LMFDB - Embedded newform 112.5.s.c.33.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,5,Mod(17,112)] 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("112.17"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	112.s (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$11.5774358654$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{6})$
Coefficient field:	6.0.11337408.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} + 18x^{4} + 81x^{2} + 12 $ x^6 + 18x^4 + 81x^2 + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			33.2
Root			$0.391571i$ of defining polynomial
Character	$\chi$	$=$	112.33
Dual form			112.5.s.c.17.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.35901 - 0.784623i) q^{3} +(38.3327 + 22.1314i) q^{5} +(-42.3967 + 24.5667i) q^{7} +(-39.2687 + 68.0154i) q^{9} +(11.7557 + 20.3615i) q^{11} -136.269i q^{13} +69.4592 q^{15} +(-227.333 + 131.251i) q^{17} +(387.162 + 223.528i) q^{19} +(-38.3418 + 66.6517i) q^{21} +(-374.585 + 648.800i) q^{23} +(667.098 + 1155.45i) q^{25} +250.353i q^{27} +406.524 q^{29} +(584.199 - 337.287i) q^{31} +(31.9521 + 18.4476i) q^{33} +(-2168.87 + 3.40909i) q^{35} +(-372.666 + 645.476i) q^{37} +(-106.920 - 185.191i) q^{39} -2476.20i q^{41} +2636.68 q^{43} +(-3010.55 + 1738.14i) q^{45} +(579.099 + 334.343i) q^{47} +(1193.96 - 2083.09i) q^{49} +(-205.964 + 356.741i) q^{51} +(1014.61 + 1757.36i) q^{53} +1040.68i q^{55} +701.541 q^{57} +(1014.22 - 585.561i) q^{59} +(-2022.05 - 1167.43i) q^{61} +(-6.04891 - 3848.33i) q^{63} +(3015.83 - 5223.58i) q^{65} +(-3779.79 - 6546.78i) q^{67} +1175.63i q^{69} -7575.36 q^{71} +(3284.18 - 1896.12i) q^{73} +(1813.18 + 1046.84i) q^{75} +(-998.615 - 574.460i) q^{77} +(3897.33 - 6750.37i) q^{79} +(-2984.33 - 5169.02i) q^{81} +2181.41i q^{83} -11619.0 q^{85} +(552.469 - 318.968i) q^{87} +(8050.38 + 4647.89i) q^{89} +(3347.69 + 5777.37i) q^{91} +(529.287 - 916.751i) q^{93} +(9893.98 + 17136.9i) q^{95} +9981.90i q^{97} -1846.52 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 9 q^{3} - 27 q^{5} - 66 q^{7} + 90 q^{9} - 135 q^{11} + 486 q^{15} - 1107 q^{17} + 747 q^{19} + 2169 q^{21} - 243 q^{23} + 1878 q^{25} - 540 q^{29} + 5355 q^{31} - 1863 q^{33} - 6021 q^{35} + 2355 q^{37}+ \cdots - 8100 q^{99}+O(q^{100}) $ 6 * q - 9 * q^3 - 27 * q^5 - 66 * q^7 + 90 * q^9 - 135 * q^11 + 486 * q^15 - 1107 * q^17 + 747 * q^19 + 2169 * q^21 - 243 * q^23 + 1878 * q^25 - 540 * q^29 + 5355 * q^31 - 1863 * q^33 - 6021 * q^35 + 2355 * q^37 - 6588 * q^39 + 948 * q^43 - 14418 * q^45 + 9747 * q^47 + 8430 * q^49 + 891 * q^51 + 6291 * q^53 - 6894 * q^57 + 2943 * q^59 + 4041 * q^61 - 15138 * q^63 + 7668 * q^65 - 1659 * q^67 - 2268 * q^71 - 8703 * q^73 + 3438 * q^75 - 10665 * q^77 + 7773 * q^79 - 7479 * q^81 - 702 * q^85 - 40014 * q^87 + 14985 * q^89 + 20952 * q^91 + 1449 * q^93 + 20655 * q^95 - 8100 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.35901	−	0.784623i	0.151001	−	0.0871803i	−0.422596	−	0.906318i	$-0.638881\pi$
$3$	1.35901	−	0.784623i	0.151001	−	0.0871803i	0.573597	+	0.819138i	$0.305548\pi$
$4$		0			0
$5$	38.3327	+	22.1314i	1.53331	+	0.885256i	0.999206	+	0.0398360i	$0.0126836\pi$
$5$	38.3327	+	22.1314i	1.53331	+	0.885256i	0.534102	+	0.845420i	$0.320650\pi$
$6$		0			0
$7$	−42.3967	+	24.5667i	−0.865238	+	0.501361i
$7$	−42.3967	+	24.5667i	−0.865238	+	0.501361i
$8$		0			0
$9$	−39.2687	+	68.0154i	−0.484799	+	0.839697i
$10$		0			0
$11$	11.7557	+	20.3615i	0.0971545	+	0.168276i	0.910506	−	0.413496i	$-0.135693\pi$
$11$	11.7557	+	20.3615i	0.0971545	+	0.168276i	−0.813351	+	0.581773i	$0.802359\pi$
$12$		0			0
$13$		−	136.269i		−	0.806328i	−0.915128	−	0.403164i	$-0.867910\pi$
$13$		−	136.269i		−	0.806328i	0.915128	−	0.403164i	$-0.132090\pi$
$14$		0			0
$15$	69.4592			0.308708
$16$		0			0
$17$	−227.333	+	131.251i	−0.786618	+	0.454154i	−0.838771	−	0.544485i	$-0.816725\pi$
$17$	−227.333	+	131.251i	−0.786618	+	0.454154i	0.0521523	+	0.998639i	$0.483392\pi$
$18$		0			0
$19$	387.162	+	223.528i	1.07247	+	0.619191i	0.928855	−	0.370442i	$-0.120794\pi$
$19$	387.162	+	223.528i	1.07247	+	0.619191i	0.143615	+	0.989634i	$0.454127\pi$
$20$		0			0
$21$	−38.3418	+	66.6517i	−0.0869429	+	0.151138i
$22$		0			0
$23$	−374.585	+	648.800i	−0.708100	+	1.22646i	0.257462	+	0.966288i	$0.417114\pi$
$23$	−374.585	+	648.800i	−0.708100	+	1.22646i	−0.965561	+	0.260176i	$0.916219\pi$
$24$		0			0
$25$	667.098	+	1155.45i	1.06736	+	1.84872i
$26$		0			0
$27$			250.353i			0.343420i
$28$		0			0
$29$	406.524			0.483382			0.241691	−	0.970353i	$-0.422298\pi$
$29$	406.524			0.483382			0.241691	+	0.970353i	$0.422298\pi$
$30$		0			0
$31$	584.199	−	337.287i	0.607907	−	0.350975i	−0.164239	−	0.986421i	$-0.552517\pi$
$31$	584.199	−	337.287i	0.607907	−	0.350975i	0.772146	+	0.635445i	$0.219183\pi$
$32$		0			0
$33$	31.9521	+	18.4476i	0.0293408	+	0.0169399i
$34$		0			0
$35$	−2168.87	+	3.40909i	−1.77051	+	0.00278293i
$36$		0			0
$37$	−372.666	+	645.476i	−0.272218	+	0.471495i	−0.969429	−	0.245370i	$-0.921090\pi$
$37$	−372.666	+	645.476i	−0.272218	+	0.471495i	0.697212	+	0.716865i	$0.254424\pi$
$38$		0			0
$39$	−106.920	−	185.191i	−0.0702959	−	0.121756i
$40$		0			0
$41$		−	2476.20i		−	1.47305i	−0.676410	−	0.736526i	$-0.736465\pi$
$41$		−	2476.20i		−	1.47305i	0.676410	−	0.736526i	$-0.263535\pi$
$42$		0			0
$43$	2636.68			1.42601			0.713003	−	0.701161i	$-0.247335\pi$
$43$	2636.68			1.42601			0.713003	+	0.701161i	$0.247335\pi$
$44$		0			0
$45$	−3010.55	+	1738.14i	−1.48669	+	0.858343i
$46$		0			0
$47$	579.099	+	334.343i	0.262154	+	0.151355i	0.625317	−	0.780371i	$-0.284970\pi$
$47$	579.099	+	334.343i	0.262154	+	0.151355i	−0.363163	+	0.931726i	$0.618303\pi$
$48$		0			0
$49$	1193.96	−	2083.09i	0.497275	−	0.867593i
$50$		0			0
$51$	−205.964	+	356.741i	−0.0791866	+	0.137155i
$52$		0			0
$53$	1014.61	+	1757.36i	0.361200	+	0.625617i	0.988159	−	0.153436i	$-0.0490338\pi$
$53$	1014.61	+	1757.36i	0.361200	+	0.625617i	−0.626959	+	0.779053i	$0.715700\pi$
$54$		0			0
$55$			1040.68i			0.344026i
$56$		0			0
$57$	701.541			0.215925
$58$		0			0
$59$	1014.22	−	585.561i	0.291359	−	0.168216i	−0.347195	−	0.937793i	$-0.612866\pi$
$59$	1014.22	−	585.561i	0.291359	−	0.168216i	0.638555	+	0.769576i	$0.279533\pi$
$60$		0			0
$61$	−2022.05	−	1167.43i	−0.543416	−	0.313741i	0.203047	−	0.979169i	$-0.434916\pi$
$61$	−2022.05	−	1167.43i	−0.543416	−	0.313741i	−0.746462	+	0.665428i	$0.768249\pi$
$62$		0			0
$63$	−6.04891	−	3848.33i	−0.00152404	−	0.969597i
$64$		0			0
$65$	3015.83	−	5223.58i	0.713807	−	1.23635i
$66$		0			0
$67$	−3779.79	−	6546.78i	−0.842011	−	1.45841i	−0.888192	−	0.459472i	$-0.848039\pi$
$67$	−3779.79	−	6546.78i	−0.842011	−	1.45841i	0.0461815	−	0.998933i	$-0.485295\pi$
