LMFDB - Embedded newform 162.8.c.h.109.1

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:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	162.109
Dual form			162.8.c.h.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.00000 + 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(-82.5000 + 142.894i) q^{5} +(254.000 + 439.941i) q^{7} -512.000 q^{8} +O(q^{10})$	\(q+(4.00000 + 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(-82.5000 + 142.894i) q^{5} +(254.000 + 439.941i) q^{7} -512.000 q^{8} -1320.00 q^{10} +(1512.00 + 2618.86i) q^{11} +(-2519.50 + 4363.90i) q^{13} +(-2032.00 + 3519.53i) q^{14} +(-2048.00 - 3547.24i) q^{16} +3189.00 q^{17} +1508.00 q^{19} +(-5280.00 - 9145.23i) q^{20} +(-12096.0 + 20950.9i) q^{22} +(-37800.0 + 65471.5i) q^{23} +(25450.0 + 44080.7i) q^{25} -40312.0 q^{26} -32512.0 q^{28} +(-41332.5 - 71590.0i) q^{29} +(87446.0 - 151461. i) q^{31} +(16384.0 - 28377.9i) q^{32} +(12756.0 + 22094.0i) q^{34} -83820.0 q^{35} -323569. q^{37} +(6032.00 + 10447.7i) q^{38} +(42240.0 - 73161.8i) q^{40} +(-154059. + 266838. i) q^{41} +(-168340. - 291573. i) q^{43} -193536. q^{44} -604800. q^{46} +(-191598. - 331857. i) q^{47} +(282740. - 489719. i) q^{49} +(-203600. + 352646. i) q^{50} +(-161248. - 279290. i) q^{52} -760206. q^{53} -498960. q^{55} +(-130048. - 225250. i) q^{56} +(330660. - 572720. i) q^{58} +(-1.11283e6 + 1.92748e6i) q^{59} +(-1.12241e6 - 1.94407e6i) q^{61} +1.39914e6 q^{62} +262144. q^{64} +(-415718. - 720044. i) q^{65} +(-736594. + 1.27582e6i) q^{67} +(-102048. + 176752. i) q^{68} +(-335280. - 580722. i) q^{70} +5.00689e6 q^{71} -5.89830e6 q^{73} +(-1.29428e6 - 2.24175e6i) q^{74} +(-48256.0 + 83581.8i) q^{76} +(-768096. + 1.33038e6i) q^{77} +(-3.51438e6 - 6.08709e6i) q^{79} +675840. q^{80} -2.46494e6 q^{82} +(-1.32560e6 - 2.29600e6i) q^{83} +(-263092. + 455690. i) q^{85} +(1.34672e6 - 2.33259e6i) q^{86} +(-774144. - 1.34086e6i) q^{88} +6.77090e6 q^{89} -2.55981e6 q^{91} +(-2.41920e6 - 4.19018e6i) q^{92} +(1.53278e6 - 2.65486e6i) q^{94} +(-124410. + 215484. i) q^{95} +(-8.08819e6 - 1.40092e7i) q^{97} +4.52383e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 8 q^{2} - 64 q^{4} - 165 q^{5} + 508 q^{7} - 1024 q^{8}+O(q^{10}) $ 2 * q + 8 * q^2 - 64 * q^4 - 165 * q^5 + 508 * q^7 - 1024 * q^8	\( 2 q + 8 q^{2} - 64 q^{4} - 165 q^{5} + 508 q^{7} - 1024 q^{8} - 2640 q^{10} + 3024 q^{11} - 5039 q^{13} - 4064 q^{14} - 4096 q^{16} + 6378 q^{17} + 3016 q^{19} - 10560 q^{20} - 24192 q^{22} - 75600 q^{23} + 50900 q^{25} - 80624 q^{26} - 65024 q^{28} - 82665 q^{29} + 174892 q^{31} + 32768 q^{32} + 25512 q^{34} - 167640 q^{35} - 647138 q^{37} + 12064 q^{38} + 84480 q^{40} - 308118 q^{41} - 336680 q^{43} - 387072 q^{44} - 1209600 q^{46} - 383196 q^{47} + 565479 q^{49} - 407200 q^{50} - 322496 q^{52} - 1520412 q^{53} - 997920 q^{55} - 260096 q^{56} + 661320 q^{58} - 2225664 q^{59} - 