LMFDB - Embedded newform 405.4.e.l.271.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,4,Mod(136,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 0]))

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

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

chi := DirichletCharacter("405.136");

S:= CuspForms(chi, 4);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$23.8957735523$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 5)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.00000 + 3.46410i) q^{2} +(-4.00000 + 6.92820i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-3.00000 - 5.19615i) q^{7} +O(q^{10})$	\(q+(2.00000 + 3.46410i) q^{2} +(-4.00000 + 6.92820i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-3.00000 - 5.19615i) q^{7} +20.0000 q^{10} +(-16.0000 - 27.7128i) q^{11} +(19.0000 - 32.9090i) q^{13} +(12.0000 - 20.7846i) q^{14} +(32.0000 + 55.4256i) q^{16} +26.0000 q^{17} +100.000 q^{19} +(20.0000 + 34.6410i) q^{20} +(64.0000 - 110.851i) q^{22} +(39.0000 - 67.5500i) q^{23} +(-12.5000 - 21.6506i) q^{25} +152.000 q^{26} +48.0000 q^{28} +(25.0000 + 43.3013i) q^{29} +(54.0000 - 93.5307i) q^{31} +(-128.000 + 221.703i) q^{32} +(52.0000 + 90.0666i) q^{34} -30.0000 q^{35} +266.000 q^{37} +(200.000 + 346.410i) q^{38} +(-11.0000 + 19.0526i) q^{41} +(-221.000 - 382.783i) q^{43} +256.000 q^{44} +312.000 q^{46} +(257.000 + 445.137i) q^{47} +(153.500 - 265.870i) q^{49} +(50.0000 - 86.6025i) q^{50} +(152.000 + 263.272i) q^{52} +2.00000 q^{53} -160.000 q^{55} +(-100.000 + 173.205i) q^{58} +(-250.000 + 433.013i) q^{59} +(259.000 + 448.601i) q^{61} +432.000 q^{62} -512.000 q^{64} +(-95.0000 - 164.545i) q^{65} +(-63.0000 + 109.119i) q^{67} +(-104.000 + 180.133i) q^{68} +(-60.0000 - 103.923i) q^{70} +412.000 q^{71} -878.000 q^{73} +(532.000 + 921.451i) q^{74} +(-400.000 + 692.820i) q^{76} +(-96.0000 + 166.277i) q^{77} +(-300.000 - 519.615i) q^{79} +320.000 q^{80} -88.0000 q^{82} +(-141.000 - 244.219i) q^{83} +(65.0000 - 112.583i) q^{85} +(884.000 - 1531.13i) q^{86} -150.000 q^{89} -228.000 q^{91} +(312.000 + 540.400i) q^{92} +(-1028.00 + 1780.55i) q^{94} +(250.000 - 433.013i) q^{95} +(-193.000 - 334.286i) q^{97} +1228.00 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 6 q^{7}+O(q^{10}) $ 2 * q + 4 * q^2 - 8 * q^4 + 5 * q^5 - 6 * q^7	$ 2 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 6 q^{7} + 40 q^{10} - 32 q^{11} + 38 q^{13} + 24 q^{14} + 64 q^{16} + 52 q^{17} + 200 q^{19} + 40 q^{20} + 128 q^{22} + 78 q^{23} - 25 q^{25} + 304 q^{26} + 96 q^{28} + 50 q^{29} + 108 q^{31} - 256 q^{32} + 104 q^{34} - 60 q^{35} + 532 q^{37} + 400 q^{38} - 22 q^{41} - 442 q^{43} + 512 q^{44} + 624 q^{46} + 514 q^{47} + 307 q^{49} + 100 q^{50} + 304 q^{52} + 4 q^{53} - 320 q^{55} - 200 q^{58} - 500 q^{59} + 518 q^{61} + 864 q^{62} - 1024 q^{64} - 190 q^{65} - 126 q^{67} - 208 q^{68} - 120 q^{70} + 824 q^{71} - 1756 q^{73} + 1064 q^{74} - 800 q^{76} - 192 q^{77} - 600 q^{79} + 640 q^{80} - 176 q^{82} - 282 q^{83} + 130 q^{85} + 1768 q^{86} - 300 q^{89} - 456 q^{91} + 624 q^{92} - 2056 q^{94} + 500 q^{95} - 386 q^{97} + 2456 q^{98}+O(q^{100}) $ 2 * q + 4 * q^2 - 8 * q^4 + 5 * q^5 - 6 * q^7 + 40 * q^10 - 32 * q^11 + 38 * q^13 + 24 * q^14 + 64 * q^16 + 52 * q^17 + 200 * q^19 + 40 * q^20 + 128 * q^22 + 78 * q^23 - 25 * q^25 + 304 * q^26 + 96 * q^28 + 50 * q^29 + 108 * q^31 - 256 * q^32 + 104 * q^34 - 60 * q^35 + 532 * q^37 + 400 * q^38 - 22 * q^41 - 442 * q^43 + 512 * q^44 + 624 * q^46 + 514 * q^47 + 307 * q^49 + 100 * q^50 + 304 * q^52 + 4 * q^53 - 320 * q^55 - 200 * q^58 - 500 * q^59 + 518 * q^61 + 864 * q^62 - 1024 * q^64 - 190 * q^65 - 126 * q^67 - 208 * q^68 - 120 * q^70 + 824 * q^71 - 1756 * q^73 + 1064 * q^74 - 800 * q^76 - 192 * q^77 - 600 * q^79 + 640 * q^80 - 176 * q^82 - 282 * q^83 + 130 * q^85 + 1768 * q^86 - 300 * q^89 - 456 * q^91 + 624 * q^92 - 2056 * q^94 + 500 * q^95 - 386 * q^97 + 2456 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/405\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$	2.00000	+	3.46410i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$2$	2.00000	+	3.46410i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$3$		0			0
$3$		0			0
$4$	−4.00000	+	6.92820i	−0.500000	+	0.866025i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$5$	2.50000	−	4.33013i	0.223607	−	0.387298i
$6$		0			0
$7$	−3.00000	−	5.19615i	−0.161985	−	0.280566i	0.773596	−	0.633680i	$-0.218456\pi$
$7$	−3.00000	−	5.19615i	−0.161985	−	0.280566i	−0.935580	+	0.353114i	$0.885123\pi$
$8$		0			0
$9$		0			0
$10$	20.0000			0.632456
$11$	−16.0000	−	27.7128i	−0.438562	−	0.759612i	0.559017	−	0.829156i	$-0.311179\pi$
$11$	−16.0000	−	27.7128i	−0.438562	−	0.759612i	−0.997579	+	0.0695447i	$0.977845\pi$
$12$		0			0
$13$	19.0000	−	32.9090i	0.405358	−	0.702100i	−0.589005	−	0.808129i	$-0.700480\pi$
$13$	19.0000	−	32.9090i	0.405358	−	0.702100i	0.994363	+	0.106029i	$0.0338136\pi$
$14$	12.0000	−	20.7846i	0.229081	−	0.396780i
$15$		0			0
$16$	32.0000	+	55.4256i	0.500000	+	0.866025i
$17$	26.0000			0.370937			0.185468	−	0.982650i	$-0.440620\pi$
$17$	26.0000			0.370937			0.185468	+	0.982650i	$0.440620\pi$
$18$		0			0
$19$	100.000			1.20745			0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000			1.20745			0.603726	+	0.797192i	$0.293682\pi$
$20$	20.0000	+	34.6410i	0.223607	+	0.387298i
$21$		0			0
$22$	64.0000	−	110.851i	0.620220	−	1.07425i
$23$	39.0000	−	67.5500i	0.353568	−	0.612398i	−0.633304	−	0.773903i	$-0.718302\pi$
$23$	39.0000	−	67.5500i	0.353568	−	0.612398i	0.986872	+	0.161506i	$0.0516350\pi$
$24$		0			0
$25$	−12.5000	−	21.6506i	−0.100000	−	0.173205i
$26$	152.000			1.14653
$27$		0			0
$28$	48.0000			0.323970
$29$	25.0000	+	43.3013i	0.160082	+	0.277270i	0.934898	−	0.354917i	$-0.115491\pi$
$29$	25.0000	+	43.3013i	0.160082	+	0.277270i	−0.774816	+	0.632187i	$0.782157\pi$
$30$		0			0
$31$	54.0000	−	93.5307i	0.312861	−	0.541891i	−0.666120	−	0.745845i	$-0.732046\pi$
$31$	54.0000	−	93.5307i	0.312861	−	0.541891i	0.978980	+	0.203954i	$0.0653793\pi$
$32$	−128.000	+	221.703i	−0.707107	+	1.22474i
$33$		0			0
$34$	52.0000	+	90.0666i	0.262292	+	0.454303i
$35$	−30.0000			−0.144884
$36$		0			0
$37$	266.000			1.18190			0.590948	−	0.806710i	$-0.298754\pi$
$37$	266.000			1.18190			0.590948	+	0.806710i	$0.298754\pi$
$38$	200.000	+	346.410i	0.853797	+	1.47882i
$39$		0			0
$40$		0			0
$41$	−11.0000	+	19.0526i	−0.0419003	+	0.0725734i	−0.886215	−	0.463274i	$-0.846675\pi$
$41$	−11.0000	+	19.0526i	−0.0419003	+	0.0725734i	0.844315	+	0.535848i	$0.180008\pi$
$42$		0			0
$43$	−221.000	−	382.783i	−0.783772	−	1.35753i	−0.929730	−	0.368242i	$-0.879960\pi$
$43$	−221.000	−	382.783i	−0.783772	−	1.35753i	0.145958	−	0.989291i	$-0.453373\pi$
$44$	256.000			0.877124
$45$		0			0
$46$	312.000			1.00004
$47$	257.000	+	445.137i	0.797602	+	1.38149i	0.921174	+	0.389152i	$0.127232\pi$
$47$	257.000	+	445.137i	0.797602	+	1.38149i	−0.123571	+	0.992336i	$0.539435\pi$
$48$		0			0
$49$	153.500	−	265.870i	0.447522	−	0.775131i
$50$	50.0000	−	86.6025i	0.141421	−	0.244949i
$51$		0			0
$52$	152.000	+	263.272i	0.405358	+	0.702100i
$53$	2.00000			0.00518342			0.00259171	−	0.999997i	$-0.499175\pi$
$53$	2.00000			0.00518342			0.00259171	+	0.999997i	$0.499175\pi$
$54$		0			0
$55$	−160.000			−0.392262
$56$		0			0
$57$		0			0
$58$	−100.000	+	173.205i	−0.226390	+	0.392120i
