LMFDB - Embedded newform 112.10.i.b.81.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.65"); S:= CuspForms(chi, 10); N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,233] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$57.6840136504$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} + 1116x^{4} - 3085x^{3} + 1245325x^{2} - 2341500x + 4410000 $ x^6 - x^5 + 1116x^4 - 3085x^3 + 1245325x^2 - 2341500x + 4410000
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.2
Root			$0.943118 + 1.63353i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.10.i.b.65.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-6.98734 + 12.1024i) q^{3} +(859.469 + 1488.64i) q^{5} +(-1802.93 + 6091.23i) q^{7} +(9743.85 + 16876.9i) q^{9} +(-35661.7 + 61767.9i) q^{11} +156547. q^{13} -24021.6 q^{15} +(-255856. + 443155. i) q^{17} +(-97247.0 - 168437. i) q^{19} +(-61120.9 - 64381.3i) q^{21} +(-54078.1 - 93666.0i) q^{23} +(-500810. + 867429. i) q^{25} -547398. q^{27} +4.21769e6 q^{29} +(1.58367e6 - 2.74299e6i) q^{31} +(-498361. - 863187. i) q^{33} +(-1.06172e7 + 2.55130e6i) q^{35} +(-7.20686e6 - 1.24827e7i) q^{37} +(-1.09385e6 + 1.89460e6i) q^{39} +7.69007e6 q^{41} +3.64544e7 q^{43} +(-1.67491e7 + 2.90102e7i) q^{45} +(9.11895e6 + 1.57945e7i) q^{47} +(-3.38525e7 - 2.19641e7i) q^{49} +(-3.57550e6 - 6.19295e6i) q^{51} +(-2.26037e7 + 3.91507e7i) q^{53} -1.22601e8 q^{55} +2.71799e6 q^{57} +(-4.89074e7 + 8.47100e7i) q^{59} +(-5.73352e7 - 9.93075e7i) q^{61} +(-1.20368e8 + 2.89242e7i) q^{63} +(1.34547e8 + 2.33042e8i) q^{65} +(1.00808e8 - 1.74605e8i) q^{67} +1.51145e6 q^{69} -2.08831e7 q^{71} +(-4.42392e6 + 7.66245e6i) q^{73} +(-6.99866e6 - 1.21220e7i) q^{75} +(-3.11947e8 - 3.28587e8i) q^{77} +(1.92244e7 + 3.32976e7i) q^{79} +(-1.87963e8 + 3.25562e8i) q^{81} -5.55300e8 q^{83} -8.79600e8 q^{85} +(-2.94704e7 + 5.10443e7i) q^{87} +(8.24594e7 + 1.42824e8i) q^{89} +(-2.82243e8 + 9.53562e8i) q^{91} +(2.21312e7 + 3.83324e7i) q^{93} +(1.67162e8 - 2.89532e8i) q^{95} +1.57025e8 q^{97} -1.38993e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 233 q^{3} - 733 q^{5} - 5012 q^{7} - 15058 q^{9} - 7339 q^{11} + 197036 q^{13} - 738238 q^{15} - 306665 q^{17} + 377991 q^{19} - 1585927 q^{21} + 2267255 q^{23} - 142612 q^{25} - 21348358 q^{27} - 13085956 q^{29}+ \cdots + 1256218868 q^{99}+O(q^{100}) $ 6 * q + 233 * q^3 - 733 * q^5 - 5012 * q^7 - 15058 * q^9 - 7339 * q^11 + 197036 * q^13 - 738238 * q^15 - 306665 * q^17 + 377991 * q^19 - 1585927 * q^21 + 2267255 * q^23 - 142612 * q^25 - 21348358 * q^27 - 13085956 * q^29 + 6654517 * q^31 + 3747977 * q^33 - 22864807 * q^35 - 22287969 * q^37 - 3151974 * q^39 + 68096196 * q^41 + 125648280 * q^43 - 86300638 * q^45 + 52703019 * q^47 + 57596070 * q^49 - 97736333 * q^51 - 12091125 * q^53 - 377435398 * q^55 - 158517670 * q^57 + 12949897 * q^59 - 160252153 * q^61 + 356021666 * q^63 + 334191270 * q^65 + 480890225 * q^67 + 1131233162 * q^69 + 74421440 * q^71 + 251382283 * q^73 - 16322324 * q^75 - 527045155 * q^77 - 286494785 * q^79 - 2165894587 * q^81 - 2295182344 * q^83 - 2389185074 * q^85 - 1412199358 * q^87 - 901243845 * q^89 - 1026798920 * q^91 + 710456889 * q^93 + 887366177 * q^95 + 629707876 * q^97 + 1256218868 * q^99

Character values

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

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	−6.98734	+	12.1024i	−0.0498042	+	0.0862634i	−0.889853	−	0.456248i	$-0.849193\pi$
$3$	−6.98734	+	12.1024i	−0.0498042	+	0.0862634i	0.840049	+	0.542511i	$0.182526\pi$
$4$		0			0
$5$	859.469	+	1488.64i	0.614986	+	1.06519i	0.990387	+	0.138324i	$0.0441716\pi$
$5$	859.469	+	1488.64i	0.614986	+	1.06519i	−0.375401	+	0.926862i	$0.622495\pi$
$6$		0			0
$7$	−1802.93	+	6091.23i	−0.283817	+	0.958878i
$7$	−1802.93	+	6091.23i	−0.283817	+	0.958878i
$8$		0			0
$9$	9743.85	+	16876.9i	0.495039	+	0.857433i
$10$		0			0
$11$	−35661.7	+	61767.9i	−0.734404	+	1.27203i	0.220580	+	0.975369i	$0.429205\pi$
$11$	−35661.7	+	61767.9i	−0.734404	+	1.27203i	−0.954984	+	0.296657i	$0.904128\pi$
$12$		0			0
$13$	156547.			1.52019			0.760097	−	0.649809i	$-0.225151\pi$
$13$	156547.			1.52019			0.760097	+	0.649809i	$0.225151\pi$
$14$		0			0
$15$	−24021.6			−0.122516
$16$		0			0
$17$	−255856.	+	443155.i	−0.742976	+	1.28687i	0.208158	+	0.978095i	$0.433253\pi$
$17$	−255856.	+	443155.i	−0.742976	+	1.28687i	−0.951134	+	0.308778i	$0.900080\pi$
$18$		0			0
$19$	−97247.0	−	168437.i	−0.171193	−	0.296514i	0.767644	−	0.640876i	$-0.221429\pi$
$19$	−97247.0	−	168437.i	−0.171193	−	0.296514i	−0.938837	+	0.344362i	$0.888095\pi$
$20$		0			0
$21$	−61120.9	−	64381.3i	−0.0685809	−	0.0722392i
$22$		0			0
$23$	−54078.1	−	93666.0i	−0.0402945	−	0.0697922i	0.845175	−	0.534490i	$-0.179496\pi$
$23$	−54078.1	−	93666.0i	−0.0402945	−	0.0697922i	−0.885469	+	0.464698i	$0.846163\pi$
$24$		0			0
$25$	−500810.	+	867429.i	−0.256415	+	0.444124i
$26$		0			0
$27$	−547398.			−0.198229
$28$		0			0
$29$	4.21769e6			1.10735			0.553674	−	0.832734i	$-0.313226\pi$
$29$	4.21769e6			1.10735			0.553674	+	0.832734i	$0.313226\pi$
$30$		0			0
$31$	1.58367e6	−	2.74299e6i	0.307990	−	0.533454i	−0.669933	−	0.742422i	$-0.733677\pi$
$31$	1.58367e6	−	2.74299e6i	0.307990	−	0.533454i	0.977922	+	0.208968i	$0.0670105\pi$
$32$		0			0
$33$	−498361.	−	863187.i	−0.0731529	−	0.126704i
$34$		0			0
$35$	−1.06172e7	+	2.55130e6i	−1.19593	+	0.287379i
$36$		0			0
$37$	−7.20686e6	−	1.24827e7i	−0.632177	−	1.09496i	−0.987106	−	0.160069i	$-0.948828\pi$
$37$	−7.20686e6	−	1.24827e7i	−0.632177	−	1.09496i	0.354929	−	0.934893i	$-0.384505\pi$
$38$		0			0
$39$	−1.09385e6	+	1.89460e6i	−0.0757121	+	0.131137i
$40$		0			0
$41$	7.69007e6			0.425014			0.212507	−	0.977160i	$-0.431837\pi$
$41$	7.69007e6			0.425014			0.212507	+	0.977160i	$0.431837\pi$
$42$		0			0
$43$	3.64544e7			1.62608			0.813040	−	0.582208i	$-0.197811\pi$
$43$	3.64544e7			1.62608			0.813040	+	0.582208i	$0.197811\pi$
$44$		0			0
$45$	−1.67491e7	+	2.90102e7i	−0.608884	+	1.05462i
$46$		0			0
$47$	9.11895e6	+	1.57945e7i	0.272587	+	0.472134i	0.969523	−	0.244999i	$-0.0787876\pi$
$47$	9.11895e6	+	1.57945e7i	0.272587	+	0.472134i	−0.696937	+	0.717132i	$0.745454\pi$
$48$		0			0
$49$	−3.38525e7	−	2.19641e7i	−0.838896	−	0.544292i
$50$		0			0
$51$	−3.57550e6	−	6.19295e6i	−0.0740067	−	0.128183i
$52$		0			0
$53$	−2.26037e7	+	3.91507e7i	−0.393494	+	0.681551i	−0.992908	−	0.118888i	$-0.962067\pi$
$53$	−2.26037e7	+	3.91507e7i	−0.393494	+	0.681551i	0.599414	+	0.800439i	$0.295400\pi$
$54$		0			0
$55$	−1.22601e8			−1.80659
$56$		0			0
$57$	2.71799e6			0.0341045
$58$		0			0
$59$	−4.89074e7	+	8.47100e7i	−0.525461	+	0.910124i	0.474100	+	0.880471i	$0.342774\pi$
$59$	−4.89074e7	+	8.47100e7i	−0.525461	+	0.910124i	−0.999560	+	0.0296532i	$0.990560\pi$
$60$		0			0
$61$	−5.73352e7	−	9.93075e7i	−0.530197	−	0.918328i	−0.999379	−	0.0352269i	$-0.988785\pi$
$61$	−5.73352e7	−	9.93075e7i	−0.530197	−	0.918328i	0.469182	−	0.883101i	$-0.344549\pi$
$62$		0			0
$63$	−1.20368e8	+	2.89242e7i	−0.962674	+	0.231328i
$64$		0			0
$65$	1.34547e8	+	2.33042e8i	0.934898	+	1.61929i
$66$		0			0
