LMFDB - Embedded newform 495.3.b.a.406.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [495,3,Mod(406,495)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("495.406"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(495, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 495 = 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	495.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,-28] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$13.4877730858$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 30x^{6} + 280x^{4} + 890x^{2} + 895 $ x^8 + 30x^6 + 280x^4 + 890*x^2 + 895
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 55)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			406.4
Root			$-1.46104i$ of defining polynomial
Character	$\chi$	$=$	495.406
Dual form			495.3.b.a.406.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.46104i q^{2} +1.86536 q^{4} -2.23607 q^{5} +4.56066i q^{7} -8.56953i q^{8} +3.26698i q^{10} +(-5.89241 + 9.28868i) q^{11} -16.5645i q^{13} +6.66330 q^{14} -5.05897 q^{16} -17.2939i q^{17} -35.8838i q^{19} -4.17108 q^{20} +(13.5711 + 8.60904i) q^{22} +29.3902 q^{23} +5.00000 q^{25} -24.2014 q^{26} +8.50728i q^{28} -8.51985i q^{29} -26.3476 q^{31} -26.8868i q^{32} -25.2670 q^{34} -10.1979i q^{35} +44.4227 q^{37} -52.4276 q^{38} +19.1620i q^{40} -52.2243i q^{41} +6.77375i q^{43} +(-10.9915 + 17.3268i) q^{44} -42.9402i q^{46} -15.0434 q^{47} +28.2004 q^{49} -7.30520i q^{50} -30.8989i q^{52} +33.1498 q^{53} +(13.1758 - 20.7701i) q^{55} +39.0827 q^{56} -12.4478 q^{58} -51.5447 q^{59} +23.1889i q^{61} +38.4948i q^{62} -59.5185 q^{64} +37.0394i q^{65} -113.668 q^{67} -32.2593i q^{68} -14.8996 q^{70} -8.00364 q^{71} +32.5342i q^{73} -64.9034i q^{74} -66.9363i q^{76} +(-42.3625 - 26.8732i) q^{77} +52.0160i q^{79} +11.3122 q^{80} -76.3018 q^{82} +43.3699i q^{83} +38.6702i q^{85} +9.89672 q^{86} +(79.5996 + 50.4951i) q^{88} -73.8028 q^{89} +75.5451 q^{91} +54.8233 q^{92} +21.9790i q^{94} +80.2386i q^{95} +22.0298 q^{97} -41.2019i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 28 q^{4} - 8 q^{11} + 88 q^{16} + 20 q^{20} + 80 q^{22} - 8 q^{23} + 40 q^{25} + 100 q^{26} + 36 q^{31} + 80 q^{34} - 88 q^{37} + 160 q^{38} - 12 q^{44} + 8 q^{47} + 172 q^{49} + 152 q^{53} - 20 q^{55}+ \cdots + 312 q^{97}+O(q^{100}) $ 8 * q - 28 * q^4 - 8 * q^11 + 88 * q^16 + 20 * q^20 + 80 * q^22 - 8 * q^23 + 40 * q^25 + 100 * q^26 + 36 * q^31 + 80 * q^34 - 88 * q^37 + 160 * q^38 - 12 * q^44 + 8 * q^47 + 172 * q^49 + 152 * q^53 - 20 * q^55 + 80 * q^56 + 160 * q^58 + 16 * q^59 - 588 * q^64 - 88 * q^67 + 60 * q^70 - 276 * q^71 - 160 * q^77 - 320 * q^80 - 520 * q^82 - 420 * q^86 - 520 * q^88 - 444 * q^89 - 40 * q^91 + 368 * q^92 + 312 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/495\mathbb{Z}\right)^\times$.

$n$	$46$	$56$	$397$
$\chi(n)$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.46104i		−	0.730520i	−0.930906	−	0.365260i	$-0.880980\pi$
$2$		−	1.46104i		−	0.730520i	0.930906	−	0.365260i	$-0.119020\pi$
$3$		0			0
$3$		0			0
$4$	1.86536			0.466341
$5$	−2.23607			−0.447214
$5$	−2.23607			−0.447214
$6$		0			0
$7$			4.56066i			0.651522i	0.945452	+	0.325761i	$0.105621\pi$
$7$			4.56066i			0.651522i	−0.945452	+	0.325761i	$0.894379\pi$
$8$		−	8.56953i		−	1.07119i
$9$		0			0
$10$			3.26698i			0.326698i
$11$	−5.89241	+	9.28868i	−0.535673	+	0.844425i
$11$	−5.89241	+	9.28868i	−0.535673	+	0.844425i
$12$		0			0
$13$		−	16.5645i		−	1.27420i	−0.770783	−	0.637098i	$-0.780135\pi$
$13$		−	16.5645i		−	1.27420i	0.770783	−	0.637098i	$-0.219865\pi$
$14$	6.66330			0.475950
$15$		0			0
$16$	−5.05897			−0.316185
$17$		−	17.2939i		−	1.01729i	−0.860978	−	0.508643i	$-0.830147\pi$
$17$		−	17.2939i		−	1.01729i	0.860978	−	0.508643i	$-0.169853\pi$
$18$		0			0
$19$		−	35.8838i		−	1.88862i	−0.329057	−	0.944310i	$-0.606731\pi$
$19$		−	35.8838i		−	1.88862i	0.329057	−	0.944310i	$-0.393269\pi$
$20$	−4.17108			−0.208554
$21$		0			0
$22$	13.5711	+	8.60904i	0.616869	+	0.391320i
$23$	29.3902			1.27783			0.638917	−	0.769276i	$-0.279383\pi$
$23$	29.3902			1.27783			0.638917	+	0.769276i	$0.279383\pi$
$24$		0			0
$25$	5.00000			0.200000
$26$	−24.2014			−0.930825
$27$		0			0
$28$			8.50728i			0.303831i
$29$		−	8.51985i		−	0.293788i	−0.989152	−	0.146894i	$-0.953072\pi$
$29$		−	8.51985i		−	0.293788i	0.989152	−	0.146894i	$-0.0469276\pi$
$30$		0			0
$31$	−26.3476			−0.849921			−0.424961	−	0.905212i	$-0.639712\pi$
$31$	−26.3476			−0.849921			−0.424961	+	0.905212i	$0.639712\pi$
$32$		−	26.8868i		−	0.840211i
$33$		0			0
$34$	−25.2670			−0.743147
$35$		−	10.1979i		−	0.291370i
$36$		0			0
$37$	44.4227			1.20061			0.600307	−	0.799769i	$-0.295045\pi$
$37$	44.4227			1.20061			0.600307	+	0.799769i	$0.295045\pi$
$38$	−52.4276			−1.37967
$39$		0			0
$40$			19.1620i			0.479051i
$41$		−	52.2243i		−	1.27376i	−0.770961	−	0.636882i	$-0.780224\pi$
$41$		−	52.2243i		−	1.27376i	0.770961	−	0.636882i	$-0.219776\pi$
$42$		0			0
$43$			6.77375i			0.157529i	0.996893	+	0.0787646i	$0.0250975\pi$
$43$			6.77375i			0.157529i	−0.996893	+	0.0787646i	$0.974902\pi$
$44$	−10.9915	+	17.3268i	−0.249806	+	0.393790i
$45$		0			0
$46$		−	42.9402i		−	0.933482i
$47$	−15.0434			−0.320072			−0.160036	−	0.987111i	$-0.551161\pi$
$47$	−15.0434			−0.320072			−0.160036	+	0.987111i	$0.551161\pi$
$48$		0			0
$49$	28.2004			0.575519
$50$		−	7.30520i		−	0.146104i
$51$		0			0
$52$		−	30.8989i		−	0.594209i
$53$	33.1498			0.625468			0.312734	−	0.949841i	$-0.398755\pi$
$53$	33.1498			0.625468			0.312734	+	0.949841i	$0.398755\pi$
$54$		0			0
$55$	13.1758	−	20.7701i	0.239560	−	0.377638i
$56$	39.0827			0.697905
$57$		0			0
$58$	−12.4478			−0.214618
$59$	−51.5447			−0.873639			−0.436819	−	0.899549i	$-0.643895\pi$
$59$	−51.5447			−0.873639			−0.436819	+	0.899549i	$0.643895\pi$
$60$		0			0
$61$			23.1889i			0.380146i	0.981770	+	0.190073i	$0.0608725\pi$
$61$			23.1889i			0.380146i	−0.981770	+	0.190073i	$0.939127\pi$
