LMFDB - Embedded newform 784.2.x.g.373.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [784,2,Mod(165,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.165"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([0, 3, 8])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 784 = 2^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	784.x (of order $12$, degree $4$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,2,4,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$6.26027151847$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label			373.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	784.373
Dual form			784.2.x.g.557.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.366025 - 1.36603i) q^{2} +(2.73205 + 0.732051i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(2.73205 - 0.732051i) q^{5} -4.00000i q^{6} +(2.00000 + 2.00000i) q^{8} +(4.33013 + 2.50000i) q^{9} +(-2.00000 - 3.46410i) q^{10} +(-1.09808 + 4.09808i) q^{11} +(-5.46410 + 1.46410i) q^{12} +8.00000 q^{15} +(2.00000 - 3.46410i) q^{16} +(3.00000 + 5.19615i) q^{17} +(1.83013 - 6.83013i) q^{18} +(-1.46410 - 5.46410i) q^{19} +(-4.00000 + 4.00000i) q^{20} +6.00000 q^{22} +(-1.73205 - 1.00000i) q^{23} +(4.00000 + 6.92820i) q^{24} +(2.59808 - 1.50000i) q^{25} +(4.00000 + 4.00000i) q^{27} +(-1.00000 + 1.00000i) q^{29} +(-2.92820 - 10.9282i) q^{30} +(-2.00000 - 3.46410i) q^{31} +(-5.46410 - 1.46410i) q^{32} +(-6.00000 + 10.3923i) q^{33} +(6.00000 - 6.00000i) q^{34} -10.0000 q^{36} +(-4.09808 + 1.09808i) q^{37} +(-6.92820 + 4.00000i) q^{38} +(6.92820 + 4.00000i) q^{40} +2.00000i q^{41} +(-5.00000 - 5.00000i) q^{43} +(-2.19615 - 8.19615i) q^{44} +(13.6603 + 3.66025i) q^{45} +(-0.732051 + 2.73205i) q^{46} +(4.00000 - 6.92820i) q^{47} +(8.00000 - 8.00000i) q^{48} +(-3.00000 - 3.00000i) q^{50} +(4.39230 + 16.3923i) q^{51} +(2.56218 - 9.56218i) q^{53} +(4.00000 - 6.92820i) q^{54} +12.0000i q^{55} -16.0000i q^{57} +(1.73205 + 1.00000i) q^{58} +(0.732051 - 2.73205i) q^{59} +(-13.8564 + 8.00000i) q^{60} +(2.19615 + 8.19615i) q^{61} +(-4.00000 + 4.00000i) q^{62} +8.00000i q^{64} +(16.3923 + 4.39230i) q^{66} +(6.83013 + 1.83013i) q^{67} +(-10.3923 - 6.00000i) q^{68} +(-4.00000 - 4.00000i) q^{69} -8.00000i q^{71} +(3.66025 + 13.6603i) q^{72} +(-8.66025 + 5.00000i) q^{73} +(3.00000 + 5.19615i) q^{74} +(8.19615 - 2.19615i) q^{75} +(8.00000 + 8.00000i) q^{76} +(7.00000 - 12.1244i) q^{79} +(2.92820 - 10.9282i) q^{80} +(0.500000 + 0.866025i) q^{81} +(2.73205 - 0.732051i) q^{82} +(12.0000 + 12.0000i) q^{85} +(-5.00000 + 8.66025i) q^{86} +(-3.46410 + 2.00000i) q^{87} +(-10.3923 + 6.00000i) q^{88} +(-5.19615 - 3.00000i) q^{89} -20.0000i q^{90} +4.00000 q^{92} +(-2.92820 - 10.9282i) q^{93} +(-10.9282 - 2.92820i) q^{94} +(-8.00000 - 13.8564i) q^{95} +(-13.8564 - 8.00000i) q^{96} -6.00000 q^{97} +(-15.0000 + 15.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{3} + 4 q^{5} + 8 q^{8} - 8 q^{10} + 6 q^{11} - 8 q^{12} + 32 q^{15} + 8 q^{16} + 12 q^{17} - 10 q^{18} + 8 q^{19} - 16 q^{20} + 24 q^{22} + 16 q^{24} + 16 q^{27} - 4 q^{29} + 16 q^{30}+ \cdots - 60 q^{99}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^5 + 8 * q^8 - 8 * q^10 + 6 * q^11 - 8 * q^12 + 32 * q^15 + 8 * q^16 + 12 * q^17 - 10 * q^18 + 8 * q^19 - 16 * q^20 + 24 * q^22 + 16 * q^24 + 16 * q^27 - 4 * q^29 + 16 * q^30 - 8 * q^31 - 8 * q^32 - 24 * q^33 + 24 * q^34 - 40 * q^36 - 6 * q^37 - 20 * q^43 + 12 * q^44 + 20 * q^45 + 4 * q^46 + 16 * q^47 + 32 * q^48 - 12 * q^50 - 24 * q^51 - 14 * q^53 + 16 * q^54 - 4 * q^59 - 12 * q^61 - 16 * q^62 + 24 * q^66 + 10 * q^67 - 16 * q^69 - 20 * q^72 + 12 * q^74 + 12 * q^75 + 32 * q^76 + 28 * q^79 - 16 * q^80 + 2 * q^81 + 4 * q^82 + 48 * q^85 - 20 * q^86 + 16 * q^92 + 16 * q^93 - 16 * q^94 - 32 * q^95 - 24 * q^97 - 60 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/784\mathbb{Z}\right)^\times$.

$n$	$197$	$687$	$689$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$e\left(\frac{1}{3}\right)$

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 $

$2$	−0.366025	−	1.36603i	−0.258819	−	0.965926i
$2$	−0.366025	−	1.36603i	−0.258819	−	0.965926i
$3$	2.73205	+	0.732051i	1.57735	+	0.422650i	0.938104	−	0.346353i	$-0.112580\pi$
$3$	2.73205	+	0.732051i	1.57735	+	0.422650i	0.639246	+	0.769002i	$0.279247\pi$
$4$	−1.73205	+	1.00000i	−0.866025	+	0.500000i
$5$	2.73205	−	0.732051i	1.22181	−	0.327383i	0.410425	−	0.911894i	$-0.365380\pi$
$5$	2.73205	−	0.732051i	1.22181	−	0.327383i	0.811386	+	0.584511i	$0.198714\pi$
$6$		−	4.00000i		−	1.63299i
$7$		0			0
$7$		0			0
$8$	2.00000	+	2.00000i	0.707107	+	0.707107i
$9$	4.33013	+	2.50000i	1.44338	+	0.833333i
$10$	−2.00000	−	3.46410i	−0.632456	−	1.09545i
$11$	−1.09808	+	4.09808i	−0.331082	+	1.23562i	0.576972	+	0.816764i	$0.304234\pi$
$11$	−1.09808	+	4.09808i	−0.331082	+	1.23562i	−0.908054	+	0.418852i	$0.862432\pi$
$12$	−5.46410	+	1.46410i	−1.57735	+	0.422650i
$13$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$13$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$14$		0			0
$15$	8.00000			2.06559
$16$	2.00000	−	3.46410i	0.500000	−	0.866025i
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	0.957892	+	0.287129i	$0.0927008\pi$
$17$	3.00000	+	5.19615i	0.727607	+	1.26025i	−0.230285	+	0.973123i	$0.573966\pi$
$18$	1.83013	−	6.83013i	0.431365	−	1.60988i
$19$	−1.46410	−	5.46410i	−0.335888	−	1.25355i	−0.902903	−	0.429844i	$-0.858569\pi$
$19$	−1.46410	−	5.46410i	−0.335888	−	1.25355i	0.567015	−	0.823707i	$-0.308098\pi$
$20$	−4.00000	+	4.00000i	−0.894427	+	0.894427i
$21$		0			0
$22$	6.00000			1.27920
$23$	−1.73205	−	1.00000i	−0.361158	−	0.208514i	0.308431	−	0.951247i	$-0.400196\pi$
$23$	−1.73205	−	1.00000i	−0.361158	−	0.208514i	−0.669588	+	0.742732i	$0.733529\pi$
$24$	4.00000	+	6.92820i	0.816497	+	1.41421i
$25$	2.59808	−	1.50000i	0.519615	−	0.300000i
$26$		0			0
$27$	4.00000	+	4.00000i	0.769800	+	0.769800i
$28$		0			0
$29$	−1.00000	+	1.00000i	−0.185695	+	0.185695i	−0.793832	−	0.608137i	$-0.791917\pi$
$29$	−1.00000	+	1.00000i	−0.185695	+	0.185695i	0.608137	+	0.793832i	$0.291917\pi$
$30$	−2.92820	−	10.9282i	−0.534614	−	1.99521i
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	0.628619	−	0.777714i	$-0.283621\pi$
$31$	−2.00000	−	3.46410i	−0.359211	−	0.622171i	−0.987829	+	0.155543i	$0.950287\pi$
$32$	−5.46410	−	1.46410i	−0.965926	−	0.258819i
$33$	−6.00000	+	10.3923i	−1.04447	+	1.80907i
$34$	6.00000	−	6.00000i	1.02899	−	1.02899i
$35$		0			0
$36$	−10.0000			−1.66667
$37$	−4.09808	+	1.09808i	−0.673720	+	0.180523i	−0.579430	−	0.815022i	$-0.696725\pi$
$37$	−4.09808	+	1.09808i	−0.673720	+	0.180523i	−0.0942898	+	0.995545i	$0.530058\pi$
$38$	−6.92820	+	4.00000i	−1.12390	+	0.648886i
$39$		0			0
$40$	6.92820	+	4.00000i	1.09545	+	0.632456i
$41$			2.00000i			0.312348i	0.987730	+	0.156174i	$0.0499160\pi$
$41$			2.00000i			0.312348i	−0.987730	+	0.156174i	$0.950084\pi$
$42$		0			0
$43$	−5.00000	−	5.00000i	−0.762493	−	0.762493i	0.214280	−	0.976772i	$-0.431260\pi$
$43$	−5.00000	−	5.00000i	−0.762493	−	0.762493i	−0.976772	+	0.214280i	$0.931260\pi$
$44$	−2.19615	−	8.19615i	−0.331082	−	1.23562i
$45$	13.6603	+	3.66025i	2.03635	+	0.545638i
$46$	−0.732051	+	2.73205i	−0.107935	+	0.402819i
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	−0.411606	−	0.911362i	$-0.635032\pi$
$47$	4.00000	−	6.92820i	0.583460	−	1.01058i	0.995066	−	0.0992202i	$-0.0316348\pi$
$48$	8.00000	−	8.00000i	1.15470	−	1.15470i
$49$		0			0
$50$	−3.00000	−	3.00000i	−0.424264	−	0.424264i
$51$	4.39230	+	16.3923i	0.615046	+	2.29538i
$52$		0			0
$53$	2.56218	−	9.56218i	0.351942	−	1.31347i	−0.532347	−	0.846526i	$-0.678690\pi$
$53$	2.56218	−	9.56218i	0.351942	−	1.31347i	0.884289	−	0.466940i	$-0.154644\pi$
$54$	4.00000	−	6.92820i	0.544331	−	0.942809i
$55$			12.0000i			1.61808i
$56$		0			0
$57$		−	16.0000i		−	2.11925i
$58$	1.73205	+	1.00000i	0.227429	+	0.131306i
$59$	0.732051	−	2.73205i	0.0953049	−	0.355683i	−0.901761	−	0.432236i	$-0.857725\pi$
$59$	0.732051	−	2.73205i	0.0953049	−	0.355683i	0.997066	+	0.0765531i	$0.0243915\pi$
$60$	−13.8564	+	8.00000i	−1.78885	+	1.03280i
