LMFDB - Embedded newform 84.12.i.b.37.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,12,Mod(25,84)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(84, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("84.25");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	84.i (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$64.5408271670$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 581500324 x^{14} - 481772282104 x^{13} + \cdots + 79\!\cdots\!77 $ x^16 - 581500324x^14 - 481772282104x^13 + 132272376701859942x^12 + 189765040124256066208x^11 - 14668309022748191106194508x^10 - 25034330150368988089577197656x^9 + 804524023197576594568815873370179x^8 + 1124828921377792402289540847268007968x^7 - 19980431516277460534216470290766973558668x^6 + 592727132887361668148704829618109324342008x^5 + 215459975341822734071937522290785139860745873306x^4 - 372150170517680072482976606267991582419751904690704x^3 - 31534379943215460069637664338468383174583531082850128x^2 + 257934680050232001165280545701170549413209663811915341424x + 79978257504936591964642108900330900905737463001503418421877
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{42}\cdot 3^{15}\cdot 7^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.4
Root			$2396.04 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.37
Dual form			84.12.i.b.25.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-121.500 - 210.444i) q^{3} +(-1332.77 + 2308.42i) q^{5} +(43648.5 - 8493.30i) q^{7} +(-29524.5 + 51137.9i) q^{9} +O(q^{10})$	\(q+(-121.500 - 210.444i) q^{3} +(-1332.77 + 2308.42i) q^{5} +(43648.5 - 8493.30i) q^{7} +(-29524.5 + 51137.9i) q^{9} +(262991. + 455515. i) q^{11} -1.10180e6 q^{13} +647726. q^{15} +(-222021. - 384552. i) q^{17} +(-1.58519e6 + 2.74562e6i) q^{19} +(-7.09066e6 - 8.15363e6i) q^{21} +(2.38816e7 - 4.13641e7i) q^{23} +(2.08615e7 + 3.61332e7i) q^{25} +1.43489e7 q^{27} -1.65232e8 q^{29} +(-8.05562e7 - 1.39527e8i) q^{31} +(6.39069e7 - 1.10690e8i) q^{33} +(-3.85672e7 + 1.12079e8i) q^{35} +(-1.73311e8 + 3.00184e8i) q^{37} +(1.33868e8 + 2.31866e8i) q^{39} -1.07369e9 q^{41} +1.10803e9 q^{43} +(-7.86987e7 - 1.36310e8i) q^{45} +(-4.49033e7 + 7.77748e7i) q^{47} +(1.83305e9 - 7.41439e8i) q^{49} +(-5.39512e7 + 9.34462e7i) q^{51} +(1.97953e8 + 3.42864e8i) q^{53} -1.40203e9 q^{55} +7.70401e8 q^{57} +(3.65401e9 + 6.32894e9i) q^{59} +(-6.05057e9 + 1.04799e10i) q^{61} +(-8.54370e8 + 2.48285e9i) q^{63} +(1.46844e9 - 2.54341e9i) q^{65} +(8.63592e9 + 1.49578e10i) q^{67} -1.16064e10 q^{69} -1.31065e10 q^{71} +(-7.39348e9 - 1.28059e10i) q^{73} +(5.06935e9 - 8.78037e9i) q^{75} +(1.53480e10 + 1.76489e10i) q^{77} +(-1.45406e10 + 2.51850e10i) q^{79} +(-1.74339e9 - 3.01964e9i) q^{81} +2.43778e9 q^{83} +1.18361e9 q^{85} +(2.00757e10 + 3.47721e10i) q^{87} +(-4.46315e10 + 7.73040e10i) q^{89} +(-4.80917e10 + 9.35787e9i) q^{91} +(-1.95752e10 + 3.39052e10i) q^{93} +(-4.22537e9 - 7.31856e9i) q^{95} +6.63296e10 q^{97} -3.10588e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9}+O(q^{10}) $ 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9	$ 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9} - 222796 q^{11} + 2703176 q^{13} + 1047816 q^{15} + 5114600 q^{17} + 6910556 q^{19} - 18340668 q^{21} - 51387712 q^{23} - 191456372 q^{25} + 229582512 q^{27} + 118854616 q^{29} + 164659160 q^{31} - 54139428 q^{33} + 55239344 q^{35} + 75658364 q^{37} - 328435884 q^{39} - 1815568608 q^{41} + 10754408 q^{43} - 127309644 q^{45} - 1034359464 q^{47} + 4123496848 q^{49} + 1242847800 q^{51} - 665159988 q^{53} - 1264543896 q^{55} - 3358530216 q^{57} + 1040514580 q^{59} - 14391208024 q^{61} + 1474099236 q^{63} - 20938150200 q^{65} - 33307097284 q^{67} + 24974428032 q^{69} + 65848902896 q^{71} + 17709749204 q^{73} - 46523898396 q^{75} + 8594484604 q^{77} - 26626784032 q^{79} - 27894275208 q^{81} - 210306955048 q^{83} - 25867402032 q^{85} - 14440835844 q^{87} - 55951560072 q^{89} + 66078280292 q^{91} + 40012175880 q^{93} + 106810047392 q^{95} - 156216030712 q^{97} + 26311762008 q^{99}+O(q^{100}) $ 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9 - 222796 * q^11 + 2703176 * q^13 + 1047816 * q^15 + 5114600 * q^17 + 6910556 * q^19 - 18340668 * q^21 - 51387712 * q^23 - 191456372 * q^25 + 229582512 * q^27 + 118854616 * q^29 + 164659160 * q^31 - 54139428 * q^33 + 55239344 * q^35 + 75658364 * q^37 - 328435884 * q^39 - 1815568608 * q^41 + 10754408 * q^43 - 127309644 * q^45 - 1034359464 * q^47 + 4123496848 * q^49 + 1242847800 * q^51 - 665159988 * q^53 - 1264543896 * q^55 - 3358530216 * q^57 + 1040514580 * q^59 - 14391208024 * q^61 + 1474099236 * q^63 - 20938150200 * q^65 - 33307097284 * q^67 + 24974428032 * q^69 + 65848902896 * q^71 + 17709749204 * q^73 - 46523898396 * q^75 + 8594484604 * q^77 - 26626784032 * q^79 - 27894275208 * q^81 - 210306955048 * q^83 - 25867402032 * q^85 - 14440835844 * q^87 - 55951560072 * q^89 + 66078280292 * q^91 + 40012175880 * q^93 + 106810047392 * q^95 - 156216030712 * q^97 + 26311762008 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$3$	−121.500	−	210.444i	−0.288675	−	0.500000i
$3$	−121.500	−	210.444i	−0.288675	−	0.500000i
$4$		0			0
$5$	−1332.77	+	2308.42i	−0.190730	+	0.330355i	−0.945492	−	0.325644i	$-0.894419\pi$
$5$	−1332.77	+	2308.42i	−0.190730	+	0.330355i	0.754762	+	0.655999i	$0.227752\pi$
$6$		0			0
$7$	43648.5	−	8493.30i	0.981590	−	0.191002i
$7$	43648.5	−	8493.30i	0.981590	−	0.191002i
$8$		0			0
$9$	−29524.5	+	51137.9i	−0.166667	+	0.288675i
$10$		0			0
$11$	262991.	+	455515.i	0.492359	+	0.852791i	0.999961	−	0.00880080i	$-0.00280142\pi$
$11$	262991.	+	455515.i	0.492359	+	0.852791i	−0.507602	+	0.861591i	$0.669468\pi$
$12$		0			0
$13$	−1.10180e6			−0.823024			−0.411512	−	0.911404i	$-0.634999\pi$
$13$	−1.10180e6			−0.823024			−0.411512	+	0.911404i	$0.634999\pi$
$14$		0			0
$15$	647726.			0.220236
$16$		0			0
$17$	−222021.	−	384552.i	−0.0379250	−	0.0656880i	0.846440	−	0.532484i	$-0.178741\pi$
$17$	−222021.	−	384552.i	−0.0379250	−	0.0656880i	−0.884365	+	0.466796i	$0.845408\pi$
$18$		0			0
$19$	−1.58519e6	+	2.74562e6i	−0.146871	+	0.254388i	−0.930069	−	0.367384i	$-0.880253\pi$
$19$	−1.58519e6	+	2.74562e6i	−0.146871	+	0.254388i	0.783199	+	0.621772i	$0.213587\pi$
$20$		0			0
$21$	−7.09066e6	−	8.15363e6i	−0.378861	−	0.435657i
$22$		0			0
$23$	2.38816e7	−	4.13641e7i	0.773677	−	1.34005i	−0.161857	−	0.986814i	$-0.551748\pi$
$23$	2.38816e7	−	4.13641e7i	0.773677	−	1.34005i	0.935535	−	0.353234i	$-0.114918\pi$
$24$		0			0
$25$	2.08615e7	+	3.61332e7i	0.427244	+	0.740008i
$26$		0			0
$27$	1.43489e7			0.192450
$28$		0			0
$29$	−1.65232e8			−1.49591			−0.747954	−	0.663750i	$-0.768964\pi$
$29$	−1.65232e8			−1.49591			−0.747954	+	0.663750i	$0.768964\pi$
$30$		0			0
$31$	−8.05562e7	−	1.39527e8i	−0.505370	−	0.875327i	−0.999981	−	0.00621203i	$-0.998023\pi$
$31$	−8.05562e7	−	1.39527e8i	−0.505370	−	0.875327i	0.494611	−	0.869115i	$-0.335311\pi$
$32$		0			0
$33$	6.39069e7	−	1.10690e8i	0.284264	−	0.492359i
$34$		0			0
$35$	−3.85672e7	+	1.12079e8i	−0.124121	+	0.360703i
$36$		0			0
$37$	−1.73311e8	+	3.00184e8i	−0.410882	+	0.711669i	−0.994986	−	0.100010i	$-0.968113\pi$
$37$	−1.73311e8	+	3.00184e8i	−0.410882	+	0.711669i	0.584104	+	0.811679i	$0.301446\pi$
$38$		0			0
$39$	1.33868e8	+	2.31866e8i	0.237586	+	0.411512i
$40$		0			0
$41$	−1.07369e9			−1.44734			−0.723668	−	0.690148i	$-0.757545\pi$
