LMFDB - Embedded newform 405.3.d.a.404.11

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [405,3,Mod(404,405)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(405, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("405.404"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	405.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [20] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$11.0354507066$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 3 x^{18} - 19 x^{16} + 66 x^{14} + 109 x^{12} - 813 x^{10} + 981 x^{8} + 5346 x^{6} + \cdots + 59049 $ x^20 - 3x^18 - 19x^16 + 66x^14 + 109x^12 - 813x^10 + 981x^8 + 5346x^6 - 13851x^4 - 19683*x^2 + 59049
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{20} $
Twist minimal:	no (minimal twist has level 45)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			404.11
Root			$-0.315300 - 1.70311i$ of defining polynomial
Character	$\chi$	$=$	405.404
Dual form			405.3.d.a.404.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.528791 q^{2} -3.72038 q^{4} +(-3.86156 - 3.17622i) q^{5} -2.76658i q^{7} -4.08247 q^{8} +(-2.04196 - 1.67955i) q^{10} +9.22820i q^{11} +13.5883i q^{13} -1.46294i q^{14} +12.7227 q^{16} +12.2161 q^{17} +20.2664 q^{19} +(14.3665 + 11.8167i) q^{20} +4.87979i q^{22} -2.37127 q^{23} +(4.82331 + 24.5303i) q^{25} +7.18539i q^{26} +10.2927i q^{28} -34.9123i q^{29} +29.4466 q^{31} +23.0576 q^{32} +6.45977 q^{34} +(-8.78726 + 10.6833i) q^{35} +64.3630i q^{37} +10.7167 q^{38} +(15.7647 + 12.9668i) q^{40} +39.8597i q^{41} +67.5964i q^{43} -34.3324i q^{44} -1.25391 q^{46} -93.3802 q^{47} +41.3460 q^{49} +(2.55053 + 12.9714i) q^{50} -50.5538i q^{52} -9.82656 q^{53} +(29.3108 - 35.6353i) q^{55} +11.2945i q^{56} -18.4613i q^{58} -58.5035i q^{59} -15.5057 q^{61} +15.5711 q^{62} -38.6984 q^{64} +(43.1595 - 52.4722i) q^{65} +15.5649i q^{67} -45.4486 q^{68} +(-4.64663 + 5.64925i) q^{70} +53.1970i q^{71} -23.6547i q^{73} +34.0346i q^{74} -75.3988 q^{76} +25.5306 q^{77} +34.5385 q^{79} +(-49.1297 - 40.4102i) q^{80} +21.0775i q^{82} +75.3460 q^{83} +(-47.1733 - 38.8010i) q^{85} +35.7444i q^{86} -37.6739i q^{88} -29.1344i q^{89} +37.5932 q^{91} +8.82203 q^{92} -49.3786 q^{94} +(-78.2601 - 64.3706i) q^{95} -62.3713i q^{97} +21.8634 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) $ 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

Character values

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

$n$	$82$	$326$
$\chi(n)$	$-1$	$-1$

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 $

$2$	0.528791			0.264396			0.132198	−	0.991223i	$-0.457797\pi$
$2$	0.528791			0.264396			0.132198	+	0.991223i	$0.457797\pi$
$3$		0			0
$3$		0			0
$4$	−3.72038			−0.930095
$5$	−3.86156	−	3.17622i	−0.772312	−	0.635243i
$5$	−3.86156	−	3.17622i	−0.772312	−	0.635243i
$6$		0			0
$7$		−	2.76658i		−	0.395226i	−0.980280	−	0.197613i	$-0.936681\pi$
$7$		−	2.76658i		−	0.395226i	0.980280	−	0.197613i	$-0.0633189\pi$
$8$	−4.08247			−0.510309
$9$		0			0
$10$	−2.04196	−	1.67955i	−0.204196	−	0.167955i
$11$			9.22820i			0.838928i	0.907772	+	0.419464i	$0.137782\pi$
$11$			9.22820i			0.838928i	−0.907772	+	0.419464i	$0.862218\pi$
$12$		0			0
$13$			13.5883i			1.04526i	0.852561	+	0.522628i	$0.175048\pi$
$13$			13.5883i			1.04526i	−0.852561	+	0.522628i	$0.824952\pi$
$14$		−	1.46294i		−	0.104496i
$15$		0			0
$16$	12.7227			0.795172
$17$	12.2161			0.718595			0.359297	−	0.933223i	$-0.383016\pi$
$17$	12.2161			0.718595			0.359297	+	0.933223i	$0.383016\pi$
$18$		0			0
$19$	20.2664			1.06665			0.533327	−	0.845909i	$-0.320941\pi$
$19$	20.2664			1.06665			0.533327	+	0.845909i	$0.320941\pi$
$20$	14.3665	+	11.8167i	0.718324	+	0.590836i
$21$		0			0
$22$			4.87979i			0.221809i
$23$	−2.37127			−0.103099			−0.0515494	−	0.998670i	$-0.516416\pi$
$23$	−2.37127			−0.103099			−0.0515494	+	0.998670i	$0.516416\pi$
$24$		0			0
$25$	4.82331	+	24.5303i	0.192933	+	0.981212i
$26$			7.18539i			0.276361i
$27$		0			0
$28$			10.2927i			0.367598i
$29$		−	34.9123i		−	1.20387i	−0.798544	−	0.601936i	$-0.794396\pi$
$29$		−	34.9123i		−	1.20387i	0.798544	−	0.601936i	$-0.205604\pi$
$30$		0			0
$31$	29.4466			0.949890			0.474945	−	0.880015i	$-0.342468\pi$
$31$	29.4466			0.949890			0.474945	+	0.880015i	$0.342468\pi$
$32$	23.0576			0.720549
$33$		0			0
$34$	6.45977			0.189993
$35$	−8.78726	+	10.6833i	−0.251065	+	0.305238i
$36$		0			0
$37$			64.3630i			1.73954i	0.493457	+	0.869770i	$0.335733\pi$
$37$			64.3630i			1.73954i	−0.493457	+	0.869770i	$0.664267\pi$
$38$	10.7167			0.282019
$39$		0			0
$40$	15.7647	+	12.9668i	0.394118	+	0.324170i
$41$			39.8597i			0.972188i	0.873906	+	0.486094i	$0.161579\pi$
$41$			39.8597i			0.972188i	−0.873906	+	0.486094i	$0.838421\pi$
$42$		0			0
$43$			67.5964i			1.57201i	0.618221	+	0.786004i	$0.287854\pi$
$43$			67.5964i			1.57201i	−0.618221	+	0.786004i	$0.712146\pi$
$44$		−	34.3324i		−	0.780282i
$45$		0			0
$46$	−1.25391			−0.0272588
$47$	−93.3802			−1.98681			−0.993406	−	0.114649i	$-0.963426\pi$
$47$	−93.3802			−1.98681			−0.993406	+	0.114649i	$0.963426\pi$
$48$		0			0
$49$	41.3460			0.843796
$50$	2.55053	+	12.9714i	0.0510105	+	0.259428i
$51$		0			0
$52$		−	50.5538i		−	0.972188i
$53$	−9.82656			−0.185407			−0.0927034	−	0.995694i	$-0.529551\pi$
$53$	−9.82656			−0.185407			−0.0927034	+	0.995694i	$0.529551\pi$
$54$		0			0
$55$	29.3108	−	35.6353i	0.532923	−	0.647914i
$56$			11.2945i			0.201687i
$57$		0			0
$58$		−	18.4613i		−	0.318299i
$59$		−	58.5035i		−	0.991584i	−0.868441	−	0.495792i	$-0.834878\pi$
$59$		−	58.5035i		−	0.991584i	0.868441	−	0.495792i	$-0.165122\pi$
$60$		0			0
$61$	−15.5057			−0.254191			−0.127096	−	0.991890i	$-0.540566\pi$
$61$	−15.5057			−0.254191			−0.127096	+	0.991890i	$0.540566\pi$
$62$	15.5711			0.251147
$63$		0			0
$64$	−38.6984			−0.604662
$65$	43.1595	−	52.4722i	0.663992	−	0.807264i
$66$		0			0
$67$			15.5649i			0.232311i	0.993231	+	0.116156i	$0.0370572\pi$
$67$			15.5649i			0.232311i	−0.993231	+	0.116156i	$0.962943\pi$
$68$	−45.4486			−0.668361
$69$		0			0
$70$	−4.64663	+	5.64925i	−0.0663804	+	0.0807036i
$71$			53.1970i			0.749254i	0.927176	+	0.374627i	$0.122229\pi$
