LMFDB - Embedded newform 405.4.e.w.271.6

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			271.6
Root			$-2.61824 - 2.61824i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.4.e.w.136.6

$q$-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 $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

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

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	0.957790	+	0.287468i	$0.0928136\pi$
$2$	2.05867	+	3.56572i	0.727850	+	1.26067i	−0.229940	+	0.973205i	$0.573853\pi$
$3$		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	0.998871	+	0.0474955i	$0.0151240\pi$
$7$	10.0115	+	17.3404i	0.540568	+	0.936291i	−0.458303	+	0.888796i	$0.651543\pi$
$8$	−3.92177			−0.173319
$9$		0			0
$10$	−20.5867			−0.651009
$11$	0.839133	+	1.45342i	0.0230007	+	0.0398385i	0.877297	−	0.479948i	$-0.159345\pi$
$11$	0.839133	+	1.45342i	0.0230007	+	0.0398385i	−0.854296	+	0.519787i	$0.826011\pi$
$12$		0			0
$13$	−5.55615	+	9.62353i	−0.118538	+	0.205314i	−0.919189	−	0.393818i	$-0.871154\pi$
$13$	−5.55615	+	9.62353i	−0.118538	+	0.205314i	0.800650	+	0.599132i	$0.204488\pi$
$14$	−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	−0.283614	+	0.958939i	$0.591533\pi$
$23$	−107.246	+	185.755i	−0.972272	+	1.68402i	−0.688658	+	0.725086i	$0.741800\pi$
$24$		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	0.962478	−	0.271360i	$-0.0874733\pi$
$29$	38.4544	+	66.6050i	0.246235	+	0.426491i	−0.716243	+	0.697851i	$0.754140\pi$
$30$		0			0
$31$	136.769	−	236.890i	0.792399	−	1.37247i	−0.132079	−	0.991239i	$-0.542165\pi$
$31$	136.769	−	236.890i	0.792399	−	1.37247i	0.924478	−	0.381236i	$-0.124501\pi$
$32$	−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	−0.810744	−	0.585401i	$-0.800937\pi$
$41$	26.6729	−	46.1987i	0.101600	−	0.175976i	0.912344	+	0.409424i	$0.134270\pi$
$42$		0			0
$43$	−147.635	−	255.711i	−0.523584	−	0.906874i	−0.999623	−	0.0274502i	$-0.991261\pi$
$43$	−147.635	−	255.711i	−0.523584	−	0.906874i	0.476039	−	0.879424i	$-0.342072\pi$
$44$	−15.0247			−0.0514785
$45$		0			0
$46$	−883.134			−2.83067
$47$	97.4916	+	168.860i	0.302566	+	0.524060i	0.976716	−	0.214534i	$-0.0688234\pi$
$47$	97.4916	+	168.860i	0.302566	+	0.524060i	−0.674150	+	0.738594i	$0.735490\pi$
$48$		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	0.468507	+	0.883460i	$0.344792\pi$
$59$	−240.572	+	416.684i	−0.530845	+	0.919451i	−0.999352	+	0.0359910i	$0.988541\pi$
$60$		0			0
$61$	−337.927	−	585.307i	−0.709297	−	1.22854i	−0.965118	−	0.261814i	$-0.915679\pi$
$61$	−337.927	−	585.307i	−0.709297	−	1.22854i	0.255821	−	0.966724i	$-0.417654\pi$
$62$	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	0.0932823	+	0.995640i	$0.470264\pi$
$67$	−447.295	+	774.737i	−0.815608	+	1.41267i	−0.908890	+	0.417035i	$0.863069\pi$
$68$	−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	0.844001	−	0.536342i	$-0.180194\pi$
$79$	−29.8321	−	51.6707i	−0.0424858	−	0.0735875i	−0.886486	+	0.462755i	$0.846861\pi$
$80$	−277.364			−0.387628
$81$		0			0
$82$	219.643			0.295798
$83$	−371.438	−	643.349i	−0.491212	−	0.850803i	0.508737	−	0.860922i	$-0.330112\pi$