$41$	−1.07369e9			−1.44734			−0.723668	+	0.690148i	$0.757545\pi$
$42$		0			0
$43$	1.10803e9			1.14941			0.574706	−	0.818360i	$-0.305117\pi$
$43$	1.10803e9			1.14941			0.574706	+	0.818360i	$0.305117\pi$
$44$		0			0
$45$	−7.86987e7	−	1.36310e8i	−0.0635768	−	0.110118i
$46$		0			0
$47$	−4.49033e7	+	7.77748e7i	−0.0285588	+	0.0494653i	−0.879952	−	0.475063i	$-0.842425\pi$
$47$	−4.49033e7	+	7.77748e7i	−0.0285588	+	0.0494653i	0.851393	+	0.524529i	$0.175758\pi$
$48$		0			0
$49$	1.83305e9	−	7.41439e8i	0.927037	−	0.374970i
$50$		0			0
$51$	−5.39512e7	+	9.34462e7i	−0.0218960	+	0.0379250i
$52$		0			0
$53$	1.97953e8	+	3.42864e8i	0.0650196	+	0.112617i	0.896703	−	0.442633i	$-0.145956\pi$
$53$	1.97953e8	+	3.42864e8i	0.0650196	+	0.112617i	−0.831683	+	0.555251i	$0.812622\pi$
$54$		0			0
$55$	−1.40203e9			−0.375631
$56$		0			0
$57$	7.70401e8			0.169592
$58$		0			0
$59$	3.65401e9	+	6.32894e9i	0.665402	+	1.15251i	0.979176	+	0.203012i	$0.0650732\pi$
$59$	3.65401e9	+	6.32894e9i	0.665402	+	1.15251i	−0.313774	+	0.949498i	$0.601593\pi$
$60$		0			0
$61$	−6.05057e9	+	1.04799e10i	−0.917238	+	1.58870i	−0.113647	+	0.993521i	$0.536253\pi$
$61$	−6.05057e9	+	1.04799e10i	−0.917238	+	1.58870i	−0.803591	+	0.595182i	$0.797080\pi$
$62$		0			0
$63$	−8.54370e8	+	2.48285e9i	−0.108461	+	0.315194i
$64$		0			0
$65$	1.46844e9	−	2.54341e9i	0.156976	−	0.271890i
$66$		0			0
$67$	8.63592e9	+	1.49578e10i	0.781442	+	1.35350i	0.931101	+	0.364760i	$0.118849\pi$
$67$	8.63592e9	+	1.49578e10i	0.781442	+	1.35350i	−0.149659	+	0.988738i	$0.547818\pi$
$68$		0			0
$69$	−1.16064e10			−0.893366
$70$		0			0
$71$	−1.31065e10			−0.862113			−0.431056	−	0.902325i	$-0.641859\pi$
$71$	−1.31065e10			−0.862113			−0.431056	+	0.902325i	$0.641859\pi$
$72$		0			0
$73$	−7.39348e9	−	1.28059e10i	−0.417420	−	0.722993i	0.578259	−	0.815853i	$-0.303732\pi$
$73$	−7.39348e9	−	1.28059e10i	−0.417420	−	0.722993i	−0.995679	+	0.0928606i	$0.970399\pi$
$74$		0			0
$75$	5.06935e9	−	8.78037e9i	0.246669	−	0.427244i
$76$		0			0
$77$	1.53480e10	+	1.76489e10i	0.646179	+	0.743049i
$78$		0			0
$79$	−1.45406e10	+	2.51850e10i	−0.531658	+	0.920859i	0.467659	+	0.883909i	$0.345098\pi$
$79$	−1.45406e10	+	2.51850e10i	−0.531658	+	0.920859i	−0.999317	+	0.0369500i	$0.988236\pi$
$80$		0			0
$81$	−1.74339e9	−	3.01964e9i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	2.43778e9			0.0679305			0.0339652	−	0.999423i	$-0.489186\pi$
$83$	2.43778e9			0.0679305			0.0339652	+	0.999423i	$0.489186\pi$
$84$		0			0
$85$	1.18361e9			0.0289338
$86$		0			0
$87$	2.00757e10	+	3.47721e10i	0.431832	+	0.747954i
$88$		0			0
$89$	−4.46315e10	+	7.73040e10i	−0.847220	+	1.46743i	0.0364584	+	0.999335i	$0.488392\pi$
$89$	−4.46315e10	+	7.73040e10i	−0.847220	+	1.46743i	−0.883679	+	0.468094i	$0.844941\pi$
$90$		0			0
$91$	−4.80917e10	+	9.35787e9i	−0.807872	+	0.157199i
$92$		0			0
$93$	−1.95752e10	+	3.39052e10i	−0.291776	+	0.505370i
$94$		0			0
$95$	−4.22537e9	−	7.31856e9i	−0.0560254	−	0.0970389i
$96$		0			0
$97$	6.63296e10			0.784265			0.392132	−	0.919909i	$-0.371738\pi$
$97$	6.63296e10			0.784265			0.392132	+	0.919909i	$0.371738\pi$
$98$		0			0
$99$	−3.10588e10			−0.328239
$100$		0			0
$101$	1.40959e10	+	2.44149e10i	0.133452	+	0.231146i	0.925005	−	0.379954i	$-0.124060\pi$
$101$	1.40959e10	+	2.44149e10i	0.133452	+	0.231146i	−0.791553	+	0.611101i	$0.790727\pi$
$102$		0			0
$103$	−7.47509e10	+	1.29472e11i	−0.635348	+	1.10045i	0.351093	+	0.936340i	$0.385810\pi$
$103$	−7.47509e10	+	1.29472e11i	−0.635348	+	1.10045i	−0.986441	+	0.164114i	$0.947523\pi$
$104$		0			0
$105$	2.82722e10	−	5.50133e9i	0.216182	−	0.0420655i
$106$		0			0
$107$	−1.31786e11	+	2.28259e11i	−0.908358	+	1.57332i	−0.0920128	+	0.995758i	$0.529330\pi$
$107$	−1.31786e11	+	2.28259e11i	−0.908358	+	1.57332i	−0.816345	+	0.577564i	$0.804003\pi$
$108$		0			0
$109$	2.35713e10	+	4.08266e10i	0.146736	+	0.254155i	0.930019	−	0.367511i	$-0.119790\pi$
$109$	2.35713e10	+	4.08266e10i	0.146736	+	0.254155i	−0.783283	+	0.621665i	$0.786456\pi$
$110$		0			0
$111$	8.42293e10			0.474446
$112$		0			0
$113$	7.07560e10			0.361270			0.180635	−	0.983550i	$-0.442185\pi$
$113$	7.07560e10			0.361270			0.180635	+	0.983550i	$0.442185\pi$
$114$		0			0
$115$	6.36572e10	+	1.10258e11i	0.295128	+	0.511176i
$116$		0			0
$117$	3.25299e10	−	5.63435e10i	0.137171	−	0.237586i
$118$		0			0
$119$	−1.29570e10	−	1.48994e10i	−0.0497733	−	0.0572350i
$120$		0			0
$121$	4.32686e9	−	7.49434e9i	0.0151654	−	0.0262672i
$122$		0			0
$123$	1.30454e11	+	2.25953e11i	0.417810	+	0.723668i
$124$		0			0
$125$	−2.41368e11			−0.707414
$126$		0			0
$127$	−9.35541e10			−0.251271			−0.125635	−	0.992076i	$-0.540097\pi$
$127$	−9.35541e10			−0.251271			−0.125635	+	0.992076i	$0.540097\pi$
$128$		0			0
$129$	−1.34626e11	−	2.33179e11i	−0.331806	−	0.574706i
$130$		0			0
$131$	3.03929e10	−	5.26420e10i	0.0688303	−	0.119218i	−0.829556	−	0.558423i	$-0.811407\pi$
$131$	3.03929e10	−	5.26420e10i	0.0688303	−	0.119218i	0.898387	+	0.439205i	$0.144740\pi$
$132$		0			0
$133$	−4.58716e10	+	1.33306e11i	−0.0955784	+	0.277757i
$134$		0			0
$135$	−1.91238e10	+	3.31234e10i	−0.0367061	+	0.0635768i
$136$		0			0
$137$	1.72230e11	+	2.98311e11i	0.304891	+	0.528087i	0.977237	−	0.212150i	$-0.0680465\pi$
$137$	1.72230e11	+	2.98311e11i	0.304891	+	0.528087i	−0.672346	+	0.740237i	$0.734713\pi$
$138$		0			0
$139$	3.54339e11			0.579211			0.289606	−	0.957146i	$-0.406476\pi$
$139$	3.54339e11			0.579211			0.289606	+	0.957146i	$0.406476\pi$
$140$		0			0
$141$	2.18230e10			0.0329769
$142$		0			0
$143$	−2.89763e11	−	5.01884e11i	−0.405223	−	0.701867i
$144$		0			0
$145$	2.20216e11	−	3.81425e11i	0.285315	−	0.494180i
$146$		0			0
$147$	−3.78748e11	−	2.95671e11i	−0.455098	−	0.355274i
$148$		0			0
$149$	1.87394e11	−	3.24576e11i	0.209041	−	0.362069i	−0.742372	−	0.669988i	$-0.766299\pi$
$149$	1.87394e11	−	3.24576e11i	0.209041	−	0.362069i	0.951413	+	0.307919i	$0.0996325\pi$
$150$		0			0
$151$	−2.29692e11	−	3.97838e11i	−0.238107	−	0.412413i	0.722064	−	0.691826i	$-0.243194\pi$
$151$	−2.29692e11	−	3.97838e11i	−0.238107	−	0.412413i	−0.960171	+	0.279413i	$0.909860\pi$
$152$		0			0
$153$	2.62203e10			0.0252833
$154$		0			0
$155$	4.29451e11			0.385558
$156$		0			0
$157$	−6.37048e11	−	1.10340e12i	−0.532996	−	0.923176i	−0.999257	−	0.0385292i	$-0.987733\pi$
$157$	−6.37048e11	−	1.10340e12i	−0.532996	−	0.923176i	0.466261	−	0.884647i	$-0.345601\pi$
$158$		0			0
$159$	4.81025e10	−	8.33160e10i	0.0375391	−	0.0650196i
$160$		0			0
$161$	6.91077e11	−	2.00831e12i	0.503482	−	1.46315i
$162$		0			0
$163$	8.84049e11	−	1.53122e12i	0.601790	−	1.04233i	−0.390761	−	0.920492i	$-0.627788\pi$
$163$	8.84049e11	−	1.53122e12i	0.601790	−	1.04233i	0.992550	−	0.121838i	$-0.0388787\pi$
$164$		0			0
$165$	1.70346e11	+	2.95048e11i	0.108435	+	0.187816i
$166$		0			0
$167$	−1.36833e12			−0.815174			−0.407587	−	0.913166i	$-0.633630\pi$
$167$	−1.36833e12			−0.815174			−0.407587	+	0.913166i	$0.633630\pi$
$168$		0			0
$169$	−5.78208e11			−0.322632
$170$		0			0
$171$	−9.36037e10	−	1.62126e11i	−0.0489569	−	0.0847959i
$172$		0			0
$173$	−1.14381e12	+	1.98114e12i	−0.561179	+	0.971990i	0.436215	+	0.899842i	$0.356319\pi$
$173$	−1.14381e12	+	1.98114e12i	−0.561179	+	0.971990i	−0.997394	+	0.0721478i	$0.977015\pi$
$174$		0			0
$175$	1.21746e12	+	1.39998e12i	0.560721	+	0.644780i
$176$		0			0
$177$	8.87925e11	−	1.53793e12i	0.384170	−	0.665402i
$178$		0			0
