LMFDB - Embedded newform 108.6.i.a.13.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [108,6,Mod(13,108)]

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, 8]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("108.13");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 108 = 2^{2} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	108.i (of order $9$, 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:	$17.3214525398$
Analytic rank:	$0$
Dimension:	$90$
Relative dimension:	$15$ over $\Q(\zeta_{9})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label			13.1
Character	$\chi$	$=$	108.13
Dual form			108.6.i.a.25.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-15.3241 + 2.85854i) q^{3} +(-63.0232 - 52.8828i) q^{5} +(-226.162 - 82.3161i) q^{7} +(226.658 - 87.6091i) q^{9} +O(q^{10})$	\(q+(-15.3241 + 2.85854i) q^{3} +(-63.0232 - 52.8828i) q^{5} +(-226.162 - 82.3161i) q^{7} +(226.658 - 87.6091i) q^{9} +(168.350 - 141.262i) q^{11} +(-58.2732 + 330.484i) q^{13} +(1116.94 + 630.228i) q^{15} +(-410.244 + 710.563i) q^{17} +(-855.680 - 1482.08i) q^{19} +(3701.03 + 614.931i) q^{21} +(-1786.98 + 650.409i) q^{23} +(632.690 + 3588.16i) q^{25} +(-3222.89 + 1990.44i) q^{27} +(1107.28 + 6279.68i) q^{29} +(9610.47 - 3497.93i) q^{31} +(-2176.01 + 2645.95i) q^{33} +(9900.34 + 17147.9i) q^{35} +(-1684.51 + 2917.66i) q^{37} +(-51.7140 - 5230.95i) q^{39} +(2481.63 - 14074.0i) q^{41} +(3340.31 - 2802.85i) q^{43} +(-18917.7 - 6464.87i) q^{45} +(-8981.66 - 3269.06i) q^{47} +(31498.2 + 26430.2i) q^{49} +(4255.45 - 12061.4i) q^{51} +3604.98 q^{53} -18080.3 q^{55} +(17349.1 + 20265.6i) q^{57} +(-11533.6 - 9677.86i) q^{59} +(-9907.01 - 3605.86i) q^{61} +(-58472.9 + 1156.26i) q^{63} +(21149.5 - 17746.5i) q^{65} +(7832.20 - 44418.6i) q^{67} +(25524.8 - 15075.1i) q^{69} +(-35278.7 + 61104.5i) q^{71} +(16056.7 + 27811.0i) q^{73} +(-19952.3 - 53176.9i) q^{75} +(-49702.4 + 18090.2i) q^{77} +(-9989.66 - 56654.2i) q^{79} +(43698.3 - 39714.5i) q^{81} +(11506.6 + 65257.3i) q^{83} +(63431.4 - 23087.1i) q^{85} +(-34918.7 - 93065.4i) q^{87} +(13254.0 + 22956.7i) q^{89} +(40383.3 - 69946.0i) q^{91} +(-137273. + 81074.5i) q^{93} +(-24448.9 + 138656. i) q^{95} +(119917. - 100623. i) q^{97} +(25781.9 - 46767.1i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) $ 90 * q - 87 * q^5 + 330 * q^9	$ 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) $ 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{4}{9}\right)$	$1$

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
$3$	−15.3241	+	2.85854i	−0.983043	+	0.183375i
$3$	−15.3241	+	2.85854i	−0.983043	+	0.183375i
$4$		0			0
$5$	−63.0232	−	52.8828i	−1.12739	−	0.945996i	−0.128440	−	0.991717i	$-0.540997\pi$
$5$	−63.0232	−	52.8828i	−1.12739	−	0.945996i	−0.998954	+	0.0457214i	$0.985441\pi$
$6$		0			0
$7$	−226.162	−	82.3161i	−1.74451	−	0.634950i	−0.745027	−	0.667035i	$-0.767563\pi$
$7$	−226.162	−	82.3161i	−1.74451	−	0.634950i	−0.999485	+	0.0320844i	$0.989785\pi$
$8$		0			0
$9$	226.658	−	87.6091i	0.932747	−	0.360531i
$10$		0			0
$11$	168.350	−	141.262i	0.419498	−	0.352001i	−0.408474	−	0.912770i	$-0.633939\pi$
$11$	168.350	−	141.262i	0.419498	−	0.352001i	0.827972	+	0.560769i	$0.189494\pi$
$12$		0			0
$13$	−58.2732	+	330.484i	−0.0956336	+	0.542365i	0.898918	+	0.438117i	$0.144355\pi$
$13$	−58.2732	+	330.484i	−0.0956336	+	0.542365i	−0.994551	+	0.104248i	$0.966756\pi$
$14$		0			0
$15$	1116.94	+	630.228i	1.28175	+	0.723219i
$16$		0			0
$17$	−410.244	+	710.563i	−0.344286	+	0.596321i	−0.985224	−	0.171272i	$-0.945212\pi$
$17$	−410.244	+	710.563i	−0.344286	+	0.596321i	0.640938	+	0.767593i	$0.278546\pi$
$18$		0			0
$19$	−855.680	−	1482.08i	−0.543785	−	0.941864i	−0.998682	−	0.0513199i	$-0.983657\pi$
$19$	−855.680	−	1482.08i	−0.543785	−	0.941864i	0.454897	−	0.890544i	$-0.349676\pi$
$20$		0			0
$21$	3701.03	+	614.931i	1.83136	+	0.304283i
$22$		0			0
$23$	−1786.98	+	650.409i	−0.704371	+	0.256370i	−0.669276	−	0.743014i	$-0.733396\pi$
$23$	−1786.98	+	650.409i	−0.704371	+	0.256370i	−0.0350948	+	0.999384i	$0.511173\pi$
$24$		0			0
$25$	632.690	+	3588.16i	0.202461	+	1.14821i
$26$		0			0
$27$	−3222.89	+	1990.44i	−0.850818	+	0.525461i
$28$		0			0
$29$	1107.28	+	6279.68i	0.244490	+	1.38657i	0.821674	+	0.569958i	$0.193040\pi$
$29$	1107.28	+	6279.68i	0.244490	+	1.38657i	−0.577184	+	0.816614i	$0.695848\pi$
$30$		0			0
$31$	9610.47	−	3497.93i	1.79614	−	0.653742i	0.797407	−	0.603441i	$-0.206204\pi$
$31$	9610.47	−	3497.93i	1.79614	−	0.653742i	0.998734	−	0.0503008i	$-0.0160180\pi$
$32$		0			0
$33$	−2176.01	+	2645.95i	−0.347837	+	0.422958i
$34$		0			0
$35$	9900.34	+	17147.9i	1.36609	+	2.36614i
$36$		0			0
$37$	−1684.51	+	2917.66i	−0.202288	+	0.350373i	−0.949265	−	0.314477i	$-0.898171\pi$
$37$	−1684.51	+	2917.66i	−0.202288	+	0.350373i	0.746977	+	0.664849i	$0.231504\pi$
$38$		0			0
$39$	−51.7140	−	5230.95i	−0.00544436	−	0.550705i
$40$		0			0
$41$	2481.63	−	14074.0i	0.230557	−	1.30755i	−0.621215	−	0.783640i	$-0.713361\pi$
$41$	2481.63	−	14074.0i	0.230557	−	1.30755i	0.851772	−	0.523913i	$-0.175528\pi$
$42$		0			0
$43$	3340.31	−	2802.85i	0.275496	−	0.231169i	−0.494562	−	0.869142i	$-0.664672\pi$
$43$	3340.31	−	2802.85i	0.275496	−	0.231169i	0.770058	+	0.637974i	$0.220227\pi$
$44$		0			0
$45$	−18917.7	−	6464.87i	−1.39263	−	0.475914i
$46$		0			0
$47$	−8981.66	−	3269.06i	−0.593078	−	0.215863i	0.0280046	−	0.999608i	$-0.491085\pi$
$47$	−8981.66	−	3269.06i	−0.593078	−	0.215863i	−0.621083	+	0.783745i	$0.713307\pi$
$48$		0			0
$49$	31498.2	+	26430.2i	1.87411	+	1.57257i
$50$		0			0
$51$	4255.45	−	12061.4i	0.229098	−	0.649343i
$52$		0			0
$53$	3604.98			0.176284			0.0881420	−	0.996108i	$-0.471907\pi$
$53$	3604.98			0.176284			0.0881420	+	0.996108i	$0.471907\pi$
$54$		0			0
$55$	−18080.3			−0.805931
$56$		0			0
$57$	17349.1	+	20265.6i	0.707279	+	0.826176i
$58$		0			0
$59$	−11533.6	−	9677.86i	−0.431356	−	0.361951i	0.401107	−	0.916031i	$-0.368626\pi$
$59$	−11533.6	−	9677.86i	−0.431356	−	0.361951i	−0.832463	+	0.554080i	$0.813070\pi$
$60$		0			0
$61$	−9907.01	−	3605.86i	−0.340893	−	0.124075i	0.165900	−	0.986143i	$-0.446947\pi$
$61$	−9907.01	−	3605.86i	−0.340893	−	0.124075i	−0.506793	+	0.862068i	$0.669169\pi$
$62$		0			0
$63$	−58472.9	+	1156.26i	−1.85611	+	0.0367031i
$64$		0			0
$65$	21149.5	−	17746.5i	0.620892	−	0.520990i
$66$		0			0
$67$	7832.20	−	44418.6i	0.213156	−	1.20887i	−0.670923	−	0.741527i	$-0.734102\pi$
$67$	7832.20	−	44418.6i	0.213156	−	1.20887i	0.884079	−	0.467338i	$-0.154787\pi$
$68$		0			0
$69$	25524.8	−	15075.1i	0.645415	−	0.381187i
$70$		0			0
$71$	−35278.7	+	61104.5i	−0.830551	+	1.43856i	0.0670504	+	0.997750i	$0.478641\pi$
$71$	−35278.7	+	61104.5i	−0.830551	+	1.43856i	−0.897602	+	0.440807i	$0.854692\pi$
$72$		0			0
$73$	16056.7	+	27811.0i	0.352654	+	0.610815i	0.986714	−	0.162470i	$-0.0519459\pi$
$73$	16056.7	+	27811.0i	0.352654	+	0.610815i	−0.634060	+	0.773284i	$0.718613\pi$
$74$		0			0
$75$	−19952.3	−	53176.9i	−0.409581	−	1.09162i
$76$		0			0
$77$	−49702.4	+	18090.2i	−0.955323	+	0.347709i
$78$		0			0
$79$	−9989.66	−	56654.2i	−0.180087	−	1.02133i	−0.932106	−	0.362185i	$-0.882031\pi$