2244815 q^{61} + 2798272 q^{62} + 524288 q^{64} - 831435 q^{65} - 1473188 q^{67} - 204096 q^{68} - 670560 q^{70} + 10013784 q^{71} - 11796602 q^{73} - 2588552 q^{74} - 96512 q^{76} - 1536192 q^{77} - 7028768 q^{79} + 1351680 q^{80} - 4929888 q^{82} - 2651196 q^{83} - 526185 q^{85} + 2693440 q^{86} - 1548288 q^{88} + 13541802 q^{89} - 5119624 q^{91} - 4838400 q^{92} + 3065568 q^{94} - 248820 q^{95} - 16176386 q^{97} + 9047664 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 - 64 * q^4 - 165 * q^5 + 508 * q^7 - 1024 * q^8 - 2640 * q^10 + 3024 * q^11 - 5039 * q^13 - 4064 * q^14 - 4096 * q^16 + 6378 * q^17 + 3016 * q^19 - 10560 * q^20 - 24192 * q^22 - 75600 * q^23 + 50900 * q^25 - 80624 * q^26 - 65024 * q^28 - 82665 * q^29 + 174892 * q^31 + 32768 * q^32 + 25512 * q^34 - 167640 * q^35 - 647138 * q^37 + 12064 * q^38 + 84480 * q^40 - 308118 * q^41 - 336680 * q^43 - 387072 * q^44 - 1209600 * q^46 - 383196 * q^47 + 565479 * q^49 - 407200 * q^50 - 322496 * q^52 - 1520412 * q^53 - 997920 * q^55 - 260096 * q^56 + 661320 * q^58 - 2225664 * q^59 - 2244815 * q^61 + 2798272 * q^62 + 524288 * q^64 - 831435 * q^65 - 1473188 * q^67 - 204096 * q^68 - 670560 * q^70 + 10013784 * q^71 - 11796602 * q^73 - 2588552 * q^74 - 96512 * q^76 - 1536192 * q^77 - 7028768 * q^79 + 1351680 * q^80 - 4929888 * q^82 - 2651196 * q^83 - 526185 * q^85 + 2693440 * q^86 - 1548288 * q^88 + 13541802 * q^89 - 5119624 * q^91 - 4838400 * q^92 + 3065568 * q^94 - 248820 * q^95 - 16176386 * q^97 + 9047664 * q^98

Display 100 coefficients 10 coefficients

Character values

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

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{3}\right)$

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
$5$	−82.5000	+	142.894i	−0.295161	+	0.511234i	−0.975022	−	0.222107i	$-0.928707\pi$
$5$	−82.5000	+	142.894i	−0.295161	+	0.511234i	0.679861	+	0.733341i	$0.262040\pi$
$6$		0			0
$7$	254.000	+	439.941i	0.279892	+	0.484787i	0.971358	−	0.237622i	$-0.0763680\pi$
$7$	254.000	+	439.941i	0.279892	+	0.484787i	−0.691466	+	0.722409i	$0.743035\pi$
$8$	−512.000			−0.353553
$9$		0			0
$10$	−1320.00			−0.417421
$11$	1512.00	+	2618.86i	0.342513	+	0.593250i	0.984899	−	0.173131i	$-0.0553885\pi$
$11$	1512.00	+	2618.86i	0.342513	+	0.593250i	−0.642385	+	0.766382i	$0.722055\pi$
$12$		0			0
$13$	−2519.50	+	4363.90i	−0.318063	+	0.550901i	−0.980084	−	0.198585i	$-0.936366\pi$
$13$	−2519.50	+	4363.90i	−0.318063	+	0.550901i	0.662021	+	0.749485i	$0.269699\pi$
$14$	−2032.00	+	3519.53i	−0.197914	+	0.342796i
$15$		0			0
$16$	−2048.00	−	3547.24i	−0.125000	−	0.216506i
$17$	3189.00			0.157428			0.0787142	−	0.996897i	$-0.474919\pi$
$17$	3189.00			0.157428			0.0787142	+	0.996897i	$0.474919\pi$
$18$		0			0
$19$	1508.00			0.0504387			0.0252193	−	0.999682i	$-0.491972\pi$
$19$	1508.00			0.0504387			0.0252193	+	0.999682i	$0.491972\pi$