$179$	−1.28321e12	−	2.22258e12i	−0.521922	−	0.903995i	−0.999675	−	0.0255010i	$-0.991882\pi$
$179$	−1.28321e12	−	2.22258e12i	−0.521922	−	0.903995i	0.477753	−	0.878494i	$-0.341451\pi$
$180$		0			0
$181$	4.37998e12			1.67587			0.837934	−	0.545771i	$-0.183763\pi$
$181$	4.37998e12			1.67587			0.837934	+	0.545771i	$0.183763\pi$
$182$		0			0
$183$	2.94058e12			1.05914
$184$		0			0
$185$	−4.61968e11	−	8.00152e11i	−0.156735	−	0.271474i
$186$		0			0
$187$	1.16779e11	−	2.02268e11i	0.0373454	−	0.0646842i
$188$		0			0
$189$	6.26308e11	−	1.21870e11i	0.188907	−	0.0367583i
$190$		0			0
$191$	−8.48818e10	+	1.47020e11i	−0.0241619	+	0.0418496i	−0.877854	−	0.478929i	$-0.841025\pi$
$191$	−8.48818e10	+	1.47020e11i	−0.0241619	+	0.0418496i	0.853692	+	0.520779i	$0.174358\pi$
$192$		0			0
$193$	−1.61580e11	−	2.79865e11i	−0.0434333	−	0.0752286i	0.843492	−	0.537142i	$-0.180496\pi$
$193$	−1.61580e11	−	2.79865e11i	−0.0434333	−	0.0752286i	−0.886925	+	0.461914i	$0.847163\pi$
$194$		0			0
$195$	−7.13661e11			−0.181260
$196$		0			0
$197$	8.26787e11			0.198531			0.0992657	−	0.995061i	$-0.468351\pi$
$197$	8.26787e11			0.198531			0.0992657	+	0.995061i	$0.468351\pi$
$198$		0			0
$199$	−2.12706e12	−	3.68418e12i	−0.483157	−	0.836853i	0.516656	−	0.856193i	$-0.327177\pi$
$199$	−2.12706e12	−	3.68418e12i	−0.483157	−	0.836853i	−0.999813	+	0.0193401i	$0.993843\pi$
$200$		0			0
$201$	2.09853e12	−	3.63476e12i	0.451166	−	0.781442i
$202$		0			0
$203$	−7.21213e12	+	1.40336e12i	−1.46837	+	0.285721i
$204$		0			0
$205$	1.43099e12	−	2.47854e12i	0.276051	−	0.478134i
$206$		0			0
$207$	1.41018e12	+	2.44251e12i	0.257892	+	0.446683i
$208$		0			0
$209$	−1.66756e12			−0.289253
$210$		0			0
$211$	6.58675e12			1.08422			0.542110	−	0.840307i	$-0.317626\pi$
$211$	6.58675e12			1.08422			0.542110	+	0.840307i	$0.317626\pi$
$212$		0			0
$213$	1.59243e12	+	2.75818e12i	0.248871	+	0.431056i
$214$		0			0
$215$	−1.47675e12	+	2.55780e12i	−0.219228	+	0.379713i
$216$		0			0
$217$	−4.70120e12	−	5.40597e12i	−0.663255	−	0.762685i
$218$		0			0
$219$	−1.79662e12	+	3.11183e12i	−0.240998	+	0.417420i
$220$		0			0
$221$	2.44622e11	+	4.23698e11i	0.0312132	+	0.0540628i
$222$		0			0
$223$	−7.33282e11			−0.0890419			−0.0445209	−	0.999008i	$-0.514176\pi$
$223$	−7.33282e11			−0.0890419			−0.0445209	+	0.999008i	$0.514176\pi$
$224$		0			0
$225$	−2.46370e12			−0.284829
$226$		0			0
$227$	5.85905e11	+	1.01482e12i	0.0645186	+	0.111749i	0.896480	−	0.443084i	$-0.146115\pi$
$227$	5.85905e11	+	1.01482e12i	0.0645186	+	0.111749i	−0.831962	+	0.554833i	$0.812782\pi$
$228$		0			0
$229$	−7.12391e12	+	1.23390e13i	−0.747521	+	1.29474i	0.201487	+	0.979491i	$0.435423\pi$
$229$	−7.12391e12	+	1.23390e13i	−0.747521	+	1.29474i	−0.949008	+	0.315253i	$0.897911\pi$
$230$		0			0
$231$	1.84932e12	−	5.37423e12i	0.184989	−	0.537589i
$232$		0			0
$233$	−7.16183e12	+	1.24047e13i	−0.683229	+	1.18339i	0.290760	+	0.956796i	$0.406092\pi$
$233$	−7.16183e12	+	1.24047e13i	−0.683229	+	1.18339i	−0.973990	+	0.226592i	$0.927242\pi$
$234$		0			0
$235$	−1.19692e11	−	2.07312e11i	−0.0108941	−	0.0188691i
$236$		0			0
$237$	7.06672e12			0.613906
$238$		0			0
$239$	−1.00149e13			−0.830728			−0.415364	−	0.909655i	$-0.636346\pi$
$239$	−1.00149e13			−0.830728			−0.415364	+	0.909655i	$0.636346\pi$
$240$		0			0
$241$	6.58907e12	+	1.14126e13i	0.522072	+	0.904255i	0.999670	+	0.0256770i	$0.00817415\pi$
$241$	6.58907e12	+	1.14126e13i	0.522072	+	0.904255i	−0.477598	+	0.878578i	$0.658493\pi$
$242$		0			0
$243$	−4.23644e11	+	7.33773e11i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	−7.31483e11	+	5.21963e12i	−0.0529408	+	0.377769i
$246$		0			0
$247$	1.74655e12	−	3.02511e12i	0.120878	−	0.209367i
$248$		0			0
$249$	−2.96190e11	−	5.13016e11i	−0.0196098	−	0.0339652i
$250$		0			0
$251$	−2.25032e13			−1.42574			−0.712868	−	0.701298i	$-0.752604\pi$
$251$	−2.25032e13			−1.42574			−0.712868	+	0.701298i	$0.752604\pi$
$252$		0			0
$253$	2.51226e13			1.52371
$254$		0			0
$255$	−1.43809e11	−	2.49084e11i	−0.00835247	−	0.0144669i
$256$		0			0
$257$	−6.53000e12	+	1.13103e13i	−0.363313	+	0.629276i	−0.988504	−	0.151195i	$-0.951688\pi$
$257$	−6.53000e12	+	1.13103e13i	−0.363313	+	0.629276i	0.625191	+	0.780472i	$0.285021\pi$
$258$		0			0
$259$	−5.01523e12	+	1.45746e13i	−0.267388	+	0.777046i
$260$		0			0
$261$	4.87839e12	−	8.44962e12i	0.249318	−	0.431832i
$262$		0			0
$263$	1.14385e13	+	1.98121e13i	0.560547	+	0.970897i	0.997449	+	0.0713872i	$0.0227426\pi$
$263$	1.14385e13	+	1.98121e13i	0.560547	+	0.970897i	−0.436901	+	0.899509i	$0.643924\pi$
$264$		0			0
$265$	−1.05530e12			−0.0496048
$266$		0			0
$267$	2.16909e13			0.978286
$268$		0			0
$269$	−6.89726e12	−	1.19464e13i	−0.298565	−	0.517130i	0.677243	−	0.735760i	$-0.263175\pi$
$269$	−6.89726e12	−	1.19464e13i	−0.298565	−	0.517130i	−0.975808	+	0.218630i	$0.929841\pi$
$270$		0			0
$271$	−5.69926e11	+	9.87141e11i	−0.0236858	+	0.0410250i	−0.877625	−	0.479347i	$-0.840873\pi$
$271$	−5.69926e11	+	9.87141e11i	−0.0236858	+	0.0410250i	0.853940	+	0.520372i	$0.174207\pi$
$272$		0			0
$273$	7.81245e12	+	8.98363e12i	0.311812	+	0.358556i
$274$		0			0
$275$	−1.09728e13	+	1.90054e13i	−0.420715	+	0.728699i
$276$		0			0
$277$	2.28587e13	+	3.95924e13i	0.842195	+	1.45873i	0.888035	+	0.459777i	$0.152070\pi$
$277$	2.28587e13	+	3.95924e13i	0.842195	+	1.45873i	−0.0458392	+	0.998949i	$0.514596\pi$
$278$		0			0
$279$	9.51352e12			0.336913
$280$		0			0
$281$	−3.91820e13			−1.33414			−0.667072	−	0.744994i	$-0.732452\pi$
$281$	−3.91820e13			−1.33414			−0.667072	+	0.744994i	$0.732452\pi$
$282$		0			0
$283$	−1.93169e12	−	3.34578e12i	−0.0632574	−	0.109565i	0.832662	−	0.553781i	$-0.186816\pi$
$283$	−1.93169e12	−	3.34578e12i	−0.0632574	−	0.109565i	−0.895920	+	0.444216i	$0.853482\pi$
$284$		0			0
$285$	−1.02677e12	+	1.77841e12i	−0.0323463	+	0.0560254i
$286$		0			0
$287$	−4.68651e13	+	9.11921e12i	−1.42069	+	0.276444i
$288$		0			0
$289$	1.70374e13	−	2.95096e13i	0.497123	−	0.861043i
$290$		0			0
$291$	−8.05904e12	−	1.39587e13i	−0.226398	−	0.392132i
$292$		0			0
$293$	−9.78633e12			−0.264757			−0.132379	−	0.991199i	$-0.542261\pi$
$293$	−9.78633e12			−0.264757			−0.132379	+	0.991199i	$0.542261\pi$
$294$		0			0
$295$	−1.94798e13			−0.507649
$296$		0			0
$297$	3.77364e12	+	6.53614e12i	0.0947545	+	0.164120i
$298$		0			0
$299$	−2.63126e13	+	4.55748e13i	−0.636755	+	1.10289i
$300$		0			0
$301$	4.83639e13	−	9.41084e12i	1.12825	−	0.219539i
$302$		0			0
$303$	3.42531e12	−	5.93282e12i	0.0770488	−	0.133452i
$304$		0			0
$305$	−1.61280e13	−	2.79346e13i	−0.349890	−	0.606028i
$306$		0			0
$307$	9.68832e12			0.202762			0.101381	−	0.994848i	$-0.467674\pi$
$307$	9.68832e12			0.202762			0.101381	+	0.994848i	$0.467674\pi$
$308$		0			0
$309$	3.63289e13			0.733636
$310$		0			0
$311$	−1.64052e13	−	2.84146e13i	−0.319741	−	0.553808i	0.660693	−	0.750656i	$-0.270263\pi$
$311$	−1.64052e13	−	2.84146e13i	−0.319741	−	0.553808i	−0.980434	+	0.196849i	$0.936929\pi$
$312$		0			0
$313$	4.65288e13	−	8.05902e13i	0.875443	−	1.51631i	0.0191533	−	0.999817i	$-0.493903\pi$
$313$	4.65288e13	−	8.05902e13i	0.875443	−	1.51631i	0.856290	−	0.516496i	$-0.172764\pi$
$314$		0			0
$315$	−4.59280e12	−	5.28132e12i	−0.0834391	−	0.0959476i
$316$		0			0
$317$	1.88239e13	−	3.26039e13i	0.330281	−	0.572063i	−0.652286	−	0.757973i	$-0.726190\pi$
$317$	1.88239e13	−	3.26039e13i	0.330281	−	0.572063i	0.982567	+	0.185910i	$0.0595232\pi$
$318$		0			0
$319$	−4.34546e13	−	7.52656e13i	−0.736524	−	1.27570i