$61$	2.19615	+	8.19615i	0.281189	+	1.04941i	0.951580	+	0.307402i	$0.0994596\pi$
$61$	2.19615	+	8.19615i	0.281189	+	1.04941i	−0.670391	+	0.742008i	$0.733874\pi$
$62$	−4.00000	+	4.00000i	−0.508001	+	0.508001i
$63$		0			0
$64$			8.00000i			1.00000i
$65$		0			0
$66$	16.3923	+	4.39230i	2.01775	+	0.540655i
$67$	6.83013	+	1.83013i	0.834433	+	0.223586i	0.650647	−	0.759381i	$-0.274498\pi$
$67$	6.83013	+	1.83013i	0.834433	+	0.223586i	0.183786	+	0.982966i	$0.441165\pi$
$68$	−10.3923	−	6.00000i	−1.26025	−	0.727607i
$69$	−4.00000	−	4.00000i	−0.481543	−	0.481543i
$70$		0			0
$71$		−	8.00000i		−	0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$		−	8.00000i		−	0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$	3.66025	+	13.6603i	0.431365	+	1.60988i
$73$	−8.66025	+	5.00000i	−1.01361	+	0.585206i	−0.912245	−	0.409644i	$-0.865653\pi$
$73$	−8.66025	+	5.00000i	−1.01361	+	0.585206i	−0.101361	+	0.994850i	$0.532320\pi$
$74$	3.00000	+	5.19615i	0.348743	+	0.604040i
$75$	8.19615	−	2.19615i	0.946410	−	0.253590i
$76$	8.00000	+	8.00000i	0.917663	+	0.917663i
$77$		0			0
$78$		0			0
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	−0.139895	−	0.990166i	$-0.544677\pi$
$79$	7.00000	−	12.1244i	0.787562	−	1.36410i	0.927457	−	0.373930i	$-0.121990\pi$
$80$	2.92820	−	10.9282i	0.327383	−	1.22181i
$81$	0.500000	+	0.866025i	0.0555556	+	0.0962250i
$82$	2.73205	−	0.732051i	0.301705	−	0.0808415i
$83$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$83$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$84$		0			0
$85$	12.0000	+	12.0000i	1.30158	+	1.30158i
$86$	−5.00000	+	8.66025i	−0.539164	+	0.933859i
$87$	−3.46410	+	2.00000i	−0.371391	+	0.214423i
$88$	−10.3923	+	6.00000i	−1.10782	+	0.639602i
$89$	−5.19615	−	3.00000i	−0.550791	−	0.317999i	0.198650	−	0.980071i	$-0.436344\pi$
$89$	−5.19615	−	3.00000i	−0.550791	−	0.317999i	−0.749441	+	0.662071i	$0.769678\pi$
$90$		−	20.0000i		−	2.10819i
$91$		0			0
$92$	4.00000			0.417029
$93$	−2.92820	−	10.9282i	−0.303641	−	1.13320i
$94$	−10.9282	−	2.92820i	−1.12716	−	0.302021i
$95$	−8.00000	−	13.8564i	−0.820783	−	1.42164i
$96$	−13.8564	−	8.00000i	−1.41421	−	0.816497i
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$	−15.0000	+	15.0000i	−1.50756	+	1.50756i
$100$	−3.00000	+	5.19615i	−0.300000	+	0.519615i
$101$	0.732051	−	2.73205i	0.0728418	−	0.271849i	−0.919893	−	0.392168i	$-0.871725\pi$
$101$	0.732051	−	2.73205i	0.0728418	−	0.271849i	0.992735	+	0.120319i	$0.0383918\pi$
$102$	20.7846	−	12.0000i	2.05798	−	1.18818i
$103$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$103$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$104$		0			0
$105$		0			0
$106$	−14.0000			−1.35980
$107$	−6.83013	+	1.83013i	−0.660293	+	0.176925i	−0.573378	−	0.819291i	$-0.694367\pi$
$107$	−6.83013	+	1.83013i	−0.660293	+	0.176925i	−0.0869149	+	0.996216i	$0.527701\pi$
$108$	−10.9282	−	2.92820i	−1.05157	−	0.281766i
$109$	4.09808	+	1.09808i	0.392525	+	0.105177i	0.449682	−	0.893189i	$-0.351537\pi$
$109$	4.09808	+	1.09808i	0.392525	+	0.105177i	−0.0571579	+	0.998365i	$0.518204\pi$
$110$	16.3923	−	4.39230i	1.56294	−	0.418790i
$111$	−12.0000			−1.13899
$112$		0			0
$113$	−12.0000			−1.12887			−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000			−1.12887			−0.564433	+	0.825479i	$0.690905\pi$
$114$	−21.8564	+	5.85641i	−2.04704	+	0.548503i
$115$	−5.46410	−	1.46410i	−0.509530	−	0.136528i
$116$	0.732051	−	2.73205i	0.0679692	−	0.253665i
$117$		0			0
$118$	−4.00000			−0.368230
$119$		0			0
$120$	16.0000	+	16.0000i	1.46059	+	1.46059i
$121$	−6.06218	−	3.50000i	−0.551107	−	0.318182i
$122$	10.3923	−	6.00000i	0.940875	−	0.543214i
$123$	−1.46410	+	5.46410i	−0.132014	+	0.492681i
$124$	6.92820	+	4.00000i	0.622171	+	0.359211i
$125$	−4.00000	+	4.00000i	−0.357771	+	0.357771i
$126$		0			0
$127$	8.00000			0.709885			0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000			0.709885			0.354943	+	0.934888i	$0.384500\pi$
$128$	10.9282	−	2.92820i	0.965926	−	0.258819i
$129$	−10.0000	−	17.3205i	−0.880451	−	1.52499i
$130$		0			0
$131$	4.39230	+	16.3923i	0.383757	+	1.43220i	0.840116	+	0.542406i	$0.182487\pi$
$131$	4.39230	+	16.3923i	0.383757	+	1.43220i	−0.456359	+	0.889796i	$0.650847\pi$
$132$		−	24.0000i		−	2.08893i
$133$		0			0
$134$		−	10.0000i		−	0.863868i
$135$	13.8564	+	8.00000i	1.19257	+	0.688530i
$136$	−4.39230	+	16.3923i	−0.376637	+	1.40563i
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.0146279	−	0.999893i	$-0.495344\pi$
$137$	10.3923	−	6.00000i	0.887875	−	0.512615i	0.873247	+	0.487278i	$0.162010\pi$
$138$	−4.00000	+	6.92820i	−0.340503	+	0.589768i
$139$	−12.0000	−	12.0000i	−1.01783	−	1.01783i	−0.999838	−	0.0179885i	$-0.994274\pi$
$139$	−12.0000	−	12.0000i	−1.01783	−	1.01783i	−0.0179885	−	0.999838i	$-0.505726\pi$
$140$		0			0
$141$	16.0000	−	16.0000i	1.34744	−	1.34744i
$142$	−10.9282	+	2.92820i	−0.917074	+	0.245729i
$143$		0			0
$144$	17.3205	−	10.0000i	1.44338	−	0.833333i
$145$	−2.00000	+	3.46410i	−0.166091	+	0.287678i
$146$	10.0000	+	10.0000i	0.827606	+	0.827606i
$147$		0			0
$148$	6.00000	−	6.00000i	0.493197	−	0.493197i
$149$	17.7583	−	4.75833i	1.45482	−	0.389818i	0.557122	−	0.830431i	$-0.311906\pi$
$149$	17.7583	−	4.75833i	1.45482	−	0.389818i	0.897697	+	0.440613i	$0.145239\pi$
$150$	−6.00000	−	10.3923i	−0.489898	−	0.848528i
$151$	−5.19615	+	3.00000i	−0.422857	+	0.244137i	−0.696299	−	0.717752i	$-0.745171\pi$
$151$	−5.19615	+	3.00000i	−0.422857	+	0.244137i	0.273442	+	0.961888i	$0.411838\pi$
$152$	8.00000	−	13.8564i	0.648886	−	1.12390i
$153$			30.0000i			2.42536i
$154$		0			0
$155$	−8.00000	−	8.00000i	−0.642575	−	0.642575i
$156$		0			0
$157$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$157$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$158$	−19.1244	−	5.12436i	−1.52145	−	0.407672i
$159$	14.0000	−	24.2487i	1.11027	−	1.92305i
$160$	−16.0000			−1.26491
$161$		0			0
$162$	1.00000	−	1.00000i	0.0785674	−	0.0785674i
$163$	−1.83013	−	6.83013i	−0.143347	−	0.534977i	−0.999823	−	0.0187913i	$-0.994018\pi$
$163$	−1.83013	−	6.83013i	−0.143347	−	0.534977i	0.856477	−	0.516185i	$-0.172648\pi$
$164$	−2.00000	−	3.46410i	−0.156174	−	0.270501i
$165$	−8.78461	+	32.7846i	−0.683881	+	2.55228i
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$			13.0000i			1.00000i
$170$	12.0000	−	20.7846i	0.920358	−	1.59411i
$171$	7.32051	−	27.3205i	0.559813	−	2.08925i
$172$	13.6603	+	3.66025i	1.04158	+	0.279092i
$173$	5.85641	+	21.8564i	0.445254	+	1.66171i	0.715264	+	0.698854i	$0.246306\pi$
$173$	5.85641	+	21.8564i	0.445254	+	1.66171i	−0.270010	+	0.962858i	$0.587027\pi$
$174$	4.00000	+	4.00000i	0.303239	+	0.303239i
$175$		0			0
$176$	12.0000	+	12.0000i	0.904534	+	0.904534i
$177$	4.00000	−	6.92820i	0.300658	−	0.520756i
$178$	−2.19615	+	8.19615i	−0.164609	+	0.614328i
$179$	−6.83013	−	1.83013i	−0.510508	−	0.136790i	−0.00563529	−	0.999984i	$-0.501794\pi$
$179$	−6.83013	−	1.83013i	−0.510508	−	0.136790i	−0.504872	+	0.863194i	$0.668460\pi$
$180$	−27.3205	+	7.32051i	−2.03635	+	0.545638i
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	0.839962	−	0.542645i	$-0.182577\pi$
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	−0.542645	+	0.839962i	$0.682577\pi$
$182$		0			0
$183$			24.0000i			1.77413i
$184$	−1.46410	−	5.46410i	−0.107935	−	0.402819i
$185$	−10.3923	+	6.00000i	−0.764057	+	0.441129i
$186$	−13.8564	+	8.00000i	−1.01600	+	0.586588i
$187$	−24.5885	+	6.58846i	−1.79809	+	0.481796i
$188$			16.0000i			1.16692i
$189$		0			0
$190$	−16.0000	+	16.0000i	−1.16076	+	1.16076i
$191$	−11.0000	+	19.0526i	−0.795932	+	1.37859i	0.126314	+	0.991990i	$0.459685\pi$
$191$	−11.0000	+	19.0526i	−0.795932	+	1.37859i	−0.922246	+	0.386604i	$0.873648\pi$
$192$	−5.85641	+	21.8564i	−0.422650	+	1.57735i
$193$	12.0000	+	20.7846i	0.863779	+	1.49611i	0.868255	+	0.496119i	$0.165242\pi$
$193$	12.0000	+	20.7846i	0.863779	+	1.49611i	−0.00447566	+	0.999990i	$0.501425\pi$
$194$	2.19615	+	8.19615i	0.157675	+	0.588449i
$195$		0			0
$196$		0			0
$197$	−11.0000	−	11.0000i	−0.783718	−	0.783718i	0.196738	−	0.980456i	$-0.436965\pi$
$197$	−11.0000	−	11.0000i	−0.783718	−	0.783718i	−0.980456	+	0.196738i	$0.936965\pi$
$198$	25.9808	+	15.0000i	1.84637	+	1.06600i
