LMFDB - Embedded newform 108.5.k.a.5.10

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

$q$-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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

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

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
$3$	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	−0.982255	−	0.187548i	$-0.939946\pi$
$5$	−8.75895	−	24.0650i	−0.350358	−	0.962600i	0.631898	−	0.775052i	$-0.282276\pi$
$6$		0			0
$7$	−0.632293	−	3.58591i	−0.0129039	−	0.0731819i	0.977676	−	0.210119i	$-0.0673852\pi$
$7$	−0.632293	−	3.58591i	−0.0129039	−	0.0731819i	−0.990580	+	0.136937i	$0.956274\pi$
$8$		0			0
$9$	16.4774	−	79.3063i	0.203424	−	0.979091i
$10$		0			0
$11$	−22.6368	+	62.1941i	−0.187081	+	0.514001i	−0.997406	−	0.0719797i	$-0.977068\pi$
$11$	−22.6368	+	62.1941i	−0.187081	+	0.514001i	0.810325	+	0.585981i	$0.199291\pi$
$12$		0			0
$13$	−236.237	−	198.227i	−1.39785	−	1.17294i	−0.962042	−	0.272901i	$-0.912017\pi$
$13$	−236.237	−	198.227i	−1.39785	−	1.17294i	−0.435812	−	0.900038i	$-0.643539\pi$
$14$		0			0
$15$	−197.836	−	118.255i	−0.879271	−	0.525579i
$16$		0			0
$17$	205.336	+	118.551i	0.710505	+	0.410210i	0.811248	−	0.584702i	$-0.198789\pi$
$17$	205.336	+	118.551i	0.710505	+	0.410210i	−0.100743	+	0.994912i	$0.532122\pi$
$18$		0			0
$19$	25.2661	+	43.7622i	0.0699892	+	0.121225i	0.898896	−	0.438161i	$-0.144370\pi$
$19$	25.2661	+	43.7622i	0.0699892	+	0.121225i	−0.828907	+	0.559386i	$0.811037\pi$
$20$		0			0
$21$	−24.7819	−	21.4430i	−0.0561947	−	0.0486236i
$22$		0			0
$23$	−330.376	−	58.2542i	−0.624530	−	0.110121i	−0.147577	−	0.989051i	$-0.547147\pi$
$23$	−330.376	−	58.2542i	−0.624530	−	0.110121i	−0.476952	+	0.878929i	$0.658259\pi$
$24$		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	0.471368	−	0.881936i	$-0.343760\pi$
$29$	−402.857	−	480.107i	−0.479022	−	0.570876i	−0.950390	+	0.311060i	$0.899316\pi$
$30$		0			0
$31$	111.849	−	634.330i	0.116389	−	0.660073i	−0.869665	−	0.493643i	$-0.835665\pi$
$31$	111.849	−	634.330i	0.116389	−	0.660073i	0.986053	−	0.166430i	$-0.0532239\pi$
$32$		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	−0.248416	−	0.968654i	$-0.579910\pi$
$37$	978.384	−	1694.61i	0.714671	−	1.23785i	0.963086	−	0.269193i	$-0.0867568\pi$
$38$		0			0
$39$	−2775.15	−	42.0759i	−1.82456	−	0.0276633i
$40$		0			0
$41$	−724.003	+	862.834i	−0.430698	+	0.513286i	−0.937124	−	0.348998i	$-0.886522\pi$
$41$	−724.003	+	862.834i	−0.430698	+	0.513286i	0.506426	+	0.862284i	$0.330967\pi$
$42$		0			0
$43$	296.362	+	107.867i	0.160282	+	0.0583380i	0.420915	−	0.907100i	$-0.361709\pi$
$43$	296.362	+	107.867i	0.160282	+	0.0583380i	−0.260633	+	0.965438i	$0.583931\pi$
$44$		0			0
$45$	−2052.83	+	298.112i	−1.01374	+	0.147216i
$46$		0			0
$47$	3610.79	−	636.681i	1.63458	−	0.288221i	0.720411	−	0.693547i	$-0.243953\pi$
$47$	3610.79	−	636.681i	1.63458	−	0.288221i	0.914172	+	0.405326i	$0.132842\pi$
$48$		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	$0.246404\pi$
$53$			3927.39i			1.39815i	−0.715051	+	0.699073i	$0.753596\pi$
$54$		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	0.997748	−	0.0670710i	$-0.0213654\pi$
$59$	662.491	+	1820.18i	0.190316	+	0.522890i	−0.807432	+	0.589961i	$0.799143\pi$
$60$		0			0
$61$	633.851	+	3594.75i	0.170344	+	0.966070i	0.943382	+	0.331710i	$0.107625\pi$
$61$	633.851	+	3594.75i	0.170344	+	0.966070i	−0.773037	+	0.634361i	$0.781263\pi$
$62$		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	0.492548	−	0.870285i	$-0.336066\pi$
$67$	−1252.37	−	1050.86i	−0.278986	−	0.234097i	−0.771534	+	0.636188i	$0.780510\pi$
$68$		0			0
$69$	−2637.34	+	1469.81i	−0.553946	+	0.308720i
$70$		0			0
$71$	6356.50	+	3669.93i	1.26096	+	0.728016i	0.973260	−	0.229705i	$-0.0737761\pi$
$71$	6356.50	+	3669.93i	1.26096	+	0.728016i	0.287700	+	0.957721i	$0.407109\pi$