$68$		0			0
$69$			1175.63i			0.246929i
$70$		0			0
$71$	−7575.36			−1.50275			−0.751375	−	0.659876i	$-0.770609\pi$
$71$	−7575.36			−1.50275			−0.751375	+	0.659876i	$0.770609\pi$
$72$		0			0
$73$	3284.18	−	1896.12i	0.616284	−	0.355812i	−0.159137	−	0.987257i	$-0.550871\pi$
$73$	3284.18	−	1896.12i	0.616284	−	0.355812i	0.775421	+	0.631445i	$0.217538\pi$
$74$		0			0
$75$	1813.18	+	1046.84i	0.322343	+	0.186105i
$76$		0			0
$77$	−998.615	−	574.460i	−0.168429	−	0.0968898i
$78$		0			0
$79$	3897.33	−	6750.37i	0.624472	−	1.08162i	−0.364171	−	0.931332i	$-0.618647\pi$
$79$	3897.33	−	6750.37i	0.624472	−	1.08162i	0.988643	−	0.150285i	$-0.0480192\pi$
$80$		0			0
$81$	−2984.33	−	5169.02i	−0.454860	−	0.787840i
$82$		0			0
$83$			2181.41i			0.316651i	0.987387	+	0.158326i	$0.0506096\pi$
$83$			2181.41i			0.316651i	−0.987387	+	0.158326i	$0.949390\pi$
$84$		0			0
$85$	−11619.0			−1.60817
$86$		0			0
$87$	552.469	−	318.968i	0.0729910	−	0.0421414i
$88$		0			0
$89$	8050.38	+	4647.89i	1.01633	+	0.586781i	0.913040	−	0.407870i	$-0.133728\pi$
$89$	8050.38	+	4647.89i	1.01633	+	0.586781i	0.103294	+	0.994651i	$0.467062\pi$
$90$		0			0
$91$	3347.69	+	5777.37i	0.404261	+	0.697666i
$92$		0			0
$93$	529.287	−	916.751i	0.0611963	−	0.105995i
$94$		0			0
$95$	9893.98	+	17136.9i	1.09629	+	1.89882i
$96$		0			0
$97$			9981.90i			1.06089i	0.847720	+	0.530444i	$0.177975\pi$
$97$			9981.90i			1.06089i	−0.847720	+	0.530444i	$0.822025\pi$
$98$		0			0
$99$	−1846.52			−0.188402
$100$		0			0
$101$	−675.552	+	390.030i	−0.0662241	+	0.0382345i	−0.532746	−	0.846275i	$-0.678840\pi$
$101$	−675.552	+	390.030i	−0.0662241	+	0.0382345i	0.466522	+	0.884509i	$0.345507\pi$
$102$		0			0
$103$	−2383.90	−	1376.35i	−0.224706	−	0.129734i	0.383422	−	0.923573i	$-0.374746\pi$
$103$	−2383.90	−	1376.35i	−0.224706	−	0.129734i	−0.608127	+	0.793840i	$0.708079\pi$
$104$		0			0
$105$	−2944.84	+	1706.38i	−0.267106	+	0.154774i
$106$		0			0
$107$	7429.46	−	12868.2i	0.648917	−	1.12396i	−0.334464	−	0.942408i	$-0.608555\pi$
$107$	7429.46	−	12868.2i	0.648917	−	1.12396i	0.983382	−	0.181550i	$-0.0581113\pi$
$108$		0			0
$109$	−1378.33	−	2387.33i	−0.116011	−	0.200937i	0.802172	−	0.597092i	$-0.203677\pi$
$109$	−1378.33	−	2387.33i	−0.116011	−	0.200937i	−0.918183	+	0.396155i	$0.870344\pi$
$110$		0			0
$111$			1169.61i			0.0949281i
$112$		0			0
$113$	6264.79			0.490625			0.245312	−	0.969444i	$-0.421109\pi$
$113$	6264.79			0.490625			0.245312	+	0.969444i	$0.421109\pi$
$114$		0			0
$115$	−28717.7	+	16580.2i	−2.17147	+	1.25370i
$116$		0			0
$117$	9268.43	+	5351.13i	0.677071	+	0.390907i
$118$		0			0
$119$	6413.76	−	11149.4i	0.452917	−	0.787331i
$120$		0			0
$121$	7044.11	−	12200.8i	0.481122	−	0.833328i
$122$		0			0
$123$	−1942.88	−	3365.17i	−0.128421	−	0.222432i
$124$		0			0
$125$			31391.0i			2.00902i
$126$		0			0
$127$	−12855.0			−0.797010			−0.398505	−	0.917166i	$-0.630471\pi$
$127$	−12855.0			−0.797010			−0.398505	+	0.917166i	$0.630471\pi$
$128$		0			0
$129$	3583.27	−	2068.80i	0.215328	−	0.124320i
$130$		0			0
$131$	21412.5	+	12362.5i	1.24774	+	0.720383i	0.970658	−	0.240464i	$-0.0772995\pi$
$131$	21412.5	+	12362.5i	1.24774	+	0.720383i	0.277082	+	0.960846i	$0.410633\pi$
$132$		0			0
$133$	−21905.7	+	34.4320i	−1.23838	+	0.00194652i
$134$		0			0
$135$	−5540.67	+	9596.73i	−0.304015	+	0.526569i
$136$		0			0
$137$	−661.078	−	1145.02i	−0.0352218	−	0.0610059i	0.847877	−	0.530193i	$-0.177880\pi$
$137$	−661.078	−	1145.02i	−0.0352218	−	0.0610059i	−0.883099	+	0.469187i	$0.844547\pi$
$138$		0			0
$139$			5119.78i			0.264985i	0.991184	+	0.132493i	$0.0422981\pi$
$139$			5119.78i			0.264985i	−0.991184	+	0.132493i	$0.957702\pi$
$140$		0			0
$141$	1049.33			0.0527807
$142$		0			0
$143$	2774.64	−	1601.94i	0.135686	−	0.0783384i
$144$		0			0
$145$	15583.2	+	8996.95i	0.741173	+	0.427917i
$146$		0			0
$147$	−11.8445	−	3767.74i	−0.000548128	−	0.174360i
$148$		0			0
$149$	−5104.93	+	8842.00i	−0.229942	+	0.398270i	−0.957791	−	0.287467i	$-0.907187\pi$
$149$	−5104.93	+	8842.00i	−0.229942	+	0.398270i	0.727849	+	0.685737i	$0.240520\pi$
$150$		0			0
$151$	14672.1	+	25412.8i	0.643483	+	1.11455i	0.984650	+	0.174543i	$0.0558448\pi$
$151$	14672.1	+	25412.8i	0.643483	+	1.11455i	−0.341166	+	0.940003i	$0.610822\pi$
$152$		0			0
$153$		−	20616.2i		−	0.880695i
$154$		0			0
$155$	29858.6			1.24281
$156$		0			0
$157$	−26033.1	+	15030.2i	−1.05615	+	0.609770i	−0.924366	−	0.381508i	$-0.875405\pi$
$157$	−26033.1	+	15030.2i	−1.05615	+	0.609770i	−0.131787	+	0.991278i	$0.542072\pi$
$158$		0			0
$159$	2757.73	+	1592.17i	0.109083	+	0.0629791i
$160$		0			0
$161$	−57.7005	−	36709.3i	−0.00222602	−	1.41620i
$162$		0			0
$163$	15439.7	−	26742.3i	0.581115	−	1.00652i	−0.414232	−	0.910171i	$-0.635950\pi$
$163$	15439.7	−	26742.3i	0.581115	−	1.00652i	0.995348	−	0.0963500i	$-0.0307168\pi$
$164$		0			0
$165$	816.541	+	1414.29i	0.0299923	+	0.0519482i
$166$		0			0
$167$		−	12858.8i		−	0.461070i	−0.973064	−	0.230535i	$-0.925952\pi$
$167$		−	12858.8i		−	0.461070i	0.973064	−	0.230535i	$-0.0740477\pi$
$168$		0			0
$169$	9991.63			0.349835
$170$		0			0
$171$	−30406.7	+	17555.3i	−1.03987	+	0.600367i
$172$		0			0
$173$	13588.8	+	7845.51i	0.454035	+	0.262137i	0.709533	−	0.704672i	$-0.248906\pi$
$173$	13588.8	+	7845.51i	0.454035	+	0.262137i	−0.255498	+	0.966810i	$0.582239\pi$
$174$		0			0
$175$	−56668.2	−	32598.8i	−1.85039	−	1.06445i
$176$		0			0
$177$	918.890	−	1591.56i	0.0293303	−	0.0508016i
$178$		0			0
$179$	−15993.2	−	27701.0i	−0.499148	−	0.864550i	0.500851	−	0.865533i	$-0.333020\pi$
$179$	−15993.2	−	27701.0i	−0.499148	−	0.864550i	−1.00000		0.000983488i	$0.999687\pi$
$180$		0			0
$181$		−	47967.6i		−	1.46417i	−0.681215	−	0.732083i	$-0.738548\pi$
$181$		−	47967.6i		−	1.46417i	0.681215	−	0.732083i	$-0.261452\pi$
$182$		0			0
$183$	−3663.97			−0.109408
$184$		0			0
$185$	−28570.6	+	16495.2i	−0.834787	+	0.481965i
$186$		0			0
$187$	−5344.91	−	3085.88i	−0.152847	−	0.0882462i
$188$		0			0
$189$	−6150.35	−	10614.2i	−0.172177	−	0.297141i
$190$		0			0
$191$	2227.93	−	3858.88i	0.0610709	−	0.105778i	−0.833873	−	0.551956i	$-0.813882\pi$
$191$	2227.93	−	3858.88i	0.0610709	−	0.105778i	0.894944	+	0.446178i	$0.147215\pi$
$192$		0			0
$193$	14699.9	+	25460.9i	0.394638	+	0.683533i	0.993055	−	0.117652i	$-0.0375366\pi$
$193$	14699.9	+	25460.9i	0.394638	+	0.683533i	−0.598417	+	0.801185i	$0.704203\pi$
$194$		0			0
$195$		−	9465.17i		−	0.248920i
$196$		0			0
$197$	23816.0			0.613671			0.306836	−	0.951763i	$-0.400730\pi$
$197$	23816.0			0.613671			0.306836	+	0.951763i	$0.400730\pi$
$198$		0			0
$199$	42280.9	−	24410.9i	1.06767	−	0.616421i	0.140128	−	0.990133i	$-0.455249\pi$