$199$	−17.3205	+	10.0000i	−1.22782	+	0.708881i	−0.966573	−	0.256391i	$-0.917466\pi$
$199$	−17.3205	+	10.0000i	−1.22782	+	0.708881i	−0.261245	+	0.965272i	$0.584133\pi$
$200$	8.19615	+	2.19615i	0.579555	+	0.155291i
$201$	17.3205	+	10.0000i	1.22169	+	0.705346i
$202$	−4.00000			−0.281439
$203$		0			0
$204$	−24.0000	−	24.0000i	−1.68034	−	1.68034i
$205$	1.46410	+	5.46410i	0.102257	+	0.381629i
$206$		0			0
$207$	−5.00000	−	8.66025i	−0.347524	−	0.601929i
$208$		0			0
$209$	24.0000			1.66011
$210$		0			0
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	−0.945425	−	0.325840i	$-0.894353\pi$
$211$	−9.00000	+	9.00000i	−0.619586	+	0.619586i	0.325840	+	0.945425i	$0.394353\pi$
$212$	5.12436	+	19.1244i	0.351942	+	1.31347i
$213$	5.85641	−	21.8564i	0.401274	−	1.49758i
$214$	5.00000	+	8.66025i	0.341793	+	0.592003i
$215$	−17.3205	−	10.0000i	−1.18125	−	0.681994i
$216$			16.0000i			1.08866i
$217$		0			0
$218$		−	6.00000i		−	0.406371i
$219$	−27.3205	+	7.32051i	−1.84615	+	0.494674i
$220$	−12.0000	−	20.7846i	−0.809040	−	1.40130i
$221$		0			0
$222$	4.39230	+	16.3923i	0.294792	+	1.10018i
$223$	4.00000			0.267860			0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000			0.267860			0.133930	+	0.990991i	$0.457240\pi$
$224$		0			0
$225$	15.0000			1.00000
$226$	4.39230	+	16.3923i	0.292172	+	1.09040i
$227$	−5.46410	−	1.46410i	−0.362665	−	0.0971758i	0.0728849	−	0.997340i	$-0.476779\pi$
$227$	−5.46410	−	1.46410i	−0.362665	−	0.0971758i	−0.435550	+	0.900165i	$0.643446\pi$
$228$	16.0000	+	27.7128i	1.05963	+	1.83533i
$229$	−16.3923	+	4.39230i	−1.08323	+	0.290252i	−0.755920	−	0.654664i	$-0.772810\pi$
$229$	−16.3923	+	4.39230i	−1.08323	+	0.290252i	−0.327314	+	0.944916i	$0.606143\pi$
$230$			8.00000i			0.527504i
$231$		0			0
$232$	−4.00000			−0.262613
$233$	6.92820	+	4.00000i	0.453882	+	0.262049i	0.709468	−	0.704737i	$-0.248935\pi$
$233$	6.92820	+	4.00000i	0.453882	+	0.262049i	−0.255586	+	0.966786i	$0.582269\pi$
$234$		0			0
$235$	5.85641	−	21.8564i	0.382030	−	1.42575i
$236$	1.46410	+	5.46410i	0.0953049	+	0.355683i
$237$	28.0000	−	28.0000i	1.81880	−	1.81880i
$238$		0			0
$239$	−8.00000			−0.517477			−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000			−0.517477			−0.258738	+	0.965947i	$0.583307\pi$
$240$	16.0000	−	27.7128i	1.03280	−	1.78885i
$241$	11.0000	+	19.0526i	0.708572	+	1.22728i	0.965387	+	0.260822i	$0.0839937\pi$
$241$	11.0000	+	19.0526i	0.708572	+	1.22728i	−0.256814	+	0.966461i	$0.582673\pi$
$242$	−2.56218	+	9.56218i	−0.164703	+	0.614680i
$243$	−3.66025	−	13.6603i	−0.234805	−	0.876306i
$244$	−12.0000	−	12.0000i	−0.768221	−	0.768221i
$245$		0			0
$246$	8.00000			0.510061
$247$		0			0
$248$	2.92820	−	10.9282i	0.185941	−	0.693942i
$249$		0			0
$250$	6.92820	+	4.00000i	0.438178	+	0.252982i
$251$	4.00000	+	4.00000i	0.252478	+	0.252478i	0.821986	−	0.569508i	$-0.192866\pi$
$251$	4.00000	+	4.00000i	0.252478	+	0.252478i	−0.569508	+	0.821986i	$0.692866\pi$
$252$		0			0
$253$	6.00000	−	6.00000i	0.377217	−	0.377217i
$254$	−2.92820	−	10.9282i	−0.183732	−	0.685696i
$255$	24.0000	+	41.5692i	1.50294	+	2.60317i
$256$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$257$	5.00000	−	8.66025i	0.311891	−	0.540212i	−0.666880	−	0.745165i	$-0.732371\pi$
$257$	5.00000	−	8.66025i	0.311891	−	0.540212i	0.978772	+	0.204953i	$0.0657041\pi$
$258$	−20.0000	+	20.0000i	−1.24515	+	1.24515i
$259$		0			0
$260$		0			0
$261$	−6.83013	+	1.83013i	−0.422774	+	0.113282i
$262$	20.7846	−	12.0000i	1.28408	−	0.741362i
$263$	20.7846	−	12.0000i	1.28163	−	0.739952i	0.304487	−	0.952517i	$-0.401515\pi$
$263$	20.7846	−	12.0000i	1.28163	−	0.739952i	0.977147	+	0.212565i	$0.0681817\pi$
$264$	−32.7846	+	8.78461i	−2.01775	+	0.540655i
$265$		−	28.0000i		−	1.72003i
$266$		0			0
$267$	−12.0000	−	12.0000i	−0.734388	−	0.734388i
$268$	−13.6603	+	3.66025i	−0.834433	+	0.223586i
$269$	21.8564	+	5.85641i	1.33261	+	0.357071i	0.853687	−	0.520786i	$-0.174361\pi$
$269$	21.8564	+	5.85641i	1.33261	+	0.357071i	0.478922	+	0.877858i	$0.341028\pi$
$270$	5.85641	−	21.8564i	0.356410	−	1.33014i
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	−0.999870	−	0.0161307i	$-0.994865\pi$
$271$	−8.00000	+	13.8564i	−0.485965	+	0.841717i	0.513905	+	0.857847i	$0.328199\pi$
$272$	24.0000			1.45521
$273$		0			0
$274$	−12.0000	−	12.0000i	−0.724947	−	0.724947i
$275$	3.29423	+	12.2942i	0.198649	+	0.741370i
$276$	10.9282	+	2.92820i	0.657801	+	0.176257i
$277$	0.366025	−	1.36603i	0.0219923	−	0.0820765i	−0.954057	−	0.299624i	$-0.903139\pi$
$277$	0.366025	−	1.36603i	0.0219923	−	0.0820765i	0.976050	+	0.217547i	$0.0698056\pi$
$278$	−12.0000	+	20.7846i	−0.719712	+	1.24658i
$279$		−	20.0000i		−	1.19737i
$280$		0			0
$281$			24.0000i			1.43172i	0.698244	+	0.715860i	$0.253965\pi$
$281$			24.0000i			1.43172i	−0.698244	+	0.715860i	$0.746035\pi$
$282$	−27.7128	−	16.0000i	−1.65027	−	0.952786i
$283$	−2.19615	+	8.19615i	−0.130548	+	0.487211i	−0.999977	−	0.00684749i	$-0.997820\pi$
$283$	−2.19615	+	8.19615i	−0.130548	+	0.487211i	0.869429	+	0.494058i	$0.164487\pi$
$284$	8.00000	+	13.8564i	0.474713	+	0.822226i
$285$	−11.7128	−	43.7128i	−0.693807	−	2.58932i
$286$		0			0
$287$		0			0
$288$	−20.0000	−	20.0000i	−1.17851	−	1.17851i
$289$	−9.50000	+	16.4545i	−0.558824	+	0.967911i
$290$	5.46410	+	1.46410i	0.320863	+	0.0859750i
$291$	−16.3923	−	4.39230i	−0.960934	−	0.257481i
$292$	10.0000	−	17.3205i	0.585206	−	1.01361i
$293$	−10.0000	−	10.0000i	−0.584206	−	0.584206i	0.351850	−	0.936056i	$-0.385553\pi$
$293$	−10.0000	−	10.0000i	−0.584206	−	0.584206i	−0.936056	+	0.351850i	$0.885553\pi$
$294$		0			0
$295$		−	8.00000i		−	0.465778i
$296$	−10.3923	−	6.00000i	−0.604040	−	0.348743i
$297$	−20.7846	+	12.0000i	−1.20605	+	0.696311i
$298$	−13.0000	−	22.5167i	−0.753070	−	1.30436i
$299$		0			0
$300$	−12.0000	+	12.0000i	−0.692820	+	0.692820i
$301$		0			0
$302$	6.00000	+	6.00000i	0.345261	+	0.345261i
$303$	4.00000	−	6.92820i	0.229794	−	0.398015i
$304$	−21.8564	−	5.85641i	−1.25355	−	0.335888i
$305$	12.0000	+	20.7846i	0.687118	+	1.19012i
$306$	40.9808	−	10.9808i	2.34271	−	0.627728i
$307$	−8.00000	+	8.00000i	−0.456584	+	0.456584i	−0.897532	−	0.440948i	$-0.854642\pi$
$307$	−8.00000	+	8.00000i	−0.456584	+	0.456584i	0.440948	+	0.897532i	$0.354642\pi$
$308$		0			0
$309$		0			0
$310$	−8.00000	+	13.8564i	−0.454369	+	0.786991i
$311$	−3.46410	+	2.00000i	−0.196431	+	0.113410i	−0.594990	−	0.803733i	$-0.702844\pi$
$311$	−3.46410	+	2.00000i	−0.196431	+	0.113410i	0.398559	+	0.917143i	$0.369511\pi$
$312$		0			0
$313$	1.73205	+	1.00000i	0.0979013	+	0.0565233i	0.548151	−	0.836379i	$-0.315332\pi$
$313$	1.73205	+	1.00000i	0.0979013	+	0.0565233i	−0.450250	+	0.892903i	$0.648665\pi$
$314$		0			0
$315$		0			0
$316$			28.0000i			1.57512i
$317$	−5.49038	−	20.4904i	−0.308371	−	1.15085i	−0.930005	−	0.367547i	$-0.880198\pi$
$317$	−5.49038	−	20.4904i	−0.308371	−	1.15085i	0.621634	−	0.783307i	$-0.286469\pi$
$318$	−38.2487	−	10.2487i	−2.14488	−	0.574719i
$319$	−3.00000	−	5.19615i	−0.167968	−	0.290929i
$320$	5.85641	+	21.8564i	0.327383	+	1.22181i
$321$	−20.0000			−1.11629
$322$		0			0
$323$	24.0000	−	24.0000i	1.33540	−	1.33540i
$324$	−1.73205	−	1.00000i	−0.0962250	−	0.0555556i
$325$		0			0
$326$	−8.66025	+	5.00000i	−0.479647	+	0.276924i
$327$	10.3923	+	6.00000i	0.574696	+	0.331801i
$328$	−4.00000	+	4.00000i	−0.220863	+	0.220863i
$329$		0			0
$330$	48.0000			2.64231
$331$	20.4904	−	5.49038i	1.12625	−	0.301779i	0.352842	−	0.935683i	$-0.385215\pi$
$331$	20.4904	−	5.49038i	1.12625	−	0.301779i	0.773411	+	0.633904i	$0.218549\pi$
$332$		0			0
$333$	−20.4904	−	5.49038i	−1.12287	−	0.300871i
$334$		0			0
$335$	20.0000			1.09272
$336$		0			0
$337$	−8.00000			−0.435788			−0.217894	−	0.975972i	$-0.569919\pi$
$337$	−8.00000			−0.435788			−0.217894	+	0.975972i	$0.569919\pi$
$338$	17.7583	−	4.75833i	0.965926	−	0.258819i
$339$	−32.7846	−	8.78461i	−1.78062	−	0.477115i
$340$	−32.7846	−	8.78461i	−1.77800	−	0.476412i
$341$	16.3923	−	4.39230i	0.887693	−	0.237857i
$342$	−40.0000			−2.16295
$343$		0			0
$344$		−	20.0000i		−	1.07833i
$345$	−13.8564	−	8.00000i	−0.746004	−	0.430706i
$346$	27.7128	−	16.0000i	1.48985	−	0.860165i
