LMFDB - Embedded newform 756.2.bi.b.307.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(307,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2, 3]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("756.307");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.bi (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			307.1
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	756.307
Dual form			756.2.bi.b.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(1.50000 + 0.866025i) q^{5} +(-2.59808 + 0.500000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(1.50000 + 0.866025i) q^{5} +(-2.59808 + 0.500000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-1.73205 - 1.73205i) q^{10} +(-0.866025 + 0.500000i) q^{11} +(1.50000 + 0.866025i) q^{13} +(3.73205 + 0.267949i) q^{14} +(2.00000 + 3.46410i) q^{16} +3.46410i q^{17} -6.92820 q^{19} +(1.73205 + 3.00000i) q^{20} +(1.36603 - 0.366025i) q^{22} +(4.33013 + 2.50000i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-1.73205 - 1.73205i) q^{26} +(-5.00000 - 1.73205i) q^{28} +(2.50000 + 4.33013i) q^{29} +(-4.33013 + 7.50000i) q^{31} +(-1.46410 - 5.46410i) q^{32} +(1.26795 - 4.73205i) q^{34} +(-4.33013 - 1.50000i) q^{35} +(9.46410 + 2.53590i) q^{38} +(-1.26795 - 4.73205i) q^{40} +(-7.50000 - 4.33013i) q^{41} +(-0.866025 + 0.500000i) q^{43} -2.00000 q^{44} +(-5.00000 - 5.00000i) q^{46} +(-2.59808 - 4.50000i) q^{47} +(6.50000 - 2.59808i) q^{49} +(0.732051 + 2.73205i) q^{50} +(1.73205 + 3.00000i) q^{52} -4.00000 q^{53} -1.73205 q^{55} +(6.19615 + 4.19615i) q^{56} +(-1.83013 - 6.83013i) q^{58} +(-4.33013 + 7.50000i) q^{59} +(-4.50000 + 2.59808i) q^{61} +(8.66025 - 8.66025i) q^{62} +8.00000i q^{64} +(1.50000 + 2.59808i) q^{65} +(12.9904 + 7.50000i) q^{67} +(-3.46410 + 6.00000i) q^{68} +(5.36603 + 3.63397i) q^{70} +4.00000i q^{71} +10.3923i q^{73} +(-12.0000 - 6.92820i) q^{76} +(2.00000 - 1.73205i) q^{77} +(-2.59808 + 1.50000i) q^{79} +6.92820i q^{80} +(8.66025 + 8.66025i) q^{82} +(0.866025 + 1.50000i) q^{83} +(-3.00000 + 5.19615i) q^{85} +(1.36603 - 0.366025i) q^{86} +(2.73205 + 0.732051i) q^{88} +17.3205i q^{89} +(-4.33013 - 1.50000i) q^{91} +(5.00000 + 8.66025i) q^{92} +(1.90192 + 7.09808i) q^{94} +(-10.3923 - 6.00000i) q^{95} +(-4.50000 + 2.59808i) q^{97} +(-9.83013 + 1.16987i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8	$ 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} + 6 q^{13} + 8 q^{14} + 8 q^{16} + 2 q^{22} - 4 q^{25} - 20 q^{28} + 10 q^{29} + 8 q^{32} + 12 q^{34} + 24 q^{38} - 12 q^{40} - 30 q^{41} - 8 q^{44} - 20 q^{46} + 26 q^{49} - 4 q^{50} - 16 q^{53} + 4 q^{56} + 10 q^{58} - 18 q^{61} + 6 q^{65} + 18 q^{70} - 48 q^{76} + 8 q^{77} - 12 q^{85} + 2 q^{86} + 4 q^{88} + 20 q^{92} + 18 q^{94} - 18 q^{97} - 22 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8 + 6 * q^13 + 8 * q^14 + 8 * q^16 + 2 * q^22 - 4 * q^25 - 20 * q^28 + 10 * q^29 + 8 * q^32 + 12 * q^34 + 24 * q^38 - 12 * q^40 - 30 * q^41 - 8 * q^44 - 20 * q^46 + 26 * q^49 - 4 * q^50 - 16 * q^53 + 4 * q^56 + 10 * q^58 - 18 * q^61 + 6 * q^65 + 18 * q^70 - 48 * q^76 + 8 * q^77 - 12 * q^85 + 2 * q^86 + 4 * q^88 + 20 * q^92 + 18 * q^94 - 18 * q^97 - 22 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{1}{3}\right)$	$-1$	$-1$

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 $

$2$	−1.36603	−	0.366025i	−0.965926	−	0.258819i
$2$	−1.36603	−	0.366025i	−0.965926	−	0.258819i
$3$		0			0
$3$		0			0
$4$	1.73205	+	1.00000i	0.866025	+	0.500000i
$5$	1.50000	+	0.866025i	0.670820	+	0.387298i	0.796387	−	0.604787i	$-0.206742\pi$
$5$	1.50000	+	0.866025i	0.670820	+	0.387298i	−0.125567	+	0.992085i	$0.540075\pi$
$6$		0			0
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$7$	−2.59808	+	0.500000i	−0.981981	+	0.188982i
$8$	−2.00000	−	2.00000i	−0.707107	−	0.707107i
$9$		0			0
$10$	−1.73205	−	1.73205i	−0.547723	−	0.547723i
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i	−0.624844	−	0.780750i	$-0.714837\pi$
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i	0.363727	+	0.931505i	$0.381504\pi$
$12$		0			0
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	0.693375	−	0.720577i	$-0.256123\pi$
$13$	1.50000	+	0.866025i	0.416025	+	0.240192i	−0.277350	+	0.960769i	$0.589456\pi$
$14$	3.73205	+	0.267949i	0.997433	+	0.0716124i
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$			3.46410i			0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$			3.46410i			0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$		0			0
$19$	−6.92820			−1.58944			−0.794719	−	0.606977i	$-0.792382\pi$
$19$	−6.92820			−1.58944			−0.794719	+	0.606977i	$0.792382\pi$
$20$	1.73205	+	3.00000i	0.387298	+	0.670820i
$21$		0			0
$22$	1.36603	−	0.366025i	0.291238	−	0.0780369i
$23$	4.33013	+	2.50000i	0.902894	+	0.521286i	0.878138	−	0.478407i	$-0.158786\pi$
$23$	4.33013	+	2.50000i	0.902894	+	0.521286i	0.0247559	+	0.999694i	$0.492119\pi$
$24$		0			0
$25$	−1.00000	−	1.73205i	−0.200000	−	0.346410i
$26$	−1.73205	−	1.73205i	−0.339683	−	0.339683i
$27$		0			0
$28$	−5.00000	−	1.73205i	−0.944911	−	0.327327i
$29$	2.50000	+	4.33013i	0.464238	+	0.804084i	0.999167	−	0.0408130i	$-0.0129948\pi$
$29$	2.50000	+	4.33013i	0.464238	+	0.804084i	−0.534928	+	0.844897i	$0.679661\pi$
$30$		0			0
$31$	−4.33013	+	7.50000i	−0.777714	+	1.34704i	0.155543	+	0.987829i	$0.450287\pi$
$31$	−4.33013	+	7.50000i	−0.777714	+	1.34704i	−0.933257	+	0.359211i	$0.883046\pi$
$32$	−1.46410	−	5.46410i	−0.258819	−	0.965926i
$33$		0			0
$34$	1.26795	−	4.73205i	0.217451	−	0.811540i
$35$	−4.33013	−	1.50000i	−0.731925	−	0.253546i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$	9.46410	+	2.53590i	1.53528	+	0.411377i
$39$		0			0
$40$	−1.26795	−	4.73205i	−0.200480	−	0.748203i
$41$	−7.50000	−	4.33013i	−1.17130	−	0.676252i	−0.217317	−	0.976101i	$-0.569730\pi$
$41$	−7.50000	−	4.33013i	−1.17130	−	0.676252i	−0.953987	+	0.299849i	$0.903064\pi$
$42$		0			0
$43$	−0.866025	+	0.500000i	−0.132068	+	0.0762493i	−0.564578	−	0.825380i	$-0.690961\pi$
$43$	−0.866025	+	0.500000i	−0.132068	+	0.0762493i	0.432511	+	0.901629i	$0.357628\pi$
$44$	−2.00000			−0.301511
$45$		0			0
$46$	−5.00000	−	5.00000i	−0.737210	−	0.737210i
$47$	−2.59808	−	4.50000i	−0.378968	−	0.656392i	0.611944	−	0.790901i	$-0.290388\pi$
$47$	−2.59808	−	4.50000i	−0.378968	−	0.656392i	−0.990912	+	0.134509i	$0.957054\pi$
$48$		0			0
$49$	6.50000	−	2.59808i	0.928571	−	0.371154i
$50$	0.732051	+	2.73205i	0.103528	+	0.386370i
$51$		0			0
$52$	1.73205	+	3.00000i	0.240192	+	0.416025i
$53$	−4.00000			−0.549442			−0.274721	−	0.961524i	$-0.588586\pi$
$53$	−4.00000			−0.549442			−0.274721	+	0.961524i	$0.588586\pi$
$54$		0			0
$55$	−1.73205			−0.233550
$56$	6.19615	+	4.19615i	0.827996	+	0.560734i
$57$		0			0
$58$	−1.83013	−	6.83013i	−0.240307	−	0.896840i
$59$	−4.33013	+	7.50000i	−0.563735	+	0.976417i	0.433432	+	0.901186i	$0.357303\pi$
$59$	−4.33013	+	7.50000i	−0.563735	+	0.976417i	−0.997166	+	0.0752304i	$0.976031\pi$
$60$		0			0
$61$	−4.50000	+	2.59808i	−0.576166	+	0.332650i	−0.759608	−	0.650381i	$-0.774609\pi$
$61$	−4.50000	+	2.59808i	−0.576166	+	0.332650i	0.183442	+	0.983030i	$0.441276\pi$
$62$	8.66025	−	8.66025i	1.09985	−	1.09985i
$63$		0			0
$64$			8.00000i			1.00000i
$65$	1.50000	+	2.59808i	0.186052	+	0.322252i
$66$		0			0
$67$	12.9904	+	7.50000i	1.58703	+	0.916271i	0.993793	+	0.111241i	$0.0354825\pi$
$67$	12.9904	+	7.50000i	1.58703	+	0.916271i	0.593234	+	0.805030i	$0.297851\pi$