$67$	1.00808e8	−	1.74605e8i	0.611166	−	1.05857i	−0.379878	−	0.925036i	$-0.624034\pi$
$67$	1.00808e8	−	1.74605e8i	0.611166	−	1.05857i	0.991044	−	0.133534i	$-0.0426325\pi$
$68$		0			0
$69$	1.51145e6			0.00802735
$70$		0			0
$71$	−2.08831e7			−0.0975288			−0.0487644	−	0.998810i	$-0.515528\pi$
$71$	−2.08831e7			−0.0975288			−0.0487644	+	0.998810i	$0.515528\pi$
$72$		0			0
$73$	−4.42392e6	+	7.66245e6i	−0.0182328	+	0.0315802i	−0.874998	−	0.484127i	$-0.839137\pi$
$73$	−4.42392e6	+	7.66245e6i	−0.0182328	+	0.0315802i	0.856765	+	0.515707i	$0.172471\pi$
$74$		0			0
$75$	−6.99866e6	−	1.21220e7i	−0.0255411	−	0.0442385i
$76$		0			0
$77$	−3.11947e8	−	3.28587e8i	−1.01128	−	1.06523i
$78$		0			0
$79$	1.92244e7	+	3.32976e7i	0.0555304	+	0.0961814i	0.892454	−	0.451138i	$-0.148982\pi$
$79$	1.92244e7	+	3.32976e7i	0.0555304	+	0.0961814i	−0.836924	+	0.547319i	$0.815648\pi$
$80$		0			0
$81$	−1.87963e8	+	3.25562e8i	−0.485166	+	0.840333i
$82$		0			0
$83$	−5.55300e8			−1.28433			−0.642164	−	0.766567i	$-0.721963\pi$
$83$	−5.55300e8			−1.28433			−0.642164	+	0.766567i	$0.721963\pi$
$84$		0			0
$85$	−8.79600e8			−1.82768
$86$		0			0
$87$	−2.94704e7	+	5.10443e7i	−0.0551506	+	0.0955236i
$88$		0			0
$89$	8.24594e7	+	1.42824e8i	0.139311	+	0.241294i	0.927236	−	0.374478i	$-0.122178\pi$
$89$	8.24594e7	+	1.42824e8i	0.139311	+	0.241294i	−0.787925	+	0.615771i	$0.788845\pi$
$90$		0			0
$91$	−2.82243e8	+	9.53562e8i	−0.431457	+	1.45768i
$92$		0			0
$93$	2.21312e7	+	3.83324e7i	0.0306784	+	0.0531365i
$94$		0			0
$95$	1.67162e8	−	2.89532e8i	0.210562	−	0.364704i
$96$		0			0
$97$	1.57025e8			0.180093			0.0900466	−	0.995938i	$-0.471298\pi$
$97$	1.57025e8			0.180093			0.0900466	+	0.995938i	$0.471298\pi$
$98$		0			0
$99$	−1.38993e9			−1.45424
$100$		0			0
$101$	−8.13868e7	+	1.40966e8i	−0.0778230	+	0.134793i	−0.902310	−	0.431087i	$-0.858130\pi$
$101$	−8.13868e7	+	1.40966e8i	−0.0778230	+	0.134793i	0.824487	+	0.565880i	$0.191464\pi$
$102$		0			0
$103$	6.77028e8	+	1.17265e9i	0.592705	+	1.02660i	0.993866	+	0.110588i	$0.0352735\pi$
$103$	6.77028e8	+	1.17265e9i	0.592705	+	1.02660i	−0.401161	+	0.916008i	$0.631393\pi$
$104$		0			0
$105$	4.33093e7	−	1.46321e8i	0.0347720	−	0.117478i
$106$		0			0
$107$	4.39302e6	+	7.60893e6i	0.00323993	+	0.00561173i	0.867641	−	0.497192i	$-0.165635\pi$
$107$	4.39302e6	+	7.60893e6i	0.00323993	+	0.00561173i	−0.864401	+	0.502803i	$0.832302\pi$
$108$		0			0
$109$	−3.61434e8	+	6.26023e8i	−0.245251	+	0.424787i	−0.962202	−	0.272337i	$-0.912204\pi$
$109$	−3.61434e8	+	6.26023e8i	−0.245251	+	0.424787i	0.716951	+	0.697123i	$0.245537\pi$
$110$		0			0
$111$	2.01427e8			0.125940
$112$		0			0
$113$	−2.94124e9			−1.69698			−0.848492	−	0.529209i	$-0.822489\pi$
$113$	−2.94124e9			−1.69698			−0.848492	+	0.529209i	$0.822489\pi$
$114$		0			0
$115$	9.29568e7	−	1.61006e8i	0.0495611	−	0.0858424i
$116$		0			0
$117$	1.52537e9	+	2.64202e9i	0.752556	+	1.30346i
$118$		0			0
$119$	−2.23807e9	−	2.35745e9i	−1.02309	−	1.07766i
$120$		0			0
$121$	−1.36454e9	−	2.36346e9i	−0.578699	−	1.00234i
$122$		0			0
$123$	−5.37331e7	+	9.30685e7i	−0.0211675	+	0.0366631i
$124$		0			0
$125$	1.63558e9			0.599205
$126$		0			0
$127$	6.47761e8			0.220952			0.110476	−	0.993879i	$-0.464762\pi$
$127$	6.47761e8			0.220952			0.110476	+	0.993879i	$0.464762\pi$
$128$		0			0
$129$	−2.54719e8	+	4.41187e8i	−0.0809857	+	0.140271i
$130$		0			0
$131$	−1.47377e9	−	2.55264e9i	−0.437228	−	0.757301i	0.560246	−	0.828326i	$-0.310706\pi$
$131$	−1.47377e9	−	2.55264e9i	−0.437228	−	0.757301i	−0.997475	+	0.0710247i	$0.977373\pi$
$132$		0			0
$133$	1.20132e9	−	2.88674e8i	0.332909	−	0.0799972i
$134$		0			0
$135$	−4.70472e8	−	8.14881e8i	−0.121908	−	0.211150i
$136$		0			0
$137$	3.85668e9	−	6.67997e9i	0.935344	−	1.62006i	0.161325	−	0.986901i	$-0.448423\pi$
$137$	3.85668e9	−	6.67997e9i	0.935344	−	1.62006i	0.774019	−	0.633162i	$-0.218243\pi$
$138$		0			0
$139$	5.40626e9			1.22837			0.614187	−	0.789161i	$-0.289484\pi$
$139$	5.40626e9			1.22837			0.614187	+	0.789161i	$0.289484\pi$
$140$		0			0
$141$	−2.54869e8			−0.0543038
$142$		0			0
$143$	−5.58273e9	+	9.66957e9i	−1.11644	+	1.93373i
$144$		0			0
$145$	3.62497e9	+	6.27864e9i	0.681003	+	1.17953i
$146$		0			0
$147$	5.02358e8	−	2.56226e8i	0.0887331	−	0.0452580i
$148$		0			0
$149$	−1.55301e7	−	2.68989e7i	−0.00258129	−	0.00447092i	0.864732	−	0.502234i	$-0.167488\pi$
$149$	−1.55301e7	−	2.68989e7i	−0.00258129	−	0.00447092i	−0.867313	+	0.497763i	$0.834155\pi$
$150$		0			0
$151$	4.08350e9	−	7.07283e9i	0.639199	−	1.10713i	−0.346409	−	0.938083i	$-0.612599\pi$
$151$	4.08350e9	−	7.07283e9i	0.639199	−	1.10713i	0.985609	−	0.169043i	$-0.0540675\pi$
$152$		0			0
$153$	−9.97208e9			−1.47121
$154$		0			0
$155$	5.44445e9			0.757637
$156$		0			0
$157$	−7.72057e7	+	1.33724e8i	−0.0101415	+	0.0175655i	−0.871052	−	0.491192i	$-0.836562\pi$
$157$	−7.72057e7	+	1.33724e8i	−0.0101415	+	0.0175655i	0.860910	+	0.508757i	$0.169895\pi$
$158$		0			0
$159$	−3.15879e8	−	5.47119e8i	−0.0391953	−	0.0678882i
$160$		0			0
$161$	6.68040e8	−	1.60528e8i	0.0783585	−	0.0188294i
$162$		0			0
$163$	5.72335e9	+	9.91314e9i	0.635048	+	1.09994i	0.986505	+	0.163731i	$0.0523529\pi$
$163$	5.72335e9	+	9.91314e9i	0.635048	+	1.09994i	−0.351457	+	0.936204i	$0.614314\pi$
$164$		0			0
$165$	8.56652e8	−	1.48376e9i	0.0899759	−	0.155843i
$166$		0			0
$167$	1.71056e10			1.70182			0.850911	−	0.525310i	$-0.176051\pi$
$167$	1.71056e10			1.70182			0.850911	+	0.525310i	$0.176051\pi$
$168$		0			0
$169$	1.39024e10			1.31099
$170$		0			0
$171$	1.89512e9	−	3.28245e9i	0.169494	−	0.293572i
$172$		0			0
$173$	9.09679e8	+	1.57561e9i	0.0772113	+	0.133734i	0.902046	−	0.431640i	$-0.142065\pi$
$173$	9.09679e8	+	1.57561e9i	0.0772113	+	0.133734i	−0.824834	+	0.565374i	$0.808732\pi$
$174$		0			0
$175$	−4.38078e9	−	4.61447e9i	−0.353086	−	0.371921i
$176$		0			0
$177$	−6.83465e8	−	1.18380e9i	−0.0523403	−	0.0906561i
$178$		0			0
$179$	−1.18157e9	+	2.04654e9i	−0.0860241	+	0.148998i	−0.905827	−	0.423647i	$-0.860750\pi$
$179$	−1.18157e9	+	2.04654e9i	−0.0860241	+	0.148998i	0.819803	+	0.572646i	$0.194083\pi$
$180$		0			0
$181$	1.65554e10			1.14653			0.573265	−	0.819370i	$-0.305677\pi$
$181$	1.65554e10			1.14653			0.573265	+	0.819370i	$0.305677\pi$
$182$		0			0
$183$	1.60248e9			0.105624
$184$		0			0
$185$	1.23881e10	−	2.14569e10i	0.777559	−	1.34677i
$186$		0			0
$187$	−1.82485e10	−	3.16073e10i	−1.09129	−	1.89017i
$188$		0			0
$189$	9.86922e8	−	3.33433e9i	0.0562606	−	0.190077i
$190$		0			0
$191$	−1.68265e9	−	2.91443e9i	−0.0914835	−	0.158454i	0.816652	−	0.577130i	$-0.195828\pi$
$191$	−1.68265e9	−	2.91443e9i	−0.0914835	−	0.158454i	−0.908136	+	0.418676i	$0.862494\pi$
$192$		0			0
$193$	2.36192e9	−	4.09097e9i	0.122534	−	0.212236i	−0.798232	−	0.602350i	$-0.794231\pi$
$193$	2.36192e9	−	4.09097e9i	0.122534	−	0.212236i	0.920766	+	0.390114i	$0.127564\pi$
$194$		0			0
$195$	−3.76051e9			−0.186248
$196$		0			0
$197$	−1.66982e10			−0.789897			−0.394949	−	0.918703i	$-0.629238\pi$
$197$	−1.66982e10			−0.789897			−0.394949	+	0.918703i	$0.629238\pi$
$198$		0			0
$199$	1.30913e10	−	2.26748e10i	0.591759	−	1.02496i	−0.402237	−	0.915536i	$-0.631767\pi$