$83$	−371.438	−	643.349i	−0.491212	−	0.850803i	−0.999949	+	0.0101184i	$0.996779\pi$
$84$		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	0.994623	+	0.103562i	$0.0330240\pi$
$97$	560.783	+	971.305i	0.586999	+	1.01671i	−0.407624	+	0.913150i	$0.633643\pi$
$98$	−238.465			−0.245802
$99$		0			0
$100$	223.812			0.223812
$101$	879.236	+	1522.88i	0.866211	+	1.50032i	0.865840	+	0.500321i	$0.166785\pi$
$101$	879.236	+	1522.88i	0.866211	+	1.50032i	0.000370864		1.00000i	$0.499882\pi$
$102$		0			0
$103$	−528.242	+	914.941i	−0.505332	+	0.875261i	0.494649	+	0.869093i	$0.335297\pi$
$103$	−528.242	+	914.941i	−0.505332	+	0.875261i	−0.999981	+	0.00616773i	$0.998037\pi$
$104$	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	−0.994991	−	0.0999606i	$-0.968128\pi$
$113$	−493.609	+	854.955i	−0.410927	+	0.711747i	0.584064	+	0.811708i	$0.301462\pi$
$114$		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	−0.769110	−	0.639117i	$-0.779300\pi$
$131$	253.298	−	438.724i	0.168937	−	0.292607i	0.938046	+	0.346510i	$0.112633\pi$
$132$		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	0.672852	−	0.739777i	$-0.265069\pi$
$137$	−487.861	−	845.001i	−0.304240	−	0.526958i	−0.977092	+	0.212819i	$0.931736\pi$
$138$		0			0
$139$	944.413	−	1635.77i	0.576288	−	0.998160i	−0.419612	−	0.907703i	$-0.637834\pi$
$139$	944.413	−	1635.77i	0.576288	−	0.998160i	0.995900	−	0.0904566i	$-0.0288327\pi$
$140$	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	0.254997	+	0.966942i	$0.417926\pi$
$149$	−1291.15	+	2236.33i	−0.709898	+	1.22958i	−0.964895	+	0.262637i	$0.915408\pi$
$150$		0			0
$151$	−1268.96	−	2197.91i	−0.683885	−	1.18452i	−0.973786	−	0.227467i	$-0.926956\pi$
$151$	−1268.96	−	2197.91i	−0.683885	−	1.18452i	0.289901	−	0.957057i	$-0.406378\pi$
$152$	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	−0.879065	−	0.476702i	$-0.841832\pi$
$157$	−52.5175	+	90.9630i	−0.0266965	+	0.0462397i	0.852368	+	0.522942i	$0.175165\pi$
$158$	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	0.330748	+	0.943719i	$0.392699\pi$
$167$	−1406.90	+	2436.82i	−0.651911	+	1.12914i	−0.982659	+	0.185424i	$0.940634\pi$
$168$		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	0.960314	+	0.278922i	$0.0899772\pi$
$173$	1642.22	+	2844.41i	0.721711	+	1.25004i	−0.238603	+	0.971117i	$0.576689\pi$
$174$		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	0.837593	−	0.546295i	$-0.183962\pi$
$191$	−143.357	−	248.302i	−0.0543087	−	0.0940654i	−0.891902	+	0.452229i	$0.850629\pi$
$192$		0			0
$193$	2325.35	−	4027.63i	0.867267	−	1.50215i	0.00248836	−	0.999997i	$-0.499208\pi$
$193$	2325.35	−	4027.63i	0.867267	−	1.50215i	0.864779	−	0.502153i	$-0.167459\pi$
$194$	−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	−0.764539	−	0.644577i	$-0.777034\pi$
$211$	539.279	−	934.059i	0.175950	−	0.304755i	0.940490	+	0.339822i	$0.110367\pi$
$212$	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	0.936332	+	0.351115i	$0.114197\pi$
$223$	2571.64	+	4454.21i	0.772241	+	1.33756i	−0.164091	+	0.986445i	$0.552469\pi$
$224$	−5201.44			−1.55150
$225$		0			0
