83186 + 8.36903i) q^{14} +(10.3428 - 17.9142i) q^{16} +2.66659 q^{17} +125.599 q^{19} +(9.25397 - 16.0283i) q^{20} +(-9.21990 - 15.9693i) q^{22} +(-65.9874 - 114.294i) q^{23} +(-12.5000 + 21.6506i) q^{25} +71.7704 q^{26} +17.2535 q^{28} +(35.6053 - 61.6702i) q^{29} +(-6.71252 - 11.6264i) q^{31} +(-75.5985 - 130.940i) q^{32} +(-2.76427 + 4.78785i) q^{34} -23.3056 q^{35} +283.849 q^{37} +(-130.200 + 225.513i) q^{38} +(60.6511 + 105.051i) q^{40} +(-191.726 - 332.079i) q^{41} +(169.588 - 293.734i) q^{43} -32.9223 q^{44} +273.618 q^{46} +(-39.1434 + 67.7983i) q^{47} +(160.637 + 278.231i) q^{49} +(-25.9158 - 44.8874i) q^{50} +(64.0692 - 110.971i) q^{52} +254.626 q^{53} +44.4705 q^{55} +(-56.5404 + 97.9309i) q^{56} +(73.8191 + 127.858i) q^{58} +(16.4098 + 28.4226i) q^{59} +(-93.4914 + 161.932i) q^{61} +27.8336 q^{62} +478.955 q^{64} +(-86.5429 + 149.897i) q^{65} +(-203.307 - 352.138i) q^{67} +(4.93530 + 8.54819i) q^{68} +(24.1593 - 41.8451i) q^{70} +966.124 q^{71} +276.177 q^{73} +(-294.246 + 509.649i) q^{74} +(232.458 + 402.629i) q^{76} +(20.7282 + 35.9024i) q^{77} +(573.426 - 993.204i) q^{79} -103.428 q^{80} +794.996 q^{82} +(-89.0225 + 154.191i) q^{83} +(-6.66647 - 11.5467i) q^{85} +(351.599 + 608.988i) q^{86} +(107.887 - 186.867i) q^{88} +806.486 q^{89} -161.355 q^{91} +(244.258 - 423.068i) q^{92} +(-81.1544 - 140.564i) q^{94} +(-313.998 - 543.860i) q^{95} +(619.023 - 1072.18i) q^{97} -666.085 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8}+O(q^{10}) \) 12 * q - 4 * q^2 - 34 * q^4 - 30 * q^5 - 40 * q^7 + 132 * q^8	\( 12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8} + 40 q^{10} - 88 q^{11} - 20 q^{13} - 180 q^{14} - 58 q^{16} + 248 q^{17} - 92 q^{19} - 170 q^{20} + 74 q^{22} - 210 q^{23} - 150 q^{25} + 8 q^{26} + 704 q^{28} - 296 q^{29} + 104 q^{31} - 722 q^{32} + 428 q^{34} + 400 q^{35} - 408 q^{37} + 20 q^{38} - 330 q^{40} - 344 q^{41} - 512 q^{43} + 1432 q^{44} - 372 q^{46} - 238 q^{47} - 68 q^{49} - 100 q^{50} + 468 q^{52} + 1700 q^{53} + 880 q^{55} - 2316 q^{56} - 890 q^{58} - 1840 q^{59} + 364 q^{61} + 2076 q^{62} - 1980 q^{64} - 100 q^{65} - 88 q^{67} - 236 q^{68} - 900 q^{70} + 2728 q^{71} + 1672 q^{73} - 1316 q^{74} + 2106 q^{76} - 840 q^{77} + 680 q^{79} + 580 q^{80} + 3484 q^{82} - 2148 q^{83} - 620 q^{85} - 2872 q^{86} - 1296 q^{88} + 6000 q^{89} - 6116 q^{91} - 1002 q^{92} + 3662 q^{94} + 230 q^{95} + 612 q^{97} + 3964 q^{98}+O(q^{100}) \) 12 * q - 4 * q^2 - 34 * q^4 - 30 * q^5 - 40 * q^7 + 132 * q^8 + 40 * q^10 - 88 * q^11 - 20 * q^13 - 180 * q^14 - 58 * q^16 + 248 * q^17 - 92 * q^19 - 170 * q^20 + 74 * q^22 - 210 * q^23 - 150 * q^25 + 8 * q^26 + 704 * q^28 - 296 * q^29 + 104 * q^31 - 722 * q^32 + 428 * q^34 + 400 * q^35 - 408 * q^37 + 20 * q^38 - 330 * q^40 - 344 * q^41 - 512 * q^43 + 1432 * q^44 - 372 * q^46 - 238 * q^47 - 68 * q^49 - 100 * q^50 + 468 * q^52 + 1700 * q^53 + 880 * q^55 - 2316 * q^56 - 890 * q^58 - 1840 * q^59 + 364 * q^61 + 2076 * q^62 - 1980 * q^64 - 100 * q^65 - 88 * q^67 - 236 * q^68 - 900 * q^70 + 2728 * q^71 + 1672 * q^73 - 1316 * q^74 + 2106 * q^76 - 840 * q^77 + 680 * q^79 + 580 * q^80 + 3484 * q^82 - 2148 * q^83 - 620 * q^85 - 2872 * q^86 - 1296 * q^88 + 6000 * q^89 - 6116 * q^91 - 1002 * q^92 + 3662 * q^94 + 230 * q^95 + 612 * q^97 + 3964 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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