$199$	1.30913e10	−	2.26748e10i	0.591759	−	1.02496i	0.993996	−	0.109421i	$-0.0348996\pi$
$200$		0			0
$201$	1.40876e9	+	2.44005e9i	0.0608773	+	0.105443i
$202$		0			0
$203$	−7.60421e9	+	2.56909e10i	−0.314284	+	1.06181i
$204$		0			0
$205$	6.60937e9	+	1.14478e10i	0.261377	+	0.452719i
$206$		0			0
$207$	1.05386e9	−	1.82534e9i	0.0398947	−	0.0690997i
$208$		0			0
$209$	1.38720e10			0.502898
$210$		0			0
$211$	3.90431e10			1.35604			0.678021	−	0.735042i	$-0.262838\pi$
$211$	3.90431e10			1.35604			0.678021	+	0.735042i	$0.262838\pi$
$212$		0			0
$213$	1.45917e8	−	2.52736e8i	0.00485734	−	0.00841317i
$214$		0			0
$215$	3.13314e10	+	5.42676e10i	1.00002	+	1.73208i
$216$		0			0
$217$	1.38529e10	+	1.45919e10i	0.424104	+	0.446728i
$218$		0			0
$219$	−6.18228e7	−	1.07080e8i	−0.00181614	−	0.00314565i
$220$		0			0
$221$	−4.00534e10	+	6.93745e10i	−1.12947	+	1.95630i
$222$		0			0
$223$	−4.53210e10			−1.22723			−0.613617	−	0.789603i	$-0.710286\pi$
$223$	−4.53210e10			−1.22723			−0.613617	+	0.789603i	$0.710286\pi$
$224$		0			0
$225$	−1.95193e10			−0.507742
$226$		0			0
$227$	3.75422e9	−	6.50249e9i	0.0938432	−	0.162541i	−0.815282	−	0.579064i	$-0.803418\pi$
$227$	3.75422e9	−	6.50249e9i	0.0938432	−	0.162541i	0.909125	+	0.416523i	$0.136751\pi$
$228$		0			0
$229$	−2.09237e10	−	3.62409e10i	−0.502781	−	0.870842i	−0.999995	−	0.00321410i	$-0.998977\pi$
$229$	−2.09237e10	−	3.62409e10i	−0.502781	−	0.870842i	0.497214	−	0.867628i	$-0.334356\pi$
$230$		0			0
$231$	6.15638e9	−	1.47936e9i	0.142256	−	0.0341838i
$232$		0			0
$233$	−1.52461e10	−	2.64070e10i	−0.338888	−	0.586971i	0.645336	−	0.763899i	$-0.276717\pi$
$233$	−1.52461e10	−	2.64070e10i	−0.338888	−	0.586971i	−0.984224	+	0.176928i	$0.943384\pi$
$234$		0			0
$235$	−1.56749e10	+	2.71497e10i	−0.335274	+	0.580711i
$236$		0			0
$237$	−5.37309e8			−0.0110626
$238$		0			0
$239$	−4.29290e10			−0.851060			−0.425530	−	0.904944i	$-0.639912\pi$
$239$	−4.29290e10			−0.851060			−0.425530	+	0.904944i	$0.639912\pi$
$240$		0			0
$241$	4.20533e9	−	7.28385e9i	0.0803015	−	0.139086i	−0.823078	−	0.567928i	$-0.807745\pi$
$241$	4.20533e9	−	7.28385e9i	0.0803015	−	0.139086i	0.903379	+	0.428842i	$0.141078\pi$
$242$		0			0
$243$	−8.01395e9	−	1.38806e10i	−0.147441	−	0.255375i
$244$		0			0
$245$	3.60163e9	−	6.92718e10i	0.0638633	−	1.22831i
$246$		0			0
$247$	−1.52237e10	−	2.63682e10i	−0.260246	−	0.450760i
$248$		0			0
$249$	3.88007e9	−	6.72047e9i	0.0639650	−	0.110791i
$250$		0			0
$251$	−8.14744e10			−1.29566			−0.647828	−	0.761787i	$-0.724322\pi$
$251$	−8.14744e10			−1.29566			−0.647828	+	0.761787i	$0.724322\pi$
$252$		0			0
$253$	7.71407e9			0.118370
$254$		0			0
$255$	6.14606e9	−	1.06453e10i	0.0910262	−	0.157662i
$256$		0			0
$257$	−1.90519e10	−	3.29988e10i	−0.272420	−	0.471845i	0.697061	−	0.717012i	$-0.254491\pi$
$257$	−1.90519e10	−	3.29988e10i	−0.272420	−	0.471845i	−0.969481	+	0.245167i	$0.921157\pi$
$258$		0			0
$259$	8.90281e10	−	2.13933e10i	1.22936	−	0.295412i
$260$		0			0
$261$	4.10966e10	+	7.11813e10i	0.548180	+	0.949476i
$262$		0			0
$263$	−2.83241e10	+	4.90588e10i	−0.365053	+	0.632290i	−0.988785	−	0.149349i	$-0.952282\pi$
$263$	−2.83241e10	+	4.90588e10i	−0.365053	+	0.632290i	0.623732	+	0.781638i	$0.285616\pi$
$264$		0			0
$265$	−7.77086e10			−0.967972
$266$		0			0
$267$	−2.30469e9			−0.0277531
$268$		0			0
$269$	−7.10675e10	+	1.23092e11i	−0.827534	+	1.43333i	0.0724338	+	0.997373i	$0.476923\pi$
$269$	−7.10675e10	+	1.23092e11i	−0.827534	+	1.43333i	−0.899967	+	0.435957i	$0.856410\pi$
$270$		0			0
$271$	−1.86332e10	−	3.22737e10i	−0.209859	−	0.363486i	0.741811	−	0.670609i	$-0.233967\pi$
$271$	−1.86332e10	−	3.22737e10i	−0.209859	−	0.363486i	−0.951670	+	0.307123i	$0.900634\pi$
$272$		0			0
$273$	−9.56829e9	−	1.00787e10i	−0.104256	−	0.109818i
$274$		0			0
$275$	−3.57195e10	−	6.18680e10i	−0.376624	−	0.652333i
$276$		0			0
$277$	−7.86972e10	+	1.36308e11i	−0.803157	+	1.39111i	0.114371	+	0.993438i	$0.463515\pi$
$277$	−7.86972e10	+	1.36308e11i	−0.803157	+	1.39111i	−0.917528	+	0.397671i	$0.869819\pi$
$278$		0			0
$279$	6.17240e10			0.609867
$280$		0			0
$281$	9.66888e10			0.925119			0.462560	−	0.886588i	$-0.346931\pi$
$281$	9.66888e10			0.925119			0.462560	+	0.886588i	$0.346931\pi$
$282$		0			0
$283$	−9.95631e10	+	1.72448e11i	−0.922698	+	1.59816i	−0.127475	+	0.991842i	$0.540687\pi$
$283$	−9.95631e10	+	1.72448e11i	−0.922698	+	1.59816i	−0.795223	+	0.606318i	$0.792646\pi$
$284$		0			0
$285$	2.33603e9	+	4.04612e9i	0.0209738	+	0.0363276i
$286$		0			0
$287$	−1.38647e10	+	4.68420e10i	−0.120626	+	0.407536i
$288$		0			0
$289$	−7.16304e10	−	1.24067e11i	−0.604028	−	1.04621i
$290$		0			0
$291$	−1.09719e9	+	1.90039e9i	−0.00896940	+	0.0155355i
$292$		0			0
$293$	7.28364e10			0.577357			0.288679	−	0.957426i	$-0.406784\pi$
$293$	7.28364e10			0.577357			0.288679	+	0.957426i	$0.406784\pi$
$294$		0			0
$295$	−1.68137e11			−1.29260
$296$		0			0
$297$	1.95212e10	−	3.38116e10i	0.145580	−	0.252152i
$298$		0			0
$299$	−8.46575e9	−	1.46631e10i	−0.0612555	−	0.106098i
$300$		0			0
$301$	−6.57248e10	+	2.22052e11i	−0.461509	+	1.55921i
$302$		0			0
$303$	−1.13735e9	−	1.96996e9i	−0.00775183	−	0.0134266i
$304$		0			0
$305$	9.85557e10	−	1.70703e11i	0.652127	−	1.12952i
$306$		0			0
$307$	1.17824e11			0.757027			0.378514	−	0.925596i	$-0.376435\pi$
$307$	1.17824e11			0.757027			0.378514	+	0.925596i	$0.376435\pi$
$308$		0			0
$309$	−1.89225e10			−0.118077
$310$		0			0
$311$	9.06944e10	−	1.57087e11i	0.549742	−	0.952180i	−0.448550	−	0.893758i	$-0.648060\pi$
$311$	9.06944e10	−	1.57087e11i	0.549742	−	0.952180i	0.998292	−	0.0584227i	$-0.0186071\pi$
$312$		0			0
$313$	−1.42218e11	−	2.46328e11i	−0.837537	−	1.45066i	−0.891948	−	0.452138i	$-0.850662\pi$
$313$	−1.42218e11	−	2.46328e11i	−0.837537	−	1.45066i	0.0544112	−	0.998519i	$-0.482672\pi$
$314$		0			0
$315$	−1.46511e11	−	1.54326e11i	−0.838439	−	0.883164i
$316$		0			0
$317$	5.08743e10	+	8.81169e10i	0.282964	+	0.490109i	0.972114	−	0.234511i	$-0.0753488\pi$
$317$	5.08743e10	+	8.81169e10i	0.282964	+	0.490109i	−0.689149	+	0.724620i	$0.742016\pi$
$318$		0			0
$319$	−1.50410e11	+	2.60518e11i	−0.813241	+	1.40857i
$320$		0			0
$321$	−1.22782e8			−0.000645449
$322$		0			0
$323$	9.95248e10			0.508768
$324$		0			0
$325$	−7.84003e10	+	1.35793e11i	−0.389801	+	0.675154i
$326$		0			0
$327$	−5.05093e9	−	8.74846e9i	−0.0244290	−	0.0423123i
$328$		0			0
$329$	−1.12649e11	+	2.70692e10i	−0.530084	+	0.127378i
$330$		0			0
$331$	1.51983e11	+	2.63242e11i	0.695935	+	1.20540i	0.969864	+	0.243645i	$0.0783432\pi$
$331$	1.51983e11	+	2.63242e11i	0.695935	+	1.20540i	−0.273929	+	0.961750i	$0.588323\pi$
$332$		0			0
$333$	1.40445e11	−	2.43258e11i	0.625904	−	1.08410i
$334$		0			0
$335$	3.46566e11			1.50343
$336$		0			0
$337$	−1.72473e10			−0.0728427			−0.0364213	−	0.999337i	$-0.511596\pi$
$337$	−1.72473e10			−0.0728427			−0.0364213	+	0.999337i	$0.511596\pi$
$338$		0			0
$339$	2.05514e10	−	3.55961e10i	0.0845169	−	0.146388i
$340$		0			0
$341$	1.12953e11	+	1.95639e11i	0.452378	+	0.783541i
$342$		0			0
$343$	1.94822e11	−	1.66603e11i	0.760003	−	0.649920i
