LMFDB - Embedded newform 108.5.k.a.5.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,5,Mod(5,108)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(108, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([0, 5]))

N = Newforms(chi, 5, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("108.5");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	108.k (of order $18$, degree $6$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$11.1639560131$
Analytic rank:	$0$
Dimension:	$72$
Relative dimension:	$12$ over $\Q(\zeta_{18})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label			5.12
Character	$\chi$	$=$	108.5
Dual form			108.5.k.a.65.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(8.73547 + 2.16600i) q^{3} +(9.54164 + 26.2154i) q^{5} +(-0.541932 - 3.07345i) q^{7} +(71.6168 + 37.8421i) q^{9} +O(q^{10})$	\(q+(8.73547 + 2.16600i) q^{3} +(9.54164 + 26.2154i) q^{5} +(-0.541932 - 3.07345i) q^{7} +(71.6168 + 37.8421i) q^{9} +(-44.3351 + 121.810i) q^{11} +(-30.3898 - 25.5001i) q^{13} +(26.5679 + 249.671i) q^{15} +(-72.0247 - 41.5835i) q^{17} +(32.6516 + 56.5542i) q^{19} +(1.92308 - 28.0219i) q^{21} +(346.437 + 61.0862i) q^{23} +(-117.428 + 98.5341i) q^{25} +(543.640 + 485.691i) q^{27} +(421.220 + 501.991i) q^{29} +(-18.7225 + 106.181i) q^{31} +(-651.128 + 968.034i) q^{33} +(75.4009 - 43.5328i) q^{35} +(-745.268 + 1290.84i) q^{37} +(-210.236 - 288.580i) q^{39} +(1342.26 - 1599.64i) q^{41} +(-1405.92 - 511.712i) q^{43} +(-308.706 + 2238.54i) q^{45} +(2360.91 - 416.292i) q^{47} +(2247.05 - 817.859i) q^{49} +(-539.100 - 519.257i) q^{51} -2343.67i q^{53} -3616.32 q^{55} +(162.730 + 564.751i) q^{57} +(-978.002 - 2687.04i) q^{59} +(-1139.50 - 6462.44i) q^{61} +(77.4945 - 240.619i) q^{63} +(378.527 - 1040.00i) q^{65} +(-6613.79 - 5549.63i) q^{67} +(2893.98 + 1284.00i) q^{69} +(-426.072 - 245.993i) q^{71} +(-2004.31 - 3471.57i) q^{73} +(-1239.22 + 606.391i) q^{75} +(398.403 + 70.2491i) q^{77} +(-5374.08 + 4509.39i) q^{79} +(3696.94 + 5420.27i) q^{81} +(1605.97 + 1913.93i) q^{83} +(402.895 - 2284.93i) q^{85} +(2592.24 + 5297.49i) q^{87} +(12923.1 - 7461.14i) q^{89} +(-61.9041 + 107.221i) q^{91} +(-393.538 + 886.984i) q^{93} +(-1171.04 + 1395.60i) q^{95} +(15873.9 + 5777.63i) q^{97} +(-7784.68 + 7045.89i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{5}{18}\right)$	$1$

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
$3$	8.73547	+	2.16600i	0.970608	+	0.240667i
$3$	8.73547	+	2.16600i	0.970608	+	0.240667i
$4$		0			0
$5$	9.54164	+	26.2154i	0.381666	+	1.04862i	0.970655	+	0.240477i	$0.0773037\pi$
$5$	9.54164	+	26.2154i	0.381666	+	1.04862i	−0.588989	+	0.808141i	$0.700474\pi$
$6$		0			0
$7$	−0.541932	−	3.07345i	−0.0110598	−	0.0627235i	0.978778	−	0.204922i	$-0.0656940\pi$
$7$	−0.541932	−	3.07345i	−0.0110598	−	0.0627235i	−0.989838	+	0.142198i	$0.954583\pi$
$8$		0			0
$9$	71.6168	+	37.8421i	0.884159	+	0.467187i
$10$		0			0
$11$	−44.3351	+	121.810i	−0.366406	+	1.00669i	0.610312	+	0.792161i	$0.291044\pi$
$11$	−44.3351	+	121.810i	−0.366406	+	1.00669i	−0.976717	+	0.214530i	$0.931178\pi$
$12$		0			0
$13$	−30.3898	−	25.5001i	−0.179821	−	0.150888i	0.548434	−	0.836194i	$-0.315224\pi$
$13$	−30.3898	−	25.5001i	−0.179821	−	0.150888i	−0.728256	+	0.685306i	$0.759669\pi$
$14$		0			0
$15$	26.5679	+	249.671i	0.118080	+	1.10965i
$16$		0			0
$17$	−72.0247	−	41.5835i	−0.249220	−	0.143887i	0.370187	−	0.928957i	$-0.379294\pi$
$17$	−72.0247	−	41.5835i	−0.249220	−	0.143887i	−0.619407	+	0.785070i	$0.712627\pi$
$18$		0			0
$19$	32.6516	+	56.5542i	0.0904476	+	0.156660i	0.907700	−	0.419621i	$-0.137837\pi$
$19$	32.6516	+	56.5542i	0.0904476	+	0.156660i	−0.817252	+	0.576281i	$0.804504\pi$
$20$		0			0
$21$	1.92308	−	28.0219i	0.00436072	−	0.0635417i
$22$		0			0
$23$	346.437	+	61.0862i	0.654890	+	0.115475i	0.491214	−	0.871039i	$-0.336553\pi$
$23$	346.437	+	61.0862i	0.654890	+	0.115475i	0.163677	+	0.986514i	$0.447665\pi$
$24$		0			0
$25$	−117.428	+	98.5341i	−0.187885	+	0.157655i
$26$		0			0
$27$	543.640	+	485.691i	0.745735	+	0.666243i
$28$		0			0
$29$	421.220	+	501.991i	0.500856	+	0.596897i	0.955944	−	0.293549i	$-0.0948365\pi$
$29$	421.220	+	501.991i	0.500856	+	0.596897i	−0.455088	+	0.890447i	$0.650392\pi$
$30$		0			0
$31$	−18.7225	+	106.181i	−0.0194823	+	0.110490i	−0.992998	−	0.118130i	$-0.962310\pi$
$31$	−18.7225	+	106.181i	−0.0194823	+	0.110490i	0.973516	+	0.228620i	$0.0734212\pi$
$32$		0			0
$33$	−651.128	+	968.034i	−0.597914	+	0.888920i
$34$		0			0
$35$	75.4009	−	43.5328i	0.0615518	−	0.0355369i
$36$		0			0
$37$	−745.268	+	1290.84i	−0.544389	+	0.942909i	0.454256	+	0.890871i	$0.349905\pi$
$37$	−745.268	+	1290.84i	−0.544389	+	0.942909i	−0.998645	+	0.0520382i	$0.983428\pi$
$38$		0			0
$39$	−210.236	−	288.580i	−0.138222	−	0.189730i
$40$		0			0
$41$	1342.26	−	1599.64i	0.798486	−	0.951599i	−0.201123	−	0.979566i	$-0.564459\pi$
$41$	1342.26	−	1599.64i	0.798486	−	0.951599i	0.999609	+	0.0279673i	$0.00890341\pi$
$42$		0			0
$43$	−1405.92	−	511.712i	−0.760366	−	0.276750i	−0.0674048	−	0.997726i	$-0.521472\pi$
$43$	−1405.92	−	511.712i	−0.760366	−	0.276750i	−0.692961	+	0.720975i	$0.743694\pi$
$44$		0			0
$45$	−308.706	+	2238.54i	−0.152447	+	1.10545i
$46$		0			0
$47$	2360.91	−	416.292i	1.06877	−	0.188453i	0.388524	−	0.921439i	$-0.372985\pi$
$47$	2360.91	−	416.292i	1.06877	−	0.188453i	0.680244	+	0.732986i	$0.261874\pi$
$48$		0			0
$49$	2247.05	−	817.859i	0.935881	−	0.340633i
$50$		0			0
$51$	−539.100	−	519.257i	−0.207266	−	0.199637i
$52$		0			0
$53$		−	2343.67i		−	0.834344i	−0.908828	−	0.417172i	$-0.863021\pi$
$53$		−	2343.67i		−	0.834344i	0.908828	−	0.417172i	$-0.136979\pi$
$54$		0			0
$55$	−3616.32			−1.19548
$56$		0			0
$57$	162.730	+	564.751i	0.0500863	+	0.173823i
$58$		0			0
$59$	−978.002	−	2687.04i	−0.280954	−	0.771916i	−0.997249	−	0.0741183i	$-0.976386\pi$
$59$	−978.002	−	2687.04i	−0.280954	−	0.771916i	0.716295	−	0.697797i	$-0.245836\pi$
$60$		0			0
$61$	−1139.50	−	6462.44i	−0.306236	−	1.73675i	−0.617630	−	0.786469i	$-0.711907\pi$
$61$	−1139.50	−	6462.44i	−0.306236	−	1.73675i	0.311394	−	0.950281i	$-0.399204\pi$
$62$		0			0
$63$	77.4945	−	240.619i	0.0195249	−	0.0606245i
$64$		0			0
$65$	378.527	−	1040.00i	0.0895923	−	0.246153i
$66$		0			0
$67$	−6613.79	−	5549.63i	−1.47333	−	1.23627i	−0.912971	−	0.408024i	$-0.866218\pi$
$67$	−6613.79	−	5549.63i	−1.47333	−	1.23627i	−0.560361	−	0.828249i	$-0.689337\pi$
$68$		0			0
$69$	2893.98	+	1284.00i	0.607851	+	0.269691i
$70$		0			0
$71$	−426.072	−	245.993i	−0.0845214	−	0.0487985i	0.457144	−	0.889393i	$-0.348873\pi$
$71$	−426.072	−	245.993i	−0.0845214	−	0.0487985i	−0.541665	+	0.840594i	$0.682206\pi$
$72$		0			0
$73$	−2004.31	−	3471.57i	−0.376114	−	0.651449i	0.614379	−	0.789011i	$-0.289407\pi$
$73$	−2004.31	−	3471.57i	−0.376114	−	0.651449i	−0.990493	+	0.137562i	$0.956073\pi$