$79$	−9989.66	−	56654.2i	−0.180087	−	1.02133i	0.752019	−	0.659141i	$-0.229080\pi$
$80$		0			0
$81$	43698.3	−	39714.5i	0.740034	−	0.672569i
$82$		0			0
$83$	11506.6	+	65257.3i	0.183338	+	1.03976i	0.928072	+	0.372400i	$0.121465\pi$
$83$	11506.6	+	65257.3i	0.183338	+	1.03976i	−0.744734	+	0.667361i	$0.767424\pi$
$84$		0			0
$85$	63431.4	−	23087.1i	0.952263	−	0.346595i
$86$		0			0
$87$	−34918.7	−	93065.4i	−0.494607	−	1.31823i
$88$		0			0
$89$	13254.0	+	22956.7i	0.177367	+	0.307209i	0.940978	−	0.338468i	$-0.109909\pi$
$89$	13254.0	+	22956.7i	0.177367	+	0.307209i	−0.763611	+	0.645677i	$0.776575\pi$
$90$		0			0
$91$	40383.3	−	69946.0i	0.511209	−	0.885440i
$92$		0			0
$93$	−137273.	+	81074.5i	−1.64580	+	0.972024i
$94$		0			0
$95$	−24448.9	+	138656.i	−0.277939	+	1.57627i
$96$		0			0
$97$	119917.	−	100623.i	1.29406	−	1.08584i	0.302917	−	0.953017i	$-0.402040\pi$
$97$	119917.	−	100623.i	1.29406	−	1.08584i	0.991140	−	0.132825i	$-0.0424048\pi$
$98$		0			0
$99$	25781.9	−	46767.1i	0.264379	−	0.479570i
$100$		0			0
$101$	−77361.1	−	28157.2i	−0.754605	−	0.274654i	−0.0640625	−	0.997946i	$-0.520406\pi$
$101$	−77361.1	−	28157.2i	−0.754605	−	0.274654i	−0.690542	+	0.723292i	$0.742628\pi$
$102$		0			0
$103$	−78208.2	−	65624.5i	−0.726372	−	0.609499i	0.202768	−	0.979227i	$-0.435006\pi$
$103$	−78208.2	−	65624.5i	−0.726372	−	0.609499i	−0.929140	+	0.369728i	$0.879451\pi$
$104$		0			0
$105$	−200732.	−	234476.i	−1.77682	−	2.07551i
$106$		0			0
$107$	−190615.			−1.60953			−0.804764	−	0.593595i	$-0.797708\pi$
$107$	−190615.			−1.60953			−0.804764	+	0.593595i	$0.797708\pi$
$108$		0			0
$109$	−14119.9			−0.113832			−0.0569161	−	0.998379i	$-0.518127\pi$
$109$	−14119.9			−0.113832			−0.0569161	+	0.998379i	$0.518127\pi$
$110$		0			0
$111$	17473.4	−	49525.8i	0.134608	−	0.381526i
$112$		0			0
$113$	110748.	+	92928.5i	0.815905	+	0.684625i	0.952009	−	0.306069i	$-0.0990139\pi$
$113$	110748.	+	92928.5i	0.815905	+	0.684625i	−0.136105	+	0.990694i	$0.543458\pi$
$114$		0			0
$115$	147017.	+	53509.8i	1.03663	+	0.377302i
$116$		0			0
$117$	15745.3	+	80011.9i	0.106338	+	0.540369i
$118$		0			0
$119$	151272.	−	126932.i	0.979245	−	0.821684i
$120$		0			0
$121$	−19579.6	+	111041.i	−0.121574	+	0.689480i
$122$		0			0
$123$	2202.30	+	222766.i	0.0131254	+	1.32766i
$124$		0			0
$125$	21329.5	−	36943.7i	0.122097	−	0.211478i
$126$		0			0
$127$	159320.	+	275950.i	0.876517	+	1.51817i	0.855138	+	0.518400i	$0.173472\pi$
$127$	159320.	+	275950.i	0.876517	+	1.51817i	0.0213780	+	0.999771i	$0.493195\pi$
$128$		0			0
$129$	−43175.3	+	52499.6i	−0.228434	+	0.277768i
$130$		0			0
$131$	221278.	−	80538.7i	1.12658	−	0.410040i	0.289528	−	0.957169i	$-0.406502\pi$
$131$	221278.	−	80538.7i	1.12658	−	0.410040i	0.837048	+	0.547129i	$0.184279\pi$
$132$		0			0
$133$	71522.9	+	405626.i	0.350603	+	1.98837i
$134$		0			0
$135$	308377.	+	44991.5i	1.45629	+	0.212469i
$136$		0			0
$137$	70289.8	+	398633.i	0.319957	+	1.81456i	0.542974	+	0.839749i	$0.317298\pi$
$137$	70289.8	+	398633.i	0.319957	+	1.81456i	−0.223018	+	0.974814i	$0.571591\pi$
$138$		0			0
$139$	52815.3	−	19223.2i	0.231859	−	0.0843896i	−0.223478	−	0.974709i	$-0.571741\pi$
$139$	52815.3	−	19223.2i	0.231859	−	0.0843896i	0.455337	+	0.890319i	$0.349519\pi$
$140$		0			0
$141$	146981.	+	24421.0i	0.622605	+	0.103447i
$142$		0			0
$143$	36874.6	+	63868.6i	0.150795	+	0.261184i
$144$		0			0
$145$	262303.	−	454322.i	1.03605	−	1.79450i
$146$		0			0
$147$	−558235.	−	314980.i	−2.13071	−	1.20224i
$148$		0			0
$149$	−19461.6	+	110372.i	−0.0718147	+	0.407282i	0.927616	+	0.373536i	$0.121855\pi$
$149$	−19461.6	+	110372.i	−0.0718147	+	0.407282i	−0.999430	+	0.0337457i	$0.989256\pi$
$150$		0			0
$151$	−176577.	+	148166.i	−0.630220	+	0.528818i	−0.900997	−	0.433824i	$-0.857164\pi$
$151$	−176577.	+	148166.i	−0.630220	+	0.528818i	0.270777	+	0.962642i	$0.412719\pi$
$152$		0			0
$153$	−30733.0	+	196995.i	−0.106139	+	0.680343i
$154$		0			0
$155$	−790663.	−	287778.i	−2.64340	−	0.962118i
$156$		0			0
$157$	100041.	+	83944.5i	0.323914	+	0.271796i	0.790215	−	0.612830i	$-0.209969\pi$
$157$	100041.	+	83944.5i	0.323914	+	0.271796i	−0.466301	+	0.884626i	$0.654414\pi$
$158$		0			0
$159$	−55243.1	+	10305.0i	−0.173295	+	0.0323261i
$160$		0			0
$161$	457687.			1.39156
$162$		0			0
$163$	−423823.			−1.24944			−0.624721	−	0.780848i	$-0.714787\pi$
$163$	−423823.			−1.24944			−0.624721	+	0.780848i	$0.714787\pi$
$164$		0			0
$165$	277064.	−	51683.1i	0.792265	−	0.147788i
$166$		0			0
$167$	315212.	+	264494.i	0.874604	+	0.733880i	0.965062	−	0.262020i	$-0.0843887\pi$
$167$	315212.	+	264494.i	0.874604	+	0.733880i	−0.0904583	+	0.995900i	$0.528833\pi$
$168$		0			0
$169$	243077.	+	88473.0i	0.654678	+	0.238283i
$170$		0			0
$171$	−323790.	−	260960.i	−0.846786	−	0.682469i
$172$		0			0
$173$	−212227.	+	178080.i	−0.539121	+	0.452376i	−0.871237	−	0.490862i	$-0.836682\pi$
$173$	−212227.	+	178080.i	−0.539121	+	0.452376i	0.332116	+	0.943239i	$0.392237\pi$
$174$		0			0
$175$	152273.	−	863586.i	0.375863	−	2.13162i
$176$		0			0
$177$	204407.	+	115335.i	0.490414	+	0.276713i
$178$		0			0
$179$	−368973.	+	639080.i	−0.860720	+	1.49081i	0.0105146	+	0.999945i	$0.496653\pi$
$179$	−368973.	+	639080.i	−0.860720	+	1.49081i	−0.871235	+	0.490866i	$0.836680\pi$
$180$		0			0
$181$	−303627.	−	525898.i	−0.688881	−	1.19318i	−0.972200	−	0.234152i	$-0.924769\pi$
$181$	−303627.	−	525898.i	−0.688881	−	1.19318i	0.283319	−	0.959026i	$-0.408565\pi$
$182$		0			0
$183$	162124.	+	26937.0i	0.357865	+	0.0594596i
$184$		0			0
$185$	260457.	−	94798.7i	0.559509	−	0.203645i
$186$		0			0
$187$	31311.2	+	177575.i	0.0654781	+	0.371345i
$188$		0			0
$189$	892741.	−	184865.i	1.81790	−	0.376445i
$190$		0			0
$191$	−52904.2	−	300034.i	−0.104932	−	0.595097i	−0.991248	−	0.132016i	$-0.957855\pi$
$191$	−52904.2	−	300034.i	−0.104932	−	0.595097i	0.886316	−	0.463081i	$-0.153256\pi$
$192$		0			0
$193$	370303.	−	134779.i	0.715590	−	0.260453i	0.0415376	−	0.999137i	$-0.486774\pi$
$193$	370303.	−	134779.i	0.715590	−	0.260453i	0.674052	+	0.738683i	$0.264552\pi$
$194$		0			0
$195$	−273368.	+	332406.i	−0.514827	+	0.626012i
$196$		0			0
$197$	−66774.1	−	115656.i	−0.122586	−	0.212326i	0.798200	−	0.602392i	$-0.205786\pi$
$197$	−66774.1	−	115656.i	−0.122586	−	0.212326i	−0.920787	+	0.390066i	$0.872452\pi$
$198$		0			0
$199$	−17470.7	+	30260.1i	−0.0312735	+	0.0541673i	−0.881239	−	0.472672i	$-0.843290\pi$
$199$	−17470.7	+	30260.1i	−0.0312735	+	0.0541673i	0.849965	+	0.526839i	$0.176623\pi$
$200$		0			0
$201$	6950.61	+	703065.i	0.0121348	+	1.22745i
$202$		0			0
$203$	266495.	−	1.51137e6i	0.453889	−	2.57413i
$204$		0			0
$205$	−900675.	+	755756.i	−1.49687	+	1.25602i
$206$		0			0
$207$	−348052.	+	303976.i	−0.564570	+	0.493076i
$208$		0			0
$209$	−353415.	−	128633.i	−0.559654	−	0.203697i
$210$		0			0
$211$	−13712.1	−	11505.8i	−0.0212031	−	0.0177915i	0.632125	−	0.774867i	$-0.282183\pi$
$211$	−13712.1	−	11505.8i	−0.0212031	−	0.0177915i	−0.653328	+	0.757075i	$0.726628\pi$
$212$		0			0
$213$	365946.	−	1.03722e6i	0.552672	−	1.56647i
$214$		0			0
$215$	−358740.			−0.529277
$216$		0			0
$217$	−2.46146e6			−3.54848