$62$			38.4948i			0.620884i
$63$		0			0
$64$	−59.5185			−0.929976
$65$			37.0394i			0.569837i
$66$		0			0
$67$	−113.668			−1.69653			−0.848265	−	0.529572i	$-0.822353\pi$
$67$	−113.668			−1.69653			−0.848265	+	0.529572i	$0.822353\pi$
$68$		−	32.2593i		−	0.474402i
$69$		0			0
$70$	−14.8996			−0.212851
$71$	−8.00364			−0.112727			−0.0563637	−	0.998410i	$-0.517951\pi$
$71$	−8.00364			−0.112727			−0.0563637	+	0.998410i	$0.517951\pi$
$72$		0			0
$73$			32.5342i			0.445674i	0.974856	+	0.222837i	$0.0715318\pi$
$73$			32.5342i			0.445674i	−0.974856	+	0.222837i	$0.928468\pi$
$74$		−	64.9034i		−	0.877073i
$75$		0			0
$76$		−	66.9363i		−	0.880741i
$77$	−42.3625	−	26.8732i	−0.550162	−	0.349003i
$78$		0			0
$79$			52.0160i			0.658431i	0.944255	+	0.329215i	$0.106784\pi$
$79$			52.0160i			0.658431i	−0.944255	+	0.329215i	$0.893216\pi$
$80$	11.3122			0.141402
$81$		0			0
$82$	−76.3018			−0.930510
$83$			43.3699i			0.522529i	0.965267	+	0.261264i	$0.0841394\pi$
$83$			43.3699i			0.522529i	−0.965267	+	0.261264i	$0.915861\pi$
$84$		0			0
$85$			38.6702i			0.454944i
$86$	9.89672			0.115078
$87$		0			0
$88$	79.5996	+	50.4951i	0.904541	+	0.573808i
$89$	−73.8028			−0.829245			−0.414623	−	0.909993i	$-0.636086\pi$
$89$	−73.8028			−0.829245			−0.414623	+	0.909993i	$0.636086\pi$
$90$		0			0
$91$	75.5451			0.830166
$92$	54.8233			0.595906
$93$		0			0
$94$			21.9790i			0.233819i
$95$			80.2386i			0.844617i
$96$		0			0
$97$	22.0298			0.227111			0.113556	−	0.993532i	$-0.463776\pi$
$97$	22.0298			0.227111			0.113556	+	0.993532i	$0.463776\pi$
$98$		−	41.2019i		−	0.420428i
$99$		0			0
$100$	9.32682			0.0932682
$101$		−	163.013i		−	1.61399i	−0.590561	−	0.806993i	$-0.701094\pi$
$101$		−	163.013i		−	1.61399i	0.590561	−	0.806993i	$-0.298906\pi$
$102$		0			0
$103$	42.6423			0.414003			0.207001	−	0.978341i	$-0.433630\pi$
$103$	42.6423			0.414003			0.207001	+	0.978341i	$0.433630\pi$
$104$	−141.950			−1.36491
$105$		0			0
$106$		−	48.4332i		−	0.456917i
$107$			131.098i			1.22522i	0.790387	+	0.612608i	$0.209879\pi$
$107$			131.098i			1.22522i	−0.790387	+	0.612608i	$0.790121\pi$
$108$		0			0
$109$			54.5201i			0.500184i	0.968222	+	0.250092i	$0.0804609\pi$
$109$			54.5201i			0.500184i	−0.968222	+	0.250092i	$0.919539\pi$
$110$	−30.3460	−	19.2504i	−0.275872	−	0.175004i
$111$		0			0
$112$		−	23.0722i		−	0.206002i
$113$	117.985			1.04412			0.522058	−	0.852910i	$-0.325164\pi$
$113$	117.985			1.04412			0.522058	+	0.852910i	$0.325164\pi$
$114$		0			0
$115$	−65.7184			−0.571464
$116$		−	15.8926i		−	0.137005i
$117$		0			0
$118$			75.3088i			0.638210i
$119$	78.8713			0.662784
$120$		0			0
$121$	−51.5591	−	109.465i	−0.426108	−	0.904672i
$122$	33.8799			0.277704
$123$		0			0
$124$	−49.1478			−0.396353
$125$	−11.1803			−0.0894427
$126$		0			0
$127$		−	113.764i		−	0.895782i	−0.894088	−	0.447891i	$-0.852175\pi$
$127$		−	113.764i		−	0.895782i	0.894088	−	0.447891i	$-0.147825\pi$
$128$		−	20.5882i		−	0.160845i
$129$		0			0
$130$	54.1161			0.416277
$131$			115.599i			0.882435i	0.897400	+	0.441218i	$0.145453\pi$
$131$			115.599i			0.882435i	−0.897400	+	0.441218i	$0.854547\pi$
$132$		0			0
$133$	163.654			1.23048
$134$			166.073i			1.23935i
$135$		0			0
$136$	−148.200			−1.08971
$137$	216.355			1.57924			0.789618	−	0.613598i	$-0.210279\pi$
$137$	216.355			1.57924			0.789618	+	0.613598i	$0.210279\pi$
$138$		0			0
$139$		−	5.08013i		−	0.0365477i	−0.999833	−	0.0182738i	$-0.994183\pi$
$139$		−	5.08013i		−	0.0365477i	0.999833	−	0.0182738i	$-0.00581707\pi$
$140$		−	19.0229i		−	0.135878i
$141$		0			0
$142$			11.6936i			0.0823495i
$143$	153.863	+	97.6050i	1.07596	+	0.682552i
$144$		0			0
$145$			19.0510i			0.131386i
$146$	47.5338			0.325574
$147$		0			0
$148$	82.8646			0.559896
$149$			187.947i			1.26139i	0.776030	+	0.630696i	$0.217231\pi$
$149$			187.947i			1.26139i	−0.776030	+	0.630696i	$0.782769\pi$
$150$		0			0
$151$			136.741i			0.905571i	0.891620	+	0.452785i	$0.149570\pi$
$151$			136.741i			0.905571i	−0.891620	+	0.452785i	$0.850430\pi$
$152$	−307.507			−2.02307
$153$		0			0
$154$	−39.2629	+	61.8932i	−0.254954	+	0.401904i
$155$	58.9149			0.380096
$156$		0			0
$157$	256.057			1.63094			0.815470	−	0.578800i	$-0.196479\pi$
$157$	256.057			1.63094			0.815470	+	0.578800i	$0.196479\pi$
$158$	75.9975			0.480997
$159$		0			0
$160$			60.1206i			0.375754i
$161$			134.038i			0.832537i
$162$		0			0
$163$	−25.1370			−0.154215			−0.0771075	−	0.997023i	$-0.524568\pi$
$163$	−25.1370			−0.154215			−0.0771075	+	0.997023i	$0.524568\pi$
$164$		−	97.4174i		−	0.594008i
$165$		0			0
$166$	63.3651			0.381718
$167$			4.21038i			0.0252119i	0.999921	+	0.0126059i	$0.00401270\pi$
$167$			4.21038i			0.0252119i	−0.999921	+	0.0126059i	$0.995987\pi$
$168$		0			0
$169$	−105.384			−0.623573
$170$	56.4987			0.332346
$171$		0			0
$172$			12.6355i			0.0734623i
$173$			168.015i			0.971185i	0.874185	+	0.485592i	$0.161396\pi$
$173$			168.015i			0.971185i	−0.874185	+	0.485592i	$0.838604\pi$
$174$		0			0
$175$			22.8033i			0.130304i
$176$	29.8095	−	46.9911i	0.169372	−	0.266995i
$177$		0			0
$178$			107.829i			0.605780i
$179$	34.5542			0.193040			0.0965200	−	0.995331i	$-0.469229\pi$
$179$	34.5542			0.193040			0.0965200	+	0.995331i	$0.469229\pi$
$180$		0			0
$181$	−107.349			−0.593091			−0.296545	−	0.955019i	$-0.595835\pi$
$181$	−107.349			−0.593091			−0.296545	+	0.955019i	$0.595835\pi$
$182$		−	110.374i		−	0.606453i
$183$		0			0
$184$		−	251.860i		−	1.36880i
$185$	−99.3323			−0.536931
$186$		0			0
$187$	160.637	+	101.902i	0.859022	+	0.544933i
$188$	−28.0614			−0.149263
$189$		0			0
$190$	117.232			0.617009
$191$	75.4470			0.395011			0.197505	−	0.980302i	$-0.436716\pi$
$191$	75.4470			0.395011			0.197505	+	0.980302i	$0.436716\pi$
$192$		0			0
$193$			167.064i			0.865617i	0.901486	+	0.432809i	$0.142477\pi$
$193$			167.064i			0.865617i	−0.901486	+	0.432809i	$0.857523\pi$
$194$		−	32.1864i		−	0.165909i
$195$		0			0
