LMFDB - Embedded newform 756.2.bi.a.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.a.559.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(-1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})$	\(q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(-1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(1.73205 + 1.73205i) q^{10} +(-0.866025 + 0.500000i) q^{11} +(-1.50000 - 0.866025i) q^{13} +(3.09808 - 2.09808i) q^{14} +(2.00000 + 3.46410i) q^{16} -3.46410i q^{17} +6.92820 q^{19} +(-1.73205 - 3.00000i) q^{20} +(1.36603 - 0.366025i) q^{22} +(4.33013 + 2.50000i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(1.73205 + 1.73205i) q^{26} +(-5.00000 + 1.73205i) q^{28} +(2.50000 + 4.33013i) q^{29} +(4.33013 - 7.50000i) q^{31} +(-1.46410 - 5.46410i) q^{32} +(-1.26795 + 4.73205i) q^{34} +(4.33013 - 1.50000i) q^{35} +(-9.46410 - 2.53590i) q^{38} +(1.26795 + 4.73205i) q^{40} +(7.50000 + 4.33013i) q^{41} +(-0.866025 + 0.500000i) q^{43} -2.00000 q^{44} +(-5.00000 - 5.00000i) q^{46} +(2.59808 + 4.50000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(0.732051 + 2.73205i) q^{50} +(-1.73205 - 3.00000i) q^{52} -4.00000 q^{53} +1.73205 q^{55} +(7.46410 - 0.535898i) q^{56} +(-1.83013 - 6.83013i) q^{58} +(4.33013 - 7.50000i) q^{59} +(4.50000 - 2.59808i) q^{61} +(-8.66025 + 8.66025i) q^{62} +8.00000i q^{64} +(1.50000 + 2.59808i) q^{65} +(12.9904 + 7.50000i) q^{67} +(3.46410 - 6.00000i) q^{68} +(-6.46410 + 0.464102i) q^{70} +4.00000i q^{71} -10.3923i q^{73} +(12.0000 + 6.92820i) q^{76} +(0.500000 - 2.59808i) q^{77} +(-2.59808 + 1.50000i) q^{79} -6.92820i q^{80} +(-8.66025 - 8.66025i) q^{82} +(-0.866025 - 1.50000i) q^{83} +(-3.00000 + 5.19615i) q^{85} +(1.36603 - 0.366025i) q^{86} +(2.73205 + 0.732051i) q^{88} -17.3205i q^{89} +(4.33013 - 1.50000i) q^{91} +(5.00000 + 8.66025i) q^{92} +(-1.90192 - 7.09808i) q^{94} +(-10.3923 - 6.00000i) q^{95} +(4.50000 - 2.59808i) q^{97} +(-1.16987 + 9.83013i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 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} - 4 q^{49} - 4 q^{50} - 16 q^{53} + 16 q^{56} + 10 q^{58} + 18 q^{61} + 6 q^{65} - 12 q^{70} + 48 q^{76} + 2 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 + 2 * 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 - 4 * q^49 - 4 * q^50 - 16 * q^53 + 16 * q^56 + 10 * q^58 + 18 * q^61 + 6 * q^65 - 12 * q^70 + 48 * q^76 + 2 * 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	0.125567	−	0.992085i	$-0.459925\pi$
$5$	−1.50000	−	0.866025i	−0.670820	−	0.387298i	−0.796387	+	0.604787i	$0.793258\pi$
$6$		0			0
$7$	−1.73205	+	2.00000i	−0.654654	+	0.755929i
$7$	−1.73205	+	2.00000i	−0.654654	+	0.755929i
$8$	−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	0.277350	−	0.960769i	$-0.410544\pi$
$13$	−1.50000	−	0.866025i	−0.416025	−	0.240192i	−0.693375	+	0.720577i	$0.743877\pi$
$14$	3.09808	−	2.09808i	0.827996	−	0.560734i
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		0			0
$19$	6.92820			1.58944			0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820			1.58944			0.794719	+	0.606977i	$0.207618\pi$
$20$	−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	$-0.549713\pi$
$31$	4.33013	−	7.50000i	0.777714	−	1.34704i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	−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	0.953987	−	0.299849i	$-0.0969363\pi$