$226$	−4064.71			−1.19637
$227$	−1454.83	−	2519.84i	−0.425377	−	0.736774i	0.571079	−	0.820895i	$-0.306525\pi$
$227$	−1454.83	−	2519.84i	−0.425377	−	0.736774i	−0.996456	+	0.0841209i	$0.973192\pi$
$228$		0			0
$229$	−862.910	+	1494.60i	−0.249007	+	0.431293i	−0.963251	−	0.268604i	$-0.913438\pi$
$229$	−862.910	+	1494.60i	−0.249007	+	0.431293i	0.714243	+	0.699897i	$0.246771\pi$
$230$	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	−0.843609	−	0.536958i	$-0.819573\pi$
$239$	159.672	−	276.561i	0.0432149	−	0.0748504i	0.886824	+	0.462108i	$0.152907\pi$
$240$		0			0
$241$	−1822.91	−	3157.38i	−0.487238	−	0.843920i	0.512655	−	0.858595i	$-0.328662\pi$
$241$	−1822.91	−	3157.38i	−0.487238	−	0.843920i	−0.999892	+	0.0146745i	$0.995329\pi$
$242$	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	−0.551828	−	0.833958i	$-0.686069\pi$
$257$	1838.83	−	3184.94i	0.446315	−	0.773040i	0.998143	+	0.0609179i	$0.0194028\pi$
$258$		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	0.724514	−	0.689260i	$-0.242064\pi$
$263$	−1000.86	−	1733.53i	−0.234659	−	0.406442i	−0.959174	+	0.282818i	$0.908731\pi$
$264$		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	−0.998579	−	0.0532826i	$-0.983032\pi$
$277$	−2514.56	−	4355.34i	−0.545434	−	0.944719i	0.453146	−	0.891436i	$-0.350302\pi$
$278$	7776.94			1.67780
$279$		0			0
$280$	392.626			0.0837996
$281$	−2904.16	−	5030.14i	−0.616539	−	1.06788i	−0.990112	−	0.140276i	$-0.955201\pi$
$281$	−2904.16	−	5030.14i	−0.616539	−	1.06788i	0.373574	−	0.927600i	$-0.378132\pi$
$282$		0			0
$283$	−106.512	+	184.483i	−0.0223726	+	0.0387505i	−0.876995	−	0.480500i	$-0.840455\pi$
$283$	−106.512	+	184.483i	−0.0223726	+	0.0387505i	0.854622	+	0.519250i	$0.173789\pi$
$284$	−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	−0.629452	−	0.777039i	$-0.716721\pi$
$293$	1796.55	−	3111.71i	0.358210	−	0.620437i	0.987662	+	0.156602i	$0.0500540\pi$
$294$		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	−0.783363	−	0.621564i	$-0.786498\pi$
$311$	804.080	−	1392.71i	0.146608	−	0.253933i	0.929972	+	0.367631i	$0.119831\pi$
$312$		0			0
$313$	−1390.27	−	2408.02i	−0.251063	−	0.434855i	0.712755	−	0.701413i	$-0.247447\pi$
$313$	−1390.27	−	2408.02i	−0.251063	−	0.434855i	−0.963819	+	0.266558i	$0.914114\pi$
$314$	−432.465			−0.0777242
$315$		0			0
$316$	534.144			0.0950885
$317$	−2767.85	−	4794.05i	−0.490403	−	0.849403i	0.509536	−	0.860449i	$-0.329817\pi$
$317$	−2767.85	−	4794.05i	−0.490403	−	0.849403i	−0.999939	+	0.0110461i	$0.996484\pi$
$318$		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	0.920615	+	0.390471i	$0.127688\pi$
$331$	4808.37	+	8328.35i	0.798466	+	1.38298i	−0.122149	+	0.992512i	$0.538979\pi$
$332$	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	−0.769413	−	0.638751i	$-0.779451\pi$
$337$	1042.23	−	1805.19i	0.168468	−	0.291795i	0.937881	+	0.346956i	$0.112785\pi$
$338$	−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	−0.526628	−	0.850096i	$-0.676544\pi$
$347$	3056.71	−	5294.38i	0.472890	−	0.819070i	0.999519	+	0.0310257i	$0.00987737\pi$
$348$		0			0
$349$	−1094.53	−	1895.78i	−0.167877	−	0.290771i	0.769797	−	0.638289i	$-0.220358\pi$