$68$	−3.46410	+	6.00000i	−0.420084	+	0.727607i
$69$		0			0
$70$	5.36603	+	3.63397i	0.641363	+	0.434343i
$71$			4.00000i			0.474713i	0.971423	+	0.237356i	$0.0762809\pi$
$71$			4.00000i			0.474713i	−0.971423	+	0.237356i	$0.923719\pi$
$72$		0			0
$73$			10.3923i			1.21633i	0.793812	+	0.608164i	$0.208094\pi$
$73$			10.3923i			1.21633i	−0.793812	+	0.608164i	$0.791906\pi$
$74$		0			0
$75$		0			0
$76$	−12.0000	−	6.92820i	−1.37649	−	0.794719i
$77$	2.00000	−	1.73205i	0.227921	−	0.197386i
$78$		0			0
$79$	−2.59808	+	1.50000i	−0.292306	+	0.168763i	−0.638982	−	0.769222i	$-0.720644\pi$
$79$	−2.59808	+	1.50000i	−0.292306	+	0.168763i	0.346675	+	0.937985i	$0.387311\pi$
$80$			6.92820i			0.774597i
$81$		0			0
$82$	8.66025	+	8.66025i	0.956365	+	0.956365i
$83$	0.866025	+	1.50000i	0.0950586	+	0.164646i	0.909633	−	0.415413i	$-0.136363\pi$
$83$	0.866025	+	1.50000i	0.0950586	+	0.164646i	−0.814574	+	0.580059i	$0.803029\pi$
$84$		0			0
$85$	−3.00000	+	5.19615i	−0.325396	+	0.563602i
$86$	1.36603	−	0.366025i	0.147302	−	0.0394695i
$87$		0			0
$88$	2.73205	+	0.732051i	0.291238	+	0.0780369i
$89$			17.3205i			1.83597i	0.396615	+	0.917985i	$0.370185\pi$
$89$			17.3205i			1.83597i	−0.396615	+	0.917985i	$0.629815\pi$
$90$		0			0
$91$	−4.33013	−	1.50000i	−0.453921	−	0.157243i
$92$	5.00000	+	8.66025i	0.521286	+	0.902894i
$93$		0			0
$94$	1.90192	+	7.09808i	0.196168	+	0.732111i
$95$	−10.3923	−	6.00000i	−1.06623	−	0.615587i
$96$		0			0
$97$	−4.50000	+	2.59808i	−0.456906	+	0.263795i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−4.50000	+	2.59808i	−0.456906	+	0.263795i	0.253837	+	0.967247i	$0.418307\pi$
$98$	−9.83013	+	1.16987i	−0.992993	+	0.118175i
$99$		0			0
$100$		−	4.00000i		−	0.400000i
$101$	−10.5000	+	6.06218i	−1.04479	+	0.603209i	−0.921186	−	0.389123i	$-0.872778\pi$
$101$	−10.5000	+	6.06218i	−1.04479	+	0.603209i	−0.123603	+	0.992332i	$0.539445\pi$
$102$		0			0
$103$	4.33013	−	7.50000i	0.426660	−	0.738997i	−0.569914	−	0.821705i	$-0.693023\pi$
$103$	4.33013	−	7.50000i	0.426660	−	0.738997i	0.996574	+	0.0827075i	$0.0263567\pi$
$104$	−1.26795	−	4.73205i	−0.124333	−	0.464016i
$105$		0			0
$106$	5.46410	+	1.46410i	0.530720	+	0.142206i
$107$			2.00000i			0.193347i	0.995316	+	0.0966736i	$0.0308203\pi$
$107$			2.00000i			0.193347i	−0.995316	+	0.0966736i	$0.969180\pi$
$108$		0			0
$109$	12.0000			1.14939			0.574696	−	0.818367i	$-0.305120\pi$
$109$	12.0000			1.14939			0.574696	+	0.818367i	$0.305120\pi$
$110$	2.36603	+	0.633975i	0.225592	+	0.0604471i
$111$		0			0
$112$	−6.92820	−	8.00000i	−0.654654	−	0.755929i
$113$	9.50000	−	16.4545i	0.893685	−	1.54791i	0.0582609	−	0.998301i	$-0.481444\pi$
$113$	9.50000	−	16.4545i	0.893685	−	1.54791i	0.835424	−	0.549606i	$-0.185222\pi$
$114$		0			0
$115$	4.33013	+	7.50000i	0.403786	+	0.699379i
$116$			10.0000i			0.928477i
$117$		0			0
$118$	8.66025	−	8.66025i	0.797241	−	0.797241i
$119$	−1.73205	−	9.00000i	−0.158777	−	0.825029i
$120$		0			0
$121$	−5.00000	+	8.66025i	−0.454545	+	0.787296i
$122$	7.09808	−	1.90192i	0.642630	−	0.172192i
$123$		0			0
$124$	−15.0000	+	8.66025i	−1.34704	+	0.777714i
$125$		−	12.1244i		−	1.08444i
$126$		0			0
$127$		−	18.0000i		−	1.59724i	−0.601834	−	0.798621i	$-0.705563\pi$
$127$		−	18.0000i		−	1.59724i	0.601834	−	0.798621i	$-0.294437\pi$
$128$	2.92820	−	10.9282i	0.258819	−	0.965926i
$129$		0			0
$130$	−1.09808	−	4.09808i	−0.0963077	−	0.359425i
$131$	6.06218	−	10.5000i	0.529655	−	0.917389i	−0.469747	−	0.882801i	$-0.655655\pi$
$131$	6.06218	−	10.5000i	0.529655	−	0.917389i	0.999402	−	0.0345880i	$-0.0110119\pi$
$132$		0			0
$133$	18.0000	−	3.46410i	1.56080	−	0.300376i
$134$	−15.0000	−	15.0000i	−1.29580	−	1.29580i
$135$		0			0
$136$	6.92820	−	6.92820i	0.594089	−	0.594089i
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	0.952835	−	0.303488i	$-0.0981512\pi$
$137$	2.50000	+	4.33013i	0.213589	+	0.369948i	−0.739246	+	0.673436i	$0.764818\pi$
$138$		0			0
$139$	4.33013	−	7.50000i	0.367277	−	0.636142i	−0.621862	−	0.783127i	$-0.713624\pi$
$139$	4.33013	−	7.50000i	0.367277	−	0.636142i	0.989139	+	0.146985i	$0.0469569\pi$
$140$	−6.00000	−	6.92820i	−0.507093	−	0.585540i
$141$		0			0
$142$	1.46410	−	5.46410i	0.122865	−	0.458537i
$143$	−1.73205			−0.144841
$144$		0			0
$145$			8.66025i			0.719195i
$146$	3.80385	−	14.1962i	0.314809	−	1.17488i
$147$		0			0
$148$		0			0
$149$	−3.50000	+	6.06218i	−0.286731	+	0.496633i	−0.973028	−	0.230689i	$-0.925902\pi$
$149$	−3.50000	+	6.06218i	−0.286731	+	0.496633i	0.686296	+	0.727322i	$0.259235\pi$
$150$		0			0
$151$	−9.52628	+	5.50000i	−0.775238	+	0.447584i	−0.834740	−	0.550645i	$-0.814382\pi$
$151$	−9.52628	+	5.50000i	−0.775238	+	0.447584i	0.0595022	+	0.998228i	$0.481049\pi$
$152$	13.8564	+	13.8564i	1.12390	+	1.12390i
$153$		0			0
$154$	−3.36603	+	1.63397i	−0.271242	+	0.131669i
$155$	−12.9904	+	7.50000i	−1.04341	+	0.602414i
$156$		0			0
$157$	4.50000	+	2.59808i	0.359139	+	0.207349i	0.668703	−	0.743530i	$-0.266850\pi$
$157$	4.50000	+	2.59808i	0.359139	+	0.207349i	−0.309564	+	0.950879i	$0.600183\pi$
$158$	4.09808	−	1.09808i	0.326025	−	0.0873583i
$159$		0			0
$160$	2.53590	−	9.46410i	0.200480	−	0.748203i
$161$	−12.5000	−	4.33013i	−0.985138	−	0.341262i
$162$		0			0
$163$			6.00000i			0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$			6.00000i			0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	−8.66025	−	15.0000i	−0.676252	−	1.17130i
$165$		0			0
$166$	−0.633975	−	2.36603i	−0.0492060	−	0.183639i
$167$	6.06218	−	10.5000i	0.469105	−	0.812514i	−0.530271	−	0.847828i	$-0.677910\pi$
$167$	6.06218	−	10.5000i	0.469105	−	0.812514i	0.999376	+	0.0353139i	$0.0112431\pi$
$168$		0			0
$169$	−5.00000	−	8.66025i	−0.384615	−	0.666173i
$170$	6.00000	−	6.00000i	0.460179	−	0.460179i
$171$		0			0
$172$	−2.00000			−0.152499
$173$	7.50000	−	4.33013i	0.570214	−	0.329213i	−0.187021	−	0.982356i	$-0.559883\pi$
$173$	7.50000	−	4.33013i	0.570214	−	0.329213i	0.757235	+	0.653143i	$0.226550\pi$
$174$		0			0
$175$	3.46410	+	4.00000i	0.261861	+	0.302372i
$176$	−3.46410	−	2.00000i	−0.261116	−	0.150756i
$177$		0			0
$178$	6.33975	−	23.6603i	0.475184	−	1.77341i
$179$		−	2.00000i		−	0.149487i	−0.997203	−	0.0747435i	$-0.976186\pi$
$179$		−	2.00000i		−	0.149487i	0.997203	−	0.0747435i	$-0.0238138\pi$
$180$		0			0
$181$		−	17.3205i		−	1.28742i	−0.765268	−	0.643712i	$-0.777394\pi$
$181$		−	17.3205i		−	1.28742i	0.765268	−	0.643712i	$-0.222606\pi$
$182$	5.36603	+	3.63397i	0.397756	+	0.269368i
$183$		0			0
$184$	−3.66025	−	13.6603i	−0.269838	−	1.00705i
$185$		0			0
$186$		0			0
$187$	−1.73205	−	3.00000i	−0.126660	−	0.219382i
$188$		−	10.3923i		−	0.757937i
$189$		0			0
$190$	12.0000	+	12.0000i	0.870572	+	0.870572i
$191$	4.33013	−	2.50000i	0.313317	−	0.180894i	−0.335093	−	0.942185i	$-0.608768\pi$
$191$	4.33013	−	2.50000i	0.313317	−	0.180894i	0.648410	+	0.761291i	$0.275434\pi$
$192$		0			0
$193$	−4.50000	+	7.79423i	−0.323917	+	0.561041i	−0.981293	−	0.192522i	$-0.938333\pi$
$193$	−4.50000	+	7.79423i	−0.323917	+	0.561041i	0.657376	+	0.753563i	$0.271667\pi$
$194$	7.09808	−	1.90192i	0.509612	−	0.136550i
$195$		0			0
$196$	13.8564	+	2.00000i	0.989743	+	0.142857i
$197$	−2.00000			−0.142494			−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		0			0
$199$	17.3205			1.22782			0.613909	−	0.789377i	$-0.289596\pi$
$199$	17.3205			1.22782			0.613909	+	0.789377i	$0.289596\pi$
$200$	−1.46410	+	5.46410i	−0.103528	+	0.386370i
