LMFDB - Embedded newform 405.4.e.w.136.6

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			136.6
Root			$-2.61824 + 2.61824i$ of defining polynomial
Character	$\chi$	$=$	405.136
Dual form			405.4.e.w.271.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.05867 - 3.56572i) q^{2} +(-4.47625 - 7.75309i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(10.0115 - 17.3404i) q^{7} -3.92177 q^{8} +O(q^{10})$	\(q+(2.05867 - 3.56572i) q^{2} +(-4.47625 - 7.75309i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(10.0115 - 17.3404i) q^{7} -3.92177 q^{8} -20.5867 q^{10} +(0.839133 - 1.45342i) q^{11} +(-5.55615 - 9.62353i) q^{13} +(-41.2206 - 71.3962i) q^{14} +(27.7364 - 48.0408i) q^{16} +8.98682 q^{17} -50.6419 q^{19} +(-22.3812 + 38.7655i) q^{20} +(-3.45500 - 5.98423i) q^{22} +(-107.246 - 185.755i) q^{23} +(-12.5000 + 21.6506i) q^{25} -45.7531 q^{26} -179.255 q^{28} +(38.4544 - 66.6050i) q^{29} +(136.769 + 236.890i) q^{31} +(-129.887 - 224.971i) q^{32} +(18.5009 - 32.0445i) q^{34} -100.115 q^{35} -137.283 q^{37} +(-104.255 + 180.575i) q^{38} +(9.80442 + 16.9818i) q^{40} +(26.6729 + 46.1987i) q^{41} +(-147.635 + 255.711i) q^{43} -15.0247 q^{44} -883.134 q^{46} +(97.4916 - 168.860i) q^{47} +(-28.9587 - 50.1579i) q^{49} +(51.4668 + 89.1431i) q^{50} +(-49.7414 + 86.1547i) q^{52} -450.173 q^{53} -8.39133 q^{55} +(-39.2626 + 68.0048i) q^{56} +(-158.330 - 274.236i) q^{58} +(-240.572 - 416.684i) q^{59} +(-337.927 + 585.307i) q^{61} +1126.25 q^{62} -625.798 q^{64} +(-27.7807 + 48.1177i) q^{65} +(-447.295 - 774.737i) q^{67} +(-40.2273 - 69.6757i) q^{68} +(-206.103 + 356.981i) q^{70} +721.947 q^{71} +915.175 q^{73} +(-282.621 + 489.513i) q^{74} +(226.686 + 392.632i) q^{76} +(-16.8019 - 29.1017i) q^{77} +(-29.8321 + 51.6707i) q^{79} -277.364 q^{80} +219.643 q^{82} +(-371.438 + 643.349i) q^{83} +(-22.4671 - 38.9141i) q^{85} +(607.863 + 1052.85i) q^{86} +(-3.29089 + 5.69998i) q^{88} +1540.72 q^{89} -222.501 q^{91} +(-960.116 + 1662.97i) q^{92} +(-401.406 - 695.256i) q^{94} +(126.605 + 219.286i) q^{95} +(560.783 - 971.305i) q^{97} -238.465 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.05867	−	3.56572i	0.727850	−	1.26067i	−0.229940	−	0.973205i	$-0.573853\pi$
$2$	2.05867	−	3.56572i	0.727850	−	1.26067i	0.957790	−	0.287468i	$-0.0928136\pi$
$3$		0			0
$3$		0			0
$4$	−4.47625	−	7.75309i	−0.559531	−	0.969137i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$5$	−2.50000	−	4.33013i	−0.223607	−	0.387298i
$6$		0			0
$7$	10.0115	−	17.3404i	0.540568	−	0.936291i	−0.458303	−	0.888796i	$-0.651543\pi$
$7$	10.0115	−	17.3404i	0.540568	−	0.936291i	0.998871	−	0.0474955i	$-0.0151240\pi$
$8$	−3.92177			−0.173319
$9$		0			0
$10$	−20.5867			−0.651009
$11$	0.839133	−	1.45342i	0.0230007	−	0.0398385i	−0.854296	−	0.519787i	$-0.826011\pi$
$11$	0.839133	−	1.45342i	0.0230007	−	0.0398385i	0.877297	+	0.479948i	$0.159345\pi$
$12$		0			0
$13$	−5.55615	−	9.62353i	−0.118538	−	0.205314i	0.800650	−	0.599132i	$-0.204488\pi$
$13$	−5.55615	−	9.62353i	−0.118538	−	0.205314i	−0.919189	+	0.393818i	$0.871154\pi$
$14$	−41.2206	−	71.3962i	−0.786905	−	1.36296i
$15$		0			0
$16$	27.7364	−	48.0408i	0.433381	−	0.750638i
$17$	8.98682			0.128213			0.0641066	−	0.997943i	$-0.479580\pi$
$17$	8.98682			0.128213			0.0641066	+	0.997943i	$0.479580\pi$
$18$		0			0
$19$	−50.6419			−0.611477			−0.305738	−	0.952116i	$-0.598903\pi$
$19$	−50.6419			−0.611477			−0.305738	+	0.952116i	$0.598903\pi$
$20$	−22.3812	+	38.7655i	−0.250230	+	0.433411i
$21$		0			0
$22$	−3.45500	−	5.98423i	−0.0334822	−	0.0579928i
$23$	−107.246	−	185.755i	−0.972272	−	1.68402i	−0.688658	−	0.725086i	$-0.741800\pi$
$23$	−107.246	−	185.755i	−0.972272	−	1.68402i	−0.283614	−	0.958939i	$-0.591533\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	−45.7531			−0.345113
$27$		0			0
$28$	−179.255			−1.20986
$29$	38.4544	−	66.6050i	0.246235	−	0.426491i	−0.716243	−	0.697851i	$-0.754140\pi$
$29$	38.4544	−	66.6050i	0.246235	−	0.426491i	0.962478	+	0.271360i	$0.0874733\pi$
$30$		0			0
$31$	136.769	+	236.890i	0.792399	+	1.37247i	0.924478	+	0.381236i	$0.124501\pi$
$31$	136.769	+	236.890i	0.792399	+	1.37247i	−0.132079	+	0.991239i	$0.542165\pi$
$32$	−129.887	−	224.971i	−0.717532	−	1.24280i
$33$		0			0
$34$	18.5009	−	32.0445i	0.0933200	−	0.161635i
$35$	−100.115			−0.483499
$36$		0			0
$37$	−137.283			−0.609978			−0.304989	−	0.952356i	$-0.598653\pi$
$37$	−137.283			−0.609978			−0.304989	+	0.952356i	$0.598653\pi$
$38$	−104.255	+	180.575i	−0.445063	+	0.770872i
$39$		0			0
$40$	9.80442	+	16.9818i	0.0387554	+	0.0671263i
$41$	26.6729	+	46.1987i	0.101600	+	0.175976i	0.912344	−	0.409424i	$-0.134270\pi$
$41$	26.6729	+	46.1987i	0.101600	+	0.175976i	−0.810744	+	0.585401i	$0.800937\pi$
$42$		0			0
$43$	−147.635	+	255.711i	−0.523584	+	0.906874i	0.476039	+	0.879424i	$0.342072\pi$
$43$	−147.635	+	255.711i	−0.523584	+	0.906874i	−0.999623	+	0.0274502i	$0.991261\pi$
$44$	−15.0247			−0.0514785
$45$		0			0
$46$	−883.134			−2.83067
$47$	97.4916	−	168.860i	0.302566	−	0.524060i	−0.674150	−	0.738594i	$-0.735490\pi$
$47$	97.4916	−	168.860i	0.302566	−	0.524060i	0.976716	+	0.214534i	$0.0688234\pi$
$48$		0			0
$49$	−28.9587	−	50.1579i	−0.0844276	−	0.146233i
$50$	51.4668	+	89.1431i	0.145570	+	0.252135i
$51$		0			0
$52$	−49.7414	+	86.1547i	−0.132652	+	0.229760i
$53$	−450.173			−1.16672			−0.583359	−	0.812214i	$-0.698262\pi$
$53$	−450.173			−1.16672			−0.583359	+	0.812214i	$0.698262\pi$
$54$		0			0
$55$	−8.39133			−0.0205725
$56$	−39.2626	+	68.0048i	−0.0936909	+	0.162277i
$57$		0			0
$58$	−158.330	−	274.236i	−0.358444	−	0.620843i
$59$	−240.572	−	416.684i	−0.530845	−	0.919451i	−0.999352	−	0.0359910i	$-0.988541\pi$
$59$	−240.572	−	416.684i	−0.530845	−	0.919451i	0.468507	−	0.883460i	$-0.344792\pi$
$60$		0			0
$61$	−337.927	+	585.307i	−0.709297	+	1.22854i	0.255821	+	0.966724i	$0.417654\pi$
$61$	−337.927	+	585.307i	−0.709297	+	1.22854i	−0.965118	+	0.261814i	$0.915679\pi$
$62$	1126.25			2.30699
$63$		0			0
$64$	−625.798			−1.22226
$65$	−27.7807	+	48.1177i	−0.0530120	+	0.0918194i
$66$		0			0
$67$	−447.295	−	774.737i	−0.815608	−	1.41267i	−0.908890	−	0.417035i	$-0.863069\pi$
$67$	−447.295	−	774.737i	−0.815608	−	1.41267i	0.0932823	−	0.995640i	$-0.470264\pi$
$68$	−40.2273	−	69.6757i	−0.0717393	−	0.124256i
$69$		0			0
$70$	−206.103	+	356.981i	−0.351915	+	0.609534i
$71$	721.947			1.20675			0.603376	−	0.797457i	$-0.293822\pi$
$71$	721.947			1.20675			0.603376	+	0.797457i	$0.293822\pi$
$72$		0			0
$73$	915.175			1.46730			0.733651	−	0.679526i	$-0.237815\pi$
$73$	915.175			1.46730			0.733651	+	0.679526i	$0.237815\pi$
$74$	−282.621	+	489.513i	−0.443973	+	0.768983i
$75$		0			0
$76$	226.686	+	392.632i	0.342140	+	0.592604i
$77$	−16.8019	−	29.1017i	−0.0248669	−	0.0430708i
$78$		0			0
$79$	−29.8321	+	51.6707i	−0.0424858	+	0.0735875i	−0.886486	−	0.462755i	$-0.846861\pi$
$79$	−29.8321	+	51.6707i	−0.0424858	+	0.0735875i	0.844001	+	0.536342i	$0.180194\pi$