$344$		0			0
$345$	1.29904e9	+	2.25001e9i	0.00493671	+	0.00855062i
$346$		0			0
$347$	1.58417e11	−	2.74386e11i	0.586569	−	1.01597i	−0.408109	−	0.912933i	$-0.633812\pi$
$347$	1.58417e11	−	2.74386e11i	0.586569	−	1.01597i	0.994678	−	0.103034i	$-0.0328551\pi$
$348$		0			0
$349$	9.59821e10			0.346318			0.173159	−	0.984894i	$-0.444602\pi$
$349$	9.59821e10			0.346318			0.173159	+	0.984894i	$0.444602\pi$
$350$		0			0
$351$	−8.56934e10			−0.301346
$352$		0			0
$353$	1.31095e11	−	2.27063e11i	0.449365	−	0.778324i	−0.548979	−	0.835836i	$-0.684983\pi$
$353$	1.31095e11	−	2.27063e11i	0.449365	−	0.778324i	0.998345	+	0.0575122i	$0.0183168\pi$
$354$		0			0
$355$	−1.79484e10	−	3.10875e10i	−0.0599788	−	0.103886i
$356$		0			0
$357$	4.41691e10	−	1.06137e10i	0.143917	−	0.0345828i
$358$		0			0
$359$	−5.42664e10	−	9.39922e10i	−0.172427	−	0.298653i	0.766841	−	0.641838i	$-0.221828\pi$
$359$	−5.42664e10	−	9.39922e10i	−0.172427	−	0.298653i	−0.939268	+	0.343185i	$0.888494\pi$
$360$		0			0
$361$	1.42430e11	−	2.46696e11i	0.441386	−	0.764503i
$362$		0			0
$363$	3.81381e10			0.115287
$364$		0			0
$365$	−1.52089e10			−0.0448517
$366$		0			0
$367$	1.06268e11	−	1.84061e11i	0.305777	−	0.529621i	−0.671657	−	0.740862i	$-0.734417\pi$
$367$	1.06268e11	−	1.84061e11i	0.305777	−	0.529621i	0.977434	+	0.211241i	$0.0677505\pi$
$368$		0			0
$369$	7.49309e10	+	1.29784e11i	0.210398	+	0.364421i
$370$		0			0
$371$	−1.97723e11	−	2.08270e11i	−0.541844	−	0.570748i
$372$		0			0
$373$	1.58915e11	+	2.75249e11i	0.425085	+	0.736269i	0.996428	−	0.0844421i	$-0.0269108\pi$
$373$	1.58915e11	+	2.75249e11i	0.425085	+	0.736269i	−0.571343	+	0.820711i	$0.693577\pi$
$374$		0			0
$375$	−1.14283e10	+	1.97944e10i	−0.0298430	+	0.0516895i
$376$		0			0
$377$	6.60266e11			1.68338
$378$		0			0
$379$	−6.09315e11			−1.51693			−0.758465	−	0.651714i	$-0.774050\pi$
$379$	−6.09315e11			−1.51693			−0.758465	+	0.651714i	$0.774050\pi$
$380$		0			0
$381$	−4.52613e9	+	7.83948e9i	−0.0110043	+	0.0190601i
$382$		0			0
$383$	1.70032e11	+	2.94503e11i	0.403771	+	0.699352i	0.994178	−	0.107754i	$-0.0343659\pi$
$383$	1.70032e11	+	2.94503e11i	0.403771	+	0.699352i	−0.590407	+	0.807106i	$0.701033\pi$
$384$		0			0
$385$	2.21040e11	−	7.46788e11i	0.512742	−	1.73230i
$386$		0			0
$387$	3.55206e11	+	6.15236e11i	0.804973	+	1.39425i
$388$		0			0
$389$	−3.77442e11	+	6.53749e11i	−0.835751	+	1.44756i	0.0576663	+	0.998336i	$0.481634\pi$
$389$	−3.77442e11	+	6.53749e11i	−0.835751	+	1.44756i	−0.893417	+	0.449227i	$0.851699\pi$
$390$		0			0
$391$	5.53447e10			0.119751
$392$		0			0
$393$	4.11908e10			0.0871032
$394$		0			0
$395$	−3.30455e10	+	5.72365e10i	−0.0683008	+	0.118300i
$396$		0			0
$397$	1.34694e11	+	2.33297e11i	0.272140	+	0.471360i	0.969409	−	0.245449i	$-0.0789354\pi$
$397$	1.34694e11	+	2.33297e11i	0.272140	+	0.471360i	−0.697270	+	0.716809i	$0.745602\pi$
$398$		0			0
$399$	−4.90036e9	+	1.65559e10i	−0.00967942	+	0.0327020i
$400$		0			0
$401$	1.78083e11	+	3.08449e11i	0.343932	+	0.595708i	0.985159	−	0.171643i	$-0.0549076\pi$
$401$	1.78083e11	+	3.08449e11i	0.343932	+	0.595708i	−0.641227	+	0.767351i	$0.721574\pi$
$402$		0			0
$403$	2.47918e11	−	4.29406e11i	0.468204	−	0.810953i
$404$		0			0
$405$	−6.46195e11			−1.19348
$406$		0			0
$407$	1.02804e12			1.85709
$408$		0			0
$409$	4.37201e10	−	7.57254e10i	0.0772549	−	0.133809i	−0.824810	−	0.565410i	$-0.808718\pi$
$409$	4.37201e10	−	7.57254e10i	0.0772549	−	0.133809i	0.902065	+	0.431601i	$0.142051\pi$
$410$		0			0
$411$	5.38959e10	+	9.33505e10i	0.0931682	+	0.161372i
$412$		0			0
$413$	−4.27811e11	−	4.50632e11i	−0.723564	−	0.762162i
$414$		0			0
$415$	−4.77263e11	−	8.26643e11i	−0.789843	−	1.36805i
$416$		0			0
$417$	−3.77754e10	+	6.54289e10i	−0.0611782	+	0.105964i
$418$		0			0
$419$	−4.46048e9			−0.00706999			−0.00353499	−	0.999994i	$-0.501125\pi$
$419$	−4.46048e9			−0.00706999			−0.00353499	+	0.999994i	$0.501125\pi$
$420$		0			0
$421$	−9.34684e10			−0.145009			−0.0725046	−	0.997368i	$-0.523099\pi$
$421$	−9.34684e10			−0.145009			−0.0725046	+	0.997368i	$0.523099\pi$
$422$		0			0
$423$	−1.77707e11	+	3.07798e11i	−0.269882	+	0.467449i
$424$		0			0
$425$	−2.56270e11	−	4.43873e11i	−0.381020	−	0.659947i
$426$		0			0
$427$	7.08276e11	−	1.70197e11i	1.03104	−	0.247757i
$428$		0			0
$429$	−7.80169e10	−	1.35129e11i	−0.111207	−	0.192616i
$430$		0			0
$431$	−2.49562e11	+	4.32254e11i	−0.348362	+	0.603381i	−0.985959	−	0.166990i	$-0.946595\pi$
$431$	−2.49562e11	+	4.32254e11i	−0.348362	+	0.603381i	0.637597	+	0.770370i	$0.279929\pi$
$432$		0			0
$433$	1.04168e12			1.42410			0.712049	−	0.702130i	$-0.247768\pi$
$433$	1.04168e12			1.42410			0.712049	+	0.702130i	$0.247768\pi$
$434$		0			0
$435$	−1.01316e11			−0.135667
$436$		0			0
$437$	−1.05179e10	+	1.82175e10i	−0.0137963	+	0.0238958i
$438$		0			0
$439$	−1.00140e11	−	1.73447e11i	−0.128681	−	0.222883i	0.794485	−	0.607284i	$-0.207741\pi$
$439$	−1.00140e11	−	1.73447e11i	−0.128681	−	0.222883i	−0.923166	+	0.384402i	$0.874408\pi$
$440$		0			0
$441$	4.08319e10	−	7.85339e11i	0.0514074	−	0.988743i
$442$		0			0
$443$	1.13051e11	+	1.95810e11i	0.139463	+	0.241556i	0.927293	−	0.374335i	$-0.122129\pi$
$443$	1.13051e11	+	1.95810e11i	0.139463	+	0.241556i	−0.787831	+	0.615892i	$0.788796\pi$
$444$		0			0
$445$	−1.41743e11	+	2.45505e11i	−0.171349	+	0.296784i
$446$		0			0
$447$	4.34056e8			0.000514236
$448$		0			0
$449$	−1.20089e12			−1.39443			−0.697213	−	0.716864i	$-0.745577\pi$
$449$	−1.20089e12			−1.39443			−0.697213	+	0.716864i	$0.745577\pi$
$450$		0			0
$451$	−2.74241e11	+	4.74999e11i	−0.312132	+	0.540628i
$452$		0			0
$453$	5.70656e10	+	9.88405e10i	0.0636697	+	0.110279i
$454$		0			0
$455$	−1.66209e12	+	3.99397e11i	−1.81804	+	0.436872i
$456$		0			0
$457$	−1.64864e11	−	2.85552e11i	−0.176808	−	0.306241i	0.763977	−	0.645243i	$-0.223244\pi$
$457$	−1.64864e11	−	2.85552e11i	−0.176808	−	0.306241i	−0.940786	+	0.339002i	$0.889911\pi$
$458$		0			0
$459$	1.40055e11	−	2.42582e11i	0.147279	−	0.255095i
$460$		0			0
$461$	1.38928e12			1.43263			0.716317	−	0.697775i	$-0.245827\pi$
$461$	1.38928e12			1.43263			0.716317	+	0.697775i	$0.245827\pi$
$462$		0			0
$463$	6.88123e7			6.95908e−5			3.47954e−5	−	1.00000i	$-0.499989\pi$
$463$	6.88123e7			6.95908e−5			3.47954e−5		1.00000i	$0.499989\pi$
$464$		0			0
$465$	−3.80422e10	+	6.58910e10i	−0.0377335	+	0.0653564i
$466$		0			0
$467$	−5.39451e11	−	9.34356e11i	−0.524839	−	0.909048i	−0.999582	−	0.0289230i	$-0.990792\pi$
$467$	−5.39451e11	−	9.34356e11i	−0.524839	−	0.909048i	0.474743	−	0.880125i	$-0.342541\pi$
$468$		0			0
$469$	8.81808e11	+	9.28846e11i	0.841581	+	0.886474i
$470$		0			0
$471$	−1.07893e9	−	1.86875e9i	−0.00101018	−	0.00174968i
$472$		0			0
$473$	−1.30003e12	+	2.25171e12i	−1.19420	+	2.06842i
$474$		0			0
$475$	1.94809e11			0.175585
$476$		0			0
$477$	−8.80988e11			−0.779179
$478$		0			0
$479$	−3.11905e11	+	5.40235e11i	−0.270715	+	0.468892i	−0.969045	−	0.246884i	$-0.920593\pi$
$479$	−3.11905e11	+	5.40235e11i	−0.270715	+	0.468892i	0.698330	+	0.715776i	$0.253927\pi$
$480$		0			0
$481$	−1.12821e12	−	1.95412e12i	−0.961032	−	1.66456i
