LMFDB - Embedded newform 162.8.c.p.55.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [162,8,Mod(55,162)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(162, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4]))

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

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

chi := DirichletCharacter("162.55");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	162.c (of order $3$, degree $2$, not 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:	$50.6063741284$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{329})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 83x^{2} + 82x + 6724 $ x^4 - x^3 + 83x^2 + 82x + 6724
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 54)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			55.2
Root			$-4.28459 + 7.42113i$ of defining polynomial
Character	$\chi$	$=$	162.55
Dual form			162.8.c.p.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.00000 - 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(232.868 + 403.339i) q^{5} +(24.8678 - 43.0723i) q^{7} -512.000 q^{8} +3725.89 q^{10} +(-151.443 + 262.306i) q^{11} +(2757.36 + 4775.88i) q^{13} +(-198.943 - 344.579i) q^{14} +(-2048.00 + 3547.24i) q^{16} -22646.7 q^{17} +52805.5 q^{19} +(14903.5 - 25813.7i) q^{20} +(1211.54 + 2098.45i) q^{22} +(-10009.6 - 17337.2i) q^{23} +(-69392.3 + 120191. i) q^{25} +44117.7 q^{26} -3183.08 q^{28} +(-125140. + 216748. i) q^{29} +(-131183. - 227216. i) q^{31} +(16384.0 + 28377.9i) q^{32} +(-90586.9 + 156901. i) q^{34} +23163.7 q^{35} -460621. q^{37} +(211222. - 365847. i) q^{38} +(-119228. - 206510. i) q^{40} +(120202. + 208196. i) q^{41} +(-284188. + 492228. i) q^{43} +19384.6 q^{44} -160154. q^{46} +(-437616. + 757973. i) q^{47} +(410535. + 711067. i) q^{49} +(555139. + 961529. i) q^{50} +(176471. - 305656. i) q^{52} +811041. q^{53} -141064. q^{55} +(-12732.3 + 22053.0i) q^{56} +(1.00112e6 + 1.73399e6i) q^{58} +(646016. + 1.11893e6i) q^{59} +(814260. - 1.41034e6i) q^{61} -2.09893e6 q^{62} +262144. q^{64} +(-1.28420e6 + 2.22430e6i) q^{65} +(921587. + 1.59623e6i) q^{67} +(724695. + 1.25521e6i) q^{68} +(92654.6 - 160483. i) q^{70} +4.56707e6 q^{71} -1.91494e6 q^{73} +(-1.84248e6 + 3.19127e6i) q^{74} +(-1.68978e6 - 2.92678e6i) q^{76} +(7532.09 + 13046.0i) q^{77} +(905441. - 1.56827e6i) q^{79} -1.90765e6 q^{80} +1.92323e6 q^{82} +(-1.54464e6 + 2.67540e6i) q^{83} +(-5.27369e6 - 9.13430e6i) q^{85} +(2.27350e6 + 3.93782e6i) q^{86} +(77538.6 - 134301. i) q^{88} -4.10492e6 q^{89} +274278. q^{91} +(-640617. + 1.10958e6i) q^{92} +(3.50093e6 + 6.06378e6i) q^{94} +(1.22967e7 + 2.12985e7i) q^{95} +(-3.95811e6 + 6.85566e6i) q^{97} +6.56855e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 16 q^{2} - 128 q^{4} - 48 q^{5} - 880 q^{7} - 2048 q^{8} - 768 q^{10} + 7230 q^{11} - 8560 q^{13} + 7040 q^{14} - 8192 q^{16} - 51408 q^{17} + 74096 q^{19} - 3072 q^{20} - 57840 q^{22} - 59628 q^{23}+ \cdots + 12483264 q^{98}+O(q^{100}) $ 4 * q + 16 * q^2 - 128 * q^4 - 48 * q^5 - 880 * q^7 - 2048 * q^8 - 768 * q^10 + 7230 * q^11 - 8560 * q^13 + 7040 * q^14 - 8192 * q^16 - 51408 * q^17 + 74096 * q^19 - 3072 * q^20 - 57840 * q^22 - 59628 * q^23 - 324584 * q^25 - 136960 * q^26 + 112640 * q^28 - 275280 * q^29 - 245584 * q^31 + 65536 * q^32 - 205632 * q^34 + 1001604 * q^35 - 142120 * q^37 + 296384 * q^38 + 24576 * q^40 + 494520 * q^41 - 825280 * q^43 - 925440 * q^44 - 954048 * q^46 - 927708 * q^47 + 780204 * q^49 + 2596672 * q^50 - 547840 * q^52 + 2764224 * q^53 - 8021952 * q^55 + 450560 * q^56 + 2202240 * q^58 + 1588920 * q^59 + 3021968 * q^61 - 3929344 * q^62 + 1048576 * q^64 - 9799080 * q^65 + 1882160 * q^67 + 1645056 * q^68 + 4006416 * q^70 + 6788880 * q^71 + 1790180 * q^73 - 568480 * q^74 - 2371072 * q^76 + 7018656 * q^77 + 369920 * q^79 + 393216 * q^80 + 7912320 * q^82 - 9352050 * q^83 - 8976744 * q^85 + 6602240 * q^86 - 3701760 * q^88 + 1825920 * q^89 + 26720080 * q^91 - 3816192 * q^92 + 7421664 * q^94 + 32688588 * q^95 - 1359790 * q^97 + 12483264 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/162\mathbb{Z}\right)^\times$.

$n$	$83$
$\chi(n)$	$e\left(\frac{2}{3}\right)$

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 $

$2$	4.00000	−	6.92820i	0.353553	−	0.612372i
$2$	4.00000	−	6.92820i	0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−32.0000	−	55.4256i	−0.250000	−	0.433013i
$5$	232.868	+	403.339i	0.833133	+	1.44303i	0.895541	+	0.444978i	$0.146789\pi$
$5$	232.868	+	403.339i	0.833133	+	1.44303i	−0.0624081	+	0.998051i	$0.519878\pi$
$6$		0			0
$7$	24.8678	−	43.0723i	0.0274028	−	0.0474630i	−0.851999	−	0.523544i	$-0.824610\pi$
$7$	24.8678	−	43.0723i	0.0274028	−	0.0474630i	0.879402	+	0.476081i	$0.157943\pi$
$8$	−512.000			−0.353553
$9$		0			0
$10$	3725.89			1.17823
$11$	−151.443	+	262.306i	−0.0343063	+	0.0594202i	−0.882669	−	0.469996i	$-0.844256\pi$
$11$	−151.443	+	262.306i	−0.0343063	+	0.0594202i	0.848362	+	0.529416i	$0.177589\pi$
$12$		0			0
$13$	2757.36	+	4775.88i	0.348090	+	0.602909i	0.985910	−	0.167276i	$-0.0534971\pi$
$13$	2757.36	+	4775.88i	0.348090	+	0.602909i	−0.637820	+	0.770185i	$0.720164\pi$
$14$	−198.943	−	344.579i	−0.0193767	−	0.0335614i
$15$		0			0
$16$	−2048.00	+	3547.24i	−0.125000	+	0.216506i
$17$	−22646.7			−1.11798			−0.558990	−	0.829174i	$-0.688811\pi$
$17$	−22646.7			−1.11798			−0.558990	+	0.829174i	$0.688811\pi$
$18$		0			0
$19$	52805.5			1.76621			0.883103	−	0.469179i	$-0.155450\pi$
$19$	52805.5			1.76621			0.883103	+	0.469179i	$0.155450\pi$
$20$	14903.5	−	25813.7i	0.416567	−	0.721515i
$21$		0			0
$22$	1211.54	+	2098.45i	0.0242582	+	0.0420164i
$23$	−10009.6	−	17337.2i	−0.171542	−	0.297120i	0.767417	−	0.641148i	$-0.221542\pi$
$23$	−10009.6	−	17337.2i	−0.171542	−	0.297120i	−0.938959	+	0.344028i	$0.888208\pi$
$24$		0			0
$25$	−69392.3	+	120191.i	−0.888222	+	1.53845i
$26$	44117.7			0.492273
$27$		0			0
$28$	−3183.08			−0.0274028
$29$	−125140.	+	216748.i	−0.952800	+	1.65030i	−0.213476	+	0.976948i	$0.568479\pi$
$29$	−125140.	+	216748.i	−0.952800	+	1.65030i	−0.739324	+	0.673350i	$0.764855\pi$
$30$		0			0
$31$	−131183.	−	227216.i	−0.790884	−	1.36985i	−0.925420	−	0.378942i	$-0.876288\pi$
$31$	−131183.	−	227216.i	−0.790884	−	1.36985i	0.134536	−	0.990909i	$-0.457045\pi$
$32$	16384.0	+	28377.9i	0.0883883	+	0.153093i
$33$		0			0
$34$	−90586.9	+	156901.i	−0.395266	+	0.684620i
$35$	23163.7			0.0913207
$36$		0			0
$37$	−460621.			−1.49499			−0.747493	−	0.664269i	$-0.768743\pi$
$37$	−460621.			−1.49499			−0.747493	+	0.664269i	$0.768743\pi$
$38$	211222.	−	365847.i	0.624448	−	1.08158i
$39$		0			0
$40$	−119228.	−	206510.i	−0.294557	−	0.510188i
$41$	120202.	+	208196.i	0.272375	+	0.471768i	0.969470	−	0.245212i	$-0.0788575\pi$
$41$	120202.	+	208196.i	0.272375	+	0.471768i	−0.697094	+	0.716979i	$0.745524\pi$
$42$		0			0
$43$	−284188.	+	492228.i	−0.545087	+	0.944119i	0.453514	+	0.891249i	$0.350170\pi$
$43$	−284188.	+	492228.i	−0.545087	+	0.944119i	−0.998601	+	0.0528698i	$0.983163\pi$
$44$	19384.6			0.0343063
$45$		0			0
$46$	−160154.			−0.242597
$47$	−437616.	+	757973.i	−0.614824	+	1.06491i	0.375592	+	0.926785i	$0.377440\pi$
$47$	−437616.	+	757973.i	−0.614824	+	1.06491i	−0.990415	+	0.138121i	$0.955894\pi$
$48$		0			0
$49$	410535.	+	711067.i	0.498498	+	0.863424i
$50$	555139.	+	961529.i	0.628068	+	1.08785i
$51$		0			0
$52$	176471.	−	305656.i	0.174045	−	0.301455i
$53$	811041.			0.748303			0.374151	−	0.927368i	$-0.377934\pi$
$53$	811041.			0.748303			0.374151	+	0.927368i	$0.377934\pi$
$54$		0			0
$55$	−141064.			−0.114327
$56$	−12732.3	+	22053.0i	−0.00968835	+	0.0167807i
$57$		0			0
$58$	1.00112e6	+	1.73399e6i	0.673731	+	1.16694i