$199$	42280.9	−	24410.9i	1.06767	−	0.616421i	0.927545	+	0.373713i	$0.121915\pi$
$200$		0			0
$201$	−10273.5	−	5931.41i	−0.254288	−	0.146813i
$202$		0			0
$203$	−17235.3	+	9986.94i	−0.418241	+	0.242349i
$204$		0			0
$205$	54801.8	−	94919.4i	1.30403	−	2.25864i
$206$		0			0
$207$	−29418.9	−	50955.1i	−0.686572	−	1.18918i
$208$		0			0
$209$			10510.9i			0.240629i
$210$		0			0
$211$	9582.07			0.215226			0.107613	−	0.994193i	$-0.465679\pi$
$211$	9582.07			0.215226			0.107613	+	0.994193i	$0.465679\pi$
$212$		0			0
$213$	−10295.0	+	5943.80i	−0.226916	+	0.131010i
$214$		0			0
$215$	101071.	+	58353.5i	2.18651	+	1.26238i
$216$		0			0
$217$	−16482.1	+	28651.7i	−0.350019	+	0.608458i
$218$		0			0
$219$	2975.48	−	5153.68i	0.0620396	−	0.107456i
$220$		0			0
$221$	17885.5	+	30978.5i	0.366197	+	0.634273i
$222$		0			0
$223$			54978.9i			1.10557i	0.833324	+	0.552785i	$0.186435\pi$
$223$			54978.9i			1.10557i	−0.833324	+	0.552785i	$0.813565\pi$
$224$		0			0
$225$	−104784.			−2.06981
$226$		0			0
$227$	7518.24	−	4340.66i	0.145903	−	0.0842372i	−0.425271	−	0.905066i	$-0.639821\pi$
$227$	7518.24	−	4340.66i	0.145903	−	0.0842372i	0.571174	+	0.820829i	$0.306488\pi$
$228$		0			0
$229$	−6262.86	−	3615.86i	−0.119427	−	0.0689511i	0.439097	−	0.898440i	$-0.355299\pi$
$229$	−6262.86	−	3615.86i	−0.119427	−	0.0689511i	−0.558523	+	0.829489i	$0.688632\pi$
$230$		0			0
$231$	−1807.86	+	2.84164i	−0.0338798	+	5.32531e-5i
$232$		0			0
$233$	−4422.96	+	7660.79i	−0.0814707	+	0.141111i	−0.903882	−	0.427782i	$-0.859295\pi$
$233$	−4422.96	+	7660.79i	−0.0814707	+	0.141111i	0.822411	+	0.568894i	$0.192628\pi$
$234$		0			0
$235$	14799.0	+	25632.5i	0.267976	+	0.464147i
$236$		0			0
$237$		−	12231.7i		−	0.217767i
$238$		0			0
$239$	−88491.7			−1.54920			−0.774599	−	0.632453i	$-0.782048\pi$
$239$	−88491.7			−1.54920			−0.774599	+	0.632453i	$0.782048\pi$
$240$		0			0
$241$	−53365.8	+	30810.7i	−0.918816	+	0.530479i	−0.883257	−	0.468889i	$-0.844654\pi$
$241$	−53365.8	+	30810.7i	−0.918816	+	0.530479i	−0.0355589	+	0.999368i	$0.511321\pi$
$242$		0			0
$243$	−25673.3	−	14822.5i	−0.434779	−	0.251020i
$244$		0			0
$245$	91869.3	−	53426.6i	1.53052	−	0.890072i
$246$		0			0
$247$	30460.0	−	52758.4i	0.499271	−	0.864763i
$248$		0			0
$249$	1711.59	+	2964.55i	0.0276058	+	0.0478146i
$250$		0			0
$251$			57796.3i			0.917386i	0.888595	+	0.458693i	$0.151682\pi$
$251$			57796.3i			0.917386i	−0.888595	+	0.458693i	$0.848318\pi$
$252$		0			0
$253$	−17614.0			−0.275180
$254$		0			0
$255$	−15790.3	+	9116.56i	−0.242835	+	0.140201i
$256$		0			0
$257$	60734.3	+	35065.0i	0.919534	+	0.530893i	0.883486	−	0.468457i	$-0.155190\pi$
$257$	60734.3	+	35065.0i	0.919534	+	0.530893i	0.0360479	+	0.999350i	$0.488523\pi$
$258$		0			0
$259$	−57.4050	−	36521.2i	−0.000855756	−	0.544435i
$260$		0			0
$261$	−15963.7	+	27649.9i	−0.234343	+	0.405894i
$262$		0			0
$263$	11120.9	+	19262.0i	0.160779	+	0.278477i	0.935148	−	0.354257i	$-0.115266\pi$
$263$	11120.9	+	19262.0i	0.160779	+	0.278477i	−0.774369	+	0.632734i	$0.781933\pi$
$264$		0			0
$265$			89819.0i			1.27902i
$266$		0			0
$267$	14587.4			0.204623
$268$		0			0
$269$	40332.3	−	23285.9i	0.557376	−	0.321801i	−0.194715	−	0.980860i	$-0.562378\pi$
$269$	40332.3	−	23285.9i	0.557376	−	0.321801i	0.752092	+	0.659058i	$0.229045\pi$
$270$		0			0
$271$	−71958.3	−	41545.1i	−0.979811	−	0.565694i	−0.0775980	−	0.996985i	$-0.524725\pi$
$271$	−71958.3	−	41545.1i	−0.979811	−	0.565694i	−0.902213	+	0.431291i	$0.858058\pi$
$272$		0			0
$273$	9082.59	+	5224.82i	0.121866	+	0.0701045i
$274$		0			0
$275$	−15684.4	+	27166.2i	−0.207397	+	0.359222i
$276$		0			0
$277$	4514.06	+	7818.59i	0.0588313	+	0.101899i	0.893941	−	0.448185i	$-0.147929\pi$
$277$	4514.06	+	7818.59i	0.0588313	+	0.101899i	−0.835110	+	0.550083i	$0.814596\pi$
$278$		0			0
$279$			52979.4i			0.680610i
$280$		0			0
$281$	64188.4			0.812912			0.406456	−	0.913670i	$-0.366764\pi$
$281$	64188.4			0.812912			0.406456	+	0.913670i	$0.366764\pi$
$282$		0			0
$283$	−19563.7	+	11295.1i	−0.244274	+	0.141032i	−0.617140	−	0.786854i	$-0.711709\pi$
$283$	−19563.7	+	11295.1i	−0.244274	+	0.141032i	0.372865	+	0.927885i	$0.378375\pi$
$284$		0			0
$285$	26892.0	+	15526.1i	0.331080	+	0.191149i
$286$		0			0
$287$	60832.0	+	104983.i	0.738530	+	1.27454i
$288$		0			0
$289$	−7307.06	+	12656.2i	−0.0874877	+	0.151533i
$290$		0			0
$291$	7832.02	+	13565.5i	0.0924885	+	0.160195i
$292$		0			0
$293$			130473.i			1.51980i	0.650040	+	0.759900i	$0.274752\pi$
$293$			130473.i			1.51980i	−0.650040	+	0.759900i	$0.725248\pi$
$294$		0			0
$295$	51837.2			0.595658
$296$		0			0
$297$	−5097.56	+	2943.08i	−0.0577896	+	0.0333648i
$298$		0			0
$299$	88411.6	+	51044.5i	0.988933	+	0.570961i
$300$		0			0
$301$	−111787.	+	64774.5i	−1.23383	+	0.714943i
$302$		0			0
$303$	−612.053	+	1060.11i	−0.00666659	+	0.0115469i
$304$		0			0
$305$	−51673.7	−	89501.6i	−0.555482	−	0.962124i
$306$		0			0
$307$			56173.9i			0.596015i	0.954563	+	0.298008i	$0.0963221\pi$
$307$			56173.9i			0.596015i	−0.954563	+	0.298008i	$0.903678\pi$
$308$		0			0
$309$	−4319.65			−0.0452410
$310$		0			0
$311$	−40087.2	+	23144.4i	−0.414462	+	0.239290i	−0.692705	−	0.721221i	$-0.743581\pi$
$311$	−40087.2	+	23144.4i	−0.414462	+	0.239290i	0.278243	+	0.960511i	$0.410248\pi$
$312$		0			0
$313$	−119814.	−	69174.4i	−1.22298	−	0.706085i	−0.257424	−	0.966299i	$-0.582874\pi$
$313$	−119814.	−	69174.4i	−1.22298	−	0.706085i	−0.965551	+	0.260213i	$0.916207\pi$
$314$		0			0
$315$	84937.1	−	147651.i	0.856005	−	1.48804i
$316$		0			0
$317$	29152.7	−	50493.9i	0.290108	−	0.502482i	−0.683727	−	0.729738i	$-0.739642\pi$
$317$	29152.7	−	50493.9i	0.290108	−	0.502482i	0.973835	+	0.227256i	$0.0729754\pi$
$318$		0			0
$319$	4778.97	+	8277.42i	0.0469627	+	0.0813418i
$320$		0			0
$321$		−	23317.3i		−	0.226291i
$322$		0			0
$323$	−117353.			−1.12483
$324$		0			0
$325$	157452.	−	90905.1i	1.49067	−	0.860640i
$326$		0			0
$327$	−3746.31	−	2162.93i	−0.0350355	−	0.0202278i
$328$		0			0
$329$	−32765.6	+	51.5018i	−0.302709	+	0.000475806i
$330$		0			0
$331$	494.602	−	856.675i	0.00451440	−	0.00781916i	−0.863759	−	0.503904i	$-0.831896\pi$
$331$	494.602	−	856.675i	0.00451440	−	0.00781916i	0.868274	+	0.496085i	$0.165230\pi$
$332$		0			0
$333$	−29268.2	−	50694.1i	−0.263942	−	0.457161i
$334$		0			0
$335$		−	334608.i		−	2.98158i
$336$		0			0
$337$	−55958.7			−0.492728			−0.246364	−	0.969177i	$-0.579236\pi$
$337$	−55958.7			−0.492728			−0.246364	+	0.969177i	$0.579236\pi$
$338$		0			0
$339$	8513.89	−	4915.49i	0.0740847	−	0.0427728i
$340$		0			0
$341$	13735.3	+	7930.09i	0.118122	+	0.0681976i
$342$		0			0
$343$	554.769	+	117648.i	0.00471546	+	0.999989i
$344$		0			0
$345$	−26018.3	+	45065.1i	−0.218596	+	0.378619i
$346$		0			0
