LMFDB - Embedded newform 112.10.i.d.65.2

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 = [12,0,0] 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:	$12$
Relative dimension:	$6$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - x^{11} + 7779 x^{10} + 365650 x^{9} + 45150527 x^{8} + 2129694927 x^{7} + 167292926543 x^{6} + \cdots + 68\!\cdots\!56 $ x^12 - x^11 + 7779x^10 + 365650x^9 + 45150527x^8 + 2129694927x^7 + 167292926543x^6 + 8284983835540x^5 + 421203033146961x^4 + 13701568631176769x^3 + 374859098811070993x^2 + 5870183155171733832x + 68823015636381281856
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{4}\cdot 7^{7} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.2
Root			$-19.1194 - 33.1157i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.10.i.d.81.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-60.0993 - 104.095i) q^{3} +(501.604 - 868.804i) q^{5} +(2319.36 + 5913.90i) q^{7} +(2617.65 - 4533.90i) q^{9} +(-53.5365 - 92.7280i) q^{11} +85099.9 q^{13} -120584. q^{15} +(37986.8 + 65795.1i) q^{17} +(-323268. + 559916. i) q^{19} +(476216. - 596855. i) q^{21} +(-1.10595e6 + 1.91556e6i) q^{23} +(473349. + 819864. i) q^{25} -2.99514e6 q^{27} -1.99729e6 q^{29} +(-759827. - 1.31606e6i) q^{31} +(-6435.02 + 11145.8i) q^{33} +(6.30142e6 + 951367. i) q^{35} +(-1.51057e6 + 2.61639e6i) q^{37} +(-5.11444e6 - 8.85848e6i) q^{39} -3.75255e6 q^{41} +3.76132e7 q^{43} +(-2.62605e6 - 4.54844e6i) q^{45} +(284531. - 492823. i) q^{47} +(-2.95948e7 + 2.74329e7i) q^{49} +(4.56596e6 - 7.90848e6i) q^{51} +(3.73658e7 + 6.47195e7i) q^{53} -107417. q^{55} +7.77127e7 q^{57} +(5.26990e7 + 9.12773e7i) q^{59} +(7.59359e7 - 1.31525e8i) q^{61} +(3.28843e7 + 4.96476e6i) q^{63} +(4.26865e7 - 7.39351e7i) q^{65} +(2.75442e7 + 4.77079e7i) q^{67} +2.65867e8 q^{69} -7.22461e7 q^{71} +(2.14721e7 + 3.71908e7i) q^{73} +(5.68959e7 - 9.85466e7i) q^{75} +(424213. - 531679. i) q^{77} +(-3.32044e7 + 5.75117e7i) q^{79} +(1.28483e8 + 2.22539e8i) q^{81} +5.46511e8 q^{83} +7.62174e7 q^{85} +(1.20036e8 + 2.07908e8i) q^{87} +(-5.51884e8 + 9.55891e8i) q^{89} +(1.97377e8 + 5.03272e8i) q^{91} +(-9.13302e7 + 1.58189e8i) q^{93} +(3.24305e8 + 5.61713e8i) q^{95} +2.20561e8 q^{97} -560559. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 966 q^{5} - 7696 q^{7} - 21620 q^{9} + 47640 q^{11} + 103432 q^{13} + 400280 q^{15} - 402234 q^{17} + 519960 q^{19} - 1025858 q^{21} + 683124 q^{23} - 1134656 q^{25} + 7641648 q^{27} + 4252680 q^{29}+ \cdots - 8151941296 q^{99}+O(q^{100}) $ 12 * q - 966 * q^5 - 7696 * q^7 - 21620 * q^9 + 47640 * q^11 + 103432 * q^13 + 400280 * q^15 - 402234 * q^17 + 519960 * q^19 - 1025858 * q^21 + 683124 * q^23 - 1134656 * q^25 + 7641648 * q^27 + 4252680 * q^29 + 1040564 * q^31 + 5260178 * q^33 + 552600 * q^35 - 4886646 * q^37 - 12290272 * q^39 + 34183128 * q^41 - 55245072 * q^43 - 17040100 * q^45 + 26858748 * q^47 + 12089772 * q^49 - 57449552 * q^51 - 21717558 * q^53 - 54952184 * q^55 - 72948300 * q^57 - 31298904 * q^59 - 155592542 * q^61 + 400205088 * q^63 + 7729020 * q^65 - 212196992 * q^67 - 249902212 * q^69 - 648519648 * q^71 - 549596530 * q^73 + 647801672 * q^75 - 519615786 * q^77 - 227611204 * q^79 + 363916186 * q^81 - 989338224 * q^83 + 2712621236 * q^85 + 860715296 * q^87 - 1746541986 * q^89 + 2592411872 * q^91 + 1354546774 * q^93 - 1586101596 * q^95 + 2306715320 * q^97 - 8151941296 * q^99

Character values

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

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

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
$3$	−60.0993	−	104.095i	−0.428375	−	0.741967i	0.568354	−	0.822784i	$-0.307580\pi$
$3$	−60.0993	−	104.095i	−0.428375	−	0.741967i	−0.996729	+	0.0808172i	$0.974247\pi$
$4$		0			0
$5$	501.604	−	868.804i	0.358919	−	0.621666i	−0.628862	−	0.777517i	$-0.716479\pi$
$5$	501.604	−	868.804i	0.358919	−	0.621666i	0.987780	+	0.155852i	$0.0498122\pi$
$6$		0			0
$7$	2319.36	+	5913.90i	0.365113	+	0.930963i
$7$	2319.36	+	5913.90i	0.365113	+	0.930963i
$8$		0			0
$9$	2617.65	−	4533.90i	0.132990	−	0.230346i
$10$		0			0
$11$	−53.5365	−	92.7280i	−0.00110251	−	0.00190961i	0.865474	−	0.500954i	$-0.167018\pi$
$11$	−53.5365	−	92.7280i	−0.00110251	−	0.00190961i	−0.866576	+	0.499045i	$0.833684\pi$
$12$		0			0
$13$	85099.9			0.826388			0.413194	−	0.910643i	$-0.364413\pi$
$13$	85099.9			0.826388			0.413194	+	0.910643i	$0.364413\pi$
$14$		0			0
$15$	−120584.			−0.615007
$16$		0			0
$17$	37986.8	+	65795.1i	0.110310	+	0.191062i	0.915895	−	0.401418i	$-0.131482\pi$
$17$	37986.8	+	65795.1i	0.110310	+	0.191062i	−0.805586	+	0.592479i	$0.798149\pi$
$18$		0			0
$19$	−323268.	+	559916.i	−0.569077	+	0.985670i	0.427580	+	0.903977i	$0.359366\pi$
$19$	−323268.	+	559916.i	−0.569077	+	0.985670i	−0.996658	+	0.0816931i	$0.973967\pi$
$20$		0			0
$21$	476216.	−	596855.i	0.534339	−	0.669703i
$22$		0			0
$23$	−1.10595e6	+	1.91556e6i	−0.824062	+	1.42732i	0.0785726	+	0.996908i	$0.474964\pi$
$23$	−1.10595e6	+	1.91556e6i	−0.824062	+	1.42732i	−0.902634	+	0.430408i	$0.858370\pi$
$24$		0			0
$25$	473349.	+	819864.i	0.242355	+	0.419771i
$26$		0			0
$27$	−2.99514e6			−1.08463
$28$		0			0
$29$	−1.99729e6			−0.524386			−0.262193	−	0.965015i	$-0.584446\pi$
$29$	−1.99729e6			−0.524386			−0.262193	+	0.965015i	$0.584446\pi$
$30$		0			0
$31$	−759827.	−	1.31606e6i	−0.147770	−	0.255946i	0.782633	−	0.622484i	$-0.213876\pi$
$31$	−759827.	−	1.31606e6i	−0.147770	−	0.255946i	−0.930403	+	0.366538i	$0.880543\pi$
$32$		0			0
$33$	−6435.02	+	11145.8i	−0.000944576	+	0.00163605i
$34$		0			0
$35$	6.30142e6	+	951367.i	0.709794	+	0.107162i
$36$		0			0
$37$	−1.51057e6	+	2.61639e6i	−0.132506	+	0.229507i	−0.924642	−	0.380838i	$-0.875636\pi$
$37$	−1.51057e6	+	2.61639e6i	−0.132506	+	0.229507i	0.792136	+	0.610344i	$0.208969\pi$
$38$		0			0
$39$	−5.11444e6	−	8.85848e6i	−0.354004	−	0.613152i
$40$		0			0
$41$	−3.75255e6			−0.207395			−0.103698	−	0.994609i	$-0.533067\pi$
$41$	−3.75255e6			−0.207395			−0.103698	+	0.994609i	$0.533067\pi$
$42$		0			0
$43$	3.76132e7			1.67777			0.838885	−	0.544309i	$-0.183208\pi$
$43$	3.76132e7			1.67777			0.838885	+	0.544309i	$0.183208\pi$
$44$		0			0
$45$	−2.62605e6	−	4.54844e6i	−0.0954654	−	0.165351i
$46$		0			0
$47$	284531.	−	492823.i	0.00850530	−	0.0147316i	−0.861741	−	0.507348i	$-0.830626\pi$
$47$	284531.	−	492823.i	0.00850530	−	0.0147316i	0.870247	+	0.492616i	$0.163959\pi$
$48$		0			0
$49$	−2.95948e7	+	2.74329e7i	−0.733386	+	0.679813i
$50$		0			0
$51$	4.56596e6	−	7.90848e6i	0.0945076	−	0.163692i
$52$		0			0
$53$	3.73658e7	+	6.47195e7i	0.650479	+	1.12666i	0.983007	+	0.183569i	$0.0587651\pi$
$53$	3.73658e7	+	6.47195e7i	0.650479	+	1.12666i	−0.332528	+	0.943093i	$0.607902\pi$
$54$		0			0
$55$	−107417.			−0.00158285
$56$		0			0
$57$	7.77127e7			0.975113
$58$		0			0
$59$	5.26990e7	+	9.12773e7i	0.566198	+	0.980684i	0.996937	+	0.0782071i	$0.0249195\pi$
$59$	5.26990e7	+	9.12773e7i	0.566198	+	0.980684i	−0.430739	+	0.902476i	$0.641747\pi$
$60$		0			0
$61$	7.59359e7	−	1.31525e8i	0.702203	−	1.21625i	−0.265488	−	0.964114i	$-0.585533\pi$
