LMFDB - Embedded newform 112.6.p.b.47.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,Mod(31,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 1])) N = Newforms(chi, 6, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); S:= CuspForms(chi, 6); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,-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:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} + 691 x^{12} - 8602 x^{11} + 416261 x^{10} - 3521447 x^{9} + 66162087 x^{8} + \cdots + 17213603549184 $ x^14 - 3x^13 + 691x^12 - 8602x^11 + 416261x^10 - 3521447x^9 + 66162087x^8 - 31847546x^7 + 3819865057x^6 + 14320412589x^5 + 280505627703x^4 + 1315235310354x^3 + 6574080917124x^2 + 11537944725888*x + 17213603549184
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{20}\cdot 3^{6}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			47.5
Root			$-2.79973 + 4.84928i$ of defining polynomial
Character	$\chi$	$=$	112.47
Dual form			112.6.p.b.31.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.77385 + 8.26855i) q^{3} +(88.0882 + 50.8577i) q^{5} +(12.6490 - 129.023i) q^{7} +(75.9208 - 131.499i) q^{9} +(496.703 - 286.771i) q^{11} -25.1634i q^{13} +971.148i q^{15} +(-1129.13 + 651.905i) q^{17} +(425.421 - 736.851i) q^{19} +(1127.22 - 511.349i) q^{21} +(-1903.24 - 1098.84i) q^{23} +(3610.52 + 6253.60i) q^{25} +3769.83 q^{27} +7137.60 q^{29} +(-1633.03 - 2828.49i) q^{31} +(4742.36 + 2738.01i) q^{33} +(7676.05 - 10722.1i) q^{35} +(-6764.09 + 11715.7i) q^{37} +(208.065 - 120.126i) q^{39} -169.841i q^{41} +17478.2i q^{43} +(13375.4 - 7722.31i) q^{45} +(-325.545 + 563.860i) q^{47} +(-16487.0 - 3264.02i) q^{49} +(-10780.6 - 6224.19i) q^{51} +(4905.03 + 8495.76i) q^{53} +58338.1 q^{55} +8123.59 q^{57} +(-20726.3 - 35899.1i) q^{59} +(10084.1 + 5822.05i) q^{61} +(-16006.1 - 11458.9i) q^{63} +(1279.75 - 2216.60i) q^{65} +(-26548.8 + 15328.0i) q^{67} -20982.7i q^{69} +32773.7i q^{71} +(-18007.7 + 10396.7i) q^{73} +(-34472.1 + 59707.4i) q^{75} +(-30717.4 - 67713.5i) q^{77} +(11149.2 + 6437.02i) q^{79} +(-452.167 - 783.177i) q^{81} -120956. q^{83} -132618. q^{85} +(34073.8 + 59017.6i) q^{87} +(-86406.2 - 49886.6i) q^{89} +(-3246.67 - 318.291i) q^{91} +(15591.7 - 27005.6i) q^{93} +(74949.2 - 43271.9i) q^{95} -46717.2i q^{97} -87087.6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 9 q^{3} + 33 q^{5} - 28 q^{7} - 538 q^{9} + 333 q^{11} + 801 q^{17} - 2135 q^{19} + 2017 q^{21} + 2667 q^{23} + 5434 q^{25} + 17910 q^{27} + 684 q^{29} + 3119 q^{31} + 29013 q^{33} - 2247 q^{35}+ \cdots - 124833 q^{95}+O(q^{100}) $ 14 * q - 9 * q^3 + 33 * q^5 - 28 * q^7 - 538 * q^9 + 333 * q^11 + 801 * q^17 - 2135 * q^19 + 2017 * q^21 + 2667 * q^23 + 5434 * q^25 + 17910 * q^27 + 684 * q^29 + 3119 * q^31 + 29013 * q^33 - 2247 * q^35 + 3431 * q^37 + 2424 * q^39 - 28086 * q^45 + 8409 * q^47 - 24706 * q^49 + 24657 * q^51 + 1707 * q^53 - 56622 * q^55 + 68450 * q^57 - 49305 * q^59 - 21843 * q^61 + 164142 * q^63 + 17448 * q^65 - 73263 * q^67 - 77043 * q^73 - 147834 * q^75 + 22287 * q^77 + 206523 * q^79 - 145567 * q^81 - 305400 * q^83 - 176454 * q^85 - 95034 * q^87 + 171381 * q^89 + 174816 * q^91 + 292873 * q^93 - 124833 * q^95

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
$3$	4.77385	+	8.26855i	0.306242	+	0.530428i	0.977537	−	0.210763i	$-0.0675948\pi$
$3$	4.77385	+	8.26855i	0.306242	+	0.530428i	−0.671295	+	0.741191i	$0.734262\pi$
$4$		0			0
$5$	88.0882	+	50.8577i	1.57577	+	0.909771i	0.995440	+	0.0953882i	$0.0304092\pi$
$5$	88.0882	+	50.8577i	1.57577	+	0.909771i	0.580329	+	0.814382i	$0.302924\pi$
$6$		0			0
$7$	12.6490	−	129.023i	0.0975685	−	0.995229i
$7$	12.6490	−	129.023i	0.0975685	−	0.995229i
$8$		0			0
$9$	75.9208	−	131.499i	0.312431	−	0.541147i
$10$		0			0
$11$	496.703	−	286.771i	1.23770	−	0.714585i	0.269075	−	0.963119i	$-0.413282\pi$
$11$	496.703	−	286.771i	1.23770	−	0.714585i	0.968623	+	0.248534i	$0.0799488\pi$
$12$		0			0
$13$		−	25.1634i		−	0.0412963i	−0.999787	−	0.0206482i	$-0.993427\pi$
$13$		−	25.1634i		−	0.0412963i	0.999787	−	0.0206482i	$-0.00657298\pi$
$14$		0			0
$15$			971.148i			1.11444i
$16$		0			0
$17$	−1129.13	+	651.905i	−0.947594	+	0.547094i	−0.892333	−	0.451378i	$-0.850933\pi$
$17$	−1129.13	+	651.905i	−0.947594	+	0.547094i	−0.0552616	+	0.998472i	$0.517599\pi$
$18$		0			0
$19$	425.421	−	736.851i	0.270356	−	0.468270i	−0.698597	−	0.715515i	$-0.746192\pi$
$19$	425.421	−	736.851i	0.270356	−	0.468270i	0.968953	+	0.247246i	$0.0795254\pi$
$20$		0			0
$21$	1127.22	−	511.349i	0.557776	−	0.253028i
$22$		0			0
$23$	−1903.24	−	1098.84i	−0.750195	−	0.433125i	0.0755694	−	0.997141i	$-0.475923\pi$
$23$	−1903.24	−	1098.84i	−0.750195	−	0.433125i	−0.825764	+	0.564015i	$0.809256\pi$
$24$		0			0
$25$	3610.52	+	6253.60i	1.15536	+	2.00115i
$26$		0			0
$27$	3769.83			0.995204
$28$		0			0
$29$	7137.60			1.57600			0.788002	−	0.615673i	$-0.211116\pi$
$29$	7137.60			1.57600			0.788002	+	0.615673i	$0.211116\pi$
$30$		0			0
$31$	−1633.03	−	2828.49i	−0.305204	−	0.528628i	0.672103	−	0.740458i	$-0.265391\pi$
$31$	−1633.03	−	2828.49i	−0.305204	−	0.528628i	−0.977307	+	0.211829i	$0.932058\pi$
$32$		0			0
$33$	4742.36	+	2738.01i	0.758071	+	0.437673i
$34$		0			0
$35$	7676.05	−	10722.1i	1.05918	−	1.47949i
$36$		0			0
$37$	−6764.09	+	11715.7i	−0.812278	+	1.40691i	0.0989877	+	0.995089i	$0.468440\pi$
$37$	−6764.09	+	11715.7i	−0.812278	+	1.40691i	−0.911266	+	0.411818i	$0.864894\pi$
$38$		0			0
$39$	208.065	−	120.126i	0.0219047	−	0.0126467i
$40$		0			0
$41$		−	169.841i		−	0.0157791i	−0.999969	−	0.00788957i	$-0.997489\pi$
$41$		−	169.841i		−	0.0157791i	0.999969	−	0.00788957i	$-0.00251135\pi$
$42$		0			0
$43$			17478.2i			1.44154i	0.693175	+	0.720770i	$0.256211\pi$
$43$			17478.2i			1.44154i	−0.693175	+	0.720770i	$0.743789\pi$
$44$		0			0
$45$	13375.4	−	7722.31i	0.984638	−	0.568481i
$46$		0			0
$47$	−325.545	+	563.860i	−0.0214964	+	0.0372329i	−0.876573	−	0.481268i	$-0.840176\pi$
$47$	−325.545	+	563.860i	−0.0214964	+	0.0372329i	0.855077	+	0.518501i	$0.173510\pi$
$48$		0			0
$49$	−16487.0	−	3264.02i	−0.980961	−	0.194206i
$50$		0			0
$51$	−10780.6	−	6224.19i	−0.580387	−	0.335087i
$52$		0			0
$53$	4905.03	+	8495.76i	0.239857	+	0.415444i	0.960673	−	0.277682i	$-0.0895662\pi$
$53$	4905.03	+	8495.76i	0.239857	+	0.415444i	−0.720816	+	0.693126i	$0.756233\pi$
$54$		0			0
$55$	58338.1			2.60043
$56$		0			0
$57$	8123.59			0.331177
$58$		0			0
$59$	−20726.3	−	35899.1i	−0.775162	−	1.34262i	−0.934703	−	0.355429i	$-0.884335\pi$
$59$	−20726.3	−	35899.1i	−0.775162	−	1.34262i	0.159542	−	0.987191i	$-0.448998\pi$
$60$		0			0
$61$	10084.1	+	5822.05i	0.346986	+	0.200332i	0.663357	−	0.748303i	$-0.269131\pi$
$61$	10084.1	+	5822.05i	0.346986	+	0.200332i	−0.316371	+	0.948636i	$0.602464\pi$
$62$		0			0
$63$	−16006.1	−	11458.9i	−0.508081	−	0.363739i