$218$		0			0
$219$	−325553.	−	380281.i	−0.458682	−	0.535789i
$220$		0			0
$221$	−210923.	−	176986.i	−0.290498	−	0.243757i
$222$		0			0
$223$	−15050.3	−	5477.85i	−0.0202667	−	0.00737646i	0.331867	−	0.943326i	$-0.392322\pi$
$223$	−15050.3	−	5477.85i	−0.0202667	−	0.00737646i	−0.352134	+	0.935950i	$0.614544\pi$
$224$		0			0
$225$	457760.	+	757855.i	0.602811	+	0.997998i
$226$		0			0
$227$	727213.	−	610204.i	0.936693	−	0.785979i	−0.0403139	−	0.999187i	$-0.512836\pi$
$227$	727213.	−	610204.i	0.936693	−	0.785979i	0.977007	+	0.213208i	$0.0683913\pi$
$228$		0			0
$229$	−122367.	+	693978.i	−0.154197	+	0.874495i	0.805319	+	0.592841i	$0.201994\pi$
$229$	−122367.	+	693978.i	−0.154197	+	0.874495i	−0.959516	+	0.281653i	$0.909117\pi$
$230$		0			0
$231$	709934.	−	419292.i	0.875362	−	0.516996i
$232$		0			0
$233$	370147.	−	641113.i	0.446667	−	0.773650i	−0.551499	−	0.834175i	$-0.685944\pi$
$233$	370147.	−	641113.i	0.446667	−	0.773650i	0.998167	+	0.0605248i	$0.0192774\pi$
$234$		0			0
$235$	393176.	+	681002.i	0.464427	+	0.804412i
$236$		0			0
$237$	315031.	+	839620.i	0.364319	+	0.970984i
$238$		0			0
$239$	229796.	−	83638.9i	0.260224	−	0.0947139i	−0.208614	−	0.977998i	$-0.566895\pi$
$239$	229796.	−	83638.9i	0.260224	−	0.0947139i	0.468838	+	0.883284i	$0.344673\pi$
$240$		0			0
$241$	−7512.60	−	42606.1i	−0.00833197	−	0.0472529i	0.980358	−	0.197225i	$-0.0631928\pi$
$241$	−7512.60	−	42606.1i	−0.00833197	−	0.0472529i	−0.988690	+	0.149972i	$0.952082\pi$
$242$		0			0
$243$	−556112.	+	733504.i	−0.604153	+	0.796868i
$244$		0			0
$245$	−587421.	−	3.33143e6i	−0.625222	−	3.54581i
$246$		0			0
$247$	539667.	−	196423.i	0.562839	−	0.204856i
$248$		0			0
$249$	−362869.	−	967118.i	−0.370896	−	0.988511i
$250$		0			0
$251$	532299.	+	921968.i	0.533299	+	0.923701i	0.999244	+	0.0388874i	$0.0123814\pi$
$251$	532299.	+	921968.i	0.533299	+	0.923701i	−0.465944	+	0.884814i	$0.654285\pi$
$252$		0			0
$253$	−208960.	+	361929.i	−0.205240	+	0.355486i
$254$		0			0
$255$	−906035.	+	535111.i	−0.872559	+	0.515340i
$256$		0			0
$257$	−1582.07	+	8972.39i	−0.00149415	+	0.00847375i	−0.985546	−	0.169410i	$-0.945814\pi$
$257$	−1582.07	+	8972.39i	−0.00149415	+	0.00847375i	0.984052	+	0.177883i	$0.0569250\pi$
$258$		0			0
$259$	621142.	−	521200.i	0.575363	−	0.482787i
$260$		0			0
$261$	801130.	+	1.32633e6i	0.727950	+	1.20517i
$262$		0			0
$263$	2.08671e6	+	759502.i	1.86026	+	0.677079i	0.978799	+	0.204821i	$0.0656614\pi$
$263$	2.08671e6	+	759502.i	1.86026	+	0.677079i	0.881460	+	0.472258i	$0.156561\pi$
$264$		0			0
$265$	−227197.	−	190641.i	−0.198742	−	0.166764i
$266$		0			0
$267$	−268729.	−	313903.i	−0.230694	−	0.269475i
$268$		0			0
$269$	−1.91645e6			−1.61480			−0.807398	−	0.590008i	$-0.799125\pi$
$269$	−1.91645e6			−1.61480			−0.807398	+	0.590008i	$0.799125\pi$
$270$		0			0
$271$	−741161.			−0.613041			−0.306521	−	0.951864i	$-0.599165\pi$
$271$	−741161.			−0.613041			−0.306521	+	0.951864i	$0.599165\pi$
$272$		0			0
$273$	−418896.	+	1.18730e6i	−0.340173	+	0.964169i
$274$		0			0
$275$	613384.	+	514691.i	0.489104	+	0.410407i
$276$		0			0
$277$	−394725.	−	143668.i	−0.309097	−	0.112502i	0.182814	−	0.983147i	$-0.441479\pi$
$277$	−394725.	−	143668.i	−0.309097	−	0.112502i	−0.491911	+	0.870645i	$0.663702\pi$
$278$		0			0
$279$	1.87184e6	−	1.63480e6i	1.43965	−	1.25734i
$280$		0			0
$281$	331494.	−	278157.i	0.250444	−	0.210147i	−0.508920	−	0.860814i	$-0.669955\pi$
$281$	331494.	−	278157.i	0.250444	−	0.210147i	0.759363	+	0.650667i	$0.225511\pi$
$282$		0			0
$283$	331108.	−	1.87781e6i	0.245756	−	1.39375i	−0.572976	−	0.819572i	$-0.694211\pi$
$283$	331108.	−	1.87781e6i	0.245756	−	1.39375i	0.818731	−	0.574177i	$-0.194678\pi$
$284$		0			0
$285$	−21696.9	−	2.19468e6i	−0.0158229	−	1.60051i
$286$		0			0
$287$	−1.71977e6	+	2.97873e6i	−1.23244	+	2.13465i
$288$		0			0
$289$	373329.	+	646625.i	0.262934	+	0.455415i
$290$		0			0
$291$	−1.55000e6	+	1.88474e6i	−1.07300	+	1.30473i
$292$		0			0
$293$	1.51213e6	−	550371.i	1.02901	−	0.374530i	0.228307	−	0.973589i	$-0.426681\pi$
$293$	1.51213e6	−	550371.i	1.02901	−	0.374530i	0.800704	+	0.599060i	$0.204459\pi$
$294$		0			0
$295$	215094.	+	1.21986e6i	0.143904	+	0.816122i
$296$		0			0
$297$	−261399.	+	790363.i	−0.171954	+	0.519919i
$298$		0			0
$299$	−110816.	−	628471.i	−0.0716846	−	0.406544i
$300$		0			0
$301$	−986170.	+	358936.i	−0.627387	+	0.228350i
$302$		0			0
$303$	1.26598e6	+	210344.i	0.792173	+	0.131621i
$304$		0			0
$305$	433684.	+	751163.i	0.266946	+	0.462365i
$306$		0			0
$307$	−1.00986e6	+	1.74913e6i	−0.611526	+	1.05919i	0.379458	+	0.925209i	$0.376111\pi$
$307$	−1.00986e6	+	1.74913e6i	−0.611526	+	1.05919i	−0.990983	+	0.133984i	$0.957223\pi$
$308$		0			0
$309$	1.38606e6	+	782077.i	0.825822	+	0.465965i
$310$		0			0
$311$	−105533.	+	598510.i	−0.0618712	+	0.350889i	0.938118	+	0.346315i	$0.112567\pi$
$311$	−105533.	+	598510.i	−0.0618712	+	0.350889i	−0.999990	+	0.00457428i	$0.998544\pi$
$312$		0			0
$313$	366716.	−	307711.i	0.211577	−	0.177534i	−0.530840	−	0.847472i	$-0.678124\pi$
$313$	366716.	−	307711.i	0.211577	−	0.177534i	0.742417	+	0.669938i	$0.233679\pi$
$314$		0			0
$315$	3.74630e6	+	3.01934e6i	2.12729	+	1.71449i
$316$		0			0
$317$	−1.78953e6	−	651335.i	−1.00021	−	0.364046i	−0.210544	−	0.977584i	$-0.567523\pi$
$317$	−1.78953e6	−	651335.i	−1.00021	−	0.364046i	−0.789665	+	0.613538i	$0.789746\pi$
$318$		0			0
$319$	1.07349e6	+	900765.i	0.590638	+	0.495604i
$320$		0			0
$321$	2.92101e6	−	544881.i	1.58223	−	0.295147i
$322$		0			0
$323$	1.40415e6			0.748871
$324$		0			0
$325$	−1.22270e6			−0.642113
$326$		0			0
$327$	216375.	−	40362.2i	0.111902	−	0.0208740i
$328$		0			0
$329$	1.76221e6	+	1.47867e6i	0.897570	+	0.753150i
$330$		0			0
$331$	924442.	+	336470.i	0.463778	+	0.168801i	0.563332	−	0.826231i	$-0.309519\pi$
$331$	924442.	+	336470.i	0.463778	+	0.168801i	−0.0995539	+	0.995032i	$0.531742\pi$
$332$		0			0
$333$	−126194.	+	808888.i	−0.0623630	+	0.399740i
$334$		0			0
$335$	−2.84259e6	+	2.38522e6i	−1.38389	+	1.16122i
$336$		0			0
$337$	92135.3	−	522525.i	0.0441928	−	0.250630i	−0.954706	−	0.297551i	$-0.903830\pi$
$337$	92135.3	−	522525.i	0.0441928	−	0.250630i	0.998899	+	0.0469218i	$0.0149412\pi$
$338$		0			0
$339$	−1.96275e6	−	1.10747e6i	−0.927613	−	0.523399i
$340$		0			0
$341$	1.12379e6	−	1.94647e6i	0.523361	−	0.906487i
$342$		0			0
$343$	−2.92554e6	−	5.06719e6i	−1.34268	−	2.32558i
$344$		0			0
$345$	−2.40587e6	−	399738.i	−1.08824	−	0.180812i
$346$		0			0
$347$	663333.	−	241433.i	0.295738	−	0.107640i	−0.189890	−	0.981805i	$-0.560813\pi$
$347$	663333.	−	241433.i	0.295738	−	0.107640i	0.485628	+	0.874165i	$0.338591\pi$
$348$		0			0
$349$	637662.	+	3.61636e6i	0.280238	+	1.58931i	0.721816	+	0.692086i	$0.243308\pi$
$349$	637662.	+	3.61636e6i	0.280238	+	1.58931i	−0.441578	+	0.897223i	$0.645581\pi$
$350$		0			0
$351$	−470000.	−	1.18110e6i	−0.203625	−	0.511706i
$352$		0			0
$353$	−511663.	−	2.90179e6i	−0.218548	−	1.23945i	−0.874642	−	0.484770i	$-0.838903\pi$
$353$	−511663.	−	2.90179e6i	−0.218548	−	1.23945i	0.656094	−	0.754680i	$-0.272208\pi$
$354$		0			0
$355$	5.45475e6	−	1.98537e6i	2.29723	−	0.836122i
$356$		0			0
