LMFDB - Embedded newform 405.4.e.d.271.1

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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 15)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			271.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	405.271
Dual form			405.4.e.d.136.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-10.0000 - 17.3205i) q^{7} -21.0000 q^{8} +O(q^{10})$	\(q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-10.0000 - 17.3205i) q^{7} -21.0000 q^{8} -15.0000 q^{10} +(12.0000 + 20.7846i) q^{11} +(-37.0000 + 64.0859i) q^{13} +(-30.0000 + 51.9615i) q^{14} +(35.5000 + 61.4878i) q^{16} +54.0000 q^{17} -124.000 q^{19} +(2.50000 + 4.33013i) q^{20} +(36.0000 - 62.3538i) q^{22} +(60.0000 - 103.923i) q^{23} +(-12.5000 - 21.6506i) q^{25} +222.000 q^{26} +20.0000 q^{28} +(39.0000 + 67.5500i) q^{29} +(-100.000 + 173.205i) q^{31} +(22.5000 - 38.9711i) q^{32} +(-81.0000 - 140.296i) q^{34} -100.000 q^{35} -70.0000 q^{37} +(186.000 + 322.161i) q^{38} +(-52.5000 + 90.9327i) q^{40} +(-165.000 + 285.788i) q^{41} +(-46.0000 - 79.6743i) q^{43} -24.0000 q^{44} -360.000 q^{46} +(12.0000 + 20.7846i) q^{47} +(-28.5000 + 49.3634i) q^{49} +(-37.5000 + 64.9519i) q^{50} +(-37.0000 - 64.0859i) q^{52} +450.000 q^{53} +120.000 q^{55} +(210.000 + 363.731i) q^{56} +(117.000 - 202.650i) q^{58} +(-12.0000 + 20.7846i) q^{59} +(161.000 + 278.860i) q^{61} +600.000 q^{62} +433.000 q^{64} +(185.000 + 320.429i) q^{65} +(98.0000 - 169.741i) q^{67} +(-27.0000 + 46.7654i) q^{68} +(150.000 + 259.808i) q^{70} -288.000 q^{71} -430.000 q^{73} +(105.000 + 181.865i) q^{74} +(62.0000 - 107.387i) q^{76} +(240.000 - 415.692i) q^{77} +(260.000 + 450.333i) q^{79} +355.000 q^{80} +990.000 q^{82} +(-78.0000 - 135.100i) q^{83} +(135.000 - 233.827i) q^{85} +(-138.000 + 239.023i) q^{86} +(-252.000 - 436.477i) q^{88} +1026.00 q^{89} +1480.00 q^{91} +(60.0000 + 103.923i) q^{92} +(36.0000 - 62.3538i) q^{94} +(-310.000 + 536.936i) q^{95} +(143.000 + 247.683i) q^{97} +171.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8}+O(q^{10}) $ 2 * q - 3 * q^2 - q^4 + 5 * q^5 - 20 * q^7 - 42 * q^8	$ 2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8} - 30 q^{10} + 24 q^{11} - 74 q^{13} - 60 q^{14} + 71 q^{16} + 108 q^{17} - 248 q^{19} + 5 q^{20} + 72 q^{22} + 120 q^{23} - 25 q^{25} + 444 q^{26} + 40 q^{28} + 78 q^{29} - 200 q^{31} + 45 q^{32} - 162 q^{34} - 200 q^{35} - 140 q^{37} + 372 q^{38} - 105 q^{40} - 330 q^{41} - 92 q^{43} - 48 q^{44} - 720 q^{46} + 24 q^{47} - 57 q^{49} - 75 q^{50} - 74 q^{52} + 900 q^{53} + 240 q^{55} + 420 q^{56} + 234 q^{58} - 24 q^{59} + 322 q^{61} + 1200 q^{62} + 866 q^{64} + 370 q^{65} + 196 q^{67} - 54 q^{68} + 300 q^{70} - 576 q^{71} - 860 q^{73} + 210 q^{74} + 124 q^{76} + 480 q^{77} + 520 q^{79} + 710 q^{80} + 1980 q^{82} - 156 q^{83} + 270 q^{85} - 276 q^{86} - 504 q^{88} + 2052 q^{89} + 2960 q^{91} + 120 q^{92} + 72 q^{94} - 620 q^{95} + 286 q^{97} + 342 q^{98}+O(q^{100}) $ 2 * q - 3 * q^2 - q^4 + 5 * q^5 - 20 * q^7 - 42 * q^8 - 30 * q^10 + 24 * q^11 - 74 * q^13 - 60 * q^14 + 71 * q^16 + 108 * q^17 - 248 * q^19 + 5 * q^20 + 72 * q^22 + 120 * q^23 - 25 * q^25 + 444 * q^26 + 40 * q^28 + 78 * q^29 - 200 * q^31 + 45 * q^32 - 162 * q^34 - 200 * q^35 - 140 * q^37 + 372 * q^38 - 105 * q^40 - 330 * q^41 - 92 * q^43 - 48 * q^44 - 720 * q^46 + 24 * q^47 - 57 * q^49 - 75 * q^50 - 74 * q^52 + 900 * q^53 + 240 * q^55 + 420 * q^56 + 234 * q^58 - 24 * q^59 + 322 * q^61 + 1200 * q^62 + 866 * q^64 + 370 * q^65 + 196 * q^67 - 54 * q^68 + 300 * q^70 - 576 * q^71 - 860 * q^73 + 210 * q^74 + 124 * q^76 + 480 * q^77 + 520 * q^79 + 710 * q^80 + 1980 * q^82 - 156 * q^83 + 270 * q^85 - 276 * q^86 - 504 * q^88 + 2052 * q^89 + 2960 * q^91 + 120 * q^92 + 72 * q^94 - 620 * q^95 + 286 * q^97 + 342 * 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 $

$2$	−1.50000	−	2.59808i	−0.530330	−	0.918559i	−0.999374	−	0.0353837i	$-0.988735\pi$
$2$	−1.50000	−	2.59808i	−0.530330	−	0.918559i	0.469044	−	0.883175i	$-0.344599\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	+	0.866025i	−0.0625000	+	0.108253i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	−10.0000	−	17.3205i	−0.539949	−	0.935220i	−0.998906	−	0.0467610i	$-0.985110\pi$
$7$	−10.0000	−	17.3205i	−0.539949	−	0.935220i	0.458957	−	0.888459i	$-0.348223\pi$
$8$	−21.0000			−0.928078
$9$		0			0
$10$	−15.0000			−0.474342
$11$	12.0000	+	20.7846i	0.328921	+	0.569709i	0.982298	−	0.187324i	$-0.0599815\pi$
$11$	12.0000	+	20.7846i	0.328921	+	0.569709i	−0.653377	+	0.757033i	$0.726648\pi$
$12$		0			0
$13$	−37.0000	+	64.0859i	−0.789381	+	1.36725i	0.136966	+	0.990576i	$0.456265\pi$
$13$	−37.0000	+	64.0859i	−0.789381	+	1.36725i	−0.926347	+	0.376672i	$0.877068\pi$
$14$	−30.0000	+	51.9615i	−0.572703	+	0.991950i
$15$		0			0
$16$	35.5000	+	61.4878i	0.554688	+	0.960747i
$17$	54.0000			0.770407			0.385204	−	0.922832i	$-0.374131\pi$
$17$	54.0000			0.770407			0.385204	+	0.922832i	$0.374131\pi$
$18$		0			0
$19$	−124.000			−1.49724			−0.748620	−	0.663000i	$-0.769283\pi$
$19$	−124.000			−1.49724			−0.748620	+	0.663000i	$0.769283\pi$
$20$	2.50000	+	4.33013i	0.0279508	+	0.0484123i
$21$		0			0
$22$	36.0000	−	62.3538i	0.348874	−	0.604267i
$23$	60.0000	−	103.923i	0.543951	−	0.942150i	−0.454721	−	0.890634i	$-0.650261\pi$
$23$	60.0000	−	103.923i	0.543951	−	0.942150i	0.998672	−	0.0515165i	$-0.0164055\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	222.000			1.67453
$27$		0			0
$28$	20.0000			0.134987
$29$	39.0000	+	67.5500i	0.249728	+	0.432542i	0.963450	−	0.267887i	$-0.0863254\pi$
$29$	39.0000	+	67.5500i	0.249728	+	0.432542i	−0.713722	+	0.700429i	$0.752992\pi$
$30$		0			0
$31$	−100.000	+	173.205i	−0.579372	+	1.00350i	0.416180	+	0.909282i	$0.363369\pi$
$31$	−100.000	+	173.205i	−0.579372	+	1.00350i	−0.995551	+	0.0942192i	$0.969965\pi$
$32$	22.5000	−	38.9711i	0.124296	−	0.215287i
$33$		0			0
$34$	−81.0000	−	140.296i	−0.408570	−	0.707664i
$35$	−100.000			−0.482945
$36$		0			0
$37$	−70.0000			−0.311025			−0.155513	−	0.987834i	$-0.549703\pi$
$37$	−70.0000			−0.311025			−0.155513	+	0.987834i	$0.549703\pi$
$38$	186.000	+	322.161i	0.794031	+	1.37530i
$39$		0			0
$40$	−52.5000	+	90.9327i	−0.207524	+	0.359443i
$41$	−165.000	+	285.788i	−0.628504	+	1.08860i	0.359348	+	0.933204i	$0.382999\pi$
$41$	−165.000	+	285.788i	−0.628504	+	1.08860i	−0.987852	+	0.155397i	$0.950334\pi$
$42$		0			0
$43$	−46.0000	−	79.6743i	−0.163138	−	0.282563i	0.772854	−	0.634583i	$-0.218828\pi$
$43$	−46.0000	−	79.6743i	−0.163138	−	0.282563i	−0.935992	+	0.352020i	$0.885495\pi$
$44$	−24.0000			−0.0822304
$45$		0			0
$46$	−360.000			−1.15389
$47$	12.0000	+	20.7846i	0.0372421	+	0.0645053i	0.884046	−	0.467401i	$-0.154809\pi$
$47$	12.0000	+	20.7846i	0.0372421	+	0.0645053i	−0.846804	+	0.531906i	$0.821476\pi$
$48$		0			0
$49$	−28.5000	+	49.3634i	−0.0830904	+	0.143917i
$50$	−37.5000	+	64.9519i	−0.106066	+	0.183712i
$51$		0			0
$52$	−37.0000	−	64.0859i	−0.0986726	−	0.170906i
$53$	450.000			1.16627			0.583134	−	0.812376i	$-0.301826\pi$
$53$	450.000			1.16627			0.583134	+	0.812376i	$0.301826\pi$
$54$		0			0
$55$	120.000			0.294196
$56$	210.000	+	363.731i	0.501115	+	0.867956i
$57$		0			0
$58$	117.000	−	202.650i	0.264877	−	0.458780i
$59$	−12.0000	+	20.7846i	−0.0264791	+	0.0458631i	−0.878961	−	0.476893i	$-0.841763\pi$
$59$	−12.0000	+	20.7846i	−0.0264791	+	0.0458631i	0.852482	+	0.522756i	$0.175096\pi$
$60$		0			0
