LMFDB - Embedded newform 84.9.p.b.53.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [84,9,Mod(53,84)]

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, 3, 4]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("84.53");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 84 = 2^{2} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	84.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:	$34.2198032451$
Analytic rank:	$0$
Dimension:	$40$
Relative dimension:	$20$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			53.6
Character	$\chi$	$=$	84.53
Dual form			84.9.p.b.65.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-67.0937 + 45.3810i) q^{3} +(701.327 - 404.911i) q^{5} +(1952.36 - 1397.53i) q^{7} +(2442.13 - 6089.56i) q^{9} +O(q^{10})$	\(q+(-67.0937 + 45.3810i) q^{3} +(701.327 - 404.911i) q^{5} +(1952.36 - 1397.53i) q^{7} +(2442.13 - 6089.56i) q^{9} +(1114.61 + 643.521i) q^{11} -24570.2 q^{13} +(-28679.3 + 58993.9i) q^{15} +(35201.9 + 20323.8i) q^{17} +(-39959.6 - 69212.1i) q^{19} +(-67569.6 + 182366. i) q^{21} +(-293149. + 169249. i) q^{23} +(132594. - 229659. i) q^{25} +(112499. + 519397. i) q^{27} -1.09251e6i q^{29} +(521266. - 902859. i) q^{31} +(-103987. + 7405.99i) q^{33} +(803366. - 1.77066e6i) q^{35} +(-184949. - 320341. i) q^{37} +(1.64851e6 - 1.11502e6i) q^{39} -2.77625e6i q^{41} +4.17741e6 q^{43} +(-753002. - 5.25962e6i) q^{45} +(-7.07575e6 + 4.08519e6i) q^{47} +(1.85862e6 - 5.45697e6i) q^{49} +(-3.28414e6 + 233898. i) q^{51} +(-6.31437e6 - 3.64560e6i) q^{53} +1.04228e6 q^{55} +(5.82196e6 + 2.83029e6i) q^{57} +(1.54219e7 + 8.90382e6i) q^{59} +(-1.05761e7 - 1.83184e7i) q^{61} +(-3.74244e6 - 1.53020e7i) q^{63} +(-1.72318e7 + 9.94876e6i) q^{65} +(5.90653e6 - 1.02304e7i) q^{67} +(1.19877e7 - 2.46589e7i) q^{69} -2.35033e6i q^{71} +(1.95490e7 - 3.38599e7i) q^{73} +(1.52596e6 + 2.14259e7i) q^{75} +(3.07546e6 - 301319. i) q^{77} +(9.37450e6 + 1.62371e7i) q^{79} +(-3.11188e7 - 2.97430e7i) q^{81} -3.15449e7i q^{83} +3.29174e7 q^{85} +(4.95792e7 + 7.33005e7i) q^{87} +(-7.65369e6 + 4.41886e6i) q^{89} +(-4.79699e7 + 3.43377e7i) q^{91} +(5.99900e6 + 8.42317e7i) q^{93} +(-5.60495e7 - 3.23602e7i) q^{95} -1.17592e8 q^{97} +(6.64078e6 - 5.21593e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	$ 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) $ 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * 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
$3$	−67.0937	+	45.3810i	−0.828317	+	0.560259i
$3$	−67.0937	+	45.3810i	−0.828317	+	0.560259i
$4$		0			0
$5$	701.327	−	404.911i	1.12212	−	0.647858i	0.180181	−	0.983634i	$-0.442332\pi$
$5$	701.327	−	404.911i	1.12212	−	0.647858i	0.941942	+	0.335776i	$0.108998\pi$
$6$		0			0
$7$	1952.36	−	1397.53i	0.813144	−	0.582062i
$7$	1952.36	−	1397.53i	0.813144	−	0.582062i
$8$		0			0
$9$	2442.13	−	6089.56i	0.372219	−	0.928145i
$10$		0			0
$11$	1114.61	+	643.521i	0.0761295	+	0.0439534i	0.537582	−	0.843212i	$-0.319338\pi$
$11$	1114.61	+	643.521i	0.0761295	+	0.0439534i	−0.461452	+	0.887165i	$0.652671\pi$
$12$		0			0
$13$	−24570.2			−0.860272			−0.430136	−	0.902764i	$-0.641534\pi$
$13$	−24570.2			−0.860272			−0.430136	+	0.902764i	$0.641534\pi$
$14$		0			0
$15$	−28679.3	+	58993.9i	−0.566505	+	1.16531i
$16$		0			0
$17$	35201.9	+	20323.8i	0.421474	+	0.243338i	0.695708	−	0.718325i	$-0.255091\pi$
$17$	35201.9	+	20323.8i	0.421474	+	0.243338i	−0.274234	+	0.961663i	$0.588424\pi$
$18$		0			0
$19$	−39959.6	−	69212.1i	−0.306625	−	0.531090i	0.670997	−	0.741460i	$-0.265866\pi$
$19$	−39959.6	−	69212.1i	−0.306625	−	0.531090i	−0.977622	+	0.210370i	$0.932533\pi$
$20$		0			0
$21$	−67569.6	+	182366.i	−0.347436	+	0.937704i
$22$		0			0
$23$	−293149.	+	169249.i	−1.04755	+	0.604806i	−0.921964	−	0.387277i	$-0.873416\pi$
$23$	−293149.	+	169249.i	−1.04755	+	0.604806i	−0.125590	+	0.992082i	$0.540083\pi$
$24$		0			0
$25$	132594.	−	229659.i	0.339440	−	0.587927i
$26$		0			0
$27$	112499.	+	519397.i	0.211687	+	0.977338i
$28$		0			0
$29$		−	1.09251e6i		−	1.54466i	−0.635221	−	0.772331i	$-0.719091\pi$
$29$		−	1.09251e6i		−	1.54466i	0.635221	−	0.772331i	$-0.280909\pi$
$30$		0			0
$31$	521266.	−	902859.i	0.564433	−	0.977627i	−0.432669	−	0.901553i	$-0.642428\pi$
$31$	521266.	−	902859.i	0.564433	−	0.977627i	0.997102	−	0.0760739i	$-0.0242385\pi$
$32$		0			0
$33$	−103987.	+	7405.99i	−0.0876846	+	0.00624492i
$34$		0			0
$35$	803366.	−	1.77066e6i	0.535354	−	1.17995i
$36$		0			0
$37$	−184949.	−	320341.i	−0.0986837	−	0.170925i	0.812456	−	0.583022i	$-0.198130\pi$
$37$	−184949.	−	320341.i	−0.0986837	−	0.170925i	−0.911140	+	0.412097i	$0.864797\pi$
$38$		0			0
$39$	1.64851e6	−	1.11502e6i	0.712578	−	0.481976i
$40$		0			0
$41$		−	2.77625e6i		−	0.982480i	−0.871024	−	0.491240i	$-0.836544\pi$
$41$		−	2.77625e6i		−	0.982480i	0.871024	−	0.491240i	$-0.163456\pi$
$42$		0			0
$43$	4.17741e6			1.22189			0.610946	−	0.791672i	$-0.290789\pi$
$43$	4.17741e6			1.22189			0.610946	+	0.791672i	$0.290789\pi$
$44$		0			0
$45$	−753002.	−	5.25962e6i	−0.183631	−	1.28264i
$46$		0			0
$47$	−7.07575e6	+	4.08519e6i	−1.45004	+	0.837184i	−0.998483	−	0.0550566i	$-0.982466\pi$
$47$	−7.07575e6	+	4.08519e6i	−1.45004	+	0.837184i	−0.451561	+	0.892240i	$0.649133\pi$
$48$		0			0
$49$	1.85862e6	−	5.45697e6i	0.322408	−	0.946601i
$50$		0			0
$51$	−3.28414e6	+	233898.i	−0.485446	+	0.0345736i
$52$		0			0
$53$	−6.31437e6	−	3.64560e6i	−0.800251	−	0.462025i	0.0433076	−	0.999062i	$-0.486210\pi$
$53$	−6.31437e6	−	3.64560e6i	−0.800251	−	0.462025i	−0.843559	+	0.537036i	$0.819544\pi$
$54$		0			0
$55$	1.04228e6			0.113902
$56$		0			0
$57$	5.82196e6	+	2.83029e6i	0.551531	+	0.268121i
$58$		0			0
$59$	1.54219e7	+	8.90382e6i	1.27271	+	0.734799i	0.975497	−	0.220013i	$-0.0706098\pi$
$59$	1.54219e7	+	8.90382e6i	1.27271	+	0.734799i	0.297212	+	0.954812i	$0.403943\pi$
$60$		0			0
$61$	−1.05761e7	−	1.83184e7i	−0.763848	−	1.32302i	−0.940854	−	0.338813i	$-0.889974\pi$
$61$	−1.05761e7	−	1.83184e7i	−0.763848	−	1.32302i	0.177006	−	0.984210i	$-0.443359\pi$
$62$		0			0
$63$	−3.74244e6	−	1.53020e7i	−0.237570	−	0.971370i
$64$		0			0
$65$	−1.72318e7	+	9.94876e6i	−0.965331	+	0.557334i
$66$		0			0
$67$	5.90653e6	−	1.02304e7i	0.293112	−	0.507684i	−0.681432	−	0.731881i	$-0.738643\pi$
$67$	5.90653e6	−	1.02304e7i	0.293112	−	0.507684i	0.974544	+	0.224197i	$0.0719759\pi$
$68$		0			0
$69$	1.19877e7	−	2.46589e7i	0.528859	−	1.08787i
$70$		0			0
$71$		−	2.35033e6i		−	0.0924903i	−0.998930	−	0.0462451i	$-0.985274\pi$
$71$		−	2.35033e6i		−	0.0924903i	0.998930	−	0.0462451i	$-0.0147255\pi$
$72$		0			0
$73$	1.95490e7	−	3.38599e7i	0.688388	−	1.19232i	−0.283971	−	0.958833i	$-0.591652\pi$
$73$	1.95490e7	−	3.38599e7i	0.688388	−	1.19232i	0.972359	−	0.233490i	$-0.0750146\pi$
$74$		0			0
$75$	1.52596e6	+	2.14259e7i	0.0482278	+	0.677164i
$76$		0			0
$77$	3.07546e6	−	301319.i	0.0874878	−	0.00857164i
$78$		0			0
$79$	9.37450e6	+	1.62371e7i	0.240680	+	0.416870i	0.960908	−	0.276867i	$-0.0892962\pi$
$79$	9.37450e6	+	1.62371e7i	0.240680	+	0.416870i	−0.720228	+	0.693737i	$0.755963\pi$
$80$		0			0
