LMFDB - Embedded newform 459.2.d.b.271.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [459,2,Mod(271,459)] 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("459.271"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(459, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 459 = 3^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	459.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$3.66513345278$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.897122304.10
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 51x^{4} - 104x^{2} + 169 $ x^8 - 8x^6 + 51x^4 - 104*x^2 + 169
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.5
Root			$-1.30421 + 0.752986i$ of defining polynomial
Character	$\chi$	$=$	459.271
Dual form			459.2.d.b.271.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.50597 q^{2} +0.267949 q^{4} -1.73205i q^{5} -4.11439i q^{7} -2.60842 q^{8} -2.60842i q^{10} -2.00000i q^{11} -4.46410 q^{13} -6.19615i q^{14} -4.46410 q^{16} +(4.11439 - 0.267949i) q^{17} +7.19615 q^{19} -0.464102i q^{20} -3.01194i q^{22} +8.46410i q^{23} +2.00000 q^{25} -6.72281 q^{26} -1.10245i q^{28} +1.73205i q^{29} -6.02388i q^{31} -1.50597 q^{32} +(6.19615 - 0.403524i) q^{34} -7.12633 q^{35} -9.33123i q^{37} +10.8372 q^{38} +4.51791i q^{40} +6.66025i q^{41} +0.267949 q^{43} -0.535898i q^{44} +12.7467i q^{46} +9.33123 q^{47} -9.92820 q^{49} +3.01194 q^{50} -1.19615 q^{52} -1.90949 q^{53} -3.46410 q^{55} +10.7321i q^{56} +2.60842i q^{58} -7.12633 q^{59} -1.10245i q^{61} -9.07180i q^{62} +6.66025 q^{64} +7.73205i q^{65} +4.26795 q^{67} +(1.10245 - 0.0717968i) q^{68} -10.7321 q^{70} +5.53590i q^{71} -7.12633i q^{73} -14.0526i q^{74} +1.92820 q^{76} -8.22878 q^{77} +10.1383i q^{79} +7.73205i q^{80} +10.0302i q^{82} +(-0.464102 - 7.12633i) q^{85} +0.403524 q^{86} +5.21684i q^{88} -2.20489 q^{89} +18.3671i q^{91} +2.26795i q^{92} +14.0526 q^{94} -12.4641i q^{95} +13.1502i q^{97} -14.9516 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 16 q^{4} - 8 q^{13} - 8 q^{16} + 16 q^{19} + 16 q^{25} + 8 q^{34} + 16 q^{43} - 24 q^{49} + 32 q^{52} - 16 q^{64} + 48 q^{67} - 72 q^{70} - 40 q^{76} + 24 q^{85} - 40 q^{94}+O(q^{100}) $ 8 * q + 16 * q^4 - 8 * q^13 - 8 * q^16 + 16 * q^19 + 16 * q^25 + 8 * q^34 + 16 * q^43 - 24 * q^49 + 32 * q^52 - 16 * q^64 + 48 * q^67 - 72 * q^70 - 40 * q^76 + 24 * q^85 - 40 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/459\mathbb{Z}\right)^\times$.

$n$	$137$	$190$
$\chi(n)$	$1$	$-1$

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 $

$2$	1.50597			1.06488			0.532441	−	0.846467i	$-0.321275\pi$
$2$	1.50597			1.06488			0.532441	+	0.846467i	$0.321275\pi$
$3$		0			0
$3$		0			0
$4$	0.267949			0.133975
$5$		−	1.73205i		−	0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$		−	1.73205i		−	0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$		0			0
$7$		−	4.11439i		−	1.55509i	−0.628825	−	0.777547i	$-0.716464\pi$
$7$		−	4.11439i		−	1.55509i	0.628825	−	0.777547i	$-0.283536\pi$
$8$	−2.60842			−0.922215
$9$		0			0
$10$		−	2.60842i		−	0.824854i
$11$		−	2.00000i		−	0.603023i	−0.953463	−	0.301511i	$-0.902509\pi$
$11$		−	2.00000i		−	0.603023i	0.953463	−	0.301511i	$-0.0974911\pi$
$12$		0			0
$13$	−4.46410			−1.23812			−0.619060	−	0.785344i	$-0.712486\pi$
$13$	−4.46410			−1.23812			−0.619060	+	0.785344i	$0.712486\pi$
$14$		−	6.19615i		−	1.65599i
$15$		0			0
$16$	−4.46410			−1.11603
$17$	4.11439	−	0.267949i	0.997886	−	0.0649872i
$17$	4.11439	−	0.267949i	0.997886	−	0.0649872i
$18$		0			0
$19$	7.19615			1.65091			0.825455	−	0.564467i	$-0.190918\pi$
$19$	7.19615			1.65091			0.825455	+	0.564467i	$0.190918\pi$
$20$		−	0.464102i		−	0.103776i
$21$		0			0
$22$		−	3.01194i		−	0.642148i
$23$			8.46410i			1.76489i	0.470418	+	0.882444i	$0.344103\pi$
$23$			8.46410i			1.76489i	−0.470418	+	0.882444i	$0.655897\pi$
$24$		0			0
$25$	2.00000			0.400000
$26$	−6.72281			−1.31845
$27$		0			0
$28$		−	1.10245i		−	0.208343i
$29$			1.73205i			0.321634i	0.986984	+	0.160817i	$0.0514129\pi$
$29$			1.73205i			0.321634i	−0.986984	+	0.160817i	$0.948587\pi$
$30$		0			0
$31$		−	6.02388i		−	1.08192i	−0.841048	−	0.540961i	$-0.818061\pi$
$31$		−	6.02388i		−	1.08192i	0.841048	−	0.540961i	$-0.181939\pi$
$32$	−1.50597			−0.266221
$33$		0			0
$34$	6.19615	−	0.403524i	1.06263	−	0.0692038i
$35$	−7.12633			−1.20457
$36$		0			0
$37$		−	9.33123i		−	1.53404i	−0.641621	−	0.767022i	$-0.721738\pi$
$37$		−	9.33123i		−	1.53404i	0.641621	−	0.767022i	$-0.278262\pi$
$38$	10.8372			1.75803
$39$		0			0
$40$			4.51791i			0.714345i
$41$			6.66025i			1.04016i	0.854118	+	0.520078i	$0.174097\pi$
$41$			6.66025i			1.04016i	−0.854118	+	0.520078i	$0.825903\pi$
$42$		0			0
$43$	0.267949			0.0408619			0.0204309	−	0.999791i	$-0.493496\pi$
$43$	0.267949			0.0408619			0.0204309	+	0.999791i	$0.493496\pi$
$44$		−	0.535898i		−	0.0807897i
$45$		0			0
$46$			12.7467i			1.87940i
$47$	9.33123			1.36110			0.680550	−	0.732702i	$-0.261741\pi$
$47$	9.33123			1.36110			0.680550	+	0.732702i	$0.261741\pi$
$48$		0			0
$49$	−9.92820			−1.41831
$50$	3.01194			0.425953
$51$		0			0
$52$	−1.19615			−0.165876
$53$	−1.90949			−0.262289			−0.131145	−	0.991363i	$-0.541865\pi$
$53$	−1.90949			−0.262289			−0.131145	+	0.991363i	$0.541865\pi$
$54$		0			0
$55$	−3.46410			−0.467099
$56$			10.7321i			1.43413i
$57$		0			0
$58$			2.60842i			0.342502i
$59$	−7.12633			−0.927769			−0.463885	−	0.885896i	$-0.653545\pi$
$59$	−7.12633			−0.927769			−0.463885	+	0.885896i	$0.653545\pi$
$60$		0			0
$61$		−	1.10245i		−	0.141154i	−0.997506	−	0.0705770i	$-0.977516\pi$
$61$		−	1.10245i		−	0.141154i	0.997506	−	0.0705770i	$-0.0224840\pi$
$62$		−	9.07180i		−	1.15212i
$63$		0			0
$64$	6.66025			0.832532
$65$			7.73205i			0.959043i
$66$		0			0
