LMFDB - Embedded newform 252.12.k.d.109.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,12,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	252.k (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:	$193.622481501$
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:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.4
Root			$-438.744 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	252.109
Dual form			252.12.k.d.37.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-84.6221 - 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +O(q^{10})$	$q+(-84.6221 - 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +(143939. - 249309. i) q^{11} +2.48841e6 q^{13} +(3.58228e6 - 6.20469e6i) q^{17} +(-1.63429e6 - 2.83067e6i) q^{19} +(-1.56616e7 - 2.71268e7i) q^{23} +(2.43997e7 - 4.22616e7i) q^{25} -5.63075e7 q^{29} +(-8.04560e7 + 1.39354e8i) q^{31} +(7.18924e6 - 2.22543e6i) q^{35} +(1.48880e8 + 2.57868e8i) q^{37} -1.39570e9 q^{41} -4.76296e8 q^{43} +(1.19265e9 + 2.06573e9i) q^{47} +(-1.78321e9 - 8.54381e8i) q^{49} +(2.34079e9 - 4.05437e9i) q^{53} -4.87215e7 q^{55} +(8.21849e8 - 1.42348e9i) q^{59} +(-1.55428e9 - 2.69209e9i) q^{61} +(-2.10575e8 - 3.64726e8i) q^{65} +(-5.33960e9 + 9.24846e9i) q^{67} +2.59617e10 q^{71} +(-2.45048e8 + 4.24435e8i) q^{73} +(9.39252e9 + 8.69760e9i) q^{77} +(-4.61089e9 - 7.98630e9i) q^{79} +3.48255e10 q^{83} -1.21256e9 q^{85} +(-1.81904e10 - 3.15067e10i) q^{89} +(-2.45151e10 + 1.07903e11i) q^{91} +(-2.76593e8 + 4.79074e8i) q^{95} -1.51725e11 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2156 q^{5} + 50512 q^{7}+O(q^{10}) $ 16 * q + 2156 * q^5 + 50512 * q^7	$ 16 q + 2156 q^{5} + 50512 q^{7} + 222796 q^{11} + 2703176 q^{13} - 5114600 q^{17} + 6910556 q^{19} + 51387712 q^{23} - 191456372 q^{25} - 118854616 q^{29} + 164659160 q^{31} - 55239344 q^{35} + 75658364 q^{37} + 1815568608 q^{41} + 10754408 q^{43} + 1034359464 q^{47} + 4123496848 q^{49} + 665159988 q^{53} - 1264543896 q^{55} - 1040514580 q^{59} - 14391208024 q^{61} + 20938150200 q^{65} - 33307097284 q^{67} - 65848902896 q^{71} + 17709749204 q^{73} - 8594484604 q^{77} - 26626784032 q^{79} + 210306955048 q^{83} - 25867402032 q^{85} + 55951560072 q^{89} + 66078280292 q^{91} - 106810047392 q^{95} - 156216030712 q^{97}+O(q^{100}) $ 16 * q + 2156 * q^5 + 50512 * q^7 + 222796 * q^11 + 2703176 * q^13 - 5114600 * q^17 + 6910556 * q^19 + 51387712 * q^23 - 191456372 * q^25 - 118854616 * q^29 + 164659160 * q^31 - 55239344 * q^35 + 75658364 * q^37 + 1815568608 * q^41 + 10754408 * q^43 + 1034359464 * q^47 + 4123496848 * q^49 + 665159988 * q^53 - 1264543896 * q^55 - 1040514580 * q^59 - 14391208024 * q^61 + 20938150200 * q^65 - 33307097284 * q^67 - 65848902896 * q^71 + 17709749204 * q^73 - 8594484604 * q^77 - 26626784032 * q^79 + 210306955048 * q^83 - 25867402032 * q^85 + 55951560072 * q^89 + 66078280292 * q^91 - 106810047392 * q^95 - 156216030712 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−84.6221	−	146.570i	−0.0121101	−	0.0209754i	0.859907	−	0.510451i	$-0.170522\pi$
$5$	−84.6221	−	146.570i	−0.0121101	−	0.0209754i	−0.872017	+	0.489476i	$0.837188\pi$
$6$		0			0
$7$	−9851.70	+	43362.1i	−0.221550	+	0.975149i
$7$	−9851.70	+	43362.1i	−0.221550	+	0.975149i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	143939.	−	249309.i	0.269474	−	0.466743i	−0.699252	−	0.714875i	$-0.746483\pi$
$11$	143939.	−	249309.i	0.269474	−	0.466743i	0.968726	+	0.248132i	$0.0798167\pi$
$12$		0			0
$13$	2.48841e6			1.85881			0.929404	−	0.369065i	$-0.120322\pi$
$13$	2.48841e6			1.85881			0.929404	+	0.369065i	$0.120322\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.58228e6	−	6.20469e6i	0.611914	−	1.05987i	−0.379004	−	0.925395i	$-0.623733\pi$
$17$	3.58228e6	−	6.20469e6i	0.611914	−	1.05987i	0.990918	−	0.134470i	$-0.0429333\pi$
$18$		0			0
$19$	−1.63429e6	−	2.83067e6i	−0.151420	−	0.262267i	0.780330	−	0.625368i	$-0.215051\pi$
$19$	−1.63429e6	−	2.83067e6i	−0.151420	−	0.262267i	−0.931750	+	0.363101i	$0.881718\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.56616e7	−	2.71268e7i	−0.507381	−	0.878810i	−0.999963	−	0.00854416i	$-0.997280\pi$
$23$	−1.56616e7	−	2.71268e7i	−0.507381	−	0.878810i	0.492582	−	0.870266i	$-0.336053\pi$
$24$		0			0
$25$	2.43997e7	−	4.22616e7i	0.499707	−	0.865517i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−5.63075e7			−0.509773			−0.254887	−	0.966971i	$-0.582038\pi$
$29$	−5.63075e7			−0.509773			−0.254887	+	0.966971i	$0.582038\pi$
$30$		0			0
$31$	−8.04560e7	+	1.39354e8i	−0.504742	+	0.874238i	0.495243	+	0.868754i	$0.335079\pi$
$31$	−8.04560e7	+	1.39354e8i	−0.504742	+	0.874238i	−0.999985	+	0.00548373i	$0.998254\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	7.18924e6	−	2.22543e6i	0.0231371	−	0.00716208i
$36$		0			0
$37$	1.48880e8	+	2.57868e8i	0.352962	+	0.611348i	0.986767	−	0.162145i	$-0.0518413\pi$
$37$	1.48880e8	+	2.57868e8i	0.352962	+	0.611348i	−0.633805	+	0.773493i	$0.718508\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.39570e9			−1.88140			−0.940702	−	0.339235i	$-0.889832\pi$
$41$	−1.39570e9			−1.88140			−0.940702	+	0.339235i	$0.889832\pi$
$42$		0			0
$43$	−4.76296e8			−0.494083			−0.247042	−	0.969005i	$-0.579458\pi$
$43$	−4.76296e8			−0.494083			−0.247042	+	0.969005i	$0.579458\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.19265e9	+	2.06573e9i	0.758533	+	1.31382i	0.943599	+	0.331091i	$0.107417\pi$
$47$	1.19265e9	+	2.06573e9i	0.758533	+	1.31382i	−0.185066	+	0.982726i	$0.559250\pi$
$48$		0			0
$49$	−1.78321e9	−	8.54381e8i	−0.901831	−	0.432089i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.34079e9	−	4.05437e9i	0.768856	−	1.33170i	−0.169327	−	0.985560i	$-0.554159\pi$
$53$	2.34079e9	−	4.05437e9i	0.768856	−	1.33170i	0.938183	−	0.346139i	$-0.112507\pi$
$54$		0			0
$55$	−4.87215e7			−0.0130535
$56$		0			0
$57$		0			0
$58$		0			0
$59$	8.21849e8	−	1.42348e9i	0.149660	−	0.259219i	−0.781442	−	0.623978i	$-0.785515\pi$
$59$	8.21849e8	−	1.42348e9i	0.149660	−	0.259219i	0.931102	+	0.364759i	$0.118849\pi$
$60$		0			0
$61$	−1.55428e9	−	2.69209e9i	−0.235621	−	0.408108i	0.723832	−	0.689977i	$-0.242379\pi$
$61$	−1.55428e9	−	2.69209e9i	−0.235621	−	0.408108i	−0.959453	+	0.281868i	$0.909046\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.10575e8	−	3.64726e8i	−0.0225104	−	0.0389891i
$66$		0			0
$67$	−5.33960e9	+	9.24846e9i	−0.483167	+	0.836870i	−0.999813	−	0.0193290i	$-0.993847\pi$