$71$			53.1970i			0.749254i	−0.927176	+	0.374627i	$0.877771\pi$
$72$		0			0
$73$		−	23.6547i		−	0.324037i	−0.986788	−	0.162019i	$-0.948200\pi$
$73$		−	23.6547i		−	0.324037i	0.986788	−	0.162019i	$-0.0518005\pi$
$74$			34.0346i			0.459927i
$75$		0			0
$76$	−75.3988			−0.992090
$77$	25.5306			0.331566
$78$		0			0
$79$	34.5385			0.437196			0.218598	−	0.975815i	$-0.429852\pi$
$79$	34.5385			0.437196			0.218598	+	0.975815i	$0.429852\pi$
$80$	−49.1297	−	40.4102i	−0.614121	−	0.505127i
$81$		0			0
$82$			21.0775i			0.257042i
$83$	75.3460			0.907783			0.453892	−	0.891057i	$-0.350035\pi$
$83$	75.3460			0.907783			0.453892	+	0.891057i	$0.350035\pi$
$84$		0			0
$85$	−47.1733	−	38.8010i	−0.554980	−	0.456482i
$86$			35.7444i			0.415632i
$87$		0			0
$88$		−	37.6739i		−	0.428112i
$89$		−	29.1344i		−	0.327352i	−0.986514	−	0.163676i	$-0.947665\pi$
$89$		−	29.1344i		−	0.327352i	0.986514	−	0.163676i	$-0.0523352\pi$
$90$		0			0
$91$	37.5932			0.413112
$92$	8.82203			0.0958916
$93$		0			0
$94$	−49.3786			−0.525304
$95$	−78.2601	−	64.3706i	−0.823790	−	0.677585i
$96$		0			0
$97$		−	62.3713i		−	0.643003i	−0.946909	−	0.321502i	$-0.895812\pi$
$97$		−	62.3713i		−	0.643003i	0.946909	−	0.321502i	$-0.104188\pi$
$98$	21.8634			0.223096
$99$		0			0
$100$	−17.9446	−	91.2620i	−0.179446	−	0.912620i
$101$			137.102i			1.35745i	0.734394	+	0.678723i	$0.237466\pi$
$101$			137.102i			1.35745i	−0.734394	+	0.678723i	$0.762534\pi$
$102$		0			0
$103$			108.803i			1.05634i	0.849137	+	0.528172i	$0.177122\pi$
$103$			108.803i			1.05634i	−0.849137	+	0.528172i	$0.822878\pi$
$104$		−	55.4739i		−	0.533403i
$105$		0			0
$106$	−5.19620			−0.0490207
$107$	64.0002			0.598133			0.299067	−	0.954232i	$-0.403325\pi$
$107$	64.0002			0.598133			0.299067	+	0.954232i	$0.403325\pi$
$108$		0			0
$109$	−14.8135			−0.135904			−0.0679520	−	0.997689i	$-0.521646\pi$
$109$	−14.8135			−0.135904			−0.0679520	+	0.997689i	$0.521646\pi$
$110$	15.4993	−	18.8436i	0.140902	−	0.171306i
$111$		0			0
$112$		−	35.1985i		−	0.314272i
$113$	82.8066			0.732802			0.366401	−	0.930457i	$-0.380590\pi$
$113$	82.8066			0.732802			0.366401	+	0.930457i	$0.380590\pi$
$114$		0			0
$115$	9.15681	+	7.53166i	0.0796244	+	0.0654927i
$116$			129.887i			1.11972i
$117$		0			0
$118$		−	30.9361i		−	0.262170i
$119$		−	33.7969i		−	0.284007i
$120$		0			0
$121$	35.8402			0.296200
$122$	−8.19926			−0.0672070
$123$		0			0
$124$	−109.553			−0.883488
$125$	59.2880	−	110.045i	0.474304	−	0.880361i
$126$		0			0
$127$		−	36.7291i		−	0.289206i	−0.989490	−	0.144603i	$-0.953810\pi$
$127$		−	36.7291i		−	0.289206i	0.989490	−	0.144603i	$-0.0461904\pi$
$128$	−112.694			−0.880418
$129$		0			0
$130$	22.8223	−	27.7468i	0.175557	−	0.213437i
$131$			57.9363i			0.442262i	0.975244	+	0.221131i	$0.0709748\pi$
$131$			57.9363i			0.442262i	−0.975244	+	0.221131i	$0.929025\pi$
$132$		0			0
$133$		−	56.0688i		−	0.421570i
$134$			8.23056i			0.0614221i
$135$		0			0
$136$	−49.8719			−0.366705
$137$	57.5791			0.420285			0.210143	−	0.977671i	$-0.432607\pi$
$137$	57.5791			0.420285			0.210143	+	0.977671i	$0.432607\pi$
$138$		0			0
$139$	−60.6491			−0.436325			−0.218162	−	0.975913i	$-0.570006\pi$
$139$	−60.6491			−0.436325			−0.218162	+	0.975913i	$0.570006\pi$
$140$	32.6919	−	39.7460i	0.233514	−	0.283900i
$141$		0			0
$142$			28.1301i			0.198099i
$143$	−125.396			−0.876894
$144$		0			0
$145$	−110.889	+	134.816i	−0.764751	+	0.929765i
$146$		−	12.5084i		−	0.0856740i
$147$		0			0
$148$		−	239.455i		−	1.61794i
$149$			235.682i			1.58176i	0.611973	+	0.790879i	$0.290376\pi$
$149$			235.682i			1.58176i	−0.611973	+	0.790879i	$0.709624\pi$
$150$		0			0
$151$	−14.8368			−0.0982569			−0.0491285	−	0.998792i	$-0.515644\pi$
$151$	−14.8368			−0.0982569			−0.0491285	+	0.998792i	$0.515644\pi$
$152$	−82.7371			−0.544323
$153$		0			0
$154$	13.5003			0.0876646
$155$	−113.710	−	93.5287i	−0.733612	−	0.603411i
$156$		0			0
$157$		−	147.208i		−	0.937632i	−0.883296	−	0.468816i	$-0.844681\pi$
$157$		−	147.208i		−	0.937632i	0.883296	−	0.468816i	$-0.155319\pi$
$158$	18.2636			0.115593
$159$		0			0
$160$	−89.0382	−	73.2358i	−0.556488	−	0.457723i
$161$			6.56031i			0.0407473i
$162$		0			0
$163$			9.62130i			0.0590264i	0.999564	+	0.0295132i	$0.00939571\pi$
$163$			9.62130i			0.0590264i	−0.999564	+	0.0295132i	$0.990604\pi$
$164$		−	148.293i		−	0.904227i
$165$		0			0
$166$	39.8423			0.240014
$167$	−83.0786			−0.497477			−0.248738	−	0.968571i	$-0.580016\pi$
$167$	−83.0786			−0.497477			−0.248738	+	0.968571i	$0.580016\pi$
$168$		0			0
$169$	−15.6427			−0.0925605
$170$	−24.9448	−	20.5176i	−0.146734	−	0.120692i
$171$		0			0
$172$		−	251.484i		−	1.46212i
$173$	0.439157			0.00253848			0.00126924	−	0.999999i	$-0.499596\pi$
$173$	0.439157			0.00253848			0.00126924	+	0.999999i	$0.499596\pi$
$174$		0			0
$175$	67.8651	−	13.3441i	0.387801	−	0.0762519i
$176$			117.408i			0.667091i
$177$		0			0
$178$		−	15.4060i		−	0.0865505i
$179$			102.699i			0.573740i	0.957970	+	0.286870i	$0.0926148\pi$
$179$			102.699i			0.573740i	−0.957970	+	0.286870i	$0.907385\pi$
$180$		0			0
$181$	−120.426			−0.665337			−0.332669	−	0.943044i	$-0.607949\pi$
$181$	−120.426			−0.665337			−0.332669	+	0.943044i	$0.607949\pi$
$182$	19.8790			0.109225
$183$		0			0
$184$	9.68064			0.0526122
$185$	204.431	−	248.542i	1.10503	−	1.34347i
$186$		0			0
$187$			112.733i			0.602849i
$188$	347.410			1.84792
$189$		0			0
$190$	−41.3833	−	34.0386i	−0.217807	−	0.179150i
$191$			146.365i			0.766307i	0.923685	+	0.383153i	$0.125162\pi$
$191$			146.365i			0.766307i	−0.923685	+	0.383153i	$0.874838\pi$
$192$		0			0
$193$			130.403i			0.675663i	0.941207	+	0.337831i	$0.109693\pi$
$193$			130.403i			0.675663i	−0.941207	+	0.337831i	$0.890307\pi$
$194$		−	32.9814i		−	0.170007i
$195$		0			0
$196$	−153.823			−0.784811
$197$	−186.652			−0.947470			−0.473735	−	0.880668i	$-0.657094\pi$
$197$	−186.652			−0.947470			−0.473735	+	0.880668i	$0.657094\pi$
$198$		0			0
$199$	45.1917			0.227094			0.113547	−	0.993533i	$-0.463779\pi$
$199$	45.1917			0.227094			0.113547	+	0.993533i	$0.463779\pi$
$200$	−19.6910	−	100.144i	−0.0984551	−	0.500721i
$201$		0			0