$67$	4.26795			0.521413			0.260706	−	0.965418i	$-0.416045\pi$
$67$	4.26795			0.521413			0.260706	+	0.965418i	$0.416045\pi$
$68$	1.10245	−	0.0717968i	0.133691	−	0.00870664i
$69$		0			0
$70$	−10.7321			−1.28273
$71$			5.53590i			0.656990i	0.944506	+	0.328495i	$0.106541\pi$
$71$			5.53590i			0.656990i	−0.944506	+	0.328495i	$0.893459\pi$
$72$		0			0
$73$		−	7.12633i		−	0.834074i	−0.908889	−	0.417037i	$-0.863069\pi$
$73$		−	7.12633i		−	0.834074i	0.908889	−	0.417037i	$-0.136931\pi$
$74$		−	14.0526i		−	1.63358i
$75$		0			0
$76$	1.92820			0.221180
$77$	−8.22878			−0.937756
$78$		0			0
$79$			10.1383i			1.14064i	0.821421	+	0.570322i	$0.193182\pi$
$79$			10.1383i			1.14064i	−0.821421	+	0.570322i	$0.806818\pi$
$80$			7.73205i			0.864470i
$81$		0			0
$82$			10.0302i			1.10764i
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$		0			0
$85$	−0.464102	−	7.12633i	−0.0503389	−	0.772959i
$86$	0.403524			0.0435131
$87$		0			0
$88$			5.21684i			0.556117i
$89$	−2.20489			−0.233718			−0.116859	−	0.993148i	$-0.537283\pi$
$89$	−2.20489			−0.233718			−0.116859	+	0.993148i	$0.537283\pi$
$90$		0			0
$91$			18.3671i			1.92539i
$92$			2.26795i			0.236450i
$93$		0			0
$94$	14.0526			1.44941
$95$		−	12.4641i		−	1.27879i
$96$		0			0
$97$			13.1502i			1.33520i	0.744519	+	0.667601i	$0.232679\pi$
$97$			13.1502i			1.33520i	−0.744519	+	0.667601i	$0.767321\pi$
$98$	−14.9516			−1.51034
$99$		0			0
$100$	0.535898			0.0535898
$101$	12.3432			1.22819			0.614096	−	0.789232i	$-0.289521\pi$
$101$	12.3432			1.22819			0.614096	+	0.789232i	$0.289521\pi$
$102$		0			0
$103$	−6.80385			−0.670403			−0.335202	−	0.942146i	$-0.608804\pi$
$103$	−6.80385			−0.670403			−0.335202	+	0.942146i	$0.608804\pi$
$104$	11.6442			1.14181
$105$		0			0
$106$	−2.87564			−0.279307
$107$			3.00000i			0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$			3.00000i			0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$		0			0
$109$		−	13.1502i		−	1.25956i	−0.776773	−	0.629781i	$-0.783145\pi$
$109$		−	13.1502i		−	1.25956i	0.776773	−	0.629781i	$-0.216855\pi$
$110$	−5.21684			−0.497406
$111$		0			0
$112$			18.3671i			1.73552i
$113$		−	11.4641i		−	1.07845i	−0.842161	−	0.539226i	$-0.818717\pi$
$113$		−	11.4641i		−	1.07845i	0.842161	−	0.539226i	$-0.181283\pi$
$114$		0			0
$115$	14.6603			1.36708
$116$			0.464102i			0.0430908i
$117$		0			0
$118$	−10.7321			−0.987965
$119$	−1.10245	−	16.9282i	−0.101061	−	1.55181i
$120$		0			0
$121$	7.00000			0.636364
$122$		−	1.66025i		−	0.150312i
$123$		0			0
$124$		−	1.61410i		−	0.144950i
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$	−10.9282			−0.969721			−0.484861	−	0.874591i	$-0.661130\pi$
$127$	−10.9282			−0.969721			−0.484861	+	0.874591i	$0.661130\pi$
$128$	13.0421			1.15277
$129$		0			0
$130$			11.6442i			1.02127i
$131$			1.00000i			0.0873704i	0.999045	+	0.0436852i	$0.0139099\pi$
$131$			1.00000i			0.0873704i	−0.999045	+	0.0436852i	$0.986090\pi$
$132$		0			0
$133$		−	29.6078i		−	2.56732i
$134$	6.42741			0.555244
$135$		0			0
$136$	−10.7321	+	0.698924i	−0.920266	+	0.0599322i
$137$	−2.20489			−0.188377			−0.0941884	−	0.995554i	$-0.530026\pi$
$137$	−2.20489			−0.188377			−0.0941884	+	0.995554i	$0.530026\pi$
$138$		0			0
$139$			16.4576i			1.39591i	0.716141	+	0.697956i	$0.245907\pi$
$139$			16.4576i			1.39591i	−0.716141	+	0.697956i	$0.754093\pi$
$140$	−1.90949			−0.161382
$141$		0			0
$142$			8.33690i			0.699617i
$143$			8.92820i			0.746614i
$144$		0			0
$145$	3.00000			0.249136
$146$		−	10.7321i		−	0.888191i
$147$		0			0
$148$		−	2.50029i		−	0.205523i
$149$	1.90949			0.156432			0.0782160	−	0.996936i	$-0.475078\pi$
$149$	1.90949			0.156432			0.0782160	+	0.996936i	$0.475078\pi$
$150$		0			0
$151$	−1.73205			−0.140952			−0.0704761	−	0.997513i	$-0.522452\pi$
$151$	−1.73205			−0.140952			−0.0704761	+	0.997513i	$0.522452\pi$
$152$	−18.7706			−1.52249
$153$		0			0
$154$	−12.3923			−0.998600
$155$	−10.4337			−0.838053
$156$		0			0
$157$	9.07180			0.724008			0.362004	−	0.932177i	$-0.382093\pi$
$157$	9.07180			0.724008			0.362004	+	0.932177i	$0.382093\pi$
$158$			15.2679i			1.21465i
$159$		0			0
$160$			2.60842i			0.206214i
$161$	34.8246			2.74456
$162$		0			0
$163$		−	6.02388i		−	0.471827i	−0.971774	−	0.235914i	$-0.924192\pi$
$163$		−	6.02388i		−	0.471827i	0.971774	−	0.235914i	$-0.0758082\pi$
$164$			1.78461i			0.139355i
$165$		0			0
$166$		0			0
$167$		−	6.92820i		−	0.536120i	−0.963402	−	0.268060i	$-0.913617\pi$
$167$		−	6.92820i		−	0.536120i	0.963402	−	0.268060i	$-0.0863826\pi$
$168$		0			0
$169$	6.92820			0.532939
$170$	−0.698924	−	10.7321i	−0.0536050	−	0.823111i
$171$		0			0
$172$	0.0717968			0.00547445
$173$			14.9282i			1.13497i	0.823384	+	0.567485i	$0.192084\pi$
$173$			14.9282i			1.13497i	−0.823384	+	0.567485i	$0.807916\pi$
$174$		0			0
$175$		−	8.22878i		−	0.622037i
$176$			8.92820i			0.672989i
$177$		0			0
$178$	−3.32051			−0.248883
$179$	6.02388			0.450246			0.225123	−	0.974330i	$-0.427722\pi$
$179$	6.02388			0.450246			0.225123	+	0.974330i	$0.427722\pi$
$180$		0			0
$181$			6.02388i			0.447752i	0.974618	+	0.223876i	$0.0718710\pi$
$181$			6.02388i			0.447752i	−0.974618	+	0.223876i	$0.928129\pi$
$182$			27.6603i			2.05031i
$183$		0			0
$184$		−	22.0779i		−	1.62761i
$185$	−16.1622			−1.18827
$186$		0			0
$187$	−0.535898	−	8.22878i	−0.0391888	−	0.601748i
$188$	2.50029			0.182353
$189$		0			0
$190$		−	18.7706i		−	1.36176i
$191$	−22.4814			−1.62670			−0.813350	−	0.581775i	$-0.802359\pi$
$191$	−22.4814			−1.62670			−0.813350	+	0.581775i	$0.802359\pi$
$192$		0			0
$193$		−	10.4337i		−	0.751032i	−0.926816	−	0.375516i	$-0.877465\pi$
$193$		−	10.4337i		−	0.751032i	0.926816	−	0.375516i	$-0.122535\pi$
$194$			19.8038i			1.42183i
$195$		0			0
$196$	−2.66025			−0.190018