$80$	−277.364			−0.387628
$81$		0			0
$82$	219.643			0.295798
$83$	−371.438	+	643.349i	−0.491212	+	0.850803i	−0.999949	−	0.0101184i	$-0.996779\pi$
$83$	−371.438	+	643.349i	−0.491212	+	0.850803i	0.508737	+	0.860922i	$0.330112\pi$
$84$		0			0
$85$	−22.4671	−	38.9141i	−0.0286693	−	0.0496568i
$86$	607.863	+	1052.85i	0.762181	+	1.32014i
$87$		0			0
$88$	−3.29089	+	5.69998i	−0.00398647	+	0.00690477i
$89$	1540.72			1.83501			0.917505	−	0.397723i	$-0.130200\pi$
$89$	1540.72			1.83501			0.917505	+	0.397723i	$0.130200\pi$
$90$		0			0
$91$	−222.501			−0.256312
$92$	−960.116	+	1662.97i	−1.08803	+	1.88453i
$93$		0			0
$94$	−401.406	−	695.256i	−0.440445	−	0.762874i
$95$	126.605	+	219.286i	0.136730	+	0.236824i
$96$		0			0
$97$	560.783	−	971.305i	0.586999	−	1.01671i	−0.407624	−	0.913150i	$-0.633643\pi$
$97$	560.783	−	971.305i	0.586999	−	1.01671i	0.994623	−	0.103562i	$-0.0330240\pi$
$98$	−238.465			−0.245802
$99$		0			0
$100$	223.812			0.223812
$101$	879.236	−	1522.88i	0.866211	−	1.50032i	0.000370864	−	1.00000i	$-0.499882\pi$
$101$	879.236	−	1522.88i	0.866211	−	1.50032i	0.865840	−	0.500321i	$-0.166785\pi$
$102$		0			0
$103$	−528.242	−	914.941i	−0.505332	−	0.875261i	−0.999981	−	0.00616773i	$-0.998037\pi$
$103$	−528.242	−	914.941i	−0.505332	−	0.875261i	0.494649	−	0.869093i	$-0.335297\pi$
$104$	21.7899	+	37.7413i	0.0205450	+	0.0355850i
$105$		0			0
$106$	−926.759	+	1605.19i	−0.849196	+	1.47085i
$107$	1470.03			1.32816			0.664081	−	0.747661i	$-0.268823\pi$
$107$	1470.03			1.32816			0.664081	+	0.747661i	$0.268823\pi$
$108$		0			0
$109$	918.889			0.807464			0.403732	−	0.914877i	$-0.367713\pi$
$109$	918.889			0.807464			0.403732	+	0.914877i	$0.367713\pi$
$110$	−17.2750	+	29.9212i	−0.0149737	+	0.0259352i
$111$		0			0
$112$	−555.363	−	961.917i	−0.468544	−	0.811541i
$113$	−493.609	−	854.955i	−0.410927	−	0.711747i	0.584064	−	0.811708i	$-0.301462\pi$
$113$	−493.609	−	854.955i	−0.410927	−	0.711747i	−0.994991	+	0.0999606i	$0.968128\pi$
$114$		0			0
$115$	−536.228	+	928.774i	−0.434813	+	0.753119i
$116$	−688.526			−0.551104
$117$		0			0
$118$	−1981.04			−1.54550
$119$	89.9712	−	155.835i	0.0693080	−	0.120045i
$120$		0			0
$121$	664.092	+	1150.24i	0.498942	+	0.864193i
$122$	1391.36	+	2409.91i	1.03252	+	1.78838i
$123$		0			0
$124$	1224.42	−	2120.76i	0.886744	−	1.53589i
$125$	125.000			0.0894427
$126$		0			0
$127$	1480.10			1.03415			0.517077	−	0.855939i	$-0.327020\pi$
$127$	1480.10			1.03415			0.517077	+	0.855939i	$0.327020\pi$
$128$	−249.214	+	431.651i	−0.172091	+	0.298070i
$129$		0			0
$130$	114.383	+	198.117i	0.0771695	+	0.133662i
$131$	253.298	+	438.724i	0.168937	+	0.292607i	0.938046	−	0.346510i	$-0.112633\pi$
$131$	253.298	+	438.724i	0.168937	+	0.292607i	−0.769110	+	0.639117i	$0.779300\pi$
$132$		0			0
$133$	−507.000	+	878.149i	−0.330545	+	0.572520i
$134$	−3683.33			−2.37456
$135$		0			0
$136$	−35.2442			−0.0222218
$137$	−487.861	+	845.001i	−0.304240	+	0.526958i	−0.977092	−	0.212819i	$-0.931736\pi$
$137$	−487.861	+	845.001i	−0.304240	+	0.526958i	0.672852	+	0.739777i	$0.265069\pi$
$138$		0			0
$139$	944.413	+	1635.77i	0.576288	+	0.998160i	0.995900	+	0.0904566i	$0.0288327\pi$
$139$	944.413	+	1635.77i	0.576288	+	0.998160i	−0.419612	+	0.907703i	$0.637834\pi$
$140$	448.138	+	776.198i	0.270533	+	0.468576i
$141$		0			0
$142$	1486.25	−	2574.26i	0.878334	−	1.52132i
$143$	−18.6494			−0.0109059
$144$		0			0
$145$	−384.544			−0.220239
$146$	1884.04	−	3263.26i	1.06798	−	1.84979i
$147$		0			0
$148$	614.513	+	1064.37i	0.341302	+	0.591152i
$149$	−1291.15	−	2236.33i	−0.709898	−	1.22958i	−0.964895	−	0.262637i	$-0.915408\pi$
$149$	−1291.15	−	2236.33i	−0.709898	−	1.22958i	0.254997	−	0.966942i	$-0.417926\pi$
$150$		0			0
$151$	−1268.96	+	2197.91i	−0.683885	+	1.18452i	0.289901	+	0.957057i	$0.406378\pi$
$151$	−1268.96	+	2197.91i	−0.683885	+	1.18452i	−0.973786	+	0.227467i	$0.926956\pi$
$152$	198.606			0.105981
$153$		0			0
$154$	−138.358			−0.0723976
$155$	683.843	−	1184.45i	0.354372	−	0.613789i
$156$		0			0
$157$	−52.5175	−	90.9630i	−0.0266965	−	0.0462397i	0.852368	−	0.522942i	$-0.175165\pi$
$157$	−52.5175	−	90.9630i	−0.0266965	−	0.0462397i	−0.879065	+	0.476702i	$0.841832\pi$
$158$	122.829	+	212.746i	0.0618465	+	0.107121i
$159$		0			0
$160$	−649.436	+	1124.86i	−0.320890	+	0.555798i
$161$	−4294.74			−2.10232
$162$		0			0
$163$	2228.88			1.07104			0.535519	−	0.844523i	$-0.320116\pi$
$163$	2228.88			1.07104			0.535519	+	0.844523i	$0.320116\pi$
$164$	238.789	−	413.594i	0.113697	−	0.196929i
$165$		0			0
$166$	1529.34	+	2648.89i	0.715057	+	1.23851i
$167$	−1406.90	−	2436.82i	−0.651911	−	1.12914i	−0.982659	−	0.185424i	$-0.940634\pi$
$167$	−1406.90	−	2436.82i	−0.651911	−	1.12914i	0.330748	−	0.943719i	$-0.392699\pi$
$168$		0			0
$169$	1036.76	−	1795.72i	0.471897	−	0.817350i
$170$	−185.009			−0.0834679
$171$		0			0
$172$	2643.40			1.17185
$173$	1642.22	−	2844.41i	0.721711	−	1.25004i	−0.238603	−	0.971117i	$-0.576689\pi$
$173$	1642.22	−	2844.41i	0.721711	−	1.25004i	0.960314	−	0.278922i	$-0.0899772\pi$
$174$		0			0
$175$	250.286	+	433.509i	0.108114	+	0.187258i
$176$	−46.5490	−	80.6253i	−0.0199362	−	0.0345304i
$177$		0			0
$178$	3171.83	−	5493.78i	1.33561	−	2.31335i
$179$	1268.62			0.529726			0.264863	−	0.964286i	$-0.414673\pi$
$179$	1268.62			0.529726			0.264863	+	0.964286i	$0.414673\pi$
$180$		0			0
$181$	1080.10			0.443554			0.221777	−	0.975097i	$-0.428814\pi$
$181$	1080.10			0.443554			0.221777	+	0.975097i	$0.428814\pi$
$182$	−458.056	+	793.376i	−0.186557	+	0.323126i
$183$		0			0
$184$	420.592	+	728.487i	0.168513	+	0.291874i
$185$	343.208	+	594.453i	0.136395	+	0.236243i
$186$		0			0
$187$	7.54114	−	13.0616i	0.00294900	−	0.00510782i
$188$	−1745.59			−0.677181
$189$		0			0
$190$	1042.55			0.398077
$191$	−143.357	+	248.302i	−0.0543087	+	0.0940654i	−0.891902	−	0.452229i	$-0.850629\pi$
$191$	−143.357	+	248.302i	−0.0543087	+	0.0940654i	0.837593	+	0.546295i	$0.183962\pi$
$192$		0			0
$193$	2325.35	+	4027.63i	0.867267	+	1.50215i	0.864779	+	0.502153i	$0.167459\pi$
$193$	2325.35	+	4027.63i	0.867267	+	1.50215i	0.00248836	+	0.999997i	$0.499208\pi$
$194$	−2308.93	−	3999.19i	−0.854494	−	1.48003i
$195$		0			0
$196$	−259.252	+	449.038i	−0.0944797	+	0.163644i
$197$	1694.82			0.612947			0.306474	−	0.951879i	$-0.400851\pi$
$197$	1694.82			0.612947			0.306474	+	0.951879i	$0.400851\pi$
$198$		0			0
$199$	−3249.77			−1.15764			−0.578819	−	0.815456i	$-0.696486\pi$
$199$	−3249.77			−1.15764			−0.578819	+	0.815456i	$0.696486\pi$
$200$	49.0221	−	84.9088i	0.0173319	−	0.0300198i
$201$		0			0
$202$	−3620.12	−	6270.23i	−1.26094	−	2.18402i
$203$	−769.970	−	1333.63i	−0.266213	−	0.461095i
$204$		0			0
$205$	133.364	−	230.994i	0.0454369	−	0.0786990i
$206$	−4349.90			−1.47122
$207$		0			0
$208$	−616.430			−0.205489
$209$	−42.4953	+	73.6041i	−0.0140644	+	0.0243603i
$210$		0			0
$211$	539.279	+	934.059i	0.175950	+	0.304755i	0.940490	−	0.339822i	$-0.110367\pi$
$211$	539.279	+	934.059i	0.175950	+	0.304755i	−0.764539	+	0.644577i	$0.777034\pi$
