LMFDB - Embedded newform 84.12.i.b.25.1

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			25.1
Root			$13166.9 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.25
Dual form			84.12.i.b.37.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-121.500 + 210.444i) q^{3} +(-6718.22 - 11636.3i) q^{5} +(41445.4 - 16112.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})$	\(q+(-121.500 + 210.444i) q^{3} +(-6718.22 - 11636.3i) q^{5} +(41445.4 - 16112.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-425643. + 737235. i) q^{11} +2.13303e6 q^{13} +3.26505e6 q^{15} +(1.28105e6 - 2.21884e6i) q^{17} +(8.87791e6 + 1.53770e7i) q^{19} +(-1.64485e6 + 1.06796e7i) q^{21} +(-1.81716e7 - 3.14742e7i) q^{23} +(-6.58548e7 + 1.14064e8i) q^{25} +1.43489e7 q^{27} +571626. q^{29} +(1.07907e8 - 1.86901e8i) q^{31} +(-1.03431e8 - 1.79148e8i) q^{33} +(-4.65927e8 - 3.74024e8i) q^{35} +(-9.87524e7 - 1.71044e8i) q^{37} +(-2.59163e8 + 4.48883e8i) q^{39} -2.72342e8 q^{41} +1.11575e9 q^{43} +(-3.96704e8 + 6.87111e8i) q^{45} +(3.98664e8 + 6.90506e8i) q^{47} +(1.45811e9 - 1.33557e9i) q^{49} +(3.11295e8 + 5.39178e8i) q^{51} +(-6.76715e8 + 1.17210e9i) q^{53} +1.14382e10 q^{55} -4.31466e9 q^{57} +(3.81442e8 - 6.60677e8i) q^{59} +(-2.04039e9 - 3.53405e9i) q^{61} +(-2.04761e9 - 1.64372e9i) q^{63} +(-1.43301e10 - 2.48205e10i) q^{65} +(1.08784e9 - 1.88419e9i) q^{67} +8.83141e9 q^{69} +1.82655e10 q^{71} +(1.03329e10 - 1.78972e10i) q^{73} +(-1.60027e10 - 2.77175e10i) q^{75} +(-5.76230e9 + 3.74131e10i) q^{77} +(-1.28553e10 - 2.22661e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +1.36264e10 q^{83} -3.44255e10 q^{85} +(-6.94525e7 + 1.20295e8i) q^{87} +(5.18461e9 + 8.98001e9i) q^{89} +(8.84041e10 - 3.43682e10i) q^{91} +(2.62214e10 + 4.54168e10i) q^{93} +(1.19287e11 - 2.06612e11i) q^{95} -1.14522e11 q^{97} +5.02675e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−6718.22	−	11636.3i	−0.961433	−	1.66525i	−0.718908	−	0.695106i	$-0.755358\pi$
$5$	−6718.22	−	11636.3i	−0.961433	−	1.66525i	−0.242525	−	0.970145i	$-0.577976\pi$
$6$		0			0
$7$	41445.4	−	16112.4i	0.932045	−	0.362344i
$7$	41445.4	−	16112.4i	0.932045	−	0.362344i
$8$		0			0
$9$	−29524.5	−	51137.9i	−0.166667	−	0.288675i
$10$		0			0
$11$	−425643.	+	737235.i	−0.796866	+	1.38021i	0.124781	+	0.992184i	$0.460177\pi$
$11$	−425643.	+	737235.i	−0.796866	+	1.38021i	−0.921647	+	0.388028i	$0.873156\pi$
$12$		0			0
$13$	2.13303e6			1.59334			0.796669	−	0.604416i	$-0.206593\pi$
$13$	2.13303e6			1.59334			0.796669	+	0.604416i	$0.206593\pi$
$14$		0			0
$15$	3.26505e6			1.11017
$16$		0			0
$17$	1.28105e6	−	2.21884e6i	0.218825	−	0.379016i	−0.735624	−	0.677390i	$-0.763111\pi$
$17$	1.28105e6	−	2.21884e6i	0.218825	−	0.379016i	0.954449	+	0.298374i	$0.0964443\pi$
$18$		0			0
$19$	8.87791e6	+	1.53770e7i	0.822557	+	1.42471i	0.903772	+	0.428014i	$0.140787\pi$
$19$	8.87791e6	+	1.53770e7i	0.822557	+	1.42471i	−0.0812152	+	0.996697i	$0.525880\pi$
$20$		0			0
$21$	−1.64485e6	+	1.06796e7i	−0.0878862	+	0.570622i
$22$		0			0
$23$	−1.81716e7	−	3.14742e7i	−0.588695	−	1.01965i	−0.994404	−	0.105648i	$-0.966308\pi$
$23$	−1.81716e7	−	3.14742e7i	−0.588695	−	1.01965i	0.405708	−	0.914003i	$-0.367025\pi$
$24$		0			0
$25$	−6.58548e7	+	1.14064e8i	−1.34871	+	2.33603i
$26$		0			0
$27$	1.43489e7			0.192450
$28$		0			0
$29$	571626.			0.00517515			0.00258757	−	0.999997i	$-0.499176\pi$
$29$	571626.			0.00517515			0.00258757	+	0.999997i	$0.499176\pi$
$30$		0			0
$31$	1.07907e8	−	1.86901e8i	0.676956	−	1.17252i	−0.298936	−	0.954273i	$-0.596632\pi$
$31$	1.07907e8	−	1.86901e8i	0.676956	−	1.17252i	0.975893	−	0.218250i	$-0.0700348\pi$
$32$		0			0
$33$	−1.03431e8	−	1.79148e8i	−0.460071	−	0.796866i
$34$		0			0
$35$	−4.65927e8	−	3.74024e8i	−1.49949	−	1.20372i
$36$		0			0
$37$	−9.87524e7	−	1.71044e8i	−0.234120	−	0.405508i	0.724897	−	0.688858i	$-0.241887\pi$
$37$	−9.87524e7	−	1.71044e8i	−0.234120	−	0.405508i	−0.959017	+	0.283350i	$0.908554\pi$
$38$		0			0
$39$	−2.59163e8	+	4.48883e8i	−0.459957	+	0.796669i
$40$		0			0
$41$	−2.72342e8			−0.367116			−0.183558	−	0.983009i	$-0.558762\pi$
$41$	−2.72342e8			−0.367116			−0.183558	+	0.983009i	$0.558762\pi$
$42$		0			0
$43$	1.11575e9			1.15741			0.578707	−	0.815536i	$-0.303558\pi$
$43$	1.11575e9			1.15741			0.578707	+	0.815536i	$0.303558\pi$
$44$		0			0
$45$	−3.96704e8	+	6.87111e8i	−0.320478	+	0.555084i
$46$		0			0
$47$	3.98664e8	+	6.90506e8i	0.253553	+	0.439167i	0.964501	−	0.264077i	$-0.0850675\pi$
$47$	3.98664e8	+	6.90506e8i	0.253553	+	0.439167i	−0.710949	+	0.703244i	$0.751734\pi$
$48$		0			0
$49$	1.45811e9	−	1.33557e9i	0.737414	−	0.675441i
$50$		0			0
$51$	3.11295e8	+	5.39178e8i	0.126339	+	0.218825i
$52$		0			0
$53$	−6.76715e8	+	1.17210e9i	−0.222274	+	0.384990i	−0.955498	−	0.294997i	$-0.904681\pi$
$53$	−6.76715e8	+	1.17210e9i	−0.222274	+	0.384990i	0.733224	+	0.679987i	$0.238015\pi$
$54$		0			0
$55$	1.14382e10			3.06453
$56$		0			0
$57$	−4.31466e9			−0.949807
$58$		0			0
$59$	3.81442e8	−	6.60677e8i	0.0694613	−	0.120310i	−0.829203	−	0.558948i	$-0.811205\pi$
$59$	3.81442e8	−	6.60677e8i	0.0694613	−	0.120310i	0.898664	+	0.438637i	$0.144539\pi$
$60$		0			0
$61$	−2.04039e9	−	3.53405e9i	−0.309313	−	0.535746i	0.668899	−	0.743353i	$-0.266766\pi$
$61$	−2.04039e9	−	3.53405e9i	−0.309313	−	0.535746i	−0.978212	+	0.207607i	$0.933432\pi$
$62$		0			0
$63$	−2.04761e9	−	1.64372e9i	−0.259940	−	0.208667i
$64$		0			0
$65$	−1.43301e10	−	2.48205e10i	−1.53189	−	2.65331i
$66$		0			0
$67$	1.08784e9	−	1.88419e9i	0.0984358	−	0.170496i	−0.812602	−	0.582820i	$-0.801949\pi$
$67$	1.08784e9	−	1.88419e9i	0.0984358	−	0.170496i	0.911037	+	0.412324i	$0.135283\pi$
$68$		0			0
$69$	8.83141e9			0.679767
$70$		0			0
$71$	1.82655e10			1.20147			0.600733	−	0.799450i	$-0.294876\pi$
$71$	1.82655e10			1.20147			0.600733	+	0.799450i	$0.294876\pi$
$72$		0			0
$73$	1.03329e10	−	1.78972e10i	0.583375	−	1.01043i	−0.411701	−	0.911319i	$-0.635065\pi$
$73$	1.03329e10	−	1.78972e10i	0.583375	−	1.01043i	0.995076	−	0.0991158i	$-0.0316014\pi$
$74$		0			0
$75$	−1.60027e10	−	2.77175e10i	−0.778676	−	1.34871i
$76$		0			0
$77$	−5.76230e9	+	3.74131e10i	−0.242603	+	1.57516i
$78$		0			0
$79$	−1.28553e10	−	2.22661e10i	−0.470039	−	0.814132i	0.529374	−	0.848389i	$-0.322427\pi$
$79$	−1.28553e10	−	2.22661e10i	−0.470039	−	0.814132i	−0.999413	+	0.0342567i	$0.989094\pi$
$80$		0			0
$81$	−1.74339e9	+	3.01964e9i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	1.36264e10			0.379710			0.189855	−	0.981812i	$-0.439198\pi$
$83$	1.36264e10			0.379710			0.189855	+	0.981812i	$0.439198\pi$
$84$		0			0
$85$	−3.44255e10			−0.841542
$86$		0			0
$87$	−6.94525e7	+	1.20295e8i	−0.00149394	+	0.00258757i
$88$		0			0
$89$	5.18461e9	+	8.98001e9i	0.0984173	+	0.170464i	0.911030	−	0.412341i	$-0.135289\pi$
