LMFDB - Embedded newform 108.5.k.a.5.2

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			5.2
Character	$\chi$	$=$	108.5
Dual form			108.5.k.a.65.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.09115 - 3.94122i) q^{3} +(14.0198 + 38.5191i) q^{5} +(-13.2730 - 75.2747i) q^{7} +(49.9335 + 63.7781i) q^{9} +O(q^{10})$	\(q+(-8.09115 - 3.94122i) q^{3} +(14.0198 + 38.5191i) q^{5} +(-13.2730 - 75.2747i) q^{7} +(49.9335 + 63.7781i) q^{9} +(-14.4803 + 39.7844i) q^{11} +(-99.4208 - 83.4239i) q^{13} +(38.3759 - 366.919i) q^{15} +(25.0688 + 14.4735i) q^{17} +(-340.107 - 589.082i) q^{19} +(-189.281 + 661.371i) q^{21} +(-925.342 - 163.163i) q^{23} +(-808.388 + 678.318i) q^{25} +(-152.656 - 712.837i) q^{27} +(-743.151 - 885.653i) q^{29} +(-216.511 + 1227.89i) q^{31} +(273.962 - 264.832i) q^{33} +(2713.43 - 1566.60i) q^{35} +(951.756 - 1648.49i) q^{37} +(475.637 + 1066.84i) q^{39} +(663.134 - 790.292i) q^{41} +(989.699 + 360.221i) q^{43} +(-1756.61 + 2817.55i) q^{45} +(747.698 - 131.839i) q^{47} +(-3233.91 + 1177.05i) q^{49} +(-145.793 - 215.909i) q^{51} -202.878i q^{53} -1735.47 q^{55} +(430.153 + 6106.79i) q^{57} +(1284.25 + 3528.44i) q^{59} +(-211.588 - 1199.97i) q^{61} +(4138.11 - 4605.26i) q^{63} +(1819.55 - 4999.18i) q^{65} +(-2491.41 - 2090.54i) q^{67} +(6844.03 + 4967.15i) q^{69} +(2946.68 + 1701.27i) q^{71} +(-1292.56 - 2238.78i) q^{73} +(9214.19 - 2302.34i) q^{75} +(3186.96 + 561.947i) q^{77} +(-5812.54 + 4877.30i) q^{79} +(-1574.28 + 6369.33i) q^{81} +(-5284.48 - 6297.80i) q^{83} +(-206.046 + 1168.54i) q^{85} +(2522.40 + 10094.9i) q^{87} +(-11385.8 + 6573.58i) q^{89} +(-4960.10 + 8591.15i) q^{91} +(6591.22 - 9081.76i) q^{93} +(17922.7 - 21359.4i) q^{95} +(-3367.63 - 1225.72i) q^{97} +(-3260.43 + 1063.05i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) $ 72 * q + 9 * q^5 - 102 * q^9	$ 72 q + 9 q^{5} - 102 q^{9} + 18 q^{11} - 225 q^{15} - 282 q^{21} - 1278 q^{23} + 441 q^{25} + 54 q^{27} + 1854 q^{29} - 1665 q^{31} - 45 q^{33} - 2673 q^{35} + 6951 q^{39} - 5472 q^{41} + 1260 q^{43} + 5553 q^{45} + 5103 q^{47} - 5904 q^{49} + 1899 q^{51} + 1107 q^{57} - 10944 q^{59} + 8352 q^{61} - 11985 q^{63} + 8757 q^{65} + 378 q^{67} + 5607 q^{69} - 19764 q^{71} + 6111 q^{73} - 3453 q^{75} - 5679 q^{77} - 5652 q^{79} - 20466 q^{81} - 20061 q^{83} + 26100 q^{85} + 40545 q^{87} + 15633 q^{89} - 6039 q^{91} + 40179 q^{93} + 48024 q^{95} - 37530 q^{97} + 12177 q^{99}+O(q^{100}) $ 72 * q + 9 * q^5 - 102 * q^9 + 18 * q^11 - 225 * q^15 - 282 * q^21 - 1278 * q^23 + 441 * q^25 + 54 * q^27 + 1854 * q^29 - 1665 * q^31 - 45 * q^33 - 2673 * q^35 + 6951 * q^39 - 5472 * q^41 + 1260 * q^43 + 5553 * q^45 + 5103 * q^47 - 5904 * q^49 + 1899 * q^51 + 1107 * q^57 - 10944 * q^59 + 8352 * q^61 - 11985 * q^63 + 8757 * q^65 + 378 * q^67 + 5607 * q^69 - 19764 * q^71 + 6111 * q^73 - 3453 * q^75 - 5679 * q^77 - 5652 * q^79 - 20466 * q^81 - 20061 * q^83 + 26100 * q^85 + 40545 * q^87 + 15633 * q^89 - 6039 * q^91 + 40179 * q^93 + 48024 * q^95 - 37530 * q^97 + 12177 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−8.09115	−	3.94122i	−0.899017	−	0.437914i
$3$	−8.09115	−	3.94122i	−0.899017	−	0.437914i
$4$		0			0
$5$	14.0198	+	38.5191i	0.560792	+	1.54076i	0.818480	+	0.574535i	$0.194817\pi$
$5$	14.0198	+	38.5191i	0.560792	+	1.54076i	−0.257688	+	0.966228i	$0.582961\pi$
$6$		0			0
$7$	−13.2730	−	75.2747i	−0.270877	−	1.53622i	−0.751762	−	0.659435i	$-0.770796\pi$
$7$	−13.2730	−	75.2747i	−0.270877	−	1.53622i	0.480885	−	0.876784i	$-0.340315\pi$
$8$		0			0
$9$	49.9335	+	63.7781i	0.616463	+	0.787384i
$10$		0			0
$11$	−14.4803	+	39.7844i	−0.119672	+	0.328797i	−0.985036	−	0.172347i	$-0.944865\pi$
$11$	−14.4803	+	39.7844i	−0.119672	+	0.328797i	0.865364	+	0.501144i	$0.167087\pi$
$12$		0			0
$13$	−99.4208	−	83.4239i	−0.588289	−	0.493633i	0.299369	−	0.954138i	$-0.403224\pi$
$13$	−99.4208	−	83.4239i	−0.588289	−	0.493633i	−0.887657	+	0.460505i	$0.847668\pi$
$14$		0			0
$15$	38.3759	−	366.919i	0.170560	−	1.63075i
$16$		0			0
$17$	25.0688	+	14.4735i	0.0867434	+	0.0500813i	0.542744	−	0.839898i	$-0.317385\pi$
$17$	25.0688	+	14.4735i	0.0867434	+	0.0500813i	−0.456001	+	0.889979i	$0.650719\pi$
$18$		0			0
$19$	−340.107	−	589.082i	−0.942124	−	1.63181i	−0.761409	−	0.648271i	$-0.775492\pi$
$19$	−340.107	−	589.082i	−0.942124	−	1.63181i	−0.180715	−	0.983536i	$-0.557841\pi$
$20$		0			0
$21$	−189.281	+	661.371i	−0.429208	+	1.49971i
$22$		0			0
$23$	−925.342	−	163.163i	−1.74923	−	0.308436i	−0.794800	−	0.606872i	$-0.792424\pi$
$23$	−925.342	−	163.163i	−1.74923	−	0.308436i	−0.954430	+	0.298436i	$0.903535\pi$
$24$		0			0
$25$	−808.388	+	678.318i	−1.29342	+	1.08531i
$26$		0			0
$27$	−152.656	−	712.837i	−0.209405	−	0.977829i
$28$		0			0
$29$	−743.151	−	885.653i	−0.883652	−	1.05310i	−0.998218	−	0.0596798i	$-0.980992\pi$
$29$	−743.151	−	885.653i	−0.883652	−	1.05310i	0.114566	−	0.993416i	$-0.463452\pi$
$30$		0			0
$31$	−216.511	+	1227.89i	−0.225297	+	1.27773i	0.636818	+	0.771014i	$0.280250\pi$
$31$	−216.511	+	1227.89i	−0.225297	+	1.27773i	−0.862116	+	0.506711i	$0.830861\pi$
$32$		0			0
$33$	273.962	−	264.832i	0.251572	−	0.243188i
$34$		0			0
$35$	2713.43	−	1566.60i	2.21504	−	1.27886i
$36$		0			0
$37$	951.756	−	1648.49i	0.695220	−	1.20416i	−0.274886	−	0.961477i	$-0.588640\pi$
$37$	951.756	−	1648.49i	0.695220	−	1.20416i	0.970106	−	0.242680i	$-0.0780265\pi$
$38$		0			0
$39$	475.637	+	1066.84i	0.312713	+	0.701404i
$40$		0			0
$41$	663.134	−	790.292i	0.394488	−	0.470132i	−0.531843	−	0.846843i	$-0.678500\pi$
$41$	663.134	−	790.292i	0.394488	−	0.470132i	0.926331	+	0.376711i	$0.122945\pi$
$42$		0			0
$43$	989.699	+	360.221i	0.535262	+	0.194819i	0.595486	−	0.803365i	$-0.296959\pi$
$43$	989.699	+	360.221i	0.535262	+	0.194819i	−0.0602245	+	0.998185i	$0.519182\pi$
$44$		0			0
$45$	−1756.61	+	2817.55i	−0.867464	+	1.39138i
$46$		0			0
$47$	747.698	−	131.839i	0.338478	−	0.0596828i	−0.00182574	−	0.999998i	$-0.500581\pi$
$47$	747.698	−	131.839i	0.338478	−	0.0596828i	0.340304	+	0.940315i	$0.389470\pi$
$48$		0			0
$49$	−3233.91	+	1177.05i	−1.34690	+	0.490232i
$50$		0			0
$51$	−145.793	−	215.909i	−0.0560525	−	0.0830100i
$52$		0			0
$53$		−	202.878i		−	0.0722244i	−0.999348	−	0.0361122i	$-0.988503\pi$
$53$		−	202.878i		−	0.0722244i	0.999348	−	0.0361122i	$-0.0114974\pi$
$54$		0			0
$55$	−1735.47			−0.573709
$56$		0			0
$57$	430.153	+	6106.79i	0.132395	+	1.87959i
$58$		0			0
$59$	1284.25	+	3528.44i	0.368931	+	1.01363i	0.975769	+	0.218804i	$0.0702154\pi$
$59$	1284.25	+	3528.44i	0.368931	+	1.01363i	−0.606838	+	0.794825i	$0.707562\pi$
$60$		0			0
$61$	−211.588	−	1199.97i	−0.0568631	−	0.322487i	0.943086	−	0.332548i	$-0.107908\pi$
$61$	−211.588	−	1199.97i	−0.0568631	−	0.322487i	−0.999949	+	0.0100614i	$0.996797\pi$
$62$		0			0
$63$	4138.11	−	4605.26i	1.04261	−	1.16031i
$64$		0			0
