LMFDB - Embedded newform 84.12.i.b.25.5

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			25.5
Root			$-438.744 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	84.25
Dual form			84.12.i.b.37.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-121.500 + 210.444i) q^{3} +(84.6221 + 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})$	\(q+(-121.500 + 210.444i) q^{3} +(84.6221 + 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-143939. + 249309. i) q^{11} +2.48841e6 q^{13} -41126.3 q^{15} +(-3.58228e6 + 6.20469e6i) q^{17} +(-1.63429e6 - 2.83067e6i) q^{19} +(-7.92832e6 - 7.34173e6i) q^{21} +(1.56616e7 + 2.71268e7i) q^{23} +(2.43997e7 - 4.22616e7i) q^{25} +1.43489e7 q^{27} +5.63075e7 q^{29} +(-8.04560e7 + 1.39354e8i) q^{31} +(-3.49771e7 - 6.05821e7i) q^{33} +(-7.18924e6 + 2.22543e6i) q^{35} +(1.48880e8 + 2.57868e8i) q^{37} +(-3.02342e8 + 5.23672e8i) q^{39} +1.39570e9 q^{41} -4.76296e8 q^{43} +(4.99685e6 - 8.65480e6i) q^{45} +(-1.19265e9 - 2.06573e9i) q^{47} +(-1.78321e9 - 8.54381e8i) q^{49} +(-8.70493e8 - 1.50774e9i) q^{51} +(-2.34079e9 + 4.05437e9i) q^{53} -4.87215e7 q^{55} +7.94263e8 q^{57} +(-8.21849e8 + 1.42348e9i) q^{59} +(-1.55428e9 - 2.69209e9i) q^{61} +(2.50831e9 - 7.76448e8i) q^{63} +(2.10575e8 + 3.64726e8i) q^{65} +(-5.33960e9 + 9.24846e9i) q^{67} -7.61156e9 q^{69} -2.59617e10 q^{71} +(-2.45048e8 + 4.24435e8i) q^{73} +(5.92914e9 + 1.02696e10i) q^{75} +(-9.39252e9 - 8.69760e9i) q^{77} +(-4.61089e9 - 7.98630e9i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} -3.48255e10 q^{83} -1.21256e9 q^{85} +(-6.84136e9 + 1.18496e10i) q^{87} +(1.81904e10 + 3.15067e10i) q^{89} +(-2.45151e10 + 1.07903e11i) q^{91} +(-1.95508e10 - 3.38630e10i) q^{93} +(2.76593e8 - 4.79074e8i) q^{95} -1.51725e11 q^{97} +1.69989e10 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	84.6221	+	146.570i	0.0121101	+	0.0209754i	0.872017	−	0.489476i	$-0.162812\pi$
$5$	84.6221	+	146.570i	0.0121101	+	0.0209754i	−0.859907	+	0.510451i	$0.829478\pi$
$6$		0			0
$7$	−9851.70	+	43362.1i	−0.221550	+	0.975149i
$7$	−9851.70	+	43362.1i	−0.221550	+	0.975149i
$8$		0			0
$9$	−29524.5	−	51137.9i	−0.166667	−	0.288675i
$10$		0			0
$11$	−143939.	+	249309.i	−0.269474	+	0.466743i	−0.968726	−	0.248132i	$-0.920183\pi$
$11$	−143939.	+	249309.i	−0.269474	+	0.466743i	0.699252	+	0.714875i	$0.253517\pi$
$12$		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
$15$	−41126.3			−0.0139836
$16$		0			0
$17$	−3.58228e6	+	6.20469e6i	−0.611914	+	1.05987i	0.379004	+	0.925395i	$0.376267\pi$
$17$	−3.58228e6	+	6.20469e6i	−0.611914	+	1.05987i	−0.990918	+	0.134470i	$0.957067\pi$
$18$		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
$21$	−7.92832e6	−	7.34173e6i	−0.423618	−	0.392276i
$22$		0			0
$23$	1.56616e7	+	2.71268e7i	0.507381	+	0.878810i	0.999963	+	0.00854416i	$0.00271972\pi$
$23$	1.56616e7	+	2.71268e7i	0.507381	+	0.878810i	−0.492582	+	0.870266i	$0.663947\pi$
$24$		0			0
$25$	2.43997e7	−	4.22616e7i	0.499707	−	0.865517i
$26$		0			0
$27$	1.43489e7			0.192450
$28$		0			0
$29$	5.63075e7			0.509773			0.254887	−	0.966971i	$-0.417962\pi$
$29$	5.63075e7			0.509773			0.254887	+	0.966971i	$0.417962\pi$
$30$		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
$33$	−3.49771e7	−	6.05821e7i	−0.155581	−	0.269474i
$34$		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
$39$	−3.02342e8	+	5.23672e8i	−0.536591	+	0.929404i
$40$		0			0
$41$	1.39570e9			1.88140			0.940702	−	0.339235i	$-0.110168\pi$
$41$	1.39570e9			1.88140			0.940702	+	0.339235i	$0.110168\pi$
$42$		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
$45$	4.99685e6	−	8.65480e6i	0.00403671	−	0.00699178i
$46$		0			0
$47$	−1.19265e9	−	2.06573e9i	−0.758533	−	1.31382i	−0.943599	−	0.331091i	$-0.892583\pi$
$47$	−1.19265e9	−	2.06573e9i	−0.758533	−	1.31382i	0.185066	−	0.982726i	$-0.440750\pi$
$48$		0			0
$49$	−1.78321e9	−	8.54381e8i	−0.901831	−	0.432089i
$50$		0			0
$51$	−8.70493e8	−	1.50774e9i	−0.353288	−	0.611914i
$52$		0			0
$53$	−2.34079e9	+	4.05437e9i	−0.768856	+	1.33170i	0.169327	+	0.985560i	$0.445841\pi$
$53$	−2.34079e9	+	4.05437e9i	−0.768856	+	1.33170i	−0.938183	+	0.346139i	$0.887493\pi$
$54$		0			0
$55$	−4.87215e7			−0.0130535
$56$		0			0
$57$	7.94263e8			0.174845
$58$		0			0
$59$	−8.21849e8	+	1.42348e9i	−0.149660	+	0.259219i	−0.931102	−	0.364759i	$-0.881151\pi$
$59$	−8.21849e8	+	1.42348e9i	−0.149660	+	0.259219i	0.781442	+	0.623978i	$0.214485\pi$
$60$		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
$63$	2.50831e9	−	7.76448e8i	0.318426	−	0.0985688i
$64$		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
$69$	−7.61156e9			−0.585873
$70$		0			0
$71$	−2.59617e10			−1.70770			−0.853852	−	0.520516i	$-0.825739\pi$
$71$	−2.59617e10			−1.70770			−0.853852	+	0.520516i	$0.825739\pi$
$72$		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
