LMFDB - Embedded newform 112.10.p.b.31.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,10,Mod(31,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("112.31");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.p (of order $6$, degree $2$, 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:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.8
Character	$\chi$	$=$	112.31
Dual form			112.10.p.b.47.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(41.2793 - 71.4979i) q^{3} +(-557.845 + 322.072i) q^{5} +(3078.62 - 5556.59i) q^{7} +(6433.53 + 11143.2i) q^{9} +O(q^{10})$	\(q+(41.2793 - 71.4979i) q^{3} +(-557.845 + 322.072i) q^{5} +(3078.62 - 5556.59i) q^{7} +(6433.53 + 11143.2i) q^{9} +(-16615.5 - 9592.95i) q^{11} +86715.5i q^{13} +53179.7i q^{15} +(241549. + 139458. i) q^{17} +(-217224. - 376243. i) q^{19} +(-270201. - 449487. i) q^{21} +(-852491. + 492186. i) q^{23} +(-769101. + 1.33212e6i) q^{25} +2.68729e6 q^{27} +2.79727e6 q^{29} +(912073. - 1.57976e6i) q^{31} +(-1.37175e6 + 791981. i) q^{33} +(72229.0 + 4.09126e6i) q^{35} +(6.88423e6 + 1.19238e7i) q^{37} +(6.19998e6 + 3.57956e6i) q^{39} +7.10745e6i q^{41} -8.93184e6i q^{43} +(-7.17783e6 - 4.14412e6i) q^{45} +(1.14962e7 + 1.99119e7i) q^{47} +(-2.13978e7 - 3.42133e7i) q^{49} +(1.99419e7 - 1.15135e7i) q^{51} +(2.88806e7 - 5.00227e7i) q^{53} +1.23585e7 q^{55} -3.58675e7 q^{57} +(3.90730e7 - 6.76765e7i) q^{59} +(-1.48740e6 + 858751. i) q^{61} +(8.17246e7 - 1.44281e6i) q^{63} +(-2.79287e7 - 4.83739e7i) q^{65} +(2.44149e8 + 1.40959e8i) q^{67} +8.12684e7i q^{69} +1.63814e8i q^{71} +(1.74661e8 + 1.00840e8i) q^{73} +(6.34960e7 + 1.09978e8i) q^{75} +(-1.04457e8 + 6.27924e7i) q^{77} +(3.99982e8 - 2.30930e8i) q^{79} +(-1.57016e7 + 2.71960e7i) q^{81} +7.74335e8 q^{83} -1.79662e8 q^{85} +(1.15469e8 - 1.99999e8i) q^{87} +(5.30912e8 - 3.06522e8i) q^{89} +(4.81843e8 + 2.66964e8i) q^{91} +(-7.52996e7 - 1.30423e8i) q^{93} +(2.42355e8 + 1.39924e8i) q^{95} +1.06547e9i q^{97} -2.46866e8i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 1704 q^{5} - 80428 q^{9}+O(q^{10}) $ 24 * q + 1704 * q^5 - 80428 * q^9	$ 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57} - 593611308 q^{61} + 160281180 q^{65} - 1676922516 q^{73} + 1645751688 q^{77} - 528698056 q^{81} + 1370082456 q^{85} - 941603484 q^{89} - 1432348316 q^{93}+O(q^{100}) $ 24 * q + 1704 * q^5 - 80428 * q^9 - 1578672 * q^17 + 1219540 * q^21 + 8001240 * q^25 - 6709416 * q^29 - 29129772 * q^33 - 11130084 * q^37 + 57023292 * q^45 - 12671904 * q^49 - 93652164 * q^53 + 742621544 * q^57 - 593611308 * q^61 + 160281180 * q^65 - 1676922516 * q^73 + 1645751688 * q^77 - 528698056 * q^81 + 1370082456 * q^85 - 941603484 * q^89 - 1432348316 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	41.2793	−	71.4979i	0.294230	−	0.509622i	−0.680575	−	0.732678i	$-0.738270\pi$
$3$	41.2793	−	71.4979i	0.294230	−	0.509622i	0.974806	+	0.223057i	$0.0716035\pi$
$4$		0			0
$5$	−557.845	+	322.072i	−0.399162	+	0.230456i	−0.686122	−	0.727486i	$-0.740688\pi$
$5$	−557.845	+	322.072i	−0.399162	+	0.230456i	0.286960	+	0.957942i	$0.407355\pi$
$6$		0			0
$7$	3078.62	−	5556.59i	0.484635	−	0.874716i
$7$	3078.62	−	5556.59i	0.484635	−	0.874716i
$8$		0			0
$9$	6433.53	+	11143.2i	0.326857	+	0.566133i
$10$		0			0
$11$	−16615.5	−	9592.95i	−0.342173	−	0.197554i	0.319060	−	0.947735i	$-0.396633\pi$
$11$	−16615.5	−	9592.95i	−0.342173	−	0.197554i	−0.661233	+	0.750181i	$0.729966\pi$
$12$		0			0
$13$			86715.5i			0.842077i	0.907043	+	0.421038i	$0.138334\pi$
$13$			86715.5i			0.842077i	−0.907043	+	0.421038i	$0.861666\pi$
$14$		0			0
$15$			53179.7i			0.271229i
$16$		0			0
$17$	241549.	+	139458.i	0.701430	+	0.404971i	0.807880	−	0.589347i	$-0.200615\pi$
$17$	241549.	+	139458.i	0.701430	+	0.404971i	−0.106450	+	0.994318i	$0.533948\pi$
$18$		0			0
$19$	−217224.	−	376243.i	−0.382399	−	0.662335i	0.609005	−	0.793166i	$-0.291569\pi$
$19$	−217224.	−	376243.i	−0.382399	−	0.662335i	−0.991405	+	0.130831i	$0.958235\pi$
$20$		0			0
$21$	−270201.	−	449487.i	−0.303180	−	0.504348i
$22$		0			0
$23$	−852491.	+	492186.i	−0.635206	+	0.366736i	−0.782765	−	0.622317i	$-0.786192\pi$
$23$	−852491.	+	492186.i	−0.635206	+	0.366736i	0.147560	+	0.989053i	$0.452858\pi$
$24$		0			0
$25$	−769101.	+	1.33212e6i	−0.393780	+	0.682047i
$26$		0			0
$27$	2.68729e6			0.973145
$28$		0			0
$29$	2.79727e6			0.734418			0.367209	−	0.930138i	$-0.380313\pi$
$29$	2.79727e6			0.734418			0.367209	+	0.930138i	$0.380313\pi$
$30$		0			0
$31$	912073.	−	1.57976e6i	0.177379	−	0.307229i	−0.763603	−	0.645686i	$-0.776572\pi$
$31$	912073.	−	1.57976e6i	0.177379	−	0.307229i	0.940982	+	0.338457i	$0.109905\pi$
$32$		0			0
$33$	−1.37175e6	+	791981.i	−0.201355	+	0.116252i
$34$		0			0
$35$	72229.0	+	4.09126e6i	0.00813589	+	0.460840i
$36$		0			0
$37$	6.88423e6	+	1.19238e7i	0.603876	+	1.04594i	0.992228	+	0.124432i	$0.0397110\pi$
$37$	6.88423e6	+	1.19238e7i	0.603876	+	1.04594i	−0.388352	+	0.921511i	$0.626956\pi$
$38$		0			0
$39$	6.19998e6	+	3.57956e6i	0.429141	+	0.247764i
$40$		0			0
$41$			7.10745e6i			0.392814i	0.980522	+	0.196407i	$0.0629273\pi$
$41$			7.10745e6i			0.392814i	−0.980522	+	0.196407i	$0.937073\pi$
$42$		0			0
$43$		−	8.93184e6i		−	0.398412i	−0.979958	−	0.199206i	$-0.936164\pi$
$43$		−	8.93184e6i		−	0.398412i	0.979958	−	0.199206i	$-0.0638363\pi$
$44$		0			0
$45$	−7.17783e6	−	4.14412e6i	−0.260938	−	0.150653i
$46$		0			0
$47$	1.14962e7	+	1.99119e7i	0.343647	+	0.595214i	0.985107	−	0.171943i	$-0.0550044\pi$
$47$	1.14962e7	+	1.99119e7i	0.343647	+	0.595214i	−0.641460	+	0.767156i	$0.721671\pi$
$48$		0			0
$49$	−2.13978e7	−	3.42133e7i	−0.530257	−	0.847837i
$50$		0			0
$51$	1.99419e7	−	1.15135e7i	0.412764	−	0.238309i
$52$		0			0
$53$	2.88806e7	−	5.00227e7i	0.502765	−	0.870815i	−0.497230	−	0.867619i	$-0.665649\pi$
$53$	2.88806e7	−	5.00227e7i	0.502765	−	0.870815i	0.999995	−	0.00319570i	$-0.00101722\pi$
$54$		0			0
$55$	1.23585e7			0.182110
$56$		0			0
$57$	−3.58675e7			−0.450054
$58$		0			0
$59$	3.90730e7	−	6.76765e7i	0.419801	−	0.727116i	−0.576118	−	0.817366i	$-0.695433\pi$
$59$	3.90730e7	−	6.76765e7i	0.419801	−	0.727116i	0.995919	+	0.0902501i	$0.0287666\pi$
$60$		0			0
$61$	−1.48740e6	+	858751.i	−0.0137545	+	0.00794115i	−0.506861	−	0.862028i	$-0.669194\pi$
$61$	−1.48740e6	+	858751.i	−0.0137545	+	0.00794115i	0.493107	+	0.869969i	$0.335861\pi$
$62$		0			0
$63$	8.17246e7	−	1.44281e6i	0.653613	−	0.0115392i
$64$		0			0
$65$	−2.79287e7	−	4.83739e7i	−0.194062	−	0.336125i
$66$		0			0
$67$	2.44149e8	+	1.40959e8i	1.48019	+	0.854589i	0.999748	−	0.0224363i	$-0.00714230\pi$
$67$	2.44149e8	+	1.40959e8i	1.48019	+	0.854589i	0.480444	+	0.877026i	$0.340476\pi$
$68$		0			0
$69$			8.12684e7i			0.431619i
$70$		0			0
$71$			1.63814e8i			0.765048i	0.923945	+	0.382524i	$0.124945\pi$
$71$			1.63814e8i			0.765048i	−0.923945	+	0.382524i	$0.875055\pi$
$72$		0			0
$73$	1.74661e8	+	1.00840e8i	0.719850	+	0.415606i	0.814697	−	0.579886i	$-0.196903\pi$