$61$	161.000	+	278.860i	0.337933	+	0.585318i	0.984044	−	0.177926i	$-0.0569388\pi$
$61$	161.000	+	278.860i	0.337933	+	0.585318i	−0.646110	+	0.763244i	$0.723605\pi$
$62$	600.000			1.22903
$63$		0			0
$64$	433.000			0.845703
$65$	185.000	+	320.429i	0.353022	+	0.611452i
$66$		0			0
$67$	98.0000	−	169.741i	0.178696	−	0.309510i	−0.762738	−	0.646707i	$-0.776146\pi$
$67$	98.0000	−	169.741i	0.178696	−	0.309510i	0.941434	+	0.337197i	$0.109479\pi$
$68$	−27.0000	+	46.7654i	−0.0481505	+	0.0833990i
$69$		0			0
$70$	150.000	+	259.808i	0.256120	+	0.443614i
$71$	−288.000			−0.481399			−0.240699	−	0.970600i	$-0.577377\pi$
$71$	−288.000			−0.481399			−0.240699	+	0.970600i	$0.577377\pi$
$72$		0			0
$73$	−430.000			−0.689420			−0.344710	−	0.938709i	$-0.612023\pi$
$73$	−430.000			−0.689420			−0.344710	+	0.938709i	$0.612023\pi$
$74$	105.000	+	181.865i	0.164946	+	0.285695i
$75$		0			0
$76$	62.0000	−	107.387i	0.0935775	−	0.162081i
$77$	240.000	−	415.692i	0.355202	−	0.615228i
$78$		0			0
$79$	260.000	+	450.333i	0.370282	+	0.641347i	0.989609	−	0.143786i	$-0.0459277\pi$
$79$	260.000	+	450.333i	0.370282	+	0.641347i	−0.619327	+	0.785133i	$0.712594\pi$
$80$	355.000			0.496128
$81$		0			0
$82$	990.000			1.33326
$83$	−78.0000	−	135.100i	−0.103152	−	0.178664i	0.809830	−	0.586665i	$-0.199559\pi$
$83$	−78.0000	−	135.100i	−0.103152	−	0.178664i	−0.912982	+	0.408001i	$0.866226\pi$
$84$		0			0
$85$	135.000	−	233.827i	0.172268	−	0.298377i
$86$	−138.000	+	239.023i	−0.173034	+	0.299704i
$87$		0			0
$88$	−252.000	−	436.477i	−0.305265	−	0.528734i
$89$	1026.00			1.22198			0.610988	−	0.791640i	$-0.290773\pi$
$89$	1026.00			1.22198			0.610988	+	0.791640i	$0.290773\pi$
$90$		0			0
$91$	1480.00			1.70490
$92$	60.0000	+	103.923i	0.0679938	+	0.117769i
$93$		0			0
$94$	36.0000	−	62.3538i	0.0395012	−	0.0684182i
$95$	−310.000	+	536.936i	−0.334793	+	0.579878i
$96$		0			0
$97$	143.000	+	247.683i	0.149685	+	0.259262i	0.931111	−	0.364736i	$-0.118841\pi$
$97$	143.000	+	247.683i	0.149685	+	0.259262i	−0.781426	+	0.623998i	$0.785507\pi$
$98$	171.000			0.176261
$99$		0			0
$100$	25.0000			0.0250000
$101$	867.000	+	1501.69i	0.854156	+	1.47944i	0.877426	+	0.479712i	$0.159259\pi$
$101$	867.000	+	1501.69i	0.854156	+	1.47944i	−0.0232704	+	0.999729i	$0.507408\pi$
$102$		0			0
$103$	−226.000	+	391.443i	−0.216198	+	0.374467i	−0.953643	−	0.300941i	$-0.902699\pi$
$103$	−226.000	+	391.443i	−0.216198	+	0.374467i	0.737444	+	0.675408i	$0.236032\pi$
$104$	777.000	−	1345.80i	0.732607	−	1.26891i
$105$		0			0
$106$	−675.000	−	1169.13i	−0.618508	−	1.07129i
$107$	−1404.00			−1.26850			−0.634251	−	0.773127i	$-0.718692\pi$
$107$	−1404.00			−1.26850			−0.634251	+	0.773127i	$0.718692\pi$
$108$		0			0
$109$	−1474.00			−1.29526			−0.647631	−	0.761954i	$-0.724240\pi$
$109$	−1474.00			−1.29526			−0.647631	+	0.761954i	$0.724240\pi$
$110$	−180.000	−	311.769i	−0.156021	−	0.270237i
$111$		0			0
$112$	710.000	−	1229.76i	0.599006	−	1.03751i
$113$	−543.000	+	940.504i	−0.452046	+	0.782966i	−0.998513	−	0.0545145i	$-0.982639\pi$
$113$	−543.000	+	940.504i	−0.452046	+	0.782966i	0.546467	+	0.837480i	$0.315972\pi$
$114$		0			0
$115$	−300.000	−	519.615i	−0.243262	−	0.421342i
$116$	−78.0000			−0.0624321
$117$		0			0
$118$	72.0000			0.0561707
$119$	−540.000	−	935.307i	−0.415981	−	0.720500i
$120$		0			0
$121$	377.500	−	653.849i	0.283621	−	0.491247i
$122$	483.000	−	836.581i	0.358433	−	0.620823i
$123$		0			0
$124$	−100.000	−	173.205i	−0.0724215	−	0.125438i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	1244.00			0.869190			0.434595	−	0.900626i	$-0.356891\pi$
$127$	1244.00			0.869190			0.434595	+	0.900626i	$0.356891\pi$
$128$	−829.500	−	1436.74i	−0.572798	−	0.992115i
$129$		0			0
$130$	555.000	−	961.288i	0.374436	−	0.648543i
$131$	−1164.00	+	2016.11i	−0.776329	+	1.34464i	0.157715	+	0.987485i	$0.449587\pi$
$131$	−1164.00	+	2016.11i	−0.776329	+	1.34464i	−0.934044	+	0.357157i	$0.883746\pi$
$132$		0			0
$133$	1240.00	+	2147.74i	0.808433	+	1.40025i
$134$	−588.000			−0.379071
$135$		0			0
$136$	−1134.00			−0.714998
$137$	−1059.00	−	1834.24i	−0.660412	−	1.14387i	−0.980507	−	0.196482i	$-0.937048\pi$
$137$	−1059.00	−	1834.24i	−0.660412	−	1.14387i	0.320095	−	0.947385i	$-0.396285\pi$
$138$		0			0
$139$	−1162.00	+	2012.64i	−0.709062	+	1.22813i	0.256144	+	0.966639i	$0.417548\pi$
$139$	−1162.00	+	2012.64i	−0.709062	+	1.22813i	−0.965206	+	0.261492i	$0.915785\pi$
$140$	50.0000	−	86.6025i	0.0301841	−	0.0522804i
$141$		0			0
$142$	432.000	+	748.246i	0.255300	+	0.442193i
$143$	−1776.00			−1.03858
$144$		0			0
$145$	390.000			0.223364
$146$	645.000	+	1117.17i	0.365620	+	0.633273i
$147$		0			0
$148$	35.0000	−	60.6218i	0.0194391	−	0.0336695i
$149$	−129.000	+	223.435i	−0.0709268	+	0.122849i	−0.899308	−	0.437316i	$-0.855929\pi$
$149$	−129.000	+	223.435i	−0.0709268	+	0.122849i	0.828381	+	0.560165i	$0.189262\pi$
$150$		0			0
$151$	404.000	+	699.749i	0.217729	+	0.377117i	0.954113	−	0.299446i	$-0.0968018\pi$
$151$	404.000	+	699.749i	0.217729	+	0.377117i	−0.736384	+	0.676563i	$0.763468\pi$
$152$	2604.00			1.38955
$153$		0			0
$154$	−1440.00			−0.753497
$155$	500.000	+	866.025i	0.259103	+	0.448780i
$156$		0			0
$157$	−1189.00	+	2059.41i	−0.604411	+	1.04687i	0.387733	+	0.921772i	$0.373258\pi$
$157$	−1189.00	+	2059.41i	−0.604411	+	1.04687i	−0.992144	+	0.125099i	$0.960075\pi$
$158$	780.000	−	1351.00i	0.392743	−	0.680252i
$159$		0			0
$160$	−112.500	−	194.856i	−0.0555869	−	0.0962794i
$161$	−2400.00			−1.17482
$162$		0			0
$163$	−52.0000			−0.0249874			−0.0124937	−	0.999922i	$-0.503977\pi$
$163$	−52.0000			−0.0249874			−0.0124937	+	0.999922i	$0.503977\pi$
$164$	−165.000	−	285.788i	−0.0785630	−	0.136075i
$165$		0			0
$166$	−234.000	+	405.300i	−0.109409	+	0.189502i
$167$	1860.00	−	3221.61i	0.861863	−	1.49279i	−0.00826564	−	0.999966i	$-0.502631\pi$
$167$	1860.00	−	3221.61i	0.861863	−	1.49279i	0.870129	−	0.492825i	$-0.164036\pi$
$168$		0			0
$169$	−1639.50	−	2839.70i	−0.746245	−	1.29253i
$170$	−810.000			−0.365436
$171$		0			0
$172$	92.0000			0.0407845
$173$	−213.000	−	368.927i	−0.0936075	−	0.162133i	0.815419	−	0.578871i	$-0.196507\pi$
$173$	−213.000	−	368.927i	−0.0936075	−	0.162133i	−0.909027	+	0.416738i	$0.863173\pi$
$174$		0			0
$175$	−250.000	+	433.013i	−0.107990	+	0.187044i
$176$	−852.000	+	1475.71i	−0.364897	+	0.632021i
$177$		0			0
$178$	−1539.00	−	2665.63i	−0.648050	−	1.12246i
$179$	−1440.00			−0.601289			−0.300644	−	0.953736i	$-0.597202\pi$
$179$	−1440.00			−0.601289			−0.300644	+	0.953736i	$0.597202\pi$
$180$		0			0
$181$	−3130.00			−1.28537			−0.642683	−	0.766133i	$-0.722179\pi$
$181$	−3130.00			−1.28537			−0.642683	+	0.766133i	$0.722179\pi$
$182$	−2220.00	−	3845.15i	−0.904161	−	1.56605i
$183$		0			0
$184$	−1260.00	+	2182.38i	−0.504828	+	0.874389i
$185$	−175.000	+	303.109i	−0.0695473	+	0.120460i
$186$		0			0
$187$	648.000	+	1122.37i	0.253403	+	0.438908i
$188$	−24.0000			−0.00931053
$189$		0			0
$190$	1860.00			0.710203
$191$	−1788.00	−	3096.91i	−0.677357	−	1.17322i	−0.975774	−	0.218781i	$-0.929792\pi$
$191$	−1788.00	−	3096.91i	−0.677357	−	1.17322i	0.298417	−	0.954436i	$-0.403541\pi$
$192$		0			0
$193$	−1333.00	+	2308.82i	−0.497158	+	0.861102i	−0.999995	−	0.00327888i	$-0.998956\pi$
$193$	−1333.00	+	2308.82i	−0.497158	+	0.861102i	0.502837	+	0.864381i	$0.332290\pi$
$194$	429.000	−	743.050i	0.158765	−	0.274989i
$195$		0			0
$196$	−28.5000	−	49.3634i	−0.0103863	−	0.0179896i
$197$	−2718.00			−0.982992			−0.491496	−	0.870880i	$-0.663550\pi$
