LMFDB - Embedded newform 108.5.k.a.5.8

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			5.8
Character	$\chi$	$=$	108.5
Dual form			108.5.k.a.65.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.78712 + 7.62125i) q^{3} +(-3.22217 - 8.85284i) q^{5} +(-16.2288 - 92.0381i) q^{7} +(-35.1670 + 72.9677i) q^{9} +O(q^{10})$	\(q+(4.78712 + 7.62125i) q^{3} +(-3.22217 - 8.85284i) q^{5} +(-16.2288 - 92.0381i) q^{7} +(-35.1670 + 72.9677i) q^{9} +(74.5484 - 204.820i) q^{11} +(-77.8115 - 65.2916i) q^{13} +(52.0448 - 66.9366i) q^{15} +(135.945 + 78.4880i) q^{17} +(237.980 + 412.193i) q^{19} +(623.756 - 564.281i) q^{21} +(50.4332 + 8.89274i) q^{23} +(410.787 - 344.691i) q^{25} +(-724.454 + 81.2889i) q^{27} +(-936.902 - 1116.56i) q^{29} +(62.7625 - 355.944i) q^{31} +(1917.86 - 412.346i) q^{33} +(-762.507 + 440.234i) q^{35} +(-499.310 + 864.830i) q^{37} +(125.111 - 905.580i) q^{39} +(1273.34 - 1517.51i) q^{41} +(739.364 + 269.107i) q^{43} +(759.286 + 76.2131i) q^{45} +(-280.621 + 49.4811i) q^{47} +(-5951.44 + 2166.15i) q^{49} +(52.6093 + 1411.80i) q^{51} -3261.43i q^{53} -2053.45 q^{55} +(-2002.19 + 3786.92i) q^{57} +(1215.30 + 3339.02i) q^{59} +(744.052 + 4219.73i) q^{61} +(7286.53 + 2052.52i) q^{63} +(-327.294 + 899.234i) q^{65} +(5409.04 + 4538.72i) q^{67} +(173.656 + 426.935i) q^{69} +(-6345.67 - 3663.67i) q^{71} +(851.462 + 1474.78i) q^{73} +(4593.47 + 1480.63i) q^{75} +(-20061.1 - 3537.31i) q^{77} +(-4665.20 + 3914.57i) q^{79} +(-4087.57 - 5132.10i) q^{81} +(1610.83 + 1919.72i) q^{83} +(256.803 - 1456.40i) q^{85} +(4024.49 - 12485.5i) q^{87} +(9812.53 - 5665.26i) q^{89} +(-4746.53 + 8221.23i) q^{91} +(3013.19 - 1225.62i) q^{93} +(2882.27 - 3434.95i) q^{95} +(-6396.61 - 2328.18i) q^{97} +(12323.6 + 12642.5i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	4.78712	+	7.62125i	0.531902	+	0.846806i
$3$	4.78712	+	7.62125i	0.531902	+	0.846806i
$4$		0			0
$5$	−3.22217	−	8.85284i	−0.128887	−	0.354114i	0.858418	−	0.512951i	$-0.171448\pi$
$5$	−3.22217	−	8.85284i	−0.128887	−	0.354114i	−0.987305	+	0.158837i	$0.949226\pi$
$6$		0			0
$7$	−16.2288	−	92.0381i	−0.331200	−	1.87833i	−0.461939	−	0.886912i	$-0.652846\pi$
$7$	−16.2288	−	92.0381i	−0.331200	−	1.87833i	0.130739	−	0.991417i	$-0.458265\pi$
$8$		0			0
$9$	−35.1670	+	72.9677i	−0.434160	+	0.900836i
$10$		0			0
$11$	74.5484	−	204.820i	0.616102	−	1.69273i	−0.100228	−	0.994965i	$-0.531957\pi$
$11$	74.5484	−	204.820i	0.616102	−	1.69273i	0.716330	−	0.697762i	$-0.245821\pi$
$12$		0			0
$13$	−77.8115	−	65.2916i	−0.460423	−	0.386341i	0.382863	−	0.923805i	$-0.374938\pi$
$13$	−77.8115	−	65.2916i	−0.460423	−	0.386341i	−0.843287	+	0.537464i	$0.819382\pi$
$14$		0			0
$15$	52.0448	−	66.9366i	0.231310	−	0.297496i
$16$		0			0
$17$	135.945	+	78.4880i	0.470399	+	0.271585i	0.716407	−	0.697683i	$-0.245786\pi$
$17$	135.945	+	78.4880i	0.470399	+	0.271585i	−0.246008	+	0.969268i	$0.579119\pi$
$18$		0			0
$19$	237.980	+	412.193i	0.659223	+	1.14181i	0.980817	+	0.194931i	$0.0624482\pi$
$19$	237.980	+	412.193i	0.659223	+	1.14181i	−0.321594	+	0.946878i	$0.604219\pi$
$20$		0			0
$21$	623.756	−	564.281i	1.41441	−	1.27955i
$22$		0			0
$23$	50.4332	+	8.89274i	0.0953369	+	0.0168105i	0.221113	−	0.975248i	$-0.429031\pi$
$23$	50.4332	+	8.89274i	0.0953369	+	0.0168105i	−0.125776	+	0.992059i	$0.540142\pi$
$24$		0			0
$25$	410.787	−	344.691i	0.657260	−	0.551506i
$26$		0			0
$27$	−724.454	+	81.2889i	−0.993764	+	0.111507i
$28$		0			0
$29$	−936.902	−	1116.56i	−1.11403	−	1.32765i	−0.939323	−	0.343033i	$-0.888546\pi$
$29$	−936.902	−	1116.56i	−1.11403	−	1.32765i	−0.174710	−	0.984620i	$-0.555899\pi$
$30$		0			0
$31$	62.7625	−	355.944i	0.0653095	−	0.370389i	−0.934583	−	0.355745i	$-0.884227\pi$
$31$	62.7625	−	355.944i	0.0653095	−	0.370389i	0.999893	−	0.0146442i	$-0.00466157\pi$
$32$		0			0
$33$	1917.86	−	412.346i	1.76112	−	0.378646i
$34$		0			0
$35$	−762.507	+	440.234i	−0.622455	+	0.359374i
$36$		0			0
$37$	−499.310	+	864.830i	−0.364726	+	0.631724i	−0.988732	−	0.149696i	$-0.952171\pi$
$37$	−499.310	+	864.830i	−0.364726	+	0.631724i	0.624006	+	0.781419i	$0.285504\pi$
$38$		0			0
$39$	125.111	−	905.580i	0.0822556	−	0.595385i
$40$		0			0
$41$	1273.34	−	1517.51i	0.757489	−	0.902740i	−0.240198	−	0.970724i	$-0.577212\pi$
$41$	1273.34	−	1517.51i	0.757489	−	0.902740i	0.997686	+	0.0679839i	$0.0216567\pi$
$42$		0			0
$43$	739.364	+	269.107i	0.399872	+	0.145542i	0.534125	−	0.845405i	$-0.320641\pi$
$43$	739.364	+	269.107i	0.399872	+	0.145542i	−0.134253	+	0.990947i	$0.542863\pi$
$44$		0			0
$45$	759.286	+	76.2131i	0.374956	+	0.0376361i
$46$		0			0
$47$	−280.621	+	49.4811i	−0.127035	+	0.0223998i	−0.236804	−	0.971557i	$-0.576100\pi$
$47$	−280.621	+	49.4811i	−0.127035	+	0.0223998i	0.109769	+	0.993957i	$0.464989\pi$
$48$		0			0
$49$	−5951.44	+	2166.15i	−2.47873	+	0.902185i
$50$		0			0
$51$	52.6093	+	1411.80i	0.0202265	+	0.542793i
$52$		0			0
$53$		−	3261.43i		−	1.16106i	−0.814237	−	0.580532i	$-0.802845\pi$
$53$		−	3261.43i		−	1.16106i	0.814237	−	0.580532i	$-0.197155\pi$
$54$		0			0
$55$	−2053.45			−0.678825
$56$		0			0
$57$	−2002.19	+	3786.92i	−0.616247	+	1.16556i
$58$		0			0
$59$	1215.30	+	3339.02i	0.349125	+	0.959213i	0.982647	+	0.185488i	$0.0593866\pi$
$59$	1215.30	+	3339.02i	0.349125	+	0.959213i	−0.633522	+	0.773725i	$0.718391\pi$
$60$		0			0
$61$	744.052	+	4219.73i	0.199960	+	1.13403i	0.905176	+	0.425038i	$0.139739\pi$
$61$	744.052	+	4219.73i	0.199960	+	1.13403i	−0.705215	+	0.708993i	$0.749150\pi$
$62$		0			0
$63$	7286.53	+	2052.52i	1.83586	+	0.517138i
$64$		0			0
$65$	−327.294	+	899.234i	−0.0774661	+	0.212836i
$66$		0			0
$67$	5409.04	+	4538.72i	1.20495	+	1.01108i	0.999475	+	0.0324148i	$0.0103198\pi$
$67$	5409.04	+	4538.72i	1.20495	+	1.01108i	0.205479	+	0.978661i	$0.434125\pi$
$68$		0			0
$69$	173.656	+	426.935i	0.0364747	+	0.0896733i
$70$		0			0
$71$	−6345.67	−	3663.67i	−1.25881	−	0.726775i	−0.285967	−	0.958239i	$-0.592315\pi$
$71$	−6345.67	−	3663.67i	−1.25881	−	0.726775i	−0.972843	+	0.231465i	$0.925648\pi$
$72$		0			0
$73$	851.462	+	1474.78i	0.159779	+	0.276745i	0.934789	−	0.355204i	$-0.115589\pi$
$73$	851.462	+	1474.78i	0.159779	+	0.276745i	−0.775010	+	0.631949i	$0.782255\pi$
$74$		0			0
$75$	4593.47	+	1480.63i	0.816617	+	0.263224i
$76$		0			0
$77$	−20061.1	−	3537.31i	−3.38355	−	0.596611i
$78$		0			0
$79$	−4665.20	+	3914.57i	−0.747509	+	0.627234i	−0.934843	−	0.355062i	$-0.884460\pi$
$79$	−4665.20	+	3914.57i	−0.747509	+	0.627234i	0.187334	+	0.982296i	$0.440015\pi$
$80$		0			0
$81$	−4087.57	−	5132.10i	−0.623010	−	0.782214i
$82$		0			0
$83$	1610.83	+	1919.72i	0.233827	+	0.278664i	0.870180	−	0.492734i	$-0.164002\pi$
$83$	1610.83	+	1919.72i	0.233827	+	0.278664i	−0.636353	+	0.771398i	$0.719558\pi$
