LMFDB - Embedded newform 108.5.k.a.41.1

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			41.1
Character	$\chi$	$=$	108.41
Dual form			108.5.k.a.29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.99629 + 0.258464i) q^{3} +(-25.3131 + 30.1670i) q^{5} +(-27.6070 - 10.0481i) q^{7} +(80.8664 - 4.65043i) q^{9} +O(q^{10})$	\(q+(-8.99629 + 0.258464i) q^{3} +(-25.3131 + 30.1670i) q^{5} +(-27.6070 - 10.0481i) q^{7} +(80.8664 - 4.65043i) q^{9} +(15.7890 + 18.8166i) q^{11} +(25.9745 - 147.308i) q^{13} +(219.927 - 277.934i) q^{15} +(172.593 + 99.6466i) q^{17} +(-129.877 - 224.954i) q^{19} +(250.958 + 83.2604i) q^{21} +(-211.930 - 582.273i) q^{23} +(-160.764 - 911.738i) q^{25} +(-726.295 + 62.7377i) q^{27} +(1271.79 - 224.250i) q^{29} +(992.428 - 361.214i) q^{31} +(-146.906 - 165.199i) q^{33} +(1001.94 - 578.471i) q^{35} +(-1125.43 + 1949.31i) q^{37} +(-195.600 + 1331.94i) q^{39} +(-461.110 - 81.3061i) q^{41} +(1135.15 - 952.502i) q^{43} +(-1906.69 + 2557.21i) q^{45} +(829.841 - 2279.97i) q^{47} +(-1178.09 - 988.536i) q^{49} +(-1578.45 - 851.841i) q^{51} +4450.35i q^{53} -967.312 q^{55} +(1226.56 + 1990.19i) q^{57} +(793.356 - 945.485i) q^{59} +(-4790.10 - 1743.45i) q^{61} +(-2279.21 - 684.171i) q^{63} +(3786.36 + 4512.41i) q^{65} +(379.378 - 2151.56i) q^{67} +(2057.08 + 5183.52i) q^{69} +(4243.91 + 2450.22i) q^{71} +(-3145.50 - 5448.16i) q^{73} +(1681.93 + 8160.70i) q^{75} +(-246.816 - 678.121i) q^{77} +(-732.250 - 4152.80i) q^{79} +(6517.75 - 752.128i) q^{81} +(-6036.15 + 1064.34i) q^{83} +(-7374.91 + 2684.25i) q^{85} +(-11383.4 + 2346.13i) q^{87} +(7061.95 - 4077.22i) q^{89} +(-2197.25 + 3805.75i) q^{91} +(-8834.81 + 3506.10i) q^{93} +(10073.8 + 1776.28i) q^{95} +(1708.86 - 1433.90i) q^{97} +(1364.31 + 1448.21i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/108\mathbb{Z}\right)^\times$.

$n$	$29$	$55$
$\chi(n)$	$e\left(\frac{17}{18}\right)$	$1$

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
$3$	−8.99629	+	0.258464i	−0.999588	+	0.0287182i
$3$	−8.99629	+	0.258464i	−0.999588	+	0.0287182i
$4$		0			0
$5$	−25.3131	+	30.1670i	−1.01253	+	1.20668i	−0.0342406	+	0.999414i	$0.510901\pi$
$5$	−25.3131	+	30.1670i	−1.01253	+	1.20668i	−0.978284	+	0.207267i	$0.933543\pi$
$6$		0			0
$7$	−27.6070	−	10.0481i	−0.563408	−	0.205064i	0.0445856	−	0.999006i	$-0.485803\pi$
$7$	−27.6070	−	10.0481i	−0.563408	−	0.205064i	−0.607994	+	0.793942i	$0.708025\pi$
$8$		0			0
$9$	80.8664	−	4.65043i	0.998351	−	0.0574128i
$10$		0			0
$11$	15.7890	+	18.8166i	0.130488	+	0.155509i	0.827332	−	0.561713i	$-0.189858\pi$
$11$	15.7890	+	18.8166i	0.130488	+	0.155509i	−0.696844	+	0.717223i	$0.745413\pi$
$12$		0			0
$13$	25.9745	−	147.308i	0.153695	−	0.871648i	−0.806274	−	0.591542i	$-0.798519\pi$
$13$	25.9745	−	147.308i	0.153695	−	0.871648i	0.959969	−	0.280106i	$-0.0903695\pi$
$14$		0			0
$15$	219.927	−	277.934i	0.977454	−	1.23526i
$16$		0			0
$17$	172.593	+	99.6466i	0.597208	+	0.344798i	0.767942	−	0.640519i	$-0.221281\pi$
$17$	172.593	+	99.6466i	0.597208	+	0.344798i	−0.170735	+	0.985317i	$0.554614\pi$
$18$		0			0
$19$	−129.877	−	224.954i	−0.359771	−	0.623142i	0.628151	−	0.778091i	$-0.283812\pi$
$19$	−129.877	−	224.954i	−0.359771	−	0.623142i	−0.987922	+	0.154949i	$0.950479\pi$
$20$		0			0
$21$	250.958	+	83.2604i	0.569065	+	0.188799i
$22$		0			0
$23$	−211.930	−	582.273i	−0.400624	−	1.10071i	−0.961977	−	0.273129i	$-0.911941\pi$
$23$	−211.930	−	582.273i	−0.400624	−	1.10071i	0.561353	−	0.827576i	$-0.310281\pi$
$24$		0			0
$25$	−160.764	−	911.738i	−0.257222	−	1.45878i
$26$		0			0
$27$	−726.295	+	62.7377i	−0.996290	+	0.0860599i
$28$		0			0
$29$	1271.79	−	224.250i	1.51223	−	0.266647i	0.644857	−	0.764303i	$-0.276917\pi$
$29$	1271.79	−	224.250i	1.51223	−	0.266647i	0.867375	+	0.497656i	$0.165806\pi$
$30$		0			0
$31$	992.428	−	361.214i	1.03270	−	0.375873i	0.230594	−	0.973050i	$-0.425933\pi$
$31$	992.428	−	361.214i	1.03270	−	0.375873i	0.802110	+	0.597177i	$0.203711\pi$
$32$		0			0
$33$	−146.906	−	165.199i	−0.134900	−	0.151698i
$34$		0			0
$35$	1001.94	−	578.471i	0.817911	−	0.472221i
$36$		0			0
$37$	−1125.43	+	1949.31i	−0.822084	+	1.42389i	0.0820437	+	0.996629i	$0.473855\pi$
$37$	−1125.43	+	1949.31i	−0.822084	+	1.42389i	−0.904128	+	0.427262i	$0.859478\pi$
$38$		0			0
$39$	−195.600	+	1331.94i	−0.128599	+	0.875702i
$40$		0			0
$41$	−461.110	−	81.3061i	−0.274307	−	0.0483677i	0.0348026	−	0.999394i	$-0.488920\pi$
$41$	−461.110	−	81.3061i	−0.274307	−	0.0483677i	−0.309109	+	0.951027i	$0.600031\pi$
$42$		0			0
$43$	1135.15	−	952.502i	0.613925	−	0.515144i	−0.281962	−	0.959425i	$-0.590985\pi$
$43$	1135.15	−	952.502i	0.613925	−	0.515144i	0.895887	+	0.444281i	$0.146541\pi$
$44$		0			0
$45$	−1906.69	+	2557.21i	−0.941576	+	1.26282i
$46$		0			0
$47$	829.841	−	2279.97i	0.375664	−	1.03213i	−0.597471	−	0.801891i	$-0.703828\pi$
$47$	829.841	−	2279.97i	0.375664	−	1.03213i	0.973135	−	0.230237i	$-0.0739502\pi$
$48$		0			0
$49$	−1178.09	−	988.536i	−0.490667	−	0.411718i
$50$		0			0
$51$	−1578.45	−	851.841i	−0.606863	−	0.327505i
$52$		0			0
$53$			4450.35i			1.58432i	0.610314	+	0.792160i	$0.291043\pi$
$53$			4450.35i			1.58432i	−0.610314	+	0.792160i	$0.708957\pi$
$54$		0			0
$55$	−967.312			−0.319773
$56$		0			0
$57$	1226.56	+	1990.19i	0.377519	+	0.612553i
$58$		0			0
$59$	793.356	−	945.485i	0.227910	−	0.271613i	−0.639955	−	0.768412i	$-0.721047\pi$
$59$	793.356	−	945.485i	0.227910	−	0.271613i	0.867865	+	0.496799i	$0.165491\pi$
$60$		0			0
$61$	−4790.10	−	1743.45i	−1.28732	−	0.468544i	−0.394470	−	0.918909i	$-0.629072\pi$
$61$	−4790.10	−	1743.45i	−1.28732	−	0.468544i	−0.892845	+	0.450364i	$0.851294\pi$
$62$		0			0
$63$	−2279.21	−	684.171i	−0.574252	−	0.172379i
$64$		0			0
$65$	3786.36	+	4512.41i	0.896180	+	1.06803i
$66$		0			0
$67$	379.378	−	2151.56i	0.0845127	−	0.479296i	−0.912948	−	0.408076i	$-0.866200\pi$
$67$	379.378	−	2151.56i	0.0845127	−	0.479296i	0.997461	−	0.0712195i	$-0.0226891\pi$
$68$		0			0
$69$	2057.08	+	5183.52i	0.432069	+	1.08875i
$70$		0			0
$71$	4243.91	+	2450.22i	0.841879	+	0.486059i	0.857903	−	0.513812i	$-0.171767\pi$
$71$	4243.91	+	2450.22i	0.841879	+	0.486059i	−0.0160233	+	0.999872i	$0.505101\pi$
$72$		0			0
$73$	−3145.50	−	5448.16i	−0.590261	−	1.02236i	−0.994197	−	0.107574i	$-0.965692\pi$
$73$	−3145.50	−	5448.16i	−0.590261	−	1.02236i	0.403936	−	0.914787i	$-0.367642\pi$
$74$		0			0
$75$	1681.93	+	8160.70i	0.299010	+	1.45079i
$76$		0			0
$77$	−246.816	−	678.121i	−0.0416286	−	0.114374i
$78$		0			0
$79$	−732.250	−	4152.80i	−0.117329	−	0.665406i	−0.985571	−	0.169264i	$-0.945861\pi$
$79$	−732.250	−	4152.80i	−0.117329	−	0.665406i	0.868242	−	0.496142i	$-0.165250\pi$
$80$		0			0
$81$	6517.75	−	752.128i	0.993408	−	0.114636i
$82$		0			0
