LMFDB - Embedded newform 112.10.p.a.31.12

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,10,Mod(31,112)] 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([3, 0, 1])) N = Newforms(chi, 10, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("112.31"); 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:traces = [24,0,-162] 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:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.12
Character	$\chi$	$=$	112.31
Dual form			112.10.p.a.47.12

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(127.679 - 221.147i) q^{3} +(-241.245 + 139.283i) q^{5} +(-2213.93 + 5954.17i) q^{7} +(-22762.6 - 39426.0i) q^{9} +(-1893.22 - 1093.05i) q^{11} +16268.8i q^{13} +71134.4i q^{15} +(158301. + 91395.1i) q^{17} +(-240118. - 415897. i) q^{19} +(1.03407e6 + 1.24983e6i) q^{21} +(-1.61434e6 + 932040. i) q^{23} +(-937763. + 1.62425e6i) q^{25} -6.59903e6 q^{27} -2.16004e6 q^{29} +(-3.50603e6 + 6.07262e6i) q^{31} +(-483451. + 279121. i) q^{33} +(-295214. - 1.74478e6i) q^{35} +(7.40082e6 + 1.28186e7i) q^{37} +(3.59780e6 + 2.07719e6i) q^{39} +1.28601e7i q^{41} -2.79515e7i q^{43} +(1.09827e7 + 6.34089e6i) q^{45} +(2.56866e7 + 4.44906e7i) q^{47} +(-3.05506e7 - 2.63642e7i) q^{49} +(4.04236e7 - 2.33386e7i) q^{51} +(4.26903e6 - 7.39418e6i) q^{53} +608975. q^{55} -1.22633e8 q^{57} +(-3.25838e6 + 5.64368e6i) q^{59} +(7.16467e7 - 4.13653e7i) q^{61} +(2.85144e8 - 4.82459e7i) q^{63} +(-2.26597e6 - 3.92477e6i) q^{65} +(2.34278e8 + 1.35260e8i) q^{67} +4.76009e8i q^{69} -4.13896e8i q^{71} +(2.15848e7 + 1.24620e7i) q^{73} +(2.39466e8 + 4.14767e8i) q^{75} +(1.06997e7 - 8.85262e6i) q^{77} +(-2.52310e8 + 1.45671e8i) q^{79} +(-3.94525e8 + 6.83337e8i) q^{81} -6.54289e8 q^{83} -5.09192e7 q^{85} +(-2.75792e8 + 4.77686e8i) q^{87} +(-7.12797e8 + 4.11534e8i) q^{89} +(-9.68672e7 - 3.60180e7i) q^{91} +(8.95295e8 + 1.55070e9i) q^{93} +(1.15855e8 + 6.68889e7i) q^{95} +6.66674e8i q^{97} +9.95228e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) $ 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

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
$3$	127.679	−	221.147i	0.910071	−	1.57629i	0.0961097	−	0.995371i	$-0.469360\pi$
$3$	127.679	−	221.147i	0.910071	−	1.57629i	0.813961	−	0.580919i	$-0.197307\pi$
$4$		0			0
$5$	−241.245	+	139.283i	−0.172621	+	0.0996629i	−0.583821	−	0.811882i	$-0.698443\pi$
$5$	−241.245	+	139.283i	−0.172621	+	0.0996629i	0.411200	+	0.911545i	$0.365110\pi$
$6$		0			0
$7$	−2213.93	+	5954.17i	−0.348516	+	0.937303i
$7$	−2213.93	+	5954.17i	−0.348516	+	0.937303i
$8$		0			0
$9$	−22762.6	−	39426.0i	−1.15646	−	2.00305i
$10$		0			0
$11$	−1893.22	−	1093.05i	−0.0389883	−	0.0225099i	0.480379	−	0.877061i	$-0.340499\pi$
$11$	−1893.22	−	1093.05i	−0.0389883	−	0.0225099i	−0.519367	+	0.854551i	$0.673832\pi$
$12$		0			0
$13$			16268.8i			0.157983i	0.996875	+	0.0789915i	$0.0251700\pi$
$13$			16268.8i			0.157983i	−0.996875	+	0.0789915i	$0.974830\pi$
$14$		0			0
$15$			71134.4i			0.362801i
$16$		0			0
$17$	158301.	+	91395.1i	0.459688	+	0.265401i	0.711913	−	0.702268i	$-0.247829\pi$
$17$	158301.	+	91395.1i	0.459688	+	0.265401i	−0.252225	+	0.967669i	$0.581162\pi$
$18$		0			0
$19$	−240118.	−	415897.i	−0.422702	−	0.732141i	0.573501	−	0.819205i	$-0.305585\pi$
$19$	−240118.	−	415897.i	−0.422702	−	0.732141i	−0.996203	+	0.0870637i	$0.972252\pi$
$20$		0			0
$21$	1.03407e6	+	1.24983e6i	1.16029	+	1.40237i
$22$		0			0
$23$	−1.61434e6	+	932040.i	−1.20287	+	0.694479i	−0.961193	−	0.275877i	$-0.911032\pi$
$23$	−1.61434e6	+	932040.i	−1.20287	+	0.694479i	−0.241680	+	0.970356i	$0.577698\pi$
$24$		0			0
$25$	−937763.	+	1.62425e6i	−0.480135	+	0.831618i
$26$		0			0
$27$	−6.59903e6			−2.38970
$28$		0			0
$29$	−2.16004e6			−0.567113			−0.283557	−	0.958955i	$-0.591514\pi$
$29$	−2.16004e6			−0.567113			−0.283557	+	0.958955i	$0.591514\pi$
$30$		0			0
$31$	−3.50603e6	+	6.07262e6i	−0.681848	+	1.18100i	0.292568	+	0.956245i	$0.405490\pi$
$31$	−3.50603e6	+	6.07262e6i	−0.681848	+	1.18100i	−0.974416	+	0.224751i	$0.927843\pi$
$32$		0			0
$33$	−483451.	+	279121.i	−0.0709643	+	0.0409713i
$34$		0			0
$35$	−295214.	−	1.74478e6i	−0.0332530	−	0.196532i
$36$		0			0
$37$	7.40082e6	+	1.28186e7i	0.649190	+	1.12443i	0.983317	+	0.181902i	$0.0582253\pi$
$37$	7.40082e6	+	1.28186e7i	0.649190	+	1.12443i	−0.334127	+	0.942528i	$0.608441\pi$
$38$		0			0
$39$	3.59780e6	+	2.07719e6i	0.249027	+	0.143776i
$40$		0			0
$41$			1.28601e7i			0.710748i	0.934724	+	0.355374i	$0.115647\pi$
$41$			1.28601e7i			0.710748i	−0.934724	+	0.355374i	$0.884353\pi$
$42$		0			0
$43$		−	2.79515e7i		−	1.24680i	−0.781903	−	0.623400i	$-0.785751\pi$
$43$		−	2.79515e7i		−	1.24680i	0.781903	−	0.623400i	$-0.214249\pi$
$44$		0			0
$45$	1.09827e7	+	6.34089e6i	0.399259	+	0.230512i
$46$		0			0
$47$	2.56866e7	+	4.44906e7i	0.767833	+	1.32993i	0.938736	+	0.344638i	$0.111998\pi$
$47$	2.56866e7	+	4.44906e7i	0.767833	+	1.32993i	−0.170903	+	0.985288i	$0.554668\pi$
$48$		0			0
$49$	−3.05506e7	−	2.63642e7i	−0.757073	−	0.653330i
$50$		0			0
$51$	4.04236e7	−	2.33386e7i	0.836698	−	0.483068i
$52$		0			0
$53$	4.26903e6	−	7.39418e6i	0.0743170	−	0.128721i	−0.826472	−	0.562978i	$-0.809656\pi$
$53$	4.26903e6	−	7.39418e6i	0.0743170	−	0.128721i	0.900789	+	0.434257i	$0.142989\pi$
$54$		0			0
$55$	608975.			0.00897361
$56$		0			0
$57$	−1.22633e8			−1.53876
$58$		0			0
$59$	−3.25838e6	+	5.64368e6i	−0.0350080	+	0.0606357i	−0.882999	−	0.469376i	$-0.844479\pi$
$59$	−3.25838e6	+	5.64368e6i	−0.0350080	+	0.0606357i	0.847991	+	0.530011i	$0.177812\pi$
$60$		0			0
$61$	7.16467e7	−	4.13653e7i	0.662540	−	0.382518i	−0.130704	−	0.991421i	$-0.541724\pi$
$61$	7.16467e7	−	4.13653e7i	0.662540	−	0.382518i	0.793244	+	0.608904i	$0.208390\pi$
$62$		0			0
$63$	2.85144e8	−	4.82459e7i	2.28051	−	0.385858i
$64$		0			0
$65$	−2.26597e6	−	3.92477e6i	−0.0157450	−	0.0272712i
$66$		0			0
$67$	2.34278e8	+	1.35260e8i	1.42035	+	0.820037i	0.996328	−	0.0856162i	$-0.0272859\pi$
$67$	2.34278e8	+	1.35260e8i	1.42035	+	0.820037i	0.424018	+	0.905654i	$0.360619\pi$
$68$		0			0
$69$			4.76009e8i			2.52810i
$70$		0			0
$71$		−	4.13896e8i		−	1.93299i	−0.256695	−	0.966493i	$-0.582634\pi$
$71$		−	4.13896e8i		−	1.93299i	0.256695	−	0.966493i	$-0.417366\pi$
$72$		0			0
$73$	2.15848e7	+	1.24620e7i	0.0889603	+	0.0513612i	0.543820	−	0.839202i	$-0.316977\pi$
$73$	2.15848e7	+	1.24620e7i	0.0889603	+	0.0513612i	−0.454860	+	0.890563i	$0.650311\pi$
$74$		0			0
$75$	2.39466e8	+	4.14767e8i	0.873913	+	1.51366i
$76$		0			0
$77$	1.06997e7	−	8.85262e6i	0.0346867	−	0.0286988i
$78$		0			0
$79$	−2.52310e8	+	1.45671e8i	−0.728808	+	0.420777i	−0.817986	−	0.575238i	$-0.804909\pi$
$79$	−2.52310e8	+	1.45671e8i	−0.728808	+	0.420777i	0.0891782	+	0.996016i	$0.471576\pi$
$80$		0			0
$81$	−3.94525e8	+	6.83337e8i	−1.01834	+	1.76381i
$82$		0			0
$83$	−6.54289e8			−1.51328			−0.756638	−	0.653834i	$-0.773159\pi$
