LMFDB - Embedded newform 900.3.f.g.199.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(199,900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(900, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 1]))

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

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

chi := DirichletCharacter("900.199");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	900.f (of order $2$, degree $1$, 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:	$24.5232237924$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{12} + 25x^{8} - 16x^{4} + 256 $ x^16 - x^12 + 25x^8 - 16x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{20}\cdot 5^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			199.9
Root			$1.26588 - 0.630504i$ of defining polynomial
Character	$\chi$	$=$	900.199
Dual form			900.3.f.g.199.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.635381 - 1.89639i) q^{2} +(-3.19258 - 2.40986i) q^{4} -10.1035 q^{7} +(-6.59853 + 4.52320i) q^{8} +O(q^{10})$	\(q+(0.635381 - 1.89639i) q^{2} +(-3.19258 - 2.40986i) q^{4} -10.1035 q^{7} +(-6.59853 + 4.52320i) q^{8} -10.6555i q^{11} +11.7703i q^{13} +(-6.41959 + 19.1602i) q^{14} +(4.38516 + 15.3873i) q^{16} +16.6320i q^{17} +0.464096i q^{19} +(-20.2071 - 6.77033i) q^{22} +42.1327 q^{23} +(22.3211 + 7.47864i) q^{26} +(32.2564 + 24.3481i) q^{28} -19.5536 q^{29} -9.17534i q^{31} +(31.9666 + 1.46084i) q^{32} +(31.5407 + 10.5676i) q^{34} -23.5407i q^{37} +(0.880107 + 0.294878i) q^{38} -44.0525 q^{41} +58.3007 q^{43} +(-25.6784 + 34.0187i) q^{44} +(26.7703 - 79.9000i) q^{46} +41.6433 q^{47} +53.0813 q^{49} +(28.3648 - 37.5777i) q^{52} +80.2381i q^{53} +(66.6685 - 45.7003i) q^{56} +(-12.4240 + 37.0813i) q^{58} +63.9333i q^{59} +80.3923 q^{61} +(-17.4000 - 5.82983i) q^{62} +(23.0813 - 59.6930i) q^{64} +22.5275 q^{67} +(40.0807 - 53.0989i) q^{68} +61.4860i q^{71} +137.703i q^{73} +(-44.6422 - 14.9573i) q^{74} +(1.11841 - 1.48167i) q^{76} +107.659i q^{77} -138.665i q^{79} +(-27.9901 + 83.5407i) q^{82} -86.2233 q^{83} +(37.0431 - 110.561i) q^{86} +(48.1972 + 70.3110i) q^{88} +127.212 q^{89} -118.922i q^{91} +(-134.512 - 101.534i) q^{92} +(26.4593 - 78.9719i) q^{94} -15.0000i q^{97} +(33.7269 - 100.663i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{4}+O(q^{10}) $ 16 * q - 8 * q^4	$ 16 q - 8 q^{4} - 16 q^{16} + 160 q^{34} + 256 q^{46} + 160 q^{49} + 80 q^{61} - 320 q^{64} - 456 q^{76} + 768 q^{94}+O(q^{100}) $ 16 * q - 8 * q^4 - 16 * q^16 + 160 * q^34 + 256 * q^46 + 160 * q^49 + 80 * q^61 - 320 * q^64 - 456 * q^76 + 768 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/900\mathbb{Z}\right)^\times$.

$n$	$101$	$451$	$577$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0.635381	−	1.89639i	0.317690	−	0.948194i
$2$	0.635381	−	1.89639i	0.317690	−	0.948194i
$3$		0			0
$3$		0			0
$4$	−3.19258	−	2.40986i	−0.798146	−	0.602465i
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−10.1035			−1.44336			−0.721681	−	0.692226i	$-0.756630\pi$
$7$	−10.1035			−1.44336			−0.721681	+	0.692226i	$0.756630\pi$
$8$	−6.59853	+	4.52320i	−0.824817	+	0.565400i
$9$		0			0
$10$		0			0
$11$		−	10.6555i		−	0.968686i	−0.874878	−	0.484343i	$-0.839059\pi$
$11$		−	10.6555i		−	0.968686i	0.874878	−	0.484343i	$-0.160941\pi$
$12$		0			0
$13$			11.7703i			0.905410i	0.891660	+	0.452705i	$0.149541\pi$
$13$			11.7703i			0.905410i	−0.891660	+	0.452705i	$0.850459\pi$
$14$	−6.41959	+	19.1602i	−0.458542	+	1.36859i
$15$		0			0
$16$	4.38516	+	15.3873i	0.274073	+	0.961709i
$17$			16.6320i			0.978350i	0.872186	+	0.489175i	$0.162702\pi$
$17$			16.6320i			0.978350i	−0.872186	+	0.489175i	$0.837298\pi$
$18$		0			0
$19$			0.464096i			0.0244261i	0.999925	+	0.0122131i	$0.00388763\pi$
$19$			0.464096i			0.0244261i	−0.999925	+	0.0122131i	$0.996112\pi$
$20$		0			0
$21$		0			0
$22$	−20.2071	−	6.77033i	−0.918503	−	0.307742i
$23$	42.1327			1.83186			0.915929	−	0.401340i	$-0.131456\pi$
$23$	42.1327			1.83186			0.915929	+	0.401340i	$0.131456\pi$
$24$		0			0
$25$		0			0
$26$	22.3211	+	7.47864i	0.858505	+	0.287640i
$27$		0			0
$28$	32.2564	+	24.3481i	1.15201	+	0.869574i
$29$	−19.5536			−0.674264			−0.337132	−	0.941457i	$-0.609457\pi$
$29$	−19.5536			−0.674264			−0.337132	+	0.941457i	$0.609457\pi$
$30$		0			0
$31$		−	9.17534i		−	0.295979i	−0.988989	−	0.147989i	$-0.952720\pi$
$31$		−	9.17534i		−	0.295979i	0.988989	−	0.147989i	$-0.0472801\pi$
$32$	31.9666	+	1.46084i	0.998957	+	0.0456514i
$33$		0			0
$34$	31.5407	+	10.5676i	0.927666	+	0.310813i
$35$		0			0
$36$		0			0
$37$		−	23.5407i		−	0.636234i	−0.948051	−	0.318117i	$-0.896950\pi$
$37$		−	23.5407i		−	0.636234i	0.948051	−	0.318117i	$-0.103050\pi$
$38$	0.880107	+	0.294878i	0.0231607	+	0.00775994i
$39$		0			0
$40$		0			0
$41$	−44.0525			−1.07445			−0.537226	−	0.843439i	$-0.680528\pi$
$41$	−44.0525			−1.07445			−0.537226	+	0.843439i	$0.680528\pi$
$42$		0			0
$43$	58.3007			1.35583			0.677915	−	0.735140i	$-0.262884\pi$
$43$	58.3007			1.35583			0.677915	+	0.735140i	$0.262884\pi$
$44$	−25.6784	+	34.0187i	−0.583599	+	0.773152i
$45$		0			0
$46$	26.7703	−	79.9000i	0.581964	−	1.73696i
$47$	41.6433			0.886027			0.443014	−	0.896515i	$-0.353909\pi$
$47$	41.6433			0.886027			0.443014	+	0.896515i	$0.353909\pi$
$48$		0			0
$49$	53.0813			1.08329
$50$		0			0
$51$		0			0
$52$	28.3648	−	37.5777i	0.545477	−	0.722649i
$53$			80.2381i			1.51393i	0.653458	+	0.756963i	$0.273318\pi$
$53$			80.2381i			1.51393i	−0.653458	+	0.756963i	$0.726682\pi$
$54$		0			0
$55$		0			0
$56$	66.6685	−	45.7003i	1.19051	−	0.816077i
$57$		0			0
$58$	−12.4240	+	37.0813i	−0.214207	+	0.639333i
$59$			63.9333i			1.08361i	0.840503	+	0.541807i	$0.182260\pi$
$59$			63.9333i			1.08361i	−0.840503	+	0.541807i	$0.817740\pi$
$60$		0			0
$61$	80.3923			1.31791			0.658953	−	0.752184i	$-0.270999\pi$
$61$	80.3923			1.31791			0.658953	+	0.752184i	$0.270999\pi$
$62$	−17.4000	−	5.82983i	−0.280645	−	0.0940296i
$63$		0			0
$64$	23.0813	−	59.6930i	0.360646	−	0.932703i
$65$		0			0
$66$		0			0
$67$	22.5275			0.336232			0.168116	−	0.985767i	$-0.446232\pi$
$67$	22.5275			0.336232			0.168116	+	0.985767i	$0.446232\pi$
$68$	40.0807	−	53.0989i	0.589421	−	0.780866i
