LMFDB - Embedded newform 63.8.p.a.17.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [63,8,Mod(17,63)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(63, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 1]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("63.17");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 63 = 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	63.p (of order $6$, 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:	$19.6802566055$
Analytic rank:	$0$
Dimension:	$36$
Relative dimension:	$18$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.12
Character	$\chi$	$=$	63.17
Dual form			63.8.p.a.26.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(6.63630 - 3.83147i) q^{2} +(-34.6397 + 59.9977i) q^{4} +(-101.591 - 175.960i) q^{5} +(-81.3062 - 903.843i) q^{7} +1511.74i q^{8} +(-1348.37 - 778.484i) q^{10} +(3728.43 + 2152.61i) q^{11} +9584.21i q^{13} +(-4002.62 - 5686.65i) q^{14} +(1358.31 + 2352.66i) q^{16} +(-15657.1 + 27118.9i) q^{17} +(-31502.4 + 18187.9i) q^{19} +14076.3 q^{20} +32990.7 q^{22} +(-85080.4 + 49121.2i) q^{23} +(18421.1 - 31906.3i) q^{25} +(36721.6 + 63603.7i) q^{26} +(57044.9 + 26430.6i) q^{28} +30169.2i q^{29} +(239293. + 138156. i) q^{31} +(-149550. - 86342.7i) q^{32} +239959. i q^{34} +(-150781. + 106129. i) q^{35} +(-107721. - 186579. i) q^{37} +(-139373. + 241401. i) q^{38} +(266006. - 153579. i) q^{40} -142227. q^{41} +885738. q^{43} +(-258303. + 149132. i) q^{44} +(-376413. + 651966. i) q^{46} +(127756. + 221279. i) q^{47} +(-810322. + 146976. i) q^{49} -282320. i q^{50} +(-575030. - 331994. i) q^{52} +(-179581. - 103681. i) q^{53} -874742. i q^{55} +(1.36638e6 - 122914. i) q^{56} +(115592. + 200212. i) q^{58} +(-375721. + 650768. i) q^{59} +(-778906. + 449702. i) q^{61} +2.11736e6 q^{62} -1.67101e6 q^{64} +(1.68644e6 - 973667. i) q^{65} +(1.15568e6 - 2.00169e6i) q^{67} +(-1.08471e6 - 1.87878e6i) q^{68} +(-593996. + 1.28201e6i) q^{70} -1.18333e6i q^{71} +(-1.22477e6 - 707118. i) q^{73} +(-1.42974e6 - 825462. i) q^{74} -2.52009e6i q^{76} +(1.64248e6 - 3.54494e6i) q^{77} +(-2.10977e6 - 3.65423e6i) q^{79} +(275983. - 478017. i) q^{80} +(-943858. + 544937. i) q^{82} -4.02734e6 q^{83} +6.36247e6 q^{85} +(5.87802e6 - 3.39368e6i) q^{86} +(-3.25419e6 + 5.63642e6i) q^{88} +(3.83319e6 + 6.63929e6i) q^{89} +(8.66262e6 - 779255. i) q^{91} -6.80617e6i q^{92} +(1.69565e6 + 978985. i) q^{94} +(6.40070e6 + 3.69544e6i) q^{95} -7.42958e6i q^{97} +(-4.81440e6 + 4.08010e6i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q + 1024 q^{4} - 2074 q^{7} + 1248 q^{10} - 59708 q^{16} - 105330 q^{19} + 544 q^{22} - 56250 q^{25} + 556220 q^{28} + 86862 q^{31} + 591034 q^{37} - 3036324 q^{40} - 837332 q^{43} + 3896752 q^{46} + 6626454 q^{49}+ \cdots + 96093708 q^{94}+O(q^{100}) $ 36 * q + 1024 * q^4 - 2074 * q^7 + 1248 * q^10 - 59708 * q^16 - 105330 * q^19 + 544 * q^22 - 56250 * q^25 + 556220 * q^28 + 86862 * q^31 + 591034 * q^37 - 3036324 * q^40 - 837332 * q^43 + 3896752 * q^46 + 6626454 * q^49 - 11085900 * q^52 - 134084 * q^58 + 4324644 * q^61 - 11555464 * q^64 + 8750706 * q^67 + 8823972 * q^70 - 744402 * q^73 + 1136238 * q^79 - 29281932 * q^82 + 18719472 * q^85 - 4365916 * q^88 + 21779742 * q^91 + 96093708 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/63\mathbb{Z}\right)^\times$.

$n$	$10$	$29$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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 $

$2$	6.63630	−	3.83147i	0.586572	−	0.338657i	−0.177169	−	0.984180i	$-0.556694\pi$
$2$	6.63630	−	3.83147i	0.586572	−	0.338657i	0.763741	+	0.645523i	$0.223361\pi$
$3$		0			0
$3$		0			0
$4$	−34.6397	+	59.9977i	−0.270622	+	0.468732i
$5$	−101.591	−	175.960i	−0.363462	−	0.629535i	0.625066	−	0.780572i	$-0.285072\pi$
$5$	−101.591	−	175.960i	−0.363462	−	0.629535i	−0.988528	+	0.151037i	$0.951739\pi$
$6$		0			0
$7$	−81.3062	−	903.843i	−0.0895943	−	0.995978i
$7$	−81.3062	−	903.843i	−0.0895943	−	0.995978i
$8$			1511.74i			1.04391i
$9$		0			0
$10$	−1348.37	−	778.484i	−0.426393	−	0.246178i
$11$	3728.43	+	2152.61i	0.844602	+	0.487631i	0.858826	−	0.512268i	$-0.171194\pi$
$11$	3728.43	+	2152.61i	0.844602	+	0.487631i	−0.0142238	+	0.999899i	$0.504528\pi$
$12$		0			0
$13$			9584.21i			1.20991i	0.796258	+	0.604957i	$0.206810\pi$
$13$			9584.21i			1.20991i	−0.796258	+	0.604957i	$0.793190\pi$
$14$	−4002.62	−	5686.65i	−0.389849	−	0.553871i
$15$		0			0
$16$	1358.31	+	2352.66i	0.0829046	+	0.143595i
$17$	−15657.1	+	27118.9i	−0.772931	+	1.33876i	0.163019	+	0.986623i	$0.447877\pi$
$17$	−15657.1	+	27118.9i	−0.772931	+	1.33876i	−0.935950	+	0.352133i	$0.885457\pi$
$18$		0			0
$19$	−31502.4	+	18187.9i	−1.05367	+	0.608338i	−0.923675	−	0.383177i	$-0.874830\pi$
$19$	−31502.4	+	18187.9i	−1.05367	+	0.608338i	−0.129997	+	0.991514i	$0.541497\pi$
$20$	14076.3			0.393444
$21$		0			0
$22$	32990.7			0.660560
$23$	−85080.4	+	49121.2i	−1.45808	+	0.841825i	−0.998917	−	0.0465257i	$-0.985185\pi$
$23$	−85080.4	+	49121.2i	−1.45808	+	0.841825i	−0.459166	+	0.888350i	$0.651852\pi$
$24$		0			0
$25$	18421.1	−	31906.3i	0.235791	−	0.408401i
$26$	36721.6	+	63603.7i	0.409746	+	0.709701i
$27$		0			0
$28$	57044.9	+	26430.6i	0.491093	+	0.227538i
$29$			30169.2i			0.229705i	0.993383	+	0.114852i	$0.0366395\pi$
$29$			30169.2i			0.229705i	−0.993383	+	0.114852i	$0.963360\pi$
$30$		0			0
$31$	239293.	+	138156.i	1.44266	+	0.832922i	0.998027	−	0.0627900i	$-0.0199998\pi$
$31$	239293.	+	138156.i	1.44266	+	0.832922i	0.444636	+	0.895712i	$0.353333\pi$
$32$	−149550.	−	86342.7i	−0.806792	−	0.465801i
$33$		0			0
$34$			239959.i			1.04703i
$35$	−150781.	+	106129.i	−0.594439	+	0.418403i
$36$		0			0
$37$	−107721.	−	186579.i	−0.349619	−	0.605559i	0.636562	−	0.771225i	$-0.280356\pi$
$37$	−107721.	−	186579.i	−0.349619	−	0.605559i	−0.986182	+	0.165667i	$0.947022\pi$
$38$	−139373.	+	241401.i	−0.412036	+	0.713668i
$39$		0			0
$40$	266006.	−	153579.i	0.657176	−	0.379421i
$41$	−142227.			−0.322283			−0.161141	−	0.986931i	$-0.551518\pi$
$41$	−142227.			−0.322283			−0.161141	+	0.986931i	$0.551518\pi$
$42$		0			0
$43$	885738.			1.69889			0.849445	−	0.527676i	$-0.176937\pi$
$43$	885738.			1.69889			0.849445	+	0.527676i	$0.176937\pi$
$44$	−258303.	+	149132.i	−0.457136	+	0.263928i
$45$		0			0
$46$	−376413.	+	651966.i	−0.570180	+	0.987581i
$47$	127756.	+	221279.i	0.179489	+	0.310884i	0.941706	−	0.336438i	$-0.109222\pi$
$47$	127756.	+	221279.i	0.179489	+	0.310884i	−0.762217	+	0.647322i	$0.775889\pi$
$48$		0			0
$49$	−810322.	+	146976.i	−0.983946	+	0.178468i
$50$		−	282320.i		−	0.319409i
$51$		0			0
$52$	−575030.	−	331994.i	−0.567125	−	0.327430i
$53$	−179581.	−	103681.i	−0.165690	−	0.0956610i	0.414862	−	0.909884i	$-0.363830\pi$
$53$	−179581.	−	103681.i	−0.165690	−	0.0956610i	−0.580552	+	0.814223i	$0.697163\pi$
$54$		0			0
$55$		−	874742.i		−	0.708942i
$56$	1.36638e6	−	122914.i	1.03971	−	0.0935282i
$57$		0			0
$58$	115592.	+	200212.i	0.0777913	+	0.134738i
$59$	−375721.	+	650768.i	−0.238168	+	0.412519i	−0.960189	−	0.279353i	$-0.909880\pi$
$59$	−375721.	+	650768.i	−0.238168	+	0.412519i	0.722021	+	0.691871i	$0.243214\pi$
$60$		0			0
$61$	−778906.	+	449702.i	−0.439370	+	0.253671i	−0.703331	−	0.710863i	$-0.748305\pi$
$61$	−778906.	+	449702.i	−0.439370	+	0.253671i	0.263960	+	0.964534i	$0.414971\pi$
$62$	2.11736e6			1.12830
$63$		0			0
$64$	−1.67101e6			−0.796797
$65$	1.68644e6	−	973667.i	0.761683	−	0.439758i
$66$		0			0
$67$	1.15568e6	−	2.00169e6i	0.469434	−	0.813083i	−0.529956	−	0.848025i	$-0.677791\pi$
$67$	1.15568e6	−	2.00169e6i	0.469434	−	0.813083i	0.999389	+	0.0349423i	$0.0111247\pi$
$68$	−1.08471e6	−	1.87878e6i	−0.418345	−	0.724594i
$69$		0			0
