LMFDB - Embedded newform 408.2.ba.b.121.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [408,2,Mod(25,408)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(408, base_ring=CyclotomicField(8)) chi = DirichletCharacter(H, H._module([0, 0, 0, 5])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("408.25"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 408 = 2^{3} \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	408.ba (of order $8$, degree $4$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$3.25789640247$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$4$ over $\Q(\zeta_{8})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 8 x^{15} + 40 x^{14} - 112 x^{13} + 166 x^{12} - 328 x^{11} + 728 x^{10} - 368 x^{9} + \cdots + 12689 $ x^16 - 8x^15 + 40x^14 - 112x^13 + 166x^12 - 328x^11 + 728x^10 - 368x^9 + 210x^8 - 2712x^7 + 104x^6 + 11840x^5 - 6586x^4 - 32760x^3 + 63448x^2 - 46336*x + 12689
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			121.3
Root			$2.30326 - 0.259235i$ of defining polynomial
Character	$\chi$	$=$	408.121
Dual form			408.2.ba.b.145.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.923880 - 0.382683i) q^{3} +(0.116792 + 0.281961i) q^{5} +(1.76284 - 4.25587i) q^{7} +(0.707107 - 0.707107i) q^{9} +(-1.38172 - 0.572328i) q^{11} +0.482039i q^{13} +(0.215804 + 0.215804i) q^{15} +(-3.68244 - 1.85462i) q^{17} +(0.608716 + 0.608716i) q^{19} -4.60653i q^{21} +(7.19822 + 2.98160i) q^{23} +(3.46967 - 3.46967i) q^{25} +(0.382683 - 0.923880i) q^{27} +(1.46885 + 3.54612i) q^{29} +(2.31123 - 0.957343i) q^{31} -1.49557 q^{33} +1.40588 q^{35} +(-9.91949 + 4.10879i) q^{37} +(0.184468 + 0.445346i) q^{39} +(-1.00000 + 2.41422i) q^{41} +(6.58440 - 6.58440i) q^{43} +(0.281961 + 0.116792i) q^{45} +4.07637i q^{47} +(-10.0551 - 10.0551i) q^{49} +(-4.11187 - 0.304232i) q^{51} +(2.85393 + 2.85393i) q^{53} -0.456435i q^{55} +(0.795325 + 0.329435i) q^{57} +(-4.56250 + 4.56250i) q^{59} +(-4.08150 + 9.85361i) q^{61} +(-1.76284 - 4.25587i) q^{63} +(-0.135916 + 0.0562983i) q^{65} +8.51383 q^{67} +7.79129 q^{69} +(-13.2795 + 5.50056i) q^{71} +(1.02252 + 2.46859i) q^{73} +(1.87777 - 4.53335i) q^{75} +(-4.87151 + 4.87151i) q^{77} +(8.89610 + 3.68488i) q^{79} -1.00000i q^{81} +(0.329900 + 0.329900i) q^{83} +(0.0928492 - 1.25491i) q^{85} +(2.71409 + 2.71409i) q^{87} +7.33259i q^{89} +(2.05150 + 0.849758i) q^{91} +(1.76894 - 1.76894i) q^{93} +(-0.100541 + 0.242727i) q^{95} +(-1.02663 - 2.47851i) q^{97} +(-1.38172 + 0.572328i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{11} - 8 q^{15} - 16 q^{17} + 8 q^{19} + 8 q^{23} + 24 q^{25} + 8 q^{31} + 24 q^{33} + 16 q^{35} - 8 q^{37} + 24 q^{39} - 8 q^{41} + 8 q^{43} + 8 q^{45} - 40 q^{49} - 8 q^{51} - 24 q^{53} + 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) $ 16 * q + 8 * q^11 - 8 * q^15 - 16 * q^17 + 8 * q^19 + 8 * q^23 + 24 * q^25 + 8 * q^31 + 24 * q^33 + 16 * q^35 - 8 * q^37 + 24 * q^39 - 8 * q^41 + 8 * q^43 + 8 * q^45 - 40 * q^49 - 8 * q^51 - 24 * q^53 + 8 * q^57 - 32 * q^59 - 48 * q^61 + 48 * q^65 - 24 * q^69 - 32 * q^71 - 40 * q^73 + 16 * q^79 - 48 * q^83 + 32 * q^85 - 16 * q^87 + 16 * q^91 + 24 * q^93 - 56 * q^95 - 40 * q^97 + 8 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/408\mathbb{Z}\right)^\times$.

$n$	$103$	$137$	$205$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{7}{8}\right)$

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			0
$2$		0			0
$3$	0.923880	−	0.382683i	0.533402	−	0.220942i
$3$	0.923880	−	0.382683i	0.533402	−	0.220942i
$4$		0			0
$5$	0.116792	+	0.281961i	0.0522310	+	0.126097i	0.947841	−	0.318743i	$-0.103261\pi$
$5$	0.116792	+	0.281961i	0.0522310	+	0.126097i	−0.895610	+	0.444840i	$0.853261\pi$
$6$		0			0
$7$	1.76284	−	4.25587i	0.666291	−	1.60857i	−0.121475	−	0.992595i	$-0.538762\pi$
$7$	1.76284	−	4.25587i	0.666291	−	1.60857i	0.787766	−	0.615975i	$-0.211238\pi$
$8$		0			0
$9$	0.707107	−	0.707107i	0.235702	−	0.235702i
$10$		0			0
$11$	−1.38172	−	0.572328i	−0.416605	−	0.172563i	0.164527	−	0.986373i	$-0.447390\pi$
$11$	−1.38172	−	0.572328i	−0.416605	−	0.172563i	−0.581132	+	0.813809i	$0.697390\pi$
$12$		0			0
$13$			0.482039i			0.133693i	0.997763	+	0.0668467i	$0.0212939\pi$
$13$			0.482039i			0.133693i	−0.997763	+	0.0668467i	$0.978706\pi$
$14$		0			0
$15$	0.215804	+	0.215804i	0.0557203	+	0.0557203i
$16$		0			0
$17$	−3.68244	−	1.85462i	−0.893124	−	0.449811i
$17$	−3.68244	−	1.85462i	−0.893124	−	0.449811i
$18$		0			0
$19$	0.608716	+	0.608716i	0.139649	+	0.139649i	0.773475	−	0.633826i	$-0.218517\pi$
$19$	0.608716	+	0.608716i	0.139649	+	0.139649i	−0.633826	+	0.773475i	$0.718517\pi$
$20$		0			0
$21$		−	4.60653i		−	1.00523i
$22$		0			0
$23$	7.19822	+	2.98160i	1.50093	+	0.621706i	0.973663	−	0.227992i	$-0.0732160\pi$
$23$	7.19822	+	2.98160i	1.50093	+	0.621706i	0.527269	+	0.849698i	$0.323216\pi$
$24$		0			0
$25$	3.46967	−	3.46967i	0.693934	−	0.693934i
$26$		0			0
$27$	0.382683	−	0.923880i	0.0736475	−	0.177801i
$28$		0			0
$29$	1.46885	+	3.54612i	0.272759	+	0.658499i	0.999599	−	0.0283080i	$-0.00901192\pi$
$29$	1.46885	+	3.54612i	0.272759	+	0.658499i	−0.726840	+	0.686807i	$0.759012\pi$
$30$		0			0
$31$	2.31123	−	0.957343i	0.415109	−	0.171944i	−0.165347	−	0.986235i	$-0.552874\pi$
$31$	2.31123	−	0.957343i	0.415109	−	0.171944i	0.580456	+	0.814292i	$0.302874\pi$
$32$		0			0
$33$	−1.49557			−0.260345
$34$		0			0
$35$	1.40588			0.237637
$36$		0			0
$37$	−9.91949	+	4.10879i	−1.63075	+	0.675481i	−0.995316	−	0.0966712i	$-0.969180\pi$
$37$	−9.91949	+	4.10879i	−1.63075	+	0.675481i	−0.635438	+	0.772152i	$0.719180\pi$
$38$		0			0
$39$	0.184468	+	0.445346i	0.0295386	+	0.0713124i
$40$		0			0
$41$	−1.00000	+	2.41422i	−0.156174	+	0.377038i	−0.982528	−	0.186113i	$-0.940411\pi$
$41$	−1.00000	+	2.41422i	−0.156174	+	0.377038i	0.826354	+	0.563151i	$0.190411\pi$
$42$		0			0
$43$	6.58440	−	6.58440i	1.00411	−	1.00411i	0.00411989	−	0.999992i	$-0.498689\pi$
$43$	6.58440	−	6.58440i	1.00411	−	1.00411i	0.999992	−	0.00411989i	$-0.00131140\pi$
$44$		0			0
$45$	0.281961	+	0.116792i	0.0420323	+	0.0174103i
$46$		0			0
$47$			4.07637i			0.594600i	0.954784	+	0.297300i	$0.0960862\pi$
$47$			4.07637i			0.594600i	−0.954784	+	0.297300i	$0.903914\pi$
$48$		0			0
$49$	−10.0551	−	10.0551i	−1.43644	−	1.43644i
$50$		0			0
$51$	−4.11187	−	0.304232i	−0.575776	−	0.0426010i
$52$		0			0
$53$	2.85393	+	2.85393i	0.392018	+	0.392018i	0.875406	−	0.483388i	$-0.160594\pi$
$53$	2.85393	+	2.85393i	0.392018	+	0.392018i	−0.483388	+	0.875406i	$0.660594\pi$
$54$		0			0
$55$		−	0.456435i		−	0.0615457i
$56$		0			0
$57$	0.795325	+	0.329435i	0.105343	+	0.0436347i
$58$		0			0
$59$	−4.56250	+	4.56250i	−0.593987	+	0.593987i	−0.938706	−	0.344719i	$-0.887974\pi$
$59$	−4.56250	+	4.56250i	−0.593987	+	0.593987i	0.344719	+	0.938706i	$0.387974\pi$