$73$	1.74661e8	+	1.00840e8i	0.719850	+	0.415606i	−0.0948475	+	0.995492i	$0.530236\pi$
$74$		0			0
$75$	6.34960e7	+	1.09978e8i	0.231724	+	0.401357i
$76$		0			0
$77$	−1.04457e8	+	6.27924e7i	−0.338633	+	0.203563i
$78$		0			0
$79$	3.99982e8	−	2.30930e8i	1.15536	−	0.667050i	0.205176	−	0.978725i	$-0.434223\pi$
$79$	3.99982e8	−	2.30930e8i	1.15536	−	0.667050i	0.950189	+	0.311675i	$0.100890\pi$
$80$		0			0
$81$	−1.57016e7	+	2.71960e7i	−0.0405287	+	0.0701977i
$82$		0			0
$83$	7.74335e8			1.79092			0.895462	−	0.445137i	$-0.146845\pi$
$83$	7.74335e8			1.79092			0.895462	+	0.445137i	$0.146845\pi$
$84$		0			0
$85$	−1.79662e8			−0.373312
$86$		0			0
$87$	1.15469e8	−	1.99999e8i	0.216088	−	0.374275i
$88$		0			0
$89$	5.30912e8	−	3.06522e8i	0.896949	−	0.517854i	0.0207398	−	0.999785i	$-0.493398\pi$
$89$	5.30912e8	−	3.06522e8i	0.896949	−	0.517854i	0.876209	+	0.481931i	$0.160065\pi$
$90$		0			0
$91$	4.81843e8	+	2.66964e8i	0.736578	+	0.408100i
$92$		0			0
$93$	−7.52996e7	−	1.30423e8i	−0.104380	−	0.180792i
$94$		0			0
$95$	2.42355e8	+	1.39924e8i	0.305278	+	0.176253i
$96$		0			0
$97$			1.06547e9i			1.22200i	0.791632	+	0.610998i	$0.209232\pi$
$97$			1.06547e9i			1.22200i	−0.791632	+	0.610998i	$0.790768\pi$
$98$		0			0
$99$		−	2.46866e8i		−	0.258287i
$100$		0			0
$101$	−7.18595e8	−	4.14881e8i	−0.687129	−	0.396714i	0.115407	−	0.993318i	$-0.463183\pi$
$101$	−7.18595e8	−	4.14881e8i	−0.687129	−	0.396714i	−0.802535	+	0.596604i	$0.796516\pi$
$102$		0			0
$103$	6.55579e8	+	1.13550e9i	0.573928	+	0.994072i	0.996157	+	0.0875831i	$0.0279143\pi$
$103$	6.55579e8	+	1.13550e9i	0.573928	+	0.994072i	−0.422229	+	0.906489i	$0.638752\pi$
$104$		0			0
$105$	2.95498e8	+	1.63720e8i	0.237248	+	0.131447i
$106$		0			0
$107$	5.11073e8	−	2.95068e8i	0.376926	−	0.217618i	−0.299554	−	0.954079i	$-0.596838\pi$
$107$	5.11073e8	−	2.95068e8i	0.376926	−	0.217618i	0.676480	+	0.736461i	$0.263505\pi$
$108$		0			0
$109$	−1.85531e8	+	3.21349e8i	−0.125892	+	0.218051i	−0.922081	−	0.386997i	$-0.873512\pi$
$109$	−1.85531e8	+	3.21349e8i	−0.125892	+	0.218051i	0.796189	+	0.605047i	$0.206846\pi$
$110$		0			0
$111$	1.13671e9			0.710714
$112$		0			0
$113$	2.06081e9			1.18901			0.594505	−	0.804092i	$-0.297348\pi$
$113$	2.06081e9			1.18901			0.594505	+	0.804092i	$0.297348\pi$
$114$		0			0
$115$	3.17039e8	−	5.49127e8i	0.169033	−	0.292774i
$116$		0			0
$117$	−9.66289e8	+	5.57887e8i	−0.476728	+	0.275239i
$118$		0			0
$119$	1.51855e9	−	9.12848e8i	0.694172	−	0.417289i
$120$		0			0
$121$	−9.94924e8	−	1.72326e9i	−0.421945	−	0.730830i
$122$		0			0
$123$	5.08168e8	+	2.93391e8i	0.200186	+	0.115578i
$124$		0			0
$125$		−	2.24892e9i		−	0.823908i
$126$		0			0
$127$			2.24612e9i			0.766153i	0.923717	+	0.383077i	$0.125136\pi$
$127$			2.24612e9i			0.766153i	−0.923717	+	0.383077i	$0.874864\pi$
$128$		0			0
$129$	−6.38608e8	−	3.68700e8i	−0.203039	−	0.117225i
$130$		0			0
$131$	−1.72041e8	−	2.97984e8i	−0.0510401	−	0.0884041i	0.839377	−	0.543550i	$-0.182920\pi$
$131$	−1.72041e8	−	2.97984e8i	−0.0510401	−	0.0884041i	−0.890417	+	0.455146i	$0.849587\pi$
$132$		0			0
$133$	−2.75938e9	+	4.87154e7i	−0.764679	+	0.0135000i
$134$		0			0
$135$	−1.49909e9	+	8.65502e8i	−0.388442	+	0.224267i
$136$		0			0
$137$	1.27310e9	−	2.20508e9i	0.308760	−	0.534788i	−0.669332	−	0.742964i	$-0.733419\pi$
$137$	1.27310e9	−	2.20508e9i	0.308760	−	0.534788i	0.978091	+	0.208176i	$0.0667527\pi$
$138$		0			0
$139$	−2.65774e9			−0.603874			−0.301937	−	0.953328i	$-0.597633\pi$
$139$	−2.65774e9			−0.603874			−0.301937	+	0.953328i	$0.597633\pi$
$140$		0			0
$141$	1.89822e9			0.404445
$142$		0			0
$143$	8.31858e8	−	1.44082e9i	0.166355	−	0.288136i
$144$		0			0
$145$	−1.56044e9	+	9.00923e8i	−0.293152	+	0.169251i
$146$		0			0
$147$	−3.32946e9	+	1.17597e8i	−0.588094	+	0.0207714i
$148$		0			0
$149$	1.64386e9	+	2.84725e9i	0.273229	+	0.473246i	0.969687	−	0.244352i	$-0.0785751\pi$
$149$	1.64386e9	+	2.84725e9i	0.273229	+	0.473246i	−0.696458	+	0.717597i	$0.745242\pi$
$150$		0			0
$151$	−8.35894e9	−	4.82603e9i	−1.30844	−	0.755430i	−0.326607	−	0.945160i	$-0.605905\pi$
$151$	−8.35894e9	−	4.82603e9i	−1.30844	−	0.755430i	−0.981836	+	0.189731i	$0.939239\pi$
$152$		0			0
$153$			3.58883e9i			0.529471i
$154$		0			0
$155$			1.17501e9i			0.163512i
$156$		0			0
$157$	4.02105e9	+	2.32156e9i	0.528191	+	0.304951i	0.740280	−	0.672299i	$-0.234693\pi$
$157$	4.02105e9	+	2.32156e9i	0.528191	+	0.304951i	−0.212088	+	0.977251i	$0.568026\pi$
$158$		0			0
$159$	−2.38435e9	−	4.12981e9i	−0.295857	−	0.512440i
$160$		0			0
$161$	1.10379e8	+	6.25220e9i	0.0129470	+	0.733358i
$162$		0			0
$163$	−5.89019e9	+	3.40070e9i	−0.653560	+	0.377333i	−0.789819	−	0.613340i	$-0.789825\pi$
$163$	−5.89019e9	+	3.40070e9i	−0.653560	+	0.377333i	0.136259	+	0.990673i	$0.456492\pi$
$164$		0			0
$165$	5.10150e8	−	8.83607e8i	0.0535822	−	0.0928071i
$166$		0			0
$167$	−1.50462e10			−1.49693			−0.748466	−	0.663174i	$-0.769209\pi$
$167$	−1.50462e10			−1.49693			−0.748466	+	0.663174i	$0.769209\pi$
$168$		0			0
$169$	3.08492e9			0.290906
$170$		0			0
$171$	2.79504e9	−	4.84115e9i	0.249980	−	0.432978i
$172$		0			0
$173$	−2.84836e9	+	1.64450e9i	−0.241762	+	0.139581i	−0.615986	−	0.787757i	$-0.711242\pi$
$173$	−2.84836e9	+	1.64450e9i	−0.241762	+	0.139581i	0.374224	+	0.927338i	$0.377909\pi$
$174$		0			0
$175$	5.03429e9	+	8.37468e9i	0.405758	+	0.674990i
$176$		0			0
$177$	−3.22582e9	−	5.58728e9i	−0.247036	−	0.427879i
$178$		0			0
$179$	−1.47876e10	−	8.53762e9i	−1.07661	−	0.621582i	−0.146631	−	0.989191i	$-0.546843\pi$
$179$	−1.47876e10	−	8.53762e9i	−1.07661	−	0.621582i	−0.929980	+	0.367610i	$0.880176\pi$
$180$		0			0
$181$			2.33064e9i			0.161407i	0.996738	+	0.0807034i	$0.0257166\pi$
$181$			2.33064e9i			0.161407i	−0.996738	+	0.0807034i	$0.974283\pi$
$182$		0			0
$183$			1.41795e8i			0.00934610i
$184$		0			0
$185$	−7.68067e9	−	4.43444e9i	−0.482088	−	0.278334i
$186$		0			0
$187$	−2.67563e9	−	4.63433e9i	−0.160007	−	0.277140i
$188$		0			0
$189$	8.27315e9	−	1.49322e10i	0.471620	−	0.851226i
$190$		0			0
$191$	−2.26841e10	+	1.30967e10i	−1.23331	+	0.712052i	−0.967718	−	0.252034i	$-0.918901\pi$
$191$	−2.26841e10	+	1.30967e10i	−1.23331	+	0.712052i	−0.265592	+	0.964086i	$0.585567\pi$
$192$		0			0
$193$	−3.27470e9	+	5.67195e9i	−0.169889	+	0.294256i	−0.938381	−	0.345604i	$-0.887674\pi$
$193$	−3.27470e9	+	5.67195e9i	−0.169889	+	0.294256i	0.768492	+	0.639859i	$0.221007\pi$
$194$		0			0
$195$	−4.61151e9			−0.228395
$196$		0			0
$197$	−1.61293e10			−0.762987			−0.381493	−	0.924372i	$-0.624590\pi$
$197$	−1.61293e10			−0.762987			−0.381493	+	0.924372i	$0.624590\pi$
$198$		0			0
$199$	−3.07349e9	+	5.32344e9i	−0.138929	+	0.240632i	−0.927091	−	0.374835i	$-0.877699\pi$
$199$	−3.07349e9	+	5.32344e9i	−0.138929	+	0.240632i	0.788163	+	0.615467i	$0.211033\pi$
$200$		0			0
$201$	2.01566e10	−	1.16374e10i	0.871034	−	0.502892i
$202$		0			0
$203$	8.61173e9	−	1.55433e10i	0.355925	−	0.642408i
$204$		0			0
$205$	−2.28911e9	−	3.96486e9i	−0.0905263	−	0.156796i
$206$		0			0
