LMFDB - Embedded newform 810.3.b.b.809.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(809,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("810.809");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.b (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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 2x^{14} - 230x^{12} - 96x^{10} + 25551x^{8} - 7776x^{6} - 1509030x^{4} + 1062882x^{2} + 43046721 $ x^16 + 2x^14 - 230x^12 - 96x^10 + 25551x^8 - 7776x^6 - 1509030x^4 + 1062882*x^2 + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.7
Root			$2.87096 - 0.870383i$ of defining polynomial
Character	$\chi$	$=$	810.809
Dual form			810.3.b.b.809.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +(4.25978 - 2.61807i) q^{5} -4.80818i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +(4.25978 - 2.61807i) q^{5} -4.80818i q^{7} -2.82843 q^{8} +(-6.02424 + 3.70251i) q^{10} -4.90332i q^{11} +14.9323i q^{13} +6.79979i q^{14} +4.00000 q^{16} +6.09874 q^{17} +14.5280 q^{19} +(8.51956 - 5.23614i) q^{20} +6.93435i q^{22} +42.3965 q^{23} +(11.2914 - 22.3048i) q^{25} -21.1174i q^{26} -9.61635i q^{28} -40.0051i q^{29} -43.1593 q^{31} -5.65685 q^{32} -8.62493 q^{34} +(-12.5881 - 20.4818i) q^{35} +49.3976i q^{37} -20.5457 q^{38} +(-12.0485 + 7.40502i) q^{40} -32.8725i q^{41} +31.5395i q^{43} -9.80665i q^{44} -59.9577 q^{46} -4.72351 q^{47} +25.8814 q^{49} +(-15.9685 + 31.5437i) q^{50} +29.8645i q^{52} +19.9361 q^{53} +(-12.8372 - 20.8871i) q^{55} +13.5996i q^{56} +56.5757i q^{58} -75.5886i q^{59} -3.65935 q^{61} +61.0365 q^{62} +8.00000 q^{64} +(39.0937 + 63.6082i) q^{65} -59.3429i q^{67} +12.1975 q^{68} +(17.8023 + 28.9656i) q^{70} -73.3308i q^{71} -45.6159i q^{73} -69.8588i q^{74} +29.0560 q^{76} -23.5761 q^{77} -124.381 q^{79} +(17.0391 - 10.4723i) q^{80} +46.4887i q^{82} +83.6935 q^{83} +(25.9793 - 15.9669i) q^{85} -44.6036i q^{86} +13.8687i q^{88} +138.123i q^{89} +71.7970 q^{91} +84.7929 q^{92} +6.68005 q^{94} +(61.8860 - 38.0353i) q^{95} +10.0793i q^{97} -36.6019 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 32 q^{4}+O(q^{10}) $ 16 * q + 32 * q^4	$ 16 q + 32 q^{4} - 16 q^{10} + 64 q^{16} + 144 q^{19} - 12 q^{25} - 56 q^{31} + 272 q^{34} - 32 q^{40} - 56 q^{46} - 24 q^{49} + 20 q^{55} - 136 q^{61} + 128 q^{64} + 224 q^{70} + 288 q^{76} - 840 q^{79} + 272 q^{85} - 168 q^{91} + 328 q^{94}+O(q^{100}) $ 16 * q + 32 * q^4 - 16 * q^10 + 64 * q^16 + 144 * q^19 - 12 * q^25 - 56 * q^31 + 272 * q^34 - 32 * q^40 - 56 * q^46 - 24 * q^49 + 20 * q^55 - 136 * q^61 + 128 * q^64 + 224 * q^70 + 288 * q^76 - 840 * q^79 + 272 * q^85 - 168 * q^91 + 328 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$-1$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$	4.25978	−	2.61807i	0.851956	−	0.523614i
$5$	4.25978	−	2.61807i	0.851956	−	0.523614i
$6$		0			0
$7$		−	4.80818i		−	0.686882i	−0.939174	−	0.343441i	$-0.888407\pi$
$7$		−	4.80818i		−	0.686882i	0.939174	−	0.343441i	$-0.111593\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$	−6.02424	+	3.70251i	−0.602424	+	0.370251i
$11$		−	4.90332i		−	0.445757i	−0.974846	−	0.222878i	$-0.928455\pi$
$11$		−	4.90332i		−	0.445757i	0.974846	−	0.222878i	$-0.0715453\pi$
$12$		0			0
$13$			14.9323i			1.14864i	0.818632	+	0.574318i	$0.194733\pi$
$13$			14.9323i			1.14864i	−0.818632	+	0.574318i	$0.805267\pi$
$14$			6.79979i			0.485699i
$15$		0			0
$16$	4.00000			0.250000
$17$	6.09874			0.358750			0.179375	−	0.983781i	$-0.442593\pi$
$17$	6.09874			0.358750			0.179375	+	0.983781i	$0.442593\pi$
$18$		0			0
$19$	14.5280			0.764630			0.382315	−	0.924032i	$-0.375127\pi$
$19$	14.5280			0.764630			0.382315	+	0.924032i	$0.375127\pi$
$20$	8.51956	−	5.23614i	0.425978	−	0.261807i
$21$		0			0
$22$			6.93435i			0.315198i
$23$	42.3965			1.84332			0.921662	−	0.387993i	$-0.126831\pi$
$23$	42.3965			1.84332			0.921662	+	0.387993i	$0.126831\pi$
$24$		0			0
$25$	11.2914	−	22.3048i	0.451657	−	0.892192i
$26$		−	21.1174i		−	0.812209i
$27$		0			0
$28$		−	9.61635i		−	0.343441i
$29$		−	40.0051i		−	1.37949i	−0.724054	−	0.689743i	$-0.757724\pi$
$29$		−	40.0051i		−	1.37949i	0.724054	−	0.689743i	$-0.242276\pi$
$30$		0			0
$31$	−43.1593			−1.39224			−0.696118	−	0.717928i	$-0.745091\pi$
$31$	−43.1593			−1.39224			−0.696118	+	0.717928i	$0.745091\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$	−8.62493			−0.253674
$35$	−12.5881	−	20.4818i	−0.359661	−	0.585193i
$36$		0			0
$37$			49.3976i			1.33507i	0.744578	+	0.667535i	$0.232651\pi$
$37$			49.3976i			1.33507i	−0.744578	+	0.667535i	$0.767349\pi$
$38$	−20.5457			−0.540675
$39$		0			0
$40$	−12.0485	+	7.40502i	−0.301212	+	0.185125i
$41$		−	32.8725i		−	0.801768i	−0.916129	−	0.400884i	$-0.868703\pi$
$41$		−	32.8725i		−	0.801768i	0.916129	−	0.400884i	$-0.131297\pi$
$42$		0			0
$43$			31.5395i			0.733477i	0.930324	+	0.366739i	$0.119526\pi$
$43$			31.5395i			0.733477i	−0.930324	+	0.366739i	$0.880474\pi$
$44$		−	9.80665i		−	0.222878i
$45$		0			0
$46$	−59.9577			−1.30343
$47$	−4.72351			−0.100500			−0.0502501	−	0.998737i	$-0.516002\pi$
$47$	−4.72351			−0.100500			−0.0502501	+	0.998737i	$0.516002\pi$
$48$		0			0
$49$	25.8814			0.528192
$50$	−15.9685	+	31.5437i	−0.319370	+	0.630875i
$51$		0			0
$52$			29.8645i			0.574318i
$53$	19.9361			0.376153			0.188076	−	0.982154i	$-0.439775\pi$
$53$	19.9361			0.376153			0.188076	+	0.982154i	$0.439775\pi$
$54$		0			0
$55$	−12.8372	−	20.8871i	−0.233404	−	0.379765i
$56$			13.5996i			0.242850i
$57$		0			0
$58$			56.5757i			0.975444i
$59$		−	75.5886i		−	1.28116i	−0.767890	−	0.640582i	$-0.778693\pi$
$59$		−	75.5886i		−	1.28116i	0.767890	−	0.640582i	$-0.221307\pi$
$60$		0			0
$61$	−3.65935			−0.0599893			−0.0299946	−	0.999550i	$-0.509549\pi$
$61$	−3.65935			−0.0599893			−0.0299946	+	0.999550i	$0.509549\pi$
$62$	61.0365			0.984459
$63$		0			0
$64$	8.00000			0.125000
$65$	39.0937	+	63.6082i	0.601442	+	0.978587i
$66$		0			0
$67$		−	59.3429i		−	0.885715i	−0.896592	−	0.442858i	$-0.853965\pi$
$67$		−	59.3429i		−	0.885715i	0.896592	−	0.442858i	$-0.146035\pi$
$68$	12.1975			0.179375
$69$		0			0
$70$	17.8023	+	28.9656i	0.254319	+	0.413794i
