LMFDB - Embedded newform 108.5.k.a.65.10

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			65.10
Character	$\chi$	$=$	108.65
Dual form			108.5.k.a.5.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.98131 + 5.67991i) q^{3} +(-8.75895 + 24.0650i) q^{5} +(-0.632293 + 3.58591i) q^{7} +(16.4774 + 79.3063i) q^{9} +O(q^{10})$	\(q+(6.98131 + 5.67991i) q^{3} +(-8.75895 + 24.0650i) q^{5} +(-0.632293 + 3.58591i) q^{7} +(16.4774 + 79.3063i) q^{9} +(-22.6368 - 62.1941i) q^{11} +(-236.237 + 198.227i) q^{13} +(-197.836 + 118.255i) q^{15} +(205.336 - 118.551i) q^{17} +(25.2661 - 43.7622i) q^{19} +(-24.7819 + 21.4430i) q^{21} +(-330.376 + 58.2542i) q^{23} +(-23.6276 - 19.8259i) q^{25} +(-335.419 + 647.252i) q^{27} +(-402.857 + 480.107i) q^{29} +(111.849 + 634.330i) q^{31} +(195.222 - 562.771i) q^{33} +(-80.7568 - 46.6249i) q^{35} +(978.384 + 1694.61i) q^{37} +(-2775.15 + 42.0759i) q^{39} +(-724.003 - 862.834i) q^{41} +(296.362 - 107.867i) q^{43} +(-2052.83 - 298.112i) q^{45} +(3610.79 + 636.681i) q^{47} +(2243.74 + 816.656i) q^{49} +(2106.87 + 338.649i) q^{51} -3927.39i q^{53} +1694.98 q^{55} +(424.955 - 162.008i) q^{57} +(662.491 - 1820.18i) q^{59} +(633.851 - 3594.75i) q^{61} +(-294.804 + 8.94148i) q^{63} +(-2701.14 - 7421.31i) q^{65} +(-1252.37 + 1050.86i) q^{67} +(-2637.34 - 1469.81i) q^{69} +(6356.50 - 3669.93i) q^{71} +(-832.121 + 1441.28i) q^{73} +(-52.3423 - 272.614i) q^{75} +(237.336 - 41.8487i) q^{77} +(8242.86 + 6916.58i) q^{79} +(-6017.99 + 2613.52i) q^{81} +(4039.89 - 4814.55i) q^{83} +(1054.40 + 5979.79i) q^{85} +(-5539.43 + 1063.58i) q^{87} +(4492.85 + 2593.95i) q^{89} +(-561.452 - 972.463i) q^{91} +(-2822.08 + 5063.75i) q^{93} +(831.832 + 991.339i) q^{95} +(-3954.60 + 1439.36i) q^{97} +(4559.39 - 2820.04i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{13}{18}\right)$	$1$

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
$3$	6.98131	+	5.67991i	0.775701	+	0.631101i
$3$	6.98131	+	5.67991i	0.775701	+	0.631101i
$4$		0			0
$5$	−8.75895	+	24.0650i	−0.350358	+	0.962600i	0.631898	+	0.775052i	$0.282276\pi$
$5$	−8.75895	+	24.0650i	−0.350358	+	0.962600i	−0.982255	+	0.187548i	$0.939946\pi$
$6$		0			0
$7$	−0.632293	+	3.58591i	−0.0129039	+	0.0731819i	−0.990580	−	0.136937i	$-0.956274\pi$
$7$	−0.632293	+	3.58591i	−0.0129039	+	0.0731819i	0.977676	+	0.210119i	$0.0673852\pi$
$8$		0			0
$9$	16.4774	+	79.3063i	0.203424	+	0.979091i
$10$		0			0
$11$	−22.6368	−	62.1941i	−0.187081	−	0.514001i	0.810325	−	0.585981i	$-0.199291\pi$
$11$	−22.6368	−	62.1941i	−0.187081	−	0.514001i	−0.997406	+	0.0719797i	$0.977068\pi$
$12$		0			0
$13$	−236.237	+	198.227i	−1.39785	+	1.17294i	−0.435812	+	0.900038i	$0.643539\pi$
$13$	−236.237	+	198.227i	−1.39785	+	1.17294i	−0.962042	+	0.272901i	$0.912017\pi$
$14$		0			0
$15$	−197.836	+	118.255i	−0.879271	+	0.525579i
$16$		0			0
$17$	205.336	−	118.551i	0.710505	−	0.410210i	−0.100743	−	0.994912i	$-0.532122\pi$
$17$	205.336	−	118.551i	0.710505	−	0.410210i	0.811248	+	0.584702i	$0.198789\pi$
$18$		0			0
$19$	25.2661	−	43.7622i	0.0699892	−	0.121225i	−0.828907	−	0.559386i	$-0.811037\pi$
$19$	25.2661	−	43.7622i	0.0699892	−	0.121225i	0.898896	+	0.438161i	$0.144370\pi$
$20$		0			0
$21$	−24.7819	+	21.4430i	−0.0561947	+	0.0486236i
$22$		0			0
$23$	−330.376	+	58.2542i	−0.624530	+	0.110121i	−0.476952	−	0.878929i	$-0.658259\pi$
$23$	−330.376	+	58.2542i	−0.624530	+	0.110121i	−0.147577	+	0.989051i	$0.547147\pi$
$24$		0			0
$25$	−23.6276	−	19.8259i	−0.0378042	−	0.0317215i
$26$		0			0
$27$	−335.419	+	647.252i	−0.460108	+	0.887863i
$28$		0			0
$29$	−402.857	+	480.107i	−0.479022	+	0.570876i	−0.950390	−	0.311060i	$-0.899316\pi$
$29$	−402.857	+	480.107i	−0.479022	+	0.570876i	0.471368	+	0.881936i	$0.343760\pi$
$30$		0			0
$31$	111.849	+	634.330i	0.116389	+	0.660073i	0.986053	+	0.166430i	$0.0532239\pi$
$31$	111.849	+	634.330i	0.116389	+	0.660073i	−0.869665	+	0.493643i	$0.835665\pi$
$32$		0			0
$33$	195.222	−	562.771i	0.179267	−	0.516778i
$34$		0			0
$35$	−80.7568	−	46.6249i	−0.0659239	−	0.0380612i
$36$		0			0
$37$	978.384	+	1694.61i	0.714671	+	1.23785i	0.963086	+	0.269193i	$0.0867568\pi$
$37$	978.384	+	1694.61i	0.714671	+	1.23785i	−0.248416	+	0.968654i	$0.579910\pi$
$38$		0			0
$39$	−2775.15	+	42.0759i	−1.82456	+	0.0276633i
$40$		0			0
$41$	−724.003	−	862.834i	−0.430698	−	0.513286i	0.506426	−	0.862284i	$-0.330967\pi$
$41$	−724.003	−	862.834i	−0.430698	−	0.513286i	−0.937124	+	0.348998i	$0.886522\pi$
$42$		0			0
$43$	296.362	−	107.867i	0.160282	−	0.0583380i	−0.260633	−	0.965438i	$-0.583931\pi$
$43$	296.362	−	107.867i	0.160282	−	0.0583380i	0.420915	+	0.907100i	$0.361709\pi$
$44$		0			0
$45$	−2052.83	−	298.112i	−1.01374	−	0.147216i
$46$		0			0
$47$	3610.79	+	636.681i	1.63458	+	0.288221i	0.914172	−	0.405326i	$-0.132842\pi$
$47$	3610.79	+	636.681i	1.63458	+	0.288221i	0.720411	+	0.693547i	$0.243953\pi$
$48$		0			0
$49$	2243.74	+	816.656i	0.934504	+	0.340131i
$50$		0			0
$51$	2106.87	+	338.649i	0.810023	+	0.130200i
$52$		0			0
$53$		−	3927.39i		−	1.39815i	−0.715051	−	0.699073i	$-0.753596\pi$
$53$		−	3927.39i		−	1.39815i	0.715051	−	0.699073i	$-0.246404\pi$
$54$		0			0
$55$	1694.98			0.560323
$56$		0			0
$57$	424.955	−	162.008i	0.130796	−	0.0498640i
$58$		0			0
$59$	662.491	−	1820.18i	0.190316	−	0.522890i	−0.807432	−	0.589961i	$-0.799143\pi$
$59$	662.491	−	1820.18i	0.190316	−	0.522890i	0.997748	+	0.0670710i	$0.0213654\pi$
$60$		0			0
$61$	633.851	−	3594.75i	0.170344	−	0.966070i	−0.773037	−	0.634361i	$-0.781263\pi$
$61$	633.851	−	3594.75i	0.170344	−	0.966070i	0.943382	−	0.331710i	$-0.107625\pi$
$62$		0			0
$63$	−294.804	+	8.94148i	−0.0742766	+	0.00225283i
$64$		0			0
$65$	−2701.14	−	7421.31i	−0.639322	−	1.75652i
$66$		0			0
$67$	−1252.37	+	1050.86i	−0.278986	+	0.234097i	−0.771534	−	0.636188i	$-0.780510\pi$
$67$	−1252.37	+	1050.86i	−0.278986	+	0.234097i	0.492548	+	0.870285i	$0.336066\pi$
$68$		0			0
$69$	−2637.34	−	1469.81i	−0.553946	−	0.308720i
$70$		0			0
$71$	6356.50	−	3669.93i	1.26096	−	0.728016i	0.287700	−	0.957721i	$-0.407109\pi$
$71$	6356.50	−	3669.93i	1.26096	−	0.728016i	0.973260	+	0.229705i	$0.0737761\pi$
$72$		0			0
$73$	−832.121	+	1441.28i	−0.156150	+	0.270459i	−0.933477	−	0.358637i	$-0.883242\pi$
$73$	−832.121	+	1441.28i	−0.156150	+	0.270459i	0.777327	+	0.629096i	$0.216575\pi$
$74$		0			0
$75$	−52.3423	−	272.614i	−0.00930530	−	0.0484646i
$76$		0			0
$77$	237.336	−	41.8487i	0.0400296	−	0.00705830i
$78$		0			0
$79$	8242.86	+	6916.58i	1.32076	+	1.10825i	0.986143	+	0.165895i	$0.0530513\pi$
$79$	8242.86	+	6916.58i	1.32076	+	1.10825i	0.334617	+	0.942354i	$0.391393\pi$
$80$		0			0
$81$	−6017.99	+	2613.52i	−0.917237	+	0.398341i
$82$		0			0
$83$	4039.89	−	4814.55i	0.586426	−	0.698875i	−0.388489	−	0.921453i	$-0.627003\pi$