$74$		0			0
$75$	−1239.22	+	606.391i	−0.220305	+	0.107803i
$76$		0			0
$77$	398.403	+	70.2491i	0.0671956	+	0.0118484i
$78$		0			0
$79$	−5374.08	+	4509.39i	−0.861093	+	0.722543i	−0.962203	−	0.272332i	$-0.912205\pi$
$79$	−5374.08	+	4509.39i	−0.861093	+	0.722543i	0.101110	+	0.994875i	$0.467761\pi$
$80$		0			0
$81$	3696.94	+	5420.27i	0.563473	+	0.826135i
$82$		0			0
$83$	1605.97	+	1913.93i	0.233122	+	0.277823i	0.869905	−	0.493219i	$-0.164180\pi$
$83$	1605.97	+	1913.93i	0.233122	+	0.277823i	−0.636784	+	0.771043i	$0.719735\pi$
$84$		0			0
$85$	402.895	−	2284.93i	0.0557641	−	0.316254i
$86$		0			0
$87$	2592.24	+	5297.49i	0.342481	+	0.699893i
$88$		0			0
$89$	12923.1	−	7461.14i	1.63150	−	0.941945i	0.647864	−	0.761756i	$-0.275662\pi$
$89$	12923.1	−	7461.14i	1.63150	−	0.941945i	0.983632	−	0.180189i	$-0.0576710\pi$
$90$		0			0
$91$	−61.9041	+	107.221i	−0.00747543	+	0.0129478i
$92$		0			0
$93$	−393.538	+	886.984i	−0.0455009	+	0.102553i
$94$		0			0
$95$	−1171.04	+	1395.60i	−0.129756	+	0.154637i
$96$		0			0
$97$	15873.9	+	5777.63i	1.68710	+	0.614054i	0.994255	−	0.107034i	$-0.0341352\pi$
$97$	15873.9	+	5777.63i	1.68710	+	0.614054i	0.692844	+	0.721088i	$0.256357\pi$
$98$		0			0
$99$	−7784.68	+	7045.89i	−0.794274	+	0.718895i
$100$		0			0
$101$	3976.62	−	701.186i	0.389827	−	0.0687370i	0.0247004	−	0.999695i	$-0.492137\pi$
$101$	3976.62	−	701.186i	0.389827	−	0.0687370i	0.365127	+	0.930958i	$0.381026\pi$
$102$		0			0
$103$	−12081.1	+	4397.16i	−1.13876	+	0.414475i	−0.841465	−	0.540311i	$-0.818307\pi$
$103$	−12081.1	+	4397.16i	−1.13876	+	0.414475i	−0.297295	+	0.954786i	$0.596084\pi$
$104$		0			0
$105$	752.955	−	216.960i	0.0682952	−	0.0196789i
$106$		0			0
$107$		−	6203.55i		−	0.541842i	−0.962602	−	0.270921i	$-0.912672\pi$
$107$		−	6203.55i		−	0.541842i	0.962602	−	0.270921i	$-0.0873283\pi$
$108$		0			0
$109$	−6362.04			−0.535480			−0.267740	−	0.963491i	$-0.586277\pi$
$109$	−6362.04			−0.535480			−0.267740	+	0.963491i	$0.586277\pi$
$110$		0			0
$111$	−9306.24	+	9661.86i	−0.755315	+	0.784178i
$112$		0			0
$113$	2688.37	+	7386.23i	0.210539	+	0.578450i	0.999345	−	0.0361920i	$-0.0115228\pi$
$113$	2688.37	+	7386.23i	0.210539	+	0.578450i	−0.788806	+	0.614642i	$0.789301\pi$
$114$		0			0
$115$	1704.18	+	9664.86i	0.128860	+	0.730802i
$116$		0			0
$117$	−1211.45	−	2976.25i	−0.0884977	−	0.217419i
$118$		0			0
$119$	−88.7723	+	243.900i	−0.00626879	+	0.0172233i
$120$		0			0
$121$	−1656.33	−	1389.83i	−0.113130	−	0.0949271i
$122$		0			0
$123$	15190.1	−	11066.3i	1.00404	−	0.731460i
$124$		0			0
$125$	11396.6	+	6579.83i	0.729382	+	0.421109i
$126$		0			0
$127$	2979.36	+	5160.40i	0.184721	+	0.319946i	0.943482	−	0.331423i	$-0.107529\pi$
$127$	2979.36	+	5160.40i	0.184721	+	0.319946i	−0.758762	+	0.651368i	$0.774195\pi$
$128$		0			0
$129$	−11173.0	−	7515.26i	−0.671412	−	0.451611i
$130$		0			0
$131$	−22916.5	−	4040.80i	−1.33538	−	0.235464i	−0.540048	−	0.841634i	$-0.681594\pi$
$131$	−22916.5	−	4040.80i	−1.33538	−	0.235464i	−0.795335	+	0.606171i	$0.792705\pi$
$132$		0			0
$133$	156.122	−	131.002i	0.00882592	−	0.00740583i
$134$		0			0
$135$	−7545.39	+	18886.1i	−0.414013	+	1.03627i
$136$		0			0
$137$	8869.93	+	10570.8i	0.472584	+	0.563204i	0.948700	−	0.316179i	$-0.102400\pi$
$137$	8869.93	+	10570.8i	0.472584	+	0.563204i	−0.476115	+	0.879383i	$0.657956\pi$
$138$		0			0
$139$	3753.73	−	21288.5i	0.194283	−	1.10183i	−0.719154	−	0.694850i	$-0.755471\pi$
$139$	3753.73	−	21288.5i	0.194283	−	1.10183i	0.913437	−	0.406981i	$-0.133418\pi$
$140$		0			0
$141$	21525.3	+	1477.23i	1.08271	+	0.0743038i
$142$		0			0
$143$	4453.49	−	2571.23i	0.217785	−	0.125738i
$144$		0			0
$145$	−9140.77	+	15832.3i	−0.434757	+	0.753022i
$146$		0			0
$147$	21400.5	−	2277.26i	0.990352	−	0.105385i
$148$		0			0
$149$	−22455.8	+	26761.7i	−1.01148	+	1.20543i	−0.0329171	+	0.999458i	$0.510480\pi$
$149$	−22455.8	+	26761.7i	−1.01148	+	1.20543i	−0.978558	+	0.205971i	$0.933965\pi$
$150$		0			0
$151$	39773.2	+	14476.3i	1.74436	+	0.634896i	0.999478	−	0.0322972i	$-0.0102823\pi$
$151$	39773.2	+	14476.3i	1.74436	+	0.634896i	0.744885	+	0.667193i	$0.232505\pi$
$152$		0			0
$153$	−3584.57	−	5703.65i	−0.153128	−	0.243652i
$154$		0			0
$155$	−2962.21	+	522.318i	−0.123297	+	0.0217406i
$156$		0			0
$157$	15090.4	−	5492.45i	0.612211	−	0.222826i	−0.0172596	−	0.999851i	$-0.505494\pi$
$157$	15090.4	−	5492.45i	0.612211	−	0.222826i	0.629470	+	0.777025i	$0.283272\pi$
$158$		0			0
$159$	5076.40	−	20473.1i	0.200799	−	0.809820i
$160$		0			0
$161$		−	1097.86i		−	0.0423541i
$162$		0			0
$163$	−10729.0			−0.403815			−0.201908	−	0.979405i	$-0.564714\pi$
$163$	−10729.0			−0.403815			−0.201908	+	0.979405i	$0.564714\pi$
$164$		0			0
$165$	−31590.3	−	7832.97i	−1.16034	−	0.287712i
$166$		0			0
$167$	−9638.45	−	26481.4i	−0.345600	−	0.949529i	−0.983738	−	0.179607i	$-0.942517\pi$
$167$	−9638.45	−	26481.4i	−0.345600	−	0.949529i	0.638138	−	0.769922i	$-0.279705\pi$
$168$		0			0
$169$	−4686.28	−	26577.2i	−0.164080	−	0.930542i
$170$		0			0
$171$	198.271	+	5285.84i	0.00678060	+	0.180768i
$172$		0			0
$173$	−12892.4	+	35421.5i	−0.430765	+	1.18352i	0.514579	+	0.857443i	$0.327948\pi$
$173$	−12892.4	+	35421.5i	−0.430765	+	1.18352i	−0.945344	+	0.326074i	$0.894274\pi$
$174$		0			0
$175$	366.478	+	307.511i	0.0119666	+	0.0100412i
$176$		0			0
$177$	−2723.17	−	25590.9i	−0.0869216	−	0.816844i
$178$		0			0
$179$	−47834.6	−	27617.3i	−1.49292	−	0.861938i	−0.492953	−	0.870056i	$-0.664082\pi$
$179$	−47834.6	−	27617.3i	−1.49292	−	0.861938i	−0.999967	+	0.00811870i	$0.997416\pi$
$180$		0			0
$181$	5248.19	+	9090.13i	0.160196	+	0.277468i	0.934939	−	0.354809i	$-0.115454\pi$
$181$	5248.19	+	9090.13i	0.160196	+	0.277468i	−0.774743	+	0.632277i	$0.782121\pi$
$182$		0			0
$183$	4043.59	−	58920.7i	0.120744	−	1.75940i
$184$		0			0
$185$	−40951.1	−	7220.78i	−1.19653	−	0.210980i
$186$		0			0
$187$	8258.49	−	6929.70i	0.236166	−	0.198167i
$188$		0			0
$189$	1198.13	−	1934.06i	0.0335414	−	0.0541436i
$190$		0			0
$191$	25833.2	+	30786.8i	0.708127	+	0.843913i	0.993420	−	0.114527i	$-0.0365353\pi$
$191$	25833.2	+	30786.8i	0.708127	+	0.843913i	−0.285293	+	0.958440i	$0.592091\pi$
$192$		0			0
$193$	8845.50	−	50165.3i	0.237469	−	1.34676i	−0.599881	−	0.800089i	$-0.704785\pi$
$193$	8845.50	−	50165.3i	0.237469	−	1.34676i	0.837350	−	0.546667i	$-0.184104\pi$
$194$		0			0
$195$	5559.25	−	8264.95i	0.146200	−	0.217356i
$196$		0			0
$197$	24425.4	−	14102.0i	0.629376	−	0.363370i	−0.151135	−	0.988513i	$-0.548293\pi$
$197$	24425.4	−	14102.0i	0.629376	−	0.363370i	0.780510	+	0.625143i	$0.214959\pi$
$198$		0			0
$199$	−25117.8	+	43505.4i	−0.634273	+	1.09859i	0.352396	+	0.935851i	$0.385367\pi$
$199$	−25117.8	+	43505.4i	−0.634273	+	1.09859i	−0.986669	+	0.162742i	$0.947966\pi$
$200$		0			0
$201$	−45754.0	−	62804.1i	−1.13250	−	1.55452i
$202$		0			0