$83$	−6.54289e8			−1.51328			−0.756638	+	0.653834i	$0.773159\pi$
$84$		0			0
$85$	−5.09192e7			−0.105803
$86$		0			0
$87$	−2.75792e8	+	4.77686e8i	−0.516113	+	0.893935i
$88$		0			0
$89$	−7.12797e8	+	4.11534e8i	−1.20423	+	0.695265i	−0.961494	−	0.274826i	$-0.911380\pi$
$89$	−7.12797e8	+	4.11534e8i	−1.20423	+	0.695265i	−0.242740	+	0.970091i	$0.578046\pi$
$90$		0			0
$91$	−9.68672e7	−	3.60180e7i	−0.148078	−	0.0550596i
$92$		0			0
$93$	8.95295e8	+	1.55070e9i	1.24106	+	2.14958i
$94$		0			0
$95$	1.15855e8	+	6.68889e7i	0.145935	+	0.0842554i
$96$		0			0
$97$			6.66674e8i			0.764612i	0.924036	+	0.382306i	$0.124870\pi$
$97$			6.66674e8i			0.764612i	−0.924036	+	0.382306i	$0.875130\pi$
$98$		0			0
$99$			9.95228e7i			0.104127i
$100$		0			0
$101$	4.46790e8	+	2.57954e8i	0.427226	+	0.246659i	0.698164	−	0.715938i	$-0.254001\pi$
$101$	4.46790e8	+	2.57954e8i	0.427226	+	0.246659i	−0.270938	+	0.962597i	$0.587334\pi$
$102$		0			0
$103$	1.37475e7	+	2.38114e7i	0.0120353	+	0.0208458i	0.871980	−	0.489541i	$-0.162836\pi$
$103$	1.37475e7	+	2.38114e7i	0.0120353	+	0.0208458i	−0.859945	+	0.510387i	$0.829502\pi$
$104$		0			0
$105$	−4.23546e8	−	1.57487e8i	−0.340055	−	0.126442i
$106$		0			0
$107$	1.30600e9	−	7.54021e8i	0.963201	−	0.556105i	0.0660445	−	0.997817i	$-0.478962\pi$
$107$	1.30600e9	−	7.54021e8i	0.963201	−	0.556105i	0.897157	+	0.441712i	$0.145629\pi$
$108$		0			0
$109$	−5.79402e8	+	1.00355e9i	−0.393152	+	0.680959i	−0.992863	−	0.119257i	$-0.961949\pi$
$109$	−5.79402e8	+	1.00355e9i	−0.393152	+	0.680959i	0.599711	+	0.800216i	$0.295282\pi$
$110$		0			0
$111$	3.77973e9			2.36324
$112$		0			0
$113$	−4.28060e8			−0.246974			−0.123487	−	0.992346i	$-0.539408\pi$
$113$	−4.28060e8			−0.246974			−0.123487	+	0.992346i	$0.539408\pi$
$114$		0			0
$115$	2.59635e8	−	4.49701e8i	0.138428	−	0.239764i
$116$		0			0
$117$	6.41413e8	−	3.70320e8i	0.316447	−	0.182701i
$118$		0			0
$119$	−8.94649e8	+	7.40208e8i	−0.408970	+	0.338370i
$120$		0			0
$121$	−1.17658e9	−	2.03790e9i	−0.498987	−	0.864270i
$122$		0			0
$123$	2.84397e9	+	1.64197e9i	1.12034	+	0.646831i
$124$		0			0
$125$		−	1.06653e9i		−	0.390732i
$126$		0			0
$127$			1.54451e9i			0.526834i	0.964682	+	0.263417i	$0.0848494\pi$
$127$			1.54451e9i			0.526834i	−0.964682	+	0.263417i	$0.915151\pi$
$128$		0			0
$129$	−6.18139e9	−	3.56883e9i	−1.96532	−	1.13468i
$130$		0			0
$131$	−2.79315e9	−	4.83789e9i	−0.828656	−	1.43527i	−0.899093	−	0.437758i	$-0.855773\pi$
$131$	−2.79315e9	−	4.83789e9i	−0.828656	−	1.43527i	0.0704374	−	0.997516i	$-0.477560\pi$
$132$		0			0
$133$	3.00793e9	−	5.08937e8i	0.833557	−	0.141037i
$134$		0			0
$135$	1.59199e9	−	9.19133e8i	0.412513	−	0.238164i
$136$		0			0
$137$	−3.18074e9	+	5.50921e9i	−0.771411	+	1.33612i	0.165378	+	0.986230i	$0.447116\pi$
$137$	−3.18074e9	+	5.50921e9i	−0.771411	+	1.33612i	−0.936790	+	0.349893i	$0.886218\pi$
$138$		0			0
$139$	5.47109e9			1.24310			0.621552	−	0.783373i	$-0.286502\pi$
$139$	5.47109e9			1.24310			0.621552	+	0.783373i	$0.286502\pi$
$140$		0			0
$141$	1.31186e10			2.79513
$142$		0			0
$143$	1.77827e7	−	3.08005e7i	0.00355619	−	0.00615949i
$144$		0			0
$145$	5.21099e8	−	3.00856e8i	0.0978958	−	0.0565201i
$146$		0			0
$147$	−9.73107e9	+	3.39002e9i	−1.71883	+	0.598789i
$148$		0			0
$149$	3.08690e9	+	5.34667e9i	0.513079	+	0.888679i	0.999885	+	0.0151690i	$0.00482864\pi$
$149$	3.08690e9	+	5.34667e9i	0.513079	+	0.888679i	−0.486806	+	0.873510i	$0.661838\pi$
$150$		0			0
$151$	−5.22159e9	−	3.01469e9i	−0.817348	−	0.471896i	0.0321533	−	0.999483i	$-0.489764\pi$
$151$	−5.22159e9	−	3.01469e9i	−0.817348	−	0.471896i	−0.849501	+	0.527587i	$0.823097\pi$
$152$		0			0
$153$		−	8.32156e9i		−	1.22770i
$154$		0			0
$155$		−	1.95332e9i		−	0.271820i
$156$		0			0
$157$	−9.90264e9	−	5.71729e9i	−1.30078	−	0.751003i	−0.320239	−	0.947337i	$-0.603763\pi$
$157$	−9.90264e9	−	5.71729e9i	−1.30078	−	0.751003i	−0.980537	+	0.196333i	$0.937097\pi$
$158$		0			0
$159$	−1.09014e9	−	1.88817e9i	−0.135267	−	0.234290i
$160$		0			0
$161$	−1.97548e9	−	1.16755e10i	−0.231716	−	1.36949i
$162$		0			0
$163$	−8.47879e9	+	4.89523e9i	−0.940784	+	0.543162i	−0.890206	−	0.455558i	$-0.849440\pi$
$163$	−8.47879e9	+	4.89523e9i	−0.940784	+	0.543162i	−0.0505780	+	0.998720i	$0.516106\pi$
$164$		0			0
$165$	7.77536e7	−	1.34673e8i	0.00816663	−	0.0141450i
$166$		0			0
$167$	−1.57291e10			−1.56488			−0.782439	−	0.622727i	$-0.786025\pi$
$167$	−1.57291e10			−1.56488			−0.782439	+	0.622727i	$0.786025\pi$
$168$		0			0
$169$	1.03398e10			0.975041
$170$		0			0
$171$	−1.09314e10	+	1.89338e10i	−0.977675	+	1.69338i
$172$		0			0
$173$	4.97796e9	−	2.87402e9i	0.422517	−	0.243940i	−0.273637	−	0.961833i	$-0.588227\pi$
$173$	4.97796e9	−	2.87402e9i	0.422517	−	0.243940i	0.696153	+	0.717893i	$0.254893\pi$
$174$		0			0
$175$	−7.59493e9	−	9.17958e9i	−0.612143	−	0.739864i
$176$		0			0
$177$	8.32056e8	+	1.44116e9i	0.0637196	+	0.110366i
$178$		0			0
$179$	1.54921e9	+	8.94437e8i	0.112790	+	0.0651195i	0.555334	−	0.831628i	$-0.312590\pi$
$179$	1.54921e9	+	8.94437e8i	0.112790	+	0.0651195i	−0.442544	+	0.896747i	$0.645924\pi$
$180$		0			0
$181$			9.22046e9i			0.638556i	0.947661	+	0.319278i	$0.103440\pi$
$181$			9.22046e9i			0.638556i	−0.947661	+	0.319278i	$0.896560\pi$
$182$		0			0
$183$		−	2.11260e10i		−	1.39247i
$184$		0			0
$185$	−3.57083e9	−	2.06162e9i	−0.224128	−	0.129400i
$186$		0			0
$187$	−1.99799e8	−	3.46062e8i	−0.0119483	−	0.0206951i
$188$		0			0
$189$	1.46098e10	−	3.92917e10i	0.832849	−	2.23987i
$190$		0			0
$191$	1.33490e10	−	7.70707e9i	0.725771	−	0.419024i	−0.0911022	−	0.995842i	$-0.529039\pi$
$191$	1.33490e10	−	7.70707e9i	0.725771	−	0.419024i	0.816873	+	0.576818i	$0.195706\pi$
$192$		0			0
$193$	−1.52019e10	+	2.63304e10i	−0.788658	+	1.36600i	0.138131	+	0.990414i	$0.455891\pi$
$193$	−1.52019e10	+	2.63304e10i	−0.788658	+	1.36600i	−0.926789	+	0.375582i	$0.877443\pi$
$194$		0			0
$195$	−1.15727e9			−0.0573164
$196$		0			0
$197$	−9.89295e9			−0.467980			−0.233990	−	0.972239i	$-0.575178\pi$
$197$	−9.89295e9			−0.467980			−0.233990	+	0.972239i	$0.575178\pi$
$198$		0			0
$199$	−5.09431e9	+	8.82360e9i	−0.230275	+	0.398847i	−0.957889	−	0.287139i	$-0.907296\pi$
$199$	−5.09431e9	+	8.82360e9i	−0.230275	+	0.398847i	0.727614	+	0.685987i	$0.240629\pi$
$200$		0			0
$201$	5.98249e10	−	3.45399e10i	2.58523	−	1.49258i
$202$		0			0
$203$	4.78217e9	−	1.28612e10i	0.197648	−	0.531557i
$204$		0			0
$205$	−1.79119e9	−	3.10243e9i	−0.0708352	−	0.122690i
$206$		0			0
$207$	7.34931e10	+	4.24313e10i	2.78215	+	1.60627i
$208$		0			0
$209$			1.04985e9i			0.0380599i
$210$		0			0
$211$		−	5.36994e10i		−	1.86508i	−0.361063	−	0.932541i	$-0.617586\pi$
$211$		−	5.36994e10i		−	1.86508i	0.361063	−	0.932541i	$-0.382414\pi$
$212$		0			0
$213$	−9.15320e10	−	5.28460e10i	−3.04694	−	1.75915i
$214$		0			0
$215$	3.89317e9	+	6.74316e9i	0.124260	+	0.215224i
$216$		0			0
$217$	−2.83953e10	−	3.43198e10i	−0.869315	−	1.05069i
$218$		0			0
