LMFDB - Embedded newform 810.4.e.x.541.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [810,4,Mod(271,810)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("810.271"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(810, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([2, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	810.e (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,2,0,-4,5,0,34,-16,0,20,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$47.7915471046$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			541.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	810.541
Dual form			810.4.e.x.271.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} +(17.0000 - 29.4449i) q^{7} -8.00000 q^{8} +10.0000 q^{10} +(-24.0000 + 41.5692i) q^{11} +(35.0000 + 60.6218i) q^{13} +(-34.0000 - 58.8897i) q^{14} +(-8.00000 + 13.8564i) q^{16} +27.0000 q^{17} +119.000 q^{19} +(10.0000 - 17.3205i) q^{20} +(48.0000 + 83.1384i) q^{22} +(-25.5000 - 44.1673i) q^{23} +(-12.5000 + 21.6506i) q^{25} +140.000 q^{26} -136.000 q^{28} +(-15.0000 + 25.9808i) q^{29} +(66.5000 + 115.181i) q^{31} +(16.0000 + 27.7128i) q^{32} +(27.0000 - 46.7654i) q^{34} +170.000 q^{35} +218.000 q^{37} +(119.000 - 206.114i) q^{38} +(-20.0000 - 34.6410i) q^{40} +(78.0000 + 135.100i) q^{41} +(44.0000 - 76.2102i) q^{43} +192.000 q^{44} -102.000 q^{46} +(-258.000 + 446.869i) q^{47} +(-406.500 - 704.079i) q^{49} +(25.0000 + 43.3013i) q^{50} +(140.000 - 242.487i) q^{52} +639.000 q^{53} -240.000 q^{55} +(-136.000 + 235.559i) q^{56} +(30.0000 + 51.9615i) q^{58} +(-327.000 - 566.381i) q^{59} +(-230.500 + 399.238i) q^{61} +266.000 q^{62} +64.0000 q^{64} +(-175.000 + 303.109i) q^{65} +(-91.0000 - 157.617i) q^{67} +(-54.0000 - 93.5307i) q^{68} +(170.000 - 294.449i) q^{70} -900.000 q^{71} +704.000 q^{73} +(218.000 - 377.587i) q^{74} +(-238.000 - 412.228i) q^{76} +(816.000 + 1413.35i) q^{77} +(687.500 - 1190.78i) q^{79} -80.0000 q^{80} +312.000 q^{82} +(457.500 - 792.413i) q^{83} +(67.5000 + 116.913i) q^{85} +(-88.0000 - 152.420i) q^{86} +(192.000 - 332.554i) q^{88} +1116.00 q^{89} +2380.00 q^{91} +(-102.000 + 176.669i) q^{92} +(516.000 + 893.738i) q^{94} +(297.500 + 515.285i) q^{95} +(8.00000 - 13.8564i) q^{97} -1626.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 4 q^{4} + 5 q^{5} + 34 q^{7} - 16 q^{8} + 20 q^{10} - 48 q^{11} + 70 q^{13} - 68 q^{14} - 16 q^{16} + 54 q^{17} + 238 q^{19} + 20 q^{20} + 96 q^{22} - 51 q^{23} - 25 q^{25} + 280 q^{26}+ \cdots - 3252 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 - 4 * q^4 + 5 * q^5 + 34 * q^7 - 16 * q^8 + 20 * q^10 - 48 * q^11 + 70 * q^13 - 68 * q^14 - 16 * q^16 + 54 * q^17 + 238 * q^19 + 20 * q^20 + 96 * q^22 - 51 * q^23 - 25 * q^25 + 280 * q^26 - 272 * q^28 - 30 * q^29 + 133 * q^31 + 32 * q^32 + 54 * q^34 + 340 * q^35 + 436 * q^37 + 238 * q^38 - 40 * q^40 + 156 * q^41 + 88 * q^43 + 384 * q^44 - 204 * q^46 - 516 * q^47 - 813 * q^49 + 50 * q^50 + 280 * q^52 + 1278 * q^53 - 480 * q^55 - 272 * q^56 + 60 * q^58 - 654 * q^59 - 461 * q^61 + 532 * q^62 + 128 * q^64 - 350 * q^65 - 182 * q^67 - 108 * q^68 + 340 * q^70 - 1800 * q^71 + 1408 * q^73 + 436 * q^74 - 476 * q^76 + 1632 * q^77 + 1375 * q^79 - 160 * q^80 + 624 * q^82 + 915 * q^83 + 135 * q^85 - 176 * q^86 + 384 * q^88 + 2232 * q^89 + 4760 * q^91 - 204 * q^92 + 1032 * q^94 + 595 * q^95 + 16 * q^97 - 3252 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$2$	1.00000	−	1.73205i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−2.00000	−	3.46410i	−0.250000	−	0.433013i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$5$	2.50000	+	4.33013i	0.223607	+	0.387298i
$6$		0			0
$7$	17.0000	−	29.4449i	0.917914	−	1.58987i	0.115335	−	0.993327i	$-0.463206\pi$
$7$	17.0000	−	29.4449i	0.917914	−	1.58987i	0.802578	−	0.596547i	$-0.203461\pi$
$8$	−8.00000			−0.353553
$9$		0			0
$10$	10.0000			0.316228
$11$	−24.0000	+	41.5692i	−0.657843	+	1.13942i	0.323330	+	0.946286i	$0.395198\pi$
$11$	−24.0000	+	41.5692i	−0.657843	+	1.13942i	−0.981173	+	0.193131i	$0.938136\pi$
$12$		0			0
$13$	35.0000	+	60.6218i	0.746712	+	1.29334i	0.949391	+	0.314098i	$0.101702\pi$
$13$	35.0000	+	60.6218i	0.746712	+	1.29334i	−0.202679	+	0.979245i	$0.564965\pi$
$14$	−34.0000	−	58.8897i	−0.649063	−	1.12421i
$15$		0			0
$16$	−8.00000	+	13.8564i	−0.125000	+	0.216506i
$17$	27.0000			0.385204			0.192602	−	0.981277i	$-0.438307\pi$
$17$	27.0000			0.385204			0.192602	+	0.981277i	$0.438307\pi$
$18$		0			0
$19$	119.000			1.43687			0.718433	−	0.695596i	$-0.244859\pi$
$19$	119.000			1.43687			0.718433	+	0.695596i	$0.244859\pi$
$20$	10.0000	−	17.3205i	0.111803	−	0.193649i
$21$		0			0
$22$	48.0000	+	83.1384i	0.465165	+	0.805690i
$23$	−25.5000	−	44.1673i	−0.231179	−	0.400414i	0.726976	−	0.686663i	$-0.240925\pi$
$23$	−25.5000	−	44.1673i	−0.231179	−	0.400414i	−0.958155	+	0.286249i	$0.907592\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.100000	+	0.173205i
$26$	140.000			1.05601
$27$		0			0
$28$	−136.000			−0.917914
$29$	−15.0000	+	25.9808i	−0.0960493	+	0.166362i	−0.910046	−	0.414507i	$-0.863954\pi$
$29$	−15.0000	+	25.9808i	−0.0960493	+	0.166362i	0.813997	+	0.580869i	$0.197287\pi$
$30$		0			0
$31$	66.5000	+	115.181i	0.385282	+	0.667329i	0.991808	−	0.127735i	$-0.0407707\pi$
$31$	66.5000	+	115.181i	0.385282	+	0.667329i	−0.606526	+	0.795064i	$0.707437\pi$
$32$	16.0000	+	27.7128i	0.0883883	+	0.153093i
$33$		0			0
$34$	27.0000	−	46.7654i	0.136190	−	0.235888i
$35$	170.000			0.821007
$36$		0			0
$37$	218.000			0.968621			0.484311	−	0.874896i	$-0.339070\pi$
$37$	218.000			0.968621			0.484311	+	0.874896i	$0.339070\pi$
$38$	119.000	−	206.114i	0.508009	−	0.879898i
$39$		0			0
$40$	−20.0000	−	34.6410i	−0.0790569	−	0.136931i
$41$	78.0000	+	135.100i	0.297111	+	0.514611i	0.975474	−	0.220116i	$-0.0706435\pi$
$41$	78.0000	+	135.100i	0.297111	+	0.514611i	−0.678363	+	0.734727i	$0.737310\pi$
$42$		0			0
$43$	44.0000	−	76.2102i	0.156045	−	0.270278i	−0.777394	−	0.629014i	$-0.783459\pi$
$43$	44.0000	−	76.2102i	0.156045	−	0.270278i	0.933439	+	0.358736i	$0.116792\pi$
$44$	192.000			0.657843
$45$		0			0
$46$	−102.000			−0.326937
$47$	−258.000	+	446.869i	−0.800706	+	1.38686i	0.118447	+	0.992960i	$0.462209\pi$
$47$	−258.000	+	446.869i	−0.800706	+	1.38686i	−0.919152	+	0.393903i	$0.871125\pi$
$48$		0			0
$49$	−406.500	−	704.079i	−1.18513	−	2.05271i
$50$	25.0000	+	43.3013i	0.0707107	+	0.122474i
$51$		0			0
$52$	140.000	−	242.487i	0.373356	−	0.646671i
$53$	639.000			1.65610			0.828051	−	0.560653i	$-0.189450\pi$
$53$	639.000			1.65610			0.828051	+	0.560653i	$0.189450\pi$
$54$		0			0
$55$	−240.000			−0.588393
$56$	−136.000	+	235.559i	−0.324532	+	0.562105i
$57$		0			0
$58$	30.0000	+	51.9615i	0.0679171	+	0.117636i
$59$	−327.000	−	566.381i	−0.721555	−	1.24977i	−0.960376	−	0.278707i	$-0.910094\pi$
$59$	−327.000	−	566.381i	−0.721555	−	1.24977i	0.238821	−	0.971064i	$-0.423239\pi$
$60$		0			0
$61$	−230.500	+	399.238i	−0.483811	+	0.837986i	−0.999827	−	0.0185931i	$-0.994081\pi$
$61$	−230.500	+	399.238i	−0.483811	+	0.837986i	0.516016	+	0.856579i	$0.327415\pi$
$62$	266.000			0.544872
$63$		0			0
$64$	64.0000			0.125000
$65$	−175.000	+	303.109i	−0.333940	+	0.578400i
$66$		0			0