$197$		−	16.6603i		−	1.18699i	−0.804836	−	0.593497i	$-0.797747\pi$
$197$		−	16.6603i		−	1.18699i	0.804836	−	0.593497i	$-0.202253\pi$
$198$		0			0
$199$			20.2765i			1.43737i	0.695338	+	0.718683i	$0.255255\pi$
$199$			20.2765i			1.43737i	−0.695338	+	0.718683i	$0.744745\pi$
$200$	−5.21684			−0.368886
$201$		0			0
$202$	18.5885			1.30788
$203$	7.12633			0.500170
$204$		0			0
$205$	11.5359			0.805702
$206$	−10.2464			−0.713900
$207$		0			0
$208$	19.9282			1.38177
$209$		−	14.3923i		−	0.995537i
$210$		0			0
$211$			0.295400i			0.0203362i	0.999948	+	0.0101681i	$0.00323666\pi$
$211$			0.295400i			0.0203362i	−0.999948	+	0.0101681i	$0.996763\pi$
$212$	−0.511648			−0.0351401
$213$		0			0
$214$			4.51791i			0.308838i
$215$		−	0.464102i		−	0.0316515i
$216$		0			0
$217$	−24.7846			−1.68249
$218$		−	19.8038i		−	1.34129i
$219$		0			0
$220$	−0.928203			−0.0625794
$221$	−18.3671	+	1.19615i	−1.23550	+	0.0804619i
$222$		0			0
$223$	7.58846			0.508161			0.254080	−	0.967183i	$-0.418227\pi$
$223$	7.58846			0.508161			0.254080	+	0.967183i	$0.418227\pi$
$224$			6.19615i			0.413998i
$225$		0			0
$226$		−	17.2646i		−	1.14842i
$227$			17.7846i			1.18041i	0.807255	+	0.590203i	$0.200952\pi$
$227$			17.7846i			1.18041i	−0.807255	+	0.590203i	$0.799048\pi$
$228$		0			0
$229$	5.39230			0.356334			0.178167	−	0.984000i	$-0.442983\pi$
$229$	5.39230			0.356334			0.178167	+	0.984000i	$0.442983\pi$
$230$	22.0779			1.45577
$231$		0			0
$232$		−	4.51791i		−	0.296616i
$233$			22.6603i			1.48452i	0.670111	+	0.742261i	$0.266247\pi$
$233$			22.6603i			1.48452i	−0.670111	+	0.742261i	$0.733753\pi$
$234$		0			0
$235$		−	16.1622i		−	1.05430i
$236$	−1.90949			−0.124298
$237$		0			0
$238$	−1.66025	−	25.4934i	−0.107618	−	1.65249i
$239$	27.4029			1.77255			0.886273	−	0.463164i	$-0.153286\pi$
$239$	27.4029			1.77255			0.886273	+	0.463164i	$0.153286\pi$
$240$		0			0
$241$			8.22878i			0.530062i	0.964240	+	0.265031i	$0.0853822\pi$
$241$			8.22878i			0.530062i	−0.964240	+	0.265031i	$0.914618\pi$
$242$	10.5418			0.677652
$243$		0			0
$244$		−	0.295400i		−	0.0189110i
$245$			17.1962i			1.09862i
$246$		0			0
$247$	−32.1244			−2.04402
$248$			15.7128i			0.997765i
$249$		0			0
$250$		−	18.2589i		−	1.15480i
$251$	10.4337			0.658568			0.329284	−	0.944231i	$-0.393193\pi$
$251$	10.4337			0.658568			0.329284	+	0.944231i	$0.393193\pi$
$252$		0			0
$253$	16.9282			1.06427
$254$	−16.4576			−1.03264
$255$		0			0
$256$	6.32051			0.395032
$257$	7.93338			0.494871			0.247435	−	0.968904i	$-0.420412\pi$
$257$	7.93338			0.494871			0.247435	+	0.968904i	$0.420412\pi$
$258$		0			0
$259$	−38.3923			−2.38558
$260$			2.07180i			0.128487i
$261$		0			0
$262$			1.50597i			0.0930392i
$263$	−14.2527			−0.878857			−0.439428	−	0.898278i	$-0.644819\pi$
$263$	−14.2527			−0.878857			−0.439428	+	0.898278i	$0.644819\pi$
$264$		0			0
$265$			3.30734i			0.203168i
$266$		−	44.5885i		−	2.73389i
$267$		0			0
$268$	1.14359			0.0698561
$269$		−	17.3205i		−	1.05605i	−0.849229	−	0.528025i	$-0.822933\pi$
$269$		−	17.3205i		−	1.05605i	0.849229	−	0.528025i	$-0.177067\pi$
$270$		0			0
$271$	−12.0000			−0.728948			−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000			−0.728948			−0.364474	+	0.931214i	$0.618751\pi$
$272$	−18.3671	+	1.19615i	−1.11367	+	0.0725274i
$273$		0			0
$274$	−3.32051			−0.200599
$275$		−	4.00000i		−	0.241209i
$276$		0			0
$277$		−	11.5361i		−	0.693138i	−0.938024	−	0.346569i	$-0.887347\pi$
$277$		−	11.5361i		−	0.693138i	0.938024	−	0.346569i	$-0.112653\pi$
$278$			24.7846i			1.48648i
$279$		0			0
$280$	18.5885			1.11087
$281$	4.11439			0.245444			0.122722	−	0.992441i	$-0.460838\pi$
$281$	4.11439			0.245444			0.122722	+	0.992441i	$0.460838\pi$
$282$		0			0
$283$			7.93338i			0.471590i	0.971803	+	0.235795i	$0.0757695\pi$
$283$			7.93338i			0.471590i	−0.971803	+	0.235795i	$0.924231\pi$
$284$			1.48334i			0.0880200i
$285$		0			0
$286$			13.4456i			0.795056i
$287$	27.4029			1.61754
$288$		0			0
$289$	16.8564	−	2.20489i	0.991553	−	0.129700i
$290$	4.51791			0.265301
$291$		0			0
$292$		−	1.90949i		−	0.111745i
$293$	−28.8007			−1.68256			−0.841278	−	0.540602i	$-0.818196\pi$
$293$	−28.8007			−1.68256			−0.841278	+	0.540602i	$0.818196\pi$
$294$		0			0
$295$			12.3432i			0.718647i
$296$			24.3397i			1.41472i
$297$		0			0
$298$	2.87564			0.166582
$299$		−	37.7846i		−	2.18514i
$300$		0			0
$301$		−	1.10245i		−	0.0635440i
$302$	−2.60842			−0.150098
$303$		0			0
$304$	−32.1244			−1.84246
$305$	−1.90949			−0.109337
$306$		0			0
$307$	21.5885			1.23212			0.616059	−	0.787700i	$-0.288728\pi$
$307$	21.5885			1.23212			0.616059	+	0.787700i	$0.288728\pi$
$308$	−2.20489			−0.125636
$309$		0			0
$310$	−15.7128			−0.892428
$311$			33.7128i			1.91168i	0.293890	+	0.955839i	$0.405050\pi$
$311$			33.7128i			1.91168i	−0.293890	+	0.955839i	$0.594950\pi$
$312$		0			0
$313$		−	20.2765i		−	1.14610i	−0.819521	−	0.573049i	$-0.805760\pi$
$313$		−	20.2765i		−	1.14610i	0.819521	−	0.573049i	$-0.194240\pi$
$314$	13.6619			0.770984
$315$		0			0
$316$			2.71654i			0.152817i
$317$		−	11.5885i		−	0.650873i	−0.945564	−	0.325436i	$-0.894489\pi$
$317$		−	11.5885i		−	0.650873i	0.945564	−	0.325436i	$-0.105511\pi$
$318$		0			0
$319$	3.46410			0.193952
$320$		−	11.5359i		−	0.644876i
$321$		0			0
$322$	52.4449			2.92264
$323$	29.6078	−	1.92820i	1.64742	−	0.107288i
$324$		0			0
$325$	−8.92820			−0.495248
$326$		−	9.07180i		−	0.502440i
$327$		0			0
$328$		−	17.3727i		−	0.959249i
$329$		−	38.3923i		−	2.11664i
$330$		0			0
$331$	22.1244			1.21606			0.608032	−	0.793912i	$-0.291959\pi$
$331$	22.1244			1.21606			0.608032	+	0.793912i	$0.291959\pi$
$332$		0			0
$333$		0			0
$334$		−	10.4337i		−	0.570905i
$335$		−	7.39230i		−	0.403885i
$336$		0			0
$337$		−	2.71654i		−	0.147979i	−0.997259	−	0.0739897i	$-0.976427\pi$
