LMFDB - Embedded newform 84.12.i.b.25.3

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			25.3
Root			$2410.13 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.25
Dual form			84.12.i.b.37.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-121.500 + 210.444i) q^{3} +(-1339.81 - 2320.63i) q^{5} +(-28039.7 - 34512.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})$	\(q+(-121.500 + 210.444i) q^{3} +(-1339.81 - 2320.63i) q^{5} +(-28039.7 - 34512.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-239334. + 414539. i) q^{11} -1.17059e6 q^{13} +651149. q^{15} +(-2.40839e6 + 4.17145e6i) q^{17} +(-4.55281e6 - 7.88570e6i) q^{19} +(1.06697e7 - 1.70754e6i) q^{21} +(-4.21644e6 - 7.30309e6i) q^{23} +(2.08239e7 - 3.60680e7i) q^{25} +1.43489e7 q^{27} +3.65821e7 q^{29} +(-1.20947e7 + 2.09487e7i) q^{31} +(-5.81582e7 - 1.00733e8i) q^{33} +(-4.25223e7 + 1.11310e8i) q^{35} +(-1.66789e8 - 2.88888e8i) q^{37} +(1.42226e8 - 2.46343e8i) q^{39} +5.86887e7 q^{41} +8.57155e8 q^{43} +(-7.91147e7 + 1.37031e8i) q^{45} +(4.75883e8 + 8.24254e8i) q^{47} +(-4.04880e8 + 1.93543e9i) q^{49} +(-5.85239e8 - 1.01366e9i) q^{51} +(-9.09651e8 + 1.57556e9i) q^{53} +1.28265e9 q^{55} +2.21267e9 q^{57} +(8.43761e8 - 1.46144e9i) q^{59} +(3.82941e9 + 6.63273e9i) q^{61} +(-9.37033e8 + 2.45285e9i) q^{63} +(1.56837e9 + 2.71649e9i) q^{65} +(-2.92346e9 + 5.06358e9i) q^{67} +2.04919e9 q^{69} +1.03360e10 q^{71} +(4.61314e9 - 7.99018e9i) q^{73} +(5.06020e9 + 8.76452e9i) q^{75} +(2.10176e10 - 3.36355e9i) q^{77} +(9.94865e9 + 1.72316e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +4.78020e10 q^{83} +1.29072e10 q^{85} +(-4.44472e9 + 7.69849e9i) q^{87} +(-9.70762e9 - 1.68141e10i) q^{89} +(3.28229e10 + 4.03997e10i) q^{91} +(-2.93902e9 - 5.09054e9i) q^{93} +(-1.21998e10 + 2.11307e10i) q^{95} +8.32856e10 q^{97} +2.82649e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/84\mathbb{Z}\right)^\times$.

$n$	$29$	$43$	$73$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−1339.81	−	2320.63i	−0.191739	−	0.332101i	0.754088	−	0.656773i	$-0.228079\pi$
$5$	−1339.81	−	2320.63i	−0.191739	−	0.332101i	−0.945826	+	0.324673i	$0.894746\pi$
$6$		0			0
$7$	−28039.7	−	34512.4i	−0.630571	−	0.776132i
$7$	−28039.7	−	34512.4i	−0.630571	−	0.776132i
$8$		0			0
$9$	−29524.5	−	51137.9i	−0.166667	−	0.288675i
$10$		0			0
$11$	−239334.	+	414539.i	−0.448069	+	0.776079i	−0.998260	−	0.0589601i	$-0.981222\pi$
$11$	−239334.	+	414539.i	−0.448069	+	0.776079i	0.550191	+	0.835039i	$0.314555\pi$
$12$		0			0
$13$	−1.17059e6			−0.874410			−0.437205	−	0.899362i	$-0.644032\pi$
$13$	−1.17059e6			−0.874410			−0.437205	+	0.899362i	$0.644032\pi$
$14$		0			0
$15$	651149.			0.221401
$16$		0			0
$17$	−2.40839e6	+	4.17145e6i	−0.411394	+	0.712555i	−0.995042	−	0.0994514i	$-0.968291\pi$
$17$	−2.40839e6	+	4.17145e6i	−0.411394	+	0.712555i	0.583649	+	0.812006i	$0.301625\pi$
$18$		0			0
$19$	−4.55281e6	−	7.88570e6i	−0.421827	−	0.730626i	0.574291	−	0.818651i	$-0.305278\pi$
$19$	−4.55281e6	−	7.88570e6i	−0.421827	−	0.730626i	−0.996118	+	0.0880250i	$0.971944\pi$
$20$		0			0
$21$	1.06697e7	−	1.70754e6i	0.570096	−	0.0912354i
$22$		0			0
$23$	−4.21644e6	−	7.30309e6i	−0.136598	−	0.236594i	0.789609	−	0.613610i	$-0.210283\pi$
$23$	−4.21644e6	−	7.30309e6i	−0.136598	−	0.236594i	−0.926207	+	0.377016i	$0.876950\pi$
$24$		0			0
$25$	2.08239e7	−	3.60680e7i	0.426473	−	0.738672i
$26$		0			0
$27$	1.43489e7			0.192450
$28$		0			0
$29$	3.65821e7			0.331192			0.165596	−	0.986194i	$-0.447045\pi$
$29$	3.65821e7			0.331192			0.165596	+	0.986194i	$0.447045\pi$
$30$		0			0
$31$	−1.20947e7	+	2.09487e7i	−0.0758765	+	0.131422i	−0.901467	−	0.432848i	$-0.857509\pi$
$31$	−1.20947e7	+	2.09487e7i	−0.0758765	+	0.131422i	0.825591	+	0.564270i	$0.190842\pi$
$32$		0			0
$33$	−5.81582e7	−	1.00733e8i	−0.258693	−	0.448069i
$34$		0			0
$35$	−4.25223e7	+	1.11310e8i	−0.136849	+	0.358227i
$36$		0			0
$37$	−1.66789e8	−	2.88888e8i	−0.395420	−	0.684888i	0.597734	−	0.801694i	$-0.296068\pi$
$37$	−1.66789e8	−	2.88888e8i	−0.395420	−	0.684888i	−0.993155	+	0.116806i	$0.962734\pi$
$38$		0			0
$39$	1.42226e8	−	2.46343e8i	0.252420	−	0.437205i
$40$		0			0
$41$	5.86887e7			0.0791122			0.0395561	−	0.999217i	$-0.487406\pi$
$41$	5.86887e7			0.0791122			0.0395561	+	0.999217i	$0.487406\pi$
$42$		0			0
$43$	8.57155e8			0.889166			0.444583	−	0.895738i	$-0.353352\pi$
$43$	8.57155e8			0.889166			0.444583	+	0.895738i	$0.353352\pi$
$44$		0			0
$45$	−7.91147e7	+	1.37031e8i	−0.0639128	+	0.110700i
$46$		0			0
$47$	4.75883e8	+	8.24254e8i	0.302665	+	0.524231i	0.976739	−	0.214433i	$-0.0687904\pi$
$47$	4.75883e8	+	8.24254e8i	0.302665	+	0.524231i	−0.674074	+	0.738664i	$0.735457\pi$
$48$		0			0
$49$	−4.04880e8	+	1.93543e9i	−0.204761	+	0.978812i
$50$		0			0
$51$	−5.85239e8	−	1.01366e9i	−0.237518	−	0.411394i
$52$		0			0
$53$	−9.09651e8	+	1.57556e9i	−0.298784	+	0.517509i	−0.975858	−	0.218406i	$-0.929914\pi$
$53$	−9.09651e8	+	1.57556e9i	−0.298784	+	0.517509i	0.677074	+	0.735915i	$0.263248\pi$
$54$		0			0
$55$	1.28265e9			0.343649
$56$		0			0
$57$	2.21267e9			0.487084
$58$		0			0
$59$	8.43761e8	−	1.46144e9i	0.153650	−	0.266130i	−0.778916	−	0.627128i	$-0.784230\pi$
$59$	8.43761e8	−	1.46144e9i	0.153650	−	0.266130i	0.932567	+	0.360998i	$0.117564\pi$
$60$		0			0
$61$	3.82941e9	+	6.63273e9i	0.580521	+	1.00549i	0.995418	+	0.0956231i	$0.0304843\pi$
$61$	3.82941e9	+	6.63273e9i	0.580521	+	1.00549i	−0.414897	+	0.909868i	$0.636182\pi$
$62$		0			0
$63$	−9.37033e8	+	2.45285e9i	−0.118955	+	0.311385i
$64$		0			0
$65$	1.56837e9	+	2.71649e9i	0.167658	+	0.290392i
$66$		0			0
$67$	−2.92346e9	+	5.06358e9i	−0.264537	+	0.458191i	−0.967442	−	0.253093i	$-0.918552\pi$
$67$	−2.92346e9	+	5.06358e9i	−0.264537	+	0.458191i	0.702906	+	0.711283i	$0.251886\pi$
$68$		0			0
$69$	2.04919e9			0.157729
$70$		0			0
$71$	1.03360e10			0.679877			0.339939	−	0.940448i	$-0.389594\pi$
$71$	1.03360e10			0.679877			0.339939	+	0.940448i	$0.389594\pi$
$72$		0			0
$73$	4.61314e9	−	7.99018e9i	0.260448	−	0.451109i	−0.705913	−	0.708298i	$-0.749463\pi$
$73$	4.61314e9	−	7.99018e9i	0.260448	−	0.451109i	0.966361	+	0.257190i	$0.0827965\pi$
$74$		0			0
$75$	5.06020e9	+	8.76452e9i	0.246224	+	0.426473i
$76$		0			0
$77$	2.10176e10	−	3.36355e9i	0.884879	−	0.141612i
$78$		0			0
$79$	9.94865e9	+	1.72316e10i	0.363760	+	0.630051i	0.988576	−	0.150721i	$-0.0481594\pi$
$79$	9.94865e9	+	1.72316e10i	0.363760	+	0.630051i	−0.624816	+	0.780772i	$0.714826\pi$
$80$		0			0
$81$	−1.74339e9	+	3.01964e9i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	4.78020e10			1.33204			0.666018	−	0.745935i	$-0.267997\pi$
$83$	4.78020e10			1.33204			0.666018	+	0.745935i	$0.267997\pi$
$84$		0			0
$85$	1.29072e10			0.315520
$86$		0			0
$87$	−4.44472e9	+	7.69849e9i	−0.0956068	+	0.165596i
