LMFDB - Embedded newform 810.3.b.c.809.2

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:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			809.2
Character	$\chi$	$=$	810.809
Dual form			810.3.b.c.809.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +2.00000 q^{4} +(-4.81687 + 1.34080i) q^{5} +0.364778i q^{7} -2.82843 q^{8} +O(q^{10})$	\(q-1.41421 q^{2} +2.00000 q^{4} +(-4.81687 + 1.34080i) q^{5} +0.364778i q^{7} -2.82843 q^{8} +(6.81209 - 1.89618i) q^{10} -9.61330i q^{11} +7.65513i q^{13} -0.515874i q^{14} +4.00000 q^{16} -12.3400 q^{17} +5.36509 q^{19} +(-9.63374 + 2.68160i) q^{20} +13.5953i q^{22} -12.0661 q^{23} +(21.4045 - 12.9169i) q^{25} -10.8260i q^{26} +0.729555i q^{28} -31.3949i q^{29} +4.91964 q^{31} -5.65685 q^{32} +17.4513 q^{34} +(-0.489094 - 1.75709i) q^{35} +40.2567i q^{37} -7.58739 q^{38} +(13.6242 - 3.79235i) q^{40} +63.2951i q^{41} -53.8139i q^{43} -19.2266i q^{44} +17.0640 q^{46} +28.2335 q^{47} +48.8669 q^{49} +(-30.2706 + 18.2673i) q^{50} +15.3103i q^{52} +41.6903 q^{53} +(12.8895 + 46.3060i) q^{55} -1.03175i q^{56} +44.3991i q^{58} +112.254i q^{59} +45.9575 q^{61} -6.95743 q^{62} +8.00000 q^{64} +(-10.2640 - 36.8738i) q^{65} -46.1412i q^{67} -24.6799 q^{68} +(0.691683 + 2.48490i) q^{70} -125.617i q^{71} +59.0826i q^{73} -56.9316i q^{74} +10.7302 q^{76} +3.50672 q^{77} +50.6871 q^{79} +(-19.2675 + 5.36320i) q^{80} -89.5128i q^{82} +59.0365 q^{83} +(59.4400 - 16.5454i) q^{85} +76.1044i q^{86} +27.1905i q^{88} +8.93057i q^{89} -2.79242 q^{91} -24.1322 q^{92} -39.9282 q^{94} +(-25.8430 + 7.19351i) q^{95} +131.306i q^{97} -69.1083 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 48 q^{4}+O(q^{10}) $ 24 * q + 48 * q^4	$ 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) $ 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * 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
$5$	−4.81687	+	1.34080i	−0.963374	+	0.268160i
$5$	−4.81687	+	1.34080i	−0.963374	+	0.268160i
$6$		0			0
$7$			0.364778i			0.0521111i	0.999660	+	0.0260556i	$0.00829468\pi$
$7$			0.364778i			0.0521111i	−0.999660	+	0.0260556i	$0.991705\pi$
$8$	−2.82843			−0.353553
$9$		0			0
$10$	6.81209	−	1.89618i	0.681209	−	0.189618i
$11$		−	9.61330i		−	0.873937i	−0.899477	−	0.436968i	$-0.856052\pi$
$11$		−	9.61330i		−	0.873937i	0.899477	−	0.436968i	$-0.143948\pi$
$12$		0			0
$13$			7.65513i			0.588856i	0.955674	+	0.294428i	$0.0951292\pi$
$13$			7.65513i			0.588856i	−0.955674	+	0.294428i	$0.904871\pi$
$14$		−	0.515874i		−	0.0368481i
$15$		0			0
$16$	4.00000			0.250000
$17$	−12.3400			−0.725879			−0.362940	−	0.931813i	$-0.618227\pi$
$17$	−12.3400			−0.725879			−0.362940	+	0.931813i	$0.618227\pi$
$18$		0			0
$19$	5.36509			0.282373			0.141187	−	0.989983i	$-0.454908\pi$
$19$	5.36509			0.282373			0.141187	+	0.989983i	$0.454908\pi$
$20$	−9.63374	+	2.68160i	−0.481687	+	0.134080i
$21$		0			0
$22$			13.5953i			0.617966i
$23$	−12.0661			−0.524613			−0.262307	−	0.964985i	$-0.584483\pi$
$23$	−12.0661			−0.524613			−0.262307	+	0.964985i	$0.584483\pi$
$24$		0			0
$25$	21.4045	−	12.9169i	0.856181	−	0.516677i
$26$		−	10.8260i		−	0.416384i
$27$		0			0
$28$			0.729555i			0.0260556i
$29$		−	31.3949i		−	1.08258i	−0.840835	−	0.541292i	$-0.817936\pi$
$29$		−	31.3949i		−	1.08258i	0.840835	−	0.541292i	$-0.182064\pi$
$30$		0			0
$31$	4.91964			0.158698			0.0793491	−	0.996847i	$-0.474716\pi$
$31$	4.91964			0.158698			0.0793491	+	0.996847i	$0.474716\pi$
$32$	−5.65685			−0.176777
$33$		0			0
$34$	17.4513			0.513274
$35$	−0.489094	−	1.75709i	−0.0139741	−	0.0502025i
$36$		0			0
$37$			40.2567i			1.08802i	0.839079	+	0.544010i	$0.183095\pi$
$37$			40.2567i			1.08802i	−0.839079	+	0.544010i	$0.816905\pi$
$38$	−7.58739			−0.199668
$39$		0			0
$40$	13.6242	−	3.79235i	0.340604	−	0.0948088i
$41$			63.2951i			1.54378i	0.635754	+	0.771892i	$0.280689\pi$
$41$			63.2951i			1.54378i	−0.635754	+	0.771892i	$0.719311\pi$
$42$		0			0
$43$		−	53.8139i		−	1.25149i	−0.780029	−	0.625743i	$-0.784796\pi$
$43$		−	53.8139i		−	1.25149i	0.780029	−	0.625743i	$-0.215204\pi$
$44$		−	19.2266i		−	0.436968i
$45$		0			0
$46$	17.0640			0.370957
$47$	28.2335			0.600713			0.300356	−	0.953827i	$-0.402894\pi$
$47$	28.2335			0.600713			0.300356	+	0.953827i	$0.402894\pi$
$48$		0			0
$49$	48.8669			0.997284
$50$	−30.2706	+	18.2673i	−0.605411	+	0.365346i
$51$		0			0
$52$			15.3103i			0.294428i
$53$	41.6903			0.786610			0.393305	−	0.919408i	$-0.371332\pi$
$53$	41.6903			0.786610			0.393305	+	0.919408i	$0.371332\pi$
$54$		0			0
$55$	12.8895	+	46.3060i	0.234355	+	0.841928i
$56$		−	1.03175i		−	0.0184241i
$57$		0			0
$58$			44.3991i			0.765502i
$59$			112.254i			1.90261i	0.308248	+	0.951306i	$0.400257\pi$
$59$			112.254i			1.90261i	−0.308248	+	0.951306i	$0.599743\pi$
$60$		0			0
$61$	45.9575			0.753401			0.376701	−	0.926335i	$-0.377059\pi$
$61$	45.9575			0.753401			0.376701	+	0.926335i	$0.377059\pi$
$62$	−6.95743			−0.112217
$63$		0			0
$64$	8.00000			0.125000
$65$	−10.2640	−	36.8738i	−0.157908	−	0.567289i
$66$		0			0
$67$		−	46.1412i		−	0.688674i	−0.938846	−	0.344337i	$-0.888104\pi$
$67$		−	46.1412i		−	0.688674i	0.938846	−	0.344337i	$-0.111896\pi$
$68$	−24.6799			−0.362940
$69$		0			0
$70$	0.691683	+	2.48490i	0.00988118	+	0.0354985i
$71$		−	125.617i		−	1.76926i	−0.466295	−	0.884629i	$-0.654412\pi$
$71$		−	125.617i		−	1.76926i	0.466295	−	0.884629i	$-0.345588\pi$
$72$		0			0
$73$			59.0826i			0.809351i	0.914460	+	0.404676i	$0.132616\pi$
$73$			59.0826i			0.809351i	−0.914460	+	0.404676i	$0.867384\pi$
$74$		−	56.9316i		−	0.769346i
$75$		0			0
$76$	10.7302			0.141187
$77$	3.50672			0.0455418
$78$		0			0
$79$	50.6871			0.641609			0.320804	−	0.947145i	$-0.396047\pi$
$79$	50.6871			0.641609			0.320804	+	0.947145i	$0.396047\pi$
$80$	−19.2675	+	5.36320i	−0.240844	+	0.0670400i
$81$		0			0
$82$		−	89.5128i		−	1.09162i
$83$	59.0365			0.711284			0.355642	−	0.934622i	$-0.384262\pi$
$83$	59.0365			0.711284			0.355642	+	0.934622i	$0.384262\pi$
$84$		0			0