$89$	5.18461e9	+	8.98001e9i	0.0984173	+	0.170464i	−0.812612	+	0.582804i	$0.801955\pi$
$90$		0			0
$91$	8.84041e10	−	3.43682e10i	1.48506	−	0.577336i
$92$		0			0
$93$	2.62214e10	+	4.54168e10i	0.390841	+	0.676956i
$94$		0			0
$95$	1.19287e11	−	2.06612e11i	1.58167	−	2.73953i
$96$		0			0
$97$	−1.14522e11			−1.35408			−0.677040	−	0.735947i	$-0.736737\pi$
$97$	−1.14522e11			−1.35408			−0.677040	+	0.735947i	$0.736737\pi$
$98$		0			0
$99$	5.02675e10			0.531244
$100$		0			0
$101$	1.25331e10	−	2.17079e10i	0.118656	−	0.205518i	−0.800579	−	0.599227i	$-0.795475\pi$
$101$	1.25331e10	−	2.17079e10i	0.118656	−	0.205518i	0.919235	+	0.393708i	$0.128808\pi$
$102$		0			0
$103$	−3.90267e10	−	6.75962e10i	−0.331709	−	0.574536i	0.651138	−	0.758959i	$-0.274292\pi$
$103$	−3.90267e10	−	6.75962e10i	−0.331709	−	0.574536i	−0.982847	+	0.184423i	$0.940958\pi$
$104$		0			0
$105$	1.35321e11	−	5.26078e10i	1.03473	−	0.402262i
$106$		0			0
$107$	−1.14704e11	−	1.98672e11i	−0.790617	−	1.36939i	−0.925585	−	0.378539i	$-0.876427\pi$
$107$	−1.14704e11	−	1.98672e11i	−0.790617	−	1.36939i	0.134969	−	0.990850i	$-0.456907\pi$
$108$		0			0
$109$	1.04619e11	−	1.81206e11i	0.651278	−	1.12805i	−0.331535	−	0.943443i	$-0.607567\pi$
$109$	1.04619e11	−	1.81206e11i	0.651278	−	1.12805i	0.982813	−	0.184603i	$-0.0591001\pi$
$110$		0			0
$111$	4.79937e10			0.270338
$112$		0			0
$113$	−2.19133e11			−1.11886			−0.559432	−	0.828877i	$-0.688981\pi$
$113$	−2.19133e11			−1.11886			−0.559432	+	0.828877i	$0.688981\pi$
$114$		0			0
$115$	−2.44162e11	+	4.22901e11i	−1.13198	+	1.96065i
$116$		0			0
$117$	−6.29766e10	−	1.09079e11i	−0.265556	−	0.459957i
$118$		0			0
$119$	1.73427e10	−	1.12601e11i	0.0666205	−	0.432549i
$120$		0			0
$121$	−2.19687e11	−	3.80510e11i	−0.769991	−	1.33366i
$122$		0			0
$123$	3.30896e10	−	5.73128e10i	0.105977	−	0.183558i
$124$		0			0
$125$	1.11363e12			3.26390
$126$		0			0
$127$	2.57513e11			0.691639			0.345819	−	0.938301i	$-0.387601\pi$
$127$	2.57513e11			0.691639			0.345819	+	0.938301i	$0.387601\pi$
$128$		0			0
$129$	−1.35563e11	+	2.34802e11i	−0.334117	+	0.578707i
$130$		0			0
$131$	−2.19906e11	−	3.80889e11i	−0.498018	−	0.862593i	0.501979	−	0.864880i	$-0.332606\pi$
$131$	−2.19906e11	−	3.80889e11i	−0.498018	−	0.862593i	−0.999997	+	0.00228653i	$0.999272\pi$
$132$		0			0
$133$	6.15708e11	+	4.94261e11i	1.28289	+	1.02985i
$134$		0			0
$135$	−9.63991e10	−	1.66968e11i	−0.185028	−	0.320478i
$136$		0			0
$137$	−2.51907e11	+	4.36315e11i	−0.445940	+	0.772391i	−0.998117	−	0.0613357i	$-0.980464\pi$
$137$	−2.51907e11	+	4.36315e11i	−0.445940	+	0.772391i	0.552177	+	0.833727i	$0.313797\pi$
$138$		0			0
$139$	1.93201e11			0.315812			0.157906	−	0.987454i	$-0.449526\pi$
$139$	1.93201e11			0.315812			0.157906	+	0.987454i	$0.449526\pi$
$140$		0			0
$141$	−1.93751e11			−0.292778
$142$		0			0
$143$	−9.07907e11	+	1.57254e12i	−1.26968	+	2.19915i
$144$		0			0
$145$	−3.84031e9	−	6.65161e9i	−0.00497556	−	0.00861792i
$146$		0			0
$147$	1.03902e11	+	4.69122e11i	0.124847	+	0.563690i
$148$		0			0
$149$	3.50276e10	+	6.06696e10i	0.0390738	+	0.0676778i	0.884901	−	0.465779i	$-0.154226\pi$
$149$	3.50276e10	+	6.06696e10i	0.0390738	+	0.0676778i	−0.845827	+	0.533457i	$0.820893\pi$
$150$		0			0
$151$	5.29667e11	−	9.17410e11i	0.549073	−	0.951022i	−0.449266	−	0.893398i	$-0.648314\pi$
$151$	5.29667e11	−	9.17410e11i	0.549073	−	0.951022i	0.998338	−	0.0576234i	$-0.0183523\pi$
$152$		0			0
$153$	−1.51289e11			−0.145883
$154$		0			0
$155$	−2.89977e12			−2.60339
$156$		0			0
$157$	−2.29713e11	+	3.97875e11i	−0.192193	+	0.332888i	−0.945977	−	0.324234i	$-0.894893\pi$
$157$	−2.29713e11	+	3.97875e11i	−0.192193	+	0.332888i	0.753784	+	0.657123i	$0.228227\pi$
$158$		0			0
$159$	−1.64442e11	−	2.84821e11i	−0.128330	−	0.222274i
$160$		0			0
$161$	−1.26025e12	−	1.01167e12i	−0.918154	−	0.737049i
$162$		0			0
$163$	−1.42888e11	−	2.47489e11i	−0.0972665	−	0.168470i	0.813286	−	0.581864i	$-0.197677\pi$
$163$	−1.42888e11	−	2.47489e11i	−0.0972665	−	0.168470i	−0.910552	+	0.413394i	$0.864343\pi$
$164$		0			0
$165$	−1.38975e12	+	2.40711e12i	−0.884655	+	1.53227i
$166$		0			0
$167$	1.47522e12			0.878850			0.439425	−	0.898279i	$-0.355182\pi$
$167$	1.47522e12			0.878850			0.439425	+	0.898279i	$0.355182\pi$
$168$		0			0
$169$	2.75764e12			1.53873
$170$		0			0
$171$	5.24232e11	−	9.07996e11i	0.274186	−	0.474903i
$172$		0			0
$173$	7.02485e11	+	1.21674e12i	0.344654	+	0.596959i	0.985291	−	0.170886i	$-0.0546628\pi$
$173$	7.02485e11	+	1.21674e12i	0.344654	+	0.596959i	−0.640637	+	0.767844i	$0.721330\pi$
$174$		0			0
$175$	−8.91535e11	+	5.78850e12i	−0.410610	+	2.66598i
$176$		0			0
$177$	9.26905e10	+	1.60545e11i	0.0401035	+	0.0694613i
$178$		0			0
$179$	1.25836e12	−	2.17955e12i	0.511816	−	0.886491i	−0.488090	−	0.872793i	$-0.662306\pi$
$179$	1.25836e12	−	2.17955e12i	0.511816	−	0.886491i	0.999906	−	0.0136982i	$-0.00436041\pi$
$180$		0			0
$181$	−3.11043e12			−1.19011			−0.595056	−	0.803684i	$-0.702870\pi$
$181$	−3.11043e12			−1.19011			−0.595056	+	0.803684i	$0.702870\pi$
$182$		0			0
$183$	9.91627e11			0.357164
$184$		0			0
$185$	−1.32688e12	+	2.29822e12i	−0.450181	+	0.779737i
$186$		0			0
$187$	1.09054e12	+	1.88887e12i	0.348748	+	0.604049i
$188$		0			0
$189$	5.94696e11	−	2.31195e11i	0.179372	−	0.0697331i
$190$		0			0
$191$	−5.63090e11	−	9.75301e11i	−0.160286	−	0.277623i	0.774686	−	0.632347i	$-0.217908\pi$
$191$	−5.63090e11	−	9.75301e11i	−0.160286	−	0.277623i	−0.934971	+	0.354724i	$0.884575\pi$
$192$		0			0
$193$	−1.08140e12	+	1.87305e12i	−0.290685	+	0.503482i	−0.973972	−	0.226669i	$-0.927217\pi$
$193$	−1.08140e12	+	1.87305e12i	−0.290685	+	0.503482i	0.683287	+	0.730150i	$0.260550\pi$
$194$		0			0
$195$	6.96445e12			1.76887
$196$		0			0
$197$	6.02832e12			1.44755			0.723773	−	0.690038i	$-0.242406\pi$
$197$	6.02832e12			1.44755			0.723773	+	0.690038i	$0.242406\pi$
$198$		0			0
$199$	−1.43126e12	+	2.47901e12i	−0.325106	+	0.563100i	−0.981534	−	0.191289i	$-0.938733\pi$
$199$	−1.43126e12	+	2.47901e12i	−0.325106	+	0.563100i	0.656428	+	0.754389i	$0.272067\pi$
$200$		0			0
$201$	2.64345e11	+	4.57859e11i	0.0568319	+	0.0984358i
$202$		0			0
$203$	2.36912e10	−	9.21026e9i	0.00482347	−	0.00187518i
$204$		0			0
$205$	1.82965e12	+	3.16905e12i	0.352958	+	0.611341i
$206$		0			0
$207$	−1.07302e12	+	1.85852e12i	−0.196232	+	0.339883i
$208$		0			0
$209$	−1.51153e13			−2.62187
$210$		0			0
$211$	−3.22306e12			−0.530535			−0.265268	−	0.964175i	$-0.585460\pi$
$211$	−3.22306e12			−0.530535			−0.265268	+	0.964175i	$0.585460\pi$
$212$		0			0
$213$	−2.21926e12	+	3.84387e12i	−0.346833	+	0.600733i
$214$		0			0
$215$	−7.49582e12	−	1.29831e13i	−1.11278	−	1.92738i
$216$		0			0
$217$	1.46083e12	−	9.48480e12i	0.206097	−	1.33813i