$71$		−	73.3308i		−	1.03283i	−0.856339	−	0.516414i	$-0.827266\pi$
$71$		−	73.3308i		−	1.03283i	0.856339	−	0.516414i	$-0.172734\pi$
$72$		0			0
$73$		−	45.6159i		−	0.624876i	−0.949938	−	0.312438i	$-0.898854\pi$
$73$		−	45.6159i		−	0.624876i	0.949938	−	0.312438i	$-0.101146\pi$
$74$		−	69.8588i		−	0.944037i
$75$		0			0
$76$	29.0560			0.382315
$77$	−23.5761			−0.306182
$78$		0			0
$79$	−124.381			−1.57445			−0.787224	−	0.616667i	$-0.788482\pi$
$79$	−124.381			−1.57445			−0.787224	+	0.616667i	$0.788482\pi$
$80$	17.0391	−	10.4723i	0.212989	−	0.130903i
$81$		0			0
$82$			46.4887i			0.566935i
$83$	83.6935			1.00835			0.504177	−	0.863600i	$-0.331796\pi$
$83$	83.6935			1.00835			0.504177	+	0.863600i	$0.331796\pi$
$84$		0			0
$85$	25.9793	−	15.9669i	0.305639	−	0.187846i
$86$		−	44.6036i		−	0.518647i
$87$		0			0
$88$			13.8687i			0.157599i
$89$			138.123i			1.55195i	0.630765	+	0.775974i	$0.282741\pi$
$89$			138.123i			1.55195i	−0.630765	+	0.775974i	$0.717259\pi$
$90$		0			0
$91$	71.7970			0.788978
$92$	84.7929			0.921662
$93$		0			0
$94$	6.68005			0.0710644
$95$	61.8860	−	38.0353i	0.651431	−	0.400371i
$96$		0			0
$97$			10.0793i			0.103910i	0.998649	+	0.0519549i	$0.0165452\pi$
$97$			10.0793i			0.103910i	−0.998649	+	0.0519549i	$0.983455\pi$
$98$	−36.6019			−0.373488
$99$		0			0
$100$	22.5829	−	44.6096i	0.225829	−	0.446096i
$101$		−	155.933i		−	1.54389i	−0.635691	−	0.771943i	$-0.719285\pi$
$101$		−	155.933i		−	1.54389i	0.635691	−	0.771943i	$-0.280715\pi$
$102$		0			0
$103$		−	159.388i		−	1.54746i	−0.633518	−	0.773728i	$-0.718390\pi$
$103$		−	159.388i		−	1.54746i	0.633518	−	0.773728i	$-0.281610\pi$
$104$		−	42.2348i		−	0.406104i
$105$		0			0
$106$	−28.1939			−0.265980
$107$	55.2076			0.515959			0.257979	−	0.966150i	$-0.416943\pi$
$107$	55.2076			0.515959			0.257979	+	0.966150i	$0.416943\pi$
$108$		0			0
$109$	101.451			0.930743			0.465371	−	0.885115i	$-0.345921\pi$
$109$	101.451			0.930743			0.465371	+	0.885115i	$0.345921\pi$
$110$	18.1546	+	29.5388i	0.165042	+	0.268534i
$111$		0			0
$112$		−	19.2327i		−	0.171721i
$113$	−155.980			−1.38035			−0.690176	−	0.723641i	$-0.742467\pi$
$113$	−155.980			−1.38035			−0.690176	+	0.723641i	$0.742467\pi$
$114$		0			0
$115$	180.600	−	110.997i	1.57043	−	0.965191i
$116$		−	80.0102i		−	0.689743i
$117$		0			0
$118$			106.898i			0.905919i
$119$		−	29.3238i		−	0.246419i
$120$		0			0
$121$	96.9574			0.801301
$122$	5.17510			0.0424188
$123$		0			0
$124$	−86.3186			−0.696118
$125$	−10.2965	−	124.575i	−0.0823722	−	0.996602i
$126$		0			0
$127$		−	115.909i		−	0.912668i	−0.889809	−	0.456334i	$-0.849162\pi$
$127$		−	115.909i		−	0.912668i	0.889809	−	0.456334i	$-0.150838\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	−55.2869	−	89.9556i	−0.425284	−	0.691966i
$131$		−	66.7763i		−	0.509743i	−0.966975	−	0.254872i	$-0.917967\pi$
$131$		−	66.7763i		−	0.509743i	0.966975	−	0.254872i	$-0.0820332\pi$
$132$		0			0
$133$		−	69.8531i		−	0.525211i
$134$			83.9235i			0.626295i
$135$		0			0
$136$	−17.2499			−0.126837
$137$	−71.6499			−0.522992			−0.261496	−	0.965205i	$-0.584216\pi$
$137$	−71.6499			−0.522992			−0.261496	+	0.965205i	$0.584216\pi$
$138$		0			0
$139$	115.435			0.830465			0.415233	−	0.909715i	$-0.363700\pi$
$139$	115.435			0.830465			0.415233	+	0.909715i	$0.363700\pi$
$140$	−25.1763	−	40.9635i	−0.179831	−	0.292597i
$141$		0			0
$142$			103.705i			0.730320i
$143$	73.2178			0.512012
$144$		0			0
$145$	−104.736	−	170.413i	−0.722318	−	1.17526i
$146$			64.5107i			0.441854i
$147$		0			0
$148$			98.7952i			0.667535i
$149$			178.379i			1.19717i	0.801058	+	0.598587i	$0.204271\pi$
$149$			178.379i			1.19717i	−0.801058	+	0.598587i	$0.795729\pi$
$150$		0			0
$151$	268.628			1.77899			0.889497	−	0.456941i	$-0.151055\pi$
$151$	268.628			1.77899			0.889497	+	0.456941i	$0.151055\pi$
$152$	−41.0913			−0.270338
$153$		0			0
$154$	33.3416			0.216504
$155$	−183.849	+	112.994i	−1.18612	+	0.728994i
$156$		0			0
$157$			259.211i			1.65102i	0.564385	+	0.825512i	$0.309113\pi$
$157$			259.211i			1.65102i	−0.564385	+	0.825512i	$0.690887\pi$
$158$	175.902			1.11330
$159$		0			0
$160$	−24.0969	+	14.8100i	−0.150606	+	0.0925627i
$161$		−	203.850i		−	1.26615i
$162$		0			0
$163$			1.67785i			0.0102936i	0.999987	+	0.00514679i	$0.00163828\pi$
$163$			1.67785i			0.0102936i	−0.999987	+	0.00514679i	$0.998362\pi$
$164$		−	65.7450i		−	0.400884i
$165$		0			0
$166$	−118.360			−0.713015
$167$	180.166			1.07884			0.539420	−	0.842037i	$-0.318644\pi$
$167$	180.166			1.07884			0.539420	+	0.842037i	$0.318644\pi$
$168$		0			0
$169$	−53.9728			−0.319366
$170$	−36.7403	+	22.5807i	−0.216119	+	0.132827i
$171$		0			0
$172$			63.0791i			0.366739i
$173$	83.6891			0.483752			0.241876	−	0.970307i	$-0.422237\pi$
$173$	83.6891			0.483752			0.241876	+	0.970307i	$0.422237\pi$
$174$		0			0
$175$	−107.245	−	54.2912i	−0.612831	−	0.310235i
$176$		−	19.6133i		−	0.111439i
$177$		0			0
$178$		−	195.336i		−	1.09739i
$179$			50.5590i			0.282453i	0.989977	+	0.141226i	$0.0451045\pi$
$179$			50.5590i			0.282453i	−0.989977	+	0.141226i	$0.954895\pi$
$180$		0			0
$181$	−191.874			−1.06008			−0.530038	−	0.847974i	$-0.677822\pi$
$181$	−191.874			−1.06008			−0.530038	+	0.847974i	$0.677822\pi$
$182$	−101.536			−0.557892
$183$		0			0
$184$	−119.915			−0.651714
$185$	129.326	+	210.423i	0.699061	+	1.13742i
$186$		0			0
$187$		−	29.9041i		−	0.159915i
$188$	−9.44702			−0.0502501
$189$		0			0
$190$	−87.5200	+	53.7900i	−0.460631	+	0.283105i
$191$			113.164i			0.592481i	0.955113	+	0.296241i	$0.0957331\pi$
$191$			113.164i			0.592481i	−0.955113	+	0.296241i	$0.904267\pi$
$192$		0			0
$193$		−	118.486i		−	0.613918i	−0.951723	−	0.306959i	$-0.900688\pi$
$193$		−	118.486i		−	0.613918i	0.951723	−	0.306959i	$-0.0993115\pi$
$194$		−	14.2542i		−	0.0734754i
$195$		0			0
$196$	51.7629			0.264096
$197$	−247.866			−1.25820			−0.629102	−	0.777323i	$-0.716577\pi$
$197$	−247.866			−1.25820			−0.629102	+	0.777323i	$0.716577\pi$
$198$		0			0
$199$	83.8620			0.421417			0.210708	−	0.977549i	$-0.432423\pi$
$199$	83.8620			0.421417			0.210708	+	0.977549i	$0.432423\pi$
$200$	−31.9370	+	63.0875i	−0.159685	+	0.315437i
$201$		0			0
