LMFDB - Embedded newform 900.3.c.r.451.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [900,3,Mod(451,900)]

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, 0]))

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

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

chi := DirichletCharacter("900.451");

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.c (of order $2$, degree $1$, 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:	$8$
Coefficient field:	8.0.6080256576.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} + 7x^{6} - 12x^{5} + 12x^{4} - 48x^{3} + 112x^{2} - 192x + 256 $ x^8 - 3x^7 + 7x^6 - 12x^5 + 12x^4 - 48x^3 + 112x^2 - 192*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 60)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			451.3
Root			$0.670410 + 1.88429i$ of defining polynomial
Character	$\chi$	$=$	900.451
Dual form			900.3.c.r.451.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.29664 - 1.52274i) q^{2} +(-0.637459 + 3.94888i) q^{4} -0.837253i q^{7} +(6.83966 - 4.14959i) q^{8} +O(q^{10})$	\(q+(-1.29664 - 1.52274i) q^{2} +(-0.637459 + 3.94888i) q^{4} -0.837253i q^{7} +(6.83966 - 4.14959i) q^{8} -15.7955i q^{11} +5.18655 q^{13} +(-1.27492 + 1.08561i) q^{14} +(-15.1873 - 5.03449i) q^{16} +27.3586 q^{17} +17.9667i q^{19} +(-24.0524 + 20.4811i) q^{22} +19.1101i q^{23} +(-6.72508 - 7.89776i) q^{26} +(3.30621 + 0.533714i) q^{28} -45.6495 q^{29} -13.6243i q^{31} +(12.0262 + 29.6542i) q^{32} +(-35.4743 - 41.6600i) q^{34} +15.5597 q^{37} +(27.3586 - 23.2964i) q^{38} -13.2990 q^{41} +27.9430i q^{43} +(62.3746 + 10.0690i) q^{44} +(29.0997 - 24.7789i) q^{46} -55.6558i q^{47} +48.2990 q^{49} +(-3.30621 + 20.4811i) q^{52} +15.5597 q^{53} +(-3.47425 - 5.72653i) q^{56} +(59.1909 + 69.5122i) q^{58} -87.6625i q^{59} +38.0000 q^{61} +(-20.7462 + 17.6658i) q^{62} +(29.5619 - 56.7635i) q^{64} -92.2015i q^{67} +(-17.4400 + 108.036i) q^{68} -130.707i q^{71} +54.7173 q^{73} +(-20.1752 - 23.6933i) q^{74} +(-70.9485 - 11.4531i) q^{76} -13.2249 q^{77} -13.6243i q^{79} +(17.2440 + 20.2509i) q^{82} -59.0048i q^{83} +(42.5498 - 36.2319i) q^{86} +(-65.5448 - 108.036i) q^{88} +39.8007 q^{89} -4.34246i q^{91} +(-75.4635 - 12.1819i) q^{92} +(-84.7492 + 72.1654i) q^{94} +168.821 q^{97} +(-62.6263 - 73.5467i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 10 q^{4}+O(q^{10}) $ 8 * q + 10 * q^4	$ 8 q + 10 q^{4} + 20 q^{14} - 46 q^{16} - 84 q^{26} - 184 q^{29} - 12 q^{34} + 256 q^{41} + 348 q^{44} + 112 q^{46} + 24 q^{49} + 244 q^{56} + 304 q^{61} + 10 q^{64} - 252 q^{74} - 24 q^{76} + 280 q^{86} + 560 q^{89} - 376 q^{94}+O(q^{100}) $ 8 * q + 10 * q^4 + 20 * q^14 - 46 * q^16 - 84 * q^26 - 184 * q^29 - 12 * q^34 + 256 * q^41 + 348 * q^44 + 112 * q^46 + 24 * q^49 + 244 * q^56 + 304 * q^61 + 10 * q^64 - 252 * q^74 - 24 * q^76 + 280 * q^86 + 560 * q^89 - 376 * 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 $

$2$	−1.29664	−	1.52274i	−0.648319	−	0.761369i
$2$	−1.29664	−	1.52274i	−0.648319	−	0.761369i
$3$		0			0
$3$		0			0
$4$	−0.637459	+	3.94888i	−0.159365	+	0.987220i
$5$		0			0
$5$		0			0
$6$		0			0
$7$		−	0.837253i		−	0.119608i	−0.998210	−	0.0598038i	$-0.980952\pi$
$7$		−	0.837253i		−	0.119608i	0.998210	−	0.0598038i	$-0.0190475\pi$
$8$	6.83966	−	4.14959i	0.854957	−	0.518698i
$9$		0			0
$10$		0			0
$11$		−	15.7955i		−	1.43596i	−0.696066	−	0.717978i	$-0.745068\pi$
$11$		−	15.7955i		−	1.43596i	0.696066	−	0.717978i	$-0.254932\pi$
$12$		0			0
$13$	5.18655			0.398966			0.199483	−	0.979901i	$-0.436074\pi$
$13$	5.18655			0.398966			0.199483	+	0.979901i	$0.436074\pi$
$14$	−1.27492	+	1.08561i	−0.0910655	+	0.0775439i
$15$		0			0
$16$	−15.1873	−	5.03449i	−0.949206	−	0.314656i
$17$	27.3586			1.60933			0.804666	−	0.593728i	$-0.202344\pi$
$17$	27.3586			1.60933			0.804666	+	0.593728i	$0.202344\pi$
$18$		0			0
$19$			17.9667i			0.945618i	0.881165	+	0.472809i	$0.156760\pi$
$19$			17.9667i			0.945618i	−0.881165	+	0.472809i	$0.843240\pi$
$20$		0			0
$21$		0			0
$22$	−24.0524	+	20.4811i	−1.09329	+	0.930958i
$23$			19.1101i			0.830874i	0.909622	+	0.415437i	$0.136371\pi$
$23$			19.1101i			0.830874i	−0.909622	+	0.415437i	$0.863629\pi$
$24$		0			0
$25$		0			0
$26$	−6.72508	−	7.89776i	−0.258657	−	0.303760i
$27$		0			0
$28$	3.30621	+	0.533714i	0.118079	+	0.0190612i
$29$	−45.6495			−1.57412			−0.787060	−	0.616876i	$-0.788398\pi$
$29$	−45.6495			−1.57412			−0.787060	+	0.616876i	$0.788398\pi$
$30$		0			0
$31$		−	13.6243i		−	0.439493i	−0.975557	−	0.219747i	$-0.929477\pi$
$31$		−	13.6243i		−	0.439493i	0.975557	−	0.219747i	$-0.0705230\pi$
$32$	12.0262	+	29.6542i	0.375819	+	0.926693i
$33$		0			0
$34$	−35.4743	−	41.6600i	−1.04336	−	1.22529i
$35$		0			0
$36$		0			0
$37$	15.5597			0.420531			0.210266	−	0.977644i	$-0.432567\pi$
$37$	15.5597			0.420531			0.210266	+	0.977644i	$0.432567\pi$
$38$	27.3586	−	23.2964i	0.719964	−	0.613062i
$39$		0			0
$40$		0			0
$41$	−13.2990			−0.324366			−0.162183	−	0.986761i	$-0.551853\pi$
$41$	−13.2990			−0.324366			−0.162183	+	0.986761i	$0.551853\pi$
$42$		0			0
$43$			27.9430i			0.649837i	0.945742	+	0.324918i	$0.105337\pi$
$43$			27.9430i			0.649837i	−0.945742	+	0.324918i	$0.894663\pi$
$44$	62.3746	+	10.0690i	1.41760	+	0.228841i
$45$		0			0
$46$	29.0997	−	24.7789i	0.632601	−	0.538672i
$47$		−	55.6558i		−	1.18417i	−0.805877	−	0.592083i	$-0.798306\pi$
$47$		−	55.6558i		−	1.18417i	0.805877	−	0.592083i	$-0.201694\pi$
$48$		0			0
$49$	48.2990			0.985694
$50$		0			0
$51$		0			0
$52$	−3.30621	+	20.4811i	−0.0635810	+	0.393867i
$53$	15.5597			0.293578			0.146789	−	0.989168i	$-0.453106\pi$
$53$	15.5597			0.293578			0.146789	+	0.989168i	$0.453106\pi$
$54$		0			0
$55$		0			0
$56$	−3.47425	−	5.72653i	−0.0620403	−	0.102259i
$57$		0			0
$58$	59.1909	+	69.5122i	1.02053	+	1.19849i
$59$		−	87.6625i		−	1.48581i	−0.669400	−	0.742903i	$-0.733449\pi$
$59$		−	87.6625i		−	1.48581i	0.669400	−	0.742903i	$-0.266551\pi$
$60$		0			0
$61$	38.0000			0.622951			0.311475	−	0.950254i	$-0.399177\pi$
$61$	38.0000			0.622951			0.311475	+	0.950254i	$0.399177\pi$
$62$	−20.7462	+	17.6658i	−0.334616	+	0.284932i
$63$		0			0
$64$	29.5619	−	56.7635i	0.461904	−	0.886930i
$65$		0			0
$66$		0			0
$67$		−	92.2015i		−	1.37614i	−0.725643	−	0.688071i	$-0.758458\pi$