$83$	4039.89	−	4814.55i	0.586426	−	0.698875i	0.974915	+	0.222578i	$0.0714473\pi$
$84$		0			0
$85$	1054.40	+	5979.79i	0.145937	+	0.827652i
$86$		0			0
$87$	−5539.43	+	1063.58i	−0.731858	+	0.140518i
$88$		0			0
$89$	4492.85	+	2593.95i	0.567207	+	0.327477i	0.756033	−	0.654533i	$-0.227135\pi$
$89$	4492.85	+	2593.95i	0.567207	+	0.327477i	−0.188826	+	0.982011i	$0.560468\pi$
$90$		0			0
$91$	−561.452	−	972.463i	−0.0678000	−	0.117433i
$92$		0			0
$93$	−2822.08	+	5063.75i	−0.326289	+	0.585472i
$94$		0			0
$95$	831.832	+	991.339i	0.0921698	+	0.109844i
$96$		0			0
$97$	−3954.60	+	1439.36i	−0.420300	+	0.152977i	−0.543507	−	0.839404i	$-0.682904\pi$
$97$	−3954.60	+	1439.36i	−0.420300	+	0.152977i	0.123208	+	0.992381i	$0.460682\pi$
$98$		0			0
$99$	4559.39	−	2820.04i	0.465197	−	0.287730i
$100$		0			0
$101$	−15898.8	−	2803.39i	−1.55856	−	0.274815i	−0.673104	−	0.739548i	$-0.735039\pi$
$101$	−15898.8	−	2803.39i	−1.55856	−	0.274815i	−0.885451	+	0.464732i	$0.846151\pi$
$102$		0			0
$103$	13317.2	+	4847.05i	1.25527	+	0.456881i	0.882179	−	0.470914i	$-0.156076\pi$
$103$	13317.2	+	4847.05i	1.25527	+	0.456881i	0.373091	+	0.927795i	$0.378298\pi$
$104$		0			0
$105$	−298.963	−	784.194i	−0.0271168	−	0.0711287i
$106$		0			0
$107$			5472.24i			0.477967i	0.971024	+	0.238984i	$0.0768142\pi$
$107$			5472.24i			0.477967i	−0.971024	+	0.238984i	$0.923186\pi$
$108$		0			0
$109$	−8864.19			−0.746081			−0.373041	−	0.927815i	$-0.621685\pi$
$109$	−8864.19			−0.746081			−0.373041	+	0.927815i	$0.621685\pi$
$110$		0			0
$111$	−2794.83	+	17387.7i	−0.226835	+	1.41123i
$112$		0			0
$113$	−1907.44	+	5240.66i	−0.149381	+	0.410421i	−0.991702	−	0.128555i	$-0.958966\pi$
$113$	−1907.44	+	5240.66i	−0.149381	+	0.410421i	0.842322	+	0.538975i	$0.181188\pi$
$114$		0			0
$115$	1491.86	−	8460.75i	0.112806	−	0.639754i
$116$		0			0
$117$	−19613.2	−	15468.9i	−1.43277	−	1.13002i
$118$		0			0
$119$	295.280	+	811.275i	0.0208516	+	0.0572894i
$120$		0			0
$121$	7859.97	−	6595.30i	0.536847	−	0.450468i
$122$		0			0
$123$	−153.678	−	10136.0i	−0.0101578	−	0.669970i
$124$		0			0
$125$	−13177.5	+	7608.01i	−0.843357	+	0.486912i
$126$		0			0
$127$	−12709.1	+	22012.8i	−0.787967	+	1.36480i	0.139244	+	0.990258i	$0.455533\pi$
$127$	−12709.1	+	22012.8i	−0.787967	+	1.36480i	−0.927211	+	0.374540i	$0.877800\pi$
$128$		0			0
$129$	2681.67	+	930.255i	0.161148	+	0.0559014i
$130$		0			0
$131$	−6066.88	+	1069.75i	−0.353527	+	0.0623364i	−0.347592	−	0.937646i	$-0.613000\pi$
$131$	−6066.88	+	1069.75i	−0.353527	+	0.0623364i	−0.00593554	+	0.999982i	$0.501889\pi$
$132$		0			0
$133$	140.952	+	118.272i	0.00796832	+	0.00668622i
$134$		0			0
$135$	−12638.2	−	13741.1i	−0.693454	−	0.753970i
$136$		0			0
$137$	13079.1	−	15587.0i	0.696845	−	0.830467i	−0.295321	−	0.955398i	$-0.595426\pi$
$137$	13079.1	−	15587.0i	0.696845	−	0.830467i	0.992165	+	0.124931i	$0.0398709\pi$
$138$		0			0
$139$	1564.03	+	8870.06i	0.0809498	+	0.459089i	0.998157	+	0.0606794i	$0.0193267\pi$
$139$	1564.03	+	8870.06i	0.0809498	+	0.459089i	−0.917207	+	0.398410i	$0.869562\pi$
$140$		0			0
$141$	21591.8	+	24953.8i	1.08605	+	1.25516i
$142$		0			0
$143$	17676.2	+	10205.4i	0.864404	+	0.499064i
$144$		0			0
$145$	−8025.16	−	13900.0i	−0.381696	−	0.661117i
$146$		0			0
$147$	11025.7	+	18445.6i	0.510238	+	0.853606i
$148$		0			0
$149$	9361.36	+	11156.4i	0.421664	+	0.502519i	0.934498	−	0.355968i	$-0.115849\pi$
$149$	9361.36	+	11156.4i	0.421664	+	0.502519i	−0.512834	+	0.858488i	$0.671404\pi$
$150$		0			0
$151$	−4398.99	+	1601.10i	−0.192930	+	0.0702207i	−0.436678	−	0.899618i	$-0.643845\pi$
$151$	−4398.99	+	1601.10i	−0.192930	+	0.0702207i	0.243748	+	0.969839i	$0.421623\pi$
$152$		0			0
$153$	12785.2	+	14331.0i	0.546167	+	0.612202i
$154$		0			0
$155$	−16244.8	−	2864.40i	−0.676164	−	0.119226i
$156$		0			0
$157$	−4131.63	−	1503.79i	−0.167619	−	0.0610082i	0.256848	−	0.966452i	$-0.417316\pi$
$157$	−4131.63	−	1503.79i	−0.167619	−	0.0610082i	−0.424466	+	0.905444i	$0.639538\pi$
$158$		0			0
$159$	22307.2	−	27418.3i	0.882370	−	1.08454i
$160$		0			0
$161$		−	1221.53i		−	0.0471252i
$162$		0			0
$163$	−48545.4			−1.82714			−0.913572	−	0.406677i	$-0.866688\pi$
$163$	−48545.4			−1.82714			−0.913572	+	0.406677i	$0.866688\pi$
$164$		0			0
$165$	11833.2	+	9627.31i	0.434643	+	0.353620i
$166$		0			0
$167$	−14935.4	+	41034.8i	−0.535532	+	1.47136i	0.316868	+	0.948470i	$0.397369\pi$
$167$	−14935.4	+	41034.8i	−0.535532	+	1.47136i	−0.852399	+	0.522891i	$0.824853\pi$
$168$		0			0
$169$	11554.7	−	65530.0i	0.404562	−	2.29439i
$170$		0			0
$171$	3886.94	+	1282.68i	0.132928	+	0.0438657i
$172$		0			0
$173$	−7728.43	−	21233.7i	−0.258225	−	0.709469i	−0.999277	−	0.0380204i	$-0.987895\pi$
$173$	−7728.43	−	21233.7i	−0.258225	−	0.709469i	0.741051	−	0.671448i	$-0.234327\pi$
$174$		0			0
$175$	86.0336	−	72.1907i	0.00280926	−	0.00235725i
$176$		0			0
$177$	14963.5	−	8944.35i	0.477625	−	0.285497i
$178$		0			0
$179$	47920.0	−	27666.6i	1.49558	−	0.863476i	0.495597	−	0.868553i	$-0.334949\pi$
$179$	47920.0	−	27666.6i	1.49558	−	0.863476i	0.999987	+	0.00507689i	$0.00161603\pi$
$180$		0			0
$181$	20301.3	−	35162.8i	0.619678	−	1.07331i	−0.369866	−	0.929085i	$-0.620596\pi$
$181$	20301.3	−	35162.8i	0.619678	−	1.07331i	0.989544	−	0.144229i	$-0.0460702\pi$
$182$		0			0
$183$	24842.9	−	21495.8i	0.741824	−	0.641877i
$184$		0			0
$185$	−49350.4	+	8701.82i	−1.44194	+	0.254253i
$186$		0			0
$187$	−12021.3	−	10087.1i	−0.343770	−	0.288458i
$188$		0			0
$189$	−2108.90	−	1612.04i	−0.0590382	−	0.0451285i
$190$		0			0
$191$	−10391.8	+	12384.5i	−0.284856	+	0.339478i	−0.889430	−	0.457071i	$-0.848899\pi$
$191$	−10391.8	+	12384.5i	−0.284856	+	0.339478i	0.604575	+	0.796549i	$0.293343\pi$
$192$		0			0
$193$	5546.95	+	31458.3i	0.148915	+	0.844542i	0.964140	+	0.265396i	$0.0855027\pi$
$193$	5546.95	+	31458.3i	0.148915	+	0.844542i	−0.815224	+	0.579146i	$0.803386\pi$
$194$		0			0
$195$	23294.9	−	67152.7i	0.612620	−	1.76601i
$196$		0			0
$197$	50661.7	+	29249.5i	1.30541	+	0.753679i	0.981326	−	0.192350i	$-0.0616108\pi$
$197$	50661.7	+	29249.5i	1.30541	+	0.753679i	0.324083	+	0.946028i	$0.394944\pi$
$198$		0			0
$199$	−8677.79	−	15030.4i	−0.219131	−	0.379545i	0.735412	−	0.677620i	$-0.236989\pi$
$199$	−8677.79	−	15030.4i	−0.219131	−	0.379545i	−0.954542	+	0.298075i	$0.903655\pi$
$200$		0			0
$201$	−14712.0	+	223.057i	−0.364148	+	0.00552109i
$202$		0			0
$203$	−1466.90	−	1748.18i	−0.0355965	−	0.0424223i
$204$		0			0
$205$	27105.6	−	9865.63i	0.644988	−	0.234756i
$206$		0			0
$207$	−10063.7	−	25241.0i	−0.234863	−	0.589070i
$208$		0			0
$209$	−3293.69	−	580.767i	−0.0754033	−	0.0132956i
$210$		0			0
$211$	6345.41	+	2309.54i	0.142526	+	0.0518753i	0.412298	−	0.911049i	$-0.364726\pi$