$347$	−4.75833	+	17.7583i	−0.255441	+	0.953317i	0.712404	+	0.701769i	$0.247606\pi$
$347$	−4.75833	+	17.7583i	−0.255441	+	0.953317i	−0.967845	+	0.251548i	$0.919060\pi$
$348$	4.00000	−	6.92820i	0.214423	−	0.371391i
$349$	16.0000	−	16.0000i	0.856460	−	0.856460i	−0.134459	−	0.990919i	$-0.542930\pi$
$349$	16.0000	−	16.0000i	0.856460	−	0.856460i	0.990919	+	0.134459i	$0.0429296\pi$
$350$		0			0
$351$		0			0
$352$	12.0000	−	20.7846i	0.639602	−	1.10782i
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	0.520689	−	0.853746i	$-0.325675\pi$
$353$	−9.00000	−	15.5885i	−0.479022	−	0.829690i	−0.999711	+	0.0240566i	$0.992342\pi$
$354$	−10.9282	−	2.92820i	−0.580827	−	0.155632i
$355$	−5.85641	−	21.8564i	−0.310826	−	1.16002i
$356$	12.0000			0.635999
$357$		0			0
$358$			10.0000i			0.528516i
$359$	−19.0526	−	11.0000i	−1.00556	−	0.580558i	−0.0956683	−	0.995413i	$-0.530499\pi$
$359$	−19.0526	−	11.0000i	−1.00556	−	0.580558i	−0.909887	+	0.414855i	$0.863832\pi$
$360$	20.0000	+	34.6410i	1.05409	+	1.82574i
$361$	−11.2583	+	6.50000i	−0.592544	+	0.342105i
$362$	4.00000	−	6.92820i	0.210235	−	0.364138i
$363$	−14.0000	−	14.0000i	−0.734809	−	0.734809i
$364$		0			0
$365$	−20.0000	+	20.0000i	−1.04685	+	1.04685i
$366$	32.7846	−	8.78461i	1.71368	−	0.459179i
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	0.995697	−	0.0926670i	$-0.0295392\pi$
$367$	8.00000	+	13.8564i	0.417597	+	0.723299i	−0.578101	+	0.815966i	$0.696206\pi$
$368$	−6.92820	+	4.00000i	−0.361158	+	0.208514i
$369$	−5.00000	+	8.66025i	−0.260290	+	0.450835i
$370$	12.0000	+	12.0000i	0.623850	+	0.623850i
$371$		0			0
$372$	16.0000	+	16.0000i	0.829561	+	0.829561i
$373$	−6.83013	+	1.83013i	−0.353651	+	0.0947604i	−0.431271	−	0.902223i	$-0.641935\pi$
$373$	−6.83013	+	1.83013i	−0.353651	+	0.0947604i	0.0776200	+	0.996983i	$0.475268\pi$
$374$	18.0000	+	31.1769i	0.930758	+	1.61212i
$375$	−13.8564	+	8.00000i	−0.715542	+	0.413118i
$376$	21.8564	−	5.85641i	1.12716	−	0.302021i
$377$		0			0
$378$		0			0
$379$	17.0000	+	17.0000i	0.873231	+	0.873231i	0.992823	−	0.119592i	$-0.0381586\pi$
$379$	17.0000	+	17.0000i	0.873231	+	0.873231i	−0.119592	+	0.992823i	$0.538159\pi$
$380$	27.7128	+	16.0000i	1.42164	+	0.820783i
$381$	21.8564	+	5.85641i	1.11974	+	0.300033i
$382$	30.0526	+	8.05256i	1.53762	+	0.412005i
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	−0.585973	−	0.810331i	$-0.699287\pi$
$383$	8.00000	−	13.8564i	0.408781	−	0.708029i	0.994753	+	0.102302i	$0.0326207\pi$
$384$	32.0000			1.63299
$385$		0			0
$386$	24.0000	−	24.0000i	1.22157	−	1.22157i
$387$	−9.15064	−	34.1506i	−0.465153	−	1.73597i
$388$	10.3923	−	6.00000i	0.527589	−	0.304604i
$389$	6.95448	−	25.9545i	0.352606	−	1.31594i	−0.530864	−	0.847457i	$-0.678132\pi$
$389$	6.95448	−	25.9545i	0.352606	−	1.31594i	0.883470	−	0.468487i	$-0.155201\pi$
$390$		0			0
$391$		−	12.0000i		−	0.606866i
$392$		0			0
$393$			48.0000i			2.42128i
$394$	−11.0000	+	19.0526i	−0.554172	+	0.959854i
$395$	10.2487	−	38.2487i	0.515669	−	1.92450i
$396$	10.9808	−	40.9808i	0.551804	−	2.05936i
$397$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$397$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$398$	20.0000	+	20.0000i	1.00251	+	1.00251i
$399$		0			0
$400$		−	12.0000i		−	0.600000i
$401$	2.00000	−	3.46410i	0.0998752	−	0.172989i	−0.811758	−	0.583994i	$-0.801489\pi$
$401$	2.00000	−	3.46410i	0.0998752	−	0.172989i	0.911633	+	0.411005i	$0.134822\pi$
$402$	7.32051	−	27.3205i	0.365114	−	1.36262i
$403$		0			0
$404$	1.46410	+	5.46410i	0.0728418	+	0.271849i
$405$	2.00000	+	2.00000i	0.0993808	+	0.0993808i
$406$		0			0
$407$		−	18.0000i		−	0.892227i
$408$	−24.0000	+	41.5692i	−1.18818	+	2.05798i
$409$	8.66025	−	5.00000i	0.428222	−	0.247234i	−0.270367	−	0.962757i	$-0.587145\pi$
$409$	8.66025	−	5.00000i	0.428222	−	0.247234i	0.698589	+	0.715523i	$0.253812\pi$
$410$	6.92820	−	4.00000i	0.342160	−	0.197546i
$411$	32.7846	−	8.78461i	1.61715	−	0.433313i
$412$		0			0
$413$		0			0
$414$	−10.0000	+	10.0000i	−0.491473	+	0.491473i
$415$		0			0
$416$		0			0
$417$	−24.0000	−	41.5692i	−1.17529	−	2.03565i
$418$	−8.78461	−	32.7846i	−0.429669	−	1.60355i
$419$	−10.0000	+	10.0000i	−0.488532	+	0.488532i	−0.907843	−	0.419311i	$-0.862272\pi$
$419$	−10.0000	+	10.0000i	−0.488532	+	0.488532i	0.419311	+	0.907843i	$0.362272\pi$
$420$		0			0
$421$	−25.0000	−	25.0000i	−1.21843	−	1.21843i	−0.968183	−	0.250242i	$-0.919490\pi$
$421$	−25.0000	−	25.0000i	−1.21843	−	1.21843i	−0.250242	−	0.968183i	$-0.580510\pi$
$422$	15.5885	+	9.00000i	0.758834	+	0.438113i
$423$	34.6410	−	20.0000i	1.68430	−	0.972433i
$424$	24.2487	−	14.0000i	1.17762	−	0.679900i
$425$	15.5885	+	9.00000i	0.756151	+	0.436564i
$426$	−32.0000			−1.55041
$427$		0			0
$428$	10.0000	−	10.0000i	0.483368	−	0.483368i
$429$		0			0
$430$	−7.32051	+	27.3205i	−0.353026	+	1.31751i
$431$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$431$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$432$	21.8564	−	5.85641i	1.05157	−	0.281766i
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$	−8.00000	+	8.00000i	−0.383571	+	0.383571i
$436$	−8.19615	+	2.19615i	−0.392525	+	0.105177i
$437$	−2.92820	+	10.9282i	−0.140075	+	0.522767i
$438$	20.0000	+	34.6410i	0.955637	+	1.65521i
$439$	20.7846	+	12.0000i	0.991995	+	0.572729i	0.905870	−	0.423556i	$-0.139218\pi$
$439$	20.7846	+	12.0000i	0.991995	+	0.572729i	0.0861252	+	0.996284i	$0.472552\pi$
$440$	−24.0000	+	24.0000i	−1.14416	+	1.14416i
$441$		0			0
$442$		0			0
$443$	23.2224	−	6.22243i	1.10333	−	0.295637i	0.339211	−	0.940710i	$-0.389840\pi$
$443$	23.2224	−	6.22243i	1.10333	−	0.295637i	0.764120	+	0.645074i	$0.223173\pi$
$444$	20.7846	−	12.0000i	0.986394	−	0.569495i
$445$	−16.3923	−	4.39230i	−0.777070	−	0.208215i
$446$	−1.46410	−	5.46410i	−0.0693272	−	0.258733i
$447$	52.0000			2.45952
$448$		0			0
$449$	34.0000			1.60456			0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000			1.60456			0.802280	+	0.596948i	$0.203620\pi$
$450$	−5.49038	−	20.4904i	−0.258819	−	0.965926i
$451$	−8.19615	−	2.19615i	−0.385942	−	0.103413i
$452$	20.7846	−	12.0000i	0.977626	−	0.564433i
$453$	−16.3923	+	4.39230i	−0.770178	+	0.206368i
$454$			8.00000i			0.375459i
$455$		0			0
$456$	32.0000	−	32.0000i	1.49854	−	1.49854i
$457$	22.5167	+	13.0000i	1.05328	+	0.608114i	0.923567	−	0.383437i	$-0.125260\pi$
$457$	22.5167	+	13.0000i	1.05328	+	0.608114i	0.129718	+	0.991551i	$0.458593\pi$
$458$	12.0000	+	20.7846i	0.560723	+	0.971201i
$459$	−8.78461	+	32.7846i	−0.410030	+	1.53025i
$460$	10.9282	−	2.92820i	0.509530	−	0.136528i
$461$	−28.0000	+	28.0000i	−1.30409	+	1.30409i	−0.378481	+	0.925609i	$0.623553\pi$
$461$	−28.0000	+	28.0000i	−1.30409	+	1.30409i	−0.925609	+	0.378481i	$0.876447\pi$
$462$		0			0
$463$	14.0000			0.650635			0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000			0.650635			0.325318	+	0.945605i	$0.394529\pi$
$464$	1.46410	+	5.46410i	0.0679692	+	0.253665i
$465$	−16.0000	−	27.7128i	−0.741982	−	1.28515i
$466$	2.92820	−	10.9282i	0.135646	−	0.506239i
$467$	−5.85641	−	21.8564i	−0.271002	−	1.01139i	−0.958473	−	0.285182i	$-0.907946\pi$
$467$	−5.85641	−	21.8564i	−0.271002	−	1.01139i	0.687471	−	0.726212i	$-0.258721\pi$
$468$		0			0
$469$		0			0
$470$	−32.0000			−1.47605
$471$		0			0
$472$	6.92820	−	4.00000i	0.318896	−	0.184115i
$473$	25.9808	−	15.0000i	1.19460	−	0.689701i
$474$	−48.4974	−	28.0000i	−2.22756	−	1.28608i
$475$	−12.0000	−	12.0000i	−0.550598	−	0.550598i
$476$		0			0
$477$	35.0000	−	35.0000i	1.60254	−	1.60254i
$478$	2.92820	+	10.9282i	0.133933	+	0.499844i
$479$	4.00000	+	6.92820i	0.182765	+	0.316558i	0.942821	−	0.333300i	$-0.108162\pi$
$479$	4.00000	+	6.92820i	0.182765	+	0.316558i	−0.760056	+	0.649857i	$0.774829\pi$
$480$	−43.7128	−	11.7128i	−1.99521	−	0.534614i
$481$		0			0
$482$	22.0000	−	22.0000i	1.00207	−	1.00207i
$483$		0			0
$484$	14.0000			0.636364
$485$	−16.3923	+	4.39230i	−0.744336	+	0.199444i
$486$	−17.3205	+	10.0000i	−0.785674	+	0.453609i
$487$	−1.73205	+	1.00000i	−0.0784867	+	0.0453143i	−0.538730	−	0.842479i	$-0.681096\pi$
$487$	−1.73205	+	1.00000i	−0.0784867	+	0.0453143i	0.460243	+	0.887793i	$0.347762\pi$
$488$	−12.0000	+	20.7846i	−0.543214	+	0.940875i
$489$		−	20.0000i		−	0.904431i