$59$	646016.	+	1.11893e6i	0.409507	+	0.709286i	0.994834	−	0.101510i	$-0.0323675\pi$
$59$	646016.	+	1.11893e6i	0.409507	+	0.709286i	−0.585328	+	0.810797i	$0.699034\pi$
$60$		0			0
$61$	814260.	−	1.41034e6i	0.459313	−	0.795554i	−0.539612	−	0.841914i	$-0.681429\pi$
$61$	814260.	−	1.41034e6i	0.459313	−	0.795554i	0.998925	+	0.0463604i	$0.0147623\pi$
$62$	−2.09893e6			−1.11848
$63$		0			0
$64$	262144.			0.125000
$65$	−1.28420e6	+	2.22430e6i	−0.580010	+	1.00461i
$66$		0			0
$67$	921587.	+	1.59623e6i	0.374347	+	0.648388i	0.990229	−	0.139450i	$-0.0445336\pi$
$67$	921587.	+	1.59623e6i	0.374347	+	0.648388i	−0.615882	+	0.787838i	$0.711200\pi$
$68$	724695.	+	1.25521e6i	0.279495	+	0.484099i
$69$		0			0
$70$	92654.6	−	160483.i	0.0322867	−	0.0559223i
$71$	4.56707e6			1.51438			0.757188	−	0.653197i	$-0.226573\pi$
$71$	4.56707e6			1.51438			0.757188	+	0.653197i	$0.226573\pi$
$72$		0			0
$73$	−1.91494e6			−0.576136			−0.288068	−	0.957610i	$-0.593013\pi$
$73$	−1.91494e6			−0.576136			−0.288068	+	0.957610i	$0.593013\pi$
$74$	−1.84248e6	+	3.19127e6i	−0.528558	+	0.915489i
$75$		0			0
$76$	−1.68978e6	−	2.92678e6i	−0.441552	−	0.764790i
$77$	7532.09	+	13046.0i	0.00188017	+	0.00325656i
$78$		0			0
$79$	905441.	−	1.56827e6i	0.206617	−	0.357871i	−0.744030	−	0.668146i	$-0.767088\pi$
$79$	905441.	−	1.56827e6i	0.206617	−	0.357871i	0.950647	+	0.310276i	$0.100421\pi$
$80$	−1.90765e6			−0.416567
$81$		0			0
$82$	1.92323e6			0.385197
$83$	−1.54464e6	+	2.67540e6i	−0.296520	+	0.513588i	−0.975337	−	0.220719i	$-0.929160\pi$
$83$	−1.54464e6	+	2.67540e6i	−0.296520	+	0.513588i	0.678817	+	0.734307i	$0.262493\pi$
$84$		0			0
$85$	−5.27369e6	−	9.13430e6i	−0.931426	−	1.61328i
$86$	2.27350e6	+	3.93782e6i	0.385435	+	0.667593i
$87$		0			0
$88$	77538.6	−	134301.i	0.0121291	−	0.0210082i
$89$	−4.10492e6			−0.617219			−0.308610	−	0.951189i	$-0.599864\pi$
$89$	−4.10492e6			−0.617219			−0.308610	+	0.951189i	$0.599864\pi$
$90$		0			0
$91$	274278.			0.0381545
$92$	−640617.	+	1.10958e6i	−0.0857711	+	0.148560i
$93$		0			0
$94$	3.50093e6	+	6.06378e6i	0.434746	+	0.753002i
$95$	1.22967e7	+	2.12985e7i	1.47149	+	2.54869i
$96$		0			0
$97$	−3.95811e6	+	6.85566e6i	−0.440339	+	0.762690i	−0.997714	−	0.0675709i	$-0.978475\pi$
$97$	−3.95811e6	+	6.85566e6i	−0.440339	+	0.762690i	0.557375	+	0.830261i	$0.311808\pi$
$98$	6.56855e6			0.704983
$99$		0			0
$100$	8.88222e6			0.888222
$101$	−4.81629e6	+	8.34207e6i	−0.465145	+	0.805654i	−0.999208	−	0.0397901i	$-0.987331\pi$
$101$	−4.81629e6	+	8.34207e6i	−0.465145	+	0.805654i	0.534063	+	0.845445i	$0.320664\pi$
$102$		0			0
$103$	−4.59008e6	−	7.95025e6i	−0.413894	−	0.716886i	0.581417	−	0.813606i	$-0.302498\pi$
$103$	−4.59008e6	−	7.95025e6i	−0.413894	−	0.716886i	−0.995312	+	0.0967193i	$0.969165\pi$
$104$	−1.41177e6	−	2.44525e6i	−0.123068	−	0.213161i
$105$		0			0
$106$	3.24416e6	−	5.61906e6i	0.264565	−	0.458240i
$107$	3.82175e6			0.301591			0.150796	−	0.988565i	$-0.451816\pi$
$107$	3.82175e6			0.301591			0.150796	+	0.988565i	$0.451816\pi$
$108$		0			0
$109$	1.90995e7			1.41264			0.706318	−	0.707895i	$-0.250355\pi$
$109$	1.90995e7			1.41264			0.706318	+	0.707895i	$0.250355\pi$
$110$	−564258.	+	977323.i	−0.0404206	+	0.0700106i
$111$		0			0
$112$	101859.	+	176424.i	0.00685069	+	0.0118658i
$113$	−115135.	−	199420.i	−0.00750643	−	0.0130015i	0.862248	−	0.506487i	$-0.169056\pi$
$113$	−115135.	−	199420.i	−0.00750643	−	0.0130015i	−0.869754	+	0.493485i	$0.835723\pi$
$114$		0			0
$115$	4.66185e6	−	8.07456e6i	0.285835	−	0.495081i
$116$	1.60179e7			0.952800
$117$		0			0
$118$	1.03363e7			0.579130
$119$	−563174.	+	975447.i	−0.0306358	+	0.0530627i
$120$		0			0
$121$	9.69772e6	+	1.67969e7i	0.497646	+	0.861948i
$122$	−6.51408e6	−	1.12827e7i	−0.324783	−	0.562541i
$123$		0			0
$124$	−8.39573e6	+	1.45418e7i	−0.395442	+	0.684926i
$125$	−2.82514e7			−1.29376
$126$		0			0
$127$	−3.92443e7			−1.70006			−0.850028	−	0.526737i	$-0.823415\pi$
$127$	−3.92443e7			−1.70006			−0.850028	+	0.526737i	$0.823415\pi$
$128$	1.04858e6	−	1.81619e6i	0.0441942	−	0.0765466i
$129$		0			0
$130$	1.02736e7	+	1.77944e7i	0.410129	+	0.710365i
$131$	1.06388e7	+	1.84270e7i	0.413470	+	0.716151i	0.995266	−	0.0971836i	$-0.0309834\pi$
$131$	1.06388e7	+	1.84270e7i	0.413470	+	0.716151i	−0.581797	+	0.813334i	$0.697650\pi$
$132$		0			0
$133$	1.31316e6	−	2.27446e6i	0.0483990	−	0.0838295i
$134$	1.47454e7			0.529407
$135$		0			0
$136$	1.15951e7			0.395266
$137$	−9.11520e6	+	1.57880e7i	−0.302862	+	0.524572i	−0.976783	−	0.214231i	$-0.931275\pi$
$137$	−9.11520e6	+	1.57880e7i	−0.302862	+	0.524572i	0.673921	+	0.738803i	$0.264609\pi$
$138$		0			0
$139$	−1.08085e7	−	1.87208e7i	−0.341360	−	0.591253i	0.643326	−	0.765593i	$-0.277554\pi$
$139$	−1.08085e7	−	1.87208e7i	−0.341360	−	0.591253i	−0.984686	+	0.174340i	$0.944221\pi$
$140$	−741237.	−	1.28386e6i	−0.0228302	−	0.0395430i
$141$		0			0
$142$	1.82683e7	−	3.16416e7i	0.535413	−	0.927362i
$143$	−1.67032e6			−0.0477666
$144$		0			0
$145$	−1.16564e8			−3.17524
$146$	−7.65976e6	+	1.32671e7i	−0.203695	+	0.352810i
$147$		0			0
$148$	1.47399e7	+	2.55302e7i	0.373747	+	0.647348i
$149$	−3.50300e7	−	6.06738e7i	−0.867538	−	1.50262i	−0.864505	−	0.502624i	$-0.832368\pi$
$149$	−3.50300e7	−	6.06738e7i	−0.867538	−	1.50262i	−0.00303238	−	0.999995i	$-0.500965\pi$
$150$		0			0
$151$	−3.10773e7	+	5.38275e7i	−0.734555	+	1.27229i	0.220364	+	0.975418i	$0.429276\pi$
$151$	−3.10773e7	+	5.38275e7i	−0.734555	+	1.27229i	−0.954918	+	0.296868i	$0.904058\pi$
$152$	−2.70364e7			−0.624448
$153$		0			0
$154$	120513.			0.00265897
$155$	6.10968e7	−	1.05823e8i	1.31782	−	2.28254i
$156$		0			0
$157$	−2.66627e7	−	4.61811e7i	−0.549864	−	0.952392i	−0.998283	−	0.0585686i	$-0.981346\pi$
$157$	−2.66627e7	−	4.61811e7i	−0.549864	−	0.952392i	0.448420	−	0.893823i	$-0.351987\pi$
$158$	−7.24353e6	−	1.25462e7i	−0.146100	−	0.253053i
$159$		0			0
$160$	−7.63061e6	+	1.32166e7i	−0.147279	+	0.255094i
$161$	−995672.			−0.0188029
$162$		0			0
$163$	8.41147e6			0.152130			0.0760650	−	0.997103i	$-0.475764\pi$
$163$	8.41147e6			0.152130			0.0760650	+	0.997103i	$0.475764\pi$
$164$	7.69292e6	−	1.33245e7i	0.136188	−	0.235884i
$165$		0			0
$166$	1.23571e7	+	2.14032e7i	0.209671	+	0.363161i
$167$	−1.43922e6	−	2.49281e6i	−0.0239122	−	0.0414172i	0.853822	−	0.520566i	$-0.174279\pi$
$167$	−1.43922e6	−	2.49281e6i	−0.0239122	−	0.0414172i	−0.877734	+	0.479148i	$0.840946\pi$
$168$		0			0
$169$	1.61682e7	−	2.80042e7i	0.257667	−	0.446293i
$170$	−8.43791e7			−1.31724
$171$		0			0
$172$	3.63761e7			0.545087
$173$	1.22544e7	−	2.12253e7i	0.179941	−	0.311668i	−0.761919	−	0.647673i	$-0.775743\pi$
$173$	1.22544e7	−	2.12253e7i	0.179941	−	0.311668i	0.941860	+	0.336005i	$0.109076\pi$
$174$		0			0
$175$	3.45127e6	+	5.97778e6i	0.0486795	+	0.0843154i
$176$	−620309.	−	1.07441e6i	−0.00857657	−	0.0148551i
$177$		0			0
$178$	−1.64197e7	+	2.84397e7i	−0.218220	+	0.377968i
$179$	8.25306e7			1.07555			0.537773	−	0.843089i	$-0.319266\pi$
$179$	8.25306e7			1.07555			0.537773	+	0.843089i	$0.319266\pi$
$180$		0			0
$181$	4.49477e7			0.563420			0.281710	−	0.959500i	$-0.409098\pi$
$181$	4.49477e7			0.563420			0.281710	+	0.959500i	$0.409098\pi$
$182$	1.09711e6	−	1.90025e6i	0.0134897	−	0.0233648i
$183$		0			0
$184$	5.12494e6	+	8.87665e6i	0.0606494	+	0.105048i
$185$	−1.07264e8	−	1.85786e8i	−1.24552	−	2.15731i
$186$		0			0
$187$	3.42968e6	−	5.94037e6i	0.0383537	−	0.0664306i
$188$	5.60148e7			0.614824
$189$		0			0
$190$	1.96747e8			2.08099
$191$	−2.22501e6	+	3.85383e6i	−0.0231055	+	0.0400199i	−0.877347	−	0.479857i	$-0.840689\pi$
$191$	−2.22501e6	+	3.85383e6i	−0.0231055	+	0.0400199i	0.854241	+	0.519876i	$0.174022\pi$
$192$		0			0
$193$	−3.35159e7	−	5.80512e7i	−0.335583	−	0.581247i	0.648014	−	0.761629i	$-0.275600\pi$