$212$	2015.09	+	3490.24i	0.652815	+	1.13071i
$213$		0			0
$214$	3026.31	−	5241.72i	0.966702	−	1.67438i
$215$	1476.35			0.468308
$216$		0			0
$217$	5477.01			1.71338
$218$	1891.69	−	3276.50i	0.587713	−	1.01795i
$219$		0			0
$220$	37.5617	+	65.0588i	0.0115110	+	0.0199376i
$221$	−49.9321	−	86.4850i	−0.0151982	−	0.0263240i
$222$		0			0
$223$	2571.64	−	4454.21i	0.772241	−	1.33756i	−0.164091	−	0.986445i	$-0.552469\pi$
$223$	2571.64	−	4454.21i	0.772241	−	1.33756i	0.936332	−	0.351115i	$-0.114197\pi$
$224$	−5201.44			−1.55150
$225$		0			0
$226$	−4064.71			−1.19637
$227$	−1454.83	+	2519.84i	−0.425377	+	0.736774i	−0.996456	−	0.0841209i	$-0.973192\pi$
$227$	−1454.83	+	2519.84i	−0.425377	+	0.736774i	0.571079	+	0.820895i	$0.306525\pi$
$228$		0			0
$229$	−862.910	−	1494.60i	−0.249007	−	0.431293i	0.714243	−	0.699897i	$-0.246771\pi$
$229$	−862.910	−	1494.60i	−0.249007	−	0.431293i	−0.963251	+	0.268604i	$0.913438\pi$
$230$	2207.83	+	3824.08i	0.632958	+	1.09631i
$231$		0			0
$232$	−150.809	+	261.209i	−0.0426772	+	0.0739191i
$233$	1087.48			0.305765			0.152882	−	0.988244i	$-0.451144\pi$
$233$	1087.48			0.305765			0.152882	+	0.988244i	$0.451144\pi$
$234$		0			0
$235$	−974.916			−0.270623
$236$	−2153.72	+	3730.36i	−0.594049	+	1.02892i
$237$		0			0
$238$	−370.442	−	641.625i	−0.100892	−	0.174749i
$239$	159.672	+	276.561i	0.0432149	+	0.0748504i	0.886824	−	0.462108i	$-0.152907\pi$
$239$	159.672	+	276.561i	0.0432149	+	0.0748504i	−0.843609	+	0.536958i	$0.819573\pi$
$240$		0			0
$241$	−1822.91	+	3157.38i	−0.487238	+	0.843920i	−0.999892	−	0.0146745i	$-0.995329\pi$
$241$	−1822.91	+	3157.38i	−0.487238	+	0.843920i	0.512655	+	0.858595i	$0.328662\pi$
$242$	5468.58			1.45262
$243$		0			0
$244$	6050.58			1.58750
$245$	−144.793	+	250.789i	−0.0377572	+	0.0653973i
$246$		0			0
$247$	281.374	+	487.354i	0.0724834	+	0.125545i
$248$	−536.375	−	929.028i	−0.137338	−	0.237876i
$249$		0			0
$250$	257.334	−	445.715i	0.0651009	−	0.112758i
$251$	−572.874			−0.144062			−0.0720309	−	0.997402i	$-0.522948\pi$
$251$	−572.874			−0.144062			−0.0720309	+	0.997402i	$0.522948\pi$
$252$		0			0
$253$	−359.973			−0.0894519
$254$	3047.04	−	5277.62i	0.752709	−	1.30373i
$255$		0			0
$256$	−1477.09	−	2558.40i	−0.360618	−	0.624609i
$257$	1838.83	+	3184.94i	0.446315	+	0.773040i	0.998143	−	0.0609179i	$-0.0194028\pi$
$257$	1838.83	+	3184.94i	0.446315	+	0.773040i	−0.551828	+	0.833958i	$0.686069\pi$
$258$		0			0
$259$	−1374.40	+	2380.54i	−0.329735	+	0.571117i
$260$	497.414			0.118647
$261$		0			0
$262$	2085.83			0.491842
$263$	−1000.86	+	1733.53i	−0.234659	+	0.406442i	−0.959174	−	0.282818i	$-0.908731\pi$
$263$	−1000.86	+	1733.53i	−0.234659	+	0.406442i	0.724514	+	0.689260i	$0.242064\pi$
$264$		0			0
$265$	1125.43	+	1949.31i	0.260886	+	0.451868i
$266$	2087.49	+	3615.64i	0.481174	+	0.833418i
$267$		0			0
$268$	−4004.41	+	6935.83i	−0.912716	+	1.58087i
$269$	−20.1629			−0.00457009			−0.00228504	−	0.999997i	$-0.500727\pi$
$269$	−20.1629			−0.00457009			−0.00228504	+	0.999997i	$0.500727\pi$
$270$		0			0
$271$	4733.12			1.06095			0.530474	−	0.847701i	$-0.322014\pi$
$271$	4733.12			1.06095			0.530474	+	0.847701i	$0.322014\pi$
$272$	249.262	−	431.734i	0.0555651	−	0.0962416i
$273$		0			0
$274$	2008.69	+	3479.16i	0.442881	+	0.767093i
$275$	20.9783	+	36.3355i	0.00460015	+	0.00796769i
$276$		0			0
$277$	−2514.56	+	4355.34i	−0.545434	+	0.944719i	0.453146	+	0.891436i	$0.350302\pi$
$277$	−2514.56	+	4355.34i	−0.545434	+	0.944719i	−0.998579	+	0.0532826i	$0.983032\pi$
$278$	7776.94			1.67780
$279$		0			0
$280$	392.626			0.0837996
$281$	−2904.16	+	5030.14i	−0.616539	+	1.06788i	0.373574	+	0.927600i	$0.378132\pi$
$281$	−2904.16	+	5030.14i	−0.616539	+	1.06788i	−0.990112	+	0.140276i	$0.955201\pi$
$282$		0			0
$283$	−106.512	−	184.483i	−0.0223726	−	0.0387505i	0.854622	−	0.519250i	$-0.173789\pi$
$283$	−106.512	−	184.483i	−0.0223726	−	0.0387505i	−0.876995	+	0.480500i	$0.840455\pi$
$284$	−3231.62	−	5597.32i	−0.675215	−	1.16951i
$285$		0			0
$286$	−38.3930	+	66.4986i	−0.00793785	+	0.0137488i
$287$	1068.14			0.219687
$288$		0			0
$289$	−4832.24			−0.983561
$290$	−791.650	+	1371.18i	−0.160301	+	0.277649i
$291$		0			0
$292$	−4096.55	−	7095.43i	−0.821002	−	1.42202i
$293$	1796.55	+	3111.71i	0.358210	+	0.620437i	0.987662	−	0.156602i	$-0.0500540\pi$
$293$	1796.55	+	3111.71i	0.358210	+	0.620437i	−0.629452	+	0.777039i	$0.716721\pi$
$294$		0			0
$295$	−1202.86	+	2083.42i	−0.237401	+	0.411191i
$296$	538.392			0.105721
$297$		0			0
$298$	−10632.2			−2.06680
$299$	−1191.75	+	2064.16i	−0.230503	+	0.399243i
$300$		0			0
$301$	2956.08	+	5120.09i	0.566066	+	0.980455i
$302$	5224.75	+	9049.53i	0.995531	+	1.72431i
$303$		0			0
$304$	−1404.62	+	2432.88i	−0.265002	+	0.458997i
$305$	3379.27			0.634415
$306$		0			0
$307$	2403.23			0.446774			0.223387	−	0.974730i	$-0.428289\pi$
$307$	2403.23			0.446774			0.223387	+	0.974730i	$0.428289\pi$
$308$	−150.419	+	260.533i	−0.0278277	+	0.0481989i
$309$		0			0
$310$	−2815.61	−	4876.79i	−0.515859	−	0.893493i
$311$	804.080	+	1392.71i	0.146608	+	0.253933i	0.929972	−	0.367631i	$-0.119831\pi$
$311$	804.080	+	1392.71i	0.146608	+	0.253933i	−0.783363	+	0.621564i	$0.786498\pi$
$312$		0			0
$313$	−1390.27	+	2408.02i	−0.251063	+	0.434855i	−0.963819	−	0.266558i	$-0.914114\pi$
$313$	−1390.27	+	2408.02i	−0.251063	+	0.434855i	0.712755	+	0.701413i	$0.247447\pi$
$314$	−432.465			−0.0777242
$315$		0			0
$316$	534.144			0.0950885
$317$	−2767.85	+	4794.05i	−0.490403	+	0.849403i	−0.999939	−	0.0110461i	$-0.996484\pi$
$317$	−2767.85	+	4794.05i	−0.490403	+	0.849403i	0.509536	+	0.860449i	$0.329817\pi$
$318$		0			0
$319$	−64.5368	−	111.781i	−0.0113272	−	0.0196192i
$320$	1564.49	+	2709.78i	0.273306	+	0.473380i
$321$		0			0
$322$	−8841.46	+	15313.9i	−1.53017	+	2.65033i
$323$	−455.110			−0.0783994
$324$		0			0
$325$	277.807			0.0474153
$326$	4588.53	−	7947.56i	0.779555	−	1.35023i
$327$		0			0
$328$	−104.605	−	181.181i	−0.0176092	−	0.0305001i
$329$	−1952.07	−	3381.08i	−0.327115	−	0.566580i
$330$		0			0
$331$	4808.37	−	8328.35i	0.798466	−	1.38298i	−0.122149	−	0.992512i	$-0.538979\pi$
$331$	4808.37	−	8328.35i	0.798466	−	1.38298i	0.920615	−	0.390471i	$-0.127688\pi$
$332$	6650.59			1.09939
$333$		0			0
$334$	−11585.4			−1.89797
$335$	−2236.47	+	3873.69i	−0.364751	+	0.631767i
$336$		0			0
$337$	1042.23	+	1805.19i	0.168468	+	0.291795i	0.937881	−	0.346956i	$-0.112785\pi$
$337$	1042.23	+	1805.19i	0.168468	+	0.291795i	−0.769413	+	0.638751i	$0.779451\pi$
$338$	−4268.69	−	7393.58i	−0.686941	−	1.18982i
$339$		0			0
$340$	−201.136	+	348.378i	−0.0320828	+	0.0555690i
$341$	459.068			0.0729030
$342$		0			0
$343$	5708.19			0.898581
$344$	578.990	−	1002.84i	0.0907472	−	0.157179i
$345$		0			0
$346$	−6761.59	−	11711.4i	−1.05059	−	1.81968i
$347$	3056.71	+	5294.38i	0.472890	+	0.819070i	0.999519	−	0.0310257i	$-0.00987737\pi$
$347$	3056.71	+	5294.38i	0.472890	+	0.819070i	−0.526628	+	0.850096i	$0.676544\pi$