$61$	7.59359e7	−	1.31525e8i	0.702203	−	1.21625i	0.967691	−	0.252138i	$-0.0811336\pi$
$62$		0			0
$63$	3.28843e7	+	4.96476e6i	0.263000	+	0.0397069i
$64$		0			0
$65$	4.26865e7	−	7.39351e7i	0.296606	−	0.513737i
$66$		0			0
$67$	2.75442e7	+	4.77079e7i	0.166991	+	0.289237i	0.937361	−	0.348361i	$-0.113262\pi$
$67$	2.75442e7	+	4.77079e7i	0.166991	+	0.289237i	−0.770370	+	0.637598i	$0.779928\pi$
$68$		0			0
$69$	2.65867e8			1.41203
$70$		0			0
$71$	−7.22461e7			−0.337405			−0.168703	−	0.985667i	$-0.553958\pi$
$71$	−7.22461e7			−0.337405			−0.168703	+	0.985667i	$0.553958\pi$
$72$		0			0
$73$	2.14721e7	+	3.71908e7i	0.0884956	+	0.153279i	0.906875	−	0.421399i	$-0.138461\pi$
$73$	2.14721e7	+	3.71908e7i	0.0884956	+	0.153279i	−0.818380	+	0.574678i	$0.805127\pi$
$74$		0			0
$75$	5.68959e7	−	9.85466e7i	0.207637	−	0.359638i
$76$		0			0
$77$	424213.	−	531679.i	0.00137523	−	0.00172362i
$78$		0			0
$79$	−3.32044e7	+	5.75117e7i	−0.0959122	+	0.166125i	−0.909989	−	0.414632i	$-0.863910\pi$
$79$	−3.32044e7	+	5.75117e7i	−0.0959122	+	0.166125i	0.814077	+	0.580757i	$0.197243\pi$
$80$		0			0
$81$	1.28483e8	+	2.22539e8i	0.331637	+	0.574412i
$82$		0			0
$83$	5.46511e8			1.26400			0.632000	−	0.774968i	$-0.282234\pi$
$83$	5.46511e8			1.26400			0.632000	+	0.774968i	$0.282234\pi$
$84$		0			0
$85$	7.62174e7			0.158369
$86$		0			0
$87$	1.20036e8	+	2.07908e8i	0.224634	+	0.389077i
$88$		0			0
$89$	−5.51884e8	+	9.55891e8i	−0.932379	+	1.61493i	−0.153137	+	0.988205i	$0.548938\pi$
$89$	−5.51884e8	+	9.55891e8i	−0.932379	+	1.61493i	−0.779242	+	0.626723i	$0.784396\pi$
$90$		0			0
$91$	1.97377e8	+	5.03272e8i	0.301725	+	0.769337i
$92$		0			0
$93$	−9.13302e7	+	1.58189e8i	−0.126602	+	0.219281i
$94$		0			0
$95$	3.24305e8	+	5.61713e8i	0.408505	+	0.707551i
$96$		0			0
$97$	2.20561e8			0.252963			0.126481	−	0.991969i	$-0.459632\pi$
$97$	2.20561e8			0.252963			0.126481	+	0.991969i	$0.459632\pi$
$98$		0			0
$99$	−560559.			−0.000586493
$100$		0			0
$101$	4.05449e8	+	7.02259e8i	0.387695	+	0.671508i	0.992139	−	0.125140i	$-0.0399380\pi$
$101$	4.05449e8	+	7.02259e8i	0.387695	+	0.671508i	−0.604444	+	0.796648i	$0.706605\pi$
$102$		0			0
$103$	9.18456e8	−	1.59081e9i	0.804064	−	1.39268i	−0.112857	−	0.993611i	$-0.536000\pi$
$103$	9.18456e8	−	1.59081e9i	0.804064	−	1.39268i	0.916921	−	0.399069i	$-0.130666\pi$
$104$		0			0
$105$	−2.79678e8	−	7.13123e8i	−0.224547	−	0.572549i
$106$		0			0
$107$	−4.42462e7	+	7.66367e7i	−0.0326324	+	0.0565210i	−0.881880	−	0.471473i	$-0.843722\pi$
$107$	−4.42462e7	+	7.66367e7i	−0.0326324	+	0.0565210i	0.849248	+	0.527994i	$0.177056\pi$
$108$		0			0
$109$	2.65241e8	+	4.59411e8i	0.179979	+	0.311732i	0.941873	−	0.335969i	$-0.109064\pi$
$109$	2.65241e8	+	4.59411e8i	0.179979	+	0.311732i	−0.761894	+	0.647701i	$0.775730\pi$
$110$		0			0
$111$	3.63138e8			0.227048
$112$		0			0
$113$	3.08486e9			1.77985			0.889924	−	0.456108i	$-0.150757\pi$
$113$	3.08486e9			1.77985			0.889924	+	0.456108i	$0.150757\pi$
$114$		0			0
$115$	1.10950e9	+	1.92171e9i	0.591542	+	1.02458i
$116$		0			0
$117$	2.22761e8	−	3.85834e8i	0.109901	−	0.190355i
$118$		0			0
$119$	−3.01001e8	+	3.77253e8i	−0.137596	+	0.172453i
$120$		0			0
$121$	1.17897e9	−	2.04203e9i	0.499998	−	0.866021i
$122$		0			0
$123$	2.25525e8	+	3.90621e8i	0.0888428	+	0.153880i
$124$		0			0
$125$	2.90913e9			1.06578
$126$		0			0
$127$	−4.97087e9			−1.69557			−0.847785	−	0.530340i	$-0.822064\pi$
$127$	−4.97087e9			−1.69557			−0.847785	+	0.530340i	$0.822064\pi$
$128$		0			0
$129$	−2.26053e9	−	3.91535e9i	−0.718714	−	1.24485i
$130$		0			0
$131$	2.22067e9	−	3.84631e9i	0.658815	−	1.14110i	−0.322108	−	0.946703i	$-0.604391\pi$
$131$	2.22067e9	−	3.84631e9i	0.658815	−	1.14110i	0.980923	−	0.194398i	$-0.0622753\pi$
$132$		0			0
$133$	−4.06106e9	−	6.13125e8i	−1.12540	−	0.169909i
$134$		0			0
$135$	−1.50238e9	+	2.60219e9i	−0.389293	+	0.674276i
$136$		0			0
$137$	2.40438e9	+	4.16450e9i	0.583123	+	1.01000i	0.995107	+	0.0988069i	$0.0315026\pi$
$137$	2.40438e9	+	4.16450e9i	0.583123	+	1.01000i	−0.411984	+	0.911191i	$0.635164\pi$
$138$		0			0
$139$	8.95323e8			0.203429			0.101715	−	0.994814i	$-0.467567\pi$
$139$	8.95323e8			0.203429			0.101715	+	0.994814i	$0.467567\pi$
$140$		0			0
$141$	−6.84006e7			−0.0145738
$142$		0			0
$143$	−4.55595e6	−	7.89114e6i	−0.000911102	−	0.00157808i
$144$		0			0
$145$	−1.00185e9	+	1.73526e9i	−0.188212	+	0.325993i
$146$		0			0
$147$	4.63425e9	+	1.43197e9i	0.818562	+	0.252933i
$148$		0			0
$149$	−4.14527e9	+	7.17982e9i	−0.688993	+	1.19337i	0.283171	+	0.959069i	$0.408614\pi$
$149$	−4.14527e9	+	7.17982e9i	−0.688993	+	1.19337i	−0.972164	+	0.234301i	$0.924720\pi$
$150$		0			0
$151$	−4.18563e9	−	7.24973e9i	−0.655186	−	1.13482i	−0.981847	−	0.189674i	$-0.939257\pi$
$151$	−4.18563e9	−	7.24973e9i	−0.655186	−	1.13482i	0.326661	−	0.945142i	$-0.394077\pi$
$152$		0			0
$153$	3.97744e8			0.0586804
$154$		0			0
$155$	−1.52453e9			−0.212150
$156$		0			0
$157$	−4.95278e9	−	8.57847e9i	−0.650580	−	1.12684i	−0.982982	−	0.183700i	$-0.941192\pi$
$157$	−4.95278e9	−	8.57847e9i	−0.650580	−	1.12684i	0.332402	−	0.943138i	$-0.392141\pi$
$158$		0			0
$159$	4.49132e9	−	7.77920e9i	0.557297	−	0.965267i
$160$		0			0
$161$	−1.38935e10	−	2.09760e9i	−1.62965	−	0.246040i
$162$		0			0
$163$	−6.08514e9	+	1.05398e10i	−0.675191	+	1.16947i	0.301222	+	0.953554i	$0.402605\pi$
$163$	−6.08514e9	+	1.05398e10i	−0.675191	+	1.16947i	−0.976413	+	0.215911i	$0.930728\pi$
$164$		0			0
$165$	6.45566e6	+	1.11815e7i	0.000678052	+	0.00117442i
$166$		0			0
$167$	−4.27831e9			−0.425645			−0.212823	−	0.977091i	$-0.568266\pi$
$167$	−4.27831e9			−0.425645			−0.212823	+	0.977091i	$0.568266\pi$
$168$		0			0
$169$	−3.36251e9			−0.317083
$170$		0			0
$171$	1.69240e9	+	2.93132e9i	0.151363	+	0.262169i
$172$		0			0
$173$	−8.71112e9	+	1.50881e10i	−0.739378	+	1.28064i	0.213398	+	0.976965i	$0.431547\pi$
$173$	−8.71112e9	+	1.50881e10i	−0.739378	+	1.28064i	−0.952776	+	0.303675i	$0.901786\pi$
$174$		0			0
$175$	−3.75073e9	+	4.70090e9i	−0.302304	+	0.378887i
$176$		0			0
$177$	6.33435e9	−	1.09714e10i	0.485090	−	0.840200i
$178$		0			0
$179$	−1.23974e10	−	2.14728e10i	−0.902590	−	1.56333i	−0.824120	−	0.566415i	$-0.808330\pi$
$179$	−1.23974e10	−	2.14728e10i	−0.902590	−	1.56333i	−0.0784692	−	0.996917i	$-0.525003\pi$
$180$		0			0
$181$	6.66502e9			0.461581			0.230791	−	0.973003i	$-0.425869\pi$
$181$	6.66502e9			0.461581			0.230791	+	0.973003i	$0.425869\pi$
$182$		0			0
$183$	−1.82548e10			−1.20322
$184$		0			0
$185$	1.51542e9	+	2.62479e9i	0.0951175	+	0.164748i
$186$		0			0
$187$	4.06737e6	−	7.04489e6i	0.000243235	−	0.000421295i
$188$		0			0
$189$	−6.94682e9	−	1.77130e10i	−0.396011	−	1.00975i
$190$		0			0
$191$	−1.43315e9	+	2.48229e9i	−0.0779186	+	0.134959i	−0.902352	−	0.431000i	$-0.858161\pi$
$191$	−1.43315e9	+	2.48229e9i	−0.0779186	+	0.134959i	0.824433	+	0.565959i	$0.191494\pi$
$192$		0			0
$193$	−1.29890e10	−	2.24977e10i	−0.673859	−	1.16716i	−0.976801	−	0.214149i	$-0.931302\pi$
$193$	−1.29890e10	−	2.24977e10i	−0.673859	−	1.16716i	0.302942	−	0.953009i	$-0.402031\pi$
$194$		0			0
$195$	−1.02617e10			−0.508234
$196$		0			0
$197$	2.60941e10			1.23437			0.617183	−	0.786820i	$-0.288274\pi$