$85$	59.4400	−	16.5454i	0.699294	−	0.194652i
$86$			76.1044i			0.884935i
$87$		0			0
$88$			27.1905i			0.308983i
$89$			8.93057i			0.100344i	0.998741	+	0.0501718i	$0.0159769\pi$
$89$			8.93057i			0.100344i	−0.998741	+	0.0501718i	$0.984023\pi$
$90$		0			0
$91$	−2.79242			−0.0306860
$92$	−24.1322			−0.262307
$93$		0			0
$94$	−39.9282			−0.424768
$95$	−25.8430	+	7.19351i	−0.272031	+	0.0757212i
$96$		0			0
$97$			131.306i			1.35367i	0.736134	+	0.676835i	$0.236649\pi$
$97$			131.306i			1.35367i	−0.736134	+	0.676835i	$0.763351\pi$
$98$	−69.1083			−0.705187
$99$		0			0
$100$	42.8090	−	25.8338i	0.428090	−	0.258338i
$101$			10.9479i			0.108395i	0.998530	+	0.0541975i	$0.0172601\pi$
$101$			10.9479i			0.108395i	−0.998530	+	0.0541975i	$0.982740\pi$
$102$		0			0
$103$			52.4574i			0.509295i	0.967034	+	0.254648i	$0.0819595\pi$
$103$			52.4574i			0.509295i	−0.967034	+	0.254648i	$0.918041\pi$
$104$		−	21.6520i		−	0.208192i
$105$		0			0
$106$	−58.9590			−0.556217
$107$	189.947			1.77521			0.887603	−	0.460608i	$-0.152369\pi$
$107$	189.947			1.77521			0.887603	+	0.460608i	$0.152369\pi$
$108$		0			0
$109$	142.655			1.30876			0.654382	−	0.756164i	$-0.272929\pi$
$109$	142.655			1.30876			0.654382	+	0.756164i	$0.272929\pi$
$110$	−18.2285	−	65.4866i	−0.165714	−	0.595333i
$111$		0			0
$112$			1.45911i			0.0130278i
$113$	81.6780			0.722814			0.361407	−	0.932408i	$-0.382297\pi$
$113$	81.6780			0.722814			0.361407	+	0.932408i	$0.382297\pi$
$114$		0			0
$115$	58.1209	−	16.1782i	0.505399	−	0.140680i
$116$		−	62.7898i		−	0.541292i
$117$		0			0
$118$		−	158.751i		−	1.34535i
$119$		−	4.50134i		−	0.0378264i
$120$		0			0
$121$	28.5844			0.236235
$122$	−64.9937			−0.532735
$123$		0			0
$124$	9.83929			0.0793491
$125$	−85.7838	+	90.9183i	−0.686271	+	0.727346i
$126$		0			0
$127$		−	83.4011i		−	0.656701i	−0.944556	−	0.328351i	$-0.893507\pi$
$127$		−	83.4011i		−	0.656701i	0.944556	−	0.328351i	$-0.106493\pi$
$128$	−11.3137			−0.0883883
$129$		0			0
$130$	14.5155	+	52.1474i	0.111658	+	0.401134i
$131$			113.896i			0.869431i	0.900568	+	0.434716i	$0.143151\pi$
$131$			113.896i			0.869431i	−0.900568	+	0.434716i	$0.856849\pi$
$132$		0			0
$133$			1.95707i			0.0147148i
$134$			65.2535i			0.486966i
$135$		0			0
$136$	34.9027			0.256637
$137$	152.734			1.11485			0.557425	−	0.830227i	$-0.311789\pi$
$137$	152.734			1.11485			0.557425	+	0.830227i	$0.311789\pi$
$138$		0			0
$139$	100.550			0.723382			0.361691	−	0.932298i	$-0.382199\pi$
$139$	100.550			0.723382			0.361691	+	0.932298i	$0.382199\pi$
$140$	−0.978187	−	3.51418i	−0.00698705	−	0.0251013i
$141$		0			0
$142$			177.650i			1.25105i
$143$	73.5911			0.514623
$144$		0			0
$145$	42.0943	+	151.225i	0.290305	+	1.04293i
$146$		−	83.5555i		−	0.572298i
$147$		0			0
$148$			80.5135i			0.544010i
$149$		−	25.9785i		−	0.174352i	−0.996193	−	0.0871761i	$-0.972216\pi$
$149$		−	25.9785i		−	0.174352i	0.996193	−	0.0871761i	$-0.0277843\pi$
$150$		0			0
$151$	−155.954			−1.03281			−0.516403	−	0.856346i	$-0.672729\pi$
$151$	−155.954			−1.03281			−0.516403	+	0.856346i	$0.672729\pi$
$152$	−15.1748			−0.0998340
$153$		0			0
$154$	−4.95925			−0.0322029
$155$	−23.6973	+	6.59625i	−0.152886	+	0.0425565i
$156$		0			0
$157$		−	252.704i		−	1.60958i	−0.593560	−	0.804790i	$-0.702278\pi$
$157$		−	252.704i		−	1.60958i	0.593560	−	0.804790i	$-0.297722\pi$
$158$	−71.6824			−0.453686
$159$		0			0
$160$	27.2483	−	7.58471i	0.170302	−	0.0474044i
$161$		−	4.40144i		−	0.0273382i
$162$		0			0
$163$			156.387i			0.959430i	0.877424	+	0.479715i	$0.159260\pi$
$163$			156.387i			0.959430i	−0.877424	+	0.479715i	$0.840740\pi$
$164$			126.590i			0.771892i
$165$		0			0
$166$	−83.4903			−0.502953
$167$	320.062			1.91654			0.958268	−	0.285870i	$-0.0922825\pi$
$167$	320.062			1.91654			0.958268	+	0.285870i	$0.0922825\pi$
$168$		0			0
$169$	110.399			0.653248
$170$	−84.0608	+	23.3987i	−0.494475	+	0.137640i
$171$		0			0
$172$		−	107.628i		−	0.625743i
$173$	39.6720			0.229318			0.114659	−	0.993405i	$-0.463422\pi$
$173$	39.6720			0.229318			0.114659	+	0.993405i	$0.463422\pi$
$174$		0			0
$175$	4.71180	+	7.80789i	0.0269246	+	0.0446165i
$176$		−	38.4532i		−	0.218484i
$177$		0			0
$178$		−	12.6297i		−	0.0709536i
$179$			51.3204i			0.286706i	0.989672	+	0.143353i	$0.0457884\pi$
$179$			51.3204i			0.286706i	−0.989672	+	0.143353i	$0.954212\pi$
$180$		0			0
$181$	−298.058			−1.64673			−0.823363	−	0.567515i	$-0.807905\pi$
$181$	−298.058			−1.64673			−0.823363	+	0.567515i	$0.807905\pi$
$182$	3.94908			0.0216982
$183$		0			0
$184$	34.1281			0.185479
$185$	−53.9762	−	193.912i	−0.291763	−	1.04817i
$186$		0			0
$187$			118.628i			0.634373i
$188$	56.4670			0.300356
$189$		0			0
$190$	36.5475	−	10.1732i	0.192355	−	0.0535430i
$191$			214.897i			1.12511i	0.826759	+	0.562557i	$0.190182\pi$
$191$			214.897i			1.12511i	−0.826759	+	0.562557i	$0.809818\pi$
$192$		0			0
$193$		−	175.722i		−	0.910475i	−0.890370	−	0.455237i	$-0.849554\pi$
$193$		−	175.722i		−	0.910475i	0.890370	−	0.455237i	$-0.150446\pi$
$194$		−	185.695i		−	0.957190i
$195$		0			0
$196$	97.7339			0.498642
$197$	147.580			0.749136			0.374568	−	0.927200i	$-0.377791\pi$
$197$	147.580			0.749136			0.374568	+	0.927200i	$0.377791\pi$
$198$		0			0
$199$	−192.973			−0.969716			−0.484858	−	0.874593i	$-0.661129\pi$
$199$	−192.973			−0.969716			−0.484858	+	0.874593i	$0.661129\pi$
$200$	−60.5411	+	36.5346i	−0.302706	+	0.182673i
$201$		0			0
$202$		−	15.4827i		−	0.0766469i
$203$	11.4522			0.0564146
$204$		0			0
$205$	−84.8660	−	304.884i	−0.413981	−	1.48724i
$206$		−	74.1860i		−	0.360126i
$207$		0			0
$208$			30.6205i			0.147214i
$209$		−	51.5763i		−	0.246776i
$210$		0			0
$211$	−383.899			−1.81942			−0.909712	−	0.415239i	$-0.863698\pi$
$211$	−383.899			−1.81942			−0.909712	+	0.415239i	$0.863698\pi$
$212$	83.3806			0.393305
$213$		0			0
$214$	−268.626			−1.25526
$215$	72.1537	+	259.215i	0.335598	+	1.20565i
$216$		0			0
$217$			1.79458i			0.00826994i
$218$	−201.745			−0.925436
$219$		0			0