$203$	1314.57	−	1566.64i	0.0319001	−	0.0380170i
$204$		0			0
$205$	54742.5	+	19924.6i	1.30262	+	0.474114i
$206$		0			0
$207$	22499.1	+	17484.7i	0.525079	+	0.408054i
$208$		0			0
$209$	−8336.46	+	1469.94i	−0.190849	+	0.0336518i
$210$		0			0
$211$	17352.0	−	6315.62i	0.389749	−	0.141857i	−0.139710	−	0.990192i	$-0.544617\pi$
$211$	17352.0	−	6315.62i	0.389749	−	0.141857i	0.529459	+	0.848335i	$0.322395\pi$
$212$		0			0
$213$	−3189.12	−	3071.74i	−0.0702929	−	0.0677057i
$214$		0			0
$215$		−	41739.3i		−	0.902959i
$216$		0			0
$217$	336.487			0.00714577
$218$		0			0
$219$	−9989.17	−	34667.2i	−0.208277	−	0.722820i
$220$		0			0
$221$	1128.44	+	3100.35i	0.0231043	+	0.0634785i
$222$		0			0
$223$	11371.6	+	64491.7i	0.228672	+	1.29686i	0.855540	+	0.517737i	$0.173226\pi$
$223$	11371.6	+	64491.7i	0.228672	+	1.29686i	−0.626868	+	0.779126i	$0.715663\pi$
$224$		0			0
$225$	−12138.6	+	2612.96i	−0.239775	+	0.0516140i
$226$		0			0
$227$	−21863.2	+	60068.6i	−0.424289	+	1.16572i	0.524941	+	0.851139i	$0.324088\pi$
$227$	−21863.2	+	60068.6i	−0.424289	+	1.16572i	−0.949229	+	0.314585i	$0.898135\pi$
$228$		0			0
$229$	1269.42	+	1065.17i	0.0242067	+	0.0203118i	0.654811	−	0.755793i	$-0.272748\pi$
$229$	1269.42	+	1065.17i	0.0242067	+	0.0203118i	−0.630604	+	0.776105i	$0.717193\pi$
$230$		0			0
$231$	3328.07	+	1476.60i	0.0623690	+	0.0276719i
$232$		0			0
$233$	−18385.1	−	10614.6i	−0.338652	−	0.195521i	0.321024	−	0.947071i	$-0.395973\pi$
$233$	−18385.1	−	10614.6i	−0.338652	−	0.195521i	−0.659676	+	0.751550i	$0.729306\pi$
$234$		0			0
$235$	33440.2	+	57920.1i	0.605526	+	1.04880i
$236$		0			0
$237$	−56712.5	+	27751.3i	−1.00968	+	0.494069i
$238$		0			0
$239$	−18699.3	−	3297.20i	−0.327363	−	0.0577230i	0.00755117	−	0.999971i	$-0.497596\pi$
$239$	−18699.3	−	3297.20i	−0.327363	−	0.0577230i	−0.334915	+	0.942248i	$0.608707\pi$
$240$		0			0
$241$	25092.7	−	21055.3i	0.432030	−	0.362516i	−0.400687	−	0.916215i	$-0.631228\pi$
$241$	25092.7	−	21055.3i	0.432030	−	0.362516i	0.832717	+	0.553699i	$0.186784\pi$
$242$		0			0
$243$	20554.2	+	55356.2i	0.348088	+	0.937462i
$244$		0			0
$245$	42881.1	+	51103.7i	0.714387	+	0.851373i
$246$		0			0
$247$	449.862	−	2551.29i	0.00737369	−	0.0418183i
$248$		0			0
$249$	9883.37	+	20197.6i	0.159407	+	0.325762i
$250$		0			0
$251$	27978.7	−	16153.5i	0.444099	−	0.256401i	−0.261236	−	0.965275i	$-0.584130\pi$
$251$	27978.7	−	16153.5i	0.444099	−	0.256401i	0.705335	+	0.708874i	$0.250797\pi$
$252$		0			0
$253$	−22800.2	+	39491.1i	−0.356203	+	0.616962i
$254$		0			0
$255$	8468.66	−	19087.3i	0.130237	−	0.293538i
$256$		0			0
$257$	−45702.6	+	54466.3i	−0.691950	+	0.824634i	−0.991590	−	0.129418i	$-0.958689\pi$
$257$	−45702.6	+	54466.3i	−0.691950	+	0.824634i	0.299640	+	0.954052i	$0.403134\pi$
$258$		0			0
$259$	4371.23	+	1591.00i	0.0651634	+	0.0237175i
$260$		0			0
$261$	11170.1	+	51890.9i	0.163974	+	0.761745i
$262$		0			0
$263$	45451.4	−	8014.31i	0.657107	−	0.115866i	0.164854	−	0.986318i	$-0.447285\pi$
$263$	45451.4	−	8014.31i	0.657107	−	0.115866i	0.492253	+	0.870452i	$0.336174\pi$
$264$		0			0
$265$	61440.4	−	22362.5i	0.874907	−	0.318440i
$266$		0			0
$267$	129050.	−	37185.1i	1.81024	−	0.521611i
$268$		0			0
$269$			62867.5i			0.868804i	0.900719	+	0.434402i	$0.143040\pi$
$269$			62867.5i			0.868804i	−0.900719	+	0.434402i	$0.856960\pi$
$270$		0			0
$271$	−89938.3			−1.22463			−0.612317	−	0.790612i	$-0.709762\pi$
$271$	−89938.3			−1.22463			−0.612317	+	0.790612i	$0.709762\pi$
$272$		0			0
$273$	−773.002	+	802.541i	−0.0103718	+	0.0107682i
$274$		0			0
$275$	−6796.21	−	18672.4i	−0.0898672	−	0.246908i
$276$		0			0
$277$	2359.61	+	13382.0i	0.0307525	+	0.174406i	0.996316	−	0.0857630i	$-0.0273328\pi$
$277$	2359.61	+	13382.0i	0.0307525	+	0.174406i	−0.965563	+	0.260169i	$0.916222\pi$
$278$		0			0
$279$	−5358.95	+	6895.82i	−0.0688448	+	0.0885885i
$280$		0			0
$281$	−12110.1	+	33272.1i	−0.153368	+	0.421374i	−0.992453	−	0.122626i	$-0.960869\pi$
$281$	−12110.1	+	33272.1i	−0.153368	+	0.421374i	0.839085	+	0.544000i	$0.183091\pi$
$282$		0			0
$283$	−87306.7	−	73259.0i	−1.09012	−	0.914720i	−0.0933993	−	0.995629i	$-0.529773\pi$
$283$	−87306.7	−	73259.0i	−1.09012	−	0.914720i	−0.996722	+	0.0809087i	$0.974218\pi$
$284$		0			0
$285$	−13252.5	+	9654.70i	−0.163158	+	0.118864i
$286$		0			0
$287$	−5643.82	−	3258.46i	−0.0685187	−	0.0395593i
$288$		0			0
$289$	−38302.1	−	66341.2i	−0.458593	−	0.794306i
$290$		0			0
$291$	126152.	+	84853.3i	1.48973	+	1.00204i
$292$		0			0
$293$	−70486.8	−	12428.7i	−0.821056	−	0.144774i	−0.252687	−	0.967548i	$-0.581314\pi$
$293$	−70486.8	−	12428.7i	−0.821056	−	0.144774i	−0.568369	+	0.822774i	$0.692425\pi$
$294$		0			0
$295$	61110.1	−	51277.5i	0.702214	−	0.589227i
$296$		0			0
$297$	−83264.2	+	44687.5i	−0.943943	+	0.506609i
$298$		0			0
$299$	−8970.46	−	10690.6i	−0.100340	−	0.119580i
$300$		0			0
$301$	−810.809	+	4598.33i	−0.00894923	+	0.0507536i
$302$		0			0
$303$	36256.5	+	2488.20i	0.394912	+	0.0271019i
$304$		0			0
$305$	158543.	−	91534.9i	1.70431	−	0.983982i
$306$		0			0
$307$	−8937.59	+	15480.4i	−0.0948296	+	0.164250i	−0.909537	−	0.415622i	$-0.863564\pi$
$307$	−8937.59	+	15480.4i	−0.0948296	+	0.164250i	0.814708	+	0.579872i	$0.196897\pi$
$308$		0			0
$309$	−115058.	+	12243.5i	−1.20504	+	0.128230i
$310$		0			0
$311$	−3233.99	+	3854.12i	−0.0334363	+	0.0398478i	−0.782503	−	0.622646i	$-0.786058\pi$
$311$	−3233.99	+	3854.12i	−0.0334363	+	0.0398478i	0.749067	+	0.662494i	$0.230502\pi$
$312$		0			0
$313$	−47171.3	−	17169.0i	−0.481493	−	0.175249i	0.0898588	−	0.995955i	$-0.471358\pi$
$313$	−47171.3	−	17169.0i	−0.481493	−	0.175249i	−0.571351	+	0.820706i	$0.693581\pi$
$314$		0			0
$315$	7047.35	−	264.346i	0.0710239	−	0.00266410i
$316$		0			0
$317$	43193.0	−	7616.09i	0.429828	−	0.0757903i	0.0454512	−	0.998967i	$-0.485527\pi$
$317$	43193.0	−	7616.09i	0.429828	−	0.0757903i	0.384377	+	0.923176i	$0.374416\pi$
$318$		0			0
$319$	−79822.1	+	29052.9i	−0.784408	+	0.285501i
$320$		0			0
$321$	13436.9	−	54190.9i	0.130404	−	0.525916i
$322$		0			0
$323$		−	5431.07i		−	0.0520571i
$324$		0			0
$325$	6081.26			0.0575740
$326$		0			0
$327$	−55575.4	−	13780.2i	−0.519741	−	0.128873i
$328$		0			0
$329$	−2558.90	−	7030.53i	−0.0236408	−	0.0649526i
$330$		0			0
$331$	24127.4	+	136833.i	0.220219	+	1.24892i	0.871618	+	0.490186i	$0.163071\pi$
$331$	24127.4	+	136833.i	0.220219	+	1.24892i	−0.651399	+	0.758735i	$0.725818\pi$
$332$		0			0
$333$	−102222.	+	64243.5i	−0.921841	+	0.579350i
$334$		0			0
$335$	82379.5	−	226336.i	0.734057	−	2.01680i
$336$		0			0
$337$	−103874.	−	87160.2i	−0.914629	−	0.767465i	0.0583648	−	0.998295i	$-0.481411\pi$
$337$	−103874.	−	87160.2i	−0.914629	−	0.767465i	−0.972994	+	0.230830i	$0.925856\pi$
$338$		0			0
$339$	7485.54	+	70345.2i	0.0651364	+	0.612118i
$340$		0			0
$341$	−12103.8	−	6988.11i	−0.104091	−	0.0600967i
