LMFDB - Embedded newform 336.4.bl.i.31.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(31,336)] 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("336.31"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 0, 0, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	336.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-12,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} + 23x^{6} + 48x^{5} + 422x^{4} + 384x^{3} + 1247x^{2} - 637x + 2401 $ x^8 - x^7 + 23x^6 + 48x^5 + 422x^4 + 384x^3 + 1247x^2 - 637x + 2401
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.2
Root			$-1.54953 - 2.68386i$ of defining polynomial
Character	$\chi$	$=$	336.31
Dual form			336.4.bl.i.271.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.50000 + 2.59808i) q^{3} +(-9.43293 + 5.44610i) q^{5} +(-4.29373 - 18.0157i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(22.0517 + 12.7316i) q^{11} -79.4730i q^{13} -32.6766i q^{15} +(84.2226 + 48.6259i) q^{17} +(36.5404 + 63.2898i) q^{19} +(53.2466 + 15.8681i) q^{21} +(-128.326 + 74.0891i) q^{23} +(-3.17995 + 5.50784i) q^{25} +27.0000 q^{27} +265.419 q^{29} +(-84.1099 + 145.683i) q^{31} +(-66.1552 + 38.1948i) q^{33} +(138.617 + 146.556i) q^{35} +(-24.5880 - 42.5877i) q^{37} +(206.477 + 119.210i) q^{39} -179.225i q^{41} +430.737i q^{43} +(84.8963 + 49.0149i) q^{45} +(155.813 + 269.876i) q^{47} +(-306.128 + 154.709i) q^{49} +(-252.668 + 145.878i) q^{51} +(-74.5436 + 129.113i) q^{53} -277.350 q^{55} -219.242 q^{57} +(369.223 - 639.514i) q^{59} +(759.047 - 438.236i) q^{61} +(-121.096 + 114.537i) q^{63} +(432.818 + 749.663i) q^{65} +(669.575 + 386.579i) q^{67} -444.534i q^{69} +322.302i q^{71} +(361.044 + 208.449i) q^{73} +(-9.53985 - 16.5235i) q^{75} +(134.684 - 451.943i) q^{77} +(414.770 - 239.468i) q^{79} +(-40.5000 + 70.1481i) q^{81} +108.903 q^{83} -1059.29 q^{85} +(-398.128 + 689.578i) q^{87} +(759.402 - 438.441i) q^{89} +(-1431.76 + 341.235i) q^{91} +(-252.330 - 437.048i) q^{93} +(-689.365 - 398.005i) q^{95} +286.024i q^{97} -229.169i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 12 q^{3} + 18 q^{5} - 12 q^{7} - 36 q^{9} + 150 q^{11} + 192 q^{17} - 66 q^{19} + 90 q^{21} - 492 q^{23} + 322 q^{25} + 216 q^{27} - 372 q^{29} + 156 q^{31} - 450 q^{33} + 36 q^{35} + 554 q^{37} - 198 q^{39}+ \cdots + 108 q^{95}+O(q^{100}) $ 8 * q - 12 * q^3 + 18 * q^5 - 12 * q^7 - 36 * q^9 + 150 * q^11 + 192 * q^17 - 66 * q^19 + 90 * q^21 - 492 * q^23 + 322 * q^25 + 216 * q^27 - 372 * q^29 + 156 * q^31 - 450 * q^33 + 36 * q^35 + 554 * q^37 - 198 * q^39 - 162 * q^45 + 840 * q^47 - 340 * q^49 - 576 * q^51 + 186 * q^53 + 156 * q^55 + 396 * q^57 - 126 * q^59 + 1632 * q^61 - 162 * q^63 + 936 * q^65 + 582 * q^67 - 1386 * q^73 + 966 * q^75 + 3030 * q^77 - 1260 * q^79 - 324 * q^81 + 756 * q^83 - 5688 * q^85 + 558 * q^87 + 948 * q^89 - 4074 * q^91 + 468 * q^93 + 108 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.50000	+	2.59808i	−0.288675	+	0.500000i
$3$	−1.50000	+	2.59808i	−0.288675	+	0.500000i
$4$		0			0
$5$	−9.43293	+	5.44610i	−0.843706	+	0.487114i	−0.858522	−	0.512776i	$-0.828617\pi$
$5$	−9.43293	+	5.44610i	−0.843706	+	0.487114i	0.0148159	+	0.999890i	$0.495284\pi$
$6$		0			0
$7$	−4.29373	−	18.0157i	−0.231839	−	0.972754i
$7$	−4.29373	−	18.0157i	−0.231839	−	0.972754i
$8$		0			0
$9$	−4.50000	−	7.79423i	−0.166667	−	0.288675i
$10$		0			0
$11$	22.0517	+	12.7316i	0.604441	+	0.348974i	0.770787	−	0.637093i	$-0.219863\pi$
$11$	22.0517	+	12.7316i	0.604441	+	0.348974i	−0.166346	+	0.986068i	$0.553197\pi$
$12$		0			0
$13$		−	79.4730i		−	1.69553i	−0.530374	−	0.847764i	$-0.677949\pi$
$13$		−	79.4730i		−	1.69553i	0.530374	−	0.847764i	$-0.322051\pi$
$14$		0			0
$15$		−	32.6766i		−	0.562471i
$16$		0			0
$17$	84.2226	+	48.6259i	1.20159	+	0.693736i	0.960908	−	0.276869i	$-0.0892967\pi$
$17$	84.2226	+	48.6259i	1.20159	+	0.693736i	0.240679	+	0.970605i	$0.422630\pi$
$18$		0			0
$19$	36.5404	+	63.2898i	0.441207	+	0.764193i	0.997779	−	0.0666061i	$-0.0212171\pi$
$19$	36.5404	+	63.2898i	0.441207	+	0.764193i	−0.556572	+	0.830799i	$0.687884\pi$
$20$		0			0
$21$	53.2466	+	15.8681i	0.553303	+	0.164890i
$22$		0			0
$23$	−128.326	+	74.0891i	−1.16338	+	0.671680i	−0.952113	−	0.305747i	$-0.901094\pi$
$23$	−128.326	+	74.0891i	−1.16338	+	0.671680i	−0.211271	+	0.977427i	$0.567760\pi$
$24$		0			0
$25$	−3.17995	+	5.50784i	−0.0254396	+	0.0440627i
$26$		0			0
$27$	27.0000			0.192450
$28$		0			0
$29$	265.419			1.69955			0.849776	−	0.527145i	$-0.176737\pi$
$29$	265.419			1.69955			0.849776	+	0.527145i	$0.176737\pi$
$30$		0			0
$31$	−84.1099	+	145.683i	−0.487309	+	0.844044i	−0.999894	−	0.0145929i	$-0.995355\pi$
$31$	−84.1099	+	145.683i	−0.487309	+	0.844044i	0.512585	+	0.858637i	$0.328688\pi$
$32$		0			0
$33$	−66.1552	+	38.1948i	−0.348974	+	0.201480i
$34$		0			0
$35$	138.617	+	146.556i	0.669447	+	0.707787i
$36$		0			0
$37$	−24.5880	−	42.5877i	−0.109250	−	0.189226i	0.806217	−	0.591620i	$-0.201512\pi$
$37$	−24.5880	−	42.5877i	−0.109250	−	0.189226i	−0.915467	+	0.402394i	$0.868178\pi$
$38$		0			0
$39$	206.477	+	119.210i	0.847764	+	0.489457i
$40$		0			0
$41$		−	179.225i		−	0.682689i	−0.939938	−	0.341344i	$-0.889118\pi$
$41$		−	179.225i		−	0.682689i	0.939938	−	0.341344i	$-0.110882\pi$
$42$		0			0
$43$			430.737i			1.52760i	0.645453	+	0.763800i	$0.276668\pi$
$43$			430.737i			1.52760i	−0.645453	+	0.763800i	$0.723332\pi$
$44$		0			0
$45$	84.8963	+	49.0149i	0.281235	+	0.162371i
$46$		0			0
$47$	155.813	+	269.876i	0.483567	+	0.837563i	0.999822	−	0.0188723i	$-0.00600758\pi$
$47$	155.813	+	269.876i	0.483567	+	0.837563i	−0.516255	+	0.856435i	$0.672674\pi$
$48$		0			0
$49$	−306.128	+	154.709i	−0.892501	+	0.451045i
$50$		0			0
$51$	−252.668	+	145.878i	−0.693736	+	0.400529i
$52$		0			0
$53$	−74.5436	+	129.113i	−0.193195	+	0.334624i	−0.946307	−	0.323268i	$-0.895218\pi$
$53$	−74.5436	+	129.113i	−0.193195	+	0.334624i	0.753112	+	0.657892i	$0.228552\pi$
$54$		0			0
$55$	−277.350			−0.679961
$56$		0			0
$57$	−219.242			−0.509462
$58$		0			0
$59$	369.223	−	639.514i	0.814725	−	1.41115i	−0.0947994	−	0.995496i	$-0.530221\pi$
$59$	369.223	−	639.514i	0.814725	−	1.41115i	0.909525	−	0.415649i	$-0.136446\pi$
$60$		0			0
$61$	759.047	−	438.236i	1.59321	−	0.919842i	0.600463	−	0.799653i	$-0.294983\pi$
$61$	759.047	−	438.236i	1.59321	−	0.919842i	0.992751	−	0.120189i	$-0.0383502\pi$
$62$		0			0
$63$	−121.096	+	114.537i	−0.242170	+	0.229052i
$64$		0			0
$65$	432.818	+	749.663i	0.825915	+	1.43053i
$66$		0			0
$67$	669.575	+	386.579i	1.22092	+	0.704898i	0.965114	−	0.261830i	$-0.0843259\pi$