$69$		0			0
$70$		0			0
$71$			61.4860i			0.866000i	0.901394	+	0.433000i	$0.142545\pi$
$71$			61.4860i			0.866000i	−0.901394	+	0.433000i	$0.857455\pi$
$72$		0			0
$73$			137.703i			1.88635i	0.332301	+	0.943173i	$0.392175\pi$
$73$			137.703i			1.88635i	−0.332301	+	0.943173i	$0.607825\pi$
$74$	−44.6422	−	14.9573i	−0.603274	−	0.202125i
$75$		0			0
$76$	1.11841	−	1.48167i	0.0147159	−	0.0194956i
$77$			107.659i			1.39816i
$78$		0			0
$79$		−	138.665i		−	1.75525i	−0.479347	−	0.877626i	$-0.659126\pi$
$79$		−	138.665i		−	1.75525i	0.479347	−	0.877626i	$-0.340874\pi$
$80$		0			0
$81$		0			0
$82$	−27.9901	+	83.5407i	−0.341343	+	1.01879i
$83$	−86.2233			−1.03883			−0.519417	−	0.854521i	$-0.673851\pi$
$83$	−86.2233			−1.03883			−0.519417	+	0.854521i	$0.673851\pi$
$84$		0			0
$85$		0			0
$86$	37.0431	−	110.561i	0.430734	−	1.28559i
$87$		0			0
$88$	48.1972	+	70.3110i	0.547695	+	0.798989i
$89$	127.212			1.42935			0.714676	−	0.699456i	$-0.246574\pi$
$89$	127.212			1.42935			0.714676	+	0.699456i	$0.246574\pi$
$90$		0			0
$91$		−	118.922i		−	1.30683i
$92$	−134.512	−	101.534i	−1.46209	−	1.10363i
$93$		0			0
$94$	26.4593	−	78.9719i	0.281482	−	0.840126i
$95$		0			0
$96$		0			0
$97$		−	15.0000i		−	0.154639i	−0.997006	−	0.0773196i	$-0.975364\pi$
$97$		−	15.0000i		−	0.154639i	0.997006	−	0.0773196i	$-0.0246362\pi$
$98$	33.7269	−	100.663i	0.344152	−	1.02717i
$99$		0			0
$100$		0			0
$101$	−195.764			−1.93825			−0.969127	−	0.246563i	$-0.920699\pi$
$101$	−195.764			−1.93825			−0.969127	+	0.246563i	$0.920699\pi$
$102$		0			0
$103$	15.5661			0.151127			0.0755636	−	0.997141i	$-0.475924\pi$
$103$	15.5661			0.151127			0.0755636	+	0.997141i	$0.475924\pi$
$104$	−53.2396	−	77.6669i	−0.511919	−	0.746797i
$105$		0			0
$106$	152.163	+	50.9817i	1.43550	+	0.480960i
$107$	51.3199			0.479625			0.239813	−	0.970819i	$-0.422914\pi$
$107$	51.3199			0.479625			0.239813	+	0.970819i	$0.422914\pi$
$108$		0			0
$109$	46.8516			0.429832			0.214916	−	0.976633i	$-0.431052\pi$
$109$	46.8516			0.429832			0.214916	+	0.976633i	$0.431052\pi$
$110$		0			0
$111$		0			0
$112$	−44.3056	−	155.466i	−0.395586	−	1.38809i
$113$			115.526i			1.02235i	0.859477	+	0.511175i	$0.170790\pi$
$113$			115.526i			1.02235i	−0.859477	+	0.511175i	$0.829210\pi$
$114$		0			0
$115$		0			0
$116$	62.4266	+	47.1215i	0.538161	+	0.406220i
$117$		0			0
$118$	121.242	+	40.6220i	1.02748	+	0.344254i
$119$		−	168.041i		−	1.41211i
$120$		0			0
$121$	7.45934			0.0616474
$122$	51.0797	−	152.455i	0.418686	−	1.24963i
$123$		0			0
$124$	−22.1113	+	29.2930i	−0.178317	+	0.236234i
$125$		0			0
$126$		0			0
$127$	−161.656			−1.27289			−0.636443	−	0.771324i	$-0.719595\pi$
$127$	−161.656			−1.27289			−0.636443	+	0.771324i	$0.719595\pi$
$128$	−98.5357	−	81.6989i	−0.769810	−	0.638273i
$129$		0			0
$130$		0			0
$131$			197.561i			1.50810i	0.656818	+	0.754049i	$0.271902\pi$
$131$			197.561i			1.50810i	−0.656818	+	0.754049i	$0.728098\pi$
$132$		0			0
$133$		−	4.68901i		−	0.0352557i
$134$	14.3136	−	42.7210i	0.106818	−	0.318813i
$135$		0			0
$136$	−75.2297	−	109.747i	−0.553159	−	0.806960i
$137$			76.4182i			0.557797i	0.960321	+	0.278899i	$0.0899693\pi$
$137$			76.4182i			0.557797i	−0.960321	+	0.278899i	$0.910031\pi$
$138$		0			0
$139$			223.206i			1.60580i	0.596115	+	0.802899i	$0.296710\pi$
$139$			223.206i			1.60580i	−0.596115	+	0.802899i	$0.703290\pi$
$140$		0			0
$141$		0			0
$142$	116.601	+	39.0670i	0.821137	+	0.275120i
$143$	125.419			0.877058
$144$		0			0
$145$		0			0
$146$	261.139	+	87.4940i	1.78862	+	0.599274i
$147$		0			0
$148$	−56.7297	+	75.1555i	−0.383308	+	0.507807i
$149$	166.320			1.11624			0.558119	−	0.829761i	$-0.311523\pi$
$149$	166.320			1.11624			0.558119	+	0.829761i	$0.311523\pi$
$150$		0			0
$151$			112.995i			0.748313i	0.927366	+	0.374156i	$0.122068\pi$
$151$			112.995i			0.748313i	−0.927366	+	0.374156i	$0.877932\pi$
$152$	−2.09920	−	3.06236i	−0.0138105	−	0.0201471i
$153$		0			0
$154$	204.163	+	68.4042i	1.32573	+	0.444183i
$155$		0			0
$156$		0			0
$157$		−	75.9330i		−	0.483649i	−0.970320	−	0.241825i	$-0.922254\pi$
$157$		−	75.9330i		−	0.483649i	0.970320	−	0.241825i	$-0.0777459\pi$
$158$	−262.962	−	88.1050i	−1.66432	−	0.557626i
$159$		0			0
$160$		0			0
$161$	−425.689			−2.64403
$162$		0			0
$163$	184.184			1.12996			0.564982	−	0.825103i	$-0.308883\pi$
$163$	184.184			1.12996			0.564982	+	0.825103i	$0.308883\pi$
$164$	140.641	+	106.160i	0.857568	+	0.647319i
$165$		0			0
$166$	−54.7846	+	163.513i	−0.330028	+	0.985017i
$167$	−193.268			−1.15729			−0.578647	−	0.815578i	$-0.696419\pi$
$167$	−193.268			−1.15729			−0.578647	+	0.815578i	$0.696419\pi$
$168$		0			0
$169$	30.4593			0.180233
$170$		0			0
$171$		0			0
$172$	−186.130	−	140.496i	−1.08215	−	0.816840i
$173$			56.8646i			0.328697i	0.986402	+	0.164348i	$0.0525522\pi$
$173$			56.8646i			0.328697i	−0.986402	+	0.164348i	$0.947448\pi$
$174$		0			0
$175$		0			0
$176$	163.961	−	46.7263i	0.931594	−	0.265490i
$177$		0			0
$178$	80.8282	−	241.244i	0.454091	−	1.35530i
$179$		−	272.150i		−	1.52039i	−0.649695	−	0.760195i	$-0.725104\pi$
$179$		−	272.150i		−	1.52039i	0.649695	−	0.760195i	$-0.274896\pi$
$180$		0			0
$181$	−50.2297			−0.277512			−0.138756	−	0.990327i	$-0.544310\pi$
$181$	−50.2297			−0.277512			−0.138756	+	0.990327i	$0.544310\pi$
$182$	−225.522	−	75.5607i	−1.23913	−	0.415169i
$183$		0			0
$184$	−278.014	+	190.575i	−1.51095	+	1.03573i
$185$		0			0
$186$		0			0
$187$	177.223			0.947714
$188$	−132.950	−	100.354i	−0.707179	−	0.533800i
$189$		0			0
$190$		0			0
$191$		−	194.247i		−	1.01700i	−0.861062	−	0.508500i	$-0.830200\pi$
$191$		−	194.247i		−	1.01700i	0.861062	−	0.508500i	$-0.169800\pi$
$192$		0			0
$193$		−	22.7033i		−	0.117634i	−0.998269	−	0.0588168i	$-0.981267\pi$
$193$		−	22.7033i		−	0.117634i	0.998269	−	0.0588168i	$-0.0187328\pi$
$194$	−28.4458	−	9.53071i	−0.146628	−	0.0491274i
$195$		0			0
$196$	−169.466	−	127.918i	−0.864625	−	0.652645i
$197$			154.633i			0.784938i	0.919765	+	0.392469i	$0.128379\pi$
$197$			154.633i			0.784938i	−0.919765	+	0.392469i	$0.871621\pi$
$198$		0			0
$199$		−	236.094i		−	1.18640i	−0.805054	−	0.593201i	$-0.797864\pi$
$199$		−	236.094i		−	1.18640i	0.805054	−	0.593201i	$-0.202136\pi$