$342$		0			0
$343$	−7478.00	−	12952.3i	−0.0635619	−	0.110092i
$344$		0			0
$345$	−6047.36	+	88118.3i	−0.0508075	+	0.740334i
$346$		0			0
$347$	23696.0	+	4178.25i	0.196796	+	0.0347005i	0.271177	−	0.962529i	$-0.412587\pi$
$347$	23696.0	+	4178.25i	0.196796	+	0.0347005i	−0.0743812	+	0.997230i	$0.523698\pi$
$348$		0			0
$349$	39974.2	−	33542.4i	0.328193	−	0.275387i	−0.463770	−	0.885956i	$-0.653504\pi$
$349$	39974.2	−	33542.4i	0.328193	−	0.275387i	0.791963	+	0.610569i	$0.209059\pi$
$350$		0			0
$351$	−4135.97	−	28623.0i	−0.0335709	−	0.232327i
$352$		0			0
$353$	6742.14	+	8034.97i	0.0541064	+	0.0644814i	0.792419	−	0.609977i	$-0.208821\pi$
$353$	6742.14	+	8034.97i	0.0541064	+	0.0644814i	−0.738313	+	0.674459i	$0.764377\pi$
$354$		0			0
$355$	2383.39	−	13516.8i	0.0189120	−	0.107255i
$356$		0			0
$357$	−1303.76	+	1938.30i	−0.0102296	+	0.0152084i
$358$		0			0
$359$	193420.	−	111671.i	1.50077	−	0.866467i	0.500766	−	0.865583i	$-0.333052\pi$
$359$	193420.	−	111671.i	1.50077	−	0.866467i	1.00000		0.000884863i	$-0.000281661\pi$
$360$		0			0
$361$	63028.2	−	109168.i	0.483638	−	0.837686i
$362$		0			0
$363$	−11458.5	−	15728.4i	−0.0869587	−	0.119364i
$364$		0			0
$365$	71884.3	−	85668.4i	0.539571	−	0.643035i
$366$		0			0
$367$	−114748.	−	41764.8i	−0.851946	−	0.310083i	−0.121113	−	0.992639i	$-0.538646\pi$
$367$	−114748.	−	41764.8i	−0.851946	−	0.310083i	−0.730834	+	0.682556i	$0.760868\pi$
$368$		0			0
$369$	156662.	−	63767.2i	1.15056	−	0.468322i
$370$		0			0
$371$	−7203.16	+	1270.11i	−0.0523329	+	0.00922771i
$372$		0			0
$373$	−85335.1	+	31059.4i	−0.613352	+	0.223242i	−0.629969	−	0.776620i	$-0.716933\pi$
$373$	−85335.1	+	31059.4i	−0.613352	+	0.223242i	0.0166172	+	0.999862i	$0.494710\pi$
$374$		0			0
$375$	85302.7	+	82163.0i	0.606597	+	0.584270i
$376$		0			0
$377$		−	25996.6i		−	0.182908i
$378$		0			0
$379$	−122906.			−0.855645			−0.427823	−	0.903863i	$-0.640719\pi$
$379$	−122906.			−0.855645			−0.427823	+	0.903863i	$0.640719\pi$
$380$		0			0
$381$	14848.7	+	51531.9i	0.102291	+	0.354998i
$382$		0			0
$383$	−1901.62	−	5224.66i	−0.0129636	−	0.0356173i	0.933042	−	0.359767i	$-0.117144\pi$
$383$	−1901.62	−	5224.66i	−0.0129636	−	0.0356173i	−0.946006	+	0.324150i	$0.894922\pi$
$384$		0			0
$385$	1959.80	+	11114.6i	0.0132218	+	0.0749846i
$386$		0			0
$387$	−81323.0	−	89850.0i	−0.542990	−	0.599924i
$388$		0			0
$389$	−45262.3	+	124357.i	−0.299114	+	0.821810i	0.695534	+	0.718493i	$0.255168\pi$
$389$	−45262.3	+	124357.i	−0.299114	+	0.821810i	−0.994649	+	0.103317i	$0.967054\pi$
$390$		0			0
$391$	−22411.8	−	18805.8i	−0.146597	−	0.123009i
$392$		0			0
$393$	−191434.	−	84935.5i	−1.23946	−	0.549926i
$394$		0			0
$395$	−169493.	−	97856.9i	−1.08632	−	0.627187i
$396$		0			0
$397$	−101449.	−	175715.i	−0.643676	−	1.11488i	−0.984605	−	0.174791i	$-0.944075\pi$
$397$	−101449.	−	175715.i	−0.643676	−	1.11488i	0.340929	−	0.940089i	$-0.389258\pi$
$398$		0			0
$399$	1647.55	−	806.200i	0.0103488	−	0.00506404i
$400$		0			0
$401$	168219.	+	29661.5i	1.04613	+	0.184461i	0.670195	−	0.742185i	$-0.266211\pi$
$401$	168219.	+	29661.5i	1.04613	+	0.184461i	0.375935	+	0.926646i	$0.377322\pi$
$402$		0			0
$403$	3276.59	−	2749.39i	0.0201749	−	0.0169288i
$404$		0			0
$405$	−106820.	+	148635.i	−0.651241	+	0.906174i
$406$		0			0
$407$	−124196.	−	148011.i	−0.749751	−	0.893519i
$408$		0			0
$409$	12770.1	−	72422.8i	0.0763391	−	0.432941i	−0.922552	−	0.385872i	$-0.873901\pi$
$409$	12770.1	−	72422.8i	0.0763391	−	0.432941i	0.998892	−	0.0470691i	$-0.0149881\pi$
$410$		0			0
$411$	54586.7	+	111553.i	0.323149	+	0.660386i
$412$		0			0
$413$	−7728.47	+	4462.04i	−0.0453099	+	0.0261597i
$414$		0			0
$415$	−34850.8	+	60363.3i	−0.202356	+	0.350491i
$416$		0			0
$417$	78901.6	−	177834.i	0.453747	−	1.02269i
$418$		0			0
$419$	−14771.8	+	17604.3i	−0.0841404	+	0.100275i	−0.806472	−	0.591272i	$-0.798626\pi$
$419$	−14771.8	+	17604.3i	−0.0841404	+	0.100275i	0.722332	+	0.691547i	$0.243070\pi$
$420$		0			0
$421$	−143751.	−	52321.0i	−0.811046	−	0.295197i	−0.0969904	−	0.995285i	$-0.530922\pi$
$421$	−143751.	−	52321.0i	−0.811046	−	0.295197i	−0.714056	+	0.700089i	$0.753144\pi$
$422$		0			0
$423$	184834.	+	59528.3i	1.03300	+	0.332692i
$424$		0			0
$425$	12555.1	−	2213.81i	0.0695094	−	0.0122564i
$426$		0			0
$427$	−19244.5	+	7004.42i	−0.105548	+	0.0384164i
$428$		0			0
$429$	44472.6	−	12814.6i	0.241645	−	0.0696289i
$430$		0			0
$431$			350782.i			1.88835i	0.329444	+	0.944175i	$0.393139\pi$
$431$			350782.i			1.88835i	−0.329444	+	0.944175i	$0.606861\pi$
$432$		0			0
$433$	262879.			1.40210			0.701051	−	0.713111i	$-0.252715\pi$
$433$	262879.			1.40210			0.701051	+	0.713111i	$0.252715\pi$
$434$		0			0
$435$	−114142.	+	118503.i	−0.603206	+	0.626257i
$436$		0			0
$437$	7857.04	+	21587.0i	0.0411430	+	0.113039i
$438$		0			0
$439$	−29423.4	−	166868.i	−0.152674	−	0.865855i	−0.960882	−	0.276958i	$-0.910674\pi$
$439$	−29423.4	−	166868.i	−0.152674	−	0.865855i	0.808208	−	0.588897i	$-0.200438\pi$
$440$		0			0
$441$	191876.	+	26460.7i	0.986606	+	0.136058i
$442$		0			0
$443$	15829.7	−	43491.8i	0.0806614	−	0.221615i	−0.892806	−	0.450442i	$-0.851267\pi$
$443$	15829.7	−	43491.8i	0.0806614	−	0.221615i	0.973467	+	0.228826i	$0.0734888\pi$
$444$		0			0
$445$	318905.	+	267593.i	1.61043	+	1.35131i
$446$		0			0
$447$	−254128.	+	185137.i	−1.27185	+	0.926570i
$448$		0			0
$449$	−63267.2	−	36527.3i	−0.313824	−	0.181186i	0.334813	−	0.942285i	$-0.391327\pi$
$449$	−63267.2	−	36527.3i	−0.313824	−	0.181186i	−0.648636	+	0.761099i	$0.724660\pi$
$450$		0			0
$451$	135342.	+	234420.i	0.665396	+	1.15250i
$452$		0			0
$453$	316082.	+	212606.i	1.54029	+	1.03605i
$454$		0			0
$455$	−3401.51	−	599.778i	−0.0164304	−	0.00289713i
$456$		0			0
$457$	−98911.9	+	82997.0i	−0.473605	+	0.397402i	−0.848108	−	0.529824i	$-0.822258\pi$
$457$	−98911.9	+	82997.0i	−0.473605	+	0.397402i	0.374503	+	0.927226i	$0.377813\pi$
$458$		0			0
$459$	−18958.8	−	57588.2i	−0.0899882	−	0.273343i
$460$		0			0
$461$	228366.	+	272156.i	1.07456	+	1.28061i	0.957796	+	0.287450i	$0.0928074\pi$
$461$	228366.	+	272156.i	1.07456	+	1.28061i	0.116763	+	0.993160i	$0.462748\pi$
$462$		0			0
$463$	−5698.10	+	32315.6i	−0.0265808	+	0.150747i	−0.995210	−	0.0977652i	$-0.968831\pi$
$463$	−5698.10	+	32315.6i	−0.0265808	+	0.150747i	0.968629	+	0.248512i	$0.0799417\pi$
$464$		0			0
$465$	−27007.7	−	1853.48i	−0.124905	−	0.00857198i
$466$		0			0
$467$	−154413.	+	89150.2i	−0.708026	+	0.408779i	−0.810330	−	0.585974i	$-0.800712\pi$
$467$	−154413.	+	89150.2i	−0.708026	+	0.408779i	0.102304	+	0.994753i	$0.467379\pi$
$468$		0			0
$469$	−13472.3	+	23334.7i	−0.0612485	+	0.106086i
$470$		0			0
$471$	143718.	−	15293.3i	0.647843	−	0.0689380i
$472$		0			0
$473$	124663.	−	148567.i	0.557204	−	0.664050i
$474$		0			0
$475$	−9406.74	−	3423.77i	−0.0416919	−	0.0151746i