$347$	65322.2	+	113141.i	0.542502	+	0.939641i	0.998760	+	0.0497936i	$0.0158563\pi$
$347$	65322.2	+	113141.i	0.542502	+	0.939641i	−0.456257	+	0.889848i	$0.650810\pi$
$348$		0			0
$349$		−	110948.i		−	0.910899i	−0.890262	−	0.455450i	$-0.849478\pi$
$349$		−	110948.i		−	0.910899i	0.890262	−	0.455450i	$-0.150522\pi$
$350$		0			0
$351$	34115.5			0.276910
$352$		0			0
$353$	14727.9	−	8503.18i	0.118193	−	0.0682389i	−0.439738	−	0.898126i	$-0.644929\pi$
$353$	14727.9	−	8503.18i	0.118193	−	0.0682389i	0.557931	+	0.829887i	$0.311595\pi$
$354$		0			0
$355$	−290384.	−	167653.i	−2.30418	−	1.33032i
$356$		0			0
$357$	−31.7265	−	20184.5i	−0.000248935	−	0.158373i
$358$		0			0
$359$	35732.4	−	61890.3i	0.277251	−	0.480212i	−0.693450	−	0.720505i	$-0.743910\pi$
$359$	35732.4	−	61890.3i	0.277251	−	0.480212i	0.970701	+	0.240293i	$0.0772434\pi$
$360$		0			0
$361$	34769.0	+	60221.8i	0.266795	+	0.462103i
$362$		0			0
$363$		−	22107.9i		−	0.167777i
$364$		0			0
$365$	167855.			1.25994
$366$		0			0
$367$	−147543.	+	85183.8i	−1.09543	+	0.632448i	−0.935017	−	0.354602i	$-0.884617\pi$
$367$	−147543.	+	85183.8i	−1.09543	+	0.632448i	−0.160415	+	0.987050i	$0.551283\pi$
$368$		0			0
$369$	168420.	+	97237.2i	1.23692	+	0.714134i
$370$		0			0
$371$	−86188.6	−	49580.5i	−0.626184	−	0.360216i
$372$		0			0
$373$	−75346.0	+	130503.i	−0.541555	+	0.938001i	0.457260	+	0.889333i	$0.348831\pi$
$373$	−75346.0	+	130503.i	−0.541555	+	0.938001i	−0.998815	+	0.0486680i	$0.984502\pi$
$374$		0			0
$375$	24630.1	+	42660.5i	0.175147	+	0.303364i
$376$		0			0
$377$		−	55396.8i		−	0.389764i
$378$		0			0
$379$	211477.			1.47226			0.736129	−	0.676841i	$-0.236652\pi$
$379$	211477.			1.47226			0.736129	+	0.676841i	$0.236652\pi$
$380$		0			0
$381$	−17470.0	+	10086.3i	−0.120349	+	0.0694836i
$382$		0			0
$383$	102769.	+	59333.6i	0.700590	+	0.404486i	0.807567	−	0.589776i	$-0.200784\pi$
$383$	102769.	+	59333.6i	0.700590	+	0.404486i	−0.106977	+	0.994261i	$0.534117\pi$
$384$		0			0
$385$	−25566.0	−	44121.4i	−0.172481	−	0.297665i
$386$		0			0
$387$	−103539.	+	179335.i	−0.691326	+	1.19741i
$388$		0			0
$389$	−107858.	−	186816.i	−0.712778	−	1.23457i	−0.963810	−	0.266588i	$-0.914104\pi$
$389$	−107858.	−	186816.i	−0.712778	−	1.23457i	0.251033	−	0.967979i	$-0.419230\pi$
$390$		0			0
$391$		−	196658.i		−	1.28635i
$392$		0			0
$393$	38799.6			0.251213
$394$		0			0
$395$	298790.	−	172507.i	1.91502	−	1.10564i
$396$		0			0
$397$	−134011.	−	77371.1i	−0.850273	−	0.490905i	0.0104699	−	0.999945i	$-0.496667\pi$
$397$	−134011.	−	77371.1i	−0.850273	−	0.490905i	−0.860743	+	0.509040i	$0.830001\pi$
$398$		0			0
$399$	−29743.0	+	17234.5i	−0.186827	+	0.108256i
$400$		0			0
$401$	−100029.	+	173255.i	−0.622067	+	1.07745i	0.367033	+	0.930208i	$0.380374\pi$
$401$	−100029.	+	173255.i	−0.622067	+	1.07745i	−0.989100	+	0.147244i	$0.952960\pi$
$402$		0			0
$403$	−45962.0	−	79608.5i	−0.283001	−	0.490173i
$404$		0			0
$405$		−	264190.i		−	1.61067i
$406$		0			0
$407$	−17523.8			−0.105789
$408$		0			0
$409$	190382.	−	109917.i	1.13810	−	0.657080i	0.192137	−	0.981368i	$-0.438458\pi$
$409$	190382.	−	109917.i	1.13810	−	0.657080i	0.945958	+	0.324288i	$0.105125\pi$
$410$		0			0
$411$	−1796.82	−	1037.39i	−0.0106370	−	0.00614129i
$412$		0			0
$413$	−28614.4	+	49741.9i	−0.167758	+	0.291623i
$414$		0			0
$415$	−48277.7	+	83619.4i	−0.280318	+	0.485524i
$416$		0			0
$417$	4017.10	+	6957.81i	0.0231015	+	0.0400130i
$418$		0			0
$419$			31359.8i			0.178626i	0.996004	+	0.0893132i	$0.0284672\pi$
$419$			31359.8i			0.178626i	−0.996004	+	0.0893132i	$0.971533\pi$
$420$		0			0
$421$	−73690.7			−0.415766			−0.207883	−	0.978154i	$-0.566657\pi$
$421$	−73690.7			−0.415766			−0.207883	+	0.978154i	$0.566657\pi$
$422$		0			0
$423$	−45481.0	+	26258.5i	−0.254184	+	0.146753i
$424$		0			0
$425$	−303306.	−	175114.i	−1.67920	−	0.969489i
$426$		0			0
$427$	114408.	−	179.830i	0.627481	−	0.000986291i
$428$		0			0
$429$	2513.84	−	4354.10i	0.0136591	−	0.0236583i
$430$		0			0
$431$	−132586.	−	229646.i	−0.713746	−	1.23624i	−0.963441	−	0.267920i	$-0.913664\pi$
$431$	−132586.	−	229646.i	−0.713746	−	1.23624i	0.249695	−	0.968325i	$-0.419670\pi$
$432$		0			0
$433$		−	124873.i		−	0.666027i	−0.942922	−	0.333014i	$-0.891934\pi$
$433$		−	124873.i		−	0.666027i	0.942922	−	0.333014i	$-0.108066\pi$
$434$		0			0
$435$	28236.8			0.149224
$436$		0			0
$437$	−290050.	+	167460.i	−1.51883	+	0.876898i
$438$		0			0
$439$	120989.	+	69852.8i	0.627792	+	0.362456i	0.779896	−	0.625909i	$-0.215272\pi$
$439$	120989.	+	69852.8i	0.627792	+	0.362456i	−0.152105	+	0.988364i	$0.548605\pi$
$440$		0			0
$441$	94797.1	+	163008.i	0.487437	+	0.838169i
$442$		0			0
$443$	−110595.	+	191556.i	−0.563544	+	0.976087i	0.433639	+	0.901087i	$0.357229\pi$
$443$	−110595.	+	191556.i	−0.563544	+	0.976087i	−0.997183	+	0.0750005i	$0.976104\pi$
$444$		0			0
$445$	205729.	+	356332.i	1.03890	+	1.79943i
$446$		0			0
$447$			16021.8i			0.0801855i
$448$		0			0
$449$	255878.			1.26923			0.634615	−	0.772828i	$-0.281159\pi$
$449$	255878.			1.26923			0.634615	+	0.772828i	$0.281159\pi$
$450$		0			0
$451$	50419.0	−	29109.4i	0.247880	−	0.143114i
$452$		0			0
$453$	39878.9	+	23024.1i	0.194333	+	0.112198i
$454$		0			0
$455$	464.555	+	295551.i	0.00224396	+	1.42761i
$456$		0			0
$457$	159964.	−	277065.i	0.765929	−	1.32663i	−0.173825	−	0.984777i	$-0.555613\pi$
$457$	159964.	−	277065.i	0.765929	−	1.32663i	0.939754	−	0.341852i	$-0.111054\pi$
$458$		0			0
$459$	−32859.0	−	56913.5i	−0.155966	−	0.270141i
$460$		0			0
$461$			369557.i			1.73892i	0.494002	+	0.869461i	$0.335534\pi$
$461$			369557.i			1.73892i	−0.494002	+	0.869461i	$0.664466\pi$
$462$		0			0
$463$	−32715.0			−0.152611			−0.0763053	−	0.997084i	$-0.524312\pi$
$463$	−32715.0			−0.152611			−0.0763053	+	0.997084i	$0.524312\pi$
$464$		0			0
$465$	40578.0	−	23427.7i	0.187666	−	0.108349i
$466$		0			0
$467$	−265325.	−	153186.i	−1.21659	−	0.702400i	−0.252405	−	0.967622i	$-0.581221\pi$
$467$	−265325.	−	153186.i	−1.21659	−	0.702400i	−0.964187	+	0.265222i	$0.914555\pi$
$468$		0			0
$469$	321083.	+	184705.i	1.45973	+	0.839717i
$470$		0			0
$471$	−23586.1	+	40852.3i	−0.106320	+	0.184151i
$472$		0			0
$473$	30996.0	+	53686.7i	0.138543	+	0.239963i
$474$		0			0
$475$			596460.i			2.64359i
$476$		0			0
$477$	−159370.			−0.700438
$478$		0			0
$479$	226801.	−	130944.i	0.988493	−	0.570707i	0.0836697	−	0.996494i	$-0.473336\pi$
$479$	226801.	−	130944.i	0.988493	−	0.570707i	0.904824	+	0.425787i	$0.140003\pi$
$480$		0			0
$481$	87958.7	+	50783.0i	0.380180	+	0.219497i
$482$		0			0
$483$	−28881.3	−	49842.8i	−0.123801	−	0.213653i
$484$		0			0
$485$	−220913.	+	382633.i	−0.939158	+	1.62667i
$486$		0			0