$200$		0			0
$201$		0			0
$202$	−124.384	+	371.244i	−0.615765	+	1.83784i
$203$	197.561			0.973206
$204$		0			0
$205$		0			0
$206$	9.89040	−	29.5194i	0.0480116	−	0.143298i
$207$		0			0
$208$	−181.114	+	51.6148i	−0.870741	+	0.248148i
$209$	4.94520			0.0236612
$210$		0			0
$211$			115.209i			0.546015i	0.962012	+	0.273007i	$0.0880183\pi$
$211$			115.209i			0.546015i	−0.962012	+	0.273007i	$0.911982\pi$
$212$	193.362	−	256.167i	0.912087	−	1.20833i
$213$		0			0
$214$	32.6077	−	97.3225i	0.152372	−	0.454778i
$215$		0			0
$216$		0			0
$217$			92.7033i			0.427204i
$218$	29.7686	−	88.8489i	0.136553	−	0.407564i
$219$		0			0
$220$		0			0
$221$	−195.764			−0.885808
$222$		0			0
$223$	−425.170			−1.90659			−0.953296	−	0.302039i	$-0.902333\pi$
$223$	−425.170			−1.90659			−0.953296	+	0.302039i	$0.902333\pi$
$224$	−322.976	−	14.7597i	−1.44186	−	0.0658915i
$225$		0			0
$226$	219.081	+	73.4027i	0.969386	+	0.324791i
$227$	178.208			0.785055			0.392528	−	0.919740i	$-0.371601\pi$
$227$	178.208			0.785055			0.392528	+	0.919740i	$0.371601\pi$
$228$		0			0
$229$	−331.636			−1.44819			−0.724097	−	0.689699i	$-0.757743\pi$
$229$	−331.636			−1.44819			−0.724097	+	0.689699i	$0.757743\pi$
$230$		0			0
$231$		0			0
$232$	129.025	−	88.4451i	0.556144	−	0.381229i
$233$			182.952i			0.785200i	0.919709	+	0.392600i	$0.128424\pi$
$233$			182.952i			0.785200i	−0.919709	+	0.392600i	$0.871576\pi$
$234$		0			0
$235$		0			0
$236$	154.070	−	204.112i	0.652840	−	0.864882i
$237$		0			0
$238$	−318.672	−	106.770i	−1.33896	−	0.448615i
$239$			210.664i			0.881438i	0.897645	+	0.440719i	$0.145276\pi$
$239$			210.664i			0.881438i	−0.897645	+	0.440719i	$0.854724\pi$
$240$		0			0
$241$	−3.62198			−0.0150290			−0.00751448	−	0.999972i	$-0.502392\pi$
$241$	−3.62198			−0.0150290			−0.00751448	+	0.999972i	$0.502392\pi$
$242$	4.73952	−	14.1458i	0.0195848	−	0.0584538i
$243$		0			0
$244$	−256.659	−	193.734i	−1.05188	−	0.793992i
$245$		0			0
$246$		0			0
$247$	−5.46257			−0.0221157
$248$	41.5019	+	60.5438i	0.167346	+	0.244128i
$249$		0			0
$250$		0			0
$251$			250.839i			0.999357i	0.866211	+	0.499678i	$0.166548\pi$
$251$			250.839i			0.999357i	−0.866211	+	0.499678i	$0.833452\pi$
$252$		0			0
$253$		−	448.947i		−	1.77450i
$254$	−102.713	+	306.564i	−0.404384	+	1.20694i
$255$		0			0
$256$	−217.541	+	134.952i	−0.849768	+	0.527157i
$257$			360.060i			1.40101i	0.713647	+	0.700505i	$0.247042\pi$
$257$			360.060i			1.40101i	−0.713647	+	0.700505i	$0.752958\pi$
$258$		0			0
$259$			237.844i			0.918316i
$260$		0			0
$261$		0			0
$262$	374.652	+	125.526i	1.42997	+	0.479108i
$263$	−191.310			−0.727416			−0.363708	−	0.931513i	$-0.618489\pi$
$263$	−191.310			−0.727416			−0.363708	+	0.931513i	$0.618489\pi$
$264$		0			0
$265$		0			0
$266$	−8.89219	−	2.97931i	−0.0334293	−	0.0112004i
$267$		0			0
$268$	−71.9210	−	54.2882i	−0.268362	−	0.202568i
$269$	156.656			0.582365			0.291183	−	0.956667i	$-0.405951\pi$
$269$	156.656			0.582365			0.291183	+	0.956667i	$0.405951\pi$
$270$		0			0
$271$		−	30.4172i		−	0.112241i	−0.998424	−	0.0561203i	$-0.982127\pi$
$271$		−	30.4172i		−	0.112241i	0.998424	−	0.0561203i	$-0.0178730\pi$
$272$	−255.922	+	72.9339i	−0.940888	+	0.268139i
$273$		0			0
$274$	144.919	+	48.5547i	0.528900	+	0.177207i
$275$		0			0
$276$		0			0
$277$			438.421i			1.58275i	0.611333	+	0.791373i	$0.290634\pi$
$277$			438.421i			1.58275i	−0.611333	+	0.791373i	$0.709366\pi$
$278$	423.285	+	141.821i	1.52261	+	0.510146i
$279$		0			0
$280$		0			0
$281$	239.589			0.852630			0.426315	−	0.904575i	$-0.359812\pi$
$281$	239.589			0.852630			0.426315	+	0.904575i	$0.359812\pi$
$282$		0			0
$283$	−235.523			−0.832238			−0.416119	−	0.909310i	$-0.636610\pi$
$283$	−235.523			−0.832238			−0.416119	+	0.909310i	$0.636610\pi$
$284$	148.173	−	196.299i	0.521734	−	0.691194i
$285$		0			0
$286$	79.6890	−	237.844i	0.278633	−	0.831622i
$287$	445.086			1.55082
$288$		0			0
$289$	12.3780			0.0428305
$290$		0			0
$291$		0			0
$292$	331.845	−	439.629i	1.13646	−	1.50558i
$293$			271.955i			0.928173i	0.885790	+	0.464086i	$0.153617\pi$
$293$			271.955i			0.928173i	−0.885790	+	0.464086i	$0.846383\pi$
$294$		0			0
$295$		0			0
$296$	106.479	+	155.334i	0.359727	+	0.524777i
$297$		0			0
$298$	105.676	−	315.407i	0.354618	−	1.05841i
$299$			495.916i			1.65858i
$300$		0			0
$301$	−589.043			−1.95695
$302$	214.283	+	71.7950i	0.709546	+	0.237732i
$303$		0			0
$304$	−7.14121	+	2.03514i	−0.0234908	+	0.00669454i
$305$		0			0
$306$		0			0
$307$	316.352			1.03046			0.515230	−	0.857052i	$-0.327706\pi$
$307$	316.352			1.03046			0.515230	+	0.857052i	$0.327706\pi$
$308$	259.442	−	343.709i	0.842344	−	1.11594i
$309$		0			0
$310$		0			0
$311$			298.355i			0.959342i	0.877448	+	0.479671i	$0.159244\pi$
$311$			298.355i			0.959342i	−0.877448	+	0.479671i	$0.840756\pi$
$312$		0			0
$313$		−	191.651i		−	0.612302i	−0.951983	−	0.306151i	$-0.900959\pi$
$313$		−	191.651i		−	0.612302i	0.951983	−	0.306151i	$-0.0990413\pi$
$314$	−143.998	−	48.2464i	−0.458594	−	0.153651i
$315$		0			0
$316$	−334.163	+	442.699i	−1.05748	+	1.40095i
$317$		−	232.847i		−	0.734534i	−0.930115	−	0.367267i	$-0.880293\pi$
$317$		−	232.847i		−	0.734534i	0.930115	−	0.367267i	$-0.119707\pi$
$318$		0			0
$319$			208.355i			0.653150i
$320$		0			0
$321$		0			0
$322$	−270.475	+	807.273i	−0.839984	+	2.50706i
$323$	−7.71883			−0.0238973
$324$		0			0
$325$		0			0
$326$	117.027	−	349.285i	0.358979	−	1.07143i
$327$		0			0
$328$	290.682	−	199.258i	0.886225	−	0.607495i
$329$	−420.744			−1.27886
$330$		0			0
$331$		−	200.214i		−	0.604877i	−0.953169	−	0.302438i	$-0.902199\pi$
$331$		−	200.214i		−	0.604877i	0.953169	−	0.302438i	$-0.0978007\pi$
$332$	275.275	+	207.786i	0.829141	+	0.625861i
$333$		0			0
$334$	−122.799	+	366.512i	−0.367661	+	1.09734i
$335$		0			0
$336$		0			0
$337$		−	451.029i		−	1.33836i	−0.743099	−	0.669182i	$-0.766645\pi$
$337$		−	451.029i		−	1.33836i	0.743099	−	0.669182i	$-0.233355\pi$
$338$	19.3533	−	57.7628i	0.0572582	−	0.170896i
$339$		0			0
$340$		0			0
$341$	−97.7682			−0.286710
$342$		0			0
$343$	−41.2357			−0.120221
$344$	−384.699	+	263.706i	−1.11831	+	0.766586i
$345$		0			0
$346$	107.837	+	36.1307i	0.311669	+	0.104424i