$72$		0			0
$73$	−832.121	−	1441.28i	−0.156150	−	0.270459i	0.777327	−	0.629096i	$-0.216575\pi$
$73$	−832.121	−	1441.28i	−0.156150	−	0.270459i	−0.933477	+	0.358637i	$0.883242\pi$
$74$		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	0.334617	−	0.942354i	$-0.391393\pi$
$79$	8242.86	−	6916.58i	1.32076	−	1.10825i	0.986143	−	0.165895i	$-0.0530513\pi$
$80$		0			0
$81$	−6017.99	−	2613.52i	−0.917237	−	0.398341i
$82$		0			0
$83$	4039.89	+	4814.55i	0.586426	+	0.698875i	0.974915	−	0.222578i	$-0.0714473\pi$
$83$	4039.89	+	4814.55i	0.586426	+	0.698875i	−0.388489	+	0.921453i	$0.627003\pi$
$84$		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	−0.188826	−	0.982011i	$-0.560468\pi$
$89$	4492.85	−	2593.95i	0.567207	−	0.327477i	0.756033	+	0.654533i	$0.227135\pi$
$90$		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	0.123208	−	0.992381i	$-0.460682\pi$
$97$	−3954.60	−	1439.36i	−0.420300	−	0.152977i	−0.543507	+	0.839404i	$0.682904\pi$
$98$		0			0
$99$	4559.39	+	2820.04i	0.465197	+	0.287730i
$100$		0			0
$101$	−15898.8	+	2803.39i	−1.55856	+	0.274815i	−0.885451	−	0.464732i	$-0.846151\pi$
$101$	−15898.8	+	2803.39i	−1.55856	+	0.274815i	−0.673104	+	0.739548i	$0.735039\pi$
$102$		0			0
$103$	13317.2	−	4847.05i	1.25527	−	0.456881i	0.373091	−	0.927795i	$-0.378298\pi$
$103$	13317.2	−	4847.05i	1.25527	−	0.456881i	0.882179	+	0.470914i	$0.156076\pi$
$104$		0			0
$105$	−298.963	+	784.194i	−0.0271168	+	0.0711287i
$106$		0			0
$107$		−	5472.24i		−	0.477967i	−0.971024	−	0.238984i	$-0.923186\pi$
$107$		−	5472.24i		−	0.477967i	0.971024	−	0.238984i	$-0.0768142\pi$
$108$		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	0.842322	−	0.538975i	$-0.181188\pi$
$113$	−1907.44	−	5240.66i	−0.149381	−	0.410421i	−0.991702	+	0.128555i	$0.958966\pi$
$114$		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	−0.927211	−	0.374540i	$-0.877800\pi$
$127$	−12709.1	−	22012.8i	−0.787967	−	1.36480i	0.139244	−	0.990258i	$-0.455533\pi$
$128$		0			0
$129$	2681.67	−	930.255i	0.161148	−	0.0559014i
$130$		0			0
$131$	−6066.88	−	1069.75i	−0.353527	−	0.0623364i	−0.00593554	−	0.999982i	$-0.501889\pi$
$131$	−6066.88	−	1069.75i	−0.353527	−	0.0623364i	−0.347592	+	0.937646i	$0.613000\pi$
$132$		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	0.992165	−	0.124931i	$-0.0398709\pi$
$137$	13079.1	+	15587.0i	0.696845	+	0.830467i	−0.295321	+	0.955398i	$0.595426\pi$
$138$		0			0
$139$	1564.03	−	8870.06i	0.0809498	−	0.459089i	−0.917207	−	0.398410i	$-0.869562\pi$
$139$	1564.03	−	8870.06i	0.0809498	−	0.459089i	0.998157	−	0.0606794i	$-0.0193267\pi$
$140$		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	−0.512834	−	0.858488i	$-0.671404\pi$
$149$	9361.36	−	11156.4i	0.421664	−	0.502519i	0.934498	+	0.355968i	$0.115849\pi$
$150$		0			0
$151$	−4398.99	−	1601.10i	−0.192930	−	0.0702207i	0.243748	−	0.969839i	$-0.421623\pi$
$151$	−4398.99	−	1601.10i	−0.192930	−	0.0702207i	−0.436678	+	0.899618i	$0.643845\pi$
$152$		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	−0.424466	−	0.905444i	$-0.639538\pi$
$157$	−4131.63	+	1503.79i	−0.167619	+	0.0610082i	0.256848	+	0.966452i	$0.417316\pi$
$158$		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	−0.852399	−	0.522891i	$-0.824853\pi$
$167$	−14935.4	−	41034.8i	−0.535532	−	1.47136i	0.316868	−	0.948470i	$-0.397369\pi$
$168$		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	0.741051	+	0.671448i	$0.234327\pi$
$173$	−7728.43	+	21233.7i	−0.258225	+	0.709469i	−0.999277	+	0.0380204i	$0.987895\pi$
$174$		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	0.999987	−	0.00507689i	$-0.00161603\pi$
$179$	47920.0	+	27666.6i	1.49558	+	0.863476i	0.495597	+	0.868553i	$0.334949\pi$
$180$		0			0
$181$	20301.3	+	35162.8i	0.619678	+	1.07331i	0.989544	+	0.144229i	$0.0460702\pi$
$181$	20301.3	+	35162.8i	0.619678	+	1.07331i	−0.369866	+	0.929085i	$0.620596\pi$