$337$		−	2.71654i		−	0.147979i	0.997259	−	0.0739897i	$-0.0235732\pi$
$338$	10.4337			0.567517
$339$		0			0
$340$	−0.124356	−	1.90949i	−0.00674413	−	0.103557i
$341$	−12.0478			−0.652423
$342$		0			0
$343$			12.0478i			0.650518i
$344$	−0.698924			−0.0376834
$345$		0			0
$346$			22.4814i			1.20861i
$347$		−	0.856406i		−	0.0459743i	−0.999736	−	0.0229872i	$-0.992682\pi$
$347$		−	0.856406i		−	0.0459743i	0.999736	−	0.0229872i	$-0.00731768\pi$
$348$		0			0
$349$	−17.2487			−0.923302			−0.461651	−	0.887062i	$-0.652743\pi$
$349$	−17.2487			−0.923302			−0.461651	+	0.887062i	$0.652743\pi$
$350$		−	12.3923i		−	0.662397i
$351$		0			0
$352$			3.01194i			0.160537i
$353$	−14.2527			−0.758593			−0.379296	−	0.925275i	$-0.623834\pi$
$353$	−14.2527			−0.758593			−0.379296	+	0.925275i	$0.623834\pi$
$354$		0			0
$355$	9.58846			0.508902
$356$	−0.590800			−0.0313123
$357$		0			0
$358$	9.07180			0.479459
$359$	−27.4029			−1.44627			−0.723134	−	0.690707i	$-0.757299\pi$
$359$	−27.4029			−1.44627			−0.723134	+	0.690707i	$0.757299\pi$
$360$		0			0
$361$	32.7846			1.72551
$362$			9.07180i			0.476803i
$363$		0			0
$364$			4.92144i			0.257953i
$365$	−12.3432			−0.646071
$366$		0			0
$367$			22.7768i			1.18894i	0.804117	+	0.594471i	$0.202638\pi$
$367$			22.7768i			1.18894i	−0.804117	+	0.594471i	$0.797362\pi$
$368$		−	37.7846i		−	1.96966i
$369$		0			0
$370$	−24.3397			−1.26536
$371$			7.85641i			0.407884i
$372$		0			0
$373$	−19.2487			−0.996660			−0.498330	−	0.866987i	$-0.666053\pi$
$373$	−19.2487			−0.996660			−0.498330	+	0.866987i	$0.666053\pi$
$374$	−0.807048	−	12.3923i	−0.0417314	−	0.640791i
$375$		0			0
$376$	−24.3397			−1.25523
$377$		−	7.73205i		−	0.398221i
$378$		0			0
$379$			34.8246i			1.78882i	0.447248	+	0.894410i	$0.352404\pi$
$379$			34.8246i			1.78882i	−0.447248	+	0.894410i	$0.647596\pi$
$380$		−	3.33975i		−	0.171325i
$381$		0			0
$382$	−33.8564			−1.73224
$383$	−20.2765			−1.03608			−0.518042	−	0.855355i	$-0.673339\pi$
$383$	−20.2765			−1.03608			−0.518042	+	0.855355i	$0.673339\pi$
$384$		0			0
$385$			14.2527i			0.726383i
$386$		−	15.7128i		−	0.799761i
$387$		0			0
$388$			3.52359i			0.178883i
$389$	−10.4337			−0.529008			−0.264504	−	0.964385i	$-0.585208\pi$
$389$	−10.4337			−0.529008			−0.264504	+	0.964385i	$0.585208\pi$
$390$		0			0
$391$	2.26795	+	34.8246i	0.114695	+	1.76116i
$392$	25.8969			1.30799
$393$		0			0
$394$		−	25.0899i		−	1.26401i
$395$	17.5600			0.883540
$396$		0			0
$397$			2.20489i			0.110660i	0.998468	+	0.0553302i	$0.0176212\pi$
$397$			2.20489i			0.110660i	−0.998468	+	0.0553302i	$0.982379\pi$
$398$			30.5359i			1.53063i
$399$		0			0
$400$	−8.92820			−0.446410
$401$			13.0526i			0.651814i	0.945402	+	0.325907i	$0.105670\pi$
$401$			13.0526i			0.651814i	−0.945402	+	0.325907i	$0.894330\pi$
$402$		0			0
$403$			26.8912i			1.33955i
$404$	3.30734			0.164546
$405$		0			0
$406$	10.7321			0.532623
$407$	−18.6625			−0.925063
$408$		0			0
$409$	−4.07180			−0.201337			−0.100669	−	0.994920i	$-0.532098\pi$
$409$	−4.07180			−0.201337			−0.100669	+	0.994920i	$0.532098\pi$
$410$	17.3727			0.857978
$411$		0			0
$412$	−1.82309			−0.0898170
$413$			29.3205i			1.44277i
$414$		0			0
$415$		0			0
$416$	6.72281			0.329613
$417$		0			0
$418$		−	21.6744i		−	1.06013i
$419$			6.07180i			0.296627i	0.988940	+	0.148313i	$0.0473844\pi$
$419$			6.07180i			0.296627i	−0.988940	+	0.148313i	$0.952616\pi$
$420$		0			0
$421$	−4.00000			−0.194948			−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000			−0.194948			−0.0974740	+	0.995238i	$0.531076\pi$
$422$			0.444864i			0.0216556i
$423$		0			0
$424$	4.98076			0.241887
$425$	8.22878	−	0.535898i	0.399154	−	0.0259949i
$426$		0			0
$427$	−4.53590			−0.219508
$428$			0.803848i			0.0388554i
$429$		0			0
$430$		−	0.698924i		−	0.0337051i
$431$		−	27.3923i		−	1.31944i	−0.751511	−	0.659720i	$-0.770675\pi$
$431$		−	27.3923i		−	1.31944i	0.751511	−	0.659720i	$-0.229325\pi$
$432$		0			0
$433$	−17.0000			−0.816968			−0.408484	−	0.912766i	$-0.633942\pi$
$433$	−17.0000			−0.816968			−0.408484	+	0.912766i	$0.633942\pi$
$434$	−37.3249			−1.79165
$435$		0			0
$436$		−	3.52359i		−	0.168749i
$437$			60.9090i			2.91367i
$438$		0			0
$439$			8.52418i			0.406837i	0.979092	+	0.203418i	$0.0652052\pi$
$439$			8.52418i			0.406837i	−0.979092	+	0.203418i	$0.934795\pi$
$440$	9.03583			0.430766
$441$		0			0
$442$	−27.6603	+	1.80137i	−1.31566	+	0.0856825i
$443$	5.51224			0.261894			0.130947	−	0.991389i	$-0.458198\pi$
$443$	5.51224			0.261894			0.130947	+	0.991389i	$0.458198\pi$
$444$		0			0
$445$			3.81899i			0.181037i
$446$	11.4280			0.541131
$447$		0			0
$448$		−	27.4029i		−	1.29466i
$449$			13.0718i			0.616896i	0.951241	+	0.308448i	$0.0998096\pi$
$449$			13.0718i			0.616896i	−0.951241	+	0.308448i	$0.900190\pi$
$450$		0			0
$451$	13.3205			0.627238
$452$		−	3.07180i		−	0.144485i
$453$		0			0
$454$			26.7831i			1.25699i
$455$	31.8127			1.49140
$456$		0			0
$457$	9.85641			0.461063			0.230532	−	0.973065i	$-0.425953\pi$
$457$	9.85641			0.461063			0.230532	+	0.973065i	$0.425953\pi$
$458$	8.12066			0.379453
$459$		0			0
$460$	3.92820			0.183153
$461$	18.6625			0.869197			0.434599	−	0.900624i	$-0.356890\pi$
$461$	18.6625			0.869197			0.434599	+	0.900624i	$0.356890\pi$
$462$		0			0
$463$	25.3205			1.17674			0.588372	−	0.808590i	$-0.299769\pi$
$463$	25.3205			1.17674			0.588372	+	0.808590i	$0.299769\pi$
$464$		−	7.73205i		−	0.358951i
$465$		0			0
$466$			34.1257i			1.58084i
$467$	2.71654			0.125707			0.0628533	−	0.998023i	$-0.479980\pi$
$467$	2.71654			0.125707			0.0628533	+	0.998023i	$0.479980\pi$
$468$		0			0
$469$		−	17.5600i		−	0.810846i
$470$		−	24.3397i		−	1.12271i
$471$		0			0
$472$	18.5885			0.855603
$473$		−	0.535898i		−	0.0246406i
$474$		0			0