$70$	−593996.	+	1.28201e6i	−0.206986	+	0.446735i
$71$		−	1.18333e6i		−	0.392375i	−0.980566	−	0.196188i	$-0.937144\pi$
$71$		−	1.18333e6i		−	0.392375i	0.980566	−	0.196188i	$-0.0628562\pi$
$72$		0			0
$73$	−1.22477e6	−	707118.i	−0.368488	−	0.212746i	0.304310	−	0.952573i	$-0.401574\pi$
$73$	−1.22477e6	−	707118.i	−0.368488	−	0.212746i	−0.672797	+	0.739827i	$0.734907\pi$
$74$	−1.42974e6	−	825462.i	−0.410154	−	0.236802i
$75$		0			0
$76$		−	2.52009e6i		−	0.658519i
$77$	1.64248e6	−	3.54494e6i	0.409999	−	0.884894i
$78$		0			0
$79$	−2.10977e6	−	3.65423e6i	−0.481438	−	0.833875i	0.518335	−	0.855178i	$-0.326552\pi$
$79$	−2.10977e6	−	3.65423e6i	−0.481438	−	0.833875i	−0.999773	+	0.0213027i	$0.993219\pi$
$80$	275983.	−	478017.i	0.0602654	−	0.104383i
$81$		0			0
$82$	−943858.	+	544937.i	−0.189042	+	0.109143i
$83$	−4.02734e6			−0.773117			−0.386558	−	0.922265i	$-0.626336\pi$
$83$	−4.02734e6			−0.773117			−0.386558	+	0.922265i	$0.626336\pi$
$84$		0			0
$85$	6.36247e6			1.12372
$86$	5.87802e6	−	3.39368e6i	0.996521	−	0.575342i
$87$		0			0
$88$	−3.25419e6	+	5.63642e6i	−0.509042	+	0.881687i
$89$	3.83319e6	+	6.63929e6i	0.576363	+	0.998289i	0.995892	+	0.0905476i	$0.0288617\pi$
$89$	3.83319e6	+	6.63929e6i	0.576363	+	0.998289i	−0.419530	+	0.907742i	$0.637805\pi$
$90$		0			0
$91$	8.66262e6	−	779255.i	1.20505	−	0.108401i
$92$		−	6.80617e6i		−	0.911267i
$93$		0			0
$94$	1.69565e6	+	978985.i	0.210566	+	0.121571i
$95$	6.40070e6	+	3.69544e6i	0.765940	+	0.442216i
$96$		0			0
$97$		−	7.42958e6i		−	0.826538i	−0.910609	−	0.413269i	$-0.864387\pi$
$97$		−	7.42958e6i		−	0.826538i	0.910609	−	0.413269i	$-0.135613\pi$
$98$	−4.81440e6	+	4.08010e6i	−0.516715	+	0.437905i
$99$		0			0
$100$	1.27620e6	+	2.21045e6i	0.127620	+	0.221045i
$101$	−2.41174e6	+	4.17726e6i	−0.232920	+	0.403428i	−0.958666	−	0.284534i	$-0.908161\pi$
$101$	−2.41174e6	+	4.17726e6i	−0.232920	+	0.403428i	0.725746	+	0.687962i	$0.241495\pi$
$102$		0			0
$103$	−1.28033e7	+	7.39201e6i	−1.15450	+	0.666549i	−0.949979	−	0.312314i	$-0.898896\pi$
$103$	−1.28033e7	+	7.39201e6i	−1.15450	+	0.666549i	−0.204518	+	0.978863i	$0.565563\pi$
$104$	−1.44888e7			−1.26304
$105$		0			0
$106$	−1.58901e6			−0.129585
$107$	−1.66277e7	+	9.60000e6i	−1.31217	+	0.757579i	−0.982454	−	0.186503i	$-0.940285\pi$
$107$	−1.66277e7	+	9.60000e6i	−1.31217	+	0.757579i	−0.329711	+	0.944082i	$0.606951\pi$
$108$		0			0
$109$	−3.91522e6	+	6.78136e6i	−0.289577	+	0.501562i	−0.973709	−	0.227797i	$-0.926848\pi$
$109$	−3.91522e6	+	6.78136e6i	−0.289577	+	0.501562i	0.684132	+	0.729358i	$0.260181\pi$
$110$	−3.35155e6	−	5.80505e6i	−0.240088	−	0.415845i
$111$		0			0
$112$	2.01600e6	−	1.41898e6i	0.135590	−	0.0954365i
$113$			2.28548e7i			1.49006i	0.667032	+	0.745029i	$0.267564\pi$
$113$			2.28548e7i			1.49006i	−0.667032	+	0.745029i	$0.732436\pi$
$114$		0			0
$115$	1.72868e7	+	9.98052e6i	1.05992	+	0.611943i
$116$	−1.81008e6	−	1.04505e6i	−0.107670	−	0.0621633i
$117$		0			0
$118$			5.75825e6i			0.322629i
$119$	2.57843e7	+	1.19466e7i	1.40262	+	0.649878i
$120$		0			0
$121$	−476103.	−	824634.i	−0.0244316	−	0.0423168i
$122$	−3.44604e6	+	5.96871e6i	−0.171815	+	0.297592i
$123$		0			0
$124$	−1.65781e7	+	9.57136e6i	−0.780834	+	0.450814i
$125$	−2.33592e7			−1.06973
$126$		0			0
$127$	3.26982e7			1.41648			0.708240	−	0.705971i	$-0.249489\pi$
$127$	3.26982e7			1.41648			0.708240	+	0.705971i	$0.249489\pi$
$128$	8.05310e6	−	4.64946e6i	0.339413	−	0.195960i
$129$		0			0
$130$	7.46115e6	−	1.29231e7i	0.297855	−	0.515899i
$131$	−2.55808e6	−	4.43072e6i	−0.0994178	−	0.172197i	0.812026	−	0.583621i	$-0.198365\pi$
$131$	−2.55808e6	−	4.43072e6i	−0.0994178	−	0.172197i	−0.911444	+	0.411425i	$0.865031\pi$
$132$		0			0
$133$	1.90003e7	+	2.69944e7i	0.700294	+	0.994931i
$134$		−	1.77118e7i		−	0.635909i
$135$		0			0
$136$	−4.09968e7	−	2.36695e7i	−1.39754	−	0.806868i
$137$	1.02151e7	+	5.89768e6i	0.339406	+	0.195956i	0.660009	−	0.751257i	$-0.270552\pi$
$137$	1.02151e7	+	5.89768e6i	0.339406	+	0.195956i	−0.320603	+	0.947214i	$0.603886\pi$
$138$		0			0
$139$		−	1.30571e7i		−	0.412379i	−0.978512	−	0.206189i	$-0.933894\pi$
$139$		−	1.30571e7i		−	0.412379i	0.978512	−	0.206189i	$-0.0661063\pi$
$140$	−1.14449e6	−	1.27227e7i	−0.0352503	−	0.391862i
$141$		0			0
$142$	−4.53389e6	−	7.85293e6i	−0.132881	−	0.230156i
$143$	−2.06311e7	+	3.57341e7i	−0.589992	+	1.02190i
$144$		0			0
$145$	5.30858e6	−	3.06491e6i	0.144607	−	0.0834890i
$146$	−1.08372e7			−0.288192
$147$		0			0
$148$	1.49257e7			0.378459
$149$	6.80993e7	−	3.93171e7i	1.68652	−	0.973711i	0.729364	−	0.684125i	$-0.239816\pi$
$149$	6.80993e7	−	3.93171e7i	1.68652	−	0.973711i	0.957152	−	0.289585i	$-0.0935174\pi$
$150$		0			0
$151$	7.58072e6	−	1.31302e7i	0.179181	−	0.310350i	−0.762419	−	0.647083i	$-0.775989\pi$
$151$	7.58072e6	−	1.31302e7i	0.179181	−	0.310350i	0.941600	+	0.336733i	$0.109322\pi$
$152$	−2.74954e7	−	4.76234e7i	−0.635049	−	1.09994i
$153$		0			0
$154$	−2.68235e6	−	2.98184e7i	−0.0591824	−	0.657903i
$155$		−	5.61415e7i		−	1.21094i
$156$		0			0
$157$	6.29277e6	+	3.63313e6i	0.129776	+	0.0749259i	0.563482	−	0.826128i	$-0.309461\pi$
$157$	6.29277e6	+	3.63313e6i	0.129776	+	0.0749259i	−0.433707	+	0.901054i	$0.642795\pi$
$158$	−2.80021e7	−	1.61670e7i	−0.564796	−	0.326085i
$159$		0			0
$160$			3.50865e7i			0.677205i
$161$	5.13154e7	+	7.29055e7i	0.969075	+	1.37680i
$162$		0			0
$163$	−3.59595e7	−	6.22836e7i	−0.650364	−	1.12646i	−0.983035	−	0.183420i	$-0.941283\pi$
$163$	−3.59595e7	−	6.22836e7i	−0.650364	−	1.12646i	0.332671	−	0.943043i	$-0.392050\pi$
$164$	4.92668e6	−	8.53326e6i	0.0872170	−	0.151064i
$165$		0			0
$166$	−2.67266e7	+	1.54306e7i	−0.453488	+	0.261822i
$167$	4.92749e7			0.818688			0.409344	−	0.912380i	$-0.365758\pi$
$167$	4.92749e7			0.818688			0.409344	+	0.912380i	$0.365758\pi$
$168$		0			0
$169$	−2.91086e7			−0.463892
$170$	4.22233e7	−	2.43776e7i	0.659145	−	0.380557i
$171$		0			0
$172$	−3.06817e7	+	5.31422e7i	−0.459758	+	0.796324i
$173$	−1.47003e7	−	2.54616e7i	−0.215856	−	0.373874i	0.737681	−	0.675149i	$-0.235921\pi$
$173$	−1.47003e7	−	2.54616e7i	−0.215856	−	0.373874i	−0.953537	+	0.301276i	$0.902588\pi$
$174$		0			0
$175$	−3.03361e7	−	1.40556e7i	−0.427884	−	0.198252i
$176$			1.16957e7i			0.161707i
$177$		0			0
$178$	5.08765e7	+	2.93735e7i	0.676156	+	0.390379i
$179$	−1.06254e7	−	6.13456e6i	−0.138471	−	0.0799462i	0.429164	−	0.903227i	$-0.358808\pi$
$179$	−1.06254e7	−	6.13456e6i	−0.138471	−	0.0799462i	−0.567635	+	0.823280i	$0.692142\pi$
$180$		0			0
$181$		−	2.63116e7i		−	0.329816i	−0.986309	−	0.164908i	$-0.947267\pi$
$181$		−	2.63116e7i		−	0.329816i	0.986309	−	0.164908i	$-0.0527328\pi$
$182$	5.45021e7	−	3.83620e7i	0.670136	−	0.471684i
$183$		0			0
$184$	−7.42585e7	−	1.28619e8i	−0.878787	−	1.52210i
$185$	−2.18870e7	+	3.79094e7i	−0.254147	+	0.440195i
$186$		0			0
$187$	−1.16753e8	+	6.74074e7i	−1.30564	+	0.753810i
$188$	−1.77017e7			−0.194295
$189$		0			0
$190$	5.66359e7			0.599038
$191$	2.75328e7	−	1.58961e7i	0.285913	−	0.165072i	−0.350184	−	0.936681i	$-0.613881\pi$
$191$	2.75328e7	−	1.58961e7i	0.285913	−	0.165072i	0.636097	+	0.771609i	$0.280548\pi$
$192$		0			0
$193$	1.59404e7	−	2.76095e7i	0.159606	−	0.276445i	−0.775121	−	0.631813i	$-0.782311\pi$
$193$	1.59404e7	−	2.76095e7i	0.159606	−	0.276445i	0.934727	+	0.355368i	$0.115644\pi$
$194$	−2.84662e7	−	4.93049e7i	−0.279913	−	0.484824i
$195$		0			0
$196$	1.92511e7	−	5.37086e7i	0.182624	−	0.509504i
$197$			1.73350e8i			1.61544i	0.589564	+	0.807722i	$0.299300\pi$
$197$			1.73350e8i			1.61544i	−0.589564	+	0.807722i	$0.700700\pi$
$198$		0			0
$199$	1.52428e8	+	8.80043e7i	1.37113	+	0.791623i	0.991070	−	0.133339i	$-0.0425700\pi$
$199$	1.52428e8	+	8.80043e7i	1.37113	+	0.791623i	0.380060	+	0.924962i	$0.375903\pi$