$320$		0			0
$321$	6.40478e13			1.04888
$322$		0			0
$323$	1.40778e12			0.0222803
$324$		0			0
$325$	−2.29851e13	−	3.98114e13i	−0.351632	−	0.609044i
$326$		0			0
$327$	5.72782e12	−	9.92087e12i	0.0847182	−	0.146736i
$328$		0			0
$329$	−1.29940e12	+	3.77613e12i	−0.0185851	+	0.0540094i
$330$		0			0
$331$	−5.00419e13	+	8.66751e13i	−0.692276	+	1.19906i	0.278814	+	0.960345i	$0.410059\pi$
$331$	−5.00419e13	+	8.66751e13i	−0.692276	+	1.19906i	−0.971090	+	0.238713i	$0.923275\pi$
$332$		0			0
$333$	−1.02339e13	−	1.77256e13i	−0.136961	−	0.237223i
$334$		0			0
$335$	−4.60387e13			−0.596179
$336$		0			0
$337$	8.73473e13			1.09467			0.547337	−	0.836912i	$-0.315642\pi$
$337$	8.73473e13			1.09467			0.547337	+	0.836912i	$0.315642\pi$
$338$		0			0
$339$	−8.59685e12	−	1.48902e13i	−0.104290	−	0.180635i
$340$		0			0
$341$	4.23712e13	−	7.33890e13i	0.497647	−	0.861950i
$342$		0			0
$343$	7.37128e13	−	4.79314e13i	0.838350	−	0.545133i
$344$		0			0
$345$	1.54687e13	−	2.67926e13i	0.170392	−	0.295128i
$346$		0			0
$347$	−3.55238e13	−	6.15290e13i	−0.379059	−	0.656550i	0.611866	−	0.790961i	$-0.290419\pi$
$347$	−3.55238e13	−	6.15290e13i	−0.379059	−	0.656550i	−0.990926	+	0.134411i	$0.957086\pi$
$348$		0			0
$349$	−7.50390e13			−0.775796			−0.387898	−	0.921702i	$-0.626799\pi$
$349$	−7.50390e13			−0.775796			−0.387898	+	0.921702i	$0.626799\pi$
$350$		0			0
$351$	−1.58096e13			−0.158391
$352$		0			0
$353$	−6.79326e12	−	1.17663e13i	−0.0659655	−	0.114256i	0.831156	−	0.556039i	$-0.187679\pi$
$353$	−6.79326e12	−	1.17663e13i	−0.0659655	−	0.114256i	−0.897122	+	0.441783i	$0.854346\pi$
$354$		0			0
$355$	1.74679e13	−	3.02552e13i	0.164431	−	0.284803i
$356$		0			0
$357$	−1.56122e12	+	4.53701e12i	−0.0142492	+	0.0414090i
$358$		0			0
$359$	−7.02740e13	+	1.21718e14i	−0.621978	+	1.07730i	0.367139	+	0.930166i	$0.380337\pi$
$359$	−7.02740e13	+	1.21718e14i	−0.621978	+	1.07730i	−0.989117	+	0.147131i	$0.952996\pi$
$360$		0			0
$361$	5.32195e13	+	9.21789e13i	0.456858	+	0.791301i
$362$		0			0
$363$	−2.10285e12			−0.0175115
$364$		0			0
$365$	3.94152e13			0.318459
$366$		0			0
$367$	−1.75641e13	−	3.04219e13i	−0.137709	−	0.238519i	0.788920	−	0.614496i	$-0.210641\pi$
$367$	−1.75641e13	−	3.04219e13i	−0.137709	−	0.238519i	−0.926629	+	0.375977i	$0.877307\pi$
$368$		0			0
$369$	3.17003e13	−	5.49065e13i	0.241223	−	0.417810i
$370$		0			0
$371$	1.15524e13	+	1.32842e13i	0.0853326	+	0.0981251i
$372$		0			0
$373$	8.94455e13	−	1.54924e14i	0.641446	−	1.11102i	−0.343664	−	0.939093i	$-0.611668\pi$
$373$	8.94455e13	−	1.54924e14i	0.641446	−	1.11102i	0.985110	−	0.171924i	$-0.0549985\pi$
$374$		0			0
$375$	2.93262e13	+	5.07944e13i	0.204213	+	0.353707i
$376$		0			0
$377$	1.82052e14			1.23117
$378$		0			0
$379$	−3.33296e13			−0.218935			−0.109467	−	0.993990i	$-0.534915\pi$
$379$	−3.33296e13			−0.218935			−0.109467	+	0.993990i	$0.534915\pi$
$380$		0			0
$381$	1.13668e13	+	1.96879e13i	0.0725357	+	0.125635i
$382$		0			0
$383$	1.37824e13	−	2.38718e13i	0.0854539	−	0.148010i	−0.820131	−	0.572176i	$-0.806099\pi$
$383$	1.37824e13	−	2.38718e13i	0.0854539	−	0.148010i	0.905585	+	0.424166i	$0.139433\pi$
$384$		0			0
$385$	−6.11964e13	+	1.19078e13i	−0.368716	+	0.0717462i
$386$		0			0
$387$	−3.27141e13	+	5.66624e13i	−0.191569	+	0.331806i
$388$		0			0
$389$	−1.16390e14	−	2.01594e14i	−0.662513	−	1.14751i	−0.979953	−	0.199228i	$-0.936156\pi$
$389$	−1.16390e14	−	2.01594e14i	−0.662513	−	1.14751i	0.317440	−	0.948278i	$-0.397177\pi$
$390$		0			0
$391$	−2.12089e13			−0.117367
$392$		0			0
$393$	−1.47709e13			−0.0794784
$394$		0			0
$395$	−3.87584e13	−	6.71316e13i	−0.202807	−	0.351272i
$396$		0			0
$397$	7.15857e13	−	1.23990e14i	0.364316	−	0.631014i	−0.624350	−	0.781145i	$-0.714636\pi$
$397$	7.15857e13	−	1.23990e14i	0.364316	−	0.631014i	0.988666	+	0.150131i	$0.0479694\pi$
$398$		0			0
$399$	3.36268e13	−	6.54324e12i	0.166470	−	0.0323923i
$400$		0			0
$401$	−1.44586e14	+	2.50431e14i	−0.696359	+	1.20613i	0.273362	+	0.961911i	$0.411864\pi$
$401$	−1.44586e14	+	2.50431e14i	−0.696359	+	1.20613i	−0.969721	+	0.244217i	$0.921469\pi$
$402$		0			0
$403$	8.87564e13	+	1.53731e14i	0.415932	+	0.720415i
$404$		0			0
$405$	9.29416e12			0.0423845
$406$		0			0
$407$	−1.82318e14			−0.809206
$408$		0			0
$409$	4.66276e13	+	8.07613e13i	0.201449	+	0.348919i	0.948995	−	0.315290i	$-0.102102\pi$
$409$	4.66276e13	+	8.07613e13i	0.201449	+	0.348919i	−0.747547	+	0.664209i	$0.768768\pi$
$410$		0			0
$411$	4.18518e13	−	7.24895e13i	0.176029	−	0.304891i
$412$		0			0
$413$	2.13246e14	+	2.45214e14i	0.873283	+	1.00420i
$414$		0			0
$415$	−3.24900e12	+	5.62742e12i	−0.0129564	+	0.0224412i
$416$		0			0
$417$	−4.30521e13	−	7.45685e13i	−0.167204	−	0.289606i
$418$		0			0
$419$	−4.50519e14			−1.70426			−0.852130	−	0.523330i	$-0.824689\pi$
$419$	−4.50519e14			−1.70426			−0.852130	+	0.523330i	$0.824689\pi$
$420$		0			0
$421$	1.82722e14			0.673347			0.336673	−	0.941621i	$-0.390698\pi$
$421$	1.82722e14			0.673347			0.336673	+	0.941621i	$0.390698\pi$
$422$		0			0
$423$	−2.65150e12	−	4.59253e12i	−0.00951961	−	0.0164884i
$424$		0			0
$425$	9.26341e12	−	1.60447e13i	0.0324065	−	0.0561296i
$426$		0			0
$427$	−1.75089e14	+	5.08821e14i	−0.596907	+	1.73465i
$428$		0			0
$429$	−7.04123e13	+	1.21958e14i	−0.233956	+	0.405223i
$430$		0			0
$431$	7.13121e13	+	1.23516e14i	0.230961	+	0.400036i	0.958091	−	0.286463i	$-0.0924797\pi$
$431$	7.13121e13	+	1.23516e14i	0.230961	+	0.400036i	−0.727130	+	0.686500i	$0.759146\pi$
$432$		0			0
$433$	−1.67830e14			−0.529892			−0.264946	−	0.964263i	$-0.585354\pi$
$433$	−1.67830e14			−0.529892			−0.264946	+	0.964263i	$0.585354\pi$
$434$		0			0
$435$	−1.07025e14			−0.329454
$436$		0			0
$437$	7.57135e13	+	1.31140e14i	0.227261	+	0.393628i
$438$		0			0
$439$	−2.55296e14	+	4.42186e14i	−0.747291	+	1.29435i	0.201826	+	0.979421i	$0.435313\pi$
$439$	−2.55296e14	+	4.42186e14i	−0.747291	+	1.29435i	−0.949117	+	0.314925i	$0.898021\pi$
$440$		0			0
$441$	−1.62044e13	+	1.15629e14i	−0.0462615	+	0.330108i
$442$		0			0
$443$	1.38404e14	−	2.39723e14i	0.385415	−	0.667559i	−0.606411	−	0.795151i	$-0.707391\pi$
$443$	1.38404e14	−	2.39723e14i	0.385415	−	0.667559i	0.991827	+	0.127592i	$0.0407248\pi$
$444$		0			0
$445$	−1.18967e14	−	2.06057e14i	−0.323181	−	0.559766i
$446$		0			0
$447$	−9.10735e13			−0.241380
$448$		0			0
$449$	7.18277e14			1.85754			0.928768	−	0.370661i	$-0.120869\pi$
$449$	7.18277e14			1.85754			0.928768	+	0.370661i	$0.120869\pi$
$450$		0			0
$451$	−2.82372e14	−	4.89083e14i	−0.712609	−	1.23428i
$452$		0			0
$453$	−5.58151e13	+	9.66746e13i	−0.137471	+	0.238107i
$454$		0			0
$455$	4.24932e13	−	1.23488e14i	0.102154	−	0.296867i
$456$		0			0
$457$	−2.69134e14	+	4.66153e14i	−0.631581	+	1.09393i	0.355648	+	0.934620i	$0.384260\pi$
$457$	−2.69134e14	+	4.66153e14i	−0.631581	+	1.09393i	−0.987229	+	0.159310i	$0.949073\pi$
$458$		0			0
$459$	−3.18576e12	−	5.51791e12i	−0.00729867	−	0.0126417i
$460$		0			0
$461$	3.08940e14			0.691065			0.345532	−	0.938407i	$-0.387698\pi$
$461$	3.08940e14			0.691065			0.345532	+	0.938407i	$0.387698\pi$
$462$		0			0
$463$	−3.55334e14			−0.776142			−0.388071	−	0.921630i	$-0.626858\pi$
$463$	−3.55334e14			−0.776142			−0.388071	+	0.921630i	$0.626858\pi$
$464$		0			0
$465$	−5.21783e13	−	9.03755e13i	−0.111301	−	0.192779i
$466$		0			0
$467$	1.09007e14	−	1.88806e14i	0.227097	−	0.393344i	−0.729849	−	0.683608i	$-0.760410\pi$
