LMFDB - Embedded newform 108.5.k.a.5.5

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			5.5
Character	$\chi$	$=$	108.5
Dual form			108.5.k.a.65.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.33607 + 8.69153i) q^{3} +(15.1037 + 41.4971i) q^{5} +(8.53787 + 48.4207i) q^{7} +(-70.0855 - 40.6081i) q^{9} +O(q^{10})$	\(q+(-2.33607 + 8.69153i) q^{3} +(15.1037 + 41.4971i) q^{5} +(8.53787 + 48.4207i) q^{7} +(-70.0855 - 40.6081i) q^{9} +(43.7103 - 120.093i) q^{11} +(6.33166 + 5.31289i) q^{13} +(-395.957 + 34.3341i) q^{15} +(51.3995 + 29.6755i) q^{17} +(195.065 + 337.863i) q^{19} +(-440.795 - 38.9071i) q^{21} +(-956.638 - 168.681i) q^{23} +(-1015.11 + 851.777i) q^{25} +(516.672 - 514.287i) q^{27} +(342.577 + 408.267i) q^{29} +(267.572 - 1517.48i) q^{31} +(941.682 + 660.456i) q^{33} +(-1880.36 + 1085.63i) q^{35} +(157.509 - 272.813i) q^{37} +(-60.9684 + 42.6205i) q^{39} +(-1523.53 + 1815.67i) q^{41} +(2059.19 + 749.486i) q^{43} +(626.568 - 3521.68i) q^{45} +(1944.70 - 342.903i) q^{47} +(-15.4624 + 5.62784i) q^{49} +(-377.999 + 377.416i) q^{51} +2837.46i q^{53} +5643.70 q^{55} +(-3392.23 + 906.144i) q^{57} +(-1440.52 - 3957.81i) q^{59} +(-170.879 - 969.103i) q^{61} +(1367.89 - 3740.29i) q^{63} +(-124.838 + 342.990i) q^{65} +(3050.44 + 2559.62i) q^{67} +(3700.88 - 7920.60i) q^{69} +(-2358.33 - 1361.58i) q^{71} +(4483.19 + 7765.12i) q^{73} +(-5031.88 - 10812.7i) q^{75} +(6188.18 + 1091.14i) q^{77} +(2155.80 - 1808.93i) q^{79} +(3262.96 + 5692.08i) q^{81} +(-1372.40 - 1635.57i) q^{83} +(-455.124 + 2581.14i) q^{85} +(-4348.75 + 2023.77i) q^{87} +(-925.164 + 534.144i) q^{89} +(-203.195 + 351.944i) q^{91} +(12564.1 + 5870.55i) q^{93} +(-11074.1 + 13197.6i) q^{95} +(15343.7 + 5584.66i) q^{97} +(-7940.21 + 6641.79i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.33607	+	8.69153i	−0.259564	+	0.965726i
$3$	−2.33607	+	8.69153i	−0.259564	+	0.965726i
$4$		0			0
$5$	15.1037	+	41.4971i	0.604148	+	1.65988i	0.742777	+	0.669539i	$0.233508\pi$
$5$	15.1037	+	41.4971i	0.604148	+	1.65988i	−0.138629	+	0.990344i	$0.544270\pi$
$6$		0			0
$7$	8.53787	+	48.4207i	0.174242	+	0.988177i	0.939015	+	0.343876i	$0.111740\pi$
$7$	8.53787	+	48.4207i	0.174242	+	0.988177i	−0.764773	+	0.644300i	$0.777149\pi$
$8$		0			0
$9$	−70.0855	−	40.6081i	−0.865253	−	0.501335i
$10$		0			0
$11$	43.7103	−	120.093i	0.361242	−	0.992504i	−0.617349	−	0.786689i	$-0.711793\pi$
$11$	43.7103	−	120.093i	0.361242	−	0.992504i	0.978591	−	0.205815i	$-0.0659845\pi$
$12$		0			0
$13$	6.33166	+	5.31289i	0.0374654	+	0.0314372i	0.661328	−	0.750097i	$-0.269993\pi$
$13$	6.33166	+	5.31289i	0.0374654	+	0.0314372i	−0.623863	+	0.781534i	$0.714438\pi$
$14$		0			0
$15$	−395.957	+	34.3341i	−1.75981	+	0.152596i
$16$		0			0
$17$	51.3995	+	29.6755i	0.177853	+	0.102683i	0.586283	−	0.810106i	$-0.300590\pi$
$17$	51.3995	+	29.6755i	0.177853	+	0.102683i	−0.408431	+	0.912789i	$0.633924\pi$
$18$		0			0
$19$	195.065	+	337.863i	0.540347	+	0.935909i	0.998884	+	0.0472332i	$0.0150404\pi$
$19$	195.065	+	337.863i	0.540347	+	0.935909i	−0.458537	+	0.888675i	$0.651626\pi$
$20$		0			0
$21$	−440.795	−	38.9071i	−0.999535	−	0.0882246i
$22$		0			0
$23$	−956.638	−	168.681i	−1.80839	−	0.318868i	−0.835388	−	0.549661i	$-0.814757\pi$
$23$	−956.638	−	168.681i	−1.80839	−	0.318868i	−0.973003	+	0.230793i	$0.925868\pi$
$24$		0			0
$25$	−1015.11	+	851.777i	−1.62417	+	1.36284i
$26$		0			0
$27$	516.672	−	514.287i	0.708741	−	0.705469i
$28$		0			0
$29$	342.577	+	408.267i	0.407344	+	0.485454i	0.930245	−	0.366940i	$-0.119594\pi$
$29$	342.577	+	408.267i	0.407344	+	0.485454i	−0.522900	+	0.852394i	$0.675150\pi$
$30$		0			0
$31$	267.572	−	1517.48i	0.278431	−	1.57906i	−0.449417	−	0.893322i	$-0.648368\pi$
$31$	267.572	−	1517.48i	0.278431	−	1.57906i	0.727848	−	0.685739i	$-0.240521\pi$
$32$		0			0
$33$	941.682	+	660.456i	0.864722	+	0.606479i
$34$		0			0
$35$	−1880.36	+	1085.63i	−1.53499	+	0.886227i
$36$		0			0
$37$	157.509	−	272.813i	0.115054	−	0.199279i	−0.802747	−	0.596319i	$-0.796629\pi$
$37$	157.509	−	272.813i	0.115054	−	0.199279i	0.917801	+	0.397040i	$0.129963\pi$
$38$		0			0
$39$	−60.9684	+	42.6205i	−0.0400844	+	0.0280214i
$40$		0			0
$41$	−1523.53	+	1815.67i	−0.906323	+	1.08011i	0.0901271	+	0.995930i	$0.471273\pi$
$41$	−1523.53	+	1815.67i	−0.906323	+	1.08011i	−0.996450	+	0.0841836i	$0.973172\pi$
$42$		0			0
$43$	2059.19	+	749.486i	1.11368	+	0.405346i	0.832342	−	0.554262i	$-0.187000\pi$
$43$	2059.19	+	749.486i	1.11368	+	0.405346i	0.281338	+	0.959609i	$0.409222\pi$
$44$		0			0
$45$	626.568	−	3521.68i	0.309416	−	1.73910i
$46$		0			0
$47$	1944.70	−	342.903i	0.880352	−	0.155230i	0.284839	−	0.958575i	$-0.408060\pi$
$47$	1944.70	−	342.903i	0.880352	−	0.155230i	0.595513	+	0.803345i	$0.296949\pi$
$48$		0			0
$49$	−15.4624	+	5.62784i	−0.00643997	+	0.00234396i
$50$		0			0
$51$	−377.999	+	377.416i	−0.145328	+	0.145104i
$52$		0			0
$53$			2837.46i			1.01013i	0.863081	+	0.505065i	$0.168531\pi$
$53$			2837.46i			1.01013i	−0.863081	+	0.505065i	$0.831469\pi$
$54$		0			0
$55$	5643.70			1.86568
$56$		0			0
$57$	−3392.23	+	906.144i	−1.04409	+	0.278899i
$58$		0			0
$59$	−1440.52	−	3957.81i	−0.413825	−	1.13697i	−0.955140	−	0.296154i	$-0.904296\pi$
$59$	−1440.52	−	3957.81i	−0.413825	−	1.13697i	0.541315	−	0.840820i	$-0.317927\pi$
$60$		0			0
$61$	−170.879	−	969.103i	−0.0459229	−	0.260442i	0.953199	−	0.302344i	$-0.0977692\pi$
$61$	−170.879	−	969.103i	−0.0459229	−	0.260442i	−0.999122	+	0.0419026i	$0.986658\pi$
$62$		0			0
$63$	1367.89	−	3740.29i	0.344644	−	0.942377i
$64$		0			0
$65$	−124.838	+	342.990i	−0.0295475	+	0.0811810i
$66$		0			0
$67$	3050.44	+	2559.62i	0.679536	+	0.570199i	0.915871	−	0.401473i	$-0.131502\pi$
$67$	3050.44	+	2559.62i	0.679536	+	0.570199i	−0.236334	+	0.971672i	$0.575946\pi$
$68$		0			0
$69$	3700.88	−	7920.60i	0.777332	−	1.66364i
$70$		0			0
$71$	−2358.33	−	1361.58i	−0.467830	−	0.270102i	0.247501	−	0.968888i	$-0.420391\pi$
$71$	−2358.33	−	1361.58i	−0.467830	−	0.270102i	−0.715331	+	0.698786i	$0.753724\pi$
$72$		0			0
$73$	4483.19	+	7765.12i	0.841282	+	1.45714i	0.888811	+	0.458274i	$0.151532\pi$
$73$	4483.19	+	7765.12i	0.841282	+	1.45714i	−0.0475290	+	0.998870i	$0.515135\pi$
$74$		0			0
$75$	−5031.88	−	10812.7i	−0.894556	−	1.92225i
$76$		0			0
$77$	6188.18	+	1091.14i	1.04371	+	0.184035i
$78$		0			0
$79$	2155.80	−	1808.93i	0.345426	−	0.289847i	−0.453524	−	0.891244i	$-0.649834\pi$
$79$	2155.80	−	1808.93i	0.345426	−	0.289847i	0.798950	+	0.601397i	$0.205389\pi$
$80$		0			0
$81$	3262.96	+	5692.08i	0.497327	+	0.867563i
$82$		0			0
$83$	−1372.40	−	1635.57i	−0.199217	−	0.237417i	0.657182	−	0.753732i	$-0.271748\pi$
$83$	−1372.40	−	1635.57i	−0.199217	−	0.237417i	−0.856399	+	0.516314i	$0.827304\pi$