$482$		0			0
$483$	−2.72504e9	+	9.20657e9i	−0.00227830	+	0.00769725i
$484$		0			0
$485$	1.34958e11	+	2.33755e11i	0.110755	+	0.191833i
$486$		0			0
$487$	−7.48463e11	+	1.29638e12i	−0.602962	+	1.04436i	0.389408	+	0.921065i	$0.372679\pi$
$487$	−7.48463e11	+	1.29638e12i	−0.602962	+	1.04436i	−0.992370	+	0.123295i	$0.960654\pi$
$488$		0			0
$489$	−1.59964e11			−0.126512
$490$		0			0
$491$	−6.91050e11			−0.536590			−0.268295	−	0.963337i	$-0.586460\pi$
$491$	−6.91050e11			−0.536590			−0.268295	+	0.963337i	$0.586460\pi$
$492$		0			0
$493$	−1.07912e12	+	1.86909e12i	−0.822733	+	1.42501i
$494$		0			0
$495$	−1.19460e12	−	2.06911e12i	−0.894334	−	1.54903i
$496$		0			0
$497$	3.76508e10	−	1.27204e11i	0.0276803	−	0.0935182i
$498$		0			0
$499$	6.87459e11	+	1.19071e12i	0.496357	+	0.859716i	0.999991	−	0.00420123i	$-0.00133730\pi$
$499$	6.87459e11	+	1.19071e12i	0.496357	+	0.859716i	−0.503634	+	0.863917i	$0.668004\pi$
$500$		0			0
$501$	−1.19523e11	+	2.07019e11i	−0.0847579	+	0.146805i
$502$		0			0
$503$	2.02197e12			1.40837			0.704187	−	0.710015i	$-0.251312\pi$
$503$	2.02197e12			1.40837			0.704187	+	0.710015i	$0.251312\pi$
$504$		0			0
$505$	−2.79798e11			−0.191440
$506$		0			0
$507$	−9.71409e10	+	1.68253e11i	−0.0652930	+	0.113091i
$508$		0			0
$509$	5.74038e11	+	9.94263e11i	0.379062	+	0.656555i	0.990926	−	0.134408i	$-0.0429133\pi$
$509$	5.74038e11	+	9.94263e11i	0.379062	+	0.656555i	−0.611864	+	0.790963i	$0.709580\pi$
$510$		0			0
$511$	−3.86977e10	−	4.07620e10i	−0.0251068	−	0.0264461i
$512$		0			0
$513$	5.32328e10	+	9.22020e10i	0.0339353	+	0.0587776i
$514$		0			0
$515$	−1.16377e12	+	2.01571e12i	−0.729011	+	1.26268i
$516$		0			0
$517$	−1.30079e12			−0.800755
$518$		0			0
$519$	−2.54250e10			−0.0153818
$520$		0			0
$521$	6.50321e11	−	1.12639e12i	0.386686	−	0.669759i	−0.605316	−	0.795985i	$-0.706953\pi$
$521$	6.50321e11	−	1.12639e12i	0.386686	−	0.669759i	0.992002	+	0.126226i	$0.0402865\pi$
$522$		0			0
$523$	−1.12086e12	−	1.94139e12i	−0.655079	−	1.13463i	−0.981874	−	0.189535i	$-0.939302\pi$
$523$	−1.12086e12	−	1.94139e12i	−0.655079	−	1.13463i	0.326794	−	0.945095i	$-0.394032\pi$
$524$		0			0
$525$	8.64562e10	−	2.07752e10i	0.0496683	−	0.0119352i
$526$		0			0
$527$	8.10380e11	+	1.40362e12i	0.457658	+	0.792687i
$528$		0			0
$529$	8.94727e11	−	1.54971e12i	0.496753	−	0.860401i
$530$		0			0
$531$	−1.90618e12			−1.04049
$532$		0			0
$533$	1.20386e12			0.646104
$534$		0			0
$535$	−7.55132e9	+	1.30793e10i	−0.00398502	+	0.00690227i
$536$		0			0
$537$	−1.65120e10	−	2.85997e10i	−0.00856873	−	0.0148415i
$538$		0			0
$539$	2.56392e12	−	1.30772e12i	1.30844	−	0.667367i
$540$		0			0
$541$	5.81162e10	+	1.00660e11i	0.0291682	+	0.0505208i	0.880241	−	0.474527i	$-0.157381\pi$
$541$	5.81162e10	+	1.00660e11i	0.0291682	+	0.0505208i	−0.851073	+	0.525048i	$0.824047\pi$
$542$		0			0
$543$	−1.15678e11	+	2.00360e11i	−0.0571021	+	0.0989037i
$544$		0			0
$545$	−1.24257e12			−0.603303
$546$		0			0
$547$	−1.58473e12			−0.756854			−0.378427	−	0.925631i	$-0.623535\pi$
$547$	−1.58473e12			−0.756854			−0.378427	+	0.925631i	$0.623535\pi$
$548$		0			0
$549$	1.11733e12	−	1.93528e12i	0.524937	−	0.909217i
$550$		0			0
$551$	−4.10158e11	−	7.10414e11i	−0.189570	−	0.328344i
$552$		0			0
$553$	−2.37484e11	+	5.70667e10i	−0.107987	+	0.0259490i
$554$		0			0
$555$	1.73120e11	+	2.99853e11i	0.0774515	+	0.134150i
$556$		0			0
$557$	−7.55286e11	+	1.30819e12i	−0.332478	+	0.575869i	−0.982997	−	0.183621i	$-0.941218\pi$
$557$	−7.55286e11	+	1.30819e12i	−0.332478	+	0.575869i	0.650519	+	0.759490i	$0.274551\pi$
$558$		0			0
$559$	5.70682e12			2.47196
$560$		0			0
$561$	5.10034e11			0.217403
$562$		0			0
$563$	−3.03012e11	+	5.24832e11i	−0.127108	+	0.220157i	−0.922555	−	0.385866i	$-0.873903\pi$
$563$	−3.03012e11	+	5.24832e11i	−0.127108	+	0.220157i	0.795447	+	0.606023i	$0.207236\pi$
$564$		0			0
$565$	−2.52790e12	−	4.37846e12i	−1.04362	−	1.80760i
$566$		0			0
$567$	−1.64419e12	−	1.73189e12i	−0.668079	−	0.703716i
$568$		0			0
$569$	1.60132e11	+	2.77356e11i	0.0640431	+	0.110926i	0.896269	−	0.443511i	$-0.146267\pi$
$569$	1.60132e11	+	2.77356e11i	0.0640431	+	0.110926i	−0.832226	+	0.554436i	$0.812934\pi$
$570$		0			0
$571$	−1.52614e12	+	2.64335e12i	−0.600802	+	1.04062i	0.391898	+	0.920009i	$0.371819\pi$
$571$	−1.52614e12	+	2.64335e12i	−0.600802	+	1.04062i	−0.992700	+	0.120611i	$0.961515\pi$
$572$		0			0
$573$	4.70289e10			0.0182251
$574$		0			0
$575$	1.08331e11			0.0413285
$576$		0			0
$577$	−1.41483e12	+	2.45055e12i	−0.531389	+	0.920392i	0.467940	+	0.883760i	$0.344996\pi$
$577$	−1.41483e12	+	2.45055e12i	−0.531389	+	0.920392i	−0.999329	+	0.0366320i	$0.988337\pi$
$578$		0			0
$579$	3.30071e10	+	5.71700e10i	0.0122055	+	0.0211405i
$580$		0			0
$581$	1.00117e12	−	3.38246e12i	0.364514	−	1.23151i
$582$		0			0
$583$	−1.61217e12	−	2.79236e12i	−0.577967	−	1.00107i
$584$		0			0
$585$	−2.62201e12	+	4.54146e12i	−0.925622	+	1.60322i
$586$		0			0
$587$	2.10206e12			0.730756			0.365378	−	0.930859i	$-0.380940\pi$
$587$	2.10206e12			0.730756			0.365378	+	0.930859i	$0.380940\pi$
$588$		0			0
$589$	−6.16027e11			−0.210902
$590$		0			0
$591$	1.16676e11	−	2.02088e11i	0.0393402	−	0.0681393i
$592$		0			0
$593$	2.96534e12	+	5.13612e12i	0.984756	+	1.70565i	0.643014	+	0.765854i	$0.277684\pi$
$593$	2.96534e12	+	5.13612e12i	0.984756	+	1.70565i	0.341742	+	0.939794i	$0.388983\pi$
$594$		0			0
$595$	1.58586e12	−	5.35784e12i	0.518726	−	1.75252i
$596$		0			0
$597$	1.82947e11	+	3.16874e11i	0.0589442	+	0.102094i
$598$		0			0
$599$	−1.57607e12	+	2.72984e12i	−0.500213	+	0.866395i	0.499787	+	0.866149i	$0.333412\pi$
$599$	−1.57607e12	+	2.72984e12i	−0.500213	+	0.866395i	−1.00000		0.000246421i	$0.999922\pi$
$600$		0			0
$601$	6.20966e12			1.94148			0.970740	−	0.240132i	$-0.0771909\pi$
$601$	6.20966e12			1.94148			0.970740	+	0.240132i	$0.0771909\pi$
$602$		0			0
$603$	3.92904e12			1.21020
$604$		0			0
$605$	2.34556e12	−	4.06264e12i	0.711784	−	1.23285i
$606$		0			0
$607$	2.27519e12	+	3.94075e12i	0.680251	+	1.17823i	0.974904	+	0.222625i	$0.0714627\pi$
$607$	2.27519e12	+	3.94075e12i	0.680251	+	1.17823i	−0.294653	+	0.955604i	$0.595204\pi$
$608$		0			0
$609$	−2.57789e11	−	2.71541e11i	−0.0759428	−	0.0799939i
$610$		0			0
$611$	1.42754e12	+	2.47258e12i	0.414385	+	0.717735i
$612$		0			0
$613$	3.48631e11	−	6.03847e11i	0.0997227	−	0.172725i	−0.811847	−	0.583870i	$-0.801538\pi$
$613$	3.48631e11	−	6.03847e11i	0.0997227	−	0.172725i	0.911570	+	0.411145i	$0.134871\pi$
$614$		0			0
$615$	−1.84728e11			−0.0520708
$616$		0			0
$617$	3.02922e12			0.841488			0.420744	−	0.907179i	$-0.361769\pi$
$617$	3.02922e12			0.841488			0.420744	+	0.907179i	$0.361769\pi$
$618$		0			0
$619$	1.58193e12	−	2.73998e12i	0.433090	−	0.750135i	−0.564047	−	0.825743i	$-0.690756\pi$
$619$	1.58193e12	−	2.73998e12i	0.433090	−	0.750135i	0.997138	+	0.0756079i	$0.0240897\pi$
$620$		0			0
$621$	2.96022e10	+	5.12726e10i	0.00798753	+	0.0138348i
$622$		0			0
$623$	−1.01864e12	+	2.44777e11i	−0.270910	+	0.0650991i
$624$		0			0
$625$	2.38387e12	+	4.12899e12i	0.624918	+	1.08239i
$626$		0			0
$627$	−9.69283e10	+	1.67885e11i	−0.0250465	+	0.0433818i