$64$		0			0
$65$	1279.75	−	2216.60i	0.0375702	−	0.0650734i
$66$		0			0
$67$	−26548.8	+	15328.0i	−0.722534	+	0.417155i	−0.815685	−	0.578497i	$-0.803639\pi$
$67$	−26548.8	+	15328.0i	−0.722534	+	0.417155i	0.0931507	+	0.995652i	$0.470306\pi$
$68$		0			0
$69$		−	20982.7i		−	0.530565i
$70$		0			0
$71$			32773.7i			0.771578i	0.922587	+	0.385789i	$0.126071\pi$
$71$			32773.7i			0.771578i	−0.922587	+	0.385789i	$0.873929\pi$
$72$		0			0
$73$	−18007.7	+	10396.7i	−0.395504	+	0.228344i	−0.684542	−	0.728973i	$-0.739998\pi$
$73$	−18007.7	+	10396.7i	−0.395504	+	0.228344i	0.289038	+	0.957318i	$0.406664\pi$
$74$		0			0
$75$	−34472.1	+	59707.4i	−0.707644	+	1.22567i
$76$		0			0
$77$	−30717.4	−	67713.5i	−0.590416	−	1.30151i
$78$		0			0
$79$	11149.2	+	6437.02i	0.200991	+	0.116042i	0.597118	−	0.802154i	$-0.296312\pi$
$79$	11149.2	+	6437.02i	0.200991	+	0.116042i	−0.396126	+	0.918196i	$0.629646\pi$
$80$		0			0
$81$	−452.167	−	783.177i	−0.00765749	−	0.0132632i
$82$		0			0
$83$	−120956.			−1.92723			−0.963614	−	0.267297i	$-0.913870\pi$
$83$	−120956.			−1.92723			−0.963614	+	0.267297i	$0.913870\pi$
$84$		0			0
$85$	−132618.			−1.99092
$86$		0			0
$87$	34073.8	+	59017.6i	0.482639	+	0.835956i
$88$		0			0
$89$	−86406.2	−	49886.6i	−1.15630	−	0.667589i	−0.205884	−	0.978576i	$-0.566007\pi$
$89$	−86406.2	−	49886.6i	−1.15630	−	0.667589i	−0.950414	+	0.310987i	$0.899340\pi$
$90$		0			0
$91$	−3246.67	−	318.291i	−0.0410993	−	0.00402922i
$92$		0			0
$93$	15591.7	−	27005.6i	0.186933	−	0.323777i
$94$		0			0
$95$	74949.2	−	43271.9i	0.852036	−	0.491923i
$96$		0			0
$97$		−	46717.2i		−	0.504136i	−0.967710	−	0.252068i	$-0.918889\pi$
$97$		−	46717.2i		−	0.504136i	0.967710	−	0.252068i	$-0.0811106\pi$
$98$		0			0
$99$		−	87087.6i		−	0.893035i
$100$		0			0
$101$	65121.0	−	37597.6i	0.635210	−	0.366739i	−0.147557	−	0.989054i	$-0.547141\pi$
$101$	65121.0	−	37597.6i	0.635210	−	0.366739i	0.782767	+	0.622315i	$0.213808\pi$
$102$		0			0
$103$	−6802.39	+	11782.1i	−0.0631783	+	0.109428i	−0.895884	−	0.444287i	$-0.853457\pi$
$103$	−6802.39	+	11782.1i	−0.0631783	+	0.109428i	0.832706	+	0.553715i	$0.186790\pi$
$104$		0			0
$105$	125301.	+	12284.0i	1.10912	+	0.108734i
$106$		0			0
$107$	171323.	+	98913.6i	1.44663	+	0.835212i	0.998279	−	0.0586455i	$-0.0186782\pi$
$107$	171323.	+	98913.6i	1.44663	+	0.835212i	0.448351	+	0.893858i	$0.352012\pi$
$108$		0			0
$109$	31302.3	+	54217.2i	0.252354	+	0.437090i	0.964173	−	0.265273i	$-0.0854620\pi$
$109$	31302.3	+	54217.2i	0.252354	+	0.437090i	−0.711820	+	0.702362i	$0.752129\pi$
$110$		0			0
$111$	−129163.			−0.995016
$112$		0			0
$113$	−21630.5			−0.159357			−0.0796783	−	0.996821i	$-0.525389\pi$
$113$	−21630.5			−0.159357			−0.0796783	+	0.996821i	$0.525389\pi$
$114$		0			0
$115$	−111769.	−	193589.i	−0.788089	−	1.36501i
$116$		0			0
$117$	−3308.95	−	1910.43i	−0.0223474	−	0.0129023i
$118$		0			0
$119$	69828.5	+	153930.i	0.452028	+	0.996452i
$120$		0			0
$121$	83950.1	−	145406.i	0.521264	−	0.902856i
$122$		0			0
$123$	1404.34	−	810.796i	0.00836969	−	0.00483224i
$124$		0			0
$125$			416630.i			2.38493i
$126$		0			0
$127$		−	162179.i		−	0.892245i	−0.894972	−	0.446123i	$-0.852805\pi$
$127$		−	162179.i		−	0.892245i	0.894972	−	0.446123i	$-0.147195\pi$
$128$		0			0
$129$	−144520.	+	83438.4i	−0.764632	+	0.441461i
$130$		0			0
$131$	−23348.6	+	40440.9i	−0.118873	+	0.205894i	−0.919321	−	0.393508i	$-0.871261\pi$
$131$	−23348.6	+	40440.9i	−0.118873	+	0.205894i	0.800448	+	0.599402i	$0.204595\pi$
$132$		0			0
$133$	−89689.8	−	64209.6i	−0.439657	−	0.314754i
$134$		0			0
$135$	332077.	+	191725.i	1.56821	+	0.905407i
$136$		0			0
$137$	−92841.2	−	160806.i	−0.422610	−	0.731981i	0.573584	−	0.819147i	$-0.305553\pi$
$137$	−92841.2	−	160806.i	−0.422610	−	0.731981i	−0.996194	+	0.0871652i	$0.972219\pi$
$138$		0			0
$139$	−233047.			−1.02307			−0.511536	−	0.859262i	$-0.670923\pi$
$139$	−233047.			−1.02307			−0.511536	+	0.859262i	$0.670923\pi$
$140$		0			0
$141$	−6216.40			−0.0263324
$142$		0			0
$143$	−7216.15	−	12498.7i	−0.0295097	−	0.0511124i
$144$		0			0
$145$	628738.	+	363002.i	2.48342	+	1.43380i
$146$		0			0
$147$	−51717.8	−	151906.i	−0.197400	−	0.579803i
$148$		0			0
$149$	−5242.93	+	9081.01i	−0.0193468	+	0.0335096i	−0.875537	−	0.483152i	$-0.839492\pi$
$149$	−5242.93	+	9081.01i	−0.0193468	+	0.0335096i	0.856190	+	0.516661i	$0.172825\pi$
$150$		0			0
$151$	168952.	−	97544.7i	0.603007	−	0.348146i	−0.167217	−	0.985920i	$-0.553478\pi$
$151$	168952.	−	97544.7i	0.603007	−	0.348146i	0.770223	+	0.637774i	$0.220145\pi$
$152$		0			0
$153$			197972.i			0.683717i
$154$		0			0
$155$		−	332209.i		−	1.11066i
$156$		0			0
$157$	310448.	−	179237.i	1.00517	−	0.580335i	0.0953958	−	0.995439i	$-0.469588\pi$
$157$	310448.	−	179237.i	1.00517	−	0.580335i	0.909774	+	0.415105i	$0.136255\pi$
$158$		0			0
$159$	−46831.7	+	81114.9i	−0.146909	+	0.254453i
$160$		0			0
$161$	−165849.	+	231663.i	−0.504254	+	0.704356i
$162$		0			0
$163$	−93743.9	−	54123.1i	−0.276359	−	0.159556i	0.355415	−	0.934709i	$-0.384340\pi$
$163$	−93743.9	−	54123.1i	−0.276359	−	0.159556i	−0.631774	+	0.775153i	$0.717673\pi$
$164$		0			0
$165$	278497.	+	482372.i	0.796363	+	1.37934i
$166$		0			0
$167$	419205.			1.16315			0.581574	−	0.813493i	$-0.302437\pi$
$167$	419205.			1.16315			0.581574	+	0.813493i	$0.302437\pi$
$168$		0			0
$169$	370660.			0.998295
$170$		0			0
$171$	−64596.6	−	111885.i	−0.168935	−	0.292604i
$172$		0			0
$173$	−26114.7	−	15077.3i	−0.0663392	−	0.0383009i	0.466464	−	0.884540i	$-0.345528\pi$
$173$	−26114.7	−	15077.3i	−0.0663392	−	0.0383009i	−0.532803	+	0.846239i	$0.678861\pi$
$174$		0			0
$175$	852529.	−	386739.i	2.10433	−	0.954603i
$176$		0			0
$177$	197889.	−	342753.i	0.474775	−	0.822334i
$178$		0			0
$179$	−667579.	+	385427.i	−1.55729	+	0.899103i	−0.559777	+	0.828643i	$0.689113\pi$
$179$	−667579.	+	385427.i	−1.55729	+	0.899103i	−0.997515	+	0.0704593i	$0.977554\pi$
$180$		0			0
$181$		−	664929.i		−	1.50862i	−0.656520	−	0.754308i	$-0.727972\pi$
$181$		−	664929.i		−	1.50862i	0.656520	−	0.754308i	$-0.272028\pi$
$182$		0			0
$183$			111174.i			0.245401i
$184$		0			0
$185$	−1.19167e6	+	688012.i	−2.55993	+	1.47797i
$186$		0			0
$187$	−373895.	+	647605.i	−0.781890	+	1.35427i
$188$		0			0
$189$	47684.4	−	486395.i	0.0971005	−	0.990455i
$190$		0			0
$191$	488504.	+	282038.i	0.968912	+	0.559402i	0.898904	−	0.438145i	$-0.144364\pi$
$191$	488504.	+	282038.i	0.968912	+	0.559402i	0.0700078	+	0.997546i	$0.477698\pi$
$192$		0			0
$193$	−286373.	−	496013.i	−0.553400	−	0.958516i	−0.998026	−	0.0628001i	$-0.979997\pi$
$193$	−286373.	−	496013.i	−0.553400	−	0.958516i	0.444627	−	0.895716i	$-0.353336\pi$
$194$		0			0
$195$	24437.4			0.0460223
$196$		0			0
$197$	−436595.			−0.801517			−0.400759	−	0.916184i	$-0.631253\pi$
$197$	−436595.			−0.801517			−0.400759	+	0.916184i	$0.631253\pi$