$487$	41127.7	+	71235.3i	0.173411	+	0.300357i	0.939610	−	0.342246i	$-0.111188\pi$
$487$	41127.7	+	71235.3i	0.173411	+	0.300357i	−0.766199	+	0.642603i	$0.777854\pi$
$488$		0			0
$489$		−	48457.2i		−	0.202647i
$490$		0			0
$491$	102923.			0.426924			0.213462	−	0.976951i	$-0.431526\pi$
$491$	102923.			0.426924			0.213462	+	0.976951i	$0.431526\pi$
$492$		0			0
$493$	−92416.2	+	53356.5i	−0.380237	+	0.219530i
$494$		0			0
$495$	−70782.3	−	40866.2i	−0.288878	−	0.166784i
$496$		0			0
$497$	321170.	−	186101.i	1.30024	−	0.753420i
$498$		0			0
$499$	75028.2	−	129953.i	0.301317	−	0.521896i	−0.675118	−	0.737710i	$-0.735907\pi$
$499$	75028.2	−	129953.i	0.301317	−	0.521896i	0.976434	+	0.215814i	$0.0692405\pi$
$500$		0			0
$501$	−10089.3	−	17475.2i	−0.0401962	−	0.0696219i
$502$		0			0
$503$			91806.6i			0.362859i	0.983404	+	0.181430i	$0.0580725\pi$
$503$			91806.6i			0.362859i	−0.983404	+	0.181430i	$0.941928\pi$
$504$		0			0
$505$	−34527.6			−0.135389
$506$		0			0
$507$	13578.7	−	7839.66i	0.0528253	−	0.0304987i
$508$		0			0
$509$	−412600.	−	238215.i	−1.59255	−	0.919460i	−0.992867	−	0.119224i	$-0.961959\pi$
$509$	−412600.	−	238215.i	−1.59255	−	0.919460i	−0.599684	−	0.800237i	$-0.704707\pi$
$510$		0			0
$511$	−92656.9	+	161071.i	−0.354843	+	0.616843i
$512$		0			0
$513$	−55961.0	+	96927.3i	−0.212643	+	0.368308i
$514$		0			0
$515$	−60921.0	−	105518.i	−0.229696	−	0.397844i
$516$		0			0
$517$			15721.7i			0.0588192i
$518$		0			0
$519$	24623.1			0.0914128
$520$		0			0
$521$	147768.	−	85314.0i	0.544384	−	0.314300i	−0.202470	−	0.979289i	$-0.564897\pi$
$521$	147768.	−	85314.0i	0.544384	−	0.314300i	0.746854	+	0.664988i	$0.231563\pi$
$522$		0			0
$523$	−85040.5	−	49098.2i	−0.310901	−	0.179499i	0.336428	−	0.941709i	$-0.390781\pi$
$523$	−85040.5	−	49098.2i	−0.310901	−	0.179499i	−0.647330	+	0.762210i	$0.724114\pi$
$524$		0			0
$525$	−102590.	+	161.254i	−0.372209	+	0.000585048i
$526$		0			0
$527$	−88538.3	+	153353.i	−0.318794	+	0.552167i
$528$		0			0
$529$	−140707.	−	243711.i	−0.502810	−	0.870892i
$530$		0			0
$531$			91977.0i			0.326205i
$532$		0			0
$533$	−337430.			−1.18776
$534$		0			0
$535$	569582.	−	328849.i	1.98998	−	1.14892i
$536$		0			0
$537$	−43469.7	−	25097.3i	−0.150743	−	0.0870318i
$538$		0			0
$539$	56450.5	−	177.461i	0.194308	−	0.000610838i
$540$		0			0
$541$	52833.7	−	91510.6i	0.180516	−	0.312663i	−0.761540	−	0.648118i	$-0.775557\pi$
$541$	52833.7	−	91510.6i	0.180516	−	0.312663i	0.942056	+	0.335454i	$0.108890\pi$
$542$		0			0
$543$	−37636.4	−	65188.2i	−0.127646	−	0.221090i
$544$		0			0
$545$		−	122017.i		−	0.410798i
$546$		0			0
$547$	127121.			0.424855			0.212428	−	0.977177i	$-0.431863\pi$
$547$	127121.			0.424855			0.212428	+	0.977177i	$0.431863\pi$
$548$		0			0
$549$	158807.	−	91687.0i	0.526895	−	0.304203i
$550$		0			0
$551$	157391.	+	90869.5i	0.518413	+	0.299306i
$552$		0			0
$553$	600.340	+	381938.i	0.00196312	+	1.24894i
$554$		0			0
$555$	−25885.1	+	44834.3i	−0.0840356	+	0.145554i
$556$		0			0
$557$	−15246.5	−	26407.6i	−0.0491427	−	0.0851176i	0.840408	−	0.541955i	$-0.182316\pi$
$557$	−15246.5	−	26407.6i	−0.0491427	−	0.0851176i	−0.889550	+	0.456837i	$0.848982\pi$
$558$		0			0
$559$		−	359300.i		−	1.14983i
$560$		0			0
$561$	−9685.01			−0.0307733
$562$		0			0
$563$	−459404.	+	265237.i	−1.44936	+	0.836791i	−0.998444	−	0.0557725i	$-0.982238\pi$
$563$	−459404.	+	265237.i	−1.44936	+	0.836791i	−0.450921	+	0.892564i	$0.648905\pi$
$564$		0			0
$565$	240146.	+	138649.i	0.752279	+	0.434329i
$566$		0			0
$567$	253511.	+	145834.i	0.788554	+	0.453621i
$568$		0			0
$569$	169366.	−	293351.i	0.523121	−	0.906072i	−0.476517	−	0.879165i	$-0.658101\pi$
$569$	169366.	−	293351.i	0.523121	−	0.906072i	0.999638	−	0.0269067i	$-0.00856571\pi$
$570$		0			0
$571$	−53455.6	−	92587.8i	−0.163954	−	0.283976i	0.772330	−	0.635222i	$-0.219091\pi$
$571$	−53455.6	−	92587.8i	−0.163954	−	0.283976i	−0.936283	+	0.351246i	$0.885758\pi$
$572$		0			0
$573$		−	6992.33i		−	0.0212967i
$574$		0			0
$575$	−999538.			−3.02318
$576$		0			0
$577$	−104337.	+	60238.8i	−0.313390	+	0.180936i	−0.648443	−	0.761264i	$-0.724579\pi$
$577$	−104337.	+	60238.8i	−0.313390	+	0.180936i	0.335052	+	0.942200i	$0.391246\pi$
$578$		0			0
$579$	39954.4	+	23067.7i	0.119181	+	0.0688093i
$580$		0			0
$581$	−53590.0	−	92484.6i	−0.158757	−	0.273979i
$582$		0			0
$583$	−23854.9	+	41317.9i	−0.0701844	+	0.121563i
$584$		0			0
$585$	236856.	+	410247.i	0.692106	+	1.19876i
$586$		0			0
$587$		−	525818.i		−	1.52602i	−0.646389	−	0.763008i	$-0.723722\pi$
$587$		−	525818.i		−	1.52602i	0.646389	−	0.763008i	$-0.276278\pi$
$588$		0			0
$589$	301573.			0.869283
$590$		0			0
$591$	32366.0	−	18686.5i	0.0926648	−	0.0535000i
$592$		0			0
$593$	−64283.0	−	37113.8i	−0.182804	−	0.105542i	0.405805	−	0.913960i	$-0.366991\pi$
$593$	−64283.0	−	37113.8i	−0.182804	−	0.105542i	−0.588610	+	0.808417i	$0.700324\pi$
$594$		0			0
$595$	492609.	−	285441.i	1.39145	−	0.806274i
$596$		0			0
$597$	38306.7	−	66349.1i	0.107480	−	0.186160i
$598$		0			0
$599$	10990.7	+	19036.5i	0.0306318	+	0.0530558i	0.880935	−	0.473237i	$-0.156915\pi$
$599$	10990.7	+	19036.5i	0.0306318	+	0.0530558i	−0.850303	+	0.526293i	$0.823581\pi$
$600$		0			0
$601$		−	77418.2i		−	0.214336i	−0.994241	−	0.107168i	$-0.965822\pi$
$601$		−	77418.2i		−	0.214336i	0.994241	−	0.107168i	$-0.0341782\pi$
$602$		0			0
$603$	593710.			1.63282
$604$		0			0
$605$	540039.	−	311792.i	1.47542	−	0.851832i
$606$		0			0
$607$	−155765.	−	89931.2i	−0.422760	−	0.244081i	0.273498	−	0.961873i	$-0.411819\pi$
$607$	−155765.	−	89931.2i	−0.422760	−	0.244081i	−0.696257	+	0.717792i	$0.745153\pi$
$608$		0			0
$609$	−15586.9	+	27095.5i	−0.0420266	+	0.0730572i
$610$		0			0
$611$	45560.7	−	78913.5i	0.122042	−	0.211382i
$612$		0			0
$613$	−314637.	−	544968.i	−0.837316	−	1.45027i	−0.892131	−	0.451777i	$-0.850790\pi$
$613$	−314637.	−	544968.i	−0.837316	−	1.45027i	0.0548147	−	0.998497i	$-0.482543\pi$
$614$		0			0
$615$		−	171995.i		−	0.454742i
$616$		0			0
$617$	−52877.1			−0.138899			−0.0694493	−	0.997585i	$-0.522124\pi$
$617$	−52877.1			−0.138899			−0.0694493	+	0.997585i	$0.522124\pi$
$618$		0			0
$619$	−447040.	+	258099.i	−1.16672	+	0.673603i	−0.952904	−	0.303271i	$-0.901921\pi$
$619$	−447040.	+	258099.i	−1.16672	+	0.673603i	−0.213811	+	0.976875i	$0.568588\pi$
$620$		0			0
$621$	−162429.	−	93778.6i	−0.421193	−	0.243176i
$622$		0			0
$623$	−455493.	+	715.955i	−1.17356	+	0.00184463i
$624$		0			0
$625$	−277790.	+	481147.i	−0.711143	+	1.23174i
$626$		0			0
$627$	8247.09	+	14284.4i	0.0209781	+	0.0363351i
$628$		0			0
$629$		−	195651.i		−	0.494515i
$630$		0			0
$631$	−193439.			−0.485831			−0.242915	−	0.970047i	$-0.578104\pi$
$631$	−193439.			−0.485831			−0.242915	+	0.970047i	$0.578104\pi$