$196$	52.6040			0.268388
$197$		−	362.777i		−	1.84151i	−0.390145	−	0.920754i	$-0.627575\pi$
$197$		−	362.777i		−	1.84151i	0.390145	−	0.920754i	$-0.372425\pi$
$198$		0			0
$199$	337.735			1.69716			0.848580	−	0.529068i	$-0.177458\pi$
$199$	337.735			1.69716			0.848580	+	0.529068i	$0.177458\pi$
$200$		−	42.8476i		−	0.214238i
$201$		0			0
$202$	−238.168			−1.17905
$203$	38.8561			0.191409
$204$		0			0
$205$			116.777i			0.569645i
$206$		−	62.3021i		−	0.302437i
$207$		0			0
$208$			83.7994i			0.402882i
$209$	333.313	+	211.442i	1.59480	+	1.01168i
$210$		0			0
$211$		−	317.915i		−	1.50670i	−0.657617	−	0.753352i	$-0.728436\pi$
$211$		−	317.915i		−	1.50670i	0.657617	−	0.753352i	$-0.271564\pi$
$212$	61.8364			0.291681
$213$		0			0
$214$	191.539			0.895044
$215$		−	15.1466i		−	0.0704492i
$216$		0			0
$217$		−	120.162i		−	0.553743i
$218$	79.6560			0.365395
$219$		0			0
$220$	24.5777	−	38.7438i	0.111717	−	0.176108i
$221$	−286.465			−1.29622
$222$		0			0
$223$	292.643			1.31230			0.656151	−	0.754630i	$-0.272183\pi$
$223$	292.643			1.31230			0.656151	+	0.754630i	$0.272183\pi$
$224$	122.621			0.547416
$225$		0			0
$226$		−	172.381i		−	0.762747i
$227$		−	9.54908i		−	0.0420664i	−0.999779	−	0.0210332i	$-0.993304\pi$
$227$		−	9.54908i		−	0.0420664i	0.999779	−	0.0210332i	$-0.00669557\pi$
$228$		0			0
$229$	−355.912			−1.55420			−0.777100	−	0.629377i	$-0.783310\pi$
$229$	−355.912			−1.55420			−0.777100	+	0.629377i	$0.783310\pi$
$230$			96.0172i			0.417466i
$231$		0			0
$232$	−73.0111			−0.314703
$233$			270.961i			1.16292i	0.813573	+	0.581462i	$0.197519\pi$
$233$			270.961i			1.16292i	−0.813573	+	0.581462i	$0.802481\pi$
$234$		0			0
$235$	33.6381			0.143141
$236$	−96.1496			−0.407413
$237$		0			0
$238$		−	115.234i		−	0.484177i
$239$		−	79.1374i		−	0.331119i	−0.986200	−	0.165560i	$-0.947057\pi$
$239$		−	79.1374i		−	0.331119i	0.986200	−	0.165560i	$-0.0529430\pi$
$240$		0			0
$241$		−	146.567i		−	0.608164i	−0.952646	−	0.304082i	$-0.901650\pi$
$241$		−	146.567i		−	0.608164i	0.952646	−	0.304082i	$-0.0983496\pi$
$242$	−159.933	+	75.3299i	−0.660881	+	0.311281i
$243$		0			0
$244$			43.2558i			0.177278i
$245$	−63.0581			−0.257380
$246$		0			0
$247$	−594.398			−2.40647
$248$			225.786i			0.910428i
$249$		0			0
$250$			16.3349i			0.0653397i
$251$	−52.7485			−0.210154			−0.105077	−	0.994464i	$-0.533509\pi$
$251$	−52.7485			−0.210154			−0.105077	+	0.994464i	$0.533509\pi$
$252$		0			0
$253$	−173.179	+	272.996i	−0.684501	+	1.07903i
$254$	−166.214			−0.654386
$255$		0			0
$256$	−268.154			−1.04748
$257$	−41.8668			−0.162906			−0.0814529	−	0.996677i	$-0.525956\pi$
$257$	−41.8668			−0.162906			−0.0814529	+	0.996677i	$0.525956\pi$
$258$		0			0
$259$			202.597i			0.782227i
$260$			69.0920i			0.265738i
$261$		0			0
$262$	168.895			0.644636
$263$		−	289.448i		−	1.10056i	−0.834979	−	0.550282i	$-0.814520\pi$
$263$		−	289.448i		−	1.10056i	0.834979	−	0.550282i	$-0.185480\pi$
$264$		0			0
$265$	−74.1252			−0.279718
$266$		−	239.104i		−	0.898888i
$267$		0			0
$268$	−212.031			−0.791161
$269$	−297.640			−1.10647			−0.553234	−	0.833026i	$-0.686607\pi$
$269$	−297.640			−1.10647			−0.553234	+	0.833026i	$0.686607\pi$
$270$		0			0
$271$			277.488i			1.02394i	0.859003	+	0.511970i	$0.171084\pi$
$271$			277.488i			1.02394i	−0.859003	+	0.511970i	$0.828916\pi$
$272$			87.4890i			0.321651i
$273$		0			0
$274$		−	316.104i		−	1.15366i
$275$	−29.4620	+	46.4434i	−0.107135	+	0.168885i
$276$		0			0
$277$			22.6740i			0.0818556i	0.999162	+	0.0409278i	$0.0130314\pi$
$277$			22.6740i			0.0818556i	−0.999162	+	0.0409278i	$0.986969\pi$
$278$	−7.42227			−0.0266988
$279$		0			0
$280$	−87.3915			−0.312112
$281$			182.887i			0.650845i	0.945569	+	0.325422i	$0.105506\pi$
$281$			182.887i			0.650845i	−0.945569	+	0.325422i	$0.894494\pi$
$282$		0			0
$283$			367.233i			1.29764i	0.760941	+	0.648821i	$0.224738\pi$
$283$			367.233i			1.29764i	−0.760941	+	0.648821i	$0.775262\pi$
$284$	−14.9297			−0.0525694
$285$		0			0
$286$	142.605	−	224.799i	0.498618	−	0.786012i
$287$	238.177			0.829886
$288$		0			0
$289$	−10.0774			−0.0348699
$290$	27.8342			0.0959800
$291$		0			0
$292$			60.6881i			0.207836i
$293$			10.3180i			0.0352150i	0.999845	+	0.0176075i	$0.00560493\pi$
$293$			10.3180i			0.0352150i	−0.999845	+	0.0176075i	$0.994395\pi$
$294$		0			0
$295$	115.257			0.390703
$296$		−	380.682i		−	1.28609i
$297$		0			0
$298$	274.599			0.921472
$299$		−	486.834i		−	1.62821i
$300$		0			0
$301$	−30.8928			−0.102634
$302$	199.784			0.661537
$303$		0			0
$304$			181.535i			0.597154i
$305$		−	51.8520i		−	0.170007i
$306$		0			0
$307$		−	459.671i		−	1.49730i	−0.662966	−	0.748649i	$-0.730703\pi$
$307$		−	459.671i		−	1.49730i	0.662966	−	0.748649i	$-0.269297\pi$
$308$	−79.0214	−	50.1283i	−0.256563	−	0.162754i
$309$		0			0
$310$		−	86.0770i		−	0.277668i
$311$	−101.131			−0.325181			−0.162590	−	0.986694i	$-0.551985\pi$
$311$	−101.131			−0.325181			−0.162590	+	0.986694i	$0.551985\pi$
$312$		0			0
$313$	288.760			0.922556			0.461278	−	0.887256i	$-0.347391\pi$
$313$	288.760			0.922556			0.461278	+	0.887256i	$0.347391\pi$
$314$		−	374.110i		−	1.19143i
$315$		0			0
$316$			97.0288i			0.307053i
$317$	−242.929			−0.766338			−0.383169	−	0.923678i	$-0.625167\pi$
$317$	−242.929			−0.766338			−0.383169	+	0.923678i	$0.625167\pi$
$318$		0			0
$319$	79.1381	+	50.2024i	0.248082	+	0.157374i
$320$	133.087			0.415898
$321$		0			0
$322$	195.835			0.608185
$323$	−620.569			−1.92127
$324$		0			0
$325$		−	82.8227i		−	0.254839i
$326$			36.7262i			0.112657i
$327$		0			0
$328$	−447.538			−1.36445
$329$		−	68.6078i		−	0.208534i
$330$		0			0
$331$	−241.395			−0.729291			−0.364646	−	0.931146i	$-0.618810\pi$
$331$	−241.395			−0.729291			−0.364646	+	0.931146i	$0.618810\pi$
$332$			80.9006i			0.243676i
$333$		0			0
$334$	6.15154			0.0184178
$335$	254.168			0.758711
$336$		0			0
$337$		−	527.662i		−	1.56576i	−0.622171	−	0.782881i	$-0.713749\pi$
