LMFDB - Embedded newform 270.3.g.e.163.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [270,3,Mod(163,270)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(270, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("270.163");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 270 = 2 \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	270.g (of order $4$, degree $2$, 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:	$7.35696713773$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.122645643264.38
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 24x^{6} + 164x^{4} + 336x^{2} + 100 $ x^8 + 24x^6 + 164x^4 + 336*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			163.2
Root			$3.72938i$ of defining polynomial
Character	$\chi$	$=$	270.163
Dual form			270.3.g.e.217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(1.97347 + 4.59406i) q^{5} +(-6.46083 + 6.46083i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-6.56753 - 2.62059i) q^{10} +12.3232 q^{11} +(-13.0758 - 13.0758i) q^{13} -12.9217i q^{14} -4.00000 q^{16} +(-12.7500 + 12.7500i) q^{17} +13.6021i q^{19} +(9.18813 - 3.94694i) q^{20} +(-12.3232 + 12.3232i) q^{22} +(-12.0221 - 12.0221i) q^{23} +(-17.2108 + 18.1325i) q^{25} +26.1515 q^{26} +(12.9217 + 12.9217i) q^{28} +33.7903i q^{29} -36.9772 q^{31} +(4.00000 - 4.00000i) q^{32} -25.5000i q^{34} +(-42.4317 - 16.9312i) q^{35} +(6.86120 - 6.86120i) q^{37} +(-13.6021 - 13.6021i) q^{38} +(-5.24119 + 13.1351i) q^{40} -0.973558 q^{41} +(-49.2083 - 49.2083i) q^{43} -24.6464i q^{44} +24.0442 q^{46} +(-53.1311 + 53.1311i) q^{47} -34.4847i q^{49} +(-0.921664 - 35.3433i) q^{50} +(-26.1515 + 26.1515i) q^{52} +(47.4841 + 47.4841i) q^{53} +(24.3195 + 56.6135i) q^{55} -25.8433 q^{56} +(-33.7903 - 33.7903i) q^{58} -53.9887i q^{59} +107.631 q^{61} +(36.9772 - 36.9772i) q^{62} +8.00000i q^{64} +(34.2662 - 85.8755i) q^{65} +(22.6879 - 22.6879i) q^{67} +(25.5000 + 25.5000i) q^{68} +(59.3629 - 25.5005i) q^{70} -36.8169 q^{71} +(85.0530 + 85.0530i) q^{73} +13.7224i q^{74} +27.2043 q^{76} +(-79.6181 + 79.6181i) q^{77} +29.5091i q^{79} +(-7.89388 - 18.3763i) q^{80} +(0.973558 - 0.973558i) q^{82} +(70.7126 + 70.7126i) q^{83} +(-83.7360 - 33.4126i) q^{85} +98.4165 q^{86} +(24.6464 + 24.6464i) q^{88} +30.9687i q^{89} +168.961 q^{91} +(-24.0442 + 24.0442i) q^{92} -106.262i q^{94} +(-62.4891 + 26.8434i) q^{95} +(9.50991 - 9.50991i) q^{97} +(34.4847 + 34.4847i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{2} + 12 q^{5} - 8 q^{7} + 16 q^{8} - 16 q^{10} - 40 q^{11} - 24 q^{13} - 32 q^{16} + 8 q^{17} + 8 q^{20} + 40 q^{22} - 16 q^{23} + 16 q^{25} + 48 q^{26} + 16 q^{28} - 64 q^{31} + 32 q^{32} - 124 q^{35}+ \cdots + 128 q^{98}+O(q^{100}) $ 8 * q - 8 * q^2 + 12 * q^5 - 8 * q^7 + 16 * q^8 - 16 * q^10 - 40 * q^11 - 24 * q^13 - 32 * q^16 + 8 * q^17 + 8 * q^20 + 40 * q^22 - 16 * q^23 + 16 * q^25 + 48 * q^26 + 16 * q^28 - 64 * q^31 + 32 * q^32 - 124 * q^35 - 24 * q^37 + 8 * q^38 + 16 * q^40 + 56 * q^41 - 72 * q^43 + 32 * q^46 + 80 * q^50 - 48 * q^52 - 80 * q^53 - 96 * q^55 - 32 * q^56 - 88 * q^58 + 216 * q^61 + 64 * q^62 + 156 * q^65 + 136 * q^67 - 16 * q^68 + 152 * q^70 - 56 * q^71 + 368 * q^73 - 16 * q^76 - 320 * q^77 - 48 * q^80 - 56 * q^82 + 168 * q^83 - 56 * q^85 + 144 * q^86 - 80 * q^88 - 312 * q^91 - 32 * q^92 + 4 * q^95 - 312 * q^97 + 128 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/270\mathbb{Z}\right)^\times$.

$n$	$191$	$217$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$2$	−1.00000	+	1.00000i	−0.500000	+	0.500000i
$3$		0			0
$3$		0			0
$4$		−	2.00000i		−	0.500000i
$5$	1.97347	+	4.59406i	0.394694	+	0.918813i
$5$	1.97347	+	4.59406i	0.394694	+	0.918813i
$6$		0			0
$7$	−6.46083	+	6.46083i	−0.922976	+	0.922976i	−0.997239	−	0.0742627i	$-0.976340\pi$
$7$	−6.46083	+	6.46083i	−0.922976	+	0.922976i	0.0742627	+	0.997239i	$0.476340\pi$
$8$	2.00000	+	2.00000i	0.250000	+	0.250000i
$9$		0			0
$10$	−6.56753	−	2.62059i	−0.656753	−	0.262059i
$11$	12.3232			1.12029			0.560145	−	0.828394i	$-0.310745\pi$
$11$	12.3232			1.12029			0.560145	+	0.828394i	$0.310745\pi$
$12$		0			0
$13$	−13.0758	−	13.0758i	−1.00583	−	1.00583i	−0.999983	−	0.00584502i	$-0.998139\pi$
$13$	−13.0758	−	13.0758i	−1.00583	−	1.00583i	−0.00584502	−	0.999983i	$-0.501861\pi$
$14$		−	12.9217i		−	0.922976i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	−12.7500	+	12.7500i	−0.750000	+	0.750000i	−0.974479	−	0.224479i	$-0.927932\pi$
$17$	−12.7500	+	12.7500i	−0.750000	+	0.750000i	0.224479	+	0.974479i	$0.427932\pi$
$18$		0			0
$19$			13.6021i			0.715902i	0.933740	+	0.357951i	$0.116525\pi$
$19$			13.6021i			0.715902i	−0.933740	+	0.357951i	$0.883475\pi$
$20$	9.18813	−	3.94694i	0.459406	−	0.197347i
$21$		0			0
$22$	−12.3232	+	12.3232i	−0.560145	+	0.560145i
$23$	−12.0221	−	12.0221i	−0.522701	−	0.522701i	0.395685	−	0.918386i	$-0.370507\pi$
$23$	−12.0221	−	12.0221i	−0.522701	−	0.522701i	−0.918386	+	0.395685i	$0.870507\pi$
$24$		0			0
$25$	−17.2108	+	18.1325i	−0.688433	+	0.725300i
$26$	26.1515			1.00583
$27$		0			0
$28$	12.9217	+	12.9217i	0.461488	+	0.461488i
$29$			33.7903i			1.16518i	0.812766	+	0.582591i	$0.197961\pi$
$29$			33.7903i			1.16518i	−0.812766	+	0.582591i	$0.802039\pi$
$30$		0			0
$31$	−36.9772			−1.19281			−0.596407	−	0.802682i	$-0.703406\pi$
$31$	−36.9772			−1.19281			−0.596407	+	0.802682i	$0.703406\pi$
$32$	4.00000	−	4.00000i	0.125000	−	0.125000i
$33$		0			0
$34$		−	25.5000i		−	0.750000i
$35$	−42.4317	−	16.9312i	−1.21234	−	0.483749i
$36$		0			0
$37$	6.86120	−	6.86120i	0.185438	−	0.185438i	−0.608283	−	0.793720i	$-0.708141\pi$
$37$	6.86120	−	6.86120i	0.185438	−	0.185438i	0.793720	+	0.608283i	$0.208141\pi$
$38$	−13.6021	−	13.6021i	−0.357951	−	0.357951i
$39$		0			0
$40$	−5.24119	+	13.1351i	−0.131030	+	0.328377i
$41$	−0.973558			−0.0237453			−0.0118727	−	0.999930i	$-0.503779\pi$
$41$	−0.973558			−0.0237453			−0.0118727	+	0.999930i	$0.503779\pi$
$42$		0			0
$43$	−49.2083	−	49.2083i	−1.14438	−	1.14438i	−0.987640	−	0.156738i	$-0.949902\pi$
$43$	−49.2083	−	49.2083i	−1.14438	−	1.14438i	−0.156738	−	0.987640i	$-0.550098\pi$
$44$		−	24.6464i		−	0.560145i
$45$		0			0
$46$	24.0442			0.522701
$47$	−53.1311	+	53.1311i	−1.13045	+	1.13045i	−0.140346	+	0.990102i	$0.544822\pi$
$47$	−53.1311	+	53.1311i	−1.13045	+	1.13045i	−0.990102	+	0.140346i	$0.955178\pi$
$48$		0			0
$49$		−	34.4847i		−	0.703769i
$50$	−0.921664	−	35.3433i	−0.0184333	−	0.706866i
$51$		0			0
$52$	−26.1515	+	26.1515i	−0.502914	+	0.502914i
$53$	47.4841	+	47.4841i	0.895926	+	0.895926i	0.995073	−	0.0991473i	$-0.0316115\pi$
$53$	47.4841	+	47.4841i	0.895926	+	0.895926i	−0.0991473	+	0.995073i	$0.531611\pi$
$54$		0			0
$55$	24.3195	+	56.6135i	0.442172	+	1.02934i
$56$	−25.8433			−0.461488
$57$		0			0
$58$	−33.7903	−	33.7903i	−0.582591	−	0.582591i
$59$		−	53.9887i		−	0.915062i	−0.889194	−	0.457531i	$-0.848734\pi$
$59$		−	53.9887i		−	0.915062i	0.889194	−	0.457531i	$-0.151266\pi$
$60$		0			0
$61$	107.631			1.76444			0.882222	−	0.470834i	$-0.156047\pi$
$61$	107.631			1.76444			0.882222	+	0.470834i	$0.156047\pi$
