LMFDB - Embedded newform 72.8.f.a.35.25

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			35.25
Character	$\chi$	$=$	72.35
Dual form			72.8.f.a.35.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(10.7674 - 3.47333i) q^{2} +(103.872 - 74.7971i) q^{4} +527.660 q^{5} +1312.48i q^{7} +(858.632 - 1166.15i) q^{8} +O(q^{10})$	\(q+(10.7674 - 3.47333i) q^{2} +(103.872 - 74.7971i) q^{4} +527.660 q^{5} +1312.48i q^{7} +(858.632 - 1166.15i) q^{8} +(5681.50 - 1832.74i) q^{10} +961.214i q^{11} +10755.9i q^{13} +(4558.69 + 14132.0i) q^{14} +(5194.78 - 15538.7i) q^{16} -11544.1i q^{17} -46908.6 q^{19} +(54809.1 - 39467.5i) q^{20} +(3338.61 + 10349.7i) q^{22} +8879.25 q^{23} +200300. q^{25} +(37358.6 + 115812. i) q^{26} +(98170.0 + 136330. i) q^{28} +109471. q^{29} -239590. i q^{31} +(1963.17 - 185353. i) q^{32} +(-40096.4 - 124299. i) q^{34} +692545. i q^{35} -366153. i q^{37} +(-505081. + 162929. i) q^{38} +(453066. - 615330. i) q^{40} +40704.6i q^{41} +225674. q^{43} +(71896.1 + 99843.2i) q^{44} +(95606.0 - 30840.5i) q^{46} -713906. q^{47} -899069. q^{49} +(2.15670e6 - 695708. i) q^{50} +(804507. + 1.11723e6i) q^{52} -786052. q^{53} +507194. i q^{55} +(1.53055e6 + 1.12694e6i) q^{56} +(1.17871e6 - 380228. i) q^{58} -1.19024e6i q^{59} -1.03215e6i q^{61} +(-832174. - 2.57975e6i) q^{62} +(-622655. - 2.00259e6i) q^{64} +5.67543e6i q^{65} -1.72692e6 q^{67} +(-863464. - 1.19911e6i) q^{68} +(2.40544e6 + 7.45688e6i) q^{70} +82775.2 q^{71} -2.92308e6 q^{73} +(-1.27177e6 - 3.94251e6i) q^{74} +(-4.87248e6 + 3.50863e6i) q^{76} -1.26158e6 q^{77} +1.21761e6i q^{79} +(2.74108e6 - 8.19913e6i) q^{80} +(141380. + 438281. i) q^{82} +1.39552e6i q^{83} -6.09135e6i q^{85} +(2.42991e6 - 783839. i) q^{86} +(1.12092e6 + 825329. i) q^{88} -1.64007e6i q^{89} -1.41169e7 q^{91} +(922305. - 664142. i) q^{92} +(-7.68689e6 + 2.47963e6i) q^{94} -2.47518e7 q^{95} +175296. q^{97} +(-9.68060e6 + 3.12276e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	10.7674	−	3.47333i	0.951709	−	0.307002i
$2$	10.7674	−	3.47333i	0.951709	−	0.307002i
$3$		0			0
$3$		0			0
$4$	103.872	−	74.7971i	0.811500	−	0.584353i
$5$	527.660			1.88781			0.943907	−	0.330212i	$-0.107120\pi$
$5$	527.660			1.88781			0.943907	+	0.330212i	$0.107120\pi$
$6$		0			0
$7$			1312.48i			1.44627i	0.690705	+	0.723137i	$0.257301\pi$
$7$			1312.48i			1.44627i	−0.690705	+	0.723137i	$0.742699\pi$
$8$	858.632	−	1166.15i	0.592914	−	0.805266i
$9$		0			0
$10$	5681.50	−	1832.74i	1.79665	−	0.579562i
$11$			961.214i			0.217744i	0.994056	+	0.108872i	$0.0347238\pi$
$11$			961.214i			0.217744i	−0.994056	+	0.108872i	$0.965276\pi$
$12$		0			0
$13$			10755.9i			1.35782i	0.734220	+	0.678911i	$0.237548\pi$
$13$			10755.9i			1.35782i	−0.734220	+	0.678911i	$0.762452\pi$
$14$	4558.69	+	14132.0i	0.444009	+	1.37643i
$15$		0			0
$16$	5194.78	−	15538.7i	0.317064	−	0.948404i
$17$		−	11544.1i		−	0.569886i	−0.958544	−	0.284943i	$-0.908025\pi$
$17$		−	11544.1i		−	0.569886i	0.958544	−	0.284943i	$-0.0919747\pi$
$18$		0			0
$19$	−46908.6			−1.56897			−0.784485	−	0.620148i	$-0.787072\pi$
$19$	−46908.6			−1.56897			−0.784485	+	0.620148i	$0.787072\pi$
$20$	54809.1	−	39467.5i	1.53196	−	1.10315i
$21$		0			0
$22$	3338.61	+	10349.7i	0.0668477	+	0.207229i
$23$	8879.25			0.152170			0.0760849	−	0.997101i	$-0.475758\pi$
$23$	8879.25			0.152170			0.0760849	+	0.997101i	$0.475758\pi$
$24$		0			0
$25$	200300.			2.56384
$26$	37358.6	+	115812.i	0.416854	+	1.29225i
$27$		0			0
$28$	98170.0	+	136330.i	0.845134	+	1.17365i
$29$	109471.			0.833498			0.416749	−	0.909022i	$-0.363169\pi$
$29$	109471.			0.833498			0.416749	+	0.909022i	$0.363169\pi$
$30$		0			0
$31$		−	239590.i		−	1.44445i	−0.691659	−	0.722224i	$-0.743120\pi$
$31$		−	239590.i		−	1.44445i	0.691659	−	0.722224i	$-0.256880\pi$
$32$	1963.17	−	185353.i	0.0105909	−	0.999944i
$33$		0			0
$34$	−40096.4	−	124299.i	−0.174956	−	0.542366i
$35$			692545.i			2.73030i
$36$		0			0
$37$		−	366153.i		−	1.18838i	−0.804323	−	0.594192i	$-0.797472\pi$
$37$		−	366153.i		−	1.18838i	0.804323	−	0.594192i	$-0.202528\pi$
$38$	−505081.	+	162929.i	−1.49320	+	0.481676i
$39$		0			0
$40$	453066.	−	615330.i	1.11931	−	1.52019i
$41$			40704.6i			0.0922359i	0.998936	+	0.0461180i	$0.0146850\pi$
$41$			40704.6i			0.0922359i	−0.998936	+	0.0461180i	$0.985315\pi$
$42$		0			0
$43$	225674.			0.432854			0.216427	−	0.976299i	$-0.430560\pi$
$43$	225674.			0.432854			0.216427	+	0.976299i	$0.430560\pi$
$44$	71896.1	+	99843.2i	0.127239	+	0.176699i
$45$		0			0
$46$	95606.0	−	30840.5i	0.144821	−	0.0467164i
$47$	−713906.			−1.00300			−0.501498	−	0.865159i	$-0.667217\pi$
$47$	−713906.			−1.00300			−0.501498	+	0.865159i	$0.667217\pi$
$48$		0			0
$49$	−899069.			−1.09171
$50$	2.15670e6	−	695708.i	2.44003	−	0.787104i
$51$		0			0
$52$	804507.	+	1.11723e6i	0.793447	+	1.10187i
$53$	−786052.			−0.725246			−0.362623	−	0.931936i	$-0.618119\pi$
$53$	−786052.			−0.725246			−0.362623	+	0.931936i	$0.618119\pi$
$54$		0			0
$55$			507194.i			0.411060i
$56$	1.53055e6	+	1.12694e6i	1.16463	+	0.857517i
$57$		0			0
$58$	1.17871e6	−	380228.i	0.793248	−	0.255885i
$59$		−	1.19024e6i		−	0.754491i	−0.926113	−	0.377246i	$-0.876871\pi$
$59$		−	1.19024e6i		−	0.754491i	0.926113	−	0.377246i	$-0.123129\pi$
$60$		0			0
$61$		−	1.03215e6i		−	0.582220i	−0.956690	−	0.291110i	$-0.905975\pi$
$61$		−	1.03215e6i		−	0.582220i	0.956690	−	0.291110i	$-0.0940246\pi$
$62$	−832174.	−	2.57975e6i	−0.443448	−	1.37469i
$63$		0			0
$64$	−622655.	−	2.00259e6i	−0.296905	−	0.954907i
$65$			5.67543e6i			2.56332i
$66$		0			0
$67$	−1.72692e6			−0.701474			−0.350737	−	0.936474i	$-0.614069\pi$
$67$	−1.72692e6			−0.701474			−0.350737	+	0.936474i	$0.614069\pi$
$68$	−863464.	−	1.19911e6i	−0.333015	−	0.462463i
$69$		0			0
$70$	2.40544e6	+	7.45688e6i	0.838206	+	2.59845i
$71$	82775.2			0.0274471			0.0137235	−	0.999906i	$-0.495632\pi$
$71$	82775.2			0.0274471			0.0137235	+	0.999906i	$0.495632\pi$
$72$		0			0
$73$	−2.92308e6			−0.879449			−0.439724	−	0.898133i	$-0.644924\pi$
$73$	−2.92308e6			−0.879449			−0.439724	+	0.898133i	$0.644924\pi$
$74$	−1.27177e6	−	3.94251e6i	−0.364836	−	1.13100i
$75$		0			0
$76$	−4.87248e6	+	3.50863e6i	−1.27322	+	0.916831i
$77$	−1.26158e6			−0.314917
$78$		0			0
$79$			1.21761e6i			0.277851i	0.990303	+	0.138926i	$0.0443649\pi$
$79$			1.21761e6i			0.277851i	−0.990303	+	0.138926i	$0.955635\pi$
$80$	2.74108e6	−	8.19913e6i	0.598558	−	1.79041i
$81$		0			0
$82$	141380.	+	438281.i	0.0283166	+	0.0877817i
$83$			1.39552e6i			0.267894i	0.990989	+	0.133947i	$0.0427652\pi$
$83$			1.39552e6i			0.267894i	−0.990989	+	0.133947i	$0.957235\pi$
$84$		0			0
$85$		−	6.09135e6i		−	1.07584i
$86$	2.42991e6	−	783839.i	0.411951	−	0.132887i
$87$		0			0