$83$	−6036.15	+	1064.34i	−0.876202	+	0.154498i	−0.593620	−	0.804745i	$-0.702302\pi$
$83$	−6036.15	+	1064.34i	−0.876202	+	0.154498i	−0.282581	+	0.959243i	$0.591191\pi$
$84$		0			0
$85$	−7374.91	+	2684.25i	−1.02075	+	0.371522i
$86$		0			0
$87$	−11383.4	+	2346.13i	−1.50395	+	0.309966i
$88$		0			0
$89$	7061.95	−	4077.22i	0.891548	−	0.514736i	0.0170997	−	0.999854i	$-0.494557\pi$
$89$	7061.95	−	4077.22i	0.891548	−	0.514736i	0.874449	+	0.485118i	$0.161223\pi$
$90$		0			0
$91$	−2197.25	+	3805.75i	−0.265336	+	0.459576i
$92$		0			0
$93$	−8834.81	+	3506.10i	−1.02148	+	0.405376i
$94$		0			0
$95$	10073.8	+	1776.28i	1.11621	+	0.196818i
$96$		0			0
$97$	1708.86	−	1433.90i	0.181619	−	0.152397i	−0.547445	−	0.836841i	$-0.684400\pi$
$97$	1708.86	−	1433.90i	0.181619	−	0.152397i	0.729065	+	0.684445i	$0.239955\pi$
$98$		0			0
$99$	1364.31	+	1448.21i	0.139201	+	0.147761i
$100$		0			0
$101$	−1570.11	+	4313.84i	−0.153917	+	0.422884i	−0.992554	−	0.121807i	$-0.961131\pi$
$101$	−1570.11	+	4313.84i	−0.153917	+	0.422884i	0.838637	+	0.544691i	$0.183353\pi$
$102$		0			0
$103$	−9520.09	−	7988.31i	−0.897360	−	0.752975i	0.0723124	−	0.997382i	$-0.476962\pi$
$103$	−9520.09	−	7988.31i	−0.897360	−	0.752975i	−0.969673	+	0.244407i	$0.921407\pi$
$104$		0			0
$105$	−8864.24	+	5463.06i	−0.804013	+	0.495516i
$106$		0			0
$107$		−	17618.5i		−	1.53887i	−0.638724	−	0.769436i	$-0.720538\pi$
$107$		−	17618.5i		−	1.53887i	0.638724	−	0.769436i	$-0.279462\pi$
$108$		0			0
$109$	23518.9			1.97953			0.989767	−	0.142691i	$-0.0455754\pi$
$109$	23518.9			1.97953			0.989767	+	0.142691i	$0.0455754\pi$
$110$		0			0
$111$	9620.89	−	17827.4i	0.780853	−	1.44691i
$112$		0			0
$113$	11590.6	−	13813.2i	0.907717	−	1.08177i	−0.0886034	−	0.996067i	$-0.528240\pi$
$113$	11590.6	−	13813.2i	0.907717	−	1.08177i	0.996320	−	0.0857079i	$-0.0273152\pi$
$114$		0			0
$115$	22930.1	+	8345.86i	1.73384	+	0.631067i
$116$		0			0
$117$	1415.41	−	12033.1i	0.103398	−	0.879034i
$118$		0			0
$119$	−3763.51	−	4485.18i	−0.265766	−	0.316728i
$120$		0			0
$121$	2437.61	−	13824.4i	0.166492	−	0.944224i
$122$		0			0
$123$	4169.29	+	612.272i	0.275583	+	0.0404701i
$124$		0			0
$125$	10258.7	+	5922.84i	0.656555	+	0.379062i
$126$		0			0
$127$	−9957.61	−	17247.1i	−0.617373	−	1.06932i	−0.989963	−	0.141325i	$-0.954864\pi$
$127$	−9957.61	−	17247.1i	−0.617373	−	1.06932i	0.372590	−	0.927996i	$-0.378470\pi$
$128$		0			0
$129$	−9965.93	+	8862.38i	−0.598878	+	0.532563i
$130$		0			0
$131$	−3024.44	−	8309.58i	−0.176239	−	0.484213i	0.819849	−	0.572580i	$-0.194057\pi$
$131$	−3024.44	−	8309.58i	−0.176239	−	0.484213i	−0.996088	+	0.0883670i	$0.971835\pi$
$132$		0			0
$133$	1325.16	+	7515.34i	0.0749142	+	0.424860i
$134$		0			0
$135$	16492.2	−	23498.2i	0.904922	−	1.28934i
$136$		0			0
$137$	−16794.2	+	2961.27i	−0.894785	+	0.157775i	−0.602087	−	0.798430i	$-0.705664\pi$
$137$	−16794.2	+	2961.27i	−0.894785	+	0.157775i	−0.292698	+	0.956205i	$0.594553\pi$
$138$		0			0
$139$	−15479.4	+	5634.06i	−0.801172	+	0.291603i	−0.709972	−	0.704230i	$-0.751292\pi$
$139$	−15479.4	+	5634.06i	−0.801172	+	0.291603i	−0.0912000	+	0.995833i	$0.529070\pi$
$140$		0			0
$141$	−6876.20	+	20725.8i	−0.345868	+	1.04249i
$142$		0			0
$143$	3181.96	−	1837.11i	0.155605	−	0.0898385i
$144$		0			0
$145$	−25427.9	+	44042.5i	−1.20941	+	2.09477i
$146$		0			0
$147$	10853.9	+	8588.66i	0.502288	+	0.397457i
$148$		0			0
$149$	−16943.7	−	2987.63i	−0.763194	−	0.134572i	−0.221515	−	0.975157i	$-0.571100\pi$
$149$	−16943.7	−	2987.63i	−0.763194	−	0.134572i	−0.541679	+	0.840585i	$0.682211\pi$
$150$		0			0
$151$	21558.4	−	18089.6i	0.945502	−	0.793370i	−0.0330327	−	0.999454i	$-0.510517\pi$
$151$	21558.4	−	18089.6i	0.945502	−	0.793370i	0.978534	+	0.206084i	$0.0660721\pi$
$152$		0			0
$153$	14420.4	+	7255.43i	0.616019	+	0.309942i
$154$		0			0
$155$	−14224.7	+	39082.1i	−0.592079	+	1.62672i
$156$		0			0
$157$	−25228.4	−	21169.1i	−1.02350	−	0.858823i	−0.0334410	−	0.999441i	$-0.510647\pi$
$157$	−25228.4	−	21169.1i	−1.02350	−	0.858823i	−0.990064	+	0.140618i	$0.955091\pi$
$158$		0			0
$159$	−1150.26	−	40036.7i	−0.0454988	−	1.58367i
$160$		0			0
$161$			18204.3i			0.702300i
$162$		0			0
$163$	19799.5			0.745212			0.372606	−	0.927990i	$-0.378464\pi$
$163$	19799.5			0.745212			0.372606	+	0.927990i	$0.378464\pi$
$164$		0			0
$165$	8702.22	−	250.015i	0.319641	−	0.00918330i
$166$		0			0
$167$	−27954.3	+	33314.7i	−1.00234	+	1.19454i	−0.0214942	+	0.999769i	$0.506842\pi$
$167$	−27954.3	+	33314.7i	−1.00234	+	1.19454i	−0.980848	+	0.194776i	$0.937602\pi$
$168$		0			0
$169$	5813.44	+	2115.92i	0.203545	+	0.0740842i
$170$		0			0
$171$	−11548.9	−	17587.3i	−0.394954	−	0.601459i
$172$		0			0
$173$	12197.7	+	14536.7i	0.407555	+	0.485706i	0.930308	−	0.366779i	$-0.119539\pi$
$173$	12197.7	+	14536.7i	0.407555	+	0.485706i	−0.522753	+	0.852484i	$0.675095\pi$
$174$		0			0
$175$	−4723.05	+	26785.7i	−0.154222	+	0.874636i
$176$		0			0
$177$	−6892.89	+	8710.91i	−0.220016	+	0.278046i
$178$		0			0
$179$	15338.0	+	8855.40i	0.478699	+	0.276377i	0.719874	−	0.694105i	$-0.244199\pi$
$179$	15338.0	+	8855.40i	0.478699	+	0.276377i	−0.241175	+	0.970482i	$0.577533\pi$
$180$		0			0
$181$	26541.9	+	45971.9i	0.810168	+	1.40325i	0.912746	+	0.408526i	$0.133957\pi$
$181$	26541.9	+	45971.9i	0.810168	+	1.40325i	−0.102579	+	0.994725i	$0.532709\pi$
$182$		0			0
$183$	43543.7	+	14446.5i	1.30024	+	0.431382i
$184$		0			0
$185$	−30316.5	−	83294.0i	−0.885801	−	2.43372i
$186$		0			0
$187$	850.063	+	4820.95i	0.0243090	+	0.137863i
$188$		0			0
$189$	20681.2	+	5565.91i	0.578966	+	0.155816i
$190$		0			0
$191$	−57550.4	+	10147.7i	−1.57755	+	0.278164i	−0.892744	−	0.450565i	$-0.851223\pi$
$191$	−57550.4	+	10147.7i	−1.57755	+	0.278164i	−0.684802	+	0.728729i	$0.740111\pi$
$192$		0			0
$193$	23447.3	−	8534.12i	0.629475	−	0.229110i	−0.00752797	−	0.999972i	$-0.502396\pi$
$193$	23447.3	−	8534.12i	0.629475	−	0.229110i	0.637003	+	0.770862i	$0.280174\pi$
$194$		0			0
$195$	−35229.5	−	39616.3i	−0.926483	−	1.04185i
$196$		0			0
$197$	1584.31	−	914.700i	0.0408232	−	0.0235693i	−0.479450	−	0.877569i	$-0.659164\pi$
$197$	1584.31	−	914.700i	0.0408232	−	0.0235693i	0.520273	+	0.854000i	$0.325830\pi$
$198$		0			0
$199$	−37552.8	+	65043.4i	−0.948279	+	1.64247i	−0.199230	+	0.979953i	$0.563844\pi$
$199$	−37552.8	+	65043.4i	−0.948279	+	1.64247i	−0.749049	+	0.662515i	$0.769489\pi$
$200$		0			0
$201$	−2856.89	+	19454.1i	−0.0707134	+	0.481525i
$202$		0			0
$203$	−37363.5	−	6588.20i	−0.906684	−	0.159873i
$204$		0			0
$205$	14124.9	−	11852.2i	0.336107	−	0.282027i
$206$		0			0
$207$	−19845.8	−	46100.8i	−0.463158	−	1.07589i
$208$		0			0
$209$	2182.25	−	5995.67i	0.0499587	−	0.137260i
$210$		0			0