$349$	−1094.53	−	1895.78i	−0.167877	−	0.290771i	−0.937673	+	0.347519i	$0.887024\pi$
$350$	2061.03			0.314762
$351$		0			0
$352$	−435.971			−0.0660151
$353$	1250.10	+	2165.24i	0.188488	+	0.326471i	0.944746	−	0.327802i	$-0.106308\pi$
$353$	1250.10	+	2165.24i	0.188488	+	0.326471i	−0.756258	+	0.654273i	$0.772975\pi$
$354$		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	0.993637	−	0.112630i	$-0.0359273\pi$
$367$	2807.21	+	4862.23i	0.399278	+	0.691571i	−0.594359	+	0.804200i	$0.702594\pi$
$368$	−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	−0.908168	−	0.418606i	$-0.862519\pi$
$373$	−659.587	+	1142.44i	−0.0915607	+	0.158588i	0.816607	+	0.577194i	$0.195852\pi$
$374$	−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	−0.931075	−	0.364828i	$-0.881128\pi$
$383$	−1121.22	+	1942.02i	−0.149587	+	0.259092i	0.781488	+	0.623920i	$0.214461\pi$
$384$		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	−0.951340	−	0.308142i	$-0.900293\pi$
$389$	−5696.89	−	9867.30i	−0.742529	−	1.28610i	0.208811	−	0.977956i	$-0.433041\pi$
$390$		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	−0.998133	−	0.0610707i	$-0.980548\pi$
$401$	−3582.82	+	6205.62i	−0.446178	+	0.772803i	0.551955	+	0.833874i	$0.313882\pi$
$402$		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	−0.289062	−	0.957310i	$-0.593343\pi$
$409$	5662.05	−	9806.96i	0.684524	−	1.18563i	0.973586	−	0.228321i	$-0.0733234\pi$
$410$	−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	0.211964	+	0.977277i	$0.432014\pi$
$419$	−6349.90	+	10998.4i	−0.740365	+	1.28235i	−0.952329	+	0.305072i	$0.901319\pi$
$420$		0			0
$421$	8581.39	+	14863.4i	0.993424	+	1.72066i	0.595867	+	0.803083i	$0.296809\pi$
$421$	8581.39	+	14863.4i	0.993424	+	1.72066i	0.397557	+	0.917577i	$0.369858\pi$
$422$	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	0.759998	−	0.649926i	$-0.225200\pi$
$439$	−1681.90	−	2913.13i	−0.182853	−	0.316711i	−0.942851	+	0.333214i	$0.891867\pi$
$440$	32.9089			0.00356561
$441$		0			0
$442$	−411.175			−0.0442480
$443$	3947.03	+	6836.46i	0.423316	+	0.733205i	0.996262	−	0.0863883i	$-0.0275326\pi$
$443$	3947.03	+	6836.46i	0.423316	+	0.733205i	−0.572945	+	0.819594i	$0.694199\pi$
$444$		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	−0.945443	−	0.325787i	$-0.894371\pi$
$457$	−7374.65	−	12773.3i	−0.754862	−	1.30746i	0.190581	−	0.981671i	$-0.438963\pi$
$458$	−7105.79			−0.724960
$459$		0			0
$460$	9601.16			0.973166
$461$	6239.75	+	10807.6i	0.630400	+	1.09188i	0.987470	+	0.157807i	$0.0504424\pi$
$461$	6239.75	+	10807.6i	0.630400	+	1.09188i	−0.357070	+	0.934078i	$0.616224\pi$
$462$		0			0
$463$	4839.56	−	8382.37i	0.485774	−	0.841386i	−0.514092	−	0.857735i	$-0.671871\pi$
$463$	4839.56	−	8382.37i	0.485774	−	0.841386i	0.999866	+	0.0163492i	$0.00520436\pi$
$464$	−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	0.912290	−	0.409545i	$-0.134313\pi$
$479$	1063.74	+	1842.45i	0.101469	+	0.175749i	−0.810821	+	0.585294i	$0.800979\pi$
$480$		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	−0.399243	−	0.916845i	$-0.630727\pi$