$348$		0			0
$349$	−1094.53	+	1895.78i	−0.167877	+	0.290771i	−0.937673	−	0.347519i	$-0.887024\pi$
$349$	−1094.53	+	1895.78i	−0.167877	+	0.290771i	0.769797	+	0.638289i	$0.220358\pi$
$350$	2061.03			0.314762
$351$		0			0
$352$	−435.971			−0.0660151
$353$	1250.10	−	2165.24i	0.188488	−	0.326471i	−0.756258	−	0.654273i	$-0.772975\pi$
$353$	1250.10	−	2165.24i	0.188488	−	0.326471i	0.944746	+	0.327802i	$0.106308\pi$
$354$		0			0
$355$	−1804.87	−	3126.12i	−0.269838	−	0.467373i
$356$	−6896.65	−	11945.3i	−1.02675	−	1.77838i
$357$		0			0
$358$	2611.67	−	4523.54i	0.385561	−	0.667812i
$359$	841.607			0.123728			0.0618639	−	0.998085i	$-0.480296\pi$
$359$	841.607			0.123728			0.0618639	+	0.998085i	$0.480296\pi$
$360$		0			0
$361$	−4294.40			−0.626096
$362$	2223.57	−	3851.34i	0.322841	−	0.559176i
$363$		0			0
$364$	995.969	+	1725.07i	0.143415	+	0.248402i
$365$	−2287.94	−	3962.82i	−0.328099	−	0.568284i
$366$		0			0
$367$	2807.21	−	4862.23i	0.399278	−	0.691571i	−0.594359	−	0.804200i	$-0.702594\pi$
$367$	2807.21	−	4862.23i	0.399278	−	0.691571i	0.993637	+	0.112630i	$0.0359273\pi$
$368$	−11898.4			−1.68546
$369$		0			0
$370$	2826.21			0.397101
$371$	−4506.89	+	7806.17i	−0.630691	+	1.09239i
$372$		0			0
$373$	−659.587	−	1142.44i	−0.0915607	−	0.158588i	0.816607	−	0.577194i	$-0.195852\pi$
$373$	−659.587	−	1142.44i	−0.0915607	−	0.158588i	−0.908168	+	0.418606i	$0.862519\pi$
$374$	−31.0495	−	53.7792i	−0.00429286	−	0.00743545i
$375$		0			0
$376$	−382.339	+	662.231i	−0.0524405	+	0.0908297i
$377$	−854.634			−0.116753
$378$		0			0
$379$	−5002.07			−0.677939			−0.338970	−	0.940797i	$-0.610078\pi$
$379$	−5002.07			−0.677939			−0.338970	+	0.940797i	$0.610078\pi$
$380$	1133.43	−	1963.16i	0.153010	−	0.265021i
$381$		0			0
$382$	590.251	+	1022.34i	0.0790572	+	0.136931i
$383$	−1121.22	−	1942.02i	−0.149587	−	0.259092i	0.781488	−	0.623920i	$-0.214461\pi$
$383$	−1121.22	−	1942.02i	−0.149587	−	0.259092i	−0.931075	+	0.364828i	$0.881128\pi$
$384$		0			0
$385$	−84.0095	+	145.509i	−0.0111208	+	0.0192618i
$386$	19148.5			2.52496
$387$		0			0
$388$	−10040.8			−1.31378
$389$	−5696.89	+	9867.30i	−0.742529	+	1.28610i	0.208811	+	0.977956i	$0.433041\pi$
$389$	−5696.89	+	9867.30i	−0.742529	+	1.28610i	−0.951340	+	0.308142i	$0.900293\pi$
$390$		0			0
$391$	−963.797	−	1669.35i	−0.124658	−	0.215914i
$392$	113.569	+	196.707i	0.0146329	+	0.0253450i
$393$		0			0
$394$	3489.07	−	6043.24i	0.446134	−	0.772726i
$395$	298.321			0.0380004
$396$		0			0
$397$	−14926.0			−1.88694			−0.943472	−	0.331454i	$-0.892461\pi$
$397$	−14926.0			−1.88694			−0.943472	+	0.331454i	$0.892461\pi$
$398$	−6690.20	+	11587.8i	−0.842586	+	1.45940i
$399$		0			0
$400$	693.409	+	1201.02i	0.0866762	+	0.150128i
$401$	−3582.82	−	6205.62i	−0.446178	−	0.772803i	0.551955	−	0.833874i	$-0.313882\pi$
$401$	−3582.82	−	6205.62i	−0.446178	−	0.772803i	−0.998133	+	0.0610707i	$0.980548\pi$
$402$		0			0
$403$	1519.81	−	2632.39i	0.187859	−	0.325382i
$404$	−15742.7			−1.93869
$405$		0			0
$406$	−6340.46			−0.775053
$407$	−115.199	+	199.530i	−0.0140299	+	0.0243006i
$408$		0			0
$409$	5662.05	+	9806.96i	0.684524	+	1.18563i	0.973586	+	0.228321i	$0.0733234\pi$
$409$	5662.05	+	9806.96i	0.684524	+	1.18563i	−0.289062	+	0.957310i	$0.593343\pi$
$410$	−549.106	−	951.080i	−0.0661425	−	0.114562i
$411$		0			0
$412$	−4729.08	+	8191.01i	−0.565498	+	0.979471i
$413$	−9633.93			−1.14783
$414$		0			0
$415$	3714.38			0.439353
$416$	−1443.35	+	2499.95i	−0.170110	+	0.294639i
$417$		0			0
$418$	174.968	+	303.053i	0.0204736	+	0.0354613i
$419$	−6349.90	−	10998.4i	−0.740365	−	1.28235i	−0.952329	−	0.305072i	$-0.901319\pi$
$419$	−6349.90	−	10998.4i	−0.740365	−	1.28235i	0.211964	−	0.977277i	$-0.432014\pi$
$420$		0			0
$421$	8581.39	−	14863.4i	0.993424	−	1.72066i	0.397557	−	0.917577i	$-0.369858\pi$
$421$	8581.39	−	14863.4i	0.993424	−	1.72066i	0.595867	−	0.803083i	$-0.296809\pi$
$422$	4440.79			0.512261
$423$		0			0
$424$	1765.48			0.202215
$425$	−112.335	+	194.570i	−0.0128213	+	0.0222072i
$426$		0			0
$427$	6766.29	+	11719.6i	0.766847	+	1.32822i
$428$	−6580.23	−	11397.3i	−0.743148	−	1.28717i
$429$		0			0
$430$	3039.32	−	5264.25i	0.340858	−	0.590383i
$431$	11130.9			1.24399			0.621994	−	0.783022i	$-0.286323\pi$
$431$	11130.9			1.24399			0.621994	+	0.783022i	$0.286323\pi$
$432$		0			0
$433$	7675.55			0.851878			0.425939	−	0.904752i	$-0.359944\pi$
$433$	7675.55			0.851878			0.425939	+	0.904752i	$0.359944\pi$
$434$	11275.4	−	19529.5i	1.24708	−	2.16001i
$435$		0			0
$436$	−4113.17	−	7124.23i	−0.451801	−	0.782543i
$437$	5431.12	+	9406.98i	0.594522	+	1.02974i
$438$		0			0
$439$	−1681.90	+	2913.13i	−0.182853	+	0.316711i	−0.942851	−	0.333214i	$-0.891867\pi$
$439$	−1681.90	+	2913.13i	−0.182853	+	0.316711i	0.759998	+	0.649926i	$0.225200\pi$
$440$	32.9089			0.00356561
$441$		0			0
$442$	−411.175			−0.0442480
$443$	3947.03	−	6836.46i	0.423316	−	0.733205i	−0.572945	−	0.819594i	$-0.694199\pi$
$443$	3947.03	−	6836.46i	0.423316	−	0.733205i	0.996262	+	0.0863883i	$0.0275326\pi$
$444$		0			0
$445$	−3851.80	−	6671.51i	−0.410321	−	0.710697i
$446$	−10588.3	−	18339.5i	−1.12415	−	1.94709i
$447$		0			0
$448$	−6265.15	+	10851.6i	−0.660715	+	1.14439i
$449$	−2244.52			−0.235914			−0.117957	−	0.993019i	$-0.537634\pi$
$449$	−2244.52			−0.235914			−0.117957	+	0.993019i	$0.537634\pi$
$450$		0			0
$451$	89.5283			0.00934750
$452$	−4419.03	+	7653.99i	−0.459853	+	0.796489i
$453$		0			0
$454$	5990.04	+	10375.1i	0.619221	+	1.07252i
$455$	556.252	+	963.456i	0.0573131	+	0.0992693i
$456$		0			0
$457$	−7374.65	+	12773.3i	−0.754862	+	1.30746i	0.190581	+	0.981671i	$0.438963\pi$
$457$	−7374.65	+	12773.3i	−0.754862	+	1.30746i	−0.945443	+	0.325787i	$0.894371\pi$
$458$	−7105.79			−0.724960
$459$		0			0
$460$	9601.16			0.973166
$461$	6239.75	−	10807.6i	0.630400	−	1.09188i	−0.357070	−	0.934078i	$-0.616224\pi$
$461$	6239.75	−	10807.6i	0.630400	−	1.09188i	0.987470	−	0.157807i	$-0.0504424\pi$
$462$		0			0
$463$	4839.56	+	8382.37i	0.485774	+	0.841386i	0.999866	−	0.0163492i	$-0.00520436\pi$
$463$	4839.56	+	8382.37i	0.485774	+	0.841386i	−0.514092	+	0.857735i	$0.671871\pi$
$464$	−2133.17	−	3694.76i	−0.213427	−	0.369666i
$465$		0			0
$466$	2238.76	−	3877.65i	0.222551	−	0.385469i
$467$	−6714.33			−0.665315			−0.332658	−	0.943048i	$-0.607945\pi$
$467$	−6714.33			−0.665315			−0.332658	+	0.943048i	$0.607945\pi$
$468$		0			0
$469$	−17912.3			−1.76357
$470$	−2007.03	+	3476.28i	−0.196973	+	0.341168i
$471$		0			0
$472$	943.469	+	1634.14i	0.0920057	+	0.159359i
$473$	247.771	+	429.152i	0.0240857	+	0.0417176i
$474$		0			0
$475$	633.024	−	1096.43i	0.0611477	−	0.105911i
$476$	−1610.93			−0.155120
$477$		0			0
$478$	1314.85			0.125816
$479$	1063.74	−	1842.45i	0.101469	−	0.175749i	−0.810821	−	0.585294i	$-0.800979\pi$
$479$	1063.74	−	1842.45i	0.101469	−	0.175749i	0.912290	+	0.409545i	$0.134313\pi$
$480$		0			0
$481$	762.765	+	1321.15i	0.0723058	+	0.125237i
$482$	7505.56	+	13000.0i	0.709272	+	1.22849i