$200$	4.82341e7	+	2.78480e7i	0.426333	+	0.246144i
$201$		0			0
$202$			3.69621e7i			0.315520i
$203$	2.72682e7	−	2.45294e6i	0.228781	−	0.0205802i
$204$		0			0
$205$	1.44489e7	+	2.50262e7i	0.117138	+	0.202888i
$206$	−5.66445e7	+	9.81112e7i	−0.451463	+	0.781958i
$207$		0			0
$208$	−2.25484e7	+	1.30183e7i	−0.173738	+	0.100307i
$209$	−1.56606e8			−1.18658
$210$		0			0
$211$	2.87790e7			0.210905			0.105453	−	0.994424i	$-0.466371\pi$
$211$	2.87790e7			0.210905			0.105453	+	0.994424i	$0.466371\pi$
$212$	1.24413e7	−	7.18297e6i	0.0896787	−	0.0517760i
$213$		0			0
$214$	−7.35642e7	+	1.27417e8i	−0.513119	+	0.888749i
$215$	−8.99827e7	−	1.55855e8i	−0.617482	−	1.06951i
$216$		0			0
$217$	1.05415e8	−	2.27517e8i	0.700318	−	1.51149i
$218$			6.00042e7i			0.392269i
$219$		0			0
$220$	5.24825e7	+	3.03008e7i	0.332304	+	0.191856i
$221$	−2.59913e8	−	1.50061e8i	−1.61978	−	0.935180i
$222$		0			0
$223$			2.62590e8i			1.58566i	0.609442	+	0.792831i	$0.291394\pi$
$223$			2.62590e8i			1.58566i	−0.609442	+	0.792831i	$0.708606\pi$
$224$	−6.58809e7	+	1.42190e8i	−0.391644	+	0.845280i
$225$		0			0
$226$	8.75676e7	+	1.51671e8i	0.504619	+	0.874026i
$227$	−1.05849e8	+	1.83335e8i	−0.600612	+	1.04029i	0.392116	+	0.919916i	$0.371743\pi$
$227$	−1.05849e8	+	1.83335e8i	−0.600612	+	1.04029i	−0.992728	+	0.120375i	$0.961590\pi$
$228$		0			0
$229$	1.48331e8	−	8.56391e7i	0.816223	−	0.471246i	−0.0328895	−	0.999459i	$-0.510471\pi$
$229$	1.48331e8	−	8.56391e7i	0.816223	−	0.471246i	0.849112	+	0.528213i	$0.177138\pi$
$230$	1.52960e8			0.828956
$231$		0			0
$232$	−4.56079e7			−0.239791
$233$	3.17369e8	−	1.83233e8i	1.64369	−	0.948983i	0.664180	−	0.747572i	$-0.268781\pi$
$233$	3.17369e8	−	1.83233e8i	1.64369	−	0.948983i	0.979507	−	0.201411i	$-0.0645526\pi$
$234$		0			0
$235$	2.59576e7	−	4.49599e7i	0.130475	−	0.225989i
$236$	−2.60297e7	−	4.50847e7i	−0.128907	−	0.223274i
$237$		0			0
$238$	2.16885e8	−	1.95102e7i	1.04282	−	0.0938083i
$239$			1.87637e8i			0.889047i	0.895767	+	0.444524i	$0.146627\pi$
$239$			1.87637e8i			0.889047i	−0.895767	+	0.444524i	$0.853373\pi$
$240$		0			0
$241$	1.89388e8	+	1.09343e8i	0.871550	+	0.503190i	0.867863	−	0.496803i	$-0.165493\pi$
$241$	1.89388e8	+	1.09343e8i	0.871550	+	0.503190i	0.00368718	+	0.999993i	$0.498826\pi$
$242$	−6.31912e6	−	3.64835e6i	−0.0286618	−	0.0165479i
$243$		0			0
$244$		−	6.23101e7i		−	0.274596i
$245$	1.08183e8	+	1.27653e8i	0.469979	+	0.554562i
$246$		0			0
$247$	−1.74317e8	−	3.01925e8i	−0.736037	−	1.27485i
$248$	−2.08856e8	+	3.61749e8i	−0.869493	+	1.50601i
$249$		0			0
$250$	−1.55019e8	+	8.95002e7i	−0.627472	+	0.362271i
$251$	−3.26802e8			−1.30445			−0.652224	−	0.758026i	$-0.726164\pi$
$251$	−3.26802e8			−1.30445			−0.652224	+	0.758026i	$0.726164\pi$
$252$		0			0
$253$	−4.22956e8			−1.64200
$254$	2.16995e8	−	1.25282e8i	0.830868	−	0.479702i
$255$		0			0
$256$	1.42573e8	−	2.46943e8i	0.531125	−	0.919936i
$257$	−2.29354e8	−	3.97253e8i	−0.842830	−	1.45983i	−0.887492	−	0.460823i	$-0.847554\pi$
$257$	−2.29354e8	−	3.97253e8i	−0.842830	−	1.45983i	0.0446617	−	0.999002i	$-0.485779\pi$
$258$		0			0
$259$	−1.59880e8	+	1.12533e8i	−0.571799	+	0.402468i
$260$			1.34910e8i			0.476033i
$261$		0			0
$262$	−3.39523e7	−	1.96024e7i	−0.116631	−	0.0673371i
$263$	1.79191e8	+	1.03456e8i	0.607393	+	0.350679i	0.771945	−	0.635690i	$-0.219284\pi$
$263$	1.79191e8	+	1.03456e8i	0.607393	+	0.350679i	−0.164551	+	0.986369i	$0.552618\pi$
$264$		0			0
$265$			4.21323e7i			0.139077i
$266$	2.29520e8	+	1.06344e8i	0.747714	+	0.346439i
$267$		0			0
$268$	8.00645e7	+	1.38676e8i	0.254079	+	0.440077i
$269$	2.86680e8	−	4.96544e8i	0.897975	−	1.55534i	0.0678961	−	0.997692i	$-0.478371\pi$
$269$	2.86680e8	−	4.96544e8i	0.897975	−	1.55534i	0.830079	−	0.557646i	$-0.188295\pi$
$270$		0			0
$271$	−1.61433e8	+	9.32034e7i	−0.492720	+	0.284472i	−0.725702	−	0.688009i	$-0.758485\pi$
$271$	−1.61433e8	+	9.32034e7i	−0.492720	+	0.284472i	0.232982	+	0.972481i	$0.425152\pi$
$272$	−8.50688e7			−0.256318
$273$		0			0
$274$	9.03871e7			0.265448
$275$	1.37364e8	−	7.93072e7i	0.398298	−	0.229958i
$276$		0			0
$277$	−2.36522e8	+	4.09669e8i	−0.668641	+	1.15812i	0.309643	+	0.950853i	$0.399790\pi$
$277$	−2.36522e8	+	4.09669e8i	−0.668641	+	1.15812i	−0.978284	+	0.207267i	$0.933543\pi$
$278$	−5.00280e7	−	8.66511e7i	−0.139655	−	0.241890i
$279$		0			0
$280$	−1.60439e8	−	2.27941e8i	−0.436774	−	0.620539i
$281$			2.65063e7i			0.0712650i	0.999365	+	0.0356325i	$0.0113446\pi$
$281$			2.65063e7i			0.0712650i	−0.999365	+	0.0356325i	$0.988655\pi$
$282$		0			0
$283$	220075.	+	127060.i	0.000577190	+	0.000333241i	0.500289	−	0.865859i	$-0.333227\pi$
$283$	220075.	+	127060.i	0.000577190	+	0.000333241i	−0.499711	+	0.866192i	$0.666561\pi$
$284$	7.09970e7	+	4.09902e7i	0.183919	+	0.106186i
$285$		0			0
$286$			3.16190e8i			0.799220i
$287$	1.15639e7	+	1.28550e8i	0.0288747	+	0.320987i
$288$		0			0
$289$	−2.85121e8	−	4.93845e8i	−0.694844	−	1.20351i
$290$	2.34862e7	−	4.06793e7i	0.0565483	−	0.0979446i
$291$		0			0
$292$	8.48509e7	−	4.89887e7i	0.199442	−	0.115148i
$293$	5.08362e8			1.18069			0.590346	−	0.807151i	$-0.298991\pi$
$293$	5.08362e8			1.18069			0.590346	+	0.807151i	$0.298991\pi$
$294$		0			0
$295$	1.52679e8			0.346260
$296$	2.82059e8	−	1.62847e8i	0.632147	−	0.364970i
$297$		0			0
$298$	3.01285e8	−	5.21841e8i	0.659509	−	1.14230i
$299$	−4.70788e8	−	8.15429e8i	−1.01854	−	1.76416i
$300$		0			0
$301$	−7.20159e7	−	8.00568e8i	−0.152211	−	1.69206i
$302$		−	1.16181e8i		−	0.242723i
$303$		0			0
$304$	−8.55799e7	−	4.94096e7i	−0.174709	−	0.100868i
$305$	1.58259e8	+	9.13711e7i	0.319389	+	0.184399i
$306$		0			0
$307$			2.89078e8i			0.570204i	0.958497	+	0.285102i	$0.0920274\pi$
$307$			2.89078e8i			0.570204i	−0.958497	+	0.285102i	$0.907973\pi$
$308$	1.55793e8	+	2.21340e8i	0.303823	+	0.431652i
$309$		0			0
$310$	−2.15105e8	−	3.72572e8i	−0.410094	−	0.710304i
$311$	−1.43038e8	+	2.47749e8i	−0.269644	+	0.467036i	−0.968770	−	0.247962i	$-0.920239\pi$
$311$	−1.43038e8	+	2.47749e8i	−0.269644	+	0.467036i	0.699126	+	0.714998i	$0.253573\pi$
$312$		0			0
$313$	−5.80645e8	+	3.35235e8i	−1.07030	+	0.617938i	−0.928263	−	0.371924i	$-0.878698\pi$
$313$	−5.80645e8	+	3.35235e8i	−1.07030	+	0.617938i	−0.142036	+	0.989861i	$0.545365\pi$
$314$	5.56809e7			0.101497
$315$		0			0
$316$	2.92327e8			0.521152
$317$	−1.77425e8	+	1.02437e8i	−0.312830	+	0.180612i	−0.648192	−	0.761477i	$-0.724475\pi$
$317$	−1.77425e8	+	1.02437e8i	−0.312830	+	0.180612i	0.335362	+	0.942089i	$0.391141\pi$
$318$		0			0
$319$	−6.49425e7	+	1.12484e8i	−0.112011	+	0.194009i
$320$	1.69759e8	+	2.94031e8i	0.289606	+	0.501612i
$321$		0			0
$322$	6.19880e8	+	2.87209e8i	1.03469	+	0.479406i
$323$		−	1.13908e9i		−	1.88081i
$324$		0			0
$325$	3.05797e8	+	1.76552e8i	0.494130	+	0.285286i
$326$	−4.77276e8	−	2.75555e8i	−0.762970	−	0.440501i
$327$		0			0
$328$		−	2.15010e8i		−	0.336434i
$329$	1.89615e8	−	1.33463e8i	0.293553	−	0.206621i
$330$		0			0
$331$	1.46132e8	+	2.53108e8i	0.221487	+	0.383626i	0.955260	−	0.295769i	$-0.0955758\pi$
$331$	1.46132e8	+	2.53108e8i	0.221487	+	0.383626i	−0.733773	+	0.679395i	$0.762242\pi$
$332$	1.39506e8	−	2.41631e8i	0.209223	−	0.362384i
$333$		0			0
$334$	3.27003e8	−	1.88795e8i	0.480219	−	0.277255i
$335$	−4.69624e8			−0.682486
$336$		0			0
$337$	4.81745e8			0.685665			0.342833	−	0.939396i	$-0.388614\pi$
$337$	4.81745e8			0.685665			0.342833	+	0.939396i	$0.388614\pi$
$338$	−1.93173e8	+	1.11529e8i	−0.272106	+	0.157101i
$339$		0			0
$340$	−2.20394e8	+	3.81734e8i	−0.304105	+	0.526725i
$341$	5.94793e8	+	1.03021e9i	0.812317	+	1.40697i
$342$		0			0
$343$	1.98727e8	+	7.20454e8i	0.265906	+	0.963999i
$344$			1.33900e9i			1.77349i
$345$		0			0