$88$	1.12092e6	+	825329.i	0.175342	+	0.129103i
$89$		−	1.64007e6i		−	0.246602i	−0.992369	−	0.123301i	$-0.960652\pi$
$89$		−	1.64007e6i		−	0.246602i	0.992369	−	0.123301i	$-0.0393480\pi$
$90$		0			0
$91$	−1.41169e7			−1.96378
$92$	922305.	−	664142.i	0.123486	−	0.0889209i
$93$		0			0
$94$	−7.68689e6	+	2.47963e6i	−0.954559	+	0.307921i
$95$	−2.47518e7			−2.96192
$96$		0			0
$97$	175296.			0.0195017			0.00975083	−	0.999952i	$-0.496896\pi$
$97$	175296.			0.0195017			0.00975083	+	0.999952i	$0.496896\pi$
$98$	−9.68060e6	+	3.12276e6i	−1.03899	+	0.335157i
$99$		0			0
$100$	2.08056e7	−	1.49819e7i	2.08056	−	1.49819i
$101$	−1.77576e6			−0.171498			−0.0857489	−	0.996317i	$-0.527328\pi$
$101$	−1.77576e6			−0.171498			−0.0857489	+	0.996317i	$0.527328\pi$
$102$		0			0
$103$			8.36138e6i			0.753959i	0.926222	+	0.376979i	$0.123037\pi$
$103$			8.36138e6i			0.753959i	−0.926222	+	0.376979i	$0.876963\pi$
$104$	1.25429e7	+	9.23532e6i	1.09341	+	0.805073i
$105$		0			0
$106$	−8.46370e6	+	2.73022e6i	−0.690223	+	0.222652i
$107$			2.16063e7i			1.70505i	0.522684	+	0.852526i	$0.324931\pi$
$107$			2.16063e7i			1.70505i	−0.522684	+	0.852526i	$0.675069\pi$
$108$		0			0
$109$		−	7.42811e6i		−	0.549396i	−0.961531	−	0.274698i	$-0.911422\pi$
$109$		−	7.42811e6i		−	0.549396i	0.961531	−	0.274698i	$-0.0885779\pi$
$110$	1.76165e6	+	5.46114e6i	0.126196	+	0.391209i
$111$		0			0
$112$	2.03942e7	+	6.81806e6i	1.37165	+	0.458561i
$113$		−	1.32692e7i		−	0.865106i	−0.901609	−	0.432553i	$-0.857613\pi$
$113$		−	1.32692e7i		−	0.865106i	0.901609	−	0.432553i	$-0.142387\pi$
$114$		0			0
$115$	4.68522e6			0.287268
$116$	1.13709e7	−	8.18809e6i	0.676384	−	0.487057i
$117$		0			0
$118$	−4.13411e6	−	1.28158e7i	−0.231630	−	0.718056i
$119$	1.51514e7			0.824212
$120$		0			0
$121$	1.85632e7			0.952588
$122$	−3.58498e6	−	1.11135e7i	−0.178743	−	0.554104i
$123$		0			0
$124$	−1.79206e7	−	2.48867e7i	−0.844067	−	1.17217i
$125$	6.44669e7			2.95224
$126$		0			0
$127$			3.28675e7i			1.42381i	0.702273	+	0.711907i	$0.252168\pi$
$127$			3.28675e7i			1.42381i	−0.702273	+	0.711907i	$0.747832\pi$
$128$	−1.36600e7	−	1.93999e7i	−0.575725	−	0.817643i
$129$		0			0
$130$	1.97126e7	+	6.11094e7i	0.786943	+	2.43953i
$131$		−	2.89771e7i		−	1.12617i	−0.826398	−	0.563086i	$-0.809614\pi$
$131$		−	2.89771e7i		−	1.12617i	0.826398	−	0.563086i	$-0.190386\pi$
$132$		0			0
$133$		−	6.15667e7i		−	2.26916i
$134$	−1.85944e7	+	5.99818e6i	−0.667599	+	0.215354i
$135$		0			0
$136$	−1.34621e7	−	9.91212e6i	−0.458910	−	0.337894i
$137$			5.00356e7i			1.66248i	0.555911	+	0.831242i	$0.312369\pi$
$137$			5.00356e7i			1.66248i	−0.555911	+	0.831242i	$0.687631\pi$
$138$		0			0
$139$	3.52978e7			1.11480			0.557399	−	0.830245i	$-0.311800\pi$
$139$	3.52978e7			1.11480			0.557399	+	0.830245i	$0.311800\pi$
$140$	5.18004e7	+	7.19360e7i	1.59546	+	2.21564i
$141$		0			0
$142$	891270.	−	287505.i	0.0261216	−	0.00842630i
$143$	−1.03387e7			−0.295658
$144$		0			0
$145$	5.77633e7			1.57349
$146$	−3.14738e7	+	1.01528e7i	−0.836979	+	0.269992i
$147$		0			0
$148$	−2.73872e7	−	3.80331e7i	−0.694436	−	0.964374i
$149$	−2.26425e7			−0.560754			−0.280377	−	0.959890i	$-0.590459\pi$
$149$	−2.26425e7			−0.560754			−0.280377	+	0.959890i	$0.590459\pi$
$150$		0			0
$151$			9.18770e6i			0.217164i	0.994088	+	0.108582i	$0.0346310\pi$
$151$			9.18770e6i			0.217164i	−0.994088	+	0.108582i	$0.965369\pi$
$152$	−4.02772e7	+	5.47024e7i	−0.930264	+	1.26344i
$153$		0			0
$154$	−1.35839e7	+	4.38187e6i	−0.299710	+	0.0966802i
$155$		−	1.26422e8i		−	2.72685i
$156$		0			0
$157$			1.49030e7i			0.307345i	0.988122	+	0.153672i	$0.0491101\pi$
$157$			1.49030e7i			0.307345i	−0.988122	+	0.153672i	$0.950890\pi$
$158$	4.22915e6	+	1.31104e7i	0.0853008	+	0.264434i
$159$		0			0
$160$	1.03589e6	−	9.78036e7i	0.0199936	−	1.88771i
$161$			1.16539e7i			0.220079i
$162$		0			0
$163$	−4.56940e7			−0.826423			−0.413212	−	0.910635i	$-0.635593\pi$
$163$	−4.56940e7			−0.826423			−0.413212	+	0.910635i	$0.635593\pi$
$164$	3.04459e6	+	4.22807e6i	0.0538983	+	0.0748494i
$165$		0			0
$166$	4.84710e6	+	1.50261e7i	0.0822439	+	0.254957i
$167$	−2.75768e7			−0.458181			−0.229090	−	0.973405i	$-0.573575\pi$
$167$	−2.75768e7			−0.458181			−0.229090	+	0.973405i	$0.573575\pi$
$168$		0			0
$169$	−5.29398e7			−0.843683
$170$	−2.11573e7	−	6.55878e7i	−0.330285	−	1.02389i
$171$		0			0
$172$	2.34412e7	−	1.68798e7i	0.351261	−	0.252939i
$173$	9.78654e7			1.43704			0.718519	−	0.695508i	$-0.244821\pi$
$173$	9.78654e7			1.43704			0.718519	+	0.695508i	$0.244821\pi$
$174$		0			0
$175$			2.62890e8i			3.70802i
$176$	1.49360e7	+	4.99329e6i	0.206509	+	0.0690387i
$177$		0			0
$178$	−5.69649e6	−	1.76592e7i	−0.0757072	−	0.234693i
$179$			2.60015e7i			0.338855i	0.985543	+	0.169427i	$0.0541918\pi$
$179$			2.60015e7i			0.338855i	−0.985543	+	0.169427i	$0.945808\pi$
$180$		0			0
$181$		−	8.07924e7i		−	1.01273i	−0.862318	−	0.506367i	$-0.830988\pi$
$181$		−	8.07924e7i		−	1.01273i	0.862318	−	0.506367i	$-0.169012\pi$
$182$	−1.52001e8	+	4.90325e7i	−1.86895	+	0.602885i
$183$		0			0
$184$	7.62400e6	−	1.03545e7i	0.0902237	−	0.122537i
$185$		−	1.93205e8i		−	2.24345i
$186$		0			0
$187$	1.10963e7			0.124089
$188$	−7.41549e7	+	5.33982e7i	−0.813930	+	0.586103i
$189$		0			0
$190$	−2.66511e8	+	8.59710e7i	−2.81889	+	0.909315i
$191$	−1.15025e8			−1.19447			−0.597236	−	0.802065i	$-0.703735\pi$
$191$	−1.15025e8			−1.19447			−0.597236	+	0.802065i	$0.703735\pi$
$192$		0			0
$193$	−1.06745e7			−0.106880			−0.0534400	−	0.998571i	$-0.517019\pi$
$193$	−1.06745e7			−0.106880			−0.0534400	+	0.998571i	$0.517019\pi$
$194$	1.88748e6	−	608862.i	0.0185599	−	0.00598705i
$195$		0			0
$196$	−9.33881e7	+	6.72478e7i	−0.885922	+	0.637943i
$197$	5.17071e7			0.481857			0.240929	−	0.970543i	$-0.422548\pi$
$197$	5.17071e7			0.481857			0.240929	+	0.970543i	$0.422548\pi$
$198$		0			0
$199$		−	2.11722e8i		−	1.90450i	−0.305321	−	0.952249i	$-0.598764\pi$
$199$		−	2.11722e8i		−	1.90450i	0.305321	−	0.952249i	$-0.401236\pi$
$200$	1.71984e8	−	2.33580e8i	1.52014	−	2.06457i
$201$		0			0
$202$	−1.91202e7	+	6.16779e6i	−0.163216	+	0.0526502i
$203$			1.43678e8i			1.20547i
$204$		0			0
$205$			2.14782e7i			0.174124i
$206$	2.90418e7	+	9.00299e7i	0.231467	+	0.717549i
$207$		0			0
$208$	1.67131e8	+	5.58742e7i	1.28776	+	0.430517i
$209$		−	4.50892e7i		−	0.341633i
$210$		0			0
$211$	−3.28972e7			−0.241085			−0.120542	−	0.992708i	$-0.538463\pi$
$211$	−3.28972e7			−0.241085			−0.120542	+	0.992708i	$0.538463\pi$
$212$	−8.16487e7	+	5.87944e7i	−0.588537	+	0.423800i
$213$		0			0
$214$	7.50459e7	+	2.32643e8i	0.523454	+	1.62271i
$215$	1.19079e8			0.817148
$216$		0			0
$217$	3.14457e8			2.08907
$218$	−2.58003e7	−	7.99811e7i	−0.168666	−	0.522865i
$219$		0			0
$220$	3.79367e7	+	5.26833e7i	0.240204	+	0.333575i