$193$	−3.35159e7	−	5.80512e7i	−0.335583	−	0.581247i	−0.983597	+	0.180382i	$0.942267\pi$
$194$	3.16649e7	+	5.48452e7i	0.311367	+	0.539303i
$195$		0			0
$196$	2.62742e7	−	4.55083e7i	0.249249	−	0.431712i
$197$	9.80246e7			0.913489			0.456745	−	0.889598i	$-0.349015\pi$
$197$	9.80246e7			0.913489			0.456745	+	0.889598i	$0.349015\pi$
$198$		0			0
$199$	5.48349e7			0.493254			0.246627	−	0.969110i	$-0.420678\pi$
$199$	5.48349e7			0.493254			0.246627	+	0.969110i	$0.420678\pi$
$200$	3.55289e7	−	6.15378e7i	0.314034	−	0.543923i
$201$		0			0
$202$	3.85303e7	+	6.67365e7i	0.328907	+	0.569684i
$203$	6.22390e6	+	1.07801e7i	0.0522187	+	0.0904455i
$204$		0			0
$205$	−5.59823e7	+	9.69642e7i	−0.453850	+	0.786091i
$206$	−7.34412e7			−0.585335
$207$		0			0
$208$	−2.25883e7			−0.174045
$209$	−7.99700e6	+	1.38512e7i	−0.0605920	+	0.104948i
$210$		0			0
$211$	−5.81377e7	−	1.00697e8i	−0.426058	−	0.737955i	0.570460	−	0.821325i	$-0.306765\pi$
$211$	−5.81377e7	−	1.00697e8i	−0.426058	−	0.737955i	−0.996519	+	0.0833704i	$0.973432\pi$
$212$	−2.59533e7	−	4.49525e7i	−0.187076	−	0.324025i
$213$		0			0
$214$	1.52870e7	−	2.64778e7i	0.106629	−	0.184686i
$215$	−2.64713e8			−1.81652
$216$		0			0
$217$	−1.30490e7			−0.0866897
$218$	7.63981e7	−	1.32325e8i	0.499442	−	0.865059i
$219$		0			0
$220$	4.51406e6	+	7.81858e6i	0.0285817	+	0.0495050i
$221$	−6.24451e7	−	1.08158e8i	−0.389157	−	0.674040i
$222$		0			0
$223$	6.60745e7	−	1.14444e8i	0.398994	−	0.691078i	−0.594608	−	0.804016i	$-0.702693\pi$
$223$	6.60745e7	−	1.14444e8i	0.398994	−	0.691078i	0.993602	+	0.112938i	$0.0360260\pi$
$224$	1.62974e6			0.00968835
$225$		0			0
$226$	−1.84216e6			−0.0106157
$227$	2.82306e7	−	4.88969e7i	0.160188	−	0.277454i	−0.774748	−	0.632270i	$-0.782123\pi$
$227$	2.82306e7	−	4.88969e7i	0.160188	−	0.277454i	0.934936	+	0.354816i	$0.115457\pi$
$228$		0			0
$229$	−8.89369e7	−	1.54043e8i	−0.489393	−	0.847654i	0.510532	−	0.859859i	$-0.329448\pi$
$229$	−8.89369e7	−	1.54043e8i	−0.489393	−	0.847654i	−0.999926	+	0.0122046i	$0.996115\pi$
$230$	−3.72948e7	−	6.45965e7i	−0.202116	−	0.350075i
$231$		0			0
$232$	6.40715e7	−	1.10975e8i	0.336866	−	0.583468i
$233$	1.00997e8			0.523076			0.261538	−	0.965193i	$-0.415770\pi$
$233$	1.00997e8			0.523076			0.261538	+	0.965193i	$0.415770\pi$
$234$		0			0
$235$	−4.07627e8			−2.04892
$236$	4.13450e7	−	7.16116e7i	0.204753	−	0.354643i
$237$		0			0
$238$	4.50540e6	+	7.80357e6i	0.0216627	+	0.0375210i
$239$	6.07131e7	+	1.05158e8i	0.287667	+	0.498254i	0.973252	−	0.229739i	$-0.0737871\pi$
$239$	6.07131e7	+	1.05158e8i	0.287667	+	0.498254i	−0.685586	+	0.727992i	$0.740454\pi$
$240$		0			0
$241$	1.14124e8	−	1.97669e8i	0.525191	−	0.909657i	−0.474379	−	0.880321i	$-0.657327\pi$
$241$	1.14124e8	−	1.97669e8i	0.525191	−	0.909657i	0.999570	−	0.0293364i	$-0.00933942\pi$
$242$	1.55163e8			0.703778
$243$		0			0
$244$	−1.04225e8			−0.459313
$245$	−1.91201e8	+	3.31169e8i	−0.830631	+	1.43869i
$246$		0			0
$247$	1.45604e8	+	2.52193e8i	0.614798	+	1.06486i
$248$	6.71659e7	+	1.16335e8i	0.279620	+	0.484315i
$249$		0			0
$250$	−1.13006e8	+	1.95731e8i	−0.457414	+	0.792265i
$251$	−3.32724e8			−1.32808			−0.664042	−	0.747695i	$-0.731160\pi$
$251$	−3.32724e8			−1.32808			−0.664042	+	0.747695i	$0.731160\pi$
$252$		0			0
$253$	6.06354e6			0.0235399
$254$	−1.56977e8	+	2.71892e8i	−0.601061	+	1.04107i
$255$		0			0
$256$	−8.38861e6	−	1.45295e7i	−0.0312500	−	0.0541266i
$257$	2.31212e8	+	4.00470e8i	0.849657	+	1.47165i	0.881515	+	0.472157i	$0.156524\pi$
$257$	2.31212e8	+	4.00470e8i	0.849657	+	1.47165i	−0.0318576	+	0.999492i	$0.510142\pi$
$258$		0			0
$259$	−1.14546e7	+	1.98400e7i	−0.0409668	+	0.0709566i
$260$	1.64377e8			0.580010
$261$		0			0
$262$	1.70221e8			0.584735
$263$	6.65315e7	−	1.15236e8i	0.225519	−	0.390610i	−0.730956	−	0.682424i	$-0.760926\pi$
$263$	6.65315e7	−	1.15236e8i	0.225519	−	0.390610i	0.956475	+	0.291815i	$0.0942590\pi$
$264$		0			0
$265$	1.88865e8	+	3.27124e8i	0.623436	+	1.07982i
$266$	−1.05053e7	−	1.81956e7i	−0.0342232	−	0.0592764i
$267$		0			0
$268$	5.89815e7	−	1.02159e8i	0.187173	−	0.324194i
$269$	3.65232e8			1.14403			0.572013	−	0.820245i	$-0.306163\pi$
$269$	3.65232e8			1.14403			0.572013	+	0.820245i	$0.306163\pi$
$270$		0			0
$271$	1.75400e8			0.535348			0.267674	−	0.963510i	$-0.413745\pi$
$271$	1.75400e8			0.535348			0.267674	+	0.963510i	$0.413745\pi$
$272$	4.63805e7	−	8.03333e7i	0.139747	−	0.242050i
$273$		0			0
$274$	7.29216e7	+	1.26304e8i	0.214156	+	0.370928i
$275$	−2.10179e7	−	3.64041e7i	−0.0609432	−	0.105557i
$276$		0			0
$277$	−9.21421e7	+	1.59595e8i	−0.260483	+	0.451169i	−0.966370	−	0.257155i	$-0.917215\pi$
$277$	−9.21421e7	+	1.59595e8i	−0.260483	+	0.451169i	0.705888	+	0.708324i	$0.250548\pi$
$278$	−1.72936e8			−0.482756
$279$		0			0
$280$	−1.18598e7			−0.0322867
$281$	−4.36306e7	+	7.55704e7i	−0.117306	+	0.203179i	−0.918699	−	0.394958i	$-0.870759\pi$
$281$	−4.36306e7	+	7.55704e7i	−0.117306	+	0.203179i	0.801393	+	0.598138i	$0.204092\pi$
$282$		0			0
$283$	−9.38103e7	−	1.62484e8i	−0.246036	−	0.426146i	0.716387	−	0.697703i	$-0.245795\pi$
$283$	−9.38103e7	−	1.62484e8i	−0.246036	−	0.426146i	−0.962422	+	0.271557i	$0.912461\pi$
$284$	−1.46146e8	−	2.53133e8i	−0.378594	−	0.655744i
$285$		0			0
$286$	−6.68130e6	+	1.15723e7i	−0.0168881	+	0.0292510i
$287$	1.19566e7			0.0298554
$288$		0			0
$289$	1.02535e8			0.249879
$290$	−4.66256e8	+	8.07579e8i	−1.12262	+	1.94443i
$291$		0			0
$292$	6.12781e7	+	1.06137e8i	0.144034	+	0.249474i
$293$	3.05073e8	+	5.28401e8i	0.708543	+	1.22723i	0.965397	+	0.260783i	$0.0839807\pi$
$293$	3.05073e8	+	5.28401e8i	0.708543	+	1.22723i	−0.256854	+	0.966450i	$0.582686\pi$
$294$		0			0
$295$	−3.00873e8	+	5.21127e8i	−0.682347	+	1.18186i
$296$	2.35838e8			0.528558
$297$		0			0
$298$	−5.60480e8			−1.22688
$299$	5.52003e7	−	9.56097e7i	0.119424	−	0.206849i
$300$		0			0
$301$	1.41343e7	+	2.44813e7i	0.0298738	+	0.0517430i
$302$	2.48619e8	+	4.30620e8i	0.519409	+	0.899642i
$303$		0			0
$304$	−1.08146e8	+	1.87314e8i	−0.220776	+	0.382395i
$305$	7.58460e8			1.53068
$306$		0			0
$307$	7.70231e8			1.51928			0.759638	−	0.650346i	$-0.225376\pi$
$307$	7.70231e8			1.51928			0.759638	+	0.650346i	$0.225376\pi$
$308$	482054.	−	834942.i	0.000940087	−	0.00162828i
$309$		0			0
$310$	−4.88774e8	−	8.46581e8i	−0.931842	−	1.61400i
$311$	4.36357e8	+	7.55792e8i	0.822584	+	1.42476i	0.903752	+	0.428057i	$0.140802\pi$
$311$	4.36357e8	+	7.55792e8i	0.822584	+	1.42476i	−0.0811678	+	0.996700i	$0.525865\pi$
$312$		0			0
$313$	7.71953e6	−	1.33706e7i	0.0142294	−	0.0246460i	−0.858823	−	0.512272i	$-0.828804\pi$
$313$	7.71953e6	−	1.33706e7i	0.0142294	−	0.0246460i	0.873052	+	0.487626i	$0.162137\pi$
$314$	−4.26603e8			−0.777625
$315$		0			0
$316$	−1.15896e8			−0.206617
$317$	1.27944e8	−	2.21605e8i	0.225586	−	0.390726i	−0.730909	−	0.682475i	$-0.760904\pi$
$317$	1.27944e8	−	2.21605e8i	0.225586	−	0.390726i	0.956495	+	0.291749i	$0.0942371\pi$
$318$		0			0
$319$	−3.79029e7	−	6.56498e7i	−0.0653740	−	0.113231i
$320$	6.10449e7	+	1.05733e8i	0.104142	+	0.180379i
$321$		0			0
$322$	−3.98269e6	+	6.89822e6i	−0.00664784	+	0.0115144i
$323$	−1.19587e9			−1.97458
$324$		0			0
$325$	−7.65358e8			−1.23672
$326$	3.36459e7	−	5.82763e7i	0.0537861	−	0.0931603i
$327$		0			0
$328$	−6.15433e7	−	1.06596e8i	−0.0962992	−	0.166795i
$329$	2.17651e7	+	3.76983e7i	0.0336958	+	0.0583628i
$330$		0			0
$331$	−1.99183e8	+	3.44995e8i	−0.301893	+	0.522895i	−0.976565	−	0.215223i	$-0.930952\pi$
$331$	−1.99183e8	+	3.44995e8i	−0.301893	+	0.522895i	0.674671	+	0.738118i	$0.264285\pi$
$332$	1.97714e8			0.296520
$333$		0			0
$334$	−2.30276e7			−0.0338170
$335$	−4.29216e8	+	7.43423e8i	−0.623762	+	1.08039i
$336$		0			0