$84$		0			0
$85$	−455.124	+	2581.14i	−0.0629930	+	0.357251i
$86$		0			0
$87$	−4348.75	+	2023.77i	−0.574548	+	0.267377i
$88$		0			0
$89$	−925.164	+	534.144i	−0.116799	+	0.0674339i	−0.557261	−	0.830337i	$-0.688148\pi$
$89$	−925.164	+	534.144i	−0.116799	+	0.0674339i	0.440462	+	0.897771i	$0.354814\pi$
$90$		0			0
$91$	−203.195	+	351.944i	−0.0245375	+	0.0425001i
$92$		0			0
$93$	12564.1	+	5870.55i	1.45267	+	0.678755i
$94$		0			0
$95$	−11074.1	+	13197.6i	−1.22705	+	1.46234i
$96$		0			0
$97$	15343.7	+	5584.66i	1.63075	+	0.593544i	0.985387	−	0.170332i	$-0.0544841\pi$
$97$	15343.7	+	5584.66i	1.63075	+	0.593544i	0.645363	+	0.763876i	$0.276706\pi$
$98$		0			0
$99$	−7940.21	+	6641.79i	−0.810143	+	0.677664i
$100$		0			0
$101$	−15575.1	+	2746.31i	−1.52682	+	0.269219i	−0.873109	−	0.487526i	$-0.837900\pi$
$101$	−15575.1	+	2746.31i	−1.52682	+	0.269219i	−0.653710	+	0.756745i	$0.726788\pi$
$102$		0			0
$103$	−10316.3	+	3754.83i	−0.972412	+	0.353929i	−0.778885	−	0.627166i	$-0.784215\pi$
$103$	−10316.3	+	3754.83i	−0.972412	+	0.353929i	−0.193526	+	0.981095i	$0.561992\pi$
$104$		0			0
$105$	−5043.10	−	18879.3i	−0.457424	−	1.71241i
$106$		0			0
$107$		−	3099.49i		−	0.270721i	−0.990796	−	0.135361i	$-0.956781\pi$
$107$		−	3099.49i		−	0.270721i	0.990796	−	0.135361i	$-0.0432193\pi$
$108$		0			0
$109$	−2838.56			−0.238916			−0.119458	−	0.992839i	$-0.538116\pi$
$109$	−2838.56			−0.238916			−0.119458	+	0.992839i	$0.538116\pi$
$110$		0			0
$111$	2003.22	+	2006.31i	0.162585	+	0.162836i
$112$		0			0
$113$	7212.14	+	19815.2i	0.564816	+	1.55182i	0.812489	+	0.582976i	$0.198112\pi$
$113$	7212.14	+	19815.2i	0.564816	+	1.55182i	−0.247673	+	0.968844i	$0.579666\pi$
$114$		0			0
$115$	−7449.01	−	42245.4i	−0.563252	−	3.19436i
$116$		0			0
$117$	−228.011	−	629.473i	−0.0166565	−	0.0459839i
$118$		0			0
$119$	−998.065	+	2742.16i	−0.0704799	+	0.193642i
$120$		0			0
$121$	−1296.09	−	1087.55i	−0.0885246	−	0.0742810i
$122$		0			0
$123$	−12221.9	−	17483.3i	−0.807845	−	1.15562i
$124$		0			0
$125$	−26775.7	−	15458.9i	−1.71364	−	0.989372i
$126$		0			0
$127$	−5070.21	−	8781.86i	−0.314354	−	0.544476i	0.664946	−	0.746891i	$-0.268454\pi$
$127$	−5070.21	−	8781.86i	−0.314354	−	0.544476i	−0.979300	+	0.202415i	$0.935121\pi$
$128$		0			0
$129$	−11324.6	+	16146.7i	−0.680525	+	0.970297i
$130$		0			0
$131$	7455.32	+	1314.57i	0.434434	+	0.0766024i	0.386589	−	0.922252i	$-0.373653\pi$
$131$	7455.32	+	1314.57i	0.434434	+	0.0766024i	0.0478455	+	0.998855i	$0.484764\pi$
$132$		0			0
$133$	−14694.1	+	12329.8i	−0.830692	+	0.697033i
$134$		0			0
$135$	29145.1	+	13672.7i	1.59918	+	0.750219i
$136$		0			0
$137$	2071.34	+	2468.52i	0.110359	+	0.131521i	0.818396	−	0.574654i	$-0.194863\pi$
$137$	2071.34	+	2468.52i	0.110359	+	0.131521i	−0.708037	+	0.706175i	$0.750419\pi$
$138$		0			0
$139$	−5631.07	+	31935.4i	−0.291448	+	1.65289i	0.389849	+	0.920879i	$0.372527\pi$
$139$	−5631.07	+	31935.4i	−0.291448	+	1.65289i	−0.681298	+	0.732007i	$0.738584\pi$
$140$		0			0
$141$	−1562.61	+	17703.5i	−0.0785980	+	0.890471i
$142$		0			0
$143$	914.800	−	528.160i	0.0447357	−	0.0258282i
$144$		0			0
$145$	−11767.7	+	20382.3i	−0.559701	+	0.969430i
$146$		0			0
$147$	−12.7933	−	147.539i	−0.000592037	−	0.00682765i
$148$		0			0
$149$	3566.54	−	4250.44i	0.160648	−	0.191453i	−0.679716	−	0.733476i	$-0.737897\pi$
$149$	3566.54	−	4250.44i	0.160648	−	0.191453i	0.840364	+	0.542023i	$0.182341\pi$
$150$		0			0
$151$	40345.3	+	14684.5i	1.76945	+	0.644029i	0.999986	+	0.00534447i	$0.00170121\pi$
$151$	40345.3	+	14684.5i	1.76945	+	0.644029i	0.769469	+	0.638684i	$0.220521\pi$
$152$		0			0
$153$	−2397.29	−	4167.06i	−0.102409	−	0.178011i
$154$		0			0
$155$	67012.2	−	11816.1i	2.78927	−	0.491824i
$156$		0			0
$157$	−417.986	+	152.134i	−0.0169575	+	0.00617203i	−0.350485	−	0.936568i	$-0.613983\pi$
$157$	−417.986	+	152.134i	−0.0169575	+	0.00617203i	0.333527	+	0.942740i	$0.391761\pi$
$158$		0			0
$159$	−24661.8	−	6628.51i	−0.975509	−	0.262193i
$160$		0			0
$161$		−	47761.2i		−	1.84257i
$162$		0			0
$163$	−697.311			−0.0262453			−0.0131226	−	0.999914i	$-0.504177\pi$
$163$	−697.311			−0.0262453			−0.0131226	+	0.999914i	$0.504177\pi$
$164$		0			0
$165$	−13184.1	+	49052.4i	−0.484264	+	1.80174i
$166$		0			0
$167$	3648.98	+	10025.5i	0.130839	+	0.359479i	0.987762	−	0.155966i	$-0.0498490\pi$
$167$	3648.98	+	10025.5i	0.130839	+	0.359479i	−0.856923	+	0.515445i	$0.827627\pi$
$168$		0			0
$169$	−4947.70	−	28059.8i	−0.173233	−	0.982452i
$170$		0			0
$171$	48.7322	−	31600.5i	0.00166657	−	1.08069i
$172$		0			0
$173$	18317.0	−	50325.5i	0.612015	−	1.68150i	−0.113712	−	0.993514i	$-0.536274\pi$
$173$	18317.0	−	50325.5i	0.612015	−	1.68150i	0.725727	−	0.687983i	$-0.241504\pi$
$174$		0			0
$175$	−49910.4	−	41879.8i	−1.62973	−	1.36750i
$176$		0			0
$177$	37764.6	−	3274.63i	1.20542	−	0.104524i
$178$		0			0
$179$	6224.26	+	3593.58i	0.194259	+	0.112156i	0.593975	−	0.804484i	$-0.297558\pi$
$179$	6224.26	+	3593.58i	0.194259	+	0.112156i	−0.399716	+	0.916639i	$0.630891\pi$
$180$		0			0
$181$	22245.3	+	38530.1i	0.679019	+	1.17610i	0.975277	+	0.220987i	$0.0709279\pi$
$181$	22245.3	+	38530.1i	0.679019	+	1.17610i	−0.296258	+	0.955108i	$0.595739\pi$
$182$		0			0
$183$	8822.18	+	778.696i	0.263435	+	0.0232523i
$184$		0			0
$185$	13699.9	+	2415.67i	0.400290	+	0.0705820i
$186$		0			0
$187$	5810.51	−	4875.59i	0.166162	−	0.139426i
$188$		0			0
$189$	29313.4	+	20626.7i	0.820621	+	0.577438i
$190$		0			0
$191$	10413.0	+	12409.7i	0.285436	+	0.340170i	0.889642	−	0.456659i	$-0.150954\pi$
$191$	10413.0	+	12409.7i	0.285436	+	0.340170i	−0.604206	+	0.796828i	$0.706509\pi$
$192$		0			0
$193$	7032.75	−	39884.7i	0.188804	−	1.07076i	−0.732167	−	0.681126i	$-0.761491\pi$
$193$	7032.75	−	39884.7i	0.188804	−	1.07076i	0.920970	−	0.389633i	$-0.127398\pi$
$194$		0			0
$195$	−2689.47	−	1886.28i	−0.0707291	−	0.0496064i
$196$		0			0
$197$	−1269.74	+	733.085i	−0.0327177	+	0.0188896i	−0.516270	−	0.856426i	$-0.672680\pi$
$197$	−1269.74	+	733.085i	−0.0327177	+	0.0188896i	0.483552	+	0.875316i	$0.339346\pi$
$198$		0			0
$199$	25177.4	−	43608.5i	0.635776	−	1.10120i	−0.350574	−	0.936535i	$-0.614014\pi$
$199$	25177.4	−	43608.5i	0.635776	−	1.10120i	0.986350	−	0.164661i	$-0.0526531\pi$
$200$		0			0
$201$	−29373.1	+	20533.5i	−0.727039	+	0.508243i
$202$		0			0
$203$	−16843.7	+	20073.5i	−0.408738	+	0.487115i
$204$		0			0
$205$	−98356.0	−	35798.7i	−2.34042	−	0.851842i
$206$		0			0
$207$	60196.7	+	50669.4i	1.40486	+	1.18251i
$208$		0			0
$209$	49101.4	−	8657.89i	1.12409	−	0.198207i
$210$		0			0
$211$	−62282.9	+	22669.1i	−1.39895	+	0.509178i	−0.927868	−	0.372909i	$-0.878360\pi$
$211$	−62282.9	+	22669.1i	−1.39895	+	0.509178i	−0.471087	+	0.882087i	$0.656138\pi$
$212$		0			0