$65$	1819.55	−	4999.18i	0.430664	−	1.18324i
$66$		0			0
$67$	−2491.41	−	2090.54i	−0.555003	−	0.465702i	0.321628	−	0.946866i	$-0.395770\pi$
$67$	−2491.41	−	2090.54i	−0.555003	−	0.465702i	−0.876631	+	0.481164i	$0.840214\pi$
$68$		0			0
$69$	6844.03	+	4967.15i	1.43752	+	1.04330i
$70$		0			0
$71$	2946.68	+	1701.27i	0.584543	+	0.337486i	0.762937	−	0.646473i	$-0.223757\pi$
$71$	2946.68	+	1701.27i	0.584543	+	0.337486i	−0.178394	+	0.983959i	$0.557090\pi$
$72$		0			0
$73$	−1292.56	−	2238.78i	−0.242552	−	0.420112i	0.718889	−	0.695125i	$-0.244651\pi$
$73$	−1292.56	−	2238.78i	−0.242552	−	0.420112i	−0.961440	+	0.275013i	$0.911318\pi$
$74$		0			0
$75$	9214.19	−	2302.34i	1.63808	−	0.409305i
$76$		0			0
$77$	3186.96	+	561.947i	0.537520	+	0.0947793i
$78$		0			0
$79$	−5812.54	+	4877.30i	−0.931347	+	0.781493i	−0.976059	−	0.217507i	$-0.930207\pi$
$79$	−5812.54	+	4877.30i	−0.931347	+	0.781493i	0.0447118	+	0.999000i	$0.485763\pi$
$80$		0			0
$81$	−1574.28	+	6369.33i	−0.239946	+	0.970786i
$82$		0			0
$83$	−5284.48	−	6297.80i	−0.767090	−	0.914182i	0.231184	−	0.972910i	$-0.425740\pi$
$83$	−5284.48	−	6297.80i	−0.767090	−	0.914182i	−0.998274	+	0.0587278i	$0.981296\pi$
$84$		0			0
$85$	−206.046	+	1168.54i	−0.0285185	+	0.161736i
$86$		0			0
$87$	2522.40	+	10094.9i	0.333254	+	1.33371i
$88$		0			0
$89$	−11385.8	+	6573.58i	−1.43742	+	0.829892i	−0.997670	−	0.0682303i	$-0.978265\pi$
$89$	−11385.8	+	6573.58i	−1.43742	+	0.829892i	−0.439746	+	0.898122i	$0.644931\pi$
$90$		0			0
$91$	−4960.10	+	8591.15i	−0.598974	+	1.03745i
$92$		0			0
$93$	6591.22	−	9081.76i	0.762079	−	1.05004i
$94$		0			0
$95$	17922.7	−	21359.4i	1.98589	−	2.36669i
$96$		0			0
$97$	−3367.63	−	1225.72i	−0.357916	−	0.130271i	0.156803	−	0.987630i	$-0.449881\pi$
$97$	−3367.63	−	1225.72i	−0.357916	−	0.130271i	−0.514718	+	0.857359i	$0.672104\pi$
$98$		0			0
$99$	−3260.43	+	1063.05i	−0.332663	+	0.108463i
$100$		0			0
$101$	4409.08	−	777.440i	0.432221	−	0.0762122i	0.0466949	−	0.998909i	$-0.485131\pi$
$101$	4409.08	−	777.440i	0.432221	−	0.0762122i	0.385526	+	0.922697i	$0.374020\pi$
$102$		0			0
$103$	5907.83	−	2150.28i	0.556870	−	0.202684i	−0.0482263	−	0.998836i	$-0.515357\pi$
$103$	5907.83	−	2150.28i	0.556870	−	0.202684i	0.605096	+	0.796152i	$0.293135\pi$
$104$		0			0
$105$	−28129.1	+	1981.37i	−2.55139	+	0.179716i
$106$		0			0
$107$		−	1788.65i		−	0.156228i	−0.996944	−	0.0781138i	$-0.975110\pi$
$107$		−	1788.65i		−	0.156228i	0.996944	−	0.0781138i	$-0.0248898\pi$
$108$		0			0
$109$	1062.30			0.0894121			0.0447060	−	0.999000i	$-0.485765\pi$
$109$	1062.30			0.0894121			0.0447060	+	0.999000i	$0.485765\pi$
$110$		0			0
$111$	−14197.9	+	9587.11i	−1.15233	+	0.778111i
$112$		0			0
$113$	1289.08	+	3541.72i	0.100954	+	0.277369i	0.979879	−	0.199590i	$-0.0639611\pi$
$113$	1289.08	+	3541.72i	0.100954	+	0.277369i	−0.878925	+	0.476959i	$0.841739\pi$
$114$		0			0
$115$	−6688.23	−	37930.9i	−0.505726	−	2.86812i
$116$		0			0
$117$	356.186	−	10506.5i	0.0260198	−	0.767515i
$118$		0			0
$119$	756.751	−	2079.16i	0.0534391	−	0.146823i
$120$		0			0
$121$	9842.54	+	8258.87i	0.672259	+	0.564092i
$122$		0			0
$123$	−8480.24	+	3780.82i	−0.560529	+	0.249905i
$124$		0			0
$125$	−15274.5	−	8818.74i	−0.977568	−	0.564399i
$126$		0			0
$127$	3488.64	+	6042.49i	0.216296	+	0.374635i	0.953673	−	0.300846i	$-0.0972691\pi$
$127$	3488.64	+	6042.49i	0.216296	+	0.374635i	−0.737377	+	0.675482i	$0.763936\pi$
$128$		0			0
$129$	−6588.10	−	6815.23i	−0.395895	−	0.409544i
$130$		0			0
$131$	−16625.3	−	2931.49i	−0.968785	−	0.170823i	−0.333202	−	0.942856i	$-0.608129\pi$
$131$	−16625.3	−	2931.49i	−0.968785	−	0.170823i	−0.635583	+	0.772033i	$0.719240\pi$
$132$		0			0
$133$	−39828.8	+	33420.3i	−2.25161	+	1.88933i
$134$		0			0
$135$	25317.6	−	15874.0i	1.38917	−	0.871003i
$136$		0			0
$137$	11323.7	+	13495.0i	0.603317	+	0.719006i	0.978107	−	0.208104i	$-0.0667294\pi$
$137$	11323.7	+	13495.0i	0.603317	+	0.719006i	−0.374789	+	0.927110i	$0.622285\pi$
$138$		0			0
$139$	−2285.93	+	12964.2i	−0.118313	+	0.670989i	0.866743	+	0.498755i	$0.166209\pi$
$139$	−2285.93	+	12964.2i	−0.118313	+	0.670989i	−0.985056	+	0.172233i	$0.944902\pi$
$140$		0			0
$141$	−6569.35	−	1880.11i	−0.330434	−	0.0945683i
$142$		0			0
$143$	4758.62	−	2747.39i	0.232707	−	0.134353i
$144$		0			0
$145$	23695.7	−	41042.2i	1.12703	−	1.95207i
$146$		0			0
$147$	30805.1	+	3221.89i	1.42557	+	0.149099i
$148$		0			0
$149$	8675.93	−	10339.6i	0.390790	−	0.465726i	−0.534399	−	0.845232i	$-0.679462\pi$
$149$	8675.93	−	10339.6i	0.390790	−	0.465726i	0.925189	+	0.379507i	$0.123906\pi$
$150$		0			0
$151$	5618.82	+	2045.08i	0.246429	+	0.0896926i	0.462281	−	0.886733i	$-0.347031\pi$
$151$	5618.82	+	2045.08i	0.246429	+	0.0896926i	−0.215853	+	0.976426i	$0.569253\pi$
$152$		0			0
$153$	328.684	+	2321.55i	0.0140409	+	0.0991736i
$154$		0			0
$155$	−50332.8	+	8875.03i	−2.09502	+	0.369408i
$156$		0			0
$157$	7950.66	−	2893.80i	0.322555	−	0.117400i	−0.175668	−	0.984450i	$-0.556208\pi$
$157$	7950.66	−	2893.80i	0.322555	−	0.117400i	0.498223	+	0.867049i	$0.333986\pi$
$158$		0			0
$159$	−799.589	+	1641.52i	−0.0316280	+	0.0649310i
$160$		0			0
$161$			71820.5i			2.77075i
$162$		0			0
$163$	27677.0			1.04170			0.520852	−	0.853647i	$-0.325614\pi$
$163$	27677.0			1.04170			0.520852	+	0.853647i	$0.325614\pi$
$164$		0			0
$165$	14042.0	+	6839.88i	0.515775	+	0.251235i
$166$		0			0
$167$	3153.79	+	8664.97i	0.113084	+	0.310695i	0.983305	−	0.181967i	$-0.0582463\pi$
$167$	3153.79	+	8664.97i	0.113084	+	0.310695i	−0.870221	+	0.492661i	$0.836024\pi$
$168$		0			0
$169$	−2034.63	−	11539.0i	−0.0712380	−	0.404011i
$170$		0			0
$171$	20587.8	−	51106.3i	0.704073	−	1.74776i
$172$		0			0
$173$	−3872.82	+	10640.5i	−0.129400	+	0.355524i	−0.987426	−	0.158083i	$-0.949469\pi$
$173$	−3872.82	+	10640.5i	−0.129400	+	0.355524i	0.858026	+	0.513607i	$0.171691\pi$
$174$		0			0
$175$	61789.9	+	51847.9i	2.01763	+	1.69299i
$176$		0			0
$177$	3515.33	−	33610.7i	0.112207	−	1.07283i
$178$		0			0
$179$	17264.7	+	9967.80i	0.538833	+	0.311095i	0.744606	−	0.667505i	$-0.232638\pi$
$179$	17264.7	+	9967.80i	0.538833	+	0.311095i	−0.205773	+	0.978600i	$0.565971\pi$
$180$		0			0
$181$	−18973.5	−	32863.1i	−0.579150	−	1.00312i	−0.995577	−	0.0939484i	$-0.970051\pi$
$181$	−18973.5	−	32863.1i	−0.579150	−	1.00312i	0.416427	−	0.909169i	$-0.363282\pi$
$182$		0			0
$183$	−3017.37	+	10543.1i	−0.0901004	+	0.314822i
$184$		0			0
$185$	76841.8	+	13549.3i	2.24519	+	0.395888i
$186$		0			0
$187$	−938.825	+	787.768i	−0.0268473	+	0.0225276i
$188$		0			0
$189$	−51632.4	+	20952.6i	−1.44544	+	0.586564i
$190$		0			0
$191$	−43742.9	−	52130.7i	−1.19906	−	1.42898i	−0.875802	−	0.482671i	$-0.839667\pi$
$191$	−43742.9	−	52130.7i	−1.19906	−	1.42898i	−0.323257	−	0.946311i	$-0.604778\pi$
$192$		0			0
$193$	−4087.08	+	23179.0i	−0.109723	+	0.622272i	0.879505	+	0.475890i	$0.157874\pi$