$67$		−	92.2015i		−	1.37614i	0.725643	−	0.688071i	$-0.241542\pi$
$68$	−17.4400	+	108.036i	−0.256471	+	1.58876i
$69$		0			0
$70$		0			0
$71$		−	130.707i		−	1.84094i	−0.390816	−	0.920469i	$-0.627807\pi$
$71$		−	130.707i		−	1.84094i	0.390816	−	0.920469i	$-0.372193\pi$
$72$		0			0
$73$	54.7173			0.749552			0.374776	−	0.927115i	$-0.377720\pi$
$73$	54.7173			0.749552			0.374776	+	0.927115i	$0.377720\pi$
$74$	−20.1752	−	23.6933i	−0.272638	−	0.320179i
$75$		0			0
$76$	−70.9485	−	11.4531i	−0.933533	−	0.150698i
$77$	−13.2249			−0.171751
$78$		0			0
$79$		−	13.6243i		−	0.172459i	−0.996275	−	0.0862297i	$-0.972518\pi$
$79$		−	13.6243i		−	0.172459i	0.996275	−	0.0862297i	$-0.0274819\pi$
$80$		0			0
$81$		0			0
$82$	17.2440	+	20.2509i	0.210293	+	0.246962i
$83$		−	59.0048i		−	0.710901i	−0.934695	−	0.355451i	$-0.884327\pi$
$83$		−	59.0048i		−	0.710901i	0.934695	−	0.355451i	$-0.115673\pi$
$84$		0			0
$85$		0			0
$86$	42.5498	−	36.2319i	0.494766	−	0.421302i
$87$		0			0
$88$	−65.5448	−	108.036i	−0.744828	−	1.22768i
$89$	39.8007			0.447198			0.223599	−	0.974681i	$-0.428219\pi$
$89$	39.8007			0.447198			0.223599	+	0.974681i	$0.428219\pi$
$90$		0			0
$91$		−	4.34246i		−	0.0477193i
$92$	−75.4635	−	12.1819i	−0.820255	−	0.132412i
$93$		0			0
$94$	−84.7492	+	72.1654i	−0.901587	+	0.767717i
$95$		0			0
$96$		0			0
$97$	168.821			1.74043			0.870214	−	0.492675i	$-0.163981\pi$
$97$	168.821			1.74043			0.870214	+	0.492675i	$0.163981\pi$
$98$	−62.6263	−	73.5467i	−0.639044	−	0.750477i
$99$		0			0
$100$		0			0
$101$	−44.5498			−0.441087			−0.220544	−	0.975377i	$-0.570783\pi$
$101$	−44.5498			−0.441087			−0.220544	+	0.975377i	$0.570783\pi$
$102$		0			0
$103$		−	126.466i		−	1.22782i	−0.789375	−	0.613911i	$-0.789595\pi$
$103$		−	126.466i		−	1.22782i	0.789375	−	0.613911i	$-0.210405\pi$
$104$	35.4743	−	21.5220i	0.341099	−	0.206943i
$105$		0			0
$106$	−20.1752	−	23.6933i	−0.190333	−	0.223521i
$107$			104.383i			0.975546i	0.872971	+	0.487773i	$0.162191\pi$
$107$			104.383i			0.975546i	−0.872971	+	0.487773i	$0.837809\pi$
$108$		0			0
$109$	0.501656			0.00460235			0.00230117	−	0.999997i	$-0.499268\pi$
$109$	0.501656			0.00460235			0.00230117	+	0.999997i	$0.499268\pi$
$110$		0			0
$111$		0			0
$112$	−4.21515	+	12.7156i	−0.0376352	+	0.113532i
$113$	16.9855			0.150314			0.0751572	−	0.997172i	$-0.476054\pi$
$113$	16.9855			0.150314			0.0751572	+	0.997172i	$0.476054\pi$
$114$		0			0
$115$		0			0
$116$	29.0997	−	180.264i	0.250859	−	1.55400i
$117$		0			0
$118$	−133.487	+	113.667i	−1.13125	+	0.963276i
$119$		−	22.9061i		−	0.192488i
$120$		0			0
$121$	−128.498			−1.06197
$122$	−49.2723	−	57.8640i	−0.403871	−	0.474295i
$123$		0			0
$124$	53.8007	+	8.68492i	0.433876	+	0.0700397i
$125$		0			0
$126$		0			0
$127$		−	8.45598i		−	0.0665825i	−0.999446	−	0.0332913i	$-0.989401\pi$
$127$		−	8.45598i		−	0.0665825i	0.999446	−	0.0332913i	$-0.0105989\pi$
$128$	−124.767	+	28.5867i	−0.974742	+	0.223334i
$129$		0			0
$130$		0			0
$131$			51.7290i			0.394878i	0.980315	+	0.197439i	$0.0632624\pi$
$131$			51.7290i			0.394878i	−0.980315	+	0.197439i	$0.936738\pi$
$132$		0			0
$133$	15.0427			0.113103
$134$	−140.399	+	119.552i	−1.04775	+	0.892179i
$135$		0			0
$136$	187.124	−	113.527i	1.37591	−	0.834757i
$137$	−53.8083			−0.392762			−0.196381	−	0.980528i	$-0.562919\pi$
$137$	−53.8083			−0.392762			−0.196381	+	0.980528i	$0.562919\pi$
$138$		0			0
$139$			13.6243i			0.0980165i	0.998798	+	0.0490082i	$0.0156061\pi$
$139$			13.6243i			0.0980165i	−0.998798	+	0.0490082i	$0.984394\pi$
$140$		0			0
$141$		0			0
$142$	−199.032	+	169.479i	−1.40163	+	1.19352i
$143$		−	81.9243i		−	0.572897i
$144$		0			0
$145$		0			0
$146$	−70.9485	−	83.3200i	−0.485949	−	0.570685i
$147$		0			0
$148$	−9.91864	+	61.4432i	−0.0670178	+	0.415157i
$149$	−33.6495			−0.225836			−0.112918	−	0.993604i	$-0.536020\pi$
$149$	−33.6495			−0.225836			−0.112918	+	0.993604i	$0.536020\pi$
$150$		0			0
$151$		−	139.988i		−	0.927076i	−0.886077	−	0.463538i	$-0.846580\pi$
$151$		−	139.988i		−	0.927076i	0.886077	−	0.463538i	$-0.153420\pi$
$152$	74.5546	+	122.886i	0.490490	+	0.808463i
$153$		0			0
$154$	17.1478	+	20.1380i	0.111350	+	0.130766i
$155$		0			0
$156$		0			0
$157$	−21.2631			−0.135434			−0.0677170	−	0.997705i	$-0.521572\pi$
$157$	−21.2631			−0.135434			−0.0677170	+	0.997705i	$0.521572\pi$
$158$	−20.7462	+	17.6658i	−0.131305	+	0.111809i
$159$		0			0
$160$		0			0
$161$	16.0000			0.0993789
$162$		0			0
$163$		−	210.211i		−	1.28964i	−0.764335	−	0.644819i	$-0.776933\pi$
$163$		−	210.211i		−	1.28964i	0.764335	−	0.644819i	$-0.223067\pi$
$164$	8.47757	−	52.5162i	0.0516925	−	0.320221i
$165$		0			0
$166$	−89.8488	+	76.5079i	−0.541258	+	0.460891i
$167$		−	238.384i		−	1.42745i	−0.700426	−	0.713725i	$-0.747006\pi$
$167$		−	238.384i		−	1.42745i	0.700426	−	0.713725i	$-0.252994\pi$
$168$		0			0
$169$	−142.100			−0.840826
$170$		0			0
$171$		0			0
$172$	−110.343	−	17.8125i	−0.641532	−	0.103561i
$173$	2.33481			0.0134960			0.00674800	−	0.999977i	$-0.497852\pi$
$173$	2.33481			0.0134960			0.00674800	+	0.999977i	$0.497852\pi$
$174$		0			0
$175$		0			0
$176$	−79.5224	+	239.891i	−0.451832	+	1.36302i
$177$		0			0
$178$	−51.6071	−	60.6060i	−0.289927	−	0.340483i
$179$			227.054i			1.26846i	0.773145	+	0.634229i	$0.218682\pi$
$179$			227.054i			1.26846i	−0.773145	+	0.634229i	$0.781318\pi$
$180$		0			0
$181$	114.096			0.630367			0.315183	−	0.949031i	$-0.397934\pi$
$181$	114.096			0.630367			0.315183	+	0.949031i	$0.397934\pi$
$182$	−6.61243	+	5.63060i	−0.0363320	+	0.0309374i
$183$		0			0
$184$	79.2990	+	130.707i	0.430973	+	0.710362i
$185$		0			0
$186$		0			0
$187$		−	432.144i		−	2.31093i
$188$	219.778	+	35.4783i	1.16903	+	0.188714i
$189$		0			0
$190$		0			0
$191$			139.392i			0.729798i	0.931047	+	0.364899i	$0.118897\pi$
$191$			139.392i			0.729798i	−0.931047	+	0.364899i	$0.881103\pi$
$192$		0			0
$193$	−182.046			−0.943245			−0.471623	−	0.881801i	$-0.656331\pi$
$193$	−182.046			−0.943245			−0.471623	+	0.881801i	$0.656331\pi$
$194$	−218.900	−	257.071i	−1.12835	−	1.32511i
$195$		0			0
$196$	−30.7886	+	190.727i	−0.157085	+	0.973097i
$197$	258.027			1.30978			0.654890	−	0.755724i	$-0.272715\pi$
$197$	258.027			1.30978			0.654890	+	0.755724i	$0.272715\pi$
