LMFDB - Embedded newform 81.4.c.c.55.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [81,4,Mod(28,81)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(81, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

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

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

chi := DirichletCharacter("81.28");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 81 = 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	81.c (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:	$4.77915471046$
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 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	81.55
Dual form			81.4.c.c.28.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 2.59808i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(7.50000 + 12.9904i) q^{5} +(12.5000 - 21.6506i) q^{7} +21.0000 q^{8} +O(q^{10})$	\(q+(1.50000 - 2.59808i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(7.50000 + 12.9904i) q^{5} +(12.5000 - 21.6506i) q^{7} +21.0000 q^{8} +45.0000 q^{10} +(-7.50000 + 12.9904i) q^{11} +(-10.0000 - 17.3205i) q^{13} +(-37.5000 - 64.9519i) q^{14} +(35.5000 - 61.4878i) q^{16} -72.0000 q^{17} +2.00000 q^{19} +(7.50000 - 12.9904i) q^{20} +(22.5000 + 38.9711i) q^{22} +(57.0000 + 98.7269i) q^{23} +(-50.0000 + 86.6025i) q^{25} -60.0000 q^{26} -25.0000 q^{28} +(15.0000 - 25.9808i) q^{29} +(-50.5000 - 87.4686i) q^{31} +(-22.5000 - 38.9711i) q^{32} +(-108.000 + 187.061i) q^{34} +375.000 q^{35} -430.000 q^{37} +(3.00000 - 5.19615i) q^{38} +(157.500 + 272.798i) q^{40} +(-15.0000 - 25.9808i) q^{41} +(-55.0000 + 95.2628i) q^{43} +15.0000 q^{44} +342.000 q^{46} +(-165.000 + 285.788i) q^{47} +(-141.000 - 244.219i) q^{49} +(150.000 + 259.808i) q^{50} +(-10.0000 + 17.3205i) q^{52} -621.000 q^{53} -225.000 q^{55} +(262.500 - 454.663i) q^{56} +(-45.0000 - 77.9423i) q^{58} +(-330.000 - 571.577i) q^{59} +(188.000 - 325.626i) q^{61} -303.000 q^{62} +433.000 q^{64} +(150.000 - 259.808i) q^{65} +(125.000 + 216.506i) q^{67} +(36.0000 + 62.3538i) q^{68} +(562.500 - 974.279i) q^{70} +360.000 q^{71} +785.000 q^{73} +(-645.000 + 1117.17i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(187.500 + 324.760i) q^{77} +(-244.000 + 422.620i) q^{79} +1065.00 q^{80} -90.0000 q^{82} +(244.500 - 423.486i) q^{83} +(-540.000 - 935.307i) q^{85} +(165.000 + 285.788i) q^{86} +(-157.500 + 272.798i) q^{88} +450.000 q^{89} -500.000 q^{91} +(57.0000 - 98.7269i) q^{92} +(495.000 + 857.365i) q^{94} +(15.0000 + 25.9808i) q^{95} +(552.500 - 956.958i) q^{97} -846.000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} - q^{4} + 15 q^{5} + 25 q^{7} + 42 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 - q^4 + 15 * q^5 + 25 * q^7 + 42 * q^8	$ 2 q + 3 q^{2} - q^{4} + 15 q^{5} + 25 q^{7} + 42 q^{8} + 90 q^{10} - 15 q^{11} - 20 q^{13} - 75 q^{14} + 71 q^{16} - 144 q^{17} + 4 q^{19} + 15 q^{20} + 45 q^{22} + 114 q^{23} - 100 q^{25} - 120 q^{26} - 50 q^{28} + 30 q^{29} - 101 q^{31} - 45 q^{32} - 216 q^{34} + 750 q^{35} - 860 q^{37} + 6 q^{38} + 315 q^{40} - 30 q^{41} - 110 q^{43} + 30 q^{44} + 684 q^{46} - 330 q^{47} - 282 q^{49} + 300 q^{50} - 20 q^{52} - 1242 q^{53} - 450 q^{55} + 525 q^{56} - 90 q^{58} - 660 q^{59} + 376 q^{61} - 606 q^{62} + 866 q^{64} + 300 q^{65} + 250 q^{67} + 72 q^{68} + 1125 q^{70} + 720 q^{71} + 1570 q^{73} - 1290 q^{74} - 2 q^{76} + 375 q^{77} - 488 q^{79} + 2130 q^{80} - 180 q^{82} + 489 q^{83} - 1080 q^{85} + 330 q^{86} - 315 q^{88} + 900 q^{89} - 1000 q^{91} + 114 q^{92} + 990 q^{94} + 30 q^{95} + 1105 q^{97} - 1692 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 - q^4 + 15 * q^5 + 25 * q^7 + 42 * q^8 + 90 * q^10 - 15 * q^11 - 20 * q^13 - 75 * q^14 + 71 * q^16 - 144 * q^17 + 4 * q^19 + 15 * q^20 + 45 * q^22 + 114 * q^23 - 100 * q^25 - 120 * q^26 - 50 * q^28 + 30 * q^29 - 101 * q^31 - 45 * q^32 - 216 * q^34 + 750 * q^35 - 860 * q^37 + 6 * q^38 + 315 * q^40 - 30 * q^41 - 110 * q^43 + 30 * q^44 + 684 * q^46 - 330 * q^47 - 282 * q^49 + 300 * q^50 - 20 * q^52 - 1242 * q^53 - 450 * q^55 + 525 * q^56 - 90 * q^58 - 660 * q^59 + 376 * q^61 - 606 * q^62 + 866 * q^64 + 300 * q^65 + 250 * q^67 + 72 * q^68 + 1125 * q^70 + 720 * q^71 + 1570 * q^73 - 1290 * q^74 - 2 * q^76 + 375 * q^77 - 488 * q^79 + 2130 * q^80 - 180 * q^82 + 489 * q^83 - 1080 * q^85 + 330 * q^86 - 315 * q^88 + 900 * q^89 - 1000 * q^91 + 114 * q^92 + 990 * q^94 + 30 * q^95 + 1105 * q^97 - 1692 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/81\mathbb{Z}\right)^\times$.

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

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 $

$2$	1.50000	−	2.59808i	0.530330	−	0.918559i	−0.469044	−	0.883175i	$-0.655401\pi$
$2$	1.50000	−	2.59808i	0.530330	−	0.918559i	0.999374	−	0.0353837i	$-0.0112653\pi$
$3$		0			0
$3$		0			0
$4$	−0.500000	−	0.866025i	−0.0625000	−	0.108253i
$5$	7.50000	+	12.9904i	0.670820	+	1.16190i	0.977672	+	0.210138i	$0.0673912\pi$
$5$	7.50000	+	12.9904i	0.670820	+	1.16190i	−0.306851	+	0.951757i	$0.599275\pi$
$6$		0			0
$7$	12.5000	−	21.6506i	0.674937	−	1.16902i	−0.301551	−	0.953450i	$-0.597504\pi$
$7$	12.5000	−	21.6506i	0.674937	−	1.16902i	0.976487	−	0.215574i	$-0.0691624\pi$
$8$	21.0000			0.928078
$9$		0			0
$10$	45.0000			1.42302
$11$	−7.50000	+	12.9904i	−0.205576	+	0.356068i	−0.950316	−	0.311287i	$-0.899240\pi$
$11$	−7.50000	+	12.9904i	−0.205576	+	0.356068i	0.744740	+	0.667355i	$0.232573\pi$
$12$		0			0
$13$	−10.0000	−	17.3205i	−0.213346	−	0.369527i	0.739413	−	0.673252i	$-0.235103\pi$
$13$	−10.0000	−	17.3205i	−0.213346	−	0.369527i	−0.952760	+	0.303725i	$0.901770\pi$
$14$	−37.5000	−	64.9519i	−0.715878	−	1.23994i
$15$		0			0
$16$	35.5000	−	61.4878i	0.554688	−	0.960747i
$17$	−72.0000			−1.02721			−0.513605	−	0.858027i	$-0.671690\pi$
$17$	−72.0000			−1.02721			−0.513605	+	0.858027i	$0.671690\pi$
$18$		0			0
$19$	2.00000			0.0241490			0.0120745	−	0.999927i	$-0.496156\pi$
$19$	2.00000			0.0241490			0.0120745	+	0.999927i	$0.496156\pi$
$20$	7.50000	−	12.9904i	0.0838525	−	0.145237i
$21$		0			0
$22$	22.5000	+	38.9711i	0.218046	+	0.377667i
$23$	57.0000	+	98.7269i	0.516753	+	0.895043i	0.999811	+	0.0194541i	$0.00619282\pi$
$23$	57.0000	+	98.7269i	0.516753	+	0.895043i	−0.483058	+	0.875589i	$0.660474\pi$
$24$		0			0
$25$	−50.0000	+	86.6025i	−0.400000	+	0.692820i
$26$	−60.0000			−0.452576
$27$		0			0
$28$	−25.0000			−0.168734
$29$	15.0000	−	25.9808i	0.0960493	−	0.166362i	−0.813997	−	0.580869i	$-0.802713\pi$
$29$	15.0000	−	25.9808i	0.0960493	−	0.166362i	0.910046	+	0.414507i	$0.136046\pi$
$30$		0			0
$31$	−50.5000	−	87.4686i	−0.292583	−	0.506768i	0.681837	−	0.731504i	$-0.261181\pi$
$31$	−50.5000	−	87.4686i	−0.292583	−	0.506768i	−0.974420	+	0.224736i	$0.927848\pi$
$32$	−22.5000	−	38.9711i	−0.124296	−	0.215287i
$33$		0			0
$34$	−108.000	+	187.061i	−0.544760	+	0.943552i
$35$	375.000			1.81104
$36$		0			0
$37$	−430.000			−1.91058			−0.955291	−	0.295666i	$-0.904458\pi$
$37$	−430.000			−1.91058			−0.955291	+	0.295666i	$0.904458\pi$
$38$	3.00000	−	5.19615i	0.0128070	−	0.0221823i
$39$		0			0
$40$	157.500	+	272.798i	0.622573	+	1.07833i
$41$	−15.0000	−	25.9808i	−0.0571367	−	0.0989637i	0.836042	−	0.548665i	$-0.184864\pi$
$41$	−15.0000	−	25.9808i	−0.0571367	−	0.0989637i	−0.893179	+	0.449701i	$0.851530\pi$
$42$		0			0
$43$	−55.0000	+	95.2628i	−0.195056	+	0.337847i	−0.946919	−	0.321472i	$-0.895822\pi$
$43$	−55.0000	+	95.2628i	−0.195056	+	0.337847i	0.751863	+	0.659320i	$0.229156\pi$
$44$	15.0000			0.0513940
$45$		0			0
$46$	342.000			1.09620