$211$	53061.3	+	44523.7i	1.19183	+	1.00006i	0.999826	+	0.0186653i	$0.00594171\pi$
$211$	53061.3	+	44523.7i	1.19183	+	1.00006i	0.192000	+	0.981395i	$0.438503\pi$
$212$		0			0
$213$	−38812.8	−	20946.0i	−0.855491	−	0.461681i
$214$		0			0
$215$			58354.8i			1.26241i
$216$		0			0
$217$	−31027.5			−0.658912
$218$		0			0
$219$	29706.0	+	48200.3i	0.619378	+	1.00499i
$220$		0			0
$221$	19161.8	−	22836.2i	0.392330	−	0.467561i
$222$		0			0
$223$	50266.6	+	18295.5i	1.01081	+	0.367905i	0.793744	−	0.608252i	$-0.208129\pi$
$223$	50266.6	+	18295.5i	1.01081	+	0.367905i	0.217067	+	0.976157i	$0.430351\pi$
$224$		0			0
$225$	−17240.4	−	72981.3i	−0.340551	−	1.44161i
$226$		0			0
$227$	−41479.3	−	49433.0i	−0.804969	−	0.959325i	0.194799	−	0.980843i	$-0.437594\pi$
$227$	−41479.3	−	49433.0i	−0.804969	−	0.959325i	−0.999768	+	0.0215183i	$0.993150\pi$
$228$		0			0
$229$	5176.04	−	29354.8i	0.0987021	−	0.559767i	−0.894848	−	0.446371i	$-0.852716\pi$
$229$	5176.04	−	29354.8i	0.0987021	−	0.559767i	0.993550	−	0.113396i	$-0.0361728\pi$
$230$		0			0
$231$	2395.70	+	6036.78i	0.0448960	+	0.113131i
$232$		0			0
$233$	36745.1	+	21214.8i	0.676842	+	0.390775i	0.798664	−	0.601777i	$-0.205540\pi$
$233$	36745.1	+	21214.8i	0.676842	+	0.390775i	−0.121822	+	0.992552i	$0.538874\pi$
$234$		0			0
$235$	47774.0	+	82747.0i	0.865080	+	1.49836i
$236$		0			0
$237$	7660.88	+	37170.5i	0.136390	+	0.661762i
$238$		0			0
$239$	−16787.1	−	46122.3i	−0.293887	−	0.807449i	−0.995489	−	0.0948791i	$-0.969754\pi$
$239$	−16787.1	−	46122.3i	−0.293887	−	0.807449i	0.701602	−	0.712569i	$-0.252469\pi$
$240$		0			0
$241$	−12866.3	−	72968.3i	−0.221523	−	1.25632i	−0.869221	−	0.494423i	$-0.835379\pi$
$241$	−12866.3	−	72968.3i	−0.221523	−	1.25632i	0.647698	−	0.761897i	$-0.275732\pi$
$242$		0			0
$243$	−58441.1	+	8450.96i	−0.989706	+	0.143118i
$244$		0			0
$245$	59642.3	−	10516.6i	0.993625	−	0.175203i
$246$		0			0
$247$	−36511.2	+	13289.0i	−0.598456	+	0.217820i
$248$		0			0
$249$	54027.9	−	11135.2i	0.871403	−	0.179597i
$250$		0			0
$251$	8436.89	−	4871.04i	0.133917	−	0.0773168i	−0.431545	−	0.902091i	$-0.642031\pi$
$251$	8436.89	−	4871.04i	0.133917	−	0.0773168i	0.565462	+	0.824775i	$0.308698\pi$
$252$		0			0
$253$	7610.25	−	13181.3i	0.118894	−	0.205930i
$254$		0			0
$255$	65653.0	−	26054.4i	1.00966	−	0.400683i
$256$		0			0
$257$	−6076.74	−	1071.49i	−0.0920035	−	0.0162227i	0.127457	−	0.991844i	$-0.459319\pi$
$257$	−6076.74	−	1071.49i	−0.0920035	−	0.0162227i	−0.219460	+	0.975621i	$0.570430\pi$
$258$		0			0
$259$	50656.7	−	42506.0i	0.755157	−	0.633652i
$260$		0			0
$261$	101802.	−	24048.7i	1.49443	−	0.353029i
$262$		0			0
$263$	−28561.4	+	78471.8i	−0.412922	+	1.13449i	0.542708	+	0.839922i	$0.317399\pi$
$263$	−28561.4	+	78471.8i	−0.412922	+	1.13449i	−0.955629	+	0.294572i	$0.904823\pi$
$264$		0			0
$265$	−134254.	−	112652.i	−1.91177	−	1.60416i
$266$		0			0
$267$	−62477.6	+	38505.1i	−0.876398	+	0.540127i
$268$		0			0
$269$		−	2129.98i		−	0.0294355i	−0.999892	−	0.0147177i	$-0.995315\pi$
$269$		−	2129.98i		−	0.0294355i	0.999892	−	0.0147177i	$-0.00468497\pi$
$270$		0			0
$271$	−3403.65			−0.0463453			−0.0231727	−	0.999731i	$-0.507377\pi$
$271$	−3403.65			−0.0463453			−0.0231727	+	0.999731i	$0.507377\pi$
$272$		0			0
$273$	18783.5	−	34805.5i	0.252029	−	0.467007i
$274$		0			0
$275$	14617.5	−	17420.5i	0.193290	−	0.230354i
$276$		0			0
$277$	−8558.30	−	3114.97i	−0.111539	−	0.0405970i	0.285648	−	0.958335i	$-0.407791\pi$
$277$	−8558.30	−	3114.97i	−0.111539	−	0.0405970i	−0.397187	+	0.917738i	$0.630014\pi$
$278$		0			0
$279$	78574.3	−	33825.3i	1.00942	−	0.434544i
$280$		0			0
$281$	−81.2245	−	96.7995i	−0.00102867	−	0.00122592i	0.765530	−	0.643400i	$-0.222477\pi$
$281$	−81.2245	−	96.7995i	−0.00102867	−	0.00122592i	−0.766559	+	0.642174i	$0.778032\pi$
$282$		0			0
$283$	9704.17	−	55035.1i	0.121167	−	0.687174i	−0.862343	−	0.506324i	$-0.831004\pi$
$283$	9704.17	−	55035.1i	0.121167	−	0.687174i	0.983511	−	0.180850i	$-0.0578849\pi$
$284$		0			0
$285$	−91086.0	−	13376.2i	−1.12140	−	0.164681i
$286$		0			0
$287$	11912.9	+	6877.90i	0.144628	+	0.0835011i
$288$		0			0
$289$	−21901.6	−	37934.7i	−0.262229	−	0.454193i
$290$		0			0
$291$	−15002.8	+	13341.5i	−0.177168	+	0.157550i
$292$		0			0
$293$	−11469.3	−	31511.5i	−0.133598	−	0.367058i	0.854797	−	0.518962i	$-0.173682\pi$
$293$	−11469.3	−	31511.5i	−0.133598	−	0.367058i	−0.988395	+	0.151905i	$0.951459\pi$
$294$		0			0
$295$	8440.13	+	47866.4i	0.0969851	+	0.550030i
$296$		0			0
$297$	−12648.0	−	12675.9i	−0.143387	−	0.143703i
$298$		0			0
$299$	−91278.6	+	16094.9i	−1.02100	+	0.180030i
$300$		0			0
$301$	−40908.9	+	14889.6i	−0.451528	+	0.164343i
$302$		0			0
$303$	13010.2	−	39214.3i	0.141709	−	0.427130i
$304$		0			0
$305$	173847.	−	100371.i	1.86882	−	1.07896i
$306$		0			0
$307$	60296.2	−	104436.i	0.639754	−	1.10809i	−0.345732	−	0.938333i	$-0.612369\pi$
$307$	60296.2	−	104436.i	0.639754	−	1.10809i	0.985486	−	0.169754i	$-0.0542973\pi$
$308$		0			0
$309$	87710.2	+	69404.5i	0.918614	+	0.726893i
$310$		0			0
$311$	158476.	+	27943.7i	1.63849	+	0.288910i	0.915612	−	0.402064i	$-0.131707\pi$
$311$	158476.	+	27943.7i	1.63849	+	0.288910i	0.722879	+	0.690974i	$0.242818\pi$
$312$		0			0
$313$	40180.2	−	33715.2i	0.410131	−	0.344141i	−0.414263	−	0.910157i	$-0.635961\pi$
$313$	40180.2	−	33715.2i	0.410131	−	0.344141i	0.824394	+	0.566016i	$0.191516\pi$
$314$		0			0
$315$	78333.2	−	51438.3i	0.789451	−	0.518401i
$316$		0			0
$317$	−43794.7	+	120325.i	−0.435816	+	1.19739i	0.506374	+	0.862314i	$0.330986\pi$
$317$	−43794.7	+	120325.i	−0.435816	+	1.19739i	−0.942190	+	0.335080i	$0.891237\pi$
$318$		0			0
$319$	24299.9	+	20390.1i	0.238794	+	0.200372i
$320$		0			0
$321$	4553.76	+	158501.i	0.0441937	+	1.53824i
$322$		0			0
$323$		−	51767.4i		−	0.496194i
$324$		0			0
$325$	−138482.			−1.31108
$326$		0			0
$327$	−211582.	+	6078.78i	−1.97872	+	0.0568487i
$328$		0			0
$329$	−45818.9	+	54604.8i	−0.423304	+	0.504474i
$330$		0			0
$331$	20984.5	+	7637.74i	0.191533	+	0.0697122i	0.436006	−	0.899944i	$-0.356393\pi$
$331$	20984.5	+	7637.74i	0.191533	+	0.0697122i	−0.244473	+	0.969656i	$0.578615\pi$
$332$		0			0
$333$	−81944.6	+	162867.i	−0.738978	+	1.46874i
$334$		0			0
$335$	55302.8	+	65907.4i	0.492785	+	0.587279i
$336$		0			0
$337$	−105.587	+	598.812i	−0.000929714	+	0.00527267i	−0.985269	−	0.171011i	$-0.945297\pi$
$337$	−105.587	+	598.812i	−0.000929714	+	0.00527267i	0.984339	+	0.176284i	$0.0564077\pi$
$338$		0			0
$339$	−100702.	+	127263.i	−0.876276	+	1.10740i
$340$		0			0
$341$	22466.3	+	12970.9i	0.193207	+	0.111548i
$342$		0			0
$343$	57859.8	+	100216.i	0.491800	+	0.851823i
$344$		0			0
$345$	−208442.	−	69155.1i	−1.75125	−	0.581014i
$346$		0			0
$347$	−48459.4	−	133141.i	−0.402457	−	1.10574i	−0.961068	−	0.276312i	$-0.910888\pi$