$60$		0			0
$61$	−4.08150	+	9.85361i	−0.522583	+	1.26163i	0.413711	+	0.910408i	$0.364232\pi$
$61$	−4.08150	+	9.85361i	−0.522583	+	1.26163i	−0.936294	+	0.351218i	$0.885768\pi$
$62$		0			0
$63$	−1.76284	−	4.25587i	−0.222097	−	0.536190i
$64$		0			0
$65$	−0.135916	+	0.0562983i	−0.0168583	+	0.00698295i
$66$		0			0
$67$	8.51383			1.04013			0.520065	−	0.854127i	$-0.325908\pi$
$67$	8.51383			1.04013			0.520065	+	0.854127i	$0.325908\pi$
$68$		0			0
$69$	7.79129			0.937962
$70$		0			0
$71$	−13.2795	+	5.50056i	−1.57599	+	0.652796i	−0.987772	−	0.155906i	$-0.950170\pi$
$71$	−13.2795	+	5.50056i	−1.57599	+	0.652796i	−0.588218	+	0.808702i	$0.700170\pi$
$72$		0			0
$73$	1.02252	+	2.46859i	0.119677	+	0.288926i	0.972354	−	0.233513i	$-0.0750221\pi$
$73$	1.02252	+	2.46859i	0.119677	+	0.288926i	−0.852676	+	0.522439i	$0.825022\pi$
$74$		0			0
$75$	1.87777	−	4.53335i	0.216827	−	0.523466i
$76$		0			0
$77$	−4.87151	+	4.87151i	−0.555161	+	0.555161i
$78$		0			0
$79$	8.89610	+	3.68488i	1.00089	+	0.414582i	0.822123	−	0.569310i	$-0.192789\pi$
$79$	8.89610	+	3.68488i	1.00089	+	0.414582i	0.178766	+	0.983892i	$0.442789\pi$
$80$		0			0
$81$		−	1.00000i		−	0.111111i
$82$		0			0
$83$	0.329900	+	0.329900i	0.0362112	+	0.0362112i	0.724981	−	0.688769i	$-0.241849\pi$
$83$	0.329900	+	0.329900i	0.0362112	+	0.0362112i	−0.688769	+	0.724981i	$0.741849\pi$
$84$		0			0
$85$	0.0928492	−	1.25491i	0.0100709	−	0.136114i
$86$		0			0
$87$	2.71409	+	2.71409i	0.290981	+	0.290981i
$88$		0			0
$89$			7.33259i			0.777253i	0.921395	+	0.388626i	$0.127050\pi$
$89$			7.33259i			0.777253i	−0.921395	+	0.388626i	$0.872950\pi$
$90$		0			0
$91$	2.05150	+	0.849758i	0.215055	+	0.0890788i
$92$		0			0
$93$	1.76894	−	1.76894i	0.183430	−	0.183430i
$94$		0			0
$95$	−0.100541	+	0.242727i	−0.0103153	+	0.0249033i
$96$		0			0
$97$	−1.02663	−	2.47851i	−0.104239	−	0.251654i	0.863152	−	0.504945i	$-0.168487\pi$
$97$	−1.02663	−	2.47851i	−0.104239	−	0.251654i	−0.967390	+	0.253291i	$0.918487\pi$
$98$		0			0
$99$	−1.38172	+	0.572328i	−0.138868	+	0.0575212i
$100$		0			0
$101$	−0.478543			−0.0476168			−0.0238084	−	0.999717i	$-0.507579\pi$
$101$	−0.478543			−0.0476168			−0.0238084	+	0.999717i	$0.507579\pi$
$102$		0			0
$103$	−11.7997			−1.16266			−0.581331	−	0.813667i	$-0.697468\pi$
$103$	−11.7997			−1.16266			−0.581331	+	0.813667i	$0.697468\pi$
$104$		0			0
$105$	1.29886	−	0.538006i	0.126756	−	0.0525040i
$106$		0			0
$107$	3.20308	+	7.73292i	0.309654	+	0.747570i	0.999716	+	0.0238203i	$0.00758296\pi$
$107$	3.20308	+	7.73292i	0.309654	+	0.747570i	−0.690063	+	0.723750i	$0.742417\pi$
$108$		0			0
$109$	−3.40018	+	8.20877i	−0.325679	+	0.786258i	0.673225	+	0.739438i	$0.264909\pi$
$109$	−3.40018	+	8.20877i	−0.325679	+	0.786258i	−0.998903	+	0.0468198i	$0.985091\pi$
$110$		0			0
$111$	−7.59205	+	7.59205i	−0.720606	+	0.720606i
$112$		0			0
$113$	−10.8462	−	4.49265i	−1.02033	−	0.422633i	−0.191115	−	0.981568i	$-0.561210\pi$
$113$	−10.8462	−	4.49265i	−1.02033	−	0.422633i	−0.829212	+	0.558934i	$0.811210\pi$
$114$		0			0
$115$			2.37784i			0.221735i
$116$		0			0
$117$	0.340853	+	0.340853i	0.0315119	+	0.0315119i
$118$		0			0
$119$	−14.3846	+	12.4026i	−1.31863	+	1.13695i
$120$		0			0
$121$	−6.19658	−	6.19658i	−0.563325	−	0.563325i
$122$		0			0
$123$			2.61313i			0.235618i
$124$		0			0
$125$	2.79335	+	1.15704i	0.249845	+	0.103489i
$126$		0			0
$127$	1.71765	−	1.71765i	0.152417	−	0.152417i	−0.626780	−	0.779197i	$-0.715627\pi$
$127$	1.71765	−	1.71765i	0.152417	−	0.152417i	0.779197	+	0.626780i	$0.215627\pi$
$128$		0			0
$129$	3.56345	−	8.60293i	0.313744	−	0.757446i
$130$		0			0
$131$	−7.44022	−	17.9623i	−0.650055	−	1.56937i	−0.812697	−	0.582687i	$-0.802001\pi$
$131$	−7.44022	−	17.9623i	−0.650055	−	1.56937i	0.162642	−	0.986685i	$-0.447999\pi$
$132$		0			0
$133$	3.66369	−	1.51755i	0.317682	−	0.131588i
$134$		0			0
$135$	0.305192			0.0262668
$136$		0			0
$137$	11.3855			0.972728			0.486364	−	0.873756i	$-0.338323\pi$
$137$	11.3855			0.972728			0.486364	+	0.873756i	$0.338323\pi$
$138$		0			0
$139$	0.434226	−	0.179862i	0.0368306	−	0.0152557i	−0.364192	−	0.931324i	$-0.618655\pi$
$139$	0.434226	−	0.179862i	0.0368306	−	0.0152557i	0.401023	+	0.916068i	$0.368655\pi$
$140$		0			0
$141$	1.55996	+	3.76608i	0.131372	+	0.317161i
$142$		0			0
$143$	0.275884	−	0.666044i	0.0230706	−	0.0556974i
$144$		0			0
$145$	−0.828318	+	0.828318i	−0.0687881	+	0.0687881i
$146$		0			0
$147$	−13.1376	−	5.44179i	−1.08357	−	0.448831i
$148$		0			0
$149$			7.98133i			0.653856i	0.945049	+	0.326928i	$0.106013\pi$
$149$			7.98133i			0.653856i	−0.945049	+	0.326928i	$0.893987\pi$
$150$		0			0
$151$	−6.32113	−	6.32113i	−0.514406	−	0.514406i	0.401467	−	0.915873i	$-0.368500\pi$
$151$	−6.32113	−	6.32113i	−0.514406	−	0.514406i	−0.915873	+	0.401467i	$0.868500\pi$
$152$		0			0
$153$	−3.91529	+	1.29247i	−0.316533	+	0.104490i
$154$		0			0
$155$	0.539867	+	0.539867i	0.0433632	+	0.0433632i
$156$		0			0
$157$			17.6946i			1.41218i	0.708123	+	0.706090i	$0.249542\pi$
$157$			17.6946i			1.41218i	−0.708123	+	0.706090i	$0.750458\pi$
$158$		0			0
$159$	3.72884	+	1.54454i	0.295716	+	0.122490i
$160$		0			0
$161$	25.3786	−	25.3786i	2.00012	−	2.00012i
$162$		0			0
$163$	−5.18199	+	12.5104i	−0.405885	+	0.979892i	0.580324	+	0.814386i	$0.302926\pi$
$163$	−5.18199	+	12.5104i	−0.405885	+	0.979892i	−0.986209	+	0.165507i	$0.947074\pi$
$164$		0			0
$165$	−0.174670	−	0.421691i	−0.0135981	−	0.0328286i
$166$		0			0
$167$	17.8198	−	7.38120i	1.37894	−	0.571174i	0.434740	−	0.900556i	$-0.356840\pi$
$167$	17.8198	−	7.38120i	1.37894	−	0.571174i	0.944197	+	0.329382i	$0.106840\pi$
$168$		0			0
$169$	12.7676			0.982126
$170$		0			0
$171$	0.860854			0.0658311
$172$		0			0
$173$	20.3010	−	8.40895i	1.54346	−	0.639321i	0.561338	−	0.827586i	$-0.310287\pi$
$173$	20.3010	−	8.40895i	1.54346	−	0.639321i	0.982118	+	0.188266i	$0.0602866\pi$
$174$		0			0
$175$	−8.65001	−	20.8830i	−0.653879	−	1.57860i
$176$		0			0
$177$	−2.46921	+	5.96119i	−0.185597	+	0.448071i
$178$		0			0
$179$	−10.5117	+	10.5117i	−0.785685	+	0.785685i	−0.980784	−	0.195099i	$-0.937497\pi$
$179$	−10.5117	+	10.5117i	−0.785685	+	0.785685i	0.195099	+	0.980784i	$0.437497\pi$
$180$		0			0
$181$	15.0642	+	6.23981i	1.11972	+	0.463801i	0.864273	−	0.503023i	$-0.167779\pi$
$181$	15.0642	+	6.23981i	1.11972	+	0.463801i	0.255443	+	0.966824i	$0.417779\pi$
$182$		0			0
$183$			10.6655i			0.788414i
$184$		0			0
$185$	−2.31704	−	2.31704i	−0.170352	−	0.170352i
$186$		0			0
$187$	4.02667	+	4.67013i	0.294459	+	0.341514i
$188$		0			0
$189$	−3.25731	−	3.25731i	−0.236934	−	0.236934i
$190$		0			0
$191$		−	17.6217i		−	1.27506i	−0.770425	−	0.637530i	$-0.779956\pi$
$191$		−	17.6217i		−	1.27506i	0.770425	−	0.637530i	$-0.220044\pi$
$192$		0			0