$88$		0			0
$89$	−9.70762e9	−	1.68141e10i	−0.184276	−	0.319175i	0.759057	−	0.651025i	$-0.225661\pi$
$89$	−9.70762e9	−	1.68141e10i	−0.184276	−	0.319175i	−0.943332	+	0.331850i	$0.892327\pi$
$90$		0			0
$91$	3.28229e10	+	4.03997e10i	0.551377	+	0.678657i
$92$		0			0
$93$	−2.93902e9	−	5.09054e9i	−0.0438073	−	0.0758765i
$94$		0			0
$95$	−1.21998e10	+	2.11307e10i	−0.161761	+	0.280178i
$96$		0			0
$97$	8.32856e10			0.984749			0.492374	−	0.870383i	$-0.336129\pi$
$97$	8.32856e10			0.984749			0.492374	+	0.870383i	$0.336129\pi$
$98$		0			0
$99$	2.82649e10			0.298713
$100$		0			0
$101$	−7.46928e10	+	1.29372e11i	−0.707149	+	1.22482i	0.258761	+	0.965941i	$0.416686\pi$
$101$	−7.46928e10	+	1.29372e11i	−0.707149	+	1.22482i	−0.965910	+	0.258877i	$0.916648\pi$
$102$		0			0
$103$	3.62922e10	+	6.28600e10i	0.308467	+	0.534281i	0.978027	−	0.208477i	$-0.0668507\pi$
$103$	3.62922e10	+	6.28600e10i	0.308467	+	0.534281i	−0.669560	+	0.742758i	$0.733517\pi$
$104$		0			0
$105$	−1.82580e10	−	2.24727e10i	−0.139609	−	0.171836i
$106$		0			0
$107$	1.26123e11	+	2.18451e11i	0.869326	+	1.50572i	0.862687	+	0.505738i	$0.168780\pi$
$107$	1.26123e11	+	2.18451e11i	0.869326	+	1.50572i	0.00663879	+	0.999978i	$0.497887\pi$
$108$		0			0
$109$	6.84739e10	−	1.18600e11i	0.426265	−	0.738312i	−0.570273	−	0.821455i	$-0.693162\pi$
$109$	6.84739e10	−	1.18600e11i	0.426265	−	0.738312i	0.996538	+	0.0831430i	$0.0264958\pi$
$110$		0			0
$111$	8.10597e10			0.456592
$112$		0			0
$113$	2.33958e11			1.19456			0.597279	−	0.802034i	$-0.296249\pi$
$113$	2.33958e11			1.19456			0.597279	+	0.802034i	$0.296249\pi$
$114$		0			0
$115$	−1.12985e10	+	1.95696e10i	−0.0523821	+	0.0907284i
$116$		0			0
$117$	3.45610e10	+	5.98614e10i	0.145735	+	0.252420i
$118$		0			0
$119$	2.11497e11	−	3.38470e10i	0.812449	−	0.130020i
$120$		0			0
$121$	2.80940e10	+	4.86603e10i	0.0984679	+	0.170551i
$122$		0			0
$123$	−7.13068e9	+	1.23507e10i	−0.0228377	+	0.0395561i
$124$		0			0
$125$	−2.42442e11			−0.710562
$126$		0			0
$127$	−2.05552e11			−0.552078			−0.276039	−	0.961146i	$-0.589022\pi$
$127$	−2.05552e11			−0.552078			−0.276039	+	0.961146i	$0.589022\pi$
$128$		0			0
$129$	−1.04144e11	+	1.80383e11i	−0.256680	+	0.444583i
$130$		0			0
$131$	−1.71371e10	−	2.96824e10i	−0.0388102	−	0.0672213i	0.845968	−	0.533234i	$-0.179023\pi$
$131$	−1.71371e10	−	2.96824e10i	−0.0388102	−	0.0672213i	−0.884778	+	0.466013i	$0.845690\pi$
$132$		0			0
$133$	−1.44495e11	+	3.78241e11i	−0.301070	+	0.788105i
$134$		0			0
$135$	−1.92249e10	−	3.32984e10i	−0.0369001	−	0.0639128i
$136$		0			0
$137$	7.70171e10	−	1.33397e11i	0.136340	−	0.236148i	−0.789768	−	0.613405i	$-0.789799\pi$
$137$	7.70171e10	−	1.33397e11i	0.136340	−	0.236148i	0.926109	+	0.377257i	$0.123133\pi$
$138$		0			0
$139$	−1.11480e11			−0.182228			−0.0911138	−	0.995840i	$-0.529043\pi$
$139$	−1.11480e11			−0.182228			−0.0911138	+	0.995840i	$0.529043\pi$
$140$		0			0
$141$	−2.31279e11			−0.349487
$142$		0			0
$143$	2.80161e11	−	4.85254e11i	0.391796	−	0.678611i
$144$		0			0
$145$	−4.90132e10	−	8.48933e10i	−0.0635022	−	0.109989i
$146$		0			0
$147$	−3.58107e11	−	3.20359e11i	−0.430297	−	0.384939i
$148$		0			0
$149$	−8.33795e11	−	1.44418e12i	−0.930112	−	1.61100i	−0.783127	−	0.621862i	$-0.786377\pi$
$149$	−8.33795e11	−	1.44418e12i	−0.930112	−	1.61100i	−0.146985	−	0.989139i	$-0.546957\pi$
$150$		0			0
$151$	3.71218e11	−	6.42968e11i	0.384818	−	0.666524i	−0.606926	−	0.794758i	$-0.707598\pi$
$151$	3.71218e11	−	6.42968e11i	0.384818	−	0.666524i	0.991744	+	0.128234i	$0.0409309\pi$
$152$		0			0
$153$	2.84426e11			0.274262
$154$		0			0
$155$	6.48188e10			0.0581938
$156$		0			0
$157$	−4.37190e11	+	7.57235e11i	−0.365782	+	0.633552i	−0.988901	−	0.148574i	$-0.952532\pi$
$157$	−4.37190e11	+	7.57235e11i	−0.365782	+	0.633552i	0.623120	+	0.782126i	$0.285865\pi$
$158$		0			0
$159$	−2.21045e11	−	3.82861e11i	−0.172503	−	0.298784i
$160$		0			0
$161$	−1.33819e11	+	3.50296e11i	−0.0974937	+	0.255207i
$162$		0			0
$163$	−5.70121e11	−	9.87478e11i	−0.388092	−	0.672196i	0.604101	−	0.796908i	$-0.293533\pi$
$163$	−5.70121e11	−	9.87478e11i	−0.388092	−	0.672196i	−0.992193	+	0.124712i	$0.960199\pi$
$164$		0			0
$165$	−1.55842e11	+	2.69927e11i	−0.0992028	+	0.171824i
$166$		0			0
$167$	1.67533e12			0.998068			0.499034	−	0.866582i	$-0.333688\pi$
$167$	1.67533e12			0.998068			0.499034	+	0.866582i	$0.333688\pi$
$168$		0			0
$169$	−4.21888e11			−0.235408
$170$		0			0
$171$	−2.68839e11	+	4.65643e11i	−0.140609	+	0.243542i
$172$		0			0
$173$	−1.84651e12	−	3.19825e12i	−0.905937	−	1.56913i	−0.819655	−	0.572858i	$-0.805835\pi$
$173$	−1.84651e12	−	3.19825e12i	−0.905937	−	1.56913i	−0.0862817	−	0.996271i	$-0.527498\pi$
$174$		0			0
$175$	−1.82869e12	+	2.92654e11i	−0.842228	+	0.134786i
$176$		0			0
$177$	2.05034e11	+	3.55129e11i	0.0887100	+	0.153650i
$178$		0			0
$179$	−8.10662e11	+	1.40411e12i	−0.329722	+	0.571096i	−0.982457	−	0.186491i	$-0.940288\pi$
$179$	−8.10662e11	+	1.40411e12i	−0.329722	+	0.571096i	0.652734	+	0.757587i	$0.273622\pi$
$180$		0			0
$181$	−3.72161e12			−1.42396			−0.711982	−	0.702198i	$-0.752202\pi$
$181$	−3.72161e12			−1.42396			−0.711982	+	0.702198i	$0.752202\pi$
$182$		0			0
$183$	−1.86109e12			−0.670328
$184$		0			0
$185$	−4.46934e11	+	7.74112e11i	−0.151635	+	0.262639i
$186$		0			0
$187$	−1.15282e12	−	1.99674e12i	−0.368666	−	0.638548i
$188$		0			0
$189$	−4.02339e11	−	4.95215e11i	−0.121353	−	0.149367i
$190$		0			0
$191$	1.15479e12	+	2.00015e12i	0.328714	+	0.569349i	0.982257	−	0.187540i	$-0.0600516\pi$
$191$	1.15479e12	+	2.00015e12i	0.328714	+	0.569349i	−0.653543	+	0.756889i	$0.726718\pi$
$192$		0			0
$193$	6.67031e11	−	1.15533e12i	0.179300	−	0.310557i	−0.762341	−	0.647176i	$-0.775950\pi$
$193$	6.67031e11	−	1.15533e12i	0.179300	−	0.310557i	0.941641	+	0.336619i	$0.109283\pi$
$194$		0			0
$195$	−7.62227e11			−0.193595
$196$		0			0
$197$	3.95053e12			0.948618			0.474309	−	0.880359i	$-0.342698\pi$
$197$	3.95053e12			0.948618			0.474309	+	0.880359i	$0.342698\pi$
$198$		0			0
$199$	−5.18976e11	+	8.98893e11i	−0.117884	+	0.204181i	−0.918929	−	0.394423i	$-0.870944\pi$
$199$	−5.18976e11	+	8.98893e11i	−0.117884	+	0.204181i	0.801045	+	0.598604i	$0.204278\pi$
$200$		0			0
$201$	−7.10401e11	−	1.23045e12i	−0.152730	−	0.264537i
$202$		0			0
$203$	−1.02575e12	−	1.26253e12i	−0.208840	−	0.257048i
$204$		0			0
$205$	−7.86320e10	−	1.36195e11i	−0.0151689	−	0.0262732i
$206$		0			0
$207$	−2.48977e11	+	4.31240e11i	−0.0455325	+	0.0788647i
$208$		0			0
$209$	4.35857e12			0.756031
$210$		0			0
$211$	4.91335e12			0.808769			0.404385	−	0.914589i	$-0.367486\pi$
$211$	4.91335e12			0.808769			0.404385	+	0.914589i	$0.367486\pi$
$212$		0			0
$213$	−1.25582e12	+	2.17515e12i	−0.196264	+	0.339939i
$214$		0			0