$67$	−5.33960e9	+	9.24846e9i	−0.483167	+	0.836870i	0.516646	+	0.856199i	$0.327180\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.59617e10			1.70770			0.853852	−	0.520516i	$-0.174261\pi$
$71$	2.59617e10			1.70770			0.853852	+	0.520516i	$0.174261\pi$
$72$		0			0
$73$	−2.45048e8	+	4.24435e8i	−0.0138349	+	0.0239627i	−0.872860	−	0.487971i	$-0.837737\pi$
$73$	−2.45048e8	+	4.24435e8i	−0.0138349	+	0.0239627i	0.859025	+	0.511933i	$0.171071\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.39252e9	+	8.69760e9i	0.395442	+	0.366185i
$78$		0			0
$79$	−4.61089e9	−	7.98630e9i	−0.168592	−	0.292009i	0.769333	−	0.638848i	$-0.220589\pi$
$79$	−4.61089e9	−	7.98630e9i	−0.168592	−	0.292009i	−0.937925	+	0.346838i	$0.887255\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.48255e10			0.970437			0.485219	−	0.874393i	$-0.338740\pi$
$83$	3.48255e10			0.970437			0.485219	+	0.874393i	$0.338740\pi$
$84$		0			0
$85$	−1.21256e9			−0.0296414
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1.81904e10	−	3.15067e10i	−0.345301	−	0.598078i	0.640108	−	0.768285i	$-0.278890\pi$
$89$	−1.81904e10	−	3.15067e10i	−0.345301	−	0.598078i	−0.985408	+	0.170207i	$0.945556\pi$
$90$		0			0
$91$	−2.45151e10	+	1.07903e11i	−0.411819	+	1.81261i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.76593e8	+	4.79074e8i	−0.00366743	+	0.00635218i
$96$		0			0
$97$	−1.51725e11			−1.79396			−0.896978	−	0.442076i	$-0.854242\pi$
$97$	−1.51725e11			−1.79396			−0.896978	+	0.442076i	$0.854242\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	7.68508e10	−	1.33110e11i	0.727581	−	1.26021i	−0.230322	−	0.973114i	$-0.573978\pi$
$101$	7.68508e10	−	1.33110e11i	0.727581	−	1.26021i	0.957903	−	0.287092i	$-0.0926886\pi$
$102$		0			0
$103$	−8.88312e10	−	1.53860e11i	−0.755024	−	1.30774i	−0.945362	−	0.326022i	$-0.894292\pi$
$103$	−8.88312e10	−	1.53860e11i	−0.755024	−	1.30774i	0.190338	−	0.981719i	$-0.439042\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.46571e9	+	2.53868e9i	0.0101027	+	0.0174984i	0.871033	−	0.491225i	$-0.163451\pi$
$107$	1.46571e9	+	2.53868e9i	0.0101027	+	0.0174984i	−0.860930	+	0.508724i	$0.830117\pi$
$108$		0			0
$109$	1.37623e10	−	2.38371e10i	0.0856734	−	0.148391i	−0.820005	−	0.572357i	$-0.806029\pi$
$109$	1.37623e10	−	2.38371e10i	0.0856734	−	0.148391i	0.905678	+	0.423966i	$0.139362\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	7.29842e10			0.372647			0.186323	−	0.982488i	$-0.440343\pi$
$113$	7.29842e10			0.372647			0.186323	+	0.982488i	$0.440343\pi$
$114$		0			0
$115$	−2.65064e9	+	4.59105e9i	−0.0122889	+	0.0212850i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.33757e11	+	2.16462e11i	0.897957	+	0.831520i
$120$		0			0
$121$	1.01219e11	+	1.75317e11i	0.354767	+	0.614475i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.65229e10			−0.0484263
$126$		0			0
$127$	−4.83730e11			−1.29922			−0.649610	−	0.760268i	$-0.725068\pi$
$127$	−4.83730e11			−1.29922			−0.649610	+	0.760268i	$0.725068\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.84860e11	−	3.20188e11i	−0.418651	−	0.725124i	0.577153	−	0.816636i	$-0.304164\pi$
$131$	−1.84860e11	−	3.20188e11i	−0.418651	−	0.725124i	−0.995804	+	0.0915115i	$0.970830\pi$
$132$		0			0
$133$	1.38844e11	−	4.29792e10i	0.289297	−	0.0895517i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.08974e11	+	5.35159e11i	−0.546964	+	0.947370i	0.451516	+	0.892263i	$0.350883\pi$
$137$	−3.08974e11	+	5.35159e11i	−0.546964	+	0.947370i	−0.998480	+	0.0551072i	$0.982450\pi$
$138$		0			0
$139$	8.57532e11			1.40175			0.700873	−	0.713287i	$-0.252794\pi$
$139$	8.57532e11			1.40175			0.700873	+	0.713287i	$0.252794\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	3.58179e11	−	6.20384e11i	0.500901	−	0.867586i
$144$		0			0
$145$	4.76485e9	+	8.25297e9i	0.00617342	+	0.0106927i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.33560e11	−	5.77744e11i	−0.372092	−	0.644482i	0.617795	−	0.786339i	$-0.288026\pi$
$149$	−3.33560e11	−	5.77744e11i	−0.372092	−	0.644482i	−0.989887	+	0.141857i	$0.954693\pi$
$150$		0			0
$151$	−2.61750e11	+	4.53365e11i	−0.271340	+	0.469975i	−0.969205	−	0.246254i	$-0.920800\pi$
$151$	−2.61750e11	+	4.53365e11i	−0.271340	+	0.469975i	0.697865	+	0.716229i	$0.254134\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.72334e10			0.0244499
$156$		0			0
$157$	8.68955e11	−	1.50507e12i	0.727024	−	1.25924i	−0.231111	−	0.972927i	$-0.574236\pi$
$157$	8.68955e11	−	1.50507e12i	0.727024	−	1.25924i	0.958135	−	0.286316i	$-0.0924307\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.33057e12	−	4.11877e11i	0.969381	−	0.300072i
$162$		0			0
$163$	−6.47918e11	−	1.12223e12i	−0.441050	−	0.763922i	0.556717	−	0.830702i	$-0.312061\pi$
$163$	−6.47918e11	−	1.12223e12i	−0.441050	−	0.763922i	−0.997768	+	0.0667802i	$0.978727\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2.54401e12			−1.51558			−0.757789	−	0.652500i	$-0.773720\pi$
$167$	−2.54401e12			−1.51558			−0.757789	+	0.652500i	$0.773720\pi$
$168$		0			0
$169$	4.40005e12			2.45516
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−6.74851e11	−	1.16888e12i	−0.331096	−	0.573476i	0.651631	−	0.758536i	$-0.274085\pi$
$173$	−6.74851e11	−	1.16888e12i	−0.331096	−	0.573476i	−0.982727	+	0.185061i	$0.940752\pi$
$174$		0			0
$175$	1.59217e12	+	1.47437e12i	0.733298	+	0.679044i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.63584e12	+	2.83337e12i	−0.665350	+	1.15242i	0.313840	+	0.949476i	$0.398385\pi$
$179$	−1.63584e12	+	2.83337e12i	−0.665350	+	1.15242i	−0.979190	+	0.202945i	$0.934949\pi$
$180$		0			0
$181$	−4.17692e12			−1.59817			−0.799087	−	0.601216i	$-0.794683\pi$
$181$	−4.17692e12			−1.59817			−0.799087	+	0.601216i	$0.794683\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	2.51971e10	−	4.36427e10i	0.00854882	−	0.0148070i
$186$		0			0
$187$	−1.03126e12	−	1.78619e12i	−0.329790	−	0.571213i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.45917e12	−	2.52736e12i	−0.415359	−	0.719423i	0.580107	−	0.814540i	$-0.303011\pi$
$191$	−1.45917e12	−	2.52736e12i	−0.415359	−	0.719423i	−0.995466	+	0.0951173i	$0.969677\pi$
$192$		0			0
$193$	2.76848e12	−	4.79515e12i	0.744178	−	1.28895i	−0.206400	−	0.978468i	$-0.566175\pi$
$193$	2.76848e12	−	4.79515e12i	0.744178	−	1.28895i	0.950578	−	0.310486i	$-0.100492\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.28547e11			−0.0788920			−0.0394460	−	0.999222i	$-0.512559\pi$
