LMFDB - Embedded newform 72.8.f.a.35.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [72,8,Mod(35,72)]

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, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("72.35");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 72 = 2^{3} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	72.f (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:	$22.4917218349$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			35.8
Character	$\chi$	$=$	72.35
Dual form			72.8.f.a.35.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-8.14946 + 7.84769i) q^{2} +(4.82744 - 127.909i) q^{4} +294.266 q^{5} +328.633i q^{7} +(964.449 + 1080.27i) q^{8} +O(q^{10})$	\(q+(-8.14946 + 7.84769i) q^{2} +(4.82744 - 127.909i) q^{4} +294.266 q^{5} +328.633i q^{7} +(964.449 + 1080.27i) q^{8} +(-2398.11 + 2309.31i) q^{10} -2680.12i q^{11} +1486.01i q^{13} +(-2579.01 - 2678.18i) q^{14} +(-16337.4 - 1234.95i) q^{16} -35589.5i q^{17} +16097.0 q^{19} +(1420.55 - 37639.2i) q^{20} +(21032.7 + 21841.5i) q^{22} -11593.2 q^{23} +8467.20 q^{25} +(-11661.8 - 12110.2i) q^{26} +(42035.1 + 1586.46i) q^{28} +204036. q^{29} +149568. i q^{31} +(142832. - 118147. i) q^{32} +(279296. + 290036. i) q^{34} +96705.4i q^{35} -61595.2i q^{37} +(-131182. + 126324. i) q^{38} +(283804. + 317887. i) q^{40} -232923. i q^{41} +619370. q^{43} +(-342811. - 12938.1i) q^{44} +(94478.0 - 90979.6i) q^{46} +1.07968e6 q^{47} +715543. q^{49} +(-69003.1 + 66448.0i) q^{50} +(190074. + 7173.63i) q^{52} +713736. q^{53} -788666. i q^{55} +(-355013. + 316950. i) q^{56} +(-1.66278e6 + 1.60121e6i) q^{58} -1.30458e6i q^{59} -1.52680e6i q^{61} +(-1.17376e6 - 1.21890e6i) q^{62} +(-236829. + 2.08374e6i) q^{64} +437282. i q^{65} -4.81113e6 q^{67} +(-4.55222e6 - 171807. i) q^{68} +(-758914. - 788097. i) q^{70} +4.36754e6 q^{71} -1.23348e6 q^{73} +(483380. + 501968. i) q^{74} +(77707.2 - 2.05895e6i) q^{76} +880774. q^{77} -5.34474e6i q^{79} +(-4.80753e6 - 363402. i) q^{80} +(1.82791e6 + 1.89820e6i) q^{82} +6.96837e6i q^{83} -1.04728e7i q^{85} +(-5.04753e6 + 4.86063e6i) q^{86} +(2.89526e6 - 2.58483e6i) q^{88} -166492. i q^{89} -488352. q^{91} +(-55965.3 + 1.48287e6i) q^{92} +(-8.79884e6 + 8.47302e6i) q^{94} +4.73678e6 q^{95} +7.05299e6 q^{97} +(-5.83129e6 + 5.61536e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 52 q^{4}+O(q^{10}) $ 28 * q + 52 * q^4	$ 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) $ 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * 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 $

$2$	−8.14946	+	7.84769i	−0.720317	+	0.693645i
$2$	−8.14946	+	7.84769i	−0.720317	+	0.693645i
$3$		0			0
$3$		0			0
$4$	4.82744	−	127.909i	0.0377144	−	0.999289i
$5$	294.266			1.05280			0.526398	−	0.850238i	$-0.323542\pi$
$5$	294.266			1.05280			0.526398	+	0.850238i	$0.323542\pi$
$6$		0			0
$7$			328.633i			0.362133i	0.983471	+	0.181066i	$0.0579549\pi$
$7$			328.633i			0.362133i	−0.983471	+	0.181066i	$0.942045\pi$
$8$	964.449	+	1080.27i	0.665985	+	0.745965i
$9$		0			0
$10$	−2398.11	+	2309.31i	−0.758348	+	0.730266i
$11$		−	2680.12i		−	0.607126i	−0.952811	−	0.303563i	$-0.901824\pi$
$11$		−	2680.12i		−	0.607126i	0.952811	−	0.303563i	$-0.0981763\pi$
$12$		0			0
$13$			1486.01i			0.187595i	0.995591	+	0.0937973i	$0.0299005\pi$
$13$			1486.01i			0.187595i	−0.995591	+	0.0937973i	$0.970099\pi$
$14$	−2579.01	−	2678.18i	−0.251192	−	0.260851i
$15$		0			0
$16$	−16337.4	−	1234.95i	−0.997155	−	0.0753752i
$17$		−	35589.5i		−	1.75692i	−0.477819	−	0.878458i	$-0.658573\pi$
$17$		−	35589.5i		−	1.75692i	0.477819	−	0.878458i	$-0.341427\pi$
$18$		0			0
$19$	16097.0			0.538402			0.269201	−	0.963084i	$-0.413240\pi$
$19$	16097.0			0.538402			0.269201	+	0.963084i	$0.413240\pi$
$20$	1420.55	−	37639.2i	0.0397056	−	1.05205i
$21$		0			0
$22$	21032.7	+	21841.5i	0.421130	+	0.437324i
$23$	−11593.2			−0.198680			−0.0993401	−	0.995054i	$-0.531673\pi$
$23$	−11593.2			−0.198680			−0.0993401	+	0.995054i	$0.531673\pi$
$24$		0			0
$25$	8467.20			0.108380
$26$	−11661.8	−	12110.2i	−0.130124	−	0.135128i
$27$		0			0
$28$	42035.1	+	1586.46i	0.361875	+	0.0136576i
$29$	204036.			1.55351			0.776753	−	0.629806i	$-0.216865\pi$
$29$	204036.			1.55351			0.776753	+	0.629806i	$0.216865\pi$
$30$		0			0
$31$			149568.i			0.901722i	0.892594	+	0.450861i	$0.148883\pi$
$31$			149568.i			0.901722i	−0.892594	+	0.450861i	$0.851117\pi$
$32$	142832.	−	118147.i	0.770552	−	0.637377i
$33$		0			0
$34$	279296.	+	290036.i	1.21868	+	1.26554i
$35$			96705.4i			0.381252i
$36$		0			0
$37$		−	61595.2i		−	0.199913i	−0.994992	−	0.0999564i	$-0.968130\pi$
$37$		−	61595.2i		−	0.199913i	0.994992	−	0.0999564i	$-0.0318703\pi$
$38$	−131182.	+	126324.i	−0.387820	+	0.373459i
$39$		0			0
$40$	283804.	+	317887.i	0.701146	+	0.785350i
$41$		−	232923.i		−	0.527799i	−0.964550	−	0.263900i	$-0.914991\pi$
$41$		−	232923.i		−	0.527799i	0.964550	−	0.263900i	$-0.0850088\pi$
$42$		0			0
$43$	619370.			1.18798			0.593992	−	0.804471i	$-0.297551\pi$
$43$	619370.			1.18798			0.593992	+	0.804471i	$0.297551\pi$
$44$	−342811.	−	12938.1i	−0.606694	−	0.0228974i
$45$		0			0
$46$	94478.0	−	90979.6i	0.143113	−	0.137813i
$47$	1.07968e6			1.51689			0.758445	−	0.651738i	$-0.225960\pi$
$47$	1.07968e6			1.51689			0.758445	+	0.651738i	$0.225960\pi$
$48$		0			0
$49$	715543.			0.868860
$50$	−69003.1	+	66448.0i	−0.0780681	+	0.0751773i
$51$		0			0
$52$	190074.	+	7173.63i	0.187461	+	0.00707502i
$53$	713736.			0.658524			0.329262	−	0.944239i	$-0.393200\pi$
$53$	713736.			0.658524			0.329262	+	0.944239i	$0.393200\pi$
$54$		0			0
$55$		−	788666.i		−	0.639180i
$56$	−355013.	+	316950.i	−0.270139	+	0.241175i
$57$		0			0
$58$	−1.66278e6	+	1.60121e6i	−1.11902	+	1.07758i
$59$		−	1.30458e6i		−	0.826965i	−0.910512	−	0.413483i	$-0.864312\pi$
$59$		−	1.30458e6i		−	0.826965i	0.910512	−	0.413483i	$-0.135688\pi$
$60$		0			0
$61$		−	1.52680e6i		−	0.861248i	−0.902531	−	0.430624i	$-0.858293\pi$
$61$		−	1.52680e6i		−	0.861248i	0.902531	−	0.430624i	$-0.141707\pi$
$62$	−1.17376e6	−	1.21890e6i	−0.625474	−	0.649526i
$63$		0			0
$64$	−236829.	+	2.08374e6i	−0.112929	+	0.993603i
$65$			437282.i			0.197499i
$66$		0			0
$67$	−4.81113e6			−1.95427			−0.977137	−	0.212610i	$-0.931804\pi$
$67$	−4.81113e6			−1.95427			−0.977137	+	0.212610i	$0.931804\pi$
$68$	−4.55222e6	−	171807.i	−1.75567	−	0.0662611i
$69$		0			0
$70$	−758914.	−	788097.i	−0.264454	−	0.274623i
$71$	4.36754e6			1.44821			0.724107	−	0.689688i	$-0.242252\pi$
$71$	4.36754e6			1.44821			0.724107	+	0.689688i	$0.242252\pi$
$72$		0			0
$73$	−1.23348e6			−0.371108			−0.185554	−	0.982634i	$-0.559408\pi$
$73$	−1.23348e6			−0.371108			−0.185554	+	0.982634i	$0.559408\pi$
$74$	483380.	+	501968.i	0.138668	+	0.144001i
$75$		0			0
$76$	77707.2	−	2.05895e6i	0.0203055	−	0.538019i
$77$	880774.			0.219860
$78$		0			0
$79$		−	5.34474e6i		−	1.21964i	−0.792540	−	0.609820i	$-0.791242\pi$