$193$	−4087.08	+	23179.0i	−0.109723	+	0.622272i	−0.989228	+	0.146382i	$0.953237\pi$
$194$		0			0
$195$	−34425.2	+	33277.9i	−0.905330	+	0.875159i
$196$		0			0
$197$	−9679.76	+	5588.61i	−0.249420	+	0.144003i	−0.619499	−	0.784998i	$-0.712664\pi$
$197$	−9679.76	+	5588.61i	−0.249420	+	0.144003i	0.370078	+	0.929001i	$0.379331\pi$
$198$		0			0
$199$	18006.3	−	31187.8i	0.454693	−	0.787551i	−0.543978	−	0.839100i	$-0.683082\pi$
$199$	18006.3	−	31187.8i	0.454693	−	0.787551i	0.998670	+	0.0515490i	$0.0164158\pi$
$200$		0			0
$201$	11919.1	+	26734.1i	0.295019	+	0.661718i
$202$		0			0
$203$	−56803.5	+	67695.7i	−1.37842	+	1.64274i
$204$		0			0
$205$	39738.4	+	14463.6i	0.945588	+	0.344166i
$206$		0			0
$207$	−35799.4	−	67163.8i	−0.835478	−	1.56745i
$208$		0			0
$209$	28361.2	−	5000.84i	0.649279	−	0.114485i
$210$		0			0
$211$	−15217.5	+	5538.72i	−0.341805	+	0.124407i	−0.507219	−	0.861817i	$-0.669326\pi$
$211$	−15217.5	+	5538.72i	−0.341805	+	0.124407i	0.165413	+	0.986224i	$0.447104\pi$
$212$		0			0
$213$	−17137.0	−	25378.7i	−0.377725	−	0.559385i
$214$		0			0
$215$			43172.5i			0.933965i
$216$		0			0
$217$	95303.1			2.02389
$218$		0			0
$219$	1634.77	+	23208.5i	0.0340855	+	0.483904i
$220$		0			0
$221$	−1284.93	−	3530.31i	−0.0263084	−	0.0722816i
$222$		0			0
$223$	11489.0	+	65157.3i	0.231032	+	1.31025i	0.850811	+	0.525472i	$0.176111\pi$
$223$	11489.0	+	65157.3i	0.231032	+	1.31025i	−0.619779	+	0.784776i	$0.712778\pi$
$224$		0			0
$225$	−83627.4	−	17686.6i	−1.65190	−	0.349365i
$226$		0			0
$227$	21727.7	−	59696.4i	0.421660	−	1.15850i	−0.529097	−	0.848562i	$-0.677469\pi$
$227$	21727.7	−	59696.4i	0.421660	−	1.15850i	0.950756	−	0.309939i	$-0.100309\pi$
$228$		0			0
$229$	−44843.4	−	37628.1i	−0.855121	−	0.717532i	0.105790	−	0.994389i	$-0.466263\pi$
$229$	−44843.4	−	37628.1i	−0.855121	−	0.717532i	−0.960911	+	0.276856i	$0.910707\pi$
$230$		0			0
$231$	−23571.4	−	17107.3i	−0.441735	−	0.320596i
$232$		0			0
$233$	5317.36	+	3069.98i	0.0979454	+	0.0565488i	0.548173	−	0.836365i	$-0.315324\pi$
$233$	5317.36	+	3069.98i	0.0979454	+	0.0565488i	−0.450227	+	0.892914i	$0.648657\pi$
$234$		0			0
$235$	15560.9	+	26952.3i	0.281773	+	0.488045i
$236$		0			0
$237$	66252.6	−	16554.5i	1.17952	−	0.294726i
$238$		0			0
$239$	68273.2	+	12038.4i	1.19524	+	0.210753i	0.735639	−	0.677374i	$-0.236882\pi$
$239$	68273.2	+	12038.4i	1.19524	+	0.210753i	0.459599	+	0.888126i	$0.347993\pi$
$240$		0			0
$241$	61153.9	−	51314.3i	1.05291	−	0.883495i	0.0595115	−	0.998228i	$-0.481046\pi$
$241$	61153.9	−	51314.3i	1.05291	−	0.883495i	0.993396	+	0.114733i	$0.0366012\pi$
$242$		0			0
$243$	37840.7	−	45330.6i	0.640836	−	0.767678i
$244$		0			0
$245$	−90677.5	−	108065.i	−1.51066	−	1.80034i
$246$		0			0
$247$	−15329.9	+	86940.1i	−0.251272	+	1.42504i
$248$		0			0
$249$	17936.5	+	71783.8i	0.289294	+	1.15778i
$250$		0			0
$251$	−61753.5	+	35653.4i	−0.980199	+	0.565918i	−0.902330	−	0.431046i	$-0.858145\pi$
$251$	−61753.5	+	35653.4i	−0.980199	+	0.565918i	−0.0778687	+	0.996964i	$0.524812\pi$
$252$		0			0
$253$	19890.6	−	34451.5i	0.310747	−	0.538230i
$254$		0			0
$255$	6272.64	−	8642.80i	0.0964651	−	0.132915i
$256$		0			0
$257$	25901.3	−	30867.9i	0.392152	−	0.467349i	−0.533458	−	0.845826i	$-0.679108\pi$
$257$	25901.3	−	30867.9i	0.392152	−	0.467349i	0.925610	+	0.378478i	$0.123552\pi$
$258$		0			0
$259$	−136722.	−	49762.8i	−2.03817	−	0.741832i
$260$		0			0
$261$	19377.1	−	91620.6i	0.284451	−	1.34497i
$262$		0			0
$263$	−102694.	+	18107.8i	−1.48469	+	0.261790i	−0.856449	−	0.516232i	$-0.827334\pi$
$263$	−102694.	+	18107.8i	−1.48469	+	0.261790i	−0.628237	+	0.778022i	$0.716223\pi$
$264$		0			0
$265$	7814.69	−	2844.31i	0.111281	−	0.0405029i
$266$		0			0
$267$	118032.	−	8313.98i	1.65568	−	0.116624i
$268$		0			0
$269$			79054.1i			1.09250i	0.837624	+	0.546248i	$0.183944\pi$
$269$			79054.1i			1.09250i	−0.837624	+	0.546248i	$0.816056\pi$
$270$		0			0
$271$	−138085.			−1.88021			−0.940107	−	0.340878i	$-0.889276\pi$
$271$	−138085.			−1.88021			−0.940107	+	0.340878i	$0.889276\pi$
$272$		0			0
$273$	73992.6	−	49963.5i	0.992803	−	0.670390i
$274$		0			0
$275$	−15280.7	−	41983.5i	−0.202059	−	0.555154i
$276$		0			0
$277$	−15081.1	−	85529.3i	−0.196550	−	1.11469i	−0.910193	−	0.414183i	$-0.864067\pi$
$277$	−15081.1	−	85529.3i	−0.196550	−	1.11469i	0.713643	−	0.700510i	$-0.247044\pi$
$278$		0			0
$279$	−89123.8	+	47504.4i	−1.14495	+	0.610275i
$280$		0			0
$281$	−23913.1	+	65700.6i	−0.302847	+	0.832064i	0.691156	+	0.722706i	$0.257102\pi$
$281$	−23913.1	+	65700.6i	−0.302847	+	0.832064i	−0.994002	+	0.109358i	$0.965120\pi$
$282$		0			0
$283$	31541.7	+	26466.6i	0.393833	+	0.330465i	0.818104	−	0.575070i	$-0.195025\pi$
$283$	31541.7	+	26466.6i	0.393833	+	0.330465i	−0.424271	+	0.905535i	$0.639470\pi$
$284$		0			0
$285$	−229197.	+	102185.i	−2.82176	+	1.25805i
$286$		0			0
$287$	−68290.8	−	39427.7i	−0.829084	−	0.478672i
$288$		0			0
$289$	−41341.5	−	71605.6i	−0.494984	−	0.857337i
$290$		0			0
$291$	22417.2	+	23190.0i	0.264725	+	0.273852i
$292$		0			0
$293$	46384.3	+	8178.80i	0.540300	+	0.0952695i	0.437137	−	0.899395i	$-0.355992\pi$
$293$	46384.3	+	8178.80i	0.540300	+	0.0952695i	0.103163	+	0.994664i	$0.467104\pi$
$294$		0			0
$295$	−117908.	+	98936.1i	−1.35487	+	1.13687i
$296$		0			0
$297$	30570.3	+	4248.78i	0.346567	+	0.0481672i
$298$		0			0
$299$	78386.6	+	93417.5i	0.876797	+	1.04493i
$300$		0			0
$301$	13979.3	−	79280.5i	0.154295	−	0.875051i
$302$		0			0
$303$	−38738.6	−	11086.8i	−0.421948	−	0.120759i
$304$		0			0
$305$	43255.5	−	24973.6i	0.464988	−	0.268461i
$306$		0			0
$307$	38793.6	−	67192.5i	0.411608	−	0.712925i	−0.583458	−	0.812143i	$-0.698301\pi$
$307$	38793.6	−	67192.5i	0.411608	−	0.712925i	0.995066	+	0.0992180i	$0.0316341\pi$
$308$		0			0
$309$	−56275.9	−	5885.87i	−0.589394	−	0.0616444i
$310$		0			0
$311$	83544.6	−	99564.5i	0.863769	−	1.02940i	−0.135485	−	0.990779i	$-0.543259\pi$
$311$	83544.6	−	99564.5i	0.863769	−	1.02940i	0.999254	−	0.0386201i	$-0.0122962\pi$
$312$		0			0
$313$	−135922.	−	49471.7i	−1.38740	−	0.504973i	−0.462989	−	0.886364i	$-0.653223\pi$
$313$	−135922.	−	49471.7i	−1.38740	−	0.504973i	−0.924414	+	0.381391i	$0.875445\pi$
$314$		0			0
$315$	235406.	+	94831.4i	2.37244	+	0.955721i
$316$		0			0
$317$	127720.	−	22520.6i	1.27099	−	0.224110i	0.502839	−	0.864380i	$-0.332289\pi$
$317$	127720.	−	22520.6i	1.27099	−	0.224110i	0.768150	+	0.640270i	$0.221178\pi$
$318$		0			0
$319$	45996.3	−	16741.3i	0.452003	−	0.164516i
$320$		0			0
$321$	−7049.47	+	14472.2i	−0.0684142	+	0.140451i
$322$		0			0
$323$		−	19690.1i		−	0.188731i
$324$		0			0
$325$	136958.			1.29665
$326$		0			0
$327$	−8595.27	−	4186.78i	−0.0803830	−	0.0391548i
$328$		0			0
$329$	−19848.3	−	54532.9i	−0.183372	−	0.503810i
$330$		0			0
$331$	16860.1	+	95618.2i	0.153887	+	0.872739i	0.959796	+	0.280699i	$0.0905663\pi$