$202$			72.4984i			0.358903i
$203$	−96.5877			−0.475802
$204$		0			0
$205$	126.603	−	153.921i	0.617576	−	0.750833i
$206$			57.5343i			0.279293i
$207$		0			0
$208$			172.881i			0.831158i
$209$			187.023i			0.894846i
$210$		0			0
$211$	−138.211			−0.655027			−0.327514	−	0.944846i	$-0.606211\pi$
$211$	−138.211			−0.655027			−0.327514	+	0.944846i	$0.606211\pi$
$212$	36.5585			0.172446
$213$		0			0
$214$	33.8428			0.158144
$215$	214.701	−	261.028i	0.998608	−	1.21408i
$216$		0			0
$217$		−	81.4664i		−	0.375421i
$218$	−7.83327			−0.0359324
$219$		0			0
$220$	−109.047	+	132.577i	−0.495669	+	0.602622i
$221$			165.997i			0.751116i
$222$		0			0
$223$			58.2052i			0.261010i	0.991448	+	0.130505i	$0.0416598\pi$
$223$			58.2052i			0.261010i	−0.991448	+	0.130505i	$0.958340\pi$
$224$		−	63.7906i		−	0.284780i
$225$		0			0
$226$	43.7874			0.193750
$227$	374.581			1.65014			0.825069	−	0.565032i	$-0.191136\pi$
$227$	374.581			1.65014			0.825069	+	0.565032i	$0.191136\pi$
$228$		0			0
$229$	67.6836			0.295562			0.147781	−	0.989020i	$-0.452787\pi$
$229$	67.6836			0.295562			0.147781	+	0.989020i	$0.452787\pi$
$230$	4.84204	+	3.98268i	0.0210523	+	0.0173160i
$231$		0			0
$232$			142.528i			0.614346i
$233$	−282.378			−1.21192			−0.605961	−	0.795495i	$-0.707211\pi$
$233$	−282.378			−1.21192			−0.605961	+	0.795495i	$0.707211\pi$
$234$		0			0
$235$	360.593	+	296.596i	1.53444	+	1.26211i
$236$			217.655i			0.922267i
$237$		0			0
$238$		−	17.8715i		−	0.0750903i
$239$		−	368.390i		−	1.54138i	−0.637209	−	0.770691i	$-0.719911\pi$
$239$		−	368.390i		−	1.54138i	0.637209	−	0.770691i	$-0.280089\pi$
$240$		0			0
$241$	371.108			1.53987			0.769934	−	0.638123i	$-0.220289\pi$
$241$	371.108			1.53987			0.769934	+	0.638123i	$0.220289\pi$
$242$	18.9520			0.0783141
$243$		0			0
$244$	57.6870			0.236422
$245$	−159.660	−	131.324i	−0.651674	−	0.536016i
$246$		0			0
$247$			275.387i			1.11493i
$248$	−120.215			−0.484737
$249$		0			0
$250$	31.3510	−	58.1909i	0.125404	−	0.232764i
$251$			306.449i			1.22091i	0.792049	+	0.610457i	$0.209014\pi$
$251$			306.449i			1.22091i	−0.792049	+	0.610457i	$0.790986\pi$
$252$		0			0
$253$		−	21.8826i		−	0.0864924i
$254$		−	19.4220i		−	0.0764647i
$255$		0			0
$256$	95.2020			0.371883
$257$	−305.948			−1.19046			−0.595230	−	0.803555i	$-0.702939\pi$
$257$	−305.948			−1.19046			−0.595230	+	0.803555i	$0.702939\pi$
$258$		0			0
$259$	178.066			0.687512
$260$	−160.570	+	195.216i	−0.617575	+	0.750832i
$261$		0			0
$262$			30.6362i			0.116932i
$263$	−146.685			−0.557739			−0.278869	−	0.960329i	$-0.589960\pi$
$263$	−146.685			−0.557739			−0.278869	+	0.960329i	$0.589960\pi$
$264$		0			0
$265$	37.9459	+	31.2113i	0.143192	+	0.117778i
$266$		−	29.6487i		−	0.111461i
$267$		0			0
$268$		−	57.9072i		−	0.216072i
$269$		−	276.133i		−	1.02652i	−0.858234	−	0.513258i	$-0.828438\pi$
$269$		−	276.133i		−	1.02652i	0.858234	−	0.513258i	$-0.171562\pi$
$270$		0			0
$271$	304.822			1.12480			0.562402	−	0.826864i	$-0.309877\pi$
$271$	304.822			1.12480			0.562402	+	0.826864i	$0.309877\pi$
$272$	155.422			0.571406
$273$		0			0
$274$	30.4473			0.111122
$275$	−226.371	+	44.5105i	−0.823166	+	0.161856i
$276$		0			0
$277$			410.560i			1.48217i	0.671414	+	0.741083i	$0.265687\pi$
$277$			410.560i			1.48217i	−0.671414	+	0.741083i	$0.734313\pi$
$278$	−32.0707			−0.115362
$279$		0			0
$280$	35.8737	−	43.6144i	0.128120	−	0.155766i
$281$		−	103.834i		−	0.369517i	−0.982784	−	0.184758i	$-0.940850\pi$
$281$		−	103.834i		−	0.369517i	0.982784	−	0.184758i	$-0.0591502\pi$
$282$		0			0
$283$			281.472i			0.994602i	0.867578	+	0.497301i	$0.165676\pi$
$283$			281.472i			0.994602i	−0.867578	+	0.497301i	$0.834324\pi$
$284$		−	197.913i		−	0.696877i
$285$		0			0
$286$	−66.3082			−0.231847
$287$	110.275			0.384234
$288$		0			0
$289$	−139.767			−0.483621
$290$	−58.6371	+	71.2895i	−0.202197	+	0.245826i
$291$		0			0
$292$			88.0045i			0.301385i
$293$	408.669			1.39477			0.697387	−	0.716695i	$-0.254346\pi$
$293$	408.669			1.39477			0.697387	+	0.716695i	$0.254346\pi$
$294$		0			0
$295$	−185.820	+	225.915i	−0.629897	+	0.765813i
$296$		−	262.760i		−	0.887703i
$297$		0			0
$298$			124.626i			0.418210i
$299$		−	32.2216i		−	0.107765i
$300$		0			0
$301$	187.011			0.621299
$302$	−7.84557			−0.0259787
$303$		0			0
$304$	257.845			0.848173
$305$	59.8761	+	49.2493i	0.196315	+	0.161473i
$306$		0			0
$307$			174.133i			0.567208i	0.958941	+	0.283604i	$0.0915301\pi$
$307$			174.133i			0.567208i	−0.958941	+	0.283604i	$0.908470\pi$
$308$	−94.9835			−0.308388
$309$		0			0
$310$	−60.1288	−	49.4572i	−0.193964	−	0.159539i
$311$			274.326i			0.882076i	0.897488	+	0.441038i	$0.145390\pi$
$311$			274.326i			0.882076i	−0.897488	+	0.441038i	$0.854610\pi$
$312$		0			0
$313$		−	213.111i		−	0.680866i	−0.940269	−	0.340433i	$-0.889426\pi$
$313$		−	213.111i		−	0.680866i	0.940269	−	0.340433i	$-0.110574\pi$
$314$		−	77.8424i		−	0.247906i
$315$		0			0
$316$	−128.496			−0.406634
$317$	−242.124			−0.763798			−0.381899	−	0.924204i	$-0.624730\pi$
$317$	−242.124			−0.763798			−0.381899	+	0.924204i	$0.624730\pi$
$318$		0			0
$319$	322.178			1.00996
$320$	149.436	+	122.914i	0.466988	+	0.384107i
$321$		0			0
$322$			3.46904i			0.0107734i
$323$	247.577			0.766493
$324$		0			0
$325$	−333.326	+	65.5408i	−1.02562	+	0.201664i
$326$			5.08766i			0.0156063i
$327$		0			0
$328$		−	162.726i		−	0.496116i
$329$			258.344i			0.785240i
$330$		0			0
$331$	−198.929			−0.600995			−0.300498	−	0.953783i	$-0.597153\pi$
$331$	−198.929			−0.600995			−0.300498	+	0.953783i	$0.597153\pi$
$332$	−280.316			−0.844325
$333$		0			0
$334$	−43.9313			−0.131531
$335$	49.4374	−	60.1047i	0.147574	−	0.179417i
$336$		0			0
$337$		−	315.248i		−	0.935455i	−0.883873	−	0.467728i	$-0.845073\pi$
$337$		−	315.248i		−	0.935455i	0.883873	−	0.467728i	$-0.154927\pi$
$338$	−8.27174			−0.0244726
$339$		0			0
$340$	175.502	+	144.354i	0.516184	+	0.424572i
$341$			271.739i			0.796889i
$342$		0			0
$343$		−	249.950i		−	0.728716i
$344$		−	275.960i		−	0.802210i
$345$		0			0
$346$	0.232222			0.000671163
$347$	506.005			1.45823			0.729114	−	0.684392i	$-0.239932\pi$
$347$	506.005			1.45823			0.729114	+	0.684392i	$0.239932\pi$