$79$		−	5.34474e6i		−	1.21964i	0.792540	−	0.609820i	$-0.208758\pi$
$80$	−4.80753e6	−	363402.i	−1.04980	−	0.0793547i
$81$		0			0
$82$	1.82791e6	+	1.89820e6i	0.366105	+	0.380183i
$83$			6.96837e6i			1.33770i	0.743398	+	0.668849i	$0.233213\pi$
$83$			6.96837e6i			1.33770i	−0.743398	+	0.668849i	$0.766787\pi$
$84$		0			0
$85$		−	1.04728e7i		−	1.84968i
$86$	−5.04753e6	+	4.86063e6i	−0.855726	+	0.824039i
$87$		0			0
$88$	2.89526e6	−	2.58483e6i	0.452895	−	0.404337i
$89$		−	166492.i		−	0.0250338i	−0.999922	−	0.0125169i	$-0.996016\pi$
$89$		−	166492.i		−	0.0250338i	0.999922	−	0.0125169i	$-0.00398436\pi$
$90$		0			0
$91$	−488352.			−0.0679341
$92$	−55965.3	+	1.48287e6i	−0.00749310	+	0.198539i
$93$		0			0
$94$	−8.79884e6	+	8.47302e6i	−1.09264	+	1.05218i
$95$	4.73678e6			0.566827
$96$		0			0
$97$	7.05299e6			0.784644			0.392322	−	0.919828i	$-0.371672\pi$
$97$	7.05299e6			0.784644			0.392322	+	0.919828i	$0.371672\pi$
$98$	−5.83129e6	+	5.61536e6i	−0.625855	+	0.602680i
$99$		0			0
$100$	40875.0	−	1.08303e6i	0.00408750	−	0.108303i
$101$	5.23205e6			0.505298			0.252649	−	0.967558i	$-0.418698\pi$
$101$	5.23205e6			0.505298			0.252649	+	0.967558i	$0.418698\pi$
$102$		0			0
$103$			1.46478e7i			1.32081i	0.750909	+	0.660406i	$0.229616\pi$
$103$			1.46478e7i			1.32081i	−0.750909	+	0.660406i	$0.770384\pi$
$104$	−1.60530e6	+	1.43318e6i	−0.139939	+	0.124935i
$105$		0			0
$106$	−5.81656e6	+	5.60118e6i	−0.474347	+	0.456782i
$107$			7.81950e6i			0.617072i	0.951213	+	0.308536i	$0.0998391\pi$
$107$			7.81950e6i			0.617072i	−0.951213	+	0.308536i	$0.900161\pi$
$108$		0			0
$109$		−	1.32338e7i		−	0.978794i	−0.872061	−	0.489397i	$-0.837217\pi$
$109$		−	1.32338e7i		−	0.978794i	0.872061	−	0.489397i	$-0.162783\pi$
$110$	6.18920e6	+	6.42720e6i	0.443364	+	0.460413i
$111$		0			0
$112$	405844.	−	5.36901e6i	0.0272958	−	0.361103i
$113$		−	2.35216e7i		−	1.53353i	−0.641927	−	0.766765i	$-0.721865\pi$
$113$		−	2.35216e7i		−	1.53353i	0.641927	−	0.766765i	$-0.278135\pi$
$114$		0			0
$115$	−3.41147e6			−0.209170
$116$	984970.	−	2.60980e7i	0.0585896	−	1.55240i
$117$		0			0
$118$	1.02379e7	+	1.06316e7i	0.573620	+	0.595677i
$119$	1.16959e7			0.636237
$120$		0			0
$121$	1.23042e7			0.631398
$122$	1.19819e7	+	1.24426e7i	0.597400	+	0.620372i
$123$		0			0
$124$	1.91311e7	+	722031.i	0.901080	+	0.0340079i
$125$	−2.04979e7			−0.938694
$126$		0			0
$127$			2.98987e7i			1.29521i	0.761978	+	0.647603i	$0.224228\pi$
$127$			2.98987e7i			1.29521i	−0.761978	+	0.647603i	$0.775772\pi$
$128$	−1.44225e7	−	1.88399e7i	−0.607863	−	0.794042i
$129$		0			0
$130$	−3.43165e6	−	3.56361e6i	−0.136994	−	0.142262i
$131$		−	9.48915e6i		−	0.368789i	−0.982852	−	0.184394i	$-0.940968\pi$
$131$		−	9.48915e6i		−	0.368789i	0.982852	−	0.184394i	$-0.0590324\pi$
$132$		0			0
$133$			5.28999e6i			0.194973i
$134$	3.92081e7	−	3.77563e7i	1.40770	−	1.35557i
$135$		0			0
$136$	3.84464e7	−	3.43243e7i	1.31060	−	1.17008i
$137$		−	1.01971e7i		−	0.338809i	−0.985547	−	0.169404i	$-0.945816\pi$
$137$		−	1.01971e7i		−	0.338809i	0.985547	−	0.169404i	$-0.0541844\pi$
$138$		0			0
$139$	−4.97951e7			−1.57266			−0.786330	−	0.617807i	$-0.788021\pi$
$139$	−4.97951e7			−1.57266			−0.786330	+	0.617807i	$0.788021\pi$
$140$	1.23695e7	+	466840.i	0.380981	+	0.0143787i
$141$		0			0
$142$	−3.55931e7	+	3.42751e7i	−1.04317	+	1.00455i
$143$	3.98268e6			0.113894
$144$		0			0
$145$	6.00406e7			1.63553
$146$	1.00522e7	−	9.67994e6i	0.267316	−	0.257417i
$147$		0			0
$148$	−7.87857e6	−	297347.i	−0.199771	−	0.00753960i
$149$	−6.45647e7			−1.59898			−0.799490	−	0.600680i	$-0.794897\pi$
$149$	−6.45647e7			−1.59898			−0.799490	+	0.600680i	$0.794897\pi$
$150$		0			0
$151$			6.38149e7i			1.50835i	0.656672	+	0.754177i	$0.271964\pi$
$151$			6.38149e7i			1.50835i	−0.656672	+	0.754177i	$0.728036\pi$
$152$	1.55247e7	+	1.73891e7i	0.358567	+	0.401629i
$153$		0			0
$154$	−7.17784e6	+	6.91205e6i	−0.158369	+	0.152505i
$155$			4.40127e7i			0.949329i
$156$		0			0
$157$			1.74778e7i			0.360444i	0.983626	+	0.180222i	$0.0576816\pi$
$157$			1.74778e7i			0.360444i	−0.983626	+	0.180222i	$0.942318\pi$
$158$	4.19438e7	+	4.35567e7i	0.845996	+	0.878528i
$159$		0			0
$160$	4.20307e7	−	3.47665e7i	0.811234	−	0.671028i
$161$		−	3.80989e6i		−	0.0719486i
$162$		0			0
$163$	52066.5			0.000941677			0.000470839	−	1.00000i	$-0.499850\pi$
$163$	52066.5			0.000941677			0.000470839		1.00000i	$0.499850\pi$
$164$	−2.97929e7	−	1.12442e6i	−0.527424	−	0.0199056i
$165$		0			0
$166$	−5.46856e7	−	5.67885e7i	−0.927887	−	0.963567i
$167$	1.62528e7			0.270036			0.135018	−	0.990843i	$-0.456891\pi$
$167$	1.62528e7			0.270036			0.135018	+	0.990843i	$0.456891\pi$
$168$		0			0
$169$	6.05403e7			0.964808
$170$	8.21871e7	+	8.53475e7i	1.28302	+	1.33235i
$171$		0			0
$172$	2.98998e6	−	7.92230e7i	0.0448041	−	1.18714i
$173$	−1.18628e8			−1.74191			−0.870954	−	0.491364i	$-0.836498\pi$
$173$	−1.18628e8			−1.74191			−0.870954	+	0.491364i	$0.836498\pi$
$174$		0			0
$175$			2.78260e6i			0.0392480i
$176$	−3.30980e6	+	4.37861e7i	−0.0457622	+	0.605399i
$177$		0			0
$178$	1.30657e6	+	1.35682e6i	0.0173646	+	0.0180323i
$179$			4.22356e7i			0.550419i	0.961384	+	0.275210i	$0.0887472\pi$
$179$			4.22356e7i			0.550419i	−0.961384	+	0.275210i	$0.911253\pi$
$180$		0			0
$181$		−	1.16870e8i		−	1.46497i	−0.680781	−	0.732487i	$-0.738360\pi$
$181$		−	1.16870e8i		−	1.46497i	0.680781	−	0.732487i	$-0.261640\pi$
$182$	3.97981e6	−	3.83244e6i	0.0489342	−	0.0471222i
$183$		0			0
$184$	−1.11810e7	−	1.25238e7i	−0.132318	−	0.148208i
$185$		−	1.81253e7i		−	0.210468i
$186$		0			0
$187$	−9.53841e7			−1.06667
$188$	5.21211e6	−	1.38101e8i	0.0572086	−	1.51581i
$189$		0			0
$190$	−3.86022e7	+	3.71728e7i	−0.408296	+	0.393177i
$191$	−1.46598e8			−1.52234			−0.761171	−	0.648551i	$-0.775375\pi$
$191$	−1.46598e8			−1.52234			−0.761171	+	0.648551i	$0.775375\pi$
$192$		0			0
$193$	−9.92822e7			−0.994079			−0.497039	−	0.867728i	$-0.665580\pi$
$193$	−9.92822e7			−0.994079			−0.497039	+	0.867728i	$0.665580\pi$
$194$	−5.74781e7	+	5.53497e7i	−0.565192	+	0.544264i
$195$		0			0
$196$	3.45425e6	−	9.15244e7i	0.0327685	−	0.868242i
$197$	2.90394e7			0.270618			0.135309	−	0.990803i	$-0.456797\pi$
$197$	2.90394e7			0.270618			0.135309	+	0.990803i	$0.456797\pi$
$198$		0			0
$199$			1.90948e8i			1.71763i	0.512288	+	0.858814i	$0.328798\pi$
$199$			1.90948e8i			1.71763i	−0.512288	+	0.858814i	$0.671202\pi$
$200$	8.16618e6	+	9.14689e6i	0.0721796	+	0.0808479i
$201$		0			0
$202$	−4.26384e7	+	4.10595e7i	−0.363975	+	0.350497i
$203$			6.70528e7i			0.562576i
$204$		0			0
$205$		−	6.85412e7i		−	0.555665i
$206$	−1.14951e8	−	1.19371e8i	−0.916174	−	0.951403i
$207$		0			0
$208$	1.83514e6	−	2.42775e7i	0.0141400	−	0.187061i
$209$		−	4.31417e7i		−	0.326878i
$210$		0			0
$211$	−1.21971e8			−0.893854			−0.446927	−	0.894570i	$-0.647482\pi$
$211$	−1.21971e8			−0.893854			−0.446927	+	0.894570i	$0.647482\pi$
$212$	3.44552e6	−	9.12932e7i	0.0248359	−	0.658056i