$47$	−165.000	+	285.788i	−0.512079	+	0.886947i	0.487823	+	0.872943i	$0.337791\pi$
$47$	−165.000	+	285.788i	−0.512079	+	0.886947i	−0.999902	+	0.0140045i	$0.995542\pi$
$48$		0			0
$49$	−141.000	−	244.219i	−0.411079	−	0.712009i
$50$	150.000	+	259.808i	0.424264	+	0.734847i
$51$		0			0
$52$	−10.0000	+	17.3205i	−0.0266683	+	0.0461908i
$53$	−621.000			−1.60945			−0.804726	−	0.593647i	$-0.797688\pi$
$53$	−621.000			−1.60945			−0.804726	+	0.593647i	$0.797688\pi$
$54$		0			0
$55$	−225.000			−0.551618
$56$	262.500	−	454.663i	0.626394	−	1.08495i
$57$		0			0
$58$	−45.0000	−	77.9423i	−0.101876	−	0.176454i
$59$	−330.000	−	571.577i	−0.728175	−	1.26124i	−0.957654	−	0.287923i	$-0.907035\pi$
$59$	−330.000	−	571.577i	−0.728175	−	1.26124i	0.229478	−	0.973314i	$-0.426298\pi$
$60$		0			0
$61$	188.000	−	325.626i	0.394605	−	0.683477i	−0.598445	−	0.801164i	$-0.704215\pi$
$61$	188.000	−	325.626i	0.394605	−	0.683477i	0.993051	+	0.117687i	$0.0375479\pi$
$62$	−303.000			−0.620662
$63$		0			0
$64$	433.000			0.845703
$65$	150.000	−	259.808i	0.286234	−	0.495772i
$66$		0			0
$67$	125.000	+	216.506i	0.227928	+	0.394783i	0.957194	−	0.289447i	$-0.0934716\pi$
$67$	125.000	+	216.506i	0.227928	+	0.394783i	−0.729266	+	0.684230i	$0.760138\pi$
$68$	36.0000	+	62.3538i	0.0642006	+	0.111199i
$69$		0			0
$70$	562.500	−	974.279i	0.960452	−	1.66355i
$71$	360.000			0.601748			0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000			0.601748			0.300874	+	0.953664i	$0.402722\pi$
$72$		0			0
$73$	785.000			1.25859			0.629297	−	0.777165i	$-0.283343\pi$
$73$	785.000			1.25859			0.629297	+	0.777165i	$0.283343\pi$
$74$	−645.000	+	1117.17i	−1.01324	+	1.75498i
$75$		0			0
$76$	−1.00000	−	1.73205i	−0.00150931	−	0.00261421i
$77$	187.500	+	324.760i	0.277501	+	0.480647i
$78$		0			0
$79$	−244.000	+	422.620i	−0.347496	+	0.601880i	−0.985804	−	0.167901i	$-0.946301\pi$
$79$	−244.000	+	422.620i	−0.347496	+	0.601880i	0.638308	+	0.769781i	$0.279634\pi$
$80$	1065.00			1.48838
$81$		0			0
$82$	−90.0000			−0.121205
$83$	244.500	−	423.486i	0.323342	−	0.560044i	−0.657834	−	0.753163i	$-0.728527\pi$
$83$	244.500	−	423.486i	0.323342	−	0.560044i	0.981175	+	0.193119i	$0.0618603\pi$
$84$		0			0
$85$	−540.000	−	935.307i	−0.689073	−	1.19351i
$86$	165.000	+	285.788i	0.206888	+	0.358341i
$87$		0			0
$88$	−157.500	+	272.798i	−0.190790	+	0.330459i
$89$	450.000			0.535954			0.267977	−	0.963425i	$-0.413645\pi$
$89$	450.000			0.535954			0.267977	+	0.963425i	$0.413645\pi$
$90$		0			0
$91$	−500.000			−0.575981
$92$	57.0000	−	98.7269i	0.0645941	−	0.111880i
$93$		0			0
$94$	495.000	+	857.365i	0.543142	+	0.940750i
$95$	15.0000	+	25.9808i	0.0161997	+	0.0280586i
$96$		0			0
$97$	552.500	−	956.958i	0.578329	−	1.00169i	−0.417343	−	0.908749i	$-0.637038\pi$
$97$	552.500	−	956.958i	0.578329	−	1.00169i	0.995671	−	0.0929453i	$-0.0296282\pi$
$98$	−846.000			−0.872030
$99$		0			0
$100$	100.000			0.100000
$101$	712.500	−	1234.09i	0.701945	−	1.21580i	−0.265838	−	0.964018i	$-0.585649\pi$
$101$	712.500	−	1234.09i	0.701945	−	1.21580i	0.967783	−	0.251786i	$-0.0810179\pi$
$102$		0			0
$103$	530.000	+	917.987i	0.507014	+	0.878174i	0.999967	+	0.00811820i	$0.00258413\pi$
$103$	530.000	+	917.987i	0.507014	+	0.878174i	−0.492953	+	0.870056i	$0.664083\pi$
$104$	−210.000	−	363.731i	−0.198002	−	0.342949i
$105$		0			0
$106$	−931.500	+	1613.41i	−0.853540	+	1.47838i
$107$	−1485.00			−1.34169			−0.670843	−	0.741600i	$-0.734067\pi$
$107$	−1485.00			−1.34169			−0.670843	+	0.741600i	$0.734067\pi$
$108$		0			0
$109$	−862.000			−0.757474			−0.378737	−	0.925504i	$-0.623641\pi$
$109$	−862.000			−0.757474			−0.378737	+	0.925504i	$0.623641\pi$
$110$	−337.500	+	584.567i	−0.292540	+	0.506694i
$111$		0			0
$112$	−887.500	−	1537.20i	−0.748758	−	1.29689i
$113$	345.000	+	597.558i	0.287211	+	0.497464i	0.973143	−	0.230201i	$-0.0739385\pi$
$113$	345.000	+	597.558i	0.287211	+	0.497464i	−0.685932	+	0.727666i	$0.740605\pi$
$114$		0			0
$115$	−855.000	+	1480.90i	−0.693297	+	1.20083i
$116$	−30.0000			−0.0240123
$117$		0			0
$118$	−1980.00			−1.54469
$119$	−900.000	+	1558.85i	−0.693301	+	1.20083i
$120$		0			0
$121$	553.000	+	957.824i	0.415477	+	0.719627i
$122$	−564.000	−	976.877i	−0.418542	−	0.724937i
$123$		0			0
$124$	−50.5000	+	87.4686i	−0.0365729	+	0.0633460i
$125$	375.000			0.268328
$126$		0			0
$127$	1865.00			1.30309			0.651543	−	0.758611i	$-0.274122\pi$
$127$	1865.00			1.30309			0.651543	+	0.758611i	$0.274122\pi$
$128$	829.500	−	1436.74i	0.572798	−	0.992115i
$129$		0			0
$130$	−450.000	−	779.423i	−0.303597	−	0.525845i
$131$	−577.500	−	1000.26i	−0.385163	−	0.667123i	0.606628	−	0.794986i	$-0.292522\pi$
$131$	−577.500	−	1000.26i	−0.385163	−	0.667123i	−0.991792	+	0.127863i	$0.959188\pi$
$132$		0			0
$133$	25.0000	−	43.3013i	0.0162991	−	0.0282308i
$134$	750.000			0.483508
$135$		0			0
$136$	−1512.00			−0.953330
$137$	−1389.00	+	2405.82i	−0.866206	+	1.50031i	−0.000362133		1.00000i	$0.500115\pi$
$137$	−1389.00	+	2405.82i	−0.866206	+	1.50031i	−0.865844	+	0.500314i	$0.833218\pi$
$138$		0			0
$139$	962.000	+	1666.23i	0.587020	+	1.01675i	0.994620	+	0.103588i	$0.0330323\pi$
$139$	962.000	+	1666.23i	0.587020	+	1.01675i	−0.407600	+	0.913160i	$0.633634\pi$
$140$	−187.500	−	324.760i	−0.113190	−	0.196051i
$141$		0			0
$142$	540.000	−	935.307i	0.319125	−	0.552741i
$143$	300.000			0.175435
$144$		0			0
$145$	450.000			0.257727
$146$	1177.50	−	2039.49i	0.667470	−	1.15609i
$147$		0			0
$148$	215.000	+	372.391i	0.119411	+	0.206827i
$149$	727.500	+	1260.07i	0.399994	+	0.692810i	0.993725	−	0.111854i	$-0.0356788\pi$
$149$	727.500	+	1260.07i	0.399994	+	0.692810i	−0.593731	+	0.804664i	$0.702345\pi$
$150$		0			0
$151$	363.500	−	629.600i	0.195902	−	0.339312i	−0.751294	−	0.659968i	$-0.770570\pi$
$151$	363.500	−	629.600i	0.195902	−	0.339312i	0.947196	+	0.320656i	$0.103903\pi$
$152$	42.0000			0.0224122
$153$		0			0
$154$	1125.00			0.588669
$155$	757.500	−	1312.03i	0.392541	−	0.679901i
$156$		0			0
$157$	−1630.00	−	2823.24i	−0.828587	−	1.43515i	−0.899147	−	0.437647i	$-0.855812\pi$
$157$	−1630.00	−	2823.24i	−0.828587	−	1.43515i	0.0705597	−	0.997508i	$-0.477521\pi$
$158$	732.000	+	1267.86i	0.368575	+	0.638390i
$159$		0			0
$160$	337.500	−	584.567i	0.166761	−	0.288838i
$161$	2850.00			1.39510
$162$		0			0
$163$	2540.00			1.22054			0.610270	−	0.792193i	$-0.291061\pi$
$163$	2540.00			1.22054			0.610270	+	0.792193i	$0.291061\pi$
$164$	−15.0000	+	25.9808i	−0.00714209	+	0.0123705i
$165$		0			0
$166$	−733.500	−	1270.46i	−0.342956	−	0.594017i
$167$	1749.00	+	3029.36i	0.810429	+	1.40370i	0.912564	+	0.408934i	$0.134099\pi$
$167$	1749.00	+	3029.36i	0.810429	+	1.40370i	−0.102135	+	0.994771i	$0.532567\pi$
$168$		0			0
$169$	898.500	−	1556.25i	0.408967	−	0.708351i
$170$	−3240.00			−1.46175
$171$		0			0
$172$	110.000			0.0487641
$173$	−574.500	+	995.063i	−0.252476	+	0.437302i	−0.964207	−	0.265150i	$-0.914578\pi$
$173$	−574.500	+	995.063i	−0.252476	+	0.437302i	0.711731	+	0.702453i	$0.247912\pi$
$174$		0			0
$175$	1250.00	+	2165.06i	0.539949	+	0.935220i
$176$	532.500	+	922.317i	0.228061	+	0.395013i
$177$		0			0
$178$	675.000	−	1169.13i	0.284233	−	0.492305i
$179$	−315.000			−0.131532			−0.0657659	−	0.997835i	$-0.520949\pi$
$179$	−315.000			−0.131532			−0.0657659	+	0.997835i	$0.520949\pi$
$180$		0			0
$181$	1136.00			0.466509			0.233255	−	0.972416i	$-0.425062\pi$
$181$	1136.00			0.466509			0.233255	+	0.972416i	$0.425062\pi$
$182$	−750.000	+	1299.04i	−0.305460	+	0.529072i
$183$		0			0
$184$	1197.00	+	2073.26i	0.479587	+	0.830669i
$185$	−3225.00	−	5585.86i	−1.28166	−	2.21990i
$186$		0			0
$187$	540.000	−	935.307i	0.211170	−	0.365756i