$215$	−1.14843e12	−	1.98914e12i	−0.170487	−	0.295293i
$216$		0			0
$217$	1.06212e12	−	1.69977e11i	0.149846	−	0.0239807i
$218$		0			0
$219$	1.12099e12	+	1.94161e12i	0.150370	+	0.260448i
$220$		0			0
$221$	2.81923e12	−	4.88305e12i	0.359727	−	0.623065i
$222$		0			0
$223$	−8.25284e12			−1.00214			−0.501068	−	0.865408i	$-0.667059\pi$
$223$	−8.25284e12			−1.00214			−0.501068	+	0.865408i	$0.667059\pi$
$224$		0			0
$225$	−2.45926e12			−0.284315
$226$		0			0
$227$	6.55247e12	−	1.13492e13i	0.721544	−	1.24975i	−0.238837	−	0.971060i	$-0.576766\pi$
$227$	6.55247e12	−	1.13492e13i	0.721544	−	1.24975i	0.960381	−	0.278691i	$-0.0899005\pi$
$228$		0			0
$229$	3.18856e12	+	5.52274e12i	0.334579	+	0.579508i	0.983404	−	0.181430i	$-0.0580724\pi$
$229$	3.18856e12	+	5.52274e12i	0.334579	+	0.579508i	−0.648825	+	0.760938i	$0.724739\pi$
$230$		0			0
$231$	−1.84580e12	+	4.83170e12i	−0.184637	+	0.483319i
$232$		0			0
$233$	3.04721e12	+	5.27793e12i	0.290700	+	0.503507i	0.973975	−	0.226653i	$-0.0727785\pi$
$233$	3.04721e12	+	5.27793e12i	0.290700	+	0.503507i	−0.683275	+	0.730161i	$0.739445\pi$
$234$		0			0
$235$	1.27519e12	−	2.20869e12i	0.116065	−	0.201031i
$236$		0			0
$237$	−4.83504e12			−0.420034
$238$		0			0
$239$	1.63982e13			1.36022			0.680108	−	0.733112i	$-0.261933\pi$
$239$	1.63982e13			1.36022			0.680108	+	0.733112i	$0.261933\pi$
$240$		0			0
$241$	−7.93321e12	+	1.37407e13i	−0.628573	+	1.08872i	0.359266	+	0.933235i	$0.383027\pi$
$241$	−7.93321e12	+	1.37407e13i	−0.628573	+	1.08872i	−0.987838	+	0.155484i	$0.950306\pi$
$242$		0			0
$243$	−4.23644e11	−	7.33773e11i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	5.03387e12	−	1.65354e12i	0.364325	−	0.119675i
$246$		0			0
$247$	5.32946e12	+	9.23089e12i	0.368850	+	0.638867i
$248$		0			0
$249$	−5.80794e12	+	1.00596e13i	−0.384526	+	0.666018i
$250$		0			0
$251$	2.61416e13			1.65625			0.828125	−	0.560544i	$-0.189408\pi$
$251$	2.61416e13			1.65625			0.828125	+	0.560544i	$0.189408\pi$
$252$		0			0
$253$	4.03656e12			0.244821
$254$		0			0
$255$	−1.56822e12	+	2.71624e12i	−0.0910828	+	0.157760i
$256$		0			0
$257$	9.97426e12	+	1.72759e13i	0.554943	+	0.961190i	0.997908	+	0.0646509i	$0.0205934\pi$
$257$	9.97426e12	+	1.72759e13i	0.554943	+	0.961190i	−0.442965	+	0.896539i	$0.646073\pi$
$258$		0			0
$259$	−5.29348e12	+	1.38566e13i	−0.282223	+	0.738769i
$260$		0			0
$261$	−1.08007e12	−	1.87073e12i	−0.0551986	−	0.0956068i
$262$		0			0
$263$	1.90133e13	−	3.29321e13i	0.931755	−	1.61385i	0.151434	−	0.988467i	$-0.451611\pi$
$263$	1.90133e13	−	3.29321e13i	0.931755	−	1.61385i	0.780321	−	0.625379i	$-0.215056\pi$
$264$		0			0
$265$	4.87505e12			0.229154
$266$		0			0
$267$	4.71790e12			0.212783
$268$		0			0
$269$	−6.11726e12	+	1.05954e13i	−0.264801	+	0.458649i	−0.967511	−	0.252827i	$-0.918640\pi$
$269$	−6.11726e12	+	1.05954e13i	−0.264801	+	0.458649i	0.702710	+	0.711476i	$0.251973\pi$
$270$		0			0
$271$	−4.65635e12	−	8.06503e12i	−0.193515	−	0.335177i	0.752898	−	0.658137i	$-0.228655\pi$
$271$	−4.65635e12	−	8.06503e12i	−0.193515	−	0.335177i	−0.946413	+	0.322960i	$0.895322\pi$
$272$		0			0
$273$	−1.24899e13	+	1.99882e12i	−0.498497	+	0.0797771i
$274$		0			0
$275$	9.96773e12	+	1.72646e13i	0.382179	+	0.661953i
$276$		0			0
$277$	−1.90071e13	+	3.29212e13i	−0.700288	+	1.21293i	0.268077	+	0.963397i	$0.413612\pi$
$277$	−1.90071e13	+	3.29212e13i	−0.700288	+	1.21293i	−0.968365	+	0.249537i	$0.919722\pi$
$278$		0			0
$279$	1.42837e12			0.0505844
$280$		0			0
$281$	3.06527e13			1.04372			0.521860	−	0.853031i	$-0.325238\pi$
$281$	3.06527e13			1.04372			0.521860	+	0.853031i	$0.325238\pi$
$282$		0			0
$283$	3.55669e12	−	6.16037e12i	0.116472	−	0.201735i	−0.801895	−	0.597465i	$-0.796175\pi$
$283$	3.55669e12	−	6.16037e12i	0.116472	−	0.201735i	0.918367	+	0.395729i	$0.129508\pi$
$284$		0			0
$285$	−2.96456e12	−	5.13477e12i	−0.0933928	−	0.161761i
$286$		0			0
$287$	−1.64561e12	−	2.02549e12i	−0.0498859	−	0.0614015i
$288$		0			0
$289$	5.53527e12	+	9.58736e12i	0.161510	+	0.279744i
$290$		0			0
$291$	−1.01192e13	+	1.75270e13i	−0.284273	+	0.492374i
$292$		0			0
$293$	4.95279e12			0.133992			0.0669959	−	0.997753i	$-0.478659\pi$
$293$	4.95279e12			0.133992			0.0669959	+	0.997753i	$0.478659\pi$
$294$		0			0
$295$	−4.52193e12			−0.117843
$296$		0			0
$297$	−3.43419e12	+	5.94818e12i	−0.0862310	+	0.149356i
$298$		0			0
$299$	4.93571e12	+	8.54890e12i	0.119442	+	0.206880i
$300$		0			0
$301$	−2.40343e13	−	2.95824e13i	−0.560682	−	0.690110i
$302$		0			0
$303$	−1.81503e13	−	3.14373e13i	−0.408273	−	0.707149i
$304$		0			0
$305$	1.02614e13	−	1.77733e13i	0.222616	−	0.385583i
$306$		0			0
$307$	2.23542e13			0.467842			0.233921	−	0.972256i	$-0.424844\pi$
$307$	2.23542e13			0.467842			0.233921	+	0.972256i	$0.424844\pi$
$308$		0			0
$309$	−1.76380e13			−0.356187
$310$		0			0
$311$	2.65812e13	−	4.60399e13i	0.518074	−	0.897330i	−0.481706	−	0.876333i	$-0.659983\pi$
$311$	2.65812e13	−	4.60399e13i	0.518074	−	0.897330i	0.999780	−	0.0209972i	$-0.00668410\pi$
$312$		0			0
$313$	−2.79998e13	−	4.84971e13i	−0.526819	−	0.912476i	−0.999512	−	0.0312493i	$-0.990051\pi$
$313$	−2.79998e13	−	4.84971e13i	−0.526819	−	0.912476i	0.472693	−	0.881227i	$-0.343282\pi$
$314$		0			0
$315$	6.94760e12	−	1.11186e12i	0.126220	−	0.0201996i
$316$		0			0
$317$	−3.41827e13	−	5.92062e13i	−0.599764	−	1.03882i	−0.992856	−	0.119322i	$-0.961928\pi$
$317$	−3.41827e13	−	5.92062e13i	−0.599764	−	1.03882i	0.393092	−	0.919499i	$-0.371406\pi$
$318$		0			0
$319$	−8.75535e12	+	1.51647e13i	−0.148397	+	0.257031i
$320$		0			0
$321$	−6.12956e13			−1.00381
$322$		0			0
$323$	4.38598e13			0.694148
$324$		0			0
$325$	−2.43761e13	+	4.22207e13i	−0.372912	+	0.645902i
$326$		0			0
$327$	1.66392e13	+	2.88199e13i	0.246104	+	0.426265i
$328$		0			0
$329$	1.51033e13	−	3.95357e13i	0.216021	−	0.565473i
$330$		0			0
$331$	−1.92107e13	−	3.32739e13i	−0.265759	−	0.460309i	0.702003	−	0.712174i	$-0.252289\pi$
$331$	−1.92107e13	−	3.32739e13i	−0.265759	−	0.460309i	−0.967762	+	0.251865i	$0.918956\pi$
$332$		0			0
$333$	−9.84875e12	+	1.70585e13i	−0.131807	+	0.228296i
$334$		0			0
$335$	1.56676e13			0.202887
$336$		0			0
$337$	−1.59403e14			−1.99771			−0.998855	−	0.0478440i	$-0.984765\pi$
$337$	−1.59403e14			−1.99771			−0.998855	+	0.0478440i	$0.984765\pi$
$338$		0			0
$339$	−2.84259e13	+	4.92352e13i	−0.344839	+	0.597279i
$340$		0			0
$341$	−5.78938e12	−	1.00275e13i	−0.0679959	−	0.117772i
$342$		0			0
$343$	7.81490e13	−	4.02955e13i	0.888803	−	0.458289i
$344$		0			0
$345$	−2.74553e12	−	4.75541e12i	−0.0302428	−	0.0523821i
$346$		0			0
$347$	3.76248e13	−	6.51680e13i	0.401478	−	0.695380i	−0.592426	−	0.805625i	$-0.701830\pi$
$347$	3.76248e13	−	6.51680e13i	0.401478	−	0.695380i	0.993905	+	0.110244i	$0.0351633\pi$
$348$		0			0