$346$	−1.95111e8	−	1.12647e8i	−0.253230	−	0.146202i
$347$	−2.69954e8	−	1.55858e8i	−0.346846	−	0.200252i	0.316449	−	0.948609i	$-0.397509\pi$
$347$	−2.69954e8	−	1.55858e8i	−0.346846	−	0.200252i	−0.663295	+	0.748358i	$0.730843\pi$
$348$		0			0
$349$			3.79572e8i			0.477975i	0.971023	+	0.238988i	$0.0768155\pi$
$349$			3.79572e8i			0.477975i	−0.971023	+	0.238988i	$0.923184\pi$
$350$	−2.55173e8	+	2.29544e7i	−0.318124	+	0.0286172i
$351$		0			0
$352$	−3.71725e8	−	6.43846e8i	−0.454279	−	0.786834i
$353$	−3.53235e8	+	6.11821e8i	−0.427417	+	0.740308i	−0.996643	−	0.0818730i	$-0.973910\pi$
$353$	−3.53235e8	+	6.11821e8i	−0.427417	+	0.740308i	0.569225	+	0.822181i	$0.307243\pi$
$354$		0			0
$355$	−2.08219e8	+	1.20215e8i	−0.247014	+	0.142614i
$356$	−5.31122e8			−0.623907
$357$		0			0
$358$	−9.40176e7			−0.108298
$359$	−3.27619e7	+	1.89151e7i	−0.0373713	+	0.0215763i	−0.518569	−	0.855036i	$-0.673535\pi$
$359$	−3.27619e7	+	1.89151e7i	−0.0373713	+	0.0215763i	0.481198	+	0.876612i	$0.340202\pi$
$360$		0			0
$361$	2.14663e8	−	3.71808e8i	0.240150	−	0.415952i
$362$	−1.00812e8	−	1.74612e8i	−0.111695	−	0.193461i
$363$		0			0
$364$	−2.53317e8	+	5.46730e8i	−0.275302	+	0.594180i
$365$			2.87347e8i			0.309301i
$366$		0			0
$367$	8.63337e7	+	4.98448e7i	0.0911694	+	0.0526367i	0.544892	−	0.838507i	$-0.316571\pi$
$367$	8.63337e7	+	4.98448e7i	0.0911694	+	0.0526367i	−0.453722	+	0.891143i	$0.649904\pi$
$368$	−2.31131e8	−	1.33444e8i	−0.241764	−	0.139582i
$369$		0			0
$370$			3.35437e8i			0.344275i
$371$	−7.91106e7	+	1.70743e8i	−0.0804314	+	0.173594i
$372$		0			0
$373$	6.52577e8	+	1.13030e9i	0.651105	+	1.12775i	0.982855	+	0.184379i	$0.0590275\pi$
$373$	6.52577e8	+	1.13030e9i	0.651105	+	1.12775i	−0.331750	+	0.943367i	$0.607639\pi$
$374$	−5.16539e8	+	8.94672e8i	−0.510567	+	0.884328i
$375$		0			0
$376$	−3.34517e8	+	1.93133e8i	−0.324534	+	0.187370i
$377$	−2.89148e8			−0.277923
$378$		0			0
$379$	−1.49723e8			−0.141270			−0.0706352	−	0.997502i	$-0.522503\pi$
$379$	−1.49723e8			−0.141270			−0.0706352	+	0.997502i	$0.522503\pi$
$380$	−4.43436e8	+	2.56018e8i	−0.414561	+	0.239347i
$381$		0			0
$382$	1.21811e8	−	2.10982e8i	0.111806	−	0.193653i
$383$	2.24323e8	+	3.88539e8i	0.204023	+	0.353378i	0.949821	−	0.312794i	$-0.101265\pi$
$383$	2.24323e8	+	3.88539e8i	0.204023	+	0.353378i	−0.745798	+	0.666172i	$0.767932\pi$
$384$		0			0
$385$	−7.90630e8	+	7.11219e7i	−0.706091	+	0.0635171i
$386$		−	2.44300e8i		−	0.216206i
$387$		0			0
$388$	4.45757e8	+	2.57358e8i	0.387425	+	0.223680i
$389$	1.31524e9	+	7.59357e8i	1.13288	+	0.654067i	0.944657	−	0.328060i	$-0.106395\pi$
$389$	1.31524e9	+	7.59357e8i	1.13288	+	0.654067i	0.188220	+	0.982127i	$0.439728\pi$
$390$		0			0
$391$		−	3.07639e9i		−	2.60269i
$392$	−2.22190e8	−	1.22500e9i	−0.186304	−	1.02715i
$393$		0			0
$394$	6.64185e8	+	1.15040e9i	0.547082	+	0.947574i
$395$	−4.28666e8	+	7.42472e8i	−0.349969	+	0.606164i
$396$		0			0
$397$	1.36885e9	−	7.90303e8i	1.09796	−	0.633909i	0.162278	−	0.986745i	$-0.448116\pi$
$397$	1.36885e9	−	7.90303e8i	1.09796	−	0.633909i	0.935685	+	0.352836i	$0.114783\pi$
$398$	1.34874e9			1.07236
$399$		0			0
$400$	1.00086e8			0.0781925
$401$	−2.88198e8	+	1.66391e8i	−0.223196	+	0.128862i	−0.607429	−	0.794374i	$-0.707799\pi$
$401$	−2.88198e8	+	1.66391e8i	−0.223196	+	0.128862i	0.384233	+	0.923236i	$0.374466\pi$
$402$		0			0
$403$	−1.32412e9	+	2.29344e9i	−1.00776	+	1.74550i
$404$	−1.67084e8	−	2.89398e8i	−0.126067	−	0.218354i
$405$		0			0
$406$	1.71562e8	−	1.20756e8i	0.127227	−	0.0895502i
$407$		−	9.27529e8i		−	0.681941i
$408$		0			0
$409$	−8.71218e8	−	5.02998e8i	−0.629645	−	0.363525i	0.150970	−	0.988538i	$-0.451760\pi$
$409$	−8.71218e8	−	5.02998e8i	−0.629645	−	0.363525i	−0.780614	+	0.625013i	$0.785094\pi$
$410$	1.91775e8	+	1.10721e8i	0.137419	+	0.0793390i
$411$		0			0
$412$		−	1.02423e9i		−	0.721532i
$413$	6.18740e8	+	2.86681e8i	0.432198	+	0.200251i
$414$		0			0
$415$	4.09141e8	+	7.08652e8i	0.280999	+	0.486704i
$416$	8.27527e8	−	1.43332e9i	0.563580	−	0.976149i
$417$		0			0
$418$	−1.03928e9	+	6.00031e8i	−0.696013	+	0.401843i
$419$	−2.34842e9			−1.55965			−0.779825	−	0.625997i	$-0.784692\pi$
$419$	−2.34842e9			−1.55965			−0.779825	+	0.625997i	$0.784692\pi$
$420$		0			0
$421$	1.77244e9			1.15767			0.578834	−	0.815445i	$-0.303508\pi$
$421$	1.77244e9			1.15767			0.578834	+	0.815445i	$0.303508\pi$
$422$	1.90986e8	−	1.10266e8i	0.123711	−	0.0714247i
$423$		0			0
$424$	1.56739e8	−	2.71480e8i	0.0998613	−	0.172965i
$425$	5.76844e8	+	9.99123e8i	0.364500	+	0.631332i
$426$		0			0
$427$	4.69790e8	+	6.67446e8i	0.292016	+	0.414876i
$428$		−	1.33016e9i		−	0.820072i
$429$		0			0
$430$	−1.19431e9	−	6.89532e8i	−0.724395	−	0.418230i
$431$	−5.03292e8	−	2.90576e8i	−0.302795	−	0.174819i	0.340903	−	0.940099i	$-0.389267\pi$
$431$	−5.03292e8	−	2.90576e8i	−0.302795	−	0.174819i	−0.643698	+	0.765280i	$0.722601\pi$
$432$		0			0
$433$		−	1.71570e9i		−	1.01563i	−0.861467	−	0.507814i	$-0.830454\pi$
$433$		−	1.71570e9i		−	1.01563i	0.861467	−	0.507814i	$-0.169546\pi$
$434$	−1.72155e8	−	1.91376e9i	−0.101089	−	1.12376i
$435$		0			0
$436$	−2.71244e8	−	4.69808e8i	−0.156732	−	0.271468i
$437$	1.78682e9	−	3.09487e9i	1.02423	−	1.77401i
$438$		0			0
$439$	1.58466e9	−	9.14906e8i	0.893946	−	0.516120i	0.0187148	−	0.999825i	$-0.494043\pi$
$439$	1.58466e9	−	9.14906e8i	0.893946	−	0.516120i	0.875231	+	0.483705i	$0.160709\pi$
$440$	1.32238e9			0.740070
$441$		0			0
$442$	−2.29982e9			−1.26682
$443$	−1.23432e9	+	7.12636e8i	−0.674551	+	0.389452i	−0.797799	−	0.602923i	$-0.794002\pi$
$443$	−1.23432e9	+	7.12636e8i	−0.674551	+	0.389452i	0.123248	+	0.992376i	$0.460669\pi$
$444$		0			0
$445$	7.78834e8	−	1.34898e9i	0.418972	−	0.725681i
$446$	1.00610e9	+	1.74262e9i	0.536996	+	0.930104i
$447$		0			0
$448$	1.35863e8	+	1.51033e9i	0.0713885	+	0.793593i
$449$		−	4.59393e8i		−	0.239509i	−0.992804	−	0.119755i	$-0.961789\pi$
$449$		−	4.59393e8i		−	0.239509i	0.992804	−	0.119755i	$-0.0382108\pi$
$450$		0			0
$451$	−5.30282e8	−	3.06159e8i	−0.272201	−	0.157155i
$452$	−1.37124e9	−	7.91683e8i	−0.698438	−	0.403243i
$453$		0			0
$454$			1.62222e9i			0.813607i
$455$	−1.01716e9	−	1.44511e9i	−0.506232	−	0.719220i
$456$		0			0
$457$	−1.31427e9	−	2.27639e9i	−0.644139	−	1.11568i	−0.984500	−	0.175386i	$-0.943883\pi$
$457$	−1.31427e9	−	2.27639e9i	−0.644139	−	1.11568i	0.340361	−	0.940295i	$-0.389451\pi$
$458$	6.56248e8	−	1.13665e9i	0.319182	−	0.552840i
$459$		0			0
$460$	−1.19762e9	+	6.91444e8i	−0.573674	+	0.331211i
$461$	2.53767e9			1.20637			0.603186	−	0.797601i	$-0.293898\pi$
$461$	2.53767e9			1.20637			0.603186	+	0.797601i	$0.293898\pi$
$462$		0			0
$463$	−7.97346e8			−0.373348			−0.186674	−	0.982422i	$-0.559771\pi$
$463$	−7.97346e8			−0.373348			−0.186674	+	0.982422i	$0.559771\pi$
$464$	−7.09778e7	+	4.09790e7i	−0.0329845	+	0.0190436i
$465$		0			0
$466$	1.40411e9	−	2.43198e9i	0.642760	−	1.11329i
$467$	1.98580e9	+	3.43950e9i	0.902248	+	1.56274i	0.824569	+	0.565761i	$0.191418\pi$
$467$	1.98580e9	+	3.43950e9i	0.902248	+	1.56274i	0.0776793	+	0.996978i	$0.475249\pi$
$468$		0			0
$469$	−1.90318e9	−	8.81800e8i	−0.851872	−	0.394698i
$470$		−	3.97823e8i		−	0.176745i
$471$		0			0
$472$	−9.83791e8	−	5.67992e8i	−0.430632	−	0.248625i
$473$	3.30241e9	+	1.90665e9i	1.43489	+	0.828432i
$474$		0			0
$475$			1.34017e9i			0.573761i
$476$	−1.60993e9	+	1.13317e9i	−0.684199	+	0.481582i
$477$		0			0
$478$	7.18924e8	+	1.24521e9i	0.301082	+	0.521490i
$479$	3.05274e8	−	5.28750e8i	0.126916	−	0.219825i	−0.795564	−	0.605869i	$-0.792826\pi$
$479$	3.05274e8	−	5.28750e8i	0.126916	−	0.219825i	0.922480	+	0.386044i	$0.126159\pi$
$480$		0			0
$481$	1.78821e9	−	1.03242e9i	0.732674	−	0.423009i
$482$	1.67578e9			0.681636
$483$		0			0
$484$	6.59682e7			0.0264470