$84$		0			0
$85$	256.803	−	1456.40i	0.0355437	−	0.201578i
$86$		0			0
$87$	4024.49	−	12485.5i	0.531708	−	1.64955i
$88$		0			0
$89$	9812.53	−	5665.26i	1.23880	−	0.715221i	0.269950	−	0.962874i	$-0.412993\pi$
$89$	9812.53	−	5665.26i	1.23880	−	0.715221i	0.968849	+	0.247653i	$0.0796594\pi$
$90$		0			0
$91$	−4746.53	+	8221.23i	−0.573183	+	0.992782i
$92$		0			0
$93$	3013.19	−	1225.62i	0.348386	−	0.141706i
$94$		0			0
$95$	2882.27	−	3434.95i	0.319365	−	0.380604i
$96$		0			0
$97$	−6396.61	−	2328.18i	−0.679840	−	0.247441i	−0.0210609	−	0.999778i	$-0.506704\pi$
$97$	−6396.61	−	2328.18i	−0.679840	−	0.247441i	−0.658779	+	0.752337i	$0.728927\pi$
$98$		0			0
$99$	12323.6	+	12642.5i	1.25738	+	1.28992i
$100$		0			0
$101$	9985.15	−	1760.65i	0.978840	−	0.172596i	0.338734	−	0.940882i	$-0.390001\pi$
$101$	9985.15	−	1760.65i	0.978840	−	0.172596i	0.640106	+	0.768286i	$0.278890\pi$
$102$		0			0
$103$	6493.17	−	2363.32i	0.612044	−	0.222766i	−0.0173536	−	0.999849i	$-0.505524\pi$
$103$	6493.17	−	2363.32i	0.612044	−	0.222766i	0.629397	+	0.777084i	$0.283302\pi$
$104$		0			0
$105$	−7005.34	−	3703.81i	−0.635405	−	0.335946i
$106$		0			0
$107$		−	6794.89i		−	0.593492i	−0.954956	−	0.296746i	$-0.904099\pi$
$107$		−	6794.89i		−	0.593492i	0.954956	−	0.296746i	$-0.0959015\pi$
$108$		0			0
$109$	1303.17			0.109685			0.0548426	−	0.998495i	$-0.482534\pi$
$109$	1303.17			0.109685			0.0548426	+	0.998495i	$0.482534\pi$
$110$		0			0
$111$	−8981.34	+	334.679i	−0.728946	+	0.0271633i
$112$		0			0
$113$	2799.78	+	7692.32i	0.219264	+	0.602422i	0.999741	−	0.0227625i	$-0.00724614\pi$
$113$	2799.78	+	7692.32i	0.219264	+	0.602422i	−0.780477	+	0.625184i	$0.785024\pi$
$114$		0			0
$115$	−83.7785	−	475.131i	−0.00633486	−	0.0359267i
$116$		0			0
$117$	7500.57	−	3381.62i	0.547927	−	0.247032i
$118$		0			0
$119$	5017.66	−	13785.9i	0.354330	−	0.973512i
$120$		0			0
$121$	−25178.1	−	21126.9i	−1.71970	−	1.44300i
$122$		0			0
$123$	17660.9	+	2439.95i	1.16736	+	0.161276i
$124$		0			0
$125$	−9474.39	−	5470.04i	−0.606361	−	0.350083i
$126$		0			0
$127$	7118.80	+	12330.1i	0.441367	+	0.764469i	0.997791	−	0.0664288i	$-0.0211605\pi$
$127$	7118.80	+	12330.1i	0.441367	+	0.764469i	−0.556425	+	0.830898i	$0.687827\pi$
$128$		0			0
$129$	1488.50	+	6923.13i	0.0894475	+	0.416028i
$130$		0			0
$131$	2925.80	+	515.898i	0.170492	+	0.0300623i	0.258242	−	0.966080i	$-0.416857\pi$
$131$	2925.80	+	515.898i	0.170492	+	0.0300623i	−0.0877506	+	0.996142i	$0.527968\pi$
$132$		0			0
$133$	34075.3	−	28592.6i	1.92636	−	1.61640i
$134$		0			0
$135$	3053.95	+	6151.55i	0.167569	+	0.337534i
$136$		0			0
$137$	10598.0	+	12630.2i	0.564652	+	0.672926i	0.970524	−	0.241004i	$-0.0774766\pi$
$137$	10598.0	+	12630.2i	0.564652	+	0.672926i	−0.405872	+	0.913930i	$0.633032\pi$
$138$		0			0
$139$	−669.020	+	3794.20i	−0.0346266	+	0.196377i	−0.997214	−	0.0745944i	$-0.976234\pi$
$139$	−669.020	+	3794.20i	−0.0346266	+	0.196377i	0.962587	+	0.270971i	$0.0873449\pi$
$140$		0			0
$141$	−1720.47	−	1901.81i	−0.0865386	−	0.0956598i
$142$		0			0
$143$	−19173.7	+	11070.0i	−0.937637	+	0.541345i
$144$		0			0
$145$	−6865.84	+	11892.0i	−0.326556	+	0.565612i
$146$		0			0
$147$	−44999.0	−	34987.8i	−2.08242	−	1.61913i
$148$		0			0
$149$	930.055	−	1108.40i	0.0418925	−	0.0499255i	−0.744691	−	0.667409i	$-0.767403\pi$
$149$	930.055	−	1108.40i	0.0418925	−	0.0499255i	0.786584	+	0.617483i	$0.211848\pi$
$150$		0			0
$151$	35146.0	+	12792.1i	1.54142	+	0.561033i	0.966386	−	0.257094i	$-0.0827649\pi$
$151$	35146.0	+	12792.1i	1.54142	+	0.561033i	0.575038	+	0.818126i	$0.304987\pi$
$152$		0			0
$153$	−10507.9	+	7159.43i	−0.448882	+	0.305841i
$154$		0			0
$155$	−3353.34	+	591.285i	−0.139577	+	0.0246112i
$156$		0			0
$157$	−17397.8	+	6332.28i	−0.705822	+	0.256898i	−0.669895	−	0.742456i	$-0.733661\pi$
$157$	−17397.8	+	6332.28i	−0.705822	+	0.256898i	−0.0359274	+	0.999354i	$0.511439\pi$
$158$		0			0
$159$	24856.2	−	15612.9i	0.983196	−	0.617573i
$160$		0			0
$161$		−	4786.10i		−	0.184642i
$162$		0			0
$163$	7731.09			0.290982			0.145491	−	0.989360i	$-0.453524\pi$
$163$	7731.09			0.290982			0.145491	+	0.989360i	$0.453524\pi$
$164$		0			0
$165$	−9830.09	−	15649.8i	−0.361069	−	0.574833i
$166$		0			0
$167$	12185.8	+	33480.3i	0.436941	+	1.20048i	0.941472	+	0.337091i	$0.109443\pi$
$167$	12185.8	+	33480.3i	0.436941	+	1.20048i	−0.504531	+	0.863393i	$0.668335\pi$
$168$		0			0
$169$	−3167.93	−	17966.2i	−0.110918	−	0.629047i
$170$		0			0
$171$	−38445.8	+	2869.26i	−1.31479	+	0.0981245i
$172$		0			0
$173$	−9745.05	+	26774.3i	−0.325606	+	0.894594i	0.663604	+	0.748084i	$0.269026\pi$
$173$	−9745.05	+	26774.3i	−0.325606	+	0.894594i	−0.989209	+	0.146510i	$0.953196\pi$
$174$		0			0
$175$	−38391.3	−	32214.2i	−1.25359	−	1.05189i
$176$		0			0
$177$	−19629.7	+	25246.4i	−0.626567	+	0.805849i
$178$		0			0
$179$	−970.180	−	560.134i	−0.0302793	−	0.0174818i	0.484784	−	0.874634i	$-0.338898\pi$
$179$	−970.180	−	560.134i	−0.0302793	−	0.0174818i	−0.515063	+	0.857152i	$0.672232\pi$
$180$		0			0
$181$	14022.2	+	24287.1i	0.428014	+	0.741341i	0.996697	−	0.0812156i	$-0.0258802\pi$
$181$	14022.2	+	24287.1i	0.428014	+	0.741341i	−0.568683	+	0.822557i	$0.692547\pi$
$182$		0			0
$183$	−28597.8	+	25871.0i	−0.853945	+	0.772521i
$184$		0			0
$185$	9265.06	+	1633.68i	0.270710	+	0.0477336i
$186$		0			0
$187$	26210.4	−	21993.1i	0.749533	−	0.628933i
$188$		0			0
$189$	19238.7	+	65358.1i	0.538582	+	1.82968i
$190$		0			0
$191$	28168.4	+	33569.8i	0.772140	+	0.920201i	0.998550	−	0.0538315i	$-0.0171434\pi$
$191$	28168.4	+	33569.8i	0.772140	+	0.920201i	−0.226410	+	0.974032i	$0.572699\pi$
$192$		0			0
$193$	4122.59	−	23380.4i	0.110677	−	0.627678i	−0.878124	−	0.478434i	$-0.841205\pi$
$193$	4122.59	−	23380.4i	0.110677	−	0.627678i	0.988800	−	0.149245i	$-0.0476842\pi$
$194$		0			0
$195$	−8420.09	+	1810.35i	−0.221436	+	0.0476094i
$196$		0			0
$197$	44291.2	−	25571.6i	1.14126	−	0.658908i	0.194519	−	0.980899i	$-0.437685\pi$
$197$	44291.2	−	25571.6i	1.14126	−	0.658908i	0.946743	+	0.321991i	$0.104352\pi$
$198$		0			0
$199$	−4204.89	+	7283.09i	−0.106181	+	0.183912i	−0.914220	−	0.405218i	$-0.867196\pi$
$199$	−4204.89	+	7283.09i	−0.106181	+	0.183912i	0.808039	+	0.589129i	$0.200529\pi$
$200$		0			0
$201$	−8697.03	+	62951.0i	−0.215268	+	1.55816i
$202$		0			0
$203$	−87560.9	+	104351.i	−2.12480	+	2.53224i
$204$		0			0
$205$	−17537.2	−	6383.00i	−0.417303	−	0.151886i
$206$		0			0
$207$	−2422.46	+	3367.26i	−0.0565349	+	0.0785844i
$208$		0			0
$209$	102166.	−	18014.7i	2.33892	−	0.412414i
$210$		0			0
$211$	30234.2	−	11004.4i	0.679101	−	0.247172i	0.0206394	−	0.999787i	$-0.493430\pi$
$211$	30234.2	−	11004.4i	0.679101	−	0.247172i	0.658461	+	0.752615i	$0.271208\pi$
