LMFDB - Embedded newform 72.7.b.d.19.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,7,Mod(19,72)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(72, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("72.19");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	72.b (of order $2$, degree $1$, 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:	$16.5638940206$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 78x^{10} + 3408x^{8} + 73216x^{6} + 13959168x^{4} + 1308622848x^{2} + 68719476736 $ x^12 + 78x^10 + 3408x^8 + 73216x^6 + 13959168x^4 + 1308622848*x^2 + 68719476736
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{30}\cdot 3^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.1
Root			$7.38480 - 3.07648i$ of defining polynomial
Character	$\chi$	$=$	72.19
Dual form			72.7.b.d.19.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.38480 - 3.07648i) q^{2} +(45.0705 + 45.4384i) q^{4} -70.4379i q^{5} +87.7768i q^{7} +(-193.046 - 474.212i) q^{8} +O(q^{10})$	\(q+(-7.38480 - 3.07648i) q^{2} +(45.0705 + 45.4384i) q^{4} -70.4379i q^{5} +87.7768i q^{7} +(-193.046 - 474.212i) q^{8} +(-216.701 + 520.170i) q^{10} +407.191 q^{11} +1892.73i q^{13} +(270.044 - 648.214i) q^{14} +(-33.3024 + 4095.86i) q^{16} +2510.66 q^{17} +437.774 q^{19} +(3200.59 - 3174.67i) q^{20} +(-3007.03 - 1252.72i) q^{22} -14020.6i q^{23} +10663.5 q^{25} +(5822.94 - 13977.4i) q^{26} +(-3988.44 + 3956.14i) q^{28} -31543.7i q^{29} -38494.1i q^{31} +(12846.8 - 30144.7i) q^{32} +(-18540.7 - 7724.02i) q^{34} +6182.82 q^{35} -55016.0i q^{37} +(-3232.87 - 1346.80i) q^{38} +(-33402.5 + 13597.7i) q^{40} +45788.9 q^{41} +89987.5 q^{43} +(18352.3 + 18502.1i) q^{44} +(-43134.2 + 103539. i) q^{46} +30207.1i q^{47} +109944. q^{49} +(-78747.8 - 32806.1i) q^{50} +(-86002.5 + 85306.1i) q^{52} +44085.2i q^{53} -28681.7i q^{55} +(41624.8 - 16944.9i) q^{56} +(-97043.8 + 232944. i) q^{58} -274121. q^{59} -233924. i q^{61} +(-118426. + 284271. i) q^{62} +(-187611. + 183089. i) q^{64} +133320. q^{65} +275950. q^{67} +(113157. + 114081. i) q^{68} +(-45658.9 - 19021.3i) q^{70} +481892. i q^{71} +280545. q^{73} +(-169256. + 406282. i) q^{74} +(19730.7 + 19891.7i) q^{76} +35742.0i q^{77} -701125. i q^{79} +(288504. + 2345.76i) q^{80} +(-338142. - 140869. i) q^{82} -740057. q^{83} -176846. i q^{85} +(-664539. - 276845. i) q^{86} +(-78606.6 - 193095. i) q^{88} +1.17451e6 q^{89} -166137. q^{91} +(637074. - 631916. i) q^{92} +(92931.7 - 223073. i) q^{94} -30835.9i q^{95} -105663. q^{97} +(-811916. - 338242. i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 156 q^{4}+O(q^{10}) $ 12 * q - 156 * q^4	$ 12 q - 156 q^{4} + 1416 q^{10} - 1464 q^{16} + 3936 q^{19} + 15888 q^{22} - 47796 q^{25} + 11256 q^{28} + 50016 q^{34} + 70896 q^{40} - 340704 q^{43} + 213696 q^{46} - 304644 q^{49} + 548016 q^{52} - 38616 q^{58} + 206544 q^{64} - 962112 q^{67} + 1074480 q^{70} - 1069560 q^{73} + 1064352 q^{76} - 694944 q^{82} - 3072672 q^{88} + 775008 q^{91} + 3752256 q^{94} - 86952 q^{97}+O(q^{100}) $ 12 * q - 156 * q^4 + 1416 * q^10 - 1464 * q^16 + 3936 * q^19 + 15888 * q^22 - 47796 * q^25 + 11256 * q^28 + 50016 * q^34 + 70896 * q^40 - 340704 * q^43 + 213696 * q^46 - 304644 * q^49 + 548016 * q^52 - 38616 * q^58 + 206544 * q^64 - 962112 * q^67 + 1074480 * q^70 - 1069560 * q^73 + 1064352 * q^76 - 694944 * q^82 - 3072672 * q^88 + 775008 * q^91 + 3752256 * q^94 - 86952 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/72\mathbb{Z}\right)^\times$.

$n$	$37$	$55$	$65$
$\chi(n)$	$-1$	$-1$	$1$

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 $

$2$	−7.38480	−	3.07648i	−0.923100	−	0.384561i
$2$	−7.38480	−	3.07648i	−0.923100	−	0.384561i
$3$		0			0
$3$		0			0
$4$	45.0705	+	45.4384i	0.704226	+	0.709976i
$5$		−	70.4379i		−	0.563503i	−0.959487	−	0.281752i	$-0.909085\pi$
$5$		−	70.4379i		−	0.563503i	0.959487	−	0.281752i	$-0.0909155\pi$
$6$		0			0
$7$			87.7768i			0.255909i	0.991780	+	0.127955i	$0.0408412\pi$
$7$			87.7768i			0.255909i	−0.991780	+	0.127955i	$0.959159\pi$
$8$	−193.046	−	474.212i	−0.377043	−	0.926196i
$9$		0			0
$10$	−216.701	+	520.170i	−0.216701	+	0.520170i
$11$	407.191			0.305929			0.152964	−	0.988232i	$-0.451118\pi$
$11$	407.191			0.305929			0.152964	+	0.988232i	$0.451118\pi$
$12$		0			0
$13$			1892.73i			0.861505i	0.902470	+	0.430752i	$0.141752\pi$
$13$			1892.73i			0.861505i	−0.902470	+	0.430752i	$0.858248\pi$
$14$	270.044	−	648.214i	0.0984125	−	0.236230i
$15$		0			0
$16$	−33.3024	+	4095.86i	−0.00813048	+	0.999967i
$17$	2510.66			0.511025			0.255512	−	0.966806i	$-0.417756\pi$
$17$	2510.66			0.511025			0.255512	+	0.966806i	$0.417756\pi$
$18$		0			0
$19$	437.774			0.0638247			0.0319124	−	0.999491i	$-0.489840\pi$
$19$	437.774			0.0638247			0.0319124	+	0.999491i	$0.489840\pi$
$20$	3200.59	−	3174.67i	0.400074	−	0.396834i
$21$		0			0
$22$	−3007.03	−	1252.72i	−0.282403	−	0.117648i
$23$		−	14020.6i		−	1.15235i	−0.817327	−	0.576173i	$-0.804545\pi$
$23$		−	14020.6i		−	1.15235i	0.817327	−	0.576173i	$-0.195455\pi$
$24$		0			0
$25$	10663.5			0.682464
$26$	5822.94	−	13977.4i	0.331301	−	0.795255i
$27$		0			0
$28$	−3988.44	+	3956.14i	−0.181689	+	0.180218i
$29$		−	31543.7i		−	1.29336i	−0.762762	−	0.646679i	$-0.776157\pi$
$29$		−	31543.7i		−	1.29336i	0.762762	−	0.646679i	$-0.223843\pi$
$30$		0			0
$31$		−	38494.1i		−	1.29214i	−0.763279	−	0.646069i	$-0.776412\pi$
$31$		−	38494.1i		−	1.29214i	0.763279	−	0.646069i	$-0.223588\pi$
$32$	12846.8	−	30144.7i	0.392053	−	0.919943i
$33$		0			0
$34$	−18540.7	−	7724.02i	−0.471727	−	0.196520i
$35$	6182.82			0.144206
$36$		0			0
$37$		−	55016.0i		−	1.08613i	−0.839689	−	0.543067i	$-0.817263\pi$
$37$		−	55016.0i		−	1.08613i	0.839689	−	0.543067i	$-0.182737\pi$
$38$	−3232.87	−	1346.80i	−0.0589166	−	0.0245445i
$39$		0			0
$40$	−33402.5	+	13597.7i	−0.521915	+	0.212465i
$41$	45788.9			0.664368			0.332184	−	0.943215i	$-0.392214\pi$
$41$	45788.9			0.664368			0.332184	+	0.943215i	$0.392214\pi$
$42$		0			0
$43$	89987.5			1.13182			0.565909	−	0.824468i	$-0.308525\pi$
$43$	89987.5			1.13182			0.565909	+	0.824468i	$0.308525\pi$
$44$	18352.3	+	18502.1i	0.215443	+	0.217202i
$45$		0			0
$46$	−43134.2	+	103539.i	−0.443147	+	1.06373i
$47$			30207.1i			0.290948i	0.989362	+	0.145474i	$0.0464707\pi$
$47$			30207.1i			0.290948i	−0.989362	+	0.145474i	$0.953529\pi$
$48$		0			0
$49$	109944.			0.934511
$50$	−78747.8	−	32806.1i	−0.629982	−	0.262449i
$51$		0			0
$52$	−86002.5	+	85306.1i	−0.611647	+	0.606694i
$53$			44085.2i			0.296118i	0.988979	+	0.148059i	$0.0473026\pi$
$53$			44085.2i			0.296118i	−0.988979	+	0.148059i	$0.952697\pi$
$54$		0			0
$55$		−	28681.7i		−	0.172392i
$56$	41624.8	−	16944.9i	0.237022	−	0.0964886i
$57$		0			0
$58$	−97043.8	+	232944.i	−0.497375	+	1.19390i
$59$	−274121.			−1.33471			−0.667353	−	0.744741i	$-0.732573\pi$
$59$	−274121.			−1.33471			−0.667353	+	0.744741i	$0.732573\pi$
$60$		0			0
$61$		−	233924.i		−	1.03059i	−0.857014	−	0.515294i	$-0.827683\pi$