$211$	6345.41	+	2309.54i	0.142526	+	0.0518753i	−0.269772	+	0.962924i	$0.586948\pi$
$212$		0			0
$213$	65221.5	+	10483.4i	1.43758	+	0.231070i
$214$		0			0
$215$			8076.75i			0.174727i
$216$		0			0
$217$	−2345.37			−0.0498072
$218$		0			0
$219$	−13995.6	+	5335.62i	−0.291812	+	0.111249i
$220$		0			0
$221$	−25008.1	+	68709.2i	−0.512031	+	1.40679i
$222$		0			0
$223$	14045.2	−	79654.1i	0.282434	−	1.60176i	−0.431876	−	0.901933i	$-0.642148\pi$
$223$	14045.2	−	79654.1i	0.282434	−	1.60176i	0.714310	−	0.699830i	$-0.246741\pi$
$224$		0			0
$225$	1183.00	−	2200.50i	0.0233679	−	0.0434666i
$226$		0			0
$227$	−4625.90	−	12709.6i	−0.0897728	−	0.246649i	0.886679	−	0.462386i	$-0.153007\pi$
$227$	−4625.90	−	12709.6i	−0.0897728	−	0.246649i	−0.976451	+	0.215738i	$0.930784\pi$
$228$		0			0
$229$	−14498.7	+	12165.9i	−0.276477	+	0.231992i	−0.770473	−	0.637472i	$-0.779980\pi$
$229$	−14498.7	+	12165.9i	−0.276477	+	0.231992i	0.493996	+	0.869464i	$0.335536\pi$
$230$		0			0
$231$	1894.61	+	1055.89i	0.0355055	+	0.0197876i
$232$		0			0
$233$	−46376.2	+	26775.3i	−0.854247	+	0.493200i	−0.862081	−	0.506770i	$-0.830839\pi$
$233$	−46376.2	+	26775.3i	−0.854247	+	0.493200i	0.00783474	+	0.999969i	$0.497506\pi$
$234$		0			0
$235$	−46948.5	+	81317.1i	−0.850131	+	1.47247i
$236$		0			0
$237$	18260.4	+	95105.5i	0.325098	+	1.69320i
$238$		0			0
$239$	−41032.0	+	7235.05i	−0.718335	+	0.126662i	−0.520855	−	0.853645i	$-0.674387\pi$
$239$	−41032.0	+	7235.05i	−0.718335	+	0.126662i	−0.197480	+	0.980307i	$0.563276\pi$
$240$		0			0
$241$	−71951.4	−	60374.4i	−1.23881	−	1.03949i	−0.997616	−	0.0690098i	$-0.978016\pi$
$241$	−71951.4	−	60374.4i	−1.23881	−	1.03949i	−0.241196	−	0.970477i	$-0.577540\pi$
$242$		0			0
$243$	−56858.0	−	15935.9i	−0.962895	−	0.269875i
$244$		0			0
$245$	−39305.6	+	46842.6i	−0.654821	+	0.780386i
$246$		0			0
$247$	2706.03	+	15346.7i	0.0443546	+	0.251548i
$248$		0			0
$249$	55549.9	−	10665.7i	0.895952	−	0.172024i
$250$		0			0
$251$	53554.3	+	30919.6i	0.850054	+	0.490779i	0.860669	−	0.509165i	$-0.170046\pi$
$251$	53554.3	+	30919.6i	0.850054	+	0.490779i	−0.0106150	+	0.999944i	$0.503379\pi$
$252$		0			0
$253$	11101.7	+	19228.8i	0.173440	+	0.300407i
$254$		0			0
$255$	−26603.6	+	47735.6i	−0.409128	+	0.734112i
$256$		0			0
$257$	6611.38	+	7879.13i	0.100098	+	0.119292i	0.813768	−	0.581190i	$-0.197413\pi$
$257$	6611.38	+	7879.13i	0.100098	+	0.119292i	−0.713670	+	0.700482i	$0.752968\pi$
$258$		0			0
$259$	−6695.35	+	2436.91i	−0.0998099	+	0.0363278i
$260$		0			0
$261$	−44713.5	−	24038.3i	−0.656384	−	0.352876i
$262$		0			0
$263$	64659.5	+	11401.2i	0.934805	+	0.164831i	0.620247	−	0.784407i	$-0.287032\pi$
$263$	64659.5	+	11401.2i	0.934805	+	0.164831i	0.314558	+	0.949238i	$0.398144\pi$
$264$		0			0
$265$	94512.7	+	34399.8i	1.34586	+	0.489851i
$266$		0			0
$267$	16632.6	+	43628.1i	0.233312	+	0.611989i
$268$		0			0
$269$		−	78857.5i		−	1.08978i	−0.838508	−	0.544890i	$-0.816572\pi$
$269$		−	78857.5i		−	1.08978i	0.838508	−	0.544890i	$-0.183428\pi$
$270$		0			0
$271$	−85836.1			−1.16878			−0.584388	−	0.811474i	$-0.698666\pi$
$271$	−85836.1			−1.16878			−0.584388	+	0.811474i	$0.698666\pi$
$272$		0			0
$273$	1603.83	−	9978.06i	0.0215195	−	0.133882i
$274$		0			0
$275$	−698.202	+	1918.29i	−0.00923242	+	0.0253659i
$276$		0			0
$277$	−12768.2	+	72412.3i	−0.166407	+	0.943741i	0.781195	+	0.624287i	$0.214611\pi$
$277$	−12768.2	+	72412.3i	−0.166407	+	0.943741i	−0.947602	+	0.319454i	$0.896501\pi$
$278$		0			0
$279$	−48463.4	+	19322.4i	−0.622595	+	0.248230i
$280$		0			0
$281$	−50983.9	−	140077.i	−0.645685	−	1.77401i	−0.633082	−	0.774084i	$-0.718211\pi$
$281$	−50983.9	−	140077.i	−0.645685	−	1.77401i	−0.0126027	−	0.999921i	$-0.504012\pi$
$282$		0			0
$283$	−43283.8	+	36319.5i	−0.540447	+	0.453489i	−0.871691	−	0.490056i	$-0.836976\pi$
$283$	−43283.8	+	36319.5i	−0.540447	+	0.453489i	0.331244	+	0.943545i	$0.392532\pi$
$284$		0			0
$285$	176.566	+	11645.6i	0.00217379	+	0.143374i
$286$		0			0
$287$	3551.83	−	2050.65i	0.0431209	−	0.0248959i
$288$		0			0
$289$	−13651.9	+	23645.9i	−0.163455	+	0.283113i
$290$		0			0
$291$	−35783.7	−	12413.2i	−0.422570	−	0.146587i
$292$		0			0
$293$	−8976.28	+	1582.76i	−0.104559	+	0.0184366i	−0.225683	−	0.974201i	$-0.572461\pi$
$293$	−8976.28	+	1582.76i	−0.104559	+	0.0184366i	0.121124	+	0.992637i	$0.461350\pi$
$294$		0			0
$295$	37999.9	+	31885.7i	0.436655	+	0.366397i
$296$		0			0
$297$	47848.1	+	6209.38i	0.542440	+	0.0703939i
$298$		0			0
$299$	66499.6	−	79251.2i	0.743835	−	0.886469i
$300$		0			0
$301$	199.414	+	1130.93i	0.00220101	+	0.0124825i
$302$		0			0
$303$	−95071.6	−	109875.i	−1.03554	−	1.19678i
$304$		0			0
$305$	80955.7	+	46739.8i	0.870258	+	0.502444i
$306$		0			0
$307$	−55967.7	−	96939.0i	−0.593829	−	1.02854i	−0.993711	−	0.111975i	$-0.964282\pi$
$307$	−55967.7	−	96939.0i	−0.593829	−	1.02854i	0.399882	−	0.916566i	$-0.369051\pi$
$308$		0			0
$309$	65440.4	+	109479.i	0.685376	+	1.14660i
$310$		0			0
$311$	15433.3	+	18392.6i	0.159565	+	0.190162i	0.839903	−	0.542736i	$-0.182612\pi$
$311$	15433.3	+	18392.6i	0.159565	+	0.190162i	−0.680338	+	0.732898i	$0.738167\pi$
$312$		0			0
$313$	166946.	−	60763.3i	1.70407	−	0.620230i	0.707790	−	0.706423i	$-0.249692\pi$
$313$	166946.	−	60763.3i	1.70407	−	0.620230i	0.996278	+	0.0861930i	$0.0274702\pi$
$314$		0			0
$315$	2367.00	−	7172.78i	0.0238548	−	0.0722880i
$316$		0			0
$317$	−168850.	−	29772.9i	−1.68029	−	0.296280i	−0.749544	−	0.661954i	$-0.769727\pi$
$317$	−168850.	−	29772.9i	−1.68029	−	0.296280i	−0.930743	+	0.365674i	$0.880838\pi$
$318$		0			0
$319$	38979.2	+	14187.3i	0.383047	+	0.139418i
$320$		0			0
$321$	−31081.8	+	38203.4i	−0.301645	+	0.370760i
$322$		0			0
$323$		−	11981.3i		−	0.114841i
$324$		0			0
$325$	9511.75			0.0900521
$326$		0			0
$327$	−61883.7	−	50347.8i	−0.578736	−	0.470852i
$328$		0			0
$329$	−4566.16	+	12545.4i	−0.0421851	+	0.115903i
$330$		0			0
$331$	−21564.0	+	122296.i	−0.196822	+	1.11623i	0.712978	+	0.701186i	$0.247346\pi$
$331$	−21564.0	+	122296.i	−0.196822	+	1.11623i	−0.909800	+	0.415047i	$0.863765\pi$
$332$		0			0
$333$	−118272.	+	105515.i	−1.06658	+	0.951535i
$334$		0			0
$335$	−14319.6	−	39342.7i	−0.127597	−	0.350570i
$336$		0			0
$337$	144069.	−	120888.i	1.26856	−	1.06445i	0.273845	−	0.961774i	$-0.411704\pi$
$337$	144069.	−	120888.i	1.26856	−	1.06445i	0.994715	−	0.102674i	$-0.0327400\pi$
$338$		0			0
$339$	−43082.9	+	25752.6i	−0.374892	+	0.224089i
$340$		0			0
$341$	36919.7	−	21315.6i	0.317504	−	0.183311i
$342$		0			0
$343$	−8718.45	+	15100.8i	−0.0741056	+	0.128355i
$344$		0			0
$345$	58471.4	−	50593.5i	0.491253	−	0.425066i
$346$		0			0
$347$	126703.	−	22341.2i	1.05227	−	0.185544i	0.379347	−	0.925254i	$-0.376149\pi$