$212$		0			0
$213$	−2455.70	−	65900.3i	−0.0541273	−	1.45254i
$214$		0			0
$215$		−	7412.58i		−	0.160359i
$216$		0			0
$217$	−33778.9			−0.717342
$218$		0			0
$219$	−7163.58	+	13549.1i	−0.149363	+	0.282503i
$220$		0			0
$221$	−5453.50	−	14983.4i	−0.111658	−	0.306778i
$222$		0			0
$223$	−3467.79	−	19666.8i	−0.0697337	−	0.395480i	−0.999618	−	0.0276281i	$-0.991205\pi$
$223$	−3467.79	−	19666.8i	−0.0697337	−	0.395480i	0.929885	−	0.367852i	$-0.119907\pi$
$224$		0			0
$225$	10705.2	+	42096.0i	0.211461	+	0.831525i
$226$		0			0
$227$	3080.37	−	8463.24i	0.0597793	−	0.164242i	−0.906208	−	0.422833i	$-0.861036\pi$
$227$	3080.37	−	8463.24i	0.0597793	−	0.164242i	0.965987	+	0.258590i	$0.0832580\pi$
$228$		0			0
$229$	1739.18	+	1459.35i	0.0331646	+	0.0278284i	0.659219	−	0.751951i	$-0.270887\pi$
$229$	1739.18	+	1459.35i	0.0331646	+	0.0278284i	−0.626055	+	0.779779i	$0.715331\pi$
$230$		0			0
$231$	−69076.0	−	169824.i	−1.29450	−	3.18255i
$232$		0			0
$233$	−34404.1	−	19863.2i	−0.633721	−	0.365879i	0.148471	−	0.988917i	$-0.452565\pi$
$233$	−34404.1	−	19863.2i	−0.633721	−	0.365879i	−0.782192	+	0.623038i	$0.785898\pi$
$234$		0			0
$235$	1342.26	+	2324.86i	0.0243052	+	0.0420979i
$236$		0			0
$237$	−52166.8	−	16815.2i	−0.928747	−	0.299367i
$238$		0			0
$239$	−21056.6	−	3712.84i	−0.368631	−	0.0649995i	−0.0137356	−	0.999906i	$-0.504372\pi$
$239$	−21056.6	−	3712.84i	−0.368631	−	0.0649995i	−0.354895	+	0.934906i	$0.615483\pi$
$240$		0			0
$241$	23758.0	−	19935.4i	0.409050	−	0.343234i	−0.414929	−	0.909854i	$-0.636194\pi$
$241$	23758.0	−	19935.4i	0.409050	−	0.343234i	0.823979	+	0.566620i	$0.191749\pi$
$242$		0			0
$243$	19545.4	−	55720.4i	0.331002	−	0.943630i
$244$		0			0
$245$	38353.1	+	45707.4i	0.638952	+	0.761473i
$246$		0			0
$247$	8395.18	−	47611.4i	0.137606	−	0.780400i
$248$		0			0
$249$	−6919.40	+	21466.5i	−0.111601	+	0.346228i
$250$		0			0
$251$	82656.9	−	47722.0i	1.31199	−	0.757479i	0.329567	−	0.944132i	$-0.393097\pi$
$251$	82656.9	−	47722.0i	1.31199	−	0.757479i	0.982426	+	0.186653i	$0.0597640\pi$
$252$		0			0
$253$	5581.12	−	9666.79i	0.0871928	−	0.151022i
$254$		0			0
$255$	12329.0	−	5014.82i	0.189604	−	0.0771214i
$256$		0			0
$257$	−53966.9	+	64315.2i	−0.817073	+	0.973750i	−0.999956	−	0.00938502i	$-0.997013\pi$
$257$	−53966.9	+	64315.2i	−0.817073	+	0.973750i	0.182883	+	0.983135i	$0.441457\pi$
$258$		0			0
$259$	87700.5	+	31920.4i	1.30738	+	0.475848i
$260$		0			0
$261$	114421.	−	29097.7i	1.67967	−	0.427147i
$262$		0			0
$263$	−31439.9	+	5543.70i	−0.454537	+	0.0801471i	−0.396229	−	0.918152i	$-0.629681\pi$
$263$	−31439.9	+	5543.70i	−0.454537	+	0.0801471i	−0.0583075	+	0.998299i	$0.518570\pi$
$264$		0			0
$265$	−28872.9	+	10508.9i	−0.411149	+	0.149646i
$266$		0			0
$267$	90150.1	+	47663.4i	1.26457	+	0.668594i
$268$		0			0
$269$		−	48870.4i		−	0.675370i	−0.941259	−	0.337685i	$-0.890356\pi$
$269$		−	48870.4i		−	0.675370i	0.941259	−	0.337685i	$-0.109644\pi$
$270$		0			0
$271$	59047.3			0.804010			0.402005	−	0.915637i	$-0.368313\pi$
$271$	59047.3			0.804010			0.402005	+	0.915637i	$0.368313\pi$
$272$		0			0
$273$	−85378.3	+	3181.52i	−1.14557	+	0.0426884i
$274$		0			0
$275$	−39976.2	−	109834.i	−0.528610	−	1.45235i
$276$		0			0
$277$	6145.10	+	34850.6i	0.0800884	+	0.454204i	0.998309	+	0.0581339i	$0.0185150\pi$
$277$	6145.10	+	34850.6i	0.0800884	+	0.454204i	−0.918220	+	0.396070i	$0.870374\pi$
$278$		0			0
$279$	23765.2	+	17097.1i	0.305305	+	0.219641i
$280$		0			0
$281$	−9557.52	+	26259.1i	−0.121041	+	0.332557i	−0.985385	−	0.170344i	$-0.945512\pi$
$281$	−9557.52	+	26259.1i	−0.121041	+	0.332557i	0.864344	+	0.502902i	$0.167734\pi$
$282$		0			0
$283$	4019.76	+	3372.98i	0.0501912	+	0.0421154i	0.667538	−	0.744576i	$-0.267348\pi$
$283$	4019.76	+	3372.98i	0.0501912	+	0.0421154i	−0.617347	+	0.786691i	$0.711793\pi$
$284$		0			0
$285$	39976.4	+	5522.95i	0.492169	+	0.0679957i
$286$		0			0
$287$	−160333.	−	92568.4i	−1.94652	−	1.12383i
$288$		0			0
$289$	−29439.8	−	50991.2i	−0.352483	−	0.610519i
$290$		0			0
$291$	−12877.7	−	59895.4i	−0.152073	−	0.707307i
$292$		0			0
$293$	132502.	+	23363.7i	1.54344	+	0.272149i	0.879594	−	0.475725i	$-0.157814\pi$
$293$	132502.	+	23363.7i	1.54344	+	0.272149i	0.663841	+	0.747874i	$0.268925\pi$
$294$		0			0
$295$	25643.9	−	21517.8i	0.294673	−	0.247260i
$296$		0			0
$297$	−37357.2	+	154442.i	−0.423508	+	1.75087i
$298$		0			0
$299$	−3343.66	−	3984.82i	−0.0374007	−	0.0445725i
$300$		0			0
$301$	12769.1	−	72416.9i	0.140937	−	0.799295i
$302$		0			0
$303$	61218.5	+	67670.9i	0.666802	+	0.737083i
$304$		0			0
$305$	34959.2	−	20183.7i	0.375804	−	0.216970i
$306$		0			0
$307$	−81998.8	+	142026.i	−0.870023	+	1.50692i	−0.00805046	+	0.999968i	$0.502563\pi$
$307$	−81998.8	+	142026.i	−0.870023	+	1.50692i	−0.861972	+	0.506956i	$0.830771\pi$
$308$		0			0
$309$	49095.1	+	38172.6i	0.514187	+	0.399792i
$310$		0			0
$311$	19542.3	−	23289.6i	0.202048	−	0.240792i	−0.655500	−	0.755195i	$-0.727542\pi$
$311$	19542.3	−	23289.6i	0.202048	−	0.240792i	0.857548	+	0.514403i	$0.171987\pi$
$312$		0			0
$313$	11927.4	+	4341.21i	0.121746	+	0.0443121i	0.402175	−	0.915563i	$-0.368254\pi$
$313$	11927.4	+	4341.21i	0.121746	+	0.0443121i	−0.280429	+	0.959875i	$0.590477\pi$
$314$		0			0
$315$	−5307.78	−	71120.1i	−0.0534924	−	0.716755i
$316$		0			0
$317$	140231.	−	24726.4i	1.39548	−	0.246061i	0.575196	−	0.818016i	$-0.304926\pi$
$317$	140231.	−	24726.4i	1.39548	−	0.246061i	0.820285	+	0.571955i	$0.193815\pi$
$318$		0			0
$319$	−298537.	+	108659.i	−2.93371	+	1.06778i
$320$		0			0
$321$	51785.6	−	32527.9i	0.502572	−	0.315680i
$322$		0			0
$323$			74714.2i			0.716140i
$324$		0			0
$325$	−54469.5			−0.515687
$326$		0			0
$327$	6238.43	+	9931.78i	0.0583418	+	0.0928820i
$328$		0			0
$329$	9108.29	+	25024.8i	0.0841482	+	0.231195i
$330$		0			0
$331$	−23354.9	−	132452.i	−0.213168	−	1.20893i	−0.884058	−	0.467377i	$-0.845199\pi$
$331$	−23354.9	−	132452.i	−0.213168	−	1.20893i	0.670890	−	0.741557i	$-0.265912\pi$
$332$		0			0
$333$	−45545.4	−	66846.9i	−0.410730	−	0.602827i
$334$		0			0
$335$	22751.7	−	62509.9i	0.202733	−	0.557005i
$336$		0			0
$337$	−11763.7	−	9870.88i	−0.103582	−	0.0869152i	0.589526	−	0.807749i	$-0.299315\pi$
$337$	−11763.7	−	9870.88i	−0.103582	−	0.0869152i	−0.693108	+	0.720834i	$0.743759\pi$
$338$		0			0
$339$	−45222.2	+	58161.9i	−0.393507	+	0.506103i
$340$		0			0
$341$	−68225.5	−	39390.0i	−0.586730	−	0.338748i
$342$		0			0
$343$	183756.	+	318275.i	1.56190	+	2.70530i
$344$		0			0
$345$	3220.04	−	2913.01i	0.0270535	−	0.0244739i
$346$		0			0
$347$	−131462.	−	23180.4i	−1.09180	−	0.192513i	−0.401371	−	0.915916i	$-0.631466\pi$
$347$	−131462.	−	23180.4i	−1.09180	−	0.192513i	−0.690427	+	0.723402i	$0.742577\pi$