$182$		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	0.604575	−	0.796549i	$-0.293343\pi$
$191$	−10391.8	−	12384.5i	−0.284856	−	0.339478i	−0.889430	+	0.457071i	$0.848899\pi$
$192$		0			0
$193$	5546.95	−	31458.3i	0.148915	−	0.844542i	−0.815224	−	0.579146i	$-0.803386\pi$
$193$	5546.95	−	31458.3i	0.148915	−	0.844542i	0.964140	−	0.265396i	$-0.0855027\pi$
$194$		0			0
$195$	23294.9	+	67152.7i	0.612620	+	1.76601i
$196$		0			0
$197$	50661.7	−	29249.5i	1.30541	−	0.753679i	0.324083	−	0.946028i	$-0.394944\pi$
$197$	50661.7	−	29249.5i	1.30541	−	0.753679i	0.981326	+	0.192350i	$0.0616108\pi$
$198$		0			0
$199$	−8677.79	+	15030.4i	−0.219131	+	0.379545i	−0.954542	−	0.298075i	$-0.903655\pi$
$199$	−8677.79	+	15030.4i	−0.219131	+	0.379545i	0.735412	+	0.677620i	$0.236989\pi$
$200$		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	−0.269772	−	0.962924i	$-0.586948\pi$
$211$	6345.41	−	2309.54i	0.142526	−	0.0518753i	0.412298	+	0.911049i	$0.364726\pi$
$212$		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	0.714310	+	0.699830i	$0.246741\pi$
$223$	14045.2	+	79654.1i	0.282434	+	1.60176i	−0.431876	+	0.901933i	$0.642148\pi$
$224$		0			0
$225$	1183.00	+	2200.50i	0.0233679	+	0.0434666i
$226$		0			0
$227$	−4625.90	+	12709.6i	−0.0897728	+	0.246649i	−0.976451	−	0.215738i	$-0.930784\pi$
$227$	−4625.90	+	12709.6i	−0.0897728	+	0.246649i	0.886679	+	0.462386i	$0.153007\pi$
$228$		0			0
$229$	−14498.7	−	12165.9i	−0.276477	−	0.231992i	0.493996	−	0.869464i	$-0.335536\pi$
$229$	−14498.7	−	12165.9i	−0.276477	−	0.231992i	−0.770473	+	0.637472i	$0.779980\pi$
$230$		0			0
$231$	1894.61	−	1055.89i	0.0355055	−	0.0197876i
$232$		0			0
$233$	−46376.2	−	26775.3i	−0.854247	−	0.493200i	0.00783474	−	0.999969i	$-0.497506\pi$
$233$	−46376.2	−	26775.3i	−0.854247	−	0.493200i	−0.862081	+	0.506770i	$0.830839\pi$
$234$		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	−0.197480	−	0.980307i	$-0.563276\pi$
$239$	−41032.0	−	7235.05i	−0.718335	−	0.126662i	−0.520855	+	0.853645i	$0.674387\pi$
$240$		0			0
$241$	−71951.4	+	60374.4i	−1.23881	+	1.03949i	−0.241196	+	0.970477i	$0.577540\pi$
$241$	−71951.4	+	60374.4i	−1.23881	+	1.03949i	−0.997616	+	0.0690098i	$0.978016\pi$
$242$		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	−0.0106150	−	0.999944i	$-0.503379\pi$
$251$	53554.3	−	30919.6i	0.850054	−	0.490779i	0.860669	+	0.509165i	$0.170046\pi$
$252$		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	−0.713670	−	0.700482i	$-0.752968\pi$
$257$	6611.38	−	7879.13i	0.100098	−	0.119292i	0.813768	+	0.581190i	$0.197413\pi$
$258$		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	0.314558	−	0.949238i	$-0.398144\pi$
$263$	64659.5	−	11401.2i	0.934805	−	0.164831i	0.620247	+	0.784407i	$0.287032\pi$
$264$		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	$0.183428\pi$
$269$			78857.5i			1.08978i	−0.838508	+	0.544890i	$0.816572\pi$
$270$		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	−0.947602	−	0.319454i	$-0.896501\pi$
$277$	−12768.2	−	72412.3i	−0.166407	−	0.943741i	0.781195	−	0.624287i	$-0.214611\pi$
$278$		0			0
$279$	−48463.4	−	19322.4i	−0.622595	−	0.248230i
$280$		0			0
$281$	−50983.9	+	140077.i	−0.645685	+	1.77401i	−0.0126027	+	0.999921i	$0.504012\pi$
$281$	−50983.9	+	140077.i	−0.645685	+	1.77401i	−0.633082	+	0.774084i	$0.718211\pi$
$282$		0			0
$283$	−43283.8	−	36319.5i	−0.540447	−	0.453489i	0.331244	−	0.943545i	$-0.392532\pi$
$283$	−43283.8	−	36319.5i	−0.540447	−	0.453489i	−0.871691	+	0.490056i	$0.836976\pi$
$284$		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	0.121124	−	0.992637i	$-0.461350\pi$
$293$	−8976.28	−	1582.76i	−0.104559	−	0.0184366i	−0.225683	+	0.974201i	$0.572461\pi$
$294$		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	0.399882	+	0.916566i	$0.369051\pi$
$307$	−55967.7	+	96939.0i	−0.593829	+	1.02854i	−0.993711	+	0.111975i	$0.964282\pi$
$308$		0			0