$197$	−3.28547e11			−0.0788920			−0.0394460	+	0.999222i	$0.512559\pi$
$198$		0			0
$199$	−1.41328e12	+	2.44787e12i	−0.321023	+	0.556027i	−0.980699	−	0.195522i	$-0.937360\pi$
$199$	−1.41328e12	+	2.44787e12i	−0.321023	+	0.556027i	0.659677	+	0.751549i	$0.270693\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	5.54724e11	−	2.44161e12i	0.112940	−	0.497105i
$204$		0			0
$205$	1.18107e11	+	2.04568e11i	0.0227840	+	0.0394631i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.40947e11			−0.163215
$210$		0			0
$211$	4.59968e12			0.757137			0.378568	−	0.925573i	$-0.376416\pi$
$211$	4.59968e12			0.757137			0.378568	+	0.925573i	$0.376416\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	4.03051e10	+	6.98105e10i	0.00598341	+	0.0103636i
$216$		0			0
$217$	−5.25005e12	−	4.86161e12i	−0.740687	−	0.685886i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	8.91419e12	−	1.54398e13i	1.13743	−	1.97009i
$222$		0			0
$223$	5.12583e12			0.622426			0.311213	−	0.950340i	$-0.399265\pi$
$223$	5.12583e12			0.622426			0.311213	+	0.950340i	$0.399265\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	7.71055e12	−	1.33551e13i	0.849069	−	1.47063i	−0.0329705	−	0.999456i	$-0.510497\pi$
$227$	7.71055e12	−	1.33551e13i	0.849069	−	1.47063i	0.882040	−	0.471175i	$-0.156170\pi$
$228$		0			0
$229$	−1.68621e12	−	2.92060e12i	−0.176936	−	0.306463i	0.763893	−	0.645342i	$-0.223285\pi$
$229$	−1.68621e12	−	2.92060e12i	−0.176936	−	0.306463i	−0.940830	+	0.338880i	$0.889952\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−3.52941e12	−	6.11311e12i	−0.336701	−	0.583183i	0.647109	−	0.762397i	$-0.275978\pi$
$233$	−3.52941e12	−	6.11311e12i	−0.336701	−	0.583183i	−0.983810	+	0.179215i	$0.942644\pi$
$234$		0			0
$235$	2.01849e11	−	3.49612e11i	0.0183719	−	0.0318210i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.38390e12			0.612487			0.306244	−	0.951953i	$-0.400928\pi$
$239$	7.38390e12			0.612487			0.306244	+	0.951953i	$0.400928\pi$
$240$		0			0
$241$	−3.35164e12	+	5.80522e12i	−0.265561	+	0.459965i	−0.967710	−	0.252065i	$-0.918890\pi$
$241$	−3.35164e12	+	5.80522e12i	−0.265561	+	0.459965i	0.702149	+	0.712030i	$0.252224\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.56729e10	+	3.33665e11i	0.00185807	+	0.0241489i
$246$		0			0
$247$	−4.06678e12	−	7.04387e12i	−0.281461	−	0.487504i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.93352e13			1.22502			0.612509	−	0.790464i	$-0.290160\pi$
$251$	1.93352e13			1.22502			0.612509	+	0.790464i	$0.290160\pi$
$252$		0			0
$253$	−9.01726e12			−0.546905
$254$		0			0
$255$		0			0
$256$		0			0
$257$	9.86761e12	+	1.70912e13i	0.549009	+	0.950912i	0.998343	+	0.0575480i	$0.0183282\pi$
$257$	9.86761e12	+	1.70912e13i	0.549009	+	0.950912i	−0.449333	+	0.893364i	$0.648338\pi$
$258$		0			0
$259$	−1.26484e13	+	3.91532e12i	−0.674354	+	0.208746i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.57662e13	−	2.73078e13i	0.772627	−	1.33823i	−0.163492	−	0.986545i	$-0.552276\pi$
$263$	1.57662e13	−	2.73078e13i	0.772627	−	1.33823i	0.936119	−	0.351685i	$-0.114391\pi$
$264$		0			0
$265$	−7.92330e11			−0.0372438
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1.20365e13	−	2.08477e13i	0.521028	−	0.902447i	−0.478673	−	0.877993i	$-0.658882\pi$
$269$	1.20365e13	−	2.08477e13i	0.521028	−	0.902447i	0.999701	−	0.0244535i	$-0.00778457\pi$
$270$		0			0
$271$	−9.29904e12	−	1.61064e13i	−0.386462	−	0.669372i	0.605509	−	0.795839i	$-0.292970\pi$
$271$	−9.29904e12	−	1.61064e13i	−0.386462	−	0.669372i	−0.991971	+	0.126467i	$0.959636\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.02413e12	−	1.21661e13i	−0.269316	−	0.466470i
$276$		0			0
$277$	1.44061e13	−	2.49521e13i	0.530772	−	0.919323i	−0.468584	−	0.883419i	$-0.655236\pi$
$277$	1.44061e13	−	2.49521e13i	0.530772	−	0.919323i	0.999355	−	0.0359041i	$-0.0114311\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−3.61357e13			−1.23042			−0.615208	−	0.788365i	$-0.710928\pi$
$281$	−3.61357e13			−1.23042			−0.615208	+	0.788365i	$0.710928\pi$
$282$		0			0
$283$	−7.12729e11	+	1.23448e12i	−0.0233399	+	0.0404259i	−0.877459	−	0.479651i	$-0.840763\pi$
$283$	−7.12729e11	+	1.23448e12i	−0.0233399	+	0.0404259i	0.854120	+	0.520077i	$0.174097\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.37501e13	−	6.05206e13i	0.416825	−	1.83465i
$288$		0			0
$289$	−8.52946e12	−	1.47735e13i	−0.248876	−	0.431066i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.28945e13			0.889922			0.444961	−	0.895550i	$-0.353218\pi$
$293$	3.28945e13			0.889922			0.444961	+	0.895550i	$0.353218\pi$
$294$		0			0
$295$	−2.78186e11			−0.00724961
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.89727e13	−	6.75027e13i	−0.943124	−	1.63354i
$300$		0			0
$301$	4.69232e12	−	2.06532e13i	0.109464	−	0.481805i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.63052e11	+	4.55620e11i	−0.00570681	+	0.00988448i
$306$		0			0
$307$	1.20907e13			0.253040			0.126520	−	0.991964i	$-0.459619\pi$
$307$	1.20907e13			0.253040			0.126520	+	0.991964i	$0.459619\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2.99473e12	−	5.18703e12i	0.0583682	−	0.101097i	−0.835365	−	0.549696i	$-0.814744\pi$
$311$	2.99473e12	−	5.18703e12i	0.0583682	−	0.101097i	0.893733	+	0.448599i	$0.148077\pi$
$312$		0			0
$313$	1.28228e13	+	2.22098e13i	0.241263	+	0.417879i	0.961074	−	0.276290i	$-0.0891052\pi$
$313$	1.28228e13	+	2.22098e13i	0.241263	+	0.417879i	−0.719812	+	0.694169i	$0.755772\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	9.63494e12	+	1.66882e13i	0.169053	+	0.292808i	0.938087	−	0.346399i	$-0.112596\pi$
$317$	9.63494e12	+	1.66882e13i	0.169053	+	0.292808i	−0.769034	+	0.639208i	$0.779262\pi$
$318$		0			0
$319$	−8.10482e12	+	1.40380e13i	−0.137371	+	0.237933i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2.34179e13			−0.370624
$324$		0			0
$325$	6.07167e13	−	1.05164e14i	0.928858	−	1.60883i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−1.01324e14	+	3.13648e13i	−1.44922	+	0.448606i
$330$		0			0
$331$	3.34435e13	+	5.79259e13i	0.462656	+	0.801343i	0.999092	−	0.0425974i	$-0.0135633\pi$
$331$	3.34435e13	+	5.79259e13i	0.462656	+	0.801343i	−0.536437	+	0.843941i	$0.680230\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.80739e12			0.0234049
$336$		0			0
$337$	−1.48408e13			−0.185992			−0.0929958	−	0.995667i	$-0.529644\pi$