$59$	−250.000	+	433.013i	−0.551648	+	0.955482i	0.446508	+	0.894780i	$0.352667\pi$
$59$	−250.000	+	433.013i	−0.551648	+	0.955482i	−0.998156	+	0.0607026i	$0.980666\pi$
$60$		0			0
$61$	259.000	+	448.601i	0.543632	+	0.941598i	0.998692	+	0.0511373i	$0.0162846\pi$
$61$	259.000	+	448.601i	0.543632	+	0.941598i	−0.455060	+	0.890461i	$0.650382\pi$
$62$	432.000			0.884904
$63$		0			0
$64$	−512.000			−1.00000
$65$	−95.0000	−	164.545i	−0.181282	−	0.313989i
$66$		0			0
$67$	−63.0000	+	109.119i	−0.114876	+	0.198971i	−0.917730	−	0.397205i	$-0.869980\pi$
$67$	−63.0000	+	109.119i	−0.114876	+	0.198971i	0.802854	+	0.596175i	$0.203314\pi$
$68$	−104.000	+	180.133i	−0.185468	+	0.321241i
$69$		0			0
$70$	−60.0000	−	103.923i	−0.102448	−	0.177445i
$71$	412.000			0.688668			0.344334	−	0.938847i	$-0.388105\pi$
$71$	412.000			0.688668			0.344334	+	0.938847i	$0.388105\pi$
$72$		0			0
$73$	−878.000			−1.40770			−0.703850	−	0.710348i	$-0.748537\pi$
$73$	−878.000			−1.40770			−0.703850	+	0.710348i	$0.748537\pi$
$74$	532.000	+	921.451i	0.835726	+	1.44752i
$75$		0			0
$76$	−400.000	+	692.820i	−0.603726	+	1.04568i
$77$	−96.0000	+	166.277i	−0.142081	+	0.246091i
$78$		0			0
$79$	−300.000	−	519.615i	−0.427249	−	0.740016i	0.569379	−	0.822075i	$-0.307184\pi$
$79$	−300.000	−	519.615i	−0.427249	−	0.740016i	−0.996627	+	0.0820590i	$0.973850\pi$
$80$	320.000			0.447214
$81$		0			0
$82$	−88.0000			−0.118512
$83$	−141.000	−	244.219i	−0.186467	−	0.322970i	0.757603	−	0.652716i	$-0.226370\pi$
$83$	−141.000	−	244.219i	−0.186467	−	0.322970i	−0.944070	+	0.329745i	$0.893037\pi$
$84$		0			0
$85$	65.0000	−	112.583i	0.0829440	−	0.143663i
$86$	884.000	−	1531.13i	1.10842	−	1.91984i
$87$		0			0
$88$		0			0
$89$	−150.000			−0.178651			−0.0893257	−	0.996002i	$-0.528471\pi$
$89$	−150.000			−0.178651			−0.0893257	+	0.996002i	$0.528471\pi$
$90$		0			0
$91$	−228.000			−0.262647
$92$	312.000	+	540.400i	0.353568	+	0.612398i
$93$		0			0
$94$	−1028.00	+	1780.55i	−1.12798	+	1.95372i
$95$	250.000	−	433.013i	0.269994	−	0.467644i
$96$		0			0
$97$	−193.000	−	334.286i	−0.202022	−	0.349913i	0.747157	−	0.664647i	$-0.231418\pi$
$97$	−193.000	−	334.286i	−0.202022	−	0.349913i	−0.949180	+	0.314734i	$0.898085\pi$
$98$	1228.00			1.26578
$99$		0			0
$100$	200.000			0.200000
$101$	−351.000	−	607.950i	−0.345800	−	0.598943i	0.639699	−	0.768626i	$-0.279059\pi$
$101$	−351.000	−	607.950i	−0.345800	−	0.598943i	−0.985499	+	0.169682i	$0.945726\pi$
$102$		0			0
$103$	299.000	−	517.883i	0.286032	−	0.495423i	−0.686827	−	0.726821i	$-0.740997\pi$
$103$	299.000	−	517.883i	0.286032	−	0.495423i	0.972859	+	0.231399i	$0.0743301\pi$
$104$		0			0
$105$		0			0
$106$	4.00000	+	6.92820i	0.00366523	+	0.00634836i
$107$	−1194.00			−1.07877			−0.539385	−	0.842059i	$-0.681343\pi$
$107$	−1194.00			−1.07877			−0.539385	+	0.842059i	$0.681343\pi$
$108$		0			0
$109$	−550.000			−0.483307			−0.241653	−	0.970363i	$-0.577690\pi$
$109$	−550.000			−0.483307			−0.241653	+	0.970363i	$0.577690\pi$
$110$	−320.000	−	554.256i	−0.277371	−	0.480421i
$111$		0			0
$112$	192.000	−	332.554i	0.161985	−	0.280566i
$113$	−781.000	+	1352.73i	−0.650180	+	1.12614i	0.332899	+	0.942962i	$0.391973\pi$
$113$	−781.000	+	1352.73i	−0.650180	+	1.12614i	−0.983079	+	0.183182i	$0.941360\pi$
$114$		0			0
$115$	−195.000	−	337.750i	−0.158120	−	0.273873i
$116$	−400.000			−0.320164
$117$		0			0
$118$	−2000.00			−1.56030
$119$	−78.0000	−	135.100i	−0.0600861	−	0.104072i
$120$		0			0
$121$	153.500	−	265.870i	0.115327	−	0.199752i
$122$	−1036.00	+	1794.40i	−0.768812	+	1.33162i
$123$		0			0
$124$	432.000	+	748.246i	0.312861	+	0.541891i
$125$	−125.000			−0.0894427
$126$		0			0
$127$	1846.00			1.28981			0.644906	−	0.764262i	$-0.276897\pi$
$127$	1846.00			1.28981			0.644906	+	0.764262i	$0.276897\pi$
$128$		0			0
$129$		0			0
$130$	380.000	−	658.179i	0.256371	−	0.444047i
$131$	1104.00	−	1912.18i	0.736312	−	1.27533i	−0.217833	−	0.975986i	$-0.569899\pi$
$131$	1104.00	−	1912.18i	0.736312	−	1.27533i	0.954145	−	0.299344i	$-0.0967680\pi$
$132$		0			0
$133$	−300.000	−	519.615i	−0.195589	−	0.338770i
$134$	−504.000			−0.324918
$135$		0			0
$136$		0			0
$137$	1167.00	+	2021.30i	0.727763	+	1.26052i	0.957827	+	0.287347i	$0.0927733\pi$
$137$	1167.00	+	2021.30i	0.727763	+	1.26052i	−0.230064	+	0.973176i	$0.573893\pi$
$138$		0			0
$139$	350.000	−	606.218i	0.213573	−	0.369919i	−0.739257	−	0.673423i	$-0.764823\pi$
$139$	350.000	−	606.218i	0.213573	−	0.369919i	0.952830	+	0.303504i	$0.0981566\pi$
$140$	120.000	−	207.846i	0.0724418	−	0.125473i
$141$		0			0
$142$	824.000	+	1427.21i	0.486962	+	0.843442i
$143$	−1216.00			−0.711098
$144$		0			0
$145$	250.000			0.143182
$146$	−1756.00	−	3041.48i	−0.995394	−	1.72407i
$147$		0			0
$148$	−1064.00	+	1842.90i	−0.590948	+	1.02355i
$149$	−1025.00	+	1775.35i	−0.563566	+	0.976124i	0.433616	+	0.901098i	$0.357237\pi$
$149$	−1025.00	+	1775.35i	−0.563566	+	0.976124i	−0.997182	+	0.0750264i	$0.976096\pi$
$150$		0			0
$151$	−926.000	−	1603.88i	−0.499052	−	0.864383i	0.500948	−	0.865478i	$-0.332985\pi$
$151$	−926.000	−	1603.88i	−0.499052	−	0.864383i	−0.999999	+	0.00109462i	$0.999652\pi$
$152$		0			0
$153$		0			0
$154$	−768.000			−0.401865
$155$	−270.000	−	467.654i	−0.139916	−	0.242341i
$156$		0			0
$157$	1247.00	−	2159.87i	0.633894	−	1.09794i	−0.352854	−	0.935678i	$-0.614789\pi$
$157$	1247.00	−	2159.87i	0.633894	−	1.09794i	0.986748	−	0.162259i	$-0.0518780\pi$
$158$	1200.00	−	2078.46i	0.604221	−	1.04654i
$159$		0			0
$160$	640.000	+	1108.51i	0.316228	+	0.547723i
$161$	−468.000			−0.229090
$162$		0			0
$163$	2762.00			1.32722			0.663609	−	0.748080i	$-0.269024\pi$
$163$	2762.00			1.32722			0.663609	+	0.748080i	$0.269024\pi$
$164$	−88.0000	−	152.420i	−0.0419003	−	0.0725734i
$165$		0			0
$166$	564.000	−	976.877i	0.263704	−	0.456749i
$167$	−1563.00	+	2707.20i	−0.724243	+	1.25443i	0.235042	+	0.971985i	$0.424477\pi$
$167$	−1563.00	+	2707.20i	−0.724243	+	1.25443i	−0.959285	+	0.282440i	$0.908856\pi$
$168$		0			0
$169$	376.500	+	652.117i	0.171370	+	0.296822i
$170$	520.000			0.234601
$171$		0			0
$172$	3536.00			1.56754
$173$	39.0000	+	67.5500i	0.0171394	+	0.0296863i	0.874468	−	0.485083i	$-0.161211\pi$
$173$	39.0000	+	67.5500i	0.0171394	+	0.0296863i	−0.857328	+	0.514770i	$0.827877\pi$
$174$		0			0
$175$	−75.0000	+	129.904i	−0.0323970	+	0.0561132i
$176$	1024.00	−	1773.62i	0.438562	−	0.759612i
$177$		0			0
$178$	−300.000	−	519.615i	−0.126326	−	0.218802i
$179$	−1300.00			−0.542830			−0.271415	−	0.962462i	$-0.587492\pi$
$179$	−1300.00			−0.542830			−0.271415	+	0.962462i	$0.587492\pi$
$180$		0			0
$181$	1742.00			0.715369			0.357685	−	0.933842i	$-0.383566\pi$
$181$	1742.00			0.715369			0.357685	+	0.933842i	$0.383566\pi$
$182$	−456.000	−	789.815i	−0.185720	−	0.321676i
$183$		0			0
$184$		0			0
$185$	665.000	−	1151.81i	0.264280	−	0.457746i
$186$		0			0
$187$	−416.000	−	720.533i	−0.162679	−	0.281768i
$188$	−4112.00			−1.59520
$189$		0			0
$190$	2000.00			0.763659
$191$	−1886.00	−	3266.65i	−0.714483	−	1.23752i	−0.963159	−	0.268933i	$-0.913329\pi$
$191$	−1886.00	−	3266.65i	−0.714483	−	1.23752i	0.248676	−	0.968587i	$-0.420004\pi$
$192$		0			0
$193$	179.000	−	310.037i	0.0667601	−	0.115632i	−0.830713	−	0.556700i	$-0.812067\pi$
$193$	179.000	−	310.037i	0.0667601	−	0.115632i	0.897473	+	0.441069i	$0.145400\pi$
$194$	772.000	−	1337.14i	0.285703	−	0.494852i
$195$		0			0
$196$	1228.00	+	2126.96i	0.447522	+	0.775131i
$197$	−2214.00			−0.800716			−0.400358	−	0.916359i	$-0.631114\pi$