$201$		0			0
$202$	16.5622	−	4.43782i	1.16531	−	0.312244i
$203$	−8.66025	−	10.0000i	−0.607831	−	0.701862i
$204$		0			0
$205$	−7.50000	−	12.9904i	−0.523823	−	0.907288i
$206$	−8.66025	+	8.66025i	−0.603388	+	0.603388i
$207$		0			0
$208$			6.92820i			0.480384i
$209$	6.00000	−	3.46410i	0.415029	−	0.239617i
$210$		0			0
$211$	−0.866025	−	0.500000i	−0.0596196	−	0.0344214i	0.469894	−	0.882723i	$-0.344292\pi$
$211$	−0.866025	−	0.500000i	−0.0596196	−	0.0344214i	−0.529514	+	0.848301i	$0.677626\pi$
$212$	−6.92820	−	4.00000i	−0.475831	−	0.274721i
$213$		0			0
$214$	0.732051	−	2.73205i	0.0500420	−	0.186759i
$215$	−1.73205			−0.118125
$216$		0			0
$217$	7.50000	−	21.6506i	0.509133	−	1.46974i
$218$	−16.3923	−	4.39230i	−1.11023	−	0.297484i
$219$		0			0
$220$	−3.00000	−	1.73205i	−0.202260	−	0.116775i
$221$	−3.00000	+	5.19615i	−0.201802	+	0.349531i
$222$		0			0
$223$	−0.866025	−	1.50000i	−0.0579934	−	0.100447i	0.835571	−	0.549382i	$-0.185137\pi$
$223$	−0.866025	−	1.50000i	−0.0579934	−	0.100447i	−0.893565	+	0.448935i	$0.851804\pi$
$224$	6.53590	+	13.4641i	0.436698	+	0.899608i
$225$		0			0
$226$	−19.0000	+	19.0000i	−1.26386	+	1.26386i
$227$	2.59808	+	4.50000i	0.172440	+	0.298675i	0.939272	−	0.343172i	$-0.111501\pi$
$227$	2.59808	+	4.50000i	0.172440	+	0.298675i	−0.766832	+	0.641848i	$0.778168\pi$
$228$		0			0
$229$	−1.50000	−	0.866025i	−0.0991228	−	0.0572286i	0.449619	−	0.893220i	$-0.351560\pi$
$229$	−1.50000	−	0.866025i	−0.0991228	−	0.0572286i	−0.548742	+	0.835992i	$0.684893\pi$
$230$	−3.16987	−	11.8301i	−0.209015	−	0.780055i
$231$		0			0
$232$	3.66025	−	13.6603i	0.240307	−	0.896840i
$233$	14.0000			0.917170			0.458585	−	0.888650i	$-0.348356\pi$
$233$	14.0000			0.917170			0.458585	+	0.888650i	$0.348356\pi$
$234$		0			0
$235$		−	9.00000i		−	0.587095i
$236$	−15.0000	+	8.66025i	−0.976417	+	0.563735i
$237$		0			0
$238$	−0.928203	+	12.9282i	−0.0601665	+	0.838011i
$239$	21.6506	+	12.5000i	1.40046	+	0.808558i	0.994440	−	0.105305i	$-0.0335819\pi$
$239$	21.6506	+	12.5000i	1.40046	+	0.808558i	0.406023	+	0.913863i	$0.366915\pi$
$240$		0			0
$241$	−25.5000	+	14.7224i	−1.64260	+	0.948355i	−0.662695	+	0.748890i	$0.730587\pi$
$241$	−25.5000	+	14.7224i	−1.64260	+	0.948355i	−0.979905	+	0.199465i	$0.936079\pi$
$242$	10.0000	−	10.0000i	0.642824	−	0.642824i
$243$		0			0
$244$	−10.3923			−0.665299
$245$	12.0000	+	1.73205i	0.766652	+	0.110657i
$246$		0			0
$247$	−10.3923	−	6.00000i	−0.661247	−	0.381771i
$248$	23.6603	−	6.33975i	1.50243	−	0.402574i
$249$		0			0
$250$	−4.43782	+	16.5622i	−0.280673	+	1.04748i
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$		0			0
$253$	−5.00000			−0.314347
$254$	−6.58846	+	24.5885i	−0.413397	+	1.54282i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	25.5000	+	14.7224i	1.59065	+	0.918360i	0.993196	+	0.116454i	$0.0371528\pi$
$257$	25.5000	+	14.7224i	1.59065	+	0.918360i	0.597450	+	0.801906i	$0.296181\pi$
$258$		0			0
$259$		0			0
$260$			6.00000i			0.372104i
$261$		0			0
$262$	−12.1244	+	12.1244i	−0.749045	+	0.749045i
$263$	16.4545	−	9.50000i	1.01463	−	0.585795i	0.102084	−	0.994776i	$-0.467449\pi$
$263$	16.4545	−	9.50000i	1.01463	−	0.585795i	0.912543	+	0.408981i	$0.134116\pi$
$264$		0			0
$265$	−6.00000	−	3.46410i	−0.368577	−	0.212798i
$266$	−25.8564	−	1.85641i	−1.58536	−	0.113824i
$267$		0			0
$268$	15.0000	+	25.9808i	0.916271	+	1.58703i
$269$		−	10.3923i		−	0.633630i	−0.948487	−	0.316815i	$-0.897387\pi$
$269$		−	10.3923i		−	0.633630i	0.948487	−	0.316815i	$-0.102613\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$	−12.0000	+	6.92820i	−0.727607	+	0.420084i
$273$		0			0
$274$	−1.83013	−	6.83013i	−0.110562	−	0.412623i
$275$	1.73205	+	1.00000i	0.104447	+	0.0603023i
$276$		0			0
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	−0.971521	−	0.236953i	$-0.923851\pi$
$277$	−11.5000	−	19.9186i	−0.690968	−	1.19679i	0.280553	−	0.959839i	$-0.409482\pi$
$278$	−8.66025	+	8.66025i	−0.519408	+	0.519408i
$279$		0			0
$280$	5.66025	+	11.6603i	0.338265	+	0.696833i
$281$	0.500000	+	0.866025i	0.0298275	+	0.0516627i	0.880554	−	0.473946i	$-0.157171\pi$
$281$	0.500000	+	0.866025i	0.0298275	+	0.0516627i	−0.850726	+	0.525609i	$0.823838\pi$
$282$		0			0
$283$	7.79423	−	13.5000i	0.463319	−	0.802492i	−0.535805	−	0.844342i	$-0.679992\pi$
$283$	7.79423	−	13.5000i	0.463319	−	0.802492i	0.999124	+	0.0418500i	$0.0133252\pi$
$284$	−4.00000	+	6.92820i	−0.237356	+	0.411113i
$285$		0			0
$286$	2.36603	+	0.633975i	0.139906	+	0.0374877i
$287$	21.6506	+	7.50000i	1.27800	+	0.442711i
$288$		0			0
$289$	5.00000			0.294118
$290$	3.16987	−	11.8301i	0.186141	−	0.694689i
$291$		0			0
$292$	−10.3923	+	18.0000i	−0.608164	+	1.05337i
$293$	−22.5000	−	12.9904i	−1.31446	−	0.758906i	−0.331632	−	0.943409i	$-0.607599\pi$
$293$	−22.5000	−	12.9904i	−1.31446	−	0.758906i	−0.982832	+	0.184503i	$0.940933\pi$
$294$		0			0
$295$	−12.9904	+	7.50000i	−0.756329	+	0.436667i
$296$		0			0
$297$		0			0
$298$	7.00000	−	7.00000i	0.405499	−	0.405499i
$299$	4.33013	+	7.50000i	0.250418	+	0.433736i
$300$		0			0
$301$	2.00000	−	1.73205i	0.115278	−	0.0998337i
$302$	15.0263	−	4.02628i	0.864665	−	0.231686i
$303$		0			0
$304$	−13.8564	−	24.0000i	−0.794719	−	1.37649i
$305$	−9.00000			−0.515339
$306$		0			0
$307$	13.8564			0.790827			0.395413	−	0.918503i	$-0.370601\pi$
$307$	13.8564			0.790827			0.395413	+	0.918503i	$0.370601\pi$
$308$	5.19615	−	1.00000i	0.296078	−	0.0569803i
$309$		0			0
$310$	20.4904	−	5.49038i	1.16378	−	0.311833i
$311$	−12.9904	+	22.5000i	−0.736617	+	1.27586i	0.217393	+	0.976084i	$0.430245\pi$
$311$	−12.9904	+	22.5000i	−0.736617	+	1.27586i	−0.954010	+	0.299774i	$0.903089\pi$
$312$		0			0
$313$	−7.50000	+	4.33013i	−0.423925	+	0.244753i	−0.696755	−	0.717309i	$-0.745374\pi$
$313$	−7.50000	+	4.33013i	−0.423925	+	0.244753i	0.272830	+	0.962062i	$0.412040\pi$
$314$	−5.19615	−	5.19615i	−0.293236	−	0.293236i
$315$		0			0
$316$	−6.00000			−0.337526
$317$	−11.5000	−	19.9186i	−0.645904	−	1.11874i	−0.984092	−	0.177660i	$-0.943147\pi$
$317$	−11.5000	−	19.9186i	−0.645904	−	1.11874i	0.338188	−	0.941079i	$-0.390186\pi$
$318$		0			0
$319$	−4.33013	−	2.50000i	−0.242441	−	0.139973i
$320$	−6.92820	+	12.0000i	−0.387298	+	0.670820i
$321$		0			0
$322$	15.4904	+	10.4904i	0.863245	+	0.584606i
$323$		−	24.0000i		−	1.33540i
$324$		0			0
$325$		−	3.46410i		−	0.192154i
$326$	2.19615	−	8.19615i	0.121634	−	0.453943i
$327$		0			0
$328$	6.33975	+	23.6603i	0.350054	+	1.30642i
$329$	9.00000	+	10.3923i	0.496186	+	0.572946i
$330$		0			0
$331$	25.1147	−	14.5000i	1.38043	−	0.796992i	0.388221	−	0.921567i	$-0.373090\pi$
$331$	25.1147	−	14.5000i	1.38043	−	0.796992i	0.992210	+	0.124574i	$0.0397566\pi$
$332$			3.46410i			0.190117i
$333$		0			0
$334$	−12.1244	+	12.1244i	−0.663415	+	0.663415i
$335$	12.9904	+	22.5000i	0.709740	+	1.22931i
$336$		0			0
$337$	−7.50000	+	12.9904i	−0.408551	+	0.707631i	−0.994728	−	0.102552i	$-0.967299\pi$
$337$	−7.50000	+	12.9904i	−0.408551	+	0.707631i	0.586177	+	0.810183i	$0.300632\pi$
$338$	3.66025	+	13.6603i	0.199092	+	0.743020i
$339$		0			0
$340$	−10.3923	+	6.00000i	−0.563602	+	0.325396i
$341$		−	8.66025i		−	0.468979i
$342$		0			0
$343$	−15.5885	+	10.0000i	−0.841698	+	0.539949i
$344$	2.73205	+	0.732051i	0.147302	+	0.0394695i
$345$		0			0
$346$	−11.8301	+	3.16987i	−0.635992	+	0.170413i
$347$	4.33013	+	2.50000i	0.232453	+	0.134207i	0.611703	−	0.791087i	$-0.290485\pi$