$337$	−5.81304e8	−	1.00685e9i	−0.827368	−	1.43304i	−0.900096	−	0.435691i	$-0.856504\pi$
$337$	−5.81304e8	−	1.00685e9i	−0.827368	−	1.43304i	0.0727283	−	0.997352i	$-0.476829\pi$
$338$	−1.29346e8	−	2.24034e8i	−0.182198	−	0.315576i
$339$		0			0
$340$	−3.37516e8	+	5.84595e8i	−0.465713	+	0.806639i
$341$	7.94670e7			0.108529
$342$		0			0
$343$	8.17959e7			0.109446
$344$	1.45504e8	−	2.52021e8i	0.192717	−	0.333796i
$345$		0			0
$346$	−9.80353e7	−	1.69802e8i	−0.127238	−	0.220382i
$347$	−5.74010e8	−	9.94214e8i	−0.737507	−	1.27740i	−0.953615	−	0.301030i	$-0.902670\pi$
$347$	−5.74010e8	−	9.94214e8i	−0.737507	−	1.27740i	0.216108	−	0.976369i	$-0.430664\pi$
$348$		0			0
$349$	6.87991e8	−	1.19164e9i	0.866351	−	1.50056i	0.000651345	−	1.00000i	$-0.499793\pi$
$349$	6.87991e8	−	1.19164e9i	0.866351	−	1.50056i	0.865700	−	0.500564i	$-0.166874\pi$
$350$	5.52204e7			0.0688432
$351$		0			0
$352$	−9.92494e6			−0.0121291
$353$	−2.22495e8	+	3.85373e8i	−0.269221	+	0.466304i	−0.968661	−	0.248387i	$-0.920099\pi$
$353$	−2.22495e8	+	3.85373e8i	−0.269221	+	0.466304i	0.699440	+	0.714691i	$0.253433\pi$
$354$		0			0
$355$	1.06352e9	+	1.84208e9i	1.26168	+	2.18529i
$356$	1.31357e8	+	2.27518e8i	0.154305	+	0.267264i
$357$		0			0
$358$	3.30122e8	−	5.71789e8i	0.380263	−	0.658635i
$359$	1.14314e9			1.30397			0.651984	−	0.758233i	$-0.273937\pi$
$359$	1.14314e9			1.30397			0.651984	+	0.758233i	$0.273937\pi$
$360$		0			0
$361$	1.89455e9			2.11949
$362$	1.79791e8	−	3.11407e8i	0.199199	−	0.345023i
$363$		0			0
$364$	−8.77689e6	−	1.52020e7i	−0.00953863	−	0.0165214i
$365$	−4.45928e8	−	7.72370e8i	−0.479998	−	0.831381i
$366$		0			0
$367$	3.75522e8	−	6.50423e8i	0.396555	−	0.686854i	−0.596743	−	0.802432i	$-0.703539\pi$
$367$	3.75522e8	−	6.50423e8i	0.396555	−	0.686854i	0.993298	+	0.115578i	$0.0368722\pi$
$368$	8.19990e7			0.0857711
$369$		0			0
$370$	−1.71622e9			−1.76144
$371$	2.01688e7	−	3.49334e7i	0.0205056	−	0.0355167i
$372$		0			0
$373$	6.65173e8	+	1.15211e9i	0.663672	+	1.14951i	0.979644	+	0.200745i	$0.0643362\pi$
$373$	6.65173e8	+	1.15211e9i	0.663672	+	1.14951i	−0.315972	+	0.948769i	$0.602330\pi$
$374$	−2.74374e7	−	4.75230e7i	−0.0271202	−	0.0469735i
$375$		0			0
$376$	2.24059e8	−	3.88082e8i	0.217373	−	0.376501i
$377$	−1.38022e9			−1.32664
$378$		0			0
$379$	1.06374e9			1.00369			0.501845	−	0.864958i	$-0.332655\pi$
$379$	1.06374e9			1.00369			0.501845	+	0.864958i	$0.332655\pi$
$380$	7.86989e8	−	1.36310e9i	0.735743	−	1.27434i
$381$		0			0
$382$	1.78001e7	+	3.08306e7i	0.0163380	+	0.0282983i
$383$	1.05087e8	+	1.82017e8i	0.0955774	+	0.165545i	0.909849	−	0.414939i	$-0.136197\pi$
$383$	1.05087e8	+	1.82017e8i	0.0955774	+	0.165545i	−0.814272	+	0.580483i	$0.802864\pi$
$384$		0			0
$385$	−3.50796e6	+	6.07597e6i	−0.00313287	+	0.00542629i
$386$	−5.36254e8			−0.474586
$387$		0			0
$388$	5.06639e8			0.440339
$389$	−1.97731e8	+	3.42480e8i	−0.170314	+	0.294993i	−0.938530	−	0.345198i	$-0.887812\pi$
$389$	−1.97731e8	+	3.42480e8i	−0.170314	+	0.294993i	0.768216	+	0.640191i	$0.221145\pi$
$390$		0			0
$391$	2.26686e8	+	3.92631e8i	0.191781	+	0.332174i
$392$	−2.10194e8	−	3.64066e8i	−0.176246	−	0.305267i
$393$		0			0
$394$	3.92099e8	−	6.79135e8i	0.322967	−	0.559396i
$395$	8.43392e8			0.688557
$396$		0			0
$397$	1.60421e8			0.128675			0.0643375	−	0.997928i	$-0.479507\pi$
$397$	1.60421e8			0.128675			0.0643375	+	0.997928i	$0.479507\pi$
$398$	2.19339e8	−	3.79907e8i	0.174392	−	0.302055i
$399$		0			0
$400$	−2.84231e8	−	4.92303e8i	−0.222056	−	0.384611i
$401$	−1.40643e8	−	2.43602e8i	−0.108922	−	0.188658i	0.806412	−	0.591354i	$-0.201406\pi$
$401$	−1.40643e8	−	2.43602e8i	−0.108922	−	0.188658i	−0.915334	+	0.402696i	$0.868073\pi$
$402$		0			0
$403$	7.23438e8	−	1.25303e9i	0.550597	−	0.953662i
$404$	6.16486e8			0.465145
$405$		0			0
$406$	9.95824e7			0.0738484
$407$	6.97576e7	−	1.20824e8i	0.0512874	−	0.0888324i
$408$		0			0
$409$	−2.07274e8	−	3.59010e8i	−0.149801	−	0.259463i	0.781353	−	0.624089i	$-0.214530\pi$
$409$	−2.07274e8	−	3.59010e8i	−0.149801	−	0.259463i	−0.931154	+	0.364627i	$0.881197\pi$
$410$	4.47858e8	+	7.75713e8i	0.320920	+	0.555850i
$411$		0			0
$412$	−2.93765e8	+	5.08816e8i	−0.206947	+	0.358443i
$413$	6.42600e7			0.0448865
$414$		0			0
$415$	−1.43879e9			−0.988163
$416$	−9.03531e7	+	1.56496e8i	−0.0615341	+	0.106580i
$417$		0			0
$418$	6.39760e7	+	1.10810e8i	0.0428450	+	0.0742097i
$419$	−1.29962e9	−	2.25101e9i	−0.863114	−	1.49496i	−0.868907	−	0.494975i	$-0.835177\pi$
$419$	−1.29962e9	−	2.25101e9i	−0.863114	−	1.49496i	0.00579285	−	0.999983i	$-0.498156\pi$
$420$		0			0
$421$	−2.89282e8	+	5.01051e8i	−0.188944	+	0.327261i	−0.944899	−	0.327363i	$-0.893840\pi$
$421$	−2.89282e8	+	5.01051e8i	−0.188944	+	0.327261i	0.755954	+	0.654625i	$0.227173\pi$
$422$	−9.30203e8			−0.602538
$423$		0			0
$424$	−4.15253e8			−0.264565
$425$	1.57151e9	−	2.72193e9i	0.993014	−	1.71995i
$426$		0			0
$427$	−4.04978e7	−	7.01442e7i	−0.0251729	−	0.0436008i
$428$	−1.22296e8	−	2.11823e8i	−0.0753978	−	0.130593i
$429$		0			0
$430$	−1.05885e9	+	1.83398e9i	−0.642237	+	1.11239i
$431$	7.69740e7			0.0463099			0.0231549	−	0.999732i	$-0.492629\pi$
$431$	7.69740e7			0.0463099			0.0231549	+	0.999732i	$0.492629\pi$
$432$		0			0
$433$	−2.75020e8			−0.162801			−0.0814005	−	0.996681i	$-0.525939\pi$
$433$	−2.75020e8			−0.162801			−0.0814005	+	0.996681i	$0.525939\pi$
$434$	−5.21959e7	+	9.04059e7i	−0.0306494	+	0.0530864i
$435$		0			0
$436$	−6.11185e8	−	1.05860e9i	−0.353159	−	0.611689i
$437$	−5.28564e8	−	9.15500e8i	−0.302979	−	0.524775i
$438$		0			0
$439$	−7.61497e8	+	1.31895e9i	−0.429579	+	0.744052i	−0.996836	−	0.0794888i	$-0.974671\pi$
$439$	−7.61497e8	+	1.31895e9i	−0.429579	+	0.744052i	0.567257	+	0.823541i	$0.308005\pi$
$440$	7.22250e7			0.0404206
$441$		0			0
$442$	−9.99121e8			−0.550351
$443$	1.04029e9	−	1.80183e9i	0.568512	−	0.984691i	−0.428202	−	0.903683i	$-0.640853\pi$
$443$	1.04029e9	−	1.80183e9i	0.568512	−	0.984691i	0.996713	−	0.0810079i	$-0.0258139\pi$
$444$		0			0
$445$	−9.55903e8	−	1.65567e9i	−0.514226	−	0.890665i
$446$	−5.28596e8	−	9.15555e8i	−0.282132	−	0.488666i
$447$		0			0
$448$	6.51895e6	−	1.12912e7i	0.00342535	−	0.00593288i
$449$	−1.89838e9			−0.989741			−0.494870	−	0.868967i	$-0.664784\pi$
$449$	−1.89838e9			−0.989741			−0.494870	+	0.868967i	$0.664784\pi$
$450$		0			0
$451$	−7.28147e7			−0.0373767
$452$	−7.36865e6	+	1.27629e7i	−0.00375322	+	0.00650076i
$453$		0			0
$454$	−2.25845e8	−	3.91175e8i	−0.113270	−	0.196189i
$455$	6.38705e7	+	1.10627e8i	0.0317878	+	0.0550581i
$456$		0			0
$457$	−3.43032e8	+	5.94149e8i	−0.168123	+	0.291198i	−0.937760	−	0.347284i	$-0.887104\pi$
$457$	−3.43032e8	+	5.94149e8i	−0.168123	+	0.291198i	0.769637	+	0.638482i	$0.220437\pi$
$458$	−1.42299e9			−0.692107
$459$		0			0
$460$	−5.96717e8			−0.285835
$461$	7.60483e8	−	1.31720e9i	0.361523	−	0.626177i	−0.626688	−	0.779270i	$-0.715590\pi$
$461$	7.60483e8	−	1.31720e9i	0.361523	−	0.626177i	0.988212	+	0.153093i	$0.0489234\pi$
$462$		0			0
$463$	2.50143e8	+	4.33260e8i	0.117126	+	0.202869i	0.918628	−	0.395124i	$-0.129298\pi$
$463$	2.50143e8	+	4.33260e8i	0.117126	+	0.202869i	−0.801501	+	0.597993i	$0.795965\pi$
$464$	−5.12572e8	−	8.87800e8i	−0.238200	−	0.412575i
$465$		0			0
$466$	4.03990e8	−	6.99730e8i	0.184935	−	0.320317i
$467$	4.03333e9			1.83255			0.916274	−	0.400553i	$-0.131182\pi$
$467$	4.03333e9			1.83255			0.916274	+	0.400553i	$0.131182\pi$
$468$		0			0
$469$	9.16714e7			0.0410326
$470$	−1.63051e9	+	2.82412e9i	−0.724403	+	1.25470i
$471$		0			0
$472$	−3.30760e8	−	5.72893e8i	−0.144782	−	0.250771i
$473$	−8.60763e7	−	1.49089e8i	−0.0373998	−	0.0647784i
$474$		0			0
$475$	−3.66430e9	+	6.34675e9i	−1.56878	+	2.71721i
$476$	7.20863e7			0.0306358
$477$		0			0
$478$	9.71410e8			0.406822