$309$	65440.4	−	109479.i	0.685376	−	1.14660i
$310$		0			0
$311$	15433.3	−	18392.6i	0.159565	−	0.190162i	−0.680338	−	0.732898i	$-0.738167\pi$
$311$	15433.3	−	18392.6i	0.159565	−	0.190162i	0.839903	+	0.542736i	$0.182612\pi$
$312$		0			0
$313$	166946.	+	60763.3i	1.70407	+	0.620230i	0.996278	−	0.0861930i	$-0.0274702\pi$
$313$	166946.	+	60763.3i	1.70407	+	0.620230i	0.707790	+	0.706423i	$0.249692\pi$
$314$		0			0
$315$	2367.00	+	7172.78i	0.0238548	+	0.0722880i
$316$		0			0
$317$	−168850.	+	29772.9i	−1.68029	+	0.296280i	−0.930743	−	0.365674i	$-0.880838\pi$
$317$	−168850.	+	29772.9i	−1.68029	+	0.296280i	−0.749544	+	0.661954i	$0.769727\pi$
$318$		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	−0.909800	−	0.415047i	$-0.863765\pi$
$331$	−21564.0	−	122296.i	−0.196822	−	1.11623i	0.712978	−	0.701186i	$-0.247346\pi$
$332$		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	0.994715	+	0.102674i	$0.0327400\pi$
$337$	144069.	+	120888.i	1.26856	+	1.06445i	0.273845	+	0.961774i	$0.411704\pi$
$338$		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	0.672925	−	0.739710i	$-0.265037\pi$
$347$	126703.	+	22341.2i	1.05227	+	0.185544i	0.379347	+	0.925254i	$0.376149\pi$
$348$		0			0
$349$	36525.7	−	30648.7i	0.299881	−	0.251630i	−0.480414	−	0.877042i	$-0.659514\pi$
$349$	36525.7	−	30648.7i	0.299881	−	0.251630i	0.780295	+	0.625412i	$0.215069\pi$
$350$		0			0
$351$	−49064.1	+	219394.i	−0.398244	+	1.78078i
$352$		0			0
$353$	94657.4	+	112808.i	0.759635	+	0.905298i	0.997825	−	0.0659209i	$-0.0209985\pi$
$353$	94657.4	+	112808.i	0.759635	+	0.905298i	−0.238190	+	0.971219i	$0.576554\pi$
$354$		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	−0.225134	−	0.974328i	$-0.572282\pi$
$359$	65225.6	−	37658.0i	0.506092	−	0.292192i	0.731226	+	0.682136i	$0.238949\pi$
$360$		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	0.707394	−	0.706820i	$-0.249871\pi$
$367$	107072.	+	38970.8i	0.794954	+	0.289339i	0.0875599	+	0.996159i	$0.472093\pi$
$368$		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	−0.149264	−	0.988797i	$-0.547690\pi$
$373$	51753.2	−	18836.6i	0.371980	−	0.135390i	0.521244	+	0.853408i	$0.325468\pi$
$374$		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	0.804028	+	0.594591i	$0.202686\pi$
$383$	83657.1	+	229846.i	0.570303	+	1.56689i	−0.233726	+	0.972303i	$0.575092\pi$
$384$		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	−0.708500	−	0.705711i	$-0.750628\pi$
$389$	43560.0	−	119680.i	0.287865	−	0.790903i	0.996365	−	0.0851915i	$-0.0271502\pi$
$390$		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	0.978096	−	0.208154i	$-0.0667454\pi$
$397$	48666.8	+	84293.3i	0.308782	+	0.534825i	−0.669314	+	0.742979i	$0.733412\pi$
$398$		0			0
$399$	312.251	−	1626.29i	0.00196136	−	0.0102153i
$400$		0			0
$401$	−20496.9	−	3614.15i	−0.127467	−	0.0224759i	0.109551	−	0.993981i	$-0.465059\pi$
$401$	−20496.9	−	3614.15i	−0.127467	−	0.0224759i	−0.237018	+	0.971505i	$0.576170\pi$
$402$		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	0.322702	+	0.946501i	$0.395409\pi$
$409$	−50897.1	+	288652.i	−0.304261	+	1.72555i	−0.626963	+	0.779049i	$0.715702\pi$
$410$		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	−0.865906	−	0.500207i	$-0.833257\pi$
$419$	−39138.7	+	46643.7i	−0.222935	+	0.265684i	0.642971	+	0.765891i	$0.277702\pi$
$420$		0			0
$421$	−210167.	−	76494.4i	−1.18577	−	0.431584i	−0.327533	−	0.944840i	$-0.606217\pi$
$421$	−210167.	−	76494.4i	−1.18577	−	0.431584i	−0.858236	+	0.513256i	$0.828439\pi$
$422$		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	$0.228782\pi$
$431$			244624.i			1.31687i	−0.752635	+	0.658437i	$0.771218\pi$
$432$		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	−0.997597	−	0.0692860i	$-0.977928\pi$
$439$	−16161.5	−	91656.5i	−0.0838597	−	0.475592i	0.913737	−	0.406306i	$-0.133183\pi$
$440$		0			0
$441$	−27795.0	−	191399.i	−0.142919	−	0.984155i