$207$	−1.09691e10	−	6.33299e9i	−0.415243	−	0.239741i
$208$		0			0
$209$			8.33529e9i			0.302178i
$210$		0			0
$211$			1.27006e10i			0.441115i	0.975374	+	0.220557i	$0.0707877\pi$
$211$			1.27006e10i			0.441115i	−0.975374	+	0.220557i	$0.929212\pi$
$212$		0			0
$213$	1.17124e10	+	6.76214e9i	0.389885	+	0.225100i
$214$		0			0
$215$	2.87670e9	+	4.98259e9i	0.0918166	+	0.159031i
$216$		0			0
$217$	−5.97014e9	−	9.93149e9i	−0.182774	−	0.304050i
$218$		0			0
$219$	1.44197e10	−	8.32524e9i	0.423603	−	0.244567i
$220$		0			0
$221$	−1.20932e10	+	2.09460e10i	−0.341017	+	0.590658i
$222$		0			0
$223$	−8.07397e9			−0.218633			−0.109316	−	0.994007i	$-0.534866\pi$
$223$	−8.07397e9			−0.218633			−0.109316	+	0.994007i	$0.534866\pi$
$224$		0			0
$225$	−1.97922e10			−0.514839
$226$		0			0
$227$	3.09567e10	−	5.36185e10i	0.773816	−	1.34029i	−0.161641	−	0.986850i	$-0.551679\pi$
$227$	3.09567e10	−	5.36185e10i	0.773816	−	1.34029i	0.935457	−	0.353439i	$-0.114988\pi$
$228$		0			0
$229$	4.30471e9	−	2.48532e9i	0.103439	−	0.0597205i	−0.447388	−	0.894340i	$-0.647646\pi$
$229$	4.30471e9	−	2.48532e9i	0.103439	−	0.0597205i	0.550827	+	0.834619i	$0.314312\pi$
$230$		0			0
$231$	1.77612e8	+	1.00605e10i	0.00410411	+	0.232469i
$232$		0			0
$233$	4.86434e9	+	8.42528e9i	0.108124	+	0.187276i	0.915010	−	0.403431i	$-0.132182\pi$
$233$	4.86434e9	+	8.42528e9i	0.108124	+	0.187276i	−0.806886	+	0.590707i	$0.798849\pi$
$234$		0			0
$235$	−1.28262e10	−	7.40519e9i	−0.274341	−	0.158391i
$236$		0			0
$237$		−	3.81305e10i		−	0.785065i
$238$		0			0
$239$			6.07667e10i			1.20469i	0.798236	+	0.602345i	$0.205767\pi$
$239$			6.07667e10i			1.20469i	−0.798236	+	0.602345i	$0.794233\pi$
$240$		0			0
$241$	7.04935e9	+	4.06995e9i	0.134609	+	0.0777163i	0.565792	−	0.824548i	$-0.308571\pi$
$241$	7.04935e9	+	4.06995e9i	0.134609	+	0.0777163i	−0.431183	+	0.902264i	$0.641904\pi$
$242$		0			0
$243$	2.77433e10	+	4.80528e10i	0.510422	+	0.884077i
$244$		0			0
$245$	2.29558e10	+	1.21941e10i	0.407048	+	0.216223i
$246$		0			0
$247$	3.26261e10	−	1.88367e10i	0.557737	−	0.322010i
$248$		0			0
$249$	3.19640e10	−	5.53633e10i	0.526944	−	0.912694i
$250$		0			0
$251$	9.71124e10			1.54434			0.772170	−	0.635416i	$-0.219171\pi$
$251$	9.71124e10			1.54434			0.772170	+	0.635416i	$0.219171\pi$
$252$		0			0
$253$	1.88861e10			0.289800
$254$		0			0
$255$	−7.41634e9	+	1.28455e10i	−0.109840	+	0.190248i
$256$		0			0
$257$	−1.16470e11	+	6.72438e10i	−1.66538	+	0.961508i	−0.695303	+	0.718717i	$0.744730\pi$
$257$	−1.16470e11	+	6.72438e10i	−1.66538	+	0.961508i	−0.970079	+	0.242791i	$0.921937\pi$
$258$		0			0
$259$	8.74498e10	−	1.54388e9i	1.20756	−	0.0213189i
$260$		0			0
$261$	1.79963e10	+	3.11706e10i	0.240050	+	0.415779i
$262$		0			0
$263$	4.59408e10	+	2.65239e10i	0.592103	+	0.341851i	0.765929	−	0.642926i	$-0.222280\pi$
$263$	4.59408e10	+	2.65239e10i	0.592103	+	0.341851i	−0.173826	+	0.984776i	$0.555613\pi$
$264$		0			0
$265$			3.72066e10i			0.463461i
$266$		0			0
$267$		−	5.06122e10i		−	0.609473i
$268$		0			0
$269$	6.86711e10	+	3.96473e10i	0.799630	+	0.461666i	0.843342	−	0.537378i	$-0.180585\pi$
$269$	6.86711e10	+	3.96473e10i	0.799630	+	0.461666i	−0.0437119	+	0.999044i	$0.513918\pi$
$270$		0			0
$271$	2.52467e10	+	4.37285e10i	0.284343	+	0.492496i	0.972450	−	0.233113i	$-0.0748912\pi$
$271$	2.52467e10	+	4.37285e10i	0.284343	+	0.492496i	−0.688107	+	0.725610i	$0.741558\pi$
$272$		0			0
$273$	3.89775e10	−	2.34306e10i	0.424700	−	0.255301i
$274$		0			0
$275$	2.55580e10	−	1.47559e10i	0.269482	−	0.155585i
$276$		0			0
$277$	−3.25830e10	+	5.64354e10i	−0.332531	+	0.575961i	−0.983007	−	0.183566i	$-0.941236\pi$
$277$	−3.25830e10	+	5.64354e10i	−0.332531	+	0.575961i	0.650476	+	0.759526i	$0.274569\pi$
$278$		0			0
$279$	2.34714e10			0.231910
$280$		0			0
$281$	−1.88520e9			−0.0180376			−0.00901879	−	0.999959i	$-0.502871\pi$
$281$	−1.88520e9			−0.0180376			−0.00901879	+	0.999959i	$0.502871\pi$
$282$		0			0
$283$	4.51498e10	−	7.82018e10i	0.418424	−	0.724732i	−0.577357	−	0.816492i	$-0.695916\pi$
$283$	4.51498e10	−	7.82018e10i	0.418424	−	0.724732i	0.995781	+	0.0917596i	$0.0292491\pi$
$284$		0			0
$285$	2.00085e10	−	1.15519e10i	0.179644	−	0.103718i
$286$		0			0
$287$	3.94932e10	+	2.18812e10i	0.343601	+	0.190371i
$288$		0			0
$289$	−2.03968e10	−	3.53283e10i	−0.171997	−	0.297908i
$290$		0			0
$291$	7.61791e10	+	4.39820e10i	0.622755	+	0.359548i
$292$		0			0
$293$		−	9.59512e10i		−	0.760582i	−0.924867	−	0.380291i	$-0.875824\pi$
$293$		−	9.59512e10i		−	0.760582i	0.924867	−	0.380291i	$-0.124176\pi$
$294$		0			0
$295$			5.03374e10i			0.386983i
$296$		0			0
$297$	−4.46506e10	−	2.57790e10i	−0.332984	−	0.192248i
$298$		0			0
$299$	−4.26802e10	−	7.39242e10i	−0.308820	−	0.534892i
$300$		0			0
$301$	−4.96306e10	−	2.74977e10i	−0.348498	−	0.193085i
$302$		0			0
$303$	−5.93263e10	+	3.42520e10i	−0.404348	+	0.233450i
$304$		0			0
$305$	5.53160e8	−	9.58101e8i	0.00366017	−	0.00633960i
$306$		0			0
$307$	7.08353e9			0.0455121			0.0227561	−	0.999741i	$-0.492756\pi$
$307$	7.08353e9			0.0455121			0.0227561	+	0.999741i	$0.492756\pi$
$308$		0			0
$309$	1.08247e11			0.675467
$310$		0			0
$311$	−5.84987e10	+	1.01323e11i	−0.354588	+	0.614165i	−0.987047	−	0.160429i	$-0.948712\pi$
$311$	−5.84987e10	+	1.01323e11i	−0.354588	+	0.614165i	0.632459	+	0.774594i	$0.282046\pi$
$312$		0			0
$313$	−3.10688e10	+	1.79376e10i	−0.182968	+	0.105637i	−0.588686	−	0.808362i	$-0.700355\pi$
$313$	−3.10688e10	+	1.79376e10i	−0.182968	+	0.105637i	0.405718	+	0.913998i	$0.367021\pi$
$314$		0			0
$315$	−4.51250e10	+	2.71261e10i	−0.258238	+	0.155235i
$316$		0			0
$317$	−1.19759e11	−	2.07428e11i	−0.666101	−	1.15372i	−0.978985	−	0.203930i	$-0.934629\pi$
$317$	−1.19759e11	−	2.07428e11i	−0.666101	−	1.15372i	0.312884	−	0.949791i	$-0.398705\pi$
$318$		0			0
$319$	−4.64780e10	−	2.68341e10i	−0.251298	−	0.145087i
$320$		0			0
$321$		−	4.87209e10i		−	0.256119i
$322$		0			0
$323$		−	1.21175e11i		−	0.619442i
$324$		0			0
$325$	−1.15516e11	−	6.66930e10i	−0.574336	−	0.331593i
$326$		0			0
$327$	1.53172e10	+	2.65301e10i	0.0740822	+	0.128314i
$328$		0			0
$329$	1.46035e11	−	2.57817e9i	0.687187	−	0.0121319i
$330$		0			0
$331$	1.81432e11	−	1.04750e11i	0.830781	−	0.479652i	−0.0233387	−	0.999728i	$-0.507430\pi$
$331$	1.81432e11	−	1.04750e11i	0.830781	−	0.479652i	0.854120	+	0.520076i	$0.174096\pi$
$332$		0			0
$333$	−8.85798e10	+	1.53425e11i	−0.394762	+	0.683748i
$334$		0			0
$335$	−1.81596e11			−0.787781
$336$		0			0
$337$	8.44844e10			0.356814			0.178407	−	0.983957i	$-0.442906\pi$
$337$	8.44844e10			0.356814			0.178407	+	0.983957i	$0.442906\pi$
$338$		0			0
$339$	8.50691e10	−	1.47344e11i	0.349843	−	0.605946i
$340$		0			0
$341$	−3.03091e10	+	1.74989e10i	−0.121389	+	0.0700837i
$342$		0			0
$343$	−2.55985e11	+	1.35691e10i	−0.998598	+	0.0529332i
$344$		0			0
$345$	−2.61743e10	−	4.53352e10i	−0.0994693	−	0.172286i
$346$		0			0
$347$	−5.80056e10	−	3.34896e10i	−0.214777	−	0.124001i	0.388753	−	0.921342i	$-0.372906\pi$