$197$	−2214.00			−0.800716			−0.400358	+	0.916359i	$0.631114\pi$
$198$		0			0
$199$	−2600.00			−0.926176			−0.463088	−	0.886312i	$-0.653259\pi$
$199$	−2600.00			−0.926176			−0.463088	+	0.886312i	$0.653259\pi$
$200$		0			0
$201$		0			0
$202$	1404.00	−	2431.80i	0.489035	−	0.847034i
$203$	150.000	−	259.808i	0.0518618	−	0.0898272i
$204$		0			0
$205$	55.0000	+	95.2628i	0.0187384	+	0.0324558i
$206$	2392.00			0.809022
$207$		0			0
$208$	2432.00			0.810716
$209$	−1600.00	−	2771.28i	−0.529542	−	0.917194i
$210$		0			0
$211$	584.000	−	1011.52i	0.190541	−	0.330027i	−0.754888	−	0.655853i	$-0.772309\pi$
$211$	584.000	−	1011.52i	0.190541	−	0.330027i	0.945430	+	0.325826i	$0.105642\pi$
$212$	−8.00000	+	13.8564i	−0.00259171	+	0.00448897i
$213$		0			0
$214$	−2388.00	−	4136.14i	−0.762805	−	1.32122i
$215$	−2210.00			−0.701027
$216$		0			0
$217$	−648.000			−0.202715
$218$	−1100.00	−	1905.26i	−0.341750	−	0.591928i
$219$		0			0
$220$	640.000	−	1108.51i	0.196131	−	0.339709i
$221$	494.000	−	855.633i	0.150362	−	0.260435i
$222$		0			0
$223$	3239.00	+	5610.11i	0.972643	+	1.68467i	0.687502	+	0.726182i	$0.258707\pi$
$223$	3239.00	+	5610.11i	0.972643	+	1.68467i	0.285141	+	0.958486i	$0.407959\pi$
$224$	1536.00			0.458162
$225$		0			0
$226$	−6248.00			−1.83899
$227$	−323.000	−	559.452i	−0.0944417	−	0.163578i	0.814934	−	0.579554i	$-0.196773\pi$
$227$	−323.000	−	559.452i	−0.0944417	−	0.163578i	−0.909375	+	0.415976i	$0.863440\pi$
$228$		0			0
$229$	−1875.00	+	3247.60i	−0.541063	+	0.937149i	0.457780	+	0.889065i	$0.348645\pi$
$229$	−1875.00	+	3247.60i	−0.541063	+	0.937149i	−0.998843	+	0.0480836i	$0.984689\pi$
$230$	780.000	−	1351.00i	0.223616	−	0.387314i
$231$		0			0
$232$		0			0
$233$	1482.00			0.416691			0.208346	−	0.978055i	$-0.433192\pi$
$233$	1482.00			0.416691			0.208346	+	0.978055i	$0.433192\pi$
$234$		0			0
$235$	2570.00			0.713397
$236$	−2000.00	−	3464.10i	−0.551648	−	0.955482i
$237$		0			0
$238$	312.000	−	540.400i	0.0849746	−	0.147180i
$239$	−700.000	+	1212.44i	−0.189453	+	0.328142i	−0.945068	−	0.326874i	$-0.894005\pi$
$239$	−700.000	+	1212.44i	−0.189453	+	0.328142i	0.755615	+	0.655016i	$0.227338\pi$
$240$		0			0
$241$	−1511.00	−	2617.13i	−0.403867	−	0.699519i	0.590321	−	0.807168i	$-0.299001\pi$
$241$	−1511.00	−	2617.13i	−0.403867	−	0.699519i	−0.994189	+	0.107649i	$0.965668\pi$
$242$	1228.00			0.326194
$243$		0			0
$244$	−4144.00			−1.08726
$245$	−767.500	−	1329.35i	−0.200138	−	0.346649i
$246$		0			0
$247$	1900.00	−	3290.90i	0.489450	−	0.847752i
$248$		0			0
$249$		0			0
$250$	−250.000	−	433.013i	−0.0632456	−	0.109545i
$251$	−1248.00			−0.313837			−0.156918	−	0.987612i	$-0.550156\pi$
$251$	−1248.00			−0.313837			−0.156918	+	0.987612i	$0.550156\pi$
$252$		0			0
$253$	−2496.00			−0.620246
$254$	3692.00	+	6394.73i	0.912034	+	1.57969i
$255$		0			0
$256$	−2048.00	+	3547.24i	−0.500000	+	0.866025i
$257$	−1053.00	+	1823.85i	−0.255581	+	0.442679i	−0.965053	−	0.262054i	$-0.915600\pi$
$257$	−1053.00	+	1823.85i	−0.255581	+	0.442679i	0.709472	+	0.704734i	$0.248933\pi$
$258$		0			0
$259$	−798.000	−	1382.18i	−0.191449	−	0.331600i
$260$	1520.00			0.362563
$261$		0			0
$262$	8832.00			2.08261
$263$	1819.00	+	3150.60i	0.426480	+	0.738686i	0.996557	−	0.0829055i	$-0.0264200\pi$
$263$	1819.00	+	3150.60i	0.426480	+	0.738686i	−0.570077	+	0.821591i	$0.693087\pi$
$264$		0			0
$265$	5.00000	−	8.66025i	0.00115905	−	0.00200753i
$266$	1200.00	−	2078.46i	0.276604	−	0.479093i
$267$		0			0
$268$	−504.000	−	872.954i	−0.114876	−	0.198971i
$269$	−6550.00			−1.48461			−0.742306	−	0.670061i	$-0.766268\pi$
$269$	−6550.00			−1.48461			−0.742306	+	0.670061i	$0.766268\pi$
$270$		0			0
$271$	−4388.00			−0.983587			−0.491793	−	0.870712i	$-0.663658\pi$
$271$	−4388.00			−0.983587			−0.491793	+	0.870712i	$0.663658\pi$
$272$	832.000	+	1441.07i	0.185468	+	0.321241i
$273$		0			0
$274$	−4668.00	+	8085.21i	−1.02921	+	1.78265i
$275$	−400.000	+	692.820i	−0.0877124	+	0.151922i
$276$		0			0
$277$	−273.000	−	472.850i	−0.0592165	−	0.102566i	0.834897	−	0.550406i	$-0.185527\pi$
$277$	−273.000	−	472.850i	−0.0592165	−	0.102566i	−0.894114	+	0.447840i	$0.852194\pi$
$278$	2800.00			0.604075
$279$		0			0
$280$		0			0
$281$	3429.00	+	5939.20i	0.727961	+	1.26087i	0.957744	+	0.287623i	$0.0928651\pi$
$281$	3429.00	+	5939.20i	0.727961	+	1.26087i	−0.229783	+	0.973242i	$0.573802\pi$
$282$		0			0
$283$	−4641.00	+	8038.45i	−0.974837	+	1.68847i	−0.294364	+	0.955693i	$0.595108\pi$
$283$	−4641.00	+	8038.45i	−0.974837	+	1.68847i	−0.680473	+	0.732774i	$0.738225\pi$
$284$	−1648.00	+	2854.42i	−0.344334	+	0.596404i
$285$		0			0
$286$	−2432.00	−	4212.35i	−0.502822	−	0.870914i
$287$	132.000			0.0271488
$288$		0			0
$289$	−4237.00			−0.862406
$290$	500.000	+	866.025i	0.101245	+	0.175361i
$291$		0			0
$292$	3512.00	−	6082.96i	0.703850	−	1.21910i
$293$	−2421.00	+	4193.30i	−0.482718	+	0.836092i	−0.999803	−	0.0198420i	$-0.993684\pi$
$293$	−2421.00	+	4193.30i	−0.482718	+	0.836092i	0.517085	+	0.855934i	$0.327017\pi$
$294$		0			0
$295$	1250.00	+	2165.06i	0.246704	+	0.427305i
$296$		0			0
$297$		0			0
$298$	−8200.00			−1.59400
$299$	−1482.00	−	2566.90i	−0.286643	−	0.496480i
$300$		0			0
$301$	−1326.00	+	2296.70i	−0.253918	+	0.439799i
$302$	3704.00	−	6415.52i	0.705766	−	1.22242i
$303$		0			0
$304$	3200.00	+	5542.56i	0.603726	+	1.04568i
$305$	2590.00			0.486239
$306$		0			0
$307$	−2594.00			−0.482239			−0.241120	−	0.970495i	$-0.577515\pi$
$307$	−2594.00			−0.482239			−0.241120	+	0.970495i	$0.577515\pi$
$308$	−768.000	−	1330.22i	−0.142081	−	0.246091i
$309$		0			0
$310$	1080.00	−	1870.61i	0.197871	−	0.342722i
$311$	−3666.00	+	6349.70i	−0.668424	+	1.15774i	0.309921	+	0.950762i	$0.399697\pi$
$311$	−3666.00	+	6349.70i	−0.668424	+	1.15774i	−0.978345	+	0.206982i	$0.933636\pi$
$312$		0			0
$313$	−781.000	−	1352.73i	−0.141037	−	0.244284i	0.786850	−	0.617144i	$-0.211710\pi$
$313$	−781.000	−	1352.73i	−0.141037	−	0.244284i	−0.927888	+	0.372860i	$0.878377\pi$
$314$	9976.00			1.79292
$315$		0			0
$316$	4800.00			0.854497
$317$	−713.000	−	1234.95i	−0.126328	−	0.218807i	0.795923	−	0.605398i	$-0.206986\pi$
$317$	−713.000	−	1234.95i	−0.126328	−	0.218807i	−0.922251	+	0.386591i	$0.873653\pi$
$318$		0			0
$319$	800.000	−	1385.64i	0.140412	−	0.243201i
$320$	−1280.00	+	2217.03i	−0.223607	+	0.387298i
$321$		0			0
$322$	−936.000	−	1621.20i	−0.161991	−	0.280577i
$323$	2600.00			0.447888
$324$		0			0
$325$	−950.000			−0.162143
$326$	5524.00	+	9567.85i	0.938485	+	1.62550i
$327$		0			0
$328$		0			0
$329$	1542.00	−	2670.82i	0.258399	−	0.447560i
$330$		0			0
$331$	2004.00	+	3471.03i	0.332779	+	0.576390i	0.983056	−	0.183308i	$-0.0586804\pi$
$331$	2004.00	+	3471.03i	0.332779	+	0.576390i	−0.650277	+	0.759697i	$0.725347\pi$
$332$	2256.00			0.372934
$333$		0			0
$334$	−12504.0			−2.04847
$335$	315.000	+	545.596i	0.0513740	+	0.0889824i
$336$		0			0
$337$	−4433.00	+	7678.18i	−0.716561	+	1.24112i	0.245794	+	0.969322i	$0.420951\pi$
$337$	−4433.00	+	7678.18i	−0.716561	+	1.24112i	−0.962355	+	0.271797i	$0.912382\pi$
$338$	−1506.00	+	2608.47i	−0.242354	+	0.419769i
$339$		0			0
$340$	520.000	+	900.666i	0.0829440	+	0.143663i
$341$	−3456.00			−0.548835
$342$		0			0
$343$	−3900.00			−0.613936
$344$		0			0
$345$		0			0
$346$	−156.000	+	270.200i	−0.0242388	+	0.0419828i
$347$	857.000	−	1484.37i	0.132583	−	0.229640i	−0.792089	−	0.610406i	$-0.791006\pi$
$347$	857.000	−	1484.37i	0.132583	−	0.229640i	0.924671	+	0.380766i	$0.124340\pi$