$193$	3.04507	+	1.26131i	0.219189	+	0.0907911i	0.489576	−	0.871961i	$-0.337152\pi$
$193$	3.04507	+	1.26131i	0.219189	+	0.0907911i	−0.270387	+	0.962752i	$0.587152\pi$
$194$		0			0
$195$	−0.104026	+	0.104026i	−0.00744944	+	0.00744944i
$196$		0			0
$197$	0.807382	−	1.94919i	0.0575236	−	0.138874i	−0.892505	−	0.451038i	$-0.851054\pi$
$197$	0.807382	−	1.94919i	0.0575236	−	0.138874i	0.950028	+	0.312164i	$0.101054\pi$
$198$		0			0
$199$	−8.78628	−	21.2119i	−0.622843	−	1.50367i	−0.848351	−	0.529434i	$-0.822404\pi$
$199$	−8.78628	−	21.2119i	−0.622843	−	1.50367i	0.225509	−	0.974241i	$-0.427596\pi$
$200$		0			0
$201$	7.86575	−	3.25810i	0.554807	−	0.229809i
$202$		0			0
$203$	17.6812			1.24098
$204$		0			0
$205$	−0.797509			−0.0557004
$206$		0			0
$207$	7.19822	−	2.98160i	0.500311	−	0.207235i
$208$		0			0
$209$	−0.492691	−	1.18946i	−0.0340802	−	0.0822768i
$210$		0			0
$211$	3.78592	−	9.14001i	0.260633	−	0.629225i	−0.738345	−	0.674424i	$-0.764392\pi$
$211$	3.78592	−	9.14001i	0.260633	−	0.629225i	0.998978	+	0.0451990i	$0.0143922\pi$
$212$		0			0
$213$	−10.1637	+	10.1637i	−0.696406	+	0.696406i
$214$		0			0
$215$	2.62555	+	1.08754i	0.179061	+	0.0741695i
$216$		0			0
$217$		−	11.5239i		−	0.782297i
$218$		0			0
$219$	1.88938	+	1.88938i	0.127672	+	0.127672i
$220$		0			0
$221$	0.893997	−	1.77508i	0.0601367	−	0.119405i
$222$		0			0
$223$	8.51112	+	8.51112i	0.569947	+	0.569947i	0.932113	−	0.362167i	$-0.117963\pi$
$223$	8.51112	+	8.51112i	0.569947	+	0.569947i	−0.362167	+	0.932113i	$0.617963\pi$
$224$		0			0
$225$		−	4.90686i		−	0.327124i
$226$		0			0
$227$	−25.2633	−	10.4644i	−1.67678	−	0.694546i	−0.677618	−	0.735414i	$-0.736988\pi$
$227$	−25.2633	−	10.4644i	−1.67678	−	0.694546i	−0.999165	+	0.0408684i	$0.986988\pi$
$228$		0			0
$229$	12.0421	−	12.0421i	0.795763	−	0.795763i	−0.186661	−	0.982424i	$-0.559767\pi$
$229$	12.0421	−	12.0421i	0.795763	−	0.795763i	0.982424	+	0.186661i	$0.0597666\pi$
$230$		0			0
$231$	−2.63644	+	6.36494i	−0.173465	+	0.418782i
$232$		0			0
$233$	−9.87784	−	23.8472i	−0.647119	−	1.56228i	−0.816886	−	0.576799i	$-0.804302\pi$
$233$	−9.87784	−	23.8472i	−0.647119	−	1.56228i	0.169767	−	0.985484i	$-0.445698\pi$
$234$		0			0
$235$	−1.14938	+	0.476088i	−0.0749772	+	0.0310566i
$236$		0			0
$237$	9.62907			0.625475
$238$		0			0
$239$	10.6595			0.689503			0.344752	−	0.938694i	$-0.387963\pi$
$239$	10.6595			0.689503			0.344752	+	0.938694i	$0.387963\pi$
$240$		0			0
$241$	−15.7902	+	6.54053i	−1.01714	+	0.421312i	−0.828054	−	0.560648i	$-0.810552\pi$
$241$	−15.7902	+	6.54053i	−1.01714	+	0.421312i	−0.189084	+	0.981961i	$0.560552\pi$
$242$		0			0
$243$	−0.382683	−	0.923880i	−0.0245492	−	0.0592669i
$244$		0			0
$245$	1.66079	−	4.00951i	0.106104	−	0.256158i
$246$		0			0
$247$	−0.293425	+	0.293425i	−0.0186702	+	0.0186702i
$248$		0			0
$249$	0.431035	+	0.178541i	0.0273157	+	0.0113145i
$250$		0			0
$251$		−	17.8022i		−	1.12367i	−0.827250	−	0.561834i	$-0.810096\pi$
$251$		−	17.8022i		−	1.12367i	0.827250	−	0.561834i	$-0.189904\pi$
$252$		0			0
$253$	−8.23949	−	8.23949i	−0.518012	−	0.518012i
$254$		0			0
$255$	−0.394452	−	1.19492i	−0.0247015	−	0.0748287i
$256$		0			0
$257$	−7.48656	−	7.48656i	−0.466999	−	0.466999i	0.433942	−	0.900941i	$-0.357122\pi$
$257$	−7.48656	−	7.48656i	−0.466999	−	0.466999i	−0.900941	+	0.433942i	$0.857122\pi$
$258$		0			0
$259$			49.4593i			3.07325i
$260$		0			0
$261$	3.54612	+	1.46885i	0.219500	+	0.0909197i
$262$		0			0
$263$	4.23145	−	4.23145i	0.260922	−	0.260922i	−0.564507	−	0.825429i	$-0.690933\pi$
$263$	4.23145	−	4.23145i	0.260922	−	0.260922i	0.825429	+	0.564507i	$0.190933\pi$
$264$		0			0
$265$	−0.471381	+	1.13801i	−0.0289567	+	0.0699076i
$266$		0			0
$267$	2.80606	+	6.77443i	0.171728	+	0.414588i
$268$		0			0
$269$	−8.18727	+	3.39128i	−0.499186	+	0.206770i	−0.618047	−	0.786141i	$-0.712076\pi$
$269$	−8.18727	+	3.39128i	−0.499186	+	0.206770i	0.118861	+	0.992911i	$0.462076\pi$
$270$		0			0
$271$	30.2531			1.83774			0.918871	−	0.394558i	$-0.129102\pi$
$271$	30.2531			1.83774			0.918871	+	0.394558i	$0.129102\pi$
$272$		0			0
$273$	2.22052			0.134392
$274$		0			0
$275$	−6.77992	+	2.80833i	−0.408844	+	0.169349i
$276$		0			0
$277$	7.29819	+	17.6194i	0.438506	+	1.05865i	0.976465	+	0.215676i	$0.0691955\pi$
$277$	7.29819	+	17.6194i	0.438506	+	1.05865i	−0.537959	+	0.842971i	$0.680804\pi$
$278$		0			0
$279$	0.957343	−	2.31123i	0.0573146	−	0.138370i
$280$		0			0
$281$	6.44462	−	6.44462i	0.384454	−	0.384454i	−0.488250	−	0.872704i	$-0.662365\pi$
$281$	6.44462	−	6.44462i	0.384454	−	0.384454i	0.872704	+	0.488250i	$0.162365\pi$
$282$		0			0
$283$	−17.4877	−	7.24366i	−1.03954	−	0.430591i	−0.203392	−	0.979097i	$-0.565197\pi$
$283$	−17.4877	−	7.24366i	−1.03954	−	0.430591i	−0.836146	+	0.548507i	$0.815197\pi$
$284$		0			0
$285$			0.262726i			0.0155626i
$286$		0			0
$287$	8.51178	+	8.51178i	0.502434	+	0.502434i
$288$		0			0
$289$	10.1208	+	13.6590i	0.595341	+	0.803473i
$290$		0			0
$291$	−1.89697	−	1.89697i	−0.111202	−	0.111202i
$292$		0			0
$293$			16.4534i			0.961217i	0.876935	+	0.480608i	$0.159584\pi$
$293$			16.4534i			0.961217i	−0.876935	+	0.480608i	$0.840416\pi$
$294$		0			0
$295$	−1.81931	−	0.753583i	−0.105924	−	0.0438753i
$296$		0			0
$297$	−1.05752	+	1.05752i	−0.0613638	+	0.0613638i
$298$		0			0
$299$	−1.43725	+	3.46982i	−0.0831181	+	0.200665i
$300$		0			0
$301$	−16.4151	−	39.6296i	−0.946152	−	2.28421i
$302$		0			0
$303$	−0.442116	+	0.183130i	−0.0253989	+	0.0105206i
$304$		0			0
$305$	−3.25502			−0.186382
$306$		0			0
$307$	19.7615			1.12785			0.563925	−	0.825826i	$-0.309291\pi$
$307$	19.7615			1.12785			0.563925	+	0.825826i	$0.309291\pi$
$308$		0			0
$309$	−10.9015	+	4.51556i	−0.620166	+	0.256881i
$310$		0			0
$311$	−4.61187	−	11.1340i	−0.261515	−	0.631353i	0.737518	−	0.675328i	$-0.235998\pi$
$311$	−4.61187	−	11.1340i	−0.261515	−	0.631353i	−0.999033	+	0.0439747i	$0.985998\pi$
$312$		0			0
$313$	−9.74292	+	23.5215i	−0.550702	+	1.32951i	0.366250	+	0.930516i	$0.380642\pi$
$313$	−9.74292	+	23.5215i	−0.550702	+	1.32951i	−0.916952	+	0.398997i	$0.869358\pi$
$314$		0			0
$315$	0.994105	−	0.994105i	0.0560115	−	0.0560115i
$316$		0			0
$317$	−11.1706	−	4.62701i	−0.627403	−	0.259879i	0.0462463	−	0.998930i	$-0.485274\pi$
$317$	−11.1706	−	4.62701i	−0.627403	−	0.259879i	−0.673649	+	0.739051i	$0.735274\pi$
$318$		0			0
$319$		−	5.74043i		−	0.321402i
$320$		0			0
$321$	5.91852	+	5.91852i	0.330340	+	0.330340i
$322$		0			0
$323$	−1.11263	−	3.37050i	−0.0619083	−	0.187539i
$324$		0			0
$325$	1.67252	+	1.67252i	0.0927745	+	0.0927745i
$326$		0			0
$327$			8.88511i			0.491348i
$328$		0			0
$329$	17.3485	+	7.18600i	0.956455	+	0.396177i
$330$		0			0
$331$	−3.55301	+	3.55301i	−0.195291	+	0.195291i	−0.797978	−	0.602687i	$-0.794097\pi$
$331$	−3.55301	+	3.55301i	−0.195291	+	0.195291i	0.602687	+	0.797978i	$0.294097\pi$