$347$	−128.845			−0.371313			−0.185656	−	0.982615i	$-0.559441\pi$
$347$	−128.845			−0.371313			−0.185656	+	0.982615i	$0.559441\pi$
$348$		0			0
$349$	−0.756045			−0.00216632			−0.00108316	−	0.999999i	$-0.500345\pi$
$349$	−0.756045			−0.00216632			−0.00108316	+	0.999999i	$0.500345\pi$
$350$		0			0
$351$		0			0
$352$	15.5661	−	340.622i	0.0442219	−	0.967676i
$353$			55.7392i			0.157902i	0.996879	+	0.0789508i	$0.0251570\pi$
$353$			55.7392i			0.157902i	−0.996879	+	0.0789508i	$0.974843\pi$
$354$		0			0
$355$		0			0
$356$	−406.136	−	306.564i	−1.14083	−	0.861134i
$357$		0			0
$358$	−516.102	−	172.919i	−1.44162	−	0.483013i
$359$			194.247i			0.541078i	0.962709	+	0.270539i	$0.0872019\pi$
$359$			194.247i			0.541078i	−0.962709	+	0.270539i	$0.912798\pi$
$360$		0			0
$361$	360.785			0.999403
$362$	−31.9150	+	95.2550i	−0.0881629	+	0.263135i
$363$		0			0
$364$	−286.585	+	379.668i	−0.787321	+	1.04304i
$365$		0			0
$366$		0			0
$367$	−24.0264			−0.0654671			−0.0327335	−	0.999464i	$-0.510421\pi$
$367$	−24.0264			−0.0654671			−0.0327335	+	0.999464i	$0.510421\pi$
$368$	184.759	+	648.311i	0.502062	+	1.76171i
$369$		0			0
$370$		0			0
$371$		−	810.688i		−	2.18514i
$372$		0			0
$373$		−	136.986i		−	0.367254i	−0.982996	−	0.183627i	$-0.941216\pi$
$373$		−	136.986i		−	0.367254i	0.982996	−	0.183627i	$-0.0587838\pi$
$374$	112.604	−	336.083i	0.301080	−	0.898617i
$375$		0			0
$376$	−274.785	+	188.361i	−0.730810	+	0.500960i
$377$		−	230.153i		−	0.610485i
$378$		0			0
$379$			103.356i			0.272707i	0.990660	+	0.136353i	$0.0435382\pi$
$379$			103.356i			0.272707i	−0.990660	+	0.136353i	$0.956462\pi$
$380$		0			0
$381$		0			0
$382$	−368.368	−	123.421i	−0.964314	−	0.323091i
$383$	19.3533			0.0505308			0.0252654	−	0.999681i	$-0.491957\pi$
$383$	19.3533			0.0505308			0.0252654	+	0.999681i	$0.491957\pi$
$384$		0			0
$385$		0			0
$386$	−43.0543	−	14.4252i	−0.111540	−	0.0373711i
$387$		0			0
$388$	−36.1479	+	47.8887i	−0.0931646	+	0.123425i
$389$	−273.978			−0.704314			−0.352157	−	0.935941i	$-0.614552\pi$
$389$	−273.978			−0.704314			−0.352157	+	0.935941i	$0.614552\pi$
$390$		0			0
$391$			700.750i			1.79220i
$392$	−350.259	+	240.097i	−0.893518	+	0.612493i
$393$		0			0
$394$	293.244	+	98.2507i	0.744274	+	0.249367i
$395$		0			0
$396$		0			0
$397$			218.230i			0.549697i	0.961488	+	0.274848i	$0.0886277\pi$
$397$			218.230i			0.549697i	−0.961488	+	0.274848i	$0.911372\pi$
$398$	−447.726	−	150.010i	−1.12494	−	0.376908i
$399$		0			0
$400$		0			0
$401$	396.472			0.988709			0.494355	−	0.869260i	$-0.335404\pi$
$401$	396.472			0.988709			0.494355	+	0.869260i	$0.335404\pi$
$402$		0			0
$403$	107.997			0.267982
$404$	624.991	+	471.763i	1.54701	+	1.16773i
$405$		0			0
$406$	125.526	−	374.652i	0.309178	−	0.922789i
$407$	−250.839			−0.616311
$408$		0			0
$409$	−301.976			−0.738327			−0.369164	−	0.929364i	$-0.620356\pi$
$409$	−301.976			−0.738327			−0.369164	+	0.929364i	$0.620356\pi$
$410$		0			0
$411$		0			0
$412$	−49.6960	−	37.5121i	−0.120621	−	0.0910488i
$413$		−	645.952i		−	1.56405i
$414$		0			0
$415$		0			0
$416$	−17.1946	+	376.258i	−0.0413332	+	0.904466i
$417$		0			0
$418$	3.14209	−	9.37802i	0.00751695	−	0.0224355i
$419$		−	420.461i		−	1.00349i	−0.865017	−	0.501743i	$-0.832692\pi$
$419$		−	420.461i		−	1.00349i	0.865017	−	0.501743i	$-0.167308\pi$
$420$		0			0
$421$	−197.378			−0.468831			−0.234416	−	0.972136i	$-0.575318\pi$
$421$	−197.378			−0.468831			−0.234416	+	0.972136i	$0.575318\pi$
$422$	218.481	+	73.2017i	0.517728	+	0.173464i
$423$		0			0
$424$	−362.933	−	529.454i	−0.855974	−	1.24871i
$425$		0			0
$426$		0			0
$427$	−812.246			−1.90222
$428$	−163.843	−	123.674i	−0.382811	−	0.288957i
$429$		0			0
$430$		0			0
$431$			236.869i			0.549581i	0.961504	+	0.274790i	$0.0886084\pi$
$431$			236.869i			0.549581i	−0.961504	+	0.274790i	$0.911392\pi$
$432$		0			0
$433$			364.378i			0.841520i	0.907172	+	0.420760i	$0.138236\pi$
$433$			364.378i			0.841520i	−0.907172	+	0.420760i	$0.861764\pi$
$434$	175.802	+	58.9019i	0.405073	+	0.135719i
$435$		0			0
$436$	−149.578	−	112.906i	−0.343068	−	0.258958i
$437$			19.5536i			0.0447452i
$438$		0			0
$439$		−	72.3679i		−	0.164847i	−0.996597	−	0.0824236i	$-0.973734\pi$
$439$		−	72.3679i		−	0.164847i	0.996597	−	0.0824236i	$-0.0262660\pi$
$440$		0			0
$441$		0			0
$442$	−124.384	+	371.244i	−0.281413	+	0.839918i
$443$	240.295			0.542428			0.271214	−	0.962519i	$-0.412575\pi$
$443$	240.295			0.542428			0.271214	+	0.962519i	$0.412575\pi$
$444$		0			0
$445$		0			0
$446$	−270.145	+	806.287i	−0.605706	+	1.80782i
$447$		0			0
$448$	−233.203	+	603.110i	−0.520542	+	1.34623i
$449$	498.959			1.11127			0.555633	−	0.831427i	$-0.312476\pi$
$449$	498.959			1.11127			0.555633	+	0.831427i	$0.312476\pi$
$450$		0			0
$451$			469.403i			1.04081i
$452$	278.400	−	368.825i	0.615930	−	0.815984i
$453$		0			0
$454$	113.230	−	337.951i	0.249405	−	0.744385i
$455$		0			0
$456$		0			0
$457$			171.866i			0.376074i	0.982162	+	0.188037i	$0.0602125\pi$
$457$			171.866i			0.376074i	−0.982162	+	0.188037i	$0.939787\pi$
$458$	−210.715	+	628.911i	−0.460077	+	1.37317i
$459$		0			0
$460$		0			0
$461$	831.825			1.80439			0.902196	−	0.431326i	$-0.141954\pi$
$461$	831.825			1.80439			0.902196	+	0.431326i	$0.141954\pi$
$462$		0			0
$463$	370.011			0.799160			0.399580	−	0.916698i	$-0.369156\pi$
$463$	370.011			0.799160			0.399580	+	0.916698i	$0.369156\pi$
$464$	−85.7460	−	300.879i	−0.184797	−	0.648445i
$465$		0			0
$466$	346.947	+	116.244i	0.744522	+	0.249450i
$467$	−527.770			−1.13013			−0.565065	−	0.825047i	$-0.691149\pi$
$467$	−527.770			−1.13013			−0.565065	+	0.825047i	$0.691149\pi$
$468$		0			0
$469$	−227.608			−0.485304
$470$		0			0
$471$		0			0
$472$	−289.183	−	421.866i	−0.612676	−	0.893784i
$473$		−	621.226i		−	1.31337i
$474$		0			0
$475$		0			0
$476$	−404.956	+	536.486i	−0.850748	+	1.12707i
$477$		0			0
$478$	399.500	+	133.852i	0.835775	+	0.280024i
$479$		−	532.777i		−	1.11227i	−0.831092	−	0.556135i	$-0.812284\pi$
$479$		−	532.777i		−	1.11227i	0.831092	−	0.556135i	$-0.187716\pi$
$480$		0			0
$481$	277.081			0.576053
$482$	−2.30134	+	6.86868i	−0.00477455	+	0.0142504i
$483$		0			0