$483$		0			0
$484$	5945.28	−	10297.5i	0.558347	−	0.967086i
$485$	−5607.83			−0.525028
$486$		0			0
$487$	−6549.04			−0.609375			−0.304687	−	0.952452i	$-0.598552\pi$
$487$	−6549.04			−0.609375			−0.304687	+	0.952452i	$0.598552\pi$
$488$	1325.27	−	2295.44i	0.122935	−	0.212929i
$489$		0			0
$490$	596.163	+	1032.59i	0.0549631	+	0.0951989i
$491$	6466.86	+	11200.9i	0.594389	+	1.02951i	0.993633	+	0.112668i	$0.0359396\pi$
$491$	6466.86	+	11200.9i	0.594389	+	1.02951i	−0.399243	+	0.916845i	$0.630727\pi$
$492$		0			0
$493$	345.583	−	598.567i	0.0315705	−	0.0546818i
$494$	2317.03			0.211028
$495$		0			0
$496$	15173.9			1.37364
$497$	7227.74	−	12518.8i	0.652331	−	1.12987i
$498$		0			0
$499$	−2560.33	−	4434.62i	−0.229692	−	0.397837i	0.728025	−	0.685550i	$-0.240438\pi$
$499$	−2560.33	−	4434.62i	−0.229692	−	0.397837i	−0.957717	+	0.287713i	$0.907105\pi$
$500$	−559.531	−	969.137i	−0.0500460	−	0.0866822i
$501$		0			0
$502$	−1179.36	+	2042.71i	−0.104855	+	0.181615i
$503$	−10809.6			−0.958204			−0.479102	−	0.877759i	$-0.659038\pi$
$503$	−10809.6			−0.958204			−0.479102	+	0.877759i	$0.659038\pi$
$504$		0			0
$505$	−8792.36			−0.774762
$506$	−741.067	+	1283.57i	−0.0651076	+	0.112770i
$507$		0			0
$508$	−6625.29	−	11475.3i	−0.578642	−	1.00224i
$509$	7378.46	+	12779.9i	0.642524	+	1.11288i	0.984868	+	0.173309i	$0.0554459\pi$
$509$	7378.46	+	12779.9i	0.642524	+	1.11288i	−0.342344	+	0.939575i	$0.611221\pi$
$510$		0			0
$511$	9162.24	−	15869.5i	0.793177	−	1.37382i
$512$	−16150.8			−1.39409
$513$		0			0
$514$	15142.2			1.29940
$515$	−2641.21	+	4574.71i	−0.225991	+	0.391428i
$516$		0			0
$517$	−163.617	−	283.393i	−0.0139185	−	0.0241075i
$518$	5658.89	+	9801.48i	0.479995	+	0.831375i
$519$		0			0
$520$	108.950	−	188.706i	0.00918800	−	0.0159141i
$521$	1742.68			0.146542			0.0732709	−	0.997312i	$-0.476656\pi$
$521$	1742.68			0.146542			0.0732709	+	0.997312i	$0.476656\pi$
$522$		0			0
$523$	10304.8			0.861565			0.430782	−	0.902456i	$-0.358238\pi$
$523$	10304.8			0.861565			0.430782	+	0.902456i	$0.358238\pi$
$524$	2267.65	−	3927.68i	0.189051	−	0.327446i
$525$		0			0
$526$	4120.87	+	7137.55i	0.341594	+	0.591658i
$527$	1229.11	+	2128.89i	0.101596	+	0.175969i
$528$		0			0
$529$	−16919.7	+	29305.9i	−1.39063	+	2.40863i
$530$	9267.59			0.759544
$531$		0			0
$532$	9077.83			0.739800
$533$	296.397	−	513.374i	0.0240870	−	0.0417199i
$534$		0			0
$535$	−3675.08	−	6365.42i	−0.296986	−	0.514395i
$536$	1754.19	+	3038.34i	0.141361	+	0.244844i
$537$		0			0
$538$	−41.5087	+	71.8953i	−0.00332634	+	0.00576138i
$539$	−97.2007			−0.00776759
$540$		0			0
$541$	3966.86			0.315247			0.157623	−	0.987499i	$-0.449617\pi$
$541$	3966.86			0.315247			0.157623	+	0.987499i	$0.449617\pi$
$542$	9743.94	−	16877.0i	0.772211	−	1.33751i
$543$		0			0
$544$	−1167.27	−	2021.78i	−0.0919971	−	0.159344i
$545$	−2297.22	−	3978.90i	−0.180554	−	0.312729i
$546$		0			0
$547$	2904.36	−	5030.50i	0.227023	−	0.393215i	−0.729902	−	0.683552i	$-0.760434\pi$
$547$	2904.36	−	5030.50i	0.227023	−	0.393215i	0.956924	+	0.290337i	$0.0937675\pi$
$548$	8735.16			0.680926
$549$		0			0
$550$	172.750			0.0133929
$551$	−1947.41	+	3373.01i	−0.150567	+	0.260789i
$552$		0			0
$553$	597.326	+	1034.60i	0.0459329	+	0.0795581i
$554$	10353.3	+	17932.4i	0.793988	+	1.37523i
$555$		0			0
$556$	8454.85	−	14644.2i	0.644902	−	1.11700i
$557$	3563.91			0.271109			0.135554	−	0.990770i	$-0.456718\pi$
$557$	3563.91			0.271109			0.135554	+	0.990770i	$0.456718\pi$
$558$		0			0
$559$	3281.13			0.248259
$560$	−2776.82	+	4809.59i	−0.209539	+	0.362932i
$561$		0			0
$562$	11957.4	+	20710.8i	0.897495	+	1.55451i
$563$	255.772	+	443.010i	0.0191466	+	0.0331628i	0.875440	−	0.483327i	$-0.160572\pi$
$563$	255.772	+	443.010i	0.0191466	+	0.0331628i	−0.856293	+	0.516490i	$0.827238\pi$
$564$		0			0
$565$	−2468.04	+	4274.78i	−0.183772	+	0.318303i
$566$	−877.089			−0.0651357
$567$		0			0
$568$	−2831.31			−0.209153
$569$	−4058.27	+	7029.13i	−0.299001	+	0.517885i	−0.975908	−	0.218184i	$-0.929987\pi$
$569$	−4058.27	+	7029.13i	−0.299001	+	0.517885i	0.676907	+	0.736069i	$0.263320\pi$
$570$		0			0
$571$	−7539.06	−	13058.0i	−0.552539	−	0.957026i	−0.998090	−	0.0617692i	$-0.980326\pi$
$571$	−7539.06	−	13058.0i	−0.552539	−	0.957026i	0.445551	−	0.895256i	$-0.353008\pi$
$572$	83.4794	+	144.591i	0.00610218	+	0.0105693i
$573$		0			0
$574$	2198.94	−	3808.68i	0.159899	−	0.276953i
$575$	5362.28			0.388909
$576$		0			0
$577$	27101.5			1.95537			0.977685	−	0.210077i	$-0.0673716\pi$
$577$	27101.5			1.95537			0.977685	+	0.210077i	$0.0673716\pi$
$578$	−9947.98	+	17230.4i	−0.715885	+	1.23995i
$579$		0			0
$580$	1721.32	+	2981.41i	0.123231	+	0.213442i
$581$	7437.26	+	12881.7i	0.531067	+	0.919834i
$582$		0			0
$583$	−377.755	+	654.292i	−0.0268354	+	0.0464803i
$584$	−3589.10			−0.254312
$585$		0			0
$586$	14794.0			1.04289
$587$	−51.0411	+	88.4057i	−0.00358891	+	0.00621617i	−0.867814	−	0.496889i	$-0.834476\pi$
$587$	−51.0411	+	88.4057i	−0.00358891	+	0.00621617i	0.864225	+	0.503105i	$0.167809\pi$
$588$		0			0
$589$	−6926.22	−	11996.6i	−0.484533	−	0.839236i
$590$	4952.59	+	8578.15i	0.345585	+	0.598571i
$591$		0			0
$592$	−3807.73	+	6595.19i	−0.264353	+	0.457872i
$593$	16467.2			1.14035			0.570174	−	0.821524i	$-0.306876\pi$
$593$	16467.2			1.14035			0.570174	+	0.821524i	$0.306876\pi$
$594$		0			0
$595$	−899.712			−0.0619909
$596$	−11559.0	+	20020.7i	−0.794420	+	1.37598i
$597$		0			0
$598$	4906.82	+	8498.87i	0.335543	+	0.581178i
$599$	−11608.1	−	20105.8i	−0.791809	−	1.37145i	−0.924846	−	0.380342i	$-0.875806\pi$
$599$	−11608.1	−	20105.8i	−0.791809	−	1.37145i	0.133038	−	0.991111i	$-0.457527\pi$
$600$		0			0
$601$	839.147	−	1453.45i	0.0569543	−	0.0986477i	−0.836143	−	0.548512i	$-0.815194\pi$
$601$	839.147	−	1453.45i	0.0569543	−	0.0986477i	0.893097	+	0.449865i	$0.148528\pi$
$602$	24342.4			1.64804
$603$		0			0
$604$	22720.8			1.53062
$605$	3320.46	−	5751.20i	0.223134	−	0.386479i
$606$		0			0
$607$	−9653.64	−	16720.6i	−0.645517	−	1.11807i	−0.984182	−	0.177162i	$-0.943308\pi$
$607$	−9653.64	−	16720.6i	−0.645517	−	1.11807i	0.338664	−	0.940907i	$-0.390025\pi$
$608$	6577.74	+	11393.0i	0.438754	+	0.759944i
$609$		0			0
$610$	6956.80	−	12049.5i	0.461759	−	0.799789i
$611$	−2166.71			−0.143463
$612$		0			0
$613$	13659.9			0.900032			0.450016	−	0.893020i	$-0.351418\pi$
$613$	13659.9			0.900032			0.450016	+	0.893020i	$0.351418\pi$
$614$	4947.46	−	8569.26i	0.325185	−	0.563236i
$615$		0			0
$616$	65.8931	+	114.130i	0.00430992	+	0.00746500i
$617$	7301.84	+	12647.2i	0.476436	+	0.825211i	0.999635	−	0.0269990i	$-0.00859510\pi$
$617$	7301.84	+	12647.2i	0.476436	+	0.825211i	−0.523200	+	0.852210i	$0.675262\pi$
$618$		0			0
$619$	13194.1	−	22852.9i	0.856732	−	1.48390i	−0.0182965	−	0.999833i	$-0.505824\pi$
$619$	13194.1	−	22852.9i	0.856732	−	1.48390i	0.875029	−	0.484071i	$-0.160842\pi$
$620$	−12244.2			−0.793128
$621$		0			0
$622$	6621.35			0.426836
