LMFDB - Embedded newform 76.8.e.a.49.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [76,8,Mod(45,76)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(76, base_ring=CyclotomicField(6))

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

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

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

chi := DirichletCharacter("76.45");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 76 = 2^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	76.e (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$23.7412619368$
Analytic rank:	$0$
Dimension:	$22$
Relative dimension:	$11$ over $\Q(\zeta_{3})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			49.8
Character	$\chi$	$=$	76.49
Dual form			76.8.e.a.45.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(20.2338 - 35.0460i) q^{3} +(-90.2780 + 156.366i) q^{5} +231.633 q^{7} +(274.684 + 475.766i) q^{9} +O(q^{10})$	\(q+(20.2338 - 35.0460i) q^{3} +(-90.2780 + 156.366i) q^{5} +231.633 q^{7} +(274.684 + 475.766i) q^{9} -4274.66 q^{11} +(-3994.94 - 6919.44i) q^{13} +(3653.34 + 6327.78i) q^{15} +(-15465.6 + 26787.3i) q^{17} +(-29139.8 - 6689.17i) q^{19} +(4686.83 - 8117.82i) q^{21} +(2412.95 + 4179.35i) q^{23} +(22762.2 + 39425.4i) q^{25} +110734. q^{27} +(97558.8 + 168977. i) q^{29} -240512. q^{31} +(-86492.8 + 149810. i) q^{33} +(-20911.4 + 36219.6i) q^{35} +137040. q^{37} -323332. q^{39} +(-288875. + 500346. i) q^{41} +(-395948. + 685803. i) q^{43} -99191.6 q^{45} +(-322935. - 559339. i) q^{47} -769889. q^{49} +(625858. + 1.08402e6i) q^{51} +(284590. + 492924. i) q^{53} +(385908. - 668412. i) q^{55} +(-824039. + 885886. i) q^{57} +(917429. - 1.58903e6i) q^{59} +(451671. + 782317. i) q^{61} +(63625.8 + 110203. i) q^{63} +1.44262e6 q^{65} +(-952079. - 1.64905e6i) q^{67} +195293. q^{69} +(-1.01191e6 + 1.75267e6i) q^{71} +(2.40496e6 - 4.16551e6i) q^{73} +1.84227e6 q^{75} -990153. q^{77} +(1.10602e6 - 1.91568e6i) q^{79} +(1.63985e6 - 2.84030e6i) q^{81} -320945. q^{83} +(-2.79241e6 - 4.83660e6i) q^{85} +7.89596e6 q^{87} +(-3.24510e6 - 5.62069e6i) q^{89} +(-925361. - 1.60277e6i) q^{91} +(-4.86649e6 + 8.42900e6i) q^{93} +(3.67664e6 - 3.95259e6i) q^{95} +(646270. - 1.11937e6i) q^{97} +(-1.17418e6 - 2.03374e6i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	$ 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) $ 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/76\mathbb{Z}\right)^\times$.

$n$	$21$	$39$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	20.2338	−	35.0460i	0.432667	−	0.749401i	−0.564435	−	0.825478i	$-0.690906\pi$
$3$	20.2338	−	35.0460i	0.432667	−	0.749401i	0.997102	+	0.0760762i	$0.0242392\pi$
$4$		0			0
$5$	−90.2780	+	156.366i	−0.322989	+	0.559433i	−0.981103	−	0.193485i	$-0.938021\pi$
$5$	−90.2780	+	156.366i	−0.322989	+	0.559433i	0.658115	+	0.752918i	$0.271354\pi$
$6$		0			0
$7$	231.633			0.255245			0.127623	−	0.991823i	$-0.459265\pi$
$7$	231.633			0.255245			0.127623	+	0.991823i	$0.459265\pi$
$8$		0			0
$9$	274.684	+	475.766i	0.125598	+	0.217543i
$10$		0			0
$11$	−4274.66			−0.968339			−0.484169	−	0.874974i	$-0.660878\pi$
$11$	−4274.66			−0.968339			−0.484169	+	0.874974i	$0.660878\pi$
$12$		0			0
$13$	−3994.94	−	6919.44i	−0.504323	−	0.873513i	−0.999988	−	0.00499907i	$-0.998409\pi$
$13$	−3994.94	−	6919.44i	−0.504323	−	0.873513i	0.495664	−	0.868514i	$-0.334925\pi$
$14$		0			0
$15$	3653.34	+	6327.78i	0.279493	+	0.484096i
$16$		0			0
$17$	−15465.6	+	26787.3i	−0.763478	+	1.32238i	0.177570	+	0.984108i	$0.443176\pi$
$17$	−15465.6	+	26787.3i	−0.763478	+	1.32238i	−0.941048	+	0.338274i	$0.890157\pi$
$18$		0			0
$19$	−29139.8	−	6689.17i	−0.974650	−	0.223736i
$19$	−29139.8	−	6689.17i	−0.974650	−	0.223736i
$20$		0			0
$21$	4686.83	−	8117.82i	0.110436	−	0.191281i
$22$		0			0
$23$	2412.95	+	4179.35i	0.0413524	+	0.0716245i	0.885961	−	0.463760i	$-0.153500\pi$
$23$	2412.95	+	4179.35i	0.0413524	+	0.0716245i	−0.844608	+	0.535385i	$0.820167\pi$
$24$		0			0
$25$	22762.2	+	39425.4i	0.291357	+	0.504645i
$26$		0			0
$27$	110734.			1.08270
$28$		0			0
$29$	97558.8	+	168977.i	0.742803	+	1.28657i	0.951214	+	0.308531i	$0.0998374\pi$
$29$	97558.8	+	168977.i	0.742803	+	1.28657i	−0.208411	+	0.978041i	$0.566829\pi$
$30$		0			0
$31$	−240512.			−1.45001			−0.725005	−	0.688743i	$-0.758163\pi$
$31$	−240512.			−1.45001			−0.725005	+	0.688743i	$0.758163\pi$
$32$		0			0
$33$	−86492.8	+	149810.i	−0.418968	+	0.725674i
$34$		0			0
$35$	−20911.4	+	36219.6i	−0.0824412	+	0.142792i
$36$		0			0
$37$	137040.			0.444776			0.222388	−	0.974958i	$-0.428615\pi$
$37$	137040.			0.444776			0.222388	+	0.974958i	$0.428615\pi$
$38$		0			0
$39$	−323332.			−0.872816
$40$		0			0
$41$	−288875.	+	500346.i	−0.654585	+	1.13377i	0.327413	+	0.944881i	$0.393823\pi$
$41$	−288875.	+	500346.i	−0.654585	+	1.13377i	−0.981998	+	0.188893i	$0.939510\pi$
$42$		0			0
$43$	−395948.	+	685803.i	−0.759450	+	1.31541i	0.183682	+	0.982986i	$0.441198\pi$
$43$	−395948.	+	685803.i	−0.759450	+	1.31541i	−0.943132	+	0.332420i	$0.892135\pi$
$44$		0			0
$45$	−99191.6			−0.162267
$46$		0			0
$47$	−322935.	−	559339.i	−0.453703	−	0.785837i	0.544909	−	0.838495i	$-0.316564\pi$
$47$	−322935.	−	559339.i	−0.453703	−	0.785837i	−0.998613	+	0.0526578i	$0.983231\pi$
$48$		0			0
$49$	−769889.			−0.934850
$50$		0			0
$51$	625858.	+	1.08402e6i	0.660663	+	1.14430i
$52$		0			0
$53$	284590.	+	492924.i	0.262575	+	0.454794i	0.966925	−	0.255059i	$-0.0820949\pi$
$53$	284590.	+	492924.i	0.262575	+	0.454794i	−0.704350	+	0.709853i	$0.748762\pi$
$54$		0			0
$55$	385908.	−	668412.i	0.312762	−	0.541720i
$56$		0			0
$57$	−824039.	+	885886.i	−0.589367	+	0.633601i
$58$		0			0
$59$	917429.	−	1.58903e6i	0.581554	−	1.00728i	−0.413741	−	0.910395i	$-0.635778\pi$
$59$	917429.	−	1.58903e6i	0.581554	−	1.00728i	0.995295	−	0.0968872i	$-0.0308886\pi$
$60$		0			0
$61$	451671.	+	782317.i	0.254781	+	0.441294i	0.964836	−	0.262852i	$-0.0846632\pi$
$61$	451671.	+	782317.i	0.254781	+	0.441294i	−0.710055	+	0.704146i	$0.751330\pi$
$62$		0			0
$63$	63625.8	+	110203.i	0.0320584	+	0.0555267i
$64$		0			0
$65$	1.44262e6			0.651562
$66$		0			0
$67$	−952079.	−	1.64905e6i	−0.386733	−	0.669841i	0.605275	−	0.796016i	$-0.293063\pi$
$67$	−952079.	−	1.64905e6i	−0.386733	−	0.669841i	−0.992008	+	0.126175i	$0.959730\pi$
$68$		0			0
$69$	195293.			0.0715674
$70$		0			0
$71$	−1.01191e6	+	1.75267e6i	−0.335533	+	0.581161i	−0.983587	−	0.180434i	$-0.942250\pi$
$71$	−1.01191e6	+	1.75267e6i	−0.335533	+	0.581161i	0.648054	+	0.761595i	$0.275583\pi$
$72$		0			0
$73$	2.40496e6	−	4.16551e6i	0.723565	−	1.25325i	−0.235997	−	0.971754i	$-0.575835\pi$
$73$	2.40496e6	−	4.16551e6i	0.723565	−	1.25325i	0.959562	−	0.281498i	$-0.0908313\pi$