$218$		0			0
$219$	2.51090e12	+	4.34901e12i	0.336812	+	0.583375i
$220$		0			0
$221$	2.73251e12	−	4.73285e12i	0.348662	−	0.603900i
$222$		0			0
$223$	−1.12361e13			−1.36440			−0.682198	−	0.731167i	$-0.738976\pi$
$223$	−1.12361e13			−1.36440			−0.682198	+	0.731167i	$0.738976\pi$
$224$		0			0
$225$	7.77732e12			0.899138
$226$		0			0
$227$	7.58401e12	−	1.31359e13i	0.835135	−	1.44650i	−0.0587856	−	0.998271i	$-0.518723\pi$
$227$	7.58401e12	−	1.31359e13i	0.835135	−	1.44650i	0.893921	−	0.448226i	$-0.147944\pi$
$228$		0			0
$229$	−1.82140e12	−	3.15476e12i	−0.191122	−	0.331033i	0.754500	−	0.656300i	$-0.227879\pi$
$229$	−1.82140e12	−	3.15476e12i	−0.191122	−	0.331033i	−0.945622	+	0.325267i	$0.894546\pi$
$230$		0			0
$231$	−7.17324e12	−	5.75833e12i	−0.717546	−	0.576011i
$232$		0			0
$233$	6.03650e12	+	1.04555e13i	0.575874	+	0.997443i	0.995946	+	0.0899524i	$0.0286715\pi$
$233$	6.03650e12	+	1.04555e13i	0.575874	+	0.997443i	−0.420072	+	0.907491i	$0.637995\pi$
$234$		0			0
$235$	5.35662e12	−	9.27794e12i	0.487548	−	0.844458i
$236$		0			0
$237$	6.24769e12			0.542755
$238$		0			0
$239$	−1.62209e12			−0.134551			−0.0672756	−	0.997734i	$-0.521431\pi$
$239$	−1.62209e12			−0.134551			−0.0672756	+	0.997734i	$0.521431\pi$
$240$		0			0
$241$	7.66226e12	−	1.32714e13i	0.607104	−	1.05154i	−0.384611	−	0.923079i	$-0.625664\pi$
$241$	7.66226e12	−	1.32714e13i	0.607104	−	1.05154i	0.991715	−	0.128456i	$-0.0410022\pi$
$242$		0			0
$243$	−4.23644e11	−	7.33773e11i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−2.53369e13	−	7.99435e12i	−1.83375	−	0.578588i
$246$		0			0
$247$	1.89368e13	+	3.27995e13i	1.31061	+	2.27005i
$248$		0			0
$249$	−1.65561e12	+	2.86760e12i	−0.109613	+	0.189855i
$250$		0			0
$251$	1.41336e13			0.895461			0.447731	−	0.894169i	$-0.352232\pi$
$251$	1.41336e13			0.895461			0.447731	+	0.894169i	$0.352232\pi$
$252$		0			0
$253$	3.09385e13			1.87645
$254$		0			0
$255$	4.18269e12	−	7.24464e12i	0.242932	−	0.420771i
$256$		0			0
$257$	−7.51915e12	−	1.30235e13i	−0.418347	−	0.724598i	0.577427	−	0.816443i	$-0.304057\pi$
$257$	−7.51915e12	−	1.30235e13i	−0.418347	−	0.724598i	−0.995773	+	0.0918448i	$0.970724\pi$
$258$		0			0
$259$	−6.84876e12	−	5.49785e12i	−0.365143	−	0.293119i
$260$		0			0
$261$	−1.68770e10	−	2.92318e10i	−0.000862525	−	0.00149394i
$262$		0			0
$263$	−1.49881e13	+	2.59601e13i	−0.734496	+	1.27218i	0.220448	+	0.975399i	$0.429248\pi$
$263$	−1.49881e13	+	2.59601e13i	−0.734496	+	1.27218i	−0.954944	+	0.296786i	$0.904085\pi$
$264$		0			0
$265$	1.81853e13			0.854806
$266$		0			0
$267$	−2.51972e12			−0.113642
$268$		0			0
$269$	2.23799e13	−	3.87631e13i	0.968769	−	1.67796i	0.269642	−	0.962961i	$-0.413095\pi$
$269$	2.23799e13	−	3.87631e13i	0.968769	−	1.67796i	0.699127	−	0.714997i	$-0.253572\pi$
$270$		0			0
$271$	5.30406e12	+	9.18689e12i	0.220433	+	0.381801i	0.954940	−	0.296800i	$-0.0959196\pi$
$271$	5.30406e12	+	9.18689e12i	0.220433	+	0.381801i	−0.734506	+	0.678602i	$0.762586\pi$
$272$		0			0
$273$	−3.50851e12	+	2.27799e13i	−0.140032	+	0.909194i
$274$		0			0
$275$	−5.60612e13	−	9.71009e13i	−2.14948	−	3.72300i
$276$		0			0
$277$	2.06423e13	−	3.57535e13i	0.760536	−	1.31729i	−0.182039	−	0.983291i	$-0.558270\pi$
$277$	2.06423e13	−	3.57535e13i	0.760536	−	1.31729i	0.942575	−	0.333996i	$-0.108397\pi$
$278$		0			0
$279$	−1.27436e13			−0.451304
$280$		0			0
$281$	5.69959e12			0.194070			0.0970350	−	0.995281i	$-0.469064\pi$
$281$	5.69959e12			0.194070			0.0970350	+	0.995281i	$0.469064\pi$
$282$		0			0
$283$	−2.13160e11	+	3.69205e11i	−0.00698041	+	0.0120904i	−0.869495	−	0.493943i	$-0.835555\pi$
$283$	−2.13160e11	+	3.69205e11i	−0.00698041	+	0.0120904i	0.862514	+	0.506033i	$0.168889\pi$
$284$		0			0
$285$	2.89869e13	+	5.02067e13i	0.913176	+	1.58167i
$286$		0			0
$287$	−1.12873e13	+	4.38808e12i	−0.342169	+	0.133022i
$288$		0			0
$289$	1.38538e13	+	2.39954e13i	0.404231	+	0.700149i
$290$		0			0
$291$	1.39144e13	−	2.41005e13i	0.390889	−	0.677040i
$292$		0			0
$293$	−1.60966e13			−0.435473			−0.217737	−	0.976008i	$-0.569867\pi$
$293$	−1.60966e13			−0.435473			−0.217737	+	0.976008i	$0.569867\pi$
$294$		0			0
$295$	−1.02504e13			−0.267129
$296$		0			0
$297$	−6.10751e12	+	1.05785e13i	−0.153357	+	0.265622i
$298$		0			0
$299$	−3.87606e13	−	6.71353e13i	−0.937991	−	1.62465i
$300$		0			0
$301$	4.62425e13	−	1.79773e13i	1.07876	−	0.419382i
$302$		0			0
$303$	3.04554e12	+	5.27503e12i	0.0685062	+	0.118656i
$304$		0			0
$305$	−2.74155e13	+	4.74850e13i	−0.594767	+	1.03017i
$306$		0			0
$307$	−5.67881e13			−1.18849			−0.594246	−	0.804283i	$-0.702550\pi$
$307$	−5.67881e13			−1.18849			−0.594246	+	0.804283i	$0.702550\pi$
$308$		0			0
$309$	1.89670e13			0.383024
$310$		0			0
$311$	−2.92156e13	+	5.06029e13i	−0.569420	+	0.986264i	0.427203	+	0.904156i	$0.359499\pi$
$311$	−2.92156e13	+	5.06029e13i	−0.569420	+	0.986264i	−0.996623	+	0.0821088i	$0.973835\pi$
$312$		0			0
$313$	7.17860e12	+	1.24337e13i	0.135066	+	0.233941i	0.925623	−	0.378448i	$-0.123542\pi$
$313$	7.17860e12	+	1.24337e13i	0.135066	+	0.233941i	−0.790557	+	0.612389i	$0.790209\pi$
$314$		0			0
$315$	−5.37053e12	+	3.48694e13i	−0.0975684	+	0.633486i
$316$		0			0
$317$	−1.47676e13	−	2.55782e13i	−0.259109	−	0.448790i	0.706894	−	0.707319i	$-0.250096\pi$
$317$	−1.47676e13	−	2.55782e13i	−0.259109	−	0.448790i	−0.966004	+	0.258529i	$0.916762\pi$
$318$		0			0
$319$	−2.43308e11	+	4.21422e11i	−0.00412390	+	0.00714281i
$320$		0			0
$321$	5.57459e13			0.912926
$322$		0			0
$323$	4.54921e13			0.719983
$324$		0			0
$325$	−1.40470e14	+	2.43301e14i	−2.14895	+	3.72208i
$326$		0			0
$327$	2.54225e13	+	4.40331e13i	0.376015	+	0.651278i
$328$		0			0
$329$	2.76485e13	+	2.21948e13i	0.395452	+	0.317449i
$330$		0			0
$331$	5.55541e13	+	9.62225e13i	0.768532	+	1.33114i	0.938359	+	0.345663i	$0.112346\pi$
$331$	5.55541e13	+	9.62225e13i	0.768532	+	1.33114i	−0.169827	+	0.985474i	$0.554321\pi$
$332$		0			0
$333$	−5.83123e12	+	1.01000e13i	−0.0780400	+	0.135169i
$334$		0			0
$335$	−2.92334e13			−0.378558
$336$		0			0
$337$	−8.99428e13			−1.12720			−0.563601	−	0.826047i	$-0.690585\pi$
$337$	−8.99428e13			−1.12720			−0.563601	+	0.826047i	$0.690585\pi$
$338$		0			0
$339$	2.66247e13	−	4.61153e13i	0.322988	−	0.559432i
$340$		0			0
$341$	9.18597e13	+	1.59106e14i	1.07889	+	1.86869i
$342$		0			0
$343$	3.89127e13	−	7.88467e13i	0.442561	−	0.896738i
$344$		0			0
$345$	−5.93313e13	−	1.02765e14i	−0.653550	−	1.13198i
$346$		0			0
$347$	5.22864e13	−	9.05626e13i	0.557926	−	0.966355i	−0.439744	−	0.898123i	$-0.644931\pi$
$347$	5.22864e13	−	9.05626e13i	0.557926	−	0.966355i	0.997669	−	0.0682322i	$-0.0217359\pi$
$348$		0			0
$349$	5.39858e13			0.558136			0.279068	−	0.960271i	$-0.409975\pi$