$332$		0			0
$333$	−4.10879	+	9.91949i	−0.225160	+	0.543585i
$334$		0			0
$335$	0.994348	+	2.40057i	0.0543270	+	0.131157i
$336$		0			0
$337$	−21.1770	+	8.77178i	−1.15358	+	0.477830i	−0.875733	−	0.482795i	$-0.839622\pi$
$337$	−21.1770	+	8.77178i	−1.15358	+	0.477830i	−0.277849	+	0.960625i	$0.589622\pi$
$338$		0			0
$339$	−11.7399			−0.637622
$340$		0			0
$341$	−3.74139			−0.202608
$342$		0			0
$343$	−30.7277	+	12.7278i	−1.65914	+	0.687239i
$344$		0			0
$345$	0.909961	+	2.19684i	0.0489907	+	0.118274i
$346$		0			0
$347$	9.08405	−	21.9308i	0.487658	−	1.17731i	−0.468238	−	0.883602i	$-0.655111\pi$
$347$	9.08405	−	21.9308i	0.487658	−	1.17731i	0.955895	−	0.293707i	$-0.0948890\pi$
$348$		0			0
$349$	0.813560	−	0.813560i	0.0435488	−	0.0435488i	−0.684997	−	0.728546i	$-0.740197\pi$
$349$	0.813560	−	0.813560i	0.0435488	−	0.0435488i	0.728546	+	0.684997i	$0.240197\pi$
$350$		0			0
$351$	0.445346	+	0.184468i	0.0237708	+	0.00984619i
$352$		0			0
$353$			26.9955i			1.43683i	0.695616	+	0.718413i	$0.255131\pi$
$353$			26.9955i			1.43683i	−0.695616	+	0.718413i	$0.744869\pi$
$354$		0			0
$355$	−3.10189	−	3.10189i	−0.164631	−	0.164631i
$356$		0			0
$357$	−8.54334	+	16.9633i	−0.452161	+	0.897792i
$358$		0			0
$359$	−1.99896	−	1.99896i	−0.105501	−	0.105501i	0.652386	−	0.757887i	$-0.273768\pi$
$359$	−1.99896	−	1.99896i	−0.105501	−	0.105501i	−0.757887	+	0.652386i	$0.773768\pi$
$360$		0			0
$361$		−	18.2589i		−	0.960996i
$362$		0			0
$363$	−8.09622	−	3.35356i	−0.424941	−	0.176016i
$364$		0			0
$365$	−0.576623	+	0.576623i	−0.0301818	+	0.0301818i
$366$		0			0
$367$	−6.60126	+	15.9369i	−0.344583	+	0.831897i	0.652657	+	0.757654i	$0.273654\pi$
$367$	−6.60126	+	15.9369i	−0.344583	+	0.831897i	−0.997240	+	0.0742438i	$0.976346\pi$
$368$		0			0
$369$	1.00000	+	2.41422i	0.0520581	+	0.125679i
$370$		0			0
$371$	17.1770	−	7.11495i	0.891785	−	0.369390i
$372$		0			0
$373$	1.88240			0.0974672			0.0487336	−	0.998812i	$-0.484481\pi$
$373$	1.88240			0.0974672			0.0487336	+	0.998812i	$0.484481\pi$
$374$		0			0
$375$	3.02350			0.156133
$376$		0			0
$377$	−1.70937	+	0.708044i	−0.0880370	+	0.0364661i
$378$		0			0
$379$	8.53921	+	20.6155i	0.438630	+	1.05895i	0.976422	+	0.215869i	$0.0692584\pi$
$379$	8.53921	+	20.6155i	0.438630	+	1.05895i	−0.537792	+	0.843077i	$0.680742\pi$
$380$		0			0
$381$	0.929586	−	2.24422i	0.0476241	−	0.114975i
$382$		0			0
$383$	−14.5167	+	14.5167i	−0.741770	+	0.741770i	−0.972918	−	0.231149i	$-0.925752\pi$
$383$	−14.5167	+	14.5167i	−0.741770	+	0.741770i	0.231149	+	0.972918i	$0.425752\pi$
$384$		0			0
$385$	−1.94253	−	0.804623i	−0.0990006	−	0.0410074i
$386$		0			0
$387$		−	9.31175i		−	0.473343i
$388$		0			0
$389$	−18.9186	−	18.9186i	−0.959213	−	0.959213i	0.0399875	−	0.999200i	$-0.487268\pi$
$389$	−18.9186	−	18.9186i	−0.959213	−	0.959213i	−0.999200	+	0.0399875i	$0.987268\pi$
$390$		0			0
$391$	−20.9773	−	24.3295i	−1.06087	−	1.23040i
$392$		0			0
$393$	−13.7477	−	13.7477i	−0.693482	−	0.693482i
$394$		0			0
$395$			2.93872i			0.147863i
$396$		0			0
$397$	8.39311	+	3.47654i	0.421238	+	0.174482i	0.583225	−	0.812310i	$-0.301790\pi$
$397$	8.39311	+	3.47654i	0.421238	+	0.174482i	−0.161988	+	0.986793i	$0.551790\pi$
$398$		0			0
$399$	2.80406	−	2.80406i	0.140379	−	0.140379i
$400$		0			0
$401$	−0.957292	+	2.31111i	−0.0478049	+	0.115411i	−0.945978	−	0.324231i	$-0.894895\pi$
$401$	−0.957292	+	2.31111i	−0.0478049	+	0.115411i	0.898173	+	0.439642i	$0.144895\pi$
$402$		0			0
$403$	0.461476	+	1.11410i	0.0229878	+	0.0554974i
$404$		0			0
$405$	0.281961	−	0.116792i	0.0140108	−	0.00580345i
$406$		0			0
$407$	16.0576			0.795944
$408$		0			0
$409$	−2.84589			−0.140720			−0.0703600	−	0.997522i	$-0.522415\pi$
$409$	−2.84589			−0.140720			−0.0703600	+	0.997522i	$0.522415\pi$
$410$		0			0
$411$	10.5188	−	4.35704i	0.518855	−	0.214917i
$412$		0			0
$413$	11.3745	+	27.4604i	0.559701	+	1.35124i
$414$		0			0
$415$	−0.0544892	+	0.131549i	−0.00267477	+	0.00645747i
$416$		0			0
$417$	0.332342	−	0.332342i	0.0162749	−	0.0162749i
$418$		0			0
$419$	−2.76137	−	1.14380i	−0.134902	−	0.0558782i	0.314211	−	0.949353i	$-0.398260\pi$
$419$	−2.76137	−	1.14380i	−0.134902	−	0.0558782i	−0.449113	+	0.893475i	$0.648260\pi$
$420$		0			0
$421$			28.5480i			1.39134i	0.718359	+	0.695672i	$0.244893\pi$
$421$			28.5480i			1.39134i	−0.718359	+	0.695672i	$0.755107\pi$
$422$		0			0
$423$	2.88243	+	2.88243i	0.140149	+	0.140149i
$424$		0			0
$425$	−19.2118	+	6.34196i	−0.931909	+	0.307630i
$426$		0			0
$427$	34.7407	+	34.7407i	1.68122	+	1.68122i
$428$		0			0
$429$		−	0.720921i		−	0.0348064i
$430$		0			0
$431$	20.5780	+	8.52369i	0.991208	+	0.410572i	0.818566	−	0.574413i	$-0.194770\pi$
$431$	20.5780	+	8.52369i	0.991208	+	0.410572i	0.172642	+	0.984985i	$0.444770\pi$
$432$		0			0
$433$	−1.70878	+	1.70878i	−0.0821186	+	0.0821186i	−0.746973	−	0.664854i	$-0.768494\pi$
$433$	−1.70878	+	1.70878i	−0.0821186	+	0.0821186i	0.664854	+	0.746973i	$0.268494\pi$
$434$		0			0
$435$	−0.448283	+	1.08225i	−0.0214935	+	0.0518899i
$436$		0			0
$437$	2.56672	+	6.19661i	0.122783	+	0.296424i
$438$		0			0
$439$	−11.4570	+	4.74566i	−0.546815	+	0.226498i	−0.638950	−	0.769248i	$-0.720631\pi$
$439$	−11.4570	+	4.74566i	−0.546815	+	0.226498i	0.0921351	+	0.995747i	$0.470631\pi$
$440$		0			0
$441$	−14.2201			−0.677146
$442$		0			0
$443$	35.9225			1.70673			0.853366	−	0.521312i	$-0.174557\pi$
$443$	35.9225			1.70673			0.853366	+	0.521312i	$0.174557\pi$
$444$		0			0
$445$	−2.06750	+	0.856389i	−0.0980091	+	0.0405967i
$446$		0			0
$447$	3.05432	+	7.37379i	0.144465	+	0.348768i
$448$		0			0
$449$	14.0143	−	33.8336i	0.661378	−	1.59671i	−0.134267	−	0.990945i	$-0.542868\pi$
$449$	14.0143	−	33.8336i	0.661378	−	1.59671i	0.795645	−	0.605763i	$-0.207132\pi$
$450$		0			0
$451$	2.76345	−	2.76345i	0.130126	−	0.130126i
$452$		0			0
$453$	−8.25896	−	3.42097i	−0.388040	−	0.160731i
$454$		0			0
$455$			0.677687i			0.0317705i
$456$		0			0
$457$	6.37867	+	6.37867i	0.298382	+	0.298382i	0.840380	−	0.541998i	$-0.182332\pi$
$457$	6.37867	+	6.37867i	0.298382	+	0.298382i	−0.541998	+	0.840380i	$0.682332\pi$
$458$		0			0
$459$	−3.12265	+	2.69240i	−0.145753	+	0.125671i
$460$		0			0
$461$	−25.4220	−	25.4220i	−1.18402	−	1.18402i	−0.978693	−	0.205329i	$-0.934174\pi$
$461$	−25.4220	−	25.4220i	−1.18402	−	1.18402i	−0.205329	−	0.978693i	$-0.565826\pi$
$462$		0			0
$463$		−	15.1138i		−	0.702399i	−0.936301	−	0.351199i	$-0.885774\pi$
$463$		−	15.1138i		−	0.702399i	0.936301	−	0.351199i	$-0.114226\pi$
$464$		0			0
$465$	0.705370	+	0.292174i	0.0327108	+	0.0135492i
$466$		0			0
$467$	−11.2571	+	11.2571i	−0.520917	+	0.520917i	−0.917848	−	0.396931i	$-0.870075\pi$
$467$	−11.2571	+	11.2571i	−0.520917	+	0.520917i	0.396931	+	0.917848i	$0.370075\pi$
$468$		0			0
$469$	15.0085	−	36.2338i	0.693029	−	1.67312i
$470$		0			0