$41$	7.50000	+	4.33013i	1.17130	+	0.676252i	0.217317	+	0.976101i	$0.430270\pi$
$42$		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	0.990912	−	0.134509i	$-0.0429456\pi$
$47$	2.59808	+	4.50000i	0.378968	+	0.656392i	−0.611944	+	0.790901i	$0.709612\pi$
$48$		0			0
$49$	−1.00000	−	6.92820i	−0.142857	−	0.989743i
$50$	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
$56$	7.46410	−	0.535898i	0.997433	−	0.0716124i
$57$		0			0
$58$	−1.83013	−	6.83013i	−0.240307	−	0.896840i
$59$	4.33013	−	7.50000i	0.563735	−	0.976417i	−0.433432	−	0.901186i	$-0.642697\pi$
$59$	4.33013	−	7.50000i	0.563735	−	0.976417i	0.997166	−	0.0752304i	$-0.0239692\pi$
$60$		0			0
$61$	4.50000	−	2.59808i	0.576166	−	0.332650i	−0.183442	−	0.983030i	$-0.558724\pi$
$61$	4.50000	−	2.59808i	0.576166	−	0.332650i	0.759608	+	0.650381i	$0.225391\pi$
$62$	−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
$70$	−6.46410	+	0.464102i	−0.772608	+	0.0554708i
$71$			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	$-0.791906\pi$
$73$		−	10.3923i		−	1.21633i	0.793812	−	0.608164i	$-0.208094\pi$
$74$		0			0
$75$		0			0
$76$	12.0000	+	6.92820i	1.37649	+	0.794719i
$77$	0.500000	−	2.59808i	0.0569803	−	0.296078i
$78$		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	0.814574	−	0.580059i	$-0.196971\pi$
$83$	−0.866025	−	1.50000i	−0.0950586	−	0.164646i	−0.909633	+	0.415413i	$0.863637\pi$
$84$		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	$-0.629815\pi$
$89$		−	17.3205i		−	1.83597i	0.396615	−	0.917985i	$-0.370185\pi$
$90$		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	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	4.50000	−	2.59808i	0.456906	−	0.263795i	0.710742	+	0.703452i	$0.248359\pi$
$98$	−1.16987	+	9.83013i	−0.118175	+	0.992993i
$99$		0			0
$100$		−	4.00000i		−	0.400000i
$101$	10.5000	−	6.06218i	1.04479	−	0.603209i	0.123603	−	0.992332i	$-0.460555\pi$
$101$	10.5000	−	6.06218i	1.04479	−	0.603209i	0.921186	+	0.389123i	$0.127222\pi$
$102$		0			0
$103$	−4.33013	+	7.50000i	−0.426660	+	0.738997i	−0.996574	−	0.0827075i	$-0.973643\pi$
$103$	−4.33013	+	7.50000i	−0.426660	+	0.738997i	0.569914	+	0.821705i	$0.306977\pi$
$104$	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
$112$	−10.3923	−	2.00000i	−0.981981	−	0.188982i
$113$	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
$119$	6.92820	+	6.00000i	0.635107	+	0.550019i
$120$		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	$0.344345\pi$
$131$	−6.06218	+	10.5000i	−0.529655	+	0.917389i	−0.999402	+	0.0345880i	$0.988988\pi$
$132$		0			0
$133$	−12.0000	+	13.8564i	−1.04053	+	1.20150i
$134$	−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	−0.989139	−	0.146985i	$-0.953043\pi$
$139$	−4.33013	+	7.50000i	−0.367277	+	0.636142i	0.621862	+	0.783127i	$0.286376\pi$
$140$	9.00000	+	1.73205i	0.760639	+	0.146385i
$141$		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
$154$	−1.63397	+	3.36603i	−0.131669	+	0.271242i
$155$	−12.9904	+	7.50000i	−1.04341	+	0.602414i
$156$		0			0
$157$	−4.50000	−	2.59808i	−0.359139	−	0.207349i	0.309564	−	0.950879i	$-0.399817\pi$
$157$	−4.50000	−	2.59808i	−0.359139	−	0.207349i	−0.668703	+	0.743530i	$0.733150\pi$
$158$	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	−0.999376	−	0.0353139i	$-0.988757\pi$
$167$	−6.06218	+	10.5000i	−0.469105	+	0.812514i	0.530271	+	0.847828i	$0.322090\pi$