$67$	−91.0000	−	157.617i	−0.165932	−	0.287402i	0.771054	−	0.636770i	$-0.219730\pi$
$67$	−91.0000	−	157.617i	−0.165932	−	0.287402i	−0.936986	+	0.349368i	$0.886396\pi$
$68$	−54.0000	−	93.5307i	−0.0963009	−	0.166798i
$69$		0			0
$70$	170.000	−	294.449i	0.290270	−	0.502762i
$71$	−900.000			−1.50437			−0.752186	−	0.658951i	$-0.771000\pi$
$71$	−900.000			−1.50437			−0.752186	+	0.658951i	$0.771000\pi$
$72$		0			0
$73$	704.000			1.12873			0.564363	−	0.825527i	$-0.309122\pi$
$73$	704.000			1.12873			0.564363	+	0.825527i	$0.309122\pi$
$74$	218.000	−	377.587i	0.342459	−	0.593157i
$75$		0			0
$76$	−238.000	−	412.228i	−0.359217	−	0.622182i
$77$	816.000	+	1413.35i	1.20769	+	2.09177i
$78$		0			0
$79$	687.500	−	1190.78i	0.979111	−	1.69587i	0.313471	−	0.949598i	$-0.398508\pi$
$79$	687.500	−	1190.78i	0.979111	−	1.69587i	0.665640	−	0.746273i	$-0.268159\pi$
$80$	−80.0000			−0.111803
$81$		0			0
$82$	312.000			0.420178
$83$	457.500	−	792.413i	0.605026	−	1.04794i	−0.387022	−	0.922071i	$-0.626496\pi$
$83$	457.500	−	792.413i	0.605026	−	1.04794i	0.992047	−	0.125865i	$-0.0401706\pi$
$84$		0			0
$85$	67.5000	+	116.913i	0.0861342	+	0.149189i
$86$	−88.0000	−	152.420i	−0.110341	−	0.191115i
$87$		0			0
$88$	192.000	−	332.554i	0.232583	−	0.402845i
$89$	1116.00			1.32917			0.664583	−	0.747215i	$-0.268609\pi$
$89$	1116.00			1.32917			0.664583	+	0.747215i	$0.268609\pi$
$90$		0			0
$91$	2380.00			2.74167
$92$	−102.000	+	176.669i	−0.115590	+	0.200207i
$93$		0			0
$94$	516.000	+	893.738i	0.566184	+	0.980660i
$95$	297.500	+	515.285i	0.321293	+	0.556496i
$96$		0			0
$97$	8.00000	−	13.8564i	0.00837399	−	0.0145042i	−0.861808	−	0.507235i	$-0.830668\pi$
$97$	8.00000	−	13.8564i	0.00837399	−	0.0145042i	0.870182	+	0.492730i	$0.164001\pi$
$98$	−1626.00			−1.67603
$99$		0			0
$100$	100.000			0.100000
$101$	174.000	−	301.377i	0.171422	−	0.296912i	−0.767495	−	0.641055i	$-0.778497\pi$
$101$	174.000	−	301.377i	0.171422	−	0.296912i	0.938917	+	0.344143i	$0.111830\pi$
$102$		0			0
$103$	206.000	+	356.802i	0.197066	+	0.341328i	0.947576	−	0.319531i	$-0.103525\pi$
$103$	206.000	+	356.802i	0.197066	+	0.341328i	−0.750510	+	0.660859i	$0.770192\pi$
$104$	−280.000	−	484.974i	−0.264002	−	0.457266i
$105$		0			0
$106$	639.000	−	1106.78i	0.585520	−	1.01415i
$107$	900.000			0.813143			0.406571	−	0.913619i	$-0.366724\pi$
$107$	900.000			0.813143			0.406571	+	0.913619i	$0.366724\pi$
$108$		0			0
$109$	−115.000			−0.101055			−0.0505275	−	0.998723i	$-0.516090\pi$
$109$	−115.000			−0.101055			−0.0505275	+	0.998723i	$0.516090\pi$
$110$	−240.000	+	415.692i	−0.208028	+	0.360315i
$111$		0			0
$112$	272.000	+	471.118i	0.229478	+	0.397468i
$113$	483.000	+	836.581i	0.402096	+	0.696450i	0.993979	−	0.109574i	$-0.0349486\pi$
$113$	483.000	+	836.581i	0.402096	+	0.696450i	−0.591883	+	0.806024i	$0.701615\pi$
$114$		0			0
$115$	127.500	−	220.836i	0.103386	−	0.179071i
$116$	120.000			0.0960493
$117$		0			0
$118$	−1308.00			−1.02043
$119$	459.000	−	795.011i	0.353584	−	0.612425i
$120$		0			0
$121$	−486.500	−	842.643i	−0.365515	−	0.633090i
$122$	461.000	+	798.475i	0.342106	+	0.592546i
$123$		0			0
$124$	266.000	−	460.726i	0.192641	−	0.333664i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	1406.00			0.982381			0.491190	−	0.871052i	$-0.336562\pi$
$127$	1406.00			0.982381			0.491190	+	0.871052i	$0.336562\pi$
$128$	64.0000	−	110.851i	0.0441942	−	0.0765466i
$129$		0			0
$130$	350.000	+	606.218i	0.236131	+	0.408991i
$131$	123.000	+	213.042i	0.0820348	+	0.142088i	0.904124	−	0.427271i	$-0.140525\pi$
$131$	123.000	+	213.042i	0.0820348	+	0.142088i	−0.822089	+	0.569359i	$0.807191\pi$
$132$		0			0
$133$	2023.00	−	3503.94i	1.31892	−	2.28444i
$134$	−364.000			−0.234663
$135$		0			0
$136$	−216.000			−0.136190
$137$	259.500	−	449.467i	0.161829	−	0.280296i	−0.773696	−	0.633557i	$-0.781594\pi$
$137$	259.500	−	449.467i	0.161829	−	0.280296i	0.935525	+	0.353261i	$0.114927\pi$
$138$		0			0
$139$	−658.000	−	1139.69i	−0.401517	−	0.695447i	0.592392	−	0.805650i	$-0.298184\pi$
$139$	−658.000	−	1139.69i	−0.401517	−	0.695447i	−0.993909	+	0.110202i	$0.964850\pi$
$140$	−340.000	−	588.897i	−0.205252	−	0.355506i
$141$		0			0
$142$	−900.000	+	1558.85i	−0.531876	+	0.921235i
$143$	−3360.00			−1.96488
$144$		0			0
$145$	−150.000			−0.0859091
$146$	704.000	−	1219.36i	0.399065	−	0.691200i
$147$		0			0
$148$	−436.000	−	755.174i	−0.242155	−	0.419425i
$149$	186.000	+	322.161i	0.102267	+	0.177131i	0.912618	−	0.408813i	$-0.134057\pi$
$149$	186.000	+	322.161i	0.102267	+	0.177131i	−0.810352	+	0.585944i	$0.800724\pi$
$150$		0			0
$151$	728.000	−	1260.93i	0.392343	−	0.679558i	−0.600415	−	0.799689i	$-0.704998\pi$
$151$	728.000	−	1260.93i	0.392343	−	0.679558i	0.992758	+	0.120130i	$0.0383313\pi$
$152$	−952.000			−0.508009
$153$		0			0
$154$	3264.00			1.70793
$155$	−332.500	+	575.907i	−0.172304	+	0.298438i
$156$		0			0
$157$	−478.000	−	827.920i	−0.242984	−	0.420861i	0.718579	−	0.695446i	$-0.244793\pi$
$157$	−478.000	−	827.920i	−0.242984	−	0.420861i	−0.961563	+	0.274584i	$0.911460\pi$
$158$	−1375.00	−	2381.57i	−0.692336	−	1.19916i
$159$		0			0
$160$	−80.0000	+	138.564i	−0.0395285	+	0.0684653i
$161$	−1734.00			−0.848810
$162$		0			0
$163$	−2446.00			−1.17537			−0.587686	−	0.809089i	$-0.699961\pi$
$163$	−2446.00			−1.17537			−0.587686	+	0.809089i	$0.699961\pi$
$164$	312.000	−	540.400i	0.148556	−	0.257306i
$165$		0			0
$166$	−915.000	−	1584.83i	−0.427818	−	0.741002i
$167$	−1555.50	−	2694.21i	−0.720768	−	1.24841i	−0.960692	−	0.277615i	$-0.910456\pi$
$167$	−1555.50	−	2694.21i	−0.720768	−	1.24841i	0.239925	−	0.970791i	$-0.422877\pi$
$168$		0			0
$169$	−1351.50	+	2340.87i	−0.615157	+	1.06548i
$170$	270.000			0.121812
$171$		0			0
$172$	−352.000			−0.156045
$173$	−1198.50	+	2075.86i	−0.526707	+	0.912283i	0.472809	+	0.881165i	$0.343240\pi$
$173$	−1198.50	+	2075.86i	−0.526707	+	0.912283i	−0.999516	+	0.0311179i	$0.990093\pi$
$174$		0			0
$175$	425.000	+	736.122i	0.183583	+	0.317975i
$176$	−384.000	−	665.108i	−0.164461	−	0.284854i
$177$		0			0
$178$	1116.00	−	1932.97i	0.469931	−	0.813945i
$179$	540.000			0.225483			0.112742	−	0.993624i	$-0.464037\pi$
$179$	540.000			0.225483			0.112742	+	0.993624i	$0.464037\pi$
$180$		0			0
$181$	2333.00			0.958069			0.479035	−	0.877796i	$-0.340987\pi$
$181$	2333.00			0.958069			0.479035	+	0.877796i	$0.340987\pi$
$182$	2380.00	−	4122.28i	0.969326	−	1.67892i
$183$		0			0
$184$	204.000	+	353.338i	0.0817341	+	0.141568i
$185$	545.000	+	943.968i	0.216590	+	0.375145i
$186$		0			0
$187$	−648.000	+	1122.37i	−0.253403	+	0.438908i
$188$	2064.00			0.800706
$189$		0			0
$190$	1190.00			0.454377
$191$	−1365.00	+	2364.25i	−0.517110	+	0.895660i	0.482693	+	0.875790i	$0.339659\pi$
$191$	−1365.00	+	2364.25i	−0.517110	+	0.895660i	−0.999803	+	0.0198705i	$0.993675\pi$
$192$		0			0
$193$	2285.00	+	3957.74i	0.852217	+	1.47608i	0.879203	+	0.476447i	$0.158076\pi$
$193$	2285.00	+	3957.74i	0.852217	+	1.47608i	−0.0269858	+	0.999636i	$0.508591\pi$
$194$	−16.0000	−	27.7128i	−0.00592130	−	0.0102560i
$195$		0			0
$196$	−1626.00	+	2816.31i	−0.592566	+	1.02635i
$197$	−675.000			−0.244121			−0.122060	−	0.992523i	$-0.538950\pi$
$197$	−675.000			−0.244121			−0.122060	+	0.992523i	$0.538950\pi$
$198$		0			0
$199$	−3112.00			−1.10856			−0.554281	−	0.832330i	$-0.687007\pi$