$337$		−	527.662i		−	1.56576i	0.622171	−	0.782881i	$-0.286251\pi$
$338$			153.970i			0.455532i
$339$		0			0
$340$			72.1340i			0.212159i
$341$	155.251	−	244.734i	0.455280	−	0.717695i
$342$		0			0
$343$			352.085i			1.02649i
$344$	58.0479			0.168744
$345$		0			0
$346$	245.476			0.709470
$347$			214.531i			0.618244i	0.951022	+	0.309122i	$0.100035\pi$
$347$			214.531i			0.618244i	−0.951022	+	0.309122i	$0.899965\pi$
$348$		0			0
$349$			76.4945i			0.219182i	0.993977	+	0.109591i	$0.0349541\pi$
$349$			76.4945i			0.219182i	−0.993977	+	0.109591i	$0.965046\pi$
$350$	33.3165			0.0951900
$351$		0			0
$352$	249.743	+	158.428i	0.709496	+	0.450079i
$353$	176.689			0.500535			0.250268	−	0.968177i	$-0.419481\pi$
$353$	176.689			0.500535			0.250268	+	0.968177i	$0.419481\pi$
$354$		0			0
$355$	17.8967			0.0504132
$356$	−137.669			−0.386711
$357$		0			0
$358$		−	50.4850i		−	0.141020i
$359$			357.810i			0.996686i	0.866980	+	0.498343i	$0.166058\pi$
$359$			357.810i			0.996686i	−0.866980	+	0.498343i	$0.833942\pi$
$360$		0			0
$361$	−926.646			−2.56689
$362$			156.842i			0.433265i
$363$		0			0
$364$	140.919			0.387140
$365$		−	72.7487i		−	0.199312i
$366$		0			0
$367$	−314.687			−0.857458			−0.428729	−	0.903433i	$-0.641038\pi$
$367$	−314.687			−0.857458			−0.428729	+	0.903433i	$0.641038\pi$
$368$	−148.684			−0.404032
$369$		0			0
$370$			145.128i			0.392239i
$371$			151.185i			0.407506i
$372$		0			0
$373$		−	420.645i		−	1.12773i	−0.825865	−	0.563867i	$-0.809313\pi$
$373$		−	420.645i		−	1.12773i	0.825865	−	0.563867i	$-0.190687\pi$
$374$	148.883	−	234.697i	0.398084	−	0.627532i
$375$		0			0
$376$			128.915i			0.342859i
$377$	−141.127			−0.374343
$378$		0			0
$379$	401.667			1.05981			0.529904	−	0.848058i	$-0.322228\pi$
$379$	401.667			1.05981			0.529904	+	0.848058i	$0.322228\pi$
$380$			149.674i			0.393879i
$381$		0			0
$382$		−	110.231i		−	0.288563i
$383$	−60.0864			−0.156884			−0.0784418	−	0.996919i	$-0.524994\pi$
$383$	−60.0864			−0.156884			−0.0784418	+	0.996919i	$0.524994\pi$
$384$		0			0
$385$	94.7253	+	60.0904i	0.246040	+	0.156079i
$386$	244.087			0.632350
$387$		0			0
$388$	41.0936			0.105911
$389$	−411.464			−1.05775			−0.528874	−	0.848700i	$-0.677386\pi$
$389$	−411.464			−1.05775			−0.528874	+	0.848700i	$0.677386\pi$
$390$		0			0
$391$		−	508.269i		−	1.29992i
$392$		−	241.664i		−	0.616491i
$393$		0			0
$394$	−530.031			−1.34526
$395$		−	116.311i		−	0.294459i
$396$		0			0
$397$	−237.517			−0.598279			−0.299140	−	0.954209i	$-0.596700\pi$
$397$	−237.517			−0.598279			−0.299140	+	0.954209i	$0.596700\pi$
$398$		−	493.444i		−	1.23981i
$399$		0			0
$400$	−25.2948			−0.0632371
$401$	−101.501			−0.253120			−0.126560	−	0.991959i	$-0.540394\pi$
$401$	−101.501			−0.253120			−0.126560	+	0.991959i	$0.540394\pi$
$402$		0			0
$403$			436.435i			1.08297i
$404$		−	304.078i		−	0.752667i
$405$		0			0
$406$		−	56.7703i		−	0.139828i
$407$	−261.757	+	412.629i	−0.643137	+	1.01383i
$408$		0			0
$409$		−	694.176i		−	1.69725i	−0.528993	−	0.848626i	$-0.677430\pi$
$409$		−	694.176i		−	1.69725i	0.528993	−	0.848626i	$-0.322570\pi$
$410$	170.616			0.416137
$411$		0			0
$412$	79.5433			0.193066
$413$		−	235.078i		−	0.569195i
$414$		0			0
$415$		−	96.9780i		−	0.233682i
$416$	−445.367			−1.07059
$417$		0			0
$418$	308.925	−	486.983i	0.739055	−	1.16503i
$419$	416.354			0.993686			0.496843	−	0.867841i	$-0.334493\pi$
$419$	416.354			0.993686			0.496843	+	0.867841i	$0.334493\pi$
$420$		0			0
$421$	539.341			1.28110			0.640548	−	0.767918i	$-0.278707\pi$
$421$	539.341			1.28110			0.640548	+	0.767918i	$0.278707\pi$
$422$	−464.486			−1.10068
$423$		0			0
$424$		−	284.078i		−	0.669995i
$425$		−	86.4693i		−	0.203457i
$426$		0			0
$427$	−105.757			−0.247674
$428$			244.546i			0.571368i
$429$		0			0
$430$	−22.1297			−0.0514645
$431$			343.493i			0.796967i	0.917176	+	0.398483i	$0.130463\pi$
$431$			343.493i			0.796967i	−0.917176	+	0.398483i	$0.869537\pi$
$432$		0			0
$433$	18.0353			0.0416520			0.0208260	−	0.999783i	$-0.493370\pi$
$433$	18.0353			0.0416520			0.0208260	+	0.999783i	$0.493370\pi$
$434$	−175.562			−0.404520
$435$		0			0
$436$			101.700i			0.233256i
$437$		−	1054.63i		−	2.41334i
$438$		0			0
$439$			693.839i			1.58050i	0.612786	+	0.790249i	$0.290049\pi$
$439$			693.839i			1.58050i	−0.612786	+	0.790249i	$0.709951\pi$
$440$	−177.990	−	112.911i	−0.404523	−	0.256615i
$441$		0			0
$442$			418.536i			0.946914i
$443$	359.314			0.811092			0.405546	−	0.914075i	$-0.367081\pi$
$443$	359.314			0.811092			0.405546	+	0.914075i	$0.367081\pi$
$444$		0			0
$445$	165.028			0.370850
$446$		−	427.564i		−	0.958663i
$447$		0			0
$448$		−	271.443i		−	0.605900i
$449$	567.882			1.26477			0.632385	−	0.774654i	$-0.282076\pi$
$449$	567.882			1.26477			0.632385	+	0.774654i	$0.282076\pi$
$450$		0			0
$451$	485.095	+	307.727i	1.07560	+	0.682322i
$452$	220.085			0.486914
$453$		0			0
$454$	−13.9516			−0.0307303
$455$	−168.924			−0.371262
$456$		0			0
$457$			797.218i			1.74446i	0.489096	+	0.872230i	$0.337327\pi$
$457$			797.218i			1.74446i	−0.489096	+	0.872230i	$0.662673\pi$
$458$			520.001i			1.13537i
$459$		0			0
$460$	−122.589			−0.266497
$461$		−	87.9144i		−	0.190704i	−0.995444	−	0.0953518i	$-0.969602\pi$
$461$		−	87.9144i		−	0.190704i	0.995444	−	0.0953518i	$-0.0303976\pi$
$462$		0			0
$463$	−50.9282			−0.109996			−0.0549981	−	0.998486i	$-0.517515\pi$
$463$	−50.9282			−0.109996			−0.0549981	+	0.998486i	$0.517515\pi$
$464$			43.1016i			0.0928914i
$465$		0			0
$466$	395.885			0.849539
$467$	463.064			0.991572			0.495786	−	0.868445i	$-0.334880\pi$
$467$	463.064			0.991572			0.495786	+	0.868445i	$0.334880\pi$
$468$		0			0
$469$		−	518.398i		−	1.10533i
$470$		−	49.1466i		−	0.104567i
$471$		0			0
$472$			441.714i			0.935834i
$473$	−62.9192	−	39.9137i	−0.133022	−	0.0843841i
$474$		0			0
$475$		−	179.419i		−	0.377724i
$476$	147.124			0.309083
$477$		0			0
$478$	−115.623			−0.241889
$479$		−	182.184i		−	0.380343i	−0.981751	−	0.190171i	$-0.939096\pi$