$484$	−23.8146	−	17.9760i	−0.0492036	−	0.0371404i
$485$		0			0
$486$		0			0
$487$	495.073			1.01658			0.508288	−	0.861187i	$-0.330278\pi$
$487$	495.073			1.01658			0.508288	+	0.861187i	$0.330278\pi$
$488$	−530.471	+	363.631i	−1.08703	+	0.745144i
$489$		0			0
$490$		0			0
$491$		−	518.094i		−	1.05518i	−0.849499	−	0.527590i	$-0.823096\pi$
$491$		−	518.094i		−	1.05518i	0.849499	−	0.527590i	$-0.176904\pi$
$492$		0			0
$493$		−	325.215i		−	0.659666i
$494$	−3.47081	+	10.3592i	−0.00702593	+	0.0209699i
$495$		0			0
$496$	141.184	−	40.2354i	0.284645	−	0.0811197i
$497$		−	621.226i		−	1.24995i
$498$		0			0
$499$		−	676.579i		−	1.35587i	−0.735122	−	0.677935i	$-0.762875\pi$
$499$		−	676.579i		−	1.35587i	0.735122	−	0.677935i	$-0.237125\pi$
$500$		0			0
$501$		0			0
$502$	475.688	+	159.378i	0.947585	+	0.317486i
$503$	−35.2804			−0.0701399			−0.0350700	−	0.999385i	$-0.511165\pi$
$503$	−35.2804			−0.0701399			−0.0350700	+	0.999385i	$0.511165\pi$
$504$		0			0
$505$		0			0
$506$	−851.379	−	285.252i	−1.68257	−	0.563740i
$507$		0			0
$508$	516.102	+	389.569i	1.01595	+	0.766869i
$509$	−538.520			−1.05800			−0.528998	−	0.848623i	$-0.677432\pi$
$509$	−538.520			−1.05800			−0.528998	+	0.848623i	$0.677432\pi$
$510$		0			0
$511$		−	1391.29i		−	2.72268i
$512$	117.700	+	498.288i	0.229884	+	0.973218i
$513$		0			0
$514$	682.813	+	228.775i	1.32843	+	0.445088i
$515$		0			0
$516$		0			0
$517$		−	443.732i		−	0.858282i
$518$	451.044	+	151.121i	0.870742	+	0.291740i
$519$		0			0
$520$		0			0
$521$	73.7237			0.141504			0.0707521	−	0.997494i	$-0.477460\pi$
$521$	73.7237			0.141504			0.0707521	+	0.997494i	$0.477460\pi$
$522$		0			0
$523$	−465.440			−0.889942			−0.444971	−	0.895545i	$-0.646786\pi$
$523$	−465.440			−0.889942			−0.444971	+	0.895545i	$0.646786\pi$
$524$	476.094	−	630.729i	0.908576	−	1.20368i
$525$		0			0
$526$	−121.555	+	362.799i	−0.231093	+	0.689732i
$527$	152.604			0.289571
$528$		0			0
$529$	1246.17			2.35570
$530$		0			0
$531$		0			0
$532$	−11.2999	+	14.9701i	−0.0212403	+	0.0281392i
$533$		−	518.512i		−	0.972819i
$534$		0			0
$535$		0			0
$536$	−148.649	+	101.897i	−0.277330	+	0.190106i
$537$		0			0
$538$	99.5364	−	297.081i	0.185012	−	0.552196i
$539$		−	565.610i		−	1.04937i
$540$		0			0
$541$	−48.9857			−0.0905466			−0.0452733	−	0.998975i	$-0.514416\pi$
$541$	−48.9857			−0.0905466			−0.0452733	+	0.998975i	$0.514416\pi$
$542$	−57.6828	−	19.3265i	−0.106426	−	0.0356577i
$543$		0			0
$544$	−24.2967	+	531.668i	−0.0446631	+	0.977330i
$545$		0			0
$546$		0			0
$547$	500.536			0.915056			0.457528	−	0.889195i	$-0.348735\pi$
$547$	500.536			0.915056			0.457528	+	0.889195i	$0.348735\pi$
$548$	184.157	−	243.971i	0.336053	−	0.445203i
$549$		0			0
$550$		0			0
$551$		−	9.07478i		−	0.0164696i
$552$		0			0
$553$			1401.00i			2.53346i
$554$	831.417	+	278.564i	1.50075	+	0.502824i
$555$		0			0
$556$	537.895	−	712.603i	0.967436	−	1.28166i
$557$		−	771.368i		−	1.38486i	−0.721484	−	0.692431i	$-0.756540\pi$
$557$		−	771.368i		−	1.38486i	0.721484	−	0.692431i	$-0.243460\pi$
$558$		0			0
$559$			686.218i			1.22758i
$560$		0			0
$561$		0			0
$562$	152.230	−	454.354i	0.270872	−	0.808459i
$563$	−535.602			−0.951335			−0.475667	−	0.879625i	$-0.657793\pi$
$563$	−535.602			−0.951335			−0.475667	+	0.879625i	$0.657793\pi$
$564$		0			0
$565$		0			0
$566$	−149.647	+	446.644i	−0.264394	+	0.789123i
$567$		0			0
$568$	−278.114	−	405.718i	−0.489637	−	0.714292i
$569$	−102.941			−0.180915			−0.0904575	−	0.995900i	$-0.528833\pi$
$569$	−102.941			−0.180915			−0.0904575	+	0.995900i	$0.528833\pi$
$570$		0			0
$571$		−	642.662i		−	1.12550i	−0.826626	−	0.562752i	$-0.809743\pi$
$571$		−	642.662i		−	1.12550i	0.826626	−	0.562752i	$-0.190257\pi$
$572$	−400.411	−	302.243i	−0.700020	−	0.528396i
$573$		0			0
$574$	282.799	−	844.056i	0.492681	−	1.47048i
$575$		0			0
$576$		0			0
$577$		−	397.919i		−	0.689634i	−0.938670	−	0.344817i	$-0.887941\pi$
$577$		−	397.919i		−	0.689634i	0.938670	−	0.344817i	$-0.112059\pi$
$578$	7.86476	−	23.4735i	0.0136068	−	0.0406117i
$579$		0			0
$580$		0			0
$581$	871.159			1.49941
$582$		0			0
$583$	854.981			1.46652
$584$	−622.860	−	908.640i	−1.06654	−	1.55589i
$585$		0			0
$586$	515.732	+	172.795i	0.880089	+	0.294872i
$587$	−813.625			−1.38607			−0.693036	−	0.720903i	$-0.743727\pi$
$587$	−813.625			−1.38607			−0.693036	+	0.720903i	$0.743727\pi$
$588$		0			0
$589$	4.25824			0.00722961
$590$		0			0
$591$		0			0
$592$	362.228	−	103.230i	0.611872	−	0.174374i
$593$		−	731.362i		−	1.23333i	−0.787227	−	0.616663i	$-0.788484\pi$
$593$		−	731.362i		−	1.23333i	0.787227	−	0.616663i	$-0.211516\pi$
$594$		0			0
$595$		0			0
$596$	−530.989	−	400.807i	−0.890921	−	0.672494i
$597$		0			0
$598$	940.450	+	315.096i	1.57266	+	0.526916i
$599$			752.516i			1.25629i	0.778098	+	0.628143i	$0.216185\pi$
$599$			752.516i			1.25629i	−0.778098	+	0.628143i	$0.783815\pi$
$600$		0			0
$601$	1013.81			1.68688			0.843439	−	0.537226i	$-0.180528\pi$
$601$	1013.81			1.68688			0.843439	+	0.537226i	$0.180528\pi$
$602$	−374.267	+	1117.05i	−0.621705	+	1.85557i
$603$		0			0
$604$	272.302	−	360.747i	0.450832	−	0.597262i
$605$		0			0
$606$		0			0
$607$	−432.276			−0.712151			−0.356075	−	0.934457i	$-0.615885\pi$
$607$	−432.276			−0.712151			−0.356075	+	0.934457i	$0.615885\pi$
$608$	−0.677973	+	14.8356i	−0.00111509	+	0.0244007i
$609$		0			0
$610$		0			0
$611$			490.155i			0.802218i
$612$		0			0
$613$		−	213.971i		−	0.349056i	−0.984652	−	0.174528i	$-0.944160\pi$
$613$		−	213.971i		−	0.349056i	0.984652	−	0.174528i	$-0.0558400\pi$
$614$	201.004	−	599.926i	0.327368	−	0.977077i
$615$		0			0
$616$	−486.962	−	710.389i	−0.790522	−	1.15323i
$617$		−	814.966i		−	1.32085i	−0.750891	−	0.660426i	$-0.770376\pi$
$617$		−	814.966i		−	1.32085i	0.750891	−	0.660426i	$-0.229624\pi$
$618$		0			0
$619$		−	1035.45i		−	1.67278i	−0.548133	−	0.836391i	$-0.684661\pi$
$619$		−	1035.45i		−	1.67278i	0.548133	−	0.836391i	$-0.315339\pi$
$620$		0			0
$621$		0			0
$622$	565.798	+	189.569i	0.909643	+	0.304774i
$623$	−1285.29			−2.06307
$624$		0			0
$625$		0			0
$626$	−363.444	−	121.771i	−0.580581	−	0.194522i
$627$		0			0
$628$	−182.988	+	242.422i	−0.291382	+	0.386023i
$629$	391.527			0.622460
$630$		0			0