$490$		0			0
$491$	−11.0000	−	11.0000i	−0.496423	−	0.496423i	0.413900	−	0.910323i	$-0.364166\pi$
$491$	−11.0000	−	11.0000i	−0.496423	−	0.496423i	−0.910323	+	0.413900i	$0.864166\pi$
$492$	−2.92820	−	10.9282i	−0.132014	−	0.492681i
$493$	−8.19615	−	2.19615i	−0.369136	−	0.0989097i
$494$		0			0
$495$	−30.0000	+	51.9615i	−1.34840	+	2.33550i
$496$	−16.0000			−0.718421
$497$		0			0
$498$		0			0
$499$	−5.49038	−	20.4904i	−0.245783	−	0.917275i	−0.972988	−	0.230855i	$-0.925848\pi$
$499$	−5.49038	−	20.4904i	−0.245783	−	0.917275i	0.727205	−	0.686420i	$-0.240819\pi$
$500$	2.92820	−	10.9282i	0.130953	−	0.488724i
$501$		0			0
$502$	4.00000	−	6.92820i	0.178529	−	0.309221i
$503$			8.00000i			0.356702i	0.983967	+	0.178351i	$0.0570763\pi$
$503$			8.00000i			0.356702i	−0.983967	+	0.178351i	$0.942924\pi$
$504$		0			0
$505$		−	8.00000i		−	0.355995i
$506$	−10.3923	−	6.00000i	−0.461994	−	0.266733i
$507$	−9.51666	+	35.5167i	−0.422650	+	1.57735i
$508$	−13.8564	+	8.00000i	−0.614779	+	0.354943i
$509$	−4.39230	−	16.3923i	−0.194685	−	0.726576i	−0.992348	−	0.123472i	$-0.960597\pi$
$509$	−4.39230	−	16.3923i	−0.194685	−	0.726576i	0.797663	−	0.603104i	$-0.206070\pi$
$510$	48.0000	−	48.0000i	2.12548	−	2.12548i
$511$		0			0
$512$	−16.0000	+	16.0000i	−0.707107	+	0.707107i
$513$	16.0000	−	27.7128i	0.706417	−	1.22355i
$514$	−13.6603	−	3.66025i	−0.602528	−	0.161447i
$515$		0			0
$516$	34.6410	+	20.0000i	1.52499	+	0.880451i
$517$	24.0000	+	24.0000i	1.05552	+	1.05552i
$518$		0			0
$519$			64.0000i			2.80929i
$520$		0			0
$521$	5.19615	−	3.00000i	0.227648	−	0.131432i	−0.381839	−	0.924229i	$-0.624709\pi$
$521$	5.19615	−	3.00000i	0.227648	−	0.131432i	0.609486	+	0.792797i	$0.291376\pi$
$522$	5.00000	+	8.66025i	0.218844	+	0.379049i
$523$	−35.5167	+	9.51666i	−1.55304	+	0.416135i	−0.930450	−	0.366418i	$-0.880584\pi$
$523$	−35.5167	+	9.51666i	−1.55304	+	0.416135i	−0.622585	+	0.782552i	$0.713917\pi$
$524$	−24.0000	−	24.0000i	−1.04844	−	1.04844i
$525$		0			0
$526$	−24.0000	−	24.0000i	−1.04645	−	1.04645i
$527$	12.0000	−	20.7846i	0.522728	−	0.905392i
$528$	24.0000	+	41.5692i	1.04447	+	1.80907i
$529$	−9.50000	−	16.4545i	−0.413043	−	0.715412i
$530$	−38.2487	+	10.2487i	−1.66142	+	0.445176i
$531$	10.0000	−	10.0000i	0.433963	−	0.433963i
$532$		0			0
$533$		0			0
$534$	−12.0000	+	20.7846i	−0.519291	+	0.899438i
$535$	−17.3205	+	10.0000i	−0.748831	+	0.432338i
$536$	10.0000	+	17.3205i	0.431934	+	0.748132i
$537$	−17.3205	−	10.0000i	−0.747435	−	0.431532i
$538$		−	32.0000i		−	1.37962i
$539$		0			0
$540$	−32.0000			−1.37706
$541$	−10.6147	−	39.6147i	−0.456363	−	1.70317i	−0.684049	−	0.729436i	$-0.739783\pi$
$541$	−10.6147	−	39.6147i	−0.456363	−	1.70317i	0.227686	−	0.973735i	$-0.426884\pi$
$542$	21.8564	+	5.85641i	0.938813	+	0.251554i
$543$	8.00000	+	13.8564i	0.343313	+	0.594635i
$544$	−8.78461	−	32.7846i	−0.376637	−	1.40563i
$545$	12.0000			0.514024
$546$		0			0
$547$	19.0000	−	19.0000i	0.812381	−	0.812381i	−0.172609	−	0.984990i	$-0.555220\pi$
$547$	19.0000	−	19.0000i	0.812381	−	0.812381i	0.984990	+	0.172609i	$0.0552197\pi$
$548$	−12.0000	+	20.7846i	−0.512615	+	0.887875i
$549$	−10.9808	+	40.9808i	−0.468648	+	1.74902i
$550$	15.5885	−	9.00000i	0.664694	−	0.383761i
$551$	6.92820	+	4.00000i	0.295151	+	0.170406i
$552$		−	16.0000i		−	0.681005i
$553$		0			0
$554$	−2.00000			−0.0849719
$555$	−32.7846	+	8.78461i	−1.39163	+	0.372886i
$556$	32.7846	+	8.78461i	1.39038	+	0.372550i
$557$	−20.4904	−	5.49038i	−0.868205	−	0.232635i	−0.202894	−	0.979201i	$-0.565035\pi$
$557$	−20.4904	−	5.49038i	−0.868205	−	0.232635i	−0.665312	+	0.746566i	$0.731701\pi$
$558$	−27.3205	+	7.32051i	−1.15657	+	0.309902i
$559$		0			0
$560$		0			0
$561$	−72.0000			−3.03984
$562$	32.7846	−	8.78461i	1.38294	−	0.370556i
$563$	30.0526	+	8.05256i	1.26656	+	0.339375i	0.828714	−	0.559673i	$-0.189073\pi$
$563$	30.0526	+	8.05256i	1.26656	+	0.339375i	0.437851	+	0.899048i	$0.355740\pi$
$564$	−11.7128	+	43.7128i	−0.493198	+	1.84064i
$565$	−32.7846	+	8.78461i	−1.37926	+	0.369571i
$566$	12.0000			0.504398
$567$		0			0
$568$	16.0000	−	16.0000i	0.671345	−	0.671345i
$569$	39.8372	+	23.0000i	1.67006	+	0.964210i	0.967600	+	0.252488i	$0.0812488\pi$
$569$	39.8372	+	23.0000i	1.67006	+	0.964210i	0.702461	+	0.711722i	$0.252085\pi$
$570$	−55.4256	+	32.0000i	−2.32152	+	1.34033i
$571$	−7.68653	+	28.6865i	−0.321671	+	1.20049i	0.595944	+	0.803026i	$0.296778\pi$
$571$	−7.68653	+	28.6865i	−0.321671	+	1.20049i	−0.917616	+	0.397468i	$0.869889\pi$
$572$		0			0
$573$	−44.0000	+	44.0000i	−1.83813	+	1.83813i
$574$		0			0
$575$	−6.00000			−0.250217
$576$	−20.0000	+	34.6410i	−0.833333	+	1.44338i
$577$	13.0000	+	22.5167i	0.541197	+	0.937381i	0.998836	+	0.0482425i	$0.0153620\pi$
$577$	13.0000	+	22.5167i	0.541197	+	0.937381i	−0.457639	+	0.889138i	$0.651305\pi$
$578$	25.9545	+	6.95448i	1.07956	+	0.289268i
$579$	17.5692	+	65.5692i	0.730152	+	2.72496i
$580$		−	8.00000i		−	0.332182i
$581$		0			0
$582$			24.0000i			0.994832i
$583$	36.3731	+	21.0000i	1.50642	+	0.869731i
$584$	−27.3205	−	7.32051i	−1.13053	−	0.302925i
$585$		0			0
$586$	−10.0000	+	17.3205i	−0.413096	+	0.715504i
$587$	8.00000	+	8.00000i	0.330195	+	0.330195i	0.852661	−	0.522465i	$-0.174988\pi$
$587$	8.00000	+	8.00000i	0.330195	+	0.330195i	−0.522465	+	0.852661i	$0.674988\pi$
$588$		0			0
$589$	−16.0000	+	16.0000i	−0.659269	+	0.659269i
$590$	−10.9282	+	2.92820i	−0.449907	+	0.120552i
$591$	−22.0000	−	38.1051i	−0.904959	−	1.56744i
$592$	−4.39230	+	16.3923i	−0.180523	+	0.673720i
$593$	11.0000	−	19.0526i	0.451716	−	0.782395i	−0.546777	−	0.837278i	$-0.684145\pi$
$593$	11.0000	−	19.0526i	0.451716	−	0.782395i	0.998493	+	0.0548835i	$0.0174787\pi$
$594$	24.0000	+	24.0000i	0.984732	+	0.984732i
$595$		0			0
$596$	−26.0000	+	26.0000i	−1.06500	+	1.06500i
$597$	−54.6410	+	14.6410i	−2.23631	+	0.599217i
$598$		0			0
$599$	−6.92820	+	4.00000i	−0.283079	+	0.163436i	−0.634816	−	0.772663i	$-0.718924\pi$
$599$	−6.92820	+	4.00000i	−0.283079	+	0.163436i	0.351738	+	0.936099i	$0.385591\pi$
$600$	20.7846	+	12.0000i	0.848528	+	0.489898i
$601$		−	10.0000i		−	0.407909i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$		−	10.0000i		−	0.407909i	0.978980	−	0.203954i	$-0.0653794\pi$
$602$		0			0
$603$	25.0000	+	25.0000i	1.01808	+	1.01808i
$604$	6.00000	−	10.3923i	0.244137	−	0.422857i
$605$	−19.1244	−	5.12436i	−0.777516	−	0.208335i
$606$	−10.9282	−	2.92820i	−0.443928	−	0.118950i
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	−0.994424	−	0.105453i	$-0.966371\pi$
$607$	−10.0000	+	17.3205i	−0.405887	+	0.703018i	0.588537	+	0.808470i	$0.299704\pi$
$608$			32.0000i			1.29777i
$609$		0			0
$610$	24.0000	−	24.0000i	0.971732	−	0.971732i
$611$		0			0
$612$	−30.0000	−	51.9615i	−1.21268	−	2.10042i
$613$	−5.49038	+	20.4904i	−0.221754	+	0.827599i	0.761924	+	0.647666i	$0.224255\pi$
$613$	−5.49038	+	20.4904i	−0.221754	+	0.827599i	−0.983679	+	0.179933i	$0.942412\pi$
$614$	13.8564	+	8.00000i	0.559199	+	0.322854i
$615$			16.0000i			0.645182i
$616$		0			0
$617$		−	6.00000i		−	0.241551i	−0.992680	−	0.120775i	$-0.961462\pi$
$617$		−	6.00000i		−	0.241551i	0.992680	−	0.120775i	$-0.0385381\pi$
$618$		0			0
$619$	6.58846	−	24.5885i	0.264812	−	0.988294i	−0.697553	−	0.716534i	$-0.745728\pi$
$619$	6.58846	−	24.5885i	0.264812	−	0.988294i	0.962365	−	0.271760i	$-0.0876057\pi$
$620$	21.8564	+	5.85641i	0.877774	+	0.235199i
$621$	−2.92820	−	10.9282i	−0.117505	−	0.438534i
$622$	4.00000	+	4.00000i	0.160385	+	0.160385i
$623$		0			0
$624$		0			0
$625$	−15.5000	+	26.8468i	−0.620000	+	1.07387i
$626$	0.732051	−	2.73205i	0.0292586	−	0.109195i
$627$	65.5692	+	17.5692i	2.61858	+	0.701647i
$628$		0			0
$629$	−18.0000	−	18.0000i	−0.717707	−	0.717707i
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$	38.2487	−	10.2487i	1.52145	−	0.407672i
$633$	−31.1769	+	18.0000i	−1.23917	+	0.715436i
$634$	−25.9808	+	15.0000i	−1.03183	+	0.595726i
$635$	21.8564	−	5.85641i	0.867345	−	0.232404i
$636$			56.0000i			2.22054i
$637$		0			0
$638$	−6.00000	+	6.00000i	−0.237542	+	0.237542i
$639$	20.0000	−	34.6410i	0.791188	−	1.37038i
$640$	27.7128	−	16.0000i	1.09545	−	0.632456i