$197$	2.60941e10			1.23437			0.617183	+	0.786820i	$0.288274\pi$
$198$		0			0
$199$	1.19791e10	+	2.07484e10i	0.541485	+	0.937879i	0.998819	+	0.0485840i	$0.0154709\pi$
$199$	1.19791e10	+	2.07484e10i	0.541485	+	0.937879i	−0.457335	+	0.889295i	$0.651196\pi$
$200$		0			0
$201$	3.31077e9	−	5.73442e9i	0.143069	−	0.247804i
$202$		0			0
$203$	−4.63244e9	−	1.18118e10i	−0.191460	−	0.488184i
$204$		0			0
$205$	−1.88229e9	+	3.26023e9i	−0.0744380	+	0.128930i
$206$		0			0
$207$	5.78997e9	+	1.00285e10i	0.219184	+	0.379638i
$208$		0			0
$209$	6.92265e7			0.00250966
$210$		0			0
$211$	−1.80028e10			−0.625272			−0.312636	−	0.949873i	$-0.601212\pi$
$211$	−1.80028e10			−0.625272			−0.312636	+	0.949873i	$0.601212\pi$
$212$		0			0
$213$	4.34194e9	+	7.52046e9i	0.144536	+	0.250343i
$214$		0			0
$215$	1.88669e10	−	3.26785e10i	0.602183	−	1.04301i
$216$		0			0
$217$	6.02073e9	−	7.54595e9i	0.184323	−	0.231018i
$218$		0			0
$219$	2.58092e9	−	4.47028e9i	0.0758185	−	0.131322i
$220$		0			0
$221$	3.23268e9	+	5.59916e9i	0.0911584	+	0.157891i
$222$		0			0
$223$	6.64392e10			1.79909			0.899544	−	0.436830i	$-0.143899\pi$
$223$	6.64392e10			1.79909			0.899544	+	0.436830i	$0.143899\pi$
$224$		0			0
$225$	4.95624e9			0.128923
$226$		0			0
$227$	−2.69985e9	−	4.67628e9i	−0.0674875	−	0.116892i	0.830307	−	0.557306i	$-0.188165\pi$
$227$	−2.69985e9	−	4.67628e9i	−0.0674875	−	0.116892i	−0.897795	+	0.440414i	$0.854832\pi$
$228$		0			0
$229$	−1.91113e10	+	3.31017e10i	−0.459230	+	0.795410i	−0.998920	−	0.0464539i	$-0.985208\pi$
$229$	−1.91113e10	+	3.31017e10i	−0.459230	+	0.795410i	0.539690	+	0.841864i	$0.318541\pi$
$230$		0			0
$231$	−8.08401e7	−	1.22050e7i	−0.00186798	−	0.000282022i
$232$		0			0
$233$	−3.37400e10	+	5.84394e10i	−0.749969	+	1.29898i	0.197868	+	0.980229i	$0.436598\pi$
$233$	−3.37400e10	+	5.84394e10i	−0.749969	+	1.29898i	−0.947837	+	0.318756i	$0.896735\pi$
$234$		0			0
$235$	−2.85444e8	−	4.94404e8i	−0.00610543	−	0.0105749i
$236$		0			0
$237$	7.98225e9			0.164346
$238$		0			0
$239$	5.07144e10			1.00541			0.502703	−	0.864459i	$-0.332339\pi$
$239$	5.07144e10			1.00541			0.502703	+	0.864459i	$0.332339\pi$
$240$		0			0
$241$	1.46290e9	+	2.53382e9i	0.0279344	+	0.0483837i	0.879655	−	0.475613i	$-0.157774\pi$
$241$	1.46290e9	+	2.53382e9i	0.0279344	+	0.0483837i	−0.851720	+	0.523997i	$0.824440\pi$
$242$		0			0
$243$	−1.40332e10	+	2.43063e10i	−0.258184	+	0.447188i
$244$		0			0
$245$	8.98896e9	+	3.94725e10i	0.159390	+	0.699918i
$246$		0			0
$247$	−2.75100e10	+	4.76488e10i	−0.470278	+	0.814546i
$248$		0			0
$249$	−3.28449e10	−	5.68891e10i	−0.541466	−	0.937847i
$250$		0			0
$251$	−7.14342e9			−0.113599			−0.0567995	−	0.998386i	$-0.518090\pi$
$251$	−7.14342e9			−0.113599			−0.0567995	+	0.998386i	$0.518090\pi$
$252$		0			0
$253$	2.36835e8			0.00363415
$254$		0			0
$255$	−4.58061e9	−	7.93386e9i	−0.0678411	−	0.117504i
$256$		0			0
$257$	3.86294e10	−	6.69081e10i	0.552356	−	0.956708i	−0.445748	−	0.895158i	$-0.647062\pi$
$257$	3.86294e10	−	6.69081e10i	0.552356	−	0.956708i	0.998104	−	0.0615497i	$-0.0196043\pi$
$258$		0			0
$259$	−1.89766e10	−	2.86503e9i	−0.262042	−	0.0395622i
$260$		0			0
$261$	−5.22821e9	+	9.05553e9i	−0.0697382	+	0.120790i
$262$		0			0
$263$	−1.23540e10	−	2.13977e10i	−0.159223	−	0.275782i	0.775366	−	0.631513i	$-0.217566\pi$
$263$	−1.23540e10	−	2.13977e10i	−0.159223	−	0.275782i	−0.934589	+	0.355730i	$0.884232\pi$
$264$		0			0
$265$	7.49714e10			0.933876
$266$		0			0
$267$	1.32671e11			1.59763
$268$		0			0
$269$	6.73244e10	+	1.16609e11i	0.783948	+	1.35784i	0.929625	+	0.368506i	$0.120131\pi$
$269$	6.73244e10	+	1.16609e11i	0.783948	+	1.35784i	−0.145677	+	0.989332i	$0.546536\pi$
$270$		0			0
$271$	−6.70894e10	+	1.16202e11i	−0.755600	+	1.30874i	0.189475	+	0.981886i	$0.439321\pi$
$271$	−6.70894e10	+	1.16202e11i	−0.755600	+	1.30874i	−0.945075	+	0.326853i	$0.894012\pi$
$272$		0			0
$273$	4.05259e10	−	5.07923e10i	0.441571	−	0.553434i
$274$		0			0
$275$	5.06829e7	−	8.77854e7i	0.000534398	−	0.000925604i
$276$		0			0
$277$	3.43120e10	+	5.94301e10i	0.350176	+	0.606523i	0.986280	−	0.165080i	$-0.0527883\pi$
$277$	3.43120e10	+	5.94301e10i	0.350176	+	0.606523i	−0.636104	+	0.771604i	$0.719455\pi$
$278$		0			0
$279$	−7.95584e9			−0.0786080
$280$		0			0
$281$	5.32135e9			0.0509147			0.0254573	−	0.999676i	$-0.491896\pi$
$281$	5.32135e9			0.0509147			0.0254573	+	0.999676i	$0.491896\pi$
$282$		0			0
$283$	1.09353e10	+	1.89405e10i	0.101343	+	0.175531i	0.912238	−	0.409660i	$-0.134353\pi$
$283$	1.09353e10	+	1.89405e10i	0.101343	+	0.175531i	−0.810895	+	0.585191i	$0.801019\pi$
$284$		0			0
$285$	3.89810e10	−	6.75171e10i	0.349986	−	0.606194i
$286$		0			0
$287$	−8.70350e9	−	2.21922e10i	−0.0757226	−	0.193077i
$288$		0			0
$289$	5.64079e10	−	9.77014e10i	0.475664	−	0.823874i
$290$		0			0
$291$	−1.32556e10	−	2.29593e10i	−0.108363	−	0.187690i
$292$		0			0
$293$	3.63609e10			0.288224			0.144112	−	0.989561i	$-0.453967\pi$
$293$	3.63609e10			0.288224			0.144112	+	0.989561i	$0.453967\pi$
$294$		0			0
$295$	1.05736e11			0.812876
$296$		0			0
$297$	1.60350e8	+	2.77734e8i	0.00119582	+	0.00207121i
$298$		0			0
$299$	−9.41161e10	+	1.63014e11i	−0.680994	+	1.17952i
$300$		0			0
$301$	8.72385e10	+	2.22441e11i	0.612575	+	1.56194i
$302$		0			0
$303$	4.87344e10	−	8.44105e10i	0.332158	−	0.575314i
$304$		0			0
$305$	−7.61795e10	−	1.31947e11i	−0.504068	−	0.873071i
$306$		0			0
$307$	2.08878e11			1.34205			0.671026	−	0.741434i	$-0.265854\pi$
$307$	2.08878e11			1.34205			0.671026	+	0.741434i	$0.265854\pi$
$308$		0			0
$309$	−2.20794e11			−1.37776
$310$		0			0
$311$	−2.10562e10	−	3.64704e10i	−0.127632	−	0.221065i	0.795127	−	0.606443i	$-0.207404\pi$
$311$	−2.10562e10	−	3.64704e10i	−0.127632	−	0.221065i	−0.922759	+	0.385379i	$0.874071\pi$
$312$		0			0
$313$	5.65662e10	−	9.79756e10i	0.333125	−	0.576990i	−0.649998	−	0.759936i	$-0.725230\pi$
$313$	5.65662e10	−	9.79756e10i	0.333125	−	0.576990i	0.983123	+	0.182946i	$0.0585635\pi$
$314$		0			0
$315$	2.08083e10	−	2.60796e10i	0.119080	−	0.149246i
$316$		0			0
$317$	−1.34834e11	+	2.33540e11i	−0.749953	+	1.29896i	0.197892	+	0.980224i	$0.436590\pi$
$317$	−1.34834e11	+	2.33540e11i	−0.749953	+	1.29896i	−0.947845	+	0.318732i	$0.896743\pi$
$318$		0			0
$319$	1.06928e8	+	1.85205e8i	0.000578142	+	0.00100137i
$320$		0			0
$321$	1.06367e10			0.0559156
$322$		0			0
$323$	−4.91197e10			−0.251098
$324$		0			0
$325$	4.02819e10	+	6.97704e10i	0.200279	+	0.346893i
$326$		0			0
$327$	3.18816e10	−	5.52205e10i	0.154197	−	0.267077i
$328$		0			0
$329$	3.57443e9	+	5.39656e8i	0.0168200	+	0.00253943i
$330$		0			0
$331$	1.54240e11	−	2.67152e11i	0.706272	−	1.22330i	−0.259959	−	0.965620i	$-0.583709\pi$
$331$	1.54240e11	−	2.67152e11i	0.706272	−	1.22330i	0.966231	−	0.257679i	$-0.0829577\pi$
$332$		0			0
$333$	7.90830e9	+	1.36976e10i	0.0352439	+	0.0610443i
$334$		0			0
$335$	5.52651e10			0.239745
$336$		0			0
$337$	−2.09034e11			−0.882843			−0.441421	−	0.897300i	$-0.645526\pi$
$337$	−2.09034e11			−0.882843			−0.441421	+	0.897300i	$0.645526\pi$
$338$		0			0
$339$	−1.85398e11	−	3.21119e11i	−0.762442	−	1.32059i
$340$		0			0