$168$		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	−0.757235	−	0.653143i	$-0.773450\pi$
$173$	−7.50000	+	4.33013i	−0.570214	+	0.329213i	0.187021	+	0.982356i	$0.440117\pi$
$174$		0			0
$175$	5.19615	+	1.00000i	0.392792	+	0.0755929i
$176$	−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	$0.222606\pi$
$181$			17.3205i			1.28742i	−0.765268	+	0.643712i	$0.777394\pi$
$182$	−6.46410	+	0.464102i	−0.479151	+	0.0344015i
$183$		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
$196$	5.19615	−	13.0000i	0.371154	−	0.928571i
$197$	−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	$-0.710404\pi$
$199$	−17.3205			−1.22782			−0.613909	+	0.789377i	$0.710404\pi$
$200$	−1.46410	+	5.46410i	−0.103528	+	0.386370i
$201$		0			0
$202$	−16.5622	+	4.43782i	−1.16531	+	0.312244i
$203$	−12.9904	−	2.50000i	−0.911746	−	0.175466i
$204$		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	0.893565	−	0.448935i	$-0.148196\pi$
$223$	0.866025	+	1.50000i	0.0579934	+	0.100447i	−0.835571	+	0.549382i	$0.814863\pi$
$224$	13.4641	+	6.53590i	0.899608	+	0.436698i
$225$		0			0
$226$	−19.0000	+	19.0000i	−1.26386	+	1.26386i
$227$	−2.59808	−	4.50000i	−0.172440	−	0.298675i	0.766832	−	0.641848i	$-0.221832\pi$
$227$	−2.59808	−	4.50000i	−0.172440	−	0.298675i	−0.939272	+	0.343172i	$0.888499\pi$
$228$		0			0
$229$	1.50000	+	0.866025i	0.0991228	+	0.0572286i	0.548742	−	0.835992i	$-0.315107\pi$
$229$	1.50000	+	0.866025i	0.0991228	+	0.0572286i	−0.449619	+	0.893220i	$0.648440\pi$
$230$	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
$238$	−7.26795	−	10.7321i	−0.471111	−	0.695656i
$239$	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	$-0.269413\pi$
$241$	25.5000	−	14.7224i	1.64260	−	0.948355i	0.979905	−	0.199465i	$-0.0639205\pi$
$242$	10.0000	−	10.0000i	0.642824	−	0.642824i
$243$		0			0
$244$	10.3923			0.665299
$245$	−4.50000	+	11.2583i	−0.287494	+	0.719268i
$246$		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	$-0.962847\pi$
$257$	−25.5000	−	14.7224i	−1.59065	−	0.918360i	−0.597450	−	0.801906i	$-0.703819\pi$
$258$		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
$266$	21.4641	−	14.5359i	1.31605	−	0.891253i
$267$		0			0
$268$	15.0000	+	25.9808i	0.916271	+	1.58703i
$269$			10.3923i			0.633630i	0.948487	+	0.316815i	$0.102613\pi$
$269$			10.3923i			0.633630i	−0.948487	+	0.316815i	$0.897387\pi$
$270$		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
$280$	−11.6603	−	5.66025i	−0.696833	−	0.338265i
$281$	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	−0.999124	−	0.0418500i	$-0.986675\pi$
$283$	−7.79423	+	13.5000i	−0.463319	+	0.802492i	0.535805	+	0.844342i	$0.320008\pi$
$284$	−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	0.982832	−	0.184503i	$-0.0590674\pi$
$293$	22.5000	+	12.9904i	1.31446	+	0.758906i	0.331632	+	0.943409i	$0.392401\pi$
$294$		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
$301$	0.500000	−	2.59808i	0.0288195	−	0.149751i
$302$	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	$-0.629399\pi$
$307$	−13.8564			−0.790827			−0.395413	+	0.918503i	$0.629399\pi$
$308$	3.46410	−	4.00000i	0.197386	−	0.227921i
$309$		0			0
$310$	20.4904	−	5.49038i	1.16378	−	0.311833i
$311$	12.9904	−	22.5000i	0.736617	−	1.27586i	−0.217393	−	0.976084i	$-0.569755\pi$
$311$	12.9904	−	22.5000i	0.736617	−	1.27586i	0.954010	−	0.299774i	$-0.0969112\pi$
$312$		0			0
$313$	7.50000	−	4.33013i	0.423925	−	0.244753i	−0.272830	−	0.962062i	$-0.587960\pi$