$198$		0			0
$199$		−	256.474i		−	1.28881i	−0.764683	−	0.644407i	$-0.777104\pi$
$199$		−	256.474i		−	1.28881i	0.764683	−	0.644407i	$-0.222896\pi$
$200$		0			0
$201$		0			0
$202$	57.7650	+	67.8377i	0.285965	+	0.335830i
$203$			38.2202i			0.188277i
$204$		0			0
$205$		0			0
$206$	−192.574	+	163.980i	−0.934825	+	0.796020i
$207$		0			0
$208$	−78.7697	−	26.1117i	−0.378700	−	0.125537i
$209$	283.794			1.35787
$210$		0			0
$211$			211.855i			1.00405i	0.864852	+	0.502027i	$0.167412\pi$
$211$			211.855i			1.00405i	−0.864852	+	0.502027i	$0.832588\pi$
$212$	−9.91864	+	61.4432i	−0.0467860	+	0.289826i
$213$		0			0
$214$	158.949	−	135.348i	0.742750	−	0.632465i
$215$		0			0
$216$		0			0
$217$	−11.4070			−0.0525667
$218$	−0.650466	−	0.763890i	−0.00298379	−	0.00350408i
$219$		0			0
$220$		0			0
$221$	141.897			0.642068
$222$		0			0
$223$			349.843i			1.56880i	0.620255	+	0.784401i	$0.287029\pi$
$223$			349.843i			1.56880i	−0.620255	+	0.784401i	$0.712971\pi$
$224$	24.8281	−	10.0690i	0.110840	−	0.0449508i
$225$		0			0
$226$	−22.0241	−	25.8645i	−0.0974517	−	0.114445i
$227$		−	185.554i		−	0.817418i	−0.912665	−	0.408709i	$-0.865979\pi$
$227$		−	185.554i		−	0.817418i	0.912665	−	0.408709i	$-0.134021\pi$
$228$		0			0
$229$	−263.897			−1.15239			−0.576194	−	0.817313i	$-0.695463\pi$
$229$	−263.897			−1.15239			−0.576194	+	0.817313i	$0.695463\pi$
$230$		0			0
$231$		0			0
$232$	−312.227	+	189.427i	−1.34581	+	0.816494i
$233$	−58.4780			−0.250978			−0.125489	−	0.992095i	$-0.540050\pi$
$233$	−58.4780			−0.250978			−0.125489	+	0.992095i	$0.540050\pi$
$234$		0			0
$235$		0			0
$236$	346.169	+	55.8812i	1.46682	+	0.236785i
$237$		0			0
$238$	−34.8800	+	29.7009i	−0.146555	+	0.124794i
$239$			113.337i			0.474212i	0.971484	+	0.237106i	$0.0761989\pi$
$239$			113.337i			0.474212i	−0.971484	+	0.237106i	$0.923801\pi$
$240$		0			0
$241$	−77.7940			−0.322797			−0.161398	−	0.986889i	$-0.551600\pi$
$241$	−77.7940			−0.322797			−0.161398	+	0.986889i	$0.551600\pi$
$242$	166.616	+	195.669i	0.688495	+	0.808551i
$243$		0			0
$244$	−24.2234	+	150.057i	−0.0992763	+	0.614989i
$245$		0			0
$246$		0			0
$247$			93.1855i			0.377269i
$248$	−56.5351	−	93.1855i	−0.227964	−	0.375748i
$249$		0			0
$250$		0			0
$251$			106.226i			0.423212i	0.977355	+	0.211606i	$0.0678693\pi$
$251$			106.226i			0.423212i	−0.977355	+	0.211606i	$0.932131\pi$
$252$		0			0
$253$	301.854			1.19310
$254$	−12.8762	+	10.9644i	−0.0506939	+	0.0431667i
$255$		0			0
$256$	205.308	+	152.921i	0.801983	+	0.597346i
$257$	381.078			1.48279			0.741397	−	0.671067i	$-0.234164\pi$
$257$	381.078			1.48279			0.741397	+	0.671067i	$0.234164\pi$
$258$		0			0
$259$		−	13.0274i		−	0.0502988i
$260$		0			0
$261$		0			0
$262$	78.7697	−	67.0738i	0.300648	−	0.256007i
$263$			11.4914i			0.0436934i	0.999761	+	0.0218467i	$0.00695458\pi$
$263$			11.4914i			0.0436934i	−0.999761	+	0.0218467i	$0.993045\pi$
$264$		0			0
$265$		0			0
$266$	−19.5050	−	22.9061i	−0.0733269	−	0.0861132i
$267$		0			0
$268$	364.093	+	58.7746i	1.35855	+	0.219308i
$269$	−77.9518			−0.289784			−0.144892	−	0.989447i	$-0.546283\pi$
$269$	−77.9518			−0.289784			−0.144892	+	0.989447i	$0.546283\pi$
$270$		0			0
$271$			86.6851i			0.319871i	0.987127	+	0.159936i	$0.0511287\pi$
$271$			86.6851i			0.319871i	−0.987127	+	0.159936i	$0.948871\pi$
$272$	−415.504	−	137.737i	−1.52759	−	0.506386i
$273$		0			0
$274$	69.7700	+	81.9360i	0.254635	+	0.299036i
$275$		0			0
$276$		0			0
$277$	287.328			1.03729			0.518643	−	0.854991i	$-0.326437\pi$
$277$	287.328			1.03729			0.518643	+	0.854991i	$0.326437\pi$
$278$	20.7462	−	17.6658i	0.0746267	−	0.0635459i
$279$		0			0
$280$		0			0
$281$	224.598			0.799281			0.399641	−	0.916672i	$-0.369135\pi$
$281$	224.598			0.799281			0.399641	+	0.916672i	$0.369135\pi$
$282$		0			0
$283$		−	84.1224i		−	0.297252i	−0.988893	−	0.148626i	$-0.952515\pi$
$283$		−	84.1224i		−	0.297252i	0.988893	−	0.148626i	$-0.0474851\pi$
$284$	516.145	+	83.3200i	1.81741	+	0.293380i
$285$		0			0
$286$	−124.749	+	106.226i	−0.436186	+	0.371420i
$287$			11.1346i			0.0387967i
$288$		0			0
$289$	459.495			1.58995
$290$		0			0
$291$		0			0
$292$	−34.8800	+	216.072i	−0.119452	+	0.739972i
$293$	246.620			0.841706			0.420853	−	0.907129i	$-0.361731\pi$
$293$	246.620			0.841706			0.420853	+	0.907129i	$0.361731\pi$
$294$		0			0
$295$		0			0
$296$	106.423	−	64.5661i	0.359536	−	0.218129i
$297$		0			0
$298$	43.6312	+	51.2394i	0.146414	+	0.171944i
$299$			99.1156i			0.331490i
$300$		0			0
$301$	23.3954			0.0777255
$302$	−213.166	+	181.514i	−0.705846	+	0.601041i
$303$		0			0
$304$	90.4535	−	272.866i	0.297544	−	0.897586i
$305$		0			0
$306$		0			0
$307$			115.811i			0.377236i	0.982051	+	0.188618i	$0.0604008\pi$
$307$			115.811i			0.377236i	−0.982051	+	0.188618i	$0.939599\pi$
$308$	8.43030	−	52.2233i	0.0273711	−	0.169556i
$309$		0			0
$310$		0			0
$311$			203.767i			0.655201i	0.944816	+	0.327600i	$0.106240\pi$
$311$			203.767i			0.655201i	−0.944816	+	0.327600i	$0.893760\pi$
$312$		0			0
$313$	−99.0614			−0.316490			−0.158245	−	0.987400i	$-0.550584\pi$
$313$	−99.0614			−0.316490			−0.158245	+	0.987400i	$0.550584\pi$
$314$	27.5706	+	32.3782i	0.0878045	+	0.103115i
$315$		0			0
$316$	53.8007	+	8.68492i	0.170255	+	0.0274839i
$317$	−471.192			−1.48641			−0.743206	−	0.669063i	$-0.766696\pi$
$317$	−471.192			−1.48641			−0.743206	+	0.669063i	$0.766696\pi$
$318$		0			0
$319$			721.057i			2.26037i
$320$		0			0
$321$		0			0
$322$	−20.7462	−	24.3638i	−0.0644292	−	0.0756640i
$323$			491.546i			1.52181i
$324$		0			0
$325$		0			0
$326$	−320.096	+	272.568i	−0.981891	+	0.836098i
$327$		0			0
$328$	−90.9607	+	55.1854i	−0.277319	+	0.168248i
$329$	−46.5980			−0.141635
$330$		0			0
$331$		−	270.695i		−	0.817810i	−0.912577	−	0.408905i	$-0.865911\pi$
$331$		−	270.695i		−	0.817810i	0.912577	−	0.408905i	$-0.134089\pi$
$332$	233.003	+	37.6131i	0.701816	+	0.113293i
$333$		0			0
$334$	−362.997	+	309.098i	−1.08682	+	0.925444i
$335$		0			0
$336$		0			0
$337$	377.317			1.11964			0.559818	−	0.828615i	$-0.310871\pi$
$337$	377.317			1.11964			0.559818	+	0.828615i	$0.310871\pi$
$338$	184.252	+	216.380i	0.545124	+	0.640179i
$339$		0			0
$340$		0			0
$341$	−215.203			−0.631093
$342$		0			0
$343$		−	81.4639i		−	0.237504i
$344$	115.952	+	191.121i	0.337069	+	0.555583i