$467$	1.09007e14	−	1.88806e14i	0.227097	−	0.393344i	0.956947	+	0.290264i	$0.0937432\pi$
$468$		0			0
$469$	5.03986e14	+	5.79540e14i	1.02558	+	1.17932i
$470$		0			0
$471$	−1.54803e14	+	2.68126e14i	−0.307725	+	0.532996i
$472$		0			0
$473$	2.91403e14	+	5.04724e14i	0.565923	+	0.980207i
$474$		0			0
$475$	−1.32278e14			−0.250999
$476$		0			0
$477$	−2.33778e13			−0.0433464
$478$		0			0
$479$	3.15790e14	+	5.46965e14i	0.572207	+	0.991092i	0.996339	+	0.0854915i	$0.0272460\pi$
$479$	3.15790e14	+	5.46965e14i	0.572207	+	0.991092i	−0.424132	+	0.905601i	$0.639421\pi$
$480$		0			0
$481$	1.90954e14	−	3.30741e14i	0.338166	−	0.585721i
$482$		0			0
$483$	−5.06604e14	+	9.85770e13i	−0.876919	+	0.170634i
$484$		0			0
$485$	−8.84020e13	+	1.53117e14i	−0.149583	+	0.259085i
$486$		0			0
$487$	−1.36327e14	−	2.36125e14i	−0.225513	−	0.390600i	0.730960	−	0.682420i	$-0.239072\pi$
$487$	−1.36327e14	−	2.36125e14i	−0.225513	−	0.390600i	−0.956473	+	0.291820i	$0.905739\pi$
$488$		0			0
$489$	−4.29648e14			−0.694887
$490$		0			0
$491$	1.03331e15			1.63411			0.817055	−	0.576560i	$-0.195605\pi$
$491$	1.03331e15			1.63411			0.817055	+	0.576560i	$0.195605\pi$
$492$		0			0
$493$	3.66850e13	+	6.35403e13i	0.0567324	+	0.0982633i
$494$		0			0
$495$	4.13941e13	−	7.16968e13i	0.0626052	−	0.108435i
$496$		0			0
$497$	−5.72077e14	+	1.11317e14i	−0.846241	+	0.164665i
$498$		0			0
$499$	−1.89945e14	+	3.28995e14i	−0.274837	+	0.476031i	−0.970094	−	0.242730i	$-0.921957\pi$
$499$	−1.89945e14	+	3.28995e14i	−0.274837	+	0.476031i	0.695257	+	0.718761i	$0.255291\pi$
$500$		0			0
$501$	1.66252e14	+	2.87957e14i	0.235321	+	0.407587i
$502$		0			0
$503$	2.88084e14			0.398928			0.199464	−	0.979905i	$-0.436080\pi$
$503$	2.88084e14			0.398928			0.199464	+	0.979905i	$0.436080\pi$
$504$		0			0
$505$	−7.51465e13			−0.101814
$506$		0			0
$507$	7.02523e13	+	1.21681e14i	0.0931358	+	0.161316i
$508$		0			0
$509$	7.02577e14	−	1.21690e15i	0.911478	−	1.57873i	0.0995016	−	0.995037i	$-0.468275\pi$
$509$	7.02577e14	−	1.21690e15i	0.911478	−	1.57873i	0.811977	−	0.583690i	$-0.198392\pi$
$510$		0			0
$511$	−4.31478e14	−	4.96162e14i	−0.547828	−	0.629954i
$512$		0			0
$513$	−2.27457e13	+	3.93967e13i	−0.0282653	+	0.0489569i
$514$		0			0
$515$	−1.99251e14	−	3.45113e14i	−0.242360	−	0.419780i
$516$		0			0
$517$	−4.72368e13			−0.0562448
$518$		0			0
$519$	5.55893e14			0.647993
$520$		0			0
$521$	−8.31749e14	−	1.44063e15i	−0.949259	−	1.64417i	−0.746990	−	0.664835i	$-0.768502\pi$
$521$	−8.31749e14	−	1.44063e15i	−0.949259	−	1.64417i	−0.202269	−	0.979330i	$-0.564832\pi$
$522$		0			0
$523$	5.23224e13	−	9.06250e13i	0.0584693	−	0.101272i	−0.835309	−	0.549781i	$-0.814711\pi$
$523$	5.23224e13	−	9.06250e13i	0.0584693	−	0.101272i	0.893778	+	0.448509i	$0.148045\pi$
$524$		0			0
$525$	1.46695e14	−	4.26305e14i	0.160524	−	0.466492i
$526$		0			0
$527$	−3.57704e13	+	6.19561e13i	−0.0383323	+	0.0663936i
$528$		0			0
$529$	−6.64255e14	−	1.15052e15i	−0.697154	−	1.20751i
$530$		0			0
$531$	−4.31532e14			−0.443601
$532$		0			0
$533$	1.18299e15			1.19119
$534$		0			0
$535$	−3.51279e14	−	6.08434e14i	−0.346503	−	0.600161i
$536$		0			0
$537$	−3.11820e14	+	5.40087e14i	−0.301332	+	0.521922i
$538$		0			0
$539$	8.19814e14	+	6.39991e14i	0.776206	+	0.605948i
$540$		0			0
$541$	9.40174e13	−	1.62843e14i	0.0872215	−	0.151072i	−0.819114	−	0.573630i	$-0.805535\pi$
$541$	9.40174e13	−	1.62843e14i	0.0872215	−	0.151072i	0.906336	+	0.422558i	$0.138868\pi$
$542$		0			0
$543$	−5.32168e14	−	9.21741e14i	−0.483782	−	0.837934i
$544$		0			0
$545$	−1.25660e14			−0.111948
$546$		0			0
$547$	2.21322e15			1.93238			0.966192	−	0.257824i	$-0.0830056\pi$
$547$	2.21322e15			1.93238			0.966192	+	0.257824i	$0.0830056\pi$
$548$		0			0
$549$	−3.57280e14	−	6.18827e14i	−0.305746	−	0.529568i
$550$		0			0
$551$	2.61923e14	−	4.53665e14i	0.219705	−	0.380541i
$552$		0			0
$553$	−4.20770e14	+	1.22279e15i	−0.345985	+	1.00545i
$554$		0			0
$555$	−1.12258e14	+	1.94437e14i	−0.0904913	+	0.156735i
$556$		0			0
$557$	−4.08922e14	−	7.08274e14i	−0.323174	−	0.559755i	0.657967	−	0.753047i	$-0.271417\pi$
$557$	−4.08922e14	−	7.08274e14i	−0.323174	−	0.559755i	−0.981141	+	0.193292i	$0.938083\pi$
$558$		0			0
$559$	−1.22082e15			−0.945993
$560$		0			0
$561$	−5.67548e13			−0.0431228
$562$		0			0
$563$	−1.06151e15	−	1.83858e15i	−0.790909	−	1.36989i	−0.925405	−	0.378981i	$-0.876275\pi$
$563$	−1.06151e15	−	1.83858e15i	−0.790909	−	1.36989i	0.134495	−	0.990914i	$-0.457059\pi$
$564$		0			0
$565$	−9.43014e13	+	1.63335e14i	−0.0689051	+	0.119347i
$566$		0			0
$567$	−1.01743e14	−	1.16996e14i	−0.0729119	−	0.0838423i
$568$		0			0
$569$	4.39692e14	−	7.61570e14i	0.309052	−	0.535294i	−0.669103	−	0.743169i	$-0.733322\pi$
$569$	4.39692e14	−	7.61570e14i	0.309052	−	0.535294i	0.978155	+	0.207876i	$0.0666549\pi$
$570$		0			0
$571$	3.97661e14	+	6.88769e14i	0.274166	+	0.474870i	0.969924	−	0.243406i	$-0.0782648\pi$
$571$	3.97661e14	+	6.88769e14i	0.274166	+	0.474870i	−0.695758	+	0.718276i	$0.744931\pi$
$572$		0			0
$573$	4.12526e13			0.0278998
$574$		0			0
$575$	1.99282e15			1.32220
$576$		0			0
$577$	−8.87379e14	−	1.53699e15i	−0.577620	−	1.00047i	−0.995752	−	0.0920805i	$-0.970648\pi$
$577$	−8.87379e14	−	1.53699e15i	−0.577620	−	1.00047i	0.418132	−	0.908386i	$-0.362685\pi$
$578$		0			0
$579$	−3.92640e13	+	6.80072e13i	−0.0250762	+	0.0434333i
$580$		0			0
$581$	1.06405e14	−	2.07048e13i	0.0666799	−	0.0129748i
$582$		0			0
$583$	−1.04120e14	+	1.80341e14i	−0.0640260	+	0.110896i
$584$		0			0
$585$	8.67098e13	+	1.50186e14i	0.0523252	+	0.0906299i
$586$		0			0
$587$	1.91619e15			1.13482			0.567412	−	0.823434i	$-0.307945\pi$
$587$	1.91619e15			1.13482			0.567412	+	0.823434i	$0.307945\pi$
$588$		0			0
$589$	5.10786e14			0.296896
$590$		0			0
$591$	−1.00455e14	−	1.73992e14i	−0.0573111	−	0.0992657i
$592$		0			0
$593$	2.37770e14	−	4.11829e14i	0.133154	−	0.230630i	−0.791736	−	0.610863i	$-0.790823\pi$
$593$	2.37770e14	−	4.11829e14i	0.133154	−	0.230630i	0.924891	+	0.380233i	$0.124156\pi$
$594$		0			0
$595$	5.16629e13	−	1.00528e13i	0.0284011	−	0.00552640i
$596$		0			0
$597$	−5.16877e14	+	8.95256e14i	−0.278951	+	0.483157i
$598$		0			0
$599$	1.06635e15	+	1.84697e15i	0.565006	+	0.978618i	0.997049	+	0.0767657i	$0.0244593\pi$
$599$	1.06635e15	+	1.84697e15i	0.565006	+	0.978618i	−0.432044	+	0.901853i	$0.642207\pi$
$600$		0			0
$601$	3.19165e12			0.00166037			0.000830187	−	1.00000i	$-0.499736\pi$
$601$	3.19165e12			0.00166037			0.000830187		1.00000i	$0.499736\pi$
$602$		0			0
$603$	−1.01988e15			−0.520962
$604$		0			0
$605$	1.15334e13	+	1.99765e13i	0.00578500	+	0.0100199i
$606$		0			0
$607$	6.33721e14	−	1.09764e15i	0.312148	−	0.540656i	−0.666679	−	0.745345i	$-0.732285\pi$
$607$	6.33721e14	−	1.09764e15i	0.312148	−	0.540656i	0.978827	+	0.204689i	$0.0656183\pi$
$608$		0			0
$609$	1.17160e15	+	1.34724e15i	0.566742	+	0.651704i
$610$		0			0
$611$	4.94743e13	−	8.56919e13i	0.0235046	−	0.0407111i
$612$		0			0
$613$	5.75304e14	+	9.96455e14i	0.268451	+	0.464970i	0.968462	−	0.249162i	$-0.0801550\pi$
$613$	5.75304e14	+	9.96455e14i	0.268451	+	0.464970i	−0.700011	+	0.714132i	$0.746822\pi$
$614$		0			0
$615$	−6.95460e14			−0.318756
$616$		0			0
$617$	−1.65065e14			−0.0743169			−0.0371584	−	0.999309i	$-0.511831\pi$
$617$	−1.65065e14			−0.0743169			−0.0371584	+	0.999309i	$0.511831\pi$
$618$		0			0
$619$	−1.48618e15	−	2.57414e15i	−0.657313	−	1.13850i	−0.981308	−	0.192441i	$-0.938360\pi$