$475$	14.3923			0.660364
$476$	−0.295400	−	4.53590i	−0.0135396	−	0.207903i
$477$		0			0
$478$	41.2679			1.88755
$479$			40.1769i			1.83573i	0.396893	+	0.917865i	$0.370089\pi$
$479$			40.1769i			1.83573i	−0.396893	+	0.917865i	$0.629911\pi$
$480$		0			0
$481$			41.6555i			1.89933i
$482$			12.3923i			0.564454i
$483$		0			0
$484$	1.87564			0.0852566
$485$	22.7768			1.03424
$486$		0			0
$487$			29.0961i			1.31847i	0.751936	+	0.659236i	$0.229120\pi$
$487$			29.0961i			1.31847i	−0.751936	+	0.659236i	$0.770880\pi$
$488$			2.87564i			0.130174i
$489$		0			0
$490$			25.8969i			1.16990i
$491$	−40.5531			−1.83014			−0.915068	−	0.403300i	$-0.867863\pi$
$491$	−40.5531			−1.83014			−0.915068	+	0.403300i	$0.867863\pi$
$492$		0			0
$493$	0.464102	+	7.12633i	0.0209021	+	0.320954i
$494$	−48.3784			−2.17665
$495$		0			0
$496$			26.8912i			1.20745i
$497$	22.7768			1.02168
$498$		0			0
$499$		−	1.90949i		−	0.0854807i	−0.999086	−	0.0427404i	$-0.986391\pi$
$499$		−	1.90949i		−	0.0854807i	0.999086	−	0.0427404i	$-0.0136088\pi$
$500$		−	3.24871i		−	0.145287i
$501$		0			0
$502$	15.7128			0.701297
$503$		−	3.67949i		−	0.164060i	−0.996630	−	0.0820302i	$-0.973860\pi$
$503$		−	3.67949i		−	0.164060i	0.996630	−	0.0820302i	$-0.0261404\pi$
$504$		0			0
$505$		−	21.3790i		−	0.951353i
$506$	25.4934			1.13332
$507$		0			0
$508$	−2.92820			−0.129918
$509$	24.6863			1.09420			0.547101	−	0.837066i	$-0.315731\pi$
$509$	24.6863			1.09420			0.547101	+	0.837066i	$0.315731\pi$
$510$		0			0
$511$	−29.3205			−1.29706
$512$	−16.5657			−0.732107
$513$		0			0
$514$	11.9474			0.526979
$515$			11.7846i			0.519292i
$516$		0			0
$517$		−	18.6625i		−	0.820774i
$518$	−57.8177			−2.54036
$519$		0			0
$520$		−	20.1684i		−	0.884444i
$521$			34.3923i			1.50675i	0.657589	+	0.753377i	$0.271576\pi$
$521$			34.3923i			1.50675i	−0.657589	+	0.753377i	$0.728424\pi$
$522$		0			0
$523$	−6.92820			−0.302949			−0.151475	−	0.988461i	$-0.548402\pi$
$523$	−6.92820			−0.302949			−0.151475	+	0.988461i	$0.548402\pi$
$524$			0.267949i			0.0117054i
$525$		0			0
$526$	−21.4641			−0.935879
$527$	−1.61410	−	24.7846i	−0.0703111	−	1.07963i
$528$		0			0
$529$	−48.6410			−2.11483
$530$			4.98076i			0.216350i
$531$		0			0
$532$		−	7.93338i		−	0.343956i
$533$		−	29.7321i		−	1.28784i
$534$		0			0
$535$	5.19615			0.224649
$536$	−11.1326			−0.480855
$537$		0			0
$538$		−	26.0842i		−	1.12457i
$539$			19.8564i			0.855276i
$540$		0			0
$541$		−	12.0478i		−	0.517974i	−0.965881	−	0.258987i	$-0.916611\pi$
$541$		−	12.0478i		−	0.517974i	0.965881	−	0.258987i	$-0.0833887\pi$
$542$	−18.0717			−0.776244
$543$		0			0
$544$	−6.19615	+	0.403524i	−0.265658	+	0.0173009i
$545$	−22.7768			−0.975653
$546$		0			0
$547$		−	6.61468i		−	0.282823i	−0.989951	−	0.141412i	$-0.954836\pi$
$547$		−	6.61468i		−	0.282823i	0.989951	−	0.141412i	$-0.0451641\pi$
$548$	−0.590800			−0.0252377
$549$		0			0
$550$		−	6.02388i		−	0.256859i
$551$			12.4641i			0.530989i
$552$		0			0
$553$	41.7128			1.77381
$554$		−	17.3731i		−	0.738111i
$555$		0			0
$556$			4.40979i			0.187017i
$557$	−26.5958			−1.12690			−0.563451	−	0.826150i	$-0.690527\pi$
$557$	−26.5958			−1.12690			−0.563451	+	0.826150i	$0.690527\pi$
$558$		0			0
$559$	−1.19615			−0.0505919
$560$	31.8127			1.34433
$561$		0			0
$562$	6.19615			0.261369
$563$	29.6078			1.24782			0.623909	−	0.781497i	$-0.285543\pi$
$563$	29.6078			1.24782			0.623909	+	0.781497i	$0.285543\pi$
$564$		0			0
$565$	−19.8564			−0.835365
$566$			11.9474i			0.502188i
$567$		0			0
$568$		−	14.4399i		−	0.605886i
$569$	2.50029			0.104818			0.0524089	−	0.998626i	$-0.483310\pi$
$569$	2.50029			0.104818			0.0524089	+	0.998626i	$0.483310\pi$
$570$		0			0
$571$		−	14.5481i		−	0.608818i	−0.952541	−	0.304409i	$-0.901541\pi$
$571$		−	14.5481i		−	0.608818i	0.952541	−	0.304409i	$-0.0984589\pi$
$572$			2.39230i			0.100027i
$573$		0			0
$574$	41.2679			1.72249
$575$			16.9282i			0.705955i
$576$		0			0
$577$	−37.8564			−1.57598			−0.787991	−	0.615686i	$-0.788879\pi$
$577$	−37.8564			−1.57598			−0.787991	+	0.615686i	$0.788879\pi$
$578$	25.3853	−	3.32051i	1.05589	−	0.138115i
$579$		0			0
$580$	0.803848			0.0333780
$581$		0			0
$582$		0			0
$583$			3.81899i			0.158166i
$584$			18.5885i			0.769196i
$585$		0			0
$586$	−43.3731			−1.79172
$587$	−13.7410			−0.567152			−0.283576	−	0.958950i	$-0.591521\pi$
$587$	−13.7410			−0.567152			−0.283576	+	0.958950i	$0.591521\pi$
$588$		0			0
$589$		−	43.3488i		−	1.78616i
$590$			18.5885i			0.765275i
$591$		0			0
$592$			41.6555i			1.71203i
$593$	26.3004			1.08003			0.540015	−	0.841656i	$-0.318419\pi$
$593$	26.3004			1.08003			0.540015	+	0.841656i	$0.318419\pi$
$594$		0			0
$595$	−29.3205	+	1.90949i	−1.20202	+	0.0782817i
$596$	0.511648			0.0209579
$597$		0			0
$598$		−	56.9025i		−	2.32692i
$599$	27.4029			1.11965			0.559826	−	0.828610i	$-0.310868\pi$
$599$	27.4029			1.11965			0.559826	+	0.828610i	$0.310868\pi$
$600$		0			0
$601$			19.7649i			0.806227i	0.915150	+	0.403114i	$0.132072\pi$
$601$			19.7649i			0.806227i	−0.915150	+	0.403114i	$0.867928\pi$
$602$		−	1.66025i		−	0.0676669i
$603$		0			0
$604$	−0.464102			−0.0188840
$605$		−	12.1244i		−	0.492925i
$606$		0			0
$607$		−	30.7102i		−	1.24649i	−0.782027	−	0.623245i	$-0.785814\pi$
$607$		−	30.7102i		−	1.24649i	0.782027	−	0.623245i	$-0.214186\pi$
$608$	−10.8372			−0.439506
$609$		0			0
$610$	−2.87564			−0.116431
$611$	−41.6555			−1.68520
$612$		0			0
$613$	−22.1769			−0.895717			−0.447859	−	0.894104i	$-0.647813\pi$
$613$	−22.1769			−0.895717			−0.447859	+	0.894104i	$0.647813\pi$
$614$	32.5116			1.31206
$615$		0			0
$616$	21.4641			0.864813
$617$		−	40.5167i		−	1.63114i	−0.578659	−	0.815570i	$-0.696424\pi$
$617$		−	40.5167i		−	1.63114i	0.578659	−	0.815570i	$-0.303576\pi$
$618$		0			0