$75$	5.92914e9	+	1.02696e10i	0.288506	+	0.499707i
$76$		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
$81$	−1.74339e9	+	3.01964e9i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−3.48255e10			−0.970437			−0.485219	−	0.874393i	$-0.661260\pi$
$83$	−3.48255e10			−0.970437			−0.485219	+	0.874393i	$0.661260\pi$
$84$		0			0
$85$	−1.21256e9			−0.0296414
$86$		0			0
$87$	−6.84136e9	+	1.18496e10i	−0.147159	+	0.254887i
$88$		0			0
$89$	1.81904e10	+	3.15067e10i	0.345301	+	0.598078i	0.985408	−	0.170207i	$-0.0544437\pi$
$89$	1.81904e10	+	3.15067e10i	0.345301	+	0.598078i	−0.640108	+	0.768285i	$0.721110\pi$
$90$		0			0
$91$	−2.45151e10	+	1.07903e11i	−0.411819	+	1.81261i
$92$		0			0
$93$	−1.95508e10	−	3.38630e10i	−0.291413	−	0.504742i
$94$		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
$99$	1.69989e10			0.179650
$100$		0			0
$101$	−7.68508e10	+	1.33110e11i	−0.727581	+	1.26021i	0.230322	+	0.973114i	$0.426022\pi$
$101$	−7.68508e10	+	1.33110e11i	−0.727581	+	1.26021i	−0.957903	+	0.287092i	$0.907311\pi$
$102$		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
$105$	4.05164e8	−	1.78332e9i	0.00309806	−	0.0136361i
$106$		0			0
$107$	−1.46571e9	−	2.53868e9i	−0.0101027	−	0.0174984i	0.860930	−	0.508724i	$-0.169883\pi$
$107$	−1.46571e9	−	2.53868e9i	−0.0101027	−	0.0174984i	−0.871033	+	0.491225i	$0.836549\pi$
$108$		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
$111$	−7.23558e10			−0.407565
$112$		0			0
$113$	−7.29842e10			−0.372647			−0.186323	−	0.982488i	$-0.559657\pi$
$113$	−7.29842e10			−0.372647			−0.186323	+	0.982488i	$0.559657\pi$
$114$		0			0
$115$	−2.65064e9	+	4.59105e9i	−0.0122889	+	0.0212850i
$116$		0			0
$117$	−7.34692e10	−	1.27252e11i	−0.309801	−	0.536591i
$118$		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
$123$	−1.69578e11	+	2.93718e11i	−0.543114	+	0.940702i
$124$		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
$129$	5.78699e10	−	1.00234e11i	0.142630	−	0.247042i
$130$		0			0
$131$	1.84860e11	+	3.20188e11i	0.418651	+	0.725124i	0.995804	−	0.0915115i	$-0.0291698\pi$
$131$	1.84860e11	+	3.20188e11i	0.418651	+	0.725124i	−0.577153	+	0.816636i	$0.695836\pi$
$132$		0			0
$133$	1.38844e11	−	4.29792e10i	0.289297	−	0.0895517i
$134$		0			0
$135$	1.21423e9	+	2.10312e9i	0.00233059	+	0.00403671i
$136$		0			0
$137$	3.08974e11	−	5.35159e11i	0.546964	−	0.947370i	−0.451516	−	0.892263i	$-0.649117\pi$
$137$	3.08974e11	−	5.35159e11i	0.546964	−	0.947370i	0.998480	−	0.0551072i	$-0.0175501\pi$
$138$		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
$141$	5.79627e11			0.875878
$142$		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
$147$	3.96460e11	−	2.71460e11i	0.476381	−	0.326182i
$148$		0			0
$149$	3.33560e11	+	5.77744e11i	0.372092	+	0.644482i	0.989887	−	0.141857i	$-0.0453073\pi$
$149$	3.33560e11	+	5.77744e11i	0.372092	+	0.644482i	−0.617795	+	0.786339i	$0.711974\pi$
$150$		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
$153$	4.23060e11			0.407942
$154$		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
$159$	−5.68812e11	−	9.85211e11i	−0.443899	−	0.768856i
$160$		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
$165$	5.91967e9	−	1.02532e10i	0.00376821	−	0.00652674i
$166$		0			0
$167$	2.54401e12			1.51558			0.757789	−	0.652500i	$-0.226280\pi$
$167$	2.54401e12			1.51558			0.757789	+	0.652500i	$0.226280\pi$
$168$		0			0
$169$	4.40005e12			2.45516
$170$		0			0
$171$	−9.65030e10	+	1.67148e11i	−0.0504733	+	0.0874224i
$172$		0			0
$173$	6.74851e11	+	1.16888e12i	0.331096	+	0.573476i	0.982727	−	0.185061i	$-0.0592482\pi$
$173$	6.74851e11	+	1.16888e12i	0.331096	+	0.573476i	−0.651631	+	0.758536i	$0.725915\pi$
$174$		0			0
$175$	1.59217e12	+	1.47437e12i	0.733298	+	0.679044i
$176$		0			0
$177$	−1.99709e11	−	3.45907e11i	−0.0864063	−	0.149660i
$178$		0			0
$179$	1.63584e12	−	2.83337e12i	0.665350	−	1.15242i	−0.313840	−	0.949476i	$-0.601615\pi$
$179$	1.63584e12	−	2.83337e12i	0.665350	−	1.15242i	0.979190	−	0.202945i	$-0.0650512\pi$
$180$		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
$183$	7.55379e11			0.272072
$184$		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
$189$	−1.41361e11	+	6.22199e11i	−0.0426374	+	0.187668i
$190$		0			0
$191$	1.45917e12	+	2.52736e12i	0.415359	+	0.719423i	0.995466	−	0.0951173i	$-0.0303226\pi$
$191$	1.45917e12	+	2.52736e12i	0.415359	+	0.719423i	−0.580107	+	0.814540i	$0.696989\pi$
$192$		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
$195$	−1.02339e11			−0.0259928
$196$		0			0
$197$	3.28547e11			0.0788920			0.0394460	−	0.999222i	$-0.487441\pi$
$197$	3.28547e11			0.0788920			0.0394460	+	0.999222i	$0.487441\pi$
$198$		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
$201$	−1.29752e12	−	2.24738e12i	−0.278957	−	0.483167i