$347$	126703.	−	22341.2i	1.05227	−	0.185544i	0.672925	+	0.739710i	$0.265037\pi$
$348$		0			0
$349$	36525.7	+	30648.7i	0.299881	+	0.251630i	0.780295	−	0.625412i	$-0.215069\pi$
$349$	36525.7	+	30648.7i	0.299881	+	0.251630i	−0.480414	+	0.877042i	$0.659514\pi$
$350$		0			0
$351$	−49064.1	−	219394.i	−0.398244	−	1.78078i
$352$		0			0
$353$	94657.4	−	112808.i	0.759635	−	0.905298i	−0.238190	−	0.971219i	$-0.576554\pi$
$353$	94657.4	−	112808.i	0.759635	−	0.905298i	0.997825	+	0.0659209i	$0.0209985\pi$
$354$		0			0
$355$	32640.6	+	185114.i	0.259001	+	1.46887i
$356$		0			0
$357$	−2546.52	+	7340.92i	−0.0199807	+	0.0575989i
$358$		0			0
$359$	65225.6	+	37658.0i	0.506092	+	0.292192i	0.731226	−	0.682136i	$-0.238949\pi$
$359$	65225.6	+	37658.0i	0.506092	+	0.292192i	−0.225134	+	0.974328i	$0.572282\pi$
$360$		0			0
$361$	63883.7	+	110650.i	0.490203	+	0.849057i
$362$		0			0
$363$	92333.6	−	1399.93i	0.700723	−	0.0106241i
$364$		0			0
$365$	−27395.8	−	32649.1i	−0.205636	−	0.245067i
$366$		0			0
$367$	107072.	−	38970.8i	0.794954	−	0.289339i	0.0875599	−	0.996159i	$-0.472093\pi$
$367$	107072.	−	38970.8i	0.794954	−	0.289339i	0.707394	+	0.706820i	$0.249871\pi$
$368$		0			0
$369$	56498.5	−	71635.3i	0.414939	−	0.526107i
$370$		0			0
$371$	14083.3	+	2483.26i	0.102319	+	0.0180416i
$372$		0			0
$373$	51753.2	+	18836.6i	0.371980	+	0.135390i	0.521244	−	0.853408i	$-0.325468\pi$
$373$	51753.2	+	18836.6i	0.371980	+	0.135390i	−0.149264	+	0.988797i	$0.547690\pi$
$374$		0			0
$375$	−135209.	−	21732.8i	−0.961484	−	0.154545i
$376$		0			0
$377$		−	193276.i		−	1.35986i
$378$		0			0
$379$	129313.			0.900249			0.450125	−	0.892966i	$-0.351380\pi$
$379$	129313.			0.900249			0.450125	+	0.892966i	$0.351380\pi$
$380$		0			0
$381$	−213757.	+	81491.8i	−1.47255	+	0.561389i
$382$		0			0
$383$	83657.1	−	229846.i	0.570303	−	1.56689i	−0.233726	−	0.972303i	$-0.575092\pi$
$383$	83657.1	−	229846.i	0.570303	−	1.56689i	0.804028	−	0.594591i	$-0.202686\pi$
$384$		0			0
$385$	−1071.72	+	6078.03i	−0.00723037	+	0.0410055i
$386$		0			0
$387$	13437.8	+	21726.0i	0.0897234	+	0.145063i
$388$		0			0
$389$	43560.0	+	119680.i	0.287865	+	0.790903i	0.996365	+	0.0851915i	$0.0271502\pi$
$389$	43560.0	+	119680.i	0.287865	+	0.790903i	−0.708500	+	0.705711i	$0.750628\pi$
$390$		0			0
$391$	−60932.0	+	51128.0i	−0.398558	+	0.334430i
$392$		0			0
$393$	−48430.9	−	26991.0i	−0.313572	−	0.174757i
$394$		0			0
$395$	−238646.	+	137783.i	−1.52954	+	0.883080i
$396$		0			0
$397$	48666.8	−	84293.3i	0.308782	−	0.534825i	−0.669314	−	0.742979i	$-0.733412\pi$
$397$	48666.8	−	84293.3i	0.308782	−	0.534825i	0.978096	+	0.208154i	$0.0667454\pi$
$398$		0			0
$399$	312.251	+	1626.29i	0.00196136	+	0.0102153i
$400$		0			0
$401$	−20496.9	+	3614.15i	−0.127467	+	0.0224759i	−0.237018	−	0.971505i	$-0.576170\pi$
$401$	−20496.9	+	3614.15i	−0.127467	+	0.0224759i	0.109551	+	0.993981i	$0.465059\pi$
$402$		0			0
$403$	−152164.	−	127681.i	−0.936919	−	0.786168i
$404$		0			0
$405$	−10183.0	−	167715.i	−0.0620821	−	1.02249i
$406$		0			0
$407$	83247.4	−	99210.3i	0.502553	−	0.598919i
$408$		0			0
$409$	−50897.1	−	288652.i	−0.304261	−	1.72555i	−0.626963	−	0.779049i	$-0.715702\pi$
$409$	−50897.1	−	288652.i	−0.304261	−	1.72555i	0.322702	−	0.946501i	$-0.395409\pi$
$410$		0			0
$411$	179842.	−	34530.0i	1.06465	−	0.204415i
$412$		0			0
$413$	6108.11	+	3526.52i	0.0358102	+	0.0206750i
$414$		0			0
$415$	80477.0	+	139390.i	0.467278	+	0.809350i
$416$		0			0
$417$	−39462.1	+	70808.2i	−0.226939	+	0.407203i
$418$		0			0
$419$	−39138.7	−	46643.7i	−0.222935	−	0.265684i	0.642971	−	0.765891i	$-0.277702\pi$
$419$	−39138.7	−	46643.7i	−0.222935	−	0.265684i	−0.865906	+	0.500207i	$0.833257\pi$
$420$		0			0
$421$	−210167.	+	76494.4i	−1.18577	+	0.431584i	−0.858236	−	0.513256i	$-0.828439\pi$
$421$	−210167.	+	76494.4i	−1.18577	+	0.431584i	−0.327533	+	0.944840i	$0.606217\pi$
$422$		0			0
$423$	9003.53	+	296850.i	0.0503190	+	1.65904i
$424$		0			0
$425$	−7201.98	−	1269.90i	−0.0398725	−	0.00703060i
$426$		0			0
$427$	12489.7	+	4545.87i	0.0685007	+	0.0249322i
$428$		0			0
$429$	65437.5	+	171646.i	0.355559	+	0.932650i
$430$		0			0
$431$		−	244624.i		−	1.31687i	−0.752635	−	0.658437i	$-0.771218\pi$
$431$		−	244624.i		−	1.31687i	0.752635	−	0.658437i	$-0.228782\pi$
$432$		0			0
$433$	−119238.			−0.635971			−0.317986	−	0.948096i	$-0.603006\pi$
$433$	−119238.			−0.635971			−0.317986	+	0.948096i	$0.603006\pi$
$434$		0			0
$435$	22924.5	−	142622.i	0.121149	−	0.753718i
$436$		0			0
$437$	−5797.98	+	15929.8i	−0.0303609	+	0.0834158i
$438$		0			0
$439$	−16161.5	+	91656.5i	−0.0838597	+	0.475592i	0.913737	+	0.406306i	$0.133183\pi$
$439$	−16161.5	+	91656.5i	−0.0838597	+	0.475592i	−0.997597	+	0.0692860i	$0.977928\pi$
$440$		0			0
$441$	−27795.0	+	191399.i	−0.142919	+	0.984155i
$442$		0			0
$443$	94532.4	+	259726.i	0.481696	+	1.32345i	0.908038	+	0.418887i	$0.137580\pi$
$443$	94532.4	+	259726.i	0.481696	+	1.32345i	−0.426342	+	0.904562i	$0.640198\pi$
$444$		0			0
$445$	−101776.	+	85400.2i	−0.513955	+	0.431259i
$446$		0			0
$447$	1987.06	+	131058.i	0.00994477	+	0.655917i
$448$		0			0
$449$	−84746.9	+	48928.6i	−0.420369	+	0.242700i	−0.695235	−	0.718782i	$-0.744700\pi$
$449$	−84746.9	+	48928.6i	−0.420369	+	0.242700i	0.274866	+	0.961483i	$0.411366\pi$
$450$		0			0
$451$	−37274.0	+	64560.5i	−0.183254	+	0.317405i
$452$		0			0
$453$	−39804.8	−	13808.1i	−0.193972	−	0.0672878i
$454$		0			0
$455$	28320.1	−	4993.59i	0.136795	−	0.0241207i
$456$		0			0
$457$	132440.	+	111130.i	0.634141	+	0.532107i	0.902212	−	0.431292i	$-0.141942\pi$
$457$	132440.	+	111130.i	0.634141	+	0.532107i	−0.268072	+	0.963399i	$0.586387\pi$
$458$		0			0
$459$	7858.62	+	172668.i	0.0373010	+	0.819572i
$460$		0			0
$461$	−207164.	+	246888.i	−0.974793	+	1.16171i	0.0120331	+	0.999928i	$0.496170\pi$
$461$	−207164.	+	246888.i	−0.974793	+	1.16171i	−0.986826	+	0.161785i	$0.948275\pi$
$462$		0			0
$463$	42566.1	+	241404.i	0.198564	+	1.12611i	0.907250	+	0.420591i	$0.138177\pi$
$463$	42566.1	+	241404.i	0.198564	+	1.12611i	−0.708686	+	0.705524i	$0.750712\pi$
$464$		0			0
$465$	−97140.7	−	112266.i	−0.449257	−	0.519211i
$466$		0			0
$467$	199445.	+	115149.i	0.914510	+	0.527992i	0.881879	−	0.471475i	$-0.156278\pi$
$467$	199445.	+	115149.i	0.914510	+	0.527992i	0.0326303	+	0.999467i	$0.489612\pi$
$468$		0			0
$469$	−2976.43	−	5155.33i	−0.0135316	−	0.0234375i
$470$		0			0
$471$	−20302.8	−	33965.7i	−0.0915197	−	0.153108i
$472$		0			0
$473$	−13417.4	−	15990.2i	−0.0599715	−	0.0714713i
$474$		0			0
$475$	−1464.60	+	533.072i	−0.00649131	+	0.00236264i
$476$		0			0
$477$	311467.	−	64713.0i	1.36891	−	0.284416i
$478$		0			0
$479$	133421.	+	23525.7i	0.581505	+	0.102535i	0.456660	−	0.889641i	$-0.349046\pi$