$348$		0			0
$349$	−81464.1	+	68356.5i	−0.668830	+	0.561215i	−0.912719	−	0.408588i	$-0.866021\pi$
$349$	−81464.1	+	68356.5i	−0.668830	+	0.561215i	0.243889	+	0.969803i	$0.421577\pi$
$350$		0			0
$351$	61678.3	+	40975.5i	0.500632	+	0.332591i
$352$		0			0
$353$	−128967.	−	153697.i	−1.03497	−	1.23343i	−0.971893	−	0.235422i	$-0.924353\pi$
$353$	−128967.	−	153697.i	−1.03497	−	1.23343i	−0.0630783	−	0.998009i	$-0.520092\pi$
$354$		0			0
$355$	−11987.1	+	67982.2i	−0.0951168	+	0.539434i
$356$		0			0
$357$	129086.	−	27754.0i	1.01284	−	0.217765i
$358$		0			0
$359$	−138211.	+	79796.1i	−1.07239	+	0.619145i	−0.928834	−	0.370497i	$-0.879188\pi$
$359$	−138211.	+	79796.1i	−1.07239	+	0.619145i	−0.143557	+	0.989642i	$0.545854\pi$
$360$		0			0
$361$	−48108.1	+	83325.7i	−0.369151	+	0.639388i
$362$		0			0
$363$	40483.1	−	293026.i	0.307228	−	2.22378i
$364$		0			0
$365$	10312.4	−	12289.8i	0.0774059	−	0.0922488i
$366$		0			0
$367$	19730.1	+	7181.15i	0.146486	+	0.0533165i	0.414223	−	0.910176i	$-0.364053\pi$
$367$	19730.1	+	7181.15i	0.146486	+	0.0533165i	−0.267737	+	0.963492i	$0.586276\pi$
$368$		0			0
$369$	65949.5	+	146279.i	0.484349	+	1.07431i
$370$		0			0
$371$	−300176.	+	52929.1i	−2.18086	+	0.384545i
$372$		0			0
$373$	126485.	−	46036.6i	0.909117	−	0.330892i	0.155217	−	0.987880i	$-0.450392\pi$
$373$	126485.	−	46036.6i	0.909117	−	0.330892i	0.753900	+	0.656989i	$0.228170\pi$
$374$		0			0
$375$	−3666.48	−	98392.5i	−0.0260728	−	0.699680i
$376$		0			0
$377$			148053.i			1.04168i
$378$		0			0
$379$	−14232.6			−0.0990847			−0.0495424	−	0.998772i	$-0.515776\pi$
$379$	−14232.6			−0.0990847			−0.0495424	+	0.998772i	$0.515776\pi$
$380$		0			0
$381$	−59892.4	+	113280.i	−0.412593	+	0.780375i
$382$		0			0
$383$	29653.3	+	81471.7i	0.202151	+	0.555405i	0.998797	−	0.0490416i	$-0.0156167\pi$
$383$	29653.3	+	81471.7i	0.202151	+	0.555405i	−0.796646	+	0.604446i	$0.793394\pi$
$384$		0			0
$385$	33325.0	+	188995.i	0.224827	+	1.27506i
$386$		0			0
$387$	−45637.3	+	44486.0i	−0.304718	+	0.297031i
$388$		0			0
$389$	57297.2	−	157423.i	0.378647	−	1.04032i	−0.593271	−	0.805003i	$-0.702164\pi$
$389$	57297.2	−	157423.i	0.378647	−	1.04032i	0.971918	−	0.235321i	$-0.0756140\pi$
$390$		0			0
$391$	6158.18	+	5167.33i	0.0402809	+	0.0337997i
$392$		0			0
$393$	10074.4	+	24768.0i	0.0652279	+	0.160363i
$394$		0			0
$395$	49687.1	+	28686.9i	0.318456	+	0.183861i
$396$		0			0
$397$	−70460.9	−	122042.i	−0.447062	−	0.774333i	0.551132	−	0.834418i	$-0.314196\pi$
$397$	−70460.9	−	122042.i	−0.447062	−	0.774333i	−0.998193	+	0.0600849i	$0.980863\pi$
$398$		0			0
$399$	381034.	+	122820.i	2.39341	+	0.771480i
$400$		0			0
$401$	−279401.	−	49266.0i	−1.73756	−	0.306379i	−0.787008	−	0.616943i	$-0.788371\pi$
$401$	−279401.	−	49266.0i	−1.73756	−	0.306379i	−0.950552	+	0.310564i	$0.899482\pi$
$402$		0			0
$403$	−28123.8	+	23598.7i	−0.173166	+	0.145304i
$404$		0			0
$405$	−32262.9	+	52723.1i	−0.196695	+	0.321434i
$406$		0			0
$407$	139912.	+	166740.i	0.844627	+	1.00659i
$408$		0			0
$409$	−29298.1	+	166158.i	−0.175143	+	0.993286i	0.762836	+	0.646593i	$0.223807\pi$
$409$	−29298.1	+	166158.i	−0.175143	+	0.993286i	−0.937979	+	0.346693i	$0.887305\pi$
$410$		0			0
$411$	−45523.9	+	141232.i	−0.269498	+	0.836082i
$412$		0			0
$413$	287594.	−	166043.i	1.68609	−	0.973463i
$414$		0			0
$415$	11804.6	−	20446.1i	0.0685416	−	0.118718i
$416$		0			0
$417$	−32119.2	+	13064.5i	−0.184711	+	0.0751314i
$418$		0			0
$419$	209146.	−	249251.i	1.19130	−	1.41974i	0.307713	−	0.951479i	$-0.400436\pi$
$419$	209146.	−	249251.i	1.19130	−	1.41974i	0.883590	−	0.468261i	$-0.155119\pi$
$420$		0			0
$421$	−186899.	−	68025.7i	−1.05449	−	0.383804i	−0.244136	−	0.969741i	$-0.578504\pi$
$421$	−186899.	−	68025.7i	−1.05449	−	0.383804i	−0.810356	+	0.585938i	$0.800726\pi$
$422$		0			0
$423$	6258.07	−	22216.4i	0.0349752	−	0.124163i
$424$		0			0
$425$	82898.7	−	14617.3i	0.458955	−	0.0809261i
$426$		0			0
$427$	376301.	−	136962.i	2.06386	−	0.751182i
$428$		0			0
$429$	−176154.	−	93134.7i	−0.957146	−	0.506054i
$430$		0			0
$431$			42286.8i			0.227641i	0.993501	+	0.113820i	$0.0363088\pi$
$431$			42286.8i			0.227641i	−0.993501	+	0.113820i	$0.963691\pi$
$432$		0			0
$433$	−133410.			−0.711564			−0.355782	−	0.934569i	$-0.615785\pi$
$433$	−133410.			−0.711564			−0.355782	+	0.934569i	$0.615785\pi$
$434$		0			0
$435$	−123499.	+	4602.06i	−0.652659	+	0.0243206i
$436$		0			0
$437$	8336.55	+	22904.5i	0.0436540	+	0.119938i
$438$		0			0
$439$	−1918.30	−	10879.2i	−0.00995375	−	0.0564505i	0.979426	−	0.201802i	$-0.0646798\pi$
$439$	−1918.30	−	10879.2i	−0.00995375	−	0.0564505i	−0.989380	+	0.145352i	$0.953569\pi$
$440$		0			0
$441$	51235.2	−	510439.i	0.263446	−	2.62462i
$442$		0			0
$443$	−14620.6	+	40169.8i	−0.0745004	+	0.204688i	−0.971353	−	0.237642i	$-0.923626\pi$
$443$	−14620.6	+	40169.8i	−0.0745004	+	0.204688i	0.896853	+	0.442330i	$0.145848\pi$
$444$		0			0
$445$	−81771.3	−	68614.3i	−0.412934	−	0.346493i
$446$		0			0
$447$	12899.6	+	1782.16i	0.0645599	+	0.00891930i
$448$		0			0
$449$	−217766.	−	125727.i	−1.08018	−	0.623643i	−0.149236	−	0.988802i	$-0.547681\pi$
$449$	−217766.	−	125727.i	−1.08018	−	0.623643i	−0.930945	+	0.365159i	$0.881015\pi$
$450$		0			0
$451$	−215890.	−	373933.i	−1.06140	−	1.83840i
$452$		0			0
$453$	70756.4	+	329094.i	0.344802	+	1.60370i
$454$		0			0
$455$	88075.4	+	15530.1i	0.425434	+	0.0750154i
$456$		0			0
$457$	−210621.	+	176732.i	−1.00848	+	0.846218i	−0.988137	−	0.153575i	$-0.950921\pi$
$457$	−210621.	+	176732.i	−1.00848	+	0.846218i	−0.0203462	+	0.999793i	$0.506477\pi$
$458$		0			0
$459$	−104866.	−	45810.1i	−0.497749	−	0.217438i
$460$		0			0
$461$	−108791.	−	129652.i	−0.511906	−	0.610066i	0.446741	−	0.894663i	$-0.352585\pi$
$461$	−108791.	−	129652.i	−0.511906	−	0.610066i	−0.958647	+	0.284597i	$0.908140\pi$
$462$		0			0
$463$	−36940.9	+	209502.i	−0.172324	+	0.977298i	0.768864	+	0.639413i	$0.220823\pi$
$463$	−36940.9	+	209502.i	−0.172324	+	0.977298i	−0.941187	+	0.337885i	$0.890289\pi$
$464$		0			0
$465$	−20559.2	−	22726.1i	−0.0950824	−	0.105104i
$466$		0			0
$467$	291740.	−	168436.i	1.33771	−	0.772326i	0.351241	−	0.936285i	$-0.385760\pi$
$467$	291740.	−	168436.i	1.33771	−	0.772326i	0.986467	+	0.163958i	$0.0524263\pi$
$468$		0			0
$469$	329953.	−	571496.i	1.50005	−	2.59817i
$470$		0			0
$471$	−131545.	−	102280.i	−0.592971	−	0.461049i
$472$		0			0
$473$	110237.	−	131375.i	0.492724	−	0.587206i
$474$		0			0
$475$	239838.	+	87294.0i	1.06300	+	0.386899i
$476$		0			0
$477$	237979.	+	114695.i	1.04593	+	0.504088i
$478$		0			0
$479$	46210.0	−	8148.07i	0.201403	−	0.0355127i	−0.0720366	−	0.997402i	$-0.522950\pi$
$479$	46210.0	−	8148.07i	0.201403	−	0.0355127i	0.273439	+	0.961889i	$0.411839\pi$