$188$	330.000			0.128020
$189$		0			0
$190$	90.0000			0.0343647
$191$	1230.00	−	2130.42i	0.465967	−	0.807078i	−0.533278	−	0.845940i	$-0.679040\pi$
$191$	1230.00	−	2130.42i	0.465967	−	0.807078i	0.999245	+	0.0388619i	$0.0123732\pi$
$192$		0			0
$193$	−482.500	−	835.715i	−0.179954	−	0.311689i	0.761911	−	0.647682i	$-0.224262\pi$
$193$	−482.500	−	835.715i	−0.179954	−	0.311689i	−0.941865	+	0.335993i	$0.890928\pi$
$194$	−1657.50	−	2870.87i	−0.613410	−	1.06246i
$195$		0			0
$196$	−141.000	+	244.219i	−0.0513848	+	0.0890012i
$197$	−2493.00			−0.901619			−0.450809	−	0.892620i	$-0.648865\pi$
$197$	−2493.00			−0.901619			−0.450809	+	0.892620i	$0.648865\pi$
$198$		0			0
$199$	−511.000			−0.182029			−0.0910146	−	0.995850i	$-0.529011\pi$
$199$	−511.000			−0.182029			−0.0910146	+	0.995850i	$0.529011\pi$
$200$	−1050.00	+	1818.65i	−0.371231	+	0.642991i
$201$		0			0
$202$	−2137.50	−	3702.26i	−0.744525	−	1.28955i
$203$	−375.000	−	649.519i	−0.129654	−	0.224568i
$204$		0			0
$205$	225.000	−	389.711i	0.0766570	−	0.132774i
$206$	3180.00			1.07554
$207$		0			0
$208$	−1420.00			−0.473362
$209$	−15.0000	+	25.9808i	−0.00496446	+	0.00859869i
$210$		0			0
$211$	1043.00	+	1806.53i	0.340299	+	0.589415i	0.984488	−	0.175451i	$-0.0561384\pi$
$211$	1043.00	+	1806.53i	0.340299	+	0.589415i	−0.644189	+	0.764866i	$0.722805\pi$
$212$	310.500	+	537.802i	0.100591	+	0.174228i
$213$		0			0
$214$	−2227.50	+	3858.14i	−0.711536	+	1.23242i
$215$	−1650.00			−0.523391
$216$		0			0
$217$	−2525.00			−0.789899
$218$	−1293.00	+	2239.54i	−0.401711	+	0.695784i
$219$		0			0
$220$	112.500	+	194.856i	0.0344761	+	0.0597144i
$221$	720.000	+	1247.08i	0.219151	+	0.379581i
$222$		0			0
$223$	−2620.00	+	4537.97i	−0.786763	+	1.36271i	0.141177	+	0.989984i	$0.454911\pi$
$223$	−2620.00	+	4537.97i	−0.786763	+	1.36271i	−0.927940	+	0.372729i	$0.878422\pi$
$224$	−1125.00			−0.335568
$225$		0			0
$226$	2070.00			0.609267
$227$	1194.00	−	2068.07i	0.349113	−	0.604681i	−0.636979	−	0.770881i	$-0.719816\pi$
$227$	1194.00	−	2068.07i	0.349113	−	0.604681i	0.986092	+	0.166200i	$0.0531497\pi$
$228$		0			0
$229$	−91.0000	−	157.617i	−0.0262596	−	0.0454830i	0.852597	−	0.522569i	$-0.175026\pi$
$229$	−91.0000	−	157.617i	−0.0262596	−	0.0454830i	−0.878857	+	0.477086i	$0.841693\pi$
$230$	2565.00	+	4442.71i	0.735353	+	1.27367i
$231$		0			0
$232$	315.000	−	545.596i	0.0891412	−	0.154397i
$233$	−450.000			−0.126526			−0.0632628	−	0.997997i	$-0.520151\pi$
$233$	−450.000			−0.126526			−0.0632628	+	0.997997i	$0.520151\pi$
$234$		0			0
$235$	−4950.00			−1.37405
$236$	−330.000	+	571.577i	−0.0910219	+	0.157655i
$237$		0			0
$238$	2700.00	+	4676.54i	0.735357	+	1.27368i
$239$	2595.00	+	4494.67i	0.702329	+	1.21647i	0.967647	+	0.252308i	$0.0811897\pi$
$239$	2595.00	+	4494.67i	0.702329	+	1.21647i	−0.265318	+	0.964161i	$0.585477\pi$
$240$		0			0
$241$	1133.00	−	1962.41i	0.302834	−	0.524524i	−0.673943	−	0.738783i	$-0.735401\pi$
$241$	1133.00	−	1962.41i	0.302834	−	0.524524i	0.976777	+	0.214260i	$0.0687340\pi$
$242$	3318.00			0.881360
$243$		0			0
$244$	−376.000			−0.0986514
$245$	2115.00	−	3663.29i	0.551520	−	0.955261i
$246$		0			0
$247$	−20.0000	−	34.6410i	−0.00515210	−	0.00892370i
$248$	−1060.50	−	1836.84i	−0.271540	−	0.470320i
$249$		0			0
$250$	562.500	−	974.279i	0.142302	−	0.246475i
$251$	2880.00			0.724239			0.362119	−	0.932132i	$-0.382053\pi$
$251$	2880.00			0.724239			0.362119	+	0.932132i	$0.382053\pi$
$252$		0			0
$253$	−1710.00			−0.424928
$254$	2797.50	−	4845.41i	0.691066	−	1.19696i
$255$		0			0
$256$	−756.500	−	1310.30i	−0.184692	−	0.319897i
$257$	−2094.00	−	3626.91i	−0.508250	−	0.880314i	−0.999954	−	0.00955209i	$-0.996959\pi$
$257$	−2094.00	−	3626.91i	−0.508250	−	0.880314i	0.491705	−	0.870762i	$-0.336374\pi$
$258$		0			0
$259$	−5375.00	+	9309.77i	−1.28952	+	2.23352i
$260$	−300.000			−0.0715585
$261$		0			0
$262$	−3465.00			−0.817055
$263$	−1515.00	+	2624.06i	−0.355205	+	0.615233i	−0.987153	−	0.159778i	$-0.948922\pi$
$263$	−1515.00	+	2624.06i	−0.355205	+	0.615233i	0.631948	+	0.775011i	$0.282256\pi$
$264$		0			0
$265$	−4657.50	−	8067.03i	−1.07965	−	1.87001i
$266$	−75.0000	−	129.904i	−0.0172878	−	0.0299433i
$267$		0			0
$268$	125.000	−	216.506i	0.0284910	−	0.0493479i
$269$	−3510.00			−0.795571			−0.397785	−	0.917479i	$-0.630221\pi$
$269$	−3510.00			−0.795571			−0.397785	+	0.917479i	$0.630221\pi$
$270$		0			0
$271$	2999.00			0.672237			0.336119	−	0.941820i	$-0.390886\pi$
$271$	2999.00			0.672237			0.336119	+	0.941820i	$0.390886\pi$
$272$	−2556.00	+	4427.12i	−0.569780	+	0.986889i
$273$		0			0
$274$	4167.00	+	7217.46i	0.918751	+	1.59132i
$275$	−750.000	−	1299.04i	−0.164461	−	0.284854i
$276$		0			0
$277$	3860.00	−	6685.72i	0.837274	−	1.45020i	−0.0548913	−	0.998492i	$-0.517481\pi$
$277$	3860.00	−	6685.72i	0.837274	−	1.45020i	0.892165	−	0.451709i	$-0.149185\pi$
$278$	5772.00			1.24526
$279$		0			0
$280$	7875.00			1.68079
$281$	−3720.00	+	6443.23i	−0.789739	+	1.36787i	0.136388	+	0.990655i	$0.456451\pi$
$281$	−3720.00	+	6443.23i	−0.789739	+	1.36787i	−0.926127	+	0.377212i	$0.876883\pi$
$282$		0			0
$283$	−415.000	−	718.801i	−0.0871703	−	0.150983i	0.819144	−	0.573588i	$-0.194449\pi$
$283$	−415.000	−	718.801i	−0.0871703	−	0.150983i	−0.906314	+	0.422605i	$0.861116\pi$
$284$	−180.000	−	311.769i	−0.0376093	−	0.0651412i
$285$		0			0
$286$	450.000	−	779.423i	0.0930387	−	0.161148i
$287$	−750.000			−0.154255
$288$		0			0
$289$	271.000			0.0551598
$290$	675.000	−	1169.13i	0.136681	−	0.236738i
$291$		0			0
$292$	−392.500	−	679.830i	−0.0786621	−	0.136247i
$293$	273.000	+	472.850i	0.0544329	+	0.0942805i	0.891958	−	0.452118i	$-0.149332\pi$
$293$	273.000	+	472.850i	0.0544329	+	0.0942805i	−0.837525	+	0.546399i	$0.815998\pi$
$294$		0			0
$295$	4950.00	−	8573.65i	0.976950	−	1.69213i
$296$	−9030.00			−1.77317
$297$		0			0
$298$	4365.00			0.848516
$299$	1140.00	−	1974.54i	0.220495	−	0.381908i
$300$		0			0
$301$	1375.00	+	2381.57i	0.263301	+	0.456051i
$302$	−1090.50	−	1888.80i	−0.207786	−	0.359895i
$303$		0			0
$304$	71.0000	−	122.976i	0.0133952	−	0.0232011i
$305$	5640.00			1.05884
$306$		0			0
$307$	−5560.00			−1.03364			−0.516818	−	0.856096i	$-0.672883\pi$
$307$	−5560.00			−1.03364			−0.516818	+	0.856096i	$0.672883\pi$
$308$	187.500	−	324.760i	0.0346877	−	0.0600808i
$309$		0			0
$310$	−2272.50	−	3936.09i	−0.416353	−	0.721144i
$311$	−4335.00	−	7508.44i	−0.790403	−	1.36902i	−0.925718	−	0.378215i	$-0.876538\pi$
$311$	−4335.00	−	7508.44i	−0.790403	−	1.36902i	0.135315	−	0.990803i	$-0.456795\pi$
$312$		0			0
$313$	−2282.50	+	3953.41i	−0.412187	+	0.713929i	−0.995129	−	0.0985852i	$-0.968568\pi$
$313$	−2282.50	+	3953.41i	−0.412187	+	0.713929i	0.582942	+	0.812514i	$0.301902\pi$
$314$	−9780.00			−1.75770
$315$		0			0
$316$	488.000			0.0868739
$317$	2116.50	−	3665.89i	0.374998	−	0.649516i	−0.615328	−	0.788271i	$-0.710977\pi$
$317$	2116.50	−	3665.89i	0.374998	−	0.649516i	0.990327	+	0.138755i	$0.0443099\pi$
$318$		0			0
$319$	225.000	+	389.711i	0.0394909	+	0.0684002i
$320$	3247.50	+	5624.83i	0.567315	+	0.982618i
$321$		0			0
$322$	4275.00	−	7404.52i	0.739865	−	1.28148i
$323$	−144.000			−0.0248061
$324$		0			0
$325$	2000.00			0.341354
$326$	3810.00	−	6599.11i	0.647290	−	1.12114i
$327$		0			0
$328$	−315.000	−	545.596i	−0.0530273	−	0.0918460i
$329$	4125.00	+	7144.71i	0.691242	+	1.19727i
$330$		0			0
$331$	−271.000	+	469.386i	−0.0450015	+	0.0779449i	−0.887649	−	0.460521i	$-0.847663\pi$
$331$	−271.000	+	469.386i	−0.0450015	+	0.0779449i	0.842647	+	0.538466i	$0.180996\pi$
$332$	−489.000			−0.0808354
$333$		0			0
$334$	10494.0			1.71918