$476$		0			0
$477$	88689.5	−	167846.i	0.389794	−	0.737692i
$478$		0			0
$479$	−100124.	+	17654.6i	−0.436383	+	0.0769462i	−0.387524	−	0.921860i	$-0.626670\pi$
$479$	−100124.	+	17654.6i	−0.436383	+	0.0769462i	−0.0488591	+	0.998806i	$0.515559\pi$
$480$		0			0
$481$	55565.2	−	20224.1i	0.240167	−	0.0874135i
$482$		0			0
$483$	2377.97	−	9590.34i	0.0101933	−	0.0411093i
$484$		0			0
$485$			471270.i			2.00348i
$486$		0			0
$487$	−126703.			−0.534229			−0.267115	−	0.963665i	$-0.586070\pi$
$487$	−126703.			−0.534229			−0.267115	+	0.963665i	$0.586070\pi$
$488$		0			0
$489$	−93722.6	−	23239.0i	−0.391946	−	0.0971851i
$490$		0			0
$491$	38725.2	+	106397.i	0.160632	+	0.441332i	0.993732	−	0.111790i	$-0.0356585\pi$
$491$	38725.2	+	106397.i	0.160632	+	0.441332i	−0.833100	+	0.553122i	$0.813436\pi$
$492$		0			0
$493$	−9463.74	−	53671.5i	−0.0389376	−	0.220826i
$494$		0			0
$495$	−258990.	−	136849.i	−1.05699	−	0.558512i
$496$		0			0
$497$	−525.145	+	1442.82i	−0.00212602	+	0.00584118i
$498$		0			0
$499$	293152.	+	245983.i	1.17731	+	0.987881i	0.999993	+	0.00372771i	$0.00118657\pi$
$499$	293152.	+	245983.i	1.17731	+	0.987881i	0.177318	+	0.984154i	$0.443258\pi$
$500$		0			0
$501$	−26837.5	−	252204.i	−0.106922	−	1.00479i
$502$		0			0
$503$	−357988.	−	206684.i	−1.41492	−	0.816905i	−0.419074	−	0.907952i	$-0.637645\pi$
$503$	−357988.	−	206684.i	−1.41492	−	0.816905i	−0.995847	+	0.0910468i	$0.970979\pi$
$504$		0			0
$505$	56325.4	+	97558.5i	0.220862	+	0.382545i
$506$		0			0
$507$	16629.5	−	242315.i	0.0646940	−	0.942680i
$508$		0			0
$509$	40339.3	+	7112.92i	0.155702	+	0.0274544i	0.250956	−	0.967999i	$-0.419255\pi$
$509$	40339.3	+	7112.92i	0.155702	+	0.0274544i	−0.0952539	+	0.995453i	$0.530366\pi$
$510$		0			0
$511$	−9583.50	+	8041.51i	−0.0367014	+	0.0307961i
$512$		0			0
$513$	−9717.16	+	46603.8i	−0.0369237	+	0.177087i
$514$		0			0
$515$	−230547.	−	274755.i	−0.869251	−	1.03593i
$516$		0			0
$517$	−53962.7	+	306038.i	−0.201889	+	1.14497i
$518$		0			0
$519$	−189344.	+	281498.i	−0.702937	+	1.04506i
$520$		0			0
$521$	171490.	−	99009.5i	0.631775	−	0.364755i	−0.149664	−	0.988737i	$-0.547819\pi$
$521$	171490.	−	99009.5i	0.631775	−	0.364755i	0.781439	+	0.623982i	$0.214486\pi$
$522$		0			0
$523$	54197.2	−	93872.3i	0.198141	−	0.343190i	−0.749785	−	0.661682i	$-0.769843\pi$
$523$	54197.2	−	93872.3i	0.198141	−	0.343190i	0.947926	+	0.318492i	$0.103176\pi$
$524$		0			0
$525$	2535.29	+	3480.05i	0.00919831	+	0.0126260i
$526$		0			0
$527$	5763.84	−	6869.08i	0.0207535	−	0.0247330i
$528$		0			0
$529$	−146677.	−	53386.2i	−0.524146	−	0.190773i
$530$		0			0
$531$	31641.9	−	229447.i	0.112221	−	0.813754i
$532$		0			0
$533$	−81581.8	+	14385.1i	−0.287170	+	0.0506358i
$534$		0			0
$535$	162629.	−	59192.0i	0.568185	−	0.206803i
$536$		0			0
$537$	−358039.	−	344860.i	−1.24160	−	1.19590i
$538$		0			0
$539$			309972.i			1.06695i
$540$		0			0
$541$	570303.			1.94855			0.974274	−	0.225368i	$-0.0723584\pi$
$541$	570303.			1.94855			0.974274	+	0.225368i	$0.0723584\pi$
$542$		0			0
$543$	26156.1	+	90774.2i	0.0887103	+	0.307867i
$544$		0			0
$545$	−60704.3	−	166784.i	−0.204374	−	0.561514i
$546$		0			0
$547$	−30332.4	−	172024.i	−0.101375	−	0.574929i	−0.992606	−	0.121379i	$-0.961268\pi$
$547$	−30332.4	−	172024.i	−0.101375	−	0.574929i	0.891231	−	0.453550i	$-0.149843\pi$
$548$		0			0
$549$	162945.	−	505941.i	0.540626	−	1.67863i
$550$		0			0
$551$	−14636.2	+	40212.6i	−0.0482086	+	0.132452i
$552$		0			0
$553$	16771.8	+	14073.2i	0.0548440	+	0.0460196i
$554$		0			0
$555$	−342087.	−	151777.i	−1.11058	−	0.492743i
$556$		0			0
$557$	−381382.	−	220191.i	−1.22928	−	0.709723i	−0.262398	−	0.964960i	$-0.584513\pi$
$557$	−381382.	−	220191.i	−1.22928	−	0.709723i	−0.966879	+	0.255237i	$0.917847\pi$
$558$		0			0
$559$	29676.9	+	51401.8i	0.0949717	+	0.164496i
$560$		0			0
$561$	87151.5	−	42646.2i	0.276917	−	0.135505i
$562$		0			0
$563$	438021.	+	77234.9i	1.38190	+	0.243667i	0.814687	−	0.579901i	$-0.196909\pi$
$563$	438021.	+	77234.9i	1.38190	+	0.243667i	0.567218	+	0.823568i	$0.308020\pi$
$564$		0			0
$565$	−167982.	+	140953.i	−0.526217	+	0.441549i
$566$		0			0
$567$	14655.4	−	14299.8i	0.0455861	−	0.0444799i
$568$		0			0
$569$	−388030.	−	462436.i	−1.19851	−	1.42833i	−0.876374	−	0.481632i	$-0.840044\pi$
$569$	−388030.	−	462436.i	−1.19851	−	1.42833i	−0.322134	−	0.946694i	$-0.604400\pi$
$570$		0			0
$571$	−31098.8	+	176370.i	−0.0953831	+	0.540944i	0.899246	+	0.437443i	$0.144116\pi$
$571$	−31098.8	+	176370.i	−0.0953831	+	0.540944i	−0.994629	+	0.103502i	$0.966995\pi$
$572$		0			0
$573$	158981.	+	324892.i	0.484212	+	0.989532i
$574$		0			0
$575$	−46700.6	+	26962.6i	−0.141249	+	0.0815504i
$576$		0			0
$577$	283964.	−	491839.i	0.852925	−	1.47731i	−0.0256315	−	0.999671i	$-0.508160\pi$
$577$	283964.	−	491839.i	0.852925	−	1.47731i	0.878557	−	0.477638i	$-0.158507\pi$
$578$		0			0
$579$	185928.	−	419058.i	0.554610	−	1.25002i
$580$		0			0
$581$	5012.03	−	5973.10i	0.0148478	−	0.0176949i
$582$		0			0
$583$	285482.	+	103907.i	0.839926	+	0.305708i
$584$		0			0
$585$	66464.6	−	60156.9i	0.194213	−	0.175782i
$586$		0			0
$587$	−295856.	+	52167.4i	−0.858626	+	0.151399i	−0.585592	−	0.810606i	$-0.699138\pi$
$587$	−295856.	+	52167.4i	−0.858626	+	0.151399i	−0.273033	+	0.962005i	$0.588027\pi$
$588$		0			0
$589$	−6616.28	+	2408.13i	−0.0190714	+	0.00694143i
$590$		0			0
$591$	243913.	−	70282.3i	0.698328	−	0.201220i
$592$		0			0
$593$			452524.i			1.28686i	0.765503	+	0.643432i	$0.222490\pi$
$593$			452524.i			1.28686i	−0.765503	+	0.643432i	$0.777510\pi$
$594$		0			0
$595$	−7240.97			−0.0204533
$596$		0			0
$597$	−313649.	+	325635.i	−0.880026	+	0.913654i
$598$		0			0
$599$	222816.	+	612183.i	0.621003	+	1.70619i	0.704520	+	0.709684i	$0.251162\pi$
$599$	222816.	+	612183.i	0.621003	+	1.70619i	−0.0835172	+	0.996506i	$0.526615\pi$
$600$		0			0
$601$	22312.8	+	126542.i	0.0617739	+	0.350337i	0.999991	+	0.00428948i	$0.00136539\pi$
$601$	22312.8	+	126542.i	0.0617739	+	0.350337i	−0.938217	+	0.346048i	$0.887524\pi$
$602$		0			0
$603$	−263649.	−	647727.i	−0.725089	−	1.78138i
$604$		0			0
$605$	20630.8	−	56682.7i	0.0563645	−	0.154860i
$606$		0			0
$607$	−304661.	−	255641.i	−0.826875	−	0.693831i	0.127696	−	0.991813i	$-0.459242\pi$
$607$	−304661.	−	255641.i	−0.826875	−	0.693831i	−0.954571	+	0.297983i	$0.903686\pi$
$608$		0			0
$609$	14876.8	−	10838.0i	0.0401119	−	0.0292223i
$610$		0			0
$611$	−82363.0	−	47552.3i	−0.220623	−	0.127376i
$612$		0			0
$613$	−267072.	−	462583.i	−0.710736	−	1.23103i	−0.964581	−	0.263785i	$-0.915029\pi$
$613$	−267072.	−	462583.i	−0.710736	−	1.23103i	0.253846	−	0.967245i	$-0.418304\pi$
$614$		0			0
$615$	435045.	+	292624.i	1.15023	+	0.773676i
$616$		0			0
$617$	−528258.	−	93146.1i	−1.38764	−	0.244678i	−0.570582	−	0.821240i	$-0.693283\pi$