$480$		0			0
$481$	95318.2	−	34693.0i	0.411989	−	0.149952i
$482$		0			0
$483$	36476.0	−	22911.6i	0.156356	−	0.0982113i
$484$		0			0
$485$			64130.0i			0.272632i
$486$		0			0
$487$	303643.			1.28028			0.640141	−	0.768257i	$-0.278876\pi$
$487$	303643.			1.28028			0.640141	+	0.768257i	$0.278876\pi$
$488$		0			0
$489$	37009.7	+	58920.6i	0.154774	+	0.246405i
$490$		0			0
$491$	94072.6	+	258462.i	0.390212	+	1.07210i	0.966905	+	0.255137i	$0.0821206\pi$
$491$	94072.6	+	258462.i	0.390212	+	1.07210i	−0.576693	+	0.816961i	$0.695657\pi$
$492$		0			0
$493$	−39731.1	−	225326.i	−0.163469	−	0.927081i
$494$		0			0
$495$	72213.5	−	149835.i	0.294719	−	0.611510i
$496$		0			0
$497$	−234215.	+	643500.i	−0.948204	+	2.60517i
$498$		0			0
$499$	122094.	+	102449.i	0.490336	+	0.411441i	0.854147	−	0.520032i	$-0.174080\pi$
$499$	122094.	+	102449.i	0.490336	+	0.411441i	−0.363811	+	0.931473i	$0.618524\pi$
$500$		0			0
$501$	−196827.	+	253146.i	−0.784168	+	1.00854i
$502$		0			0
$503$	−231464.	−	133636.i	−0.914846	−	0.528187i	−0.0328591	−	0.999460i	$-0.510461\pi$
$503$	−231464.	−	133636.i	−0.914846	−	0.528187i	−0.881987	+	0.471273i	$0.843795\pi$
$504$		0			0
$505$	−47760.6	−	82723.8i	−0.187278	−	0.324375i
$506$		0			0
$507$	121760.	−	110150.i	0.473683	−	0.428518i
$508$		0			0
$509$	−49139.8	−	8664.68i	−0.189670	−	0.0334439i	0.0780063	−	0.996953i	$-0.475145\pi$
$509$	−49139.8	−	8664.68i	−0.189670	−	0.0334439i	−0.267676	+	0.963509i	$0.586256\pi$
$510$		0			0
$511$	121917.	−	102301.i	0.466900	−	0.391775i
$512$		0			0
$513$	−205912.	−	279269.i	−0.782432	−	1.06118i
$514$		0			0
$515$	−41844.2	−	49868.0i	−0.157769	−	0.188021i
$516$		0			0
$517$	−10785.1	+	61165.5i	−0.0403501	+	0.228837i
$518$		0			0
$519$	−250704.	+	53902.3i	−0.930738	+	0.200112i
$520$		0			0
$521$	184784.	−	106685.i	0.680752	−	0.393032i	−0.119386	−	0.992848i	$-0.538093\pi$
$521$	184784.	−	106685.i	0.680752	−	0.393032i	0.800138	+	0.599816i	$0.204759\pi$
$522$		0			0
$523$	30678.3	−	53136.3i	0.112157	−	0.194262i	−0.804483	−	0.593976i	$-0.797557\pi$
$523$	30678.3	−	53136.3i	0.112157	−	0.194262i	0.916640	+	0.399714i	$0.130891\pi$
$524$		0			0
$525$	61728.2	−	446803.i	0.223957	−	1.62105i
$526$		0			0
$527$	36469.6	−	43462.7i	0.131314	−	0.156493i
$528$		0			0
$529$	−260500.	−	94814.3i	−0.930886	−	0.338815i
$530$		0			0
$531$	−286379.	−	28745.3i	−1.01567	−	0.101948i
$532$		0			0
$533$	−198161.	+	34941.1i	−0.697531	+	0.122994i
$534$		0			0
$535$	−60154.1	+	21894.3i	−0.210164	+	0.0764933i
$536$		0			0
$537$	−375.449	−	10075.4i	−0.00130197	−	0.0349393i
$538$		0			0
$539$			1.38046e6i			4.75165i
$540$		0			0
$541$	−68196.0			−0.233005			−0.116502	−	0.993190i	$-0.537168\pi$
$541$	−68196.0			−0.233005			−0.116502	+	0.993190i	$0.537168\pi$
$542$		0			0
$543$	−117972.	+	223132.i	−0.400111	+	0.756765i
$544$		0			0
$545$	−4199.03	−	11536.8i	−0.0141370	−	0.0388410i
$546$		0			0
$547$	87643.7	+	497052.i	0.292918	+	1.66122i	0.675547	+	0.737317i	$0.263908\pi$
$547$	87643.7	+	497052.i	0.292918	+	1.66122i	−0.382629	+	0.923902i	$0.624981\pi$
$548$		0			0
$549$	−334070.	−	94103.3i	−1.10839	−	0.312220i
$550$		0			0
$551$	237273.	−	651902.i	0.781529	−	2.14723i
$552$		0			0
$553$	436000.	+	365848.i	1.42573	+	1.19633i
$554$		0			0
$555$	31902.3	+	78432.0i	0.103570	+	0.254629i
$556$		0			0
$557$	316880.	+	182951.i	1.02137	+	0.589691i	0.914502	−	0.404582i	$-0.132583\pi$
$557$	316880.	+	182951.i	1.02137	+	0.589691i	0.106873	+	0.994273i	$0.465916\pi$
$558$		0			0
$559$	−39960.6	−	69213.9i	−0.127882	−	0.221498i
$560$		0			0
$561$	293088.	+	94472.3i	0.931262	+	0.300178i
$562$		0			0
$563$	486748.	+	85826.8i	1.53563	+	0.270773i	0.876555	−	0.481302i	$-0.159836\pi$
$563$	486748.	+	85826.8i	1.53563	+	0.270773i	0.659077	+	0.752075i	$0.270947\pi$
$564$		0			0
$565$	59077.6	−	49572.0i	0.185066	−	0.155288i
$566$		0			0
$567$	−406013.	+	459500.i	−1.26291	+	1.42929i
$568$		0			0
$569$	62776.8	+	74814.5i	0.193899	+	0.231079i	0.854230	−	0.519895i	$-0.174029\pi$
$569$	62776.8	+	74814.5i	0.193899	+	0.231079i	−0.660332	+	0.750974i	$0.729584\pi$
$570$		0			0
$571$	101941.	−	578137.i	0.312664	−	1.77320i	−0.272367	−	0.962193i	$-0.587806\pi$
$571$	101941.	−	578137.i	0.312664	−	1.77320i	0.585031	−	0.811011i	$-0.301082\pi$
$572$		0			0
$573$	−120998.	+	375382.i	−0.368528	+	1.14331i
$574$		0			0
$575$	23782.6	−	13730.9i	0.0719322	−	0.0415301i
$576$		0			0
$577$	−118959.	+	206042.i	−0.357309	+	0.618878i	−0.987510	−	0.157554i	$-0.949639\pi$
$577$	−118959.	+	206042.i	−0.357309	+	0.618878i	0.630201	+	0.776432i	$0.282972\pi$
$578$		0			0
$579$	197923.	−	80505.4i	0.590391	−	0.240142i
$580$		0			0
$581$	150545.	−	179413.i	0.445980	−	0.531498i
$582$		0			0
$583$	−668006.	−	243134.i	−1.96536	−	0.715334i
$584$		0			0
$585$	−54105.1	−	55505.3i	−0.158098	−	0.162189i
$586$		0			0
$587$	−356147.	+	62798.3i	−1.03360	+	0.182252i	−0.664617	−	0.747184i	$-0.731405\pi$
$587$	−356147.	+	62798.3i	−1.03360	+	0.182252i	−0.368984	+	0.929436i	$0.620294\pi$
$588$		0			0
$589$	161654.	−	58837.1i	0.465967	−	0.169598i
$590$		0			0
$591$	406915.	+	215141.i	1.16501	+	0.615952i
$592$		0			0
$593$			434745.i			1.23630i	0.786059	+	0.618151i	$0.212118\pi$
$593$			434745.i			1.23630i	−0.786059	+	0.618151i	$0.787882\pi$
$594$		0			0
$595$	−138212.			−0.390403
$596$		0			0
$597$	−75635.6	+	2818.47i	−0.212216	+	0.00790797i
$598$		0			0
$599$	−48494.3	−	133237.i	−0.135157	−	0.371340i	0.853589	−	0.520947i	$-0.174421\pi$
$599$	−48494.3	−	133237.i	−0.135157	−	0.371340i	−0.988745	+	0.149608i	$0.952199\pi$
$600$		0			0
$601$	−92872.1	−	526704.i	−0.257120	−	1.45820i	−0.790571	−	0.612370i	$-0.790216\pi$
$601$	−92872.1	−	526704.i	−0.257120	−	1.45820i	0.533451	−	0.845831i	$-0.320895\pi$
$602$		0			0
$603$	−521399.	+	235072.i	−1.43396	+	0.646497i
$604$		0			0
$605$	−105905.	+	290972.i	−0.289339	+	0.794952i
$606$		0			0
$607$	−288297.	−	241910.i	−0.782462	−	0.656564i	0.161405	−	0.986888i	$-0.448397\pi$
$607$	−288297.	−	241910.i	−0.782462	−	0.656564i	−0.943867	+	0.330324i	$0.892842\pi$
$608$		0			0
$609$	−1.21445e6	−	167783.i	−3.27450	−	0.452390i
$610$		0			0
$611$	25066.2	+	14472.0i	0.0671440	+	0.0387656i
$612$		0			0
$613$	62517.2	+	108283.i	0.166371	+	0.288164i	0.937141	−	0.348950i	$-0.113462\pi$
$613$	62517.2	+	108283.i	0.166371	+	0.288164i	−0.770770	+	0.637113i	$0.780128\pi$
$614$		0			0
$615$	−35306.0	−	164211.i	−0.0933466	−	0.434163i
$616$		0			0
$617$	130509.	+	23012.2i	0.342823	+	0.0604489i	0.342409	−	0.939551i	$-0.388757\pi$
$617$	130509.	+	23012.2i	0.342823	+	0.0604489i	0.000413857		1.00000i	$0.499868\pi$
$618$		0			0
$619$	−69930.6	+	58678.7i	−0.182510	+	0.153144i	−0.729465	−	0.684018i	$-0.760231\pi$