$197$	−2718.00			−0.982992			−0.491496	+	0.870880i	$0.663550\pi$
$198$		0			0
$199$	−3832.00			−1.36504			−0.682521	−	0.730866i	$-0.739116\pi$
$199$	−3832.00			−1.36504			−0.682521	+	0.730866i	$0.739116\pi$
$200$	262.500	+	454.663i	0.0928078	+	0.160748i
$201$		0			0
$202$	2601.00	−	4505.06i	0.905969	−	1.56918i
$203$	780.000	−	1351.00i	0.269681	−	0.467101i
$204$		0			0
$205$	825.000	+	1428.94i	0.281076	+	0.486837i
$206$	1356.00			0.458626
$207$		0			0
$208$	−5254.00			−1.75144
$209$	−1488.00	−	2577.29i	−0.492474	−	0.852990i
$210$		0			0
$211$	−550.000	+	952.628i	−0.179448	+	0.310813i	−0.941692	−	0.336477i	$-0.890765\pi$
$211$	−550.000	+	952.628i	−0.179448	+	0.310813i	0.762244	+	0.647290i	$0.224098\pi$
$212$	−225.000	+	389.711i	−0.0728918	+	0.126252i
$213$		0			0
$214$	2106.00	+	3647.70i	0.672725	+	1.16519i
$215$	−460.000			−0.145915
$216$		0			0
$217$	4000.00			1.25133
$218$	2211.00	+	3829.56i	0.686917	+	1.18977i
$219$		0			0
$220$	−60.0000	+	103.923i	−0.0183873	+	0.0318477i
$221$	−1998.00	+	3460.64i	−0.608145	+	1.05334i
$222$		0			0
$223$	−982.000	−	1700.87i	−0.294886	−	0.510758i	0.680072	−	0.733145i	$-0.261948\pi$
$223$	−982.000	−	1700.87i	−0.294886	−	0.510758i	−0.974958	+	0.222387i	$0.928615\pi$
$224$	−900.000			−0.268454
$225$		0			0
$226$	3258.00			0.958933
$227$	−330.000	−	571.577i	−0.0964884	−	0.167123i	0.813740	−	0.581228i	$-0.197428\pi$
$227$	−330.000	−	571.577i	−0.0964884	−	0.167123i	−0.910229	+	0.414106i	$0.864094\pi$
$228$		0			0
$229$	953.000	−	1650.64i	0.275004	−	0.476322i	−0.695132	−	0.718882i	$-0.744654\pi$
$229$	953.000	−	1650.64i	0.275004	−	0.476322i	0.970136	+	0.242561i	$0.0779873\pi$
$230$	−900.000	+	1558.85i	−0.258018	+	0.446901i
$231$		0			0
$232$	−819.000	−	1418.55i	−0.231767	−	0.401433i
$233$	−1458.00			−0.409943			−0.204972	−	0.978768i	$-0.565710\pi$
$233$	−1458.00			−0.409943			−0.204972	+	0.978768i	$0.565710\pi$
$234$		0			0
$235$	120.000			0.0333104
$236$	−12.0000	−	20.7846i	−0.00330989	−	0.00573289i
$237$		0			0
$238$	−1620.00	+	2805.92i	−0.441214	+	0.764206i
$239$	−588.000	+	1018.45i	−0.159140	+	0.275639i	−0.934559	−	0.355808i	$-0.884206\pi$
$239$	−588.000	+	1018.45i	−0.159140	+	0.275639i	0.775419	+	0.631448i	$0.217539\pi$
$240$		0			0
$241$	−433.000	−	749.978i	−0.115734	−	0.200458i	0.802339	−	0.596869i	$-0.203589\pi$
$241$	−433.000	−	749.978i	−0.115734	−	0.200458i	−0.918073	+	0.396411i	$0.870255\pi$
$242$	−2265.00			−0.601652
$243$		0			0
$244$	−322.000			−0.0844834
$245$	142.500	+	246.817i	0.0371591	+	0.0643615i
$246$		0			0
$247$	4588.00	−	7946.65i	1.18189	−	2.04710i
$248$	2100.00	−	3637.31i	0.537702	−	0.931327i
$249$		0			0
$250$	187.500	+	324.760i	0.0474342	+	0.0821584i
$251$	432.000			0.108636			0.0543179	−	0.998524i	$-0.482702\pi$
$251$	432.000			0.108636			0.0543179	+	0.998524i	$0.482702\pi$
$252$		0			0
$253$	2880.00			0.715668
$254$	−1866.00	−	3232.01i	−0.460958	−	0.798402i
$255$		0			0
$256$	−756.500	+	1310.30i	−0.184692	+	0.319897i
$257$	−1263.00	+	2187.58i	−0.306552	+	0.530963i	−0.977606	−	0.210445i	$-0.932509\pi$
$257$	−1263.00	+	2187.58i	−0.306552	+	0.530963i	0.671054	+	0.741409i	$0.265842\pi$
$258$		0			0
$259$	700.000	+	1212.44i	0.167938	+	0.290877i
$260$	−370.000			−0.0882555
$261$		0			0
$262$	6984.00			1.64684
$263$	−2724.00	−	4718.11i	−0.638666	−	1.10620i	−0.985726	−	0.168358i	$-0.946153\pi$
$263$	−2724.00	−	4718.11i	−0.638666	−	1.10620i	0.347060	−	0.937843i	$-0.387180\pi$
$264$		0			0
$265$	1125.00	−	1948.56i	0.260786	−	0.451694i
$266$	3720.00	−	6443.23i	0.857473	−	1.48519i
$267$		0			0
$268$	98.0000	+	169.741i	0.0223370	+	0.0386887i
$269$	−2574.00			−0.583418			−0.291709	−	0.956507i	$-0.594224\pi$
$269$	−2574.00			−0.583418			−0.291709	+	0.956507i	$0.594224\pi$
$270$		0			0
$271$	−3184.00			−0.713706			−0.356853	−	0.934161i	$-0.616150\pi$
$271$	−3184.00			−0.713706			−0.356853	+	0.934161i	$0.616150\pi$
$272$	1917.00	+	3320.34i	0.427335	+	0.740166i
$273$		0			0
$274$	−3177.00	+	5502.73i	−0.700473	+	1.21325i
$275$	300.000	−	519.615i	0.0657843	−	0.113942i
$276$		0			0
$277$	−1981.00	−	3431.19i	−0.429699	−	0.744261i	0.567147	−	0.823617i	$-0.308047\pi$
$277$	−1981.00	−	3431.19i	−0.429699	−	0.744261i	−0.996846	+	0.0793553i	$0.974714\pi$
$278$	6972.00			1.50415
$279$		0			0
$280$	2100.00			0.448211
$281$	4143.00	+	7175.89i	0.879540	+	1.52341i	0.851847	+	0.523791i	$0.175483\pi$
$281$	4143.00	+	7175.89i	0.879540	+	1.52341i	0.0276929	+	0.999616i	$0.491184\pi$
$282$		0			0
$283$	1358.00	−	2352.12i	0.285246	−	0.494061i	−0.687423	−	0.726258i	$-0.741258\pi$
$283$	1358.00	−	2352.12i	0.285246	−	0.494061i	0.972669	+	0.232197i	$0.0745912\pi$
$284$	144.000	−	249.415i	0.0300874	−	0.0521129i
$285$		0			0
$286$	2664.00	+	4614.18i	0.550789	+	0.953994i
$287$	6600.00			1.35744
$288$		0			0
$289$	−1997.00			−0.406473
$290$	−585.000	−	1013.25i	−0.118456	−	0.205173i
$291$		0			0
$292$	215.000	−	372.391i	0.0430888	−	0.0746320i
$293$	−3009.00	+	5211.74i	−0.599958	+	1.03916i	0.392869	+	0.919595i	$0.371483\pi$
$293$	−3009.00	+	5211.74i	−0.599958	+	1.03916i	−0.992827	+	0.119563i	$0.961851\pi$
$294$		0			0
$295$	60.0000	+	103.923i	0.0118418	+	0.0205106i
$296$	1470.00			0.288655
$297$		0			0
$298$	774.000			0.150458
$299$	4440.00	+	7690.31i	0.858769	+	1.48743i
$300$		0			0
$301$	−920.000	+	1593.49i	−0.176172	+	0.305140i
$302$	1212.00	−	2099.25i	0.230936	−	0.399993i
$303$		0			0
$304$	−4402.00	−	7624.49i	−0.830500	−	1.43847i
$305$	1610.00			0.302257
$306$		0			0
$307$	9236.00			1.71702			0.858512	−	0.512793i	$-0.171389\pi$
$307$	9236.00			1.71702			0.858512	+	0.512793i	$0.171389\pi$
$308$	240.000	+	415.692i	0.0444002	+	0.0769034i
$309$		0			0
$310$	1500.00	−	2598.08i	0.274820	−	0.476003i
$311$	−768.000	+	1330.22i	−0.140030	+	0.242539i	−0.927508	−	0.373804i	$-0.878053\pi$
$311$	−768.000	+	1330.22i	−0.140030	+	0.242539i	0.787478	+	0.616343i	$0.211387\pi$
$312$		0			0
$313$	3671.00	+	6358.36i	0.662930	+	1.14823i	0.979842	+	0.199774i	$0.0640208\pi$
$313$	3671.00	+	6358.36i	0.662930	+	1.14823i	−0.316912	+	0.948455i	$0.602646\pi$
$314$	7134.00			1.28215
$315$		0			0
$316$	−520.000			−0.0925705
$317$	1947.00	+	3372.30i	0.344967	+	0.597500i	0.985348	−	0.170558i	$-0.0545569\pi$
$317$	1947.00	+	3372.30i	0.344967	+	0.597500i	−0.640381	+	0.768057i	$0.721224\pi$
$318$		0			0
$319$	−936.000	+	1621.20i	−0.164282	+	0.284545i
$320$	1082.50	−	1874.94i	0.189105	−	0.327539i
$321$		0			0
$322$	3600.00	+	6235.38i	0.623044	+	1.07914i
$323$	−6696.00			−1.15348
$324$		0			0
$325$	1850.00			0.315752
$326$	78.0000	+	135.100i	0.0132516	+	0.0229524i
$327$		0			0
$328$	3465.00	−	6001.56i	0.583301	−	1.01031i
$329$	240.000	−	415.692i	0.0402177	−	0.0696591i
$330$		0			0
$331$	−1846.00	−	3197.37i	−0.306542	−	0.530946i	0.671062	−	0.741402i	$-0.265839\pi$
$331$	−1846.00	−	3197.37i	−0.306542	−	0.530946i	−0.977603	+	0.210456i	$0.932505\pi$
$332$	156.000			0.0257880
$333$		0			0
$334$	−11160.0			−1.82829
$335$	−490.000	−	848.705i	−0.0799151	−	0.138417i
$336$		0			0
$337$	4499.00	−	7792.50i	0.727229	−	1.25960i	−0.230821	−	0.972996i	$-0.574141\pi$
$337$	4499.00	−	7792.50i	0.727229	−	1.25960i	0.958050	−	0.286601i	$-0.0925255\pi$
$338$	−4918.50	+	8519.09i	−0.791512	+	1.37094i
$339$		0			0
$340$	135.000	+	233.827i	0.0215335	+	0.0372972i
$341$	−4800.00			−0.762271
$342$		0			0
$343$	−5720.00			−0.900440
$344$	966.000	+	1673.16i	0.151405	+	0.262241i
$345$		0			0
$346$	−639.000	+	1106.78i	−0.0992857	+	0.171968i