$641$	−22.0000	−	38.1051i	−0.868948	−	1.50506i	−0.863073	−	0.505079i	$-0.831463\pi$
$641$	−22.0000	−	38.1051i	−0.868948	−	1.50506i	−0.00587459	−	0.999983i	$-0.501870\pi$
$642$	7.32051	+	27.3205i	0.288917	+	1.07825i
$643$	26.0000	−	26.0000i	1.02534	−	1.02534i	0.0256694	−	0.999670i	$-0.491828\pi$
$643$	26.0000	−	26.0000i	1.02534	−	1.02534i	0.999670	−	0.0256694i	$-0.00817173\pi$
$644$		0			0
$645$	−40.0000	−	40.0000i	−1.57500	−	1.57500i
$646$	−41.5692	−	24.0000i	−1.63552	−	0.944267i
$647$	−20.7846	+	12.0000i	−0.817127	+	0.471769i	−0.849425	−	0.527710i	$-0.823051\pi$
$647$	−20.7846	+	12.0000i	−0.817127	+	0.471769i	0.0322975	+	0.999478i	$0.489718\pi$
$648$	−0.732051	+	2.73205i	−0.0287577	+	0.107325i
$649$	10.3923	+	6.00000i	0.407934	+	0.235521i
$650$		0			0
$651$		0			0
$652$	10.0000	+	10.0000i	0.391630	+	0.391630i
$653$	9.15064	+	34.1506i	0.358092	+	1.33642i	0.876548	+	0.481314i	$0.159840\pi$
$653$	9.15064	+	34.1506i	0.358092	+	1.33642i	−0.518456	+	0.855104i	$0.673493\pi$
$654$	4.39230	−	16.3923i	0.171753	−	0.640990i
$655$	24.0000	+	41.5692i	0.937758	+	1.62424i
$656$	6.92820	+	4.00000i	0.270501	+	0.156174i
$657$	−50.0000			−1.95069
$658$		0			0
$659$	17.0000	−	17.0000i	0.662226	−	0.662226i	−0.293678	−	0.955904i	$-0.594879\pi$
$659$	17.0000	−	17.0000i	0.662226	−	0.662226i	0.955904	+	0.293678i	$0.0948794\pi$
$660$	−17.5692	−	65.5692i	−0.683881	−	2.55228i
$661$	10.9808	−	40.9808i	0.427102	−	1.59397i	−0.332187	−	0.943214i	$-0.607787\pi$
$661$	10.9808	−	40.9808i	0.427102	−	1.59397i	0.759289	−	0.650753i	$-0.225547\pi$
$662$	−15.0000	−	25.9808i	−0.582992	−	1.00977i
$663$		0			0
$664$		0			0
$665$		0			0
$666$			30.0000i			1.16248i
$667$	2.73205	−	0.732051i	0.105785	−	0.0283451i
$668$		0			0
$669$	10.9282	+	2.92820i	0.422509	+	0.113211i
$670$	−7.32051	−	27.3205i	−0.282816	−	1.05548i
$671$	−36.0000			−1.38976
$672$		0			0
$673$	26.0000			1.00223			0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000			1.00223			0.501113	+	0.865382i	$0.332924\pi$
$674$	2.92820	+	10.9282i	0.112790	+	0.420939i
$675$	16.3923	+	4.39230i	0.630940	+	0.169060i
$676$	−13.0000	−	22.5167i	−0.500000	−	0.866025i
$677$	−10.9282	+	2.92820i	−0.420005	+	0.112540i	−0.462631	−	0.886551i	$-0.653094\pi$
$677$	−10.9282	+	2.92820i	−0.420005	+	0.112540i	0.0426257	+	0.999091i	$0.486428\pi$
$678$			48.0000i			1.84343i
$679$		0			0
$680$			48.0000i			1.84072i
$681$	−13.8564	−	8.00000i	−0.530979	−	0.306561i
$682$	−12.0000	−	20.7846i	−0.459504	−	0.795884i
$683$	1.83013	−	6.83013i	0.0700279	−	0.261348i	−0.922032	−	0.387113i	$-0.873472\pi$
$683$	1.83013	−	6.83013i	0.0700279	−	0.261348i	0.992060	+	0.125766i	$0.0401388\pi$
$684$	14.6410	+	54.6410i	0.559813	+	2.08925i
$685$	24.0000	−	24.0000i	0.916993	−	0.916993i
$686$		0			0
$687$	−48.0000			−1.83131
$688$	−27.3205	+	7.32051i	−1.04158	+	0.279092i
$689$		0			0
$690$	−5.85641	+	21.8564i	−0.222950	+	0.832059i
$691$	4.39230	+	16.3923i	0.167091	+	0.623593i	0.997764	+	0.0668322i	$0.0212892\pi$
$691$	4.39230	+	16.3923i	0.167091	+	0.623593i	−0.830673	+	0.556760i	$0.812044\pi$
$692$	−32.0000	−	32.0000i	−1.21646	−	1.21646i
$693$		0			0
$694$	26.0000			0.986947
$695$	−41.5692	−	24.0000i	−1.57681	−	0.910372i
$696$	−10.9282	−	2.92820i	−0.414232	−	0.110993i
$697$	−10.3923	+	6.00000i	−0.393637	+	0.227266i
$698$	−27.7128	−	16.0000i	−1.04895	−	0.605609i
$699$	16.0000	+	16.0000i	0.605176	+	0.605176i
$700$		0			0
$701$	13.0000	−	13.0000i	0.491003	−	0.491003i	−0.417619	−	0.908622i	$-0.637135\pi$
$701$	13.0000	−	13.0000i	0.491003	−	0.491003i	0.908622	+	0.417619i	$0.137135\pi$
$702$		0			0
$703$	12.0000	+	20.7846i	0.452589	+	0.783906i
$704$	−32.7846	−	8.78461i	−1.23562	−	0.331082i
$705$	32.0000	−	55.4256i	1.20519	−	2.08745i
$706$	−18.0000	+	18.0000i	−0.677439	+	0.677439i
$707$		0			0
$708$			16.0000i			0.601317i
$709$	42.3468	−	11.3468i	1.59037	−	0.426138i	0.648252	−	0.761426i	$-0.275501\pi$
$709$	42.3468	−	11.3468i	1.59037	−	0.426138i	0.942115	+	0.335289i	$0.108834\pi$
$710$	−27.7128	+	16.0000i	−1.04004	+	0.600469i
$711$	60.6218	−	35.0000i	2.27349	−	1.31260i
$712$	−4.39230	−	16.3923i	−0.164609	−	0.614328i
$713$			8.00000i			0.299602i
$714$		0			0
$715$		0			0
$716$	13.6603	−	3.66025i	0.510508	−	0.136790i
$717$	−21.8564	−	5.85641i	−0.816242	−	0.218712i
$718$	−8.05256	+	30.0526i	−0.300519	+	1.12155i
$719$	4.00000	−	6.92820i	0.149175	−	0.258378i	−0.781748	−	0.623595i	$-0.785672\pi$
$719$	4.00000	−	6.92820i	0.149175	−	0.258378i	0.930923	+	0.365216i	$0.119005\pi$
$720$	40.0000	−	40.0000i	1.49071	−	1.49071i
$721$		0			0
$722$	13.0000	+	13.0000i	0.483810	+	0.483810i
$723$	16.1051	+	60.1051i	0.598956	+	2.23533i
$724$	−10.9282	−	2.92820i	−0.406143	−	0.108826i
$725$	−1.09808	+	4.09808i	−0.0407815	+	0.152199i
$726$	−14.0000	+	24.2487i	−0.519589	+	0.899954i
$727$			44.0000i			1.63187i	0.578144	+	0.815935i	$0.303777\pi$
$727$			44.0000i			1.63187i	−0.578144	+	0.815935i	$0.696223\pi$
$728$		0			0
$729$		−	43.0000i		−	1.59259i
$730$	34.6410	+	20.0000i	1.28212	+	0.740233i
$731$	10.9808	−	40.9808i	0.406138	−	1.51573i
$732$	−24.0000	−	41.5692i	−0.887066	−	1.53644i
$733$	−0.732051	−	2.73205i	−0.0270389	−	0.100911i	0.951088	−	0.308921i	$-0.0999679\pi$
$733$	−0.732051	−	2.73205i	−0.0270389	−	0.100911i	−0.978127	+	0.208010i	$0.933301\pi$
$734$	16.0000	−	16.0000i	0.590571	−	0.590571i
$735$		0			0
$736$	8.00000	+	8.00000i	0.294884	+	0.294884i
$737$	−15.0000	+	25.9808i	−0.552532	+	0.957014i
$738$	13.6603	+	3.66025i	0.502841	+	0.134736i
$739$	6.83013	+	1.83013i	0.251250	+	0.0673223i	0.382246	−	0.924061i	$-0.375151\pi$
$739$	6.83013	+	1.83013i	0.251250	+	0.0673223i	−0.130996	+	0.991383i	$0.541817\pi$
$740$	12.0000	−	20.7846i	0.441129	−	0.764057i
$741$		0			0
$742$		0			0
$743$		−	6.00000i		−	0.220119i	−0.993925	−	0.110059i	$-0.964896\pi$
$743$		−	6.00000i		−	0.220119i	0.993925	−	0.110059i	$-0.0351041\pi$
$744$	16.0000	−	27.7128i	0.586588	−	1.01600i
$745$	45.0333	−	26.0000i	1.64989	−	0.952566i
$746$	5.00000	+	8.66025i	0.183063	+	0.317074i
$747$		0			0
$748$	36.0000	−	36.0000i	1.31629	−	1.31629i
$749$		0			0
$750$	16.0000	+	16.0000i	0.584237	+	0.584237i
$751$	−20.0000	+	34.6410i	−0.729810	+	1.26407i	0.227153	+	0.973859i	$0.427058\pi$
$751$	−20.0000	+	34.6410i	−0.729810	+	1.26407i	−0.956963	+	0.290209i	$0.906275\pi$
$752$	−16.0000	−	27.7128i	−0.583460	−	1.01058i
$753$	8.00000	+	13.8564i	0.291536	+	0.504956i
$754$		0			0
$755$	−12.0000	+	12.0000i	−0.436725	+	0.436725i
$756$		0			0
$757$	33.0000	+	33.0000i	1.19941	+	1.19941i	0.974345	+	0.225061i	$0.0722580\pi$
$757$	33.0000	+	33.0000i	1.19941	+	1.19941i	0.225061	+	0.974345i	$0.427742\pi$
$758$	17.0000	−	29.4449i	0.617468	−	1.06949i
$759$	20.7846	−	12.0000i	0.754434	−	0.435572i
$760$	11.7128	−	43.7128i	0.424868	−	1.58563i
$761$	−15.5885	−	9.00000i	−0.565081	−	0.326250i	0.190101	−	0.981764i	$-0.439118\pi$
$761$	−15.5885	−	9.00000i	−0.565081	−	0.326250i	−0.755182	+	0.655515i	$0.772452\pi$
$762$		−	32.0000i		−	1.15924i
$763$		0			0
$764$		−	44.0000i		−	1.59186i
$765$	21.9615	+	81.9615i	0.794021	+	2.96333i
$766$	−21.8564	−	5.85641i	−0.789704	−	0.211601i
$767$		0			0
$768$	−11.7128	−	43.7128i	−0.422650	−	1.57735i
$769$	−30.0000			−1.08183			−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000			−1.08183			−0.540914	+	0.841078i	$0.681921\pi$
$770$		0			0
$771$	20.0000	−	20.0000i	0.720282	−	0.720282i
$772$	−41.5692	−	24.0000i	−1.49611	−	0.863779i
$773$	−10.9808	+	40.9808i	−0.394951	+	1.47398i	0.426914	+	0.904292i	$0.359601\pi$
$773$	−10.9808	+	40.9808i	−0.394951	+	1.47398i	−0.821864	+	0.569683i	$0.807066\pi$
$774$	−43.3013	+	25.0000i	−1.55643	+	0.898606i
$775$	−10.3923	−	6.00000i	−0.373303	−	0.215526i
$776$	−12.0000	−	12.0000i	−0.430775	−	0.430775i
$777$		0			0
$778$	−38.0000			−1.36237
$779$	10.9282	−	2.92820i	0.391544	−	0.104914i
$780$		0			0
$781$	32.7846	+	8.78461i	1.17313	+	0.314338i
$782$	−16.3923	+	4.39230i	−0.586188	+	0.157069i
$783$	−8.00000			−0.285897
$784$		0			0
$785$		0			0
$786$	65.5692	−	17.5692i	2.33878	−	0.626673i
$787$	16.3923	+	4.39230i	0.584323	+	0.156569i	0.538857	−	0.842397i	$-0.318856\pi$
$787$	16.3923	+	4.39230i	0.584323	+	0.156569i	0.0454654	+	0.998966i	$0.485523\pi$