$619$	−69930.6	+	58678.7i	−0.182510	+	0.153144i	0.546956	+	0.837162i	$0.315787\pi$
$620$		0			0
$621$	−37259.4	−	2342.71i	−0.0966168	−	0.00607486i
$622$		0			0
$623$	−680666.	−	811186.i	−1.75371	−	2.08999i
$624$		0			0
$625$	40301.4	−	228561.i	0.103172	−	0.585115i
$626$		0			0
$627$	626377.	+	692397.i	1.59331	+	1.76125i
$628$		0			0
$629$	−135758.	+	78379.7i	−0.343133	+	0.198108i
$630$		0			0
$631$	129807.	−	224832.i	0.326017	−	0.564677i	−0.655701	−	0.755021i	$-0.727627\pi$
$631$	129807.	−	224832.i	0.326017	−	0.564677i	0.981718	+	0.190343i	$0.0609601\pi$
$632$		0			0
$633$	228602.	+	177744.i	0.570522	+	0.443595i
$634$		0			0
$635$	86218.7	−	102751.i	0.213823	−	0.254824i
$636$		0			0
$637$	604521.	+	220028.i	1.48982	+	0.542249i
$638$		0			0
$639$	490487.	−	334188.i	1.20123	−	0.818445i
$640$		0			0
$641$	61880.0	−	10911.1i	0.150603	−	0.0265554i	−0.0978382	−	0.995202i	$-0.531193\pi$
$641$	61880.0	−	10911.1i	0.150603	−	0.0265554i	0.248441	+	0.968647i	$0.420082\pi$
$642$		0			0
$643$	−374496.	+	136305.i	−0.905786	+	0.329679i	−0.752569	−	0.658514i	$-0.771186\pi$
$643$	−374496.	+	136305.i	−0.905786	+	0.329679i	−0.153217	+	0.988193i	$0.548963\pi$
$644$		0			0
$645$	56493.2	−	35484.9i	0.135793	−	0.0852952i
$646$		0			0
$647$			397193.i			0.948840i	0.880299	+	0.474420i	$0.157342\pi$
$647$			397193.i			0.948840i	−0.880299	+	0.474420i	$0.842658\pi$
$648$		0			0
$649$	774497.			1.83878
$650$		0			0
$651$	−161704.	−	257438.i	−0.381556	−	0.607450i
$652$		0			0
$653$	−58683.0	−	161230.i	−0.137622	−	0.378112i	0.851667	−	0.524083i	$-0.175592\pi$
$653$	−58683.0	−	161230.i	−0.137622	−	0.378112i	−0.989289	+	0.145971i	$0.953369\pi$
$654$		0			0
$655$	−4860.28	−	27564.0i	−0.0113287	−	0.0642480i
$656$		0			0
$657$	−137554.	+	10265.9i	−0.318672	+	0.0237829i
$658$		0			0
$659$	−31406.5	+	86288.6i	−0.0723184	+	0.198693i	−0.970585	−	0.240757i	$-0.922604\pi$
$659$	−31406.5	+	86288.6i	−0.0723184	+	0.198693i	0.898267	+	0.439450i	$0.144827\pi$
$660$		0			0
$661$	601557.	+	504767.i	1.37681	+	1.15528i	0.970376	+	0.241601i	$0.0776726\pi$
$661$	601557.	+	504767.i	1.37681	+	1.15528i	0.406435	+	0.913680i	$0.366772\pi$
$662$		0			0
$663$	88085.4	−	113290.i	0.200390	−	0.257729i
$664$		0			0
$665$	−362922.	−	209533.i	−0.820673	−	0.473816i
$666$		0			0
$667$	−37321.7	−	64643.1i	−0.0838900	−	0.145302i
$668$		0			0
$669$	133285.	−	120576.i	0.297803	−	0.269407i
$670$		0			0
$671$	919753.	+	162177.i	2.04280	+	0.360201i
$672$		0			0
$673$	−428014.	+	359147.i	−0.944992	+	0.792942i	−0.978447	−	0.206498i	$-0.933793\pi$
$673$	−428014.	+	359147.i	−0.944992	+	0.792942i	0.0334553	+	0.999440i	$0.489349\pi$
$674$		0			0
$675$	−269577.	+	283105.i	−0.591664	+	0.621356i
$676$		0			0
$677$	466782.	+	556289.i	1.01844	+	1.21373i	0.976701	+	0.214604i	$0.0688460\pi$
$677$	466782.	+	556289.i	1.01844	+	1.21373i	0.0417409	+	0.999128i	$0.486710\pi$
$678$		0			0
$679$	−110472.	+	626515.i	−0.239613	+	1.35891i
$680$		0			0
$681$	79246.5	−	17038.3i	0.170878	−	0.0367394i
$682$		0			0
$683$	−38854.8	+	22432.8i	−0.0832920	+	0.0480886i	−0.541068	−	0.840979i	$-0.681980\pi$
$683$	−38854.8	+	22432.8i	−0.0832920	+	0.0480886i	0.457776	+	0.889068i	$0.348646\pi$
$684$		0			0
$685$	77664.3	−	134519.i	0.165516	−	0.286682i
$686$		0			0
$687$	−2796.38	+	20240.8i	−0.00592492	+	0.0428859i
$688$		0			0
$689$	−212944.	+	253777.i	−0.448567	+	0.534581i
$690$		0			0
$691$	−599696.	−	218272.i	−1.25596	−	0.457131i	−0.373547	−	0.927611i	$-0.621859\pi$
$691$	−599696.	−	218272.i	−1.25596	−	0.457131i	−0.882411	+	0.470480i	$0.844081\pi$
$692$		0			0
$693$	963596.	−	1.33941e6i	2.00645	−	2.78900i
$694$		0			0
$695$	35745.2	−	6302.84i	0.0740027	−	0.0130487i
$696$		0			0
$697$	292210.	−	106356.i	0.601492	−	0.218925i
$698$		0			0
$699$	−13314.0	−	357289.i	−0.0272492	−	0.731250i
$700$		0			0
$701$		−	260097.i		−	0.529296i	−0.964345	−	0.264648i	$-0.914744\pi$
$701$		−	260097.i		−	0.529296i	0.964345	−	0.264648i	$-0.0852558\pi$
$702$		0			0
$703$	−475302.			−0.961743
$704$		0			0
$705$	−11292.8	+	21359.1i	−0.0227207	+	0.0429738i
$706$		0			0
$707$	−324094.	−	890441.i	−0.648384	−	1.78142i
$708$		0			0
$709$	165213.	+	936967.i	0.328663	+	1.86394i	0.482575	+	0.875855i	$0.339702\pi$
$709$	165213.	+	936967.i	0.328663	+	1.86394i	−0.153912	+	0.988085i	$0.549187\pi$
$710$		0			0
$711$	−121576.	−	478073.i	−0.240497	−	0.945703i
$712$		0			0
$713$	6330.62	−	17393.2i	0.0124528	−	0.0342138i
$714$		0			0
$715$	159782.	+	134073.i	0.312547	+	0.262258i
$716$		0			0
$717$	−72503.8	−	178251.i	−0.141034	−	0.346732i
$718$		0			0
$719$	−451724.	−	260803.i	−0.873808	−	0.504493i	−0.00519598	−	0.999987i	$-0.501654\pi$
$719$	−451724.	−	260803.i	−0.873808	−	0.504493i	−0.868612	+	0.495493i	$0.834987\pi$
$720$		0			0
$721$	−322892.	−	559265.i	−0.621136	−	1.07584i
$722$		0			0
$723$	265665.	+	85633.0i	0.508227	+	0.163819i
$724$		0			0
$725$	−769735.	−	135725.i	−1.46442	−	0.258216i
$726$		0			0
$727$	497415.	−	417381.i	0.941131	−	0.789703i	−0.0366508	−	0.999328i	$-0.511669\pi$
$727$	497415.	−	417381.i	0.941131	−	0.789703i	0.977782	+	0.209626i	$0.0672245\pi$
$728$		0			0
$729$	518225.	−	117780.i	0.975132	−	0.221624i
$730$		0			0
$731$	79391.4	+	94615.0i	0.148573	+	0.177062i
$732$		0			0
$733$	−11853.3	+	67223.2i	−0.0220612	+	0.125116i	−0.993849	−	0.110740i	$-0.964678\pi$
$733$	−11853.3	+	67223.2i	−0.0220612	+	0.125116i	0.971788	+	0.235855i	$0.0757891\pi$
$734$		0			0
$735$	−164747.	+	511106.i	−0.304960	+	0.946098i
$736$		0			0
$737$	1.33286e6	−	769524.i	2.45385	−	1.41673i
$738$		0			0
$739$	287118.	−	497303.i	0.525741	−	0.910610i	−0.473810	−	0.880627i	$-0.657122\pi$
$739$	287118.	−	497303.i	0.525741	−	0.910610i	0.999550	−	0.0299825i	$-0.00954516\pi$
$740$		0			0
$741$	403047.	−	163940.i	0.734040	−	0.298571i
$742$		0			0
$743$	−61286.3	+	73038.2i	−0.111016	+	0.132304i	−0.818691	−	0.574235i	$-0.805300\pi$
$743$	−61286.3	+	73038.2i	−0.111016	+	0.132304i	0.707675	+	0.706538i	$0.249744\pi$
$744$		0			0
$745$	−12809.3	−	4662.19i	−0.0230787	−	0.00839996i
$746$		0			0
$747$	−196726.	+	50028.3i	−0.352549	+	0.0896549i
$748$		0			0
$749$	−625389.	+	110273.i	−1.11477	+	0.196565i
$750$		0			0
$751$	−98404.2	+	35816.2i	−0.174475	+	0.0635038i	−0.427781	−	0.903883i	$-0.640704\pi$
$751$	−98404.2	+	35816.2i	−0.174475	+	0.0635038i	0.253306	+	0.967386i	$0.418482\pi$
$752$		0			0
$753$	759389.	+	401498.i	1.33929	+	0.708098i
$754$		0			0
$755$		−	352361.i		−	0.618150i
$756$		0			0
$757$	−957516.			−1.67091			−0.835457	−	0.549556i	$-0.814797\pi$
$757$	−957516.			−1.67091			−0.835457	+	0.549556i	$0.814797\pi$
$758$		0			0
$759$	100391.	−	3740.94i	0.174265	−	0.00649377i
$760$		0			0