$220$	25.7790	+	92.6121i	0.117177	+	0.420964i
$221$		−	94.4640i		−	0.427439i
$222$		0			0
$223$		−	345.549i		−	1.54955i	−0.632239	−	0.774773i	$-0.717864\pi$
$223$		−	345.549i		−	1.54955i	0.632239	−	0.774773i	$-0.282136\pi$
$224$		−	2.06349i		−	0.00921203i
$225$		0			0
$226$	−115.510			−0.511107
$227$	272.796			1.20174			0.600872	−	0.799346i	$-0.294820\pi$
$227$	272.796			1.20174			0.600872	+	0.799346i	$0.294820\pi$
$228$		0			0
$229$	0.104243			0.000455208			0.000227604	−	1.00000i	$-0.499928\pi$
$229$	0.104243			0.000455208			0.000227604		1.00000i	$0.499928\pi$
$230$	−82.1953	+	22.8795i	−0.357371	+	0.0994759i
$231$		0			0
$232$			88.7982i			0.382751i
$233$	−317.302			−1.36181			−0.680905	−	0.732372i	$-0.738413\pi$
$233$	−317.302			−1.36181			−0.680905	+	0.732372i	$0.738413\pi$
$234$		0			0
$235$	−135.997	+	37.8554i	−0.578711	+	0.161087i
$236$			224.508i			0.951306i
$237$		0			0
$238$			6.36586i			0.0267473i
$239$		−	193.025i		−	0.807634i	−0.914840	−	0.403817i	$-0.867683\pi$
$239$		−	193.025i		−	0.807634i	0.914840	−	0.403817i	$-0.132317\pi$
$240$		0			0
$241$	−314.726			−1.30592			−0.652959	−	0.757393i	$-0.726473\pi$
$241$	−314.726			−1.30592			−0.652959	+	0.757393i	$0.726473\pi$
$242$	−40.4245			−0.167043
$243$		0			0
$244$	91.9149			0.376701
$245$	−235.386	+	65.5207i	−0.960758	+	0.267432i
$246$		0			0
$247$			41.0705i			0.166277i
$248$	−13.9149			−0.0561083
$249$		0			0
$250$	121.317	−	128.578i	0.485267	−	0.514312i
$251$		−	57.1694i		−	0.227767i	−0.993494	−	0.113883i	$-0.963671\pi$
$251$		−	57.1694i		−	0.227767i	0.993494	−	0.113883i	$-0.0363290\pi$
$252$		0			0
$253$			115.995i			0.458479i
$254$			117.947i			0.464358i
$255$		0			0
$256$	16.0000			0.0625000
$257$	71.7120			0.279035			0.139517	−	0.990220i	$-0.455445\pi$
$257$	71.7120			0.279035			0.139517	+	0.990220i	$0.455445\pi$
$258$		0			0
$259$	−14.6848			−0.0566979
$260$	−20.5280	−	73.7476i	−0.0789538	−	0.283645i
$261$		0			0
$262$		−	161.073i		−	0.614781i
$263$	−228.470			−0.868707			−0.434353	−	0.900743i	$-0.643023\pi$
$263$	−228.470			−0.868707			−0.434353	+	0.900743i	$0.643023\pi$
$264$		0			0
$265$	−200.817	+	55.8983i	−0.757800	+	0.210937i
$266$		−	2.76771i		−	0.0104049i
$267$		0			0
$268$		−	92.2823i		−	0.344337i
$269$			8.79629i			0.0326999i	0.999866	+	0.0163500i	$0.00520459\pi$
$269$			8.79629i			0.0326999i	−0.999866	+	0.0163500i	$0.994795\pi$
$270$		0			0
$271$	0.783006			0.00288932			0.00144466	−	0.999999i	$-0.499540\pi$
$271$	0.783006			0.00288932			0.00144466	+	0.999999i	$0.499540\pi$
$272$	−49.3598			−0.181470
$273$		0			0
$274$	−215.999			−0.788318
$275$	−124.174	−	205.768i	−0.451543	−	0.748248i
$276$		0			0
$277$			472.182i			1.70463i	0.523032	+	0.852313i	$0.324801\pi$
$277$			472.182i			1.70463i	−0.523032	+	0.852313i	$0.675199\pi$
$278$	−142.199			−0.511509
$279$		0			0
$280$	1.38337	+	4.96979i	0.00494059	+	0.0177493i
$281$			237.444i			0.844998i	0.906363	+	0.422499i	$0.138847\pi$
$281$			237.444i			0.844998i	−0.906363	+	0.422499i	$0.861153\pi$
$282$		0			0
$283$			266.347i			0.941155i	0.882359	+	0.470577i	$0.155954\pi$
$283$			266.347i			0.941155i	−0.882359	+	0.470577i	$0.844046\pi$
$284$		−	251.235i		−	0.884629i
$285$		0			0
$286$	−104.074			−0.363894
$287$	−23.0886			−0.0804482
$288$		0			0
$289$	−136.726			−0.473099
$290$	−59.5303	−	213.865i	−0.205277	−	0.737465i
$291$		0			0
$292$			118.165i			0.404676i
$293$	−331.586			−1.13169			−0.565847	−	0.824511i	$-0.691451\pi$
$293$	−331.586			−1.13169			−0.565847	+	0.824511i	$0.691451\pi$
$294$		0			0
$295$	−150.510	−	540.714i	−0.510204	−	1.83293i
$296$		−	113.863i		−	0.384673i
$297$		0			0
$298$			36.7391i			0.123286i
$299$		−	92.3676i		−	0.308922i
$300$		0			0
$301$	19.6301			0.0652163
$302$	220.552			0.730304
$303$		0			0
$304$	21.4604			0.0705933
$305$	−221.371	+	61.6197i	−0.725807	+	0.202032i
$306$		0			0
$307$		−	414.087i		−	1.34882i	−0.738358	−	0.674409i	$-0.764399\pi$
$307$		−	414.087i		−	1.34882i	0.738358	−	0.674409i	$-0.235601\pi$
$308$	7.01344			0.0227709
$309$		0			0
$310$	33.5130	−	9.32851i	0.108107	−	0.0300920i
$311$		−	493.339i		−	1.58630i	−0.609027	−	0.793149i	$-0.708440\pi$
$311$		−	493.339i		−	1.58630i	0.609027	−	0.793149i	$-0.291560\pi$
$312$		0			0
$313$		−	110.095i		−	0.351742i	−0.984413	−	0.175871i	$-0.943726\pi$
$313$		−	110.095i		−	0.351742i	0.984413	−	0.175871i	$-0.0562741\pi$
$314$			357.377i			1.13814i
$315$		0			0
$316$	101.374			0.320804
$317$	−99.5759			−0.314120			−0.157060	−	0.987589i	$-0.550202\pi$
$317$	−99.5759			−0.314120			−0.157060	+	0.987589i	$0.550202\pi$
$318$		0			0
$319$	−301.809			−0.946109
$320$	−38.5350	+	10.7264i	−0.120422	+	0.0335200i
$321$		0			0
$322$			6.22458i			0.0193310i
$323$	−66.2050			−0.204969
$324$		0			0
$325$	98.8807	+	163.854i	0.304248	+	0.504167i
$326$		−	221.165i		−	0.678419i
$327$		0			0
$328$		−	179.026i		−	0.545810i
$329$			10.2990i			0.0313038i
$330$		0			0
$331$	198.808			0.600627			0.300314	−	0.953841i	$-0.402909\pi$
$331$	198.808			0.600627			0.300314	+	0.953841i	$0.402909\pi$
$332$	118.073			0.355642
$333$		0			0
$334$	−452.636			−1.35520
$335$	61.8660	+	222.256i	0.184675	+	0.663451i
$336$		0			0
$337$		−	324.790i		−	0.963768i	−0.876235	−	0.481884i	$-0.839953\pi$
$337$		−	324.790i		−	0.963768i	0.876235	−	0.481884i	$-0.160047\pi$
$338$	−156.128			−0.461916
$339$		0			0
$340$	118.880	−	33.0908i	0.349647	−	0.0973259i
$341$		−	47.2940i		−	0.138692i
$342$		0			0
$343$			35.6997i			0.104081i
$344$			152.209i			0.442467i
$345$		0			0
$346$	−56.1047			−0.162152
$347$	441.359			1.27193			0.635963	−	0.771719i	$-0.280603\pi$
$347$	441.359			1.27193			0.635963	+	0.771719i	$0.280603\pi$
$348$		0			0
$349$	−0.134775			−0.000386176			−0.000193088	−	1.00000i	$-0.500061\pi$
$349$	−0.134775			−0.000386176			−0.000193088		1.00000i	$0.500061\pi$
$350$	−6.66350	−	11.0420i	−0.0190386	−	0.0315486i
$351$		0			0
$352$			54.3811i			0.154492i
$353$	−522.249			−1.47946			−0.739729	−	0.672905i	$-0.765046\pi$