$341$	−8.13570e7	+	1.40915e8i	−0.000325837	+	0.000564366i
$342$		0			0
$343$	−2.30876e11	−	1.11394e11i	−0.900649	−	0.434547i
$344$		0			0
$345$	1.33360e11	−	2.30986e11i	0.506804	−	0.877810i
$346$		0			0
$347$	2.35120e11	+	4.07241e11i	0.870578	+	1.50789i	0.861400	+	0.507928i	$0.169588\pi$
$347$	2.35120e11	+	4.07241e11i	0.870578	+	1.50789i	0.00917879	+	0.999958i	$0.497078\pi$
$348$		0			0
$349$	−1.77598e11			−0.640801			−0.320401	−	0.947282i	$-0.603818\pi$
$349$	−1.77598e11			−0.640801			−0.320401	+	0.947282i	$0.603818\pi$
$350$		0			0
$351$	−2.54886e11			−0.896323
$352$		0			0
$353$	−2.31027e11	−	4.00150e11i	−0.791910	−	1.37163i	−0.924783	−	0.380496i	$-0.875753\pi$
$353$	−2.31027e11	−	4.00150e11i	−0.791910	−	1.37163i	0.132872	−	0.991133i	$-0.457580\pi$
$354$		0			0
$355$	−3.62389e10	+	6.27677e10i	−0.121101	+	0.209753i
$356$		0			0
$357$	5.73601e10	+	8.66003e9i	0.186897	+	0.0282171i
$358$		0			0
$359$	−9.84861e10	+	1.70583e11i	−0.312932	+	0.542014i	−0.978996	−	0.203881i	$-0.934645\pi$
$359$	−9.84861e10	+	1.70583e11i	−0.312932	+	0.542014i	0.666064	+	0.745895i	$0.267978\pi$
$360$		0			0
$361$	−4.76601e10	−	8.25498e10i	−0.147697	−	0.255819i
$362$		0			0
$363$	−2.83421e11			−0.856745
$364$		0			0
$365$	4.30820e10			0.127051
$366$		0			0
$367$	−6.65243e10	−	1.15223e11i	−0.191418	−	0.331546i	0.754302	−	0.656527i	$-0.227975\pi$
$367$	−6.65243e10	−	1.15223e11i	−0.191418	−	0.331546i	−0.945720	+	0.324981i	$0.894642\pi$
$368$		0			0
$369$	−9.82284e9	+	1.70137e10i	−0.0275815	+	0.0477726i
$370$		0			0
$371$	−2.96080e11	+	3.71085e11i	−0.811383	+	1.01693i
$372$		0			0
$373$	4.10561e10	−	7.11113e10i	0.109822	−	0.190217i	−0.805876	−	0.592084i	$-0.798305\pi$
$373$	4.10561e10	−	7.11113e10i	0.109822	−	0.190217i	0.915698	+	0.401867i	$0.131639\pi$
$374$		0			0
$375$	−1.74836e11	−	3.02826e11i	−0.456553	−	0.790773i
$376$		0			0
$377$	−1.69969e11			−0.433346
$378$		0			0
$379$	−4.02028e11			−1.00088			−0.500438	−	0.865773i	$-0.666828\pi$
$379$	−4.02028e11			−1.00088			−0.500438	+	0.865773i	$0.666828\pi$
$380$		0			0
$381$	2.98746e11	+	5.17443e11i	0.726339	+	1.25806i
$382$		0			0
$383$	−1.47591e11	+	2.55634e11i	−0.350481	+	0.607050i	−0.986334	−	0.164760i	$-0.947315\pi$
$383$	−1.47591e11	+	2.55634e11i	−0.350481	+	0.607050i	0.635853	+	0.771810i	$0.280648\pi$
$384$		0			0
$385$	−2.49138e8	−	6.35251e8i	−0.000577918	−	0.00147357i
$386$		0			0
$387$	9.84581e10	−	1.70534e11i	0.223127	−	0.386467i
$388$		0			0
$389$	−1.15300e11	−	1.99705e11i	−0.255303	−	0.442198i	0.709675	−	0.704529i	$-0.248842\pi$
$389$	−1.15300e11	−	1.99705e11i	−0.255303	−	0.442198i	−0.964978	+	0.262332i	$0.915509\pi$
$390$		0			0
$391$	−1.68046e11			−0.363607
$392$		0			0
$393$	−5.33843e11			−1.12888
$394$		0			0
$395$	3.33110e10	+	5.76963e10i	0.0688494	+	0.119251i
$396$		0			0
$397$	−3.25570e11	+	5.63903e11i	−0.657789	+	1.13932i	0.323398	+	0.946263i	$0.395175\pi$
$397$	−3.25570e11	+	5.63903e11i	−0.657789	+	1.13932i	−0.981187	+	0.193061i	$0.938159\pi$
$398$		0			0
$399$	1.80244e11	+	4.59585e11i	0.356026	+	0.907794i
$400$		0			0
$401$	3.22241e11	−	5.58138e11i	0.622345	−	1.07793i	−0.366703	−	0.930338i	$-0.619513\pi$
$401$	3.22241e11	−	5.58138e11i	0.622345	−	1.07793i	0.989048	−	0.147595i	$-0.0471534\pi$
$402$		0			0
$403$	−6.46612e10	−	1.11997e11i	−0.122116	−	0.211510i
$404$		0			0
$405$	2.57790e11			0.476123
$406$		0			0
$407$	3.23484e8			0.000584356
$408$		0			0
$409$	−3.14975e11	−	5.45553e11i	−0.556572	−	0.964011i	−0.997779	−	0.0666060i	$-0.978783\pi$
$409$	−3.14975e11	−	5.45553e11i	−0.556572	−	0.964011i	0.441207	−	0.897405i	$-0.354550\pi$
$410$		0			0
$411$	2.89003e11	−	5.00567e11i	0.499590	−	0.865315i
$412$		0			0
$413$	−4.17577e11	+	5.23361e11i	−0.706254	+	0.885169i
$414$		0			0
$415$	2.74132e11	−	4.74811e11i	0.453674	−	0.785786i
$416$		0			0
$417$	−5.38083e10	−	9.31987e10i	−0.0871439	−	0.150938i
$418$		0			0
$419$	2.03533e11			0.322605			0.161303	−	0.986905i	$-0.448430\pi$
$419$	2.03533e11			0.322605			0.161303	+	0.986905i	$0.448430\pi$
$420$		0			0
$421$	6.89974e11			1.07044			0.535221	−	0.844712i	$-0.320228\pi$
$421$	6.89974e11			1.07044			0.535221	+	0.844712i	$0.320228\pi$
$422$		0			0
$423$	−1.48961e9	−	2.58007e9i	−0.00226224	−	0.00391832i
$424$		0			0
$425$	−3.59621e10	+	6.22881e10i	−0.0534680	+	0.0926094i
$426$		0			0
$427$	9.53947e11	+	1.44024e11i	1.38867	+	0.209657i
$428$		0			0
$429$	−5.47619e8	+	9.48504e8i	−0.000780586	+	0.00135201i
$430$		0			0
$431$	−4.51272e11	−	7.81626e11i	−0.629927	−	1.09107i	−0.987566	−	0.157206i	$-0.949751\pi$
$431$	−4.51272e11	−	7.81626e11i	−0.629927	−	1.09107i	0.357638	−	0.933860i	$-0.383582\pi$
$432$		0			0
$433$	1.36194e11			0.186192			0.0930962	−	0.995657i	$-0.470324\pi$
$433$	1.36194e11			0.186192			0.0930962	+	0.995657i	$0.470324\pi$
$434$		0			0
$435$	2.40842e11			0.322501
$436$		0			0
$437$	−7.15035e11	−	1.23848e12i	−0.937909	−	1.62451i
$438$		0			0
$439$	1.14239e11	−	1.97867e11i	0.146799	−	0.254263i	−0.783244	−	0.621715i	$-0.786436\pi$
$439$	1.14239e11	−	1.97867e11i	0.146799	−	0.254263i	0.930043	+	0.367452i	$0.119770\pi$
$440$		0			0
$441$	4.69094e10	+	2.05989e11i	0.0590590	+	0.259341i
$442$		0			0
$443$	1.58733e11	−	2.74934e11i	0.195817	−	0.339166i	−0.751351	−	0.659903i	$-0.770597\pi$
$443$	1.58733e11	−	2.74934e11i	0.195817	−	0.339166i	0.947168	+	0.320737i	$0.103931\pi$
$444$		0			0
$445$	5.53654e11	+	9.58958e11i	0.669297	+	1.15926i
$446$		0			0
$447$	9.96512e11			1.18059
$448$		0			0
$449$	2.61398e11			0.303524			0.151762	−	0.988417i	$-0.451505\pi$
$449$	2.61398e11			0.303524			0.151762	+	0.988417i	$0.451505\pi$
$450$		0			0
$451$	2.00898e8	+	3.47966e8i	0.000228656	+	0.000396043i
$452$		0			0
$453$	−5.03107e11	+	8.71407e11i	−0.561330	+	0.972253i
$454$		0			0
$455$	5.36250e11	+	8.09612e10i	0.586565	+	0.0885576i
$456$		0			0
$457$	3.48607e11	−	6.03806e11i	0.373864	−	0.647552i	−0.616292	−	0.787518i	$-0.711366\pi$
$457$	3.48607e11	−	6.03806e11i	0.373864	−	0.647552i	0.990156	+	0.139966i	$0.0446993\pi$
$458$		0			0
$459$	−1.13776e11	−	1.97066e11i	−0.119645	−	0.207231i
$460$		0			0
$461$	1.30791e12			1.34872			0.674362	−	0.738400i	$-0.264419\pi$
$461$	1.30791e12			1.34872			0.674362	+	0.738400i	$0.264419\pi$
$462$		0			0
$463$	−9.10787e11			−0.921090			−0.460545	−	0.887636i	$-0.652346\pi$
$463$	−9.10787e11			−0.921090			−0.460545	+	0.887636i	$0.652346\pi$
$464$		0			0
$465$	9.16232e10	+	1.58696e11i	0.0908798	+	0.157408i
$466$		0			0
$467$	−4.68071e11	+	8.10724e11i	−0.455393	+	0.788764i	−0.998711	−	0.0507635i	$-0.983835\pi$
$467$	−4.68071e11	+	8.10724e11i	−0.455393	+	0.788764i	0.543318	+	0.839527i	$0.317168\pi$
$468$		0			0
$469$	−2.18255e11	+	2.73545e11i	−0.208298	+	0.261067i
$470$		0			0
$471$	−5.95318e11	+	1.03112e12i	−0.557384	+	0.965418i
$472$		0			0
$473$	−2.01368e9	−	3.48780e9i	−0.00184976	−	0.00320388i
$474$		0			0
$475$	−6.12074e11			−0.551674
$476$		0			0
$477$	3.91242e11			0.346029
$478$		0			0
$479$	8.09674e11	+	1.40240e12i	0.702749	+	1.21720i	0.967498	+	0.252880i	$0.0813778\pi$
$479$	8.09674e11	+	1.40240e12i	0.702749	+	1.21720i	−0.264748	+	0.964318i	$0.585289\pi$
$480$		0			0