$81$	−3.11188e7	−	2.97430e7i	−0.722906	−	0.690946i
$82$		0			0
$83$		−	3.15449e7i		−	0.664687i	−0.943158	−	0.332343i	$-0.892161\pi$
$83$		−	3.15449e7i		−	0.664687i	0.943158	−	0.332343i	$-0.107839\pi$
$84$		0			0
$85$	3.29174e7			0.630594
$86$		0			0
$87$	4.95792e7	+	7.33005e7i	0.865411	+	1.27947i
$88$		0			0
$89$	−7.65369e6	+	4.41886e6i	−0.121986	+	0.0704287i	−0.559751	−	0.828661i	$-0.689103\pi$
$89$	−7.65369e6	+	4.41886e6i	−0.121986	+	0.0704287i	0.437765	+	0.899089i	$0.355770\pi$
$90$		0			0
$91$	−4.79699e7	+	3.43377e7i	−0.699526	+	0.500732i
$92$		0			0
$93$	5.99900e6	+	8.42317e7i	0.0801950	+	1.12601i
$94$		0			0
$95$	−5.60495e7	−	3.23602e7i	−0.688141	−	0.397299i
$96$		0			0
$97$	−1.17592e8			−1.32828			−0.664141	−	0.747607i	$-0.731202\pi$
$97$	−1.17592e8			−1.32828			−0.664141	+	0.747607i	$0.731202\pi$
$98$		0			0
$99$	6.64078e6	−	5.21593e6i	0.0691319	−	0.0542989i
$100$		0			0
$101$	1.32453e8	+	7.64720e7i	1.27285	+	0.734880i	0.975523	−	0.219895i	$-0.0705716\pi$
$101$	1.32453e8	+	7.64720e7i	1.27285	+	0.734880i	0.297327	+	0.954776i	$0.403905\pi$
$102$		0			0
$103$	−3.91745e7	−	6.78523e7i	−0.348061	−	0.602859i	0.637844	−	0.770165i	$-0.279826\pi$
$103$	−3.91745e7	−	6.78523e7i	−0.348061	−	0.602859i	−0.985905	+	0.167306i	$0.946493\pi$
$104$		0			0
$105$	2.64535e7	+	1.55258e8i	0.217633	+	1.27731i
$106$		0			0
$107$	8.92504e7	−	5.15288e7i	0.680887	−	0.393110i	−0.119302	−	0.992858i	$-0.538066\pi$
$107$	8.92504e7	−	5.15288e7i	0.680887	−	0.393110i	0.800189	+	0.599748i	$0.204732\pi$
$108$		0			0
$109$	−5.43401e7	+	9.41198e7i	−0.384959	+	0.666768i	−0.991763	−	0.128083i	$-0.959118\pi$
$109$	−5.43401e7	+	9.41198e7i	−0.384959	+	0.666768i	0.606805	+	0.794851i	$0.292451\pi$
$110$		0			0
$111$	2.69463e7	+	1.30997e7i	0.177504	+	0.0862918i
$112$		0			0
$113$		−	2.58454e7i		−	0.158514i	−0.996854	−	0.0792572i	$-0.974745\pi$
$113$		−	2.58454e7i		−	0.158514i	0.996854	−	0.0792572i	$-0.0252548\pi$
$114$		0			0
$115$	−1.37062e8	+	2.37398e8i	−0.783656	+	1.35733i
$116$		0			0
$117$	−6.00036e7	+	1.49622e8i	−0.320209	+	0.798458i
$118$		0			0
$119$	9.71300e7	−	9.51633e6i	0.484357	−	0.0474550i
$120$		0			0
$121$	−1.06351e8	−	1.84206e8i	−0.496136	−	0.859333i
$122$		0			0
$123$	1.25989e8	+	1.86269e8i	0.550444	+	0.813805i
$124$		0			0
$125$			1.01582e8i			0.416081i
$126$		0			0
$127$	8.21235e7			0.315684			0.157842	−	0.987464i	$-0.449546\pi$
$127$	8.21235e7			0.315684			0.157842	+	0.987464i	$0.449546\pi$
$128$		0			0
$129$	−2.80278e8	+	1.89575e8i	−1.01211	+	0.684577i
$130$		0			0
$131$	−3.76294e8	+	2.17254e8i	−1.27774	+	0.737703i	−0.976432	−	0.215824i	$-0.930756\pi$
$131$	−3.76294e8	+	2.17254e8i	−1.27774	+	0.737703i	−0.301307	+	0.953527i	$0.597423\pi$
$132$		0			0
$133$	−1.74742e8	−	7.92821e7i	−0.558457	−	0.253378i
$134$		0			0
$135$	2.89208e8	+	3.18715e8i	0.870714	+	0.959550i
$136$		0			0
$137$	7.54413e7	+	4.35561e7i	0.214154	+	0.123642i	0.603241	−	0.797559i	$-0.293876\pi$
$137$	7.54413e7	+	4.35561e7i	0.214154	+	0.123642i	−0.389086	+	0.921201i	$0.627209\pi$
$138$		0			0
$139$	2.42039e8			0.648376			0.324188	−	0.945993i	$-0.394909\pi$
$139$	2.42039e8			0.648376			0.324188	+	0.945993i	$0.394909\pi$
$140$		0			0
$141$	2.89348e8	−	5.95195e8i	0.732057	−	1.50585i
$142$		0			0
$143$	−2.73863e7	−	1.58115e7i	−0.0654921	−	0.0378119i
$144$		0			0
$145$	−4.42369e8	−	7.66206e8i	−1.00072	−	1.73330i
$146$		0			0
$147$	1.22941e8	+	4.50474e8i	0.263286	+	0.964718i
$148$		0			0
$149$	5.16102e8	−	2.97971e8i	1.04711	−	0.604546i	0.125268	−	0.992123i	$-0.460021\pi$
$149$	5.16102e8	−	2.97971e8i	1.04711	−	0.604546i	0.921838	+	0.387577i	$0.126688\pi$
$150$		0			0
$151$	3.63368e8	−	6.29373e8i	0.698939	−	1.21060i	−0.269895	−	0.962890i	$-0.586989\pi$
$151$	3.63368e8	−	6.29373e8i	0.698939	−	1.21060i	0.968835	−	0.247709i	$-0.0796776\pi$
$152$		0			0
$153$	2.09731e8	−	1.64731e8i	0.382733	−	0.300614i
$154$		0			0
$155$		−	8.44265e8i		−	1.46269i
$156$		0			0
$157$	2.30584e8	−	3.99382e8i	0.379516	−	0.657340i	−0.611476	−	0.791263i	$-0.709424\pi$
$157$	2.30584e8	−	3.99382e8i	0.379516	−	0.657340i	0.990992	+	0.133922i	$0.0427573\pi$
$158$		0			0
$159$	5.89095e8	−	4.19555e7i	0.921716	−	0.0656449i
$160$		0			0
$161$	−3.35800e8	+	7.40120e8i	−0.499778	+	1.10154i
$162$		0			0
$163$	−3.48066e8	−	6.02868e8i	−0.493073	−	0.854028i	0.506895	−	0.862008i	$-0.330793\pi$
$163$	−3.48066e8	−	6.02868e8i	−0.493073	−	0.854028i	−0.999968	+	0.00798005i	$0.997460\pi$
$164$		0			0
$165$	−6.99301e7	+	4.72995e7i	−0.0943471	+	0.0638148i
$166$		0			0
$167$			1.02172e9i			1.31361i	0.754060	+	0.656805i	$0.228093\pi$
$167$			1.02172e9i			1.31361i	−0.754060	+	0.656805i	$0.771907\pi$
$168$		0			0
$169$	−2.12034e8			−0.259931
$170$		0			0
$171$	−5.19058e8	+	7.43118e7i	−0.607060	+	0.0869108i
$172$		0			0
$173$	−5.61654e8	+	3.24271e8i	−0.627024	+	0.362013i	−0.779599	−	0.626279i	$-0.784577\pi$
$173$	−5.61654e8	+	3.24271e8i	−0.627024	+	0.362013i	0.152575	+	0.988292i	$0.451244\pi$
$174$		0			0
$175$	−6.20850e7	−	6.33680e8i	−0.0661964	−	0.675644i
$176$		0			0
$177$	−1.43878e9	+	1.02470e8i	−1.46588	+	0.104401i
$178$		0			0
$179$	−1.17623e9	−	6.79096e8i	−1.14572	−	0.661483i	−0.197882	−	0.980226i	$-0.563406\pi$
$179$	−1.17623e9	−	6.79096e8i	−1.14572	−	0.661483i	−0.947841	+	0.318743i	$0.896739\pi$
$180$		0			0
$181$	−4.87644e8			−0.454348			−0.227174	−	0.973854i	$-0.572949\pi$
$181$	−4.87644e8			−0.454348			−0.227174	+	0.973854i	$0.572949\pi$
$182$		0			0
$183$	1.54090e9	+	7.49092e8i	1.37394	+	0.667930i
$184$		0			0
$185$	−2.59420e8	−	1.49776e8i	−0.221471	−	0.127866i
$186$		0			0
$187$	2.61576e7	+	4.53064e7i	0.0213911	+	0.0370504i
$188$		0			0
$189$	9.45512e8	+	8.56829e8i	0.741003	+	0.671502i
$190$		0			0
$191$	1.52710e9	−	8.81672e8i	1.14745	−	0.662481i	0.199186	−	0.979962i	$-0.436170\pi$
$191$	1.52710e9	−	8.81672e8i	1.14745	−	0.662481i	0.948265	+	0.317481i	$0.102837\pi$
$192$		0			0
$193$	1.13004e9	−	1.95729e9i	0.814450	−	1.41067i	−0.0952722	−	0.995451i	$-0.530372\pi$
$193$	1.13004e9	−	1.95729e9i	0.814450	−	1.41067i	0.909722	−	0.415218i	$-0.136295\pi$
$194$		0			0
$195$	7.04658e8	−	1.44949e9i	0.487349	−	1.00249i
$196$		0			0
$197$			4.26915e7i			0.0283450i	0.999900	+	0.0141725i	$0.00451140\pi$
$197$			4.26915e7i			0.0283450i	−0.999900	+	0.0141725i	$0.995489\pi$
$198$		0			0
$199$	−4.76158e8	+	8.24730e8i	−0.303626	+	0.525895i	−0.976954	−	0.213448i	$-0.931531\pi$
$199$	−4.76158e8	+	8.24730e8i	−0.303626	+	0.525895i	0.673329	+	0.739343i	$0.264864\pi$
$200$		0			0
$201$	6.79755e7	+	9.54440e8i	0.0416455	+	0.584742i
$202$		0			0
$203$	−1.52682e9	−	2.13297e9i	−0.899089	−	1.25603i
$204$		0			0
$205$	−1.12414e9	−	1.94706e9i	−0.636507	−	1.10246i
$206$		0			0
$207$	3.14748e8	+	2.19847e9i	0.171428	+	1.19740i
$208$		0			0
$209$		−	1.02860e8i		−	0.0539088i
$210$		0			0
$211$	1.45001e9			0.731547			0.365774	−	0.930704i	$-0.380804\pi$
$211$	1.45001e9			0.731547			0.365774	+	0.930704i	$0.380804\pi$
$212$		0			0
$213$	1.06660e8	+	1.57693e8i	0.0518185	+	0.0766113i
$214$		0			0
$215$	2.92973e9	−	1.69148e9i	1.37111	−	0.791613i
$216$		0			0
$217$	−2.44075e8	−	2.49119e9i	−0.110074	−	1.12349i