$349$	−1.81299e14			−1.87437			−0.937187	−	0.348828i	$-0.886580\pi$
$349$	−1.81299e14			−1.87437			−0.937187	+	0.348828i	$0.886580\pi$
$350$		0			0
$351$	−1.67966e13			−0.168280
$352$		0			0
$353$	−6.08571e12	+	1.05408e13i	−0.0590949	+	0.102355i	−0.894059	−	0.447948i	$-0.852155\pi$
$353$	−6.08571e12	+	1.05408e13i	−0.0590949	+	0.102355i	0.834964	+	0.550304i	$0.185488\pi$
$354$		0			0
$355$	−1.38483e13	−	2.39859e13i	−0.130359	−	0.225788i
$356$		0			0
$357$	−1.85740e13	+	4.86208e13i	−0.169524	+	0.443758i
$358$		0			0
$359$	−4.61043e13	−	7.98551e13i	−0.408058	−	0.706778i	0.586614	−	0.809867i	$-0.300461\pi$
$359$	−4.61043e13	−	7.98551e13i	−0.408058	−	0.706778i	−0.994672	+	0.103089i	$0.967127\pi$
$360$		0			0
$361$	1.67890e13	−	2.90794e13i	0.144123	−	0.249629i
$362$		0			0
$363$	−1.36537e13			−0.113701
$364$		0			0
$365$	−2.47230e13			−0.199751
$366$		0			0
$367$	−3.47460e13	+	6.01818e13i	−0.272421	+	0.471847i	−0.969481	−	0.245165i	$-0.921158\pi$
$367$	−3.47460e13	+	6.01818e13i	−0.272421	+	0.471847i	0.697060	+	0.717013i	$0.254491\pi$
$368$		0			0
$369$	−1.73276e12	−	3.00122e12i	−0.0131854	−	0.0228377i
$370$		0			0
$371$	7.98826e13	−	1.27840e13i	0.590060	−	0.0944303i
$372$		0			0
$373$	2.75190e13	+	4.76643e13i	0.197349	+	0.341818i	0.947668	−	0.319258i	$-0.103434\pi$
$373$	2.75190e13	+	4.76643e13i	0.197349	+	0.341818i	−0.750319	+	0.661076i	$0.770100\pi$
$374$		0			0
$375$	2.94566e13	−	5.10204e13i	0.205122	−	0.355281i
$376$		0			0
$377$	−4.28225e13			−0.289597
$378$		0			0
$379$	1.64609e14			1.08128			0.540640	−	0.841254i	$-0.318182\pi$
$379$	1.64609e14			1.08128			0.540640	+	0.841254i	$0.318182\pi$
$380$		0			0
$381$	2.49745e13	−	4.32571e13i	0.159371	−	0.276039i
$382$		0			0
$383$	2.37086e13	+	4.10645e13i	0.146998	+	0.254609i	0.930117	−	0.367264i	$-0.119705\pi$
$383$	2.37086e13	+	4.10645e13i	0.146998	+	0.254609i	−0.783118	+	0.621873i	$0.786372\pi$
$384$		0			0
$385$	−3.59652e13	−	4.42674e13i	−0.216695	−	0.266717i
$386$		0			0
$387$	−2.53071e13	−	4.38331e13i	−0.148194	−	0.256680i
$388$		0			0
$389$	−1.47659e14	+	2.55752e14i	−0.840496	+	1.45578i	0.0489795	+	0.998800i	$0.484403\pi$
$389$	−1.47659e14	+	2.55752e14i	−0.840496	+	1.45578i	−0.889476	+	0.456982i	$0.848930\pi$
$390$		0			0
$391$	4.06194e13			0.224782
$392$		0			0
$393$	8.32865e12			0.0448142
$394$		0			0
$395$	2.66587e13	−	4.61742e13i	0.139494	−	0.241610i
$396$		0			0
$397$	1.02252e14	+	1.77106e14i	0.520386	+	0.901335i	0.999719	+	0.0237017i	$0.00754518\pi$
$397$	1.02252e14	+	1.77106e14i	0.520386	+	0.901335i	−0.479333	+	0.877633i	$0.659121\pi$
$398$		0			0
$399$	−6.20424e13	−	7.63643e13i	−0.307141	−	0.378042i
$400$		0			0
$401$	−7.18349e13	−	1.24422e14i	−0.345972	−	0.599242i	0.639558	−	0.768743i	$-0.279118\pi$
$401$	−7.18349e13	−	1.24422e14i	−0.345972	−	0.599242i	−0.985530	+	0.169502i	$0.945784\pi$
$402$		0			0
$403$	1.41579e13	−	2.45223e13i	0.0663472	−	0.114917i
$404$		0			0
$405$	9.34328e12			0.0426086
$406$		0			0
$407$	1.59674e14			0.708703
$408$		0			0
$409$	−5.12511e13	+	8.87695e13i	−0.221424	+	0.383518i	−0.955241	−	0.295830i	$-0.904404\pi$
$409$	−5.12511e13	+	8.87695e13i	−0.221424	+	0.383518i	0.733816	+	0.679348i	$0.237737\pi$
$410$		0			0
$411$	1.87151e13	+	3.24156e13i	0.0787160	+	0.136340i
$412$		0			0
$413$	−7.40964e13	+	1.18580e13i	−0.303439	+	0.0485610i
$414$		0			0
$415$	−6.40457e13	−	1.10930e14i	−0.255403	−	0.442371i
$416$		0			0
$417$	1.35448e13	−	2.34603e13i	0.0526046	−	0.0911138i
$418$		0			0
$419$	2.06951e14			0.782871			0.391436	−	0.920206i	$-0.371979\pi$
$419$	2.06951e14			0.782871			0.391436	+	0.920206i	$0.371979\pi$
$420$		0			0
$421$	−2.28895e13			−0.0843498			−0.0421749	−	0.999110i	$-0.513429\pi$
$421$	−2.28895e13			−0.0843498			−0.0421749	+	0.999110i	$0.513429\pi$
$422$		0			0
$423$	2.81004e13	−	4.86714e13i	0.100888	−	0.174744i
$424$		0			0
$425$	1.00304e14	+	1.73732e14i	0.350896	+	0.607770i
$426$		0			0
$427$	1.21536e14	−	3.18142e14i	0.414335	−	1.08459i
$428$		0			0
$429$	6.80792e13	+	1.17917e14i	0.226204	+	0.391796i
$430$		0			0
$431$	−6.89096e13	+	1.19355e14i	−0.223180	+	0.386559i	−0.955772	−	0.294109i	$-0.904977\pi$
$431$	−6.89096e13	+	1.19355e14i	−0.223180	+	0.386559i	0.732592	+	0.680668i	$0.238310\pi$
$432$		0			0
$433$	1.37606e14			0.434463			0.217231	−	0.976120i	$-0.430297\pi$
$433$	1.37606e14			0.434463			0.217231	+	0.976120i	$0.430297\pi$
$434$		0			0
$435$	2.38204e13			0.0733261
$436$		0			0
$437$	−3.83933e13	+	6.64992e13i	−0.115241	+	0.199604i
$438$		0			0
$439$	2.24449e14	+	3.88757e14i	0.656996	+	1.13795i	0.981389	+	0.192028i	$0.0615066\pi$
$439$	2.24449e14	+	3.88757e14i	0.656996	+	1.13795i	−0.324393	+	0.945922i	$0.605160\pi$
$440$		0			0
$441$	1.10928e14	−	3.64379e13i	0.316686	−	0.104026i
$442$		0			0
$443$	9.09994e13	+	1.57616e14i	0.253407	+	0.438913i	0.964462	−	0.264223i	$-0.0851156\pi$
$443$	9.09994e13	+	1.57616e14i	0.253407	+	0.438913i	−0.711055	+	0.703137i	$0.751782\pi$
$444$		0			0
$445$	−2.60128e13	+	4.50555e13i	−0.0706654	+	0.122396i
$446$		0			0
$447$	4.05225e14			1.07400
$448$		0			0
$449$	7.82493e13			0.202360			0.101180	−	0.994868i	$-0.467738\pi$
$449$	7.82493e13			0.202360			0.101180	+	0.994868i	$0.467738\pi$
$450$		0			0
$451$	−1.40462e13	+	2.43288e13i	−0.0354478	+	0.0613973i
$452$		0			0
$453$	9.02059e13	+	1.56241e14i	0.222175	+	0.384818i
$454$		0			0
$455$	4.97760e13	−	1.30298e14i	0.119662	−	0.313238i
$456$		0			0
$457$	6.37712e12	+	1.10455e13i	0.0149653	+	0.0259206i	0.873411	−	0.486984i	$-0.161903\pi$
$457$	6.37712e12	+	1.10455e13i	0.0149653	+	0.0259206i	−0.858446	+	0.512904i	$0.828570\pi$
$458$		0			0
$459$	−3.45578e13	+	5.98558e13i	−0.0791728	+	0.137131i
$460$		0			0
$461$	−4.11130e14			−0.919653			−0.459826	−	0.888009i	$-0.652088\pi$
$461$	−4.11130e14			−0.919653			−0.459826	+	0.888009i	$0.652088\pi$
$462$		0			0
$463$	8.91432e14			1.94712			0.973560	−	0.228430i	$-0.0733593\pi$
$463$	8.91432e14			1.94712			0.973560	+	0.228430i	$0.0733593\pi$
$464$		0			0
$465$	−7.87549e12	+	1.36407e13i	−0.0167991	+	0.0290969i
$466$		0			0
$467$	9.36710e13	+	1.62243e14i	0.195147	+	0.338005i	0.946949	−	0.321384i	$-0.104148\pi$
$467$	9.36710e13	+	1.62243e14i	0.195147	+	0.338005i	−0.751802	+	0.659389i	$0.770815\pi$
$468$		0			0
$469$	2.56729e14	−	4.10857e13i	0.522425	−	0.0836064i
$470$		0			0
$471$	−1.06237e14	−	1.84008e14i	−0.211184	−	0.365782i
$472$		0			0
$473$	−2.05147e14	+	3.55324e14i	−0.398408	+	0.690063i
$474$		0			0
$475$	−3.79228e14			−0.719591
$476$		0			0
$477$	1.07428e14			0.199189
$478$		0			0
$479$	−3.54805e14	+	6.14540e14i	−0.642901	+	1.11354i	0.341881	+	0.939743i	$0.388936\pi$
$479$	−3.54805e14	+	6.14540e14i	−0.642901	+	1.11354i	−0.984782	+	0.173794i	$0.944397\pi$
$480$		0			0
$481$	1.95241e14	+	3.38168e14i	0.345759	+	0.598873i