$485$	−1.30731e9	+	7.54776e8i	−0.520335	+	0.300415i
$486$		0			0
$487$	−1.09923e9	+	1.90391e9i	−0.431256	+	0.746958i	−0.996982	−	0.0776358i	$-0.975263\pi$
$487$	−1.09923e9	+	1.90391e9i	−0.431256	+	0.746958i	0.565725	+	0.824594i	$0.308596\pi$
$488$	−6.79832e8	−	1.17750e9i	−0.264809	−	0.458662i
$489$		0			0
$490$	1.20703e9	+	4.32644e8i	0.463483	+	0.166128i
$491$		−	3.07424e8i		−	0.117207i	−0.998281	−	0.0586033i	$-0.981335\pi$
$491$		−	3.07424e8i		−	0.117207i	0.998281	−	0.0586033i	$-0.0186647\pi$
$492$		0			0
$493$	−8.18155e8	−	4.72362e8i	−0.307519	−	0.177546i
$494$	−2.31363e9	−	1.33578e9i	−0.863476	−	0.498528i
$495$		0			0
$496$			7.50635e8i			0.276212i
$497$	−1.06954e9	+	9.62120e7i	−0.390797	+	0.0351546i
$498$		0			0
$499$	1.18719e9	+	2.05628e9i	0.427730	+	0.740850i	0.996671	−	0.0815282i	$-0.0259801\pi$
$499$	1.18719e9	+	2.05628e9i	0.427730	+	0.740850i	−0.568941	+	0.822378i	$0.692647\pi$
$500$	8.09156e8	−	1.40150e9i	0.289492	−	0.501416i
$501$		0			0
$502$	−2.16876e9	+	1.25213e9i	−0.765153	+	0.441761i
$503$	−1.50595e9			−0.527623			−0.263812	−	0.964574i	$-0.584980\pi$
$503$	−1.50595e9			−0.527623			−0.263812	+	0.964574i	$0.584980\pi$
$504$		0			0
$505$	9.80042e8			0.338630
$506$	−2.80686e9	+	1.62054e9i	−0.963151	+	0.556075i
$507$		0			0
$508$	−1.13265e9	+	1.96182e9i	−0.383331	+	0.663950i
$509$	5.18772e8	+	8.98540e8i	0.174367	+	0.302013i	0.939942	−	0.341334i	$-0.110879\pi$
$509$	5.18772e8	+	8.98540e8i	0.174367	+	0.302013i	−0.765575	+	0.643347i	$0.777545\pi$
$510$		0			0
$511$	−5.39543e8	+	1.16449e9i	−0.178876	+	0.386066i
$512$		−	9.94793e8i		−	0.327558i
$513$		0			0
$514$	−3.04412e9	−	1.75753e9i	−0.988761	−	0.570861i
$515$	2.60140e9	+	1.50192e9i	0.839232	+	0.484531i
$516$		0			0
$517$			1.10003e9i			0.350098i
$518$	−6.29841e8	+	1.35938e9i	−0.199103	+	0.429720i
$519$		0			0
$520$	1.47193e9	+	2.54946e9i	0.459067	+	0.795127i
$521$	−1.15239e9	+	1.99599e9i	−0.356999	+	0.618340i	−0.987458	−	0.157882i	$-0.949533\pi$
$521$	−1.15239e9	+	1.99599e9i	−0.356999	+	0.618340i	0.630459	+	0.776222i	$0.282867\pi$
$522$		0			0
$523$	3.91811e9	−	2.26212e9i	1.19763	−	0.691449i	0.237600	−	0.971363i	$-0.423639\pi$
$523$	3.91811e9	−	2.26212e9i	1.19763	−	0.691449i	0.960025	+	0.279914i	$0.0903058\pi$
$524$	3.54444e8			0.107619
$525$		0			0
$526$	1.58555e9			0.475040
$527$	−7.49329e9	+	4.32625e9i	−2.23016	+	1.28758i
$528$		0			0
$529$	3.12337e9	−	5.40984e9i	0.917338	−	1.58888i
$530$	1.61428e8	+	2.79602e8i	0.0470993	+	0.0815784i
$531$		0			0
$532$	−2.27777e9	+	2.04899e8i	−0.655871	+	0.0589996i
$533$		−	1.36313e9i		−	0.389935i
$534$		0			0
$535$	3.37844e9	+	1.95054e9i	0.953845	+	0.550703i
$536$	3.02603e9	+	1.74708e9i	0.848784	+	0.490046i
$537$		0			0
$538$		−	4.39362e9i		−	1.21642i
$539$	−3.33761e9	−	1.19632e9i	−0.918069	−	0.329068i
$540$		0			0
$541$	2.15400e9	+	3.73084e9i	0.584864	+	1.01301i	0.994892	+	0.100942i	$0.0321856\pi$
$541$	2.15400e9	+	3.73084e9i	0.584864	+	1.01301i	−0.410028	+	0.912073i	$0.634481\pi$
$542$	−7.14212e8	+	1.23705e9i	−0.192677	+	0.333726i
$543$		0			0
$544$	4.68304e9	−	2.70376e9i	1.24719	−	0.720064i
$545$	1.59100e9			0.421001
$546$		0			0
$547$	−3.29693e9			−0.861298			−0.430649	−	0.902519i	$-0.641715\pi$
$547$	−3.29693e9			−0.861298			−0.430649	+	0.902519i	$0.641715\pi$
$548$	−7.07694e8	+	4.08587e8i	−0.183702	+	0.106060i
$549$		0			0
$550$	6.07726e8	−	1.05261e9i	0.155754	−	0.269773i
$551$	−5.48714e8	−	9.50400e8i	−0.139738	−	0.242034i
$552$		0			0
$553$	−3.13131e9	+	2.20401e9i	−0.787387	+	0.554212i
$554$			3.62491e9i			0.905761i
$555$		0			0
$556$	7.83398e8	+	4.52295e8i	0.193295	+	0.111599i
$557$	−1.68019e9	−	9.70056e8i	−0.411969	−	0.237850i	0.279666	−	0.960097i	$-0.409776\pi$
$557$	−1.68019e9	−	9.70056e8i	−0.411969	−	0.237850i	−0.691635	+	0.722247i	$0.743109\pi$
$558$		0			0
$559$			8.48909e9i			2.05551i
$560$	−4.54491e8	−	2.10580e8i	−0.109362	−	0.0506709i
$561$		0			0
$562$	1.01558e8	+	1.75903e8i	0.0241344	+	0.0418020i
$563$	3.54664e9	−	6.14296e9i	0.837602	−	1.45077i	−0.0542922	−	0.998525i	$-0.517290\pi$
$563$	3.54664e9	−	6.14296e9i	0.837602	−	1.45077i	0.891894	−	0.452244i	$-0.149376\pi$
$564$		0			0
$565$	4.02154e9	−	2.32184e9i	0.938044	−	0.541580i
$566$	1.94731e6			0.000451417
$567$		0			0
$568$	1.78889e9			0.409604
$569$	5.71838e9	−	3.30151e9i	1.30131	−	0.751310i	0.320679	−	0.947188i	$-0.396089\pi$
$569$	5.71838e9	−	3.30151e9i	1.30131	−	0.751310i	0.980628	+	0.195877i	$0.0627554\pi$
$570$		0			0
$571$	−1.13120e9	+	1.95929e9i	−0.254280	+	0.440426i	−0.964700	−	0.263352i	$-0.915172\pi$
$571$	−1.13120e9	+	1.95929e9i	−0.254280	+	0.440426i	0.710420	+	0.703778i	$0.248505\pi$
$572$	−1.42931e9	−	2.47563e9i	−0.319330	−	0.553096i
$573$		0			0
$574$	5.69279e8	+	8.08793e8i	0.125642	+	0.178503i
$575$			3.61947e9i			0.793977i
$576$		0			0
$577$	3.16618e9	+	1.82800e9i	0.686153	+	0.396150i	0.802169	−	0.597097i	$-0.203679\pi$
$577$	3.16618e9	+	1.82800e9i	0.686153	+	0.396150i	−0.116016	+	0.993247i	$0.537013\pi$
$578$	−3.78430e9	−	2.18487e9i	−0.815152	−	0.470628i
$579$		0			0
$580$			4.24670e8i			0.0903760i
$581$	3.27448e8	+	3.64008e9i	0.0692668	+	0.770007i
$582$		0			0
$583$	−4.46371e8	−	7.73138e8i	−0.0932946	−	0.161591i
$584$	1.06898e9	−	1.85153e9i	0.222088	−	0.384667i
$585$		0			0
$586$	3.37364e9	−	1.94777e9i	0.692560	−	0.399850i
$587$	−3.24417e9			−0.662019			−0.331009	−	0.943627i	$-0.607389\pi$
$587$	−3.24417e9			−0.662019			−0.331009	+	0.943627i	$0.607389\pi$
$588$		0			0
$589$	−1.00511e10			−2.02679
$590$	1.01322e9	−	5.84985e8i	0.203106	−	0.117264i
$591$		0			0
$592$	2.92638e8	−	5.06863e8i	0.0579701	−	0.100407i
$593$	4.47929e9	+	7.75836e9i	0.882101	+	1.52784i	0.849001	+	0.528391i	$0.177204\pi$
$593$	4.47929e9	+	7.75836e9i	0.882101	+	1.52784i	0.0330992	+	0.999452i	$0.489462\pi$
$594$		0			0
$595$	−5.17308e8	−	5.75068e9i	−0.100679	−	1.11921i
$596$			5.44773e9i			1.05403i
$597$		0			0
$598$	−6.24858e9	−	3.60762e9i	−1.19489	−	0.689869i
$599$	−5.26078e9	−	3.03731e9i	−1.00013	−	0.577425i	−0.0918429	−	0.995774i	$-0.529276\pi$
$599$	−5.26078e9	−	3.03731e9i	−1.00013	−	0.577425i	−0.908287	+	0.418348i	$0.862609\pi$
$600$		0			0
$601$		−	9.23837e9i		−	1.73594i	−0.496616	−	0.867970i	$-0.665424\pi$
$601$		−	9.23837e9i		−	1.73594i	0.496616	−	0.867970i	$-0.334576\pi$
$602$	−3.54527e9	−	5.03688e9i	−0.662311	−	0.940966i
$603$		0			0
$604$	5.25187e8	+	9.09651e8i	0.0969806	+	0.167975i
$605$	−9.67353e7	+	1.67550e8i	−0.0177599	+	0.0307611i
$606$		0			0
$607$	−7.86096e9	+	4.53853e9i	−1.42664	+	0.823673i	−0.996854	−	0.0792567i	$-0.974745\pi$
$607$	−7.86096e9	+	4.53853e9i	−1.42664	+	0.823673i	−0.429789	+	0.902929i	$0.641412\pi$
$608$	6.28157e9			1.13346
$609$		0			0
$610$	1.40034e9			0.249793
$611$	−2.12079e9	+	1.22444e9i	−0.376143	+	0.217166i
$612$		0			0
$613$	−1.57364e9	+	2.72562e9i	−0.275926	+	0.477919i	−0.970368	−	0.241630i	$-0.922318\pi$
$613$	−1.57364e9	+	2.72562e9i	−0.275926	+	0.477919i	0.694442	+	0.719549i	$0.255651\pi$
$614$	1.10759e9	+	1.91841e9i	0.193104	+	0.334465i
$615$		0			0
$616$	5.35903e9	+	2.48300e9i	0.923748	+	0.428001i
$617$		−	1.10094e10i		−	1.88698i	−0.331406	−	0.943488i	$-0.607523\pi$
$617$		−	1.10094e10i		−	1.88698i	0.331406	−	0.943488i	$-0.392477\pi$
$618$		0			0
$619$	−1.81934e9	−	1.05040e9i	−0.308316	−	0.178007i	0.337857	−	0.941198i	$-0.390298\pi$
$619$	−1.81934e9	−	1.05040e9i	−0.308316	−	0.178007i	−0.646173	+	0.763191i	$0.723631\pi$
$620$	3.36836e9	+	1.94472e9i	0.567607	+	0.327708i
$621$		0			0
$622$			2.19218e9i			0.365267i
$623$	5.68921e9	−	4.00442e9i	0.942636	−	0.663486i
$624$		0			0
$625$	9.33930e8	+	1.61761e9i	0.153015	+	0.265030i
$626$	−2.56889e9	+	4.44945e9i	−0.418538	+	0.724929i
$627$		0			0