$219$	5.51188e9	−	3.18229e9i	0.161920	−	0.0934848i
$220$		0			0
$221$	−1.48689e9	+	2.57537e9i	−0.0419289	+	0.0726229i
$222$		0			0
$223$	−3.27063e9			−0.0885645			−0.0442823	−	0.999019i	$-0.514100\pi$
$223$	−3.27063e9			−0.0885645			−0.0442823	+	0.999019i	$0.514100\pi$
$224$		0			0
$225$	8.53836e10			2.22102
$226$		0			0
$227$	−2.28370e10	+	3.95548e10i	−0.570850	+	0.988742i	0.425629	+	0.904898i	$0.360053\pi$
$227$	−2.28370e10	+	3.95548e10i	−0.570850	+	0.988742i	−0.996479	+	0.0838437i	$0.973280\pi$
$228$		0			0
$229$	−4.11021e10	+	2.37303e10i	−0.987653	+	0.570222i	−0.904572	−	0.426321i	$-0.859809\pi$
$229$	−4.11021e10	+	2.37303e10i	−0.987653	+	0.570222i	−0.0830813	+	0.996543i	$0.526476\pi$
$230$		0			0
$231$	−5.91604e8	−	3.49650e9i	−0.0136703	−	0.0807942i
$232$		0			0
$233$	3.65781e10	+	6.33551e10i	0.813054	+	1.40825i	0.910717	+	0.413031i	$0.135530\pi$
$233$	3.65781e10	+	6.33551e10i	0.813054	+	1.40825i	−0.0976627	+	0.995220i	$0.531137\pi$
$234$		0			0
$235$	−1.23936e10	−	7.15543e9i	−0.265088	−	0.153049i
$236$		0			0
$237$			7.43970e10i			1.53175i
$238$		0			0
$239$		−	5.90869e10i		−	1.17139i	−0.810533	−	0.585694i	$-0.800822\pi$
$239$		−	5.90869e10i		−	1.17139i	0.810533	−	0.585694i	$-0.199178\pi$
$240$		0			0
$241$	4.28335e10	+	2.47299e10i	0.817913	+	0.472222i	0.849696	−	0.527273i	$-0.176785\pi$
$241$	4.28335e10	+	2.47299e10i	0.817913	+	0.472222i	−0.0317834	+	0.999495i	$0.510119\pi$
$242$		0			0
$243$	3.58010e10	+	6.20092e10i	0.658669	+	1.14085i
$244$		0			0
$245$	1.10423e10	+	2.10507e9i	0.195800	+	0.0373266i
$246$		0			0
$247$	6.76615e9	−	3.90644e9i	0.115666	−	0.0667797i
$248$		0			0
$249$	−8.35392e10	+	1.44694e11i	−1.37719	+	2.38536i
$250$		0			0
$251$	−5.82999e10			−0.927120			−0.463560	−	0.886065i	$-0.653428\pi$
$251$	−5.82999e10			−0.927120			−0.463560	+	0.886065i	$0.653428\pi$
$252$		0			0
$253$	4.07507e9			0.0625307
$254$		0			0
$255$	−6.50133e9	+	1.12606e10i	−0.0962879	+	0.166775i
$256$		0			0
$257$	−1.75820e10	+	1.01510e10i	−0.251403	+	0.145147i	−0.620406	−	0.784280i	$-0.713032\pi$
$257$	−1.75820e10	+	1.01510e10i	−0.251403	+	0.145147i	0.369004	+	0.929428i	$0.379699\pi$
$258$		0			0
$259$	−9.27089e10	+	1.56862e10i	−1.28018	+	0.216605i
$260$		0			0
$261$	4.91680e10	+	8.51615e10i	0.655843	+	1.13595i
$262$		0			0
$263$	−1.17235e10	−	6.76858e9i	−0.151098	−	0.0872362i	0.422545	−	0.906342i	$-0.361137\pi$
$263$	−1.17235e10	−	6.76858e9i	−0.151098	−	0.0872362i	−0.573643	+	0.819106i	$0.694470\pi$
$264$		0			0
$265$			2.37842e9i			0.0296266i
$266$		0			0
$267$			2.10178e11i			2.53096i
$268$		0			0
$269$	−1.26526e11	−	7.30498e10i	−1.47331	−	0.850617i	−0.473762	−	0.880653i	$-0.657104\pi$
$269$	−1.26526e11	−	7.30498e10i	−1.47331	−	0.850617i	−0.999549	+	0.0300360i	$0.990438\pi$
$270$		0			0
$271$	−6.55977e10	−	1.13619e11i	−0.738800	−	1.27964i	−0.953036	−	0.302857i	$-0.902059\pi$
$271$	−6.55977e10	−	1.13619e11i	−0.738800	−	1.27964i	0.214236	−	0.976782i	$-0.431274\pi$
$272$		0			0
$273$	−2.03332e10	+	1.68232e10i	−0.221551	+	0.183306i
$274$		0			0
$275$	3.55079e9	−	2.05005e9i	0.0374393	−	0.0216156i
$276$		0			0
$277$	−5.51358e10	+	9.54980e10i	−0.562697	+	0.974620i	0.434563	+	0.900641i	$0.356903\pi$
$277$	−5.51358e10	+	9.54980e10i	−0.562697	+	0.974620i	−0.997260	+	0.0739781i	$0.976431\pi$
$278$		0			0
$279$	3.19225e11			3.15412
$280$		0			0
$281$	5.31385e10			0.508430			0.254215	−	0.967148i	$-0.418183\pi$
$281$	5.31385e10			0.508430			0.254215	+	0.967148i	$0.418183\pi$
$282$		0			0
$283$	2.83116e10	−	4.90372e10i	0.262377	−	0.454450i	−0.704496	−	0.709708i	$-0.748827\pi$
$283$	2.83116e10	−	4.90372e10i	0.262377	−	0.454450i	0.966873	+	0.255258i	$0.0821603\pi$
$284$		0			0
$285$	2.95846e10	−	1.70807e10i	0.265622	−	0.153357i
$286$		0			0
$287$	−7.65710e10	−	2.84713e10i	−0.666186	−	0.247707i
$288$		0			0
$289$	−4.25878e10	−	7.37643e10i	−0.359124	−	0.622022i
$290$		0			0
$291$	1.47433e11	+	8.51206e10i	1.20525	+	0.695851i
$292$		0			0
$293$			6.16280e10i			0.488510i	0.969711	+	0.244255i	$0.0785434\pi$
$293$			6.16280e10i			0.488510i	−0.969711	+	0.244255i	$0.921457\pi$
$294$		0			0
$295$		−	1.81535e9i		−	0.0139560i
$296$		0			0
$297$	1.24934e10	+	7.21309e9i	0.0931703	+	0.0537919i
$298$		0			0
$299$	−1.51632e10	−	2.62634e10i	−0.109716	−	0.190034i
$300$		0			0
$301$	1.66428e11	+	6.18826e10i	1.16863	+	0.434530i
$302$		0			0
$303$	1.14092e11	−	6.58709e10i	0.777611	−	0.448954i
$304$		0			0
$305$	−1.15230e10	+	1.99584e10i	−0.0762456	+	0.132061i
$306$		0			0
$307$	6.35064e10			0.408032			0.204016	−	0.978968i	$-0.434600\pi$
$307$	6.35064e10			0.408032			0.204016	+	0.978968i	$0.434600\pi$
$308$		0			0
$309$	7.02111e9			0.0438120
$310$		0			0
$311$	1.39140e11	−	2.40997e11i	0.843393	−	1.46080i	−0.0436161	−	0.999048i	$-0.513888\pi$
$311$	1.39140e11	−	2.40997e11i	0.843393	−	1.46080i	0.887009	−	0.461751i	$-0.152779\pi$
$312$		0			0
$313$	−1.08773e11	+	6.28001e10i	−0.640578	+	0.369838i	−0.784837	−	0.619702i	$-0.787253\pi$
$313$	−1.08773e11	+	6.28001e10i	−0.640578	+	0.369838i	0.144259	+	0.989540i	$0.453920\pi$
$314$		0			0
$315$	−6.20697e10	+	5.13548e10i	−0.355208	+	0.293889i
$316$		0			0
$317$	2.69664e10	+	4.67073e10i	0.149988	+	0.259787i	0.931223	−	0.364450i	$-0.118743\pi$
$317$	2.69664e10	+	4.67073e10i	0.149988	+	0.259787i	−0.781235	+	0.624237i	$0.785410\pi$
$318$		0			0
$319$	4.08943e9	+	2.36103e9i	0.0221108	+	0.0127657i
$320$		0			0
$321$		−	3.85092e11i		−	2.02438i
$322$		0			0
$323$		−	8.77826e10i		−	0.448742i
$324$		0			0
$325$	−2.64246e10	−	1.52563e10i	−0.131381	−	0.0758531i
$326$		0			0
$327$	1.47955e11	+	2.56266e11i	0.715593	+	1.23944i
$328$		0			0
$329$	−3.21773e11	+	5.44435e10i	−1.51415	+	0.256191i
$330$		0			0
$331$	−2.66395e11	+	1.53803e11i	−1.21983	+	0.704271i	−0.964882	−	0.262683i	$-0.915393\pi$
$331$	−2.66395e11	+	1.53803e11i	−1.21983	+	0.704271i	−0.254951	+	0.966954i	$0.582059\pi$
$332$		0			0
$333$	3.36923e11	−	5.83569e11i	1.50152	−	2.60072i
$334$		0			0
$335$	−7.53579e10			−0.326909
$336$		0			0
$337$	−1.95120e11			−0.824077			−0.412038	−	0.911166i	$-0.635183\pi$
$337$	−1.95120e11			−0.824077			−0.412038	+	0.911166i	$0.635183\pi$
$338$		0			0
$339$	−5.46545e10	+	9.46644e10i	−0.224764	+	0.389303i
$340$		0			0
$341$	1.32754e10	−	7.66454e9i	0.0531682	−	0.0306967i
$342$		0			0
$343$	2.24614e11	−	1.23535e11i	0.876221	−	0.481910i
$344$		0			0
$345$	−6.63000e10	−	1.14835e11i	−0.251958	−	0.436404i
$346$		0			0
$347$	−4.35584e11	−	2.51484e11i	−1.61283	−	0.931169i	−0.988710	−	0.149841i	$-0.952124\pi$
$347$	−4.35584e11	−	2.51484e11i	−1.61283	−	0.931169i	−0.624121	−	0.781327i	$-0.714543\pi$
$348$		0			0
$349$			7.30865e10i			0.263708i	0.991269	+	0.131854i	$0.0420929\pi$
$349$			7.30865e10i			0.263708i	−0.991269	+	0.131854i	$0.957907\pi$
$350$		0			0
$351$		−	1.07358e11i		−	0.377532i
$352$		0			0
$353$	3.44276e11	+	1.98768e11i	1.18010	+	0.681333i	0.956039	−	0.293241i	$-0.0947339\pi$