$347$	−5.80056e10	−	3.34896e10i	−0.214777	−	0.124001i	−0.603529	+	0.797341i	$0.706239\pi$
$348$		0			0
$349$			5.48730e11i			1.97991i	0.141400	+	0.989953i	$0.454840\pi$
$349$			5.48730e11i			1.97991i	−0.141400	+	0.989953i	$0.545160\pi$
$350$		0			0
$351$			2.33030e11i			0.819463i
$352$		0			0
$353$	3.92229e11	+	2.26454e11i	1.34448	+	0.776235i	0.987461	−	0.157863i	$-0.0504605\pi$
$353$	3.92229e11	+	2.26454e11i	1.34448	+	0.776235i	0.357017	+	0.934098i	$0.383794\pi$
$354$		0			0
$355$	−5.27600e10	−	9.13830e10i	−0.176310	−	0.305378i
$356$		0			0
$357$	−2.58205e9	−	1.46255e11i	−0.00841313	−	0.476544i
$358$		0			0
$359$	−4.19507e11	+	2.42203e11i	−1.33295	+	0.769580i	−0.985751	−	0.168211i	$-0.946201\pi$
$359$	−4.19507e11	+	2.42203e11i	−1.33295	+	0.769580i	−0.347200	+	0.937791i	$0.612868\pi$
$360$		0			0
$361$	6.69711e10	−	1.15997e11i	0.207542	−	0.359472i
$362$		0			0
$363$	−1.64279e11			−0.496596
$364$		0			0
$365$	−1.29911e11			−0.383115
$366$		0			0
$367$	−2.85084e11	+	4.93780e11i	−0.820306	+	1.42081i	0.0851491	+	0.996368i	$0.472863\pi$
$367$	−2.85084e11	+	4.93780e11i	−0.820306	+	1.42081i	−0.905455	+	0.424443i	$0.860470\pi$
$368$		0			0
$369$	−7.91998e10	+	4.57260e10i	−0.222385	+	0.128394i
$370$		0			0
$371$	−1.89043e11	−	3.14479e11i	−0.518058	−	0.861804i
$372$		0			0
$373$	−1.75432e11	−	3.03856e11i	−0.469265	−	0.812790i	0.530118	−	0.847924i	$-0.322148\pi$
$373$	−1.75432e11	−	3.03856e11i	−0.469265	−	0.812790i	−0.999383	+	0.0351337i	$0.988814\pi$
$374$		0			0
$375$	−1.60793e11	−	9.28339e10i	−0.419881	−	0.242419i
$376$		0			0
$377$			2.42567e11i			0.618437i
$378$		0			0
$379$		−	5.99850e11i		−	1.49337i	−0.665180	−	0.746683i	$-0.731645\pi$
$379$		−	5.99850e11i		−	1.49337i	0.665180	−	0.746683i	$-0.268355\pi$
$380$		0			0
$381$	1.60593e11	+	9.27183e10i	0.390448	+	0.225425i
$382$		0			0
$383$	−1.95296e11	−	3.38262e11i	−0.463765	−	0.803265i	0.535380	−	0.844612i	$-0.320169\pi$
$383$	−1.95296e11	−	3.38262e11i	−0.463765	−	0.803265i	−0.999145	+	0.0413466i	$0.986835\pi$
$384$		0			0
$385$	3.80471e10	−	6.86711e10i	0.0882568	−	0.159294i
$386$		0			0
$387$	9.95293e10	−	5.74633e10i	0.225555	−	0.130224i
$388$		0			0
$389$	1.75740e11	−	3.04390e11i	0.389132	−	0.673997i	−0.603201	−	0.797589i	$-0.706108\pi$
$389$	1.75740e11	−	3.04390e11i	0.389132	−	0.673997i	0.992333	+	0.123593i	$0.0394416\pi$
$390$		0			0
$391$	−2.74557e11			−0.594070
$392$		0			0
$393$	−2.84070e10			−0.0600701
$394$		0			0
$395$	−1.48752e11	+	2.57646e11i	−0.307452	+	0.532522i
$396$		0			0
$397$	−4.36541e11	+	2.52037e11i	−0.881998	+	0.509222i	−0.871317	−	0.490721i	$-0.836733\pi$
$397$	−4.36541e11	+	2.52037e11i	−0.881998	+	0.509222i	−0.0106816	+	0.999943i	$0.503400\pi$
$398$		0			0
$399$	−1.10422e11	+	1.99301e11i	−0.218112	+	0.393669i
$400$		0			0
$401$	−1.50926e11	−	2.61411e11i	−0.291483	−	0.504864i	0.682678	−	0.730720i	$-0.260815\pi$
$401$	−1.50926e11	−	2.61411e11i	−0.291483	−	0.504864i	−0.974161	+	0.225856i	$0.927482\pi$
$402$		0			0
$403$	1.36989e11	+	7.90909e10i	0.258711	+	0.149367i
$404$		0			0
$405$		−	2.02282e10i		−	0.0373603i
$406$		0			0
$407$		−	2.64160e11i		−	0.477191i
$408$		0			0
$409$	2.19951e11	+	1.26989e11i	0.388662	+	0.224394i	0.681580	−	0.731744i	$-0.261293\pi$
$409$	2.19951e11	+	1.26989e11i	0.388662	+	0.224394i	−0.292918	+	0.956137i	$0.594626\pi$
$410$		0			0
$411$	−1.05106e11	−	1.82048e11i	−0.181693	−	0.314701i
$412$		0			0
$413$	−2.55759e11	−	4.25463e11i	−0.432570	−	0.719593i
$414$		0			0
$415$	−4.31959e11	+	2.49392e11i	−0.714869	+	0.412730i
$416$		0			0
$417$	−1.09710e11	+	1.90023e11i	−0.177678	+	0.307747i
$418$		0			0
$419$	4.35591e11			0.690424			0.345212	−	0.938525i	$-0.387807\pi$
$419$	4.35591e11			0.690424			0.345212	+	0.938525i	$0.387807\pi$
$420$		0			0
$421$	9.42122e11			1.46163			0.730815	−	0.682575i	$-0.239140\pi$
$421$	9.42122e11			1.46163			0.730815	+	0.682575i	$0.239140\pi$
$422$		0			0
$423$	−1.47922e11	+	2.56208e11i	−0.224647	+	0.389100i
$424$		0			0
$425$	−3.71551e11	+	2.14515e11i	−0.552418	+	0.318939i
$426$		0			0
$427$	1.92586e8	+	1.09087e10i	0.000280350	+	0.0158798i
$428$		0			0
$429$	−6.86771e10	−	1.18952e11i	−0.0978935	−	0.169557i
$430$		0			0
$431$	4.25630e9	+	2.45738e9i	0.00594135	+	0.00343024i	0.502968	−	0.864305i	$-0.332241\pi$
$431$	4.25630e9	+	2.45738e9i	0.00594135	+	0.00343024i	−0.497026	+	0.867735i	$0.665575\pi$
$432$		0			0
$433$		−	1.05250e11i		−	0.143888i	−0.997409	−	0.0719441i	$-0.977080\pi$
$433$		−	1.05250e11i		−	0.143888i	0.997409	−	0.0719441i	$-0.0229203\pi$
$434$		0			0
$435$			1.48758e11i			0.199195i
$436$		0			0
$437$	3.70363e11	+	2.13829e11i	0.485804	+	0.280479i
$438$		0			0
$439$	4.26448e11	+	7.38629e11i	0.547994	+	0.949153i	0.998412	+	0.0563352i	$0.0179416\pi$
$439$	4.26448e11	+	7.38629e11i	0.547994	+	0.949153i	−0.450418	+	0.892818i	$0.648725\pi$
$440$		0			0
$441$	2.43582e11	−	4.58552e11i	0.306670	−	0.577318i
$442$		0			0
$443$	1.33326e12	−	7.69760e11i	1.64475	−	0.949595i	0.665635	−	0.746278i	$-0.268161\pi$
$443$	1.33326e12	−	7.69760e11i	1.64475	−	0.949595i	0.979113	−	0.203317i	$-0.0651724\pi$
$444$		0			0
$445$	−1.97445e11	+	3.41984e11i	−0.238685	+	0.413415i
$446$		0			0
$447$	2.71429e11			0.321568
$448$		0			0
$449$	−4.68631e11			−0.544155			−0.272078	−	0.962275i	$-0.587711\pi$
$449$	−4.68631e11			−0.544155			−0.272078	+	0.962275i	$0.587711\pi$
$450$		0			0
$451$	6.81815e10	−	1.18094e11i	0.0776018	−	0.134410i
$452$		0			0
$453$	−6.90103e11	+	3.98431e11i	−0.769967	+	0.444540i
$454$		0			0
$455$	−3.54776e11	+	6.26337e9i	−0.388063	+	0.00685105i
$456$		0			0
$457$	−7.21160e11	−	1.24909e12i	−0.773408	−	1.33958i	−0.935685	−	0.352837i	$-0.885217\pi$
$457$	−7.21160e11	−	1.24909e12i	−0.773408	−	1.33958i	0.162277	−	0.986745i	$-0.448116\pi$
$458$		0			0
$459$	6.49111e11	+	3.74764e11i	0.682593	+	0.394095i
$460$		0			0
$461$			4.70316e11i			0.484993i	0.970152	+	0.242497i	$0.0779664\pi$
$461$			4.70316e11i			0.484993i	−0.970152	+	0.242497i	$0.922034\pi$
$462$		0			0
$463$			5.23273e11i			0.529192i	0.964359	+	0.264596i	$0.0852387\pi$
$463$			5.23273e11i			0.529192i	−0.964359	+	0.264596i	$0.914761\pi$
$464$		0			0
$465$	8.40110e10	+	4.85038e10i	0.0833294	+	0.0481102i
$466$		0			0
$467$	−2.44390e11	−	4.23296e11i	−0.237770	−	0.411831i	0.722304	−	0.691576i	$-0.243083\pi$
$467$	−2.44390e11	−	4.23296e11i	−0.237770	−	0.411831i	−0.960074	+	0.279745i	$0.909750\pi$
$468$		0			0
$469$	1.53490e12	−	9.22675e11i	1.46488	−	0.880584i
$470$		0			0
$471$	3.31973e11	−	1.91665e11i	0.310820	−	0.179452i
$472$		0			0
$473$	−8.56827e10	+	1.48407e11i	−0.0787078	+	0.136326i
$474$		0			0
$475$	6.68270e11			0.602325
$476$		0			0
$477$	7.43217e11			0.657330
$478$		0			0
$479$	3.99513e11	−	6.91978e11i	0.346754	−	0.600596i	−0.638917	−	0.769276i	$-0.720617\pi$
$479$	3.99513e11	−	6.91978e11i	0.346754	−	0.600596i	0.985671	+	0.168680i	$0.0539505\pi$
$480$		0			0
$481$	−1.03398e12	+	5.96970e11i	−0.880765	+	0.508510i
$482$		0			0
$483$	4.51575e11	+	2.50195e11i	0.377544	+	0.209178i