$631$		−	67.7270i		−	0.107333i	−0.998559	−	0.0536664i	$-0.982909\pi$
$631$		−	67.7270i		−	0.107333i	0.998559	−	0.0536664i	$-0.0170907\pi$
$632$	627.209	+	914.985i	0.992419	+	1.44776i
$633$		0			0
$634$	−441.569	−	147.947i	−0.696481	−	0.233355i
$635$		0			0
$636$		0			0
$637$			624.785i			0.980824i
$638$	395.122	+	132.385i	0.619313	+	0.207499i
$639$		0			0
$640$		0			0
$641$	851.379			1.32820			0.664102	−	0.747642i	$-0.268814\pi$
$641$	851.379			1.32820			0.664102	+	0.747642i	$0.268814\pi$
$642$		0			0
$643$	562.800			0.875272			0.437636	−	0.899152i	$-0.355816\pi$
$643$	562.800			0.875272			0.437636	+	0.899152i	$0.355816\pi$
$644$	1359.05	+	1025.85i	2.11032	+	1.59294i
$645$		0			0
$646$	−4.90440	+	14.6379i	−0.00759195	+	0.0226593i
$647$	−331.188			−0.511883			−0.255942	−	0.966692i	$-0.582385\pi$
$647$	−331.188			−0.511883			−0.255942	+	0.966692i	$0.582385\pi$
$648$		0			0
$649$	681.244			1.04968
$650$		0			0
$651$		0			0
$652$	−588.023	−	443.857i	−0.901875	−	0.680763i
$653$			804.177i			1.23151i	0.787937	+	0.615756i	$0.211149\pi$
$653$			804.177i			1.23151i	−0.787937	+	0.615756i	$0.788851\pi$
$654$		0			0
$655$		0			0
$656$	−193.177	−	677.851i	−0.294478	−	1.03331i
$657$		0			0
$658$	−267.333	+	797.895i	−0.406281	+	1.21261i
$659$			1061.53i			1.61081i	0.592722	+	0.805407i	$0.298053\pi$
$659$			1061.53i			1.61081i	−0.592722	+	0.805407i	$0.701947\pi$
$660$		0			0
$661$	−117.646			−0.177982			−0.0889910	−	0.996032i	$-0.528364\pi$
$661$	−117.646			−0.177982			−0.0889910	+	0.996032i	$0.528364\pi$
$662$	−379.684	−	127.212i	−0.573541	−	0.192164i
$663$		0			0
$664$	568.947	−	390.005i	0.856848	−	0.587357i
$665$		0			0
$666$		0			0
$667$	−823.848			−1.23516
$668$	617.025	+	465.749i	0.923690	+	0.697229i
$669$		0			0
$670$		0			0
$671$		−	856.624i		−	1.27664i
$672$		0			0
$673$			984.785i			1.46328i	0.681693	+	0.731638i	$0.261244\pi$
$673$			984.785i			1.46328i	−0.681693	+	0.731638i	$0.738756\pi$
$674$	−855.326	−	286.575i	−1.26903	−	0.425185i
$675$		0			0
$676$	−97.2440	−	73.4027i	−0.143852	−	0.108584i
$677$			99.5646i			0.147067i	0.997293	+	0.0735337i	$0.0234276\pi$
$677$			99.5646i			0.147067i	−0.997293	+	0.0735337i	$0.976572\pi$
$678$		0			0
$679$			151.553i			0.223200i
$680$		0			0
$681$		0			0
$682$	−62.1201	+	185.407i	−0.0910851	+	0.271857i
$683$	−434.807			−0.636614			−0.318307	−	0.947988i	$-0.603114\pi$
$683$	−434.807			−0.636614			−0.318307	+	0.947988i	$0.603114\pi$
$684$		0			0
$685$		0			0
$686$	−26.2004	+	78.1990i	−0.0381930	+	0.113993i
$687$		0			0
$688$	255.658	+	897.093i	0.371596	+	1.30391i
$689$	−944.429			−1.37072
$690$		0			0
$691$			1153.87i			1.66986i	0.550358	+	0.834929i	$0.314491\pi$
$691$			1153.87i			1.66986i	−0.550358	+	0.834929i	$0.685509\pi$
$692$	137.036	−	181.545i	0.198028	−	0.262348i
$693$		0			0
$694$	−81.8659	+	244.341i	−0.117962	+	0.352077i
$695$		0			0
$696$		0			0
$697$		−	732.679i		−	1.05119i
$698$	−0.480376	+	1.43375i	−0.000688218	+	0.00205409i
$699$		0			0
$700$		0			0
$701$	479.405			0.683887			0.341944	−	0.939720i	$-0.388915\pi$
$701$	479.405			0.683887			0.341944	+	0.939720i	$0.388915\pi$
$702$		0			0
$703$	10.9251			0.0155407
$704$	−636.061	−	245.944i	−0.903496	−	0.349352i
$705$		0			0
$706$	105.703	+	35.4156i	0.149721	+	0.0501638i
$707$	1977.90			2.79760
$708$		0			0
$709$	−932.880			−1.31577			−0.657884	−	0.753119i	$-0.728549\pi$
$709$	−932.880			−1.31577			−0.657884	+	0.753119i	$0.728549\pi$
$710$		0			0
$711$		0			0
$712$	−839.415	+	575.407i	−1.17895	+	0.808155i
$713$		−	386.582i		−	0.542191i
$714$		0			0
$715$		0			0
$716$	−655.842	+	868.860i	−0.915981	+	1.21349i
$717$		0			0
$718$	368.368	+	123.421i	0.513047	+	0.171895i
$719$		−	141.836i		−	0.197268i	−0.995124	−	0.0986341i	$-0.968553\pi$
$719$		−	141.836i		−	0.197268i	0.995124	−	0.0986341i	$-0.0314473\pi$
$720$		0			0
$721$	−157.273			−0.218131
$722$	229.236	−	684.188i	0.317501	−	0.947629i
$723$		0			0
$724$	160.362	+	121.046i	0.221495	+	0.167191i
$725$		0			0
$726$		0			0
$727$	891.575			1.22638			0.613188	−	0.789937i	$-0.289887\pi$
$727$	891.575			1.22638			0.613188	+	0.789937i	$0.289887\pi$
$728$	537.907	+	784.710i	0.738884	+	1.07790i
$729$		0			0
$730$		0			0
$731$			969.655i			1.32648i
$732$		0			0
$733$		−	614.593i		−	0.838463i	−0.907879	−	0.419232i	$-0.862300\pi$
$733$		−	614.593i		−	0.838463i	0.907879	−	0.419232i	$-0.137700\pi$
$734$	−15.2659	+	45.5634i	−0.0207983	+	0.0620755i
$735$		0			0
$736$	1346.84	+	61.5494i	1.82995	+	0.0836269i
$737$		−	240.043i		−	0.325703i
$738$		0			0
$739$		−	817.138i		−	1.10573i	−0.833269	−	0.552867i	$-0.813534\pi$
$739$		−	817.138i		−	1.10573i	0.833269	−	0.552867i	$-0.186466\pi$
$740$		0			0
$741$		0			0
$742$	−1537.38	−	515.096i	−2.07194	−	0.694199i
$743$	868.860			1.16939			0.584697	−	0.811252i	$-0.301213\pi$
$743$	868.860			1.16939			0.584697	+	0.811252i	$0.301213\pi$
$744$		0			0
$745$		0			0
$746$	−259.778	−	87.0381i	−0.348228	−	0.116673i
$747$		0			0
$748$	−565.798	−	427.081i	−0.756414	−	0.570964i
$749$	−518.512			−0.692273
$750$		0			0
$751$			100.822i			0.134250i	0.997745	+	0.0671252i	$0.0213827\pi$
$751$			100.822i			0.134250i	−0.997745	+	0.0671252i	$0.978617\pi$
$752$	182.613	+	640.779i	0.242836	+	0.852100i
$753$		0			0
$754$	−436.459	−	146.235i	−0.578859	−	0.193945i
$755$		0			0
$756$		0			0
$757$		−	33.6363i		−	0.0444336i	−0.999753	−	0.0222168i	$-0.992928\pi$
$757$		−	33.6363i		−	0.0444336i	0.999753	−	0.0222168i	$-0.00707241\pi$
$758$	196.003	+	65.6703i	0.258579	+	0.0866363i
$759$		0			0
$760$		0			0
$761$	685.059			0.900209			0.450105	−	0.892976i	$-0.351387\pi$
$761$	685.059			0.900209			0.450105	+	0.892976i	$0.351387\pi$
$762$		0			0
$763$	−473.367			−0.620402
$764$	−468.108	+	620.150i	−0.612707	+	0.811714i
$765$		0			0
$766$	12.2967	−	36.7013i	0.0160531	−	0.0479130i
$767$	−752.516			−0.981116
$768$		0			0
$769$	−23.6505			−0.0307549			−0.0153775	−	0.999882i	$-0.504895\pi$
$769$	−23.6505			−0.0307549			−0.0153775	+	0.999882i	$0.504895\pi$
$770$		0			0
$771$		0			0
$772$	−54.7117	+	72.4821i	−0.0708701	+	0.0938888i
$773$		−	947.350i		−	1.22555i	−0.790257	−	0.612775i	$-0.790053\pi$
$773$		−	947.350i		−	1.22555i	0.790257	−	0.612775i	$-0.209947\pi$