$67$	669.575	+	386.579i	1.22092	+	0.704898i	0.255806	+	0.966728i	$0.417659\pi$
$68$		0			0
$69$		−	444.534i		−	0.775589i
$70$		0			0
$71$			322.302i			0.538736i	0.963037	+	0.269368i	$0.0868148\pi$
$71$			322.302i			0.538736i	−0.963037	+	0.269368i	$0.913185\pi$
$72$		0			0
$73$	361.044	+	208.449i	0.578864	+	0.334207i	0.760682	−	0.649125i	$-0.224865\pi$
$73$	361.044	+	208.449i	0.578864	+	0.334207i	−0.181818	+	0.983332i	$0.558198\pi$
$74$		0			0
$75$	−9.53985	−	16.5235i	−0.0146876	−	0.0254396i
$76$		0			0
$77$	134.684	−	451.943i	0.199333	−	0.668879i
$78$		0			0
$79$	414.770	−	239.468i	0.590700	−	0.341041i	−0.174674	−	0.984626i	$-0.555887\pi$
$79$	414.770	−	239.468i	0.590700	−	0.341041i	0.765374	+	0.643585i	$0.222554\pi$
$80$		0			0
$81$	−40.5000	+	70.1481i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	108.903			0.144020			0.0720099	−	0.997404i	$-0.477059\pi$
$83$	108.903			0.144020			0.0720099	+	0.997404i	$0.477059\pi$
$84$		0			0
$85$	−1059.29			−1.35172
$86$		0			0
$87$	−398.128	+	689.578i	−0.490618	+	0.849776i
$88$		0			0
$89$	759.402	−	438.441i	0.904454	−	0.522187i	0.0258117	−	0.999667i	$-0.491783\pi$
$89$	759.402	−	438.441i	0.904454	−	0.522187i	0.878643	+	0.477480i	$0.158450\pi$
$90$		0			0
$91$	−1431.76	+	341.235i	−1.64933	+	0.393090i
$92$		0			0
$93$	−252.330	−	437.048i	−0.281348	−	0.487309i
$94$		0			0
$95$	−689.365	−	398.005i	−0.744499	−	0.429836i
$96$		0			0
$97$			286.024i			0.299396i	0.988732	+	0.149698i	$0.0478301\pi$
$97$			286.024i			0.299396i	−0.988732	+	0.149698i	$0.952170\pi$
$98$		0			0
$99$		−	229.169i		−	0.232650i
$100$		0			0
$101$	888.772	+	513.132i	0.875605	+	0.505531i	0.869207	−	0.494449i	$-0.164630\pi$
$101$	888.772	+	513.132i	0.875605	+	0.505531i	0.00639800	+	0.999980i	$0.497963\pi$
$102$		0			0
$103$	463.844	+	803.402i	0.443727	+	0.768559i	0.997963	−	0.0638016i	$-0.0203225\pi$
$103$	463.844	+	803.402i	0.443727	+	0.768559i	−0.554235	+	0.832360i	$0.686989\pi$
$104$		0			0
$105$	−588.691	+	140.304i	−0.547146	+	0.130403i
$106$		0			0
$107$	−352.842	+	203.713i	−0.318790	+	0.184053i	−0.650853	−	0.759204i	$-0.725589\pi$
$107$	−352.842	+	203.713i	−0.318790	+	0.184053i	0.332063	+	0.943257i	$0.392255\pi$
$108$		0			0
$109$	−309.065	+	535.315i	−0.271587	+	0.470403i	−0.969268	−	0.246005i	$-0.920882\pi$
$109$	−309.065	+	535.315i	−0.271587	+	0.470403i	0.697681	+	0.716408i	$0.254215\pi$
$110$		0			0
$111$	147.528			0.126151
$112$		0			0
$113$	853.435			0.710482			0.355241	−	0.934775i	$-0.384399\pi$
$113$	853.435			0.710482			0.355241	+	0.934775i	$0.384399\pi$
$114$		0			0
$115$	806.993	−	1397.75i	0.654370	−	1.13340i
$116$		0			0
$117$	−619.431	+	357.629i	−0.489457	+	0.282588i
$118$		0			0
$119$	514.399	−	1726.11i	0.396260	−	1.32968i
$120$		0			0
$121$	−341.314	−	591.172i	−0.256434	−	0.444157i
$122$		0			0
$123$	465.640	+	268.837i	0.341344	+	0.197075i
$124$		0			0
$125$		−	1430.80i		−	1.02380i
$126$		0			0
$127$		−	1392.04i		−	0.972630i	−0.873784	−	0.486315i	$-0.838341\pi$
$127$		−	1392.04i		−	0.972630i	0.873784	−	0.486315i	$-0.161659\pi$
$128$		0			0
$129$	−1119.09	−	646.105i	−0.763800	−	0.440980i
$130$		0			0
$131$	−386.357	−	669.189i	−0.257680	−	0.446316i	0.707940	−	0.706273i	$-0.249625\pi$
$131$	−386.357	−	669.189i	−0.257680	−	0.446316i	−0.965620	+	0.259957i	$0.916292\pi$
$132$		0			0
$133$	983.313	−	930.048i	0.641083	−	0.606356i
$134$		0			0
$135$	−254.689	+	147.045i	−0.162371	+	0.0937452i
$136$		0			0
$137$	−1030.87	+	1785.52i	−0.642868	+	1.11348i	0.341921	+	0.939729i	$0.388922\pi$
$137$	−1030.87	+	1785.52i	−0.642868	+	1.11348i	−0.984789	+	0.173752i	$0.944411\pi$
$138$		0			0
$139$	−434.251			−0.264983			−0.132492	−	0.991184i	$-0.542298\pi$
$139$	−434.251			−0.264983			−0.132492	+	0.991184i	$0.542298\pi$
$140$		0			0
$141$	−934.878			−0.558375
$142$		0			0
$143$	1011.82	−	1752.52i	0.591695	−	1.02485i
$144$		0			0
$145$	−2503.67	+	1445.50i	−1.43392	+	0.827875i
$146$		0			0
$147$	57.2471	−	1027.41i	0.0321202	−	0.576456i
$148$		0			0
$149$	1393.52	+	2413.65i	0.766187	+	1.32707i	0.939617	+	0.342229i	$0.111182\pi$
$149$	1393.52	+	2413.65i	0.766187	+	1.32707i	−0.173430	+	0.984846i	$0.555485\pi$
$150$		0			0
$151$	−1904.02	−	1099.29i	−1.02614	−	0.592442i	−0.110263	−	0.993902i	$-0.535169\pi$
$151$	−1904.02	−	1099.29i	−1.02614	−	0.592442i	−0.915876	+	0.401460i	$0.868503\pi$
$152$		0			0
$153$		−	875.266i		−	0.462491i
$154$		0			0
$155$		−	1832.28i		−	0.949500i
$156$		0			0
$157$	−2589.70	−	1495.17i	−1.31644	−	0.760046i	−0.333285	−	0.942826i	$-0.608157\pi$
$157$	−2589.70	−	1495.17i	−1.31644	−	0.760046i	−0.983154	+	0.182780i	$0.941490\pi$
$158$		0			0
$159$	−223.631	−	387.340i	−0.111541	−	0.193195i
$160$		0			0
$161$	1885.76	+	1993.76i	0.923098	+	0.975965i
$162$		0			0
$163$	−510.329	+	294.639i	−0.245227	+	0.141582i	−0.617577	−	0.786510i	$-0.711886\pi$
$163$	−510.329	+	294.639i	−0.245227	+	0.141582i	0.372350	+	0.928093i	$0.378552\pi$
$164$		0			0
$165$	416.025	−	720.576i	0.196288	−	0.339981i
$166$		0			0
$167$	−762.386			−0.353265			−0.176632	−	0.984277i	$-0.556520\pi$
$167$	−762.386			−0.353265			−0.176632	+	0.984277i	$0.556520\pi$
$168$		0			0
$169$	−4118.96			−1.87481
$170$		0			0
$171$	328.863	−	569.608i	0.147069	−	0.254731i
$172$		0			0
$173$	56.7491	−	32.7641i	0.0249396	−	0.0143989i	−0.487478	−	0.873135i	$-0.662083\pi$
$173$	56.7491	−	32.7641i	0.0249396	−	0.0143989i	0.512418	+	0.858736i	$0.328750\pi$
$174$		0			0
$175$	112.881	+	33.6398i	0.0487601	+	0.0145310i
$176$		0			0
$177$	1107.67	+	1918.54i	0.470382	+	0.814725i
$178$		0			0
$179$	2143.29	+	1237.43i	0.894955	+	0.516703i	0.875560	−	0.483109i	$-0.160493\pi$
$179$	2143.29	+	1237.43i	0.894955	+	0.516703i	0.0193950	+	0.999812i	$0.493826\pi$
$180$		0			0
$181$		−	3991.94i		−	1.63933i	−0.572843	−	0.819665i	$-0.694160\pi$
$181$		−	3991.94i		−	1.63933i	0.572843	−	0.819665i	$-0.305840\pi$
$182$		0			0
$183$			2629.42i			1.06214i
$184$		0			0
$185$	463.874	+	267.818i	0.184350	+	0.106434i
$186$		0			0
$187$	1238.17	+	2144.57i	0.484192	+	0.838646i
$188$		0			0
$189$	−115.931	−	486.423i	−0.0446175	−	0.187207i
$190$		0			0
$191$	2882.28	−	1664.08i	1.09191	−	0.630413i	0.157823	−	0.987467i	$-0.449552\pi$
$191$	2882.28	−	1664.08i	1.09191	−	0.630413i	0.934083	+	0.357055i	$0.116219\pi$
$192$		0			0
$193$	−665.967	+	1153.49i	−0.248380	+	0.430207i	−0.963077	−	0.269228i	$-0.913232\pi$
$193$	−665.967	+	1153.49i	−0.248380	+	0.430207i	0.714696	+	0.699435i	$0.246565\pi$
$194$		0			0
$195$	−2596.91			−0.953685
$196$		0			0