$335$	−1875.00	+	3247.60i	−0.305798	+	0.529657i
$336$		0			0
$337$	−2845.00	−	4927.68i	−0.459872	−	0.796522i	0.539081	−	0.842254i	$-0.318772\pi$
$337$	−2845.00	−	4927.68i	−0.459872	−	0.796522i	−0.998954	+	0.0457312i	$0.985438\pi$
$338$	−2695.50	−	4668.74i	−0.433775	−	0.751320i
$339$		0			0
$340$	−540.000	+	935.307i	−0.0861342	+	0.149189i
$341$	1515.00			0.240592
$342$		0			0
$343$	1525.00			0.240065
$344$	−1155.00	+	2000.52i	−0.181027	+	0.313549i
$345$		0			0
$346$	1723.50	+	2985.19i	0.267792	+	0.463829i
$347$	2527.50	+	4377.76i	0.391018	+	0.677263i	0.992584	−	0.121560i	$-0.0387896\pi$
$347$	2527.50	+	4377.76i	0.391018	+	0.677263i	−0.601566	+	0.798823i	$0.705456\pi$
$348$		0			0
$349$	−811.000	+	1404.69i	−0.124389	+	0.215448i	−0.921494	−	0.388393i	$-0.873030\pi$
$349$	−811.000	+	1404.69i	−0.124389	+	0.215448i	0.797105	+	0.603841i	$0.206364\pi$
$350$	7500.00			1.14541
$351$		0			0
$352$	675.000			0.102209
$353$	15.0000	−	25.9808i	0.00226167	−	0.00391733i	−0.864892	−	0.501957i	$-0.832613\pi$
$353$	15.0000	−	25.9808i	0.00226167	−	0.00391733i	0.867154	+	0.498040i	$0.165947\pi$
$354$		0			0
$355$	2700.00	+	4676.54i	0.403665	+	0.699169i
$356$	−225.000	−	389.711i	−0.0334971	−	0.0580187i
$357$		0			0
$358$	−472.500	+	818.394i	−0.0697553	+	0.120820i
$359$	7470.00			1.09819			0.549097	−	0.835759i	$-0.314972\pi$
$359$	7470.00			1.09819			0.549097	+	0.835759i	$0.314972\pi$
$360$		0			0
$361$	−6855.00			−0.999417
$362$	1704.00	−	2951.41i	0.247404	−	0.428516i
$363$		0			0
$364$	250.000	+	433.013i	0.0359988	+	0.0623517i
$365$	5887.50	+	10197.4i	0.844290	+	1.46235i
$366$		0			0
$367$	687.500	−	1190.78i	0.0977853	−	0.169369i	−0.812982	−	0.582288i	$-0.802157\pi$
$367$	687.500	−	1190.78i	0.0977853	−	0.169369i	0.910768	+	0.412919i	$0.135491\pi$
$368$	8094.00			1.14655
$369$		0			0
$370$	−19350.0			−2.71881
$371$	−7762.50	+	13445.0i	−1.08628	+	1.88149i
$372$		0			0
$373$	2420.00	+	4191.56i	0.335933	+	0.581852i	0.983664	−	0.180017i	$-0.0576152\pi$
$373$	2420.00	+	4191.56i	0.335933	+	0.581852i	−0.647731	+	0.761869i	$0.724282\pi$
$374$	−1620.00	−	2805.92i	−0.223979	−	0.387943i
$375$		0			0
$376$	−3465.00	+	6001.56i	−0.475249	+	0.823156i
$377$	−600.000			−0.0819670
$378$		0			0
$379$	1892.00			0.256426			0.128213	−	0.991747i	$-0.459076\pi$
$379$	1892.00			0.256426			0.128213	+	0.991747i	$0.459076\pi$
$380$	15.0000	−	25.9808i	0.00202496	−	0.00350733i
$381$		0			0
$382$	−3690.00	−	6391.27i	−0.494233	−	0.856036i
$383$	−5352.00	−	9269.94i	−0.714032	−	1.23674i	−0.963332	−	0.268314i	$-0.913534\pi$
$383$	−5352.00	−	9269.94i	−0.714032	−	1.23674i	0.249299	−	0.968427i	$-0.419800\pi$
$384$		0			0
$385$	−2812.50	+	4871.39i	−0.372307	+	0.644855i
$386$	−2895.00			−0.381740
$387$		0			0
$388$	−1105.00			−0.144582
$389$	3907.50	−	6767.99i	0.509301	−	0.882135i	−0.490641	−	0.871362i	$-0.663237\pi$
$389$	3907.50	−	6767.99i	0.509301	−	0.882135i	0.999942	−	0.0107735i	$-0.00342936\pi$
$390$		0			0
$391$	−4104.00	−	7108.34i	−0.530814	−	0.919396i
$392$	−2961.00	−	5128.60i	−0.381513	−	0.660800i
$393$		0			0
$394$	−3739.50	+	6477.00i	−0.478156	+	0.828190i
$395$	−7320.00			−0.932428
$396$		0			0
$397$	4700.00			0.594172			0.297086	−	0.954851i	$-0.403985\pi$
$397$	4700.00			0.594172			0.297086	+	0.954851i	$0.403985\pi$
$398$	−766.500	+	1327.62i	−0.0965356	+	0.167205i
$399$		0			0
$400$	3550.00	+	6148.78i	0.443750	+	0.768598i
$401$	−1050.00	−	1818.65i	−0.130759	−	0.226482i	0.793210	−	0.608948i	$-0.208408\pi$
$401$	−1050.00	−	1818.65i	−0.130759	−	0.226482i	−0.923970	+	0.382466i	$0.875075\pi$
$402$		0			0
$403$	−1010.00	+	1749.37i	−0.124843	+	0.216234i
$404$	−1425.00			−0.175486
$405$		0			0
$406$	−2250.00			−0.275038
$407$	3225.00	−	5585.86i	0.392770	−	0.680297i
$408$		0			0
$409$	5376.50	+	9312.37i	0.650002	+	1.12584i	0.983122	+	0.182952i	$0.0585653\pi$
$409$	5376.50	+	9312.37i	0.650002	+	1.12584i	−0.333120	+	0.942884i	$0.608101\pi$
$410$	−675.000	−	1169.13i	−0.0813070	−	0.140828i
$411$		0			0
$412$	530.000	−	917.987i	0.0633768	−	0.109772i
$413$	−16500.0			−1.96589
$414$		0			0
$415$	7335.00			0.867617
$416$	−450.000	+	779.423i	−0.0530362	+	0.0918614i
$417$		0			0
$418$	45.0000	+	77.9423i	0.00526560	+	0.00912029i
$419$	1470.00	+	2546.11i	0.171394	+	0.296864i	0.938908	−	0.344169i	$-0.111839\pi$
$419$	1470.00	+	2546.11i	0.171394	+	0.296864i	−0.767513	+	0.641033i	$0.778506\pi$
$420$		0			0
$421$	−4348.00	+	7530.96i	−0.503346	+	0.871820i	0.496647	+	0.867953i	$0.334564\pi$
$421$	−4348.00	+	7530.96i	−0.503346	+	0.871820i	−0.999993	+	0.00386764i	$0.998769\pi$
$422$	6258.00			0.721883
$423$		0			0
$424$	−13041.0			−1.49370
$425$	3600.00	−	6235.38i	0.410884	−	0.711672i
$426$		0			0
$427$	−4700.00	−	8140.64i	−0.532667	−	0.922607i
$428$	742.500	+	1286.05i	0.0838554	+	0.145242i
$429$		0			0
$430$	−2475.00	+	4286.83i	−0.277570	+	0.480765i
$431$	−8370.00			−0.935426			−0.467713	−	0.883880i	$-0.654922\pi$
$431$	−8370.00			−0.935426			−0.467713	+	0.883880i	$0.654922\pi$
$432$		0			0
$433$	−5155.00			−0.572133			−0.286066	−	0.958210i	$-0.592348\pi$
$433$	−5155.00			−0.572133			−0.286066	+	0.958210i	$0.592348\pi$
$434$	−3787.50	+	6560.14i	−0.418907	+	0.725569i
$435$		0			0
$436$	431.000	+	746.514i	0.0473421	+	0.0819989i
$437$	114.000	+	197.454i	0.0124791	+	0.0216144i
$438$		0			0
$439$	5493.50	−	9515.02i	0.597245	−	1.03446i	−0.395981	−	0.918259i	$-0.629595\pi$
$439$	5493.50	−	9515.02i	0.597245	−	1.03446i	0.993226	−	0.116200i	$-0.0370712\pi$
$440$	−4725.00			−0.511944
$441$		0			0
$442$	4320.00			0.464890
$443$	978.000	−	1693.95i	0.104890	−	0.181674i	−0.808803	−	0.588079i	$-0.799884\pi$
$443$	978.000	−	1693.95i	0.104890	−	0.181674i	0.913693	+	0.406405i	$0.133218\pi$
$444$		0			0
$445$	3375.00	+	5845.67i	0.359529	+	0.622722i
$446$	7860.00	+	13613.9i	0.834488	+	1.44538i
$447$		0			0
$448$	5412.50	−	9374.72i	0.570796	−	0.988648i
$449$	8730.00			0.917582			0.458791	−	0.888544i	$-0.348283\pi$
$449$	8730.00			0.917582			0.458791	+	0.888544i	$0.348283\pi$
$450$		0			0
$451$	450.000			0.0469838
$452$	345.000	−	597.558i	0.0359014	−	0.0621831i
$453$		0			0
$454$	−3582.00	−	6204.21i	−0.370290	−	0.641361i
$455$	−3750.00	−	6495.19i	−0.386380	−	0.669229i
$456$		0			0
$457$	4332.50	−	7504.11i	0.443470	−	0.768113i	−0.554474	−	0.832201i	$-0.687081\pi$
$457$	4332.50	−	7504.11i	0.443470	−	0.768113i	0.997944	+	0.0640883i	$0.0204139\pi$
$458$	−546.000			−0.0557050
$459$		0			0
$460$	1710.00			0.173324
$461$	−4912.50	+	8508.70i	−0.496308	+	0.859630i	−0.999991	−	0.00425805i	$-0.998645\pi$
$461$	−4912.50	+	8508.70i	−0.496308	+	0.859630i	0.503683	+	0.863889i	$0.331978\pi$
$462$		0			0
$463$	2622.50	+	4542.30i	0.263235	+	0.455937i	0.967100	−	0.254397i	$-0.0818771\pi$
$463$	2622.50	+	4542.30i	0.263235	+	0.455937i	−0.703865	+	0.710334i	$0.748544\pi$
$464$	−1065.00	−	1844.63i	−0.106555	−	0.184558i
$465$		0			0
$466$	−675.000	+	1169.13i	−0.0671004	+	0.116221i
$467$	11007.0			1.09067			0.545335	−	0.838218i	$-0.316402\pi$
$467$	11007.0			1.09067			0.545335	+	0.838218i	$0.316402\pi$
$468$		0			0
$469$	6250.00			0.615348
$470$	−7425.00	+	12860.5i	−0.728702	+	1.26215i
$471$		0			0
$472$	−6930.00	−	12003.1i	−0.675803	−	1.17053i
$473$	−825.000	−	1428.94i	−0.0801978	−	0.138907i
$474$		0			0
$475$	−100.000	+	173.205i	−0.00965961	+	0.0167309i
$476$	1800.00			0.173325
$477$		0			0
$478$	15570.0			1.48986
$479$	8475.00	−	14679.1i	0.808419	−	1.40022i	−0.105540	−	0.994415i	$-0.533657\pi$
$479$	8475.00	−	14679.1i	0.808419	−	1.40022i	0.913959	−	0.405807i	$-0.133010\pi$
$480$		0			0
$481$	4300.00	+	7447.82i	0.407616	+	0.706011i