$199$	−3112.00			−1.10856			−0.554281	+	0.832330i	$0.687007\pi$
$200$	100.000	−	173.205i	0.0353553	−	0.0612372i
$201$		0			0
$202$	−348.000	−	602.754i	−0.121214	−	0.209949i
$203$	510.000	+	883.346i	0.176330	+	0.305412i
$204$		0			0
$205$	−390.000	+	675.500i	−0.132872	+	0.230141i
$206$	824.000			0.278693
$207$		0			0
$208$	−1120.00			−0.373356
$209$	−2856.00	+	4946.74i	−0.945233	+	1.63719i
$210$		0			0
$211$	−1220.50	−	2113.97i	−0.398212	−	0.689723i	0.595294	−	0.803508i	$-0.297036\pi$
$211$	−1220.50	−	2113.97i	−0.398212	−	0.689723i	−0.993505	+	0.113785i	$0.963702\pi$
$212$	−1278.00	−	2213.56i	−0.414025	−	0.717113i
$213$		0			0
$214$	900.000	−	1558.85i	0.287489	−	0.497946i
$215$	440.000			0.139571
$216$		0			0
$217$	4522.00			1.41462
$218$	−115.000	+	199.186i	−0.0357284	+	0.0618833i
$219$		0			0
$220$	480.000	+	831.384i	0.147098	+	0.254781i
$221$	945.000	+	1636.79i	0.287636	+	0.498200i
$222$		0			0
$223$	1709.00	−	2960.07i	0.513198	−	0.888885i	−0.486685	−	0.873578i	$-0.661794\pi$
$223$	1709.00	−	2960.07i	0.513198	−	0.888885i	0.999883	−	0.0153071i	$-0.00487260\pi$
$224$	1088.00			0.324532
$225$		0			0
$226$	1932.00			0.568649
$227$	−2188.50	+	3790.59i	−0.639894	+	1.10833i	0.345562	+	0.938396i	$0.387688\pi$
$227$	−2188.50	+	3790.59i	−0.639894	+	1.10833i	−0.985456	+	0.169932i	$0.945645\pi$
$228$		0			0
$229$	−2093.50	−	3626.05i	−0.604115	−	1.04636i	−0.992191	−	0.124731i	$-0.960193\pi$
$229$	−2093.50	−	3626.05i	−0.604115	−	1.04636i	0.388075	−	0.921628i	$-0.373140\pi$
$230$	−255.000	−	441.673i	−0.0731052	−	0.126622i
$231$		0			0
$232$	120.000	−	207.846i	0.0339586	−	0.0588180i
$233$	1098.00			0.308723			0.154361	−	0.988014i	$-0.450668\pi$
$233$	1098.00			0.308723			0.154361	+	0.988014i	$0.450668\pi$
$234$		0			0
$235$	−2580.00			−0.716173
$236$	−1308.00	+	2265.52i	−0.360778	+	0.624885i
$237$		0			0
$238$	−918.000	−	1590.02i	−0.250021	−	0.433050i
$239$	3237.00	+	5606.65i	0.876084	+	1.51742i	0.855603	+	0.517632i	$0.173186\pi$
$239$	3237.00	+	5606.65i	0.876084	+	1.51742i	0.0204809	+	0.999790i	$0.493480\pi$
$240$		0			0
$241$	−1625.50	+	2815.45i	−0.434472	+	0.752527i	−0.997252	−	0.0740793i	$-0.976398\pi$
$241$	−1625.50	+	2815.45i	−0.434472	+	0.752527i	0.562781	+	0.826606i	$0.309732\pi$
$242$	−1946.00			−0.516916
$243$		0			0
$244$	1844.00			0.483811
$245$	2032.50	−	3520.39i	0.530007	−	0.917999i
$246$		0			0
$247$	4165.00	+	7213.99i	1.07293	+	1.85836i
$248$	−532.000	−	921.451i	−0.136218	−	0.235936i
$249$		0			0
$250$	−125.000	+	216.506i	−0.0316228	+	0.0547723i
$251$	−1728.00			−0.434543			−0.217272	−	0.976111i	$-0.569716\pi$
$251$	−1728.00			−0.434543			−0.217272	+	0.976111i	$0.569716\pi$
$252$		0			0
$253$	2448.00			0.608318
$254$	1406.00	−	2435.26i	0.347324	−	0.601583i
$255$		0			0
$256$	−128.000	−	221.703i	−0.0312500	−	0.0541266i
$257$	−2734.50	−	4736.29i	−0.663710	−	1.14958i	−0.979633	−	0.200794i	$-0.935648\pi$
$257$	−2734.50	−	4736.29i	−0.663710	−	1.14958i	0.315924	−	0.948785i	$-0.397686\pi$
$258$		0			0
$259$	3706.00	−	6418.98i	0.889111	−	1.53998i
$260$	1400.00			0.333940
$261$		0			0
$262$	492.000			0.116015
$263$	−1608.00	+	2785.14i	−0.377010	+	0.653000i	−0.990626	−	0.136605i	$-0.956381\pi$
$263$	−1608.00	+	2785.14i	−0.377010	+	0.653000i	0.613616	+	0.789605i	$0.289714\pi$
$264$		0			0
$265$	1597.50	+	2766.95i	0.370316	+	0.641406i
$266$	−4046.00	−	7007.88i	−0.932617	−	1.61534i
$267$		0			0
$268$	−364.000	+	630.466i	−0.0829658	+	0.143701i
$269$	8010.00			1.81553			0.907766	−	0.419476i	$-0.137786\pi$
$269$	8010.00			1.81553			0.907766	+	0.419476i	$0.137786\pi$
$270$		0			0
$271$	−3805.00			−0.852905			−0.426453	−	0.904510i	$-0.640237\pi$
$271$	−3805.00			−0.852905			−0.426453	+	0.904510i	$0.640237\pi$
$272$	−216.000	+	374.123i	−0.0481505	+	0.0833990i
$273$		0			0
$274$	−519.000	−	898.934i	−0.114430	−	0.198199i
$275$	−600.000	−	1039.23i	−0.131569	−	0.227883i
$276$		0			0
$277$	−1612.00	+	2792.07i	−0.349660	+	0.605628i	−0.986189	−	0.165624i	$-0.947036\pi$
$277$	−1612.00	+	2792.07i	−0.349660	+	0.605628i	0.636529	+	0.771252i	$0.280369\pi$
$278$	−2632.00			−0.567830
$279$		0			0
$280$	−1360.00			−0.290270
$281$	−2265.00	+	3923.10i	−0.480849	+	0.832855i	−0.999759	−	0.0219742i	$-0.993005\pi$
$281$	−2265.00	+	3923.10i	−0.480849	+	0.832855i	0.518910	+	0.854829i	$0.326338\pi$
$282$		0			0
$283$	1646.00	+	2850.96i	0.345740	+	0.598840i	0.985488	−	0.169745i	$-0.0542944\pi$
$283$	1646.00	+	2850.96i	0.345740	+	0.598840i	−0.639748	+	0.768585i	$0.720961\pi$
$284$	1800.00	+	3117.69i	0.376093	+	0.651412i
$285$		0			0
$286$	−3360.00	+	5819.69i	−0.694689	+	1.20324i
$287$	5304.00			1.09089
$288$		0			0
$289$	−4184.00			−0.851618
$290$	−150.000	+	259.808i	−0.0303735	+	0.0526084i
$291$		0			0
$292$	−1408.00	−	2438.73i	−0.282181	−	0.488753i
$293$	−3976.50	−	6887.50i	−0.792866	−	1.37328i	−0.924186	−	0.381943i	$-0.875255\pi$
$293$	−3976.50	−	6887.50i	−0.792866	−	1.37328i	0.131320	−	0.991340i	$-0.458078\pi$
$294$		0			0
$295$	1635.00	−	2831.90i	0.322689	−	0.558914i
$296$	−1744.00			−0.342459
$297$		0			0
$298$	744.000			0.144627
$299$	1785.00	−	3091.71i	0.345248	−	0.597987i
$300$		0			0
$301$	−1496.00	−	2591.15i	−0.286472	−	0.496184i
$302$	−1456.00	−	2521.87i	−0.277428	−	0.480520i
$303$		0			0
$304$	−952.000	+	1648.91i	−0.179608	+	0.311091i
$305$	−2305.00			−0.432734
$306$		0			0
$307$	−5290.00			−0.983441			−0.491720	−	0.870753i	$-0.663632\pi$
$307$	−5290.00			−0.983441			−0.491720	+	0.870753i	$0.663632\pi$
$308$	3264.00	−	5653.41i	0.603843	−	1.04589i
$309$		0			0
$310$	665.000	+	1151.81i	0.121837	+	0.211028i
$311$	2679.00	+	4640.16i	0.488464	+	0.846044i	0.999912	−	0.0132703i	$-0.00422421\pi$
$311$	2679.00	+	4640.16i	0.488464	+	0.846044i	−0.511448	+	0.859314i	$0.670891\pi$
$312$		0			0
$313$	−2800.00	+	4849.74i	−0.505640	+	0.875794i	0.494339	+	0.869269i	$0.335410\pi$
$313$	−2800.00	+	4849.74i	−0.505640	+	0.875794i	−0.999979	+	0.00652494i	$0.997923\pi$
$314$	−1912.00			−0.343632
$315$		0			0
$316$	−5500.00			−0.979111
$317$	3670.50	−	6357.49i	0.650334	−	1.12641i	−0.332708	−	0.943030i	$-0.607962\pi$
$317$	3670.50	−	6357.49i	0.650334	−	1.12641i	0.983042	−	0.183381i	$-0.0587043\pi$
$318$		0			0
$319$	−720.000	−	1247.08i	−0.126371	−	0.218881i
$320$	160.000	+	277.128i	0.0279508	+	0.0484123i
$321$		0			0
$322$	−1734.00	+	3003.38i	−0.300100	+	0.519788i
$323$	3213.00			0.553486
$324$		0			0
$325$	−1750.00			−0.298685
$326$	−2446.00	+	4236.60i	−0.415556	+	0.719765i
$327$		0			0
$328$	−624.000	−	1080.80i	−0.105045	−	0.181943i
$329$	8772.00	+	15193.5i	1.46996	+	2.54604i
$330$		0			0
$331$	−190.000	+	329.090i	−0.0315509	+	0.0546477i	−0.881370	−	0.472427i	$-0.843378\pi$
$331$	−190.000	+	329.090i	−0.0315509	+	0.0546477i	0.849819	+	0.527075i	$0.176711\pi$
$332$	−3660.00			−0.605026
$333$		0			0
$334$	−6222.00			−1.01932
$335$	455.000	−	788.083i	0.0742069	−	0.128530i
$336$		0			0
$337$	−217.000	−	375.855i	−0.0350764	−	0.0607541i	0.847954	−	0.530069i	$-0.177834\pi$
$337$	−217.000	−	375.855i	−0.0350764	−	0.0607541i	−0.883031	+	0.469315i	$0.844501\pi$
$338$	2703.00	+	4681.73i	0.434982	+	0.753410i
$339$		0			0
$340$	270.000	−	467.654i	0.0430671	−	0.0745944i
$341$	−6384.00			−1.01382
$342$		0			0
$343$	−15980.0			−2.51557
$344$	−352.000	+	609.682i	−0.0551703	+	0.0955577i