$353$	−522.249			−1.47946			−0.739729	+	0.672905i	$0.765046\pi$
$354$		0			0
$355$	168.428	+	605.083i	0.474444	+	1.70446i
$356$			17.8611i			0.0501718i
$357$		0			0
$358$		−	72.5780i		−	0.202732i
$359$			146.484i			0.408034i	0.978967	+	0.204017i	$0.0653998\pi$
$359$			146.484i			0.408034i	−0.978967	+	0.204017i	$0.934600\pi$
$360$		0			0
$361$	−332.216			−0.920265
$362$	421.517			1.16441
$363$		0			0
$364$	−5.58484			−0.0153430
$365$	−79.2179	−	284.593i	−0.217035	−	0.779708i
$366$		0			0
$367$			339.675i			0.925546i	0.886477	+	0.462773i	$0.153145\pi$
$367$			339.675i			0.925546i	−0.886477	+	0.462773i	$0.846855\pi$
$368$	−48.2644			−0.131153
$369$		0			0
$370$	76.3339	+	274.232i	0.206308	+	0.741168i
$371$			15.2077i			0.0409911i
$372$		0			0
$373$		−	436.780i		−	1.17099i	−0.810676	−	0.585496i	$-0.800900\pi$
$373$		−	436.780i		−	1.17099i	0.810676	−	0.585496i	$-0.199100\pi$
$374$		−	167.765i		−	0.448569i
$375$		0			0
$376$	−79.8564			−0.212384
$377$	240.332			0.637486
$378$		0			0
$379$	677.751			1.78826			0.894130	−	0.447807i	$-0.147795\pi$
$379$	677.751			1.78826			0.894130	+	0.447807i	$0.147795\pi$
$380$	−51.6859	+	14.3870i	−0.136016	+	0.0378606i
$381$		0			0
$382$		−	303.910i		−	0.795575i
$383$	145.114			0.378888			0.189444	−	0.981892i	$-0.439331\pi$
$383$	145.114			0.378888			0.189444	+	0.981892i	$0.439331\pi$
$384$		0			0
$385$	−16.8914	+	4.70181i	−0.0438738	+	0.0122125i
$386$			248.508i			0.643803i
$387$		0			0
$388$			262.612i			0.676835i
$389$		−	31.9693i		−	0.0821833i	−0.999155	−	0.0410916i	$-0.986916\pi$
$389$		−	31.9693i		−	0.0821833i	0.999155	−	0.0410916i	$-0.0130836\pi$
$390$		0			0
$391$	148.895			0.380806
$392$	−138.217			−0.352593
$393$		0			0
$394$	−208.709			−0.529719
$395$	−244.153	+	67.9612i	−0.618109	+	0.172054i
$396$		0			0
$397$		−	399.771i		−	1.00698i	−0.864001	−	0.503490i	$-0.832049\pi$
$397$		−	399.771i		−	1.00698i	0.864001	−	0.503490i	$-0.167951\pi$
$398$	272.906			0.685693
$399$		0			0
$400$	85.6181	−	51.6677i	0.214045	−	0.129169i
$401$			79.5032i			0.198262i	0.995074	+	0.0991311i	$0.0316063\pi$
$401$			79.5032i			0.198262i	−0.995074	+	0.0991311i	$0.968394\pi$
$402$		0			0
$403$			37.6605i			0.0934504i
$404$			21.8958i			0.0541975i
$405$		0			0
$406$	−16.1958			−0.0398912
$407$	387.000			0.950860
$408$		0			0
$409$	458.917			1.12205			0.561024	−	0.827800i	$-0.310408\pi$
$409$	458.917			1.12205			0.561024	+	0.827800i	$0.310408\pi$
$410$	120.019	+	431.172i	0.292729	+	1.05164i
$411$		0			0
$412$			104.915i			0.254648i
$413$	−40.9478			−0.0991472
$414$		0			0
$415$	−284.371	+	79.1561i	−0.685232	+	0.190738i
$416$		−	43.3040i		−	0.104096i
$417$		0			0
$418$			72.9398i			0.174497i
$419$			737.930i			1.76117i	0.473889	+	0.880585i	$0.342850\pi$
$419$			737.930i			1.76117i	−0.473889	+	0.880585i	$0.657150\pi$
$420$		0			0
$421$	−102.556			−0.243601			−0.121801	−	0.992555i	$-0.538867\pi$
$421$	−102.556			−0.243601			−0.121801	+	0.992555i	$0.538867\pi$
$422$	542.915			1.28653
$423$		0			0
$424$	−117.918			−0.278109
$425$	−264.131	+	159.394i	−0.621484	+	0.375045i
$426$		0			0
$427$			16.7643i			0.0392606i
$428$	379.894			0.887603
$429$		0			0
$430$	−102.041	−	366.585i	−0.237304	−	0.852523i
$431$			251.755i			0.584118i	0.956400	+	0.292059i	$0.0943403\pi$
$431$			251.755i			0.584118i	−0.956400	+	0.292059i	$0.905660\pi$
$432$		0			0
$433$			153.868i			0.355353i	0.984089	+	0.177676i	$0.0568581\pi$
$433$			153.868i			0.355353i	−0.984089	+	0.177676i	$0.943142\pi$
$434$		−	2.53791i		−	0.00584773i
$435$		0			0
$436$	285.310			0.654382
$437$	−64.7357			−0.148137
$438$		0			0
$439$	492.876			1.12272			0.561362	−	0.827570i	$-0.310277\pi$
$439$	492.876			1.12272			0.561362	+	0.827570i	$0.310277\pi$
$440$	−36.4570	−	130.973i	−0.0828569	−	0.297667i
$441$		0			0
$442$			133.592i			0.302245i
$443$	516.580			1.16610			0.583048	−	0.812438i	$-0.301860\pi$
$443$	516.580			1.16610			0.583048	+	0.812438i	$0.301860\pi$
$444$		0			0
$445$	−11.9741	−	43.0174i	−0.0269081	−	0.0966684i
$446$			488.680i			1.09569i
$447$		0			0
$448$			2.91822i			0.00651389i
$449$		−	612.611i		−	1.36439i	−0.731170	−	0.682195i	$-0.761026\pi$
$449$		−	612.611i		−	1.36439i	0.731170	−	0.682195i	$-0.238974\pi$
$450$		0			0
$451$	608.475			1.34917
$452$	163.356			0.361407
$453$		0			0
$454$	−385.791			−0.849761
$455$	13.4507	−	3.74408i	0.0295621	−	0.00822874i
$456$		0			0
$457$			882.866i			1.93187i	0.258781	+	0.965936i	$0.416679\pi$
$457$			882.866i			1.93187i	−0.258781	+	0.965936i	$0.583321\pi$
$458$	−0.147421			−0.000321880
$459$		0			0
$460$	116.242	−	32.3564i	0.252699	−	0.0703401i
$461$		−	343.354i		−	0.744802i	−0.928072	−	0.372401i	$-0.878535\pi$
$461$		−	343.354i		−	0.744802i	0.928072	−	0.372401i	$-0.121465\pi$
$462$		0			0
$463$			654.610i			1.41384i	0.707292	+	0.706922i	$0.249917\pi$
$463$			654.610i			1.41384i	−0.707292	+	0.706922i	$0.750083\pi$
$464$		−	125.580i		−	0.270646i
$465$		0			0
$466$	448.732			0.962945
$467$	861.410			1.84456			0.922280	−	0.386522i	$-0.126324\pi$
$467$	861.410			1.84456			0.922280	+	0.386522i	$0.126324\pi$
$468$		0			0
$469$	16.8313			0.0358876
$470$	192.329	−	53.5357i	0.409211	−	0.113906i
$471$		0			0
$472$		−	317.503i		−	0.672675i
$473$	−517.330			−1.09372
$474$		0			0
$475$	114.837	−	69.3004i	0.241763	−	0.145896i
$476$		−	9.00268i		−	0.0189132i
$477$		0			0
$478$			272.978i			0.571084i
$479$		−	359.641i		−	0.750817i	−0.926859	−	0.375409i	$-0.877502\pi$
$479$		−	359.641i		−	0.750817i	0.926859	−	0.375409i	$-0.122498\pi$
$480$		0			0
$481$	−308.171			−0.640687
$482$	445.090			0.923424
$483$		0			0
$484$	57.1688			0.118117
$485$	−176.055	−	632.485i	−0.363000	−	1.30409i
$486$		0			0
$487$		−	771.071i		−	1.58331i	−0.610970	−	0.791654i	$-0.709220\pi$
$487$		−	771.071i		−	1.58331i	0.610970	−	0.791654i	$-0.290780\pi$
$488$	−129.987			−0.266368
$489$		0			0
$490$	332.886	−	92.6603i	0.679359	−	0.189103i
$491$			158.509i			0.322829i	0.986887	+	0.161415i	$0.0516057\pi$