$197$	−970.738			−0.351077			−0.175539	−	0.984473i	$-0.556167\pi$
$197$	−970.738			−0.351077			−0.175539	+	0.984473i	$0.556167\pi$
$198$		0			0
$199$	−976.072	+	1690.61i	−0.347698	+	0.602231i	−0.985840	−	0.167688i	$-0.946370\pi$
$199$	−976.072	+	1690.61i	−0.347698	+	0.602231i	0.638142	+	0.769919i	$0.279703\pi$
$200$		0			0
$201$	−2008.73	+	1159.74i	−0.704898	+	0.406973i
$202$		0			0
$203$	−1139.63	−	4781.69i	−0.394023	−	1.65325i
$204$		0			0
$205$	976.078	+	1690.62i	0.332547	+	0.575989i
$206$		0			0
$207$	1154.93	+	666.802i	0.387795	+	0.223893i
$208$		0			0
$209$			1860.87i			0.615880i
$210$		0			0
$211$			1405.77i			0.458660i	0.973349	+	0.229330i	$0.0736536\pi$
$211$			1405.77i			0.458660i	−0.973349	+	0.229330i	$0.926346\pi$
$212$		0			0
$213$	−837.366	−	483.454i	−0.269368	−	0.155520i
$214$		0			0
$215$	−2345.84	−	4063.11i	−0.744115	−	1.28885i
$216$		0			0
$217$	2985.71	+	889.774i	0.934025	+	0.278349i
$218$		0			0
$219$	−1083.13	+	625.347i	−0.334207	+	0.192955i
$220$		0			0
$221$	3864.45	−	6693.42i	1.17625	−	2.03732i
$222$		0			0
$223$	−1116.97			−0.335415			−0.167708	−	0.985837i	$-0.553636\pi$
$223$	−1116.97			−0.335415			−0.167708	+	0.985837i	$0.553636\pi$
$224$		0			0
$225$	57.2391			0.0169597
$226$		0			0
$227$	−2137.24	+	3701.80i	−0.624904	+	1.08237i	0.363655	+	0.931534i	$0.381529\pi$
$227$	−2137.24	+	3701.80i	−0.624904	+	1.08237i	−0.988559	+	0.150832i	$0.951805\pi$
$228$		0			0
$229$	3468.05	−	2002.28i	1.00077	−	0.577792i	0.0922906	−	0.995732i	$-0.470581\pi$
$229$	3468.05	−	2002.28i	1.00077	−	0.577792i	0.908475	+	0.417940i	$0.137248\pi$
$230$		0			0
$231$	972.156	+	1027.83i	0.276897	+	0.292755i
$232$		0			0
$233$	1058.98	+	1834.21i	0.297752	+	0.515722i	0.975621	−	0.219461i	$-0.0704297\pi$
$233$	1058.98	+	1834.21i	0.297752	+	0.515722i	−0.677869	+	0.735183i	$0.737096\pi$
$234$		0			0
$235$	−2939.54	−	1697.15i	−0.815977	−	0.471105i
$236$		0			0
$237$			1436.81i			0.393800i
$238$		0			0
$239$			3845.74i			1.04084i	0.853911	+	0.520419i	$0.174224\pi$
$239$			3845.74i			1.04084i	−0.853911	+	0.520419i	$0.825776\pi$
$240$		0			0
$241$	1048.94	+	605.606i	0.280366	+	0.161869i	0.633589	−	0.773670i	$-0.281581\pi$
$241$	1048.94	+	605.606i	0.280366	+	0.161869i	−0.353223	+	0.935539i	$0.614914\pi$
$242$		0			0
$243$	−121.500	−	210.444i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	2045.12	−	3126.56i	0.533298	−	0.815300i
$246$		0			0
$247$	5029.83	−	2903.97i	1.29571	−	0.748079i
$248$		0			0
$249$	−163.354	+	282.938i	−0.0415750	+	0.0720099i
$250$		0			0
$251$	4071.69			1.02392			0.511958	−	0.859010i	$-0.328920\pi$
$251$	4071.69			1.02392			0.511958	+	0.859010i	$0.328920\pi$
$252$		0			0
$253$	−3773.09			−0.937596
$254$		0			0
$255$	1588.93	−	2752.11i	0.390207	−	0.675858i
$256$		0			0
$257$	−1202.04	+	694.000i	−0.291757	+	0.168446i	−0.638734	−	0.769428i	$-0.720541\pi$
$257$	−1202.04	+	694.000i	−0.291757	+	0.168446i	0.346977	+	0.937874i	$0.387208\pi$
$258$		0			0
$259$	−661.671	+	625.829i	−0.158742	+	0.150143i
$260$		0			0
$261$	−1194.38	−	2068.73i	−0.283259	−	0.490618i
$262$		0			0
$263$	576.693	+	332.954i	0.135211	+	0.0780639i	0.566079	−	0.824351i	$-0.308460\pi$
$263$	576.693	+	332.954i	0.135211	+	0.0780639i	−0.430869	+	0.902415i	$0.641793\pi$
$264$		0			0
$265$		−	1623.89i		−	0.376433i
$266$		0			0
$267$			2630.65i			0.602970i
$268$		0			0
$269$	7380.73	+	4261.26i	1.67290	+	0.965851i	0.966001	+	0.258539i	$0.0832411\pi$
$269$	7380.73	+	4261.26i	1.67290	+	0.965851i	0.706902	+	0.707312i	$0.250092\pi$
$270$		0			0
$271$	2135.11	+	3698.12i	0.478593	+	0.828948i	0.999699	−	0.0245443i	$-0.00781348\pi$
$271$	2135.11	+	3698.12i	0.478593	+	0.828948i	−0.521105	+	0.853492i	$0.674480\pi$
$272$		0			0
$273$	1261.08	−	4231.67i	0.279576	−	0.938141i
$274$		0			0
$275$	−140.247	+	80.9716i	−0.0307535	+	0.0177555i
$276$		0			0
$277$	−1561.52	+	2704.64i	−0.338710	+	0.586664i	−0.984190	−	0.177114i	$-0.943324\pi$
$277$	−1561.52	+	2704.64i	−0.338710	+	0.586664i	0.645480	+	0.763777i	$0.276657\pi$
$278$		0			0
$279$	1513.98			0.324873
$280$		0			0
$281$	−7317.30			−1.55343			−0.776714	−	0.629853i	$-0.783115\pi$
$281$	−7317.30			−1.55343			−0.776714	+	0.629853i	$0.783115\pi$
$282$		0			0
$283$	1273.25	−	2205.34i	0.267446	−	0.463229i	−0.700756	−	0.713401i	$-0.747154\pi$
$283$	1273.25	−	2205.34i	0.267446	−	0.463229i	0.968201	+	0.250172i	$0.0804871\pi$
$284$		0			0
$285$	2068.10	−	1194.02i	0.429836	−	0.248166i
$286$		0			0
$287$	−3228.86	+	769.543i	−0.664088	+	0.158274i
$288$		0			0
$289$	2272.46	+	3936.01i	0.462540	+	0.801143i
$290$		0			0
$291$	−743.113	−	429.037i	−0.149698	−	0.0864281i
$292$		0			0
$293$		−	7570.74i		−	1.50951i	−0.656005	−	0.754757i	$-0.727755\pi$
$293$		−	7570.74i		−	1.50951i	0.656005	−	0.754757i	$-0.272245\pi$
$294$		0			0
$295$			8043.31i			1.58746i
$296$		0			0
$297$	595.397	+	343.753i	0.116325	+	0.0671601i
$298$		0			0
$299$	5888.08	+	10198.5i	1.13885	+	1.97255i
$300$		0			0
$301$	7760.01	−	1849.47i	1.48598	−	0.354158i
$302$		0			0
$303$	−2666.31	+	1539.40i	−0.505531	+	0.291868i
$304$		0			0
$305$	−4773.36	+	8267.70i	−0.896136	+	1.55215i
$306$		0			0
$307$	−7774.87			−1.44539			−0.722696	−	0.691167i	$-0.757097\pi$
$307$	−7774.87			−1.44539			−0.722696	+	0.691167i	$0.757097\pi$
$308$		0			0
$309$	−2783.07			−0.512372
$310$		0			0
$311$	−3474.91	+	6018.72i	−0.633582	+	1.09740i	0.353232	+	0.935536i	$0.385083\pi$
$311$	−3474.91	+	6018.72i	−0.633582	+	1.09740i	−0.986814	+	0.161860i	$0.948251\pi$
$312$		0			0
$313$	−1815.98	+	1048.46i	−0.327940	+	0.189336i	−0.654926	−	0.755693i	$-0.727300\pi$
$313$	−1815.98	+	1048.46i	−0.327940	+	0.189336i	0.326986	+	0.945029i	$0.393967\pi$
$314$		0			0
$315$	518.514	−	1739.92i	0.0927460	−	0.311217i
$316$		0			0
$317$	1268.93	+	2197.85i	0.224827	+	0.389412i	0.956268	−	0.292493i	$-0.0944849\pi$
$317$	1268.93	+	2197.85i	0.224827	+	0.389412i	−0.731440	+	0.681905i	$0.761152\pi$
$318$		0			0
$319$	5852.94	+	3379.20i	1.02728	+	0.593100i
$320$		0			0
$321$		−	1222.28i		−	0.212527i
$322$		0			0
$323$			7107.23i			1.22433i
$324$		0			0
$325$	437.724	+	252.720i	0.0747095	+	0.0431335i
$326$		0			0
$327$	−927.194	−	1605.95i	−0.156801	−	0.271587i
$328$		0			0
$329$	4192.97	−	3965.85i	0.702633	−	0.664572i
$330$		0			0
$331$	5243.13	−	3027.12i	0.870659	−	0.502675i	0.00309202	−	0.999995i	$-0.499016\pi$
$331$	5243.13	−	3027.12i	0.870659	−	0.502675i	0.867567	+	0.497320i	$0.165682\pi$
$332$		0			0
$333$	−221.292	+	383.289i	−0.0364166	+	0.0630754i
$334$		0			0
$335$	−8421.40			−1.37346
$336$		0			0
$337$	6968.11			1.12634			0.563171	−	0.826340i	$-0.309581\pi$