$619$			6.61468i			0.265867i	0.991125	+	0.132933i	$0.0424396\pi$
$619$			6.61468i			0.265867i	−0.991125	+	0.132933i	$0.957560\pi$
$620$	−2.79569			−0.112278
$621$		0			0
$622$			50.7705i			2.03571i
$623$			9.07180i			0.363454i
$624$		0			0
$625$	−11.0000			−0.440000
$626$		−	30.5359i		−	1.22046i
$627$		0			0
$628$	2.43078			0.0969987
$629$	−2.50029	−	38.3923i	−0.0996933	−	1.53080i
$630$		0			0
$631$	23.0526			0.917708			0.458854	−	0.888512i	$-0.348260\pi$
$631$	23.0526			0.917708			0.458854	+	0.888512i	$0.348260\pi$
$632$		−	26.4449i		−	1.05192i
$633$		0			0
$634$		−	17.4519i		−	0.693103i
$635$			18.9282i			0.751143i
$636$		0			0
$637$	44.3205			1.75604
$638$	5.21684			0.206537
$639$		0			0
$640$		−	22.5896i		−	0.892931i
$641$			8.80385i			0.347731i	0.984769	+	0.173866i	$0.0556258\pi$
$641$			8.80385i			0.347731i	−0.984769	+	0.173866i	$0.944374\pi$
$642$		0			0
$643$			32.6197i			1.28640i	0.765700	+	0.643198i	$0.222393\pi$
$643$			32.6197i			1.28640i	−0.765700	+	0.643198i	$0.777607\pi$
$644$	9.33123			0.367702
$645$		0			0
$646$	44.5885	−	2.90382i	1.75431	−	0.114249i
$647$	−27.4029			−1.07732			−0.538659	−	0.842524i	$-0.681069\pi$
$647$	−27.4029			−1.07732			−0.538659	+	0.842524i	$0.681069\pi$
$648$		0			0
$649$			14.2527i			0.559466i
$650$	−13.4456			−0.527380
$651$		0			0
$652$		−	1.61410i		−	0.0632128i
$653$			4.24871i			0.166265i	0.996539	+	0.0831325i	$0.0264925\pi$
$653$			4.24871i			0.166265i	−0.996539	+	0.0831325i	$0.973508\pi$
$654$		0			0
$655$	1.73205			0.0676768
$656$		−	29.7321i		−	1.16084i
$657$		0			0
$658$		−	57.8177i		−	2.25397i
$659$	−23.0722			−0.898767			−0.449384	−	0.893339i	$-0.648356\pi$
$659$	−23.0722			−0.898767			−0.449384	+	0.893339i	$0.648356\pi$
$660$		0			0
$661$	5.53590			0.215321			0.107661	−	0.994188i	$-0.465664\pi$
$661$	5.53590			0.215321			0.107661	+	0.994188i	$0.465664\pi$
$662$	33.3186			1.29497
$663$		0			0
$664$		0			0
$665$	−51.2822			−1.98864
$666$		0			0
$667$	−14.6603			−0.567647
$668$		−	1.85641i		−	0.0718265i
$669$		0			0
$670$		−	11.1326i		−	0.430090i
$671$	−2.20489			−0.0851190
$672$		0			0
$673$		−	48.7819i		−	1.88040i	−0.340618	−	0.940202i	$-0.610636\pi$
$673$		−	48.7819i		−	1.88040i	0.340618	−	0.940202i	$-0.389364\pi$
$674$		−	4.09103i		−	0.157581i
$675$		0			0
$676$	1.85641			0.0714002
$677$		−	10.5167i		−	0.404188i	−0.979366	−	0.202094i	$-0.935225\pi$
$677$		−	10.5167i		−	0.404188i	0.979366	−	0.202094i	$-0.0647747\pi$
$678$		0			0
$679$	54.1051			2.07636
$680$	1.21057	+	18.5885i	0.0464233	+	0.712835i
$681$		0			0
$682$	−18.1436			−0.694754
$683$		−	2.78461i		−	0.106550i	−0.998580	−	0.0532751i	$-0.983034\pi$
$683$		−	2.78461i		−	0.106550i	0.998580	−	0.0532751i	$-0.0169660\pi$
$684$		0			0
$685$			3.81899i			0.145916i
$686$			18.1436i			0.692726i
$687$		0			0
$688$	−1.19615			−0.0456029
$689$	8.52418			0.324745
$690$		0			0
$691$		−	20.2765i		−	0.771356i	−0.922633	−	0.385678i	$-0.873968\pi$
$691$		−	20.2765i		−	0.771356i	0.922633	−	0.385678i	$-0.126032\pi$
$692$			4.00000i			0.152057i
$693$		0			0
$694$		−	1.28972i		−	0.0489572i
$695$	28.5053			1.08127
$696$		0			0
$697$	1.78461	+	27.4029i	0.0675969	+	1.03796i
$698$	−25.9761			−0.983208
$699$		0			0
$700$		−	2.20489i		−	0.0833372i
$701$	41.1439			1.55398			0.776992	−	0.629511i	$-0.216745\pi$
$701$	41.1439			1.55398			0.776992	+	0.629511i	$0.216745\pi$
$702$		0			0
$703$		−	67.1489i		−	2.53257i
$704$		−	13.3205i		−	0.502036i
$705$		0			0
$706$	−21.4641			−0.807812
$707$		−	50.7846i		−	1.90995i
$708$		0			0
$709$		−	34.5292i		−	1.29677i	−0.761312	−	0.648386i	$-0.775444\pi$
$709$		−	34.5292i		−	1.29677i	0.761312	−	0.648386i	$-0.224556\pi$
$710$	14.4399			0.541921
$711$		0			0
$712$	5.75129			0.215539
$713$	50.9868			1.90947
$714$		0			0
$715$	15.4641			0.578325
$716$	1.61410			0.0603216
$717$		0			0
$718$	−41.2679			−1.54011
$719$		−	43.3923i		−	1.61826i	−0.587630	−	0.809130i	$-0.699939\pi$
$719$		−	43.3923i		−	1.61826i	0.587630	−	0.809130i	$-0.300061\pi$
$720$		0			0
$721$			27.9937i			1.04254i
$722$	49.3727			1.83746
$723$		0			0
$724$			1.61410i			0.0599874i
$725$			3.46410i			0.128654i
$726$		0			0
$727$	25.1962			0.934474			0.467237	−	0.884132i	$-0.345250\pi$
$727$	25.1962			0.934474			0.467237	+	0.884132i	$0.345250\pi$
$728$		−	47.9090i		−	1.77562i
$729$		0			0
$730$	−18.5885			−0.687990
$731$	1.10245	−	0.0717968i	0.0407755	−	0.00265550i
$732$		0			0
$733$	32.4641			1.19909			0.599544	−	0.800341i	$-0.295348\pi$
$733$	32.4641			1.19909			0.599544	+	0.800341i	$0.295348\pi$
$734$			34.3013i			1.26608i
$735$		0			0
$736$		−	12.7467i		−	0.469849i
$737$		−	8.53590i		−	0.314424i
$738$		0			0
$739$	7.19615			0.264715			0.132357	−	0.991202i	$-0.457745\pi$
$739$	7.19615			0.264715			0.132357	+	0.991202i	$0.457745\pi$
$740$	−4.33064			−0.159197
$741$		0			0
$742$			11.8315i			0.434349i
$743$		−	40.6410i		−	1.49097i	−0.666520	−	0.745487i	$-0.732217\pi$
$743$		−	40.6410i		−	1.49097i	0.666520	−	0.745487i	$-0.267783\pi$
$744$		0			0
$745$		−	3.30734i		−	0.121172i
$746$	−28.9880			−1.06133
$747$		0			0
$748$	−0.143594	−	2.20489i	−0.00525030	−	0.0806189i
$749$	12.3432			0.451010
$750$		0			0
$751$		−	8.22878i		−	0.300272i	−0.988665	−	0.150136i	$-0.952029\pi$
$751$		−	8.22878i		−	0.300272i	0.988665	−	0.150136i	$-0.0479712\pi$
$752$	−41.6555			−1.51902
$753$		0			0
$754$		−	11.6442i		−	0.424058i
$755$			3.00000i			0.109181i
$756$		0			0
$757$	−24.1769			−0.878725			−0.439362	−	0.898310i	$-0.644796\pi$
$757$	−24.1769			−0.878725			−0.439362	+	0.898310i	$0.644796\pi$
$758$			52.4449i			1.90488i
$759$		0			0
$760$			32.5116i			1.17932i
$761$	−18.9579			−0.687222			−0.343611	−	0.939112i	$-0.611650\pi$
$761$	−18.9579			−0.687222			−0.343611	+	0.939112i	$0.611650\pi$
$762$		0			0
$763$	−54.1051			−1.95874