$313$	7.50000	−	4.33013i	0.423925	−	0.244753i	0.696755	+	0.717309i	$0.254626\pi$
$314$	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
$322$	18.6603	−	1.33975i	1.03990	−	0.0746611i
$323$		−	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
$329$	−13.5000	−	2.59808i	−0.744279	−	0.143237i
$330$		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	−0.961530	−	0.274700i	$-0.911421\pi$
$349$	−22.5000	+	12.9904i	−1.20440	+	0.695359i	−0.242867	+	0.970059i	$0.578088\pi$
$350$	−6.73205	−	3.26795i	−0.359843	−	0.174679i
$351$		0			0
$352$	4.00000	+	4.00000i	0.213201	+	0.213201i
$353$	7.50000	−	4.33013i	0.399185	−	0.230469i	−0.286947	−	0.957946i	$-0.592641\pi$
$353$	7.50000	−	4.33013i	0.399185	−	0.230469i	0.686132	+	0.727477i	$0.259307\pi$
$354$		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
$364$	9.00000	+	1.73205i	0.471728	+	0.0907841i
$365$	−9.00000	+	15.5885i	−0.471082	+	0.815937i
$366$		0			0
$367$	−11.2583	−	19.5000i	−0.587680	−	1.01789i	−0.994535	−	0.104399i	$-0.966708\pi$
$367$	−11.2583	−	19.5000i	−0.587680	−	1.01789i	0.406855	−	0.913493i	$-0.366625\pi$
$368$			20.0000i			1.04257i
$369$		0			0
$370$		0			0
$371$	6.92820	−	8.00000i	0.359694	−	0.415339i
$372$		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	−0.843051	−	0.537833i	$-0.819243\pi$
$383$	0.866025	−	1.50000i	0.0442518	−	0.0766464i	0.887303	+	0.461187i	$0.152576\pi$
$384$		0			0
$385$	−3.00000	+	3.46410i	−0.152894	+	0.176547i
$386$	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
$392$	−11.8564	+	15.8564i	−0.598839	+	0.800869i
$393$		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	$0.143126\pi$
$397$			17.3205i			0.869291i	−0.900602	+	0.434646i	$0.856874\pi$
$398$	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
$406$	16.8301	+	8.16987i	0.835265	+	0.405464i
$407$		0			0
$408$		0			0
$409$	13.5000	+	7.79423i	0.667532	+	0.385400i	0.795141	−	0.606425i	$-0.207397\pi$
$409$	13.5000	+	7.79423i	0.667532	+	0.385400i	−0.127609	+	0.991825i	$0.540730\pi$
$410$	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	$-0.614489\pi$
$419$	12.9904	−	22.5000i	0.634622	−	1.09920i	0.986595	−	0.163187i	$-0.0521774\pi$
$420$		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
$427$	−2.59808	+	13.5000i	−0.125730	+	0.653311i
$428$	−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	$-0.863367\pi$
$433$		−	17.3205i		−	0.832370i	0.909280	−	0.416185i	$-0.136633\pi$
$434$	−2.32051	−	32.3205i	−0.111388	−	1.55143i
$435$		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	0.998669	−	0.0515804i	$-0.0164258\pi$
$439$	9.52628	+	16.5000i	0.454665	+	0.787502i	−0.544004	+	0.839082i	$0.683092\pi$
$440$	−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
$448$	−16.0000	−	13.8564i	−0.755929	−	0.654654i
$449$	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
$455$	−7.79423	−	1.50000i	−0.365399	−	0.0703211i
$456$		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	−0.724174	−	0.689618i	$-0.757779\pi$
$461$	−10.5000	+	6.06218i	−0.489034	+	0.282344i	0.235140	+	0.971962i	$0.424445\pi$
$462$		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	$-0.659689\pi$
$467$	−20.7846			−0.961797			−0.480899	+	0.876776i	$0.659689\pi$
$468$		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	$-0.895841\pi$
$479$	−16.4545	−	28.5000i	−0.751825	−	1.30220i	0.195113	−	0.980781i	$-0.437493\pi$