$221$	1.24166e8			0.773805
$222$		0			0
$223$			2.58450e8i			1.56067i	0.625365	+	0.780333i	$0.284950\pi$
$223$			2.58450e8i			1.56067i	−0.625365	+	0.780333i	$0.715050\pi$
$224$	2.43273e8	+	2.57663e6i	1.44619	+	0.0153173i
$225$		0			0
$226$	−4.60882e7	−	1.42874e8i	−0.265589	−	0.823329i
$227$		−	9.78780e7i		−	0.555386i	−0.960670	−	0.277693i	$-0.910430\pi$
$227$		−	9.78780e7i		−	0.555386i	0.960670	−	0.277693i	$-0.0895698\pi$
$228$		0			0
$229$		−	3.04204e7i		−	0.167394i	−0.996491	−	0.0836972i	$-0.973327\pi$
$229$		−	3.04204e7i		−	0.167394i	0.996491	−	0.0836972i	$-0.0266729\pi$
$230$	5.04475e7	−	1.62733e7i	0.273396	−	0.0881919i
$231$		0			0
$232$	9.39950e7	−	1.27659e8i	0.494193	−	0.671187i
$233$			3.03447e8i			1.57158i	0.618493	+	0.785790i	$0.287743\pi$
$233$			3.03447e8i			1.57158i	−0.618493	+	0.785790i	$0.712257\pi$
$234$		0			0
$235$	−3.76700e8			−1.89347
$236$	−8.90269e7	−	1.23633e8i	−0.440889	−	0.612270i
$237$		0			0
$238$	1.63141e8	−	5.26258e7i	0.784410	−	0.253034i
$239$	−2.46584e8			−1.16835			−0.584173	−	0.811629i	$-0.698581\pi$
$239$	−2.46584e8			−1.16835			−0.584173	+	0.811629i	$0.698581\pi$
$240$		0			0
$241$	−3.24247e8			−1.49216			−0.746082	−	0.665854i	$-0.768067\pi$
$241$	−3.24247e8			−1.49216			−0.746082	+	0.665854i	$0.768067\pi$
$242$	1.99877e8	−	6.44762e7i	0.906586	−	0.292446i
$243$		0			0
$244$	−7.72016e7	−	1.07211e8i	−0.340222	−	0.472471i
$245$	−4.74403e8			−2.06094
$246$		0			0
$247$		−	5.04541e8i		−	2.13038i
$248$	−2.79397e8	−	2.05719e8i	−1.16316	−	0.856434i
$249$		0			0
$250$	6.94138e8	−	2.23915e8i	2.80967	−	0.906343i
$251$			1.31952e8i			0.526692i	0.964701	+	0.263346i	$0.0848261\pi$
$251$			1.31952e8i			0.526692i	−0.964701	+	0.263346i	$0.915174\pi$
$252$		0			0
$253$			8.53486e6i			0.0331340i
$254$	1.14160e8	+	3.53896e8i	0.437114	+	1.35506i
$255$		0			0
$256$	−2.14464e8	−	1.61440e8i	−0.798941	−	0.601410i
$257$			4.25505e8i			1.56365i	0.623500	+	0.781824i	$0.285710\pi$
$257$			4.25505e8i			1.56365i	−0.623500	+	0.781824i	$0.714290\pi$
$258$		0			0
$259$	4.80570e8			1.71873
$260$	4.24506e8	+	5.89518e8i	1.49788	+	2.08013i
$261$		0			0
$262$	−1.00647e8	−	3.12006e8i	−0.345737	−	1.07179i
$263$	2.98275e8			1.01105			0.505524	−	0.862813i	$-0.331299\pi$
$263$	2.98275e8			1.01105			0.505524	+	0.862813i	$0.331299\pi$
$264$		0			0
$265$	−4.14768e8			−1.36913
$266$	−2.13841e8	−	6.62911e8i	−0.696636	−	2.15958i
$267$		0			0
$268$	−1.79379e8	+	1.29169e8i	−0.569246	+	0.409908i
$269$	−3.01605e8			−0.944726			−0.472363	−	0.881404i	$-0.656599\pi$
$269$	−3.01605e8			−0.944726			−0.472363	+	0.881404i	$0.656599\pi$
$270$		0			0
$271$		−	8.93027e7i		−	0.272566i	−0.990670	−	0.136283i	$-0.956484\pi$
$271$		−	8.93027e7i		−	0.272566i	0.990670	−	0.136283i	$-0.0435157\pi$
$272$	−1.79380e8	−	5.99689e7i	−0.540483	−	0.180690i
$273$		0			0
$274$	1.73790e8	+	5.38751e8i	0.510385	+	1.58220i
$275$			1.92531e8i			0.558260i
$276$		0			0
$277$		−	2.27412e8i		−	0.642886i	−0.946929	−	0.321443i	$-0.895832\pi$
$277$		−	2.27412e8i		−	0.642886i	0.946929	−	0.321443i	$-0.104168\pi$
$278$	3.80064e8	−	1.22601e8i	1.06096	−	0.342245i
$279$		0			0
$280$	8.07611e8	+	5.94641e8i	2.19861	+	1.61883i
$281$			1.42040e8i			0.381889i	0.981601	+	0.190945i	$0.0611551\pi$
$281$			1.42040e8i			0.381889i	−0.981601	+	0.190945i	$0.938845\pi$
$282$		0			0
$283$	6.43886e8			1.68871			0.844357	−	0.535781i	$-0.179983\pi$
$283$	6.43886e8			1.68871			0.844357	+	0.535781i	$0.179983\pi$
$284$	8.59802e6	−	6.19135e6i	0.0222733	−	0.0160388i
$285$		0			0
$286$	−1.11320e8	+	3.59096e7i	−0.281380	+	0.0907674i
$287$	−5.34241e7			−0.133398
$288$		0			0
$289$	2.77073e8			0.675230
$290$	6.21958e8	−	2.00631e8i	1.49750	−	0.483064i
$291$		0			0
$292$	−3.03626e8	+	2.18638e8i	−0.713673	+	0.513908i
$293$	2.93386e8			0.681402			0.340701	−	0.940172i	$-0.389336\pi$
$293$	2.93386e8			0.681402			0.340701	+	0.940172i	$0.389336\pi$
$294$		0			0
$295$		−	6.28045e8i		−	1.42434i
$296$	−4.26990e8	−	3.14391e8i	−0.956965	−	0.704611i
$297$		0			0
$298$	−2.43800e8	+	7.86448e7i	−0.533674	+	0.172152i
$299$			9.55039e7i			0.206620i
$300$		0			0
$301$			2.96193e8i			0.626026i
$302$	3.19119e7	+	9.89272e7i	0.0666697	+	0.206677i
$303$		0			0
$304$	−2.43679e8	+	7.28896e8i	−0.497464	+	1.48802i
$305$		−	5.44623e8i		−	1.09912i
$306$		0			0
$307$	−7.74955e8			−1.52859			−0.764297	−	0.644865i	$-0.776914\pi$
$307$	−7.74955e8			−1.52859			−0.764297	+	0.644865i	$0.776914\pi$
$308$	−1.31043e8	+	9.43624e7i	−0.255555	+	0.184023i
$309$		0			0
$310$	−4.39105e8	−	1.36123e9i	−0.837148	−	2.59517i
$311$	9.49879e8			1.79063			0.895317	−	0.445429i	$-0.146949\pi$
$311$	9.49879e8			1.79063			0.895317	+	0.445429i	$0.146949\pi$
$312$		0			0
$313$	−7.46781e8			−1.37654			−0.688269	−	0.725456i	$-0.741629\pi$
$313$	−7.46781e8			−1.37654			−0.688269	+	0.725456i	$0.741629\pi$
$314$	5.17631e7	+	1.60466e8i	0.0943554	+	0.292503i
$315$		0			0
$316$	9.10735e7	+	1.26475e8i	0.162363	+	0.225476i
$317$	−6.87425e8			−1.21204			−0.606021	−	0.795448i	$-0.707235\pi$
$317$	−6.87425e8			−1.21204			−0.606021	+	0.795448i	$0.707235\pi$
$318$		0			0
$319$			1.05225e8i			0.181489i
$320$	−3.28550e8	−	1.05668e9i	−0.560502	−	1.80269i
$321$		0			0
$322$	4.04777e7	+	1.25481e8i	0.0675648	+	0.209452i
$323$			5.41516e8i			0.894134i
$324$		0			0
$325$			2.15440e9i			3.48124i
$326$	−4.92003e8	+	1.58710e8i	−0.786514	+	0.253713i
$327$		0			0
$328$	4.74676e7	+	3.49503e7i	0.0742744	+	0.0546880i
$329$		−	9.36990e8i		−	1.45061i
$330$		0			0
$331$	−4.77550e8			−0.723803			−0.361902	−	0.932216i	$-0.617872\pi$
$331$	−4.77550e8			−0.723803			−0.361902	+	0.932216i	$0.617872\pi$
$332$	1.04381e8	+	1.44955e8i	0.156544	+	0.217396i
$333$		0			0
$334$	−2.96930e8	+	9.57834e7i	−0.436055	+	0.140662i
$335$	−9.11229e8			−1.32425
$336$		0			0
$337$	3.20237e8			0.455792			0.227896	−	0.973685i	$-0.426815\pi$
$337$	3.20237e8			0.455792			0.227896	+	0.973685i	$0.426815\pi$
$338$	−5.70022e8	+	1.83877e8i	−0.802940	+	0.259012i
$339$		0			0
$340$	−4.55616e8	−	6.32721e8i	−0.628669	−	0.873043i
$341$	2.30297e8			0.314520
$342$		0			0
$343$		−	9.91271e7i		−	0.132637i
$344$	1.93771e8	−	2.63169e8i	0.256645	−	0.348562i
$345$		0			0
$346$	1.05375e9	−	3.39919e8i	1.36764	−	0.441173i
$347$		−	6.18689e8i		−	0.794913i	−0.917621	−	0.397456i	$-0.869893\pi$
$347$		−	6.18689e8i		−	0.794913i	0.917621	−	0.397456i	$-0.130107\pi$
$348$		0			0
$349$			5.94067e7i			0.0748078i	0.999300	+	0.0374039i	$0.0119088\pi$
$349$			5.94067e7i			0.0748078i	−0.999300	+	0.0374039i	$0.988091\pi$
$350$	9.13105e8	+	2.83064e9i	1.13837	+	3.52895i
$351$		0			0
$352$	1.78164e8	+	1.88703e6i	0.217732	+	0.00230610i
$353$		−	1.32577e9i		−	1.60420i	−0.597192	−	0.802098i	$-0.703717\pi$