$347$	4.33013	+	2.50000i	0.232453	+	0.134207i	−0.379250	+	0.925294i	$0.623818\pi$
$348$		0			0
$349$	22.5000	−	12.9904i	1.20440	−	0.695359i	0.242867	−	0.970059i	$-0.421912\pi$
$349$	22.5000	−	12.9904i	1.20440	−	0.695359i	0.961530	+	0.274700i	$0.0885786\pi$
$350$	−3.26795	−	6.73205i	−0.174679	−	0.359843i
$351$		0			0
$352$	4.00000	+	4.00000i	0.213201	+	0.213201i
$353$	−7.50000	+	4.33013i	−0.399185	+	0.230469i	−0.686132	−	0.727477i	$-0.740693\pi$
$353$	−7.50000	+	4.33013i	−0.399185	+	0.230469i	0.286947	+	0.957946i	$0.407359\pi$
$354$		0			0
$355$	−3.46410	+	6.00000i	−0.183855	+	0.318447i
$356$	−17.3205	+	30.0000i	−0.917985	+	1.59000i
$357$		0			0
$358$	−0.732051	+	2.73205i	−0.0386901	+	0.144393i
$359$		−	10.0000i		−	0.527780i	−0.964553	−	0.263890i	$-0.914994\pi$
$359$		−	10.0000i		−	0.527780i	0.964553	−	0.263890i	$-0.0850056\pi$
$360$		0			0
$361$	29.0000			1.52632
$362$	−6.33975	+	23.6603i	−0.333210	+	1.24356i
$363$		0			0
$364$	−6.00000	−	6.92820i	−0.314485	−	0.363137i
$365$	−9.00000	+	15.5885i	−0.471082	+	0.815937i
$366$		0			0
$367$	11.2583	+	19.5000i	0.587680	+	1.01789i	0.994535	+	0.104399i	$0.0332919\pi$
$367$	11.2583	+	19.5000i	0.587680	+	1.01789i	−0.406855	+	0.913493i	$0.633375\pi$
$368$			20.0000i			1.04257i
$369$		0			0
$370$		0			0
$371$	10.3923	−	2.00000i	0.539542	−	0.103835i
$372$		0			0
$373$	−0.500000	+	0.866025i	−0.0258890	+	0.0448411i	−0.878680	−	0.477412i	$-0.841575\pi$
$373$	−0.500000	+	0.866025i	−0.0258890	+	0.0448411i	0.852791	+	0.522253i	$0.174908\pi$
$374$	1.26795	+	4.73205i	0.0655641	+	0.244689i
$375$		0			0
$376$	−3.80385	+	14.1962i	−0.196168	+	0.732111i
$377$			8.66025i			0.446026i
$378$		0			0
$379$		−	8.00000i		−	0.410932i	−0.978664	−	0.205466i	$-0.934129\pi$
$379$		−	8.00000i		−	0.410932i	0.978664	−	0.205466i	$-0.0658711\pi$
$380$	−12.0000	−	20.7846i	−0.615587	−	1.06623i
$381$		0			0
$382$	−6.83013	+	1.83013i	−0.349460	+	0.0936374i
$383$	−0.866025	+	1.50000i	−0.0442518	+	0.0766464i	−0.887303	−	0.461187i	$-0.847424\pi$
$383$	−0.866025	+	1.50000i	−0.0442518	+	0.0766464i	0.843051	+	0.537833i	$0.180757\pi$
$384$		0			0
$385$	4.50000	−	0.866025i	0.229341	−	0.0441367i
$386$	9.00000	−	9.00000i	0.458088	−	0.458088i
$387$		0			0
$388$	−10.3923			−0.527589
$389$	−8.50000	−	14.7224i	−0.430967	−	0.746457i	0.565990	−	0.824412i	$-0.308494\pi$
$389$	−8.50000	−	14.7224i	−0.430967	−	0.746457i	−0.996957	+	0.0779554i	$0.975161\pi$
$390$		0			0
$391$	−8.66025	+	15.0000i	−0.437968	+	0.758583i
$392$	−18.1962	−	7.80385i	−0.919044	−	0.394154i
$393$		0			0
$394$	2.73205	+	0.732051i	0.137639	+	0.0368802i
$395$	−5.19615			−0.261447
$396$		0			0
$397$		−	17.3205i		−	0.869291i	−0.900602	−	0.434646i	$-0.856874\pi$
$397$		−	17.3205i		−	0.869291i	0.900602	−	0.434646i	$-0.143126\pi$
$398$	−23.6603	−	6.33975i	−1.18598	−	0.317783i
$399$		0			0
$400$	4.00000	−	6.92820i	0.200000	−	0.346410i
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	−0.796828	−	0.604206i	$-0.793490\pi$
$401$	2.50000	−	4.33013i	0.124844	−	0.216236i	0.921672	+	0.387970i	$0.126824\pi$
$402$		0			0
$403$	−12.9904	+	7.50000i	−0.647097	+	0.373602i
$404$	−24.2487			−1.20642
$405$		0			0
$406$	8.16987	+	16.8301i	0.405464	+	0.835265i
$407$		0			0
$408$		0			0
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	0.127609	−	0.991825i	$-0.459270\pi$
$409$	−13.5000	−	7.79423i	−0.667532	−	0.385400i	−0.795141	+	0.606425i	$0.792603\pi$
$410$	5.49038	+	20.4904i	0.271151	+	1.01195i
$411$		0			0
$412$	15.0000	−	8.66025i	0.738997	−	0.426660i
$413$	7.50000	−	21.6506i	0.369051	−	1.06536i
$414$		0			0
$415$			3.00000i			0.147264i
$416$	2.53590	−	9.46410i	0.124333	−	0.464016i
$417$		0			0
$418$	−9.46410	+	2.53590i	−0.462904	+	0.124035i
$419$	−12.9904	+	22.5000i	−0.634622	+	1.09920i	0.351974	+	0.936010i	$0.385511\pi$
$419$	−12.9904	+	22.5000i	−0.634622	+	1.09920i	−0.986595	+	0.163187i	$0.947823\pi$
$420$		0			0
$421$	5.50000	+	9.52628i	0.268054	+	0.464282i	0.968359	−	0.249561i	$-0.0802862\pi$
$421$	5.50000	+	9.52628i	0.268054	+	0.464282i	−0.700306	+	0.713843i	$0.746953\pi$
$422$	1.00000	+	1.00000i	0.0486792	+	0.0486792i
$423$		0			0
$424$	8.00000	+	8.00000i	0.388514	+	0.388514i
$425$	6.00000	−	3.46410i	0.291043	−	0.168034i
$426$		0			0
$427$	10.3923	−	9.00000i	0.502919	−	0.435541i
$428$	−2.00000	+	3.46410i	−0.0966736	+	0.167444i
$429$		0			0
$430$	2.36603	+	0.633975i	0.114100	+	0.0305730i
$431$			26.0000i			1.25238i	0.779672	+	0.626188i	$0.215386\pi$
$431$			26.0000i			1.25238i	−0.779672	+	0.626188i	$0.784614\pi$
$432$		0			0
$433$			17.3205i			0.832370i	0.909280	+	0.416185i	$0.136633\pi$
$433$			17.3205i			0.832370i	−0.909280	+	0.416185i	$0.863367\pi$
$434$	−18.1699	+	26.8301i	−0.872182	+	1.28789i
$435$		0			0
$436$	20.7846	+	12.0000i	0.995402	+	0.574696i
$437$	−30.0000	−	17.3205i	−1.43509	−	0.828552i
$438$		0			0
$439$	−9.52628	−	16.5000i	−0.454665	−	0.787502i	0.544004	−	0.839082i	$-0.316908\pi$
$439$	−9.52628	−	16.5000i	−0.454665	−	0.787502i	−0.998669	+	0.0515804i	$0.983574\pi$
$440$	3.46410	+	3.46410i	0.165145	+	0.165145i
$441$		0			0
$442$	6.00000	−	6.00000i	0.285391	−	0.285391i
$443$	−30.3109	+	17.5000i	−1.44011	+	0.831450i	−0.997857	−	0.0654382i	$-0.979155\pi$
$443$	−30.3109	+	17.5000i	−1.44011	+	0.831450i	−0.442257	+	0.896888i	$0.645822\pi$
$444$		0			0
$445$	−15.0000	+	25.9808i	−0.711068	+	1.23161i
$446$	0.633975	+	2.36603i	0.0300196	+	0.112035i
$447$		0			0
$448$	−4.00000	−	20.7846i	−0.188982	−	0.981981i
$449$	20.0000			0.943858			0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000			0.943858			0.471929	+	0.881636i	$0.343558\pi$
$450$		0			0
$451$	8.66025			0.407795
$452$	32.9090	−	19.0000i	1.54791	−	0.893685i
$453$		0			0
$454$	−1.90192	−	7.09808i	−0.0892617	−	0.333129i
$455$	−5.19615	−	6.00000i	−0.243599	−	0.281284i
$456$		0			0
$457$	7.50000	+	12.9904i	0.350835	+	0.607664i	0.986396	−	0.164386i	$-0.0525644\pi$
$457$	7.50000	+	12.9904i	0.350835	+	0.607664i	−0.635561	+	0.772051i	$0.719231\pi$
$458$	1.73205	+	1.73205i	0.0809334	+	0.0809334i
$459$		0			0
$460$			17.3205i			0.807573i
$461$	10.5000	−	6.06218i	0.489034	−	0.282344i	−0.235140	−	0.971962i	$-0.575555\pi$
$461$	10.5000	−	6.06218i	0.489034	−	0.282344i	0.724174	+	0.689618i	$0.242221\pi$
$462$		0			0
$463$	33.7750	+	19.5000i	1.56966	+	0.906242i	0.996208	+	0.0870004i	$0.0277281\pi$
$463$	33.7750	+	19.5000i	1.56966	+	0.906242i	0.573449	+	0.819242i	$0.305605\pi$
$464$	−10.0000	+	17.3205i	−0.464238	+	0.804084i
$465$		0			0
$466$	−19.1244	−	5.12436i	−0.885919	−	0.237381i
$467$	20.7846			0.961797			0.480899	−	0.876776i	$-0.340311\pi$
$467$	20.7846			0.961797			0.480899	+	0.876776i	$0.340311\pi$
$468$		0			0
$469$	−37.5000	−	12.9904i	−1.73159	−	0.599840i
$470$	−3.29423	+	12.2942i	−0.151951	+	0.567090i
$471$		0			0
$472$	23.6603	−	6.33975i	1.08905	−	0.291810i
$473$	0.500000	−	0.866025i	0.0229900	−	0.0398199i
$474$		0			0
$475$	6.92820	+	12.0000i	0.317888	+	0.550598i
$476$	6.00000	−	17.3205i	0.275010	−	0.793884i
$477$		0			0
$478$	−25.0000	−	25.0000i	−1.14347	−	1.14347i
$479$	16.4545	+	28.5000i	0.751825	+	1.30220i	0.946938	+	0.321417i	$0.104159\pi$
$479$	16.4545	+	28.5000i	0.751825	+	1.30220i	−0.195113	+	0.980781i	$0.562507\pi$
$480$		0			0
$481$		0			0
$482$	40.2224	−	10.7776i	1.83208	−	0.490905i
$483$		0			0
$484$	−17.3205	+	10.0000i	−0.787296	+	0.454545i
$485$	−9.00000			−0.408669
$486$		0			0
$487$		−	10.0000i		−	0.453143i	−0.973995	−	0.226572i	$-0.927248\pi$