$353$	3.44276e11	+	1.98768e11i	1.18010	+	0.681333i	0.224065	+	0.974574i	$0.428067\pi$
$354$		0			0
$355$	5.76487e10	+	9.98505e10i	0.192647	+	0.333674i
$356$		0			0
$357$	4.94667e10	+	2.92359e11i	0.161178	+	0.952596i
$358$		0			0
$359$	−2.44544e11	+	1.41188e11i	−0.777021	+	0.448613i	−0.835373	−	0.549683i	$-0.814749\pi$
$359$	−2.44544e11	+	1.41188e11i	−0.777021	+	0.448613i	0.0583526	+	0.998296i	$0.481415\pi$
$360$		0			0
$361$	4.60301e10	−	7.97265e10i	0.142646	−	0.247070i
$362$		0			0
$363$	−6.00903e11			−1.81645
$364$		0			0
$365$	−6.94299e9			−0.0204752
$366$		0			0
$367$	3.13774e10	−	5.43472e10i	0.0902857	−	0.156379i	−0.817346	−	0.576148i	$-0.804555\pi$
$367$	3.13774e10	−	5.43472e10i	0.0902857	−	0.156379i	0.907631	+	0.419768i	$0.137889\pi$
$368$		0			0
$369$	5.07020e11	−	2.92728e11i	1.42366	−	0.821951i
$370$		0			0
$371$	3.45748e10	+	4.17887e10i	0.0947497	+	0.114519i
$372$		0			0
$373$	1.08093e10	+	1.87223e10i	0.0289140	+	0.0500805i	0.880120	−	0.474751i	$-0.157462\pi$
$373$	1.08093e10	+	1.87223e10i	0.0289140	+	0.0500805i	−0.851206	+	0.524831i	$0.824128\pi$
$374$		0			0
$375$	−2.35861e11	−	1.36174e11i	−0.615907	−	0.355594i
$376$		0			0
$377$		−	3.51412e10i		−	0.0895943i
$378$		0			0
$379$		−	1.43819e11i		−	0.358048i	−0.983845	−	0.179024i	$-0.942706\pi$
$379$		−	1.43819e11i		−	0.358048i	0.983845	−	0.179024i	$-0.0572939\pi$
$380$		0			0
$381$	3.41564e11	+	1.97202e11i	0.830442	+	0.479456i
$382$		0			0
$383$	−4.22507e10	−	7.31804e10i	−0.100332	−	0.173780i	0.811489	−	0.584367i	$-0.198657\pi$
$383$	−4.22507e10	−	7.31804e10i	−0.100332	−	0.173780i	−0.911821	+	0.410587i	$0.865324\pi$
$384$		0			0
$385$	−1.34823e9	+	3.62594e9i	−0.00312745	+	0.00841099i
$386$		0			0
$387$	−1.10201e12	+	6.36248e11i	−2.49740	+	1.44187i
$388$		0			0
$389$	2.57391e11	−	4.45814e11i	0.569928	−	0.987145i	−0.426644	−	0.904420i	$-0.640304\pi$
$389$	2.57391e11	−	4.45814e11i	0.569928	−	0.987145i	0.996572	−	0.0827252i	$-0.0263624\pi$
$390$		0			0
$391$	−3.40735e11			−0.737262
$392$		0			0
$393$	−1.42651e12			−3.01654
$394$		0			0
$395$	4.05791e10	−	7.02851e10i	0.0838717	−	0.145270i
$396$		0			0
$397$	4.75818e11	−	2.74714e11i	0.961355	−	0.555039i	0.0647653	−	0.997901i	$-0.479370\pi$
$397$	4.75818e11	−	2.74714e11i	0.961355	−	0.555039i	0.896590	+	0.442862i	$0.146037\pi$
$398$		0			0
$399$	2.71501e11	−	7.30176e11i	0.536281	−	1.44228i
$400$		0			0
$401$	−1.82033e11	−	3.15291e11i	−0.351561	−	0.608922i	0.634962	−	0.772544i	$-0.281016\pi$
$401$	−1.82033e11	−	3.15291e11i	−0.351561	−	0.608922i	−0.986523	+	0.163621i	$0.947683\pi$
$402$		0			0
$403$	−9.87942e10	−	5.70388e10i	−0.186577	−	0.107720i
$404$		0			0
$405$		−	2.19802e11i		−	0.405961i
$406$		0			0
$407$		−	3.23579e10i		−	0.0584529i
$408$		0			0
$409$	1.46225e11	+	8.44231e10i	0.258385	+	0.149179i	0.623598	−	0.781746i	$-0.285670\pi$
$409$	1.46225e11	+	8.44231e10i	0.258385	+	0.149179i	−0.365213	+	0.930924i	$0.619004\pi$
$410$		0			0
$411$	8.12231e11	+	1.40683e12i	1.40408	+	2.43194i
$412$		0			0
$413$	−2.63896e10	−	3.18957e10i	−0.0446331	−	0.0539456i
$414$		0			0
$415$	1.57844e11	−	9.11314e10i	0.261223	−	0.150817i
$416$		0			0
$417$	6.98546e11	−	1.20992e12i	1.13131	−	1.95949i
$418$		0			0
$419$	2.37545e11			0.376516			0.188258	−	0.982120i	$-0.439716\pi$
$419$	2.37545e11			0.376516			0.188258	+	0.982120i	$0.439716\pi$
$420$		0			0
$421$	−7.71208e11			−1.19647			−0.598235	−	0.801321i	$-0.704131\pi$
$421$	−7.71208e11			−1.19647			−0.598235	+	0.801321i	$0.704131\pi$
$422$		0			0
$423$	1.16939e12	−	2.02544e12i	1.77594	−	3.07601i
$424$		0			0
$425$	−2.96898e11	+	1.71414e11i	−0.441424	+	0.254857i
$426$		0			0
$427$	8.76748e10	+	5.18177e11i	0.127629	+	0.754314i
$428$		0			0
$429$	−4.54096e9	−	7.86517e9i	−0.00647276	−	0.0112112i
$430$		0			0
$431$	5.51930e11	+	3.18657e11i	0.770436	+	0.444811i	0.833030	−	0.553228i	$-0.186604\pi$
$431$	5.51930e11	+	3.18657e11i	0.770436	+	0.444811i	−0.0625943	+	0.998039i	$0.519937\pi$
$432$		0			0
$433$		−	6.49769e11i		−	0.888307i	−0.895951	−	0.444154i	$-0.853504\pi$
$433$		−	6.49769e11i		−	0.888307i	0.895951	−	0.444154i	$-0.146496\pi$
$434$		0			0
$435$		−	1.53653e11i		−	0.205749i
$436$		0			0
$437$	7.75266e11	+	4.47600e11i	1.01691	+	0.587115i
$438$		0			0
$439$	1.04153e11	+	1.80398e11i	0.133838	+	0.231815i	0.925153	−	0.379594i	$-0.123936\pi$
$439$	1.04153e11	+	1.80398e11i	0.133838	+	0.231815i	−0.791315	+	0.611409i	$0.790603\pi$
$440$		0			0
$441$	−3.44024e11	+	1.80461e12i	−0.433127	+	2.27200i
$442$		0			0
$443$	8.06607e11	−	4.65695e11i	0.995051	−	0.574493i	0.0882706	−	0.996097i	$-0.471866\pi$
$443$	8.06607e11	−	4.65695e11i	0.995051	−	0.574493i	0.906780	+	0.421604i	$0.138533\pi$
$444$		0			0
$445$	1.14639e11	−	1.98561e11i	0.138584	−	0.240035i
$446$		0			0
$447$	1.57654e12			1.86775
$448$		0			0
$449$	1.15814e12			1.34479			0.672393	−	0.740194i	$-0.265267\pi$
$449$	1.15814e12			1.34479			0.672393	+	0.740194i	$0.265267\pi$
$450$		0			0
$451$	1.40567e10	−	2.43470e10i	0.0159989	−	0.0277109i
$452$		0			0
$453$	−1.33338e12	+	7.69828e11i	−1.48769	+	0.858918i
$454$		0			0
$455$	2.83855e10	−	4.80278e9i	0.0310488	−	0.00525341i
$456$		0			0
$457$	3.62628e11	+	6.28090e11i	0.388901	+	0.673596i	0.992302	−	0.123842i	$-0.0395216\pi$
$457$	3.62628e11	+	6.28090e11i	0.388901	+	0.673596i	−0.603401	+	0.797438i	$0.706188\pi$
$458$		0			0
$459$	−1.04463e12	−	6.03119e11i	−1.09852	−	0.634229i
$460$		0			0
$461$		−	9.90960e11i		−	1.02189i	−0.859615	−	0.510943i	$-0.829296\pi$
$461$		−	9.90960e11i		−	1.02189i	0.859615	−	0.510943i	$-0.170704\pi$
$462$		0			0
$463$			9.22866e11i			0.933306i	0.884440	+	0.466653i	$0.154540\pi$
$463$			9.22866e11i			0.933306i	−0.884440	+	0.466653i	$0.845460\pi$
$464$		0			0
$465$	−4.31972e11	−	2.49399e11i	−0.428467	−	0.247375i
$466$		0			0
$467$	4.17837e10	+	7.23716e10i	0.0406519	+	0.0704112i	0.885635	−	0.464381i	$-0.153723\pi$
$467$	4.17837e10	+	7.23716e10i	0.0406519	+	0.0704112i	−0.844984	+	0.534792i	$0.820390\pi$
$468$		0			0
$469$	−1.32404e12	+	1.09547e12i	−1.26364	+	1.04550i
$470$		0			0
$471$	−2.52873e12	+	1.45996e12i	−2.36760	+	1.36693i
$472$		0			0
$473$	−3.05524e10	+	5.29184e10i	−0.0280654	+	0.0486106i
$474$		0			0
$475$	9.00697e11			0.811815
$476$		0			0
$477$	−3.88697e11			−0.343778
$478$		0			0
$479$	−4.34619e10	+	7.52782e10i	−0.0377224	+	0.0653371i	−0.884270	−	0.466976i	$-0.845344\pi$
$479$	−4.34619e10	+	7.52782e10i	−0.0377224	+	0.0653371i	0.846548	+	0.532313i	$0.178677\pi$
$480$		0			0
$481$	−2.08543e11	+	1.20402e11i	−0.177641	+	0.102561i
$482$		0			0
$483$	−2.83424e12	−	1.05385e12i	−2.36960	−	0.881084i
$484$		0			0
$485$	−9.28565e10	−	1.60832e11i	−0.0762034	−	0.131988i
$486$		0			0
$487$	5.60706e11	+	3.23724e11i	0.451705	+	0.260792i	0.708550	−	0.705661i	$-0.249350\pi$
$487$	5.60706e11	+	3.23724e11i	0.451705	+	0.260792i	−0.256845	+	0.966453i	$0.582683\pi$
$488$		0			0
$489$			2.50008e12i			1.97726i