$353$		−	1.32577e9i		−	1.60420i	0.597192	−	0.802098i	$-0.296283\pi$
$354$		0			0
$355$	4.36772e7			0.0518150
$356$	−1.22672e8	−	1.70357e8i	−0.144102	−	0.200117i
$357$		0			0
$358$	9.03119e7	+	2.79968e8i	0.104029	+	0.322491i
$359$	8.17907e8			0.932982			0.466491	−	0.884526i	$-0.345518\pi$
$359$	8.17907e8			0.932982			0.466491	+	0.884526i	$0.345518\pi$
$360$		0			0
$361$	1.30654e9			1.46166
$362$	−2.80619e8	−	8.69921e8i	−0.310911	−	0.963828i
$363$		0			0
$364$	−1.46635e9	+	1.05590e9i	−1.59361	+	1.14754i
$365$	−1.54239e9			−1.66024
$366$		0			0
$367$		−	1.06854e9i		−	1.12839i	−0.825642	−	0.564195i	$-0.809187\pi$
$367$		−	1.06854e9i		−	1.12839i	0.825642	−	0.564195i	$-0.190813\pi$
$368$	4.61257e7	−	1.37972e8i	0.0482476	−	0.144319i
$369$		0			0
$370$	−6.71063e8	−	2.08030e9i	−0.688743	−	2.13511i
$371$		−	1.03168e9i		−	1.04891i
$372$		0			0
$373$		−	2.96640e7i		−	0.0295970i	−0.999890	−	0.0147985i	$-0.995289\pi$
$373$		−	2.96640e7i		−	0.0295970i	0.999890	−	0.0147985i	$-0.00471069\pi$
$374$	1.19478e8	−	3.85412e7i	0.118097	−	0.0380956i
$375$		0			0
$376$	−6.12983e8	+	8.32521e8i	−0.594690	+	0.807677i
$377$			1.17745e9i			1.13174i
$378$		0			0
$379$	9.95464e8			0.939265			0.469633	−	0.882862i	$-0.344386\pi$
$379$	9.95464e8			0.939265			0.469633	+	0.882862i	$0.344386\pi$
$380$	−2.57101e9	+	1.85136e9i	−2.40360	+	1.73081i
$381$		0			0
$382$	−1.23852e9	+	3.99520e8i	−1.13679	+	0.366705i
$383$	1.46129e9			1.32905			0.664523	−	0.747268i	$-0.268635\pi$
$383$	1.46129e9			1.32905			0.664523	+	0.747268i	$0.268635\pi$
$384$		0			0
$385$	−6.65684e8			−0.594505
$386$	−1.14936e8	+	3.70760e7i	−0.101719	+	0.0328123i
$387$		0			0
$388$	1.82084e7	−	1.31117e7i	0.0158256	−	0.0113959i
$389$	1.67137e9			1.43963			0.719813	−	0.694169i	$-0.244228\pi$
$389$	1.67137e9			1.43963			0.719813	+	0.694169i	$0.244228\pi$
$390$		0			0
$391$		−	1.02503e8i		−	0.0867195i
$392$	−7.71970e8	+	1.04845e9i	−0.647290	+	0.879116i
$393$		0			0
$394$	5.56749e8	−	1.79596e8i	0.458588	−	0.147931i
$395$			6.42483e8i			0.524531i
$396$		0			0
$397$		−	4.73007e8i		−	0.379403i	−0.981842	−	0.189701i	$-0.939248\pi$
$397$		−	4.73007e8i		−	0.379403i	0.981842	−	0.189701i	$-0.0607520\pi$
$398$	−7.35381e8	−	2.27969e9i	−0.584684	−	1.81253i
$399$		0			0
$400$	1.04051e9	−	3.11239e9i	0.812902	−	2.43156i
$401$			1.63188e9i			1.26381i	0.775045	+	0.631906i	$0.217727\pi$
$401$			1.63188e9i			1.26381i	−0.775045	+	0.631906i	$0.782273\pi$
$402$		0			0
$403$	2.57699e9			1.96131
$404$	−1.84451e8	+	1.32822e8i	−0.139171	+	0.100215i
$405$		0			0
$406$	4.99042e8	+	1.54704e9i	0.370081	+	1.14725i
$407$	3.51952e8			0.258763
$408$		0			0
$409$	−3.20095e8			−0.231339			−0.115669	−	0.993288i	$-0.536901\pi$
$409$	−3.20095e8			−0.231339			−0.115669	+	0.993288i	$0.536901\pi$
$410$	7.46008e7	+	2.31263e8i	0.0534564	+	0.165716i
$411$		0			0
$412$	6.25407e8	+	8.68513e8i	0.440578	+	0.611837i
$413$	1.56218e9			1.09120
$414$		0			0
$415$			7.36360e8i			0.505734i
$416$	1.99363e9	+	2.11156e7i	1.35775	+	0.0143806i
$417$		0			0
$418$	−1.56609e8	−	4.85491e8i	−0.104882	−	0.325136i
$419$		−	2.17937e9i		−	1.44738i	−0.690128	−	0.723688i	$-0.742446\pi$
$419$		−	2.17937e9i		−	1.44738i	0.690128	−	0.723688i	$-0.257554\pi$
$420$		0			0
$421$			1.38985e8i			0.0907783i	0.998969	+	0.0453892i	$0.0144528\pi$
$421$			1.38985e8i			0.0907783i	−0.998969	+	0.0453892i	$0.985547\pi$
$422$	−3.54216e8	+	1.14263e8i	−0.229443	+	0.0740135i
$423$		0			0
$424$	−6.74929e8	+	9.16653e8i	−0.430009	+	0.584016i
$425$		−	2.31228e9i		−	1.46110i
$426$		0			0
$427$	1.35468e9			0.842050
$428$	1.61609e9	+	2.24429e9i	0.996352	+	1.38365i
$429$		0			0
$430$	1.28217e9	−	4.13601e8i	0.777687	−	0.250866i
$431$	2.50706e9			1.50832			0.754161	−	0.656690i	$-0.228044\pi$
$431$	2.50706e9			1.50832			0.754161	+	0.656690i	$0.228044\pi$
$432$		0			0
$433$	−7.58853e8			−0.449211			−0.224605	−	0.974450i	$-0.572109\pi$
$433$	−7.58853e8			−0.449211			−0.224605	+	0.974450i	$0.572109\pi$
$434$	3.38588e9	−	1.09221e9i	1.98819	−	0.641348i
$435$		0			0
$436$	−5.55601e8	−	7.71572e8i	−0.321041	−	0.445835i
$437$	−4.16513e8			−0.238750
$438$		0			0
$439$			2.63015e9i			1.48373i	0.670548	+	0.741866i	$0.266059\pi$
$439$			2.63015e9i			1.48373i	−0.670548	+	0.741866i	$0.733941\pi$
$440$	5.91464e8	+	4.35493e8i	0.331012	+	0.243723i
$441$		0			0
$442$	1.33694e9	−	4.31271e8i	0.736437	−	0.237559i
$443$			1.63076e9i			0.891206i	0.895231	+	0.445603i	$0.147011\pi$
$443$			1.63076e9i			0.891206i	−0.895231	+	0.445603i	$0.852989\pi$
$444$		0			0
$445$		−	8.65397e8i		−	0.465538i
$446$	8.97682e8	+	2.78283e9i	0.479127	+	1.48530i
$447$		0			0
$448$	2.62836e9	−	8.17224e8i	1.38106	−	0.429406i
$449$			1.68143e9i			0.876628i	0.898822	+	0.438314i	$0.144424\pi$
$449$			1.68143e9i			0.876628i	−0.898822	+	0.438314i	$0.855576\pi$
$450$		0			0
$451$	−3.91258e7			−0.0200838
$452$	−9.92496e8	−	1.37829e9i	−0.505527	−	0.702033i
$453$		0			0
$454$	−3.39963e8	−	1.05389e9i	−0.170504	−	0.528566i
$455$	−7.44891e9			−3.70726
$456$		0			0
$457$	6.30718e8			0.309121			0.154561	−	0.987983i	$-0.450604\pi$
$457$	6.30718e8			0.309121			0.154561	+	0.987983i	$0.450604\pi$
$458$	−1.05660e8	−	3.27548e8i	−0.0513904	−	0.159311i
$459$		0			0
$460$	4.86663e8	−	3.50441e8i	0.233118	−	0.167866i
$461$	−1.44175e9			−0.685390			−0.342695	−	0.939447i	$-0.611340\pi$
$461$	−1.44175e9			−0.685390			−0.342695	+	0.939447i	$0.611340\pi$
$462$		0			0
$463$		−	1.16830e9i		−	0.547044i	−0.961866	−	0.273522i	$-0.911811\pi$
$463$		−	1.16830e9i		−	0.547044i	0.961866	−	0.273522i	$-0.0881887\pi$
$464$	5.68676e8	−	1.70103e9i	0.264272	−	0.790493i
$465$		0			0
$466$	1.05397e9	+	3.26732e9i	0.482478	+	1.49569i
$467$			8.97005e8i			0.407555i	0.979017	+	0.203777i	$0.0653218\pi$
$467$			8.97005e8i			0.407555i	−0.979017	+	0.203777i	$0.934678\pi$
$468$		0			0
$469$		−	2.26656e9i		−	1.01452i
$470$	−4.05606e9	+	1.30840e9i	−1.80203	+	0.581298i
$471$		0			0
$472$	−1.38800e9	−	1.02198e9i	−0.607566	−	0.447349i
$473$			2.16921e8i			0.0942513i
$474$		0			0
$475$	−9.39579e9			−4.02259
$476$	1.57381e9	−	1.13328e9i	0.668848	−	0.481630i
$477$		0			0
$478$	−2.65505e9	+	8.56466e8i	−1.11193	+	0.358684i
$479$	8.31795e8			0.345813			0.172907	−	0.984938i	$-0.444684\pi$
$479$	8.31795e8			0.345813			0.172907	+	0.984938i	$0.444684\pi$
$480$		0			0
$481$	3.93829e9			1.61362
$482$	−3.49128e9	+	1.12622e9i	−1.42011	+	0.458097i
$483$		0			0
$484$	1.92820e9	−	1.38848e9i	0.773025	−	0.556647i
$485$	9.24969e7			0.0368155
$486$		0			0
$487$			1.00377e9i			0.393805i	0.980423	+	0.196902i	$0.0630882\pi$
$487$			1.00377e9i			0.393805i	−0.980423	+	0.196902i	$0.936912\pi$
$488$	−1.20364e9	−	8.86234e8i	−0.468842	−	0.345207i
$489$		0			0
$490$	−5.10807e9	+	1.64776e9i	−1.96142	+	0.632713i