$774$		0			0
$775$		0			0
$776$	67.8480	+	98.9780i	0.0874330	+	0.127549i
$777$		0			0
$778$	−174.080	+	519.569i	−0.223754	+	0.667827i
$779$		−	20.4446i		−	0.0262447i
$780$		0			0
$781$	655.167			0.838882
$782$	1328.89	+	445.243i	1.69935	+	0.569364i
$783$		0			0
$784$	232.770	+	816.780i	0.296901	+	1.04181i
$785$		0			0
$786$		0			0
$787$	−1054.73			−1.34019			−0.670096	−	0.742275i	$-0.733747\pi$
$787$	−1054.73			−1.34019			−0.670096	+	0.742275i	$0.733747\pi$
$788$	372.643	−	493.678i	0.472897	−	0.626495i
$789$		0			0
$790$		0			0
$791$		−	1167.22i		−	1.47562i
$792$		0			0
$793$			946.244i			1.19325i
$794$	413.848	+	138.659i	0.521220	+	0.174633i
$795$		0			0
$796$	−568.953	+	753.749i	−0.714765	+	0.946921i
$797$			810.475i			1.01691i	0.861089	+	0.508454i	$0.169783\pi$
$797$			810.475i			1.01691i	−0.861089	+	0.508454i	$0.830217\pi$
$798$		0			0
$799$			692.609i			0.866845i
$800$		0			0
$801$		0			0
$802$	251.911	−	751.866i	0.314103	−	0.937489i
$803$	1467.30			1.82728
$804$		0			0
$805$		0			0
$806$	68.6191	−	204.804i	0.0851353	−	0.254099i
$807$		0			0
$808$	1291.75	−	885.478i	1.59870	−	1.09589i
$809$	768.673			0.950152			0.475076	−	0.879945i	$-0.342421\pi$
$809$	768.673			0.950152			0.475076	+	0.879945i	$0.342421\pi$
$810$		0			0
$811$		−	744.199i		−	0.917632i	−0.888531	−	0.458816i	$-0.848274\pi$
$811$		−	744.199i		−	0.917632i	0.888531	−	0.458816i	$-0.151726\pi$
$812$	−630.729	−	476.094i	−0.776760	−	0.586322i
$813$		0			0
$814$	−159.378	+	475.688i	−0.195796	+	0.584383i
$815$		0			0
$816$		0			0
$817$			27.0571i			0.0331177i
$818$	−191.870	+	572.664i	−0.234559	+	0.700078i
$819$		0			0
$820$		0			0
$821$	−205.427			−0.250215			−0.125108	−	0.992143i	$-0.539928\pi$
$821$	−205.427			−0.250215			−0.125108	+	0.992143i	$0.539928\pi$
$822$		0			0
$823$	1287.79			1.56475			0.782375	−	0.622808i	$-0.214008\pi$
$823$	1287.79			1.56475			0.782375	+	0.622808i	$0.214008\pi$
$824$	−102.713	+	70.4086i	−0.124652	+	0.0854473i
$825$		0			0
$826$	−1224.98	−	410.425i	−1.48302	−	0.496883i
$827$	−255.621			−0.309094			−0.154547	−	0.987985i	$-0.549392\pi$
$827$	−255.621			−0.309094			−0.154547	+	0.987985i	$0.549392\pi$
$828$		0			0
$829$	14.0286			0.0169223			0.00846114	−	0.999964i	$-0.497307\pi$
$829$	14.0286			0.0169223			0.00846114	+	0.999964i	$0.497307\pi$
$830$		0			0
$831$		0			0
$832$	702.606	+	271.675i	0.844479	+	0.326532i
$833$			882.846i			1.05984i
$834$		0			0
$835$		0			0
$836$	−15.7880	−	11.9172i	−0.0188851	−	0.0142551i
$837$		0			0
$838$	−797.357	−	267.153i	−0.951500	−	0.318798i
$839$			333.636i			0.397659i	0.980034	+	0.198829i	$0.0637140\pi$
$839$			333.636i			0.397659i	−0.980034	+	0.198829i	$0.936286\pi$
$840$		0			0
$841$	−458.655			−0.545369
$842$	−125.410	+	374.306i	−0.148943	+	0.444543i
$843$		0			0
$844$	277.638	−	367.815i	0.328955	−	0.435799i
$845$		0			0
$846$		0			0
$847$	−75.3657			−0.0889795
$848$	−1234.65	+	351.857i	−1.45596	+	0.414926i
$849$		0			0
$850$		0			0
$851$		−	991.832i		−	1.16549i
$852$		0			0
$853$		−	1563.83i		−	1.83333i	−0.399661	−	0.916663i	$-0.630872\pi$
$853$		−	1563.83i		−	1.83333i	0.399661	−	0.916663i	$-0.369128\pi$
$854$	−516.086	+	1540.33i	−0.604316	+	1.80367i
$855$		0			0
$856$	−338.636	+	232.130i	−0.395603	+	0.271180i
$857$			1283.36i			1.49750i	0.662854	+	0.748749i	$0.269345\pi$
$857$			1283.36i			1.49750i	−0.662854	+	0.748749i	$0.730655\pi$
$858$		0			0
$859$			1448.91i			1.68674i	0.537330	+	0.843372i	$0.319433\pi$
$859$			1448.91i			1.68674i	−0.537330	+	0.843372i	$0.680567\pi$
$860$		0			0
$861$		0			0
$862$	449.196	+	150.502i	0.521109	+	0.174597i
$863$	755.677			0.875640			0.437820	−	0.899063i	$-0.355751\pi$
$863$	755.677			0.875640			0.437820	+	0.899063i	$0.355751\pi$
$864$		0			0
$865$		0			0
$866$	691.002	+	231.519i	0.797924	+	0.267343i
$867$		0			0
$868$	223.402	−	295.963i	0.257375	−	0.340971i
$869$	−1477.55			−1.70029
$870$		0			0
$871$			265.157i			0.304428i
$872$	−309.152	+	211.919i	−0.354532	+	0.243027i
$873$		0			0
$874$	37.0813	+	12.4240i	0.0424271	+	0.0142151i
$875$		0			0
$876$		0			0
$877$		−	248.612i		−	0.283480i	−0.989904	−	0.141740i	$-0.954730\pi$
$877$		−	248.612i		−	0.283480i	0.989904	−	0.141740i	$-0.0452697\pi$
$878$	−137.238	−	45.9812i	−0.156307	−	0.0523704i
$879$		0			0
$880$		0			0
$881$	−460.306			−0.522481			−0.261240	−	0.965274i	$-0.584132\pi$
$881$	−460.306			−0.522481			−0.261240	+	0.965274i	$0.584132\pi$
$882$		0			0
$883$	−554.195			−0.627628			−0.313814	−	0.949485i	$-0.601607\pi$
$883$	−554.195			−0.627628			−0.313814	+	0.949485i	$0.601607\pi$
$884$	624.991	+	471.763i	0.707004	+	0.533668i
$885$		0			0
$886$	152.679	−	455.694i	0.172324	−	0.514327i
$887$	251.593			0.283644			0.141822	−	0.989892i	$-0.454704\pi$
$887$	251.593			0.283644			0.141822	+	0.989892i	$0.454704\pi$
$888$		0			0
$889$	1633.30			1.83723
$890$		0			0
$891$		0			0
$892$	1357.39	+	1024.60i	1.52174	+	1.14865i
$893$			19.3265i			0.0216422i
$894$		0			0
$895$		0			0
$896$	995.558	+	825.448i	1.11111	+	0.921259i
$897$		0			0
$898$	317.029	−	946.220i	0.353039	−	1.05370i
$899$			179.411i			0.199568i
$900$		0			0
$901$	−1334.52			−1.48115
$902$	890.171	+	298.250i	0.986886	+	0.330654i
$903$		0			0
$904$	−522.545	−	762.299i	−0.578037	−	0.843251i
$905$		0			0
$906$		0			0
$907$	447.842			0.493762			0.246881	−	0.969046i	$-0.420594\pi$
$907$	447.842			0.493762			0.246881	+	0.969046i	$0.420594\pi$
$908$	−568.942	−	429.455i	−0.626588	−	0.472968i
$909$		0			0
$910$		0			0
$911$			96.7664i			0.106220i	0.998589	+	0.0531100i	$0.0169134\pi$
$911$			96.7664i			0.106220i	−0.998589	+	0.0531100i	$0.983087\pi$
$912$		0			0
$913$			918.756i			1.00630i
$914$	325.925	+	109.200i	0.356592	+	0.119475i
$915$		0			0
$916$	1058.78	+	799.196i	1.15587	+	0.872485i
$917$		−	1996.06i		−	2.17673i
$918$		0			0
$919$			234.739i			0.255429i	0.991811	+	0.127715i	$0.0407641\pi$
$919$			234.739i			0.255429i	−0.991811	+	0.127715i	$0.959236\pi$
$920$		0			0
$921$		0			0
$922$	528.526	−	1577.46i	0.573238	−	1.71092i
$923$	−723.711			−0.784085
$924$		0			0
$925$		0			0
$926$	235.098	−	701.685i	0.253886	−	0.757759i
$927$		0			0
$928$	−625.064	−	28.5648i	−0.673561	−	0.0307811i