$202$			220.522i			1.09169i
$203$	−192.352			−0.947545
$204$		0			0
$205$	−86.0624	−	140.029i	−0.419817	−	0.683071i
$206$			225.409i			1.09422i
$207$		0			0
$208$			59.7291i			0.287159i
$209$		−	71.2354i		−	0.340839i
$210$		0			0
$211$	−80.5908			−0.381947			−0.190973	−	0.981595i	$-0.561164\pi$
$211$	−80.5908			−0.381947			−0.190973	+	0.981595i	$0.561164\pi$
$212$	39.8722			0.188076
$213$		0			0
$214$	−78.0753			−0.364838
$215$	82.5727	+	134.351i	0.384059	+	0.624890i
$216$		0			0
$217$			207.518i			0.956302i
$218$	−143.473			−0.658135
$219$		0			0
$220$	−25.6745	−	41.7741i	−0.116702	−	0.189882i
$221$			91.0681i			0.412073i
$222$		0			0
$223$			32.8692i			0.147396i	0.997281	+	0.0736979i	$0.0234800\pi$
$223$			32.8692i			0.147396i	−0.997281	+	0.0736979i	$0.976520\pi$
$224$			27.1992i			0.121425i
$225$		0			0
$226$	220.589			0.976056
$227$	91.6646			0.403809			0.201904	−	0.979405i	$-0.435287\pi$
$227$	91.6646			0.403809			0.201904	+	0.979405i	$0.435287\pi$
$228$		0			0
$229$	73.6946			0.321811			0.160905	−	0.986970i	$-0.448559\pi$
$229$	73.6946			0.321811			0.160905	+	0.986970i	$0.448559\pi$
$230$	−255.406	+	156.973i	−1.11046	+	0.682493i
$231$		0			0
$232$			113.151i			0.487722i
$233$	−13.3749			−0.0574029			−0.0287014	−	0.999588i	$-0.509137\pi$
$233$	−13.3749			−0.0574029			−0.0287014	+	0.999588i	$0.509137\pi$
$234$		0			0
$235$	−20.1211	+	12.3665i	−0.0856217	+	0.0526233i
$236$		−	151.177i		−	0.640582i
$237$		0			0
$238$			41.4702i			0.174244i
$239$			223.710i			0.936026i	0.883722	+	0.468013i	$0.155030\pi$
$239$			223.710i			0.936026i	−0.883722	+	0.468013i	$0.844970\pi$
$240$		0			0
$241$	−215.351			−0.893571			−0.446786	−	0.894641i	$-0.647431\pi$
$241$	−215.351			−0.893571			−0.446786	+	0.894641i	$0.647431\pi$
$242$	−137.118			−0.566605
$243$		0			0
$244$	−7.31869			−0.0299946
$245$	110.249	−	67.7594i	0.449997	−	0.276569i
$246$		0			0
$247$			216.936i			0.878282i
$248$	122.073			0.492230
$249$		0			0
$250$	14.5615	+	176.176i	0.0582459	+	0.704704i
$251$			5.69286i			0.0226807i	0.999936	+	0.0113404i	$0.00360982\pi$
$251$			5.69286i			0.0226807i	−0.999936	+	0.0113404i	$0.996390\pi$
$252$		0			0
$253$		−	207.884i		−	0.821675i
$254$			163.920i			0.645354i
$255$		0			0
$256$	16.0000			0.0625000
$257$	−96.4638			−0.375346			−0.187673	−	0.982232i	$-0.560094\pi$
$257$	−96.4638			−0.375346			−0.187673	+	0.982232i	$0.560094\pi$
$258$		0			0
$259$	237.512			0.917036
$260$	78.1875	+	127.216i	0.300721	+	0.489294i
$261$		0			0
$262$			94.4360i			0.360443i
$263$	−233.601			−0.888218			−0.444109	−	0.895973i	$-0.646480\pi$
$263$	−233.601			−0.888218			−0.444109	+	0.895973i	$0.646480\pi$
$264$		0			0
$265$	84.9233	−	52.1941i	0.320465	−	0.196959i
$266$			98.7872i			0.371380i
$267$		0			0
$268$		−	118.686i		−	0.442858i
$269$			492.222i			1.82982i	0.403655	+	0.914911i	$0.367740\pi$
$269$			492.222i			1.82982i	−0.403655	+	0.914911i	$0.632260\pi$
$270$		0			0
$271$	183.942			0.678752			0.339376	−	0.940651i	$-0.389784\pi$
$271$	183.942			0.678752			0.339376	+	0.940651i	$0.389784\pi$
$272$	24.3950			0.0896874
$273$		0			0
$274$	101.328			0.369811
$275$	−109.368	−	55.3655i	−0.397700	−	0.201329i
$276$		0			0
$277$			226.087i			0.816199i	0.912937	+	0.408100i	$0.133808\pi$
$277$			226.087i			0.816199i	−0.912937	+	0.408100i	$0.866192\pi$
$278$	−163.249			−0.587228
$279$		0			0
$280$	35.6046	+	57.9312i	0.127159	+	0.206897i
$281$			150.264i			0.534749i	0.963593	+	0.267374i	$0.0861560\pi$
$281$			150.264i			0.534749i	−0.963593	+	0.267374i	$0.913844\pi$
$282$		0			0
$283$		−	316.153i		−	1.11715i	−0.829455	−	0.558574i	$-0.811349\pi$
$283$		−	316.153i		−	1.11715i	0.829455	−	0.558574i	$-0.188651\pi$
$284$		−	146.662i		−	0.516414i
$285$		0			0
$286$	−103.546			−0.362047
$287$	−158.057			−0.550720
$288$		0			0
$289$	−251.805			−0.871299
$290$	148.119	+	241.000i	0.510756	+	0.831035i
$291$		0			0
$292$		−	91.2319i		−	0.312438i
$293$	−171.722			−0.586081			−0.293041	−	0.956100i	$-0.594667\pi$
$293$	−171.722			−0.586081			−0.293041	+	0.956100i	$0.594667\pi$
$294$		0			0
$295$	−197.896	−	321.991i	−0.670835	−	1.09149i
$296$		−	139.718i		−	0.472019i
$297$		0			0
$298$		−	252.266i		−	0.846530i
$299$			633.076i			2.11731i
$300$		0			0
$301$	151.648			0.503813
$302$	−379.898			−1.25794
$303$		0			0
$304$	58.1119			0.191158
$305$	−15.5880	+	9.58042i	−0.0511082	+	0.0314112i
$306$		0			0
$307$			226.678i			0.738365i	0.929357	+	0.369182i	$0.120362\pi$
$307$			226.678i			0.738365i	−0.929357	+	0.369182i	$0.879638\pi$
$308$	−47.1521			−0.153091
$309$		0			0
$310$	260.002	−	159.798i	0.838716	−	0.515477i
$311$		−	348.817i		−	1.12160i	−0.827952	−	0.560800i	$-0.810494\pi$
$311$		−	348.817i		−	1.12160i	0.827952	−	0.560800i	$-0.189506\pi$
$312$		0			0
$313$			510.550i			1.63115i	0.578652	+	0.815574i	$0.303579\pi$
$313$			510.550i			1.63115i	−0.578652	+	0.815574i	$0.696421\pi$
$314$		−	366.579i		−	1.16745i
$315$		0			0
$316$	−248.763			−0.787224
$317$	−200.685			−0.633077			−0.316538	−	0.948580i	$-0.602521\pi$
$317$	−200.685			−0.633077			−0.316538	+	0.948580i	$0.602521\pi$
$318$		0			0
$319$	−196.158			−0.614915
$320$	34.0782	−	20.9446i	0.106494	−	0.0654517i
$321$		0			0
$322$			288.287i			0.895302i
$323$	88.6024			0.274311
$324$		0			0
$325$	333.061	+	168.607i	1.02480	+	0.518790i
$326$		−	2.37284i		−	0.00727866i
$327$		0			0
$328$			92.9774i			0.283468i
$329$			22.7115i			0.0690318i
$330$		0			0
$331$	−23.3687			−0.0706002			−0.0353001	−	0.999377i	$-0.511239\pi$
$331$	−23.3687			−0.0706002			−0.0353001	+	0.999377i	$0.511239\pi$
$332$	167.387			0.504177
$333$		0			0
$334$	−254.794			−0.762855
$335$	−155.364	−	252.788i	−0.463773	−	0.754590i
$336$		0			0
$337$			413.512i			1.22704i	0.789680	+	0.613519i	$0.210247\pi$
$337$			413.512i			1.22704i	−0.789680	+	0.613519i	$0.789753\pi$
$338$	76.3291			0.225826
$339$		0			0
$340$	51.9586	−	31.9339i	0.152819	−	0.0939231i
$341$			211.624i			0.620598i
$342$		0			0
$343$		−	360.043i		−	1.04969i
$344$		−	89.2072i		−	0.259323i
$345$		0			0
$346$	−118.354			−0.342064
$347$	−145.826			−0.420248			−0.210124	−	0.977675i	$-0.567387\pi$