$491$			158.509i			0.322829i	−0.986887	+	0.161415i	$0.948394\pi$
$492$		0			0
$493$			387.412i			0.785825i
$494$		−	58.0825i		−	0.117576i
$495$		0			0
$496$	19.6786			0.0396745
$497$	45.8224			0.0921980
$498$		0			0
$499$	634.106			1.27075			0.635377	−	0.772202i	$-0.280845\pi$
$499$	634.106			1.27075			0.635377	+	0.772202i	$0.280845\pi$
$500$	−171.568	+	181.837i	−0.343135	+	0.363673i
$501$		0			0
$502$			80.8498i			0.161055i
$503$	−212.791			−0.423043			−0.211522	−	0.977373i	$-0.567842\pi$
$503$	−212.791			−0.423043			−0.211522	+	0.977373i	$0.567842\pi$
$504$		0			0
$505$	−14.6789	−	52.7346i	−0.0290672	−	0.104425i
$506$		−	164.042i		−	0.324193i
$507$		0			0
$508$		−	166.802i		−	0.328351i
$509$			982.342i			1.92995i	0.262350	+	0.964973i	$0.415502\pi$
$509$			982.342i			1.92995i	−0.262350	+	0.964973i	$0.584498\pi$
$510$		0			0
$511$	−21.5520			−0.0421762
$512$	−22.6274			−0.0441942
$513$		0			0
$514$	−101.416			−0.197307
$515$	−70.3348	−	252.681i	−0.136573	−	0.490642i
$516$		0			0
$517$		−	271.417i		−	0.524985i
$518$	20.7674			0.0400915
$519$		0			0
$520$	29.0310	+	104.295i	0.0558288	+	0.200567i
$521$			148.759i			0.285525i	0.989757	+	0.142763i	$0.0455986\pi$
$521$			148.759i			0.285525i	−0.989757	+	0.142763i	$0.954401\pi$
$522$		0			0
$523$		−	896.212i		−	1.71360i	−0.515650	−	0.856799i	$-0.672450\pi$
$523$		−	896.212i		−	1.71360i	0.515650	−	0.856799i	$-0.327550\pi$
$524$			227.791i			0.434716i
$525$		0			0
$526$	323.105			0.614268
$527$	−60.7082			−0.115196
$528$		0			0
$529$	−383.409			−0.724781
$530$	283.998	−	79.0522i	0.535845	−	0.149155i
$531$		0			0
$532$			3.91413i			0.00735739i
$533$	−484.533			−0.909067
$534$		0			0
$535$	−914.951	+	254.681i	−1.71019	+	0.476039i
$536$			130.507i			0.243483i
$537$		0			0
$538$		−	12.4398i		−	0.0231224i
$539$		−	469.773i		−	0.871563i
$540$		0			0
$541$	513.074			0.948381			0.474190	−	0.880422i	$-0.342741\pi$
$541$	513.074			0.948381			0.474190	+	0.880422i	$0.342741\pi$
$542$	−1.10734			−0.00204306
$543$		0			0
$544$	69.8053			0.128319
$545$	−687.152	+	191.272i	−1.26083	+	0.350958i
$546$		0			0
$547$			565.339i			1.03353i	0.856129	+	0.516763i	$0.172863\pi$
$547$			565.339i			1.03353i	−0.856129	+	0.516763i	$0.827137\pi$
$548$	305.469			0.557425
$549$		0			0
$550$	175.609	+	291.000i	0.319289	+	0.529091i
$551$		−	168.437i		−	0.305693i
$552$		0			0
$553$			18.4895i			0.0334349i
$554$		−	667.766i		−	1.20535i
$555$		0			0
$556$	201.100			0.361691
$557$	−487.883			−0.875913			−0.437956	−	0.898996i	$-0.644298\pi$
$557$	−487.883			−0.875913			−0.437956	+	0.898996i	$0.644298\pi$
$558$		0			0
$559$	411.953			0.736946
$560$	−1.95637	−	7.02835i	−0.00349353	−	0.0125506i
$561$		0			0
$562$		−	335.797i		−	0.597504i
$563$	−290.343			−0.515707			−0.257854	−	0.966184i	$-0.583015\pi$
$563$	−290.343			−0.515707			−0.257854	+	0.966184i	$0.583015\pi$
$564$		0			0
$565$	−393.432	+	109.514i	−0.696340	+	0.193830i
$566$		−	376.671i		−	0.665497i
$567$		0			0
$568$			355.300i			0.625527i
$569$		−	585.008i		−	1.02813i	−0.857750	−	0.514067i	$-0.828138\pi$
$569$		−	585.008i		−	1.02813i	0.857750	−	0.514067i	$-0.171862\pi$
$570$		0			0
$571$	649.475			1.13743			0.568717	−	0.822533i	$-0.307440\pi$
$571$	649.475			1.13743			0.568717	+	0.822533i	$0.307440\pi$
$572$	147.182			0.257312
$573$		0			0
$574$	32.6523			0.0568855
$575$	−258.269	+	155.857i	−0.449164	+	0.271055i
$576$		0			0
$577$		−	362.217i		−	0.627760i	−0.949463	−	0.313880i	$-0.898371\pi$
$577$		−	362.217i		−	0.627760i	0.949463	−	0.313880i	$-0.101629\pi$
$578$	193.359			0.334532
$579$		0			0
$580$	84.1885	+	302.451i	0.145153	+	0.521467i
$581$			21.5352i			0.0370658i
$582$		0			0
$583$		−	400.782i		−	0.687447i
$584$		−	167.111i		−	0.286149i
$585$		0			0
$586$	468.933			0.800228
$587$	259.474			0.442034			0.221017	−	0.975270i	$-0.429062\pi$
$587$	259.474			0.442034			0.221017	+	0.975270i	$0.429062\pi$
$588$		0			0
$589$	26.3943			0.0448121
$590$	212.854	+	764.685i	0.360769	+	1.29608i
$591$		0			0
$592$			161.027i			0.272005i
$593$	−207.377			−0.349709			−0.174854	−	0.984594i	$-0.555945\pi$
$593$	−207.377			−0.349709			−0.174854	+	0.984594i	$0.555945\pi$
$594$		0			0
$595$	6.03539	+	21.6824i	0.0101435	+	0.0364410i
$596$		−	51.9570i		−	0.0871761i
$597$		0			0
$598$			130.628i			0.218441i
$599$			770.031i			1.28553i	0.766065	+	0.642764i	$0.222212\pi$
$599$			770.031i			1.28553i	−0.766065	+	0.642764i	$0.777788\pi$
$600$		0			0
$601$	698.337			1.16196			0.580979	−	0.813919i	$-0.302670\pi$
$601$	698.337			1.16196			0.580979	+	0.813919i	$0.302670\pi$
$602$	−27.7612			−0.0461149
$603$		0			0
$604$	−311.908			−0.516403
$605$	−137.687	+	38.3260i	−0.227583	+	0.0633487i
$606$		0			0
$607$			964.614i			1.58915i	0.607166	+	0.794575i	$0.292306\pi$
$607$			964.614i			1.58915i	−0.607166	+	0.794575i	$0.707694\pi$
$608$	−30.3495			−0.0499170
$609$		0			0
$610$	313.066	−	87.1435i	0.513223	−	0.142858i
$611$			216.131i			0.353734i
$612$		0			0
$613$			123.585i			0.201607i	0.994906	+	0.100803i	$0.0321413\pi$
$613$			123.585i			0.201607i	−0.994906	+	0.100803i	$0.967859\pi$
$614$			585.607i			0.953758i
$615$		0			0
$616$	−9.91850			−0.0161015
$617$	−125.354			−0.203166			−0.101583	−	0.994827i	$-0.532391\pi$
$617$	−125.354			−0.203166			−0.101583	+	0.994827i	$0.532391\pi$
$618$		0			0
$619$	176.072			0.284445			0.142223	−	0.989835i	$-0.454575\pi$
$619$	176.072			0.284445			0.142223	+	0.989835i	$0.454575\pi$
$620$	−47.3946	+	13.1925i	−0.0764429	+	0.0212782i
$621$		0			0
$622$			697.686i			1.12168i
$623$	−3.25767			−0.00522901
$624$		0			0
$625$	291.307	−	552.961i	0.466090	−	0.884737i
$626$			155.698i			0.248719i
$627$		0			0
$628$		−	505.408i		−	0.804790i
$629$		−	496.766i		−	0.789771i
$630$		0			0
$631$	−154.760			−0.245262			−0.122631	−	0.992452i	$-0.539133\pi$
$631$	−154.760			−0.245262			−0.122631	+	0.992452i	$0.539133\pi$
$632$	−143.365			−0.226843
$633$		0			0
$634$	140.822			0.222116
$635$	111.824	+	401.732i	0.176101	+	0.632649i
$636$		0			0