$331$	16860.1	+	95618.2i	0.153887	+	0.872739i	−0.805908	+	0.592040i	$0.798323\pi$
$332$		0			0
$333$	152662.	−	21613.8i	1.37671	−	0.194914i
$334$		0			0
$335$	45596.6	−	125276.i	0.406296	−	1.11629i
$336$		0			0
$337$	104364.	+	87572.1i	0.918951	+	0.771091i	0.973801	−	0.227403i	$-0.0730236\pi$
$337$	104364.	+	87572.1i	0.918951	+	0.771091i	−0.0548500	+	0.998495i	$0.517468\pi$
$338$		0			0
$339$	3528.56	−	33737.2i	0.0307042	−	0.293569i
$340$		0			0
$341$	−45715.9	−	26394.1i	−0.393150	−	0.226985i
$342$		0			0
$343$	39764.0	+	68873.4i	0.337989	+	0.585414i
$344$		0			0
$345$	−95378.4	+	333264.i	−0.801331	+	2.79995i
$346$		0			0
$347$	25871.4	+	4561.82i	0.214863	+	0.0378861i	0.280043	−	0.959987i	$-0.409651\pi$
$347$	25871.4	+	4561.82i	0.214863	+	0.0378861i	−0.0651805	+	0.997873i	$0.520762\pi$
$348$		0			0
$349$	69050.2	−	57940.0i	0.566910	−	0.475694i	−0.313709	−	0.949519i	$-0.601572\pi$
$349$	69050.2	−	57940.0i	0.566910	−	0.475694i	0.880619	+	0.473825i	$0.157127\pi$
$350$		0			0
$351$	−44290.5	+	83606.0i	−0.359498	+	0.678615i
$352$		0			0
$353$	35190.7	+	41938.7i	0.282409	+	0.336562i	0.888537	−	0.458805i	$-0.151722\pi$
$353$	35190.7	+	41938.7i	0.282409	+	0.336562i	−0.606128	+	0.795367i	$0.707278\pi$
$354$		0			0
$355$	−24219.4	+	137355.i	−0.192179	+	1.08990i
$356$		0			0
$357$	−14317.4	+	13840.2i	−0.112338	+	0.108594i
$358$		0			0
$359$	209960.	−	121221.i	1.62910	−	0.940562i	0.644739	−	0.764403i	$-0.276966\pi$
$359$	209960.	−	121221.i	1.62910	−	0.940562i	0.984362	−	0.176159i	$-0.0563673\pi$
$360$		0			0
$361$	−166185.	+	287840.i	−1.27520	+	2.20870i
$362$		0			0
$363$	−47087.4	−	105615.i	−0.357348	−	0.801519i
$364$		0			0
$365$	68114.2	−	81175.3i	0.511272	−	0.609310i
$366$		0			0
$367$	−229564.	−	83554.5i	−1.70440	−	0.620351i	−0.708086	−	0.706126i	$-0.750441\pi$
$367$	−229564.	−	83554.5i	−1.70440	−	0.620351i	−0.996315	+	0.0857748i	$0.972663\pi$
$368$		0			0
$369$	83516.0	+	2831.31i	0.613362	+	0.0207938i
$370$		0			0
$371$	−15271.6	+	2692.80i	−0.110952	+	0.0195639i
$372$		0			0
$373$	−23536.5	+	8566.57i	−0.169170	+	0.0615729i	−0.425217	−	0.905092i	$-0.639802\pi$
$373$	−23536.5	+	8566.57i	−0.169170	+	0.0615729i	0.256047	+	0.966664i	$0.417580\pi$
$374$		0			0
$375$	88831.7	+	131554.i	0.631692	+	0.935495i
$376$		0			0
$377$			150049.i			1.05572i
$378$		0			0
$379$	−55835.8			−0.388717			−0.194359	−	0.980931i	$-0.562263\pi$
$379$	−55835.8			−0.388717			−0.194359	+	0.980931i	$0.562263\pi$
$380$		0			0
$381$	−4412.28	−	62640.2i	−0.0303958	−	0.431522i
$382$		0			0
$383$	−74077.0	−	203525.i	−0.504993	−	1.38746i	−0.886344	−	0.463027i	$-0.846763\pi$
$383$	−74077.0	−	203525.i	−0.504993	−	1.38746i	0.381351	−	0.924430i	$-0.375459\pi$
$384$		0			0
$385$	23034.8	+	130637.i	0.155405	+	0.881343i
$386$		0			0
$387$	26445.0	+	81108.2i	0.176572	+	0.541555i
$388$		0			0
$389$	−21052.0	+	57839.9i	−0.139121	+	0.382233i	−0.989613	−	0.143755i	$-0.954082\pi$
$389$	−21052.0	+	57839.9i	−0.139121	+	0.382233i	0.850492	+	0.525988i	$0.176304\pi$
$390$		0			0
$391$	−20835.7	−	17483.2i	−0.136287	−	0.114358i
$392$		0			0
$393$	122964.	+	89243.2i	0.796148	+	0.577817i
$394$		0			0
$395$	−269360.	−	155515.i	−1.72639	−	0.996730i
$396$		0			0
$397$	−45460.4	−	78739.7i	−0.288438	−	0.499589i	0.684999	−	0.728544i	$-0.259803\pi$
$397$	−45460.4	−	78739.7i	−0.288438	−	0.499589i	−0.973437	+	0.228955i	$0.926469\pi$
$398$		0			0
$399$	453978.	−	113435.i	2.85160	−	0.712526i
$400$		0			0
$401$	−45351.9	−	7996.77i	−0.282038	−	0.0497308i	0.0308399	−	0.999524i	$-0.490182\pi$
$401$	−45351.9	−	7996.77i	−0.282038	−	0.0497308i	−0.312877	+	0.949794i	$0.601293\pi$
$402$		0			0
$403$	123961.	−	104016.i	0.763267	−	0.640457i
$404$		0			0
$405$	−267412.	+	28656.8i	−1.63031	+	0.174710i
$406$		0			0
$407$	51802.5	+	61735.8i	0.312724	+	0.372690i
$408$		0			0
$409$	20626.2	−	116977.i	0.123303	−	0.699284i	−0.858999	−	0.511977i	$-0.828913\pi$
$409$	20626.2	−	116977.i	0.123303	−	0.699284i	0.982302	−	0.187306i	$-0.0599757\pi$
$410$		0			0
$411$	−38434.6	−	153819.i	−0.227530	−	0.910599i
$412$		0			0
$413$	248557.	−	143504.i	1.45722	−	0.841327i
$414$		0			0
$415$	168498.	−	291847.i	0.978361	−	1.69457i
$416$		0			0
$417$	69590.5	−	95885.7i	0.400201	−	0.551419i
$418$		0			0
$419$	−52759.1	+	62875.9i	−0.300517	+	0.358143i	−0.895079	−	0.445907i	$-0.852881\pi$
$419$	−52759.1	+	62875.9i	−0.300517	+	0.358143i	0.594562	+	0.804050i	$0.297326\pi$
$420$		0			0
$421$	−87248.8	−	31756.0i	−0.492261	−	0.179168i	0.0839492	−	0.996470i	$-0.473247\pi$
$421$	−87248.8	−	31756.0i	−0.492261	−	0.179168i	−0.576210	+	0.817302i	$0.695469\pi$
$422$		0			0
$423$	45743.7	+	41103.5i	0.255653	+	0.229720i
$424$		0			0
$425$	−30083.0	+	5304.44i	−0.166549	+	0.0293671i
$426$		0			0
$427$	−87519.3	+	31854.4i	−0.480007	+	0.174708i
$428$		0			0
$429$	−49330.8	+	3474.78i	−0.268042	+	0.0188805i
$430$		0			0
$431$			220842.i			1.18885i	0.804151	+	0.594425i	$0.202620\pi$
$431$			220842.i			1.18885i	−0.804151	+	0.594425i	$0.797380\pi$
$432$		0			0
$433$	−289818.			−1.54579			−0.772894	−	0.634535i	$-0.781191\pi$
$433$	−289818.			−1.54579			−0.772894	+	0.634535i	$0.781191\pi$
$434$		0			0
$435$	−353482.	+	238689.i	−1.86805	+	1.26140i
$436$		0			0
$437$	218599.	+	600596.i	1.14468	+	3.14499i
$438$		0			0
$439$	26622.1	+	150981.i	0.138138	+	0.783419i	0.972623	+	0.232389i	$0.0746542\pi$
$439$	26622.1	+	150981.i	0.138138	+	0.783419i	−0.834485	+	0.551031i	$0.814235\pi$
$440$		0			0
$441$	−236550.	−	147478.i	−1.21632	−	0.758318i
$442$		0			0
$443$	13976.1	−	38399.0i	0.0712161	−	0.195665i	−0.898978	−	0.437994i	$-0.855689\pi$
$443$	13976.1	−	38399.0i	0.0712161	−	0.195665i	0.970194	+	0.242329i	$0.0779114\pi$
$444$		0			0
$445$	−412834.	−	346409.i	−2.08476	−	1.74932i
$446$		0			0
$447$	−110949.	+	49465.3i	−0.555274	+	0.247563i
$448$		0			0
$449$	22218.8	+	12828.0i	0.110212	+	0.0636307i	0.554093	−	0.832455i	$-0.313065\pi$
$449$	22218.8	+	12828.0i	0.110212	+	0.0636307i	−0.443881	+	0.896086i	$0.646399\pi$
$450$		0			0
$451$	21838.9	+	37826.1i	0.107369	+	0.185968i
$452$		0			0
$453$	−37402.6	−	38692.1i	−0.182266	−	0.188550i
$454$		0			0
$455$	−400463.	−	70612.4i	−1.93437	−	0.341082i
$456$		0			0
$457$	266155.	−	223330.i	1.27439	−	1.06934i	0.280397	−	0.959884i	$-0.409534\pi$
$457$	266155.	−	223330.i	1.27439	−	1.06934i	0.993992	−	0.109455i	$-0.0349106\pi$
$458$		0			0
$459$	6490.33	−	20079.5i	0.0308064	−	0.0953075i
$460$		0			0
$461$	−229113.	−	273047.i	−1.07807	−	1.28480i	−0.956347	−	0.292235i	$-0.905601\pi$
$461$	−229113.	−	273047.i	−1.07807	−	1.28480i	−0.121727	−	0.992564i	$-0.538843\pi$
$462$		0			0
$463$	31344.4	−	177763.i	0.146217	−	0.829239i	−0.820165	−	0.572128i	$-0.806118\pi$
$463$	31344.4	−	177763.i	0.146217	−	0.829239i	0.966382	−	0.257111i	$-0.0827708\pi$
$464$		0			0
$465$	442229.	+	126563.i	2.04523	+	0.585332i
$466$		0			0