$619$	−1.48618e15	−	2.57414e15i	−0.657313	−	1.13850i	0.323995	−	0.946059i	$-0.394974\pi$
$620$		0			0
$621$	3.42675e14	−	5.93530e14i	0.148894	−	0.257892i
$622$		0			0
$623$	−1.29153e15	+	3.75327e15i	−0.551342	+	1.60223i
$624$		0			0
$625$	−6.96942e14	+	1.20714e15i	−0.292319	+	0.506310i
$626$		0			0
$627$	2.02609e14	+	3.50929e14i	0.0835000	+	0.144626i
$628$		0			0
$629$	1.53915e14			0.0623309
$630$		0			0
$631$	4.28671e15			1.70594			0.852968	−	0.521964i	$-0.174800\pi$
$631$	4.28671e15			1.70594			0.852968	+	0.521964i	$0.174800\pi$
$632$		0			0
$633$	−8.00290e14	−	1.38614e15i	−0.312987	−	0.542110i
$634$		0			0
$635$	1.24686e14	−	2.15962e14i	0.0479250	−	0.0830085i
$636$		0			0
$637$	−2.01965e15	+	8.16914e14i	−0.762973	+	0.308610i
$638$		0			0
$639$	3.86961e14	−	6.70237e14i	0.143685	−	0.248871i
$640$		0			0
$641$	−1.32146e15	−	2.28884e15i	−0.482321	−	0.835404i	0.517473	−	0.855699i	$-0.326873\pi$
$641$	−1.32146e15	−	2.28884e15i	−0.482321	−	0.835404i	−0.999794	+	0.0202954i	$0.993539\pi$
$642$		0			0
$643$	−2.07428e15			−0.744228			−0.372114	−	0.928187i	$-0.621367\pi$
$643$	−2.07428e15			−0.744228			−0.372114	+	0.928187i	$0.621367\pi$
$644$		0			0
$645$	7.17700e14			0.253142
$646$		0			0
$647$	−1.01990e15	−	1.76651e15i	−0.353657	−	0.612552i	0.633230	−	0.773964i	$-0.281729\pi$
$647$	−1.01990e15	−	1.76651e15i	−0.353657	−	0.612552i	−0.986887	+	0.161412i	$0.948395\pi$
$648$		0			0
$649$	−1.92195e15	+	3.32891e15i	−0.655233	+	1.13490i
$650$		0			0
$651$	−5.66459e14	+	1.64617e15i	−0.189877	+	0.551796i
$652$		0			0
$653$	−2.93480e15	+	5.08322e15i	−0.967289	+	1.67539i	−0.263953	+	0.964536i	$0.585026\pi$
$653$	−2.93480e15	+	5.08322e15i	−0.967289	+	1.67539i	−0.703336	+	0.710858i	$0.748307\pi$
$654$		0			0
$655$	8.10133e13	+	1.40319e14i	0.0262561	+	0.0454768i
$656$		0			0
$657$	8.73155e14			0.278280
$658$		0			0
$659$	4.41751e15			1.38455			0.692274	−	0.721634i	$-0.256609\pi$
$659$	4.41751e15			1.38455			0.692274	+	0.721634i	$0.256609\pi$
$660$		0			0
$661$	−5.04436e13	−	8.73709e13i	−0.0155488	−	0.0269314i	0.858146	−	0.513405i	$-0.171616\pi$
$661$	−5.04436e13	−	8.73709e13i	−0.0155488	−	0.0269314i	−0.873695	+	0.486474i	$0.838283\pi$
$662$		0			0
$663$	5.94432e13	−	1.02959e14i	0.0180209	−	0.0312132i
$664$		0			0
$665$	−2.46590e14	−	2.83557e14i	−0.0735286	−	0.0845514i
$666$		0			0
$667$	−3.94600e15	+	6.83467e15i	−1.15735	+	2.00459i
$668$		0			0
$669$	8.90938e13	+	1.54315e14i	0.0257042	+	0.0445209i
$670$		0			0
$671$	−6.36499e15			−1.80644
$672$		0			0
$673$	2.37415e15			0.662866			0.331433	−	0.943479i	$-0.392468\pi$
$673$	2.37415e15			0.662866			0.331433	+	0.943479i	$0.392468\pi$
$674$		0			0
$675$	2.99340e14	+	5.18472e14i	0.0822231	+	0.142415i
$676$		0			0
$677$	−1.65318e15	+	2.86340e15i	−0.446769	+	0.773827i	−0.998174	−	0.0604114i	$-0.980759\pi$
$677$	−1.65318e15	+	2.86340e15i	−0.446769	+	0.773827i	0.551405	+	0.834238i	$0.314092\pi$
$678$		0			0
$679$	2.89519e15	−	5.63357e14i	0.769826	−	0.149796i
$680$		0			0
$681$	1.42375e14	−	2.46601e14i	0.0372498	−	0.0645186i
$682$		0			0
$683$	1.56905e15	+	2.71767e15i	0.403946	+	0.699654i	0.994198	−	0.107564i	$-0.0343052\pi$
$683$	1.56905e15	+	2.71767e15i	0.403946	+	0.699654i	−0.590253	+	0.807219i	$0.700972\pi$
$684$		0			0
$685$	−9.18170e14			−0.232608
$686$		0			0
$687$	3.46222e15			0.863163
$688$		0			0
$689$	−2.18103e14	−	3.77766e14i	−0.0535127	−	0.0926867i
$690$		0			0
$691$	3.74263e15	−	6.48243e15i	0.903749	−	1.56534i	0.0811608	−	0.996701i	$-0.474137\pi$
$691$	3.74263e15	−	6.48243e15i	0.903749	−	1.56534i	0.822588	−	0.568638i	$-0.192529\pi$
$692$		0			0
$693$	−1.35567e15	+	2.63791e14i	−0.322196	+	0.0626942i
$694$		0			0
$695$	−4.72251e14	+	8.17963e14i	−0.110473	+	0.191345i
$696$		0			0
$697$	2.38383e14	+	4.12892e14i	0.0548903	+	0.0950727i
$698$		0			0
$699$	3.48065e15			0.788925
$700$		0			0
$701$	−4.02232e15			−0.897487			−0.448743	−	0.893661i	$-0.648128\pi$
$701$	−4.02232e15			−0.897487			−0.448743	+	0.893661i	$0.648128\pi$
$702$		0			0
$703$	−5.49461e14	−	9.51695e14i	−0.120693	−	0.209047i
$704$		0			0
$705$	−2.90850e13	+	5.03768e13i	−0.00628969	+	0.0108941i
$706$		0			0
$707$	8.22630e14	+	9.45952e14i	0.175145	+	0.201401i
$708$		0			0
$709$	−6.43811e14	+	1.11511e15i	−0.134960	+	0.233757i	−0.925582	−	0.378547i	$-0.876424\pi$
$709$	−6.43811e14	+	1.11511e15i	−0.134960	+	0.233757i	0.790622	+	0.612304i	$0.209757\pi$
$710$		0			0
$711$	−8.58606e14	−	1.48715e15i	−0.177219	−	0.306953i
$712$		0			0
$713$	−7.69524e15			−1.56397
$714$		0			0
$715$	1.54475e15			0.309153
$716$		0			0
$717$	1.21681e15	+	2.10758e15i	0.239810	+	0.415364i
$718$		0			0
$719$	−9.68896e14	+	1.67818e15i	−0.188048	+	0.325708i	−0.944599	−	0.328226i	$-0.893549\pi$
$719$	−9.68896e14	+	1.67818e15i	−0.188048	+	0.325708i	0.756552	+	0.653934i	$0.226883\pi$
$720$		0			0
$721$	−2.16312e15	+	6.28615e15i	−0.413462	+	1.20155i
$722$		0			0
$723$	1.60114e15	−	2.77326e15i	0.301418	−	0.522072i
$724$		0			0
$725$	−3.44699e15	−	5.97036e15i	−0.639118	−	1.10698i
$726$		0			0
$727$	8.16531e15			1.49119			0.745596	−	0.666398i	$-0.232165\pi$
$727$	8.16531e15			1.49119			0.745596	+	0.666398i	$0.232165\pi$
$728$		0			0
$729$	2.05891e14			0.0370370
$730$		0			0
$731$	−2.46007e14	−	4.26096e14i	−0.0435914	−	0.0755026i
$732$		0			0
$733$	4.09691e15	−	7.09605e15i	0.715129	−	1.23864i	−0.247781	−	0.968816i	$-0.579701\pi$
$733$	4.09691e15	−	7.09605e15i	0.715129	−	1.23864i	0.962910	−	0.269824i	$-0.0869655\pi$
$734$		0			0
$735$	1.18732e15	−	4.80249e14i	0.204167	−	0.0825822i
$736$		0			0
$737$	−4.54234e15	+	7.86757e15i	−0.769500	+	1.33281i
$738$		0			0
$739$	−2.53817e14	−	4.39624e14i	−0.0423620	−	0.0733731i	0.844067	−	0.536238i	$-0.180155\pi$
$739$	−2.53817e14	−	4.39624e14i	−0.0423620	−	0.0733731i	−0.886429	+	0.462865i	$0.846822\pi$
$740$		0			0
$741$	−8.48823e14			−0.139578
$742$		0			0
$743$	7.56283e15			1.22531			0.612655	−	0.790350i	$-0.290101\pi$
$743$	7.56283e15			1.22531			0.612655	+	0.790350i	$0.290101\pi$
$744$		0			0
$745$	4.99506e14	+	8.65169e14i	0.0797409	+	0.138115i
$746$		0			0
$747$	−7.19742e13	+	1.24663e14i	−0.0113217	+	0.0196098i
$748$		0			0
$749$	−3.81357e15	+	1.10825e16i	−0.591128	+	1.71785i
$750$		0			0
$751$	−1.98511e15	+	3.43832e15i	−0.303226	+	0.525202i	−0.976865	−	0.213858i	$-0.931397\pi$
$751$	−1.98511e15	+	3.43832e15i	−0.303226	+	0.525202i	0.673639	+	0.739061i	$0.264730\pi$
$752$		0			0
$753$	2.73414e15	+	4.73567e15i	0.411575	+	0.712868i
$754$		0			0
$755$	1.22450e15			0.181657
$756$		0			0
$757$	5.28767e15			0.773103			0.386551	−	0.922268i	$-0.373666\pi$
$757$	5.28767e15			0.773103			0.386551	+	0.922268i	$0.373666\pi$
$758$		0			0
$759$	−3.05240e15	−	5.28690e15i	−0.439857	−	0.761854i
$760$		0			0
$761$	7.14612e14	−	1.23774e15i	0.101497	−	0.175798i	−0.810804	−	0.585317i	$-0.800970\pi$
$761$	7.14612e14	−	1.23774e15i	0.101497	−	0.175798i	0.912302	+	0.409519i	$0.134303\pi$
$762$		0			0
$763$	1.37560e15	+	1.58182e15i	0.192579	+	0.221449i
$764$		0			0
$765$	−3.49456e13	+	6.05275e13i	−0.00482230	+	0.00835247i
$766$		0			0
$767$	−4.02597e15	−	6.97319e15i	−0.547642	−	0.948543i
$768$		0			0
$769$	−5.17246e15			−0.693589			−0.346795	−	0.937941i	$-0.612730\pi$
$769$	−5.17246e15			−0.693589			−0.346795	+	0.937941i	$0.612730\pi$
$770$		0			0
$771$	3.17358e15			0.419518
$772$		0			0
$773$	−6.71899e15	−	1.16376e16i	−0.875622	−	1.51662i	−0.856098	−	0.516813i	$-0.827118\pi$