$349$	5.39858e13			0.558136			0.279068	+	0.960271i	$0.409975\pi$
$350$		0			0
$351$	3.06066e13			0.306638
$352$		0			0
$353$	1.09068e13	−	1.88911e13i	0.105910	−	0.183441i	−0.808200	−	0.588909i	$-0.799558\pi$
$353$	1.09068e13	−	1.88911e13i	0.105910	−	0.183441i	0.914110	+	0.405467i	$0.132891\pi$
$354$		0			0
$355$	−1.22712e14	−	2.12543e14i	−1.15513	−	2.00074i
$356$		0			0
$357$	2.15892e13	+	1.73307e13i	0.197043	+	0.158176i
$358$		0			0
$359$	−6.99484e13	−	1.21154e14i	−0.619096	−	1.07231i	−0.989651	−	0.143495i	$-0.954166\pi$
$359$	−6.99484e13	−	1.21154e14i	−0.619096	−	1.07231i	0.370555	−	0.928811i	$-0.379168\pi$
$360$		0			0
$361$	−9.93895e13	+	1.72148e14i	−0.853200	+	1.47779i
$362$		0			0
$363$	1.06768e14			0.889109
$364$		0			0
$365$	−2.77675e14			−2.24350
$366$		0			0
$367$	−1.60223e13	+	2.77514e13i	−0.125621	+	0.217581i	−0.921975	−	0.387249i	$-0.873426\pi$
$367$	−1.60223e13	+	2.77514e13i	−0.125621	+	0.217581i	0.796355	+	0.604830i	$0.206759\pi$
$368$		0			0
$369$	8.04077e12	+	1.39270e13i	0.0611860	+	0.105977i
$370$		0			0
$371$	−9.16129e12	+	5.94818e13i	−0.0676706	+	0.439367i
$372$		0			0
$373$	9.32722e13	+	1.61552e14i	0.668888	+	1.15855i	0.978215	+	0.207593i	$0.0665629\pi$
$373$	9.32722e13	+	1.61552e14i	0.668888	+	1.15855i	−0.309327	+	0.950956i	$0.600104\pi$
$374$		0			0
$375$	−1.35306e14	+	2.34357e14i	−0.942206	+	1.63195i
$376$		0			0
$377$	1.21929e12			0.00824576
$378$		0			0
$379$	2.85750e13			0.187703			0.0938515	−	0.995586i	$-0.470082\pi$
$379$	2.85750e13			0.187703			0.0938515	+	0.995586i	$0.470082\pi$
$380$		0			0
$381$	−3.12879e13	+	5.41922e13i	−0.199659	+	0.345819i
$382$		0			0
$383$	−1.81568e13	−	3.14486e13i	−0.112576	−	0.194988i	0.804232	−	0.594315i	$-0.202577\pi$
$383$	−1.81568e13	−	3.14486e13i	−0.112576	−	0.194988i	−0.916808	+	0.399328i	$0.869244\pi$
$384$		0			0
$385$	4.74062e14	−	1.84297e14i	2.85628	−	1.11041i
$386$		0			0
$387$	−3.29418e13	−	5.70569e13i	−0.192902	−	0.334117i
$388$		0			0
$389$	8.04933e13	−	1.39419e14i	0.458181	−	0.793593i	−0.540684	−	0.841226i	$-0.681835\pi$
$389$	8.04933e13	−	1.39419e14i	0.458181	−	0.793593i	0.998865	+	0.0476332i	$0.0151678\pi$
$390$		0			0
$391$	−9.31149e13			−0.515285
$392$		0			0
$393$	1.06874e14			0.575062
$394$		0			0
$395$	−1.72730e14	+	2.99177e14i	−0.903823	+	1.56547i
$396$		0			0
$397$	1.79387e14	+	3.10707e14i	0.912940	+	1.58126i	0.809889	+	0.586583i	$0.199527\pi$
$397$	1.79387e14	+	3.10707e14i	0.912940	+	1.58126i	0.103052	+	0.994676i	$0.467139\pi$
$398$		0			0
$399$	−1.78823e14	+	6.95196e13i	−0.885262	+	0.344157i
$400$		0			0
$401$	−8.22260e13	−	1.42420e14i	−0.396018	−	0.685923i	0.597213	−	0.802083i	$-0.296275\pi$
$401$	−8.22260e13	−	1.42420e14i	−0.396018	−	0.685923i	−0.993231	+	0.116160i	$0.962942\pi$
$402$		0			0
$403$	2.30169e14	−	3.98664e14i	1.07862	−	1.86823i
$404$		0			0
$405$	4.68499e13			0.213652
$406$		0			0
$407$	1.68133e14			0.746249
$408$		0			0
$409$	3.13369e13	−	5.42771e13i	0.135387	−	0.234498i	−0.790358	−	0.612645i	$-0.790105\pi$
$409$	3.13369e13	−	5.42771e13i	0.135387	−	0.234498i	0.925745	+	0.378148i	$0.123439\pi$
$410$		0			0
$411$	−6.12133e13	−	1.06025e14i	−0.257464	−	0.445940i
$412$		0			0
$413$	5.16392e12	−	3.35280e13i	0.0211473	−	0.137304i
$414$		0			0
$415$	−9.15452e13	−	1.58561e14i	−0.365066	−	0.632312i
$416$		0			0
$417$	−2.34739e13	+	4.06580e13i	−0.0911669	+	0.157906i
$418$		0			0
$419$	8.23062e13			0.311355			0.155677	−	0.987808i	$-0.450244\pi$
$419$	8.23062e13			0.311355			0.155677	+	0.987808i	$0.450244\pi$
$420$		0			0
$421$	−6.99072e13			−0.257614			−0.128807	−	0.991670i	$-0.541115\pi$
$421$	−6.99072e13			−0.257614			−0.128807	+	0.991670i	$0.541115\pi$
$422$		0			0
$423$	2.35407e13	−	4.07737e13i	0.0845176	−	0.146389i
$424$		0			0
$425$	1.68726e14	+	2.92243e14i	0.590261	+	1.02236i
$426$		0			0
$427$	−1.41506e14	−	1.13595e14i	−0.482417	−	0.387261i
$428$		0			0
$429$	−2.20621e14	−	3.82128e14i	−0.733048	−	1.26968i
$430$		0			0
$431$	−1.23307e14	+	2.13574e14i	−0.399358	+	0.691708i	−0.993647	−	0.112544i	$-0.964100\pi$
$431$	−1.23307e14	+	2.13574e14i	−0.399358	+	0.691708i	0.594289	+	0.804252i	$0.297434\pi$
$432$		0			0
$433$	−4.21634e14			−1.33123			−0.665614	−	0.746296i	$-0.731830\pi$
$433$	−4.21634e14			−1.33123			−0.665614	+	0.746296i	$0.731830\pi$
$434$		0			0
$435$	1.86639e12			0.00574528
$436$		0			0
$437$	3.22652e14	−	5.58850e14i	0.968471	−	1.67744i
$438$		0			0
$439$	−4.27459e13	−	7.40380e13i	−0.125124	−	0.216720i	0.796658	−	0.604431i	$-0.206599\pi$
$439$	−4.27459e13	−	7.40380e13i	−0.125124	−	0.216720i	−0.921781	+	0.387710i	$0.873266\pi$
$440$		0			0
$441$	−1.11348e14	−	3.51327e13i	−0.317885	−	0.100300i
$442$		0			0
$443$	−2.86748e14	−	4.96662e14i	−0.798509	−	1.38306i	−0.920587	−	0.390537i	$-0.872289\pi$
$443$	−2.86748e14	−	4.96662e14i	−0.798509	−	1.38306i	0.122078	−	0.992520i	$-0.461044\pi$
$444$		0			0
$445$	6.96627e13	−	1.20659e14i	0.189243	−	0.327779i
$446$		0			0
$447$	−1.70234e13			−0.0451185
$448$		0			0
$449$	6.33244e14			1.63763			0.818815	−	0.574057i	$-0.194631\pi$
$449$	6.33244e14			1.63763			0.818815	+	0.574057i	$0.194631\pi$
$450$		0			0
$451$	1.15920e14	−	2.00780e14i	0.292543	−	0.506699i
$452$		0			0
$453$	1.28709e14	+	2.22931e14i	0.317007	+	0.549073i
$454$		0			0
$455$	−9.93836e14	−	7.97803e14i	−2.38920	−	1.91793i
$456$		0			0
$457$	1.99228e13	+	3.45073e13i	0.0467532	+	0.0809788i	0.888455	−	0.458964i	$-0.151779\pi$
$457$	1.99228e13	+	3.45073e13i	0.0467532	+	0.0809788i	−0.841702	+	0.539943i	$0.818446\pi$
$458$		0			0
$459$	1.83816e13	−	3.18379e13i	0.0421129	−	0.0729416i
$460$		0			0
$461$	8.47695e14			1.89620			0.948101	−	0.317970i	$-0.103001\pi$
$461$	8.47695e14			1.89620			0.948101	+	0.317970i	$0.103001\pi$
$462$		0			0
$463$	7.99938e14			1.74727			0.873636	−	0.486579i	$-0.161756\pi$
$463$	7.99938e14			1.74727			0.873636	+	0.486579i	$0.161756\pi$
$464$		0			0
$465$	3.52322e14	−	6.10240e14i	0.751535	−	1.30170i
$466$		0			0
$467$	4.29125e14	+	7.43267e14i	0.894008	+	1.54847i	0.835028	+	0.550207i	$0.185451\pi$
$467$	4.29125e14	+	7.43267e14i	0.894008	+	1.54847i	0.0589797	+	0.998259i	$0.481215\pi$
$468$		0			0
$469$	1.47270e13	−	9.56187e13i	0.0299685	−	0.194577i
$470$		0			0
$471$	−5.58203e13	−	9.66836e13i	−0.110963	−	0.192193i
$472$		0			0
$473$	−4.74909e14	+	8.22566e14i	−0.922304	+	1.59748i
$474$		0			0
$475$	−2.33861e15			−4.43755
$476$		0			0
$477$	7.99187e13			0.148183
$478$		0			0
$479$	−6.28415e13	+	1.08845e14i	−0.113868	+	0.197225i	−0.917327	−	0.398135i	$-0.869657\pi$
$479$	−6.28415e13	+	1.08845e14i	−0.113868	+	0.197225i	0.803459	+	0.595360i	$0.202991\pi$
$480$		0			0
$481$	−2.10642e14	−	3.64842e14i	−0.373032	−	0.646111i
$482$		0			0
$483$	3.66021e14	−	1.42295e14i	0.633573	−	0.246309i
$484$		0			0