$61$		−	233924.i		−	1.03059i	0.857014	−	0.515294i	$-0.172317\pi$
$62$	−118426.	+	284271.i	−0.496905	+	1.19277i
$63$		0			0
$64$	−187611.	+	183089.i	−0.715678	+	0.698431i
$65$	133320.			0.485461
$66$		0			0
$67$	275950.			0.917501			0.458751	−	0.888565i	$-0.348297\pi$
$67$	275950.			0.917501			0.458751	+	0.888565i	$0.348297\pi$
$68$	113157.	+	114081.i	0.359877	+	0.362815i
$69$		0			0
$70$	−45658.9	−	19021.3i	−0.133116	−	0.0554558i
$71$			481892.i			1.34640i	0.739460	+	0.673201i	$0.235081\pi$
$71$			481892.i			1.34640i	−0.739460	+	0.673201i	$0.764919\pi$
$72$		0			0
$73$	280545.			0.721164			0.360582	−	0.932727i	$-0.382578\pi$
$73$	280545.			0.721164			0.360582	+	0.932727i	$0.382578\pi$
$74$	−169256.	+	406282.i	−0.417685	+	1.00261i
$75$		0			0
$76$	19730.7	+	19891.7i	0.0449470	+	0.0453140i
$77$			35742.0i			0.0782900i
$78$		0			0
$79$		−	701125.i		−	1.42205i	−0.703167	−	0.711024i	$-0.748232\pi$
$79$		−	701125.i		−	1.42205i	0.703167	−	0.711024i	$-0.251768\pi$
$80$	288504.	+	2345.76i	0.563485	+	0.00458155i
$81$		0			0
$82$	−338142.	−	140869.i	−0.613278	−	0.255490i
$83$	−740057.			−1.29429			−0.647144	−	0.762368i	$-0.724037\pi$
$83$	−740057.			−1.29429			−0.647144	+	0.762368i	$0.724037\pi$
$84$		0			0
$85$		−	176846.i		−	0.287964i
$86$	−664539.	−	276845.i	−1.04478	−	0.435253i
$87$		0			0
$88$	−78606.6	−	193095.i	−0.115348	−	0.283350i
$89$	1.17451e6			1.66604			0.833021	−	0.553241i	$-0.186609\pi$
$89$	1.17451e6			1.66604			0.833021	+	0.553241i	$0.186609\pi$
$90$		0			0
$91$	−166137.			−0.220467
$92$	637074.	−	631916.i	0.818138	−	0.811513i
$93$		0			0
$94$	92931.7	−	223073.i	0.111887	−	0.268574i
$95$		−	30835.9i		−	0.0359654i
$96$		0			0
$97$	−105663.			−0.115773			−0.0578867	−	0.998323i	$-0.518436\pi$
$97$	−105663.			−0.115773			−0.0578867	+	0.998323i	$0.518436\pi$
$98$	−811916.	−	338242.i	−0.862646	−	0.359376i
$99$		0			0
$100$	480609.	+	484533.i	0.480609	+	0.484533i
$101$		−	1.55533e6i		−	1.50959i	−0.655961	−	0.754795i	$-0.727736\pi$
$101$		−	1.55533e6i		−	1.50959i	0.655961	−	0.754795i	$-0.272264\pi$
$102$		0			0
$103$			829617.i			0.759217i	0.925147	+	0.379609i	$0.123941\pi$
$103$			829617.i			0.759217i	−0.925147	+	0.379609i	$0.876059\pi$
$104$	897554.	−	365383.i	0.797922	−	0.324824i
$105$		0			0
$106$	135627.	−	325560.i	0.113875	−	0.273346i
$107$	−1.27006e6			−1.03675			−0.518373	−	0.855155i	$-0.673462\pi$
$107$	−1.27006e6			−1.03675			−0.518373	+	0.855155i	$0.673462\pi$
$108$		0			0
$109$			1.53794e6i			1.18757i	0.804623	+	0.593786i	$0.202367\pi$
$109$			1.53794e6i			1.18757i	−0.804623	+	0.593786i	$0.797633\pi$
$110$	−88238.9	+	211809.i	−0.0662952	+	0.159135i
$111$		0			0
$112$	−359522.	−	2923.18i	−0.255901	−	0.00208066i
$113$	2.21064e6			1.53208			0.766042	−	0.642790i	$-0.222223\pi$
$113$	2.21064e6			1.53208			0.766042	+	0.642790i	$0.222223\pi$
$114$		0			0
$115$	−987582.			−0.649352
$116$	1.43330e6	−	1.42169e6i	0.918253	−	0.910817i
$117$		0			0
$118$	2.02433e6	+	843328.i	1.23207	+	0.513276i
$119$			220378.i			0.130776i
$120$		0			0
$121$	−1.60576e6			−0.906407
$122$	−719663.	+	1.72748e6i	−0.396323	+	0.951335i
$123$		0			0
$124$	1.74911e6	−	1.73495e6i	0.917386	−	0.909957i
$125$		−	1.85171e6i		−	0.948074i
$126$		0			0
$127$			2.92447e6i			1.42770i	0.700299	+	0.713849i	$0.253050\pi$
$127$			2.92447e6i			1.42770i	−0.700299	+	0.713849i	$0.746950\pi$
$128$	1.94874e6	−	774897.i	0.929231	−	0.369500i
$129$		0			0
$130$	−984539.	−	410156.i	−0.448129	−	0.186689i
$131$	−3.11430e6			−1.38531			−0.692654	−	0.721270i	$-0.743559\pi$
$131$	−3.11430e6			−1.38531			−0.692654	+	0.721270i	$0.743559\pi$
$132$		0			0
$133$			38426.4i			0.0163333i
$134$	−2.03784e6	−	848957.i	−0.846945	−	0.352835i
$135$		0			0
$136$	−484673.	−	1.19059e6i	−0.192678	−	0.473309i
$137$	2.90658e6			1.13037			0.565184	−	0.824965i	$-0.308805\pi$
$137$	2.90658e6			1.13037			0.565184	+	0.824965i	$0.308805\pi$
$138$		0			0
$139$	−1.18587e6			−0.441564			−0.220782	−	0.975323i	$-0.570861\pi$
$139$	−1.18587e6			−0.441564			−0.220782	+	0.975323i	$0.570861\pi$
$140$	278663.	+	280938.i	0.101553	+	0.102382i
$141$		0			0
$142$	1.48253e6	−	3.55868e6i	0.517773	−	1.24286i
$143$			770702.i			0.263559i
$144$		0			0
$145$	−2.22187e6			−0.728812
$146$	−2.07177e6	−	863093.i	−0.665707	−	0.277331i
$147$		0			0
$148$	2.49984e6	−	2.47960e6i	0.771129	−	0.764885i
$149$			2.36000e6i			0.713433i	0.934213	+	0.356716i	$0.116104\pi$
$149$			2.36000e6i			0.713433i	−0.934213	+	0.356716i	$0.883896\pi$
$150$		0			0
$151$		−	1.41988e6i		−	0.412402i	−0.978510	−	0.206201i	$-0.933890\pi$
$151$		−	1.41988e6i		−	0.412402i	0.978510	−	0.206201i	$-0.0661101\pi$
$152$	−84510.4	−	207598.i	−0.0240646	−	0.0591142i
$153$		0			0
$154$	109960.	−	263947.i	0.0301072	−	0.0722695i
$155$	−2.71144e6			−0.728124
$156$		0			0
$157$			476199.i			0.123052i	0.998105	+	0.0615262i	$0.0195968\pi$
$157$			476199.i			0.123052i	−0.998105	+	0.0615262i	$0.980403\pi$
$158$	−2.15700e6	+	5.17767e6i	−0.546864	+	1.31269i
$159$		0			0
$160$	−2.12333e6	−	904902.i	−0.518391	−	0.220923i
$161$	1.23068e6			0.294896
$162$		0			0
$163$	−1.43617e6			−0.331621			−0.165811	−	0.986158i	$-0.553024\pi$
$163$	−1.43617e6			−0.331621			−0.165811	+	0.986158i	$0.553024\pi$
$164$	2.06373e6	+	2.08058e6i	0.467866	+	0.471685i
$165$		0			0
$166$	5.46517e6	+	2.27677e6i	1.19476	+	0.497732i
$167$			9.03000e6i			1.93882i	0.245440	+	0.969412i	$0.421067\pi$
$167$			9.03000e6i			1.93882i	−0.245440	+	0.969412i	$0.578933\pi$
$168$		0			0
$169$	1.24440e6			0.257810
$170$	−544064.	+	1.30597e6i	−0.110740	+	0.265820i
$171$		0			0
$172$	4.05578e6	+	4.08889e6i	0.797056	+	0.803563i
$173$			76231.1i			0.0147229i	0.999973	+	0.00736146i	$0.00234325\pi$
$173$			76231.1i			0.0147229i	−0.999973	+	0.00736146i	$0.997657\pi$
$174$		0			0
$175$			936008.i			0.174649i
$176$	−13560.5	+	1.66780e6i	−0.00248735	+	0.305919i
$177$		0			0
$178$	−8.67351e6	−	3.61336e6i	−1.53792	−	0.640694i
$179$	−902306.			−0.157324			−0.0786620	−	0.996901i	$-0.525065\pi$
$179$	−902306.			−0.157324			−0.0786620	+	0.996901i	$0.525065\pi$
$180$		0			0
$181$			3.31042e6i			0.558274i	0.960251	+	0.279137i	$0.0900483\pi$
$181$			3.31042e6i			0.558274i	−0.960251	+	0.279137i	$0.909952\pi$
$182$	1.22689e6	+	511119.i	0.203513	+	0.0847829i
$183$		0			0
$184$	−6.64874e6	+	2.70662e6i	−1.06730	+	0.434484i
$185$	−3.87521e6			−0.612041
$186$		0			0
$187$	1.02232e6			0.156337
$188$	−1.37256e6	+	1.36145e6i	−0.206566	+	0.204893i
$189$		0			0
$190$	−94866.1	+	227717.i	−0.0138309	+	0.0331997i
$191$		−	8.82736e6i		−	1.26687i	−0.773797	−	0.633433i	$-0.781645\pi$
$191$		−	8.82736e6i		−	1.26687i	0.773797	−	0.633433i	$-0.218355\pi$
$192$		0			0
$193$	4.70843e6			0.654944			0.327472	−	0.944861i	$-0.393803\pi$
$193$	4.70843e6			0.654944			0.327472	+	0.944861i	$0.393803\pi$
$194$	780302.	+	325071.i	0.106870	+	0.0445219i
$195$		0			0
$196$	4.95524e6	+	4.99569e6i	0.658107	+	0.663480i
$197$		−	1.73840e6i		−	0.227380i	−0.993516	−	0.113690i	$-0.963733\pi$
$197$		−	1.73840e6i		−	0.227380i	0.993516	−	0.113690i	$-0.0362671\pi$