$213$		0			0
$214$	−6.13651e7	−	6.37247e7i	−0.428029	−	0.444488i
$215$	1.82259e8			1.25071
$216$		0			0
$217$	−4.91530e7			−0.326543
$218$	1.03855e8	+	1.07848e8i	0.678935	+	0.705042i
$219$		0			0
$220$	−1.00877e8	−	3.80724e6i	−0.638726	−	0.0241063i
$221$	5.28864e7			0.329588
$222$		0			0
$223$		−	2.53524e8i		−	1.53092i	−0.643483	−	0.765460i	$-0.722511\pi$
$223$		−	2.53524e8i		−	1.53092i	0.643483	−	0.765460i	$-0.277489\pi$
$224$	3.88269e7	+	4.69394e7i	0.230815	+	0.279042i
$225$		0			0
$226$	1.84590e8	+	1.91688e8i	1.06373	+	1.10463i
$227$			7.31259e7i			0.414936i	0.978242	+	0.207468i	$0.0665222\pi$
$227$			7.31259e7i			0.414936i	−0.978242	+	0.207468i	$0.933478\pi$
$228$		0			0
$229$			4.41665e7i			0.243035i	0.992589	+	0.121518i	$0.0387761\pi$
$229$			4.41665e7i			0.243035i	−0.992589	+	0.121518i	$0.961224\pi$
$230$	2.78016e7	−	2.67721e7i	0.150669	−	0.145089i
$231$		0			0
$232$	1.96782e8	+	2.20414e8i	1.03461	+	1.15886i
$233$		−	1.86368e8i		−	0.965219i	−0.875836	−	0.482609i	$-0.839689\pi$
$233$		−	1.86368e8i		−	0.965219i	0.875836	−	0.482609i	$-0.160311\pi$
$234$		0			0
$235$	3.17714e8			1.59698
$236$	−1.66867e8	−	6.29777e6i	−0.826377	−	0.0311885i
$237$		0			0
$238$	−9.53152e7	+	9.17858e7i	−0.458293	+	0.441323i
$239$	6.04324e7			0.286337			0.143168	−	0.989698i	$-0.454271\pi$
$239$	6.04324e7			0.286337			0.143168	+	0.989698i	$0.454271\pi$
$240$		0			0
$241$	2.53357e8			1.16593			0.582965	−	0.812497i	$-0.301892\pi$
$241$	2.53357e8			1.16593			0.582965	+	0.812497i	$0.301892\pi$
$242$	−1.00272e8	+	9.65592e7i	−0.454807	+	0.437966i
$243$		0			0
$244$	−1.95292e8	−	7.37055e6i	−0.860636	−	0.0324815i
$245$	2.10560e8			0.914732
$246$		0			0
$247$			2.39203e7i			0.101001i
$248$	−1.61574e8	+	1.44251e8i	−0.672653	+	0.600533i
$249$		0			0
$250$	1.67047e8	−	1.60861e8i	0.676158	−	0.651120i
$251$		−	4.32231e8i		−	1.72527i	−0.505825	−	0.862636i	$-0.668812\pi$
$251$		−	4.32231e8i		−	1.72527i	0.505825	−	0.862636i	$-0.331188\pi$
$252$		0			0
$253$			3.10710e7i			0.120624i
$254$	−2.34635e8	−	2.43658e8i	−0.898412	−	0.932959i
$255$		0			0
$256$	2.65385e8	+	4.03516e7i	0.988637	+	0.150321i
$257$			3.53732e8i			1.29990i	0.759978	+	0.649948i	$0.225210\pi$
$257$			3.53732e8i			1.29990i	−0.759978	+	0.649948i	$0.774790\pi$
$258$		0			0
$259$	2.02422e7			0.0723950
$260$	5.59322e7	+	2.11095e6i	0.197358	+	0.00744855i
$261$		0			0
$262$	7.44679e7	+	7.73315e7i	0.255808	+	0.265645i
$263$	3.41375e8			1.15714			0.578571	−	0.815632i	$-0.303611\pi$
$263$	3.41375e8			1.15714			0.578571	+	0.815632i	$0.303611\pi$
$264$		0			0
$265$	2.10028e8			0.693292
$266$	−4.15142e7	−	4.31106e7i	−0.135242	−	0.140442i
$267$		0			0
$268$	−2.32255e7	+	6.15387e8i	−0.0737043	+	1.95288i
$269$	4.19128e8			1.31284			0.656422	−	0.754394i	$-0.272069\pi$
$269$	4.19128e8			1.31284			0.656422	+	0.754394i	$0.272069\pi$
$270$		0			0
$271$		−	3.31720e8i		−	1.01246i	−0.862397	−	0.506232i	$-0.831038\pi$
$271$		−	3.31720e8i		−	1.01246i	0.862397	−	0.506232i	$-0.168962\pi$
$272$	−4.39512e7	+	5.81440e8i	−0.132428	+	1.75192i
$273$		0			0
$274$	8.00237e7	+	8.31009e7i	0.235013	+	0.244050i
$275$		−	2.26931e7i		−	0.0658005i
$276$		0			0
$277$			1.71149e8i			0.483834i	0.970297	+	0.241917i	$0.0777761\pi$
$277$			1.71149e8i			0.483834i	−0.970297	+	0.241917i	$0.922224\pi$
$278$	4.05803e8	−	3.90776e8i	1.13281	−	1.09087i
$279$		0			0
$280$	−1.04468e8	+	9.32674e7i	−0.284401	+	0.253908i
$281$			3.60548e8i			0.969374i	0.874688	+	0.484687i	$0.161066\pi$
$281$			3.60548e8i			0.969374i	−0.874688	+	0.484687i	$0.838934\pi$
$282$		0			0
$283$	7.88163e7			0.206711			0.103356	−	0.994644i	$-0.467042\pi$
$283$	7.88163e7			0.206711			0.103356	+	0.994644i	$0.467042\pi$
$284$	2.10841e7	−	5.58647e8i	0.0546185	−	1.44718i
$285$		0			0
$286$	−3.24567e7	+	3.12548e7i	−0.0820395	+	0.0790017i
$287$	7.65462e7			0.191134
$288$		0			0
$289$	−8.56276e8			−2.08675
$290$	−4.89299e8	+	4.71180e8i	−1.17810	+	1.13447i
$291$		0			0
$292$	−5.95454e6	+	1.57773e8i	−0.0139961	+	0.370844i
$293$	1.16444e7			0.0270447			0.0135223	−	0.999909i	$-0.495696\pi$
$293$	1.16444e7			0.0270447			0.0135223	+	0.999909i	$0.495696\pi$
$294$		0			0
$295$		−	3.83892e8i		−	0.870626i
$296$	6.65396e7	−	5.94054e7i	0.149128	−	0.133139i
$297$		0			0
$298$	5.26167e8	−	5.06684e8i	1.15177	−	1.10912i
$299$		−	1.72276e7i		−	0.0372713i
$300$		0			0
$301$			2.03545e8i			0.430208i
$302$	−5.00800e8	−	5.20057e8i	−1.04626	−	1.08649i
$303$		0			0
$304$	−2.62982e8	−	1.98789e7i	−0.536870	−	0.0405821i
$305$		−	4.49285e8i		−	0.906719i
$306$		0			0
$307$	3.71307e8			0.732401			0.366201	−	0.930536i	$-0.380658\pi$
$307$	3.71307e8			0.732401			0.366201	+	0.930536i	$0.380658\pi$
$308$	4.25189e6	−	1.12659e8i	0.00829191	−	0.219704i
$309$		0			0
$310$	−3.45398e8	−	3.58680e8i	−0.658497	−	0.683819i
$311$	−5.55339e8			−1.04688			−0.523440	−	0.852063i	$-0.675352\pi$
$311$	−5.55339e8			−1.04688			−0.523440	+	0.852063i	$0.675352\pi$
$312$		0			0
$313$	−6.90438e8			−1.27268			−0.636340	−	0.771408i	$-0.719553\pi$
$313$	−6.90438e8			−1.27268			−0.636340	+	0.771408i	$0.719553\pi$
$314$	−1.37160e8	−	1.42435e8i	−0.250020	−	0.259634i
$315$		0			0
$316$	−6.83640e8	−	2.58014e7i	−1.21877	−	0.0459980i
$317$	6.87240e7			0.121172			0.0605858	−	0.998163i	$-0.480703\pi$
$317$	6.87240e7			0.121172			0.0605858	+	0.998163i	$0.480703\pi$
$318$		0			0
$319$		−	5.46839e8i		−	0.943174i
$320$	−6.96905e7	+	6.13172e8i	−0.118891	+	1.04606i
$321$		0			0
$322$	2.98989e7	+	3.10486e7i	0.0499068	+	0.0518258i
$323$		−	5.72883e8i		−	0.945927i
$324$		0			0
$325$			1.25824e7i			0.0203315i
$326$	−424314.	+	408602.i	−0.000678306	+	0.000653189i
$327$		0			0
$328$	2.51620e8	−	2.24642e8i	0.393720	−	0.351506i
$329$			3.54820e8i			0.549315i
$330$		0			0
$331$	−1.13679e8			−0.172299			−0.0861496	−	0.996282i	$-0.527456\pi$
$331$	−1.13679e8			−0.172299			−0.0861496	+	0.996282i	$0.527456\pi$
$332$	8.91317e8	+	3.36394e7i	1.33675	+	0.0504505i
$333$		0			0
$334$	−1.32452e8	+	1.27547e8i	−0.194511	+	0.187309i
$335$	−1.41575e9			−2.05745
$336$		0			0
$337$	1.16559e8			0.165898			0.0829492	−	0.996554i	$-0.473566\pi$
$337$	1.16559e8			0.165898			0.0829492	+	0.996554i	$0.473566\pi$
$338$	−4.93371e8	+	4.75102e8i	−0.694968	+	0.669234i
$339$		0			0
$340$	−1.33956e9	−	5.05567e7i	−1.84836	−	0.0697594i
$341$	4.00859e8			0.547459
$342$		0			0
$343$			5.05795e8i			0.676776i
$344$	5.97351e8	+	6.69089e8i	0.791179	+	0.886195i
$345$		0			0
$346$	9.66753e8	−	9.30955e8i	1.25473	−	1.20827i
$347$			7.74986e8i			0.995729i	0.867255	+	0.497864i	$0.165882\pi$
$347$			7.74986e8i			0.995729i	−0.867255	+	0.497864i	$0.834118\pi$
$348$		0			0
$349$			2.75593e8i			0.347039i	0.984830	+	0.173520i	$0.0555140\pi$
$349$			2.75593e8i			0.347039i	−0.984830	+	0.173520i	$0.944486\pi$
$350$	−2.18370e7	−	2.26767e7i	−0.0272242	−	0.0282710i
$351$		0			0
$352$	−3.16647e8	−	3.82807e8i	−0.386969	−	0.467822i