$345$		0			0
$346$	2397.00	+	4151.73i	0.372438	+	0.645081i
$347$	4002.00	+	6931.67i	0.619131	+	1.07237i	0.989645	+	0.143540i	$0.0458485\pi$
$347$	4002.00	+	6931.67i	0.619131	+	1.07237i	−0.370513	+	0.928827i	$0.620818\pi$
$348$		0			0
$349$	−554.500	+	960.422i	−0.0850479	+	0.147307i	−0.905412	−	0.424535i	$-0.860438\pi$
$349$	−554.500	+	960.422i	−0.0850479	+	0.147307i	0.820364	+	0.571842i	$0.193771\pi$
$350$	1700.00			0.259625
$351$		0			0
$352$	−1536.00			−0.232583
$353$	−3831.00	+	6635.49i	−0.577630	+	1.00049i	0.418120	+	0.908392i	$0.362689\pi$
$353$	−3831.00	+	6635.49i	−0.577630	+	1.00049i	−0.995750	+	0.0920935i	$0.970644\pi$
$354$		0			0
$355$	−2250.00	−	3897.11i	−0.336388	−	0.582640i
$356$	−2232.00	−	3865.94i	−0.332291	−	0.575546i
$357$		0			0
$358$	540.000	−	935.307i	0.0797204	−	0.138080i
$359$	8478.00			1.24638			0.623192	−	0.782069i	$-0.285836\pi$
$359$	8478.00			1.24638			0.623192	+	0.782069i	$0.285836\pi$
$360$		0			0
$361$	7302.00			1.06459
$362$	2333.00	−	4040.87i	0.338729	−	0.586695i
$363$		0			0
$364$	−4760.00	−	8244.56i	−0.685417	−	1.18718i
$365$	1760.00	+	3048.41i	0.252391	+	0.437154i
$366$		0			0
$367$	−6643.00	+	11506.0i	−0.944855	+	1.63654i	−0.188814	+	0.982013i	$0.560464\pi$
$367$	−6643.00	+	11506.0i	−0.944855	+	1.63654i	−0.756041	+	0.654524i	$0.772869\pi$
$368$	816.000			0.115590
$369$		0			0
$370$	2180.00			0.306305
$371$	10863.0	−	18815.3i	1.52016	−	2.63299i
$372$		0			0
$373$	−1540.00	−	2667.36i	−0.213775	−	0.370270i	0.739118	−	0.673576i	$-0.235243\pi$
$373$	−1540.00	−	2667.36i	−0.213775	−	0.370270i	−0.952893	+	0.303307i	$0.901909\pi$
$374$	1296.00	+	2244.74i	0.179183	+	0.310355i
$375$		0			0
$376$	2064.00	−	3574.95i	0.283092	−	0.490330i
$377$	−2100.00			−0.286885
$378$		0			0
$379$	10109.0			1.37009			0.685045	−	0.728500i	$-0.259782\pi$
$379$	10109.0			1.37009			0.685045	+	0.728500i	$0.259782\pi$
$380$	1190.00	−	2061.14i	0.160647	−	0.278248i
$381$		0			0
$382$	2730.00	+	4728.50i	0.365652	+	0.633327i
$383$	−4363.50	−	7557.80i	−0.582153	−	1.00832i	−0.995224	−	0.0976191i	$-0.968877\pi$
$383$	−4363.50	−	7557.80i	−0.582153	−	1.00832i	0.413071	−	0.910699i	$-0.364456\pi$
$384$		0			0
$385$	−4080.00	+	7066.77i	−0.540094	+	0.935470i
$386$	9140.00			1.20522
$387$		0			0
$388$	−64.0000			−0.00837399
$389$	−1356.00	+	2348.66i	−0.176740	+	0.306123i	−0.940762	−	0.339067i	$-0.889889\pi$
$389$	−1356.00	+	2348.66i	−0.176740	+	0.306123i	0.764022	+	0.645190i	$0.223222\pi$
$390$		0			0
$391$	−688.500	−	1192.52i	−0.0890510	−	0.154241i
$392$	3252.00	+	5632.63i	0.419007	+	0.725742i
$393$		0			0
$394$	−675.000	+	1169.13i	−0.0863097	+	0.149493i
$395$	6875.00			0.875744
$396$		0			0
$397$	−8818.00			−1.11477			−0.557384	−	0.830255i	$-0.688195\pi$
$397$	−8818.00			−1.11477			−0.557384	+	0.830255i	$0.688195\pi$
$398$	−3112.00	+	5390.14i	−0.391936	+	0.678853i
$399$		0			0
$400$	−200.000	−	346.410i	−0.0250000	−	0.0433013i
$401$	1653.00	+	2863.08i	0.205853	+	0.356547i	0.950404	−	0.311018i	$-0.100670\pi$
$401$	1653.00	+	2863.08i	0.205853	+	0.356547i	−0.744551	+	0.667565i	$0.767337\pi$
$402$		0			0
$403$	−4655.00	+	8062.70i	−0.575390	+	0.996604i
$404$	−1392.00			−0.171422
$405$		0			0
$406$	2040.00			0.249368
$407$	−5232.00	+	9062.09i	−0.637201	+	1.10366i
$408$		0			0
$409$	−3200.50	−	5543.43i	−0.386930	−	0.670183i	0.605105	−	0.796146i	$-0.293131\pi$
$409$	−3200.50	−	5543.43i	−0.386930	−	0.670183i	−0.992035	+	0.125963i	$0.959798\pi$
$410$	780.000	+	1351.00i	0.0939548	+	0.162734i
$411$		0			0
$412$	824.000	−	1427.21i	0.0985329	−	0.170664i
$413$	−22236.0			−2.64930
$414$		0			0
$415$	4575.00			0.541152
$416$	−1120.00	+	1939.90i	−0.132001	+	0.228633i
$417$		0			0
$418$	5712.00	+	9893.47i	0.668381	+	1.15767i
$419$	−1128.00	−	1953.75i	−0.131519	−	0.227797i	0.792743	−	0.609556i	$-0.208652\pi$
$419$	−1128.00	−	1953.75i	−0.131519	−	0.227797i	−0.924262	+	0.381758i	$0.875319\pi$
$420$		0			0
$421$	−905.500	+	1568.37i	−0.104825	+	0.181562i	−0.913667	−	0.406464i	$-0.866762\pi$
$421$	−905.500	+	1568.37i	−0.104825	+	0.181562i	0.808842	+	0.588027i	$0.200095\pi$
$422$	−4882.00			−0.563156
$423$		0			0
$424$	−5112.00			−0.585520
$425$	−337.500	+	584.567i	−0.0385204	+	0.0667192i
$426$		0			0
$427$	7837.00	+	13574.1i	0.888194	+	1.53840i
$428$	−1800.00	−	3117.69i	−0.203286	−	0.352101i
$429$		0			0
$430$	440.000	−	762.102i	0.0493458	−	0.0854694i
$431$	5454.00			0.609536			0.304768	−	0.952427i	$-0.401421\pi$
$431$	5454.00			0.609536			0.304768	+	0.952427i	$0.401421\pi$
$432$		0			0
$433$	2990.00			0.331848			0.165924	−	0.986139i	$-0.446939\pi$
$433$	2990.00			0.331848			0.165924	+	0.986139i	$0.446939\pi$
$434$	4522.00	−	7832.33i	0.500145	−	0.866277i
$435$		0			0
$436$	230.000	+	398.372i	0.0252638	+	0.0437581i
$437$	−3034.50	−	5255.91i	−0.332174	−	0.575341i
$438$		0			0
$439$	−4685.50	+	8115.52i	−0.509400	+	0.882307i	0.490540	+	0.871418i	$0.336799\pi$
$439$	−4685.50	+	8115.52i	−0.509400	+	0.882307i	−0.999941	+	0.0108887i	$0.996534\pi$
$440$	1920.00			0.208028
$441$		0			0
$442$	3780.00			0.406779
$443$	3085.50	−	5344.24i	0.330918	−	0.573166i	−0.651774	−	0.758413i	$-0.725975\pi$
$443$	3085.50	−	5344.24i	0.330918	−	0.573166i	0.982692	+	0.185247i	$0.0593084\pi$
$444$		0			0
$445$	2790.00	+	4832.42i	0.297211	+	0.514784i
$446$	−3418.00	−	5920.15i	−0.362886	−	0.628536i
$447$		0			0
$448$	1088.00	−	1884.47i	0.114739	−	0.198734i
$449$	−4122.00			−0.433250			−0.216625	−	0.976255i	$-0.569505\pi$
$449$	−4122.00			−0.433250			−0.216625	+	0.976255i	$0.569505\pi$
$450$		0			0
$451$	−7488.00			−0.781810
$452$	1932.00	−	3346.32i	0.201048	−	0.348225i
$453$		0			0
$454$	4377.00	+	7581.19i	0.452473	+	0.783706i
$455$	5950.00	+	10305.7i	0.613056	+	1.06184i
$456$		0			0
$457$	−3538.00	+	6128.00i	−0.362146	+	0.627255i	−0.988314	−	0.152433i	$-0.951289\pi$
$457$	−3538.00	+	6128.00i	−0.362146	+	0.627255i	0.626168	+	0.779688i	$0.284622\pi$
$458$	−8374.00			−0.854348
$459$		0			0
$460$	−1020.00			−0.103386
$461$	381.000	−	659.911i	0.0384923	−	0.0666706i	−0.846137	−	0.532965i	$-0.821078\pi$
$461$	381.000	−	659.911i	0.0384923	−	0.0666706i	0.884630	+	0.466294i	$0.154411\pi$
$462$		0			0
$463$	−4411.00	−	7640.08i	−0.442757	−	0.766878i	0.555136	−	0.831760i	$-0.312666\pi$
$463$	−4411.00	−	7640.08i	−0.442757	−	0.766878i	−0.997893	+	0.0648819i	$0.979333\pi$
$464$	−240.000	−	415.692i	−0.0240123	−	0.0415906i
$465$		0			0
$466$	1098.00	−	1901.79i	0.109150	−	0.189053i
$467$	4977.00			0.493165			0.246583	−	0.969122i	$-0.420692\pi$
$467$	4977.00			0.493165			0.246583	+	0.969122i	$0.420692\pi$
$468$		0			0
$469$	−6188.00			−0.609244
$470$	−2580.00	+	4468.69i	−0.253205	+	0.438565i
$471$		0			0
$472$	2616.00	+	4531.04i	0.255108	+	0.441861i
$473$	2112.00	+	3658.09i	0.205306	+	0.355601i
$474$		0			0
$475$	−1487.50	+	2576.43i	−0.143687	+	0.248873i
$476$	−3672.00			−0.353584
$477$		0			0
$478$	12948.0			1.23897
$479$	5052.00	−	8750.32i	0.481903	−	0.834681i	−0.517881	−	0.855453i	$-0.673279\pi$
$479$	5052.00	−	8750.32i	0.481903	−	0.834681i	0.999784	+	0.0207715i	$0.00661224\pi$
$480$		0			0
$481$	7630.00	+	13215.5i	0.723281	+	1.25276i
$482$	3251.00	+	5630.90i	0.307218	+	0.532117i
$483$		0			0
$484$	−1946.00	+	3370.57i	−0.182757	+	0.316545i