$484$		0			0
$485$	−3.43159e11	−	5.94369e11i	−0.281616	−	0.487774i
$486$		0			0
$487$	2.10937e12	+	1.21784e12i	1.69931	+	0.981095i	0.946417	+	0.322947i	$0.104674\pi$
$487$	2.10937e12	+	1.21784e12i	1.69931	+	0.981095i	0.752889	+	0.658148i	$0.228660\pi$
$488$		0			0
$489$			5.61515e11i			0.444091i
$490$		0			0
$491$		−	6.93279e11i		−	0.538321i	−0.963095	−	0.269160i	$-0.913254\pi$
$491$		−	6.93279e11i		−	0.538321i	0.963095	−	0.269160i	$-0.0867462\pi$
$492$		0			0
$493$	6.75676e11	+	3.90102e11i	0.515143	+	0.297418i
$494$		0			0
$495$	7.95088e10	+	1.37713e11i	0.0595239	+	0.103098i
$496$		0			0
$497$	9.10248e11	+	5.04322e11i	0.669200	+	0.370769i
$498$		0			0
$499$	−2.73905e11	+	1.58139e11i	−0.197764	+	0.114179i	−0.595612	−	0.803272i	$-0.703090\pi$
$499$	−2.73905e11	+	1.58139e11i	−0.197764	+	0.114179i	0.397848	+	0.917451i	$0.369757\pi$
$500$		0			0
$501$	−6.21096e11	+	1.07577e12i	−0.440442	+	0.762868i
$502$		0			0
$503$	4.65003e10			0.0323892			0.0161946	−	0.999869i	$-0.494845\pi$
$503$	4.65003e10			0.0323892			0.0161946	+	0.999869i	$0.494845\pi$
$504$		0			0
$505$	5.34487e11			0.365701
$506$		0			0
$507$	1.27343e11	−	2.20565e11i	0.0855934	−	0.148252i
$508$		0			0
$509$	−1.29875e12	+	7.49833e11i	−0.857621	+	0.495147i	−0.863215	−	0.504837i	$-0.831553\pi$
$509$	−1.29875e12	+	7.49833e11i	−0.857621	+	0.495147i	0.00559421	+	0.999984i	$0.498219\pi$
$510$		0			0
$511$	1.09804e12	−	6.60068e11i	0.712402	−	0.428247i
$512$		0			0
$513$	−5.83745e11	−	1.01108e12i	−0.372130	−	0.644548i
$514$		0			0
$515$	−7.31423e11	−	4.22287e11i	−0.458180	−	0.264530i
$516$		0			0
$517$		−	4.41128e11i		−	0.271555i
$518$		0			0
$519$			2.71536e11i			0.164276i
$520$		0			0
$521$	2.81600e12	+	1.62582e12i	1.67441	+	0.966724i	0.965119	+	0.261813i	$0.0843203\pi$
$521$	2.81600e12	+	1.62582e12i	1.67441	+	0.966724i	0.709296	+	0.704911i	$0.249013\pi$
$522$		0			0
$523$	1.07319e12	+	1.85882e12i	0.627217	+	1.08637i	0.988108	+	0.153764i	$0.0491395\pi$
$523$	1.07319e12	+	1.85882e12i	0.627217	+	1.08637i	−0.360890	+	0.932608i	$0.617527\pi$
$524$		0			0
$525$	8.06585e11	−	1.42398e10i	0.463375	−	0.00818065i
$526$		0			0
$527$	4.40620e11	−	2.54392e11i	0.248838	−	0.143667i
$528$		0			0
$529$	−4.16083e11	+	7.20676e11i	−0.231009	+	0.400120i
$530$		0			0
$531$	1.00551e12			0.548860
$532$		0			0
$533$	−6.16327e11			−0.330779
$534$		0			0
$535$	−1.90066e11	+	3.29205e11i	−0.100303	+	0.173730i
$536$		0			0
$537$	−1.22084e12	+	7.04855e11i	−0.633543	+	0.365776i
$538$		0			0
$539$	2.73284e10	+	7.73738e11i	0.0139465	+	0.394861i
$540$		0			0
$541$	−8.48475e11	−	1.46960e12i	−0.425845	−	0.737585i	0.570654	−	0.821191i	$-0.306690\pi$
$541$	−8.48475e11	−	1.46960e12i	−0.425845	−	0.737585i	−0.996499	+	0.0836056i	$0.973356\pi$
$542$		0			0
$543$	1.66636e11	+	9.62073e10i	0.0822563	+	0.0474907i
$544$		0			0
$545$		−	2.39017e11i		−	0.116050i
$546$		0			0
$547$			8.09019e11i			0.386381i	0.981161	+	0.193191i	$0.0618835\pi$
$547$			8.09019e11i			0.386381i	−0.981161	+	0.193191i	$0.938116\pi$
$548$		0			0
$549$	−1.91385e10	−	1.10496e10i	−0.00899150	−	0.00519124i
$550$		0			0
$551$	−6.07635e11	−	1.05245e12i	−0.280841	−	0.486431i
$552$		0			0
$553$	−5.17891e10	−	2.93348e12i	−0.0235492	−	1.33389i
$554$		0			0
$555$	−6.34106e11	+	3.66101e11i	−0.283690	+	0.163788i
$556$		0			0
$557$	7.07997e11	−	1.22629e12i	0.311662	−	0.539814i	−0.667061	−	0.745003i	$-0.732448\pi$
$557$	7.07997e11	−	1.22629e12i	0.311662	−	0.539814i	0.978722	+	0.205190i	$0.0657812\pi$
$558$		0			0
$559$	7.74529e11			0.335494
$560$		0			0
$561$	−4.41793e11			−0.188315
$562$		0			0
$563$	9.71638e11	−	1.68293e12i	0.407584	−	0.705956i	−0.587035	−	0.809562i	$-0.699705\pi$
$563$	9.71638e11	−	1.68293e12i	0.407584	−	0.705956i	0.994618	+	0.103606i	$0.0330381\pi$
$564$		0			0
$565$	−1.14962e12	+	6.63731e11i	−0.474608	+	0.274015i
$566$		0			0
$567$	1.02778e11	+	1.70974e11i	0.0417614	+	0.0694713i
$568$		0			0
$569$	−2.21505e12	−	3.83658e12i	−0.885887	−	1.53440i	−0.844693	−	0.535250i	$-0.820217\pi$
$569$	−2.21505e12	−	3.83658e12i	−0.885887	−	1.53440i	−0.0411936	−	0.999151i	$-0.513116\pi$
$570$		0			0
$571$	−2.75136e12	−	1.58850e12i	−1.08314	−	0.625353i	−0.151400	−	0.988473i	$-0.548378\pi$
$571$	−2.75136e12	−	1.58850e12i	−1.08314	−	0.625353i	−0.931742	+	0.363120i	$0.881712\pi$
$572$		0			0
$573$			2.16249e12i			0.838029i
$574$		0			0
$575$		−	1.51416e12i		−	0.577653i
$576$		0			0
$577$	3.26620e12	+	1.88574e12i	1.22674	+	0.708257i	0.966346	−	0.257246i	$-0.0828151\pi$
$577$	3.26620e12	+	1.88574e12i	1.22674	+	0.708257i	0.260391	+	0.965503i	$0.416148\pi$
$578$		0			0
$579$	2.70355e11	+	4.68269e11i	0.0999726	+	0.173158i
$580$		0			0
$581$	2.38388e12	−	4.30266e12i	0.867945	−	1.56655i
$582$		0			0
$583$	−9.59731e11	+	5.54101e11i	−0.344065	+	0.198646i
$584$		0			0
$585$	3.59360e11	−	6.22430e11i	0.126861	−	0.219730i
$586$		0			0
$587$	−4.29228e12			−1.49216			−0.746082	−	0.665854i	$-0.768068\pi$
$587$	−4.29228e12			−1.49216			−0.746082	+	0.665854i	$0.768068\pi$
$588$		0			0
$589$	−7.92498e11			−0.271318
$590$		0			0
$591$	−6.65806e11	+	1.15321e12i	−0.224494	+	0.388834i
$592$		0			0
$593$	−4.77252e12	+	2.75541e12i	−1.58490	+	0.915042i	−0.590770	+	0.806840i	$0.701176\pi$
$593$	−4.77252e12	+	2.75541e12i	−1.58490	+	0.915042i	−0.994129	+	0.108201i	$0.965491\pi$
$594$		0			0
$595$	−5.53112e11	+	9.98310e11i	−0.180920	+	0.326542i
$596$		0			0
$597$	2.53743e11	+	4.39496e11i	0.0817541	+	0.141602i
$598$		0			0
$599$	−5.05088e12	−	2.91612e12i	−1.60305	−	0.925519i	−0.990874	−	0.134794i	$-0.956963\pi$
$599$	−5.05088e12	−	2.91612e12i	−1.60305	−	0.925519i	−0.612172	−	0.790724i	$-0.709704\pi$
$600$		0			0
$601$			2.21379e12i			0.692150i	0.938207	+	0.346075i	$0.112486\pi$
$601$			2.21379e12i			0.692150i	−0.938207	+	0.346075i	$0.887514\pi$
$602$		0			0
$603$			3.62747e12i			1.11731i
$604$		0			0
$605$	1.11003e12	+	6.40875e11i	0.336849	+	0.194480i
$606$		0			0
$607$	−1.72195e12	−	2.98251e12i	−0.514840	−	0.891728i	−0.999852	−	0.0172210i	$-0.994518\pi$
$607$	−1.72195e12	−	2.98251e12i	−0.514840	−	0.891728i	0.485012	−	0.874507i	$-0.338815\pi$
$608$		0			0
$609$	−7.55826e11	−	1.25734e12i	−0.222661	−	0.370403i
$610$		0			0
$611$	−1.72667e12	+	9.96895e11i	−0.501216	+	0.289377i
$612$		0			0
$613$	−1.48665e12	+	2.57496e12i	−0.425243	+	0.736543i	−0.996443	−	0.0842683i	$-0.973145\pi$
$613$	−1.48665e12	+	2.57496e12i	−0.425243	+	0.736543i	0.571200	+	0.820811i	$0.306478\pi$
$614$		0			0
$615$	−3.77972e11			−0.106542
$616$		0			0
$617$	6.80602e12			1.89064			0.945322	−	0.326138i	$-0.105747\pi$
$617$	6.80602e12			1.89064			0.945322	+	0.326138i	$0.105747\pi$
$618$		0			0
$619$	−8.00244e11	+	1.38606e12i	−0.219086	+	0.379468i	−0.954529	−	0.298119i	$-0.903641\pi$
$619$	−8.00244e11	+	1.38606e12i	−0.219086	+	0.379468i	0.735443	+	0.677587i	$0.236974\pi$
$620$		0			0
$621$	−2.29089e12	+	1.32265e12i	−0.618147	+	0.356888i
$622$		0			0
$623$	−6.87417e10	−	3.89373e12i	−0.0182820	−	1.03555i
$624$		0			0