$345$		0			0
$346$	−3.02740	−	3.55530i	−0.00874971	−	0.0102754i
$347$			462.222i			1.33205i	0.745929	+	0.666025i	$0.232006\pi$
$347$			462.222i			1.33205i	−0.745929	+	0.666025i	$0.767994\pi$
$348$		0			0
$349$	−200.598			−0.574779			−0.287390	−	0.957814i	$-0.592787\pi$
$349$	−200.598			−0.574779			−0.287390	+	0.957814i	$0.592787\pi$
$350$		0			0
$351$		0			0
$352$	468.403	−	189.960i	1.33069	−	0.539660i
$353$	−250.897			−0.710757			−0.355379	−	0.934722i	$-0.615648\pi$
$353$	−250.897			−0.710757			−0.355379	+	0.934722i	$0.615648\pi$
$354$		0			0
$355$		0			0
$356$	−25.3713	+	157.168i	−0.0712676	+	0.441483i
$357$		0			0
$358$	345.744	−	294.407i	0.965764	−	0.822366i
$359$		−	215.601i		−	0.600560i	−0.953851	−	0.300280i	$-0.902920\pi$
$359$		−	215.601i		−	0.600560i	0.953851	−	0.300280i	$-0.0970801\pi$
$360$		0			0
$361$	38.1960			0.105806
$362$	−147.942	−	173.739i	−0.408679	−	0.479941i
$363$		0			0
$364$	17.1478	+	2.76814i	0.0471095	+	0.00760478i
$365$		0			0
$366$		0			0
$367$			67.0637i			0.182735i	0.995817	+	0.0913675i	$0.0291238\pi$
$367$			67.0637i			0.182735i	−0.995817	+	0.0913675i	$0.970876\pi$
$368$	96.2097	−	290.231i	0.261439	−	0.788670i
$369$		0			0
$370$		0			0
$371$		−	13.0274i		−	0.0351142i
$372$		0			0
$373$	567.402			1.52119			0.760593	−	0.649230i	$-0.224909\pi$
$373$	567.402			1.52119			0.760593	+	0.649230i	$0.224909\pi$
$374$	−658.042	+	560.334i	−1.75947	+	1.49822i
$375$		0			0
$376$	−230.949	−	380.667i	−0.614225	−	1.01241i
$377$	−236.764			−0.628020
$378$		0			0
$379$		−	240.298i		−	0.634031i	−0.948420	−	0.317016i	$-0.897319\pi$
$379$		−	240.298i		−	0.634031i	0.948420	−	0.317016i	$-0.102681\pi$
$380$		0			0
$381$		0			0
$382$	212.257	−	180.740i	0.555646	−	0.473142i
$383$		−	670.068i		−	1.74952i	−0.484553	−	0.874762i	$-0.661018\pi$
$383$		−	670.068i		−	1.74952i	0.484553	−	0.874762i	$-0.338982\pi$
$384$		0			0
$385$		0			0
$386$	236.048	+	277.209i	0.611524	+	0.718157i
$387$		0			0
$388$	−107.617	+	666.655i	−0.277363	+	1.71818i
$389$	−474.640			−1.22015			−0.610077	−	0.792342i	$-0.708861\pi$
$389$	−474.640			−1.22015			−0.610077	+	0.792342i	$0.708861\pi$
$390$		0			0
$391$			522.826i			1.33715i
$392$	330.349	−	200.421i	0.842726	−	0.511278i
$393$		0			0
$394$	−334.567	−	392.907i	−0.849156	−	0.997226i
$395$		0			0
$396$		0			0
$397$	−499.460			−1.25809			−0.629043	−	0.777371i	$-0.716553\pi$
$397$	−499.460			−1.25809			−0.629043	+	0.777371i	$0.716553\pi$
$398$	−390.542	+	332.554i	−0.981262	+	0.835562i
$399$		0			0
$400$		0			0
$401$	−344.694			−0.859587			−0.429793	−	0.902927i	$-0.641414\pi$
$401$	−344.694			−0.859587			−0.429793	+	0.902927i	$0.641414\pi$
$402$		0			0
$403$		−	70.6631i		−	0.175343i
$404$	28.3987	−	175.922i	0.0702938	−	0.435450i
$405$		0			0
$406$	58.1993	−	49.5578i	0.143348	−	0.122063i
$407$		−	245.773i		−	0.603864i
$408$		0			0
$409$	501.890			1.22712			0.613558	−	0.789650i	$-0.289738\pi$
$409$	501.890			1.22712			0.613558	+	0.789650i	$0.289738\pi$
$410$		0			0
$411$		0			0
$412$	499.397	+	80.6166i	1.21213	+	0.195671i
$413$	−73.3957			−0.177714
$414$		0			0
$415$		0			0
$416$	62.3746	+	153.803i	0.149939	+	0.369719i
$417$		0			0
$418$	−367.978	−	432.144i	−0.880331	−	1.03384i
$419$		−	218.369i		−	0.521167i	−0.965451	−	0.260584i	$-0.916085\pi$
$419$		−	218.369i		−	0.521167i	0.965451	−	0.260584i	$-0.0839150\pi$
$420$		0			0
$421$	281.698			0.669116			0.334558	−	0.942375i	$-0.391413\pi$
$421$	281.698			0.669116			0.334558	+	0.942375i	$0.391413\pi$
$422$	322.600	−	274.700i	0.764455	−	0.650947i
$423$		0			0
$424$	106.423	−	64.5661i	0.250997	−	0.152279i
$425$		0			0
$426$		0			0
$427$		−	31.8156i		−	0.0745097i
$428$	−412.197	−	66.5401i	−0.963078	−	0.155468i
$429$		0			0
$430$		0			0
$431$			441.081i			1.02339i	0.859167	+	0.511694i	$0.170982\pi$
$431$			441.081i			1.02339i	−0.859167	+	0.511694i	$0.829018\pi$
$432$		0			0
$433$	−123.443			−0.285089			−0.142544	−	0.989788i	$-0.545528\pi$
$433$	−123.443			−0.285089			−0.142544	+	0.989788i	$0.545528\pi$
$434$	14.7907	+	17.3698i	0.0340800	+	0.0400227i
$435$		0			0
$436$	−0.319785	+	1.98098i	−0.000733451	+	0.00454353i
$437$	−343.346			−0.785690
$438$		0			0
$439$		−	330.728i		−	0.753368i	−0.926342	−	0.376684i	$-0.877064\pi$
$439$		−	330.728i		−	0.753368i	0.926342	−	0.376684i	$-0.122936\pi$
$440$		0			0
$441$		0			0
$442$	−183.989	−	216.072i	−0.416265	−	0.488850i
$443$		−	154.952i		−	0.349780i	−0.984588	−	0.174890i	$-0.944043\pi$
$443$		−	154.952i		−	0.349780i	0.984588	−	0.174890i	$-0.0559569\pi$
$444$		0			0
$445$		0			0
$446$	532.718	−	453.619i	1.19444	−	1.01708i
$447$		0			0
$448$	−47.5254	−	24.7508i	−0.106084	−	0.0552473i
$449$	−95.8970			−0.213579			−0.106790	−	0.994282i	$-0.534057\pi$
$449$	−95.8970			−0.213579			−0.106790	+	0.994282i	$0.534057\pi$
$450$		0			0
$451$			210.065i			0.465775i
$452$	−10.8276	+	67.0738i	−0.0239548	+	0.148393i
$453$		0			0
$454$	−282.550	+	240.596i	−0.622356	+	0.529948i
$455$		0			0
$456$		0			0
$457$	−485.718			−1.06284			−0.531420	−	0.847108i	$-0.678341\pi$
$457$	−485.718			−1.06284			−0.531420	+	0.847108i	$0.678341\pi$
$458$	342.179	+	401.846i	0.747116	+	0.877393i
$459$		0			0
$460$		0			0
$461$	−353.650			−0.767136			−0.383568	−	0.923513i	$-0.625305\pi$
$461$	−353.650			−0.767136			−0.383568	+	0.923513i	$0.625305\pi$
$462$		0			0
$463$			421.720i			0.910842i	0.890276	+	0.455421i	$0.150511\pi$
$463$			421.720i			0.910842i	−0.890276	+	0.455421i	$0.849489\pi$
$464$	693.292	+	229.822i	1.49416	+	0.495306i
$465$		0			0
$466$	75.8248	+	89.0466i	0.162714	+	0.191087i
$467$			640.974i			1.37254i	0.727349	+	0.686268i	$0.240752\pi$
$467$			640.974i			1.37254i	−0.727349	+	0.686268i	$0.759248\pi$
$468$		0			0
$469$	−77.1960			−0.164597
$470$		0			0
$471$		0			0
$472$	−363.763	−	599.582i	−0.770684	−	1.27030i
$473$	441.374			0.933137
$474$		0			0
$475$		0			0
$476$	90.4535	+	14.6017i	0.190028	+	0.0306758i
$477$		0			0
$478$	172.582	−	146.957i	0.361050	−	0.307441i
$479$		−	221.137i		−	0.461664i	−0.972994	−	0.230832i	$-0.925855\pi$
$479$		−	221.137i		−	0.461664i	0.972994	−	0.230832i	$-0.0741448\pi$
$480$		0			0
$481$	80.7010			0.167778
$482$	100.871	+	118.460i	0.209275	+	0.245767i
$483$		0			0
$484$	81.9124	−	507.424i	0.169240	−	1.04840i