$337$	−1.48408e13			−0.185992			−0.0929958	+	0.995667i	$0.529644\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.31614e13	+	4.01168e13i	0.272030	+	0.471169i
$342$		0			0
$343$	5.46154e13	−	6.89068e13i	0.621152	−	0.783690i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	6.92423e13	−	1.19931e14i	0.738855	−	1.27974i	−0.214156	−	0.976800i	$-0.568700\pi$
$347$	6.92423e13	−	1.19931e14i	0.738855	−	1.27974i	0.953011	−	0.302936i	$-0.0979667\pi$
$348$		0			0
$349$	1.38407e14			1.43093			0.715463	−	0.698651i	$-0.246216\pi$
$349$	1.38407e14			1.43093			0.715463	+	0.698651i	$0.246216\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	5.08428e13	−	8.80624e13i	0.493707	−	0.855125i	−0.506267	−	0.862377i	$-0.668975\pi$
$353$	5.08428e13	−	8.80624e13i	0.493707	−	0.855125i	0.999974	+	0.00725175i	$0.00230832\pi$
$354$		0			0
$355$	−2.19693e12	−	3.80520e12i	−0.0206805	−	0.0358197i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−4.26384e13	−	7.38518e13i	−0.377382	−	0.653645i	0.613299	−	0.789851i	$-0.289842\pi$
$359$	−4.26384e13	−	7.38518e13i	−0.377382	−	0.653645i	−0.990680	+	0.136207i	$0.956509\pi$
$360$		0			0
$361$	5.29033e13	−	9.16313e13i	0.454144	−	0.786600i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8.29458e10			0.000670168
$366$		0			0
$367$	−3.58256e13	+	6.20517e13i	−0.280886	+	0.486509i	−0.971603	−	0.236616i	$-0.923962\pi$
$367$	−3.58256e13	+	6.20517e13i	−0.280886	+	0.486509i	0.690717	+	0.723125i	$0.257295\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.52745e14	+	1.41444e14i	1.12826	+	1.04479i
$372$		0			0
$373$	−9.39443e13	−	1.62716e14i	−0.673708	−	1.16690i	−0.976845	−	0.213950i	$-0.931367\pi$
$373$	−9.39443e13	−	1.62716e14i	−0.673708	−	1.16690i	0.303137	−	0.952947i	$-0.401966\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.40116e14			−0.947570
$378$		0			0
$379$	1.29326e14			0.849516			0.424758	−	0.905307i	$-0.360359\pi$
$379$	1.29326e14			0.849516			0.424758	+	0.905307i	$0.360359\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−1.21897e14	−	2.11132e14i	−0.755790	−	1.30907i	−0.944981	−	0.327126i	$-0.893920\pi$
$383$	−1.21897e14	−	2.11132e14i	−0.755790	−	1.30907i	0.189191	−	0.981940i	$-0.439413\pi$
$384$		0			0
$385$	4.79990e11	−	2.11267e12i	0.00289200	−	0.0127291i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−1.06560e14	+	1.84568e14i	−0.606559	+	1.05059i	0.385244	+	0.922815i	$0.374117\pi$
$389$	−1.06560e14	+	1.84568e14i	−0.606559	+	1.05059i	−0.991803	+	0.127776i	$0.959216\pi$
$390$		0			0
$391$	−2.24417e14			−1.24189
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−7.80367e11	+	1.35163e12i	−0.00408333	+	0.00707254i
$396$		0			0
$397$	−3.57142e13	−	6.18588e13i	−0.181758	−	0.314814i	0.760721	−	0.649078i	$-0.224845\pi$
$397$	−3.57142e13	−	6.18588e13i	−0.181758	−	0.314814i	−0.942479	+	0.334265i	$0.891512\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.73100e13	+	6.46228e13i	0.179693	+	0.311237i	0.941775	−	0.336243i	$-0.109156\pi$
$401$	3.73100e13	+	6.46228e13i	0.179693	+	0.311237i	−0.762082	+	0.647480i	$0.775823\pi$
$402$		0			0
$403$	−2.00208e14	+	3.46770e14i	−0.938217	+	1.62504i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.57184e13			0.380457
$408$		0			0
$409$	−7.84760e13	+	1.35924e14i	−0.339046	+	0.587245i	−0.984254	−	0.176762i	$-0.943438\pi$
$409$	−7.84760e13	+	1.35924e14i	−0.339046	+	0.587245i	0.645207	+	0.764007i	$0.276771\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.36287e13	+	4.96609e13i	0.219620	+	0.203371i
$414$		0			0
$415$	−2.94700e12	−	5.10436e12i	−0.0117521	−	0.0203553i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5.96915e13			0.225806			0.112903	−	0.993606i	$-0.463985\pi$
$419$	5.96915e13			0.225806			0.112903	+	0.993606i	$0.463985\pi$
$420$		0			0
$421$	2.52328e14			0.929853			0.464927	−	0.885349i	$-0.346081\pi$
$421$	2.52328e14			0.929853			0.464927	+	0.885349i	$0.346081\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.74813e14	−	3.02785e14i	−0.611555	−	1.05924i
$426$		0			0
$427$	1.32047e14	−	4.08751e13i	0.450168	−	0.139349i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−6.39033e13	+	1.10684e14i	−0.206966	+	0.358475i	−0.950757	−	0.309937i	$-0.899692\pi$
$431$	−6.39033e13	+	1.10684e14i	−0.206966	+	0.358475i	0.743792	+	0.668412i	$0.233026\pi$
$432$		0			0
$433$	−3.97113e14			−1.25381			−0.626903	−	0.779098i	$-0.715678\pi$
$433$	−3.97113e14			−1.25381			−0.626903	+	0.779098i	$0.715678\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−5.11912e13	+	8.86658e13i	−0.153655	+	0.266139i
$438$		0			0
$439$	1.50834e14	+	2.61252e14i	0.441513	+	0.764723i	0.997802	−	0.0662662i	$-0.0211086\pi$
$439$	1.50834e14	+	2.61252e14i	0.441513	+	0.764723i	−0.556289	+	0.830989i	$0.687775\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2.51908e14	+	4.36317e14i	0.701489	+	1.21502i	0.967944	+	0.251168i	$0.0808146\pi$
$443$	2.51908e14	+	4.36317e14i	0.701489	+	1.21502i	−0.266454	+	0.963848i	$0.585852\pi$
$444$		0			0
$445$	−3.07862e12	+	5.33232e12i	−0.00836327	+	0.0144856i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	4.47372e14			1.15695			0.578474	−	0.815701i	$-0.303649\pi$
$449$	4.47372e14			1.15695			0.578474	+	0.815701i	$0.303649\pi$
$450$		0			0
$451$	−2.00896e14	+	3.47961e14i	−0.506990	+	0.878133i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.78898e13	−	5.53779e12i	0.0430074	−	0.0133129i
$456$		0			0
$457$	−2.59567e14	−	4.49583e14i	−0.609131	−	1.05505i	−0.991384	−	0.130988i	$-0.958185\pi$
$457$	−2.59567e14	−	4.49583e14i	−0.609131	−	1.05505i	0.382253	−	0.924058i	$-0.375148\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	6.55113e14			1.46542			0.732709	−	0.680543i	$-0.238256\pi$
$461$	6.55113e14			1.46542			0.732709	+	0.680543i	$0.238256\pi$
$462$		0			0
$463$	−1.51172e13			−0.0330199			−0.0165099	−	0.999864i	$-0.505256\pi$
$463$	−1.51172e13			−0.0330199			−0.0165099	+	0.999864i	$0.505256\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.12796e14	+	1.95369e14i	0.234992	+	0.407017i	0.959270	−	0.282490i	$-0.0911604\pi$
$467$	1.12796e14	+	1.95369e14i	0.234992	+	0.407017i	−0.724279	+	0.689507i	$0.757827\pi$
$468$		0			0
$469$	−3.48428e14	−	3.22649e14i	−0.709027	−	0.656569i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−6.85573e13	+	1.18745e14i	−0.133143	+	0.230610i
$474$		0			0
$475$	−1.59505e14			−0.302662