$198$		0			0
$199$	−380385.	−	658847.i	−0.680912	−	1.17937i	−0.974703	−	0.223505i	$-0.928250\pi$
$199$	−380385.	−	658847.i	−0.680912	−	1.17937i	0.293790	−	0.955870i	$-0.405083\pi$
$200$		0			0
$201$	−253480.	−	146347.i	−0.442541	−	0.255501i
$202$		0			0
$203$	90283.2	−	920917.i	0.153768	−	1.56848i
$204$		0			0
$205$	8637.73	−	14961.0i	0.0143554	−	0.0248643i
$206$		0			0
$207$	−288991.	+	166849.i	−0.468768	+	0.270644i
$208$		0			0
$209$		−	487995.i		−	0.772768i
$210$		0			0
$211$			790222.i			1.22192i	0.791661	+	0.610961i	$0.209217\pi$
$211$			790222.i			1.22192i	−0.791661	+	0.610961i	$0.790783\pi$
$212$		0			0
$213$	−270991.	+	156457.i	−0.409266	+	0.236290i
$214$		0			0
$215$	−888903.	+	1.53963e6i	−1.31147	+	2.27153i
$216$		0			0
$217$	−385597.	+	174921.i	−0.555884	+	0.252170i
$218$		0			0
$219$	−171932.	−	99264.9i	−0.242240	−	0.139857i
$220$		0			0
$221$	16404.2	+	28412.8i	0.0225930	+	0.0391322i
$222$		0			0
$223$	−583111.			−0.785216			−0.392608	−	0.919706i	$-0.628427\pi$
$223$	−583111.			−0.785216			−0.392608	+	0.919706i	$0.628427\pi$
$224$		0			0
$225$	1.09645e6			1.44389
$226$		0			0
$227$	−36799.4	−	63738.4i	−0.0473998	−	0.0820988i	0.841352	−	0.540487i	$-0.181760\pi$
$227$	−36799.4	−	63738.4i	−0.0473998	−	0.0820988i	−0.888752	+	0.458389i	$0.848427\pi$
$228$		0			0
$229$	766067.	+	442289.i	0.965335	+	0.557336i	0.897811	−	0.440381i	$-0.145157\pi$
$229$	766067.	+	442289.i	0.965335	+	0.557336i	0.0675238	+	0.997718i	$0.478490\pi$
$230$		0			0
$231$	413252.	−	577242.i	0.509548	−	0.711751i
$232$		0			0
$233$	333981.	−	578472.i	0.403025	−	0.698059i	−0.591065	−	0.806624i	$-0.701292\pi$
$233$	333981.	−	578472.i	0.403025	−	0.698059i	0.994089	+	0.108565i	$0.0346255\pi$
$234$		0			0
$235$	−57353.2	+	33112.9i	−0.0677467	+	0.0391136i
$236$		0			0
$237$			122917.i			0.142149i
$238$		0			0
$239$			320522.i			0.362963i	0.983394	+	0.181482i	$0.0580893\pi$
$239$			320522.i			0.362963i	−0.983394	+	0.181482i	$0.941911\pi$
$240$		0			0
$241$	−534192.	+	308416.i	−0.592455	+	0.342054i	−0.766067	−	0.642760i	$-0.777789\pi$
$241$	−534192.	+	308416.i	−0.592455	+	0.342054i	0.173613	+	0.984814i	$0.444456\pi$
$242$		0			0
$243$	462351.	−	800816.i	0.502292	−	0.869995i
$244$		0			0
$245$	−1.28631e6	−	1.12601e6i	−1.36908	−	1.19847i
$246$		0			0
$247$	−18541.7	−	10705.1i	−0.0193378	−	0.0111647i
$248$		0			0
$249$	−577427.	−	1.00013e6i	−0.590199	−	1.02226i
$250$		0			0
$251$	−349034.			−0.349691			−0.174845	−	0.984596i	$-0.555943\pi$
$251$	−349034.			−0.349691			−0.174845	+	0.984596i	$0.555943\pi$
$252$		0			0
$253$	−1.26046e6			−1.23802
$254$		0			0
$255$	−633096.	−	1.09655e6i	−0.609704	−	1.05604i
$256$		0			0
$257$	144724.	+	83556.6i	0.136681	+	0.0789129i	0.566781	−	0.823868i	$-0.308188\pi$
$257$	144724.	+	83556.6i	0.136681	+	0.0789129i	−0.430100	+	0.902781i	$0.641522\pi$
$258$		0			0
$259$	1.42604e6	+	1.02092e6i	1.32094	+	0.945672i
$260$		0			0
$261$	541892.	−	938585.i	0.492393	−	0.852849i
$262$		0			0
$263$	−948936.	+	547868.i	−0.845956	+	0.488413i	−0.859284	−	0.511498i	$-0.829091\pi$
$263$	−948936.	+	547868.i	−0.845956	+	0.488413i	0.0133286	+	0.999911i	$0.495757\pi$
$264$		0			0
$265$			997834.i			0.872858i
$266$		0			0
$267$		−	952605.i		−	0.817776i
$268$		0			0
$269$	−228858.	+	132131.i	−0.192834	+	0.111333i	−0.593309	−	0.804975i	$-0.702179\pi$
$269$	−228858.	+	132131.i	−0.192834	+	0.111333i	0.400474	+	0.916308i	$0.368845\pi$
$270$		0			0
$271$	−860139.	+	1.48980e6i	−0.711452	+	1.23227i	0.252861	+	0.967503i	$0.418629\pi$
$271$	−860139.	+	1.48980e6i	−0.711452	+	1.23227i	−0.964312	+	0.264767i	$0.914705\pi$
$272$		0			0
$273$	−12867.3	−	28364.7i	−0.0104491	−	0.0230341i
$274$		0			0
$275$	3.58670e6	+	2.07078e6i	2.85999	+	1.65121i
$276$		0			0
$277$	512463.	+	887613.i	0.401295	+	0.695063i	0.993882	−	0.110443i	$-0.0352270\pi$
$277$	512463.	+	887613.i	0.401295	+	0.695063i	−0.592588	+	0.805506i	$0.701894\pi$
$278$		0			0
$279$	−495923.			−0.381420
$280$		0			0
$281$	−520974.			−0.393595			−0.196798	−	0.980444i	$-0.563054\pi$
$281$	−520974.			−0.393595			−0.196798	+	0.980444i	$0.563054\pi$
$282$		0			0
$283$	−517444.	−	896240.i	−0.384059	−	0.665209i	0.607580	−	0.794259i	$-0.292141\pi$
$283$	−517444.	−	896240.i	−0.384059	−	0.665209i	−0.991638	+	0.129050i	$0.958807\pi$
$284$		0			0
$285$	715592.	+	413147.i	0.521859	+	0.301295i
$286$		0			0
$287$	−21913.5	−	2148.31i	−0.0157039	−	0.00153955i
$288$		0			0
$289$	140031.	−	242541.i	0.0986234	−	0.170821i
$290$		0			0
$291$	386283.	−	223021.i	0.267407	−	0.154388i
$292$		0			0
$293$			2.08248e6i			1.41714i	0.705643	+	0.708568i	$0.250658\pi$
$293$			2.08248e6i			1.41714i	−0.705643	+	0.708568i	$0.749342\pi$
$294$		0			0
$295$		−	4.21638e6i		−	2.82088i
$296$		0			0
$297$	1.87248e6	−	1.08108e6i	1.23176	−	0.711158i
$298$		0			0
$299$	−27650.5	+	47892.0i	−0.0178865	+	0.0309803i
$300$		0			0
$301$	2.25510e6	+	221081.i	1.43466	+	0.140649i
$302$		0			0
$303$	621755.	+	358971.i	0.389057	+	0.224622i
$304$		0			0
$305$	592192.	+	1.02571e6i	0.364513	+	0.631355i
$306$		0			0
$307$	−1.31236e6			−0.794709			−0.397354	−	0.917665i	$-0.630072\pi$
$307$	−1.31236e6			−0.794709			−0.397354	+	0.917665i	$0.630072\pi$
$308$		0			0
$309$	−129894.			−0.0773916
$310$		0			0
$311$	133380.	+	231020.i	0.0781967	+	0.135441i	0.902472	−	0.430749i	$-0.141750\pi$
$311$	133380.	+	231020.i	0.0781967	+	0.135441i	−0.824275	+	0.566189i	$0.808417\pi$
$312$		0			0
$313$	2.04046e6	+	1.17806e6i	1.17725	+	0.679683i	0.955376	−	0.295393i	$-0.0954505\pi$
$313$	2.04046e6	+	1.17806e6i	1.17725	+	0.679683i	0.221870	+	0.975076i	$0.428784\pi$
$314$		0			0
$315$	−827173.	−	1.82342e6i	−0.469699	−	1.03541i
$316$		0			0
$317$	−336374.	+	582616.i	−0.188007	+	0.325637i	−0.944586	−	0.328265i	$-0.893536\pi$
$317$	−336374.	+	582616.i	−0.188007	+	0.325637i	0.756579	+	0.653903i	$0.226869\pi$
$318$		0			0
$319$	3.54527e6	−	2.04686e6i	1.95062	−	1.12619i
$320$		0			0
$321$			1.88879e6i			1.02311i
$322$		0			0
$323$			1.10934e6i			0.591639i
$324$		0			0
$325$	157362.	−	90852.9i	0.0826401	−	0.0477123i
$326$		0			0
$327$	−298865.	+	517649.i	−0.154563	+	0.267711i
$328$		0			0
$329$	68633.2	+	49135.1i	0.0349578	+	0.0250266i
$330$		0			0
$331$	1.94594e6	+	1.12349e6i	0.976247	+	0.563636i	0.901135	−	0.433539i	$-0.142735\pi$
$331$	1.94594e6	+	1.12349e6i	0.976247	+	0.563636i	0.0751118	+	0.997175i	$0.476069\pi$
$332$		0			0
$333$	1.02707e6	+	1.77894e6i	0.507562	+	0.879123i
$334$		0			0
$335$	−3.11818e6			−1.51806
$336$		0			0
$337$	−1.09294e6			−0.524231			−0.262116	−	0.965036i	$-0.584420\pi$
$337$	−1.09294e6			−0.524231			−0.262116	+	0.965036i	$0.584420\pi$
$338$		0			0
$339$	−103261.	−	178853.i	−0.0488017	−	0.0845271i
$340$		0			0
$341$	−1.62226e6	−	936612.i	−0.755500	−	0.436188i
$342$		0			0