$74$		0			0
$75$	1.84227e6			0.504242
$76$		0			0
$77$	−990153.			−0.247164
$78$		0			0
$79$	1.10602e6	−	1.91568e6i	0.252388	−	0.437149i	−0.711795	−	0.702387i	$-0.752117\pi$
$79$	1.10602e6	−	1.91568e6i	0.252388	−	0.437149i	0.964183	+	0.265239i	$0.0854508\pi$
$80$		0			0
$81$	1.63985e6	−	2.84030e6i	0.342852	−	0.593837i
$82$		0			0
$83$	−320945.			−0.0616108			−0.0308054	−	0.999525i	$-0.509807\pi$
$83$	−320945.			−0.0616108			−0.0308054	+	0.999525i	$0.509807\pi$
$84$		0			0
$85$	−2.79241e6	−	4.83660e6i	−0.493189	−	0.854229i
$86$		0			0
$87$	7.89596e6			1.28555
$88$		0			0
$89$	−3.24510e6	−	5.62069e6i	−0.487937	−	0.845132i	0.511967	−	0.859005i	$-0.328917\pi$
$89$	−3.24510e6	−	5.62069e6i	−0.487937	−	0.845132i	−0.999904	+	0.0138736i	$0.995584\pi$
$90$		0			0
$91$	−925361.	−	1.60277e6i	−0.128726	−	0.222960i
$92$		0			0
$93$	−4.86649e6	+	8.42900e6i	−0.627372	+	1.08664i
$94$		0			0
$95$	3.67664e6	−	3.95259e6i	0.439966	−	0.472987i
$96$		0			0
$97$	646270.	−	1.11937e6i	0.0718974	−	0.124530i	−0.827835	−	0.560971i	$-0.810428\pi$
$97$	646270.	−	1.11937e6i	0.0718974	−	0.124530i	0.899733	+	0.436441i	$0.143761\pi$
$98$		0			0
$99$	−1.17418e6	−	2.03374e6i	−0.121622	−	0.210655i
$100$		0			0
$101$	−1.25640e6	−	2.17614e6i	−0.121339	−	0.210166i	0.798957	−	0.601389i	$-0.205386\pi$
$101$	−1.25640e6	−	2.17614e6i	−0.121339	−	0.210166i	−0.920296	+	0.391223i	$0.872052\pi$
$102$		0			0
$103$	−7.13067e6			−0.642984			−0.321492	−	0.946912i	$-0.604184\pi$
$103$	−7.13067e6			−0.642984			−0.321492	+	0.946912i	$0.604184\pi$
$104$		0			0
$105$	846235.	+	1.46572e6i	0.0713392	+	0.123563i
$106$		0			0
$107$	2.34863e7			1.85341			0.926705	−	0.375789i	$-0.122628\pi$
$107$	2.34863e7			1.85341			0.926705	+	0.375789i	$0.122628\pi$
$108$		0			0
$109$	−643393.	+	1.11439e6i	−0.0475865	+	0.0824222i	−0.888838	−	0.458222i	$-0.848486\pi$
$109$	−643393.	+	1.11439e6i	−0.0475865	+	0.0824222i	0.841251	+	0.540645i	$0.181820\pi$
$110$		0			0
$111$	2.77284e6	−	4.80271e6i	0.192440	−	0.333316i
$112$		0			0
$113$	−1.53109e7			−0.998219			−0.499110	−	0.866539i	$-0.666340\pi$
$113$	−1.53109e7			−0.998219			−0.499110	+	0.866539i	$0.666340\pi$
$114$		0			0
$115$	−871346.			−0.0534255
$116$		0			0
$117$	2.19469e6	−	3.80132e6i	0.126684	−	0.219424i
$118$		0			0
$119$	−3.58235e6	+	6.20481e6i	−0.194874	+	0.337532i
$120$		0			0
$121$	−1.21444e6			−0.0623200
$122$		0			0
$123$	1.16901e7	+	2.02478e7i	0.566435	+	0.981093i
$124$		0			0
$125$	−2.23257e7			−1.02240
$126$		0			0
$127$	1.65115e7	+	2.85987e7i	0.715275	+	1.23889i	0.962853	+	0.270025i	$0.0870321\pi$
$127$	1.65115e7	+	2.85987e7i	0.715275	+	1.23889i	−0.247578	+	0.968868i	$0.579635\pi$
$128$		0			0
$129$	1.60231e7	+	2.77528e7i	0.657178	+	1.13827i
$130$		0			0
$131$	4.53185e6	−	7.84940e6i	0.176127	−	0.305061i	−0.764424	−	0.644714i	$-0.776976\pi$
$131$	4.53185e6	−	7.84940e6i	0.176127	−	0.305061i	0.940551	+	0.339653i	$0.110310\pi$
$132$		0			0
$133$	−6.74974e6	−	1.54943e6i	−0.248775	−	0.0571074i
$134$		0			0
$135$	−9.99689e6	+	1.73151e7i	−0.349701	+	0.605700i
$136$		0			0
$137$	−1.23593e7	−	2.14070e7i	−0.410651	−	0.711269i	0.584310	−	0.811531i	$-0.301365\pi$
$137$	−1.23593e7	−	2.14070e7i	−0.410651	−	0.711269i	−0.994961	+	0.100262i	$0.968032\pi$
$138$		0			0
$139$	−1.68706e7	−	2.92208e7i	−0.532819	−	0.922870i	−0.999266	−	0.0383203i	$-0.987799\pi$
$139$	−1.68706e7	−	2.92208e7i	−0.532819	−	0.922870i	0.466446	−	0.884549i	$-0.345534\pi$
$140$		0			0
$141$	−2.61368e7			−0.785210
$142$		0			0
$143$	1.70770e7	+	2.95783e7i	0.488356	+	0.845857i
$144$		0			0
$145$	−3.52297e7			−0.959667
$146$		0			0
$147$	−1.55778e7	+	2.69816e7i	−0.404479	+	0.700578i
$148$		0			0
$149$	−2.35083e7	+	4.07175e7i	−0.582195	+	1.00839i	0.413024	+	0.910720i	$0.364473\pi$
$149$	−2.35083e7	+	4.07175e7i	−0.582195	+	1.00839i	−0.995219	+	0.0976709i	$0.968861\pi$
$150$		0			0
$151$	−1.79017e7			−0.423131			−0.211565	−	0.977364i	$-0.567856\pi$
$151$	−1.79017e7			−0.423131			−0.211565	+	0.977364i	$0.567856\pi$
$152$		0			0
$153$	−1.69926e7			−0.383566
$154$		0			0
$155$	2.17130e7	−	3.76080e7i	0.468337	−	0.811183i
$156$		0			0
$157$	−3.11728e7	+	5.39929e7i	−0.642876	+	1.11349i	0.341912	+	0.939732i	$0.388925\pi$
$157$	−3.11728e7	+	5.39929e7i	−0.642876	+	1.11349i	−0.984788	+	0.173761i	$0.944408\pi$
$158$		0			0
$159$	2.30334e7			0.454431
$160$		0			0
$161$	558919.	+	968077.i	0.0105550	+	0.0182818i
$162$		0			0
$163$	6.02075e7			1.08891			0.544457	−	0.838789i	$-0.316736\pi$
$163$	6.02075e7			1.08891			0.544457	+	0.838789i	$0.316736\pi$
$164$		0			0
$165$	−1.56168e7	−	2.70491e7i	−0.270644	−	0.468769i
$166$		0			0
$167$	6.04738e6	+	1.04744e7i	0.100475	+	0.174028i	0.911881	−	0.410456i	$-0.134630\pi$
$167$	6.04738e6	+	1.04744e7i	0.100475	+	0.174028i	−0.811405	+	0.584484i	$0.801297\pi$
$168$		0			0
$169$	−544879.	+	943758.i	−0.00868353	+	0.0150403i
$170$		0			0
$171$	−4.82174e6	−	1.57011e7i	−0.0737424	−	0.240129i
$172$		0			0
$173$	5.73748e7	−	9.93760e7i	0.842480	−	1.45922i	−0.0453118	−	0.998973i	$-0.514428\pi$
$173$	5.73748e7	−	9.93760e7i	0.842480	−	1.45922i	0.887792	−	0.460245i	$-0.152239\pi$
$174$		0			0
$175$	5.27249e6	+	9.13222e6i	0.0743674	+	0.128808i
$176$		0			0
$177$	−3.71262e7	−	6.43045e7i	−0.503239	−	0.871635i
$178$		0			0
$179$	4.34383e7			0.566092			0.283046	−	0.959106i	$-0.408655\pi$
$179$	4.34383e7			0.566092			0.283046	+	0.959106i	$0.408655\pi$
$180$		0			0
$181$	7.45663e7	+	1.29153e8i	0.934690	+	1.61893i	0.775185	+	0.631734i	$0.217656\pi$
$181$	7.45663e7	+	1.29153e8i	0.934690	+	1.61893i	0.159505	+	0.987197i	$0.449010\pi$
$182$		0			0
$183$	3.65561e7			0.440942
$184$		0			0
$185$	−1.23717e7	+	2.14284e7i	−0.143657	+	0.248822i
$186$		0			0
$187$	6.61103e7	−	1.14506e8i	0.739305	−	1.28051i
$188$		0			0
$189$	2.56498e7			0.276355
$190$		0			0
$191$	6.31695e7			0.655980			0.327990	−	0.944681i	$-0.393629\pi$
$191$	6.31695e7			0.655980			0.327990	+	0.944681i	$0.393629\pi$
$192$		0			0
$193$	4.42125e7	−	7.65783e7i	0.442685	−	0.766752i	−0.555203	−	0.831715i	$-0.687359\pi$
$193$	4.42125e7	−	7.65783e7i	0.442685	−	0.766752i	0.997888	+	0.0649626i	$0.0206928\pi$
$194$		0			0
$195$	2.91898e7	−	5.05582e7i	0.281910	−	0.488282i
$196$		0			0
$197$	−1.28557e8			−1.19802			−0.599009	−	0.800742i	$-0.704439\pi$
$197$	−1.28557e8			−1.19802			−0.599009	+	0.800742i	$0.704439\pi$
$198$		0			0
$199$	−7.53075e7	−	1.30436e8i	−0.677411	−	1.17331i	−0.975758	−	0.218853i	$-0.929768\pi$
$199$	−7.53075e7	−	1.30436e8i	−0.677411	−	1.17331i	0.298347	−	0.954458i	$-0.403565\pi$
$200$		0			0
$201$	−7.70568e7			−0.669306
$202$		0			0
$203$	2.25979e7	+	3.91406e7i	0.189597	+	0.328391i
$204$		0			0
$205$	−5.21581e7	−	9.03404e7i	−0.422847	−	0.732392i
$206$		0			0