$617$	−528258.	−	93146.1i	−1.38764	−	0.244678i	−0.817053	+	0.576563i	$0.804394\pi$
$618$		0			0
$619$	−71460.3	+	59962.3i	−0.186502	+	0.156494i	−0.731258	−	0.682101i	$-0.761067\pi$
$619$	−71460.3	+	59962.3i	−0.186502	+	0.156494i	0.544756	+	0.838594i	$0.316622\pi$
$620$		0			0
$621$	158668.	+	201470.i	0.411440	+	0.522430i
$622$		0			0
$623$	−29934.9	−	35675.0i	−0.0771262	−	0.0919154i
$624$		0			0
$625$	−80387.6	+	455901.i	−0.205792	+	1.16711i
$626$		0			0
$627$	−76006.8	−	5216.17i	−0.193338	−	0.0132684i
$628$		0			0
$629$	107355.	−	61981.7i	0.271346	−	0.156661i
$630$		0			0
$631$	257879.	−	446660.i	0.647676	−	1.12181i	−0.336001	−	0.941862i	$-0.609074\pi$
$631$	257879.	−	446660.i	0.647676	−	1.12181i	0.983677	−	0.179946i	$-0.0575923\pi$
$632$		0			0
$633$	165258.	−	17585.3i	0.412434	−	0.0438877i
$634$		0			0
$635$	−106854.	+	127344.i	−0.264999	+	0.315814i
$636$		0			0
$637$	−89142.9	−	32445.4i	−0.219689	−	0.0799602i
$638$		0			0
$639$	−21205.1	−	33740.7i	−0.0519323	−	0.0826329i
$640$		0			0
$641$	−323390.	+	57022.3i	−0.787064	+	0.138781i	−0.552713	−	0.833371i	$-0.686408\pi$
$641$	−323390.	+	57022.3i	−0.787064	+	0.138781i	−0.234351	+	0.972152i	$0.575296\pi$
$642$		0			0
$643$	−122155.	+	44460.8i	−0.295454	+	0.107536i	−0.485494	−	0.874240i	$-0.661360\pi$
$643$	−122155.	+	44460.8i	−0.295454	+	0.107536i	0.190040	+	0.981776i	$0.439138\pi$
$644$		0			0
$645$	90407.4	−	364612.i	0.217313	−	0.876419i
$646$		0			0
$647$			667031.i			1.59345i	0.604345	+	0.796723i	$0.293435\pi$
$647$			667031.i			1.59345i	−0.604345	+	0.796723i	$0.706565\pi$
$648$		0			0
$649$	370667.			0.880024
$650$		0			0
$651$	2939.37	+	728.833i	0.00693574	+	0.00171975i
$652$		0			0
$653$	−129771.	−	356543.i	−0.304335	−	0.836153i	−0.993734	−	0.111771i	$-0.964348\pi$
$653$	−129771.	−	356543.i	−0.304335	−	0.836153i	0.689399	−	0.724382i	$-0.257875\pi$
$654$		0			0
$655$	−112730.	−	639322.i	−0.262758	−	1.49017i
$656$		0			0
$657$	−12170.9	−	324470.i	−0.0281962	−	0.751700i
$658$		0			0
$659$	228126.	−	626770.i	0.525295	−	1.44324i	−0.339257	−	0.940694i	$-0.610176\pi$
$659$	228126.	−	626770.i	0.525295	−	1.44324i	0.864552	−	0.502543i	$-0.167602\pi$
$660$		0			0
$661$	254247.	+	213339.i	0.581906	+	0.488277i	0.885572	−	0.464501i	$-0.153766\pi$
$661$	254247.	+	213339.i	0.581906	+	0.488277i	−0.303666	+	0.952778i	$0.598211\pi$
$662$		0			0
$663$	3142.04	+	29527.2i	0.00714800	+	0.0671731i
$664$		0			0
$665$	4923.92	+	2842.83i	0.0111344	+	0.00642846i
$666$		0			0
$667$	115262.	+	199639.i	0.259079	+	0.448738i
$668$		0			0
$669$	−40352.8	+	587996.i	−0.0901617	+	1.31378i
$670$		0			0
$671$	837708.	+	147711.i	1.86058	+	0.328070i
$672$		0			0
$673$	−558140.	+	468335.i	−1.23229	+	1.03401i	−0.234203	+	0.972188i	$0.575248\pi$
$673$	−558140.	+	468335.i	−1.23229	+	1.03401i	−0.998087	+	0.0618264i	$0.980307\pi$
$674$		0			0
$675$	−111696.	−	3466.81i	−0.245149	−	0.00760891i
$676$		0			0
$677$	−148547.	−	177031.i	−0.324105	−	0.386254i	0.579248	−	0.815152i	$-0.303346\pi$
$677$	−148547.	−	177031.i	−0.324105	−	0.386254i	−0.903353	+	0.428898i	$0.858902\pi$
$678$		0			0
$679$	9154.69	−	51918.8i	0.0198566	−	0.112612i
$680$		0			0
$681$	−321094.	+	477372.i	−0.692369	+	1.02935i
$682$		0			0
$683$	192239.	−	110989.i	0.412098	−	0.237925i	−0.279593	−	0.960119i	$-0.590199\pi$
$683$	192239.	−	110989.i	0.412098	−	0.237925i	0.691691	+	0.722194i	$0.256866\pi$
$684$		0			0
$685$	−192484.	+	333392.i	−0.410216	+	0.710516i
$686$		0			0
$687$	8781.82	+	12054.3i	0.0186068	+	0.0255405i
$688$		0			0
$689$	−59763.8	+	71223.8i	−0.125893	+	0.150033i
$690$		0			0
$691$	660421.	+	240374.i	1.38314	+	0.503420i	0.923127	−	0.384494i	$-0.125624\pi$
$691$	660421.	+	240374.i	1.38314	+	0.503420i	0.460008	+	0.887915i	$0.347846\pi$
$692$		0			0
$693$	25874.0	+	20107.4i	0.0538761	+	0.0418688i
$694$		0			0
$695$	593903.	−	104721.i	1.22955	−	0.216803i
$696$		0			0
$697$	−163194.	+	59397.8i	−0.335922	+	0.122266i
$698$		0			0
$699$	−137611.	−	132546.i	−0.281643	−	0.271277i
$700$		0			0
$701$		−	16939.1i		−	0.0344710i	−0.999851	−	0.0172355i	$-0.994514\pi$
$701$		−	16939.1i		−	0.0344710i	0.999851	−	0.0172355i	$-0.00548649\pi$
$702$		0			0
$703$	−97336.8			−0.196955
$704$		0			0
$705$	166660.	+	578391.i	0.335316	+	1.16371i
$706$		0			0
$707$	−4310.12	−	11842.0i	−0.00862285	−	0.0236911i
$708$		0			0
$709$	−30941.8	−	175479.i	−0.0615535	−	0.349087i	−0.999993	−	0.00364447i	$-0.998840\pi$
$709$	−30941.8	−	175479.i	−0.0615535	−	0.349087i	0.938440	−	0.345443i	$-0.112271\pi$
$710$		0			0
$711$	−555520.	+	119582.i	−1.09891	+	0.236551i
$712$		0			0
$713$	−12972.3	+	35641.2i	−0.0255176	+	0.0701089i
$714$		0			0
$715$	109899.	+	92216.5i	0.214973	+	0.180383i
$716$		0			0
$717$	−156206.	−	69305.4i	−0.303849	−	0.134812i
$718$		0			0
$719$	−187920.	−	108496.i	−0.363509	−	0.209872i	0.307110	−	0.951674i	$-0.400638\pi$
$719$	−187920.	−	108496.i	−0.363509	−	0.209872i	−0.670619	+	0.741802i	$0.733971\pi$
$720$		0			0
$721$	20061.6	+	34747.7i	0.0385918	+	0.0668430i
$722$		0			0
$723$	264803.	−	129577.i	0.506578	−	0.247886i
$724$		0			0
$725$	−98926.4	−	17443.4i	−0.188207	−	0.0331860i
$726$		0			0
$727$	−113085.	+	94889.8i	−0.213962	+	0.179536i	−0.743470	−	0.668770i	$-0.766821\pi$
$727$	−113085.	+	94889.8i	−0.213962	+	0.179536i	0.529507	+	0.848305i	$0.322377\pi$
$728$		0			0
$729$	59649.0	+	528083.i	0.112240	+	0.993681i
$730$		0			0
$731$	79981.9	+	95318.7i	0.149678	+	0.178379i
$732$		0			0
$733$	−124786.	+	707699.i	−0.232252	+	1.31717i	0.616073	+	0.787689i	$0.288723\pi$
$733$	−124786.	+	707699.i	−0.232252	+	1.31717i	−0.848325	+	0.529476i	$0.822388\pi$
$734$		0			0
$735$	263895.	+	539295.i	0.488492	+	0.998279i
$736$		0			0
$737$	969221.	−	559580.i	1.78438	−	1.03021i
$738$		0			0
$739$	−34667.1	+	60045.2i	−0.0634788	+	0.109948i	−0.896018	−	0.444017i	$-0.853553\pi$
$739$	−34667.1	+	60045.2i	−0.0634788	+	0.109948i	0.832539	+	0.553966i	$0.186886\pi$
$740$		0			0
$741$	9455.86	−	21312.3i	0.0172213	−	0.0388145i
$742$		0			0
$743$	349329.	−	416315.i	0.632787	−	0.754126i	−0.350425	−	0.936591i	$-0.613963\pi$
$743$	349329.	−	416315.i	0.632787	−	0.754126i	0.983212	+	0.182464i	$0.0584074\pi$
$744$		0			0
$745$	−915835.	−	333337.i	−1.65008	−	0.600580i
$746$		0			0
$747$	42587.8	+	197843.i	0.0763210	+	0.354551i
$748$		0			0
$749$	−19066.3	+	3361.91i	−0.0339862	+	0.00599269i
$750$		0			0
$751$	133159.	−	48466.0i	0.236097	−	0.0859324i	−0.221262	−	0.975214i	$-0.571018\pi$
$751$	133159.	−	48466.0i	0.236097	−	0.0859324i	0.457359	+	0.889282i	$0.348795\pi$
$752$		0			0
$753$	279396.	−	80506.5i	0.492754	−	0.141985i
$754$		0			0
$755$			1.18080e6i			2.07149i
$756$		0			0
$757$	−497775.			−0.868643			−0.434322	−	0.900758i	$-0.643012\pi$
$757$	−497775.			−0.868643			−0.434322	+	0.900758i	$0.643012\pi$
$758$		0			0
$759$	−284708.	+	295588.i	−0.494216	+	0.513101i
$760$		0			0