$764$	−6.02388			−0.217937
$765$		0			0
$766$	−30.5359			−1.10331
$767$	31.8127			1.14869
$768$		0			0
$769$	−37.9282			−1.36773			−0.683863	−	0.729610i	$-0.739701\pi$
$769$	−37.9282			−1.36773			−0.683863	+	0.729610i	$0.739701\pi$
$770$			21.4641i			0.773513i
$771$		0			0
$772$		−	2.79569i		−	0.100619i
$773$	3.81899			0.137360			0.0686798	−	0.997639i	$-0.478121\pi$
$773$	3.81899			0.137360			0.0686798	+	0.997639i	$0.478121\pi$
$774$		0			0
$775$		−	12.0478i		−	0.432769i
$776$		−	34.3013i		−	1.23134i
$777$		0			0
$778$	−15.7128			−0.563332
$779$			47.9282i			1.71721i
$780$		0			0
$781$	11.0718			0.396180
$782$	3.41547	+	52.4449i	0.122137	+	1.87542i
$783$		0			0
$784$	44.3205			1.58288
$785$		−	15.7128i		−	0.560814i
$786$		0			0
$787$			46.5770i			1.66029i	0.557547	+	0.830145i	$0.311742\pi$
$787$			46.5770i			1.66029i	−0.557547	+	0.830145i	$0.688258\pi$
$788$		−	4.46410i		−	0.159027i
$789$		0			0
$790$	26.4449			0.940866
$791$	−47.1678			−1.67709
$792$		0			0
$793$			4.92144i			0.174765i
$794$			3.32051i			0.117840i
$795$		0			0
$796$			5.43308i			0.192571i
$797$	−51.2822			−1.81651			−0.908254	−	0.418420i	$-0.862584\pi$
$797$	−51.2822			−1.81651			−0.908254	+	0.418420i	$0.862584\pi$
$798$		0			0
$799$	38.3923	−	2.50029i	1.35822	−	0.0884541i
$800$	−3.01194			−0.106488
$801$		0			0
$802$			19.6568i			0.694105i
$803$	−14.2527			−0.502966
$804$		0			0
$805$		−	60.3180i		−	2.12593i
$806$			40.4974i			1.42646i
$807$		0			0
$808$	−32.1962			−1.13266
$809$			26.1244i			0.918483i	0.888311	+	0.459242i	$0.151879\pi$
$809$			26.1244i			0.918483i	−0.888311	+	0.459242i	$0.848121\pi$
$810$		0			0
$811$		−	34.8246i		−	1.22286i	−0.791299	−	0.611429i	$-0.790595\pi$
$811$		−	34.8246i		−	1.22286i	0.791299	−	0.611429i	$-0.209405\pi$
$812$	1.90949			0.0670101
$813$		0			0
$814$	−28.1051			−0.985084
$815$	−10.4337			−0.365476
$816$		0			0
$817$	1.92820			0.0674593
$818$	−6.13201			−0.214401
$819$		0			0
$820$	3.09103			0.107944
$821$			28.5359i			0.995910i	0.867203	+	0.497955i	$0.165915\pi$
$821$			28.5359i			0.995910i	−0.867203	+	0.497955i	$0.834085\pi$
$822$		0			0
$823$			4.40979i			0.153716i	0.997042	+	0.0768578i	$0.0244887\pi$
$823$			4.40979i			0.153716i	−0.997042	+	0.0768578i	$0.975511\pi$
$824$	17.7473			0.618256
$825$		0			0
$826$			44.1558i			1.53638i
$827$		−	41.7846i		−	1.45299i	−0.687170	−	0.726497i	$-0.741147\pi$
$827$		−	41.7846i		−	1.45299i	0.687170	−	0.726497i	$-0.258853\pi$
$828$		0			0
$829$	18.3205			0.636298			0.318149	−	0.948041i	$-0.396939\pi$
$829$	18.3205			0.636298			0.318149	+	0.948041i	$0.396939\pi$
$830$		0			0
$831$		0			0
$832$	−29.7321			−1.03077
$833$	−40.8485	+	2.66025i	−1.41532	+	0.0921723i
$834$		0			0
$835$	−12.0000			−0.415277
$836$		−	3.85641i		−	0.133377i
$837$		0			0
$838$			9.14395i			0.315873i
$839$			13.5359i			0.467311i	0.972319	+	0.233656i	$0.0750688\pi$
$839$			13.5359i			0.467311i	−0.972319	+	0.233656i	$0.924931\pi$
$840$		0			0
$841$	26.0000			0.896552
$842$	−6.02388			−0.207597
$843$		0			0
$844$			0.0791522i			0.00272453i
$845$		−	12.0000i		−	0.412813i
$846$		0			0
$847$		−	28.8007i		−	0.989605i
$848$	8.52418			0.292722
$849$		0			0
$850$	12.3923	−	0.807048i	0.425053	−	0.0276815i
$851$	78.9805			2.70742
$852$		0			0
$853$			28.5053i			0.976004i	0.872843	+	0.488002i	$0.162274\pi$
$853$			28.5053i			0.976004i	−0.872843	+	0.488002i	$0.837726\pi$
$854$	−6.83093			−0.233750
$855$		0			0
$856$		−	7.82526i		−	0.267462i
$857$		−	24.5359i		−	0.838130i	−0.907956	−	0.419065i	$-0.862358\pi$
$857$		−	24.5359i		−	0.838130i	0.907956	−	0.419065i	$-0.137642\pi$
$858$		0			0
$859$	30.1436			1.02849			0.514243	−	0.857644i	$-0.328073\pi$
$859$	30.1436			1.02849			0.514243	+	0.857644i	$0.328073\pi$
$860$		−	0.124356i		−	0.00424049i
$861$		0			0
$862$		−	41.2520i		−	1.40505i
$863$	−10.4337			−0.355166			−0.177583	−	0.984106i	$-0.556828\pi$
$863$	−10.4337			−0.355166			−0.177583	+	0.984106i	$0.556828\pi$
$864$		0			0
$865$	25.8564			0.879144
$866$	−25.6015			−0.869975
$867$		0			0
$868$	−6.64102			−0.225411
$869$	20.2765			0.687835
$870$		0			0
$871$	−19.0526			−0.645571
$872$			34.3013i			1.16159i
$873$		0			0
$874$			91.7271i			3.10272i
$875$	−49.8843			−1.68640
$876$		0			0
$877$		−	38.3482i		−	1.29493i	−0.762097	−	0.647463i	$-0.775830\pi$
$877$		−	38.3482i		−	1.29493i	0.762097	−	0.647463i	$-0.224170\pi$
$878$			12.8372i			0.433233i
$879$		0			0
$880$	15.4641			0.521295
$881$			43.9808i			1.48175i	0.671643	+	0.740875i	$0.265589\pi$
$881$			43.9808i			1.48175i	−0.671643	+	0.740875i	$0.734411\pi$
$882$		0			0
$883$	−48.5167			−1.63272			−0.816358	−	0.577546i	$-0.804010\pi$
$883$	−48.5167			−1.63272			−0.816358	+	0.577546i	$0.804010\pi$
$884$	−4.92144	+	0.320508i	−0.165526	+	0.0107799i
$885$		0			0
$886$	8.30127			0.278887
$887$			26.0000i			0.872995i	0.899706	+	0.436497i	$0.143781\pi$
$887$			26.0000i			0.872995i	−0.899706	+	0.436497i	$0.856219\pi$
$888$		0			0
$889$			44.9629i			1.50801i
$890$			5.75129i			0.192784i
$891$		0			0
$892$	2.03332			0.0680806
$893$	67.1489			2.24705
$894$		0			0
$895$		−	10.4337i		−	0.348759i
$896$		−	53.6603i		−	1.79266i
$897$		0			0
$898$			19.6857i			0.656922i
$899$	10.4337			0.347983
$900$		0			0
$901$	−7.85641	+	0.511648i	−0.261735	+	0.0170455i
$902$	20.0603			0.667935
$903$		0			0
$904$			29.9032i			0.994565i
$905$	10.4337			0.346827
$906$		0			0
$907$		−	28.8007i		−	0.956312i	−0.878275	−	0.478156i	$-0.841305\pi$
$907$		−	28.8007i		−	0.956312i	0.878275	−	0.478156i	$-0.158695\pi$
$908$			4.76537i			0.158144i
$909$		0			0
$910$	47.9090			1.58817
$911$			7.24871i			0.240161i	0.992764	+	0.120080i	$0.0383152\pi$
$911$			7.24871i			0.240161i	−0.992764	+	0.120080i	$0.961685\pi$
$912$		0			0
$913$		0			0