$213$	17343.5	−	17316.8i	0.382276	−	0.381687i
$214$		0			0
$215$			96770.6i			2.09347i
$216$		0			0
$217$	75761.8			1.60891
$218$		0			0
$219$	−77963.9	+	20825.9i	−1.62557	+	0.434226i
$220$		0			0
$221$	167.781	+	460.975i	0.00343525	+	0.00943828i
$222$		0			0
$223$	−8376.59	−	47506.0i	−0.168445	−	0.955297i	−0.945441	−	0.325793i	$-0.894369\pi$
$223$	−8376.59	−	47506.0i	−0.168445	−	0.955297i	0.776997	−	0.629505i	$-0.216742\pi$
$224$		0			0
$225$	105733.	−	18475.6i	2.08856	−	0.364950i
$226$		0			0
$227$	28224.9	−	77547.3i	0.547748	−	1.50493i	−0.288995	−	0.957331i	$-0.593321\pi$
$227$	28224.9	−	77547.3i	0.547748	−	1.50493i	0.836743	−	0.547595i	$-0.184457\pi$
$228$		0			0
$229$	−51383.6	−	43116.0i	−0.979837	−	0.822181i	0.00422786	−	0.999991i	$-0.498654\pi$
$229$	−51383.6	−	43116.0i	−0.979837	−	0.822181i	−0.984065	+	0.177810i	$0.943099\pi$
$230$		0			0
$231$	−23939.7	+	51235.7i	−0.448637	+	0.960172i
$232$		0			0
$233$	−74125.1	−	42796.1i	−1.36538	−	0.788302i	−0.375046	−	0.927006i	$-0.622373\pi$
$233$	−74125.1	−	42796.1i	−1.36538	−	0.788302i	−0.990334	+	0.138704i	$0.955706\pi$
$234$		0			0
$235$	43601.6	+	75520.2i	0.789527	+	1.36750i
$236$		0			0
$237$	10686.3	+	22963.0i	0.190252	+	0.408820i
$238$		0			0
$239$	−41173.0	−	7259.90i	−0.720802	−	0.127097i	−0.198799	−	0.980040i	$-0.563704\pi$
$239$	−41173.0	−	7259.90i	−0.720802	−	0.127097i	−0.522003	+	0.852943i	$0.674815\pi$
$240$		0			0
$241$	14342.0	−	12034.4i	0.246931	−	0.207200i	−0.510918	−	0.859629i	$-0.670695\pi$
$241$	14342.0	−	12034.4i	0.246931	−	0.207200i	0.757850	+	0.652429i	$0.226250\pi$
$242$		0			0
$243$	−57095.5	+	15063.0i	−0.966916	+	0.255093i
$244$		0			0
$245$	−467.078	−	556.642i	−0.00778139	−	0.00927350i
$246$		0			0
$247$	−559.943	+	3175.59i	−0.00917804	+	0.0520512i
$248$		0			0
$249$	17421.6	−	8107.50i	0.280990	−	0.130764i
$250$		0			0
$251$	10300.5	−	5946.97i	0.163497	−	0.0943949i	−0.416019	−	0.909356i	$-0.636575\pi$
$251$	10300.5	−	5946.97i	0.163497	−	0.0943949i	0.579516	+	0.814961i	$0.303242\pi$
$252$		0			0
$253$	−62072.4	+	107513.i	−0.969744	+	1.67965i
$254$		0			0
$255$	−21370.8	−	9985.46i	−0.328656	−	0.153563i
$256$		0			0
$257$	31637.2	−	37703.7i	0.478996	−	0.570845i	−0.471387	−	0.881926i	$-0.656247\pi$
$257$	31637.2	−	37703.7i	0.478996	−	0.570845i	0.950383	+	0.311081i	$0.100691\pi$
$258$		0			0
$259$	14554.6	+	5297.44i	0.216970	+	0.0789708i
$260$		0			0
$261$	−7430.70	−	42525.0i	−0.109081	−	0.624257i
$262$		0			0
$263$	1880.97	−	331.666i	0.0271939	−	0.00479501i	−0.160035	−	0.987111i	$-0.551161\pi$
$263$	1880.97	−	331.666i	0.0271939	−	0.00479501i	0.187229	+	0.982316i	$0.440050\pi$
$264$		0			0
$265$	−117746.	+	42856.1i	−1.67670	+	0.610268i
$266$		0			0
$267$	−2481.28	−	9288.89i	−0.0348059	−	0.130299i
$268$		0			0
$269$		−	34061.9i		−	0.470721i	−0.971908	−	0.235361i	$-0.924373\pi$
$269$		−	34061.9i		−	0.470721i	0.971908	−	0.235361i	$-0.0756271\pi$
$270$		0			0
$271$	−4544.07			−0.0618737			−0.0309369	−	0.999521i	$-0.509849\pi$
$271$	−4544.07			−0.0618737			−0.0309369	+	0.999521i	$0.509849\pi$
$272$		0			0
$273$	−2584.25	−	2588.24i	−0.0346745	−	0.0347280i
$274$		0			0
$275$	57921.8	+	159139.i	0.765908	+	2.10431i
$276$		0			0
$277$	9712.10	+	55080.0i	0.126577	+	0.717852i	0.980359	+	0.197222i	$0.0631918\pi$
$277$	9712.10	+	55080.0i	0.126577	+	0.717852i	−0.853782	+	0.520630i	$0.825697\pi$
$278$		0			0
$279$	−80374.9	+	95487.6i	−1.03255	+	1.22670i
$280$		0			0
$281$	−11916.1	+	32739.2i	−0.150911	+	0.414625i	−0.991995	−	0.126280i	$-0.959696\pi$
$281$	−11916.1	+	32739.2i	−0.150911	+	0.414625i	0.841083	+	0.540906i	$0.181918\pi$
$282$		0			0
$283$	77800.4	+	65282.3i	0.971424	+	0.815121i	0.982773	−	0.184814i	$-0.0591683\pi$
$283$	77800.4	+	65282.3i	0.971424	+	0.815121i	−0.0113496	+	0.999936i	$0.503613\pi$
$284$		0			0
$285$	−88837.6	−	127082.i	−1.09372	−	1.56456i
$286$		0			0
$287$	−100924.	−	58268.3i	−1.22526	−	0.707406i
$288$		0			0
$289$	−39999.2	−	69280.7i	−0.478912	−	0.829500i
$290$		0			0
$291$	−84383.3	+	120314.i	−0.996484	+	1.42079i
$292$		0			0
$293$	−81181.4	−	14314.5i	−0.945630	−	0.166740i	−0.320490	−	0.947252i	$-0.603847\pi$
$293$	−81181.4	−	14314.5i	−0.945630	−	0.166740i	−0.625141	+	0.780512i	$0.714959\pi$
$294$		0			0
$295$	142480.	−	119555.i	1.63723	−	1.37380i
$296$		0			0
$297$	−39178.4	−	84528.3i	−0.444154	−	0.958273i
$298$		0			0
$299$	−5160.92	−	6150.55i	−0.0577278	−	0.0687973i
$300$		0			0
$301$	−18709.5	+	106107.i	−0.206504	+	1.17114i
$302$		0			0
$303$	12514.9	−	141787.i	0.136315	−	1.54437i
$304$		0			0
$305$	37634.0	−	21728.0i	0.404558	−	0.233572i
$306$		0			0
$307$	39583.0	−	68559.8i	0.419983	−	0.727433i	−0.575954	−	0.817482i	$-0.695369\pi$
$307$	39583.0	−	68559.8i	0.419983	−	0.727433i	0.995937	+	0.0900495i	$0.0287025\pi$
$308$		0			0
$309$	−8535.57	−	98436.2i	−0.0893955	−	1.03095i
$310$		0			0
$311$	8502.04	−	10132.3i	0.0879027	−	0.104758i	−0.720300	−	0.693662i	$-0.755996\pi$
$311$	8502.04	−	10132.3i	0.0879027	−	0.104758i	0.808203	+	0.588904i	$0.200440\pi$
$312$		0			0
$313$	18431.7	+	6708.58i	0.188138	+	0.0684766i	0.434371	−	0.900734i	$-0.356971\pi$
$313$	18431.7	+	6708.58i	0.188138	+	0.0684766i	−0.246233	+	0.969211i	$0.579193\pi$
$314$		0			0
$315$	175871.	+	271.217i	1.77245	+	0.00273335i
$316$		0			0
$317$	6921.15	−	1220.39i	0.0688747	−	0.0121445i	−0.139105	−	0.990278i	$-0.544422\pi$
$317$	6921.15	−	1220.39i	0.0688747	−	0.0121445i	0.207979	+	0.978133i	$0.433311\pi$
$318$		0			0
$319$	64004.1	−	23295.6i	0.628965	−	0.228925i
$320$		0			0
$321$	26939.3	+	7240.63i	0.261443	+	0.0702694i
$322$		0			0
$323$			23154.6i			0.221939i
$324$		0			0
$325$	−10952.7			−0.103694
$326$		0			0
$327$	6631.09	−	24671.4i	0.0620139	−	0.230727i
$328$		0			0
$329$	33207.2	+	91235.9i	0.306789	+	0.842896i
$330$		0			0
$331$	−329.445	−	1868.37i	−0.00300695	−	0.0170533i	0.983267	−	0.182168i	$-0.0583116\pi$
$331$	−329.445	−	1868.37i	−0.00300695	−	0.0170533i	−0.986274	+	0.165115i	$0.947200\pi$
$332$		0			0
$333$	−22117.5	+	12724.1i	−0.199457	+	0.114747i
$334$		0			0
$335$	−60143.9	+	165244.i	−0.535923	+	1.47244i
$336$		0			0
$337$	96024.8	+	80574.4i	0.845519	+	0.709475i	0.958798	−	0.284088i	$-0.0916909\pi$
$337$	96024.8	+	80574.4i	0.845519	+	0.709475i	−0.113279	+	0.993563i	$0.536135\pi$
$338$		0			0
$339$	−189072.	+	16394.8i	−1.64524	+	0.142662i
$340$		0			0
$341$	−170543.	−	98462.9i	−1.46664	−	0.846767i
$342$		0			0
$343$	58621.2	+	101535.i	0.498272	+	0.863032i
$344$		0			0
$345$	384579.	+	33945.1i	3.23108	+	0.285193i
$346$		0			0
$347$	191137.	+	33702.5i	1.58739	+	0.279901i	0.896496	−	0.443051i	$-0.146104\pi$
$347$	191137.	+	33702.5i	1.58739	+	0.279901i	0.690898	+	0.722952i	$0.257215\pi$
$348$		0			0