$479$	133421.	+	23525.7i	0.581505	+	0.102535i	0.124845	+	0.992176i	$0.460157\pi$
$480$		0			0
$481$	−567048.	−	206389.i	−2.45092	−	0.892063i
$482$		0			0
$483$	6938.19	−	8527.90i	0.0297408	−	0.0365551i
$484$		0			0
$485$		−	107775.i		−	0.458177i
$486$		0			0
$487$	253009.			1.06679			0.533393	−	0.845868i	$-0.320917\pi$
$487$	253009.			1.06679			0.533393	+	0.845868i	$0.320917\pi$
$488$		0			0
$489$	−338910.	−	275733.i	−1.41732	−	1.15311i
$490$		0			0
$491$	−86121.8	+	236618.i	−0.357232	+	0.981487i	0.622754	+	0.782418i	$0.286014\pi$
$491$	−86121.8	+	236618.i	−0.357232	+	0.981487i	−0.979986	+	0.199069i	$0.936208\pi$
$492$		0			0
$493$	−25804.1	+	146342.i	−0.106168	+	0.602110i
$494$		0			0
$495$	27928.7	+	134422.i	0.113983	+	0.548607i
$496$		0			0
$497$	9140.86	+	25114.3i	0.0370062	+	0.101674i
$498$		0			0
$499$	49665.0	−	41673.9i	0.199457	−	0.167364i	−0.537589	−	0.843207i	$-0.680665\pi$
$499$	49665.0	−	41673.9i	0.199457	−	0.167364i	0.737046	+	0.675843i	$0.236220\pi$
$500$		0			0
$501$	−337343.	+	201645.i	−1.34399	+	0.803362i
$502$		0			0
$503$	−420415.	+	242727.i	−1.66166	+	0.959360i	−0.689737	+	0.724060i	$0.742274\pi$
$503$	−420415.	+	242727.i	−1.66166	+	0.959360i	−0.971923	+	0.235300i	$0.924393\pi$
$504$		0			0
$505$	206721.	−	358051.i	0.810589	−	1.40398i
$506$		0			0
$507$	452871.	−	391855.i	1.76181	−	1.52444i
$508$		0			0
$509$	−287186.	+	50638.7i	−1.10848	+	0.195455i	−0.697778	−	0.716314i	$-0.745828\pi$
$509$	−287186.	+	50638.7i	−1.10848	+	0.195455i	−0.410703	+	0.911769i	$0.634717\pi$
$510$		0			0
$511$	−4642.14	−	3895.22i	−0.0177777	−	0.0149173i
$512$		0			0
$513$	19850.4	+	31032.2i	0.0754284	+	0.117917i
$514$		0			0
$515$	−233289.	+	278023.i	−0.879587	+	1.04825i
$516$		0			0
$517$	−42139.1	−	238983.i	−0.157654	−	0.894098i
$518$		0			0
$519$	66650.8	−	192136.i	0.247440	−	0.713302i
$520$		0			0
$521$	−41930.4	−	24208.5i	−0.154473	−	0.0891853i	0.420771	−	0.907167i	$-0.361760\pi$
$521$	−41930.4	−	24208.5i	−0.154473	−	0.0891853i	−0.575244	+	0.817982i	$0.695093\pi$
$522$		0			0
$523$	−155771.	−	269804.i	−0.569488	−	0.986382i	−0.996617	−	0.0821910i	$-0.973808\pi$
$523$	−155771.	−	269804.i	−0.569488	−	0.986382i	0.427129	−	0.904191i	$-0.359525\pi$
$524$		0			0
$525$	1010.66	−	15.3233i	0.00366681	−	5.55948e-5i
$526$		0			0
$527$	98167.0	+	116991.i	0.353463	+	0.421241i
$528$		0			0
$529$	−157210.	+	57219.7i	−0.561782	+	0.204472i
$530$		0			0
$531$	155268.	+	22548.0i	0.550672	+	0.0799686i
$532$		0			0
$533$	342073.	+	60316.7i	1.20411	+	0.212316i
$534$		0			0
$535$	−131690.	−	47931.1i	−0.460091	−	0.167460i
$536$		0			0
$537$	491688.	+	79031.8i	1.70507	+	0.274065i
$538$		0			0
$539$		−	158034.i		−	0.543968i
$540$		0			0
$541$	333475.			1.13938			0.569690	−	0.821860i	$-0.307063\pi$
$541$	333475.			1.13938			0.569690	+	0.821860i	$0.307063\pi$
$542$		0			0
$543$	341451.	−	130173.i	1.15805	−	0.441492i
$544$		0			0
$545$	77641.0	−	213317.i	0.261395	−	0.718178i
$546$		0			0
$547$	58942.6	−	334280.i	0.196995	−	1.11721i	−0.712554	−	0.701617i	$-0.752462\pi$
$547$	58942.6	−	334280.i	0.196995	−	1.11721i	0.909549	−	0.415596i	$-0.136427\pi$
$548$		0			0
$549$	295530.	−	8963.51i	0.980522	−	0.0297395i
$550$		0			0
$551$	10831.9	+	29760.3i	0.0356780	+	0.0980245i
$552$		0			0
$553$	−30014.2	+	25184.9i	−0.0981468	+	0.0823549i
$554$		0			0
$555$	−393956.	−	219556.i	−1.27897	−	0.712786i
$556$		0			0
$557$	−211847.	+	122310.i	−0.682828	+	0.394231i	−0.800920	−	0.598772i	$-0.795656\pi$
$557$	−211847.	+	122310.i	−0.682828	+	0.394231i	0.118092	+	0.993003i	$0.462322\pi$
$558$		0			0
$559$	−48629.6	+	84229.0i	−0.155624	+	0.269549i
$560$		0			0
$561$	−26630.8	−	138701.i	−0.0846173	−	0.440711i
$562$		0			0
$563$	465712.	−	82117.5i	1.46927	−	0.259071i	0.618986	−	0.785402i	$-0.287544\pi$
$563$	465712.	−	82117.5i	1.46927	−	0.259071i	0.850280	+	0.526331i	$0.176433\pi$
$564$		0			0
$565$	−109409.	−	91805.3i	−0.342734	−	0.287588i
$566$		0			0
$567$	−5566.71	−	23232.5i	−0.0173154	−	0.0722653i
$568$		0			0
$569$	−190547.	+	227085.i	−0.588542	+	0.701397i	−0.975325	−	0.220774i	$-0.929142\pi$
$569$	−190547.	+	227085.i	−0.588542	+	0.701397i	0.386783	+	0.922171i	$0.373586\pi$
$570$		0			0
$571$	−96134.7	−	545207.i	−0.294855	−	1.67220i	−0.667792	−	0.744347i	$-0.732761\pi$
$571$	−96134.7	−	545207.i	−0.294855	−	1.67220i	0.372938	−	0.927856i	$-0.378350\pi$
$572$		0			0
$573$	−142891.	+	27435.4i	−0.435208	+	0.0835607i
$574$		0			0
$575$	8960.94	+	5173.60i	0.0271030	+	0.0156479i
$576$		0			0
$577$	127612.	+	221030.i	0.383300	+	0.663895i	0.991532	−	0.129864i	$-0.0414542\pi$
$577$	127612.	+	221030.i	0.383300	+	0.663895i	−0.608232	+	0.793760i	$0.708121\pi$
$578$		0			0
$579$	−139955.	+	251127.i	−0.417477	+	0.749092i
$580$		0			0
$581$	14710.2	+	17530.9i	0.0435778	+	0.0519340i
$582$		0			0
$583$	−244261.	+	88903.6i	−0.718648	+	0.261567i
$584$		0			0
$585$	544049.	−	336501.i	1.58974	−	0.983273i
$586$		0			0
$587$	390885.	+	68923.6i	1.13442	+	0.200029i	0.709163	−	0.705045i	$-0.249073\pi$
$587$	390885.	+	68923.6i	1.13442	+	0.200029i	0.425255	+	0.905073i	$0.360184\pi$
$588$		0			0
$589$	30585.6	+	11132.3i	0.0881631	+	0.0320887i
$590$		0			0
$591$	187550.	+	491953.i	0.536961	+	1.40847i
$592$		0			0
$593$		−	141628.i		−	0.402754i	−0.979514	−	0.201377i	$-0.935458\pi$
$593$		−	141628.i		−	0.402754i	0.979514	−	0.201377i	$-0.0645416\pi$
$594$		0			0
$595$	−22109.7			−0.0624523
$596$		0			0
$597$	24788.8	−	154221.i	0.0695515	−	0.432707i
$598$		0			0
$599$	205094.	−	563492.i	0.571611	−	1.57049i	−0.230348	−	0.973108i	$-0.573987\pi$
$599$	205094.	−	563492.i	0.571611	−	1.57049i	0.801959	−	0.597379i	$-0.203791\pi$
$600$		0			0
$601$	−67002.3	+	379989.i	−0.185499	+	1.05202i	0.739814	+	0.672811i	$0.234913\pi$
$601$	−67002.3	+	379989.i	−0.185499	+	1.05202i	−0.925313	+	0.379204i	$0.876198\pi$
$602$		0			0
$603$	−103976.	−	82005.3i	−0.285955	−	0.225532i
$604$		0			0
$605$	89870.9	+	246918.i	0.245532	+	0.674594i
$606$		0			0
$607$	109301.	−	91714.6i	0.296652	−	0.248921i	−0.482297	−	0.876008i	$-0.660197\pi$
$607$	109301.	−	91714.6i	0.296652	−	0.248921i	0.778949	+	0.627087i	$0.215753\pi$
$608$		0			0
$609$	−311.366	−	20536.4i	−0.000839529	−	0.0553720i
$610$		0			0
$611$	−979212.	+	565348.i	−2.62297	+	1.51438i
$612$		0			0
$613$	67410.6	−	116759.i	0.179394	−	0.310719i	−0.762279	−	0.647248i	$-0.775920\pi$
$613$	67410.6	−	116759.i	0.179394	−	0.310719i	0.941673	+	0.336529i	$0.109253\pi$
$614$		0			0
$615$	245268.	+	85082.2i	0.648472	+	0.224951i
$616$		0			0
$617$	550083.	−	96994.6i	1.44497	−	0.254787i	0.604481	−	0.796619i	$-0.293380\pi$
$617$	550083.	−	96994.6i	1.44497	−	0.254787i	0.840487	+	0.541832i	$0.182269\pi$
$618$		0			0
$619$	206114.	+	172950.i	0.537931	+	0.451377i	0.870830	−	0.491585i	$-0.163582\pi$