$207$	−1.32560e6	+	2.29600e6i	−0.0103876	+	0.0179918i
$208$		0			0
$209$	1.24563e8	+	2.85940e7i	0.943791	+	0.216652i
$210$		0			0
$211$	−1.95225e7	+	3.38140e7i	−0.143069	+	0.247804i	−0.928651	−	0.370955i	$-0.879031\pi$
$211$	−1.95225e7	+	3.38140e7i	−0.143069	+	0.247804i	0.785582	+	0.618758i	$0.212364\pi$
$212$		0			0
$213$	4.09495e7	+	7.09266e7i	0.290349	+	0.502898i
$214$		0			0
$215$	−7.14909e7	−	1.23826e8i	−0.490587	−	0.849722i
$216$		0			0
$217$	−5.57106e7			−0.370108
$218$		0			0
$219$	−9.73231e7	−	1.68569e8i	−0.626126	−	1.08448i
$220$		0			0
$221$	2.47137e8			1.54016
$222$		0			0
$223$	−9.21672e7	+	1.59638e8i	−0.556557	+	0.963985i	0.441224	+	0.897397i	$0.354545\pi$
$223$	−9.21672e7	+	1.59638e8i	−0.556557	+	0.963985i	−0.997781	+	0.0665876i	$0.978789\pi$
$224$		0			0
$225$	−1.25048e7	+	2.16590e7i	−0.0731879	+	0.126765i
$226$		0			0
$227$	1.20797e8			0.685435			0.342718	−	0.939438i	$-0.388653\pi$
$227$	1.20797e8			0.685435			0.342718	+	0.939438i	$0.388653\pi$
$228$		0			0
$229$	−3.02797e8			−1.66620			−0.833100	−	0.553122i	$-0.813436\pi$
$229$	−3.02797e8			−1.66620			−0.833100	+	0.553122i	$0.813436\pi$
$230$		0			0
$231$	−2.00346e7	+	3.47009e7i	−0.106940	+	0.185225i
$232$		0			0
$233$	−5.63267e7	+	9.75607e7i	−0.291722	+	0.505277i	−0.974217	−	0.225613i	$-0.927561\pi$
$233$	−5.63267e7	+	9.75607e7i	−0.291722	+	0.505277i	0.682495	+	0.730890i	$0.260895\pi$
$234$		0			0
$235$	1.16616e8			0.586164
$236$		0			0
$237$	−4.47581e7	−	7.75233e7i	−0.218400	−	0.378280i
$238$		0			0
$239$	3.51151e8			1.66380			0.831899	−	0.554927i	$-0.187254\pi$
$239$	3.51151e8			1.66380			0.831899	+	0.554927i	$0.187254\pi$
$240$		0			0
$241$	−8.72490e7	−	1.51120e8i	−0.401514	−	0.695443i	0.592395	−	0.805648i	$-0.298183\pi$
$241$	−8.72490e7	−	1.51120e8i	−0.401514	−	0.695443i	−0.993909	+	0.110205i	$0.964849\pi$
$242$		0			0
$243$	5.47272e7	+	9.47903e7i	0.244670	+	0.423781i
$244$		0			0
$245$	6.95041e7	−	1.20385e8i	0.301946	−	0.522986i
$246$		0			0
$247$	7.01264e7	+	2.28354e8i	0.296102	+	0.964205i
$248$		0			0
$249$	−6.49394e6	+	1.12478e7i	−0.0266570	+	0.0461712i
$250$		0			0
$251$	2.34125e7	+	4.05517e7i	0.0934524	+	0.161864i	0.908962	−	0.416880i	$-0.136876\pi$
$251$	2.34125e7	+	4.05517e7i	0.0934524	+	0.161864i	−0.815509	+	0.578744i	$0.803543\pi$
$252$		0			0
$253$	−1.03146e7	−	1.78653e7i	−0.0400432	−	0.0693568i
$254$		0			0
$255$	−2.26005e8			−0.853547
$256$		0			0
$257$	1.79401e8	+	3.10732e8i	0.659265	+	1.14188i	0.980806	+	0.194985i	$0.0624657\pi$
$257$	1.79401e8	+	3.10732e8i	0.659265	+	1.14188i	−0.321542	+	0.946895i	$0.604201\pi$
$258$		0			0
$259$	3.17430e7			0.113527
$260$		0			0
$261$	−5.35956e7	+	9.28303e7i	−0.186590	+	0.323183i
$262$		0			0
$263$	−1.75785e8	+	3.04469e8i	−0.595850	+	1.03204i	0.397577	+	0.917569i	$0.369851\pi$
$263$	−1.75785e8	+	3.04469e8i	−0.595850	+	1.03204i	−0.993426	+	0.114473i	$0.963482\pi$
$264$		0			0
$265$	−1.02769e8			−0.339235
$266$		0			0
$267$	−2.62644e8			−0.844457
$268$		0			0
$269$	−3.62222e7	+	6.27387e7i	−0.113460	+	0.196518i	−0.917163	−	0.398512i	$-0.869527\pi$
$269$	−3.62222e7	+	6.27387e7i	−0.113460	+	0.196518i	0.803703	+	0.595030i	$0.202860\pi$
$270$		0			0
$271$	−2.08845e8	+	3.61730e8i	−0.637428	+	1.10406i	0.348567	+	0.937284i	$0.386668\pi$
$271$	−2.08845e8	+	3.61730e8i	−0.637428	+	1.10406i	−0.985995	+	0.166774i	$0.946665\pi$
$272$		0			0
$273$	−7.48944e7			−0.222782
$274$		0			0
$275$	−9.73009e7	−	1.68530e8i	−0.282132	−	0.488667i
$276$		0			0
$277$	3.75868e8			1.06257			0.531283	−	0.847194i	$-0.321710\pi$
$277$	3.75868e8			1.06257			0.531283	+	0.847194i	$0.321710\pi$
$278$		0			0
$279$	−6.60648e7	−	1.14428e8i	−0.182119	−	0.315439i
$280$		0			0
$281$	1.23185e8	+	2.13362e8i	0.331195	+	0.573647i	0.982747	−	0.184957i	$-0.0592147\pi$
$281$	1.23185e8	+	2.13362e8i	0.331195	+	0.573647i	−0.651551	+	0.758605i	$0.725881\pi$
$282$		0			0
$283$	−1.42331e8	+	2.46524e8i	−0.373289	+	0.646556i	−0.990069	−	0.140579i	$-0.955103\pi$
$283$	−1.42331e8	+	2.46524e8i	−0.373289	+	0.646556i	0.616780	+	0.787136i	$0.288437\pi$
$284$		0			0
$285$	−6.41300e7	−	2.08828e8i	−0.164098	−	0.534357i
$286$		0			0
$287$	−6.69129e7	+	1.15897e8i	−0.167080	+	0.289390i
$288$		0			0
$289$	−2.73202e8	−	4.73200e8i	−0.665796	−	1.15319i
$290$		0			0
$291$	−2.61531e7	−	4.52984e7i	−0.0622153	−	0.107760i
$292$		0			0
$293$	4.38930e8			1.01943			0.509716	−	0.860343i	$-0.329750\pi$
$293$	4.38930e8			1.01943			0.509716	+	0.860343i	$0.329750\pi$
$294$		0			0
$295$	1.65647e8	+	2.86910e8i	0.375671	+	0.650681i
$296$		0			0
$297$	−4.73352e8			−1.04842
$298$		0			0
$299$	1.92792e7	−	3.33926e7i	0.0417100	−	0.0722438i
$300$		0			0
$301$	−9.17148e7	+	1.58855e8i	−0.193846	+	0.335751i
$302$		0			0
$303$	−1.01687e8			−0.209998
$304$		0			0
$305$	−1.63104e8			−0.329166
$306$		0			0
$307$	−4.36631e7	+	7.56266e7i	−0.0861251	+	0.149173i	−0.905870	−	0.423556i	$-0.860782\pi$
$307$	−4.36631e7	+	7.56266e7i	−0.0861251	+	0.149173i	0.819745	+	0.572729i	$0.194115\pi$
$308$		0			0
$309$	−1.44281e8	+	2.49902e8i	−0.278198	+	0.481853i
$310$		0			0
$311$	7.79745e7			0.146991			0.0734956	−	0.997296i	$-0.476585\pi$
$311$	7.79745e7			0.146991			0.0734956	+	0.997296i	$0.476585\pi$
$312$		0			0
$313$	2.93111e8	+	5.07684e8i	0.540291	+	0.935811i	0.998887	+	0.0471663i	$0.0150191\pi$
$313$	2.93111e8	+	5.07684e8i	0.540291	+	0.935811i	−0.458596	+	0.888645i	$0.651648\pi$
$314$		0			0
$315$	−2.29761e7			−0.0414179
$316$		0			0
$317$	1.75593e8	+	3.04136e8i	0.309599	+	0.536241i	0.978275	−	0.207313i	$-0.0664718\pi$
$317$	1.75593e8	+	3.04136e8i	0.309599	+	0.536241i	−0.668676	+	0.743554i	$0.733139\pi$
$318$		0			0
$319$	−4.17031e8	−	7.22319e8i	−0.719285	−	1.24584i
$320$		0			0
$321$	4.75218e8	−	8.23102e8i	0.801910	−	1.38895i
$322$		0			0
$323$	6.29850e8	−	6.77122e8i	1.03999	−	1.11804i
$324$		0			0
$325$	1.81868e8	−	3.15004e8i	0.293876	−	0.509008i
$326$		0			0
$327$	2.60366e7	+	4.50967e7i	0.0411782	+	0.0713227i
$328$		0			0
$329$	−7.48023e7	−	1.29561e8i	−0.115806	−	0.200581i
$330$		0			0
$331$	−8.37649e8			−1.26959			−0.634796	−	0.772680i	$-0.718916\pi$
$331$	−8.37649e8			−1.26959			−0.634796	+	0.772680i	$0.718916\pi$
$332$		0			0
$333$	3.76426e7	+	6.51989e7i	0.0558631	+	0.0967577i
$334$		0			0
$335$	3.43807e8			0.499641
$336$		0			0
$337$	2.64010e8	−	4.57278e8i	0.375764	−	0.650843i	−0.614677	−	0.788779i	$-0.710714\pi$
$337$	2.64010e8	−	4.57278e8i	0.375764	−	0.650843i	0.990441	+	0.137936i	$0.0440469\pi$
$338$		0			0
$339$	−3.09798e8	+	5.36586e8i	−0.431897	+	0.748067i
$340$		0			0
$341$	1.02811e9			1.40410
$342$		0			0
$343$	−3.69092e8			−0.493861
$344$		0			0
$345$	−1.76307e7	+	3.05372e7i	−0.0231154	+	0.0400371i
$346$		0			0
$347$	3.25656e8	−	5.64052e8i	0.418413	−	0.724713i	−0.577367	−	0.816485i	$-0.695920\pi$