$348$		0			0
$349$	−561.673			−1.60938			−0.804689	−	0.593696i	$-0.797668\pi$
$349$	−561.673			−1.60938			−0.804689	+	0.593696i	$0.797668\pi$
$350$	35.8865	−	7.05624i	0.102533	−	0.0201607i
$351$		0			0
$352$			212.780i			0.604488i
$353$	241.799			0.684984			0.342492	−	0.939521i	$-0.388729\pi$
$353$	241.799			0.684984			0.342492	+	0.939521i	$0.388729\pi$
$354$		0			0
$355$	168.965	−	205.424i	0.475958	−	0.578658i
$356$			108.391i			0.304469i
$357$		0			0
$358$			54.3066i			0.151694i
$359$			361.904i			1.00809i	0.863678	+	0.504044i	$0.168155\pi$
$359$			361.904i			1.00809i	−0.863678	+	0.504044i	$0.831845\pi$
$360$		0			0
$361$	49.7285			0.137752
$362$	−63.6802			−0.175912
$363$		0			0
$364$	−139.861			−0.384234
$365$	−75.1325	+	91.3441i	−0.205842	+	0.250258i
$366$		0			0
$367$		−	262.812i		−	0.716108i	−0.933701	−	0.358054i	$-0.883440\pi$
$367$		−	262.812i		−	0.716108i	0.933701	−	0.358054i	$-0.116560\pi$
$368$	−30.1691			−0.0819812
$369$		0			0
$370$	108.101	−	131.427i	0.292165	−	0.355207i
$371$			27.1860i			0.0732776i
$372$		0			0
$373$		−	312.640i		−	0.838178i	−0.907945	−	0.419089i	$-0.862350\pi$
$373$		−	312.640i		−	0.838178i	0.907945	−	0.419089i	$-0.137650\pi$
$374$			59.6121i			0.159391i
$375$		0			0
$376$	381.222			1.01389
$377$	474.400			1.25835
$378$		0			0
$379$	256.518			0.676830			0.338415	−	0.940997i	$-0.390109\pi$
$379$	256.518			0.676830			0.338415	+	0.940997i	$0.390109\pi$
$380$	291.157	+	239.483i	0.766203	+	0.630218i
$381$		0			0
$382$			77.3963i			0.202608i
$383$	−38.8290			−0.101381			−0.0506906	−	0.998714i	$-0.516142\pi$
$383$	−38.8290			−0.101381			−0.0506906	+	0.998714i	$0.516142\pi$
$384$		0			0
$385$	−98.5879	−	81.0906i	−0.256073	−	0.210625i
$386$			68.9559i			0.178642i
$387$		0			0
$388$			232.045i			0.598054i
$389$		−	307.000i		−	0.789203i	−0.918852	−	0.394601i	$-0.870883\pi$
$389$		−	307.000i		−	0.789203i	0.918852	−	0.394601i	$-0.129117\pi$
$390$		0			0
$391$	−28.9677			−0.0740862
$392$	−168.794			−0.430597
$393$		0			0
$394$	−98.6997			−0.250507
$395$	−133.372	−	109.702i	−0.337652	−	0.277726i
$396$		0			0
$397$		−	47.0129i		−	0.118420i	−0.998246	−	0.0592102i	$-0.981142\pi$
$397$		−	47.0129i		−	0.118420i	0.998246	−	0.0592102i	$-0.0188582\pi$
$398$	23.8970			0.0600426
$399$		0			0
$400$	61.3658	+	312.093i	0.153414	+	0.780232i
$401$		−	568.644i		−	1.41806i	−0.705176	−	0.709032i	$-0.749132\pi$
$401$		−	568.644i		−	1.41806i	0.705176	−	0.709032i	$-0.250868\pi$
$402$		0			0
$403$			400.130i			0.992878i
$404$		−	510.072i		−	1.26255i
$405$		0			0
$406$	−51.0747			−0.125800
$407$	−593.955			−1.45935
$408$		0			0
$409$	−562.140			−1.37443			−0.687213	−	0.726456i	$-0.741166\pi$
$409$	−562.140			−1.37443			−0.687213	+	0.726456i	$0.741166\pi$
$410$	66.9466	−	81.3919i	0.163284	−	0.198517i
$411$		0			0
$412$		−	404.790i		−	0.982501i
$413$	−161.855			−0.391900
$414$		0			0
$415$	−290.953	−	239.315i	−0.701092	−	0.576663i
$416$			313.314i			0.753158i
$417$		0			0
$418$			98.8960i			0.236593i
$419$			208.057i			0.496557i	0.968689	+	0.248279i	$0.0798648\pi$
$419$			208.057i			0.496557i	−0.968689	+	0.248279i	$0.920135\pi$
$420$		0			0
$421$	−394.471			−0.936987			−0.468493	−	0.883467i	$-0.655203\pi$
$421$	−394.471			−0.936987			−0.468493	+	0.883467i	$0.655203\pi$
$422$	−73.0846			−0.173186
$423$		0			0
$424$	40.1166			0.0946147
$425$	58.9221	+	299.665i	0.138640	+	0.705094i
$426$		0			0
$427$			42.8977i			0.100463i
$428$	−238.105			−0.556321
$429$		0			0
$430$	113.532	−	138.029i	0.264027	−	0.320998i
$431$		−	216.515i		−	0.502355i	−0.967941	−	0.251177i	$-0.919182\pi$
$431$		−	216.515i		−	0.502355i	0.967941	−	0.251177i	$-0.0808177\pi$
$432$		0			0
$433$		−	614.024i		−	1.41807i	−0.705173	−	0.709035i	$-0.749131\pi$
$433$		−	614.024i		−	1.41807i	0.705173	−	0.709035i	$-0.250869\pi$
$434$		−	43.0787i		−	0.0992597i
$435$		0			0
$436$	55.1120			0.126404
$437$	−48.0572			−0.109971
$438$		0			0
$439$	519.701			1.18383			0.591915	−	0.806000i	$-0.298372\pi$
$439$	519.701			1.18383			0.591915	+	0.806000i	$0.298372\pi$
$440$	−119.660	+	145.480i	−0.271955	+	0.330636i
$441$		0			0
$442$			87.7775i			0.198592i
$443$	501.649			1.13239			0.566196	−	0.824271i	$-0.308415\pi$
$443$	501.649			1.13239			0.566196	+	0.824271i	$0.308415\pi$
$444$		0			0
$445$	−92.5370	+	112.504i	−0.207948	+	0.252818i
$446$			30.7784i			0.0690099i
$447$		0			0
$448$			107.062i			0.238978i
$449$			166.389i			0.370576i	0.982684	+	0.185288i	$0.0593218\pi$
$449$			166.389i			0.370576i	−0.982684	+	0.185288i	$0.940678\pi$
$450$		0			0
$451$	−367.834			−0.815596
$452$	−308.072			−0.681575
$453$		0			0
$454$	198.075			0.436289
$455$	−145.169	−	119.404i	−0.319052	−	0.262427i
$456$		0			0
$457$		−	832.042i		−	1.82066i	−0.413883	−	0.910330i	$-0.635828\pi$
$457$		−	832.042i		−	1.82066i	0.413883	−	0.910330i	$-0.364172\pi$
$458$	35.7905			0.0781452
$459$		0			0
$460$	−34.0668	−	28.0207i	−0.0740582	−	0.0609145i
$461$		−	434.267i		−	0.942012i	−0.882130	−	0.471006i	$-0.843891\pi$
$461$		−	434.267i		−	0.942012i	0.882130	−	0.471006i	$-0.156109\pi$
$462$		0			0
$463$		−	174.053i		−	0.375924i	−0.982176	−	0.187962i	$-0.939812\pi$
$463$		−	174.053i		−	0.375924i	0.982176	−	0.187962i	$-0.0601882\pi$
$464$		−	444.180i		−	0.957285i
$465$		0			0
$466$	−149.319			−0.320427
$467$	−134.443			−0.287886			−0.143943	−	0.989586i	$-0.545978\pi$
$467$	−134.443			−0.287886			−0.143943	+	0.989586i	$0.545978\pi$
$468$		0			0
$469$	43.0615			0.0918155
$470$	190.679	+	156.837i	0.405699	+	0.333696i
$471$		0			0
$472$			238.839i			0.506014i
$473$	−623.793			−1.31880
$474$		0			0
$475$	97.7514	+	497.142i	0.205792	+	1.04661i
$476$			125.737i			0.264154i
$477$		0			0
$478$		−	194.802i		−	0.407535i
$479$			712.567i			1.48761i	0.668395	+	0.743806i	$0.266982\pi$
$479$			712.567i			1.48761i	−0.668395	+	0.743806i	$0.733018\pi$
$480$		0			0
$481$	−874.586			−1.81827
$482$	196.239			0.407134
$483$		0			0
$484$	−133.339			−0.275494
$485$	−198.105	+	240.851i	−0.408463	+	0.496599i
$486$		0			0
$487$			208.661i			0.428462i	0.976783	+	0.214231i	$0.0687246\pi$
$487$			208.661i			0.428462i	−0.976783	+	0.214231i	$0.931275\pi$