$481$	−1.28550e11	+	2.22655e11i	−0.109501	+	0.189661i
$482$		0			0
$483$	6.16641e11	+	1.57231e12i	0.515549	+	1.31455i
$484$		0			0
$485$	1.10635e11	−	1.91625e11i	0.0907931	−	0.157258i
$486$		0			0
$487$	2.63528e11	+	4.56444e11i	0.212298	+	0.367712i	0.952433	−	0.304747i	$-0.0985718\pi$
$487$	2.63528e11	+	4.56444e11i	0.212298	+	0.367712i	−0.740135	+	0.672458i	$0.765238\pi$
$488$		0			0
$489$	1.46285e12			1.15694
$490$		0			0
$491$	−2.12833e12			−1.65262			−0.826308	−	0.563218i	$-0.809563\pi$
$491$	−2.12833e12			−1.65262			−0.826308	+	0.563218i	$0.809563\pi$
$492$		0			0
$493$	−7.58709e10	−	1.31412e11i	−0.0578448	−	0.100190i
$494$		0			0
$495$	−2.81179e8	+	4.87016e8i	−0.000210503	+	0.000364603i
$496$		0			0
$497$	−1.67565e11	−	4.27256e11i	−0.123191	−	0.314112i
$498$		0			0
$499$	5.05306e11	−	8.75215e11i	0.364839	−	0.631920i	−0.623911	−	0.781495i	$-0.714457\pi$
$499$	5.05306e11	−	8.75215e11i	0.364839	−	0.631920i	0.988750	+	0.149575i	$0.0477906\pi$
$500$		0			0
$501$	2.57123e11	+	4.45350e11i	0.182336	+	0.315815i
$502$		0			0
$503$	1.51372e12			1.05436			0.527180	−	0.849754i	$-0.323249\pi$
$503$	1.51372e12			1.05436			0.527180	+	0.849754i	$0.323249\pi$
$504$		0			0
$505$	8.13500e11			0.556604
$506$		0			0
$507$	2.02084e11	+	3.50021e11i	0.135830	+	0.235265i
$508$		0			0
$509$	−1.21438e12	+	2.10337e12i	−0.801907	+	1.38894i	0.116452	+	0.993196i	$0.462848\pi$
$509$	−1.21438e12	+	2.10337e12i	−0.801907	+	1.38894i	−0.918359	+	0.395748i	$0.870485\pi$
$510$		0			0
$511$	−1.70141e11	+	2.13242e11i	−0.110386	+	0.138350i
$512$		0			0
$513$	9.68233e11	−	1.67703e12i	0.617237	−	1.06909i
$514$		0			0
$515$	−9.21402e11	−	1.59592e12i	−0.577187	−	0.999718i
$516$		0			0
$517$	−6.09313e7			−3.75088e−5
$518$		0			0
$519$	2.09413e12			1.26692
$520$		0			0
$521$	1.18874e12	+	2.05897e12i	0.706836	+	1.22428i	0.966025	+	0.258449i	$0.0832115\pi$
$521$	1.18874e12	+	2.05897e12i	0.706836	+	1.22428i	−0.259189	+	0.965827i	$0.583455\pi$
$522$		0			0
$523$	2.29128e11	−	3.96861e11i	0.133912	−	0.231943i	−0.791269	−	0.611468i	$-0.790579\pi$
$523$	2.29128e11	−	3.96861e11i	0.133912	−	0.231943i	0.925181	+	0.379525i	$0.123913\pi$
$524$		0			0
$525$	7.14756e11	+	1.07912e11i	0.410621	+	0.0619942i
$526$		0			0
$527$	5.77269e10	−	9.99859e10i	0.0326009	−	0.0564665i
$528$		0			0
$529$	−1.54567e12	−	2.67718e12i	−0.858155	−	1.48637i
$530$		0			0
$531$	5.51789e11			0.301195
$532$		0			0
$533$	−3.19341e11			−0.171389
$534$		0			0
$535$	4.43882e10	+	7.68826e10i	0.0234248	+	0.0405729i
$536$		0			0
$537$	−1.49014e12	+	2.58101e12i	−0.773293	+	1.33938i
$538$		0			0
$539$	4.12820e9	+	1.27560e9i	0.00210674	+	0.000650976i
$540$		0			0
$541$	−4.26542e11	+	7.38792e11i	−0.214079	+	0.370795i	−0.952987	−	0.303010i	$-0.902008\pi$
$541$	−4.26542e11	+	7.38792e11i	−0.214079	+	0.370795i	0.738908	+	0.673806i	$0.235342\pi$
$542$		0			0
$543$	−4.00563e11	−	6.93796e11i	−0.197730	−	0.342478i
$544$		0			0
$545$	5.32184e11			0.258391
$546$		0			0
$547$	9.09540e11			0.434389			0.217195	−	0.976128i	$-0.430309\pi$
$547$	9.09540e11			0.434389			0.217195	+	0.976128i	$0.430309\pi$
$548$		0			0
$549$	−3.97547e11	−	6.88571e11i	−0.186772	−	0.323499i
$550$		0			0
$551$	6.45661e11	−	1.11832e12i	0.298416	−	0.516872i
$552$		0			0
$553$	−4.17132e11	−	6.29771e10i	−0.189675	−	0.0286365i
$554$		0			0
$555$	1.82152e11	−	3.15496e11i	0.0814919	−	0.141148i
$556$		0			0
$557$	−1.78178e12	−	3.08613e12i	−0.784341	−	1.35852i	−0.929392	−	0.369094i	$-0.879668\pi$
$557$	−1.78178e12	−	3.08613e12i	−0.784341	−	1.35852i	0.145051	−	0.989424i	$-0.453665\pi$
$558$		0			0
$559$	3.20088e12			1.38649
$560$		0			0
$561$	−9.77784e8			−0.000416783
$562$		0			0
$563$	−5.62089e11	−	9.73567e11i	−0.235786	−	0.408393i	0.723715	−	0.690099i	$-0.242433\pi$
$563$	−5.62089e11	−	9.73567e11i	−0.235786	−	0.408393i	−0.959501	+	0.281706i	$0.909100\pi$
$564$		0			0
$565$	1.54738e12	−	2.68014e12i	0.638821	−	1.10647i
$566$		0			0
$567$	−1.01807e12	+	1.27598e12i	−0.413672	+	0.518467i
$568$		0			0
$569$	−1.94913e12	+	3.37599e12i	−0.779534	+	1.35019i	0.152676	+	0.988276i	$0.451211\pi$
$569$	−1.94913e12	+	3.37599e12i	−0.779534	+	1.35019i	−0.932211	+	0.361917i	$0.882123\pi$
$570$		0			0
$571$	−7.73495e11	−	1.33973e12i	−0.304505	−	0.527419i	0.672646	−	0.739965i	$-0.265158\pi$
$571$	−7.73495e11	−	1.33973e12i	−0.304505	−	0.527419i	−0.977151	+	0.212546i	$0.931825\pi$
$572$		0			0
$573$	3.44525e11			0.133513
$574$		0			0
$575$	−2.09400e12			−0.798861
$576$		0			0
$577$	−4.16268e11	−	7.20997e11i	−0.156344	−	0.270796i	0.777203	−	0.629249i	$-0.216638\pi$
$577$	−4.16268e11	−	7.20997e11i	−0.156344	−	0.270796i	−0.933548	+	0.358453i	$0.883304\pi$
$578$		0			0
$579$	−1.56126e12	+	2.70419e12i	−0.577328	+	0.999962i
$580$		0			0
$581$	1.26756e12	+	3.23201e12i	0.461503	+	1.17674i
$582$		0			0
$583$	4.00087e9	−	6.92972e9i	0.00143432	−	0.00248432i
$584$		0			0
$585$	−2.23476e11	−	3.87072e11i	−0.0788914	−	0.136644i
$586$		0			0
$587$	1.39886e12			0.486299			0.243150	−	0.969989i	$-0.421819\pi$
$587$	1.39886e12			0.486299			0.243150	+	0.969989i	$0.421819\pi$
$588$		0			0
$589$	9.82510e11			0.336371
$590$		0			0
$591$	−1.56824e12	−	2.71626e12i	−0.528771	−	0.915858i
$592$		0			0
$593$	−8.13971e11	+	1.40984e12i	−0.270310	+	0.468191i	−0.968941	−	0.247291i	$-0.920460\pi$
$593$	−8.13971e11	+	1.40984e12i	−0.270310	+	0.468191i	0.698631	+	0.715482i	$0.253793\pi$
$594$		0			0
$595$	1.76776e11	+	4.50742e11i	0.0578224	+	0.147435i
$596$		0			0
$597$	1.43987e12	−	2.49393e12i	0.463917	−	0.803527i
$598$		0			0
$599$	−1.52515e12	−	2.64164e12i	−0.484052	−	0.838402i	0.515780	−	0.856721i	$-0.327502\pi$
$599$	−1.52515e12	−	2.64164e12i	−0.484052	−	0.838402i	−0.999832	+	0.0183186i	$0.994169\pi$
$600$		0			0
$601$	5.50287e12			1.72050			0.860249	−	0.509875i	$-0.170308\pi$
$601$	5.50287e12			1.72050			0.860249	+	0.509875i	$0.170308\pi$
$602$		0			0
$603$	2.88404e11			0.0888327
$604$		0			0
$605$	−1.18275e12	−	2.04858e12i	−0.358917	−	0.621663i
$606$		0			0
$607$	−2.65974e12	+	4.60681e12i	−0.795226	+	1.37737i	0.127470	+	0.991842i	$0.459314\pi$
$607$	−2.65974e12	+	4.60681e12i	−0.795226	+	1.37737i	−0.922696	+	0.385529i	$0.874019\pi$
$608$		0			0
$609$	−9.51142e11	+	1.19209e12i	−0.280200	+	0.351183i
$610$		0			0
$611$	2.42136e10	−	4.19392e10i	0.00702868	−	0.0121740i
$612$		0			0
$613$	−2.99609e12	−	5.18939e12i	−0.857005	−	1.48438i	−0.874773	−	0.484534i	$-0.838989\pi$
$613$	−2.99609e12	−	5.18939e12i	−0.857005	−	1.48438i	0.0177677	−	0.999842i	$-0.494344\pi$
$614$		0			0
$615$	4.52498e11			0.127549
$616$		0			0
$617$	−6.91302e12			−1.92037			−0.960184	−	0.279369i	$-0.909875\pi$
$617$	−6.91302e12			−1.92037			−0.960184	+	0.279369i	$0.909875\pi$
$618$		0			0
$619$	−9.61950e10	−	1.66615e11i	−0.0263357	−	0.0456147i	0.852557	−	0.522634i	$-0.175051\pi$
$619$	−9.61950e10	−	1.66615e11i	−0.0263357	−	0.0456147i	−0.878893	+	0.477019i	$0.841717\pi$
$620$		0			0
$621$	3.31248e12	−	5.73738e12i	0.893800	−	1.54811i
$622$		0			0
$623$	−6.93306e12	−	1.04673e12i	−1.84386	−	0.278380i
$624$		0			0
$625$	5.34721e11	−	9.26163e11i	0.140174	−	0.242788i
$626$		0			0