$347$	3.25656e8	−	5.64052e8i	0.418413	−	0.724713i	0.995780	+	0.0917720i	$0.0292531\pi$
$348$		0			0
$349$	6.72067e8			0.846298			0.423149	−	0.906060i	$-0.360925\pi$
$349$	6.72067e8			0.846298			0.423149	+	0.906060i	$0.360925\pi$
$350$		0			0
$351$	−4.42378e8	−	7.66221e8i	−0.546032	−	0.945756i
$352$		0			0
$353$	9.38977e8			1.13617			0.568085	−	0.822970i	$-0.307684\pi$
$353$	9.38977e8			1.13617			0.568085	+	0.822970i	$0.307684\pi$
$354$		0			0
$355$	−1.82706e8	−	3.16456e8i	−0.216747	−	0.375417i
$356$		0			0
$357$	1.44969e8	+	2.51094e8i	0.168631	+	0.292078i
$358$		0			0
$359$	5.30529e8	−	9.18903e8i	0.605171	−	1.04819i	−0.386853	−	0.922141i	$-0.626438\pi$
$359$	5.30529e8	−	9.18903e8i	0.605171	−	1.04819i	0.992024	−	0.126046i	$-0.0402288\pi$
$360$		0			0
$361$	8.04382e8	+	3.89842e8i	0.899885	+	0.436128i
$362$		0			0
$363$	−2.45728e7	+	4.25613e7i	−0.0269638	+	0.0467027i
$364$		0			0
$365$	4.34230e8	+	7.52109e8i	0.467407	+	0.809572i
$366$		0			0
$367$	−4.50804e8	−	7.80815e8i	−0.476054	−	0.824550i	0.523569	−	0.851983i	$-0.324600\pi$
$367$	−4.50804e8	−	7.80815e8i	−0.476054	−	0.824550i	−0.999624	+	0.0274330i	$0.991267\pi$
$368$		0			0
$369$	−3.17396e8			−0.328859
$370$		0			0
$371$	6.59204e7	+	1.14177e8i	0.0670210	+	0.116084i
$372$		0			0
$373$	−1.22701e9			−1.22424			−0.612122	−	0.790764i	$-0.709684\pi$
$373$	−1.22701e9			−1.22424			−0.612122	+	0.790764i	$0.709684\pi$
$374$		0			0
$375$	−4.51734e8	+	7.82426e8i	−0.442357	+	0.766186i
$376$		0			0
$377$	7.79484e8	−	1.35011e9i	0.749225	−	1.29770i
$378$		0			0
$379$	−1.48301e9			−1.39929			−0.699645	−	0.714491i	$-0.746658\pi$
$379$	−1.48301e9			−1.39929			−0.699645	+	0.714491i	$0.746658\pi$
$380$		0			0
$381$	1.33636e9			1.23790
$382$		0			0
$383$	2.27944e8	−	3.94810e8i	0.207315	−	0.359081i	−0.743553	−	0.668678i	$-0.766861\pi$
$383$	2.27944e8	−	3.94810e8i	0.207315	−	0.359081i	0.950868	+	0.309597i	$0.100194\pi$
$384$		0			0
$385$	8.93891e7	−	1.54826e8i	0.0798311	−	0.138271i
$386$		0			0
$387$	−4.35042e8			−0.381543
$388$		0			0
$389$	−4.25573e8	−	7.37115e8i	−0.366565	−	0.634909i	0.622461	−	0.782651i	$-0.286133\pi$
$389$	−4.25573e8	−	7.37115e8i	−0.366565	−	0.634909i	−0.989026	+	0.147742i	$0.952800\pi$
$390$		0			0
$391$	−1.49271e8			−0.126287
$392$		0			0
$393$	−1.83394e8	−	3.17647e8i	−0.152409	−	0.263980i
$394$		0			0
$395$	1.99699e8	+	3.45888e8i	0.163037	+	0.282388i
$396$		0			0
$397$	−7.37987e8	+	1.27823e9i	−0.591946	+	1.02528i	0.402024	+	0.915629i	$0.368307\pi$
$397$	−7.37987e8	+	1.27823e9i	−0.591946	+	1.02528i	−0.993970	+	0.109651i	$0.965027\pi$
$398$		0			0
$399$	−1.90875e8	+	2.05200e8i	−0.150433	+	0.161724i
$400$		0			0
$401$	−6.56885e8	+	1.13776e9i	−0.508726	+	0.881140i	0.491222	+	0.871034i	$0.336550\pi$
$401$	−6.56885e8	+	1.13776e9i	−0.508726	+	0.881140i	−0.999949	+	0.0101060i	$0.996783\pi$
$402$		0			0
$403$	9.60833e8	+	1.66421e9i	0.731274	+	1.26660i
$404$		0			0
$405$	2.96085e8	+	5.12834e8i	0.221474	+	0.383605i
$406$		0			0
$407$	−5.85799e8			−0.430694
$408$		0			0
$409$	−5.25544e8	−	9.10269e8i	−0.379820	−	0.657867i	0.611216	−	0.791464i	$-0.290681\pi$
$409$	−5.25544e8	−	9.10269e8i	−0.379820	−	0.657867i	−0.991036	+	0.133597i	$0.957347\pi$
$410$		0			0
$411$	−1.00031e9			−0.710701
$412$		0			0
$413$	2.12507e8	−	3.68073e8i	0.148439	−	0.257104i
$414$		0			0
$415$	2.89743e7	−	5.01849e7i	0.0198996	−	0.0344671i
$416$		0			0
$417$	−1.36543e9			−0.922133
$418$		0			0
$419$	4.42327e8			0.293761			0.146881	−	0.989154i	$-0.453077\pi$
$419$	4.42327e8			0.293761			0.146881	+	0.989154i	$0.453077\pi$
$420$		0			0
$421$	−7.88591e8	+	1.36588e9i	−0.515068	+	0.892124i	0.484779	+	0.874637i	$0.338900\pi$
$421$	−7.88591e8	+	1.36588e9i	−0.515068	+	0.892124i	−0.999847	+	0.0174873i	$0.994433\pi$
$422$		0			0
$423$	1.77410e8	−	3.07282e8i	0.113969	−	0.197400i
$424$		0			0
$425$	−1.40813e9			−0.889778
$426$		0			0
$427$	1.04622e8	+	1.81210e8i	0.0650317	+	0.112638i
$428$		0			0
$429$	1.38214e9			0.845182
$430$		0			0
$431$	1.20485e8	+	2.08687e8i	0.0724877	+	0.125552i	0.899991	−	0.435908i	$-0.143573\pi$
$431$	1.20485e8	+	2.08687e8i	0.0724877	+	0.125552i	−0.827503	+	0.561461i	$0.810240\pi$
$432$		0			0
$433$	3.09599e8	+	5.36242e8i	0.183270	+	0.317434i	0.942992	−	0.332815i	$-0.107998\pi$
$433$	3.09599e8	+	5.36242e8i	0.183270	+	0.317434i	−0.759722	+	0.650248i	$0.774665\pi$
$434$		0			0
$435$	−7.12832e8	+	1.23466e9i	−0.415217	+	0.719176i
$436$		0			0
$437$	−4.23564e7	−	1.37926e8i	−0.0242792	−	0.0790608i
$438$		0			0
$439$	8.79216e8	−	1.52285e9i	0.495986	−	0.859074i	−0.504003	−	0.863702i	$-0.668140\pi$
$439$	8.79216e8	−	1.52285e9i	0.495986	−	0.859074i	0.999989	+	0.00462831i	$0.00147324\pi$
$440$		0			0
$441$	−2.11476e8	−	3.66287e8i	−0.117416	−	0.203370i
$442$		0			0
$443$	1.62487e9	+	2.81435e9i	0.887984	+	1.53803i	0.842255	+	0.539079i	$0.181228\pi$
$443$	1.62487e9	+	2.81435e9i	0.887984	+	1.53803i	0.0457290	+	0.998954i	$0.485439\pi$
$444$		0			0
$445$	1.17185e9			0.630392
$446$		0			0
$447$	9.51324e8	+	1.64774e9i	0.503793	+	0.872595i
$448$		0			0
$449$	2.08405e9			1.08654			0.543269	−	0.839558i	$-0.317186\pi$
$449$	2.08405e9			1.08654			0.543269	+	0.839558i	$0.317186\pi$
$450$		0			0
$451$	1.23484e9	−	2.13881e9i	0.633860	−	1.09788i
$452$		0			0
$453$	−3.62220e8	+	6.27383e8i	−0.183075	+	0.317095i
$454$		0			0
$455$	3.34159e8			0.166308
$456$		0			0
$457$	−2.85892e9			−1.40119			−0.700593	−	0.713561i	$-0.747081\pi$
$457$	−2.85892e9			−1.40119			−0.700593	+	0.713561i	$0.747081\pi$
$458$		0			0
$459$	−1.71258e9	+	2.96627e9i	−0.826620	+	1.43175i
$460$		0			0
$461$	−1.40910e9	+	2.44063e9i	−0.669865	+	1.16024i	0.308077	+	0.951362i	$0.400315\pi$
$461$	−1.40910e9	+	2.44063e9i	−0.669865	+	1.16024i	−0.977942	+	0.208879i	$0.933019\pi$
$462$		0			0
$463$	3.80732e7			0.0178273			0.00891365	−	0.999960i	$-0.497163\pi$
$463$	3.80732e7			0.0178273			0.00891365	+	0.999960i	$0.497163\pi$
$464$		0			0
$465$	−8.78674e8	−	1.52191e9i	−0.405268	−	0.701945i
$466$		0			0
$467$	−2.56216e9			−1.16412			−0.582058	−	0.813147i	$-0.697752\pi$
$467$	−2.56216e9			−1.16412			−0.582058	+	0.813147i	$0.697752\pi$
$468$		0			0
$469$	−2.20533e8	−	3.81974e8i	−0.0987117	−	0.170974i
$470$		0			0
$471$	1.26149e9	+	2.18497e9i	0.556302	+	0.963544i
$472$		0			0
$473$	1.69255e9	−	2.93158e9i	0.735405	−	1.27376i
$474$		0			0
$475$	−3.99564e8	−	1.30111e9i	−0.171064	−	0.557039i
$476$		0			0
$477$	−1.56344e8	+	2.70796e8i	−0.0659580	+	0.114243i
$478$		0			0
$479$	1.20105e9	+	2.08027e9i	0.499328	+	0.864861i	1.00000		0.000776255i	$-0.000247090\pi$
$479$	1.20105e9	+	2.08027e9i	0.499328	+	0.864861i	−0.500672	+	0.865637i	$0.666914\pi$
$480$		0			0
$481$	−5.47467e8	−	9.48240e8i	−0.224311	−	0.388517i