$202$		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
$207$	9.24805e11	−	1.60181e12i	0.169127	−	0.292937i
$208$		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
$213$	3.15435e12	−	5.46349e12i	0.492972	−	0.853852i
$214$		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
$219$	−5.95466e10	−	1.03138e11i	−0.00798756	−	0.0138349i
$220$		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
$225$	−2.88156e12			−0.333138
$226$		0			0
$227$	−7.71055e12	+	1.33551e13i	−0.849069	+	1.47063i	0.0329705	+	0.999456i	$0.489503\pi$
$227$	−7.71055e12	+	1.33551e13i	−0.849069	+	1.47063i	−0.882040	+	0.471175i	$0.843830\pi$
$228$		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
$231$	2.97155e12	−	9.19842e11i	0.297247	−	0.0920127i
$232$		0			0
$233$	3.52941e12	+	6.11311e12i	0.336701	+	0.583183i	0.983810	−	0.179215i	$-0.0573556\pi$
$233$	3.52941e12	+	6.11311e12i	0.336701	+	0.583183i	−0.647109	+	0.762397i	$0.724022\pi$
$234$		0			0
$235$	2.01849e11	−	3.49612e11i	0.0183719	−	0.0318210i
$236$		0			0
$237$	2.24089e12			0.194673
$238$		0			0
$239$	−7.38390e12			−0.612487			−0.306244	−	0.951953i	$-0.599072\pi$
$239$	−7.38390e12			−0.612487			−0.306244	+	0.951953i	$0.599072\pi$
$240$		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
$243$	−4.23644e11	−	7.33773e11i	−0.0320750	−	0.0555556i
$244$		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
$249$	4.23129e12	−	7.32882e12i	0.280141	−	0.485219i
$250$		0			0
$251$	−1.93352e13			−1.22502			−0.612509	−	0.790464i	$-0.709840\pi$
$251$	−1.93352e13			−1.22502			−0.612509	+	0.790464i	$0.709840\pi$
$252$		0			0
$253$	−9.01726e12			−0.546905
$254$		0			0
$255$	1.47326e11	−	2.55176e11i	0.00855674	−	0.0148207i
$256$		0			0
$257$	−9.86761e12	−	1.70912e13i	−0.549009	−	0.950912i	−0.998343	−	0.0575480i	$-0.981672\pi$
$257$	−9.86761e12	−	1.70912e13i	−0.549009	−	0.950912i	0.449333	−	0.893364i	$-0.351662\pi$
$258$		0			0
$259$	−1.26484e13	+	3.91532e12i	−0.674354	+	0.208746i
$260$		0			0
$261$	−1.66245e12	−	2.87945e12i	−0.0849622	−	0.147159i
$262$		0			0
$263$	−1.57662e13	+	2.73078e13i	−0.772627	+	1.33823i	0.163492	+	0.986545i	$0.447724\pi$
$263$	−1.57662e13	+	2.73078e13i	−0.772627	+	1.33823i	−0.936119	+	0.351685i	$0.885609\pi$
$264$		0			0
$265$	−7.92330e11			−0.0372438
$266$		0			0
$267$	−8.84054e12			−0.398719
$268$		0			0
$269$	−1.20365e13	+	2.08477e13i	−0.521028	+	0.902447i	0.478673	+	0.877993i	$0.341118\pi$
$269$	−1.20365e13	+	2.08477e13i	−0.521028	+	0.902447i	−0.999701	+	0.0244535i	$0.992215\pi$
$270$		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
$273$	−1.97289e13	−	1.82693e13i	−0.787425	−	0.729166i
$274$		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
$279$	9.50169e12			0.336494
$280$		0			0
$281$	3.61357e13			1.23042			0.615208	−	0.788365i	$-0.289072\pi$
$281$	3.61357e13			1.23042			0.615208	+	0.788365i	$0.289072\pi$
$282$		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
$285$	6.72122e10	+	1.16415e11i	0.00211739	+	0.00366743i
$286$		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
$291$	1.84345e13	−	3.19296e13i	0.517870	−	0.896978i
$292$		0			0
$293$	−3.28945e13			−0.889922			−0.444961	−	0.895550i	$-0.646782\pi$
$293$	−3.28945e13			−0.889922			−0.444961	+	0.895550i	$0.646782\pi$
$294$		0			0
$295$	−2.78186e11			−0.00724961
$296$		0			0
$297$	−2.06536e12	+	3.57731e12i	−0.0518604	+	0.0898248i
$298$		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
$303$	−1.86748e13	−	3.23456e13i	−0.420069	−	0.727581i
$304$		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
$309$	4.31720e13			0.871827
$310$		0			0
$311$	−2.99473e12	+	5.18703e12i	−0.0583682	+	0.101097i	−0.893733	−	0.448599i	$-0.851923\pi$
$311$	−2.99473e12	+	5.18703e12i	−0.0583682	+	0.101097i	0.835365	+	0.549696i	$0.185256\pi$
$312$		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
$315$	3.26063e11	+	3.01938e11i	0.00592370	+	0.00548542i
$316$		0			0
$317$	−9.63494e12	−	1.66882e13i	−0.169053	−	0.292808i	0.769034	−	0.639208i	$-0.220738\pi$
$317$	−9.63494e12	−	1.66882e13i	−0.169053	−	0.292808i	−0.938087	+	0.346399i	$0.887404\pi$
$318$		0			0
$319$	−8.10482e12	+	1.40380e13i	−0.137371	+	0.237933i
$320$		0			0
$321$	7.12335e11			0.0116656
$322$		0			0
$323$	2.34179e13			0.370624
$324$		0			0
$325$	6.07167e13	−	1.05164e14i	0.928858	−	1.60883i
$326$		0			0
$327$	3.34425e12	+	5.79240e12i	0.0494636	+	0.0856734i
$328$		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
$333$	8.79123e12	−	1.52269e13i	0.117654	−	0.203783i
$334$		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
$339$	8.86757e12	−	1.53591e13i	0.107574	−	0.186323i
$340$		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
$345$	−6.44106e11	−	1.11562e12i	−0.00709500	−	0.0122889i
$346$		0			0
$347$	−6.92423e13	+	1.19931e14i	−0.738855	+	1.27974i	0.214156	+	0.976800i	$0.431300\pi$