$347$	−2622.00	+	4541.44i	−0.405638	+	0.702585i	−0.994395	−	0.105724i	$-0.966284\pi$
$347$	−2622.00	+	4541.44i	−0.405638	+	0.702585i	0.588758	+	0.808310i	$0.299617\pi$
$348$		0			0
$349$	−3151.00	−	5457.69i	−0.483293	−	0.837088i	0.516523	−	0.856273i	$-0.327226\pi$
$349$	−3151.00	−	5457.69i	−0.483293	−	0.837088i	−0.999816	+	0.0191856i	$0.993893\pi$
$350$	1500.00			0.229081
$351$		0			0
$352$	1080.00			0.163535
$353$	−1707.00	−	2956.61i	−0.257378	−	0.445792i	0.708161	−	0.706051i	$-0.249525\pi$
$353$	−1707.00	−	2956.61i	−0.257378	−	0.445792i	−0.965539	+	0.260259i	$0.916192\pi$
$354$		0			0
$355$	−720.000	+	1247.08i	−0.107644	+	0.186445i
$356$	−513.000	+	888.542i	−0.0763734	+	0.132283i
$357$		0			0
$358$	2160.00	+	3741.23i	0.318881	+	0.552319i
$359$	4824.00			0.709195			0.354597	−	0.935019i	$-0.384618\pi$
$359$	4824.00			0.709195			0.354597	+	0.935019i	$0.384618\pi$
$360$		0			0
$361$	8517.00			1.24173
$362$	4695.00	+	8131.98i	0.681668	+	1.18068i
$363$		0			0
$364$	−740.000	+	1281.72i	−0.106556	+	0.184561i
$365$	−1075.00	+	1861.95i	−0.154159	+	0.267011i
$366$		0			0
$367$	1754.00	+	3038.02i	0.249477	+	0.432107i	0.963381	−	0.268137i	$-0.0864080\pi$
$367$	1754.00	+	3038.02i	0.249477	+	0.432107i	−0.713904	+	0.700244i	$0.753075\pi$
$368$	8520.00			1.20689
$369$		0			0
$370$	1050.00			0.147532
$371$	−4500.00	−	7794.23i	−0.629726	−	1.09072i
$372$		0			0
$373$	−5401.00	+	9354.81i	−0.749740	+	1.29859i	0.198207	+	0.980160i	$0.436488\pi$
$373$	−5401.00	+	9354.81i	−0.749740	+	1.29859i	−0.947947	+	0.318428i	$0.896845\pi$
$374$	1944.00	−	3367.11i	0.268775	−	0.465532i
$375$		0			0
$376$	−252.000	−	436.477i	−0.0345636	−	0.0598659i
$377$	−5772.00			−0.788523
$378$		0			0
$379$	1460.00			0.197876			0.0989382	−	0.995094i	$-0.468455\pi$
$379$	1460.00			0.197876			0.0989382	+	0.995094i	$0.468455\pi$
$380$	−310.000	−	536.936i	−0.0418491	−	0.0724848i
$381$		0			0
$382$	−5364.00	+	9290.72i	−0.718445	+	1.24438i
$383$	2436.00	−	4219.28i	0.324997	−	0.562911i	−0.656515	−	0.754313i	$-0.727970\pi$
$383$	2436.00	−	4219.28i	0.324997	−	0.562911i	0.981512	+	0.191402i	$0.0613034\pi$
$384$		0			0
$385$	−1200.00	−	2078.46i	−0.158851	−	0.275138i
$386$	7998.00			1.05463
$387$		0			0
$388$	−286.000			−0.0374213
$389$	7023.00	+	12164.2i	0.915373	+	1.58547i	0.806354	+	0.591434i	$0.201438\pi$
$389$	7023.00	+	12164.2i	0.915373	+	1.58547i	0.109020	+	0.994040i	$0.465229\pi$
$390$		0			0
$391$	3240.00	−	5611.84i	0.419064	−	0.725839i
$392$	598.500	−	1036.63i	0.0771143	−	0.133566i
$393$		0			0
$394$	4077.00	+	7061.57i	0.521310	+	0.902936i
$395$	2600.00			0.331190
$396$		0			0
$397$	−2734.00			−0.345631			−0.172816	−	0.984954i	$-0.555286\pi$
$397$	−2734.00			−0.345631			−0.172816	+	0.984954i	$0.555286\pi$
$398$	5748.00	+	9955.83i	0.723923	+	1.25387i
$399$		0			0
$400$	887.500	−	1537.20i	0.110937	−	0.192149i
$401$	7971.00	−	13806.2i	0.992650	−	1.71932i	0.391520	−	0.920170i	$-0.371949\pi$
$401$	7971.00	−	13806.2i	0.992650	−	1.71932i	0.601130	−	0.799151i	$-0.294717\pi$
$402$		0			0
$403$	−7400.00	−	12817.2i	−0.914690	−	1.58429i
$404$	−1734.00			−0.213539
$405$		0			0
$406$	−4680.00			−0.572080
$407$	−840.000	−	1454.92i	−0.102303	−	0.177194i
$408$		0			0
$409$	−4357.00	+	7546.55i	−0.526748	+	0.912354i	0.472767	+	0.881188i	$0.343255\pi$
$409$	−4357.00	+	7546.55i	−0.526748	+	0.912354i	−0.999514	+	0.0311660i	$0.990078\pi$
$410$	2475.00	−	4286.83i	0.298126	−	0.516369i
$411$		0			0
$412$	−226.000	−	391.443i	−0.0270248	−	0.0468083i
$413$	480.000			0.0571895
$414$		0			0
$415$	−780.000			−0.0922619
$416$	1665.00	+	2883.86i	0.196234	+	0.339887i
$417$		0			0
$418$	−4464.00	+	7731.87i	−0.522348	+	0.904733i
$419$	−5988.00	+	10371.5i	−0.698169	+	1.20926i	0.270931	+	0.962599i	$0.412668\pi$
$419$	−5988.00	+	10371.5i	−0.698169	+	1.20926i	−0.969101	+	0.246666i	$0.920665\pi$
$420$		0			0
$421$	−5527.00	−	9573.04i	−0.639833	−	1.10822i	−0.985469	−	0.169854i	$-0.945670\pi$
$421$	−5527.00	−	9573.04i	−0.639833	−	1.10822i	0.345637	−	0.938368i	$-0.387663\pi$
$422$	3300.00			0.380667
$423$		0			0
$424$	−9450.00			−1.08239
$425$	−675.000	−	1169.13i	−0.0770407	−	0.133438i
$426$		0			0
$427$	3220.00	−	5577.20i	0.364934	−	0.632084i
$428$	702.000	−	1215.90i	0.0792814	−	0.137319i
$429$		0			0
$430$	690.000	+	1195.12i	0.0773832	+	0.134032i
$431$	720.000			0.0804668			0.0402334	−	0.999190i	$-0.487190\pi$
$431$	720.000			0.0804668			0.0402334	+	0.999190i	$0.487190\pi$
$432$		0			0
$433$	−15622.0			−1.73382			−0.866912	−	0.498462i	$-0.833898\pi$
$433$	−15622.0			−1.73382			−0.866912	+	0.498462i	$0.833898\pi$
$434$	−6000.00	−	10392.3i	−0.663616	−	1.14942i
$435$		0			0
$436$	737.000	−	1276.52i	0.0809539	−	0.140216i
$437$	−7440.00	+	12886.5i	−0.814424	+	1.41062i
$438$		0			0
$439$	4940.00	+	8556.33i	0.537069	+	0.930231i	0.999060	+	0.0433464i	$0.0138019\pi$
$439$	4940.00	+	8556.33i	0.537069	+	0.930231i	−0.461991	+	0.886885i	$0.652865\pi$
$440$	−2520.00			−0.273037
$441$		0			0
$442$	11988.0			1.29007
$443$	8058.00	+	13956.9i	0.864215	+	1.49686i	0.867825	+	0.496870i	$0.165518\pi$
$443$	8058.00	+	13956.9i	0.864215	+	1.49686i	−0.00361002	+	0.999993i	$0.501149\pi$
$444$		0			0
$445$	2565.00	−	4442.71i	0.273242	−	0.473269i
$446$	−2946.00	+	5102.62i	−0.312774	+	0.541740i
$447$		0			0
$448$	−4330.00	−	7499.78i	−0.456637	−	0.790918i
$449$	9018.00			0.947852			0.473926	−	0.880565i	$-0.342836\pi$
$449$	9018.00			0.947852			0.473926	+	0.880565i	$0.342836\pi$
$450$		0			0
$451$	−7920.00			−0.826914
$452$	−543.000	−	940.504i	−0.0565057	−	0.0978707i
$453$		0			0
$454$	−990.000	+	1714.73i	−0.102341	+	0.177261i
$455$	3700.00	−	6408.59i	0.381228	−	0.660306i
$456$		0			0
$457$	1835.00	+	3178.31i	0.187829	+	0.325329i	0.944526	−	0.328437i	$-0.106522\pi$
$457$	1835.00	+	3178.31i	0.187829	+	0.325329i	−0.756697	+	0.653765i	$0.773188\pi$
$458$	−5718.00			−0.583372
$459$		0			0
$460$	600.000			0.0608155
$461$	−8781.00	−	15209.1i	−0.887141	−	1.53657i	−0.843240	−	0.537537i	$-0.819355\pi$
$461$	−8781.00	−	15209.1i	−0.887141	−	1.53657i	−0.0439008	−	0.999036i	$-0.513979\pi$
$462$		0			0
$463$	−586.000	+	1014.98i	−0.0588202	+	0.101879i	−0.893936	−	0.448195i	$-0.852067\pi$
$463$	−586.000	+	1014.98i	−0.0588202	+	0.101879i	0.835116	+	0.550074i	$0.185401\pi$
$464$	−2769.00	+	4796.05i	−0.277042	+	0.479851i
$465$		0			0
$466$	2187.00	+	3788.00i	0.217405	+	0.376557i
$467$	6876.00			0.681335			0.340667	−	0.940184i	$-0.389347\pi$
$467$	6876.00			0.681335			0.340667	+	0.940184i	$0.389347\pi$
$468$		0			0
$469$	−3920.00			−0.385946
$470$	−180.000	−	311.769i	−0.0176655	−	0.0305975i
$471$		0			0
$472$	252.000	−	436.477i	0.0245747	−	0.0425646i
$473$	1104.00	−	1912.18i	0.107319	−	0.185882i
$474$		0			0
$475$	1550.00	+	2684.68i	0.149724	+	0.259329i
$476$	1080.00			0.103995
$477$		0			0
$478$	3528.00			0.337588
$479$	−1140.00	−	1974.54i	−0.108743	−	0.188349i	0.806518	−	0.591209i	$-0.201349\pi$
$479$	−1140.00	−	1974.54i	−0.108743	−	0.188349i	−0.915261	+	0.402861i	$0.868016\pi$
$480$		0			0
$481$	2590.00	−	4486.01i	0.245517	−	0.425248i
$482$	−1299.00	+	2249.93i	−0.122755	+	0.212618i
$483$		0			0
$484$	377.500	+	653.849i	0.0354527	+	0.0614058i
$485$	1430.00			0.133882
$486$		0			0
$487$	−3076.00			−0.286215			−0.143108	−	0.989707i	$-0.545710\pi$
$487$	−3076.00			−0.286215			−0.143108	+	0.989707i	$0.545710\pi$
$488$	−3381.00	−	5856.06i	−0.313628	−	0.543220i
$489$		0			0
$490$	427.500	−	740.452i	0.0394132	−	0.0682657i