$482$		0			0
$483$	4.52363e7			0.0182672
$484$		0			0
$485$	1.16688e8	+	2.02110e8i	0.0464441	+	0.0804435i
$486$		0			0
$487$	−1.11604e9			−0.437851			−0.218926	−	0.975742i	$-0.570255\pi$
$487$	−1.11604e9			−0.437851			−0.218926	+	0.975742i	$0.570255\pi$
$488$		0			0
$489$	1.21823e9	−	2.11003e9i	0.471138	−	0.816034i
$490$		0			0
$491$	1.72765e9	−	2.99238e9i	0.658674	−	1.14086i	−0.322285	−	0.946643i	$-0.604451\pi$
$491$	1.72765e9	−	2.99238e9i	0.658674	−	1.14086i	0.980959	−	0.194215i	$-0.0622158\pi$
$492$		0			0
$493$	−6.03523e9			−2.26845
$494$		0			0
$495$	4.24010e8			0.157130
$496$		0			0
$497$	−2.34391e8	+	4.05977e8i	−0.0856433	+	0.148338i
$498$		0			0
$499$	1.56731e8	−	2.71467e8i	0.0564682	−	0.0978058i	−0.836409	−	0.548105i	$-0.815349\pi$
$499$	1.56731e8	−	2.71467e8i	0.0564682	−	0.0978058i	0.892878	+	0.450299i	$0.148683\pi$
$500$		0			0
$501$	4.89447e8			0.173890
$502$		0			0
$503$	3.58080e8	+	6.20213e8i	0.125456	+	0.217296i	0.921911	−	0.387401i	$-0.126627\pi$
$503$	3.58080e8	+	6.20213e8i	0.125456	+	0.217296i	−0.796455	+	0.604698i	$0.793294\pi$
$504$		0			0
$505$	4.53700e8			0.156765
$506$		0			0
$507$	2.20500e7	+	3.81917e7i	0.00751416	+	0.0130149i
$508$		0			0
$509$	1.59794e9	+	2.76771e9i	0.537091	+	0.930269i	0.999059	+	0.0433727i	$0.0138103\pi$
$509$	1.59794e9	+	2.76771e9i	0.537091	+	0.930269i	−0.461968	+	0.886897i	$0.652856\pi$
$510$		0			0
$511$	5.57068e8	−	9.64870e8i	0.184686	−	0.319886i
$512$		0			0
$513$	−3.22678e9	−	7.40722e8i	−1.05526	−	0.242239i
$514$		0			0
$515$	6.43743e8	−	1.11500e9i	0.207677	−	0.359706i
$516$		0			0
$517$	1.38044e9	+	2.39098e9i	0.439339	+	0.760957i
$518$		0			0
$519$	−2.32182e9	−	4.02152e9i	−0.729027	−	1.26271i
$520$		0			0
$521$	−3.90643e9			−1.21018			−0.605088	−	0.796159i	$-0.706862\pi$
$521$	−3.90643e9			−1.21018			−0.605088	+	0.796159i	$0.706862\pi$
$522$		0			0
$523$	−4.98340e8	−	8.63150e8i	−0.152324	−	0.263834i	0.779757	−	0.626082i	$-0.215343\pi$
$523$	−4.98340e8	−	8.63150e8i	−0.152324	−	0.263834i	−0.932082	+	0.362248i	$0.882009\pi$
$524$		0			0
$525$	4.26731e8			0.128705
$526$		0			0
$527$	3.71967e9	−	6.44266e9i	1.10705	−	1.91747i
$528$		0			0
$529$	1.69077e9	−	2.92850e9i	0.496580	−	0.860102i
$530$		0			0
$531$	1.00801e9			0.292169
$532$		0			0
$533$	4.61615e9			1.32049
$534$		0			0
$535$	−2.12030e9	+	3.67246e9i	−0.598630	+	1.03686i
$536$		0			0
$537$	8.78923e8	−	1.52234e9i	0.244929	−	0.424230i
$538$		0			0
$539$	3.29102e9			0.905251
$540$		0			0
$541$	9.33447e8	+	1.61678e9i	0.253454	+	0.438995i	0.964474	−	0.264176i	$-0.0851000\pi$
$541$	9.33447e8	+	1.61678e9i	0.253454	+	0.438995i	−0.711020	+	0.703171i	$0.751767\pi$
$542$		0			0
$543$	6.03505e9			1.61764
$544$		0			0
$545$	−1.16168e8	−	2.01210e8i	−0.0307398	−	0.0532428i
$546$		0			0
$547$	1.27252e9	+	2.20406e9i	0.332436	+	0.575796i	0.982989	−	0.183665i	$-0.0587962\pi$
$547$	1.27252e9	+	2.20406e9i	0.332436	+	0.575796i	−0.650553	+	0.759461i	$0.725463\pi$
$548$		0			0
$549$	−2.48133e8	+	4.29779e8i	−0.0640002	+	0.110852i
$550$		0			0
$551$	−1.71253e9	−	5.57654e9i	−0.436121	−	1.42015i
$552$		0			0
$553$	2.56191e8	−	4.43736e8i	0.0644208	−	0.111580i
$554$		0			0
$555$	5.00654e8	+	8.67158e8i	0.124312	+	0.215314i
$556$		0			0
$557$	1.13465e9	+	1.96527e9i	0.278207	+	0.481868i	0.970939	−	0.239326i	$-0.0769266\pi$
$557$	1.13465e9	+	1.96527e9i	0.278207	+	0.481868i	−0.692732	+	0.721195i	$0.743593\pi$
$558$		0			0
$559$	6.32717e9			1.53203
$560$		0			0
$561$	−2.67533e9	−	4.63381e9i	−0.639746	−	1.10807i
$562$		0			0
$563$	−1.89355e9			−0.447195			−0.223597	−	0.974682i	$-0.571780\pi$
$563$	−1.89355e9			−0.447195			−0.223597	+	0.974682i	$0.571780\pi$
$564$		0			0
$565$	1.38224e9	−	2.39411e9i	0.322413	−	0.558436i
$566$		0			0
$567$	3.79843e8	−	6.57908e8i	0.0875112	−	0.151574i
$568$		0			0
$569$	−7.07111e9			−1.60914			−0.804572	−	0.593855i	$-0.797605\pi$
$569$	−7.07111e9			−1.60914			−0.804572	+	0.593855i	$0.797605\pi$
$570$		0			0
$571$	5.26263e9			1.18298			0.591489	−	0.806313i	$-0.298540\pi$
$571$	5.26263e9			1.18298			0.591489	+	0.806313i	$0.298540\pi$
$572$		0			0
$573$	1.27816e9	−	2.21384e9i	0.283821	−	0.491593i
$574$		0			0
$575$	−1.09848e8	+	1.90263e8i	−0.0240966	+	0.0417366i
$576$		0			0
$577$	−5.36095e9			−1.16179			−0.580893	−	0.813980i	$-0.697297\pi$
$577$	−5.36095e9			−1.16179			−0.580893	+	0.813980i	$0.697297\pi$
$578$		0			0
$579$	−1.78918e9	−	3.09895e9i	−0.383070	−	0.663497i
$580$		0			0
$581$	−7.43414e7			−0.0157259
$582$		0			0
$583$	−1.21652e9	−	2.10708e9i	−0.254262	−	0.440394i
$584$		0			0
$585$	3.96265e8	+	6.86351e8i	0.0818352	+	0.141743i
$586$		0			0
$587$	−1.40535e9	+	2.43413e9i	−0.286781	+	0.496719i	−0.973040	−	0.230638i	$-0.925919\pi$
$587$	−1.40535e9	+	2.43413e9i	−0.286781	+	0.496719i	0.686259	+	0.727358i	$0.259252\pi$
$588$		0			0
$589$	7.00847e9	+	1.60883e9i	1.41325	+	0.324419i
$590$		0			0
$591$	−2.60120e9	+	4.50541e9i	−0.518343	+	0.897796i
$592$		0			0
$593$	4.19587e9	+	7.26746e9i	0.826287	+	1.43117i	0.900932	+	0.433960i	$0.142884\pi$
$593$	4.19587e9	+	7.26746e9i	0.826287	+	1.43117i	−0.0746455	+	0.997210i	$0.523783\pi$
$594$		0			0
$595$	−6.46815e8	−	1.12032e9i	−0.125884	−	0.218038i
$596$		0			0
$597$	−6.09504e9			−1.17237
$598$		0			0
$599$	1.95182e9	+	3.38065e9i	0.371061	+	0.642697i	0.989729	−	0.142956i	$-0.0456607\pi$
$599$	1.95182e9	+	3.38065e9i	0.371061	+	0.642697i	−0.618668	+	0.785653i	$0.712327\pi$
$600$		0			0
$601$	−7.29646e9			−1.37104			−0.685522	−	0.728052i	$-0.740426\pi$
$601$	−7.29646e9			−1.37104			−0.685522	+	0.728052i	$0.740426\pi$
$602$		0			0
$603$	5.23041e8	−	9.05933e8i	0.0971460	−	0.168262i
$604$		0			0
$605$	1.09637e8	−	1.89897e8i	0.0201286	−	0.0348638i
$606$		0			0
$607$	−3.53830e9			−0.642147			−0.321074	−	0.947054i	$-0.604044\pi$
$607$	−3.53830e9			−0.642147			−0.321074	+	0.947054i	$0.604044\pi$
$608$		0			0
$609$	1.82897e9			0.328129
$610$		0			0
$611$	−2.58021e9	+	4.46905e9i	−0.457626	+	0.792632i
$612$		0			0
$613$	−8.19153e7	+	1.41882e8i	−0.0143633	+	0.0248779i	−0.873118	−	0.487509i	$-0.837905\pi$
$613$	−8.19153e7	+	1.41882e8i	−0.0143633	+	0.0248779i	0.858754	+	0.512387i	$0.171239\pi$
$614$		0			0
$615$	−4.22143e9			−0.731808
$616$		0			0
$617$	−1.90994e9	−	3.30812e9i	−0.327358	−	0.567000i	0.654629	−	0.755950i	$-0.272825\pi$
$617$	−1.90994e9	−	3.30812e9i	−0.327358	−	0.567000i	−0.981987	+	0.188950i	$0.939491\pi$
$618$		0			0
$619$	3.98991e8			0.0676154			0.0338077	−	0.999428i	$-0.489237\pi$
$619$	3.98991e8			0.0676154			0.0338077	+	0.999428i	$0.489237\pi$
$620$		0			0
$621$	2.67197e8	+	4.62798e8i	0.0447724	+	0.0775481i
$622$		0			0
$623$	−7.51674e8	−	1.30194e9i	−0.124544	−	0.215716i
$624$		0			0
$625$	2.37217e8	−	4.10872e8i	0.0388657	−	0.0673173i