$628$	−4.35959e8	+	2.51701e8i	−0.0702403	+	0.0405533i
$629$	6.74642e9			1.08093
$630$		0			0
$631$	−7.15955e9			−1.13444			−0.567222	−	0.823565i	$-0.691982\pi$
$631$	−7.15955e9			−1.13444			−0.567222	+	0.823565i	$0.691982\pi$
$632$	5.52424e9	−	3.18942e9i	0.870488	−	0.502577i
$633$		0			0
$634$	−7.84965e8	+	1.35960e9i	−0.122331	+	0.211884i
$635$	−3.32183e9	−	5.75358e9i	−0.514837	−	0.891724i
$636$		0			0
$637$	−1.40865e9	−	7.76629e9i	−0.215931	−	1.19049i
$638$			9.95301e8i			0.151734i
$639$		0			0
$640$	−1.63624e9	−	9.44684e8i	−0.246727	−	0.142448i
$641$	4.96416e9	+	2.86606e9i	0.744463	+	0.429816i	0.823690	−	0.567041i	$-0.191912\pi$
$641$	4.96416e9	+	2.86606e9i	0.744463	+	0.429816i	−0.0792269	+	0.996857i	$0.525245\pi$
$642$		0			0
$643$			4.97543e9i			0.738061i	0.929417	+	0.369030i	$0.120310\pi$
$643$			4.97543e9i			0.738061i	−0.929417	+	0.369030i	$0.879690\pi$
$644$	−6.15171e9	+	5.53384e8i	−0.907602	+	0.0816443i
$645$		0			0
$646$	−4.36435e9	−	7.55928e9i	−0.636951	−	1.10323i
$647$	2.91869e9	−	5.05532e9i	0.423665	−	0.733810i	−0.572629	−	0.819814i	$-0.694077\pi$
$647$	2.91869e9	−	5.05532e9i	0.423665	−	0.733810i	0.996295	+	0.0860044i	$0.0274099\pi$
$648$		0			0
$649$	−2.80170e9	+	1.61756e9i	−0.402314	+	0.232276i
$650$	2.70582e9			0.386457
$651$		0			0
$652$	4.98249e9			0.704012
$653$	2.00968e8	−	1.16029e8i	0.0282443	−	0.0163068i	−0.485811	−	0.874064i	$-0.661476\pi$
$653$	2.00968e8	−	1.16029e8i	0.0282443	−	0.0163068i	0.514056	+	0.857757i	$0.328142\pi$
$654$		0			0
$655$	−5.19754e8	+	9.00240e8i	−0.0722692	+	0.125174i
$656$	−1.93188e8	−	3.34611e8i	−0.0267187	−	0.0462782i
$657$		0			0
$658$	7.46982e8	−	1.61220e9i	0.102216	−	0.220612i
$659$		−	1.47334e9i		−	0.200541i	−0.994960	−	0.100270i	$-0.968029\pi$
$659$		−	1.47334e9i		−	0.200541i	0.994960	−	0.100270i	$-0.0319708\pi$
$660$		0			0
$661$	6.53789e9	+	3.77465e9i	0.880506	+	0.508360i	0.870825	−	0.491593i	$-0.163585\pi$
$661$	6.53789e9	+	3.77465e9i	0.880506	+	0.508360i	0.00968068	+	0.999953i	$0.496918\pi$
$662$	1.93955e9	+	1.11980e9i	0.259836	+	0.150016i
$663$		0			0
$664$		−	6.08829e9i		−	0.807062i
$665$	2.81969e9	−	6.08569e9i	0.371813	−	0.802479i
$666$		0			0
$667$	−1.48195e9	−	2.56681e9i	−0.193371	−	0.334929i
$668$	−1.70687e9	+	2.95638e9i	−0.221555	+	0.383745i
$669$		0			0
$670$	−3.11657e9	+	1.79935e9i	−0.400327	+	0.231129i
$671$	−3.87213e9			−0.494791
$672$		0			0
$673$	2.47252e9			0.312671			0.156336	−	0.987704i	$-0.450032\pi$
$673$	2.47252e9			0.312671			0.156336	+	0.987704i	$0.450032\pi$
$674$	3.19700e9	−	1.84579e9i	0.402192	−	0.232206i
$675$		0			0
$676$	1.00831e9	−	1.74645e9i	0.125540	−	0.217441i
$677$	−2.50932e9	−	4.34627e9i	−0.310811	−	0.538340i	0.667727	−	0.744406i	$-0.267267\pi$
$677$	−2.50932e9	−	4.34627e9i	−0.310811	−	0.538340i	−0.978538	+	0.206066i	$0.933934\pi$
$678$		0			0
$679$	−6.71517e9	+	6.04070e8i	−0.823214	+	0.0740531i
$680$			9.61841e9i			1.17306i
$681$		0			0
$682$	7.89445e9	+	4.55786e9i	0.952964	+	0.550194i
$683$	−4.40892e9	−	2.54549e9i	−0.529493	−	0.305703i	0.211317	−	0.977418i	$-0.432225\pi$
$683$	−4.40892e9	−	2.54549e9i	−0.529493	−	0.305703i	−0.740810	+	0.671715i	$0.765558\pi$
$684$		0			0
$685$		−	2.39660e9i		−	0.284891i
$686$	4.07921e9	+	4.01973e9i	0.482438	+	0.475403i
$687$		0			0
$688$	1.20311e9	+	2.08384e9i	0.140846	+	0.243952i
$689$	9.93703e8	−	1.72114e9i	0.115742	−	0.200470i
$690$		0			0
$691$	9.87316e9	−	5.70027e9i	1.13837	−	0.657238i	0.192343	−	0.981328i	$-0.438391\pi$
$691$	9.87316e9	−	5.70027e9i	1.13837	−	0.657238i	0.946026	+	0.324090i	$0.105058\pi$
$692$	2.03685e9			0.233662
$693$		0			0
$694$	−2.38866e9			−0.271267
$695$	−2.29754e9	+	1.32648e9i	−0.259607	+	0.149884i
$696$		0			0
$697$	2.22686e9	−	3.85703e9i	0.249102	−	0.431458i
$698$	1.45432e9	+	2.51896e9i	0.161870	+	0.280367i
$699$		0			0
$700$	1.89414e9	−	1.33321e9i	0.208722	−	0.146912i
$701$			5.86499e8i			0.0643064i	0.999483	+	0.0321532i	$0.0102364\pi$
$701$			5.86499e8i			0.0643064i	−0.999483	+	0.0321532i	$0.989764\pi$
$702$		0			0
$703$	6.78695e9	+	3.91845e9i	0.736768	+	0.425373i
$704$	−6.23023e9	−	3.59703e9i	−0.672977	−	0.388543i
$705$		0			0
$706$			5.41364e9i			0.578992i
$707$	3.97168e9	+	1.84020e9i	0.422674	+	0.195838i
$708$		0			0
$709$	−3.17119e9	−	5.49266e9i	−0.334165	−	0.578790i	0.649159	−	0.760652i	$-0.275121\pi$
$709$	−3.17119e9	−	5.49266e9i	−0.334165	−	0.578790i	−0.983324	+	0.181862i	$0.941787\pi$
$710$	−9.21203e8	+	1.59557e9i	−0.0965943	+	0.167306i
$711$		0			0
$712$	−1.00369e10	+	5.79479e9i	−1.04212	+	0.601669i
$713$	−2.71456e10			−2.80470
$714$		0			0
$715$	8.38371e9			0.857759
$716$	7.36119e8	−	4.24999e8i	0.0749467	−	0.0432705i
$717$		0			0
$718$	−1.44945e8	+	2.51052e8i	−0.0146140	+	0.0253121i
$719$	−8.28036e9	−	1.43420e10i	−0.830803	−	1.43899i	−0.897402	−	0.441214i	$-0.854548\pi$
$719$	−8.28036e9	−	1.43420e10i	−0.830803	−	1.43899i	0.0665986	−	0.997780i	$-0.478785\pi$
$720$		0			0
$721$	7.72220e9	+	1.09712e10i	0.767305	+	1.09013i
$722$		−	3.28990e9i		−	0.325314i
$723$		0			0
$724$	1.57863e9	+	9.11425e8i	0.154595	+	0.0892557i
$725$	9.62588e8	+	5.55750e8i	0.0938118	+	0.0541623i
$726$		0			0
$727$			4.61865e9i			0.445804i	0.974841	+	0.222902i	$0.0715530\pi$
$727$			4.61865e9i			0.445804i	−0.974841	+	0.222902i	$0.928447\pi$
$728$	1.17803e9	+	1.30956e10i	0.113161	+	1.25796i
$729$		0			0
$730$	1.10096e9	+	1.90692e9i	0.104747	+	0.181427i
$731$	−1.38681e10	+	2.40203e10i	−1.31313	+	2.27440i
$732$		0			0
$733$	−7.18124e9	+	4.14609e9i	−0.673496	+	0.388843i	−0.797400	−	0.603451i	$-0.793792\pi$
$733$	−7.18124e9	+	4.14609e9i	−0.673496	+	0.388843i	0.123904	+	0.992294i	$0.460459\pi$
$734$	7.63915e8			0.0713032
$735$		0			0
$736$	1.69650e10			1.56849
$737$	8.61772e9	−	4.97545e9i	0.792969	−	0.457821i
$738$		0			0
$739$	−3.06492e9	+	5.30860e9i	−0.279360	+	0.483866i	−0.971226	−	0.238160i	$-0.923456\pi$
$739$	−3.06492e9	+	5.30860e9i	−0.279360	+	0.483866i	0.691866	+	0.722026i	$0.256789\pi$
$740$	−1.51632e9	−	2.62633e9i	−0.137556	−	0.238253i
$741$		0			0
$742$	1.29196e8	+	1.43621e9i	0.0116101	+	0.129064i
$743$		−	6.48159e9i		−	0.579723i	−0.957069	−	0.289862i	$-0.906391\pi$
$743$		−	6.48159e9i		−	0.579723i	0.957069	−	0.289862i	$-0.0936093\pi$
$744$		0			0
$745$	−1.38365e10	−	7.98852e9i	−1.22597	−	0.707814i
$746$	8.66140e9	+	5.00066e9i	0.763839	+	0.441003i
$747$		0			0
$748$		−	9.33988e9i		−	0.815992i
$749$	1.00288e10	+	1.42483e10i	0.872095	+	1.23901i
$750$		0			0
$751$	−2.40290e9	−	4.16195e9i	−0.207012	−	0.358556i	0.743760	−	0.668447i	$-0.233041\pi$
$751$	−2.40290e9	−	4.16195e9i	−0.207012	−	0.358556i	−0.950772	+	0.309891i	$0.899707\pi$
$752$	−3.47063e8	+	6.01132e8i	−0.0297609	+	0.0515474i
$753$		0			0
$754$	−1.91887e9	+	1.10786e9i	−0.163022	+	0.0941207i
$755$	−3.08052e9			−0.260502
$756$		0			0
$757$	−2.66592e8			−0.0223364			−0.0111682	−	0.999938i	$-0.503555\pi$
$757$	−2.66592e8			−0.0223364			−0.0111682	+	0.999938i	$0.503555\pi$
$758$	−9.93607e8	+	5.73659e8i	−0.0828652	+	0.0478423i
$759$		0			0
$760$	−5.58655e9	+	9.67619e9i	−0.461632	+	0.799570i
$761$	1.16854e10	+	2.02396e10i	0.961160	+	1.66478i	0.719597	+	0.694392i	$0.244327\pi$
$761$	1.16854e10	+	2.02396e10i	0.961160	+	1.66478i	0.241563	+	0.970385i	$0.422340\pi$
$762$		0			0
$763$	6.44762e9	+	2.98738e9i	0.525489	+	0.243475i
$764$			2.20254e9i			0.178689i
$765$		0			0
$766$	2.97735e9	+	1.71898e9i	0.239348	+	0.138188i
$767$	−6.23709e9	−	3.60099e9i	−0.499113	−	0.288163i
$768$		0			0
$769$			1.30604e10i			1.03565i	0.855486	+	0.517826i	$0.173259\pi$
$769$			1.30604e10i			1.03565i	−0.855486	+	0.517826i	$0.826741\pi$
$770$	−4.97435e9	+	3.50126e9i	−0.392662	+	0.276380i
$771$		0			0
$772$	1.10434e9	+	1.91277e9i	0.0863857	+	0.149624i