$357$	−1.95527e6	+	2.37754e6i	−0.811964	+	0.987320i
$358$		0			0
$359$	−393491.	−	681547.i	−0.161138	−	0.279100i	0.774139	−	0.633016i	$-0.218183\pi$
$359$	−393491.	−	681547.i	−0.161138	−	0.279100i	−0.935277	+	0.353916i	$0.884850\pi$
$360$		0			0
$361$	−226328.	+	392012.i	−0.0914052	+	0.158319i
$362$		0			0
$363$	−17375.7	−	1.75758e6i	−0.00692112	−	0.700082i
$364$		0			0
$365$	458778.	−	2.60186e6i	0.180248	−	1.02224i
$366$		0			0
$367$	−3.55611e6	+	2.98393e6i	−1.37819	+	1.15644i	−0.408319	+	0.912839i	$0.633885\pi$
$367$	−3.55611e6	+	2.98393e6i	−1.37819	+	1.15644i	−0.969875	+	0.243603i	$0.921671\pi$
$368$		0			0
$369$	−670534.	−	3.40740e6i	−0.256363	−	1.30274i
$370$		0			0
$371$	−815308.	−	296748.i	−0.307530	−	0.111932i
$372$		0			0
$373$	−2.39819e6	−	2.01232e6i	−0.892507	−	0.748903i	0.0762041	−	0.997092i	$-0.475720\pi$
$373$	−2.39819e6	−	2.01232e6i	−0.892507	−	0.748903i	−0.968711	+	0.248190i	$0.920164\pi$
$374$		0			0
$375$	−221250.	+	627101.i	−0.0812467	+	0.230282i
$376$		0			0
$377$	−2.13986e6			−0.775410
$378$		0			0
$379$	4.75541e6			1.70055			0.850276	−	0.526337i	$-0.176435\pi$
$379$	4.75541e6			1.70055			0.850276	+	0.526337i	$0.176435\pi$
$380$		0			0
$381$	−3.23025e6	−	3.77327e6i	−1.14005	−	1.33170i
$382$		0			0
$383$	2.71957e6	+	2.28199e6i	0.947333	+	0.794907i	0.978846	−	0.204597i	$-0.0655883\pi$
$383$	2.71957e6	+	2.28199e6i	0.947333	+	0.794907i	−0.0315136	+	0.999503i	$0.510033\pi$
$384$		0			0
$385$	4.08906e6	+	1.48830e6i	1.40596	+	0.511726i
$386$		0			0
$387$	511551.	−	927929.i	0.173625	−	0.314947i
$388$		0			0
$389$	−2.49158e6	+	2.09068e6i	−0.834835	+	0.700510i	−0.956396	−	0.292074i	$-0.905655\pi$
$389$	−2.49158e6	+	2.09068e6i	−0.834835	+	0.700510i	0.121560	+	0.992584i	$0.461210\pi$
$390$		0			0
$391$	270942.	−	1.53659e6i	0.0896262	−	0.508296i
$392$		0			0
$393$	−3.16067e6	+	1.86672e6i	−1.03228	+	0.609673i
$394$		0			0
$395$	−2.36645e6	+	4.09881e6i	−0.763141	+	1.32180i
$396$		0			0
$397$	−1.86316e6	−	3.22709e6i	−0.593299	−	1.02762i	−0.993784	−	0.111321i	$-0.964492\pi$
$397$	−1.86316e6	−	3.22709e6i	−0.593299	−	1.02762i	0.400485	−	0.916303i	$-0.368842\pi$
$398$		0			0
$399$	−2.25552e6	−	6.01142e6i	−0.709276	−	1.89036i
$400$		0			0
$401$	733607.	−	267011.i	0.227826	−	0.0829217i	−0.225585	−	0.974223i	$-0.572429\pi$
$401$	733607.	−	267011.i	0.227826	−	0.0829217i	0.453411	+	0.891302i	$0.350207\pi$
$402$		0			0
$403$	595975.	+	3.37994e6i	0.182795	+	1.03668i
$404$		0			0
$405$	−4.85422e6	+	192052.i	−1.47056	+	0.0581811i
$406$		0			0
$407$	128568.	+	729144.i	0.0384721	+	0.218186i
$408$		0			0
$409$	1.87953e6	−	684093.i	0.555573	−	0.202212i	−0.0489479	−	0.998801i	$-0.515587\pi$
$409$	1.87953e6	−	684093.i	0.555573	−	0.202212i	0.604521	+	0.796589i	$0.293365\pi$
$410$		0			0
$411$	−2.21664e6	−	5.90778e6i	−0.647277	−	1.72512i
$412$		0			0
$413$	1.81182e6	+	3.13816e6i	0.522685	+	0.905317i
$414$		0			0
$415$	2.72580e6	−	4.72123e6i	0.776916	−	1.34566i
$416$		0			0
$417$	−754399.	+	445553.i	−0.212452	+	0.125476i
$418$		0			0
$419$	487374.	−	2.76403e6i	0.135621	−	0.769145i	−0.838804	−	0.544433i	$-0.816745\pi$
$419$	487374.	−	2.76403e6i	0.135621	−	0.769145i	0.974425	−	0.224712i	$-0.0721441\pi$
$420$		0			0
$421$	1.99487e6	−	1.67389e6i	0.548540	−	0.460280i	−0.325906	−	0.945402i	$-0.605669\pi$
$421$	1.99487e6	−	1.67389e6i	0.548540	−	0.460280i	0.874446	+	0.485122i	$0.161225\pi$
$422$		0			0
$423$	−2.32216e6	+	45918.9i	−0.631017	+	0.0124779i
$424$		0			0
$425$	−2.80917e6	−	1.02246e6i	−0.754408	−	0.274582i
$426$		0			0
$427$	1.94377e6	+	1.63101e6i	0.515910	+	0.432900i
$428$		0			0
$429$	−747641.	−	873323.i	−0.196133	−	0.229104i
$430$		0			0
$431$	5.53924e6			1.43634			0.718170	−	0.695868i	$-0.244980\pi$
$431$	5.53924e6			1.43634			0.718170	+	0.695868i	$0.244980\pi$
$432$		0			0
$433$	393258.			0.100799			0.0503997	−	0.998729i	$-0.483950\pi$
$433$	393258.			0.100799			0.0503997	+	0.998729i	$0.483950\pi$
$434$		0			0
$435$	−2.72086e6	+	7.71188e6i	−0.689420	+	1.95406i
$436$		0			0
$437$	2.49305e6	+	2.09192e6i	0.624492	+	0.524011i
$438$		0			0
$439$	1.15339e6	+	419801.i	0.285638	+	0.103964i	0.480866	−	0.876794i	$-0.340322\pi$
$439$	1.15339e6	+	419801.i	0.285638	+	0.103964i	−0.195228	+	0.980758i	$0.562545\pi$
$440$		0			0
$441$	9.45484e6	+	3.23106e6i	2.31504	+	0.791132i
$442$		0			0
$443$	−3.89660e6	+	3.26963e6i	−0.943358	+	0.791571i	−0.978166	−	0.207823i	$-0.933362\pi$
$443$	−3.89660e6	+	3.26963e6i	−0.943358	+	0.791571i	0.0348089	+	0.999394i	$0.488918\pi$
$444$		0			0
$445$	378700.	−	2.14771e6i	0.0906557	−	0.514134i
$446$		0			0
$447$	−17271.0	−	1.74699e6i	−0.00408836	−	0.413544i
$448$		0			0
$449$	−1.40447e6	+	2.43262e6i	−0.328774	+	0.569454i	−0.982269	−	0.187477i	$-0.939969\pi$
$449$	−1.40447e6	+	2.43262e6i	−0.328774	+	0.569454i	0.653495	+	0.756931i	$0.273302\pi$
$450$		0			0
$451$	−1.57035e6	−	2.71992e6i	−0.363542	−	0.629673i
$452$		0			0
$453$	2.28235e6	−	2.77527e6i	0.522562	−	0.635417i
$454$		0			0
$455$	−6.24402e6	+	2.27264e6i	−1.41396	+	0.514638i
$456$		0			0
$457$	−121385.	−	688409.i	−0.0271878	−	0.154190i	0.968192	−	0.250210i	$-0.0804997\pi$
$457$	−121385.	−	688409.i	−0.0271878	−	0.154190i	−0.995379	+	0.0960203i	$0.969389\pi$
$458$		0			0
$459$	−92162.0	−	3.10663e6i	−0.0204183	−	0.688269i
$460$		0			0
$461$	369940.	+	2.09804e6i	0.0810736	+	0.459791i	0.998135	+	0.0610451i	$0.0194433\pi$
$461$	369940.	+	2.09804e6i	0.0810736	+	0.459791i	−0.917061	+	0.398746i	$0.869446\pi$
$462$		0			0
$463$	−2.94568e6	+	1.07214e6i	−0.638607	+	0.232434i	−0.640973	−	0.767563i	$-0.721469\pi$
$463$	−2.94568e6	+	1.07214e6i	−0.638607	+	0.232434i	0.00236619	+	0.999997i	$0.499247\pi$
$464$		0			0
$465$	1.29388e7	+	2.14980e6i	2.77500	+	0.461070i
$466$		0			0
$467$	−200336.	−	346993.i	−0.0425077	−	0.0736255i	0.843989	−	0.536361i	$-0.180201\pi$
$467$	−200336.	−	346993.i	−0.0425077	−	0.0736255i	−0.886496	+	0.462735i	$0.846868\pi$
$468$		0			0
$469$	−5.42771e6	+	9.40107e6i	−1.13942	+	1.97354i
$470$		0			0
$471$	−1.77300e6	−	1.00040e6i	−0.368262	−	0.207789i
$472$		0			0
$473$	166403.	−	943718.i	0.0341986	−	0.193950i
$474$		0			0
$475$	4.77657e6	−	4.00802e6i	0.971365	−	0.815072i
$476$		0			0
$477$	817096.	−	315829.i	0.164428	−	0.0635559i
$478$		0			0
$479$	−1.11926e6	−	407379.i	−0.222892	−	0.0811259i	0.228161	−	0.973624i	$-0.426729\pi$
$479$	−1.11926e6	−	407379.i	−0.222892	−	0.0811259i	−0.451052	+	0.892498i	$0.648951\pi$
$480$		0			0
$481$	−866077.	−	726725.i	−0.170684	−	0.143221i
$482$		0			0
$483$	−7.01365e6	+	1.30831e6i	−1.36797	+	0.255178i
$484$		0			0
$485$	−1.28788e7			−2.48611
$486$		0			0
$487$	2.71938e6			0.519574			0.259787	−	0.965666i	$-0.416348\pi$
$487$	2.71938e6			0.519574			0.259787	+	0.965666i	$0.416348\pi$
$488$		0			0
$489$	6.49472e6	−	1.21151e6i	1.22825	−	0.229116i
$490$		0			0
$491$	4.01156e6	+	3.36610e6i	0.750947	+	0.630120i	0.935753	−	0.352655i	$-0.114721\pi$
$491$	4.01156e6	+	3.36610e6i	0.750947	+	0.630120i	−0.184806	+	0.982775i	$0.559166\pi$
$492$		0			0
$493$	−4.91636e6	−	1.78941e6i	−0.911016	−	0.331583i
$494$		0			0
$495$	−4.09803e6	+	1.58400e6i	−0.751730	+	0.290564i
$496$		0			0