$353$			7.91797e8i			0.958081i	0.877793	+	0.479041i	$0.159015\pi$
$353$			7.91797e8i			0.958081i	−0.877793	+	0.479041i	$0.840985\pi$
$354$		0			0
$355$	1.28522e9			1.52467
$356$	−2.12958e7	−	803729.i	−0.0250160	−	0.000944136i
$357$		0			0
$358$	−3.31452e8	−	3.44198e8i	−0.381795	−	0.396477i
$359$	−4.47158e8			−0.510071			−0.255035	−	0.966932i	$-0.582087\pi$
$359$	−4.47158e8			−0.510071			−0.255035	+	0.966932i	$0.582087\pi$
$360$		0			0
$361$	−6.34759e8			−0.710124
$362$	9.17163e8	+	9.52431e8i	1.01617	+	1.05525i
$363$		0			0
$364$	−2.35749e6	+	6.24646e7i	−0.00256210	+	0.0678858i
$365$	−3.62970e8			−0.390702
$366$		0			0
$367$		−	6.59899e8i		−	0.696861i	−0.937335	−	0.348431i	$-0.886715\pi$
$367$		−	6.59899e8i		−	0.696861i	0.937335	−	0.348431i	$-0.113285\pi$
$368$	1.89402e8	+	1.43169e7i	0.198115	+	0.0149755i
$369$		0			0
$370$	1.42242e8	+	1.47712e8i	0.145990	+	0.151603i
$371$			2.34557e8i			0.238473i
$372$		0			0
$373$			1.64257e9i			1.63886i	0.573177	+	0.819432i	$0.305711\pi$
$373$			1.64257e9i			1.63886i	−0.573177	+	0.819432i	$0.694289\pi$
$374$	7.77329e8	−	7.48545e8i	0.768341	−	0.739890i
$375$		0			0
$376$	1.04130e9	+	1.16635e9i	1.01022	+	1.13155i
$377$			3.03199e8i			0.291429i
$378$		0			0
$379$	−4.91034e8			−0.463313			−0.231656	−	0.972798i	$-0.574414\pi$
$379$	−4.91034e8			−0.463313			−0.231656	+	0.972798i	$0.574414\pi$
$380$	2.28666e7	−	6.05877e8i	0.0213776	−	0.566424i
$381$		0			0
$382$	1.19470e9	−	1.15046e9i	1.09657	−	1.05596i
$383$	−2.15258e8			−0.195778			−0.0978889	−	0.995197i	$-0.531209\pi$
$383$	−2.15258e8			−0.195778			−0.0978889	+	0.995197i	$0.531209\pi$
$384$		0			0
$385$	2.59182e8			0.231468
$386$	8.09097e8	−	7.79136e8i	0.716052	−	0.689537i
$387$		0			0
$388$	3.40479e7	−	9.02141e8i	0.0295924	−	0.784085i
$389$	−1.68469e9			−1.45109			−0.725547	−	0.688172i	$-0.758413\pi$
$389$	−1.68469e9			−1.45109			−0.725547	+	0.688172i	$0.758413\pi$
$390$		0			0
$391$			4.12595e8i			0.349064i
$392$	6.90105e8	+	7.72982e8i	0.578647	+	0.648139i
$393$		0			0
$394$	−2.36656e8	+	2.27892e8i	−0.194931	+	0.187712i
$395$		−	1.57277e9i		−	1.28403i
$396$		0			0
$397$			8.46867e8i			0.679280i	0.940556	+	0.339640i	$0.110305\pi$
$397$			8.46867e8i			0.679280i	−0.940556	+	0.339640i	$0.889695\pi$
$398$	−1.49850e9	−	1.55612e9i	−1.19142	−	1.23724i
$399$		0			0
$400$	−1.38332e8	−	1.04565e7i	−0.108072	−	0.00816917i
$401$		−	3.67705e8i		−	0.284770i	−0.989811	−	0.142385i	$-0.954523\pi$
$401$		−	3.67705e8i		−	0.284770i	0.989811	−	0.142385i	$-0.0454771\pi$
$402$		0			0
$403$	−2.22260e8			−0.169158
$404$	2.52574e7	−	6.69226e8i	0.0190570	−	0.504938i
$405$		0			0
$406$	−5.26210e8	−	5.46444e8i	−0.390227	−	0.405233i
$407$	−1.65082e8			−0.121372
$408$		0			0
$409$	2.91851e8			0.210925			0.105463	−	0.994423i	$-0.466368\pi$
$409$	2.91851e8			0.210925			0.105463	+	0.994423i	$0.466368\pi$
$410$	5.37890e8	+	5.58574e8i	0.385434	+	0.400255i
$411$		0			0
$412$	1.87358e9	+	7.07112e7i	1.31987	+	0.0498136i
$413$	4.28726e8			0.299471
$414$		0			0
$415$			2.05055e9i			1.40832i
$416$	1.75567e8	+	2.12250e8i	0.119568	+	0.144551i
$417$		0			0
$418$	3.38563e8	+	3.51582e8i	0.226737	+	0.235456i
$419$		−	8.75874e8i		−	0.581691i	−0.956770	−	0.290845i	$-0.906063\pi$
$419$		−	8.75874e8i		−	0.581691i	0.956770	−	0.290845i	$-0.0939366\pi$
$420$		0			0
$421$		−	3.93556e8i		−	0.257051i	−0.991706	−	0.128526i	$-0.958976\pi$
$421$		−	3.93556e8i		−	0.257051i	0.991706	−	0.128526i	$-0.0410244\pi$
$422$	9.93995e8	−	9.57188e8i	0.643859	−	0.620017i
$423$		0			0
$424$	6.88362e8	+	7.71030e8i	0.438567	+	0.491236i
$425$		−	3.01344e8i		−	0.190415i
$426$		0			0
$427$	5.01758e8			0.311886
$428$	1.00018e9	+	3.77482e7i	0.616633	+	0.0232725i
$429$		0			0
$430$	−1.48532e9	+	1.43032e9i	−0.900905	+	0.867545i
$431$	2.03454e9			1.22404			0.612020	−	0.790842i	$-0.290357\pi$
$431$	2.03454e9			1.22404			0.612020	+	0.790842i	$0.290357\pi$
$432$		0			0
$433$	1.34016e9			0.793318			0.396659	−	0.917966i	$-0.370169\pi$
$433$	1.34016e9			0.793318			0.396659	+	0.917966i	$0.370169\pi$
$434$	4.00570e8	−	3.85737e8i	0.235215	−	0.226505i
$435$		0			0
$436$	−1.69272e9	−	6.38854e7i	−0.978098	−	0.0369146i
$437$	−1.86615e8			−0.106970
$438$		0			0
$439$		−	1.78225e9i		−	1.00541i	−0.864459	−	0.502704i	$-0.832339\pi$
$439$		−	1.78225e9i		−	1.00541i	0.864459	−	0.502704i	$-0.167661\pi$
$440$	8.51974e8	−	7.60628e8i	0.476806	−	0.425684i
$441$		0			0
$442$	−4.30996e8	+	4.15036e8i	−0.237408	+	0.228617i
$443$		−	7.08329e8i		−	0.387099i	−0.981090	−	0.193550i	$-0.938000\pi$
$443$		−	7.08329e8i		−	0.387099i	0.981090	−	0.193550i	$-0.0620000\pi$
$444$		0			0
$445$		−	4.89927e7i		−	0.0263555i
$446$	1.98958e9	+	2.06609e9i	1.06191	+	1.10275i
$447$		0			0
$448$	−6.84785e8	−	7.78297e7i	−0.359816	−	0.0408952i
$449$		−	1.08114e9i		−	0.563665i	−0.959464	−	0.281832i	$-0.909058\pi$
$449$		−	1.08114e9i		−	0.563665i	0.959464	−	0.281832i	$-0.0909422\pi$
$450$		0			0
$451$	−6.24260e8			−0.320441
$452$	−3.00862e9	−	1.13549e8i	−1.53244	−	0.0578362i
$453$		0			0
$454$	−5.73870e8	−	5.95937e8i	−0.287818	−	0.298885i
$455$	−1.43705e8			−0.0715208
$456$		0			0
$457$	7.45370e8			0.365313			0.182657	−	0.983177i	$-0.441530\pi$
$457$	7.45370e8			0.365313			0.182657	+	0.983177i	$0.441530\pi$
$458$	−3.46605e8	−	3.59933e8i	−0.168580	−	0.175062i
$459$		0			0
$460$	−1.64687e7	+	4.36357e8i	−0.00788871	+	0.209021i
$461$	1.30861e8			0.0622097			0.0311048	−	0.999516i	$-0.490097\pi$
$461$	1.30861e8			0.0622097			0.0311048	+	0.999516i	$0.490097\pi$
$462$		0			0
$463$			3.29457e9i			1.54264i	0.636447	+	0.771321i	$0.280403\pi$
$463$			3.29457e9i			1.54264i	−0.636447	+	0.771321i	$0.719597\pi$
$464$	−3.33341e9	−	2.51973e8i	−1.54909	−	0.117096i
$465$		0			0
$466$	1.46256e9	+	1.51880e9i	0.669519	+	0.695264i
$467$			3.84572e9i			1.74731i	0.486549	+	0.873653i	$0.338255\pi$
$467$			3.84572e9i			1.74731i	−0.486549	+	0.873653i	$0.661745\pi$
$468$		0			0
$469$		−	1.58110e9i		−	0.707707i
$470$	−2.58919e9	+	2.49332e9i	−1.15033	+	1.10773i
$471$		0			0
$472$	1.40930e9	−	1.25820e9i	0.616887	−	0.550746i
$473$		−	1.65998e9i		−	0.721257i
$474$		0			0
$475$	1.36296e8			0.0583521
$476$	5.64613e7	−	1.49601e9i	0.0239953	−	0.635785i
$477$		0			0
$478$	−4.92492e8	+	4.74255e8i	−0.206253	+	0.198616i
$479$	1.27149e9			0.528616			0.264308	−	0.964438i	$-0.414857\pi$
$479$	1.27149e9			0.528616			0.264308	+	0.964438i	$0.414857\pi$
$480$		0			0
$481$	9.15311e7			0.0375026
$482$	−2.06472e9	+	1.98827e9i	−0.839840	+	0.808741i
$483$		0			0
$484$	5.93976e7	−	1.57381e9i	0.0238128	−	0.630948i
$485$	2.07545e9			0.826070
$486$		0			0
$487$		−	2.64160e9i		−	1.03637i	−0.855268	−	0.518185i	$-0.826608\pi$
$487$		−	2.64160e9i		−	1.03637i	0.855268	−	0.518185i	$-0.173392\pi$
$488$	1.64936e9	−	1.47252e9i	0.642461	−	0.573578i
$489$		0			0
$490$	−1.71595e9	+	1.65241e9i	−0.658898	+	0.634499i
$491$			1.97319e9i			0.752285i	0.926562	+	0.376143i	$0.122750\pi$
$491$			1.97319e9i			0.752285i	−0.926562	+	0.376143i	$0.877250\pi$