$480$		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
$490$	10.2679	−	13.7321i	0.463859	−	0.620351i
$491$	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
$497$	−8.00000	−	6.92820i	−0.358849	−	0.310772i
$498$		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	$-0.373808\pi$
$503$	17.3205			0.772283			0.386142	+	0.922440i	$0.373808\pi$
$504$		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	−0.0898481	−	0.995955i	$-0.528638\pi$
$509$	−22.5000	−	12.9904i	−0.997295	−	0.575789i	−0.907447	+	0.420167i	$0.861972\pi$
$510$		0			0
$511$	20.7846	+	18.0000i	0.919457	+	0.796273i
$512$	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	$-0.322131\pi$
$523$	24.2487			1.06032			0.530161	+	0.847897i	$0.322131\pi$
$524$	−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
$553$	1.50000	−	7.79423i	0.0637865	−	0.331444i
$554$	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
$560$	13.8564	+	12.0000i	0.585540	+	0.507093i
$561$		0			0
$562$	−0.366025	−	1.36603i	−0.0154398	−	0.0576223i
$563$	−0.866025	+	1.50000i	−0.0364986	+	0.0632175i	−0.883698	−	0.468058i	$-0.844954\pi$
$563$	−0.866025	+	1.50000i	−0.0364986	+	0.0632175i	0.847199	+	0.531276i	$0.178287\pi$
$564$		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
$574$	32.3205	−	2.32051i	1.34903	−	0.0968561i
$575$		−	10.0000i		−	0.417029i
$576$		0			0
$577$		−	38.1051i		−	1.58634i	−0.609002	−	0.793168i	$-0.708430\pi$
$577$		−	38.1051i		−	1.58634i	0.609002	−	0.793168i	$-0.291570\pi$
$578$	−6.83013	−	1.83013i	−0.284096	−	0.0761232i
$579$		0			0
$580$	8.66025	−	15.0000i	0.359597	−	0.622841i
$581$	4.50000	+	0.866025i	0.186691	+	0.0359288i
$582$		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	$0.0134629\pi$
$587$	12.9904	+	22.5000i	0.536170	+	0.928674i	−0.462935	+	0.886392i	$0.653204\pi$
$588$		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	$-0.778876\pi$
$593$		−	31.1769i		−	1.28028i	0.768257	−	0.640141i	$-0.221124\pi$
$594$		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	−0.405405	−	0.914137i	$-0.632869\pi$
$601$	4.50000	−	2.59808i	0.183559	−	0.105978i	0.588964	+	0.808160i	$0.299536\pi$
$602$	−1.63397	+	3.36603i	−0.0665958	+	0.137189i
$603$		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	−0.940422	−	0.340009i	$-0.889570\pi$
$607$	−4.33013	+	7.50000i	−0.175754	+	0.304416i	0.764668	+	0.644425i	$0.222903\pi$
$608$	−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
$616$	−6.19615	+	4.19615i	−0.249650	+	0.169068i
$617$	−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	0.765787	−	0.643094i	$-0.222350\pi$
$619$	−4.33013	−	7.50000i	−0.174042	−	0.301450i	−0.939829	+	0.341644i	$0.889016\pi$
$620$	−30.0000			−1.20483
$621$		0			0
$622$	−25.9808	+	25.9808i	−1.04173	+	1.04173i
$623$	34.6410	+	30.0000i	1.38786	+	1.20192i
$624$		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
$637$	−4.50000	+	11.2583i	−0.178296	+	0.446071i
$638$	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	−0.848444	−	0.529285i	$-0.822460\pi$
$643$	0.866025	−	1.50000i	0.0341527	−	0.0591542i	0.882597	+	0.470131i	$0.155793\pi$
$644$	−25.9808	−	5.00000i	−1.02379	−	0.197028i
$645$		0			0
$646$	−8.78461	+	32.7846i	−0.345626	+	1.28989i
$647$	−34.6410			−1.36188			−0.680939	−	0.732340i	$-0.738428\pi$
$647$	−34.6410			−1.36188			−0.680939	+	0.732340i	$0.738428\pi$
$648$		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