$20$	−5280.00	−	9145.23i	−0.147580	−	0.255617i
$21$		0			0
$22$	−12096.0	+	20950.9i	−0.242193	+	0.419491i
$23$	−37800.0	+	65471.5i	−0.647805	+	1.12203i	0.335841	+	0.941919i	$0.390980\pi$
$23$	−37800.0	+	65471.5i	−0.647805	+	1.12203i	−0.983646	+	0.180113i	$0.942354\pi$
$24$		0			0
$25$	25450.0	+	44080.7i	0.325760	+	0.564233i
$26$	−40312.0			−0.449808
$27$		0			0
$28$	−32512.0			−0.279892
$29$	−41332.5	−	71590.0i	−0.314701	−	0.545079i	0.664673	−	0.747135i	$-0.268571\pi$
$29$	−41332.5	−	71590.0i	−0.314701	−	0.545079i	−0.979374	+	0.202056i	$0.935238\pi$
$30$		0			0
$31$	87446.0	−	151461.i	0.527198	−	0.913134i	−0.472299	−	0.881438i	$-0.656576\pi$
$31$	87446.0	−	151461.i	0.527198	−	0.913134i	0.999498	−	0.0316960i	$-0.0100908\pi$
$32$	16384.0	−	28377.9i	0.0883883	−	0.153093i
$33$		0			0
$34$	12756.0	+	22094.0i	0.0556594	+	0.0964049i
$35$	−83820.0			−0.330453
$36$		0			0
$37$	−323569.			−1.05017			−0.525087	−	0.851049i	$-0.675967\pi$
$37$	−323569.			−1.05017			−0.525087	+	0.851049i	$0.675967\pi$
$38$	6032.00	+	10447.7i	0.0178328	+	0.0308873i
$39$		0			0
$40$	42240.0	−	73161.8i	0.104355	−	0.180748i
$41$	−154059.	+	266838.i	−0.349095	+	0.604650i	−0.986089	−	0.166219i	$-0.946844\pi$
$41$	−154059.	+	266838.i	−0.349095	+	0.604650i	0.636994	+	0.770869i	$0.280178\pi$
$42$		0			0
$43$	−168340.	−	291573.i	−0.322885	−	0.559253i	0.658197	−	0.752846i	$-0.271319\pi$
$43$	−168340.	−	291573.i	−0.322885	−	0.559253i	−0.981082	+	0.193593i	$0.937986\pi$
$44$	−193536.			−0.342513
$45$		0			0
$46$	−604800.			−0.916135
$47$	−191598.	−	331857.i	−0.269184	−	0.466240i	0.699468	−	0.714664i	$-0.253420\pi$
$47$	−191598.	−	331857.i	−0.269184	−	0.466240i	−0.968651	+	0.248425i	$0.920087\pi$
$48$		0			0
$49$	282740.	−	489719.i	0.343321	−	0.594649i
$50$	−203600.	+	352646.i	−0.230347	+	0.398973i
$51$		0			0
$52$	−161248.	−	279290.i	−0.159031	−	0.275450i
$53$	−760206.			−0.701400			−0.350700	−	0.936488i	$-0.614056\pi$
$53$	−760206.			−0.701400			−0.350700	+	0.936488i	$0.614056\pi$
$54$		0			0
$55$	−498960.			−0.404386
$56$	−130048.	−	225250.i	−0.0989568	−	0.171398i
$57$		0			0
$58$	330660.	−	572720.i	0.222527	−	0.385429i
$59$	−1.11283e6	+	1.92748e6i	−0.705420	+	1.22182i	0.261120	+	0.965306i	$0.415908\pi$
$59$	−1.11283e6	+	1.92748e6i	−0.705420	+	1.22182i	−0.966540	+	0.256516i	$0.917425\pi$
$60$		0			0
$61$	−1.12241e6	−	1.94407e6i	−0.633135	−	1.09662i	−0.986907	−	0.161290i	$-0.948435\pi$
$61$	−1.12241e6	−	1.94407e6i	−0.633135	−	1.09662i	0.353772	−	0.935332i	$-0.384899\pi$
$62$	1.39914e6			0.745571
$63$		0			0
$64$	262144.			0.125000
$65$	−415718.	−	720044.i	−0.187759	−	0.325209i
$66$		0			0
$67$	−736594.	+	1.27582e6i	−0.299203	+	0.518235i	−0.975954	−	0.217977i	$-0.930054\pi$
$67$	−736594.	+	1.27582e6i	−0.299203	+	0.518235i	0.676751	+	0.736212i	$0.263388\pi$
$68$	−102048.	+	176752.i	−0.0393571	+	0.0681685i
$69$		0			0
$70$	−335280.	−	580722.i	−0.116833	−	0.202360i