$349$	153791.	−	129046.i	1.26264	−	1.05948i	0.267248	−	0.963628i	$-0.413886\pi$
$349$	153791.	−	129046.i	1.26264	−	1.05948i	0.995395	−	0.0958560i	$-0.0305588\pi$
$350$		0			0
$351$	6003.74	−	511.268i	0.0487313	−	0.00414987i
$352$		0			0
$353$	−226.590	−	270.039i	−0.00181841	−	0.00216709i	0.765134	−	0.643871i	$-0.222672\pi$
$353$	−226.590	−	270.039i	−0.00181841	−	0.00216709i	−0.766953	+	0.641703i	$0.778228\pi$
$354$		0			0
$355$	20882.2	−	118429.i	0.165699	−	0.939725i
$356$		0			0
$357$	−21502.0	−	15080.6i	−0.168711	−	0.118327i
$358$		0			0
$359$	14162.8	−	8176.91i	0.109891	−	0.0634454i	−0.444047	−	0.896003i	$-0.646458\pi$
$359$	14162.8	−	8176.91i	0.109891	−	0.0634454i	0.553938	+	0.832558i	$0.313124\pi$
$360$		0			0
$361$	−10940.5	+	18949.4i	−0.0839501	+	0.145406i
$362$		0			0
$363$	12480.2	−	8724.41i	0.0947128	−	0.0662099i
$364$		0			0
$365$	−254517.	+	303321.i	−1.91043	+	2.27676i
$366$		0			0
$367$	−61422.1	−	22355.8i	−0.456029	−	0.165981i	0.103784	−	0.994600i	$-0.466905\pi$
$367$	−61422.1	−	22355.8i	−0.456029	−	0.165981i	−0.559813	+	0.828619i	$0.689127\pi$
$368$		0			0
$369$	180508.	−	65384.6i	1.32570	−	0.480201i
$370$		0			0
$371$	−137391.	+	24225.8i	−0.998187	+	0.176007i
$372$		0			0
$373$	126454.	−	46025.4i	0.908896	−	0.330811i	0.155084	−	0.987901i	$-0.450435\pi$
$373$	126454.	−	46025.4i	0.908896	−	0.330811i	0.753812	+	0.657090i	$0.228213\pi$
$374$		0			0
$375$	196912.	−	196608.i	1.40026	−	1.39810i
$376$		0			0
$377$			4405.08i			0.0309935i
$378$		0			0
$379$	−34222.9			−0.238253			−0.119126	−	0.992879i	$-0.538009\pi$
$379$	−34222.9			−0.238253			−0.119126	+	0.992879i	$0.538009\pi$
$380$		0			0
$381$	88172.2	−	23552.8i	0.607410	−	0.162253i
$382$		0			0
$383$	−45187.3	−	124151.i	−0.308048	−	0.846356i	−0.993037	−	0.117802i	$-0.962415\pi$
$383$	−45187.3	−	124151.i	−0.308048	−	0.846356i	0.684989	−	0.728554i	$-0.259807\pi$
$384$		0			0
$385$	48185.1	+	273271.i	0.325081	+	1.84363i
$386$		0			0
$387$	−113885.	−	136148.i	−0.760401	−	0.909054i
$388$		0			0
$389$	78137.1	−	214680.i	0.516366	−	1.41871i	−0.358129	−	0.933672i	$-0.616585\pi$
$389$	78137.1	−	214680.i	0.516366	−	1.41871i	0.874496	−	0.485033i	$-0.161192\pi$
$390$		0			0
$391$	−44165.0	−	37058.9i	−0.288885	−	0.242403i
$392$		0			0
$393$	−28841.9	+	61727.2i	−0.186740	+	0.399661i
$394$		0			0
$395$	107626.	+	62137.9i	0.689800	+	0.398256i
$396$		0			0
$397$	−59361.6	−	102817.i	−0.376638	−	0.652357i	0.613932	−	0.789359i	$-0.289587\pi$
$397$	−59361.6	−	102817.i	−0.376638	−	0.652357i	−0.990571	+	0.137002i	$0.956253\pi$
$398$		0			0
$399$	−72838.5	−	156518.i	−0.457526	−	0.983145i
$400$		0			0
$401$	−29720.7	−	5240.56i	−0.184829	−	0.0325903i	0.0804674	−	0.996757i	$-0.474359\pi$
$401$	−29720.7	−	5240.56i	−0.184829	−	0.0325903i	−0.265296	+	0.964167i	$0.585470\pi$
$402$		0			0
$403$	9756.37	−	8186.57i	0.0600728	−	0.0504071i
$404$		0			0
$405$	−186922.	+	221375.i	−1.13959	+	1.34964i
$406$		0			0
$407$	−25878.2	−	30840.5i	−0.156223	−	0.186180i
$408$		0			0
$409$	−30526.3	+	173123.i	−0.182485	+	1.03493i	0.746659	+	0.665207i	$0.231657\pi$
$409$	−30526.3	+	173123.i	−0.182485	+	1.03493i	−0.929144	+	0.369718i	$0.879454\pi$
$410$		0			0
$411$	−26294.0	+	12236.4i	−0.155659	+	0.0724388i
$412$		0			0
$413$	179341.	−	103542.i	1.05143	−	0.607041i
$414$		0			0
$415$	47142.9	−	81653.9i	0.273729	−	0.474112i
$416$		0			0
$417$	−264413.	−	123546.i	−1.52058	−	0.710488i
$418$		0			0
$419$	113503.	−	135267.i	0.646514	−	0.770486i	−0.338870	−	0.940833i	$-0.610045\pi$
$419$	113503.	−	135267.i	0.646514	−	0.770486i	0.985384	+	0.170348i	$0.0544891\pi$
$420$		0			0
$421$	−56524.7	−	20573.3i	−0.318914	−	0.116075i	0.177603	−	0.984102i	$-0.443166\pi$
$421$	−56524.7	−	20573.3i	−0.318914	−	0.116075i	−0.496517	+	0.868027i	$0.665388\pi$
$422$		0			0
$423$	−150220.	−	54938.0i	−0.839550	−	0.307038i
$424$		0			0
$425$	−77452.9	+	13657.0i	−0.428805	+	0.0756099i
$426$		0			0
$427$	45465.7	−	16548.1i	0.249361	−	0.0907598i
$428$		0			0
$429$	2453.48	+	9184.83i	0.0133312	+	0.0499065i
$430$		0			0
$431$			340028.i			1.83046i	0.402932	+	0.915230i	$0.367991\pi$
$431$			340028.i			1.83046i	−0.402932	+	0.915230i	$0.632009\pi$
$432$		0			0
$433$	207455.			1.10649			0.553246	−	0.833018i	$-0.313389\pi$
$433$	207455.			1.10649			0.553246	+	0.833018i	$0.313389\pi$
$434$		0			0
$435$	−149663.	−	149894.i	−0.790926	−	0.792147i
$436$		0			0
$437$	−129616.	−	356117.i	−0.678727	−	1.86479i
$438$		0			0
$439$	31086.7	+	176301.i	0.161304	+	0.914801i	0.952794	+	0.303618i	$0.0981948\pi$
$439$	31086.7	+	176301.i	0.161304	+	0.914801i	−0.791490	+	0.611183i	$0.790694\pi$
$440$		0			0
$441$	1312.22	+	233.468i	0.00674731	+	0.00120046i
$442$		0			0
$443$	−91577.4	+	251607.i	−0.466639	+	1.28208i	0.453769	+	0.891119i	$0.350079\pi$
$443$	−91577.4	+	251607.i	−0.466639	+	1.28208i	−0.920408	+	0.390960i	$0.872143\pi$
$444$		0			0
$445$	−36138.8	−	30324.0i	−0.182496	−	0.153132i
$446$		0			0
$447$	28611.2	+	40928.1i	0.143193	+	0.204836i
$448$		0			0
$449$	−67183.1	−	38788.2i	−0.333248	−	0.192401i	0.324034	−	0.946045i	$-0.394961\pi$
$449$	−67183.1	−	38788.2i	−0.333248	−	0.192401i	−0.657282	+	0.753645i	$0.728294\pi$
$450$		0			0
$451$	151456.	+	262329.i	0.744616	+	1.28971i
$452$		0			0
$453$	−221881.	+	316359.i	−1.08124	+	1.54164i
$454$		0			0
$455$	−17673.6	−	3116.34i	−0.0853695	−	0.0150530i
$456$		0			0
$457$	−109698.	+	92047.6i	−0.525250	+	0.440737i	−0.866458	−	0.499251i	$-0.833609\pi$
$457$	−109698.	+	92047.6i	−0.525250	+	0.440737i	0.341207	+	0.939988i	$0.389164\pi$
$458$		0			0
$459$	41818.4	−	11101.6i	0.198492	−	0.0526938i
$460$		0			0
$461$	126706.	+	151002.i	0.596204	+	0.710529i	0.976786	−	0.214219i	$-0.0687205\pi$
$461$	126706.	+	151002.i	0.596204	+	0.710529i	−0.380581	+	0.924747i	$0.624276\pi$
$462$		0			0
$463$	16939.0	−	96065.6i	0.0790177	−	0.448132i	−0.919470	−	0.393160i	$-0.871382\pi$
$463$	16939.0	−	96065.6i	0.0790177	−	0.448132i	0.998488	−	0.0549721i	$-0.0175070\pi$
$464$		0			0
$465$	−53845.8	+	610042.i	−0.249027	+	2.82133i
$466$		0			0
$467$	35247.4	−	20350.1i	0.161619	−	0.0933108i	−0.417009	−	0.908902i	$-0.636922\pi$
$467$	35247.4	−	20350.1i	0.161619	−	0.0933108i	0.578628	+	0.815592i	$0.303588\pi$
$468$		0			0
$469$	−97894.3	+	169558.i	−0.445053	+	0.770855i
$470$		0			0
$471$	−345.835	−	3988.33i	−0.00155893	−	0.0179783i
$472$		0			0
$473$	180016.	−	214535.i	0.804616	−	0.958904i
$474$		0			0
$475$	−485796.	−	176815.i	−2.15311	−	0.783669i
$476$		0			0
$477$	115224.	−	198865.i	0.506413	−	0.874018i
$478$		0			0
$479$	172260.	−	30374.0i	0.750779	−	0.132383i	0.214852	−	0.976647i	$-0.431073\pi$
$479$	172260.	−	30374.0i	0.750779	−	0.132383i	0.535927	+	0.844264i	$0.319962\pi$
$480$		0			0