$761$	130892.	+	359622.i	0.226018	+	0.620979i	0.999924	−	0.0123215i	$-0.00392215\pi$
$761$	130892.	+	359622.i	0.226018	+	0.620979i	−0.773906	+	0.633300i	$0.781700\pi$
$762$		0			0
$763$	3447.80	+	19553.4i	0.00592233	+	0.0335872i
$764$		0			0
$765$	115321.	−	148393.i	0.197054	−	0.253566i
$766$		0			0
$767$	−38798.4	+	106598.i	−0.0659513	+	0.181200i
$768$		0			0
$769$	16103.1	+	13512.1i	0.0272305	+	0.0228491i	0.656301	−	0.754499i	$-0.272120\pi$
$769$	16103.1	+	13512.1i	0.0272305	+	0.0228491i	−0.629071	+	0.777348i	$0.716564\pi$
$770$		0			0
$771$	−517208.	+	376796.i	−0.870075	+	0.633867i
$772$		0			0
$773$	−337649.	−	194942.i	−0.565076	−	0.326247i	0.190104	−	0.981764i	$-0.439117\pi$
$773$	−337649.	−	194942.i	−0.565076	−	0.326247i	−0.755180	+	0.655517i	$0.772451\pi$
$774$		0			0
$775$	−8263.86	−	14313.4i	−0.0137588	−	0.0238309i
$776$		0			0
$777$	34738.6	+	23366.2i	0.0575401	+	0.0387031i
$778$		0			0
$779$	134293.	+	23679.5i	0.221299	+	0.0390209i
$780$		0			0
$781$	48854.3	−	40993.6i	0.0800941	−	0.0672069i
$782$		0			0
$783$	−14820.1	+	477485.i	−0.0241729	+	0.778819i
$784$		0			0
$785$	287974.	+	343194.i	0.467319	+	0.556930i
$786$		0			0
$787$	160183.	−	908442.i	0.258622	−	1.46672i	−0.527978	−	0.849258i	$-0.677050\pi$
$787$	160183.	−	908442.i	0.258622	−	1.46672i	0.786600	−	0.617463i	$-0.211839\pi$
$788$		0			0
$789$	414398.	+	28439.2i	0.665678	+	0.0456840i
$790$		0			0
$791$	21244.3	−	12265.4i	0.0339539	−	0.0196033i
$792$		0			0
$793$	−130164.	+	225450.i	−0.206987	+	0.358512i
$794$		0			0
$795$	585148.	−	62266.5i	0.925830	−	0.0985190i
$796$		0			0
$797$	591632.	−	705079.i	0.931397	−	1.11000i	−0.0623181	−	0.998056i	$-0.519849\pi$
$797$	591632.	−	705079.i	0.931397	−	1.11000i	0.993715	−	0.111939i	$-0.0357062\pi$
$798$		0			0
$799$	−187354.	−	68191.5i	−0.293475	−	0.106816i
$800$		0			0
$801$	1.20786e6	−	45306.6i	1.88257	−	0.0706149i
$802$		0			0
$803$	511732.	−	90232.2i	0.793618	−	0.139936i
$804$		0			0
$805$	28780.9	−	10475.4i	0.0444133	−	0.0161651i
$806$		0			0
$807$	−136171.	+	549177.i	−0.209093	+	0.843268i
$808$		0			0
$809$		−	497784.i		−	0.760578i	−0.924868	−	0.380289i	$-0.875824\pi$
$809$		−	497784.i		−	0.760578i	0.924868	−	0.380289i	$-0.124176\pi$
$810$		0			0
$811$	275127.			0.418304			0.209152	−	0.977883i	$-0.432930\pi$
$811$	275127.			0.418304			0.209152	+	0.977883i	$0.432930\pi$
$812$		0			0
$813$	−785654.	−	194807.i	−1.18864	−	0.294729i
$814$		0			0
$815$	−102372.	−	281265.i	−0.154122	−	0.423448i
$816$		0			0
$817$	−16965.9	−	96218.7i	−0.0254176	−	0.144150i
$818$		0			0
$819$	−8490.85	+	5336.25i	−0.0126585	+	0.00795551i
$820$		0			0
$821$	23033.1	−	63282.9i	0.0341717	−	0.0938859i	−0.921434	−	0.388536i	$-0.872981\pi$
$821$	23033.1	−	63282.9i	0.0341717	−	0.0938859i	0.955605	+	0.294650i	$0.0952031\pi$
$822$		0			0
$823$	−972832.	−	816303.i	−1.43628	−	1.20518i	−0.941883	−	0.335942i	$-0.890945\pi$
$823$	−972832.	−	816303.i	−1.43628	−	1.20518i	−0.494394	−	0.869238i	$-0.664610\pi$
$824$		0			0
$825$	−18923.5	−	177833.i	−0.0278031	−	0.261279i
$826$		0			0
$827$	−1.12901e6	−	651837.i	−1.65078	−	0.953077i	−0.976753	−	0.214369i	$-0.931231\pi$
$827$	−1.12901e6	−	651837.i	−1.65078	−	0.953077i	−0.674025	−	0.738708i	$-0.735436\pi$
$828$		0			0
$829$	−442947.	−	767206.i	−0.644529	−	1.11636i	−0.984410	−	0.175888i	$-0.943720\pi$
$829$	−442947.	−	767206.i	−0.644529	−	1.11636i	0.339881	−	0.940468i	$-0.389613\pi$
$830$		0			0
$831$	−8373.20	+	122009.i	−0.0121252	+	0.176681i
$832$		0			0
$833$	−195853.	−	34534.1i	−0.282253	−	0.0497689i
$834$		0			0
$835$	602255.	−	505352.i	0.863789	−	0.724805i
$836$		0			0
$837$	−61749.3	+	48630.7i	−0.0881417	+	0.0694160i
$838$		0			0
$839$	137520.	+	163890.i	0.195362	+	0.232824i	0.854829	−	0.518910i	$-0.173662\pi$
$839$	137520.	+	163890.i	0.195362	+	0.232824i	−0.659466	+	0.751734i	$0.729218\pi$
$840$		0			0
$841$	48249.9	−	273639.i	0.0682188	−	0.386888i
$842$		0			0
$843$	−177855.	+	264417.i	−0.250271	+	0.372078i
$844$		0			0
$845$	652018.	−	376443.i	0.913159	−	0.527213i
$846$		0			0
$847$	−3373.95	+	5843.85i	−0.00470296	+	0.00814577i
$848$		0			0
$849$	−603986.	−	829059.i	−0.837937	−	1.15019i
$850$		0			0
$851$	−337041.	+	401670.i	−0.465397	+	0.554639i
$852$		0			0
$853$	−998450.	−	363406.i	−1.37223	−	0.499452i	−0.452419	−	0.891805i	$-0.649439\pi$
$853$	−998450.	−	363406.i	−1.37223	−	0.499452i	−0.919815	+	0.392353i	$0.871661\pi$
$854$		0			0
$855$	−136679.	+	55633.3i	−0.186969	+	0.0761032i
$856$		0			0
$857$	−730137.	+	128743.i	−0.994129	+	0.175292i	−0.646971	−	0.762515i	$-0.723965\pi$
$857$	−730137.	+	128743.i	−0.994129	+	0.175292i	−0.347158	+	0.937806i	$0.612853\pi$
$858$		0			0
$859$	−317020.	+	115386.i	−0.429636	+	0.156375i	−0.547781	−	0.836622i	$-0.684527\pi$
$859$	−317020.	+	115386.i	−0.429636	+	0.156375i	0.118145	+	0.992996i	$0.462305\pi$
$860$		0			0
$861$	−42243.6	−	40688.7i	−0.0569842	−	0.0548868i
$862$		0			0
$863$		−	783845.i		−	1.05247i	−0.850340	−	0.526234i	$-0.823604\pi$
$863$		−	783845.i		−	1.05247i	0.850340	−	0.526234i	$-0.176396\pi$
$864$		0			0
$865$	−1.05160e6			−1.40546
$866$		0			0
$867$	−190892.	−	662484.i	−0.253950	−	0.881328i
$868$		0			0
$869$	−311027.	−	854539.i	−0.411868	−	1.13160i
$870$		0			0
$871$	59475.9	+	337304.i	0.0783979	+	0.444617i
$872$		0			0
$873$	918202.	+	1.01448e6i	1.20479	+	1.33111i
$874$		0			0
$875$	14046.6	−	38592.7i	0.0183466	−	0.0504068i
$876$		0			0
$877$	544116.	+	456568.i	0.707444	+	0.593616i	0.923881	−	0.382680i	$-0.124999\pi$
$877$	544116.	+	456568.i	0.707444	+	0.593616i	−0.216436	+	0.976297i	$0.569443\pi$
$878$		0			0
$879$	−588815.	−	261246.i	−0.762081	−	0.338120i
$880$		0			0
$881$	54907.6	+	31700.9i	0.0707425	+	0.0408432i	0.534954	−	0.844881i	$-0.320329\pi$
$881$	54907.6	+	31700.9i	0.0707425	+	0.0408432i	−0.464212	+	0.885724i	$0.653662\pi$
$882$		0			0
$883$	87691.3	+	151886.i	0.112470	+	0.194803i	0.916765	−	0.399426i	$-0.130791\pi$
$883$	87691.3	+	151886.i	0.112470	+	0.194803i	−0.804296	+	0.594229i	$0.797457\pi$
$884$		0			0
$885$	644893.	−	315568.i	0.823382	−	0.402909i
$886$		0			0
$887$	914676.	+	161282.i	1.16257	+	0.204993i	0.721458	−	0.692458i	$-0.243472\pi$
$887$	914676.	+	161282.i	1.16257	+	0.204993i	0.441114	+	0.897451i	$0.354583\pi$
$888$		0			0
$889$	14245.6	−	11953.5i	0.0180251	−	0.0151249i
$890$		0			0
$891$	−824145.	+	210015.i	−1.03812	+	0.264543i
$892$		0			0
$893$	100630.	+	119927.i	0.126190	+	0.150388i
$894$		0			0
$895$	267580.	−	1.51752e6i	0.334047	−	1.89447i
$896$		0			0
$897$	−55205.3	−	112817.i	−0.0686114	−	0.140214i
$898$		0			0
$899$	−61188.0	+	35326.9i	−0.0757088	+	0.0437105i
$900$		0			0
$901$	−97458.0	+	168802.i	−0.120052	+	0.207935i
$902$		0			0
$903$	−17042.8	+	38412.3i	−0.0209009	+	0.0471081i
$904$		0			0
$905$	−188225.	+	224318.i	−0.229816	+	0.273885i