$482$	−3399.00	−	5887.24i	−0.321204	−	0.556341i
$483$		0			0
$484$	553.000	−	957.824i	0.0519346	−	0.0899534i
$485$	16575.0			1.55182
$486$		0			0
$487$	10640.0			0.990030			0.495015	−	0.868885i	$-0.335163\pi$
$487$	10640.0			0.990030			0.495015	+	0.868885i	$0.335163\pi$
$488$	3948.00	−	6838.14i	0.366225	−	0.634319i
$489$		0			0
$490$	−6345.00	−	10989.9i	−0.584975	−	1.01321i
$491$	817.500	+	1415.95i	0.0751390	+	0.130145i	0.901147	−	0.433514i	$-0.142727\pi$
$491$	817.500	+	1415.95i	0.0751390	+	0.130145i	−0.826008	+	0.563659i	$0.809393\pi$
$492$		0			0
$493$	−1080.00	+	1870.61i	−0.0986628	+	0.170889i
$494$	−120.000			−0.0109293
$495$		0			0
$496$	−7171.00			−0.649168
$497$	4500.00	−	7794.23i	0.406142	−	0.703459i
$498$		0			0
$499$	7901.00	+	13684.9i	0.708812	+	1.22770i	0.965298	+	0.261151i	$0.0841020\pi$
$499$	7901.00	+	13684.9i	0.708812	+	1.22770i	−0.256486	+	0.966548i	$0.582565\pi$
$500$	−187.500	−	324.760i	−0.0167705	−	0.0290474i
$501$		0			0
$502$	4320.00	−	7482.46i	0.384086	−	0.665256i
$503$	7866.00			0.697272			0.348636	−	0.937258i	$-0.386645\pi$
$503$	7866.00			0.697272			0.348636	+	0.937258i	$0.386645\pi$
$504$		0			0
$505$	21375.0			1.88351
$506$	−2565.00	+	4442.71i	−0.225352	+	0.390321i
$507$		0			0
$508$	−932.500	−	1615.14i	−0.0814429	−	0.141063i
$509$	−5977.50	−	10353.3i	−0.520527	−	0.901578i	−0.999715	−	0.0238665i	$-0.992402\pi$
$509$	−5977.50	−	10353.3i	−0.520527	−	0.901578i	0.479189	−	0.877712i	$-0.340931\pi$
$510$		0			0
$511$	9812.50	−	16995.7i	0.849471	−	1.47133i
$512$	8733.00			0.753804
$513$		0			0
$514$	−12564.0			−1.07816
$515$	−7950.00	+	13769.8i	−0.680231	+	1.17819i
$516$		0			0
$517$	−2475.00	−	4286.83i	−0.210542	−	0.364670i
$518$	16125.0	+	27929.3i	1.36774	+	2.36900i
$519$		0			0
$520$	3150.00	−	5455.96i	0.265647	−	0.460115i
$521$	−19260.0			−1.61957			−0.809785	−	0.586727i	$-0.800416\pi$
$521$	−19260.0			−1.61957			−0.809785	+	0.586727i	$0.800416\pi$
$522$		0			0
$523$	−18520.0			−1.54842			−0.774209	−	0.632930i	$-0.781852\pi$
$523$	−18520.0			−1.54842			−0.774209	+	0.632930i	$0.781852\pi$
$524$	−577.500	+	1000.26i	−0.0481454	+	0.0833903i
$525$		0			0
$526$	4545.00	+	7872.17i	0.376752	+	0.652553i
$527$	3636.00	+	6297.74i	0.300544	+	0.520557i
$528$		0			0
$529$	−414.500	+	717.935i	−0.0340676	+	0.0590067i
$530$	−27945.0			−2.29029
$531$		0			0
$532$	−50.0000			−0.00407476
$533$	−300.000	+	519.615i	−0.0243798	+	0.0422271i
$534$		0			0
$535$	−11137.5	−	19290.7i	−0.900030	−	1.55890i
$536$	2625.00	+	4546.63i	0.211535	+	0.366389i
$537$		0			0
$538$	−5265.00	+	9119.25i	−0.421915	+	0.730778i
$539$	4230.00			0.338032
$540$		0			0
$541$	8372.00			0.665324			0.332662	−	0.943046i	$-0.392053\pi$
$541$	8372.00			0.665324			0.332662	+	0.943046i	$0.392053\pi$
$542$	4498.50	−	7791.63i	0.356508	−	0.617489i
$543$		0			0
$544$	1620.00	+	2805.92i	0.127678	+	0.221145i
$545$	−6465.00	−	11197.7i	−0.508129	−	0.880105i
$546$		0			0
$547$	−8560.00	+	14826.4i	−0.669103	+	1.15892i	0.309053	+	0.951045i	$0.399988\pi$
$547$	−8560.00	+	14826.4i	−0.669103	+	1.15892i	−0.978155	+	0.207875i	$0.933345\pi$
$548$	2778.00			0.216552
$549$		0			0
$550$	−4500.00			−0.348874
$551$	30.0000	−	51.9615i	0.00231950	−	0.00401749i
$552$		0			0
$553$	6100.00	+	10565.5i	0.469075	+	0.812461i
$554$	−11580.0	−	20057.1i	−0.888063	−	1.53817i
$555$		0			0
$556$	962.000	−	1666.23i	0.0733775	−	0.127094i
$557$	−10575.0			−0.804447			−0.402224	−	0.915541i	$-0.631763\pi$
$557$	−10575.0			−0.804447			−0.402224	+	0.915541i	$0.631763\pi$
$558$		0			0
$559$	2200.00			0.166458
$560$	13312.5	−	23057.9i	1.00456	−	1.73996i
$561$		0			0
$562$	11160.0	+	19329.7i	0.837644	+	1.45084i
$563$	5227.50	+	9054.30i	0.391319	+	0.677785i	0.992624	−	0.121235i	$-0.0386854\pi$
$563$	5227.50	+	9054.30i	0.391319	+	0.677785i	−0.601304	+	0.799020i	$0.705352\pi$
$564$		0			0
$565$	−5175.00	+	8963.36i	−0.385334	+	0.667419i
$566$	−2490.00			−0.184916
$567$		0			0
$568$	7560.00			0.558469
$569$	−12270.0	+	21252.3i	−0.904016	+	1.56580i	−0.0817836	+	0.996650i	$0.526062\pi$
$569$	−12270.0	+	21252.3i	−0.904016	+	1.56580i	−0.822233	+	0.569152i	$0.807272\pi$
$570$		0			0
$571$	−12322.0	−	21342.3i	−0.903082	−	1.56418i	−0.823472	−	0.567357i	$-0.807966\pi$
$571$	−12322.0	−	21342.3i	−0.903082	−	1.56418i	−0.0796100	−	0.996826i	$-0.525367\pi$
$572$	−150.000	−	259.808i	−0.0109647	−	0.0189914i
$573$		0			0
$574$	−1125.00	+	1948.56i	−0.0818059	+	0.141692i
$575$	−11400.0			−0.826805
$576$		0			0
$577$	−9610.00			−0.693361			−0.346681	−	0.937983i	$-0.612691\pi$
$577$	−9610.00			−0.693361			−0.346681	+	0.937983i	$0.612691\pi$
$578$	406.500	−	704.079i	0.0292529	−	0.0506675i
$579$		0			0
$580$	−225.000	−	389.711i	−0.0161080	−	0.0278998i
$581$	−6112.50	−	10587.2i	−0.436470	−	0.755989i
$582$		0			0
$583$	4657.50	−	8067.03i	0.330864	−	0.573074i
$584$	16485.0			1.16807
$585$		0			0
$586$	1638.00			0.115470
$587$	2008.50	−	3478.82i	0.141226	−	0.244611i	−0.786733	−	0.617294i	$-0.788229\pi$
$587$	2008.50	−	3478.82i	0.141226	−	0.244611i	0.927959	+	0.372683i	$0.121562\pi$
$588$		0			0
$589$	−101.000	−	174.937i	−0.00706559	−	0.0122380i
$590$	−14850.0	−	25721.0i	−1.03621	−	1.79477i
$591$		0			0
$592$	−15265.0	+	26439.8i	−1.05978	+	1.83559i
$593$	−594.000			−0.0411343			−0.0205672	−	0.999788i	$-0.506547\pi$
$593$	−594.000			−0.0411343			−0.0205672	+	0.999788i	$0.506547\pi$
$594$		0			0
$595$	−27000.0			−1.86032
$596$	727.500	−	1260.07i	0.0499993	−	0.0866013i
$597$		0			0
$598$	−3420.00	−	5923.61i	−0.233870	−	0.405075i
$599$	4395.00	+	7612.36i	0.299791	+	0.519253i	0.976088	−	0.217376i	$-0.0697497\pi$
$599$	4395.00	+	7612.36i	0.299791	+	0.519253i	−0.676297	+	0.736629i	$0.736416\pi$
$600$		0			0
$601$	−4685.50	+	8115.52i	−0.318013	+	0.550814i	−0.980073	−	0.198636i	$-0.936349\pi$
$601$	−4685.50	+	8115.52i	−0.318013	+	0.550814i	0.662061	+	0.749450i	$0.269682\pi$
$602$	8250.00			0.558546
$603$		0			0
$604$	−727.000			−0.0489755
$605$	−8295.00	+	14367.4i	−0.557421	+	0.965481i
$606$		0			0
$607$	7280.00	+	12609.3i	0.486798	+	0.843158i	0.999885	−	0.0151784i	$-0.00483163\pi$
$607$	7280.00	+	12609.3i	0.486798	+	0.843158i	−0.513087	+	0.858336i	$0.671498\pi$
$608$	−45.0000	−	77.9423i	−0.00300163	−	0.00519898i
$609$		0			0
$610$	8460.00	−	14653.1i	0.561533	−	0.972604i
$611$	6600.00			0.437001
$612$		0			0
$613$	−18250.0			−1.20246			−0.601232	−	0.799074i	$-0.705323\pi$
$613$	−18250.0			−1.20246			−0.601232	+	0.799074i	$0.705323\pi$
$614$	−8340.00	+	14445.3i	−0.548168	+	0.949454i
$615$		0			0
$616$	3937.50	+	6819.95i	0.257543	+	0.446077i
$617$	9831.00	+	17027.8i	0.641461	+	1.11104i	0.985107	+	0.171943i	$0.0550045\pi$
$617$	9831.00	+	17027.8i	0.641461	+	1.11104i	−0.343646	+	0.939099i	$0.611662\pi$
$618$		0			0
$619$	−6022.00	+	10430.4i	−0.391025	+	0.677276i	−0.992585	−	0.121552i	$-0.961213\pi$
$619$	−6022.00	+	10430.4i	−0.391025	+	0.677276i	0.601560	+	0.798828i	$0.294546\pi$
$620$	−1515.00			−0.0981353
$621$		0			0
$622$	−26010.0			−1.67670
$623$	5625.00	−	9742.79i	0.361735	−	0.626543i
$624$		0			0
$625$	9062.50	+	15696.7i	0.580000	+	1.00459i
$626$	6847.50	+	11860.2i	0.437190	+	0.757236i
$627$		0			0
$628$	−1630.00	+	2823.24i	−0.103573	+	0.179394i
$629$	30960.0			1.96257
$630$		0			0
$631$	14879.0			0.938706			0.469353	−	0.883011i	$-0.344487\pi$
$631$	14879.0			0.938706			0.469353	+	0.883011i	$0.344487\pi$
$632$	−5124.00	+	8875.03i	−0.322503	+	0.558591i
$633$		0			0
$634$	−6349.50	−	10997.7i	−0.397746	−	0.688916i
$635$	13987.5	+	24227.1i	0.874137	+	1.51405i
$636$		0			0
$637$	−2820.00	+	4884.38i	−0.175404	+	0.303809i
$638$	1350.00			0.0837727