$632$		0			0
$633$	13022.1	−	7518.31i	0.0324993	−	0.0187635i
$634$		0			0
$635$	−492766.	−	284499.i	−1.22206	−	0.705558i
$636$		0			0
$637$	−283862.	−	162700.i	−0.699565	−	0.400967i
$638$		0			0
$639$	297475.	−	515242.i	0.728532	−	1.26185i
$640$		0			0
$641$	−376563.	−	652226.i	−0.916477	−	1.58738i	−0.804725	−	0.593648i	$-0.797687\pi$
$641$	−376563.	−	652226.i	−0.916477	−	1.58738i	−0.111751	−	0.993736i	$-0.535646\pi$
$642$		0			0
$643$		−	7110.23i		−	0.0171974i	−0.999963	−	0.00859868i	$-0.997263\pi$
$643$		−	7110.23i		−	0.0171974i	0.999963	−	0.00859868i	$-0.00273708\pi$
$644$		0			0
$645$	183142.			0.440219
$646$		0			0
$647$	341808.	−	197343.i	0.816534	−	0.471426i	−0.0326859	−	0.999466i	$-0.510406\pi$
$647$	341808.	−	197343.i	0.816534	−	0.471426i	0.849220	+	0.528040i	$0.177073\pi$
$648$		0			0
$649$	23845.8	+	13767.4i	0.0566137	+	0.0326860i
$650$		0			0
$651$	81.5306	+	51870.0i	0.000192380	+	0.122392i
$652$		0			0
$653$	−115578.	+	200186.i	−0.271049	+	0.469470i	−0.969131	−	0.246548i	$-0.920704\pi$
$653$	−115578.	+	200186.i	−0.271049	+	0.469470i	0.698082	+	0.716018i	$0.254037\pi$
$654$		0			0
$655$	547198.	+	947775.i	1.27545	+	2.20914i
$656$		0			0
$657$			297833.i			0.689989i
$658$		0			0
$659$	772498.			1.77880			0.889399	−	0.457131i	$-0.151123\pi$
$659$	772498.			1.77880			0.889399	+	0.457131i	$0.151123\pi$
$660$		0			0
$661$	−195287.	+	112749.i	−0.446962	+	0.258053i	−0.706546	−	0.707667i	$-0.749748\pi$
$661$	−195287.	+	112749.i	−0.446962	+	0.258053i	0.259584	+	0.965720i	$0.416414\pi$
$662$		0			0
$663$	48612.9	+	28066.7i	0.110592	+	0.0638504i
$664$		0			0
$665$	−840468.	−	483484.i	−1.90054	−	1.09330i
$666$		0			0
$667$	−152278.	+	263753.i	−0.342282	+	0.592851i
$668$		0			0
$669$	43137.7	+	74716.6i	0.0963839	+	0.166942i
$670$		0			0
$671$		−	54895.8i		−	0.121925i
$672$		0			0
$673$	59564.6			0.131510			0.0657549	−	0.997836i	$-0.479054\pi$
$673$	59564.6			0.131510			0.0657549	+	0.997836i	$0.479054\pi$
$674$		0			0
$675$	−289270.	+	167010.i	−0.634887	+	0.366552i
$676$		0			0
$677$	687131.	+	396715.i	1.49921	+	0.865569i	1.00000		0.000912440i	$-0.000290439\pi$
$677$	687131.	+	396715.i	1.49921	+	0.865569i	0.499210	+	0.866481i	$0.333624\pi$
$678$		0			0
$679$	−245222.	−	423199.i	−0.531888	−	0.917921i
$680$		0			0
$681$	6811.56	−	11798.0i	0.0146876	−	0.0254397i
$682$		0			0
$683$	−46304.3	−	80201.3i	−0.0992612	−	0.171925i	0.812118	−	0.583493i	$-0.198315\pi$
$683$	−46304.3	−	80201.3i	−0.0992612	−	0.171925i	−0.911379	+	0.411568i	$0.864981\pi$
$684$		0			0
$685$		−	58522.3i		−	0.124721i
$686$		0			0
$687$	−11348.4			−0.0240447
$688$		0			0
$689$	239474.	−	138261.i	0.504453	−	0.291246i
$690$		0			0
$691$	162273.	+	93688.2i	0.339852	+	0.196213i	0.660206	−	0.751084i	$-0.270469\pi$
$691$	162273.	+	93688.2i	0.339852	+	0.196213i	−0.320355	+	0.947298i	$0.603802\pi$
$692$		0			0
$693$	78286.5	−	45362.9i	0.163012	−	0.0944571i
$694$		0			0
$695$	−113308.	+	196255.i	−0.234580	+	0.406304i
$696$		0			0
$697$	325003.	+	562921.i	0.668993	+	1.15873i
$698$		0			0
$699$			13881.4i			0.0284105i
$700$		0			0
$701$	326056.			0.663522			0.331761	−	0.943363i	$-0.392357\pi$
$701$	326056.			0.663522			0.331761	+	0.943363i	$0.392357\pi$
$702$		0			0
$703$	−288564.	+	166603.i	−0.583891	+	0.337110i
$704$		0			0
$705$	40223.8	+	23223.2i	0.0809290	+	0.0467244i
$706$		0			0
$707$	19059.4	−	33132.0i	0.0381303	−	0.0662841i
$708$		0			0
$709$	−32372.6	+	56070.9i	−0.0643998	+	0.111544i	−0.896428	−	0.443190i	$-0.853847\pi$
$709$	−32372.6	+	56070.9i	−0.0643998	+	0.111544i	0.832028	+	0.554734i	$0.187180\pi$
$710$		0			0
$711$	306086.	+	530157.i	0.605487	+	1.04873i
$712$		0			0
$713$			505371.i			0.994102i
$714$		0			0
$715$	141813.			0.277398
$716$		0			0
$717$	−120261.	+	69432.6i	−0.233930	+	0.135060i
$718$		0			0
$719$	616355.	+	355853.i	1.19227	+	0.688355i	0.958820	−	0.284014i	$-0.0916664\pi$
$719$	616355.	+	355853.i	1.19227	+	0.688355i	0.233446	+	0.972370i	$0.425000\pi$
$720$		0			0
$721$	134882.	−	212.011i	0.259468	−	0.000407838i
$722$		0			0
$723$	−48349.6	+	83744.0i	−0.0924946	+	0.160205i
$724$		0			0
$725$	271191.	+	469717.i	0.515941	+	0.893635i
$726$		0			0
$727$		−	763134.i		−	1.44388i	−0.691954	−	0.721941i	$-0.743250\pi$
$727$		−	763134.i		−	1.44388i	0.691954	−	0.721941i	$-0.256750\pi$
$728$		0			0
$729$	436942.			0.822183
$730$		0			0
$731$	−599404.	+	346066.i	−1.12172	+	0.647626i
$732$		0			0
$733$	−17848.0	−	10304.6i	−0.0332187	−	0.0191788i	0.483299	−	0.875456i	$-0.339439\pi$
$733$	−17848.0	−	10304.6i	−0.0332187	−	0.0191788i	−0.516517	+	0.856277i	$0.672772\pi$
$734$		0			0
$735$	82931.3	−	144690.i	0.153513	−	0.267832i
$736$		0			0
$737$	88868.0	−	153924.i	0.163610	−	0.283381i
$738$		0			0
$739$	−393194.	−	681032.i	−0.719976	−	1.24703i	−0.961009	−	0.276518i	$-0.910819\pi$
$739$	−393194.	−	681032.i	−0.719976	−	1.24703i	0.241033	−	0.970517i	$-0.422514\pi$
$740$		0			0
$741$		−	95598.6i		−	0.174107i
$742$		0			0
$743$	−740479.			−1.34133			−0.670664	−	0.741761i	$-0.733991\pi$
$743$	−740479.			−1.34133			−0.670664	+	0.741761i	$0.733991\pi$
$744$		0			0
$745$	−391372.	+	225959.i	−0.705143	+	0.407114i
$746$		0			0
$747$	−148370.	−	85661.3i	−0.265891	−	0.153512i
$748$		0			0
$749$	1144.42	+	728086.i	0.00203997	+	1.29783i
$750$		0			0
$751$	−247292.	+	428322.i	−0.438460	+	0.759436i	−0.997571	−	0.0696574i	$-0.977809\pi$
$751$	−247292.	+	428322.i	−0.438460	+	0.759436i	0.559111	+	0.829093i	$0.311143\pi$
$752$		0			0
$753$	45348.3	+	78545.5i	0.0799780	+	0.138526i
$754$		0			0
$755$			1.29885e6i			2.27859i
$756$		0			0
$757$	−139399.			−0.243258			−0.121629	−	0.992576i	$-0.538812\pi$
$757$	−139399.			−0.243258			−0.121629	+	0.992576i	$0.538812\pi$
$758$		0			0
$759$	−23937.5	+	13820.3i	−0.0415524	+	0.0239903i
$760$		0			0
$761$	−207340.	−	119708.i	−0.358025	−	0.206706i	0.310189	−	0.950675i	$-0.399608\pi$
$761$	−207340.	−	119708.i	−0.358025	−	0.206706i	−0.668214	+	0.743969i	$0.732941\pi$
$762$		0			0
$763$	117085.	+	67354.1i	0.201119	+	0.115695i
$764$		0			0
$765$	456265.	−	790274.i	0.779640	−	1.35038i
$766$		0			0
$767$	−79794.1	−	138208.i	−0.135638	−	0.234931i
$768$		0			0
$769$			98701.5i			0.166906i	0.996512	+	0.0834528i	$0.0265948\pi$
$769$			98701.5i			0.166906i	−0.996512	+	0.0834528i	$0.973405\pi$
$770$		0			0
$771$	110051.			0.185134
$772$		0			0
$773$	−507921.	+	293248.i	−0.850036	+	0.490769i	−0.860663	−	0.509175i	$-0.829951\pi$
$773$	−507921.	+	293248.i	−0.850036	+	0.490769i	0.0106268	+	0.999944i	$0.496617\pi$
$774$		0			0
$775$	779435.	+	450007.i	1.29771	+	0.749232i
$776$		0			0
$777$	−28733.4	−	49587.5i	−0.0475932	−	0.0821354i
$778$		0			0