$490$		0			0
$491$			2.28257e12i			1.77238i	0.463319	+	0.886191i	$0.346658\pi$
$491$			2.28257e12i			1.77238i	−0.463319	+	0.886191i	$0.653342\pi$
$492$		0			0
$493$	−3.41936e11	−	1.97417e11i	−0.260695	−	0.150513i
$494$		0			0
$495$	−1.38618e10	−	2.40094e10i	−0.0103776	−	0.0179746i
$496$		0			0
$497$	2.46441e12	+	9.16337e11i	1.81179	+	0.673677i
$498$		0			0
$499$	−1.10496e10	+	6.37948e9i	−0.00797799	+	0.00460609i	−0.503984	−	0.863713i	$-0.668133\pi$
$499$	−1.10496e10	+	6.37948e9i	−0.00797799	+	0.00460609i	0.496006	+	0.868319i	$0.334800\pi$
$500$		0			0
$501$	−2.00829e12	+	3.47846e12i	−1.42415	+	2.46670i
$502$		0			0
$503$	1.22362e12			0.852297			0.426149	−	0.904653i	$-0.359870\pi$
$503$	1.22362e12			0.852297			0.426149	+	0.904653i	$0.359870\pi$
$504$		0			0
$505$	−1.43715e11			−0.0983309
$506$		0			0
$507$	1.32018e12	−	2.28662e12i	0.887357	−	1.53695i
$508$		0			0
$509$	2.27225e12	−	1.31189e12i	1.50047	−	0.866296i	0.500469	−	0.865754i	$-0.333161\pi$
$509$	2.27225e12	−	1.31189e12i	1.50047	−	0.866296i	1.00000		0.000541866i	$-0.000172481\pi$
$510$		0			0
$511$	−1.21988e11	+	1.00930e11i	−0.0791451	+	0.0654825i
$512$		0			0
$513$	1.58455e12	+	2.74452e12i	1.01013	+	1.74960i
$514$		0			0
$515$	−6.63306e9	−	3.82960e9i	−0.00415510	−	0.00239895i
$516$		0			0
$517$		−	1.12307e11i		−	0.0691354i
$518$		0			0
$519$		−	1.46782e12i		−	0.888011i
$520$		0			0
$521$	−5.42575e11	−	3.13256e11i	−0.322619	−	0.186264i	0.329940	−	0.944002i	$-0.392971\pi$
$521$	−5.42575e11	−	3.13256e11i	−0.322619	−	0.186264i	−0.652559	+	0.757738i	$0.726305\pi$
$522$		0			0
$523$	8.90020e11	+	1.54156e12i	0.520166	+	0.900955i	0.999725	+	0.0234450i	$0.00746345\pi$
$523$	8.90020e11	+	1.54156e12i	0.520166	+	0.900955i	−0.479559	+	0.877510i	$0.659203\pi$
$524$		0			0
$525$	−2.99976e12	+	5.07555e11i	−1.72333	+	0.291586i
$526$		0			0
$527$	−1.11001e12	+	6.40867e11i	−0.626875	+	0.361926i
$528$		0			0
$529$	8.36819e11	−	1.44941e12i	0.464602	−	0.804714i
$530$		0			0
$531$	2.96677e11			0.161941
$532$		0			0
$533$	−2.09218e11			−0.112286
$534$		0			0
$535$	−2.10045e11	+	3.63808e11i	−0.110846	+	0.191991i
$536$		0			0
$537$	3.95604e11	−	2.28402e11i	0.205294	−	0.118527i
$538$		0			0
$539$	2.90216e10	+	8.33068e10i	0.0148106	+	0.0425139i
$540$		0			0
$541$	5.77772e11	+	1.00073e12i	0.289980	+	0.502261i	0.973805	−	0.227386i	$-0.0730179\pi$
$541$	5.77772e11	+	1.00073e12i	0.289980	+	0.502261i	−0.683824	+	0.729647i	$0.739685\pi$
$542$		0			0
$543$	2.03908e12	+	1.17726e12i	1.00655	+	0.581131i
$544$		0			0
$545$		−	3.22803e11i		−	0.156731i
$546$		0			0
$547$		−	6.09900e11i		−	0.291283i	−0.989337	−	0.145642i	$-0.953475\pi$
$547$		−	6.09900e11i		−	0.291283i	0.989337	−	0.145642i	$-0.0465246\pi$
$548$		0			0
$549$	−3.26173e12	−	1.88316e12i	−1.53240	−	0.884732i
$550$		0			0
$551$	5.18664e11	+	8.98353e11i	0.239720	+	0.415207i
$552$		0			0
$553$	−3.08754e11	−	1.82480e12i	−0.140394	−	0.829761i
$554$		0			0
$555$	−9.11842e11	+	5.26452e11i	−0.407945	+	0.235527i
$556$		0			0
$557$	3.16523e11	−	5.48233e11i	0.139334	−	0.241333i	−0.787911	−	0.615789i	$-0.788837\pi$
$557$	3.16523e11	−	5.48233e11i	0.139334	−	0.241333i	0.927245	+	0.374456i	$0.122171\pi$
$558$		0			0
$559$	4.54737e11			0.196973
$560$		0			0
$561$	−1.02041e11			−0.0434953
$562$		0			0
$563$	1.71355e12	−	2.96796e12i	0.718803	−	1.24500i	−0.242671	−	0.970109i	$-0.578024\pi$
$563$	1.71355e12	−	2.96796e12i	0.718803	−	1.24500i	0.961474	−	0.274895i	$-0.0886430\pi$
$564$		0			0
$565$	1.03268e11	−	5.96216e10i	0.0426330	−	0.0246142i
$566$		0			0
$567$	−3.19525e12	−	3.86193e12i	−1.29832	−	1.56921i
$568$		0			0
$569$	5.21698e11	+	9.03608e11i	0.208648	+	0.361389i	0.951289	−	0.308301i	$-0.0997604\pi$
$569$	5.21698e11	+	9.03608e11i	0.208648	+	0.361389i	−0.742641	+	0.669690i	$0.766427\pi$
$570$		0			0
$571$	2.05456e12	+	1.18620e12i	0.808830	+	0.466978i	0.846549	−	0.532310i	$-0.178676\pi$
$571$	2.05456e12	+	1.18620e12i	0.808830	+	0.466978i	−0.0377192	+	0.999288i	$0.512009\pi$
$572$		0			0
$573$		−	3.93614e12i		−	1.52537i
$574$		0			0
$575$		−	3.49613e12i		−	1.33377i
$576$		0			0
$577$	−3.01236e12	−	1.73918e12i	−1.13140	−	0.653212i	−0.187111	−	0.982339i	$-0.559912\pi$
$577$	−3.01236e12	−	1.73918e12i	−1.13140	−	0.653212i	−0.944286	+	0.329126i	$0.893246\pi$
$578$		0			0
$579$	3.88193e12	+	6.72370e12i	1.43547	+	2.48631i
$580$		0			0
$581$	1.44855e12	−	3.89575e12i	0.527401	−	1.41840i
$582$		0			0
$583$	−1.61645e10	+	9.33255e9i	−0.00579499	+	0.00334574i
$584$		0			0
$585$	−1.03159e11	+	1.78676e11i	−0.0364170	+	0.0630761i
$586$		0			0
$587$	−1.86579e12			−0.648621			−0.324311	−	0.945951i	$-0.605132\pi$
$587$	−1.86579e12			−0.648621			−0.324311	+	0.945951i	$0.605132\pi$
$588$		0			0
$589$	3.36745e12			1.15287
$590$		0			0
$591$	−1.26313e12	+	2.18780e12i	−0.425896	+	0.737673i
$592$		0			0
$593$	3.43264e12	−	1.98184e12i	1.13994	−	0.658145i	0.193525	−	0.981095i	$-0.438008\pi$
$593$	3.43264e12	−	1.98184e12i	1.13994	−	0.658145i	0.946416	+	0.322950i	$0.104675\pi$
$594$		0			0
$595$	1.12732e11	−	3.03181e11i	0.0368739	−	0.0991690i
$596$		0			0
$597$	1.30088e12	+	2.25318e12i	0.419133	+	0.725959i
$598$		0			0
$599$	1.96176e12	+	1.13262e12i	0.622623	+	0.359472i	0.777889	−	0.628401i	$-0.216290\pi$
$599$	1.96176e12	+	1.13262e12i	0.622623	+	0.359472i	−0.155267	+	0.987873i	$0.549624\pi$
$600$		0			0
$601$			1.28723e12i			0.402459i	0.979544	+	0.201229i	$0.0644936\pi$
$601$			1.28723e12i			0.402459i	−0.979544	+	0.201229i	$0.935506\pi$
$602$		0			0
$603$		−	1.23155e13i		−	3.79336i
$604$		0			0
$605$	5.67691e11	+	3.27757e11i	0.172271	+	0.0994609i
$606$		0			0
$607$	2.89183e11	+	5.00880e11i	0.0864618	+	0.149756i	0.906013	−	0.423249i	$-0.139111\pi$
$607$	2.89183e11	+	5.00880e11i	0.0864618	+	0.149756i	−0.819551	+	0.573006i	$0.805777\pi$
$608$		0			0
$609$	−2.23364e12	−	2.69968e12i	−0.658014	−	0.795305i
$610$		0			0
$611$	−7.23808e11	+	4.17891e11i	−0.210106	+	0.121305i
$612$		0			0
$613$	−2.56502e12	+	4.44274e12i	−0.733699	+	1.27080i	0.221593	+	0.975139i	$0.428874\pi$
$613$	−2.56502e12	+	4.44274e12i	−0.733699	+	1.27080i	−0.955292	+	0.295665i	$0.904459\pi$
$614$		0			0
$615$	−9.14792e11			−0.257860
$616$		0			0
$617$	1.51024e12			0.419531			0.209765	−	0.977752i	$-0.432730\pi$
$617$	1.51024e12			0.419531			0.209765	+	0.977752i	$0.432730\pi$
$618$		0			0
$619$	−2.39636e12	+	4.15062e12i	−0.656062	+	1.13633i	0.325565	+	0.945520i	$0.394446\pi$
$619$	−2.39636e12	+	4.15062e12i	−0.656062	+	1.13633i	−0.981626	+	0.190813i	$0.938888\pi$
$620$		0			0
$621$	1.06531e13	−	6.15056e12i	2.87450	−	1.65960i
$622$		0			0
$623$	−8.72256e11	−	5.15522e12i	−0.231979	−	1.37104i
$624$		0			0
$625$	−1.68302e12	−	2.91507e12i	−0.441193	−	0.764169i
$626$		0			0
$627$	2.32171e11	+	1.34044e11i	0.0599935	+	0.0346373i
$628$		0			0
$629$			2.70559e12i			0.689183i
$630$		0			0
$631$			3.49632e12i			0.877969i	0.898495	+	0.438984i	$0.144662\pi$