$773$	−6.71899e15	−	1.16376e16i	−0.875622	−	1.51662i	−0.0195238	−	0.999809i	$-0.506215\pi$
$774$		0			0
$775$	3.36105e15	−	5.82151e15i	0.431833	−	0.747956i
$776$		0			0
$777$	3.67648e15	−	7.15384e14i	0.465711	−	0.0906200i
$778$		0			0
$779$	1.70201e15	−	2.94796e15i	0.212571	−	0.368185i
$780$		0			0
$781$	−3.44688e15	−	5.97018e15i	−0.424469	−	0.735202i
$782$		0			0
$783$	−2.37090e15			−0.287888
$784$		0			0
$785$	3.39615e15			0.406634
$786$		0			0
$787$	−4.09229e14	−	7.08805e14i	−0.0483176	−	0.0836885i	0.840855	−	0.541260i	$-0.182053\pi$
$787$	−4.09229e14	−	7.08805e14i	−0.0483176	−	0.0836885i	−0.889173	+	0.457572i	$0.848719\pi$
$788$		0			0
$789$	2.77956e15	−	4.81433e15i	0.323632	−	0.560547i
$790$		0			0
$791$	3.08839e15	−	6.00952e14i	0.354619	−	0.0690031i
$792$		0			0
$793$	6.66649e15	−	1.15467e16i	0.754909	−	1.30754i
$794$		0			0
$795$	1.28219e14	+	2.22082e14i	0.0143197	+	0.0248024i
$796$		0			0
$797$	−5.45305e15			−0.600646			−0.300323	−	0.953838i	$-0.597094\pi$
$797$	−5.45305e15			−0.600646			−0.300323	+	0.953838i	$0.597094\pi$
$798$		0			0
$799$	3.98780e13			0.00433237
$800$		0			0
$801$	−2.63545e15	−	4.56473e15i	−0.282407	−	0.489143i
$802$		0			0
$803$	3.88884e15	−	6.73568e15i	0.411041	−	0.711944i
$804$		0			0
$805$	3.71499e15	+	4.27192e15i	0.387330	+	0.445395i
$806$		0			0
$807$	−1.67603e15	+	2.90298e15i	−0.172377	+	0.298565i
$808$		0			0
$809$	−1.18150e15	−	2.04641e15i	−0.119871	−	0.207623i	0.799845	−	0.600206i	$-0.204915\pi$
$809$	−1.18150e15	−	2.04641e15i	−0.119871	−	0.207623i	−0.919717	+	0.392583i	$0.871582\pi$
$810$		0			0
$811$	−1.42723e15			−0.142849			−0.0714247	−	0.997446i	$-0.522755\pi$
$811$	−1.42723e15			−0.142849			−0.0714247	+	0.997446i	$0.522755\pi$
$812$		0			0
$813$	2.76984e14			0.0273500
$814$		0			0
$815$	2.35647e15	+	4.08152e15i	0.229559	+	0.397608i
$816$		0			0
$817$	−1.75644e15	+	3.04224e15i	−0.168815	+	0.292396i
$818$		0			0
$819$	9.41341e14	−	2.73560e15i	0.0892659	−	0.259412i
$820$		0			0
$821$	6.71415e15	−	1.16293e16i	0.628209	−	1.08809i	−0.359702	−	0.933067i	$-0.617122\pi$
$821$	6.71415e15	−	1.16293e16i	0.628209	−	1.08809i	0.987911	−	0.155022i	$-0.0495449\pi$
$822$		0			0
$823$	−5.39480e15	−	9.34408e15i	−0.498054	−	0.862655i	0.501943	−	0.864901i	$-0.332619\pi$
$823$	−5.39480e15	−	9.34408e15i	−0.498054	−	0.862655i	−0.999997	+	0.00224537i	$0.999285\pi$
$824$		0			0
$825$	5.33278e15			0.485799
$826$		0			0
$827$	−1.09410e16			−0.983505			−0.491753	−	0.870735i	$-0.663644\pi$
$827$	−1.09410e16			−0.983505			−0.491753	+	0.870735i	$0.663644\pi$
$828$		0			0
$829$	−1.00194e15	−	1.73541e15i	−0.0888773	−	0.153940i	0.818160	−	0.574991i	$-0.194995\pi$
$829$	−1.00194e15	−	1.73541e15i	−0.0888773	−	0.153940i	−0.907037	+	0.421051i	$0.861661\pi$
$830$		0			0
$831$	5.55467e15	−	9.62096e15i	0.486242	−	0.842195i
$832$		0			0
$833$	−6.92099e14	−	5.40290e14i	−0.0597890	−	0.0466745i
$834$		0			0
$835$	1.82367e15	−	3.15869e15i	0.155478	−	0.269297i
$836$		0			0
$837$	−1.15589e15	−	2.00207e15i	−0.0972585	−	0.168457i
$838$		0			0
$839$	−2.04104e16			−1.69497			−0.847483	−	0.530822i	$-0.821883\pi$
$839$	−2.04104e16			−1.69497			−0.847483	+	0.530822i	$0.821883\pi$
$840$		0			0
$841$	1.51011e16			1.23774
$842$		0			0
$843$	4.76062e15	+	8.24563e15i	0.385134	+	0.667072i
$844$		0			0
$845$	7.70618e14	−	1.33475e15i	0.0615357	−	0.106583i
$846$		0			0
$847$	1.25209e14	−	3.63866e14i	0.00986911	−	0.0286802i
$848$		0			0
$849$	−4.69400e14	+	8.13025e14i	−0.0365217	+	0.0632574i
$850$		0			0
$851$	8.27790e15	+	1.43377e16i	0.635781	+	1.10120i
$852$		0			0
$853$	8.92536e15			0.676715			0.338358	−	0.941018i	$-0.390129\pi$
$853$	8.92536e15			0.676715			0.338358	+	0.941018i	$0.390129\pi$
$854$		0			0
$855$	4.99008e14			0.0373503
$856$		0			0
$857$	−3.45345e15	−	5.98155e15i	−0.255187	−	0.441997i	0.709759	−	0.704444i	$-0.248804\pi$
$857$	−3.45345e15	−	5.98155e15i	−0.255187	−	0.441997i	−0.964946	+	0.262447i	$0.915470\pi$
$858$		0			0
$859$	9.64574e14	−	1.67069e15i	0.0703677	−	0.121880i	−0.828695	−	0.559701i	$-0.810916\pi$
$859$	9.64574e14	−	1.67069e15i	0.0703677	−	0.121880i	0.899062	+	0.437820i	$0.144249\pi$
$860$		0			0
$861$	7.61320e15	+	8.75451e15i	0.548340	+	0.630543i
$862$		0			0
$863$	−1.08319e16	+	1.87614e16i	−0.770273	+	1.33415i	0.167140	+	0.985933i	$0.446547\pi$
$863$	−1.08319e16	+	1.87614e16i	−0.770273	+	1.33415i	−0.937413	+	0.348219i	$0.886787\pi$
$864$		0			0
$865$	−3.04888e15	−	5.28081e15i	−0.214068	−	0.370776i
$866$		0			0
$867$	−8.28016e15			−0.574029
$868$		0			0
$869$	−1.52962e16			−1.04707
$870$		0			0
$871$	−9.51501e15	−	1.64805e16i	−0.643146	−	1.11396i
$872$		0			0
$873$	−1.95835e15	+	3.39196e15i	−0.130711	+	0.226398i
$874$		0			0
$875$	−1.05353e16	+	2.05001e15i	−0.694391	+	0.135117i
$876$		0			0
$877$	−9.84189e15	+	1.70467e16i	−0.640591	+	1.10954i	0.344710	+	0.938709i	$0.387977\pi$
$877$	−9.84189e15	+	1.70467e16i	−0.640591	+	1.10954i	−0.985301	+	0.170827i	$0.945356\pi$
$878$		0			0
$879$	1.18904e15	+	2.05948e15i	0.0764289	+	0.132379i
$880$		0			0
$881$	1.36545e16			0.866780			0.433390	−	0.901206i	$-0.357317\pi$
$881$	1.36545e16			0.866780			0.433390	+	0.901206i	$0.357317\pi$
$882$		0			0
$883$	1.49342e15			0.0936266			0.0468133	−	0.998904i	$-0.485093\pi$
$883$	1.49342e15			0.0936266			0.0468133	+	0.998904i	$0.485093\pi$
$884$		0			0
$885$	2.36680e15	+	4.09941e15i	0.146546	+	0.253825i
$886$		0			0
$887$	9.38670e15	−	1.62582e16i	0.574028	−	0.994246i	−0.422118	−	0.906541i	$-0.638713\pi$
$887$	9.38670e15	−	1.62582e16i	0.574028	−	0.994246i	0.996146	−	0.0877050i	$-0.0279533\pi$
$888$		0			0
$889$	−4.08350e15	+	7.94583e14i	−0.246645	+	0.0479932i
$890$		0			0
$891$	9.16994e14	−	1.58828e15i	0.0547065	−	0.0947545i
$892$		0			0
$893$	−1.42360e14	−	2.46575e14i	−0.00838891	−	0.0145300i
$894$		0			0
$895$	6.84088e15			0.398185
$896$		0			0
$897$	1.27879e16			0.735261
$898$		0			0
$899$	1.33105e16	+	2.30544e16i	0.755988	+	1.30941i
$900$		0			0
$901$	8.78995e13	−	1.52246e14i	0.00493174	−	0.00854202i
$902$		0			0
$903$	−7.85667e15	−	9.03448e15i	−0.435467	−	0.500749i
$904$		0			0
$905$	−5.83750e15	+	1.01109e16i	−0.319639	+	0.553631i
$906$		0			0
$907$	1.19397e16	+	2.06802e16i	0.645882	+	1.11870i	0.984097	+	0.177632i	$0.0568437\pi$
$907$	1.19397e16	+	2.06802e16i	0.645882	+	1.11870i	−0.338215	+	0.941069i	$0.609823\pi$
$908$		0			0
$909$	−1.66470e15			−0.0889683
$910$		0			0
$911$	2.54913e16			1.34599			0.672995	−	0.739647i	$-0.265008\pi$
$911$	2.54913e16			1.34599			0.672995	+	0.739647i	$0.265008\pi$
$912$		0			0
$913$	6.41115e14	+	1.11044e15i	0.0334462	+	0.0579305i
$914$		0			0
$915$	−3.91911e15	+	6.78810e15i	−0.202009	+	0.349890i
$916$		0			0
$917$	8.79499e14	−	2.55588e15i	0.0447924	−	0.130169i
$918$		0			0
$919$	−5.65752e15	+	9.79911e15i	−0.284702	+	0.493119i	−0.972537	−	0.232749i	$-0.925228\pi$
$919$	−5.65752e15	+	9.79911e15i	−0.284702	+	0.493119i	0.687835	+	0.725867i	$0.258561\pi$
$920$		0			0
$921$	−1.17713e15	−	2.03885e15i	−0.0585325	−	0.101381i
$922$		0			0
$923$	1.44406e16			0.709539
$924$		0			0
$925$	−1.44621e16			−0.702188
$926$		0			0
$927$	−4.41396e15	−	7.64521e15i	−0.211783	−	0.366818i
$928$		0			0
$929$	2.74021e15	−	4.74619e15i	0.129927	−	0.225039i	−0.793721	−	0.608281i	$-0.791859\pi$
$929$	2.74021e15	−	4.74619e15i	0.129927	−	0.225039i	0.923648	+	0.383242i	$0.125193\pi$
$930$		0			0
$931$	−8.70020e14	+	6.20820e15i	−0.0407668	+	0.290899i
$932$		0			0