$337$	6968.11			1.12634			0.563171	+	0.826340i	$0.309581\pi$
$338$		0			0
$339$	−1280.15	+	2217.29i	−0.205098	+	0.355241i
$340$		0			0
$341$	−3709.54	+	2141.70i	−0.589099	+	0.340117i
$342$		0			0
$343$	4101.61	+	4850.82i	0.645673	+	0.763614i
$344$		0			0
$345$	2420.98	+	4193.26i	0.377801	+	0.654370i
$346$		0			0
$347$	1075.12	+	620.718i	0.166326	+	0.0960285i	0.580852	−	0.814009i	$-0.302719\pi$
$347$	1075.12	+	620.718i	0.166326	+	0.0960285i	−0.414526	+	0.910037i	$0.636053\pi$
$348$		0			0
$349$		−	485.911i		−	0.0745279i	−0.999305	−	0.0372639i	$-0.988136\pi$
$349$		−	485.911i		−	0.0745279i	0.999305	−	0.0372639i	$-0.0118642\pi$
$350$		0			0
$351$		−	2145.77i		−	0.326304i
$352$		0			0
$353$	1925.08	+	1111.44i	0.290259	+	0.167581i	0.638059	−	0.769988i	$-0.279738\pi$
$353$	1925.08	+	1111.44i	0.290259	+	0.167581i	−0.347800	+	0.937569i	$0.613071\pi$
$354$		0			0
$355$	−1755.29	−	3040.25i	−0.262426	−	0.454535i
$356$		0			0
$357$	3712.97	+	3925.62i	0.550451	+	0.581976i
$358$		0			0
$359$	5131.15	−	2962.47i	0.754350	−	0.435524i	−0.0729136	−	0.997338i	$-0.523230\pi$
$359$	5131.15	−	2962.47i	0.754350	−	0.435524i	0.827263	+	0.561814i	$0.189896\pi$
$360$		0			0
$361$	759.104	−	1314.81i	0.110673	−	0.191691i
$362$		0			0
$363$	2047.88			0.296104
$364$		0			0
$365$	−4540.94			−0.651188
$366$		0			0
$367$	2516.61	−	4358.89i	0.357945	−	0.619979i	−0.629673	−	0.776861i	$-0.716811\pi$
$367$	2516.61	−	4358.89i	0.357945	−	0.619979i	0.987617	+	0.156882i	$0.0501442\pi$
$368$		0			0
$369$	−1396.92	+	806.512i	−0.197075	+	0.113781i
$370$		0			0
$371$	2646.13	+	788.575i	0.370297	+	0.110352i
$372$		0			0
$373$	−1103.31	−	1910.99i	−0.153156	−	0.265274i	0.779230	−	0.626738i	$-0.215610\pi$
$373$	−1103.31	−	1910.99i	−0.153156	−	0.265274i	−0.932386	+	0.361464i	$0.882277\pi$
$374$		0			0
$375$	3717.32	+	2146.20i	0.511898	+	0.295545i
$376$		0			0
$377$		−	21093.6i		−	2.88164i
$378$		0			0
$379$		−	6545.89i		−	0.887176i	−0.896231	−	0.443588i	$-0.853705\pi$
$379$		−	6545.89i		−	0.887176i	0.896231	−	0.443588i	$-0.146295\pi$
$380$		0			0
$381$	3616.64	+	2088.07i	0.486315	+	0.280774i
$382$		0			0
$383$	−4887.35	−	8465.14i	−0.652042	−	1.12937i	−0.982627	−	0.185593i	$-0.940579\pi$
$383$	−4887.35	−	8465.14i	−0.652042	−	1.12937i	0.330585	−	0.943776i	$-0.392754\pi$
$384$		0			0
$385$	1190.86	+	4996.64i	0.157642	+	0.661435i
$386$		0			0
$387$	3357.26	−	1938.32i	0.440980	−	0.254600i
$388$		0			0
$389$	−4893.17	+	8475.22i	−0.637773	+	1.10465i	0.348148	+	0.937440i	$0.386811\pi$
$389$	−4893.17	+	8475.22i	−0.637773	+	1.10465i	−0.985920	+	0.167215i	$0.946523\pi$
$390$		0			0
$391$	−14410.6			−1.86388
$392$		0			0
$393$	2318.14			0.297544
$394$		0			0
$395$	−2608.33	+	4517.76i	−0.332252	+	0.575477i
$396$		0			0
$397$	−1955.05	+	1128.75i	−0.247157	+	0.142696i	−0.618462	−	0.785815i	$-0.712244\pi$
$397$	−1955.05	+	1128.75i	−0.247157	+	0.142696i	0.371305	+	0.928511i	$0.378911\pi$
$398$		0			0
$399$	941.366	+	3949.79i	0.118113	+	0.495581i
$400$		0			0
$401$	46.2874	+	80.1720i	0.00576429	+	0.00998404i	0.868893	−	0.495000i	$-0.164832\pi$
$401$	46.2874	+	80.1720i	0.00576429	+	0.00998404i	−0.863129	+	0.504984i	$0.831498\pi$
$402$		0			0
$403$	11577.8	+	6684.47i	1.43110	+	0.826246i
$404$		0			0
$405$		−	882.268i		−	0.108248i
$406$		0			0
$407$		−	1252.18i		−	0.152502i
$408$		0			0
$409$	2269.62	+	1310.37i	0.274390	+	0.158419i	0.630881	−	0.775879i	$-0.282694\pi$
$409$	2269.62	+	1310.37i	0.274390	+	0.158419i	−0.356491	+	0.934299i	$0.616027\pi$
$410$		0			0
$411$	−3092.60	−	5356.55i	−0.371160	−	0.642868i
$412$		0			0
$413$	−13106.6	−	3905.91i	−1.56158	−	0.465368i
$414$		0			0
$415$	−1027.27	+	593.096i	−0.121510	+	0.0701541i
$416$		0			0
$417$	651.376	−	1128.22i	0.0764941	−	0.132492i
$418$		0			0
$419$	5213.75			0.607896			0.303948	−	0.952689i	$-0.401695\pi$
$419$	5213.75			0.607896			0.303948	+	0.952689i	$0.401695\pi$
$420$		0			0
$421$	−8214.13			−0.950908			−0.475454	−	0.879740i	$-0.657716\pi$
$421$	−8214.13			−0.950908			−0.475454	+	0.879740i	$0.657716\pi$
$422$		0			0
$423$	1402.32	−	2428.88i	0.161189	−	0.279188i
$424$		0			0
$425$	−535.647	+	309.256i	−0.0611358	+	0.0352967i
$426$		0			0
$427$	−11154.3	−	11793.1i	−1.26415	−	1.33655i
$428$		0			0
$429$	3035.45	+	5257.56i	0.341615	+	0.591695i
$430$		0			0
$431$	−12724.4	−	7346.43i	−1.42207	−	0.821032i	−0.425594	−	0.904914i	$-0.639935\pi$
$431$	−12724.4	−	7346.43i	−1.42207	−	0.821032i	−0.996476	+	0.0838817i	$0.973268\pi$
$432$		0			0
$433$		−	15413.6i		−	1.71069i	−0.518060	−	0.855344i	$-0.673346\pi$
$433$		−	15413.6i		−	1.71069i	0.518060	−	0.855344i	$-0.326654\pi$
$434$		0			0
$435$		−	8672.98i		−	0.955948i
$436$		0			0
$437$	−9378.16	−	5414.48i	−1.02659	−	0.592700i
$438$		0			0
$439$	−3683.14	−	6379.39i	−0.400426	−	0.693557i	0.593352	−	0.804943i	$-0.297804\pi$
$439$	−3683.14	−	6379.39i	−0.400426	−	0.693557i	−0.993777	+	0.111386i	$0.964471\pi$
$440$		0			0
$441$	2583.41	+	1689.84i	0.278956	+	0.182469i
$442$		0			0
$443$	3155.93	−	1822.08i	0.338471	−	0.195416i	−0.321125	−	0.947037i	$-0.604061\pi$
$443$	3155.93	−	1822.08i	0.338471	−	0.195416i	0.659596	+	0.751621i	$0.270727\pi$
$444$		0			0
$445$	−4775.59	+	8271.56i	−0.508729	+	0.881145i
$446$		0			0
$447$	−8361.14			−0.884717
$448$		0			0
$449$	−8345.46			−0.877164			−0.438582	−	0.898691i	$-0.644519\pi$
$449$	−8345.46			−0.877164			−0.438582	+	0.898691i	$0.644519\pi$
$450$		0			0
$451$	2281.82	−	3952.22i	0.238241	−	0.412645i
$452$		0			0
$453$	5712.07	−	3297.86i	0.592442	−	0.342047i
$454$		0			0
$455$	11647.3	−	11016.4i	1.20007	−	1.13507i
$456$		0			0
$457$	1489.03	+	2579.07i	0.152415	+	0.263991i	0.932115	−	0.362163i	$-0.117962\pi$
$457$	1489.03	+	2579.07i	0.152415	+	0.263991i	−0.779700	+	0.626154i	$0.784628\pi$
$458$		0			0
$459$	2274.01	+	1312.90i	0.231245	+	0.133510i
$460$		0			0
$461$			5112.61i			0.516525i	0.966075	+	0.258262i	$0.0831499\pi$
$461$			5112.61i			0.516525i	−0.966075	+	0.258262i	$0.916850\pi$
$462$		0			0
$463$			9641.97i			0.967819i	0.875118	+	0.483910i	$0.160784\pi$
$463$			9641.97i			0.967819i	−0.875118	+	0.483910i	$0.839216\pi$
$464$		0			0
$465$	4760.41	+	2748.43i	0.474750	+	0.274097i
$466$		0			0
$467$	2236.56	+	3873.83i	0.221618	+	0.383853i	0.955299	−	0.295640i	$-0.0955329\pi$
$467$	2236.56	+	3873.83i	0.221618	+	0.383853i	−0.733682	+	0.679493i	$0.762200\pi$
$468$		0			0
$469$	4089.51	−	13722.7i	0.402635	−	1.35108i
$470$		0			0
$471$	7769.11	−	4485.50i	0.760046	−	0.438813i
$472$		0			0
$473$	−5483.96	+	9498.50i	−0.533093	+	0.923344i