$773$	−7.20449e9	+	1.24785e10i	−0.561016	+	0.971708i	0.436392	+	0.899756i	$0.356256\pi$
$773$	−7.20449e9	+	1.24785e10i	−0.561016	+	0.971708i	−0.997408	+	0.0719514i	$0.977077\pi$
$774$		0			0
$775$	8.81611e9	−	5.08998e9i	0.680332	−	0.392790i
$776$	1.12316e10			0.862830
$777$		0			0
$778$	1.16378e10			0.886018
$779$	4.48047e9	−	2.58680e9i	0.339580	−	0.196057i
$780$		0			0
$781$	2.54725e9	−	4.41197e9i	0.191334	−	0.331401i
$782$	−1.17871e10	−	2.04158e10i	−0.881420	−	1.52666i
$783$		0			0
$784$	−1.44645e9	−	1.70677e9i	−0.107201	−	0.126494i
$785$		−	1.47637e9i		−	0.108931i
$786$		0			0
$787$	−1.25825e10	−	7.26451e9i	−0.920144	−	0.531245i	−0.0364627	−	0.999335i	$-0.511609\pi$
$787$	−1.25825e10	−	7.26451e9i	−0.920144	−	0.531245i	−0.883681	+	0.468090i	$0.844942\pi$
$788$	−1.04006e10	−	6.00478e9i	−0.757210	−	0.437175i
$789$		0			0
$790$			6.56969e9i			0.474078i
$791$	2.06572e10	−	1.85824e9i	1.48407	−	0.133501i
$792$		0			0
$793$	−4.31004e9	−	7.46520e9i	−0.306920	−	0.531600i
$794$	6.05605e9	−	1.04894e10i	0.429356	−	0.743666i
$795$		0			0
$796$	−1.05601e10	+	6.09688e9i	−0.742117	+	0.428462i
$797$	−1.98952e10			−1.39202			−0.696010	−	0.718032i	$-0.745043\pi$
$797$	−1.98952e10			−1.39202			−0.696010	+	0.718032i	$0.745043\pi$
$798$		0			0
$799$	−8.00115e9			−0.554930
$800$	−5.50976e9	+	3.18106e9i	−0.380468	+	0.219663i
$801$		0			0
$802$	−1.27505e9	+	2.20845e9i	−0.0872802	+	0.151174i
$803$	−3.04430e9	−	5.27289e9i	−0.207484	−	0.359372i
$804$		0			0
$805$	7.61530e9	−	1.64360e10i	0.514519	−	1.11048i
$806$			2.02933e10i			1.36515i
$807$		0			0
$808$	−6.31493e9	−	3.64593e9i	−0.421142	−	0.243147i
$809$	7.27450e9	+	4.19994e9i	0.483040	+	0.278883i	0.721683	−	0.692224i	$-0.243369\pi$
$809$	7.27450e9	+	4.19994e9i	0.483040	+	0.278883i	−0.238642	+	0.971108i	$0.576702\pi$
$810$		0			0
$811$			8.93491e9i			0.588190i	0.955776	+	0.294095i	$0.0950182\pi$
$811$			8.93491e9i			0.588190i	−0.955776	+	0.294095i	$0.904982\pi$
$812$	−7.97390e8	+	1.72100e9i	−0.0522667	+	0.112806i
$813$		0			0
$814$	−3.55380e9	−	6.15536e9i	−0.230944	−	0.400008i
$815$	−7.30630e9	+	1.26549e10i	−0.472765	+	0.818854i
$816$		0			0
$817$	−2.79028e10	+	1.61097e10i	−1.79007	+	1.03350i
$818$	−7.70889e9			−0.492442
$819$		0			0
$820$	−2.00202e9			−0.126800
$821$	2.29606e10	−	1.32563e10i	1.44805	−	0.836030i	0.449681	−	0.893189i	$-0.351538\pi$
$821$	2.29606e10	−	1.32563e10i	1.44805	−	0.836030i	0.998365	+	0.0571599i	$0.0182045\pi$
$822$		0			0
$823$	1.19616e10	−	2.07180e10i	0.747977	−	1.29553i	−0.200814	−	0.979629i	$-0.564359\pi$
$823$	1.19616e10	−	2.07180e10i	0.747977	−	1.29553i	0.948791	−	0.315905i	$-0.102308\pi$
$824$	−1.11748e10	−	1.93553e10i	−0.695816	−	1.20519i
$825$		0			0
$826$	5.20456e9	−	4.68181e8i	0.321332	−	0.0289057i
$827$		−	1.66831e10i		−	1.02567i	−0.858487	−	0.512836i	$-0.828595\pi$
$827$		−	1.66831e10i		−	1.02567i	0.858487	−	0.512836i	$-0.171405\pi$
$828$		0			0
$829$	2.73726e10	+	1.58036e10i	1.66869	+	0.963416i	0.968348	+	0.249605i	$0.0803009\pi$
$829$	2.73726e10	+	1.58036e10i	1.66869	+	0.963416i	0.700338	+	0.713811i	$0.253032\pi$
$830$	5.43036e9	+	3.13522e9i	0.329652	+	0.190324i
$831$		0			0
$832$		−	1.60153e10i		−	0.964056i
$833$	8.70147e9	−	2.42763e10i	0.521597	−	1.45521i
$834$		0			0
$835$	−5.00588e9	−	8.67044e9i	−0.297562	−	0.515393i
$836$	5.42478e9	−	9.39599e9i	0.321115	−	0.556187i
$837$		0			0
$838$	−1.55848e10	+	8.99791e9i	−0.914847	+	0.528187i
$839$	−4.44026e9			−0.259562			−0.129781	−	0.991543i	$-0.541427\pi$
$839$	−4.44026e9			−0.259562			−0.129781	+	0.991543i	$0.541427\pi$
$840$		0			0
$841$	1.63397e10			0.947236
$842$	1.17624e10	−	6.79105e9i	0.679055	−	0.392053i
$843$		0			0
$844$	−9.96897e8	+	1.72668e9i	−0.0570757	+	0.0988581i
$845$	2.95716e9	+	5.12195e9i	0.168607	+	0.292036i
$846$		0			0
$847$	−7.06630e8	+	4.97370e8i	−0.0399577	+	0.0281247i
$848$		−	5.63325e8i		−	0.0317229i
$849$		0			0
$850$	7.65622e9	+	4.42032e9i	0.427610	+	0.246881i
$851$	1.83300e10	+	1.05828e10i	1.01955	+	0.588637i
$852$		0			0
$853$		−	1.08633e10i		−	0.599297i	−0.954050	−	0.299649i	$-0.903131\pi$
$853$		−	1.08633e10i		−	0.599297i	0.954050	−	0.299649i	$-0.0968694\pi$
$854$	5.67496e9	+	2.62938e9i	0.311789	+	0.144461i
$855$		0			0
$856$	−1.45127e10	−	2.51367e10i	−0.790843	−	1.36978i
$857$	3.97818e9	−	6.89041e9i	0.215899	−	0.373949i	−0.737651	−	0.675182i	$-0.764065\pi$
$857$	3.97818e9	−	6.89041e9i	0.215899	−	0.373949i	0.953550	+	0.301233i	$0.0973983\pi$
$858$		0			0
$859$	−1.18829e9	+	6.86061e8i	−0.0639657	+	0.0369306i	−0.531642	−	0.846969i	$-0.678425\pi$
$859$	−1.18829e9	+	6.86061e8i	−0.0639657	+	0.0369306i	0.467676	+	0.883900i	$0.345091\pi$
$860$	1.24679e10			0.668418
$861$		0			0
$862$	−4.45333e9			−0.236815
$863$	−1.35439e9	+	7.81958e8i	−0.0717309	+	0.0414138i	−0.535436	−	0.844576i	$-0.679853\pi$
$863$	−1.35439e9	+	7.81958e8i	−0.0717309	+	0.0414138i	0.463706	+	0.885989i	$0.346520\pi$
$864$		0			0
$865$	−2.98682e9	+	5.17333e9i	−0.156911	+	0.271778i
$866$	−6.57367e9	−	1.13859e10i	−0.343950	−	0.595738i
$867$		0			0
$868$	9.99891e9	+	1.42058e10i	0.518960	+	0.737303i
$869$		−	1.81661e10i		−	0.939056i
$870$		0			0
$871$	1.91846e10	+	1.10762e10i	0.983761	+	0.567975i
$872$	−1.02517e10	−	5.91880e9i	−0.523584	−	0.302291i
$873$		0			0
$874$		−	2.73846e10i		−	1.38745i
$875$	1.89925e9	+	2.11131e10i	0.0958415	+	1.06543i
$876$		0			0
$877$	2.04854e9	+	3.54817e9i	0.102552	+	0.177626i	0.912736	−	0.408551i	$-0.133966\pi$
$877$	2.04854e9	+	3.54817e9i	0.102552	+	0.177626i	−0.810183	+	0.586177i	$0.800632\pi$
$878$	7.01087e9	−	1.21432e10i	0.349576	−	0.605483i
$879$		0			0
$880$	2.05797e9	−	1.18817e9i	0.101800	−	0.0587745i
$881$	−3.07305e9			−0.151410			−0.0757048	−	0.997130i	$-0.524121\pi$
$881$	−3.07305e9			−0.151410			−0.0757048	+	0.997130i	$0.524121\pi$
$882$		0			0
$883$	2.92284e9			0.142870			0.0714352	−	0.997445i	$-0.477242\pi$
$883$	2.92284e9			0.142870			0.0714352	+	0.997445i	$0.477242\pi$
$884$	1.80066e10	−	1.03961e10i	0.876697	−	0.506161i
$885$		0			0
$886$	−5.46088e9	+	9.45853e9i	−0.263782	+	0.456884i
$887$	1.22352e10	+	2.11919e10i	0.588677	+	1.01962i	0.994406	+	0.105625i	$0.0336843\pi$
$887$	1.22352e10	+	2.11919e10i	0.588677	+	1.01962i	−0.405729	+	0.913993i	$0.632982\pi$
$888$		0			0
$889$	−2.65856e9	−	2.95540e10i	−0.126909	−	1.41078i
$890$		−	1.19363e10i		−	0.567552i
$891$		0			0
$892$	−1.57548e10	−	9.09602e9i	−0.743250	−	0.429116i
$893$	−8.04921e9	−	4.64722e9i	−0.378245	−	0.218380i
$894$		0			0
$895$			2.49286e9i			0.116230i
$896$	−4.85715e9	−	6.90071e9i	−0.225581	−	0.320491i
$897$		0			0
$898$	−1.76015e9	−	3.04867e9i	−0.0811115	−	0.140489i
$899$	−4.16805e9	+	7.21928e9i	−0.191326	+	0.331387i
$900$		0			0
$901$	5.62345e9	−	3.24670e9i	0.256133	−	0.147879i
$902$	−4.69215e9			−0.212887
$903$		0			0
$904$	−3.45505e10			−1.55548
$905$	−4.62980e9	+	2.67301e9i	−0.207631	+	0.119876i
$906$		0			0
$907$	−2.10777e10	+	3.65077e10i	−0.937990	+	1.62465i	−0.168776	+	0.985654i	$0.553982\pi$
$907$	−2.10777e10	+	3.65077e10i	−0.937990	+	1.62465i	−0.769214	+	0.638992i	$0.779352\pi$
$908$	−7.33312e9	−	1.27013e10i	−0.325078	−	0.563052i
$909$		0			0
$910$	−1.22871e10	−	5.69298e9i	−0.540510	−	0.250435i
$911$			2.63292e10i			1.15378i	0.816822	+	0.576890i	$0.195734\pi$
$911$			2.63292e10i			1.15378i	−0.816822	+	0.576890i	$0.804266\pi$
$912$		0			0
$913$	−1.50157e10	−	8.66930e9i	−0.652976	−	0.376996i
$914$	−1.74438e10	−	1.00712e10i	−0.755667	−	0.436284i
$915$		0			0
$916$			1.18660e10i			0.510119i
$917$	−3.79669e9	+	2.67234e9i	−0.162597	+	0.114446i
$918$		0			0
$919$	5.64393e9	+	9.77557e9i	0.239871	+	0.415468i	0.960677	−	0.277668i	$-0.0895617\pi$
$919$	5.64393e9	+	9.77557e9i	0.239871	+	0.415468i	−0.720806	+	0.693136i	$0.756228\pi$