$492$		0			0
$493$		−	7.26153e9i		−	2.72938i
$494$	−1.87719e8	−	1.94937e8i	−0.0700589	−	0.0727529i
$495$		0			0
$496$	1.84708e8	−	2.44355e9i	0.0679674	−	0.899157i
$497$			1.43532e9i			0.524446i
$498$		0			0
$499$	−4.39385e9			−1.58304			−0.791522	−	0.611140i	$-0.790711\pi$
$499$	−4.39385e9			−1.58304			−0.791522	+	0.611140i	$0.790711\pi$
$500$	−9.89524e7	+	2.62186e9i	−0.0354023	+	0.938026i
$501$		0			0
$502$	3.39201e9	+	3.52245e9i	1.19673	+	1.24274i
$503$	−1.42589e8			−0.0499574			−0.0249787	−	0.999688i	$-0.507952\pi$
$503$	−1.42589e8			−0.0499574			−0.0249787	+	0.999688i	$0.507952\pi$
$504$		0			0
$505$	1.53961e9			0.531975
$506$	−2.43836e8	−	2.53212e8i	−0.0836701	−	0.0868875i
$507$		0			0
$508$	3.82430e9	+	1.44334e8i	1.29428	+	0.0488479i
$509$	−3.09362e8			−0.103981			−0.0519906	−	0.998648i	$-0.516557\pi$
$509$	−3.09362e8			−0.103981			−0.0519906	+	0.998648i	$0.516557\pi$
$510$		0			0
$511$		−	4.05361e8i		−	0.134391i
$512$	−2.47941e9	+	1.75382e9i	−0.816402	+	0.577484i
$513$		0			0
$514$	−2.77598e9	−	2.88273e9i	−0.901666	−	0.936338i
$515$			4.31033e9i			1.39055i
$516$		0			0
$517$		−	2.89368e9i		−	0.920943i
$518$	−1.64963e8	+	1.58855e8i	−0.0521474	+	0.0502164i
$519$		0			0
$520$	−4.72384e8	+	4.21736e8i	−0.147327	+	0.131531i
$521$		−	2.29488e8i		−	0.0710932i	−0.999368	−	0.0355466i	$-0.988683\pi$
$521$		−	2.29488e8i		−	0.0710932i	0.999368	−	0.0355466i	$-0.0113172\pi$
$522$		0			0
$523$	−3.09139e9			−0.944926			−0.472463	−	0.881351i	$-0.656635\pi$
$523$	−3.09139e9			−0.944926			−0.472463	+	0.881351i	$0.656635\pi$
$524$	−1.21375e9	−	4.58083e7i	−0.368526	−	0.0139087i
$525$		0			0
$526$	−2.78202e9	+	2.67901e9i	−0.833510	+	0.802645i
$527$	5.32305e9			1.58425
$528$		0			0
$529$	−3.27042e9			−0.960526
$530$	−1.71161e9	+	1.64823e9i	−0.499390	+	0.480898i
$531$		0			0
$532$	6.76637e8	+	2.55372e7i	0.194834	+	0.00735329i
$533$	3.46126e8			0.0990123
$534$		0			0
$535$			2.30101e9i			0.649651i
$536$	−4.64009e9	−	5.19734e9i	−1.30152	−	1.45782i
$537$		0			0
$538$	−3.41566e9	+	3.28918e9i	−0.945665	+	0.910648i
$539$		−	1.91774e9i		−	0.527508i
$540$		0			0
$541$			3.25383e9i			0.883497i	0.897139	+	0.441748i	$0.145642\pi$
$541$			3.25383e9i			0.883497i	−0.897139	+	0.441748i	$0.854358\pi$
$542$	2.60324e9	+	2.70334e9i	0.702290	+	0.729296i
$543$		0			0
$544$	−4.20478e9	−	5.08334e9i	−1.11982	−	1.35380i
$545$		−	3.89425e9i		−	1.03047i
$546$		0			0
$547$	1.49458e9			0.390449			0.195224	−	0.980759i	$-0.437457\pi$
$547$	1.49458e9			0.390449			0.195224	+	0.980759i	$0.437457\pi$
$548$	−1.30430e9	−	4.92259e7i	−0.338568	−	0.0127780i
$549$		0			0
$550$	1.78088e8	+	1.84936e8i	0.0456421	+	0.0473972i
$551$	3.28435e9			0.836410
$552$		0			0
$553$	1.75646e9			0.441672
$554$	−1.34313e9	−	1.39478e9i	−0.335609	−	0.348514i
$555$		0			0
$556$	−2.40383e8	+	6.36923e9i	−0.0593119	+	1.57154i
$557$	−4.01514e9			−0.984483			−0.492241	−	0.870459i	$-0.663822\pi$
$557$	−4.01514e9			−0.984483			−0.492241	+	0.870459i	$0.663822\pi$
$558$		0			0
$559$			9.20391e8i			0.222859i
$560$	1.19426e8	−	1.57991e9i	0.0287369	−	0.380168i
$561$		0			0
$562$	−2.82947e9	−	2.93827e9i	−0.672401	−	0.698257i
$563$		−	2.87925e9i		−	0.679987i	−0.940428	−	0.339994i	$-0.889575\pi$
$563$		−	2.87925e9i		−	0.679987i	0.940428	−	0.339994i	$-0.110425\pi$
$564$		0			0
$565$		−	6.92160e9i		−	1.61450i
$566$	−6.42311e8	+	6.18526e8i	−0.148898	+	0.143384i
$567$		0			0
$568$	4.21227e9	+	4.71814e9i	0.964488	+	1.08032i
$569$		−	1.91695e9i		−	0.436233i	−0.975923	−	0.218116i	$-0.930009\pi$
$569$		−	1.91695e9i		−	0.436233i	0.975923	−	0.218116i	$-0.0699912\pi$
$570$		0			0
$571$	−3.99834e9			−0.898779			−0.449389	−	0.893336i	$-0.648358\pi$
$571$	−3.99834e9			−0.898779			−0.449389	+	0.893336i	$0.648358\pi$
$572$	1.92262e7	−	5.09420e8i	0.00429543	−	0.113813i
$573$		0			0
$574$	−6.23810e8	+	6.00711e8i	−0.137677	+	0.132579i
$575$	−9.81616e7			−0.0215330
$576$		0			0
$577$	8.39033e9			1.81829			0.909146	−	0.416477i	$-0.136735\pi$
$577$	8.39033e9			1.81829			0.909146	+	0.416477i	$0.136735\pi$
$578$	6.97819e9	−	6.71979e9i	1.50313	−	1.44747i
$579$		0			0
$580$	2.89843e8	−	7.67973e9i	0.0616829	−	1.63436i
$581$	−2.29004e9			−0.484424
$582$		0			0
$583$		−	1.91289e9i		−	0.399808i
$584$	−1.18962e9	−	1.33249e9i	−0.247153	−	0.276834i
$585$		0			0
$586$	−9.48959e7	+	9.13820e7i	−0.0194808	+	0.0187594i
$587$		−	4.84669e9i		−	0.989036i	−0.869167	−	0.494518i	$-0.835345\pi$
$587$		−	4.84669e9i		−	0.989036i	0.869167	−	0.494518i	$-0.164655\pi$
$588$		0			0
$589$			2.40759e9i			0.485489i
$590$	3.01266e9	+	3.12851e9i	0.603905	+	0.627127i
$591$		0			0
$592$	−7.60668e7	+	1.00630e9i	−0.0150685	+	0.199344i
$593$			3.59521e9i			0.708000i	0.935245	+	0.354000i	$0.115179\pi$
$593$			3.59521e9i			0.708000i	−0.935245	+	0.354000i	$0.884821\pi$
$594$		0			0
$595$	3.44170e9			0.669828
$596$	−3.11682e8	+	8.25840e9i	−0.0603046	+	1.59784i
$597$		0			0
$598$	1.35197e8	+	1.40395e8i	0.0258530	+	0.0268472i
$599$	−3.60041e8			−0.0684476			−0.0342238	−	0.999414i	$-0.510896\pi$
$599$	−3.60041e8			−0.0684476			−0.0342238	+	0.999414i	$0.510896\pi$
$600$		0			0
$601$	5.21055e9			0.979091			0.489545	−	0.871978i	$-0.337163\pi$
$601$	5.21055e9			0.979091			0.489545	+	0.871978i	$0.337163\pi$
$602$	−1.59736e9	−	1.65879e9i	−0.298412	−	0.309887i
$603$		0			0
$604$	8.16250e9	+	3.08063e8i	1.50728	+	0.0568867i
$605$	3.62069e9			0.664733
$606$		0			0
$607$			7.88101e9i			1.43028i	0.698980	+	0.715141i	$0.253637\pi$
$607$			7.88101e9i			1.43028i	−0.698980	+	0.715141i	$0.746363\pi$
$608$	2.29917e9	−	1.90180e9i	0.414866	−	0.343165i
$609$		0			0
$610$	3.52585e9	+	3.66143e9i	0.628941	+	0.653126i
$611$			1.60442e9i			0.284560i
$612$		0			0
$613$			8.47669e9i			1.48633i	0.669109	+	0.743164i	$0.266676\pi$
$613$			8.47669e9i			1.48633i	−0.669109	+	0.743164i	$0.733324\pi$
$614$	−3.02595e9	+	2.91390e9i	−0.527561	+	0.508026i
$615$		0			0
$616$	8.49462e8	+	9.51477e8i	0.146424	+	0.164008i
$617$			1.30414e9i			0.223525i	0.993735	+	0.111762i	$0.0356495\pi$
$617$			1.30414e9i			0.223525i	−0.993735	+	0.111762i	$0.964350\pi$
$618$		0			0
$619$	−3.50616e9			−0.594175			−0.297087	−	0.954850i	$-0.596015\pi$
$619$	−3.50616e9			−0.594175			−0.297087	+	0.954850i	$0.596015\pi$
$620$	5.62962e9	+	2.12469e8i	0.948654	+	0.0358034i
$621$		0			0
$622$	4.52571e9	−	4.35813e9i	0.754086	−	0.726163i
$623$	5.47146e7			0.00906557
$624$		0			0
$625$	−6.69332e9			−1.09663
$626$	5.62670e9	−	5.41835e9i	0.916734	−	0.882788i
$627$		0			0
$628$	2.23556e9	+	8.43730e7i	0.360187	+	0.0135939i
$629$	−2.19214e9			−0.351230
$630$		0			0
$631$			2.18649e9i			0.346453i	0.984882	+	0.173226i	$0.0554193\pi$
$631$			2.18649e9i			0.346453i	−0.984882	+	0.173226i	$0.944581\pi$
$632$	5.77378e9	−	5.15473e9i	0.909809	−	0.812261i
$633$		0			0
$634$	−5.60064e8	+	5.39325e8i	−0.0872821	+	0.0840501i
$635$			8.79814e9i			1.36359i
$636$		0			0
$637$			1.06330e9i			0.162993i