$497$	1.30086e7	−	1.09155e7i	2.36232	−	1.98222i
$498$		0			0
$499$	810071.	−	4.59414e6i	0.145637	−	0.825948i	−0.821217	−	0.570617i	$-0.806704\pi$
$499$	810071.	−	4.59414e6i	0.145637	−	0.825948i	0.966854	−	0.255332i	$-0.0821847\pi$
$500$		0			0
$501$	−5.58641e6	−	3.15210e6i	−0.994349	−	0.561055i
$502$		0			0
$503$	911890.	−	1.57944e6i	0.160702	−	0.278345i	−0.774418	−	0.632674i	$-0.781957\pi$
$503$	911890.	−	1.57944e6i	0.160702	−	0.278345i	0.935121	+	0.354329i	$0.115291\pi$
$504$		0			0
$505$	3.38652e6	+	5.86563e6i	0.590915	+	1.02350i
$506$		0			0
$507$	−3.97785e6	−	660925.i	−0.687272	−	0.114191i
$508$		0			0
$509$	−5.53701e6	+	2.01531e6i	−0.947286	+	0.344784i	−0.769039	−	0.639202i	$-0.779265\pi$
$509$	−5.53701e6	+	2.01531e6i	−0.947286	+	0.344784i	−0.178247	+	0.983986i	$0.557043\pi$
$510$		0			0
$511$	−1.34211e6	−	7.61151e6i	−0.227372	−	1.28949i
$512$		0			0
$513$	5.70777e6	+	3.07341e6i	0.957575	+	0.515617i
$514$		0			0
$515$	1.45853e6	+	8.27173e6i	0.242324	+	1.37429i
$516$		0			0
$517$	−1.97385e6	+	718423.i	−0.324779	+	0.118210i
$518$		0			0
$519$	2.74315e6	−	3.33558e6i	0.447025	−	0.543567i
$520$		0			0
$521$	301045.	+	521425.i	0.0485889	+	0.0841584i	0.889297	−	0.457330i	$-0.151194\pi$
$521$	301045.	+	521425.i	0.0485889	+	0.0841584i	−0.840708	+	0.541489i	$0.817861\pi$
$522$		0			0
$523$	−1.23159e6	+	2.13318e6i	−0.196885	+	0.341015i	−0.947517	−	0.319706i	$-0.896416\pi$
$523$	−1.23159e6	+	2.13318e6i	−0.196885	+	0.341015i	0.750632	+	0.660721i	$0.229749\pi$
$524$		0			0
$525$	135134.	+	1.36690e7i	0.0213976	+	2.16440i
$526$		0			0
$527$	−1.45714e6	+	8.26385e6i	−0.228546	+	1.29615i
$528$		0			0
$529$	−2.16024e6	+	1.81266e6i	−0.335632	+	0.281629i
$530$		0			0
$531$	−3.46205e6	−	1.18311e6i	−0.532841	−	0.182091i
$532$		0			0
$533$	4.50663e6	+	1.64028e6i	0.687122	+	0.250092i
$534$		0			0
$535$	1.20132e7	+	1.00803e7i	1.81457	+	1.52261i
$536$		0			0
$537$	3.82735e6	−	1.08481e7i	0.572747	−	1.62337i
$538$		0			0
$539$	9.03630e6			1.33973
$540$		0			0
$541$	1.18840e7			1.74570			0.872851	−	0.487987i	$-0.162269\pi$
$541$	1.18840e7			1.74570			0.872851	+	0.487987i	$0.162269\pi$
$542$		0			0
$543$	6.15612e6	+	7.19099e6i	0.895999	+	1.04662i
$544$		0			0
$545$	889881.	+	746699.i	0.128334	+	0.107685i
$546$		0			0
$547$	5.46555e6	+	1.98930e6i	0.781027	+	0.284270i	0.701601	−	0.712570i	$-0.252469\pi$
$547$	5.46555e6	+	1.98930e6i	0.781027	+	0.284270i	0.0794259	+	0.996841i	$0.474691\pi$
$548$		0			0
$549$	−2.56140e6	+	50649.8i	−0.362700	+	0.00717211i
$550$		0			0
$551$	8.35952e6	−	7.01447e6i	1.17301	−	0.984274i
$552$		0			0
$553$	−2.40427e6	+	1.36353e7i	−0.334327	+	1.89606i
$554$		0			0
$555$	−3.72029e6	+	2.19723e6i	−0.512678	+	0.302792i
$556$		0			0
$557$	−596669.	+	1.03346e6i	−0.0814884	+	0.141142i	−0.903889	−	0.427766i	$-0.859301\pi$
$557$	−596669.	+	1.03346e6i	−0.0814884	+	0.141142i	0.822401	+	0.568908i	$0.192634\pi$
$558$		0			0
$559$	731647.	+	1.26725e6i	0.0990312	+	0.171527i
$560$		0			0
$561$	−987421.	−	2.63167e6i	−0.132463	−	0.353041i
$562$		0			0
$563$	−2.79396e6	+	1.01692e6i	−0.371491	+	0.135212i	−0.521018	−	0.853546i	$-0.674447\pi$
$563$	−2.79396e6	+	1.01692e6i	−0.371491	+	0.135212i	0.149526	+	0.988758i	$0.452225\pi$
$564$		0			0
$565$	−2.06537e6	−	1.17133e7i	−0.272193	−	1.54368i
$566$		0			0
$567$	−1.31520e7	+	5.38483e6i	−1.71805	+	0.703420i
$568$		0			0
$569$	1.30892e6	+	7.42327e6i	0.169486	+	0.961202i	0.944318	+	0.329035i	$0.106723\pi$
$569$	1.30892e6	+	7.42327e6i	0.169486	+	0.961202i	−0.774832	+	0.632167i	$0.782166\pi$
$570$		0			0
$571$	−774796.	+	282003.i	−0.0994483	+	0.0361962i	−0.391265	−	0.920278i	$-0.627962\pi$
$571$	−774796.	+	282003.i	−0.0994483	+	0.0361962i	0.291817	+	0.956474i	$0.405740\pi$
$572$		0			0
$573$	1.66837e6	+	4.44654e6i	0.212278	+	0.565764i
$574$		0			0
$575$	−3.46438e6	−	6.00049e6i	−0.436975	−	0.756862i
$576$		0			0
$577$	−4.14037e6	+	7.17133e6i	−0.517726	+	0.896727i	0.482062	+	0.876137i	$0.339888\pi$
$577$	−4.14037e6	+	7.17133e6i	−0.517726	+	0.896727i	−0.999788	+	0.0205901i	$0.993445\pi$
$578$		0			0
$579$	−5.28930e6	+	3.12390e6i	−0.655695	+	0.387258i
$580$		0			0
$581$	2.76937e6	−	1.57059e7i	0.340362	−	1.93029i
$582$		0			0
$583$	606897.	−	509247.i	0.0739509	−	0.0620521i
$584$		0			0
$585$	3.23893e6	−	5.87527e6i	0.391302	−	0.709803i
$586$		0			0
$587$	−7.18127e6	−	2.61377e6i	−0.860213	−	0.313092i	−0.126016	−	0.992028i	$-0.540219\pi$
$587$	−7.18127e6	−	2.61377e6i	−0.860213	−	0.313092i	−0.734197	+	0.678936i	$0.762441\pi$
$588$		0			0
$589$	−1.34077e7	−	1.12504e7i	−1.59245	−	1.33623i
$590$		0			0
$591$	1.35386e6	+	1.58145e6i	0.159443	+	0.186246i
$592$		0			0
$593$	6.23804e6			0.728470			0.364235	−	0.931307i	$-0.381330\pi$
$593$	6.23804e6			0.728470			0.364235	+	0.931307i	$0.381330\pi$
$594$		0			0
$595$	−1.62462e7			−1.88131
$596$		0			0
$597$	181223.	−	513649.i	0.0208103	−	0.0589836i
$598$		0			0
$599$	−1.01835e7	−	8.54496e6i	−1.15966	−	0.973068i	−0.159757	−	0.987156i	$-0.551071\pi$
$599$	−1.01835e7	−	8.54496e6i	−1.15966	−	0.973068i	−0.999901	+	0.0140884i	$0.995515\pi$
$600$		0			0
$601$	−6.11905e6	−	2.22715e6i	−0.691031	−	0.251515i	−0.0274545	−	0.999623i	$-0.508740\pi$
$601$	−6.11905e6	−	2.22715e6i	−0.691031	−	0.251515i	−0.663577	+	0.748108i	$0.730962\pi$
$602$		0			0
$603$	−2.11625e6	−	1.07540e7i	−0.237014	−	1.20441i
$604$		0			0
$605$	7.10615e6	−	5.96277e6i	0.789307	−	0.662307i
$606$		0			0
$607$	−273519.	+	1.55120e6i	−0.0301312	+	0.170882i	−0.996160	−	0.0875522i	$-0.972096\pi$
$607$	−273519.	+	1.55120e6i	−0.0301312	+	0.170882i	0.966029	+	0.258435i	$0.0832066\pi$
$608$		0			0
$609$	236499.	+	2.39222e7i	0.0258396	+	2.61371i
$610$		0			0
$611$	1.60376e6	−	2.77779e6i	0.173795	−	0.301021i
$612$		0			0
$613$	1.15257e6	+	1.99631e6i	0.123884	+	0.214574i	0.921296	−	0.388861i	$-0.127132\pi$
$613$	1.15257e6	+	1.99631e6i	0.123884	+	0.214574i	−0.797412	+	0.603435i	$0.793798\pi$
$614$		0			0
$615$	1.16417e7	−	1.41559e7i	1.24116	−	1.50921i
$616$		0			0
$617$	−1.36819e7	+	4.97980e6i	−1.44688	+	0.526622i	−0.941719	−	0.336401i	$-0.890790\pi$
$617$	−1.36819e7	+	4.97980e6i	−1.44688	+	0.526622i	−0.505164	+	0.863023i	$0.668568\pi$
$618$		0			0
$619$	1.16209e6	+	6.59054e6i	0.121903	+	0.691344i	0.983099	+	0.183072i	$0.0586043\pi$
$619$	1.16209e6	+	6.59054e6i	0.121903	+	0.691344i	−0.861197	+	0.508272i	$0.830285\pi$
$620$		0			0
$621$	4.46466e6	−	5.65309e6i	0.464579	−	0.588243i
$622$		0			0
$623$	−1.10785e6	−	6.28294e6i	−0.114357	−	0.648548i
$624$		0			0
$625$	7.40139e6	−	2.69389e6i	0.757903	−	0.275854i
$626$		0			0
$627$	5.78348e6	+	960932.i	0.587517	+	0.0976167i
$628$		0			0
$629$	−1.38212e6	−	2.39390e6i	−0.139290	−	0.241257i
$630$		0			0
$631$	3.61450e6	−	6.26051e6i	0.361389	−	0.625945i	−0.626800	−	0.779180i	$-0.715636\pi$
$631$	3.61450e6	−	6.26051e6i	0.361389	−	0.625945i	0.988190	+	0.153235i	$0.0489692\pi$
$632$		0			0
$633$	243016.	+	137120.i	0.0241060	+	0.0136017i
$634$		0			0
$635$	4.55215e6	−	2.58165e7i	0.448004	−	2.54076i
$636$		0			0
$637$	−1.05702e7	+	8.86949e6i	−1.03214	+	0.866064i
$638$		0			0
$639$	−2.64287e6	+	1.69405e7i	−0.256049	+	1.64125i
$640$		0			0