$62$	36.9772	−	36.9772i	0.596407	−	0.596407i
$63$		0			0
$64$			8.00000i			0.125000i
$65$	34.2662	−	85.8755i	0.527173	−	1.32116i
$66$		0			0
$67$	22.6879	−	22.6879i	0.338625	−	0.338625i	−0.517225	−	0.855850i	$-0.673035\pi$
$67$	22.6879	−	22.6879i	0.338625	−	0.338625i	0.855850	+	0.517225i	$0.173035\pi$
$68$	25.5000	+	25.5000i	0.375000	+	0.375000i
$69$		0			0
$70$	59.3629	−	25.5005i	0.848042	−	0.364293i
$71$	−36.8169			−0.518548			−0.259274	−	0.965804i	$-0.583483\pi$
$71$	−36.8169			−0.518548			−0.259274	+	0.965804i	$0.583483\pi$
$72$		0			0
$73$	85.0530	+	85.0530i	1.16511	+	1.16511i	0.983341	+	0.181768i	$0.0581820\pi$
$73$	85.0530	+	85.0530i	1.16511	+	1.16511i	0.181768	+	0.983341i	$0.441818\pi$
$74$			13.7224i			0.185438i
$75$		0			0
$76$	27.2043			0.357951
$77$	−79.6181	+	79.6181i	−1.03400	+	1.03400i
$78$		0			0
$79$			29.5091i			0.373533i	0.982404	+	0.186767i	$0.0598008\pi$
$79$			29.5091i			0.373533i	−0.982404	+	0.186767i	$0.940199\pi$
$80$	−7.89388	−	18.3763i	−0.0986735	−	0.229703i
$81$		0			0
$82$	0.973558	−	0.973558i	0.0118727	−	0.0118727i
$83$	70.7126	+	70.7126i	0.851959	+	0.851959i	0.990374	−	0.138416i	$-0.0442009\pi$
$83$	70.7126	+	70.7126i	0.851959	+	0.851959i	−0.138416	+	0.990374i	$0.544201\pi$
$84$		0			0
$85$	−83.7360	−	33.4126i	−0.985130	−	0.393089i
$86$	98.4165			1.14438
$87$		0			0
$88$	24.6464	+	24.6464i	0.280073	+	0.280073i
$89$			30.9687i			0.347963i	0.984749	+	0.173982i	$0.0556633\pi$
$89$			30.9687i			0.347963i	−0.984749	+	0.173982i	$0.944337\pi$
$90$		0			0
$91$	168.961			1.85671
$92$	−24.0442	+	24.0442i	−0.261350	+	0.261350i
$93$		0			0
$94$		−	106.262i		−	1.13045i
$95$	−62.4891	+	26.8434i	−0.657780	+	0.282562i
$96$		0			0
$97$	9.50991	−	9.50991i	0.0980403	−	0.0980403i	−0.656385	−	0.754426i	$-0.727915\pi$
$97$	9.50991	−	9.50991i	0.0980403	−	0.0980403i	0.754426	+	0.656385i	$0.227915\pi$
$98$	34.4847	+	34.4847i	0.351885	+	0.351885i
$99$		0			0
$100$	36.2650	+	34.4217i	0.362650	+	0.344217i
$101$	125.544			1.24301			0.621503	−	0.783412i	$-0.286522\pi$
$101$	125.544			1.24301			0.621503	+	0.783412i	$0.286522\pi$
$102$		0			0
$103$	−19.9733	−	19.9733i	−0.193916	−	0.193916i	0.603470	−	0.797386i	$-0.293784\pi$
$103$	−19.9733	−	19.9733i	−0.193916	−	0.193916i	−0.797386	+	0.603470i	$0.793784\pi$
$104$		−	52.3031i		−	0.502914i
$105$		0			0
$106$	−94.9681			−0.895926
$107$	−51.5206	+	51.5206i	−0.481501	+	0.481501i	−0.905611	−	0.424110i	$-0.860587\pi$
$107$	−51.5206	+	51.5206i	−0.481501	+	0.481501i	0.424110	+	0.905611i	$0.360587\pi$
$108$		0			0
$109$		−	45.8585i		−	0.420720i	−0.977624	−	0.210360i	$-0.932536\pi$
$109$		−	45.8585i		−	0.420720i	0.977624	−	0.210360i	$-0.0674636\pi$
$110$	−80.9330	−	32.2941i	−0.735754	−	0.293582i
$111$		0			0
$112$	25.8433	−	25.8433i	0.230744	−	0.230744i
$113$	77.3377	+	77.3377i	0.684404	+	0.684404i	0.960989	−	0.276585i	$-0.0892028\pi$
$113$	77.3377	+	77.3377i	0.684404	+	0.684404i	−0.276585	+	0.960989i	$0.589203\pi$
$114$		0			0
$115$	31.5051	−	78.9557i	0.273957	−	0.686571i
$116$	67.5805			0.582591
$117$		0			0
$118$	53.9887	+	53.9887i	0.457531	+	0.457531i
$119$		−	164.751i		−	1.38446i
$120$		0			0
$121$	30.8611			0.255050
$122$	−107.631	+	107.631i	−0.882222	+	0.882222i
$123$		0			0
$124$			73.9545i			0.596407i
$125$	−117.267	−	43.2837i	−0.938135	−	0.346270i
$126$		0			0
$127$	71.9103	−	71.9103i	0.566223	−	0.566223i	−0.364845	−	0.931068i	$-0.618878\pi$
$127$	71.9103	−	71.9103i	0.566223	−	0.566223i	0.931068	+	0.364845i	$0.118878\pi$
$128$	−8.00000	−	8.00000i	−0.0625000	−	0.0625000i
$129$		0			0
$130$	51.6093	+	120.142i	0.396994	+	0.924167i
$131$	198.637			1.51631			0.758155	−	0.652074i	$-0.226101\pi$
$131$	198.637			1.51631			0.758155	+	0.652074i	$0.226101\pi$
$132$		0			0
$133$	−87.8812	−	87.8812i	−0.660761	−	0.660761i
$134$			45.3758i			0.338625i
$135$		0			0
$136$	−51.0000			−0.375000
$137$	111.144	−	111.144i	0.811272	−	0.811272i	−0.173553	−	0.984825i	$-0.555525\pi$
$137$	111.144	−	111.144i	0.811272	−	0.811272i	0.984825	+	0.173553i	$0.0555246\pi$
$138$		0			0
$139$			162.695i			1.17047i	0.810864	+	0.585234i	$0.198997\pi$
$139$			162.695i			1.17047i	−0.810864	+	0.585234i	$0.801003\pi$
$140$	−33.8624	+	84.8635i	−0.241874	+	0.606168i
$141$		0			0
$142$	36.8169	−	36.8169i	0.259274	−	0.259274i
$143$	−161.135	−	161.135i	−1.12682	−	1.12682i
$144$		0			0
$145$	−155.235	+	66.6841i	−1.07058	+	0.459890i
$146$	−170.106			−1.16511
$147$		0			0
$148$	−13.7224	−	13.7224i	−0.0927189	−	0.0927189i
$149$		−	144.268i		−	0.968245i	−0.875000	−	0.484122i	$-0.839139\pi$
$149$		−	144.268i		−	0.968245i	0.875000	−	0.484122i	$-0.160861\pi$
$150$		0			0
$151$	−35.2866			−0.233686			−0.116843	−	0.993150i	$-0.537277\pi$
$151$	−35.2866			−0.233686			−0.116843	+	0.993150i	$0.537277\pi$
$152$	−27.2043	+	27.2043i	−0.178976	+	0.178976i
$153$		0			0
$154$		−	159.236i		−	1.03400i
$155$	−72.9735	−	169.876i	−0.470797	−	1.09597i
$156$		0			0
$157$	−61.4290	+	61.4290i	−0.391268	+	0.391268i	−0.875139	−	0.483871i	$-0.839230\pi$
$157$	−61.4290	+	61.4290i	−0.391268	+	0.391268i	0.483871	+	0.875139i	$0.339230\pi$
$158$	−29.5091	−	29.5091i	−0.186767	−	0.186767i
$159$		0			0
$160$	26.2701	+	10.4824i	0.164188	+	0.0655148i
$161$	155.346			0.964881
$162$		0			0
$163$	−72.7564	−	72.7564i	−0.446358	−	0.446358i	0.447784	−	0.894142i	$-0.352213\pi$
$163$	−72.7564	−	72.7564i	−0.446358	−	0.446358i	−0.894142	+	0.447784i	$0.852213\pi$
$164$			1.94712i			0.0118727i
$165$		0			0
$166$	−141.425			−0.851959
$167$	−151.643	+	151.643i	−0.908044	+	0.908044i	−0.996114	−	0.0880707i	$-0.971930\pi$
$167$	−151.643	+	151.643i	−0.908044	+	0.908044i	0.0880707	+	0.996114i	$0.471930\pi$
$168$		0			0
$169$			172.951i			1.02338i
$170$	117.149	−	50.3235i	0.689109	−	0.296020i
$171$		0			0
$172$	−98.4165	+	98.4165i	−0.572189	+	0.572189i
$173$	217.572	+	217.572i	1.25764	+	1.25764i	0.952217	+	0.305422i	$0.0987975\pi$
$173$	217.572	+	217.572i	1.25764	+	1.25764i	0.305422	+	0.952217i	$0.401203\pi$
$174$		0			0
$175$	−5.95472	−	228.347i	−0.0340270	−	1.30484i
$176$	−49.2928			−0.280073
$177$		0			0
$178$	−30.9687	−	30.9687i	−0.173982	−	0.173982i
$179$		−	106.382i		−	0.594315i	−0.954828	−	0.297157i	$-0.903961\pi$
$179$		−	106.382i		−	0.594315i	0.954828	−	0.297157i	$-0.0960386\pi$
$180$		0			0
$181$	−347.724			−1.92113			−0.960565	−	0.278056i	$-0.910310\pi$
$181$	−347.724			−1.92113			−0.960565	+	0.278056i	$0.910310\pi$
$182$	−168.961	+	168.961i	−0.928355	+	0.928355i
$183$		0			0
$184$		−	48.0885i		−	0.261350i
$185$	45.0611	+	17.9804i	0.243574	+	0.0971913i
$186$		0			0
$187$	−157.121	+	157.121i	−0.840218	+	0.840218i
$188$	106.262	+	106.262i	0.565224	+	0.565224i
$189$		0			0
$190$	35.6457	−	89.3325i	0.187609	−	0.470171i
$191$	−206.695			−1.08217			−0.541086	−	0.840967i	$-0.681987\pi$
$191$	−206.695			−1.08217			−0.541086	+	0.840967i	$0.681987\pi$
$192$		0			0
$193$	121.798	+	121.798i	0.631075	+	0.631075i	0.948338	−	0.317262i	$-0.102764\pi$
$193$	121.798	+	121.798i	0.631075	+	0.631075i	−0.317262	+	0.948338i	$0.602764\pi$
$194$			19.0198i			0.0980403i
$195$		0			0