$487$		−	10.0000i		−	0.453143i	0.973995	−	0.226572i	$-0.0727517\pi$
$488$	14.1962	+	3.80385i	0.642630	+	0.172192i
$489$		0			0
$490$	−15.7583	−	6.75833i	−0.711889	−	0.305310i
$491$	16.4545	+	9.50000i	0.742580	+	0.428729i	0.823007	−	0.568032i	$-0.192295\pi$
$491$	16.4545	+	9.50000i	0.742580	+	0.428729i	−0.0804264	+	0.996761i	$0.525628\pi$
$492$		0			0
$493$	−15.0000	+	8.66025i	−0.675566	+	0.390038i
$494$	12.0000	+	12.0000i	0.539906	+	0.539906i
$495$		0			0
$496$	−34.6410			−1.55543
$497$	−2.00000	−	10.3923i	−0.0897123	−	0.466159i
$498$		0			0
$499$	−6.06218	−	3.50000i	−0.271380	−	0.156682i	0.358134	−	0.933670i	$-0.383413\pi$
$499$	−6.06218	−	3.50000i	−0.271380	−	0.156682i	−0.629515	+	0.776989i	$0.716746\pi$
$500$	12.1244	−	21.0000i	0.542218	−	0.939149i
$501$		0			0
$502$		0			0
$503$	−17.3205			−0.772283			−0.386142	−	0.922440i	$-0.626192\pi$
$503$	−17.3205			−0.772283			−0.386142	+	0.922440i	$0.626192\pi$
$504$		0			0
$505$	−21.0000			−0.934488
$506$	6.83013	+	1.83013i	0.303636	+	0.0813591i
$507$		0			0
$508$	18.0000	−	31.1769i	0.798621	−	1.38325i
$509$	22.5000	+	12.9904i	0.997295	+	0.575789i	0.907447	−	0.420167i	$-0.138028\pi$
$509$	22.5000	+	12.9904i	0.997295	+	0.575789i	0.0898481	+	0.995955i	$0.471362\pi$
$510$		0			0
$511$	−5.19615	−	27.0000i	−0.229864	−	1.19441i
$512$	16.0000	−	16.0000i	0.707107	−	0.707107i
$513$		0			0
$514$	−29.4449	−	29.4449i	−1.29876	−	1.29876i
$515$	12.9904	−	7.50000i	0.572425	−	0.330489i
$516$		0			0
$517$	4.50000	+	2.59808i	0.197910	+	0.114263i
$518$		0			0
$519$		0			0
$520$	2.19615	−	8.19615i	0.0963077	−	0.359425i
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−24.2487			−1.06032			−0.530161	−	0.847897i	$-0.677869\pi$
$523$	−24.2487			−1.06032			−0.530161	+	0.847897i	$0.677869\pi$
$524$	21.0000	−	12.1244i	0.917389	−	0.529655i
$525$		0			0
$526$	−25.9545	+	6.95448i	−1.13167	+	0.303230i
$527$	−25.9808	−	15.0000i	−1.13174	−	0.653410i
$528$		0			0
$529$	1.00000	+	1.73205i	0.0434783	+	0.0753066i
$530$	6.92820	+	6.92820i	0.300942	+	0.300942i
$531$		0			0
$532$	34.6410	+	12.0000i	1.50188	+	0.520266i
$533$	−7.50000	−	12.9904i	−0.324861	−	0.562676i
$534$		0			0
$535$	−1.73205	+	3.00000i	−0.0748831	+	0.129701i
$536$	−10.9808	−	40.9808i	−0.474297	−	1.77010i
$537$		0			0
$538$	−3.80385	+	14.1962i	−0.163996	+	0.612040i
$539$	−4.33013	+	5.50000i	−0.186512	+	0.236902i
$540$		0			0
$541$	26.0000			1.11783			0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000			1.11783			0.558914	+	0.829226i	$0.311218\pi$
$542$		0			0
$543$		0			0
$544$	18.9282	−	5.07180i	0.811540	−	0.217451i
$545$	18.0000	+	10.3923i	0.771035	+	0.445157i
$546$		0			0
$547$	−23.3827	+	13.5000i	−0.999771	+	0.577218i	−0.908181	−	0.418578i	$-0.862529\pi$
$547$	−23.3827	+	13.5000i	−0.999771	+	0.577218i	−0.0915908	+	0.995797i	$0.529195\pi$
$548$			10.0000i			0.427179i
$549$		0			0
$550$	−2.00000	−	2.00000i	−0.0852803	−	0.0852803i
$551$	−17.3205	−	30.0000i	−0.737878	−	1.27804i
$552$		0			0
$553$	6.00000	−	5.19615i	0.255146	−	0.220963i
$554$	8.41858	+	31.4186i	0.357671	+	1.33485i
$555$		0			0
$556$	15.0000	−	8.66025i	0.636142	−	0.367277i
$557$	40.0000			1.69485			0.847427	−	0.530912i	$-0.178150\pi$
$557$	40.0000			1.69485			0.847427	+	0.530912i	$0.178150\pi$
$558$		0			0
$559$	−1.73205			−0.0732579
$560$	−3.46410	−	18.0000i	−0.146385	−	0.760639i
$561$		0			0
$562$	−0.366025	−	1.36603i	−0.0154398	−	0.0576223i
$563$	0.866025	−	1.50000i	0.0364986	−	0.0632175i	−0.847199	−	0.531276i	$-0.821713\pi$
$563$	0.866025	−	1.50000i	0.0364986	−	0.0632175i	0.883698	+	0.468058i	$0.155046\pi$
$564$		0			0
$565$	28.5000	−	16.4545i	1.19900	−	0.692245i
$566$	−15.5885	+	15.5885i	−0.655232	+	0.655232i
$567$		0			0
$568$	8.00000	−	8.00000i	0.335673	−	0.335673i
$569$	3.50000	+	6.06218i	0.146728	+	0.254140i	0.930016	−	0.367519i	$-0.119793\pi$
$569$	3.50000	+	6.06218i	0.146728	+	0.254140i	−0.783289	+	0.621658i	$0.786459\pi$
$570$		0			0
$571$	−7.79423	−	4.50000i	−0.326178	−	0.188319i	0.327965	−	0.944690i	$-0.393637\pi$
$571$	−7.79423	−	4.50000i	−0.326178	−	0.188319i	−0.654143	+	0.756371i	$0.726971\pi$
$572$	−3.00000	−	1.73205i	−0.125436	−	0.0724207i
$573$		0			0
$574$	−26.8301	−	18.1699i	−1.11987	−	0.758396i
$575$		−	10.0000i		−	0.417029i
$576$		0			0
$577$			38.1051i			1.58634i	0.609002	+	0.793168i	$0.291570\pi$
$577$			38.1051i			1.58634i	−0.609002	+	0.793168i	$0.708430\pi$
$578$	−6.83013	−	1.83013i	−0.284096	−	0.0761232i
$579$		0			0
$580$	−8.66025	+	15.0000i	−0.359597	+	0.622841i
$581$	−3.00000	−	3.46410i	−0.124461	−	0.143715i
$582$		0			0
$583$	3.46410	−	2.00000i	0.143468	−	0.0828315i
$584$	20.7846	−	20.7846i	0.860073	−	0.860073i
$585$		0			0
$586$	25.9808	+	25.9808i	1.07326	+	1.07326i
$587$	−12.9904	−	22.5000i	−0.536170	−	0.928674i	−0.999106	−	0.0422823i	$-0.986537\pi$
$587$	−12.9904	−	22.5000i	−0.536170	−	0.928674i	0.462935	−	0.886392i	$-0.346796\pi$
$588$		0			0
$589$	30.0000	−	51.9615i	1.23613	−	2.14104i
$590$	20.4904	−	5.49038i	0.843576	−	0.226035i
$591$		0			0
$592$		0			0
$593$			31.1769i			1.28028i	0.768257	+	0.640141i	$0.221124\pi$
$593$			31.1769i			1.28028i	−0.768257	+	0.640141i	$0.778876\pi$
$594$		0			0
$595$	5.19615	−	15.0000i	0.213021	−	0.614940i
$596$	−12.1244	+	7.00000i	−0.496633	+	0.286731i
$597$		0			0
$598$	−3.16987	−	11.8301i	−0.129626	−	0.483770i
$599$	−14.7224	−	8.50000i	−0.601542	−	0.347301i	0.168106	−	0.985769i	$-0.446235\pi$
$599$	−14.7224	−	8.50000i	−0.601542	−	0.347301i	−0.769648	+	0.638468i	$0.779568\pi$
$600$		0			0
$601$	−4.50000	+	2.59808i	−0.183559	+	0.105978i	−0.588964	−	0.808160i	$-0.700464\pi$
$601$	−4.50000	+	2.59808i	−0.183559	+	0.105978i	0.405405	+	0.914137i	$0.367131\pi$
$602$	−3.36603	+	1.63397i	−0.137189	+	0.0665958i
$603$		0			0
$604$	−22.0000			−0.895167
$605$	−15.0000	+	8.66025i	−0.609837	+	0.352089i
$606$		0			0
$607$	4.33013	−	7.50000i	0.175754	−	0.304416i	−0.764668	−	0.644425i	$-0.777097\pi$
$607$	4.33013	−	7.50000i	0.175754	−	0.304416i	0.940422	+	0.340009i	$0.110430\pi$
$608$	10.1436	+	37.8564i	0.411377	+	1.53528i
$609$		0			0
$610$	12.2942	+	3.29423i	0.497779	+	0.133379i
$611$		−	9.00000i		−	0.364101i
$612$		0			0
$613$	−40.0000			−1.61558			−0.807792	−	0.589467i	$-0.799338\pi$
$613$	−40.0000			−1.61558			−0.807792	+	0.589467i	$0.799338\pi$
$614$	−18.9282	−	5.07180i	−0.763880	−	0.204681i
$615$		0			0
$616$	−7.46410	−	0.535898i	−0.300737	−	0.0215920i
$617$	−17.5000	+	30.3109i	−0.704523	+	1.22027i	0.262340	+	0.964976i	$0.415506\pi$
$617$	−17.5000	+	30.3109i	−0.704523	+	1.22027i	−0.966863	+	0.255295i	$0.917827\pi$
$618$		0			0
$619$	4.33013	+	7.50000i	0.174042	+	0.301450i	0.939829	−	0.341644i	$-0.110984\pi$
$619$	4.33013	+	7.50000i	0.174042	+	0.301450i	−0.765787	+	0.643094i	$0.777650\pi$
$620$	−30.0000			−1.20483
$621$		0			0
$622$	25.9808	−	25.9808i	1.04173	−	1.04173i
$623$	−8.66025	−	45.0000i	−0.346966	−	1.80289i
$624$		0			0
$625$	5.50000	−	9.52628i	0.220000	−	0.381051i
$626$	11.8301	−	3.16987i	0.472827	−	0.126694i
$627$		0			0
$628$	5.19615	+	9.00000i	0.207349	+	0.359139i
$629$		0			0
$630$		0			0
$631$			6.00000i			0.238856i	0.992843	+	0.119428i	$0.0381061\pi$
$631$			6.00000i			0.238856i	−0.992843	+	0.119428i	$0.961894\pi$
$632$	8.19615	+	2.19615i	0.326025	+	0.0873583i
$633$		0			0
$634$	8.41858	+	31.4186i	0.334345	+	1.24779i
$635$	15.5885	−	27.0000i	0.618609	−	1.07146i
$636$		0			0
$637$	12.0000	+	1.73205i	0.475457	+	0.0686264i