$198$		0			0
$199$		−	3.96340e6i		−	0.502931i	−0.967866	−	0.251466i	$-0.919087\pi$
$199$		−	3.96340e6i		−	0.502931i	0.967866	−	0.251466i	$-0.0809125\pi$
$200$	−2.05854e6	−	5.05676e6i	−0.257318	−	0.632095i
$201$		0			0
$202$	−4.78496e6	+	1.14858e7i	−0.580529	+	1.39350i
$203$	2.76881e6			0.330982
$204$		0			0
$205$		−	3.22528e6i		−	0.374374i
$206$	2.55230e6	−	6.12656e6i	0.291965	−	0.700833i
$207$		0			0
$208$	−7.75235e6	−	63032.4i	−0.861476	−	0.00700445i
$209$	178258.			0.0195258
$210$		0			0
$211$	1.08409e7			1.15403			0.577017	−	0.816732i	$-0.304216\pi$
$211$	1.08409e7			1.15403			0.577017	+	0.816732i	$0.304216\pi$
$212$	−2.00316e6	+	1.98694e6i	−0.210237	+	0.208534i
$213$		0			0
$214$	9.37912e6	+	3.90731e6i	0.957020	+	0.398692i
$215$		−	6.33853e6i		−	0.637784i
$216$		0			0
$217$	3.37889e6			0.330670
$218$	4.73145e6	−	1.13574e7i	0.456693	−	1.09625i
$219$		0			0
$220$	1.30325e6	−	1.29270e6i	0.122394	−	0.121403i
$221$			4.75200e6i			0.440250i
$222$		0			0
$223$		−	1.09680e7i		−	0.989036i	−0.869167	−	0.494518i	$-0.835345\pi$
$223$		−	1.09680e7i		−	0.989036i	0.869167	−	0.494518i	$-0.164655\pi$
$224$	2.64600e6	+	1.12765e6i	0.235422	+	0.100330i
$225$		0			0
$226$	−1.63251e7	−	6.80100e6i	−1.41427	−	0.589179i
$227$	1.96401e7			1.67906			0.839531	−	0.543311i	$-0.182830\pi$
$227$	1.96401e7			1.67906			0.839531	+	0.543311i	$0.182830\pi$
$228$		0			0
$229$		−	2.19907e7i		−	1.83119i	−0.402102	−	0.915595i	$-0.631720\pi$
$229$		−	2.19907e7i		−	1.83119i	0.402102	−	0.915595i	$-0.368280\pi$
$230$	7.29310e6	+	3.03828e6i	0.599416	+	0.249715i
$231$		0			0
$232$	−1.49584e7	+	6.08938e6i	−1.19790	+	0.487651i
$233$	−8.24811e6			−0.652058			−0.326029	−	0.945360i	$-0.605711\pi$
$233$	−8.24811e6			−0.652058			−0.326029	+	0.945360i	$0.605711\pi$
$234$		0			0
$235$	2.12773e6			0.163950
$236$	−1.23548e7	−	1.24556e7i	−0.939936	−	0.947609i
$237$		0			0
$238$	677990.	−	1.62745e6i	0.0502912	−	0.120719i
$239$		−	7.28997e6i		−	0.533989i	−0.963698	−	0.266994i	$-0.913969\pi$
$239$		−	7.28997e6i		−	0.533989i	0.963698	−	0.266994i	$-0.0860305\pi$
$240$		0			0
$241$	6.35534e6			0.454033			0.227017	−	0.973891i	$-0.427103\pi$
$241$	6.35534e6			0.454033			0.227017	+	0.973891i	$0.427103\pi$
$242$	1.18582e7	+	4.94008e6i	0.836705	+	0.348569i
$243$		0			0
$244$	1.06291e7	−	1.05431e7i	0.731692	−	0.725767i
$245$		−	7.74424e6i		−	0.526600i
$246$		0			0
$247$			828586.i			0.0549853i
$248$	−1.82544e7	+	7.43112e6i	−1.19677	+	0.487191i
$249$		0			0
$250$	−5.69675e6	+	1.36745e7i	−0.364592	+	0.875167i
$251$	1.52326e7			0.963278			0.481639	−	0.876370i	$-0.340042\pi$
$251$	1.52326e7			0.963278			0.481639	+	0.876370i	$0.340042\pi$
$252$		0			0
$253$		−	5.70907e6i		−	0.352536i
$254$	8.99710e6	−	2.15967e7i	0.549037	−	1.31791i
$255$		0			0
$256$	−1.67750e7	−	272805.i	−0.999868	−	0.0162604i
$257$	−2.49259e7			−1.46842			−0.734211	−	0.678921i	$-0.762448\pi$
$257$	−2.49259e7			−1.46842			−0.734211	+	0.678921i	$0.762448\pi$
$258$		0			0
$259$	4.82913e6			0.277952
$260$	6.00878e6	+	6.05784e6i	0.341874	+	0.344665i
$261$		0			0
$262$	2.29985e7	+	9.58109e6i	1.27878	+	0.532735i
$263$		−	9.24561e6i		−	0.508239i	−0.967173	−	0.254120i	$-0.918214\pi$
$263$		−	9.24561e6i		−	0.508239i	0.967173	−	0.254120i	$-0.0817857\pi$
$264$		0			0
$265$	3.10527e6			0.166864
$266$	118218.	−	283771.i	0.00628115	−	0.0150773i
$267$		0			0
$268$	1.24372e7	+	1.25388e7i	0.646129	+	0.651403i
$269$			1.04075e7i			0.534673i	0.963603	+	0.267336i	$0.0861435\pi$
$269$			1.04075e7i			0.534673i	−0.963603	+	0.267336i	$0.913857\pi$
$270$		0			0
$271$			3.59018e6i			0.180388i	0.995924	+	0.0901941i	$0.0287487\pi$
$271$			3.59018e6i			0.180388i	−0.995924	+	0.0901941i	$0.971251\pi$
$272$	−83611.2	+	1.02833e7i	−0.00415487	+	0.511008i
$273$		0			0
$274$	−2.14645e7	−	8.94204e6i	−1.04344	−	0.434695i
$275$	4.34209e6			0.208785
$276$		0			0
$277$			3.18610e7i			1.49906i	0.661968	+	0.749532i	$0.269722\pi$
$277$			3.18610e7i			1.49906i	−0.661968	+	0.749532i	$0.730278\pi$
$278$	8.75743e6	+	3.64832e6i	0.407608	+	0.169808i
$279$		0			0
$280$	−1.19357e6	−	2.93197e6i	−0.0543717	−	0.133563i
$281$	−2.14679e7			−0.967544			−0.483772	−	0.875194i	$-0.660734\pi$
$281$	−2.14679e7			−0.967544			−0.483772	+	0.875194i	$0.660734\pi$
$282$		0			0
$283$	−4.37178e6			−0.192885			−0.0964427	−	0.995339i	$-0.530746\pi$
$283$	−4.37178e6			−0.192885			−0.0964427	+	0.995339i	$0.530746\pi$
$284$	−2.18964e7	+	2.17191e7i	−0.955913	+	0.948172i
$285$		0			0
$286$	2.37105e6	−	5.69148e6i	0.101354	−	0.243291i
$287$			4.01921e6i			0.170018i
$288$		0			0
$289$	−1.78341e7			−0.738854
$290$	1.64081e7	+	6.83556e6i	0.672766	+	0.280272i
$291$		0			0
$292$	1.26443e7	+	1.27475e7i	0.507863	+	0.512009i
$293$			4.57949e7i			1.82060i	0.413952	+	0.910299i	$0.364148\pi$
$293$			4.57949e7i			1.82060i	−0.413952	+	0.910299i	$0.635852\pi$
$294$		0			0
$295$			1.93085e7i			0.752112i
$296$	−2.60893e7	+	1.06206e7i	−1.00597	+	0.409519i
$297$		0			0
$298$	7.26050e6	−	1.74281e7i	0.274358	−	0.658569i
$299$	2.65372e7			0.992752
$300$		0			0
$301$			7.89881e6i			0.289643i
$302$	−4.36824e6	+	1.04855e7i	−0.158594	+	0.380688i
$303$		0			0
$304$	−14578.9	+	1.79306e6i	−0.000518925	+	0.0638226i
$305$	−1.64771e7			−0.580739
$306$		0			0
$307$	−2.72059e7			−0.940261			−0.470130	−	0.882597i	$-0.655793\pi$
$307$	−2.72059e7			−0.940261			−0.470130	+	0.882597i	$0.655793\pi$
$308$	−1.62406e6	+	1.61091e6i	−0.0555840	+	0.0551339i
$309$		0			0
$310$	2.00235e7	+	8.34171e6i	0.672131	+	0.280008i
$311$			2.61783e7i			0.870281i	0.900363	+	0.435141i	$0.143301\pi$
$311$			2.61783e7i			0.870281i	−0.900363	+	0.435141i	$0.856699\pi$
$312$		0			0
$313$	−9.34650e6			−0.304801			−0.152400	−	0.988319i	$-0.548700\pi$
$313$	−9.34650e6			−0.304801			−0.152400	+	0.988319i	$0.548700\pi$
$314$	1.46502e6	−	3.51664e6i	0.0473211	−	0.113590i
$315$		0			0
$316$	3.18580e7	−	3.16001e7i	1.00962	−	1.00144i
$317$		−	3.10625e6i		−	0.0975122i	−0.998811	−	0.0487561i	$-0.984474\pi$
$317$		−	3.10625e6i		−	0.0975122i	0.998811	−	0.0487561i	$-0.0155257\pi$
$318$		0			0
$319$		−	1.28443e7i		−	0.395676i
$320$	1.28964e7	+	1.32149e7i	0.393568	+	0.403287i
$321$		0			0
$322$	−9.08835e6	−	3.78618e6i	−0.272218	−	0.113405i
$323$	1.09910e6			0.0326160
$324$		0			0
$325$			2.01831e7i			0.587946i
$326$	1.06058e7	+	4.41835e6i	0.306120	+	0.127528i
$327$		0			0
$328$	−8.83936e6	−	2.17137e7i	−0.250495	−	0.615335i
$329$	−2.65148e6			−0.0744563
$330$		0			0
$331$	−3.12401e7			−0.861446			−0.430723	−	0.902484i	$-0.641741\pi$
$331$	−3.12401e7			−0.861446			−0.430723	+	0.902484i	$0.641741\pi$
$332$	−3.33547e7	−	3.36270e7i	−0.911472	−	0.918913i
$333$		0			0
$334$	2.77806e7	−	6.66847e7i	0.745595	−	1.78973i
$335$		−	1.94374e7i		−	0.517015i
$336$		0			0
$337$	−3.53358e6			−0.0923261			−0.0461631	−	0.998934i	$-0.514699\pi$
$337$	−3.53358e6			−0.0923261			−0.0461631	+	0.998934i	$0.514699\pi$
$338$	−9.18963e6	−	3.82837e6i	−0.237984	−	0.0991434i
$339$		0			0
$340$	8.03560e6	−	7.97053e6i	0.204447	−	0.202792i