$481$	2446.72	−	890.534i	0.0105753	−	0.00384911i
$482$		0			0
$483$	415118.	+	111574.i	1.77942	+	0.478264i
$484$		0			0
$485$			721068.i			3.06544i
$486$		0			0
$487$	−126602.			−0.533804			−0.266902	−	0.963724i	$-0.586000\pi$
$487$	−126602.			−0.533804			−0.266902	+	0.963724i	$0.586000\pi$
$488$		0			0
$489$	1628.97	−	6060.70i	0.00681232	−	0.0253458i
$490$		0			0
$491$	−50828.3	−	139650.i	−0.210835	−	0.579264i	0.788526	−	0.615001i	$-0.210844\pi$
$491$	−50828.3	−	139650.i	−0.210835	−	0.579264i	−0.999361	+	0.0357367i	$0.988622\pi$
$492$		0			0
$493$	5492.73	+	31150.8i	0.0225993	+	0.128167i
$494$		0			0
$495$	−395541.	−	229180.i	−1.61429	−	0.935333i
$496$		0			0
$497$	45793.6	−	125817.i	0.185393	−	0.509362i
$498$		0			0
$499$	−241396.	−	202555.i	−0.969457	−	0.813471i	0.0130085	−	0.999915i	$-0.495859\pi$
$499$	−241396.	−	202555.i	−0.969457	−	0.813471i	−0.982466	+	0.186444i	$0.940304\pi$
$500$		0			0
$501$	−95661.2	+	8294.95i	−0.381119	+	0.0330475i
$502$		0			0
$503$	−246833.	−	142509.i	−0.975589	−	0.563256i	−0.0746533	−	0.997210i	$-0.523785\pi$
$503$	−246833.	−	142509.i	−0.975589	−	0.563256i	−0.900935	+	0.433953i	$0.857118\pi$
$504$		0			0
$505$	−349205.	−	604841.i	−1.36930	−	2.37169i
$506$		0			0
$507$	255441.	+	22546.7i	0.993745	+	0.0877135i
$508$		0			0
$509$	−270821.	−	47753.0i	−1.04531	−	0.184317i	−0.375481	−	0.926830i	$-0.622523\pi$
$509$	−270821.	−	47753.0i	−1.04531	−	0.184317i	−0.669832	+	0.742513i	$0.733634\pi$
$510$		0			0
$511$	−337715.	+	283377.i	−1.29333	+	1.08523i
$512$		0			0
$513$	274543.	+	74244.8i	1.04322	+	0.282118i
$514$		0			0
$515$	−311629.	−	371385.i	−1.17496	−	1.40026i
$516$		0			0
$517$	43823.1	−	248533.i	0.163954	−	0.929829i
$518$		0			0
$519$	394616.	+	276767.i	1.46501	+	1.02749i
$520$		0			0
$521$	312798.	−	180594.i	1.15236	−	0.665315i	0.202899	−	0.979200i	$-0.434964\pi$
$521$	312798.	−	180594.i	1.15236	−	0.665315i	0.949461	+	0.313884i	$0.101630\pi$
$522$		0			0
$523$	164895.	−	285607.i	0.602845	−	1.04416i	−0.389544	−	0.921008i	$-0.627367\pi$
$523$	164895.	−	285607.i	0.602845	−	1.04416i	0.992388	−	0.123149i	$-0.0392994\pi$
$524$		0			0
$525$	480594.	−	335964.i	1.74365	−	1.21892i
$526$		0			0
$527$	58785.0	−	70057.2i	0.211663	−	0.252250i
$528$		0			0
$529$	623739.	+	227023.i	2.22891	+	0.811255i
$530$		0			0
$531$	−59759.3	+	335882.i	−0.211942	+	1.19124i
$532$		0			0
$533$	−19292.9	+	3401.86i	−0.0679116	+	0.0119746i
$534$		0			0
$535$	128620.	−	46813.7i	0.449366	−	0.163556i
$536$		0			0
$537$	−45774.0	+	45703.5i	−0.158734	+	0.158490i
$538$		0			0
$539$			2102.92i			0.00723843i
$540$		0			0
$541$	−458131.			−1.56529			−0.782645	−	0.622468i	$-0.786130\pi$
$541$	−458131.			−1.56529			−0.782645	+	0.622468i	$0.786130\pi$
$542$		0			0
$543$	−386852.	+	103337.i	−1.31203	+	0.350474i
$544$		0			0
$545$	−42872.8	−	117792.i	−0.144341	−	0.396573i
$546$		0			0
$547$	−100576.	−	570394.i	−0.336139	−	1.90634i	−0.415685	−	0.909509i	$-0.636458\pi$
$547$	−100576.	−	570394.i	−0.336139	−	1.90634i	0.0795459	−	0.996831i	$-0.474653\pi$
$548$		0			0
$549$	−27377.3	+	74859.2i	−0.0908335	+	0.248371i
$550$		0			0
$551$	−71113.5	+	195383.i	−0.234233	+	0.643551i
$552$		0			0
$553$	105996.	+	88940.9i	0.346607	+	0.290838i
$554$		0			0
$555$	−52999.9	+	113430.i	−0.172064	+	0.368250i
$556$		0			0
$557$	254413.	+	146885.i	0.820027	+	0.473443i	0.850426	−	0.526095i	$-0.176344\pi$
$557$	254413.	+	146885.i	0.820027	+	0.473443i	−0.0303989	+	0.999538i	$0.509678\pi$
$558$		0			0
$559$	9056.18	+	15685.8i	0.0289815	+	0.0501975i
$560$		0			0
$561$	28802.6	+	61892.0i	0.0915179	+	0.196657i
$562$		0			0
$563$	353649.	+	62357.8i	1.11572	+	0.196732i	0.700961	−	0.713199i	$-0.252755\pi$
$563$	353649.	+	62357.8i	1.11572	+	0.196732i	0.414760	+	0.909931i	$0.363866\pi$
$564$		0			0
$565$	−713342.	+	598565.i	−2.23461	+	1.87506i
$566$		0			0
$567$	−247756.	+	206593.i	−0.770651	+	0.642613i
$568$		0			0
$569$	311809.	+	371600.i	0.963085	+	1.14776i	0.988973	+	0.148095i	$0.0473141\pi$
$569$	311809.	+	371600.i	0.963085	+	1.14776i	−0.0258884	+	0.999665i	$0.508241\pi$
$570$		0			0
$571$	−9613.31	+	54519.8i	−0.0294850	+	0.167218i	−0.995995	−	0.0894125i	$-0.971501\pi$
$571$	−9613.31	+	54519.8i	−0.0294850	+	0.167218i	0.966510	+	0.256630i	$0.0826122\pi$
$572$		0			0
$573$	−132185.	+	61514.9i	−0.402600	+	0.187358i
$574$		0			0
$575$	1.11477e6	−	643613.i	3.37170	−	1.94665i
$576$		0			0
$577$	60962.7	−	105591.i	0.183110	−	0.317156i	−0.759828	−	0.650124i	$-0.774717\pi$
$577$	60962.7	−	105591.i	0.183110	−	0.317156i	0.942938	+	0.332968i	$0.108050\pi$
$578$		0			0
$579$	330230.	+	154299.i	0.985053	+	0.460263i
$580$		0			0
$581$	67477.9	−	80417.0i	0.199898	−	0.238230i
$582$		0			0
$583$	340759.	+	124026.i	1.00256	+	0.364901i
$584$		0			0
$585$	22677.5	−	18969.2i	0.0662649	−	0.0554289i
$586$		0			0
$587$	−365201.	+	64394.8i	−1.05988	+	0.186885i	−0.676302	−	0.736624i	$-0.736419\pi$
$587$	−365201.	+	64394.8i	−1.05988	+	0.186885i	−0.383576	+	0.923509i	$0.625308\pi$
$588$		0			0
$589$	564894.	−	205604.i	1.62831	−	0.592655i
$590$		0			0
$591$	−3405.43	−	12748.5i	−0.00974982	−	0.0364994i
$592$		0			0
$593$		−	172744.i		−	0.491240i	−0.969366	−	0.245620i	$-0.921008\pi$
$593$		−	172744.i		−	0.491240i	0.969366	−	0.245620i	$-0.0789916\pi$
$594$		0			0
$595$	−128866.			−0.364003
$596$		0			0
$597$	320208.	+	320702.i	0.898429	+	0.899816i
$598$		0			0
$599$	−118262.	−	324923.i	−0.329605	−	0.905581i	−0.988212	−	0.153094i	$-0.951076\pi$
$599$	−118262.	−	324923.i	−0.329605	−	0.905581i	0.658607	−	0.752487i	$-0.271146\pi$
$600$		0			0
$601$	13755.4	+	78010.7i	0.0380824	+	0.215976i	0.997910	−	0.0646132i	$-0.0205813\pi$
$601$	13755.4	+	78010.7i	0.0380824	+	0.215976i	−0.959828	+	0.280589i	$0.909470\pi$
$602$		0			0
$603$	−109850.	−	303265.i	−0.302111	−	0.834042i
$604$		0			0
$605$	25554.3	−	70209.9i	0.0698157	−	0.191817i
$606$		0			0
$607$	410746.	+	344657.i	1.11480	+	0.935426i	0.998330	−	0.0577680i	$-0.0183984\pi$
$607$	410746.	+	344657.i	1.11480	+	0.935426i	0.116468	+	0.993194i	$0.462843\pi$
$608$		0			0
$609$	−135122.	−	193291.i	−0.364326	−	0.521166i
$610$		0			0
$611$	14135.0	+	8160.83i	0.0378628	+	0.0218601i
$612$		0			0
$613$	−199705.	−	345899.i	−0.531457	−	0.920511i	−0.999326	−	0.0367129i	$-0.988311\pi$
$613$	−199705.	−	345899.i	−0.531457	−	0.920511i	0.467869	−	0.883798i	$-0.345022\pi$
$614$		0			0
$615$	540912.	−	771236.i	1.43013	−	2.03909i
$616$		0			0
$617$	299092.	+	52738.1i	0.785661	+	0.138533i	0.552065	−	0.833801i	$-0.313840\pi$
$617$	299092.	+	52738.1i	0.785661	+	0.138533i	0.233595	+	0.972334i	$0.424951\pi$
$618$		0			0
$619$	148396.	−	124519.i	0.387293	−	0.324978i	−0.428264	−	0.903654i	$-0.640875\pi$