$619$	206114.	+	172950.i	0.537931	+	0.451377i	−0.332899	+	0.942963i	$0.608027\pi$
$620$		0			0
$621$	73109.3	−	233376.i	0.189579	−	0.605164i
$622$		0			0
$623$	−12142.5	+	14470.8i	−0.0312846	+	0.0372835i
$624$		0			0
$625$	−71013.6	−	402738.i	−0.181795	−	1.03101i
$626$		0			0
$627$	−19695.6	−	22762.4i	−0.0500995	−	0.0579005i
$628$		0			0
$629$	401795.	+	231976.i	1.01555	+	0.586330i
$630$		0			0
$631$	−263153.	−	455795.i	−0.660922	−	1.14475i	−0.980374	−	0.197148i	$-0.936832\pi$
$631$	−263153.	−	455795.i	−0.660922	−	1.14475i	0.319451	−	0.947603i	$-0.396501\pi$
$632$		0			0
$633$	31181.3	+	52165.0i	0.0778192	+	0.130188i
$634$		0			0
$635$	−418421.	−	498654.i	−1.03769	−	1.23666i
$636$		0			0
$637$	−691939.	+	251845.i	−1.70525	+	0.620661i
$638$		0			0
$639$	395787.	+	443640.i	0.969303	+	1.08650i
$640$		0			0
$641$	−505241.	−	89087.6i	−1.22965	−	0.216821i	−0.479176	−	0.877719i	$-0.659065\pi$
$641$	−505241.	−	89087.6i	−1.22965	−	0.216821i	−0.750476	+	0.660898i	$0.770176\pi$
$642$		0			0
$643$	−20896.4	−	7605.67i	−0.0505417	−	0.0183957i	0.316626	−	0.948551i	$-0.397450\pi$
$643$	−20896.4	−	7605.67i	−0.0505417	−	0.0183957i	−0.367167	+	0.930155i	$0.619672\pi$
$644$		0			0
$645$	−45875.2	+	56386.3i	−0.110270	+	0.135536i
$646$		0			0
$647$		−	341326.i		−	0.815381i	−0.913120	−	0.407690i	$-0.866334\pi$
$647$		−	341326.i		−	0.815381i	0.913120	−	0.407690i	$-0.133666\pi$
$648$		0			0
$649$	−128201.			−0.304370
$650$		0			0
$651$	−16373.8	−	13321.5i	−0.0386355	−	0.0314334i
$652$		0			0
$653$	−25633.4	+	70427.1i	−0.0601145	+	0.165163i	−0.966114	−	0.258115i	$-0.916899\pi$
$653$	−25633.4	+	70427.1i	−0.0601145	+	0.165163i	0.906000	+	0.423278i	$0.139121\pi$
$654$		0			0
$655$	27395.8	−	155369.i	0.0638560	−	0.362145i
$656$		0			0
$657$	−128013.	−	42244.1i	−0.296568	−	0.0978667i
$658$		0			0
$659$	97130.3	+	266863.i	0.223658	+	0.614495i	0.999872	−	0.0159741i	$-0.00508493\pi$
$659$	97130.3	+	266863.i	0.223658	+	0.614495i	−0.776215	+	0.630469i	$0.782863\pi$
$660$		0			0
$661$	118548.	−	99473.7i	0.271326	−	0.227670i	−0.496964	−	0.867771i	$-0.665552\pi$
$661$	118548.	−	99473.7i	0.271326	−	0.227670i	0.768291	+	0.640101i	$0.221108\pi$
$662$		0			0
$663$	−564851.	+	337636.i	−1.28501	+	0.768108i
$664$		0			0
$665$	−4080.82	+	2356.06i	−0.00922792	+	0.00532774i
$666$		0			0
$667$	105126.	−	182084.i	0.236298	−	0.409279i
$668$		0			0
$669$	550481.	−	476314.i	1.22996	−	1.06424i
$670$		0			0
$671$	−237920.	+	41951.8i	−0.528429	+	0.0931763i
$672$		0			0
$673$	606326.	+	508768.i	1.33868	+	1.12328i	0.981963	+	0.189071i	$0.0605478\pi$
$673$	606326.	+	508768.i	1.33868	+	1.12328i	0.356715	+	0.934213i	$0.383897\pi$
$674$		0			0
$675$	20757.5	−	8643.03i	0.0455583	−	0.0189696i
$676$		0			0
$677$	319117.	−	380309.i	0.696262	−	0.829773i	−0.295836	−	0.955239i	$-0.595598\pi$
$677$	319117.	−	380309.i	0.696262	−	0.829773i	0.992098	+	0.125466i	$0.0400426\pi$
$678$		0			0
$679$	−2660.94	−	15090.9i	−0.00577159	−	0.0327323i
$680$		0			0
$681$	39894.3	−	115004.i	0.0860233	−	0.247981i
$682$		0			0
$683$	54865.2	+	31676.5i	0.117613	+	0.0679040i	0.557653	−	0.830074i	$-0.311702\pi$
$683$	54865.2	+	31676.5i	0.117613	+	0.0679040i	−0.440039	+	0.897978i	$0.645036\pi$
$684$		0			0
$685$	260543.	+	451274.i	0.555263	+	0.961744i
$686$		0			0
$687$	−170321.	+	2582.35i	−0.360874	+	0.00547144i
$688$		0			0
$689$	778514.	+	927796.i	1.63994	+	1.95440i
$690$		0			0
$691$	205774.	−	74895.6i	0.430958	−	0.156856i	−0.117428	−	0.993081i	$-0.537465\pi$
$691$	205774.	−	74895.6i	0.430958	−	0.156856i	0.548385	+	0.836226i	$0.315243\pi$
$692$		0			0
$693$	7229.53	+	18132.7i	0.0150537	+	0.0377568i
$694$		0			0
$695$	−227157.	−	40054.0i	−0.470281	−	0.0829232i
$696$		0			0
$697$	−250953.	−	91339.6i	−0.516568	−	0.188015i
$698$		0			0
$699$	−475848.	−	76485.7i	−0.973899	−	0.156540i
$700$		0			0
$701$		−	484481.i		−	0.985919i	−0.870052	−	0.492959i	$-0.835915\pi$
$701$		−	484481.i		−	0.985919i	0.870052	−	0.492959i	$-0.164085\pi$
$702$		0			0
$703$	98879.8			0.200077
$704$		0			0
$705$	−789636.	+	301037.i	−1.58872	+	0.605678i
$706$		0			0
$707$	20105.4	−	55239.2i	0.0402230	−	0.110512i
$708$		0			0
$709$	−68351.0	+	387638.i	−0.135973	+	0.771140i	0.838205	+	0.545356i	$0.183606\pi$
$709$	−68351.0	+	387638.i	−0.135973	+	0.771140i	−0.974177	+	0.225784i	$0.927506\pi$
$710$		0			0
$711$	−412708.	+	767678.i	−0.816402	+	1.51859i
$712$		0			0
$713$	−73904.8	−	203052.i	−0.145376	−	0.399418i
$714$		0			0
$715$	−400417.	+	335990.i	−0.783249	+	0.657224i
$716$		0			0
$717$	−327552.	−	182548.i	−0.637150	−	0.355090i
$718$		0			0
$719$	−643290.	+	371404.i	−1.24437	+	0.718437i	−0.969981	−	0.243182i	$-0.921809\pi$
$719$	−643290.	+	371404.i	−1.24437	+	0.718437i	−0.274389	+	0.961619i	$0.588475\pi$
$720$		0			0
$721$	−25801.4	+	44689.4i	−0.0496333	+	0.0859674i
$722$		0			0
$723$	−159394.	−	830170.i	−0.304927	−	1.58815i
$724$		0			0
$725$	19037.1	−	3356.76i	0.0362181	−	0.00638622i
$726$		0			0
$727$	511540.	+	429233.i	0.967856	+	0.812128i	0.982213	−	0.187769i	$-0.0601256\pi$
$727$	511540.	+	429233.i	0.967856	+	0.812128i	−0.0143568	+	0.999897i	$0.504570\pi$
$728$		0			0
$729$	−306429.	−	434201.i	−0.576600	−	0.817026i
$730$		0			0
$731$	48066.0	−	57282.9i	0.0899505	−	0.107199i
$732$		0			0
$733$	90190.3	+	511495.i	0.167862	+	0.951991i	0.946065	+	0.323978i	$0.105020\pi$
$733$	90190.3	+	511495.i	0.167862	+	0.951991i	−0.778203	+	0.628013i	$0.783868\pi$
$734$		0			0
$735$	−540467.	+	103771.i	−1.00045	+	0.192088i
$736$		0			0
$737$	93707.0	+	54101.8i	0.172519	+	0.0996039i
$738$		0			0
$739$	−18612.0	−	32237.0i	−0.0340804	−	0.0590290i	0.848482	−	0.529224i	$-0.177517\pi$
$739$	−18612.0	−	32237.0i	−0.0340804	−	0.0590290i	−0.882563	+	0.470195i	$0.844184\pi$
$740$		0			0
$741$	−68276.0	+	122510.i	−0.124346	+	0.223118i
$742$		0			0
$743$	−533099.	−	635323.i	−0.965674	−	1.15085i	−0.988517	−	0.151107i	$-0.951716\pi$
$743$	−533099.	−	635323.i	−0.965674	−	1.15085i	0.0228434	−	0.999739i	$-0.492728\pi$
$744$		0			0
$745$	−350475.	+	127563.i	−0.631458	+	0.229832i
$746$		0			0
$747$	448391.	+	241058.i	0.803555	+	0.431996i
$748$		0			0
$749$	−19623.0	−	3460.06i	−0.0349785	−	0.00616766i
$750$		0			0
$751$	410674.	+	149473.i	0.728144	+	0.265023i	0.679379	−	0.733787i	$-0.262249\pi$
$751$	410674.	+	149473.i	0.728144	+	0.265023i	0.0487651	+	0.998810i	$0.484471\pi$
$752$		0			0
$753$	198259.	+	520042.i	0.349657	+	0.917167i
$754$		0			0
$755$		−	119886.i		−	0.210317i
$756$		0			0
$757$	515657.			0.899848			0.449924	−	0.893067i	$-0.351451\pi$
$757$	515657.			0.899848			0.449924	+	0.893067i	$0.351451\pi$
$758$		0			0
$759$	−31712.9	+	197299.i	−0.0550494	+	0.342484i
$760$		0			0