$196$	−68.9694			−0.351885
$197$	−147.354	+	147.354i	−0.747988	+	0.747988i	−0.974101	−	0.226113i	$-0.927398\pi$
$197$	−147.354	+	147.354i	−0.747988	+	0.747988i	0.226113	+	0.974101i	$0.427398\pi$
$198$		0			0
$199$			244.458i			1.22843i	0.789139	+	0.614215i	$0.210527\pi$
$199$			244.458i			1.22843i	−0.789139	+	0.614215i	$0.789473\pi$
$200$	−70.6866	+	1.84333i	−0.353433	+	0.00921664i
$201$		0			0
$202$	−125.544	+	125.544i	−0.621503	+	0.621503i
$203$	−218.313	−	218.313i	−1.07543	−	1.07543i
$204$		0			0
$205$	−1.92129	−	4.47259i	−0.00937214	−	0.0218175i
$206$	39.9466			0.193916
$207$		0			0
$208$	52.3031	+	52.3031i	0.251457	+	0.251457i
$209$			167.622i			0.802018i
$210$		0			0
$211$	−28.2709			−0.133985			−0.0669927	−	0.997753i	$-0.521340\pi$
$211$	−28.2709			−0.133985			−0.0669927	+	0.997753i	$0.521340\pi$
$212$	94.9681	−	94.9681i	0.447963	−	0.447963i
$213$		0			0
$214$		−	103.041i		−	0.481501i
$215$	128.955	−	323.177i	0.599790	−	1.50315i
$216$		0			0
$217$	238.904	−	238.904i	1.10094	−	1.10094i
$218$	45.8585	+	45.8585i	0.210360	+	0.210360i
$219$		0			0
$220$	113.227	−	48.6389i	0.514668	−	0.221086i
$221$	333.432			1.50874
$222$		0			0
$223$	8.47057	+	8.47057i	0.0379846	+	0.0379846i	0.725844	−	0.687859i	$-0.241449\pi$
$223$	8.47057	+	8.47057i	0.0379846	+	0.0379846i	−0.687859	+	0.725844i	$0.741449\pi$
$224$			51.6867i			0.230744i
$225$		0			0
$226$	−154.675			−0.684404
$227$	260.270	−	260.270i	1.14656	−	1.14656i	0.159338	−	0.987224i	$-0.449064\pi$
$227$	260.270	−	260.270i	1.14656	−	1.14656i	0.987224	−	0.159338i	$-0.0509358\pi$
$228$		0			0
$229$		−	52.4929i		−	0.229227i	−0.993410	−	0.114613i	$-0.963437\pi$
$229$		−	52.4929i		−	0.229227i	0.993410	−	0.114613i	$-0.0365629\pi$
$230$	47.4506	+	110.461i	0.206307	+	0.480264i
$231$		0			0
$232$	−67.5805	+	67.5805i	−0.291295	+	0.291295i
$233$	221.397	+	221.397i	0.950201	+	0.950201i	0.998817	−	0.0486170i	$-0.0154814\pi$
$233$	221.397	+	221.397i	0.950201	+	0.950201i	−0.0486170	+	0.998817i	$0.515481\pi$
$234$		0			0
$235$	−348.940	−	139.235i	−1.48485	−	0.592489i
$236$	−107.977			−0.457531
$237$		0			0
$238$	164.751	+	164.751i	0.692232	+	0.692232i
$239$			3.75966i			0.0157308i	0.999969	+	0.00786540i	$0.00250366\pi$
$239$			3.75966i			0.0157308i	−0.999969	+	0.00786540i	$0.997496\pi$
$240$		0			0
$241$	137.954			0.572425			0.286213	−	0.958166i	$-0.407604\pi$
$241$	137.954			0.572425			0.286213	+	0.958166i	$0.407604\pi$
$242$	−30.8611	+	30.8611i	−0.127525	+	0.127525i
$243$		0			0
$244$		−	215.262i		−	0.882222i
$245$	158.425	−	68.0545i	0.646632	−	0.277774i
$246$		0			0
$247$	177.858	−	177.858i	0.720074	−	0.720074i
$248$	−73.9545	−	73.9545i	−0.298203	−	0.298203i
$249$		0			0
$250$	160.551	−	73.9832i	0.642202	−	0.295933i
$251$	−242.805			−0.967352			−0.483676	−	0.875247i	$-0.660699\pi$
$251$	−242.805			−0.967352			−0.483676	+	0.875247i	$0.660699\pi$
$252$		0			0
$253$	−148.151	−	148.151i	−0.585577	−	0.585577i
$254$			143.821i			0.566223i
$255$		0			0
$256$	16.0000			0.0625000
$257$	107.024	−	107.024i	0.416437	−	0.416437i	−0.467537	−	0.883974i	$-0.654859\pi$
$257$	107.024	−	107.024i	0.416437	−	0.416437i	0.883974	+	0.467537i	$0.154859\pi$
$258$		0			0
$259$			88.6581i			0.342309i
$260$	−171.751	−	68.5325i	−0.660581	−	0.263587i
$261$		0			0
$262$	−198.637	+	198.637i	−0.758155	+	0.758155i
$263$	153.982	+	153.982i	0.585483	+	0.585483i	0.936405	−	0.350922i	$-0.114132\pi$
$263$	153.982	+	153.982i	0.585483	+	0.585483i	−0.350922	+	0.936405i	$0.614132\pi$
$264$		0			0
$265$	−124.436	+	311.853i	−0.469571	+	1.17680i
$266$	175.762			0.660761
$267$		0			0
$268$	−45.3758	−	45.3758i	−0.169313	−	0.169313i
$269$			265.824i			0.988193i	0.869407	+	0.494097i	$0.164501\pi$
$269$			265.824i			0.988193i	−0.869407	+	0.494097i	$0.835499\pi$
$270$		0			0
$271$	−303.438			−1.11970			−0.559849	−	0.828595i	$-0.689141\pi$
$271$	−303.438			−1.11970			−0.559849	+	0.828595i	$0.689141\pi$
$272$	51.0000	−	51.0000i	0.187500	−	0.187500i
$273$		0			0
$274$			222.289i			0.811272i
$275$	−212.092	+	223.450i	−0.771245	+	0.812546i
$276$		0			0
$277$	215.146	−	215.146i	0.776699	−	0.776699i	−0.202569	−	0.979268i	$-0.564929\pi$
$277$	215.146	−	215.146i	0.776699	−	0.776699i	0.979268	+	0.202569i	$0.0649291\pi$
$278$	−162.695	−	162.695i	−0.585234	−	0.585234i
$279$		0			0
$280$	−51.0010	−	118.726i	−0.182147	−	0.424021i
$281$	−153.456			−0.546107			−0.273053	−	0.961999i	$-0.588034\pi$
$281$	−153.456			−0.546107			−0.273053	+	0.961999i	$0.588034\pi$
$282$		0			0
$283$	252.110	+	252.110i	0.890847	+	0.890847i	0.994603	−	0.103756i	$-0.0330862\pi$
$283$	252.110	+	252.110i	0.890847	+	0.890847i	−0.103756	+	0.994603i	$0.533086\pi$
$284$			73.6338i			0.259274i
$285$		0			0
$286$	322.270			1.12682
$287$	6.29000	−	6.29000i	0.0219164	−	0.0219164i
$288$		0			0
$289$		−	36.1249i		−	0.125000i
$290$	88.5505	−	221.919i	0.305347	−	0.765237i
$291$		0			0
$292$	170.106	−	170.106i	0.582555	−	0.582555i
$293$	−331.076	−	331.076i	−1.12995	−	1.12995i	−0.990184	−	0.139766i	$-0.955365\pi$
$293$	−331.076	−	331.076i	−1.12995	−	1.12995i	−0.139766	−	0.990184i	$-0.544635\pi$
$294$		0			0
$295$	248.027	−	106.545i	0.840771	−	0.361170i
$296$	27.4448			0.0927189
$297$		0			0
$298$	144.268	+	144.268i	0.484122	+	0.484122i
$299$			314.397i			1.05149i
$300$		0			0
$301$	635.853			2.11247
$302$	35.2866	−	35.2866i	0.116843	−	0.116843i
$303$		0			0
$304$		−	54.4086i		−	0.178976i
$305$	212.407	+	494.464i	0.696416	+	1.62119i
$306$		0			0
$307$	207.650	−	207.650i	0.676386	−	0.676386i	−0.282795	−	0.959180i	$-0.591262\pi$
$307$	207.650	−	207.650i	0.676386	−	0.676386i	0.959180	+	0.282795i	$0.0912615\pi$
$308$	159.236	+	159.236i	0.517001	+	0.517001i
$309$		0			0
$310$	242.849	+	96.9023i	0.783384	+	0.312588i
$311$	83.9014			0.269779			0.134890	−	0.990861i	$-0.456932\pi$
$311$	83.9014			0.269779			0.134890	+	0.990861i	$0.456932\pi$
$312$		0			0
$313$	−160.209	−	160.209i	−0.511850	−	0.511850i	0.403243	−	0.915093i	$-0.367883\pi$
$313$	−160.209	−	160.209i	−0.511850	−	0.511850i	−0.915093	+	0.403243i	$0.867883\pi$
$314$		−	122.858i		−	0.391268i
$315$		0			0
$316$	59.0182			0.186767
$317$	−192.318	+	192.318i	−0.606680	+	0.606680i	−0.942077	−	0.335397i	$-0.891130\pi$
$317$	−192.318	+	192.318i	−0.606680	+	0.606680i	0.335397	+	0.942077i	$0.391130\pi$
$318$		0			0
$319$			416.404i			1.30534i
$320$	−36.7525	+	15.7878i	−0.114852	+	0.0493368i
$321$		0			0
$322$	−155.346	+	155.346i	−0.482440	+	0.482440i
$323$	−173.427	−	173.427i	−0.536927	−	0.536927i
$324$		0			0
$325$	462.141	−	12.0515i	1.42197	−	0.0370814i
$326$	145.513			0.446358
$327$		0			0
$328$	−1.94712	−	1.94712i	−0.00593633	−	0.00593633i
$329$		−	686.542i		−	2.08675i
$330$		0			0
$331$	53.1278			0.160507			0.0802535	−	0.996774i	$-0.474427\pi$
$331$	53.1278			0.160507			0.0802535	+	0.996774i	$0.474427\pi$
$332$	141.425	−	141.425i	0.425979	−	0.425979i
$333$		0			0
$334$		−	303.287i		−	0.908044i
$335$	149.003	+	59.4557i	0.444786	+	0.177480i
$336$		0			0
$337$	104.914	−	104.914i	0.311317	−	0.311317i	−0.534103	−	0.845420i	$-0.679350\pi$