$482$		0			0
$483$	−5.74587e13	−	7.07224e13i	−0.0994595	−	0.122419i
$484$		0			0
$485$	−1.11587e14	−	1.93275e14i	−0.188814	−	0.327036i
$486$		0			0
$487$	1.72334e14	−	2.98491e14i	0.285077	−	0.493768i	−0.687551	−	0.726136i	$-0.741314\pi$
$487$	1.72334e14	−	2.98491e14i	0.285077	−	0.493768i	0.972628	+	0.232368i	$0.0746475\pi$
$488$		0			0
$489$	2.77079e14			0.448130
$490$		0			0
$491$	2.50906e12			0.00396791			0.00198396	−	0.999998i	$-0.499368\pi$
$491$	2.50906e12			0.00396791			0.00198396	+	0.999998i	$0.499368\pi$
$492$		0			0
$493$	−8.81039e13	+	1.52601e14i	−0.136250	+	0.235992i
$494$		0			0
$495$	−3.78697e13	−	6.55922e13i	−0.0572748	−	0.0992028i
$496$		0			0
$497$	−2.89817e14	−	3.56719e14i	−0.428711	−	0.527674i
$498$		0			0
$499$	3.62370e12	+	6.27644e12i	0.00524324	+	0.00908155i	0.868635	−	0.495452i	$-0.164998\pi$
$499$	3.62370e12	+	6.27644e12i	0.00524324	+	0.00908155i	−0.863392	+	0.504534i	$0.831664\pi$
$500$		0			0
$501$	−2.03553e14	+	3.52564e14i	−0.288118	+	0.499034i
$502$		0			0
$503$	−1.22748e15			−1.69977			−0.849887	−	0.526965i	$-0.823330\pi$
$503$	−1.22748e15			−1.69977			−0.849887	+	0.526965i	$0.823330\pi$
$504$		0			0
$505$	4.00297e14			0.542351
$506$		0			0
$507$	5.12594e13	−	8.87839e13i	0.0679563	−	0.117704i
$508$		0			0
$509$	−1.64697e13	−	2.85263e13i	−0.0213667	−	0.0370082i	0.855144	−	0.518390i	$-0.173468\pi$
$509$	−1.64697e13	−	2.85263e13i	−0.0213667	−	0.0370082i	−0.876511	+	0.481382i	$0.840135\pi$
$510$		0			0
$511$	−4.05111e14	+	6.48320e13i	−0.514350	+	0.0823142i
$512$		0			0
$513$	−6.53278e13	−	1.13151e14i	−0.0811807	−	0.140609i
$514$		0			0
$515$	9.72497e13	−	1.68441e14i	0.118290	−	0.204884i
$516$		0			0
$517$	−4.55581e14			−0.542459
$518$		0			0
$519$	8.97403e14			1.04609
$520$		0			0
$521$	−4.69615e14	+	8.13397e14i	−0.535962	+	0.928314i	0.463154	+	0.886278i	$0.346718\pi$
$521$	−4.69615e14	+	8.13397e14i	−0.535962	+	0.928314i	−0.999116	+	0.0420361i	$0.986616\pi$
$522$		0			0
$523$	−4.77717e14	−	8.27430e14i	−0.533840	−	0.924639i	−0.999219	−	0.0395267i	$-0.987415\pi$
$523$	−4.77717e14	−	8.27430e14i	−0.533840	−	0.924639i	0.465378	−	0.885112i	$-0.345918\pi$
$524$		0			0
$525$	1.60598e14	−	4.20394e14i	0.175737	−	0.460024i
$526$		0			0
$527$	−5.82577e13	−	1.00905e14i	−0.0624303	−	0.108132i
$528$		0			0
$529$	4.40848e14	−	7.63571e14i	0.462682	−	0.801389i
$530$		0			0
$531$	−9.96465e13			−0.102433
$532$		0			0
$533$	−6.87002e13			−0.0691765
$534$		0			0
$535$	3.37962e14	−	5.85367e14i	0.333366	−	0.577408i
$536$		0			0
$537$	−1.96991e14	−	3.41198e14i	−0.190365	−	0.329722i
$538$		0			0
$539$	−7.05410e14	−	6.31053e14i	−0.667888	−	0.597486i
$540$		0			0
$541$	−5.37113e14	−	9.30307e14i	−0.498289	−	0.863061i	0.501709	−	0.865036i	$-0.332705\pi$
$541$	−5.37113e14	−	9.30307e14i	−0.498289	−	0.863061i	−0.999998	+	0.00197505i	$0.999371\pi$
$542$		0			0
$543$	4.52176e14	−	7.83191e14i	0.411063	−	0.711982i
$544$		0			0
$545$	−3.66969e14			−0.326926
$546$		0			0
$547$	1.38381e15			1.20822			0.604111	−	0.796900i	$-0.293528\pi$
$547$	1.38381e15			1.20822			0.604111	+	0.796900i	$0.293528\pi$
$548$		0			0
$549$	2.26123e14	−	3.91656e14i	0.193507	−	0.335164i
$550$		0			0
$551$	−1.66551e14	−	2.88475e14i	−0.139706	−	0.241977i
$552$		0			0
$553$	3.15745e14	−	8.26519e14i	0.259626	−	0.679618i
$554$		0			0
$555$	−1.08605e14	−	1.88109e14i	−0.0875463	−	0.151635i
$556$		0			0
$557$	−7.14824e14	+	1.23811e15i	−0.564931	+	0.978490i	0.432125	+	0.901814i	$0.357764\pi$
$557$	−7.14824e14	+	1.23811e15i	−0.564931	+	0.978490i	−0.997056	+	0.0766761i	$0.975569\pi$
$558$		0			0
$559$	−1.00337e15			−0.777495
$560$		0			0
$561$	5.60271e14			0.425699
$562$		0			0
$563$	−1.27988e15	+	2.21682e15i	−0.953615	+	1.65171i	−0.216108	+	0.976370i	$0.569336\pi$
$563$	−1.27988e15	+	2.21682e15i	−0.953615	+	1.65171i	−0.737507	+	0.675340i	$0.763997\pi$
$564$		0			0
$565$	−3.13461e14	−	5.42930e14i	−0.229043	−	0.396714i
$566$		0			0
$567$	1.53099e14	−	2.45013e13i	0.109715	−	0.0175583i
$568$		0			0
$569$	−1.13256e15	−	1.96165e15i	−0.796056	−	1.37881i	−0.922166	−	0.386794i	$-0.873583\pi$
$569$	−1.13256e15	−	1.96165e15i	−0.796056	−	1.37881i	0.126110	−	0.992016i	$-0.459751\pi$
$570$		0			0
$571$	−5.49376e14	+	9.51547e14i	−0.378766	+	0.656042i	−0.990883	−	0.134725i	$-0.956985\pi$
$571$	−5.49376e14	+	9.51547e14i	−0.378766	+	0.656042i	0.612117	+	0.790767i	$0.290318\pi$
$572$		0			0
$573$	−5.61226e14			−0.379566
$574$		0			0
$575$	−3.51211e14			−0.233021
$576$		0			0
$577$	−1.22669e15	+	2.12470e15i	−0.798490	+	1.38303i	0.122109	+	0.992517i	$0.461034\pi$
$577$	−1.22669e15	+	2.12470e15i	−0.798490	+	1.38303i	−0.920599	+	0.390509i	$0.872299\pi$
$578$		0			0
$579$	1.62089e14	+	2.80746e14i	0.103519	+	0.179300i
$580$		0			0
$581$	−1.34035e15	−	1.64976e15i	−0.839943	−	1.03384i
$582$		0			0
$583$	−4.35421e14	−	7.54172e14i	−0.267752	−	0.463760i
$584$		0			0
$585$	9.26105e13	−	1.60406e14i	0.0558860	−	0.0967974i
$586$		0			0
$587$	−3.57451e14			−0.211693			−0.105847	−	0.994382i	$-0.533755\pi$
$587$	−3.57451e14			−0.211693			−0.105847	+	0.994382i	$0.533755\pi$
$588$		0			0
$589$	2.20260e14			0.128027
$590$		0			0
$591$	−4.79990e14	+	8.31366e14i	−0.273842	+	0.474309i
$592$		0			0
$593$	3.47719e14	+	6.02267e14i	0.194728	+	0.337278i	0.946811	−	0.321790i	$-0.104284\pi$
$593$	3.47719e14	+	6.02267e14i	0.194728	+	0.337278i	−0.752084	+	0.659068i	$0.770951\pi$
$594$		0			0
$595$	−3.61913e14	−	4.45457e14i	−0.198958	−	0.244885i
$596$		0			0
$597$	−1.26111e14	−	2.18431e14i	−0.0680604	−	0.117884i
$598$		0			0
$599$	1.18753e14	−	2.05685e14i	0.0629210	−	0.108982i	−0.832849	−	0.553500i	$-0.813292\pi$
$599$	1.18753e14	−	2.05685e14i	0.0629210	−	0.108982i	0.895770	+	0.444518i	$0.146625\pi$
$600$		0			0
$601$	2.93732e15			1.52806			0.764032	−	0.645178i	$-0.223217\pi$
$601$	2.93732e15			1.52806			0.764032	+	0.645178i	$0.223217\pi$
$602$		0			0
$603$	3.45255e14			0.176358
$604$		0			0
$605$	7.52816e13	−	1.30391e14i	0.0377602	−	0.0654026i
$606$		0			0
$607$	8.62651e14	+	1.49416e15i	0.424910	+	0.735966i	0.996412	−	0.0846345i	$-0.0269723\pi$
$607$	8.62651e14	+	1.49416e15i	0.424910	+	0.735966i	−0.571502	+	0.820601i	$0.693639\pi$
$608$		0			0
$609$	3.90322e14	−	6.24652e13i	0.188811	−	0.0302164i
$610$		0			0
$611$	−5.57062e14	−	9.64860e14i	−0.264653	−	0.458393i
$612$		0			0
$613$	−5.07686e14	+	8.79338e14i	−0.236899	+	0.410320i	−0.959823	−	0.280607i	$-0.909464\pi$
$613$	−5.07686e14	+	8.79338e14i	−0.236899	+	0.410320i	0.722924	+	0.690927i	$0.242798\pi$
$614$		0			0
$615$	3.82151e13			0.0175155
$616$		0			0
$617$	4.29465e15			1.93357			0.966784	−	0.255595i	$-0.0822714\pi$
$617$	4.29465e15			1.93357			0.966784	+	0.255595i	$0.0822714\pi$
$618$		0			0
$619$	−8.18875e14	+	1.41833e15i	−0.362175	+	0.627306i	−0.988319	−	0.152402i	$-0.951299\pi$