$625$	−7.77837e11	−	1.34725e12i	−0.203905	−	0.353174i
$626$		0			0
$627$	5.95956e11	+	3.44075e11i	0.153996	+	0.0889097i
$628$		0			0
$629$			3.84025e12i			0.978208i
$630$		0			0
$631$			1.97228e12i			0.495264i	0.968854	+	0.247632i	$0.0796524\pi$
$631$			1.97228e12i			0.495264i	−0.968854	+	0.247632i	$0.920348\pi$
$632$		0			0
$633$	9.08063e11	+	5.24271e11i	0.224802	+	0.129789i
$634$		0			0
$635$	−7.23412e11	−	1.25299e12i	−0.176565	−	0.305819i
$636$		0			0
$637$	2.96682e12	−	1.85552e12i	0.713944	−	0.446517i
$638$		0			0
$639$	−1.82541e12	+	1.05390e12i	−0.433120	+	0.250062i
$640$		0			0
$641$	1.27130e11	−	2.20196e11i	0.0297433	−	0.0515168i	−0.850771	−	0.525537i	$-0.823864\pi$
$641$	1.27130e11	−	2.20196e11i	0.0297433	−	0.0515168i	0.880514	+	0.474020i	$0.157198\pi$
$642$		0			0
$643$	−5.33463e12			−1.23071			−0.615354	−	0.788251i	$-0.710987\pi$
$643$	−5.33463e12			−1.23071			−0.615354	+	0.788251i	$0.710987\pi$
$644$		0			0
$645$	4.74993e11			0.108061
$646$		0			0
$647$	2.81386e12	−	4.87374e12i	0.631296	−	1.09344i	−0.355991	−	0.934489i	$-0.615857\pi$
$647$	2.81386e12	−	4.87374e12i	0.631296	−	1.09344i	0.987287	−	0.158947i	$-0.0508099\pi$
$648$		0			0
$649$	−1.29843e12	+	7.49652e11i	−0.287289	+	0.165866i
$650$		0			0
$651$	−9.56524e11	+	1.68869e10i	−0.208728	+	0.00368499i
$652$		0			0
$653$	−4.21385e12	−	7.29861e12i	−0.906922	−	1.57083i	−0.818316	−	0.574768i	$-0.805092\pi$
$653$	−4.21385e12	−	7.29861e12i	−0.906922	−	1.57083i	−0.0886057	−	0.996067i	$-0.528241\pi$
$654$		0			0
$655$	1.91945e11	+	1.10819e11i	0.0407465	+	0.0235250i
$656$		0			0
$657$			2.59504e12i			0.543375i
$658$		0			0
$659$		−	2.09492e12i		−	0.432696i	−0.976316	−	0.216348i	$-0.930585\pi$
$659$		−	2.09492e12i		−	0.432696i	0.976316	−	0.216348i	$-0.0694145\pi$
$660$		0			0
$661$	4.24987e12	+	2.45366e12i	0.865903	+	0.499929i	0.865985	−	0.500071i	$-0.166693\pi$
$661$	4.24987e12	+	2.45366e12i	0.865903	+	0.499929i	−8.17675e−5		1.00000i	$0.500026\pi$
$662$		0			0
$663$	9.98397e11	+	1.72927e12i	0.200675	+	0.347579i
$664$		0			0
$665$	1.52362e12	−	9.15896e11i	0.302120	−	0.181614i
$666$		0			0
$667$	−2.38465e12	+	1.37678e12i	−0.466507	+	0.269338i
$668$		0			0
$669$	−3.33288e11	+	5.77272e11i	−0.0643283	+	0.111420i
$670$		0			0
$671$	3.29518e10			0.00627521
$672$		0			0
$673$	−2.56322e12			−0.481636			−0.240818	−	0.970570i	$-0.577416\pi$
$673$	−2.56322e12			−0.481636			−0.240818	+	0.970570i	$0.577416\pi$
$674$		0			0
$675$	−2.06680e12	+	3.57980e12i	−0.383205	+	0.663731i
$676$		0			0
$677$	−3.58879e12	+	2.07199e12i	−0.656598	+	0.379087i	−0.790979	−	0.611843i	$-0.790429\pi$
$677$	−3.58879e12	+	2.07199e12i	−0.656598	+	0.379087i	0.134382	+	0.990930i	$0.457095\pi$
$678$		0			0
$679$	5.92040e12	+	3.28019e12i	1.06890	+	0.592222i
$680$		0			0
$681$	−2.55574e12	−	4.42667e12i	−0.455360	−	0.788707i
$682$		0			0
$683$	−6.28386e12	−	3.62799e12i	−1.10493	−	0.637929i	−0.167415	−	0.985886i	$-0.553542\pi$
$683$	−6.28386e12	−	3.62799e12i	−1.10493	−	0.637929i	−0.937510	+	0.347957i	$0.886875\pi$
$684$		0			0
$685$			1.64012e12i			0.284622i
$686$		0			0
$687$		−	4.10370e11i		−	0.0702862i
$688$		0			0
$689$	4.33774e12	+	2.50440e12i	0.733293	+	0.423367i
$690$		0			0
$691$	3.57039e12	+	6.18410e12i	0.595750	+	1.03187i	0.993441	+	0.114350i	$0.0364786\pi$
$691$	3.57039e12	+	6.18410e12i	0.595750	+	1.03187i	−0.397690	+	0.917520i	$0.630188\pi$
$692$		0			0
$693$	−1.37173e12	−	7.60007e11i	−0.225928	−	0.125175i
$694$		0			0
$695$	1.48261e12	−	8.55985e11i	0.241043	−	0.139166i
$696$		0			0
$697$	−9.91192e11	+	1.71680e12i	−0.159078	+	0.275531i
$698$		0			0
$699$	8.03187e11			0.127253
$700$		0			0
$701$	6.16424e12			0.964159			0.482080	−	0.876127i	$-0.339882\pi$
$701$	6.16424e12			0.964159			0.482080	+	0.876127i	$0.339882\pi$
$702$		0			0
$703$	2.99084e12	−	5.18029e12i	0.461843	−	0.799936i
$704$		0			0
$705$	−1.05891e12	+	6.11362e11i	−0.161439	+	0.0932068i
$706$		0			0
$707$	−4.51761e12	+	2.71568e12i	−0.680019	+	0.408781i
$708$		0			0
$709$	−6.57925e12	−	1.13956e13i	−0.977842	−	1.69367i	−0.670219	−	0.742163i	$-0.733800\pi$
$709$	−6.57925e12	−	1.13956e13i	−0.977842	−	1.69367i	−0.307623	−	0.951508i	$-0.599533\pi$
$710$		0			0
$711$	5.14660e12	+	2.97139e12i	0.755279	+	0.436060i
$712$		0			0
$713$			1.79564e12i			0.260205i
$714$		0			0
$715$			1.07167e12i			0.153351i
$716$		0			0
$717$	4.34469e12	+	2.50841e12i	0.613936	+	0.354456i
$718$		0			0
$719$	−5.42801e12	−	9.40159e12i	−0.757462	−	1.31196i	−0.944141	−	0.329542i	$-0.893106\pi$
$719$	−5.42801e12	−	9.40159e12i	−0.757462	−	1.31196i	0.186679	−	0.982421i	$-0.440228\pi$
$720$		0			0
$721$	8.32776e12	−	1.47022e11i	1.14768	−	0.0202616i
$722$		0			0
$723$	5.81985e11	−	3.36009e11i	0.0792118	−	0.0457329i
$724$		0			0
$725$	−2.15138e12	+	3.72631e12i	−0.289199	+	0.500908i
$726$		0			0
$727$	7.79083e12			1.03438			0.517189	−	0.855871i	$-0.326979\pi$
$727$	7.79083e12			1.03438			0.517189	+	0.855871i	$0.326979\pi$
$728$		0			0
$729$	3.96279e12			0.519669
$730$		0			0
$731$	1.24562e12	−	2.15747e12i	0.161345	−	0.279458i
$732$		0			0
$733$	−1.12127e13	+	6.47365e12i	−1.43464	+	0.828288i	−0.997470	−	0.0710929i	$-0.977351\pi$
$733$	−1.12127e13	+	6.47365e12i	−1.43464	+	0.828288i	−0.437167	+	0.899381i	$0.644018\pi$
$734$		0			0
$735$	1.81945e12	−	1.13793e12i	0.229958	−	0.143821i
$736$		0			0
$737$	−2.70443e12	−	4.68422e12i	−0.337655	−	0.584835i
$738$		0			0
$739$	6.71838e12	+	3.87886e12i	0.828637	+	0.478414i	0.853386	−	0.521280i	$-0.174545\pi$
$739$	6.71838e12	+	3.87886e12i	0.828637	+	0.478414i	−0.0247488	+	0.999694i	$0.507879\pi$
$740$		0			0
$741$		−	3.11027e12i		−	0.378980i
$742$		0			0
$743$		−	6.21088e12i		−	0.747659i	−0.927497	−	0.373829i	$-0.878045\pi$
$743$		−	6.21088e12i		−	0.747659i	0.927497	−	0.373829i	$-0.121955\pi$
$744$		0			0
$745$	−1.83404e12	−	1.05888e12i	−0.218125	−	0.125934i
$746$		0			0
$747$	4.98171e12	+	8.62857e12i	0.585377	+	1.01390i
$748$		0			0
$749$	−6.61729e10	−	3.74823e12i	−0.00768267	−	0.435169i
$750$		0			0
$751$	7.22548e12	−	4.17163e12i	0.828871	−	0.478549i	−0.0245949	−	0.999698i	$-0.507830\pi$
$751$	7.22548e12	−	4.17163e12i	0.828871	−	0.478549i	0.853466	+	0.521149i	$0.174496\pi$
$752$		0			0
$753$	4.00874e12	−	6.94333e12i	0.454391	−	0.787029i
$754$		0			0
$755$	6.21733e12			0.696374
$756$		0			0
$757$	5.31088e12			0.587808			0.293904	−	0.955835i	$-0.405045\pi$
$757$	5.31088e12			0.587808			0.293904	+	0.955835i	$0.405045\pi$
$758$		0			0
$759$	7.79604e11	−	1.35031e12i	0.0852680	−	0.147688i
$760$		0			0
$761$	−8.83629e12	+	5.10164e12i	−0.955078	+	0.551415i	−0.894655	−	0.446758i	$-0.852579\pi$
$761$	−8.83629e12	+	5.10164e12i	−0.955078	+	0.551415i	−0.0604236	+	0.998173i	$0.519245\pi$
$762$		0			0
$763$	1.21442e12	+	2.02023e12i	0.129721	+	0.215795i
$764$		0			0
$765$	−1.15586e12	−	2.00201e12i	−0.122020	−	0.211344i
$766$		0			0
$767$	5.86860e12	+	3.38824e12i	0.612288	+	0.353504i
$768$		0			0
$769$			3.47256e12i			0.358081i	0.983842	+	0.179041i	$0.0572994\pi$