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.28941e14	−	2.23332e14i	0.233639	−	0.404674i	−0.725237	−	0.688499i	$-0.758270\pi$
$479$	1.28941e14	−	2.23332e14i	0.233639	−	0.404674i	0.958876	+	0.283825i	$0.0916034\pi$
$480$		0			0
$481$	3.70476e14	+	6.41683e14i	0.656088	+	1.13638i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.28393e13	+	2.22382e13i	0.0217250	+	0.0376289i
$486$		0			0
$487$	−1.49469e14	+	2.58887e14i	−0.247253	+	0.428254i	−0.962763	−	0.270349i	$-0.912861\pi$
$487$	−1.49469e14	+	2.58887e14i	−0.247253	+	0.428254i	0.715510	+	0.698603i	$0.246194\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−7.76606e14			−1.22815			−0.614076	−	0.789247i	$-0.710471\pi$
$491$	−7.76606e14			−1.22815			−0.614076	+	0.789247i	$0.710471\pi$
$492$		0			0
$493$	−2.01709e14	+	3.49370e14i	−0.311937	+	0.540291i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−2.55767e14	+	1.12575e15i	−0.378342	+	1.66527i
$498$		0			0
$499$	6.88471e13	+	1.19247e14i	0.0996169	+	0.172541i	0.911526	−	0.411242i	$-0.134905\pi$
$499$	6.88471e13	+	1.19247e14i	0.0996169	+	0.172541i	−0.811909	+	0.583784i	$0.801572\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−7.47083e14			−1.03453			−0.517267	−	0.855824i	$-0.673051\pi$
$503$	−7.47083e14			−1.03453			−0.517267	+	0.855824i	$0.673051\pi$
$504$		0			0
$505$	−2.60131e13			−0.0352444
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.15602e14	+	1.23946e15i	0.928376	+	1.60799i	0.786039	+	0.618176i	$0.212128\pi$
$509$	7.15602e14	+	1.23946e15i	0.928376	+	1.60799i	0.142337	+	0.989818i	$0.454538\pi$
$510$		0			0
$511$	−1.59903e13	−	1.48072e13i	−0.0203021	−	0.0188000i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.50342e13	+	2.60399e13i	−0.0182869	+	0.0316738i
$516$		0			0
$517$	6.86673e14			0.817621
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.34528e14	+	2.33009e14i	−0.153534	+	0.265929i	−0.932524	−	0.361107i	$-0.882399\pi$
$521$	−1.34528e14	+	2.33009e14i	−0.153534	+	0.265929i	0.778990	+	0.627036i	$0.215732\pi$
$522$		0			0
$523$	5.20397e13	+	9.01353e13i	0.0581534	+	0.100725i	0.893637	−	0.448791i	$-0.148145\pi$
$523$	5.20397e13	+	9.01353e13i	0.0581534	+	0.100725i	−0.835483	+	0.549516i	$0.814812\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.76431e14	+	9.98408e14i	0.617716	+	1.06992i
$528$		0			0
$529$	−1.41696e13	+	2.45424e13i	−0.0148714	+	0.0257580i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.47309e15			−3.49717
$534$		0			0
$535$	2.48063e11	−	4.29657e11i	0.000244690	−	0.000423815i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−4.69678e14	+	3.21593e14i	−0.444695	+	0.304487i
$540$		0			0
$541$	4.91782e14	+	8.51791e14i	0.456234	+	0.790220i	0.998758	−	0.0498200i	$-0.0158648\pi$
$541$	4.91782e14	+	8.51791e14i	0.456234	+	0.790220i	−0.542524	+	0.840040i	$0.682531\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−4.65839e12			−0.00415006
$546$		0			0
$547$	1.61612e15			1.41105			0.705527	−	0.708683i	$-0.250710\pi$
$547$	1.61612e15			1.41105			0.705527	+	0.708683i	$0.250710\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	9.20225e13	+	1.59388e14i	0.0771899	+	0.133697i
$552$		0			0
$553$	3.91728e14	−	1.21259e14i	0.322104	−	0.0997073i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1.67288e14	−	2.89751e14i	0.132209	−	0.228992i	−0.792319	−	0.610107i	$-0.791126\pi$
$557$	1.67288e14	−	2.89751e14i	0.132209	−	0.228992i	0.924528	+	0.381115i	$0.124460\pi$
$558$		0			0
$559$	−1.18522e15			−0.918405
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.53096e14	−	2.65170e14i	0.114069	−	0.197573i	−0.803338	−	0.595523i	$-0.796945\pi$
$563$	1.53096e14	−	2.65170e14i	0.114069	−	0.197573i	0.917407	+	0.397950i	$0.130278\pi$
$564$		0			0
$565$	−6.17607e12	−	1.06973e13i	−0.00451280	−	0.00781639i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−9.44055e13	−	1.63515e14i	−0.0663559	−	0.114932i	0.830939	−	0.556364i	$-0.187804\pi$
$569$	−9.44055e13	−	1.63515e14i	−0.0663559	−	0.114932i	−0.897295	+	0.441432i	$0.854471\pi$
$570$		0			0
$571$	7.11167e14	−	1.23178e15i	0.490313	−	0.849247i	−0.509625	−	0.860397i	$-0.670216\pi$
$571$	7.11167e14	−	1.23178e15i	0.490313	−	0.849247i	0.999938	+	0.0111499i	$0.00354920\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.52856e15			−1.01417
$576$		0			0
$577$	−7.83726e14	+	1.35745e15i	−0.510149	+	0.883605i	0.489781	+	0.871845i	$0.337077\pi$
$577$	−7.83726e14	+	1.35745e15i	−0.510149	+	0.883605i	−0.999931	+	0.0117595i	$0.996257\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−3.43090e14	+	1.51011e15i	−0.215001	+	0.946321i
$582$		0			0
$583$	−6.73860e14	−	1.16716e15i	−0.414374	−	0.717717i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	8.80723e14			0.521591			0.260795	−	0.965394i	$-0.416015\pi$
$587$	8.80723e14			0.521591			0.260795	+	0.965394i	$0.416015\pi$
$588$		0			0
$589$	5.25952e14			0.305712
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−5.84464e14	−	1.01232e15i	−0.327308	−	0.566915i	0.654668	−	0.755916i	$-0.272808\pi$
$593$	−5.84464e14	−	1.01232e15i	−0.327308	−	0.566915i	−0.981977	+	0.189001i	$0.939475\pi$
$594$		0			0
$595$	1.19458e13	−	5.25791e13i	0.00656706	−	0.0289048i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.09149e14	+	5.35461e14i	−0.163802	+	0.283714i	−0.936229	−	0.351390i	$-0.885709\pi$
$599$	−3.09149e14	+	5.35461e14i	−0.163802	+	0.283714i	0.772427	+	0.635104i	$0.219043\pi$
$600$		0			0
$601$	1.60739e15			0.836202			0.418101	−	0.908401i	$-0.362696\pi$
$601$	1.60739e15			0.836202			0.418101	+	0.908401i	$0.362696\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1.71308e13	−	2.96713e13i	0.00859255	−	0.0148827i
$606$		0			0
$607$	7.44943e14	+	1.29028e15i	0.366932	+	0.635544i	0.989084	−	0.147352i	$-0.0470749\pi$
$607$	7.44943e14	+	1.29028e15i	0.366932	+	0.635544i	−0.622152	+	0.782896i	$0.713742\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	2.96780e15	+	5.14039e15i	1.40997	+	2.44213i
$612$		0			0
$613$	−3.20198e14	+	5.54599e14i	−0.149412	+	0.258790i	−0.931010	−	0.364993i	$-0.881071\pi$
$613$	−3.20198e14	+	5.54599e14i	−0.149412	+	0.258790i	0.781598	+	0.623782i	$0.214405\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−2.78051e14			−0.125186			−0.0625932	−	0.998039i	$-0.519937\pi$
$617$	−2.78051e14			−0.125186			−0.0625932	+	0.998039i	$0.519937\pi$
$618$		0			0
$619$	−1.25743e15	+	2.17793e15i	−0.556140	+	0.963263i	0.441674	+	0.897176i	$0.354385\pi$