$627$	−4.16047e9	−	7.20614e9i	−0.00107507	−	0.00186208i
$628$		0			0
$629$	−2.29528e11			−0.0584665
$630$		0			0
$631$	−1.56565e12			−0.393153			−0.196577	−	0.980488i	$-0.562982\pi$
$631$	−1.56565e12			−0.393153			−0.196577	+	0.980488i	$0.562982\pi$
$632$		0			0
$633$	1.08196e12	+	1.87400e12i	0.267851	+	0.463931i
$634$		0			0
$635$	−2.49341e12	+	4.31871e12i	−0.608572	+	1.05408i
$636$		0			0
$637$	−2.51851e12	+	2.33454e12i	−0.606061	+	0.561789i
$638$		0			0
$639$	−1.89115e11	+	3.27556e11i	−0.0448716	+	0.0777199i
$640$		0			0
$641$	2.06150e12	+	3.57063e12i	0.482306	+	0.835379i	0.999794	−	0.0203120i	$-0.00646594\pi$
$641$	2.06150e12	+	3.57063e12i	0.482306	+	0.835379i	−0.517488	+	0.855691i	$0.673133\pi$
$642$		0			0
$643$	3.15626e12			0.728153			0.364077	−	0.931369i	$-0.381385\pi$
$643$	3.15626e12			0.728153			0.364077	+	0.931369i	$0.381385\pi$
$644$		0			0
$645$	−4.53556e12			−1.03184
$646$		0			0
$647$	−8.11738e11	−	1.40597e12i	−0.182115	−	0.315433i	0.760485	−	0.649355i	$-0.224961\pi$
$647$	−8.11738e11	−	1.40597e12i	−0.182115	−	0.315433i	−0.942601	+	0.333922i	$0.891628\pi$
$648$		0			0
$649$	5.64264e9	−	9.77334e9i	0.00124848	−	0.00216243i
$650$		0			0
$651$	−1.14734e12	−	1.73221e11i	−0.250367	−	0.0377996i
$652$		0			0
$653$	−1.88231e12	+	3.26026e12i	−0.405118	+	0.701685i	−0.994335	−	0.106290i	$-0.966103\pi$
$653$	−1.88231e12	+	3.26026e12i	−0.405118	+	0.701685i	0.589217	+	0.807975i	$0.299436\pi$
$654$		0			0
$655$	−2.22780e12	−	3.85866e12i	−0.472922	−	0.819125i
$656$		0			0
$657$	2.24825e11			0.0470762
$658$		0			0
$659$	2.33436e12			0.482152			0.241076	−	0.970506i	$-0.422500\pi$
$659$	2.33436e12			0.482152			0.241076	+	0.970506i	$0.422500\pi$
$660$		0			0
$661$	7.27600e11	+	1.26024e12i	0.148247	+	0.256772i	0.930580	−	0.366089i	$-0.119304\pi$
$661$	7.27600e11	+	1.26024e12i	0.148247	+	0.256772i	−0.782333	+	0.622861i	$0.785970\pi$
$662$		0			0
$663$	3.88563e11	−	6.73011e11i	0.0780999	−	0.135273i
$664$		0			0
$665$	−2.56973e12	+	3.22072e12i	−0.509554	+	0.638639i
$666$		0			0
$667$	2.20890e12	−	3.82594e12i	0.432126	−	0.748465i
$668$		0			0
$669$	−3.99295e12	−	6.91599e12i	−0.770684	−	1.33486i
$670$		0			0
$671$	−1.62614e10			−0.00309675
$672$		0			0
$673$	−1.00036e13			−1.87970			−0.939850	−	0.341586i	$-0.889036\pi$
$673$	−1.00036e13			−1.87970			−0.939850	+	0.341586i	$0.889036\pi$
$674$		0			0
$675$	−1.41775e12	−	2.45561e12i	−0.262865	−	0.455295i
$676$		0			0
$677$	−6.01698e10	+	1.04217e11i	−0.0110085	+	0.0190673i	−0.871477	−	0.490436i	$-0.836838\pi$
$677$	−6.01698e10	+	1.04217e11i	−0.0110085	+	0.0190673i	0.860469	+	0.509503i	$0.170171\pi$
$678$		0			0
$679$	5.11561e11	+	1.30438e12i	0.0923599	+	0.235499i
$680$		0			0
$681$	−3.24518e11	+	5.62082e11i	−0.0578199	+	0.100147i
$682$		0			0
$683$	−5.21161e11	−	9.02677e11i	−0.0916387	−	0.158723i	0.816562	−	0.577258i	$-0.195877\pi$
$683$	−5.21161e11	−	9.02677e11i	−0.0916387	−	0.158723i	−0.908201	+	0.418535i	$0.862544\pi$
$684$		0			0
$685$	4.82418e12			0.837175
$686$		0			0
$687$	4.59430e12			0.786890
$688$		0			0
$689$	3.17983e12	+	5.50762e12i	0.537548	+	0.931060i
$690$		0			0
$691$	9.76538e10	−	1.69141e11i	0.0162944	−	0.0282227i	−0.857763	−	0.514045i	$-0.828146\pi$
$691$	9.76538e10	−	1.69141e11i	0.0162944	−	0.0282227i	0.874058	+	0.485822i	$0.161480\pi$
$692$		0			0
$693$	−1.30014e9	−	3.31509e9i	−0.000214136	−	0.000546004i
$694$		0			0
$695$	4.49098e11	−	7.77860e11i	0.0730145	−	0.126465i
$696$		0			0
$697$	−1.42547e11	−	2.46899e11i	−0.0228777	−	0.0396253i
$698$		0			0
$699$	8.11100e12			1.28507
$700$		0			0
$701$	4.54818e11			0.0711389			0.0355694	−	0.999367i	$-0.488676\pi$
$701$	4.54818e11			0.0711389			0.0355694	+	0.999367i	$0.488676\pi$
$702$		0			0
$703$	−9.76640e11	−	1.69159e12i	−0.150812	−	0.261214i
$704$		0			0
$705$	−3.43100e10	+	5.94267e10i	−0.00523082	+	0.00906005i
$706$		0			0
$707$	−3.21270e12	+	4.02658e12i	−0.483597	+	0.606106i
$708$		0			0
$709$	1.83880e12	−	3.18490e12i	0.273292	−	0.473355i	−0.696411	−	0.717643i	$-0.745221\pi$
$709$	1.83880e12	−	3.18490e12i	0.273292	−	0.473355i	0.969703	+	0.244288i	$0.0785542\pi$
$710$		0			0
$711$	1.73835e11	+	3.01091e11i	0.0255108	+	0.0441860i
$712$		0			0
$713$	3.36132e12			0.487087
$714$		0			0
$715$	−9.14114e9			−0.00130805
$716$		0			0
$717$	−3.04790e12	−	5.27912e12i	−0.430690	−	0.745977i
$718$		0			0
$719$	−3.39116e12	+	5.87366e12i	−0.473226	+	0.819651i	−0.999530	−	0.0306453i	$-0.990244\pi$
$719$	−3.39116e12	+	5.87366e12i	−0.473226	+	0.819651i	0.526305	+	0.850296i	$0.323577\pi$
$720$		0			0
$721$	1.15381e13	+	1.74199e12i	1.59011	+	0.240069i
$722$		0			0
$723$	1.75839e11	−	3.04562e11i	0.0239327	−	0.0414527i
$724$		0			0
$725$	−9.45417e11	−	1.63751e12i	−0.127087	−	0.220122i
$726$		0			0
$727$	1.34259e13			1.78253			0.891266	−	0.453480i	$-0.149818\pi$
$727$	1.34259e13			1.78253			0.891266	+	0.453480i	$0.149818\pi$
$728$		0			0
$729$	8.43141e12			1.10567
$730$		0			0
$731$	1.42881e12	+	2.47477e12i	0.185074	+	0.320557i
$732$		0			0
$733$	5.97864e11	−	1.03553e12i	0.0764952	−	0.132494i	−0.825240	−	0.564782i	$-0.808960\pi$
$733$	5.97864e11	−	1.03553e12i	0.0764952	−	0.132494i	0.901736	+	0.432288i	$0.142294\pi$
$734$		0			0
$735$	3.56866e12	−	3.30798e12i	0.451037	−	0.418090i
$736$		0			0
$737$	2.94924e9	−	5.10823e9i	0.000368219	−	0.000637774i
$738$		0			0
$739$	3.16512e12	+	5.48216e12i	0.390383	+	0.676163i	0.992500	−	0.122245i	$-0.0390093\pi$
$739$	3.16512e12	+	5.48216e12i	0.390383	+	0.676163i	−0.602117	+	0.798408i	$0.705676\pi$
$740$		0			0
$741$	6.61334e12			0.805821
$742$		0			0
$743$	−6.40429e12			−0.770942			−0.385471	−	0.922720i	$-0.625961\pi$
$743$	−6.40429e12			−0.770942			−0.385471	+	0.922720i	$0.625961\pi$
$744$		0			0
$745$	4.15857e12	+	7.20286e12i	0.494585	+	0.856646i
$746$		0			0
$747$	1.43057e12	−	2.47782e12i	0.168100	−	0.291157i
$748$		0			0
$749$	−5.55845e11	−	8.39196e10i	−0.0645335	−	0.00974306i
$750$		0			0
$751$	−6.65261e11	+	1.15227e12i	−0.0763155	+	0.132182i	−0.901658	−	0.432451i	$-0.857649\pi$
$751$	−6.65261e11	+	1.15227e12i	−0.0763155	+	0.132182i	0.825342	+	0.564633i	$0.190982\pi$
$752$		0			0
$753$	4.29314e11	+	7.43594e11i	0.0486629	+	0.0842866i
$754$		0			0
$755$	−8.39812e12			−0.940635
$756$		0			0
$757$	1.19794e13			1.32588			0.662941	−	0.748671i	$-0.269308\pi$
$757$	1.19794e13			1.32588			0.662941	+	0.748671i	$0.269308\pi$
$758$		0			0
$759$	−1.42336e10	−	2.46533e10i	−0.00155678	−	0.00269642i
$760$		0			0
$761$	−4.40736e12	+	7.63377e12i	−0.476373	+	0.825103i	−0.999634	−	0.0270701i	$-0.991382\pi$
$761$	−4.40736e12	+	7.63377e12i	−0.476373	+	0.825103i	0.523260	+	0.852173i	$0.324716\pi$
$762$		0			0
$763$	−2.10172e12	+	2.63415e12i	−0.224499	+	0.281371i
$764$		0			0
$765$	1.99510e11	−	3.45562e11i	0.0210615	−	0.0364796i
$766$		0			0
$767$	4.48468e12	+	7.76769e12i	0.467899	+	0.810425i
$768$		0			0
$769$	8.94607e12			0.922494			0.461247	−	0.887272i	$-0.347402\pi$
$769$	8.94607e12			0.922494			0.461247	+	0.887272i	$0.347402\pi$
$770$		0			0
$771$	−9.28640e12			−0.946461
$772$		0			0
$773$	−8.53874e12	−	1.47895e13i	−0.860174	−	1.48986i	−0.871761	−	0.489932i	$-0.837022\pi$