$347$	−6.92423e13	+	1.19931e14i	−0.738855	+	1.27974i	−0.953011	+	0.302936i	$0.902033\pi$
$348$		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
$351$	3.57060e13			0.357728
$352$		0			0
$353$	−5.08428e13	+	8.80624e13i	−0.493707	+	0.855125i	−0.999974	−	0.00725175i	$-0.997692\pi$
$353$	−5.08428e13	+	8.80624e13i	−0.493707	+	0.855125i	0.506267	+	0.862377i	$0.331025\pi$
$354$		0			0
$355$	−2.19693e12	−	3.80520e12i	−0.0206805	−	0.0358197i
$356$		0			0
$357$	7.39545e13	−	2.28926e13i	0.674978	−	0.208939i
$358$		0			0
$359$	4.26384e13	+	7.38518e13i	0.377382	+	0.653645i	0.990680	−	0.136207i	$-0.0434911\pi$
$359$	4.26384e13	+	7.38518e13i	0.377382	+	0.653645i	−0.613299	+	0.789851i	$0.710158\pi$
$360$		0			0
$361$	5.29033e13	−	9.16313e13i	0.454144	−	0.786600i
$362$		0			0
$363$	−4.91925e13			−0.409650
$364$		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
$369$	−4.12074e13	−	7.13734e13i	−0.313567	−	0.543114i
$370$		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
$375$	−2.00753e12	+	3.47715e12i	−0.0139795	+	0.0242131i
$376$		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
$381$	5.87732e13	−	1.01798e14i	0.375052	−	0.649610i
$382$		0			0
$383$	1.21897e14	+	2.11132e14i	0.755790	+	1.30907i	0.944981	+	0.327126i	$0.106080\pi$
$383$	1.21897e14	+	2.11132e14i	0.755790	+	1.30907i	−0.189191	+	0.981940i	$0.560587\pi$
$384$		0			0
$385$	4.79990e11	−	2.11267e12i	0.00289200	−	0.0127291i
$386$		0			0
$387$	1.40624e13	+	2.43568e13i	0.0823472	+	0.142630i
$388$		0			0
$389$	1.06560e14	−	1.84568e14i	0.606559	−	1.05059i	−0.385244	−	0.922815i	$-0.625883\pi$
$389$	1.06560e14	−	1.84568e14i	0.606559	−	1.05059i	0.991803	−	0.127776i	$-0.0407839\pi$
$390$		0			0
$391$	−2.24417e14			−1.24189
$392$		0			0
$393$	−8.98422e13			−0.483416
$394$		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
$399$	−7.82484e12	+	3.44409e13i	−0.0387369	+	0.170500i
$400$		0			0
$401$	−3.73100e13	−	6.46228e13i	−0.179693	−	0.311237i	0.762082	−	0.647480i	$-0.224177\pi$
$401$	−3.73100e13	−	6.46228e13i	−0.179693	−	0.311237i	−0.941775	+	0.336243i	$0.890844\pi$
$402$		0			0
$403$	−2.00208e14	+	3.46770e14i	−0.938217	+	1.62504i
$404$		0			0
$405$	−5.90118e11			−0.00269114
$406$		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
$411$	7.50807e13	+	1.30044e14i	0.315790	+	0.546964i
$412$		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
$417$	−1.04190e14	+	1.80463e14i	−0.404649	+	0.700873i
$418$		0			0
$419$	−5.96915e13			−0.225806			−0.112903	−	0.993606i	$-0.536015\pi$
$419$	−5.96915e13			−0.225806			−0.112903	+	0.993606i	$0.536015\pi$
$420$		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
$423$	−7.04247e13	+	1.21979e14i	−0.252844	+	0.437939i
$424$		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
$429$	−8.70375e13	−	1.50753e14i	−0.289195	−	0.500901i
$430$		0			0
$431$	6.39033e13	−	1.10684e14i	0.206966	−	0.358475i	−0.743792	−	0.668412i	$-0.766974\pi$
$431$	6.39033e13	−	1.10684e14i	0.206966	−	0.358475i	0.950757	+	0.309937i	$0.100308\pi$
$432$		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
$435$	−2.31572e12			−0.00712845
$436$		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
$441$	8.95724e12	+	1.16415e14i	0.0255718	+	0.332351i
$442$		0			0
$443$	−2.51908e14	−	4.36317e14i	−0.701489	−	1.21502i	−0.967944	−	0.251168i	$-0.919185\pi$
$443$	−2.51908e14	−	4.36317e14i	−0.701489	−	1.21502i	0.266454	−	0.963848i	$-0.414148\pi$
$444$		0			0
$445$	−3.07862e12	+	5.33232e12i	−0.00836327	+	0.0144856i
$446$		0			0
$447$	−1.62110e14			−0.429655
$448$		0			0
$449$	−4.47372e14			−1.15695			−0.578474	−	0.815701i	$-0.696351\pi$
$449$	−4.47372e14			−1.15695			−0.578474	+	0.815701i	$0.696351\pi$
$450$		0			0
$451$	−2.00896e14	+	3.47961e14i	−0.506990	+	0.878133i
$452$		0			0
$453$	−6.36054e13	−	1.10168e14i	−0.156658	−	0.271340i
$454$		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
$459$	−5.14018e13	+	8.90304e13i	−0.117763	+	0.203971i
$460$		0			0
$461$	−6.55113e14			−1.46542			−0.732709	−	0.680543i	$-0.761744\pi$
$461$	−6.55113e14			−1.46542			−0.732709	+	0.680543i	$0.761744\pi$
$462$		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
$465$	3.30886e12	−	5.73111e12i	0.00705809	−	0.0122250i
$466$		0			0
$467$	−1.12796e14	−	1.95369e14i	−0.234992	−	0.407017i	0.724279	−	0.689507i	$-0.242173\pi$
$467$	−1.12796e14	−	1.95369e14i	−0.234992	−	0.407017i	−0.959270	+	0.282490i	$0.908840\pi$
$468$		0			0
$469$	−3.48428e14	−	3.22649e14i	−0.709027	−	0.656569i
$470$		0			0
$471$	2.11156e14	+	3.65733e14i	0.419748	+	0.727024i
$472$		0			0
$473$	6.85573e13	−	1.18745e14i	0.133143	−	0.230610i
$474$		0			0
$475$	−1.59505e14			−0.302662
$476$		0			0
$477$	2.76443e14			0.512571
$478$		0			0
$479$	−1.28941e14	+	2.23332e14i	−0.233639	+	0.404674i	−0.958876	−	0.283825i	$-0.908397\pi$