$467$	36379.9	−	21003.9i	0.166812	−	0.0963090i	−0.414270	−	0.910154i	$-0.635963\pi$
$467$	36379.9	−	21003.9i	0.166812	−	0.0963090i	0.581082	+	0.813845i	$0.302630\pi$
$468$		0			0
$469$	−124296.	+	215288.i	−0.565083	+	0.978753i
$470$		0			0
$471$	−75735.1	−	7921.11i	−0.341394	−	0.0357062i
$472$		0			0
$473$	−28662.4	+	34158.5i	−0.128112	+	0.152678i
$474$		0			0
$475$	674523.	+	245506.i	2.98958	+	1.08812i
$476$		0			0
$477$	12939.2	−	10130.4i	0.0568683	−	0.0445237i
$478$		0			0
$479$	70992.0	−	12517.8i	0.309413	−	0.0545578i	−0.0167857	−	0.999859i	$-0.505343\pi$
$479$	70992.0	−	12517.8i	0.309413	−	0.0545578i	0.326199	+	0.945301i	$0.394232\pi$
$480$		0			0
$481$	−232148.	+	84494.9i	−1.00340	+	0.365208i
$482$		0			0
$483$	283061.	−	581111.i	1.21335	−	2.49095i
$484$		0			0
$485$		−	146902.i		−	0.624518i
$486$		0			0
$487$	109686.			0.462480			0.231240	−	0.972897i	$-0.425722\pi$
$487$	109686.			0.462480			0.231240	+	0.972897i	$0.425722\pi$
$488$		0			0
$489$	−223939.	−	109081.i	−0.936509	−	0.456176i
$490$		0			0
$491$	38757.8	+	106486.i	0.160767	+	0.441703i	0.993755	−	0.111588i	$-0.0355938\pi$
$491$	38757.8	+	106486.i	0.160767	+	0.441703i	−0.832988	+	0.553292i	$0.813372\pi$
$492$		0			0
$493$	−5811.44	−	32958.3i	−0.0239106	−	0.135603i
$494$		0			0
$495$	−86658.2	−	110685.i	−0.353671	−	0.451729i
$496$		0			0
$497$	88951.2	−	244391.i	0.360113	−	0.989403i
$498$		0			0
$499$	7589.09	+	6368.00i	0.0304781	+	0.0255742i	0.657899	−	0.753106i	$-0.271445\pi$
$499$	7589.09	+	6368.00i	0.0304781	+	0.0255742i	−0.627421	+	0.778680i	$0.715890\pi$
$500$		0			0
$501$	8632.76	−	82539.4i	0.0343933	−	0.328841i
$502$		0			0
$503$	−137742.	−	79525.2i	−0.544414	−	0.314318i	0.202452	−	0.979292i	$-0.435109\pi$
$503$	−137742.	−	79525.2i	−0.544414	−	0.314318i	−0.746866	+	0.664975i	$0.768442\pi$
$504$		0			0
$505$	91760.8	+	158934.i	0.359811	+	0.623211i
$506$		0			0
$507$	−29015.1	+	101382.i	−0.112878	+	0.394409i
$508$		0			0
$509$	−283839.	−	50048.5i	−1.09556	−	0.193177i	−0.403475	−	0.914991i	$-0.632198\pi$
$509$	−283839.	−	50048.5i	−1.09556	−	0.193177i	−0.692088	+	0.721814i	$0.743309\pi$
$510$		0			0
$511$	−151367.	+	127012.i	−0.579682	+	0.486411i
$512$		0			0
$513$	−368000.	+	332368.i	−1.39834	+	1.26295i
$514$		0			0
$515$	165653.	+	197418.i	0.624576	+	0.744341i
$516$		0			0
$517$	−5581.78	+	31655.8i	−0.0208829	+	0.118433i
$518$		0			0
$519$	73272.0	−	70830.1i	0.272022	−	0.262956i
$520$		0			0
$521$	−23240.6	+	13418.0i	−0.0856195	+	0.0494324i	−0.542198	−	0.840250i	$-0.682408\pi$
$521$	−23240.6	+	13418.0i	−0.0856195	+	0.0494324i	0.456579	+	0.889683i	$0.349075\pi$
$522$		0			0
$523$	−15425.6	+	26717.8i	−0.0563946	+	0.0976783i	−0.892844	−	0.450365i	$-0.851294\pi$
$523$	−15425.6	+	26717.8i	−0.0563946	+	0.0976783i	0.836450	+	0.548043i	$0.184627\pi$
$524$		0			0
$525$	−295607.	−	663037.i	−1.07250	−	2.40557i
$526$		0			0
$527$	−23199.6	+	27648.2i	−0.0835332	+	0.0995510i
$528$		0			0
$529$	566672.	+	206252.i	2.02498	+	0.737032i
$530$		0			0
$531$	−160910.	+	258095.i	−0.570682	+	0.915355i
$532$		0			0
$533$	−131859.	+	23250.2i	−0.464145	+	0.0818414i
$534$		0			0
$535$	68897.2	−	25076.5i	0.240710	−	0.0876112i
$536$		0			0
$537$	−100406.	−	148695.i	−0.348187	−	0.515642i
$538$		0			0
$539$		−	145703.i		−	0.501524i
$540$		0			0
$541$	179058.			0.611786			0.305893	−	0.952066i	$-0.401045\pi$
$541$	179058.			0.611786			0.305893	+	0.952066i	$0.401045\pi$
$542$		0			0
$543$	23996.9	+	340680.i	0.0813872	+	1.15544i
$544$		0			0
$545$	14893.3	+	40919.0i	0.0501416	+	0.137763i
$546$		0			0
$547$	−13041.3	−	73961.0i	−0.0435860	−	0.247188i	0.955228	−	0.295869i	$-0.0956093\pi$
$547$	−13041.3	−	73961.0i	−0.0435860	−	0.247188i	−0.998814	+	0.0486811i	$0.984498\pi$
$548$		0			0
$549$	65966.7	−	73413.6i	0.218867	−	0.243574i
$550$		0			0
$551$	−268972.	+	738994.i	−0.885939	+	2.43410i
$552$		0			0
$553$	444287.	+	372801.i	1.45282	+	1.21906i
$554$		0			0
$555$	−568338.	−	412480.i	−1.84510	−	1.33911i
$556$		0			0
$557$	−142602.	−	82331.2i	−0.459637	−	0.265371i	0.252255	−	0.967661i	$-0.418828\pi$
$557$	−142602.	−	82331.2i	−0.459637	−	0.265371i	−0.711891	+	0.702290i	$0.752161\pi$
$558$		0			0
$559$	−68345.6	−	118378.i	−0.218719	−	0.378833i
$560$		0			0
$561$	10700.9	−	2673.83i	0.0340014	−	0.00849588i
$562$		0			0
$563$	−187372.	−	33038.7i	−0.591136	−	0.104233i	−0.129925	−	0.991524i	$-0.541474\pi$
$563$	−187372.	−	33038.7i	−0.591136	−	0.104233i	−0.461211	+	0.887291i	$0.652585\pi$
$564$		0			0
$565$	−118351.	+	99308.5i	−0.370746	+	0.311093i
$566$		0			0
$567$	500345.	+	33963.8i	1.55634	+	0.105645i
$568$		0			0
$569$	277003.	+	330120.i	0.855579	+	1.01964i	0.999548	+	0.0300575i	$0.00956905\pi$
$569$	277003.	+	330120.i	0.855579	+	1.01964i	−0.143969	+	0.989582i	$0.545987\pi$
$570$		0			0
$571$	69768.2	−	395675.i	0.213986	−	1.21357i	−0.668671	−	0.743558i	$-0.733137\pi$
$571$	69768.2	−	395675.i	0.213986	−	1.21357i	0.882657	−	0.470017i	$-0.155752\pi$
$572$		0			0
$573$	148472.	+	594198.i	0.452203	+	1.80976i
$574$		0			0
$575$	858711.	−	495777.i	2.59724	−	1.49952i
$576$		0			0
$577$	−194165.	+	336303.i	−0.583202	+	1.01014i	0.411895	+	0.911231i	$0.364867\pi$
$577$	−194165.	+	336303.i	−0.583202	+	1.01014i	−0.995097	+	0.0989042i	$0.968466\pi$
$578$		0			0
$579$	124423.	−	171437.i	0.371144	−	0.511384i
$580$		0			0
$581$	−403924.	+	481378.i	−1.19660	+	1.42605i
$582$		0			0
$583$	8071.40	+	2937.75i	0.0237472	+	0.00864326i
$584$		0			0
$585$	409695.	−	133579.i	1.19715	−	0.390326i
$586$		0			0
$587$	306677.	−	54075.5i	0.890032	−	0.156937i	0.290108	−	0.956994i	$-0.406309\pi$
$587$	306677.	−	54075.5i	0.890032	−	0.156937i	0.599924	+	0.800057i	$0.295198\pi$
$588$		0			0
$589$	796967.	−	290072.i	2.29726	−	0.836134i
$590$		0			0
$591$	100346.	−	7068.24i	0.287294	−	0.0202365i
$592$		0			0
$593$			665766.i			1.89327i	0.322311	+	0.946634i	$0.395540\pi$
$593$			665766.i			1.89327i	−0.322311	+	0.946634i	$0.604460\pi$
$594$		0			0
$595$	90696.7			0.256187
$596$		0			0
$597$	−268610.	+	181378.i	−0.753655	+	0.508905i
$598$		0			0
$599$	−157004.	−	431365.i	−0.437580	−	1.20224i	−0.941062	−	0.338235i	$-0.890170\pi$
$599$	−157004.	−	431365.i	−0.437580	−	1.20224i	0.503482	−	0.864006i	$-0.332052\pi$
$600$		0			0
$601$	−77878.2	−	441669.i	−0.215609	−	1.22278i	−0.879846	−	0.475258i	$-0.842355\pi$
$601$	−77878.2	−	441669.i	−0.215609	−	1.22278i	0.664237	−	0.747522i	$-0.268756\pi$
$602$		0			0
$603$	8925.73	−	263285.i	0.0245476	−	0.724088i
$604$		0			0
$605$	−180134.	+	494913.i	−0.492135	+	1.35213i
$606$		0			0
$607$	−223553.	−	187584.i	−0.606742	−	0.509117i	0.286863	−	0.957972i	$-0.407388\pi$
$607$	−223553.	−	187584.i	−0.606742	−	0.509117i	−0.893605	+	0.448855i	$0.851832\pi$
$608$		0			0
$609$	726410.	−	323862.i	1.95861	−	0.873222i
$610$		0			0
$611$	−85335.3	−	49268.4i	−0.228584	−	0.131973i
$612$		0			0