$485$		0			0
$486$		0			0
$487$			889.949i			1.82741i	0.406377	+	0.913705i	$0.366792\pi$
$487$			889.949i			1.82741i	−0.406377	+	0.913705i	$0.633208\pi$
$488$	259.907	−	157.684i	0.532596	−	0.323123i
$489$		0			0
$490$		0			0
$491$		−	552.843i		−	1.12595i	−0.826473	−	0.562977i	$-0.809656\pi$
$491$		−	552.843i		−	1.12595i	0.826473	−	0.562977i	$-0.190344\pi$
$492$		0			0
$493$	−1248.91			−2.53328
$494$	141.897	−	120.828i	0.287241	−	0.244591i
$495$		0			0
$496$	−68.5914	+	206.916i	−0.138289	+	0.417169i
$497$	−109.435			−0.220190
$498$		0			0
$499$			533.302i			1.06874i	0.845250	+	0.534371i	$0.179451\pi$
$499$			533.302i			1.06874i	−0.845250	+	0.534371i	$0.820549\pi$
$500$		0			0
$501$		0			0
$502$	161.755	−	137.737i	0.322220	−	0.274376i
$503$			574.914i			1.14297i	0.820612	+	0.571485i	$0.193633\pi$
$503$			574.914i			1.14297i	−0.820612	+	0.571485i	$0.806367\pi$
$504$		0			0
$505$		0			0
$506$	−391.395	−	459.644i	−0.773509	−	0.908388i
$507$		0			0
$508$	33.3917	+	5.39034i	0.0657316	+	0.0106109i
$509$	207.547			0.407753			0.203877	−	0.978997i	$-0.434646\pi$
$509$	207.547			0.407753			0.203877	+	0.978997i	$0.434646\pi$
$510$		0			0
$511$		−	45.8122i		−	0.0896521i
$512$	−33.3518	−	510.913i	−0.0651403	−	0.997876i
$513$		0			0
$514$	−494.120	−	580.282i	−0.961324	−	1.12895i
$515$		0			0
$516$		0			0
$517$	−879.112			−1.70041
$518$	−19.8373	+	16.8918i	−0.0382959	+	0.0326096i
$519$		0			0
$520$		0			0
$521$	712.900			1.36833			0.684165	−	0.729327i	$-0.260167\pi$
$521$	712.900			1.36833			0.684165	+	0.729327i	$0.260167\pi$
$522$		0			0
$523$		−	139.548i		−	0.266822i	−0.991061	−	0.133411i	$-0.957407\pi$
$523$		−	139.548i		−	0.266822i	0.991061	−	0.133411i	$-0.0425931\pi$
$524$	−204.272	−	32.9751i	−0.389831	−	0.0629296i
$525$		0			0
$526$	17.4983	−	14.9002i	0.0332668	−	0.0283273i
$527$		−	372.742i		−	0.707290i
$528$		0			0
$529$	163.804			0.309648
$530$		0			0
$531$		0			0
$532$	−9.58911	+	59.4019i	−0.0180246	+	0.111658i
$533$	−68.9760			−0.129411
$534$		0			0
$535$		0			0
$536$	−382.598	−	630.627i	−0.713802	−	1.17654i
$537$		0			0
$538$	101.075	+	118.700i	0.187872	+	0.220632i
$539$		−	762.908i		−	1.41541i
$540$		0			0
$541$	−946.688			−1.74988			−0.874942	−	0.484227i	$-0.839101\pi$
$541$	−946.688			−1.74988			−0.874942	+	0.484227i	$0.839101\pi$
$542$	131.999	−	112.399i	0.243540	−	0.207379i
$543$		0			0
$544$	329.021	+	811.298i	0.604818	+	1.49136i
$545$		0			0
$546$		0			0
$547$			50.3388i			0.0920271i	0.998941	+	0.0460136i	$0.0146517\pi$
$547$			50.3388i			0.0920271i	−0.998941	+	0.0460136i	$0.985348\pi$
$548$	34.3006	−	212.483i	0.0625923	−	0.387742i
$549$		0			0
$550$		0			0
$551$		−	820.173i		−	1.48852i
$552$		0			0
$553$	−11.4070			−0.0206275
$554$	−372.561	−	437.525i	−0.672492	−	0.789757i
$555$		0			0
$556$	−53.8007	−	8.68492i	−0.0967638	−	0.0156204i
$557$	790.157			1.41859			0.709297	−	0.704910i	$-0.249013\pi$
$557$	790.157			1.41859			0.709297	+	0.704910i	$0.249013\pi$
$558$		0			0
$559$			144.928i			0.259263i
$560$		0			0
$561$		0			0
$562$	−291.222	−	342.004i	−0.518189	−	0.608548i
$563$		−	354.133i		−	0.629010i	−0.949256	−	0.314505i	$-0.898162\pi$
$563$		−	354.133i		−	0.629010i	0.949256	−	0.314505i	$-0.101838\pi$
$564$		0			0
$565$		0			0
$566$	−128.096	+	109.076i	−0.226319	+	0.192714i
$567$		0			0
$568$	−542.378	−	893.989i	−0.954891	−	1.57392i
$569$	55.4983			0.0975366			0.0487683	−	0.998810i	$-0.484470\pi$
$569$	55.4983			0.0975366			0.0487683	+	0.998810i	$0.484470\pi$
$570$		0			0
$571$		−	791.134i		−	1.38552i	−0.721167	−	0.692762i	$-0.756394\pi$
$571$		−	791.134i		−	1.38552i	0.721167	−	0.692762i	$-0.243606\pi$
$572$	323.509	+	52.2233i	0.565575	+	0.0912995i
$573$		0			0
$574$	16.9551	−	14.4376i	0.0295386	−	0.0251526i
$575$		0			0
$576$		0			0
$577$	−201.759			−0.349668			−0.174834	−	0.984598i	$-0.555939\pi$
$577$	−201.759			−0.349668			−0.174834	+	0.984598i	$0.555939\pi$
$578$	−595.799	−	699.690i	−1.03079	−	1.21054i
$579$		0			0
$580$		0			0
$581$	−49.4020			−0.0850292
$582$		0			0
$583$		−	245.773i		−	0.421566i
$584$	374.248	−	227.054i	0.640835	−	0.388791i
$585$		0			0
$586$	−319.777	−	375.537i	−0.545694	−	0.640848i
$587$			444.556i			0.757335i	0.925533	+	0.378668i	$0.123618\pi$
$587$			444.556i			0.757335i	−0.925533	+	0.378668i	$0.876382\pi$
$588$		0			0
$589$	244.784			0.415593
$590$		0			0
$591$		0			0
$592$	−236.309	−	78.3350i	−0.399171	−	0.132323i
$593$	−563.908			−0.950942			−0.475471	−	0.879731i	$-0.657722\pi$
$593$	−563.908			−0.950942			−0.475471	+	0.879731i	$0.657722\pi$
$594$		0			0
$595$		0			0
$596$	21.4502	−	132.878i	0.0359902	−	0.222949i
$597$		0			0
$598$	150.927	−	128.517i	0.252386	−	0.214911i
$599$			845.034i			1.41074i	0.708839	+	0.705371i	$0.249219\pi$
$599$			845.034i			1.41074i	−0.708839	+	0.705371i	$0.750781\pi$
$600$		0			0
$601$	672.296			1.11863			0.559314	−	0.828956i	$-0.311065\pi$
$601$	672.296			1.11863			0.559314	+	0.828956i	$0.311065\pi$
$602$	−30.3353	−	35.6250i	−0.0503909	−	0.0591777i
$603$		0			0
$604$	552.797	+	89.2368i	0.915227	+	0.147743i
$605$		0			0
$606$		0			0
$607$			882.664i			1.45414i	0.686562	+	0.727071i	$0.259119\pi$
$607$			882.664i			1.45414i	−0.686562	+	0.727071i	$0.740881\pi$
$608$	−532.789	+	216.072i	−0.876298	+	0.355381i
$609$		0			0
$610$		0			0
$611$		−	288.662i		−	0.472442i
$612$		0			0
$613$	−469.374			−0.765701			−0.382850	−	0.923810i	$-0.625057\pi$
$613$	−469.374			−0.765701			−0.382850	+	0.923810i	$0.625057\pi$
$614$	176.350	−	150.166i	0.287216	−	0.244569i
$615$		0			0
$616$	−90.4535	+	54.8777i	−0.146840	+	0.0890871i
$617$	218.994			0.354934			0.177467	−	0.984127i	$-0.443210\pi$
$617$	218.994			0.354934			0.177467	+	0.984127i	$0.443210\pi$
$618$		0			0
$619$		−	879.610i		−	1.42102i	−0.703689	−	0.710509i	$-0.748465\pi$
$619$		−	879.610i		−	1.42102i	0.703689	−	0.710509i	$-0.251535\pi$
$620$		0			0
$621$		0			0
$622$	310.284	−	264.213i	0.498849	−	0.424779i
$623$		−	33.3232i		−	0.0534884i
$624$		0			0
$625$		0			0
$626$	128.447	+	150.845i	0.205187	+	0.240966i
$627$		0			0
$628$	13.5544	−	83.9656i	0.0215834	−	0.133703i
$629$	425.691			0.676774
$630$		0			0
$631$		−	635.566i		−	1.00724i	−0.863926	−	0.503618i	$-0.832002\pi$
$631$		−	635.566i		−	1.00724i	0.863926	−	0.503618i	$-0.167998\pi$