$474$		0			0
$475$	−464.786			−0.0448965
$476$		0			0
$477$	1341.78			0.128797
$478$		0			0
$479$	6167.17	−	10681.9i	0.588278	−	1.01893i	−0.406180	−	0.913793i	$-0.633139\pi$
$479$	6167.17	−	10681.9i	0.588278	−	1.01893i	0.994458	−	0.105135i	$-0.0335274\pi$
$480$		0			0
$481$	−3384.57	+	1954.08i	−0.320838	+	0.185236i
$482$		0			0
$483$	−8008.58	+	1908.71i	−0.754458	+	0.179812i
$484$		0			0
$485$	−1557.72	−	2698.05i	−0.145840	−	0.252602i
$486$		0			0
$487$	−5019.55	−	2898.04i	−0.467059	−	0.269657i	0.247949	−	0.968773i	$-0.420244\pi$
$487$	−5019.55	−	2898.04i	−0.467059	−	0.269657i	−0.715008	+	0.699117i	$0.753577\pi$
$488$		0			0
$489$		−	1767.83i		−	0.163485i
$490$		0			0
$491$			20606.3i			1.89399i	0.321249	+	0.946995i	$0.395897\pi$
$491$			20606.3i			1.89399i	−0.321249	+	0.946995i	$0.604103\pi$
$492$		0			0
$493$	22354.2	+	12906.2i	2.04216	+	1.17904i
$494$		0			0
$495$	1248.08	+	2161.73i	0.113327	+	0.196288i
$496$		0			0
$497$	5806.49	−	1383.88i	0.524058	−	0.124900i
$498$		0			0
$499$	3322.27	−	1918.11i	0.298047	−	0.172077i	−0.343518	−	0.939146i	$-0.611619\pi$
$499$	3322.27	−	1918.11i	0.298047	−	0.172077i	0.641565	+	0.767069i	$0.278285\pi$
$500$		0			0
$501$	1143.58	−	1980.74i	0.101979	−	0.176632i
$502$		0			0
$503$	−8841.62			−0.783754			−0.391877	−	0.920018i	$-0.628174\pi$
$503$	−8841.62			−0.783754			−0.391877	+	0.920018i	$0.628174\pi$
$504$		0			0
$505$	−11178.3			−0.985004
$506$		0			0
$507$	6178.45	−	10701.4i	0.541212	−	0.937406i
$508$		0			0
$509$	−4806.90	+	2775.27i	−0.418590	+	0.241673i	−0.694474	−	0.719518i	$-0.744363\pi$
$509$	−4806.90	+	2775.27i	−0.418590	+	0.241673i	0.275884	+	0.961191i	$0.411029\pi$
$510$		0			0
$511$	2205.12	−	7399.47i	0.190898	−	0.640574i
$512$		0			0
$513$	986.590	+	1708.82i	0.0849103	+	0.147069i
$514$		0			0
$515$	−8750.82	−	5052.29i	−0.748751	−	0.432292i
$516$		0			0
$517$			7934.98i			0.675010i
$518$		0			0
$519$			196.585i			0.0166264i
$520$		0			0
$521$	−5733.95	−	3310.50i	−0.482167	−	0.278379i	0.239152	−	0.970982i	$-0.423130\pi$
$521$	−5733.95	−	3310.50i	−0.482167	−	0.278379i	−0.721319	+	0.692603i	$0.756464\pi$
$522$		0			0
$523$	−9893.41	−	17135.9i	−0.827168	−	1.43270i	−0.900251	−	0.435371i	$-0.856617\pi$
$523$	−9893.41	−	17135.9i	−0.827168	−	1.43270i	0.0730834	−	0.997326i	$-0.476716\pi$
$524$		0			0
$525$	−256.720	+	242.814i	−0.0213413	+	0.0201853i
$526$		0			0
$527$	−14167.9	+	8179.84i	−1.17109	+	0.676128i
$528$		0			0
$529$	4894.88	−	8478.19i	0.402308	−	0.696818i
$530$		0			0
$531$	−6646.02			−0.543150
$532$		0			0
$533$	−14243.6			−1.15752
$534$		0			0
$535$	2218.89	−	3843.23i	0.179310	−	0.310574i
$536$		0			0
$537$	−6429.87	+	3712.29i	−0.516703	+	0.298318i
$538$		0			0
$539$	−8720.34	−	485.898i	−0.696868	−	0.0388295i
$540$		0			0
$541$	10024.3	+	17362.7i	0.796636	+	1.37981i	0.921795	+	0.387678i	$0.126723\pi$
$541$	10024.3	+	17362.7i	0.796636	+	1.37981i	−0.125159	+	0.992137i	$0.539944\pi$
$542$		0			0
$543$	10371.4	+	5987.91i	0.819665	+	0.473234i
$544$		0			0
$545$		−	6732.79i		−	0.529176i
$546$		0			0
$547$		−	17711.1i		−	1.38441i	−0.721703	−	0.692203i	$-0.756640\pi$
$547$		−	17711.1i		−	1.38441i	0.721703	−	0.692203i	$-0.243360\pi$
$548$		0			0
$549$	−6831.42	−	3944.12i	−0.531071	−	0.306614i
$550$		0			0
$551$	9698.49	+	16798.3i	0.749854	+	1.29879i
$552$		0			0
$553$	−6095.08	−	6444.15i	−0.468696	−	0.495539i
$554$		0			0
$555$	−1391.62	+	803.453i	−0.106434	+	0.0614499i
$556$		0			0
$557$	−7318.17	+	12675.4i	−0.556698	+	0.964230i	0.441071	+	0.897472i	$0.354599\pi$
$557$	−7318.17	+	12675.4i	−0.556698	+	0.964230i	−0.997769	+	0.0667574i	$0.978735\pi$
$558$		0			0
$559$	34232.0			2.59009
$560$		0			0
$561$	−7429.02			−0.559097
$562$		0			0
$563$	−5753.21	+	9964.85i	−0.430673	+	0.745948i	−0.996931	−	0.0782802i	$-0.975057\pi$
$563$	−5753.21	+	9964.85i	−0.430673	+	0.745948i	0.566258	+	0.824228i	$0.308390\pi$
$564$		0			0
$565$	−8050.39	+	4647.89i	−0.599438	+	0.346086i
$566$		0			0
$567$	1437.66	+	428.438i	0.106483	+	0.0317331i
$568$		0			0
$569$	−11575.8	−	20049.9i	−0.852872	−	1.47722i	−0.878605	−	0.477549i	$-0.841525\pi$
$569$	−11575.8	−	20049.9i	−0.852872	−	1.47722i	0.0257334	−	0.999669i	$-0.491808\pi$
$570$		0			0
$571$	8344.39	+	4817.63i	0.611562	+	0.353085i	0.773576	−	0.633703i	$-0.218466\pi$
$571$	8344.39	+	4817.63i	0.611562	+	0.353085i	−0.162015	+	0.986788i	$0.551799\pi$
$572$		0			0
$573$			9984.50i			0.727938i
$574$		0			0
$575$		−	942.398i		−	0.0683491i
$576$		0			0
$577$	−7479.16	−	4318.09i	−0.539621	−	0.311550i	0.205304	−	0.978698i	$-0.434182\pi$
$577$	−7479.16	−	4318.09i	−0.539621	−	0.311550i	−0.744925	+	0.667148i	$0.767515\pi$
$578$		0			0
$579$	−1997.90	−	3460.47i	−0.143402	−	0.248380i
$580$		0			0
$581$	−467.599	−	1961.96i	−0.0333895	−	0.140096i
$582$		0			0
$583$	−3287.63	+	1898.12i	−0.233550	+	0.134840i
$584$		0			0
$585$	3895.36	−	6746.97i	0.275305	−	0.476842i
$586$		0			0
$587$	4919.56			0.345915			0.172958	−	0.984929i	$-0.444668\pi$
$587$	4919.56			0.345915			0.172958	+	0.984929i	$0.444668\pi$
$588$		0			0
$589$	−12293.6			−0.860017
$590$		0			0
$591$	1456.11	−	2522.05i	0.101347	−	0.175539i
$592$		0			0
$593$	14041.6	−	8106.92i	0.972377	−	0.561402i	0.0724169	−	0.997374i	$-0.476929\pi$
$593$	14041.6	−	8106.92i	0.972377	−	0.561402i	0.899960	+	0.435972i	$0.143595\pi$
$594$		0			0
$595$	4548.29	+	19083.7i	0.313381	+	1.31489i
$596$		0			0
$597$	−2928.22	−	5071.82i	−0.200744	−	0.347698i
$598$		0			0
$599$	−18592.9	−	10734.6i	−1.26825	−	0.732226i	−0.293595	−	0.955930i	$-0.594852\pi$
$599$	−18592.9	−	10734.6i	−1.26825	−	0.732226i	−0.974657	+	0.223704i	$0.928185\pi$
$600$		0			0
$601$			14575.2i			0.989246i	0.869108	+	0.494623i	$0.164694\pi$
$601$			14575.2i			0.989246i	−0.869108	+	0.494623i	$0.835306\pi$
$602$		0			0
$603$		−	6958.43i		−	0.469932i
$604$		0			0
$605$	6439.17	+	3717.66i	0.432710	+	0.249825i
$606$		0			0
$607$	1896.41	+	3284.68i	0.126809	+	0.219639i	0.922438	−	0.386144i	$-0.126193\pi$
$607$	1896.41	+	3284.68i	0.126809	+	0.219639i	−0.795630	+	0.605783i	$0.792860\pi$
$608$		0			0
$609$	14132.6	+	4211.68i	0.940367	+	0.280239i
$610$		0			0
$611$	21447.9	−	12382.9i	1.42011	−	0.819901i
$612$		0			0
$613$	10096.0	−	17486.8i	0.665211	−	1.15218i	−0.314018	−	0.949417i	$-0.601675\pi$
$613$	10096.0	−	17486.8i	0.665211	−	1.15218i	0.979228	−	0.202761i	$-0.0649916\pi$
$614$		0			0
$615$	−5856.47			−0.383993
$616$		0			0
$617$	3228.67			0.210667			0.105333	−	0.994437i	$-0.466409\pi$
$617$	3228.67			0.210667			0.105333	+	0.994437i	$0.466409\pi$
$618$		0			0