$658$	17.4904	+	8.49038i	0.681846	+	0.330990i
$659$	−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	0.638716	−	0.769443i	$-0.279466\pi$
$661$	7.50000	+	4.33013i	0.291716	+	0.168422i	−0.346999	+	0.937865i	$0.612799\pi$
$662$	−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	−0.411027	−	0.911623i	$-0.634830\pi$
$677$	4.50000	−	2.59808i	0.172949	−	0.0998522i	0.583976	+	0.811771i	$0.301496\pi$
$678$		0			0
$679$	−2.59808	+	13.5000i	−0.0997050	+	0.518082i
$680$	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
$686$	−17.6340	−	19.3660i	−0.673268	−	0.739398i
$687$		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	0.936558	−	0.350513i	$-0.113993\pi$
$691$	4.33013	+	7.50000i	0.164726	+	0.285313i	−0.771832	+	0.635826i	$0.780659\pi$
$692$	−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
$700$	8.00000	+	6.92820i	0.302372	+	0.261861i
$701$	−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
$707$	−6.06218	+	31.5000i	−0.227992	+	1.18468i
$708$		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	$-0.276463\pi$
$719$	34.6410			1.29189			0.645946	+	0.763383i	$0.276463\pi$
$720$		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	$-0.982807\pi$
$727$	−14.7224	−	25.5000i	−0.546025	−	0.945743i	0.452517	−	0.891756i	$-0.350526\pi$
$728$	−11.6603	−	5.66025i	−0.432158	−	0.209783i
$729$		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	0.0945954	−	0.995516i	$-0.469844\pi$
$733$	−19.5000	−	11.2583i	−0.720249	−	0.415836i	−0.814844	+	0.579680i	$0.803178\pi$
$734$	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
$742$	−12.3923	+	8.39230i	−0.454936	+	0.308091i
$743$	−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
$749$	−4.00000	−	3.46410i	−0.146157	−	0.126576i
$750$		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	0.809903	−	0.586564i	$-0.199520\pi$
$761$	19.5000	+	11.2583i	0.706874	+	0.408114i	−0.103028	+	0.994678i	$0.532853\pi$
$762$		0			0
$763$	−20.7846	+	24.0000i	−0.752453	+	0.868858i
$764$	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	0.0360609	−	0.999350i	$-0.488519\pi$
$769$	−22.5000	−	12.9904i	−0.811371	−	0.468445i	−0.847432	+	0.530904i	$0.821852\pi$
$770$	5.36603	−	3.63397i	0.193378	−	0.130959i
$771$		0			0
$772$	−15.5885	+	9.00000i	−0.561041	+	0.323917i
$773$			24.2487i			0.872166i	0.899907	+	0.436083i	$0.143635\pi$
$773$			24.2487i			0.872166i	−0.899907	+	0.436083i	$0.856365\pi$
$774$		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	−0.737517	−	0.675329i	$-0.764002\pi$
$787$	6.06218	−	10.5000i	0.216093	−	0.374285i	0.953610	+	0.301044i	$0.0973351\pi$
$788$	−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	0.418179	−	0.908365i	$-0.362668\pi$
$797$	−4.50000	−	2.59808i	−0.159398	−	0.0920286i	−0.577577	+	0.816336i	$0.696002\pi$
$798$		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
$805$	22.5000	+	4.33013i	0.793021	+	0.152617i
$806$	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	$-0.381095\pi$
$811$	20.7846			0.729846			0.364923	+	0.931038i	$0.381095\pi$
$812$	−20.0000	−	17.3205i	−0.701862	−	0.607831i
$813$		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
$826$	−2.32051	−	32.3205i	−0.0807408	−	1.12457i
$827$		−	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	$-0.942238\pi$
$829$		−	10.3923i		−	0.360940i	0.983581	−	0.180470i	$-0.0577618\pi$
$830$	1.09808	−	4.09808i	0.0381148	−	0.142246i