$485$	7.69383e14	+	1.33261e15i	1.30186	+	2.25488i
$486$		0			0
$487$	4.87117e14	−	8.43711e14i	0.805794	−	1.39568i	−0.109960	−	0.993936i	$-0.535072\pi$
$487$	4.87117e14	−	8.43711e14i	0.805794	−	1.39568i	0.915754	−	0.401739i	$-0.131594\pi$
$488$		0			0
$489$	6.94434e13			0.112314
$490$		0			0
$491$	1.14406e15			1.80926			0.904628	−	0.426201i	$-0.140148\pi$
$491$	1.14406e15			1.80926			0.904628	+	0.426201i	$0.140148\pi$
$492$		0			0
$493$	7.32281e11	−	1.26835e12i	0.00113245	−	0.00196146i
$494$		0			0
$495$	−3.37708e14	−	5.84928e14i	−0.510756	−	0.884655i
$496$		0			0
$497$	7.57021e14	−	2.94301e14i	1.11982	−	0.435343i
$498$		0			0
$499$	6.44591e14	+	1.11646e15i	0.932677	+	1.61544i	0.778725	+	0.627366i	$0.215867\pi$
$499$	6.44591e14	+	1.11646e15i	0.932677	+	1.61544i	0.153952	+	0.988078i	$0.450800\pi$
$500$		0			0
$501$	−1.79239e14	+	3.10451e14i	−0.253702	+	0.439425i
$502$		0			0
$503$	1.01920e15			1.41135			0.705677	−	0.708534i	$-0.250643\pi$
$503$	1.01920e15			1.41135			0.705677	+	0.708534i	$0.250643\pi$
$504$		0			0
$505$	−3.36800e14			−0.456320
$506$		0			0
$507$	−3.35054e14	+	5.80330e14i	−0.444192	+	0.769363i
$508$		0			0
$509$	−1.28883e14	−	2.23232e14i	−0.167205	−	0.289607i	0.770231	−	0.637765i	$-0.220141\pi$
$509$	−1.28883e14	−	2.23232e14i	−0.167205	−	0.289607i	−0.937436	+	0.348158i	$0.886807\pi$
$510$		0			0
$511$	1.39886e14	−	9.08242e14i	0.177607	−	1.15315i
$512$		0			0
$513$	1.27388e14	+	2.20643e14i	0.158301	+	0.274186i
$514$		0			0
$515$	−5.24379e14	+	9.08252e14i	−0.637831	+	1.10476i
$516$		0			0
$517$	−6.78753e14			−0.808191
$518$		0			0
$519$	−3.41408e14			−0.397972
$520$		0			0
$521$	−1.50755e14	+	2.61115e14i	−0.172053	+	0.298005i	−0.939138	−	0.343541i	$-0.888373\pi$
$521$	−1.50755e14	+	2.61115e14i	−0.172053	+	0.298005i	0.767084	+	0.641546i	$0.221707\pi$
$522$		0			0
$523$	−1.19275e14	−	2.06590e14i	−0.133287	−	0.230860i	0.791655	−	0.610969i	$-0.209220\pi$
$523$	−1.19275e14	−	2.06590e14i	−0.133287	−	0.230860i	−0.924942	+	0.380109i	$0.875887\pi$
$524$		0			0
$525$	−1.10983e15	−	8.90921e14i	−1.21446	−	0.974906i
$526$		0			0
$527$	−2.76468e14	−	4.78857e14i	−0.296270	−	0.513154i
$528$		0			0
$529$	−1.84011e14	+	3.18716e14i	−0.193124	+	0.334501i
$530$		0			0
$531$	−4.50476e13			−0.0463075
$532$		0			0
$533$	−5.80913e14			−0.584940
$534$		0			0
$535$	−1.54121e15	+	2.66945e15i	−1.52025	+	2.63315i
$536$		0			0
$537$	3.05782e14	+	5.29630e14i	0.295497	+	0.511816i
$538$		0			0
$539$	3.63993e14	+	1.64344e15i	0.344632	+	1.55602i
$540$		0			0
$541$	−7.15031e14	−	1.23847e15i	−0.663346	−	1.14895i	−0.979731	−	0.200318i	$-0.935803\pi$
$541$	−7.15031e14	−	1.23847e15i	−0.663346	−	1.14895i	0.316385	−	0.948631i	$-0.397531\pi$
$542$		0			0
$543$	3.77917e14	−	6.54571e14i	0.343556	−	0.595056i
$544$		0			0
$545$	−2.81142e15			−2.50464
$546$		0			0
$547$	9.19792e14			0.803081			0.401540	−	0.915841i	$-0.368475\pi$
$547$	9.19792e14			0.803081			0.401540	+	0.915841i	$0.368475\pi$
$548$		0			0
$549$	−1.20483e14	+	2.08682e14i	−0.103104	+	0.178582i
$550$		0			0
$551$	5.07484e12	+	8.78989e12i	0.00425686	+	0.00737309i
$552$		0			0
$553$	−8.91553e14	−	7.15696e14i	−0.733093	−	0.588491i
$554$		0			0
$555$	−3.22432e14	−	5.58469e14i	−0.259912	−	0.450181i
$556$		0			0
$557$	−5.08145e14	+	8.80133e14i	−0.401591	+	0.695576i	−0.993918	−	0.110121i	$-0.964876\pi$
$557$	−5.08145e14	+	8.80133e14i	−0.401591	+	0.695576i	0.592327	+	0.805698i	$0.298209\pi$
$558$		0			0
$559$	2.37992e15			1.84415
$560$		0			0
$561$	−5.30001e14			−0.402700
$562$		0			0
$563$	−6.05790e14	+	1.04926e15i	−0.451363	+	0.781784i	−0.998471	−	0.0552782i	$-0.982395\pi$
$563$	−6.05790e14	+	1.04926e15i	−0.451363	+	0.781784i	0.547108	+	0.837062i	$0.315729\pi$
$564$		0			0
$565$	1.47219e15	+	2.54990e15i	1.07571	+	1.86319i
$566$		0			0
$567$	−2.36018e13	+	1.53240e14i	−0.0169137	+	0.109816i
$568$		0			0
$569$	−7.24341e14	−	1.25460e15i	−0.509126	−	0.881833i	−0.999944	−	0.0105706i	$-0.996635\pi$
$569$	−7.24341e14	−	1.25460e15i	−0.509126	−	0.881833i	0.490818	−	0.871262i	$-0.336698\pi$
$570$		0			0
$571$	−2.13063e14	+	3.69036e14i	−0.146896	+	0.254431i	−0.930079	−	0.367360i	$-0.880262\pi$
$571$	−2.13063e14	+	3.69036e14i	−0.146896	+	0.254431i	0.783183	+	0.621792i	$0.213595\pi$
$572$		0			0
$573$	2.73662e14			0.185082
$574$		0			0
$575$	4.78676e15			3.17591
$576$		0			0
$577$	2.04431e14	−	3.54086e14i	0.133070	−	0.230484i	−0.791788	−	0.610795i	$-0.790850\pi$
$577$	2.04431e14	−	3.54086e14i	0.133070	−	0.230484i	0.924859	+	0.380311i	$0.124183\pi$
$578$		0			0
$579$	−2.62781e14	−	4.55150e14i	−0.167827	−	0.290685i
$580$		0			0
$581$	5.64751e14	−	2.19554e14i	0.353907	−	0.137586i
$582$		0			0
$583$	−5.76077e14	−	9.97795e14i	−0.354245	−	0.613570i
$584$		0			0
$585$	−8.46180e14	+	1.46563e15i	−0.510629	+	0.884436i
$586$		0			0
$587$	−7.59585e14			−0.449849			−0.224925	−	0.974376i	$-0.572214\pi$
$587$	−7.59585e14			−0.449849			−0.224925	+	0.974376i	$0.572214\pi$
$588$		0			0
$589$	3.83196e15			2.22734
$590$		0			0
$591$	−7.32441e14	+	1.26863e15i	−0.417870	+	0.723773i
$592$		0			0
$593$	1.64766e15	+	2.85384e15i	0.922715	+	1.59819i	0.795196	+	0.606353i	$0.207368\pi$
$593$	1.64766e15	+	2.85384e15i	0.922715	+	1.59819i	0.127519	+	0.991836i	$0.459299\pi$
$594$		0			0
$595$	−1.42678e15	+	5.54676e14i	−0.784354	+	0.304927i
$596$		0			0
$597$	−3.47795e14	−	6.02399e14i	−0.187700	−	0.325106i
$598$		0			0
$599$	1.26409e15	−	2.18947e15i	0.669777	−	1.16009i	−0.308189	−	0.951325i	$-0.599723\pi$
$599$	1.26409e15	−	2.18947e15i	0.669777	−	1.16009i	0.977966	−	0.208763i	$-0.0669437\pi$
$600$		0			0
$601$	1.40595e15			0.731410			0.365705	−	0.930731i	$-0.380828\pi$
$601$	1.40595e15			0.731410			0.365705	+	0.930731i	$0.380828\pi$
$602$		0			0
$603$	−1.28472e14			−0.0656239
$604$		0			0
$605$	−2.95182e15	+	5.11270e15i	−1.48059	+	2.56446i
$606$		0			0
$607$	−1.31451e15	−	2.27680e15i	−0.647480	−	1.12147i	−0.983723	−	0.179694i	$-0.942489\pi$
$607$	−1.31451e15	−	2.27680e15i	−0.647480	−	1.12147i	0.336242	−	0.941776i	$-0.390844\pi$
$608$		0			0
$609$	−9.40240e11	+	6.10473e12i	−0.000454824	+	0.00295305i
$610$		0			0
$611$	8.50361e14	+	1.47287e15i	0.403995	+	0.699741i
$612$		0			0
$613$	−2.36386e14	+	4.09433e14i	−0.110304	+	0.191051i	−0.915893	−	0.401423i	$-0.868516\pi$
$613$	−2.36386e14	+	4.09433e14i	−0.110304	+	0.191051i	0.805589	+	0.592475i	$0.201849\pi$
$614$		0			0
$615$	−8.89212e14			−0.407560
$616$		0			0
$617$	6.41641e14			0.288884			0.144442	−	0.989513i	$-0.453861\pi$
$617$	6.41641e14			0.288884			0.144442	+	0.989513i	$0.453861\pi$
$618$		0			0
$619$	−1.06581e15	+	1.84603e15i	−0.471390	+	0.816471i	−0.999464	−	0.0327268i	$-0.989581\pi$
$619$	−1.06581e15	+	1.84603e15i	−0.471390	+	0.816471i	0.528074	+	0.849198i	$0.322914\pi$