$638$	5.00000	+	5.00000i	0.197952	+	0.197952i
$639$		0			0
$640$	13.8564	−	13.8564i	0.547723	−	0.547723i
$641$	−2.50000	−	4.33013i	−0.0987441	−	0.171030i	0.812421	−	0.583071i	$-0.198149\pi$
$641$	−2.50000	−	4.33013i	−0.0987441	−	0.171030i	−0.911165	+	0.412042i	$0.864816\pi$
$642$		0			0
$643$	−0.866025	+	1.50000i	−0.0341527	+	0.0591542i	−0.882597	−	0.470131i	$-0.844207\pi$
$643$	−0.866025	+	1.50000i	−0.0341527	+	0.0591542i	0.848444	+	0.529285i	$0.177540\pi$
$644$	−17.3205	−	20.0000i	−0.682524	−	0.788110i
$645$		0			0
$646$	−8.78461	+	32.7846i	−0.345626	+	1.28989i
$647$	34.6410			1.36188			0.680939	−	0.732340i	$-0.261572\pi$
$647$	34.6410			1.36188			0.680939	+	0.732340i	$0.261572\pi$
$648$		0			0
$649$		−	8.66025i		−	0.339945i
$650$	−1.26795	+	4.73205i	−0.0497331	+	0.185606i
$651$		0			0
$652$	−6.00000	+	10.3923i	−0.234978	+	0.406994i
$653$	−0.500000	+	0.866025i	−0.0195665	+	0.0338902i	−0.875643	−	0.482959i	$-0.839562\pi$
$653$	−0.500000	+	0.866025i	−0.0195665	+	0.0338902i	0.856076	+	0.516849i	$0.172895\pi$
$654$		0			0
$655$	18.1865	−	10.5000i	0.710607	−	0.410269i
$656$		−	34.6410i		−	1.35250i
$657$		0			0
$658$	−8.49038	−	17.4904i	−0.330990	−	0.681846i
$659$	−14.7224	+	8.50000i	−0.573505	+	0.331113i	−0.758548	−	0.651617i	$-0.774091\pi$
$659$	−14.7224	+	8.50000i	−0.573505	+	0.331113i	0.185043	+	0.982730i	$0.440757\pi$
$660$		0			0
$661$	−7.50000	−	4.33013i	−0.291716	−	0.168422i	0.346999	−	0.937865i	$-0.387201\pi$
$661$	−7.50000	−	4.33013i	−0.291716	−	0.168422i	−0.638716	+	0.769443i	$0.720534\pi$
$662$	−39.6147	+	10.6147i	−1.53967	+	0.412553i
$663$		0			0
$664$	1.26795	−	4.73205i	0.0492060	−	0.183639i
$665$	30.0000	+	10.3923i	1.16335	+	0.402996i
$666$		0			0
$667$			25.0000i			0.968004i
$668$	21.0000	−	12.1244i	0.812514	−	0.469105i
$669$		0			0
$670$	−9.50962	−	35.4904i	−0.367389	−	1.37111i
$671$	2.59808	−	4.50000i	0.100298	−	0.173721i
$672$		0			0
$673$	19.5000	+	33.7750i	0.751670	+	1.30193i	0.947013	+	0.321195i	$0.104085\pi$
$673$	19.5000	+	33.7750i	0.751670	+	1.30193i	−0.195343	+	0.980735i	$0.562582\pi$
$674$	15.0000	−	15.0000i	0.577778	−	0.577778i
$675$		0			0
$676$		−	20.0000i		−	0.769231i
$677$	−4.50000	+	2.59808i	−0.172949	+	0.0998522i	−0.583976	−	0.811771i	$-0.698504\pi$
$677$	−4.50000	+	2.59808i	−0.172949	+	0.0998522i	0.411027	+	0.911623i	$0.365170\pi$
$678$		0			0
$679$	10.3923	−	9.00000i	0.398820	−	0.345388i
$680$	16.3923	−	4.39230i	0.628616	−	0.168437i
$681$		0			0
$682$	−3.16987	+	11.8301i	−0.121381	+	0.452999i
$683$			14.0000i			0.535695i	0.963461	+	0.267848i	$0.0863124\pi$
$683$			14.0000i			0.535695i	−0.963461	+	0.267848i	$0.913688\pi$
$684$		0			0
$685$			8.66025i			0.330891i
$686$	24.9545	−	7.95448i	0.952767	−	0.303704i
$687$		0			0
$688$	−3.46410	−	2.00000i	−0.132068	−	0.0762493i
$689$	−6.00000	−	3.46410i	−0.228582	−	0.131972i
$690$		0			0
$691$	−4.33013	−	7.50000i	−0.164726	−	0.285313i	0.771832	−	0.635826i	$-0.219341\pi$
$691$	−4.33013	−	7.50000i	−0.164726	−	0.285313i	−0.936558	+	0.350513i	$0.886007\pi$
$692$	17.3205			0.658427
$693$		0			0
$694$	−5.00000	−	5.00000i	−0.189797	−	0.189797i
$695$	12.9904	−	7.50000i	0.492753	−	0.284491i
$696$		0			0
$697$	15.0000	−	25.9808i	0.568166	−	0.984092i
$698$	−35.4904	+	9.50962i	−1.34333	+	0.359944i
$699$		0			0
$700$	2.00000	+	10.3923i	0.0755929	+	0.392792i
$701$	−44.0000			−1.66186			−0.830929	−	0.556379i	$-0.812190\pi$
$701$	−44.0000			−1.66186			−0.830929	+	0.556379i	$0.812190\pi$
$702$		0			0
$703$		0			0
$704$	−4.00000	−	6.92820i	−0.150756	−	0.261116i
$705$		0			0
$706$	11.8301	−	3.16987i	0.445233	−	0.119300i
$707$	24.2487	−	21.0000i	0.911967	−	0.789786i
$708$		0			0
$709$	−7.50000	−	12.9904i	−0.281668	−	0.487864i	0.690127	−	0.723688i	$-0.257554\pi$
$709$	−7.50000	−	12.9904i	−0.281668	−	0.487864i	−0.971796	+	0.235824i	$0.924221\pi$
$710$	6.92820	−	6.92820i	0.260011	−	0.260011i
$711$		0			0
$712$	34.6410	−	34.6410i	1.29823	−	1.29823i
$713$	−37.5000	+	21.6506i	−1.40439	+	0.810823i
$714$		0			0
$715$	−2.59808	−	1.50000i	−0.0971625	−	0.0560968i
$716$	2.00000	−	3.46410i	0.0747435	−	0.129460i
$717$		0			0
$718$	−3.66025	+	13.6603i	−0.136599	+	0.509796i
$719$	−34.6410			−1.29189			−0.645946	−	0.763383i	$-0.723537\pi$
$719$	−34.6410			−1.29189			−0.645946	+	0.763383i	$0.723537\pi$
$720$		0			0
$721$	−7.50000	+	21.6506i	−0.279315	+	0.806312i
$722$	−39.6147	−	10.6147i	−1.47431	−	0.395040i
$723$		0			0
$724$	17.3205	−	30.0000i	0.643712	−	1.11494i
$725$	5.00000	−	8.66025i	0.185695	−	0.321634i
$726$		0			0
$727$	14.7224	+	25.5000i	0.546025	+	0.945743i	0.998542	+	0.0539868i	$0.0171929\pi$
$727$	14.7224	+	25.5000i	0.546025	+	0.945743i	−0.452517	+	0.891756i	$0.649474\pi$
$728$	5.66025	+	11.6603i	0.209783	+	0.432158i
$729$		0			0
$730$	18.0000	−	18.0000i	0.666210	−	0.666210i
$731$	−1.73205	−	3.00000i	−0.0640622	−	0.110959i
$732$		0			0
$733$	19.5000	+	11.2583i	0.720249	+	0.415836i	0.814844	−	0.579680i	$-0.196822\pi$
$733$	19.5000	+	11.2583i	0.720249	+	0.415836i	−0.0945954	+	0.995516i	$0.530156\pi$
$734$	−8.24167	−	30.7583i	−0.304206	−	1.13531i
$735$		0			0
$736$	7.32051	−	27.3205i	0.269838	−	1.00705i
$737$	−15.0000			−0.552532
$738$		0			0
$739$		0			0		1.00000			$0$
$739$		0			0		−1.00000			$\pi$
$740$		0			0
$741$		0			0
$742$	−14.9282	−	1.07180i	−0.548032	−	0.0393469i
$743$	−25.1147	−	14.5000i	−0.921370	−	0.531953i	−0.0372984	−	0.999304i	$-0.511875\pi$
$743$	−25.1147	−	14.5000i	−0.921370	−	0.531953i	−0.884072	+	0.467351i	$0.845209\pi$
$744$		0			0
$745$	−10.5000	+	6.06218i	−0.384690	+	0.222101i
$746$	1.00000	−	1.00000i	0.0366126	−	0.0366126i
$747$		0			0
$748$		−	6.92820i		−	0.253320i
$749$	−1.00000	−	5.19615i	−0.0365392	−	0.189863i
$750$		0			0
$751$	−30.3109	−	17.5000i	−1.10606	−	0.638584i	−0.168254	−	0.985744i	$-0.553813\pi$
$751$	−30.3109	−	17.5000i	−1.10606	−	0.638584i	−0.937806	+	0.347160i	$0.887146\pi$
$752$	10.3923	−	18.0000i	0.378968	−	0.656392i
$753$		0			0
$754$	3.16987	−	11.8301i	0.115440	−	0.430828i
$755$	−19.0526			−0.693394
$756$		0			0
$757$	30.0000			1.09037			0.545184	−	0.838316i	$-0.316460\pi$
$757$	30.0000			1.09037			0.545184	+	0.838316i	$0.316460\pi$
$758$	−2.92820	+	10.9282i	−0.106357	+	0.396930i
$759$		0			0
$760$	8.78461	+	32.7846i	0.318651	+	1.18922i
$761$	−19.5000	−	11.2583i	−0.706874	−	0.408114i	0.103028	−	0.994678i	$-0.467147\pi$
$761$	−19.5000	−	11.2583i	−0.706874	−	0.408114i	−0.809903	+	0.586564i	$0.800480\pi$
$762$		0			0
$763$	−31.1769	+	6.00000i	−1.12868	+	0.217215i
$764$	10.0000			0.361787
$765$		0			0
$766$	1.73205	−	1.73205i	0.0625815	−	0.0625815i
$767$	−12.9904	+	7.50000i	−0.469055	+	0.270809i
$768$		0			0
$769$	22.5000	+	12.9904i	0.811371	+	0.468445i	0.847432	−	0.530904i	$-0.178148\pi$
$769$	22.5000	+	12.9904i	0.811371	+	0.468445i	−0.0360609	+	0.999350i	$0.511481\pi$
$770$	−6.46410	−	0.464102i	−0.232950	−	0.0167251i
$771$		0			0
$772$	−15.5885	+	9.00000i	−0.561041	+	0.323917i
$773$		−	24.2487i		−	0.872166i	−0.899907	−	0.436083i	$-0.856365\pi$
$773$		−	24.2487i		−	0.872166i	0.899907	−	0.436083i	$-0.143635\pi$
$774$		0			0
$775$	17.3205			0.622171
$776$	14.1962	+	3.80385i	0.509612	+	0.136550i
$777$		0			0
$778$	6.22243	+	23.2224i	0.223085	+	0.832565i
$779$	51.9615	+	30.0000i	1.86171	+	1.07486i
$780$		0			0
$781$	−2.00000	−	3.46410i	−0.0715656	−	0.123955i
$782$	17.3205	−	17.3205i	0.619380	−	0.619380i
$783$		0			0
$784$	22.0000	+	17.3205i	0.785714	+	0.618590i