$491$			6.35648e7i			0.0242344i	0.999927	+	0.0121172i	$0.00385711\pi$
$491$			6.35648e7i			0.0242344i	−0.999927	+	0.0121172i	$0.996143\pi$
$492$		0			0
$493$		−	1.26374e9i		−	0.474999i
$494$	−1.75244e9	−	5.43258e9i	−0.654031	−	2.02750i
$495$		0			0
$496$	−3.72290e9	−	1.24461e9i	−1.36992	−	0.457983i
$497$			1.08641e8i			0.0396960i
$498$		0			0
$499$	−3.66659e9			−1.32102			−0.660512	−	0.750816i	$-0.729661\pi$
$499$	−3.66659e9			−1.32102			−0.660512	+	0.750816i	$0.729661\pi$
$500$	6.69630e9	−	4.82194e9i	2.39574	−	1.72515i
$501$		0			0
$502$	4.58311e8	+	1.42077e9i	0.161695	+	0.501257i
$503$	−2.44858e9			−0.857880			−0.428940	−	0.903333i	$-0.641113\pi$
$503$	−2.44858e9			−0.857880			−0.428940	+	0.903333i	$0.641113\pi$
$504$		0			0
$505$	−9.36996e8			−0.323756
$506$	2.96444e7	+	9.18979e7i	0.0101722	+	0.0315340i
$507$		0			0
$508$	2.45839e9	+	3.41401e9i	0.832010	+	1.15543i
$509$	4.07396e9			1.36932			0.684659	−	0.728863i	$-0.259951\pi$
$509$	4.07396e9			1.36932			0.684659	+	0.728863i	$0.259951\pi$
$510$		0			0
$511$		−	3.83649e9i		−	1.27192i
$512$	−2.86994e9	−	9.93374e8i	−0.944993	−	0.327091i
$513$		0			0
$514$	1.47792e9	+	4.58156e9i	0.480042	+	1.48814i
$515$			4.41196e9i			1.42333i
$516$		0			0
$517$		−	6.86217e8i		−	0.218396i
$518$	5.17447e9	−	1.66918e9i	1.63573	−	0.527653i
$519$		0			0
$520$	6.61840e9	+	4.87311e9i	2.06415	+	1.51983i
$521$			3.19867e9i			0.990917i	0.868632	+	0.495459i	$0.165000\pi$
$521$			3.19867e9i			0.990917i	−0.868632	+	0.495459i	$0.835000\pi$
$522$		0			0
$523$	5.00703e9			1.53047			0.765235	−	0.643752i	$-0.222623\pi$
$523$	5.00703e9			1.53047			0.765235	+	0.643752i	$0.222623\pi$
$524$	−2.16740e9	−	3.00991e9i	−0.658082	−	0.913889i
$525$		0			0
$526$	3.21163e9	−	1.03601e9i	0.962223	−	0.310393i
$527$	−2.76584e9			−0.823172
$528$		0			0
$529$	−3.32598e9			−0.976844
$530$	−4.46596e9	+	1.44063e9i	−1.30301	+	0.420325i
$531$		0			0
$532$	−4.60501e9	−	6.39505e9i	−1.32599	−	1.84142i
$533$	−4.37813e8			−0.125240
$534$		0			0
$535$			1.14008e10i			3.21882i
$536$	−1.48279e9	+	2.01385e9i	−0.415914	+	0.564873i
$537$		0			0
$538$	−3.24749e9	+	1.04757e9i	−0.899104	+	0.290033i
$539$		−	8.64198e8i		−	0.237713i
$540$		0			0
$541$		−	4.70918e9i		−	1.27866i	−0.768932	−	0.639330i	$-0.779212\pi$
$541$		−	4.70918e9i		−	1.27866i	0.768932	−	0.639330i	$-0.220788\pi$
$542$	−3.10178e8	−	9.61554e8i	−0.0836783	−	0.259404i
$543$		0			0
$544$	−2.13974e9	−	2.26630e7i	−0.569854	−	0.00603561i
$545$		−	3.91952e9i		−	1.03716i
$546$		0			0
$547$	−2.77317e9			−0.724472			−0.362236	−	0.932086i	$-0.617987\pi$
$547$	−2.77317e9			−0.724472			−0.362236	+	0.932086i	$0.617987\pi$
$548$	3.74252e9	+	5.19730e9i	0.971477	+	1.34910i
$549$		0			0
$550$	6.68724e8	+	2.07305e9i	0.171387	+	0.531301i
$551$	−5.13511e9			−1.30773
$552$		0			0
$553$	−1.59809e9			−0.401849
$554$	−7.89876e8	−	2.44862e9i	−0.197367	−	0.611840i
$555$		0			0
$556$	3.66645e9	−	2.64017e9i	0.904658	−	0.651435i
$557$	−1.23669e9			−0.303227			−0.151614	−	0.988440i	$-0.548447\pi$
$557$	−1.23669e9			−0.303227			−0.151614	+	0.988440i	$0.548447\pi$
$558$		0			0
$559$			2.42731e9i			0.587739i
$560$	1.07612e10	+	3.59762e9i	2.58942	+	0.865679i
$561$		0			0
$562$	4.93350e8	+	1.52939e9i	0.117241	+	0.363447i
$563$		−	4.49885e9i		−	1.06248i	−0.847220	−	0.531242i	$-0.821725\pi$
$563$		−	4.49885e9i		−	1.06248i	0.847220	−	0.531242i	$-0.178275\pi$
$564$		0			0
$565$		−	7.00161e9i		−	1.63316i
$566$	6.93295e9	−	2.23643e9i	1.60716	−	0.518438i
$567$		0			0
$568$	7.10734e7	−	9.65282e7i	0.0162738	−	0.0221022i
$569$			5.60760e9i			1.27610i	0.769995	+	0.638049i	$0.220258\pi$
$569$			5.60760e9i			1.27610i	−0.769995	+	0.638049i	$0.779742\pi$
$570$		0			0
$571$	3.27460e9			0.736091			0.368046	−	0.929808i	$-0.380027\pi$
$571$	3.27460e9			0.736091			0.368046	+	0.929808i	$0.380027\pi$
$572$	−1.07390e9	+	7.73304e8i	−0.239926	+	0.172768i
$573$		0			0
$574$	−5.75237e8	+	1.85559e8i	−0.126956	+	0.0409535i
$575$	1.77851e9			0.390139
$576$		0			0
$577$	−6.54667e9			−1.41875			−0.709373	−	0.704833i	$-0.751022\pi$
$577$	−6.54667e9			−1.41875			−0.709373	+	0.704833i	$0.751022\pi$
$578$	2.98334e9	−	9.62365e8i	0.642622	−	0.207297i
$579$		0			0
$580$	5.99999e9	−	4.32053e9i	1.27689	−	0.919473i
$581$	−1.83160e9			−0.387448
$582$		0			0
$583$		−	7.55564e8i		−	0.157918i
$584$	−2.50985e9	+	3.40875e9i	−0.521438	+	0.708190i
$585$		0			0
$586$	3.15899e9	−	1.01903e9i	0.648496	−	0.209191i
$587$		−	3.23096e9i		−	0.659323i	−0.944099	−	0.329662i	$-0.893065\pi$
$587$		−	3.23096e9i		−	0.659323i	0.944099	−	0.329662i	$-0.106935\pi$
$588$		0			0
$589$			1.12388e10i			2.26630i
$590$	−2.18141e9	−	6.76238e9i	−0.437275	−	1.35556i
$591$		0			0
$592$	−5.68953e9	−	1.90209e9i	−1.12707	−	0.376794i
$593$			4.14696e8i			0.0816655i	0.999166	+	0.0408328i	$0.0130011\pi$
$593$			4.14696e8i			0.0816655i	−0.999166	+	0.0408328i	$0.986999\pi$
$594$		0			0
$595$	7.99480e9			1.55596
$596$	−2.35192e9	+	1.69359e9i	−0.455052	+	0.327678i
$597$		0			0
$598$	3.31716e8	+	1.02832e9i	0.0634326	+	0.196642i
$599$	8.31761e9			1.58127			0.790633	−	0.612291i	$-0.209752\pi$
$599$	8.31761e9			1.58127			0.790633	+	0.612291i	$0.209752\pi$
$600$		0			0
$601$	4.33439e9			0.814456			0.407228	−	0.913327i	$-0.366495\pi$
$601$	4.33439e9			0.814456			0.407228	+	0.913327i	$0.366495\pi$
$602$	1.02878e9	+	3.18922e9i	0.192191	+	0.595794i
$603$		0			0
$604$	6.87213e8	+	9.54344e8i	0.126900	+	0.176228i
$605$	9.79508e9			1.79831
$606$		0			0
$607$		−	3.06356e9i		−	0.555989i	−0.960583	−	0.277994i	$-0.910330\pi$
$607$		−	3.06356e9i		−	0.555989i	0.960583	−	0.277994i	$-0.0896697\pi$
$608$	−9.20894e7	+	8.69466e9i	−0.0166168	+	1.56888i
$609$		0			0
$610$	−1.89165e9	−	5.86415e9i	−0.337433	−	1.04605i
$611$		−	7.67867e9i		−	1.36189i
$612$		0			0
$613$		−	5.72765e9i		−	1.00430i	−0.864779	−	0.502152i	$-0.832542\pi$
$613$		−	5.72765e9i		−	1.00430i	0.864779	−	0.502152i	$-0.167458\pi$
$614$	−8.34421e9	+	2.69167e9i	−1.45478	+	0.469281i
$615$		0			0
$616$	−1.08323e9	+	1.47119e9i	−0.186719	+	0.253592i
$617$			6.41667e9i			1.09980i	0.835232	+	0.549898i	$0.185333\pi$
$617$			6.41667e9i			1.09980i	−0.835232	+	0.549898i	$0.814667\pi$
$618$		0			0
$619$	−9.67083e9			−1.63888			−0.819439	−	0.573167i	$-0.805715\pi$
$619$	−9.67083e9			−1.63888			−0.819439	+	0.573167i	$0.805715\pi$
$620$	−9.45600e9	−	1.31317e10i	−1.59344	−	2.21284i
$621$		0			0
$622$	1.02277e10	−	3.29924e9i	1.70416	−	0.549728i
$623$	2.15256e9			0.356654
$624$		0			0
$625$	1.83682e10			3.00944
$626$	−8.04086e9	+	2.59382e9i	−1.31006	+	0.422600i
$627$		0			0
$628$	1.11470e9	+	1.54801e9i	0.179598	+	0.249410i
$629$	−4.22691e9			−0.677244
$630$		0			0
$631$			2.36943e9i			0.375441i	0.982223	+	0.187720i	$0.0601098\pi$