$637$			374.083i			0.587257i
$638$	426.822			0.669000
$639$		0			0
$640$	54.4967	−	15.1694i	0.0851511	−	0.0237022i
$641$		−	919.930i		−	1.43515i	−0.696482	−	0.717574i	$-0.745252\pi$
$641$		−	919.930i		−	1.43515i	0.696482	−	0.717574i	$-0.254748\pi$
$642$		0			0
$643$			628.524i			0.977487i	0.872428	+	0.488743i	$0.162545\pi$
$643$			628.524i			0.977487i	−0.872428	+	0.488743i	$0.837455\pi$
$644$		−	8.80289i		−	0.0136691i
$645$		0			0
$646$	93.6280			0.144935
$647$	−423.162			−0.654037			−0.327019	−	0.945018i	$-0.606044\pi$
$647$	−423.162			−0.654037			−0.327019	+	0.945018i	$0.606044\pi$
$648$		0			0
$649$	1079.13			1.66276
$650$	−139.838	−	231.725i	−0.215136	−	0.356500i
$651$		0			0
$652$			312.774i			0.479715i
$653$	−210.642			−0.322575			−0.161288	−	0.986907i	$-0.551565\pi$
$653$	−210.642			−0.322575			−0.161288	+	0.986907i	$0.551565\pi$
$654$		0			0
$655$	−152.711	−	548.620i	−0.233147	−	0.837588i
$656$			253.180i			0.385946i
$657$		0			0
$658$		−	14.5649i		−	0.0221351i
$659$		−	389.064i		−	0.590385i	−0.955438	−	0.295192i	$-0.904616\pi$
$659$		−	389.064i		−	0.590385i	0.955438	−	0.295192i	$-0.0953837\pi$
$660$		0			0
$661$	526.996			0.797271			0.398636	−	0.917109i	$-0.369484\pi$
$661$	526.996			0.797271			0.398636	+	0.917109i	$0.369484\pi$
$662$	−281.156			−0.424708
$663$		0			0
$664$	−166.981			−0.251477
$665$	−2.62403	−	9.42694i	−0.00394591	−	0.0141758i
$666$		0			0
$667$			378.814i			0.567937i
$668$	640.123			0.958268
$669$		0			0
$670$	−87.4918	−	314.318i	−0.130585	−	0.469131i
$671$		−	441.803i		−	0.658425i
$672$		0			0
$673$		−	376.968i		−	0.560130i	−0.959981	−	0.280065i	$-0.909644\pi$
$673$		−	376.968i		−	0.560130i	0.959981	−	0.280065i	$-0.0903561\pi$
$674$			459.322i			0.681487i
$675$		0			0
$676$	220.798			0.326624
$677$	302.675			0.447083			0.223542	−	0.974694i	$-0.428238\pi$
$677$	302.675			0.447083			0.223542	+	0.974694i	$0.428238\pi$
$678$		0			0
$679$	−47.8975			−0.0705413
$680$	−168.122	+	46.7974i	−0.247238	+	0.0688198i
$681$		0			0
$682$			66.8838i			0.0980702i
$683$	665.773			0.974777			0.487389	−	0.873185i	$-0.337950\pi$
$683$	665.773			0.974777			0.487389	+	0.873185i	$0.337950\pi$
$684$		0			0
$685$	−735.702	+	204.786i	−1.07402	+	0.298958i
$686$		−	50.4870i		−	0.0735962i
$687$		0			0
$688$		−	215.256i		−	0.312872i
$689$			319.145i			0.463200i
$690$		0			0
$691$	711.331			1.02942			0.514711	−	0.857364i	$-0.327899\pi$
$691$	711.331			1.02942			0.514711	+	0.857364i	$0.327899\pi$
$692$	79.3440			0.114659
$693$		0			0
$694$	−624.175			−0.899388
$695$	−484.337	+	134.818i	−0.696888	+	0.193982i
$696$		0			0
$697$		−	781.059i		−	1.12060i
$698$	0.190601			0.000273068
$699$		0			0
$700$	9.42361	+	15.6158i	0.0134623	+	0.0223083i
$701$			792.935i			1.13115i	0.824698	+	0.565574i	$0.191345\pi$
$701$			792.935i			1.13115i	−0.824698	+	0.565574i	$0.808655\pi$
$702$		0			0
$703$			215.981i			0.307228i
$704$		−	76.9064i		−	0.109242i
$705$		0			0
$706$	738.571			1.04613
$707$	−3.99355			−0.00564859
$708$		0			0
$709$	1211.24			1.70838			0.854188	−	0.519964i	$-0.174055\pi$
$709$	1211.24			1.70838			0.854188	+	0.519964i	$0.174055\pi$
$710$	−238.193	−	855.716i	−0.335483	−	1.20523i
$711$		0			0
$712$		−	25.2595i		−	0.0354768i
$713$	−59.3609			−0.0832551
$714$		0			0
$715$	−354.479	+	98.6709i	−0.495775	+	0.138001i
$716$			102.641i			0.143353i
$717$		0			0
$718$		−	207.160i		−	0.288524i
$719$		−	239.649i		−	0.333309i	−0.986015	−	0.166654i	$-0.946704\pi$
$719$		−	239.649i		−	0.333309i	0.986015	−	0.166654i	$-0.0532964\pi$
$720$		0			0
$721$	−19.1353			−0.0265399
$722$	469.824			0.650726
$723$		0			0
$724$	−596.115			−0.823363
$725$	−405.525	−	671.993i	−0.559345	−	0.926887i
$726$		0			0
$727$		−	1106.88i		−	1.52252i	−0.648444	−	0.761262i	$-0.724580\pi$
$727$		−	1106.88i		−	1.52252i	0.648444	−	0.761262i	$-0.275420\pi$
$728$	7.89816			0.0108491
$729$		0			0
$730$	112.031	+	402.476i	0.153467	+	0.551337i
$731$			664.061i			0.908428i
$732$		0			0
$733$			284.047i			0.387513i	0.981050	+	0.193756i	$0.0620672\pi$
$733$			284.047i			0.387513i	−0.981050	+	0.193756i	$0.937933\pi$
$734$		−	480.373i		−	0.654460i
$735$		0			0
$736$	68.2562			0.0927394
$737$	−443.569			−0.601857
$738$		0			0
$739$	−851.158			−1.15177			−0.575885	−	0.817531i	$-0.695342\pi$
$739$	−851.158			−1.15177			−0.575885	+	0.817531i	$0.695342\pi$
$740$	−107.952	−	387.823i	−0.145882	−	0.524085i
$741$		0			0
$742$		−	21.5069i		−	0.0289851i
$743$	938.698			1.26339			0.631695	−	0.775217i	$-0.282360\pi$
$743$	938.698			1.26339			0.631695	+	0.775217i	$0.282360\pi$
$744$		0			0
$745$	34.8319	+	125.135i	0.0467543	+	0.167967i
$746$			617.700i			0.828016i
$747$		0			0
$748$			237.255i			0.317186i
$749$			69.2885i			0.0925080i
$750$		0			0
$751$	−706.414			−0.940632			−0.470316	−	0.882498i	$-0.655860\pi$
$751$	−706.414			−0.940632			−0.470316	+	0.882498i	$0.655860\pi$
$752$	112.934			0.150178
$753$		0			0
$754$	−339.881			−0.450771
$755$	751.209	−	209.103i	0.994979	−	0.276957i
$756$		0			0
$757$			624.266i			0.824658i	0.911035	+	0.412329i	$0.135284\pi$
$757$			624.266i			0.824658i	−0.911035	+	0.412329i	$0.864716\pi$
$758$	−958.484			−1.26449
$759$		0			0
$760$	73.0949	−	20.3463i	0.0961776	−	0.0267715i
$761$		−	784.248i		−	1.03055i	−0.857025	−	0.515275i	$-0.827690\pi$
$761$		−	784.248i		−	1.03055i	0.857025	−	0.515275i	$-0.172310\pi$
$762$		0			0
$763$			52.0374i			0.0682011i
$764$			429.793i			0.562557i
$765$		0			0
$766$	−205.222			−0.267914
$767$	−859.320			−1.12037
$768$		0			0
$769$	676.912			0.880250			0.440125	−	0.897937i	$-0.354934\pi$
$769$	676.912			0.880250			0.440125	+	0.897937i	$0.354934\pi$
$770$	23.8881	−	6.64936i	0.0310235	−	0.00863553i
$771$		0			0
$772$		−	351.443i		−	0.455237i
$773$	−1033.43			−1.33691			−0.668453	−	0.743754i	$-0.733043\pi$
$773$	−1033.43			−1.33691			−0.668453	+	0.743754i	$0.733043\pi$
$774$		0			0
$775$	105.303	−	63.5466i	0.135874	−	0.0819956i
$776$		−	371.390i		−	0.478595i
$777$		0			0