$479$		−	182.184i		−	0.380343i	0.981751	−	0.190171i	$-0.0609044\pi$
$480$		0			0
$481$		−	735.842i		−	1.52982i
$482$	−214.141			−0.444276
$483$		0			0
$484$	−96.1765	−	204.193i	−0.198712	−	0.421886i
$485$	−49.2602			−0.101567
$486$		0			0
$487$	366.810			0.753203			0.376602	−	0.926375i	$-0.377093\pi$
$487$	366.810			0.753203			0.376602	+	0.926375i	$0.377093\pi$
$488$	198.718			0.407209
$489$		0			0
$490$			92.1303i			0.188021i
$491$		−	42.0181i		−	0.0855766i	−0.999084	−	0.0427883i	$-0.986376\pi$
$491$		−	42.0181i		−	0.0855766i	0.999084	−	0.0427883i	$-0.0136241\pi$
$492$		0			0
$493$	−147.341			−0.298866
$494$			868.439i			1.75797i
$495$		0			0
$496$	133.291			0.268733
$497$		−	36.5018i		−	0.0734444i
$498$		0			0
$499$	−229.963			−0.460848			−0.230424	−	0.973090i	$-0.574011\pi$
$499$	−229.963			−0.460848			−0.230424	+	0.973090i	$0.574011\pi$
$500$	−20.8554			−0.0417108
$501$		0			0
$502$			77.0677i			0.153521i
$503$		−	9.38400i		−	0.0186561i	−0.999956	−	0.00932803i	$-0.997031\pi$
$503$		−	9.38400i		−	0.0186561i	0.999956	−	0.00932803i	$-0.00296925\pi$
$504$		0			0
$505$			364.507i			0.721796i
$506$	398.858	+	253.021i	0.788256	+	0.500042i
$507$		0			0
$508$		−	212.212i		−	0.417740i
$509$	584.355			1.14804			0.574022	−	0.818840i	$-0.305382\pi$
$509$	584.355			1.14804			0.574022	+	0.818840i	$0.305382\pi$
$510$		0			0
$511$	−148.377			−0.290367
$512$			309.431i			0.604357i
$513$		0			0
$514$			61.1691i			0.119006i
$515$	−95.3510			−0.185148
$516$		0			0
$517$	88.6418	−	139.733i	0.171454	−	0.270277i
$518$	296.002			0.571432
$519$		0			0
$520$	317.410			0.610405
$521$	−121.060			−0.232361			−0.116180	−	0.993228i	$-0.537065\pi$
$521$	−121.060			−0.232361			−0.116180	+	0.993228i	$0.537065\pi$
$522$		0			0
$523$		−	446.636i		−	0.853988i	−0.904254	−	0.426994i	$-0.859572\pi$
$523$		−	446.636i		−	0.853988i	0.904254	−	0.426994i	$-0.140428\pi$
$524$			215.634i			0.411516i
$525$		0			0
$526$	−422.896			−0.803984
$527$			455.651i			0.864613i
$528$		0			0
$529$	334.782			0.632858
$530$			108.300i			0.204339i
$531$		0			0
$532$	305.273			0.573822
$533$	−865.072			−1.62302
$534$		0			0
$535$		−	293.144i		−	0.547933i
$536$			974.077i			1.81731i
$537$		0			0
$538$			434.864i			0.808297i
$539$	−166.168	+	261.945i	−0.308290	+	0.485983i
$540$		0			0
$541$			472.750i			0.873846i	0.899499	+	0.436923i	$0.143932\pi$
$541$			472.750i			0.873846i	−0.899499	+	0.436923i	$0.856068\pi$
$542$	405.421			0.748009
$543$		0			0
$544$	−464.976			−0.854735
$545$		−	121.911i		−	0.223689i
$546$		0			0
$547$			24.2342i			0.0443039i	0.999755	+	0.0221520i	$0.00705176\pi$
$547$			24.2342i			0.0443039i	−0.999755	+	0.0221520i	$0.992948\pi$
$548$	403.582			0.736463
$549$		0			0
$550$	67.8556	+	43.0452i	0.123374	+	0.0782640i
$551$	−305.724			−0.554854
$552$		0			0
$553$	−237.227			−0.428982
$554$	33.1276			0.0597972
$555$		0			0
$556$		−	9.47629i		−	0.0170437i
$557$		−	597.483i		−	1.07268i	−0.844002	−	0.536340i	$-0.819807\pi$
$557$		−	597.483i		−	1.07268i	0.844002	−	0.536340i	$-0.180193\pi$
$558$		0			0
$559$	112.204			0.200723
$560$			51.5910i			0.0921268i
$561$		0			0
$562$	267.206			0.475455
$563$			963.269i			1.71096i	0.517838	+	0.855479i	$0.326737\pi$
$563$			963.269i			1.71096i	−0.517838	+	0.855479i	$0.673263\pi$
$564$		0			0
$565$	−263.823			−0.466943
$566$	536.541			0.947953
$567$		0			0
$568$			68.5874i			0.120753i
$569$		−	136.737i		−	0.240310i	−0.992755	−	0.120155i	$-0.961661\pi$
$569$		−	136.737i		−	0.240310i	0.992755	−	0.120155i	$-0.0383392\pi$
$570$		0			0
$571$			829.845i			1.45332i	0.686998	+	0.726659i	$0.258928\pi$
$571$			829.845i			1.45332i	−0.686998	+	0.726659i	$0.741072\pi$
$572$	287.010	+	182.069i	0.501765	+	0.318302i
$573$		0			0
$574$		−	347.986i		−	0.606248i
$575$	146.951			0.255567
$576$		0			0
$577$	636.305			1.10278			0.551391	−	0.834247i	$-0.314097\pi$
$577$	636.305			1.10278			0.551391	+	0.834247i	$0.314097\pi$
$578$			14.7235i			0.0254732i
$579$		0			0
$580$			35.5370i			0.0612706i
$581$	−197.795			−0.340439
$582$		0			0
$583$	−195.332	+	307.918i	−0.335046	+	0.528161i
$584$	278.803			0.477402
$585$		0			0
$586$	15.0750			0.0257253
$587$	455.242			0.775540			0.387770	−	0.921756i	$-0.373246\pi$
$587$	455.242			0.775540			0.387770	+	0.921756i	$0.373246\pi$
$588$		0			0
$589$			945.450i			1.60518i
$590$		−	168.396i		−	0.285416i
$591$		0			0
$592$	−224.733			−0.379617
$593$		−	219.359i		−	0.369914i	−0.982747	−	0.184957i	$-0.940785\pi$
$593$		−	219.359i		−	0.369914i	0.982747	−	0.184957i	$-0.0592146\pi$
$594$		0			0
$595$	−176.362			−0.296406
$596$			350.590i			0.588239i
$597$		0			0
$598$	−711.284			−1.18944
$599$	260.846			0.435470			0.217735	−	0.976008i	$-0.430133\pi$
$599$	260.846			0.435470			0.217735	+	0.976008i	$0.430133\pi$
$600$		0			0
$601$			314.178i			0.522759i	0.965236	+	0.261379i	$0.0841773\pi$
$601$			314.178i			0.522759i	−0.965236	+	0.261379i	$0.915823\pi$
$602$			45.1355i			0.0749760i
$603$		0			0
$604$			255.072i			0.422305i
$605$	115.290	+	244.772i	0.190561	+	0.404582i
$606$		0			0
$607$		−	423.627i		−	0.697903i	−0.937141	−	0.348951i	$-0.886538\pi$
$607$		−	423.627i		−	0.697903i	0.937141	−	0.348951i	$-0.113462\pi$
$608$	−964.799			−1.58684
$609$		0			0
$610$	−75.7578			−0.124193
$611$			249.187i			0.407835i
$612$		0			0
$613$		−	144.866i		−	0.236323i	−0.992994	−	0.118162i	$-0.962300\pi$
$613$		−	144.866i		−	0.236323i	0.992994	−	0.118162i	$-0.0377001\pi$
$614$	−671.597			−1.09381
$615$		0			0
$616$	−230.291	+	363.026i	−0.373849	+	0.589328i
$617$	−988.195			−1.60161			−0.800806	−	0.598923i	$-0.795595\pi$
$617$	−988.195			−1.60161			−0.800806	+	0.598923i	$0.795595\pi$
$618$		0			0
$619$	822.930			1.32945			0.664725	−	0.747088i	$-0.268549\pi$
$619$	822.930			1.32945			0.664725	+	0.747088i	$0.268549\pi$
$620$	109.898			0.177254
$621$		0			0
$622$			147.757i			0.237551i
$623$		−	336.589i		−	0.540272i
$624$		0			0