$218$		0			0
$219$	2.24980e8	+	3.15894e9i	0.0978066	+	1.37330i
$220$		0			0
$221$	−8.64919e8	−	4.99361e8i	−0.362582	−	0.209337i
$222$		0			0
$223$	−1.59542e9			−0.645143			−0.322572	−	0.946545i	$-0.604547\pi$
$223$	−1.59542e9			−0.645143			−0.322572	+	0.946545i	$0.604547\pi$
$224$		0			0
$225$	−1.07471e9	−	1.36829e9i	−0.419335	−	0.533886i
$226$		0			0
$227$	3.60754e9	+	2.08281e9i	1.35865	+	0.784417i	0.989442	−	0.144930i	$-0.0462956\pi$
$227$	3.60754e9	+	2.08281e9i	1.35865	+	0.784417i	0.369208	+	0.929347i	$0.379629\pi$
$228$		0			0
$229$	9.61610e8	+	1.66556e9i	0.349669	+	0.605644i	0.986191	−	0.165615i	$-0.0529608\pi$
$229$	9.61610e8	+	1.66556e9i	0.349669	+	0.605644i	−0.636522	+	0.771259i	$0.719627\pi$
$230$		0			0
$231$	−1.92670e8	+	1.59784e8i	−0.0676653	+	0.0561159i
$232$		0			0
$233$	−3.38848e9	+	1.95634e9i	−1.14969	+	0.663774i	−0.948812	−	0.315842i	$-0.897713\pi$
$233$	−3.38848e9	+	1.95634e9i	−1.14969	+	0.663774i	−0.200878	+	0.979616i	$0.564380\pi$
$234$		0			0
$235$	−3.30828e9	+	5.73010e9i	−1.08475	+	1.87885i
$236$		0			0
$237$	−1.36583e9	−	6.63984e8i	−0.432915	−	0.210457i
$238$		0			0
$239$			4.78953e9i			1.46792i	0.679194	+	0.733959i	$0.262330\pi$
$239$			4.78953e9i			1.46792i	−0.679194	+	0.733959i	$0.737670\pi$
$240$		0			0
$241$	1.04556e8	−	1.81097e8i	0.0309943	−	0.0536836i	−0.850112	−	0.526602i	$-0.823466\pi$
$241$	1.04556e8	−	1.81097e8i	0.0309943	−	0.0536836i	0.881106	+	0.472918i	$0.156799\pi$
$242$		0			0
$243$	3.43764e9	+	5.83364e8i	0.985905	+	0.167307i
$244$		0			0
$245$	−9.06090e8	−	4.57969e9i	−0.251482	−	1.27108i
$246$		0			0
$247$	9.81818e8	+	1.70056e9i	0.263781	+	0.456882i
$248$		0			0
$249$	1.43154e9	+	2.11646e9i	0.372397	+	0.550571i
$250$		0			0
$251$			4.46779e9i			1.12563i	0.826581	+	0.562817i	$0.190283\pi$
$251$			4.46779e9i			1.12563i	−0.826581	+	0.562817i	$0.809717\pi$
$252$		0			0
$253$	−4.35662e8			−0.106333
$254$		0			0
$255$	−2.20855e9	+	1.49382e9i	−0.522332	+	0.353296i
$256$		0			0
$257$	−3.03909e9	+	1.75462e9i	−0.696644	+	0.402208i	−0.806096	−	0.591784i	$-0.798424\pi$
$257$	−3.03909e9	+	1.75462e9i	−0.696644	+	0.402208i	0.109452	+	0.993992i	$0.465090\pi$
$258$		0			0
$259$	−8.08774e8	−	3.66949e8i	−0.179733	−	0.0815468i
$260$		0			0
$261$	−6.65290e9	−	2.66805e9i	−1.43367	−	0.574952i
$262$		0			0
$263$	6.00484e9	+	3.46689e9i	1.25510	+	0.724632i	0.972118	−	0.234493i	$-0.0753431\pi$
$263$	6.00484e9	+	3.46689e9i	1.25510	+	0.724632i	0.282982	+	0.959125i	$0.408676\pi$
$264$		0			0
$265$	−5.90458e9			−1.19731
$266$		0			0
$267$	3.12982e8	−	6.43810e8i	0.0615849	−	0.126681i
$268$		0			0
$269$	5.76159e9	+	3.32645e9i	1.10036	+	0.635291i	0.936314	−	0.351163i	$-0.114214\pi$
$269$	5.76159e9	+	3.32645e9i	1.10036	+	0.635291i	0.164041	+	0.986453i	$0.447547\pi$
$270$		0			0
$271$	3.62650e9	+	6.28128e9i	0.672373	+	1.16458i	0.977229	+	0.212186i	$0.0680584\pi$
$271$	3.62650e9	+	6.28128e9i	0.672373	+	1.16458i	−0.304856	+	0.952399i	$0.598608\pi$
$272$		0			0
$273$	1.66020e9	−	4.48077e9i	0.298889	−	0.806681i
$274$		0			0
$275$	2.95581e8	−	1.70654e8i	0.0516827	−	0.0298390i
$276$		0			0
$277$	−2.76342e9	+	4.78638e9i	−0.469383	+	0.812995i	−0.999387	−	0.0349995i	$-0.988857\pi$
$277$	−2.76342e9	+	4.78638e9i	−0.469383	+	0.812995i	0.530004	+	0.847995i	$0.322190\pi$
$278$		0			0
$279$	−4.22501e9	−	5.37917e9i	−0.697287	−	0.887767i
$280$		0			0
$281$		−	5.02314e9i		−	0.805656i	−0.915276	−	0.402828i	$-0.868027\pi$
$281$		−	5.02314e9i		−	0.805656i	0.915276	−	0.402828i	$-0.131973\pi$
$282$		0			0
$283$	−3.01214e9	+	5.21718e9i	−0.469601	+	0.813373i	−0.999396	−	0.0347527i	$-0.988936\pi$
$283$	−3.01214e9	+	5.21718e9i	−0.469601	+	0.813373i	0.529795	+	0.848126i	$0.322269\pi$
$284$		0			0
$285$	5.22911e9	−	3.72419e8i	0.792589	−	0.0564484i
$286$		0			0
$287$	−3.87990e9	−	5.42025e9i	−0.571864	−	0.798898i
$288$		0			0
$289$	−2.66176e9	−	4.61031e9i	−0.381573	−	0.660904i
$290$		0			0
$291$	7.88967e9	−	5.33644e9i	1.10024	−	0.744182i
$292$		0			0
$293$			5.65732e9i			0.767609i	0.923414	+	0.383805i	$0.125386\pi$
$293$			5.65732e9i			0.767609i	−0.923414	+	0.383805i	$0.874614\pi$
$294$		0			0
$295$	1.44210e10			1.90418
$296$		0			0
$297$	−2.08850e8	+	6.51322e8i	−0.0268417	+	0.0837085i
$298$		0			0
$299$	7.20273e9	−	4.15850e9i	0.901182	−	0.520298i
$300$		0			0
$301$	8.15580e9	−	5.83806e9i	0.993575	−	0.711217i
$302$		0			0
$303$	−1.23572e10	+	8.80080e8i	−1.46605	+	0.104412i
$304$		0			0
$305$	−1.48346e10	−	8.56477e9i	−1.71426	−	0.989730i
$306$		0			0
$307$	8.08970e9			0.910708			0.455354	−	0.890311i	$-0.349513\pi$
$307$	8.08970e9			0.910708			0.455354	+	0.890311i	$0.349513\pi$
$308$		0			0
$309$	5.70757e9	+	2.77468e9i	0.626062	+	0.304354i
$310$		0			0
$311$	−1.38692e10	−	8.00737e9i	−1.48255	−	0.855949i	−0.482744	−	0.875761i	$-0.660360\pi$
$311$	−1.38692e10	−	8.00737e9i	−1.48255	−	0.855949i	−0.999804	+	0.0198119i	$0.993693\pi$
$312$		0			0
$313$	4.58737e9	+	7.94555e9i	0.477954	+	0.827841i	0.999681	−	0.0252721i	$-0.00804520\pi$
$313$	4.58737e9	+	7.94555e9i	0.477954	+	0.827841i	−0.521727	+	0.853113i	$0.674712\pi$
$314$		0			0
$315$	−8.82061e9	−	9.21632e9i	−0.895893	−	0.936085i
$316$		0			0
$317$	1.21210e10	−	6.99804e9i	1.20033	−	0.693010i	0.239700	−	0.970847i	$-0.422951\pi$
$317$	1.21210e10	−	6.99804e9i	1.20033	−	0.693010i	0.960628	+	0.277837i	$0.0896177\pi$
$318$		0			0
$319$	7.03053e8	−	1.21772e9i	0.0678931	−	0.117594i
$320$		0			0
$321$	−3.64971e9	+	7.50753e9i	−0.343747	+	0.707094i
$322$		0			0
$323$		−	3.24853e9i		−	0.298454i
$324$		0			0
$325$	−3.25786e9	+	5.64277e9i	−0.292011	+	0.505777i
$326$		0			0
$327$	−6.25374e8	−	8.78085e9i	−0.0546952	−	0.767972i
$328$		0			0
$329$	−8.10524e9	+	1.78643e10i	−0.691803	+	1.52477i
$330$		0			0
$331$	−5.48598e9	−	9.50200e9i	−0.457027	−	0.791595i	0.541775	−	0.840524i	$-0.317753\pi$
$331$	−5.48598e9	−	9.50200e9i	−0.457027	−	0.791595i	−0.998802	+	0.0489289i	$0.984419\pi$
$332$		0			0
$333$	−2.40241e9	+	3.43945e8i	−0.195375	+	0.0279712i
$334$		0			0
$335$		−	9.56648e9i		−	0.759579i
$336$		0			0
$337$	1.94045e10			1.50447			0.752233	−	0.658897i	$-0.228977\pi$
$337$	1.94045e10			1.50447			0.752233	+	0.658897i	$0.228977\pi$
$338$		0			0
$339$	1.17289e9	+	1.73406e9i	0.0888092	+	0.131300i
$340$		0			0
$341$	1.16202e9	−	6.70891e8i	0.0859400	−	0.0496175i
$342$		0			0
$343$	−3.99759e9	−	1.32514e10i	−0.288817	−	0.957384i
$344$		0			0
$345$	−1.57738e9	−	2.21479e10i	−0.111342	−	1.56335i
$346$		0			0
$347$	1.31408e10	+	7.58685e9i	0.906367	+	0.523291i	0.879261	−	0.476341i	$-0.158037\pi$
$347$	1.31408e10	+	7.58685e9i	0.906367	+	0.523291i	0.0271067	+	0.999633i	$0.491371\pi$
$348$		0			0
$349$	−1.72031e9			−0.115959			−0.0579797	−	0.998318i	$-0.518466\pi$
$349$	−1.72031e9			−0.115959			−0.0579797	+	0.998318i	$0.518466\pi$
$350$		0			0
$351$	−2.76413e9	−	1.27617e10i	−0.182108	−	0.840776i
$352$		0			0
$353$	1.34031e10	+	7.73830e9i	0.863192	+	0.498364i	0.865080	−	0.501634i	$-0.167268\pi$
$353$	1.34031e10	+	7.73830e9i	0.863192	+	0.498364i	−0.00188763	+	0.999998i	$0.500601\pi$
$354$		0			0
$355$	−9.51676e8	−	1.64835e9i	−0.0599205	−	0.103785i