$761$	−308545.	−	847721.i	−0.532782	−	1.46381i	−0.855747	−	0.517395i	$-0.826902\pi$
$761$	−308545.	−	847721.i	−0.532782	−	1.46381i	0.322965	−	0.946411i	$-0.395320\pi$
$762$		0			0
$763$	−21148.9	−	119941.i	−0.0363277	−	0.206025i
$764$		0			0
$765$	97239.5	+	69955.6i	0.166157	+	0.119536i
$766$		0			0
$767$	123445.	−	339163.i	0.209838	−	0.576525i
$768$		0			0
$769$	152004.	+	127546.i	0.257040	+	0.215682i	0.762197	−	0.647345i	$-0.224121\pi$
$769$	152004.	+	127546.i	0.257040	+	0.215682i	−0.505157	+	0.863028i	$0.668565\pi$
$770$		0			0
$771$	−748508.	−	103410.i	−1.25918	−	0.173962i
$772$		0			0
$773$	−811231.	−	468364.i	−1.35764	−	0.783835i	−0.368337	−	0.929692i	$-0.620073\pi$
$773$	−811231.	−	468364.i	−1.35764	−	0.783835i	−0.989306	+	0.145857i	$0.953406\pi$
$774$		0			0
$775$	−96908.7	−	167851.i	−0.161346	−	0.279460i
$776$		0			0
$777$	176560.	+	821194.i	0.292448	+	1.36020i
$778$		0			0
$779$	928534.	+	163726.i	1.53011	+	0.269800i
$780$		0			0
$781$	−1.22345e6	+	1.02660e6i	−2.00579	+	1.68305i
$782$		0			0
$783$	769506.	+	732734.i	1.25513	+	1.19515i
$784$		0			0
$785$	112117.	+	133616.i	0.181942	+	0.216830i
$786$		0			0
$787$	77480.7	−	439415.i	0.125096	−	0.709455i	−0.856155	−	0.516720i	$-0.827153\pi$
$787$	77480.7	−	439415.i	0.125096	−	0.709455i	0.981251	−	0.192736i	$-0.0617360\pi$
$788$		0			0
$789$	−192756.	−	213073.i	−0.309638	−	0.342274i
$790$		0			0
$791$	662550.	−	382523.i	1.05893	−	0.611371i
$792$		0			0
$793$	217617.	−	376924.i	0.346056	−	0.599387i
$794$		0			0
$795$	−218309.	−	169741.i	−0.345412	−	0.268566i
$796$		0			0
$797$	−611514.	+	728774.i	−0.962697	+	1.14730i	0.0263439	+	0.999653i	$0.491614\pi$
$797$	−611514.	+	728774.i	−0.962697	+	1.14730i	−0.989041	+	0.147644i	$0.952831\pi$
$798$		0			0
$799$	−42032.8	−	15298.7i	−0.0658407	−	0.0239641i
$800$		0			0
$801$	68304.6	+	915228.i	0.106460	+	1.42647i
$802$		0			0
$803$	365539.	−	64454.3i	0.566894	−	0.0999587i
$804$		0			0
$805$	−42370.6	+	15421.6i	−0.0653841	+	0.0237979i
$806$		0			0
$807$	372454.	−	233949.i	0.571907	−	0.359231i
$808$		0			0
$809$			1.06450e6i			1.62647i	0.581934	+	0.813236i	$0.302296\pi$
$809$			1.06450e6i			1.62647i	−0.581934	+	0.813236i	$0.697704\pi$
$810$		0			0
$811$	533022.			0.810407			0.405204	−	0.914226i	$-0.367201\pi$
$811$	533022.			0.810407			0.405204	+	0.914226i	$0.367201\pi$
$812$		0			0
$813$	282667.	+	450014.i	0.427655	+	0.680840i
$814$		0			0
$815$	−24910.9	−	68442.2i	−0.0375037	−	0.103041i
$816$		0			0
$817$	65029.8	+	368802.i	0.0974246	+	0.552522i
$818$		0			0
$819$	−432963.	−	635459.i	−0.645481	−	0.947370i
$820$		0			0
$821$	200472.	−	550794.i	0.297419	−	0.817151i	−0.697511	−	0.716575i	$-0.745709\pi$
$821$	200472.	−	550794.i	0.297419	−	0.817151i	0.994929	−	0.100577i	$-0.0320688\pi$
$822$		0			0
$823$	−317081.	−	266063.i	−0.468135	−	0.392812i	0.377979	−	0.925814i	$-0.376619\pi$
$823$	−317081.	−	266063.i	−0.468135	−	0.392812i	−0.846114	+	0.533003i	$0.821064\pi$
$824$		0			0
$825$	645699.	−	830455.i	0.948685	−	1.22014i
$826$		0			0
$827$	−174769.	−	100903.i	−0.255536	−	0.147534i	0.366760	−	0.930315i	$-0.380467\pi$
$827$	−174769.	−	100903.i	−0.255536	−	0.147534i	−0.622297	+	0.782782i	$0.713800\pi$
$828$		0			0
$829$	526401.	+	911754.i	0.765963	+	1.32669i	0.939736	+	0.341901i	$0.111071\pi$
$829$	526401.	+	911754.i	0.765963	+	1.32669i	−0.173773	+	0.984786i	$0.555596\pi$
$830$		0			0
$831$	−236188.	+	213667.i	−0.342023	+	0.309411i
$832$		0			0
$833$	−979086.	−	172639.i	−1.41101	−	0.248800i
$834$		0			0
$835$	257131.	−	215759.i	0.368792	−	0.309453i
$836$		0			0
$837$	−16534.2	+	262967.i	−0.0236011	+	0.375361i
$838$		0			0
$839$	−762993.	−	909299.i	−1.08392	−	1.29176i	−0.953858	−	0.300257i	$-0.902928\pi$
$839$	−762993.	−	909299.i	−1.08392	−	1.29176i	−0.130060	−	0.991506i	$-0.541517\pi$
$840$		0			0
$841$	−246095.	+	1.39567e6i	−0.347945	+	1.97329i
$842$		0			0
$843$	−245880.	+	52865.0i	−0.345993	+	0.0743898i
$844$		0			0
$845$	−148844.	+	85935.4i	−0.208458	+	0.120353i
$846$		0			0
$847$	−1.53587e6	+	2.66021e6i	−2.14086	+	3.70808i
$848$		0			0
$849$	−6463.25	+	46782.5i	−0.00896676	+	0.0649034i
$850$		0			0
$851$	−32872.5	+	39175.9i	−0.0453914	+	0.0540954i
$852$		0			0
$853$	76302.3	+	27771.8i	0.104867	+	0.0381685i	0.393921	−	0.919144i	$-0.371118\pi$
$853$	76302.3	+	27771.8i	0.104867	+	0.0381685i	−0.289054	+	0.957313i	$0.593341\pi$
$854$		0			0
$855$	149280.	+	331109.i	0.204206	+	0.452938i
$856$		0			0
$857$	−28309.6	+	4991.75i	−0.0385454	+	0.00679659i	−0.192888	−	0.981221i	$-0.561785\pi$
$857$	−28309.6	+	4991.75i	−0.0385454	+	0.00679659i	0.154342	+	0.988017i	$0.450674\pi$
$858$		0			0
$859$	−66049.9	+	24040.2i	−0.0895129	+	0.0325800i	−0.386388	−	0.922336i	$-0.626278\pi$
$859$	−66049.9	+	24040.2i	−0.0895129	+	0.0325800i	0.296876	+	0.954916i	$0.404055\pi$
$860$		0			0
$861$	−62047.1	−	1.66508e6i	−0.0836980	−	2.24609i
$862$		0			0
$863$			471153.i			0.632616i	0.948657	+	0.316308i	$0.102443\pi$
$863$			471153.i			0.632616i	−0.948657	+	0.316308i	$0.897557\pi$
$864$		0			0
$865$	268429.			0.358754
$866$		0			0
$867$	247685.	−	468469.i	0.329504	−	0.623221i
$868$		0			0
$869$	453999.	+	1.24735e6i	0.601194	+	1.65177i
$870$		0			0
$871$	−124545.	−	706330.i	−0.164169	−	0.931046i
$872$		0			0
$873$	394831.	−	384871.i	0.518063	−	0.504995i
$874$		0			0
$875$	−349694.	+	960777.i	−0.456744	+	1.25489i
$876$		0			0
$877$	−184519.	−	154830.i	−0.239907	−	0.201306i	0.514905	−	0.857247i	$-0.327827\pi$
$877$	−184519.	−	154830.i	−0.239907	−	0.201306i	−0.754812	+	0.655942i	$0.772272\pi$
$878$		0			0
$879$	456244.	+	1.12168e6i	0.590499	+	1.45175i
$880$		0			0
$881$	−718428.	−	414784.i	−0.925617	−	0.534405i	−0.0401942	−	0.999192i	$-0.512798\pi$
$881$	−718428.	−	414784.i	−0.925617	−	0.534405i	−0.885423	+	0.464787i	$0.846131\pi$
$882$		0			0
$883$	654981.	+	1.13446e6i	0.840054	+	1.45502i	0.889848	+	0.456257i	$0.150810\pi$
$883$	654981.	+	1.13446e6i	0.840054	+	1.45502i	−0.0497943	+	0.998759i	$0.515857\pi$
$884$		0			0
$885$	286753.	+	92430.4i	0.366118	+	0.118013i
$886$		0			0
$887$	−708159.	−	124868.i	−0.900085	−	0.158709i	−0.295587	−	0.955316i	$-0.595515\pi$
$887$	−708159.	−	124868.i	−0.900085	−	0.158709i	−0.604498	+	0.796607i	$0.706626\pi$
$888$		0			0
$889$	1.01931e6	−	855304.i	1.28974	−	1.08222i
$890$		0			0
$891$	−1.35588e6	+	454626.i	−1.70791	+	0.572662i
$892$		0			0
$893$	−87177.8	−	103894.i	−0.109321	−	0.130284i
$894$		0			0
$895$	−1832.69	+	10393.7i	−0.00228793	+	0.0129755i
$896$		0			0
$897$	14362.8	−	44558.7i	0.0178507	−	0.0553794i
$898$		0			0
$899$	−456233.	+	263406.i	−0.564505	+	0.325917i
$900$		0			0
$901$	255983.	−	443376.i	0.315328	−	0.546163i
$902$		0			0
$903$	613035.	−	249352.i	0.751813	−	0.305800i