$914$	14.8435			0.490978
$915$		0			0
$916$	1.44486			0.0477396
$917$	4.11439			0.135869
$918$		0			0
$919$	37.9808			1.25287			0.626435	−	0.779474i	$-0.284513\pi$
$919$	37.9808			1.25287			0.626435	+	0.779474i	$0.284513\pi$
$920$	−38.2401			−1.26074
$921$		0			0
$922$	28.1051			0.925593
$923$		−	24.7128i		−	0.813432i
$924$		0			0
$925$		−	18.6625i		−	0.613618i
$926$	38.1320			1.25309
$927$		0			0
$928$		−	2.60842i		−	0.0856255i
$929$			18.1244i			0.594641i	0.954778	+	0.297320i	$0.0960930\pi$
$929$			18.1244i			0.594641i	−0.954778	+	0.297320i	$0.903907\pi$
$930$		0			0
$931$	−71.4449			−2.34151
$932$			6.07180i			0.198888i
$933$		0			0
$934$	4.09103			0.133863
$935$	−14.2527	+	0.928203i	−0.466112	+	0.0303555i
$936$		0			0
$937$	42.8564			1.40006			0.700029	−	0.714115i	$-0.253170\pi$
$937$	42.8564			1.40006			0.700029	+	0.714115i	$0.253170\pi$
$938$		−	26.4449i		−	0.863455i
$939$		0			0
$940$		−	4.33064i		−	0.141250i
$941$		−	10.3923i		−	0.338779i	−0.985549	−	0.169390i	$-0.945820\pi$
$941$		−	10.3923i		−	0.338779i	0.985549	−	0.169390i	$-0.0541797\pi$
$942$		0			0
$943$	−56.3731			−1.83576
$944$	31.8127			1.03541
$945$		0			0
$946$		−	0.807048i		−	0.0262394i
$947$			14.7846i			0.480435i	0.970719	+	0.240218i	$0.0772188\pi$
$947$			14.7846i			0.480435i	−0.970719	+	0.240218i	$0.922781\pi$
$948$		0			0
$949$			31.8127i			1.03268i
$950$	21.6744			0.703210
$951$		0			0
$952$	2.87564	+	44.1558i	0.0932002	+	1.43110i
$953$	−38.9390			−1.26136			−0.630679	−	0.776044i	$-0.717224\pi$
$953$	−38.9390			−1.26136			−0.630679	+	0.776044i	$0.717224\pi$
$954$		0			0
$955$			38.9390i			1.26004i
$956$	7.34258			0.237476
$957$		0			0
$958$			60.5053i			1.95484i
$959$			9.07180i			0.292944i
$960$		0			0
$961$	−5.28719			−0.170554
$962$			62.7321i			2.02256i
$963$		0			0
$964$			2.20489i			0.0710149i
$965$	−18.0717			−0.581747
$966$		0			0
$967$	−27.5885			−0.887185			−0.443592	−	0.896229i	$-0.646296\pi$
$967$	−27.5885			−0.887185			−0.443592	+	0.896229i	$0.646296\pi$
$968$	−18.2589			−0.586864
$969$		0			0
$970$	34.3013			1.10135
$971$	−17.5600			−0.563527			−0.281764	−	0.959484i	$-0.590919\pi$
$971$	−17.5600			−0.563527			−0.281764	+	0.959484i	$0.590919\pi$
$972$		0			0
$973$	67.7128			2.17077
$974$			43.8179i			1.40402i
$975$		0			0
$976$			4.92144i			0.157531i
$977$	10.1383			0.324352			0.162176	−	0.986762i	$-0.448149\pi$
$977$	10.1383			0.324352			0.162176	+	0.986762i	$0.448149\pi$
$978$		0			0
$979$			4.40979i			0.140937i
$980$			4.60770i			0.147187i
$981$		0			0
$982$	−61.0718			−1.94888
$983$		−	22.3205i		−	0.711914i	−0.934502	−	0.355957i	$-0.884155\pi$
$983$		−	22.3205i		−	0.711914i	0.934502	−	0.355957i	$-0.115845\pi$
$984$		0			0
$985$	−28.8564			−0.919442
$986$	0.698924	+	10.7321i	0.0222583	+	0.341778i
$987$		0			0
$988$	−8.60770			−0.273847
$989$			2.26795i			0.0721166i
$990$		0			0
$991$			34.2338i			1.08747i	0.839256	+	0.543736i	$0.182991\pi$
$991$			34.2338i			1.08747i	−0.839256	+	0.543736i	$0.817009\pi$
$992$			9.07180i			0.288030i
$993$		0			0
$994$	34.3013			1.08797
$995$	35.1200			1.11338
$996$		0			0
$997$			4.40979i			0.139659i	0.997559	+	0.0698297i	$0.0222456\pi$
$997$			4.40979i			0.139659i	−0.997559	+	0.0698297i	$0.977754\pi$
$998$		−	2.87564i		−	0.0910269i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	459.2.d.b.271.5	yes	8
3.2	odd	2	inner	459.2.d.b.271.4	yes	8
17.4	even	4		7803.2.a.bi.1.2		4
17.13	even	4		7803.2.a.bh.1.2		4
17.16	even	2	inner	459.2.d.b.271.6	yes	8
51.38	odd	4		7803.2.a.bh.1.3		4
51.47	odd	4		7803.2.a.bi.1.3		4
51.50	odd	2	inner	459.2.d.b.271.3	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.d.b.271.3	✓	8	51.50	odd	2	inner
459.2.d.b.271.4	yes	8	3.2	odd	2	inner
459.2.d.b.271.5	yes	8	1.1	even	1	trivial
459.2.d.b.271.6	yes	8	17.16	even	2	inner
7803.2.a.bh.1.2		4	17.13	even	4
7803.2.a.bh.1.3		4	51.38	odd	4
7803.2.a.bi.1.2		4	17.4	even	4
7803.2.a.bi.1.3		4	51.47	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 459 = 3^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	459.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.50597 q^{2} +0.267949 q^{4} -1.73205i q^{5} -4.11439i q^{7} -2.60842 q^{8} -2.60842i q^{10} -2.00000i q^{11} -4.46410 q^{13} -6.19615i q^{14} -4.46410 q^{16} +(4.11439 - 0.267949i) q^{17} +7.19615 q^{19} -0.464102i q^{20} -3.01194i q^{22} +8.46410i q^{23} +2.00000 q^{25} -6.72281 q^{26} -1.10245i q^{28} +1.73205i q^{29} -6.02388i q^{31} -1.50597 q^{32} +(6.19615 - 0.403524i) q^{34} -7.12633 q^{35} -9.33123i q^{37} +10.8372 q^{38} +4.51791i q^{40} +6.66025i q^{41} +0.267949 q^{43} -0.535898i q^{44} +12.7467i q^{46} +9.33123 q^{47} -9.92820 q^{49} +3.01194 q^{50} -1.19615 q^{52} -1.90949 q^{53} -3.46410 q^{55} +10.7321i q^{56} +2.60842i q^{58} -7.12633 q^{59} -1.10245i q^{61} -9.07180i q^{62} +6.66025 q^{64} +7.73205i q^{65} +4.26795 q^{67} +(1.10245 - 0.0717968i) q^{68} -10.7321 q^{70} +5.53590i q^{71} -7.12633i q^{73} -14.0526i q^{74} +1.92820 q^{76} -8.22878 q^{77} +10.1383i q^{79} +7.73205i q^{80} +10.0302i q^{82} +(-0.464102 - 7.12633i) q^{85} +0.403524 q^{86} +5.21684i q^{88} -2.20489 q^{89} +18.3671i q^{91} +2.26795i q^{92} +14.0526 q^{94} -12.4641i q^{95} +13.1502i q^{97} -14.9516 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 16 q^{4} - 8 q^{13} - 8 q^{16} + 16 q^{19} + 16 q^{25} + 8 q^{34} + 16 q^{43} - 24 q^{49} + 32 q^{52} - 16 q^{64} + 48 q^{67} - 72 q^{70} - 40 q^{76} + 24 q^{85} - 40 q^{94}+O(q^{100}) \) 8 * q + 16 * q^4 - 8 * q^13 - 8 * q^16 + 16 * q^19 + 16 * q^25 + 8 * q^34 + 16 * q^43 - 24 * q^49 + 32 * q^52 - 16 * q^64 + 48 * q^67 - 72 * q^70 - 40 * q^76 + 24 * q^85 - 40 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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