$619$	−1.25743e15	+	2.17793e15i	−0.556140	+	0.963263i	−0.997814	+	0.0660874i	$0.978948\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	1.54540e15	−	4.78379e14i	0.659717	−	0.204215i
$624$		0			0
$625$	−1.19000e15	−	2.06113e15i	−0.499120	−	0.864502i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.13332e15			0.863928
$630$		0			0
$631$	3.79204e15			1.50908			0.754539	−	0.656256i	$-0.227861\pi$
$631$	3.79204e15			1.50908			0.754539	+	0.656256i	$0.227861\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.09342e13	+	7.09002e13i	0.0157337	+	0.0272516i
$636$		0			0
$637$	−4.43738e15	−	2.12605e15i	−1.67633	−	0.803170i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	9.30652e14	−	1.61194e15i	0.339679	−	0.588341i	−0.644694	−	0.764441i	$-0.723015\pi$
$641$	9.30652e14	−	1.61194e15i	0.339679	−	0.588341i	0.984372	+	0.176100i	$0.0563484\pi$
$642$		0			0
$643$	−2.46490e15			−0.884380			−0.442190	−	0.896921i	$-0.645798\pi$
$643$	−2.46490e15			−0.884380			−0.442190	+	0.896921i	$0.645798\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9.99898e14	−	1.73187e15i	0.346723	−	0.600541i	−0.638943	−	0.769254i	$-0.720628\pi$
$647$	9.99898e14	−	1.73187e15i	0.346723	−	0.600541i	0.985665	+	0.168713i	$0.0539612\pi$
$648$		0			0
$649$	−2.36592e14	−	4.09789e14i	−0.0806591	−	0.139706i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.80003e15	−	4.84979e15i	−0.922869	−	1.59846i	−0.794954	−	0.606670i	$-0.792505\pi$
$653$	−2.80003e15	−	4.84979e15i	−0.922869	−	1.59846i	−0.127915	−	0.991785i	$-0.540828\pi$
$654$		0			0
$655$	−3.12866e13	+	5.41899e13i	−0.0101398	+	0.0175627i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.96747e14			0.0616649			0.0308324	−	0.999525i	$-0.490184\pi$
$659$	1.96747e14			0.0616649			0.0308324	+	0.999525i	$0.490184\pi$
$660$		0			0
$661$	−2.71632e15	+	4.70481e15i	−0.837286	+	1.45022i	0.0548702	+	0.998493i	$0.482525\pi$
$661$	−2.71632e15	+	4.70481e15i	−0.837286	+	1.45022i	−0.892156	+	0.451728i	$0.850808\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.80487e13	−	1.67134e13i	−0.00538180	−	0.00498362i
$666$		0			0
$667$	8.81868e14	+	1.52744e15i	0.258649	+	0.447994i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−8.94882e14			−0.253976
$672$		0			0
$673$	3.16858e15			0.884672			0.442336	−	0.896849i	$-0.354150\pi$
$673$	3.16858e15			0.884672			0.442336	+	0.896849i	$0.354150\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−1.05566e15	−	1.82845e15i	−0.285289	−	0.494135i	0.687390	−	0.726288i	$-0.258756\pi$
$677$	−1.05566e15	−	1.82845e15i	−0.285289	−	0.494135i	−0.972679	+	0.232154i	$0.925423\pi$
$678$		0			0
$679$	1.49475e15	−	6.57910e15i	0.397451	−	1.74937i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	2.33152e14	−	4.03831e14i	0.0600240	−	0.103965i	−0.834452	−	0.551081i	$-0.814216\pi$
$683$	2.33152e14	−	4.03831e14i	0.0600240	−	0.103965i	0.894476	+	0.447116i	$0.147549\pi$
$684$		0			0
$685$	1.04584e14			0.0264952
$686$		0			0
$687$		0			0
$688$		0			0
$689$	5.82486e15	−	1.00889e16i	1.42916	−	2.47537i
$690$		0			0
$691$	−2.05478e15	−	3.55899e15i	−0.496177	−	0.859404i	0.503813	−	0.863813i	$-0.331930\pi$
$691$	−2.05478e15	−	3.55899e15i	−0.496177	−	0.859404i	−0.999990	+	0.00440848i	$0.998597\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.25662e13	−	1.25688e14i	−0.0169753	−	0.0294021i
$696$		0			0
$697$	−4.99980e15	+	8.65990e15i	−1.15126	+	1.99403i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.49697e15			−0.334013			−0.167006	−	0.985956i	$-0.553410\pi$
$701$	−1.49697e15			−0.334013			−0.167006	+	0.985956i	$0.553410\pi$
$702$		0			0
$703$	4.86626e14	−	8.42861e14i	0.106891	−	0.185141i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	5.01480e15	+	4.64377e15i	1.06769	+	0.988698i
$708$		0			0
$709$	−4.09130e15	−	7.08633e15i	−0.857643	−	1.48548i	−0.874171	−	0.485618i	$-0.838595\pi$
$709$	−4.09130e15	−	7.08633e15i	−0.857643	−	1.48548i	0.0165280	−	0.999863i	$-0.494739\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	5.04029e15			1.02439
$714$		0			0
$715$	−1.21239e14			−0.0242639
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−4.14413e15	−	7.17784e15i	−0.804312	−	1.39311i	−0.916755	−	0.399451i	$-0.869201\pi$
$719$	−4.14413e15	−	7.17784e15i	−0.804312	−	1.39311i	0.112443	−	0.993658i	$-0.464132\pi$
$720$		0			0
$721$	7.54684e15	−	2.33612e15i	1.44252	−	0.446531i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.37389e15	+	2.37964e15i	−0.254737	+	0.441218i
$726$		0			0
$727$	−6.11784e15			−1.11727			−0.558636	−	0.829413i	$-0.688675\pi$
$727$	−6.11784e15			−1.11727			−0.558636	+	0.829413i	$0.688675\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1.70622e15	+	2.95526e15i	−0.302336	+	0.523662i
$732$		0			0
$733$	1.89724e15	+	3.28612e15i	0.331170	+	0.573603i	0.982742	−	0.184984i	$-0.0592233\pi$
$733$	1.89724e15	+	3.28612e15i	0.331170	+	0.573603i	−0.651571	+	0.758587i	$0.725890\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.53715e15	+	2.66242e15i	0.260402	+	0.451030i
$738$		0			0
$739$	1.52876e15	−	2.64789e15i	0.255150	−	0.441933i	−0.709786	−	0.704417i	$-0.751209\pi$
$739$	1.52876e15	−	2.64789e15i	0.255150	−	0.441933i	0.964936	+	0.262484i	$0.0845419\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.20470e15			−0.357199			−0.178600	−	0.983922i	$-0.557157\pi$
$743$	−2.20470e15			−0.357199			−0.178600	+	0.983922i	$0.557157\pi$
$744$		0			0
$745$	−5.64532e13	+	9.77797e13i	−0.00901216	+	0.0156095i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1.24522e14	+	3.85459e13i	−0.0193018	+	0.00597486i
$750$		0			0
$751$	−1.87510e15	−	3.24777e15i	−0.286421	−	0.496096i	0.686532	−	0.727100i	$-0.259132\pi$
$751$	−1.87510e15	−	3.24777e15i	−0.286421	−	0.496096i	−0.972953	+	0.231004i	$0.925799\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.85995e13			0.0131439
$756$		0			0
$757$	3.28803e15			0.480738			0.240369	−	0.970682i	$-0.422731\pi$
$757$	3.28803e15			0.480738			0.240369	+	0.970682i	$0.422731\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−5.66595e15	−	9.81371e15i	−0.804742	−	1.39385i	−0.916465	−	0.400115i	$-0.868970\pi$
$761$	−5.66595e15	−	9.81371e15i	−0.804742	−	1.39385i	0.111722	−	0.993739i	$-0.464363\pi$
$762$		0			0
$763$	8.98042e14	+	8.31599e14i	0.125722	+	0.116420i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.04510e15	−	3.54222e15i	0.278189	−	0.481838i
$768$		0			0
$769$	−1.28420e15			−0.172201			−0.0861006	−	0.996286i	$-0.527441\pi$