$619$	1250.06	−	2165.17i	0.0811698	−	0.140590i	−0.822583	−	0.568645i	$-0.807468\pi$
$619$	1250.06	−	2165.17i	0.0811698	−	0.140590i	0.903753	+	0.428055i	$0.140801\pi$
$620$		0			0
$621$	−3464.80	+	2000.41i	−0.223893	+	0.129265i
$622$		0			0
$623$	−11159.5	−	11798.6i	−0.717648	−	0.758748i
$624$		0			0
$625$	7394.78	+	12808.1i	0.473266	+	0.819721i
$626$		0			0
$627$	−4834.67	−	2791.30i	−0.307940	−	0.177789i
$628$		0			0
$629$		−	4782.46i		−	0.303162i
$630$		0			0
$631$			1558.07i			0.0982977i	0.998791	+	0.0491489i	$0.0156509\pi$
$631$			1558.07i			0.0982977i	−0.998791	+	0.0491489i	$0.984349\pi$
$632$		0			0
$633$	−3652.30	−	2108.66i	−0.229330	−	0.132404i
$634$		0			0
$635$	7581.22	+	13131.1i	0.473782	+	0.820614i
$636$		0			0
$637$	12295.2	+	24328.9i	0.764760	+	1.51326i
$638$		0			0
$639$	2512.10	−	1450.36i	0.155520	−	0.0897893i
$640$		0			0
$641$	3524.57	−	6104.73i	0.217179	−	0.376166i	−0.736765	−	0.676149i	$-0.763648\pi$
$641$	3524.57	−	6104.73i	0.217179	−	0.376166i	0.953945	+	0.299983i	$0.0969810\pi$
$642$		0			0
$643$	−22807.0			−1.39879			−0.699393	−	0.714738i	$-0.746546\pi$
$643$	−22807.0			−1.39879			−0.699393	+	0.714738i	$0.746546\pi$
$644$		0			0
$645$	14075.0			0.859230
$646$		0			0
$647$	−10129.7	+	17545.1i	−0.615515	+	1.06610i	0.374779	+	0.927114i	$0.377719\pi$
$647$	−10129.7	+	17545.1i	−0.615515	+	1.06610i	−0.990294	+	0.138989i	$0.955615\pi$
$648$		0			0
$649$	16284.0	−	9401.60i	0.984907	−	0.568636i
$650$		0			0
$651$	−6790.27	+	6422.45i	−0.408804	+	0.386660i
$652$		0			0
$653$	6363.37	+	11021.7i	0.381344	+	0.660508i	0.991255	−	0.131963i	$-0.0421279\pi$
$653$	6363.37	+	11021.7i	0.381344	+	0.660508i	−0.609910	+	0.792470i	$0.708795\pi$
$654$		0			0
$655$	7288.95	+	4208.28i	0.434813	+	0.251040i
$656$		0			0
$657$		−	3752.08i		−	0.222805i
$658$		0			0
$659$		−	2518.75i		−	0.148887i	−0.997225	−	0.0744436i	$-0.976282\pi$
$659$		−	2518.75i		−	0.148887i	0.997225	−	0.0744436i	$-0.0237181\pi$
$660$		0			0
$661$	−26593.8	−	15353.9i	−1.56487	−	0.903479i	−0.996752	−	0.0805321i	$-0.974338\pi$
$661$	−26593.8	−	15353.9i	−1.56487	−	0.903479i	−0.568119	−	0.822947i	$-0.692329\pi$
$662$		0			0
$663$	11593.3	+	20080.3i	0.679108	+	1.17625i
$664$		0			0
$665$	−4210.38	+	14128.3i	−0.245521	+	0.823867i
$666$		0			0
$667$	−34060.1	+	19664.6i	−1.97723	+	1.14155i
$668$		0			0
$669$	1675.45	−	2901.96i	0.0968261	−	0.167708i
$670$		0			0
$671$	22317.8			1.28401
$672$		0			0
$673$	32957.8			1.88771			0.943856	−	0.330356i	$-0.107169\pi$
$673$	32957.8			1.88771			0.943856	+	0.330356i	$0.107169\pi$
$674$		0			0
$675$	−85.8587	+	148.712i	−0.00489585	+	0.00847987i
$676$		0			0
$677$	3400.40	−	1963.22i	0.193040	−	0.111452i	−0.400365	−	0.916356i	$-0.631117\pi$
$677$	3400.40	−	1963.22i	0.193040	−	0.111452i	0.593405	+	0.804904i	$0.297783\pi$
$678$		0			0
$679$	5152.92	−	1228.11i	0.291238	−	0.0694117i
$680$		0			0
$681$	−6411.71	−	11105.4i	−0.360789	−	0.624904i
$682$		0			0
$683$	21748.6	+	12556.6i	1.21843	+	0.703460i	0.964582	−	0.263784i	$-0.0849707\pi$
$683$	21748.6	+	12556.6i	1.21843	+	0.703460i	0.253847	+	0.967244i	$0.418304\pi$
$684$		0			0
$685$		−	22456.8i		−	1.25260i
$686$		0			0
$687$			12013.7i			0.667177i
$688$		0			0
$689$	10261.0	+	5924.21i	0.567364	+	0.327568i
$690$		0			0
$691$	3078.21	+	5331.61i	0.169465	+	0.293522i	0.938232	−	0.346007i	$-0.112463\pi$
$691$	3078.21	+	5331.61i	0.169465	+	0.293522i	−0.768767	+	0.639529i	$0.779129\pi$
$692$		0			0
$693$	−4128.62	+	983.987i	−0.226311	+	0.0539373i
$694$		0			0
$695$	4096.26	−	2364.97i	0.223568	−	0.129077i
$696$		0			0
$697$	8714.98	−	15094.8i	0.473606	−	0.820310i
$698$		0			0
$699$	−6353.90			−0.343815
$700$		0			0
$701$	20064.7			1.08107			0.540537	−	0.841320i	$-0.318221\pi$
$701$	20064.7			1.08107			0.540537	+	0.841320i	$0.318221\pi$
$702$		0			0
$703$	1796.91	−	3112.34i	0.0964036	−	0.166976i
$704$		0			0
$705$	8818.63	−	5091.44i	0.471105	−	0.271992i
$706$		0			0
$707$	5428.28	−	18215.1i	0.288757	−	0.968950i
$708$		0			0
$709$	14957.7	+	25907.4i	0.792309	+	1.37232i	0.924534	+	0.381099i	$0.124454\pi$
$709$	14957.7	+	25907.4i	0.792309	+	1.37232i	−0.132225	+	0.991220i	$0.542212\pi$
$710$		0			0
$711$	−3732.93	−	2155.21i	−0.196900	−	0.113680i
$712$		0			0
$713$		−	24926.5i		−	1.30926i
$714$		0			0
$715$			22041.8i			1.15289i
$716$		0			0
$717$	−9991.53	−	5768.61i	−0.520419	−	0.300464i
$718$		0			0
$719$	10091.2	+	17478.4i	0.523417	+	0.906585i	0.999629	+	0.0272541i	$0.00867633\pi$
$719$	10091.2	+	17478.4i	0.523417	+	0.906585i	−0.476211	+	0.879331i	$0.657990\pi$
$720$		0			0
$721$	12482.2	−	11806.0i	0.644745	−	0.609820i
$722$		0			0
$723$	−3146.82	+	1816.82i	−0.161869	+	0.0934554i
$724$		0			0
$725$	−844.018	+	1461.88i	−0.0432359	+	0.0748868i
$726$		0			0
$727$	−9294.28			−0.474148			−0.237074	−	0.971492i	$-0.576188\pi$
$727$	−9294.28			−0.474148			−0.237074	+	0.971492i	$0.576188\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−20945.0	+	36277.8i	−1.05975	+	1.83554i
$732$		0			0
$733$	−67.5292	+	38.9880i	−0.00340279	+	0.00196460i	−0.501700	−	0.865041i	$-0.667292\pi$
$733$	−67.5292	+	38.9880i	−0.00340279	+	0.00196460i	0.498298	+	0.867006i	$0.333959\pi$
$734$		0			0
$735$	5055.35	+	10003.2i	0.253700	+	0.502006i
$736$		0			0
$737$	9843.54	+	17049.5i	0.491983	+	0.852139i
$738$		0			0
$739$	15262.9	+	8812.06i	0.759751	+	0.438643i	0.829206	−	0.558943i	$-0.188793\pi$
$739$	15262.9	+	8812.06i	0.759751	+	0.438643i	−0.0694552	+	0.997585i	$0.522126\pi$
$740$		0			0
$741$			17423.8i			0.863807i
$742$		0			0
$743$		−	9692.27i		−	0.478566i	−0.970950	−	0.239283i	$-0.923088\pi$
$743$		−	9692.27i		−	0.478566i	0.970950	−	0.239283i	$-0.0769124\pi$
$744$		0			0
$745$	−26290.0	−	15178.5i	−1.29287	−	0.746441i
$746$		0			0
$747$	−490.063	−	848.814i	−0.0240033	−	0.0415750i
$748$		0			0
$749$	5185.04	+	5481.99i	0.252947	+	0.267433i
$750$		0			0
$751$	−15574.9	+	8992.16i	−0.756772	+	0.436922i	−0.828135	−	0.560528i	$-0.810598\pi$
$751$	−15574.9	+	8992.16i	−0.756772	+	0.436922i	0.0713639	+	0.997450i	$0.477265\pi$
$752$		0			0
$753$	−6107.54	+	10578.6i	−0.295579	+	0.511958i
$754$		0			0
$755$	23947.3			1.15435
$756$		0			0
$757$	−25068.7			−1.20362			−0.601809	−	0.798640i	$-0.705553\pi$
$757$	−25068.7			−1.20362			−0.601809	+	0.798640i	$0.705553\pi$
$758$		0			0
$759$	5659.63	−	9802.76i	0.270661	−	0.468798i
$760$		0			0
$761$	−8091.89	+	4671.86i	−0.385454	+	0.222542i	−0.680189	−	0.733037i	$-0.738102\pi$
$761$	−8091.89	+	4671.86i	−0.385454	+	0.222542i	0.294734	+	0.955579i	$0.404769\pi$
$762$		0			0