$491$	9456.00	−	16378.3i	0.869131	−	1.50538i	0.00624491	−	0.999981i	$-0.498012\pi$
$491$	9456.00	−	16378.3i	0.869131	−	1.50538i	0.862886	−	0.505398i	$-0.168654\pi$
$492$		0			0
$493$	2106.00	+	3647.70i	0.192392	+	0.333233i
$494$	−27528.0			−2.50717
$495$		0			0
$496$	−14200.0			−1.28548
$497$	2880.00	+	4988.31i	0.259931	+	0.450214i
$498$		0			0
$499$	−4978.00	+	8622.15i	−0.446585	+	0.773508i	−0.998161	−	0.0606167i	$-0.980693\pi$
$499$	−4978.00	+	8622.15i	−0.446585	+	0.773508i	0.551576	+	0.834125i	$0.314027\pi$
$500$	62.5000	−	108.253i	0.00559017	−	0.00968246i
$501$		0			0
$502$	−648.000	−	1122.37i	−0.0576129	−	0.0997884i
$503$	−10656.0			−0.944588			−0.472294	−	0.881441i	$-0.656574\pi$
$503$	−10656.0			−0.944588			−0.472294	+	0.881441i	$0.656574\pi$
$504$		0			0
$505$	8670.00			0.763980
$506$	−4320.00	−	7482.46i	−0.379540	−	0.657383i
$507$		0			0
$508$	−622.000	+	1077.34i	−0.0543244	+	0.0940926i
$509$	1383.00	−	2395.43i	0.120433	−	0.208596i	−0.799506	−	0.600659i	$-0.794905\pi$
$509$	1383.00	−	2395.43i	0.120433	−	0.208596i	0.919939	+	0.392063i	$0.128238\pi$
$510$		0			0
$511$	4300.00	+	7447.82i	0.372252	+	0.644759i
$512$	−8733.00			−0.753804
$513$		0			0
$514$	7578.00			0.650294
$515$	1130.00	+	1957.22i	0.0966869	+	0.167467i
$516$		0			0
$517$	−288.000	+	498.831i	−0.0244995	+	0.0424343i
$518$	2100.00	−	3637.31i	0.178125	−	0.308521i
$519$		0			0
$520$	−3885.00	−	6729.02i	−0.327632	−	0.567475i
$521$	10530.0			0.885466			0.442733	−	0.896654i	$-0.354009\pi$
$521$	10530.0			0.885466			0.442733	+	0.896654i	$0.354009\pi$
$522$		0			0
$523$	12692.0			1.06115			0.530576	−	0.847637i	$-0.321976\pi$
$523$	12692.0			1.06115			0.530576	+	0.847637i	$0.321976\pi$
$524$	−1164.00	−	2016.11i	−0.0970412	−	0.168080i
$525$		0			0
$526$	−8172.00	+	14154.3i	−0.677407	+	1.17330i
$527$	−5400.00	+	9353.07i	−0.446352	+	0.773105i
$528$		0			0
$529$	−1116.50	−	1933.83i	−0.0917646	−	0.158941i
$530$	−6750.00			−0.553210
$531$		0			0
$532$	−2480.00			−0.202108
$533$	−12210.0	−	21148.3i	−0.992259	−	1.71864i
$534$		0			0
$535$	−3510.00	+	6079.50i	−0.283646	+	0.491289i
$536$	−2058.00	+	3564.56i	−0.165843	+	0.287249i
$537$		0			0
$538$	3861.00	+	6687.45i	0.309404	+	0.535904i
$539$	−1368.00			−0.109321
$540$		0			0
$541$	18110.0			1.43920			0.719602	−	0.694386i	$-0.244324\pi$
$541$	18110.0			1.43920			0.719602	+	0.694386i	$0.244324\pi$
$542$	4776.00	+	8272.27i	0.378500	+	0.655580i
$543$		0			0
$544$	1215.00	−	2104.44i	0.0957586	−	0.165859i
$545$	−3685.00	+	6382.61i	−0.289629	+	0.501653i
$546$		0			0
$547$	−1810.00	−	3135.01i	−0.141481	−	0.245052i	0.786574	−	0.617496i	$-0.211853\pi$
$547$	−1810.00	−	3135.01i	−0.141481	−	0.245052i	−0.928054	+	0.372445i	$0.878520\pi$
$548$	2118.00			0.165103
$549$		0			0
$550$	−1800.00			−0.139550
$551$	−4836.00	−	8376.20i	−0.373903	−	0.647619i
$552$		0			0
$553$	5200.00	−	9006.66i	0.399867	−	0.692590i
$554$	−5943.00	+	10293.6i	−0.455765	+	0.789408i
$555$		0			0
$556$	−1162.00	−	2012.64i	−0.0886327	−	0.153516i
$557$	−14166.0			−1.07762			−0.538809	−	0.842428i	$-0.681125\pi$
$557$	−14166.0			−1.07762			−0.538809	+	0.842428i	$0.681125\pi$
$558$		0			0
$559$	6808.00			0.515112
$560$	−3550.00	−	6148.78i	−0.267884	−	0.463988i
$561$		0			0
$562$	12429.0	−	21527.7i	0.932893	−	1.61582i
$563$	6702.00	−	11608.2i	0.501697	−	0.868965i	−0.498301	−	0.867004i	$-0.666042\pi$
$563$	6702.00	−	11608.2i	0.501697	−	0.868965i	0.999998	−	0.00196107i	$-0.000624229\pi$
$564$		0			0
$565$	2715.00	+	4702.52i	0.202161	+	0.350153i
$566$	−8148.00			−0.605099
$567$		0			0
$568$	6048.00			0.446775
$569$	9327.00	+	16154.8i	0.687185	+	1.19024i	0.972745	+	0.231878i	$0.0744871\pi$
$569$	9327.00	+	16154.8i	0.687185	+	1.19024i	−0.285560	+	0.958361i	$0.592180\pi$
$570$		0			0
$571$	3842.00	−	6654.54i	0.281581	−	0.487712i	−0.690193	−	0.723625i	$-0.742475\pi$
$571$	3842.00	−	6654.54i	0.281581	−	0.487712i	0.971774	+	0.235913i	$0.0758079\pi$
$572$	888.000	−	1538.06i	0.0649111	−	0.112429i
$573$		0			0
$574$	−9900.00	−	17147.3i	−0.719892	−	1.24689i
$575$	−3000.00			−0.217580
$576$		0			0
$577$	−1726.00			−0.124531			−0.0622654	−	0.998060i	$-0.519833\pi$
$577$	−1726.00			−0.124531			−0.0622654	+	0.998060i	$0.519833\pi$
$578$	2995.50	+	5188.36i	0.215565	+	0.373369i
$579$		0			0
$580$	−195.000	+	337.750i	−0.0139602	+	0.0241798i
$581$	−1560.00	+	2702.00i	−0.111394	+	0.192939i
$582$		0			0
$583$	5400.00	+	9353.07i	0.383611	+	0.664434i
$584$	9030.00			0.639836
$585$		0			0
$586$	18054.0			1.27270
$587$	−5298.00	−	9176.41i	−0.372524	−	0.645231i	0.617429	−	0.786627i	$-0.288174\pi$
$587$	−5298.00	−	9176.41i	−0.372524	−	0.645231i	−0.989953	+	0.141396i	$0.954841\pi$
$588$		0			0
$589$	12400.0	−	21477.4i	0.867459	−	1.50248i
$590$	180.000	−	311.769i	0.0125601	−	0.0217548i
$591$		0			0
$592$	−2485.00	−	4304.15i	−0.172522	−	0.298816i
$593$	2862.00			0.198193			0.0990963	−	0.995078i	$-0.468405\pi$
$593$	2862.00			0.198193			0.0990963	+	0.995078i	$0.468405\pi$
$594$		0			0
$595$	−5400.00			−0.372065
$596$	−129.000	−	223.435i	−0.00886585	−	0.0153561i
$597$		0			0
$598$	13320.0	−	23070.9i	0.910862	−	1.57766i
$599$	11796.0	−	20431.3i	0.804627	−	1.39365i	−0.111916	−	0.993718i	$-0.535699\pi$
$599$	11796.0	−	20431.3i	0.804627	−	1.39365i	0.916543	−	0.399937i	$-0.130968\pi$
$600$		0			0
$601$	4787.00	+	8291.33i	0.324902	+	0.562746i	0.981492	−	0.191501i	$-0.0613355\pi$
$601$	4787.00	+	8291.33i	0.324902	+	0.562746i	−0.656591	+	0.754247i	$0.728002\pi$
$602$	5520.00			0.373718
$603$		0			0
$604$	−808.000			−0.0544322
$605$	−1887.50	−	3269.25i	−0.126839	−	0.219692i
$606$		0			0
$607$	−8722.00	+	15106.9i	−0.583221	+	1.01017i	0.411874	+	0.911241i	$0.364874\pi$
$607$	−8722.00	+	15106.9i	−0.583221	+	1.01017i	−0.995095	+	0.0989273i	$0.968459\pi$
$608$	−2790.00	+	4832.42i	−0.186101	+	0.322336i
$609$		0			0
$610$	−2415.00	−	4182.90i	−0.160296	−	0.277641i
$611$	−1776.00			−0.117593
$612$		0			0
$613$	−2374.00			−0.156419			−0.0782096	−	0.996937i	$-0.524920\pi$
$613$	−2374.00			−0.156419			−0.0782096	+	0.996937i	$0.524920\pi$
$614$	−13854.0	−	23995.8i	−0.910589	−	1.57719i
$615$		0			0
$616$	−5040.00	+	8729.54i	−0.329655	+	0.570979i
$617$	6081.00	−	10532.6i	0.396778	−	0.687239i	−0.596549	−	0.802577i	$-0.703462\pi$
$617$	6081.00	−	10532.6i	0.396778	−	0.687239i	0.993326	+	0.115338i	$0.0367951\pi$
$618$		0			0
$619$	−4402.00	−	7624.49i	−0.285834	−	0.495079i	0.686977	−	0.726679i	$-0.258937\pi$
$619$	−4402.00	−	7624.49i	−0.285834	−	0.495079i	−0.972811	+	0.231600i	$0.925604\pi$
$620$	−1000.00			−0.0647758
$621$		0			0
$622$	4608.00			0.297048
$623$	−10260.0	−	17770.8i	−0.659805	−	1.14281i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	11013.0	−	19075.1i	0.703144	−	1.21788i
$627$		0			0
$628$	−1189.00	−	2059.41i	−0.0755514	−	0.130859i
$629$	−3780.00			−0.239616
$630$		0			0
$631$	−12688.0			−0.800478			−0.400239	−	0.916411i	$-0.631073\pi$
$631$	−12688.0			−0.800478			−0.400239	+	0.916411i	$0.631073\pi$
$632$	−5460.00	−	9457.00i	−0.343651	−	0.595220i
$633$		0			0
$634$	5841.00	−	10116.9i	0.365892	−	0.633744i
$635$	3110.00	−	5386.68i	0.194357	−	0.336636i
$636$		0			0
$637$	−2109.00	−	3652.90i	−0.131180	−	0.227210i
$638$	5616.00			0.348495
$639$		0			0
$640$	−8295.00			−0.512326
$641$	4575.00	+	7924.13i	0.281906	+	0.488275i	0.971854	−	0.235583i	$-0.0757001\pi$
$641$	4575.00	+	7924.13i	0.281906	+	0.488275i	−0.689948	+	0.723859i	$0.742367\pi$
$642$		0			0