$623$	15424.9	−	26716.6i	0.991948	−	1.71810i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	5724.23	+	9914.65i	0.365473	+	0.633018i
$627$		0			0
$628$	−470.163	+	814.346i	−0.0298751	+	0.0517451i
$629$	−1233.74			−0.0782072
$630$		0			0
$631$	14241.1			0.898459			0.449229	−	0.893416i	$-0.351699\pi$
$631$	14241.1			0.898459			0.449229	+	0.893416i	$0.351699\pi$
$632$	116.995	−	202.641i	0.00736360	−	0.0127541i
$633$		0			0
$634$	11396.2	+	19738.8i	0.713880	+	1.23648i
$635$	−3700.25	−	6409.02i	−0.231244	−	0.400526i
$636$		0			0
$637$	−321.797	+	557.369i	−0.0200158	+	0.0346684i
$638$	−531.440			−0.0329779
$639$		0			0
$640$	2492.14			0.153923
$641$	−9480.23	+	16420.2i	−0.584160	+	1.01180i	0.410819	+	0.911717i	$0.365243\pi$
$641$	−9480.23	+	16420.2i	−0.584160	+	1.01180i	−0.994980	+	0.100078i	$0.968091\pi$
$642$		0			0
$643$	−5527.13	−	9573.26i	−0.338987	−	0.587143i	0.645256	−	0.763967i	$-0.276751\pi$
$643$	−5527.13	−	9573.26i	−0.338987	−	0.587143i	−0.984242	+	0.176824i	$0.943418\pi$
$644$	19224.3	+	33297.5i	1.17631	+	2.03743i
$645$		0			0
$646$	−936.922	+	1622.80i	−0.0570630	+	0.0988360i
$647$	−19310.1			−1.17335			−0.586677	−	0.809821i	$-0.699564\pi$
$647$	−19310.1			−1.17335			−0.586677	+	0.809821i	$0.699564\pi$
$648$		0			0
$649$	−807.489			−0.0488393
$650$	571.914	−	990.584i	0.0345113	−	0.0597753i
$651$		0			0
$652$	−9977.02	−	17280.7i	−0.599280	−	1.03798i
$653$	3467.13	+	6005.24i	0.207778	+	0.359883i	0.951014	−	0.309147i	$-0.100043\pi$
$653$	3467.13	+	6005.24i	0.207778	+	0.359883i	−0.743236	+	0.669029i	$0.766710\pi$
$654$		0			0
$655$	1266.49	−	2193.62i	0.0755508	−	0.130858i
$656$	2959.23			0.176126
$657$		0			0
$658$	−16074.6			−0.952363
$659$	−7939.35	+	13751.4i	−0.469307	+	0.812863i	−0.999384	−	0.0350862i	$-0.988829\pi$
$659$	−7939.35	+	13751.4i	−0.469307	+	0.812863i	0.530078	+	0.847949i	$0.322163\pi$
$660$		0			0
$661$	4709.16	+	8156.51i	0.277103	+	0.479957i	0.970664	−	0.240442i	$-0.0772923\pi$
$661$	4709.16	+	8156.51i	0.277103	+	0.479957i	−0.693561	+	0.720398i	$0.743959\pi$
$662$	−19797.7	−	34290.7i	−1.16233	−	2.01321i
$663$		0			0
$664$	1456.69	−	2523.06i	0.0851364	−	0.147461i
$665$	5070.00			0.295648
$666$		0			0
$667$	−16496.3			−0.957628
$668$	−12595.3	+	21815.6i	−0.729529	+	1.26358i
$669$		0			0
$670$	9208.32	+	15949.3i	0.530968	+	0.919664i
$671$	567.132	+	982.301i	0.0326287	+	0.0565146i
$672$		0			0
$673$	9196.17	−	15928.2i	0.526726	−	0.912316i	−0.472789	−	0.881176i	$-0.656753\pi$
$673$	9196.17	−	15928.2i	0.526726	−	0.912316i	0.999515	−	0.0311403i	$-0.00991387\pi$
$674$	8582.41			0.490478
$675$		0			0
$676$	−18563.2			−1.05617
$677$	5285.34	−	9154.49i	0.300048	−	0.519698i	−0.676099	−	0.736811i	$-0.736331\pi$
$677$	5285.34	−	9154.49i	0.300048	−	0.519698i	0.976146	+	0.217113i	$0.0696641\pi$
$678$		0			0
$679$	−11228.5	−	19448.4i	−0.634626	−	1.09920i
$680$	88.1106	+	152.612i	0.00496895	+	0.00860647i
$681$		0			0
$682$	945.070	−	1636.91i	0.0530625	−	0.0919069i
$683$	23975.0			1.34316			0.671579	−	0.740933i	$-0.265616\pi$
$683$	23975.0			1.34316			0.671579	+	0.740933i	$0.265616\pi$
$684$		0			0
$685$	4878.61			0.272120
$686$	11751.3	−	20353.8i	0.654032	−	1.13282i
$687$		0			0
$688$	8189.72	+	14185.0i	0.453823	+	0.786044i
$689$	2501.23	+	4332.26i	0.138301	+	0.239544i
$690$		0			0
$691$	−9820.49	+	17009.6i	−0.540650	+	0.936433i	0.458217	+	0.888840i	$0.348488\pi$
$691$	−9820.49	+	17009.6i	−0.540650	+	0.936433i	−0.998867	+	0.0475927i	$0.984845\pi$
$692$	−29404.0			−1.61528
$693$		0			0
$694$	25171.1			1.37677
$695$	4722.06	−	8178.85i	0.257724	−	0.446391i
$696$		0			0
$697$	239.704	+	415.180i	0.0130265	+	0.0225625i
$698$	4506.56	+	7805.59i	0.244378	+	0.423275i
$699$		0			0
$700$	2240.69	−	3880.99i	0.120986	−	0.209554i
$701$	−36098.2			−1.94495			−0.972475	−	0.233007i	$-0.925143\pi$
$701$	−36098.2			−1.94495			−0.972475	+	0.233007i	$0.925143\pi$
$702$		0			0
$703$	6952.28			0.372987
$704$	−525.128	+	909.548i	−0.0281129	+	0.0486930i
$705$		0			0
$706$	−5147.10	−	8915.04i	−0.274382	−	0.475243i
$707$	−17604.9	−	30492.5i	−0.936492	−	1.62205i
$708$		0			0
$709$	−17617.1	+	30513.6i	−0.933177	+	1.61631i	−0.155323	+	0.987864i	$0.549642\pi$
$709$	−17617.1	+	30513.6i	−0.933177	+	1.61631i	−0.777853	+	0.628446i	$0.783691\pi$
$710$	−14862.5			−0.785606
$711$		0			0
$712$	−6042.34			−0.318043
$713$	29335.7	−	50810.9i	1.54085	−	2.66884i
$714$		0			0
$715$	46.6235	+	80.7543i	0.00243863	+	0.00422383i
$716$	−5678.65	−	9835.71i	−0.296398	−	0.513377i
$717$		0			0
$718$	1732.59	−	3000.94i	0.0900553	−	0.155980i
$719$	22089.7			1.14577			0.572885	−	0.819636i	$-0.305824\pi$
$719$	22089.7			1.14577			0.572885	+	0.819636i	$0.305824\pi$
$720$		0			0
$721$	−21153.9			−1.09267
$722$	−8840.75	+	15312.6i	−0.455704	+	0.789303i
$723$		0			0
$724$	−4834.80	−	8374.12i	−0.248182	−	0.429864i
$725$	961.360	+	1665.13i	0.0492469	+	0.0852982i
$726$		0			0
$727$	9717.61	−	16831.4i	0.495744	−	0.858654i	−0.504244	−	0.863561i	$-0.668229\pi$
$727$	9717.61	−	16831.4i	0.495744	−	0.858654i	0.999988	+	0.00490721i	$0.00156202\pi$
$728$	872.596			0.0444238
$729$		0			0
$730$	−18840.4			−0.955227
$731$	−1326.77	+	2298.03i	−0.0671304	+	0.116273i
$732$		0			0
$733$	−7487.10	−	12968.0i	−0.377275	−	0.653459i	0.613390	−	0.789780i	$-0.289805\pi$
$733$	−7487.10	−	12968.0i	−0.377275	−	0.653459i	−0.990665	+	0.136321i	$0.956472\pi$
$734$	−11558.2	−	20019.5i	−0.581230	−	1.00672i
$735$		0			0
$736$	−27859.7	+	48254.3i	−1.39527	+	2.41668i
$737$	−1501.36			−0.0750384
$738$		0			0
$739$	−30663.4			−1.52635			−0.763174	−	0.646193i	$-0.776360\pi$
$739$	−30663.4			−1.52635			−0.763174	+	0.646193i	$0.776360\pi$
$740$	3072.57	−	5321.84i	0.152635	−	0.264371i
$741$		0			0
$742$	18556.4	+	32140.7i	0.918096	+	1.59019i
$743$	1707.34	+	2957.20i	0.0843018	+	0.146015i	0.905093	−	0.425213i	$-0.139801\pi$
$743$	1707.34	+	2957.20i	0.0843018	+	0.146015i	−0.820792	+	0.571228i	$0.806467\pi$
$744$		0			0
$745$	−6455.73	+	11181.6i	−0.317476	+	0.549884i
$746$	−5431.49			−0.266570
$747$		0			0
$748$	−135.024			−0.00660023
$749$	14717.2	−	25490.9i	0.717962	−	1.24355i
$750$		0			0
$751$	373.129	+	646.279i	0.0181301	+	0.0314022i	0.874948	−	0.484217i	$-0.160895\pi$
$751$	373.129	+	646.279i	0.0181301	+	0.0314022i	−0.856818	+	0.515619i	$0.827562\pi$
$752$	−5408.12	−	9367.15i	−0.262253	−	0.454235i
$753$		0			0
$754$	−1759.41	+	3047.39i	−0.0849787	+	0.147187i
$755$	12689.6			0.611685
$756$		0			0
$757$	−22929.5			−1.10091			−0.550454	−	0.834865i	$-0.685546\pi$
$757$	−22929.5			−1.10091			−0.550454	+	0.834865i	$0.685546\pi$
$758$	−10297.6	+	17836.0i	−0.493438	+	0.854660i
$759$		0			0
$760$	−496.515	−	859.989i	−0.0236980	−	0.0410461i
$761$	−10588.3	−	18339.5i	−0.504370	−	0.873595i	−0.999987	−	0.00505384i	$-0.998391\pi$
$761$	−10588.3	−	18339.5i	−0.504370	−	0.873595i	0.495617	−	0.868541i	$-0.334942\pi$
$762$		0			0
$763$	9199.42	−	15933.9i	0.436489	−	0.756021i
$764$	2566.81			0.121550
$765$		0			0
$766$	−9232.92			−0.435508