$491$	6466.86	−	11200.9i	0.594389	−	1.02951i	0.993633	−	0.112668i	$-0.0359396\pi$
$492$		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	−0.957717	−	0.287713i	$-0.907105\pi$
$499$	−2560.33	+	4434.62i	−0.229692	+	0.397837i	0.728025	+	0.685550i	$0.240438\pi$
$500$	−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	−0.342344	−	0.939575i	$-0.611221\pi$
$509$	7378.46	−	12779.9i	0.642524	−	1.11288i	0.984868	−	0.173309i	$-0.0554459\pi$
$510$		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	0.956924	−	0.290337i	$-0.0937675\pi$
$547$	2904.36	+	5030.50i	0.227023	+	0.393215i	−0.729902	+	0.683552i	$0.760434\pi$
$548$	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	−0.856293	−	0.516490i	$-0.827238\pi$
$563$	255.772	−	443.010i	0.0191466	−	0.0331628i	0.875440	+	0.483327i	$0.160572\pi$
$564$		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	0.676907	−	0.736069i	$-0.263320\pi$
$569$	−4058.27	−	7029.13i	−0.299001	−	0.517885i	−0.975908	+	0.218184i	$0.929987\pi$
$570$		0			0
$571$	−7539.06	+	13058.0i	−0.552539	+	0.957026i	0.445551	+	0.895256i	$0.353008\pi$
$571$	−7539.06	+	13058.0i	−0.552539	+	0.957026i	−0.998090	+	0.0617692i	$0.980326\pi$
$572$	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	0.864225	−	0.503105i	$-0.167809\pi$
$587$	−51.0411	−	88.4057i	−0.00358891	−	0.00621617i	−0.867814	+	0.496889i	$0.834476\pi$
$588$		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	0.133038	+	0.991111i	$0.457527\pi$
$599$	−11608.1	+	20105.8i	−0.791809	+	1.37145i	−0.924846	+	0.380342i	$0.875806\pi$
$600$		0			0
$601$	839.147	+	1453.45i	0.0569543	+	0.0986477i	0.893097	−	0.449865i	$-0.148528\pi$
$601$	839.147	+	1453.45i	0.0569543	+	0.0986477i	−0.836143	+	0.548512i	$0.815194\pi$
$602$	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	0.338664	+	0.940907i	$0.390025\pi$
$607$	−9653.64	+	16720.6i	−0.645517	+	1.11807i	−0.984182	+	0.177162i	$0.943308\pi$
$608$	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	−0.523200	−	0.852210i	$-0.675262\pi$
$617$	7301.84	−	12647.2i	0.476436	−	0.825211i	0.999635	+	0.0269990i	$0.00859510\pi$
$618$		0			0
$619$	13194.1	+	22852.9i	0.856732	+	1.48390i	0.875029	+	0.484071i	$0.160842\pi$
$619$	13194.1	+	22852.9i	0.856732	+	1.48390i	−0.0182965	+	0.999833i	$0.505824\pi$
$620$	−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	−0.994980	−	0.100078i	$-0.968091\pi$
$641$	−9480.23	−	16420.2i	−0.584160	−	1.01180i	0.410819	−	0.911717i	$-0.365243\pi$
$642$		0			0
$643$	−5527.13	+	9573.26i	−0.338987	+	0.587143i	−0.984242	−	0.176824i	$-0.943418\pi$
$643$	−5527.13	+	9573.26i	−0.338987	+	0.587143i	0.645256	+	0.763967i	$0.276751\pi$
$644$	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	−0.743236	−	0.669029i	$-0.766710\pi$
$653$	3467.13	−	6005.24i	0.207778	−	0.359883i	0.951014	+	0.309147i	$0.100043\pi$
$654$		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	0.530078	−	0.847949i	$-0.322163\pi$
$659$	−7939.35	−	13751.4i	−0.469307	−	0.812863i	−0.999384	+	0.0350862i	$0.988829\pi$
$660$		0			0
$661$	4709.16	−	8156.51i	0.277103	−	0.479957i	−0.693561	−	0.720398i	$-0.743959\pi$