$471$	6.77141	+	16.3476i	0.312010	+	0.753259i
$472$		0			0
$473$	−12.8663	+	5.32938i	−0.591591	+	0.245045i
$474$		0			0
$475$	4.22409			0.193814
$476$		0			0
$477$	4.03607			0.184799
$478$		0			0
$479$	−14.1676	+	5.86842i	−0.647335	+	0.268135i	−0.682098	−	0.731261i	$-0.738932\pi$
$479$	−14.1676	+	5.86842i	−0.647335	+	0.268135i	0.0347634	+	0.999396i	$0.488932\pi$
$480$		0			0
$481$	−1.98060	−	4.78158i	−0.0903074	−	0.218021i
$482$		0			0
$483$	13.7348	−	33.1588i	0.624956	−	1.50878i
$484$		0			0
$485$	0.578940	−	0.578940i	0.0262883	−	0.0262883i
$486$		0			0
$487$	−19.5757	−	8.10853i	−0.887060	−	0.367432i	−0.107829	−	0.994169i	$-0.534390\pi$
$487$	−19.5757	−	8.10853i	−0.887060	−	0.367432i	−0.779231	+	0.626737i	$0.784390\pi$
$488$		0			0
$489$			13.5412i			0.612354i
$490$		0			0
$491$	−8.52442	−	8.52442i	−0.384702	−	0.384702i	0.488091	−	0.872793i	$-0.337693\pi$
$491$	−8.52442	−	8.52442i	−0.384702	−	0.384702i	−0.872793	+	0.488091i	$0.837693\pi$
$492$		0			0
$493$	1.16773	−	15.7826i	0.0525920	−	0.710811i
$494$		0			0
$495$	−0.322749	−	0.322749i	−0.0145065	−	0.0145065i
$496$		0			0
$497$			66.2126i			2.97004i
$498$		0			0
$499$	−23.3352	−	9.66574i	−1.04463	−	0.432698i	−0.206655	−	0.978414i	$-0.566258\pi$
$499$	−23.3352	−	9.66574i	−1.04463	−	0.432698i	−0.837970	+	0.545716i	$0.816258\pi$
$500$		0			0
$501$	13.6387	−	13.6387i	0.609331	−	0.609331i
$502$		0			0
$503$	−16.4404	+	39.6907i	−0.733042	+	1.76972i	−0.100861	+	0.994901i	$0.532160\pi$
$503$	−16.4404	+	39.6907i	−0.733042	+	1.76972i	−0.632182	+	0.774820i	$0.717840\pi$
$504$		0			0
$505$	−0.0558900	−	0.134930i	−0.00248707	−	0.00600432i
$506$		0			0
$507$	11.7958	−	4.88596i	0.523868	−	0.216993i
$508$		0			0
$509$	−15.8473			−0.702417			−0.351209	−	0.936297i	$-0.614229\pi$
$509$	−15.8473			−0.702417			−0.351209	+	0.936297i	$0.614229\pi$
$510$		0			0
$511$	12.3085			0.544498
$512$		0			0
$513$	0.795325	−	0.329435i	0.0351145	−	0.0145449i
$514$		0			0
$515$	−1.37811	−	3.32706i	−0.0607270	−	0.146608i
$516$		0			0
$517$	2.33302	−	5.63242i	0.102606	−	0.247713i
$518$		0			0
$519$	15.5377	−	15.5377i	0.682030	−	0.682030i
$520$		0			0
$521$	−39.7582	−	16.4684i	−1.74184	−	0.721493i	−0.998624	−	0.0524404i	$-0.983300\pi$
$521$	−39.7582	−	16.4684i	−1.74184	−	0.721493i	−0.743215	−	0.669053i	$-0.766700\pi$
$522$		0			0
$523$		−	27.9070i		−	1.22029i	−0.792290	−	0.610145i	$-0.791111\pi$
$523$		−	27.9070i		−	1.22029i	0.792290	−	0.610145i	$-0.208889\pi$
$524$		0			0
$525$	−15.9831	−	15.9831i	−0.697561	−	0.697561i
$526$		0			0
$527$	−10.2865	−	0.761084i	−0.448086	−	0.0331533i
$528$		0			0
$529$	26.6609	+	26.6609i	1.15917	+	1.15917i
$530$		0			0
$531$			6.45235i			0.280008i
$532$		0			0
$533$	−1.16375	−	0.482040i	−0.0504075	−	0.0208795i
$534$		0			0
$535$	−1.80629	+	1.80629i	−0.0780927	+	0.0780927i
$536$		0			0
$537$	−5.68892	+	13.7343i	−0.245495	+	0.592677i
$538$		0			0
$539$	8.13855	+	19.6482i	0.350552	+	0.846308i
$540$		0			0
$541$	−14.9138	+	6.17749i	−0.641194	+	0.265591i	−0.679501	−	0.733675i	$-0.737804\pi$
$541$	−14.9138	+	6.17749i	−0.641194	+	0.265591i	0.0383068	+	0.999266i	$0.487804\pi$
$542$		0			0
$543$	16.3054			0.699732
$544$		0			0
$545$	−2.71167			−0.116155
$546$		0			0
$547$	−15.0547	+	6.23585i	−0.643692	+	0.266626i	−0.680558	−	0.732694i	$-0.738263\pi$
$547$	−15.0547	+	6.23585i	−0.643692	+	0.266626i	0.0368661	+	0.999320i	$0.488263\pi$
$548$		0			0
$549$	4.08150	+	9.85361i	0.174194	+	0.420542i
$550$		0			0
$551$	−1.26447	+	3.05269i	−0.0538681	+	0.130049i
$552$		0			0
$553$	31.3648	−	31.3648i	1.33377	−	1.33377i
$554$		0			0
$555$	−3.02735	−	1.25397i	−0.128504	−	0.0532281i
$556$		0			0
$557$		−	0.141857i		−	0.00601069i	−0.999995	−	0.00300535i	$-0.999043\pi$
$557$		−	0.141857i		−	0.00601069i	0.999995	−	0.00300535i	$-0.000956633\pi$
$558$		0			0
$559$	3.17394	+	3.17394i	0.134243	+	0.134243i
$560$		0			0
$561$	5.50734	+	2.77370i	0.232520	+	0.117106i
$562$		0			0
$563$	−10.2830	−	10.2830i	−0.433377	−	0.433377i	0.456398	−	0.889776i	$-0.349139\pi$
$563$	−10.2830	−	10.2830i	−0.433377	−	0.433377i	−0.889776	+	0.456398i	$0.849139\pi$
$564$		0			0
$565$		−	3.58292i		−	0.150735i
$566$		0			0
$567$	−4.25587	−	1.76284i	−0.178730	−	0.0740324i
$568$		0			0
$569$	−1.77146	+	1.77146i	−0.0742636	+	0.0742636i	−0.743263	−	0.668999i	$-0.766723\pi$
$569$	−1.77146	+	1.77146i	−0.0742636	+	0.0742636i	0.668999	+	0.743263i	$0.266723\pi$
$570$		0			0
$571$	12.0601	−	29.1156i	0.504698	−	1.21845i	−0.442201	−	0.896916i	$-0.645802\pi$
$571$	12.0601	−	29.1156i	0.504698	−	1.21845i	0.946899	−	0.321532i	$-0.104198\pi$
$572$		0			0
$573$	−6.74353	−	16.2803i	−0.281715	−	0.680120i
$574$		0			0
$575$	35.3206	−	14.6303i	1.47297	−	0.610125i
$576$		0			0
$577$	−7.44450			−0.309918			−0.154959	−	0.987921i	$-0.549525\pi$
$577$	−7.44450			−0.309918			−0.154959	+	0.987921i	$0.549525\pi$
$578$		0			0
$579$	3.29596			0.136976
$580$		0			0
$581$	1.98557	−	0.822451i	0.0823755	−	0.0341210i
$582$		0			0
$583$	−2.30996	−	5.57673i	−0.0956686	−	0.230964i
$584$		0			0
$585$	−0.0562983	+	0.135916i	−0.00232765	+	0.00561944i
$586$		0			0
$587$	23.1517	−	23.1517i	0.955574	−	0.955574i	−0.0434805	−	0.999054i	$-0.513845\pi$
$587$	23.1517	−	23.1517i	0.955574	−	0.955574i	0.999054	+	0.0434805i	$0.0138447\pi$
$588$		0			0
$589$	1.98963	+	0.824132i	0.0819814	+	0.0339578i
$590$		0			0
$591$		−	2.10979i		−	0.0867852i
$592$		0			0
$593$	−4.64593	−	4.64593i	−0.190785	−	0.190785i	0.605250	−	0.796035i	$-0.293073\pi$
$593$	−4.64593	−	4.64593i	−0.190785	−	0.190785i	−0.796035	+	0.605250i	$0.793073\pi$
$594$		0			0
$595$	−5.17706	−	2.60736i	−0.212239	−	0.106891i
$596$		0			0
$597$	−16.2349	−	16.2349i	−0.664451	−	0.664451i
$598$		0			0
$599$		−	11.4830i		−	0.469184i	−0.972094	−	0.234592i	$-0.924625\pi$
$599$		−	11.4830i		−	0.469184i	0.972094	−	0.234592i	$-0.0753754\pi$
$600$		0			0
$601$	33.7282	+	13.9707i	1.37580	+	0.569875i	0.943355	−	0.331785i	$-0.107651\pi$
$601$	33.7282	+	13.9707i	1.37580	+	0.569875i	0.432445	+	0.901660i	$0.357651\pi$
$602$		0			0
$603$	6.02019	−	6.02019i	0.245161	−	0.245161i
$604$		0			0
$605$	1.02348	−	2.47090i	0.0416105	−	0.100457i
$606$		0			0
$607$	−10.8854	−	26.2796i	−0.441824	−	1.06666i	−0.975308	−	0.220847i	$-0.929118\pi$
$607$	−10.8854	−	26.2796i	−0.441824	−	1.06666i	0.533485	−	0.845810i	$-0.320882\pi$
$608$		0			0
$609$	16.3353	−	6.76631i	0.661940	−	0.274185i
$610$		0			0
$611$	−1.96497			−0.0794942
$612$		0			0
$613$	38.1347			1.54025			0.770123	−	0.637895i	$-0.220195\pi$
$613$	38.1347			1.54025			0.770123	+	0.637895i	$0.220195\pi$
$614$		0			0
$615$	−0.736802	+	0.305194i	−0.0297107	+	0.0123066i
$616$		0			0
$617$	16.4667	+	39.7542i	0.662926	+	1.60044i	0.793196	+	0.608966i	$0.208415\pi$
$617$	16.4667	+	39.7542i	0.662926	+	1.60044i	−0.130271	+	0.991478i	$0.541585\pi$