$347$	−48459.4	−	133141.i	−0.402457	−	1.10574i	0.558611	−	0.829430i	$-0.311334\pi$
$348$		0			0
$349$	16171.1	+	91710.8i	0.132767	+	0.752956i	0.976389	+	0.216020i	$0.0693075\pi$
$349$	16171.1	+	91710.8i	0.132767	+	0.752956i	−0.843622	+	0.536937i	$0.819581\pi$
$350$		0			0
$351$	−9623.34	+	108619.i	−0.0781109	+	0.881641i
$352$		0			0
$353$	55783.0	−	9836.05i	0.447665	−	0.0789353i	0.0547288	−	0.998501i	$-0.482571\pi$
$353$	55783.0	−	9836.05i	0.447665	−	0.0789353i	0.392936	+	0.919566i	$0.371459\pi$
$354$		0			0
$355$	−181343.	+	66003.3i	−1.43894	+	0.523732i
$356$		0			0
$357$	35016.9	+	39377.3i	0.274752	+	0.308965i
$358$		0			0
$359$	−85755.7	+	49511.1i	−0.665387	+	0.384161i	−0.794326	−	0.607491i	$-0.792176\pi$
$359$	−85755.7	+	49511.1i	−0.665387	+	0.384161i	0.128940	+	0.991652i	$0.458843\pi$
$360$		0			0
$361$	31424.2	−	54428.3i	0.241129	−	0.417648i
$362$		0			0
$363$	−18356.3	+	124998.i	−0.139307	+	0.948615i
$364$		0			0
$365$	243977.	+	43019.8i	1.83132	+	0.322911i
$366$		0			0
$367$	−71780.1	+	60230.6i	−0.532932	+	0.447183i	−0.869112	−	0.494615i	$-0.835309\pi$
$367$	−71780.1	+	60230.6i	−0.532932	+	0.447183i	0.336180	+	0.941798i	$0.390865\pi$
$368$		0			0
$369$	−37666.4	−	4430.57i	−0.276631	−	0.0325392i
$370$		0			0
$371$	44717.7	−	122861.i	0.324887	−	0.892619i
$372$		0			0
$373$	−159172.	−	133561.i	−1.14406	−	0.959978i	−0.144494	−	0.989506i	$-0.546155\pi$
$373$	−159172.	−	133561.i	−1.14406	−	0.959978i	−0.999564	+	0.0295275i	$0.990600\pi$
$374$		0			0
$375$	−93820.8	−	50632.1i	−0.667170	−	0.360051i
$376$		0			0
$377$		−	193170.i		−	1.35912i
$378$		0			0
$379$	−49705.2			−0.346038			−0.173019	−	0.984918i	$-0.555352\pi$
$379$	−49705.2			−0.346038			−0.173019	+	0.984918i	$0.555352\pi$
$380$		0			0
$381$	94039.3	+	152586.i	0.647827	+	1.05115i
$382$		0			0
$383$	−89094.1	+	106178.i	−0.607367	+	0.723832i	−0.978843	−	0.204610i	$-0.934407\pi$
$383$	−89094.1	+	106178.i	−0.607367	+	0.723832i	0.371476	+	0.928442i	$0.378852\pi$
$384$		0			0
$385$	26704.6	+	9719.67i	0.180162	+	0.0655738i
$386$		0			0
$387$	87365.7	−	82304.3i	0.583337	−	0.549542i
$388$		0			0
$389$	43574.8	+	51930.5i	0.287963	+	0.343181i	0.890561	−	0.454864i	$-0.150312\pi$
$389$	43574.8	+	51930.5i	0.287963	+	0.343181i	−0.602598	+	0.798045i	$0.705868\pi$
$390$		0			0
$391$	21443.9	−	121614.i	0.140265	−	0.795484i
$392$		0			0
$393$	29356.5	+	73973.7i	0.190072	+	0.478952i
$394$		0			0
$395$	143813.	+	83030.5i	0.921731	+	0.532162i
$396$		0			0
$397$	15662.5	+	27128.3i	0.0993757	+	0.172124i	0.911426	−	0.411463i	$-0.134982\pi$
$397$	15662.5	+	27128.3i	0.0993757	+	0.172124i	−0.812051	+	0.583587i	$0.801649\pi$
$398$		0			0
$399$	−13863.9	−	67267.7i	−0.0870845	−	0.422533i
$400$		0			0
$401$	17473.9	+	48009.2i	0.108668	+	0.298563i	0.982093	−	0.188395i	$-0.0603286\pi$
$401$	17473.9	+	48009.2i	0.108668	+	0.298563i	−0.873425	+	0.486958i	$0.838106\pi$
$402$		0			0
$403$	−27432.2	−	155575.i	−0.168908	−	0.957924i
$404$		0			0
$405$	−142295.	+	215660.i	−0.867521	+	1.31480i
$406$		0			0
$407$	−54448.9	+	9600.81i	−0.328701	+	0.0579588i
$408$		0			0
$409$	220450.	−	80237.1i	1.31784	−	0.479654i	0.415074	−	0.909788i	$-0.363756\pi$
$409$	220450.	−	80237.1i	1.31784	−	0.479654i	0.902765	+	0.430133i	$0.141533\pi$
$410$		0			0
$411$	150320.	−	30981.2i	0.889885	−	0.183406i
$412$		0			0
$413$	−31402.5	+	18130.3i	−0.184105	+	0.106293i
$414$		0			0
$415$	120686.	−	209034.i	0.700746	−	1.21373i
$416$		0			0
$417$	137801.	−	54686.5i	0.792467	−	0.314491i
$418$		0			0
$419$	47825.8	+	8432.98i	0.272417	+	0.0480345i	0.308188	−	0.951326i	$-0.400278\pi$
$419$	47825.8	+	8432.98i	0.272417	+	0.0480345i	−0.0357707	+	0.999360i	$0.511389\pi$
$420$		0			0
$421$	23690.9	−	19879.0i	0.133665	−	0.112158i	−0.573504	−	0.819203i	$-0.694416\pi$
$421$	23690.9	−	19879.0i	0.133665	−	0.112158i	0.707169	+	0.707045i	$0.249972\pi$
$422$		0			0
$423$	56503.4	−	188232.i	0.315787	−	1.05199i
$424$		0			0
$425$	63104.9	−	173379.i	0.349370	−	0.959885i
$426$		0			0
$427$	114722.	+	96263.1i	0.629202	+	0.527963i
$428$		0			0
$429$	−28151.0	+	17349.6i	−0.152961	+	0.0942701i
$430$		0			0
$431$			235303.i			1.26670i	0.773866	+	0.633349i	$0.218320\pi$
$431$			235303.i			1.26670i	−0.773866	+	0.633349i	$0.781680\pi$
$432$		0			0
$433$	−280559.			−1.49640			−0.748201	−	0.663472i	$-0.769082\pi$
$433$	−280559.			−1.49640			−0.748201	+	0.663472i	$0.769082\pi$
$434$		0			0
$435$	217374.	−	402791.i	1.14876	−	2.12864i
$436$		0			0
$437$	−103460.	+	123299.i	−0.541763	+	0.645648i
$438$		0			0
$439$	−307672.	−	111984.i	−1.59647	−	0.581066i	−0.617765	−	0.786363i	$-0.711962\pi$
$439$	−307672.	−	111984.i	−1.59647	−	0.581066i	−0.978700	+	0.205297i	$0.934184\pi$
$440$		0			0
$441$	−99865.1	−	74460.7i	−0.513495	−	0.382869i
$442$		0			0
$443$	24021.1	+	28627.2i	0.122401	+	0.145872i	0.823765	−	0.566932i	$-0.191870\pi$
$443$	24021.1	+	28627.2i	0.122401	+	0.145872i	−0.701364	+	0.712803i	$0.747425\pi$
$444$		0			0
$445$	−55762.6	+	316245.i	−0.281594	+	1.59700i
$446$		0			0
$447$	153202.	+	22498.2i	0.766744	+	0.112599i
$448$		0			0
$449$	−123175.	−	71115.4i	−0.610987	−	0.352753i	0.162365	−	0.986731i	$-0.448088\pi$
$449$	−123175.	−	71115.4i	−0.610987	−	0.352753i	−0.773351	+	0.633978i	$0.781421\pi$
$450$		0			0
$451$	−5750.57	−	9960.28i	−0.0282721	−	0.0489687i
$452$		0			0
$453$	−189270.	+	168312.i	−0.922327	+	0.820196i
$454$		0			0
$455$	−59188.8	−	162620.i	−0.285902	−	0.785509i
$456$		0			0
$457$	−41844.7	−	237313.i	−0.200358	−	1.13629i	−0.904578	−	0.426307i	$-0.859814\pi$
$457$	−41844.7	−	237313.i	−0.200358	−	1.13629i	0.704220	−	0.709982i	$-0.251297\pi$
$458$		0			0
$459$	−131605.	−	61544.8i	−0.624665	−	0.292123i
$460$		0			0
$461$	111772.	−	19708.3i	0.525932	−	0.0927360i	0.0956228	−	0.995418i	$-0.469516\pi$
$461$	111772.	−	19708.3i	0.525932	−	0.0927360i	0.430309	+	0.902682i	$0.358405\pi$
$462$		0			0
$463$	281207.	−	102351.i	1.31179	−	0.477453i	0.410972	−	0.911648i	$-0.365189\pi$
$463$	281207.	−	102351.i	1.31179	−	0.477453i	0.900819	+	0.434195i	$0.142967\pi$
$464$		0			0
$465$	117868.	−	355270.i	0.545118	−	1.64306i
$466$		0			0
$467$	−302947.	+	174907.i	−1.38910	+	0.801996i	−0.993214	−	0.116304i	$-0.962895\pi$
$467$	−302947.	+	174907.i	−1.38910	+	0.801996i	−0.395884	+	0.918300i	$0.629562\pi$
$468$		0			0
$469$	−32092.6	+	55586.0i	−0.145901	+	0.252709i
$470$		0			0
$471$	232433.	+	183923.i	1.04775	+	0.829075i
$472$		0			0
$473$	35845.8	+	6320.58i	0.160220	+	0.0282510i
$474$		0			0
$475$	−184220.	+	154579.i	−0.816487	+	0.685114i
$476$		0			0
$477$	20696.1	+	359884.i	0.0909602	+	1.58171i
$478$		0			0
$479$	111988.	−	307685.i	0.488092	−	1.34102i	−0.414313	−	0.910134i	$-0.635978\pi$