$479$	−1.28941e14	+	2.23332e14i	−0.233639	+	0.404674i	0.725237	+	0.688499i	$0.241730\pi$
$480$		0			0
$481$	3.70476e14	+	6.41683e14i	0.656088	+	1.13638i
$482$		0			0
$483$	7.49869e13	−	3.30053e14i	0.129800	−	0.571314i
$484$		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
$489$	3.14888e14			0.509281
$490$		0			0
$491$	7.76606e14			1.22815			0.614076	−	0.789247i	$-0.289529\pi$
$491$	7.76606e14			1.22815			0.614076	+	0.789247i	$0.289529\pi$
$492$		0			0
$493$	−2.01709e14	+	3.49370e14i	−0.311937	+	0.540291i
$494$		0			0
$495$	1.43848e12	+	2.49152e12i	0.00217558	+	0.00376821i
$496$		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
$501$	−3.09097e14	+	5.35372e14i	−0.437510	+	0.757789i
$502$		0			0
$503$	7.47083e14			1.03453			0.517267	−	0.855824i	$-0.326949\pi$
$503$	7.47083e14			1.03453			0.517267	+	0.855824i	$0.326949\pi$
$504$		0			0
$505$	−2.60131e13			−0.0352444
$506$		0			0
$507$	−5.34606e14	+	9.25964e14i	−0.708745	+	1.22758i
$508$		0			0
$509$	−7.15602e14	−	1.23946e15i	−0.928376	−	1.60799i	−0.786039	−	0.618176i	$-0.787872\pi$
$509$	−7.15602e14	−	1.23946e15i	−0.928376	−	1.60799i	−0.142337	−	0.989818i	$-0.545462\pi$
$510$		0			0
$511$	−1.59903e13	−	1.48072e13i	−0.0203021	−	0.0188000i
$512$		0			0
$513$	−2.34502e13	−	4.06170e13i	−0.0291408	−	0.0504733i
$514$		0			0
$515$	1.50342e13	−	2.60399e13i	0.0182869	−	0.0316738i
$516$		0			0
$517$	6.86673e14			0.817621
$518$		0			0
$519$	−3.27978e14			−0.382317
$520$		0			0
$521$	1.34528e14	−	2.33009e14i	0.153534	−	0.265929i	−0.778990	−	0.627036i	$-0.784268\pi$
$521$	1.34528e14	−	2.33009e14i	0.153534	−	0.265929i	0.932524	+	0.361107i	$0.117601\pi$
$522$		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
$525$	−5.03722e14	+	1.55927e14i	−0.551207	+	0.170626i
$526$		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
$531$	9.70588e13			0.0997734
$532$		0			0
$533$	3.47309e15			3.49717
$534$		0			0
$535$	2.48063e11	−	4.29657e11i	0.000244690	−	0.000423815i
$536$		0			0
$537$	3.97510e14	+	6.88508e14i	0.384140	+	0.665350i
$538$		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
$543$	5.07496e14	−	8.79009e14i	0.461353	−	0.799087i
$544$		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
$549$	−9.17786e13	+	1.58965e14i	−0.0785404	+	0.136036i
$550$		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
$555$	−6.12290e12	−	1.06052e13i	−0.00493566	−	0.00854882i
$556$		0			0
$557$	−1.67288e14	+	2.89751e14i	−0.132209	+	0.228992i	−0.924528	−	0.381115i	$-0.875540\pi$
$557$	−1.67288e14	+	2.89751e14i	−0.132209	+	0.228992i	0.792319	+	0.610107i	$0.208874\pi$
$558$		0			0
$559$	−1.18522e15			−0.918405
$560$		0			0
$561$	5.01190e14			0.380809
$562$		0			0
$563$	−1.53096e14	+	2.65170e14i	−0.114069	+	0.197573i	−0.917407	−	0.397950i	$-0.869722\pi$
$563$	−1.53096e14	+	2.65170e14i	−0.114069	+	0.197573i	0.803338	+	0.595523i	$0.203055\pi$
$564$		0			0
$565$	−6.17607e12	−	1.06973e13i	−0.00451280	−	0.00781639i
$566$		0			0
$567$	−1.13763e14	−	1.05346e14i	−0.0815254	−	0.0754936i
$568$		0			0
$569$	9.44055e13	+	1.63515e14i	0.0663559	+	0.114932i	0.897295	−	0.441432i	$-0.145529\pi$
$569$	9.44055e13	+	1.63515e14i	0.0663559	+	0.114932i	−0.830939	+	0.556364i	$0.812196\pi$
$570$		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
$573$	−7.09159e14			−0.479615
$574$		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
$579$	6.72741e14	+	1.16522e15i	0.429651	+	0.744178i
$580$		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
$585$	1.24342e13	−	2.15367e13i	0.00750346	−	0.0129964i
$586$		0			0
$587$	−8.80723e14			−0.521591			−0.260795	−	0.965394i	$-0.583985\pi$
$587$	−8.80723e14			−0.521591			−0.260795	+	0.965394i	$0.583985\pi$
$588$		0			0
$589$	5.25952e14			0.305712
$590$		0			0
$591$	−3.99185e13	+	6.91408e13i	−0.0227742	+	0.0394460i
$592$		0			0
$593$	5.84464e14	+	1.01232e15i	0.327308	+	0.566915i	0.981977	−	0.189001i	$-0.0605251\pi$
$593$	5.84464e14	+	1.01232e15i	0.327308	+	0.566915i	−0.654668	+	0.755916i	$0.727192\pi$
$594$		0			0
$595$	1.19458e13	−	5.25791e13i	0.00656706	−	0.0289048i
$596$		0			0
$597$	−3.43426e14	−	5.94832e14i	−0.185342	−	0.321023i
$598$		0			0
$599$	3.09149e14	−	5.35461e14i	0.163802	−	0.283714i	−0.772427	−	0.635104i	$-0.780957\pi$
$599$	3.09149e14	−	5.35461e14i	0.163802	−	0.283714i	0.936229	+	0.351390i	$0.114291\pi$
$600$		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
$603$	6.30596e14			0.322111
$604$		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
$609$	−4.46423e14	−	4.13394e14i	−0.215949	−	0.199972i
$610$		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
$615$	−5.74002e13			−0.0263087
$616$		0			0
$617$	2.78051e14			0.125186			0.0625932	−	0.998039i	$-0.480063\pi$
$617$	2.78051e14			0.125186			0.0625932	+	0.998039i	$0.480063\pi$
$618$		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