$779$	553500.	−	958690.i	0.912100	−	1.57980i
$780$		0			0
$781$	−89053.6	−	154245.i	−0.145999	−	0.252877i
$782$		0			0
$783$			101775.i			0.166003i
$784$		0			0
$785$	−1.33056e6			−2.15921
$786$		0			0
$787$	129536.	−	74787.4i	0.209141	−	0.120748i	−0.391771	−	0.920063i	$-0.628137\pi$
$787$	129536.	−	74787.4i	0.209141	−	0.120748i	0.600912	+	0.799315i	$0.294804\pi$
$788$		0			0
$789$	30226.8	+	17451.4i	0.0485554	+	0.0280335i
$790$		0			0
$791$	−265606.	+	153905.i	−0.424507	+	0.245980i
$792$		0			0
$793$	−159085.	+	275544.i	−0.252978	+	0.438171i
$794$		0			0
$795$	70474.1	+	122065.i	0.111505	+	0.193133i
$796$		0			0
$797$			241092.i			0.379547i	0.981828	+	0.189774i	$0.0607754\pi$
$797$			241092.i			0.379547i	−0.981828	+	0.189774i	$0.939225\pi$
$798$		0			0
$799$	−175531.			−0.274954
$800$		0			0
$801$	−632256.	+	365033.i	−0.985436	+	0.568941i
$802$		0			0
$803$	77215.6	+	44580.4i	0.119750	+	0.0691374i
$804$		0			0
$805$	810215.	−	1.40844e6i	1.25028	−	2.17344i
$806$		0			0
$807$	36541.3	−	63291.3i	0.0561095	−	0.0971845i
$808$		0			0
$809$	−100933.	−	174820.i	−0.154218	−	0.267113i	0.778556	−	0.627575i	$-0.215952\pi$
$809$	−100933.	−	174820.i	−0.154218	−	0.267113i	−0.932774	+	0.360462i	$0.882619\pi$
$810$		0			0
$811$			1.16747e6i			1.77503i	0.460779	+	0.887515i	$0.347570\pi$
$811$			1.16747e6i			1.77503i	−0.460779	+	0.887515i	$0.652430\pi$
$812$		0			0
$813$	−130389.			−0.197270
$814$		0			0
$815$	1.18369e6	−	683402.i	1.78206	−	1.02887i
$816$		0			0
$817$	1.02082e6	+	589373.i	1.52935	+	0.882970i
$818$		0			0
$819$	−524410.	+	824.281i	−0.781814	+	0.00122887i
$820$		0			0
$821$	324585.	−	562198.i	0.481551	−	0.834071i	−0.518225	−	0.855245i	$-0.673407\pi$
$821$	324585.	−	562198.i	0.481551	−	0.834071i	0.999776	+	0.0211734i	$0.00674022\pi$
$822$		0			0
$823$	−245486.	−	425193.i	−0.362432	−	0.627750i	0.625929	−	0.779880i	$-0.284720\pi$
$823$	−245486.	−	425193.i	−0.362432	−	0.627750i	−0.988360	+	0.152130i	$0.951387\pi$
$824$		0			0
$825$			49225.3i			0.0723237i
$826$		0			0
$827$	709017.			1.03668			0.518341	−	0.855174i	$-0.326550\pi$
$827$	709017.			1.03668			0.518341	+	0.855174i	$0.326550\pi$
$828$		0			0
$829$	534645.	−	308677.i	0.777958	−	0.449154i	−0.0577480	−	0.998331i	$-0.518392\pi$
$829$	534645.	−	308677.i	0.777958	−	0.449154i	0.835706	+	0.549177i	$0.185059\pi$
$830$		0			0
$831$	12269.3	+	7083.68i	0.0177671	+	0.0102579i
$832$		0			0
$833$	1981.33	+	630262.i	0.00285540	+	0.908304i
$834$		0			0
$835$	284583.	−	492912.i	0.408165	−	0.706963i
$836$		0			0
$837$	84441.0	+	146256.i	0.120532	+	0.208768i
$838$		0			0
$839$			493961.i			0.701728i	0.936426	+	0.350864i	$0.114112\pi$
$839$			493961.i			0.701728i	−0.936426	+	0.350864i	$0.885888\pi$
$840$		0			0
$841$	−542019.			−0.766342
$842$		0			0
$843$	87232.4	−	50363.7i	0.122750	−	0.0708699i
$844$		0			0
$845$	383006.	+	221129.i	0.536404	+	0.309693i
$846$		0			0
$847$	1085.07	+	690322.i	0.00151248	+	0.962243i
$848$		0			0
$849$	−17724.8	+	30700.2i	−0.0245904	+	0.0425918i
$850$		0			0
$851$	−279190.	−	483571.i	−0.385514	−	0.667730i
$852$		0			0
$853$		−	505142.i		−	0.694249i	−0.937819	−	0.347124i	$-0.887158\pi$
$853$		−	505142.i		−	0.694249i	0.937819	−	0.347124i	$-0.112842\pi$
$854$		0			0
$855$	−1.55410e6			−2.12591
$856$		0			0
$857$	−969980.	+	560018.i	−1.32069	+	0.762501i	−0.983839	−	0.179056i	$-0.942696\pi$
$857$	−969980.	+	560018.i	−1.32069	+	0.762501i	−0.336852	+	0.941557i	$0.609362\pi$
$858$		0			0
$859$	605141.	+	349379.i	0.820107	+	0.473489i	0.850453	−	0.526050i	$-0.176328\pi$
$859$	605141.	+	349379.i	0.820107	+	0.473489i	−0.0303463	+	0.999539i	$0.509661\pi$
$860$		0			0
$861$	165043.	+	94941.9i	0.222633	+	0.128071i
$862$		0			0
$863$	479533.	−	830576.i	0.643868	−	1.11521i	−0.340693	−	0.940175i	$-0.610662\pi$
$863$	479533.	−	830576.i	0.643868	−	1.11521i	0.984562	−	0.175038i	$-0.0560049\pi$
$864$		0			0
$865$	347264.	+	601479.i	0.464117	+	0.803875i
$866$		0			0
$867$			22933.1i			0.0305088i
$868$		0			0
$869$	183263.			0.242681
$870$		0			0
$871$	−892127.	+	515069.i	−1.17595	+	0.678937i
$872$		0			0
$873$	−678923.	−	391976.i	−0.890824	−	0.514318i
$874$		0			0
$875$	−771172.	−	1.33087e6i	−1.00724	−	1.73828i
$876$		0			0
$877$	498442.	−	863326.i	0.648060	−	1.12247i	−0.335526	−	0.942031i	$-0.608914\pi$
$877$	498442.	−	863326.i	0.648060	−	1.12247i	0.983586	−	0.180442i	$-0.0577527\pi$
$878$		0			0
$879$	102372.	+	177314.i	0.132497	+	0.229491i
$880$		0			0
$881$		−	494568.i		−	0.637198i	−0.947890	−	0.318599i	$-0.896788\pi$
$881$		−	494568.i		−	0.637198i	0.947890	−	0.318599i	$-0.103212\pi$
$882$		0			0
$883$	−887278.			−1.13799			−0.568995	−	0.822341i	$-0.692667\pi$
$883$	−887278.			−1.13799			−0.568995	+	0.822341i	$0.692667\pi$
$884$		0			0
$885$	70447.1	−	40672.6i	0.0899448	−	0.0519297i
$886$		0			0
$887$	−542062.	−	312960.i	−0.688973	−	0.397779i	0.114254	−	0.993452i	$-0.463552\pi$
$887$	−542062.	−	312960.i	−0.688973	−	0.397779i	−0.803227	+	0.595673i	$0.796885\pi$
$888$		0			0
$889$	545008.	−	315804.i	0.689604	−	0.399589i
$890$		0			0
$891$	70165.8	−	121531.i	0.0883833	−	0.153084i
$892$		0			0
$893$	149470.	+	258890.i	0.187435	+	0.324647i
$894$		0			0
$895$		−	1.41581e6i		−	1.76750i
$896$		0			0
$897$	160203.			0.199106
$898$		0			0
$899$	237491.	−	137115.i	0.293851	−	0.169655i
$900$		0			0
$901$	−461309.	−	266337.i	−0.568253	−	0.328081i
$902$		0			0
$903$	−101095.	+	175739.i	−0.123981	+	0.215523i
$904$		0			0
$905$	1.06159e6	−	1.83873e6i	1.29616	−	2.24502i
$906$		0			0
$907$	−634131.	−	1.09835e6i	−0.770840	−	1.33513i	−0.937103	−	0.349052i	$-0.886504\pi$
$907$	−634131.	−	1.09835e6i	−0.770840	−	1.33513i	0.166263	−	0.986081i	$-0.446830\pi$
$908$		0			0
$909$		−	61263.9i		−	0.0741442i
$910$		0			0
$911$	1.09985e6			1.32525			0.662625	−	0.748951i	$-0.269442\pi$
$911$	1.09985e6			1.32525			0.662625	+	0.748951i	$0.269442\pi$
$912$		0			0
$913$	−44416.7	+	25644.0i	−0.0532850	+	0.0307641i
$914$		0			0
$915$	−140450.	−	81088.8i	−0.167756	−	0.0968542i
$916$		0			0
$917$	−1.21152e6	+	1904.30i	−1.44076	+	0.00226463i
$918$		0			0
$919$	430660.	−	745925.i	0.509922	−	0.883211i	−0.490012	−	0.871716i	$-0.663008\pi$
$919$	430660.	−	745925.i	0.509922	−	0.883211i	0.999934	−	0.0114950i	$-0.00365905\pi$
$920$		0			0
$921$	44075.3	+	76340.6i	0.0519608	+	0.0899987i
$922$		0			0
$923$			1.03229e6i			1.21171i
$924$		0			0
$925$	−994418.			−1.16221
$926$		0			0
$927$	187226.	−	108095.i	0.217874	−	0.125790i
$928$		0			0
$929$	−326247.	−	188359.i	−0.378021	−	0.218250i	0.298936	−	0.954273i	$-0.403368\pi$
$929$	−326247.	−	188359.i	−0.378021	−	0.218250i	−0.676957	+	0.736023i	$0.736702\pi$
$930$		0			0
$931$	927884.	−	539610.i	1.07052	−	0.622560i