$639$		0			0
$640$	24885.0			1.53698
$641$	4425.00	−	7664.32i	0.272663	−	0.472266i	−0.696880	−	0.717188i	$-0.745429\pi$
$641$	4425.00	−	7664.32i	0.272663	−	0.472266i	0.969543	+	0.244922i	$0.0787623\pi$
$642$		0			0
$643$	−9190.00	−	15917.5i	−0.563636	−	0.976247i	−0.997175	−	0.0751120i	$-0.976069\pi$
$643$	−9190.00	−	15917.5i	−0.563636	−	0.976247i	0.433539	−	0.901135i	$-0.357265\pi$
$644$	−1425.00	−	2468.17i	−0.0871939	−	0.151024i
$645$		0			0
$646$	−216.000	+	374.123i	−0.0131554	+	0.0227859i
$647$	−3888.00			−0.236249			−0.118124	−	0.992999i	$-0.537688\pi$
$647$	−3888.00			−0.236249			−0.118124	+	0.992999i	$0.537688\pi$
$648$		0			0
$649$	9900.00			0.598781
$650$	3000.00	−	5196.15i	0.181030	−	0.313554i
$651$		0			0
$652$	−1270.00	−	2199.70i	−0.0762838	−	0.132127i
$653$	−3394.50	−	5879.45i	−0.203426	−	0.352344i	0.746204	−	0.665717i	$-0.231874\pi$
$653$	−3394.50	−	5879.45i	−0.203426	−	0.352344i	−0.949630	+	0.313373i	$0.898541\pi$
$654$		0			0
$655$	8662.50	−	15003.9i	0.516751	−	0.895039i
$656$	−2130.00			−0.126772
$657$		0			0
$658$	24750.0			1.46635
$659$	14167.5	−	24538.8i	0.837462	−	1.45053i	−0.0545481	−	0.998511i	$-0.517372\pi$
$659$	14167.5	−	24538.8i	0.837462	−	1.45053i	0.892010	−	0.452016i	$-0.149295\pi$
$660$		0			0
$661$	3041.00	+	5267.17i	0.178943	+	0.309938i	0.941519	−	0.336961i	$-0.109399\pi$
$661$	3041.00	+	5267.17i	0.178943	+	0.309938i	−0.762576	+	0.646899i	$0.776066\pi$
$662$	813.000	+	1408.16i	0.0477313	+	0.0826731i
$663$		0			0
$664$	5134.50	−	8893.21i	0.300086	−	0.519765i
$665$	750.000			0.0437350
$666$		0			0
$667$	3420.00			0.198535
$668$	1749.00	−	3029.36i	0.101304	−	0.175463i
$669$		0			0
$670$	5625.00	+	9742.79i	0.324347	+	0.561786i
$671$	2820.00	+	4884.38i	0.162243	+	0.281013i
$672$		0			0
$673$	−4982.50	+	8629.94i	−0.285381	+	0.494294i	−0.972701	−	0.232060i	$-0.925453\pi$
$673$	−4982.50	+	8629.94i	−0.285381	+	0.494294i	0.687321	+	0.726354i	$0.258787\pi$
$674$	−17070.0			−0.975537
$675$		0			0
$676$	−1797.00			−0.102242
$677$	4065.00	−	7040.79i	0.230769	−	0.399704i	−0.727266	−	0.686356i	$-0.759209\pi$
$677$	4065.00	−	7040.79i	0.230769	−	0.399704i	0.958035	+	0.286652i	$0.0925425\pi$
$678$		0			0
$679$	−13812.5	−	23924.0i	−0.780670	−	1.35216i
$680$	−11340.0	−	19641.5i	−0.639513	−	1.10767i
$681$		0			0
$682$	2272.50	−	3936.09i	0.127593	−	0.220998i
$683$	−33516.0			−1.87768			−0.938839	−	0.344356i	$-0.888097\pi$
$683$	−33516.0			−1.87768			−0.938839	+	0.344356i	$0.888097\pi$
$684$		0			0
$685$	−41670.0			−2.32428
$686$	2287.50	−	3962.07i	0.127314	−	0.220514i
$687$		0			0
$688$	3905.00	+	6763.66i	0.216391	+	0.374800i
$689$	6210.00	+	10756.0i	0.343370	+	0.594735i
$690$		0			0
$691$	11042.0	−	19125.3i	0.607898	−	1.05291i	−0.383688	−	0.923463i	$-0.625346\pi$
$691$	11042.0	−	19125.3i	0.607898	−	1.05291i	0.991586	−	0.129447i	$-0.0413204\pi$
$692$	1149.00			0.0631191
$693$		0			0
$694$	15165.0			0.829475
$695$	−14430.0	+	24993.5i	−0.787570	+	1.36411i
$696$		0			0
$697$	1080.00	+	1870.61i	0.0586914	+	0.101657i
$698$	2433.00	+	4214.08i	0.131935	+	0.228518i
$699$		0			0
$700$	1250.00	−	2165.06i	0.0674937	−	0.116902i
$701$	10395.0			0.560077			0.280038	−	0.959989i	$-0.409653\pi$
$701$	10395.0			0.560077			0.280038	+	0.959989i	$0.409653\pi$
$702$		0			0
$703$	−860.000			−0.0461387
$704$	−3247.50	+	5624.83i	−0.173856	+	0.301128i
$705$		0			0
$706$	−45.0000	−	77.9423i	−0.00239886	−	0.00415495i
$707$	−17812.5	−	30852.2i	−0.947536	−	1.64118i
$708$		0			0
$709$	2402.00	−	4160.39i	0.127234	−	0.220376i	−0.795370	−	0.606124i	$-0.792723\pi$
$709$	2402.00	−	4160.39i	0.127234	−	0.220376i	0.922604	+	0.385748i	$0.126057\pi$
$710$	16200.0			0.856303
$711$		0			0
$712$	9450.00			0.497407
$713$	5757.00	−	9971.42i	0.302386	−	0.523748i
$714$		0			0
$715$	2250.00	+	3897.11i	0.117686	+	0.203837i
$716$	157.500	+	272.798i	0.00822074	+	0.0142387i
$717$		0			0
$718$	11205.0	−	19407.6i	0.582405	−	1.00876i
$719$	−10980.0			−0.569520			−0.284760	−	0.958599i	$-0.591914\pi$
$719$	−10980.0			−0.569520			−0.284760	+	0.958599i	$0.591914\pi$
$720$		0			0
$721$	26500.0			1.36881
$722$	−10282.5	+	17809.8i	−0.530021	+	0.918023i
$723$		0			0
$724$	−568.000	−	983.805i	−0.0291568	−	0.0505011i
$725$	1500.00	+	2598.08i	0.0768395	+	0.133090i
$726$		0			0
$727$	12972.5	−	22469.0i	0.661793	−	1.14626i	−0.318351	−	0.947973i	$-0.603129\pi$
$727$	12972.5	−	22469.0i	0.661793	−	1.14626i	0.980144	−	0.198286i	$-0.0635375\pi$
$728$	−10500.0			−0.534555
$729$		0			0
$730$	35325.0			1.79101
$731$	3960.00	−	6858.92i	0.200364	−	0.347040i
$732$		0			0
$733$	−9325.00	−	16151.4i	−0.469886	−	0.813867i	0.529521	−	0.848297i	$-0.322372\pi$
$733$	−9325.00	−	16151.4i	−0.469886	−	0.813867i	−0.999407	+	0.0344299i	$0.989038\pi$
$734$	−2062.50	−	3572.35i	−0.103717	−	0.179643i
$735$		0			0
$736$	2565.00	−	4442.71i	0.128461	−	0.222501i
$737$	−3750.00			−0.187426
$738$		0			0
$739$	−5128.00			−0.255259			−0.127630	−	0.991822i	$-0.540737\pi$
$739$	−5128.00			−0.255259			−0.127630	+	0.991822i	$0.540737\pi$
$740$	−3225.00	+	5585.86i	−0.160207	+	0.277487i
$741$		0			0
$742$	23287.5	+	40335.1i	1.15217	+	1.99562i
$743$	−16350.0	−	28319.0i	−0.807299	−	1.39828i	−0.914728	−	0.404070i	$-0.867595\pi$
$743$	−16350.0	−	28319.0i	−0.807299	−	1.39828i	0.107429	−	0.994213i	$-0.465738\pi$
$744$		0			0
$745$	−10912.5	+	18901.0i	−0.536648	+	0.929502i
$746$	14520.0			0.712621
$747$		0			0
$748$	−1080.00			−0.0527924
$749$	−18562.5	+	32151.2i	−0.905553	+	1.56846i
$750$		0			0
$751$	−10580.5	−	18326.0i	−0.514098	−	0.890445i	−0.999866	−	0.0163567i	$-0.994793\pi$
$751$	−10580.5	−	18326.0i	−0.514098	−	0.890445i	0.485768	−	0.874088i	$-0.338540\pi$
$752$	11715.0	+	20291.0i	0.568088	+	0.983957i
$753$		0			0
$754$	−900.000	+	1558.85i	−0.0434696	+	0.0752915i
$755$	10905.0			0.525660
$756$		0			0
$757$	7130.00			0.342331			0.171165	−	0.985242i	$-0.445247\pi$
$757$	7130.00			0.342331			0.171165	+	0.985242i	$0.445247\pi$
$758$	2838.00	−	4915.56i	0.135991	−	0.235542i
$759$		0			0
$760$	315.000	+	545.596i	0.0150345	+	0.0260406i
$761$	−1680.00	−	2909.85i	−0.0800262	−	0.138609i	0.823235	−	0.567701i	$-0.192167\pi$
$761$	−1680.00	−	2909.85i	−0.0800262	−	0.138609i	−0.903261	+	0.429092i	$0.858834\pi$
$762$		0			0
$763$	−10775.0	+	18662.8i	−0.511247	+	0.885505i
$764$	−2460.00			−0.116492
$765$		0			0
$766$	−32112.0			−1.51469
$767$	−6600.00	+	11431.5i	−0.310707	+	0.538160i
$768$		0			0
$769$	−16736.5	−	28988.5i	−0.784829	−	1.35936i	−0.929101	−	0.369826i	$-0.879417\pi$
$769$	−16736.5	−	28988.5i	−0.784829	−	1.35936i	0.144272	−	0.989538i	$-0.453916\pi$
$770$	8437.50	+	14614.2i	0.394891	+	0.683972i
$771$		0			0
$772$	−482.500	+	835.715i	−0.0224942	+	0.0389612i
$773$	3546.00			0.164995			0.0824973	−	0.996591i	$-0.473710\pi$
$773$	3546.00			0.164995			0.0824973	+	0.996591i	$0.473710\pi$
$774$		0			0
$775$	10100.0			0.468133
$776$	11602.5	−	20096.1i	0.536734	−	0.929650i
$777$		0			0
$778$	−11722.5	−	20304.0i	−0.540195	−	0.935646i
$779$	−30.0000	−	51.9615i	−0.00137980	−	0.00238988i
$780$		0			0
$781$	−2700.00	+	4676.54i	−0.123705	+	0.214263i
$782$	−24624.0			−1.12603
$783$		0			0
$784$	−20022.0			−0.912081
$785$	24450.0	−	42348.6i	1.11167	−	1.92546i
$786$		0			0
$787$	15920.0	+	27574.2i	0.721076	+	1.24894i	0.960569	+	0.278042i	$0.0896854\pi$
$787$	15920.0	+	27574.2i	0.721076	+	1.24894i	−0.239493	+	0.970898i	$0.576981\pi$
$788$	1246.50	+	2159.00i	0.0563512	+	0.0976031i
$789$		0			0
$790$	−10980.0	+	19017.9i	−0.494495	+	0.856490i
$791$	17250.0			0.775397
$792$		0			0
$793$	−7520.00			−0.336750
$794$	7050.00	−	12211.0i	0.315107	−	0.545782i