$442$		0			0
$443$	94532.4	−	259726.i	0.481696	−	1.32345i	−0.426342	−	0.904562i	$-0.640198\pi$
$443$	94532.4	−	259726.i	0.481696	−	1.32345i	0.908038	−	0.418887i	$-0.137580\pi$
$444$		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	0.274866	−	0.961483i	$-0.411366\pi$
$449$	−84746.9	−	48928.6i	−0.420369	−	0.242700i	−0.695235	+	0.718782i	$0.744700\pi$
$450$		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	−0.268072	−	0.963399i	$-0.586387\pi$
$457$	132440.	−	111130.i	0.634141	−	0.532107i	0.902212	+	0.431292i	$0.141942\pi$
$458$		0			0
$459$	7858.62	−	172668.i	0.0373010	−	0.819572i
$460$		0			0
$461$	−207164.	−	246888.i	−0.974793	−	1.16171i	−0.986826	−	0.161785i	$-0.948275\pi$
$461$	−207164.	−	246888.i	−0.974793	−	1.16171i	0.0120331	−	0.999928i	$-0.496170\pi$
$462$		0			0
$463$	42566.1	−	241404.i	0.198564	−	1.12611i	−0.708686	−	0.705524i	$-0.750712\pi$
$463$	42566.1	−	241404.i	0.198564	−	1.12611i	0.907250	−	0.420591i	$-0.138177\pi$
$464$		0			0
$465$	−97140.7	+	112266.i	−0.449257	+	0.519211i
$466$		0			0
$467$	199445.	−	115149.i	0.914510	−	0.527992i	0.0326303	−	0.999467i	$-0.489612\pi$
$467$	199445.	−	115149.i	0.914510	−	0.527992i	0.881879	+	0.471475i	$0.156278\pi$
$468$		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	0.124845	−	0.992176i	$-0.460157\pi$
$479$	133421.	−	23525.7i	0.581505	−	0.102535i	0.456660	+	0.889641i	$0.349046\pi$
$480$		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	−0.979986	−	0.199069i	$-0.936208\pi$
$491$	−86121.8	−	236618.i	−0.357232	−	0.981487i	0.622754	−	0.782418i	$-0.286014\pi$
$492$		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	0.737046	−	0.675843i	$-0.236220\pi$
$499$	49665.0	+	41673.9i	0.199457	+	0.167364i	−0.537589	+	0.843207i	$0.680665\pi$
$500$		0			0
$501$	−337343.	−	201645.i	−1.34399	−	0.803362i
$502$		0			0
$503$	−420415.	−	242727.i	−1.66166	−	0.959360i	−0.971923	−	0.235300i	$-0.924393\pi$
$503$	−420415.	−	242727.i	−1.66166	−	0.959360i	−0.689737	−	0.724060i	$-0.742274\pi$
$504$		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	−0.410703	−	0.911769i	$-0.634717\pi$
$509$	−287186.	−	50638.7i	−1.10848	−	0.195455i	−0.697778	+	0.716314i	$0.745828\pi$
$510$		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	−0.575244	−	0.817982i	$-0.695093\pi$
$521$	−41930.4	+	24208.5i	−0.154473	+	0.0891853i	0.420771	+	0.907167i	$0.361760\pi$
$522$		0			0
$523$	−155771.	+	269804.i	−0.569488	+	0.986382i	0.427129	+	0.904191i	$0.359525\pi$
$523$	−155771.	+	269804.i	−0.569488	+	0.986382i	−0.996617	+	0.0821910i	$0.973808\pi$
$524$		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	0.909549	+	0.415596i	$0.136427\pi$
$547$	58942.6	+	334280.i	0.196995	+	1.11721i	−0.712554	+	0.701617i	$0.752462\pi$
$548$		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	0.118092	−	0.993003i	$-0.462322\pi$
$557$	−211847.	−	122310.i	−0.682828	−	0.394231i	−0.800920	+	0.598772i	$0.795656\pi$
$558$		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	0.850280	−	0.526331i	$-0.176433\pi$
$563$	465712.	+	82117.5i	1.46927	+	0.259071i	0.618986	+	0.785402i	$0.287544\pi$
$564$		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	0.386783	−	0.922171i	$-0.373586\pi$
$569$	−190547.	−	227085.i	−0.588542	−	0.701397i	−0.975325	+	0.220774i	$0.929142\pi$
$570$		0			0
$571$	−96134.7	+	545207.i	−0.294855	+	1.67220i	0.372938	+	0.927856i	$0.378350\pi$
$571$	−96134.7	+	545207.i	−0.294855	+	1.67220i	−0.667792	+	0.744347i	$0.732761\pi$
$572$		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	−0.608232	−	0.793760i	$-0.708121\pi$
$577$	127612.	−	221030.i	0.383300	−	0.663895i	0.991532	+	0.129864i	$0.0414542\pi$
$578$		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	0.425255	−	0.905073i	$-0.360184\pi$
$587$	390885.	−	68923.6i	1.13442	−	0.200029i	0.709163	+	0.705045i	$0.249073\pi$
$588$		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	$0.0645416\pi$