$348$		0			0
$349$	−575.000	−	995.929i	−0.0881921	−	0.152753i	0.818555	−	0.574428i	$-0.194776\pi$
$349$	−575.000	−	995.929i	−0.0881921	−	0.152753i	−0.906747	+	0.421675i	$0.861442\pi$
$350$	−600.000			−0.0916324
$351$		0			0
$352$	8192.00			1.24044
$353$	2199.00	+	3808.78i	0.331561	+	0.574280i	0.982818	−	0.184577i	$-0.0590915\pi$
$353$	2199.00	+	3808.78i	0.331561	+	0.574280i	−0.651257	+	0.758857i	$0.725758\pi$
$354$		0			0
$355$	1030.00	−	1784.01i	0.153991	−	0.266720i
$356$	600.000	−	1039.23i	0.0893257	−	0.154717i
$357$		0			0
$358$	−2600.00	−	4503.33i	−0.383839	−	0.664828i
$359$	1800.00			0.264625			0.132312	−	0.991208i	$-0.457760\pi$
$359$	1800.00			0.264625			0.132312	+	0.991208i	$0.457760\pi$
$360$		0			0
$361$	3141.00			0.457938
$362$	3484.00	+	6034.47i	0.505842	+	0.876145i
$363$		0			0
$364$	912.000	−	1579.63i	0.131324	−	0.227459i
$365$	−2195.00	+	3801.85i	−0.314771	+	0.545200i
$366$		0			0
$367$	2937.00	+	5087.03i	0.417739	+	0.723545i	0.995712	−	0.0925111i	$-0.0294894\pi$
$367$	2937.00	+	5087.03i	0.417739	+	0.723545i	−0.577973	+	0.816056i	$0.696156\pi$
$368$	4992.00			0.707136
$369$		0			0
$370$	5320.00			0.747496
$371$	−6.00000	−	10.3923i	−0.000839635	−	0.00145429i
$372$		0			0
$373$	1039.00	−	1799.60i	0.144229	−	0.249812i	−0.784856	−	0.619678i	$-0.787263\pi$
$373$	1039.00	−	1799.60i	0.144229	−	0.249812i	0.929085	+	0.369866i	$0.120597\pi$
$374$	1664.00	−	2882.13i	0.230063	−	0.398480i
$375$		0			0
$376$		0			0
$377$	1900.00			0.259562
$378$		0			0
$379$	7900.00			1.07070			0.535351	−	0.844630i	$-0.320179\pi$
$379$	7900.00			1.07070			0.535351	+	0.844630i	$0.320179\pi$
$380$	2000.00	+	3464.10i	0.269994	+	0.467644i
$381$		0			0
$382$	7544.00	−	13066.6i	1.01043	−	1.75012i
$383$	3759.00	−	6510.78i	0.501504	−	0.868630i	−0.498495	−	0.866893i	$-0.666114\pi$
$383$	3759.00	−	6510.78i	0.501504	−	0.868630i	0.999998	−	0.00173723i	$-0.000552976\pi$
$384$		0			0
$385$	480.000	+	831.384i	0.0635404	+	0.110055i
$386$	1432.00			0.188826
$387$		0			0
$388$	3088.00			0.404045
$389$	975.000	+	1688.75i	0.127081	+	0.220111i	0.922544	−	0.385891i	$-0.126106\pi$
$389$	975.000	+	1688.75i	0.127081	+	0.220111i	−0.795464	+	0.606001i	$0.792773\pi$
$390$		0			0
$391$	1014.00	−	1756.30i	0.131151	−	0.227161i
$392$		0			0
$393$		0			0
$394$	−4428.00	−	7669.52i	−0.566191	−	0.980672i
$395$	−3000.00			−0.382143
$396$		0			0
$397$	13786.0			1.74282			0.871410	−	0.490555i	$-0.163206\pi$
$397$	13786.0			1.74282			0.871410	+	0.490555i	$0.163206\pi$
$398$	−5200.00	−	9006.66i	−0.654906	−	1.13433i
$399$		0			0
$400$	800.000	−	1385.64i	0.100000	−	0.173205i
$401$	−3201.00	+	5544.29i	−0.398629	+	0.690446i	−0.993557	−	0.113333i	$-0.963847\pi$
$401$	−3201.00	+	5544.29i	−0.398629	+	0.690446i	0.594928	+	0.803779i	$0.297181\pi$
$402$		0			0
$403$	−2052.00	−	3554.17i	−0.253641	−	0.439319i
$404$	5616.00			0.691600
$405$		0			0
$406$	1200.00			0.146687
$407$	−4256.00	−	7371.61i	−0.518334	−	0.897781i
$408$		0			0
$409$	−5575.00	+	9656.18i	−0.674000	+	1.16740i	0.302760	+	0.953067i	$0.402092\pi$
$409$	−5575.00	+	9656.18i	−0.674000	+	1.16740i	−0.976760	+	0.214335i	$0.931241\pi$
$410$	−220.000	+	381.051i	−0.0265001	+	0.0458995i
$411$		0			0
$412$	2392.00	+	4143.07i	0.286032	+	0.495423i
$413$	3000.00			0.357434
$414$		0			0
$415$	−1410.00			−0.166781
$416$	4864.00	+	8424.70i	0.573263	+	0.992920i
$417$		0			0
$418$	6400.00	−	11085.1i	0.748886	−	1.29711i
$419$	6850.00	−	11864.5i	0.798674	−	1.38334i	−0.121806	−	0.992554i	$-0.538869\pi$
$419$	6850.00	−	11864.5i	0.798674	−	1.38334i	0.920480	−	0.390790i	$-0.127798\pi$
$420$		0			0
$421$	2719.00	+	4709.45i	0.314765	+	0.545189i	0.979387	−	0.201991i	$-0.0647410\pi$
$421$	2719.00	+	4709.45i	0.314765	+	0.545189i	−0.664623	+	0.747179i	$0.731408\pi$
$422$	4672.00			0.538932
$423$		0			0
$424$		0			0
$425$	−325.000	−	562.917i	−0.0370937	−	0.0642481i
$426$		0			0
$427$	1554.00	−	2691.61i	0.176120	−	0.305049i
$428$	4776.00	−	8272.27i	0.539385	−	0.934242i
$429$		0			0
$430$	−4420.00	−	7655.66i	−0.495701	−	0.858579i
$431$	7692.00			0.859653			0.429827	−	0.902911i	$-0.358575\pi$
$431$	7692.00			0.859653			0.429827	+	0.902911i	$0.358575\pi$
$432$		0			0
$433$	−1118.00			−0.124082			−0.0620412	−	0.998074i	$-0.519761\pi$
$433$	−1118.00			−0.124082			−0.0620412	+	0.998074i	$0.519761\pi$
$434$	−1296.00	−	2244.74i	−0.143341	−	0.248274i
$435$		0			0
$436$	2200.00	−	3810.51i	0.241653	−	0.418556i
$437$	3900.00	−	6755.00i	0.426916	−	0.739440i
$438$		0			0
$439$	1300.00	+	2251.67i	0.141334	+	0.244798i	0.927999	−	0.372582i	$-0.121528\pi$
$439$	1300.00	+	2251.67i	0.141334	+	0.244798i	−0.786665	+	0.617380i	$0.788194\pi$
$440$		0			0
$441$		0			0
$442$	3952.00			0.425288
$443$	5979.00	+	10355.9i	0.641243	+	1.11067i	0.985155	+	0.171664i	$0.0549145\pi$
$443$	5979.00	+	10355.9i	0.641243	+	1.11067i	−0.343912	+	0.939002i	$0.611752\pi$
$444$		0			0
$445$	−375.000	+	649.519i	−0.0399477	+	0.0691914i
$446$	−12956.0	+	22440.5i	−1.37553	+	2.38248i
$447$		0			0
$448$	1536.00	+	2660.43i	0.161985	+	0.280566i
$449$	−17050.0			−1.79207			−0.896035	−	0.443984i	$-0.853565\pi$
$449$	−17050.0			−1.79207			−0.896035	+	0.443984i	$0.853565\pi$
$450$		0			0
$451$	704.000			0.0735035
$452$	−6248.00	−	10821.9i	−0.650180	−	1.12614i
$453$		0			0
$454$	1292.00	−	2237.81i	0.133561	−	0.231334i
$455$	−570.000	+	987.269i	−0.0587297	+	0.101723i
$456$		0			0
$457$	4747.00	+	8222.05i	0.485898	+	0.841600i	0.999869	−	0.0162080i	$-0.00515939\pi$
$457$	4747.00	+	8222.05i	0.485898	+	0.841600i	−0.513971	+	0.857808i	$0.671826\pi$
$458$	−15000.0			−1.53036
$459$		0			0
$460$	3120.00			0.316241
$461$	5709.00	+	9888.28i	0.576778	+	0.999009i	0.995846	+	0.0910539i	$0.0290235\pi$
$461$	5709.00	+	9888.28i	0.576778	+	0.999009i	−0.419068	+	0.907955i	$0.637643\pi$
$462$		0			0
$463$	−3981.00	+	6895.29i	−0.399596	+	0.692120i	−0.993676	−	0.112286i	$-0.964183\pi$
$463$	−3981.00	+	6895.29i	−0.399596	+	0.692120i	0.594080	+	0.804406i	$0.297516\pi$
$464$	−1600.00	+	2771.28i	−0.160082	+	0.277270i
$465$		0			0
$466$	2964.00	+	5133.80i	0.294645	+	0.510340i
$467$	6526.00			0.646654			0.323327	−	0.946287i	$-0.395199\pi$
$467$	6526.00			0.646654			0.323327	+	0.946287i	$0.395199\pi$
$468$		0			0
$469$	756.000			0.0744325
$470$	5140.00	+	8902.74i	0.504448	+	0.873729i
$471$		0			0
$472$		0			0
$473$	−7072.00	+	12249.1i	−0.687465	+	1.19072i
$474$		0			0
$475$	−1250.00	−	2165.06i	−0.120745	−	0.209137i
$476$	1248.00			0.120172
$477$		0			0
$478$	−5600.00			−0.535854
$479$	−8700.00	−	15068.8i	−0.829881	−	1.43740i	−0.898131	−	0.439728i	$-0.855075\pi$
$479$	−8700.00	−	15068.8i	−0.829881	−	1.43740i	0.0682495	−	0.997668i	$-0.478259\pi$
$480$		0			0
$481$	5054.00	−	8753.78i	0.479091	−	0.829809i
$482$	6044.00	−	10468.5i	0.571155	−	0.989269i
$483$		0			0
$484$	1228.00	+	2126.96i	0.115327	+	0.199752i
$485$	−1930.00			−0.180694
$486$		0			0
$487$	1166.00			0.108494			0.0542469	−	0.998528i	$-0.482724\pi$
$487$	1166.00			0.108494			0.0542469	+	0.998528i	$0.482724\pi$
$488$		0			0
$489$		0			0
$490$	3070.00	−	5317.40i	0.283038	−	0.490236i
$491$	−3536.00	+	6124.53i	−0.325005	+	0.562925i	−0.981513	−	0.191394i	$-0.938699\pi$
$491$	−3536.00	+	6124.53i	−0.325005	+	0.562925i	0.656508	+	0.754319i	$0.272033\pi$
$492$		0			0
$493$	650.000	+	1125.83i	0.0593804	+	0.102850i
$494$	15200.0			1.38437
$495$		0			0
$496$	6912.00			0.625722