$619$	−8.18875e14	+	1.41833e15i	−0.362175	+	0.627306i	0.626143	+	0.779708i	$0.284633\pi$
$620$		0			0
$621$	−6.05014e13	−	1.04791e14i	−0.0262882	−	0.0455325i
$622$		0			0
$623$	−3.08095e14	+	8.06495e14i	−0.131523	+	0.344284i
$624$		0			0
$625$	−6.91964e14	−	1.19852e15i	−0.290231	−	0.502694i
$626$		0			0
$627$	−5.29567e14	+	9.17236e14i	−0.218247	+	0.378016i
$628$		0			0
$629$	1.60678e15			0.650694
$630$		0			0
$631$	2.24346e15			0.892804			0.446402	−	0.894832i	$-0.352705\pi$
$631$	2.24346e15			0.892804			0.446402	+	0.894832i	$0.352705\pi$
$632$		0			0
$633$	−5.96972e14	+	1.03399e15i	−0.233472	+	0.404385i
$634$		0			0
$635$	2.75401e14	+	4.77008e14i	0.105855	+	0.183346i
$636$		0			0
$637$	4.73946e14	−	2.26559e15i	0.179045	−	0.855883i
$638$		0			0
$639$	−3.05164e14	−	5.28560e14i	−0.113313	−	0.196264i
$640$		0			0
$641$	9.60927e14	−	1.66437e15i	0.350729	−	0.607480i	−0.635649	−	0.771978i	$-0.719267\pi$
$641$	9.60927e14	−	1.66437e15i	0.350729	−	0.607480i	0.986377	+	0.164499i	$0.0526007\pi$
$642$		0			0
$643$	−8.48487e14			−0.304428			−0.152214	−	0.988348i	$-0.548640\pi$
$643$	−8.48487e14			−0.304428			−0.152214	+	0.988348i	$0.548640\pi$
$644$		0			0
$645$	5.58136e14			0.196862
$646$		0			0
$647$	−1.54367e15	+	2.67372e15i	−0.535280	+	0.927132i	0.463870	+	0.885903i	$0.346461\pi$
$647$	−1.54367e15	+	2.67372e15i	−0.535280	+	0.927132i	−0.999150	+	0.0412287i	$0.986873\pi$
$648$		0			0
$649$	4.03882e14	+	6.99544e14i	0.137692	+	0.238489i
$650$		0			0
$651$	−9.32772e13	+	2.44170e14i	−0.0312666	+	0.0818458i
$652$		0			0
$653$	4.33958e14	+	7.51637e14i	0.143029	+	0.247734i	0.928636	−	0.370992i	$-0.120982\pi$
$653$	4.33958e14	+	7.51637e14i	0.143029	+	0.247734i	−0.785607	+	0.618726i	$0.787649\pi$
$654$		0			0
$655$	−4.59211e13	+	7.95378e13i	−0.0148828	+	0.0257778i
$656$		0			0
$657$	−5.44802e14			−0.173632
$658$		0			0
$659$	3.99805e15			1.25308			0.626540	−	0.779389i	$-0.284471\pi$
$659$	3.99805e15			1.25308			0.626540	+	0.779389i	$0.284471\pi$
$660$		0			0
$661$	−1.60004e15	+	2.77135e15i	−0.493200	+	0.854247i	−0.999969	−	0.00783472i	$-0.997506\pi$
$661$	−1.60004e15	+	2.77135e15i	−0.493200	+	0.854247i	0.506770	+	0.862081i	$0.330839\pi$
$662$		0			0
$663$	6.85072e14	+	1.18658e15i	0.207688	+	0.359727i
$664$		0			0
$665$	1.07135e15	−	1.71454e14i	0.319457	−	0.0511244i
$666$		0			0
$667$	−1.54246e14	−	2.67163e14i	−0.0452400	−	0.0783580i
$668$		0			0
$669$	1.00272e15	−	1.73676e15i	0.289292	−	0.501068i
$670$		0			0
$671$	−3.66604e15			−1.04045
$672$		0			0
$673$	6.83323e15			1.90785			0.953923	−	0.300052i	$-0.0970041\pi$
$673$	6.83323e15			1.90785			0.953923	+	0.300052i	$0.0970041\pi$
$674$		0			0
$675$	2.98800e14	−	5.17536e14i	0.0820747	−	0.142158i
$676$		0			0
$677$	1.59834e15	+	2.76841e15i	0.431949	+	0.748158i	0.997041	−	0.0768700i	$-0.0244926\pi$
$677$	1.59834e15	+	2.76841e15i	0.431949	+	0.748158i	−0.565092	+	0.825028i	$0.691159\pi$
$678$		0			0
$679$	−2.33530e15	−	2.87438e15i	−0.620954	−	0.764295i
$680$		0			0
$681$	1.59225e15	+	2.75786e15i	0.416584	+	0.721544i
$682$		0			0
$683$	−1.63286e14	+	2.82819e14i	−0.0420372	+	0.0728106i	−0.886278	−	0.463153i	$-0.846718\pi$
$683$	−1.63286e14	+	2.82819e14i	−0.0420372	+	0.0728106i	0.844241	+	0.535963i	$0.180051\pi$
$684$		0			0
$685$	−4.12754e14			−0.104567
$686$		0			0
$687$	−1.54964e15			−0.386339
$688$		0			0
$689$	1.06482e15	−	1.84433e15i	0.261260	−	0.452515i
$690$		0			0
$691$	−1.77293e12	−	3.07081e12i	−0.000428117	−	0.000741521i	0.865811	−	0.500371i	$-0.166803\pi$
$691$	−1.77293e12	−	3.07081e12i	−0.000428117	−	0.000741521i	−0.866239	+	0.499629i	$0.833470\pi$
$692$		0			0
$693$	−7.92539e14	−	9.75488e14i	−0.188360	−	0.231841i
$694$		0			0
$695$	1.49362e14	+	2.58703e14i	0.0349401	+	0.0605180i
$696$		0			0
$697$	−1.41345e14	+	2.44817e14i	−0.0325463	+	0.0563718i
$698$		0			0
$699$	−1.48095e15			−0.335672
$700$		0			0
$701$	6.37107e15			1.42155			0.710777	−	0.703418i	$-0.248344\pi$
$701$	6.37107e15			1.42155			0.710777	+	0.703418i	$0.248344\pi$
$702$		0			0
$703$	−1.51872e15	+	2.63050e15i	−0.333598	+	0.577809i
$704$		0			0
$705$	3.09871e14	+	5.36712e14i	0.0670102	+	0.116065i
$706$		0			0
$707$	6.55928e15	−	1.04972e15i	1.39653	−	0.223494i
$708$		0			0
$709$	3.87624e13	+	6.71385e13i	0.00812562	+	0.0140740i	0.870060	−	0.492946i	$-0.164080\pi$
$709$	3.87624e13	+	6.71385e13i	0.00812562	+	0.0140740i	−0.861934	+	0.507020i	$0.830747\pi$
$710$		0			0
$711$	5.87458e14	−	1.01751e15i	0.121253	−	0.210017i
$712$		0			0
$713$	2.03987e14			0.0414582
$714$		0			0
$715$	−1.50146e15			−0.300490
$716$		0			0
$717$	−1.99238e15	+	3.45090e15i	−0.392660	+	0.680108i
$718$		0			0
$719$	2.30995e15	+	4.00095e15i	0.448326	+	0.776523i	0.998277	−	0.0586734i	$-0.0186870\pi$
$719$	2.30995e15	+	4.00095e15i	0.448326	+	0.776523i	−0.549951	+	0.835197i	$0.685354\pi$
$720$		0			0
$721$	1.15182e15	−	3.01511e15i	0.220162	−	0.576313i
$722$		0			0
$723$	−1.92777e15	−	3.33900e15i	−0.362907	−	0.628573i
$724$		0			0
$725$	7.61781e14	−	1.31944e15i	0.141244	−	0.244642i
$726$		0			0
$727$	−2.05407e14			−0.0375124			−0.0187562	−	0.999824i	$-0.505971\pi$
$727$	−2.05407e14			−0.0375124			−0.0187562	+	0.999824i	$0.505971\pi$
$728$		0			0
$729$	2.05891e14			0.0370370
$730$		0			0
$731$	−2.06436e15	+	3.57558e15i	−0.365797	+	0.633579i
$732$		0			0
$733$	4.30318e15	+	7.45332e15i	0.751134	+	1.30100i	0.947274	+	0.320426i	$0.103826\pi$
$733$	4.30318e15	+	7.45332e15i	0.751134	+	1.30100i	−0.196140	+	0.980576i	$0.562841\pi$
$734$		0			0
$735$	−2.63637e14	+	1.26025e15i	−0.0453342	+	0.216710i
$736$		0			0
$737$	−1.39937e15	−	2.42378e15i	−0.237061	−	0.410602i
$738$		0			0
$739$	−4.84230e15	+	8.38712e15i	−0.808179	+	1.39981i	0.105945	+	0.994372i	$0.466213\pi$
$739$	−4.84230e15	+	8.38712e15i	−0.808179	+	1.39981i	−0.914124	+	0.405435i	$0.867120\pi$
$740$		0			0
$741$	−2.59012e15			−0.425911
$742$		0			0
$743$	4.55474e15			0.737947			0.368974	−	0.929440i	$-0.379709\pi$
$743$	4.55474e15			0.737947			0.368974	+	0.929440i	$0.379709\pi$
$744$		0			0
$745$	−2.23426e15	+	3.86985e15i	−0.356676	+	0.617782i
$746$		0			0
$747$	−1.41133e15	−	2.44449e15i	−0.222006	−	0.384526i
$748$		0			0
$749$	4.00282e15	−	1.04781e16i	0.620463	−	1.62417i
$750$		0			0
$751$	−4.98893e15	−	8.64108e15i	−0.762058	−	1.31992i	−0.941788	−	0.336206i	$-0.890856\pi$
$751$	−4.98893e15	−	8.64108e15i	−0.762058	−	1.31992i	0.179731	−	0.983716i	$-0.442477\pi$
$752$		0			0
$753$	−3.17620e15	+	5.50134e15i	−0.478118	+	0.828125i
$754$		0			0
$755$	−1.98945e15			−0.295138
$756$		0			0
$757$	2.16914e15			0.317146			0.158573	−	0.987347i	$-0.449311\pi$
$757$	2.16914e15			0.317146			0.158573	+	0.987347i	$0.449311\pi$
$758$		0			0
$759$	−4.90442e14	+	8.49470e14i	−0.0706737	+	0.122410i