$488$	63.3014			0.129716
$489$		0			0
$490$	−84.4269	−	69.4429i	−0.172300	−	0.141720i
$491$		−	411.573i		−	0.838233i	−0.907932	−	0.419117i	$-0.862340\pi$
$491$		−	411.573i		−	0.838233i	0.907932	−	0.419117i	$-0.137660\pi$
$492$		0			0
$493$		−	426.493i		−	0.865096i
$494$			145.622i			0.294782i
$495$		0			0
$496$	374.641			0.755326
$497$	147.174			0.296125
$498$		0			0
$499$	−334.598			−0.670537			−0.335269	−	0.942123i	$-0.608827\pi$
$499$	−334.598			−0.670537			−0.335269	+	0.942123i	$0.608827\pi$
$500$	−220.574	+	409.410i	−0.441148	+	0.818819i
$501$		0			0
$502$			162.048i			0.322804i
$503$	−85.4624			−0.169905			−0.0849527	−	0.996385i	$-0.527074\pi$
$503$	−85.4624			−0.169905			−0.0849527	+	0.996385i	$0.527074\pi$
$504$		0			0
$505$	435.466	−	529.428i	0.862309	−	1.04837i
$506$		−	11.5713i		−	0.0228682i
$507$		0			0
$508$			136.646i			0.268989i
$509$			164.144i			0.322483i	0.986915	+	0.161241i	$0.0515498\pi$
$509$			164.144i			0.322483i	−0.986915	+	0.161241i	$0.948450\pi$
$510$		0			0
$511$	−65.4427			−0.128068
$512$	501.116			0.978743
$513$		0			0
$514$	−161.783			−0.314752
$515$	345.583	−	420.151i	0.671035	−	0.815828i
$516$		0			0
$517$		−	861.731i		−	1.66679i
$518$	94.1595			0.181775
$519$		0			0
$520$	−176.197	+	214.216i	−0.338841	+	0.411954i
$521$		−	954.386i		−	1.83184i	−0.401366	−	0.915918i	$-0.631464\pi$
$521$		−	954.386i		−	1.83184i	0.401366	−	0.915918i	$-0.368536\pi$
$522$		0			0
$523$			53.5836i			0.102454i	0.998687	+	0.0512271i	$0.0163132\pi$
$523$			53.5836i			0.102454i	−0.998687	+	0.0512271i	$0.983687\pi$
$524$		−	215.545i		−	0.411346i
$525$		0			0
$526$	−77.5659			−0.147464
$527$	359.723			0.682586
$528$		0			0
$529$	−523.377			−0.989371
$530$	20.0654	+	16.5042i	0.0378593	+	0.0311401i
$531$		0			0
$532$			208.597i			0.392100i
$533$	−541.627			−1.01619
$534$		0			0
$535$	−247.141	−	203.279i	−0.461945	−	0.379960i
$536$		−	63.5431i		−	0.118551i
$537$		0			0
$538$		−	146.017i		−	0.271406i
$539$			381.550i			0.707884i
$540$		0			0
$541$	−502.886			−0.929549			−0.464774	−	0.885429i	$-0.653865\pi$
$541$	−502.886			−0.929549			−0.464774	+	0.885429i	$0.653865\pi$
$542$	161.187			0.297393
$543$		0			0
$544$	281.674			0.517782
$545$	57.2034	+	47.0510i	0.104960	+	0.0863321i
$546$		0			0
$547$		−	730.024i		−	1.33460i	−0.744791	−	0.667298i	$-0.767451\pi$
$547$		−	730.024i		−	1.33460i	0.744791	−	0.667298i	$-0.232549\pi$
$548$	−214.216			−0.390905
$549$		0			0
$550$	−119.703	+	23.5368i	−0.217641	+	0.0427941i
$551$		−	707.548i		−	1.28412i
$552$		0			0
$553$		−	95.5535i		−	0.172791i
$554$			217.100i			0.391878i
$555$		0			0
$556$	225.638			0.405823
$557$	419.491			0.753125			0.376562	−	0.926391i	$-0.377106\pi$
$557$	419.491			0.753125			0.376562	+	0.926391i	$0.377106\pi$
$558$		0			0
$559$	−918.522			−1.64315
$560$	−111.798	+	135.921i	−0.199639	+	0.242717i
$561$		0			0
$562$		−	54.9066i		−	0.0976985i
$563$	635.844			1.12939			0.564693	−	0.825301i	$-0.308995\pi$
$563$	635.844			1.12939			0.564693	+	0.825301i	$0.308995\pi$
$564$		0			0
$565$	−319.763	−	263.012i	−0.565952	−	0.465507i
$566$			148.840i			0.262968i
$567$		0			0
$568$		−	217.175i		−	0.382351i
$569$			885.934i			1.55700i	0.627644	+	0.778501i	$0.284020\pi$
$569$			885.934i			1.55700i	−0.627644	+	0.778501i	$0.715980\pi$
$570$		0			0
$571$	926.192			1.62205			0.811026	−	0.585010i	$-0.198910\pi$
$571$	926.192			1.62205			0.811026	+	0.585010i	$0.198910\pi$
$572$	466.520			0.815595
$573$		0			0
$574$	58.3125			0.101590
$575$	−11.4374	−	58.1680i	−0.0198911	−	0.101162i
$576$		0			0
$577$		−	261.287i		−	0.452837i	−0.974030	−	0.226418i	$-0.927298\pi$
$577$		−	261.287i		−	0.452837i	0.974030	−	0.226418i	$-0.0727016\pi$
$578$	−73.9073			−0.127867
$579$		0			0
$580$	412.549	−	501.567i	0.711292	−	0.864770i
$581$		−	208.451i		−	0.358780i
$582$		0			0
$583$		−	90.6815i		−	0.155543i
$584$			96.5696i			0.165359i
$585$		0			0
$586$	216.100			0.368772
$587$	−603.386			−1.02792			−0.513958	−	0.857816i	$-0.671821\pi$
$587$	−603.386			−1.02792			−0.513958	+	0.857816i	$0.671821\pi$
$588$		0			0
$589$	596.778			1.01320
$590$	−98.2598	+	119.462i	−0.166542	+	0.202477i
$591$		0			0
$592$			818.874i			1.38323i
$593$	−314.000			−0.529511			−0.264756	−	0.964316i	$-0.585291\pi$
$593$	−314.000			−0.529511			−0.264756	+	0.964316i	$0.585291\pi$
$594$		0			0
$595$	−107.346	+	130.509i	−0.180414	+	0.219342i
$596$		−	876.826i		−	1.47118i
$597$		0			0
$598$		−	17.0385i		−	0.0284925i
$599$			753.712i			1.25828i	0.777290	+	0.629142i	$0.216594\pi$
$599$			753.712i			1.25828i	−0.777290	+	0.629142i	$0.783406\pi$
$600$		0			0
$601$	−337.952			−0.562315			−0.281158	−	0.959662i	$-0.590718\pi$
$601$	−337.952			−0.562315			−0.281158	+	0.959662i	$0.590718\pi$
$602$	98.8897			0.164269
$603$		0			0
$604$	55.1985			0.0913883
$605$	−138.399	−	113.836i	−0.228759	−	0.188159i
$606$		0			0
$607$			441.493i			0.727337i	0.931529	+	0.363668i	$0.118476\pi$
$607$			441.493i			0.727337i	−0.931529	+	0.363668i	$0.881524\pi$
$608$	467.294			0.768576
$609$		0			0
$610$	31.6619	+	26.0426i	0.0519048	+	0.0426928i
$611$		−	1268.88i		−	2.07673i
$612$		0			0
$613$			406.010i			0.662332i	0.943572	+	0.331166i	$0.107442\pi$
$613$			406.010i			0.662332i	−0.943572	+	0.331166i	$0.892558\pi$
$614$			92.0800i			0.149967i
$615$		0			0
$616$	−104.228			−0.169201
$617$	−895.709			−1.45172			−0.725858	−	0.687845i	$-0.758557\pi$
$617$	−895.709			−1.45172			−0.725858	+	0.687845i	$0.758557\pi$
$618$		0			0
$619$	371.622			0.600358			0.300179	−	0.953883i	$-0.402954\pi$
$619$	371.622			0.600358			0.300179	+	0.953883i	$0.402954\pi$
$620$	423.044	+	347.962i	0.682329	+	0.561230i
$621$		0			0
$622$			145.061i			0.233217i
$623$	−80.6026			−0.129378
$624$		0			0
$625$	−578.471	+	236.635i	−0.925554	+	0.378615i
$626$		−	112.691i		−	0.180018i
$627$		0			0
$628$			547.671i			0.872087i
$629$			786.266i			1.25002i
$630$		0			0
$631$	150.820			0.239017			0.119508	−	0.992833i	$-0.461868\pi$
$631$	150.820			0.239017			0.119508	+	0.992833i	$0.461868\pi$
$632$	−141.002			−0.223105
$633$		0			0
$634$	−128.033			−0.201945