$621$	2.24728e14	+	3.89240e14i	0.0976456	+	0.169127i
$622$		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
$627$	−1.14325e14	+	1.98017e14i	−0.0471162	+	0.0816076i
$628$		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
$633$	−5.58861e14	+	9.67976e14i	−0.218567	+	0.378568i
$634$		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
$639$	7.66507e14	+	1.32763e15i	0.284617	+	0.492972i
$640$		0			0
$641$	−9.30652e14	+	1.61194e15i	−0.339679	+	0.588341i	−0.984372	−	0.176100i	$-0.943652\pi$
$641$	−9.30652e14	+	1.61194e15i	−0.339679	+	0.588341i	0.644694	+	0.764441i	$0.276985\pi$
$642$		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
$645$	1.95883e13			0.00690905
$646$		0			0
$647$	−9.99898e14	+	1.73187e15i	−0.346723	+	0.600541i	−0.985665	−	0.168713i	$-0.946039\pi$
$647$	−9.99898e14	+	1.73187e15i	−0.346723	+	0.600541i	0.638943	+	0.769254i	$0.279372\pi$
$648$		0			0
$649$	−2.36592e14	−	4.09789e14i	−0.0806591	−	0.139706i
$650$		0			0
$651$	1.66098e15	−	5.14156e14i	0.556761	−	0.172345i
$652$		0			0
$653$	2.80003e15	+	4.84979e15i	0.922869	+	1.59846i	0.794954	+	0.606670i	$0.207495\pi$
$653$	2.80003e15	+	4.84979e15i	0.922869	+	1.59846i	0.127915	+	0.991785i	$0.459172\pi$
$654$		0			0
$655$	−3.12866e13	+	5.41899e13i	−0.0101398	+	0.0175627i
$656$		0			0
$657$	2.89397e13			0.00922325
$658$		0			0
$659$	−1.96747e14			−0.0616649			−0.0308324	−	0.999525i	$-0.509816\pi$
$659$	−1.96747e14			−0.0616649			−0.0308324	+	0.999525i	$0.509816\pi$
$660$		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
$663$	−2.16615e15	−	3.75188e15i	−0.656695	−	1.13743i
$664$		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
$669$	−6.22789e14	+	1.07870e15i	−0.179679	+	0.311213i
$670$		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
$675$	3.50110e14	−	6.06408e14i	0.0961686	−	0.166569i
$676$		0			0
$677$	1.05566e15	+	1.82845e15i	0.285289	+	0.494135i	0.972679	−	0.232154i	$-0.0745772\pi$
$677$	1.05566e15	+	1.82845e15i	0.285289	+	0.494135i	−0.687390	+	0.726288i	$0.741244\pi$
$678$		0			0
$679$	1.49475e15	−	6.57910e15i	0.397451	−	1.74937i
$680$		0			0
$681$	−1.87366e15	−	3.24528e15i	−0.490210	−	0.849069i
$682$		0			0
$683$	−2.33152e14	+	4.03831e14i	−0.0600240	+	0.103965i	−0.894476	−	0.447116i	$-0.852451\pi$
$683$	−2.33152e14	+	4.03831e14i	−0.0600240	+	0.103965i	0.834452	+	0.551081i	$0.185784\pi$
$684$		0			0
$685$	1.04584e14			0.0264952
$686$		0			0
$687$	8.19499e14			0.204308
$688$		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
$693$	−1.67468e14	+	7.37106e14i	−0.0398014	+	0.175185i
$694$		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
$699$	−1.71529e15			−0.388788
$700$		0			0
$701$	1.49697e15			0.334013			0.167006	−	0.985956i	$-0.446590\pi$
$701$	1.49697e15			0.334013			0.167006	+	0.985956i	$0.446590\pi$
$702$		0			0
$703$	4.86626e14	−	8.42861e14i	0.106891	−	0.185141i
$704$		0			0
$705$	4.90493e13	+	8.49558e13i	0.0106070	+	0.0183719i
$706$		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
$711$	−2.72269e14	+	4.71583e14i	−0.0561972	+	0.0973364i
$712$		0			0
$713$	−5.04029e15			−1.02439
$714$		0			0
$715$	−1.21239e14			−0.0242639
$716$		0			0
$717$	8.97143e14	−	1.55390e15i	0.176810	−	0.306244i
$718$		0			0
$719$	4.14413e15	+	7.17784e15i	0.804312	+	1.39311i	0.916755	+	0.399451i	$0.130799\pi$
$719$	4.14413e15	+	7.17784e15i	0.804312	+	1.39311i	−0.112443	+	0.993658i	$0.535868\pi$
$720$		0			0
$721$	7.54684e15	−	2.33612e15i	1.44252	−	0.446531i
$722$		0			0
$723$	−8.14450e14	−	1.41067e15i	−0.153322	−	0.265561i
$724$		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
$729$	2.05891e14			0.0370370
$730$		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
$735$	7.33371e13	+	3.51376e13i	0.0126108	+	0.00604215i
$736$		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
$741$	1.97646e15			0.325003
$742$		0			0
$743$	2.20470e15			0.357199			0.178600	−	0.983922i	$-0.442843\pi$
$743$	2.20470e15			0.357199			0.178600	+	0.983922i	$0.442843\pi$
$744$		0			0
$745$	−5.64532e13	+	9.77797e13i	−0.00901216	+	0.0156095i
$746$		0			0
$747$	1.02820e15	+	1.78090e15i	0.161740	+	0.280141i
$748$		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
$753$	2.34922e15	−	4.06897e15i	0.353632	−	0.612509i
$754$		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
$759$	1.09560e15	−	1.89763e15i	0.157878	−	0.273452i
$760$		0			0
$761$	5.66595e15	+	9.81371e15i	0.804742	+	1.39385i	0.916465	+	0.400115i	$0.131030\pi$
$761$	5.66595e15	+	9.81371e15i	0.804742	+	1.39385i	−0.111722	+	0.993739i	$0.535637\pi$
$762$		0			0
$763$	8.98042e14	+	8.31599e14i	0.125722	+	0.116420i
$764$		0			0
$765$	3.58002e13	+	6.20078e13i	0.00494023	+	0.00855674i
$766$		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
$771$	4.79566e15			0.633941
$772$		0			0