$593$			141628.i			0.402754i	−0.979514	+	0.201377i	$0.935458\pi$
$594$		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	0.801959	+	0.597379i	$0.203791\pi$
$599$	205094.	+	563492.i	0.571611	+	1.57049i	−0.230348	+	0.973108i	$0.573987\pi$
$600$		0			0
$601$	−67002.3	−	379989.i	−0.185499	−	1.05202i	−0.925313	−	0.379204i	$-0.876198\pi$
$601$	−67002.3	−	379989.i	−0.185499	−	1.05202i	0.739814	−	0.672811i	$-0.234913\pi$
$602$		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	0.778949	−	0.627087i	$-0.215753\pi$
$607$	109301.	+	91714.6i	0.296652	+	0.248921i	−0.482297	+	0.876008i	$0.660197\pi$
$608$		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	0.941673	−	0.336529i	$-0.109253\pi$
$613$	67410.6	+	116759.i	0.179394	+	0.310719i	−0.762279	+	0.647248i	$0.775920\pi$
$614$		0			0
$615$	245268.	−	85082.2i	0.648472	−	0.224951i
$616$		0			0
$617$	550083.	+	96994.6i	1.44497	+	0.254787i	0.840487	−	0.541832i	$-0.182269\pi$
$617$	550083.	+	96994.6i	1.44497	+	0.254787i	0.604481	+	0.796619i	$0.293380\pi$
$618$		0			0
$619$	206114.	−	172950.i	0.537931	−	0.451377i	−0.332899	−	0.942963i	$-0.608027\pi$
$619$	206114.	−	172950.i	0.537931	−	0.451377i	0.870830	+	0.491585i	$0.163582\pi$
$620$		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	0.319451	+	0.947603i	$0.396501\pi$
$631$	−263153.	+	455795.i	−0.660922	+	1.14475i	−0.980374	+	0.197148i	$0.936832\pi$
$632$		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	−0.750476	−	0.660898i	$-0.770176\pi$
$641$	−505241.	+	89087.6i	−1.22965	+	0.216821i	−0.479176	+	0.877719i	$0.659065\pi$
$642$		0			0
$643$	−20896.4	+	7605.67i	−0.0505417	+	0.0183957i	−0.367167	−	0.930155i	$-0.619672\pi$
$643$	−20896.4	+	7605.67i	−0.0505417	+	0.0183957i	0.316626	+	0.948551i	$0.397450\pi$
$644$		0			0
$645$	−45875.2	−	56386.3i	−0.110270	−	0.135536i
$646$		0			0
$647$			341326.i			0.815381i	0.913120	+	0.407690i	$0.133666\pi$
$647$			341326.i			0.815381i	−0.913120	+	0.407690i	$0.866334\pi$
$648$		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	0.906000	−	0.423278i	$-0.139121\pi$
$653$	−25633.4	−	70427.1i	−0.0601145	−	0.165163i	−0.966114	+	0.258115i	$0.916899\pi$
$654$		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	−0.776215	−	0.630469i	$-0.782863\pi$
$659$	97130.3	−	266863.i	0.223658	−	0.614495i	0.999872	+	0.0159741i	$0.00508493\pi$
$660$		0			0
$661$	118548.	+	99473.7i	0.271326	+	0.227670i	0.768291	−	0.640101i	$-0.221108\pi$
$661$	118548.	+	99473.7i	0.271326	+	0.227670i	−0.496964	+	0.867771i	$0.665552\pi$
$662$		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	0.356715	−	0.934213i	$-0.383897\pi$
$673$	606326.	−	508768.i	1.33868	−	1.12328i	0.981963	−	0.189071i	$-0.0605478\pi$
$674$		0			0
$675$	20757.5	+	8643.03i	0.0455583	+	0.0189696i
$676$		0			0
$677$	319117.	+	380309.i	0.696262	+	0.829773i	0.992098	−	0.125466i	$-0.0400426\pi$
$677$	319117.	+	380309.i	0.696262	+	0.829773i	−0.295836	+	0.955239i	$0.595598\pi$
$678$		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	−0.440039	−	0.897978i	$-0.645036\pi$
$683$	54865.2	−	31676.5i	0.117613	−	0.0679040i	0.557653	+	0.830074i	$0.311702\pi$
$684$		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	0.548385	−	0.836226i	$-0.315243\pi$
$691$	205774.	+	74895.6i	0.430958	+	0.156856i	−0.117428	+	0.993081i	$0.537465\pi$
$692$		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	$0.164085\pi$
$701$			484481.i			0.985919i	−0.870052	+	0.492959i	$0.835915\pi$
$702$		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	−0.974177	−	0.225784i	$-0.927506\pi$
$709$	−68351.0	−	387638.i	−0.135973	−	0.771140i	0.838205	−	0.545356i	$-0.183606\pi$
$710$		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	−0.274389	−	0.961619i	$-0.588475\pi$
$719$	−643290.	−	371404.i	−1.24437	−	0.718437i	−0.969981	+	0.243182i	$0.921809\pi$