$341$		−	1.56745e7i		−	0.395302i
$342$		0			0
$343$			1.99774e7i			0.495059i
$344$	−1.73717e7	−	4.26732e7i	−0.426744	−	1.04829i
$345$		0			0
$346$	234524.	−	562952.i	0.00566186	−	0.0135907i
$347$	−1.05723e7			−0.253035			−0.126517	−	0.991964i	$-0.540380\pi$
$347$	−1.05723e7			−0.253035			−0.126517	+	0.991964i	$0.540380\pi$
$348$		0			0
$349$		−	9.42178e6i		−	0.221644i	−0.993840	−	0.110822i	$-0.964652\pi$
$349$		−	9.42178e6i		−	0.221644i	0.993840	−	0.110822i	$-0.0353484\pi$
$350$	2.87961e6	−	6.91223e6i	0.0671630	−	0.161218i
$351$		0			0
$352$	5.23111e6	−	1.22747e7i	0.119940	−	0.281437i
$353$	5.39995e6			0.122762			0.0613812	−	0.998114i	$-0.480449\pi$
$353$	5.39995e6			0.122762			0.0613812	+	0.998114i	$0.480449\pi$
$354$		0			0
$355$	3.39435e7			0.758702
$356$	5.29357e7	+	5.33678e7i	1.17327	+	1.18285i
$357$		0			0
$358$	6.66335e6	+	2.77593e6i	0.145226	+	0.0605006i
$359$		−	4.92820e7i		−	1.06514i	−0.846387	−	0.532568i	$-0.821227\pi$
$359$		−	4.92820e7i		−	1.06514i	0.846387	−	0.532568i	$-0.178773\pi$
$360$		0			0
$361$	−4.68542e7			−0.995926
$362$	1.01845e7	−	2.44468e7i	0.214690	−	0.515342i
$363$		0			0
$364$	−7.48789e6	−	7.54902e6i	−0.155259	−	0.156526i
$365$		−	1.97610e7i		−	0.406379i
$366$		0			0
$367$			5.06013e7i			1.02368i	0.859081	+	0.511839i	$0.171036\pi$
$367$			5.06013e7i			1.02368i	−0.859081	+	0.511839i	$0.828964\pi$
$368$	5.74265e7	+	466920.i	1.15231	+	0.00936913i
$369$		0			0
$370$	2.86177e7	+	1.19220e7i	0.564975	+	0.235367i
$371$	−3.86965e6			−0.0757793
$372$		0			0
$373$		−	1.29304e7i		−	0.249164i	−0.992209	−	0.124582i	$-0.960241\pi$
$373$		−	1.29304e7i		−	0.249164i	0.992209	−	0.124582i	$-0.0397590\pi$
$374$	−7.54963e6	−	3.14515e6i	−0.144315	−	0.0601211i
$375$		0			0
$376$	1.43246e7	−	5.83136e6i	0.269475	−	0.109700i
$377$	5.97036e7			1.11423
$378$		0			0
$379$	−5.22376e7			−0.959545			−0.479773	−	0.877393i	$-0.659281\pi$
$379$	−5.22376e7			−0.959545			−0.479773	+	0.877393i	$0.659281\pi$
$380$	1.40113e6	−	1.38979e6i	0.0255346	−	0.0253278i
$381$		0			0
$382$	−2.71572e7	+	6.51883e7i	−0.487187	+	1.16944i
$383$			4.11780e7i			0.732942i	0.930430	+	0.366471i	$0.119434\pi$
$383$			4.11780e7i			0.732942i	−0.930430	+	0.366471i	$0.880566\pi$
$384$		0			0
$385$	2.51759e6			0.0441167
$386$	−3.47708e7	−	1.44854e7i	−0.604579	−	0.251866i
$387$		0			0
$388$	−4.76229e6	−	4.80117e6i	−0.0815307	−	0.0821963i
$389$		−	6.53759e7i		−	1.11063i	−0.831640	−	0.555315i	$-0.812598\pi$
$389$		−	6.53759e7i		−	1.11063i	0.831640	−	0.555315i	$-0.187402\pi$
$390$		0			0
$391$		−	3.52010e7i		−	0.588878i
$392$	−2.12243e7	−	5.21369e7i	−0.352350	−	0.865540i
$393$		0			0
$394$	−5.34818e6	+	1.28378e7i	−0.0874414	+	0.209894i
$395$	−4.93858e7			−0.801329
$396$		0			0
$397$			5.08416e7i			0.812545i	0.913752	+	0.406273i	$0.133172\pi$
$397$			5.08416e7i			0.812545i	−0.913752	+	0.406273i	$0.866828\pi$
$398$	−1.21933e7	+	2.92689e7i	−0.193407	+	0.464256i
$399$		0			0
$400$	−355121.	+	4.36762e7i	−0.00554876	+	0.682441i
$401$	5.67453e7			0.880028			0.440014	−	0.897991i	$-0.354973\pi$
$401$	5.67453e7			0.880028			0.440014	+	0.897991i	$0.354973\pi$
$402$		0			0
$403$	7.28587e7			1.11318
$404$	7.06719e7	−	7.00996e7i	1.07177	−	1.06309i
$405$		0			0
$406$	−2.04471e7	−	8.51819e6i	−0.305530	−	0.127283i
$407$		−	2.24020e7i		−	0.332280i
$408$		0			0
$409$	7.30390e7			1.06754			0.533771	−	0.845629i	$-0.320774\pi$
$409$	7.30390e7			1.06754			0.533771	+	0.845629i	$0.320774\pi$
$410$	−9.92252e6	+	2.38180e7i	−0.143969	+	0.345584i
$411$		0			0
$412$	−3.76965e7	+	3.73913e7i	−0.539026	+	0.534661i
$413$		−	2.40614e7i		−	0.341564i
$414$		0			0
$415$			5.21281e7i			0.729336i
$416$	5.70556e7	+	2.43155e7i	0.792535	+	0.337756i
$417$		0			0
$418$	−1.31640e6	−	548407.i	−0.0180243	−	0.00750886i
$419$	−4.08143e7			−0.554843			−0.277422	−	0.960748i	$-0.589480\pi$
$419$	−4.08143e7			−0.554843			−0.277422	+	0.960748i	$0.589480\pi$
$420$		0			0
$421$		−	6.52527e7i		−	0.874485i	−0.899344	−	0.437242i	$-0.855955\pi$
$421$		−	6.52527e7i		−	0.874485i	0.899344	−	0.437242i	$-0.144045\pi$
$422$	−8.00580e7	−	3.33519e7i	−1.06529	−	0.443796i
$423$		0			0
$424$	2.09057e7	−	8.51046e6i	0.274263	−	0.111649i
$425$	2.67725e7			0.348756
$426$		0			0
$427$	2.05331e7			0.263737
$428$	−5.72421e7	−	5.77095e7i	−0.730104	−	0.736064i
$429$		0			0
$430$	−1.95004e7	+	4.68088e7i	−0.245266	+	0.588738i
$431$			1.12608e8i			1.40649i	0.710946	+	0.703246i	$0.248267\pi$
$431$			1.12608e8i			1.40649i	−0.710946	+	0.703246i	$0.751733\pi$
$432$		0			0
$433$	6.09414e6			0.0750669			0.0375334	−	0.999295i	$-0.488050\pi$
$433$	6.09414e6			0.0750669			0.0375334	+	0.999295i	$0.488050\pi$
$434$	−2.49524e7	−	1.03951e7i	−0.305241	−	0.127162i
$435$		0			0
$436$	−6.98815e7	+	6.93157e7i	−0.843146	+	0.836319i
$437$		−	6.13785e6i		−	0.0735482i
$438$		0			0
$439$		−	1.48028e8i		−	1.74965i	−0.484441	−	0.874824i	$-0.660977\pi$
$439$		−	1.48028e8i		−	1.74965i	0.484441	−	0.874824i	$-0.339023\pi$
$440$	−1.36012e7	+	5.53689e6i	−0.159669	+	0.0649991i
$441$		0			0
$442$	1.46194e7	−	3.50925e7i	0.169303	−	0.406395i
$443$	−9.33873e7			−1.07418			−0.537089	−	0.843525i	$-0.680476\pi$
$443$	−9.33873e7			−1.07418			−0.537089	+	0.843525i	$0.680476\pi$
$444$		0			0
$445$		−	8.27300e7i		−	0.938821i
$446$	−3.37428e7	+	8.09964e7i	−0.380344	+	0.912979i
$447$		0			0
$448$	−1.60710e7	−	1.64679e7i	−0.178735	−	0.183148i
$449$	1.13240e8			1.25101			0.625506	−	0.780219i	$-0.284892\pi$
$449$	1.13240e8			1.25101			0.625506	+	0.780219i	$0.284892\pi$
$450$		0			0
$451$	1.86449e7			0.203249
$452$	9.96346e7	+	1.00448e8i	1.07893	+	1.08774i
$453$		0			0
$454$	−1.45038e8	−	6.04226e7i	−1.54994	−	0.645701i
$455$			1.17024e7i			0.124234i
$456$		0			0
$457$	5.88297e7			0.616379			0.308189	−	0.951325i	$-0.400277\pi$
$457$	5.88297e7			0.616379			0.308189	+	0.951325i	$0.400277\pi$
$458$	−6.76542e7	+	1.62397e8i	−0.704203	+	1.69037i
$459$		0			0
$460$	−4.45108e7	−	4.48742e7i	−0.457290	−	0.461024i
$461$			1.13126e8i			1.15467i	0.816507	+	0.577336i	$0.195908\pi$
$461$			1.13126e8i			1.15467i	−0.816507	+	0.577336i	$0.804092\pi$
$462$		0			0
$463$		−	1.85609e8i		−	1.87006i	−0.354565	−	0.935032i	$-0.615371\pi$
$463$		−	1.85609e8i		−	1.87006i	0.354565	−	0.935032i	$-0.384629\pi$
$464$	1.29199e8	+	1.05048e6i	1.29332	+	0.0105156i
$465$		0			0
$466$	6.09106e7	+	2.53752e7i	0.601915	+	0.250756i
$467$	−1.42477e8			−1.39893			−0.699463	−	0.714669i	$-0.746577\pi$
$467$	−1.42477e8			−1.39893			−0.699463	+	0.714669i	$0.746577\pi$
$468$		0			0
$469$			2.42220e7i			0.234797i
$470$	−1.57128e7	−	6.54592e6i	−0.151342	−	0.0630488i
$471$		0			0
$472$	5.29179e7	+	1.29991e8i	0.503241	+	1.23620i
$473$	3.66421e7			0.346256
$474$		0			0
$475$	4.66820e6			0.0435581
$476$	−1.00136e7	+	9.93254e6i	−0.0928476	+	0.0920958i
$477$		0			0
$478$	−2.24275e7	+	5.38350e7i	−0.205351	+	0.492925i
$479$			6.51931e7i			0.593192i	0.955003	+	0.296596i	$0.0958514\pi$
$479$			6.51931e7i			0.593192i	−0.955003	+	0.296596i	$0.904149\pi$
$480$		0			0