$356$		0			0
$357$	−6.08495e9	+	5.04634e9i	−0.374614	+	0.310673i
$358$		0			0
$359$	−1.65322e10	+	9.54488e9i	−0.995298	+	0.574636i	−0.906854	−	0.421445i	$-0.861523\pi$
$359$	−1.65322e10	+	9.54488e9i	−0.995298	+	0.574636i	−0.0884445	+	0.996081i	$0.528190\pi$
$360$		0			0
$361$	5.29823e9	−	9.17681e9i	0.311963	−	0.540335i
$362$		0			0
$363$	1.54949e10	+	7.53271e9i	0.892408	+	0.433835i
$364$		0			0
$365$		−	3.16624e10i		−	1.78391i
$366$		0			0
$367$	−1.67235e10	+	2.89660e10i	−0.921856	+	1.59670i	−0.125316	+	0.992117i	$0.539994\pi$
$367$	−1.67235e10	+	2.89660e10i	−0.921856	+	1.59670i	−0.796540	+	0.604585i	$0.793339\pi$
$368$		0			0
$369$	−1.69062e10	−	6.77996e9i	−0.911884	−	0.365697i
$370$		0			0
$371$	−1.74228e10	+	1.70700e9i	−0.919647	+	0.0901027i
$372$		0			0
$373$	7.61432e9	+	1.31884e10i	0.393365	+	0.681328i	0.992891	−	0.119027i	$-0.0379776\pi$
$373$	7.61432e9	+	1.31884e10i	0.393365	+	0.681328i	−0.599526	+	0.800355i	$0.704644\pi$
$374$		0			0
$375$	−4.60991e9	−	6.81553e9i	−0.233113	−	0.344647i
$376$		0			0
$377$			2.68432e10i			1.32883i
$378$		0			0
$379$	2.26969e10			1.10004			0.550021	−	0.835151i	$-0.314620\pi$
$379$	2.26969e10			1.10004			0.550021	+	0.835151i	$0.314620\pi$
$380$		0			0
$381$	−5.50997e9	+	3.72685e9i	−0.261487	+	0.176865i
$382$		0			0
$383$	−8.25381e9	+	4.76534e9i	−0.383583	+	0.221462i	−0.679376	−	0.733790i	$-0.737749\pi$
$383$	−8.25381e9	+	4.76534e9i	−0.383583	+	0.221462i	0.295793	+	0.955252i	$0.404416\pi$
$384$		0			0
$385$	2.03490e9	−	1.45661e9i	0.0926189	−	0.0662981i
$386$		0			0
$387$	1.02018e10	−	2.54386e10i	0.454811	−	1.13409i
$388$		0			0
$389$	7.01606e9	+	4.05072e9i	0.306404	+	0.176903i	0.645316	−	0.763915i	$-0.276726\pi$
$389$	7.01606e9	+	4.05072e9i	0.306404	+	0.176903i	−0.338912	+	0.940818i	$0.610059\pi$
$390$		0			0
$391$	−1.37592e10			−0.588689
$392$		0			0
$393$	1.53878e10	−	3.16530e10i	0.645068	−	1.32692i
$394$		0			0
$395$	1.31492e10	+	7.59168e9i	0.540145	+	0.311853i
$396$		0			0
$397$	2.37965e10	+	4.12167e10i	0.957967	+	1.65925i	0.727428	+	0.686184i	$0.240716\pi$
$397$	2.37965e10	+	4.12167e10i	0.957967	+	1.65925i	0.230539	+	0.973063i	$0.425951\pi$
$398$		0			0
$399$	1.53220e10	−	2.61062e9i	0.604537	−	0.103004i
$400$		0			0
$401$	−1.59819e10	+	9.22713e9i	−0.618087	+	0.356853i	−0.776124	−	0.630580i	$-0.782817\pi$
$401$	−1.59819e10	+	9.22713e9i	−0.618087	+	0.356853i	0.158037	+	0.987433i	$0.449484\pi$
$402$		0			0
$403$	−1.28076e10	+	2.21835e10i	−0.485566	+	0.841025i
$404$		0			0
$405$	−3.38677e10	−	8.25920e9i	−1.25882	−	0.306985i
$406$		0			0
$407$		−	4.76075e8i		−	0.0173499i
$408$		0			0
$409$	−2.50498e10	+	4.33876e10i	−0.895183	+	1.55050i	−0.0616053	+	0.998101i	$0.519622\pi$
$409$	−2.50498e10	+	4.33876e10i	−0.895183	+	1.55050i	−0.833578	+	0.552402i	$0.813711\pi$
$410$		0			0
$411$	−7.03825e9	+	5.01266e8i	−0.246659	+	0.0175671i
$412$		0			0
$413$	4.25524e10	−	4.16908e9i	1.46259	−	0.143298i
$414$		0			0
$415$	−1.27729e10	−	2.21233e10i	−0.430622	−	0.745860i
$416$		0			0
$417$	−1.62393e10	+	1.09840e10i	−0.537061	+	0.363259i
$418$		0			0
$419$		−	2.56571e10i		−	0.832438i	−0.909264	−	0.416219i	$-0.863355\pi$
$419$		−	2.56571e10i		−	0.832438i	0.909264	−	0.416219i	$-0.136645\pi$
$420$		0			0
$421$	6.01551e10			1.91489			0.957446	−	0.288614i	$-0.0931943\pi$
$421$	6.01551e10			1.91489			0.957446	+	0.288614i	$0.0931943\pi$
$422$		0			0
$423$	7.59711e9	+	5.30648e10i	0.237294	+	1.65747i
$424$		0			0
$425$	9.33510e9	−	5.38962e9i	0.286130	−	0.165197i
$426$		0			0
$427$	−4.62489e10	−	2.09836e10i	−1.39120	−	0.631202i
$428$		0			0
$429$	2.55499e9	−	1.81967e8i	0.0754327	−	0.00537234i
$430$		0			0
$431$	1.43030e10	+	8.25785e9i	0.414494	+	0.239308i	0.692719	−	0.721208i	$-0.256413\pi$
$431$	1.43030e10	+	8.25785e9i	0.414494	+	0.239308i	−0.278225	+	0.960516i	$0.589746\pi$
$432$		0			0
$433$	−2.28331e10			−0.649550			−0.324775	−	0.945791i	$-0.605289\pi$
$433$	−2.28331e10			−0.649550			−0.324775	+	0.945791i	$0.605289\pi$
$434$		0			0
$435$	6.44514e10	+	3.13324e10i	1.80001	+	0.875058i
$436$		0			0
$437$	2.34282e10	+	1.35263e10i	0.642412	+	0.370897i
$438$		0			0
$439$	−1.97887e10	−	3.42751e10i	−0.532794	−	0.922827i	−0.999267	−	0.0382911i	$-0.987809\pi$
$439$	−1.97887e10	−	3.42751e10i	−0.532794	−	0.922827i	0.466472	−	0.884536i	$-0.345525\pi$
$440$		0			0
$441$	−2.86915e10	−	2.46448e10i	−0.758577	−	0.651584i
$442$		0			0
$443$	−1.80223e10	+	1.04052e10i	−0.467945	+	0.270168i	−0.715379	−	0.698736i	$-0.753746\pi$
$443$	−1.80223e10	+	1.04052e10i	−0.467945	+	0.270168i	0.247434	+	0.968905i	$0.420413\pi$
$444$		0			0
$445$	−3.57849e9	+	6.19813e9i	−0.0912556	+	0.158059i
$446$		0			0
$447$	−2.11049e10	+	4.34132e10i	−0.528632	+	1.08741i
$448$		0			0
$449$		−	5.22570e10i		−	1.28576i	−0.765968	−	0.642878i	$-0.777740\pi$
$449$		−	5.22570e10i		−	1.28576i	0.765968	−	0.642878i	$-0.222260\pi$
$450$		0			0
$451$	1.78658e9	−	3.09444e9i	0.0431833	−	0.0747957i
$452$		0			0
$453$	4.18184e9	+	5.87170e10i	0.0993058	+	1.39435i
$454$		0			0
$455$	−1.97389e10	+	4.35055e10i	−0.460551	+	1.01508i
$456$		0			0
$457$	2.38191e9	+	4.12560e9i	0.0546086	+	0.0945849i	0.892037	−	0.451961i	$-0.149276\pi$
$457$	2.38191e9	+	4.12560e9i	0.0546086	+	0.0945849i	−0.837429	+	0.546546i	$0.815942\pi$
$458$		0			0
$459$	−6.59596e9	+	2.05702e10i	−0.148603	+	0.463434i
$460$		0			0
$461$		−	3.30654e10i		−	0.732099i	−0.930595	−	0.366050i	$-0.880710\pi$
$461$		−	3.30654e10i		−	0.732099i	0.930595	−	0.366050i	$-0.119290\pi$
$462$		0			0
$463$	−4.07220e10			−0.886147			−0.443073	−	0.896485i	$-0.646112\pi$
$463$	−4.07220e10			−0.886147			−0.443073	+	0.896485i	$0.646112\pi$
$464$		0			0
$465$	3.83136e10	+	5.66449e10i	0.819486	+	1.21157i
$466$		0			0
$467$	−1.85131e10	+	1.06885e10i	−0.389234	+	0.224724i	−0.681828	−	0.731512i	$-0.738815\pi$
$467$	−1.85131e10	+	1.06885e10i	−0.389234	+	0.224724i	0.292594	+	0.956237i	$0.405482\pi$
$468$		0			0
$469$	−2.76564e9	−	2.82280e10i	−0.0571617	−	0.583430i
$470$		0			0
$471$	2.65368e9	+	3.72602e10i	0.0539218	+	0.757114i
$472$		0			0
$473$	4.65619e9	+	2.68825e9i	0.0930221	+	0.0537063i
$474$		0			0
$475$	−2.11936e10			−0.416322
$476$		0			0
$477$	−3.76206e10	+	2.95487e10i	−0.726695	+	0.570775i
$478$		0			0
$479$	4.15918e10	+	2.40130e10i	0.790069	+	0.456147i	0.839987	−	0.542607i	$-0.182563\pi$
$479$	4.15918e10	+	2.40130e10i	0.790069	+	0.456147i	−0.0499177	+	0.998753i	$0.515896\pi$
$480$		0			0
$481$	4.54425e9	+	7.87087e9i	0.0848949	+	0.147042i
$482$		0			0
$483$	−1.10573e10	−	6.48963e10i	−0.203171	−	1.19243i
$484$		0			0
$485$	−8.24703e10	+	4.76142e10i	−1.49050	+	0.860538i
$486$		0			0
$487$	−3.00358e9	+	5.20235e9i	−0.0533977	+	0.0924876i	−0.891489	−	0.453043i	$-0.850338\pi$
$487$	−3.00358e9	+	5.20235e9i	−0.0533977	+	0.0924876i	0.838091	+	0.545530i	$0.183672\pi$
$488$		0			0
$489$	5.07118e10	+	2.46531e10i	0.886898	+	0.431157i
$490$		0			0
$491$		−	1.85500e10i		−	0.319167i	−0.987184	−	0.159583i	$-0.948985\pi$
$491$		−	1.85500e10i		−	0.319167i	0.987184	−	0.159583i	$-0.0510151\pi$
$492$		0			0
$493$	2.22040e10	−	3.84584e10i	0.375875	−	0.651034i