$619$	148396.	−	124519.i	0.387293	−	0.324978i	0.815558	+	0.578676i	$0.196430\pi$
$620$		0			0
$621$	−581019.	+	404834.i	−1.50663	+	1.04977i
$622$		0			0
$623$	−33762.5	−	40236.6i	−0.0869878	−	0.103668i
$624$		0			0
$625$	93272.8	−	528976.i	0.238778	−	1.35418i
$626$		0			0
$627$	−39454.0	+	446992.i	−0.100359	+	1.13701i
$628$		0			0
$629$	16191.8	−	9348.31i	0.0409254	−	0.0236283i
$630$		0			0
$631$	−147768.	+	255942.i	−0.371126	+	0.642809i	−0.989739	−	0.142887i	$-0.954362\pi$
$631$	−147768.	+	255942.i	−0.371126	+	0.642809i	0.618613	+	0.785696i	$0.287695\pi$
$632$		0			0
$633$	−51532.0	−	594290.i	−0.128608	−	1.48317i
$634$		0			0
$635$	287843.	−	343037.i	0.713851	−	0.850735i
$636$		0			0
$637$	−127.802	−	46.5163i	−0.000314964	−	0.000114637i
$638$		0			0
$639$	109994.	+	191195.i	0.269380	+	0.468246i
$640$		0			0
$641$	178986.	−	31560.0i	0.435614	−	0.0768105i	0.0484592	−	0.998825i	$-0.484569\pi$
$641$	178986.	−	31560.0i	0.435614	−	0.0768105i	0.387155	+	0.922015i	$0.373458\pi$
$642$		0			0
$643$	−416371.	+	151547.i	−1.00707	+	0.366543i	−0.792306	−	0.610124i	$-0.791120\pi$
$643$	−416371.	+	151547.i	−1.00707	+	0.366543i	−0.214761	+	0.976667i	$0.568897\pi$
$644$		0			0
$645$	−841085.	−	226063.i	−2.02172	−	0.543388i
$646$		0			0
$647$			242796.i			0.580005i	0.957026	+	0.290003i	$0.0936562\pi$
$647$			242796.i			0.580005i	−0.957026	+	0.290003i	$0.906344\pi$
$648$		0			0
$649$	−538271.			−1.27794
$650$		0			0
$651$	−176985.	+	658486.i	−0.417614	+	1.55376i
$652$		0			0
$653$	286708.	+	787724.i	0.672378	+	1.84734i	0.508967	+	0.860786i	$0.330028\pi$
$653$	286708.	+	787724.i	0.672378	+	1.84734i	0.163412	+	0.986558i	$0.447750\pi$
$654$		0			0
$655$	58052.0	+	329229.i	0.135311	+	0.767389i
$656$		0			0
$657$	1120.01	−	726276.i	0.00259473	−	1.68256i
$658$		0			0
$659$	37338.0	−	102585.i	0.0859766	−	0.236219i	−0.889252	−	0.457417i	$-0.848775\pi$
$659$	37338.0	−	102585.i	0.0859766	−	0.236219i	0.975229	+	0.221198i	$0.0709968\pi$
$660$		0			0
$661$	480727.	+	403378.i	1.10026	+	0.923229i	0.997443	−	0.0714712i	$-0.0227694\pi$
$661$	480727.	+	403378.i	1.10026	+	0.923229i	0.102819	+	0.994700i	$0.467214\pi$
$662$		0			0
$663$	−4398.53	+	381.404i	−0.0100065	+	0.000867677i
$664$		0			0
$665$	−733587.	−	423537.i	−1.65885	−	0.957740i
$666$		0			0
$667$	−258855.	−	448350.i	−0.581842	−	1.00778i
$668$		0			0
$669$	432468.	+	38172.1i	0.966278	+	0.0852892i
$670$		0			0
$671$	−123852.	−	21838.4i	−0.275079	−	0.0485038i
$672$		0			0
$673$	48577.6	−	40761.5i	0.107252	−	0.0899953i	−0.587585	−	0.809163i	$-0.699921\pi$
$673$	48577.6	−	40761.5i	0.107252	−	0.0899953i	0.694837	+	0.719167i	$0.255477\pi$
$674$		0			0
$675$	−86420.0	+	962146.i	−0.189673	+	2.11171i
$676$		0			0
$677$	−284775.	−	339382.i	−0.621334	−	0.740477i	0.359965	−	0.932966i	$-0.382789\pi$
$677$	−284775.	−	339382.i	−0.621334	−	0.740477i	−0.981299	+	0.192489i	$0.938344\pi$
$678$		0			0
$679$	−139410.	+	790634.i	−0.302381	+	1.71489i
$680$		0			0
$681$	608070.	+	426474.i	1.31117	+	0.919599i
$682$		0			0
$683$	37938.6	−	21903.9i	0.0813280	−	0.0469547i	−0.458785	−	0.888548i	$-0.651715\pi$
$683$	37938.6	−	21903.9i	0.0813280	−	0.0469547i	0.540113	+	0.841593i	$0.318382\pi$
$684$		0			0
$685$	−71151.6	+	123238.i	−0.151636	+	0.262642i
$686$		0			0
$687$	494780.	−	345880.i	1.04833	−	0.732846i
$688$		0			0
$689$	−15075.1	+	17965.8i	−0.0317557	+	0.0378450i
$690$		0			0
$691$	239969.	+	87341.4i	0.502572	+	0.182921i	0.580850	−	0.814010i	$-0.302720\pi$
$691$	239969.	+	87341.4i	0.502572	+	0.182921i	−0.0782785	+	0.996932i	$0.524942\pi$
$692$		0			0
$693$	−389392.	−	327763.i	−0.810813	−	0.682487i
$694$		0			0
$695$	−1.41028e6	+	248670.i	−2.91967	+	0.514817i
$696$		0			0
$697$	−132190.	+	48113.1i	−0.272102	+	0.0990370i
$698$		0			0
$699$	545126.	−	544286.i	1.11569	−	1.11397i
$700$		0			0
$701$			192790.i			0.392328i	0.980571	+	0.196164i	$0.0628485\pi$
$701$			192790.i			0.392328i	−0.980571	+	0.196164i	$0.937152\pi$
$702$		0			0
$703$	122898.			0.248676
$704$		0			0
$705$	−758243.	+	202544.i	−1.52556	+	0.407513i
$706$		0			0
$707$	−265956.	−	730708.i	−0.532073	−	1.46186i
$708$		0			0
$709$	−101457.	−	575389.i	−0.201831	−	1.14464i	−0.902349	−	0.431006i	$-0.858159\pi$
$709$	−101457.	−	575389.i	−0.201831	−	1.14464i	0.700518	−	0.713635i	$-0.252952\pi$
$710$		0			0
$711$	−224548.	+	39236.9i	−0.444191	+	0.0776167i
$712$		0			0
$713$	−511940.	+	1.40654e6i	−1.00702	+	2.76678i
$714$		0			0
$715$	35734.0	+	29984.3i	0.0698987	+	0.0586520i
$716$		0			0
$717$	159283.	−	340896.i	0.309835	−	0.663108i
$718$		0			0
$719$	105460.	+	60887.3i	0.204000	+	0.117779i	0.598520	−	0.801108i	$-0.295756\pi$
$719$	105460.	+	60887.3i	0.204000	+	0.117779i	−0.394520	+	0.918887i	$0.629089\pi$
$720$		0			0
$721$	−269891.	−	467465.i	−0.519179	−	0.899245i
$722$		0			0
$723$	71093.2	+	152767.i	0.136004	+	0.292249i
$724$		0			0
$725$	−695504.	−	122636.i	−1.32320	−	0.233315i
$726$		0			0
$727$	−31443.0	+	26383.8i	−0.0594916	+	0.0499194i	−0.672049	−	0.740507i	$-0.734585\pi$
$727$	−31443.0	+	26383.8i	−0.0594916	+	0.0499194i	0.612557	+	0.790426i	$0.290141\pi$
$728$		0			0
$729$	2458.65	−	531435.i	0.00462638	−	0.999989i
$730$		0			0
$731$	83600.2	+	99630.8i	0.156449	+	0.186449i
$732$		0			0
$733$	96146.6	−	545274.i	0.178948	−	1.01486i	−0.754540	−	0.656254i	$-0.772140\pi$
$733$	96146.6	−	545274.i	0.178948	−	1.01486i	0.933488	−	0.358609i	$-0.116749\pi$
$734$		0			0
$735$	5929.20	−	2759.27i	0.0109754	−	0.00510762i
$736$		0			0
$737$	440728.	−	254455.i	0.811402	−	0.468463i
$738$		0			0
$739$	256927.	−	445011.i	0.470459	−	0.814858i	−0.528970	−	0.848640i	$-0.677422\pi$
$739$	256927.	−	445011.i	0.470459	−	0.814858i	0.999429	+	0.0337817i	$0.0107551\pi$
$740$		0			0
$741$	−26292.7	−	12285.2i	−0.0478849	−	0.0223741i
$742$		0			0
$743$	17473.4	−	20824.0i	0.0316519	−	0.0377213i	−0.749986	−	0.661453i	$-0.769940\pi$
$743$	17473.4	−	20824.0i	0.0316519	−	0.0377213i	0.781638	+	0.623732i	$0.214384\pi$
$744$		0			0
$745$	230249.	+	83803.8i	0.414844	+	0.150991i
$746$		0			0
$747$	29768.3	+	170360.i	0.0533474	+	0.305301i
$748$		0			0
$749$	150079.	−	26463.0i	0.267520	−	0.0471711i
$750$		0			0
$751$	−369939.	+	134647.i	−0.655919	+	0.238735i	−0.648473	−	0.761237i	$-0.724592\pi$
$751$	−369939.	+	134647.i	−0.655919	+	0.238735i	−0.00744559	+	0.999972i	$0.502370\pi$
$752$		0			0
$753$	27625.7	+	103419.i	0.0487218	+	0.182395i
$754$		0			0
$755$			1.89600e6i			3.32618i
$756$		0			0
$757$	502122.			0.876228			0.438114	−	0.898919i	$-0.355647\pi$
$757$	502122.			0.876228			0.438114	+	0.898919i	$0.355647\pi$
$758$		0			0
$759$	−789443.	−	790661.i	−1.37037	−	1.37248i
$760$		0			0
$761$	284590.	+	781906.i	0.491418	+	1.35016i	0.899383	+	0.437162i	$0.144016\pi$