$481$	1.04130e8			0.935710
$482$	−4.69329e7	−	1.95521e7i	−0.419118	−	0.174603i
$483$		0			0
$484$	−7.23722e7	−	7.29630e7i	−0.638316	−	0.643527i
$485$			7.44270e6i			0.0652387i
$486$		0			0
$487$			1.33080e8i			1.15219i	0.817382	+	0.576096i	$0.195425\pi$
$487$			1.33080e8i			1.15219i	−0.817382	+	0.576096i	$0.804575\pi$
$488$	−1.10929e8	+	4.51580e7i	−0.954526	+	0.388575i
$489$		0			0
$490$	−2.38250e7	+	5.71897e7i	−0.202510	+	0.486104i
$491$	−6.06490e7			−0.512365			−0.256183	−	0.966628i	$-0.582465\pi$
$491$	−6.06490e7			−0.512365			−0.256183	+	0.966628i	$0.582465\pi$
$492$		0			0
$493$		−	7.91957e7i		−	0.660938i
$494$	2.54913e6	−	6.11894e6i	0.0211452	−	0.0507569i
$495$		0			0
$496$	1.57666e8	+	1.28195e6i	1.29209	+	0.0105057i
$497$	−4.22989e7			−0.344556
$498$		0			0
$499$	1.56394e8			1.25869			0.629346	−	0.777125i	$-0.283323\pi$
$499$	1.56394e8			1.25869			0.629346	+	0.777125i	$0.283323\pi$
$500$	8.41387e7	−	8.34574e7i	0.673109	−	0.667659i
$501$		0			0
$502$	−1.12489e8	−	4.68627e7i	−0.889202	−	0.370439i
$503$			8.52300e7i			0.669713i	0.942269	+	0.334856i	$0.108688\pi$
$503$			8.52300e7i			0.669713i	−0.942269	+	0.334856i	$0.891312\pi$
$504$		0			0
$505$	−1.09554e8			−0.850659
$506$	−1.75639e7	+	4.21603e7i	−0.135572	+	0.325426i
$507$		0			0
$508$	−1.32884e8	+	1.31807e8i	−1.01363	+	1.00542i
$509$		−	2.26292e8i		−	1.71599i	−0.513657	−	0.857996i	$-0.671709\pi$
$509$		−	2.26292e8i		−	1.71599i	0.513657	−	0.857996i	$-0.328291\pi$
$510$		0			0
$511$			2.46254e7i			0.184552i
$512$	1.23041e8	+	5.36226e7i	0.916725	+	0.399520i
$513$		0			0
$514$	1.84073e8	+	7.66841e7i	1.35550	+	0.564697i
$515$	5.84365e7			0.427822
$516$		0			0
$517$			1.23001e7i			0.0890095i
$518$	−3.56621e7	−	1.48567e7i	−0.256577	−	0.106889i
$519$		0			0
$520$	−2.57368e7	−	6.32218e7i	−0.183039	−	0.449632i
$521$	5.63569e7			0.398505			0.199253	−	0.979948i	$-0.436149\pi$
$521$	5.63569e7			0.398505			0.199253	+	0.979948i	$0.436149\pi$
$522$		0			0
$523$	−1.91273e8			−1.33705			−0.668525	−	0.743690i	$-0.733074\pi$
$523$	−1.91273e8			−1.33705			−0.668525	+	0.743690i	$0.733074\pi$
$524$	−1.40363e8	−	1.41509e8i	−0.975571	−	0.983535i
$525$		0			0
$526$	−2.84440e7	+	6.82769e7i	−0.195449	+	0.469156i
$527$		−	9.66456e7i		−	0.660314i
$528$		0			0
$529$	−4.85415e7			−0.327904
$530$	−2.29318e7	−	9.55331e6i	−0.154032	−	0.0641691i
$531$		0			0
$532$	−1.74603e6	+	1.73190e6i	−0.0115963	+	0.0115024i
$533$			8.66659e7i			0.572356i
$534$		0			0
$535$			8.94603e7i			0.584210i
$536$	−5.32711e7	−	1.30859e8i	−0.345937	−	0.849786i
$537$		0			0
$538$	3.20184e7	−	7.68570e7i	0.205614	−	0.493556i
$539$	4.47684e7			0.285894
$540$		0			0
$541$			2.32216e8i			1.46656i	0.679927	+	0.733280i	$0.262011\pi$
$541$			2.32216e8i			1.46656i	−0.679927	+	0.733280i	$0.737989\pi$
$542$	1.10451e7	−	2.65127e7i	0.0693702	−	0.166516i
$543$		0			0
$544$	3.22540e7	−	7.56831e7i	0.200349	−	0.470113i
$545$	1.08329e8			0.669200
$546$		0			0
$547$	1.00241e8			0.612467			0.306233	−	0.951956i	$-0.400931\pi$
$547$	1.00241e8			0.612467			0.306233	+	0.951956i	$0.400931\pi$
$548$	1.31001e8	+	1.32070e8i	0.796035	+	0.802534i
$549$		0			0
$550$	−3.20654e7	−	1.33584e7i	−0.192730	−	0.0802907i
$551$		−	1.38090e7i		−	0.0825482i
$552$		0			0
$553$	6.15426e7			0.363915
$554$	9.80199e7	−	2.35287e8i	0.576481	−	1.38379i
$555$		0			0
$556$	−5.34479e7	−	5.38842e7i	−0.310961	−	0.313500i
$557$		−	7.68492e6i		−	0.0444707i	−0.999753	−	0.0222353i	$-0.992922\pi$
$557$		−	7.68492e6i		−	0.0444707i	0.999753	−	0.0222353i	$-0.00707831\pi$
$558$		0			0
$559$			1.70322e8i			0.975067i
$560$	−205903.	+	2.53240e7i	−0.00117246	+	0.144201i
$561$		0			0
$562$	1.58536e8	+	6.60457e7i	0.893140	+	0.372079i
$563$	6.49758e7			0.364105			0.182052	−	0.983289i	$-0.441726\pi$
$563$	6.49758e7			0.364105			0.182052	+	0.983289i	$0.441726\pi$
$564$		0			0
$565$		−	1.55713e8i		−	0.863335i
$566$	3.22847e7	+	1.34497e7i	0.178052	+	0.0741761i
$567$		0			0
$568$	2.28519e8	−	9.30273e7i	1.24703	−	0.507651i
$569$	−1.39548e8			−0.757506			−0.378753	−	0.925498i	$-0.623647\pi$
$569$	−1.39548e8			−0.757506			−0.378753	+	0.925498i	$0.623647\pi$
$570$		0			0
$571$	−2.18147e8			−1.17177			−0.585884	−	0.810395i	$-0.699253\pi$
$571$	−2.18147e8			−1.17177			−0.585884	+	0.810395i	$0.699253\pi$
$572$	−3.50195e7	+	3.47359e7i	−0.187121	+	0.185605i
$573$		0			0
$574$	1.23650e7	−	2.96810e7i	0.0653822	−	0.156943i
$575$		−	1.49509e8i		−	0.786435i
$576$		0			0
$577$	−2.20393e8			−1.14728			−0.573641	−	0.819107i	$-0.694470\pi$
$577$	−2.20393e8			−1.14728			−0.573641	+	0.819107i	$0.694470\pi$
$578$	1.31702e8	+	5.48664e7i	0.682036	+	0.284134i
$579$		0			0
$580$	−1.00141e8	−	1.00958e8i	−0.513249	−	0.517439i
$581$		−	6.49598e7i		−	0.331220i
$582$		0			0
$583$			1.79511e7i			0.0905911i
$584$	−5.41581e7	−	1.33038e8i	−0.271910	−	0.667939i
$585$		0			0
$586$	1.40887e8	−	3.38186e8i	0.700130	−	1.68059i
$587$	2.40870e8			1.19088			0.595441	−	0.803399i	$-0.296977\pi$
$587$	2.40870e8			1.19088			0.595441	+	0.803399i	$0.296977\pi$
$588$		0			0
$589$		−	1.68517e7i		−	0.0824703i
$590$	5.94023e7	−	1.42589e8i	0.289233	−	0.694274i
$591$		0			0
$592$	2.25338e8	+	1.83217e6i	1.08610	+	0.00883080i
$593$	−1.25943e8			−0.603963			−0.301981	−	0.953314i	$-0.597648\pi$
$593$	−1.25943e8			−0.603963			−0.301981	+	0.953314i	$0.597648\pi$
$594$		0			0
$595$	1.55230e7			0.0736926
$596$	−1.07235e8	+	1.06366e8i	−0.506520	+	0.502418i
$597$		0			0
$598$	−1.95972e8	−	8.16412e7i	−0.916409	−	0.381773i
$599$		−	3.41747e8i		−	1.59010i	−0.606545	−	0.795049i	$-0.707445\pi$
$599$		−	3.41747e8i		−	1.59010i	0.606545	−	0.795049i	$-0.292555\pi$
$600$		0			0
$601$	−1.02322e8			−0.471354			−0.235677	−	0.971831i	$-0.575731\pi$
$601$	−1.02322e8			−0.471354			−0.235677	+	0.971831i	$0.575731\pi$
$602$	2.43006e7	−	5.83311e7i	0.111385	−	0.267369i
$603$		0			0
$604$	6.45172e7	−	6.39947e7i	0.292795	−	0.290425i
$605$			1.13106e8i			0.510764i
$606$		0			0
$607$		−	2.21360e8i		−	0.989767i	−0.868959	−	0.494883i	$-0.835211\pi$
$607$		−	2.21360e8i		−	0.989767i	0.868959	−	0.494883i	$-0.164789\pi$
$608$	5.62399e6	−	1.31965e7i	0.0250227	−	0.0587151i
$609$		0			0
$610$	1.21680e8	+	5.06915e7i	0.536080	+	0.223329i
$611$	−5.71738e7			−0.250653
$612$		0			0
$613$			2.74777e8i			1.19289i	0.802655	+	0.596443i	$0.203420\pi$
$613$			2.74777e8i			1.19289i	−0.802655	+	0.596443i	$0.796580\pi$
$614$	2.00910e8	+	8.36986e7i	0.867955	+	0.361587i
$615$		0			0
$616$	1.69493e7	−	6.89984e6i	0.0725119	−	0.0295187i
$617$	−1.15858e8			−0.493254			−0.246627	−	0.969111i	$-0.579322\pi$
$617$	−1.15858e8			−0.493254			−0.246627	+	0.969111i	$0.579322\pi$
$618$		0			0
$619$	−3.21899e8			−1.35721			−0.678606	−	0.734502i	$-0.737416\pi$
$619$	−3.21899e8			−1.35721			−0.678606	+	0.734502i	$0.737416\pi$
$620$	−1.22206e8	−	1.23204e8i	−0.512764	−	0.516950i
$621$		0			0
$622$	8.05370e7	−	1.93321e8i	0.334676	−	0.803356i
$623$			1.03095e8i			0.426355i
$624$		0			0
$625$	3.61867e7			0.148221
$626$	6.90220e7	+	2.87544e7i	0.281362	+	0.117214i
$627$		0			0