$773$	−6.84876e15	+	1.18624e16i	−0.892534	+	1.54591i	−0.0557073	+	0.998447i	$0.517741\pi$
$773$	−6.84876e15	+	1.18624e16i	−0.892534	+	1.54591i	−0.836827	+	0.547467i	$0.815592\pi$
$774$		0			0
$775$	3.92621e15	+	6.80040e15i	0.504445	+	0.873725i
$776$		0			0
$777$	7.12828e14	−	3.13750e15i	0.0902961	−	0.397437i
$778$		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
$783$	8.07951e14			0.0981059
$784$		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
$789$	−3.83118e15	−	6.63580e15i	−0.446076	−	0.772627i
$790$		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
$795$	9.62681e13	−	1.66741e14i	0.0107514	−	0.0186219i
$796$		0			0
$797$	2.53264e15			0.278967			0.139484	−	0.990224i	$-0.455456\pi$
$797$	2.53264e15			0.278967			0.139484	+	0.990224i	$0.455456\pi$
$798$		0			0
$799$	1.70896e16			1.85663
$800$		0			0
$801$	1.07413e15	−	1.86044e15i	0.115100	−	0.199359i
$802$		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
$807$	−2.92486e15	−	5.06600e15i	−0.300816	−	0.521028i
$808$		0			0
$809$	−6.65041e15	+	1.15188e16i	−0.674732	+	1.16867i	0.301815	+	0.953367i	$0.402408\pi$
$809$	−6.65041e15	+	1.15188e16i	−0.674732	+	1.16867i	−0.976547	+	0.215304i	$0.930926\pi$
$810$		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
$813$	4.51933e15			0.446248
$814$		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
$819$	6.24173e15	−	1.93213e15i	0.591893	−	0.183220i
$820$		0			0
$821$	9.30192e15	+	1.61114e16i	0.870333	+	1.50746i	0.861653	+	0.507498i	$0.169430\pi$
$821$	9.30192e15	+	1.61114e16i	0.870333	+	1.50746i	0.00867994	+	0.999962i	$0.497237\pi$
$822$		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
$825$	−3.41373e15			−0.310980
$826$		0			0
$827$	1.57444e16			1.41529			0.707644	−	0.706569i	$-0.249758\pi$
$827$	1.57444e16			1.41529			0.707644	+	0.706569i	$0.249758\pi$
$828$		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
$831$	3.50068e15	+	6.06336e15i	0.306441	+	0.530772i
$832$		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
$837$	−1.15446e15	+	1.99958e15i	−0.0971376	+	0.168247i
$838$		0			0
$839$	−7.98094e15			−0.662771			−0.331385	−	0.943495i	$-0.607516\pi$
$839$	−7.98094e15			−0.662771			−0.331385	+	0.943495i	$0.607516\pi$
$840$		0			0
$841$	−9.02998e15			−0.740131
$842$		0			0
$843$	−4.39049e15	+	7.60455e15i	−0.355190	+	0.615208i
$844$		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
$849$	−1.73193e14	−	2.99979e14i	−0.0134753	−	0.0233399i
$850$		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
$855$	−3.26651e13			−0.00244495
$856$		0			0
$857$	9.30298e15	−	1.61132e16i	0.687429	−	1.19066i	−0.285238	−	0.958457i	$-0.592073\pi$
$857$	9.30298e15	−	1.61132e16i	0.687429	−	1.19066i	0.972667	−	0.232205i	$-0.0745940\pi$
$858$		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
$861$	−1.10656e16	−	1.02469e16i	−0.796997	−	0.738030i
$862$		0			0
$863$	−9.15029e15	−	1.58488e16i	−0.650692	−	1.12703i	−0.982955	−	0.183845i	$-0.941146\pi$
$863$	−9.15029e15	−	1.58488e16i	−0.650692	−	1.12703i	0.332263	−	0.943187i	$-0.392188\pi$
$864$		0			0
$865$	−1.14215e14	+	1.97825e14i	−0.00801924	+	0.0138897i
$866$		0			0
$867$	4.14532e15			0.287378
$868$		0			0
$869$	2.65474e15			0.181725
$870$		0			0
$871$	−1.32871e16	+	2.30140e16i	−0.898115	+	1.55558i
$872$		0			0
$873$	4.47959e15	+	7.75889e15i	0.298993	+	0.517870i
$874$		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
$879$	3.99669e15	−	6.92247e15i	0.256898	−	0.444961i
$880$		0			0
$881$	2.75942e16			1.75166			0.875830	−	0.482620i	$-0.160315\pi$
$881$	2.75942e16			1.75166			0.875830	+	0.482620i	$0.160315\pi$
$882$		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
$885$	3.37996e13	−	5.85427e13i	0.00209278	−	0.00362481i
$886$		0			0
$887$	−5.41925e15	−	9.38642e15i	−0.331405	−	0.574011i	0.651382	−	0.758750i	$-0.274189\pi$
$887$	−5.41925e15	−	9.38642e15i	−0.331405	−	0.574011i	−0.982788	+	0.184739i	$0.940856\pi$
$888$		0			0
$889$	4.76556e15	−	2.09755e16i	0.287842	−	1.26693i
$890$		0			0
$891$	−5.01883e14	−	8.69287e14i	−0.0299416	−	0.0518604i
$892$		0			0
$893$	−3.89826e15	+	6.75198e15i	−0.229714	+	0.397876i
$894$		0			0
$895$	5.53714e14			0.0322299
$896$		0			0
$897$	−1.89407e16			−1.08903
$898$		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
$903$	3.77622e15	+	3.49683e15i	0.209303	+	0.193817i
$904$		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
$909$	9.07593e15			0.485054
$910$		0			0
$911$	−1.16631e16			−0.615833			−0.307916	−	0.951413i	$-0.599632\pi$
$911$	−1.16631e16			−0.615833			−0.307916	+	0.951413i	$0.599632\pi$
$912$		0			0
$913$	5.01273e15	−	8.68230e15i	0.261508	−	0.452945i
$914$		0			0
$915$	6.39218e13	+	1.10716e14i	0.00329483	+	0.00570681i
$916$		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
$921$	−1.46902e15	+	2.54441e15i	−0.0730464	+	0.126520i