$635$	−116.660	+	141.832i	−0.183716	+	0.223357i
$636$		0			0
$637$			561.823i			0.881983i
$638$	170.365			0.267029
$639$		0			0
$640$	435.173	+	357.939i	0.679958	+	0.559280i
$641$		−	203.184i		−	0.316980i	−0.987361	−	0.158490i	$-0.949337\pi$
$641$		−	203.184i		−	0.316980i	0.987361	−	0.158490i	$-0.0506626\pi$
$642$		0			0
$643$		−	49.3272i		−	0.0767142i	−0.999264	−	0.0383571i	$-0.987788\pi$
$643$		−	49.3272i		−	0.0767142i	0.999264	−	0.0383571i	$-0.0122124\pi$
$644$		−	24.4069i		−	0.0378988i
$645$		0			0
$646$	130.917			0.202657
$647$	−1189.39			−1.83832			−0.919159	−	0.393887i	$-0.871130\pi$
$647$	−1189.39			−1.83832			−0.919159	+	0.393887i	$0.871130\pi$
$648$		0			0
$649$	539.882			0.831867
$650$	−176.260	+	34.6574i	−0.271169	+	0.0533191i
$651$		0			0
$652$		−	35.7949i		−	0.0549002i
$653$	−341.212			−0.522530			−0.261265	−	0.965267i	$-0.584140\pi$
$653$	−341.212			−0.522530			−0.261265	+	0.965267i	$0.584140\pi$
$654$		0			0
$655$	184.018	−	223.725i	0.280944	−	0.341564i
$656$			507.125i			0.773056i
$657$		0			0
$658$			136.610i			0.207614i
$659$			372.296i			0.564941i	0.959276	+	0.282471i	$0.0911540\pi$
$659$			372.296i			0.564941i	−0.959276	+	0.282471i	$0.908846\pi$
$660$		0			0
$661$	332.442			0.502938			0.251469	−	0.967865i	$-0.419086\pi$
$661$	332.442			0.502938			0.251469	+	0.967865i	$0.419086\pi$
$662$	−105.192			−0.158900
$663$		0			0
$664$	−307.598			−0.463250
$665$	−178.086	+	216.513i	−0.267799	+	0.325583i
$666$		0			0
$667$			82.7865i			0.124118i
$668$	309.084			0.462701
$669$		0			0
$670$	26.1420	−	31.7828i	0.0390180	−	0.0474371i
$671$		−	143.089i		−	0.213248i
$672$		0			0
$673$			227.346i			0.337809i	0.985632	+	0.168905i	$0.0540230\pi$
$673$			227.346i			0.337809i	−0.985632	+	0.168905i	$0.945977\pi$
$674$		−	166.701i		−	0.247330i
$675$		0			0
$676$	58.1969			0.0860901
$677$	1061.47			1.56790			0.783952	−	0.620821i	$-0.213201\pi$
$677$	1061.47			1.56790			0.783952	+	0.620821i	$0.213201\pi$
$678$		0			0
$679$	−172.555			−0.254132
$680$	192.583	+	158.404i	0.283211	+	0.232947i
$681$		0			0
$682$			143.693i			0.210694i
$683$	633.553			0.927604			0.463802	−	0.885939i	$-0.346485\pi$
$683$	633.553			0.927604			0.463802	+	0.885939i	$0.346485\pi$
$684$		0			0
$685$	−222.345	−	182.884i	−0.324591	−	0.266983i
$686$		−	132.171i		−	0.192669i
$687$		0			0
$688$			860.011i			1.25002i
$689$		−	133.527i		−	0.193798i
$690$		0			0
$691$	1081.80			1.56555			0.782777	−	0.622303i	$-0.213803\pi$
$691$	1081.80			1.56555			0.782777	+	0.622303i	$0.213803\pi$
$692$	−1.63383			−0.00236103
$693$		0			0
$694$	267.571			0.385549
$695$	234.200	+	192.635i	0.336979	+	0.277172i
$696$		0			0
$697$			486.931i			0.698609i
$698$	−297.008			−0.425513
$699$		0			0
$700$	−252.484	+	49.6451i	−0.360691	+	0.0709216i
$701$		−	143.009i		−	0.204007i	−0.994784	−	0.102003i	$-0.967475\pi$
$701$		−	143.009i		−	0.204007i	0.994784	−	0.102003i	$-0.0325253\pi$
$702$		0			0
$703$			1304.41i			1.85549i
$704$		−	357.116i		−	0.507267i
$705$		0			0
$706$	127.861			0.181107
$707$	379.304			0.536498
$708$		0			0
$709$	849.083			1.19758			0.598789	−	0.800907i	$-0.295649\pi$
$709$	849.083			1.19758			0.598789	+	0.800907i	$0.295649\pi$
$710$	89.3473	−	108.626i	0.125841	−	0.152995i
$711$		0			0
$712$			118.940i			0.167051i
$713$	−69.8258			−0.0979324
$714$		0			0
$715$	484.224	+	398.284i	0.677236	+	0.557041i
$716$		−	382.081i		−	0.533633i
$717$		0			0
$718$			191.372i			0.266534i
$719$		−	155.493i		−	0.216263i	−0.994137	−	0.108131i	$-0.965513\pi$
$719$		−	155.493i		−	0.216263i	0.994137	−	0.108131i	$-0.0344867\pi$
$720$		0			0
$721$	301.014			0.417495
$722$	26.2960			0.0364211
$723$		0			0
$724$	448.031			0.618827
$725$	856.409	−	168.393i	1.18125	−	0.232266i
$726$		0			0
$727$			624.427i			0.858909i	0.903089	+	0.429454i	$0.141294\pi$
$727$			624.427i			0.858909i	−0.903089	+	0.429454i	$0.858706\pi$
$728$	−153.473			−0.210815
$729$		0			0
$730$	−39.7294	+	48.3020i	−0.0544238	+	0.0661671i
$731$			825.765i			1.12964i
$732$		0			0
$733$			453.649i			0.618893i	0.950917	+	0.309447i	$0.100144\pi$
$733$			453.649i			0.618893i	−0.950917	+	0.309447i	$0.899856\pi$
$734$		−	138.972i		−	0.189336i
$735$		0			0
$736$	−54.6757			−0.0742876
$737$	−143.636			−0.194892
$738$		0			0
$739$	827.126			1.11925			0.559625	−	0.828746i	$-0.310945\pi$
$739$	827.126			1.11925			0.559625	+	0.828746i	$0.310945\pi$
$740$	−760.560	+	924.669i	−1.02778	+	1.24955i
$741$		0			0
$742$			14.3757i			0.0193743i
$743$	−602.966			−0.811529			−0.405765	−	0.913978i	$-0.632995\pi$
$743$	−602.966			−0.811529			−0.405765	+	0.913978i	$0.632995\pi$
$744$		0			0
$745$	748.576	−	910.100i	1.00480	−	1.22161i
$746$		−	165.321i		−	0.221611i
$747$		0			0
$748$		−	419.409i		−	0.560707i
$749$		−	177.062i		−	0.236398i
$750$		0			0
$751$	−344.963			−0.459338			−0.229669	−	0.973269i	$-0.573764\pi$
$751$	−344.963			−0.459338			−0.229669	+	0.973269i	$0.573764\pi$
$752$	−1188.05			−1.57986
$753$		0			0
$754$	250.858			0.332704
$755$	57.2932	+	47.1248i	0.0758850	+	0.0624170i
$756$		0			0
$757$		−	1248.67i		−	1.64950i	−0.565501	−	0.824748i	$-0.691317\pi$
$757$		−	1248.67i		−	1.64950i	0.565501	−	0.824748i	$-0.308683\pi$
$758$	135.645			0.178951
$759$		0			0
$760$	319.494	+	262.791i	0.420387	+	0.345777i
$761$		−	1108.86i		−	1.45710i	−0.684990	−	0.728552i	$-0.740193\pi$
$761$		−	1108.86i		−	1.45710i	0.684990	−	0.728552i	$-0.259807\pi$
$762$		0			0
$763$			40.9829i			0.0537128i
$764$		−	544.532i		−	0.712738i
$765$		0			0
$766$	−20.5324			−0.0268047
$767$	794.964			1.03646
$768$		0			0
$769$	−1152.33			−1.49848			−0.749239	−	0.662300i	$-0.769581\pi$
$769$	−1152.33			−1.49848			−0.749239	+	0.662300i	$0.769581\pi$
$770$	−52.1324	−	42.8800i	−0.0677045	−	0.0556883i
$771$		0			0
$772$		−	485.148i		−	0.628430i
$773$	1116.83			1.44480			0.722399	−	0.691476i	$-0.243039\pi$
$773$	1116.83			1.44480			0.722399	+	0.691476i	$0.243039\pi$
$774$		0			0
$775$	142.030	+	722.334i	0.183265	+	0.932044i
$776$			254.629i			0.328130i
$777$		0			0
$778$		−	162.339i		−	0.208662i
$779$			807.815i			1.03699i
$780$		0			0
$781$	−490.913			−0.628570
$782$	−15.3179			−0.0195881
$783$		0			0