$631$			2.36943e9i			0.375441i	−0.982223	+	0.187720i	$0.939890\pi$
$632$	1.41991e9	+	1.04548e9i	0.223744	+	0.164742i
$633$		0			0
$634$	−7.40175e9	+	2.38765e9i	−1.15351	+	0.372099i
$635$			1.73429e10i			2.68790i
$636$		0			0
$637$		−	9.67026e9i		−	1.48235i
$638$	3.65480e8	+	1.13299e9i	0.0557175	+	0.172725i
$639$		0			0
$640$	−7.20783e9	−	1.02365e10i	−1.08686	−	1.54356i
$641$			6.10766e9i			0.915949i	0.888965	+	0.457975i	$0.151425\pi$
$641$			6.10766e9i			0.915949i	−0.888965	+	0.457975i	$0.848575\pi$
$642$		0			0
$643$	9.52741e9			1.41331			0.706653	−	0.707560i	$-0.250204\pi$
$643$	9.52741e9			1.41331			0.706653	+	0.707560i	$0.250204\pi$
$644$	8.71676e8	+	1.21051e9i	0.128604	+	0.178594i
$645$		0			0
$646$	1.88086e9	+	5.83070e9i	0.274501	+	0.850955i
$647$	−5.20878e9			−0.756086			−0.378043	−	0.925788i	$-0.623403\pi$
$647$	−5.20878e9			−0.756086			−0.378043	+	0.925788i	$0.623403\pi$
$648$		0			0
$649$	1.14408e9			0.164286
$650$	7.48293e9	+	2.31972e10i	1.06875	+	3.31313i
$651$		0			0
$652$	−4.74632e9	+	3.41778e9i	−0.670642	+	0.482922i
$653$	7.90741e9			1.11132			0.555659	−	0.831410i	$-0.312466\pi$
$653$	7.90741e9			1.11132			0.555659	+	0.831410i	$0.312466\pi$
$654$		0			0
$655$		−	1.52900e10i		−	2.12600i
$656$	6.32495e8	+	2.11451e8i	0.0874769	+	0.0292447i
$657$		0			0
$658$	−3.25448e9	−	1.00889e10i	−0.445339	−	1.38055i
$659$		−	1.05768e10i		−	1.43964i	−0.694158	−	0.719822i	$-0.744223\pi$
$659$		−	1.05768e10i		−	1.43964i	0.694158	−	0.719822i	$-0.255777\pi$
$660$		0			0
$661$			6.08761e9i			0.819864i	0.912116	+	0.409932i	$0.134448\pi$
$661$			6.08761e9i			0.819864i	−0.912116	+	0.409932i	$0.865552\pi$
$662$	−5.14195e9	+	1.65869e9i	−0.688850	+	0.222209i
$663$		0			0
$664$	1.62738e9	+	1.19824e9i	0.215726	+	0.158838i
$665$		−	3.24863e10i		−	4.28375i
$666$		0			0
$667$	9.72017e8			0.126833
$668$	−2.86446e9	+	2.06267e9i	−0.371813	+	0.267739i
$669$		0			0
$670$	−9.81153e9	+	3.16500e9i	−1.26030	+	0.406548i
$671$	9.92114e8			0.126775
$672$		0			0
$673$	−4.42313e9			−0.559341			−0.279671	−	0.960096i	$-0.590225\pi$
$673$	−4.42313e9			−0.559341			−0.279671	+	0.960096i	$0.590225\pi$
$674$	3.44811e9	−	1.11229e9i	0.433782	−	0.139929i
$675$		0			0
$676$	−5.49897e9	+	3.95975e9i	−0.684648	+	0.493008i
$677$	−1.04814e10			−1.29825			−0.649125	−	0.760682i	$-0.724865\pi$
$677$	−1.04814e10			−1.29825			−0.649125	+	0.760682i	$0.724865\pi$
$678$		0			0
$679$			2.30074e8i			0.0282048i
$680$	−7.10342e9	−	5.23023e9i	−0.866336	−	0.637881i
$681$		0			0
$682$	2.47969e9	−	7.99897e8i	0.299331	−	0.0965581i
$683$			3.89413e9i			0.467668i	0.972277	+	0.233834i	$0.0751273\pi$
$683$			3.89413e9i			0.467668i	−0.972277	+	0.233834i	$0.924873\pi$
$684$		0			0
$685$			2.64018e10i			3.13846i
$686$	−3.44301e8	−	1.06734e9i	−0.0407196	−	0.126231i
$687$		0			0
$688$	1.17232e9	−	3.50667e9i	0.137242	−	0.410521i
$689$		−	8.45466e9i		−	0.984756i
$690$		0			0
$691$	−7.00398e9			−0.807554			−0.403777	−	0.914857i	$-0.632303\pi$
$691$	−7.00398e9			−0.807554			−0.403777	+	0.914857i	$0.632303\pi$
$692$	1.01655e10	−	7.32005e9i	1.16616	−	0.839736i
$693$		0			0
$694$	−2.14891e9	−	6.66165e9i	−0.244040	−	0.756525i
$695$	1.86252e10			2.10453
$696$		0			0
$697$	4.69897e8			0.0525640
$698$	2.06339e8	+	6.39654e8i	0.0229661	+	0.0711952i
$699$		0			0
$700$	1.96635e10	+	2.73070e10i	2.16679	+	3.00906i
$701$	−4.72569e9			−0.518146			−0.259073	−	0.965858i	$-0.583417\pi$
$701$	−4.72569e9			−0.518146			−0.259073	+	0.965858i	$0.583417\pi$
$702$		0			0
$703$			1.71757e10i			1.86454i
$704$	1.92491e9	−	5.98505e8i	0.207925	−	0.0646492i
$705$		0			0
$706$	−4.60484e9	−	1.42751e10i	−0.492491	−	1.52673i
$707$		−	2.33065e9i		−	0.248033i
$708$		0			0
$709$			1.64991e10i			1.73859i	0.494292	+	0.869296i	$0.335427\pi$
$709$			1.64991e10i			1.73859i	−0.494292	+	0.869296i	$0.664573\pi$
$710$	4.70288e8	−	1.51705e8i	0.0493128	−	0.0159073i
$711$		0			0
$712$	−1.91256e9	−	1.40821e9i	−0.198580	−	0.146214i
$713$		−	2.12738e9i		−	0.219802i
$714$		0			0
$715$	−5.45531e9			−0.558146
$716$	1.94484e9	+	2.70083e9i	0.198011	+	0.274981i
$717$		0			0
$718$	8.80670e9	−	2.84086e9i	0.887927	−	0.286427i
$719$	−7.64417e9			−0.766971			−0.383486	−	0.923547i	$-0.625276\pi$
$719$	−7.64417e9			−0.766971			−0.383486	+	0.923547i	$0.625276\pi$
$720$		0			0
$721$	−1.09742e10			−1.09043
$722$	1.40680e10	−	4.53804e9i	1.39108	−	0.448734i
$723$		0			0
$724$	−6.04304e9	−	8.39207e9i	−0.591794	−	0.821834i
$725$	2.19270e10			2.13696
$726$		0			0
$727$		−	5.71439e9i		−	0.551568i	−0.961220	−	0.275784i	$-0.911062\pi$
$727$		−	5.71439e9i		−	0.551568i	0.961220	−	0.275784i	$-0.0889375\pi$
$728$	−1.21212e10	+	1.64624e10i	−1.16436	+	1.58137i
$729$		0			0
$730$	−1.66075e10	+	5.35723e9i	−1.58006	+	0.509695i
$731$		−	2.60520e9i		−	0.246678i
$732$		0			0
$733$		−	1.06519e10i		−	0.998997i	−0.866315	−	0.499499i	$-0.833518\pi$
$733$		−	1.06519e10i		−	0.998997i	0.866315	−	0.499499i	$-0.166482\pi$
$734$	−3.71139e9	−	1.15053e10i	−0.346418	−	1.07390i
$735$		0			0
$736$	1.74315e7	−	1.64580e9i	0.00161162	−	0.152161i
$737$		−	1.65994e9i		−	0.152742i
$738$		0			0
$739$	1.09195e10			0.995287			0.497643	−	0.867382i	$-0.334199\pi$
$739$	1.09195e10			0.995287			0.497643	+	0.867382i	$0.334199\pi$
$740$	−1.44511e10	−	2.00685e10i	−1.31097	−	1.82056i
$741$		0			0
$742$	−3.58336e9	−	1.11085e10i	−0.322016	−	0.998252i
$743$	−1.36427e9			−0.122022			−0.0610111	−	0.998137i	$-0.519433\pi$
$743$	−1.36427e9			−0.122022			−0.0610111	+	0.998137i	$0.519433\pi$
$744$		0			0
$745$	−1.19475e10			−1.05860
$746$	−1.03033e8	−	3.19403e8i	−0.00908634	−	0.0281678i
$747$		0			0
$748$	1.15260e9	−	8.29974e8i	0.100698	−	0.0725119i
$749$	−2.83580e10			−2.46597
$750$		0			0
$751$			6.79230e8i			0.0585164i	0.999572	+	0.0292582i	$0.00931450\pi$
$751$			6.79230e8i			0.0585164i	−0.999572	+	0.0292582i	$0.990686\pi$
$752$	−3.70858e9	+	1.10931e10i	−0.318014	+	0.951245i
$753$		0			0
$754$	4.08967e9	+	1.26780e10i	0.347447	+	1.07709i
$755$			4.84798e9i			0.409965i
$756$		0			0
$757$			1.77513e9i			0.148729i	0.997231	+	0.0743643i	$0.0236928\pi$
$757$			1.77513e9i			0.148729i	−0.997231	+	0.0743643i	$0.976307\pi$
$758$	1.07185e10	−	3.45757e9i	0.893907	−	0.288356i
$759$		0			0
$760$	−2.12527e10	+	2.88642e10i	−1.75617	+	2.38513i
$761$			2.17859e10i			1.79196i	0.444095	+	0.895980i	$0.353525\pi$
$761$			2.17859e10i			1.79196i	−0.444095	+	0.895980i	$0.646475\pi$
$762$		0			0
$763$	9.74927e9			0.794577
$764$	−1.19479e10	+	8.60356e9i	−0.969315	+	0.697993i
$765$		0			0
$766$	1.57342e10	−	5.07553e9i	1.26486	−	0.408019i
$767$	1.28021e10			1.02447
$768$		0			0
$769$	−1.13873e10			−0.902983			−0.451491	−	0.892276i	$-0.649108\pi$
$769$	−1.13873e10			−0.902983			−0.451491	+	0.892276i	$0.649108\pi$