$485$	80.0000			0.00748992
$486$		0			0
$487$	14924.0			1.38865			0.694323	−	0.719663i	$-0.255704\pi$
$487$	14924.0			1.38865			0.694323	+	0.719663i	$0.255704\pi$
$488$	1844.00	−	3193.90i	0.171053	−	0.296273i
$489$		0			0
$490$	−4065.00	−	7040.79i	−0.374771	−	0.649123i
$491$	573.000	+	992.465i	0.0526662	+	0.0912206i	0.891157	−	0.453696i	$-0.149895\pi$
$491$	573.000	+	992.465i	0.0526662	+	0.0912206i	−0.838490	+	0.544916i	$0.816561\pi$
$492$		0			0
$493$	−405.000	+	701.481i	−0.0369985	+	0.0640834i
$494$	16660.0			1.51735
$495$		0			0
$496$	−2128.00			−0.192641
$497$	−15300.0	+	26500.4i	−1.38088	+	2.39176i
$498$		0			0
$499$	7482.50	+	12960.1i	0.671268	+	1.16267i	0.977545	+	0.210728i	$0.0675833\pi$
$499$	7482.50	+	12960.1i	0.671268	+	1.16267i	−0.306277	+	0.951942i	$0.599083\pi$
$500$	250.000	+	433.013i	0.0223607	+	0.0387298i
$501$		0			0
$502$	−1728.00	+	2992.98i	−0.153634	+	0.266102i
$503$	−15525.0			−1.37619			−0.688097	−	0.725619i	$-0.741554\pi$
$503$	−15525.0			−1.37619			−0.688097	+	0.725619i	$0.741554\pi$
$504$		0			0
$505$	1740.00			0.153325
$506$	2448.00	−	4240.06i	0.215073	−	0.372517i
$507$		0			0
$508$	−2812.00	−	4870.53i	−0.245595	−	0.425383i
$509$	−4098.00	−	7097.94i	−0.356858	−	0.618096i	0.630576	−	0.776127i	$-0.282819\pi$
$509$	−4098.00	−	7097.94i	−0.356858	−	0.618096i	−0.987434	+	0.158031i	$0.949485\pi$
$510$		0			0
$511$	11968.0	−	20729.2i	1.03607	−	1.79453i
$512$	−512.000			−0.0441942
$513$		0			0
$514$	−10938.0			−0.938627
$515$	−1030.00	+	1784.01i	−0.0881305	+	0.152647i
$516$		0			0
$517$	−12384.0	−	21449.7i	−1.05348	−	1.82468i
$518$	−7412.00	−	12838.0i	−0.628696	−	1.08893i
$519$		0			0
$520$	1400.00	−	2424.87i	0.118066	−	0.204495i
$521$	−4932.00			−0.414731			−0.207365	−	0.978264i	$-0.566489\pi$
$521$	−4932.00			−0.414731			−0.207365	+	0.978264i	$0.566489\pi$
$522$		0			0
$523$	−5938.00			−0.496464			−0.248232	−	0.968701i	$-0.579850\pi$
$523$	−5938.00			−0.496464			−0.248232	+	0.968701i	$0.579850\pi$
$524$	492.000	−	852.169i	0.0410174	−	0.0710442i
$525$		0			0
$526$	3216.00	+	5570.28i	0.266586	+	0.461741i
$527$	1795.50	+	3109.90i	0.148412	+	0.257057i
$528$		0			0
$529$	4783.00	−	8284.40i	0.393113	−	0.680891i
$530$	6390.00			0.523705
$531$		0			0
$532$	−16184.0			−1.31892
$533$	−5460.00	+	9457.00i	−0.443713	+	0.768533i
$534$		0			0
$535$	2250.00	+	3897.11i	0.181824	+	0.314929i
$536$	728.000	+	1260.93i	0.0586657	+	0.101612i
$537$		0			0
$538$	8010.00	−	13873.7i	0.641888	−	1.11178i
$539$	39024.0			3.11852
$540$		0			0
$541$	−6730.00			−0.534834			−0.267417	−	0.963581i	$-0.586170\pi$
$541$	−6730.00			−0.534834			−0.267417	+	0.963581i	$0.586170\pi$
$542$	−3805.00	+	6590.45i	−0.301547	+	0.522296i
$543$		0			0
$544$	432.000	+	748.246i	0.0340475	+	0.0589720i
$545$	−287.500	−	497.965i	−0.0225966	−	0.0391385i
$546$		0			0
$547$	8828.00	−	15290.5i	0.690051	−	1.19520i	−0.281769	−	0.959482i	$-0.590921\pi$
$547$	8828.00	−	15290.5i	0.690051	−	1.19520i	0.971821	−	0.235722i	$-0.0757454\pi$
$548$	−2076.00			−0.161829
$549$		0			0
$550$	−2400.00			−0.186066
$551$	−1785.00	+	3091.71i	−0.138010	+	0.239040i
$552$		0			0
$553$	−23375.0	−	40486.7i	−1.79748	−	3.11333i
$554$	3224.00	+	5584.13i	0.247247	+	0.428244i
$555$		0			0
$556$	−2632.00	+	4558.76i	−0.200758	+	0.347724i
$557$	−7974.00			−0.606587			−0.303294	−	0.952897i	$-0.598086\pi$
$557$	−7974.00			−0.606587			−0.303294	+	0.952897i	$0.598086\pi$
$558$		0			0
$559$	6160.00			0.466083
$560$	−1360.00	+	2355.59i	−0.102626	+	0.177753i
$561$		0			0
$562$	4530.00	+	7846.19i	0.340012	+	0.588917i
$563$	−12666.0	−	21938.2i	−0.948150	−	1.64224i	−0.749319	−	0.662210i	$-0.769619\pi$
$563$	−12666.0	−	21938.2i	−0.948150	−	1.64224i	−0.198831	−	0.980034i	$-0.563714\pi$
$564$		0			0
$565$	−2415.00	+	4182.90i	−0.179823	+	0.311462i
$566$	6584.00			0.488951
$567$		0			0
$568$	7200.00			0.531876
$569$	−519.000	+	898.934i	−0.0382383	+	0.0662307i	−0.884511	−	0.466519i	$-0.845508\pi$
$569$	−519.000	+	898.934i	−0.0382383	+	0.0662307i	0.846273	+	0.532750i	$0.178841\pi$
$570$		0			0
$571$	−7835.50	−	13571.5i	−0.574265	−	0.994657i	−0.996121	−	0.0879937i	$-0.971954\pi$
$571$	−7835.50	−	13571.5i	−0.574265	−	0.994657i	0.421856	−	0.906663i	$-0.361379\pi$
$572$	6720.00	+	11639.4i	0.491219	+	0.850816i
$573$		0			0
$574$	5304.00	−	9186.80i	0.385688	−	0.668031i
$575$	1275.00			0.0924716
$576$		0			0
$577$	−916.000			−0.0660894			−0.0330447	−	0.999454i	$-0.510520\pi$
$577$	−916.000			−0.0660894			−0.0330447	+	0.999454i	$0.510520\pi$
$578$	−4184.00	+	7246.90i	−0.301092	+	0.521507i
$579$		0			0
$580$	300.000	+	519.615i	0.0214773	+	0.0371997i
$581$	−15555.0	−	26942.1i	−1.11072	−	1.92383i
$582$		0			0
$583$	−15336.0	+	26562.7i	−1.08945	+	1.88699i
$584$	−5632.00			−0.399065
$585$		0			0
$586$	−15906.0			−1.12128
$587$	4570.50	−	7916.34i	0.321371	−	0.556631i	−0.659400	−	0.751792i	$-0.729190\pi$
$587$	4570.50	−	7916.34i	0.321371	−	0.556631i	0.980771	+	0.195161i	$0.0625230\pi$
$588$		0			0
$589$	7913.50	+	13706.6i	0.553599	+	0.958862i
$590$	−3270.00	−	5663.81i	−0.228176	−	0.395212i
$591$		0			0
$592$	−1744.00	+	3020.70i	−0.121078	+	0.209713i
$593$	−5247.00			−0.363353			−0.181677	−	0.983358i	$-0.558152\pi$
$593$	−5247.00			−0.363353			−0.181677	+	0.983358i	$0.558152\pi$
$594$		0			0
$595$	4590.00			0.316255
$596$	744.000	−	1288.65i	0.0511333	−	0.0885654i
$597$		0			0
$598$	−3570.00	−	6183.42i	−0.244127	−	0.422841i
$599$	−12081.0	−	20924.9i	−0.824067	−	1.42733i	−0.902630	−	0.430417i	$-0.858366\pi$
$599$	−12081.0	−	20924.9i	−0.824067	−	1.42733i	0.0785628	−	0.996909i	$-0.474967\pi$
$600$		0			0
$601$	−7178.50	+	12433.5i	−0.487217	+	0.843884i	−0.999892	−	0.0146987i	$-0.995321\pi$
$601$	−7178.50	+	12433.5i	−0.487217	+	0.843884i	0.512675	+	0.858582i	$0.328654\pi$
$602$	−5984.00			−0.405132
$603$		0			0
$604$	−5824.00			−0.392343
$605$	2432.50	−	4213.21i	0.163463	−	0.283126i
$606$		0			0
$607$	−1576.00	−	2729.71i	−0.105384	−	0.182530i	0.808511	−	0.588481i	$-0.200274\pi$
$607$	−1576.00	−	2729.71i	−0.105384	−	0.182530i	−0.913895	+	0.405951i	$0.866940\pi$
$608$	1904.00	+	3297.82i	0.127002	+	0.219974i
$609$		0			0
$610$	−2305.00	+	3992.38i	−0.152995	+	0.264994i
$611$	−36120.0			−2.39159
$612$		0			0
$613$	4592.00			0.302560			0.151280	−	0.988491i	$-0.451661\pi$
$613$	4592.00			0.302560			0.151280	+	0.988491i	$0.451661\pi$
$614$	−5290.00	+	9162.55i	−0.347699	+	0.602232i
$615$		0			0
$616$	−6528.00	−	11306.8i	−0.426982	−	0.739554i
$617$	−3679.50	−	6373.08i	−0.240083	−	0.415836i	0.720655	−	0.693294i	$-0.243841\pi$
$617$	−3679.50	−	6373.08i	−0.240083	−	0.415836i	−0.960738	+	0.277458i	$0.910508\pi$
$618$		0			0
$619$	7856.00	−	13607.0i	0.510112	−	0.883540i	−0.489819	−	0.871824i	$-0.662937\pi$
$619$	7856.00	−	13607.0i	0.510112	−	0.883540i	0.999931	−	0.0117159i	$-0.00372937\pi$
$620$	2660.00			0.172304
$621$		0			0
$622$	10716.0			0.690792
$623$	18972.0	−	32860.5i	1.22006	−	2.11321i
$624$		0			0
$625$	−312.500	−	541.266i	−0.0200000	−	0.0346410i
$626$	5600.00	+	9699.48i	0.357542	+	0.619280i
$627$		0			0
$628$	−1912.00	+	3311.68i	−0.121492	+	0.210431i
$629$	5886.00			0.373116
$630$		0			0
$631$	−3175.00			−0.200309			−0.100154	−	0.994972i	$-0.531934\pi$
$631$	−3175.00			−0.200309			−0.100154	+	0.994972i	$0.531934\pi$