$643$	−12646.0	+	21903.5i	−0.775598	+	1.34338i	0.158860	+	0.987301i	$0.449218\pi$
$643$	−12646.0	+	21903.5i	−0.775598	+	1.34338i	−0.934458	+	0.356074i	$0.884115\pi$
$644$	1200.00	−	2078.46i	0.0734264	−	0.127178i
$645$		0			0
$646$	10044.0	+	17396.7i	0.611727	+	1.05954i
$647$	−2736.00			−0.166249			−0.0831246	−	0.996539i	$-0.526490\pi$
$647$	−2736.00			−0.166249			−0.0831246	+	0.996539i	$0.526490\pi$
$648$		0			0
$649$	−576.000			−0.0348382
$650$	−2775.00	−	4806.44i	−0.167453	−	0.290037i
$651$		0			0
$652$	26.0000	−	45.0333i	0.00156172	−	0.00270497i
$653$	−11109.0	+	19241.4i	−0.665741	+	1.15310i	0.313343	+	0.949640i	$0.398551\pi$
$653$	−11109.0	+	19241.4i	−0.665741	+	1.15310i	−0.979084	+	0.203457i	$0.934782\pi$
$654$		0			0
$655$	5820.00	+	10080.5i	0.347185	+	0.601342i
$656$	−23430.0			−1.39449
$657$		0			0
$658$	−1440.00			−0.0853147
$659$	−7260.00	−	12574.7i	−0.429149	−	0.743309i	0.567648	−	0.823271i	$-0.307853\pi$
$659$	−7260.00	−	12574.7i	−0.429149	−	0.743309i	−0.996798	+	0.0799625i	$0.974520\pi$
$660$		0			0
$661$	5309.00	−	9195.46i	0.312400	−	0.541092i	−0.666482	−	0.745521i	$-0.732201\pi$
$661$	5309.00	−	9195.46i	0.312400	−	0.541092i	0.978881	+	0.204429i	$0.0655339\pi$
$662$	−5538.00	+	9592.10i	−0.325137	+	0.563153i
$663$		0			0
$664$	1638.00	+	2837.10i	0.0957330	+	0.165814i
$665$	12400.0			0.723085
$666$		0			0
$667$	9360.00			0.543359
$668$	1860.00	+	3221.61i	0.107733	+	0.186599i
$669$		0			0
$670$	−1470.00	+	2546.11i	−0.0847628	+	0.146813i
$671$	−3864.00	+	6692.64i	−0.222307	+	0.385047i
$672$		0			0
$673$	−685.000	−	1186.45i	−0.0392345	−	0.0679561i	0.845741	−	0.533593i	$-0.179159\pi$
$673$	−685.000	−	1186.45i	−0.0392345	−	0.0679561i	−0.884976	+	0.465637i	$0.845825\pi$
$674$	−26994.0			−1.54269
$675$		0			0
$676$	3279.00			0.186561
$677$	6879.00	+	11914.8i	0.390519	+	0.676399i	0.992518	−	0.122098i	$-0.0389622\pi$
$677$	6879.00	+	11914.8i	0.390519	+	0.676399i	−0.601999	+	0.798497i	$0.705629\pi$
$678$		0			0
$679$	2860.00	−	4953.67i	0.161645	−	0.279977i
$680$	−2835.00	+	4910.36i	−0.159878	+	0.276917i
$681$		0			0
$682$	7200.00	+	12470.8i	0.404255	+	0.700191i
$683$	11988.0			0.671608			0.335804	−	0.941932i	$-0.390992\pi$
$683$	11988.0			0.671608			0.335804	+	0.941932i	$0.390992\pi$
$684$		0			0
$685$	−10590.0			−0.590691
$686$	8580.00	+	14861.0i	0.477530	+	0.827107i
$687$		0			0
$688$	3266.00	−	5656.88i	0.180981	−	0.313469i
$689$	−16650.0	+	28838.6i	−0.920631	+	1.59458i
$690$		0			0
$691$	−16498.0	−	28575.4i	−0.908268	−	1.57317i	−0.816469	−	0.577390i	$-0.804071\pi$
$691$	−16498.0	−	28575.4i	−0.908268	−	1.57317i	−0.0917997	−	0.995777i	$-0.529262\pi$
$692$	426.000			0.0234019
$693$		0			0
$694$	15732.0			0.860488
$695$	5810.00	+	10063.2i	0.317102	+	0.549237i
$696$		0			0
$697$	−8910.00	+	15432.6i	−0.484204	+	0.838666i
$698$	−9453.00	+	16373.1i	−0.512609	+	0.887865i
$699$		0			0
$700$	−250.000	−	433.013i	−0.0134987	−	0.0233805i
$701$	−25902.0			−1.39558			−0.697792	−	0.716300i	$-0.745834\pi$
$701$	−25902.0			−1.39558			−0.697792	+	0.716300i	$0.745834\pi$
$702$		0			0
$703$	8680.00			0.465679
$704$	5196.00	+	8999.74i	0.278170	+	0.481804i
$705$		0			0
$706$	−5121.00	+	8869.83i	−0.272991	+	0.472834i
$707$	17340.0	−	30033.8i	0.922401	−	1.59765i
$708$		0			0
$709$	13697.0	+	23723.9i	0.725531	+	1.25666i	0.958755	+	0.284234i	$0.0917392\pi$
$709$	13697.0	+	23723.9i	0.725531	+	1.25666i	−0.233224	+	0.972423i	$0.574927\pi$
$710$	4320.00			0.228347
$711$		0			0
$712$	−21546.0			−1.13409
$713$	12000.0	+	20784.6i	0.630299	+	1.09171i
$714$		0			0
$715$	−4440.00	+	7690.31i	−0.232233	+	0.402239i
$716$	720.000	−	1247.08i	0.0375805	−	0.0650914i
$717$		0			0
$718$	−7236.00	−	12533.1i	−0.376107	−	0.651437i
$719$	34848.0			1.80753			0.903763	−	0.428033i	$-0.140793\pi$
$719$	34848.0			1.80753			0.903763	+	0.428033i	$0.140793\pi$
$720$		0			0
$721$	9040.00			0.466945
$722$	−12775.5	−	22127.8i	−0.658525	−	1.14060i
$723$		0			0
$724$	1565.00	−	2710.66i	0.0803353	−	0.139145i
$725$	975.000	−	1688.75i	0.0499456	−	0.0865084i
$726$		0			0
$727$	−14014.0	−	24273.0i	−0.714925	−	1.23829i	−0.962988	−	0.269543i	$-0.913127\pi$
$727$	−14014.0	−	24273.0i	−0.714925	−	1.23829i	0.248063	−	0.968744i	$-0.420206\pi$
$728$	−31080.0			−1.58228
$729$		0			0
$730$	6450.00			0.327021
$731$	−2484.00	−	4302.41i	−0.125683	−	0.217689i
$732$		0			0
$733$	−9001.00	+	15590.2i	−0.453560	+	0.785589i	−0.998604	−	0.0528183i	$-0.983180\pi$
$733$	−9001.00	+	15590.2i	−0.453560	+	0.785589i	0.545044	+	0.838407i	$0.316513\pi$
$734$	5262.00	−	9114.05i	0.264610	−	0.458318i
$735$		0			0
$736$	−2700.00	−	4676.54i	−0.135222	−	0.234211i
$737$	4704.00			0.235107
$738$		0			0
$739$	15284.0			0.760800			0.380400	−	0.924822i	$-0.375786\pi$
$739$	15284.0			0.760800			0.380400	+	0.924822i	$0.375786\pi$
$740$	−175.000	−	303.109i	−0.00869342	−	0.0150574i
$741$		0			0
$742$	−13500.0	+	23382.7i	−0.667925	+	1.15688i
$743$	9384.00	−	16253.6i	0.463345	−	0.802538i	−0.535780	−	0.844358i	$-0.679982\pi$
$743$	9384.00	−	16253.6i	0.463345	−	0.802538i	0.999125	+	0.0418201i	$0.0133156\pi$
$744$		0			0
$745$	645.000	+	1117.17i	0.0317194	+	0.0549397i
$746$	32406.0			1.59044
$747$		0			0
$748$	−1296.00			−0.0633509
$749$	14040.0	+	24318.0i	0.684927	+	1.18633i
$750$		0			0
$751$	−4348.00	+	7530.96i	−0.211266	+	0.365923i	−0.952111	−	0.305753i	$-0.901092\pi$
$751$	−4348.00	+	7530.96i	−0.211266	+	0.365923i	0.740845	+	0.671676i	$0.234425\pi$
$752$	−852.000	+	1475.71i	−0.0413155	+	0.0715605i
$753$		0			0
$754$	8658.00	+	14996.1i	0.418177	+	0.724305i
$755$	4040.00			0.194743
$756$		0			0
$757$	−38662.0			−1.85627			−0.928134	−	0.372247i	$-0.878587\pi$
$757$	−38662.0			−1.85627			−0.928134	+	0.372247i	$0.878587\pi$
$758$	−2190.00	−	3793.19i	−0.104940	−	0.181761i
$759$		0			0
$760$	6510.00	−	11275.7i	0.310714	−	0.538172i
$761$	−11937.0	+	20675.5i	−0.568615	+	0.984870i	0.428088	+	0.903737i	$0.359187\pi$
$761$	−11937.0	+	20675.5i	−0.568615	+	0.984870i	−0.996703	+	0.0811330i	$0.974146\pi$
$762$		0			0
$763$	14740.0	+	25530.4i	0.699376	+	1.21135i
$764$	3576.00			0.169339
$765$		0			0
$766$	−14616.0			−0.689422
$767$	−888.000	−	1538.06i	−0.0418042	−	0.0724070i
$768$		0			0
$769$	−11809.0	+	20453.8i	−0.553763	+	0.959145i	0.444236	+	0.895910i	$0.353475\pi$
$769$	−11809.0	+	20453.8i	−0.553763	+	0.959145i	−0.997999	+	0.0632352i	$0.979858\pi$
$770$	−3600.00	+	6235.38i	−0.168487	+	0.291828i
$771$		0			0
$772$	−1333.00	−	2308.82i	−0.0621447	−	0.107638i
$773$	11538.0			0.536860			0.268430	−	0.963299i	$-0.413495\pi$
$773$	11538.0			0.536860			0.268430	+	0.963299i	$0.413495\pi$
$774$		0			0
$775$	5000.00			0.231749
$776$	−3003.00	−	5201.35i	−0.138919	−	0.240615i
$777$		0			0
$778$	21069.0	−	36492.6i	0.970900	−	1.68165i
$779$	20460.0	−	35437.8i	0.941021	−	1.62990i
$780$		0			0
$781$	−3456.00	−	5985.97i	−0.158342	−	0.274257i
$782$	−19440.0			−0.888968
$783$		0			0
$784$	−4047.00			−0.184357
$785$	5945.00	+	10297.0i	0.270301	+	0.468175i
$786$		0			0
$787$	7442.00	−	12889.9i	0.337076	−	0.583832i	−0.646806	−	0.762655i	$-0.723895\pi$
$787$	7442.00	−	12889.9i	0.337076	−	0.583832i	0.983881	+	0.178823i	$0.0572288\pi$
$788$	1359.00	−	2353.86i	0.0614370	−	0.106412i
$789$		0			0
$790$	−3900.00	−	6755.00i	−0.175640	−	0.304218i
$791$	21720.0			0.976327
$792$		0			0
$793$	−23828.0			−1.06703
$794$	4101.00	+	7103.14i	0.183299	+	0.317482i
$795$		0			0
$796$	1916.00	−	3318.61i	0.0853151	−	0.147770i