$638$	4.29142e9	+	4.45644e9i	0.654228	+	0.679385i
$639$		0			0
$640$	−4.24404e9	−	5.54393e9i	−0.639956	−	0.835965i
$641$		−	3.64284e9i		−	0.546308i	−0.961970	−	0.273154i	$-0.911933\pi$
$641$		−	3.64284e9i		−	0.546308i	0.961970	−	0.273154i	$-0.0880668\pi$
$642$		0			0
$643$	1.02912e10			1.52661			0.763304	−	0.646040i	$-0.223576\pi$
$643$	1.02912e10			1.52661			0.763304	+	0.646040i	$0.223576\pi$
$644$	−4.87320e8	−	1.83921e7i	−0.0718974	−	0.00271350i
$645$		0			0
$646$	4.49581e9	+	4.66869e9i	0.656137	+	0.681367i
$647$	−8.42123e9			−1.22239			−0.611196	−	0.791479i	$-0.709311\pi$
$647$	−8.42123e9			−1.22239			−0.611196	+	0.791479i	$0.709311\pi$
$648$		0			0
$649$	−3.49641e9			−0.502072
$650$	−9.87424e7	−	1.02539e8i	−0.0141029	−	0.0146452i
$651$		0			0
$652$	251348.	−	6.65977e6i	3.55148e−5	−	0.000941007i
$653$	−7.12964e9			−1.00201			−0.501004	−	0.865445i	$-0.667036\pi$
$653$	−7.12964e9			−1.00201			−0.501004	+	0.865445i	$0.667036\pi$
$654$		0			0
$655$		−	2.79233e9i		−	0.388260i
$656$	−2.87647e8	+	3.80535e9i	−0.0397830	+	0.526298i
$657$		0			0
$658$	−2.78451e9	−	2.89159e9i	−0.381030	−	0.395682i
$659$			2.28313e9i			0.310765i	0.987854	+	0.155383i	$0.0496610\pi$
$659$			2.28313e9i			0.310765i	−0.987854	+	0.155383i	$0.950339\pi$
$660$		0			0
$661$			1.04167e10i			1.40290i	0.712719	+	0.701450i	$0.247464\pi$
$661$			1.04167e10i			1.40290i	−0.712719	+	0.701450i	$0.752536\pi$
$662$	9.26424e8	−	8.92119e8i	0.124110	−	0.119514i
$663$		0			0
$664$	−7.52774e9	+	6.72064e9i	−0.997876	+	0.890886i
$665$			1.55666e9i			0.205267i
$666$		0			0
$667$	−2.36542e9			−0.308651
$668$	7.84596e7	−	2.07888e9i	0.0101842	−	0.269843i
$669$		0			0
$670$	1.15376e10	−	1.11104e10i	1.48202	−	1.42714i
$671$	−4.09201e9			−0.522887
$672$		0			0
$673$	5.55647e8			0.0702661			0.0351331	−	0.999383i	$-0.488814\pi$
$673$	5.55647e8			0.0702661			0.0351331	+	0.999383i	$0.488814\pi$
$674$	−9.49895e8	+	9.14721e8i	−0.119500	+	0.115075i
$675$		0			0
$676$	2.92255e8	−	7.74364e9i	0.0363872	−	0.964122i
$677$	3.52244e9			0.436298			0.218149	−	0.975916i	$-0.429998\pi$
$677$	3.52244e9			0.436298			0.218149	+	0.975916i	$0.429998\pi$
$678$		0			0
$679$			2.31785e9i			0.284145i
$680$	1.13135e10	−	1.01005e10i	1.37979	−	1.23186i
$681$		0			0
$682$	−3.26679e9	+	3.14582e9i	−0.394344	+	0.379742i
$683$		−	9.97428e9i		−	1.19787i	−0.800798	−	0.598934i	$-0.795591\pi$
$683$		−	9.97428e9i		−	1.19787i	0.800798	−	0.598934i	$-0.204409\pi$
$684$		0			0
$685$		−	3.00065e9i		−	0.356697i
$686$	−3.96932e9	−	4.12195e9i	−0.469442	−	0.487493i
$687$		0			0
$688$	−1.01189e10	−	7.64889e8i	−1.18460	−	0.0895445i
$689$			1.06062e9i			0.123536i
$690$		0			0
$691$	7.63799e9			0.880655			0.440328	−	0.897837i	$-0.354862\pi$
$691$	7.63799e9			0.880655			0.440328	+	0.897837i	$0.354862\pi$
$692$	−5.72669e8	+	1.51736e10i	−0.0656951	+	1.74067i
$693$		0			0
$694$	−6.08186e9	−	6.31572e9i	−0.690682	−	0.717241i
$695$	−1.46530e10			−1.65569
$696$		0			0
$697$	−8.28962e9			−0.927299
$698$	−2.16277e9	−	2.24593e9i	−0.240722	−	0.249979i
$699$		0			0
$700$	3.55920e8	+	1.34329e7i	0.0392201	+	0.00148022i
$701$	3.19869e9			0.350719			0.175359	−	0.984504i	$-0.443891\pi$
$701$	3.19869e9			0.350719			0.175359	+	0.984504i	$0.443891\pi$
$702$		0			0
$703$		−	9.91496e8i		−	0.107633i
$704$	5.58465e9	+	6.34728e8i	0.603243	+	0.0685620i
$705$		0			0
$706$	−6.21378e9	−	6.45272e9i	−0.664568	−	0.690123i
$707$			1.71942e9i			0.182985i
$708$		0			0
$709$		−	1.38637e10i		−	1.46088i	−0.682975	−	0.730442i	$-0.739314\pi$
$709$		−	1.38637e10i		−	1.46088i	0.682975	−	0.730442i	$-0.260686\pi$
$710$	−1.04738e10	+	1.00860e10i	−1.09825	+	1.05758i
$711$		0			0
$712$	1.79856e8	−	1.60573e8i	0.0186744	−	0.0166721i
$713$		−	1.73396e9i		−	0.179154i
$714$		0			0
$715$	1.17197e9			0.119907
$716$	5.40232e9	+	2.03890e8i	0.550028	+	0.0207587i
$717$		0			0
$718$	3.64410e9	−	3.50916e9i	0.367413	−	0.353808i
$719$	−1.52490e10			−1.53000			−0.765000	−	0.644030i	$-0.777261\pi$
$719$	−1.52490e10			−1.53000			−0.765000	+	0.644030i	$0.777261\pi$
$720$		0			0
$721$	−4.81374e9			−0.478309
$722$	5.17295e9	−	4.98140e9i	0.511514	−	0.492573i
$723$		0			0
$724$	−1.49488e10	−	5.64186e8i	−1.46393	−	0.0552506i
$725$	1.72761e9			0.168369
$726$		0			0
$727$		−	1.06784e10i		−	1.03071i	−0.856977	−	0.515355i	$-0.827660\pi$
$727$		−	1.06784e10i		−	1.03071i	0.856977	−	0.515355i	$-0.172340\pi$
$728$	−4.70991e8	−	5.27554e8i	−0.0452431	−	0.0506765i
$729$		0			0
$730$	2.95801e9	−	2.84847e9i	0.281429	−	0.271008i
$731$		−	2.20431e10i		−	2.08719i
$732$		0			0
$733$			2.09160e10i			1.96162i	0.194974	+	0.980808i	$0.437538\pi$
$733$			2.09160e10i			1.96162i	−0.194974	+	0.980808i	$0.562462\pi$
$734$	5.17868e9	+	5.37782e9i	0.483374	+	0.501961i
$735$		0			0
$736$	−1.65588e9	+	1.36969e9i	−0.153093	+	0.126634i
$737$			1.28944e10i			1.18649i
$738$		0			0
$739$	−6.69063e9			−0.609834			−0.304917	−	0.952379i	$-0.598629\pi$
$739$	−6.69063e9			−0.609834			−0.304917	+	0.952379i	$0.598629\pi$
$740$	−2.31839e9	−	8.74991e7i	−0.210318	−	0.00793766i
$741$		0			0
$742$	−1.84073e9	−	1.91151e9i	−0.165416	−	0.171777i
$743$	1.80945e10			1.61840			0.809198	−	0.587536i	$-0.199902\pi$
$743$	1.80945e10			1.61840			0.809198	+	0.587536i	$0.199902\pi$
$744$		0			0
$745$	−1.89992e10			−1.68340
$746$	−1.28904e10	−	1.33861e10i	−1.13679	−	1.18050i
$747$		0			0
$748$	−4.60461e8	+	1.22005e10i	−0.0402288	+	1.06591i
$749$	−2.56975e9			−0.223462
$750$		0			0
$751$		−	1.39247e10i		−	1.19963i	−0.800138	−	0.599815i	$-0.795241\pi$
$751$		−	1.39247e10i		−	1.19963i	0.800138	−	0.599815i	$-0.204759\pi$
$752$	−1.76392e10	−	1.33335e9i	−1.51257	−	0.114336i
$753$		0			0
$754$	−2.37941e9	−	2.47091e9i	−0.202148	−	0.209921i
$755$			1.87785e10i			1.58799i
$756$		0			0
$757$			1.41833e10i			1.18834i	0.804338	+	0.594171i	$0.202520\pi$
$757$			1.41833e10i			1.18834i	−0.804338	+	0.594171i	$0.797480\pi$
$758$	4.00166e9	−	3.85348e9i	0.333732	−	0.321374i
$759$		0			0
$760$	4.56838e9	+	5.11702e9i	0.377498	+	0.422834i
$761$			7.86744e9i			0.647123i	0.946207	+	0.323562i	$0.104880\pi$
$761$			7.86744e9i			0.647123i	−0.946207	+	0.323562i	$0.895120\pi$
$762$		0			0
$763$	4.34906e9			0.354454
$764$	−7.07695e8	+	1.87512e10i	−0.0574142	+	1.52126i
$765$		0			0
$766$	1.75424e9	−	1.68928e9i	0.141022	−	0.135800i
$767$	1.93861e9			0.155134
$768$		0			0
$769$	1.63912e9			0.129978			0.0649890	−	0.997886i	$-0.479299\pi$
$769$	1.63912e9			0.129978			0.0649890	+	0.997886i	$0.479299\pi$
$770$	−2.11219e9	+	2.03398e9i	−0.166731	+	0.160557i
$771$		0			0
$772$	−4.79279e8	+	1.26991e10i	−0.0374911	+	0.993372i
$773$	−2.71642e9			−0.211529			−0.105764	−	0.994391i	$-0.533729\pi$
$773$	−2.71642e9			−0.211529			−0.105764	+	0.994391i	$0.533729\pi$
$774$		0			0
$775$			1.26642e9i			0.0977288i
$776$	6.80225e9	+	7.61916e9i	0.522561	+	0.585317i
$777$		0			0
$778$	1.37293e10	−	1.32209e10i	1.04525	−	1.00654i
$779$		−	3.74935e9i		−	0.284168i