$641$	−598580.	−	217865.i	−0.0575410	−	0.0209432i	0.313089	−	0.949724i	$-0.398636\pi$
$641$	−598580.	−	217865.i	−0.0575410	−	0.0209432i	−0.370630	+	0.928780i	$0.620858\pi$
$642$		0			0
$643$	−1.12114e7	−	9.40747e6i	−1.06938	−	0.897316i	−0.0743837	−	0.997230i	$-0.523699\pi$
$643$	−1.12114e7	−	9.40747e6i	−1.06938	−	0.897316i	−0.994996	+	0.0999135i	$0.968143\pi$
$644$		0			0
$645$	5.49737e6	−	1.02547e6i	0.520302	−	0.0970563i
$646$		0			0
$647$	−1.14730e6			−0.107750			−0.0538751	−	0.998548i	$-0.517157\pi$
$647$	−1.14730e6			−0.107750			−0.0538751	+	0.998548i	$0.517157\pi$
$648$		0			0
$649$	−3.30880e6			−0.308360
$650$		0			0
$651$	3.77197e7	−	7.03616e6i	3.48831	−	0.650704i
$652$		0			0
$653$	3.85407e6	+	3.23395e6i	0.353701	+	0.296791i	0.802274	−	0.596956i	$-0.203623\pi$
$653$	3.85407e6	+	3.23395e6i	0.353701	+	0.296791i	−0.448573	+	0.893746i	$0.648068\pi$
$654$		0			0
$655$	−1.82048e7	−	6.62600e6i	−1.65799	−	0.603460i
$656$		0			0
$657$	6.07587e6	+	4.89686e6i	0.549155	+	0.442593i
$658$		0			0
$659$	−1.56509e7	+	1.31326e7i	−1.40386	+	1.17798i	−0.444508	+	0.895775i	$0.646622\pi$
$659$	−1.56509e7	+	1.31326e7i	−1.40386	+	1.17798i	−0.959354	+	0.282205i	$0.908934\pi$
$660$		0			0
$661$	−2.23917e6	+	1.26989e7i	−0.199335	+	1.13048i	0.706775	+	0.707439i	$0.250149\pi$
$661$	−2.23917e6	+	1.26989e7i	−0.199335	+	1.13048i	−0.906109	+	0.423044i	$0.860962\pi$
$662$		0			0
$663$	3.73813e6	+	2.10922e6i	0.330271	+	0.186354i
$664$		0			0
$665$	1.69430e7	−	2.93462e7i	1.48572	−	2.57335i
$666$		0			0
$667$	−6.06305e6	−	1.05015e7i	−0.527687	−	0.913981i
$668$		0			0
$669$	246291.	+	40921.5i	0.0212757	+	0.00353498i
$670$		0			0
$671$	−2.17721e6	+	792440.i	−0.186678	+	0.0679454i
$672$		0			0
$673$	−1.25408e6	−	7.11225e6i	−0.106730	−	0.605298i	−0.990515	−	0.137404i	$-0.956124\pi$
$673$	−1.25408e6	−	7.11225e6i	−0.106730	−	0.605298i	0.883785	−	0.467894i	$-0.154987\pi$
$674$		0			0
$675$	−9.18112e6	−	1.03049e7i	−0.775598	−	0.870535i
$676$		0			0
$677$	1.48904e6	+	8.44475e6i	0.124863	+	0.708133i	0.981389	+	0.192030i	$0.0615070\pi$
$677$	1.48904e6	+	8.44475e6i	0.124863	+	0.708133i	−0.856526	+	0.516104i	$0.827382\pi$
$678$		0			0
$679$	−3.54036e7	+	1.28859e7i	−2.94695	+	1.07260i
$680$		0			0
$681$	−9.39962e6	+	1.14296e7i	−0.776680	+	0.944417i
$682$		0			0
$683$	2.60561e6	+	4.51305e6i	0.213726	+	0.370185i	0.952878	−	0.303354i	$-0.0981066\pi$
$683$	2.60561e6	+	4.51305e6i	0.213726	+	0.370185i	−0.739151	+	0.673539i	$0.764773\pi$
$684$		0			0
$685$	1.66509e7	−	2.88403e7i	1.35585	−	2.34841i
$686$		0			0
$687$	−108593.	−	1.09844e7i	−0.00877833	−	0.887942i
$688$		0			0
$689$	−210074.	+	1.19139e6i	−0.0168587	+	0.0956103i
$690$		0			0
$691$	3.90781e6	−	3.27904e6i	0.311342	−	0.261247i	−0.473704	−	0.880684i	$-0.657083\pi$
$691$	3.90781e6	−	3.27904e6i	0.311342	−	0.261247i	0.785047	+	0.619437i	$0.212639\pi$
$692$		0			0
$693$	−9.68055e6	+	8.45466e6i	−0.765715	+	0.668749i
$694$		0			0
$695$	−4.34517e6	−	1.58151e6i	−0.341228	−	0.124197i
$696$		0			0
$697$	8.98242e6	+	7.53714e6i	0.700344	+	0.587658i
$698$		0			0
$699$	−3.83953e6	+	1.08826e7i	−0.297225	+	0.842439i
$700$		0			0
$701$	−7.60756e6			−0.584723			−0.292361	−	0.956308i	$-0.594441\pi$
$701$	−7.60756e6			−0.584723			−0.292361	+	0.956308i	$0.594441\pi$
$702$		0			0
$703$	5.76561e6			0.440005
$704$		0			0
$705$	−7.97175e6	−	9.31184e6i	−0.604061	−	0.705607i
$706$		0			0
$707$	1.51783e7	+	1.27361e7i	1.14203	+	0.958273i
$708$		0			0
$709$	23164.5	+	8431.18i	0.00173064	+	0.000629902i	0.342885	−	0.939377i	$-0.388596\pi$
$709$	23164.5	+	8431.18i	0.00173064	+	0.000629902i	−0.341155	+	0.940007i	$0.610818\pi$
$710$		0			0
$711$	−7.22766e6	−	1.19659e7i	−0.536196	−	0.887712i
$712$		0			0
$713$	−1.48987e7	+	1.25015e7i	−1.09755	+	0.920953i
$714$		0			0
$715$	1.05360e6	−	5.97524e6i	0.0770741	−	0.437109i
$716$		0			0
$717$	−3.28234e6	+	1.93857e6i	−0.238443	+	0.140826i
$718$		0			0
$719$	5.75788e6	−	9.97294e6i	0.415375	−	0.719450i	−0.580093	−	0.814550i	$-0.696984\pi$
$719$	5.75788e6	−	9.97294e6i	0.415375	−	0.719450i	0.995468	+	0.0950999i	$0.0303171\pi$
$720$		0			0
$721$	1.22857e7	+	2.12795e7i	0.880163	+	1.52449i
$722$		0			0
$723$	236915.	+	631425.i	0.0168557	+	0.0449238i
$724$		0			0
$725$	−2.18320e7	+	7.94618e6i	−1.54258	+	0.561453i
$726$		0			0
$727$	1.20327e6	+	6.82407e6i	0.0844357	+	0.478859i	0.997477	+	0.0709914i	$0.0226163\pi$
$727$	1.20327e6	+	6.82407e6i	0.0844357	+	0.478859i	−0.913041	+	0.407867i	$0.866273\pi$
$728$		0			0
$729$	6.42519e6	−	1.28300e7i	0.447783	−	0.894143i
$730$		0			0
$731$	621262.	+	3.52335e6i	0.0430013	+	0.243872i
$732$		0			0
$733$	2.77387e6	−	1.00960e6i	0.190689	−	0.0694051i	−0.244911	−	0.969546i	$-0.578759\pi$
$733$	2.77387e6	−	1.00960e6i	0.190689	−	0.0694051i	0.435600	+	0.900141i	$0.356536\pi$
$734$		0			0
$735$	1.85247e7	+	4.93721e7i	1.26483	+	3.37103i
$736$		0			0
$737$	−4.95612e6	−	8.58424e6i	−0.336103	−	0.582148i
$738$		0			0
$739$	−177442.	+	307339.i	−0.0119522	+	0.0207017i	−0.871940	−	0.489613i	$-0.837138\pi$
$739$	−177442.	+	307339.i	−0.0119522	+	0.0207017i	0.859987	+	0.510315i	$0.170471\pi$
$740$		0			0
$741$	−7.70845e6	+	4.55267e6i	−0.515729	+	0.304593i
$742$		0			0
$743$	1.13602e6	−	6.44267e6i	0.0754940	−	0.428148i	−0.923512	−	0.383570i	$-0.874695\pi$
$743$	1.13602e6	−	6.44267e6i	0.0754940	−	0.428148i	0.999006	−	0.0445780i	$-0.0141943\pi$
$744$		0			0
$745$	7.06333e6	−	5.92684e6i	0.466250	−	0.391230i
$746$		0			0
$747$	8.32519e6	+	1.37830e7i	0.545875	+	0.903735i
$748$		0			0
$749$	4.31099e7	+	1.56907e7i	2.80784	+	1.02197i
$750$		0			0
$751$	1.33389e7	+	1.11927e7i	0.863018	+	0.724158i	0.962616	−	0.270870i	$-0.0873113\pi$
$751$	1.33389e7	+	1.11927e7i	0.863018	+	0.724158i	−0.0995981	+	0.995028i	$0.531756\pi$
$752$		0			0
$753$	−1.07925e7	−	1.26068e7i	−0.693640	−	0.810244i
$754$		0			0
$755$	1.89639e7			1.21077
$756$		0			0
$757$	8.13417e6			0.515909			0.257955	−	0.966157i	$-0.416951\pi$
$757$	8.13417e6			0.515909			0.257955	+	0.966157i	$0.416951\pi$
$758$		0			0
$759$	2.16754e6	−	6.14357e6i	0.136572	−	0.387094i
$760$		0			0
$761$	9.61478e6	+	8.06776e6i	0.601836	+	0.505000i	0.892035	−	0.451966i	$-0.149277\pi$
$761$	9.61478e6	+	8.06776e6i	0.601836	+	0.505000i	−0.290199	+	0.956966i	$0.593722\pi$
$762$		0			0
$763$	3.19338e6	+	1.16229e6i	0.198581	+	0.0722777i
$764$		0			0
$765$	1.23546e7	−	1.07900e7i	0.763262	−	0.666607i
$766$		0			0
$767$	3.87048e6	−	3.24772e6i	0.237562	−	0.199338i
$768$		0			0
$769$	−18182.7	+	103119.i	−0.00110877	+	0.00628815i	−0.985357	−	0.170503i	$-0.945461\pi$
$769$	−18182.7	+	103119.i	−0.00110877	+	0.00628815i	0.984248	+	0.176791i	$0.0565718\pi$
$770$		0			0
$771$	−1404.00	−	142016.i	−8.50609e−5	−	0.00860405i
$772$		0			0
$773$	5.33658e6	−	9.24323e6i	0.321229	−	0.556385i	−0.659513	−	0.751693i	$-0.729237\pi$
$773$	5.33658e6	−	9.24323e6i	0.321229	−	0.556385i	0.980742	+	0.195308i	$0.0625708\pi$
$774$		0			0
$775$	1.86316e7	+	3.22708e7i	1.11428	+	1.92999i
$776$		0			0
$777$	−8.02859e6	+	9.76249e6i	−0.477075	+	0.580107i
$778$		0			0
$779$	−2.29824e7	+	8.36490e6i	−1.35691	+	0.493875i
$780$		0			0
$781$	2.69259e6	+	1.52704e7i	0.157959	+	0.895827i