$613$	47539.3	+	82340.6i	0.126512	+	0.219125i	0.922323	−	0.386420i	$-0.126288\pi$
$613$	47539.3	+	82340.6i	0.126512	+	0.219125i	−0.795811	+	0.605545i	$0.792955\pi$
$614$		0			0
$615$	−264525.	−	273645.i	−0.699385	−	0.723497i
$616$		0			0
$617$	42137.2	+	7429.92i	0.110687	+	0.0195170i	0.228717	−	0.973493i	$-0.426547\pi$
$617$	42137.2	+	7429.92i	0.110687	+	0.0195170i	−0.118030	+	0.993010i	$0.537658\pi$
$618$		0			0
$619$	−103830.	+	87124.1i	−0.270984	+	0.227382i	−0.768145	−	0.640276i	$-0.778820\pi$
$619$	−103830.	+	87124.1i	−0.270984	+	0.227382i	0.497161	+	0.867658i	$0.334376\pi$
$620$		0			0
$621$	24951.0	+	684526.i	0.0647000	+	1.77504i
$622$		0			0
$623$	645947.	+	769809.i	1.66426	+	1.98339i
$624$		0			0
$625$	11015.1	−	62469.8i	0.0281987	−	0.159923i
$626$		0			0
$627$	−249184.	−	71315.1i	−0.633848	−	0.181404i
$628$		0			0
$629$	47718.8	−	27550.5i	0.120611	−	0.0696351i
$630$		0			0
$631$	101308.	−	175471.i	0.254441	−	0.440705i	−0.710303	−	0.703896i	$-0.751442\pi$
$631$	101308.	−	175471.i	0.254441	−	0.440705i	0.964744	+	0.263192i	$0.0847752\pi$
$632$		0			0
$633$	144957.	+	15160.9i	0.361768	+	0.0378372i
$634$		0			0
$635$	−183841.	+	219094.i	−0.455928	+	0.543353i
$636$		0			0
$637$	419712.	+	152763.i	1.03436	+	0.376477i
$638$		0			0
$639$	38634.7	+	272884.i	0.0946185	+	0.668307i
$640$		0			0
$641$	584996.	−	103151.i	1.42376	−	0.251047i	0.591891	−	0.806018i	$-0.298382\pi$
$641$	584996.	−	103151.i	1.42376	−	0.251047i	0.831870	+	0.554971i	$0.187270\pi$
$642$		0			0
$643$	−239103.	+	87026.4i	−0.578313	+	0.210489i	−0.614582	−	0.788853i	$-0.710675\pi$
$643$	−239103.	+	87026.4i	−0.578313	+	0.210489i	0.0362682	+	0.999342i	$0.488453\pi$
$644$		0			0
$645$	170153.	−	349316.i	0.408996	−	0.839651i
$646$		0			0
$647$			582535.i			1.39160i	0.718237	+	0.695798i	$0.244949\pi$
$647$			582535.i			1.39160i	−0.718237	+	0.695798i	$0.755051\pi$
$648$		0			0
$649$	−158973.			−0.377429
$650$		0			0
$651$	−771112.	−	375611.i	−1.81951	−	0.886290i
$652$		0			0
$653$	−193842.	−	532577.i	−0.454592	−	1.24898i	−0.929460	−	0.368924i	$-0.879726\pi$
$653$	−193842.	−	532577.i	−0.454592	−	1.24898i	0.474868	−	0.880057i	$-0.342496\pi$
$654$		0			0
$655$	−120165.	−	681491.i	−0.280089	−	1.58846i
$656$		0			0
$657$	78242.8	−	194227.i	0.181265	−	0.449965i
$658$		0			0
$659$	−88685.0	+	243660.i	−0.204211	+	0.561065i	−0.998946	−	0.0458907i	$-0.985387\pi$
$659$	−88685.0	+	243660.i	−0.204211	+	0.561065i	0.794735	+	0.606956i	$0.207610\pi$
$660$		0			0
$661$	−515464.	−	432525.i	−1.17976	−	0.989939i	−0.999981	−	0.00623300i	$-0.998016\pi$
$661$	−515464.	−	432525.i	−1.17976	−	0.989939i	−0.179783	−	0.983706i	$-0.557540\pi$
$662$		0			0
$663$	−3517.18	+	33628.4i	−0.00800144	+	0.0765032i
$664$		0			0
$665$	−1.84571e6	−	1.06562e6i	−4.17369	−	2.40968i
$666$		0			0
$667$	543164.	+	940787.i	1.22090	+	2.11466i
$668$		0			0
$669$	163840.	−	572479.i	0.366073	−	1.27911i
$670$		0			0
$671$	50804.1	+	8958.14i	0.112838	+	0.0198963i
$672$		0			0
$673$	−94255.5	+	79089.7i	−0.208102	+	0.174618i	−0.740881	−	0.671636i	$-0.765592\pi$
$673$	−94255.5	+	79089.7i	−0.208102	+	0.174618i	0.532779	+	0.846254i	$0.321148\pi$
$674$		0			0
$675$	606936.	+	472699.i	1.33209	+	1.03747i
$676$		0			0
$677$	−438550.	−	522644.i	−0.956846	−	1.14033i	−0.990027	−	0.140876i	$-0.955008\pi$
$677$	−438550.	−	522644.i	−0.956846	−	1.14033i	0.0331810	−	0.999449i	$-0.489436\pi$
$678$		0			0
$679$	−47567.1	+	269766.i	−0.103173	+	0.585124i
$680$		0			0
$681$	−411079.	+	397379.i	−0.886403	+	0.856862i
$682$		0			0
$683$	619865.	−	357879.i	1.32879	−	0.767177i	0.343676	−	0.939088i	$-0.388328\pi$
$683$	619865.	−	357879.i	1.32879	−	0.767177i	0.985112	+	0.171912i	$0.0549943\pi$
$684$		0			0
$685$	−361060.	+	625375.i	−0.769482	+	1.33278i
$686$		0			0
$687$	214534.	+	481193.i	0.454552	+	1.01954i
$688$		0			0
$689$	−16924.9	+	20170.3i	−0.0356523	+	0.0424888i
$690$		0			0
$691$	−347045.	−	126314.i	−0.726824	−	0.264542i	−0.0480040	−	0.998847i	$-0.515286\pi$
$691$	−347045.	−	126314.i	−0.726824	−	0.264542i	−0.678820	+	0.734305i	$0.737508\pi$
$692$		0			0
$693$	123296.	+	231318.i	0.256734	+	0.481663i
$694$		0			0
$695$	−531416.	+	93703.1i	−1.10018	+	0.193992i
$696$		0			0
$697$	28062.3	−	10213.8i	0.0577640	−	0.0210244i
$698$		0			0
$699$	−30924.1	−	45796.6i	−0.0632911	−	0.0937300i
$700$		0			0
$701$		−	641748.i		−	1.30596i	−0.757377	−	0.652978i	$-0.773519\pi$
$701$		−	641748.i		−	1.30596i	0.757377	−	0.652978i	$-0.226481\pi$
$702$		0			0
$703$	−1.29480e6			−2.61993
$704$		0			0
$705$	−19680.8	−	279404.i	−0.0395972	−	0.562153i
$706$		0			0
$707$	−117043.	−	321574.i	−0.234157	−	0.643341i
$708$		0			0
$709$	136757.	+	775588.i	0.272055	+	1.54290i	0.748163	+	0.663515i	$0.230936\pi$
$709$	136757.	+	775588.i	0.272055	+	1.54290i	−0.476107	+	0.879387i	$0.657953\pi$
$710$		0			0
$711$	−601305.	−	127172.i	−1.18948	−	0.251565i
$712$		0			0
$713$	400693.	−	1.10090e6i	0.788194	−	2.16554i
$714$		0			0
$715$	172542.	+	144780.i	0.337507	+	0.283202i
$716$		0			0
$717$	−504963.	−	366485.i	−0.982248	−	0.712882i
$718$		0			0
$719$	301133.	+	173859.i	0.582506	+	0.336310i	0.762128	−	0.647426i	$-0.224154\pi$
$719$	301133.	+	173859.i	0.582506	+	0.336310i	−0.179623	+	0.983736i	$0.557488\pi$
$720$		0			0
$721$	−240276.	−	416170.i	−0.462210	−	0.800571i
$722$		0			0
$723$	−697047.	+	174170.i	−1.33348	+	0.333194i
$724$		0			0
$725$	1.20151e6	+	211858.i	2.28587	+	0.403060i
$726$		0			0
$727$	108810.	−	91302.1i	0.205872	−	0.172747i	−0.534022	−	0.845471i	$-0.679320\pi$
$727$	108810.	−	91302.1i	0.205872	−	0.172747i	0.739894	+	0.672723i	$0.234876\pi$
$728$		0			0
$729$	−484833.	+	217638.i	−0.912299	+	0.409525i
$730$		0			0
$731$	19596.9	+	23354.7i	0.0366736	+	0.0437059i
$732$		0			0
$733$	134924.	−	765191.i	0.251120	−	1.42417i	−0.554720	−	0.832037i	$-0.687175\pi$
$733$	134924.	−	765191.i	0.251120	−	1.42417i	0.805840	−	0.592134i	$-0.201714\pi$
$734$		0			0
$735$	307777.	+	1.23175e6i	0.569719	+	2.28007i
$736$		0			0
$737$	119247.	−	68847.4i	0.219540	−	0.126751i
$738$		0			0
$739$	−69727.6	+	120772.i	−0.127678	+	0.221145i	−0.922777	−	0.385335i	$-0.874086\pi$
$739$	−69727.6	+	120772.i	−0.127678	+	0.221145i	0.795099	+	0.606480i	$0.207419\pi$
$740$		0			0
$741$	466686.	−	643027.i	0.849941	−	1.17110i
$742$		0			0
$743$	105774.	−	126057.i	0.191603	−	0.228344i	−0.661687	−	0.749780i	$-0.730159\pi$
$743$	105774.	−	126057.i	0.191603	−	0.228344i	0.853290	+	0.521436i	$0.174604\pi$
$744$		0			0
$745$	519906.	+	189230.i	0.936725	+	0.340940i
$746$		0			0
$747$	137789.	−	651506.i	0.246929	−	1.16755i
$748$		0			0
$749$	−134640.	+	23740.7i	−0.240000	+	0.0423184i
$750$		0			0
$751$	404142.	−	147096.i	0.716563	−	0.260807i	0.0420967	−	0.999114i	$-0.486596\pi$
$751$	404142.	−	147096.i	0.716563	−	0.260807i	0.674466	+	0.738306i	$0.264374\pi$
$752$		0			0
$753$	640175.	−	45092.9i	1.12904	−	0.0795277i