$620$		0			0
$621$	−2.60743e14	−	4.51620e14i	−0.113294	−	0.196232i
$622$		0			0
$623$	3.59568e14	+	2.88643e14i	0.153496	+	0.123219i
$624$		0			0
$625$	−4.26605e15	−	7.38902e15i	−1.78931	−	3.09918i
$626$		0			0
$627$	1.83651e15	−	3.18092e15i	0.756869	−	1.31094i
$628$		0			0
$629$	−5.06027e14			−0.204925
$630$		0			0
$631$	−2.49076e15			−0.991219			−0.495609	−	0.868545i	$-0.665055\pi$
$631$	−2.49076e15			−0.991219			−0.495609	+	0.868545i	$0.665055\pi$
$632$		0			0
$633$	3.91601e14	−	6.78273e14i	0.153152	−	0.265268i
$634$		0			0
$635$	−1.73003e15	−	2.99650e15i	−0.664964	−	1.15175i
$636$		0			0
$637$	3.11018e15	−	2.84880e15i	1.17495	−	1.07621i
$638$		0			0
$639$	−5.39280e14	−	9.34061e14i	−0.200244	−	0.346833i
$640$		0			0
$641$	1.64178e15	−	2.84365e15i	0.599233	−	1.03790i	−0.393701	−	0.919239i	$-0.628805\pi$
$641$	1.64178e15	−	2.84365e15i	0.599233	−	1.03790i	0.992934	−	0.118664i	$-0.0378612\pi$
$642$		0			0
$643$	1.94429e15			0.697592			0.348796	−	0.937199i	$-0.386591\pi$
$643$	1.94429e15			0.697592			0.348796	+	0.937199i	$0.386591\pi$
$644$		0			0
$645$	3.64297e15			1.28492
$646$		0			0
$647$	−3.24795e14	+	5.62562e14i	−0.112625	+	0.195073i	−0.916828	−	0.399282i	$-0.869259\pi$
$647$	−3.24795e14	+	5.62562e14i	−0.112625	+	0.195073i	0.804203	+	0.594355i	$0.202593\pi$
$648$		0			0
$649$	3.24716e14	+	5.62425e14i	0.110703	+	0.191743i
$650$		0			0
$651$	1.81853e15	+	1.45983e15i	0.609572	+	0.489335i
$652$		0			0
$653$	−1.92474e15	−	3.33375e15i	−0.634380	−	1.09878i	−0.986646	−	0.162879i	$-0.947922\pi$
$653$	−1.92474e15	−	3.33375e15i	−0.634380	−	1.09878i	0.352266	−	0.935900i	$-0.385411\pi$
$654$		0			0
$655$	−2.95476e15	+	5.11779e15i	−0.957623	+	1.65865i
$656$		0			0
$657$	−1.22030e15			−0.388917
$658$		0			0
$659$	1.40489e15			0.440324			0.220162	−	0.975463i	$-0.429341\pi$
$659$	1.40489e15			0.440324			0.220162	+	0.975463i	$0.429341\pi$
$660$		0			0
$661$	−1.55113e15	+	2.68663e15i	−0.478123	+	0.828134i	−0.999685	−	0.0250796i	$-0.992016\pi$
$661$	−1.55113e15	+	2.68663e15i	−0.478123	+	0.828134i	0.521562	+	0.853213i	$0.325349\pi$
$662$		0			0
$663$	6.64000e14	+	1.15008e15i	0.201300	+	0.348662i
$664$		0			0
$665$	1.61490e15	−	1.04851e16i	0.481533	−	3.12647i
$666$		0			0
$667$	−1.03874e13	−	1.79915e13i	−0.00304659	−	0.00527684i
$668$		0			0
$669$	1.36519e15	−	2.36458e15i	0.393867	−	0.682198i
$670$		0			0
$671$	3.47390e15			0.985924
$672$		0			0
$673$	−4.65798e15			−1.30051			−0.650256	−	0.759715i	$-0.725338\pi$
$673$	−4.65798e15			−1.30051			−0.650256	+	0.759715i	$0.725338\pi$
$674$		0			0
$675$	−9.44945e14	+	1.63669e15i	−0.259559	+	0.449569i
$676$		0			0
$677$	−1.70181e15	−	2.94763e15i	−0.459912	−	0.796590i	0.539044	−	0.842277i	$-0.318786\pi$
$677$	−1.70181e15	−	2.94763e15i	−0.459912	−	0.796590i	−0.998956	+	0.0456873i	$0.985452\pi$
$678$		0			0
$679$	−4.74640e15	+	1.84522e15i	−1.26206	+	0.490642i
$680$		0			0
$681$	1.84291e15	+	3.19202e15i	0.482165	+	0.835135i
$682$		0			0
$683$	8.52660e14	−	1.47685e15i	0.219514	−	0.380209i	−0.735146	−	0.677909i	$-0.762886\pi$
$683$	8.52660e14	−	1.47685e15i	0.219514	−	0.380209i	0.954659	+	0.297700i	$0.0962195\pi$
$684$		0			0
$685$	6.76946e15			1.71497
$686$		0			0
$687$	8.85201e14			0.220689
$688$		0			0
$689$	−1.44345e15	+	2.50013e15i	−0.354157	+	0.613419i
$690$		0			0
$691$	2.28018e15	+	3.94939e15i	0.550606	+	0.953677i	0.998231	+	0.0594557i	$0.0189365\pi$
$691$	2.28018e15	+	3.94939e15i	0.550606	+	0.953677i	−0.447625	+	0.894221i	$0.647730\pi$
$692$		0			0
$693$	2.08336e15	−	8.09930e14i	0.495143	−	0.192493i
$694$		0			0
$695$	−1.29797e15	−	2.24814e15i	−0.303632	−	0.525905i
$696$		0			0
$697$	−3.48884e14	+	6.04284e14i	−0.0803341	+	0.139143i
$698$		0			0
$699$	−2.93374e15			−0.664962
$700$		0			0
$701$	−2.93440e15			−0.654741			−0.327371	−	0.944896i	$-0.606163\pi$
$701$	−2.93440e15			−0.654741			−0.327371	+	0.944896i	$0.606163\pi$
$702$		0			0
$703$	1.75343e15	−	3.03703e15i	0.385154	−	0.667106i
$704$		0			0
$705$	1.30166e15	+	2.25454e15i	0.281486	+	0.487548i
$706$		0			0
$707$	1.69671e14	−	1.10163e15i	0.0361245	−	0.234547i
$708$		0			0
$709$	−1.47790e15	−	2.55980e15i	−0.309807	−	0.536601i	0.668513	−	0.743700i	$-0.266931\pi$
$709$	−1.47790e15	−	2.55980e15i	−0.309807	−	0.536601i	−0.978320	+	0.207099i	$0.933598\pi$
$710$		0			0
$711$	−7.59094e14	+	1.31479e15i	−0.156680	+	0.271377i
$712$		0			0
$713$	−7.84339e15			−1.59408
$714$		0			0
$715$	2.43981e16			4.88284
$716$		0			0
$717$	1.97084e14	−	3.41360e14i	0.0388416	−	0.0672756i
$718$		0			0
$719$	1.47695e15	+	2.55816e15i	0.286654	+	0.496499i	0.973009	−	0.230768i	$-0.0741238\pi$
$719$	1.47695e15	+	2.55816e15i	0.286654	+	0.496499i	−0.686355	+	0.727266i	$0.740790\pi$
$720$		0			0
$721$	−2.70661e15	−	2.17273e15i	−0.517347	−	0.415301i
$722$		0			0
$723$	1.86193e15	+	3.22496e15i	0.350512	+	0.607104i
$724$		0			0
$725$	−3.76443e13	+	6.52019e13i	−0.00697976	+	0.0120893i
$726$		0			0
$727$	−5.57462e15			−1.01807			−0.509033	−	0.860747i	$-0.669997\pi$
$727$	−5.57462e15			−1.01807			−0.509033	+	0.860747i	$0.669997\pi$
$728$		0			0
$729$	2.05891e14			0.0370370
$730$		0			0
$731$	1.42932e15	−	2.47566e15i	0.253271	−	0.438678i
$732$		0			0
$733$	−4.84328e15	−	8.38880e15i	−0.845410	−	1.46429i	−0.885264	−	0.465088i	$-0.846023\pi$
$733$	−4.84328e15	−	8.38880e15i	−0.845410	−	1.46429i	0.0398541	−	0.999206i	$-0.487311\pi$
$734$		0			0
$735$	4.76080e15	−	4.36070e15i	0.818653	−	0.749852i
$736$		0			0
$737$	9.26061e14	+	1.60399e15i	0.156880	+	0.271725i
$738$		0			0
$739$	−9.18308e14	+	1.59056e15i	−0.153265	+	0.265463i	−0.932426	−	0.361361i	$-0.882312\pi$
$739$	−9.18308e14	+	1.59056e15i	−0.153265	+	0.265463i	0.779161	+	0.626824i	$0.215646\pi$
$740$		0			0
$741$	−9.20330e15			−1.51336
$742$		0			0
$743$	5.66724e13			0.00918191			0.00459095	−	0.999989i	$-0.498539\pi$
$743$	5.66724e13			0.00918191			0.00459095	+	0.999989i	$0.498539\pi$
$744$		0			0
$745$	4.70646e14	−	8.15182e14i	0.0751337	−	0.130135i
$746$		0			0
$747$	−4.02313e14	−	6.96826e14i	−0.0632850	−	0.109613i
$748$		0			0
$749$	−7.95502e15	−	6.38590e15i	−1.23308	−	0.989856i
$750$		0			0
$751$	8.39714e14	+	1.45443e15i	0.128266	+	0.222163i	0.923005	−	0.384788i	$-0.125726\pi$
$751$	8.39714e14	+	1.45443e15i	0.128266	+	0.222163i	−0.794739	+	0.606952i	$0.792392\pi$
$752$		0			0
$753$	−1.71723e15	+	2.97433e15i	−0.258497	+	0.447731i
$754$		0			0
$755$	−1.42337e16			−2.11159
$756$		0			0
$757$	5.82897e15			0.852245			0.426122	−	0.904666i	$-0.359879\pi$
$757$	5.82897e15			0.852245			0.426122	+	0.904666i	$0.359879\pi$
$758$		0			0
$759$	−3.75902e15	+	6.51082e15i	−0.541683	+	0.938223i
$760$		0			0
$761$	3.73179e15	+	6.46364e15i	0.530031	+	0.918041i	0.999386	+	0.0350311i	$0.0111530\pi$