$628$		0			0
$629$	7.37567e12			1.87877
$630$		0			0
$631$	−6.97874e12			−1.75245			−0.876223	−	0.481906i	$-0.839945\pi$
$631$	−6.97874e12			−1.75245			−0.876223	+	0.481906i	$0.839945\pi$
$632$		0			0
$633$	−2.72807e11	+	4.72516e11i	−0.0675367	+	0.116977i
$634$		0			0
$635$	5.56730e11	+	9.64285e11i	0.135882	+	0.235355i
$636$		0			0
$637$	−5.29950e12	−	3.43842e12i	−1.27529	−	0.827430i
$638$		0			0
$639$	−2.03482e11	−	3.52441e11i	−0.0482805	−	0.0836244i
$640$		0			0
$641$	1.49724e12	−	2.59330e12i	0.350293	−	0.606725i	−0.636008	−	0.771682i	$-0.719415\pi$
$641$	1.49724e12	−	2.59330e12i	0.350293	−	0.606725i	0.986301	+	0.164958i	$0.0527488\pi$
$642$		0			0
$643$	−1.46030e12			−0.336894			−0.168447	−	0.985711i	$-0.553875\pi$
$643$	−1.46030e12			−0.336894			−0.168447	+	0.985711i	$0.553875\pi$
$644$		0			0
$645$	−8.75693e11			−0.199220
$646$		0			0
$647$	2.07950e12	−	3.60181e12i	0.466542	−	0.808074i	−0.532728	−	0.846287i	$-0.678833\pi$
$647$	2.07950e12	−	3.60181e12i	0.466542	−	0.808074i	0.999270	+	0.0382124i	$0.0121663\pi$
$648$		0			0
$649$	−3.48824e12	−	6.04181e12i	−0.771801	−	1.33680i
$650$		0			0
$651$	−2.73392e11	+	6.56956e10i	−0.0596585	+	0.0143358i
$652$		0			0
$653$	−7.92472e11	−	1.37260e12i	−0.170559	−	0.295417i	0.768056	−	0.640382i	$-0.221224\pi$
$653$	−7.92472e11	−	1.37260e12i	−0.170559	−	0.295417i	−0.938615	+	0.344965i	$0.887891\pi$
$654$		0			0
$655$	2.53331e12	−	4.38783e12i	0.537778	−	0.931459i
$656$		0			0
$657$	−1.72424e11			−0.0361039
$658$		0			0
$659$	3.00036e12			0.619711			0.309855	−	0.950784i	$-0.399719\pi$
$659$	3.00036e12			0.619711			0.309855	+	0.950784i	$0.399719\pi$
$660$		0			0
$661$	−3.44920e12	+	5.97419e12i	−0.702768	+	1.21723i	0.264723	+	0.964325i	$0.414719\pi$
$661$	−3.44920e12	+	5.97419e12i	−0.702768	+	1.21723i	−0.967491	+	0.252905i	$0.918614\pi$
$662$		0			0
$663$	−5.59733e11	−	9.69487e11i	−0.112505	−	0.194864i
$664$		0			0
$665$	1.46223e12	+	1.54023e12i	0.289946	+	0.305413i
$666$		0			0
$667$	−2.28085e11	−	3.95054e11i	−0.0446200	−	0.0772841i
$668$		0			0
$669$	3.16673e11	−	5.48494e11i	0.0611215	−	0.105866i
$670$		0			0
$671$	8.17869e12			1.55752
$672$		0			0
$673$	−4.53756e12			−0.852618			−0.426309	−	0.904578i	$-0.640186\pi$
$673$	−4.53756e12			−0.852618			−0.426309	+	0.904578i	$0.640186\pi$
$674$		0			0
$675$	2.74143e11	−	4.74829e11i	0.0508288	−	0.0880380i
$676$		0			0
$677$	−5.21397e12	−	9.03085e12i	−0.953936	−	1.65227i	−0.736785	−	0.676127i	$-0.763657\pi$
$677$	−5.21397e12	−	9.03085e12i	−0.953936	−	1.65227i	−0.217151	−	0.976138i	$-0.569676\pi$
$678$		0			0
$679$	−2.83106e11	+	9.56477e11i	−0.0511135	+	0.172687i
$680$		0			0
$681$	5.24640e10	+	9.08703e10i	0.00934758	+	0.0161905i
$682$		0			0
$683$	2.01247e12	−	3.48569e12i	0.353863	−	0.612909i	−0.633060	−	0.774103i	$-0.718201\pi$
$683$	2.01247e12	−	3.48569e12i	0.353863	−	0.612909i	0.986923	+	0.161194i	$0.0515346\pi$
$684$		0			0
$685$	1.32588e13			2.30089
$686$		0			0
$687$	5.84804e11			0.100162
$688$		0			0
$689$	−3.53853e12	+	6.12892e12i	−0.598187	+	1.03609i
$690$		0			0
$691$	2.02863e12	+	3.51369e12i	0.338494	+	0.586289i	0.984150	−	0.177339i	$-0.0567491\pi$
$691$	2.02863e12	+	3.51369e12i	0.338494	+	0.586289i	−0.645655	+	0.763629i	$0.723416\pi$
$692$		0			0
$693$	2.50595e12	−	8.46638e12i	0.412737	−	1.39443i
$694$		0			0
$695$	4.64651e12	+	8.04800e12i	0.755432	+	1.30845i
$696$		0			0
$697$	−1.96755e12	+	3.40789e12i	−0.315775	+	0.546938i
$698$		0			0
$699$	4.26118e11			0.0675122
$700$		0			0
$701$	2.14575e11			0.0335621			0.0167810	−	0.999859i	$-0.494658\pi$
$701$	2.14575e11			0.0335621			0.0167810	+	0.999859i	$0.494658\pi$
$702$		0			0
$703$	−1.40169e12	+	2.42780e12i	−0.216448	+	0.374899i
$704$		0			0
$705$	−2.19052e11	−	3.79409e11i	−0.0333961	−	0.0578437i
$706$		0			0
$707$	−7.11922e11	−	7.49898e11i	−0.107163	−	0.112879i
$708$		0			0
$709$	4.27735e12	+	7.40859e12i	0.635721	+	1.10110i	0.986362	+	0.164592i	$0.0526306\pi$
$709$	4.27735e12	+	7.40859e12i	0.635721	+	1.10110i	−0.350640	+	0.936510i	$0.614036\pi$
$710$		0			0
$711$	−3.74639e11	+	6.48894e11i	−0.0549794	+	0.0952271i
$712$		0			0
$713$	−3.42566e11			−0.0496412
$714$		0			0
$715$	−1.91927e13			−2.74637
$716$		0			0
$717$	2.99959e11	−	5.19545e11i	0.0423864	−	0.0734153i
$718$		0			0
$719$	−3.24741e12	−	5.62468e12i	−0.453166	−	0.784906i	0.545415	−	0.838166i	$-0.316372\pi$
$719$	−3.24741e12	−	5.62468e12i	−0.453166	−	0.784906i	−0.998581	+	0.0532602i	$0.983039\pi$
$720$		0			0
$721$	−8.36349e12	+	2.00973e12i	−1.15260	+	0.276967i
$722$		0			0
$723$	5.87682e10	+	1.01789e11i	0.00799871	+	0.0138542i
$724$		0			0
$725$	−2.11226e12	+	3.65855e12i	−0.283940	+	0.491799i
$726$		0			0
$727$	−5.36028e12			−0.711677			−0.355838	−	0.934548i	$-0.615805\pi$
$727$	−5.36028e12			−0.711677			−0.355838	+	0.934548i	$0.615805\pi$
$728$		0			0
$729$	−7.17538e12			−0.940960
$730$		0			0
$731$	−9.32707e12	+	1.61550e13i	−1.20814	+	2.09256i
$732$		0			0
$733$	−1.73332e12	−	3.00220e12i	−0.221774	−	0.384124i	0.733573	−	0.679611i	$-0.237851\pi$
$733$	−1.73332e12	−	3.00220e12i	−0.221774	−	0.384124i	−0.955347	+	0.295487i	$0.904518\pi$
$734$		0			0
$735$	8.13191e11	+	5.27614e11i	0.102778	+	0.0666842i
$736$		0			0
$737$	7.18999e12	+	1.24534e13i	0.897686	+	1.55484i
$738$		0			0
$739$	−2.12046e12	+	3.67275e12i	−0.261536	+	0.452993i	−0.966650	−	0.256100i	$-0.917562\pi$
$739$	−2.12046e12	+	3.67275e12i	−0.261536	+	0.452993i	0.705115	+	0.709093i	$0.250896\pi$
$740$		0			0
$741$	4.25493e11			0.0518454
$742$		0			0
$743$	7.87489e12			0.947970			0.473985	−	0.880533i	$-0.342815\pi$
$743$	7.87489e12			0.947970			0.473985	+	0.880533i	$0.342815\pi$
$744$		0			0
$745$	2.66953e10	−	4.62376e10i	0.00317491	−	0.00549910i
$746$		0			0
$747$	−5.41076e12	−	9.37171e12i	−0.635793	−	1.10123i
$748$		0			0
$749$	−5.42680e10	+	1.30405e10i	−0.00630051	+	0.00151400i
$750$		0			0
$751$	−5.52671e12	−	9.57254e12i	−0.633997	−	1.09811i	−0.986727	−	0.162389i	$-0.948080\pi$
$751$	−5.52671e12	−	9.57254e12i	−0.633997	−	1.09811i	0.352730	−	0.935725i	$-0.385253\pi$
$752$		0			0
$753$	5.69290e11	−	9.86038e11i	0.0645291	−	0.111768i
$754$		0			0
$755$	1.40386e13			1.57239
$756$		0			0
$757$	−3.02707e12			−0.335035			−0.167518	−	0.985869i	$-0.553575\pi$
$757$	−3.02707e12			−0.335035			−0.167518	+	0.985869i	$0.553575\pi$
$758$		0			0
$759$	−5.39008e10	+	9.33590e10i	−0.00589532	+	0.0102110i
$760$		0			0
$761$	6.58247e12	+	1.14012e13i	0.711472	+	1.23231i	0.964304	+	0.264796i	$0.0853047\pi$
$761$	6.58247e12	+	1.14012e13i	0.711472	+	1.23231i	−0.252832	+	0.967510i	$0.581362\pi$
$762$		0			0
$763$	−3.16160e12	−	3.33025e12i	−0.337712	−	0.355727i
$764$		0			0
$765$	−8.57069e12	−	1.48449e13i	−0.904773	−	1.56711i
$766$		0			0
$767$	−7.65629e12	+	1.32611e13i	−0.798802	+	1.38357i
$768$		0			0
$769$	−5.52244e12			−0.569460			−0.284730	−	0.958608i	$-0.591904\pi$
$769$	−5.52244e12			−0.569460			−0.284730	+	0.958608i	$0.591904\pi$
$770$		0			0
$771$	5.32487e11			0.0542706
$772$		0			0
$773$	9.92941e11	−	1.71982e12i	0.100027	−	0.173251i	−0.811669	−	0.584118i	$-0.801441\pi$