$628$	−2.16378e7	+	2.14625e7i	−0.0873642	+	0.0866567i
$629$		−	1.38127e8i		−	0.555042i
$630$		0			0
$631$			1.13785e8i			0.452896i	0.974023	+	0.226448i	$0.0727113\pi$
$631$			1.13785e8i			0.452896i	−0.974023	+	0.226448i	$0.927289\pi$
$632$	−3.32482e8	+	1.35349e8i	−1.31710	+	0.536173i
$633$		0			0
$634$	−9.55633e6	+	2.29390e7i	−0.0374993	+	0.0900135i
$635$	2.05994e8			0.804513
$636$		0			0
$637$			2.08094e8i			0.805085i
$638$	−3.95154e7	+	9.48528e7i	−0.152161	+	0.365248i
$639$		0			0
$640$	−5.45822e7	−	1.37265e8i	−0.208214	−	0.523625i
$641$	−5.07320e8			−1.92623			−0.963114	−	0.269092i	$-0.913276\pi$
$641$	−5.07320e8			−1.92623			−0.963114	+	0.269092i	$0.913276\pi$
$642$		0			0
$643$	3.63907e8			1.36885			0.684427	−	0.729081i	$-0.260052\pi$
$643$	3.63907e8			1.36885			0.684427	+	0.729081i	$0.260052\pi$
$644$	5.54675e7	+	5.59203e7i	0.207674	+	0.209369i
$645$		0			0
$646$	−8.11665e6	−	3.38137e6i	−0.0301078	−	0.0125428i
$647$			3.22943e8i			1.19238i	0.802845	+	0.596188i	$0.203319\pi$
$647$			3.22943e8i			1.19238i	−0.802845	+	0.596188i	$0.796681\pi$
$648$		0			0
$649$	−1.11620e8			−0.408325
$650$	6.20929e7	−	1.49048e8i	0.226101	−	0.542733i
$651$		0			0
$652$	−6.47288e7	−	6.52572e7i	−0.233536	−	0.235443i
$653$			4.37552e8i			1.57141i	0.618600	+	0.785706i	$0.287700\pi$
$653$			4.37552e8i			1.57141i	−0.618600	+	0.785706i	$0.712300\pi$
$654$		0			0
$655$			2.19365e8i			0.780626i
$656$	−1.52488e6	+	1.87545e8i	−0.00540163	+	0.664346i
$657$		0			0
$658$	1.95807e7	+	8.15725e6i	0.0687306	+	0.0286329i
$659$	4.92366e8			1.72041			0.860205	−	0.509949i	$-0.170336\pi$
$659$	4.92366e8			1.72041			0.860205	+	0.509949i	$0.170336\pi$
$660$		0			0
$661$			2.64679e6i			0.00916465i	0.999990	+	0.00458232i	$0.00145860\pi$
$661$			2.64679e6i			0.00916465i	−0.999990	+	0.00458232i	$0.998541\pi$
$662$	2.30702e8	+	9.61096e7i	0.795201	+	0.331278i
$663$		0			0
$664$	1.42865e8	+	3.50944e8i	0.488002	+	1.19876i
$665$	2.70667e6			0.00920388
$666$		0			0
$667$	−4.42262e8			−1.49040
$668$	−4.10309e8	+	4.06986e8i	−1.37652	+	1.36537i
$669$		0			0
$670$	−5.97988e7	+	1.43541e8i	−0.198824	+	0.477257i
$671$		−	9.52517e7i		−	0.315286i
$672$		0			0
$673$	2.06245e8			0.676611			0.338305	−	0.941036i	$-0.390146\pi$
$673$	2.06245e8			0.676611			0.338305	+	0.941036i	$0.390146\pi$
$674$	2.60947e7	+	1.08710e7i	0.0852262	+	0.0355050i
$675$		0			0
$676$	5.60856e7	+	5.65435e7i	0.181556	+	0.183039i
$677$			5.38304e8i			1.73485i	0.497569	+	0.867424i	$0.334226\pi$
$677$			5.38304e8i			1.73485i	−0.497569	+	0.867424i	$0.665774\pi$
$678$		0			0
$679$		−	9.27478e6i		−	0.0296275i
$680$	−8.38625e7	+	3.41394e7i	−0.266711	+	0.108575i
$681$		0			0
$682$	−4.82222e7	+	1.15753e8i	−0.152018	+	0.364903i
$683$	−1.63210e8			−0.512254			−0.256127	−	0.966643i	$-0.582447\pi$
$683$	−1.63210e8			−0.512254			−0.256127	+	0.966643i	$0.582447\pi$
$684$		0			0
$685$		−	2.04733e8i		−	0.636967i
$686$	6.14602e7	−	1.47529e8i	0.190380	−	0.456989i
$687$		0			0
$688$	−2.99680e6	+	3.68577e8i	−0.00920223	+	1.13178i
$689$	−8.34411e7			−0.255107
$690$		0			0
$691$	3.54541e8			1.07456			0.537282	−	0.843403i	$-0.319451\pi$
$691$	3.54541e8			1.07456			0.537282	+	0.843403i	$0.319451\pi$
$692$	−3.46382e6	+	3.43577e6i	−0.0104529	+	0.0103683i
$693$		0			0
$694$	7.80742e7	+	3.25255e7i	0.233577	+	0.0973073i
$695$			8.35305e7i			0.248823i
$696$		0			0
$697$	1.14961e8			0.339508
$698$	−2.89860e7	+	6.95779e7i	−0.0852357	+	0.204600i
$699$		0			0
$700$	−4.25307e7	+	4.21863e7i	−0.123996	+	0.122992i
$701$		−	7.85410e7i		−	0.228004i	−0.993481	−	0.114002i	$-0.963633\pi$
$701$		−	7.85410e7i		−	0.228004i	0.993481	−	0.114002i	$-0.0363670\pi$
$702$		0			0
$703$		−	2.40846e7i		−	0.0693222i
$704$	−7.63934e7	+	7.45524e7i	−0.218947	+	0.213670i
$705$		0			0
$706$	−3.98775e7	−	1.66129e7i	−0.113322	−	0.0472096i
$707$	1.36522e8			0.386318
$708$		0			0
$709$			1.30092e8i			0.365017i	0.983204	+	0.182509i	$0.0584217\pi$
$709$			1.30092e8i			0.365017i	−0.983204	+	0.182509i	$0.941578\pi$
$710$	−2.50666e8	−	1.04427e8i	−0.700358	−	0.291767i
$711$		0			0
$712$	−2.26734e8	−	5.56966e8i	−0.628169	−	1.54308i
$713$	−5.39710e8			−1.48899
$714$		0			0
$715$	5.42866e7			0.148517
$716$	−4.06674e7	−	4.09994e7i	−0.110792	−	0.111696i
$717$		0			0
$718$	−1.51615e8	+	3.63938e8i	−0.409609	+	0.983227i
$719$		−	3.63590e8i		−	0.978194i	−0.872229	−	0.489097i	$-0.837326\pi$
$719$		−	3.63590e8i		−	0.978194i	0.872229	−	0.489097i	$-0.162674\pi$
$720$		0			0
$721$	−7.28212e7			−0.194291
$722$	3.46009e8	+	1.44146e8i	0.919339	+	0.382994i
$723$		0			0
$724$	−1.50420e8	+	1.49202e8i	−0.396361	+	0.393151i
$725$		−	3.36366e8i		−	0.882670i
$726$		0			0
$727$		−	4.31258e8i		−	1.12237i	−0.827692	−	0.561183i	$-0.810346\pi$
$727$		−	4.31258e8i		−	1.12237i	0.827692	−	0.561183i	$-0.189654\pi$
$728$	3.20721e7	+	7.87844e7i	0.0831254	+	0.204195i
$729$		0			0
$730$	−6.07945e7	+	1.45931e8i	−0.156277	+	0.375128i
$731$	2.25928e8			0.578387
$732$		0			0
$733$		−	7.20041e8i		−	1.82829i	−0.405387	−	0.914145i	$-0.632863\pi$
$733$		−	7.20041e8i		−	1.82829i	0.405387	−	0.914145i	$-0.367137\pi$
$734$	1.55674e8	−	3.73680e8i	0.393666	−	0.944957i
$735$		0			0
$736$	−4.22647e8	−	1.80120e8i	−1.06009	−	0.451781i
$737$	1.12365e8			0.280690
$738$		0			0
$739$	5.37268e8			1.33124			0.665622	−	0.746289i	$-0.268166\pi$
$739$	5.37268e8			1.33124			0.665622	+	0.746289i	$0.268166\pi$
$740$	−1.74658e8	−	1.76084e8i	−0.431015	−	0.434534i
$741$		0			0
$742$	2.85766e7	+	1.19049e7i	0.0699518	+	0.0291417i
$743$			1.64257e8i			0.400458i	0.979749	+	0.200229i	$0.0641685\pi$
$743$			1.64257e8i			0.400458i	−0.979749	+	0.200229i	$0.935831\pi$
$744$		0			0
$745$	1.66233e8			0.402022
$746$	−3.97802e7	+	9.54885e7i	−0.0958187	+	0.230003i
$747$		0			0
$748$	4.60765e7	+	4.64527e7i	0.110097	+	0.110996i
$749$		−	1.11482e8i		−	0.265313i
$750$		0			0
$751$		−	4.71236e7i		−	0.111255i	−0.998452	−	0.0556274i	$-0.982284\pi$
$751$		−	4.71236e7i		−	0.111255i	0.998452	−	0.0556274i	$-0.0177159\pi$
$752$	−1.23724e8	−	1.00597e6i	−0.290939	−	0.00236555i
$753$		0			0
$754$	−4.40899e8	−	1.83677e8i	−1.02855	−	0.428491i
$755$	−1.00013e8			−0.232390
$756$		0			0
$757$		−	4.95044e8i		−	1.14119i	−0.821233	−	0.570593i	$-0.806713\pi$
$757$		−	4.95044e8i		−	1.14119i	0.821233	−	0.570593i	$-0.193287\pi$
$758$	3.85764e8	+	1.60708e8i	0.885756	+	0.369003i
$759$		0			0
$760$	−1.46228e7	+	5.95274e6i	−0.0333110	+	0.0135605i
$761$	1.32572e8			0.300814			0.150407	−	0.988624i	$-0.451942\pi$
$761$	1.32572e8			0.300814			0.150407	+	0.988624i	$0.451942\pi$
$762$		0			0
$763$	−1.34995e8			−0.303910
$764$	4.01101e8	−	3.97853e8i	0.899444	−	0.892161i
$765$		0			0
$766$	1.26684e8	−	3.04092e8i	0.281860	−	0.676578i
$767$		−	5.18835e8i		−	1.14986i
$768$		0			0
$769$	1.63956e8			0.360536			0.180268	−	0.983618i	$-0.442303\pi$
$769$	1.63956e8			0.360536			0.180268	+	0.983618i	$0.442303\pi$
$770$	−1.85919e7	−	7.74533e6i	−0.0407241	−	0.0169655i
$771$		0			0
$772$	2.12211e8	+	2.13944e8i	0.461229	+	0.464994i