$626$		0			0
$627$	3.52249e9	−	3.78686e9i	0.570707	−	0.613540i
$628$		0			0
$629$	−2.11941e9	+	3.67092e9i	−0.339576	+	0.588164i
$630$		0			0
$631$	5.57308e9	+	9.65285e9i	0.883064	+	1.52951i	0.847916	+	0.530130i	$0.177857\pi$
$631$	5.57308e9	+	9.65285e9i	0.883064	+	1.52951i	0.0351480	+	0.999382i	$0.488810\pi$
$632$		0			0
$633$	7.90030e8	+	1.36837e9i	0.123803	+	0.214433i
$634$		0			0
$635$	−5.96250e9			−0.924103
$636$		0			0
$637$	3.07566e9	+	5.32720e9i	0.471466	+	0.816604i
$638$		0			0
$639$	−1.11182e9			−0.168570
$640$		0			0
$641$	1.95560e9	−	3.38720e9i	0.293276	−	0.507969i	−0.681306	−	0.731998i	$-0.738588\pi$
$641$	1.95560e9	−	3.38720e9i	0.293276	−	0.507969i	0.974582	+	0.224029i	$0.0719212\pi$
$642$		0			0
$643$	4.01613e9	−	6.95615e9i	0.595758	−	1.03188i	−0.397682	−	0.917523i	$-0.630185\pi$
$643$	4.01613e9	−	6.95615e9i	0.595758	−	1.03188i	0.993439	−	0.114359i	$-0.0364815\pi$
$644$		0			0
$645$	−5.78614e9			−0.849044
$646$		0			0
$647$	−8.37124e9			−1.21514			−0.607568	−	0.794268i	$-0.707855\pi$
$647$	−8.37124e9			−1.21514			−0.607568	+	0.794268i	$0.707855\pi$
$648$		0			0
$649$	−3.92170e9	+	6.79258e9i	−0.563142	+	0.975390i
$650$		0			0
$651$	−1.12724e9	+	1.95243e9i	−0.160134	+	0.277360i
$652$		0			0
$653$	−9.54361e9			−1.34127			−0.670636	−	0.741787i	$-0.733979\pi$
$653$	−9.54361e9			−1.34127			−0.670636	+	0.741787i	$0.733979\pi$
$654$		0			0
$655$	8.18253e8	+	1.41726e9i	0.113774	+	0.197062i
$656$		0			0
$657$	2.64241e9			0.363514
$658$		0			0
$659$	5.82405e9	+	1.00875e10i	0.792731	+	1.37305i	0.924270	+	0.381740i	$0.124675\pi$
$659$	5.82405e9	+	1.00875e10i	0.792731	+	1.37305i	−0.131538	+	0.991311i	$0.541992\pi$
$660$		0			0
$661$	4.63728e9	+	8.03200e9i	0.624537	+	1.08173i	0.988630	+	0.150367i	$0.0480456\pi$
$661$	4.63728e9	+	8.03200e9i	0.624537	+	1.08173i	−0.364093	+	0.931362i	$0.618621\pi$
$662$		0			0
$663$	5.00053e9	−	8.66118e9i	0.666376	−	1.15420i
$664$		0			0
$665$	8.51632e8	−	9.15550e8i	0.112299	−	0.120728i
$666$		0			0
$667$	−4.70809e8	+	8.15466e8i	−0.0614334	+	0.106406i
$668$		0			0
$669$	3.72979e9	+	6.46019e9i	0.481608	+	0.834169i
$670$		0			0
$671$	−1.93074e9	−	3.34414e9i	−0.246715	−	0.427322i
$672$		0			0
$673$	−2.60480e9			−0.329398			−0.164699	−	0.986344i	$-0.552665\pi$
$673$	−2.60480e9			−0.329398			−0.164699	+	0.986344i	$0.552665\pi$
$674$		0			0
$675$	2.52056e9	+	4.36575e9i	0.315453	+	0.546381i
$676$		0			0
$677$	−6.49648e9			−0.804670			−0.402335	−	0.915492i	$-0.631801\pi$
$677$	−6.49648e9			−0.804670			−0.402335	+	0.915492i	$0.631801\pi$
$678$		0			0
$679$	1.49698e8	−	2.59284e8i	0.0183515	−	0.0317857i
$680$		0			0
$681$	2.44419e9	−	4.23346e9i	0.296565	−	0.513666i
$682$		0			0
$683$	−9.34819e9			−1.12268			−0.561339	−	0.827586i	$-0.689713\pi$
$683$	−9.34819e9			−1.12268			−0.561339	+	0.827586i	$0.689713\pi$
$684$		0			0
$685$	4.46311e9			0.530543
$686$		0			0
$687$	−6.12674e9	+	1.06118e10i	−0.720910	+	1.24865i
$688$		0			0
$689$	2.27384e9	−	3.93841e9i	0.264845	−	0.458726i
$690$		0			0
$691$	1.58716e10			1.82998			0.914990	−	0.403477i	$-0.132198\pi$
$691$	1.58716e10			1.82998			0.914990	+	0.403477i	$0.132198\pi$
$692$		0			0
$693$	−2.71979e8	−	4.71081e8i	−0.0310434	−	0.0537687i
$694$		0			0
$695$	6.09219e9			0.688378
$696$		0			0
$697$	−8.93526e9	−	1.54763e10i	−0.999522	−	1.73122i
$698$		0			0
$699$	2.27941e9	+	3.94805e9i	0.252437	+	0.437233i
$700$		0			0
$701$	2.72493e9	−	4.71972e9i	0.298774	−	0.517491i	−0.677082	−	0.735908i	$-0.736756\pi$
$701$	2.72493e9	−	4.71972e9i	0.298774	−	0.517491i	0.975856	+	0.218416i	$0.0700890\pi$
$702$		0			0
$703$	−3.99331e9	−	9.16684e8i	−0.433501	−	0.0995121i
$704$		0			0
$705$	2.35958e9	−	4.08691e9i	0.253614	−	0.439272i
$706$		0			0
$707$	−2.91023e8	−	5.04066e8i	−0.0309713	−	0.0536438i
$708$		0			0
$709$	4.53765e9	+	7.85944e9i	0.478155	+	0.828189i	0.999686	−	0.0250429i	$-0.00797225\pi$
$709$	4.53765e9	+	7.85944e9i	0.478155	+	0.828189i	−0.521531	+	0.853232i	$0.674639\pi$
$710$		0			0
$711$	1.21522e9			0.126798
$712$		0			0
$713$	−5.80344e8	−	1.00519e9i	−0.0599615	−	0.103856i
$714$		0			0
$715$	−6.16672e9			−0.630933
$716$		0			0
$717$	7.10512e9	−	1.23064e10i	0.719871	−	1.24685i
$718$		0			0
$719$	−5.67020e9	+	9.82107e9i	−0.568914	+	0.985389i	0.427759	+	0.903893i	$0.359303\pi$
$719$	−5.67020e9	+	9.82107e9i	−0.568914	+	0.985389i	−0.996674	+	0.0814960i	$0.974030\pi$
$720$		0			0
$721$	−1.65170e9			−0.164119
$722$		0			0
$723$	−7.06153e9			−0.694888
$724$		0			0
$725$	−4.44132e9	+	7.69259e9i	−0.432841	+	0.749703i
$726$		0			0
$727$	−6.85872e9	+	1.18797e10i	−0.662023	+	1.14666i	0.318061	+	0.948070i	$0.396968\pi$
$727$	−6.85872e9	+	1.18797e10i	−0.662023	+	1.14666i	−0.980083	+	0.198586i	$0.936365\pi$
$728$		0			0
$729$	1.16021e10			1.10915
$730$		0			0
$731$	−1.22472e10	−	2.12127e10i	−1.15965	−	2.00857i
$732$		0			0
$733$	1.75962e10			1.65027			0.825135	−	0.564936i	$-0.191099\pi$
$733$	1.75962e10			1.65027			0.825135	+	0.564936i	$0.191099\pi$
$734$		0			0
$735$	−2.81267e9	−	4.87169e9i	−0.261284	−	0.452557i
$736$		0			0
$737$	4.06982e9	+	7.04913e9i	0.374488	+	0.648633i
$738$		0			0
$739$	7.88280e9	−	1.36534e10i	0.718497	−	1.24447i	−0.243098	−	0.970002i	$-0.578164\pi$
$739$	7.88280e9	−	1.36534e10i	0.718497	−	1.24447i	0.961595	−	0.274471i	$-0.0885029\pi$
$740$		0			0
$741$	9.42183e9	+	2.16283e9i	0.850690	+	0.195280i
$742$		0			0
$743$	6.32992e9	−	1.09637e10i	0.566157	−	0.980613i	−0.430784	−	0.902455i	$-0.641763\pi$
$743$	6.32992e9	−	1.09637e10i	0.566157	−	0.980613i	0.996941	−	0.0781579i	$-0.0249038\pi$
$744$		0			0
$745$	−4.24456e9	−	7.35179e9i	−0.376085	−	0.651398i
$746$		0			0
$747$	−8.81582e7	−	1.52695e8i	−0.00773821	−	0.0134030i
$748$		0			0
$749$	5.44021e9			0.473074
$750$		0			0
$751$	7.20743e8	+	1.24836e9i	0.0620928	+	0.107548i	0.895401	−	0.445261i	$-0.146889\pi$
$751$	7.20743e8	+	1.24836e9i	0.0620928	+	0.107548i	−0.833308	+	0.552809i	$0.813556\pi$
$752$		0			0
$753$	1.89490e9			0.161735
$754$		0			0
$755$	1.61613e9	−	2.79922e9i	0.136666	−	0.236713i
$756$		0			0
$757$	−1.84501e9	+	3.19565e9i	−0.154583	+	0.267746i	−0.932907	−	0.360117i	$-0.882737\pi$
$757$	−1.84501e9	+	3.19565e9i	−0.154583	+	0.267746i	0.778324	+	0.627863i	$0.216070\pi$
$758$		0			0
$759$	−8.34812e8			−0.0693015
$760$		0			0
$761$	−1.98618e10			−1.63370			−0.816851	−	0.576849i	$-0.804282\pi$
$761$	−1.98618e10			−1.63370			−0.816851	+	0.576849i	$0.804282\pi$
$762$		0			0
$763$	−1.49031e8	+	2.58129e8i	−0.0121462	+	0.0210379i
$764$		0			0
$765$	1.53406e9	−	2.65707e9i	0.123887	−	0.214579i
$766$		0			0
$767$	−1.46603e10			−1.17317
$768$		0			0
$769$	−7.84661e9	−	1.35907e10i	−0.622214	−	1.07771i	−0.989073	−	0.147430i	$-0.952900\pi$