$497$	−1236.00	−	2140.81i	−0.111554	−	0.193217i
$498$		0			0
$499$	−50.0000	+	86.6025i	−0.00448559	+	0.00776926i	−0.868259	−	0.496110i	$-0.834761\pi$
$499$	−50.0000	+	86.6025i	−0.00448559	+	0.00776926i	0.863774	+	0.503880i	$0.168094\pi$
$500$	500.000	−	866.025i	0.0447214	−	0.0774597i
$501$		0			0
$502$	−2496.00	−	4323.20i	−0.221916	−	0.384370i
$503$	2602.00			0.230651			0.115325	−	0.993328i	$-0.463209\pi$
$503$	2602.00			0.230651			0.115325	+	0.993328i	$0.463209\pi$
$504$		0			0
$505$	−3510.00			−0.309293
$506$	−4992.00	−	8646.40i	−0.438580	−	0.759643i
$507$		0			0
$508$	−7384.00	+	12789.5i	−0.644906	+	1.11701i
$509$	−5575.00	+	9656.18i	−0.485476	+	0.840870i	−0.999861	−	0.0166899i	$-0.994687\pi$
$509$	−5575.00	+	9656.18i	−0.485476	+	0.840870i	0.514384	+	0.857560i	$0.328021\pi$
$510$		0			0
$511$	2634.00	+	4562.22i	0.228026	+	0.394953i
$512$	−16384.0			−1.41421
$513$		0			0
$514$	−8424.00			−0.722892
$515$	−1495.00	−	2589.42i	−0.127918	−	0.221560i
$516$		0			0
$517$	8224.00	−	14244.4i	0.699596	−	1.21174i
$518$	3192.00	−	5528.71i	0.270750	−	0.468953i
$519$		0			0
$520$		0			0
$521$	−3638.00			−0.305919			−0.152959	−	0.988232i	$-0.548880\pi$
$521$	−3638.00			−0.305919			−0.152959	+	0.988232i	$0.548880\pi$
$522$		0			0
$523$	−2078.00			−0.173737			−0.0868686	−	0.996220i	$-0.527686\pi$
$523$	−2078.00			−0.173737			−0.0868686	+	0.996220i	$0.527686\pi$
$524$	8832.00	+	15297.5i	0.736312	+	1.27533i
$525$		0			0
$526$	−7276.00	+	12602.4i	−0.603134	+	1.04466i
$527$	1404.00	−	2431.80i	0.116052	−	0.201007i
$528$		0			0
$529$	3041.50	+	5268.03i	0.249979	+	0.432977i
$530$	40.0000			0.00327828
$531$		0			0
$532$	4800.00			0.391177
$533$	418.000	+	723.997i	0.0339692	+	0.0588364i
$534$		0			0
$535$	−2985.00	+	5170.17i	−0.241220	+	0.417806i
$536$		0			0
$537$		0			0
$538$	−13100.0	−	22689.9i	−1.04978	−	1.81827i
$539$	−9824.00			−0.785064
$540$		0			0
$541$	5622.00			0.446781			0.223391	−	0.974729i	$-0.428287\pi$
$541$	5622.00			0.446781			0.223391	+	0.974729i	$0.428287\pi$
$542$	−8776.00	−	15200.5i	−0.695501	−	1.20464i
$543$		0			0
$544$	−3328.00	+	5764.27i	−0.262292	+	0.454303i
$545$	−1375.00	+	2381.57i	−0.108071	+	0.187184i
$546$		0			0
$547$	−8243.00	−	14277.3i	−0.644324	−	1.11600i	−0.984457	−	0.175625i	$-0.943805\pi$
$547$	−8243.00	−	14277.3i	−0.644324	−	1.11600i	0.340133	−	0.940377i	$-0.389528\pi$
$548$	−18672.0			−1.45553
$549$		0			0
$550$	−3200.00			−0.248088
$551$	2500.00	+	4330.13i	0.193291	+	0.334791i
$552$		0			0
$553$	−1800.00	+	3117.69i	−0.138416	+	0.239743i
$554$	1092.00	−	1891.40i	0.0837448	−	0.145050i
$555$		0			0
$556$	2800.00	+	4849.74i	0.213573	+	0.369919i
$557$	11706.0			0.890483			0.445242	−	0.895410i	$-0.353118\pi$
$557$	11706.0			0.890483			0.445242	+	0.895410i	$0.353118\pi$
$558$		0			0
$559$	−16796.0			−1.27083
$560$	−960.000	−	1662.77i	−0.0724418	−	0.125473i
$561$		0			0
$562$	−13716.0	+	23756.8i	−1.02949	+	1.78313i
$563$	12519.0	−	21683.5i	0.937146	−	1.62318i	0.166383	−	0.986061i	$-0.446791\pi$
$563$	12519.0	−	21683.5i	0.937146	−	1.62318i	0.770763	−	0.637123i	$-0.219875\pi$
$564$		0			0
$565$	3905.00	+	6763.66i	0.290769	+	0.503627i
$566$	−37128.0			−2.75725
$567$		0			0
$568$		0			0
$569$	−8775.00	−	15198.7i	−0.646515	−	1.11980i	−0.983949	−	0.178448i	$-0.942892\pi$
$569$	−8775.00	−	15198.7i	−0.646515	−	1.11980i	0.337434	−	0.941349i	$-0.390441\pi$
$570$		0			0
$571$	−5356.00	+	9276.86i	−0.392542	+	0.679903i	−0.992784	−	0.119915i	$-0.961738\pi$
$571$	−5356.00	+	9276.86i	−0.392542	+	0.679903i	0.600242	+	0.799819i	$0.295071\pi$
$572$	4864.00	−	8424.70i	0.355549	−	0.615829i
$573$		0			0
$574$	264.000	+	457.261i	0.0191971	+	0.0332504i
$575$	−1950.00			−0.141427
$576$		0			0
$577$	−13654.0			−0.985136			−0.492568	−	0.870274i	$-0.663942\pi$
$577$	−13654.0			−0.985136			−0.492568	+	0.870274i	$0.663942\pi$
$578$	−8474.00	−	14677.4i	−0.609813	−	1.05623i
$579$		0			0
$580$	−1000.00	+	1732.05i	−0.0715909	+	0.123999i
$581$	−846.000	+	1465.31i	−0.0604096	+	0.104633i
$582$		0			0
$583$	−32.0000	−	55.4256i	−0.00227325	−	0.00393738i
$584$		0			0
$585$		0			0
$586$	−19368.0			−1.36533
$587$	−7083.00	−	12268.1i	−0.498035	−	0.862622i	0.501962	−	0.864890i	$-0.332612\pi$
$587$	−7083.00	−	12268.1i	−0.498035	−	0.862622i	−0.999997	+	0.00226720i	$0.999278\pi$
$588$		0			0
$589$	5400.00	−	9353.07i	0.377764	−	0.654307i
$590$	−5000.00	+	8660.25i	−0.348893	+	0.604300i
$591$		0			0
$592$	8512.00	+	14743.2i	0.590948	+	1.02355i
$593$	17842.0			1.23555			0.617777	−	0.786354i	$-0.288034\pi$
$593$	17842.0			1.23555			0.617777	+	0.786354i	$0.288034\pi$
$594$		0			0
$595$	−780.000			−0.0537427
$596$	−8200.00	−	14202.8i	−0.563566	−	0.976124i
$597$		0			0
$598$	5928.00	−	10267.6i	0.405374	−	0.702129i
$599$	8800.00	−	15242.0i	0.600264	−	1.03969i	−0.392517	−	0.919745i	$-0.628395\pi$
$599$	8800.00	−	15242.0i	0.600264	−	1.03969i	0.992781	−	0.119943i	$-0.0382712\pi$
$600$		0			0
$601$	−13651.0	−	23644.2i	−0.926516	−	1.60477i	−0.789105	−	0.614258i	$-0.789455\pi$
$601$	−13651.0	−	23644.2i	−0.926516	−	1.60477i	−0.137410	−	0.990514i	$-0.543878\pi$
$602$	−10608.0			−0.718189
$603$		0			0
$604$	14816.0			0.998103
$605$	−767.500	−	1329.35i	−0.0515757	−	0.0893318i
$606$		0			0
$607$	1897.00	−	3285.70i	0.126848	−	0.219708i	−0.795606	−	0.605815i	$-0.792847\pi$
$607$	1897.00	−	3285.70i	0.126848	−	0.219708i	0.922454	+	0.386107i	$0.126181\pi$
$608$	−12800.0	+	22170.3i	−0.853797	+	1.47882i
$609$		0			0
$610$	5180.00	+	8972.02i	0.343823	+	0.595519i
$611$	19532.0			1.29326
$612$		0			0
$613$	−13238.0			−0.872231			−0.436116	−	0.899891i	$-0.643646\pi$
$613$	−13238.0			−0.872231			−0.436116	+	0.899891i	$0.643646\pi$
$614$	−5188.00	−	8985.88i	−0.340995	−	0.590620i
$615$		0			0
$616$		0			0
$617$	5787.00	−	10023.4i	0.377595	−	0.654013i	−0.613117	−	0.789992i	$-0.710085\pi$
$617$	5787.00	−	10023.4i	0.377595	−	0.654013i	0.990712	+	0.135979i	$0.0434180\pi$
$618$		0			0
$619$	−4150.00	−	7188.01i	−0.269471	−	0.466738i	0.699254	−	0.714873i	$-0.253516\pi$
$619$	−4150.00	−	7188.01i	−0.269471	−	0.466738i	−0.968725	+	0.248135i	$0.920182\pi$
$620$	4320.00			0.279831
$621$		0			0
$622$	−29328.0			−1.89059
$623$	450.000	+	779.423i	0.0289388	+	0.0501235i
$624$		0			0
$625$	−312.500	+	541.266i	−0.0200000	+	0.0346410i
$626$	3124.00	−	5410.93i	0.199457	−	0.345470i
$627$		0			0
$628$	9976.00	+	17278.9i	0.633894	+	1.09794i
$629$	6916.00			0.438409
$630$		0			0
$631$	−7508.00			−0.473675			−0.236837	−	0.971549i	$-0.576111\pi$
$631$	−7508.00			−0.473675			−0.236837	+	0.971549i	$0.576111\pi$
$632$		0			0
$633$		0			0
$634$	2852.00	−	4939.81i	0.178655	−	0.309440i
$635$	4615.00	−	7993.41i	0.288411	−	0.499542i
$636$		0			0
$637$	−5833.00	−	10103.1i	−0.362813	−	0.628411i
$638$	6400.00			0.397145
$639$		0			0
$640$		0			0
$641$	13689.0	+	23710.0i	0.843499	+	1.46098i	0.886918	+	0.461927i	$0.152842\pi$
$641$	13689.0	+	23710.0i	0.843499	+	1.46098i	−0.0434190	+	0.999057i	$0.513825\pi$
$642$		0			0
$643$	−921.000	+	1595.22i	−0.0564863	+	0.0978372i	−0.892886	−	0.450283i	$-0.851323\pi$
$643$	−921.000	+	1595.22i	−0.0564863	+	0.0978372i	0.836400	+	0.548120i	$0.184656\pi$
$644$	1872.00	−	3242.40i	0.114545	−	0.198398i
$645$		0			0
$646$	5200.00	+	9006.66i	0.316705	+	0.548549i
$647$	−10114.0			−0.614563			−0.307282	−	0.951619i	$-0.599419\pi$
$647$	−10114.0			−0.614563			−0.307282	+	0.951619i	$0.599419\pi$
$648$		0			0
$649$	16000.0			0.967727