$773$	9.92941e11	−	1.71982e12i	0.100027	−	0.173251i	0.911695	+	0.410867i	$0.134774\pi$
$774$		0			0
$775$	1.58623e12	+	2.74744e12i	0.157946	+	0.273571i
$776$		0			0
$777$	−3.63160e11	+	1.22694e12i	−0.0357440	+	0.120761i
$778$		0			0
$779$	−7.47836e11	−	1.29529e12i	−0.0727592	−	0.126023i
$780$		0			0
$781$	7.44728e11	−	1.28991e12i	0.0716255	−	0.124059i
$782$		0			0
$783$	−2.30876e12			−0.219508
$784$		0			0
$785$	−2.65424e11			−0.0249474
$786$		0			0
$787$	7.05480e12	−	1.22193e13i	0.655538	−	1.13543i	−0.326220	−	0.945294i	$-0.605775\pi$
$787$	7.05480e12	−	1.22193e13i	0.655538	−	1.13543i	0.981759	−	0.190132i	$-0.0608916\pi$
$788$		0			0
$789$	−3.95820e11	−	6.85581e11i	−0.0363623	−	0.0629814i
$790$		0			0
$791$	5.30286e12	−	1.79158e13i	0.481632	−	1.62720i
$792$		0			0
$793$	−8.97565e12	−	1.55463e13i	−0.806003	−	1.39604i
$794$		0			0
$795$	5.42976e11	−	9.40463e11i	0.0482091	−	0.0835006i
$796$		0			0
$797$	−9.54651e12			−0.838074			−0.419037	−	0.907969i	$-0.637632\pi$
$797$	−9.54651e12			−0.838074			−0.419037	+	0.907969i	$0.637632\pi$
$798$		0			0
$799$	−9.33254e12			−0.810101
$800$		0			0
$801$	−1.60695e12	+	2.78331e12i	−0.137929	+	0.238900i
$802$		0			0
$803$	−3.15529e11	−	5.46512e11i	−0.0267805	−	0.0463852i
$804$		0			0
$805$	8.13129e11	+	8.56504e11i	0.0682461	+	0.0718866i
$806$		0			0
$807$	−9.93145e11	−	1.72018e12i	−0.0824293	−	0.142772i
$808$		0			0
$809$	−9.23775e12	+	1.60003e13i	−0.758225	+	1.31328i	0.185531	+	0.982639i	$0.440600\pi$
$809$	−9.23775e12	+	1.60003e13i	−0.758225	+	1.31328i	−0.943755	+	0.330645i	$0.892734\pi$
$810$		0			0
$811$	1.08163e13			0.877981			0.438991	−	0.898492i	$-0.355336\pi$
$811$	1.08163e13			0.877981			0.438991	+	0.898492i	$0.355336\pi$
$812$		0			0
$813$	5.20787e11			0.0418074
$814$		0			0
$815$	−9.83809e12	+	1.70401e13i	−0.781091	+	1.35289i
$816$		0			0
$817$	−3.54508e12	−	6.14026e12i	−0.278373	−	0.482156i
$818$		0			0
$819$	−1.88433e13	+	4.52799e12i	−1.46345	+	0.351664i
$820$		0			0
$821$	4.49231e12	+	7.78092e12i	0.345085	+	0.597704i	0.985369	−	0.170434i	$-0.0545168\pi$
$821$	4.49231e12	+	7.78092e12i	0.345085	+	0.597704i	−0.640284	+	0.768138i	$0.721183\pi$
$822$		0			0
$823$	−1.04734e13	+	1.81404e13i	−0.795770	+	1.37831i	0.126579	+	0.991957i	$0.459600\pi$
$823$	−1.04734e13	+	1.81404e13i	−0.795770	+	1.37831i	−0.922349	+	0.386357i	$0.873733\pi$
$824$		0			0
$825$	9.98338e11			0.0750300
$826$		0			0
$827$	7.58139e12			0.563604			0.281802	−	0.959473i	$-0.409068\pi$
$827$	7.58139e12			0.563604			0.281802	+	0.959473i	$0.409068\pi$
$828$		0			0
$829$	6.64524e12	−	1.15099e13i	0.488670	−	0.846401i	−0.511245	−	0.859435i	$-0.670816\pi$
$829$	6.64524e12	−	1.15099e13i	0.488670	−	0.846401i	0.999915	+	0.0130340i	$0.00414896\pi$
$830$		0			0
$831$	−1.09977e12	−	1.90485e12i	−0.0800012	−	0.138566i
$832$		0			0
$833$	1.83949e13	−	9.38225e12i	1.32371	−	0.675156i
$834$		0			0
$835$	1.47017e13	+	2.54641e13i	1.04660	+	1.81276i
$836$		0			0
$837$	−8.66896e11	+	1.50151e12i	−0.0610523	+	0.105746i
$838$		0			0
$839$	−6.44356e12			−0.448949			−0.224474	−	0.974480i	$-0.572067\pi$
$839$	−6.44356e12			−0.448949			−0.224474	+	0.974480i	$0.572067\pi$
$840$		0			0
$841$	3.28177e12			0.226218
$842$		0			0
$843$	−6.75597e11	+	1.17017e12i	−0.0460748	+	0.0798040i
$844$		0			0
$845$	1.19487e13	+	2.06957e13i	0.806242	+	1.39645i
$846$		0			0
$847$	1.68565e13	−	4.05059e12i	1.12536	−	0.270422i
$848$		0			0
$849$	−1.39136e12	−	2.40991e12i	−0.0919085	−	0.159190i
$850$		0			0
$851$	−7.79466e11	+	1.35008e12i	−0.0509465	+	0.0882419i
$852$		0			0
$853$	−1.11447e13			−0.720769			−0.360384	−	0.932804i	$-0.617354\pi$
$853$	−1.11447e13			−0.720769			−0.360384	+	0.932804i	$0.617354\pi$
$854$		0			0
$855$	6.51519e12			0.416946
$856$		0			0
$857$	−2.95516e12	+	5.11849e12i	−0.187140	+	0.324137i	−0.944296	−	0.329098i	$-0.893255\pi$
$857$	−2.95516e12	+	5.11849e12i	−0.187140	+	0.324137i	0.757155	+	0.653235i	$0.226589\pi$
$858$		0			0
$859$	−8.90380e12	−	1.54218e13i	−0.557964	−	0.966422i	−0.997666	−	0.0682787i	$-0.978249\pi$
$859$	−8.90380e12	−	1.54218e13i	−0.557964	−	0.966422i	0.439702	−	0.898144i	$-0.355084\pi$
$860$		0			0
$861$	−4.70024e11	−	4.95097e11i	−0.0291478	−	0.0307027i
$862$		0			0
$863$	7.15238e12	+	1.23883e13i	0.438937	+	0.760261i	0.997608	−	0.0691286i	$-0.0220219\pi$
$863$	7.15238e12	+	1.23883e13i	0.438937	+	0.760261i	−0.558671	+	0.829389i	$0.688689\pi$
$864$		0			0
$865$	−1.56368e12	+	2.70838e12i	−0.0949677	+	0.164489i
$866$		0			0
$867$	2.00202e12			0.120333
$868$		0			0
$869$	−2.74230e12			−0.163127
$870$		0			0
$871$	1.57812e13	−	2.73338e13i	0.929091	−	1.60923i
$872$		0			0
$873$	1.53003e12	+	2.65009e12i	0.0891531	+	0.154418i
$874$		0			0
$875$	−2.94883e12	+	9.96267e12i	−0.170065	+	0.574565i
$876$		0			0
$877$	−3.01728e12	−	5.22608e12i	−0.172233	−	0.298317i	0.766967	−	0.641687i	$-0.221765\pi$
$877$	−3.01728e12	−	5.22608e12i	−0.172233	−	0.298317i	−0.939200	+	0.343370i	$0.888432\pi$
$878$		0			0
$879$	−5.08933e11	+	8.81498e11i	−0.0287548	+	0.0498048i
$880$		0			0
$881$	1.54875e12			0.0866142			0.0433071	−	0.999062i	$-0.486211\pi$
$881$	1.54875e12			0.0866142			0.0433071	+	0.999062i	$0.486211\pi$
$882$		0			0
$883$	2.01052e13			1.11297			0.556487	−	0.830857i	$-0.312149\pi$
$883$	2.01052e13			1.11297			0.556487	+	0.830857i	$0.312149\pi$
$884$		0			0
$885$	1.17483e12	−	2.03487e12i	0.0643771	−	0.111504i
$886$		0			0
$887$	−1.07813e13	−	1.86737e13i	−0.584809	−	1.01292i	−0.994899	−	0.100873i	$-0.967836\pi$
$887$	−1.07813e13	−	1.86737e13i	−0.584809	−	1.01292i	0.410091	−	0.912045i	$-0.365497\pi$
$888$		0			0
$889$	−1.16787e12	+	3.94566e12i	−0.0627099	+	0.211866i
$890$		0			0
$891$	−1.34062e13	−	2.32202e13i	−0.712617	−	1.23429i
$892$		0			0
$893$	1.77358e12	−	3.07193e12i	0.0933296	−	0.161652i
$894$		0			0
$895$	−4.06208e12			−0.211614
$896$		0			0
$897$	2.36612e11			0.0122031
$898$		0			0
$899$	6.67941e12	−	1.15691e13i	0.341051	−	0.590718i
$900$		0			0
$901$	−1.15666e13	−	2.00339e13i	−0.584713	−	1.01275i
$902$		0			0
$903$	−2.22813e12	−	2.34698e12i	−0.111518	−	0.117467i
$904$		0			0
$905$	1.42288e13	+	2.46451e13i	0.705100	+	1.22127i
$906$		0			0
$907$	−1.10527e12	+	1.91438e12i	−0.0542295	+	0.0939283i	−0.891866	−	0.452300i	$-0.850604\pi$
$907$	−1.10527e12	+	1.91438e12i	−0.0542295	+	0.0939283i	0.837636	+	0.546228i	$0.183937\pi$
$908$		0			0
$909$	−3.17208e12			−0.154102
$910$		0			0
$911$	−7.06459e11			−0.0339824			−0.0169912	−	0.999856i	$-0.505409\pi$
$911$	−7.06459e11			−0.0339824			−0.0169912	+	0.999856i	$0.505409\pi$
$912$		0			0
$913$	1.98029e13	−	3.42997e13i	0.943216	−	1.63370i
$914$		0			0
$915$	1.37728e12	+	2.38553e12i	0.0649574	+	0.112509i
$916$		0			0
$917$	1.82058e13	−	4.37481e12i	0.850253	−	0.204314i
$918$		0			0
$919$	8.29830e12	+	1.43731e13i	0.383768	+	0.664706i	0.991597	−	0.129362i	$-0.0412928\pi$
$919$	8.29830e12	+	1.43731e13i	0.383768	+	0.664706i	−0.607829	+	0.794068i	$0.707959\pi$
$920$		0			0
$921$	−8.23277e11	+	1.42596e12i	−0.0377032	+	0.0653038i
$922$		0			0
$923$	−3.26919e12			−0.148263
$924$		0			0
$925$	1.44371e13			0.648398