$337$	104.914	−	104.914i	0.311317	−	0.311317i	0.845420	+	0.534103i	$0.179350\pi$
$338$	−172.951	−	172.951i	−0.511690	−	0.511690i
$339$		0			0
$340$	−66.8251	+	167.472i	−0.196544	+	0.492565i
$341$	−455.678			−1.33630
$342$		0			0
$343$	−93.7809	−	93.7809i	−0.273414	−	0.273414i
$344$		−	196.833i		−	0.572189i
$345$		0			0
$346$	−435.143			−1.25764
$347$	−227.752	+	227.752i	−0.656347	+	0.656347i	−0.954514	−	0.298167i	$-0.903625\pi$
$347$	−227.752	+	227.752i	−0.656347	+	0.656347i	0.298167	+	0.954514i	$0.403625\pi$
$348$		0			0
$349$			465.818i			1.33472i	0.744735	+	0.667361i	$0.232576\pi$
$349$			465.818i			1.33472i	−0.744735	+	0.667361i	$0.767424\pi$
$350$	234.302	+	222.393i	0.669434	+	0.635407i
$351$		0			0
$352$	49.2928	−	49.2928i	0.140036	−	0.140036i
$353$	−364.225	−	364.225i	−1.03180	−	1.03180i	−0.999478	−	0.0323201i	$-0.989710\pi$
$353$	−364.225	−	364.225i	−1.03180	−	1.03180i	−0.0323201	−	0.999478i	$-0.510290\pi$
$354$		0			0
$355$	−72.6570	−	169.139i	−0.204668	−	0.476448i
$356$	61.9374			0.173982
$357$		0			0
$358$	106.382	+	106.382i	0.297157	+	0.297157i
$359$		−	278.678i		−	0.776262i	−0.921604	−	0.388131i	$-0.873121\pi$
$359$		−	278.678i		−	0.776262i	0.921604	−	0.388131i	$-0.126879\pi$
$360$		0			0
$361$	175.982			0.487484
$362$	347.724	−	347.724i	0.960565	−	0.960565i
$363$		0			0
$364$		−	337.921i		−	0.928355i
$365$	−222.889	+	558.588i	−0.610655	+	1.53038i
$366$		0			0
$367$	121.328	−	121.328i	0.330594	−	0.330594i	−0.522218	−	0.852812i	$-0.674895\pi$
$367$	121.328	−	121.328i	0.330594	−	0.330594i	0.852812	+	0.522218i	$0.174895\pi$
$368$	48.0885	+	48.0885i	0.130675	+	0.130675i
$369$		0			0
$370$	−63.0415	+	27.0807i	−0.170383	+	0.0731912i
$371$	−613.573			−1.65384
$372$		0			0
$373$	−222.716	−	222.716i	−0.597094	−	0.597094i	0.342444	−	0.939538i	$-0.388745\pi$
$373$	−222.716	−	222.716i	−0.597094	−	0.597094i	−0.939538	+	0.342444i	$0.888745\pi$
$374$		−	314.241i		−	0.840218i
$375$		0			0
$376$	−212.524			−0.565224
$377$	441.834	−	441.834i	1.17197	−	1.17197i
$378$		0			0
$379$		−	271.908i		−	0.717435i	−0.933446	−	0.358718i	$-0.883214\pi$
$379$		−	271.908i		−	0.717435i	0.933446	−	0.358718i	$-0.116786\pi$
$380$	53.6868	+	124.978i	0.141281	+	0.328890i
$381$		0			0
$382$	206.695	−	206.695i	0.541086	−	0.541086i
$383$	192.728	+	192.728i	0.503205	+	0.503205i	0.912433	−	0.409227i	$-0.134202\pi$
$383$	192.728	+	192.728i	0.503205	+	0.503205i	−0.409227	+	0.912433i	$0.634202\pi$
$384$		0			0
$385$	−522.894	−	208.647i	−1.35817	−	0.541939i
$386$	−243.595			−0.631075
$387$		0			0
$388$	−19.0198	−	19.0198i	−0.0490202	−	0.0490202i
$389$			628.058i			1.61455i	0.590178	+	0.807273i	$0.299058\pi$
$389$			628.058i			1.61455i	−0.590178	+	0.807273i	$0.700942\pi$
$390$		0			0
$391$	306.564			0.784051
$392$	68.9694	−	68.9694i	0.175942	−	0.175942i
$393$		0			0
$394$		−	294.707i		−	0.747988i
$395$	−135.567	+	58.2354i	−0.343207	+	0.147431i
$396$		0			0
$397$	−298.678	+	298.678i	−0.752337	+	0.752337i	−0.974915	−	0.222578i	$-0.928553\pi$
$397$	−298.678	+	298.678i	−0.752337	+	0.752337i	0.222578	+	0.974915i	$0.428553\pi$
$398$	−244.458	−	244.458i	−0.614215	−	0.614215i
$399$		0			0
$400$	68.8433	−	72.5300i	0.172108	−	0.181325i
$401$	558.519			1.39282			0.696408	−	0.717647i	$-0.254781\pi$
$401$	558.519			1.39282			0.696408	+	0.717647i	$0.254781\pi$
$402$		0			0
$403$	483.505	+	483.505i	1.19977	+	1.19977i
$404$		−	251.087i		−	0.621503i
$405$		0			0
$406$	436.627			1.07543
$407$	84.5518	−	84.5518i	0.207744	−	0.207744i
$408$		0			0
$409$		−	399.027i		−	0.975617i	−0.872951	−	0.487808i	$-0.837796\pi$
$409$		−	399.027i		−	0.975617i	0.872951	−	0.487808i	$-0.162204\pi$
$410$	6.39388	+	2.55130i	0.0155948	+	0.00622268i
$411$		0			0
$412$	−39.9466	+	39.9466i	−0.0969578	+	0.0969578i
$413$	348.812	+	348.812i	0.844581	+	0.844581i
$414$		0			0
$415$	−185.309	+	464.407i	−0.446527	+	1.11905i
$416$	−104.606			−0.251457
$417$		0			0
$418$	−167.622	−	167.622i	−0.401009	−	0.401009i
$419$			678.231i			1.61869i	0.587334	+	0.809345i	$0.300177\pi$
$419$			678.231i			1.61869i	−0.587334	+	0.809345i	$0.699823\pi$
$420$		0			0
$421$	298.303			0.708558			0.354279	−	0.935140i	$-0.384726\pi$
$421$	298.303			0.708558			0.354279	+	0.935140i	$0.384726\pi$
$422$	28.2709	−	28.2709i	0.0669927	−	0.0669927i
$423$		0			0
$424$			189.936i			0.447963i
$425$	−11.7512	−	450.627i	−0.0276499	−	1.06030i
$426$		0			0
$427$	−695.386	+	695.386i	−1.62854	+	1.62854i
$428$	103.041	+	103.041i	0.240751	+	0.240751i
$429$		0			0
$430$	194.222	+	452.132i	0.451679	+	1.05147i
$431$	−135.033			−0.313302			−0.156651	−	0.987654i	$-0.550070\pi$
$431$	−135.033			−0.313302			−0.156651	+	0.987654i	$0.550070\pi$
$432$		0			0
$433$	−175.145	−	175.145i	−0.404492	−	0.404492i	0.475320	−	0.879813i	$-0.342332\pi$
$433$	−175.145	−	175.145i	−0.404492	−	0.404492i	−0.879813	+	0.475320i	$0.842332\pi$
$434$			477.807i			1.10094i
$435$		0			0
$436$	−91.7170			−0.210360
$437$	163.527	−	163.527i	0.374203	−	0.374203i
$438$		0			0
$439$		−	446.700i		−	1.01754i	−0.860902	−	0.508770i	$-0.830100\pi$
$439$		−	446.700i		−	1.01754i	0.860902	−	0.508770i	$-0.169900\pi$
$440$	−64.5881	+	161.866i	−0.146791	+	0.367877i
$441$		0			0
$442$	−333.432	+	333.432i	−0.754371	+	0.754371i
$443$	71.7531	+	71.7531i	0.161971	+	0.161971i	0.783439	−	0.621468i	$-0.213464\pi$
$443$	71.7531	+	71.7531i	0.161971	+	0.161971i	−0.621468	+	0.783439i	$0.713464\pi$
$444$		0			0
$445$	−142.272	+	61.1159i	−0.319713	+	0.137339i
$446$	−16.9411			−0.0379846
$447$		0			0
$448$	−51.6867	−	51.6867i	−0.115372	−	0.115372i
$449$			412.660i			0.919065i	0.888161	+	0.459532i	$0.151983\pi$
$449$			412.660i			0.919065i	−0.888161	+	0.459532i	$0.848017\pi$
$450$		0			0
$451$	−11.9973			−0.0266017
$452$	154.675	−	154.675i	0.342202	−	0.342202i
$453$		0			0
$454$			520.539i			1.14656i
$455$	333.439	+	776.216i	0.732832	+	1.70597i
$456$		0			0
$457$	−357.034	+	357.034i	−0.781257	+	0.781257i	−0.980043	−	0.198786i	$-0.936300\pi$
$457$	−357.034	+	357.034i	−0.781257	+	0.781257i	0.198786	+	0.980043i	$0.436300\pi$
$458$	52.4929	+	52.4929i	0.114613	+	0.114613i
$459$		0			0
$460$	−157.911	−	63.0102i	−0.343286	−	0.136979i
$461$	−711.833			−1.54411			−0.772053	−	0.635558i	$-0.780770\pi$
$461$	−711.833			−1.54411			−0.772053	+	0.635558i	$0.780770\pi$
$462$		0			0
$463$	−299.197	−	299.197i	−0.646213	−	0.646213i	0.305862	−	0.952076i	$-0.401055\pi$
$463$	−299.197	−	299.197i	−0.646213	−	0.646213i	−0.952076	+	0.305862i	$0.901055\pi$
$464$		−	135.161i		−	0.291295i
$465$		0			0
$466$	−442.793			−0.950201
$467$	578.852	−	578.852i	1.23951	−	1.23951i	0.279312	−	0.960200i	$-0.409894\pi$
$467$	578.852	−	578.852i	1.23951	−	1.23951i	0.960200	−	0.279312i	$-0.0901064\pi$
$468$		0			0
$469$			293.165i			0.625086i
$470$	488.175	−	209.705i	1.03867	−	0.446181i
$471$		0			0
$472$	107.977	−	107.977i	0.228766	−	0.228766i
$473$	−606.403	−	606.403i	−1.28204	−	1.28204i
$474$		0			0
$475$	−246.641	−	234.104i	−0.519244	−	0.492851i
$476$	−329.502			−0.692232
$477$		0			0
$478$	−3.75966	−	3.75966i	−0.00786540	−	0.00786540i