$347$	−145.826			−0.420248			−0.210124	+	0.977675i	$0.567387\pi$
$348$		0			0
$349$	−75.1353			−0.215287			−0.107644	−	0.994190i	$-0.534331\pi$
$349$	−75.1353			−0.215287			−0.107644	+	0.994190i	$0.534331\pi$
$350$	151.668	+	76.7793i	0.433337	+	0.219369i
$351$		0			0
$352$			27.7374i			0.0787994i
$353$	218.797			0.619822			0.309911	−	0.950766i	$-0.399701\pi$
$353$	218.797			0.619822			0.309911	+	0.950766i	$0.399701\pi$
$354$		0			0
$355$	−191.985	−	312.373i	−0.540803	−	0.879924i
$356$			276.247i			0.775974i
$357$		0			0
$358$		−	71.5013i		−	0.199724i
$359$		−	371.972i		−	1.03613i	−0.855340	−	0.518067i	$-0.826652\pi$
$359$		−	371.972i		−	1.03613i	0.855340	−	0.518067i	$-0.173348\pi$
$360$		0			0
$361$	−149.938			−0.415340
$362$	271.350			0.749587
$363$		0			0
$364$	143.594			0.394489
$365$	−119.426	−	194.314i	−0.327194	−	0.532366i
$366$		0			0
$367$			614.883i			1.67543i	0.546107	+	0.837715i	$0.316109\pi$
$367$			614.883i			1.67543i	−0.546107	+	0.837715i	$0.683891\pi$
$368$	169.586			0.460831
$369$		0			0
$370$	−182.895	−	297.583i	−0.494311	−	0.804278i
$371$		−	95.8563i		−	0.258373i
$372$		0			0
$373$		−	147.562i		−	0.395609i	−0.980242	−	0.197804i	$-0.936619\pi$
$373$		−	147.562i		−	0.395609i	0.980242	−	0.197804i	$-0.0633811\pi$
$374$			42.2908i			0.113077i
$375$		0			0
$376$	13.3601			0.0355322
$377$	597.367			1.58453
$378$		0			0
$379$	145.687			0.384398			0.192199	−	0.981356i	$-0.438438\pi$
$379$	145.687			0.384398			0.192199	+	0.981356i	$0.438438\pi$
$380$	123.772	−	76.0705i	0.325716	−	0.200186i
$381$		0			0
$382$		−	160.038i		−	0.418948i
$383$	94.6363			0.247092			0.123546	−	0.992339i	$-0.460573\pi$
$383$	94.6363			0.247092			0.123546	+	0.992339i	$0.460573\pi$
$384$		0			0
$385$	−100.429	+	61.7237i	−0.260854	+	0.160321i
$386$			167.565i			0.434106i
$387$		0			0
$388$			20.1585i			0.0519549i
$389$			137.838i			0.354338i	0.984180	+	0.177169i	$0.0566940\pi$
$389$			137.838i			0.354338i	−0.984180	+	0.177169i	$0.943306\pi$
$390$		0			0
$391$	258.565			0.661292
$392$	−73.2037			−0.186744
$393$		0			0
$394$	350.536			0.889685
$395$	−529.837	+	325.639i	−1.34136	+	0.824403i
$396$		0			0
$397$			735.633i			1.85298i	0.376320	+	0.926490i	$0.377189\pi$
$397$			735.633i			1.85298i	−0.376320	+	0.926490i	$0.622811\pi$
$398$	−118.599			−0.297987
$399$		0			0
$400$	45.1657	−	89.2192i	0.112914	−	0.223048i
$401$		−	99.1697i		−	0.247306i	−0.992326	−	0.123653i	$-0.960539\pi$
$401$		−	99.1697i		−	0.247306i	0.992326	−	0.123653i	$-0.0394610\pi$
$402$		0			0
$403$		−	644.467i		−	1.59917i
$404$		−	311.865i		−	0.771943i
$405$		0			0
$406$	272.026			0.670015
$407$	242.212			0.595117
$408$		0			0
$409$	−816.046			−1.99522			−0.997611	−	0.0690850i	$-0.977992\pi$
$409$	−816.046			−1.99522			−0.997611	+	0.0690850i	$0.977992\pi$
$410$	121.711	+	198.032i	0.296855	+	0.483004i
$411$		0			0
$412$		−	318.776i		−	0.773728i
$413$	−363.443			−0.880008
$414$		0			0
$415$	356.516	−	219.115i	0.859074	−	0.527989i
$416$		−	84.4697i		−	0.203052i
$417$		0			0
$418$			100.742i			0.241010i
$419$			180.693i			0.431249i	0.976476	+	0.215624i	$0.0691786\pi$
$419$			180.693i			0.431249i	−0.976476	+	0.215624i	$0.930821\pi$
$420$		0			0
$421$	−414.762			−0.985183			−0.492592	−	0.870261i	$-0.663950\pi$
$421$	−414.762			−0.985183			−0.492592	+	0.870261i	$0.663950\pi$
$422$	113.973			0.270077
$423$		0			0
$424$	−56.3878			−0.132990
$425$	68.8635	−	136.031i	0.162032	−	0.320073i
$426$		0			0
$427$			17.5948i			0.0412056i
$428$	110.415			0.257979
$429$		0			0
$430$	−116.775	−	190.002i	−0.271571	−	0.441864i
$431$			440.794i			1.02272i	0.859366	+	0.511362i	$0.170859\pi$
$431$			440.794i			1.02272i	−0.859366	+	0.511362i	$0.829141\pi$
$432$		0			0
$433$			357.721i			0.826147i	0.910698	+	0.413073i	$0.135545\pi$
$433$			357.721i			0.826147i	−0.910698	+	0.413073i	$0.864455\pi$
$434$		−	293.474i		−	0.676208i
$435$		0			0
$436$	202.902			0.465371
$437$	615.935			1.40946
$438$		0			0
$439$	193.113			0.439893			0.219947	−	0.975512i	$-0.429412\pi$
$439$	193.113			0.439893			0.219947	+	0.975512i	$0.429412\pi$
$440$	36.3092	+	59.0776i	0.0825209	+	0.134267i
$441$		0			0
$442$		−	128.790i		−	0.291380i
$443$	−58.2692			−0.131533			−0.0657665	−	0.997835i	$-0.520949\pi$
$443$	−58.2692			−0.131533			−0.0657665	+	0.997835i	$0.520949\pi$
$444$		0			0
$445$	361.616	+	588.375i	0.812621	+	1.32219i
$446$		−	46.4841i		−	0.104225i
$447$		0			0
$448$		−	38.4654i		−	0.0858603i
$449$		−	594.072i		−	1.32310i	−0.749901	−	0.661550i	$-0.769899\pi$
$449$		−	594.072i		−	1.32310i	0.749901	−	0.661550i	$-0.230101\pi$
$450$		0			0
$451$	−161.184			−0.357393
$452$	−311.960			−0.690176
$453$		0			0
$454$	−129.633			−0.285536
$455$	305.839	−	187.970i	0.672175	−	0.413120i
$456$		0			0
$457$		−	304.884i		−	0.667143i	−0.942725	−	0.333571i	$-0.891746\pi$
$457$		−	304.884i		−	0.667143i	0.942725	−	0.333571i	$-0.108254\pi$
$458$	−104.220			−0.227554
$459$		0			0
$460$	361.199	−	221.994i	0.785216	−	0.482595i
$461$		−	419.206i		−	0.909340i	−0.890660	−	0.454670i	$-0.849757\pi$
$461$		−	419.206i		−	0.909340i	0.890660	−	0.454670i	$-0.150243\pi$
$462$		0			0
$463$		−	101.562i		−	0.219357i	−0.993967	−	0.109679i	$-0.965018\pi$
$463$		−	101.562i		−	0.219357i	0.993967	−	0.109679i	$-0.0349822\pi$
$464$		−	160.020i		−	0.344871i
$465$		0			0
$466$	18.9149			0.0405899
$467$	−446.284			−0.955640			−0.477820	−	0.878458i	$-0.658573\pi$
$467$	−446.284			−0.955640			−0.477820	+	0.878458i	$0.658573\pi$
$468$		0			0
$469$	−285.331			−0.608382
$470$	28.4555	−	17.4888i	0.0605437	−	0.0372103i
$471$		0			0
$472$			213.797i			0.452960i
$473$	154.649			0.326952
$474$		0			0
$475$	164.042	−	324.044i	0.345351	−	0.682197i
$476$		−	58.6477i		−	0.123209i
$477$		0			0
$478$		−	316.374i		−	0.661871i
$479$			214.839i			0.448515i	0.974530	+	0.224258i	$0.0719957\pi$
$479$			214.839i			0.448515i	−0.974530	+	0.224258i	$0.928004\pi$
$480$		0			0
$481$	−737.618			−1.53351
$482$	304.552			0.631850
$483$		0			0
$484$	193.915			0.400650
$485$	26.3882	+	42.9354i	0.0544087	+	0.0885266i
$486$		0			0
$487$			151.238i			0.310551i	0.987871	+	0.155276i	$0.0496265\pi$