$784$	526.035			0.670963
$785$	−467.565	+	568.454i	−0.595624	+	0.724145i
$786$		0			0
$787$			1065.19i			1.35349i	0.736219	+	0.676743i	$0.236609\pi$
$787$			1065.19i			1.35349i	−0.736219	+	0.676743i	$0.763391\pi$
$788$	694.415			0.881237
$789$		0			0
$790$	−70.5262	−	58.0093i	−0.0892736	−	0.0734294i
$791$		−	229.091i		−	0.289622i
$792$		0			0
$793$		−	210.696i		−	0.265695i
$794$		−	24.8600i		−	0.0313098i
$795$		0			0
$796$	−168.130			−0.211219
$797$	188.979			0.237113			0.118557	−	0.992947i	$-0.462173\pi$
$797$	188.979			0.237113			0.118557	+	0.992947i	$0.462173\pi$
$798$		0			0
$799$	−1140.74			−1.42771
$800$	111.214	+	565.609i	0.139017	+	0.707011i
$801$		0			0
$802$		−	300.694i		−	0.374930i
$803$	218.291			0.271844
$804$		0			0
$805$	20.8370	−	25.3331i	0.0258844	−	0.0314696i
$806$			211.585i			0.262513i
$807$		0			0
$808$		−	559.715i		−	0.692717i
$809$			1105.36i			1.36633i	0.730266	+	0.683163i	$0.239396\pi$
$809$			1105.36i			1.36633i	−0.730266	+	0.683163i	$0.760604\pi$
$810$		0			0
$811$	4.70248			0.00579838			0.00289919	−	0.999996i	$-0.499077\pi$
$811$	4.70248			0.00579838			0.00289919	+	0.999996i	$0.499077\pi$
$812$	359.343			0.442541
$813$		0			0
$814$	−314.078			−0.385845
$815$	30.5593	−	37.1533i	0.0374961	−	0.0455868i
$816$		0			0
$817$			1369.94i			1.67679i
$818$	−297.255			−0.363392
$819$		0			0
$820$	−471.011	+	572.644i	−0.574404	+	0.698346i
$821$		−	876.894i		−	1.06808i	−0.845459	−	0.534040i	$-0.820673\pi$
$821$		−	876.894i		−	1.06808i	0.845459	−	0.534040i	$-0.179327\pi$
$822$		0			0
$823$		−	1145.71i		−	1.39211i	−0.717986	−	0.696057i	$-0.754936\pi$
$823$		−	1145.71i		−	1.39211i	0.717986	−	0.696057i	$-0.245064\pi$
$824$		−	444.187i		−	0.539062i
$825$		0			0
$826$	−85.5873			−0.103617
$827$	−255.412			−0.308841			−0.154421	−	0.988005i	$-0.549351\pi$
$827$	−255.412			−0.308841			−0.154421	+	0.988005i	$0.549351\pi$
$828$		0			0
$829$	−829.155			−1.00019			−0.500093	−	0.865971i	$-0.666701\pi$
$829$	−829.155			−1.00019			−0.500093	+	0.865971i	$0.666701\pi$
$830$	−153.854	−	126.548i	−0.185366	−	0.152467i
$831$		0			0
$832$		−	525.846i		−	0.632026i
$833$	505.088			0.606348
$834$		0			0
$835$	320.813	+	263.876i	0.384208	+	0.316019i
$836$		−	695.796i		−	0.832292i
$837$		0			0
$838$			110.019i			0.131288i
$839$		−	453.809i		−	0.540893i	−0.962735	−	0.270446i	$-0.912829\pi$
$839$		−	453.809i		−	0.540893i	0.962735	−	0.270446i	$-0.0871713\pi$
$840$		0			0
$841$	−377.868			−0.449308
$842$	−208.593			−0.247735
$843$		0			0
$844$	514.197			0.609238
$845$	60.4054	+	49.6847i	0.0714856	+	0.0587984i
$846$		0			0
$847$		−	99.1550i		−	0.117066i
$848$	−125.021			−0.147430
$849$		0			0
$850$	31.1575	+	158.460i	0.0366559	+	0.186424i
$851$		−	152.622i		−	0.179344i
$852$		0			0
$853$			16.2753i			0.0190800i	0.999954	+	0.00954002i	$0.00303673\pi$
$853$			16.2753i			0.0190800i	−0.999954	+	0.00954002i	$0.996963\pi$
$854$			22.6839i			0.0265620i
$855$		0			0
$856$	−261.279			−0.305232
$857$	1061.70			1.23885			0.619427	−	0.785054i	$-0.287365\pi$
$857$	1061.70			1.23885			0.619427	+	0.785054i	$0.287365\pi$
$858$		0			0
$859$	1623.74			1.89027			0.945133	−	0.326686i	$-0.105932\pi$
$859$	1623.74			1.89027			0.945133	+	0.326686i	$0.105932\pi$
$860$	−798.768	+	971.122i	−0.928800	+	1.12921i
$861$		0			0
$862$		−	114.491i		−	0.132820i
$863$	−687.358			−0.796475			−0.398238	−	0.917282i	$-0.630378\pi$
$863$	−687.358			−0.796475			−0.398238	+	0.917282i	$0.630378\pi$
$864$		0			0
$865$	−1.69583	−	1.39486i	−0.00196050	−	0.00161255i
$866$		−	324.691i		−	0.374932i
$867$		0			0
$868$			303.086i			0.349177i
$869$			318.728i			0.366776i
$870$		0			0
$871$	−211.501			−0.242825
$872$	60.4758			0.0693530
$873$		0			0
$874$	−25.4122			−0.0290758
$875$	−304.449	−	164.025i	−0.347942	−	0.187457i
$876$		0			0
$877$		−	1553.91i		−	1.77185i	−0.463832	−	0.885923i	$-0.653526\pi$
$877$		−	1553.91i		−	1.77185i	0.463832	−	0.885923i	$-0.346474\pi$
$878$	274.813			0.312999
$879$		0			0
$880$	372.913	−	453.379i	0.423765	−	0.515203i
$881$			452.136i			0.513208i	0.966517	+	0.256604i	$0.0826036\pi$
$881$			452.136i			0.513208i	−0.966517	+	0.256604i	$0.917396\pi$
$882$		0			0
$883$			1014.46i			1.14888i	0.818546	+	0.574441i	$0.194781\pi$
$883$			1014.46i			1.14888i	−0.818546	+	0.574441i	$0.805219\pi$
$884$		−	617.570i		−	0.698609i
$885$		0			0
$886$	265.268			0.299399
$887$	208.120			0.234634			0.117317	−	0.993095i	$-0.462571\pi$
$887$	208.120			0.234634			0.117317	+	0.993095i	$0.462571\pi$
$888$		0			0
$889$	−101.614			−0.114302
$890$	−48.9328	+	59.4912i	−0.0549806	+	0.0668441i
$891$		0			0
$892$		−	216.545i		−	0.242764i
$893$	−1892.48			−2.11924
$894$		0			0
$895$	326.196	−	396.580i	0.364464	−	0.443106i
$896$			311.776i			0.347964i
$897$		0			0
$898$			87.9849i			0.0979787i
$899$		−	1028.05i		−	1.14355i
$900$		0			0
$901$	−120.042			−0.133232
$902$	−194.507			−0.215640
$903$		0			0
$904$	−338.055			−0.373955
$905$	465.033	+	382.499i	0.513848	+	0.422651i
$906$		0			0
$907$		−	237.481i		−	0.261832i	−0.991393	−	0.130916i	$-0.958208\pi$
$907$		−	237.481i		−	0.261832i	0.991393	−	0.130916i	$-0.0417918\pi$
$908$	−1393.59			−1.53479
$909$		0			0
$910$	−76.7639	−	63.1399i	−0.0843559	−	0.0693845i
$911$		−	1348.08i		−	1.47979i	−0.672725	−	0.739893i	$-0.734876\pi$
$911$		−	1348.08i		−	1.47979i	0.672725	−	0.739893i	$-0.265124\pi$
$912$		0			0
$913$			695.308i			0.761564i
$914$		−	439.976i		−	0.481375i
$915$		0			0
$916$	−251.809			−0.274900
$917$	160.286			0.174793
$918$		0			0
$919$	−153.887			−0.167450			−0.0837250	−	0.996489i	$-0.526682\pi$
$919$	−153.887			−0.167450			−0.0837250	+	0.996489i	$0.526682\pi$
$920$	−37.3824	−	30.7478i	−0.0406330	−	0.0334215i
$921$		0			0
$922$		−	229.637i		−	0.249064i
$923$	−722.859			−0.783162
$924$		0			0
$925$	−1578.84	+	310.443i	−1.70686	+	0.335614i
$926$		−	92.0376i		−	0.0993926i
$927$		0			0
$928$		−	804.992i		−	0.867448i
$929$			768.443i			0.827172i	0.910465	+	0.413586i	$0.135724\pi$
$929$			768.443i			0.827172i	−0.910465	+	0.413586i	$0.864276\pi$
$930$		0			0
$931$	837.937			0.900039
$932$	1050.55			1.12720
$933$		0			0
$934$	−71.0922			−0.0761159