$922$		0			0
$923$	−6.46035e16			−3.17429
$924$		0			0
$925$	1.45306e16			0.705509
$926$		0			0
$927$	−5.24539e15	+	9.08529e15i	−0.251675	+	0.435913i
$928$		0			0
$929$	2.32445e15	+	4.02606e15i	0.110213	+	0.190895i	0.915856	−	0.401507i	$-0.131513\pi$
$929$	2.32445e15	+	4.02606e15i	0.110213	+	0.190895i	−0.805643	+	0.592401i	$0.798180\pi$
$930$		0			0
$931$	4.95815e14	+	6.44399e15i	0.0232325	+	0.301948i
$932$		0			0
$933$	−7.27720e14	−	1.26045e15i	−0.0336989	−	0.0583682i
$934$		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
$939$	−6.23189e15			−0.278586
$940$		0			0
$941$	4.60313e15	−	7.97285e15i	0.203381	−	0.352266i	−0.746235	−	0.665683i	$-0.768140\pi$
$941$	4.60313e15	−	7.97285e15i	0.203381	−	0.352266i	0.949616	+	0.313417i	$0.101474\pi$
$942$		0			0
$943$	2.18590e16	+	3.78609e16i	0.954589	+	1.65340i
$944$		0			0
$945$	−1.03158e14	+	3.19325e13i	−0.00445274	+	0.00137834i
$946$		0			0
$947$	−2.93459e15	−	5.08285e15i	−0.125205	−	0.216862i	0.796608	−	0.604496i	$-0.206626\pi$
$947$	−2.93459e15	−	5.08285e15i	−0.125205	−	0.216862i	−0.921813	+	0.387635i	$0.873292\pi$
$948$		0			0
$949$	−6.09780e14	+	1.05617e15i	−0.0257164	+	0.0445420i
$950$		0			0
$951$	4.68258e15			0.195206
$952$		0			0
$953$	1.09579e16			0.451562			0.225781	−	0.974178i	$-0.427507\pi$
$953$	1.09579e16			0.451562			0.225781	+	0.974178i	$0.427507\pi$
$954$		0			0
$955$	−2.46957e14	+	4.27742e14i	−0.0100601	+	0.0174246i
$956$		0			0
$957$	−1.96947e15	−	3.41122e15i	−0.0793111	−	0.137371i
$958$		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
$963$	−8.65487e13	+	1.49907e14i	−0.00336757	+	0.00583279i
$964$		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
$969$	−2.84527e15	+	4.92815e15i	−0.106990	+	0.185312i
$970$		0			0
$971$	−1.79678e16	−	3.11212e16i	−0.668021	−	1.15705i	−0.978457	−	0.206452i	$-0.933808\pi$
$971$	−1.79678e16	−	3.11212e16i	−0.668021	−	1.15705i	0.310435	−	0.950594i	$-0.399525\pi$
$972$		0			0
$973$	−8.44815e15	+	3.71844e16i	−0.310557	+	1.36691i
$974$		0			0
$975$	1.47542e16	+	2.55549e16i	0.536277	+	0.928858i
$976$		0			0
$977$	9.67723e15	−	1.67615e16i	0.347801	−	0.602409i	−0.638057	−	0.769989i	$-0.720262\pi$
$977$	9.67723e15	−	1.67615e16i	0.347801	−	0.602409i	0.985859	+	0.167579i	$0.0535951\pi$
$978$		0			0
$979$	−1.04732e16			−0.372199
$980$		0			0
$981$	−1.62530e15			−0.0571156
$982$		0			0
$983$	1.05812e15	−	1.83273e15i	0.0367699	−	0.0636873i	−0.847055	−	0.531506i	$-0.821626\pi$
$983$	1.05812e15	−	1.83273e15i	0.0367699	−	0.0636873i	0.883825	+	0.467818i	$0.154960\pi$
$984$		0			0
$985$	2.78023e13	+	4.81550e13i	0.000955393	+	0.00165479i
$986$		0			0
$987$	−5.71032e15	+	2.51339e16i	−0.194051	+	0.854112i
$988$		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
$993$	−1.62535e16			−0.534229
$994$		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
$999$	2.13627e15	+	3.70013e15i	0.0679275	+	0.117654i

Twists

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

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

\(f(q)\)	\(=\)	\(q+(-121.500 + 210.444i) q^{3} +(84.6221 + 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})\)	\(q+(-121.500 + 210.444i) q^{3} +(84.6221 + 146.570i) q^{5} +(-9851.70 + 43362.1i) q^{7} +(-29524.5 - 51137.9i) q^{9} +(-143939. + 249309. i) q^{11} +2.48841e6 q^{13} -41126.3 q^{15} +(-3.58228e6 + 6.20469e6i) q^{17} +(-1.63429e6 - 2.83067e6i) q^{19} +(-7.92832e6 - 7.34173e6i) q^{21} +(1.56616e7 + 2.71268e7i) q^{23} +(2.43997e7 - 4.22616e7i) q^{25} +1.43489e7 q^{27} +5.63075e7 q^{29} +(-8.04560e7 + 1.39354e8i) q^{31} +(-3.49771e7 - 6.05821e7i) q^{33} +(-7.18924e6 + 2.22543e6i) q^{35} +(1.48880e8 + 2.57868e8i) q^{37} +(-3.02342e8 + 5.23672e8i) q^{39} +1.39570e9 q^{41} -4.76296e8 q^{43} +(4.99685e6 - 8.65480e6i) q^{45} +(-1.19265e9 - 2.06573e9i) q^{47} +(-1.78321e9 - 8.54381e8i) q^{49} +(-8.70493e8 - 1.50774e9i) q^{51} +(-2.34079e9 + 4.05437e9i) q^{53} -4.87215e7 q^{55} +7.94263e8 q^{57} +(-8.21849e8 + 1.42348e9i) q^{59} +(-1.55428e9 - 2.69209e9i) q^{61} +(2.50831e9 - 7.76448e8i) q^{63} +(2.10575e8 + 3.64726e8i) q^{65} +(-5.33960e9 + 9.24846e9i) q^{67} -7.61156e9 q^{69} -2.59617e10 q^{71} +(-2.45048e8 + 4.24435e8i) q^{73} +(5.92914e9 + 1.02696e10i) q^{75} +(-9.39252e9 - 8.69760e9i) q^{77} +(-4.61089e9 - 7.98630e9i) q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} -3.48255e10 q^{83} -1.21256e9 q^{85} +(-6.84136e9 + 1.18496e10i) q^{87} +(1.81904e10 + 3.15067e10i) q^{89} +(-2.45151e10 + 1.07903e11i) q^{91} +(-1.95508e10 - 3.38630e10i) q^{93} +(2.76593e8 - 4.79074e8i) q^{95} -1.51725e11 q^{97} +1.69989e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1944 q^{3} - 2156 q^{5} + 50512 q^{7} - 472392 q^{9}+O(q^{10}) \) 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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