$632$	−56.5351	−	93.1855i	−0.0894543	−	0.147445i
$633$		0			0
$634$	610.966	+	717.502i	0.963669	+	1.13171i
$635$		0			0
$636$		0			0
$637$	250.505			0.393258
$638$	1097.98	−	934.951i	1.72097	−	1.46544i
$639$		0			0
$640$		0			0
$641$	296.309			0.462260			0.231130	−	0.972923i	$-0.425758\pi$
$641$	296.309			0.462260			0.231130	+	0.972923i	$0.425758\pi$
$642$		0			0
$643$			591.032i			0.919179i	0.888131	+	0.459590i	$0.152003\pi$
$643$			591.032i			0.919179i	−0.888131	+	0.459590i	$0.847997\pi$
$644$	−10.1993	+	63.1821i	−0.0158375	+	0.0981088i
$645$		0			0
$646$	748.495	−	637.357i	1.15866	−	0.986621i
$647$		−	166.507i		−	0.257352i	−0.991687	−	0.128676i	$-0.958927\pi$
$647$		−	166.507i		−	0.257352i	0.991687	−	0.128676i	$-0.0410728\pi$
$648$		0			0
$649$	−1384.67			−2.13355
$650$		0			0
$651$		0			0
$652$	830.098	+	134.001i	1.27316	+	0.205523i
$653$	−621.335			−0.951509			−0.475754	−	0.879578i	$-0.657825\pi$
$653$	−621.335			−0.951509			−0.475754	+	0.879578i	$0.657825\pi$
$654$		0			0
$655$		0			0
$656$	201.976	+	66.9538i	0.307890	+	0.102064i
$657$		0			0
$658$	60.4208	+	70.9565i	0.0918249	+	0.107837i
$659$			702.113i			1.06542i	0.846297	+	0.532711i	$0.178827\pi$
$659$			702.113i			1.06542i	−0.846297	+	0.532711i	$0.821173\pi$
$660$		0			0
$661$	358.193			0.541895			0.270948	−	0.962594i	$-0.412663\pi$
$661$	358.193			0.541895			0.270948	+	0.962594i	$0.412663\pi$
$662$	−412.197	+	350.994i	−0.622655	+	0.530202i
$663$		0			0
$664$	−244.846	−	403.573i	−0.368743	−	0.607790i
$665$		0			0
$666$		0			0
$667$		−	872.367i		−	1.30790i
$668$	941.351	+	151.960i	1.40921	+	0.227485i
$669$		0			0
$670$		0			0
$671$		−	600.230i		−	0.894530i
$672$		0			0
$673$	714.176			1.06118			0.530592	−	0.847628i	$-0.321970\pi$
$673$	714.176			1.06118			0.530592	+	0.847628i	$0.321970\pi$
$674$	−489.244	−	574.555i	−0.725882	−	0.852456i
$675$		0			0
$676$	90.5827	−	561.134i	0.133998	−	0.830081i
$677$	−509.833			−0.753077			−0.376538	−	0.926401i	$-0.622886\pi$
$677$	−509.833			−0.753077			−0.376538	+	0.926401i	$0.622886\pi$
$678$		0			0
$679$		−	141.346i		−	0.208168i
$680$		0			0
$681$		0			0
$682$	279.040	+	327.697i	0.409150	+	0.480494i
$683$			1263.93i			1.85055i	0.379298	+	0.925275i	$0.376166\pi$
$683$			1263.93i			1.85055i	−0.379298	+	0.925275i	$0.623834\pi$
$684$		0			0
$685$		0			0
$686$	−124.048	+	105.629i	−0.180828	+	0.153978i
$687$		0			0
$688$	140.679	−	424.378i	0.204475	−	0.616829i
$689$	80.7010			0.117128
$690$		0			0
$691$		−	512.351i		−	0.741463i	−0.928740	−	0.370731i	$-0.879107\pi$
$691$		−	512.351i		−	0.741463i	0.928740	−	0.370731i	$-0.120893\pi$
$692$	−1.48834	+	9.21987i	−0.00215078	+	0.0133235i
$693$		0			0
$694$	703.842	−	599.334i	1.01418	−	0.863594i
$695$		0			0
$696$		0			0
$697$	−363.843			−0.522012
$698$	260.103	+	305.458i	0.372640	+	0.437619i
$699$		0			0
$700$		0			0
$701$	−1092.03			−1.55781			−0.778907	−	0.627139i	$-0.784226\pi$
$701$	−1092.03			−1.55781			−0.778907	+	0.627139i	$0.784226\pi$
$702$		0			0
$703$			279.556i			0.397662i
$704$	−896.609	−	466.945i	−1.27359	−	0.663274i
$705$		0			0
$706$	325.323	+	382.051i	0.460798	+	0.541148i
$707$			37.2995i			0.0527574i
$708$		0			0
$709$	−416.887			−0.587993			−0.293997	−	0.955806i	$-0.594985\pi$
$709$	−416.887			−0.587993			−0.293997	+	0.955806i	$0.594985\pi$
$710$		0			0
$711$		0			0
$712$	272.223	−	165.156i	0.382336	−	0.231961i
$713$	260.362			0.365163
$714$		0			0
$715$		0			0
$716$	−896.609	−	144.738i	−1.25225	−	0.202147i
$717$		0			0
$718$	−328.304	+	279.556i	−0.457247	+	0.389354i
$719$			395.268i			0.549747i	0.961480	+	0.274874i	$0.0886361\pi$
$719$			395.268i			0.549747i	−0.961480	+	0.274874i	$0.911364\pi$
$720$		0			0
$721$	−105.884			−0.146857
$722$	−49.5264	−	58.1625i	−0.0685962	−	0.0805575i
$723$		0			0
$724$	−72.7317	+	450.553i	−0.100458	+	0.622310i
$725$		0			0
$726$		0			0
$727$			597.583i			0.821985i	0.911639	+	0.410993i	$0.134818\pi$
$727$			597.583i			0.821985i	−0.911639	+	0.410993i	$0.865182\pi$
$728$	−18.0194	−	29.7009i	−0.0247519	−	0.0407980i
$729$		0			0
$730$		0			0
$731$			764.482i			1.04580i
$732$		0			0
$733$	−23.8650			−0.0325580			−0.0162790	−	0.999867i	$-0.505182\pi$
$733$	−23.8650			−0.0325580			−0.0162790	+	0.999867i	$0.505182\pi$
$734$	102.120	−	86.9574i	0.139129	−	0.118471i
$735$		0			0
$736$	−566.694	+	229.822i	−0.769965	+	0.312258i
$737$	−1456.37			−1.97608
$738$		0			0
$739$			125.767i			0.170186i	0.996373	+	0.0850928i	$0.0271187\pi$
$739$			125.767i			0.170186i	−0.996373	+	0.0850928i	$0.972881\pi$
$740$		0			0
$741$		0			0
$742$	−19.8373	+	16.8918i	−0.0267349	+	0.0227652i
$743$			148.841i			0.200325i	0.994971	+	0.100162i	$0.0319362\pi$
$743$			148.841i			0.200325i	−0.994971	+	0.100162i	$0.968064\pi$
$744$		0			0
$745$		0			0
$746$	−735.715	−	864.004i	−0.986213	−	1.15818i
$747$		0			0
$748$	1706.48	+	275.474i	2.28140	+	0.368280i
$749$	87.3954			0.116683
$750$		0			0
$751$			463.390i			0.617030i	0.951219	+	0.308515i	$0.0998321\pi$
$751$			463.390i			0.617030i	−0.951219	+	0.308515i	$0.900168\pi$
$752$	−280.199	+	845.261i	−0.372605	+	1.12402i
$753$		0			0
$754$	306.997	+	360.529i	0.407157	+	0.478155i
$755$		0			0
$756$		0			0
$757$	719.363			0.950281			0.475141	−	0.879910i	$-0.342397\pi$
$757$	719.363			0.950281			0.475141	+	0.879910i	$0.342397\pi$
$758$	−365.910	+	311.579i	−0.482731	+	0.411054i
$759$		0			0
$760$		0			0
$761$	1107.49			1.45530			0.727651	−	0.685947i	$-0.240612\pi$
$761$	1107.49			1.45530			0.727651	+	0.685947i	$0.240612\pi$
$762$		0			0
$763$		−	0.420013i		−	0.000550476i
$764$	−550.440	−	88.8563i	−0.720472	−	0.116304i
$765$		0			0
$766$	−1020.34	+	868.835i	−1.33203	+	1.13425i
$767$		−	454.666i		−	0.592785i
$768$		0			0
$769$	231.691			0.301289			0.150644	−	0.988588i	$-0.451865\pi$
$769$	231.691			0.301289			0.150644	+	0.988588i	$0.451865\pi$
$770$		0			0
$771$		0			0
$772$	116.047	−	718.879i	0.150320	−	0.931190i
$773$	−519.956			−0.672647			−0.336324	−	0.941746i	$-0.609184\pi$
$773$	−519.956			−0.672647			−0.336324	+	0.941746i	$0.609184\pi$
$774$		0			0
$775$		0			0
$776$	1154.68	−	700.539i	1.48799	−	0.902756i
$777$		0			0
$778$	615.436	+	722.751i	0.791049	+	0.928986i
$779$		−	238.940i		−	0.306726i
$780$		0			0