$761$	−194402.	+	534114.i	−0.335684	+	0.922284i	0.650920	+	0.759147i	$0.274383\pi$
$761$	−194402.	+	534114.i	−0.335684	+	0.922284i	−0.986603	+	0.163137i	$0.947839\pi$
$762$		0			0
$763$	5604.77	−	31786.2i	0.00962739	−	0.0545996i
$764$		0			0
$765$	−456862.	+	182152.i	−0.780660	+	0.311250i
$766$		0			0
$767$	204303.	+	561318.i	0.347283	+	0.954153i
$768$		0			0
$769$	−61792.2	+	51849.8i	−0.104491	+	0.0876788i	−0.693537	−	0.720421i	$-0.743948\pi$
$769$	−61792.2	+	51849.8i	−0.104491	+	0.0876788i	0.589045	+	0.808100i	$0.299504\pi$
$770$		0			0
$771$	1403.34	+	92558.7i	0.00236077	+	0.155707i
$772$		0			0
$773$	418443.	−	241588.i	0.700288	−	0.404312i	−0.107167	−	0.994241i	$-0.534178\pi$
$773$	418443.	−	241588.i	0.700288	−	0.404312i	0.807455	+	0.589930i	$0.200844\pi$
$774$		0			0
$775$	9933.44	−	17205.2i	0.0165385	−	0.0286455i
$776$		0			0
$777$	−60583.7	−	21016.1i	−0.100349	−	0.0348106i
$778$		0			0
$779$	−56052.2	+	9883.51i	−0.0923672	+	0.0162868i
$780$		0			0
$781$	−372139.	−	312262.i	−0.610103	−	0.511937i
$782$		0			0
$783$	−175624.	−	421787.i	−0.286458	−	0.687970i
$784$		0			0
$785$	72377.5	−	86256.2i	0.117453	−	0.139975i
$786$		0			0
$787$	13069.5	+	74120.9i	0.0211013	+	0.119672i	0.993539	−	0.113492i	$-0.0362035\pi$
$787$	13069.5	+	74120.9i	0.0211013	+	0.119672i	−0.972438	+	0.233163i	$0.925092\pi$
$788$		0			0
$789$	386650.	+	446855.i	0.621104	+	0.717816i
$790$		0			0
$791$	−17586.5	−	10153.6i	−0.0281077	−	0.0162280i
$792$		0			0
$793$	562835.	+	974860.i	0.895025	+	1.55023i
$794$		0			0
$795$	464435.	+	776979.i	0.734836	+	1.22935i
$796$		0			0
$797$	−226447.	−	269869.i	−0.356493	−	0.424851i	0.557756	−	0.830005i	$-0.311663\pi$
$797$	−226447.	−	269869.i	−0.356493	−	0.424851i	−0.914249	+	0.405154i	$0.867218\pi$
$798$		0			0
$799$	816905.	−	297329.i	1.27961	−	0.465740i
$800$		0			0
$801$	−131686.	+	399053.i	−0.205246	+	0.621964i
$802$		0			0
$803$	108475.	+	19127.1i	0.168229	+	0.0296633i
$804$		0			0
$805$	29396.2	+	10699.3i	0.0453628	+	0.0165107i
$806$		0			0
$807$	447903.	−	550529.i	0.687760	−	0.845343i
$808$		0			0
$809$		−	1.15808e6i		−	1.76947i	−0.466094	−	0.884735i	$-0.654339\pi$
$809$		−	1.15808e6i		−	1.76947i	0.466094	−	0.884735i	$-0.345661\pi$
$810$		0			0
$811$	−351292.			−0.534105			−0.267053	−	0.963682i	$-0.586050\pi$
$811$	−351292.			−0.534105			−0.267053	+	0.963682i	$0.586050\pi$
$812$		0			0
$813$	−599249.	−	487541.i	−0.906621	−	0.737616i
$814$		0			0
$815$	425206.	−	1.16825e6i	0.640154	−	1.75881i
$816$		0			0
$817$	2767.42	−	15694.8i	0.00414601	−	0.0235132i
$818$		0			0
$819$	67871.3	−	60550.3i	0.101185	−	0.0902711i
$820$		0			0
$821$	−134780.	−	370305.i	−0.199958	−	0.549380i	0.798669	−	0.601771i	$-0.205538\pi$
$821$	−134780.	−	370305.i	−0.199958	−	0.549380i	−0.998627	+	0.0523911i	$0.983316\pi$
$822$		0			0
$823$	287702.	−	241410.i	0.424759	−	0.356415i	−0.405211	−	0.914223i	$-0.632802\pi$
$823$	287702.	−	241410.i	0.424759	−	0.356415i	0.829970	+	0.557808i	$0.188357\pi$
$824$		0			0
$825$	−15770.1	+	9426.48i	−0.0231700	+	0.0138497i
$826$		0			0
$827$	170424.	−	98394.3i	0.249184	−	0.143866i	−0.370207	−	0.928949i	$-0.620713\pi$
$827$	170424.	−	98394.3i	0.249184	−	0.143866i	0.619390	+	0.785083i	$0.287380\pi$
$828$		0			0
$829$	−61104.9	+	105837.i	−0.0889134	+	0.154003i	−0.907052	−	0.421018i	$-0.861673\pi$
$829$	−61104.9	+	105837.i	−0.0889134	+	0.154003i	0.818139	+	0.575021i	$0.195006\pi$
$830$		0			0
$831$	−500434.	+	433010.i	−0.724678	+	0.627041i
$832$		0			0
$833$	557536.	−	98308.7i	0.803495	−	0.141678i
$834$		0			0
$835$	−856684.	−	718843.i	−1.22870	−	1.03101i
$836$		0			0
$837$	−448088.	−	140372.i	−0.639605	−	0.200368i
$838$		0			0
$839$	495313.	−	590291.i	0.703648	−	0.838576i	−0.289286	−	0.957243i	$-0.593418\pi$
$839$	495313.	−	590291.i	0.703648	−	0.838576i	0.992934	+	0.118667i	$0.0378622\pi$
$840$		0			0
$841$	54609.7	+	309707.i	0.0772107	+	0.437884i
$842$		0			0
$843$	439691.	−	1.26751e6i	0.618717	−	1.78359i
$844$		0			0
$845$	1.47577e6	+	852038.i	2.06684	+	1.19329i
$846$		0			0
$847$	18680.4	+	32355.3i	0.0260386	+	0.0451003i
$848$		0			0
$849$	−508469.	+	7709.22i	−0.705422	+	0.0106954i
$850$		0			0
$851$	−421953.	−	502864.i	−0.582646	−	0.694371i
$852$		0			0
$853$	−222039.	+	80815.4i	−0.305162	+	0.111070i	−0.490062	−	0.871687i	$-0.663026\pi$
$853$	−222039.	+	80815.4i	−0.305162	+	0.111070i	0.184900	+	0.982757i	$0.440804\pi$
$854$		0			0
$855$	−64913.1	+	82304.2i	−0.0887974	+	0.112587i
$856$		0			0
$857$	230266.	+	40602.2i	0.313523	+	0.0552825i	0.328195	−	0.944610i	$-0.393560\pi$
$857$	230266.	+	40602.2i	0.313523	+	0.0552825i	−0.0146727	+	0.999892i	$0.504671\pi$
$858$		0			0
$859$	243605.	+	88665.0i	0.330141	+	0.120162i	0.501772	−	0.865000i	$-0.332681\pi$
$859$	243605.	+	88665.0i	0.330141	+	0.120162i	−0.171631	+	0.985161i	$0.554904\pi$
$860$		0			0
$861$	36443.9	+	5857.83i	0.0491607	+	0.00790188i
$862$		0			0
$863$		−	1.16509e6i		−	1.56437i	−0.623049	−	0.782183i	$-0.714106\pi$
$863$		−	1.16509e6i		−	1.56437i	0.623049	−	0.782183i	$-0.285894\pi$
$864$		0			0
$865$	578682.			0.773406
$866$		0			0
$867$	−229615.	+	87537.3i	−0.305465	+	0.116454i
$868$		0			0
$869$	243579.	−	669227.i	0.322552	−	0.886204i
$870$		0			0
$871$	87547.3	−	496505.i	0.115400	−	0.654467i
$872$		0			0
$873$	−179311.	−	289908.i	−0.235277	−	0.380392i
$874$		0			0
$875$	−18949.6	−	52063.7i	−0.0247505	−	0.0680015i
$876$		0			0
$877$	246319.	−	206686.i	0.320257	−	0.268727i	−0.468459	−	0.883485i	$-0.655191\pi$
$877$	246319.	−	206686.i	0.320257	−	0.268727i	0.788716	+	0.614758i	$0.210746\pi$
$878$		0			0
$879$	−71656.2	−	39934.7i	−0.0927418	−	0.0516860i
$880$		0			0
$881$	−427889.	+	247042.i	−0.551288	+	0.318286i	−0.749641	−	0.661844i	$-0.769774\pi$
$881$	−427889.	+	247042.i	−0.551288	+	0.318286i	0.198353	+	0.980131i	$0.436441\pi$
$882$		0			0
$883$	−181880.	+	315025.i	−0.233272	+	0.404039i	−0.958769	−	0.284186i	$-0.908277\pi$
$883$	−181880.	+	315025.i	−0.233272	+	0.404039i	0.725497	+	0.688225i	$0.241610\pi$
$884$		0			0
$885$	84181.3	+	438440.i	0.107480	+	0.559788i
$886$		0			0
$887$	−1.12707e6	+	198734.i	−1.43254	+	0.252595i	−0.835443	−	0.549578i	$-0.814789\pi$
$887$	−1.12707e6	+	198734.i	−1.43254	+	0.252595i	−0.597093	+	0.802172i	$0.703678\pi$
$888$		0			0
$889$	−70900.2	−	59492.3i	−0.0897106	−	0.0752761i
$890$		0			0
$891$	298774.	+	315122.i	0.376346	+	0.396939i
$892$		0			0
$893$	119093.	−	141930.i	0.149343	−	0.177980i
$894$		0			0
$895$	246069.	+	1.39553e6i	0.307192	+	1.74218i
$896$		0			0
$897$	914394.	−	175565.i	1.13644	−	0.218200i
$898$		0			0
$899$	−349605.	−	201845.i	−0.432572	−	0.249746i
$900$		0			0
$901$	−465595.	−	806434.i	−0.573533	−	0.993389i
$902$		0			0
$903$	−5031.41	+	9028.03i	−0.00617041	+	0.0110718i
$904$		0			0