$625$	25.0000			0.0400000
$626$		−	421.890i		−	0.673946i
$627$		0			0
$628$	477.640			0.760574
$629$		−	768.240i		−	1.22137i
$630$		0			0
$631$	−854.486			−1.35418			−0.677089	−	0.735902i	$-0.736759\pi$
$631$	−854.486			−1.35418			−0.677089	+	0.735902i	$0.736759\pi$
$632$	445.753			0.705305
$633$		0			0
$634$			354.929i			0.559825i
$635$			254.385i			0.400606i
$636$		0			0
$637$		−	467.127i		−	0.733323i
$638$	73.3477	−	115.624i	0.114965	−	0.181229i
$639$		0			0
$640$			46.0366i			0.0719322i
$641$	70.2095			0.109531			0.0547656	−	0.998499i	$-0.482559\pi$
$641$	70.2095			0.109531			0.0547656	+	0.998499i	$0.482559\pi$
$642$		0			0
$643$	−560.759			−0.872097			−0.436049	−	0.899923i	$-0.643622\pi$
$643$	−560.759			−0.872097			−0.436049	+	0.899923i	$0.643622\pi$
$644$			250.030i			0.388246i
$645$		0			0
$646$			906.676i			1.40352i
$647$	−73.2119			−0.113156			−0.0565780	−	0.998398i	$-0.518019\pi$
$647$	−73.2119			−0.113156			−0.0565780	+	0.998398i	$0.518019\pi$
$648$		0			0
$649$	303.722	−	478.782i	0.467985	−	0.737723i
$650$	−121.007			−0.186165
$651$		0			0
$652$	−46.8897			−0.0719167
$653$	52.7951			0.0808500			0.0404250	−	0.999183i	$-0.487129\pi$
$653$	52.7951			0.0808500			0.0404250	+	0.999183i	$0.487129\pi$
$654$		0			0
$655$		−	258.487i		−	0.394637i
$656$			264.201i			0.402746i
$657$		0			0
$658$	−100.239			−0.152338
$659$			466.110i			0.707299i	0.935378	+	0.353649i	$0.115059\pi$
$659$			466.110i			0.707299i	−0.935378	+	0.353649i	$0.884941\pi$
$660$		0			0
$661$	759.575			1.14913			0.574565	−	0.818459i	$-0.305171\pi$
$661$	759.575			1.14913			0.574565	+	0.818459i	$0.305171\pi$
$662$			352.688i			0.532762i
$663$		0			0
$664$	371.659			0.559728
$665$	−365.940			−0.550286
$666$		0			0
$667$		−	250.400i		−	0.375412i
$668$			7.85389i			0.0117573i
$669$		0			0
$670$		−	371.350i		−	0.554254i
$671$	−215.394	−	136.639i	−0.321005	−	0.203634i
$672$		0			0
$673$			608.412i			0.904030i	0.892010	+	0.452015i	$0.149295\pi$
$673$			608.412i			0.904030i	−0.892010	+	0.452015i	$0.850705\pi$
$674$	−770.935			−1.14382
$675$		0			0
$676$	−196.579			−0.290798
$677$			557.900i			0.824077i	0.911166	+	0.412038i	$0.135183\pi$
$677$			557.900i			0.824077i	−0.911166	+	0.412038i	$0.864817\pi$
$678$		0			0
$679$			100.470i			0.147968i
$680$	331.386			0.487332
$681$		0			0
$682$	−357.566	−	226.827i	−0.524290	−	0.332591i
$683$	785.901			1.15066			0.575330	−	0.817921i	$-0.304874\pi$
$683$	785.901			1.15066			0.575330	+	0.817921i	$0.304874\pi$
$684$		0			0
$685$	−483.785			−0.706256
$686$	514.409			0.749868
$687$		0			0
$688$		−	34.2682i		−	0.0498084i
$689$		−	549.111i		−	0.796968i
$690$		0			0
$691$	480.448			0.695293			0.347647	−	0.937626i	$-0.386981\pi$
$691$	480.448			0.695293			0.347647	+	0.937626i	$0.386981\pi$
$692$			313.409i			0.452903i
$693$		0			0
$694$	313.438			0.451640
$695$			11.3595i			0.0163446i
$696$		0			0
$697$	−903.160			−1.29578
$698$	111.762			0.160117
$699$		0			0
$700$			42.5364i			0.0607663i
$701$		−	284.947i		−	0.406486i	−0.979128	−	0.203243i	$-0.934852\pi$
$701$		−	284.947i		−	0.406486i	0.979128	−	0.203243i	$-0.0651481\pi$
$702$		0			0
$703$		−	1594.06i		−	2.26750i
$704$	350.707	−	552.848i	0.498163	−	0.785296i
$705$		0			0
$706$		−	258.150i		−	0.365651i
$707$	743.444			1.05155
$708$		0			0
$709$	−313.418			−0.442057			−0.221028	−	0.975267i	$-0.570941\pi$
$709$	−313.418			−0.442057			−0.221028	+	0.975267i	$0.570941\pi$
$710$		−	26.1478i		−	0.0368278i
$711$		0			0
$712$			632.455i			0.888280i
$713$	−774.359			−1.08606
$714$		0			0
$715$	−344.047	−	218.251i	−0.481185	−	0.305247i
$716$	64.4561			0.0900224
$717$		0			0
$718$	522.775			0.728099
$719$	1198.70			1.66718			0.833590	−	0.552384i	$-0.186282\pi$
$719$	1198.70			1.66718			0.833590	+	0.552384i	$0.186282\pi$
$720$		0			0
$721$			194.477i			0.269732i
$722$			1353.87i			1.87516i
$723$		0			0
$724$	−200.246			−0.276583
$725$		−	42.5992i		−	0.0587576i
$726$		0			0
$727$	152.540			0.209821			0.104910	−	0.994482i	$-0.466544\pi$
$727$	152.540			0.209821			0.104910	+	0.994482i	$0.466544\pi$
$728$		−	647.386i		−	0.889267i
$729$		0			0
$730$	−106.289			−0.145601
$731$	117.144			0.160252
$732$		0			0
$733$		−	1107.11i		−	1.51039i	−0.655501	−	0.755194i	$-0.727543\pi$
$733$		−	1107.11i		−	1.51039i	0.655501	−	0.755194i	$-0.272457\pi$
$734$			459.770i			0.626390i
$735$		0			0
$736$		−	790.206i		−	1.07365i
$737$	669.775	−	1055.82i	0.908786	−	1.43259i
$738$		0			0
$739$			438.677i			0.593609i	0.954938	+	0.296805i	$0.0959210\pi$
$739$			438.677i			0.593609i	−0.954938	+	0.296805i	$0.904079\pi$
$740$	−185.291			−0.250393
$741$		0			0
$742$	220.887			0.297691
$743$		−	764.284i		−	1.02865i	−0.857597	−	0.514323i	$-0.828043\pi$
$743$		−	764.284i		−	1.02865i	0.857597	−	0.514323i	$-0.171957\pi$
$744$		0			0
$745$		−	420.263i		−	0.564112i
$746$	−614.579			−0.823832
$747$		0			0
$748$	299.646	+	190.085i	0.400597	+	0.254124i
$749$	−597.893			−0.798255
$750$		0			0
$751$	1224.32			1.63025			0.815126	−	0.579283i	$-0.196668\pi$
$751$	1224.32			1.63025			0.815126	+	0.579283i	$0.196668\pi$
$752$	76.1041			0.101202
$753$		0			0
$754$			206.193i			0.273465i
$755$		−	305.763i		−	0.404984i
$756$		0			0
$757$	−1346.79			−1.77911			−0.889555	−	0.456828i	$-0.848986\pi$
$757$	−1346.79			−1.77911			−0.889555	+	0.456828i	$0.848986\pi$
$758$		−	586.852i		−	0.774211i
$759$		0			0
$760$	687.607			0.904746
$761$			642.215i			0.843909i	0.906617	+	0.421954i	$0.138656\pi$
$761$			642.215i			0.843909i	−0.906617	+	0.421954i	$0.861344\pi$
$762$		0			0
$763$	−248.647			−0.325881
$764$	140.736			0.184210
$765$		0			0
$766$			87.7886i			0.114607i
$767$			853.814i			1.11319i
$768$		0			0
$769$			786.000i			1.02211i	0.859549	+	0.511053i	$0.170744\pi$
$769$			786.000i			1.02211i	−0.859549	+	0.511053i	$0.829256\pi$
$770$	87.7944	−	138.397i	0.114019	−	0.179737i
$771$		0			0
$772$			311.635i			0.403673i
$773$	1311.57			1.69672			0.848360	−	0.529419i	$-0.177590\pi$