$343$	−629678.	+	2.08592e6i	−0.288990	+	0.957332i
$344$		0			0
$345$	1.06713e6	−	1.84833e6i	0.482693	−	0.836048i
$346$		0			0
$347$	718007.	−	414541.i	0.320114	−	0.184818i	−0.331329	−	0.943515i	$-0.607497\pi$
$347$	718007.	−	414541.i	0.320114	−	0.184818i	0.651443	+	0.758697i	$0.274164\pi$
$348$		0			0
$349$		−	2.59286e6i		−	1.13950i	−0.821817	−	0.569752i	$-0.807039\pi$
$349$		−	2.59286e6i		−	1.13950i	0.821817	−	0.569752i	$-0.192961\pi$
$350$		0			0
$351$		−	94861.7i		−	0.0410982i
$352$		0			0
$353$	−2.90101e6	+	1.67490e6i	−1.23912	+	0.715404i	−0.968913	−	0.247400i	$-0.920424\pi$
$353$	−2.90101e6	+	1.67490e6i	−1.23912	+	0.715404i	−0.270202	+	0.962804i	$0.587091\pi$
$354$		0			0
$355$	−1.66680e6	+	2.88698e6i	−0.701959	+	1.21583i
$356$		0			0
$357$	−939429.	+	1.31222e6i	−0.390115	+	0.544924i
$358$		0			0
$359$	137056.	+	79129.5i	0.0561258	+	0.0324043i	0.527800	−	0.849368i	$-0.323017\pi$
$359$	137056.	+	79129.5i	0.0561258	+	0.0324043i	−0.471675	+	0.881773i	$0.656350\pi$
$360$		0			0
$361$	876083.	+	1.51742e6i	0.353816	+	0.612827i
$362$		0			0
$363$	1.60306e6			0.638533
$364$		0			0
$365$	−2.11502e6			−0.830963
$366$		0			0
$367$	−1.20482e6	−	2.08681e6i	−0.466935	−	0.808755i	0.532351	−	0.846523i	$-0.321308\pi$
$367$	−1.20482e6	−	2.08681e6i	−0.466935	−	0.808755i	−0.999287	+	0.0377682i	$0.987975\pi$
$368$		0			0
$369$	−22333.9	−	12894.5i	−0.00853882	−	0.00492989i
$370$		0			0
$371$	1.15819e6	−	525400.i	0.436864	−	0.198178i
$372$		0			0
$373$	904919.	−	1.56736e6i	0.336773	−	0.583308i	−0.647051	−	0.762447i	$-0.723998\pi$
$373$	904919.	−	1.56736e6i	0.336773	−	0.583308i	0.983824	+	0.179139i	$0.0573311\pi$
$374$		0			0
$375$	−3.44492e6	+	1.98893e6i	−1.26503	+	0.730366i
$376$		0			0
$377$		−	179606.i		−	0.0650832i
$378$		0			0
$379$		−	2.34965e6i		−	0.840243i	−0.907468	−	0.420121i	$-0.861988\pi$
$379$		−	2.34965e6i		−	0.840243i	0.907468	−	0.420121i	$-0.138012\pi$
$380$		0			0
$381$	1.34098e6	−	774216.i	0.473271	−	0.273243i
$382$		0			0
$383$	1.71619e6	−	2.97252e6i	0.597816	−	1.03545i	−0.395327	−	0.918540i	$-0.629369\pi$
$383$	1.71619e6	−	2.97252e6i	0.597816	−	1.03545i	0.993143	−	0.116907i	$-0.0372979\pi$
$384$		0			0
$385$	737916.	−	7.52698e6i	0.253720	−	2.58803i
$386$		0			0
$387$	2.29836e6	+	1.32696e6i	0.780084	+	0.450382i
$388$		0			0
$389$	−1.02780e6	−	1.78019e6i	−0.344376	−	0.596476i	0.640864	−	0.767654i	$-0.278576\pi$
$389$	−1.02780e6	−	1.78019e6i	−0.344376	−	0.596476i	−0.985240	+	0.171178i	$0.945243\pi$
$390$		0			0
$391$	2.86535e6			0.947841
$392$		0			0
$393$	−445850.			−0.145616
$394$		0			0
$395$	654744.	+	1.13405e6i	0.211144	+	0.365712i
$396$		0			0
$397$	2.39567e6	+	1.38314e6i	0.762870	+	0.440443i	0.830325	−	0.557279i	$-0.188155\pi$
$397$	2.39567e6	+	1.38314e6i	0.762870	+	0.440443i	−0.0674553	+	0.997722i	$0.521488\pi$
$398$		0			0
$399$	102755.	−	1.04813e6i	0.0323125	−	0.329597i
$400$		0			0
$401$	−2.23802e6	+	3.87636e6i	−0.695029	+	1.20383i	0.275142	+	0.961404i	$0.411275\pi$
$401$	−2.23802e6	+	3.87636e6i	−0.695029	+	1.20383i	−0.970171	+	0.242422i	$0.922058\pi$
$402$		0			0
$403$	−71174.5	+	41092.6i	−0.0218304	+	0.0126038i
$404$		0			0
$405$		−	91984.8i		−	0.0278662i
$406$		0			0
$407$			7.75898e6i			2.32177i
$408$		0			0
$409$	2.56874e6	−	1.48306e6i	0.759297	−	0.438380i	−0.0697463	−	0.997565i	$-0.522219\pi$
$409$	2.56874e6	−	1.48306e6i	0.759297	−	0.438380i	0.829043	+	0.559184i	$0.188886\pi$
$410$		0			0
$411$	886420.	−	1.53532e6i	0.258842	−	0.448328i
$412$		0			0
$413$	−4.89398e6	+	2.22009e6i	−1.41185	+	0.640466i
$414$		0			0
$415$	−1.06548e7	−	6.15156e6i	−3.03687	−	1.75334i
$416$		0			0
$417$	−1.11253e6	−	1.92696e6i	−0.313308	−	0.542665i
$418$		0			0
$419$	842398.			0.234413			0.117207	−	0.993108i	$-0.462606\pi$
$419$	842398.			0.234413			0.117207	+	0.993108i	$0.462606\pi$
$420$		0			0
$421$	5.02112e6			1.38069			0.690344	−	0.723482i	$-0.257459\pi$
$421$	5.02112e6			1.38069			0.690344	+	0.723482i	$0.257459\pi$
$422$		0			0
$423$	49431.2	+	85617.3i	0.0134323	+	0.0232654i
$424$		0			0
$425$	−8.15350e6	−	4.70742e6i	−2.18963	−	1.26419i
$426$		0			0
$427$	878732.	−	1.22744e6i	0.233231	−	0.325784i
$428$		0			0
$429$	68897.6	−	119334.i	0.0180743	−	0.0313055i
$430$		0			0
$431$	−2.67724e6	+	1.54570e6i	−0.694214	+	0.400805i	−0.805189	−	0.593019i	$-0.797936\pi$
$431$	−2.67724e6	+	1.54570e6i	−0.694214	+	0.400805i	0.110975	+	0.993823i	$0.464603\pi$
$432$		0			0
$433$		−	185156.i		−	0.0474591i	−0.999718	−	0.0237295i	$-0.992446\pi$
$433$		−	185156.i		−	0.0474591i	0.999718	−	0.0237295i	$-0.00755405\pi$
$434$		0			0
$435$			6.93167e6i			1.75636i
$436$		0			0
$437$	−1.61936e6	+	934937.i	−0.405639	+	0.234196i
$438$		0			0
$439$	−3.59737e6	+	6.23083e6i	−0.890890	+	1.54307i	−0.0520793	+	0.998643i	$0.516585\pi$
$439$	−3.59737e6	+	6.23083e6i	−0.890890	+	1.54307i	−0.838811	+	0.544423i	$0.816748\pi$
$440$		0			0
$441$	−1.68092e6	+	1.92021e6i	−0.411577	+	0.470168i
$442$		0			0
$443$	−924451.	−	533732.i	−0.223807	−	0.129215i	0.383905	−	0.923373i	$-0.374579\pi$
$443$	−924451.	−	533732.i	−0.223807	−	0.129215i	−0.607712	+	0.794158i	$0.707912\pi$
$444$		0			0
$445$	−5.07424e6	−	8.78884e6i	−1.21471	−	2.10393i
$446$		0			0
$447$	−100116.			−0.0236992
$448$		0			0
$449$	−419650.			−0.0982361			−0.0491181	−	0.998793i	$-0.515641\pi$
$449$	−419650.			−0.0982361			−0.0491181	+	0.998793i	$0.515641\pi$
$450$		0			0
$451$	−48705.6	−	84360.5i	−0.0112755	−	0.0195298i
$452$		0			0
$453$	1.61311e6	+	931327.i	0.369332	+	0.213234i
$454$		0			0
$455$	−269805.	−	193156.i	−0.0610973	−	0.0437400i
$456$		0			0
$457$	−3.79912e6	+	6.58026e6i	−0.850927	+	1.47385i	0.0294463	+	0.999566i	$0.490626\pi$
$457$	−3.79912e6	+	6.58026e6i	−0.850927	+	1.47385i	−0.880373	+	0.474282i	$0.842708\pi$
$458$		0			0
$459$	−4.25663e6	+	2.45757e6i	−0.943049	+	0.544470i
$460$		0			0
$461$		−	7.75232e6i		−	1.69895i	−0.527633	−	0.849473i	$-0.676920\pi$
$461$		−	7.75232e6i		−	1.69895i	0.527633	−	0.849473i	$-0.323080\pi$
$462$		0			0
$463$		−	1.43184e6i		−	0.310414i	−0.987882	−	0.155207i	$-0.950396\pi$
$463$		−	1.43184e6i		−	0.310414i	0.987882	−	0.155207i	$-0.0496044\pi$
$464$		0			0
$465$	2.74688e6	−	1.58591e6i	0.589125	−	0.340132i
$466$		0			0
$467$	1.77394e6	−	3.07256e6i	0.376398	−	0.651940i	−0.614137	−	0.789199i	$-0.710496\pi$
$467$	1.77394e6	−	3.07256e6i	0.376398	−	0.651940i	0.990535	+	0.137259i	$0.0438292\pi$
$468$		0			0
$469$	1.64185e6	+	3.61930e6i	0.344668	+	0.759788i
$470$		0			0
$471$	2.96406e6	+	1.71130e6i	0.615651	+	0.355446i
$472$		0			0
$473$	5.01226e6	+	8.68148e6i	1.03010	+	1.78419i
$474$		0			0
$475$	6.14396e6			1.24944
$476$		0			0
$477$	1.48957e6			0.299755
$478$		0			0
$479$	−618914.	−	1.07199e6i	−0.123251	−	0.213477i	0.797797	−	0.602926i	$-0.205999\pi$
$479$	−618914.	−	1.07199e6i	−0.123251	−	0.213477i	−0.921048	+	0.389449i	$0.872665\pi$
$480$		0			0