$631$			3.49632e12i			0.877969i	−0.898495	+	0.438984i	$0.855338\pi$
$632$		0			0
$633$	−1.18755e13	−	6.85631e12i	−2.93991	−	1.69736i
$634$		0			0
$635$	−2.15124e11	−	3.72606e11i	−0.0525058	−	0.0909426i
$636$		0			0
$637$	4.28915e11	−	4.97022e11i	0.103215	−	0.119605i
$638$		0			0
$639$	−1.63182e13	+	9.42134e12i	−3.87186	+	2.23542i
$640$		0			0
$641$	1.16760e11	−	2.02234e11i	0.0273169	−	0.0473143i	−0.852044	−	0.523471i	$-0.824637\pi$
$641$	1.16760e11	−	2.02234e11i	0.0273169	−	0.0473143i	0.879361	+	0.476156i	$0.157970\pi$
$642$		0			0
$643$	−8.37514e11			−0.193216			−0.0966079	−	0.995323i	$-0.530799\pi$
$643$	−8.37514e11			−0.193216			−0.0966079	+	0.995323i	$0.530799\pi$
$644$		0			0
$645$	1.98831e12			0.452340
$646$		0			0
$647$	−1.75528e12	+	3.04023e12i	−0.393800	+	0.682082i	−0.992947	−	0.118557i	$-0.962173\pi$
$647$	−1.75528e12	+	3.04023e12i	−0.393800	+	0.682082i	0.599147	+	0.800639i	$0.295506\pi$
$648$		0			0
$649$	1.23377e10	−	7.12316e9i	0.00272981	−	0.00157606i
$650$		0			0
$651$	−1.12152e13	+	1.89760e12i	−2.44734	+	0.414086i
$652$		0			0
$653$	2.92944e12	+	5.07393e12i	0.630485	+	1.09203i	0.987453	+	0.157916i	$0.0504774\pi$
$653$	2.92944e12	+	5.07393e12i	0.630485	+	1.09203i	−0.356967	+	0.934117i	$0.616189\pi$
$654$		0			0
$655$	1.34767e12	+	7.78078e11i	0.286087	+	0.165172i
$656$		0			0
$657$		−	1.13467e12i		−	0.237589i
$658$		0			0
$659$		−	7.16693e12i		−	1.48030i	−0.672443	−	0.740149i	$-0.734755\pi$
$659$		−	7.16693e12i		−	1.48030i	0.672443	−	0.740149i	$-0.265245\pi$
$660$		0			0
$661$	1.68597e12	+	9.73396e11i	0.343513	+	0.198327i	0.661825	−	0.749659i	$-0.269782\pi$
$661$	1.68597e12	+	9.73396e11i	0.343513	+	0.198327i	−0.318311	+	0.947986i	$0.603116\pi$
$662$		0			0
$663$	3.79690e11	+	6.57643e11i	0.0763165	+	0.132184i
$664$		0			0
$665$	−6.54763e11	+	5.41732e11i	−0.129833	+	0.107421i
$666$		0			0
$667$	3.48703e12	−	2.01324e12i	0.682165	−	0.393848i
$668$		0			0
$669$	−4.17593e11	+	7.23291e11i	−0.0806000	+	0.139603i
$670$		0			0
$671$	−1.80858e11			−0.0344418
$672$		0			0
$673$	−3.40990e12			−0.640728			−0.320364	−	0.947294i	$-0.603805\pi$
$673$	−3.40990e12			−0.640728			−0.320364	+	0.947294i	$0.603805\pi$
$674$		0			0
$675$	6.18833e12	−	1.07185e13i	1.14738	−	1.98732i
$676$		0			0
$677$	1.80551e12	−	1.04241e12i	0.330332	−	0.190717i	−0.325656	−	0.945488i	$-0.605585\pi$
$677$	1.80551e12	−	1.04241e12i	0.330332	−	0.190717i	0.655989	+	0.754771i	$0.272252\pi$
$678$		0			0
$679$	−3.96949e12	−	1.47597e12i	−0.716673	−	0.266480i
$680$		0			0
$681$	5.83162e12	+	1.01007e13i	1.03903	+	1.79965i
$682$		0			0
$683$	−5.28482e12	−	3.05119e12i	−0.929259	−	0.536508i	−0.0426819	−	0.999089i	$-0.513590\pi$
$683$	−5.28482e12	−	3.05119e12i	−0.929259	−	0.536508i	−0.886577	+	0.462581i	$0.846924\pi$
$684$		0			0
$685$		−	1.77209e12i		−	0.307524i
$686$		0			0
$687$			1.21195e13i			2.07577i
$688$		0			0
$689$	1.20294e11	+	6.94520e10i	0.0203357	+	0.0117408i
$690$		0			0
$691$	2.18350e12	+	3.78193e12i	0.364336	+	0.631048i	0.988669	−	0.150110i	$-0.0479626\pi$
$691$	2.18350e12	+	3.78193e12i	0.364336	+	0.631048i	−0.624333	+	0.781158i	$0.714629\pi$
$692$		0			0
$693$	−5.92575e11	−	2.20337e11i	−0.0975987	−	0.0362900i
$694$		0			0
$695$	−1.31988e12	+	7.62031e11i	−0.214586	+	0.123891i
$696$		0			0
$697$	−1.17535e12	+	2.03576e12i	−0.188633	+	0.326723i
$698$		0			0
$699$	1.86811e13			2.95975
$700$		0			0
$701$	8.36857e12			1.30894			0.654471	−	0.756087i	$-0.272892\pi$
$701$	8.36857e12			1.30894			0.654471	+	0.756087i	$0.272892\pi$
$702$		0			0
$703$	3.55414e12	−	6.15596e12i	0.548828	−	0.950598i
$704$		0			0
$705$	−3.16481e12	+	1.82720e12i	−0.482499	+	0.278571i
$706$		0			0
$707$	−2.52507e12	+	2.08917e12i	−0.380089	+	0.314475i
$708$		0			0
$709$	6.49877e10	+	1.12562e11i	0.00965881	+	0.0167295i	0.870814	−	0.491612i	$-0.163592\pi$
$709$	6.49877e10	+	1.12562e11i	0.00965881	+	0.0167295i	−0.861156	+	0.508341i	$0.830259\pi$
$710$		0			0
$711$	1.14865e13	+	6.63171e12i	1.68567	+	0.973224i
$712$		0			0
$713$		−	1.30710e13i		−	1.89412i
$714$		0			0
$715$			9.90729e9i			0.00141768i
$716$		0			0
$717$	−1.30669e13	−	7.54418e12i	−1.84645	−	1.06605i
$718$		0			0
$719$	−4.27075e12	−	7.39716e12i	−0.595970	−	1.03225i	−0.993409	−	0.114622i	$-0.963434\pi$
$719$	−4.27075e12	−	7.39716e12i	−0.595970	−	1.03225i	0.397439	−	0.917628i	$-0.369899\pi$
$720$		0			0
$721$	−1.72213e11	+	2.91383e10i	−0.0237333	+	0.00401564i
$722$		0			0
$723$	1.09379e13	−	6.31501e12i	1.48872	−	0.859512i
$724$		0			0
$725$	2.02560e12	−	3.50844e12i	0.272291	−	0.471621i
$726$		0			0
$727$	7.27274e12			0.965591			0.482796	−	0.875733i	$-0.339621\pi$
$727$	7.27274e12			0.965591			0.482796	+	0.875733i	$0.339621\pi$
$728$		0			0
$729$	2.75336e12			0.361068
$730$		0			0
$731$	2.55463e12	−	4.42474e12i	0.330902	−	0.573139i
$732$		0			0
$733$	−1.12218e13	+	6.47893e12i	−1.43581	+	0.828963i	−0.997555	−	0.0698887i	$-0.977736\pi$
$733$	−1.12218e13	+	6.47893e12i	−1.43581	+	0.828963i	−0.438252	+	0.898852i	$0.644402\pi$
$734$		0			0
$735$	1.87540e12	−	2.17320e12i	0.237029	−	0.274667i
$736$		0			0
$737$	−2.95693e11	−	5.12156e11i	−0.0369179	−	0.0639438i
$738$		0			0
$739$	−2.43559e12	−	1.40619e12i	−0.300404	−	0.173438i	0.342221	−	0.939620i	$-0.388821\pi$
$739$	−2.43559e12	−	1.40619e12i	−0.300404	−	0.173438i	−0.642624	+	0.766182i	$0.722154\pi$
$740$		0			0
$741$		−	1.99509e12i		−	0.243097i
$742$		0			0
$743$		−	4.93364e12i		−	0.593906i	−0.954892	−	0.296953i	$-0.904030\pi$
$743$		−	4.93364e12i		−	0.593906i	0.954892	−	0.296953i	$-0.0959705\pi$
$744$		0			0
$745$	−1.48940e12	−	8.59906e11i	−0.177137	−	0.102270i
$746$		0			0
$747$	1.48933e13	+	2.57960e13i	1.75004	+	3.03116i
$748$		0			0
$749$	1.59817e12	+	9.44551e12i	0.185547	+	1.09662i
$750$		0			0
$751$	−1.48476e13	+	8.57227e12i	−1.70324	+	0.983368i	−0.760810	+	0.648975i	$0.775198\pi$
$751$	−1.48476e13	+	8.57227e12i	−1.70324	+	0.983368i	−0.942434	+	0.334393i	$0.891469\pi$
$752$		0			0
$753$	−7.44370e12	+	1.28929e13i	−0.843746	+	1.46141i
$754$		0			0
$755$	1.67958e12			0.188122
$756$		0			0
$757$	−1.01045e13			−1.11836			−0.559182	−	0.829045i	$-0.688885\pi$
$757$	−1.01045e13			−1.11836			−0.559182	+	0.829045i	$0.688885\pi$
$758$		0			0
$759$	5.20303e11	−	9.01191e11i	0.0569074	−	0.0985664i
$760$		0			0
$761$	1.48414e13	−	8.56870e12i	1.60415	−	0.926155i	0.613503	−	0.789692i	$-0.289760\pi$
$761$	1.48414e13	−	8.56870e12i	1.60415	−	0.926155i	0.990645	−	0.136463i	$-0.0435736\pi$
$762$		0			0
$763$	−4.69257e12	−	5.67165e12i	−0.501245	−	0.605828i
$764$		0			0
$765$	1.15905e12	+	2.00754e12i	0.122356	+	0.211927i
$766$		0			0
$767$	−9.18159e10	−	5.30099e10i	−0.00957941	−	0.00553067i
$768$		0			0
$769$			7.22943e12i			0.745479i	0.927936	+	0.372740i	$0.121582\pi$
$769$			7.22943e12i			0.745479i	−0.927936	+	0.372740i	$0.878418\pi$
$770$		0			0
$771$			5.18429e12i			0.528378i
$772$		0			0