$780$		0			0
$781$		−	1.17055e10i		−	0.879249i
$782$	−3.23792e9	−	3.36243e9i	−0.242127	−	0.251437i
$783$		0			0
$784$	−1.16901e10	−	8.83658e8i	−0.866388	−	0.0654904i
$785$			5.14311e9i			0.379474i
$786$		0			0
$787$	8.31174e9			0.607827			0.303914	−	0.952700i	$-0.401707\pi$
$787$	8.31174e9			0.607827			0.303914	+	0.952700i	$0.401707\pi$
$788$	1.40186e8	−	3.71440e9i	0.0102062	−	0.270425i
$789$		0			0
$790$	1.23426e10	+	1.28172e10i	0.890662	+	0.924911i
$791$	7.72998e9			0.555342
$792$		0			0
$793$	2.26884e9			0.161565
$794$	−6.64595e9	−	6.90151e9i	−0.471179	−	0.489297i
$795$		0			0
$796$	2.44239e10	+	9.21790e8i	1.71641	+	0.0647793i
$797$	1.72803e10			1.20906			0.604530	−	0.796582i	$-0.293361\pi$
$797$	1.72803e10			1.20906			0.604530	+	0.796582i	$0.293361\pi$
$798$		0			0
$799$		−	3.84254e10i		−	2.66505i
$800$	1.20939e9	−	1.00037e9i	0.0835126	−	0.0690791i
$801$		0			0
$802$	2.88564e9	+	2.99660e9i	0.197529	+	0.205125i
$803$			3.30586e9i			0.225310i
$804$		0			0
$805$		−	1.12112e9i		−	0.0757472i
$806$	1.81130e9	−	1.74422e9i	0.121847	−	0.117336i
$807$		0			0
$808$	5.04605e9	+	5.65205e9i	0.336520	+	0.376934i
$809$			2.23227e10i			1.48227i	0.671356	+	0.741135i	$0.265712\pi$
$809$			2.23227e10i			1.48227i	−0.671356	+	0.741135i	$0.734288\pi$
$810$		0			0
$811$	9.69423e9			0.638176			0.319088	−	0.947725i	$-0.396623\pi$
$811$	9.69423e9			0.638176			0.319088	+	0.947725i	$0.396623\pi$
$812$	8.57665e9	+	3.23694e8i	0.562175	+	0.0212172i
$813$		0			0
$814$	1.34533e9	−	1.29551e9i	0.0874266	−	0.0841893i
$815$	1.53214e7			0.000991394
$816$		0			0
$817$	9.96998e9			0.639613
$818$	−2.37843e9	+	2.29035e9i	−0.151933	+	0.146307i
$819$		0			0
$820$	−8.76703e9	−	3.30879e8i	−0.555270	−	0.0209566i
$821$	2.46999e10			1.55773			0.778867	−	0.627189i	$-0.215795\pi$
$821$	2.46999e10			1.55773			0.778867	+	0.627189i	$0.215795\pi$
$822$		0			0
$823$			2.00407e10i			1.25318i	0.779348	+	0.626591i	$0.215550\pi$
$823$			2.00407e10i			1.25318i	−0.779348	+	0.626591i	$0.784450\pi$
$824$	−1.58236e10	+	1.41270e10i	−0.985279	+	0.879640i
$825$		0			0
$826$	−3.49389e9	+	3.36451e9i	−0.215714	+	0.207727i
$827$		−	3.11012e10i		−	1.91209i	−0.293219	−	0.956045i	$-0.594727\pi$
$827$		−	3.11012e10i		−	1.91209i	0.293219	−	0.956045i	$-0.405273\pi$
$828$		0			0
$829$			4.01877e9i			0.244992i	0.992469	+	0.122496i	$0.0390899\pi$
$829$			4.01877e9i			0.244992i	−0.992469	+	0.122496i	$0.960910\pi$
$830$	−1.60921e10	−	1.67109e10i	−0.976876	−	1.01444i
$831$		0			0
$832$	−3.09645e9	−	3.51930e8i	−0.186394	−	0.0211848i
$833$		−	2.54659e10i		−	1.52651i
$834$		0			0
$835$	4.78264e9			0.284292
$836$	−5.51821e9	−	2.08264e8i	−0.326645	−	0.0123280i
$837$		0			0
$838$	6.87359e9	+	7.13790e9i	0.403487	+	0.419002i
$839$	1.51383e10			0.884932			0.442466	−	0.896785i	$-0.354104\pi$
$839$	1.51383e10			0.884932			0.442466	+	0.896785i	$0.354104\pi$
$840$		0			0
$841$	2.43806e10			1.41338
$842$	3.08851e9	+	3.20727e9i	0.178302	+	0.185158i
$843$		0			0
$844$	−5.88806e8	+	1.56011e10i	−0.0337112	+	0.893218i
$845$	1.78149e10			1.01575
$846$		0			0
$847$			4.04355e9i			0.228650i
$848$	−1.16606e10	−	8.81426e8i	−0.656651	−	0.0496364i
$849$		0			0
$850$	2.36485e9	+	2.45579e9i	0.132080	+	0.137159i
$851$			7.14083e8i			0.0397187i
$852$		0			0
$853$			1.53579e10i			0.847247i	0.905838	+	0.423624i	$0.139242\pi$
$853$			1.53579e10i			0.847247i	−0.905838	+	0.423624i	$0.860758\pi$
$854$	−4.08905e9	+	3.93764e9i	−0.224657	+	0.216338i
$855$		0			0
$856$	−8.44720e9	+	7.54151e9i	−0.460314	+	0.410961i
$857$		−	1.44454e10i		−	0.783968i	−0.919972	−	0.391984i	$-0.871789\pi$
$857$		−	1.44454e10i		−	0.783968i	0.919972	−	0.391984i	$-0.128211\pi$
$858$		0			0
$859$	−7.57453e9			−0.407736			−0.203868	−	0.978998i	$-0.565351\pi$
$859$	−7.57453e9			−0.407736			−0.203868	+	0.978998i	$0.565351\pi$
$860$	8.79847e8	−	2.33126e10i	0.0471696	−	1.24982i
$861$		0			0
$862$	−1.65804e10	+	1.59664e10i	−0.881697	+	0.849049i
$863$	−1.16885e10			−0.619045			−0.309522	−	0.950892i	$-0.600169\pi$
$863$	−1.16885e10			−0.619045			−0.309522	+	0.950892i	$0.600169\pi$
$864$		0			0
$865$	−3.49081e10			−1.83387
$866$	−1.09215e10	+	1.05171e10i	−0.571441	+	0.550281i
$867$		0			0
$868$	−2.37283e8	+	6.28710e9i	−0.0123154	+	0.326311i
$869$	−1.43245e10			−0.740475
$870$		0			0
$871$		−	7.14939e9i		−	0.366611i
$872$	1.42961e10	−	1.27633e10i	0.730146	−	0.651862i
$873$		0			0
$874$	1.52081e9	−	1.46449e9i	0.0770521	−	0.0741990i
$875$		−	6.73628e9i		−	0.339932i
$876$		0			0
$877$		−	1.94208e10i		−	0.972231i	−0.873894	−	0.486116i	$-0.838413\pi$
$877$		−	1.94208e10i		−	0.972231i	0.873894	−	0.486116i	$-0.161587\pi$
$878$	1.39865e10	+	1.45244e10i	0.697395	+	0.724212i
$879$		0			0
$880$	−9.73960e8	+	1.28847e10i	−0.0481783	+	0.637362i
$881$		−	1.00292e10i		−	0.494139i	−0.968998	−	0.247069i	$-0.920532\pi$
$881$		−	1.00292e10i		−	0.494139i	0.968998	−	0.247069i	$-0.0794675\pi$
$882$		0			0
$883$	3.58464e10			1.75220			0.876098	−	0.482133i	$-0.160138\pi$
$883$	3.58464e10			1.75220			0.876098	+	0.482133i	$0.160138\pi$
$884$	2.55306e8	−	6.76465e9i	0.0124302	−	0.329353i
$885$		0			0
$886$	5.55875e9	+	5.77250e9i	0.268509	+	0.278834i
$887$	1.96838e10			0.947059			0.473529	−	0.880778i	$-0.342980\pi$
$887$	1.96838e10			0.947059			0.473529	+	0.880778i	$0.342980\pi$
$888$		0			0
$889$	−9.82568e9			−0.469036
$890$	3.84480e8	+	3.99264e8i	0.0182814	+	0.0189843i
$891$		0			0
$892$	−3.24280e10	−	1.22387e9i	−1.52983	−	0.0577378i
$893$	1.73796e10			0.816696
$894$		0			0
$895$			1.24285e10i			0.579479i
$896$	6.19141e9	−	4.73971e9i	0.287549	−	0.220127i
$897$		0			0
$898$	8.48447e9	+	8.81073e9i	0.390983	+	0.406017i
$899$			3.05172e10i			1.40083i
$900$		0			0
$901$		−	2.54015e10i		−	1.15697i
$902$	5.08739e9	−	4.89900e9i	0.230819	−	0.222272i
$903$		0			0
$904$	2.54098e10	−	2.26854e10i	1.14396	−	1.02131i
$905$		−	3.43910e10i		−	1.54232i
$906$		0			0
$907$	−4.16289e10			−1.85255			−0.926273	−	0.376853i	$-0.877006\pi$
$907$	−4.16289e10			−1.85255			−0.926273	+	0.376853i	$0.877006\pi$
$908$	9.35346e9	+	3.53011e8i	0.414640	+	0.0156491i
$909$		0			0
$910$	1.17112e9	−	1.12775e9i	0.0515177	−	0.0496100i
$911$	7.63565e9			0.334604			0.167302	−	0.985906i	$-0.446494\pi$
$911$	7.63565e9			0.334604			0.167302	+	0.985906i	$0.446494\pi$
$912$		0			0
$913$	1.86760e10			0.812151
$914$	−6.07436e9	+	5.84943e9i	−0.263141	+	0.253397i
$915$		0			0
$916$	5.64929e9	+	2.13211e8i	0.242862	+	0.00916592i
$917$	3.11845e9			0.133551
$918$		0			0
$919$			1.38559e10i			0.588886i	0.955669	+	0.294443i	$0.0951341\pi$
$919$			1.38559e10i			0.588886i	−0.955669	+	0.294443i	$0.904866\pi$
$920$	−3.29019e9	−	3.68532e9i	−0.139304	−	0.156033i
$921$		0			0
$922$	−1.06645e9	+	1.02696e9i	−0.0448107	+	0.0431514i
$923$			6.49021e9i			0.271677i
$924$		0			0
$925$		−	5.21539e8i		−	0.0216666i
$926$	−2.58547e10	−	2.68489e10i	−1.07004	−	1.11119i
$927$		0			0
$928$	2.91429e10	−	2.41061e10i	1.19706	−	0.990169i