$782$		0			0
$783$	−1.60680e7	−	1.80348e7i	−0.936605	−	1.05125i
$784$		0			0
$785$	−1.86570e6	−	1.05809e7i	−0.108061	−	0.612842i
$786$		0			0
$787$	−1.60073e7	+	5.82618e6i	−0.921258	+	0.335311i	−0.758739	−	0.651395i	$-0.774184\pi$
$787$	−1.60073e7	+	5.82618e6i	−0.921258	+	0.335311i	−0.162519	+	0.986705i	$0.551962\pi$
$788$		0			0
$789$	−3.41481e7	−	5.67375e6i	−1.95287	−	0.324472i
$790$		0			0
$791$	−1.73974e7	−	3.01332e7i	−0.988652	−	1.71240i
$792$		0			0
$793$	1.76899e6	−	3.06398e6i	0.0998947	−	0.173023i
$794$		0			0
$795$	4.02656e6	+	2.27196e6i	0.225952	+	0.127492i
$796$		0			0
$797$	1.35853e6	−	7.70458e6i	0.0757569	−	0.429639i	−0.923214	−	0.384286i	$-0.874448\pi$
$797$	1.35853e6	−	7.70458e6i	0.0757569	−	0.429639i	0.998971	−	0.0453528i	$-0.0144412\pi$
$798$		0			0
$799$	6.00754e6	−	5.04092e6i	0.332912	−	0.279346i
$800$		0			0
$801$	5.01534e6	+	4.04212e6i	0.276197	+	0.222602i
$802$		0			0
$803$	6.63177e6	+	2.41377e6i	0.362945	+	0.132101i
$804$		0			0
$805$	−2.88449e7	−	2.42037e7i	−1.56884	−	1.31641i
$806$		0			0
$807$	2.93680e7	−	5.47825e6i	1.58741	−	0.296113i
$808$		0			0
$809$	1.58513e7			0.851519			0.425760	−	0.904836i	$-0.360007\pi$
$809$	1.58513e7			0.851519			0.425760	+	0.904836i	$0.360007\pi$
$810$		0			0
$811$	−5.28364e6			−0.282085			−0.141043	−	0.990004i	$-0.545045\pi$
$811$	−5.28364e6			−0.282085			−0.141043	+	0.990004i	$0.545045\pi$
$812$		0			0
$813$	1.13576e7	−	2.11864e6i	0.602646	−	0.112417i
$814$		0			0
$815$	2.67107e7	+	2.24130e7i	1.40861	+	1.18197i
$816$		0			0
$817$	−7.01229e6	−	2.55227e6i	−0.367540	−	0.133774i
$818$		0			0
$819$	3.02528e6	−	1.93917e7i	0.157600	−	1.01020i
$820$		0			0
$821$	−6.50492e6	+	5.45827e6i	−0.336809	+	0.282616i	−0.795467	−	0.605996i	$-0.792775\pi$
$821$	−6.50492e6	+	5.45827e6i	−0.336809	+	0.282616i	0.458658	+	0.888613i	$0.348330\pi$
$822$		0			0
$823$	−4.11551e6	+	2.33402e7i	−0.211799	+	1.20117i	0.674577	+	0.738205i	$0.264326\pi$
$823$	−4.11551e6	+	2.33402e7i	−0.211799	+	1.20117i	−0.886376	+	0.462967i	$0.846785\pi$
$824$		0			0
$825$	−1.08708e7	−	6.13380e6i	−0.556069	−	0.313758i
$826$		0			0
$827$	−1.46655e7	+	2.54013e7i	−0.745644	+	1.29149i	0.204249	+	0.978919i	$0.434525\pi$
$827$	−1.46655e7	+	2.54013e7i	−0.745644	+	1.29149i	−0.949893	+	0.312575i	$0.898808\pi$
$828$		0			0
$829$	1.26018e7	+	2.18270e7i	0.636863	+	1.10308i	0.986117	+	0.166052i	$0.0531018\pi$
$829$	1.26018e7	+	2.18270e7i	0.636863	+	1.10308i	−0.349254	+	0.937028i	$0.613565\pi$
$830$		0			0
$831$	6.45949e6	+	1.07325e6i	0.324486	+	0.0539137i
$832$		0			0
$833$	−3.17022e7	+	1.15387e7i	−1.58299	+	0.576160i
$834$		0			0
$835$	−5.87849e6	−	3.33386e7i	−0.291776	−	1.65474i
$836$		0			0
$837$	−2.40111e7	+	3.04025e7i	−1.18467	+	1.50002i
$838$		0			0
$839$	354257.	+	2.00909e6i	0.0173746	+	0.0985360i	0.992262	−	0.124163i	$-0.0396245\pi$
$839$	354257.	+	2.00909e6i	0.0173746	+	0.0985360i	−0.974887	+	0.222699i	$0.928513\pi$
$840$		0			0
$841$	−1.89341e7	+	6.89146e6i	−0.923114	+	0.335986i
$842$		0			0
$843$	−4.28474e6	+	5.21009e6i	−0.207661	+	0.252509i
$844$		0			0
$845$	−1.06408e7	−	1.84305e7i	−0.512665	−	0.887962i
$846$		0			0
$847$	1.35687e7	−	2.35016e7i	0.649873	−	1.12561i
$848$		0			0
$849$	293838.	+	2.97222e7i	0.0139907	+	1.41518i
$850$		0			0
$851$	1.11252e6	−	6.30943e6i	0.0526605	−	0.298653i
$852$		0			0
$853$	2.52089e7	−	2.11528e7i	1.18626	−	0.995393i	0.186346	−	0.982484i	$-0.440335\pi$
$853$	2.52089e7	−	2.11528e7i	1.18626	−	0.995393i	0.999917	−	0.0129087i	$-0.00410908\pi$
$854$		0			0
$855$	6.60604e6	+	3.35695e7i	0.309048	+	1.57047i
$856$		0			0
$857$	1.42697e7	+	5.19374e6i	0.663685	+	0.241562i	0.651826	−	0.758368i	$-0.274003\pi$
$857$	1.42697e7	+	5.19374e6i	0.663685	+	0.241562i	0.0118584	+	0.999930i	$0.496225\pi$
$858$		0			0
$859$	1.06657e7	+	8.94963e6i	0.493183	+	0.413830i	0.855166	−	0.518355i	$-0.173455\pi$
$859$	1.06657e7	+	8.94963e6i	0.493183	+	0.413830i	−0.361982	+	0.932185i	$0.617900\pi$
$860$		0			0
$861$	1.78392e7	−	5.05625e7i	0.820100	−	2.32445i
$862$		0			0
$863$	−3.46010e6			−0.158147			−0.0790736	−	0.996869i	$-0.525196\pi$
$863$	−3.46010e6			−0.158147			−0.0790736	+	0.996869i	$0.525196\pi$
$864$		0			0
$865$	2.27926e7			1.03575
$866$		0			0
$867$	−7.56934e6	−	8.84178e6i	−0.341988	−	0.399477i
$868$		0			0
$869$	−9.68484e6	−	8.12655e6i	−0.435054	−	0.365054i
$870$		0			0
$871$	1.42232e7	+	5.17683e6i	0.635262	+	0.231216i
$872$		0			0
$873$	1.83647e7	−	3.33128e7i	0.815547	−	1.47936i
$874$		0			0
$875$	−7.86497e6	+	6.59949e6i	−0.347278	+	0.291400i
$876$		0			0
$877$	−5.00658e6	+	2.83937e7i	−0.219807	+	1.24659i	0.652560	+	0.757737i	$0.273695\pi$
$877$	−5.00658e6	+	2.83937e7i	−0.219807	+	1.24659i	−0.872367	+	0.488851i	$0.837416\pi$
$878$		0			0
$879$	−2.15988e7	+	1.27564e7i	−0.942883	+	0.556874i
$880$		0			0
$881$	−1.11642e7	+	1.93369e7i	−0.484604	+	0.839358i	−0.999844	−	0.0176878i	$-0.994370\pi$
$881$	−1.11642e7	+	1.93369e7i	−0.484604	+	0.839358i	0.515240	+	0.857046i	$0.327703\pi$
$882$		0			0
$883$	7.06250e6	+	1.22326e7i	0.304829	+	0.527980i	0.977223	−	0.212213i	$-0.0680672\pi$
$883$	7.06250e6	+	1.22326e7i	0.304829	+	0.527980i	−0.672394	+	0.740194i	$0.734734\pi$
$884$		0			0
$885$	−6.78315e6	−	1.80784e7i	−0.291121	−	0.775895i
$886$		0			0
$887$	−2.29472e7	+	8.35211e6i	−0.979313	+	0.356441i	−0.781573	−	0.623814i	$-0.785582\pi$
$887$	−2.29472e7	+	8.35211e6i	−0.979313	+	0.356441i	−0.197740	+	0.980255i	$0.563360\pi$
$888$		0			0
$889$	−1.33169e7	−	7.55238e7i	−0.565130	−	3.20501i
$890$		0			0
$891$	1.74643e6	−	1.28588e7i	0.0736982	−	0.542634i
$892$		0			0
$893$	2.84042e6	+	1.61088e7i	0.119194	+	0.675982i
$894$		0			0
$895$	5.70502e7	−	2.07646e7i	2.38067	−	0.866494i
$896$		0			0
$897$	3.49467e6	+	9.31399e6i	0.145019	+	0.386505i
$898$		0			0
$899$	3.26073e7	+	5.64775e7i	1.34560	+	2.33065i
$900$		0			0
$901$	−1.47892e6	+	2.56156e6i	−0.0606921	+	0.105122i
$902$		0			0
$903$	1.40862e7	−	8.31939e6i	0.574874	−	0.339525i
$904$		0			0
$905$	−8.67537e6	+	4.92004e7i	−0.352100	+	1.99686i
$906$		0			0
$907$	−2.27318e7	+	1.90742e7i	−0.917520	+	0.769891i	−0.973535	−	0.228539i	$-0.926605\pi$
$907$	−2.27318e7	+	1.90742e7i	−0.917520	+	0.769891i	0.0560149	+	0.998430i	$0.482161\pi$
$908$		0			0
$909$	−2.00013e7	+	395511.i	−0.802876	+	0.0158763i
$910$		0			0
$911$	4.01060e7	+	1.45974e7i	1.60108	+	0.582746i	0.979648	−	0.200721i	$-0.0643286\pi$
$911$	4.01060e7	+	1.45974e7i	1.60108	+	0.582746i	0.621433	+	0.783467i	$0.286551\pi$
$912$		0			0
$913$	1.11555e7	+	9.36058e6i	0.442907	+	0.371643i
$914$		0			0
$915$	−8.79306e6	−	1.02712e7i	−0.347206	−	0.405573i
$916$		0			0
$917$	−5.66743e7			−2.22568
$918$		0			0
$919$	−5.77855e6			−0.225699			−0.112850	−	0.993612i	$-0.535998\pi$
$919$	−5.77855e6			−0.225699			−0.112850	+	0.993612i	$0.535998\pi$
$920$		0			0
$921$	1.04753e7	−	2.96905e7i	0.406926	−	1.15337i
$922$		0			0
$923$	−1.81382e7	−	1.52198e7i	−0.700795	−	0.588037i
$924$		0			0
$925$	−1.15348e7	−	4.19833e6i	−0.443258	−	0.161333i
$926$		0			0
$927$	−2.34758e7	−	8.02253e6i	−0.897265	−	0.306628i
$928$		0			0
$929$	5.18922e6	−	4.35427e6i	0.197271	−	0.165530i	−0.538802	−	0.842433i	$-0.681123\pi$