$785$	4.50000	+	7.79423i	0.160612	+	0.278188i
$786$		0			0
$787$	−6.06218	+	10.5000i	−0.216093	+	0.374285i	−0.953610	−	0.301044i	$-0.902665\pi$
$787$	−6.06218	+	10.5000i	−0.216093	+	0.374285i	0.737517	+	0.675329i	$0.235998\pi$
$788$	−3.46410	−	2.00000i	−0.123404	−	0.0712470i
$789$		0			0
$790$	7.09808	+	1.90192i	0.252538	+	0.0676674i
$791$	−16.4545	+	47.5000i	−0.585054	+	1.68891i
$792$		0			0
$793$	−9.00000			−0.319599
$794$	−6.33975	+	23.6603i	−0.224989	+	0.839671i
$795$		0			0
$796$	30.0000	+	17.3205i	1.06332	+	0.613909i
$797$	4.50000	+	2.59808i	0.159398	+	0.0920286i	0.577577	−	0.816336i	$-0.303998\pi$
$797$	4.50000	+	2.59808i	0.159398	+	0.0920286i	−0.418179	+	0.908365i	$0.637332\pi$
$798$		0			0
$799$	15.5885	−	9.00000i	0.551480	−	0.318397i
$800$	−8.00000	+	8.00000i	−0.282843	+	0.282843i
$801$		0			0
$802$	−5.00000	+	5.00000i	−0.176556	+	0.176556i
$803$	−5.19615	−	9.00000i	−0.183368	−	0.317603i
$804$		0			0
$805$	−15.0000	−	17.3205i	−0.528681	−	0.610468i
$806$	20.4904	−	5.49038i	0.721743	−	0.193390i
$807$		0			0
$808$	33.1244	+	8.87564i	1.16531	+	0.312244i
$809$	−8.00000			−0.281265			−0.140633	−	0.990062i	$-0.544914\pi$
$809$	−8.00000			−0.281265			−0.140633	+	0.990062i	$0.544914\pi$
$810$		0			0
$811$	−20.7846			−0.729846			−0.364923	−	0.931038i	$-0.618905\pi$
$811$	−20.7846			−0.729846			−0.364923	+	0.931038i	$0.618905\pi$
$812$	−5.00000	−	25.9808i	−0.175466	−	0.911746i
$813$		0			0
$814$		0			0
$815$	−5.19615	+	9.00000i	−0.182013	+	0.315256i
$816$		0			0
$817$	6.00000	−	3.46410i	0.209913	−	0.121194i
$818$	15.5885	+	15.5885i	0.545038	+	0.545038i
$819$		0			0
$820$		−	30.0000i		−	1.04765i
$821$	20.5000	+	35.5070i	0.715455	+	1.23920i	0.962784	+	0.270273i	$0.0871139\pi$
$821$	20.5000	+	35.5070i	0.715455	+	1.23920i	−0.247329	+	0.968932i	$0.579553\pi$
$822$		0			0
$823$	−7.79423	−	4.50000i	−0.271690	−	0.156860i	0.357966	−	0.933735i	$-0.383471\pi$
$823$	−7.79423	−	4.50000i	−0.271690	−	0.156860i	−0.629655	+	0.776875i	$0.716804\pi$
$824$	−23.6603	+	6.33975i	−0.824244	+	0.220856i
$825$		0			0
$826$	−18.1699	+	26.8301i	−0.632211	+	0.933540i
$827$		−	50.0000i		−	1.73867i	−0.494223	−	0.869335i	$-0.664547\pi$
$827$		−	50.0000i		−	1.73867i	0.494223	−	0.869335i	$-0.335453\pi$
$828$		0			0
$829$			10.3923i			0.360940i	0.983581	+	0.180470i	$0.0577618\pi$
$829$			10.3923i			0.360940i	−0.983581	+	0.180470i	$0.942238\pi$
$830$	1.09808	−	4.09808i	0.0381148	−	0.142246i
$831$		0			0
$832$	−6.92820	+	12.0000i	−0.240192	+	0.416025i
$833$	9.00000	+	22.5167i	0.311832	+	0.780156i
$834$		0			0
$835$	18.1865	−	10.5000i	0.629371	−	0.363367i
$836$	13.8564			0.479234
$837$		0			0
$838$	25.9808	−	25.9808i	0.897491	−	0.897491i
$839$	0.866025	+	1.50000i	0.0298985	+	0.0517858i	0.880587	−	0.473884i	$-0.157148\pi$
$839$	0.866025	+	1.50000i	0.0298985	+	0.0517858i	−0.850689	+	0.525669i	$0.823815\pi$
$840$		0			0
$841$	2.00000	−	3.46410i	0.0689655	−	0.119452i
$842$	−4.02628	−	15.0263i	−0.138755	−	0.517840i
$843$		0			0
$844$	−1.00000	−	1.73205i	−0.0344214	−	0.0596196i
$845$		−	17.3205i		−	0.595844i
$846$		0			0
$847$	8.66025	−	25.0000i	0.297570	−	0.859010i
$848$	−8.00000	−	13.8564i	−0.274721	−	0.475831i
$849$		0			0
$850$	−9.46410	+	2.53590i	−0.324616	+	0.0869806i
$851$		0			0
$852$		0			0
$853$	−34.5000	+	19.9186i	−1.18126	+	0.681999i	−0.956305	−	0.292370i	$-0.905556\pi$
$853$	−34.5000	+	19.9186i	−1.18126	+	0.681999i	−0.224952	+	0.974370i	$0.572223\pi$
$854$	−17.4904	+	8.49038i	−0.598509	+	0.290535i
$855$		0			0
$856$	4.00000	−	4.00000i	0.136717	−	0.136717i
$857$	−25.5000	+	14.7224i	−0.871063	+	0.502909i	−0.867701	−	0.497086i	$-0.834403\pi$
$857$	−25.5000	+	14.7224i	−0.871063	+	0.502909i	−0.00336193	+	0.999994i	$0.501070\pi$
$858$		0			0
$859$	−18.1865	+	31.5000i	−0.620517	+	1.07477i	0.368873	+	0.929480i	$0.379744\pi$
$859$	−18.1865	+	31.5000i	−0.620517	+	1.07477i	−0.989390	+	0.145286i	$0.953590\pi$
$860$	−3.00000	−	1.73205i	−0.102299	−	0.0590624i
$861$		0			0
$862$	9.51666	−	35.5167i	0.324139	−	1.20970i
$863$			10.0000i			0.340404i	0.985409	+	0.170202i	$0.0544420\pi$
$863$			10.0000i			0.340404i	−0.985409	+	0.170202i	$0.945558\pi$
$864$		0			0
$865$	15.0000			0.510015
$866$	6.33975	−	23.6603i	0.215433	−	0.804008i
$867$		0			0
$868$	34.6410	−	30.0000i	1.17579	−	1.01827i
$869$	1.50000	−	2.59808i	0.0508840	−	0.0881337i
$870$		0			0
$871$	12.9904	+	22.5000i	0.440162	+	0.762383i
$872$	−24.0000	−	24.0000i	−0.812743	−	0.812743i
$873$		0			0
$874$	34.6410	+	34.6410i	1.17175	+	1.17175i
$875$	6.06218	+	31.5000i	0.204939	+	1.06489i
$876$		0			0
$877$	−12.5000	+	21.6506i	−0.422095	+	0.731090i	−0.996144	−	0.0877308i	$-0.972038\pi$
$877$	−12.5000	+	21.6506i	−0.422095	+	0.731090i	0.574049	+	0.818821i	$0.305372\pi$
$878$	6.97372	+	26.0263i	0.235352	+	0.878344i
$879$		0			0
$880$	−3.46410	−	6.00000i	−0.116775	−	0.202260i
$881$			31.1769i			1.05038i	0.850986	+	0.525188i	$0.176005\pi$
$881$			31.1769i			1.05038i	−0.850986	+	0.525188i	$0.823995\pi$
$882$		0			0
$883$			26.0000i			0.874970i	0.899226	+	0.437485i	$0.144131\pi$
$883$			26.0000i			0.874970i	−0.899226	+	0.437485i	$0.855869\pi$
$884$	−10.3923	+	6.00000i	−0.349531	+	0.201802i
$885$		0			0
$886$	47.8109	−	12.8109i	1.60624	−	0.430390i
$887$	4.33013	−	7.50000i	0.145391	−	0.251825i	−0.784127	−	0.620600i	$-0.786889\pi$
$887$	4.33013	−	7.50000i	0.145391	−	0.251825i	0.929519	+	0.368774i	$0.120223\pi$
$888$		0			0
$889$	9.00000	+	46.7654i	0.301850	+	1.56846i
$890$	30.0000	−	30.0000i	1.00560	−	1.00560i
$891$		0			0
$892$		−	3.46410i		−	0.115987i
$893$	18.0000	+	31.1769i	0.602347	+	1.04330i
$894$		0			0
$895$	1.73205	−	3.00000i	0.0578961	−	0.100279i
$896$	−2.14359	+	29.8564i	−0.0716124	+	0.997433i
$897$		0			0
$898$	−27.3205	−	7.32051i	−0.911697	−	0.244289i
$899$	−43.3013			−1.44418
$900$		0			0
$901$		−	13.8564i		−	0.461624i
$902$	−11.8301	−	3.16987i	−0.393900	−	0.105545i
$903$		0			0
$904$	−51.9090	+	13.9090i	−1.72647	+	0.462605i
$905$	15.0000	−	25.9808i	0.498617	−	0.863630i
$906$		0			0
$907$	19.9186	−	11.5000i	0.661386	−	0.381851i	−0.131419	−	0.991327i	$-0.541953\pi$
$907$	19.9186	−	11.5000i	0.661386	−	0.381851i	0.792805	+	0.609476i	$0.208620\pi$
$908$			10.3923i			0.344881i
$909$		0			0
$910$	4.90192	+	10.0981i	0.162497	+	0.334748i
$911$	−35.5070	+	20.5000i	−1.17640	+	0.679195i	−0.955179	−	0.296028i	$-0.904338\pi$
$911$	−35.5070	+	20.5000i	−1.17640	+	0.679195i	−0.221222	+	0.975224i	$0.571004\pi$
$912$		0			0
$913$	−1.50000	−	0.866025i	−0.0496428	−	0.0286613i
$914$	−5.49038	−	20.4904i	−0.181606	−	0.677762i
$915$		0			0
$916$	−1.73205	−	3.00000i	−0.0572286	−	0.0991228i
$917$	−10.5000	+	30.3109i	−0.346741	+	1.00095i
$918$		0			0
$919$			50.0000i			1.64935i	0.565608	+	0.824674i	$0.308641\pi$
$919$			50.0000i			1.64935i	−0.565608	+	0.824674i	$0.691359\pi$
$920$	6.33975	−	23.6603i	0.209015	−	0.780055i
$921$		0			0
$922$	−16.5622	+	4.43782i	−0.545446	+	0.146152i
$923$	−3.46410	+	6.00000i	−0.114022	+	0.197492i
$924$		0			0
$925$		0			0
$926$	−39.0000	−	39.0000i	−1.28162	−	1.28162i
$927$		0			0
$928$	20.0000	−	20.0000i	0.656532	−	0.656532i
$929$	22.5000	−	12.9904i	0.738201	−	0.426201i	−0.0832138	−	0.996532i	$-0.526518\pi$
$929$	22.5000	−	12.9904i	0.738201	−	0.426201i	0.821415	+	0.570331i	$0.193185\pi$
$930$		0			0
$931$	−45.0333	+	18.0000i	−1.47591	+	0.589926i
$932$	24.2487	+	14.0000i	0.794293	+	0.458585i
$933$		0			0