$769$	−1.28420e15			−0.172201			−0.0861006	+	0.996286i	$0.527441\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.84876e15	−	1.18624e16i	0.892534	−	1.54591i	0.0557073	−	0.998447i	$-0.482259\pi$
$773$	6.84876e15	−	1.18624e16i	0.892534	−	1.54591i	0.836827	−	0.547467i	$-0.184408\pi$
$774$		0			0
$775$	3.92621e15	+	6.80040e15i	0.504445	+	0.873725i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	2.28098e15	+	3.95077e15i	0.284882	+	0.493430i
$780$		0			0
$781$	3.73689e15	−	6.47249e15i	0.460182	−	0.797059i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−2.94131e14			−0.0352174
$786$		0			0
$787$	6.07023e15	−	1.05139e16i	0.716711	−	1.24138i	−0.245585	−	0.969375i	$-0.578980\pi$
$787$	6.07023e15	−	1.05139e16i	0.716711	−	1.24138i	0.962296	−	0.272005i	$-0.0876866\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−7.19018e14	+	3.16475e15i	−0.0825599	+	0.363386i
$792$		0			0
$793$	−3.86769e15	−	6.69903e15i	−0.437975	−	0.758594i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−2.53264e15			−0.278967			−0.139484	−	0.990224i	$-0.544544\pi$
$797$	−2.53264e15			−0.278967			−0.139484	+	0.990224i	$0.544544\pi$
$798$		0			0
$799$	1.70896e16			1.85663
$800$		0			0
$801$		0			0
$802$		0			0
$803$	7.05437e13	+	1.22185e14i	0.00745628	+	0.0129147i
$804$		0			0
$805$	−1.72964e14	−	1.60167e14i	−0.0180334	−	0.0166992i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	6.65041e15	−	1.15188e16i	0.674732	−	1.16867i	−0.301815	−	0.953367i	$-0.597592\pi$
$809$	6.65041e15	−	1.15188e16i	0.674732	−	1.16867i	0.976547	−	0.215304i	$-0.0690742\pi$
$810$		0			0
$811$	9.18507e15			0.919322			0.459661	−	0.888094i	$-0.347971\pi$
$811$	9.18507e15			0.919322			0.459661	+	0.888094i	$0.347971\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1.09656e14	+	1.89930e14i	−0.0106824	+	0.0185024i
$816$		0			0
$817$	7.78403e14	+	1.34823e15i	0.0748141	+	0.129582i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.30192e15	−	1.61114e16i	−0.870333	−	1.50746i	−0.861653	−	0.507498i	$-0.830570\pi$
$821$	−9.30192e15	−	1.61114e16i	−0.870333	−	1.50746i	−0.00867994	−	0.999962i	$-0.502763\pi$
$822$		0			0
$823$	6.40393e15	−	1.10919e16i	0.591218	−	1.02402i	−0.402851	−	0.915266i	$-0.631981\pi$
$823$	6.40393e15	−	1.10919e16i	0.591218	−	1.02402i	0.994069	−	0.108754i	$-0.0346860\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.57444e16			−1.41529			−0.707644	−	0.706569i	$-0.750242\pi$
$827$	−1.57444e16			−1.41529			−0.707644	+	0.706569i	$0.750242\pi$
$828$		0			0
$829$	−8.53351e14	+	1.47805e15i	−0.0756968	+	0.131111i	−0.901389	−	0.433010i	$-0.857451\pi$
$829$	−8.53351e14	+	1.47805e15i	−0.0756968	+	0.131111i	0.825692	+	0.564121i	$0.190785\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.16891e16	+	8.00366e15i	−1.00980	+	0.691418i
$834$		0			0
$835$	2.15279e14	+	3.72875e14i	0.0183538	+	0.0317898i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	7.98094e15			0.662771			0.331385	−	0.943495i	$-0.392484\pi$
$839$	7.98094e15			0.662771			0.331385	+	0.943495i	$0.392484\pi$
$840$		0			0
$841$	−9.02998e15			−0.740131
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.72341e14	−	6.44914e14i	−0.0297323	−	0.0514979i
$846$		0			0
$847$	−8.59928e15	+	2.66191e15i	−0.677803	+	0.209814i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4.66342e15	−	8.07728e15i	0.358172	−	0.620373i
$852$		0			0
$853$	2.93699e15			0.222681			0.111340	−	0.993782i	$-0.464486\pi$
$853$	2.93699e15			0.222681			0.111340	+	0.993782i	$0.464486\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−9.30298e15	+	1.61132e16i	−0.687429	+	1.19066i	0.285238	+	0.958457i	$0.407927\pi$
$857$	−9.30298e15	+	1.61132e16i	−0.687429	+	1.19066i	−0.972667	+	0.232205i	$0.925406\pi$
$858$		0			0
$859$	3.96295e15	+	6.86403e15i	0.289106	+	0.500745i	0.973596	−	0.228276i	$-0.0733088\pi$
$859$	3.96295e15	+	6.86403e15i	0.289106	+	0.500745i	−0.684491	+	0.729021i	$0.739975\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	9.15029e15	+	1.58488e16i	0.650692	+	1.12703i	0.982955	+	0.183845i	$0.0588545\pi$
$863$	9.15029e15	+	1.58488e16i	0.650692	+	1.12703i	−0.332263	+	0.943187i	$0.607812\pi$
$864$		0			0
$865$	−1.14215e14	+	1.97825e14i	−0.00801924	+	0.0138897i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−2.65474e15			−0.181725
$870$		0			0
$871$	−1.32871e16	+	2.30140e16i	−0.898115	+	1.55558i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.62779e14	−	7.16468e14i	0.0107289	−	0.0472229i
$876$		0			0
$877$	5.33402e15	+	9.23880e15i	0.347182	+	0.601337i	0.985748	−	0.168231i	$-0.0538053\pi$
$877$	5.33402e15	+	9.23880e15i	0.347182	+	0.601337i	−0.638566	+	0.769567i	$0.720472\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−2.75942e16			−1.75166			−0.875830	−	0.482620i	$-0.839685\pi$
$881$	−2.75942e16			−1.75166			−0.875830	+	0.482620i	$0.839685\pi$
$882$		0			0
$883$	−2.38842e16			−1.49736			−0.748681	−	0.662930i	$-0.769313\pi$
$883$	−2.38842e16			−1.49736			−0.748681	+	0.662930i	$0.769313\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	5.41925e15	+	9.38642e15i	0.331405	+	0.574011i	0.982788	−	0.184739i	$-0.0591439\pi$
$887$	5.41925e15	+	9.38642e15i	0.331405	+	0.574011i	−0.651382	+	0.758750i	$0.725811\pi$
$888$		0			0
$889$	4.76556e15	−	2.09755e16i	0.287842	−	1.26693i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.89826e15	−	6.75198e15i	0.229714	−	0.397876i
$894$		0			0
$895$	5.53714e14			0.0322299
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4.53027e15	−	7.84666e15i	0.257304	−	0.445663i
$900$		0			0
$901$	−1.67707e16	−	2.90477e16i	−0.940947	−	1.62977i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	3.53460e14	+	6.12210e14i	0.0193541	+	0.0335223i
$906$		0			0
$907$	−6.13106e15	+	1.06193e16i	−0.331662	+	0.574456i	−0.982838	−	0.184471i	$-0.940943\pi$
$907$	−6.13106e15	+	1.06193e16i	−0.331662	+	0.574456i	0.651176	+	0.758927i	$0.274276\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.16631e16			0.615833			0.307916	−	0.951413i	$-0.400368\pi$
$911$	1.16631e16			0.615833			0.307916	+	0.951413i	$0.400368\pi$
$912$		0			0
$913$	5.01273e15	−	8.68230e15i	0.261508	−	0.452945i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1.57052e16	−	4.86154e15i	0.799856	−	0.247595i
$918$		0			0
$919$	−6.06280e14	−	1.05011e15i	−0.0305097	−	0.0528443i	0.850367	−	0.526189i	$-0.176380\pi$