$831$		0			0
$832$	6.92820	−	12.0000i	0.240192	−	0.416025i
$833$	−24.0000	+	3.46410i	−0.831551	+	0.120024i
$834$		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	0.850689	−	0.525669i	$-0.176185\pi$
$839$	−0.866025	−	1.50000i	−0.0298985	−	0.0517858i	−0.880587	+	0.473884i	$0.842852\pi$
$840$		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	0.224952	−	0.974370i	$-0.427777\pi$
$853$	34.5000	−	19.9186i	1.18126	−	0.681999i	0.956305	+	0.292370i	$0.0944440\pi$
$854$	8.49038	−	17.4904i	0.290535	−	0.598509i
$855$		0			0
$856$	4.00000	−	4.00000i	0.136717	−	0.136717i
$857$	25.5000	−	14.7224i	0.871063	−	0.502909i	0.00336193	−	0.999994i	$-0.498930\pi$
$857$	25.5000	−	14.7224i	0.871063	−	0.502909i	0.867701	+	0.497086i	$0.165597\pi$
$858$		0			0
$859$	18.1865	−	31.5000i	0.620517	−	1.07477i	−0.368873	−	0.929480i	$-0.620256\pi$
$859$	18.1865	−	31.5000i	0.620517	−	1.07477i	0.989390	−	0.145286i	$-0.0464103\pi$
$860$	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
$868$	−8.66025	+	45.0000i	−0.293948	+	1.52740i
$869$	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
$875$	−24.2487	−	21.0000i	−0.819756	−	0.709930i
$876$		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	$-0.823995\pi$
$881$		−	31.1769i		−	1.05038i	0.850986	−	0.525188i	$-0.176005\pi$
$882$		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	−0.929519	−	0.368774i	$-0.879777\pi$
$887$	−4.33013	+	7.50000i	−0.145391	+	0.251825i	0.784127	+	0.620600i	$0.213111\pi$
$888$		0			0
$889$	36.0000	+	31.1769i	1.20740	+	1.04564i
$890$	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
$896$	16.7846	+	24.7846i	0.560734	+	0.827996i
$897$		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
$910$	10.0981	+	4.90192i	0.334748	+	0.162497i
$911$	−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	−0.821415	−	0.570331i	$-0.806815\pi$
$929$	−22.5000	+	12.9904i	−0.738201	+	0.426201i	0.0832138	+	0.996532i	$0.473482\pi$
$930$		0			0
$931$	−6.92820	−	48.0000i	−0.227063	−	1.57314i
$932$	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	$-0.786150\pi$
$937$		−	38.1051i		−	1.24484i	0.782683	−	0.622420i	$-0.213850\pi$
$938$	55.9808	−	4.01924i	1.82784	−	0.131233i
$939$		0			0
$940$	9.00000	−	15.5885i	0.293548	−	0.508439i
$941$	4.50000	+	2.59808i	0.146696	+	0.0846949i	0.571551	−	0.820566i	$-0.306342\pi$
$941$	4.50000	+	2.59808i	0.146696	+	0.0846949i	−0.424856	+	0.905261i	$0.639675\pi$
$942$		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
$952$	−1.85641	−	25.8564i	−0.0601665	−	0.838011i
$953$	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
$959$	−12.9904	−	2.50000i	−0.419481	−	0.0807292i
$960$		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	$-0.517702\pi$
$971$	−3.46410			−0.111168			−0.0555842	+	0.998454i	$0.517702\pi$
$972$		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	0.879506	−	0.475888i	$-0.157873\pi$
$983$	0.866025	+	1.50000i	0.0276219	+	0.0478426i	−0.851884	+	0.523731i	$0.824540\pi$
$984$		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
$994$	8.39230	+	12.3923i	0.266188	+	0.393060i
$995$	25.9808	+	15.0000i	0.823646	+	0.475532i
$996$		0			0
$997$	−22.5000	+	12.9904i	−0.712582	+	0.411409i	−0.812016	−	0.583635i	$-0.801630\pi$
$997$	−22.5000	+	12.9904i	−0.712582	+	0.411409i	0.0994342	+	0.995044i	$0.468297\pi$
$998$	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.a.307.1		4