$934$	−28.3923	−	7.60770i	−0.929025	−	0.248931i
$935$		−	6.00000i		−	0.196221i
$936$		0			0
$937$			38.1051i			1.24484i	0.782683	+	0.622420i	$0.213850\pi$
$937$			38.1051i			1.24484i	−0.782683	+	0.622420i	$0.786150\pi$
$938$	46.4711	+	31.4711i	1.51734	+	1.02757i
$939$		0			0
$940$	9.00000	−	15.5885i	0.293548	−	0.508439i
$941$	−4.50000	−	2.59808i	−0.146696	−	0.0846949i	0.424856	−	0.905261i	$-0.360325\pi$
$941$	−4.50000	−	2.59808i	−0.146696	−	0.0846949i	−0.571551	+	0.820566i	$0.693658\pi$
$942$		0			0
$943$	−21.6506	−	37.5000i	−0.705042	−	1.22117i
$944$	−34.6410			−1.12747
$945$		0			0
$946$	−1.00000	+	1.00000i	−0.0325128	+	0.0325128i
$947$	30.3109	−	17.5000i	0.984972	−	0.568674i	0.0812041	−	0.996697i	$-0.474123\pi$
$947$	30.3109	−	17.5000i	0.984972	−	0.568674i	0.903767	+	0.428024i	$0.140790\pi$
$948$		0			0
$949$	−9.00000	+	15.5885i	−0.292152	+	0.506023i
$950$	−5.07180	−	18.9282i	−0.164551	−	0.614112i
$951$		0			0
$952$	−14.5359	+	21.4641i	−0.471111	+	0.695656i
$953$	40.0000			1.29573			0.647864	−	0.761756i	$-0.275663\pi$
$953$	40.0000			1.29573			0.647864	+	0.761756i	$0.275663\pi$
$954$		0			0
$955$	8.66025			0.280239
$956$	25.0000	+	43.3013i	0.808558	+	1.40046i
$957$		0			0
$958$	−12.0455	−	44.9545i	−0.389173	−	1.45241i
$959$	−8.66025	−	10.0000i	−0.279654	−	0.322917i
$960$		0			0
$961$	−22.0000	−	38.1051i	−0.709677	−	1.22920i
$962$		0			0
$963$		0			0
$964$	−58.8897			−1.89671
$965$	−13.5000	+	7.79423i	−0.434580	+	0.250905i
$966$		0			0
$967$	12.9904	+	7.50000i	0.417742	+	0.241184i	0.694111	−	0.719868i	$-0.255798\pi$
$967$	12.9904	+	7.50000i	0.417742	+	0.241184i	−0.276368	+	0.961052i	$0.589131\pi$
$968$	27.3205	−	7.32051i	0.878114	−	0.235290i
$969$		0			0
$970$	12.2942	+	3.29423i	0.394744	+	0.105771i
$971$	3.46410			0.111168			0.0555842	−	0.998454i	$-0.482298\pi$
$971$	3.46410			0.111168			0.0555842	+	0.998454i	$0.482298\pi$
$972$		0			0
$973$	−7.50000	+	21.6506i	−0.240439	+	0.694087i
$974$	−3.66025	+	13.6603i	−0.117282	+	0.437703i
$975$		0			0
$976$	−18.0000	−	10.3923i	−0.576166	−	0.332650i
$977$	−2.50000	+	4.33013i	−0.0799821	+	0.138533i	−0.903242	−	0.429132i	$-0.858820\pi$
$977$	−2.50000	+	4.33013i	−0.0799821	+	0.138533i	0.823260	+	0.567665i	$0.192153\pi$
$978$		0			0
$979$	−8.66025	−	15.0000i	−0.276783	−	0.479402i
$980$	19.0526	+	15.0000i	0.608612	+	0.479157i
$981$		0			0
$982$	−19.0000	−	19.0000i	−0.606314	−	0.606314i
$983$	−0.866025	−	1.50000i	−0.0276219	−	0.0478426i	0.851884	−	0.523731i	$-0.175460\pi$
$983$	−0.866025	−	1.50000i	−0.0276219	−	0.0478426i	−0.879506	+	0.475888i	$0.842127\pi$
$984$		0			0
$985$	−3.00000	−	1.73205i	−0.0955879	−	0.0551877i
$986$	23.6603	−	6.33975i	0.753496	−	0.201899i
$987$		0			0
$988$	−12.0000	−	20.7846i	−0.381771	−	0.661247i
$989$	−5.00000			−0.158991
$990$		0			0
$991$			34.0000i			1.08005i	0.841650	+	0.540023i	$0.181584\pi$
$991$			34.0000i			1.08005i	−0.841650	+	0.540023i	$0.818416\pi$
$992$	47.3205	+	12.6795i	1.50243	+	0.402574i
$993$		0			0
$994$	−1.07180	+	14.9282i	−0.0339953	+	0.473494i
$995$	25.9808	+	15.0000i	0.823646	+	0.475532i
$996$		0			0
$997$	22.5000	−	12.9904i	0.712582	−	0.411409i	−0.0994342	−	0.995044i	$-0.531703\pi$
$997$	22.5000	−	12.9904i	0.712582	−	0.411409i	0.812016	+	0.583635i	$0.198370\pi$
$998$	7.00000	+	7.00000i	0.221581	+	0.221581i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.bi.b.307.1		4
3.2	odd	2		252.2.bi.a.139.2	yes	4
4.3	odd	2	inner	756.2.bi.b.307.2		4
7.6	odd	2		756.2.bi.a.307.1		4
9.2	odd	6		252.2.bi.b.223.1	yes	4
9.7	even	3		756.2.bi.a.559.2		4
12.11	even	2		252.2.bi.a.139.1	✓	4
21.20	even	2		252.2.bi.b.139.2	yes	4
28.27	even	2		756.2.bi.a.307.2		4
36.7	odd	6		756.2.bi.a.559.1		4
36.11	even	6		252.2.bi.b.223.2	yes	4
63.20	even	6		252.2.bi.a.223.1	yes	4
63.34	odd	6	inner	756.2.bi.b.559.2		4
84.83	odd	2		252.2.bi.b.139.1	yes	4
252.83	odd	6		252.2.bi.a.223.2	yes	4
252.223	even	6	inner	756.2.bi.b.559.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.bi.a.139.1	✓	4	12.11	even	2
252.2.bi.a.139.2	yes	4	3.2	odd	2
252.2.bi.a.223.1	yes	4	63.20	even	6
252.2.bi.a.223.2	yes	4	252.83	odd	6
252.2.bi.b.139.1	yes	4	84.83	odd	2
252.2.bi.b.139.2	yes	4	21.20	even	2
252.2.bi.b.223.1	yes	4	9.2	odd	6
252.2.bi.b.223.2	yes	4	36.11	even	6
756.2.bi.a.307.1		4	7.6	odd	2
756.2.bi.a.307.2		4	28.27	even	2
756.2.bi.a.559.1		4	36.7	odd	6
756.2.bi.a.559.2		4	9.7	even	3
756.2.bi.b.307.1		4	1.1	even	1	trivial
756.2.bi.b.307.2		4	4.3	odd	2	inner
756.2.bi.b.559.1		4	252.223	even	6	inner
756.2.bi.b.559.2		4	63.34	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.bi (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(1.50000 + 0.866025i) q^{5} +(-2.59808 + 0.500000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(1.50000 + 0.866025i) q^{5} +(-2.59808 + 0.500000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(-1.73205 - 1.73205i) q^{10} +(-0.866025 + 0.500000i) q^{11} +(1.50000 + 0.866025i) q^{13} +(3.73205 + 0.267949i) q^{14} +(2.00000 + 3.46410i) q^{16} +3.46410i q^{17} -6.92820 q^{19} +(1.73205 + 3.00000i) q^{20} +(1.36603 - 0.366025i) q^{22} +(4.33013 + 2.50000i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(-1.73205 - 1.73205i) q^{26} +(-5.00000 - 1.73205i) q^{28} +(2.50000 + 4.33013i) q^{29} +(-4.33013 + 7.50000i) q^{31} +(-1.46410 - 5.46410i) q^{32} +(1.26795 - 4.73205i) q^{34} +(-4.33013 - 1.50000i) q^{35} +(9.46410 + 2.53590i) q^{38} +(-1.26795 - 4.73205i) q^{40} +(-7.50000 - 4.33013i) q^{41} +(-0.866025 + 0.500000i) q^{43} -2.00000 q^{44} +(-5.00000 - 5.00000i) q^{46} +(-2.59808 - 4.50000i) q^{47} +(6.50000 - 2.59808i) q^{49} +(0.732051 + 2.73205i) q^{50} +(1.73205 + 3.00000i) q^{52} -4.00000 q^{53} -1.73205 q^{55} +(6.19615 + 4.19615i) q^{56} +(-1.83013 - 6.83013i) q^{58} +(-4.33013 + 7.50000i) q^{59} +(-4.50000 + 2.59808i) q^{61} +(8.66025 - 8.66025i) q^{62} +8.00000i q^{64} +(1.50000 + 2.59808i) q^{65} +(12.9904 + 7.50000i) q^{67} +(-3.46410 + 6.00000i) q^{68} +(5.36603 + 3.63397i) q^{70} +4.00000i q^{71} +10.3923i q^{73} +(-12.0000 - 6.92820i) q^{76} +(2.00000 - 1.73205i) q^{77} +(-2.59808 + 1.50000i) q^{79} +6.92820i q^{80} +(8.66025 + 8.66025i) q^{82} +(0.866025 + 1.50000i) q^{83} +(-3.00000 + 5.19615i) q^{85} +(1.36603 - 0.366025i) q^{86} +(2.73205 + 0.732051i) q^{88} +17.3205i q^{89} +(-4.33013 - 1.50000i) q^{91} +(5.00000 + 8.66025i) q^{92} +(1.90192 + 7.09808i) q^{94} +(-10.3923 - 6.00000i) q^{95} +(-4.50000 + 2.59808i) q^{97} +(-9.83013 + 1.16987i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8	\( 4 q - 2 q^{2} + 6 q^{5} - 8 q^{8} + 6 q^{13} + 8 q^{14} + 8 q^{16} + 2 q^{22} - 4 q^{25} - 20 q^{28} + 10 q^{29} + 8 q^{32} + 12 q^{34} + 24 q^{38} - 12 q^{40} - 30 q^{41} - 8 q^{44} - 20 q^{46} + 26 q^{49} - 4 q^{50} - 16 q^{53} + 4 q^{56} + 10 q^{58} - 18 q^{61} + 6 q^{65} + 18 q^{70} - 48 q^{76} + 8 q^{77} - 12 q^{85} + 2 q^{86} + 4 q^{88} + 20 q^{92} + 18 q^{94} - 18 q^{97} - 22 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 6 * q^5 - 8 * q^8 + 6 * q^13 + 8 * q^14 + 8 * q^16 + 2 * q^22 - 4 * q^25 - 20 * q^28 + 10 * q^29 + 8 * q^32 + 12 * q^34 + 24 * q^38 - 12 * q^40 - 30 * q^41 - 8 * q^44 - 20 * q^46 + 26 * q^49 - 4 * q^50 - 16 * q^53 + 4 * q^56 + 10 * q^58 - 18 * q^61 + 6 * q^65 + 18 * q^70 - 48 * q^76 + 8 * q^77 - 12 * q^85 + 2 * q^86 + 4 * q^88 + 20 * q^92 + 18 * q^94 - 18 * q^97 - 22 * q^98

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