$769$			3.47256e12i			0.358081i	−0.983842	+	0.179041i	$0.942701\pi$
$770$		0			0
$771$			1.11031e13i			1.13162i
$772$		0			0
$773$	−2.51676e12	−	1.45305e12i	−0.253532	−	0.146377i	0.367848	−	0.929886i	$-0.380095\pi$
$773$	−2.51676e12	−	1.45305e12i	−0.253532	−	0.146377i	−0.621381	+	0.783509i	$0.713428\pi$
$774$		0			0
$775$	1.40295e12	+	2.42999e12i	0.139697	+	0.241962i
$776$		0			0
$777$	3.49949e12	−	6.31621e12i	0.344437	−	0.621673i
$778$		0			0
$779$	2.67413e12	−	1.54391e12i	0.260174	−	0.150212i
$780$		0			0
$781$	1.57146e12	−	2.72185e12i	0.151138	−	0.261779i
$782$		0			0
$783$	7.51708e12			0.714696
$784$		0			0
$785$	−2.99083e12			−0.281112
$786$		0			0
$787$	−6.76855e12	+	1.17235e13i	−0.628940	+	1.08936i	0.358824	+	0.933405i	$0.383178\pi$
$787$	−6.76855e12	+	1.17235e13i	−0.628940	+	1.08936i	−0.987765	+	0.155951i	$0.950156\pi$
$788$		0			0
$789$	3.79281e12	−	2.18978e12i	0.348429	−	0.201166i
$790$		0			0
$791$	6.34447e12	−	1.14511e13i	0.576237	−	1.04005i
$792$		0			0
$793$	−7.44671e10	−	1.28981e11i	−0.00668706	−	0.0115823i
$794$		0			0
$795$	2.66019e12	+	1.53586e12i	0.236190	+	0.136364i
$796$		0			0
$797$			8.51748e12i			0.747737i	0.927482	+	0.373869i	$0.121969\pi$
$797$			8.51748e12i			0.747737i	−0.927482	+	0.373869i	$0.878031\pi$
$798$		0			0
$799$			6.41293e12i			0.556668i
$800$		0			0
$801$	6.83128e12	+	3.94404e12i	0.586348	+	0.338528i
$802$		0			0
$803$	−1.93471e12	−	3.35102e12i	−0.164209	−	0.284418i
$804$		0			0
$805$	−2.07523e12	−	3.45221e12i	−0.174175	−	0.289745i
$806$		0			0
$807$	5.66940e12	−	3.27323e12i	0.470550	−	0.271672i
$808$		0			0
$809$	−1.00807e13	+	1.74602e13i	−0.827409	+	1.43312i	0.0726546	+	0.997357i	$0.476853\pi$
$809$	−1.00807e13	+	1.74602e13i	−0.827409	+	1.43312i	−0.900064	+	0.435758i	$0.856480\pi$
$810$		0			0
$811$	−2.59171e10			−0.00210374			−0.00105187	−	0.999999i	$-0.500335\pi$
$811$	−2.59171e10			−0.00210374			−0.00105187	+	0.999999i	$0.500335\pi$
$812$		0			0
$813$	4.16867e12			0.334649
$814$		0			0
$815$	2.19054e12	−	3.79413e12i	0.173917	−	0.301234i
$816$		0			0
$817$	−3.36055e12	+	1.94021e12i	−0.263882	+	0.152353i
$818$		0			0
$819$	1.25114e11	+	7.08679e12i	0.00971688	+	0.550392i
$820$		0			0
$821$	9.89895e12	+	1.71455e13i	0.760405	+	1.31706i	0.942642	+	0.333805i	$0.108333\pi$
$821$	9.89895e12	+	1.71455e13i	0.760405	+	1.31706i	−0.182237	+	0.983255i	$0.558334\pi$
$822$		0			0
$823$	1.29838e13	+	7.49618e12i	0.986510	+	0.569562i	0.904229	−	0.427047i	$-0.140446\pi$
$823$	1.29838e13	+	7.49618e12i	0.986510	+	0.569562i	0.0822808	+	0.996609i	$0.473780\pi$
$824$		0			0
$825$		−	2.43646e12i		−	0.183112i
$826$		0			0
$827$		−	2.64061e13i		−	1.96304i	−0.191352	−	0.981521i	$-0.561287\pi$
$827$		−	2.64061e13i		−	1.96304i	0.191352	−	0.981521i	$-0.438713\pi$
$828$		0			0
$829$	1.81810e13	+	1.04968e13i	1.33697	+	0.771902i	0.986358	−	0.164617i	$-0.0526389\pi$
$829$	1.81810e13	+	1.04968e13i	1.33697	+	0.771902i	0.350616	+	0.936519i	$0.385972\pi$
$830$		0			0
$831$	2.69001e12	+	4.65923e12i	0.195681	+	0.338930i
$832$		0			0
$833$	−3.97288e11	−	1.12483e13i	−0.0285893	−	0.809437i
$834$		0			0
$835$	8.39344e12	−	4.84595e12i	0.597518	−	0.344977i
$836$		0			0
$837$	2.45101e12	−	4.24527e12i	0.172615	−	0.298979i
$838$		0			0
$839$	−1.05193e13			−0.732921			−0.366461	−	0.930434i	$-0.619431\pi$
$839$	−1.05193e13			−0.732921			−0.366461	+	0.930434i	$0.619431\pi$
$840$		0			0
$841$	−6.68243e12			−0.460630
$842$		0			0
$843$	−7.78197e10	+	1.34788e11i	−0.00530720	+	0.00919234i
$844$		0			0
$845$	−1.72091e12	+	9.93566e11i	−0.116119	+	0.0670412i
$846$		0			0
$847$	−1.26384e13	+	2.23125e11i	−0.843759	+	0.0148961i
$848$		0			0
$849$	−3.72751e12	−	6.45624e12i	−0.246226	−	0.426476i
$850$		0			0
$851$	−1.17375e13	−	6.77664e12i	−0.767171	−	0.442926i
$852$		0			0
$853$			4.17267e12i			0.269863i	0.990855	+	0.134932i	$0.0430814\pi$
$853$			4.17267e12i			0.269863i	−0.990855	+	0.134932i	$0.956919\pi$
$854$		0			0
$855$			3.60082e12i			0.230438i
$856$		0			0
$857$	−4.18438e12	−	2.41585e12i	−0.264982	−	0.152988i	0.361623	−	0.932324i	$-0.382223\pi$
$857$	−4.18438e12	−	2.41585e12i	−0.264982	−	0.152988i	−0.626605	+	0.779337i	$0.715556\pi$
$858$		0			0
$859$	2.69958e12	+	4.67581e12i	0.169172	+	0.293014i	0.938129	−	0.346286i	$-0.112557\pi$
$859$	2.69958e12	+	4.67581e12i	0.169172	+	0.293014i	−0.768957	+	0.639300i	$0.779224\pi$
$860$		0			0
$861$	3.19471e12	−	1.92044e12i	0.198115	−	0.119093i
$862$		0			0
$863$	−1.06144e13	+	6.12824e12i	−0.651400	+	0.376086i	−0.788992	−	0.614403i	$-0.789397\pi$
$863$	−1.06144e13	+	6.12824e12i	−0.651400	+	0.376086i	0.137592	+	0.990489i	$0.456064\pi$
$864$		0			0
$865$	1.05930e12	−	1.83476e12i	0.0643347	−	0.111431i
$866$		0			0
$867$	−3.36787e12			−0.202427
$868$		0			0
$869$	−8.86120e12			−0.527113
$870$		0			0
$871$	−1.22234e13	+	2.11715e13i	−0.719630	+	1.24644i
$872$		0			0
$873$	−1.18728e13	+	6.85475e12i	−0.691812	+	0.399418i
$874$		0			0
$875$	−1.24963e13	−	6.92357e12i	−0.720686	−	0.399295i
$876$		0			0
$877$	−1.19930e13	−	2.07725e13i	−0.684588	−	1.18574i	−0.973566	−	0.228406i	$-0.926649\pi$
$877$	−1.19930e13	−	2.07725e13i	−0.684588	−	1.18574i	0.288978	−	0.957336i	$-0.406685\pi$
$878$		0			0
$879$	−6.86031e12	−	3.96080e12i	−0.387609	−	0.223786i
$880$		0			0
$881$		−	1.04639e13i		−	0.585194i	−0.956236	−	0.292597i	$-0.905481\pi$
$881$		−	1.04639e13i		−	0.585194i	0.956236	−	0.292597i	$-0.0945195\pi$
$882$		0			0
$883$			4.30650e12i			0.238397i	0.992870	+	0.119199i	$0.0380325\pi$
$883$			4.30650e12i			0.238397i	−0.992870	+	0.119199i	$0.961968\pi$
$884$		0			0
$885$	3.59902e12	+	2.07789e12i	0.197215	+	0.113862i
$886$		0			0
$887$	−1.40603e13	−	2.43531e13i	−0.762671	−	1.32099i	−0.941469	−	0.337100i	$-0.890554\pi$
$887$	−1.40603e13	−	2.43531e13i	−0.762671	−	1.32099i	0.178798	−	0.983886i	$-0.442779\pi$
$888$		0			0
$889$	1.24808e13	+	6.91494e12i	0.670167	+	0.371305i
$890$		0			0
$891$	5.21780e11	−	3.01250e11i	0.0277356	−	0.0160132i
$892$		0			0
$893$	4.99449e12	−	8.65071e12i	0.262821	−	0.455219i
$894$		0			0
$895$	1.09989e13			0.572989
$896$		0			0
$897$	−7.04723e12			−0.363457
$898$		0			0
$899$	2.55132e12	−	4.41901e12i	0.130270	−	0.225635i
$900$		0			0
$901$	1.39521e13	−	8.05527e12i	0.705309	−	0.407210i
$902$		0			0
$903$	−4.01475e12	+	2.41339e12i	−0.200939	+	0.120791i
$904$		0			0
$905$	−7.50635e11	−	1.30014e12i	−0.0371972	−	0.0644274i
$906$		0			0
$907$	−2.40984e12	−	1.39132e12i	−0.118238	−	0.0682645i	0.439715	−	0.898137i	$-0.355079\pi$
$907$	−2.40984e12	−	1.39132e12i	−0.118238	−	0.0682645i	−0.557952	+	0.829873i	$0.688413\pi$
$908$		0			0
$909$		−	1.06766e13i		−	0.518675i
$910$		0			0
$911$		−	2.52613e13i		−	1.21513i	−0.794269	−	0.607566i	$-0.792146\pi$
$911$		−	2.52613e13i		−	1.21513i	0.794269	−	0.607566i	$-0.207854\pi$
$912$		0			0
$913$	−1.28659e13	−	7.42816e12i	−0.612806	−	0.353804i
$914$		0			0
$915$	−4.56682e10	−	7.90996e10i	−0.00215387	−	0.00373060i
$916$		0			0
$917$	−2.18543e12	+	3.85825e10i	−0.102064	+	0.00180189i