$632$	−5500.00	+	9526.28i	−0.346168	+	0.599581i
$633$		0			0
$634$	−7341.00	−	12715.0i	−0.459855	−	0.796493i
$635$	3515.00	+	6088.16i	0.219667	+	0.380474i
$636$		0			0
$637$	28455.0	−	49285.5i	1.76990	−	3.06556i
$638$	−2880.00			−0.178715
$639$		0			0
$640$	640.000			0.0395285
$641$	48.0000	−	83.1384i	0.00295770	−	0.00512289i	−0.864543	−	0.502559i	$-0.832392\pi$
$641$	48.0000	−	83.1384i	0.00295770	−	0.00512289i	0.867500	+	0.497436i	$0.165725\pi$
$642$		0			0
$643$	9035.00	+	15649.1i	0.554130	+	0.959781i	0.997971	+	0.0636756i	$0.0202823\pi$
$643$	9035.00	+	15649.1i	0.554130	+	0.959781i	−0.443841	+	0.896106i	$0.646384\pi$
$644$	3468.00	+	6006.75i	0.212202	+	0.367545i
$645$		0			0
$646$	3213.00	−	5565.08i	0.195687	−	0.338940i
$647$	−1341.00			−0.0814840			−0.0407420	−	0.999170i	$-0.512972\pi$
$647$	−1341.00			−0.0814840			−0.0407420	+	0.999170i	$0.512972\pi$
$648$		0			0
$649$	31392.0			1.89868
$650$	−1750.00	+	3031.09i	−0.105601	+	0.182906i
$651$		0			0
$652$	4892.00	+	8473.19i	0.293843	+	0.508951i
$653$	−12247.5	−	21213.3i	−0.733969	−	1.27127i	−0.955174	−	0.296044i	$-0.904332\pi$
$653$	−12247.5	−	21213.3i	−0.733969	−	1.27127i	0.221205	−	0.975227i	$-0.429001\pi$
$654$		0			0
$655$	−615.000	+	1065.21i	−0.0366871	+	0.0635439i
$656$	−2496.00			−0.148556
$657$		0			0
$658$	35088.0			2.07883
$659$	−6189.00	+	10719.7i	−0.365841	+	0.633655i	−0.988911	−	0.148511i	$-0.952552\pi$
$659$	−6189.00	+	10719.7i	−0.365841	+	0.633655i	0.623070	+	0.782166i	$0.285885\pi$
$660$		0			0
$661$	12221.0	+	21167.4i	0.719125	+	1.24556i	0.961347	+	0.275341i	$0.0887906\pi$
$661$	12221.0	+	21167.4i	0.719125	+	1.24556i	−0.242221	+	0.970221i	$0.577876\pi$
$662$	380.000	+	658.179i	0.0223098	+	0.0386418i
$663$		0			0
$664$	−3660.00	+	6339.31i	−0.213909	+	0.370501i
$665$	20230.0			1.17968
$666$		0			0
$667$	1530.00			0.0888183
$668$	−6222.00	+	10776.8i	−0.360384	+	0.624203i
$669$		0			0
$670$	−910.000	−	1576.17i	−0.0524722	−	0.0908845i
$671$	−11064.0	−	19163.4i	−0.636544	−	1.10253i
$672$		0			0
$673$	−1189.00	+	2059.41i	−0.0681019	+	0.117956i	−0.898066	−	0.439861i	$-0.855028\pi$
$673$	−1189.00	+	2059.41i	−0.0681019	+	0.117956i	0.829964	+	0.557817i	$0.188361\pi$
$674$	−868.000			−0.0496055
$675$		0			0
$676$	10812.0			0.615157
$677$	2739.00	−	4744.09i	0.155492	−	0.269321i	−0.777746	−	0.628579i	$-0.783637\pi$
$677$	2739.00	−	4744.09i	0.155492	−	0.269321i	0.933238	+	0.359258i	$0.116970\pi$
$678$		0			0
$679$	−272.000	−	471.118i	−0.0153732	−	0.0266272i
$680$	−540.000	−	935.307i	−0.0304530	−	0.0527462i
$681$		0			0
$682$	−6384.00	+	11057.4i	−0.358440	+	0.620836i
$683$	−8595.00			−0.481521			−0.240760	−	0.970585i	$-0.577397\pi$
$683$	−8595.00			−0.481521			−0.240760	+	0.970585i	$0.577397\pi$
$684$		0			0
$685$	2595.00			0.144744
$686$	−15980.0	+	27678.2i	−0.889387	+	1.54046i
$687$		0			0
$688$	704.000	+	1219.36i	0.0390113	+	0.0675695i
$689$	22365.0	+	38737.3i	1.23663	+	2.14191i
$690$		0			0
$691$	15807.5	−	27379.4i	0.870254	−	1.50732i	0.00852043	−	0.999964i	$-0.497288\pi$
$691$	15807.5	−	27379.4i	0.870254	−	1.50732i	0.861734	−	0.507361i	$-0.169379\pi$
$692$	9588.00			0.526707
$693$		0			0
$694$	16008.0			0.875584
$695$	3290.00	−	5698.45i	0.179564	−	0.311014i
$696$		0			0
$697$	2106.00	+	3647.70i	0.114448	+	0.198230i
$698$	1109.00	+	1920.84i	0.0601379	+	0.104162i
$699$		0			0
$700$	1700.00	−	2944.49i	0.0917914	−	0.158987i
$701$	−29790.0			−1.60507			−0.802534	−	0.596606i	$-0.796515\pi$
$701$	−29790.0			−1.60507			−0.802534	+	0.596606i	$0.796515\pi$
$702$		0			0
$703$	25942.0			1.39178
$704$	−1536.00	+	2660.43i	−0.0822304	+	0.142427i
$705$		0			0
$706$	7662.00	+	13271.0i	0.408446	+	0.707450i
$707$	−5916.00	−	10246.8i	−0.314702	−	0.545079i
$708$		0			0
$709$	−1909.00	+	3306.48i	−0.101120	+	0.175145i	−0.912146	−	0.409865i	$-0.865576\pi$
$709$	−1909.00	+	3306.48i	−0.101120	+	0.175145i	0.811026	+	0.585010i	$0.198909\pi$
$710$	−9000.00			−0.475724
$711$		0			0
$712$	−8928.00			−0.469931
$713$	3391.50	−	5874.25i	0.178138	−	0.308545i
$714$		0			0
$715$	−8400.00	−	14549.2i	−0.439360	−	0.760993i
$716$	−1080.00	−	1870.61i	−0.0563708	−	0.0976371i
$717$		0			0
$718$	8478.00	−	14684.3i	0.440663	−	0.763251i
$719$	28314.0			1.46861			0.734307	−	0.678817i	$-0.237507\pi$
$719$	28314.0			1.46861			0.734307	+	0.678817i	$0.237507\pi$
$720$		0			0
$721$	14008.0			0.723558
$722$	7302.00	−	12647.4i	0.376388	−	0.651924i
$723$		0			0
$724$	−4666.00	−	8081.75i	−0.239517	−	0.414856i
$725$	−375.000	−	649.519i	−0.0192099	−	0.0332725i
$726$		0			0
$727$	−28.0000	+	48.4974i	−0.00142842	+	0.00247410i	−0.866739	−	0.498762i	$-0.833788\pi$
$727$	−28.0000	+	48.4974i	−0.00142842	+	0.00247410i	0.865310	+	0.501237i	$0.167121\pi$
$728$	−19040.0			−0.969326
$729$		0			0
$730$	7040.00			0.356934
$731$	1188.00	−	2057.68i	0.0601091	−	0.104112i
$732$		0			0
$733$	17216.0	+	29819.0i	0.867514	+	1.50258i	0.864529	+	0.502582i	$0.167617\pi$
$733$	17216.0	+	29819.0i	0.867514	+	1.50258i	0.00298410	+	0.999996i	$0.499050\pi$
$734$	13286.0	+	23012.0i	0.668113	+	1.15721i
$735$		0			0
$736$	816.000	−	1413.35i	0.0408671	−	0.0707838i
$737$	8736.00			0.436628
$738$		0			0
$739$	−1051.00			−0.0523162			−0.0261581	−	0.999658i	$-0.508327\pi$
$739$	−1051.00			−0.0523162			−0.0261581	+	0.999658i	$0.508327\pi$
$740$	2180.00	−	3775.87i	0.108295	−	0.187573i
$741$		0			0
$742$	−21726.0	−	37630.5i	−1.07491	−	1.86181i
$743$	19572.0	+	33899.7i	0.966389	+	1.67383i	0.705836	+	0.708375i	$0.250572\pi$
$743$	19572.0	+	33899.7i	0.966389	+	1.67383i	0.260553	+	0.965460i	$0.416095\pi$
$744$		0			0
$745$	−930.000	+	1610.81i	−0.0457350	+	0.0792153i
$746$	−6160.00			−0.302324
$747$		0			0
$748$	5184.00			0.253403
$749$	15300.0	−	26500.4i	0.746395	−	1.29279i
$750$		0			0
$751$	867.500	+	1502.55i	0.0421512	+	0.0730080i	0.886331	−	0.463052i	$-0.153246\pi$
$751$	867.500	+	1502.55i	0.0421512	+	0.0730080i	−0.844180	+	0.536060i	$0.819912\pi$
$752$	−4128.00	−	7149.91i	−0.200176	−	0.346716i
$753$		0			0
$754$	−2100.00	+	3637.31i	−0.101429	+	0.175680i
$755$	7280.00			0.350922
$756$		0			0
$757$	6698.00			0.321589			0.160795	−	0.986988i	$-0.448594\pi$
$757$	6698.00			0.321589			0.160795	+	0.986988i	$0.448594\pi$
$758$	10109.0	−	17509.3i	0.484400	−	0.839006i
$759$		0			0
$760$	−2380.00	−	4122.28i	−0.113594	−	0.196751i
$761$	19383.0	+	33572.3i	0.923302	+	1.59921i	0.794269	+	0.607567i	$0.207854\pi$
$761$	19383.0	+	33572.3i	0.923302	+	1.59921i	0.129034	+	0.991640i	$0.458812\pi$
$762$		0			0
$763$	−1955.00	+	3386.16i	−0.0927598	+	0.160665i
$764$	10920.0			0.517110
$765$		0			0
$766$	−17454.0			−0.823288
$767$	22890.0	−	39646.6i	1.07759	−	1.86644i
$768$		0			0
$769$	−11750.5	−	20352.5i	−0.551019	−	0.954393i	−0.998201	−	0.0599505i	$-0.980906\pi$
$769$	−11750.5	−	20352.5i	−0.551019	−	0.954393i	0.447182	−	0.894443i	$-0.352428\pi$
$770$	8160.00	+	14133.5i	0.381904	+	0.661477i
$771$		0			0
$772$	9140.00	−	15830.9i	0.426109	−	0.738042i
$773$	3591.00			0.167088			0.0835442	−	0.996504i	$-0.473376\pi$
$773$	3591.00			0.167088			0.0835442	+	0.996504i	$0.473376\pi$
$774$		0			0
$775$	−3325.00			−0.154113
$776$	−64.0000	+	110.851i	−0.00296065	+	0.00512800i
$777$		0			0
$778$	2712.00	+	4697.32i	0.124974	+	0.216462i
$779$	9282.00	+	16076.9i	0.426909	+	0.739428i