$781$	−2064.58			−2.64351
$782$	796.127	−	677.917i	1.01807	−	0.866901i
$783$		0			0
$784$	−733.531	−	243.161i	−0.935626	−	0.310154i
$785$		0			0
$786$		0			0
$787$			46.0288i			0.0584864i	0.999572	+	0.0292432i	$0.00930972\pi$
$787$			46.0288i			0.0584864i	−0.999572	+	0.0292432i	$0.990690\pi$
$788$	−164.481	+	1018.92i	−0.208733	+	1.29304i
$789$		0			0
$790$		0			0
$791$		−	14.2212i		−	0.0179788i
$792$		0			0
$793$	197.089			0.248536
$794$	647.619	+	760.546i	0.815641	+	0.957867i
$795$		0			0
$796$	1012.78	+	163.491i	1.27234	+	0.205391i
$797$	15.3098			0.0192093			0.00960463	−	0.999954i	$-0.496943\pi$
$797$	15.3098			0.0192093			0.00960463	+	0.999954i	$0.496943\pi$
$798$		0			0
$799$		−	1522.67i		−	1.90572i
$800$		0			0
$801$		0			0
$802$	446.944	+	524.879i	0.557287	+	0.654463i
$803$		−	864.288i		−	1.07632i
$804$		0			0
$805$		0			0
$806$	−107.601	+	91.6245i	−0.133500	+	0.113678i
$807$		0			0
$808$	−304.706	+	184.863i	−0.377111	+	0.228791i
$809$	−313.093			−0.387012			−0.193506	−	0.981099i	$-0.561986\pi$
$809$	−313.093			−0.387012			−0.193506	+	0.981099i	$0.561986\pi$
$810$		0			0
$811$			1056.89i			1.30319i	0.758566	+	0.651596i	$0.225900\pi$
$811$			1056.89i			1.30319i	−0.758566	+	0.651596i	$0.774100\pi$
$812$	−150.927	−	24.3638i	−0.185871	−	0.0300047i
$813$		0			0
$814$	−374.248	+	318.678i	−0.459764	+	0.391497i
$815$		0			0
$816$		0			0
$817$	−502.045			−0.614498
$818$	−650.770	−	764.247i	−0.795563	−	0.934288i
$819$		0			0
$820$		0			0
$821$	−308.659			−0.375955			−0.187978	−	0.982173i	$-0.560193\pi$
$821$	−308.659			−0.375955			−0.187978	+	0.982173i	$0.560193\pi$
$822$		0			0
$823$		−	109.680i		−	0.133269i	−0.997777	−	0.0666344i	$-0.978774\pi$
$823$		−	109.680i		−	0.133269i	0.997777	−	0.0666344i	$-0.0212261\pi$
$824$	−524.780	−	864.982i	−0.636869	−	1.04974i
$825$		0			0
$826$	95.1677	+	111.762i	0.115215	+	0.135306i
$827$			711.971i			0.860908i	0.902613	+	0.430454i	$0.141646\pi$
$827$			711.971i			0.860908i	−0.902613	+	0.430454i	$0.858354\pi$
$828$		0			0
$829$	118.688			0.143170			0.0715849	−	0.997435i	$-0.477194\pi$
$829$	118.688			0.143170			0.0715849	+	0.997435i	$0.477194\pi$
$830$		0			0
$831$		0			0
$832$	153.324	−	294.407i	0.184284	−	0.353854i
$833$	1321.40			1.58631
$834$		0			0
$835$		0			0
$836$	−180.907	+	1120.67i	−0.216396	+	1.34051i
$837$		0			0
$838$	−332.519	+	283.146i	−0.396800	+	0.337883i
$839$		−	1413.67i		−	1.68495i	−0.538736	−	0.842475i	$-0.681098\pi$
$839$		−	1413.67i		−	1.68495i	0.538736	−	0.842475i	$-0.318902\pi$
$840$		0			0
$841$	1242.88			1.47786
$842$	−365.260	−	428.952i	−0.433800	−	0.509444i
$843$		0			0
$844$	−836.591	−	135.049i	−0.991222	−	0.160011i
$845$		0			0
$846$		0			0
$847$			107.586i			0.127020i
$848$	−236.309	−	78.3350i	−0.278666	−	0.0923762i
$849$		0			0
$850$		0			0
$851$			297.347i			0.349409i
$852$		0			0
$853$	1308.03			1.53344			0.766722	−	0.641979i	$-0.221886\pi$
$853$	1308.03			1.53344			0.766722	+	0.641979i	$0.221886\pi$
$854$	−48.4469	+	41.2534i	−0.0567293	+	0.0483060i
$855$		0			0
$856$	433.148	+	713.947i	0.506014	+	0.834050i
$857$	719.755			0.839854			0.419927	−	0.907558i	$-0.362056\pi$
$857$	719.755			0.839854			0.419927	+	0.907558i	$0.362056\pi$
$858$		0			0
$859$			1402.44i			1.63264i	0.577601	+	0.816319i	$0.303989\pi$
$859$			1402.44i			1.63264i	−0.577601	+	0.816319i	$0.696011\pi$
$860$		0			0
$861$		0			0
$862$	671.650	−	571.922i	0.779176	−	0.663483i
$863$			72.4412i			0.0839411i	0.999119	+	0.0419706i	$0.0133636\pi$
$863$			72.4412i			0.0839411i	−0.999119	+	0.0419706i	$0.986636\pi$
$864$		0			0
$865$		0			0
$866$	160.061	+	187.972i	0.184828	+	0.217058i
$867$		0			0
$868$	7.27148	−	45.0448i	0.00837728	−	0.0518949i
$869$	−215.203			−0.247644
$870$		0			0
$871$		−	478.208i		−	0.549033i
$872$	3.43115	−	2.08166i	0.00393481	−	0.00238723i
$873$		0			0
$874$	445.196	+	522.826i	0.509378	+	0.598200i
$875$		0			0
$876$		0			0
$877$	1382.21			1.57606			0.788032	−	0.615635i	$-0.211100\pi$
$877$	1382.21			1.57606			0.788032	+	0.615635i	$0.211100\pi$
$878$	−503.613	+	428.835i	−0.573591	+	0.488423i
$879$		0			0
$880$		0			0
$881$	1131.38			1.28419			0.642097	−	0.766623i	$-0.278064\pi$
$881$	1131.38			1.28419			0.642097	+	0.766623i	$0.278064\pi$
$882$		0			0
$883$			1077.39i			1.22014i	0.792346	+	0.610072i	$0.208860\pi$
$883$			1077.39i			1.22014i	−0.792346	+	0.610072i	$0.791140\pi$
$884$	−90.4535	+	560.334i	−0.102323	+	0.633862i
$885$		0			0
$886$	−235.952	+	200.917i	−0.266311	+	0.226769i
$887$			766.896i			0.864595i	0.901731	+	0.432297i	$0.142297\pi$
$887$			766.896i			0.864595i	−0.901731	+	0.432297i	$0.857703\pi$
$888$		0			0
$889$	−7.07980			−0.00796378
$890$		0			0
$891$		0			0
$892$	−1381.49	−	223.010i	−1.54875	−	0.250011i
$893$	999.954			1.11977
$894$		0			0
$895$		0			0
$896$	23.9344	+	104.462i	0.0267124	+	0.116587i
$897$		0			0
$898$	124.344	+	146.026i	0.138467	+	0.162612i
$899$			621.942i			0.691815i
$900$		0			0
$901$	425.691			0.472465
$902$	319.873	−	272.378i	0.354627	−	0.301971i
$903$		0			0
$904$	116.175	−	70.4829i	0.128512	−	0.0779678i
$905$		0			0
$906$		0			0
$907$			1086.43i			1.19783i	0.800814	+	0.598913i	$0.204400\pi$
$907$			1086.43i			1.19783i	−0.800814	+	0.598913i	$0.795600\pi$
$908$	732.730	+	118.283i	0.806971	+	0.130268i
$909$		0			0
$910$		0			0
$911$			237.746i			0.260973i	0.991450	+	0.130486i	$0.0416539\pi$
$911$			237.746i			0.260973i	−0.991450	+	0.130486i	$0.958346\pi$
$912$		0			0
$913$	−932.012			−1.02082
$914$	629.801	+	739.621i	0.689060	+	0.809214i
$915$		0			0
$916$	168.223	−	1042.10i	0.183650	−	1.13766i
$917$	43.3103			0.0472304
$918$		0			0
$919$			344.517i			0.374882i	0.982276	+	0.187441i	$0.0600194\pi$
$919$			344.517i			0.374882i	−0.982276	+	0.187441i	$0.939981\pi$
$920$		0			0
$921$		0			0
$922$	458.555	+	538.515i	0.497349	+	0.584073i
$923$		−	677.917i		−	0.734471i
$924$		0			0
$925$		0			0
$926$	642.169	−	546.818i	0.693487	−	0.590516i
$927$		0			0
$928$	−548.991	−	1353.70i	−0.591585	−	1.45873i
$929$	1478.10			1.59106			0.795531	−	0.605913i	$-0.207192\pi$
$929$	1478.10			1.59106			0.795531	+	0.605913i	$0.207192\pi$
$930$		0			0
$931$			867.776i			0.932090i
$932$	37.2773	−	230.922i	0.0399971	−	0.247771i
$933$		0			0
$934$	976.035	−	831.111i	1.04501	−	0.889841i