$906$		0			0
$907$	763552.	+	277910.i	0.928163	+	0.337824i	0.761481	−	0.648187i	$-0.224472\pi$
$907$	763552.	+	277910.i	0.928163	+	0.337824i	0.166682	+	0.986011i	$0.446695\pi$
$908$		0			0
$909$	311328.	+	100267.i	0.376782	+	0.121348i
$910$		0			0
$911$	529460.	−	93358.1i	0.637964	−	0.112490i	0.154695	−	0.987962i	$-0.450560\pi$
$911$	529460.	−	93358.1i	0.637964	−	0.112490i	0.483269	+	0.875472i	$0.339449\pi$
$912$		0			0
$913$	−304336.	+	110769.i	−0.365100	+	0.132885i
$914$		0			0
$915$	1.58321e6	−	456195.i	1.89102	−	0.544889i
$916$		0			0
$917$			72622.6i			0.0863640i
$918$		0			0
$919$	−214963.			−0.254526			−0.127263	−	0.991869i	$-0.540619\pi$
$919$	−214963.			−0.254526			−0.127263	+	0.991869i	$0.540619\pi$
$920$		0			0
$921$	−111605.	+	115869.i	−0.131572	+	0.136600i
$922$		0			0
$923$	6675.42	+	18340.6i	0.00783566	+	0.0215283i
$924$		0			0
$925$	−39676.4	−	225016.i	−0.0463712	−	0.262984i
$926$		0			0
$927$	−1.03161e6	−	142264.i	−1.20048	−	0.165552i
$928$		0			0
$929$	−418240.	+	1.14911e6i	−0.484612	+	1.33146i	0.420887	+	0.907113i	$0.361719\pi$
$929$	−418240.	+	1.14911e6i	−0.484612	+	1.33146i	−0.905499	+	0.424348i	$0.860503\pi$
$930$		0			0
$931$	119623.	+	100376.i	0.138012	+	0.115806i
$932$		0			0
$933$	−36598.5	+	26662.7i	−0.0420436	+	0.0306296i
$934$		0			0
$935$	260464.	+	150379.i	0.297938	+	0.172014i
$936$		0			0
$937$	701240.	+	1.21458e6i	0.798707	+	1.38340i	0.920458	+	0.390841i	$0.127816\pi$
$937$	701240.	+	1.21458e6i	0.798707	+	1.38340i	−0.121751	+	0.992561i	$0.538851\pi$
$938$		0			0
$939$	−374876.	−	252152.i	−0.425164	−	0.285977i
$940$		0			0
$941$	−1.59805e6	−	281779.i	−1.80472	−	0.318222i	−0.832811	−	0.553558i	$-0.813270\pi$
$941$	−1.59805e6	−	281779.i	−1.80472	−	0.318222i	−0.971914	+	0.235336i	$0.924381\pi$
$942$		0			0
$943$	562723.	−	472180.i	0.632807	−	0.530988i
$944$		0			0
$945$	62134.5	+	12955.4i	0.0695775	+	0.0145073i
$946$		0			0
$947$	358034.	+	426688.i	0.399231	+	0.475785i	0.927785	−	0.373115i	$-0.121710\pi$
$947$	358034.	+	426688.i	0.399231	+	0.475785i	−0.528554	+	0.848899i	$0.677266\pi$
$948$		0			0
$949$	−27614.7	+	156611.i	−0.0306625	+	0.173896i
$950$		0			0
$951$	393807.	+	27026.1i	0.435435	+	0.0298829i
$952$		0			0
$953$	1.07755e6	−	622122.i	1.18645	−	0.684998i	0.228954	−	0.973437i	$-0.426470\pi$
$953$	1.07755e6	−	622122.i	1.18645	−	0.684998i	0.957498	+	0.288439i	$0.0931362\pi$
$954$		0			0
$955$	−560598.	+	970985.i	−0.614674	+	1.06465i
$956$		0			0
$957$	−760212.	+	80895.4i	−0.830063	+	0.0883283i
$958$		0			0
$959$	27681.9	−	32990.0i	0.0300994	−	0.0358711i
$960$		0			0
$961$	856902.	+	311887.i	0.927864	+	0.337715i
$962$		0			0
$963$	234756.	−	444279.i	0.253142	−	0.479075i
$964$		0			0
$965$	1.39951e6	−	246771.i	1.50287	−	0.264996i
$966$		0			0
$967$	42404.6	−	15434.0i	0.0453482	−	0.0165054i	−0.319246	−	0.947672i	$-0.603430\pi$
$967$	42404.6	−	15434.0i	0.0453482	−	0.0165054i	0.364595	+	0.931166i	$0.381208\pi$
$968$		0			0
$969$	11763.7	−	47442.9i	0.0125284	−	0.0505270i
$970$		0			0
$971$			699818.i			0.742244i	0.928584	+	0.371122i	$0.121027\pi$
$971$			699818.i			0.742244i	−0.928584	+	0.371122i	$0.878973\pi$
$972$		0			0
$973$	−67463.4			−0.0712594
$974$		0			0
$975$	53122.6	+	13172.0i	0.0558818	+	0.0138562i
$976$		0			0
$977$	164161.	+	451029.i	0.171981	+	0.472514i	0.995499	−	0.0947767i	$-0.0302137\pi$
$977$	164161.	+	451029.i	0.171981	+	0.472514i	−0.823517	+	0.567291i	$0.807991\pi$
$978$		0			0
$979$	335893.	+	1.90495e6i	0.350458	+	1.98755i
$980$		0			0
$981$	−455629.	−	240753.i	−0.473449	−	0.250169i
$982$		0			0
$983$	182645.	−	501813.i	0.189017	−	0.519320i	−0.808597	−	0.588363i	$-0.799773\pi$
$983$	182645.	−	501813.i	0.189017	−	0.519320i	0.997614	+	0.0690436i	$0.0219948\pi$
$984$		0			0
$985$	602750.	+	505767.i	0.621247	+	0.521288i
$986$		0			0
$987$	−7125.06	−	66957.6i	−0.00731399	−	0.0687330i
$988$		0			0
$989$	−455803.	−	263158.i	−0.465998	−	0.269044i
$990$		0			0
$991$	−15806.5	−	27377.7i	−0.0160949	−	0.0278772i	0.857866	−	0.513874i	$-0.171790\pi$
$991$	−15806.5	−	27377.7i	−0.0160949	−	0.0278772i	−0.873961	+	0.485997i	$0.838457\pi$
$992$		0			0
$993$	−85617.3	+	1.24756e6i	−0.0868287	+	1.26521i
$994$		0			0
$995$	−1.38018e6	−	243363.i	−1.39408	−	0.245815i
$996$		0			0
$997$	333671.	−	279983.i	0.335682	−	0.281670i	−0.459328	−	0.888267i	$-0.651910\pi$
$997$	333671.	−	279983.i	0.335682	−	0.281670i	0.795010	+	0.606596i	$0.207465\pi$
$998$		0			0
$999$	−1.03211e6	+	339784.i	−1.03418	+	0.340465i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.5.12	✓	72
3.2	odd	2		324.5.k.a.125.3		72
27.11	odd	18	inner	108.5.k.a.65.12	yes	72
27.16	even	9		324.5.k.a.197.3		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.12	✓	72	1.1	even	1	trivial
108.5.k.a.65.12	yes	72	27.11	odd	18	inner
324.5.k.a.125.3		72	3.2	odd	2
324.5.k.a.197.3		72	27.16	even	9

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 108 = 2^{2} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	108.k (of order \(18\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(8.73547 + 2.16600i) q^{3} +(9.54164 + 26.2154i) q^{5} +(-0.541932 - 3.07345i) q^{7} +(71.6168 + 37.8421i) q^{9} +O(q^{10})\)	\(q+(8.73547 + 2.16600i) q^{3} +(9.54164 + 26.2154i) q^{5} +(-0.541932 - 3.07345i) q^{7} +(71.6168 + 37.8421i) q^{9} +(-44.3351 + 121.810i) q^{11} +(-30.3898 - 25.5001i) q^{13} +(26.5679 + 249.671i) q^{15} +(-72.0247 - 41.5835i) q^{17} +(32.6516 + 56.5542i) q^{19} +(1.92308 - 28.0219i) q^{21} +(346.437 + 61.0862i) q^{23} +(-117.428 + 98.5341i) q^{25} +(543.640 + 485.691i) q^{27} +(421.220 + 501.991i) q^{29} +(-18.7225 + 106.181i) q^{31} +(-651.128 + 968.034i) q^{33} +(75.4009 - 43.5328i) q^{35} +(-745.268 + 1290.84i) q^{37} +(-210.236 - 288.580i) q^{39} +(1342.26 - 1599.64i) q^{41} +(-1405.92 - 511.712i) q^{43} +(-308.706 + 2238.54i) q^{45} +(2360.91 - 416.292i) q^{47} +(2247.05 - 817.859i) q^{49} +(-539.100 - 519.257i) q^{51} -2343.67i q^{53} -3616.32 q^{55} +(162.730 + 564.751i) q^{57} +(-978.002 - 2687.04i) q^{59} +(-1139.50 - 6462.44i) q^{61} +(77.4945 - 240.619i) q^{63} +(378.527 - 1040.00i) q^{65} +(-6613.79 - 5549.63i) q^{67} +(2893.98 + 1284.00i) q^{69} +(-426.072 - 245.993i) q^{71} +(-2004.31 - 3471.57i) q^{73} +(-1239.22 + 606.391i) q^{75} +(398.403 + 70.2491i) q^{77} +(-5374.08 + 4509.39i) q^{79} +(3696.94 + 5420.27i) q^{81} +(1605.97 + 1913.93i) q^{83} +(402.895 - 2284.93i) q^{85} +(2592.24 + 5297.49i) q^{87} +(12923.1 - 7461.14i) q^{89} +(-61.9041 + 107.221i) q^{91} +(-393.538 + 886.984i) q^{93} +(-1171.04 + 1395.60i) q^{95} +(15873.9 + 5777.63i) q^{97} +(-7784.68 + 7045.89i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 72 * q + 9 * q^5 - 102 * q^9	\( 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) \) 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more