$479$			198.574i			0.414559i	0.978282	+	0.207280i	$0.0664610\pi$
$479$			198.574i			0.414559i	−0.978282	+	0.207280i	$0.933539\pi$
$480$		0			0
$481$	−179.431			−0.373037
$482$	−137.954	+	137.954i	−0.286213	+	0.286213i
$483$		0			0
$484$		−	61.7222i		−	0.127525i
$485$	62.4567	+	24.9216i	0.128777	+	0.0513847i
$486$		0			0
$487$	452.269	−	452.269i	0.928684	−	0.928684i	−0.0689372	−	0.997621i	$-0.521961\pi$
$487$	452.269	−	452.269i	0.928684	−	0.928684i	0.997621	+	0.0689372i	$0.0219608\pi$
$488$	215.262	+	215.262i	0.441111	+	0.441111i
$489$		0			0
$490$	−90.3704	+	226.479i	−0.184429	+	0.462203i
$491$	−55.1876			−0.112398			−0.0561992	−	0.998420i	$-0.517898\pi$
$491$	−55.1876			−0.112398			−0.0561992	+	0.998420i	$0.517898\pi$
$492$		0			0
$493$	−430.826	−	430.826i	−0.873886	−	0.873886i
$494$			355.717i			0.720074i
$495$		0			0
$496$	147.909			0.298203
$497$	237.868	−	237.868i	0.478607	−	0.478607i
$498$		0			0
$499$		−	681.525i		−	1.36578i	−0.730520	−	0.682891i	$-0.760722\pi$
$499$		−	681.525i		−	1.36578i	0.730520	−	0.682891i	$-0.239278\pi$
$500$	−86.5674	+	234.534i	−0.173135	+	0.469068i
$501$		0			0
$502$	242.805	−	242.805i	0.483676	−	0.483676i
$503$	−15.2144	−	15.2144i	−0.0302473	−	0.0302473i	0.691821	−	0.722069i	$-0.256809\pi$
$503$	−15.2144	−	15.2144i	−0.0302473	−	0.0302473i	−0.722069	+	0.691821i	$0.756809\pi$
$504$		0			0
$505$	247.757	+	576.755i	0.490607	+	1.14209i
$506$	296.302			0.585577
$507$		0			0
$508$	−143.821	−	143.821i	−0.283112	−	0.283112i
$509$			385.771i			0.757899i	0.925417	+	0.378950i	$0.123715\pi$
$509$			385.771i			0.757899i	−0.925417	+	0.378950i	$0.876285\pi$
$510$		0			0
$511$	−1099.03			−2.15074
$512$	−16.0000	+	16.0000i	−0.0312500	+	0.0312500i
$513$		0			0
$514$			214.048i			0.416437i
$515$	52.3419	−	131.175i	0.101635	−	0.254710i
$516$		0			0
$517$	−654.745	+	654.745i	−1.26643	+	1.26643i
$518$	−88.6581	−	88.6581i	−0.171155	−	0.171155i
$519$		0			0
$520$	240.284	−	103.219i	0.462084	−	0.198497i
$521$	107.880			0.207064			0.103532	−	0.994626i	$-0.466986\pi$
$521$	107.880			0.207064			0.103532	+	0.994626i	$0.466986\pi$
$522$		0			0
$523$	38.9695	+	38.9695i	0.0745115	+	0.0745115i	0.743380	−	0.668869i	$-0.233221\pi$
$523$	38.9695	+	38.9695i	0.0745115	+	0.0745115i	−0.668869	+	0.743380i	$0.733221\pi$
$524$		−	397.273i		−	0.758155i
$525$		0			0
$526$	−307.964			−0.585483
$527$	471.460	−	471.460i	0.894610	−	0.894610i
$528$		0			0
$529$		−	239.937i		−	0.453567i
$530$	−187.417	−	436.289i	−0.353616	−	0.823188i
$531$		0			0
$532$	−175.762	+	175.762i	−0.330380	+	0.330380i
$533$	12.7300	+	12.7300i	0.0238837	+	0.0238837i
$534$		0			0
$535$	−338.364	−	135.015i	−0.632455	−	0.252364i
$536$	90.7516			0.169313
$537$		0			0
$538$	−265.824	−	265.824i	−0.494097	−	0.494097i
$539$		−	424.962i		−	0.788426i
$540$		0			0
$541$	579.593			1.07134			0.535668	−	0.844429i	$-0.320060\pi$
$541$	579.593			1.07134			0.535668	+	0.844429i	$0.320060\pi$
$542$	303.438	−	303.438i	0.559849	−	0.559849i
$543$		0			0
$544$			102.000i			0.187500i
$545$	210.677	−	90.5004i	0.386563	−	0.166056i
$546$		0			0
$547$	522.943	−	522.943i	0.956020	−	0.956020i	−0.0430532	−	0.999073i	$-0.513708\pi$
$547$	522.943	−	522.943i	0.956020	−	0.956020i	0.999073	+	0.0430532i	$0.0137085\pi$
$548$	−222.289	−	222.289i	−0.405636	−	0.405636i
$549$		0			0
$550$	−11.3578	−	435.543i	−0.0206506	−	0.791896i
$551$	−459.620			−0.834156
$552$		0			0
$553$	−190.653	−	190.653i	−0.344762	−	0.344762i
$554$			430.291i			0.776699i
$555$		0			0
$556$	325.390			0.585234
$557$	−590.215	+	590.215i	−1.05963	+	1.05963i	−0.0615269	+	0.998105i	$0.519597\pi$
$557$	−590.215	+	590.215i	−1.05963	+	1.05963i	−0.998105	+	0.0615269i	$0.980403\pi$
$558$		0			0
$559$			1286.87i			2.30209i
$560$	169.727	+	67.7248i	0.303084	+	0.120937i
$561$		0			0
$562$	153.456	−	153.456i	0.273053	−	0.273053i
$563$	−7.64258	−	7.64258i	−0.0135747	−	0.0135747i	0.700287	−	0.713862i	$-0.253055\pi$
$563$	−7.64258	−	7.64258i	−0.0135747	−	0.0135747i	−0.713862	+	0.700287i	$0.753055\pi$
$564$		0			0
$565$	−202.670	+	507.918i	−0.358709	+	0.898969i
$566$	−504.219			−0.890847
$567$		0			0
$568$	−73.6338	−	73.6338i	−0.129637	−	0.129637i
$569$		−	873.581i		−	1.53529i	−0.640874	−	0.767646i	$-0.721428\pi$
$569$		−	873.581i		−	1.53529i	0.640874	−	0.767646i	$-0.278572\pi$
$570$		0			0
$571$	−253.180			−0.443398			−0.221699	−	0.975115i	$-0.571160\pi$
$571$	−253.180			−0.443398			−0.221699	+	0.975115i	$0.571160\pi$
$572$	−322.270	+	322.270i	−0.563410	+	0.563410i
$573$		0			0
$574$			12.5800i			0.0219164i
$575$	424.902	−	11.0804i	0.738960	−	0.0192702i
$576$		0			0
$577$	−205.666	+	205.666i	−0.356440	+	0.356440i	−0.862499	−	0.506059i	$-0.831102\pi$
$577$	−205.666	+	205.666i	−0.356440	+	0.356440i	0.506059	+	0.862499i	$0.331102\pi$
$578$	36.1249	+	36.1249i	0.0624998	+	0.0624998i
$579$		0			0
$580$	133.368	+	310.469i	0.229945	+	0.535292i
$581$	−913.724			−1.57267
$582$		0			0
$583$	585.155	+	585.155i	1.00370	+	1.00370i
$584$			340.212i			0.582555i
$585$		0			0
$586$	662.151			1.12995
$587$	−45.0649	+	45.0649i	−0.0767716	+	0.0767716i	−0.744450	−	0.667678i	$-0.767288\pi$
$587$	−45.0649	+	45.0649i	−0.0767716	+	0.0767716i	0.667678	+	0.744450i	$0.267288\pi$
$588$		0			0
$589$		−	502.970i		−	0.853938i
$590$	−141.482	+	354.572i	−0.239801	+	0.600970i
$591$		0			0
$592$	−27.4448	+	27.4448i	−0.0463594	+	0.0463594i
$593$	−185.940	−	185.940i	−0.313559	−	0.313559i	0.532728	−	0.846287i	$-0.321167\pi$
$593$	−185.940	−	185.940i	−0.313559	−	0.313559i	−0.846287	+	0.532728i	$0.821167\pi$
$594$		0			0
$595$	756.877	−	325.132i	1.27206	−	0.546440i
$596$	−288.537			−0.484122
$597$		0			0
$598$	−314.397	−	314.397i	−0.525747	−	0.525747i
$599$		−	692.513i		−	1.15611i	−0.815996	−	0.578057i	$-0.803811\pi$
$599$		−	692.513i		−	1.15611i	0.815996	−	0.578057i	$-0.196189\pi$
$600$		0			0
$601$	982.288			1.63442			0.817212	−	0.576338i	$-0.195519\pi$
$601$	982.288			1.63442			0.817212	+	0.576338i	$0.195519\pi$
$602$	−635.853	+	635.853i	−1.05623	+	1.05623i
$603$		0			0
$604$			70.5732i			0.116843i
$605$	60.9034	+	141.778i	0.100667	+	0.234343i
$606$		0			0
$607$	709.111	−	709.111i	1.16822	−	1.16822i	0.185597	−	0.982626i	$-0.440578\pi$
$607$	709.111	−	709.111i	1.16822	−	1.16822i	0.982626	−	0.185597i	$-0.0594217\pi$
$608$	54.4086	+	54.4086i	0.0894878	+	0.0894878i
$609$		0			0
$610$	−706.871	−	282.057i	−1.15880	−	0.462389i
$611$	1389.46			2.27407
$612$		0			0
$613$	196.939	+	196.939i	0.321271	+	0.321271i	0.849254	−	0.527984i	$-0.177052\pi$
$613$	196.939	+	196.939i	0.321271	+	0.321271i	−0.527984	+	0.849254i	$0.677052\pi$
$614$			415.301i			0.676386i
$615$		0			0
$616$	−318.472			−0.517001
$617$	99.8435	−	99.8435i	0.161821	−	0.161821i	−0.621552	−	0.783373i	$-0.713498\pi$
$617$	99.8435	−	99.8435i	0.161821	−	0.161821i	0.783373	+	0.621552i	$0.213498\pi$
$618$		0			0
$619$			16.2229i			0.0262082i	0.999914	+	0.0131041i	$0.00417128\pi$
$619$			16.2229i			0.0262082i	−0.999914	+	0.0131041i	$0.995829\pi$
$620$	−339.751	+	145.947i	−0.547986	+	0.235398i
$621$		0			0
$622$	−83.9014	+	83.9014i	−0.134890	+	0.134890i