$778$			45.2114i			0.0581124i
$779$			339.584i			0.435923i
$780$		0			0
$781$	−1207.60			−1.54622
$782$	−210.569			−0.269270
$783$		0			0
$784$	195.468			0.249321
$785$	338.825	+	1217.24i	0.431625	+	1.55063i
$786$		0			0
$787$			403.270i			0.512414i	0.966622	+	0.256207i	$0.0824728\pi$
$787$			403.270i			0.512414i	−0.966622	+	0.256207i	$0.917527\pi$
$788$	295.159			0.374568
$789$		0			0
$790$	345.285	−	96.1116i	0.437069	−	0.121660i
$791$			29.7943i			0.0376666i
$792$		0			0
$793$			351.811i			0.443645i
$794$			565.362i			0.712043i
$795$		0			0
$796$	−385.947			−0.484858
$797$	−14.6224			−0.0183468			−0.00917340	−	0.999958i	$-0.502920\pi$
$797$	−14.6224			−0.0183468			−0.00917340	+	0.999958i	$0.502920\pi$
$798$		0			0
$799$	−348.400			−0.436045
$800$	−121.082	+	73.0691i	−0.151353	+	0.0913364i
$801$		0			0
$802$		−	112.434i		−	0.140193i
$803$	567.979			0.707322
$804$		0			0
$805$	5.90145	+	21.2012i	0.00733100	+	0.0263369i
$806$		−	53.2600i		−	0.0660794i
$807$		0			0
$808$		−	30.9653i		−	0.0383234i
$809$		−	225.767i		−	0.279069i	−0.990217	−	0.139535i	$-0.955439\pi$
$809$		−	225.767i		−	0.279069i	0.990217	−	0.139535i	$-0.0445607\pi$
$810$		0			0
$811$	198.448			0.244696			0.122348	−	0.992487i	$-0.460958\pi$
$811$	198.448			0.244696			0.122348	+	0.992487i	$0.460958\pi$
$812$	22.9043			0.0282073
$813$		0			0
$814$	−547.301			−0.672360
$815$	−209.684	−	753.297i	−0.257281	−	0.924290i
$816$		0			0
$817$		−	288.717i		−	0.353386i
$818$	−649.007			−0.793407
$819$		0			0
$820$	−169.732	−	609.769i	−0.206990	−	0.743621i
$821$		−	121.358i		−	0.147817i	−0.997265	−	0.0739086i	$-0.976453\pi$
$821$		−	121.358i		−	0.147817i	0.997265	−	0.0739086i	$-0.0235473\pi$
$822$		0			0
$823$			247.563i			0.300806i	0.988625	+	0.150403i	$0.0480571\pi$
$823$			247.563i			0.300806i	−0.988625	+	0.150403i	$0.951943\pi$
$824$		−	148.372i		−	0.180063i
$825$		0			0
$826$	57.9089			0.0701077
$827$	457.912			0.553703			0.276851	−	0.960913i	$-0.410709\pi$
$827$	457.912			0.553703			0.276851	+	0.960913i	$0.410709\pi$
$828$		0			0
$829$	−116.297			−0.140286			−0.0701429	−	0.997537i	$-0.522346\pi$
$829$	−116.297			−0.140286			−0.0701429	+	0.997537i	$0.522346\pi$
$830$	402.162	−	111.944i	0.484533	−	0.134872i
$831$		0			0
$832$			61.2411i			0.0736071i
$833$	−603.016			−0.723908
$834$		0			0
$835$	−1541.70	+	429.138i	−1.84634	+	0.513938i
$836$		−	103.153i		−	0.123388i
$837$		0			0
$838$		−	1043.59i		−	1.24533i
$839$			552.201i			0.658166i	0.944301	+	0.329083i	$0.106740\pi$
$839$			552.201i			0.658166i	−0.944301	+	0.329083i	$0.893260\pi$
$840$		0			0
$841$	−144.641			−0.171986
$842$	145.036			0.172252
$843$		0			0
$844$	−767.797			−0.909712
$845$	−531.778	+	148.023i	−0.629323	+	0.175175i
$846$		0			0
$847$			10.4270i			0.0123105i
$848$	166.761			0.196652
$849$		0			0
$850$	373.537	−	225.417i	0.439455	−	0.265197i
$851$		−	485.742i		−	0.570789i
$852$		0			0
$853$			275.356i			0.322809i	0.986888	+	0.161405i	$0.0516024\pi$
$853$			275.356i			0.322809i	−0.986888	+	0.161405i	$0.948398\pi$
$854$		−	23.7082i		−	0.0277614i
$855$		0			0
$856$	−537.252			−0.627630
$857$	−1623.45			−1.89434			−0.947170	−	0.320731i	$-0.896071\pi$
$857$	−1623.45			−1.89434			−0.947170	+	0.320731i	$0.896071\pi$
$858$		0			0
$859$	−1022.35			−1.19016			−0.595082	−	0.803665i	$-0.702880\pi$
$859$	−1022.35			−1.19016			−0.595082	+	0.803665i	$0.702880\pi$
$860$	144.307	+	518.430i	0.167799	+	0.602825i
$861$		0			0
$862$		−	356.035i		−	0.413034i
$863$	588.647			0.682094			0.341047	−	0.940046i	$-0.389218\pi$
$863$	588.647			0.682094			0.341047	+	0.940046i	$0.389218\pi$
$864$		0			0
$865$	−191.095	+	53.1922i	−0.220919	+	0.0614938i
$866$		−	217.602i		−	0.251272i
$867$		0			0
$868$			3.58915i			0.00413497i
$869$		−	487.270i		−	0.560725i
$870$		0			0
$871$	353.217			0.405530
$872$	−403.490			−0.462718
$873$		0			0
$874$	91.5502			0.104748
$875$	−33.1650	−	31.2920i	−0.0379028	−	0.0357623i
$876$		0			0
$877$		−	610.298i		−	0.695892i	−0.937515	−	0.347946i	$-0.886879\pi$
$877$		−	610.298i		−	0.695892i	0.937515	−	0.347946i	$-0.113121\pi$
$878$	−697.032			−0.793886
$879$		0			0
$880$	51.5580	+	185.224i	0.0585887	+	0.210482i
$881$			677.101i			0.768559i	0.923217	+	0.384280i	$0.125550\pi$
$881$			677.101i			0.768559i	−0.923217	+	0.384280i	$0.874450\pi$
$882$		0			0
$883$		−	760.076i		−	0.860788i	−0.902641	−	0.430394i	$-0.858375\pi$
$883$		−	760.076i		−	0.860788i	0.902641	−	0.430394i	$-0.141625\pi$
$884$		−	188.928i		−	0.213719i
$885$		0			0
$886$	−730.555			−0.824554
$887$	−640.123			−0.721672			−0.360836	−	0.932629i	$-0.617509\pi$
$887$	−640.123			−0.721672			−0.360836	+	0.932629i	$0.617509\pi$
$888$		0			0
$889$	30.4229			0.0342214
$890$	16.9339	+	60.8358i	0.0190269	+	0.0683549i
$891$		0			0
$892$		−	691.098i		−	0.774773i
$893$	151.475			0.169625
$894$		0			0
$895$	−68.8103	−	247.204i	−0.0768830	−	0.276205i
$896$		−	4.12699i		−	0.00460601i
$897$		0			0
$898$			866.363i			0.964769i
$899$		−	154.452i		−	0.171804i
$900$		0			0
$901$	−514.456			−0.570984
$902$	−860.514			−0.954006
$903$		0			0
$904$	−231.020			−0.255553
$905$	1435.70	−	399.635i	1.58641	−	0.441586i
$906$		0			0
$907$		−	176.147i		−	0.194208i	−0.995274	−	0.0971040i	$-0.969042\pi$
$907$		−	176.147i		−	0.194208i	0.995274	−	0.0971040i	$-0.0309579\pi$
$908$	545.591			0.600872
$909$		0			0
$910$	−19.0222	+	5.29492i	−0.0209035	+	0.00581860i
$911$			1437.08i			1.57747i	0.614733	+	0.788735i	$0.289264\pi$
$911$			1437.08i			1.57747i	−0.614733	+	0.788735i	$0.710736\pi$
$912$		0			0
$913$		−	567.536i		−	0.621617i
$914$		−	1248.56i		−	1.36604i
$915$		0			0
$916$	0.208485			0.000227604
$917$	−41.5465			−0.0453070
$918$		0			0
$919$	217.751			0.236944			0.118472	−	0.992957i	$-0.462200\pi$
$919$	217.751			0.236944			0.118472	+	0.992957i	$0.462200\pi$
$920$	−164.391	+	45.7589i	−0.178685	+	0.0497379i
$921$		0			0
$922$			485.575i			0.526654i
$923$	961.618			1.04184
$924$		0			0
$925$	519.993	+	861.676i	0.562154	+	0.931542i