$618$		0			0
$619$	1.58168	−	3.81852i	0.0635733	−	0.153479i	−0.888900	−	0.458101i	$-0.848530\pi$
$619$	1.58168	−	3.81852i	0.0635733	−	0.153479i	0.952474	+	0.304621i	$0.0985299\pi$
$620$		0			0
$621$	5.50928	−	5.50928i	0.221080	−	0.221080i
$622$		0			0
$623$	31.2066	+	12.9262i	1.25027	+	0.517877i
$624$		0			0
$625$		−	23.6115i		−	0.944462i
$626$		0			0
$627$	−0.910374	−	0.910374i	−0.0363569	−	0.0363569i
$628$		0			0
$629$	44.1482	+	3.26647i	1.76030	+	0.130243i
$630$		0			0
$631$	22.2411	+	22.2411i	0.885404	+	0.885404i	0.994077	−	0.108674i	$-0.0346604\pi$
$631$	22.2411	+	22.2411i	0.885404	+	0.885404i	−0.108674	+	0.994077i	$0.534660\pi$
$632$		0			0
$633$		−	9.89308i		−	0.393215i
$634$		0			0
$635$	0.684919	+	0.283703i	0.0271802	+	0.0112584i
$636$		0			0
$637$	4.84695	−	4.84695i	0.192043	−	0.192043i
$638$		0			0
$639$	−5.50056	+	13.2795i	−0.217599	+	0.525330i
$640$		0			0
$641$	6.17579	+	14.9097i	0.243929	+	0.588897i	0.997666	−	0.0682792i	$-0.0217509\pi$
$641$	6.17579	+	14.9097i	0.243929	+	0.588897i	−0.753737	+	0.657176i	$0.771751\pi$
$642$		0			0
$643$	−35.4000	+	14.6632i	−1.39604	+	0.578259i	−0.948721	−	0.316116i	$-0.897621\pi$
$643$	−35.4000	+	14.6632i	−1.39604	+	0.578259i	−0.447319	+	0.894374i	$0.647621\pi$
$644$		0			0
$645$	2.84187			0.111899
$646$		0			0
$647$	−24.7030			−0.971174			−0.485587	−	0.874188i	$-0.661394\pi$
$647$	−24.7030			−0.971174			−0.485587	+	0.874188i	$0.661394\pi$
$648$		0			0
$649$	8.91535	−	3.69286i	0.349958	−	0.144957i
$650$		0			0
$651$	−4.41002	−	10.6467i	−0.172842	−	0.417279i
$652$		0			0
$653$	1.56376	−	3.77524i	0.0611945	−	0.147737i	−0.890325	−	0.455327i	$-0.849523\pi$
$653$	1.56376	−	3.77524i	0.0611945	−	0.147737i	0.951519	+	0.307590i	$0.0995225\pi$
$654$		0			0
$655$	4.19571	−	4.19571i	0.163940	−	0.163940i
$656$		0			0
$657$	2.46859	+	1.02252i	0.0963088	+	0.0398924i
$658$		0			0
$659$			14.1666i			0.551851i	0.961179	+	0.275926i	$0.0889844\pi$
$659$			14.1666i			0.551851i	−0.961179	+	0.275926i	$0.911016\pi$
$660$		0			0
$661$	−35.9487	−	35.9487i	−1.39824	−	1.39824i	−0.805092	−	0.593150i	$-0.797884\pi$
$661$	−35.9487	−	35.9487i	−1.39824	−	1.39824i	−0.593150	−	0.805092i	$-0.702116\pi$
$662$		0			0
$663$	0.146651	−	1.98208i	0.00569547	−	0.0769776i
$664$		0			0
$665$	0.855779	+	0.855779i	0.0331857	+	0.0331857i
$666$		0			0
$667$			29.9053i			1.15794i
$668$		0			0
$669$	11.1203	+	4.60618i	0.429936	+	0.178085i
$670$		0			0
$671$	11.2790	−	11.2790i	0.435421	−	0.435421i
$672$		0			0
$673$	18.6293	−	44.9751i	0.718107	−	1.73366i	0.0394352	−	0.999222i	$-0.487444\pi$
$673$	18.6293	−	44.9751i	0.718107	−	1.73366i	0.678672	−	0.734442i	$-0.262556\pi$
$674$		0			0
$675$	−1.87777	−	4.53335i	−0.0722755	−	0.174489i
$676$		0			0
$677$	−22.7801	+	9.43581i	−0.875509	+	0.362648i	−0.774753	−	0.632264i	$-0.782126\pi$
$677$	−22.7801	+	9.43581i	−0.875509	+	0.362648i	−0.100755	+	0.994911i	$0.532126\pi$
$678$		0			0
$679$	−12.3580			−0.474256
$680$		0			0
$681$	−27.3448			−1.04785
$682$		0			0
$683$	−7.12343	+	2.95062i	−0.272570	+	0.112902i	−0.514782	−	0.857321i	$-0.672127\pi$
$683$	−7.12343	+	2.95062i	−0.272570	+	0.112902i	0.242212	+	0.970223i	$0.422127\pi$
$684$		0			0
$685$	1.32974	+	3.21026i	0.0508066	+	0.122658i
$686$		0			0
$687$	6.51713	−	15.7337i	0.248644	−	0.600280i
$688$		0			0
$689$	−1.37571	+	1.37571i	−0.0524102	+	0.0524102i
$690$		0			0
$691$	24.1960	+	10.0223i	0.920457	+	0.381266i	0.792050	−	0.610456i	$-0.209014\pi$
$691$	24.1960	+	10.0223i	0.920457	+	0.381266i	0.128407	+	0.991722i	$0.459014\pi$
$692$		0			0
$693$			6.88936i			0.261705i
$694$		0			0
$695$	0.101428	+	0.101428i	0.00384739	+	0.00384739i
$696$		0			0
$697$	8.15991	−	7.03562i	0.309079	−	0.266493i
$698$		0			0
$699$	−18.2519	−	18.2519i	−0.690349	−	0.690349i
$700$		0			0
$701$		−	42.5140i		−	1.60573i	−0.596159	−	0.802866i	$-0.703307\pi$
$701$		−	42.5140i		−	1.60573i	0.596159	−	0.802866i	$-0.296693\pi$
$702$		0			0
$703$	−8.53923	−	3.53707i	−0.322063	−	0.133403i
$704$		0			0
$705$	−0.879696	+	0.879696i	−0.0331313	+	0.0331313i
$706$		0			0
$707$	−0.843595	+	2.03662i	−0.0317266	+	0.0765949i
$708$		0			0
$709$	−18.6147	−	44.9397i	−0.699088	−	1.68775i	−0.725616	−	0.688099i	$-0.758445\pi$
$709$	−18.6147	−	44.9397i	−0.699088	−	1.68775i	0.0265285	−	0.999648i	$-0.491555\pi$
$710$		0			0
$711$	8.89610	−	3.68488i	0.333630	−	0.138194i
$712$		0			0
$713$	19.4912			0.729949
$714$		0			0
$715$	0.220020			0.00822826
$716$		0			0
$717$	9.84806	−	4.07920i	0.367782	−	0.152340i
$718$		0			0
$719$	2.91749	+	7.04345i	0.108804	+	0.262676i	0.968898	−	0.247459i	$-0.0795955\pi$
$719$	2.91749	+	7.04345i	0.108804	+	0.262676i	−0.860094	+	0.510135i	$0.829595\pi$
$720$		0			0
$721$	−20.8010	+	50.2182i	−0.774671	+	1.87022i
$722$		0			0
$723$	−12.0853	+	12.0853i	−0.449458	+	0.449458i
$724$		0			0
$725$	17.4003	+	7.20745i	0.646232	+	0.267678i
$726$		0			0
$727$		−	25.4311i		−	0.943189i	−0.881816	−	0.471594i	$-0.843679\pi$
$727$		−	25.4311i		−	0.943189i	0.881816	−	0.471594i	$-0.156321\pi$
$728$		0			0
$729$	−0.707107	−	0.707107i	−0.0261891	−	0.0261891i
$730$		0			0
$731$	−36.4582	+	12.0351i	−1.34846	+	0.445136i
$732$		0			0
$733$	3.15019	+	3.15019i	0.116355	+	0.116355i	0.762887	−	0.646532i	$-0.223781\pi$
$733$	3.15019	+	3.15019i	0.116355	+	0.116355i	−0.646532	+	0.762887i	$0.723781\pi$
$734$		0			0
$735$		−	4.33986i		−	0.160078i
$736$		0			0
$737$	−11.7637	−	4.87270i	−0.433323	−	0.179488i
$738$		0			0
$739$	−23.4918	+	23.4918i	−0.864161	+	0.864161i	−0.991818	−	0.127657i	$-0.959254\pi$
$739$	−23.4918	+	23.4918i	−0.864161	+	0.864161i	0.127657	+	0.991818i	$0.459254\pi$
$740$		0			0
$741$	−0.158800	+	0.383378i	−0.00583367	+	0.0140837i
$742$		0			0
$743$	4.89536	+	11.8185i	0.179593	+	0.433577i	0.987881	−	0.155210i	$-0.0496055\pi$
$743$	4.89536	+	11.8185i	0.179593	+	0.433577i	−0.808288	+	0.588787i	$0.799606\pi$
$744$		0			0
$745$	−2.25042	+	0.932156i	−0.0824492	+	0.0341516i
$746$		0			0
$747$	0.466549			0.0170701
$748$		0			0
$749$	38.5569			1.40884
$750$		0			0
$751$	−5.09007	+	2.10837i	−0.185739	+	0.0769357i	−0.473615	−	0.880732i	$-0.657051\pi$
$751$	−5.09007	+	2.10837i	−0.185739	+	0.0769357i	0.287876	+	0.957668i	$0.407051\pi$
$752$		0			0
$753$	−6.81263	−	16.4471i	−0.248266	−	0.599367i
$754$		0			0
$755$	1.04405	−	2.52057i	0.0379970	−	0.0917330i
$756$		0			0
$757$	−13.5834	+	13.5834i	−0.493698	+	0.493698i	−0.909469	−	0.415771i	$-0.863512\pi$
$757$	−13.5834	+	13.5834i	−0.493698	+	0.493698i	0.415771	+	0.909469i	$0.363512\pi$
$758$		0			0
$759$	−10.7654	−	4.45918i	−0.390760	−	0.161858i
$760$		0			0
$761$		−	1.36032i		−	0.0493115i	−0.999696	−	0.0246558i	$-0.992151\pi$
$761$		−	1.36032i		−	0.0493115i	0.999696	−	0.0246558i	$-0.00784897\pi$
$762$		0			0
$763$	28.9415	+	28.9415i	1.04775	+	1.04775i
$764$		0			0