$479$	−1.30131e9	+	2.25394e9i	−0.541012	+	0.937059i	0.457835	+	0.889037i	$0.348625\pi$
$479$	−1.30131e9	+	2.25394e9i	−0.541012	+	0.937059i	−0.998846	+	0.0480221i	$0.984708\pi$
$480$		0			0
$481$	−1.27010e9	−	2.19987e9i	−0.520389	−	0.901341i
$482$	−9.12992e8	−	1.58135e9i	−0.371366	−	0.643225i
$483$		0			0
$484$	6.20654e8	−	1.07500e9i	0.248823	−	0.430974i
$485$	−3.68687e9			−1.46744
$486$		0			0
$487$	4.35516e9			1.70865			0.854325	−	0.519739i	$-0.173971\pi$
$487$	4.35516e9			1.70865			0.854325	+	0.519739i	$0.173971\pi$
$488$	−4.16901e8	+	7.22094e8i	−0.162392	+	0.281271i
$489$		0			0
$490$	1.52961e9	+	2.64935e9i	0.587345	+	1.01731i
$491$	2.08328e8	+	3.60834e8i	0.0794258	+	0.137569i	0.903002	−	0.429636i	$-0.141358\pi$
$491$	2.08328e8	+	3.60834e8i	0.0794258	+	0.137569i	−0.823577	+	0.567205i	$0.808025\pi$
$492$		0			0
$493$	2.83400e9	−	4.90863e9i	1.06521	−	1.84500i
$494$	2.32966e9			0.869456
$495$		0			0
$496$	1.07465e9			0.395442
$497$	1.13573e8	−	1.96714e8i	0.0414981	−	0.0718768i
$498$		0			0
$499$	1.67135e9	+	2.89486e9i	0.602164	+	1.04298i	0.992493	+	0.122303i	$0.0390280\pi$
$499$	1.67135e9	+	2.89486e9i	0.602164	+	1.04298i	−0.390329	+	0.920676i	$0.627639\pi$
$500$	9.04044e8	+	1.56585e9i	0.323441	+	0.560216i
$501$		0			0
$502$	−1.33089e9	+	2.30518e9i	−0.469549	+	0.813282i
$503$	1.48681e9			0.520915			0.260458	−	0.965485i	$-0.416127\pi$
$503$	1.48681e9			0.520915			0.260458	+	0.965485i	$0.416127\pi$
$504$		0			0
$505$	−4.48624e9			−1.55011
$506$	2.42542e7	−	4.20095e7i	0.00832261	−	0.0144152i
$507$		0			0
$508$	1.25582e9	+	2.17514e9i	0.425014	+	0.736146i
$509$	−3.17171e8	−	5.49356e8i	−0.106606	−	0.184647i	0.807787	−	0.589474i	$-0.200665\pi$
$509$	−3.17171e8	−	5.49356e8i	−0.106606	−	0.184647i	−0.914393	+	0.404827i	$0.867332\pi$
$510$		0			0
$511$	−4.76204e7	+	8.24809e7i	−0.0157877	+	0.0273452i
$512$	−1.34218e8			−0.0441942
$513$		0			0
$514$	3.69939e9			1.20160
$515$	2.13776e9	−	3.70271e9i	0.689658	−	1.19452i
$516$		0			0
$517$	−1.32547e8	−	2.29579e8i	−0.0421846	−	0.0730659i
$518$	9.16370e7	+	1.58720e8i	0.0289679	+	0.0501739i
$519$		0			0
$520$	6.57510e8	−	1.13884e9i	0.205065	−	0.355182i
$521$	−5.00649e9			−1.55096			−0.775481	−	0.631371i	$-0.782493\pi$
$521$	−5.00649e9			−1.55096			−0.775481	+	0.631371i	$0.782493\pi$
$522$		0			0
$523$	−4.06385e9			−1.24217			−0.621086	−	0.783743i	$-0.713308\pi$
$523$	−4.06385e9			−1.24217			−0.621086	+	0.783743i	$0.713308\pi$
$524$	6.80884e8	−	1.17933e9i	0.206735	−	0.358075i
$525$		0			0
$526$	−5.32252e8	−	9.21888e8i	−0.159466	−	0.276203i
$527$	2.97087e9	+	5.14570e9i	0.884192	+	1.53147i
$528$		0			0
$529$	1.50203e9	−	2.60159e9i	0.441146	−	0.764088i
$530$	3.02185e9			0.881672
$531$		0			0
$532$	−1.68084e8			−0.0483990
$533$	−6.62879e8	+	1.14814e9i	−0.189622	+	0.328435i
$534$		0			0
$535$	8.89962e8	+	1.54146e9i	0.251266	+	0.435205i
$536$	−4.71852e8	−	8.17272e8i	−0.132352	−	0.229240i
$537$		0			0
$538$	1.46093e9	−	2.53040e9i	0.404474	−	0.700570i
$539$	−2.48690e8			−0.0684065
$540$		0			0
$541$	6.36182e8			0.172739			0.0863696	−	0.996263i	$-0.472473\pi$
$541$	6.36182e8			0.172739			0.0863696	+	0.996263i	$0.472473\pi$
$542$	7.01599e8	−	1.21520e9i	0.189274	−	0.327832i
$543$		0			0
$544$	−3.71044e8	−	6.42667e8i	−0.0988164	−	0.171155i
$545$	4.44767e9	+	7.70359e9i	1.17691	+	2.03847i
$546$		0			0
$547$	1.27423e9	−	2.20704e9i	0.332884	−	0.576573i	−0.650192	−	0.759770i	$-0.725311\pi$
$547$	1.27423e9	−	2.20704e9i	0.332884	−	0.576573i	0.983076	+	0.183197i	$0.0586448\pi$
$548$	1.16675e9			0.302862
$549$		0			0
$550$	−3.36287e8			−0.0861867
$551$	−6.60806e9	+	1.14455e10i	−1.68284	+	2.91477i
$552$		0			0
$553$	−4.50327e7	−	7.79989e7i	−0.0113237	−	0.0196133i
$554$	7.37136e8	+	1.27676e9i	0.184189	+	0.319025i
$555$		0			0
$556$	−6.91742e8	+	1.19813e9i	−0.170680	+	0.295626i
$557$	3.06157e9			0.750673			0.375337	−	0.926889i	$-0.377527\pi$
$557$	3.06157e9			0.750673			0.375337	+	0.926889i	$0.377527\pi$
$558$		0			0
$559$	−3.13443e9			−0.758957
$560$	−4.74392e7	+	8.21671e7i	−0.0114151	+	0.0197715i
$561$		0			0
$562$	3.49045e8	+	6.04563e8i	0.0829476	+	0.143670i
$563$	−2.12350e9	−	3.67801e9i	−0.501502	−	0.868627i	−0.999998	−	0.00173536i	$-0.999448\pi$
$563$	−2.12350e9	−	3.67801e9i	−0.501502	−	0.868627i	0.498496	−	0.866892i	$-0.333886\pi$
$564$		0			0
$565$	5.36226e7	−	9.28770e7i	0.0125077	−	0.0216640i
$566$	−1.50096e9			−0.347947
$567$		0			0
$568$	−2.33834e9			−0.535413
$569$	−2.73746e6	+	4.74142e6i	−0.000622952	+	0.00107898i	−0.866337	−	0.499460i	$-0.833532\pi$
$569$	−2.73746e6	+	4.74142e6i	−0.000622952	+	0.00107898i	0.865714	+	0.500539i	$0.166865\pi$
$570$		0			0
$571$	−2.47557e9	−	4.28782e9i	−0.556479	−	0.963851i	−0.997787	−	0.0664946i	$-0.978818\pi$
$571$	−2.47557e9	−	4.28782e9i	−0.556479	−	0.963851i	0.441307	−	0.897356i	$-0.354515\pi$
$572$	5.34504e7	+	9.25788e7i	0.0119417	+	0.0206836i
$573$		0			0
$574$	4.78265e7	−	8.28380e7i	0.0105555	−	0.0182826i
$575$	2.77837e9			0.609471
$576$		0			0
$577$	−3.55867e9			−0.771210			−0.385605	−	0.922664i	$-0.626007\pi$
$577$	−3.55867e9			−0.771210			−0.385605	+	0.922664i	$0.626007\pi$
$578$	4.10140e8	−	7.10383e8i	0.0883455	−	0.153019i
$579$		0			0
$580$	3.73005e9	+	6.46063e9i	0.793809	+	1.37492i
$581$	7.68237e7	+	1.33063e8i	0.0162510	+	0.0281475i
$582$		0			0
$583$	−1.22826e8	+	2.12741e8i	−0.0256715	+	0.0444643i
$584$	9.80449e8			0.203695
$585$		0			0
$586$	4.88116e9			1.00203
$587$	2.17427e7	−	3.76594e7i	0.00443690	−	0.00768493i	−0.863798	−	0.503838i	$-0.831921\pi$
$587$	2.17427e7	−	3.76594e7i	0.00443690	−	0.00768493i	0.868235	+	0.496153i	$0.165254\pi$
$588$		0			0
$589$	−6.92720e9	−	1.19983e10i	−1.39686	−	2.41944i
$590$	2.40698e9	+	4.16901e9i	0.482492	+	0.835701i
$591$		0			0
$592$	9.43351e8	−	1.63393e9i	0.186873	−	0.323674i
$593$	2.06982e9			0.407607			0.203804	−	0.979012i	$-0.434670\pi$
$593$	2.06982e9			0.407607			0.203804	+	0.979012i	$0.434670\pi$
$594$		0			0
$595$	−5.24581e8			−0.102095
$596$	−2.24192e9	+	3.88312e9i	−0.433769	+	0.751310i
$597$		0			0
$598$	−4.41602e8	−	7.64878e8i	−0.0844457	−	0.146264i
$599$	4.53945e9	+	7.86255e9i	0.862996	+	1.49475i	0.869023	+	0.494772i	$0.164748\pi$
$599$	4.53945e9	+	7.86255e9i	0.862996	+	1.49475i	−0.00602647	+	0.999982i	$0.501918\pi$
$600$		0			0
$601$	−1.51985e8	+	2.63246e8i	−0.0285588	+	0.0494653i	−0.879952	−	0.475063i	$-0.842425\pi$
$601$	−1.51985e8	+	2.63246e8i	−0.0285588	+	0.0494653i	0.851393	+	0.524529i	$0.175758\pi$
$602$	2.26148e8			0.0422480
$603$		0			0
$604$	3.97790e9			0.734555
$605$	−4.51657e9	+	7.82293e9i	−0.829211	+	1.43624i
$606$		0			0
$607$	3.25143e9	+	5.63164e9i	0.590084	+	1.02206i	0.994221	+	0.107357i	$0.0342388\pi$
$607$	3.25143e9	+	5.63164e9i	0.590084	+	1.02206i	−0.404136	+	0.914699i	$0.632428\pi$
$608$	8.65165e8	+	1.49851e9i	0.156112	+	0.270394i
$609$		0			0
$610$	3.03384e9	−	5.25477e9i	0.541176	−	0.937344i
$611$	−4.82665e9			−0.856055
$612$		0			0
$613$	2.20282e9			0.386249			0.193124	−	0.981174i	$-0.438138\pi$
$613$	2.20282e9			0.386249			0.193124	+	0.981174i	$0.438138\pi$
$614$	3.08092e9	−	5.33632e9i	0.537145	−	0.930363i
$615$		0			0
$616$	−3.85643e6	−	6.67954e6i	−0.000664742	−	0.00115137i
$617$	−1.99310e9	−	3.45214e9i	−0.341610	−	0.591685i	0.643122	−	0.765764i	$-0.277639\pi$
$617$	−1.99310e9	−	3.45214e9i	−0.341610	−	0.591685i	−0.984732	+	0.174078i	$0.944305\pi$
$618$		0			0
$619$	−7.54162e8	+	1.30625e9i	−0.127805	+	0.221364i	−0.922826	−	0.385217i	$-0.874126\pi$
$619$	−7.54162e8	+	1.30625e9i	−0.127805	+	0.221364i	0.795021	+	0.606582i	$0.207460\pi$
$620$	−7.82038e9			−1.31782
$621$		0			0
$622$	6.98171e9			1.16331
$623$	−1.02080e8	+	1.76808e8i	−0.0169135	+	0.0292951i
$624$		0			0
$625$	−1.15756e9	−	2.00495e9i	−0.189655	−	0.328492i