$650$	−1900.00	−	3290.90i	−0.114653	−	0.198584i
$651$		0			0
$652$	−11048.0	+	19135.7i	−0.663609	+	1.14940i
$653$	−5201.00	+	9008.40i	−0.311686	+	0.539856i	−0.978727	−	0.205165i	$-0.934227\pi$
$653$	−5201.00	+	9008.40i	−0.311686	+	0.539856i	0.667042	+	0.745020i	$0.267560\pi$
$654$		0			0
$655$	−5520.00	−	9560.92i	−0.329289	−	0.570345i
$656$	−1408.00			−0.0838006
$657$		0			0
$658$	12336.0			0.730862
$659$	−3550.00	−	6148.78i	−0.209846	−	0.363464i	0.741820	−	0.670599i	$-0.233963\pi$
$659$	−3550.00	−	6148.78i	−0.209846	−	0.363464i	−0.951666	+	0.307135i	$0.900630\pi$
$660$		0			0
$661$	3559.00	−	6164.37i	0.209424	−	0.362732i	−0.742109	−	0.670279i	$-0.766175\pi$
$661$	3559.00	−	6164.37i	0.209424	−	0.362732i	0.951533	+	0.307546i	$0.0995079\pi$
$662$	−8016.00	+	13884.1i	−0.470620	+	0.815138i
$663$		0			0
$664$		0			0
$665$	−3000.00			−0.174940
$666$		0			0
$667$	3900.00			0.226400
$668$	−12504.0	−	21657.6i	−0.724243	−	1.25443i
$669$		0			0
$670$	−1260.00	+	2182.38i	−0.0726538	+	0.125840i
$671$	8288.00	−	14355.2i	0.476833	−	0.825898i
$672$		0			0
$673$	15639.0	+	27087.5i	0.895749	+	1.55148i	0.832875	+	0.553462i	$0.186693\pi$
$673$	15639.0	+	27087.5i	0.895749	+	1.55148i	0.0628744	+	0.998021i	$0.479973\pi$
$674$	−35464.0			−2.02674
$675$		0			0
$676$	−6024.00			−0.342740
$677$	15027.0	+	26027.5i	0.853079	+	1.47758i	0.878416	+	0.477897i	$0.158601\pi$
$677$	15027.0	+	26027.5i	0.853079	+	1.47758i	−0.0253367	+	0.999679i	$0.508066\pi$
$678$		0			0
$679$	−1158.00	+	2005.71i	−0.0654491	+	0.113361i
$680$		0			0
$681$		0			0
$682$	−6912.00	−	11971.9i	−0.388085	−	0.672183i
$683$	−4518.00			−0.253113			−0.126557	−	0.991959i	$-0.540393\pi$
$683$	−4518.00			−0.253113			−0.126557	+	0.991959i	$0.540393\pi$
$684$		0			0
$685$	11670.0			0.650931
$686$	−7800.00	−	13510.0i	−0.434119	−	0.751916i
$687$		0			0
$688$	14144.0	−	24498.1i	0.783772	−	1.35753i
$689$	38.0000	−	65.8179i	0.00210114	−	0.00363928i
$690$		0			0
$691$	−14636.0	−	25350.3i	−0.805759	−	1.39562i	−0.915777	−	0.401686i	$-0.868424\pi$
$691$	−14636.0	−	25350.3i	−0.805759	−	1.39562i	0.110018	−	0.993930i	$-0.464909\pi$
$692$	−624.000			−0.0342788
$693$		0			0
$694$	6856.00			0.375000
$695$	−1750.00	−	3031.09i	−0.0955126	−	0.165433i
$696$		0			0
$697$	−286.000	+	495.367i	−0.0155424	+	0.0269202i
$698$	2300.00	−	3983.72i	0.124722	−	0.216026i
$699$		0			0
$700$	−600.000	−	1039.23i	−0.0323970	−	0.0561132i
$701$	−5798.00			−0.312393			−0.156196	−	0.987726i	$-0.549923\pi$
$701$	−5798.00			−0.312393			−0.156196	+	0.987726i	$0.549923\pi$
$702$		0			0
$703$	26600.0			1.42708
$704$	8192.00	+	14189.0i	0.438562	+	0.759612i
$705$		0			0
$706$	−8796.00	+	15235.1i	−0.468898	+	0.812155i
$707$	−2106.00	+	3647.70i	−0.112029	+	0.194039i
$708$		0			0
$709$	−4475.00	−	7750.93i	−0.237041	−	0.410567i	0.722823	−	0.691033i	$-0.242844\pi$
$709$	−4475.00	−	7750.93i	−0.237041	−	0.410567i	−0.959864	+	0.280466i	$0.909511\pi$
$710$	8240.00			0.435552
$711$		0			0
$712$		0			0
$713$	−4212.00	−	7295.40i	−0.221235	−	0.383190i
$714$		0			0
$715$	−3040.00	+	5265.43i	−0.159006	+	0.275407i
$716$	5200.00	−	9006.66i	0.271415	−	0.470105i
$717$		0			0
$718$	3600.00	+	6235.38i	0.187118	+	0.324098i
$719$	7800.00			0.404577			0.202289	−	0.979326i	$-0.435162\pi$
$719$	7800.00			0.404577			0.202289	+	0.979326i	$0.435162\pi$
$720$		0			0
$721$	−3588.00			−0.185332
$722$	6282.00	+	10880.7i	0.323811	+	0.560858i
$723$		0			0
$724$	−6968.00	+	12068.9i	−0.357685	+	0.619528i
$725$	625.000	−	1082.53i	0.0320164	−	0.0554541i
$726$		0			0
$727$	4277.00	+	7407.98i	0.218191	+	0.377919i	0.954255	−	0.298994i	$-0.0966510\pi$
$727$	4277.00	+	7407.98i	0.218191	+	0.377919i	−0.736064	+	0.676912i	$0.763318\pi$
$728$		0			0
$729$		0			0
$730$	−17560.0			−0.890308
$731$	−5746.00	−	9952.36i	−0.290730	−	0.503559i
$732$		0			0
$733$	−1441.00	+	2495.89i	−0.0726119	+	0.125768i	−0.900045	−	0.435796i	$-0.856467\pi$
$733$	−1441.00	+	2495.89i	−0.0726119	+	0.125768i	0.827433	+	0.561564i	$0.189800\pi$
$734$	−11748.0	+	20348.1i	−0.590772	+	1.02325i
$735$		0			0
$736$	9984.00	+	17292.8i	0.500021	+	0.866061i
$737$	4032.00			0.201521
$738$		0			0
$739$	18700.0			0.930840			0.465420	−	0.885090i	$-0.345903\pi$
$739$	18700.0			0.930840			0.465420	+	0.885090i	$0.345903\pi$
$740$	5320.00	+	9214.51i	0.264280	+	0.457746i
$741$		0			0
$742$	24.0000	−	41.5692i	0.00118742	−	0.00205668i
$743$	−6121.00	+	10601.9i	−0.302231	+	0.523480i	−0.976641	−	0.214878i	$-0.931065\pi$
$743$	−6121.00	+	10601.9i	−0.302231	+	0.523480i	0.674410	+	0.738357i	$0.264398\pi$
$744$		0			0
$745$	5125.00	+	8876.76i	0.252034	+	0.436536i
$746$	8312.00			0.407941
$747$		0			0
$748$	6656.00			0.325358
$749$	3582.00	+	6204.21i	0.174744	+	0.302666i
$750$		0			0
$751$	15574.0	−	26975.0i	0.756729	−	1.31069i	−0.187781	−	0.982211i	$-0.560130\pi$
$751$	15574.0	−	26975.0i	0.756729	−	1.31069i	0.944510	−	0.328482i	$-0.106537\pi$
$752$	−16448.0	+	28488.8i	−0.797602	+	1.38149i
$753$		0			0
$754$	3800.00	+	6581.79i	0.183538	+	0.317898i
$755$	−9260.00			−0.446365
$756$		0			0
$757$	−7694.00			−0.369410			−0.184705	−	0.982794i	$-0.559133\pi$
$757$	−7694.00			−0.369410			−0.184705	+	0.982794i	$0.559133\pi$
$758$	15800.0	+	27366.4i	0.757100	+	1.31134i
$759$		0			0
$760$		0			0
$761$	2259.00	−	3912.70i	0.107607	−	0.186380i	−0.807194	−	0.590287i	$-0.799015\pi$
$761$	2259.00	−	3912.70i	0.107607	−	0.186380i	0.914800	+	0.403907i	$0.132348\pi$
$762$		0			0
$763$	1650.00	+	2857.88i	0.0782883	+	0.135599i
$764$	30176.0			1.42897
$765$		0			0
$766$	30072.0			1.41847
$767$	9500.00	+	16454.5i	0.447230	+	0.774624i
$768$		0			0
$769$	19775.0	−	34251.3i	0.927314	−	1.60616i	0.139518	−	0.990220i	$-0.455445\pi$
$769$	19775.0	−	34251.3i	0.927314	−	1.60616i	0.787796	−	0.615936i	$-0.211222\pi$
$770$	−1920.00	+	3325.54i	−0.0898597	+	0.155642i
$771$		0			0
$772$	1432.00	+	2480.30i	0.0667601	+	0.115632i
$773$	22122.0			1.02933			0.514666	−	0.857391i	$-0.327916\pi$
$773$	22122.0			1.02933			0.514666	+	0.857391i	$0.327916\pi$
$774$		0			0
$775$	−2700.00			−0.125144
$776$		0			0
$777$		0			0
$778$	−3900.00	+	6755.00i	−0.179720	+	0.311283i
$779$	−1100.00	+	1905.26i	−0.0505925	+	0.0876289i
$780$		0			0
$781$	−6592.00	−	11417.7i	−0.302023	−	0.523120i
$782$	8112.00			0.370952
$783$		0			0
$784$	19648.0			0.895044
$785$	−6235.00	−	10799.3i	−0.283486	−	0.491013i
$786$		0			0
$787$	8317.00	−	14405.5i	0.376708	−	0.652477i	−0.613873	−	0.789405i	$-0.710389\pi$
$787$	8317.00	−	14405.5i	0.376708	−	0.652477i	0.990581	+	0.136928i	$0.0437228\pi$
$788$	8856.00	−	15339.0i	0.400358	−	0.693440i
$789$		0			0
$790$	−6000.00	−	10392.3i	−0.270216	−	0.468027i
$791$	9372.00			0.421277
$792$		0			0
$793$	19684.0			0.881462
$794$	27572.0	+	47756.1i	1.23236	+	2.13451i
$795$		0			0
$796$	10400.0	−	18013.3i	0.463088	−	0.802092i
$797$	−13793.0	+	23890.2i	−0.613015	+	1.06177i	0.377714	+	0.925922i	$0.376710\pi$
$797$	−13793.0	+	23890.2i	−0.613015	+	1.06177i	−0.990729	+	0.135851i	$0.956623\pi$
$798$		0			0
$799$	6682.00	+	11573.6i	0.295860	+	0.512445i
$800$	6400.00			0.282843
$801$		0			0
$802$	−25608.0			−1.12749
$803$	14048.0	+	24331.8i	0.617364	+	1.06931i
$804$		0			0
$805$	−1170.00	+	2026.50i	−0.0512262	+	0.0887264i
$806$	8208.00	−	14216.7i	0.358703	−	0.621292i
$807$		0			0
$808$		0			0
$809$	3850.00			0.167316			0.0836581	−	0.996495i	$-0.473340\pi$
$809$	3850.00			0.167316			0.0836581	+	0.996495i	$0.473340\pi$
$810$		0			0