$623$	−200.084	−	200.084i	−0.321162	−	0.321162i
$624$		0			0
$625$	−32.5747	−	624.151i	−0.0521195	−	0.998641i
$626$	320.418			0.511850
$627$		0			0
$628$	122.858	+	122.858i	0.195634	+	0.195634i
$629$			174.960i			0.278157i
$630$		0			0
$631$	−561.037			−0.889124			−0.444562	−	0.895748i	$-0.646641\pi$
$631$	−561.037			−0.889124			−0.444562	+	0.895748i	$0.646641\pi$
$632$	−59.0182	+	59.0182i	−0.0933833	+	0.0933833i
$633$		0			0
$634$		−	384.635i		−	0.606680i
$635$	472.274	+	188.448i	0.743738	+	0.296768i
$636$		0			0
$637$	−450.914	+	450.914i	−0.707871	+	0.707871i
$638$	−416.404	−	416.404i	−0.652671	−	0.652671i
$639$		0			0
$640$	20.9647	−	52.5403i	0.0327574	−	0.0820942i
$641$	−250.686			−0.391086			−0.195543	−	0.980695i	$-0.562647\pi$
$641$	−250.686			−0.391086			−0.195543	+	0.980695i	$0.562647\pi$
$642$		0			0
$643$	347.726	+	347.726i	0.540786	+	0.540786i	0.923760	−	0.382973i	$-0.125100\pi$
$643$	347.726	+	347.726i	0.540786	+	0.540786i	−0.382973	+	0.923760i	$0.625100\pi$
$644$		−	310.692i		−	0.482440i
$645$		0			0
$646$	346.855			0.536927
$647$	267.912	−	267.912i	0.414084	−	0.414084i	−0.469075	−	0.883158i	$-0.655413\pi$
$647$	267.912	−	267.912i	0.414084	−	0.414084i	0.883158	+	0.469075i	$0.155413\pi$
$648$		0			0
$649$		−	665.313i		−	1.02514i
$650$	−450.089	+	474.192i	−0.692445	+	0.729527i
$651$		0			0
$652$	−145.513	+	145.513i	−0.223179	+	0.223179i
$653$	−411.160	−	411.160i	−0.629647	−	0.629647i	0.318332	−	0.947979i	$-0.396877\pi$
$653$	−411.160	−	411.160i	−0.629647	−	0.629647i	−0.947979	+	0.318332i	$0.896877\pi$
$654$		0			0
$655$	392.004	+	912.549i	0.598479	+	1.39321i
$656$	3.89423			0.00593633
$657$		0			0
$658$	686.542	+	686.542i	1.04338	+	1.04338i
$659$			733.210i			1.11261i	0.830978	+	0.556305i	$0.187781\pi$
$659$			733.210i			1.11261i	−0.830978	+	0.556305i	$0.812219\pi$
$660$		0			0
$661$	259.433			0.392485			0.196243	−	0.980555i	$-0.437126\pi$
$661$	259.433			0.392485			0.196243	+	0.980555i	$0.437126\pi$
$662$	−53.1278	+	53.1278i	−0.0802535	+	0.0802535i
$663$		0			0
$664$			282.850i			0.425979i
$665$	230.301	−	577.162i	0.346317	−	0.867913i
$666$		0			0
$667$	406.231	−	406.231i	0.609042	−	0.609042i
$668$	303.287	+	303.287i	0.454022	+	0.454022i
$669$		0			0
$670$	−208.459	+	89.5477i	−0.311133	+	0.133653i
$671$	1326.36			1.97669
$672$		0			0
$673$	−214.601	−	214.601i	−0.318872	−	0.318872i	0.529462	−	0.848334i	$-0.322394\pi$
$673$	−214.601	−	214.601i	−0.318872	−	0.318872i	−0.848334	+	0.529462i	$0.822394\pi$
$674$			209.828i			0.311317i
$675$		0			0
$676$	345.902			0.511690
$677$	183.229	−	183.229i	0.270648	−	0.270648i	−0.558713	−	0.829361i	$-0.688705\pi$
$677$	183.229	−	183.229i	0.270648	−	0.270648i	0.829361	+	0.558713i	$0.188705\pi$
$678$		0			0
$679$			122.884i			0.180978i
$680$	−100.647	−	234.297i	−0.148010	−	0.344555i
$681$		0			0
$682$	455.678	−	455.678i	0.668149	−	0.668149i
$683$	−362.133	−	362.133i	−0.530209	−	0.530209i	0.390426	−	0.920634i	$-0.372328\pi$
$683$	−362.133	−	362.133i	−0.530209	−	0.530209i	−0.920634	+	0.390426i	$0.872328\pi$
$684$		0			0
$685$	729.944	+	291.264i	1.06561	+	0.425203i
$686$	187.562			0.273414
$687$		0			0
$688$	196.833	+	196.833i	0.286095	+	0.286095i
$689$		−	1241.78i		−	1.80229i
$690$		0			0
$691$	−404.857			−0.585901			−0.292950	−	0.956128i	$-0.594637\pi$
$691$	−404.857			−0.585901			−0.292950	+	0.956128i	$0.594637\pi$
$692$	435.143	−	435.143i	0.628819	−	0.628819i
$693$		0			0
$694$		−	455.505i		−	0.656347i
$695$	−747.432	+	321.074i	−1.07544	+	0.461977i
$696$		0			0
$697$	12.4129	−	12.4129i	0.0178090	−	0.0178090i
$698$	−465.818	−	465.818i	−0.667361	−	0.667361i
$699$		0			0
$700$	−456.695	+	11.9094i	−0.652421	+	0.0170135i
$701$	247.220			0.352667			0.176334	−	0.984330i	$-0.443576\pi$
$701$	247.220			0.352667			0.176334	+	0.984330i	$0.443576\pi$
$702$		0			0
$703$	93.3270	+	93.3270i	0.132755	+	0.132755i
$704$			98.5855i			0.140036i
$705$		0			0
$706$	728.449			1.03180
$707$	−811.116	+	811.116i	−1.14726	+	1.14726i
$708$		0			0
$709$		−	1148.58i		−	1.62000i	−0.586429	−	0.810001i	$-0.699467\pi$
$709$		−	1148.58i		−	1.62000i	0.586429	−	0.810001i	$-0.300533\pi$
$710$	241.796	+	96.4821i	0.340558	+	0.135890i
$711$		0			0
$712$	−61.9374	+	61.9374i	−0.0869908	+	0.0869908i
$713$	444.545	+	444.545i	0.623485	+	0.623485i
$714$		0			0
$715$	422.270	−	1058.26i	0.590587	−	1.48008i
$716$	−212.765			−0.297157
$717$		0			0
$718$	278.678	+	278.678i	0.388131	+	0.388131i
$719$			709.966i			0.987436i	0.869622	+	0.493718i	$0.164362\pi$
$719$			709.966i			0.987436i	−0.869622	+	0.493718i	$0.835638\pi$
$720$		0			0
$721$	258.088			0.357959
$722$	−175.982	+	175.982i	−0.243742	+	0.243742i
$723$		0			0
$724$			695.449i			0.960565i
$725$	−612.702	−	581.559i	−0.845106	−	0.802150i
$726$		0			0
$727$	−115.201	+	115.201i	−0.158461	+	0.158461i	−0.781884	−	0.623424i	$-0.785741\pi$
$727$	−115.201	+	115.201i	−0.158461	+	0.158461i	0.623424	+	0.781884i	$0.285741\pi$
$728$	337.921	+	337.921i	0.464178	+	0.464178i
$729$		0			0
$730$	−335.699	−	781.478i	−0.459862	−	1.07052i
$731$	1254.81			1.71657
$732$		0			0
$733$	632.363	+	632.363i	0.862706	+	0.862706i	0.991652	−	0.128946i	$-0.0411593\pi$
$733$	632.363	+	632.363i	0.862706	+	0.862706i	−0.128946	+	0.991652i	$0.541159\pi$
$734$			242.656i			0.330594i
$735$		0			0
$736$	−96.1770			−0.130675
$737$	279.587	−	279.587i	0.379358	−	0.379358i
$738$		0			0
$739$			1074.05i			1.45338i	0.686965	+	0.726690i	$0.258942\pi$
$739$			1074.05i			1.45338i	−0.686965	+	0.726690i	$0.741058\pi$
$740$	35.9608	−	90.1223i	0.0485957	−	0.121787i
$741$		0			0
$742$	613.573	−	613.573i	0.826918	−	0.826918i
$743$	709.169	+	709.169i	0.954467	+	0.954467i	0.999008	−	0.0445407i	$-0.0141824\pi$
$743$	709.169	+	709.169i	0.954467	+	0.954467i	−0.0445407	+	0.999008i	$0.514182\pi$
$744$		0			0
$745$	662.779	−	284.710i	0.889636	−	0.382160i
$746$	445.432			0.597094
$747$		0			0
$748$	314.241	+	314.241i	0.420109	+	0.420109i
$749$		−	665.732i		−	0.888828i
$750$		0			0
$751$	−524.769			−0.698760			−0.349380	−	0.936981i	$-0.613608\pi$
$751$	−524.769			−0.698760			−0.349380	+	0.936981i	$0.613608\pi$
$752$	212.524	−	212.524i	0.282612	−	0.282612i
$753$		0			0
$754$			883.667i			1.17197i
$755$	−69.6370	−	162.109i	−0.0922345	−	0.214714i
$756$		0			0
$757$	507.020	−	507.020i	0.669775	−	0.669775i	−0.287889	−	0.957664i	$-0.592953\pi$
$757$	507.020	−	507.020i	0.669775	−	0.669775i	0.957664	+	0.287889i	$0.0929533\pi$
$758$	271.908	+	271.908i	0.358718	+	0.358718i
$759$		0			0
$760$	−178.665	−	71.2914i	−0.235086	−	0.0938044i
$761$	239.905			0.315249			0.157625	−	0.987499i	$-0.449616\pi$
$761$	239.905			0.315249			0.157625	+	0.987499i	$0.449616\pi$
$762$		0			0
$763$	296.284	+	296.284i	0.388315	+	0.388315i
$764$			413.390i			0.541086i
$765$		0			0
$766$	−385.455			−0.503205
$767$	−705.943	+	705.943i	−0.920395	+	0.920395i
$768$		0			0
$769$			821.514i			1.06829i	0.845393	+	0.534144i	$0.179366\pi$
$769$			821.514i			1.06829i	−0.845393	+	0.534144i	$0.820634\pi$
$770$	731.541	−	314.248i	0.950053	−	0.408114i
$771$		0			0
$772$	243.595	−	243.595i	0.315538	−	0.315538i