$918$		0			0
$919$	−3.24910e13	+	1.87587e13i	−1.50260	+	0.867525i	−0.502602	+	0.864518i	$0.667624\pi$
$919$	−3.24910e13	+	1.87587e13i	−1.50260	+	0.867525i	−0.999995	+	0.00300720i	$0.999043\pi$
$920$		0			0
$921$	2.92403e11	−	5.06458e11i	0.0133910	−	0.0231939i
$922$		0			0
$923$	−1.42052e13			−0.644230
$924$		0			0
$925$	−2.11787e13			−0.951176
$926$		0			0
$927$	−8.43537e12	+	1.46105e13i	−0.375185	+	0.649839i
$928$		0			0
$929$	1.85035e13	−	1.06830e13i	0.815047	−	0.470568i	−0.0336583	−	0.999433i	$-0.510716\pi$
$929$	1.85035e13	−	1.06830e13i	0.815047	−	0.470568i	0.848706	+	0.528866i	$0.177382\pi$
$930$		0			0
$931$	−8.22440e12	+	1.54827e13i	−0.358782	+	0.675420i
$932$		0			0
$933$	4.82958e12	+	8.36507e12i	0.208661	+	0.361412i
$934$		0			0
$935$	2.98518e12	+	1.72349e12i	0.127737	+	0.0737492i
$936$		0			0
$937$			1.83690e13i			0.778496i	0.921133	+	0.389248i	$0.127265\pi$
$937$			1.83690e13i			0.778496i	−0.921133	+	0.389248i	$0.872735\pi$
$938$		0			0
$939$			2.96180e12i			0.124326i
$940$		0			0
$941$	4.78314e12	+	2.76155e12i	0.198866	+	0.114815i	0.596126	−	0.802891i	$-0.296706\pi$
$941$	4.78314e12	+	2.76155e12i	0.198866	+	0.114815i	−0.397261	+	0.917706i	$0.630039\pi$
$942$		0			0
$943$	−3.49819e12	−	6.05904e12i	−0.144059	−	0.249518i
$944$		0			0
$945$	1.94100e11	+	1.09944e13i	0.00791741	+	0.448465i
$946$		0			0
$947$	8.60981e12	−	4.97088e12i	0.347871	−	0.200844i	−0.315876	−	0.948801i	$-0.602298\pi$
$947$	8.60981e12	−	4.97088e12i	0.347871	−	0.200844i	0.663747	+	0.747957i	$0.268965\pi$
$948$		0			0
$949$	−8.74442e12	+	1.51458e13i	−0.349972	+	0.606169i
$950$		0			0
$951$	−1.97742e13			−0.783948
$952$		0			0
$953$	1.51594e13			0.595339			0.297669	−	0.954669i	$-0.403791\pi$
$953$	1.51594e13			0.595339			0.297669	+	0.954669i	$0.403791\pi$
$954$		0			0
$955$	8.43617e12	−	1.46119e13i	0.328193	−	0.568448i
$956$		0			0
$957$	−3.83716e12	+	2.21539e12i	−0.147879	+	0.0853779i
$958$		0			0
$959$	−8.33332e12	−	1.38627e13i	−0.318152	−	0.529254i
$960$		0			0
$961$	1.15561e13	+	2.00157e13i	0.437073	+	0.757033i
$962$		0			0
$963$	6.57601e12	+	3.79666e12i	0.246402	+	0.142260i
$964$		0			0
$965$		−	4.21876e12i		−	0.156607i
$966$		0			0
$967$			1.63075e13i			0.599747i	0.953979	+	0.299873i	$0.0969445\pi$
$967$			1.63075e13i			0.599747i	−0.953979	+	0.299873i	$0.903056\pi$
$968$		0			0
$969$	−8.66374e12	−	5.00201e12i	−0.315681	−	0.182259i
$970$		0			0
$971$	−1.96622e13	−	3.40560e13i	−0.709817	−	1.22944i	−0.964925	−	0.262527i	$-0.915444\pi$
$971$	−1.96622e13	−	3.40560e13i	−0.709817	−	1.22944i	0.255108	−	0.966913i	$-0.417889\pi$
$972$		0			0
$973$	−8.18218e12	+	1.47680e13i	−0.292658	+	0.528218i
$974$		0			0
$975$	−9.53683e12	+	5.50609e12i	−0.337974	+	0.195129i
$976$		0			0
$977$	4.99866e12	−	8.65794e12i	0.175521	−	0.304011i	−0.764821	−	0.644243i	$-0.777172\pi$
$977$	4.99866e12	−	8.65794e12i	0.175521	−	0.304011i	0.940341	+	0.340233i	$0.110506\pi$
$978$		0			0
$979$	−1.17618e13			−0.409216
$980$		0			0
$981$	−4.77447e12			−0.164594
$982$		0			0
$983$	−5.90189e12	+	1.02224e13i	−0.201604	+	0.349189i	−0.949046	−	0.315139i	$-0.897949\pi$
$983$	−5.90189e12	+	1.02224e13i	−0.201604	+	0.349189i	0.747441	+	0.664328i	$0.231282\pi$
$984$		0			0
$985$	8.99765e12	−	5.19479e12i	0.304555	−	0.175835i
$986$		0			0
$987$	5.84388e12	−	1.05476e13i	0.196008	−	0.353775i
$988$		0			0
$989$	4.39612e12	+	7.61431e12i	0.146112	+	0.253074i
$990$		0			0
$991$	−1.52775e13	−	8.82048e12i	−0.503178	−	0.290510i	0.226847	−	0.973930i	$-0.427158\pi$
$991$	−1.52775e13	−	8.82048e12i	−0.503178	−	0.290510i	−0.730025	+	0.683421i	$0.760492\pi$
$992$		0			0
$993$		−	1.72960e13i		−	0.564512i
$994$		0			0
$995$		−	3.95954e12i		−	0.128068i
$996$		0			0
$997$	−2.49950e13	−	1.44309e13i	−0.801171	−	0.462556i	0.0427097	−	0.999088i	$-0.486401\pi$
$997$	−2.49950e13	−	1.44309e13i	−0.801171	−	0.462556i	−0.843880	+	0.536531i	$0.819734\pi$
$998$		0			0
$999$	1.84999e13	+	3.20428e13i	0.587659	+	1.01785i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.b.31.8	yes	24
4.3	odd	2	inner	112.10.p.b.31.5	✓	24
7.5	odd	6	inner	112.10.p.b.47.5	yes	24
28.19	even	6	inner	112.10.p.b.47.8	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.b.31.5	✓	24	4.3	odd	2	inner
112.10.p.b.31.8	yes	24	1.1	even	1	trivial
112.10.p.b.47.5	yes	24	7.5	odd	6	inner
112.10.p.b.47.8	yes	24	28.19	even	6	inner

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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(41.2793 - 71.4979i) q^{3} +(-557.845 + 322.072i) q^{5} +(3078.62 - 5556.59i) q^{7} +(6433.53 + 11143.2i) q^{9} +O(q^{10})\)	\(q+(41.2793 - 71.4979i) q^{3} +(-557.845 + 322.072i) q^{5} +(3078.62 - 5556.59i) q^{7} +(6433.53 + 11143.2i) q^{9} +(-16615.5 - 9592.95i) q^{11} +86715.5i q^{13} +53179.7i q^{15} +(241549. + 139458. i) q^{17} +(-217224. - 376243. i) q^{19} +(-270201. - 449487. i) q^{21} +(-852491. + 492186. i) q^{23} +(-769101. + 1.33212e6i) q^{25} +2.68729e6 q^{27} +2.79727e6 q^{29} +(912073. - 1.57976e6i) q^{31} +(-1.37175e6 + 791981. i) q^{33} +(72229.0 + 4.09126e6i) q^{35} +(6.88423e6 + 1.19238e7i) q^{37} +(6.19998e6 + 3.57956e6i) q^{39} +7.10745e6i q^{41} -8.93184e6i q^{43} +(-7.17783e6 - 4.14412e6i) q^{45} +(1.14962e7 + 1.99119e7i) q^{47} +(-2.13978e7 - 3.42133e7i) q^{49} +(1.99419e7 - 1.15135e7i) q^{51} +(2.88806e7 - 5.00227e7i) q^{53} +1.23585e7 q^{55} -3.58675e7 q^{57} +(3.90730e7 - 6.76765e7i) q^{59} +(-1.48740e6 + 858751. i) q^{61} +(8.17246e7 - 1.44281e6i) q^{63} +(-2.79287e7 - 4.83739e7i) q^{65} +(2.44149e8 + 1.40959e8i) q^{67} +8.12684e7i q^{69} +1.63814e8i q^{71} +(1.74661e8 + 1.00840e8i) q^{73} +(6.34960e7 + 1.09978e8i) q^{75} +(-1.04457e8 + 6.27924e7i) q^{77} +(3.99982e8 - 2.30930e8i) q^{79} +(-1.57016e7 + 2.71960e7i) q^{81} +7.74335e8 q^{83} -1.79662e8 q^{85} +(1.15469e8 - 1.99999e8i) q^{87} +(5.30912e8 - 3.06522e8i) q^{89} +(4.81843e8 + 2.66964e8i) q^{91} +(-7.52996e7 - 1.30423e8i) q^{93} +(2.42355e8 + 1.39924e8i) q^{95} +1.06547e9i q^{97} -2.46866e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 1704 q^{5} - 80428 q^{9}+O(q^{10}) \) 24 * q + 1704 * q^5 - 80428 * q^9	\( 24 q + 1704 q^{5} - 80428 q^{9} - 1578672 q^{17} + 1219540 q^{21} + 8001240 q^{25} - 6709416 q^{29} - 29129772 q^{33} - 11130084 q^{37} + 57023292 q^{45} - 12671904 q^{49} - 93652164 q^{53} + 742621544 q^{57} - 593611308 q^{61} + 160281180 q^{65} - 1676922516 q^{73} + 1645751688 q^{77} - 528698056 q^{81} + 1370082456 q^{85} - 941603484 q^{89} - 1432348316 q^{93}+O(q^{100}) \) 24 * q + 1704 * q^5 - 80428 * q^9 - 1578672 * q^17 + 1219540 * q^21 + 8001240 * q^25 - 6709416 * q^29 - 29129772 * q^33 - 11130084 * q^37 + 57023292 * q^45 - 12671904 * q^49 - 93652164 * q^53 + 742621544 * q^57 - 593611308 * q^61 + 160281180 * q^65 - 1676922516 * q^73 + 1645751688 * q^77 - 528698056 * q^81 + 1370082456 * q^85 - 941603484 * q^89 - 1432348316 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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