$926$		−	925.758i		−	0.999738i
$927$		0			0
$928$			177.596i			0.191375i
$929$		−	293.708i		−	0.316155i	−0.987427	−	0.158077i	$-0.949470\pi$
$929$		−	293.708i		−	0.316155i	0.987427	−	0.158077i	$-0.0505295\pi$
$930$		0			0
$931$	262.176			0.281606
$932$	−634.604			−0.680905
$933$		0			0
$934$	−1218.22			−1.30430
$935$	−159.056	−	571.414i	−0.170113	−	0.611138i
$936$		0			0
$937$			375.806i			0.401074i	0.979686	+	0.200537i	$0.0642686\pi$
$937$			375.806i			0.401074i	−0.979686	+	0.200537i	$0.935731\pi$
$938$	−23.8030			−0.0253763
$939$		0			0
$940$	−271.994	+	75.7109i	−0.289356	+	0.0805435i
$941$		−	1111.46i		−	1.18115i	−0.806985	−	0.590573i	$-0.798902\pi$
$941$		−	1111.46i		−	1.18115i	0.806985	−	0.590573i	$-0.201098\pi$
$942$		0			0
$943$		−	763.725i		−	0.809889i
$944$			449.016i			0.475653i
$945$		0			0
$946$	731.614			0.773377
$947$	1817.23			1.91893			0.959465	−	0.281828i	$-0.0909407\pi$
$947$	1817.23			1.91893			0.959465	+	0.281828i	$0.0909407\pi$
$948$		0			0
$949$	−452.285			−0.476592
$950$	−162.404	+	98.0056i	−0.170952	+	0.103164i
$951$		0			0
$952$			12.7317i			0.0133736i
$953$	817.241			0.857545			0.428773	−	0.903412i	$-0.358946\pi$
$953$	817.241			0.857545			0.428773	+	0.903412i	$0.358946\pi$
$954$		0			0
$955$	−288.133	−	1035.13i	−0.301710	−	1.08391i
$956$		−	386.049i		−	0.403817i
$957$		0			0
$958$			508.610i			0.530908i
$959$			55.7141i			0.0580961i
$960$		0			0
$961$	−936.797			−0.974815
$962$	435.819			0.453034
$963$		0			0
$964$	−629.453			−0.652959
$965$	235.607	+	846.429i	0.244153	+	0.877128i
$966$		0			0
$967$			48.7235i			0.0503863i	0.999683	+	0.0251931i	$0.00802008\pi$
$967$			48.7235i			0.0503863i	−0.999683	+	0.0251931i	$0.991980\pi$
$968$	−80.8489			−0.0835216
$969$		0			0
$970$	248.979	+	894.468i	0.256680	+	0.922132i
$971$			260.809i			0.268598i	0.990941	+	0.134299i	$0.0428783\pi$
$971$			260.809i			0.268598i	−0.990941	+	0.134299i	$0.957122\pi$
$972$		0			0
$973$			36.6785i			0.0376963i
$974$			1090.46i			1.11957i
$975$		0			0
$976$	183.830			0.188350
$977$	−249.253			−0.255121			−0.127560	−	0.991831i	$-0.540715\pi$
$977$	−249.253			−0.255121			−0.127560	+	0.991831i	$0.540715\pi$
$978$		0			0
$979$	85.8523			0.0876939
$980$	−470.772	+	131.041i	−0.480379	+	0.133716i
$981$		0			0
$982$		−	224.166i		−	0.228275i
$983$	−588.251			−0.598424			−0.299212	−	0.954187i	$-0.596724\pi$
$983$	−588.251			−0.598424			−0.299212	+	0.954187i	$0.596724\pi$
$984$		0			0
$985$	−710.873	+	197.875i	−0.721698	+	0.200888i
$986$		−	547.883i		−	0.555662i
$987$		0			0
$988$			82.1410i			0.0831387i
$989$			649.324i			0.656546i
$990$		0			0
$991$	−598.851			−0.604289			−0.302145	−	0.953262i	$-0.597703\pi$
$991$	−598.851			−0.604289			−0.302145	+	0.953262i	$0.597703\pi$
$992$	−27.8297			−0.0280541
$993$		0			0
$994$	−64.8027			−0.0651938
$995$	929.529	−	258.739i	0.934200	−	0.260039i
$996$		0			0
$997$		−	920.900i		−	0.923671i	−0.886966	−	0.461835i	$-0.847191\pi$
$997$		−	920.900i		−	0.923671i	0.886966	−	0.461835i	$-0.152809\pi$
$998$	−896.761			−0.898559
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.b.c.809.2	yes	24
3.2	odd	2	inner	810.3.b.c.809.23	yes	24
5.4	even	2	inner	810.3.b.c.809.24	yes	24
9.2	odd	6		810.3.j.g.539.3		24
9.4	even	3		810.3.j.h.269.11		24
9.5	odd	6		810.3.j.h.269.3		24
9.7	even	3		810.3.j.g.539.11		24
15.14	odd	2	inner	810.3.b.c.809.1	✓	24
45.4	even	6		810.3.j.g.269.3		24
45.14	odd	6		810.3.j.g.269.11		24
45.29	odd	6		810.3.j.h.539.11		24
45.34	even	6		810.3.j.h.539.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.3.b.c.809.1	✓	24	15.14	odd	2	inner
810.3.b.c.809.2	yes	24	1.1	even	1	trivial
810.3.b.c.809.23	yes	24	3.2	odd	2	inner
810.3.b.c.809.24	yes	24	5.4	even	2	inner
810.3.j.g.269.3		24	45.4	even	6
810.3.j.g.269.11		24	45.14	odd	6
810.3.j.g.539.3		24	9.2	odd	6
810.3.j.g.539.11		24	9.7	even	3
810.3.j.h.269.3		24	9.5	odd	6
810.3.j.h.269.11		24	9.4	even	3
810.3.j.h.539.3		24	45.34	even	6
810.3.j.h.539.11		24	45.29	odd	6

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)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(-4.81687 + 1.34080i) q^{5} +0.364778i q^{7} -2.82843 q^{8} +O(q^{10})\)	\(q-1.41421 q^{2} +2.00000 q^{4} +(-4.81687 + 1.34080i) q^{5} +0.364778i q^{7} -2.82843 q^{8} +(6.81209 - 1.89618i) q^{10} -9.61330i q^{11} +7.65513i q^{13} -0.515874i q^{14} +4.00000 q^{16} -12.3400 q^{17} +5.36509 q^{19} +(-9.63374 + 2.68160i) q^{20} +13.5953i q^{22} -12.0661 q^{23} +(21.4045 - 12.9169i) q^{25} -10.8260i q^{26} +0.729555i q^{28} -31.3949i q^{29} +4.91964 q^{31} -5.65685 q^{32} +17.4513 q^{34} +(-0.489094 - 1.75709i) q^{35} +40.2567i q^{37} -7.58739 q^{38} +(13.6242 - 3.79235i) q^{40} +63.2951i q^{41} -53.8139i q^{43} -19.2266i q^{44} +17.0640 q^{46} +28.2335 q^{47} +48.8669 q^{49} +(-30.2706 + 18.2673i) q^{50} +15.3103i q^{52} +41.6903 q^{53} +(12.8895 + 46.3060i) q^{55} -1.03175i q^{56} +44.3991i q^{58} +112.254i q^{59} +45.9575 q^{61} -6.95743 q^{62} +8.00000 q^{64} +(-10.2640 - 36.8738i) q^{65} -46.1412i q^{67} -24.6799 q^{68} +(0.691683 + 2.48490i) q^{70} -125.617i q^{71} +59.0826i q^{73} -56.9316i q^{74} +10.7302 q^{76} +3.50672 q^{77} +50.6871 q^{79} +(-19.2675 + 5.36320i) q^{80} -89.5128i q^{82} +59.0365 q^{83} +(59.4400 - 16.5454i) q^{85} +76.1044i q^{86} +27.1905i q^{88} +8.93057i q^{89} -2.79242 q^{91} -24.1322 q^{92} -39.9282 q^{94} +(-25.8430 + 7.19351i) q^{95} +131.306i q^{97} -69.1083 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 48 q^{4}+O(q^{10}) \) 24 * q + 48 * q^4	\( 24 q + 48 q^{4} - 12 q^{10} + 96 q^{16} - 48 q^{25} - 120 q^{34} - 24 q^{40} + 72 q^{49} + 216 q^{55} + 120 q^{61} + 192 q^{64} + 192 q^{70} + 480 q^{79} + 444 q^{85} + 48 q^{91} + 96 q^{94}+O(q^{100}) \) 24 * q + 48 * q^4 - 12 * q^10 + 96 * q^16 - 48 * q^25 - 120 * q^34 - 24 * q^40 + 72 * q^49 + 216 * q^55 + 120 * q^61 + 192 * q^64 + 192 * q^70 + 480 * q^79 + 444 * q^85 + 48 * q^91 + 96 * q^94

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