$788$	30.0526	+	8.05256i	1.07058	+	0.286861i
$789$	65.5692	−	17.5692i	2.33433	−	0.625481i
$790$	−56.0000			−1.99239
$791$		0			0
$792$	−60.0000			−2.13201
$793$		0			0
$794$		0			0
$795$	20.4974	−	76.4974i	0.726969	−	2.71308i
$796$	20.0000	−	34.6410i	0.708881	−	1.22782i
$797$	8.00000	−	8.00000i	0.283375	−	0.283375i	−0.551079	−	0.834453i	$-0.685784\pi$
$797$	8.00000	−	8.00000i	0.283375	−	0.283375i	0.834453	+	0.551079i	$0.185784\pi$
$798$		0			0
$799$	48.0000			1.69812
$800$	−16.3923	+	4.39230i	−0.579555	+	0.155291i
$801$	−15.0000	−	25.9808i	−0.529999	−	0.917985i
$802$	−5.46410	−	1.46410i	−0.192944	−	0.0516992i
$803$	−10.9808	−	40.9808i	−0.387503	−	1.44618i
$804$	−40.0000			−1.41069
$805$		0			0
$806$		0			0
$807$	55.4256	+	32.0000i	1.95107	+	1.12645i
$808$	6.92820	−	4.00000i	0.243733	−	0.140720i
$809$	32.9090	−	19.0000i	1.15702	−	0.668004i	0.206430	−	0.978461i	$-0.433815\pi$
$809$	32.9090	−	19.0000i	1.15702	−	0.668004i	0.950587	+	0.310457i	$0.100482\pi$
$810$	2.00000	−	3.46410i	0.0702728	−	0.121716i
$811$	30.0000	+	30.0000i	1.05344	+	1.05344i	0.998489	+	0.0549536i	$0.0175011\pi$
$811$	30.0000	+	30.0000i	1.05344	+	1.05344i	0.0549536	+	0.998489i	$0.482499\pi$
$812$		0			0
$813$	−32.0000	+	32.0000i	−1.12229	+	1.12229i
$814$	−24.5885	+	6.58846i	−0.861825	+	0.230925i
$815$	−10.0000	−	17.3205i	−0.350285	−	0.606711i
$816$	65.5692	+	17.5692i	2.29538	+	0.615046i
$817$	−20.0000	+	34.6410i	−0.699711	+	1.21194i
$818$	−10.0000	−	10.0000i	−0.349642	−	0.349642i
$819$		0			0
$820$	−8.00000	−	8.00000i	−0.279372	−	0.279372i
$821$	9.56218	−	2.56218i	0.333722	−	0.0894206i	−0.0880668	−	0.996115i	$-0.528069\pi$
$821$	9.56218	−	2.56218i	0.333722	−	0.0894206i	0.421789	+	0.906694i	$0.361402\pi$
$822$	−24.0000	−	41.5692i	−0.837096	−	1.44989i
$823$	−34.6410	+	20.0000i	−1.20751	+	0.697156i	−0.962215	−	0.272292i	$-0.912218\pi$
$823$	−34.6410	+	20.0000i	−1.20751	+	0.697156i	−0.245295	+	0.969448i	$0.578885\pi$
$824$		0			0
$825$			36.0000i			1.25336i
$826$		0			0
$827$	−13.0000	−	13.0000i	−0.452054	−	0.452054i	0.443982	−	0.896036i	$-0.353566\pi$
$827$	−13.0000	−	13.0000i	−0.452054	−	0.452054i	−0.896036	+	0.443982i	$0.853566\pi$
$828$	17.3205	+	10.0000i	0.601929	+	0.347524i
$829$	2.73205	+	0.732051i	0.0948880	+	0.0254252i	0.305951	−	0.952047i	$-0.401026\pi$
$829$	2.73205	+	0.732051i	0.0948880	+	0.0254252i	−0.211063	+	0.977473i	$0.567692\pi$
$830$		0			0
$831$	2.00000	−	3.46410i	0.0693792	−	0.120168i
$832$		0			0
$833$		0			0
$834$	−48.0000	+	48.0000i	−1.66210	+	1.66210i
$835$		0			0
$836$	−41.5692	+	24.0000i	−1.43770	+	0.830057i
$837$	5.85641	−	21.8564i	0.202427	−	0.755468i
$838$	17.3205	+	10.0000i	0.598327	+	0.345444i
$839$		−	24.0000i		−	0.828572i	−0.910147	−	0.414286i	$-0.864031\pi$
$839$		−	24.0000i		−	0.828572i	0.910147	−	0.414286i	$-0.135969\pi$
$840$		0			0
$841$			27.0000i			0.931034i
$842$	−25.0000	+	43.3013i	−0.861557	+	1.49226i
$843$	−17.5692	+	65.5692i	−0.605116	+	2.25832i
$844$	6.58846	−	24.5885i	0.226784	−	0.846370i
$845$	9.51666	+	35.5167i	0.327383	+	1.22181i
$846$	−40.0000	−	40.0000i	−1.37523	−	1.37523i
$847$		0			0
$848$	−28.0000	−	28.0000i	−0.961524	−	0.961524i
$849$	−12.0000	+	20.7846i	−0.411839	+	0.713326i
$850$	6.58846	−	24.5885i	0.225982	−	0.843377i
$851$	8.19615	+	2.19615i	0.280960	+	0.0752831i
$852$	11.7128	+	43.7128i	0.401274	+	1.49758i
$853$	−4.00000	−	4.00000i	−0.136957	−	0.136957i	0.635304	−	0.772262i	$-0.280875\pi$
$853$	−4.00000	−	4.00000i	−0.136957	−	0.136957i	−0.772262	+	0.635304i	$0.780875\pi$
$854$		0			0
$855$		−	80.0000i		−	2.73594i
$856$	−17.3205	−	10.0000i	−0.592003	−	0.341793i
$857$	−15.5885	+	9.00000i	−0.532492	+	0.307434i	−0.742030	−	0.670366i	$-0.766137\pi$
$857$	−15.5885	+	9.00000i	−0.532492	+	0.307434i	0.209539	+	0.977800i	$0.432804\pi$
$858$		0			0
$859$	5.46410	−	1.46410i	0.186433	−	0.0499545i	−0.164395	−	0.986395i	$-0.552567\pi$
$859$	5.46410	−	1.46410i	0.186433	−	0.0499545i	0.350827	+	0.936440i	$0.385900\pi$
$860$	40.0000			1.36399
$861$		0			0
$862$		0			0
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	−0.912558	−	0.408946i	$-0.865896\pi$
$863$	−3.00000	+	5.19615i	−0.102121	+	0.176879i	0.810437	+	0.585826i	$0.199230\pi$
$864$	−16.0000	−	27.7128i	−0.544331	−	0.942809i
$865$	32.0000	+	55.4256i	1.08803	+	1.88453i
$866$	0.732051	+	2.73205i	0.0248761	+	0.0928389i
$867$	−38.0000	+	38.0000i	−1.29055	+	1.29055i
$868$		0			0
$869$	42.0000	+	42.0000i	1.42475	+	1.42475i
$870$	13.8564	+	8.00000i	0.469776	+	0.271225i
$871$		0			0
$872$	6.00000	+	10.3923i	0.203186	+	0.351928i
$873$	−25.9808	−	15.0000i	−0.879316	−	0.507673i
$874$	16.0000			0.541208
$875$		0			0
$876$	40.0000	−	40.0000i	1.35147	−	1.35147i
$877$	−7.68653	−	28.6865i	−0.259556	−	0.968675i	−0.965499	−	0.260407i	$-0.916143\pi$
$877$	−7.68653	−	28.6865i	−0.259556	−	0.968675i	0.705943	−	0.708268i	$-0.250523\pi$
$878$	8.78461	−	32.7846i	0.296466	−	1.10643i
$879$	−20.0000	−	34.6410i	−0.674583	−	1.16841i
$880$	41.5692	+	24.0000i	1.40130	+	0.809040i
$881$	30.0000			1.01073			0.505363	−	0.862907i	$-0.331359\pi$
$881$	30.0000			1.01073			0.505363	+	0.862907i	$0.331359\pi$
$882$		0			0
$883$	−17.0000	+	17.0000i	−0.572096	+	0.572096i	−0.932714	−	0.360618i	$-0.882566\pi$
$883$	−17.0000	+	17.0000i	−0.572096	+	0.572096i	0.360618	+	0.932714i	$0.382566\pi$
$884$		0			0
$885$	5.85641	−	21.8564i	0.196861	−	0.734695i
$886$	−17.0000	−	29.4449i	−0.571126	−	0.989220i
$887$	20.7846	+	12.0000i	0.697879	+	0.402921i	0.806557	−	0.591156i	$-0.201328\pi$
$887$	20.7846	+	12.0000i	0.697879	+	0.402921i	−0.108678	+	0.994077i	$0.534662\pi$
$888$	−24.0000	−	24.0000i	−0.805387	−	0.805387i
$889$		0			0
$890$			24.0000i			0.804482i
$891$	−4.09808	+	1.09808i	−0.137291	+	0.0367869i
$892$	−6.92820	+	4.00000i	−0.231973	+	0.133930i
$893$	−43.7128	−	11.7128i	−1.46279	−	0.391954i
$894$	−19.0333	−	71.0333i	−0.636569	−	2.37571i
$895$	−20.0000			−0.668526
$896$		0			0
$897$		0			0
$898$	−12.4449	−	46.4449i	−0.415290	−	1.54989i
$899$	5.46410	+	1.46410i	0.182238	+	0.0488305i
$900$	−25.9808	+	15.0000i	−0.866025	+	0.500000i
$901$	57.3731	−	15.3731i	1.91137	−	0.512151i
$902$			12.0000i			0.399556i
$903$		0			0
$904$	−24.0000	−	24.0000i	−0.798228	−	0.798228i
$905$	13.8564	+	8.00000i	0.460603	+	0.265929i
$906$	12.0000	+	20.7846i	0.398673	+	0.690522i
$907$	−3.29423	+	12.2942i	−0.109383	+	0.408223i	−0.998805	−	0.0488630i	$-0.984440\pi$
$907$	−3.29423	+	12.2942i	−0.109383	+	0.408223i	0.889422	+	0.457086i	$0.151107\pi$
$908$	10.9282	−	2.92820i	0.362665	−	0.0971758i
$909$	10.0000	−	10.0000i	0.331679	−	0.331679i
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−55.4256	−	32.0000i	−1.83533	−	1.05963i
$913$		0			0
$914$	9.51666	−	35.5167i	0.314783	−	1.17479i
$915$	17.5692	+	65.5692i	0.580820	+	2.16765i
$916$	24.0000	−	24.0000i	0.792982	−	0.792982i
$917$		0			0
$918$	48.0000			1.58424
$919$	−48.4974	−	28.0000i	−1.59978	−	0.923635i	−0.991528	−	0.129893i	$-0.958537\pi$
$919$	−48.4974	−	28.0000i	−1.59978	−	0.923635i	−0.608254	−	0.793742i	$-0.708130\pi$
$920$	−8.00000	−	13.8564i	−0.263752	−	0.456832i
$921$	−27.7128	+	16.0000i	−0.913168	+	0.527218i
$922$	48.4974	+	28.0000i	1.59718	+	0.922131i
$923$		0			0
$924$		0			0
$925$	−9.00000	+	9.00000i	−0.295918	+	0.295918i
$926$	−5.12436	−	19.1244i	−0.168397	−	0.628465i
$927$		0			0
$928$	6.92820	−	4.00000i	0.227429	−	0.131306i
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	−0.507825	−	0.861460i	$-0.669550\pi$
$929$	15.0000	−	25.9808i	0.492134	−	0.852401i	0.999959	+	0.00905914i	$0.00288365\pi$
$930$	−32.0000	+	32.0000i	−1.04932	+	1.04932i
$931$		0			0
$932$	−16.0000			−0.524097
$933$	−10.9282	+	2.92820i	−0.357773	+	0.0958651i
$934$	−27.7128	+	16.0000i	−0.906791	+	0.523536i
$935$	−62.3538	+	36.0000i	−2.03919	+	1.17733i
$936$		0			0
$937$			50.0000i			1.63343i	0.577042	+	0.816714i	$0.304207\pi$
$937$			50.0000i			1.63343i	−0.577042	+	0.816714i	$0.695793\pi$
$938$		0			0
$939$	4.00000	+	4.00000i	0.130535	+	0.130535i
$940$	11.7128	+	43.7128i	0.382030	+	1.42575i
$941$	−35.5167	−	9.51666i	−1.15781	−	0.310234i	−0.371719	−	0.928345i	$-0.621232\pi$