$797$	5667.00	−	9815.53i	0.251864	−	0.436241i	−0.712175	−	0.702002i	$-0.752290\pi$
$797$	5667.00	−	9815.53i	0.251864	−	0.436241i	0.964039	+	0.265761i	$0.0856232\pi$
$798$		0			0
$799$	648.000	+	1122.37i	0.0286916	+	0.0496953i
$800$	−1125.00			−0.0497184
$801$		0			0
$802$	−47826.0			−2.10573
$803$	−5160.00	−	8937.38i	−0.226765	−	0.392769i
$804$		0			0
$805$	−6000.00	+	10392.3i	−0.262698	+	0.455007i
$806$	−22200.0	+	38451.5i	−0.970176	+	1.68039i
$807$		0			0
$808$	−18207.0	−	31535.4i	−0.792723	−	1.37304i
$809$	44730.0			1.94391			0.971955	−	0.235167i	$-0.0755638\pi$
$809$	44730.0			1.94391			0.971955	+	0.235167i	$0.0755638\pi$
$810$		0			0
$811$	−42748.0			−1.85091			−0.925453	−	0.378862i	$-0.876316\pi$
$811$	−42748.0			−1.85091			−0.925453	+	0.378862i	$0.876316\pi$
$812$	780.000	+	1351.00i	0.0337101	+	0.0583877i
$813$		0			0
$814$	−2520.00	+	4364.77i	−0.108509	+	0.187942i
$815$	−130.000	+	225.167i	−0.00558736	+	0.00967760i
$816$		0			0
$817$	5704.00	+	9879.62i	0.244257	+	0.423065i
$818$	26142.0			1.11740
$819$		0			0
$820$	−1650.00			−0.0702689
$821$	15843.0	+	27440.9i	0.673477	+	1.16650i	0.976912	+	0.213644i	$0.0685332\pi$
$821$	15843.0	+	27440.9i	0.673477	+	1.16650i	−0.303435	+	0.952852i	$0.598133\pi$
$822$		0			0
$823$	−5518.00	+	9557.46i	−0.233713	+	0.404802i	−0.958898	−	0.283752i	$-0.908421\pi$
$823$	−5518.00	+	9557.46i	−0.233713	+	0.404802i	0.725185	+	0.688554i	$0.241754\pi$
$824$	4746.00	−	8220.31i	0.200649	−	0.347534i
$825$		0			0
$826$	−720.000	−	1247.08i	−0.0303293	−	0.0525319i
$827$	25884.0			1.08836			0.544181	−	0.838968i	$-0.316841\pi$
$827$	25884.0			1.08836			0.544181	+	0.838968i	$0.316841\pi$
$828$		0			0
$829$	15950.0			0.668234			0.334117	−	0.942532i	$-0.391562\pi$
$829$	15950.0			0.668234			0.334117	+	0.942532i	$0.391562\pi$
$830$	1170.00	+	2026.50i	0.0489293	+	0.0847480i
$831$		0			0
$832$	−16021.0	+	27749.2i	−0.667582	+	1.15629i
$833$	−1539.00	+	2665.63i	−0.0640134	+	0.110875i
$834$		0			0
$835$	−9300.00	−	16108.1i	−0.385437	−	0.667596i
$836$	2976.00			0.123119
$837$		0			0
$838$	35928.0			1.48104
$839$	−6900.00	−	11951.2i	−0.283927	−	0.491775i	0.688422	−	0.725311i	$-0.258304\pi$
$839$	−6900.00	−	11951.2i	−0.283927	−	0.491775i	−0.972348	+	0.233535i	$0.924971\pi$
$840$		0			0
$841$	9152.50	−	15852.6i	0.375272	−	0.649990i
$842$	−16581.0	+	28719.1i	−0.678645	+	1.17545i
$843$		0			0
$844$	−550.000	−	952.628i	−0.0224310	−	0.0388517i
$845$	−16395.0			−0.667462
$846$		0			0
$847$	−15100.0			−0.612565
$848$	15975.0	+	27669.5i	0.646915	+	1.12049i
$849$		0			0
$850$	−2025.00	+	3507.40i	−0.0817140	+	0.141533i
$851$	−4200.00	+	7274.61i	−0.169182	+	0.293032i
$852$		0			0
$853$	13931.0	+	24129.2i	0.559189	+	0.968544i	0.997564	+	0.0697520i	$0.0222208\pi$
$853$	13931.0	+	24129.2i	0.559189	+	0.968544i	−0.438375	+	0.898792i	$0.644446\pi$
$854$	−19320.0			−0.774141
$855$		0			0
$856$	29484.0			1.17727
$857$	3657.00	+	6334.11i	0.145765	+	0.252473i	0.929658	−	0.368423i	$-0.120102\pi$
$857$	3657.00	+	6334.11i	0.145765	+	0.252473i	−0.783893	+	0.620896i	$0.786769\pi$
$858$		0			0
$859$	14390.0	−	24924.2i	0.571572	−	0.989992i	−0.424833	−	0.905272i	$-0.639667\pi$
$859$	14390.0	−	24924.2i	0.571572	−	0.989992i	0.996405	−	0.0847201i	$-0.0269996\pi$
$860$	230.000	−	398.372i	0.00911969	−	0.0157958i
$861$		0			0
$862$	−1080.00	−	1870.61i	−0.0426740	−	0.0739135i
$863$	−32688.0			−1.28935			−0.644677	−	0.764455i	$-0.723008\pi$
$863$	−32688.0			−1.28935			−0.644677	+	0.764455i	$0.723008\pi$
$864$		0			0
$865$	−2130.00			−0.0837251
$866$	23433.0	+	40587.1i	0.919499	+	1.59262i
$867$		0			0
$868$	−2000.00	+	3464.10i	−0.0782079	+	0.135460i
$869$	−6240.00	+	10808.0i	−0.243587	+	0.421906i
$870$		0			0
$871$	7252.00	+	12560.8i	0.282118	+	0.488642i
$872$	30954.0			1.20210
$873$		0			0
$874$	44640.0			1.72766
$875$	1250.00	+	2165.06i	0.0482945	+	0.0836486i
$876$		0			0
$877$	−18325.0	+	31739.8i	−0.705577	+	1.22210i	0.260906	+	0.965364i	$0.415979\pi$
$877$	−18325.0	+	31739.8i	−0.705577	+	1.22210i	−0.966483	+	0.256731i	$0.917354\pi$
$878$	14820.0	−	25669.0i	0.569648	−	0.986659i
$879$		0			0
$880$	4260.00	+	7378.54i	0.163187	+	0.282648i
$881$	−2646.00			−0.101187			−0.0505936	−	0.998719i	$-0.516111\pi$
$881$	−2646.00			−0.101187			−0.0505936	+	0.998719i	$0.516111\pi$
$882$		0			0
$883$	10892.0			0.415113			0.207557	−	0.978223i	$-0.433449\pi$
$883$	10892.0			0.415113			0.207557	+	0.978223i	$0.433449\pi$
$884$	−1998.00	−	3460.64i	−0.0760181	−	0.131667i
$885$		0			0
$886$	24174.0	−	41870.6i	0.916638	−	1.58766i
$887$	21732.0	−	37640.9i	0.822648	−	1.42487i	−0.0810549	−	0.996710i	$-0.525829\pi$
$887$	21732.0	−	37640.9i	0.822648	−	1.42487i	0.903703	−	0.428159i	$-0.140838\pi$
$888$		0			0
$889$	−12440.0	−	21546.7i	−0.469319	−	0.812884i
$890$	−15390.0			−0.579634
$891$		0			0
$892$	1964.00			0.0737215
$893$	−1488.00	−	2577.29i	−0.0557604	−	0.0965798i
$894$		0			0
$895$	−3600.00	+	6235.38i	−0.134452	+	0.232878i
$896$	−16590.0	+	28734.7i	−0.618564	+	1.07138i
$897$		0			0
$898$	−13527.0	−	23429.5i	−0.502675	−	0.870658i
$899$	−15600.0			−0.578742
$900$		0			0
$901$	24300.0			0.898502
$902$	11880.0	+	20576.8i	0.438537	+	0.759569i
$903$		0			0
$904$	11403.0	−	19750.6i	0.419533	−	0.726653i
$905$	−7825.00	+	13553.3i	−0.287416	+	0.497820i
$906$		0			0
$907$	7442.00	+	12889.9i	0.272445	+	0.471888i	0.969487	−	0.245141i	$-0.0788344\pi$
$907$	7442.00	+	12889.9i	0.272445	+	0.471888i	−0.697042	+	0.717030i	$0.745501\pi$
$908$	660.000			0.0241221
$909$		0			0
$910$	−22200.0			−0.808706
$911$	624.000	+	1080.80i	0.0226938	+	0.0393068i	0.877149	−	0.480218i	$-0.159442\pi$
$911$	624.000	+	1080.80i	0.0226938	+	0.0393068i	−0.854455	+	0.519525i	$0.826109\pi$
$912$		0			0
$913$	1872.00	−	3242.40i	0.0678578	−	0.117533i
$914$	5505.00	−	9534.94i	0.199222	−	0.345063i
$915$		0			0
$916$	953.000	+	1650.64i	0.0343756	+	0.0595402i
$917$	46560.0			1.67671
$918$		0			0
$919$	−6640.00			−0.238339			−0.119169	−	0.992874i	$-0.538023\pi$
$919$	−6640.00			−0.238339			−0.119169	+	0.992874i	$0.538023\pi$
$920$	6300.00	+	10911.9i	0.225766	+	0.391038i
$921$		0			0
$922$	−26343.0	+	45627.4i	−0.940955	+	1.62978i
$923$	10656.0	−	18456.7i	0.380007	−	0.658192i
$924$		0			0
$925$	875.000	+	1515.54i	0.0311025	+	0.0538711i
$926$	3516.00			0.124776
$927$		0			0
$928$	3510.00			0.124161
$929$	−14973.0	−	25934.0i	−0.528792	−	0.915895i	−0.999436	−	0.0335721i	$-0.989312\pi$
$929$	−14973.0	−	25934.0i	−0.528792	−	0.915895i	0.470644	−	0.882323i	$-0.344022\pi$
$930$		0			0
$931$	3534.00	−	6121.07i	0.124406	−	0.215478i
$932$	729.000	−	1262.67i	0.0256214	−	0.0443776i
$933$		0			0
$934$	−10314.0	−	17864.4i	−0.361332	−	0.625846i
$935$	6480.00			0.226651
$936$		0			0
$937$	45002.0			1.56900			0.784499	−	0.620130i	$-0.212920\pi$
$937$	45002.0			1.56900			0.784499	+	0.620130i	$0.212920\pi$
$938$	5880.00	+	10184.5i	0.204679	+	0.354514i
$939$		0			0
$940$	−60.0000	+	103.923i	−0.00208190	+	0.00360595i
$941$	−3045.00	+	5274.09i	−0.105488	+	0.182710i	−0.913937	−	0.405855i	$-0.866974\pi$
$941$	−3045.00	+	5274.09i	−0.105488	+	0.182710i	0.808450	+	0.588566i	$0.200307\pi$
$942$		0			0
$943$	19800.0	+	34294.6i	0.683750	+	1.18429i
$944$	−1704.00			−0.0587505
$945$		0			0
$946$	−6624.00			−0.227658
$947$	−28194.0	−	48833.4i	−0.967457	−	1.67568i	−0.702863	−	0.711325i	$-0.748095\pi$
$947$	−28194.0	−	48833.4i	−0.967457	−	1.67568i	−0.264594	−	0.964360i	$-0.585238\pi$
$948$		0			0
$949$	15910.0	−	27556.9i	0.544215	−	0.942609i