$773$	−8.53874e12	−	1.47895e13i	−0.860174	−	1.48986i	0.0115869	−	0.999933i	$-0.496312\pi$
$774$		0			0
$775$	7.19327e11	−	1.24591e12i	0.0716257	−	0.124059i
$776$		0			0
$777$	8.42247e11	+	2.14756e12i	0.0828982	+	0.211374i
$778$		0			0
$779$	1.21308e12	−	2.10111e12i	0.118024	−	0.204423i
$780$		0			0
$781$	3.86781e9	+	6.69924e9i	0.000371993	+	0.000644311i
$782$		0			0
$783$	5.98218e12			0.568764
$784$		0			0
$785$	−9.93735e12			−0.934022
$786$		0			0
$787$	−5.33917e11	−	9.24772e11i	−0.0496121	−	0.0859307i	0.840153	−	0.542350i	$-0.182465\pi$
$787$	−5.33917e11	−	9.24772e11i	−0.0496121	−	0.0859307i	−0.889765	+	0.456419i	$0.849132\pi$
$788$		0			0
$789$	−1.48493e12	+	2.57198e12i	−0.136414	+	0.236276i
$790$		0			0
$791$	7.15491e12	+	1.82436e13i	0.649845	+	1.65697i
$792$		0			0
$793$	6.46214e12	−	1.11927e13i	0.580292	−	1.00510i
$794$		0			0
$795$	−4.50573e12	−	7.80415e12i	−0.400049	−	0.692905i
$796$		0			0
$797$	−4.73027e12			−0.415263			−0.207631	−	0.978207i	$-0.566575\pi$
$797$	−4.73027e12			−0.415263			−0.207631	+	0.978207i	$0.566575\pi$
$798$		0			0
$799$	4.32338e10			0.00375286
$800$		0			0
$801$	2.88927e12	+	5.00437e12i	0.247995	+	0.429539i
$802$		0			0
$803$	2.29908e9	−	3.98213e9i	0.000195135	−	0.000337983i
$804$		0			0
$805$	−8.79145e12	+	1.10186e13i	−0.737868	+	0.924792i
$806$		0			0
$807$	8.09230e12	−	1.40163e13i	0.671647	−	1.16333i
$808$		0			0
$809$	−3.89727e11	−	6.75027e11i	−0.0319884	−	0.0554055i	0.849588	−	0.527447i	$-0.176851\pi$
$809$	−3.89727e11	−	6.75027e11i	−0.0319884	−	0.0554055i	−0.881576	+	0.472041i	$0.843517\pi$
$810$		0			0
$811$	1.56934e13			1.27386			0.636931	−	0.770921i	$-0.280204\pi$
$811$	1.56934e13			1.27386			0.636931	+	0.770921i	$0.280204\pi$
$812$		0			0
$813$	1.61281e13			1.29472
$814$		0			0
$815$	6.10467e12	+	1.05736e13i	0.484677	+	0.839486i
$816$		0			0
$817$	−1.21591e13	+	2.10602e13i	−0.954780	+	1.65373i
$818$		0			0
$819$	2.79845e12	+	4.22500e11i	0.217340	+	0.0328133i
$820$		0			0
$821$	−3.44265e11	+	5.96285e11i	−0.0264453	+	0.0458047i	−0.878945	−	0.476923i	$-0.841752\pi$
$821$	−3.44265e11	+	5.96285e11i	−0.0264453	+	0.0458047i	0.852500	+	0.522727i	$0.175085\pi$
$822$		0			0
$823$	−8.08484e12	−	1.40034e13i	−0.614288	−	1.06398i	−0.990509	−	0.137448i	$-0.956110\pi$
$823$	−8.08484e12	−	1.40034e13i	−0.614288	−	1.06398i	0.376221	−	0.926530i	$-0.377223\pi$
$824$		0			0
$825$	−1.21840e10			−0.000915690
$826$		0			0
$827$	6.45408e12			0.479799			0.239899	−	0.970798i	$-0.422886\pi$
$827$	6.45408e12			0.479799			0.239899	+	0.970798i	$0.422886\pi$
$828$		0			0
$829$	3.82325e10	+	6.62206e10i	0.00281149	+	0.00486965i	0.867428	−	0.497563i	$-0.165772\pi$
$829$	3.82325e10	+	6.62206e10i	0.00281149	+	0.00486965i	−0.864616	+	0.502433i	$0.832438\pi$
$830$		0			0
$831$	4.12425e12	−	7.14341e12i	0.300013	−	0.519638i
$832$		0			0
$833$	−2.92916e12	−	9.05101e11i	−0.210786	−	0.0651320i
$834$		0			0
$835$	−2.14602e12	+	3.71701e12i	−0.152772	+	0.264609i
$836$		0			0
$837$	2.27579e12	+	3.94179e12i	0.160276	+	0.277606i
$838$		0			0
$839$	−2.31366e13			−1.61202			−0.806011	−	0.591901i	$-0.798378\pi$
$839$	−2.31366e13			−1.61202			−0.806011	+	0.591901i	$0.798378\pi$
$840$		0			0
$841$	−1.05180e13			−0.725019
$842$		0			0
$843$	−3.19809e11	−	5.53926e11i	−0.0218106	−	0.0377770i
$844$		0			0
$845$	−1.68665e12	+	2.92136e12i	−0.113807	+	0.197120i
$846$		0			0
$847$	1.48108e13	+	2.23609e12i	0.988789	+	0.149284i
$848$		0			0
$849$	1.31441e12	−	2.27662e12i	0.0868253	−	0.150386i
$850$		0			0
$851$	−3.34124e12	−	5.78719e12i	−0.218386	−	0.378255i
$852$		0			0
$853$	−9.73533e12			−0.629622			−0.314811	−	0.949154i	$-0.601941\pi$
$853$	−9.73533e12			−0.629622			−0.314811	+	0.949154i	$0.601941\pi$
$854$		0			0
$855$	3.39566e12			0.217309
$856$		0			0
$857$	−5.21409e12	−	9.03108e12i	−0.330191	−	0.571907i	0.652358	−	0.757911i	$-0.273780\pi$
$857$	−5.21409e12	−	9.03108e12i	−0.330191	−	0.571907i	−0.982549	+	0.186003i	$0.940446\pi$
$858$		0			0
$859$	5.15969e12	−	8.93684e12i	0.323336	−	0.560035i	−0.657838	−	0.753159i	$-0.728529\pi$
$859$	5.15969e12	−	8.93684e12i	0.323336	−	0.560035i	0.981174	+	0.193125i	$0.0618622\pi$
$860$		0			0
$861$	−1.78702e12	+	2.23973e12i	−0.110819	+	0.138893i
$862$		0			0
$863$	−8.54372e12	+	1.47982e13i	−0.524322	+	0.908153i	0.475277	+	0.879836i	$0.342348\pi$
$863$	−8.54372e12	+	1.47982e13i	−0.524322	+	0.908153i	−0.999599	+	0.0283164i	$0.990985\pi$
$864$		0			0
$865$	8.73907e12	+	1.51365e13i	0.530753	+	0.919292i
$866$		0			0
$867$	−1.35603e13			−0.815049
$868$		0			0
$869$	7.11060e9			0.000422977
$870$		0			0
$871$	2.34401e12	+	4.05994e12i	0.137999	+	0.239022i
$872$		0			0
$873$	5.77352e11	−	1.00000e12i	0.0336416	−	0.0582689i
$874$		0			0
$875$	6.74731e12	+	1.72043e13i	0.389130	+	0.992202i
$876$		0			0
$877$	9.95754e12	−	1.72470e13i	0.568400	−	0.984498i	−0.428324	−	0.903625i	$-0.640896\pi$
$877$	9.95754e12	−	1.72470e13i	0.568400	−	0.984498i	0.996724	−	0.0808727i	$-0.0257707\pi$
$878$		0			0
$879$	−2.18527e12	−	3.78499e12i	−0.123468	−	0.213853i
$880$		0			0
$881$	−1.66453e11			−0.00930892			−0.00465446	−	0.999989i	$-0.501482\pi$
$881$	−1.66453e11			−0.00930892			−0.00465446	+	0.999989i	$0.501482\pi$
$882$		0			0
$883$	−1.45096e13			−0.803217			−0.401608	−	0.915811i	$-0.631549\pi$
$883$	−1.45096e13			−0.803217			−0.401608	+	0.915811i	$0.631549\pi$
$884$		0			0
$885$	−6.35467e12	−	1.10066e13i	−0.348216	−	0.603127i
$886$		0			0
$887$	−6.17805e12	+	1.07007e13i	−0.335116	+	0.580438i	−0.983507	−	0.180869i	$-0.942109\pi$
$887$	−6.17805e12	+	1.07007e13i	−0.335116	+	0.580438i	0.648391	+	0.761307i	$0.275442\pi$
$888$		0			0
$889$	−1.15292e13	−	2.93972e13i	−0.619074	−	1.57851i
$890$		0			0
$891$	1.37571e10	−	2.38279e10i	0.000731267	−	0.00126659i
$892$		0			0
$893$	1.83960e11	+	3.18627e11i	0.00968035	+	0.0167669i
$894$		0			0
$895$	−2.48743e13			−1.29583
$896$		0			0
$897$	2.26252e13			1.16688
$898$		0			0
$899$	1.51760e12	+	2.62856e12i	0.0774887	+	0.134214i
$900$		0			0
$901$	−2.83882e12	+	4.91698e12i	−0.143508	+	0.248563i
$902$		0			0
$903$	1.79120e13	−	2.24496e13i	0.896497	−	1.12361i
$904$		0			0
$905$	3.34320e12	−	5.79060e12i	0.165670	−	0.286949i
$906$		0			0
$907$	−7.22112e12	−	1.25073e13i	−0.354300	−	0.613666i	0.632698	−	0.774399i	$-0.281948\pi$
$907$	−7.22112e12	−	1.25073e13i	−0.354300	−	0.613666i	−0.986998	+	0.160733i	$0.948614\pi$
$908$		0			0
$909$	4.24529e12			0.206239
$910$		0			0
$911$	−1.49161e13			−0.717499			−0.358749	−	0.933434i	$-0.616797\pi$
$911$	−1.49161e13			−0.717499			−0.358749	+	0.933434i	$0.616797\pi$
$912$		0			0
$913$	−2.92583e10	−	5.06769e10i	−0.00139358	−	0.00241374i
$914$		0			0
$915$	−9.15667e12	+	1.58598e13i	−0.431860	+	0.748003i
$916$		0			0
$917$	2.78972e13	+	4.21183e12i	1.30286	+	0.196702i
$918$		0			0
$919$	7.49253e12	−	1.29774e13i	0.346504	−	0.600163i	−0.639122	−	0.769106i	$-0.720702\pi$
$919$	7.49253e12	−	1.29774e13i	0.346504	−	0.600163i	0.985626	+	0.168943i	$0.0540353\pi$
$920$		0			0
$921$	−1.25534e13	−	2.17431e13i	−0.574901	−	0.995758i
$922$		0			0
$923$	−6.14813e12			−0.278828
$924$		0			0
$925$	−2.86012e12			−0.128453
$926$		0			0