$905$	668376.	+	796540.i	0.816063	+	0.972546i
$906$		0			0
$907$	1.30217e6	−	473949.i	1.58289	−	0.576126i	0.607062	−	0.794654i	$-0.292348\pi$
$907$	1.30217e6	−	473949.i	1.58289	−	0.576126i	0.975831	+	0.218529i	$0.0701257\pi$
$908$		0			0
$909$	−39643.8	−	1.30707e6i	−0.0479785	−	1.58187i
$910$		0			0
$911$	313406.	+	55262.0i	0.377634	+	0.0665871i	0.359244	−	0.933244i	$-0.383035\pi$
$911$	313406.	+	55262.0i	0.377634	+	0.0665871i	0.0183904	+	0.999831i	$0.494146\pi$
$912$		0			0
$913$	−390887.	−	142271.i	−0.468932	−	0.170677i
$914$		0			0
$915$	299699.	+	786126.i	0.357967	+	0.938966i
$916$		0			0
$917$		−	22431.7i		−	0.0266762i
$918$		0			0
$919$	−303248.			−0.359060			−0.179530	−	0.983753i	$-0.557458\pi$
$919$	−303248.			−0.359060			−0.179530	+	0.983753i	$0.557458\pi$
$920$		0			0
$921$	159876.	−	994653.i	0.188480	−	1.17261i
$922$		0			0
$923$	−774165.	+	2.12700e6i	−0.908721	+	2.49669i
$924$		0			0
$925$	10480.3	−	59437.0i	0.0122488	−	0.0694662i
$926$		0			0
$927$	−164970.	+	1.13600e6i	−0.191976	+	1.32196i
$928$		0			0
$929$	−319502.	−	877826.i	−0.370205	−	1.01713i	−0.975282	−	0.220964i	$-0.929080\pi$
$929$	−319502.	−	877826.i	−0.370205	−	1.01713i	0.605077	−	0.796167i	$-0.293142\pi$
$930$		0			0
$931$	92429.2	−	77557.3i	0.106638	−	0.0894795i
$932$		0			0
$933$	3275.89	+	216064.i	0.00376327	+	0.248210i
$934$		0			0
$935$	348039.	−	200941.i	0.398112	−	0.229850i
$936$		0			0
$937$	−56054.9	+	97089.9i	−0.0638461	+	0.110585i	−0.896182	−	0.443688i	$-0.853670\pi$
$937$	−56054.9	+	97089.9i	−0.0638461	+	0.110585i	0.832335	+	0.554272i	$0.187003\pi$
$938$		0			0
$939$	1.51063e6	+	524029.i	1.71328	+	0.594325i
$940$		0			0
$941$	−462012.	+	81465.1i	−0.521764	+	0.0920010i	−0.428326	−	0.903624i	$-0.640897\pi$
$941$	−462012.	+	81465.1i	−0.521764	+	0.0920010i	−0.0934373	+	0.995625i	$0.529785\pi$
$942$		0			0
$943$	289457.	+	242883.i	0.325507	+	0.273133i
$944$		0			0
$945$	57265.4	−	36631.1i	0.0641252	−	0.0410191i
$946$		0			0
$947$	−928332.	+	1.10634e6i	−1.03515	+	1.23364i	−0.0633116	+	0.997994i	$0.520166\pi$
$947$	−928332.	+	1.10634e6i	−1.03515	+	1.23364i	−0.971838	+	0.235650i	$0.924278\pi$
$948$		0			0
$949$	−89121.2	−	505432.i	−0.0989575	−	0.561216i
$950$		0			0
$951$	−1.00969e6	−	1.16691e6i	−1.11642	−	1.29025i
$952$		0			0
$953$	1.45142e6	+	837978.i	1.59811	+	0.922670i	0.991851	+	0.127405i	$0.0406647\pi$
$953$	1.45142e6	+	837978.i	1.59811	+	0.922670i	0.606261	+	0.795266i	$0.292669\pi$
$954$		0			0
$955$	−207012.	−	358554.i	−0.226980	−	0.393141i
$956$		0			0
$957$	191544.	+	320444.i	0.209143	+	0.349887i
$958$		0			0
$959$	47623.9	+	56756.0i	0.0517831	+	0.0617127i
$960$		0			0
$961$	477962.	−	173964.i	0.517543	−	0.188370i
$962$		0			0
$963$	−433984.	+	90168.1i	−0.467973	+	0.0972300i
$964$		0			0
$965$	−805630.	−	142054.i	−0.865130	−	0.152546i
$966$		0			0
$967$	1.24831e6	+	454348.i	1.33497	+	0.485888i	0.908223	−	0.418486i	$-0.137439\pi$
$967$	1.24831e6	+	454348.i	1.33497	+	0.485888i	0.426742	+	0.904373i	$0.359661\pi$
$968$		0			0
$969$	68052.4	−	83644.8i	0.0724763	−	0.0890823i
$970$		0			0
$971$			1.47439e6i			1.56378i	0.623417	+	0.781889i	$0.285744\pi$
$971$			1.47439e6i			1.56378i	−0.623417	+	0.781889i	$0.714256\pi$
$972$		0			0
$973$	−32796.2			−0.0346416
$974$		0			0
$975$	66404.5	+	54025.8i	0.0698535	+	0.0568319i
$976$		0			0
$977$	−464456.	+	1.27608e6i	−0.486582	+	1.33687i	0.417175	+	0.908826i	$0.363020\pi$
$977$	−464456.	+	1.27608e6i	−0.486582	+	1.33687i	−0.903757	+	0.428046i	$0.859202\pi$
$978$		0			0
$979$	59624.5	−	338147.i	0.0622099	−	0.352810i
$980$		0			0
$981$	−146058.	−	702987.i	−0.151771	−	0.730481i
$982$		0			0
$983$	−625431.	−	1.71836e6i	−0.647251	−	1.77831i	−0.627646	−	0.778499i	$-0.715981\pi$
$983$	−625431.	−	1.71836e6i	−0.647251	−	1.77831i	−0.0196048	−	0.999808i	$-0.506241\pi$
$984$		0			0
$985$	−1.14763e6	+	962978.i	−1.18285	+	0.992531i
$986$		0			0
$987$	−103135.	+	61648.1i	−0.105869	+	0.0632828i
$988$		0			0
$989$	−91627.2	+	52901.0i	−0.0936767	+	0.0540843i
$990$		0			0
$991$	−506647.	+	877538.i	−0.515891	+	0.893550i	0.483939	+	0.875102i	$0.339206\pi$
$991$	−506647.	+	877538.i	−0.515891	+	0.893550i	−0.999830	+	0.0184477i	$0.994128\pi$
$992$		0			0
$993$	−845172.	+	731302.i	−0.857130	+	0.741649i
$994$		0			0
$995$	437714.	−	77180.9i	0.442125	−	0.0779585i
$996$		0			0
$997$	−663978.	−	557144.i	−0.667980	−	0.560502i	0.244487	−	0.969653i	$-0.421380\pi$
$997$	−663978.	−	557144.i	−0.667980	−	0.560502i	−0.912467	+	0.409151i	$0.865825\pi$
$998$		0			0
$999$	−1.42501e6	+	64856.2i	−1.42786	+	0.0649861i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.65.10	yes	72
3.2	odd	2		324.5.k.a.197.10		72
27.5	odd	18	inner	108.5.k.a.5.10	✓	72
27.22	even	9		324.5.k.a.125.10		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.10	✓	72	27.5	odd	18	inner
108.5.k.a.65.10	yes	72	1.1	even	1	trivial
324.5.k.a.125.10		72	27.22	even	9
324.5.k.a.197.10		72	3.2	odd	2

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)

\(f(q)\)	\(=\)	\(q+(6.98131 + 5.67991i) q^{3} +(-8.75895 + 24.0650i) q^{5} +(-0.632293 + 3.58591i) q^{7} +(16.4774 + 79.3063i) q^{9} +O(q^{10})\)	\(q+(6.98131 + 5.67991i) q^{3} +(-8.75895 + 24.0650i) q^{5} +(-0.632293 + 3.58591i) q^{7} +(16.4774 + 79.3063i) q^{9} +(-22.6368 - 62.1941i) q^{11} +(-236.237 + 198.227i) q^{13} +(-197.836 + 118.255i) q^{15} +(205.336 - 118.551i) q^{17} +(25.2661 - 43.7622i) q^{19} +(-24.7819 + 21.4430i) q^{21} +(-330.376 + 58.2542i) q^{23} +(-23.6276 - 19.8259i) q^{25} +(-335.419 + 647.252i) q^{27} +(-402.857 + 480.107i) q^{29} +(111.849 + 634.330i) q^{31} +(195.222 - 562.771i) q^{33} +(-80.7568 - 46.6249i) q^{35} +(978.384 + 1694.61i) q^{37} +(-2775.15 + 42.0759i) q^{39} +(-724.003 - 862.834i) q^{41} +(296.362 - 107.867i) q^{43} +(-2052.83 - 298.112i) q^{45} +(3610.79 + 636.681i) q^{47} +(2243.74 + 816.656i) q^{49} +(2106.87 + 338.649i) q^{51} -3927.39i q^{53} +1694.98 q^{55} +(424.955 - 162.008i) q^{57} +(662.491 - 1820.18i) q^{59} +(633.851 - 3594.75i) q^{61} +(-294.804 + 8.94148i) q^{63} +(-2701.14 - 7421.31i) q^{65} +(-1252.37 + 1050.86i) q^{67} +(-2637.34 - 1469.81i) q^{69} +(6356.50 - 3669.93i) q^{71} +(-832.121 + 1441.28i) q^{73} +(-52.3423 - 272.614i) q^{75} +(237.336 - 41.8487i) q^{77} +(8242.86 + 6916.58i) q^{79} +(-6017.99 + 2613.52i) q^{81} +(4039.89 - 4814.55i) q^{83} +(1054.40 + 5979.79i) q^{85} +(-5539.43 + 1063.58i) q^{87} +(4492.85 + 2593.95i) q^{89} +(-561.452 - 972.463i) q^{91} +(-2822.08 + 5063.75i) q^{93} +(831.832 + 991.339i) q^{95} +(-3954.60 + 1439.36i) q^{97} +(4559.39 - 2820.04i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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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