$935$		0			0
$936$		0			0
$937$	−246.887			−0.263486			−0.131743	−	0.991284i	$-0.542057\pi$
$937$	−246.887			−0.263486			−0.131743	+	0.991284i	$0.542057\pi$
$938$	100.095	+	117.549i	0.106711	+	0.125319i
$939$		0			0
$940$		0			0
$941$	−1658.64			−1.76264			−0.881318	−	0.472525i	$-0.843343\pi$
$941$	−1658.64			−1.76264			−0.881318	+	0.472525i	$0.843343\pi$
$942$		0			0
$943$		−	254.145i		−	0.269507i
$944$	−441.336	+	1331.36i	−0.467517	+	1.41033i
$945$		0			0
$946$	−572.302	−	672.097i	−0.604971	−	0.710462i
$947$			6.00750i			0.00634372i	0.999995	+	0.00317186i	$0.00100964\pi$
$947$			6.00750i			0.00634372i	−0.999995	+	0.00317186i	$0.998990\pi$
$948$		0			0
$949$	283.794			0.299045
$950$		0			0
$951$		0			0
$952$	−95.0509	−	156.670i	−0.0998434	−	0.164569i
$953$	−1089.55			−1.14329			−0.571643	−	0.820503i	$-0.693694\pi$
$953$	−1089.55			−1.14329			−0.571643	+	0.820503i	$0.693694\pi$
$954$		0			0
$955$		0			0
$956$	−447.553	−	72.2475i	−0.468152	−	0.0755727i
$957$		0			0
$958$	−336.734	+	286.735i	−0.351497	+	0.299306i
$959$			45.0512i			0.0469773i
$960$		0			0
$961$	775.379			0.806846
$962$	−104.640	−	122.886i	−0.108773	−	0.127741i
$963$		0			0
$964$	49.5905	−	307.199i	0.0514424	−	0.318671i
$965$		0			0
$966$		0			0
$967$		−	1699.27i		−	1.75726i	−0.477502	−	0.878630i	$-0.658458\pi$
$967$		−	1699.27i		−	1.75726i	0.477502	−	0.878630i	$-0.341542\pi$
$968$	−878.885	+	533.215i	−0.907939	+	0.550842i
$969$		0			0
$970$		0			0
$971$			197.851i			0.203760i	0.994797	+	0.101880i	$0.0324857\pi$
$971$			197.851i			0.203760i	−0.994797	+	0.101880i	$0.967514\pi$
$972$		0			0
$973$	11.4070			0.0117235
$974$	1355.16	−	1153.94i	1.39133	−	1.18475i
$975$		0			0
$976$	−577.117	−	191.311i	−0.591309	−	0.196015i
$977$	847.868			0.867828			0.433914	−	0.900954i	$-0.357132\pi$
$977$	847.868			0.867828			0.433914	+	0.900954i	$0.357132\pi$
$978$		0			0
$979$		−	628.672i		−	0.642157i
$980$		0			0
$981$		0			0
$982$	−841.835	+	716.837i	−0.857266	+	0.729977i
$983$		−	96.0512i		−	0.0977123i	−0.998806	−	0.0488562i	$-0.984442\pi$
$983$		−	96.0512i		−	0.0977123i	0.998806	−	0.0488562i	$-0.0155576\pi$
$984$		0			0
$985$		0			0
$986$	1619.38	+	1901.76i	1.64238	+	1.92876i
$987$		0			0
$988$	−367.978	−	59.4019i	−0.372448	−	0.0601234i
$989$	−533.993			−0.539933
$990$		0			0
$991$		−	1184.45i		−	1.19520i	−0.801793	−	0.597602i	$-0.796120\pi$
$991$		−	1184.45i		−	1.19520i	0.801793	−	0.597602i	$-0.203880\pi$
$992$	404.017	−	163.849i	0.407275	−	0.165170i
$993$		0			0
$994$	141.897	+	166.640i	0.142754	+	0.167646i
$995$		0			0
$996$		0			0
$997$	−1887.35			−1.89303			−0.946517	−	0.322655i	$-0.895425\pi$
$997$	−1887.35			−1.89303			−0.946517	+	0.322655i	$0.895425\pi$
$998$	812.079	−	691.500i	0.813706	−	0.692886i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	900.3.c.r.451.3		8
3.2	odd	2		300.3.c.f.151.6		8
4.3	odd	2	inner	900.3.c.r.451.4		8
5.2	odd	4		180.3.f.h.19.7		8
5.3	odd	4		180.3.f.h.19.2		8
5.4	even	2	inner	900.3.c.r.451.6		8
12.11	even	2		300.3.c.f.151.5		8
15.2	even	4		60.3.f.b.19.2	yes	8
15.8	even	4		60.3.f.b.19.7	yes	8
15.14	odd	2		300.3.c.f.151.3		8
20.3	even	4		180.3.f.h.19.8		8
20.7	even	4		180.3.f.h.19.1		8
20.19	odd	2	inner	900.3.c.r.451.5		8
60.23	odd	4		60.3.f.b.19.1	✓	8
60.47	odd	4		60.3.f.b.19.8	yes	8
60.59	even	2		300.3.c.f.151.4		8
120.53	even	4		960.3.j.e.319.6		8
120.77	even	4		960.3.j.e.319.1		8
120.83	odd	4		960.3.j.e.319.2		8
120.107	odd	4		960.3.j.e.319.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.3.f.b.19.1	✓	8	60.23	odd	4
60.3.f.b.19.2	yes	8	15.2	even	4
60.3.f.b.19.7	yes	8	15.8	even	4
60.3.f.b.19.8	yes	8	60.47	odd	4
180.3.f.h.19.1		8	20.7	even	4
180.3.f.h.19.2		8	5.3	odd	4
180.3.f.h.19.7		8	5.2	odd	4
180.3.f.h.19.8		8	20.3	even	4
300.3.c.f.151.3		8	15.14	odd	2
300.3.c.f.151.4		8	60.59	even	2
300.3.c.f.151.5		8	12.11	even	2
300.3.c.f.151.6		8	3.2	odd	2
900.3.c.r.451.3		8	1.1	even	1	trivial
900.3.c.r.451.4		8	4.3	odd	2	inner
900.3.c.r.451.5		8	20.19	odd	2	inner
900.3.c.r.451.6		8	5.4	even	2	inner
960.3.j.e.319.1		8	120.77	even	4
960.3.j.e.319.2		8	120.83	odd	4
960.3.j.e.319.5		8	120.107	odd	4
960.3.j.e.319.6		8	120.53	even	4

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.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.29664 - 1.52274i) q^{2} +(-0.637459 + 3.94888i) q^{4} -0.837253i q^{7} +(6.83966 - 4.14959i) q^{8} +O(q^{10})\)	\(q+(-1.29664 - 1.52274i) q^{2} +(-0.637459 + 3.94888i) q^{4} -0.837253i q^{7} +(6.83966 - 4.14959i) q^{8} -15.7955i q^{11} +5.18655 q^{13} +(-1.27492 + 1.08561i) q^{14} +(-15.1873 - 5.03449i) q^{16} +27.3586 q^{17} +17.9667i q^{19} +(-24.0524 + 20.4811i) q^{22} +19.1101i q^{23} +(-6.72508 - 7.89776i) q^{26} +(3.30621 + 0.533714i) q^{28} -45.6495 q^{29} -13.6243i q^{31} +(12.0262 + 29.6542i) q^{32} +(-35.4743 - 41.6600i) q^{34} +15.5597 q^{37} +(27.3586 - 23.2964i) q^{38} -13.2990 q^{41} +27.9430i q^{43} +(62.3746 + 10.0690i) q^{44} +(29.0997 - 24.7789i) q^{46} -55.6558i q^{47} +48.2990 q^{49} +(-3.30621 + 20.4811i) q^{52} +15.5597 q^{53} +(-3.47425 - 5.72653i) q^{56} +(59.1909 + 69.5122i) q^{58} -87.6625i q^{59} +38.0000 q^{61} +(-20.7462 + 17.6658i) q^{62} +(29.5619 - 56.7635i) q^{64} -92.2015i q^{67} +(-17.4400 + 108.036i) q^{68} -130.707i q^{71} +54.7173 q^{73} +(-20.1752 - 23.6933i) q^{74} +(-70.9485 - 11.4531i) q^{76} -13.2249 q^{77} -13.6243i q^{79} +(17.2440 + 20.2509i) q^{82} -59.0048i q^{83} +(42.5498 - 36.2319i) q^{86} +(-65.5448 - 108.036i) q^{88} +39.8007 q^{89} -4.34246i q^{91} +(-75.4635 - 12.1819i) q^{92} +(-84.7492 + 72.1654i) q^{94} +168.821 q^{97} +(-62.6263 - 73.5467i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 10 q^{4}+O(q^{10}) \) 8 * q + 10 * q^4	\( 8 q + 10 q^{4} + 20 q^{14} - 46 q^{16} - 84 q^{26} - 184 q^{29} - 12 q^{34} + 256 q^{41} + 348 q^{44} + 112 q^{46} + 24 q^{49} + 244 q^{56} + 304 q^{61} + 10 q^{64} - 252 q^{74} - 24 q^{76} + 280 q^{86} + 560 q^{89} - 376 q^{94}+O(q^{100}) \) 8 * q + 10 * q^4 + 20 * q^14 - 46 * q^16 - 84 * q^26 - 184 * q^29 - 12 * q^34 + 256 * q^41 + 348 * q^44 + 112 * q^46 + 24 * q^49 + 244 * q^56 + 304 * q^61 + 10 * q^64 - 252 * q^74 - 24 * q^76 + 280 * q^86 + 560 * q^89 - 376 * q^94

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