$487$			151.238i			0.310551i	−0.987871	+	0.155276i	$0.950373\pi$
$488$	10.3502			0.0212094
$489$		0			0
$490$	−155.916	+	95.8262i	−0.318196	+	0.195564i
$491$			507.056i			1.03270i	0.856378	+	0.516350i	$0.172710\pi$
$491$			507.056i			1.03270i	−0.856378	+	0.516350i	$0.827290\pi$
$492$		0			0
$493$		−	243.981i		−	0.494890i
$494$		−	306.794i		−	0.621039i
$495$		0			0
$496$	−172.637			−0.348059
$497$	−352.588			−0.709432
$498$		0			0
$499$	933.476			1.87069			0.935346	−	0.353733i	$-0.115088\pi$
$499$	933.476			1.87069			0.935346	+	0.353733i	$0.115088\pi$
$500$	−20.5930	−	249.150i	−0.0411861	−	0.498301i
$501$		0			0
$502$		−	8.05091i		−	0.0160377i
$503$	271.226			0.539217			0.269608	−	0.962970i	$-0.413106\pi$
$503$	271.226			0.539217			0.269608	+	0.962970i	$0.413106\pi$
$504$		0			0
$505$	−408.242	−	664.238i	−0.808401	−	1.31532i
$506$			293.992i			0.581012i
$507$		0			0
$508$		−	231.818i		−	0.456334i
$509$		−	361.501i		−	0.710219i	−0.934825	−	0.355109i	$-0.884444\pi$
$509$		−	361.501i		−	0.710219i	0.934825	−	0.355109i	$-0.115556\pi$
$510$		0			0
$511$	−219.329			−0.429216
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	136.420			0.265409
$515$	−417.289	−	678.957i	−0.810269	−	1.31836i
$516$		0			0
$517$			23.1609i			0.0447986i
$518$	−335.893			−0.648443
$519$		0			0
$520$	−110.574	−	179.911i	−0.212642	−	0.345983i
$521$		−	673.089i		−	1.29192i	−0.763373	−	0.645958i	$-0.776458\pi$
$521$		−	673.089i		−	1.29192i	0.763373	−	0.645958i	$-0.223542\pi$
$522$		0			0
$523$		−	150.339i		−	0.287455i	−0.989617	−	0.143728i	$-0.954091\pi$
$523$		−	150.339i		−	0.287455i	0.989617	−	0.143728i	$-0.0459089\pi$
$524$		−	133.553i		−	0.254872i
$525$		0			0
$526$	330.362			0.628065
$527$	−263.217			−0.499464
$528$		0			0
$529$	1268.46			2.39785
$530$	−120.100	+	73.8136i	−0.226603	+	0.139271i
$531$		0			0
$532$		−	139.706i		−	0.262606i
$533$	490.861			0.920940
$534$		0			0
$535$	235.172	−	144.537i	0.439574	−	0.270163i
$536$			167.847i			0.313148i
$537$		0			0
$538$		−	696.107i		−	1.29388i
$539$		−	126.905i		−	0.235445i
$540$		0			0
$541$	713.274			1.31844			0.659218	−	0.751952i	$-0.270888\pi$
$541$	713.274			1.31844			0.659218	+	0.751952i	$0.270888\pi$
$542$	−260.133			−0.479950
$543$		0			0
$544$	−34.4997			−0.0634186
$545$	432.159	−	265.606i	0.792952	−	0.487350i
$546$		0			0
$547$			579.412i			1.05925i	0.848231	+	0.529627i	$0.177668\pi$
$547$			579.412i			1.05925i	−0.848231	+	0.529627i	$0.822332\pi$
$548$	−143.300			−0.261496
$549$		0			0
$550$	154.669	+	78.2987i	0.281217	+	0.142361i
$551$		−	581.193i		−	1.05480i
$552$		0			0
$553$			598.048i			1.08146i
$554$		−	319.736i		−	0.577140i
$555$		0			0
$556$	230.869			0.415233
$557$	727.043			1.30528			0.652642	−	0.757667i	$-0.273661\pi$
$557$	727.043			1.30528			0.652642	+	0.757667i	$0.273661\pi$
$558$		0			0
$559$	−470.957			−0.842499
$560$	−50.3526	−	81.9271i	−0.0899153	−	0.146298i
$561$		0			0
$562$		−	212.506i		−	0.378124i
$563$	239.775			0.425888			0.212944	−	0.977064i	$-0.431695\pi$
$563$	239.775			0.425888			0.212944	+	0.977064i	$0.431695\pi$
$564$		0			0
$565$	−664.439	+	408.366i	−1.17600	+	0.722772i
$566$			447.108i			0.789943i
$567$		0			0
$568$			207.411i			0.365160i
$569$			564.169i			0.991509i	0.868463	+	0.495755i	$0.165108\pi$
$569$			564.169i			0.991509i	−0.868463	+	0.495755i	$0.834892\pi$
$570$		0			0
$571$	55.4054			0.0970322			0.0485161	−	0.998822i	$-0.484551\pi$
$571$	55.4054			0.0970322			0.0485161	+	0.998822i	$0.484551\pi$
$572$	146.436			0.256006
$573$		0			0
$574$	223.526			0.389418
$575$	478.717	−	945.645i	0.832551	−	1.64460i
$576$		0			0
$577$			616.099i			1.06776i	0.845559	+	0.533881i	$0.179267\pi$
$577$			616.099i			1.06776i	−0.845559	+	0.533881i	$0.820733\pi$
$578$	356.107			0.616101
$579$		0			0
$580$	−209.472	−	340.826i	−0.361159	−	0.587630i
$581$		−	402.413i		−	0.692621i
$582$		0			0
$583$		−	97.7531i		−	0.167673i
$584$			129.021i			0.220927i
$585$		0			0
$586$	242.851			0.414422
$587$	252.812			0.430685			0.215343	−	0.976539i	$-0.430913\pi$
$587$	252.812			0.430685			0.215343	+	0.976539i	$0.430913\pi$
$588$		0			0
$589$	−627.017			−1.06455
$590$	279.868	+	455.364i	0.474352	+	0.771803i
$591$		0			0
$592$			197.590i			0.333768i
$593$	431.589			0.727805			0.363903	−	0.931437i	$-0.381444\pi$
$593$	431.589			0.727805			0.363903	+	0.931437i	$0.381444\pi$
$594$		0			0
$595$	−76.7718	−	124.913i	−0.129028	−	0.209938i
$596$			356.758i			0.598587i
$597$		0			0
$598$		−	895.304i		−	1.49716i
$599$			559.986i			0.934868i	0.884028	+	0.467434i	$0.154821\pi$
$599$			559.986i			0.934868i	−0.884028	+	0.467434i	$0.845179\pi$
$600$		0			0
$601$	537.374			0.894132			0.447066	−	0.894501i	$-0.352469\pi$
$601$	537.374			0.894132			0.447066	+	0.894501i	$0.352469\pi$
$602$	−214.462			−0.356249
$603$		0			0
$604$	537.256			0.889497
$605$	413.017	−	253.841i	0.682673	−	0.419572i
$606$		0			0
$607$			482.841i			0.795454i	0.917504	+	0.397727i	$0.130201\pi$
$607$			482.841i			0.795454i	−0.917504	+	0.397727i	$0.869799\pi$
$608$	−82.1827			−0.135169
$609$		0			0
$610$	22.0448	−	13.5488i	0.0361390	−	0.0222111i
$611$		−	70.5327i		−	0.115438i
$612$		0			0
$613$			259.741i			0.423721i	0.977300	+	0.211860i	$0.0679522\pi$
$613$			259.741i			0.423721i	−0.977300	+	0.211860i	$0.932048\pi$
$614$		−	320.571i		−	0.522103i
$615$		0			0
$616$	66.6831			0.108252
$617$	−157.281			−0.254912			−0.127456	−	0.991844i	$-0.540681\pi$
$617$	−157.281			−0.254912			−0.127456	+	0.991844i	$0.540681\pi$
$618$		0			0
$619$	−710.098			−1.14717			−0.573585	−	0.819146i	$-0.694448\pi$
$619$	−710.098			−1.14717			−0.573585	+	0.819146i	$0.694448\pi$
$620$	−367.698	+	225.988i	−0.593062	+	0.364497i
$621$		0			0
$622$			493.302i			0.793090i
$623$	664.121			1.06601
$624$		0			0
$625$	−370.007	−	503.706i	−0.592012	−	0.805929i
$626$		−	722.026i		−	1.15340i
$627$		0			0
$628$			518.421i			0.825512i
$629$			301.263i			0.478956i
$630$		0			0
$631$	−165.617			−0.262467			−0.131234	−	0.991351i	$-0.541894\pi$
$631$	−165.617			−0.262467			−0.131234	+	0.991351i	$0.541894\pi$
$632$	351.804			0.556651
$633$		0			0
$634$	283.812			0.447653
$635$	−303.457	−	493.746i	−0.477886	−	0.777553i