$780$		0			0
$781$	21600.0	−	37412.3i	0.989640	−	1.71411i
$782$	−2754.00			−0.125937
$783$		0			0
$784$	13008.0			0.592566
$785$	2390.00	−	4139.60i	0.108666	−	0.188215i
$786$		0			0
$787$	10358.0	+	17940.6i	0.469152	+	0.812596i	0.999378	−	0.0352609i	$-0.0112262\pi$
$787$	10358.0	+	17940.6i	0.469152	+	0.812596i	−0.530226	+	0.847856i	$0.677893\pi$
$788$	1350.00	+	2338.27i	0.0610302	+	0.105707i
$789$		0			0
$790$	6875.00	−	11907.8i	0.309622	−	0.536281i
$791$	32844.0			1.47636
$792$		0			0
$793$	−32270.0			−1.44507
$794$	−8818.00	+	15273.2i	−0.394130	+	0.682653i
$795$		0			0
$796$	6224.00	+	10780.3i	0.277140	+	0.480021i
$797$	−21490.5	−	37222.6i	−0.955122	−	1.65432i	−0.734088	−	0.679055i	$-0.762390\pi$
$797$	−21490.5	−	37222.6i	−0.955122	−	1.65432i	−0.221035	−	0.975266i	$-0.570943\pi$
$798$		0			0
$799$	−6966.00	+	12065.5i	−0.308435	+	0.534225i
$800$	−800.000			−0.0353553
$801$		0			0
$802$	6612.00			0.291119
$803$	−16896.0	+	29264.7i	−0.742524	+	1.28609i
$804$		0			0
$805$	−4335.00	−	7508.44i	−0.189800	−	0.328743i
$806$	9310.00	+	16125.4i	0.406862	+	0.704706i
$807$		0			0
$808$	−1392.00	+	2411.01i	−0.0606069	+	0.104974i
$809$	2268.00			0.0985644			0.0492822	−	0.998785i	$-0.484307\pi$
$809$	2268.00			0.0985644			0.0492822	+	0.998785i	$0.484307\pi$
$810$		0			0
$811$	11756.0			0.509012			0.254506	−	0.967071i	$-0.418087\pi$
$811$	11756.0			0.509012			0.254506	+	0.967071i	$0.418087\pi$
$812$	2040.00	−	3533.38i	0.0881650	−	0.152706i
$813$		0			0
$814$	10464.0	+	18124.2i	0.450569	+	0.780408i
$815$	−6115.00	−	10591.5i	−0.262821	−	0.455219i
$816$		0			0
$817$	5236.00	−	9069.02i	0.224216	−	0.388353i
$818$	−12802.0			−0.547202
$819$		0			0
$820$	3120.00			0.132872
$821$	4323.00	−	7487.66i	0.183768	−	0.318296i	−0.759393	−	0.650633i	$-0.774504\pi$
$821$	4323.00	−	7487.66i	0.183768	−	0.318296i	0.943161	+	0.332337i	$0.107837\pi$
$822$		0			0
$823$	−5392.00	−	9339.22i	−0.228376	−	0.395559i	0.728951	−	0.684566i	$-0.240008\pi$
$823$	−5392.00	−	9339.22i	−0.228376	−	0.395559i	−0.957327	+	0.289007i	$0.906675\pi$
$824$	−1648.00	−	2854.42i	−0.0696733	−	0.120678i
$825$		0			0
$826$	−22236.0	+	38513.9i	−0.936670	+	1.62236i
$827$	−42597.0			−1.79110			−0.895552	−	0.444957i	$-0.853219\pi$
$827$	−42597.0			−1.79110			−0.895552	+	0.444957i	$0.853219\pi$
$828$		0			0
$829$	−26458.0			−1.10847			−0.554237	−	0.832359i	$-0.686990\pi$
$829$	−26458.0			−1.10847			−0.554237	+	0.832359i	$0.686990\pi$
$830$	4575.00	−	7924.13i	0.191326	−	0.331386i
$831$		0			0
$832$	2240.00	+	3879.79i	0.0933390	+	0.161668i
$833$	−10975.5	−	19010.1i	−0.456517	−	0.790710i
$834$		0			0
$835$	7777.50	−	13471.0i	0.322337	−	0.558304i
$836$	22848.0			0.945233
$837$		0			0
$838$	−4512.00			−0.185996
$839$	−5748.00	+	9955.83i	−0.236523	+	0.409670i	−0.959714	−	0.280978i	$-0.909341\pi$
$839$	−5748.00	+	9955.83i	−0.236523	+	0.409670i	0.723191	+	0.690648i	$0.242675\pi$
$840$		0			0
$841$	11744.5	+	20342.1i	0.481549	+	0.834067i
$842$	1811.00	+	3136.74i	0.0741225	+	0.128384i
$843$		0			0
$844$	−4882.00	+	8455.87i	−0.199106	+	0.344862i
$845$	−13515.0			−0.550213
$846$		0			0
$847$	−33082.0			−1.34204
$848$	−5112.00	+	8854.24i	−0.207013	+	0.358557i
$849$		0			0
$850$	675.000	+	1169.13i	0.0272380	+	0.0471776i
$851$	−5559.00	−	9628.47i	−0.223925	−	0.387849i
$852$		0			0
$853$	−10774.0	+	18661.1i	−0.432467	+	0.749056i	−0.997085	−	0.0762970i	$-0.975690\pi$
$853$	−10774.0	+	18661.1i	−0.432467	+	0.749056i	0.564618	+	0.825353i	$0.309024\pi$
$854$	31348.0			1.25610
$855$		0			0
$856$	−7200.00			−0.287489
$857$	3130.50	−	5422.19i	0.124779	−	0.216124i	−0.796867	−	0.604154i	$-0.793511\pi$
$857$	3130.50	−	5422.19i	0.124779	−	0.216124i	0.921647	+	0.388030i	$0.126844\pi$
$858$		0			0
$859$	1677.50	+	2905.52i	0.0666305	+	0.115407i	0.897416	−	0.441185i	$-0.145442\pi$
$859$	1677.50	+	2905.52i	0.0666305	+	0.115407i	−0.830786	+	0.556593i	$0.812108\pi$
$860$	−880.000	−	1524.20i	−0.0348927	−	0.0604360i
$861$		0			0
$862$	5454.00	−	9446.61i	0.215503	−	0.373263i
$863$	−19701.0			−0.777091			−0.388546	−	0.921429i	$-0.627022\pi$
$863$	−19701.0			−0.777091			−0.388546	+	0.921429i	$0.627022\pi$
$864$		0			0
$865$	−11985.0			−0.471101
$866$	2990.00	−	5178.83i	0.117326	−	0.203215i
$867$		0			0
$868$	−9044.00	−	15664.7i	−0.353656	−	0.612550i
$869$	33000.0	+	57157.7i	1.28820	+	2.23123i
$870$		0			0
$871$	6370.00	−	11033.2i	0.247806	−	0.429213i
$872$	920.000			0.0357284
$873$		0			0
$874$	−12138.0			−0.469764
$875$	−2125.00	+	3680.61i	−0.0821007	+	0.142203i
$876$		0			0
$877$	−8146.00	−	14109.3i	−0.313650	−	0.543257i	0.665500	−	0.746398i	$-0.268218\pi$
$877$	−8146.00	−	14109.3i	−0.313650	−	0.543257i	−0.979150	+	0.203141i	$0.934885\pi$
$878$	9371.00	+	16231.0i	0.360200	+	0.623885i
$879$		0			0
$880$	1920.00	−	3325.54i	0.0735491	−	0.127391i
$881$	−9270.00			−0.354500			−0.177250	−	0.984166i	$-0.556720\pi$
$881$	−9270.00			−0.354500			−0.177250	+	0.984166i	$0.556720\pi$
$882$		0			0
$883$	38486.0			1.46677			0.733384	−	0.679814i	$-0.237940\pi$
$883$	38486.0			1.46677			0.733384	+	0.679814i	$0.237940\pi$
$884$	3780.00	−	6547.15i	0.143818	−	0.249100i
$885$		0			0
$886$	−6171.00	−	10688.5i	−0.233994	−	0.405290i
$887$	946.500	+	1639.39i	0.0358290	+	0.0620577i	0.883384	−	0.468650i	$-0.155259\pi$
$887$	946.500	+	1639.39i	0.0358290	+	0.0620577i	−0.847555	+	0.530708i	$0.821926\pi$
$888$		0			0
$889$	23902.0	−	41399.5i	0.901741	−	1.56186i
$890$	11160.0			0.420319
$891$		0			0
$892$	−13672.0			−0.513198
$893$	−30702.0	+	53177.4i	−1.15051	+	1.99274i
$894$		0			0
$895$	1350.00	+	2338.27i	0.0504196	+	0.0873293i
$896$	−2176.00	−	3768.94i	−0.0811329	−	0.140526i
$897$		0			0
$898$	−4122.00	+	7139.51i	−0.153177	+	0.265310i
$899$	−3990.00			−0.148024
$900$		0			0
$901$	17253.0			0.637936
$902$	−7488.00	+	12969.6i	−0.276411	+	0.478759i
$903$		0			0
$904$	−3864.00	−	6692.64i	−0.142162	−	0.246232i
$905$	5832.50	+	10102.2i	0.214231	+	0.371059i
$906$		0			0
$907$	−22438.0	+	38863.8i	−0.821435	+	1.42277i	0.0831786	+	0.996535i	$0.473493\pi$
$907$	−22438.0	+	38863.8i	−0.821435	+	1.42277i	−0.904614	+	0.426233i	$0.859841\pi$
$908$	17508.0			0.639894
$909$		0			0
$910$	23800.0			0.866992
$911$	11901.0	−	20613.1i	0.432819	−	0.749664i	−0.564296	−	0.825572i	$-0.690852\pi$
$911$	11901.0	−	20613.1i	0.432819	−	0.749664i	0.997115	+	0.0759086i	$0.0241857\pi$
$912$		0			0
$913$	21960.0	+	38035.8i	0.796024	+	1.37875i
$914$	7076.00	+	12256.0i	0.256076	+	0.443536i
$915$		0			0
$916$	−8374.00	+	14504.2i	−0.302058	+	0.523179i
$917$	8364.00			0.301204
$918$		0			0
$919$	−24784.0			−0.889607			−0.444803	−	0.895628i	$-0.646726\pi$
$919$	−24784.0			−0.889607			−0.444803	+	0.895628i	$0.646726\pi$
$920$	−1020.00	+	1766.69i	−0.0365526	+	0.0633110i
$921$		0			0
$922$	−762.000	−	1319.82i	−0.0272181	−	0.0471432i
$923$	−31500.0	−	54559.6i	−1.12333	−	1.94567i
$924$		0			0
$925$	−2725.00	+	4719.84i	−0.0968621	+	0.167770i
$926$	−17644.0			−0.626153
$927$		0			0
$928$	−960.000			−0.0339586
$929$	4530.00	−	7846.19i	0.159983	−	0.277099i	−0.774879	−	0.632109i	$-0.782189\pi$
$929$	4530.00	−	7846.19i	0.159983	−	0.277099i	0.934862	+	0.355010i	$0.115523\pi$
$930$		0			0
$931$	−48373.5	−	83785.4i	−1.70288	−	2.94947i
$932$	−2196.00	−	3803.58i	−0.0771807	−	0.133681i