$760$		0			0
$761$	−4.02973e15	−	6.97970e15i	−0.572349	−	0.991337i	−0.996324	−	0.0856631i	$-0.972699\pi$
$761$	−4.02973e15	−	6.97970e15i	−0.572349	−	0.991337i	0.423976	−	0.905674i	$-0.360634\pi$
$762$		0			0
$763$	−6.01316e15	+	9.62318e14i	−0.841818	+	0.134720i
$764$		0			0
$765$	−3.81078e14	−	6.60046e14i	−0.0525867	−	0.0910828i
$766$		0			0
$767$	−9.87695e14	+	1.71074e15i	−0.134353	+	0.232707i
$768$		0			0
$769$	−2.54259e15			−0.340943			−0.170472	−	0.985363i	$-0.554529\pi$
$769$	−2.54259e15			−0.340943			−0.170472	+	0.985363i	$0.554529\pi$
$770$		0			0
$771$	−4.84749e15			−0.640793
$772$		0			0
$773$	6.44824e15	−	1.11687e16i	0.840338	−	1.45551i	−0.0492707	−	0.998785i	$-0.515690\pi$
$773$	6.44824e15	−	1.11687e16i	0.840338	−	1.45551i	0.889609	−	0.456723i	$-0.150977\pi$
$774$		0			0
$775$	5.03719e14	+	8.72466e14i	0.0647185	+	0.112096i
$776$		0			0
$777$	−2.27289e15	−	2.79756e15i	−0.287914	−	0.354376i
$778$		0			0
$779$	−2.67199e14	−	4.62802e14i	−0.0333717	−	0.0578015i
$780$		0			0
$781$	−2.47375e15	+	4.28467e15i	−0.304632	+	0.527638i
$782$		0			0
$783$	5.24913e14			0.0637379
$784$		0			0
$785$	2.34301e15			0.280538
$786$		0			0
$787$	−2.43919e14	+	4.22480e14i	−0.0287995	+	0.0498821i	−0.880066	−	0.474852i	$-0.842502\pi$
$787$	−2.43919e14	+	4.22480e14i	−0.0287995	+	0.0498821i	0.851266	+	0.524734i	$0.175835\pi$
$788$		0			0
$789$	4.62024e15	+	8.00249e15i	0.537949	+	0.931755i
$790$		0			0
$791$	−6.56012e15	−	8.07446e15i	−0.753253	−	0.927134i
$792$		0			0
$793$	−4.48266e15	−	7.76419e15i	−0.507613	−	0.879212i
$794$		0			0
$795$	−5.92318e14	+	1.02593e15i	−0.0661509	+	0.114577i
$796$		0			0
$797$	5.38172e15			0.592789			0.296395	−	0.955066i	$-0.404216\pi$
$797$	5.38172e15			0.592789			0.296395	+	0.955066i	$0.404216\pi$
$798$		0			0
$799$	−4.58445e15			−0.498058
$800$		0			0
$801$	−5.73225e14	+	9.92855e14i	−0.0614252	+	0.106392i
$802$		0			0
$803$	2.20816e15	+	3.82465e15i	0.233397	+	0.404256i
$804$		0			0
$805$	9.92198e14	−	1.58787e14i	0.103448	−	0.0165553i
$806$		0			0
$807$	−1.48650e15	−	2.57469e15i	−0.152883	−	0.264801i
$808$		0			0
$809$	4.74273e14	−	8.21465e14i	0.0481184	−	0.0833436i	−0.840963	−	0.541093i	$-0.818011\pi$
$809$	4.74273e14	−	8.21465e14i	0.0481184	−	0.0833436i	0.889081	+	0.457749i	$0.151344\pi$
$810$		0			0
$811$	−1.52399e16			−1.52534			−0.762671	−	0.646786i	$-0.776112\pi$
$811$	−1.52399e16			−1.52534			−0.762671	+	0.646786i	$0.776112\pi$
$812$		0			0
$813$	2.26298e15			0.223452
$814$		0			0
$815$	−1.52771e15	+	2.64607e15i	−0.148825	+	0.257772i
$816$		0			0
$817$	−3.90246e15	−	6.75926e15i	−0.375074	−	0.649648i
$818$		0			0
$819$	1.09688e15	−	2.87127e15i	0.104015	−	0.272278i
$820$		0			0
$821$	−1.90529e15	−	3.30005e15i	−0.178268	−	0.308769i	0.763020	−	0.646375i	$-0.223716\pi$
$821$	−1.90529e15	−	3.30005e15i	−0.178268	−	0.308769i	−0.941287	+	0.337607i	$0.890383\pi$
$822$		0			0
$823$	8.29953e15	−	1.43752e16i	0.766221	−	1.32713i	−0.173377	−	0.984855i	$-0.555468\pi$
$823$	8.29953e15	−	1.43752e16i	0.766221	−	1.32713i	0.939599	−	0.342279i	$-0.111199\pi$
$824$		0			0
$825$	−4.84432e15			−0.441302
$826$		0			0
$827$	1.10710e16			0.995187			0.497594	−	0.867410i	$-0.334217\pi$
$827$	1.10710e16			0.995187			0.497594	+	0.867410i	$0.334217\pi$
$828$		0			0
$829$	2.11428e15	−	3.66205e15i	0.187548	−	0.324843i	−0.756884	−	0.653549i	$-0.773279\pi$
$829$	2.11428e15	−	3.66205e15i	0.187548	−	0.324843i	0.944432	+	0.328706i	$0.106613\pi$
$830$		0			0
$831$	−4.61872e15	−	7.99986e15i	−0.404312	−	0.700288i
$832$		0			0
$833$	−7.09845e15	−	6.35021e15i	−0.613220	−	0.548581i
$834$		0			0
$835$	−2.24463e15	−	3.88782e15i	−0.191368	−	0.331459i
$836$		0			0
$837$	−1.73546e14	+	3.00591e14i	−0.0146024	+	0.0252922i
$838$		0			0
$839$	−3.99192e15			−0.331506			−0.165753	−	0.986167i	$-0.553005\pi$
$839$	−3.99192e15			−0.331506			−0.165753	+	0.986167i	$0.553005\pi$
$840$		0			0
$841$	−1.08623e16			−0.890312
$842$		0			0
$843$	−3.72431e15	+	6.45069e15i	−0.301296	+	0.521860i
$844$		0			0
$845$	5.65251e14	+	9.79044e14i	0.0451367	+	0.0781791i
$846$		0			0
$847$	8.91634e14	−	2.33401e15i	0.0702794	−	0.183969i
$848$		0			0
$849$	8.64276e14	+	1.49697e15i	0.0672451	+	0.116472i
$850$		0			0
$851$	−1.40652e15	+	2.43616e15i	−0.108027	+	0.187108i
$852$		0			0
$853$	−4.36392e15			−0.330869			−0.165435	−	0.986221i	$-0.552903\pi$
$853$	−4.36392e15			−0.330869			−0.165435	+	0.986221i	$0.552903\pi$
$854$		0			0
$855$	1.44078e15			0.107841
$856$		0			0
$857$	−3.98415e15	+	6.90076e15i	−0.294403	+	0.509920i	−0.974846	−	0.222881i	$-0.928454\pi$
$857$	−3.98415e15	+	6.90076e15i	−0.294403	+	0.509920i	0.680443	+	0.732801i	$0.261787\pi$
$858$		0			0
$859$	1.00704e16	+	1.74424e16i	0.734653	+	1.27246i	0.954875	+	0.297006i	$0.0959883\pi$
$859$	1.00704e16	+	1.74424e16i	0.734653	+	1.27246i	−0.220223	+	0.975450i	$0.570678\pi$
$860$		0			0
$861$	6.26194e14	−	1.00213e14i	0.0451016	−	0.00721784i
$862$		0			0
$863$	−4.55541e15	−	7.89021e15i	−0.323943	−	0.561086i	0.657355	−	0.753581i	$-0.271675\pi$
$863$	−4.55541e15	−	7.89021e15i	−0.323943	−	0.561086i	−0.981298	+	0.192496i	$0.938342\pi$
$864$		0			0
$865$	−4.94796e15	+	8.57011e15i	−0.347406	+	0.601725i
$866$		0			0
$867$	−2.69014e15			−0.186496
$868$		0			0
$869$	−9.52421e15			−0.651959
$870$		0			0
$871$	3.42216e15	−	5.92736e15i	0.231313	−	0.400646i
$872$		0			0
$873$	−2.45897e15	−	4.25905e15i	−0.164125	−	0.284273i
$874$		0			0
$875$	6.79798e15	+	8.36723e15i	0.448060	+	0.551490i
$876$		0			0
$877$	1.47774e15	+	2.55952e15i	0.0961833	+	0.166594i	0.910102	−	0.414385i	$-0.136003\pi$
$877$	1.47774e15	+	2.55952e15i	0.0961833	+	0.166594i	−0.813919	+	0.580979i	$0.802670\pi$
$878$		0			0
$879$	−6.01764e14	+	1.04229e15i	−0.0386801	+	0.0669959i
$880$		0			0
$881$	1.08087e16			0.686130			0.343065	−	0.939312i	$-0.388535\pi$
$881$	1.08087e16			0.686130			0.343065	+	0.939312i	$0.388535\pi$
$882$		0			0
$883$	2.74783e16			1.72269			0.861344	−	0.508023i	$-0.169623\pi$
$883$	2.74783e16			1.72269			0.861344	+	0.508023i	$0.169623\pi$
$884$		0			0
$885$	5.49414e14	−	9.51613e14i	0.0340182	−	0.0589213i
$886$		0			0
$887$	1.20245e16	+	2.08271e16i	0.735339	+	1.27364i	0.954574	+	0.297973i	$0.0963105\pi$
$887$	1.20245e16	+	2.08271e16i	0.735339	+	1.27364i	−0.219235	+	0.975672i	$0.570356\pi$
$888$		0			0
$889$	5.76360e15	+	7.09407e15i	0.348124	+	0.428485i
$890$		0			0
$891$	−8.34507e14	−	1.44541e15i	−0.0497855	−	0.0862310i
$892$		0			0
$893$	4.33321e15	−	7.50534e15i	0.255345	−	0.442270i
$894$		0			0
$895$	4.34455e15			0.252882
$896$		0			0
$897$	−2.39876e15			−0.137920
$898$		0			0
$899$	−4.42451e14	+	7.66348e14i	−0.0251297	+	0.0435259i
$900$		0			0
$901$	−4.38159e15	−	7.58913e15i	−0.245836	−	0.425800i
$902$		0			0
$903$	9.14563e15	−	1.46362e15i	0.506910	−	0.0811234i