$636$		0			0
$637$			386.469i			0.606701i
$638$	277.409			0.434811
$639$		0			0
$640$	−48.1939	+	29.6201i	−0.0753030	+	0.0462814i
$641$			299.414i			0.467104i	0.972344	+	0.233552i	$0.0750349\pi$
$641$			299.414i			0.467104i	−0.972344	+	0.233552i	$0.924965\pi$
$642$		0			0
$643$			53.5070i			0.0832146i	0.999134	+	0.0416073i	$0.0132478\pi$
$643$			53.5070i			0.0832146i	−0.999134	+	0.0416073i	$0.986752\pi$
$644$		−	407.700i		−	0.633074i
$645$		0			0
$646$	−125.303			−0.193967
$647$	145.933			0.225553			0.112776	−	0.993620i	$-0.464026\pi$
$647$	145.933			0.225553			0.112776	+	0.993620i	$0.464026\pi$
$648$		0			0
$649$	−370.635			−0.571087
$650$	−471.020	−	238.446i	−0.724646	−	0.366840i
$651$		0			0
$652$			3.35571i			0.00514679i
$653$	−275.649			−0.422127			−0.211064	−	0.977472i	$-0.567693\pi$
$653$	−275.649			−0.422127			−0.211064	+	0.977472i	$0.567693\pi$
$654$		0			0
$655$	−174.825	−	284.452i	−0.266909	−	0.434279i
$656$		−	131.490i		−	0.200442i
$657$		0			0
$658$		−	32.1189i		−	0.0488129i
$659$		−	334.958i		−	0.508282i	−0.967167	−	0.254141i	$-0.918207\pi$
$659$		−	334.958i		−	0.508282i	0.967167	−	0.254141i	$-0.0817927\pi$
$660$		0			0
$661$	155.582			0.235374			0.117687	−	0.993051i	$-0.462452\pi$
$661$	155.582			0.235374			0.117687	+	0.993051i	$0.462452\pi$
$662$	33.0483			0.0499219
$663$		0			0
$664$	−236.721			−0.356507
$665$	−182.880	−	297.559i	−0.275008	−	0.447457i
$666$		0			0
$667$		−	1696.07i		−	2.54284i
$668$	360.333			0.539420
$669$		0			0
$670$	219.718	+	357.496i	0.327937	+	0.533576i
$671$			17.9430i			0.0267406i
$672$		0			0
$673$			1338.17i			1.98837i	0.107672	+	0.994186i	$0.465660\pi$
$673$			1338.17i			1.98837i	−0.107672	+	0.994186i	$0.534340\pi$
$674$		−	584.794i		−	0.867647i
$675$		0			0
$676$	−107.946			−0.159683
$677$	178.334			0.263418			0.131709	−	0.991288i	$-0.457954\pi$
$677$	178.334			0.263418			0.131709	+	0.991288i	$0.457954\pi$
$678$		0			0
$679$	48.4629			0.0713739
$680$	−73.4805	+	45.1613i	−0.108060	+	0.0664137i
$681$		0			0
$682$		−	299.282i		−	0.438829i
$683$	219.879			0.321931			0.160966	−	0.986960i	$-0.448539\pi$
$683$	219.879			0.321931			0.160966	+	0.986960i	$0.448539\pi$
$684$		0			0
$685$	−305.213	+	187.584i	−0.445566	+	0.273846i
$686$			509.178i			0.742242i
$687$		0			0
$688$			126.158i			0.183369i
$689$			297.691i			0.432063i
$690$		0			0
$691$	−90.0762			−0.130356			−0.0651782	−	0.997874i	$-0.520762\pi$
$691$	−90.0762			−0.130356			−0.0651782	+	0.997874i	$0.520762\pi$
$692$	167.378			0.241876
$693$		0			0
$694$	206.229			0.297160
$695$	491.726	−	302.216i	0.707520	−	0.434843i
$696$		0			0
$697$		−	200.481i		−	0.287634i
$698$	106.257			0.152231
$699$		0			0
$700$	−214.491	−	108.582i	−0.306415	−	0.155118i
$701$			239.302i			0.341373i	0.985325	+	0.170686i	$0.0545985\pi$
$701$			239.302i			0.341373i	−0.985325	+	0.170686i	$0.945401\pi$
$702$		0			0
$703$			717.647i			1.02084i
$704$		−	39.2266i		−	0.0557196i
$705$		0			0
$706$	−309.426			−0.438280
$707$	−749.751			−1.06047
$708$		0			0
$709$	−1227.43			−1.73122			−0.865609	−	0.500720i	$-0.833069\pi$
$709$	−1227.43			−1.73122			−0.865609	+	0.500720i	$0.833069\pi$
$710$	271.508	+	441.762i	0.382406	+	0.622200i
$711$		0			0
$712$		−	390.672i		−	0.548696i
$713$	−1829.80			−2.56634
$714$		0			0
$715$	311.892	−	191.689i	0.436212	−	0.268097i
$716$			101.118i			0.141226i
$717$		0			0
$718$			526.048i			0.732658i
$719$			177.318i			0.246617i	0.992368	+	0.123309i	$0.0393505\pi$
$719$			177.318i			0.246617i	−0.992368	+	0.123309i	$0.960649\pi$
$720$		0			0
$721$	−766.365			−1.06292
$722$	212.044			0.293690
$723$		0			0
$724$	−383.747			−0.530038
$725$	−892.305	−	451.714i	−1.23077	−	0.623054i
$726$		0			0
$727$		−	52.1000i		−	0.0716644i	−0.999358	−	0.0358322i	$-0.988592\pi$
$727$		−	52.1000i		−	0.0716644i	0.999358	−	0.0358322i	$-0.0114082\pi$
$728$	−203.073			−0.278946
$729$		0			0
$730$	168.893	+	274.801i	0.231361	+	0.376440i
$731$			192.351i			0.263135i
$732$		0			0
$733$			771.432i			1.05243i	0.850351	+	0.526216i	$0.176389\pi$
$733$			771.432i			1.05243i	−0.850351	+	0.526216i	$0.823611\pi$
$734$		−	869.576i		−	1.18471i
$735$		0			0
$736$	−239.831			−0.325857
$737$	−290.977			−0.394813
$738$		0			0
$739$	−250.154			−0.338504			−0.169252	−	0.985573i	$-0.554135\pi$
$739$	−250.154			−0.338504			−0.169252	+	0.985573i	$0.554135\pi$
$740$	258.653	+	420.846i	0.349531	+	0.568710i
$741$		0			0
$742$			135.561i			0.182697i
$743$	404.762			0.544768			0.272384	−	0.962189i	$-0.412188\pi$
$743$	404.762			0.544768			0.272384	+	0.962189i	$0.412188\pi$
$744$		0			0
$745$	467.008	+	759.855i	0.626857	+	1.01994i
$746$			208.684i			0.279738i
$747$		0			0
$748$		−	59.8082i		−	0.0799575i
$749$		−	265.448i		−	0.354403i
$750$		0			0
$751$	477.233			0.635464			0.317732	−	0.948181i	$-0.397079\pi$
$751$	477.233			0.635464			0.317732	+	0.948181i	$0.397079\pi$
$752$	−18.8940			−0.0251251
$753$		0			0
$754$	−844.804			−1.12043
$755$	1144.30	−	703.287i	1.51562	−	0.931506i
$756$		0			0
$757$			1011.95i			1.33679i	0.743806	+	0.668396i	$0.233019\pi$
$757$			1011.95i			1.33679i	−0.743806	+	0.668396i	$0.766981\pi$
$758$	−206.033			−0.271811
$759$		0			0
$760$	−175.040	+	107.580i	−0.230316	+	0.141553i
$761$		−	71.1576i		−	0.0935054i	−0.998906	−	0.0467527i	$-0.985113\pi$
$761$		−	71.1576i		−	0.0935054i	0.998906	−	0.0467527i	$-0.0148873\pi$
$762$		0			0
$763$		−	487.794i		−	0.639311i
$764$			226.328i			0.296241i
$765$		0			0
$766$	−133.836			−0.174720
$767$	1128.71			1.47159
$768$		0			0
$769$	960.734			1.24933			0.624664	−	0.780893i	$-0.285236\pi$
$769$	960.734			1.24933			0.624664	+	0.780893i	$0.285236\pi$
$770$	142.028	−	87.2906i	0.184452	−	0.113364i
$771$		0			0
$772$		−	236.973i		−	0.306959i
$773$	−1055.23			−1.36511			−0.682554	−	0.730835i	$-0.739131\pi$
$773$	−1055.23			−1.36511			−0.682554	+	0.730835i	$0.739131\pi$
$774$		0			0
$775$	−487.330	+	962.659i	−0.628813	+	1.24214i
$776$		−	28.5084i		−	0.0367377i
$777$		0			0
$778$		−	194.932i		−	0.250555i
$779$		−	477.571i		−	0.613056i
$780$		0			0
$781$	−359.565			−0.460390
$782$	−365.666			−0.467604
$783$		0			0
$784$	103.526			0.132048