$494$		0			0
$495$	2.54537e9	−	6.34700e9i	0.0423965	−	0.105718i
$496$		0			0
$497$	−3.28466e9	−	4.58869e9i	−0.0538351	−	0.0752079i
$498$		0			0
$499$	−2.61025e10	−	4.52108e10i	−0.420998	−	0.729189i	0.575040	−	0.818126i	$-0.304987\pi$
$499$	−2.61025e10	−	4.52108e10i	−0.420998	−	0.729189i	−0.996037	+	0.0889361i	$0.971653\pi$
$500$		0			0
$501$	−4.63668e10	−	6.85511e10i	−0.735963	−	1.08809i
$502$		0			0
$503$		−	8.21977e10i		−	1.28407i	−0.766676	−	0.642034i	$-0.778091\pi$
$503$		−	8.21977e10i		−	1.28407i	0.766676	−	0.642034i	$-0.221909\pi$
$504$		0			0
$505$	1.23857e11			1.90439
$506$		0			0
$507$	1.42261e10	−	9.62232e9i	0.215306	−	0.145629i
$508$		0			0
$509$	8.85830e10	−	5.11434e10i	1.31971	−	0.761937i	0.336031	−	0.941851i	$-0.390915\pi$
$509$	8.85830e10	−	5.11434e10i	1.31971	−	0.761937i	0.983682	+	0.179915i	$0.0575821\pi$
$510$		0			0
$511$	−9.15353e9	−	9.34270e10i	−0.134247	−	1.37022i
$512$		0			0
$513$	3.14532e10	−	2.85412e10i	0.454145	−	0.412101i
$514$		0			0
$515$	−5.49483e10	−	3.17244e10i	−0.781134	−	0.450988i
$516$		0			0
$517$	−1.05156e10			−0.147188
$518$		0			0
$519$	2.29677e10	−	4.72450e10i	0.316554	−	0.651157i
$520$		0			0
$521$	−4.31302e10	−	2.49012e10i	−0.585369	−	0.337963i	0.177895	−	0.984049i	$-0.443071\pi$
$521$	−4.31302e10	−	2.49012e10i	−0.585369	−	0.337963i	−0.763264	+	0.646086i	$0.776405\pi$
$522$		0			0
$523$	−1.18682e10	−	2.05563e10i	−0.158627	−	0.274750i	0.775747	−	0.631044i	$-0.217373\pi$
$523$	−1.18682e10	−	2.05563e10i	−0.158627	−	0.274750i	−0.934374	+	0.356294i	$0.884040\pi$
$524$		0			0
$525$	3.29226e10	+	3.96985e10i	0.433368	+	0.522561i
$526$		0			0
$527$	3.66991e10	−	2.11882e10i	0.475787	−	0.274696i
$528$		0			0
$529$	1.81352e10	−	3.14111e10i	0.231580	−	0.401107i
$530$		0			0
$531$	9.18825e10	−	7.21682e10i	1.15573	−	0.907753i
$532$		0			0
$533$			6.82132e10i			0.845200i
$534$		0			0
$535$	4.17291e10	−	7.22770e10i	0.509359	−	0.882236i
$536$		0			0
$537$	1.09736e11	−	7.81540e9i	1.31962	−	0.0939840i
$538$		0			0
$539$	5.58331e9	−	4.88634e9i	0.0661510	−	0.0578933i
$540$		0			0
$541$	−6.17113e10	−	1.06887e11i	−0.720403	−	1.24777i	−0.960838	−	0.277110i	$-0.910624\pi$
$541$	−6.17113e10	−	1.06887e11i	−0.720403	−	1.24777i	0.240435	−	0.970665i	$-0.422710\pi$
$542$		0			0
$543$	3.27178e10	−	2.21298e10i	0.376344	−	0.254553i
$544$		0			0
$545$			8.80116e10i			0.997594i
$546$		0			0
$547$	−2.02466e10			−0.226153			−0.113077	−	0.993586i	$-0.536071\pi$
$547$	−2.02466e10			−0.226153			−0.113077	+	0.993586i	$0.536071\pi$
$548$		0			0
$549$	−1.37379e11	+	1.96681e10i	−1.51228	+	0.216508i
$550$		0			0
$551$	−7.56149e10	+	4.36563e10i	−0.820354	+	0.473631i
$552$		0			0
$553$	4.09943e10	+	1.85995e10i	0.438352	+	0.198885i
$554$		0			0
$555$	2.42024e10	−	1.72370e9i	0.255086	−	0.0181673i
$556$		0			0
$557$	8.16565e10	+	4.71444e10i	0.848340	+	0.489789i	0.860090	−	0.510142i	$-0.170407\pi$
$557$	8.16565e10	+	4.71444e10i	0.848340	+	0.489789i	−0.0117506	+	0.999931i	$0.503740\pi$
$558$		0			0
$559$	−1.02640e11			−1.05116
$560$		0			0
$561$	−3.81106e9	−	1.85271e9i	−0.0384764	−	0.0187049i
$562$		0			0
$563$	−1.05062e11	−	6.06578e10i	−1.04572	−	0.603744i	−0.124269	−	0.992249i	$-0.539658\pi$
$563$	−1.05062e11	−	6.06578e10i	−1.04572	−	0.603744i	−0.921447	+	0.388505i	$0.872992\pi$
$564$		0			0
$565$	−1.04651e10	−	1.81260e10i	−0.102695	−	0.177873i
$566$		0			0
$567$	−1.02322e11	−	1.45795e10i	−0.990001	−	0.141062i
$568$		0			0
$569$	−1.12012e11	+	6.46699e10i	−1.06860	+	0.616955i	−0.927798	−	0.373083i	$-0.878301\pi$
$569$	−1.12012e11	+	6.46699e10i	−1.06860	+	0.616955i	−0.140799	+	0.990038i	$0.544967\pi$
$570$		0			0
$571$	1.07167e10	−	1.85619e10i	0.100813	−	0.174614i	−0.811207	−	0.584760i	$-0.801189\pi$
$571$	1.07167e10	−	1.85619e10i	0.100813	−	0.174614i	0.912020	+	0.410146i	$0.134522\pi$
$572$		0			0
$573$	−6.24476e10	+	1.28456e11i	−0.579292	+	1.19161i
$574$		0			0
$575$			8.97655e10i			0.821180i
$576$		0			0
$577$	3.30017e10	−	5.71606e10i	0.297737	−	0.515695i	−0.677881	−	0.735172i	$-0.737102\pi$
$577$	3.30017e10	−	5.71606e10i	0.297737	−	0.515695i	0.975618	+	0.219476i	$0.0704349\pi$
$578$		0			0
$579$	1.30051e10	+	1.82604e11i	0.115718	+	1.62478i
$580$		0			0
$581$	−4.40850e10	−	6.15870e10i	−0.386889	−	0.540486i
$582$		0			0
$583$	−4.69205e9	−	8.12686e9i	−0.0406151	−	0.0703475i
$584$		0			0
$585$	1.85014e10	+	1.29230e11i	0.157973	+	1.10342i
$586$		0			0
$587$			8.82785e10i			0.743537i	0.928325	+	0.371769i	$0.121248\pi$
$587$			8.82785e10i			0.743537i	−0.928325	+	0.371769i	$0.878752\pi$
$588$		0			0
$589$	−8.33184e10			−0.692277
$590$		0			0
$591$	−1.93738e9	−	2.86433e9i	−0.0158806	−	0.0234787i
$592$		0			0
$593$	1.64778e11	−	9.51345e10i	1.33254	−	0.769342i	0.346851	−	0.937920i	$-0.387251\pi$
$593$	1.64778e11	−	9.51345e10i	1.33254	−	0.769342i	0.985688	+	0.168579i	$0.0539177\pi$
$594$		0			0
$595$	6.42666e10	−	4.60031e10i	0.512764	−	0.367045i
$596$		0			0
$597$	−5.47988e9	−	7.69427e10i	−0.0431394	−	0.605717i
$598$		0			0
$599$	1.55178e11	+	8.95923e10i	1.20538	+	0.695927i	0.961747	−	0.273941i	$-0.0883272\pi$
$599$	1.55178e11	+	8.95923e10i	1.20538	+	0.695927i	0.243634	+	0.969867i	$0.421661\pi$
$600$		0			0
$601$	−4.68307e10			−0.358949			−0.179474	−	0.983763i	$-0.557440\pi$
$601$	−4.68307e10			−0.358949			−0.179474	+	0.983763i	$0.557440\pi$
$602$		0			0
$603$	−4.78742e10	−	6.09521e10i	−0.362103	−	0.461020i
$604$		0			0
$605$	−1.49174e11	−	8.61256e10i	−1.11345	−	0.642851i
$606$		0			0
$607$	1.27939e10	+	2.21598e10i	0.0942431	+	0.163234i	0.909292	−	0.416158i	$-0.136624\pi$
$607$	1.27939e10	+	2.21598e10i	0.0942431	+	0.163234i	−0.815049	+	0.579392i	$0.803290\pi$
$608$		0			0
$609$	1.99236e11	+	7.38205e10i	1.44843	+	0.536670i
$610$		0			0
$611$	1.73853e11	−	1.00374e11i	1.24743	−	0.720206i
$612$		0			0
$613$	8.19833e8	−	1.41999e9i	0.00580609	−	0.0100564i	−0.863108	−	0.505020i	$-0.831485\pi$
$613$	8.19833e8	−	1.41999e9i	0.00580609	−	0.0100564i	0.868914	+	0.494963i	$0.164819\pi$
$614$		0			0
$615$	1.63782e11	+	7.96211e10i	1.14490	+	0.556580i
$616$		0			0
$617$		−	1.24163e11i		−	0.856745i	−0.903602	−	0.428372i	$-0.859087\pi$
$617$		−	1.24163e11i		−	0.856745i	0.903602	−	0.428372i	$-0.140913\pi$
$618$		0			0
$619$	−4.46997e10	+	7.74222e10i	−0.304468	+	0.527355i	−0.977143	−	0.212584i	$-0.931812\pi$
$619$	−4.46997e10	+	7.74222e10i	−0.304468	+	0.527355i	0.672674	+	0.739939i	$0.265145\pi$
$620$		0			0
$621$	−1.20887e11	−	1.33220e11i	−0.812853	−	0.895784i
$622$		0			0
$623$	−8.76726e9	+	1.93235e10i	−0.0581985	+	0.128272i
$624$		0			0
$625$	9.29262e10	+	1.60953e11i	0.609001	+	1.05482i
$626$		0			0
$627$	4.66787e9	+	6.90123e9i	0.0302029	+	0.0446536i
$628$		0			0
$629$		−	1.50355e10i		−	0.0960540i
$630$		0			0
$631$	−3.02715e11			−1.90949			−0.954743	−	0.297431i	$-0.903870\pi$
$631$	−3.02715e11			−1.90949			−0.954743	+	0.297431i	$0.903870\pi$
$632$		0			0
$633$	−9.72868e10	+	6.58031e10i	−0.605953	+	0.409856i
$634$		0			0
$635$	5.75954e10	−	3.32527e10i	0.354236	−	0.204518i
$636$		0			0
$637$	−4.56666e10	+	1.34079e11i	−0.277358	+	0.814335i
$638$		0			0