$795$		0			0
$796$	255.500	+	442.539i	0.0113768	+	0.0197052i
$797$	−7858.50	−	13611.3i	−0.349263	−	0.604941i	0.636856	−	0.770983i	$-0.280235\pi$
$797$	−7858.50	−	13611.3i	−0.349263	−	0.604941i	−0.986119	+	0.166042i	$0.946901\pi$
$798$		0			0
$799$	11880.0	−	20576.8i	0.526013	−	0.911081i
$800$	4500.00			0.198874
$801$		0			0
$802$	−6300.00			−0.277382
$803$	−5887.50	+	10197.4i	−0.258736	+	0.448145i
$804$		0			0
$805$	21375.0	+	37022.6i	0.935863	+	1.62096i
$806$	3030.00	+	5248.11i	0.132416	+	0.229351i
$807$		0			0
$808$	14962.5	−	25915.8i	0.651459	−	1.12836i
$809$	−10530.0			−0.457621			−0.228810	−	0.973471i	$-0.573484\pi$
$809$	−10530.0			−0.457621			−0.228810	+	0.973471i	$0.573484\pi$
$810$		0			0
$811$	−26782.0			−1.15961			−0.579805	−	0.814755i	$-0.696871\pi$
$811$	−26782.0			−1.15961			−0.579805	+	0.814755i	$0.696871\pi$
$812$	−375.000	+	649.519i	−0.0162068	+	0.0280710i
$813$		0			0
$814$	−9675.00	−	16757.6i	−0.416595	−	0.721564i
$815$	19050.0	+	32995.6i	0.818764	+	1.41814i
$816$		0			0
$817$	−110.000	+	190.526i	−0.00471042	+	0.00815869i
$818$	32259.0			1.37886
$819$		0			0
$820$	−450.000			−0.0191642
$821$	5055.00	−	8755.52i	0.214885	−	0.372192i	−0.738352	−	0.674416i	$-0.764396\pi$
$821$	5055.00	−	8755.52i	0.214885	−	0.372192i	0.953237	+	0.302224i	$0.0977289\pi$
$822$		0			0
$823$	6267.50	+	10855.6i	0.265457	+	0.459785i	0.967683	−	0.252169i	$-0.0811438\pi$
$823$	6267.50	+	10855.6i	0.265457	+	0.459785i	−0.702226	+	0.711954i	$0.747810\pi$
$824$	11130.0	+	19277.7i	0.470548	+	0.815014i
$825$		0			0
$826$	−24750.0	+	42868.3i	−1.04257	+	1.80578i
$827$	−9792.00			−0.411731			−0.205865	−	0.978580i	$-0.566001\pi$
$827$	−9792.00			−0.411731			−0.205865	+	0.978580i	$0.566001\pi$
$828$		0			0
$829$	−4534.00			−0.189955			−0.0949773	−	0.995479i	$-0.530278\pi$
$829$	−4534.00			−0.189955			−0.0949773	+	0.995479i	$0.530278\pi$
$830$	11002.5	−	19056.9i	0.460123	−	0.796957i
$831$		0			0
$832$	−4330.00	−	7499.78i	−0.180428	−	0.312510i
$833$	10152.0	+	17583.8i	0.422264	+	0.731383i
$834$		0			0
$835$	−26235.0	+	45440.4i	−1.08730	+	1.88327i
$836$	30.0000			0.00124111
$837$		0			0
$838$	8820.00			0.363582
$839$	−4440.00	+	7690.31i	−0.182701	+	0.316447i	−0.942799	−	0.333361i	$-0.891817\pi$
$839$	−4440.00	+	7690.31i	−0.182701	+	0.316447i	0.760099	+	0.649808i	$0.225151\pi$
$840$		0			0
$841$	11744.5	+	20342.1i	0.481549	+	0.834067i
$842$	13044.0	+	22592.9i	0.533879	+	0.924705i
$843$		0			0
$844$	1043.00	−	1806.53i	0.0425374	−	0.0736769i
$845$	26955.0			1.09737
$846$		0			0
$847$	27650.0			1.12168
$848$	−22045.5	+	38183.9i	−0.892742	+	1.54628i
$849$		0			0
$850$	−10800.0	−	18706.1i	−0.435808	−	0.754842i
$851$	−24510.0	−	42452.6i	−0.987300	−	1.71005i
$852$		0			0
$853$	−1135.00	+	1965.88i	−0.0455588	+	0.0789102i	−0.887906	−	0.460026i	$-0.847840\pi$
$853$	−1135.00	+	1965.88i	−0.0455588	+	0.0789102i	0.842347	+	0.538936i	$0.181174\pi$
$854$	−28200.0			−1.12996
$855$		0			0
$856$	−31185.0			−1.24519
$857$	−9804.00	+	16981.0i	−0.390780	+	0.676850i	−0.992553	−	0.121817i	$-0.961128\pi$
$857$	−9804.00	+	16981.0i	−0.390780	+	0.676850i	0.601773	+	0.798667i	$0.294461\pi$
$858$		0			0
$859$	476.000	+	824.456i	0.0189068	+	0.0327475i	0.875324	−	0.483537i	$-0.160648\pi$
$859$	476.000	+	824.456i	0.0189068	+	0.0327475i	−0.856417	+	0.516284i	$0.827315\pi$
$860$	825.000	+	1428.94i	0.0327119	+	0.0566587i
$861$		0			0
$862$	−12555.0	+	21745.9i	−0.496085	+	0.859244i
$863$	17604.0			0.694377			0.347188	−	0.937795i	$-0.387136\pi$
$863$	17604.0			0.694377			0.347188	+	0.937795i	$0.387136\pi$
$864$		0			0
$865$	−17235.0			−0.677465
$866$	−7732.50	+	13393.1i	−0.303419	+	0.525538i
$867$		0			0
$868$	1262.50	+	2186.71i	0.0493687	+	0.0855091i
$869$	−3660.00	−	6339.31i	−0.142873	−	0.247464i
$870$		0			0
$871$	2500.00	−	4330.13i	0.0972552	−	0.168451i
$872$	−18102.0			−0.702994
$873$		0			0
$874$	684.000			0.0264721
$875$	4687.50	−	8118.99i	0.181104	−	0.313682i
$876$		0			0
$877$	−10945.0	−	18957.3i	−0.421421	−	0.729923i	0.574658	−	0.818394i	$-0.305135\pi$
$877$	−10945.0	−	18957.3i	−0.421421	−	0.729923i	−0.996079	+	0.0884711i	$0.971802\pi$
$878$	−16480.5	−	28545.1i	−0.633474	−	1.09721i
$879$		0			0
$880$	−7987.50	+	13834.8i	−0.305976	+	0.529965i
$881$	−23940.0			−0.915504			−0.457752	−	0.889080i	$-0.651345\pi$
$881$	−23940.0			−0.915504			−0.457752	+	0.889080i	$0.651345\pi$
$882$		0			0
$883$	−34990.0			−1.33353			−0.666765	−	0.745268i	$-0.732322\pi$
$883$	−34990.0			−1.33353			−0.666765	+	0.745268i	$0.732322\pi$
$884$	720.000	−	1247.08i	0.0273939	−	0.0474477i
$885$		0			0
$886$	−2934.00	−	5081.84i	−0.111252	−	0.192695i
$887$	−11094.0	−	19215.4i	−0.419955	−	0.727383i	0.575979	−	0.817464i	$-0.304621\pi$
$887$	−11094.0	−	19215.4i	−0.419955	−	0.727383i	−0.995934	+	0.0900808i	$0.971287\pi$
$888$		0			0
$889$	23312.5	−	40378.4i	0.879501	−	1.52334i
$890$	20250.0			0.762676
$891$		0			0
$892$	5240.00			0.196691
$893$	−330.000	+	571.577i	−0.0123662	+	0.0214189i
$894$		0			0
$895$	−2362.50	−	4091.97i	−0.0882343	−	0.152826i
$896$	−20737.5	−	35918.4i	−0.773205	−	1.33923i
$897$		0			0
$898$	13095.0	−	22681.2i	0.486621	−	0.842853i
$899$	−3030.00			−0.112410
$900$		0			0
$901$	44712.0			1.65324
$902$	675.000	−	1169.13i	0.0249169	−	0.0431573i
$903$		0			0
$904$	7245.00	+	12548.7i	0.266554	+	0.461686i
$905$	8520.00	+	14757.1i	0.312944	+	0.542035i
$906$		0			0
$907$	−18685.0	+	32363.4i	−0.684041	+	1.18479i	0.289696	+	0.957119i	$0.406446\pi$
$907$	−18685.0	+	32363.4i	−0.684041	+	1.18479i	−0.973737	+	0.227675i	$0.926888\pi$
$908$	−2388.00			−0.0872782
$909$		0			0
$910$	−22500.0			−0.819635
$911$	20355.0	−	35255.9i	0.740276	−	1.28220i	−0.212094	−	0.977249i	$-0.568028\pi$
$911$	20355.0	−	35255.9i	0.740276	−	1.28220i	0.952370	−	0.304946i	$-0.0986384\pi$
$912$		0			0
$913$	3667.50	+	6352.30i	0.132943	+	0.230263i
$914$	−12997.5	−	22512.3i	−0.470371	−	0.814706i
$915$		0			0
$916$	−91.0000	+	157.617i	−0.00328245	+	0.00568537i
$917$	−28875.0			−1.03984
$918$		0			0
$919$	20981.0			0.753100			0.376550	−	0.926396i	$-0.377110\pi$
$919$	20981.0			0.753100			0.376550	+	0.926396i	$0.377110\pi$
$920$	−17955.0	+	31099.0i	−0.643433	+	1.11446i
$921$		0			0
$922$	14737.5	+	25526.1i	0.526414	+	0.911776i
$923$	−3600.00	−	6235.38i	−0.128381	−	0.222362i
$924$		0			0
$925$	21500.0	−	37239.1i	0.764233	−	1.32369i
$926$	15735.0			0.558406
$927$		0			0
$928$	−1350.00			−0.0477542
$929$	10050.0	−	17407.1i	0.354930	−	0.614756i	−0.632176	−	0.774825i	$-0.717838\pi$
$929$	10050.0	−	17407.1i	0.354930	−	0.614756i	0.987106	+	0.160068i	$0.0511714\pi$
$930$		0			0
$931$	−282.000	−	488.438i	−0.00992715	−	0.0171943i
$932$	225.000	+	389.711i	0.00790785	+	0.0136968i
$933$		0			0
$934$	16510.5	−	28597.0i	0.578415	−	1.00185i
$935$	16200.0			0.566627
$936$		0			0
$937$	15635.0			0.545115			0.272558	−	0.962139i	$-0.412130\pi$
$937$	15635.0			0.545115			0.272558	+	0.962139i	$0.412130\pi$
$938$	9375.00	−	16238.0i	0.326338	−	0.565233i
$939$		0			0
$940$	2475.00	+	4286.83i	0.0858783	+	0.148746i
$941$	11977.5	+	20745.6i	0.414937	+	0.718691i	0.995422	−	0.0955795i	$-0.0304704\pi$
$941$	11977.5	+	20745.6i	0.414937	+	0.718691i	−0.580485	+	0.814271i	$0.697137\pi$
$942$		0			0
$943$	1710.00	−	2961.81i	0.0590512	−	0.102280i
$944$	−46860.0			−1.61564
$945$		0			0
$946$	−4950.00			−0.170125
$947$	−18196.5	+	31517.3i	−0.624400	+	1.08149i	0.364257	+	0.931299i	$0.381323\pi$
$947$	−18196.5	+	31517.3i	−0.624400	+	1.08149i	−0.988657	+	0.150194i	$0.952010\pi$
$948$		0			0
$949$	−7850.00	−	13596.6i	−0.268516	−	0.465084i