$769$	−7.84661e9	−	1.35907e10i	−0.622214	−	1.07771i	0.366859	−	0.930277i	$-0.380433\pi$
$770$		0			0
$771$	1.45199e10			1.14097
$772$		0			0
$773$	5.99537e8	+	1.03843e9i	0.0466861	+	0.0808627i	0.888424	−	0.459023i	$-0.151801\pi$
$773$	5.99537e8	+	1.03843e9i	0.0466861	+	0.0808627i	−0.841738	+	0.539886i	$0.818467\pi$
$774$		0			0
$775$	−5.47460e9	−	9.48228e9i	−0.422470	−	0.731740i
$776$		0			0
$777$	6.42282e8	−	1.11247e9i	0.0491193	−	0.0850772i
$778$		0			0
$779$	1.17646e10	−	1.26476e10i	0.891656	−	0.958579i
$780$		0			0
$781$	4.32555e9	−	7.49208e9i	0.324910	−	0.562761i
$782$		0			0
$783$	1.08031e10	+	1.87116e10i	0.804235	+	1.39298i
$784$		0			0
$785$	−5.62844e9	−	9.74874e9i	−0.415283	−	0.719291i
$786$		0			0
$787$	−1.98784e10			−1.45368			−0.726842	−	0.686805i	$-0.759013\pi$
$787$	−1.98784e10			−1.45368			−0.726842	+	0.686805i	$0.759013\pi$
$788$		0			0
$789$	7.11361e9	+	1.23211e10i	0.515609	+	0.893061i
$790$		0			0
$791$	−3.54651e9			−0.254791
$792$		0			0
$793$	3.60880e9	−	6.25062e9i	0.256984	−	0.445110i
$794$		0			0
$795$	−2.07941e9	+	3.60164e9i	−0.146776	+	0.254223i
$796$		0			0
$797$	−9.06656e9			−0.634364			−0.317182	−	0.948365i	$-0.602737\pi$
$797$	−9.06656e9			−0.634364			−0.317182	+	0.948365i	$0.602737\pi$
$798$		0			0
$799$	1.99775e10			1.38557
$800$		0			0
$801$	1.78275e9	−	3.08782e9i	0.122568	−	0.212294i
$802$		0			0
$803$	−1.02804e10	+	1.78062e10i	−0.700656	+	1.21357i
$804$		0			0
$805$	−2.01833e8			−0.0136366
$806$		0			0
$807$	1.46583e9	+	2.53889e9i	0.0981806	+	0.170054i
$808$		0			0
$809$	2.46714e10			1.63822			0.819112	−	0.573634i	$-0.194467\pi$
$809$	2.46714e10			1.63822			0.819112	+	0.573634i	$0.194467\pi$
$810$		0			0
$811$	−8.46554e9	−	1.46627e10i	−0.557291	−	0.965256i	−0.997721	−	0.0674692i	$-0.978508\pi$
$811$	−8.46554e9	−	1.46627e10i	−0.557291	−	0.965256i	0.440431	−	0.897787i	$-0.354826\pi$
$812$		0			0
$813$	8.45146e9	+	1.46384e10i	0.551588	+	0.955379i
$814$		0			0
$815$	−5.43541e9	+	9.41441e9i	−0.351707	+	0.609174i
$816$		0			0
$817$	1.61253e10	−	1.73356e10i	1.03450	−	1.11214i
$818$		0			0
$819$	5.08363e8	−	8.80510e8i	0.0323355	−	0.0560068i
$820$		0			0
$821$	3.54650e9	+	6.14271e9i	0.223665	+	0.387399i	0.955918	−	0.293633i	$-0.0948645\pi$
$821$	3.54650e9	+	6.14271e9i	0.223665	+	0.387399i	−0.732253	+	0.681033i	$0.761531\pi$
$822$		0			0
$823$	−4.63160e9	−	8.02217e9i	−0.289622	−	0.501640i	0.684098	−	0.729391i	$-0.260196\pi$
$823$	−4.63160e9	−	8.02217e9i	−0.289622	−	0.501640i	−0.973720	+	0.227751i	$0.926863\pi$
$824$		0			0
$825$	−7.87508e9			−0.488277
$826$		0			0
$827$	−5.16563e9	−	8.94714e9i	−0.317581	−	0.550066i	0.662402	−	0.749149i	$-0.269537\pi$
$827$	−5.16563e9	−	8.94714e9i	−0.317581	−	0.550066i	−0.979983	+	0.199083i	$0.936204\pi$
$828$		0			0
$829$	2.93264e10			1.78779			0.893897	−	0.448273i	$-0.147961\pi$
$829$	2.93264e10			1.78779			0.893897	+	0.448273i	$0.147961\pi$
$830$		0			0
$831$	7.60525e9	−	1.31727e10i	0.459738	−	0.796289i
$832$		0			0
$833$	1.19068e10	−	2.06232e10i	0.713737	−	1.23623i
$834$		0			0
$835$	−2.18378e9			−0.129810
$836$		0			0
$837$	−2.66330e10			−1.56993
$838$		0			0
$839$	−8.95491e9	+	1.55104e10i	−0.523473	+	0.906682i	0.476154	+	0.879362i	$0.342031\pi$
$839$	−8.95491e9	+	1.55104e10i	−0.523473	+	0.906682i	−0.999627	+	0.0273200i	$0.991303\pi$
$840$		0			0
$841$	−1.04105e10	+	1.80315e10i	−0.603512	+	1.04531i
$842$		0			0
$843$	9.96999e9			0.573190
$844$		0			0
$845$	−9.83812e7	−	1.70401e8i	−0.00560936	−	0.00971570i
$846$		0			0
$847$	−2.81305e8			−0.0159069
$848$		0			0
$849$	5.75979e9	+	9.97624e9i	0.323020	+	0.559487i
$850$		0			0
$851$	3.30671e8	+	5.72738e8i	0.0183926	+	0.0318569i
$852$		0			0
$853$	9.50501e9	−	1.64632e10i	0.524362	−	0.908222i	−0.475236	−	0.879859i	$-0.657637\pi$
$853$	9.50501e9	−	1.64632e10i	0.524362	−	0.908222i	0.999598	−	0.0283631i	$-0.00902945\pi$
$854$		0			0
$855$	2.89042e9	+	6.63510e8i	0.158154	+	0.0363050i
$856$		0			0
$857$	−1.39812e10	+	2.42161e10i	−0.758772	+	1.31423i	0.184705	+	0.982794i	$0.440867\pi$
$857$	−1.39812e10	+	2.42161e10i	−0.758772	+	1.31423i	−0.943477	+	0.331438i	$0.892466\pi$
$858$		0			0
$859$	1.54826e10	+	2.68166e10i	0.833425	+	1.44354i	0.895306	+	0.445452i	$0.146957\pi$
$859$	1.54826e10	+	2.68166e10i	0.833425	+	1.44354i	−0.0618807	+	0.998084i	$0.519710\pi$
$860$		0			0
$861$	2.70781e9	+	4.69006e9i	0.144580	+	0.250419i
$862$		0			0
$863$	−2.63768e10			−1.39696			−0.698480	−	0.715630i	$-0.746140\pi$
$863$	−2.63768e10			−1.39696			−0.698480	+	0.715630i	$0.746140\pi$
$864$		0			0
$865$	1.03594e10	+	1.79429e10i	0.544223	+	0.942622i
$866$		0			0
$867$	−2.21117e10			−1.15227
$868$		0			0
$869$	−4.72786e9	+	8.18890e9i	−0.244397	+	0.423308i
$870$		0			0
$871$	−7.60700e9	+	1.31757e10i	−0.390077	+	0.675633i
$872$		0			0
$873$	7.10080e8			0.0361208
$874$		0			0
$875$	−5.17136e9			−0.260962
$876$		0			0
$877$	−8.29210e9	+	1.43623e10i	−0.415113	+	0.718997i	−0.995440	−	0.0953867i	$-0.969591\pi$
$877$	−8.29210e9	+	1.43623e10i	−0.415113	+	0.718997i	0.580327	+	0.814383i	$0.302925\pi$
$878$		0			0
$879$	8.88123e9	−	1.53827e10i	0.441075	−	0.763964i
$880$		0			0
$881$	−1.49651e10			−0.737335			−0.368667	−	0.929561i	$-0.620186\pi$
$881$	−1.49651e10			−0.737335			−0.368667	+	0.929561i	$0.620186\pi$
$882$		0			0
$883$	−8.77910e9	−	1.52059e10i	−0.429129	−	0.743273i	0.567667	−	0.823258i	$-0.307846\pi$
$883$	−8.77910e9	−	1.52059e10i	−0.429129	−	0.743273i	−0.996796	+	0.0799853i	$0.974513\pi$
$884$		0			0
$885$	1.34067e10			0.650162
$886$		0			0
$887$	−5.67263e9	−	9.82528e9i	−0.272930	−	0.472729i	0.696681	−	0.717381i	$-0.254659\pi$
$887$	−5.67263e9	−	9.82528e9i	−0.272930	−	0.472729i	−0.969611	+	0.244653i	$0.921326\pi$
$888$		0			0
$889$	3.82461e9	+	6.62442e9i	0.182571	+	0.316221i
$890$		0			0
$891$	−7.00980e9	+	1.21413e10i	−0.331997	+	0.575035i
$892$		0			0
$893$	5.66872e9	+	1.84592e10i	0.266382	+	0.867426i
$894$		0			0
$895$	−3.92152e9	+	6.79228e9i	−0.182841	+	0.316690i
$896$		0			0
$897$	−7.80185e8	−	1.35132e9i	−0.0360931	−	0.0625150i
$898$		0			0
$899$	−2.34641e10	−	4.06410e10i	−1.07707	−	1.86554i
$900$		0			0
$901$	−1.76054e10			−0.801881
$902$		0			0
$903$	3.71148e9	+	6.42848e9i	0.167741	+	0.290537i
$904$		0			0
$905$	−2.69268e10			−1.20758
$906$		0			0
$907$	−1.77103e10	+	3.06751e10i	−0.788133	+	1.36509i	0.138976	+	0.990296i	$0.455619\pi$
$907$	−1.77103e10	+	3.06751e10i	−0.788133	+	1.36509i	−0.927109	+	0.374791i	$0.877715\pi$
$908$		0			0
$909$	6.90223e8	−	1.19550e9i	0.0304801	−	0.0527930i
$910$		0			0
$911$	−2.05155e10			−0.899016			−0.449508	−	0.893276i	$-0.648401\pi$
$911$	−2.05155e10			−0.899016			−0.449508	+	0.893276i	$0.648401\pi$
$912$		0			0
$913$	1.37193e9			0.0596601
$914$		0			0
$915$	−3.30022e9	+	5.71614e9i	−0.142419	+	0.246677i