$754$		0			0
$755$			245103.i			0.429987i
$756$		0			0
$757$	685545.			1.19631			0.598156	−	0.801380i	$-0.295900\pi$
$757$	685545.			1.19631			0.598156	+	0.801380i	$0.295900\pi$
$758$		0			0
$759$	−296719.	+	200359.i	−0.515065	+	0.347797i
$760$		0			0
$761$	−45745.1	−	125684.i	−0.0789906	−	0.217025i	0.893911	−	0.448245i	$-0.147951\pi$
$761$	−45745.1	−	125684.i	−0.0789906	−	0.217025i	−0.972902	+	0.231220i	$0.925728\pi$
$762$		0			0
$763$	−14099.9	−	79964.7i	−0.0242197	−	0.137357i
$764$		0			0
$765$	−84816.1	+	45208.3i	−0.144929	+	0.0772495i
$766$		0			0
$767$	166676.	−	457938.i	0.283323	−	0.778423i
$768$		0			0
$769$	−653305.	−	548188.i	−1.10475	−	0.926994i	−0.107013	−	0.994258i	$-0.534129\pi$
$769$	−653305.	−	548188.i	−1.10475	−	0.926994i	−0.997735	+	0.0672641i	$0.978573\pi$
$770$		0			0
$771$	−331228.	+	147674.i	−0.557210	+	0.248426i
$772$		0			0
$773$	430687.	+	248657.i	0.720780	+	0.416143i	0.815040	−	0.579405i	$-0.196715\pi$
$773$	430687.	+	248657.i	0.720780	+	0.416143i	−0.0942596	+	0.995548i	$0.530048\pi$
$774$		0			0
$775$	−657877.	−	1.13948e6i	−1.09532	−	1.89715i
$776$		0			0
$777$	910115.	+	941492.i	1.50749	+	1.55946i
$778$		0			0
$779$	−691084.	−	121857.i	−1.13882	−	0.200805i
$780$		0			0
$781$	−110353.	+	92597.1i	−0.180918	+	0.151808i
$782$		0			0
$783$	−517880.	+	664947.i	−0.844706	+	1.08458i
$784$		0			0
$785$	222933.	+	265682.i	0.361773	+	0.431144i
$786$		0			0
$787$	−19610.3	+	111216.i	−0.0316617	+	0.179563i	−0.996538	−	0.0831437i	$-0.973504\pi$
$787$	−19610.3	+	111216.i	−0.0316617	+	0.179563i	0.964876	+	0.262706i	$0.0846150\pi$
$788$		0			0
$789$	902281.	+	258228.i	1.44940	+	0.414810i
$790$		0			0
$791$	249492.	−	144044.i	0.398753	−	0.230220i
$792$		0			0
$793$	−79070.3	+	136954.i	−0.125738	+	0.217785i
$794$		0			0
$795$	−74439.9	−	7785.64i	−0.117780	−	0.0123186i
$796$		0			0
$797$	369073.	−	439844.i	0.581026	−	0.692440i	−0.392829	−	0.919612i	$-0.628503\pi$
$797$	369073.	−	439844.i	0.581026	−	0.692440i	0.973855	+	0.227172i	$0.0729479\pi$
$798$		0			0
$799$	20652.1	+	7516.75i	0.0323497	+	0.0117743i
$800$		0			0
$801$	−987782.	−	397920.i	−1.53956	−	0.620199i
$802$		0			0
$803$	107785.	−	19005.4i	0.167158	−	0.0294745i
$804$		0			0
$805$	−2.76646e6	+	1.00691e6i	−4.26907	+	1.55381i
$806$		0			0
$807$	311570.	−	639639.i	0.478419	−	0.982172i
$808$		0			0
$809$			1.09250e6i			1.66926i	0.550809	+	0.834631i	$0.314319\pi$
$809$			1.09250e6i			1.66926i	−0.550809	+	0.834631i	$0.685681\pi$
$810$		0			0
$811$	358630.			0.545261			0.272631	−	0.962119i	$-0.412106\pi$
$811$	358630.			0.545261			0.272631	+	0.962119i	$0.412106\pi$
$812$		0			0
$813$	1.11727e6	+	544223.i	1.69035	+	0.823371i
$814$		0			0
$815$	388026.	+	1.06609e6i	0.584179	+	1.60502i
$816$		0			0
$817$	−124404.	−	705528.i	−0.186375	−	1.05699i
$818$		0			0
$819$	−795603.	+	112641.i	−1.18612	+	0.167930i
$820$		0			0
$821$	320082.	−	879418.i	0.474870	−	1.30470i	−0.438926	−	0.898523i	$-0.644641\pi$
$821$	320082.	−	879418.i	0.474870	−	1.30470i	0.913797	−	0.406172i	$-0.133137\pi$
$822$		0			0
$823$	50712.9	+	42553.2i	0.0748718	+	0.0628249i	0.679456	−	0.733717i	$-0.262216\pi$
$823$	50712.9	+	42553.2i	0.0748718	+	0.0628249i	−0.604584	+	0.796542i	$0.706660\pi$
$824$		0			0
$825$	−41827.4	+	399920.i	−0.0614544	+	0.587577i
$826$		0			0
$827$	−91992.0	−	53111.6i	−0.134505	−	0.0776566i	0.431237	−	0.902238i	$-0.358077\pi$
$827$	−91992.0	−	53111.6i	−0.134505	−	0.0776566i	−0.565743	+	0.824582i	$0.691410\pi$
$828$		0			0
$829$	328215.	+	568486.i	0.477584	+	0.827200i	0.999670	−	0.0256931i	$-0.00817927\pi$
$829$	328215.	+	568486.i	0.477584	+	0.827200i	−0.522086	+	0.852893i	$0.674846\pi$
$830$		0			0
$831$	−215066.	+	751469.i	−0.311437	+	1.08820i
$832$		0			0
$833$	−98106.3	−	17298.8i	−0.141386	−	0.0249302i
$834$		0			0
$835$	−289551.	+	242962.i	−0.415291	+	0.348470i
$836$		0			0
$837$	908340.	−	33109.0i	1.29658	−	0.0472601i
$838$		0			0
$839$	−796327.	−	949026.i	−1.13127	−	1.34820i	−0.929524	−	0.368762i	$-0.879782\pi$
$839$	−796327.	−	949026.i	−1.13127	−	1.34820i	−0.201749	−	0.979437i	$-0.564663\pi$
$840$		0			0
$841$	−109290.	+	619813.i	−0.154521	+	0.876332i
$842$		0			0
$843$	452425.	−	437347.i	0.636637	−	0.615419i
$844$		0			0
$845$	415945.	−	240146.i	0.582535	−	0.336327i
$846$		0			0
$847$	491044.	−	850514.i	0.684469	−	1.18554i
$848$		0			0
$849$	−150898.	−	338458.i	−0.209348	−	0.469559i
$850$		0			0
$851$	−1.14967e6	+	1.37013e6i	−1.58751	+	1.89192i
$852$		0			0
$853$	−249307.	−	90740.3i	−0.342638	−	0.124710i	0.164969	−	0.986299i	$-0.447248\pi$
$853$	−249307.	−	90740.3i	−0.342638	−	0.124710i	−0.507607	+	0.861589i	$0.669470\pi$
$854$		0			0
$855$	2.25721e6	+	76522.4i	3.08773	+	0.104678i
$856$		0			0
$857$	−904.918	+	159.561i	−0.00123210	+	0.000217253i	−0.174264	−	0.984699i	$-0.555755\pi$
$857$	−904.918	+	159.561i	−0.00123210	+	0.000217253i	0.173032	+	0.984916i	$0.444644\pi$
$858$		0			0
$859$	1.03022e6	−	374970.i	1.39619	−	0.508172i	0.469145	−	0.883121i	$-0.344562\pi$
$859$	1.03022e6	−	374970.i	1.39619	−	0.508172i	0.927045	+	0.374949i	$0.122340\pi$
$860$		0			0
$861$	397158.	+	588165.i	0.535744	+	0.793401i
$862$		0			0
$863$		−	1.18306e6i		−	1.58849i	−0.607597	−	0.794246i	$-0.707866\pi$
$863$		−	1.18306e6i		−	1.58849i	0.607597	−	0.794246i	$-0.292134\pi$
$864$		0			0
$865$	−464157.			−0.620345
$866$		0			0
$867$	52287.0	+	742308.i	0.0695594	+	0.987521i
$868$		0			0
$869$	−109873.	−	301873.i	−0.145496	−	0.399747i
$870$		0			0
$871$	73296.6	+	415686.i	0.0966157	+	0.547935i
$872$		0			0
$873$	−89983.8	−	275985.i	−0.118069	−	0.362124i
$874$		0			0
$875$	−461090.	+	1.26683e6i	−0.602240	+	1.65464i
$876$		0			0
$877$	259766.	+	217970.i	0.337741	+	0.283398i	0.795845	−	0.605500i	$-0.207027\pi$
$877$	259766.	+	217970.i	0.337741	+	0.283398i	−0.458104	+	0.888898i	$0.651471\pi$
$878$		0			0
$879$	−343068.	−	248987.i	−0.444020	−	0.322254i
$880$		0			0
$881$	201038.	+	116069.i	0.259016	+	0.149543i	0.623885	−	0.781516i	$-0.285553\pi$
$881$	201038.	+	116069.i	0.259016	+	0.149543i	−0.364870	+	0.931059i	$0.618887\pi$
$882$		0			0
$883$	−205775.	−	356414.i	−0.263920	−	0.457123i	0.703360	−	0.710834i	$-0.251682\pi$
$883$	−205775.	−	356414.i	−0.263920	−	0.457123i	−0.967280	+	0.253711i	$0.918349\pi$
$884$		0			0
$885$	1.34394e6	−	335808.i	1.71590	−	0.428750i
$886$		0			0
$887$	804847.	+	141916.i	1.02298	+	0.180379i	0.659879	−	0.751372i	$-0.270608\pi$
$887$	804847.	+	141916.i	1.02298	+	0.180379i	0.363099	+	0.931751i	$0.381719\pi$
$888$		0			0
$889$	408542.	−	342808.i	0.516932	−	0.433758i
$890$		0			0
$891$	−230604.	−	154862.i	−0.290477	−	0.195070i
$892$		0			0
$893$	−331962.	−	395616.i	−0.416279	−	0.496102i
$894$		0			0
$895$	−141902.	+	804768.i	−0.177151	+	1.00467i
$896$		0			0
$897$	−266059.	−	1.06479e6i	−0.330668	−	1.32337i
$898$		0			0
$899$	1.24839e6	−	720757.i	1.54465	−	0.891805i