$920$	−1.50880e10	+	2.61331e10i	−0.638812	+	1.10645i
$921$		0			0
$922$	1.68407e10	−	9.72299e9i	0.707624	−	0.408547i
$923$	1.13413e10			0.474740
$924$		0			0
$925$	−7.93740e9			−0.329748
$926$	−5.29143e9	+	3.05501e9i	−0.218995	+	0.126437i
$927$		0			0
$928$	2.60489e9	−	4.51180e9i	0.106997	−	0.185324i
$929$	−8.91191e9	−	1.54359e10i	−0.364683	−	0.631650i	0.624042	−	0.781391i	$-0.285489\pi$
$929$	−8.91191e9	−	1.54359e10i	−0.364683	−	0.631650i	−0.988725	+	0.149741i	$0.952156\pi$
$930$		0			0
$931$	2.28539e10	−	1.93681e10i	0.928187	−	0.786618i
$932$			2.53886e10i			1.02726i
$933$		0			0
$934$	2.63567e10	+	1.52171e10i	1.05847	+	0.611106i
$935$	2.37221e10	+	1.36959e10i	0.949100	+	0.547963i
$936$		0			0
$937$			2.99942e10i			1.19110i	0.803317	+	0.595551i	$0.203066\pi$
$937$			2.99942e10i			1.19110i	−0.803317	+	0.595551i	$0.796934\pi$
$938$	−1.60086e10	+	1.44007e9i	−0.633351	+	0.0569738i
$939$		0			0
$940$	1.79833e9	+	3.11479e9i	0.0706189	+	0.122315i
$941$	−1.74811e10	+	3.02781e10i	−0.683918	+	1.18458i	0.289858	+	0.957070i	$0.406392\pi$
$941$	−1.74811e10	+	3.02781e10i	−0.683918	+	1.18458i	−0.973776	+	0.227511i	$0.926941\pi$
$942$		0			0
$943$	1.21007e10	−	6.98634e9i	0.469915	−	0.271306i
$944$	−2.04138e9			−0.0789809
$945$		0			0
$946$	2.92211e10			1.12222
$947$	3.40657e10	−	1.96678e10i	1.30344	−	0.752543i	0.322450	−	0.946586i	$-0.395493\pi$
$947$	3.40657e10	−	1.96678e10i	1.30344	−	0.752543i	0.980993	+	0.194043i	$0.0621602\pi$
$948$		0			0
$949$	6.77717e9	−	1.17384e10i	0.257405	−	0.445838i
$950$	5.13481e9	+	8.89375e9i	0.194308	+	0.336552i
$951$		0			0
$952$	−1.80602e10	+	3.89791e10i	−0.678412	+	1.46421i
$953$			2.56610e10i			0.960390i	0.877162	+	0.480195i	$0.159434\pi$
$953$			2.56610e10i			0.960390i	−0.877162	+	0.480195i	$0.840566\pi$
$954$		0			0
$955$	−5.59416e9	−	3.22979e9i	−0.207837	−	0.119995i
$956$	−1.12578e10	−	6.49967e9i	−0.416725	−	0.240596i
$957$		0			0
$958$		−	4.67859e9i		−	0.171924i
$959$	4.50003e9	−	9.71235e9i	0.164759	−	0.355598i
$960$		0			0
$961$	2.44179e10	+	4.22930e10i	0.887517	+	1.53722i
$962$	7.91140e9	−	1.37029e10i	0.286511	−	0.496251i
$963$		0			0
$964$	−1.31207e10	+	7.57522e9i	−0.471722	+	0.272349i
$965$	−6.47758e9			−0.232042
$966$		0			0
$967$	−3.78121e10			−1.34474			−0.672370	−	0.740216i	$-0.734723\pi$
$967$	−3.78121e10			−1.34474			−0.672370	+	0.740216i	$0.734723\pi$
$968$	1.24663e9	−	7.19744e8i	0.0441748	−	0.0255043i
$969$		0			0
$970$	−5.78381e9	+	1.00178e10i	−0.203476	+	0.352430i
$971$	2.21273e10	+	3.83257e10i	0.775643	+	1.34345i	0.934433	+	0.356140i	$0.115907\pi$
$971$	2.21273e10	+	3.83257e10i	0.775643	+	1.34345i	−0.158790	+	0.987312i	$0.550759\pi$
$972$		0			0
$973$	−1.18016e10	+	1.06163e9i	−0.410720	+	0.0369468i
$974$			1.68466e10i			0.584192i
$975$		0			0
$976$	−2.11599e9	−	1.22167e9i	−0.0728516	−	0.0420609i
$977$	−4.46379e10	−	2.57717e10i	−1.53134	−	0.884122i	−0.999300	−	0.0374057i	$-0.988091\pi$
$977$	−4.46379e10	−	2.57717e10i	−1.53134	−	0.884122i	−0.532044	−	0.846716i	$-0.678576\pi$
$978$		0			0
$979$			3.30055e10i			1.12421i
$980$	−1.14063e10	+	2.06888e9i	−0.387128	+	0.0702171i
$981$		0			0
$982$	−1.17788e9	−	2.04016e9i	−0.0396929	−	0.0687501i
$983$	−6.65368e9	+	1.15245e10i	−0.223421	+	0.386977i	−0.955845	−	0.293873i	$-0.905056\pi$
$983$	−6.65368e9	+	1.15245e10i	−0.223421	+	0.386977i	0.732423	+	0.680849i	$0.238389\pi$
$984$		0			0
$985$	3.05027e10	−	1.76107e10i	1.01698	−	0.587153i
$986$	−7.23937e9			−0.240509
$987$		0			0
$988$	2.41531e10			0.796752
$989$	−7.53589e10	+	4.35085e10i	−2.47712	+	1.43017i
$990$		0			0
$991$	2.54726e10	−	4.41199e10i	0.831412	−	1.44005i	−0.0655067	−	0.997852i	$-0.520866\pi$
$991$	2.54726e10	−	4.41199e10i	0.831412	−	1.44005i	0.896919	−	0.442196i	$-0.145800\pi$
$992$	−2.38575e10	−	4.13225e10i	−0.775952	−	1.34399i
$993$		0			0
$994$	−6.72919e9	+	4.73642e9i	−0.217325	+	0.152967i
$995$		−	3.57617e10i		−	1.15090i
$996$		0			0
$997$	2.69607e9	+	1.55658e9i	0.0861585	+	0.0497436i	0.542460	−	0.840081i	$-0.317493\pi$
$997$	2.69607e9	+	1.55658e9i	0.0861585	+	0.0497436i	−0.456302	+	0.889825i	$0.650826\pi$
$998$	1.57572e10	+	9.09740e9i	0.501789	+	0.289708i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	63.8.p.a.17.12	yes	36
3.2	odd	2	inner	63.8.p.a.17.7	✓	36
7.3	odd	6		441.8.c.a.440.10		36
7.4	even	3		441.8.c.a.440.28		36
7.5	odd	6	inner	63.8.p.a.26.7	yes	36
21.5	even	6	inner	63.8.p.a.26.12	yes	36
21.11	odd	6		441.8.c.a.440.9		36
21.17	even	6		441.8.c.a.440.27		36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.8.p.a.17.7	✓	36	3.2	odd	2	inner
63.8.p.a.17.12	yes	36	1.1	even	1	trivial
63.8.p.a.26.7	yes	36	7.5	odd	6	inner
63.8.p.a.26.12	yes	36	21.5	even	6	inner
441.8.c.a.440.9		36	21.11	odd	6
441.8.c.a.440.10		36	7.3	odd	6
441.8.c.a.440.27		36	21.17	even	6
441.8.c.a.440.28		36	7.4	even	3

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:	\( N \)	\(=\)	\( 63 = 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	63.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(6.63630 - 3.83147i) q^{2} +(-34.6397 + 59.9977i) q^{4} +(-101.591 - 175.960i) q^{5} +(-81.3062 - 903.843i) q^{7} +1511.74i q^{8} +(-1348.37 - 778.484i) q^{10} +(3728.43 + 2152.61i) q^{11} +9584.21i q^{13} +(-4002.62 - 5686.65i) q^{14} +(1358.31 + 2352.66i) q^{16} +(-15657.1 + 27118.9i) q^{17} +(-31502.4 + 18187.9i) q^{19} +14076.3 q^{20} +32990.7 q^{22} +(-85080.4 + 49121.2i) q^{23} +(18421.1 - 31906.3i) q^{25} +(36721.6 + 63603.7i) q^{26} +(57044.9 + 26430.6i) q^{28} +30169.2i q^{29} +(239293. + 138156. i) q^{31} +(-149550. - 86342.7i) q^{32} +239959. i q^{34} +(-150781. + 106129. i) q^{35} +(-107721. - 186579. i) q^{37} +(-139373. + 241401. i) q^{38} +(266006. - 153579. i) q^{40} -142227. q^{41} +885738. q^{43} +(-258303. + 149132. i) q^{44} +(-376413. + 651966. i) q^{46} +(127756. + 221279. i) q^{47} +(-810322. + 146976. i) q^{49} -282320. i q^{50} +(-575030. - 331994. i) q^{52} +(-179581. - 103681. i) q^{53} -874742. i q^{55} +(1.36638e6 - 122914. i) q^{56} +(115592. + 200212. i) q^{58} +(-375721. + 650768. i) q^{59} +(-778906. + 449702. i) q^{61} +2.11736e6 q^{62} -1.67101e6 q^{64} +(1.68644e6 - 973667. i) q^{65} +(1.15568e6 - 2.00169e6i) q^{67} +(-1.08471e6 - 1.87878e6i) q^{68} +(-593996. + 1.28201e6i) q^{70} -1.18333e6i q^{71} +(-1.22477e6 - 707118. i) q^{73} +(-1.42974e6 - 825462. i) q^{74} -2.52009e6i q^{76} +(1.64248e6 - 3.54494e6i) q^{77} +(-2.10977e6 - 3.65423e6i) q^{79} +(275983. - 478017. i) q^{80} +(-943858. + 544937. i) q^{82} -4.02734e6 q^{83} +6.36247e6 q^{85} +(5.87802e6 - 3.39368e6i) q^{86} +(-3.25419e6 + 5.63642e6i) q^{88} +(3.83319e6 + 6.63929e6i) q^{89} +(8.66262e6 - 779255. i) q^{91} -6.80617e6i q^{92} +(1.69565e6 + 978985. i) q^{94} +(6.40070e6 + 3.69544e6i) q^{95} -7.42958e6i q^{97} +(-4.81440e6 + 4.08010e6i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q + 1024 q^{4} - 2074 q^{7} + 1248 q^{10} - 59708 q^{16} - 105330 q^{19} + 544 q^{22} - 56250 q^{25} + 556220 q^{28} + 86862 q^{31} + 591034 q^{37} - 3036324 q^{40} - 837332 q^{43} + 3896752 q^{46} + 6626454 q^{49}+ \cdots + 96093708 q^{94}+O(q^{100}) \) 36 * q + 1024 * q^4 - 2074 * q^7 + 1248 * q^10 - 59708 * q^16 - 105330 * q^19 + 544 * q^22 - 56250 * q^25 + 556220 * q^28 + 86862 * q^31 + 591034 * q^37 - 3036324 * q^40 - 837332 * q^43 + 3896752 * q^46 + 6626454 * q^49 - 11085900 * q^52 - 134084 * q^58 + 4324644 * q^61 - 11555464 * q^64 + 8750706 * q^67 + 8823972 * q^70 - 744402 * q^73 + 1136238 * q^79 - 29281932 * q^82 + 18719472 * q^85 - 4365916 * q^88 + 21779742 * q^91 + 96093708 * q^94

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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