$639$	−1.43125e10	−	5.73981e9i	−0.0858444	−	0.0344266i
$640$		0			0
$641$	−5.94806e10	−	3.43411e10i	−0.352325	−	0.203415i	0.313384	−	0.949627i	$-0.398537\pi$
$641$	−5.94806e10	−	3.43411e10i	−0.352325	−	0.203415i	−0.665709	+	0.746212i	$0.731871\pi$
$642$		0			0
$643$	1.67422e11			0.979418			0.489709	−	0.871886i	$-0.337103\pi$
$643$	1.67422e11			0.979418			0.489709	+	0.871886i	$0.337103\pi$
$644$		0			0
$645$	−1.19805e11	+	2.46442e11i	−0.692208	+	1.42389i
$646$		0			0
$647$	8.78175e10	+	5.07014e10i	0.501145	+	0.289336i	0.729186	−	0.684315i	$-0.239899\pi$
$647$	8.78175e10	+	5.07014e10i	0.501145	+	0.289336i	−0.228041	+	0.973652i	$0.573232\pi$
$648$		0			0
$649$	1.14596e10	+	1.98486e10i	0.0645938	+	0.111880i
$650$		0			0
$651$	1.29429e11	+	1.56067e11i	0.720620	+	0.868933i
$652$		0			0
$653$	1.24003e11	−	7.15934e10i	0.681995	−	0.393750i	−0.118612	−	0.992941i	$-0.537844\pi$
$653$	1.24003e11	−	7.15934e10i	0.681995	−	0.393750i	0.800606	+	0.599191i	$0.204511\pi$
$654$		0			0
$655$	−1.75937e11	+	3.04731e11i	−0.955854	+	1.65559i
$656$		0			0
$657$	−1.58451e11	−	2.01735e11i	−0.850418	−	1.08273i
$658$		0			0
$659$			2.49864e10i			0.132484i	0.997804	+	0.0662418i	$0.0211009\pi$
$659$			2.49864e10i			0.132484i	−0.997804	+	0.0662418i	$0.978899\pi$
$660$		0			0
$661$	1.35081e11	−	2.33967e11i	0.707599	−	1.22560i	−0.258146	−	0.966106i	$-0.583112\pi$
$661$	1.35081e11	−	2.33967e11i	0.707599	−	1.22560i	0.965745	−	0.259492i	$-0.0835552\pi$
$662$		0			0
$663$	8.06922e10	−	5.74692e9i	0.417616	−	0.0297427i
$664$		0			0
$665$	−1.54653e11	+	1.51522e10i	−0.790811	+	0.0774798i
$666$		0			0
$667$	1.84907e11	+	3.20268e11i	0.934220	+	1.61812i
$668$		0			0
$669$	1.07043e11	−	7.24019e10i	0.534383	−	0.361448i
$670$		0			0
$671$		−	2.72238e10i		−	0.134295i
$672$		0			0
$673$	5.84302e10			0.284824			0.142412	−	0.989807i	$-0.454514\pi$
$673$	5.84302e10			0.284824			0.142412	+	0.989807i	$0.454514\pi$
$674$		0			0
$675$	1.34201e11	+	4.30323e10i	0.646458	+	0.207291i
$676$		0			0
$677$	−1.85501e11	+	1.07099e11i	−0.883063	+	0.509837i	−0.871667	−	0.490098i	$-0.836961\pi$
$677$	−1.85501e11	+	1.07099e11i	−0.883063	+	0.509837i	−0.0113962	+	0.999935i	$0.503628\pi$
$678$		0			0
$679$	−2.29582e11	+	1.64338e11i	−1.08008	+	0.773142i
$680$		0			0
$681$	−3.36563e11	+	2.39701e10i	−1.56487	+	0.111450i
$682$		0			0
$683$	−1.78222e11	−	1.02896e11i	−0.818988	−	0.472843i	0.0310794	−	0.999517i	$-0.490106\pi$
$683$	−1.78222e11	−	1.02896e11i	−0.818988	−	0.472843i	−0.850067	+	0.526674i	$0.823439\pi$
$684$		0			0
$685$	7.05453e10			0.320410
$686$		0			0
$687$	−1.40103e11	−	6.81095e10i	−0.628955	−	0.305760i
$688$		0			0
$689$	1.55146e11	+	8.95733e10i	0.688434	+	0.397468i
$690$		0			0
$691$	1.02069e11	+	1.76789e11i	0.447696	+	0.775432i	0.998236	−	0.0593771i	$-0.0189114\pi$
$691$	1.02069e11	+	1.76789e11i	0.447696	+	0.775432i	−0.550540	+	0.834809i	$0.685578\pi$
$692$		0			0
$693$	5.67577e9	−	1.94641e10i	0.0246089	−	0.0843919i
$694$		0			0
$695$	1.69749e11	−	9.80045e10i	0.727558	−	0.420056i
$696$		0			0
$697$	5.64241e10	−	9.77294e10i	0.239075	−	0.414090i
$698$		0			0
$699$	1.38565e11	−	2.85030e11i	0.580422	−	1.19394i
$700$		0			0
$701$			2.92695e11i			1.21211i	0.795421	+	0.606057i	$0.207250\pi$
$701$			2.92695e11i			1.21211i	−0.795421	+	0.606057i	$0.792750\pi$
$702$		0			0
$703$	−1.47810e10	+	2.56015e10i	−0.0605178	+	0.104820i
$704$		0			0
$705$	−3.80734e10	−	5.34587e11i	−0.154122	−	2.16402i
$706$		0			0
$707$	3.65468e11	−	3.58068e10i	1.46276	−	0.143314i
$708$		0			0
$709$	9.69687e10	+	1.67955e11i	0.383749	+	0.664672i	0.991595	−	0.129383i	$-0.0412996\pi$
$709$	9.69687e10	+	1.67955e11i	0.383749	+	0.664672i	−0.607846	+	0.794055i	$0.707966\pi$
$710$		0			0
$711$	1.21771e11	−	1.74335e10i	0.476501	−	0.0682191i
$712$		0			0
$713$			3.52896e11i			1.36549i
$714$		0			0
$715$	−2.56090e10			−0.0979869
$716$		0			0
$717$	−2.17354e11	−	3.21348e11i	−0.822415	−	1.21590i
$718$		0			0
$719$	−2.62598e11	+	1.51611e11i	−0.982599	+	0.567304i	−0.903054	−	0.429527i	$-0.858680\pi$
$719$	−2.62598e11	+	1.51611e11i	−0.982599	+	0.567304i	−0.0795455	+	0.996831i	$0.525347\pi$
$720$		0			0
$721$	−1.71309e11	−	7.77245e10i	−0.633925	−	0.287618i
$722$		0			0
$723$	1.20329e9	+	1.68953e10i	0.00440369	+	0.0618319i
$724$		0			0
$725$	−2.50904e11	−	1.44860e11i	−0.908147	−	0.524319i
$726$		0			0
$727$	−2.93281e11			−1.04990			−0.524948	−	0.851135i	$-0.675915\pi$
$727$	−2.93281e11			−1.04990			−0.524948	+	0.851135i	$0.675915\pi$
$728$		0			0
$729$	−2.57117e11	+	1.16863e11i	−0.910377	+	0.413779i
$730$		0			0
$731$	1.47053e11	+	8.49010e10i	0.514996	+	0.297333i
$732$		0			0
$733$	−2.41246e10	−	4.17850e10i	−0.0835687	−	0.144745i	0.821212	−	0.570624i	$-0.193298\pi$
$733$	−2.41246e10	−	4.17850e10i	−0.0835687	−	0.144745i	−0.904780	+	0.425878i	$0.859965\pi$
$734$		0			0
$735$	2.68624e11	+	2.66149e11i	0.920440	+	0.911959i
$736$		0			0
$737$	1.31670e10	−	7.60196e9i	0.0446289	−	0.0257665i
$738$		0			0
$739$	2.01824e11	−	3.49570e11i	0.676700	−	1.17208i	−0.299270	−	0.954169i	$-0.596743\pi$
$739$	2.01824e11	−	3.49570e11i	0.676700	−	1.17208i	0.975969	−	0.217909i	$-0.0699237\pi$
$740$		0			0
$741$	−1.43047e11	−	6.95409e10i	−0.474467	−	0.230657i
$742$		0			0
$743$			3.03948e11i			0.997342i	0.866791	+	0.498671i	$0.166178\pi$
$743$			3.03948e11i			0.997342i	−0.866791	+	0.498671i	$0.833822\pi$
$744$		0			0
$745$	2.41304e11	−	4.17951e11i	0.783320	−	1.35675i
$746$		0			0
$747$	−1.92095e11	−	7.70367e10i	−0.616926	−	0.247409i
$748$		0			0
$749$	1.02236e11	−	2.25333e11i	0.324845	−	0.715974i
$750$		0			0
$751$	1.69823e11	+	2.94141e11i	0.533870	+	0.924690i	0.999217	+	0.0395617i	$0.0125962\pi$
$751$	1.69823e11	+	2.94141e11i	0.533870	+	0.924690i	−0.465347	+	0.885128i	$0.654070\pi$
$752$		0			0
$753$	−2.02753e11	−	2.99760e11i	−0.630648	−	0.932383i
$754$		0			0
$755$		−	5.88528e11i		−	1.81125i
$756$		0			0
$757$	4.49280e11			1.36815			0.684074	−	0.729412i	$-0.260206\pi$
$757$	4.49280e11			1.36815			0.684074	+	0.729412i	$0.260206\pi$
$758$		0			0
$759$	2.92302e10	−	1.97708e10i	0.0880774	−	0.0595741i
$760$		0			0
$761$	−3.71405e11	+	2.14431e11i	−1.10741	+	0.639364i	−0.938158	−	0.346208i	$-0.887469\pi$
$761$	−3.71405e11	+	2.14431e11i	−1.10741	+	0.639364i	−0.169254	+	0.985573i	$0.554136\pi$
$762$		0			0
$763$	2.54439e10	+	2.59698e11i	0.0750734	+	0.766249i
$764$		0			0
$765$	8.03884e10	−	2.00452e11i	0.234719	−	0.585282i
$766$		0			0
$767$	−3.78919e11	−	2.18769e11i	−1.09488	−	0.632127i
$768$		0			0
$769$	3.84679e11			1.10000			0.550001	−	0.835164i	$-0.314627\pi$
$769$	3.84679e11			1.10000			0.550001	+	0.835164i	$0.314627\pi$
$770$		0			0
$771$	1.24277e11	−	2.55641e11i	0.351702	−	0.723457i
$772$		0			0
$773$	2.61563e11	+	1.51013e11i	0.732585	+	0.422958i	0.819367	−	0.573269i	$-0.194325\pi$
$773$	2.61563e11	+	1.51013e11i	0.732585	+	0.422958i	−0.0867823	+	0.996227i	$0.527658\pi$
$774$		0			0
$775$	−1.38233e11	−	2.39427e11i	−0.383182	−	0.663690i
$776$		0			0
$777$	7.09162e10	−	1.20830e10i	0.194563	−	0.0331506i
$778$		0			0
$779$	−1.92150e11	+	1.10938e11i	−0.521785	+	0.301253i