$929$			4.81860e10i			1.97181i	0.167294	+	0.985907i	$0.446497\pi$
$929$			4.81860e10i			1.97181i	−0.167294	+	0.985907i	$0.553503\pi$
$930$		0			0
$931$	1.15181e10			0.467796
$932$	−2.38381e10	−	8.99681e8i	−0.964532	−	0.0364026i
$933$		0			0
$934$	−3.01801e10	−	3.13406e10i	−1.21201	−	1.25862i
$935$	−2.80682e10			−1.12299
$936$		0			0
$937$	−5.58203e8			−0.0221668			−0.0110834	−	0.999939i	$-0.503528\pi$
$937$	−5.58203e8			−0.0221668			−0.0110834	+	0.999939i	$0.503528\pi$
$938$	1.24080e10	+	1.28851e10i	0.490897	+	0.509774i
$939$		0			0
$940$	1.53374e9	−	4.06384e10i	0.0602290	−	1.59584i
$941$	−4.60090e10			−1.80003			−0.900014	−	0.435862i	$-0.856444\pi$
$941$	−4.60090e10			−1.80003			−0.900014	+	0.435862i	$0.856444\pi$
$942$		0			0
$943$			2.70031e9i			0.104863i
$944$	−1.61108e9	+	2.13134e10i	−0.0623326	+	0.824612i
$945$		0			0
$946$	1.30270e10	+	1.35280e10i	0.500296	+	0.519534i
$947$			1.84738e10i			0.706856i	0.935462	+	0.353428i	$0.114984\pi$
$947$			1.84738e10i			0.706856i	−0.935462	+	0.353428i	$0.885016\pi$
$948$		0			0
$949$		−	1.83296e9i		−	0.0696179i
$950$	−1.11074e9	+	1.06961e9i	−0.0420320	+	0.0404756i
$951$		0			0
$952$	1.12801e10	+	1.26348e10i	0.423724	+	0.474611i
$953$			1.94712e10i			0.728732i	0.931256	+	0.364366i	$0.118714\pi$
$953$			1.94712e10i			0.728732i	−0.931256	+	0.364366i	$0.881286\pi$
$954$		0			0
$955$	−4.31388e10			−1.60272
$956$	2.91734e8	−	7.72984e9i	0.0107990	−	0.286133i
$957$		0			0
$958$	−1.03620e10	+	9.97829e9i	−0.380771	+	0.366671i
$959$	3.35110e9			0.122694
$960$		0			0
$961$	5.14205e9			0.186898
$962$	−7.45929e8	+	7.18308e8i	−0.0270137	+	0.0260134i
$963$		0			0
$964$	1.22307e9	−	3.24066e10i	0.0439724	−	1.16510i
$965$	−2.92153e10			−1.04656
$966$		0			0
$967$			1.72332e9i			0.0612876i	0.999530	+	0.0306438i	$0.00975574\pi$
$967$			1.72332e9i			0.0612876i	−0.999530	+	0.0306438i	$0.990244\pi$
$968$	1.18667e10	+	1.32918e10i	0.420501	+	0.471001i
$969$		0			0
$970$	−1.69138e10	+	1.62875e10i	−0.595033	+	0.572999i
$971$		−	1.13358e10i		−	0.397359i	−0.980064	−	0.198680i	$-0.936335\pi$
$971$		−	1.13358e10i		−	0.397359i	0.980064	−	0.198680i	$-0.0636653\pi$
$972$		0			0
$973$		−	1.63643e10i		−	0.569512i
$974$	2.07304e10	+	2.15276e10i	0.718873	+	0.746516i
$975$		0			0
$976$	−1.88552e9	+	2.49440e10i	−0.0649167	+	0.858798i
$977$		−	2.58110e10i		−	0.885470i	−0.896652	−	0.442735i	$-0.854008\pi$
$977$		−	2.58110e10i		−	0.885470i	0.896652	−	0.442735i	$-0.145992\pi$
$978$		0			0
$979$	−4.46217e8			−0.0151987
$980$	1.01647e9	−	2.69325e10i	0.0344986	−	0.914082i
$981$		0			0
$982$	−1.54850e10	−	1.60804e10i	−0.521819	−	0.541884i
$983$	−1.58165e10			−0.531097			−0.265549	−	0.964098i	$-0.585553\pi$
$983$	−1.58165e10			−0.531097			−0.265549	+	0.964098i	$0.585553\pi$
$984$		0			0
$985$	8.54530e9			0.284905
$986$	5.69863e10	+	5.91776e10i	1.89322	+	1.96602i
$987$		0			0
$988$	3.05961e9	+	1.15474e8i	0.100929	+	0.00380920i
$989$	−7.18046e9			−0.236029
$990$		0			0
$991$			2.16501e10i			0.706646i	0.935501	+	0.353323i	$0.114948\pi$
$991$			2.16501e10i			0.706646i	−0.935501	+	0.353323i	$0.885052\pi$
$992$	1.76710e10	+	2.13632e10i	0.574737	+	0.694823i
$993$		0			0
$994$	−1.12639e10	−	1.16971e10i	−0.363779	−	0.377767i
$995$			5.61894e10i			1.80831i
$996$		0			0
$997$		−	4.30355e9i		−	0.137529i	−0.997633	−	0.0687644i	$-0.978094\pi$
$997$		−	4.30355e9i		−	0.137529i	0.997633	−	0.0687644i	$-0.0219057\pi$
$998$	3.58075e10	−	3.44816e10i	1.14029	−	1.09807i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.f.a.35.8	yes	28
3.2	odd	2	inner	72.8.f.a.35.21	yes	28
4.3	odd	2		288.8.f.a.143.23		28
8.3	odd	2	inner	72.8.f.a.35.22	yes	28
8.5	even	2		288.8.f.a.143.6		28
12.11	even	2		288.8.f.a.143.5		28
24.5	odd	2		288.8.f.a.143.24		28
24.11	even	2	inner	72.8.f.a.35.7	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.7	✓	28	24.11	even	2	inner
72.8.f.a.35.8	yes	28	1.1	even	1	trivial
72.8.f.a.35.21	yes	28	3.2	odd	2	inner
72.8.f.a.35.22	yes	28	8.3	odd	2	inner
288.8.f.a.143.5		28	12.11	even	2
288.8.f.a.143.6		28	8.5	even	2
288.8.f.a.143.23		28	4.3	odd	2
288.8.f.a.143.24		28	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	72.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-8.14946 + 7.84769i) q^{2} +(4.82744 - 127.909i) q^{4} +294.266 q^{5} +328.633i q^{7} +(964.449 + 1080.27i) q^{8} +O(q^{10})\)	\(q+(-8.14946 + 7.84769i) q^{2} +(4.82744 - 127.909i) q^{4} +294.266 q^{5} +328.633i q^{7} +(964.449 + 1080.27i) q^{8} +(-2398.11 + 2309.31i) q^{10} -2680.12i q^{11} +1486.01i q^{13} +(-2579.01 - 2678.18i) q^{14} +(-16337.4 - 1234.95i) q^{16} -35589.5i q^{17} +16097.0 q^{19} +(1420.55 - 37639.2i) q^{20} +(21032.7 + 21841.5i) q^{22} -11593.2 q^{23} +8467.20 q^{25} +(-11661.8 - 12110.2i) q^{26} +(42035.1 + 1586.46i) q^{28} +204036. q^{29} +149568. i q^{31} +(142832. - 118147. i) q^{32} +(279296. + 290036. i) q^{34} +96705.4i q^{35} -61595.2i q^{37} +(-131182. + 126324. i) q^{38} +(283804. + 317887. i) q^{40} -232923. i q^{41} +619370. q^{43} +(-342811. - 12938.1i) q^{44} +(94478.0 - 90979.6i) q^{46} +1.07968e6 q^{47} +715543. q^{49} +(-69003.1 + 66448.0i) q^{50} +(190074. + 7173.63i) q^{52} +713736. q^{53} -788666. i q^{55} +(-355013. + 316950. i) q^{56} +(-1.66278e6 + 1.60121e6i) q^{58} -1.30458e6i q^{59} -1.52680e6i q^{61} +(-1.17376e6 - 1.21890e6i) q^{62} +(-236829. + 2.08374e6i) q^{64} +437282. i q^{65} -4.81113e6 q^{67} +(-4.55222e6 - 171807. i) q^{68} +(-758914. - 788097. i) q^{70} +4.36754e6 q^{71} -1.23348e6 q^{73} +(483380. + 501968. i) q^{74} +(77707.2 - 2.05895e6i) q^{76} +880774. q^{77} -5.34474e6i q^{79} +(-4.80753e6 - 363402. i) q^{80} +(1.82791e6 + 1.89820e6i) q^{82} +6.96837e6i q^{83} -1.04728e7i q^{85} +(-5.04753e6 + 4.86063e6i) q^{86} +(2.89526e6 - 2.58483e6i) q^{88} -166492. i q^{89} -488352. q^{91} +(-55965.3 + 1.48287e6i) q^{92} +(-8.79884e6 + 8.47302e6i) q^{94} +4.73678e6 q^{95} +7.05299e6 q^{97} +(-5.83129e6 + 5.61536e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 52 q^{4}+O(q^{10}) \) 28 * q + 52 * q^4	\( 28 q + 52 q^{4} + 10092 q^{10} - 1928 q^{16} - 121168 q^{19} + 59576 q^{22} + 437500 q^{25} + 46872 q^{28} - 114748 q^{34} + 1054752 q^{40} + 1505696 q^{43} - 476184 q^{46} - 2272076 q^{49} + 1468392 q^{52} + 3054996 q^{58} - 4186016 q^{64} - 776272 q^{67} + 3238872 q^{70} - 2534128 q^{73} - 21642832 q^{76} + 10334372 q^{82} + 10834016 q^{88} - 3406992 q^{91} - 22555944 q^{94} - 26311456 q^{97}+O(q^{100}) \) 28 * q + 52 * q^4 + 10092 * q^10 - 1928 * q^16 - 121168 * q^19 + 59576 * q^22 + 437500 * q^25 + 46872 * q^28 - 114748 * q^34 + 1054752 * q^40 + 1505696 * q^43 - 476184 * q^46 - 2272076 * q^49 + 1468392 * q^52 + 3054996 * q^58 - 4186016 * q^64 - 776272 * q^67 + 3238872 * q^70 - 2534128 * q^73 - 21642832 * q^76 + 10334372 * q^82 + 10834016 * q^88 - 3406992 * q^91 - 22555944 * q^94 - 26311456 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more