$773$	1311.57			1.69672			0.848360	+	0.529419i	$0.177590\pi$
$774$		0			0
$775$	−131.738			−0.169984
$776$		−	188.785i		−	0.243280i
$777$		0			0
$778$			601.166i			0.772706i
$779$	−1874.01			−2.40566
$780$		0			0
$781$	47.1607	−	74.3432i	0.0603850	−	0.0951898i
$782$	−742.601			−0.949618
$783$		0			0
$784$	−142.665			−0.181971
$785$	−572.562			−0.729378
$786$		0			0
$787$		−	658.660i		−	0.836925i	−0.908234	−	0.418462i	$-0.862569\pi$
$787$		−	658.660i		−	0.836925i	0.908234	−	0.418462i	$-0.137431\pi$
$788$		−	676.711i		−	0.858770i
$789$		0			0
$790$	−169.936			−0.215108
$791$			538.089i			0.680265i
$792$		0			0
$793$	384.114			0.484381
$794$			347.022i			0.437055i
$795$		0			0
$796$	629.998			0.791455
$797$	−1499.40			−1.88130			−0.940651	−	0.339376i	$-0.889784\pi$
$797$	−1499.40			−1.88130			−0.940651	+	0.339376i	$0.889784\pi$
$798$		0			0
$799$			260.158i			0.325605i
$800$		−	134.434i		−	0.168042i
$801$		0			0
$802$			148.297i			0.184909i
$803$	−302.200	−	191.705i	−0.376339	−	0.238736i
$804$		0			0
$805$		−	299.719i		−	0.372322i
$806$	637.649			0.791128
$807$		0			0
$808$	−1396.94			−1.72889
$809$			1400.12i			1.73068i	0.501186	+	0.865340i	$0.332897\pi$
$809$			1400.12i			1.73068i	−0.501186	+	0.865340i	$0.667103\pi$
$810$		0			0
$811$			611.795i			0.754371i	0.926138	+	0.377186i	$0.123108\pi$
$811$			611.795i			0.754371i	−0.926138	+	0.377186i	$0.876892\pi$
$812$	72.4807			0.0892620
$813$		0			0
$814$	602.867	+	382.437i	0.740622	+	0.469824i
$815$	56.2081			0.0689670
$816$		0			0
$817$	243.068			0.297513
$818$	−1014.22			−1.23988
$819$		0			0
$820$			217.832i			0.265649i
$821$		−	1249.80i		−	1.52230i	−0.648579	−	0.761148i	$-0.724636\pi$
$821$		−	1249.80i		−	1.52230i	0.648579	−	0.761148i	$-0.275364\pi$
$822$		0			0
$823$	899.845			1.09337			0.546686	−	0.837338i	$-0.315889\pi$
$823$	899.845			1.09337			0.546686	+	0.837338i	$0.315889\pi$
$824$		−	365.424i		−	0.443476i
$825$		0			0
$826$	−343.458			−0.415808
$827$		−	1331.57i		−	1.61012i	−0.593193	−	0.805060i	$-0.702133\pi$
$827$		−	1331.57i		−	1.61012i	0.593193	−	0.805060i	$-0.297867\pi$
$828$		0			0
$829$	−578.195			−0.697461			−0.348730	−	0.937223i	$-0.613387\pi$
$829$	−578.195			−0.697461			−0.348730	+	0.937223i	$0.613387\pi$
$830$	−141.689			−0.170709
$831$		0			0
$832$			985.896i			1.18497i
$833$		−	487.694i		−	0.585467i
$834$		0			0
$835$		−	9.41470i		−	0.0112751i
$836$	621.750	+	394.416i	0.743720	+	0.471789i
$837$		0			0
$838$		−	608.310i		−	0.725907i
$839$	−779.704			−0.929325			−0.464663	−	0.885488i	$-0.653824\pi$
$839$	−779.704			−0.929325			−0.464663	+	0.885488i	$0.653824\pi$
$840$		0			0
$841$	768.412			0.913689
$842$		−	787.999i		−	0.935866i
$843$		0			0
$844$		−	593.026i		−	0.702638i
$845$	235.645			0.278870
$846$		0			0
$847$	499.234	−	235.143i	0.589414	−	0.277619i
$848$	−167.704			−0.197764
$849$		0			0
$850$	−126.335			−0.148629
$851$	1305.59			1.53419
$852$		0			0
$853$		−	1330.04i		−	1.55925i	−0.626250	−	0.779623i	$-0.715411\pi$
$853$		−	1330.04i		−	1.55925i	0.626250	−	0.779623i	$-0.284589\pi$
$854$			154.515i			0.180931i
$855$		0			0
$856$	1123.45			1.31244
$857$		−	223.662i		−	0.260983i	−0.991449	−	0.130491i	$-0.958345\pi$
$857$		−	223.662i		−	0.260983i	0.991449	−	0.130491i	$-0.0416555\pi$
$858$		0			0
$859$	−1512.14			−1.76035			−0.880174	−	0.474651i	$-0.842574\pi$
$859$	−1512.14			−1.76035			−0.880174	+	0.474651i	$0.842574\pi$
$860$		−	28.2539i		−	0.0328533i
$861$		0			0
$862$	501.857			0.582200
$863$	815.242			0.944661			0.472330	−	0.881422i	$-0.343413\pi$
$863$	815.242			0.944661			0.472330	+	0.881422i	$0.343413\pi$
$864$		0			0
$865$		−	375.693i		−	0.434327i
$866$		−	26.3503i		−	0.0304276i
$867$		0			0
$868$		−	224.146i		−	0.258233i
$869$	−483.160	−	306.500i	−0.555996	−	0.352704i
$870$		0			0
$871$			1882.85i			2.16171i
$872$	467.211			0.535793
$873$		0			0
$874$	−1540.86			−1.76299
$875$		−	50.9897i		−	0.0582739i
$876$		0			0
$877$		−	138.555i		−	0.157987i	−0.996875	−	0.0789935i	$-0.974829\pi$
$877$		−	138.555i		−	0.157987i	0.996875	−	0.0789935i	$-0.0251706\pi$
$878$	1013.73			1.15459
$879$		0			0
$880$	−66.6560	+	105.075i	−0.0757455	+	0.119404i
$881$	−415.019			−0.471077			−0.235538	−	0.971865i	$-0.575685\pi$
$881$	−415.019			−0.471077			−0.235538	+	0.971865i	$0.575685\pi$
$882$		0			0
$883$	−285.257			−0.323055			−0.161527	−	0.986868i	$-0.551642\pi$
$883$	−285.257			−0.323055			−0.161527	+	0.986868i	$0.551642\pi$
$884$	−534.361			−0.604480
$885$		0			0
$886$		−	524.972i		−	0.592519i
$887$		−	405.942i		−	0.457657i	−0.973467	−	0.228829i	$-0.926510\pi$
$887$		−	405.942i		−	0.457657i	0.973467	−	0.228829i	$-0.0734895\pi$
$888$		0			0
$889$	518.840			0.583622
$890$		−	241.113i		−	0.270913i
$891$		0			0
$892$	545.886			0.611980
$893$			539.814i			0.604495i
$894$		0			0
$895$	−77.2654			−0.0863301
$896$	93.8956			0.104794
$897$		0			0
$898$		−	829.697i		−	0.923939i
$899$			224.477i			0.249697i
$900$		0			0
$901$		−	573.288i		−	0.636279i
$902$	449.601	−	708.743i	0.498449	−	0.785746i
$903$		0			0
$904$		−	1011.08i		−	1.11845i
$905$	240.041			0.265238
$906$		0			0
$907$	1577.64			1.73940			0.869700	−	0.493581i	$-0.164312\pi$
$907$	1577.64			1.73940			0.869700	+	0.493581i	$0.164312\pi$
$908$		−	17.8125i		−	0.0196173i
$909$		0			0
$910$			246.805i			0.271214i
$911$	574.512			0.630639			0.315319	−	0.948986i	$-0.397888\pi$
$911$	574.512			0.630639			0.315319	+	0.948986i	$0.397888\pi$
$912$		0			0
$913$	−402.849	−	255.553i	−0.441236	−	0.279905i
$914$	1164.77			1.27436
$915$		0			0
$916$	−663.905			−0.724787
$917$	−527.207			−0.574926
$918$		0			0
$919$			354.240i			0.385462i	0.981252	+	0.192731i	$0.0617345\pi$
$919$			354.240i			0.385462i	−0.981252	+	0.192731i	$0.938266\pi$
$920$			563.176i			0.612147i
$921$		0			0
$922$	−128.446			−0.139313
$923$			132.577i			0.143637i
$924$		0			0
$925$	222.114			0.240123
$926$			74.4082i			0.0803544i
$927$		0			0