$479$	111988.	−	307685.i	0.488092	−	1.34102i	0.902405	−	0.430888i	$-0.141800\pi$
$480$		0			0
$481$	257917.	+	216418.i	1.11478	+	0.935413i
$482$		0			0
$483$	−4705.16	−	163771.i	−0.0201688	−	0.702010i
$484$		0			0
$485$			87847.6i			0.373462i
$486$		0			0
$487$	286014.			1.20595			0.602975	−	0.797760i	$-0.293982\pi$
$487$	286014.			1.20595			0.602975	+	0.797760i	$0.293982\pi$
$488$		0			0
$489$	−178122.	+	5117.47i	−0.744905	+	0.0214012i
$490$		0			0
$491$	−81179.6	+	96746.0i	−0.336732	+	0.401301i	−0.907665	−	0.419695i	$-0.862137\pi$
$491$	−81179.6	+	96746.0i	−0.336732	+	0.401301i	0.570934	+	0.820996i	$0.306581\pi$
$492$		0			0
$493$	241847.	+	88025.3i	0.995056	+	0.362171i
$494$		0			0
$495$	−78223.0	+	4498.42i	−0.319245	+	0.0183590i
$496$		0			0
$497$	−92541.5	−	110287.i	−0.374648	−	0.446489i
$498$		0			0
$499$	11759.2	−	66689.8i	0.0472255	−	0.267829i	−0.952048	−	0.305950i	$-0.901026\pi$
$499$	11759.2	−	66689.8i	0.0472255	−	0.267829i	0.999273	+	0.0381206i	$0.0121371\pi$
$500$		0			0
$501$	242874.	−	306933.i	0.967623	−	1.22284i
$502$		0			0
$503$	−302234.	−	174495.i	−1.19456	−	0.689679i	−0.235222	−	0.971942i	$-0.575582\pi$
$503$	−302234.	−	174495.i	−1.19456	−	0.689679i	−0.959337	+	0.282262i	$0.908915\pi$
$504$		0			0
$505$	−90391.2	−	156562.i	−0.354441	−	0.613909i
$506$		0			0
$507$	−52846.3	−	17532.9i	−0.205588	−	0.0682082i
$508$		0			0
$509$	−122685.	−	337074.i	−0.473539	−	1.30104i	−0.914890	−	0.403703i	$-0.867723\pi$
$509$	−122685.	−	337074.i	−0.473539	−	1.30104i	0.441351	−	0.897334i	$-0.354499\pi$
$510$		0			0
$511$	32094.0	+	182014.i	0.122908	+	0.697048i
$512$		0			0
$513$	108443.	+	155235.i	0.412064	+	0.589869i
$514$		0			0
$515$	481967.	−	84983.7i	1.81720	−	0.320421i
$516$		0			0
$517$	56003.8	−	20383.7i	0.209525	−	0.0762610i
$518$		0			0
$519$	−113491.	−	127623.i	−0.421336	−	0.473801i
$520$		0			0
$521$	299992.	−	173200.i	1.10518	−	0.638078i	0.167605	−	0.985854i	$-0.446397\pi$
$521$	299992.	−	173200.i	1.10518	−	0.638078i	0.937577	+	0.347776i	$0.113063\pi$
$522$		0			0
$523$	−42058.5	+	72847.4i	−0.153762	+	0.266324i	−0.932608	−	0.360892i	$-0.882472\pi$
$523$	−42058.5	+	72847.4i	−0.153762	+	0.266324i	0.778845	+	0.627216i	$0.215806\pi$
$524$		0			0
$525$	35566.7	−	242193.i	0.129040	−	0.878704i
$526$		0			0
$527$	207280.	+	36549.1i	0.746339	+	0.131600i
$528$		0			0
$529$	−79757.0	+	66924.1i	−0.285008	+	0.239150i
$530$		0			0
$531$	59758.9	−	80147.4i	0.211940	−	0.284250i
$532$		0			0
$533$	−23954.1	+	65813.5i	−0.0843192	+	0.231665i
$534$		0			0
$535$	531499.	+	445980.i	1.85693	+	1.55815i
$536$		0			0
$537$	−140274.	−	75701.4i	−0.486439	−	0.262516i
$538$		0			0
$539$		−	37775.8i		−	0.130028i
$540$		0			0
$541$	425335.			1.45324			0.726619	−	0.687041i	$-0.241091\pi$
$541$	425335.			1.45324			0.726619	+	0.687041i	$0.241091\pi$
$542$		0			0
$543$	−250661.	−	406716.i	−0.850132	−	1.37941i
$544$		0			0
$545$	−595336.	+	709493.i	−2.00433	+	2.38867i
$546$		0			0
$547$	−452072.	−	164541.i	−1.51089	−	0.549919i	−0.552037	−	0.833820i	$-0.686149\pi$
$547$	−452072.	−	164541.i	−1.51089	−	0.549919i	−0.958854	+	0.283900i	$0.908372\pi$
$548$		0			0
$549$	−395466.	−	118711.i	−1.31209	−	0.393863i
$550$		0			0
$551$	−215623.	−	256969.i	−0.710217	−	0.846404i
$552$		0			0
$553$	−21512.6	+	122004.i	−0.0703465	+	0.398955i
$554$		0			0
$555$	294265.	+	741501.i	0.955328	+	2.40728i
$556$		0			0
$557$	−291717.	−	168423.i	−0.940267	−	0.542863i	−0.0502227	−	0.998738i	$-0.515993\pi$
$557$	−291717.	−	168423.i	−0.940267	−	0.542863i	−0.890044	+	0.455875i	$0.849326\pi$
$558$		0			0
$559$	−110827.	−	191958.i	−0.354667	−	0.614302i
$560$		0			0
$561$	−8893.45	−	43150.9i	−0.0282582	−	0.137108i
$562$		0			0
$563$	60246.4	+	165526.i	0.190070	+	0.522214i	0.997723	−	0.0674444i	$-0.0214845\pi$
$563$	60246.4	+	165526.i	0.190070	+	0.522214i	−0.807653	+	0.589659i	$0.799262\pi$
$564$		0			0
$565$	123307.	+	699310.i	0.386270	+	2.19065i
$566$		0			0
$567$	−187493.	−	44727.2i	−0.583202	−	0.139125i
$568$		0			0
$569$	−486349.	+	85756.5i	−1.50219	+	0.264876i	−0.863404	−	0.504513i	$-0.831672\pi$
$569$	−486349.	+	85756.5i	−1.50219	+	0.264876i	−0.638781	+	0.769389i	$0.720561\pi$
$570$		0			0
$571$	−222277.	+	80902.4i	−0.681747	+	0.248136i	−0.659597	−	0.751619i	$-0.729273\pi$
$571$	−222277.	+	80902.4i	−0.681747	+	0.248136i	−0.0221495	+	0.999755i	$0.507051\pi$
$572$		0			0
$573$	515118.	−	106166.i	1.56891	−	0.323353i
$574$		0			0
$575$	−496810.	+	286833.i	−1.50264	+	0.867549i
$576$		0			0
$577$	−220535.	+	381978.i	−0.662409	+	1.14733i	0.317572	+	0.948234i	$0.397133\pi$
$577$	−220535.	+	381978.i	−0.662409	+	1.14733i	−0.979981	+	0.199092i	$0.936201\pi$
$578$		0			0
$579$	−208733.	+	82835.7i	−0.622635	+	0.247093i
$580$		0			0
$581$	177335.	+	31268.9i	0.525341	+	0.0926318i
$582$		0			0
$583$	−83740.7	+	70266.8i	−0.246377	+	0.206735i
$584$		0			0
$585$	327174.	+	347294.i	0.956020	+	1.01481i
$586$		0			0
$587$	−4668.67	+	12827.1i	−0.0135493	+	0.0372264i	−0.946283	−	0.323340i	$-0.895194\pi$
$587$	−4668.67	+	12827.1i	−0.0135493	+	0.0372264i	0.932734	+	0.360566i	$0.117417\pi$
$588$		0			0
$589$	−210151.	−	176337.i	−0.605760	−	0.508293i
$590$		0			0
$591$	−14016.5	+	8638.40i	−0.0401295	+	0.0247319i
$592$		0			0
$593$			38180.0i			0.108574i	0.998525	+	0.0542871i	$0.0172886\pi$
$593$			38180.0i			0.108574i	−0.998525	+	0.0542871i	$0.982711\pi$
$594$		0			0
$595$	230571.			0.651284
$596$		0			0
$597$	321024.	−	594855.i	0.900719	−	1.66902i
$598$		0			0
$599$	−95924.6	+	114318.i	−0.267348	+	0.318612i	−0.882971	−	0.469428i	$-0.844460\pi$
$599$	−95924.6	+	114318.i	−0.267348	+	0.318612i	0.615623	+	0.788041i	$0.288904\pi$
$600$		0			0
$601$	227375.	+	82757.8i	0.629498	+	0.229119i	0.637013	−	0.770853i	$-0.280170\pi$
$601$	227375.	+	82757.8i	0.629498	+	0.229119i	−0.00751483	+	0.999972i	$0.502392\pi$
$602$		0			0
$603$	20673.2	−	175753.i	0.0568557	−	0.483357i
$604$		0			0
$605$	355337.	+	423474.i	0.970799	+	1.15695i
$606$		0			0
$607$	−31249.1	+	177223.i	−0.0848126	+	0.480996i	0.912584	+	0.408889i	$0.134084\pi$
$607$	−31249.1	+	177223.i	−0.0848126	+	0.480996i	−0.997397	+	0.0721076i	$0.977027\pi$
$608$		0			0
$609$	337836.	+	49612.2i	0.910901	+	0.133768i
$610$		0			0
$611$	−314304.	−	181464.i	−0.841915	−	0.486080i
$612$		0			0
$613$	−208031.	−	360320.i	−0.553614	−	0.958888i	−0.998010	−	0.0630574i	$-0.979915\pi$
$613$	−208031.	−	360320.i	−0.553614	−	0.958888i	0.444396	−	0.895831i	$-0.353418\pi$
$614$		0			0
$615$	−124008.	+	110276.i	−0.327869	+	0.291563i
$616$		0			0
$617$	86694.4	+	238191.i	0.227730	+	0.625683i	0.999953	−	0.00966797i	$-0.00307746\pi$
$617$	86694.4	+	238191.i	0.227730	+	0.625683i	−0.772223	+	0.635351i	$0.780855\pi$
$618$		0			0