$481$	294808.	+	170208.i	0.0581001	+	0.0335441i
$482$		0			0
$483$	−2.70726e6	−	265409.i	−0.528034	−	0.0517664i
$484$		0			0
$485$	2.37593e6	−	4.11523e6i	0.458648	−	0.794401i
$486$		0			0
$487$	7.44646e6	−	4.29922e6i	1.42275	−	0.821423i	0.426213	−	0.904623i	$-0.359847\pi$
$487$	7.44646e6	−	4.29922e6i	1.42275	−	0.821423i	0.996533	+	0.0831997i	$0.0265139\pi$
$488$		0			0
$489$		−	1.03350e6i		−	0.195451i
$490$		0			0
$491$		−	3.98380e6i		−	0.745750i	−0.927882	−	0.372875i	$-0.878372\pi$
$491$		−	3.98380e6i		−	0.745750i	0.927882	−	0.372875i	$-0.121628\pi$
$492$		0			0
$493$	−8.05930e6	+	4.65304e6i	−1.49341	+	0.862222i
$494$		0			0
$495$	4.42908e6	−	7.67139e6i	0.812457	−	1.40722i
$496$		0			0
$497$	4.22857e6	+	414553.i	0.767896	+	0.0752816i
$498$		0			0
$499$	−9.18700e6	−	5.30412e6i	−1.65167	−	0.953590i	−0.976387	−	0.216028i	$-0.930690\pi$
$499$	−9.18700e6	−	5.30412e6i	−1.65167	−	0.953590i	−0.675279	−	0.737562i	$-0.735977\pi$
$500$		0			0
$501$	2.00122e6	+	3.46622e6i	0.356205	+	0.616966i
$502$		0			0
$503$	6.21943e6			1.09605			0.548026	−	0.836462i	$-0.315380\pi$
$503$	6.21943e6			1.09605			0.548026	+	0.836462i	$0.315380\pi$
$504$		0			0
$505$	7.64852e6			1.33459
$506$		0			0
$507$	1.76947e6	+	3.06482e6i	0.305720	+	0.529523i
$508$		0			0
$509$	6.18513e6	+	3.57098e6i	1.05817	+	0.610933i	0.924925	−	0.380151i	$-0.124128\pi$
$509$	6.18513e6	+	3.57098e6i	1.05817	+	0.610933i	0.133242	+	0.991084i	$0.457461\pi$
$510$		0			0
$511$	1.11364e6	+	2.45492e6i	0.188666	+	0.415896i
$512$		0			0
$513$	1.60376e6	−	2.77780e6i	0.269059	−	0.466024i
$514$		0			0
$515$	−1.19842e6	+	691908.i	−0.199109	+	0.114956i
$516$		0			0
$517$			373427.i			0.0614440i
$518$		0			0
$519$		−	287908.i		−	0.0469175i
$520$		0			0
$521$	2.98507e6	−	1.72343e6i	0.481793	−	0.278164i	−0.239370	−	0.970928i	$-0.576941\pi$
$521$	2.98507e6	−	1.72343e6i	0.481793	−	0.278164i	0.721164	+	0.692765i	$0.243608\pi$
$522$		0			0
$523$	3.15469e6	−	5.46408e6i	0.504315	−	0.873499i	−0.495672	−	0.868510i	$-0.665078\pi$
$523$	3.15469e6	−	5.46408e6i	0.504315	−	0.873499i	0.999988	−	0.00498978i	$-0.00158830\pi$
$524$		0			0
$525$	7.26761e6	+	5.20294e6i	1.15078	+	0.823854i
$526$		0			0
$527$	3.68781e6	+	2.12916e6i	0.578419	+	0.333950i
$528$		0			0
$529$	−803288.	−	1.39134e6i	−0.124805	−	0.216169i
$530$		0			0
$531$	−6.29423e6			−0.968739
$532$		0			0
$533$	−4273.78			−0.000651620
$534$		0			0
$535$	1.00610e7	+	1.74262e7i	1.51970	+	2.63220i
$536$		0			0
$537$	−6.37384e6	−	3.67994e6i	−0.953818	−	0.550687i
$538$		0			0
$539$	−9.12517e6	+	3.10676e6i	−1.35291	+	0.460612i
$540$		0			0
$541$	−6.57619e6	+	1.13903e7i	−0.966009	+	1.67318i	−0.259128	+	0.965843i	$0.583435\pi$
$541$	−6.57619e6	+	1.13903e7i	−0.966009	+	1.67318i	−0.706880	+	0.707333i	$0.749898\pi$
$542$		0			0
$543$	5.49800e6	−	3.17427e6i	0.800212	−	0.462002i
$544$		0			0
$545$			6.36785e6i			0.918336i
$546$		0			0
$547$		−	3.69831e6i		−	0.528488i	−0.964456	−	0.264244i	$-0.914878\pi$
$547$		−	3.69831e6i		−	0.528488i	0.964456	−	0.264244i	$-0.0851224\pi$
$548$		0			0
$549$	1.53118e6	−	884028.i	0.216818	−	0.125180i
$550$		0			0
$551$	3.03649e6	−	5.25935e6i	0.426081	−	0.737995i
$552$		0			0
$553$	971551.	−	1.35709e6i	0.135099	−	0.188710i
$554$		0			0
$555$	−1.13777e7	−	6.56893e6i	−1.56792	−	0.905237i
$556$		0			0
$557$	−2.09933e6	−	3.63615e6i	−0.286710	−	0.496596i	0.686312	−	0.727307i	$-0.259228\pi$
$557$	−2.09933e6	−	3.63615e6i	−0.286710	−	0.496596i	−0.973022	+	0.230710i	$0.925895\pi$
$558$		0			0
$559$	439812.			0.0595303
$560$		0			0
$561$	−7.13967e6			−0.957792
$562$		0			0
$563$	5.17170e6	+	8.95765e6i	0.687642	+	1.19103i	0.972599	+	0.232491i	$0.0746875\pi$
$563$	5.17170e6	+	8.95765e6i	0.687642	+	1.19103i	−0.284956	+	0.958540i	$0.591979\pi$
$564$		0			0
$565$	−1.90539e6	−	1.10008e6i	−0.251109	−	0.144978i
$566$		0			0
$567$	−106767.	+	48433.7i	−0.0139470	+	0.00632689i
$568$		0			0
$569$	4.38456e6	−	7.59428e6i	0.567735	−	0.983345i	−0.429055	−	0.903278i	$-0.641153\pi$
$569$	4.38456e6	−	7.59428e6i	0.567735	−	0.983345i	0.996790	−	0.0800668i	$-0.0255134\pi$
$570$		0			0
$571$	4.23028e6	−	2.44235e6i	0.542974	−	0.313486i	−0.203309	−	0.979115i	$-0.565170\pi$
$571$	4.23028e6	−	2.44235e6i	0.542974	−	0.313486i	0.746283	+	0.665628i	$0.231836\pi$
$572$		0			0
$573$			5.38562e6i			0.685250i
$574$		0			0
$575$		−	1.58695e7i		−	2.00167i
$576$		0			0
$577$	2.82195e6	−	1.62925e6i	0.352866	−	0.203727i	−0.313081	−	0.949726i	$-0.601361\pi$
$577$	2.82195e6	−	1.62925e6i	0.352866	−	0.203727i	0.665947	+	0.745999i	$0.268028\pi$
$578$		0			0
$579$	2.73420e6	−	4.73578e6i	0.338949	−	0.587077i
$580$		0			0
$581$	−1.52997e6	+	1.56062e7i	−0.188037	+	1.91803i
$582$		0			0
$583$	4.87268e6	+	2.81324e6i	0.593740	+	0.342796i
$584$		0			0
$585$	−194320.	−	336572.i	−0.0234762	−	0.0406619i
$586$		0			0
$587$	−1.00160e7			−1.19977			−0.599887	−	0.800085i	$-0.704788\pi$
$587$	−1.00160e7			−1.19977			−0.599887	+	0.800085i	$0.704788\pi$
$588$		0			0
$589$	−2.77890e6			−0.330054
$590$		0			0
$591$	−2.08424e6	−	3.61000e6i	−0.245459	−	0.425147i
$592$		0			0
$593$	4.78049e6	+	2.76001e6i	0.558258	+	0.322311i	0.752446	−	0.658654i	$-0.228874\pi$
$593$	4.78049e6	+	2.76001e6i	0.558258	+	0.322311i	−0.194188	+	0.980964i	$0.562207\pi$
$594$		0			0
$595$	−1.67747e6	+	1.71108e7i	−0.194251	+	1.98142i
$596$		0			0
$597$	3.63180e6	−	6.29047e6i	0.417049	−	0.722349i
$598$		0			0
$599$	−5.65557e6	+	3.26525e6i	−0.644035	+	0.371834i	−0.786167	−	0.618014i	$-0.787938\pi$
$599$	−5.65557e6	+	3.26525e6i	−0.644035	+	0.371834i	0.142132	+	0.989848i	$0.454604\pi$
$600$		0			0
$601$			8.65166e6i			0.977042i	0.872552	+	0.488521i	$0.162464\pi$
$601$			8.65166e6i			0.977042i	−0.872552	+	0.488521i	$0.837536\pi$
$602$		0			0
$603$			4.65484e6i			0.521329i
$604$		0			0
$605$	1.47900e7	−	8.53902e6i	1.64278	−	0.948461i
$606$		0			0
$607$	1.00608e6	−	1.74259e6i	0.110831	−	0.191965i	−0.805274	−	0.592902i	$-0.797982\pi$
$607$	1.00608e6	−	1.74259e6i	0.110831	−	0.191965i	0.916106	+	0.400937i	$0.131315\pi$
$608$		0			0
$609$	8.04564e6	−	3.64981e6i	0.879058	−	0.398774i
$610$		0			0
$611$	14188.6	+	8191.81i	0.00153758	+	0.000887722i
$612$		0			0
$613$	−1.25217e6	−	2.16883e6i	−0.134590	−	0.233117i	0.790851	−	0.612009i	$-0.209638\pi$
$613$	−1.25217e6	−	2.16883e6i	−0.134590	−	0.233117i	−0.925441	+	0.378892i	$0.876305\pi$
$614$		0			0
$615$	164941.			0.0175849
$616$		0			0
$617$	1.60855e7			1.70107			0.850533	−	0.525922i	$-0.176280\pi$
$617$	1.60855e7			1.70107			0.850533	+	0.525922i	$0.176280\pi$
$618$		0			0
$619$	−8.43650e6	−	1.46124e7i	−0.884984	−	1.53284i	−0.845732	−	0.533608i	$-0.820836\pi$
$619$	−8.43650e6	−	1.46124e7i	−0.884984	−	1.53284i	−0.0392524	−	0.999229i	$-0.512498\pi$
$620$		0			0
$621$	−7.17489e6	−	4.14242e6i	−0.746597	−	0.431048i
$622$		0			0