$767$	−2673.31	+	4630.31i	−0.125851	+	0.217980i
$768$		0			0
$769$	−1387.09	−	2402.51i	−0.0650453	−	0.112662i	0.831669	−	0.555272i	$-0.187386\pi$
$769$	−1387.09	−	2402.51i	−0.0650453	−	0.112662i	−0.896714	+	0.442610i	$0.854053\pi$
$770$	345.896	+	599.109i	0.0161886	+	0.0280395i
$771$		0			0
$772$	20817.7	−	36057.3i	0.970526	−	1.68100i
$773$	−28891.7			−1.34433			−0.672163	−	0.740403i	$-0.734635\pi$
$773$	−28891.7			−1.34433			−0.672163	+	0.740403i	$0.734635\pi$
$774$		0			0
$775$	−6838.43			−0.316960
$776$	−2199.26	+	3809.23i	−0.101738	+	0.176216i
$777$		0			0
$778$	23456.0	+	40627.1i	1.08090	+	1.87217i
$779$	−1350.76	−	2339.59i	−0.0621260	−	0.107605i
$780$		0			0
$781$	605.810	−	1049.29i	0.0277562	−	0.0480751i
$782$	−7936.56			−0.362930
$783$		0			0
$784$	−3212.83			−0.146357
$785$	−262.588	+	454.815i	−0.0119390	+	0.0206790i
$786$		0			0
$787$	−4404.37	−	7628.60i	−0.199490	−	0.345527i	0.748873	−	0.662714i	$-0.230595\pi$
$787$	−4404.37	−	7628.60i	−0.199490	−	0.345527i	−0.948363	+	0.317186i	$0.897262\pi$
$788$	−7586.42	−	13140.1i	−0.342963	−	0.594029i
$789$		0			0
$790$	614.145	−	1063.73i	0.0276586	−	0.0479061i
$791$	−19767.0			−0.888537
$792$		0			0
$793$	7510.29			0.336316
$794$	−30727.8	+	53222.1i	−1.37341	+	2.37882i
$795$		0			0
$796$	14546.8	+	25195.7i	0.647734	+	1.12191i
$797$	−390.251	−	675.935i	−0.0173443	−	0.0300412i	0.857223	−	0.514945i	$-0.172188\pi$
$797$	−390.251	−	675.935i	−0.0173443	−	0.0300412i	−0.874567	+	0.484904i	$0.838854\pi$
$798$		0			0
$799$	876.139	−	1517.52i	0.0387930	−	0.0671914i
$800$	6494.36			0.287013
$801$		0			0
$802$	−29503.4			−1.29900
$803$	767.954	−	1330.13i	0.0337491	−	0.0584551i
$804$		0			0
$805$	10736.9	+	18596.8i	0.470092	+	0.814224i
$806$	−6257.59	−	10838.5i	−0.273467	−	0.473658i
$807$		0			0
$808$	−3448.16	+	5972.39i	−0.150131	+	0.260035i
$809$	−15234.0			−0.662051			−0.331025	−	0.943622i	$-0.607395\pi$
$809$	−15234.0			−0.662051			−0.331025	+	0.943622i	$0.607395\pi$
$810$		0			0
$811$	10825.2			0.468711			0.234356	−	0.972151i	$-0.424702\pi$
$811$	10825.2			0.468711			0.234356	+	0.972151i	$0.424702\pi$
$812$	−6893.15	+	11939.3i	−0.297909	+	0.515994i
$813$		0			0
$814$	474.313	+	821.533i	0.0204234	+	0.0353744i
$815$	−5572.20	−	9651.33i	−0.239492	−	0.414811i
$816$		0			0
$817$	7476.52	−	12949.7i	0.320159	−	0.554532i
$818$	46625.2			1.99292
$819$		0			0
$820$	−2387.89			−0.101693
$821$	19985.6	−	34616.1i	0.849577	−	1.47151i	−0.0320090	−	0.999488i	$-0.510191\pi$
$821$	19985.6	−	34616.1i	0.849577	−	1.47151i	0.881586	−	0.472023i	$-0.156476\pi$
$822$		0			0
$823$	−14310.3	−	24786.2i	−0.606107	−	1.04981i	−0.991876	−	0.127212i	$-0.959397\pi$
$823$	−14310.3	−	24786.2i	−0.606107	−	1.04981i	0.385769	−	0.922595i	$-0.373936\pi$
$824$	2071.64	+	3588.19i	0.0875838	+	0.151700i
$825$		0			0
$826$	−19833.1	+	34351.9i	−0.835449	+	1.44704i
$827$	−2265.62			−0.0952639			−0.0476320	−	0.998865i	$-0.515167\pi$
$827$	−2265.62			−0.0952639			−0.0476320	+	0.998865i	$0.515167\pi$
$828$		0			0
$829$	−14651.6			−0.613839			−0.306919	−	0.951736i	$-0.599298\pi$
$829$	−14651.6			−0.613839			−0.306919	+	0.951736i	$0.599298\pi$
$830$	7646.68	−	13244.4i	0.319783	−	0.553881i
$831$		0			0
$832$	3477.03	+	6022.39i	0.144885	+	0.250948i
$833$	−260.246	−	450.760i	−0.0108247	−	0.0187490i
$834$		0			0
$835$	−7034.49	+	12184.1i	−0.291543	+	0.504968i
$836$	760.879			0.0314779
$837$		0			0
$838$	−52289.4			−2.15550
$839$	−11233.2	+	19456.5i	−0.462233	+	0.800611i	−0.999072	−	0.0430736i	$-0.986285\pi$
$839$	−11233.2	+	19456.5i	−0.462233	+	0.800611i	0.536839	+	0.843685i	$0.319618\pi$
$840$		0			0
$841$	9237.02	+	15999.0i	0.378737	+	0.655992i
$842$	−35332.5	−	61197.7i	−1.44613	−	2.50477i
$843$		0			0
$844$	4827.89	−	8362.16i	0.196899	−	0.341040i
$845$	−10367.6			−0.422078
$846$		0			0
$847$	26594.1			1.07885
$848$	−12486.2	+	21626.7i	−0.505633	+	0.875783i
$849$		0			0
$850$	462.523	+	801.113i	0.0186640	+	0.0323270i
$851$	14723.0	+	25501.0i	0.593065	+	1.02722i
$852$		0			0
$853$	−22235.5	+	38513.0i	−0.892531	+	1.54591i	−0.0557007	+	0.998448i	$0.517739\pi$
$853$	−22235.5	+	38513.0i	−0.892531	+	1.54591i	−0.836831	+	0.547462i	$0.815594\pi$
$854$	55718.2			2.23260
$855$		0			0
$856$	−5765.12			−0.230196
$857$	−15740.3	+	27263.0i	−0.627396	+	1.08668i	0.360676	+	0.932691i	$0.382546\pi$
$857$	−15740.3	+	27263.0i	−0.627396	+	1.08668i	−0.988072	+	0.153991i	$0.950787\pi$
$858$		0			0
$859$	−21491.5	−	37224.5i	−0.853646	−	1.47856i	−0.877895	−	0.478853i	$-0.841053\pi$
$859$	−21491.5	−	37224.5i	−0.853646	−	1.47856i	0.0242490	−	0.999706i	$-0.492281\pi$
$860$	−6608.51	−	11446.3i	−0.262033	−	0.453854i
$861$		0			0
$862$	22914.9	−	39689.9i	0.905436	−	1.56826i
$863$	25203.4			0.994131			0.497065	−	0.867713i	$-0.334411\pi$
$863$	25203.4			0.994131			0.497065	+	0.867713i	$0.334411\pi$
$864$		0			0
$865$	−16422.2			−0.645518
$866$	15801.4	−	27368.9i	0.620039	−	1.07394i
$867$		0			0
$868$	−24516.5	−	42463.8i	−0.958691	−	1.66050i
$869$	50.0662	+	86.7173i	0.00195441	+	0.00338514i
$870$		0			0
$871$	−4970.47	+	8609.11i	−0.193362	+	0.334912i
$872$	−3603.67			−0.139949
$873$		0			0
$874$	44723.6			1.73089
$875$	1251.43	−	2167.54i	0.0483499	−	0.0837444i
$876$		0			0
$877$	−1736.36	−	3007.46i	−0.0668558	−	0.115798i	0.830660	−	0.556780i	$-0.187963\pi$
$877$	−1736.36	−	3007.46i	−0.0668558	−	0.115798i	−0.897516	+	0.440982i	$0.854630\pi$
$878$	6924.95	+	11994.4i	0.266180	+	0.461037i
$879$		0			0
$880$	−232.745	+	403.126i	−0.00891572	+	0.0154425i
$881$	−26066.1			−0.996811			−0.498405	−	0.866944i	$-0.666081\pi$
$881$	−26066.1			−0.996811			−0.498405	+	0.866944i	$0.666081\pi$
$882$		0			0
$883$	−3032.93			−0.115590			−0.0577952	−	0.998328i	$-0.518407\pi$
$883$	−3032.93			−0.115590			−0.0577952	+	0.998328i	$0.518407\pi$
$884$	−447.017	+	774.257i	−0.0170077	+	0.0294582i
$885$		0			0
$886$	−16251.3	−	28148.0i	−0.616221	−	1.06733i
$887$	−21169.7	−	36667.0i	−0.801363	−	1.38800i	−0.918719	−	0.394911i	$-0.870775\pi$
$887$	−21169.7	−	36667.0i	−0.801363	−	1.38800i	0.117357	−	0.993090i	$-0.462558\pi$
$888$		0			0
$889$	14818.0	−	25665.5i	0.559031	−	0.968270i
$890$	−31718.3			−1.19461
$891$		0			0
$892$	−46045.2			−1.72837
$893$	−4937.16	+	8551.41i	−0.185012	+	0.320450i
$894$		0			0
$895$	−3171.55	−	5493.28i	−0.118450	−	0.205162i
$896$	4989.99	+	8642.92i	0.186054	+	0.322254i
$897$		0			0
$898$	−4620.72	+	8003.32i	−0.171710	+	0.297410i
$899$	21037.4			0.780464
$900$		0			0
$901$	−4045.63			−0.149589
$902$	184.309	−	319.233i	0.00680358	−	0.0117841i
$903$		0			0
$904$	1935.82	+	3352.93i	0.0712216	+	0.123359i
$905$	−2700.25	−	4676.97i	−0.0991816	−	0.171788i
$906$		0			0
$907$	10104.9	−	17502.2i	0.369931	−	0.640739i	−0.619624	−	0.784899i	$-0.712715\pi$
$907$	10104.9	−	17502.2i	0.369931	−	0.640739i	0.989554	+	0.144160i	$0.0460481\pi$
$908$	26048.8			0.952047
$909$		0			0
$910$	4580.56			0.166861
$911$	−5627.30	+	9746.78i	−0.204655	+	0.354473i	−0.950023	−	0.312180i	$-0.898941\pi$