$763$	10971.1	+	3269.50i	0.520551	+	0.155130i
$764$		0			0
$765$	4766.79	+	8256.32i	0.225286	+	0.390207i
$766$		0			0
$767$	−50824.1	−	29343.3i	−2.39264	−	1.38139i
$768$		0			0
$769$			27967.9i			1.31150i	0.754976	+	0.655752i	$0.227648\pi$
$769$			27967.9i			1.31150i	−0.754976	+	0.655752i	$0.772352\pi$
$770$		0			0
$771$		−	4164.00i		−	0.194504i
$772$		0			0
$773$	22080.8	+	12748.4i	1.02742	+	0.593179i	0.916243	−	0.400622i	$-0.131206\pi$
$773$	22080.8	+	12748.4i	1.02742	+	0.593179i	0.111172	+	0.993801i	$0.464539\pi$
$774$		0			0
$775$	−534.930	−	926.527i	−0.0247939	−	0.0429443i
$776$		0			0
$777$	−633.445	−	2657.82i	−0.0292467	−	0.122714i
$778$		0			0
$779$	11343.1	−	6548.95i	0.521706	−	0.301207i
$780$		0			0
$781$	−4103.42	+	7107.33i	−0.188005	+	0.325634i
$782$		0			0
$783$	7166.30			0.327079
$784$		0			0
$785$	32571.3			1.48092
$786$		0			0
$787$	−5753.43	+	9965.24i	−0.260594	+	0.451363i	−0.966400	−	0.257043i	$-0.917252\pi$
$787$	−5753.43	+	9965.24i	−0.260594	+	0.451363i	0.705806	+	0.708406i	$0.250585\pi$
$788$		0			0
$789$	−1730.08	+	998.861i	−0.0780639	+	0.0450702i
$790$		0			0
$791$	−3664.42	−	15375.2i	−0.164718	−	0.691124i
$792$		0			0
$793$	−34828.0	−	60323.8i	−1.55962	−	2.70134i
$794$		0			0
$795$	4218.98	+	2435.83i	0.188216	+	0.108667i
$796$		0			0
$797$			39596.3i			1.75982i	0.475144	+	0.879908i	$0.342396\pi$
$797$			39596.3i			1.75982i	−0.475144	+	0.879908i	$0.657604\pi$
$798$		0			0
$799$			30306.2i			1.34187i
$800$		0			0
$801$	−6834.62	−	3945.97i	−0.301485	−	0.174062i
$802$		0			0
$803$	5307.77	+	9193.33i	0.233259	+	0.404017i
$804$		0			0
$805$	−28646.5	−	8536.95i	−1.25423	−	0.373774i
$806$		0			0
$807$	−22142.2	+	12783.8i	−0.965851	+	0.557634i
$808$		0			0
$809$	16562.7	−	28687.4i	0.719794	−	1.24672i	−0.241287	−	0.970454i	$-0.577569\pi$
$809$	16562.7	−	28687.4i	0.719794	−	1.24672i	0.961081	−	0.276267i	$-0.0890972\pi$
$810$		0			0
$811$	9495.37			0.411132			0.205566	−	0.978643i	$-0.434097\pi$
$811$	9495.37			0.411132			0.205566	+	0.978643i	$0.434097\pi$
$812$		0			0
$813$	−12810.7			−0.552632
$814$		0			0
$815$	3209.26	−	5558.61i	0.137933	−	0.238907i
$816$		0			0
$817$	−27261.2	+	15739.3i	−1.16738	+	0.673988i
$818$		0			0
$819$	9102.58	+	9623.90i	0.388364	+	0.410606i
$820$		0			0
$821$	−15216.5	−	26355.7i	−0.646844	−	1.12037i	−0.983872	−	0.178872i	$-0.942755\pi$
$821$	−15216.5	−	26355.7i	−0.646844	−	1.12037i	0.337028	−	0.941494i	$-0.390578\pi$
$822$		0			0
$823$	20430.7	+	11795.7i	0.865335	+	0.499602i	0.865795	−	0.500398i	$-0.166813\pi$
$823$	20430.7	+	11795.7i	0.865335	+	0.499602i	−0.000460017		1.00000i	$0.500146\pi$
$824$		0			0
$825$		−	485.830i		−	0.0205023i
$826$		0			0
$827$		−	38107.9i		−	1.60235i	−0.598432	−	0.801174i	$-0.704209\pi$
$827$		−	38107.9i		−	1.60235i	0.598432	−	0.801174i	$-0.295791\pi$
$828$		0			0
$829$	1509.16	+	871.312i	0.0632270	+	0.0365041i	0.531280	−	0.847196i	$-0.321711\pi$
$829$	1509.16	+	871.312i	0.0632270	+	0.0365041i	−0.468053	+	0.883700i	$0.655044\pi$
$830$		0			0
$831$	−4684.57	−	8113.91i	−0.195555	−	0.338710i
$832$		0			0
$833$	−33305.7	−	1855.80i	−1.38532	−	0.0771903i
$834$		0			0
$835$	7191.53	−	4152.03i	0.298052	−	0.172080i
$836$		0			0
$837$	−2270.97	+	3933.43i	−0.0937827	+	0.162436i
$838$		0			0
$839$	45398.9			1.86811			0.934054	−	0.357131i	$-0.116245\pi$
$839$	45398.9			1.86811			0.934054	+	0.357131i	$0.116245\pi$
$840$		0			0
$841$	46058.0			1.88847
$842$		0			0
$843$	10976.0	−	19010.9i	0.448436	−	0.776714i
$844$		0			0
$845$	38853.9	−	22432.3i	1.58179	−	0.913248i
$846$		0			0
$847$	−9184.85	+	8687.32i	−0.372604	+	0.352420i
$848$		0			0
$849$	3819.76	+	6616.02i	0.154410	+	0.267446i
$850$		0			0
$851$	6310.57	+	3643.41i	0.254199	+	0.146762i
$852$		0			0
$853$			24139.3i			0.968948i	0.874806	+	0.484474i	$0.160989\pi$
$853$			24139.3i			0.968948i	−0.874806	+	0.484474i	$0.839011\pi$
$854$		0			0
$855$			7164.09i			0.286558i
$856$		0			0
$857$	31092.8	+	17951.4i	1.23934	+	0.715530i	0.968958	−	0.247224i	$-0.0795184\pi$
$857$	31092.8	+	17951.4i	1.23934	+	0.715530i	0.270377	+	0.962755i	$0.412852\pi$
$858$		0			0
$859$	−10546.8	−	18267.7i	−0.418922	−	0.725594i	0.576910	−	0.816808i	$-0.304258\pi$
$859$	−10546.8	−	18267.7i	−0.418922	−	0.725594i	−0.995831	+	0.0912144i	$0.970925\pi$
$860$		0			0
$861$	2843.95	−	9543.13i	0.112569	−	0.377734i
$862$		0			0
$863$	−4036.20	+	2330.30i	−0.159205	+	0.0919169i	−0.577486	−	0.816401i	$-0.695966\pi$
$863$	−4036.20	+	2330.30i	−0.159205	+	0.0919169i	0.418281	+	0.908318i	$0.362633\pi$
$864$		0			0
$865$	−356.873	+	618.122i	−0.0140278	+	0.0242969i
$866$		0			0
$867$	−13634.8			−0.534095
$868$		0			0
$869$	12195.2			0.476058
$870$		0			0
$871$	30722.6	−	53213.2i	1.19517	−	2.07010i
$872$		0			0
$873$	2229.34	−	1287.11i	0.0864281	−	0.0498993i
$874$		0			0
$875$	−25776.8	+	6143.46i	−0.995902	+	0.237356i
$876$		0			0
$877$	−11572.7	−	20044.5i	−0.445590	−	0.771784i	0.552504	−	0.833511i	$-0.313673\pi$
$877$	−11572.7	−	20044.5i	−0.445590	−	0.771784i	−0.998093	+	0.0617268i	$0.980339\pi$
$878$		0			0
$879$	19669.4	+	11356.1i	0.754757	+	0.435759i
$880$		0			0
$881$			30585.1i			1.16962i	0.811169	+	0.584812i	$0.198832\pi$
$881$			30585.1i			1.16962i	−0.811169	+	0.584812i	$0.801168\pi$
$882$		0			0
$883$		−	45382.8i		−	1.72962i	−0.502101	−	0.864809i	$-0.667439\pi$
$883$		−	45382.8i		−	1.72962i	0.502101	−	0.864809i	$-0.332561\pi$
$884$		0			0
$885$	−20897.1	−	12065.0i	−0.793729	−	0.458259i
$886$		0			0
$887$	−13800.5	−	23903.2i	−0.522408	−	0.904838i	−0.999660	−	0.0260713i	$-0.991700\pi$
$887$	−13800.5	−	23903.2i	−0.522408	−	0.904838i	0.477252	−	0.878767i	$-0.341633\pi$
$888$		0			0
$889$	−25078.6	+	5977.06i	−0.946130	+	0.225494i
$890$		0			0
$891$	−1786.19	+	1031.26i	−0.0671601	+	0.0387749i
$892$		0			0
$893$	−11386.9	+	19722.7i	−0.426706	+	0.739077i
$894$		0			0
$895$	−26956.6			−1.00677
$896$		0			0
$897$	−35328.5			−1.31503
$898$		0			0
$899$	−22324.3	+	38666.9i	−0.828207	+	1.43450i
$900$		0			0
$901$	−12556.5	+	7249.50i	−0.464282	+	0.268053i
$902$		0			0
$903$	−6834.96	+	22935.3i	−0.251886	+	0.845226i
$904$		0			0
$905$	21740.5	+	37655.7i	0.798541	+	1.38311i
$906$		0			0
$907$	2383.18	+	1375.93i	0.0872462	+	0.0503716i	0.542988	−	0.839740i	$-0.317293\pi$
$907$	2383.18	+	1375.93i	0.0872462	+	0.0503716i	−0.455742	+	0.890112i	$0.650626\pi$
$908$		0			0
$909$		−	9236.38i		−	0.337020i
$910$		0			0
$911$		−	3191.31i		−	0.116062i	−0.998315	−	0.0580312i	$-0.981518\pi$
$911$		−	3191.31i		−	0.116062i	0.998315	−	0.0580312i	$-0.0184823\pi$
$912$		0			0