$935$	358.064	−	435.325i	0.382956	−	0.465588i
$936$		0			0
$937$		−	359.563i		−	0.383738i	−0.981421	−	0.191869i	$-0.938545\pi$
$937$		−	359.563i		−	0.383738i	0.981421	−	0.191869i	$-0.0614549\pi$
$938$	22.7705			0.0242756
$939$		0			0
$940$	−1341.54	−	1103.45i	−1.42717	−	1.17388i
$941$			211.896i			0.225182i	0.993641	+	0.112591i	$0.0359149\pi$
$941$			211.896i			0.225182i	−0.993641	+	0.112591i	$0.964085\pi$
$942$		0			0
$943$		−	94.5182i		−	0.100231i
$944$		−	744.325i		−	0.788479i
$945$		0			0
$946$	−329.856			−0.348685
$947$	−1546.93			−1.63351			−0.816754	−	0.576986i	$-0.804229\pi$
$947$	−1546.93			−1.63351			−0.816754	+	0.576986i	$0.804229\pi$
$948$		0			0
$949$	321.428			0.338702
$950$	51.6901	+	262.884i	0.0544106	+	0.276720i
$951$		0			0
$952$			137.975i			0.144931i
$953$	1453.97			1.52568			0.762838	−	0.646589i	$-0.223805\pi$
$953$	1453.97			1.52568			0.762838	+	0.646589i	$0.223805\pi$
$954$		0			0
$955$	464.885	−	565.196i	0.486791	−	0.591828i
$956$			1370.55i			1.43363i
$957$		0			0
$958$			376.799i			0.393318i
$959$		−	159.297i		−	0.166108i
$960$		0			0
$961$	−93.8983			−0.0977089
$962$	−462.473			−0.480741
$963$		0			0
$964$	−1380.66			−1.43222
$965$	414.188	−	503.559i	0.429210	−	0.521823i
$966$		0			0
$967$			382.597i			0.395654i	0.980237	+	0.197827i	$0.0633884\pi$
$967$			382.597i			0.395654i	−0.980237	+	0.197827i	$0.936612\pi$
$968$	−146.317			−0.151154
$969$		0			0
$970$	−104.756	+	127.360i	−0.107996	+	0.131299i
$971$			1474.96i			1.51901i	0.650503	+	0.759504i	$0.274558\pi$
$971$			1474.96i			1.51901i	−0.650503	+	0.759504i	$0.725442\pi$
$972$		0			0
$973$			167.791i			0.172447i
$974$			110.338i			0.113284i
$975$		0			0
$976$	−197.275			−0.202126
$977$	−1417.81			−1.45119			−0.725593	−	0.688124i	$-0.758435\pi$
$977$	−1417.81			−1.45119			−0.725593	+	0.688124i	$0.758435\pi$
$978$		0			0
$979$	268.858			0.274625
$980$	593.997	+	488.575i	0.606119	+	0.498546i
$981$		0			0
$982$		−	217.636i		−	0.221625i
$983$	−444.320			−0.452004			−0.226002	−	0.974127i	$-0.572566\pi$
$983$	−444.320			−0.452004			−0.226002	+	0.974127i	$0.572566\pi$
$984$		0			0
$985$	720.766	+	592.845i	0.731743	+	0.601874i
$986$		−	225.526i		−	0.228728i
$987$		0			0
$988$		−	1024.54i		−	1.03699i
$989$		−	160.289i		−	0.162072i
$990$		0			0
$991$	1177.77			1.18847			0.594235	−	0.804291i	$-0.297455\pi$
$991$	1177.77			1.18847			0.594235	+	0.804291i	$0.297455\pi$
$992$	678.966			0.684442
$993$		0			0
$994$	77.8243			0.0782940
$995$	−174.511	−	143.539i	−0.175387	−	0.144260i
$996$		0			0
$997$			101.208i			0.101513i	0.998711	+	0.0507563i	$0.0161632\pi$
$997$			101.208i			0.101513i	−0.998711	+	0.0507563i	$0.983837\pi$
$998$	−176.933			−0.177287
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.3.d.a.404.11		20
3.2	odd	2	inner	405.3.d.a.404.10		20
5.4	even	2	inner	405.3.d.a.404.9		20
9.2	odd	6		45.3.h.a.14.6	yes	20
9.4	even	3		45.3.h.a.29.5	yes	20
9.5	odd	6		135.3.h.a.89.6		20
9.7	even	3		135.3.h.a.44.5		20
15.14	odd	2	inner	405.3.d.a.404.12		20
45.2	even	12		225.3.j.e.176.5		20
45.4	even	6		45.3.h.a.29.6	yes	20
45.7	odd	12		675.3.j.e.476.6		20
45.13	odd	12		225.3.j.e.101.6		20
45.14	odd	6		135.3.h.a.89.5		20
45.22	odd	12		225.3.j.e.101.5		20
45.23	even	12		675.3.j.e.251.5		20
45.29	odd	6		45.3.h.a.14.5	✓	20
45.32	even	12		675.3.j.e.251.6		20
45.34	even	6		135.3.h.a.44.6		20
45.38	even	12		225.3.j.e.176.6		20
45.43	odd	12		675.3.j.e.476.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.3.h.a.14.5	✓	20	45.29	odd	6
45.3.h.a.14.6	yes	20	9.2	odd	6
45.3.h.a.29.5	yes	20	9.4	even	3
45.3.h.a.29.6	yes	20	45.4	even	6
135.3.h.a.44.5		20	9.7	even	3
135.3.h.a.44.6		20	45.34	even	6
135.3.h.a.89.5		20	45.14	odd	6
135.3.h.a.89.6		20	9.5	odd	6
225.3.j.e.101.5		20	45.22	odd	12
225.3.j.e.101.6		20	45.13	odd	12
225.3.j.e.176.5		20	45.2	even	12
225.3.j.e.176.6		20	45.38	even	12
405.3.d.a.404.9		20	5.4	even	2	inner
405.3.d.a.404.10		20	3.2	odd	2	inner
405.3.d.a.404.11		20	1.1	even	1	trivial
405.3.d.a.404.12		20	15.14	odd	2	inner
675.3.j.e.251.5		20	45.23	even	12
675.3.j.e.251.6		20	45.32	even	12
675.3.j.e.476.5		20	45.43	odd	12
675.3.j.e.476.6		20	45.7	odd	12

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	405.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.528791 q^{2} -3.72038 q^{4} +(-3.86156 - 3.17622i) q^{5} -2.76658i q^{7} -4.08247 q^{8} +(-2.04196 - 1.67955i) q^{10} +9.22820i q^{11} +13.5883i q^{13} -1.46294i q^{14} +12.7227 q^{16} +12.2161 q^{17} +20.2664 q^{19} +(14.3665 + 11.8167i) q^{20} +4.87979i q^{22} -2.37127 q^{23} +(4.82331 + 24.5303i) q^{25} +7.18539i q^{26} +10.2927i q^{28} -34.9123i q^{29} +29.4466 q^{31} +23.0576 q^{32} +6.45977 q^{34} +(-8.78726 + 10.6833i) q^{35} +64.3630i q^{37} +10.7167 q^{38} +(15.7647 + 12.9668i) q^{40} +39.8597i q^{41} +67.5964i q^{43} -34.3324i q^{44} -1.25391 q^{46} -93.3802 q^{47} +41.3460 q^{49} +(2.55053 + 12.9714i) q^{50} -50.5538i q^{52} -9.82656 q^{53} +(29.3108 - 35.6353i) q^{55} +11.2945i q^{56} -18.4613i q^{58} -58.5035i q^{59} -15.5057 q^{61} +15.5711 q^{62} -38.6984 q^{64} +(43.1595 - 52.4722i) q^{65} +15.5649i q^{67} -45.4486 q^{68} +(-4.64663 + 5.64925i) q^{70} +53.1970i q^{71} -23.6547i q^{73} +34.0346i q^{74} -75.3988 q^{76} +25.5306 q^{77} +34.5385 q^{79} +(-49.1297 - 40.4102i) q^{80} +21.0775i q^{82} +75.3460 q^{83} +(-47.1733 - 38.8010i) q^{85} +35.7444i q^{86} -37.6739i q^{88} -29.1344i q^{89} +37.5932 q^{91} +8.82203 q^{92} -49.3786 q^{94} +(-78.2601 - 64.3706i) q^{95} -62.3713i q^{97} +21.8634 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 36 q^{4} + 4 q^{10} + 52 q^{16} - 8 q^{19} - 4 q^{25} - 56 q^{31} + 8 q^{34} + 68 q^{40} + 116 q^{46} + 80 q^{49} + 36 q^{55} + 100 q^{61} + 140 q^{64} + 108 q^{70} + 192 q^{76} + 256 q^{79} + 148 q^{85}+ \cdots - 436 q^{94}+O(q^{100}) \) 20 * q + 36 * q^4 + 4 * q^10 + 52 * q^16 - 8 * q^19 - 4 * q^25 - 56 * q^31 + 8 * q^34 + 68 * q^40 + 116 * q^46 + 80 * q^49 + 36 * q^55 + 100 * q^61 + 140 * q^64 + 108 * q^70 + 192 * q^76 + 256 * q^79 + 148 * q^85 - 288 * q^91 - 436 * q^94

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