$773$	101.566	+	101.566i	0.131391	+	0.131391i	0.769744	−	0.638353i	$-0.220384\pi$
$773$	101.566	+	101.566i	0.131391	+	0.131391i	−0.638353	+	0.769744i	$0.720384\pi$
$774$		0			0
$775$	636.409	−	670.489i	0.821173	−	0.865148i
$776$	38.0396			0.0490202
$777$		0			0
$778$	−628.058	−	628.058i	−0.807273	−	0.807273i
$779$		−	13.2425i		−	0.0169993i
$780$		0			0
$781$	−453.702			−0.580924
$782$	−306.564	+	306.564i	−0.392026	+	0.392026i
$783$		0			0
$784$			137.939i			0.175942i
$785$	−403.437	−	160.980i	−0.513933	−	0.205071i
$786$		0			0
$787$	−944.082	+	944.082i	−1.19960	+	1.19960i	−0.225309	+	0.974287i	$0.572339\pi$
$787$	−944.082	+	944.082i	−1.19960	+	1.19960i	−0.974287	+	0.225309i	$0.927661\pi$
$788$	294.707	+	294.707i	0.373994	+	0.373994i
$789$		0			0
$790$	77.3314	−	193.802i	0.0978878	−	0.245319i
$791$	−999.331			−1.26338
$792$		0			0
$793$	−1407.36	−	1407.36i	−1.77473	−	1.77473i
$794$		−	597.356i		−	0.752337i
$795$		0			0
$796$	488.915			0.614215
$797$	78.1020	−	78.1020i	0.0979950	−	0.0979950i	−0.656410	−	0.754405i	$-0.727926\pi$
$797$	78.1020	−	78.1020i	0.0979950	−	0.0979950i	0.754405	+	0.656410i	$0.227926\pi$
$798$		0			0
$799$		−	1354.84i		−	1.69567i
$800$	3.68666	+	141.373i	0.00460832	+	0.176717i
$801$		0			0
$802$	−558.519	+	558.519i	−0.696408	+	0.696408i
$803$	1048.12	+	1048.12i	1.30526	+	1.30526i
$804$		0			0
$805$	306.570	+	713.668i	0.380833	+	0.886545i
$806$	−967.011			−1.19977
$807$		0			0
$808$	251.087	+	251.087i	0.310752	+	0.310752i
$809$			229.281i			0.283413i	0.989909	+	0.141707i	$0.0452589\pi$
$809$			229.281i			0.283413i	−0.989909	+	0.141707i	$0.954741\pi$
$810$		0			0
$811$	−1240.21			−1.52923			−0.764615	−	0.644487i	$-0.777071\pi$
$811$	−1240.21			−1.52923			−0.764615	+	0.644487i	$0.777071\pi$
$812$	−436.627	+	436.627i	−0.537717	+	0.537717i
$813$		0			0
$814$			169.104i			0.207744i
$815$	190.665	−	477.830i	0.233945	−	0.586295i
$816$		0			0
$817$	669.338	−	669.338i	0.819263	−	0.819263i
$818$	399.027	+	399.027i	0.487808	+	0.487808i
$819$		0			0
$820$	−8.94518	+	3.84258i	−0.0109088	+	0.00468607i
$821$	−1215.87			−1.48096			−0.740481	−	0.672077i	$-0.765402\pi$
$821$	−1215.87			−1.48096			−0.740481	+	0.672077i	$0.765402\pi$
$822$		0			0
$823$	132.661	+	132.661i	0.161192	+	0.161192i	0.783095	−	0.621903i	$-0.213640\pi$
$823$	132.661	+	132.661i	0.161192	+	0.161192i	−0.621903	+	0.783095i	$0.713640\pi$
$824$		−	79.8933i		−	0.0969578i
$825$		0			0
$826$	−697.624			−0.844581
$827$	99.9685	−	99.9685i	0.120881	−	0.120881i	−0.644079	−	0.764959i	$-0.722759\pi$
$827$	99.9685	−	99.9685i	0.120881	−	0.120881i	0.764959	+	0.644079i	$0.222759\pi$
$828$		0			0
$829$		−	475.686i		−	0.573807i	−0.957960	−	0.286903i	$-0.907374\pi$
$829$		−	475.686i		−	0.573807i	0.957960	−	0.286903i	$-0.0926259\pi$
$830$	−279.098	−	649.716i	−0.336263	−	0.782790i
$831$		0			0
$832$	104.606	−	104.606i	0.125728	−	0.125728i
$833$	439.680	+	439.680i	0.527827	+	0.527827i
$834$		0			0
$835$	−995.922	−	397.395i	−1.19272	−	0.475922i
$836$	335.244			0.401009
$837$		0			0
$838$	−678.231	−	678.231i	−0.809345	−	0.809345i
$839$			375.996i			0.448148i	0.974572	+	0.224074i	$0.0719357\pi$
$839$			375.996i			0.448148i	−0.974572	+	0.224074i	$0.928064\pi$
$840$		0			0
$841$	−300.782			−0.357648
$842$	−298.303	+	298.303i	−0.354279	+	0.354279i
$843$		0			0
$844$			56.5419i			0.0669927i
$845$	−794.549	+	341.314i	−0.940294	+	0.403922i
$846$		0			0
$847$	−199.388	+	199.388i	−0.235405	+	0.235405i
$848$	−189.936	−	189.936i	−0.223981	−	0.223981i
$849$		0			0
$850$	462.379	+	438.876i	0.543975	+	0.516325i
$851$	−164.972			−0.193857
$852$		0			0
$853$	1083.08	+	1083.08i	1.26973	+	1.26973i	0.946229	+	0.323497i	$0.104859\pi$
$853$	1083.08	+	1083.08i	1.26973	+	1.26973i	0.323497	+	0.946229i	$0.395141\pi$
$854$		−	1390.77i		−	1.62854i
$855$		0			0
$856$	−206.083			−0.240751
$857$	164.839	−	164.839i	0.192344	−	0.192344i	−0.604364	−	0.796708i	$-0.706573\pi$
$857$	164.839	−	164.839i	0.192344	−	0.192344i	0.796708	+	0.604364i	$0.206573\pi$
$858$		0			0
$859$			811.864i			0.945126i	0.881297	+	0.472563i	$0.156671\pi$
$859$			811.864i			0.945126i	−0.881297	+	0.472563i	$0.843329\pi$
$860$	−646.354	−	257.910i	−0.751574	−	0.299895i
$861$		0			0
$862$	135.033	−	135.033i	0.156651	−	0.156651i
$863$	322.861	+	322.861i	0.374114	+	0.374114i	0.868973	−	0.494859i	$-0.164780\pi$
$863$	322.861	+	322.861i	0.374114	+	0.374114i	−0.494859	+	0.868973i	$0.664780\pi$
$864$		0			0
$865$	−570.166	+	1428.91i	−0.659152	+	1.65192i
$866$	350.291			0.404492
$867$		0			0
$868$	−477.807	−	477.807i	−0.550469	−	0.550469i
$869$			363.646i			0.418465i
$870$		0			0
$871$	−593.323			−0.681197
$872$	91.7170	−	91.7170i	0.105180	−	0.105180i
$873$		0			0
$874$			327.053i			0.374203i
$875$	1037.29	−	477.993i	1.18547	−	0.546278i
$876$		0			0
$877$	10.6514	−	10.6514i	0.0121452	−	0.0121452i	−0.701008	−	0.713153i	$-0.747266\pi$
$877$	10.6514	−	10.6514i	0.0121452	−	0.0121452i	0.713153	+	0.701008i	$0.247266\pi$
$878$	446.700	+	446.700i	0.508770	+	0.508770i
$879$		0			0
$880$	−97.2778	−	226.454i	−0.110543	−	0.257334i
$881$	−63.3171			−0.0718696			−0.0359348	−	0.999354i	$-0.511441\pi$
$881$	−63.3171			−0.0718696			−0.0359348	+	0.999354i	$0.511441\pi$
$882$		0			0
$883$	111.772	+	111.772i	0.126582	+	0.126582i	0.767560	−	0.640977i	$-0.221471\pi$
$883$	111.772	+	111.772i	0.126582	+	0.126582i	−0.640977	+	0.767560i	$0.721471\pi$
$884$		−	666.864i		−	0.754371i
$885$		0			0
$886$	−143.506			−0.161971
$887$	−550.783	+	550.783i	−0.620950	+	0.620950i	−0.945774	−	0.324824i	$-0.894695\pi$
$887$	−550.783	+	550.783i	−0.620950	+	0.620950i	0.324824	+	0.945774i	$0.394695\pi$
$888$		0			0
$889$			929.201i			1.04522i
$890$	81.1564	−	203.388i	0.0911870	−	0.228526i
$891$		0			0
$892$	16.9411	−	16.9411i	0.0189923	−	0.0189923i
$893$	−722.697	−	722.697i	−0.809291	−	0.809291i
$894$		0			0
$895$	488.727	−	209.942i	0.546064	−	0.234573i
$896$	103.373			0.115372
$897$		0			0
$898$	−412.660	−	412.660i	−0.459532	−	0.459532i
$899$		−	1249.47i		−	1.38984i
$900$		0			0
$901$	−1210.84			−1.34389
$902$	11.9973	−	11.9973i	0.0133008	−	0.0133008i
$903$		0			0
$904$			309.351i			0.342202i
$905$	−686.224	−	1597.47i	−0.758258	−	1.76516i
$906$		0			0
$907$	−774.652	+	774.652i	−0.854081	+	0.854081i	−0.990633	−	0.136552i	$-0.956398\pi$
$907$	−774.652	+	774.652i	−0.854081	+	0.854081i	0.136552	+	0.990633i	$0.456398\pi$
$908$	−520.539	−	520.539i	−0.573281	−	0.573281i
$909$		0			0
$910$	−1109.65	−	442.777i	−1.21940	−	0.486568i
$911$	1468.88			1.61238			0.806189	−	0.591659i	$-0.201527\pi$
$911$	1468.88			1.61238			0.806189	+	0.591659i	$0.201527\pi$
$912$		0			0
$913$	871.405	+	871.405i	0.954441	+	0.954441i
$914$		−	714.068i		−	0.781257i
$915$		0			0
$916$	−104.986			−0.114613
$917$	−1283.36	+	1283.36i	−1.39952	+	1.39952i
$918$		0			0
$919$		−	883.246i		−	0.961095i	−0.876969	−	0.480547i	$-0.840438\pi$
$919$		−	883.246i		−	0.961095i	0.876969	−	0.480547i	$-0.159562\pi$
$920$	220.922	−	94.9012i	0.240132	−	0.103153i
$921$		0			0
$922$	711.833	−	711.833i	0.772053	−	0.772053i