$950$	4650.00	−	8054.04i	0.158806	−	0.275060i
$951$		0			0
$952$	11340.0	+	19641.5i	0.386063	+	0.668680i
$953$	10854.0			0.368936			0.184468	−	0.982839i	$-0.440944\pi$
$953$	10854.0			0.368936			0.184468	+	0.982839i	$0.440944\pi$
$954$		0			0
$955$	−17880.0			−0.605846
$956$	−588.000	−	1018.45i	−0.0198926	−	0.0344549i
$957$		0			0
$958$	−3420.00	+	5923.61i	−0.115339	+	0.199774i
$959$	−21180.0	+	36684.8i	−0.713178	+	1.23526i
$960$		0			0
$961$	−5104.50	−	8841.25i	−0.171344	−	0.296776i
$962$	−15540.0			−0.520821
$963$		0			0
$964$	866.000			0.0289336
$965$	6665.00	+	11544.1i	0.222336	+	0.385097i
$966$		0			0
$967$	21158.0	−	36646.7i	0.703615	−	1.21870i	−0.263574	−	0.964639i	$-0.584901\pi$
$967$	21158.0	−	36646.7i	0.703615	−	1.21870i	0.967189	−	0.254057i	$-0.0817652\pi$
$968$	−7927.50	+	13730.8i	−0.263223	+	0.455915i
$969$		0			0
$970$	−2145.00	−	3715.25i	−0.0710019	−	0.122979i
$971$	24480.0			0.809063			0.404532	−	0.914524i	$-0.367435\pi$
$971$	24480.0			0.809063			0.404532	+	0.914524i	$0.367435\pi$
$972$		0			0
$973$	46480.0			1.53143
$974$	4614.00	+	7991.68i	0.151789	+	0.262906i
$975$		0			0
$976$	−11431.0	+	19799.1i	−0.374895	+	0.649337i
$977$	3453.00	−	5980.77i	0.113072	−	0.195846i	−0.803935	−	0.594717i	$-0.797264\pi$
$977$	3453.00	−	5980.77i	0.113072	−	0.195846i	0.917007	+	0.398870i	$0.130598\pi$
$978$		0			0
$979$	12312.0	+	21325.0i	0.401934	+	0.696170i
$980$	−285.000			−0.00928979
$981$		0			0
$982$	−56736.0			−1.84371
$983$	−3480.00	−	6027.54i	−0.112914	−	0.195573i	0.804030	−	0.594589i	$-0.202685\pi$
$983$	−3480.00	−	6027.54i	−0.112914	−	0.195573i	−0.916944	+	0.399016i	$0.869352\pi$
$984$		0			0
$985$	−6795.00	+	11769.3i	−0.219804	+	0.380711i
$986$	6318.00	−	10943.1i	0.204063	−	0.353447i
$987$		0			0
$988$	4588.00	+	7946.65i	0.147737	+	0.255887i
$989$	−11040.0			−0.354956
$990$		0			0
$991$	47792.0			1.53195			0.765975	−	0.642870i	$-0.222256\pi$
$991$	47792.0			1.53195			0.765975	+	0.642870i	$0.222256\pi$
$992$	4500.00	+	7794.23i	0.144027	+	0.249463i
$993$		0			0
$994$	8640.00	−	14964.9i	0.275698	−	0.477524i
$995$	−9580.00	+	16593.0i	−0.305233	+	0.528678i
$996$		0			0
$997$	−4969.00	−	8606.56i	−0.157843	−	0.273393i	0.776247	−	0.630428i	$-0.217121\pi$
$997$	−4969.00	−	8606.56i	−0.157843	−	0.273393i	−0.934091	+	0.357036i	$0.883787\pi$
$998$	29868.0			0.947350
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.d.271.1		2
3.2	odd	2		405.4.e.k.271.1		2
9.2	odd	6		405.4.e.k.136.1		2
9.4	even	3		15.4.a.b.1.1	✓	1
9.5	odd	6		45.4.a.b.1.1		1
9.7	even	3	inner	405.4.e.d.136.1		2
36.23	even	6		720.4.a.r.1.1		1
36.31	odd	6		240.4.a.f.1.1		1
45.4	even	6		75.4.a.a.1.1		1
45.13	odd	12		75.4.b.a.49.1		2
45.14	odd	6		225.4.a.g.1.1		1
45.22	odd	12		75.4.b.a.49.2		2
45.23	even	12		225.4.b.d.199.2		2
45.32	even	12		225.4.b.d.199.1		2
63.13	odd	6		735.4.a.i.1.1		1
63.41	even	6		2205.4.a.c.1.1		1
72.13	even	6		960.4.a.bi.1.1		1
72.67	odd	6		960.4.a.l.1.1		1
99.76	odd	6		1815.4.a.a.1.1		1
180.67	even	12		1200.4.f.m.49.1		2
180.103	even	12		1200.4.f.m.49.2		2
180.139	odd	6		1200.4.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
15.4.a.b.1.1	✓	1	9.4	even	3
45.4.a.b.1.1		1	9.5	odd	6
75.4.a.a.1.1		1	45.4	even	6
75.4.b.a.49.1		2	45.13	odd	12
75.4.b.a.49.2		2	45.22	odd	12
225.4.a.g.1.1		1	45.14	odd	6
225.4.b.d.199.1		2	45.32	even	12
225.4.b.d.199.2		2	45.23	even	12
240.4.a.f.1.1		1	36.31	odd	6
405.4.e.d.136.1		2	9.7	even	3	inner
405.4.e.d.271.1		2	1.1	even	1	trivial
405.4.e.k.136.1		2	9.2	odd	6
405.4.e.k.271.1		2	3.2	odd	2
720.4.a.r.1.1		1	36.23	even	6
735.4.a.i.1.1		1	63.13	odd	6
960.4.a.l.1.1		1	72.67	odd	6
960.4.a.bi.1.1		1	72.13	even	6
1200.4.a.o.1.1		1	180.139	odd	6
1200.4.f.m.49.1		2	180.67	even	12
1200.4.f.m.49.2		2	180.103	even	12
1815.4.a.a.1.1		1	99.76	odd	6
2205.4.a.c.1.1		1	63.41	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-10.0000 - 17.3205i) q^{7} -21.0000 q^{8} +O(q^{10})\)	\(q+(-1.50000 - 2.59808i) q^{2} +(-0.500000 + 0.866025i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-10.0000 - 17.3205i) q^{7} -21.0000 q^{8} -15.0000 q^{10} +(12.0000 + 20.7846i) q^{11} +(-37.0000 + 64.0859i) q^{13} +(-30.0000 + 51.9615i) q^{14} +(35.5000 + 61.4878i) q^{16} +54.0000 q^{17} -124.000 q^{19} +(2.50000 + 4.33013i) q^{20} +(36.0000 - 62.3538i) q^{22} +(60.0000 - 103.923i) q^{23} +(-12.5000 - 21.6506i) q^{25} +222.000 q^{26} +20.0000 q^{28} +(39.0000 + 67.5500i) q^{29} +(-100.000 + 173.205i) q^{31} +(22.5000 - 38.9711i) q^{32} +(-81.0000 - 140.296i) q^{34} -100.000 q^{35} -70.0000 q^{37} +(186.000 + 322.161i) q^{38} +(-52.5000 + 90.9327i) q^{40} +(-165.000 + 285.788i) q^{41} +(-46.0000 - 79.6743i) q^{43} -24.0000 q^{44} -360.000 q^{46} +(12.0000 + 20.7846i) q^{47} +(-28.5000 + 49.3634i) q^{49} +(-37.5000 + 64.9519i) q^{50} +(-37.0000 - 64.0859i) q^{52} +450.000 q^{53} +120.000 q^{55} +(210.000 + 363.731i) q^{56} +(117.000 - 202.650i) q^{58} +(-12.0000 + 20.7846i) q^{59} +(161.000 + 278.860i) q^{61} +600.000 q^{62} +433.000 q^{64} +(185.000 + 320.429i) q^{65} +(98.0000 - 169.741i) q^{67} +(-27.0000 + 46.7654i) q^{68} +(150.000 + 259.808i) q^{70} -288.000 q^{71} -430.000 q^{73} +(105.000 + 181.865i) q^{74} +(62.0000 - 107.387i) q^{76} +(240.000 - 415.692i) q^{77} +(260.000 + 450.333i) q^{79} +355.000 q^{80} +990.000 q^{82} +(-78.0000 - 135.100i) q^{83} +(135.000 - 233.827i) q^{85} +(-138.000 + 239.023i) q^{86} +(-252.000 - 436.477i) q^{88} +1026.00 q^{89} +1480.00 q^{91} +(60.0000 + 103.923i) q^{92} +(36.0000 - 62.3538i) q^{94} +(-310.000 + 536.936i) q^{95} +(143.000 + 247.683i) q^{97} +171.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8}+O(q^{10}) \) 2 * q - 3 * q^2 - q^4 + 5 * q^5 - 20 * q^7 - 42 * q^8	\( 2 q - 3 q^{2} - q^{4} + 5 q^{5} - 20 q^{7} - 42 q^{8} - 30 q^{10} + 24 q^{11} - 74 q^{13} - 60 q^{14} + 71 q^{16} + 108 q^{17} - 248 q^{19} + 5 q^{20} + 72 q^{22} + 120 q^{23} - 25 q^{25} + 444 q^{26} + 40 q^{28} + 78 q^{29} - 200 q^{31} + 45 q^{32} - 162 q^{34} - 200 q^{35} - 140 q^{37} + 372 q^{38} - 105 q^{40} - 330 q^{41} - 92 q^{43} - 48 q^{44} - 720 q^{46} + 24 q^{47} - 57 q^{49} - 75 q^{50} - 74 q^{52} + 900 q^{53} + 240 q^{55} + 420 q^{56} + 234 q^{58} - 24 q^{59} + 322 q^{61} + 1200 q^{62} + 866 q^{64} + 370 q^{65} + 196 q^{67} - 54 q^{68} + 300 q^{70} - 576 q^{71} - 860 q^{73} + 210 q^{74} + 124 q^{76} + 480 q^{77} + 520 q^{79} + 710 q^{80} + 1980 q^{82} - 156 q^{83} + 270 q^{85} - 276 q^{86} - 504 q^{88} + 2052 q^{89} + 2960 q^{91} + 120 q^{92} + 72 q^{94} - 620 q^{95} + 286 q^{97} + 342 q^{98}+O(q^{100}) \) 2 * q - 3 * q^2 - q^4 + 5 * q^5 - 20 * q^7 - 42 * q^8 - 30 * q^10 + 24 * q^11 - 74 * q^13 - 60 * q^14 + 71 * q^16 + 108 * q^17 - 248 * q^19 + 5 * q^20 + 72 * q^22 + 120 * q^23 - 25 * q^25 + 444 * q^26 + 40 * q^28 + 78 * q^29 - 200 * q^31 + 45 * q^32 - 162 * q^34 - 200 * q^35 - 140 * q^37 + 372 * q^38 - 105 * q^40 - 330 * q^41 - 92 * q^43 - 48 * q^44 - 720 * q^46 + 24 * q^47 - 57 * q^49 - 75 * q^50 - 74 * q^52 + 900 * q^53 + 240 * q^55 + 420 * q^56 + 234 * q^58 - 24 * q^59 + 322 * q^61 + 1200 * q^62 + 866 * q^64 + 370 * q^65 + 196 * q^67 - 54 * q^68 + 300 * q^70 - 576 * q^71 - 860 * q^73 + 210 * q^74 + 124 * q^76 + 480 * q^77 + 520 * q^79 + 710 * q^80 + 1980 * q^82 - 156 * q^83 + 270 * q^85 - 276 * q^86 - 504 * q^88 + 2052 * q^89 + 2960 * q^91 + 120 * q^92 + 72 * q^94 - 620 * q^95 + 286 * q^97 + 342 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more