$623$	−7.52948e6	+	1.05174e7i	−0.777222	+	1.08565i
$624$		0			0
$625$	−9.90597e6	+	1.71576e7i	−1.01437	+	1.75694i
$626$		0			0
$627$	4.03501e6	−	2.32961e6i	0.409898	−	0.236654i
$628$		0			0
$629$		−	1.76382e7i		−	1.77757i
$630$		0			0
$631$			1.33841e7i			1.33818i	0.743182	+	0.669090i	$0.233316\pi$
$631$			1.33841e7i			1.33818i	−0.743182	+	0.669090i	$0.766684\pi$
$632$		0			0
$633$	−6.53399e6	+	3.77240e6i	−0.648141	+	0.374204i
$634$		0			0
$635$	8.24803e6	−	1.42860e7i	0.811738	−	1.40597i
$636$		0			0
$637$	−82133.9	+	414869.i	−0.00801999	+	0.0405101i
$638$		0			0
$639$	4.30970e6	+	2.48820e6i	0.417537	+	0.241065i
$640$		0			0
$641$	−9.50750e6	−	1.64675e7i	−0.913947	−	1.58300i	−0.808436	−	0.588585i	$-0.799685\pi$
$641$	−9.50750e6	−	1.64675e7i	−0.913947	−	1.58300i	−0.105511	−	0.994418i	$-0.533648\pi$
$642$		0			0
$643$	2.38882e6			0.227853			0.113927	−	0.993489i	$-0.463657\pi$
$643$	2.38882e6			0.227853			0.113927	+	0.993489i	$0.463657\pi$
$644$		0			0
$645$	−1.69740e7			−1.60651
$646$		0			0
$647$	1.85327e6	+	3.20996e6i	0.174052	+	0.301467i	0.939833	−	0.341635i	$-0.110981\pi$
$647$	1.85327e6	+	3.20996e6i	0.174052	+	0.301467i	−0.765781	+	0.643102i	$0.777647\pi$
$648$		0			0
$649$	−2.05896e7	−	1.18874e7i	−1.91883	−	1.10784i
$650$		0			0
$651$	−3.28713e6	−	2.35328e6i	−0.303993	−	0.217631i
$652$		0			0
$653$	−4.67699e6	+	8.10078e6i	−0.429223	+	0.743437i	−0.996804	−	0.0798808i	$-0.974546\pi$
$653$	−4.67699e6	+	8.10078e6i	−0.429223	+	0.743437i	0.567581	+	0.823318i	$0.307879\pi$
$654$		0			0
$655$	−4.11347e6	+	2.37491e6i	−0.374632	+	0.216294i
$656$		0			0
$657$			3.15731e6i			0.285367i
$658$		0			0
$659$			5.20973e6i			0.467306i	0.972320	+	0.233653i	$0.0750680\pi$
$659$			5.20973e6i			0.467306i	−0.972320	+	0.233653i	$0.924932\pi$
$660$		0			0
$661$	6.33410e6	−	3.65700e6i	0.563873	−	0.325552i	−0.190825	−	0.981624i	$-0.561116\pi$
$661$	6.33410e6	−	3.65700e6i	0.563873	−	0.325552i	0.754699	+	0.656072i	$0.227783\pi$
$662$		0			0
$663$	−156622.	+	271277.i	−0.0138378	+	0.0239679i
$664$		0			0
$665$	−4.63506e6	−	1.02175e7i	−0.406444	−	0.895967i
$666$		0			0
$667$	−1.35846e7	−	7.84306e6i	−1.18231	−	0.682607i
$668$		0			0
$669$	−2.78369e6	−	4.82148e6i	−0.240467	−	0.416500i
$670$		0			0
$671$	6.67838e6			0.572618
$672$		0			0
$673$	−1.26836e6			−0.107945			−0.0539727	−	0.998542i	$-0.517188\pi$
$673$	−1.26836e6			−0.107945			−0.0539727	+	0.998542i	$0.517188\pi$
$674$		0			0
$675$	1.36110e7	+	2.35750e7i	1.14982	+	1.99155i
$676$		0			0
$677$	−7.69422e6	−	4.44226e6i	−0.645198	−	0.372505i	0.141416	−	0.989950i	$-0.454834\pi$
$677$	−7.69422e6	−	4.44226e6i	−0.645198	−	0.372505i	−0.786614	+	0.617445i	$0.788168\pi$
$678$		0			0
$679$	−6.02761e6	−	590924.i	−0.501730	−	0.0491877i
$680$		0			0
$681$	351349.	−	608555.i	0.0290316	−	0.0502843i
$682$		0			0
$683$	5.34750e6	−	3.08738e6i	0.438631	−	0.253243i	−0.264386	−	0.964417i	$-0.585169\pi$
$683$	5.34750e6	−	3.08738e6i	0.438631	−	0.253243i	0.703017	+	0.711173i	$0.251836\pi$
$684$		0			0
$685$		−	1.88868e7i		−	1.53791i
$686$		0			0
$687$			8.44568e6i			0.682720i
$688$		0			0
$689$	213782.	−	123427.i	0.0171563	−	0.00990520i
$690$		0			0
$691$	−8.60051e6	+	1.48965e7i	−0.685218	+	1.18683i	0.288150	+	0.957585i	$0.406960\pi$
$691$	−8.60051e6	+	1.48965e7i	−0.685218	+	1.18683i	−0.973368	+	0.229248i	$0.926373\pi$
$692$		0			0
$693$	−1.12363e7	−	1.10157e6i	−0.888774	−	0.0871320i
$694$		0			0
$695$	−2.05287e7	−	1.18522e7i	−1.61212	−	0.930761i
$696$		0			0
$697$	110720.	+	191773.i	0.00863267	+	0.0149522i
$698$		0			0
$699$	6.37749e6			0.493693
$700$		0			0
$701$	−1.36656e6			−0.105035			−0.0525174	−	0.998620i	$-0.516724\pi$
$701$	−1.36656e6			−0.105035			−0.0525174	+	0.998620i	$0.516724\pi$
$702$		0			0
$703$	5.75517e6	+	9.96825e6i	0.439208	+	0.760730i
$704$		0			0
$705$	−547591.	−	316152.i	−0.0414938	−	0.0239565i
$706$		0			0
$707$	−4.02726e6	−	8.87769e6i	−0.303013	−	0.667962i
$708$		0			0
$709$	1.04862e7	−	1.81626e7i	0.783432	−	1.35694i	−0.146499	−	0.989211i	$-0.546800\pi$
$709$	1.04862e7	−	1.81626e7i	0.783432	−	1.35694i	0.929931	−	0.367734i	$-0.119866\pi$
$710$		0			0
$711$	1.69292e6	−	977407.i	0.125592	−	0.0725106i
$712$		0			0
$713$			7.17773e6i			0.528766i
$714$		0			0
$715$		−	1.46799e6i		−	0.107388i
$716$		0			0
$717$	−2.65025e6	+	1.53012e6i	−0.192526	+	0.111155i
$718$		0			0
$719$	6.61367e6	−	1.14552e7i	0.477112	−	0.826382i	−0.522544	−	0.852612i	$-0.675017\pi$
$719$	6.61367e6	−	1.14552e7i	0.477112	−	0.826382i	0.999656	+	0.0262303i	$0.00835031\pi$
$720$		0			0
$721$	1.43412e6	+	1.02670e6i	0.102742	+	0.0735536i
$722$		0			0
$723$	−5.10031e6	−	2.94466e6i	−0.362870	−	0.209503i
$724$		0			0
$725$	2.57704e7	+	4.46357e7i	1.82086	+	3.15382i
$726$		0			0
$727$	−318178.			−0.0223272			−0.0111636	−	0.999938i	$-0.503554\pi$
$727$	−318178.			−0.0223272			−0.0111636	+	0.999938i	$0.503554\pi$
$728$		0			0
$729$	8.60902e6			0.599977
$730$		0			0
$731$	−1.13941e7	−	1.97352e7i	−0.788657	−	1.36599i
$732$		0			0
$733$	2.24269e7	+	1.29482e7i	1.54173	+	0.890119i	0.998730	+	0.0503874i	$0.0160456\pi$
$733$	2.24269e7	+	1.29482e7i	1.54173	+	0.890119i	0.543002	+	0.839732i	$0.317288\pi$
$734$		0			0
$735$	3.16984e6	−	1.60113e7i	0.216431	−	1.09322i
$736$		0			0
$737$	−8.79124e6	+	1.52269e7i	−0.596186	+	1.03262i
$738$		0			0
$739$	−8.04318e6	+	4.64373e6i	−0.541772	+	0.312792i	−0.745797	−	0.666174i	$-0.767931\pi$
$739$	−8.04318e6	+	4.64373e6i	−0.541772	+	0.312792i	0.204025	+	0.978966i	$0.434598\pi$
$740$		0			0
$741$		−	204417.i		−	0.0136764i
$742$		0			0
$743$			2.54773e7i			1.69310i	0.532311	+	0.846549i	$0.321324\pi$
$743$			2.54773e7i			1.69310i	−0.532311	+	0.846549i	$0.678676\pi$
$744$		0			0
$745$	−923679.	+	533287.i	−0.0609720	+	0.0352022i
$746$		0			0
$747$	−9.18309e6	+	1.59056e7i	−0.602126	+	1.04291i
$748$		0			0
$749$	1.49292e7	−	2.08536e7i	0.972373	−	1.35824i
$750$		0			0
$751$	−2.09711e6	−	1.21077e6i	−0.135682	−	0.0783359i	0.430623	−	0.902532i	$-0.358294\pi$
$751$	−2.09711e6	−	1.21077e6i	−0.135682	−	0.0783359i	−0.566304	+	0.824196i	$0.691627\pi$
$752$		0			0
$753$	−1.66624e6	−	2.88601e6i	−0.107090	−	0.185486i
$754$		0			0
$755$	1.98436e7			1.26693
$756$		0			0
$757$	−2.09049e7			−1.32589			−0.662946	−	0.748667i	$-0.730694\pi$
$757$	−2.09049e7			−1.32589			−0.662946	+	0.748667i	$0.730694\pi$
$758$		0			0
$759$	−6.01724e6	−	1.04222e7i	−0.379134	−	0.656680i
$760$		0			0
$761$	−5.14517e6	−	2.97057e6i	−0.322061	−	0.185942i	0.330250	−	0.943894i	$-0.392867\pi$
$761$	−5.14517e6	−	2.97057e6i	−0.322061	−	0.185942i	−0.652311	+	0.757952i	$0.726200\pi$
$762$		0			0
$763$	7.39122e6	−	3.35293e6i	0.459626	−	0.208504i
$764$		0			0
$765$	−1.00684e7	+	1.74390e7i	−0.622025	+	1.07738i
$766$		0			0
$767$	−903343.	+	521545.i	−0.0554452	+	0.0320113i
$768$		0			0
$769$		−	2.31406e7i		−	1.41110i	−0.708660	−	0.705550i	$-0.750700\pi$