$765$	−0.821701	−	0.953010i	−0.0297087	−	0.0344561i
$766$		0			0
$767$	−2.19930	−	2.19930i	−0.0794121	−	0.0794121i
$768$		0			0
$769$			9.99250i			0.360339i	0.983636	+	0.180169i	$0.0576646\pi$
$769$			9.99250i			0.360339i	−0.983636	+	0.180169i	$0.942335\pi$
$770$		0			0
$771$	−9.78167	−	4.05170i	−0.352278	−	0.145918i
$772$		0			0
$773$	−7.89634	+	7.89634i	−0.284011	+	0.284011i	−0.834707	−	0.550695i	$-0.814363\pi$
$773$	−7.89634	+	7.89634i	−0.284011	+	0.284011i	0.550695	+	0.834707i	$0.314363\pi$
$774$		0			0
$775$	4.69755	−	11.3409i	0.168741	−	0.407376i
$776$		0			0
$777$	18.9272	+	45.6944i	0.679011	+	1.63928i
$778$		0			0
$779$	−2.07829	+	0.860857i	−0.0744626	+	0.0308434i
$780$		0			0
$781$	21.4967			0.769214
$782$		0			0
$783$	3.83830			0.137170
$784$		0			0
$785$	−4.98917	+	2.06658i	−0.178071	+	0.0737595i
$786$		0			0
$787$	−0.592873	−	1.43132i	−0.0211336	−	0.0510211i	0.912960	−	0.408048i	$-0.133790\pi$
$787$	−0.592873	−	1.43132i	−0.0211336	−	0.0510211i	−0.934094	+	0.357027i	$0.883790\pi$
$788$		0			0
$789$	2.29004	−	5.52865i	0.0815276	−	0.196825i
$790$		0			0
$791$	−38.2403	+	38.2403i	−1.35967	+	1.35967i
$792$		0			0
$793$	−4.74982	−	1.96744i	−0.168671	−	0.0698659i
$794$		0			0
$795$			1.23178i			0.0436866i
$796$		0			0
$797$	29.8129	+	29.8129i	1.05603	+	1.05603i	0.998334	+	0.0576922i	$0.0183742\pi$
$797$	29.8129	+	29.8129i	1.05603	+	1.05603i	0.0576922	+	0.998334i	$0.481626\pi$
$798$		0			0
$799$	7.56011	−	15.0110i	0.267457	−	0.531052i
$800$		0			0
$801$	5.18492	+	5.18492i	0.183200	+	0.183200i
$802$		0			0
$803$		−	3.99612i		−	0.141020i
$804$		0			0
$805$	10.1198	+	4.19176i	0.356676	+	0.147740i
$806$		0			0
$807$	−6.26626	+	6.26626i	−0.220583	+	0.220583i
$808$		0			0
$809$	−9.99517	+	24.1305i	−0.351412	+	0.848382i	0.645035	+	0.764153i	$0.276843\pi$
$809$	−9.99517	+	24.1305i	−0.351412	+	0.848382i	−0.996446	+	0.0842293i	$0.973157\pi$
$810$		0			0
$811$	−10.8859	−	26.2809i	−0.382255	−	0.922846i	−0.991529	−	0.129886i	$-0.958539\pi$
$811$	−10.8859	−	26.2809i	−0.382255	−	0.922846i	0.609274	−	0.792960i	$-0.291461\pi$
$812$		0			0
$813$	27.9502	−	11.5773i	0.980255	−	0.406035i
$814$		0			0
$815$	−4.13267			−0.144761
$816$		0			0
$817$	8.01605			0.280446
$818$		0			0
$819$	2.05150	−	0.849758i	0.0716851	−	0.0296929i
$820$		0			0
$821$	−21.3218	−	51.4754i	−0.744136	−	1.79650i	−0.588165	−	0.808741i	$-0.700149\pi$
$821$	−21.3218	−	51.4754i	−0.744136	−	1.79650i	−0.155971	−	0.987762i	$-0.549851\pi$
$822$		0			0
$823$	−2.03999	+	4.92497i	−0.0711095	+	0.171674i	−0.955438	−	0.295191i	$-0.904617\pi$
$823$	−2.03999	+	4.92497i	−0.0711095	+	0.171674i	0.884329	+	0.466865i	$0.154617\pi$
$824$		0			0
$825$	−5.18912	+	5.18912i	−0.180662	+	0.180662i
$826$		0			0
$827$	−21.5553	−	8.92850i	−0.749551	−	0.310474i	−0.0249928	−	0.999688i	$-0.507956\pi$
$827$	−21.5553	−	8.92850i	−0.749551	−	0.310474i	−0.724559	+	0.689213i	$0.757956\pi$
$828$		0			0
$829$			15.2115i			0.528316i	0.964479	+	0.264158i	$0.0850940\pi$
$829$			15.2115i			0.528316i	−0.964479	+	0.264158i	$0.914906\pi$
$830$		0			0
$831$	13.4853	+	13.4853i	0.467800	+	0.467800i
$832$		0			0
$833$	18.3790	+	55.6758i	0.636795	+	1.92905i
$834$		0			0
$835$	4.16242	+	4.16242i	0.144047	+	0.144047i
$836$		0			0
$837$		−	2.50166i		−	0.0864699i
$838$		0			0
$839$	−24.2106	−	10.0283i	−0.835842	−	0.346217i	−0.0766291	−	0.997060i	$-0.524416\pi$
$839$	−24.2106	−	10.0283i	−0.835842	−	0.346217i	−0.759213	+	0.650843i	$0.774416\pi$
$840$		0			0
$841$	10.0886	−	10.0886i	0.347884	−	0.347884i
$842$		0			0
$843$	3.48780	−	8.42030i	0.120126	−	0.290011i
$844$		0			0
$845$	1.49116	+	3.59998i	0.0512974	+	0.123843i
$846$		0			0
$847$	−37.2954	+	15.4483i	−1.28149	+	0.530809i
$848$		0			0
$849$	−18.9286			−0.649627
$850$		0			0
$851$	−83.6534			−2.86760
$852$		0			0
$853$	13.8416	−	5.73339i	0.473928	−	0.196307i	−0.132918	−	0.991127i	$-0.542435\pi$
$853$	13.8416	−	5.73339i	0.473928	−	0.196307i	0.606846	+	0.794820i	$0.292435\pi$
$854$		0			0
$855$	0.100541	+	0.242727i	0.00343843	+	0.00830110i
$856$		0			0
$857$	−19.2397	+	46.4487i	−0.657215	+	1.58666i	0.144873	+	0.989450i	$0.453723\pi$
$857$	−19.2397	+	46.4487i	−0.657215	+	1.58666i	−0.802088	+	0.597206i	$0.796277\pi$
$858$		0			0
$859$	−22.4275	+	22.4275i	−0.765217	+	0.765217i	−0.977260	−	0.212043i	$-0.931988\pi$
$859$	−22.4275	+	22.4275i	−0.765217	+	0.765217i	0.212043	+	0.977260i	$0.431988\pi$
$860$		0			0
$861$	11.1212	+	4.60654i	0.379009	+	0.156991i
$862$		0			0
$863$			13.0185i			0.443156i	0.975143	+	0.221578i	$0.0711207\pi$
$863$			13.0185i			0.443156i	−0.975143	+	0.221578i	$0.928879\pi$
$864$		0			0
$865$	4.74199	+	4.74199i	0.161233	+	0.161233i
$866$		0			0
$867$	14.5775	+	8.74625i	0.495077	+	0.297038i
$868$		0			0
$869$	−10.1830	−	10.1830i	−0.345434	−	0.345434i
$870$		0			0
$871$			4.10400i			0.139059i
$872$		0			0
$873$	−2.47851	−	1.02663i	−0.0838847	−	0.0347462i
$874$		0			0
$875$	9.84846	−	9.84846i	0.332939	−	0.332939i
$876$		0			0
$877$	−13.6154	+	32.8704i	−0.459758	+	1.10996i	0.508737	+	0.860922i	$0.330113\pi$
$877$	−13.6154	+	32.8704i	−0.459758	+	1.10996i	−0.968495	+	0.249033i	$0.919887\pi$
$878$		0			0
$879$	6.29644	+	15.2009i	0.212374	+	0.512715i
$880$		0			0
$881$	44.4374	−	18.4066i	1.49714	−	0.620134i	0.524279	−	0.851546i	$-0.324335\pi$
$881$	44.4374	−	18.4066i	1.49714	−	0.620134i	0.972856	+	0.231413i	$0.0743348\pi$
$882$		0			0
$883$	15.5298			0.522618			0.261309	−	0.965255i	$-0.415846\pi$
$883$	15.5298			0.522618			0.261309	+	0.965255i	$0.415846\pi$
$884$		0			0
$885$	−1.96921			−0.0661942
$886$		0			0
$887$	19.3796	−	8.02730i	0.650704	−	0.269530i	−0.0328171	−	0.999461i	$-0.510448\pi$
$887$	19.3796	−	8.02730i	0.650704	−	0.269530i	0.683521	+	0.729931i	$0.260448\pi$
$888$		0			0
$889$	−4.28216	−	10.3381i	−0.143619	−	0.346727i
$890$		0			0
$891$	−0.572328	+	1.38172i	−0.0191737	+	0.0462895i
$892$		0			0
$893$	−2.48135	+	2.48135i	−0.0830353	+	0.0830353i
$894$		0			0
$895$	−4.19159	−	1.73621i	−0.140109	−	0.0580353i
$896$		0			0
$897$			3.75571i			0.125399i
$898$		0			0
$899$	6.78971	+	6.78971i	0.226450	+	0.226450i
$900$		0			0
$901$	−5.21649	−	15.8024i	−0.173787	−	0.526454i
$902$		0			0
$903$	−30.3312	−	30.3312i	−1.00936	−	1.00936i
$904$		0			0
$905$			4.97629i			0.165417i
$906$		0			0
$907$	−6.87003	−	2.84566i	−0.228116	−	0.0944886i	0.265698	−	0.964056i	$-0.414398\pi$
$907$	−6.87003	−	2.84566i	−0.228116	−	0.0944886i	−0.493814	+	0.869568i	$0.664398\pi$
$908$		0			0
$909$	−0.338381	+	0.338381i	−0.0112234	+	0.0112234i
$910$		0			0
$911$	3.15174	−	7.60896i	0.104422	−	0.252096i	−0.863030	−	0.505153i	$-0.831436\pi$
$911$	3.15174	−	7.60896i	0.104422	−	0.252096i	0.967451	+	0.253057i	$0.0814360\pi$
$912$		0			0
$913$	−0.267019	−	0.644641i	−0.00883704	−	0.0213345i
$914$		0			0