$919$	−6.06280e14	−	1.05011e15i	−0.0305097	−	0.0528443i	−0.880877	+	0.473345i	$0.843046\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.46035e16			3.17429
$924$		0			0
$925$	1.45306e16			0.705509
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−2.32445e15	−	4.02606e15i	−0.110213	−	0.190895i	0.805643	−	0.592401i	$-0.201820\pi$
$929$	−2.32445e15	−	4.02606e15i	−0.110213	−	0.190895i	−0.915856	+	0.401507i	$0.868487\pi$
$930$		0			0
$931$	4.95815e14	+	6.44399e15i	0.0232325	+	0.301948i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.74534e14	+	3.02302e14i	−0.00798760	+	0.0138349i
$936$		0			0
$937$	−2.75328e16			−1.24532			−0.622662	−	0.782491i	$-0.713949\pi$
$937$	−2.75328e16			−1.24532			−0.622662	+	0.782491i	$0.713949\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−4.60313e15	+	7.97285e15i	−0.203381	+	0.352266i	−0.949616	−	0.313417i	$-0.898526\pi$
$941$	−4.60313e15	+	7.97285e15i	−0.203381	+	0.352266i	0.746235	+	0.665683i	$0.231860\pi$
$942$		0			0
$943$	2.18590e16	+	3.78609e16i	0.954589	+	1.65340i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.93459e15	+	5.08285e15i	0.125205	+	0.216862i	0.921813	−	0.387635i	$-0.126708\pi$
$947$	2.93459e15	+	5.08285e15i	0.125205	+	0.216862i	−0.796608	+	0.604496i	$0.793374\pi$
$948$		0			0
$949$	−6.09780e14	+	1.05617e15i	−0.0257164	+	0.0445420i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−1.09579e16			−0.451562			−0.225781	−	0.974178i	$-0.572493\pi$
$953$	−1.09579e16			−0.451562			−0.225781	+	0.974178i	$0.572493\pi$
$954$		0			0
$955$	−2.46957e14	+	4.27742e14i	−0.0100601	+	0.0174246i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.01617e16	−	1.86700e16i	−0.802647	−	0.743262i
$960$		0			0
$961$	−2.42093e14	−	4.19317e14i	−0.00952803	−	0.0165030i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.37099e14			−0.0360483
$966$		0			0
$967$	5.17427e15			0.196790			0.0983951	−	0.995147i	$-0.468629\pi$
$967$	5.17427e15			0.196790			0.0983951	+	0.995147i	$0.468629\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1.79678e16	+	3.11212e16i	0.668021	+	1.15705i	0.978457	+	0.206452i	$0.0661918\pi$
$971$	1.79678e16	+	3.11212e16i	0.668021	+	1.15705i	−0.310435	+	0.950594i	$0.600475\pi$
$972$		0			0
$973$	−8.44815e15	+	3.71844e16i	−0.310557	+	1.36691i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.67723e15	+	1.67615e16i	−0.347801	+	0.602409i	−0.985859	−	0.167579i	$-0.946405\pi$
$977$	−9.67723e15	+	1.67615e16i	−0.347801	+	0.602409i	0.638057	+	0.769989i	$0.279738\pi$
$978$		0			0
$979$	−1.04732e16			−0.372199
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.05812e15	+	1.83273e15i	−0.0367699	+	0.0636873i	−0.883825	−	0.467818i	$-0.845040\pi$
$983$	−1.05812e15	+	1.83273e15i	−0.0367699	+	0.0636873i	0.847055	+	0.531506i	$0.178374\pi$
$984$		0			0
$985$	2.78023e13	+	4.81550e13i	0.000955393	+	0.00165479i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	7.45957e15	+	1.29204e16i	0.250688	+	0.434205i
$990$		0			0
$991$	−7.42732e15	+	1.28645e16i	−0.246847	+	0.427551i	−0.962649	−	0.270752i	$-0.912728\pi$
$991$	−7.42732e15	+	1.28645e16i	−0.246847	+	0.427551i	0.715803	+	0.698303i	$0.246061\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.78378e14			0.0155505
$996$		0			0
$997$	−1.36080e16	+	2.35698e16i	−0.437494	+	0.757761i	−0.997495	−	0.0707302i	$-0.977467\pi$
$997$	−1.36080e16	+	2.35698e16i	−0.437494	+	0.757761i	0.560002	+	0.828491i	$0.310800\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.12.k.d.109.4		16
3.2	odd	2		84.12.i.b.25.5	✓	16
7.2	even	3	inner	252.12.k.d.37.4		16
21.2	odd	6		84.12.i.b.37.5	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.12.i.b.25.5	✓	16	3.2	odd	2
84.12.i.b.37.5	yes	16	21.2	odd	6
252.12.k.d.37.4		16	7.2	even	3	inner
252.12.k.d.109.4		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-84.6221 - 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +O(q^{10})\)	\(q+(-84.6221 - 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +(143939. - 249309. i) q^{11} +2.48841e6 q^{13} +(3.58228e6 - 6.20469e6i) q^{17} +(-1.63429e6 - 2.83067e6i) q^{19} +(-1.56616e7 - 2.71268e7i) q^{23} +(2.43997e7 - 4.22616e7i) q^{25} -5.63075e7 q^{29} +(-8.04560e7 + 1.39354e8i) q^{31} +(7.18924e6 - 2.22543e6i) q^{35} +(1.48880e8 + 2.57868e8i) q^{37} -1.39570e9 q^{41} -4.76296e8 q^{43} +(1.19265e9 + 2.06573e9i) q^{47} +(-1.78321e9 - 8.54381e8i) q^{49} +(2.34079e9 - 4.05437e9i) q^{53} -4.87215e7 q^{55} +(8.21849e8 - 1.42348e9i) q^{59} +(-1.55428e9 - 2.69209e9i) q^{61} +(-2.10575e8 - 3.64726e8i) q^{65} +(-5.33960e9 + 9.24846e9i) q^{67} +2.59617e10 q^{71} +(-2.45048e8 + 4.24435e8i) q^{73} +(9.39252e9 + 8.69760e9i) q^{77} +(-4.61089e9 - 7.98630e9i) q^{79} +3.48255e10 q^{83} -1.21256e9 q^{85} +(-1.81904e10 - 3.15067e10i) q^{89} +(-2.45151e10 + 1.07903e11i) q^{91} +(-2.76593e8 + 4.79074e8i) q^{95} -1.51725e11 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2156 q^{5} + 50512 q^{7}+O(q^{10}) \) 16 * q + 2156 * q^5 + 50512 * q^7	\( 16 q + 2156 q^{5} + 50512 q^{7} + 222796 q^{11} + 2703176 q^{13} - 5114600 q^{17} + 6910556 q^{19} + 51387712 q^{23} - 191456372 q^{25} - 118854616 q^{29} + 164659160 q^{31} - 55239344 q^{35} + 75658364 q^{37} + 1815568608 q^{41} + 10754408 q^{43} + 1034359464 q^{47} + 4123496848 q^{49} + 665159988 q^{53} - 1264543896 q^{55} - 1040514580 q^{59} - 14391208024 q^{61} + 20938150200 q^{65} - 33307097284 q^{67} - 65848902896 q^{71} + 17709749204 q^{73} - 8594484604 q^{77} - 26626784032 q^{79} + 210306955048 q^{83} - 25867402032 q^{85} + 55951560072 q^{89} + 66078280292 q^{91} - 106810047392 q^{95} - 156216030712 q^{97}+O(q^{100}) \) 16 * q + 2156 * q^5 + 50512 * q^7 + 222796 * q^11 + 2703176 * q^13 - 5114600 * q^17 + 6910556 * q^19 + 51387712 * q^23 - 191456372 * q^25 - 118854616 * q^29 + 164659160 * q^31 - 55239344 * q^35 + 75658364 * q^37 + 1815568608 * q^41 + 10754408 * q^43 + 1034359464 * q^47 + 4123496848 * q^49 + 665159988 * q^53 - 1264543896 * q^55 - 1040514580 * q^59 - 14391208024 * q^61 + 20938150200 * q^65 - 33307097284 * q^67 - 65848902896 * q^71 + 17709749204 * q^73 - 8594484604 * q^77 - 26626784032 * q^79 + 210306955048 * q^83 - 25867402032 * q^85 + 55951560072 * q^89 + 66078280292 * q^91 - 106810047392 * q^95 - 156216030712 * q^97

Introduction

L-functions

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Database

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