$761$	284590.	+	781906.i	0.491418	+	1.35016i	−0.407965	+	0.912997i	$0.633762\pi$
$762$		0			0
$763$	−24235.3	−	137445.i	−0.0416292	−	0.236091i
$764$		0			0
$765$	136713.	−	162419.i	0.233607	−	0.277532i
$766$		0			0
$767$	11906.5	−	32712.8i	0.0202392	−	0.0556067i
$768$		0			0
$769$	−347407.	−	291509.i	−0.587470	−	0.492946i	0.299921	−	0.953964i	$-0.403040\pi$
$769$	−347407.	−	291509.i	−0.587470	−	0.492946i	−0.887391	+	0.461018i	$0.847484\pi$
$770$		0			0
$771$	253797.	+	363055.i	0.426950	+	0.610749i
$772$		0			0
$773$	−175007.	−	101041.i	−0.292885	−	0.169097i	0.346357	−	0.938103i	$-0.387419\pi$
$773$	−175007.	−	101041.i	−0.292885	−	0.169097i	−0.639242	+	0.769005i	$0.720752\pi$
$774$		0			0
$775$	1.02094e6	+	1.76832e6i	1.69979	+	2.94413i
$776$		0			0
$777$	−80043.5	+	114127.i	−0.132582	+	0.189036i
$778$		0			0
$779$	−910636.	−	160570.i	−1.50062	−	0.264599i
$780$		0			0
$781$	−266600.	+	223704.i	−0.437077	+	0.366751i
$782$		0			0
$783$	386966.	+	34757.3i	0.631174	+	0.0566921i
$784$		0			0
$785$	−12626.3	−	15047.4i	−0.0204897	−	0.0244187i
$786$		0			0
$787$	−74203.9	+	420831.i	−0.119806	+	0.679452i	0.864452	+	0.502715i	$0.167665\pi$
$787$	−74203.9	+	420831.i	−0.119806	+	0.679452i	−0.984258	+	0.176737i	$0.943446\pi$
$788$		0			0
$789$	−1511.40	+	17123.3i	−0.00242787	+	0.0275064i
$790$		0			0
$791$	−897888.	+	518396.i	−1.43506	+	0.828531i
$792$		0			0
$793$	4066.79	−	7043.89i	0.00646704	−	0.0112012i
$794$		0			0
$795$	−97421.4	−	1.12351e6i	−0.154142	−	1.77763i
$796$		0			0
$797$	328815.	−	391867.i	0.517649	−	0.616910i	−0.442374	−	0.896831i	$-0.645864\pi$
$797$	328815.	−	391867.i	0.517649	−	0.616910i	0.960023	+	0.279920i	$0.0903081\pi$
$798$		0			0
$799$	110132.	+	40084.9i	0.172513	+	0.0627895i
$800$		0			0
$801$	86531.1	+	133.442i	0.134868	+	0.000207984i
$802$		0			0
$803$	1.12850e6	−	198985.i	1.75013	−	0.308595i
$804$		0			0
$805$	1.98195e6	−	721371.i	3.05845	−	1.11318i
$806$		0			0
$807$	296050.	+	79571.0i	0.454588	+	0.122182i
$808$		0			0
$809$			53671.4i			0.0820061i	0.999159	+	0.0410031i	$0.0130553\pi$
$809$			53671.4i			0.0820061i	−0.999159	+	0.0410031i	$0.986945\pi$
$810$		0			0
$811$	−53156.2			−0.0808187			−0.0404094	−	0.999183i	$-0.512866\pi$
$811$	−53156.2			−0.0808187			−0.0404094	+	0.999183i	$0.512866\pi$
$812$		0			0
$813$	10615.3	−	39494.9i	0.0160602	−	0.0597531i
$814$		0			0
$815$	−10532.0	−	28936.4i	−0.0158560	−	0.0435641i
$816$		0			0
$817$	148454.	+	841924.i	0.222407	+	1.26133i
$818$		0			0
$819$	28532.8	−	16414.8i	0.0425379	−	0.0244719i
$820$		0			0
$821$	343626.	−	944106.i	0.509801	−	1.40067i	−0.371643	−	0.928376i	$-0.621205\pi$
$821$	343626.	−	944106.i	0.509801	−	1.40067i	0.881443	−	0.472290i	$-0.156572\pi$
$822$		0			0
$823$	−170535.	−	143096.i	−0.251775	−	0.211265i	0.508161	−	0.861262i	$-0.330325\pi$
$823$	−170535.	−	143096.i	−0.251775	−	0.211265i	−0.759936	+	0.649998i	$0.774770\pi$
$824$		0			0
$825$	−1.51847e6	+	131669.i	−2.23099	+	0.193453i
$826$		0			0
$827$	−653828.	−	377488.i	−0.955988	−	0.551940i	−0.0610521	−	0.998135i	$-0.519446\pi$
$827$	−653828.	−	377488.i	−0.955988	−	0.551940i	−0.894936	+	0.446195i	$0.852779\pi$
$828$		0			0
$829$	310850.	+	538409.i	0.452316	+	0.783435i	0.998529	−	0.0542112i	$-0.0172644\pi$
$829$	310850.	+	538409.i	0.452316	+	0.783435i	−0.546213	+	0.837646i	$0.683931\pi$
$830$		0			0
$831$	−501418.	−	44258.0i	−0.726103	−	0.0640900i
$832$		0			0
$833$	−961.766	−	169.585i	−0.00138605	−	0.000244398i
$834$		0			0
$835$	−360916.	+	302844.i	−0.517646	+	0.434356i
$836$		0			0
$837$	−642172.	−	921647.i	−0.916644	−	1.31557i
$838$		0			0
$839$	−198649.	−	236741.i	−0.282204	−	0.336317i	0.606258	−	0.795268i	$-0.292670\pi$
$839$	−198649.	−	236741.i	−0.282204	−	0.336317i	−0.888462	+	0.458951i	$0.848225\pi$
$840$		0			0
$841$	73494.9	−	416810.i	0.103912	−	0.589314i
$842$		0			0
$843$	−256717.	−	180051.i	−0.361243	−	0.253361i
$844$		0			0
$845$	1.08967e6	−	629122.i	1.52610	−	0.881093i
$846$		0			0
$847$	41593.9	−	72042.8i	0.0579780	−	0.100421i
$848$		0			0
$849$	−749150.	+	523700.i	−1.03933	+	0.726553i
$850$		0			0
$851$	−196698.	+	234415.i	−0.271606	+	0.323688i
$852$		0			0
$853$	59039.2	+	21488.5i	0.0811414	+	0.0295330i	0.382272	−	0.924050i	$-0.375142\pi$
$853$	59039.2	+	21488.5i	0.0811414	+	0.0295330i	−0.301131	+	0.953583i	$0.597364\pi$
$854$		0			0
$855$	1.31207e6	−	475263.i	1.79483	−	0.650132i
$856$		0			0
$857$	326586.	−	57585.9i	0.444668	−	0.0784070i	0.0531691	−	0.998586i	$-0.483068\pi$
$857$	326586.	−	57585.9i	0.444668	−	0.0784070i	0.391499	+	0.920179i	$0.371957\pi$
$858$		0			0
$859$	−894089.	+	325422.i	−1.21170	+	0.441022i	−0.867293	−	0.497799i	$-0.834142\pi$
$859$	−894089.	+	325422.i	−1.21170	+	0.441022i	−0.344406	+	0.938821i	$0.611920\pi$
$860$		0			0
$861$	742206.	−	741063.i	1.00119	−	0.999651i
$862$		0			0
$863$			441486.i			0.592782i	0.955067	+	0.296391i	$0.0957831\pi$
$863$			441486.i			0.592782i	−0.955067	+	0.296391i	$0.904217\pi$
$864$		0			0
$865$	2.36502e6			3.16084
$866$		0			0
$867$	695597.	−	185810.i	0.925378	−	0.247190i
$868$		0			0
$869$	−123009.	−	337966.i	−0.162892	−	0.447541i
$870$		0			0
$871$	5715.34	+	32413.3i	0.00753366	+	0.0427255i
$872$		0			0
$873$	−848590.	−	1.01448e6i	−1.11345	−	1.33112i
$874$		0			0
$875$	519925.	−	1.42848e6i	0.679086	−	1.86577i
$876$		0			0
$877$	−478470.	−	401484.i	−0.622094	−	0.521998i	0.276367	−	0.961052i	$-0.410869\pi$
$877$	−478470.	−	401484.i	−0.622094	−	0.521998i	−0.898461	+	0.439054i	$0.855314\pi$
$878$		0			0
$879$	314061.	−	672151.i	0.406477	−	0.869940i
$880$		0			0
$881$	−974205.	−	562458.i	−1.25516	−	0.724666i	−0.283029	−	0.959111i	$-0.591339\pi$
$881$	−974205.	−	562458.i	−1.25516	−	0.724666i	−0.972129	+	0.234445i	$0.924673\pi$
$882$		0			0
$883$	49611.3	+	85929.3i	0.0636296	+	0.110210i	0.896085	−	0.443882i	$-0.146399\pi$
$883$	49611.3	+	85929.3i	0.0636296	+	0.110210i	−0.832456	+	0.554092i	$0.813066\pi$
$884$		0			0
$885$	706273.	+	1.51766e6i	0.901749	+	1.93771i
$886$		0			0
$887$	573703.	+	101159.i	0.729188	+	0.128576i	0.525904	−	0.850544i	$-0.323727\pi$
$887$	573703.	+	101159.i	0.729188	+	0.128576i	0.203285	+	0.979120i	$0.434838\pi$
$888$		0			0
$889$	381935.	−	320481.i	0.483265	−	0.405508i
$890$		0			0
$891$	826204.	−	143056.i	1.04072	−	0.180198i
$892$		0			0
$893$	495197.	+	590153.i	0.620977	+	0.740051i
$894$		0			0
$895$	−55113.6	+	312565.i	−0.0688039	+	0.390206i
$896$		0			0
$897$	65514.0	−	30488.2i	0.0814234	−	0.0378919i
$898$		0			0
$899$	711200.	−	410611.i	0.879979	−	0.508056i
$900$		0			0
$901$	−84202.9	+	145844.i	−0.103724	+	0.179655i
$902$		0			0
$903$	−878522.	−	410487.i	−1.07740	−	0.503412i
$904$		0			0
$905$	−1.26290e6	+	1.50506e6i	−1.54195	+	1.83763i
$906$		0			0