$933$	−3.98645e15	+	6.90474e15i	−0.184603	+	0.319741i
$934$		0			0
$935$	3.11280e14	+	5.39153e14i	0.0142458	+	0.0246745i
$936$		0			0
$937$	1.11059e16			0.502328			0.251164	−	0.967945i	$-0.419187\pi$
$937$	1.11059e16			0.502328			0.251164	+	0.967945i	$0.419187\pi$
$938$		0			0
$939$	−2.26130e16			−1.01087
$940$		0			0
$941$	−5.49099e15	−	9.51067e15i	−0.242609	−	0.420212i	0.718847	−	0.695168i	$-0.244670\pi$
$941$	−5.49099e15	−	9.51067e15i	−0.242609	−	0.420212i	−0.961457	+	0.274956i	$0.911337\pi$
$942$		0			0
$943$	−2.56415e16	+	4.44124e16i	−1.11977	+	1.93950i
$944$		0			0
$945$	−5.53397e14	+	1.60821e15i	−0.0238870	+	0.0694172i
$946$		0			0
$947$	−2.51676e14	+	4.35915e14i	−0.0107378	+	0.0185985i	−0.871344	−	0.490672i	$-0.836751\pi$
$947$	−2.51676e14	+	4.35915e14i	−0.0107378	+	0.0185985i	0.860607	+	0.509270i	$0.170085\pi$
$948$		0			0
$949$	8.14610e15	+	1.41095e16i	0.343547	+	0.595040i
$950$		0			0
$951$	−9.14841e15			−0.381376
$952$		0			0
$953$	1.91219e16			0.787989			0.393995	−	0.919113i	$-0.371093\pi$
$953$	1.91219e16			0.787989			0.393995	+	0.919113i	$0.371093\pi$
$954$		0			0
$955$	−2.26256e14	−	3.91886e14i	−0.00921681	−	0.0159640i
$956$		0			0
$957$	−1.05595e16	+	1.82895e16i	−0.425232	+	0.736524i
$958$		0			0
$959$	1.00512e16	+	1.15580e16i	0.400144	+	0.460130i
$960$		0			0
$961$	−2.74359e14	+	4.75204e14i	−0.0107979	+	0.0187026i
$962$		0			0
$963$	−7.78181e15	−	1.34785e16i	−0.302786	−	0.524441i
$964$		0			0
$965$	8.61396e14			0.0331362
$966$		0			0
$967$	−4.64208e16			−1.76550			−0.882749	−	0.469845i	$-0.844310\pi$
$967$	−4.64208e16			−1.76550			−0.882749	+	0.469845i	$0.844310\pi$
$968$		0			0
$969$	−1.71045e14	−	2.96259e14i	−0.00643177	−	0.0111402i
$970$		0			0
$971$	−1.39295e16	+	2.41265e16i	−0.517879	+	0.896993i	0.481905	+	0.876223i	$0.339945\pi$
$971$	−1.39295e16	+	2.41265e16i	−0.517879	+	0.896993i	−0.999784	+	0.0207694i	$0.993388\pi$
$972$		0			0
$973$	1.54663e16	−	3.00950e15i	0.568548	−	0.110630i
$974$		0			0
$975$	−5.58538e15	+	9.67417e15i	−0.203015	+	0.351632i
$976$		0			0
$977$	−4.76702e15	−	8.25672e15i	−0.171327	−	0.296748i	0.767557	−	0.640981i	$-0.221472\pi$
$977$	−4.76702e15	−	8.25672e15i	−0.171327	−	0.296748i	−0.938884	+	0.344233i	$0.888139\pi$
$978$		0			0
$979$	−4.69508e16			−1.66855
$980$		0			0
$981$	−2.78372e15			−0.0978241
$982$		0			0
$983$	1.88383e16	+	3.26290e16i	0.654634	+	1.13386i	0.981985	+	0.188957i	$0.0605105\pi$
$983$	1.88383e16	+	3.26290e16i	0.654634	+	1.13386i	−0.327352	+	0.944903i	$0.606156\pi$
$984$		0			0
$985$	−1.10192e15	+	1.90857e15i	−0.0378660	+	0.0655858i
$986$		0			0
$987$	9.52542e14	−	1.85349e14i	0.0323698	−	0.00629864i
$988$		0			0
$989$	2.64615e16	−	4.58327e16i	0.889273	−	1.54027i
$990$		0			0
$991$	−2.07442e16	−	3.59301e16i	−0.689434	−	1.19413i	−0.972021	−	0.234893i	$-0.924526\pi$
$991$	−2.07442e16	−	3.59301e16i	−0.689434	−	1.19413i	0.282588	−	0.959242i	$-0.408807\pi$
$992$		0			0
$993$	2.43203e16			0.799372
$994$		0			0
$995$	1.13395e16			0.368611
$996$		0			0
$997$	1.88055e16	+	3.25721e16i	0.604591	+	1.04718i	0.992116	+	0.125323i	$0.0399969\pi$
$997$	1.88055e16	+	3.25721e16i	0.604591	+	1.04718i	−0.387525	+	0.921859i	$0.626670\pi$
$998$		0			0
$999$	−2.48683e15	+	4.30731e15i	−0.0790743	+	0.136961i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.i.b.37.4	yes	16
3.2	odd	2		252.12.k.d.37.5		16
7.4	even	3	inner	84.12.i.b.25.4	✓	16
21.11	odd	6		252.12.k.d.109.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.b.25.4	✓	16	7.4	even	3	inner
84.12.i.b.37.4	yes	16	1.1	even	1	trivial
252.12.k.d.37.5		16	3.2	odd	2
252.12.k.d.109.5		16	21.11	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	84.i (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-121.500 - 210.444i) q^{3} +(-1332.77 + 2308.42i) q^{5} +(43648.5 - 8493.30i) q^{7} +(-29524.5 + 51137.9i) q^{9} +O(q^{10})\)	\(q+(-121.500 - 210.444i) q^{3} +(-1332.77 + 2308.42i) q^{5} +(43648.5 - 8493.30i) q^{7} +(-29524.5 + 51137.9i) q^{9} +(262991. + 455515. i) q^{11} -1.10180e6 q^{13} +647726. q^{15} +(-222021. - 384552. i) q^{17} +(-1.58519e6 + 2.74562e6i) q^{19} +(-7.09066e6 - 8.15363e6i) q^{21} +(2.38816e7 - 4.13641e7i) q^{23} +(2.08615e7 + 3.61332e7i) q^{25} +1.43489e7 q^{27} -1.65232e8 q^{29} +(-8.05562e7 - 1.39527e8i) q^{31} +(6.39069e7 - 1.10690e8i) q^{33} +(-3.85672e7 + 1.12079e8i) q^{35} +(-1.73311e8 + 3.00184e8i) q^{37} +(1.33868e8 + 2.31866e8i) q^{39} -1.07369e9 q^{41} +1.10803e9 q^{43} +(-7.86987e7 - 1.36310e8i) q^{45} +(-4.49033e7 + 7.77748e7i) q^{47} +(1.83305e9 - 7.41439e8i) q^{49} +(-5.39512e7 + 9.34462e7i) q^{51} +(1.97953e8 + 3.42864e8i) q^{53} -1.40203e9 q^{55} +7.70401e8 q^{57} +(3.65401e9 + 6.32894e9i) q^{59} +(-6.05057e9 + 1.04799e10i) q^{61} +(-8.54370e8 + 2.48285e9i) q^{63} +(1.46844e9 - 2.54341e9i) q^{65} +(8.63592e9 + 1.49578e10i) q^{67} -1.16064e10 q^{69} -1.31065e10 q^{71} +(-7.39348e9 - 1.28059e10i) q^{73} +(5.06935e9 - 8.78037e9i) q^{75} +(1.53480e10 + 1.76489e10i) q^{77} +(-1.45406e10 + 2.51850e10i) q^{79} +(-1.74339e9 - 3.01964e9i) q^{81} +2.43778e9 q^{83} +1.18361e9 q^{85} +(2.00757e10 + 3.47721e10i) q^{87} +(-4.46315e10 + 7.73040e10i) q^{89} +(-4.80917e10 + 9.35787e9i) q^{91} +(-1.95752e10 + 3.39052e10i) q^{93} +(-4.22537e9 - 7.31856e9i) q^{95} +6.63296e10 q^{97} -3.10588e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9}+O(q^{10}) \) 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9	\( 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9} - 222796 q^{11} + 2703176 q^{13} + 1047816 q^{15} + 5114600 q^{17} + 6910556 q^{19} - 18340668 q^{21} - 51387712 q^{23} - 191456372 q^{25} + 229582512 q^{27} + 118854616 q^{29} + 164659160 q^{31} - 54139428 q^{33} + 55239344 q^{35} + 75658364 q^{37} - 328435884 q^{39} - 1815568608 q^{41} + 10754408 q^{43} - 127309644 q^{45} - 1034359464 q^{47} + 4123496848 q^{49} + 1242847800 q^{51} - 665159988 q^{53} - 1264543896 q^{55} - 3358530216 q^{57} + 1040514580 q^{59} - 14391208024 q^{61} + 1474099236 q^{63} - 20938150200 q^{65} - 33307097284 q^{67} + 24974428032 q^{69} + 65848902896 q^{71} + 17709749204 q^{73} - 46523898396 q^{75} + 8594484604 q^{77} - 26626784032 q^{79} - 27894275208 q^{81} - 210306955048 q^{83} - 25867402032 q^{85} - 14440835844 q^{87} - 55951560072 q^{89} + 66078280292 q^{91} + 40012175880 q^{93} + 106810047392 q^{95} - 156216030712 q^{97} + 26311762008 q^{99}+O(q^{100}) \) 16 * q - 1944 * q^3 - 2156 * q^5 + 50512 * q^7 - 472392 * q^9 - 222796 * q^11 + 2703176 * q^13 + 1047816 * q^15 + 5114600 * q^17 + 6910556 * q^19 - 18340668 * q^21 - 51387712 * q^23 - 191456372 * q^25 + 229582512 * q^27 + 118854616 * q^29 + 164659160 * q^31 - 54139428 * q^33 + 55239344 * q^35 + 75658364 * q^37 - 328435884 * q^39 - 1815568608 * q^41 + 10754408 * q^43 - 127309644 * q^45 - 1034359464 * q^47 + 4123496848 * q^49 + 1242847800 * q^51 - 665159988 * q^53 - 1264543896 * q^55 - 3358530216 * q^57 + 1040514580 * q^59 - 14391208024 * q^61 + 1474099236 * q^63 - 20938150200 * q^65 - 33307097284 * q^67 + 24974428032 * q^69 + 65848902896 * q^71 + 17709749204 * q^73 - 46523898396 * q^75 + 8594484604 * q^77 - 26626784032 * q^79 - 27894275208 * q^81 - 210306955048 * q^83 - 25867402032 * q^85 - 14440835844 * q^87 - 55951560072 * q^89 + 66078280292 * q^91 + 40012175880 * q^93 + 106810047392 * q^95 - 156216030712 * q^97 + 26311762008 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more