$911$	−5627.30	+	9746.78i	−0.204655	+	0.354473i	0.745368	+	0.666654i	$0.232274\pi$
$912$		0			0
$913$	623.371	+	1079.71i	0.0225965	+	0.0391382i
$914$	30364.0	+	52591.9i	1.09885	+	1.90327i
$915$		0			0
$916$	−7725.20	+	13380.4i	−0.278655	+	0.482644i
$917$	10143.5			0.365287
$918$		0			0
$919$	−17825.9			−0.639851			−0.319925	−	0.947443i	$-0.603658\pi$
$919$	−17825.9			−0.639851			−0.319925	+	0.947443i	$0.603658\pi$
$920$	2102.96	−	3642.44i	0.0753615	−	0.130530i
$921$		0			0
$922$	−25691.2	−	44498.5i	−0.917673	−	1.58946i
$923$	−4011.25	−	6947.68i	−0.143046	−	0.247764i
$924$		0			0
$925$	1716.04	−	2972.26i	0.0609978	−	0.105651i
$926$	39852.3			1.41428
$927$		0			0
$928$	−19978.9			−0.706725
$929$	1042.12	−	1805.00i	0.0368039	−	0.0637461i	−0.847037	−	0.531534i	$-0.821616\pi$
$929$	1042.12	−	1805.00i	0.0368039	−	0.0637461i	0.883841	+	0.467788i	$0.154949\pi$
$930$		0			0
$931$	1466.52	+	2540.09i	0.0516255	+	0.0894180i
$932$	−4867.83	−	8431.33i	−0.171085	−	0.296328i
$933$		0			0
$934$	−13822.6	+	23941.4i	−0.484250	+	0.838745i
$935$	−75.4114			−0.00263767
$936$		0			0
$937$	−5419.51			−0.188952			−0.0944758	−	0.995527i	$-0.530117\pi$
$937$	−5419.51			−0.188952			−0.0944758	+	0.995527i	$0.530117\pi$
$938$	−36875.5	+	63870.3i	−1.28361	+	2.22328i
$939$		0			0
$940$	4363.97	+	7558.61i	0.151422	+	0.262271i
$941$	23206.9	+	40195.6i	0.803958	+	1.39250i	0.916993	+	0.398904i	$0.130609\pi$
$941$	23206.9	+	40195.6i	0.803958	+	1.39250i	−0.113035	+	0.993591i	$0.536057\pi$
$942$		0			0
$943$	5721.09	−	9909.23i	0.197566	−	0.342194i
$944$	−26690.4			−0.920232
$945$		0			0
$946$	2040.31			0.0701230
$947$	−9391.53	+	16266.6i	−0.322264	+	0.558177i	−0.980955	−	0.194237i	$-0.937777\pi$
$947$	−9391.53	+	16266.6i	−0.322264	+	0.558177i	0.658691	+	0.752413i	$0.271110\pi$
$948$		0			0
$949$	−5084.85	−	8807.22i	−0.173932	−	0.301258i
$950$	−2606.38	−	4514.38i	−0.0890126	−	0.154174i
$951$		0			0
$952$	−352.846	+	611.147i	−0.0120124	+	0.0208061i
$953$	−11796.7			−0.400979			−0.200489	−	0.979696i	$-0.564253\pi$
$953$	−11796.7			−0.400979			−0.200489	+	0.979696i	$0.564253\pi$
$954$		0			0
$955$	1433.57			0.0485752
$956$	1429.47	−	2475.91i	0.0483601	−	0.0837622i
$957$		0			0
$958$	−4379.78	−	7586.00i	−0.147708	−	0.255838i
$959$	9768.41	+	16919.4i	0.328924	+	0.569714i
$960$		0			0
$961$	−22515.8	+	38998.5i	−0.755792	+	1.30907i
$962$	6281.13			0.210511
$963$		0			0
$964$	32639.3			1.09050
$965$	11626.8	−	20138.1i	0.387854	−	0.671782i
$966$		0			0
$967$	5080.87	+	8800.33i	0.168966	+	0.292657i	0.938057	−	0.346482i	$-0.112624\pi$
$967$	5080.87	+	8800.33i	0.168966	+	0.292657i	−0.769091	+	0.639140i	$0.779291\pi$
$968$	−2604.41	−	4510.98i	−0.0864763	−	0.149781i
$969$		0			0
$970$	−11544.7	+	19996.0i	−0.382141	+	0.661888i
$971$	36614.9			1.21012			0.605060	−	0.796180i	$-0.293149\pi$
$971$	36614.9			1.21012			0.605060	+	0.796180i	$0.293149\pi$
$972$		0			0
$973$	37819.8			1.24609
$974$	−13482.3	+	23352.1i	−0.443533	+	0.768222i
$975$		0			0
$976$	18745.7	+	32468.6i	0.614791	+	1.06485i
$977$	25495.4	+	44159.4i	0.834874	+	1.44604i	0.894133	+	0.447801i	$0.147793\pi$
$977$	25495.4	+	44159.4i	0.834874	+	1.44604i	−0.0592594	+	0.998243i	$0.518874\pi$
$978$		0			0
$979$	1292.87	−	2239.32i	0.0422066	−	0.0731040i
$980$	2592.52			0.0845052
$981$		0			0
$982$	53252.5			1.73051
$983$	10891.9	−	18865.4i	0.353406	−	0.612117i	−0.633438	−	0.773794i	$-0.718357\pi$
$983$	10891.9	−	18865.4i	0.353406	−	0.612117i	0.986844	+	0.161676i	$0.0516901\pi$
$984$		0			0
$985$	−4237.04	−	7338.76i	−0.137059	−	0.237393i
$986$	−1422.88	−	2464.51i	−0.0459572	−	0.0796003i
$987$		0			0
$988$	2519.00	−	4363.04i	0.0811135	−	0.140493i
$989$	63332.8			2.03626
$990$		0			0
$991$	−50955.6			−1.63336			−0.816679	−	0.577092i	$-0.804187\pi$
$991$	−50955.6			−1.63336			−0.816679	+	0.577092i	$0.804187\pi$
$992$	35529.0	−	61538.0i	1.13714	−	1.96959i
$993$		0			0
$994$	−29759.1	−	51544.3i	−0.949599	−	1.64475i
$995$	8124.42	+	14071.9i	0.258856	+	0.448351i
$996$		0			0
$997$	9581.42	−	16595.5i	0.304360	−	0.527167i	−0.672759	−	0.739862i	$-0.734891\pi$
$997$	9581.42	−	16595.5i	0.304360	−	0.527167i	0.977119	+	0.212695i	$0.0682242\pi$
$998$	−21083.5			−0.668724
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.w.136.6		12
3.2	odd	2		405.4.e.x.136.1		12
9.2	odd	6		405.4.a.k.1.6	✓	6
9.4	even	3	inner	405.4.e.w.271.6		12
9.5	odd	6		405.4.e.x.271.1		12
9.7	even	3		405.4.a.l.1.1	yes	6
45.29	odd	6		2025.4.a.z.1.1		6
45.34	even	6		2025.4.a.y.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.6	✓	6	9.2	odd	6
405.4.a.l.1.1	yes	6	9.7	even	3
405.4.e.w.136.6		12	1.1	even	1	trivial
405.4.e.w.271.6		12	9.4	even	3	inner
405.4.e.x.136.1		12	3.2	odd	2
405.4.e.x.271.1		12	9.5	odd	6
2025.4.a.y.1.6		6	45.34	even	6
2025.4.a.z.1.1		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)

\(f(q)\)	\(=\)	\(q+(2.05867 - 3.56572i) q^{2} +(-4.47625 - 7.75309i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(10.0115 - 17.3404i) q^{7} -3.92177 q^{8} +O(q^{10})\)	\(q+(2.05867 - 3.56572i) q^{2} +(-4.47625 - 7.75309i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(10.0115 - 17.3404i) q^{7} -3.92177 q^{8} -20.5867 q^{10} +(0.839133 - 1.45342i) q^{11} +(-5.55615 - 9.62353i) q^{13} +(-41.2206 - 71.3962i) q^{14} +(27.7364 - 48.0408i) q^{16} +8.98682 q^{17} -50.6419 q^{19} +(-22.3812 + 38.7655i) q^{20} +(-3.45500 - 5.98423i) q^{22} +(-107.246 - 185.755i) q^{23} +(-12.5000 + 21.6506i) q^{25} -45.7531 q^{26} -179.255 q^{28} +(38.4544 - 66.6050i) q^{29} +(136.769 + 236.890i) q^{31} +(-129.887 - 224.971i) q^{32} +(18.5009 - 32.0445i) q^{34} -100.115 q^{35} -137.283 q^{37} +(-104.255 + 180.575i) q^{38} +(9.80442 + 16.9818i) q^{40} +(26.6729 + 46.1987i) q^{41} +(-147.635 + 255.711i) q^{43} -15.0247 q^{44} -883.134 q^{46} +(97.4916 - 168.860i) q^{47} +(-28.9587 - 50.1579i) q^{49} +(51.4668 + 89.1431i) q^{50} +(-49.7414 + 86.1547i) q^{52} -450.173 q^{53} -8.39133 q^{55} +(-39.2626 + 68.0048i) q^{56} +(-158.330 - 274.236i) q^{58} +(-240.572 - 416.684i) q^{59} +(-337.927 + 585.307i) q^{61} +1126.25 q^{62} -625.798 q^{64} +(-27.7807 + 48.1177i) q^{65} +(-447.295 - 774.737i) q^{67} +(-40.2273 - 69.6757i) q^{68} +(-206.103 + 356.981i) q^{70} +721.947 q^{71} +915.175 q^{73} +(-282.621 + 489.513i) q^{74} +(226.686 + 392.632i) q^{76} +(-16.8019 - 29.1017i) q^{77} +(-29.8321 + 51.6707i) q^{79} -277.364 q^{80} +219.643 q^{82} +(-371.438 + 643.349i) q^{83} +(-22.4671 - 38.9141i) q^{85} +(607.863 + 1052.85i) q^{86} +(-3.29089 + 5.69998i) q^{88} +1540.72 q^{89} -222.501 q^{91} +(-960.116 + 1662.97i) q^{92} +(-401.406 - 695.256i) q^{94} +(126.605 + 219.286i) q^{95} +(560.783 - 971.305i) q^{97} -238.465 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{2} - 34 q^{4} - 30 q^{5} - 40 q^{7} + 132 q^{8}+O(q^{10}) \) 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

Groups

Database

Properties

Related objects

Downloads

Learn more