$933$		0			0
$934$	4977.00	−	8620.42i	0.174360	−	0.302001i
$935$	−6480.00			−0.226651
$936$		0			0
$937$	6176.00			0.215327			0.107663	−	0.994187i	$-0.465663\pi$
$937$	6176.00			0.215327			0.107663	+	0.994187i	$0.465663\pi$
$938$	−6188.00	+	10717.9i	−0.215400	+	0.373084i
$939$		0			0
$940$	5160.00	+	8937.38i	0.179043	+	0.310112i
$941$	−2091.00	−	3621.72i	−0.0724385	−	0.125467i	0.827531	−	0.561420i	$-0.189745\pi$
$941$	−2091.00	−	3621.72i	−0.0724385	−	0.125467i	−0.899970	+	0.435953i	$0.856411\pi$
$942$		0			0
$943$	3978.00	−	6890.10i	0.137372	−	0.237935i
$944$	10464.0			0.360778
$945$		0			0
$946$	8448.00			0.290347
$947$	22039.5	−	38173.5i	0.756270	−	1.30990i	−0.188471	−	0.982079i	$-0.560353\pi$
$947$	22039.5	−	38173.5i	0.756270	−	1.30990i	0.944741	−	0.327819i	$-0.106314\pi$
$948$		0			0
$949$	24640.0	+	42677.7i	0.842833	+	1.45983i
$950$	2975.00	+	5152.85i	0.101602	+	0.175980i
$951$		0			0
$952$	−3672.00	+	6360.09i	−0.125011	+	0.216525i
$953$	−12726.0			−0.432566			−0.216283	−	0.976331i	$-0.569393\pi$
$953$	−12726.0			−0.432566			−0.216283	+	0.976331i	$0.569393\pi$
$954$		0			0
$955$	−13650.0			−0.462517
$956$	12948.0	−	22426.6i	0.438042	−	0.758711i
$957$		0			0
$958$	−10104.0	−	17500.6i	−0.340757	−	0.590209i
$959$	−8823.00	−	15281.9i	−0.297090	−	0.514575i
$960$		0			0
$961$	6051.00	−	10480.6i	0.203115	−	0.351806i
$962$	30520.0			1.02287
$963$		0			0
$964$	13004.0			0.434472
$965$	−11425.0	+	19788.7i	−0.381123	+	0.660125i
$966$		0			0
$967$	−22609.0	−	39159.9i	−0.751868	−	1.30227i	−0.946916	−	0.321480i	$-0.895820\pi$
$967$	−22609.0	−	39159.9i	−0.751868	−	1.30227i	0.195048	−	0.980794i	$-0.437514\pi$
$968$	3892.00	+	6741.14i	0.129229	+	0.223831i
$969$		0			0
$970$	80.0000	−	138.564i	0.00264809	−	0.00458662i
$971$	3978.00			0.131473			0.0657364	−	0.997837i	$-0.479060\pi$
$971$	3978.00			0.131473			0.0657364	+	0.997837i	$0.479060\pi$
$972$		0			0
$973$	−44744.0			−1.47423
$974$	14924.0	−	25849.1i	0.490961	−	0.850369i
$975$		0			0
$976$	−3688.00	−	6387.80i	−0.120953	−	0.209497i
$977$	11733.0	+	20322.2i	0.384209	+	0.665469i	0.991659	−	0.128889i	$-0.0411410\pi$
$977$	11733.0	+	20322.2i	0.384209	+	0.665469i	−0.607450	+	0.794358i	$0.707808\pi$
$978$		0			0
$979$	−26784.0	+	46391.2i	−0.874382	+	1.51447i
$980$	−16260.0			−0.530007
$981$		0			0
$982$	2292.00			0.0744813
$983$	23956.5	−	41493.9i	0.777308	−	1.34634i	−0.156180	−	0.987729i	$-0.549918\pi$
$983$	23956.5	−	41493.9i	0.777308	−	1.34634i	0.933488	−	0.358608i	$-0.116749\pi$
$984$		0			0
$985$	−1687.50	−	2922.84i	−0.0545870	−	0.0945475i
$986$	810.000	+	1402.96i	0.0261619	+	0.0453138i
$987$		0			0
$988$	16660.0	−	28856.0i	0.536463	−	0.929181i
$989$	−4488.00			−0.144297
$990$		0			0
$991$	31997.0			1.02565			0.512825	−	0.858493i	$-0.328599\pi$
$991$	31997.0			1.02565			0.512825	+	0.858493i	$0.328599\pi$
$992$	−2128.00	+	3685.80i	−0.0681089	+	0.117968i
$993$		0			0
$994$	30600.0	+	53000.8i	0.976432	+	1.69123i
$995$	−7780.00	−	13475.4i	−0.247882	−	0.429344i
$996$		0			0
$997$	22814.0	−	39515.0i	0.724701	−	1.25522i	−0.234396	−	0.972141i	$-0.575311\pi$
$997$	22814.0	−	39515.0i	0.724701	−	1.25522i	0.959097	−	0.283077i	$-0.0913553\pi$
$998$	29930.0			0.949316
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.4.e.x.541.1		2
3.2	odd	2		810.4.e.d.541.1		2
9.2	odd	6		270.4.a.k.1.1	yes	1
9.4	even	3	inner	810.4.e.x.271.1		2
9.5	odd	6		810.4.e.d.271.1		2
9.7	even	3		270.4.a.a.1.1	✓	1
36.7	odd	6		2160.4.a.j.1.1		1
36.11	even	6		2160.4.a.t.1.1		1
45.2	even	12		1350.4.c.c.649.2		2
45.7	odd	12		1350.4.c.r.649.1		2
45.29	odd	6		1350.4.a.n.1.1		1
45.34	even	6		1350.4.a.bb.1.1		1
45.38	even	12		1350.4.c.c.649.1		2
45.43	odd	12		1350.4.c.r.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.a.1.1	✓	1	9.7	even	3
270.4.a.k.1.1	yes	1	9.2	odd	6
810.4.e.d.271.1		2	9.5	odd	6
810.4.e.d.541.1		2	3.2	odd	2
810.4.e.x.271.1		2	9.4	even	3	inner
810.4.e.x.541.1		2	1.1	even	1	trivial
1350.4.a.n.1.1		1	45.29	odd	6
1350.4.a.bb.1.1		1	45.34	even	6
1350.4.c.c.649.1		2	45.38	even	12
1350.4.c.c.649.2		2	45.2	even	12
1350.4.c.r.649.1		2	45.7	odd	12
1350.4.c.r.649.2		2	45.43	odd	12
2160.4.a.j.1.1		1	36.7	odd	6
2160.4.a.t.1.1		1	36.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	810.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.73205i) q^{2} +(-2.00000 - 3.46410i) q^{4} +(2.50000 + 4.33013i) q^{5} +(17.0000 - 29.4449i) q^{7} -8.00000 q^{8} +10.0000 q^{10} +(-24.0000 + 41.5692i) q^{11} +(35.0000 + 60.6218i) q^{13} +(-34.0000 - 58.8897i) q^{14} +(-8.00000 + 13.8564i) q^{16} +27.0000 q^{17} +119.000 q^{19} +(10.0000 - 17.3205i) q^{20} +(48.0000 + 83.1384i) q^{22} +(-25.5000 - 44.1673i) q^{23} +(-12.5000 + 21.6506i) q^{25} +140.000 q^{26} -136.000 q^{28} +(-15.0000 + 25.9808i) q^{29} +(66.5000 + 115.181i) q^{31} +(16.0000 + 27.7128i) q^{32} +(27.0000 - 46.7654i) q^{34} +170.000 q^{35} +218.000 q^{37} +(119.000 - 206.114i) q^{38} +(-20.0000 - 34.6410i) q^{40} +(78.0000 + 135.100i) q^{41} +(44.0000 - 76.2102i) q^{43} +192.000 q^{44} -102.000 q^{46} +(-258.000 + 446.869i) q^{47} +(-406.500 - 704.079i) q^{49} +(25.0000 + 43.3013i) q^{50} +(140.000 - 242.487i) q^{52} +639.000 q^{53} -240.000 q^{55} +(-136.000 + 235.559i) q^{56} +(30.0000 + 51.9615i) q^{58} +(-327.000 - 566.381i) q^{59} +(-230.500 + 399.238i) q^{61} +266.000 q^{62} +64.0000 q^{64} +(-175.000 + 303.109i) q^{65} +(-91.0000 - 157.617i) q^{67} +(-54.0000 - 93.5307i) q^{68} +(170.000 - 294.449i) q^{70} -900.000 q^{71} +704.000 q^{73} +(218.000 - 377.587i) q^{74} +(-238.000 - 412.228i) q^{76} +(816.000 + 1413.35i) q^{77} +(687.500 - 1190.78i) q^{79} -80.0000 q^{80} +312.000 q^{82} +(457.500 - 792.413i) q^{83} +(67.5000 + 116.913i) q^{85} +(-88.0000 - 152.420i) q^{86} +(192.000 - 332.554i) q^{88} +1116.00 q^{89} +2380.00 q^{91} +(-102.000 + 176.669i) q^{92} +(516.000 + 893.738i) q^{94} +(297.500 + 515.285i) q^{95} +(8.00000 - 13.8564i) q^{97} -1626.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 4 q^{4} + 5 q^{5} + 34 q^{7} - 16 q^{8} + 20 q^{10} - 48 q^{11} + 70 q^{13} - 68 q^{14} - 16 q^{16} + 54 q^{17} + 238 q^{19} + 20 q^{20} + 96 q^{22} - 51 q^{23} - 25 q^{25} + 280 q^{26}+ \cdots - 3252 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 - 4 * q^4 + 5 * q^5 + 34 * q^7 - 16 * q^8 + 20 * q^10 - 48 * q^11 + 70 * q^13 - 68 * q^14 - 16 * q^16 + 54 * q^17 + 238 * q^19 + 20 * q^20 + 96 * q^22 - 51 * q^23 - 25 * q^25 + 280 * q^26 - 272 * q^28 - 30 * q^29 + 133 * q^31 + 32 * q^32 + 54 * q^34 + 340 * q^35 + 436 * q^37 + 238 * q^38 - 40 * q^40 + 156 * q^41 + 88 * q^43 + 384 * q^44 - 204 * q^46 - 516 * q^47 - 813 * q^49 + 50 * q^50 + 280 * q^52 + 1278 * q^53 - 480 * q^55 - 272 * q^56 + 60 * q^58 - 654 * q^59 - 461 * q^61 + 532 * q^62 + 128 * q^64 - 350 * q^65 - 182 * q^67 - 108 * q^68 + 340 * q^70 - 1800 * q^71 + 1408 * q^73 + 436 * q^74 - 476 * q^76 + 1632 * q^77 + 1375 * q^79 - 160 * q^80 + 624 * q^82 + 915 * q^83 + 135 * q^85 - 176 * q^86 + 384 * q^88 + 2232 * q^89 + 4760 * q^91 - 204 * q^92 + 1032 * q^94 + 595 * q^95 + 16 * q^97 - 3252 * q^98

Introduction

L-functions

Modular forms

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Database

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