$780$		0			0
$781$	1.51249e9	−	2.61971e9i	0.00406526	−	0.00704123i
$782$		0			0
$783$	5.67446e11	−	1.22906e11i	1.50966	−	0.326985i
$784$		0			0
$785$		−	3.73463e11i		−	0.983489i
$786$		0			0
$787$	3.59141e11	−	6.22051e11i	0.936195	−	1.62154i	0.163706	−	0.986509i	$-0.447655\pi$
$787$	3.59141e11	−	6.22051e11i	0.936195	−	1.62154i	0.772489	−	0.635029i	$-0.219012\pi$
$788$		0			0
$789$	−5.60218e11	+	3.98989e10i	−1.44560	+	0.102956i
$790$		0			0
$791$	−3.61197e10	−	5.04595e10i	−0.0922653	−	0.128895i
$792$		0			0
$793$	2.59858e11	+	4.50087e11i	0.657117	+	1.13816i
$794$		0			0
$795$	3.96160e11	−	2.67956e11i	0.991750	−	0.670803i
$796$		0			0
$797$		−	2.04056e11i		−	0.505727i	−0.967502	−	0.252864i	$-0.918628\pi$
$797$		−	2.04056e11i		−	0.505727i	0.967502	−	0.252864i	$-0.0813724\pi$
$798$		0			0
$799$	−3.32107e11			−0.814874
$800$		0			0
$801$	8.21763e9	+	5.73990e10i	0.0199626	+	0.139436i
$802$		0			0
$803$	4.35791e10	−	2.51604e10i	0.104813	−	0.0605139i
$804$		0			0
$805$	6.41772e10	+	6.55035e11i	0.152826	+	1.55984i
$806$		0			0
$807$	−5.37524e11	+	3.82826e10i	−1.26737	+	0.0902625i
$808$		0			0
$809$	9.51905e10	+	5.49583e10i	0.222228	+	0.128304i	0.606982	−	0.794716i	$-0.292380\pi$
$809$	9.51905e10	+	5.49583e10i	0.222228	+	0.128304i	−0.384753	+	0.923019i	$0.625713\pi$
$810$		0			0
$811$	2.33734e11			0.540305			0.270152	−	0.962818i	$-0.412926\pi$
$811$	2.33734e11			0.540305			0.270152	+	0.962818i	$0.412926\pi$
$812$		0			0
$813$	−5.28366e11	−	2.56860e11i	−1.20941	−	0.587942i
$814$		0			0
$815$	−4.88216e11	−	2.81872e11i	−1.10658	−	0.638883i
$816$		0			0
$817$	−1.66928e11	−	2.89127e11i	−0.374663	−	0.648935i
$818$		0			0
$819$	9.19526e10	+	3.75973e11i	0.204375	+	0.835643i
$820$		0			0
$821$	5.20573e11	−	3.00553e11i	1.14580	−	0.661528i	0.197940	−	0.980214i	$-0.436575\pi$
$821$	5.20573e11	−	3.00553e11i	1.14580	−	0.661528i	0.947860	+	0.318686i	$0.103241\pi$
$822$		0			0
$823$	−1.79738e11	+	3.11316e11i	−0.391779	+	0.678581i	−0.992684	−	0.120739i	$-0.961474\pi$
$823$	−1.79738e11	+	3.11316e11i	−0.391779	+	0.678581i	0.600905	+	0.799320i	$0.294807\pi$
$824$		0			0
$825$	−1.20872e10	+	2.48635e10i	−0.0260921	+	0.0536719i
$826$		0			0
$827$			5.43642e10i			0.116223i	0.998310	+	0.0581113i	$0.0185078\pi$
$827$			5.43642e10i			0.116223i	−0.998310	+	0.0581113i	$0.981492\pi$
$828$		0			0
$829$	2.74898e11	−	4.76138e11i	0.582041	−	1.00813i	−0.413196	−	0.910642i	$-0.635587\pi$
$829$	2.74898e11	−	4.76138e11i	0.582041	−	1.00813i	0.995237	−	0.0974831i	$-0.0310792\pi$
$830$		0			0
$831$	−3.18029e10	−	4.46543e11i	−0.0666903	−	0.936394i
$832$		0			0
$833$	1.76333e11	−	1.54321e11i	0.366230	−	0.320513i
$834$		0			0
$835$	4.13707e11	+	7.16561e11i	0.851033	+	1.47403i
$836$		0			0
$837$	5.27584e11	+	1.69173e11i	1.07495	+	0.344691i
$838$		0			0
$839$			3.91290e11i			0.789679i	0.918750	+	0.394840i	$0.129200\pi$
$839$			3.91290e11i			0.789679i	−0.918750	+	0.394840i	$0.870800\pi$
$840$		0			0
$841$	−6.93330e11			−1.38598
$842$		0			0
$843$	2.27955e11	+	3.37021e11i	0.451376	+	0.667339i
$844$		0			0
$845$	−1.48705e11	+	8.58550e10i	−0.291675	+	0.168399i
$846$		0			0
$847$	−4.65069e11	−	2.11007e11i	−0.903616	−	0.409980i
$848$		0			0
$849$	−3.46653e10	−	4.86734e11i	−0.0667212	−	0.936829i
$850$		0			0
$851$	1.08435e11	+	6.26051e10i	0.206753	+	0.119369i
$852$		0			0
$853$	−2.62938e11			−0.496658			−0.248329	−	0.968676i	$-0.579881\pi$
$853$	−2.62938e11			−0.496658			−0.248329	+	0.968676i	$0.579881\pi$
$854$		0			0
$855$	−3.33940e11	+	2.62289e11i	−0.624890	+	0.490813i
$856$		0			0
$857$	−1.71064e11	−	9.87636e10i	−0.317128	−	0.183094i	0.332984	−	0.942933i	$-0.391945\pi$
$857$	−1.71064e11	−	9.87636e10i	−0.317128	−	0.183094i	−0.650112	+	0.759839i	$0.725278\pi$
$858$		0			0
$859$	−5.96947e10	−	1.03394e11i	−0.109638	−	0.189899i	0.805985	−	0.591935i	$-0.201636\pi$
$859$	−5.96947e10	−	1.03394e11i	−0.109638	−	0.189899i	−0.915624	+	0.402036i	$0.868303\pi$
$860$		0			0
$861$	5.06293e11	+	1.87590e11i	0.921275	+	0.341349i
$862$		0			0
$863$	7.81431e11	−	4.51160e11i	1.40879	−	0.813368i	0.413522	−	0.910494i	$-0.364298\pi$
$863$	7.81431e11	−	4.51160e11i	1.40879	−	0.813368i	0.995272	+	0.0971260i	$0.0309650\pi$
$864$		0			0
$865$	−2.62602e11	+	4.54840e11i	−0.469065	+	0.812445i
$866$		0			0
$867$	3.87808e11	+	1.88529e11i	0.686341	+	0.333658i
$868$		0			0
$869$			2.41308e10i			0.0423148i
$870$		0			0
$871$	−1.45125e11	+	2.51364e11i	−0.252156	+	0.436747i
$872$		0			0
$873$	−2.87174e11	+	7.16082e11i	−0.494411	+	1.23284i
$874$		0			0
$875$	1.41964e11	+	1.98325e11i	0.242185	+	0.338334i
$876$		0			0
$877$	3.13531e11	+	5.43052e11i	0.530008	+	0.918001i	0.999387	+	0.0350045i	$0.0111446\pi$
$877$	3.13531e11	+	5.43052e11i	0.530008	+	0.918001i	−0.469379	+	0.882997i	$0.655522\pi$
$878$		0			0
$879$	−2.56735e11	−	3.79570e11i	−0.430060	−	0.635824i
$880$		0			0
$881$			8.66629e11i			1.43857i	0.694718	+	0.719283i	$0.255529\pi$
$881$			8.66629e11i			1.43857i	−0.694718	+	0.719283i	$0.744471\pi$
$882$		0			0
$883$	4.10650e11			0.675506			0.337753	−	0.941235i	$-0.390333\pi$
$883$	4.10650e11			0.675506			0.337753	+	0.941235i	$0.390333\pi$
$884$		0			0
$885$	−9.67560e11	+	6.54441e11i	−1.57727	+	1.06684i
$886$		0			0
$887$	−1.80385e11	+	1.04145e11i	−0.291410	+	0.168246i	−0.638578	−	0.769557i	$-0.720477\pi$
$887$	−1.80385e11	+	1.04145e11i	−0.291410	+	0.168246i	0.347167	+	0.937803i	$0.387144\pi$
$888$		0			0
$889$	1.60335e11	−	1.14770e11i	0.256697	−	0.183748i
$890$		0			0
$891$	−1.55451e10	−	5.31774e10i	−0.0246651	−	0.0843755i
$892$		0			0
$893$	5.65489e11	+	3.26485e11i	0.889239	+	0.513402i
$894$		0			0
$895$	−1.09989e12			−1.71419
$896$		0			0
$897$	−2.94541e11	+	6.05876e11i	−0.454963	+	0.935867i
$898$		0			0
$899$	−9.86382e11	−	5.69488e11i	−1.51010	−	0.871858i
$900$		0			0
$901$	−1.48185e11	−	2.56664e11i	−0.224857	−	0.389463i
$902$		0			0
$903$	−2.82266e11	+	7.61816e11i	−0.424529	+	1.14577i
$904$		0			0
$905$	−3.41998e11	+	1.97452e11i	−0.509834	+	0.294353i
$906$		0			0
$907$	1.53863e11	−	2.66498e11i	0.227355	−	0.393791i	−0.729668	−	0.683801i	$-0.760326\pi$
$907$	1.53863e11	−	2.66498e11i	0.227355	−	0.393791i	0.957023	+	0.290011i	$0.0936588\pi$
$908$		0			0
$909$	7.89148e11	−	6.19828e11i	1.15585	−	0.907853i
$910$		0			0
$911$			9.45404e11i			1.37260i	0.727318	+	0.686300i	$0.240766\pi$
$911$			9.45404e11i			1.37260i	−0.727318	+	0.686300i	$0.759234\pi$
$912$		0			0
$913$	2.02998e10	−	3.51603e10i	0.0292152	−	0.0506022i
$914$		0			0
$915$	1.38399e12	−	9.85680e10i	1.97446	−	0.140621i
$916$		0			0
$917$	−4.31043e11	+	9.50040e11i	−0.609598	+	1.34358i
$918$		0			0
$919$	−5.01341e11	−	8.68347e11i	−0.702863	−	1.21739i	−0.967457	−	0.253035i	$-0.918571\pi$
$919$	−5.01341e11	−	8.68347e11i	−0.702863	−	1.21739i	0.264594	−	0.964360i	$-0.414762\pi$
$920$		0			0
$921$	−5.42768e11	+	3.67119e11i	−0.754355	+	0.510233i
$922$		0			0
$923$			5.77482e10i			0.0795668i
$924$		0			0
$925$	−9.80923e10			−0.133989
$926$		0			0
$927$	−5.08860e11	+	7.28518e10i	−0.689095	+	0.0986555i
$928$		0			0
$929$	1.16648e12	−	6.73466e11i	1.56608	−	0.904175i	0.569458	−	0.822021i	$-0.307153\pi$