$907$	−576959.	−	209996.i	−0.701343	−	0.255268i	−0.0333590	−	0.999443i	$-0.510620\pi$
$907$	−576959.	−	209996.i	−0.701343	−	0.255268i	−0.667984	+	0.744175i	$0.732843\pi$
$908$		0			0
$909$	1.20311e6	+	439998.i	1.45605	+	0.532505i
$910$		0			0
$911$	−626058.	+	110391.i	−0.754358	+	0.133014i	−0.537586	−	0.843209i	$-0.680664\pi$
$911$	−626058.	+	110391.i	−0.754358	+	0.133014i	−0.216772	+	0.976222i	$0.569553\pi$
$912$		0			0
$913$	−256409.	+	93325.1i	−0.307603	+	0.111958i
$914$		0			0
$915$	100934.	+	377856.i	0.120558	+	0.451319i
$916$		0			0
$917$			372215.i			0.442645i
$918$		0			0
$919$	−519702.			−0.615351			−0.307676	−	0.951491i	$-0.599551\pi$
$919$	−519702.			−0.615351			−0.307676	+	0.951491i	$0.599551\pi$
$920$		0			0
$921$	503421.	+	504198.i	0.593488	+	0.594404i
$922$		0			0
$923$	−7698.20	−	21150.6i	−0.00903620	−	0.0248268i
$924$		0			0
$925$	72487.6	+	411098.i	0.0847189	+	0.480465i
$926$		0			0
$927$	875501.	+	155767.i	1.01882	+	0.181266i
$928$		0			0
$929$	−442753.	+	1.21645e6i	−0.513015	+	1.40950i	0.365064	+	0.930982i	$0.381047\pi$
$929$	−442753.	+	1.21645e6i	−0.513015	+	1.40950i	−0.878079	+	0.478515i	$0.841175\pi$
$930$		0			0
$931$	−4917.61	−	4126.36i	−0.00567355	−	0.00476067i
$932$		0			0
$933$	68204.1	+	97565.6i	0.0783515	+	0.112081i
$934$		0			0
$935$	290083.	+	167480.i	0.331817	+	0.191575i
$936$		0			0
$937$	−594922.	−	1.03044e6i	−0.677612	−	1.17366i	−0.975698	−	0.219119i	$-0.929682\pi$
$937$	−594922.	−	1.03044e6i	−0.677612	−	1.17366i	0.298086	−	0.954539i	$-0.403652\pi$
$938$		0			0
$939$	−101366.	+	144528.i	−0.114963	+	0.163916i
$940$		0			0
$941$	−94193.7	−	16608.9i	−0.106376	−	0.0187569i	0.120207	−	0.992749i	$-0.461644\pi$
$941$	−94193.7	−	16608.9i	−0.106376	−	0.0187569i	−0.226583	+	0.973992i	$0.572755\pi$
$942$		0			0
$943$	1.76374e6	−	1.47995e6i	1.98340	−	1.66427i
$944$		0			0
$945$	−413206.	+	1.52796e6i	−0.462704	+	1.71099i
$946$		0			0
$947$	584755.	+	696884.i	0.652040	+	0.777071i	0.986220	−	0.165438i	$-0.0529039\pi$
$947$	584755.	+	696884.i	0.652040	+	0.777071i	−0.334180	+	0.942509i	$0.608459\pi$
$948$		0			0
$949$	−12869.2	+	72984.8i	−0.0142896	+	0.0810401i
$950$		0			0
$951$	−5561.29	+	63006.3i	−0.00614915	+	0.0696663i
$952$		0			0
$953$	−989990.	+	571571.i	−1.09005	+	0.629339i	−0.933589	−	0.358346i	$-0.883341\pi$
$953$	−989990.	+	571571.i	−1.09005	+	0.629339i	−0.156458	+	0.987685i	$0.550008\pi$
$954$		0			0
$955$	−357693.	+	619542.i	−0.392196	+	0.679304i
$956$		0			0
$957$	52956.1	+	610714.i	0.0578219	+	0.666829i
$958$		0			0
$959$	−101843.	+	121371.i	−0.110737	+	0.131971i
$960$		0			0
$961$	−1.36332e6	−	496207.i	−1.47622	−	0.537299i
$962$		0			0
$963$	−125864.	+	217229.i	−0.135722	+	0.234242i
$964$		0			0
$965$	1.76132e6	−	310568.i	1.89140	−	0.333505i
$966$		0			0
$967$	250618.	−	91217.4i	0.268015	−	0.0975494i	−0.204517	−	0.978863i	$-0.565562\pi$
$967$	250618.	−	91217.4i	0.268015	−	0.0975494i	0.472532	+	0.881314i	$0.343340\pi$
$968$		0			0
$969$	−201249.	−	54091.0i	−0.214332	−	0.0576073i
$970$		0			0
$971$		−	529968.i		−	0.562097i	−0.959694	−	0.281049i	$-0.909318\pi$
$971$		−	529968.i		−	0.562097i	0.959694	−	0.281049i	$-0.0906822\pi$
$972$		0			0
$973$	−1.59441e6			−1.68413
$974$		0			0
$975$	25586.3	−	95195.9i	0.0269153	−	0.100140i
$976$		0			0
$977$	188766.	+	518630.i	0.197758	+	0.543336i	0.998445	−	0.0557474i	$-0.0177542\pi$
$977$	188766.	+	518630.i	0.197758	+	0.543336i	−0.800687	+	0.599083i	$0.795532\pi$
$978$		0			0
$979$	23707.7	+	134453.i	0.0247357	+	0.140283i
$980$		0			0
$981$	198942.	+	115269.i	0.206723	+	0.119777i
$982$		0			0
$983$	242469.	−	666179.i	0.250929	−	0.689421i	−0.748719	−	0.662887i	$-0.769331\pi$
$983$	242469.	−	666179.i	0.250929	−	0.689421i	0.999648	−	0.0265334i	$-0.00844684\pi$
$984$		0			0
$985$	−49598.7	−	41618.2i	−0.0511208	−	0.0428954i
$986$		0			0
$987$	−870554.	+	75487.3i	−0.893638	+	0.0774889i
$988$		0			0
$989$	−1.84348e6	−	1.06433e6i	−1.88472	−	1.08814i
$990$		0			0
$991$	−343435.	−	594847.i	−0.349702	−	0.605701i	0.636495	−	0.771281i	$-0.280384\pi$
$991$	−343435.	−	594847.i	−0.349702	−	0.605701i	−0.986196	+	0.165580i	$0.947050\pi$
$992$		0			0
$993$	17008.6	+	1501.28i	0.0172493	+	0.00152252i
$994$		0			0
$995$	2.18990e6	+	386138.i	2.21196	+	0.390028i
$996$		0			0
$997$	1.22218e6	−	1.02553e6i	1.22955	−	1.03171i	0.231282	−	0.972887i	$-0.425708\pi$
$997$	1.22218e6	−	1.02553e6i	1.22955	−	1.03171i	0.998268	−	0.0588280i	$-0.0187363\pi$
$998$		0			0
$999$	−58924.0	−	221960.i	−0.0590420	−	0.222404i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	108.5.k.a.5.5	✓	72
3.2	odd	2		324.5.k.a.125.1		72
27.11	odd	18	inner	108.5.k.a.65.5	yes	72
27.16	even	9		324.5.k.a.197.1		72

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.5.k.a.5.5	✓	72	1.1	even	1	trivial
108.5.k.a.65.5	yes	72	27.11	odd	18	inner
324.5.k.a.125.1		72	3.2	odd	2
324.5.k.a.197.1		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)

\(f(q)\)	\(=\)	\(q+(-2.33607 + 8.69153i) q^{3} +(15.1037 + 41.4971i) q^{5} +(8.53787 + 48.4207i) q^{7} +(-70.0855 - 40.6081i) q^{9} +O(q^{10})\)	\(q+(-2.33607 + 8.69153i) q^{3} +(15.1037 + 41.4971i) q^{5} +(8.53787 + 48.4207i) q^{7} +(-70.0855 - 40.6081i) q^{9} +(43.7103 - 120.093i) q^{11} +(6.33166 + 5.31289i) q^{13} +(-395.957 + 34.3341i) q^{15} +(51.3995 + 29.6755i) q^{17} +(195.065 + 337.863i) q^{19} +(-440.795 - 38.9071i) q^{21} +(-956.638 - 168.681i) q^{23} +(-1015.11 + 851.777i) q^{25} +(516.672 - 514.287i) q^{27} +(342.577 + 408.267i) q^{29} +(267.572 - 1517.48i) q^{31} +(941.682 + 660.456i) q^{33} +(-1880.36 + 1085.63i) q^{35} +(157.509 - 272.813i) q^{37} +(-60.9684 + 42.6205i) q^{39} +(-1523.53 + 1815.67i) q^{41} +(2059.19 + 749.486i) q^{43} +(626.568 - 3521.68i) q^{45} +(1944.70 - 342.903i) q^{47} +(-15.4624 + 5.62784i) q^{49} +(-377.999 + 377.416i) q^{51} +2837.46i q^{53} +5643.70 q^{55} +(-3392.23 + 906.144i) q^{57} +(-1440.52 - 3957.81i) q^{59} +(-170.879 - 969.103i) q^{61} +(1367.89 - 3740.29i) q^{63} +(-124.838 + 342.990i) q^{65} +(3050.44 + 2559.62i) q^{67} +(3700.88 - 7920.60i) q^{69} +(-2358.33 - 1361.58i) q^{71} +(4483.19 + 7765.12i) q^{73} +(-5031.88 - 10812.7i) q^{75} +(6188.18 + 1091.14i) q^{77} +(2155.80 - 1808.93i) q^{79} +(3262.96 + 5692.08i) q^{81} +(-1372.40 - 1635.57i) q^{83} +(-455.124 + 2581.14i) q^{85} +(-4348.75 + 2023.77i) q^{87} +(-925.164 + 534.144i) q^{89} +(-203.195 + 351.944i) q^{91} +(12564.1 + 5870.55i) q^{93} +(-11074.1 + 13197.6i) q^{95} +(15343.7 + 5584.66i) q^{97} +(-7940.21 + 6641.79i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 72 q + 9 q^{5} - 102 q^{9}+O(q^{10}) \) 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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