$927$	−4.80838e12	−	8.32837e12i	−0.213865	−	0.370426i
$928$		0			0
$929$	−8.63042e12	+	1.49483e13i	−0.380155	+	0.658449i	−0.991084	−	0.133237i	$-0.957463\pi$
$929$	−8.63042e12	+	1.49483e13i	−0.380155	+	0.658449i	0.610929	+	0.791686i	$0.290796\pi$
$930$		0			0
$931$	−5.79310e12	−	2.54387e13i	−0.252719	−	1.10974i
$932$		0			0
$933$	−2.53093e12	+	4.38369e12i	−0.109348	+	0.189397i
$934$		0			0
$935$	−4.08042e9	−	7.06749e9i	−0.000174603	−	0.000302422i
$936$		0			0
$937$	−1.80968e12			−0.0766961			−0.0383480	−	0.999264i	$-0.512210\pi$
$937$	−1.80968e12			−0.0766961			−0.0383480	+	0.999264i	$0.512210\pi$
$938$		0			0
$939$	−1.35984e13			−0.570810
$940$		0			0
$941$	−1.55916e13	−	2.70055e13i	−0.648244	−	1.12279i	−0.983542	−	0.180679i	$-0.942170\pi$
$941$	−1.55916e13	−	2.70055e13i	−0.648244	−	1.12279i	0.335298	−	0.942112i	$-0.391163\pi$
$942$		0			0
$943$	4.15012e12	−	7.18822e12i	0.170906	−	0.296019i
$944$		0			0
$945$	−1.88737e13	−	2.84948e12i	−0.769862	−	0.116231i
$946$		0			0
$947$	4.01232e10	−	6.94955e10i	0.00162114	−	0.00280790i	−0.865214	−	0.501403i	$-0.832817\pi$
$947$	4.01232e10	−	6.94955e10i	0.00162114	−	0.00280790i	0.866835	+	0.498595i	$0.166151\pi$
$948$		0			0
$949$	1.82727e12	+	3.16493e12i	0.0731316	+	0.126668i
$950$		0			0
$951$	3.24138e13			1.28504
$952$		0			0
$953$	−2.32667e13			−0.913727			−0.456864	−	0.889537i	$-0.651027\pi$
$953$	−2.32667e13			−0.913727			−0.456864	+	0.889537i	$0.651027\pi$
$954$		0			0
$955$	1.43775e12	+	2.49025e12i	0.0559329	+	0.0968786i
$956$		0			0
$957$	1.28526e10	−	2.22614e10i	0.000495322	−	0.000857924i
$958$		0			0
$959$	−1.90518e13	+	2.38782e13i	−0.727366	+	0.911629i
$960$		0			0
$961$	1.20651e13	−	2.08974e13i	0.456328	−	0.790383i
$962$		0			0
$963$	2.31642e11	+	4.01216e11i	0.00867959	+	0.0150335i
$964$		0			0
$965$	−2.60614e13			−0.967442
$966$		0			0
$967$	−1.75588e13			−0.645766			−0.322883	−	0.946439i	$-0.604652\pi$
$967$	−1.75588e13			−0.645766			−0.322883	+	0.946439i	$0.604652\pi$
$968$		0			0
$969$	2.95206e12	+	5.11311e12i	0.107564	+	0.186307i
$970$		0			0
$971$	1.46518e13	−	2.53777e13i	0.528939	−	0.916149i	−0.470491	−	0.882405i	$-0.655923\pi$
$971$	1.46518e13	−	2.53777e13i	0.528939	−	0.916149i	0.999430	−	0.0337447i	$-0.0107433\pi$
$972$		0			0
$973$	2.07658e12	+	5.29485e12i	0.0742745	+	0.189385i
$974$		0			0
$975$	4.84183e12	−	8.38630e12i	0.171589	−	0.297201i
$976$		0			0
$977$	5.87204e12	+	1.01707e13i	0.206188	+	0.357128i	0.950511	−	0.310692i	$-0.100561\pi$
$977$	5.87204e12	+	1.01707e13i	0.206188	+	0.357128i	−0.744323	+	0.667820i	$0.767227\pi$
$978$		0			0
$979$	1.18184e11			0.00411184
$980$		0			0
$981$	2.77723e12			0.0957417
$982$		0			0
$983$	−9.51858e12	−	1.64867e13i	−0.325148	−	0.563173i	0.656394	−	0.754418i	$-0.272081\pi$
$983$	−9.51858e12	−	1.64867e13i	−0.325148	−	0.563173i	−0.981542	+	0.191245i	$0.938747\pi$
$984$		0			0
$985$	1.30889e13	−	2.26706e13i	0.443037	−	0.767363i
$986$		0			0
$987$	−1.58645e11	−	4.04514e11i	−0.00532109	−	0.0135677i
$988$		0			0
$989$	−4.15983e13	+	7.20503e13i	−1.38259	+	2.39471i
$990$		0			0
$991$	2.61763e13	+	4.53386e13i	0.862136	+	1.49326i	0.869863	+	0.493294i	$0.164207\pi$
$991$	2.61763e13	+	4.53386e13i	0.862136	+	1.49326i	−0.00772625	+	0.999970i	$0.502459\pi$
$992$		0			0
$993$	−3.70789e13			−1.21020
$994$		0			0
$995$	2.40351e13			0.777396
$996$		0			0
$997$	2.43290e13	+	4.21391e13i	0.779824	+	1.35069i	0.932043	+	0.362348i	$0.118025\pi$
$997$	2.43290e13	+	4.21391e13i	0.779824	+	1.35069i	−0.152219	+	0.988347i	$0.548642\pi$
$998$		0			0
$999$	4.52439e12	−	7.83647e12i	0.143719	−	0.248929i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.10.i.d.65.2		12
4.3	odd	2		28.10.e.a.9.5	✓	12
7.4	even	3	inner	112.10.i.d.81.2		12
12.11	even	2		252.10.k.e.37.2		12
28.3	even	6		196.10.e.h.165.2		12
28.11	odd	6		28.10.e.a.25.5	yes	12
28.19	even	6		196.10.a.e.1.5		6
28.23	odd	6		196.10.a.f.1.2		6
28.27	even	2		196.10.e.h.177.2		12
84.11	even	6		252.10.k.e.109.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.10.e.a.9.5	✓	12	4.3	odd	2
28.10.e.a.25.5	yes	12	28.11	odd	6
112.10.i.d.65.2		12	1.1	even	1	trivial
112.10.i.d.81.2		12	7.4	even	3	inner
196.10.a.e.1.5		6	28.19	even	6
196.10.a.f.1.2		6	28.23	odd	6
196.10.e.h.165.2		12	28.3	even	6
196.10.e.h.177.2		12	28.27	even	2
252.10.k.e.37.2		12	12.11	even	2
252.10.k.e.109.2		12	84.11	even	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.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-60.0993 - 104.095i) q^{3} +(501.604 - 868.804i) q^{5} +(2319.36 + 5913.90i) q^{7} +(2617.65 - 4533.90i) q^{9} +(-53.5365 - 92.7280i) q^{11} +85099.9 q^{13} -120584. q^{15} +(37986.8 + 65795.1i) q^{17} +(-323268. + 559916. i) q^{19} +(476216. - 596855. i) q^{21} +(-1.10595e6 + 1.91556e6i) q^{23} +(473349. + 819864. i) q^{25} -2.99514e6 q^{27} -1.99729e6 q^{29} +(-759827. - 1.31606e6i) q^{31} +(-6435.02 + 11145.8i) q^{33} +(6.30142e6 + 951367. i) q^{35} +(-1.51057e6 + 2.61639e6i) q^{37} +(-5.11444e6 - 8.85848e6i) q^{39} -3.75255e6 q^{41} +3.76132e7 q^{43} +(-2.62605e6 - 4.54844e6i) q^{45} +(284531. - 492823. i) q^{47} +(-2.95948e7 + 2.74329e7i) q^{49} +(4.56596e6 - 7.90848e6i) q^{51} +(3.73658e7 + 6.47195e7i) q^{53} -107417. q^{55} +7.77127e7 q^{57} +(5.26990e7 + 9.12773e7i) q^{59} +(7.59359e7 - 1.31525e8i) q^{61} +(3.28843e7 + 4.96476e6i) q^{63} +(4.26865e7 - 7.39351e7i) q^{65} +(2.75442e7 + 4.77079e7i) q^{67} +2.65867e8 q^{69} -7.22461e7 q^{71} +(2.14721e7 + 3.71908e7i) q^{73} +(5.68959e7 - 9.85466e7i) q^{75} +(424213. - 531679. i) q^{77} +(-3.32044e7 + 5.75117e7i) q^{79} +(1.28483e8 + 2.22539e8i) q^{81} +5.46511e8 q^{83} +7.62174e7 q^{85} +(1.20036e8 + 2.07908e8i) q^{87} +(-5.51884e8 + 9.55891e8i) q^{89} +(1.97377e8 + 5.03272e8i) q^{91} +(-9.13302e7 + 1.58189e8i) q^{93} +(3.24305e8 + 5.61713e8i) q^{95} +2.20561e8 q^{97} -560559. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 966 q^{5} - 7696 q^{7} - 21620 q^{9} + 47640 q^{11} + 103432 q^{13} + 400280 q^{15} - 402234 q^{17} + 519960 q^{19} - 1025858 q^{21} + 683124 q^{23} - 1134656 q^{25} + 7641648 q^{27} + 4252680 q^{29}+ \cdots - 8151941296 q^{99}+O(q^{100}) \) 12 * q - 966 * q^5 - 7696 * q^7 - 21620 * q^9 + 47640 * q^11 + 103432 * q^13 + 400280 * q^15 - 402234 * q^17 + 519960 * q^19 - 1025858 * q^21 + 683124 * q^23 - 1134656 * q^25 + 7641648 * q^27 + 4252680 * q^29 + 1040564 * q^31 + 5260178 * q^33 + 552600 * q^35 - 4886646 * q^37 - 12290272 * q^39 + 34183128 * q^41 - 55245072 * q^43 - 17040100 * q^45 + 26858748 * q^47 + 12089772 * q^49 - 57449552 * q^51 - 21717558 * q^53 - 54952184 * q^55 - 72948300 * q^57 - 31298904 * q^59 - 155592542 * q^61 + 400205088 * q^63 + 7729020 * q^65 - 212196992 * q^67 - 249902212 * q^69 - 648519648 * q^71 - 549596530 * q^73 + 647801672 * q^75 - 519615786 * q^77 - 227611204 * q^79 + 363916186 * q^81 - 989338224 * q^83 + 2712621236 * q^85 + 860715296 * q^87 - 1746541986 * q^89 + 2592411872 * q^91 + 1354546774 * q^93 - 1586101596 * q^95 + 2306715320 * q^97 - 8151941296 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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