3.2	odd	2		252.2.bi.b.139.2	yes	4
4.3	odd	2	inner	756.2.bi.a.307.2		4
7.6	odd	2		756.2.bi.b.307.1		4
9.2	odd	6		252.2.bi.a.223.1	yes	4
9.7	even	3		756.2.bi.b.559.2		4
12.11	even	2		252.2.bi.b.139.1	yes	4
21.20	even	2		252.2.bi.a.139.2	yes	4
28.27	even	2		756.2.bi.b.307.2		4
36.7	odd	6		756.2.bi.b.559.1		4
36.11	even	6		252.2.bi.a.223.2	yes	4
63.20	even	6		252.2.bi.b.223.1	yes	4
63.34	odd	6	inner	756.2.bi.a.559.2		4
84.83	odd	2		252.2.bi.a.139.1	✓	4
252.83	odd	6		252.2.bi.b.223.2	yes	4
252.223	even	6	inner	756.2.bi.a.559.1		4

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

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)

\(f(q)\)	\(=\)	\(q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(-1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +O(q^{10})\)	\(q+(-1.36603 - 0.366025i) q^{2} +(1.73205 + 1.00000i) q^{4} +(-1.50000 - 0.866025i) q^{5} +(-1.73205 + 2.00000i) q^{7} +(-2.00000 - 2.00000i) q^{8} +(1.73205 + 1.73205i) q^{10} +(-0.866025 + 0.500000i) q^{11} +(-1.50000 - 0.866025i) q^{13} +(3.09808 - 2.09808i) q^{14} +(2.00000 + 3.46410i) q^{16} -3.46410i q^{17} +6.92820 q^{19} +(-1.73205 - 3.00000i) q^{20} +(1.36603 - 0.366025i) q^{22} +(4.33013 + 2.50000i) q^{23} +(-1.00000 - 1.73205i) q^{25} +(1.73205 + 1.73205i) q^{26} +(-5.00000 + 1.73205i) q^{28} +(2.50000 + 4.33013i) q^{29} +(4.33013 - 7.50000i) q^{31} +(-1.46410 - 5.46410i) q^{32} +(-1.26795 + 4.73205i) q^{34} +(4.33013 - 1.50000i) q^{35} +(-9.46410 - 2.53590i) q^{38} +(1.26795 + 4.73205i) q^{40} +(7.50000 + 4.33013i) q^{41} +(-0.866025 + 0.500000i) q^{43} -2.00000 q^{44} +(-5.00000 - 5.00000i) q^{46} +(2.59808 + 4.50000i) q^{47} +(-1.00000 - 6.92820i) q^{49} +(0.732051 + 2.73205i) q^{50} +(-1.73205 - 3.00000i) q^{52} -4.00000 q^{53} +1.73205 q^{55} +(7.46410 - 0.535898i) q^{56} +(-1.83013 - 6.83013i) q^{58} +(4.33013 - 7.50000i) q^{59} +(4.50000 - 2.59808i) q^{61} +(-8.66025 + 8.66025i) q^{62} +8.00000i q^{64} +(1.50000 + 2.59808i) q^{65} +(12.9904 + 7.50000i) q^{67} +(3.46410 - 6.00000i) q^{68} +(-6.46410 + 0.464102i) q^{70} +4.00000i q^{71} -10.3923i q^{73} +(12.0000 + 6.92820i) q^{76} +(0.500000 - 2.59808i) q^{77} +(-2.59808 + 1.50000i) q^{79} -6.92820i q^{80} +(-8.66025 - 8.66025i) q^{82} +(-0.866025 - 1.50000i) q^{83} +(-3.00000 + 5.19615i) q^{85} +(1.36603 - 0.366025i) q^{86} +(2.73205 + 0.732051i) q^{88} -17.3205i q^{89} +(4.33013 - 1.50000i) q^{91} +(5.00000 + 8.66025i) q^{92} +(-1.90192 - 7.09808i) q^{94} +(-10.3923 - 6.00000i) q^{95} +(4.50000 - 2.59808i) q^{97} +(-1.16987 + 9.83013i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 6 * q^5 - 8 * q^8	\( 4 q - 2 q^{2} - 6 q^{5} - 8 q^{8} - 6 q^{13} + 2 q^{14} + 8 q^{16} + 2 q^{22} - 4 q^{25} - 20 q^{28} + 10 q^{29} + 8 q^{32} - 12 q^{34} - 24 q^{38} + 12 q^{40} + 30 q^{41} - 8 q^{44} - 20 q^{46} - 4 q^{49} - 4 q^{50} - 16 q^{53} + 16 q^{56} + 10 q^{58} + 18 q^{61} + 6 q^{65} - 12 q^{70} + 48 q^{76} + 2 q^{77} - 12 q^{85} + 2 q^{86} + 4 q^{88} + 20 q^{92} - 18 q^{94} + 18 q^{97} - 22 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 6 * q^5 - 8 * q^8 - 6 * q^13 + 2 * 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 - 4 * q^49 - 4 * q^50 - 16 * q^53 + 16 * q^56 + 10 * q^58 + 18 * q^61 + 6 * q^65 - 12 * q^70 + 48 * q^76 + 2 * 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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