$941$	−35.5167	−	9.51666i	−1.15781	−	0.310234i	−0.786091	+	0.618111i	$0.787898\pi$
$942$		0			0
$943$	2.00000	−	3.46410i	0.0651290	−	0.112807i
$944$	−8.00000	−	8.00000i	−0.260378	−	0.260378i
$945$		0			0
$946$	−30.0000	−	30.0000i	−0.975384	−	0.975384i
$947$	4.75833	+	17.7583i	0.154625	+	0.577068i	0.999137	+	0.0415319i	$0.0132238\pi$
$947$	4.75833	+	17.7583i	0.154625	+	0.577068i	−0.844512	+	0.535536i	$0.820110\pi$
$948$	−20.4974	+	76.4974i	−0.665725	+	2.48452i
$949$		0			0
$950$	−12.0000	+	20.7846i	−0.389331	+	0.674342i
$951$		−	60.0000i		−	1.94563i
$952$		0			0
$953$		−	16.0000i		−	0.518291i	−0.965838	−	0.259145i	$-0.916559\pi$
$953$		−	16.0000i		−	0.518291i	0.965838	−	0.259145i	$-0.0834409\pi$
$954$	−60.6218	−	35.0000i	−1.96270	−	1.13317i
$955$	−16.1051	+	60.1051i	−0.521149	+	1.94496i
$956$	13.8564	−	8.00000i	0.448148	−	0.258738i
$957$	−4.39230	−	16.3923i	−0.141983	−	0.529888i
$958$	8.00000	−	8.00000i	0.258468	−	0.258468i
$959$		0			0
$960$			64.0000i			2.06559i
$961$	7.50000	−	12.9904i	0.241935	−	0.419045i
$962$		0			0
$963$	−34.1506	−	9.15064i	−1.10049	−	0.294875i
$964$	−38.1051	−	22.0000i	−1.22728	−	0.708572i
$965$	48.0000	+	48.0000i	1.54517	+	1.54517i
$966$		0			0
$967$		−	38.0000i		−	1.22200i	−0.791632	−	0.610999i	$-0.790768\pi$
$967$		−	38.0000i		−	1.22200i	0.791632	−	0.610999i	$-0.209232\pi$
$968$	−5.12436	−	19.1244i	−0.164703	−	0.614680i
$969$	83.1384	−	48.0000i	2.67079	−	1.54198i
$970$	12.0000	+	20.7846i	0.385297	+	0.667354i
$971$	−35.5167	+	9.51666i	−1.13978	+	0.305404i	−0.778864	−	0.627193i	$-0.784204\pi$
$971$	−35.5167	+	9.51666i	−1.13978	+	0.305404i	−0.360920	+	0.932597i	$0.617537\pi$
$972$	20.0000	+	20.0000i	0.641500	+	0.641500i
$973$		0			0
$974$	2.00000	+	2.00000i	0.0640841	+	0.0640841i
$975$		0			0
$976$	32.7846	+	8.78461i	1.04941	+	0.281189i
$977$	17.0000	+	29.4449i	0.543878	+	0.942025i	0.998677	+	0.0514302i	$0.0163780\pi$
$977$	17.0000	+	29.4449i	0.543878	+	0.942025i	−0.454798	+	0.890594i	$0.650289\pi$
$978$	−27.3205	+	7.32051i	−0.873614	+	0.234084i
$979$	18.0000	−	18.0000i	0.575282	−	0.575282i
$980$		0			0
$981$	15.0000	+	15.0000i	0.478913	+	0.478913i
$982$	−11.0000	+	19.0526i	−0.351024	+	0.607992i
$983$	41.5692	−	24.0000i	1.32585	−	0.765481i	0.341197	−	0.939992i	$-0.389168\pi$
$983$	41.5692	−	24.0000i	1.32585	−	0.765481i	0.984655	+	0.174511i	$0.0558344\pi$
$984$	−13.8564	+	8.00000i	−0.441726	+	0.255031i
$985$	−38.1051	−	22.0000i	−1.21413	−	0.700978i
$986$			12.0000i			0.382158i
$987$		0			0
$988$		0			0
$989$	3.66025	+	13.6603i	0.116389	+	0.434371i
$990$	81.9615	+	21.9615i	2.60491	+	0.697983i
$991$	5.00000	+	8.66025i	0.158830	+	0.275102i	0.934447	−	0.356102i	$-0.115894\pi$
$991$	5.00000	+	8.66025i	0.158830	+	0.275102i	−0.775617	+	0.631204i	$0.782561\pi$
$992$	5.85641	+	21.8564i	0.185941	+	0.693942i
$993$	60.0000			1.90404
$994$		0			0
$995$	−40.0000	+	40.0000i	−1.26809	+	1.26809i
$996$		0			0
$997$	10.9808	−	40.9808i	0.347764	−	1.29787i	−0.541585	−	0.840646i	$-0.682176\pi$
$997$	10.9808	−	40.9808i	0.347764	−	1.29787i	0.889350	−	0.457228i	$-0.151158\pi$
$998$	−25.9808	+	15.0000i	−0.822407	+	0.474817i
$999$	−20.7846	−	12.0000i	−0.657596	−	0.379663i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	784.2.x.g.373.1		4
7.2	even	3		112.2.m.a.85.1	yes	2
7.3	odd	6		784.2.x.b.165.1		4
7.4	even	3	inner	784.2.x.g.165.1		4
7.5	odd	6		784.2.m.c.197.1		2
7.6	odd	2		784.2.x.b.373.1		4
16.13	even	4	inner	784.2.x.g.765.1		4
28.23	odd	6		448.2.m.b.113.1		2
56.37	even	6		896.2.m.d.225.1		2
56.51	odd	6		896.2.m.a.225.1		2
112.13	odd	4		784.2.x.b.765.1		4
112.37	even	12		896.2.m.d.673.1		2
112.45	odd	12		784.2.x.b.557.1		4
112.51	odd	12		448.2.m.b.337.1		2
112.61	odd	12		784.2.m.c.589.1		2
112.93	even	12		112.2.m.a.29.1	✓	2
112.107	odd	12		896.2.m.a.673.1		2
112.109	even	12	inner	784.2.x.g.557.1		4
224.51	odd	24		7168.2.a.i.1.1		2
224.93	even	24		7168.2.a.q.1.1		2
224.163	odd	24		7168.2.a.i.1.2		2
224.205	even	24		7168.2.a.q.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.2.m.a.29.1	✓	2	112.93	even	12
112.2.m.a.85.1	yes	2	7.2	even	3
448.2.m.b.113.1		2	28.23	odd	6
448.2.m.b.337.1		2	112.51	odd	12
784.2.m.c.197.1		2	7.5	odd	6
784.2.m.c.589.1		2	112.61	odd	12
784.2.x.b.165.1		4	7.3	odd	6
784.2.x.b.373.1		4	7.6	odd	2
784.2.x.b.557.1		4	112.45	odd	12
784.2.x.b.765.1		4	112.13	odd	4
784.2.x.g.165.1		4	7.4	even	3	inner
784.2.x.g.373.1		4	1.1	even	1	trivial
784.2.x.g.557.1		4	112.109	even	12	inner
784.2.x.g.765.1		4	16.13	even	4	inner
896.2.m.a.225.1		2	56.51	odd	6
896.2.m.a.673.1		2	112.107	odd	12
896.2.m.d.225.1		2	56.37	even	6
896.2.m.d.673.1		2	112.37	even	12
7168.2.a.i.1.1		2	224.51	odd	24
7168.2.a.i.1.2		2	224.163	odd	24
7168.2.a.q.1.1		2	224.93	even	24
7168.2.a.q.1.2		2	224.205	even	24

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 784 = 2^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	784.x (of order \(12\), degree \(4\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.366025 - 1.36603i) q^{2} +(2.73205 + 0.732051i) q^{3} +(-1.73205 + 1.00000i) q^{4} +(2.73205 - 0.732051i) q^{5} -4.00000i q^{6} +(2.00000 + 2.00000i) q^{8} +(4.33013 + 2.50000i) q^{9} +(-2.00000 - 3.46410i) q^{10} +(-1.09808 + 4.09808i) q^{11} +(-5.46410 + 1.46410i) q^{12} +8.00000 q^{15} +(2.00000 - 3.46410i) q^{16} +(3.00000 + 5.19615i) q^{17} +(1.83013 - 6.83013i) q^{18} +(-1.46410 - 5.46410i) q^{19} +(-4.00000 + 4.00000i) q^{20} +6.00000 q^{22} +(-1.73205 - 1.00000i) q^{23} +(4.00000 + 6.92820i) q^{24} +(2.59808 - 1.50000i) q^{25} +(4.00000 + 4.00000i) q^{27} +(-1.00000 + 1.00000i) q^{29} +(-2.92820 - 10.9282i) q^{30} +(-2.00000 - 3.46410i) q^{31} +(-5.46410 - 1.46410i) q^{32} +(-6.00000 + 10.3923i) q^{33} +(6.00000 - 6.00000i) q^{34} -10.0000 q^{36} +(-4.09808 + 1.09808i) q^{37} +(-6.92820 + 4.00000i) q^{38} +(6.92820 + 4.00000i) q^{40} +2.00000i q^{41} +(-5.00000 - 5.00000i) q^{43} +(-2.19615 - 8.19615i) q^{44} +(13.6603 + 3.66025i) q^{45} +(-0.732051 + 2.73205i) q^{46} +(4.00000 - 6.92820i) q^{47} +(8.00000 - 8.00000i) q^{48} +(-3.00000 - 3.00000i) q^{50} +(4.39230 + 16.3923i) q^{51} +(2.56218 - 9.56218i) q^{53} +(4.00000 - 6.92820i) q^{54} +12.0000i q^{55} -16.0000i q^{57} +(1.73205 + 1.00000i) q^{58} +(0.732051 - 2.73205i) q^{59} +(-13.8564 + 8.00000i) q^{60} +(2.19615 + 8.19615i) q^{61} +(-4.00000 + 4.00000i) q^{62} +8.00000i q^{64} +(16.3923 + 4.39230i) q^{66} +(6.83013 + 1.83013i) q^{67} +(-10.3923 - 6.00000i) q^{68} +(-4.00000 - 4.00000i) q^{69} -8.00000i q^{71} +(3.66025 + 13.6603i) q^{72} +(-8.66025 + 5.00000i) q^{73} +(3.00000 + 5.19615i) q^{74} +(8.19615 - 2.19615i) q^{75} +(8.00000 + 8.00000i) q^{76} +(7.00000 - 12.1244i) q^{79} +(2.92820 - 10.9282i) q^{80} +(0.500000 + 0.866025i) q^{81} +(2.73205 - 0.732051i) q^{82} +(12.0000 + 12.0000i) q^{85} +(-5.00000 + 8.66025i) q^{86} +(-3.46410 + 2.00000i) q^{87} +(-10.3923 + 6.00000i) q^{88} +(-5.19615 - 3.00000i) q^{89} -20.0000i q^{90} +4.00000 q^{92} +(-2.92820 - 10.9282i) q^{93} +(-10.9282 - 2.92820i) q^{94} +(-8.00000 - 13.8564i) q^{95} +(-13.8564 - 8.00000i) q^{96} -6.00000 q^{97} +(-15.0000 + 15.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{3} + 4 q^{5} + 8 q^{8} - 8 q^{10} + 6 q^{11} - 8 q^{12} + 32 q^{15} + 8 q^{16} + 12 q^{17} - 10 q^{18} + 8 q^{19} - 16 q^{20} + 24 q^{22} + 16 q^{24} + 16 q^{27} - 4 q^{29} + 16 q^{30}+ \cdots - 60 q^{99}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^3 + 4 * q^5 + 8 * q^8 - 8 * q^10 + 6 * q^11 - 8 * q^12 + 32 * q^15 + 8 * q^16 + 12 * q^17 - 10 * q^18 + 8 * q^19 - 16 * q^20 + 24 * q^22 + 16 * q^24 + 16 * q^27 - 4 * q^29 + 16 * q^30 - 8 * q^31 - 8 * q^32 - 24 * q^33 + 24 * q^34 - 40 * q^36 - 6 * q^37 - 20 * q^43 + 12 * q^44 + 20 * q^45 + 4 * q^46 + 16 * q^47 + 32 * q^48 - 12 * q^50 - 24 * q^51 - 14 * q^53 + 16 * q^54 - 4 * q^59 - 12 * q^61 - 16 * q^62 + 24 * q^66 + 10 * q^67 - 16 * q^69 - 20 * q^72 + 12 * q^74 + 12 * q^75 + 32 * q^76 + 28 * q^79 - 16 * q^80 + 2 * q^81 + 4 * q^82 + 48 * q^85 - 20 * q^86 + 16 * q^92 + 16 * q^93 - 16 * q^94 - 32 * q^95 - 24 * q^97 - 60 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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