$923$	481.409	+	481.409i	0.521570	+	0.521570i
$924$		0			0
$925$	6.32372	+	242.497i	0.00683645	+	0.262159i
$926$	598.393			0.646213
$927$		0			0
$928$	135.161	+	135.161i	0.145648	+	0.145648i
$929$		−	813.460i		−	0.875629i	−0.899065	−	0.437815i	$-0.855753\pi$
$929$		−	813.460i		−	0.875629i	0.899065	−	0.437815i	$-0.144247\pi$
$930$		0			0
$931$	469.066			0.503830
$932$	442.793	−	442.793i	0.475100	−	0.475100i
$933$		0			0
$934$			1157.70i			1.23951i
$935$	−1031.90	−	411.749i	−1.10363	−	0.440374i
$936$		0			0
$937$	−1047.09	+	1047.09i	−1.11750	+	1.11750i	−0.125388	+	0.992108i	$0.540018\pi$
$937$	−1047.09	+	1047.09i	−1.11750	+	1.11750i	−0.992108	+	0.125388i	$0.959982\pi$
$938$	−293.165	−	293.165i	−0.312543	−	0.312543i
$939$		0			0
$940$	−278.470	+	697.880i	−0.296245	+	0.742426i
$941$	−675.780			−0.718151			−0.359075	−	0.933309i	$-0.616908\pi$
$941$	−675.780			−0.718151			−0.359075	+	0.933309i	$0.616908\pi$
$942$		0			0
$943$	11.7042	+	11.7042i	0.0124117	+	0.0124117i
$944$			215.955i			0.228766i
$945$		0			0
$946$	1212.81			1.28204
$947$	123.223	−	123.223i	0.130119	−	0.130119i	−0.639048	−	0.769167i	$-0.720671\pi$
$947$	123.223	−	123.223i	0.130119	−	0.130119i	0.769167	+	0.639048i	$0.220671\pi$
$948$		0			0
$949$		−	2224.27i		−	2.34380i
$950$	480.745	−	12.5366i	0.506047	−	0.0131964i
$951$		0			0
$952$	329.502	−	329.502i	0.346116	−	0.346116i
$953$	119.680	+	119.680i	0.125582	+	0.125582i	0.767104	−	0.641522i	$-0.221697\pi$
$953$	119.680	+	119.680i	0.125582	+	0.125582i	−0.641522	+	0.767104i	$0.721697\pi$
$954$		0			0
$955$	−407.907	−	949.570i	−0.427127	−	0.994314i
$956$	7.51932			0.00786540
$957$		0			0
$958$	−198.574	−	198.574i	−0.207280	−	0.207280i
$959$			1436.17i			1.49757i
$960$		0			0
$961$	406.316			0.422805
$962$	179.431	−	179.431i	0.186518	−	0.186518i
$963$		0			0
$964$		−	275.909i		−	0.286213i
$965$	−319.182	+	799.909i	−0.330758	+	0.828922i
$966$		0			0
$967$	−499.034	+	499.034i	−0.516064	+	0.516064i	−0.916378	−	0.400314i	$-0.868901\pi$
$967$	−499.034	+	499.034i	−0.516064	+	0.516064i	0.400314	+	0.916378i	$0.368901\pi$
$968$	61.7222	+	61.7222i	0.0637626	+	0.0637626i
$969$		0			0
$970$	−87.3783	+	37.5351i	−0.0900807	+	0.0386959i
$971$	−452.541			−0.466057			−0.233028	−	0.972470i	$-0.574863\pi$
$971$	−452.541			−0.466057			−0.233028	+	0.972470i	$0.574863\pi$
$972$		0			0
$973$	−1051.15	−	1051.15i	−1.08031	−	1.08031i
$974$			904.538i			0.928684i
$975$		0			0
$976$	−430.524			−0.441111
$977$	−625.196	+	625.196i	−0.639914	+	0.639914i	−0.950534	−	0.310620i	$-0.899463\pi$
$977$	−625.196	+	625.196i	−0.639914	+	0.639914i	0.310620	+	0.950534i	$0.399463\pi$
$978$		0			0
$979$			381.634i			0.389820i
$980$	−136.109	−	316.850i	−0.138887	−	0.323316i
$981$		0			0
$982$	55.1876	−	55.1876i	0.0561992	−	0.0561992i
$983$	−966.110	−	966.110i	−0.982818	−	0.982818i	0.0170369	−	0.999855i	$-0.494577\pi$
$983$	−966.110	−	966.110i	−0.982818	−	0.982818i	−0.999855	+	0.0170369i	$0.994577\pi$
$984$		0			0
$985$	−967.750	−	386.154i	−0.982487	−	0.392034i
$986$	861.652			0.873886
$987$		0			0
$988$	−355.717	−	355.717i	−0.360037	−	0.360037i
$989$			1183.18i			1.19633i
$990$		0			0
$991$	−1070.42			−1.08014			−0.540071	−	0.841620i	$-0.681603\pi$
$991$	−1070.42			−1.08014			−0.540071	+	0.841620i	$0.681603\pi$
$992$	−147.909	+	147.909i	−0.149102	+	0.149102i
$993$		0			0
$994$			475.735i			0.478607i
$995$	−1123.05	+	482.430i	−1.12870	+	0.484854i
$996$		0			0
$997$	−644.229	+	644.229i	−0.646167	+	0.646167i	−0.952064	−	0.305897i	$-0.901044\pi$
$997$	−644.229	+	644.229i	−0.646167	+	0.646167i	0.305897	+	0.952064i	$0.401044\pi$
$998$	681.525	+	681.525i	0.682891	+	0.682891i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	270.3.g.e.163.2	✓	8
3.2	odd	2		270.3.g.f.163.3	yes	8
5.2	odd	4	inner	270.3.g.e.217.2	yes	8
5.3	odd	4		1350.3.g.r.757.4		8
5.4	even	2		1350.3.g.r.1243.4		8
15.2	even	4		270.3.g.f.217.3	yes	8
15.8	even	4		1350.3.g.o.757.4		8
15.14	odd	2		1350.3.g.o.1243.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.g.e.163.2	✓	8	1.1	even	1	trivial
270.3.g.e.217.2	yes	8	5.2	odd	4	inner
270.3.g.f.163.3	yes	8	3.2	odd	2
270.3.g.f.217.3	yes	8	15.2	even	4
1350.3.g.o.757.4		8	15.8	even	4
1350.3.g.o.1243.4		8	15.14	odd	2
1350.3.g.r.757.4		8	5.3	odd	4
1350.3.g.r.1243.4		8	5.4	even	2

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:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	270.g (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{2} -2.00000i q^{4} +(1.97347 + 4.59406i) q^{5} +(-6.46083 + 6.46083i) q^{7} +(2.00000 + 2.00000i) q^{8} +(-6.56753 - 2.62059i) q^{10} +12.3232 q^{11} +(-13.0758 - 13.0758i) q^{13} -12.9217i q^{14} -4.00000 q^{16} +(-12.7500 + 12.7500i) q^{17} +13.6021i q^{19} +(9.18813 - 3.94694i) q^{20} +(-12.3232 + 12.3232i) q^{22} +(-12.0221 - 12.0221i) q^{23} +(-17.2108 + 18.1325i) q^{25} +26.1515 q^{26} +(12.9217 + 12.9217i) q^{28} +33.7903i q^{29} -36.9772 q^{31} +(4.00000 - 4.00000i) q^{32} -25.5000i q^{34} +(-42.4317 - 16.9312i) q^{35} +(6.86120 - 6.86120i) q^{37} +(-13.6021 - 13.6021i) q^{38} +(-5.24119 + 13.1351i) q^{40} -0.973558 q^{41} +(-49.2083 - 49.2083i) q^{43} -24.6464i q^{44} +24.0442 q^{46} +(-53.1311 + 53.1311i) q^{47} -34.4847i q^{49} +(-0.921664 - 35.3433i) q^{50} +(-26.1515 + 26.1515i) q^{52} +(47.4841 + 47.4841i) q^{53} +(24.3195 + 56.6135i) q^{55} -25.8433 q^{56} +(-33.7903 - 33.7903i) q^{58} -53.9887i q^{59} +107.631 q^{61} +(36.9772 - 36.9772i) q^{62} +8.00000i q^{64} +(34.2662 - 85.8755i) q^{65} +(22.6879 - 22.6879i) q^{67} +(25.5000 + 25.5000i) q^{68} +(59.3629 - 25.5005i) q^{70} -36.8169 q^{71} +(85.0530 + 85.0530i) q^{73} +13.7224i q^{74} +27.2043 q^{76} +(-79.6181 + 79.6181i) q^{77} +29.5091i q^{79} +(-7.89388 - 18.3763i) q^{80} +(0.973558 - 0.973558i) q^{82} +(70.7126 + 70.7126i) q^{83} +(-83.7360 - 33.4126i) q^{85} +98.4165 q^{86} +(24.6464 + 24.6464i) q^{88} +30.9687i q^{89} +168.961 q^{91} +(-24.0442 + 24.0442i) q^{92} -106.262i q^{94} +(-62.4891 + 26.8434i) q^{95} +(9.50991 - 9.50991i) q^{97} +(34.4847 + 34.4847i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 12 q^{5} - 8 q^{7} + 16 q^{8} - 16 q^{10} - 40 q^{11} - 24 q^{13} - 32 q^{16} + 8 q^{17} + 8 q^{20} + 40 q^{22} - 16 q^{23} + 16 q^{25} + 48 q^{26} + 16 q^{28} - 64 q^{31} + 32 q^{32} - 124 q^{35}+ \cdots + 128 q^{98}+O(q^{100}) \) 8 * q - 8 * q^2 + 12 * q^5 - 8 * q^7 + 16 * q^8 - 16 * q^10 - 40 * q^11 - 24 * q^13 - 32 * q^16 + 8 * q^17 + 8 * q^20 + 40 * q^22 - 16 * q^23 + 16 * q^25 + 48 * q^26 + 16 * q^28 - 64 * q^31 + 32 * q^32 - 124 * q^35 - 24 * q^37 + 8 * q^38 + 16 * q^40 + 56 * q^41 - 72 * q^43 + 32 * q^46 + 80 * q^50 - 48 * q^52 - 80 * q^53 - 96 * q^55 - 32 * q^56 - 88 * q^58 + 216 * q^61 + 64 * q^62 + 156 * q^65 + 136 * q^67 - 16 * q^68 + 152 * q^70 - 56 * q^71 + 368 * q^73 - 16 * q^76 - 320 * q^77 - 48 * q^80 - 56 * q^82 + 168 * q^83 - 56 * q^85 + 144 * q^86 - 80 * q^88 - 312 * q^91 - 32 * q^92 + 4 * q^95 - 312 * q^97 + 128 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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