$773$	5.40988e12	+	3.12339e12i	0.544979	+	0.314644i	0.747094	−	0.664718i	$-0.231448\pi$
$773$	5.40988e12	+	3.12339e12i	0.544979	+	0.314644i	−0.202115	+	0.979362i	$0.564782\pi$
$774$		0			0
$775$	−6.57564e12	−	1.13893e13i	−0.654758	−	1.13407i
$776$		0			0
$777$	−8.36806e12	+	2.25051e13i	−0.823626	+	2.21507i
$778$		0			0
$779$	5.34847e12	−	3.08794e12i	0.520368	−	0.300435i
$780$		0			0
$781$	−4.52410e11	+	7.83597e11i	−0.0435113	+	0.0753638i
$782$		0			0
$783$	1.42541e13			1.35523
$784$		0			0
$785$	3.18529e12			0.299389
$786$		0			0
$787$	3.95228e12	−	6.84554e12i	0.367249	−	0.636094i	−0.621885	−	0.783108i	$-0.713633\pi$
$787$	3.95228e12	−	6.84554e12i	0.367249	−	0.636094i	0.989134	+	0.147014i	$0.0469663\pi$
$788$		0			0
$789$	−2.99371e12	+	1.72842e12i	−0.275019	+	0.158782i
$790$		0			0
$791$	9.47696e11	−	2.54874e12i	0.0860746	−	0.231490i
$792$		0			0
$793$	6.72963e11	+	1.16561e12i	0.0604313	+	0.104670i
$794$		0			0
$795$	5.25980e11	+	3.03675e11i	0.0467001	+	0.0269623i
$796$		0			0
$797$			1.95186e13i			1.71350i	0.515728	+	0.856752i	$0.327521\pi$
$797$			1.95186e13i			1.71350i	−0.515728	+	0.856752i	$0.672479\pi$
$798$		0			0
$799$			9.39053e12i			0.815135i
$800$		0			0
$801$	3.24502e13	+	1.87351e13i	2.78530	+	1.60809i
$802$		0			0
$803$	−2.72433e10	−	4.71867e10i	−0.00231227	−	0.00400498i
$804$		0			0
$805$	2.10278e12	+	2.54152e12i	0.176487	+	0.213310i
$806$		0			0
$807$	−3.23095e13	+	1.86539e13i	−2.68164	+	1.54824i
$808$		0			0
$809$	4.37708e12	−	7.58132e12i	0.359266	−	0.622267i	−0.628572	−	0.777751i	$-0.716360\pi$
$809$	4.37708e12	−	7.58132e12i	0.359266	−	0.622267i	0.987838	+	0.155484i	$0.0496937\pi$
$810$		0			0
$811$	−1.92465e13			−1.56227			−0.781137	−	0.624359i	$-0.785360\pi$
$811$	−1.92465e13			−1.56227			−0.781137	+	0.624359i	$0.785360\pi$
$812$		0			0
$813$	−3.35019e13			−2.68944
$814$		0			0
$815$	1.36365e12	−	2.36191e12i	0.108266	−	0.187522i
$816$		0			0
$817$	−1.16249e13	+	6.71166e12i	−0.912833	+	0.527025i
$818$		0			0
$819$	7.84903e11	+	4.63894e12i	0.0609591	+	0.360281i
$820$		0			0
$821$	1.74055e12	+	3.01472e12i	0.133703	+	0.231581i	0.925101	−	0.379720i	$-0.123980\pi$
$821$	1.74055e12	+	3.01472e12i	0.133703	+	0.231581i	−0.791398	+	0.611301i	$0.790646\pi$
$822$		0			0
$823$	−2.41155e12	−	1.39231e12i	−0.183230	−	0.105788i	0.405579	−	0.914060i	$-0.367070\pi$
$823$	−2.41155e12	−	1.39231e12i	−0.183230	−	0.105788i	−0.588810	+	0.808272i	$0.700403\pi$
$824$		0			0
$825$		−	1.04700e12i		−	0.0786869i
$826$		0			0
$827$		−	8.99625e12i		−	0.668785i	−0.942434	−	0.334393i	$-0.891469\pi$
$827$		−	8.99625e12i		−	0.668785i	0.942434	−	0.334393i	$-0.108531\pi$
$828$		0			0
$829$	−1.07787e13	−	6.22311e12i	−0.792634	−	0.457628i	0.0482549	−	0.998835i	$-0.484634\pi$
$829$	−1.07787e13	−	6.22311e12i	−0.792634	−	0.457628i	−0.840889	+	0.541207i	$0.817967\pi$
$830$		0			0
$831$	1.40794e13	+	2.43863e13i	1.02419	+	1.77395i
$832$		0			0
$833$	−2.42663e12	−	6.96566e12i	−0.174623	−	0.501256i
$834$		0			0
$835$	3.79458e12	−	2.19080e12i	0.270131	−	0.155960i
$836$		0			0
$837$	2.31364e13	−	4.00734e13i	1.62941	−	2.82222i
$838$		0			0
$839$	−1.76180e13			−1.22752			−0.613759	−	0.789493i	$-0.710343\pi$
$839$	−1.76180e13			−1.22752			−0.613759	+	0.789493i	$0.710343\pi$
$840$		0			0
$841$	−9.84139e12			−0.678382
$842$		0			0
$843$	6.78470e12	−	1.17514e13i	0.462707	−	0.801433i
$844$		0			0
$845$	−2.49444e12	+	1.44016e12i	−0.168313	+	0.0971754i
$846$		0			0
$847$	1.47389e13	−	2.49380e12i	0.983988	−	0.166489i
$848$		0			0
$849$	−7.22963e12	−	1.25221e13i	−0.477564	−	0.827164i
$850$		0			0
$851$	−2.38949e13	−	1.37957e13i	−1.56179	−	0.901698i
$852$		0			0
$853$			6.58201e12i			0.425684i	0.977087	+	0.212842i	$0.0682720\pi$
$853$			6.58201e12i			0.425684i	−0.977087	+	0.212842i	$0.931728\pi$
$854$		0			0
$855$		−	6.09026e12i		−	0.389752i
$856$		0			0
$857$	2.34072e13	+	1.35142e13i	1.48230	+	0.855806i	0.999798	−	0.0200893i	$-0.00639504\pi$
$857$	2.34072e13	+	1.35142e13i	1.48230	+	0.855806i	0.482501	+	0.875895i	$0.339728\pi$
$858$		0			0
$859$	1.38980e13	+	2.40721e13i	0.870931	+	1.50850i	0.861035	+	0.508546i	$0.169817\pi$
$859$	1.38980e13	+	2.40721e13i	0.870931	+	1.50850i	0.00989602	+	0.999951i	$0.496850\pi$
$860$		0			0
$861$	−1.60729e13	+	1.32983e13i	−0.996735	+	0.824671i
$862$		0			0
$863$	7.83797e12	−	4.52525e12i	0.481011	−	0.277712i	−0.239827	−	0.970816i	$-0.577091\pi$
$863$	7.83797e12	−	4.52525e12i	0.481011	−	0.277712i	0.720838	+	0.693104i	$0.243757\pi$
$864$		0			0
$865$	−8.00606e11	+	1.38669e12i	−0.0486235	+	0.0842184i
$866$		0			0
$867$	−2.17504e13			−1.30732
$868$		0			0
$869$	6.36906e11			0.0378866
$870$		0			0
$871$	−2.20052e12	+	3.81142e12i	−0.129552	+	0.224391i
$872$		0			0
$873$	2.62843e13	−	1.51752e13i	1.53155	−	0.884242i
$874$		0			0
$875$	6.35032e12	+	2.36123e12i	0.366234	+	0.136176i
$876$		0			0
$877$	6.18790e12	+	1.07178e13i	0.353220	+	0.611795i	0.986812	−	0.161873i	$-0.0517534\pi$
$877$	6.18790e12	+	1.07178e13i	0.353220	+	0.611795i	−0.633592	+	0.773668i	$0.718420\pi$
$878$		0			0
$879$	1.36289e13	+	7.86863e12i	0.770034	+	0.444579i
$880$		0			0
$881$		−	1.95647e13i		−	1.09416i	−0.837080	−	0.547080i	$-0.815739\pi$
$881$		−	1.95647e13i		−	1.09416i	0.837080	−	0.547080i	$-0.184261\pi$
$882$		0			0
$883$		−	2.03572e13i		−	1.12693i	−0.826141	−	0.563463i	$-0.809469\pi$
$883$		−	2.03572e13i		−	1.12693i	0.826141	−	0.563463i	$-0.190531\pi$
$884$		0			0
$885$	−4.01459e11	−	2.31783e11i	−0.0219987	−	0.0127010i
$886$		0			0
$887$	−1.09697e13	−	1.90001e13i	−0.595029	−	1.03062i	−0.993543	−	0.113458i	$-0.963807\pi$
$887$	−1.09697e13	−	1.90001e13i	−0.595029	−	1.03062i	0.398514	−	0.917162i	$-0.369526\pi$
$888$		0			0
$889$	−9.19626e12	−	3.41944e12i	−0.493803	−	0.183610i
$890$		0			0
$891$	1.49385e12	−	8.62472e11i	0.0794065	−	0.0458454i
$892$		0			0
$893$	1.23357e13	−	2.13660e13i	0.649129	−	1.12432i
$894$		0			0
$895$	−4.98320e11			−0.0259600
$896$		0			0
$897$	−7.74410e12			−0.399397
$898$		0			0
$899$	7.57314e12	−	1.31171e13i	0.386685	−	0.669758i
$900$		0			0
$901$	1.35158e12	−	7.80337e11i	0.0683253	−	0.0394476i
$902$		0			0
$903$	3.49346e13	−	2.89039e13i	1.74848	−	1.44664i
$904$		0			0
$905$	−1.28425e12	−	2.22439e12i	−0.0636403	−	0.110228i
$906$		0			0
$907$	4.84003e9	+	2.79439e9i	0.000237473	+	0.000137105i	0.500119	−	0.865957i	$-0.333290\pi$
$907$	4.84003e9	+	2.79439e9i	0.000237473	+	0.000137105i	−0.499881	+	0.866094i	$0.666623\pi$
$908$		0			0
$909$		−	2.34868e13i		−	1.14100i
$910$		0			0
$911$			1.64389e13i			0.790752i	0.918519	+	0.395376i	$0.129386\pi$
$911$			1.64389e13i			0.790752i	−0.918519	+	0.395376i	$0.870614\pi$
$912$		0			0
$913$	1.23871e12	+	7.15172e11i	0.0590001	+	0.0340637i
$914$		0			0
$915$	2.94249e12	+	5.09654e12i	0.138778	+	0.240370i
$916$		0			0
$917$	3.49894e13	−	5.92016e12i	1.63409	−	0.276485i
$918$		0			0
$919$	−7.93400e12	+	4.58069e12i	−0.366921	+	0.211842i	−0.672112	−	0.740449i	$-0.734613\pi$