$770$	−7.16766e9	+	2.31214e9i	−0.565796	+	0.182514i
$771$		0			0
$772$	−1.10878e9	+	7.98420e8i	−0.0867330	+	0.0624556i
$773$	−2.45052e10			−1.90823			−0.954114	−	0.299445i	$-0.903199\pi$
$773$	−2.45052e10			−1.90823			−0.954114	+	0.299445i	$0.903199\pi$
$774$		0			0
$775$		−	4.79898e10i		−	3.70334i
$776$	1.50515e8	−	2.04422e8i	0.0115628	−	0.0157040i
$777$		0			0
$778$	1.79963e10	−	5.80522e9i	1.37010	−	0.441967i
$779$		−	1.90939e9i		−	0.144715i
$780$		0			0
$781$			7.95647e7i			0.00597643i
$782$	−3.56026e8	−	1.10368e9i	−0.0266230	−	0.0825318i
$783$		0			0
$784$	−4.67046e9	+	1.39703e10i	−0.346142	+	1.03538i
$785$			7.86374e9i			0.580210i
$786$		0			0
$787$	1.76625e10			1.29164			0.645818	−	0.763492i	$-0.276517\pi$
$787$	1.76625e10			1.29164			0.645818	+	0.763492i	$0.276517\pi$
$788$	5.37092e9	−	3.86754e9i	0.391027	−	0.281575i
$789$		0			0
$790$	2.23155e9	+	6.91784e9i	0.161032	+	0.499201i
$791$	1.74156e10			1.25118
$792$		0			0
$793$	1.11016e10			0.790552
$794$	−1.64291e9	−	5.09303e9i	−0.116477	−	0.361081i
$795$		0			0
$796$	−1.58362e10	−	2.19920e10i	−1.11290	−	1.54550i
$797$	−1.92078e9			−0.134392			−0.0671960	−	0.997740i	$-0.521405\pi$
$797$	−1.92078e9			−0.134392			−0.0671960	+	0.997740i	$0.521405\pi$
$798$		0			0
$799$			8.24140e9i			0.571593i
$800$	3.93223e8	−	3.71263e10i	0.0271534	−	2.56370i
$801$		0			0
$802$	5.66805e9	+	1.75710e10i	0.387992	+	1.20278i
$803$		−	2.80971e9i		−	0.191495i
$804$		0			0
$805$			6.14928e9i			0.415469i
$806$	2.77474e10	−	8.95074e9i	1.86659	−	0.602124i
$807$		0			0
$808$	−1.52472e9	+	2.07080e9i	−0.101684	+	0.138101i
$809$			1.41023e10i			0.936422i	0.883617	+	0.468211i	$0.155101\pi$
$809$			1.41023e10i			0.936422i	−0.883617	+	0.468211i	$0.844899\pi$
$810$		0			0
$811$	−1.95320e10			−1.28580			−0.642899	−	0.765951i	$-0.722268\pi$
$811$	−1.95320e10			−1.28580			−0.642899	+	0.765951i	$0.722268\pi$
$812$	1.07467e10	+	1.49242e10i	0.704418	+	0.978236i
$813$		0			0
$814$	3.78959e9	−	1.22244e9i	0.246268	−	0.0794408i
$815$	−2.41109e10			−1.56013
$816$		0			0
$817$	−1.05860e10			−0.679135
$818$	−3.44658e9	+	1.11180e9i	−0.220167	+	0.0710213i
$819$		0			0
$820$	1.60651e9	+	2.23098e9i	0.101750	+	0.141302i
$821$	1.50736e10			0.950637			0.475319	−	0.879814i	$-0.342333\pi$
$821$	1.50736e10			0.950637			0.475319	+	0.879814i	$0.342333\pi$
$822$		0			0
$823$		−	2.14319e10i		−	1.34018i	−0.742282	−	0.670088i	$-0.766257\pi$
$823$		−	2.14319e10i		−	1.34018i	0.742282	−	0.670088i	$-0.233743\pi$
$824$	9.75061e9	+	7.17934e9i	0.607137	+	0.447033i
$825$		0			0
$826$	1.68205e10	−	5.42595e9i	1.03851	−	0.335001i
$827$			4.03725e9i			0.248208i	0.992269	+	0.124104i	$0.0396057\pi$
$827$			4.03725e9i			0.248208i	−0.992269	+	0.124104i	$0.960394\pi$
$828$		0			0
$829$		−	1.60911e10i		−	0.980946i	−0.871456	−	0.490473i	$-0.836824\pi$
$829$		−	1.60911e10i		−	0.980946i	0.871456	−	0.490473i	$-0.163176\pi$
$830$	2.55762e9	+	7.92865e9i	0.155261	+	0.481311i
$831$		0			0
$832$	2.15395e10	−	6.69719e9i	1.29659	−	0.403144i
$833$			1.03789e10i			0.622150i
$834$		0			0
$835$	−1.45512e10			−0.864960
$836$	−3.37254e9	−	4.68350e9i	−0.199634	−	0.277235i
$837$		0			0
$838$	−7.56966e9	−	2.34660e10i	−0.444347	−	1.37748i
$839$	−8.27045e9			−0.483462			−0.241731	−	0.970343i	$-0.577715\pi$
$839$	−8.27045e9			−0.483462			−0.241731	+	0.970343i	$0.577715\pi$
$840$		0			0
$841$	−5.26605e9			−0.305281
$842$	4.82742e8	+	1.49651e9i	0.0278691	+	0.0863945i
$843$		0			0
$844$	−3.41709e9	+	2.46061e9i	−0.195640	+	0.140879i
$845$	−2.79342e10			−1.59272
$846$		0			0
$847$			2.43639e10i			1.37770i
$848$	−4.08336e9	+	1.22142e10i	−0.229950	+	0.687827i
$849$		0			0
$850$	−8.03131e9	−	2.48972e10i	−0.448560	−	1.39054i
$851$		−	3.25117e9i		−	0.180836i
$852$		0			0
$853$			4.07261e9i			0.224673i	0.993670	+	0.112337i	$0.0358335\pi$
$853$			4.07261e9i			0.224673i	−0.993670	+	0.112337i	$0.964167\pi$
$854$	1.45863e10	−	4.70523e9i	0.801386	−	0.258511i
$855$		0			0
$856$	2.51962e10	+	1.85519e10i	1.37302	+	1.01095i
$857$			5.23793e9i			0.284267i	0.989847	+	0.142134i	$0.0453963\pi$
$857$			5.23793e9i			0.284267i	−0.989847	+	0.142134i	$0.954604\pi$
$858$		0			0
$859$	−1.68112e10			−0.904945			−0.452472	−	0.891778i	$-0.649458\pi$
$859$	−1.68112e10			−0.904945			−0.452472	+	0.891778i	$0.649458\pi$
$860$	1.23690e10	−	8.90677e9i	0.663115	−	0.477503i
$861$		0			0
$862$	2.69944e10	−	8.70783e9i	1.43548	−	0.463057i
$863$	2.03500e10			1.07777			0.538887	−	0.842378i	$-0.318845\pi$
$863$	2.03500e10			1.07777			0.538887	+	0.842378i	$0.318845\pi$
$864$		0			0
$865$	5.16397e10			2.71286
$866$	−8.17084e9	+	2.63575e9i	−0.427518	+	0.137908i
$867$		0			0
$868$	3.26633e10	−	2.35205e10i	1.69528	−	1.22075i
$869$	−1.17038e9			−0.0605004
$870$		0			0
$871$		−	1.85745e10i		−	0.952477i
$872$	−8.66228e9	−	6.37801e9i	−0.442410	−	0.325745i
$873$		0			0
$874$	−4.48474e9	+	1.44669e9i	−0.227220	+	0.0732966i
$875$			8.46117e10i			4.26975i
$876$		0			0
$877$			2.52596e10i			1.26453i	0.774754	+	0.632263i	$0.217874\pi$
$877$			2.52596e10i			1.26453i	−0.774754	+	0.632263i	$0.782126\pi$
$878$	9.13539e9	+	2.83198e10i	0.455508	+	1.41208i
$879$		0			0
$880$	7.88112e9	+	2.63476e9i	0.389851	+	0.130332i
$881$		−	1.32056e10i		−	0.650644i	−0.945603	−	0.325322i	$-0.894527\pi$
$881$		−	1.32056e10i		−	0.650644i	0.945603	−	0.325322i	$-0.105473\pi$
$882$		0			0
$883$	1.31436e10			0.642470			0.321235	−	0.947000i	$-0.395902\pi$
$883$	1.31436e10			0.642470			0.321235	+	0.947000i	$0.395902\pi$
$884$	1.28974e10	−	9.28730e9i	0.627942	−	0.452175i
$885$		0			0
$886$	5.66418e9	+	1.75590e10i	0.273602	+	0.848168i
$887$	−3.65129e9			−0.175677			−0.0878384	−	0.996135i	$-0.527996\pi$
$887$	−3.65129e9			−0.175677			−0.0878384	+	0.996135i	$0.527996\pi$
$888$		0			0
$889$	−4.31380e10			−2.05923
$890$	−3.00581e9	−	9.31804e9i	−0.142921	−	0.443057i
$891$		0			0
$892$	1.93313e10	+	2.68457e10i	0.911979	+	1.26648i
$893$	3.34883e10			1.57367
$894$		0			0
$895$			1.37200e10i			0.639695i
$896$	2.54620e10	−	1.79285e10i	1.18254	−	0.832657i
$897$		0			0
$898$	5.84014e9	+	1.81045e10i	0.269126	+	0.834295i
$899$		−	2.62280e10i		−	1.20395i
$900$		0			0
$901$			9.07425e9i			0.413308i
$902$	−4.21282e8	+	1.35897e8i	−0.0191139	+	0.00616576i
$903$		0			0
$904$	−1.54738e10	−	1.13933e10i	−0.696640	−	0.512934i
$905$		−	4.26309e10i		−	1.91185i
$906$		0			0
$907$	2.78144e9			0.123778			0.0618890	−	0.998083i	$-0.480288\pi$
$907$	2.78144e9			0.123778			0.0618890	+	0.998083i	$0.480288\pi$
$908$	−7.32100e9	−	1.01668e10i	−0.324541	−	0.450695i
$909$		0			0
$910$	−8.02051e10	+	2.58725e10i	−3.52823	+	1.13813i
$911$	−2.30001e10			−1.00789			−0.503947	−	0.863735i	$-0.668119\pi$
$911$	−2.30001e10			−1.00789			−0.503947	+	0.863735i	$0.668119\pi$