$915$	−3.00725	+	1.24564i	−0.0994166	+	0.0411797i
$916$		0			0
$917$	−89.5612			−2.95757
$918$		0			0
$919$	−32.5311			−1.07310			−0.536550	−	0.843868i	$-0.680273\pi$
$919$	−32.5311			−1.07310			−0.536550	+	0.843868i	$0.680273\pi$
$920$		0			0
$921$	18.2573	−	7.56241i	0.601598	−	0.249190i
$922$		0			0
$923$	−2.65148	−	6.40125i	−0.0872746	−	0.210700i
$924$		0			0
$925$	−20.1612	+	48.6735i	−0.662897	+	1.60038i
$926$		0			0
$927$	−8.34367	+	8.34367i	−0.274042	+	0.274042i
$928$		0			0
$929$	−28.6894	−	11.8835i	−0.941268	−	0.389886i	−0.141326	−	0.989963i	$-0.545136\pi$
$929$	−28.6894	−	11.8835i	−0.941268	−	0.389886i	−0.799942	+	0.600077i	$0.795136\pi$
$930$		0			0
$931$		−	12.2414i		−	0.401196i
$932$		0			0
$933$	−8.52162	−	8.52162i	−0.278985	−	0.278985i
$934$		0			0
$935$	−0.846513	+	1.68080i	−0.0276839	+	0.0549680i
$936$		0			0
$937$	12.5524	+	12.5524i	0.410068	+	0.410068i	0.881762	−	0.471694i	$-0.156357\pi$
$937$	12.5524	+	12.5524i	0.410068	+	0.410068i	−0.471694	+	0.881762i	$0.656357\pi$
$938$		0			0
$939$			25.4595i			0.830839i
$940$		0			0
$941$	−39.3402	−	16.2952i	−1.28245	−	0.531210i	−0.365726	−	0.930723i	$-0.619179\pi$
$941$	−39.3402	−	16.2952i	−1.28245	−	0.531210i	−0.916728	+	0.399513i	$0.869179\pi$
$942$		0			0
$943$	−14.3965	+	14.3965i	−0.468814	+	0.468814i
$944$		0			0
$945$	0.538006	−	1.29886i	0.0175013	−	0.0422519i
$946$		0			0
$947$	9.11884	+	22.0148i	0.296322	+	0.715386i	0.999988	+	0.00485448i	$0.00154523\pi$
$947$	9.11884	+	22.0148i	0.296322	+	0.715386i	−0.703666	+	0.710531i	$0.748455\pi$
$948$		0			0
$949$	−1.18996	+	0.492896i	−0.0386276	+	0.0160001i
$950$		0			0
$951$	−12.0910			−0.392076
$952$		0			0
$953$	23.3757			0.757214			0.378607	−	0.925558i	$-0.376403\pi$
$953$	23.3757			0.757214			0.378607	+	0.925558i	$0.376403\pi$
$954$		0			0
$955$	4.96863	−	2.05807i	0.160781	−	0.0665977i
$956$		0			0
$957$	−2.19677	−	5.30346i	−0.0710113	−	0.171437i
$958$		0			0
$959$	20.0708	−	48.4552i	0.648120	−	1.56470i
$960$		0			0
$961$	−17.4950	+	17.4950i	−0.564356	+	0.564356i
$962$		0			0
$963$	7.73292	+	3.20308i	0.249190	+	0.103218i
$964$		0			0
$965$			1.00590i			0.0323812i
$966$		0			0
$967$	32.3405	+	32.3405i	1.04000	+	1.04000i	0.999166	+	0.0408360i	$0.0130021\pi$
$967$	32.3405	+	32.3405i	1.04000	+	1.04000i	0.0408360	+	0.999166i	$0.486998\pi$
$968$		0			0
$969$	−2.31777	−	2.68815i	−0.0744574	−	0.0863558i
$970$		0			0
$971$	11.9796	+	11.9796i	0.384443	+	0.384443i	0.872700	−	0.488257i	$-0.162367\pi$
$971$	11.9796	+	11.9796i	0.384443	+	0.384443i	−0.488257	+	0.872700i	$0.662367\pi$
$972$		0			0
$973$		−	2.16508i		−	0.0694092i
$974$		0			0
$975$	2.18525	+	0.905159i	0.0699839	+	0.0289883i
$976$		0			0
$977$	31.5576	−	31.5576i	1.00962	−	1.00962i	0.00966376	−	0.999953i	$-0.496924\pi$
$977$	31.5576	−	31.5576i	1.00962	−	1.00962i	0.999953	−	0.00966376i	$-0.00307612\pi$
$978$		0			0
$979$	4.19665	−	10.1316i	0.134125	−	0.323808i
$980$		0			0
$981$	3.40018	+	8.20877i	0.108560	+	0.262086i
$982$		0			0
$983$	−31.4008	+	13.0066i	−1.00153	+	0.414847i	−0.822358	−	0.568971i	$-0.807342\pi$
$983$	−31.4008	+	13.0066i	−1.00153	+	0.414847i	−0.179172	+	0.983818i	$0.557342\pi$
$984$		0			0
$985$	0.643892			0.0205161
$986$		0			0
$987$	18.7779			0.597708
$988$		0			0
$989$	67.0280	−	27.7639i	2.13137	−	0.882840i
$990$		0			0
$991$	−3.05266	−	7.36977i	−0.0969709	−	0.234108i	0.867949	−	0.496654i	$-0.165438\pi$
$991$	−3.05266	−	7.36977i	−0.0969709	−	0.234108i	−0.964920	+	0.262545i	$0.915438\pi$
$992$		0			0
$993$	−1.92287	+	4.64223i	−0.0610205	+	0.147317i
$994$		0			0
$995$	4.95477	−	4.95477i	0.157077	−	0.157077i
$996$		0			0
$997$	21.3697	+	8.85162i	0.676785	+	0.280334i	0.694482	−	0.719510i	$-0.255633\pi$
$997$	21.3697	+	8.85162i	0.676785	+	0.280334i	−0.0176971	+	0.999843i	$0.505633\pi$
$998$		0			0
$999$			10.7368i			0.339697i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	408.2.ba.b.121.3	✓	16
3.2	odd	2		1224.2.bq.d.937.3		16
4.3	odd	2		816.2.bq.e.529.1		16
17.3	odd	16		6936.2.a.bn.1.5		8
17.9	even	8	inner	408.2.ba.b.145.3	yes	16
17.14	odd	16		6936.2.a.bm.1.4		8
51.26	odd	8		1224.2.bq.d.145.3		16
68.43	odd	8		816.2.bq.e.145.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
408.2.ba.b.121.3	✓	16	1.1	even	1	trivial
408.2.ba.b.145.3	yes	16	17.9	even	8	inner
816.2.bq.e.145.1		16	68.43	odd	8
816.2.bq.e.529.1		16	4.3	odd	2
1224.2.bq.d.145.3		16	51.26	odd	8
1224.2.bq.d.937.3		16	3.2	odd	2
6936.2.a.bm.1.4		8	17.14	odd	16
6936.2.a.bn.1.5		8	17.3	odd	16

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:	\( N \)	\(=\)	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	408.ba (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.923880 - 0.382683i) q^{3} +(0.116792 + 0.281961i) q^{5} +(1.76284 - 4.25587i) q^{7} +(0.707107 - 0.707107i) q^{9} +(-1.38172 - 0.572328i) q^{11} +0.482039i q^{13} +(0.215804 + 0.215804i) q^{15} +(-3.68244 - 1.85462i) q^{17} +(0.608716 + 0.608716i) q^{19} -4.60653i q^{21} +(7.19822 + 2.98160i) q^{23} +(3.46967 - 3.46967i) q^{25} +(0.382683 - 0.923880i) q^{27} +(1.46885 + 3.54612i) q^{29} +(2.31123 - 0.957343i) q^{31} -1.49557 q^{33} +1.40588 q^{35} +(-9.91949 + 4.10879i) q^{37} +(0.184468 + 0.445346i) q^{39} +(-1.00000 + 2.41422i) q^{41} +(6.58440 - 6.58440i) q^{43} +(0.281961 + 0.116792i) q^{45} +4.07637i q^{47} +(-10.0551 - 10.0551i) q^{49} +(-4.11187 - 0.304232i) q^{51} +(2.85393 + 2.85393i) q^{53} -0.456435i q^{55} +(0.795325 + 0.329435i) q^{57} +(-4.56250 + 4.56250i) q^{59} +(-4.08150 + 9.85361i) q^{61} +(-1.76284 - 4.25587i) q^{63} +(-0.135916 + 0.0562983i) q^{65} +8.51383 q^{67} +7.79129 q^{69} +(-13.2795 + 5.50056i) q^{71} +(1.02252 + 2.46859i) q^{73} +(1.87777 - 4.53335i) q^{75} +(-4.87151 + 4.87151i) q^{77} +(8.89610 + 3.68488i) q^{79} -1.00000i q^{81} +(0.329900 + 0.329900i) q^{83} +(0.0928492 - 1.25491i) q^{85} +(2.71409 + 2.71409i) q^{87} +7.33259i q^{89} +(2.05150 + 0.849758i) q^{91} +(1.76894 - 1.76894i) q^{93} +(-0.100541 + 0.242727i) q^{95} +(-1.02663 - 2.47851i) q^{97} +(-1.38172 + 0.572328i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{11} - 8 q^{15} - 16 q^{17} + 8 q^{19} + 8 q^{23} + 24 q^{25} + 8 q^{31} + 24 q^{33} + 16 q^{35} - 8 q^{37} + 24 q^{39} - 8 q^{41} + 8 q^{43} + 8 q^{45} - 40 q^{49} - 8 q^{51} - 24 q^{53} + 8 q^{57}+ \cdots + 8 q^{99}+O(q^{100}) \) 16 * q + 8 * q^11 - 8 * q^15 - 16 * q^17 + 8 * q^19 + 8 * q^23 + 24 * q^25 + 8 * q^31 + 24 * q^33 + 16 * q^35 - 8 * q^37 + 24 * q^39 - 8 * q^41 + 8 * q^43 + 8 * q^45 - 40 * q^49 - 8 * q^51 - 24 * q^53 + 8 * q^57 - 32 * q^59 - 48 * q^61 + 48 * q^65 - 24 * q^69 - 32 * q^71 - 40 * q^73 + 16 * q^79 - 48 * q^83 + 32 * q^85 - 16 * q^87 + 16 * q^91 + 24 * q^93 - 56 * q^95 - 40 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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