$619$	7168.59	+	40655.1i	0.0187091	+	0.106105i	0.992732	−	0.120344i	$-0.0383997\pi$
$619$	7168.59	+	40655.1i	0.0187091	+	0.106105i	−0.974023	+	0.226448i	$0.927289\pi$
$620$		0			0
$621$	190454.	+	409606.i	0.493864	+	1.06214i
$622$		0			0
$623$	−235928.	+	41600.4i	−0.607859	+	0.107182i
$624$		0			0
$625$	105378.	−	38354.4i	0.269767	−	0.0981873i
$626$		0			0
$627$	−18082.5	+	54502.8i	−0.0459962	+	0.138639i
$628$		0			0
$629$	−388484.	+	224291.i	−0.981910	+	0.566906i
$630$		0			0
$631$	123269.	−	213508.i	0.309596	−	0.536236i	−0.668678	−	0.743552i	$-0.733140\pi$
$631$	123269.	−	213508.i	0.309596	−	0.536236i	0.978274	+	0.207316i	$0.0664729\pi$
$632$		0			0
$633$	−488862.	−	386833.i	−1.22005	−	0.965421i
$634$		0			0
$635$	772351.	+	136186.i	1.91543	+	0.337743i
$636$		0			0
$637$	−176220.	+	147866.i	−0.434287	+	0.364410i
$638$		0			0
$639$	354585.	+	178405.i	0.868397	+	0.436923i
$640$		0			0
$641$	−57343.1	+	157549.i	−0.139561	+	0.383441i	−0.989707	−	0.143105i	$-0.954291\pi$
$641$	−57343.1	+	157549.i	−0.139561	+	0.383441i	0.850146	+	0.526547i	$0.176513\pi$
$642$		0			0
$643$	−10320.7	−	8660.10i	−0.0249625	−	0.0209460i	0.630221	−	0.776416i	$-0.282964\pi$
$643$	−10320.7	−	8660.10i	−0.0249625	−	0.0209460i	−0.655184	+	0.755470i	$0.727409\pi$
$644$		0			0
$645$	−15082.6	−	524977.i	−0.0362541	−	1.26189i
$646$		0			0
$647$		−	570875.i		−	1.36374i	−0.731472	−	0.681872i	$-0.761166\pi$
$647$		−	570875.i		−	1.36374i	0.731472	−	0.681872i	$-0.238834\pi$
$648$		0			0
$649$	30317.2			0.0719780
$650$		0			0
$651$	279132.	−	8019.49i	0.658640	−	0.0189228i
$652$		0			0
$653$	−426029.	+	507721.i	−0.999109	+	1.19069i	−0.0174882	+	0.999847i	$0.505567\pi$
$653$	−426029.	+	507721.i	−0.999109	+	1.19069i	−0.981620	+	0.190844i	$0.938877\pi$
$654$		0			0
$655$	327233.	+	119103.i	0.762737	+	0.277614i
$656$		0			0
$657$	−279701.	−	425945.i	−0.647984	−	0.986787i
$658$		0			0
$659$	−79144.6	−	94320.9i	−0.182243	−	0.217189i	0.667187	−	0.744890i	$-0.267498\pi$
$659$	−79144.6	−	94320.9i	−0.182243	−	0.217189i	−0.849430	+	0.527702i	$0.823054\pi$
$660$		0			0
$661$	−4335.49	+	24587.8i	−0.00992282	+	0.0562751i	−0.989367	−	0.145440i	$-0.953540\pi$
$661$	−4335.49	+	24587.8i	−0.00992282	+	0.0562751i	0.979444	+	0.201715i	$0.0646514\pi$
$662$		0			0
$663$	−166483.	+	210393.i	−0.378741	+	0.478635i
$664$		0			0
$665$	−260259.	−	150261.i	−0.588522	−	0.339783i
$666$		0			0
$667$	−400105.	−	693002.i	−0.899337	−	1.55770i
$668$		0			0
$669$	−456942.	−	151600.i	−1.02096	−	0.338725i
$670$		0			0
$671$	−42825.1	−	117661.i	−0.0951160	−	0.261329i
$672$		0			0
$673$	−45851.8	−	260038.i	−0.101234	−	0.574126i	−0.992658	−	0.120956i	$-0.961404\pi$
$673$	−45851.8	−	260038.i	−0.101234	−	0.574126i	0.891424	−	0.453170i	$-0.149707\pi$
$674$		0			0
$675$	173962.	+	652105.i	0.381811	+	1.43123i
$676$		0			0
$677$	598424.	−	105518.i	1.30566	−	0.230224i	0.522821	−	0.852443i	$-0.324880\pi$
$677$	598424.	−	105518.i	1.30566	−	0.230224i	0.782843	+	0.622219i	$0.213769\pi$
$678$		0			0
$679$	−61584.5	+	22414.9i	−0.133577	+	0.0486180i
$680$		0			0
$681$	385936.	+	433993.i	0.832187	+	0.935812i
$682$		0			0
$683$	641236.	−	370218.i	1.37460	−	0.793626i	0.383097	−	0.923708i	$-0.374857\pi$
$683$	641236.	−	370218.i	1.37460	−	0.793626i	0.991503	+	0.130082i	$0.0415241\pi$
$684$		0			0
$685$	335782.	−	581591.i	0.715609	−	1.23947i
$686$		0			0
$687$	−38978.0	+	265422.i	−0.0825858	+	0.562371i
$688$		0			0
$689$	655575.	+	115596.i	1.38097	+	0.243502i
$690$		0			0
$691$	−564242.	+	473456.i	−1.18171	+	0.991569i	−0.181740	+	0.983347i	$0.558173\pi$
$691$	−564242.	+	473456.i	−1.18171	+	0.991569i	−0.999966	+	0.00822279i	$0.997383\pi$
$692$		0			0
$693$	−23112.7	−	53689.4i	−0.0481265	−	0.111795i
$694$		0			0
$695$	221871.	−	609584.i	0.459335	−	1.26201i
$696$		0			0
$697$	−71482.4	−	59980.9i	−0.147141	−	0.123466i
$698$		0			0
$699$	−336053.	−	181357.i	−0.687785	−	0.371176i
$700$		0			0
$701$		−	642240.i		−	1.30696i	−0.756945	−	0.653479i	$-0.773309\pi$
$701$		−	642240.i		−	1.30696i	0.756945	−	0.653479i	$-0.226691\pi$
$702$		0			0
$703$	584674.			1.18305
$704$		0			0
$705$	−451176.	−	732068.i	−0.907753	−	1.47290i
$706$		0			0
$707$	86692.0	−	103315.i	0.173436	−	0.206693i
$708$		0			0
$709$	−372966.	−	135749.i	−0.741954	−	0.270049i	−0.0567381	−	0.998389i	$-0.518070\pi$
$709$	−372966.	−	135749.i	−0.741954	−	0.270049i	−0.685216	+	0.728340i	$0.740292\pi$
$710$		0			0
$711$	−78526.8	−	332417.i	−0.155338	−	0.657572i
$712$		0			0
$713$	−420651.	−	501312.i	−0.827452	−	0.986119i
$714$		0			0
$715$	−25125.4	+	142493.i	−0.0491475	+	0.278729i
$716$		0			0
$717$	162943.	+	410590.i	0.316955	+	0.798676i
$718$		0			0
$719$	567107.	+	327419.i	1.09700	+	0.633354i	0.935432	−	0.353507i	$-0.115011\pi$
$719$	567107.	+	327419.i	1.09700	+	0.633354i	0.161569	+	0.986861i	$0.448344\pi$
$720$		0			0
$721$	182554.	+	316192.i	0.351172	+	0.608248i
$722$		0			0
$723$	134609.	+	653119.i	0.257511	+	1.24944i
$724$		0			0
$725$	−408915.	−	1.12349e6i	−0.777960	−	2.13743i
$726$		0			0
$727$	−12221.8	−	69313.1i	−0.0231241	−	0.131143i	0.971060	−	0.238835i	$-0.0767654\pi$
$727$	−12221.8	−	69313.1i	−0.0231241	−	0.131143i	−0.994184	+	0.107691i	$0.965654\pi$
$728$		0			0
$729$	523569.	−	91132.2i	0.985187	−	0.171481i
$730$		0			0
$731$	290832.	−	51281.6i	0.544262	−	0.0959680i
$732$		0			0
$733$	617038.	−	224583.i	1.14843	−	0.417994i	0.303478	−	0.952839i	$-0.401852\pi$
$733$	617038.	−	224583.i	1.14843	−	0.417994i	0.844950	+	0.534845i	$0.179630\pi$
$734$		0			0
$735$	−533841.	+	110025.i	−0.988184	+	0.203666i
$736$		0			0
$737$	46475.1	−	26832.4i	0.0855629	−	0.0493998i
$738$		0			0
$739$	45948.1	−	79584.4i	0.0841353	−	0.145727i	−0.820887	−	0.571091i	$-0.806521\pi$
$739$	45948.1	−	79584.4i	0.0841353	−	0.145727i	0.905022	+	0.425364i	$0.139854\pi$
$740$		0			0
$741$	325030.	−	128988.i	0.591954	−	0.234917i
$742$		0			0
$743$	842543.	+	148563.i	1.52621	+	0.269112i	0.872870	−	0.487952i	$-0.162256\pi$
$743$	842543.	+	148563.i	1.52621	+	0.269112i	0.653340	+	0.757064i	$0.273367\pi$
$744$		0			0
$745$	519025.	−	435514.i	0.935138	−	0.784674i
$746$		0			0
$747$	−483172.	+	114140.i	−0.865886	+	0.204548i
$748$		0			0
$749$	−177033.	+	486395.i	−0.315567	+	0.867013i
$750$		0			0
$751$	327108.	+	274476.i	0.579977	+	0.486659i	0.884940	−	0.465706i	$-0.154199\pi$
$751$	327108.	+	274476.i	0.579977	+	0.486659i	−0.304962	+	0.952364i	$0.598644\pi$
$752$		0			0
$753$	−74641.7	+	46001.9i	−0.131641	+	0.0811308i
$754$		0			0
$755$			1.10826e6i			1.94423i
$756$		0			0
$757$	33072.5			0.0577133			0.0288566	−	0.999584i	$-0.490813\pi$
$757$	33072.5			0.0577133			0.0288566	+	0.999584i	$0.490813\pi$
$758$		0			0
$759$	−65057.1	+	120550.i	−0.112931	+	0.209259i