$929$	1.16648e12	−	6.73466e11i	1.56608	−	0.904175i	0.996620	−	0.0821543i	$-0.0261800\pi$
$930$		0			0
$931$	−4.51958e11	+	8.94197e10i	−0.601588	+	0.119024i
$932$		0			0
$933$	1.29392e12	−	9.21530e10i	1.70757	−	0.121614i
$934$		0			0
$935$	3.66901e10	+	2.11830e10i	0.0480068	+	0.0277167i
$936$		0			0
$937$	1.75433e11			0.227590			0.113795	−	0.993504i	$-0.463699\pi$
$937$	1.75433e11			0.227590			0.113795	+	0.993504i	$0.463699\pi$
$938$		0			0
$939$	−6.68361e11	−	3.24917e11i	−0.859703	−	0.417936i
$940$		0			0
$941$	6.86868e11	+	3.96563e11i	0.876021	+	0.505771i	0.869344	−	0.494207i	$-0.164541\pi$
$941$	6.86868e11	+	3.96563e11i	0.876021	+	0.505771i	0.00667678	+	0.999978i	$0.497875\pi$
$942$		0			0
$943$	4.69879e11	+	8.13855e11i	0.594209	+	1.02920i
$944$		0			0
$945$	1.01005e12	+	2.18069e11i	1.26653	+	0.273443i
$946$		0			0
$947$	−2.71298e10	+	1.56634e10i	−0.0337323	+	0.0194754i	−0.516771	−	0.856123i	$-0.672866\pi$
$947$	−2.71298e10	+	1.56634e10i	−0.0337323	+	0.0194754i	0.483039	+	0.875599i	$0.339533\pi$
$948$		0			0
$949$	−4.80324e11	+	8.31945e11i	−0.592201	+	1.02572i
$950$		0			0
$951$	−4.95662e11	+	1.01959e12i	−0.605987	+	1.24653i
$952$		0			0
$953$		−	7.84682e11i		−	0.951310i	−0.879632	−	0.475655i	$-0.842211\pi$
$953$		−	7.84682e11i		−	0.951310i	0.879632	−	0.475655i	$-0.157789\pi$
$954$		0			0
$955$	7.13997e11	−	1.23668e12i	0.858387	−	1.48677i
$956$		0			0
$957$	8.09111e9	+	1.13607e11i	0.00964629	+	0.135443i
$958$		0			0
$959$	2.08159e11	−	2.03945e10i	0.246106	−	0.0241123i
$960$		0			0
$961$	−1.16990e11	−	2.02633e11i	−0.137169	−	0.237584i
$962$		0			0
$963$	−9.58266e10	−	6.69336e11i	−0.111425	−	0.778285i
$964$		0			0
$965$		−	1.83026e12i		−	2.11059i
$966$		0			0
$967$	−1.43127e12			−1.63687			−0.818436	−	0.574597i	$-0.805159\pi$
$967$	−1.43127e12			−1.63687			−0.818436	+	0.574597i	$0.805159\pi$
$968$		0			0
$969$	1.47422e11	+	2.17956e11i	0.167212	+	0.247214i
$970$		0			0
$971$	5.46788e11	−	3.15688e11i	0.615095	−	0.355125i	−0.159862	−	0.987139i	$-0.551105\pi$
$971$	5.46788e11	−	3.15688e11i	0.615095	−	0.355125i	0.774957	+	0.632014i	$0.217772\pi$
$972$		0			0
$973$	4.72548e11	−	3.38258e11i	0.527223	−	0.377395i
$974$		0			0
$975$	−3.74931e10	−	5.26439e11i	−0.0414890	−	0.582545i
$976$		0			0
$977$	−5.05709e10	−	2.91971e10i	−0.0555037	−	0.0320451i	0.471991	−	0.881603i	$-0.343535\pi$
$977$	−5.05709e10	−	2.91971e10i	−0.0555037	−	0.0320451i	−0.527495	+	0.849558i	$0.676869\pi$
$978$		0			0
$979$	−1.13745e10			−0.0123823
$980$		0			0
$981$	4.40443e11	+	5.60759e11i	0.475569	+	0.605481i
$982$		0			0
$983$	1.30108e11	+	7.51179e10i	0.139345	+	0.0804506i	0.568052	−	0.822993i	$-0.307697\pi$
$983$	1.30108e11	+	7.51179e10i	0.139345	+	0.0804506i	−0.428707	+	0.903444i	$0.641031\pi$
$984$		0			0
$985$	1.72863e10	+	2.99407e10i	0.0183635	+	0.0318066i
$986$		0			0
$987$	−2.66892e11	−	1.56641e12i	−0.281233	−	1.65058i
$988$		0			0
$989$	−1.22460e12	+	7.07024e11i	−1.28000	+	0.739008i
$990$		0			0
$991$	8.95580e11	−	1.55119e12i	0.928560	−	1.60831i	0.142827	−	0.989748i	$-0.454381\pi$
$991$	8.95580e11	−	1.55119e12i	0.928560	−	1.60831i	0.785733	−	0.618566i	$-0.212286\pi$
$992$		0			0
$993$	7.99285e11	+	3.88565e11i	0.822062	+	0.399638i
$994$		0			0
$995$			7.71207e11i			0.786825i
$996$		0			0
$997$	5.35305e11	−	9.27175e11i	0.541777	−	0.938385i	−0.457025	−	0.889454i	$-0.651085\pi$
$997$	5.35305e11	−	9.27175e11i	0.541777	−	0.938385i	0.998802	−	0.0489312i	$-0.0155815\pi$
$998$		0			0
$999$	1.45578e11	−	1.32100e11i	0.146162	−	0.132630i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	84.9.p.b.53.6	✓	40
3.2	odd	2	inner	84.9.p.b.53.9	yes	40
7.2	even	3	inner	84.9.p.b.65.9	yes	40
21.2	odd	6	inner	84.9.p.b.65.6	yes	40

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.9.p.b.53.6	✓	40	1.1	even	1	trivial
84.9.p.b.53.9	yes	40	3.2	odd	2	inner
84.9.p.b.65.6	yes	40	21.2	odd	6	inner
84.9.p.b.65.9	yes	40	7.2	even	3	inner

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:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	84.p (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-67.0937 + 45.3810i) q^{3} +(701.327 - 404.911i) q^{5} +(1952.36 - 1397.53i) q^{7} +(2442.13 - 6089.56i) q^{9} +O(q^{10})\)	\(q+(-67.0937 + 45.3810i) q^{3} +(701.327 - 404.911i) q^{5} +(1952.36 - 1397.53i) q^{7} +(2442.13 - 6089.56i) q^{9} +(1114.61 + 643.521i) q^{11} -24570.2 q^{13} +(-28679.3 + 58993.9i) q^{15} +(35201.9 + 20323.8i) q^{17} +(-39959.6 - 69212.1i) q^{19} +(-67569.6 + 182366. i) q^{21} +(-293149. + 169249. i) q^{23} +(132594. - 229659. i) q^{25} +(112499. + 519397. i) q^{27} -1.09251e6i q^{29} +(521266. - 902859. i) q^{31} +(-103987. + 7405.99i) q^{33} +(803366. - 1.77066e6i) q^{35} +(-184949. - 320341. i) q^{37} +(1.64851e6 - 1.11502e6i) q^{39} -2.77625e6i q^{41} +4.17741e6 q^{43} +(-753002. - 5.25962e6i) q^{45} +(-7.07575e6 + 4.08519e6i) q^{47} +(1.85862e6 - 5.45697e6i) q^{49} +(-3.28414e6 + 233898. i) q^{51} +(-6.31437e6 - 3.64560e6i) q^{53} +1.04228e6 q^{55} +(5.82196e6 + 2.83029e6i) q^{57} +(1.54219e7 + 8.90382e6i) q^{59} +(-1.05761e7 - 1.83184e7i) q^{61} +(-3.74244e6 - 1.53020e7i) q^{63} +(-1.72318e7 + 9.94876e6i) q^{65} +(5.90653e6 - 1.02304e7i) q^{67} +(1.19877e7 - 2.46589e7i) q^{69} -2.35033e6i q^{71} +(1.95490e7 - 3.38599e7i) q^{73} +(1.52596e6 + 2.14259e7i) q^{75} +(3.07546e6 - 301319. i) q^{77} +(9.37450e6 + 1.62371e7i) q^{79} +(-3.11188e7 - 2.97430e7i) q^{81} -3.15449e7i q^{83} +3.29174e7 q^{85} +(4.95792e7 + 7.33005e7i) q^{87} +(-7.65369e6 + 4.41886e6i) q^{89} +(-4.79699e7 + 3.43377e7i) q^{91} +(5.99900e6 + 8.42317e7i) q^{93} +(-5.60495e7 - 3.23602e7i) q^{95} -1.17592e8 q^{97} +(6.64078e6 - 5.21593e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9}+O(q^{10}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9	\( 40 q + 81 q^{3} - 34 q^{7} + 4771 q^{9} - 55464 q^{13} + 68482 q^{15} + 311690 q^{19} - 172343 q^{21} + 1766792 q^{25} - 3451932 q^{27} + 31596 q^{31} + 1874885 q^{33} - 1853482 q^{37} + 11217526 q^{39} - 13372600 q^{43} - 527785 q^{45} - 12653462 q^{49} - 1103461 q^{51} + 71577224 q^{55} - 17195214 q^{57} - 21761970 q^{61} + 21945045 q^{63} - 26337350 q^{67} - 5588722 q^{69} + 41115682 q^{73} - 17971730 q^{75} - 120916932 q^{79} - 24550133 q^{81} + 139250060 q^{85} - 16321046 q^{87} + 345074940 q^{91} + 25774675 q^{93} - 707216948 q^{97} - 94510994 q^{99}+O(q^{100}) \) 40 * q + 81 * q^3 - 34 * q^7 + 4771 * q^9 - 55464 * q^13 + 68482 * q^15 + 311690 * q^19 - 172343 * q^21 + 1766792 * q^25 - 3451932 * q^27 + 31596 * q^31 + 1874885 * q^33 - 1853482 * q^37 + 11217526 * q^39 - 13372600 * q^43 - 527785 * q^45 - 12653462 * q^49 - 1103461 * q^51 + 71577224 * q^55 - 17195214 * q^57 - 21761970 * q^61 + 21945045 * q^63 - 26337350 * q^67 - 5588722 * q^69 + 41115682 * q^73 - 17971730 * q^75 - 120916932 * q^79 - 24550133 * q^81 + 139250060 * q^85 - 16321046 * q^87 + 345074940 * q^91 + 25774675 * q^93 - 707216948 * q^97 - 94510994 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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