$926$		0			0
$927$	−1.31937e13	+	2.28522e13i	−0.586825	+	1.01641i
$928$		0			0
$929$	−8.93907e12	−	1.54829e13i	−0.393751	−	0.681996i	0.599190	−	0.800607i	$-0.295489\pi$
$929$	−8.93907e12	−	1.54829e13i	−0.393751	−	0.681996i	−0.992941	+	0.118610i	$0.962156\pi$
$930$		0			0
$931$	−4.07516e11	+	7.83795e12i	−0.0177775	+	0.341923i
$932$		0			0
$933$	1.26742e12	+	2.19524e12i	0.0547589	+	0.0948452i
$934$		0			0
$935$	3.13680e13	−	5.43311e13i	1.34226	−	2.32485i
$936$		0			0
$937$	4.36868e13			1.85149			0.925747	−	0.378143i	$-0.123437\pi$
$937$	4.36868e13			1.85149			0.925747	+	0.378143i	$0.123437\pi$
$938$		0			0
$939$	3.97489e12			0.166852
$940$		0			0
$941$	−3.20352e12	+	5.54866e12i	−0.133191	+	0.230693i	−0.924905	−	0.380199i	$-0.875856\pi$
$941$	−3.20352e12	+	5.54866e12i	−0.133191	+	0.230693i	0.791714	+	0.610892i	$0.209189\pi$
$942$		0			0
$943$	−4.15864e11	−	7.20298e11i	−0.0171257	−	0.0296626i
$944$		0			0
$945$	5.81185e12	−	1.39657e12i	0.237067	−	0.0569667i
$946$		0			0
$947$	−5.20865e12	−	9.02164e12i	−0.210451	−	0.364511i	0.741405	−	0.671058i	$-0.234160\pi$
$947$	−5.20865e12	−	9.02164e12i	−0.210451	−	0.364511i	−0.951856	+	0.306547i	$0.900826\pi$
$948$		0			0
$949$	−6.92550e11	+	1.19953e12i	−0.0277175	+	0.0480080i
$950$		0			0
$951$	−1.42190e12			−0.0563713
$952$		0			0
$953$	6.40900e12			0.251694			0.125847	−	0.992050i	$-0.459835\pi$
$953$	6.40900e12			0.251694			0.125847	+	0.992050i	$0.459835\pi$
$954$		0			0
$955$	2.89236e12	−	5.00972e12i	0.112522	−	0.194894i
$956$		0			0
$957$	−2.10193e12	−	3.64066e12i	−0.0810056	−	0.140306i
$958$		0			0
$959$	3.37359e13	+	3.55355e13i	1.28798	+	1.35668i
$960$		0			0
$961$	8.20382e12	+	1.42094e13i	0.310285	+	0.537429i
$962$		0			0
$963$	−8.56099e10	+	1.48281e11i	−0.00320779	+	0.00555605i
$964$		0			0
$965$	8.12000e12			0.301428
$966$		0			0
$967$	−1.24456e12			−0.0457718			−0.0228859	−	0.999738i	$-0.507285\pi$
$967$	−1.24456e12			−0.0457718			−0.0228859	+	0.999738i	$0.507285\pi$
$968$		0			0
$969$	−6.95414e11	+	1.20449e12i	−0.0253388	+	0.0438881i
$970$		0			0
$971$	1.88913e13	+	3.27208e13i	0.681988	+	1.18124i	0.974373	+	0.224937i	$0.0722175\pi$
$971$	1.88913e13	+	3.27208e13i	0.681988	+	1.18124i	−0.292386	+	0.956300i	$0.594449\pi$
$972$		0			0
$973$	−9.74713e12	+	3.29308e13i	−0.348633	+	1.17786i
$974$		0			0
$975$	−1.09562e12	−	1.89767e12i	−0.0388274	−	0.0672511i
$976$		0			0
$977$	6.39041e12	−	1.10685e13i	0.224390	−	0.388655i	−0.731746	−	0.681577i	$-0.761294\pi$
$977$	6.39041e12	−	1.10685e13i	0.224390	−	0.388655i	0.956136	+	0.292922i	$0.0946278\pi$
$978$		0			0
$979$	−1.17626e13			−0.409242
$980$		0			0
$981$	−1.40871e13			−0.485635
$982$		0			0
$983$	−1.14746e13	+	1.98746e13i	−0.391964	+	0.678902i	−0.992709	−	0.120539i	$-0.961538\pi$
$983$	−1.14746e13	+	1.98746e13i	−0.391964	+	0.678902i	0.600744	+	0.799441i	$0.294871\pi$
$984$		0			0
$985$	−1.43515e13	−	2.48576e13i	−0.485776	−	0.841388i
$986$		0			0
$987$	4.59511e11	−	1.55246e12i	0.0154123	−	0.0520708i
$988$		0			0
$989$	−1.97138e12	−	3.41454e12i	−0.0655221	−	0.113488i
$990$		0			0
$991$	1.37595e13	−	2.38322e13i	0.453182	−	0.784934i	−0.545400	−	0.838176i	$-0.683622\pi$
$991$	1.37595e13	−	2.38322e13i	0.453182	−	0.784934i	0.998582	+	0.0532423i	$0.0169556\pi$
$992$		0			0
$993$	−4.24782e12			−0.138642
$994$		0			0
$995$	4.50063e13			1.45569
$996$		0			0
$997$	−1.41861e13	+	2.45711e13i	−0.454712	+	0.787583i	−0.998672	−	0.0515277i	$-0.983591\pi$
$997$	−1.41861e13	+	2.45711e13i	−0.454712	+	0.787583i	0.543960	+	0.839111i	$0.316924\pi$
$998$		0			0
$999$	3.94502e12	+	6.83298e12i	0.125315	+	0.217053i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.i.b.81.2		6
4.3	odd	2		14.10.c.a.11.2	yes	6
7.2	even	3	inner	112.10.i.b.65.2		6
12.11	even	2		126.10.g.f.109.1		6
28.3	even	6		98.10.a.i.1.2		3
28.11	odd	6		98.10.a.j.1.2		3
28.19	even	6		98.10.c.k.79.2		6
28.23	odd	6		14.10.c.a.9.2	✓	6
28.27	even	2		98.10.c.k.67.2		6
84.23	even	6		126.10.g.f.37.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.10.c.a.9.2	✓	6	28.23	odd	6
14.10.c.a.11.2	yes	6	4.3	odd	2
98.10.a.i.1.2		3	28.3	even	6
98.10.a.j.1.2		3	28.11	odd	6
98.10.c.k.67.2		6	28.27	even	2
98.10.c.k.79.2		6	28.19	even	6
112.10.i.b.65.2		6	7.2	even	3	inner
112.10.i.b.81.2		6	1.1	even	1	trivial
126.10.g.f.37.1		6	84.23	even	6
126.10.g.f.109.1		6	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-6.98734 + 12.1024i) q^{3} +(859.469 + 1488.64i) q^{5} +(-1802.93 + 6091.23i) q^{7} +(9743.85 + 16876.9i) q^{9} +(-35661.7 + 61767.9i) q^{11} +156547. q^{13} -24021.6 q^{15} +(-255856. + 443155. i) q^{17} +(-97247.0 - 168437. i) q^{19} +(-61120.9 - 64381.3i) q^{21} +(-54078.1 - 93666.0i) q^{23} +(-500810. + 867429. i) q^{25} -547398. q^{27} +4.21769e6 q^{29} +(1.58367e6 - 2.74299e6i) q^{31} +(-498361. - 863187. i) q^{33} +(-1.06172e7 + 2.55130e6i) q^{35} +(-7.20686e6 - 1.24827e7i) q^{37} +(-1.09385e6 + 1.89460e6i) q^{39} +7.69007e6 q^{41} +3.64544e7 q^{43} +(-1.67491e7 + 2.90102e7i) q^{45} +(9.11895e6 + 1.57945e7i) q^{47} +(-3.38525e7 - 2.19641e7i) q^{49} +(-3.57550e6 - 6.19295e6i) q^{51} +(-2.26037e7 + 3.91507e7i) q^{53} -1.22601e8 q^{55} +2.71799e6 q^{57} +(-4.89074e7 + 8.47100e7i) q^{59} +(-5.73352e7 - 9.93075e7i) q^{61} +(-1.20368e8 + 2.89242e7i) q^{63} +(1.34547e8 + 2.33042e8i) q^{65} +(1.00808e8 - 1.74605e8i) q^{67} +1.51145e6 q^{69} -2.08831e7 q^{71} +(-4.42392e6 + 7.66245e6i) q^{73} +(-6.99866e6 - 1.21220e7i) q^{75} +(-3.11947e8 - 3.28587e8i) q^{77} +(1.92244e7 + 3.32976e7i) q^{79} +(-1.87963e8 + 3.25562e8i) q^{81} -5.55300e8 q^{83} -8.79600e8 q^{85} +(-2.94704e7 + 5.10443e7i) q^{87} +(8.24594e7 + 1.42824e8i) q^{89} +(-2.82243e8 + 9.53562e8i) q^{91} +(2.21312e7 + 3.83324e7i) q^{93} +(1.67162e8 - 2.89532e8i) q^{95} +1.57025e8 q^{97} -1.38993e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 233 q^{3} - 733 q^{5} - 5012 q^{7} - 15058 q^{9} - 7339 q^{11} + 197036 q^{13} - 738238 q^{15} - 306665 q^{17} + 377991 q^{19} - 1585927 q^{21} + 2267255 q^{23} - 142612 q^{25} - 21348358 q^{27} - 13085956 q^{29}+ \cdots + 1256218868 q^{99}+O(q^{100}) \) 6 * q + 233 * q^3 - 733 * q^5 - 5012 * q^7 - 15058 * q^9 - 7339 * q^11 + 197036 * q^13 - 738238 * q^15 - 306665 * q^17 + 377991 * q^19 - 1585927 * q^21 + 2267255 * q^23 - 142612 * q^25 - 21348358 * q^27 - 13085956 * q^29 + 6654517 * q^31 + 3747977 * q^33 - 22864807 * q^35 - 22287969 * q^37 - 3151974 * q^39 + 68096196 * q^41 + 125648280 * q^43 - 86300638 * q^45 + 52703019 * q^47 + 57596070 * q^49 - 97736333 * q^51 - 12091125 * q^53 - 377435398 * q^55 - 158517670 * q^57 + 12949897 * q^59 - 160252153 * q^61 + 356021666 * q^63 + 334191270 * q^65 + 480890225 * q^67 + 1131233162 * q^69 + 74421440 * q^71 + 251382283 * q^73 - 16322324 * q^75 - 527045155 * q^77 - 286494785 * q^79 - 2165894587 * q^81 - 2295182344 * q^83 - 2389185074 * q^85 - 1412199358 * q^87 - 901243845 * q^89 - 1026798920 * q^91 + 710456889 * q^93 + 887366177 * q^95 + 629707876 * q^97 + 1256218868 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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