$769$		−	2.31406e7i		−	1.41110i	0.708660	−	0.705550i	$-0.249300\pi$
$770$		0			0
$771$			1.59555e6i			0.0966659i
$772$		0			0
$773$	8.01543e6	−	4.62771e6i	0.482479	−	0.278559i	−0.238970	−	0.971027i	$-0.576810\pi$
$773$	8.01543e6	−	4.62771e6i	0.482479	−	0.278559i	0.721449	+	0.692468i	$0.243477\pi$
$774$		0			0
$775$	1.17922e7	−	2.04246e7i	0.705243	−	1.22152i
$776$		0			0
$777$	−1.63377e6	+	1.66650e7i	−0.0970822	+	0.990269i
$778$		0			0
$779$	−125148.	−	72254.0i	−0.00738889	−	0.00426598i
$780$		0			0
$781$	9.39856e6	+	1.62788e7i	0.551358	+	0.954980i
$782$		0			0
$783$	2.69075e7			1.56845
$784$		0			0
$785$	3.64624e7			2.11189
$786$		0			0
$787$	923817.	+	1.60010e6i	0.0531679	+	0.0920894i	0.891384	−	0.453248i	$-0.149735\pi$
$787$	923817.	+	1.60010e6i	0.0531679	+	0.0920894i	−0.838217	+	0.545338i	$0.816401\pi$
$788$		0			0
$789$	−9.06015e6	−	5.23088e6i	−0.518135	−	0.299145i
$790$		0			0
$791$	−273603.	+	2.79083e6i	−0.0155482	+	0.158596i
$792$		0			0
$793$	146503.	−	253750.i	0.00827299	−	0.0143292i
$794$		0			0
$795$	−8.25064e6	+	4.76351e6i	−0.462988	+	0.267306i
$796$		0			0
$797$		−	1.08580e7i		−	0.605484i	−0.953072	−	0.302742i	$-0.902098\pi$
$797$		−	1.08580e7i		−	0.605484i	0.953072	−	0.302742i	$-0.0979021\pi$
$798$		0			0
$799$		−	848896.i		−	0.0470422i
$800$		0			0
$801$	−1.31200e7	+	7.57486e6i	−0.722527	+	0.417151i
$802$		0			0
$803$	−5.96297e6	+	1.03282e7i	−0.326343	+	0.565242i
$804$		0			0
$805$	−2.63912e7	+	1.19721e7i	−1.43539	+	0.651147i
$806$		0			0
$807$	−2.18506e6	−	1.26155e6i	−0.118108	−	0.0681898i
$808$		0			0
$809$	−7.89239e6	−	1.36700e7i	−0.423972	−	0.734341i	0.572352	−	0.820008i	$-0.306031\pi$
$809$	−7.89239e6	−	1.36700e7i	−0.423972	−	0.734341i	−0.996324	+	0.0856672i	$0.972698\pi$
$810$		0			0
$811$	7.17944e6			0.383300			0.191650	−	0.981463i	$-0.438616\pi$
$811$	7.17944e6			0.383300			0.191650	+	0.981463i	$0.438616\pi$
$812$		0			0
$813$	−1.64247e7			−0.871507
$814$		0			0
$815$	−5.50515e6	−	9.53520e6i	−0.290319	−	0.502847i
$816$		0			0
$817$	1.28789e7	+	7.43561e6i	0.675029	+	0.389728i
$818$		0			0
$819$	−288344.	+	402767.i	−0.0150211	+	0.0209819i
$820$		0			0
$821$	−5.24040e6	+	9.07664e6i	−0.271336	+	0.469967i	−0.969204	−	0.246259i	$-0.920799\pi$
$821$	−5.24040e6	+	9.07664e6i	−0.271336	+	0.469967i	0.697868	+	0.716226i	$0.254132\pi$
$822$		0			0
$823$	1.78207e7	−	1.02888e7i	0.917116	−	0.529497i	0.0344021	−	0.999408i	$-0.489047\pi$
$823$	1.78207e7	−	1.02888e7i	0.917116	−	0.529497i	0.882714	+	0.469911i	$0.155714\pi$
$824$		0			0
$825$			3.95424e7i			2.02269i
$826$		0			0
$827$		−	1.47778e7i		−	0.751358i	−0.926750	−	0.375679i	$-0.877410\pi$
$827$		−	1.47778e7i		−	0.751358i	0.926750	−	0.375679i	$-0.122590\pi$
$828$		0			0
$829$	2.67357e7	−	1.54359e7i	1.35116	−	0.780091i	0.362745	−	0.931888i	$-0.381840\pi$
$829$	2.67357e7	−	1.54359e7i	1.35116	−	0.780091i	0.988412	+	0.151798i	$0.0485063\pi$
$830$		0			0
$831$	−4.89284e6	+	8.47465e6i	−0.245787	+	0.425715i
$832$		0			0
$833$	2.07438e7	−	7.06245e6i	1.03580	−	0.352649i
$834$		0			0
$835$	3.69270e7	+	2.13198e7i	1.83285	+	1.05820i
$836$		0			0
$837$	−6.15624e6	−	1.06629e7i	−0.303740	−	0.526093i
$838$		0			0
$839$	−2.42355e7			−1.18863			−0.594317	−	0.804231i	$-0.702577\pi$
$839$	−2.42355e7			−1.18863			−0.594317	+	0.804231i	$0.702577\pi$
$840$		0			0
$841$	3.04342e7			1.48379
$842$		0			0
$843$	−2.48705e6	−	4.30770e6i	−0.120536	−	0.208774i
$844$		0			0
$845$	3.26507e7	+	1.88509e7i	1.57308	+	0.908219i
$846$		0			0
$847$	−1.76989e7	−	1.26707e7i	−0.847689	−	0.606867i
$848$		0			0
$849$	4.94040e6	−	8.55702e6i	0.235230	−	0.407430i
$850$		0			0
$851$	2.57474e7	−	1.48652e7i	1.21873	−	0.703636i
$852$		0			0
$853$			2.37458e7i			1.11741i	0.829365	+	0.558707i	$0.188703\pi$
$853$			2.37458e7i			1.11741i	−0.829365	+	0.558707i	$0.811297\pi$
$854$		0			0
$855$		−	1.31409e7i		−	0.614768i
$856$		0			0
$857$	−1.12671e7	+	6.50508e6i	−0.524036	+	0.302552i	−0.738584	−	0.674161i	$-0.764505\pi$
$857$	−1.12671e7	+	6.50508e6i	−0.524036	+	0.302552i	0.214548	+	0.976713i	$0.431172\pi$
$858$		0			0
$859$	5.43583e6	−	9.41513e6i	0.251352	−	0.435355i	−0.712546	−	0.701625i	$-0.752458\pi$
$859$	5.43583e6	−	9.41513e6i	0.251352	−	0.435355i	0.963898	+	0.266270i	$0.0857914\pi$
$860$		0			0
$861$	−86848.1	−	191448.i	−0.00399257	−	0.00880123i
$862$		0			0
$863$	8.59795e6	+	4.96403e6i	0.392978	+	0.226886i	0.683450	−	0.729998i	$-0.260479\pi$
$863$	8.59795e6	+	4.96403e6i	0.392978	+	0.226886i	−0.290472	+	0.956884i	$0.593812\pi$
$864$		0			0
$865$	−1.53360e6	−	2.65627e6i	−0.0696901	−	0.120707i
$866$		0			0
$867$	2.67395e6			0.120811
$868$		0			0
$869$	7.38381e6			0.331689
$870$		0			0
$871$	385704.	+	668059.i	0.0172270	+	0.0298380i
$872$		0			0
$873$	−6.14325e6	−	3.54681e6i	−0.272811	−	0.157508i
$874$		0			0
$875$	5.37549e7	+	5.26993e6i	2.37355	+	0.232694i
$876$		0			0
$877$	1.61498e6	−	2.79722e6i	0.0709035	−	0.122808i	−0.828394	−	0.560146i	$-0.810745\pi$
$877$	1.61498e6	−	2.79722e6i	0.0709035	−	0.122808i	0.899298	+	0.437337i	$0.144078\pi$
$878$		0			0
$879$	−1.72191e7	+	9.94143e6i	−0.751688	+	0.433987i
$880$		0			0
$881$			2.54588e7i			1.10509i	0.833483	+	0.552545i	$0.186343\pi$
$881$			2.54588e7i			1.10509i	−0.833483	+	0.552545i	$0.813657\pi$
$882$		0			0
$883$		−	6.82420e6i		−	0.294544i	−0.989096	−	0.147272i	$-0.952951\pi$
$883$		−	6.82420e6i		−	0.294544i	0.989096	−	0.147272i	$-0.0470492\pi$
$884$		0			0
$885$	3.48633e7	−	2.01283e7i	1.49627	−	0.863873i
$886$		0			0
$887$	−5.76990e6	+	9.99375e6i	−0.246240	+	0.426500i	−0.962480	−	0.271354i	$-0.912529\pi$
$887$	−5.76990e6	+	9.99375e6i	−0.246240	+	0.426500i	0.716239	+	0.697855i	$0.245862\pi$
$888$		0			0
$889$	−2.09248e7	−	2.05139e6i	−0.887988	−	0.0870550i
$890$		0			0
$891$	−449185.	−	259337.i	−0.0189553	−	0.0109439i
$892$		0			0
$893$	276987.	+	479756.i	0.0116233	+	0.0201322i
$894$		0			0
$895$	−7.84077e7			−3.27191
$896$		0			0
$897$	−527997.			−0.0219104
$898$		0			0
$899$	−1.16559e7	−	2.01886e7i	−0.481002	−	0.833120i
$900$		0			0
$901$	−1.10769e7	−	6.39522e6i	−0.454574	−	0.262448i
$902$		0			0
$903$	8.93748e6	+	1.97018e7i	0.364750	+	0.804056i
$904$		0			0
$905$	3.38168e7	−	5.85724e7i	1.37249	−	2.37723i
$906$		0			0
$907$	−1.71118e6	+	987951.i	−0.0690681	+	0.0398765i	−0.534136	−	0.845398i	$-0.679363\pi$
$907$	−1.71118e6	+	987951.i	−0.0690681	+	0.0398765i	0.465068	+	0.885275i	$0.346030\pi$
$908$		0			0
$909$		−	1.14178e7i		−	0.458323i
$910$		0			0
$911$		−	4.33544e7i		−	1.73076i	−0.501114	−	0.865381i	$-0.667076\pi$
$911$		−	4.33544e7i		−	1.73076i	0.501114	−	0.865381i	$-0.332924\pi$
$912$		0			0
$913$	−6.00793e7	+	3.46868e7i	−2.38533	+	1.37717i
$914$		0			0
$915$	−5.65407e6	+	9.79313e6i	−0.223259	+	0.386695i
$916$		0			0
$917$	4.92249e6	+	3.52405e6i	0.193313	+	0.138394i
$918$		0			0