$811$	10032.0			0.434366			0.217183	−	0.976131i	$-0.430313\pi$
$811$	10032.0			0.434366			0.217183	+	0.976131i	$0.430313\pi$
$812$	1200.00	+	2078.46i	0.0518618	+	0.0898272i
$813$		0			0
$814$	17024.0	−	29486.4i	0.733035	−	1.26965i
$815$	6905.00	−	11959.8i	0.296775	−	0.514029i
$816$		0			0
$817$	−22100.0	−	38278.3i	−0.946366	−	1.63915i
$818$	−44600.0			−1.90636
$819$		0			0
$820$	−880.000			−0.0374767
$821$	−10281.0	−	17807.2i	−0.437039	−	0.756975i	0.560420	−	0.828208i	$-0.310640\pi$
$821$	−10281.0	−	17807.2i	−0.437039	−	0.756975i	−0.997460	+	0.0712339i	$0.977306\pi$
$822$		0			0
$823$	−5161.00	+	8939.11i	−0.218592	+	0.378612i	−0.954378	−	0.298602i	$-0.903480\pi$
$823$	−5161.00	+	8939.11i	−0.218592	+	0.378612i	0.735786	+	0.677214i	$0.236813\pi$
$824$		0			0
$825$		0			0
$826$	6000.00	+	10392.3i	0.252744	+	0.437766i
$827$	8846.00			0.371954			0.185977	−	0.982554i	$-0.440455\pi$
$827$	8846.00			0.371954			0.185977	+	0.982554i	$0.440455\pi$
$828$		0			0
$829$	−25350.0			−1.06205			−0.531026	−	0.847355i	$-0.678194\pi$
$829$	−25350.0			−1.06205			−0.531026	+	0.847355i	$0.678194\pi$
$830$	−2820.00	−	4884.38i	−0.117932	−	0.204264i
$831$		0			0
$832$	−9728.00	+	16849.4i	−0.405358	+	0.702100i
$833$	3991.00	−	6912.61i	0.166002	−	0.287524i
$834$		0			0
$835$	7815.00	+	13536.0i	0.323891	+	0.560996i
$836$	25600.0			1.05908
$837$		0			0
$838$	54800.0			2.25899
$839$	−23000.0	−	39837.2i	−0.946422	−	1.63925i	−0.752878	−	0.658160i	$-0.771335\pi$
$839$	−23000.0	−	39837.2i	−0.946422	−	1.63925i	−0.193544	−	0.981092i	$-0.561998\pi$
$840$		0			0
$841$	10944.5	−	18956.4i	0.448747	−	0.777253i
$842$	−10876.0	+	18837.8i	−0.445145	+	0.771013i
$843$		0			0
$844$	4672.00	+	8092.14i	0.190541	+	0.330027i
$845$	3765.00			0.153278
$846$		0			0
$847$	−1842.00			−0.0747248
$848$	64.0000	+	110.851i	0.00259171	+	0.00448897i
$849$		0			0
$850$	1300.00	−	2251.67i	0.0524584	−	0.0908606i
$851$	10374.0	−	17968.3i	0.417880	−	0.723790i
$852$		0			0
$853$	8499.00	+	14720.7i	0.341149	+	0.590888i	0.984646	−	0.174561i	$-0.0558505\pi$
$853$	8499.00	+	14720.7i	0.341149	+	0.590888i	−0.643497	+	0.765448i	$0.722517\pi$
$854$	12432.0			0.498143
$855$		0			0
$856$		0			0
$857$	13247.0	+	22944.5i	0.528015	+	0.914549i	0.999467	+	0.0326569i	$0.0103969\pi$
$857$	13247.0	+	22944.5i	0.528015	+	0.914549i	−0.471452	+	0.881892i	$0.656270\pi$
$858$		0			0
$859$	10750.0	−	18619.5i	0.426991	−	0.739570i	−0.569613	−	0.821913i	$-0.692907\pi$
$859$	10750.0	−	18619.5i	0.426991	−	0.739570i	0.996604	+	0.0823429i	$0.0262403\pi$
$860$	8840.00	−	15311.3i	0.350513	−	0.607107i
$861$		0			0
$862$	15384.0	+	26645.9i	0.607867	+	1.05286i
$863$	25762.0			1.01616			0.508082	−	0.861309i	$-0.330355\pi$
$863$	25762.0			1.01616			0.508082	+	0.861309i	$0.330355\pi$
$864$		0			0
$865$	390.000			0.0153299
$866$	−2236.00	−	3872.87i	−0.0877395	−	0.151969i
$867$		0			0
$868$	2592.00	−	4489.48i	0.101357	−	0.175556i
$869$	−9600.00	+	16627.7i	−0.374750	+	0.649086i
$870$		0			0
$871$	2394.00	+	4146.53i	0.0931316	+	0.161309i
$872$		0			0
$873$		0			0
$874$	31200.0			1.20750
$875$	375.000	+	649.519i	0.0144884	+	0.0250946i
$876$		0			0
$877$	−15273.0	+	26453.6i	−0.588064	+	1.01856i	0.406421	+	0.913686i	$0.366777\pi$
$877$	−15273.0	+	26453.6i	−0.588064	+	1.01856i	−0.994486	+	0.104872i	$0.966557\pi$
$878$	−5200.00	+	9006.66i	−0.199876	+	0.346196i
$879$		0			0
$880$	−5120.00	−	8868.10i	−0.196131	−	0.339709i
$881$	32942.0			1.25976			0.629878	−	0.776694i	$-0.283105\pi$
$881$	32942.0			1.25976			0.629878	+	0.776694i	$0.283105\pi$
$882$		0			0
$883$	−27118.0			−1.03351			−0.516757	−	0.856132i	$-0.672861\pi$
$883$	−27118.0			−1.03351			−0.516757	+	0.856132i	$0.672861\pi$
$884$	3952.00	+	6845.06i	0.150362	+	0.260435i
$885$		0			0
$886$	−23916.0	+	41423.7i	−0.906855	+	1.57072i
$887$	19317.0	−	33458.0i	0.731230	−	1.26653i	−0.225127	−	0.974329i	$-0.572280\pi$
$887$	19317.0	−	33458.0i	0.731230	−	1.26653i	0.956358	−	0.292199i	$-0.0943869\pi$
$888$		0			0
$889$	−5538.00	−	9592.10i	−0.208930	−	0.361877i
$890$	−3000.00			−0.112989
$891$		0			0
$892$	−51824.0			−1.94529
$893$	25700.0	+	44513.7i	0.963066	+	1.66808i
$894$		0			0
$895$	−3250.00	+	5629.17i	−0.121380	+	0.210237i
$896$		0			0
$897$		0			0
$898$	−34100.0	−	59062.9i	−1.26718	−	2.19483i
$899$	5400.00			0.200334
$900$		0			0
$901$	52.0000			0.00192272
$902$	1408.00	+	2438.73i	0.0519748	+	0.0900230i
$903$		0			0
$904$		0			0
$905$	4355.00	−	7543.08i	0.159961	−	0.277061i
$906$		0			0
$907$	897.000	+	1553.65i	0.0328384	+	0.0568777i	0.881978	−	0.471291i	$-0.156212\pi$
$907$	897.000	+	1553.65i	0.0328384	+	0.0568777i	−0.849139	+	0.528169i	$0.822879\pi$
$908$	5168.00			0.188883
$909$		0			0
$910$	−4560.00			−0.166113
$911$	−20866.0	−	36141.0i	−0.758860	−	1.31438i	−0.943432	−	0.331565i	$-0.892423\pi$
$911$	−20866.0	−	36141.0i	−0.758860	−	1.31438i	0.184572	−	0.982819i	$-0.440910\pi$
$912$		0			0
$913$	−4512.00	+	7815.01i	−0.163555	+	0.283285i
$914$	−18988.0	+	32888.2i	−0.687163	+	1.19020i
$915$		0			0
$916$	−15000.0	−	25980.8i	−0.541063	−	0.937149i
$917$	−13248.0			−0.477086
$918$		0			0
$919$	29200.0			1.04812			0.524058	−	0.851682i	$-0.324417\pi$
$919$	29200.0			1.04812			0.524058	+	0.851682i	$0.324417\pi$
$920$		0			0
$921$		0			0
$922$	−22836.0	+	39553.1i	−0.815687	+	1.41281i
$923$	7828.00	−	13558.5i	0.279157	−	0.483514i
$924$		0			0
$925$	−3325.00	−	5759.07i	−0.118190	−	0.204710i
$926$	−31848.0			−1.13023
$927$		0			0
$928$	−12800.0			−0.452781
$929$	24325.0	+	42132.1i	0.859071	+	1.48796i	0.872816	+	0.488050i	$0.162292\pi$
$929$	24325.0	+	42132.1i	0.859071	+	1.48796i	−0.0137443	+	0.999906i	$0.504375\pi$
$930$		0			0
$931$	15350.0	−	26587.0i	0.540361	−	0.935932i
$932$	−5928.00	+	10267.6i	−0.208346	+	0.360865i
$933$		0			0
$934$	13052.0	+	22606.7i	0.457253	+	0.791986i
$935$	−4160.00			−0.145504
$936$		0			0
$937$	−11334.0			−0.395161			−0.197580	−	0.980287i	$-0.563308\pi$
$937$	−11334.0			−0.395161			−0.197580	+	0.980287i	$0.563308\pi$
$938$	1512.00	+	2618.86i	0.0526317	+	0.0911608i
$939$		0			0
$940$	−10280.0	+	17805.5i	−0.356699	+	0.617820i
$941$	15589.0	−	27000.9i	0.540050	−	0.935394i	−0.458851	−	0.888513i	$-0.651739\pi$
$941$	15589.0	−	27000.9i	0.540050	−	0.935394i	0.998901	−	0.0468803i	$-0.0149279\pi$
$942$		0			0
$943$	858.000	+	1486.10i	0.0296292	+	0.0513193i
$944$	−32000.0			−1.10330
$945$		0			0
$946$	−56576.0			−1.94444
$947$	−2343.00	−	4058.20i	−0.0803984	−	0.139254i	0.823023	−	0.568008i	$-0.192286\pi$
$947$	−2343.00	−	4058.20i	−0.0803984	−	0.139254i	−0.903421	+	0.428754i	$0.858953\pi$
$948$		0			0
$949$	−16682.0	+	28894.1i	−0.570622	+	0.988347i
$950$	5000.00	−	8660.25i	0.170759	−	0.295764i
$951$		0			0
$952$		0			0
$953$	−598.000			−0.0203265			−0.0101632	−	0.999948i	$-0.503235\pi$
$953$	−598.000			−0.0203265			−0.0101632	+	0.999948i	$0.503235\pi$
$954$		0			0
$955$	−18860.0			−0.639053
$956$	−5600.00	−	9699.48i	−0.189453	−	0.328142i
$957$		0			0
$958$	34800.0	−	60275.4i	1.17363	−	2.03279i
$959$	7002.00	−	12127.8i	0.235773	−	0.408371i
$960$		0			0
$961$	9063.50	+	15698.4i	0.304236	+	0.526953i
$962$	40432.0			1.35507
$963$		0			0
$964$	24176.0			0.807735
$965$	−895.000	−	1550.19i	−0.0298560	−	0.0517122i
$966$		0			0
$967$	−20863.0	+	36135.8i	−0.693804	+	1.20170i	0.276778	+	0.960934i	$0.410733\pi$
$967$	−20863.0	+	36135.8i	−0.693804	+	1.20170i	−0.970582	+	0.240770i	$0.922600\pi$
$968$		0			0
$969$		0			0