$626$	−6.17562e7	−	1.06965e8i	−0.0100617	−	0.0174273i
$627$		0			0
$628$	−1.70641e9	+	2.95559e9i	−0.274932	+	0.476196i
$629$	1.04315e10			1.67136
$630$		0			0
$631$	5.99394e9			0.949751			0.474875	−	0.880053i	$-0.342493\pi$
$631$	5.99394e9			0.949751			0.474875	+	0.880053i	$0.342493\pi$
$632$	−4.63586e8	+	8.02954e8i	−0.0730500	+	0.126526i
$633$		0			0
$634$	−1.02355e9	−	1.77284e9i	−0.159513	−	0.276285i
$635$	−9.13873e9	−	1.58287e10i	−1.41637	−	2.45323i
$636$		0			0
$637$	−2.26398e9	+	3.92133e9i	−0.347044	+	0.601098i
$638$	−6.06447e8			−0.0924529
$639$		0			0
$640$	9.76718e8			0.147279
$641$	4.12807e9	−	7.15003e9i	0.619076	−	1.07227i	−0.370578	−	0.928801i	$-0.620840\pi$
$641$	4.12807e9	−	7.15003e9i	0.619076	−	1.07227i	0.989655	−	0.143471i	$-0.0458263\pi$
$642$		0			0
$643$	5.48402e9	+	9.49859e9i	0.813505	+	1.40903i	0.910396	+	0.413737i	$0.135777\pi$
$643$	5.48402e9	+	9.49859e9i	0.813505	+	1.40903i	−0.0968913	+	0.995295i	$0.530890\pi$
$644$	3.18615e7	+	5.51858e7i	0.00470074	+	0.00814191i
$645$		0			0
$646$	−4.78348e9	+	8.28524e9i	−0.698121	+	1.20918i
$647$	−2.91111e9			−0.422565			−0.211283	−	0.977425i	$-0.567764\pi$
$647$	−2.91111e9			−0.422565			−0.211283	+	0.977425i	$0.567764\pi$
$648$		0			0
$649$	−3.91337e8			−0.0561946
$650$	−3.06143e9	+	5.30255e9i	−0.437248	+	0.757336i
$651$		0			0
$652$	−2.69167e8	−	4.66211e8i	−0.0380325	−	0.0658743i
$653$	5.30674e9	+	9.19155e9i	0.745816	+	1.29179i	0.949812	+	0.312820i	$0.101274\pi$
$653$	5.30674e9	+	9.19155e9i	0.745816	+	1.29179i	−0.203996	+	0.978972i	$0.565393\pi$
$654$		0			0
$655$	−4.95488e9	+	8.58210e9i	−0.688951	+	1.19330i
$656$	−9.84694e8			−0.136188
$657$		0			0
$658$	3.48242e8			0.0476530
$659$	1.61433e9	−	2.79610e9i	0.219732	−	0.380587i	−0.734994	−	0.678074i	$-0.762815\pi$
$659$	1.61433e9	−	2.79610e9i	0.219732	−	0.380587i	0.954726	+	0.297486i	$0.0961483\pi$
$660$		0			0
$661$	−5.05736e9	−	8.75960e9i	−0.681112	−	1.17972i	−0.974642	−	0.223771i	$-0.928163\pi$
$661$	−5.05736e9	−	8.75960e9i	−0.681112	−	1.17972i	0.293530	−	0.955950i	$-0.405170\pi$
$662$	1.59346e9	+	2.75996e9i	0.213471	+	0.369742i
$663$		0			0
$664$	7.90856e8	−	1.36980e9i	0.104836	−	0.181581i
$665$	1.22317e9			0.161291
$666$		0			0
$667$	5.01041e9			0.653782
$668$	−9.21103e7	+	1.59540e8i	−0.0119561	+	0.0207086i
$669$		0			0
$670$	3.43373e9	+	5.94739e9i	0.441066	+	0.763949i
$671$	2.46627e8	+	4.27171e8i	0.0315146	+	0.0545850i
$672$		0			0
$673$	6.36689e9	−	1.10278e10i	0.805146	−	1.39455i	−0.111046	−	0.993815i	$-0.535420\pi$
$673$	6.36689e9	−	1.10278e10i	0.805146	−	1.39455i	0.916192	−	0.400739i	$-0.131247\pi$
$674$	−9.30086e9			−1.17007
$675$		0			0
$676$	−2.06953e9			−0.257667
$677$	2.30338e8	−	3.98958e8i	0.0285303	−	0.0494159i	−0.851408	−	0.524504i	$-0.824251\pi$
$677$	2.30338e8	−	3.98958e8i	0.0285303	−	0.0494159i	0.879938	+	0.475089i	$0.157584\pi$
$678$		0			0
$679$	1.96859e8	+	3.40970e8i	0.0241330	+	0.0417996i
$680$	2.70013e9	+	4.67676e9i	0.329309	+	0.570380i
$681$		0			0
$682$	3.17868e8	−	5.50563e8i	0.0383708	−	0.0664602i
$683$	−1.53853e9			−0.184771			−0.0923856	−	0.995723i	$-0.529449\pi$
$683$	−1.53853e9			−0.184771			−0.0923856	+	0.995723i	$0.529449\pi$
$684$		0			0
$685$	−8.49055e9			−1.00930
$686$	3.27183e8	−	5.66698e8i	0.0386952	−	0.0670220i
$687$		0			0
$688$	−1.16403e9	−	2.01617e9i	−0.136272	−	0.236030i
$689$	2.23633e9	+	3.87344e9i	0.260477	+	0.451159i
$690$		0			0
$691$	−4.53240e9	+	7.85035e9i	−0.522583	+	0.905140i	0.477072	+	0.878864i	$0.341698\pi$
$691$	−4.53240e9	+	7.85035e9i	−0.522583	+	0.905140i	−0.999655	+	0.0262760i	$0.991635\pi$
$692$	−1.56857e9			−0.179941
$693$		0			0
$694$	−9.18415e9			−1.04299
$695$	5.03389e9	−	8.71895e9i	0.568797	−	0.985185i
$696$		0			0
$697$	−2.72218e9	−	4.71495e9i	−0.304510	−	0.527427i
$698$	−5.50393e9	−	9.53308e9i	−0.612603	−	1.06106i
$699$		0			0
$700$	2.20881e8	−	3.82578e8i	0.0243398	−	0.0421577i
$701$	9.14932e9			1.00317			0.501586	−	0.865108i	$-0.332750\pi$
$701$	9.14932e9			1.00317			0.501586	+	0.865108i	$0.332750\pi$
$702$		0			0
$703$	−2.43233e10			−2.64046
$704$	−3.96998e7	+	6.87620e7i	−0.00428828	+	0.00742753i
$705$		0			0
$706$	1.77996e9	+	3.08298e9i	0.190368	+	0.329727i
$707$	2.39541e8	+	4.14898e8i	0.0254925	+	0.0441543i
$708$		0			0
$709$	−1.06905e8	+	1.85165e8i	−0.0112651	+	0.0195118i	−0.871603	−	0.490212i	$-0.836919\pi$
$709$	−1.06905e8	+	1.85165e8i	−0.0112651	+	0.0195118i	0.860338	+	0.509724i	$0.170253\pi$
$710$	1.70164e10			1.78428
$711$		0			0
$712$	2.10172e9			0.218220
$713$	−2.62620e9	+	4.54871e9i	−0.271340	+	0.469975i
$714$		0			0
$715$	−3.88965e8	−	6.73707e8i	−0.0397960	−	0.0689287i
$716$	−2.64098e9	−	4.57431e9i	−0.268887	−	0.465725i
$717$		0			0
$718$	4.57254e9	−	7.91987e9i	0.461022	−	0.798514i
$719$	−8.06712e9			−0.809408			−0.404704	−	0.914448i	$-0.632625\pi$
$719$	−8.06712e9			−0.809408			−0.404704	+	0.914448i	$0.632625\pi$
$720$		0			0
$721$	−4.56581e8			−0.0453674
$722$	7.57819e9	−	1.31258e10i	0.749351	−	1.29791i
$723$		0			0
$724$	−1.43833e9	−	2.49125e9i	−0.140855	−	0.243968i
$725$	−1.73675e10	−	3.00813e10i	−1.69260	−	2.93166i
$726$		0			0
$727$	−7.82244e9	+	1.35489e10i	−0.755043	+	1.30777i	0.190310	+	0.981724i	$0.439051\pi$
$727$	−7.82244e9	+	1.35489e10i	−0.755043	+	1.30777i	−0.945353	+	0.326049i	$0.894283\pi$
$728$	−1.40430e8			−0.0134897
$729$		0			0
$730$	−7.13485e9			−0.678820
$731$	6.43592e9	−	1.11473e10i	0.609397	−	1.05551i
$732$		0			0
$733$	4.90783e9	+	8.50061e9i	0.460283	+	0.797234i	0.998975	−	0.0452688i	$-0.0144144\pi$
$733$	4.90783e9	+	8.50061e9i	0.460283	+	0.797234i	−0.538691	+	0.842503i	$0.681081\pi$
$734$	−3.00417e9	−	5.20338e9i	−0.280407	−	0.485679i
$735$		0			0
$736$	3.27996e8	−	5.68106e8i	0.0303247	−	0.0525239i
$737$	−5.58270e8			−0.0513698
$738$		0			0
$739$	−3.29280e8			−0.0300131			−0.0150065	−	0.999887i	$-0.504777\pi$
$739$	−3.29280e8			−0.0300131			−0.0150065	+	0.999887i	$0.504777\pi$
$740$	−6.86488e9	+	1.18903e10i	−0.622761	+	1.07865i
$741$		0			0
$742$	−1.61351e8	−	2.79467e8i	−0.0144996	−	0.0251141i
$743$	−3.03824e9	−	5.26239e9i	−0.271745	−	0.470676i	0.697564	−	0.716523i	$-0.254267\pi$
$743$	−3.03824e9	−	5.26239e9i	−0.271745	−	0.470676i	−0.969309	+	0.245846i	$0.920934\pi$
$744$		0			0
$745$	1.63147e10	−	2.82579e10i	1.44555	−	2.50376i
$746$	1.06428e10			0.938574
$747$		0			0
$748$	−4.38999e8			−0.0383537
$749$	9.50385e7	−	1.64612e8i	0.00826444	−	0.0143144i
$750$		0			0
$751$	4.49540e9	+	7.78627e9i	0.387283	+	0.670795i	0.992083	−	0.125583i	$-0.0400802\pi$
$751$	4.49540e9	+	7.78627e9i	0.387283	+	0.670795i	−0.604800	+	0.796378i	$0.706747\pi$
$752$	−1.79248e9	−	3.10466e9i	−0.153706	−	0.266227i
$753$		0			0
$754$	−5.52087e9	+	9.56243e9i	−0.469038	+	0.812397i
$755$	−2.89476e10			−2.44793
$756$		0			0
$757$	−4.08451e9			−0.342219			−0.171109	−	0.985252i	$-0.554735\pi$
$757$	−4.08451e9			−0.342219			−0.171109	+	0.985252i	$0.554735\pi$
$758$	4.25497e9	−	7.36982e9i	0.354858	−	0.614632i
$759$		0			0
$760$	−6.29591e9	−	1.09048e10i	−0.520249	−	0.901097i
$761$	9.07760e9	+	1.57229e10i	0.746663	+	1.29326i	0.949414	+	0.314028i	$0.101679\pi$
$761$	9.07760e9	+	1.57229e10i	0.746663	+	1.29326i	−0.202751	+	0.979230i	$0.564988\pi$
$762$		0			0
$763$	4.74964e8	−	8.22661e8i	0.0387101	−	0.0670479i
$764$	2.84801e8			0.0231055
$765$		0			0
$766$	1.68140e9			0.135167
$767$	−3.56259e9	+	6.17059e9i	−0.285090	+	0.493791i
$768$		0			0
$769$	1.93456e9	+	3.35075e9i	0.153405	+	0.265705i	0.932477	−	0.361229i	$-0.117643\pi$
$769$	1.93456e9	+	3.35075e9i	0.153405	+	0.265705i	−0.779072	+	0.626934i	$0.784309\pi$
$770$	2.80637e7	+	4.86078e7i	0.00221527	+	0.00383697i
$771$		0			0
$772$	−2.14502e9	+	3.71528e9i	−0.167791	+	0.290623i