$760$		0			0
$761$	−46066.4	+	54899.8i	−0.0795454	+	0.0947985i	−0.804349	−	0.594157i	$-0.797486\pi$
$761$	−46066.4	+	54899.8i	−0.0795454	+	0.0947985i	0.724804	+	0.688955i	$0.241930\pi$
$762$		0			0
$763$	−649285.	−	236320.i	−1.11529	−	0.405931i
$764$		0			0
$765$	−583900.	+	251362.i	−0.997735	+	0.429513i
$766$		0			0
$767$	−118671.	−	141427.i	−0.201722	−	0.240403i
$768$		0			0
$769$	62603.8	−	355044.i	0.105864	−	0.600385i	−0.885008	−	0.465576i	$-0.845847\pi$
$769$	62603.8	−	355044.i	0.105864	−	0.600385i	0.990872	−	0.134808i	$-0.0430418\pi$
$770$		0			0
$771$	54945.1	+	8068.84i	0.0924315	+	0.0135738i
$772$		0			0
$773$	226732.	+	130904.i	0.379449	+	0.219075i	0.677579	−	0.735450i	$-0.263029\pi$
$773$	226732.	+	130904.i	0.379449	+	0.219075i	−0.298129	+	0.954526i	$0.596363\pi$
$774$		0			0
$775$	−488880.	−	846764.i	−0.813951	−	1.40981i
$776$		0			0
$777$	−444736.	+	395489.i	−0.736649	+	0.655078i
$778$		0			0
$779$	41597.6	+	114288.i	0.0685478	+	0.188333i
$780$		0			0
$781$	20902.3	+	118543.i	0.0342683	+	0.194345i
$782$		0			0
$783$	−909624.	+	242661.i	−1.48367	+	0.395801i
$784$		0			0
$785$	1.27722e6	−	225208.i	2.07265	−	0.365464i
$786$		0			0
$787$	−705096.	+	256634.i	−1.13841	+	0.414348i	−0.841339	−	0.540508i	$-0.818232\pi$
$787$	−705096.	+	256634.i	−1.13841	+	0.414348i	−0.297072	+	0.954855i	$0.596010\pi$
$788$		0			0
$789$	236664.	−	713337.i	0.380171	−	1.14588i
$790$		0			0
$791$	−458779.	+	264876.i	−0.733248	+	0.423341i
$792$		0			0
$793$	−381246.	+	660337.i	−0.606260	+	1.05007i
$794$		0			0
$795$	1.23690e6	+	978753.i	1.95705	+	1.54860i
$796$		0			0
$797$	−705182.	−	124343.i	−1.11016	−	0.195751i	−0.411642	−	0.911346i	$-0.635044\pi$
$797$	−705182.	−	124343.i	−1.11016	−	0.195751i	−0.698515	+	0.715595i	$0.746156\pi$
$798$		0			0
$799$	370416.	−	310816.i	0.580225	−	0.486867i
$800$		0			0
$801$	552114.	−	362551.i	0.860525	−	0.565073i
$802$		0			0
$803$	52851.8	−	145209.i	0.0819650	−	0.225197i
$804$		0			0
$805$	−549170.	−	460808.i	−0.847452	−	0.711096i
$806$		0			0
$807$	550.523	+	19161.9i	0.000845335	+	0.0294233i
$808$		0			0
$809$		−	503240.i		−	0.768915i	−0.923143	−	0.384457i	$-0.874389\pi$
$809$		−	503240.i		−	0.768915i	0.923143	−	0.384457i	$-0.125611\pi$
$810$		0			0
$811$	−933666.			−1.41955			−0.709773	−	0.704430i	$-0.751203\pi$
$811$	−933666.			−1.41955			−0.709773	+	0.704430i	$0.751203\pi$
$812$		0			0
$813$	30620.2	−	879.721i	0.0463262	−	0.00133096i
$814$		0			0
$815$	−501188.	+	597293.i	−0.754546	+	0.899233i
$816$		0			0
$817$	−361700.	−	131648.i	−0.541881	−	0.197229i
$818$		0			0
$819$	−159985.	+	317976.i	−0.238513	+	0.474052i
$820$		0			0
$821$	628774.	+	749344.i	0.932843	+	1.11172i	0.993531	+	0.113562i	$0.0362260\pi$
$821$	628774.	+	749344.i	0.932843	+	1.11172i	−0.0606883	+	0.998157i	$0.519330\pi$
$822$		0			0
$823$	−14825.8	+	84081.1i	−0.0218886	+	0.124136i	−0.993794	−	0.111235i	$-0.964519\pi$
$823$	−14825.8	+	84081.1i	−0.0218886	+	0.124136i	0.971906	+	0.235371i	$0.0756305\pi$
$824$		0			0
$825$	−127001.	+	160498.i	−0.186595	+	0.235810i
$826$		0			0
$827$	113505.	+	65532.1i	0.165960	+	0.0958171i	0.580680	−	0.814132i	$-0.302787\pi$
$827$	113505.	+	65532.1i	0.165960	+	0.0958171i	−0.414719	+	0.909949i	$0.636120\pi$
$828$		0			0
$829$	376064.	+	651363.i	0.547209	+	0.947794i	0.998464	+	0.0553988i	$0.0176430\pi$
$829$	376064.	+	651363.i	0.547209	+	0.947794i	−0.451255	+	0.892395i	$0.649024\pi$
$830$		0			0
$831$	77798.1	+	25811.1i	0.112659	+	0.0373771i
$832$		0			0
$833$	−104826.	−	288007.i	−0.151070	−	0.415062i
$834$		0			0
$835$	−297392.	−	1.68660e6i	−0.426537	−	2.41901i
$836$		0			0
$837$	−698134.	+	324611.i	−0.996525	+	0.463353i
$838$		0			0
$839$	−486809.	+	85837.6i	−0.691568	+	0.121942i	−0.508376	−	0.861135i	$-0.669754\pi$
$839$	−486809.	+	85837.6i	−0.691568	+	0.121942i	−0.183192	+	0.983077i	$0.558643\pi$
$840$		0			0
$841$	902527.	−	328493.i	1.27605	−	0.464445i
$842$		0			0
$843$	755.738	+	849.843i	0.00106345	+	0.00119587i
$844$		0			0
$845$	−210987.	+	121814.i	−0.295490	+	0.170601i
$846$		0			0
$847$	−206204.	+	357156.i	−0.287429	+	0.497842i
$848$		0			0
$849$	−73076.9	+	497620.i	−0.101383	+	0.690370i
$850$		0			0
$851$	1.37354e6	+	242193.i	1.89663	+	0.334427i
$852$		0			0
$853$	79924.2	−	67064.4i	0.109845	−	0.0921709i	−0.586211	−	0.810159i	$-0.699381\pi$
$853$	79924.2	−	67064.4i	0.109845	−	0.0921709i	0.696056	+	0.717988i	$0.254937\pi$
$854$		0			0
$855$	822893.	+	96794.1i	1.12567	+	0.132409i
$856$		0			0
$857$	137961.	−	379044.i	0.187843	−	0.516094i	−0.809646	−	0.586918i	$-0.800341\pi$
$857$	137961.	−	379044.i	0.187843	−	0.516094i	0.997489	+	0.0708248i	$0.0225631\pi$
$858$		0			0
$859$	−47511.5	−	39866.9i	−0.0643891	−	0.0540289i	0.610025	−	0.792382i	$-0.291159\pi$
$859$	−47511.5	−	39866.9i	−0.0643891	−	0.0540289i	−0.674414	+	0.738353i	$0.735604\pi$
$860$		0			0
$861$	−108949.	−	58796.5i	−0.146967	−	0.0793132i
$862$		0			0
$863$			1.21791e6i			1.63529i	0.575721	+	0.817646i	$0.304721\pi$
$863$			1.21791e6i			1.63529i	−0.575721	+	0.817646i	$0.695279\pi$
$864$		0			0
$865$	−747291.			−0.998751
$866$		0			0
$867$	206838.	+	335610.i	0.275164	+	0.446475i
$868$		0			0
$869$	66580.2	−	79347.2i	0.0881669	−	0.105073i
$870$		0			0
$871$	−307089.	−	111771.i	−0.404788	−	0.147331i
$872$		0			0
$873$	131521.	−	123901.i	0.172570	−	0.162573i
$874$		0			0
$875$	−223698.	−	266592.i	−0.292176	−	0.348202i
$876$		0			0
$877$	174422.	−	989197.i	0.226779	−	1.28613i	−0.632477	−	0.774579i	$-0.717962\pi$
$877$	174422.	−	989197.i	0.226779	−	1.28613i	0.859255	−	0.511547i	$-0.170927\pi$
$878$		0			0
$879$	111325.	+	280522.i	0.144084	+	0.363069i
$880$		0			0
$881$	1.02978e6	+	594546.i	1.32677	+	0.766008i	0.984798	−	0.173703i	$-0.0555734\pi$
$881$	1.02978e6	+	594546.i	1.32677	+	0.766008i	0.341968	+	0.939712i	$0.388907\pi$
$882$		0			0
$883$	243751.	+	422190.i	0.312627	+	0.541485i	0.978930	−	0.204196i	$-0.0654578\pi$
$883$	243751.	+	422190.i	0.312627	+	0.541485i	−0.666304	+	0.745681i	$0.732125\pi$
$884$		0			0
$885$	−88301.6	−	428438.i	−0.112741	−	0.547018i
$886$		0			0
$887$	−56179.3	−	154351.i	−0.0714051	−	0.196184i	0.898856	−	0.438243i	$-0.144399\pi$
$887$	−56179.3	−	154351.i	−0.0714051	−	0.196184i	−0.970261	+	0.242060i	$0.922177\pi$
$888$		0			0
$889$	101599.	+	576196.i	0.128554	+	0.729065i
$890$		0			0
$891$	117061.	+	110767.i	0.147455	+	0.139526i
$892$		0			0
$893$	−620667.	+	109440.i	−0.778316	+	0.137238i
$894$		0			0
$895$	−655393.	+	238544.i	−0.818194	+	0.297798i
$896$		0			0
$897$	817008.	−	168386.i	1.01541	−	0.209277i
$898$		0			0
$899$	1.18116e6	−	681940.i	1.46146	−	0.843775i
$900$		0			0
$901$	−443463.	+	768100.i	−0.546270	+	0.946168i
$902$		0			0
$903$	364180.	−	144525.i	0.446622	−	0.177242i