$970$	−3860.00	−	6685.72i	−0.127770	−	0.221305i
$971$	24312.0			0.803511			0.401756	−	0.915747i	$-0.368400\pi$
$971$	24312.0			0.803511			0.401756	+	0.915747i	$0.368400\pi$
$972$		0			0
$973$	−4200.00			−0.138382
$974$	2332.00	+	4039.14i	0.0767167	+	0.132877i
$975$		0			0
$976$	−16576.0	+	28710.5i	−0.543632	+	0.941598i
$977$	−20473.0	+	35460.3i	−0.670409	+	1.16118i	0.307380	+	0.951587i	$0.400548\pi$
$977$	−20473.0	+	35460.3i	−0.670409	+	1.16118i	−0.977788	+	0.209595i	$0.932785\pi$
$978$		0			0
$979$	2400.00	+	4156.92i	0.0783497	+	0.135706i
$980$	12280.0			0.400276
$981$		0			0
$982$	−28288.0			−0.919253
$983$	−21141.0	−	36617.3i	−0.685954	−	1.18811i	−0.973136	−	0.230232i	$-0.926052\pi$
$983$	−21141.0	−	36617.3i	−0.685954	−	1.18811i	0.287181	−	0.957876i	$-0.407282\pi$
$984$		0			0
$985$	−5535.00	+	9586.90i	−0.179045	+	0.310116i
$986$	−2600.00	+	4503.33i	−0.0839765	+	0.145452i
$987$		0			0
$988$	15200.0	+	26327.2i	0.489450	+	0.847752i
$989$	−34476.0			−1.10847
$990$		0			0
$991$	1172.00			0.0375679			0.0187840	−	0.999824i	$-0.494021\pi$
$991$	1172.00			0.0375679			0.0187840	+	0.999824i	$0.494021\pi$
$992$	13824.0	+	23943.9i	0.442452	+	0.766349i
$993$		0			0
$994$	4944.00	−	8563.26i	0.157761	−	0.273250i
$995$	−6500.00	+	11258.3i	−0.207099	+	0.358707i
$996$		0			0
$997$	15807.0	+	27378.5i	0.502119	+	0.869696i	0.999997	+	0.00244862i	$0.000779421\pi$
$997$	15807.0	+	27378.5i	0.502119	+	0.869696i	−0.497878	+	0.867247i	$0.665887\pi$
$998$	−400.000			−0.0126872
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.4.e.l.271.1		2
3.2	odd	2		405.4.e.c.271.1		2
9.2	odd	6		405.4.e.c.136.1		2
9.4	even	3		5.4.a.a.1.1	✓	1
9.5	odd	6		45.4.a.d.1.1		1
9.7	even	3	inner	405.4.e.l.136.1		2
36.23	even	6		720.4.a.u.1.1		1
36.31	odd	6		80.4.a.d.1.1		1
45.4	even	6		25.4.a.c.1.1		1
45.13	odd	12		25.4.b.a.24.2		2
45.14	odd	6		225.4.a.b.1.1		1
45.22	odd	12		25.4.b.a.24.1		2
45.23	even	12		225.4.b.c.199.1		2
45.32	even	12		225.4.b.c.199.2		2
63.4	even	3		245.4.e.f.226.1		2
63.13	odd	6		245.4.a.a.1.1		1
63.31	odd	6		245.4.e.g.226.1		2
63.40	odd	6		245.4.e.g.116.1		2
63.41	even	6		2205.4.a.q.1.1		1
63.58	even	3		245.4.e.f.116.1		2
72.13	even	6		320.4.a.g.1.1		1
72.67	odd	6		320.4.a.h.1.1		1
99.76	odd	6		605.4.a.d.1.1		1
117.103	even	6		845.4.a.b.1.1		1
144.13	even	12		1280.4.d.e.641.2		2
144.67	odd	12		1280.4.d.l.641.1		2
144.85	even	12		1280.4.d.e.641.1		2
144.139	odd	12		1280.4.d.l.641.2		2
153.67	even	6		1445.4.a.a.1.1		1
171.94	odd	6		1805.4.a.h.1.1		1
180.67	even	12		400.4.c.k.49.2		2
180.103	even	12		400.4.c.k.49.1		2
180.139	odd	6		400.4.a.m.1.1		1
315.139	odd	6		1225.4.a.k.1.1		1
360.139	odd	6		1600.4.a.s.1.1		1
360.229	even	6		1600.4.a.bi.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	9.4	even	3
25.4.a.c.1.1		1	45.4	even	6
25.4.b.a.24.1		2	45.22	odd	12
25.4.b.a.24.2		2	45.13	odd	12
45.4.a.d.1.1		1	9.5	odd	6
80.4.a.d.1.1		1	36.31	odd	6
225.4.a.b.1.1		1	45.14	odd	6
225.4.b.c.199.1		2	45.23	even	12
225.4.b.c.199.2		2	45.32	even	12
245.4.a.a.1.1		1	63.13	odd	6
245.4.e.f.116.1		2	63.58	even	3
245.4.e.f.226.1		2	63.4	even	3
245.4.e.g.116.1		2	63.40	odd	6
245.4.e.g.226.1		2	63.31	odd	6
320.4.a.g.1.1		1	72.13	even	6
320.4.a.h.1.1		1	72.67	odd	6
400.4.a.m.1.1		1	180.139	odd	6
400.4.c.k.49.1		2	180.103	even	12
400.4.c.k.49.2		2	180.67	even	12
405.4.e.c.136.1		2	9.2	odd	6
405.4.e.c.271.1		2	3.2	odd	2
405.4.e.l.136.1		2	9.7	even	3	inner
405.4.e.l.271.1		2	1.1	even	1	trivial
605.4.a.d.1.1		1	99.76	odd	6
720.4.a.u.1.1		1	36.23	even	6
845.4.a.b.1.1		1	117.103	even	6
1225.4.a.k.1.1		1	315.139	odd	6
1280.4.d.e.641.1		2	144.85	even	12
1280.4.d.e.641.2		2	144.13	even	12
1280.4.d.l.641.1		2	144.67	odd	12
1280.4.d.l.641.2		2	144.139	odd	12
1445.4.a.a.1.1		1	153.67	even	6
1600.4.a.s.1.1		1	360.139	odd	6
1600.4.a.bi.1.1		1	360.229	even	6
1805.4.a.h.1.1		1	171.94	odd	6
2205.4.a.q.1.1		1	63.41	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(2.00000 + 3.46410i) q^{2} +(-4.00000 + 6.92820i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-3.00000 - 5.19615i) q^{7} +O(q^{10})\)	\(q+(2.00000 + 3.46410i) q^{2} +(-4.00000 + 6.92820i) q^{4} +(2.50000 - 4.33013i) q^{5} +(-3.00000 - 5.19615i) q^{7} +20.0000 q^{10} +(-16.0000 - 27.7128i) q^{11} +(19.0000 - 32.9090i) q^{13} +(12.0000 - 20.7846i) q^{14} +(32.0000 + 55.4256i) q^{16} +26.0000 q^{17} +100.000 q^{19} +(20.0000 + 34.6410i) q^{20} +(64.0000 - 110.851i) q^{22} +(39.0000 - 67.5500i) q^{23} +(-12.5000 - 21.6506i) q^{25} +152.000 q^{26} +48.0000 q^{28} +(25.0000 + 43.3013i) q^{29} +(54.0000 - 93.5307i) q^{31} +(-128.000 + 221.703i) q^{32} +(52.0000 + 90.0666i) q^{34} -30.0000 q^{35} +266.000 q^{37} +(200.000 + 346.410i) q^{38} +(-11.0000 + 19.0526i) q^{41} +(-221.000 - 382.783i) q^{43} +256.000 q^{44} +312.000 q^{46} +(257.000 + 445.137i) q^{47} +(153.500 - 265.870i) q^{49} +(50.0000 - 86.6025i) q^{50} +(152.000 + 263.272i) q^{52} +2.00000 q^{53} -160.000 q^{55} +(-100.000 + 173.205i) q^{58} +(-250.000 + 433.013i) q^{59} +(259.000 + 448.601i) q^{61} +432.000 q^{62} -512.000 q^{64} +(-95.0000 - 164.545i) q^{65} +(-63.0000 + 109.119i) q^{67} +(-104.000 + 180.133i) q^{68} +(-60.0000 - 103.923i) q^{70} +412.000 q^{71} -878.000 q^{73} +(532.000 + 921.451i) q^{74} +(-400.000 + 692.820i) q^{76} +(-96.0000 + 166.277i) q^{77} +(-300.000 - 519.615i) q^{79} +320.000 q^{80} -88.0000 q^{82} +(-141.000 - 244.219i) q^{83} +(65.0000 - 112.583i) q^{85} +(884.000 - 1531.13i) q^{86} -150.000 q^{89} -228.000 q^{91} +(312.000 + 540.400i) q^{92} +(-1028.00 + 1780.55i) q^{94} +(250.000 - 433.013i) q^{95} +(-193.000 - 334.286i) q^{97} +1228.00 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 6 q^{7}+O(q^{10}) \) 2 * q + 4 * q^2 - 8 * q^4 + 5 * q^5 - 6 * q^7	\( 2 q + 4 q^{2} - 8 q^{4} + 5 q^{5} - 6 q^{7} + 40 q^{10} - 32 q^{11} + 38 q^{13} + 24 q^{14} + 64 q^{16} + 52 q^{17} + 200 q^{19} + 40 q^{20} + 128 q^{22} + 78 q^{23} - 25 q^{25} + 304 q^{26} + 96 q^{28} + 50 q^{29} + 108 q^{31} - 256 q^{32} + 104 q^{34} - 60 q^{35} + 532 q^{37} + 400 q^{38} - 22 q^{41} - 442 q^{43} + 512 q^{44} + 624 q^{46} + 514 q^{47} + 307 q^{49} + 100 q^{50} + 304 q^{52} + 4 q^{53} - 320 q^{55} - 200 q^{58} - 500 q^{59} + 518 q^{61} + 864 q^{62} - 1024 q^{64} - 190 q^{65} - 126 q^{67} - 208 q^{68} - 120 q^{70} + 824 q^{71} - 1756 q^{73} + 1064 q^{74} - 800 q^{76} - 192 q^{77} - 600 q^{79} + 640 q^{80} - 176 q^{82} - 282 q^{83} + 130 q^{85} + 1768 q^{86} - 300 q^{89} - 456 q^{91} + 624 q^{92} - 2056 q^{94} + 500 q^{95} - 386 q^{97} + 2456 q^{98}+O(q^{100}) \) 2 * q + 4 * q^2 - 8 * q^4 + 5 * q^5 - 6 * q^7 + 40 * q^10 - 32 * q^11 + 38 * q^13 + 24 * q^14 + 64 * q^16 + 52 * q^17 + 200 * q^19 + 40 * q^20 + 128 * q^22 + 78 * q^23 - 25 * q^25 + 304 * q^26 + 96 * q^28 + 50 * q^29 + 108 * q^31 - 256 * q^32 + 104 * q^34 - 60 * q^35 + 532 * q^37 + 400 * q^38 - 22 * q^41 - 442 * q^43 + 512 * q^44 + 624 * q^46 + 514 * q^47 + 307 * q^49 + 100 * q^50 + 304 * q^52 + 4 * q^53 - 320 * q^55 - 200 * q^58 - 500 * q^59 + 518 * q^61 + 864 * q^62 - 1024 * q^64 - 190 * q^65 - 126 * q^67 - 208 * q^68 - 120 * q^70 + 824 * q^71 - 1756 * q^73 + 1064 * q^74 - 800 * q^76 - 192 * q^77 - 600 * q^79 + 640 * q^80 - 176 * q^82 - 282 * q^83 + 130 * q^85 + 1768 * q^86 - 300 * q^89 - 456 * q^91 + 624 * q^92 - 2056 * q^94 + 500 * q^95 - 386 * q^97 + 2456 * q^98

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