$929$	5.18922e6	−	4.35427e6i	0.197271	−	0.165530i	0.736072	+	0.676903i	$0.236678\pi$
$930$		0			0
$931$	1.22192e7	−	6.92988e7i	0.462030	−	2.62030i
$932$		0			0
$933$	−93654.6	−	9.47330e6i	−0.00352229	−	0.356285i
$934$		0			0
$935$	7.41731e6	−	1.28472e7i	0.277471	−	0.480594i
$936$		0			0
$937$	−2.30154e7	−	3.98638e7i	−0.856384	−	1.48330i	−0.875355	−	0.483481i	$-0.839372\pi$
$937$	−2.30154e7	−	3.98638e7i	−0.856384	−	1.48330i	0.0189708	−	0.999820i	$-0.493961\pi$
$938$		0			0
$939$	−4.73999e6	+	5.76367e6i	−0.175434	+	0.213322i
$940$		0			0
$941$	−2.93760e7	+	1.06920e7i	−1.08148	+	0.393626i	−0.820458	−	0.571707i	$-0.806281\pi$
$941$	−2.93760e7	+	1.06920e7i	−1.08148	+	0.393626i	−0.261021	+	0.965333i	$0.584059\pi$
$942$		0			0
$943$	4.71925e6	+	2.67642e7i	0.172820	+	0.980110i
$944$		0			0
$945$	−6.60396e7	−	3.55598e7i	−2.40561	−	1.29533i
$946$		0			0
$947$	−1.59516e6	−	9.04659e6i	−0.0578001	−	0.327801i	0.942173	−	0.335127i	$-0.108779\pi$
$947$	−1.59516e6	−	9.04659e6i	−0.0578001	−	0.327801i	−0.999973	+	0.00732609i	$0.997668\pi$
$948$		0			0
$949$	−1.01268e7	+	3.68584e6i	−0.365010	+	0.132853i
$950$		0			0
$951$	2.92848e7	+	4.86571e6i	1.05001	+	0.174460i
$952$		0			0
$953$	1.24895e7	+	2.16324e7i	0.445464	+	0.771566i	0.998084	−	0.0618672i	$-0.0197055\pi$
$953$	1.24895e7	+	2.16324e7i	0.445464	+	0.771566i	−0.552621	+	0.833433i	$0.686372\pi$
$954$		0			0
$955$	−1.25325e7	+	2.17069e7i	−0.444660	+	0.770174i
$956$		0			0
$957$	−1.90252e7	−	1.07348e7i	−0.671504	−	0.378892i
$958$		0			0
$959$	1.69171e7	−	9.59416e7i	0.593990	−	3.36868i
$960$		0			0
$961$	5.81945e7	−	4.88310e7i	2.03270	−	1.70564i
$962$		0			0
$963$	−4.32044e7	+	1.66996e7i	−1.50128	+	0.580285i
$964$		0			0
$965$	−3.04652e7	−	1.10884e7i	−1.05314	−	0.383312i
$966$		0			0
$967$	3.02190e7	+	2.53568e7i	1.03924	+	0.872022i	0.991921	−	0.126858i	$-0.0404891\pi$
$967$	3.02190e7	+	2.53568e7i	1.03924	+	0.872022i	0.0473150	+	0.998880i	$0.484934\pi$
$968$		0			0
$969$	−2.15174e7	+	4.01381e6i	−0.736172	+	0.137324i
$970$		0			0
$971$	2.21513e7			0.753964			0.376982	−	0.926220i	$-0.376962\pi$
$971$	2.21513e7			0.753964			0.376982	+	0.926220i	$0.376962\pi$
$972$		0			0
$973$	−1.35272e7			−0.458063
$974$		0			0
$975$	1.87368e7	−	3.49513e6i	0.631224	−	0.117748i
$976$		0			0
$977$	3.67781e7	+	3.08605e7i	1.23269	+	1.03435i	0.998060	+	0.0622614i	$0.0198312\pi$
$977$	3.67781e7	+	3.08605e7i	1.23269	+	1.03435i	0.234627	+	0.972086i	$0.424613\pi$
$978$		0			0
$979$	5.47421e6	+	1.99245e6i	0.182543	+	0.0664402i
$980$		0			0
$981$	−3.20038e6	+	1.23703e6i	−0.106177	+	0.0410400i
$982$		0			0
$983$	−2.30770e7	+	1.93639e7i	−0.761720	+	0.639159i	−0.938574	−	0.345078i	$-0.887853\pi$
$983$	−2.30770e7	+	1.93639e7i	−0.761720	+	0.639159i	0.176854	+	0.984237i	$0.443408\pi$
$984$		0			0
$985$	−1.90790e6	+	1.08202e7i	−0.0626563	+	0.355341i
$986$		0			0
$987$	−3.12312e7	−	1.76220e7i	−1.02046	−	0.575787i
$988$		0			0
$989$	−4.14608e6	+	7.18122e6i	−0.134787	+	0.233457i
$990$		0			0
$991$	−2.23637e7	−	3.87350e7i	−0.723367	−	1.25291i	−0.959643	−	0.281222i	$-0.909260\pi$
$991$	−2.23637e7	−	3.87350e7i	−0.723367	−	1.25291i	0.236276	−	0.971686i	$-0.424073\pi$
$992$		0			0
$993$	−1.51281e7	−	2.51355e6i	−0.486867	−	0.0808936i
$994$		0			0
$995$	2.70129e6	−	983190.i	0.0864996	−	0.0314833i
$996$		0			0
$997$	6.29275e6	+	3.56880e7i	0.200495	+	1.13706i	0.904374	+	0.426742i	$0.140339\pi$
$997$	6.29275e6	+	3.56880e7i	0.200495	+	1.13706i	−0.703879	+	0.710320i	$0.748550\pi$
$998$		0			0
$999$	−378429.	−	1.27562e7i	−0.0119969	−	0.404398i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.6.i.a.13.1	✓	90
3.2	odd	2		324.6.i.a.253.14		90
27.2	odd	18		324.6.i.a.73.14		90
27.25	even	9	inner	108.6.i.a.25.1	yes	90

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.6.i.a.13.1	✓	90	1.1	even	1	trivial
108.6.i.a.25.1	yes	90	27.25	even	9	inner
324.6.i.a.73.14		90	27.2	odd	18
324.6.i.a.253.14		90	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	108.i (of order \(9\), degree \(6\), minimal)

\(f(q)\)	\(=\)	\(q+(-15.3241 + 2.85854i) q^{3} +(-63.0232 - 52.8828i) q^{5} +(-226.162 - 82.3161i) q^{7} +(226.658 - 87.6091i) q^{9} +O(q^{10})\)	\(q+(-15.3241 + 2.85854i) q^{3} +(-63.0232 - 52.8828i) q^{5} +(-226.162 - 82.3161i) q^{7} +(226.658 - 87.6091i) q^{9} +(168.350 - 141.262i) q^{11} +(-58.2732 + 330.484i) q^{13} +(1116.94 + 630.228i) q^{15} +(-410.244 + 710.563i) q^{17} +(-855.680 - 1482.08i) q^{19} +(3701.03 + 614.931i) q^{21} +(-1786.98 + 650.409i) q^{23} +(632.690 + 3588.16i) q^{25} +(-3222.89 + 1990.44i) q^{27} +(1107.28 + 6279.68i) q^{29} +(9610.47 - 3497.93i) q^{31} +(-2176.01 + 2645.95i) q^{33} +(9900.34 + 17147.9i) q^{35} +(-1684.51 + 2917.66i) q^{37} +(-51.7140 - 5230.95i) q^{39} +(2481.63 - 14074.0i) q^{41} +(3340.31 - 2802.85i) q^{43} +(-18917.7 - 6464.87i) q^{45} +(-8981.66 - 3269.06i) q^{47} +(31498.2 + 26430.2i) q^{49} +(4255.45 - 12061.4i) q^{51} +3604.98 q^{53} -18080.3 q^{55} +(17349.1 + 20265.6i) q^{57} +(-11533.6 - 9677.86i) q^{59} +(-9907.01 - 3605.86i) q^{61} +(-58472.9 + 1156.26i) q^{63} +(21149.5 - 17746.5i) q^{65} +(7832.20 - 44418.6i) q^{67} +(25524.8 - 15075.1i) q^{69} +(-35278.7 + 61104.5i) q^{71} +(16056.7 + 27811.0i) q^{73} +(-19952.3 - 53176.9i) q^{75} +(-49702.4 + 18090.2i) q^{77} +(-9989.66 - 56654.2i) q^{79} +(43698.3 - 39714.5i) q^{81} +(11506.6 + 65257.3i) q^{83} +(63431.4 - 23087.1i) q^{85} +(-34918.7 - 93065.4i) q^{87} +(13254.0 + 22956.7i) q^{89} +(40383.3 - 69946.0i) q^{91} +(-137273. + 81074.5i) q^{93} +(-24448.9 + 138656. i) q^{95} +(119917. - 100623. i) q^{97} +(25781.9 - 46767.1i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 90 q - 87 q^{5} + 330 q^{9}+O(q^{10}) \) 90 * q - 87 * q^5 + 330 * q^9	\( 90 q - 87 q^{5} + 330 q^{9} - 1257 q^{11} + 531 q^{15} - 3468 q^{17} + 12894 q^{21} + 8106 q^{23} + 4959 q^{25} - 17415 q^{27} + 3468 q^{29} - 6651 q^{31} + 33624 q^{33} - 8229 q^{35} - 10545 q^{39} + 68673 q^{41} + 9459 q^{43} - 53469 q^{45} - 57087 q^{47} - 5490 q^{49} + 42831 q^{51} - 4146 q^{53} + 24624 q^{57} + 5388 q^{59} + 70110 q^{61} - 98115 q^{63} - 172425 q^{65} - 15039 q^{67} + 251037 q^{69} + 67812 q^{71} - 27009 q^{73} - 75273 q^{75} + 23991 q^{77} - 216180 q^{79} + 177822 q^{81} - 76725 q^{83} - 53100 q^{85} - 201483 q^{87} - 98814 q^{89} - 90999 q^{91} + 21765 q^{93} - 143490 q^{95} - 71739 q^{97} + 13635 q^{99}+O(q^{100}) \) 90 * q - 87 * q^5 + 330 * q^9 - 1257 * q^11 + 531 * q^15 - 3468 * q^17 + 12894 * q^21 + 8106 * q^23 + 4959 * q^25 - 17415 * q^27 + 3468 * q^29 - 6651 * q^31 + 33624 * q^33 - 8229 * q^35 - 10545 * q^39 + 68673 * q^41 + 9459 * q^43 - 53469 * q^45 - 57087 * q^47 - 5490 * q^49 + 42831 * q^51 - 4146 * q^53 + 24624 * q^57 + 5388 * q^59 + 70110 * q^61 - 98115 * q^63 - 172425 * q^65 - 15039 * q^67 + 251037 * q^69 + 67812 * q^71 - 27009 * q^73 - 75273 * q^75 + 23991 * q^77 - 216180 * q^79 + 177822 * q^81 - 76725 * q^83 - 53100 * q^85 - 201483 * q^87 - 98814 * q^89 - 90999 * q^91 + 21765 * q^93 - 143490 * q^95 - 71739 * q^97 + 13635 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more