$932$		0			0
$933$	−36319.2	+	62906.7i	−0.0417227	+	0.0722659i
$934$		0			0
$935$	−136590.	−	236580.i	−0.156241	−	0.270617i
$936$		0			0
$937$			417813.i			0.475885i	0.971279	+	0.237943i	$0.0764731\pi$
$937$			417813.i			0.475885i	−0.971279	+	0.237943i	$0.923527\pi$
$938$		0			0
$939$	−217103.			−0.246227
$940$		0			0
$941$	213031.	−	122994.i	0.240582	−	0.138900i	−0.374862	−	0.927081i	$-0.622310\pi$
$941$	213031.	−	122994.i	0.240582	−	0.138900i	0.615444	+	0.788180i	$0.288977\pi$
$942$		0			0
$943$	1.60656e6	+	927546.i	1.80665	+	1.04307i
$944$		0			0
$945$	−853.478	−	542985.i	−0.000955716	−	0.608029i
$946$		0			0
$947$	−722964.	+	1.25221e6i	−0.806152	+	1.39630i	0.109359	+	0.994002i	$0.465120\pi$
$947$	−722964.	+	1.25221e6i	−0.806152	+	1.39630i	−0.915511	+	0.402294i	$0.868213\pi$
$948$		0			0
$949$	−258384.	−	447533.i	−0.286901	−	0.496928i
$950$		0			0
$951$		−	91495.4i		−	0.101167i
$952$		0			0
$953$	907377.			0.999084			0.499542	−	0.866290i	$-0.333502\pi$
$953$	907377.			0.999084			0.499542	+	0.866290i	$0.333502\pi$
$954$		0			0
$955$	170805.	−	98614.2i	0.187281	−	0.108127i
$956$		0			0
$957$	12989.3	+	7499.38i	0.0141828	+	0.00818845i
$958$		0			0
$959$	56156.9	+	32304.6i	0.0610612	+	0.0351259i
$960$		0			0
$961$	−234235.	+	405707.i	−0.253633	+	0.439305i
$962$		0			0
$963$	583491.	+	1.01064e6i	0.629189	+	1.08979i
$964$		0			0
$965$			1.30132e6i			1.39742i
$966$		0			0
$967$	50286.2			0.0537770			0.0268885	−	0.999638i	$-0.491440\pi$
$967$	50286.2			0.0537770			0.0268885	+	0.999638i	$0.491440\pi$
$968$		0			0
$969$	−159483.	+	92077.6i	−0.169851	+	0.0980633i
$970$		0			0
$971$	−677500.	−	391155.i	−0.718573	−	0.414868i	0.0956544	−	0.995415i	$-0.469506\pi$
$971$	−677500.	−	391155.i	−0.718573	−	0.414868i	−0.814227	+	0.580546i	$0.802839\pi$
$972$		0			0
$973$	−125776.	−	217062.i	−0.132853	−	0.229275i
$974$		0			0
$975$	142652.	−	247081.i	0.150062	−	0.259914i
$976$		0			0
$977$	−638714.	−	1.10629e6i	−0.669141	−	1.15899i	−0.978145	−	0.207925i	$-0.933329\pi$
$977$	−638714.	−	1.10629e6i	−0.669141	−	1.15899i	0.309004	−	0.951061i	$-0.400004\pi$
$978$		0			0
$979$			218557.i			0.228033i
$980$		0			0
$981$	216501.			0.224968
$982$		0			0
$983$	−1.36250e6	+	786638.i	−1.41003	+	0.814081i	−0.995391	−	0.0959049i	$-0.969426\pi$
$983$	−1.36250e6	+	786638.i	−1.41003	+	0.814081i	−0.414639	+	0.909986i	$0.636092\pi$
$984$		0			0
$985$	912930.	+	527080.i	0.940947	+	0.543256i
$986$		0			0
$987$	−44488.2	+	25778.6i	−0.0456679	+	0.0264621i
$988$		0			0
$989$	−987661.	+	1.71068e6i	−1.00975	+	1.74894i
$990$		0			0
$991$	162173.	+	280892.i	0.165132	+	0.286017i	0.936702	−	0.350127i	$-0.113862\pi$
$991$	162173.	+	280892.i	0.165132	+	0.286017i	−0.771570	+	0.636144i	$0.780528\pi$
$992$		0			0
$993$		−	1552.30i		−	0.00157427i
$994$		0			0
$995$	2.16099e6			2.18276
$996$		0			0
$997$	−1.57129e6	+	907185.i	−1.58076	+	0.912652i	−0.586012	+	0.810303i	$0.699303\pi$
$997$	−1.57129e6	+	907185.i	−1.58076	+	0.912652i	−0.994748	+	0.102350i	$0.967364\pi$
$998$		0			0
$999$	−161597.	−	93298.2i	−0.161921	−	0.0934851i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.5.s.c.33.2		6
4.3	odd	2		28.5.h.a.5.2	✓	6
7.2	even	3		784.5.c.e.97.4		6
7.3	odd	6	inner	112.5.s.c.17.2		6
7.5	odd	6		784.5.c.e.97.3		6
12.11	even	2		252.5.z.f.145.1		6
20.3	even	4		700.5.o.a.649.3		12
20.7	even	4		700.5.o.a.649.4		12
20.19	odd	2		700.5.s.a.201.2		6
28.3	even	6		28.5.h.a.17.2	yes	6
28.11	odd	6		196.5.h.c.129.2		6
28.19	even	6		196.5.b.a.97.4		6
28.23	odd	6		196.5.b.a.97.3		6
28.27	even	2		196.5.h.c.117.2		6
84.59	odd	6		252.5.z.f.73.1		6
140.3	odd	12		700.5.o.a.549.4		12
140.59	even	6		700.5.s.a.101.2		6
140.87	odd	12		700.5.o.a.549.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.5.h.a.5.2	✓	6	4.3	odd	2
28.5.h.a.17.2	yes	6	28.3	even	6
112.5.s.c.17.2		6	7.3	odd	6	inner
112.5.s.c.33.2		6	1.1	even	1	trivial
196.5.b.a.97.3		6	28.23	odd	6
196.5.b.a.97.4		6	28.19	even	6
196.5.h.c.117.2		6	28.27	even	2
196.5.h.c.129.2		6	28.11	odd	6
252.5.z.f.73.1		6	84.59	odd	6
252.5.z.f.145.1		6	12.11	even	2
700.5.o.a.549.3		12	140.87	odd	12
700.5.o.a.549.4		12	140.3	odd	12
700.5.o.a.649.3		12	20.3	even	4
700.5.o.a.649.4		12	20.7	even	4
700.5.s.a.101.2		6	140.59	even	6
700.5.s.a.201.2		6	20.19	odd	2
784.5.c.e.97.3		6	7.5	odd	6
784.5.c.e.97.4		6	7.2	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	112.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.35901 - 0.784623i) q^{3} +(38.3327 + 22.1314i) q^{5} +(-42.3967 + 24.5667i) q^{7} +(-39.2687 + 68.0154i) q^{9} +(11.7557 + 20.3615i) q^{11} -136.269i q^{13} +69.4592 q^{15} +(-227.333 + 131.251i) q^{17} +(387.162 + 223.528i) q^{19} +(-38.3418 + 66.6517i) q^{21} +(-374.585 + 648.800i) q^{23} +(667.098 + 1155.45i) q^{25} +250.353i q^{27} +406.524 q^{29} +(584.199 - 337.287i) q^{31} +(31.9521 + 18.4476i) q^{33} +(-2168.87 + 3.40909i) q^{35} +(-372.666 + 645.476i) q^{37} +(-106.920 - 185.191i) q^{39} -2476.20i q^{41} +2636.68 q^{43} +(-3010.55 + 1738.14i) q^{45} +(579.099 + 334.343i) q^{47} +(1193.96 - 2083.09i) q^{49} +(-205.964 + 356.741i) q^{51} +(1014.61 + 1757.36i) q^{53} +1040.68i q^{55} +701.541 q^{57} +(1014.22 - 585.561i) q^{59} +(-2022.05 - 1167.43i) q^{61} +(-6.04891 - 3848.33i) q^{63} +(3015.83 - 5223.58i) q^{65} +(-3779.79 - 6546.78i) q^{67} +1175.63i q^{69} -7575.36 q^{71} +(3284.18 - 1896.12i) q^{73} +(1813.18 + 1046.84i) q^{75} +(-998.615 - 574.460i) q^{77} +(3897.33 - 6750.37i) q^{79} +(-2984.33 - 5169.02i) q^{81} +2181.41i q^{83} -11619.0 q^{85} +(552.469 - 318.968i) q^{87} +(8050.38 + 4647.89i) q^{89} +(3347.69 + 5777.37i) q^{91} +(529.287 - 916.751i) q^{93} +(9893.98 + 17136.9i) q^{95} +9981.90i q^{97} -1846.52 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 9 q^{3} - 27 q^{5} - 66 q^{7} + 90 q^{9} - 135 q^{11} + 486 q^{15} - 1107 q^{17} + 747 q^{19} + 2169 q^{21} - 243 q^{23} + 1878 q^{25} - 540 q^{29} + 5355 q^{31} - 1863 q^{33} - 6021 q^{35} + 2355 q^{37}+ \cdots - 8100 q^{99}+O(q^{100}) \) 6 * q - 9 * q^3 - 27 * q^5 - 66 * q^7 + 90 * q^9 - 135 * q^11 + 486 * q^15 - 1107 * q^17 + 747 * q^19 + 2169 * q^21 - 243 * q^23 + 1878 * q^25 - 540 * q^29 + 5355 * q^31 - 1863 * q^33 - 6021 * q^35 + 2355 * q^37 - 6588 * q^39 + 948 * q^43 - 14418 * q^45 + 9747 * q^47 + 8430 * q^49 + 891 * q^51 + 6291 * q^53 - 6894 * q^57 + 2943 * q^59 + 4041 * q^61 - 15138 * q^63 + 7668 * q^65 - 1659 * q^67 - 2268 * q^71 - 8703 * q^73 + 3438 * q^75 - 10665 * q^77 + 7773 * q^79 - 7479 * q^81 - 702 * q^85 - 40014 * q^87 + 14985 * q^89 + 20952 * q^91 + 1449 * q^93 + 20655 * q^95 - 8100 * q^99

Introduction

L-functions

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