$950$	300.000	+	519.615i	0.0102456	+	0.0177458i
$951$		0			0
$952$	−18900.0	+	32735.8i	−0.643438	+	1.11447i
$953$	−43020.0			−1.46228			−0.731141	−	0.682227i	$-0.761012\pi$
$953$	−43020.0			−1.46228			−0.731141	+	0.682227i	$0.761012\pi$
$954$		0			0
$955$	36900.0			1.25032
$956$	2595.00	−	4494.67i	0.0877911	−	0.152059i
$957$		0			0
$958$	−25425.0	−	44037.4i	−0.857458	−	1.48516i
$959$	34725.0	+	60145.5i	1.16927	+	2.02523i
$960$		0			0
$961$	9795.00	−	16965.4i	0.328791	−	0.569482i
$962$	25800.0			0.864683
$963$		0			0
$964$	−2266.00			−0.0757084
$965$	7237.50	−	12535.7i	0.241434	−	0.418175i
$966$		0			0
$967$	21792.5	+	37745.7i	0.724715	+	1.25524i	0.959091	+	0.283097i	$0.0913620\pi$
$967$	21792.5	+	37745.7i	0.724715	+	1.25524i	−0.234376	+	0.972146i	$0.575305\pi$
$968$	11613.0	+	20114.3i	0.385595	+	0.667870i
$969$		0			0
$970$	24862.5	−	43063.1i	0.822976	−	1.42544i
$971$	43335.0			1.43222			0.716110	−	0.697987i	$-0.245921\pi$
$971$	43335.0			1.43222			0.716110	+	0.697987i	$0.245921\pi$
$972$		0			0
$973$	48100.0			1.58480
$974$	15960.0	−	27643.5i	0.525042	−	0.909400i
$975$		0			0
$976$	−13348.0	−	23119.4i	−0.437765	−	0.758232i
$977$	15195.0	+	26318.5i	0.497575	+	0.861826i	0.999996	−	0.00279748i	$-0.000890468\pi$
$977$	15195.0	+	26318.5i	0.497575	+	0.861826i	−0.502421	+	0.864623i	$0.667557\pi$
$978$		0			0
$979$	−3375.00	+	5845.67i	−0.110179	+	0.190836i
$980$	−4230.00			−0.137880
$981$		0			0
$982$	4905.00			0.159394
$983$	−29613.0	+	51291.2i	−0.960842	+	1.66423i	−0.240449	+	0.970662i	$0.577295\pi$
$983$	−29613.0	+	51291.2i	−0.960842	+	1.66423i	−0.720393	+	0.693566i	$0.756039\pi$
$984$		0			0
$985$	−18697.5	−	32385.0i	−0.604824	−	1.04759i
$986$	3240.00	+	5611.84i	0.104648	+	0.181255i
$987$		0			0
$988$	−20.0000	+	34.6410i	−0.000644013	+	0.00111546i
$989$	−12540.0			−0.403184
$990$		0			0
$991$	8399.00			0.269226			0.134613	−	0.990898i	$-0.457021\pi$
$991$	8399.00			0.269226			0.134613	+	0.990898i	$0.457021\pi$
$992$	−2272.50	+	3936.09i	−0.0727338	+	0.125979i
$993$		0			0
$994$	−13500.0	−	23382.7i	−0.430779	−	0.746131i
$995$	−3832.50	−	6638.08i	−0.122109	−	0.211499i
$996$		0			0
$997$	−6670.00	+	11552.8i	−0.211877	+	0.366981i	−0.952302	−	0.305158i	$-0.901291\pi$
$997$	−6670.00	+	11552.8i	−0.211877	+	0.366981i	0.740425	+	0.672139i	$0.234624\pi$
$998$	47406.0			1.50362
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	81.4.c.c.55.1		2
3.2	odd	2		81.4.c.a.55.1		2
9.2	odd	6		27.4.a.b.1.1	yes	1
9.4	even	3	inner	81.4.c.c.28.1		2
9.5	odd	6		81.4.c.a.28.1		2
9.7	even	3		27.4.a.a.1.1	✓	1
36.7	odd	6		432.4.a.a.1.1		1
36.11	even	6		432.4.a.n.1.1		1
45.2	even	12		675.4.b.a.649.2		2
45.7	odd	12		675.4.b.b.649.1		2
45.29	odd	6		675.4.a.a.1.1		1
45.34	even	6		675.4.a.j.1.1		1
45.38	even	12		675.4.b.a.649.1		2
45.43	odd	12		675.4.b.b.649.2		2
63.20	even	6		1323.4.a.k.1.1		1
63.34	odd	6		1323.4.a.d.1.1		1
72.11	even	6		1728.4.a.d.1.1		1
72.29	odd	6		1728.4.a.c.1.1		1
72.43	odd	6		1728.4.a.bd.1.1		1
72.61	even	6		1728.4.a.bc.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.a.1.1	✓	1	9.7	even	3
27.4.a.b.1.1	yes	1	9.2	odd	6
81.4.c.a.28.1		2	9.5	odd	6
81.4.c.a.55.1		2	3.2	odd	2
81.4.c.c.28.1		2	9.4	even	3	inner
81.4.c.c.55.1		2	1.1	even	1	trivial
432.4.a.a.1.1		1	36.7	odd	6
432.4.a.n.1.1		1	36.11	even	6
675.4.a.a.1.1		1	45.29	odd	6
675.4.a.j.1.1		1	45.34	even	6
675.4.b.a.649.1		2	45.38	even	12
675.4.b.a.649.2		2	45.2	even	12
675.4.b.b.649.1		2	45.7	odd	12
675.4.b.b.649.2		2	45.43	odd	12
1323.4.a.d.1.1		1	63.34	odd	6
1323.4.a.k.1.1		1	63.20	even	6
1728.4.a.c.1.1		1	72.29	odd	6
1728.4.a.d.1.1		1	72.11	even	6
1728.4.a.bc.1.1		1	72.61	even	6
1728.4.a.bd.1.1		1	72.43	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 81 = 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	81.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 2.59808i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(7.50000 + 12.9904i) q^{5} +(12.5000 - 21.6506i) q^{7} +21.0000 q^{8} +O(q^{10})\)	\(q+(1.50000 - 2.59808i) q^{2} +(-0.500000 - 0.866025i) q^{4} +(7.50000 + 12.9904i) q^{5} +(12.5000 - 21.6506i) q^{7} +21.0000 q^{8} +45.0000 q^{10} +(-7.50000 + 12.9904i) q^{11} +(-10.0000 - 17.3205i) q^{13} +(-37.5000 - 64.9519i) q^{14} +(35.5000 - 61.4878i) q^{16} -72.0000 q^{17} +2.00000 q^{19} +(7.50000 - 12.9904i) q^{20} +(22.5000 + 38.9711i) q^{22} +(57.0000 + 98.7269i) q^{23} +(-50.0000 + 86.6025i) q^{25} -60.0000 q^{26} -25.0000 q^{28} +(15.0000 - 25.9808i) q^{29} +(-50.5000 - 87.4686i) q^{31} +(-22.5000 - 38.9711i) q^{32} +(-108.000 + 187.061i) q^{34} +375.000 q^{35} -430.000 q^{37} +(3.00000 - 5.19615i) q^{38} +(157.500 + 272.798i) q^{40} +(-15.0000 - 25.9808i) q^{41} +(-55.0000 + 95.2628i) q^{43} +15.0000 q^{44} +342.000 q^{46} +(-165.000 + 285.788i) q^{47} +(-141.000 - 244.219i) q^{49} +(150.000 + 259.808i) q^{50} +(-10.0000 + 17.3205i) q^{52} -621.000 q^{53} -225.000 q^{55} +(262.500 - 454.663i) q^{56} +(-45.0000 - 77.9423i) q^{58} +(-330.000 - 571.577i) q^{59} +(188.000 - 325.626i) q^{61} -303.000 q^{62} +433.000 q^{64} +(150.000 - 259.808i) q^{65} +(125.000 + 216.506i) q^{67} +(36.0000 + 62.3538i) q^{68} +(562.500 - 974.279i) q^{70} +360.000 q^{71} +785.000 q^{73} +(-645.000 + 1117.17i) q^{74} +(-1.00000 - 1.73205i) q^{76} +(187.500 + 324.760i) q^{77} +(-244.000 + 422.620i) q^{79} +1065.00 q^{80} -90.0000 q^{82} +(244.500 - 423.486i) q^{83} +(-540.000 - 935.307i) q^{85} +(165.000 + 285.788i) q^{86} +(-157.500 + 272.798i) q^{88} +450.000 q^{89} -500.000 q^{91} +(57.0000 - 98.7269i) q^{92} +(495.000 + 857.365i) q^{94} +(15.0000 + 25.9808i) q^{95} +(552.500 - 956.958i) q^{97} -846.000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} - q^{4} + 15 q^{5} + 25 q^{7} + 42 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 - q^4 + 15 * q^5 + 25 * q^7 + 42 * q^8	\( 2 q + 3 q^{2} - q^{4} + 15 q^{5} + 25 q^{7} + 42 q^{8} + 90 q^{10} - 15 q^{11} - 20 q^{13} - 75 q^{14} + 71 q^{16} - 144 q^{17} + 4 q^{19} + 15 q^{20} + 45 q^{22} + 114 q^{23} - 100 q^{25} - 120 q^{26} - 50 q^{28} + 30 q^{29} - 101 q^{31} - 45 q^{32} - 216 q^{34} + 750 q^{35} - 860 q^{37} + 6 q^{38} + 315 q^{40} - 30 q^{41} - 110 q^{43} + 30 q^{44} + 684 q^{46} - 330 q^{47} - 282 q^{49} + 300 q^{50} - 20 q^{52} - 1242 q^{53} - 450 q^{55} + 525 q^{56} - 90 q^{58} - 660 q^{59} + 376 q^{61} - 606 q^{62} + 866 q^{64} + 300 q^{65} + 250 q^{67} + 72 q^{68} + 1125 q^{70} + 720 q^{71} + 1570 q^{73} - 1290 q^{74} - 2 q^{76} + 375 q^{77} - 488 q^{79} + 2130 q^{80} - 180 q^{82} + 489 q^{83} - 1080 q^{85} + 330 q^{86} - 315 q^{88} + 900 q^{89} - 1000 q^{91} + 114 q^{92} + 990 q^{94} + 30 q^{95} + 1105 q^{97} - 1692 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 - q^4 + 15 * q^5 + 25 * q^7 + 42 * q^8 + 90 * q^10 - 15 * q^11 - 20 * q^13 - 75 * q^14 + 71 * q^16 - 144 * q^17 + 4 * q^19 + 15 * q^20 + 45 * q^22 + 114 * q^23 - 100 * q^25 - 120 * q^26 - 50 * q^28 + 30 * q^29 - 101 * q^31 - 45 * q^32 - 216 * q^34 + 750 * q^35 - 860 * q^37 + 6 * q^38 + 315 * q^40 - 30 * q^41 - 110 * q^43 + 30 * q^44 + 684 * q^46 - 330 * q^47 - 282 * q^49 + 300 * q^50 - 20 * q^52 - 1242 * q^53 - 450 * q^55 + 525 * q^56 - 90 * q^58 - 660 * q^59 + 376 * q^61 - 606 * q^62 + 866 * q^64 + 300 * q^65 + 250 * q^67 + 72 * q^68 + 1125 * q^70 + 720 * q^71 + 1570 * q^73 - 1290 * q^74 - 2 * q^76 + 375 * q^77 - 488 * q^79 + 2130 * q^80 - 180 * q^82 + 489 * q^83 - 1080 * q^85 + 330 * q^86 - 315 * q^88 + 900 * q^89 - 1000 * q^91 + 114 * q^92 + 990 * q^94 + 30 * q^95 + 1105 * q^97 - 1692 * q^98

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