$904$		0			0
$905$	169828.	−	202393.i	0.207354	−	0.247115i
$906$		0			0
$907$	883888.	+	321709.i	1.07444	+	0.391065i	0.817836	−	0.575452i	$-0.195174\pi$
$907$	883888.	+	321709.i	1.07444	+	0.391065i	0.256605	+	0.966516i	$0.417396\pi$
$908$		0			0
$909$	−222677.	+	790510.i	−0.269493	+	0.956708i
$910$		0			0
$911$	522613.	−	92150.8i	0.629714	−	0.111036i	0.150323	−	0.988637i	$-0.451969\pi$
$911$	522613.	−	92150.8i	0.629714	−	0.111036i	0.479391	+	0.877601i	$0.340857\pi$
$912$		0			0
$913$	513282.	−	186819.i	0.615764	−	0.224120i
$914$		0			0
$915$	321179.	+	169811.i	0.383623	+	0.202826i
$916$		0			0
$917$		−	277658.i		−	0.330196i
$918$		0			0
$919$	−434263.			−0.514188			−0.257094	−	0.966386i	$-0.582765\pi$
$919$	−434263.			−0.514188			−0.257094	+	0.966386i	$0.582765\pi$
$920$		0			0
$921$	−1.47495e6	+	54962.4i	−1.73884	+	0.0647958i
$922$		0			0
$923$	254559.	+	699395.i	0.298803	+	0.820954i
$924$		0			0
$925$	92989.4	+	527369.i	0.108680	+	0.616355i
$926$		0			0
$927$	−55899.0	+	556903.i	−0.0650496	+	0.648067i
$928$		0			0
$929$	367718.	−	1.01030e6i	0.426072	−	1.17062i	−0.522105	−	0.852881i	$-0.674853\pi$
$929$	367718.	−	1.01030e6i	0.426072	−	1.17062i	0.948177	−	0.317742i	$-0.102925\pi$
$930$		0			0
$931$	−2.30919e6	−	1.93764e6i	−2.66416	−	2.23550i
$932$		0			0
$933$	271047.	+	37446.7i	0.311374	+	0.0430180i
$934$		0			0
$935$	−279156.	−	161171.i	−0.319319	−	0.184359i
$936$		0			0
$937$	−663860.	−	1.14984e6i	−0.756132	−	1.30966i	−0.944810	−	0.327620i	$-0.893754\pi$
$937$	−663860.	−	1.14984e6i	−0.756132	−	1.30966i	0.188678	−	0.982039i	$-0.439580\pi$
$938$		0			0
$939$	24012.3	+	111683.i	0.0272335	+	0.126665i
$940$		0			0
$941$	−39582.9	−	6979.54i	−0.0447022	−	0.00788220i	0.151252	−	0.988495i	$-0.451669\pi$
$941$	−39582.9	−	6979.54i	−0.0447022	−	0.00788220i	−0.195955	+	0.980613i	$0.562781\pi$
$942$		0			0
$943$	77713.3	−	65209.2i	0.0873921	−	0.0733307i
$944$		0			0
$945$	516615.	−	380912.i	0.578500	−	0.426541i
$946$		0			0
$947$	424659.	+	506089.i	0.473522	+	0.564321i	0.948947	−	0.315435i	$-0.102150\pi$
$947$	424659.	+	506089.i	0.473522	+	0.564321i	−0.475425	+	0.879756i	$0.657706\pi$
$948$		0			0
$949$	30036.9	−	170348.i	0.0333521	−	0.189149i
$950$		0			0
$951$	859747.	+	950364.i	0.950626	+	1.05082i
$952$		0			0
$953$	319600.	−	184521.i	0.351901	−	0.203170i	−0.313621	−	0.949548i	$-0.601542\pi$
$953$	319600.	−	184521.i	0.351901	−	0.203170i	0.665522	+	0.746378i	$0.268209\pi$
$954$		0			0
$955$	206425.	−	357539.i	0.226337	−	0.392027i
$956$		0			0
$957$	−2.25725e6	−	1.75507e6i	−2.46465	−	1.91633i
$958$		0			0
$959$	990463.	−	1.18039e6i	1.07696	−	1.28348i
$960$		0			0
$961$	745069.	+	271183.i	0.806770	+	0.293640i
$962$		0			0
$963$	495807.	+	238956.i	0.534639	+	0.257670i
$964$		0			0
$965$	−220267.	+	38839.0i	−0.236534	+	0.0417074i
$966$		0			0
$967$	−1.39074e6	+	506187.i	−1.48728	+	0.541325i	−0.952731	−	0.303814i	$-0.901740\pi$
$967$	−1.39074e6	+	506187.i	−1.48728	+	0.541325i	−0.534547	+	0.845139i	$0.679518\pi$
$968$		0			0
$969$	−569416.	+	357666.i	−0.606432	+	0.380917i
$970$		0			0
$971$		−	154010.i		−	0.163347i	−0.996659	−	0.0816735i	$-0.973974\pi$
$971$		−	154010.i		−	0.163347i	0.996659	−	0.0816735i	$-0.0260265\pi$
$972$		0			0
$973$	360069.			0.380329
$974$		0			0
$975$	−260752.	−	415125.i	−0.274295	−	0.436687i
$976$		0			0
$977$	354878.	+	975020.i	0.371783	+	1.02147i	0.974672	+	0.223641i	$0.0717943\pi$
$977$	354878.	+	975020.i	0.371783	+	1.02147i	−0.602888	+	0.797826i	$0.705983\pi$
$978$		0			0
$979$	−428851.	−	2.43214e6i	−0.447447	−	2.53760i
$980$		0			0
$981$	−45828.5	+	95089.3i	−0.0476209	+	0.0988083i
$982$		0			0
$983$	−246818.	+	678127.i	−0.255429	+	0.701784i	0.744006	+	0.668173i	$0.232923\pi$
$983$	−246818.	+	678127.i	−0.255429	+	0.701784i	−0.999435	+	0.0336118i	$0.989299\pi$
$984$		0			0
$985$	−369095.	−	309707.i	−0.380422	−	0.319212i
$986$		0			0
$987$	−147118.	+	189213.i	−0.151019	+	0.194230i
$988$		0			0
$989$	34895.4	+	20146.9i	0.0356760	+	0.0205975i
$990$		0			0
$991$	513710.	+	889772.i	0.523083	+	0.906007i	0.999639	+	0.0268629i	$0.00855175\pi$
$991$	513710.	+	889772.i	0.523083	+	0.906007i	−0.476556	+	0.879144i	$0.658115\pi$
$992$		0			0
$993$	897647.	−	812057.i	0.910348	−	0.823546i
$994$		0			0
$995$	78024.9	+	13757.9i	0.0788111	+	0.0138965i
$996$		0			0
$997$	367641.	−	308487.i	0.369857	−	0.310347i	−0.438848	−	0.898561i	$-0.644613\pi$
$997$	367641.	−	308487.i	0.369857	−	0.310347i	0.808705	+	0.588215i	$0.200169\pi$
$998$		0			0
$999$	291426.	−	667117.i	0.292009	−	0.668454i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.5.8	✓	72
3.2	odd	2		324.5.k.a.125.8		72
27.11	odd	18	inner	108.5.k.a.65.8	yes	72
27.16	even	9		324.5.k.a.197.8		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.8	✓	72	1.1	even	1	trivial
108.5.k.a.65.8	yes	72	27.11	odd	18	inner
324.5.k.a.125.8		72	3.2	odd	2
324.5.k.a.197.8		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)

\(f(q)\)	\(=\)	\(q+(4.78712 + 7.62125i) q^{3} +(-3.22217 - 8.85284i) q^{5} +(-16.2288 - 92.0381i) q^{7} +(-35.1670 + 72.9677i) q^{9} +O(q^{10})\)	\(q+(4.78712 + 7.62125i) q^{3} +(-3.22217 - 8.85284i) q^{5} +(-16.2288 - 92.0381i) q^{7} +(-35.1670 + 72.9677i) q^{9} +(74.5484 - 204.820i) q^{11} +(-77.8115 - 65.2916i) q^{13} +(52.0448 - 66.9366i) q^{15} +(135.945 + 78.4880i) q^{17} +(237.980 + 412.193i) q^{19} +(623.756 - 564.281i) q^{21} +(50.4332 + 8.89274i) q^{23} +(410.787 - 344.691i) q^{25} +(-724.454 + 81.2889i) q^{27} +(-936.902 - 1116.56i) q^{29} +(62.7625 - 355.944i) q^{31} +(1917.86 - 412.346i) q^{33} +(-762.507 + 440.234i) q^{35} +(-499.310 + 864.830i) q^{37} +(125.111 - 905.580i) q^{39} +(1273.34 - 1517.51i) q^{41} +(739.364 + 269.107i) q^{43} +(759.286 + 76.2131i) q^{45} +(-280.621 + 49.4811i) q^{47} +(-5951.44 + 2166.15i) q^{49} +(52.6093 + 1411.80i) q^{51} -3261.43i q^{53} -2053.45 q^{55} +(-2002.19 + 3786.92i) q^{57} +(1215.30 + 3339.02i) q^{59} +(744.052 + 4219.73i) q^{61} +(7286.53 + 2052.52i) q^{63} +(-327.294 + 899.234i) q^{65} +(5409.04 + 4538.72i) q^{67} +(173.656 + 426.935i) q^{69} +(-6345.67 - 3663.67i) q^{71} +(851.462 + 1474.78i) q^{73} +(4593.47 + 1480.63i) q^{75} +(-20061.1 - 3537.31i) q^{77} +(-4665.20 + 3914.57i) q^{79} +(-4087.57 - 5132.10i) q^{81} +(1610.83 + 1919.72i) q^{83} +(256.803 - 1456.40i) q^{85} +(4024.49 - 12485.5i) q^{87} +(9812.53 - 5665.26i) q^{89} +(-4746.53 + 8221.23i) q^{91} +(3013.19 - 1225.62i) q^{93} +(2882.27 - 3434.95i) q^{95} +(-6396.61 - 2328.18i) q^{97} +(12323.6 + 12642.5i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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

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