$773$			1.60949e8i			0.348457i	0.984705	+	0.174228i	$0.0557431\pi$
$773$			1.60949e8i			0.348457i	−0.984705	+	0.174228i	$0.944257\pi$
$774$		0			0
$775$		−	4.10481e8i		−	0.881837i
$776$	2.03978e7	+	5.01068e7i	0.0436515	+	0.107229i
$777$		0			0
$778$	−2.01128e8	+	4.82788e8i	−0.427104	+	1.02522i
$779$	2.00452e7			0.0424031
$780$		0			0
$781$			1.96222e8i			0.411903i
$782$	−1.08295e8	+	2.59952e8i	−0.226459	+	0.543593i
$783$		0			0
$784$	−3.66141e6	+	4.50317e8i	−0.00759802	+	0.934480i
$785$	3.35425e7			0.0693404
$786$		0			0
$787$	6.37154e8			1.30713			0.653567	−	0.756869i	$-0.273272\pi$
$787$	6.37154e8			1.30713			0.653567	+	0.756869i	$0.273272\pi$
$788$	7.89904e7	−	7.83508e7i	0.161434	−	0.160127i
$789$		0			0
$790$	3.64704e8	+	1.51935e8i	0.739707	+	0.308160i
$791$			1.94043e8i			0.392074i
$792$		0			0
$793$	4.42753e8			0.887856
$794$	1.56413e8	−	3.75455e8i	0.312473	−	0.750061i
$795$		0			0
$796$	1.80091e8	−	1.78632e8i	0.357069	−	0.354177i
$797$		−	5.37684e8i		−	1.06207i	−0.847351	−	0.531034i	$-0.821804\pi$
$797$		−	5.37684e8i		−	1.06207i	0.847351	−	0.531034i	$-0.178196\pi$
$798$		0			0
$799$			7.58399e7i			0.148682i
$800$	1.36992e8	−	3.21448e8i	0.267562	−	0.627828i
$801$		0			0
$802$	−4.19053e8	−	1.74576e8i	−0.812354	−	0.338424i
$803$	1.14236e8			0.220625
$804$		0			0
$805$		−	8.66868e7i		−	0.166175i
$806$	−5.38047e8	−	2.24149e8i	−1.02758	−	0.428086i
$807$		0			0
$808$	−7.37558e8	+	3.00250e8i	−1.39818	+	0.569180i
$809$	9.39871e8			1.77510			0.887550	−	0.460711i	$-0.152406\pi$
$809$	9.39871e8			1.77510			0.887550	+	0.460711i	$0.152406\pi$
$810$		0			0
$811$	−9.00609e8			−1.68839			−0.844197	−	0.536034i	$-0.819922\pi$
$811$	−9.00609e8			−1.68839			−0.844197	+	0.536034i	$0.819922\pi$
$812$	1.24791e8	+	1.25810e8i	0.233086	+	0.234989i
$813$		0			0
$814$	−6.89195e7	+	1.65435e8i	−0.127782	+	0.306728i
$815$			1.01161e8i			0.186870i
$816$		0			0
$817$	3.93942e7			0.0722380
$818$	−5.39378e8	−	2.24703e8i	−0.985448	−	0.410535i
$819$		0			0
$820$	1.46552e8	−	1.45365e8i	0.265796	−	0.263644i
$821$		−	1.57368e7i		−	0.0284373i	−0.999899	−	0.0142186i	$-0.995474\pi$
$821$		−	1.57368e7i		−	0.0284373i	0.999899	−	0.0142186i	$-0.00452608\pi$
$822$		0			0
$823$			8.27251e7i			0.148401i	0.997243	+	0.0742007i	$0.0236405\pi$
$823$			8.27251e7i			0.148401i	−0.997243	+	0.0742007i	$0.976359\pi$
$824$	3.93415e8	−	1.60154e8i	0.703184	−	0.286257i
$825$		0			0
$826$	−7.40246e7	+	1.77689e8i	−0.131352	+	0.315297i
$827$	−4.82714e8			−0.853441			−0.426721	−	0.904384i	$-0.640331\pi$
$827$	−4.82714e8			−0.853441			−0.426721	+	0.904384i	$0.640331\pi$
$828$		0			0
$829$			5.17481e8i			0.908304i	0.890924	+	0.454152i	$0.150058\pi$
$829$			5.17481e8i			0.908304i	−0.890924	+	0.454152i	$0.849942\pi$
$830$	1.60371e8	−	3.84955e8i	0.280474	−	0.673250i
$831$		0			0
$832$	−3.46538e8	−	3.55095e8i	−0.601701	−	0.616560i
$833$	2.76033e8			0.477558
$834$		0			0
$835$	6.36054e8			1.09253
$836$	8.03416e6	+	8.09975e6i	0.0137506	+	0.0138629i
$837$		0			0
$838$	3.01405e8	+	1.25565e8i	0.512176	+	0.213371i
$839$		−	7.46582e8i		−	1.26413i	−0.774915	−	0.632065i	$-0.782208\pi$
$839$		−	7.46582e8i		−	1.26413i	0.774915	−	0.632065i	$-0.217792\pi$
$840$		0			0
$841$	−4.00183e8			−0.672776
$842$	−2.00749e8	+	4.81878e8i	−0.336292	+	0.807237i
$843$		0			0
$844$	4.88606e8	+	4.92595e8i	0.812702	+	0.819336i
$845$		−	8.76528e7i		−	0.145277i
$846$		0			0
$847$		−	1.40948e8i		−	0.231958i
$848$	−1.80567e8	−	1.46814e6i	−0.296108	−	0.00240758i
$849$		0			0
$850$	−1.97709e8	−	8.23650e7i	−0.321936	−	0.134118i
$851$	−7.71358e8			−1.25160
$852$		0			0
$853$			4.24033e8i			0.683207i	0.939844	+	0.341603i	$0.110970\pi$
$853$			4.24033e8i			0.683207i	−0.939844	+	0.341603i	$0.889030\pi$
$854$	−1.51633e8	−	6.31697e7i	−0.243455	−	0.101423i
$855$		0			0
$856$	2.45179e8	+	6.02277e8i	0.390897	+	0.960230i
$857$	−5.00218e8			−0.794724			−0.397362	−	0.917662i	$-0.630074\pi$
$857$	−5.00218e8			−0.794724			−0.397362	+	0.917662i	$0.630074\pi$
$858$		0			0
$859$	−9.80193e8			−1.54644			−0.773219	−	0.634140i	$-0.781354\pi$
$859$	−9.80193e8			−1.54644			−0.773219	+	0.634140i	$0.781354\pi$
$860$	2.88013e8	−	2.85681e8i	0.452811	−	0.449144i
$861$		0			0
$862$	3.46437e8	−	8.31588e8i	0.540882	−	1.29833i
$863$			1.10090e9i			1.71284i	0.516280	+	0.856420i	$0.327316\pi$
$863$			1.10090e9i			1.71284i	−0.516280	+	0.856420i	$0.672684\pi$
$864$		0			0
$865$	5.36956e6			0.00829642
$866$	−4.50040e7	−	1.87485e7i	−0.0692942	−	0.0288678i
$867$		0			0
$868$	1.52288e8	+	1.53531e8i	0.232866	+	0.234767i
$869$		−	2.85492e8i		−	0.435046i
$870$		0			0
$871$			5.22299e8i			0.790432i
$872$	7.29310e8	−	2.96893e8i	1.09992	−	0.447765i
$873$		0			0
$874$	−1.88830e7	+	4.53268e7i	−0.0282837	+	0.0678923i
$875$	1.62537e8			0.242621
$876$		0			0
$877$			1.99266e8i			0.295416i	0.989031	+	0.147708i	$0.0471896\pi$
$877$			1.99266e8i			0.295416i	−0.989031	+	0.147708i	$0.952810\pi$
$878$	−4.55406e8	+	1.09316e9i	−0.672846	+	1.61510i
$879$		0			0
$880$	1.17476e8	+	955172.i	0.172386	+	0.00140163i
$881$	1.07247e9			1.56840			0.784201	−	0.620506i	$-0.213073\pi$
$881$	1.07247e9			1.56840			0.784201	+	0.620506i	$0.213073\pi$
$882$		0			0
$883$	1.06999e9			1.55416			0.777082	−	0.629399i	$-0.216699\pi$
$883$	1.06999e9			1.55416			0.777082	+	0.629399i	$0.216699\pi$
$884$	−2.15923e8	+	2.14175e8i	−0.312567	+	0.310036i
$885$		0			0
$886$	6.89646e8	+	2.87304e8i	0.991574	+	0.413087i
$887$			1.07115e9i			1.53490i	0.641111	+	0.767448i	$0.278474\pi$
$887$			1.07115e9i			1.53490i	−0.641111	+	0.767448i	$0.721526\pi$
$888$		0			0
$889$	−2.56701e8			−0.365361
$890$	−2.54517e8	+	6.10944e8i	−0.361034	+	0.866625i
$891$		0			0
$892$	4.98368e8	−	4.94332e8i	0.702192	−	0.696506i
$893$			1.32239e7i			0.0185697i
$894$		0			0
$895$			6.35566e7i			0.0886526i
$896$	6.80180e7	+	1.71054e8i	0.0945583	+	0.237799i
$897$		0			0
$898$	−8.36256e8	−	3.48382e8i	−1.15481	−	0.481090i
$899$	−1.21425e9			−1.67120
$900$		0			0
$901$			1.10683e8i			0.151324i
$902$	−1.37689e8	−	5.73606e7i	−0.187620	−	0.0781617i
$903$		0			0
$904$	−4.26755e8	−	1.04831e9i	−0.577661	−	1.41901i
$905$	2.33179e8			0.314589
$906$		0			0
$907$	−1.35384e9			−1.81445			−0.907227	−	0.420642i	$-0.861805\pi$
$907$	−1.35384e9			−1.81445			−0.907227	+	0.420642i	$0.861805\pi$
$908$	8.85190e8	+	8.92417e8i	1.18244	+	1.19209i
$909$		0			0
$910$	3.60022e7	−	8.64197e7i	0.0477754	−	0.114680i
$911$			2.30337e8i			0.304655i	0.988330	+	0.152327i	$0.0486768\pi$
$911$			2.30337e8i			0.304655i	−0.988330	+	0.152327i	$0.951323\pi$
$912$		0			0
$913$	−3.01345e8			−0.395960
$914$	−4.34445e8	−	1.80989e8i	−0.568979	−	0.237035i
$915$		0			0
$916$	9.99225e8	−	9.91133e8i	1.30010	−	1.28957i
$917$		−	2.73363e8i		−	0.354513i
$918$		0			0
$919$			5.55066e8i			0.715152i	0.933884	+	0.357576i	$0.116397\pi$
$919$			5.55066e8i			0.715152i	−0.933884	+	0.357576i	$0.883603\pi$
$920$	1.90649e8	+	4.68324e8i	0.244833	+	0.601427i
$921$		0			0
$922$	3.48030e8	−	8.35411e8i	0.444041	−	1.06588i
$923$	−9.12090e8			−1.15993
$924$		0			0