$785$	678.632	+	1104.18i	0.864499	+	1.40660i
$786$		0			0
$787$			11.3469i			0.0144179i	0.999974	+	0.00720893i	$0.00229469\pi$
$787$			11.3469i			0.0144179i	−0.999974	+	0.00720893i	$0.997705\pi$
$788$	−495.732			−0.629102
$789$		0			0
$790$	749.303	−	460.523i	0.948485	−	0.582941i
$791$			749.979i			0.948140i
$792$		0			0
$793$		−	54.6424i		−	0.0689059i
$794$		−	1040.34i		−	1.31025i
$795$		0			0
$796$	167.724			0.210708
$797$	1004.53			1.26038			0.630192	−	0.776440i	$-0.282976\pi$
$797$	1004.53			1.26038			0.630192	+	0.776440i	$0.282976\pi$
$798$		0			0
$799$	−28.8075			−0.0360544
$800$	−63.8739	+	126.175i	−0.0798424	+	0.157719i
$801$		0			0
$802$			140.247i			0.174872i
$803$	−223.670			−0.278543
$804$		0			0
$805$	−533.693	−	868.355i	−0.662972	−	1.07870i
$806$			911.413i			1.13079i
$807$		0			0
$808$			441.044i			0.545846i
$809$			634.764i			0.784628i	0.919831	+	0.392314i	$0.128325\pi$
$809$			634.764i			0.784628i	−0.919831	+	0.392314i	$0.871675\pi$
$810$		0			0
$811$	1007.27			1.24201			0.621007	−	0.783805i	$-0.286724\pi$
$811$	1007.27			1.24201			0.621007	+	0.783805i	$0.286724\pi$
$812$	−384.703			−0.473772
$813$		0			0
$814$	−342.540			−0.420811
$815$	4.39274	+	7.14729i	0.00538986	+	0.00876968i
$816$		0			0
$817$			458.206i			0.560839i
$818$	1154.06			1.41083
$819$		0			0
$820$	−172.125	−	280.059i	−0.209908	−	0.341535i
$821$			312.574i			0.380724i	0.981714	+	0.190362i	$0.0609661\pi$
$821$			312.574i			0.380724i	−0.981714	+	0.190362i	$0.939034\pi$
$822$		0			0
$823$			547.719i			0.665515i	0.943012	+	0.332758i	$0.107979\pi$
$823$			547.719i			0.665515i	−0.943012	+	0.332758i	$0.892021\pi$
$824$			450.817i			0.547108i
$825$		0			0
$826$	513.987			0.622260
$827$	−1258.40			−1.52164			−0.760820	−	0.648963i	$-0.775203\pi$
$827$	−1258.40			−1.52164			−0.760820	+	0.648963i	$0.775203\pi$
$828$		0			0
$829$	1126.79			1.35922			0.679608	−	0.733576i	$-0.262150\pi$
$829$	1126.79			1.35922			0.679608	+	0.733576i	$0.262150\pi$
$830$	−504.189	+	309.876i	−0.607457	+	0.373344i
$831$		0			0
$832$			119.458i			0.143580i
$833$	157.844			0.189489
$834$		0			0
$835$	767.469	−	471.688i	0.919124	−	0.564896i
$836$		−	142.471i		−	0.170420i
$837$		0			0
$838$		−	255.539i		−	0.304939i
$839$		−	615.550i		−	0.733671i	−0.930286	−	0.366836i	$-0.880441\pi$
$839$		−	615.550i		−	0.733671i	0.930286	−	0.366836i	$-0.119559\pi$
$840$		0			0
$841$	−759.407			−0.902981
$842$	586.562			0.696630
$843$		0			0
$844$	−161.182			−0.190973
$845$	−229.912	+	141.305i	−0.272085	+	0.167224i
$846$		0			0
$847$		−	466.188i		−	0.550400i
$848$	79.7444			0.0940382
$849$		0			0
$850$	−97.3877	+	192.377i	−0.114574	+	0.226326i
$851$			2094.28i			2.46097i
$852$		0			0
$853$		−	623.863i		−	0.731375i	−0.930738	−	0.365688i	$-0.880834\pi$
$853$		−	623.863i		−	0.731375i	0.930738	−	0.365688i	$-0.119166\pi$
$854$		−	24.8828i		−	0.0291368i
$855$		0			0
$856$	−156.151			−0.182419
$857$	−1672.77			−1.95189			−0.975943	−	0.218025i	$-0.930039\pi$
$857$	−1672.77			−1.95189			−0.975943	+	0.218025i	$0.930039\pi$
$858$		0			0
$859$	−1005.64			−1.17071			−0.585353	−	0.810779i	$-0.699044\pi$
$859$	−1005.64			−1.17071			−0.585353	+	0.810779i	$0.699044\pi$
$860$	165.145	+	268.703i	0.192029	+	0.312445i
$861$		0			0
$862$		−	623.377i		−	0.723175i
$863$	−669.746			−0.776068			−0.388034	−	0.921645i	$-0.626846\pi$
$863$	−669.746			−0.776068			−0.388034	+	0.921645i	$0.626846\pi$
$864$		0			0
$865$	356.497	−	219.104i	0.412135	−	0.253299i
$866$		−	505.895i		−	0.584174i
$867$		0			0
$868$			415.035i			0.478151i
$869$			609.882i			0.701821i
$870$		0			0
$871$	886.125			1.01736
$872$	−286.947			−0.329067
$873$		0			0
$874$	−871.064			−0.996640
$875$	−598.980	+	49.5075i	−0.684548	+	0.0565800i
$876$		0			0
$877$			241.526i			0.275400i	0.990474	+	0.137700i	$0.0439710\pi$
$877$			241.526i			0.275400i	−0.990474	+	0.137700i	$0.956029\pi$
$878$	−273.103			−0.311052
$879$		0			0
$880$	−51.3490	−	83.5483i	−0.0583511	−	0.0949412i
$881$		−	557.597i		−	0.632913i	−0.948607	−	0.316457i	$-0.897507\pi$
$881$		−	557.597i		−	0.632913i	0.948607	−	0.316457i	$-0.102493\pi$
$882$		0			0
$883$		−	996.485i		−	1.12852i	−0.825596	−	0.564261i	$-0.809161\pi$
$883$		−	996.485i		−	1.12852i	0.825596	−	0.564261i	$-0.190839\pi$
$884$			182.136i			0.206036i
$885$		0			0
$886$	82.4050			0.0930079
$887$	−939.799			−1.05953			−0.529763	−	0.848146i	$-0.677719\pi$
$887$	−939.799			−1.05953			−0.529763	+	0.848146i	$0.677719\pi$
$888$		0			0
$889$	−557.310			−0.626896
$890$	−511.403	−	832.088i	−0.574610	−	0.934930i
$891$		0			0
$892$			65.7385i			0.0736979i
$893$	−68.6231			−0.0768455
$894$		0			0
$895$	132.367	+	215.370i	0.147896	+	0.240637i
$896$			54.3983i			0.0607124i
$897$		0			0
$898$			840.145i			0.935573i
$899$			1726.59i			1.92057i
$900$		0			0
$901$	121.585			0.134945
$902$	227.949			0.252715
$903$		0			0
$904$	441.177			0.488028
$905$	−817.339	+	502.339i	−0.903137	+	0.555070i
$906$		0			0
$907$		−	1335.88i		−	1.47286i	−0.676514	−	0.736430i	$-0.736510\pi$
$907$		−	1335.88i		−	1.47286i	0.676514	−	0.736430i	$-0.263490\pi$
$908$	183.329			0.201904
$909$		0			0
$910$	−432.522	+	265.829i	−0.475299	+	0.292120i
$911$			1418.35i			1.55692i	0.627695	+	0.778459i	$0.283999\pi$
$911$			1418.35i			1.55692i	−0.627695	+	0.778459i	$0.716001\pi$
$912$		0			0
$913$		−	410.376i		−	0.449481i
$914$			431.171i			0.471741i
$915$		0			0
$916$	147.389			0.160905
$917$	−321.073			−0.350134
$918$		0			0
$919$	1152.91			1.25452			0.627262	−	0.778808i	$-0.284176\pi$
$919$	1152.91			1.25452			0.627262	+	0.778808i	$0.284176\pi$
$920$	−510.813	+	313.947i	−0.555231	+	0.341246i
$921$		0			0
$922$			592.846i			0.643000i
$923$	1095.00			1.18634
$924$		0			0
$925$	1101.80	+	557.769i	1.19114	+	0.602994i
$926$			143.631i			0.155109i
$927$		0			0
$928$			226.303i			0.243861i
$929$			1316.30i			1.41690i	0.705762	+	0.708449i	$0.250605\pi$
$929$			1316.30i			1.41690i	−0.705762	+	0.708449i	$0.749395\pi$
$930$		0			0
$931$	376.005			0.403872
$932$	−26.7497			−0.0287014
$933$		0			0
$934$	631.141			0.675740
$935$	−78.2910	−	127.385i	−0.0837337	−	0.136241i
$936$		0			0