$761$	3.73179e15	+	6.46364e15i	0.530031	+	0.918041i	−0.469355	+	0.883009i	$0.655514\pi$
$762$		0			0
$763$	1.41632e15	−	9.19582e15i	0.198279	−	1.28738i
$764$		0			0
$765$	1.01639e15	+	1.76045e15i	0.140257	+	0.242932i
$766$		0			0
$767$	8.13627e14	−	1.40924e15i	0.110675	−	0.191695i
$768$		0			0
$769$	4.93564e15			0.661833			0.330917	−	0.943660i	$-0.392642\pi$
$769$	4.93564e15			0.661833			0.330917	+	0.943660i	$0.392642\pi$
$770$		0			0
$771$	3.65431e15			0.483065
$772$		0			0
$773$	3.32146e15	−	5.75293e15i	0.432854	−	0.749725i	−0.564264	−	0.825595i	$-0.690840\pi$
$773$	3.32146e15	−	5.75293e15i	0.432854	−	0.749725i	0.997118	+	0.0758695i	$0.0241732\pi$
$774$		0			0
$775$	1.42124e16	+	2.46166e16i	1.82603	+	3.16278i
$776$		0			0
$777$	1.98912e15	−	7.73293e14i	0.251967	−	0.0979554i
$778$		0			0
$779$	−2.41783e15	−	4.18780e15i	−0.301974	−	0.523034i
$780$		0			0
$781$	−7.77458e15	+	1.34660e16i	−0.957407	+	1.65828i
$782$		0			0
$783$	8.20221e12			0.000995958
$784$		0			0
$785$	6.17305e15			0.739123
$786$		0			0
$787$	3.38350e15	−	5.86039e15i	0.399489	−	0.691935i	−0.594174	−	0.804337i	$-0.702521\pi$
$787$	3.38350e15	−	5.86039e15i	0.399489	−	0.691935i	0.993663	+	0.112401i	$0.0358542\pi$
$788$		0			0
$789$	−3.64210e15	−	6.30831e15i	−0.424061	−	0.734496i
$790$		0			0
$791$	−9.08206e15	+	3.53076e15i	−1.04283	+	0.405413i
$792$		0			0
$793$	−4.35220e15	−	7.53822e15i	−0.492840	−	0.853624i
$794$		0			0
$795$	−2.20951e15	+	3.82698e15i	−0.246761	+	0.427403i
$796$		0			0
$797$	−1.32772e16			−1.46246			−0.731232	−	0.682129i	$-0.761054\pi$
$797$	−1.32772e16			−1.46246			−0.731232	+	0.682129i	$0.761054\pi$
$798$		0			0
$799$	2.04283e15			0.221935
$800$		0			0
$801$	3.06146e14	−	5.30261e14i	0.0328058	−	0.0568212i
$802$		0			0
$803$	8.79627e15	+	1.52356e16i	0.929743	+	1.61036i
$804$		0			0
$805$	−3.30543e15	+	2.14613e16i	−0.344628	+	2.23758i
$806$		0			0
$807$	5.43831e15	+	9.41943e15i	0.559319	+	0.968769i
$808$		0			0
$809$	−3.65847e15	+	6.33665e15i	−0.371178	+	0.642900i	−0.989747	−	0.142831i	$-0.954379\pi$
$809$	−3.65847e15	+	6.33665e15i	−0.371178	+	0.642900i	0.618569	+	0.785731i	$0.287713\pi$
$810$		0			0
$811$	−8.29518e15			−0.830254			−0.415127	−	0.909763i	$-0.636263\pi$
$811$	−8.29518e15			−0.830254			−0.415127	+	0.909763i	$0.636263\pi$
$812$		0			0
$813$	−2.57777e15			−0.254534
$814$		0			0
$815$	−1.91990e15	+	3.32537e15i	−0.187030	+	0.323946i
$816$		0			0
$817$	9.90549e15	+	1.71568e16i	0.952039	+	1.64898i
$818$		0			0
$819$	−4.36760e15	−	3.50610e15i	−0.414173	−	0.332478i
$820$		0			0
$821$	−5.14344e14	−	8.90870e14i	−0.0481245	−	0.0833541i	0.840960	−	0.541098i	$-0.181991\pi$
$821$	−5.14344e14	−	8.90870e14i	−0.0481245	−	0.0833541i	−0.889084	+	0.457744i	$0.848658\pi$
$822$		0			0
$823$	−5.29428e15	+	9.16996e15i	−0.488774	+	0.846581i	−0.999917	−	0.0129147i	$-0.995889\pi$
$823$	−5.29428e15	+	9.16996e15i	−0.488774	+	0.846581i	0.511143	+	0.859496i	$0.329222\pi$
$824$		0			0
$825$	2.72458e16			2.48200
$826$		0			0
$827$	−5.40430e15			−0.485802			−0.242901	−	0.970051i	$-0.578099\pi$
$827$	−5.40430e15			−0.485802			−0.242901	+	0.970051i	$0.578099\pi$
$828$		0			0
$829$	−6.56988e15	+	1.13794e16i	−0.582784	+	1.00941i	0.412364	+	0.911019i	$0.364703\pi$
$829$	−6.56988e15	+	1.13794e16i	−0.582784	+	1.00941i	−0.995148	+	0.0983919i	$0.968630\pi$
$830$		0			0
$831$	5.01608e15	+	8.68811e15i	0.439096	+	0.760536i
$832$		0			0
$833$	−1.09550e15	−	4.94624e15i	−0.0946382	−	0.427295i
$834$		0			0
$835$	−9.91082e15	−	1.71660e16i	−0.844956	−	1.46351i
$836$		0			0
$837$	1.54835e15	−	2.68182e15i	0.130280	−	0.225652i
$838$		0			0
$839$	1.43166e16			1.18891			0.594455	−	0.804129i	$-0.297368\pi$
$839$	1.43166e16			1.18891			0.594455	+	0.804129i	$0.297368\pi$
$840$		0			0
$841$	−1.22002e16			−0.999973
$842$		0			0
$843$	−6.92500e14	+	1.19944e15i	−0.0560232	+	0.0970350i
$844$		0			0
$845$	−1.85264e16	−	3.20887e16i	−1.47938	−	2.56236i
$846$		0			0
$847$	−1.52359e16	−	1.22307e16i	−1.20091	−	0.964033i
$848$		0			0
$849$	−5.17980e13	−	8.97167e13i	−0.00403014	−	0.00698041i
$850$		0			0
$851$	−3.58898e15	+	6.21630e15i	−0.275651	+	0.477441i
$852$		0			0
$853$	1.17265e16			0.889098			0.444549	−	0.895755i	$-0.353364\pi$
$853$	1.17265e16			0.889098			0.444549	+	0.895755i	$0.353364\pi$
$854$		0			0
$855$	−1.40876e16			−1.05444
$856$		0			0
$857$	−4.25484e15	+	7.36960e15i	−0.314404	+	0.544564i	−0.979311	−	0.202362i	$-0.935138\pi$
$857$	−4.25484e15	+	7.36960e15i	−0.314404	+	0.544564i	0.664906	+	0.746927i	$0.268471\pi$
$858$		0			0
$859$	−5.63364e15	−	9.75774e15i	−0.410985	−	0.711848i	0.584012	−	0.811745i	$-0.301482\pi$
$859$	−5.63364e15	−	9.75774e15i	−0.410985	−	0.711848i	−0.994998	+	0.0998972i	$0.968149\pi$
$860$		0			0
$861$	4.47963e14	−	2.90850e15i	0.0322645	−	0.209485i
$862$		0			0
$863$	−2.68765e15	−	4.65514e15i	−0.191123	−	0.331035i	0.754500	−	0.656300i	$-0.227880\pi$
$863$	−2.68765e15	−	4.65514e15i	−0.191123	−	0.331035i	−0.945623	+	0.325266i	$0.894546\pi$
$864$		0			0
$865$	9.43890e15	−	1.63486e16i	0.662724	−	1.14787i
$866$		0			0
$867$	−6.73294e15			−0.466766
$868$		0			0
$869$	2.18871e16			1.49823
$870$		0			0
$871$	2.32039e15	−	4.01903e15i	0.156842	−	0.271657i
$872$		0			0
$873$	3.38120e15	+	5.85642e15i	0.225680	+	0.390889i
$874$		0			0
$875$	4.61549e16	−	1.79433e16i	3.04210	−	1.18265i
$876$		0			0
$877$	9.70359e15	+	1.68071e16i	0.631589	+	1.09394i	0.987227	+	0.159321i	$0.0509304\pi$
$877$	9.70359e15	+	1.68071e16i	0.631589	+	1.09394i	−0.355638	+	0.934624i	$0.615736\pi$
$878$		0			0
$879$	1.95573e15	−	3.38743e15i	0.125710	−	0.217737i
$880$		0			0
$881$	−2.05160e16			−1.30234			−0.651171	−	0.758931i	$-0.725722\pi$
$881$	−2.05160e16			−1.30234			−0.651171	+	0.758931i	$0.725722\pi$
$882$		0			0
$883$	−1.29505e16			−0.811903			−0.405951	−	0.913895i	$-0.633060\pi$
$883$	−1.29505e16			−0.811903			−0.405951	+	0.913895i	$0.633060\pi$
$884$		0			0
$885$	1.24543e15	−	2.15715e15i	0.0771136	−	0.133565i
$886$		0			0
$887$	−3.64435e15	−	6.31220e15i	−0.222864	−	0.386012i	0.732812	−	0.680431i	$-0.238207\pi$
$887$	−3.64435e15	−	6.31220e15i	−0.222864	−	0.386012i	−0.955676	+	0.294419i	$0.904874\pi$
$888$		0			0
$889$	1.06727e16	−	4.14916e15i	0.644638	−	0.250611i
$890$		0			0
$891$	−1.48412e15	−	2.57058e15i	−0.0885407	−	0.153357i
$892$		0			0
$893$	−7.07860e15	+	1.22605e16i	−0.417123	+	0.722479i
$894$		0			0
$895$	−3.38158e16			−1.96831
$896$		0			0
$897$	1.88376e16			1.08310
$898$		0			0
$899$	6.16825e13	−	1.06837e14i	0.00350335	−	0.00606798i
$900$		0			0
$901$	1.73381e15	+	3.00305e15i	0.0972781	+	0.168491i
$902$		0			0
$903$	−1.83524e15	+	1.19157e16i	−0.101721	+	0.660446i
$904$		0			0
$905$	2.08965e16	+	3.61939e16i	1.14421	+	1.98184i
$906$		0			0