$919$	2.04463e7	+	1.18047e7i	0.798596	+	0.461069i	0.842980	−	0.537945i	$-0.180799\pi$
$919$	2.04463e7	+	1.18047e7i	0.798596	+	0.461069i	−0.0443842	+	0.999015i	$0.514133\pi$
$920$		0			0
$921$	−6.26502e6	−	1.08513e7i	−0.243374	−	0.421535i
$922$		0			0
$923$	824699.			0.0318633
$924$		0			0
$925$	−9.76873e7			−3.75391
$926$		0			0
$927$	1.03288e6	+	1.78901e6i	0.0394778	+	0.0683775i
$928$		0			0
$929$	−4.52471e6	−	2.61234e6i	−0.172009	−	0.0993096i	0.411524	−	0.911399i	$-0.364997\pi$
$929$	−4.52471e6	−	2.61234e6i	−0.172009	−	0.0993096i	−0.583533	+	0.812089i	$0.698330\pi$
$930$		0			0
$931$	−9.41902e6	+	1.07599e7i	−0.356149	+	0.406849i
$932$		0			0
$933$	−1.27347e6	+	2.20571e6i	−0.0478943	+	0.0829554i
$934$		0			0
$935$	−6.58715e7	+	3.80309e7i	−2.46416	+	1.42268i
$936$		0			0
$937$			9.61409e6i			0.357733i	0.983873	+	0.178867i	$0.0572431\pi$
$937$			9.61409e6i			0.357733i	−0.983873	+	0.178867i	$0.942757\pi$
$938$		0			0
$939$			2.24955e7i			0.832592i
$940$		0			0
$941$	−3.53014e7	+	2.03813e7i	−1.29963	+	0.750339i	−0.980339	−	0.197319i	$-0.936777\pi$
$941$	−3.53014e7	+	2.03813e7i	−1.29963	+	0.750339i	−0.319287	+	0.947658i	$0.603443\pi$
$942$		0			0
$943$	−186628.	+	323249.i	−0.00683434	+	0.0118374i
$944$		0			0
$945$	2.89374e7	−	4.04206e7i	1.05409	−	1.47239i
$946$		0			0
$947$	2.10338e7	+	1.21439e7i	0.762156	+	0.440031i	0.830069	−	0.557660i	$-0.188301\pi$
$947$	2.10338e7	+	1.21439e7i	0.762156	+	0.440031i	−0.0679135	+	0.997691i	$0.521634\pi$
$948$		0			0
$949$	261617.	+	453135.i	0.00942977	+	0.0163328i
$950$		0			0
$951$	−6.42318e6			−0.230303
$952$		0			0
$953$	−6.10576e6			−0.217775			−0.108887	−	0.994054i	$-0.534729\pi$
$953$	−6.10576e6			−0.217775			−0.108887	+	0.994054i	$0.534729\pi$
$954$		0			0
$955$	2.86876e7	+	4.96884e7i	1.01785	+	1.76298i
$956$		0			0
$957$	3.38491e7	+	1.95428e7i	1.19472	+	0.689774i
$958$		0			0
$959$	−2.19220e7	+	9.94465e6i	−0.769722	+	0.349175i
$960$		0			0
$961$	8.98101e6	−	1.55556e7i	0.313701	−	0.543347i
$962$		0			0
$963$	2.60140e7	−	1.50192e7i	0.903944	−	0.521892i
$964$		0			0
$965$		−	5.82571e7i		−	2.01387i
$966$		0			0
$967$			2.26687e6i			0.0779580i	0.999240	+	0.0389790i	$0.0124105\pi$
$967$			2.26687e6i			0.0779580i	−0.999240	+	0.0389790i	$0.987589\pi$
$968$		0			0
$969$	−9.17260e6	+	5.29580e6i	−0.313822	+	0.181185i
$970$		0			0
$971$	−1.29166e7	+	2.23722e7i	−0.439643	+	0.761484i	−0.997662	−	0.0683443i	$-0.978228\pi$
$971$	−1.29166e7	+	2.23722e7i	−0.439643	+	0.761484i	0.558019	+	0.829828i	$0.311562\pi$
$972$		0			0
$973$	−2.94780e6	+	3.00685e7i	−0.0998195	+	1.01819i
$974$		0			0
$975$	1.50244e6	+	867436.i	0.0506158	+	0.0292231i
$976$		0			0
$977$	−1.54991e7	−	2.68451e7i	−0.519480	−	0.899766i	−0.999744	−	0.0226415i	$-0.992792\pi$
$977$	−1.54991e7	−	2.68451e7i	−0.519480	−	0.899766i	0.480264	−	0.877124i	$-0.340541\pi$
$978$		0			0
$979$	−5.72242e7			−1.90820
$980$		0			0
$981$	9.50598e6			0.315373
$982$		0			0
$983$	−1.68211e7	−	2.91350e7i	−0.555228	−	0.961683i	−0.997886	−	0.0649924i	$-0.979298\pi$
$983$	−1.68211e7	−	2.91350e7i	−0.555228	−	0.961683i	0.442658	−	0.896691i	$-0.354036\pi$
$984$		0			0
$985$	−3.84588e7	−	2.22042e7i	−1.26301	−	0.729197i
$986$		0			0
$987$	−78630.9	+	802060.i	−0.00256922	+	0.0262068i
$988$		0			0
$989$	1.92057e7	−	3.32653e7i	0.624367	−	1.08144i
$990$		0			0
$991$	1.72121e7	−	9.93742e6i	0.556737	−	0.321432i	−0.195098	−	0.980784i	$-0.562502\pi$
$991$	1.72121e7	−	9.93742e6i	0.556737	−	0.321432i	0.751835	+	0.659352i	$0.229169\pi$
$992$		0			0
$993$			2.14535e7i			0.690437i
$994$		0			0
$995$		−	7.73822e7i		−	2.47790i
$996$		0			0
$997$	−1.08439e7	+	6.26072e6i	−0.345499	+	0.199474i	−0.662701	−	0.748884i	$-0.730590\pi$
$997$	−1.08439e7	+	6.26072e6i	−0.345499	+	0.199474i	0.317202	+	0.948358i	$0.397257\pi$
$998$		0			0
$999$	−2.54994e7	+	4.41663e7i	−0.808382	+	1.40016i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.p.b.47.5	yes	14
4.3	odd	2		112.6.p.c.47.3	yes	14
7.2	even	3		784.6.f.d.783.5		14
7.3	odd	6		112.6.p.c.31.3	yes	14
7.5	odd	6		784.6.f.c.783.10		14
28.3	even	6	inner	112.6.p.b.31.5	✓	14
28.19	even	6		784.6.f.d.783.6		14
28.23	odd	6		784.6.f.c.783.9		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.6.p.b.31.5	✓	14	28.3	even	6	inner
112.6.p.b.47.5	yes	14	1.1	even	1	trivial
112.6.p.c.31.3	yes	14	7.3	odd	6
112.6.p.c.47.3	yes	14	4.3	odd	2
784.6.f.c.783.9		14	28.23	odd	6
784.6.f.c.783.10		14	7.5	odd	6
784.6.f.d.783.5		14	7.2	even	3
784.6.f.d.783.6		14	28.19	even	6

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.77385 + 8.26855i) q^{3} +(88.0882 + 50.8577i) q^{5} +(12.6490 - 129.023i) q^{7} +(75.9208 - 131.499i) q^{9} +(496.703 - 286.771i) q^{11} -25.1634i q^{13} +971.148i q^{15} +(-1129.13 + 651.905i) q^{17} +(425.421 - 736.851i) q^{19} +(1127.22 - 511.349i) q^{21} +(-1903.24 - 1098.84i) q^{23} +(3610.52 + 6253.60i) q^{25} +3769.83 q^{27} +7137.60 q^{29} +(-1633.03 - 2828.49i) q^{31} +(4742.36 + 2738.01i) q^{33} +(7676.05 - 10722.1i) q^{35} +(-6764.09 + 11715.7i) q^{37} +(208.065 - 120.126i) q^{39} -169.841i q^{41} +17478.2i q^{43} +(13375.4 - 7722.31i) q^{45} +(-325.545 + 563.860i) q^{47} +(-16487.0 - 3264.02i) q^{49} +(-10780.6 - 6224.19i) q^{51} +(4905.03 + 8495.76i) q^{53} +58338.1 q^{55} +8123.59 q^{57} +(-20726.3 - 35899.1i) q^{59} +(10084.1 + 5822.05i) q^{61} +(-16006.1 - 11458.9i) q^{63} +(1279.75 - 2216.60i) q^{65} +(-26548.8 + 15328.0i) q^{67} -20982.7i q^{69} +32773.7i q^{71} +(-18007.7 + 10396.7i) q^{73} +(-34472.1 + 59707.4i) q^{75} +(-30717.4 - 67713.5i) q^{77} +(11149.2 + 6437.02i) q^{79} +(-452.167 - 783.177i) q^{81} -120956. q^{83} -132618. q^{85} +(34073.8 + 59017.6i) q^{87} +(-86406.2 - 49886.6i) q^{89} +(-3246.67 - 318.291i) q^{91} +(15591.7 - 27005.6i) q^{93} +(74949.2 - 43271.9i) q^{95} -46717.2i q^{97} -87087.6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 9 q^{3} + 33 q^{5} - 28 q^{7} - 538 q^{9} + 333 q^{11} + 801 q^{17} - 2135 q^{19} + 2017 q^{21} + 2667 q^{23} + 5434 q^{25} + 17910 q^{27} + 684 q^{29} + 3119 q^{31} + 29013 q^{33} - 2247 q^{35}+ \cdots - 124833 q^{95}+O(q^{100}) \) 14 * q - 9 * q^3 + 33 * q^5 - 28 * q^7 - 538 * q^9 + 333 * q^11 + 801 * q^17 - 2135 * q^19 + 2017 * q^21 + 2667 * q^23 + 5434 * q^25 + 17910 * q^27 + 684 * q^29 + 3119 * q^31 + 29013 * q^33 - 2247 * q^35 + 3431 * q^37 + 2424 * q^39 - 28086 * q^45 + 8409 * q^47 - 24706 * q^49 + 24657 * q^51 + 1707 * q^53 - 56622 * q^55 + 68450 * q^57 - 49305 * q^59 - 21843 * q^61 + 164142 * q^63 + 17448 * q^65 - 73263 * q^67 - 77043 * q^73 - 147834 * q^75 + 22287 * q^77 + 206523 * q^79 - 145567 * q^81 - 305400 * q^83 - 176454 * q^85 - 95034 * q^87 + 171381 * q^89 + 174816 * q^91 + 292873 * q^93 - 124833 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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