$919$	−7.93400e12	+	4.58069e12i	−0.366921	+	0.211842i	0.305192	+	0.952291i	$0.401279\pi$
$920$		0			0
$921$	8.10846e12	−	1.40443e13i	0.371339	−	0.643177i
$922$		0			0
$923$	6.73359e12			0.305379
$924$		0			0
$925$	−2.77608e13			−1.24679
$926$		0			0
$927$	6.25859e11	−	1.08402e12i	0.0278367	−	0.0482146i
$928$		0			0
$929$	−1.12765e13	+	6.51049e12i	−0.496711	+	0.286776i	−0.727354	−	0.686262i	$-0.759250\pi$
$929$	−1.12765e13	+	6.51049e12i	−0.496711	+	0.286776i	0.230643	+	0.973038i	$0.425917\pi$
$930$		0			0
$931$	−3.62905e12	+	1.90365e13i	−0.158314	+	0.830448i
$932$		0			0
$933$	−3.55306e13	−	6.15408e13i	−1.53510	−	2.65886i
$934$		0			0
$935$	9.64013e10	+	5.56573e10i	0.00412506	+	0.00238161i
$936$		0			0
$937$		−	3.96658e13i		−	1.68108i	−0.541749	−	0.840540i	$-0.682238\pi$
$937$		−	3.96658e13i		−	1.68108i	0.541749	−	0.840540i	$-0.317762\pi$
$938$		0			0
$939$			3.20732e13i			1.34631i
$940$		0			0
$941$	1.10093e13	+	6.35625e12i	0.457729	+	0.264270i	0.711089	−	0.703102i	$-0.248202\pi$
$941$	1.10093e13	+	6.35625e12i	0.457729	+	0.264270i	−0.253360	+	0.967372i	$0.581536\pi$
$942$		0			0
$943$	−1.19861e13	−	2.07605e13i	−0.493600	−	0.854940i
$944$		0			0
$945$	1.94813e12	+	1.15138e13i	0.0794647	+	0.469653i
$946$		0			0
$947$	8.68822e11	−	5.01615e11i	0.0351040	−	0.0202673i	−0.482345	−	0.875981i	$-0.660215\pi$
$947$	8.68822e11	−	5.01615e11i	0.0351040	−	0.0202673i	0.517449	+	0.855714i	$0.326882\pi$
$948$		0			0
$949$	−2.02742e11	+	3.51160e11i	−0.00811420	+	0.0140542i
$950$		0			0
$951$	1.37722e13			0.546000
$952$		0			0
$953$	1.32384e13			0.519899			0.259949	−	0.965622i	$-0.416294\pi$
$953$	1.32384e13			0.519899			0.259949	+	0.965622i	$0.416294\pi$
$954$		0			0
$955$	−2.14693e12	+	3.71859e12i	−0.0835223	+	0.144665i
$956$		0			0
$957$	1.04427e12	−	6.02910e11i	0.0402448	−	0.0232353i
$958$		0			0
$959$	−2.57608e13	−	3.11357e13i	−0.983503	−	1.18871i
$960$		0			0
$961$	−1.13646e13	−	1.96841e13i	−0.429833	−	0.744493i
$962$		0			0
$963$	−5.94560e13	−	3.43269e13i	−2.22781	−	1.28622i
$964$		0			0
$965$		−	8.46945e12i		−	0.314400i
$966$		0			0
$967$			2.39024e13i			0.879066i	0.898226	+	0.439533i	$0.144856\pi$
$967$			2.39024e13i			0.879066i	−0.898226	+	0.439533i	$0.855144\pi$
$968$		0			0
$969$	−1.94129e13	−	1.12080e13i	−0.707348	−	0.408387i
$970$		0			0
$971$	2.80985e12	+	4.86680e12i	0.101437	+	0.175694i	0.912277	−	0.409574i	$-0.134323\pi$
$971$	2.80985e12	+	4.86680e12i	0.101437	+	0.175694i	−0.810840	+	0.585268i	$0.800989\pi$
$972$		0			0
$973$	−1.21126e13	+	3.25758e13i	−0.433242	+	1.16516i
$974$		0			0
$975$	−6.74777e12	+	3.89583e12i	−0.239133	+	0.138063i
$976$		0			0
$977$	1.03784e13	−	1.79759e13i	0.364422	−	0.631197i	−0.624261	−	0.781216i	$-0.714600\pi$
$977$	1.03784e13	−	1.79759e13i	0.364422	−	0.631197i	0.988683	+	0.150018i	$0.0479333\pi$
$978$		0			0
$979$	1.79931e12			0.0626014
$980$		0			0
$981$	5.27547e13			1.81866
$982$		0			0
$983$	1.16489e13	−	2.01765e13i	0.397919	−	0.689215i	−0.595550	−	0.803318i	$-0.703066\pi$
$983$	1.16489e13	−	2.01765e13i	0.397919	−	0.689215i	0.993469	+	0.114103i	$0.0363993\pi$
$984$		0			0
$985$	2.38663e12	−	1.37792e12i	0.0807833	−	0.0466403i
$986$		0			0
$987$	−2.90437e13	+	7.81105e13i	−0.974149	+	2.61988i
$988$		0			0
$989$	2.60519e13	+	4.51232e13i	0.865876	+	1.49974i
$990$		0			0
$991$	1.53189e13	+	8.84438e12i	0.504541	+	0.291297i	0.730587	−	0.682820i	$-0.239247\pi$
$991$	1.53189e13	+	8.84438e12i	0.504541	+	0.291297i	−0.226046	+	0.974117i	$0.572580\pi$
$992$		0			0
$993$			7.85501e13i			2.56375i
$994$		0			0
$995$		−	2.83820e12i		−	0.0917994i
$996$		0			0
$997$	4.60266e13	+	2.65735e13i	1.47530	+	0.851766i	0.999612	−	0.0278467i	$-0.00886502\pi$
$997$	4.60266e13	+	2.65735e13i	1.47530	+	0.851766i	0.475690	+	0.879613i	$0.342198\pi$
$998$		0			0
$999$	−4.88382e13	−	8.45903e13i	−1.55137	−	2.68705i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.p.a.31.12	✓	24
4.3	odd	2		112.10.p.c.31.1	yes	24
7.5	odd	6		112.10.p.c.47.1	yes	24
28.19	even	6	inner	112.10.p.a.47.12	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.10.p.a.31.12	✓	24	1.1	even	1	trivial
112.10.p.a.47.12	yes	24	28.19	even	6	inner
112.10.p.c.31.1	yes	24	4.3	odd	2
112.10.p.c.47.1	yes	24	7.5	odd	6

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.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(127.679 - 221.147i) q^{3} +(-241.245 + 139.283i) q^{5} +(-2213.93 + 5954.17i) q^{7} +(-22762.6 - 39426.0i) q^{9} +(-1893.22 - 1093.05i) q^{11} +16268.8i q^{13} +71134.4i q^{15} +(158301. + 91395.1i) q^{17} +(-240118. - 415897. i) q^{19} +(1.03407e6 + 1.24983e6i) q^{21} +(-1.61434e6 + 932040. i) q^{23} +(-937763. + 1.62425e6i) q^{25} -6.59903e6 q^{27} -2.16004e6 q^{29} +(-3.50603e6 + 6.07262e6i) q^{31} +(-483451. + 279121. i) q^{33} +(-295214. - 1.74478e6i) q^{35} +(7.40082e6 + 1.28186e7i) q^{37} +(3.59780e6 + 2.07719e6i) q^{39} +1.28601e7i q^{41} -2.79515e7i q^{43} +(1.09827e7 + 6.34089e6i) q^{45} +(2.56866e7 + 4.44906e7i) q^{47} +(-3.05506e7 - 2.63642e7i) q^{49} +(4.04236e7 - 2.33386e7i) q^{51} +(4.26903e6 - 7.39418e6i) q^{53} +608975. q^{55} -1.22633e8 q^{57} +(-3.25838e6 + 5.64368e6i) q^{59} +(7.16467e7 - 4.13653e7i) q^{61} +(2.85144e8 - 4.82459e7i) q^{63} +(-2.26597e6 - 3.92477e6i) q^{65} +(2.34278e8 + 1.35260e8i) q^{67} +4.76009e8i q^{69} -4.13896e8i q^{71} +(2.15848e7 + 1.24620e7i) q^{73} +(2.39466e8 + 4.14767e8i) q^{75} +(1.06997e7 - 8.85262e6i) q^{77} +(-2.52310e8 + 1.45671e8i) q^{79} +(-3.94525e8 + 6.83337e8i) q^{81} -6.54289e8 q^{83} -5.09192e7 q^{85} +(-2.75792e8 + 4.77686e8i) q^{87} +(-7.12797e8 + 4.11534e8i) q^{89} +(-9.68672e7 - 3.60180e7i) q^{91} +(8.95295e8 + 1.55070e9i) q^{93} +(1.15855e8 + 6.68889e7i) q^{95} +6.66674e8i q^{97} +9.95228e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 162 q^{3} - 852 q^{5} - 6744 q^{7} - 77884 q^{9} - 57534 q^{11} + 789336 q^{17} + 469098 q^{19} - 2104376 q^{21} + 1553682 q^{23} + 3602544 q^{25} + 6389244 q^{27} - 2462040 q^{29} - 10306686 q^{31}+ \cdots + 1433917218 q^{95}+O(q^{100}) \) 24 * q - 162 * q^3 - 852 * q^5 - 6744 * q^7 - 77884 * q^9 - 57534 * q^11 + 789336 * q^17 + 469098 * q^19 - 2104376 * q^21 + 1553682 * q^23 + 3602544 * q^25 + 6389244 * q^27 - 2462040 * q^29 - 10306686 * q^31 + 8789748 * q^33 + 21149802 * q^35 + 10422720 * q^37 + 54458664 * q^39 + 2549604 * q^45 + 12751230 * q^47 - 16137504 * q^49 - 107322414 * q^51 + 62342520 * q^53 - 422625348 * q^55 - 317200696 * q^57 - 101234394 * q^59 + 48881304 * q^61 + 726717996 * q^63 + 272864028 * q^65 - 48335310 * q^67 + 672073812 * q^73 + 950672484 * q^75 + 10755684 * q^77 + 144370122 * q^79 - 305389432 * q^81 - 568641984 * q^83 - 734871504 * q^85 + 33665484 * q^87 + 1630891500 * q^89 + 1012044192 * q^91 - 457741376 * q^93 + 1433917218 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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