$907$	6.98204e15	−	1.20933e16i	0.377696	−	0.654189i	−0.613031	−	0.790059i	$-0.710050\pi$
$907$	6.98204e15	−	1.20933e16i	0.377696	−	0.654189i	0.990727	+	0.135871i	$0.0433831\pi$
$908$		0			0
$909$	−1.48013e15			−0.0791041
$910$		0			0
$911$	−1.74544e16			−0.921623			−0.460812	−	0.887498i	$-0.652442\pi$
$911$	−1.74544e16			−0.921623			−0.460812	+	0.887498i	$0.652442\pi$
$912$		0			0
$913$	−5.79998e15	+	1.00459e16i	−0.302578	+	0.524080i
$914$		0			0
$915$	−6.66197e15	−	1.15389e16i	−0.343389	−	0.594767i
$916$		0			0
$917$	−1.52511e16	−	1.22429e16i	−0.776731	−	0.623522i
$918$		0			0
$919$	−9.96055e15	−	1.72522e16i	−0.501242	−	0.868177i	−0.999999	−	0.00143523i	$-0.999543\pi$
$919$	−9.96055e15	−	1.72522e16i	−0.501242	−	0.868177i	0.498757	−	0.866742i	$-0.333790\pi$
$920$		0			0
$921$	6.89976e15	−	1.19507e16i	0.343088	−	0.594246i
$922$		0			0
$923$	3.89608e16			1.91434
$924$		0			0
$925$	2.60133e16			1.26304
$926$		0			0
$927$	−2.30449e15	+	3.99149e15i	−0.110570	+	0.191512i
$928$		0			0
$929$	−3.73858e15	−	6.47541e15i	−0.177264	−	0.307030i	0.763679	−	0.645597i	$-0.223391\pi$
$929$	−3.73858e15	−	6.47541e15i	−0.177264	−	0.307030i	−0.940942	+	0.338567i	$0.890058\pi$
$930$		0			0
$931$	3.34820e16	+	1.05643e16i	1.56887	+	0.495013i
$932$		0			0
$933$	−7.09939e15	−	1.22965e16i	−0.328755	−	0.569420i
$934$		0			0
$935$	1.46529e16	−	2.53796e16i	0.670596	−	1.16151i
$936$		0			0
$937$	2.31704e16			1.04801			0.524006	−	0.851715i	$-0.324437\pi$
$937$	2.31704e16			1.04801			0.524006	+	0.851715i	$0.324437\pi$
$938$		0			0
$939$	−3.48880e15			−0.155961
$940$		0			0
$941$	−6.48966e15	+	1.12404e16i	−0.286734	+	0.496638i	−0.973028	−	0.230686i	$-0.925903\pi$
$941$	−6.48966e15	+	1.12404e16i	−0.286734	+	0.496638i	0.686294	+	0.727324i	$0.259236\pi$
$942$		0			0
$943$	4.94890e15	+	8.57174e15i	0.216120	+	0.374330i
$944$		0			0
$945$	−6.68555e15	−	5.36683e15i	−0.288577	−	0.231656i
$946$		0			0
$947$	−1.37270e16	−	2.37759e16i	−0.585667	−	1.01440i	−0.994792	−	0.101926i	$-0.967499\pi$
$947$	−1.37270e16	−	2.37759e16i	−0.585667	−	1.01440i	0.409125	−	0.912478i	$-0.365834\pi$
$948$		0			0
$949$	2.20404e16	−	3.81751e16i	0.929513	−	1.60996i
$950$		0			0
$951$	7.17704e15			0.299194
$952$		0			0
$953$	−4.34657e15			−0.179117			−0.0895583	−	0.995982i	$-0.528546\pi$
$953$	−4.34657e15			−0.179117			−0.0895583	+	0.995982i	$0.528546\pi$
$954$		0			0
$955$	−7.56592e15	+	1.31046e16i	−0.308208	+	0.533831i
$956$		0			0
$957$	−5.91239e13	−	1.02406e14i	−0.00238094	−	0.00412390i
$958$		0			0
$959$	−3.41028e15	+	2.21421e16i	−0.135765	+	0.881487i
$960$		0			0
$961$	−1.05836e16	−	1.83314e16i	−0.416540	−	0.721469i
$962$		0			0
$963$	−6.77313e15	+	1.17314e16i	−0.263539	+	0.456463i
$964$		0			0
$965$	2.90604e16			1.11790
$966$		0			0
$967$	−4.78536e15			−0.181999			−0.0909995	−	0.995851i	$-0.529006\pi$
$967$	−4.78536e15			−0.181999			−0.0909995	+	0.995851i	$0.529006\pi$
$968$		0			0
$969$	−5.52730e15	+	9.57356e15i	−0.207841	+	0.359992i
$970$		0			0
$971$	3.79192e15	+	6.56779e15i	0.140978	+	0.244182i	0.927865	−	0.372915i	$-0.121642\pi$
$971$	3.79192e15	+	6.56779e15i	0.140978	+	0.244182i	−0.786887	+	0.617097i	$0.788308\pi$
$972$		0			0
$973$	8.00729e15	−	3.11293e15i	0.294350	−	0.114432i
$974$		0			0
$975$	−3.41342e16	−	5.91222e16i	−1.24069	−	2.14895i
$976$		0			0
$977$	−1.61885e16	+	2.80392e16i	−0.581816	+	1.00773i	0.413448	+	0.910528i	$0.364324\pi$
$977$	−1.61885e16	+	2.80392e16i	−0.581816	+	1.00773i	−0.995264	+	0.0972072i	$0.969009\pi$
$978$		0			0
$979$	−8.82717e15			−0.313701
$980$		0			0
$981$	−1.23553e16			−0.434185
$982$		0			0
$983$	4.83330e15	−	8.37152e15i	0.167957	−	0.290911i	−0.769744	−	0.638353i	$-0.779616\pi$
$983$	4.83330e15	−	8.37152e15i	0.167957	−	0.290911i	0.937702	+	0.347442i	$0.112950\pi$
$984$		0			0
$985$	−4.04996e16	−	7.01473e16i	−1.39172	−	2.41053i
$986$		0			0
$987$	−8.03006e15	+	3.12179e15i	−0.272882	+	0.106086i
$988$		0			0
$989$	−2.02749e16	−	3.51172e16i	−0.681364	−	1.18016i
$990$		0			0
$991$	2.36964e16	−	4.10434e16i	0.787549	−	1.36407i	−0.139916	−	0.990163i	$-0.544683\pi$
$991$	2.36964e16	−	4.10434e16i	0.787549	−	1.36407i	0.927465	−	0.373911i	$-0.121984\pi$
$992$		0			0
$993$	−2.69993e16			−0.887424
$994$		0			0
$995$	3.84619e16			1.25027
$996$		0			0
$997$	−4.37128e15	+	7.57128e15i	−0.140535	+	0.243414i	−0.927698	−	0.373331i	$-0.878216\pi$
$997$	−4.37128e15	+	7.57128e15i	−0.140535	+	0.243414i	0.787163	+	0.616745i	$0.211549\pi$
$998$		0			0
$999$	−1.41699e15	−	2.45430e15i	−0.0450564	−	0.0780400i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.12.i.b.25.1	✓	16
3.2	odd	2		252.12.k.d.109.8		16
7.2	even	3	inner	84.12.i.b.37.1	yes	16
21.2	odd	6		252.12.k.d.37.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.b.25.1	✓	16	1.1	even	1	trivial
84.12.i.b.37.1	yes	16	7.2	even	3	inner
252.12.k.d.37.8		16	21.2	odd	6
252.12.k.d.109.8		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)

\(f(q)\)	\(=\)	\(q+(-121.500 + 210.444i) q^{3} +(-6718.22 - 11636.3i) q^{5} +(41445.4 - 16112.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})\)	\(q+(-121.500 + 210.444i) q^{3} +(-6718.22 - 11636.3i) q^{5} +(41445.4 - 16112.4i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-425643. + 737235. i) q^{11} +2.13303e6 q^{13} +3.26505e6 q^{15} +(1.28105e6 - 2.21884e6i) q^{17} +(8.87791e6 + 1.53770e7i) q^{19} +(-1.64485e6 + 1.06796e7i) q^{21} +(-1.81716e7 - 3.14742e7i) q^{23} +(-6.58548e7 + 1.14064e8i) q^{25} +1.43489e7 q^{27} +571626. q^{29} +(1.07907e8 - 1.86901e8i) q^{31} +(-1.03431e8 - 1.79148e8i) q^{33} +(-4.65927e8 - 3.74024e8i) q^{35} +(-9.87524e7 - 1.71044e8i) q^{37} +(-2.59163e8 + 4.48883e8i) q^{39} -2.72342e8 q^{41} +1.11575e9 q^{43} +(-3.96704e8 + 6.87111e8i) q^{45} +(3.98664e8 + 6.90506e8i) q^{47} +(1.45811e9 - 1.33557e9i) q^{49} +(3.11295e8 + 5.39178e8i) q^{51} +(-6.76715e8 + 1.17210e9i) q^{53} +1.14382e10 q^{55} -4.31466e9 q^{57} +(3.81442e8 - 6.60677e8i) q^{59} +(-2.04039e9 - 3.53405e9i) q^{61} +(-2.04761e9 - 1.64372e9i) q^{63} +(-1.43301e10 - 2.48205e10i) q^{65} +(1.08784e9 - 1.88419e9i) q^{67} +8.83141e9 q^{69} +1.82655e10 q^{71} +(1.03329e10 - 1.78972e10i) q^{73} +(-1.60027e10 - 2.77175e10i) q^{75} +(-5.76230e9 + 3.74131e10i) q^{77} +(-1.28553e10 - 2.22661e10i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +1.36264e10 q^{83} -3.44255e10 q^{85} +(-6.94525e7 + 1.20295e8i) q^{87} +(5.18461e9 + 8.98001e9i) q^{89} +(8.84041e10 - 3.43682e10i) q^{91} +(2.62214e10 + 4.54168e10i) q^{93} +(1.19287e11 - 2.06612e11i) q^{95} -1.14522e11 q^{97} +5.02675e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9}+O(q^{10}) \) 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

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