$773$	5.34479e9			0.416201			0.208100	−	0.978107i	$-0.433272\pi$
$773$	5.34479e9			0.416201			0.208100	+	0.978107i	$0.433272\pi$
$774$		0			0
$775$	3.64125e10			2.80992
$776$	2.02655e9	−	3.51010e9i	0.155683	−	0.269652i
$777$		0			0
$778$	1.58185e9	+	2.73984e9i	0.120430	+	0.208591i
$779$	6.34732e9	+	1.09939e10i	0.481071	+	0.833239i
$780$		0			0
$781$	−6.91649e8	+	1.19797e9i	−0.0519526	+	0.0899845i
$782$	3.62697e9			0.271219
$783$		0			0
$784$	−3.36310e9			−0.249249
$785$	1.24178e10	−	2.15082e10i	0.916219	−	1.58694i
$786$		0			0
$787$	9.10134e9	+	1.57640e10i	0.665570	+	1.15280i	0.979130	+	0.203233i	$0.0651449\pi$
$787$	9.10134e9	+	1.57640e10i	0.665570	+	1.15280i	−0.313560	+	0.949568i	$0.601522\pi$
$788$	−3.13679e9	−	5.43308e9i	−0.228372	−	0.395552i
$789$		0			0
$790$	3.37357e9	−	5.84319e9i	0.243442	−	0.421653i
$791$	−1.14526e7			−0.000822788
$792$		0			0
$793$	8.98082e9			0.639529
$794$	6.41683e8	−	1.11143e9i	0.0454935	−	0.0787970i
$795$		0			0
$796$	−1.75472e9	−	3.03926e9i	−0.123314	−	0.213585i
$797$	−7.84663e9	−	1.35908e10i	−0.549009	−	0.950911i	−0.998343	−	0.0575471i	$-0.981672\pi$
$797$	−7.84663e9	−	1.35908e10i	−0.549009	−	0.950911i	0.449334	−	0.893364i	$-0.351661\pi$
$798$		0			0
$799$	9.91056e9	−	1.71656e10i	0.687361	−	1.19054i
$800$	−4.54770e9			−0.314034
$801$		0			0
$802$	−2.25029e9			−0.154038
$803$	2.90003e8	−	5.02301e8i	0.0197651	−	0.0342341i
$804$		0			0
$805$	−2.31860e8	−	4.01593e8i	−0.0156654	−	0.0271332i
$806$	−5.78751e9	−	1.00243e10i	−0.389331	−	0.674341i
$807$		0			0
$808$	2.46594e9	−	4.27114e9i	0.164454	−	0.284842i
$809$	−1.94092e10			−1.28880			−0.644402	−	0.764687i	$-0.722894\pi$
$809$	−1.94092e10			−1.28880			−0.644402	+	0.764687i	$0.722894\pi$
$810$		0			0
$811$	2.10932e10			1.38858			0.694288	−	0.719698i	$-0.255720\pi$
$811$	2.10932e10			1.38858			0.694288	+	0.719698i	$0.255720\pi$
$812$	3.98329e8	−	6.89927e8i	0.0261094	−	0.0452228i
$813$		0			0
$814$	−5.58060e8	−	9.66589e8i	−0.0362657	−	0.0628140i
$815$	1.95876e9	+	3.39267e9i	0.126745	+	0.219528i
$816$		0			0
$817$	−1.50067e10	+	2.59923e10i	−0.962737	+	1.66751i
$818$	−3.31639e9			−0.211850
$819$		0			0
$820$	7.16573e9			0.453850
$821$	−1.09241e10	+	1.89211e10i	−0.688945	+	1.19329i	0.283234	+	0.959051i	$0.408593\pi$
$821$	−1.09241e10	+	1.89211e10i	−0.688945	+	1.19329i	−0.972179	+	0.234238i	$0.924741\pi$
$822$		0			0
$823$	2.17995e9	+	3.77579e9i	0.136316	+	0.236107i	0.926099	−	0.377279i	$-0.123140\pi$
$823$	2.17995e9	+	3.77579e9i	0.136316	+	0.236107i	−0.789783	+	0.613386i	$0.789807\pi$
$824$	2.35012e9	+	4.07053e9i	0.146334	+	0.253458i
$825$		0			0
$826$	2.57040e8	−	4.45206e8i	0.0158698	−	0.0274872i
$827$	8.96887e9			0.551402			0.275701	−	0.961243i	$-0.411090\pi$
$827$	8.96887e9			0.551402			0.275701	+	0.961243i	$0.411090\pi$
$828$		0			0
$829$	1.87500e10			1.14304			0.571518	−	0.820590i	$-0.306355\pi$
$829$	1.87500e10			1.14304			0.571518	+	0.820590i	$0.306355\pi$
$830$	−5.75515e9	+	9.96822e9i	−0.349368	+	0.605124i
$831$		0			0
$832$	7.22824e8	+	1.25197e9i	0.0435112	+	0.0753636i
$833$	−9.29726e9	−	1.61033e10i	−0.557311	−	0.965291i
$834$		0			0
$835$	6.70297e8	−	1.16099e9i	0.0398442	−	0.0690121i
$836$	1.02362e9			0.0605920
$837$		0			0
$838$	−2.07940e10			−1.22063
$839$	1.25531e10	−	2.17427e10i	0.733814	−	1.27100i	−0.221428	−	0.975177i	$-0.571072\pi$
$839$	1.25531e10	−	2.17427e10i	0.733814	−	1.27100i	0.955242	−	0.295826i	$-0.0955948\pi$
$840$		0			0
$841$	−2.26949e10	−	3.93087e10i	−1.31566	−	2.27878i
$842$	2.31426e9	+	4.00841e9i	0.133604	+	0.231409i
$843$		0			0
$844$	−3.72081e9	+	6.44464e9i	−0.213029	+	0.368977i
$845$	1.50602e10			0.858684
$846$		0			0
$847$	9.64644e8			0.0545475
$848$	−1.66101e9	+	2.87696e9i	−0.0935379	+	0.162012i
$849$		0			0
$850$	−1.25721e10	−	2.17755e10i	−0.702167	−	1.21619i
$851$	4.61065e9	+	7.98588e9i	0.256453	+	0.444190i
$852$		0			0
$853$	8.82920e9	−	1.52926e10i	0.487079	−	0.843646i	−0.512810	−	0.858502i	$-0.671396\pi$
$853$	8.82920e9	−	1.52926e10i	0.487079	−	0.843646i	0.999890	+	0.0148556i	$0.00472886\pi$
$854$	−6.47964e8			−0.0355999
$855$		0			0
$856$	−1.95673e9			−0.106629
$857$	−1.21984e9	+	2.11283e9i	−0.0662019	+	0.114665i	−0.897227	−	0.441571i	$-0.854421\pi$
$857$	−1.21984e9	+	2.11283e9i	−0.0662019	+	0.114665i	0.831025	+	0.556236i	$0.187755\pi$
$858$		0			0
$859$	−2.95698e9	−	5.12165e9i	−0.159174	−	0.275698i	0.775397	−	0.631474i	$-0.217550\pi$
$859$	−2.95698e9	−	5.12165e9i	−0.159174	−	0.275698i	−0.934571	+	0.355776i	$0.884216\pi$
$860$	8.47081e9	+	1.46719e10i	0.454130	+	0.786577i
$861$		0			0
$862$	3.07896e8	−	5.33292e8i	0.0163730	−	0.0283589i
$863$	2.02665e9			0.107335			0.0536673	−	0.998559i	$-0.482909\pi$
$863$	2.02665e9			0.107335			0.0536673	+	0.998559i	$0.482909\pi$
$864$		0			0
$865$	1.14146e10			0.599661
$866$	−1.10008e9	+	1.90540e9i	−0.0575588	+	0.0996948i
$867$		0			0
$868$	4.17567e8	+	7.23248e8i	0.0216724	+	0.0375377i
$869$	2.74245e8	+	4.75006e8i	0.0141765	+	0.0245544i
$870$		0			0
$871$	−5.08229e9	+	8.80278e9i	−0.260613	+	0.451394i
$872$	−9.77896e9			−0.499442
$873$		0			0
$874$	−8.45703e9			−0.428477
$875$	−7.02550e8	+	1.21685e9i	−0.0354527	+	0.0614059i
$876$		0			0
$877$	5.33945e9	+	9.24820e9i	0.267299	+	0.462976i	0.968163	−	0.250319i	$-0.0805354\pi$
$877$	5.33945e9	+	9.24820e9i	0.267299	+	0.462976i	−0.700864	+	0.713295i	$0.747202\pi$
$878$	6.09198e9	+	1.05516e10i	0.303758	+	0.526124i
$879$		0			0
$880$	2.88900e8	−	5.00389e8i	0.0142908	−	0.0247525i
$881$	4.67496e9			0.230336			0.115168	−	0.993346i	$-0.463259\pi$
$881$	4.67496e9			0.230336			0.115168	+	0.993346i	$0.463259\pi$
$882$		0			0
$883$	−2.97465e10			−1.45403			−0.727015	−	0.686622i	$-0.759093\pi$
$883$	−2.97465e10			−1.45403			−0.727015	+	0.686622i	$0.759093\pi$
$884$	−3.99648e9	+	6.92211e9i	−0.194579	+	0.337020i
$885$		0			0
$886$	−8.32228e9	−	1.44146e10i	−0.401998	−	0.696282i
$887$	7.27211e9	+	1.25957e10i	0.349887	+	0.606022i	0.986229	−	0.165385i	$-0.0528866\pi$
$887$	7.27211e9	+	1.25957e10i	0.349887	+	0.606022i	−0.636342	+	0.771407i	$0.719553\pi$
$888$		0			0
$889$	−9.75920e8	+	1.69034e9i	−0.0465863	+	0.0806898i
$890$	−1.52945e10			−0.727225
$891$		0			0
$892$	−8.45753e9			−0.398994
$893$	−2.31085e10	+	4.00251e10i	−1.08591	+	1.88084i
$894$		0			0
$895$	1.92187e10	+	3.32878e10i	0.896074	+	1.55205i
$896$	−5.21516e7	−	9.03292e7i	−0.00242209	−	0.00419518i
$897$		0			0
$898$	−7.59353e9	+	1.31524e10i	−0.349926	+	0.606090i
$899$	6.56649e10			3.01422
$900$		0			0
$901$	−1.83674e10			−0.836587
$902$	−2.91259e8	+	5.04475e8i	−0.0132147	+	0.0228885i
$903$		0			0
$904$	5.89492e7	+	1.02103e8i	0.00265392	+	0.00459673i
$905$	1.04669e10	+	1.81292e10i	0.469404	+	0.813032i
$906$		0			0
$907$	−1.78868e10	+	3.09808e10i	−0.795987	+	1.37869i	0.126224	+	0.992002i	$0.459714\pi$
$907$	−1.78868e10	+	3.09808e10i	−0.795987	+	1.37869i	−0.922211	+	0.386688i	$0.873619\pi$
$908$	−3.61352e9			−0.160188
$909$		0			0
$910$	1.02193e9			0.0449547
$911$	1.32563e9	−	2.29605e9i	0.0580908	−	0.100616i	−0.835518	−	0.549464i	$-0.814832\pi$
$911$	1.32563e9	−	2.29605e9i	0.0580908	−	0.100616i	0.893608	+	0.448848i	$0.148165\pi$
$912$		0			0
$913$	−4.67849e8	−	8.10338e8i	−0.0203450	−	0.0352386i
$914$	2.74425e9	+	4.75319e9i	0.118881	+	0.205908i
$915$		0			0
$916$	−5.69196e9	+	9.85877e9i	−0.244697	+	0.423827i
$917$	1.05826e9			0.0453209
$918$		0			0
$919$	−2.21012e9			−0.0939317			−0.0469659	−	0.998896i	$-0.514955\pi$
$919$	−2.21012e9			−0.0939317			−0.0469659	+	0.998896i	$0.514955\pi$
$920$	−2.38687e9	+	4.13417e9i	−0.101058	+	0.175038i
$921$		0			0
$922$	−6.08387e9	−	1.05376e10i	−0.255636	−	0.442774i
$923$	1.25930e10	+	2.18118e10i	0.527138	+	0.913031i
$924$		0			0
$925$	3.19635e10	−	5.53625e10i	1.32788	−	2.29996i