$904$		0			0
$905$	4.98627e15	+	8.63646e15i	0.273029	+	0.472899i
$906$		0			0
$907$	4.85838e15	−	8.41496e15i	0.262816	−	0.455210i	−0.704173	−	0.710028i	$-0.748682\pi$
$907$	4.85838e15	−	8.41496e15i	0.262816	−	0.455210i	0.966989	+	0.254818i	$0.0820155\pi$
$908$		0			0
$909$	8.82106e15			0.471433
$910$		0			0
$911$	−5.93775e15			−0.313524			−0.156762	−	0.987636i	$-0.550106\pi$
$911$	−5.93775e15			−0.313524			−0.156762	+	0.987636i	$0.550106\pi$
$912$		0			0
$913$	−1.14406e16	+	1.98158e16i	−0.596845	+	1.03377i
$914$		0			0
$915$	2.49352e15	+	4.31890e15i	0.128528	+	0.222616i
$916$		0			0
$917$	−5.43890e14	+	1.42373e15i	−0.0277000	+	0.0725096i
$918$		0			0
$919$	1.72099e15	+	2.98085e15i	0.0866052	+	0.150005i	0.906074	−	0.423119i	$-0.139065\pi$
$919$	1.72099e15	+	2.98085e15i	0.0866052	+	0.150005i	−0.819469	+	0.573124i	$0.805731\pi$
$920$		0			0
$921$	−2.71604e15	+	4.70432e15i	−0.135054	+	0.233921i
$922$		0			0
$923$	−1.20991e16			−0.594491
$924$		0			0
$925$	−1.38928e16			−0.674544
$926$		0			0
$927$	2.14302e15	−	3.71182e15i	0.102822	−	0.178094i
$928$		0			0
$929$	−6.19181e13	−	1.07245e14i	−0.00293583	−	0.00508501i	0.864554	−	0.502540i	$-0.167601\pi$
$929$	−6.19181e13	−	1.07245e14i	−0.00293583	−	0.00508501i	−0.867490	+	0.497455i	$0.834268\pi$
$930$		0			0
$931$	1.71056e16	−	5.61889e15i	0.801520	−	0.263286i
$932$		0			0
$933$	6.45922e15	+	1.11877e16i	0.299110	+	0.518074i
$934$		0			0
$935$	−3.08913e15	+	5.35053e15i	−0.141375	+	0.244868i
$936$		0			0
$937$	1.36507e16			0.617431			0.308715	−	0.951155i	$-0.400101\pi$
$937$	1.36507e16			0.617431			0.308715	+	0.951155i	$0.400101\pi$
$938$		0			0
$939$	1.36079e16			0.608318
$940$		0			0
$941$	−5.74968e15	+	9.95874e15i	−0.254039	+	0.440009i	−0.964634	−	0.263593i	$-0.915093\pi$
$941$	−5.74968e15	+	9.95874e15i	−0.254039	+	0.440009i	0.710595	+	0.703601i	$0.248426\pi$
$942$		0			0
$943$	−2.47458e14	−	4.28609e14i	−0.0108065	−	0.0187175i
$944$		0			0
$945$	−6.10149e14	+	1.59717e15i	−0.0263367	+	0.0689409i
$946$		0			0
$947$	7.87367e15	+	1.36376e16i	0.335933	+	0.581853i	0.983664	−	0.180017i	$-0.0576152\pi$
$947$	7.87367e15	+	1.36376e16i	0.335933	+	0.581853i	−0.647731	+	0.761869i	$0.724282\pi$
$948$		0			0
$949$	−5.40007e15	+	9.35320e15i	−0.227738	+	0.394454i
$950$		0			0
$951$	1.66128e16			0.692548
$952$		0			0
$953$	1.58973e15			0.0655108			0.0327554	−	0.999463i	$-0.489572\pi$
$953$	1.58973e15			0.0655108			0.0327554	+	0.999463i	$0.489572\pi$
$954$		0			0
$955$	3.09439e15	−	5.35965e15i	0.126054	−	0.218332i
$956$		0			0
$957$	−2.12755e15	−	3.68502e15i	−0.0856770	−	0.148397i
$958$		0			0
$959$	−6.76339e15	+	1.08238e15i	−0.269254	+	0.0430901i
$960$		0			0
$961$	1.24117e16	+	2.14976e16i	0.488486	+	0.846082i
$962$		0			0
$963$	7.44742e15	−	1.28993e16i	0.289775	−	0.501905i
$964$		0			0
$965$	−3.57479e15			−0.137515
$966$		0			0
$967$	1.96093e16			0.745791			0.372896	−	0.927873i	$-0.378365\pi$
$967$	1.96093e16			0.745791			0.372896	+	0.927873i	$0.378365\pi$
$968$		0			0
$969$	−5.32896e15	+	9.23003e15i	−0.200383	+	0.347074i
$970$		0			0
$971$	1.47881e14	+	2.56137e14i	0.00549802	+	0.00952285i	0.868761	−	0.495231i	$-0.164917\pi$
$971$	1.47881e14	+	2.56137e14i	0.00549802	+	0.00952285i	−0.863263	+	0.504754i	$0.831583\pi$
$972$		0			0
$973$	3.12585e15	+	3.84743e15i	0.114907	+	0.141433i
$974$		0			0
$975$	−5.92340e15	−	1.02596e16i	−0.215301	−	0.372912i
$976$		0			0
$977$	3.17338e15	−	5.49645e15i	0.114052	−	0.197543i	−0.803349	−	0.595509i	$-0.796950\pi$
$977$	3.17338e15	−	5.49645e15i	0.114052	−	0.197543i	0.917400	+	0.397966i	$0.130284\pi$
$978$		0			0
$979$	9.29347e15			0.330273
$980$		0			0
$981$	−8.08663e15			−0.284177
$982$		0			0
$983$	1.25087e16	−	2.16656e16i	0.434677	−	0.752882i	−0.562592	−	0.826734i	$-0.690196\pi$
$983$	1.25087e16	−	2.16656e16i	0.434677	−	0.752882i	0.997269	+	0.0738522i	$0.0235293\pi$
$984$		0			0
$985$	−5.29298e15	−	9.16770e15i	−0.181887	−	0.315037i
$986$		0			0
$987$	6.48499e15	+	7.98199e15i	0.220376	+	0.271248i
$988$		0			0
$989$	−3.61415e15	−	6.25988e15i	−0.121458	−	0.210371i
$990$		0			0
$991$	9.52802e15	−	1.65030e16i	0.316663	−	0.548477i	−0.663126	−	0.748507i	$-0.730771\pi$
$991$	9.52802e15	−	1.65030e16i	0.316663	−	0.548477i	0.979790	+	0.200031i	$0.0641042\pi$
$992$		0			0
$993$	9.33639e15			0.306873
$994$		0			0
$995$	2.78132e15			0.0904117
$996$		0			0
$997$	−1.01547e16	+	1.75884e16i	−0.326470	+	0.565463i	−0.981809	−	0.189872i	$-0.939193\pi$
$997$	−1.01547e16	+	1.75884e16i	−0.326470	+	0.565463i	0.655339	+	0.755335i	$0.272526\pi$
$998$		0			0
$999$	−2.39325e15	−	4.14522e15i	−0.0760987	−	0.131807i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.i.b.25.3	✓	16
3.2	odd	2		252.12.k.d.109.6		16
7.2	even	3	inner	84.12.i.b.37.3	yes	16
21.2	odd	6		252.12.k.d.37.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.b.25.3	✓	16	1.1	even	1	trivial
84.12.i.b.37.3	yes	16	7.2	even	3	inner
252.12.k.d.37.6		16	21.2	odd	6
252.12.k.d.109.6		16	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-121.500 + 210.444i) q^{3} +(-1339.81 - 2320.63i) q^{5} +(-28039.7 - 34512.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})\)	\(q+(-121.500 + 210.444i) q^{3} +(-1339.81 - 2320.63i) q^{5} +(-28039.7 - 34512.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-239334. + 414539. i) q^{11} -1.17059e6 q^{13} +651149. q^{15} +(-2.40839e6 + 4.17145e6i) q^{17} +(-4.55281e6 - 7.88570e6i) q^{19} +(1.06697e7 - 1.70754e6i) q^{21} +(-4.21644e6 - 7.30309e6i) q^{23} +(2.08239e7 - 3.60680e7i) q^{25} +1.43489e7 q^{27} +3.65821e7 q^{29} +(-1.20947e7 + 2.09487e7i) q^{31} +(-5.81582e7 - 1.00733e8i) q^{33} +(-4.25223e7 + 1.11310e8i) q^{35} +(-1.66789e8 - 2.88888e8i) q^{37} +(1.42226e8 - 2.46343e8i) q^{39} +5.86887e7 q^{41} +8.57155e8 q^{43} +(-7.91147e7 + 1.37031e8i) q^{45} +(4.75883e8 + 8.24254e8i) q^{47} +(-4.04880e8 + 1.93543e9i) q^{49} +(-5.85239e8 - 1.01366e9i) q^{51} +(-9.09651e8 + 1.57556e9i) q^{53} +1.28265e9 q^{55} +2.21267e9 q^{57} +(8.43761e8 - 1.46144e9i) q^{59} +(3.82941e9 + 6.63273e9i) q^{61} +(-9.37033e8 + 2.45285e9i) q^{63} +(1.56837e9 + 2.71649e9i) q^{65} +(-2.92346e9 + 5.06358e9i) q^{67} +2.04919e9 q^{69} +1.03360e10 q^{71} +(4.61314e9 - 7.99018e9i) q^{73} +(5.06020e9 + 8.76452e9i) q^{75} +(2.10176e10 - 3.36355e9i) q^{77} +(9.94865e9 + 1.72316e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +4.78020e10 q^{83} +1.29072e10 q^{85} +(-4.44472e9 + 7.69849e9i) q^{87} +(-9.70762e9 - 1.68141e10i) q^{89} +(3.28229e10 + 4.03997e10i) q^{91} +(-2.93902e9 - 5.09054e9i) q^{93} +(-1.21998e10 + 2.11307e10i) q^{95} +8.32856e10 q^{97} +2.82649e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9}+O(q^{10}) \) 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