$900$		0			0
$901$	2936.36	−	5085.92i	0.00361709	−	0.00626499i
$902$		0			0
$903$	−425571.	+	586375.i	−0.521911	+	0.719118i
$904$		0			0
$905$	999853.	−	1.19158e6i	1.22078	−	1.45487i
$906$		0			0
$907$	−1.10928e6	−	403746.i	−1.34843	−	0.490788i	−0.435972	−	0.899960i	$-0.643595\pi$
$907$	−1.10928e6	−	403746.i	−1.34843	−	0.490788i	−0.912457	+	0.409172i	$0.865818\pi$
$908$		0			0
$909$	269745.	+	242382.i	0.326457	+	0.293341i
$910$		0			0
$911$	760255.	−	134053.i	0.916056	−	0.161525i	0.304304	−	0.952575i	$-0.401576\pi$
$911$	760255.	−	134053.i	0.916056	−	0.161525i	0.611752	+	0.791049i	$0.290465\pi$
$912$		0			0
$913$	327075.	−	119046.i	0.392380	−	0.142814i
$914$		0			0
$915$	−448413.	+	31585.5i	−0.535594	+	0.0377264i
$916$		0			0
$917$			1.29038e6i			1.53454i
$918$		0			0
$919$	116357.			0.137772			0.0688859	−	0.997625i	$-0.478056\pi$
$919$	116357.			0.137772			0.0688859	+	0.997625i	$0.478056\pi$
$920$		0			0
$921$	−578705.	+	390771.i	−0.682242	+	0.460683i
$922$		0			0
$923$	−151035.	−	414965.i	−0.177286	−	0.487089i
$924$		0			0
$925$	348812.	+	1.97821e6i	0.407670	+	2.31201i
$926$		0			0
$927$	432139.	+	269419.i	0.502880	+	0.313523i
$928$		0			0
$929$	−220204.	+	605005.i	−0.255149	+	0.701015i	0.744301	+	0.667844i	$0.232783\pi$
$929$	−220204.	+	605005.i	−0.255149	+	0.701015i	−0.999450	+	0.0331711i	$0.989439\pi$
$930$		0			0
$931$	1.79325e6	+	1.50472e6i	2.06891	+	1.73602i
$932$		0			0
$933$	−1.06838e6	+	476324.i	−1.22733	+	0.547192i
$934$		0			0
$935$	−43506.2	−	25118.3i	−0.0497655	−	0.0287321i
$936$		0			0
$937$	−52426.8	−	90805.9i	−0.0597137	−	0.103427i	0.834623	−	0.550821i	$-0.185685\pi$
$937$	−52426.8	−	90805.9i	−0.0597137	−	0.103427i	−0.894337	+	0.447394i	$0.852352\pi$
$938$		0			0
$939$	904790.	+	935984.i	1.02616	+	1.06154i
$940$		0			0
$941$	−1.09700e6	−	193431.i	−1.23888	−	0.218448i	−0.484444	−	0.874822i	$-0.660978\pi$
$941$	−1.09700e6	−	193431.i	−1.23888	−	0.218448i	−0.754435	+	0.656374i	$0.772089\pi$
$942$		0			0
$943$	−742572.	+	623092.i	−0.835056	+	0.700695i
$944$		0			0
$945$	−1.53095e6	−	1.69508e6i	−1.71434	−	1.89813i
$946$		0			0
$947$	138946.	+	165590.i	0.154934	+	0.184643i	0.837928	−	0.545781i	$-0.183767\pi$
$947$	138946.	+	165590.i	0.154934	+	0.184643i	−0.682994	+	0.730424i	$0.739322\pi$
$948$		0			0
$949$	−58260.4	+	330411.i	−0.0646906	+	0.366878i
$950$		0			0
$951$	−1.12216e6	−	321157.i	−1.24078	−	0.355105i
$952$		0			0
$953$	−1.00796e6	+	581947.i	−1.10984	+	0.640764i	−0.938787	−	0.344499i	$-0.888049\pi$
$953$	−1.00796e6	+	581947.i	−1.10984	+	0.640764i	−0.171049	+	0.985263i	$0.554716\pi$
$954$		0			0
$955$	1.39476e6	−	2.41580e6i	1.52930	−	2.64883i
$956$		0			0
$957$	−438144.	−	45825.3i	−0.478402	−	0.0500359i
$958$		0			0
$959$	865535.	−	1.03150e6i	0.941125	−	1.12159i
$960$		0			0
$961$	−593021.	−	215842.i	−0.642130	−	0.233716i
$962$		0			0
$963$	114077.	−	89313.6i	0.123011	−	0.0963086i
$964$		0			0
$965$	−950134.	+	167534.i	−1.02031	+	0.179907i
$966$		0			0
$967$	800189.	−	291245.i	0.855736	−	0.311462i	0.123359	−	0.992362i	$-0.460633\pi$
$967$	800189.	−	291245.i	0.855736	−	0.311462i	0.732377	+	0.680900i	$0.238411\pi$
$968$		0			0
$969$	−77603.2	+	159316.i	−0.0826480	+	0.169673i
$970$		0			0
$971$			1.51916e6i			1.61125i	0.592424	+	0.805627i	$0.298171\pi$
$971$			1.51916e6i			1.61125i	−0.592424	+	0.805627i	$0.701829\pi$
$972$		0			0
$973$	1.00622e6			1.06283
$974$		0			0
$975$	−1.10815e6	−	539784.i	−1.16571	−	0.567820i
$976$		0			0
$977$	309959.	+	851606.i	0.324725	+	0.892174i	0.989423	+	0.145062i	$0.0463380\pi$
$977$	309959.	+	851606.i	0.324725	+	0.892174i	−0.664698	+	0.747112i	$0.731440\pi$
$978$		0			0
$979$	−96656.0	−	548164.i	−0.100847	−	0.571933i
$980$		0			0
$981$	53044.6	+	67751.8i	0.0551193	+	0.0704016i
$982$		0			0
$983$	334744.	−	919701.i	0.346422	−	0.951787i	−0.637065	−	0.770810i	$-0.719852\pi$
$983$	334744.	−	919701.i	0.346422	−	0.951787i	0.983487	−	0.180977i	$-0.0579259\pi$
$984$		0			0
$985$	−350976.	−	294504.i	−0.361748	−	0.303542i
$986$		0			0
$987$	−54330.2	+	519461.i	−0.0557708	+	0.533235i
$988$		0			0
$989$	−857036.	−	494810.i	−0.876206	−	0.505878i
$990$		0			0
$991$	−205501.	−	355938.i	−0.209251	−	0.362433i	0.742228	−	0.670147i	$-0.233769\pi$
$991$	−205501.	−	355938.i	−0.209251	−	0.362433i	−0.951479	+	0.307715i	$0.900436\pi$
$992$		0			0
$993$	240435.	−	840111.i	0.243837	−	0.851997i
$994$		0			0
$995$	1.45377e6	+	256339.i	1.46842	+	0.258922i
$996$		0			0
$997$	1.13815e6	−	955022.i	1.14501	−	0.960778i	0.145419	−	0.989370i	$-0.453547\pi$
$997$	1.13815e6	−	955022.i	1.14501	−	0.960778i	0.999591	+	0.0285922i	$0.00910243\pi$
$998$		0			0
$999$	−1.32040e6	−	426795.i	−1.32304	−	0.427650i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.5.2	✓	72
3.2	odd	2		324.5.k.a.125.2		72
27.11	odd	18	inner	108.5.k.a.65.2	yes	72
27.16	even	9		324.5.k.a.197.2		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.2	✓	72	1.1	even	1	trivial
108.5.k.a.65.2	yes	72	27.11	odd	18	inner
324.5.k.a.125.2		72	3.2	odd	2
324.5.k.a.197.2		72	27.16	even	9

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(-8.09115 - 3.94122i) q^{3} +(14.0198 + 38.5191i) q^{5} +(-13.2730 - 75.2747i) q^{7} +(49.9335 + 63.7781i) q^{9} +O(q^{10})\)	\(q+(-8.09115 - 3.94122i) q^{3} +(14.0198 + 38.5191i) q^{5} +(-13.2730 - 75.2747i) q^{7} +(49.9335 + 63.7781i) q^{9} +(-14.4803 + 39.7844i) q^{11} +(-99.4208 - 83.4239i) q^{13} +(38.3759 - 366.919i) q^{15} +(25.0688 + 14.4735i) q^{17} +(-340.107 - 589.082i) q^{19} +(-189.281 + 661.371i) q^{21} +(-925.342 - 163.163i) q^{23} +(-808.388 + 678.318i) q^{25} +(-152.656 - 712.837i) q^{27} +(-743.151 - 885.653i) q^{29} +(-216.511 + 1227.89i) q^{31} +(273.962 - 264.832i) q^{33} +(2713.43 - 1566.60i) q^{35} +(951.756 - 1648.49i) q^{37} +(475.637 + 1066.84i) q^{39} +(663.134 - 790.292i) q^{41} +(989.699 + 360.221i) q^{43} +(-1756.61 + 2817.55i) q^{45} +(747.698 - 131.839i) q^{47} +(-3233.91 + 1177.05i) q^{49} +(-145.793 - 215.909i) q^{51} -202.878i q^{53} -1735.47 q^{55} +(430.153 + 6106.79i) q^{57} +(1284.25 + 3528.44i) q^{59} +(-211.588 - 1199.97i) q^{61} +(4138.11 - 4605.26i) q^{63} +(1819.55 - 4999.18i) q^{65} +(-2491.41 - 2090.54i) q^{67} +(6844.03 + 4967.15i) q^{69} +(2946.68 + 1701.27i) q^{71} +(-1292.56 - 2238.78i) q^{73} +(9214.19 - 2302.34i) q^{75} +(3186.96 + 561.947i) q^{77} +(-5812.54 + 4877.30i) q^{79} +(-1574.28 + 6369.33i) q^{81} +(-5284.48 - 6297.80i) q^{83} +(-206.046 + 1168.54i) q^{85} +(2522.40 + 10094.9i) q^{87} +(-11385.8 + 6573.58i) q^{89} +(-4960.10 + 8591.15i) q^{91} +(6591.22 - 9081.76i) q^{93} +(17922.7 - 21359.4i) q^{95} +(-3367.63 - 1225.72i) q^{97} +(-3260.43 + 1063.05i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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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