$913$	2401.50	+	1386.51i	0.0870515	+	0.0502592i
$914$		0			0
$915$	−14320.1	−	24803.1i	−0.517385	−	0.896136i
$916$		0			0
$917$	−10397.0	+	9833.78i	−0.374415	+	0.354133i
$918$		0			0
$919$	−42436.7	+	24500.8i	−1.52324	+	0.879443i	−0.523617	+	0.851953i	$0.675418\pi$
$919$	−42436.7	+	24500.8i	−1.52324	+	0.879443i	−0.999622	+	0.0274893i	$0.991249\pi$
$920$		0			0
$921$	11662.3	−	20199.7i	0.417248	−	0.722696i
$922$		0			0
$923$	25614.3			0.913441
$924$		0			0
$925$	312.755			0.0111171
$926$		0			0
$927$	4174.60	−	7230.62i	0.147909	−	0.256186i
$928$		0			0
$929$	15173.9	−	8760.67i	0.535888	−	0.309395i	−0.207523	−	0.978230i	$-0.566540\pi$
$929$	15173.9	−	8760.67i	0.535888	−	0.309395i	0.743411	+	0.668835i	$0.233207\pi$
$930$		0			0
$931$	−20977.5	−	13721.7i	−0.738464	−	0.483039i
$932$		0			0
$933$	−10424.7	−	18056.1i	−0.365799	−	0.633582i
$934$		0			0
$935$	−23359.1	−	13486.4i	−0.817032	−	0.471714i
$936$		0			0
$937$			12407.7i			0.432596i	0.976327	+	0.216298i	$0.0693983\pi$
$937$			12407.7i			0.432596i	−0.976327	+	0.216298i	$0.930602\pi$
$938$		0			0
$939$		−	6290.74i		−	0.218627i
$940$		0			0
$941$	19937.6	+	11511.0i	0.690697	+	0.398774i	0.803873	−	0.594801i	$-0.202769\pi$
$941$	19937.6	+	11511.0i	0.690697	+	0.398774i	−0.113176	+	0.993575i	$0.536102\pi$
$942$		0			0
$943$	13278.6	+	22999.2i	0.458548	+	0.794229i
$944$		0			0
$945$	3742.67	+	3957.02i	0.128835	+	0.136214i
$946$		0			0
$947$	8945.69	−	5164.79i	0.306965	−	0.177226i	−0.338603	−	0.940929i	$-0.609954\pi$
$947$	8945.69	−	5164.79i	0.306965	−	0.177226i	0.645567	+	0.763703i	$0.276621\pi$
$948$		0			0
$949$	16566.1	−	28693.3i	0.566657	−	0.981479i
$950$		0			0
$951$	−7613.58			−0.259608
$952$		0			0
$953$	−27371.6			−0.930380			−0.465190	−	0.885211i	$-0.654014\pi$
$953$	−27371.6			−0.930380			−0.465190	+	0.885211i	$0.654014\pi$
$954$		0			0
$955$	−18125.5	+	31394.3i	−0.614166	+	1.06377i
$956$		0			0
$957$	−17558.8	+	10137.6i	−0.593100	+	0.342426i
$958$		0			0
$959$	36593.5	+	10905.2i	1.23219	+	0.367204i
$960$		0			0
$961$	746.562	+	1293.08i	0.0250600	+	0.0434052i
$962$		0			0
$963$	3175.58	+	1833.42i	0.106263	+	0.0613512i
$964$		0			0
$965$		−	14507.7i		−	0.483958i
$966$		0			0
$967$		−	23099.9i		−	0.768193i	−0.923293	−	0.384097i	$-0.874513\pi$
$967$		−	23099.9i		−	0.768193i	0.923293	−	0.384097i	$-0.125487\pi$
$968$		0			0
$969$	−18465.1	−	10660.9i	−0.612163	−	0.353432i
$970$		0			0
$971$	−25707.4	−	44526.5i	−0.849629	−	1.47160i	−0.881540	−	0.472109i	$-0.843493\pi$
$971$	−25707.4	−	44526.5i	−0.849629	−	1.47160i	0.0319114	−	0.999491i	$-0.489841\pi$
$972$		0			0
$973$	1864.55	+	7823.31i	0.0614335	+	0.257764i
$974$		0			0
$975$	−1313.17	+	758.161i	−0.0431335	+	0.0249032i
$976$		0			0
$977$	−21156.5	+	36644.2i	−0.692792	+	1.19995i	0.278128	+	0.960544i	$0.410286\pi$
$977$	−21156.5	+	36644.2i	−0.692792	+	1.19995i	−0.970919	+	0.239406i	$0.923047\pi$
$978$		0			0
$979$	22328.2			0.728919
$980$		0			0
$981$	5563.16			0.181058
$982$		0			0
$983$	26206.9	−	45391.7i	0.850326	−	1.47281i	−0.0305888	−	0.999532i	$-0.509738\pi$
$983$	26206.9	−	45391.7i	0.850326	−	1.47281i	0.880915	−	0.473275i	$-0.156928\pi$
$984$		0			0
$985$	9156.90	−	5286.74i	0.296206	−	0.171015i
$986$		0			0
$987$	4014.11	+	16842.4i	0.129453	+	0.543162i
$988$		0			0
$989$	−31912.9	−	55274.8i	−1.02606	−	1.77718i
$990$		0			0
$991$	−39255.1	−	22663.9i	−1.25830	−	0.726482i	−0.285558	−	0.958361i	$-0.592179\pi$
$991$	−39255.1	−	22663.9i	−1.25830	−	0.726482i	−0.972744	+	0.231880i	$0.925512\pi$
$992$		0			0
$993$			18162.7i			0.580440i
$994$		0			0
$995$		−	21263.1i		−	0.677475i
$996$		0			0
$997$	5149.95	+	2973.33i	0.163591	+	0.0944495i	0.579561	−	0.814929i	$-0.303224\pi$
$997$	5149.95	+	2973.33i	0.163591	+	0.0944495i	−0.415969	+	0.909379i	$0.636558\pi$
$998$		0			0
$999$	−663.877	−	1149.87i	−0.0210251	−	0.0364166i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.4.bl.i.31.2	✓	8
4.3	odd	2		336.4.bl.j.31.2	yes	8
7.5	odd	6		336.4.bl.j.271.2	yes	8
28.19	even	6	inner	336.4.bl.i.271.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.4.bl.i.31.2	✓	8	1.1	even	1	trivial
336.4.bl.i.271.2	yes	8	28.19	even	6	inner
336.4.bl.j.31.2	yes	8	4.3	odd	2
336.4.bl.j.271.2	yes	8	7.5	odd	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	336.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.50000 + 2.59808i) q^{3} +(-9.43293 + 5.44610i) q^{5} +(-4.29373 - 18.0157i) q^{7} +(-4.50000 - 7.79423i) q^{9} +(22.0517 + 12.7316i) q^{11} -79.4730i q^{13} -32.6766i q^{15} +(84.2226 + 48.6259i) q^{17} +(36.5404 + 63.2898i) q^{19} +(53.2466 + 15.8681i) q^{21} +(-128.326 + 74.0891i) q^{23} +(-3.17995 + 5.50784i) q^{25} +27.0000 q^{27} +265.419 q^{29} +(-84.1099 + 145.683i) q^{31} +(-66.1552 + 38.1948i) q^{33} +(138.617 + 146.556i) q^{35} +(-24.5880 - 42.5877i) q^{37} +(206.477 + 119.210i) q^{39} -179.225i q^{41} +430.737i q^{43} +(84.8963 + 49.0149i) q^{45} +(155.813 + 269.876i) q^{47} +(-306.128 + 154.709i) q^{49} +(-252.668 + 145.878i) q^{51} +(-74.5436 + 129.113i) q^{53} -277.350 q^{55} -219.242 q^{57} +(369.223 - 639.514i) q^{59} +(759.047 - 438.236i) q^{61} +(-121.096 + 114.537i) q^{63} +(432.818 + 749.663i) q^{65} +(669.575 + 386.579i) q^{67} -444.534i q^{69} +322.302i q^{71} +(361.044 + 208.449i) q^{73} +(-9.53985 - 16.5235i) q^{75} +(134.684 - 451.943i) q^{77} +(414.770 - 239.468i) q^{79} +(-40.5000 + 70.1481i) q^{81} +108.903 q^{83} -1059.29 q^{85} +(-398.128 + 689.578i) q^{87} +(759.402 - 438.441i) q^{89} +(-1431.76 + 341.235i) q^{91} +(-252.330 - 437.048i) q^{93} +(-689.365 - 398.005i) q^{95} +286.024i q^{97} -229.169i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 12 q^{3} + 18 q^{5} - 12 q^{7} - 36 q^{9} + 150 q^{11} + 192 q^{17} - 66 q^{19} + 90 q^{21} - 492 q^{23} + 322 q^{25} + 216 q^{27} - 372 q^{29} + 156 q^{31} - 450 q^{33} + 36 q^{35} + 554 q^{37} - 198 q^{39}+ \cdots + 108 q^{95}+O(q^{100}) \) 8 * q - 12 * q^3 + 18 * q^5 - 12 * q^7 - 36 * q^9 + 150 * q^11 + 192 * q^17 - 66 * q^19 + 90 * q^21 - 492 * q^23 + 322 * q^25 + 216 * q^27 - 372 * q^29 + 156 * q^31 - 450 * q^33 + 36 * q^35 + 554 * q^37 - 198 * q^39 - 162 * q^45 + 840 * q^47 - 340 * q^49 - 576 * q^51 + 186 * q^53 + 156 * q^55 + 396 * q^57 - 126 * q^59 + 1632 * q^61 - 162 * q^63 + 936 * q^65 + 582 * q^67 - 1386 * q^73 + 966 * q^75 + 3030 * q^77 - 1260 * q^79 - 324 * q^81 + 756 * q^83 - 5688 * q^85 + 558 * q^87 + 948 * q^89 - 4074 * q^91 + 468 * q^93 + 108 * q^95

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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