$937$			58.3567i			0.0622804i	0.999515	+	0.0311402i	$0.00991384\pi$
$937$			58.3567i			0.0622804i	−0.999515	+	0.0311402i	$0.990086\pi$
$938$	403.519			0.430191
$939$		0			0
$940$	−40.2422	+	24.7330i	−0.0428109	+	0.0263117i
$941$		−	411.509i		−	0.437310i	−0.975802	−	0.218655i	$-0.929833\pi$
$941$		−	411.509i		−	0.437310i	0.975802	−	0.218655i	$-0.0701670\pi$
$942$		0			0
$943$		−	1393.68i		−	1.47792i
$944$		−	302.354i		−	0.320291i
$945$		0			0
$946$	−218.706			−0.231190
$947$	35.2946			0.0372699			0.0186350	−	0.999826i	$-0.494068\pi$
$947$	35.2946			0.0372699			0.0186350	+	0.999826i	$0.494068\pi$
$948$		0			0
$949$	681.150			0.717755
$950$	−231.990	+	458.267i	−0.244200	+	0.482386i
$951$		0			0
$952$			82.9403i			0.0871222i
$953$	118.637			0.124488			0.0622440	−	0.998061i	$-0.480174\pi$
$953$	118.637			0.124488			0.0622440	+	0.998061i	$0.480174\pi$
$954$		0			0
$955$	296.271	+	482.053i	0.310231	+	0.504768i
$956$			447.421i			0.468013i
$957$		0			0
$958$		−	303.828i		−	0.317148i
$959$			344.505i			0.359234i
$960$		0			0
$961$	901.725			0.938320
$962$	1043.15			1.08436
$963$		0			0
$964$	−430.701			−0.446786
$965$	−310.205	−	504.725i	−0.321456	−	0.523031i
$966$		0			0
$967$		−	1055.06i		−	1.09107i	−0.838089	−	0.545533i	$-0.816327\pi$
$967$		−	1055.06i		−	1.09107i	0.838089	−	0.545533i	$-0.183673\pi$
$968$	−274.237			−0.283303
$969$		0			0
$970$	−37.3185	−	60.7198i	−0.0384727	−	0.0625978i
$971$		−	313.380i		−	0.322739i	−0.986894	−	0.161370i	$-0.948409\pi$
$971$		−	313.380i		−	0.322739i	0.986894	−	0.161370i	$-0.0515912\pi$
$972$		0			0
$973$		−	555.030i		−	0.570432i
$974$		−	213.883i		−	0.219593i
$975$		0			0
$976$	−14.6374			−0.0149973
$977$	295.679			0.302639			0.151320	−	0.988485i	$-0.451648\pi$
$977$	295.679			0.302639			0.151320	+	0.988485i	$0.451648\pi$
$978$		0			0
$979$	677.263			0.691791
$980$	220.498	−	135.519i	0.224998	−	0.138284i
$981$		0			0
$982$		−	717.085i		−	0.730229i
$983$	−1828.74			−1.86037			−0.930185	−	0.367092i	$-0.880354\pi$
$983$	−1828.74			−1.86037			−0.930185	+	0.367092i	$0.880354\pi$
$984$		0			0
$985$	−1055.86	+	648.931i	−1.07193	+	0.658813i
$986$			345.041i			0.349940i
$987$		0			0
$988$			433.872i			0.439141i
$989$			1337.16i			1.35204i
$990$		0			0
$991$	−1935.94			−1.95352			−0.976760	−	0.214338i	$-0.931241\pi$
$991$	−1935.94			−1.95352			−0.976760	+	0.214338i	$0.931241\pi$
$992$	244.146			0.246115
$993$		0			0
$994$	498.634			0.501644
$995$	357.233	−	219.556i	0.359029	−	0.220660i
$996$		0			0
$997$		−	1689.33i		−	1.69441i	−0.531263	−	0.847207i	$-0.678282\pi$
$997$		−	1689.33i		−	1.69441i	0.531263	−	0.847207i	$-0.321718\pi$
$998$	−1320.13			−1.32278
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.b.809.7		16
3.2	odd	2	inner	810.3.b.b.809.10		16
5.4	even	2	inner	810.3.b.b.809.9		16
9.2	odd	6		90.3.j.b.59.2	yes	16
9.4	even	3		90.3.j.b.29.7	yes	16
9.5	odd	6		270.3.j.b.89.4		16
9.7	even	3		270.3.j.b.179.8		16
15.14	odd	2	inner	810.3.b.b.809.8		16
45.2	even	12		450.3.i.e.401.5		16
45.4	even	6		90.3.j.b.29.2	✓	16
45.7	odd	12		1350.3.i.e.1151.2		16
45.13	odd	12		450.3.i.e.101.4		16
45.14	odd	6		270.3.j.b.89.8		16
45.22	odd	12		450.3.i.e.101.5		16
45.23	even	12		1350.3.i.e.251.7		16
45.29	odd	6		90.3.j.b.59.7	yes	16
45.32	even	12		1350.3.i.e.251.2		16
45.34	even	6		270.3.j.b.179.4		16
45.38	even	12		450.3.i.e.401.4		16
45.43	odd	12		1350.3.i.e.1151.7		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.j.b.29.2	✓	16	45.4	even	6
90.3.j.b.29.7	yes	16	9.4	even	3
90.3.j.b.59.2	yes	16	9.2	odd	6
90.3.j.b.59.7	yes	16	45.29	odd	6
270.3.j.b.89.4		16	9.5	odd	6
270.3.j.b.89.8		16	45.14	odd	6
270.3.j.b.179.4		16	45.34	even	6
270.3.j.b.179.8		16	9.7	even	3
450.3.i.e.101.4		16	45.13	odd	12
450.3.i.e.101.5		16	45.22	odd	12
450.3.i.e.401.4		16	45.38	even	12
450.3.i.e.401.5		16	45.2	even	12
810.3.b.b.809.7		16	1.1	even	1	trivial
810.3.b.b.809.8		16	15.14	odd	2	inner
810.3.b.b.809.9		16	5.4	even	2	inner
810.3.b.b.809.10		16	3.2	odd	2	inner
1350.3.i.e.251.2		16	45.32	even	12
1350.3.i.e.251.7		16	45.23	even	12
1350.3.i.e.1151.2		16	45.7	odd	12
1350.3.i.e.1151.7		16	45.43	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(4.25978 - 2.61807i) q^{5} -4.80818i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(4.25978 - 2.61807i) q^{5} -4.80818i q^{7} -2.82843 q^{8} +(-6.02424 + 3.70251i) q^{10} -4.90332i q^{11} +14.9323i q^{13} +6.79979i q^{14} +4.00000 q^{16} +6.09874 q^{17} +14.5280 q^{19} +(8.51956 - 5.23614i) q^{20} +6.93435i q^{22} +42.3965 q^{23} +(11.2914 - 22.3048i) q^{25} -21.1174i q^{26} -9.61635i q^{28} -40.0051i q^{29} -43.1593 q^{31} -5.65685 q^{32} -8.62493 q^{34} +(-12.5881 - 20.4818i) q^{35} +49.3976i q^{37} -20.5457 q^{38} +(-12.0485 + 7.40502i) q^{40} -32.8725i q^{41} +31.5395i q^{43} -9.80665i q^{44} -59.9577 q^{46} -4.72351 q^{47} +25.8814 q^{49} +(-15.9685 + 31.5437i) q^{50} +29.8645i q^{52} +19.9361 q^{53} +(-12.8372 - 20.8871i) q^{55} +13.5996i q^{56} +56.5757i q^{58} -75.5886i q^{59} -3.65935 q^{61} +61.0365 q^{62} +8.00000 q^{64} +(39.0937 + 63.6082i) q^{65} -59.3429i q^{67} +12.1975 q^{68} +(17.8023 + 28.9656i) q^{70} -73.3308i q^{71} -45.6159i q^{73} -69.8588i q^{74} +29.0560 q^{76} -23.5761 q^{77} -124.381 q^{79} +(17.0391 - 10.4723i) q^{80} +46.4887i q^{82} +83.6935 q^{83} +(25.9793 - 15.9669i) q^{85} -44.6036i q^{86} +13.8687i q^{88} +138.123i q^{89} +71.7970 q^{91} +84.7929 q^{92} +6.68005 q^{94} +(61.8860 - 38.0353i) q^{95} +10.0793i q^{97} -36.6019 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 32 q^{4}+O(q^{10}) \) 16 * q + 32 * q^4	\( 16 q + 32 q^{4} - 16 q^{10} + 64 q^{16} + 144 q^{19} - 12 q^{25} - 56 q^{31} + 272 q^{34} - 32 q^{40} - 56 q^{46} - 24 q^{49} + 20 q^{55} - 136 q^{61} + 128 q^{64} + 224 q^{70} + 288 q^{76} - 840 q^{79} + 272 q^{85} - 168 q^{91} + 328 q^{94}+O(q^{100}) \) 16 * q + 32 * q^4 - 16 * q^10 + 64 * q^16 + 144 * q^19 - 12 * q^25 - 56 * q^31 + 272 * q^34 - 32 * q^40 - 56 * q^46 - 24 * q^49 + 20 * q^55 - 136 * q^61 + 128 * q^64 + 224 * q^70 + 288 * q^76 - 840 * q^79 + 272 * q^85 - 168 * q^91 + 328 * q^94

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