$720$		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	−0.0143568	−	0.999897i	$-0.504570\pi$
$727$	511540.	−	429233.i	0.967856	−	0.812128i	0.982213	+	0.187769i	$0.0601256\pi$
$728$		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	−0.778203	−	0.628013i	$-0.783868\pi$
$733$	90190.3	−	511495.i	0.167862	−	0.951991i	0.946065	−	0.323978i	$-0.105020\pi$
$734$		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	−0.882563	−	0.470195i	$-0.844184\pi$
$739$	−18612.0	+	32237.0i	−0.0340804	+	0.0590290i	0.848482	+	0.529224i	$0.177517\pi$
$740$		0			0
$741$	−68276.0	−	122510.i	−0.124346	−	0.223118i
$742$		0			0
$743$	−533099.	+	635323.i	−0.965674	+	1.15085i	0.0228434	+	0.999739i	$0.492728\pi$
$743$	−533099.	+	635323.i	−0.965674	+	1.15085i	−0.988517	+	0.151107i	$0.951716\pi$
$744$		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	0.0487651	−	0.998810i	$-0.484471\pi$
$751$	410674.	−	149473.i	0.728144	−	0.265023i	0.679379	+	0.733787i	$0.262249\pi$
$752$		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	−0.986603	−	0.163137i	$-0.947839\pi$
$761$	−194402.	−	534114.i	−0.335684	−	0.922284i	0.650920	−	0.759147i	$-0.274383\pi$
$762$		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	0.589045	−	0.808100i	$-0.299504\pi$
$769$	−61792.2	−	51849.8i	−0.104491	−	0.0876788i	−0.693537	+	0.720421i	$0.743948\pi$
$770$		0			0
$771$	1403.34	−	92558.7i	0.00236077	−	0.155707i
$772$		0			0
$773$	418443.	+	241588.i	0.700288	+	0.404312i	0.807455	−	0.589930i	$-0.200844\pi$
$773$	418443.	+	241588.i	0.700288	+	0.404312i	−0.107167	+	0.994241i	$0.534178\pi$
$774$		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	−0.972438	−	0.233163i	$-0.925092\pi$
$787$	13069.5	−	74120.9i	0.0211013	−	0.119672i	0.993539	+	0.113492i	$0.0362035\pi$
$788$		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	−0.914249	−	0.405154i	$-0.867218\pi$
$797$	−226447.	+	269869.i	−0.356493	+	0.424851i	0.557756	+	0.830005i	$0.311663\pi$
$798$		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	$0.345661\pi$
$809$			1.15808e6i			1.76947i	−0.466094	+	0.884735i	$0.654339\pi$
$810$		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	−0.998627	−	0.0523911i	$-0.983316\pi$
$821$	−134780.	+	370305.i	−0.199958	+	0.549380i	0.798669	+	0.601771i	$0.205538\pi$
$822$		0			0
$823$	287702.	+	241410.i	0.424759	+	0.356415i	0.829970	−	0.557808i	$-0.188357\pi$
$823$	287702.	+	241410.i	0.424759	+	0.356415i	−0.405211	+	0.914223i	$0.632802\pi$
$824$		0			0
$825$	−15770.1	−	9426.48i	−0.0231700	−	0.0138497i
$826$		0			0
$827$	170424.	+	98394.3i	0.249184	+	0.143866i	0.619390	−	0.785083i	$-0.287380\pi$
$827$	170424.	+	98394.3i	0.249184	+	0.143866i	−0.370207	+	0.928949i	$0.620713\pi$
$828$		0			0
$829$	−61104.9	−	105837.i	−0.0889134	−	0.154003i	0.818139	−	0.575021i	$-0.195006\pi$
$829$	−61104.9	−	105837.i	−0.0889134	−	0.154003i	−0.907052	+	0.421018i	$0.861673\pi$
$830$		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	0.992934	−	0.118667i	$-0.0378622\pi$
$839$	495313.	+	590291.i	0.703648	+	0.838576i	−0.289286	+	0.957243i	$0.593418\pi$
$840$		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	0.184900	−	0.982757i	$-0.440804\pi$
$853$	−222039.	−	80815.4i	−0.305162	−	0.111070i	−0.490062	+	0.871687i	$0.663026\pi$
$854$		0			0
$855$	−64913.1	−	82304.2i	−0.0887974	−	0.112587i
$856$		0			0
$857$	230266.	−	40602.2i	0.313523	−	0.0552825i	−0.0146727	−	0.999892i	$-0.504671\pi$
$857$	230266.	−	40602.2i	0.313523	−	0.0552825i	0.328195	+	0.944610i	$0.393560\pi$
$858$		0			0
$859$	243605.	−	88665.0i	0.330141	−	0.120162i	−0.171631	−	0.985161i	$-0.554904\pi$
$859$	243605.	−	88665.0i	0.330141	−	0.120162i	0.501772	+	0.865000i	$0.332681\pi$
$860$		0			0
$861$	36443.9	−	5857.83i	0.0491607	−	0.00790188i
$862$		0			0
$863$			1.16509e6i			1.56437i	0.623049	+	0.782183i	$0.285894\pi$
$863$			1.16509e6i			1.56437i	−0.623049	+	0.782183i	$0.714106\pi$