$925$		−	5.86663e8i		−	0.741248i
$926$	−5.71023e8	+	1.37069e9i	−0.719152	+	1.72625i
$927$		0			0
$928$	−9.50875e8	−	4.05236e8i	−1.18982	−	0.507065i
$929$	−1.22721e8			−0.153063			−0.0765315	−	0.997067i	$-0.524385\pi$
$929$	−1.22721e8			−0.153063			−0.0765315	+	0.997067i	$0.524385\pi$
$930$		0			0
$931$	4.81307e7			0.0596449
$932$	−3.71746e8	−	3.74781e8i	−0.459197	−	0.462945i
$933$		0			0
$934$	1.05217e9	+	4.38329e8i	1.29135	+	0.537972i
$935$		−	7.20102e7i		−	0.0880966i
$936$		0			0
$937$	−4.62332e8			−0.561999			−0.280999	−	0.959708i	$-0.590666\pi$
$937$	−4.62332e8			−0.561999			−0.280999	+	0.959708i	$0.590666\pi$
$938$	7.45187e7	−	1.78875e8i	0.0902936	−	0.216741i
$939$		0			0
$940$	9.58977e7	+	9.66806e7i	0.115458	+	0.116401i
$941$			1.38667e9i			1.66419i	0.554633	+	0.832095i	$0.312859\pi$
$941$			1.38667e9i			1.66419i	−0.554633	+	0.832095i	$0.687141\pi$
$942$		0			0
$943$		−	6.41988e8i		−	0.765583i
$944$	9.12889e6	−	1.12276e9i	0.0108518	−	1.33466i
$945$		0			0
$946$	−2.70595e8	−	1.12729e8i	−0.319629	−	0.133156i
$947$	5.59335e8			0.658601			0.329300	−	0.944225i	$-0.393187\pi$
$947$	5.59335e8			0.658601			0.329300	+	0.944225i	$0.393187\pi$
$948$		0			0
$949$			5.30995e8i			0.621286i
$950$	−3.44737e7	−	1.43616e7i	−0.0402084	−	0.0167507i
$951$		0			0
$952$	1.04506e8	−	4.25431e7i	0.121124	−	0.0493081i
$953$	1.02355e9			1.18257			0.591287	−	0.806461i	$-0.298620\pi$
$953$	1.02355e9			1.18257			0.591287	+	0.806461i	$0.298620\pi$
$954$		0			0
$955$	−6.21781e8			−0.713884
$956$	3.31245e8	−	3.28562e8i	0.379119	−	0.376049i
$957$		0			0
$958$	2.00566e8	−	4.81438e8i	0.228118	−	0.547575i
$959$			2.55130e8i			0.289272i
$960$		0			0
$961$	−5.94289e8			−0.669619
$962$	−7.68980e8	−	3.20355e8i	−0.863754	−	0.359837i
$963$		0			0
$964$	2.86438e8	+	2.88777e8i	0.319742	+	0.322353i
$965$		−	3.31652e8i		−	0.369063i
$966$		0			0
$967$			1.09237e9i			1.20807i	0.796958	+	0.604035i	$0.206441\pi$
$967$			1.09237e9i			1.20807i	−0.796958	+	0.604035i	$0.793559\pi$
$968$	3.09984e8	+	7.61469e8i	0.341754	+	0.839511i
$969$		0			0
$970$	2.28974e7	−	5.49628e7i	0.0250882	−	0.0602218i
$971$	−1.26082e9			−1.37719			−0.688596	−	0.725145i	$-0.741773\pi$
$971$	−1.26082e9			−1.37719			−0.688596	+	0.725145i	$0.741773\pi$
$972$		0			0
$973$		−	1.04092e8i		−	0.113000i
$974$	4.09418e8	−	9.82766e8i	0.443088	−	1.06359i
$975$		0			0
$976$	9.58120e8	+	7.79023e6i	1.03055	+	0.00837917i
$977$	3.58176e8			0.384072			0.192036	−	0.981388i	$-0.438491\pi$
$977$	3.58176e8			0.384072			0.192036	+	0.981388i	$0.438491\pi$
$978$		0			0
$979$	4.78250e8			0.509691
$980$	3.51886e8	−	3.49037e8i	0.373873	−	0.370846i
$981$		0			0
$982$	4.47881e8	+	1.86586e8i	0.472964	+	0.197035i
$983$		−	4.83527e8i		−	0.509050i	−0.967066	−	0.254525i	$-0.918081\pi$
$983$		−	4.83527e8i		−	0.509050i	0.967066	−	0.254525i	$-0.0819191\pi$
$984$		0			0
$985$	−1.22450e8			−0.128129
$986$	−2.43644e8	+	5.84844e8i	−0.254171	+	0.610112i
$987$		0			0
$988$	−3.76496e7	+	3.73448e7i	−0.0390382	+	0.0387221i
$989$		−	1.26168e9i		−	1.30425i
$990$		0			0
$991$			8.18452e8i			0.840954i	0.907303	+	0.420477i	$0.138137\pi$
$991$			8.18452e8i			0.840954i	−0.907303	+	0.420477i	$0.861863\pi$
$992$	−1.16039e9	−	4.94525e8i	−1.18869	−	0.506586i
$993$		0			0
$994$	3.12369e8	+	1.30132e8i	0.318060	+	0.132503i
$995$	−2.79174e8			−0.283403
$996$		0			0
$997$		−	2.86580e8i		−	0.289175i	−0.989492	−	0.144587i	$-0.953814\pi$
$997$		−	2.86580e8i		−	0.289175i	0.989492	−	0.144587i	$-0.0461855\pi$
$998$	−1.15494e9	−	4.81145e8i	−1.16190	−	0.484043i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.7.b.d.19.1	✓	12
3.2	odd	2	inner	72.7.b.d.19.12	yes	12
4.3	odd	2		288.7.b.c.271.5		12
8.3	odd	2	inner	72.7.b.d.19.2	yes	12
8.5	even	2		288.7.b.c.271.8		12
12.11	even	2		288.7.b.c.271.7		12
24.5	odd	2		288.7.b.c.271.6		12
24.11	even	2	inner	72.7.b.d.19.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.7.b.d.19.1	✓	12	1.1	even	1	trivial
72.7.b.d.19.2	yes	12	8.3	odd	2	inner
72.7.b.d.19.11	yes	12	24.11	even	2	inner
72.7.b.d.19.12	yes	12	3.2	odd	2	inner
288.7.b.c.271.5		12	4.3	odd	2
288.7.b.c.271.6		12	24.5	odd	2
288.7.b.c.271.7		12	12.11	even	2
288.7.b.c.271.8		12	8.5	even	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:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	72.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.38480 - 3.07648i) q^{2} +(45.0705 + 45.4384i) q^{4} -70.4379i q^{5} +87.7768i q^{7} +(-193.046 - 474.212i) q^{8} +O(q^{10})\)	\(q+(-7.38480 - 3.07648i) q^{2} +(45.0705 + 45.4384i) q^{4} -70.4379i q^{5} +87.7768i q^{7} +(-193.046 - 474.212i) q^{8} +(-216.701 + 520.170i) q^{10} +407.191 q^{11} +1892.73i q^{13} +(270.044 - 648.214i) q^{14} +(-33.3024 + 4095.86i) q^{16} +2510.66 q^{17} +437.774 q^{19} +(3200.59 - 3174.67i) q^{20} +(-3007.03 - 1252.72i) q^{22} -14020.6i q^{23} +10663.5 q^{25} +(5822.94 - 13977.4i) q^{26} +(-3988.44 + 3956.14i) q^{28} -31543.7i q^{29} -38494.1i q^{31} +(12846.8 - 30144.7i) q^{32} +(-18540.7 - 7724.02i) q^{34} +6182.82 q^{35} -55016.0i q^{37} +(-3232.87 - 1346.80i) q^{38} +(-33402.5 + 13597.7i) q^{40} +45788.9 q^{41} +89987.5 q^{43} +(18352.3 + 18502.1i) q^{44} +(-43134.2 + 103539. i) q^{46} +30207.1i q^{47} +109944. q^{49} +(-78747.8 - 32806.1i) q^{50} +(-86002.5 + 85306.1i) q^{52} +44085.2i q^{53} -28681.7i q^{55} +(41624.8 - 16944.9i) q^{56} +(-97043.8 + 232944. i) q^{58} -274121. q^{59} -233924. i q^{61} +(-118426. + 284271. i) q^{62} +(-187611. + 183089. i) q^{64} +133320. q^{65} +275950. q^{67} +(113157. + 114081. i) q^{68} +(-45658.9 - 19021.3i) q^{70} +481892. i q^{71} +280545. q^{73} +(-169256. + 406282. i) q^{74} +(19730.7 + 19891.7i) q^{76} +35742.0i q^{77} -701125. i q^{79} +(288504. + 2345.76i) q^{80} +(-338142. - 140869. i) q^{82} -740057. q^{83} -176846. i q^{85} +(-664539. - 276845. i) q^{86} +(-78606.6 - 193095. i) q^{88} +1.17451e6 q^{89} -166137. q^{91} +(637074. - 631916. i) q^{92} +(92931.7 - 223073. i) q^{94} -30835.9i q^{95} -105663. q^{97} +(-811916. - 338242. i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 156 q^{4}+O(q^{10}) \) 12 * q - 156 * q^4	\( 12 q - 156 q^{4} + 1416 q^{10} - 1464 q^{16} + 3936 q^{19} + 15888 q^{22} - 47796 q^{25} + 11256 q^{28} + 50016 q^{34} + 70896 q^{40} - 340704 q^{43} + 213696 q^{46} - 304644 q^{49} + 548016 q^{52} - 38616 q^{58} + 206544 q^{64} - 962112 q^{67} + 1074480 q^{70} - 1069560 q^{73} + 1064352 q^{76} - 694944 q^{82} - 3072672 q^{88} + 775008 q^{91} + 3752256 q^{94} - 86952 q^{97}+O(q^{100}) \) 12 * q - 156 * q^4 + 1416 * q^10 - 1464 * q^16 + 3936 * q^19 + 15888 * q^22 - 47796 * q^25 + 11256 * q^28 + 50016 * q^34 + 70896 * q^40 - 340704 * q^43 + 213696 * q^46 - 304644 * q^49 + 548016 * q^52 - 38616 * q^58 + 206544 * q^64 - 962112 * q^67 + 1074480 * q^70 - 1069560 * q^73 + 1064352 * q^76 - 694944 * q^82 - 3072672 * q^88 + 775008 * q^91 + 3752256 * q^94 - 86952 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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