$904$		0			0
$905$	−2.05869e6	−	363003.i	−2.51359	−	0.443214i
$906$		0			0
$907$	−259809.	+	218006.i	−0.315820	+	0.265005i	−0.786893	−	0.617090i	$-0.788312\pi$
$907$	−259809.	+	218006.i	−0.315820	+	0.265005i	0.471072	+	0.882095i	$0.343867\pi$
$908$		0			0
$909$	−106908.	+	356146.i	−0.129384	+	0.431023i
$910$		0			0
$911$	−116224.	+	319324.i	−0.140043	+	0.384764i	−0.989810	−	0.142394i	$-0.954520\pi$
$911$	−116224.	+	319324.i	−0.140043	+	0.384764i	0.849767	+	0.527158i	$0.176742\pi$
$912$		0			0
$913$	−115332.	−	96775.3i	−0.138360	−	0.116098i
$914$		0			0
$915$	−1.53804e6	+	947897.i	−1.83707	+	1.13219i
$916$		0			0
$917$			259793.i			0.308950i
$918$		0			0
$919$	−514842.			−0.609598			−0.304799	−	0.952417i	$-0.598589\pi$
$919$	−514842.			−0.609598			−0.304799	+	0.952417i	$0.598589\pi$
$920$		0			0
$921$	−515449.	+	955122.i	−0.607668	+	1.12600i
$922$		0			0
$923$	471172.	−	561521.i	0.553065	−	0.659117i
$924$		0			0
$925$	1.95819e6	+	712721.i	2.28860	+	0.832983i
$926$		0			0
$927$	−807005.	−	601713.i	−0.939110	−	0.700213i
$928$		0			0
$929$	405667.	+	483455.i	0.470043	+	0.560176i	0.948026	−	0.318193i	$-0.103076\pi$
$929$	405667.	+	483455.i	0.470043	+	0.560176i	−0.477982	+	0.878369i	$0.658632\pi$
$930$		0			0
$931$	−69368.0	+	393405.i	−0.0800313	+	0.453880i
$932$		0			0
$933$	−1.43292e6	−	210429.i	−1.64611	−	0.241736i
$934$		0			0
$935$	−166951.	−	96389.4i	−0.190971	−	0.110257i
$936$		0			0
$937$	723432.	+	1.25302e6i	0.823984	+	1.42718i	0.902694	+	0.430284i	$0.141587\pi$
$937$	723432.	+	1.25302e6i	0.823984	+	1.42718i	−0.0787098	+	0.996898i	$0.525080\pi$
$938$		0			0
$939$	−352758.	+	313696.i	−0.400079	+	0.355777i
$940$		0			0
$941$	−352733.	−	969127.i	−0.398352	−	1.09446i	−0.963087	−	0.269191i	$-0.913244\pi$
$941$	−352733.	−	969127.i	−0.398352	−	1.09446i	0.564735	−	0.825273i	$-0.308979\pi$
$942$		0			0
$943$	50380.7	+	285723.i	0.0566553	+	0.321308i
$944$		0			0
$945$	−691414.	+	483000.i	−0.774238	+	0.540859i
$946$		0			0
$947$	−1.13226e6	+	199649.i	−1.26255	+	0.222621i	−0.764554	−	0.644560i	$-0.777041\pi$
$947$	−1.13226e6	+	199649.i	−1.26255	+	0.222621i	−0.497993	+	0.867181i	$0.665930\pi$
$948$		0			0
$949$	−884264.	+	321846.i	−0.981859	+	0.357368i
$950$		0			0
$951$	362890.	−	1.09380e6i	0.401249	−	1.20942i
$952$		0			0
$953$	−42532.6	+	24556.2i	−0.0468313	+	0.0270381i	−0.523233	−	0.852190i	$-0.675274\pi$
$953$	−42532.6	+	24556.2i	−0.0468313	+	0.0270381i	0.476402	+	0.879228i	$0.341941\pi$
$954$		0			0
$955$	1.15066e6	−	1.99299e6i	1.26165	−	2.18524i
$956$		0			0
$957$	−223879.	−	177154.i	−0.244450	−	0.193432i
$958$		0			0
$959$	493393.	+	86998.6i	0.536483	+	0.0945965i
$960$		0			0
$961$	146980.	−	123331.i	0.159152	−	0.133544i
$962$		0			0
$963$	−81933.8	−	1.42475e6i	−0.0883509	−	1.53633i
$964$		0			0
$965$	−336076.	+	923360.i	−0.360896	+	0.991554i
$966$		0			0
$967$	204190.	+	171336.i	0.218364	+	0.183229i	0.745408	−	0.666609i	$-0.232255\pi$
$967$	204190.	+	171336.i	0.218364	+	0.183229i	−0.527043	+	0.849839i	$0.676699\pi$
$968$		0			0
$969$	13380.0	+	465715.i	0.0142498	+	0.495989i
$970$		0			0
$971$		−	1.58508e6i		−	1.68117i	−0.541677	−	0.840587i	$-0.682210\pi$
$971$		−	1.58508e6i		−	1.68117i	0.541677	−	0.840587i	$-0.317790\pi$
$972$		0			0
$973$	483953.			0.511184
$974$		0			0
$975$	1.24583e6	−	35792.7i	1.31054	−	0.0376518i
$976$		0			0
$977$	−873049.	+	1.04046e6i	−0.914638	+	1.09002i	0.0809991	+	0.996714i	$0.474189\pi$
$977$	−873049.	+	1.04046e6i	−0.914638	+	1.09002i	−0.995637	+	0.0933090i	$0.970256\pi$
$978$		0			0
$979$	188221.	+	68506.9i	0.196383	+	0.0714774i
$980$		0			0
$981$	1.90188e6	−	109373.i	1.97627	−	0.113651i
$982$		0			0
$983$	653525.	+	778841.i	0.676324	+	0.806012i	0.989630	−	0.143640i	$-0.0458808\pi$
$983$	653525.	+	778841.i	0.676324	+	0.806012i	−0.313306	+	0.949652i	$0.601436\pi$
$984$		0			0
$985$	−12510.0	+	70947.8i	−0.0128939	+	0.0731251i
$986$		0			0
$987$	398086.	−	503083.i	0.408642	−	0.516423i
$988$		0			0
$989$	−795188.	−	459102.i	−0.812975	−	0.469371i
$990$		0			0
$991$	948825.	+	1.64341e6i	0.966137	+	1.67340i	0.706529	+	0.707684i	$0.250260\pi$
$991$	948825.	+	1.64341e6i	0.966137	+	1.67340i	0.259608	+	0.965714i	$0.416407\pi$
$992$		0			0
$993$	−190757.	−	63287.6i	−0.193456	−	0.0641830i
$994$		0			0
$995$	−1.01158e6	−	2.77931e6i	−1.02178	−	2.80731i
$996$		0			0
$997$	20717.2	+	117493.i	0.0208421	+	0.118201i	0.993454	−	0.114235i	$-0.0364415\pi$
$997$	20717.2	+	117493.i	0.0208421	+	0.118201i	−0.972612	+	0.232436i	$0.925330\pi$
$998$		0			0
$999$	695102.	−	1.48638e6i	0.696494	−	1.48936i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.41.1	yes	72
3.2	odd	2		324.5.k.a.233.12		72
27.2	odd	18	inner	108.5.k.a.29.1	✓	72
27.25	even	9		324.5.k.a.89.12		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.29.1	✓	72	27.2	odd	18	inner
108.5.k.a.41.1	yes	72	1.1	even	1	trivial
324.5.k.a.89.12		72	27.25	even	9
324.5.k.a.233.12		72	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-8.99629 + 0.258464i) q^{3} +(-25.3131 + 30.1670i) q^{5} +(-27.6070 - 10.0481i) q^{7} +(80.8664 - 4.65043i) q^{9} +O(q^{10})\)	\(q+(-8.99629 + 0.258464i) q^{3} +(-25.3131 + 30.1670i) q^{5} +(-27.6070 - 10.0481i) q^{7} +(80.8664 - 4.65043i) q^{9} +(15.7890 + 18.8166i) q^{11} +(25.9745 - 147.308i) q^{13} +(219.927 - 277.934i) q^{15} +(172.593 + 99.6466i) q^{17} +(-129.877 - 224.954i) q^{19} +(250.958 + 83.2604i) q^{21} +(-211.930 - 582.273i) q^{23} +(-160.764 - 911.738i) q^{25} +(-726.295 + 62.7377i) q^{27} +(1271.79 - 224.250i) q^{29} +(992.428 - 361.214i) q^{31} +(-146.906 - 165.199i) q^{33} +(1001.94 - 578.471i) q^{35} +(-1125.43 + 1949.31i) q^{37} +(-195.600 + 1331.94i) q^{39} +(-461.110 - 81.3061i) q^{41} +(1135.15 - 952.502i) q^{43} +(-1906.69 + 2557.21i) q^{45} +(829.841 - 2279.97i) q^{47} +(-1178.09 - 988.536i) q^{49} +(-1578.45 - 851.841i) q^{51} +4450.35i q^{53} -967.312 q^{55} +(1226.56 + 1990.19i) q^{57} +(793.356 - 945.485i) q^{59} +(-4790.10 - 1743.45i) q^{61} +(-2279.21 - 684.171i) q^{63} +(3786.36 + 4512.41i) q^{65} +(379.378 - 2151.56i) q^{67} +(2057.08 + 5183.52i) q^{69} +(4243.91 + 2450.22i) q^{71} +(-3145.50 - 5448.16i) q^{73} +(1681.93 + 8160.70i) q^{75} +(-246.816 - 678.121i) q^{77} +(-732.250 - 4152.80i) q^{79} +(6517.75 - 752.128i) q^{81} +(-6036.15 + 1064.34i) q^{83} +(-7374.91 + 2684.25i) q^{85} +(-11383.4 + 2346.13i) q^{87} +(7061.95 - 4077.22i) q^{89} +(-2197.25 + 3805.75i) q^{91} +(-8834.81 + 3506.10i) q^{93} +(10073.8 + 1776.28i) q^{95} +(1708.86 - 1433.90i) q^{97} +(1364.31 + 1448.21i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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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