$926$	4.00229e9			0.165642
$927$		0			0
$928$	−8.20115e9			−0.336866
$929$	−8.26820e9	+	1.43209e10i	−0.338342	+	0.586025i	−0.984121	−	0.177499i	$-0.943199\pi$
$929$	−8.26820e9	+	1.43209e10i	−0.338342	+	0.586025i	0.645779	+	0.763524i	$0.276533\pi$
$930$		0			0
$931$	2.16785e10	+	3.75482e10i	0.880451	+	1.52499i
$932$	−3.23192e9	−	5.59784e9i	−0.130769	−	0.226498i
$933$		0			0
$934$	1.61333e10	−	2.79438e10i	0.647903	−	1.12220i
$935$	3.19465e9			0.127815
$936$		0			0
$937$	1.52847e10			0.606973			0.303487	−	0.952836i	$-0.401849\pi$
$937$	1.52847e10			0.606973			0.303487	+	0.952836i	$0.401849\pi$
$938$	3.66686e8	−	6.35118e8i	0.0145072	−	0.0251272i
$939$		0			0
$940$	1.30441e10	+	2.25930e10i	0.512230	+	0.887209i
$941$	−8.12723e9	−	1.40768e10i	−0.317965	−	0.550731i	0.662099	−	0.749417i	$-0.269666\pi$
$941$	−8.12723e9	−	1.40768e10i	−0.317965	−	0.550731i	−0.980063	+	0.198686i	$0.936333\pi$
$942$		0			0
$943$	2.40636e9	−	4.16793e9i	0.0934477	−	0.161856i
$944$	−5.29216e9			−0.204753
$945$		0			0
$946$	−1.37722e9			−0.0528914
$947$	−1.38664e10	+	2.40174e10i	−0.530567	+	0.918969i	0.468797	+	0.883306i	$0.344688\pi$
$947$	−1.38664e10	+	2.40174e10i	−0.530567	+	0.918969i	−0.999364	+	0.0356631i	$0.988646\pi$
$948$		0			0
$949$	−5.28017e9	−	9.14553e9i	−0.200547	−	0.347358i
$950$	2.93144e10	+	5.07740e10i	1.10930	+	1.92136i
$951$		0			0
$952$	2.88345e8	−	4.99429e8i	0.0108314	−	0.0187605i
$953$	−3.88172e10			−1.45278			−0.726388	−	0.687285i	$-0.758802\pi$
$953$	−3.88172e10			−1.45278			−0.726388	+	0.687285i	$0.758802\pi$
$954$		0			0
$955$	−2.07253e9			−0.0769998
$956$	3.88564e9	−	6.73012e9i	0.143833	−	0.249127i
$957$		0			0
$958$	1.04105e10	+	1.80315e10i	0.382553	+	0.662601i
$959$	4.53350e8	+	7.85226e8i	0.0165985	+	0.0287494i
$960$		0			0
$961$	−2.06618e10	+	3.57873e10i	−0.750995	+	1.30076i
$962$	−2.03215e10			−0.735942
$963$		0			0
$964$	−1.46079e10			−0.525191
$965$	1.56095e10	−	2.70365e10i	0.559171	−	0.968512i
$966$		0			0
$967$	−6.37606e9	−	1.10437e10i	−0.226756	−	0.392754i	0.730089	−	0.683353i	$-0.239479\pi$
$967$	−6.37606e9	−	1.10437e10i	−0.226756	−	0.392754i	−0.956845	+	0.290599i	$0.906145\pi$
$968$	−4.96523e9	−	8.60003e9i	−0.175944	−	0.304745i
$969$		0			0
$970$	−1.47475e10	+	2.55434e10i	−0.518820	+	0.898623i
$971$	−4.92971e10			−1.72804			−0.864021	−	0.503456i	$-0.832062\pi$
$971$	−4.92971e10			−1.72804			−0.864021	+	0.503456i	$0.832062\pi$
$972$		0			0
$973$	−1.07513e9			−0.0374168
$974$	1.74207e10	−	3.01735e10i	0.604099	−	1.04633i
$975$		0			0
$976$	3.33521e9	+	5.77675e9i	0.114828	+	0.198888i
$977$	7.28714e9	+	1.26217e10i	0.249992	+	0.432999i	0.963523	−	0.267625i	$-0.0862387\pi$
$977$	7.28714e9	+	1.26217e10i	0.249992	+	0.432999i	−0.713531	+	0.700623i	$0.752905\pi$
$978$		0			0
$979$	6.21659e8	−	1.07675e9i	0.0211745	−	0.0366753i
$980$	2.44737e10			0.830631
$981$		0			0
$982$	3.33324e9			0.112325
$983$	−1.88080e10	+	3.25764e10i	−0.631546	+	1.09387i	0.355690	+	0.934604i	$0.384246\pi$
$983$	−1.88080e10	+	3.25764e10i	−0.631546	+	1.09387i	−0.987236	+	0.159265i	$0.949088\pi$
$984$		0			0
$985$	2.28268e10	+	3.95372e10i	0.761058	+	1.31819i
$986$	−2.26720e10	−	3.92691e10i	−0.753218	−	1.30461i
$987$		0			0
$988$	9.31863e9	−	1.61403e10i	0.307399	−	0.532431i
$989$	1.13785e10			0.374022
$990$		0			0
$991$	3.09606e10			1.01054			0.505268	−	0.862962i	$-0.331394\pi$
$991$	3.09606e10			1.01054			0.505268	+	0.862962i	$0.331394\pi$
$992$	4.29862e9	−	7.44542e9i	0.139810	−	0.242158i
$993$		0			0
$994$	−9.08585e8	−	1.57372e9i	−0.0293436	−	0.0508246i
$995$	1.27693e10	+	2.21170e10i	0.410947	+	0.711780i
$996$		0			0
$997$	3.91458e9	−	6.78026e9i	0.125099	−	0.216677i	−0.796673	−	0.604411i	$-0.793409\pi$
$997$	3.91458e9	−	6.78026e9i	0.125099	−	0.216677i	0.921772	+	0.387734i	$0.126742\pi$
$998$	2.67416e10			0.851589
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.8.c.p.55.2		4
3.2	odd	2		162.8.c.m.55.1		4
9.2	odd	6		54.8.a.h.1.2	yes	2
9.4	even	3	inner	162.8.c.p.109.2		4
9.5	odd	6		162.8.c.m.109.1		4
9.7	even	3		54.8.a.g.1.1	✓	2
36.7	odd	6		432.8.a.p.1.1		2
36.11	even	6		432.8.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.8.a.g.1.1	✓	2	9.7	even	3
54.8.a.h.1.2	yes	2	9.2	odd	6
162.8.c.m.55.1		4	3.2	odd	2
162.8.c.m.109.1		4	9.5	odd	6
162.8.c.p.55.2		4	1.1	even	1	trivial
162.8.c.p.109.2		4	9.4	even	3	inner
432.8.a.k.1.2		2	36.11	even	6
432.8.a.p.1.1		2	36.7	odd	6

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:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	162.c (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.00000 - 6.92820i) q^{2} +(-32.0000 - 55.4256i) q^{4} +(232.868 + 403.339i) q^{5} +(24.8678 - 43.0723i) q^{7} -512.000 q^{8} +3725.89 q^{10} +(-151.443 + 262.306i) q^{11} +(2757.36 + 4775.88i) q^{13} +(-198.943 - 344.579i) q^{14} +(-2048.00 + 3547.24i) q^{16} -22646.7 q^{17} +52805.5 q^{19} +(14903.5 - 25813.7i) q^{20} +(1211.54 + 2098.45i) q^{22} +(-10009.6 - 17337.2i) q^{23} +(-69392.3 + 120191. i) q^{25} +44117.7 q^{26} -3183.08 q^{28} +(-125140. + 216748. i) q^{29} +(-131183. - 227216. i) q^{31} +(16384.0 + 28377.9i) q^{32} +(-90586.9 + 156901. i) q^{34} +23163.7 q^{35} -460621. q^{37} +(211222. - 365847. i) q^{38} +(-119228. - 206510. i) q^{40} +(120202. + 208196. i) q^{41} +(-284188. + 492228. i) q^{43} +19384.6 q^{44} -160154. q^{46} +(-437616. + 757973. i) q^{47} +(410535. + 711067. i) q^{49} +(555139. + 961529. i) q^{50} +(176471. - 305656. i) q^{52} +811041. q^{53} -141064. q^{55} +(-12732.3 + 22053.0i) q^{56} +(1.00112e6 + 1.73399e6i) q^{58} +(646016. + 1.11893e6i) q^{59} +(814260. - 1.41034e6i) q^{61} -2.09893e6 q^{62} +262144. q^{64} +(-1.28420e6 + 2.22430e6i) q^{65} +(921587. + 1.59623e6i) q^{67} +(724695. + 1.25521e6i) q^{68} +(92654.6 - 160483. i) q^{70} +4.56707e6 q^{71} -1.91494e6 q^{73} +(-1.84248e6 + 3.19127e6i) q^{74} +(-1.68978e6 - 2.92678e6i) q^{76} +(7532.09 + 13046.0i) q^{77} +(905441. - 1.56827e6i) q^{79} -1.90765e6 q^{80} +1.92323e6 q^{82} +(-1.54464e6 + 2.67540e6i) q^{83} +(-5.27369e6 - 9.13430e6i) q^{85} +(2.27350e6 + 3.93782e6i) q^{86} +(77538.6 - 134301. i) q^{88} -4.10492e6 q^{89} +274278. q^{91} +(-640617. + 1.10958e6i) q^{92} +(3.50093e6 + 6.06378e6i) q^{94} +(1.22967e7 + 2.12985e7i) q^{95} +(-3.95811e6 + 6.85566e6i) q^{97} +6.56855e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 16 q^{2} - 128 q^{4} - 48 q^{5} - 880 q^{7} - 2048 q^{8} - 768 q^{10} + 7230 q^{11} - 8560 q^{13} + 7040 q^{14} - 8192 q^{16} - 51408 q^{17} + 74096 q^{19} - 3072 q^{20} - 57840 q^{22} - 59628 q^{23}+ \cdots + 12483264 q^{98}+O(q^{100}) \) 4 * q + 16 * q^2 - 128 * q^4 - 48 * q^5 - 880 * q^7 - 2048 * q^8 - 768 * q^10 + 7230 * q^11 - 8560 * q^13 + 7040 * q^14 - 8192 * q^16 - 51408 * q^17 + 74096 * q^19 - 3072 * q^20 - 57840 * q^22 - 59628 * q^23 - 324584 * q^25 - 136960 * q^26 + 112640 * q^28 - 275280 * q^29 - 245584 * q^31 + 65536 * q^32 - 205632 * q^34 + 1001604 * q^35 - 142120 * q^37 + 296384 * q^38 + 24576 * q^40 + 494520 * q^41 - 825280 * q^43 - 925440 * q^44 - 954048 * q^46 - 927708 * q^47 + 780204 * q^49 + 2596672 * q^50 - 547840 * q^52 + 2764224 * q^53 - 8021952 * q^55 + 450560 * q^56 + 2202240 * q^58 + 1588920 * q^59 + 3021968 * q^61 - 3929344 * q^62 + 1048576 * q^64 - 9799080 * q^65 + 1882160 * q^67 + 1645056 * q^68 + 4006416 * q^70 + 6788880 * q^71 + 1790180 * q^73 - 568480 * q^74 - 2371072 * q^76 + 7018656 * q^77 + 369920 * q^79 + 393216 * q^80 + 7912320 * q^82 - 9352050 * q^83 - 8976744 * q^85 + 6602240 * q^86 - 3701760 * q^88 + 1825920 * q^89 + 26720080 * q^91 - 3816192 * q^92 + 7421664 * q^94 + 32688588 * q^95 - 1359790 * q^97 + 12483264 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more