$661$	4709.16	−	8156.51i	0.277103	−	0.479957i	0.970664	+	0.240442i	$0.0772923\pi$
$662$	−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	0.999515	+	0.0311403i	$0.00991387\pi$
$673$	9196.17	+	15928.2i	0.526726	+	0.912316i	−0.472789	+	0.881176i	$0.656753\pi$
$674$	8582.41			0.490478
$675$		0			0
$676$	−18563.2			−1.05617
$677$	5285.34	+	9154.49i	0.300048	+	0.519698i	0.976146	−	0.217113i	$-0.0696641\pi$
$677$	5285.34	+	9154.49i	0.300048	+	0.519698i	−0.676099	+	0.736811i	$0.736331\pi$
$678$		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	−0.998867	−	0.0475927i	$-0.984845\pi$
$691$	−9820.49	−	17009.6i	−0.540650	−	0.936433i	0.458217	−	0.888840i	$-0.348488\pi$
$692$	−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	−0.777853	−	0.628446i	$-0.783691\pi$
$709$	−17617.1	−	30513.6i	−0.933177	−	1.61631i	−0.155323	−	0.987864i	$-0.549642\pi$
$710$	−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	0.999988	−	0.00490721i	$-0.00156202\pi$
$727$	9717.61	+	16831.4i	0.495744	+	0.858654i	−0.504244	+	0.863561i	$0.668229\pi$
$728$	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	−0.990665	−	0.136321i	$-0.956472\pi$
$733$	−7487.10	+	12968.0i	−0.377275	+	0.653459i	0.613390	+	0.789780i	$0.289805\pi$
$734$	−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	−0.820792	−	0.571228i	$-0.806467\pi$
$743$	1707.34	−	2957.20i	0.0843018	−	0.146015i	0.905093	+	0.425213i	$0.139801\pi$
$744$		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	−0.856818	−	0.515619i	$-0.827562\pi$
$751$	373.129	−	646.279i	0.0181301	−	0.0314022i	0.874948	+	0.484217i	$0.160895\pi$
$752$	−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	0.495617	+	0.868541i	$0.334942\pi$
$761$	−10588.3	+	18339.5i	−0.504370	+	0.873595i	−0.999987	+	0.00505384i	$0.998391\pi$
$762$		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	−0.896714	−	0.442610i	$-0.854053\pi$
$769$	−1387.09	+	2402.51i	−0.0650453	+	0.112662i	0.831669	+	0.555272i	$0.187386\pi$
$770$	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	−0.948363	−	0.317186i	$-0.897262\pi$
$787$	−4404.37	+	7628.60i	−0.199490	+	0.345527i	0.748873	+	0.662714i	$0.230595\pi$
$788$	−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	−0.874567	−	0.484904i	$-0.838854\pi$
$797$	−390.251	+	675.935i	−0.0173443	+	0.0300412i	0.857223	+	0.514945i	$0.172188\pi$
$798$		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	0.881586	+	0.472023i	$0.156476\pi$
$821$	19985.6	+	34616.1i	0.849577	+	1.47151i	−0.0320090	+	0.999488i	$0.510191\pi$
$822$		0			0
$823$	−14310.3	+	24786.2i	−0.606107	+	1.04981i	0.385769	+	0.922595i	$0.373936\pi$
$823$	−14310.3	+	24786.2i	−0.606107	+	1.04981i	−0.991876	+	0.127212i	$0.959397\pi$
$824$	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	0.536839	−	0.843685i	$-0.319618\pi$
$839$	−11233.2	−	19456.5i	−0.462233	−	0.800611i	−0.999072	+	0.0430736i	$0.986285\pi$
$840$		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	−0.836831	−	0.547462i	$-0.815594\pi$
$853$	−22235.5	−	38513.0i	−0.892531	−	1.54591i	−0.0557007	−	0.998448i	$-0.517739\pi$
$854$	55718.2			2.23260
$855$		0			0
$856$	−5765.12			−0.230196