$929$	−552.902			−0.595158			−0.297579	−	0.954697i	$-0.596179\pi$
$929$	−552.902			−0.595158			−0.297579	+	0.954697i	$0.596179\pi$
$930$		0			0
$931$			24.6348i			0.0264606i
$932$	440.887	−	584.088i	0.473055	−	0.626704i
$933$		0			0
$934$	−335.335	+	1000.86i	−0.359031	+	1.07158i
$935$		0			0
$936$		0			0
$937$			957.870i			1.02227i	0.859499	+	0.511137i	$0.170775\pi$
$937$			957.870i			1.02227i	−0.859499	+	0.511137i	$0.829225\pi$
$938$	−144.618	+	431.633i	−0.154177	+	0.460163i
$939$		0			0
$940$		0			0
$941$	−1125.13			−1.19567			−0.597837	−	0.801618i	$-0.703973\pi$
$941$	−1125.13			−1.19567			−0.597837	+	0.801618i	$0.703973\pi$
$942$		0			0
$943$	−1856.05			−1.96824
$944$	−983.763	+	280.358i	−1.04212	+	0.296989i
$945$		0			0
$946$	−1178.09	−	394.715i	−1.24533	−	0.417246i
$947$	−256.824			−0.271198			−0.135599	−	0.990764i	$-0.543296\pi$
$947$	−256.824			−0.271198			−0.135599	+	0.990764i	$0.543296\pi$
$948$		0			0
$949$	−1620.81			−1.70792
$950$		0			0
$951$		0			0
$952$	760.085	+	1108.83i	0.798409	+	1.16473i
$953$			575.377i			0.603753i	0.953347	+	0.301877i	$0.0976131\pi$
$953$			575.377i			0.603753i	−0.953347	+	0.301877i	$0.902387\pi$
$954$		0			0
$955$		0			0
$956$	507.670	−	672.561i	0.531035	−	0.703516i
$957$		0			0
$958$	−1010.35	−	338.516i	−1.05465	−	0.353357i
$959$		−	772.094i		−	0.805103i
$960$		0			0
$961$	876.813			0.912397
$962$	176.052	−	525.454i	0.183006	−	0.546210i
$963$		0			0
$964$	11.5635	+	8.72845i	0.0119953	+	0.00905441i
$965$		0			0
$966$		0			0
$967$	860.976			0.890358			0.445179	−	0.895442i	$-0.353140\pi$
$967$	860.976			0.890358			0.445179	+	0.895442i	$0.353140\pi$
$968$	−49.2207	+	33.7401i	−0.0508478	+	0.0348555i
$969$		0			0
$970$		0			0
$971$			177.116i			0.182406i	0.995832	+	0.0912030i	$0.0290712\pi$
$971$			177.116i			0.182406i	−0.995832	+	0.0912030i	$0.970929\pi$
$972$		0			0
$973$		−	2255.17i		−	2.31775i
$974$	314.560	−	938.851i	0.322957	−	0.963913i
$975$		0			0
$976$	352.534	+	1237.02i	0.361202	+	1.26744i
$977$			755.634i			0.773423i	0.922201	+	0.386711i	$0.126389\pi$
$977$			755.634i			0.773423i	−0.922201	+	0.386711i	$0.873611\pi$
$978$		0			0
$979$		−	1355.52i		−	1.38459i
$980$		0			0
$981$		0			0
$982$	−982.507	−	329.187i	−1.00052	−	0.335221i
$983$	1108.93			1.12811			0.564054	−	0.825738i	$-0.309241\pi$
$983$	1108.93			1.12811			0.564054	+	0.825738i	$0.309241\pi$
$984$		0			0
$985$		0			0
$986$	−616.735	−	206.636i	−0.625492	−	0.209570i
$987$		0			0
$988$	17.4397	+	13.1640i	0.0176515	+	0.0133239i
$989$	2456.37			2.48369
$990$		0			0
$991$		−	516.421i		−	0.521111i	−0.965459	−	0.260556i	$-0.916094\pi$
$991$		−	516.421i		−	0.521111i	0.965459	−	0.260556i	$-0.0839058\pi$
$992$	13.4037	−	293.305i	0.0135118	−	0.295670i
$993$		0			0
$994$	−1178.09	−	394.715i	−1.18520	−	0.397098i
$995$		0			0
$996$		0			0
$997$			1127.03i			1.13042i	0.824946	+	0.565212i	$0.191206\pi$
$997$			1127.03i			1.13042i	−0.824946	+	0.565212i	$0.808794\pi$
$998$	−1283.06	−	429.885i	−1.28563	−	0.430747i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.f.g.199.9		16
3.2	odd	2	inner	900.3.f.g.199.7		16
4.3	odd	2	inner	900.3.f.g.199.6		16
5.2	odd	4		900.3.c.q.451.8	yes	8
5.3	odd	4		900.3.c.p.451.1	✓	8
5.4	even	2	inner	900.3.f.g.199.8		16
12.11	even	2	inner	900.3.f.g.199.12		16
15.2	even	4		900.3.c.q.451.1	yes	8
15.8	even	4		900.3.c.p.451.8	yes	8
15.14	odd	2	inner	900.3.f.g.199.10		16
20.3	even	4		900.3.c.p.451.2	yes	8
20.7	even	4		900.3.c.q.451.7	yes	8
20.19	odd	2	inner	900.3.f.g.199.11		16
60.23	odd	4		900.3.c.p.451.7	yes	8
60.47	odd	4		900.3.c.q.451.2	yes	8
60.59	even	2	inner	900.3.f.g.199.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
900.3.c.p.451.1	✓	8	5.3	odd	4
900.3.c.p.451.2	yes	8	20.3	even	4
900.3.c.p.451.7	yes	8	60.23	odd	4
900.3.c.p.451.8	yes	8	15.8	even	4
900.3.c.q.451.1	yes	8	15.2	even	4
900.3.c.q.451.2	yes	8	60.47	odd	4
900.3.c.q.451.7	yes	8	20.7	even	4
900.3.c.q.451.8	yes	8	5.2	odd	4
900.3.f.g.199.5		16	60.59	even	2	inner
900.3.f.g.199.6		16	4.3	odd	2	inner
900.3.f.g.199.7		16	3.2	odd	2	inner
900.3.f.g.199.8		16	5.4	even	2	inner
900.3.f.g.199.9		16	1.1	even	1	trivial
900.3.f.g.199.10		16	15.14	odd	2	inner
900.3.f.g.199.11		16	20.19	odd	2	inner
900.3.f.g.199.12		16	12.11	even	2	inner

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:	\( N \)	\(=\)	\( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	900.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.635381 - 1.89639i) q^{2} +(-3.19258 - 2.40986i) q^{4} -10.1035 q^{7} +(-6.59853 + 4.52320i) q^{8} +O(q^{10})\)	\(q+(0.635381 - 1.89639i) q^{2} +(-3.19258 - 2.40986i) q^{4} -10.1035 q^{7} +(-6.59853 + 4.52320i) q^{8} -10.6555i q^{11} +11.7703i q^{13} +(-6.41959 + 19.1602i) q^{14} +(4.38516 + 15.3873i) q^{16} +16.6320i q^{17} +0.464096i q^{19} +(-20.2071 - 6.77033i) q^{22} +42.1327 q^{23} +(22.3211 + 7.47864i) q^{26} +(32.2564 + 24.3481i) q^{28} -19.5536 q^{29} -9.17534i q^{31} +(31.9666 + 1.46084i) q^{32} +(31.5407 + 10.5676i) q^{34} -23.5407i q^{37} +(0.880107 + 0.294878i) q^{38} -44.0525 q^{41} +58.3007 q^{43} +(-25.6784 + 34.0187i) q^{44} +(26.7703 - 79.9000i) q^{46} +41.6433 q^{47} +53.0813 q^{49} +(28.3648 - 37.5777i) q^{52} +80.2381i q^{53} +(66.6685 - 45.7003i) q^{56} +(-12.4240 + 37.0813i) q^{58} +63.9333i q^{59} +80.3923 q^{61} +(-17.4000 - 5.82983i) q^{62} +(23.0813 - 59.6930i) q^{64} +22.5275 q^{67} +(40.0807 - 53.0989i) q^{68} +61.4860i q^{71} +137.703i q^{73} +(-44.6422 - 14.9573i) q^{74} +(1.11841 - 1.48167i) q^{76} +107.659i q^{77} -138.665i q^{79} +(-27.9901 + 83.5407i) q^{82} -86.2233 q^{83} +(37.0431 - 110.561i) q^{86} +(48.1972 + 70.3110i) q^{88} +127.212 q^{89} -118.922i q^{91} +(-134.512 - 101.534i) q^{92} +(26.4593 - 78.9719i) q^{94} -15.0000i q^{97} +(33.7269 - 100.663i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{4}+O(q^{10}) \) 16 * q - 8 * q^4	\( 16 q - 8 q^{4} - 16 q^{16} + 160 q^{34} + 256 q^{46} + 160 q^{49} + 80 q^{61} - 320 q^{64} - 456 q^{76} + 768 q^{94}+O(q^{100}) \) 16 * q - 8 * q^4 - 16 * q^16 + 160 * q^34 + 256 * q^46 + 160 * q^49 + 80 * q^61 - 320 * q^64 - 456 * q^76 + 768 * q^94

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