$912$		0			0
$913$	−1.34139e9			−0.0583322
$914$	6.79117e9	−	2.19069e9i	0.294193	−	0.0949007i
$915$		0			0
$916$	−2.27536e9	−	3.15983e9i	−0.0978174	−	0.135841i
$917$	3.80319e10			1.62875
$918$		0			0
$919$			6.10760e9i			0.259577i	0.991542	+	0.129789i	$0.0414299\pi$
$919$			6.10760e9i			0.259577i	−0.991542	+	0.129789i	$0.958570\pi$
$920$	4.02288e9	−	5.46367e9i	0.170326	−	0.231327i
$921$		0			0
$922$	−1.55239e10	+	5.00769e9i	−0.652292	+	0.210416i
$923$			8.90318e8i			0.0372683i
$924$		0			0
$925$		−	7.33406e10i		−	3.04683i
$926$	−4.05790e9	−	1.25796e10i	−0.167944	−	0.520627i
$927$		0			0
$928$	2.14909e8	−	2.02908e10i	0.00882750	−	0.833452i
$929$		−	3.44541e10i		−	1.40989i	−0.709260	−	0.704947i	$-0.750971\pi$
$929$		−	3.44541e10i		−	1.40989i	0.709260	−	0.704947i	$-0.249029\pi$
$930$		0			0
$931$	4.21740e10			1.71286
$932$	2.26969e10	+	3.15196e10i	0.918357	+	1.27534i
$933$		0			0
$934$	3.11559e9	+	9.65837e9i	0.125120	+	0.387873i
$935$	5.85509e9			0.234257
$936$		0			0
$937$	3.31829e10			1.31773			0.658864	−	0.752262i	$-0.271037\pi$
$937$	3.31829e10			1.31773			0.658864	+	0.752262i	$0.271037\pi$
$938$	−7.87251e9	−	2.44049e10i	−0.311461	−	0.965531i
$939$		0			0
$940$	−3.91286e10	+	2.81761e10i	−1.53655	+	1.10645i
$941$	−2.21421e10			−0.866274			−0.433137	−	0.901328i	$-0.642593\pi$
$941$	−2.21421e10			−0.866274			−0.433137	+	0.901328i	$0.642593\pi$
$942$		0			0
$943$			3.61426e8i			0.0140355i
$944$	−1.84948e10	−	6.18306e9i	−0.715563	−	0.239222i
$945$		0			0
$946$	7.53437e8	+	2.33566e9i	0.0289353	+	0.0896998i
$947$		−	2.78393e10i		−	1.06521i	−0.846365	−	0.532603i	$-0.821214\pi$
$947$		−	2.78393e10i		−	1.06521i	0.846365	−	0.532603i	$-0.178786\pi$
$948$		0			0
$949$		−	3.14402e10i		−	1.19414i
$950$	−1.01168e11	+	3.26347e10i	−3.82833	+	1.23494i
$951$		0			0
$952$	1.30095e10	−	1.76688e10i	0.488687	−	0.663709i
$953$			4.27491e10i			1.59993i	0.600044	+	0.799967i	$0.295150\pi$
$953$			4.27491e10i			1.59993i	−0.600044	+	0.799967i	$0.704850\pi$
$954$		0			0
$955$	−6.06942e10			−2.25494
$956$	−2.56131e10	+	1.84437e10i	−0.948113	+	0.682726i
$957$		0			0
$958$	8.95623e9	−	2.88910e9i	0.329114	−	0.106165i
$959$	−6.56709e10			−2.40441
$960$		0			0
$961$	−2.98906e10			−1.08643
$962$	4.24050e10	−	1.36790e10i	1.53569	−	0.495383i
$963$		0			0
$964$	−3.36802e10	+	2.42528e10i	−1.21089	+	0.871950i
$965$	−5.63250e9			−0.201769
$966$		0			0
$967$			3.47491e10i			1.23581i	0.786254	+	0.617904i	$0.212018\pi$
$967$			3.47491e10i			1.23581i	−0.786254	+	0.617904i	$0.787982\pi$
$968$	1.59390e10	−	2.16475e10i	0.564803	−	0.767086i
$969$		0			0
$970$	9.95947e8	−	3.21272e8i	0.0350377	−	0.0113024i
$971$			1.10930e10i			0.388849i	0.980917	+	0.194425i	$0.0622840\pi$
$971$			1.10930e10i			0.388849i	−0.980917	+	0.194425i	$0.937716\pi$
$972$		0			0
$973$			4.63278e10i			1.61230i
$974$	3.48641e9	+	1.08079e10i	0.120899	+	0.374788i
$975$		0			0
$976$	−1.60382e10	−	5.36177e9i	−0.552180	−	0.184601i
$977$			4.68888e10i			1.60856i	0.594248	+	0.804282i	$0.297450\pi$
$977$			4.68888e10i			1.60856i	−0.594248	+	0.804282i	$0.702550\pi$
$978$		0			0
$979$	1.57645e9			0.0536960
$980$	−4.92772e10	+	3.54840e10i	−1.67246	+	1.20432i
$981$		0			0
$982$	2.20781e8	+	6.84425e8i	0.00743999	+	0.0230641i
$983$	3.96515e10			1.33144			0.665721	−	0.746200i	$-0.268124\pi$
$983$	3.96515e10			1.33144			0.665721	+	0.746200i	$0.268124\pi$
$984$		0			0
$985$	2.72838e10			0.909657
$986$	−4.38938e9	−	1.36071e10i	−0.145826	−	0.452061i
$987$		0			0
$988$	−3.77383e10	−	5.24077e10i	−1.24489	−	1.72880i
$989$	2.00381e9			0.0658674
$990$		0			0
$991$			1.80562e10i			0.589344i	0.955598	+	0.294672i	$0.0952104\pi$
$991$			1.80562e10i			0.589344i	−0.955598	+	0.294672i	$0.904790\pi$
$992$	−4.44088e10	−	4.70355e8i	−1.44437	−	0.0152980i
$993$		0			0
$994$	3.77346e8	+	1.16978e9i	0.0121867	+	0.0377790i
$995$		−	1.11717e11i		−	3.59534i
$996$		0			0
$997$			2.59063e10i			0.827891i	0.910302	+	0.413945i	$0.135850\pi$
$997$			2.59063e10i			0.827891i	−0.910302	+	0.413945i	$0.864150\pi$
$998$	−3.94795e10	+	1.27353e10i	−1.25723	+	0.405556i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	72.8.f.a.35.25	yes	28
3.2	odd	2	inner	72.8.f.a.35.4	yes	28
4.3	odd	2		288.8.f.a.143.27		28
8.3	odd	2	inner	72.8.f.a.35.3	✓	28
8.5	even	2		288.8.f.a.143.2		28
12.11	even	2		288.8.f.a.143.1		28
24.5	odd	2		288.8.f.a.143.28		28
24.11	even	2	inner	72.8.f.a.35.26	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.8.f.a.35.3	✓	28	8.3	odd	2	inner
72.8.f.a.35.4	yes	28	3.2	odd	2	inner
72.8.f.a.35.25	yes	28	1.1	even	1	trivial
72.8.f.a.35.26	yes	28	24.11	even	2	inner
288.8.f.a.143.1		28	12.11	even	2
288.8.f.a.143.2		28	8.5	even	2
288.8.f.a.143.27		28	4.3	odd	2
288.8.f.a.143.28		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)

\(f(q)\)	\(=\)	\(q+(10.7674 - 3.47333i) q^{2} +(103.872 - 74.7971i) q^{4} +527.660 q^{5} +1312.48i q^{7} +(858.632 - 1166.15i) q^{8} +O(q^{10})\)	\(q+(10.7674 - 3.47333i) q^{2} +(103.872 - 74.7971i) q^{4} +527.660 q^{5} +1312.48i q^{7} +(858.632 - 1166.15i) q^{8} +(5681.50 - 1832.74i) q^{10} +961.214i q^{11} +10755.9i q^{13} +(4558.69 + 14132.0i) q^{14} +(5194.78 - 15538.7i) q^{16} -11544.1i q^{17} -46908.6 q^{19} +(54809.1 - 39467.5i) q^{20} +(3338.61 + 10349.7i) q^{22} +8879.25 q^{23} +200300. q^{25} +(37358.6 + 115812. i) q^{26} +(98170.0 + 136330. i) q^{28} +109471. q^{29} -239590. i q^{31} +(1963.17 - 185353. i) q^{32} +(-40096.4 - 124299. i) q^{34} +692545. i q^{35} -366153. i q^{37} +(-505081. + 162929. i) q^{38} +(453066. - 615330. i) q^{40} +40704.6i q^{41} +225674. q^{43} +(71896.1 + 99843.2i) q^{44} +(95606.0 - 30840.5i) q^{46} -713906. q^{47} -899069. q^{49} +(2.15670e6 - 695708. i) q^{50} +(804507. + 1.11723e6i) q^{52} -786052. q^{53} +507194. i q^{55} +(1.53055e6 + 1.12694e6i) q^{56} +(1.17871e6 - 380228. i) q^{58} -1.19024e6i q^{59} -1.03215e6i q^{61} +(-832174. - 2.57975e6i) q^{62} +(-622655. - 2.00259e6i) q^{64} +5.67543e6i q^{65} -1.72692e6 q^{67} +(-863464. - 1.19911e6i) q^{68} +(2.40544e6 + 7.45688e6i) q^{70} +82775.2 q^{71} -2.92308e6 q^{73} +(-1.27177e6 - 3.94251e6i) q^{74} +(-4.87248e6 + 3.50863e6i) q^{76} -1.26158e6 q^{77} +1.21761e6i q^{79} +(2.74108e6 - 8.19913e6i) q^{80} +(141380. + 438281. i) q^{82} +1.39552e6i q^{83} -6.09135e6i q^{85} +(2.42991e6 - 783839. i) q^{86} +(1.12092e6 + 825329. i) q^{88} -1.64007e6i q^{89} -1.41169e7 q^{91} +(922305. - 664142. i) q^{92} +(-7.68689e6 + 2.47963e6i) q^{94} -2.47518e7 q^{95} +175296. q^{97} +(-9.68060e6 + 3.12276e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 52 q^{4}+O(q^{10}) \) 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

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