$857$	−15740.3	−	27263.0i	−0.627396	−	1.08668i	−0.988072	−	0.153991i	$-0.950787\pi$
$857$	−15740.3	−	27263.0i	−0.627396	−	1.08668i	0.360676	−	0.932691i	$-0.382546\pi$
$858$		0			0
$859$	−21491.5	+	37224.5i	−0.853646	+	1.47856i	0.0242490	+	0.999706i	$0.492281\pi$
$859$	−21491.5	+	37224.5i	−0.853646	+	1.47856i	−0.877895	+	0.478853i	$0.841053\pi$
$860$	−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	−0.897516	−	0.440982i	$-0.854630\pi$
$877$	−1736.36	+	3007.46i	−0.0668558	+	0.115798i	0.830660	+	0.556780i	$0.187963\pi$
$878$	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	0.117357	+	0.993090i	$0.462558\pi$
$887$	−21169.7	+	36667.0i	−0.801363	+	1.38800i	−0.918719	+	0.394911i	$0.870775\pi$
$888$		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	0.989554	−	0.144160i	$-0.0460481\pi$
$907$	10104.9	+	17502.2i	0.369931	+	0.640739i	−0.619624	+	0.784899i	$0.712715\pi$
$908$	26048.8			0.952047
$909$		0			0
$910$	4580.56			0.166861
$911$	−5627.30	−	9746.78i	−0.204655	−	0.354473i	0.745368	−	0.666654i	$-0.232274\pi$
$911$	−5627.30	−	9746.78i	−0.204655	−	0.354473i	−0.950023	+	0.312180i	$0.898941\pi$
$912$		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	0.883841	−	0.467788i	$-0.154949\pi$
$929$	1042.12	+	1805.00i	0.0368039	+	0.0637461i	−0.847037	+	0.531534i	$0.821616\pi$
$930$		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	−0.113035	−	0.993591i	$-0.536057\pi$
$941$	23206.9	−	40195.6i	0.803958	−	1.39250i	0.916993	−	0.398904i	$-0.130609\pi$
$942$		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	0.658691	−	0.752413i	$-0.271110\pi$
$947$	−9391.53	−	16266.6i	−0.322264	−	0.558177i	−0.980955	+	0.194237i	$0.937777\pi$
$948$		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	−0.769091	−	0.639140i	$-0.779291\pi$
$967$	5080.87	−	8800.33i	0.168966	−	0.292657i	0.938057	+	0.346482i	$0.112624\pi$
$968$	−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	−0.0592594	−	0.998243i	$-0.518874\pi$
$977$	25495.4	−	44159.4i	0.834874	−	1.44604i	0.894133	−	0.447801i	$-0.147793\pi$
$978$		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	0.986844	−	0.161676i	$-0.0516901\pi$
$983$	10891.9	+	18865.4i	0.353406	+	0.612117i	−0.633438	+	0.773794i	$0.718357\pi$
$984$		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	0.977119	−	0.212695i	$-0.0682242\pi$
$997$	9581.42	+	16595.5i	0.304360	+	0.527167i	−0.672759	+	0.739862i	$0.734891\pi$
$998$	−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.271.6		12
3.2	odd	2		405.4.e.x.271.1		12
9.2	odd	6		405.4.e.x.136.1		12
9.4	even	3		405.4.a.l.1.1	yes	6
9.5	odd	6		405.4.a.k.1.6	✓	6
9.7	even	3	inner	405.4.e.w.136.6		12
45.4	even	6		2025.4.a.y.1.6		6
45.14	odd	6		2025.4.a.z.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.a.k.1.6	✓	6	9.5	odd	6
405.4.a.l.1.1	yes	6	9.4	even	3
405.4.e.w.136.6		12	9.7	even	3	inner
405.4.e.w.271.6		12	1.1	even	1	trivial
405.4.e.x.136.1		12	9.2	odd	6
405.4.e.x.271.1		12	3.2	odd	2
2025.4.a.y.1.6		6	45.4	even	6
2025.4.a.z.1.1		6	45.14	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

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