$864$		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	0.788716	−	0.614758i	$-0.210746\pi$
$877$	246319.	+	206686.i	0.320257	+	0.268727i	−0.468459	+	0.883485i	$0.655191\pi$
$878$		0			0
$879$	−71656.2	+	39934.7i	−0.0927418	+	0.0516860i
$880$		0			0
$881$	−427889.	−	247042.i	−0.551288	−	0.318286i	0.198353	−	0.980131i	$-0.436441\pi$
$881$	−427889.	−	247042.i	−0.551288	−	0.318286i	−0.749641	+	0.661844i	$0.769774\pi$
$882$		0			0
$883$	−181880.	−	315025.i	−0.233272	−	0.404039i	0.725497	−	0.688225i	$-0.241610\pi$
$883$	−181880.	−	315025.i	−0.233272	−	0.404039i	−0.958769	+	0.284186i	$0.908277\pi$
$884$		0			0
$885$	84181.3	−	438440.i	0.107480	−	0.559788i
$886$		0			0
$887$	−1.12707e6	−	198734.i	−1.43254	−	0.252595i	−0.597093	−	0.802172i	$-0.703678\pi$
$887$	−1.12707e6	−	198734.i	−1.43254	−	0.252595i	−0.835443	+	0.549578i	$0.814789\pi$
$888$		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	0.975831	−	0.218529i	$-0.0701257\pi$
$907$	1.30217e6	+	473949.i	1.58289	+	0.576126i	0.607062	+	0.794654i	$0.292348\pi$
$908$		0			0
$909$	−39643.8	+	1.30707e6i	−0.0479785	+	1.58187i
$910$		0			0
$911$	313406.	−	55262.0i	0.377634	−	0.0665871i	0.0183904	−	0.999831i	$-0.494146\pi$
$911$	313406.	−	55262.0i	0.377634	−	0.0665871i	0.359244	+	0.933244i	$0.383035\pi$
$912$		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	0.605077	+	0.796167i	$0.293142\pi$
$929$	−319502.	+	877826.i	−0.370205	+	1.01713i	−0.975282	+	0.220964i	$0.929080\pi$
$930$		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	0.832335	−	0.554272i	$-0.187003\pi$
$937$	−56054.9	−	97089.9i	−0.0638461	−	0.110585i	−0.896182	+	0.443688i	$0.853670\pi$
$938$		0			0
$939$	1.51063e6	−	524029.i	1.71328	−	0.594325i
$940$		0			0
$941$	−462012.	−	81465.1i	−0.521764	−	0.0920010i	−0.0934373	−	0.995625i	$-0.529785\pi$
$941$	−462012.	−	81465.1i	−0.521764	−	0.0920010i	−0.428326	+	0.903624i	$0.640897\pi$
$942$		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	−0.971838	−	0.235650i	$-0.924278\pi$
$947$	−928332.	−	1.10634e6i	−1.03515	−	1.23364i	−0.0633116	−	0.997994i	$-0.520166\pi$
$948$		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	0.606261	−	0.795266i	$-0.292669\pi$
$953$	1.45142e6	−	837978.i	1.59811	−	0.922670i	0.991851	−	0.127405i	$-0.0406647\pi$
$954$		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	0.426742	−	0.904373i	$-0.359661\pi$
$967$	1.24831e6	−	454348.i	1.33497	−	0.485888i	0.908223	+	0.418486i	$0.137439\pi$
$968$		0			0
$969$	68052.4	+	83644.8i	0.0724763	+	0.0890823i
$970$		0			0
$971$		−	1.47439e6i		−	1.56378i	−0.623417	−	0.781889i	$-0.714256\pi$
$971$		−	1.47439e6i		−	1.56378i	0.623417	−	0.781889i	$-0.285744\pi$
$972$		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	−0.903757	−	0.428046i	$-0.859202\pi$
$977$	−464456.	−	1.27608e6i	−0.486582	−	1.33687i	0.417175	−	0.908826i	$-0.363020\pi$
$978$		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	−0.0196048	+	0.999808i	$0.506241\pi$
$983$	−625431.	+	1.71836e6i	−0.647251	+	1.77831i	−0.627646	+	0.778499i	$0.715981\pi$
$984$		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	−0.999830	−	0.0184477i	$-0.994128\pi$
$991$	−506647.	−	877538.i	−0.515891	−	0.893550i	0.483939	−	0.875102i	$-0.339206\pi$
$992$		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	−0.912467	−	0.409151i	$-0.865825\pi$
$997$	−663978.	+	557144.i	−0.667980	+	0.560502i	0.244487	+	0.969653i	$0.421380\pi$
$998$		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.5.10	✓	72
3.2	odd	2		324.5.k.a.125.10		72
27.11	odd	18	inner	108.5.k.a.65.10	yes	72
27.16	even	9		324.5.k.a.197.10		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.10	✓	72	1.1	even	1	trivial
108.5.k.a.65.10	yes	72	27.11	odd	18	inner
324.5.k.a.125.10		72	3.2	odd	2
324.5.k.a.197.10		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)

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