$916$		0			0
$917$	1.04973e9	−	1.81818e9i	0.0449556	−	0.0778653i
$918$		0			0
$919$	−1.13166e9			−0.0480964			−0.0240482	−	0.999711i	$-0.507656\pi$
$919$	−1.13166e9			−0.0480964			−0.0240482	+	0.999711i	$0.507656\pi$
$920$		0			0
$921$	1.76694e9	+	3.06043e9i	0.0745270	+	0.129085i
$922$		0			0
$923$	1.61700e10			0.676869
$924$		0			0
$925$	3.11934e9	+	5.40285e9i	0.129588	+	0.224454i
$926$		0			0
$927$	−1.95868e9	−	3.39253e9i	−0.0807578	−	0.139877i
$928$		0			0
$929$	9.75180e9	−	1.68906e10i	0.399052	−	0.691179i	−0.594557	−	0.804053i	$-0.702673\pi$
$929$	9.75180e9	−	1.68906e10i	0.399052	−	0.691179i	0.993609	+	0.112875i	$0.0360059\pi$
$930$		0			0
$931$	2.24344e10	+	5.14992e9i	0.911151	+	0.209159i
$932$		0			0
$933$	1.57772e9	−	2.73270e9i	0.0635982	−	0.110155i
$934$		0			0
$935$	1.19366e10	+	2.06748e10i	0.477574	+	0.827183i
$936$		0			0
$937$	−4.54820e9	−	7.87772e9i	−0.180614	−	0.312833i	0.761476	−	0.648193i	$-0.224475\pi$
$937$	−4.54820e9	−	7.87772e9i	−0.180614	−	0.312833i	−0.942090	+	0.335361i	$0.891142\pi$
$938$		0			0
$939$	2.37231e10			0.935064
$940$		0			0
$941$	2.17517e10	+	3.76750e10i	0.851000	+	1.47397i	0.880307	+	0.474404i	$0.157337\pi$
$941$	2.17517e10	+	3.76750e10i	0.851000	+	1.47397i	−0.0293072	+	0.999570i	$0.509330\pi$
$942$		0			0
$943$	−2.78816e9			−0.108275
$944$		0			0
$945$	−2.31561e9	+	4.01075e9i	−0.0892594	+	0.154602i
$946$		0			0
$947$	−2.08239e9	+	3.60680e9i	−0.0796776	+	0.138006i	−0.903111	−	0.429408i	$-0.858722\pi$
$947$	−2.08239e9	+	3.60680e9i	−0.0796776	+	0.138006i	0.823433	+	0.567413i	$0.192056\pi$
$948$		0			0
$949$	−3.84307e10			−1.45964
$950$		0			0
$951$	1.42117e10			0.535813
$952$		0			0
$953$	1.22414e10	−	2.12027e10i	0.458148	−	0.793536i	−0.540715	−	0.841206i	$-0.681846\pi$
$953$	1.22414e10	−	2.12027e10i	0.458148	−	0.793536i	0.998863	+	0.0476699i	$0.0151795\pi$
$954$		0			0
$955$	−5.70282e9	+	9.87758e9i	−0.211874	+	0.366977i
$956$		0			0
$957$	−3.37525e10			−1.24484
$958$		0			0
$959$	−2.86283e9	−	4.95857e9i	−0.104817	−	0.181548i
$960$		0			0
$961$	3.03335e10			1.10253
$962$		0			0
$963$	6.45131e9	+	1.11740e10i	0.232785	+	0.403196i
$964$		0			0
$965$	7.98283e9	+	1.38267e10i	0.285964	+	0.495304i
$966$		0			0
$967$	1.65552e10	−	2.86745e10i	0.588765	−	1.01977i	−0.405629	−	0.914038i	$-0.632948\pi$
$967$	1.65552e10	−	2.86745e10i	0.588765	−	1.01977i	0.994394	−	0.105734i	$-0.0337191\pi$
$968$		0			0
$969$	−1.09862e10	−	3.57745e10i	−0.387894	−	1.26311i
$970$		0			0
$971$	8.88358e9	−	1.53868e10i	0.311402	−	0.539363i	−0.667264	−	0.744821i	$-0.732535\pi$
$971$	8.88358e9	−	1.53868e10i	0.311402	−	0.539363i	0.978666	+	0.205457i	$0.0658682\pi$
$972$		0			0
$973$	−3.90780e9	−	6.76851e9i	−0.135999	−	0.235558i
$974$		0			0
$975$	−7.35977e9	−	1.27475e10i	−0.254301	−	0.440462i
$976$		0			0
$977$	3.44295e10			1.18114			0.590568	−	0.806988i	$-0.298904\pi$
$977$	3.44295e10			1.18114			0.590568	+	0.806988i	$0.298904\pi$
$978$		0			0
$979$	1.38717e10	+	2.40265e10i	0.472488	+	0.818374i
$980$		0			0
$981$	−7.06918e8			−0.0239071
$982$		0			0
$983$	1.12261e10	−	1.94443e10i	0.376958	−	0.652911i	−0.613660	−	0.789571i	$-0.710303\pi$
$983$	1.12261e10	−	1.94443e10i	0.376958	−	0.652911i	0.990618	+	0.136660i	$0.0436366\pi$
$984$		0			0
$985$	1.16059e10	−	2.01019e10i	0.386946	−	0.670210i
$986$		0			0
$987$	−6.05415e9			−0.200421
$988$		0			0
$989$	−3.82162e9			−0.125620
$990$		0			0
$991$	9.64683e9	−	1.67088e10i	0.314867	−	0.545365i	−0.664542	−	0.747251i	$-0.731374\pi$
$991$	9.64683e9	−	1.67088e10i	0.314867	−	0.545365i	0.979409	+	0.201885i	$0.0647068\pi$
$992$		0			0
$993$	−1.69489e10	+	2.93563e10i	−0.549311	+	0.951434i
$994$		0			0
$995$	2.71945e10			0.875184
$996$		0			0
$997$	−2.10661e10	−	3.64875e10i	−0.673210	−	1.16603i	−0.976989	−	0.213291i	$-0.931582\pi$
$997$	−2.10661e10	−	3.64875e10i	−0.673210	−	1.16603i	0.303779	−	0.952743i	$-0.401752\pi$
$998$		0			0
$999$	1.51750e10			0.481560

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	76.8.e.a.49.8	yes	22
19.7	even	3	inner	76.8.e.a.45.8	✓	22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
76.8.e.a.45.8	✓	22	19.7	even	3	inner
76.8.e.a.49.8	yes	22	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.e (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(20.2338 - 35.0460i) q^{3} +(-90.2780 + 156.366i) q^{5} +231.633 q^{7} +(274.684 + 475.766i) q^{9} +O(q^{10})\)	\(q+(20.2338 - 35.0460i) q^{3} +(-90.2780 + 156.366i) q^{5} +231.633 q^{7} +(274.684 + 475.766i) q^{9} -4274.66 q^{11} +(-3994.94 - 6919.44i) q^{13} +(3653.34 + 6327.78i) q^{15} +(-15465.6 + 26787.3i) q^{17} +(-29139.8 - 6689.17i) q^{19} +(4686.83 - 8117.82i) q^{21} +(2412.95 + 4179.35i) q^{23} +(22762.2 + 39425.4i) q^{25} +110734. q^{27} +(97558.8 + 168977. i) q^{29} -240512. q^{31} +(-86492.8 + 149810. i) q^{33} +(-20911.4 + 36219.6i) q^{35} +137040. q^{37} -323332. q^{39} +(-288875. + 500346. i) q^{41} +(-395948. + 685803. i) q^{43} -99191.6 q^{45} +(-322935. - 559339. i) q^{47} -769889. q^{49} +(625858. + 1.08402e6i) q^{51} +(284590. + 492924. i) q^{53} +(385908. - 668412. i) q^{55} +(-824039. + 885886. i) q^{57} +(917429. - 1.58903e6i) q^{59} +(451671. + 782317. i) q^{61} +(63625.8 + 110203. i) q^{63} +1.44262e6 q^{65} +(-952079. - 1.64905e6i) q^{67} +195293. q^{69} +(-1.01191e6 + 1.75267e6i) q^{71} +(2.40496e6 - 4.16551e6i) q^{73} +1.84227e6 q^{75} -990153. q^{77} +(1.10602e6 - 1.91568e6i) q^{79} +(1.63985e6 - 2.84030e6i) q^{81} -320945. q^{83} +(-2.79241e6 - 4.83660e6i) q^{85} +7.89596e6 q^{87} +(-3.24510e6 - 5.62069e6i) q^{89} +(-925361. - 1.60277e6i) q^{91} +(-4.86649e6 + 8.42900e6i) q^{93} +(3.67664e6 - 3.95259e6i) q^{95} +(646270. - 1.11937e6i) q^{97} +(-1.17418e6 - 2.03374e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9	\( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) \) 22 * q + 13 * q^3 + q^5 + 560 * q^7 - 6002 * q^9 + 472 * q^11 - 567 * q^13 + 2995 * q^15 + 5589 * q^17 + 80912 * q^19 + 44412 * q^21 - 15425 * q^23 - 32806 * q^25 + 50290 * q^27 - 18919 * q^29 + 150296 * q^31 + 314618 * q^33 + 92808 * q^35 + 350100 * q^37 + 948810 * q^39 + 698891 * q^41 + 402545 * q^43 + 1477508 * q^45 - 653621 * q^47 - 1938490 * q^49 - 1386401 * q^51 - 106763 * q^53 + 414508 * q^55 + 1267563 * q^57 + 3136737 * q^59 + 2004581 * q^61 + 1465000 * q^63 - 7397638 * q^65 + 4344391 * q^67 + 1732238 * q^69 - 133823 * q^71 - 8349685 * q^73 - 12136824 * q^75 + 9147480 * q^77 - 94679 * q^79 - 838595 * q^81 - 2884080 * q^83 - 1421409 * q^85 - 31740598 * q^87 - 7039347 * q^89 + 1520096 * q^91 - 1993628 * q^93 + 1707587 * q^95 + 13308115 * q^97 + 6011488 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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