$928$	−229.071			−0.246844
$929$	230.873			0.248518			0.124259	−	0.992250i	$-0.460345\pi$
$929$	230.873			0.248518			0.124259	+	0.992250i	$0.460345\pi$
$930$		0			0
$931$		−	1011.94i		−	1.08694i
$932$			505.442i			0.542319i
$933$		0			0
$934$		−	676.555i		−	0.724363i
$935$	−359.195	−	227.861i	−0.384166	−	0.243701i
$936$		0			0
$937$		−	414.814i		−	0.442704i	−0.975194	−	0.221352i	$-0.928953\pi$
$937$		−	414.814i		−	0.442704i	0.975194	−	0.221352i	$-0.0710470\pi$
$938$	−757.400			−0.807463
$939$		0			0
$940$	62.7472			0.0667524
$941$		−	1428.08i		−	1.51762i	−0.651315	−	0.758808i	$-0.725782\pi$
$941$		−	1428.08i		−	1.51762i	0.651315	−	0.758808i	$-0.274218\pi$
$942$		0			0
$943$		−	1534.88i		−	1.62766i
$944$	260.763			0.276232
$945$		0			0
$946$	−58.3155	+	91.9275i	−0.0616443	+	0.0971749i
$947$	−976.020			−1.03064			−0.515322	−	0.856997i	$-0.672328\pi$
$947$	−976.020			−1.03064			−0.515322	+	0.856997i	$0.672328\pi$
$948$		0			0
$949$	538.914			0.567876
$950$	−262.138			−0.275935
$951$		0			0
$952$		−	675.890i		−	0.709968i
$953$		−	1273.90i		−	1.33673i	−0.743834	−	0.668364i	$-0.766995\pi$
$953$		−	1273.90i		−	1.33673i	0.743834	−	0.668364i	$-0.233005\pi$
$954$		0			0
$955$	−168.705			−0.176654
$956$		−	147.620i		−	0.154414i
$957$		0			0
$958$	−266.178			−0.277848
$959$			986.723i			1.02891i
$960$		0			0
$961$	−266.806			−0.277634
$962$	−1075.09			−1.11756
$963$		0			0
$964$		−	273.402i		−	0.283612i
$965$		−	373.567i		−	0.387116i
$966$		0			0
$967$			1586.32i			1.64045i	0.572038	+	0.820227i	$0.306153\pi$
$967$			1586.32i			1.64045i	−0.572038	+	0.820227i	$0.693847\pi$
$968$	−938.066	+	441.837i	−0.969077	+	0.456443i
$969$		0			0
$970$			71.9710i			0.0741969i
$971$	−662.523			−0.682310			−0.341155	−	0.940007i	$-0.610818\pi$
$971$	−662.523			−0.682310			−0.341155	+	0.940007i	$0.610818\pi$
$972$		0			0
$973$	23.1687			0.0238116
$974$		−	535.924i		−	0.550230i
$975$		0			0
$976$		−	117.312i		−	0.120197i
$977$	1552.47			1.58902			0.794508	−	0.607254i	$-0.207729\pi$
$977$	1552.47			1.58902			0.794508	+	0.607254i	$0.207729\pi$
$978$		0			0
$979$	434.876	−	685.531i	0.444204	−	0.700236i
$980$	−117.626			−0.120027
$981$		0			0
$982$	−61.3901			−0.0625154
$983$	−862.208			−0.877119			−0.438560	−	0.898702i	$-0.644511\pi$
$983$	−862.208			−0.877119			−0.438560	+	0.898702i	$0.644511\pi$
$984$		0			0
$985$			811.194i			0.823547i
$986$			215.271i			0.218328i
$987$		0			0
$988$	−1108.77			−1.12224
$989$			199.082i			0.201296i
$990$		0			0
$991$	−1075.74			−1.08551			−0.542755	−	0.839891i	$-0.682619\pi$
$991$	−1075.74			−1.08551			−0.542755	+	0.839891i	$0.682619\pi$
$992$			708.401i			0.714113i
$993$		0			0
$994$	−53.3306			−0.0536526
$995$	−755.198			−0.758993
$996$		0			0
$997$			872.880i			0.875507i	0.899095	+	0.437753i	$0.144226\pi$
$997$			872.880i			0.875507i	−0.899095	+	0.437753i	$0.855774\pi$
$998$			335.985i			0.336658i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	495.3.b.a.406.4		8
3.2	odd	2		55.3.c.a.21.5	yes	8
11.10	odd	2	inner	495.3.b.a.406.5		8
12.11	even	2		880.3.j.a.241.5		8
15.2	even	4		275.3.d.c.274.7		16
15.8	even	4		275.3.d.c.274.10		16
15.14	odd	2		275.3.c.f.76.4		8
33.32	even	2		55.3.c.a.21.4	✓	8
132.131	odd	2		880.3.j.a.241.6		8
165.32	odd	4		275.3.d.c.274.9		16
165.98	odd	4		275.3.d.c.274.8		16
165.164	even	2		275.3.c.f.76.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.3.c.a.21.4	✓	8	33.32	even	2
55.3.c.a.21.5	yes	8	3.2	odd	2
275.3.c.f.76.4		8	15.14	odd	2
275.3.c.f.76.5		8	165.164	even	2
275.3.d.c.274.7		16	15.2	even	4
275.3.d.c.274.8		16	165.98	odd	4
275.3.d.c.274.9		16	165.32	odd	4
275.3.d.c.274.10		16	15.8	even	4
495.3.b.a.406.4		8	1.1	even	1	trivial
495.3.b.a.406.5		8	11.10	odd	2	inner
880.3.j.a.241.5		8	12.11	even	2
880.3.j.a.241.6		8	132.131	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 495 = 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	495.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.46104i q^{2} +1.86536 q^{4} -2.23607 q^{5} +4.56066i q^{7} -8.56953i q^{8} +3.26698i q^{10} +(-5.89241 + 9.28868i) q^{11} -16.5645i q^{13} +6.66330 q^{14} -5.05897 q^{16} -17.2939i q^{17} -35.8838i q^{19} -4.17108 q^{20} +(13.5711 + 8.60904i) q^{22} +29.3902 q^{23} +5.00000 q^{25} -24.2014 q^{26} +8.50728i q^{28} -8.51985i q^{29} -26.3476 q^{31} -26.8868i q^{32} -25.2670 q^{34} -10.1979i q^{35} +44.4227 q^{37} -52.4276 q^{38} +19.1620i q^{40} -52.2243i q^{41} +6.77375i q^{43} +(-10.9915 + 17.3268i) q^{44} -42.9402i q^{46} -15.0434 q^{47} +28.2004 q^{49} -7.30520i q^{50} -30.8989i q^{52} +33.1498 q^{53} +(13.1758 - 20.7701i) q^{55} +39.0827 q^{56} -12.4478 q^{58} -51.5447 q^{59} +23.1889i q^{61} +38.4948i q^{62} -59.5185 q^{64} +37.0394i q^{65} -113.668 q^{67} -32.2593i q^{68} -14.8996 q^{70} -8.00364 q^{71} +32.5342i q^{73} -64.9034i q^{74} -66.9363i q^{76} +(-42.3625 - 26.8732i) q^{77} +52.0160i q^{79} +11.3122 q^{80} -76.3018 q^{82} +43.3699i q^{83} +38.6702i q^{85} +9.89672 q^{86} +(79.5996 + 50.4951i) q^{88} -73.8028 q^{89} +75.5451 q^{91} +54.8233 q^{92} +21.9790i q^{94} +80.2386i q^{95} +22.0298 q^{97} -41.2019i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 28 q^{4} - 8 q^{11} + 88 q^{16} + 20 q^{20} + 80 q^{22} - 8 q^{23} + 40 q^{25} + 100 q^{26} + 36 q^{31} + 80 q^{34} - 88 q^{37} + 160 q^{38} - 12 q^{44} + 8 q^{47} + 172 q^{49} + 152 q^{53} - 20 q^{55}+ \cdots + 312 q^{97}+O(q^{100}) \) 8 * q - 28 * q^4 - 8 * q^11 + 88 * q^16 + 20 * q^20 + 80 * q^22 - 8 * q^23 + 40 * q^25 + 100 * q^26 + 36 * q^31 + 80 * q^34 - 88 * q^37 + 160 * q^38 - 12 * q^44 + 8 * q^47 + 172 * q^49 + 152 * q^53 - 20 * q^55 + 80 * q^56 + 160 * q^58 + 16 * q^59 - 588 * q^64 - 88 * q^67 + 60 * q^70 - 276 * q^71 - 160 * q^77 - 320 * q^80 - 520 * q^82 - 420 * q^86 - 520 * q^88 - 444 * q^89 - 40 * q^91 + 368 * q^92 + 312 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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