$71$	5.00689e6			1.66021			0.830107	−	0.557604i	$-0.188279\pi$
$71$	5.00689e6			1.66021			0.830107	+	0.557604i	$0.188279\pi$
$72$		0			0
$73$	−5.89830e6			−1.77459			−0.887293	−	0.461207i	$-0.847417\pi$
$73$	−5.89830e6			−1.77459			−0.887293	+	0.461207i	$0.847417\pi$
$74$	−1.29428e6	−	2.24175e6i	−0.371292	−	0.643097i
$75$		0			0
$76$	−48256.0	+	83581.8i	−0.0126097	+	0.0218406i
$77$	−768096.	+	1.33038e6i	−0.191733	+	0.332092i
$78$		0			0
$79$	−3.51438e6	−	6.08709e6i	−0.801963	−	1.38904i	−0.918322	−	0.395834i	$-0.870456\pi$
$79$	−3.51438e6	−	6.08709e6i	−0.801963	−	1.38904i	0.116359	−	0.993207i	$-0.462878\pi$
$80$	675840.			0.147580
$81$		0			0
$82$	−2.46494e6			−0.493695
$83$	−1.32560e6	−	2.29600e6i	−0.254471	−	0.440757i	0.710281	−	0.703919i	$-0.248568\pi$
$83$	−1.32560e6	−	2.29600e6i	−0.254471	−	0.440757i	−0.964752	+	0.263162i	$0.915235\pi$
$84$		0			0
$85$	−263092.	+	455690.i	−0.0464667	+	0.0804828i
$86$	1.34672e6	−	2.33259e6i	0.228314	−	0.395452i
$87$		0			0
$88$	−774144.	−	1.34086e6i	−0.121097	−	0.209746i
$89$	6.77090e6			1.01808			0.509039	−	0.860743i	$-0.330001\pi$
$89$	6.77090e6			1.01808			0.509039	+	0.860743i	$0.330001\pi$
$90$		0			0
$91$	−2.55981e6			−0.356093
$92$	−2.41920e6	−	4.19018e6i	−0.323903	−	0.561016i
$93$		0			0
$94$	1.53278e6	−	2.65486e6i	0.190341	−	0.329681i
$95$	−124410.	+	215484.i	−0.0148875	+	0.0257860i
$96$		0			0
$97$	−8.08819e6	−	1.40092e7i	−0.899809	−	1.55852i	−0.827737	−	0.561116i	$-0.810372\pi$
$97$	−8.08819e6	−	1.40092e7i	−0.899809	−	1.55852i	−0.0720719	−	0.997399i	$-0.522961\pi$
$98$	4.52383e6			0.485529
$99$		0			0
$100$	−3.25760e6			−0.325760
$101$	1.35081e6	+	2.33967e6i	0.130457	+	0.225959i	0.923853	−	0.382748i	$-0.125022\pi$
$101$	1.35081e6	+	2.33967e6i	0.130457	+	0.225959i	−0.793396	+	0.608706i	$0.791689\pi$
$102$		0			0
$103$	−1.03446e6	+	1.79173e6i	−0.0932787	+	0.161563i	−0.908889	−	0.417038i	$-0.863068\pi$
$103$	−1.03446e6	+	1.79173e6i	−0.0932787	+	0.161563i	0.815610	+	0.578602i	$0.196401\pi$
$104$	1.28998e6	−	2.23432e6i	0.112452	−	0.194773i
$105$		0			0
$106$	−3.04082e6	−	5.26686e6i	−0.247982	−	0.429518i
$107$	9.15450e6			0.722423			0.361211	−	0.932484i	$-0.382363\pi$
$107$	9.15450e6			0.722423			0.361211	+	0.932484i	$0.382363\pi$
$108$		0			0
$109$	8.68645e6			0.642465			0.321233	−	0.947000i	$-0.395903\pi$
$109$	8.68645e6			0.642465			0.321233	+	0.947000i	$0.395903\pi$
$110$	−1.99584e6	−	3.45690e6i	−0.142972	−	0.247635i
$111$		0			0
$112$	1.04038e6	−	1.80200e6i	0.0699730	−	0.121197i
$113$	−1.24888e7	+	2.16312e7i	−0.814227	+	1.41028i	0.0956551	+	0.995415i	$0.469505\pi$
$113$	−1.24888e7	+	2.16312e7i	−0.814227	+	1.41028i	−0.909882	+	0.414867i	$0.863828\pi$
$114$		0			0
$115$	−6.23700e6	−	1.08028e7i	−0.382414	−	0.662360i
$116$	5.29056e6			0.314701
$117$		0			0
$118$	−1.78053e7			−0.997614
$119$	810006.	+	1.40297e6i	0.0440630	+	0.0763193i
$120$		0			0
$121$	5.17130e6	−	8.95695e6i	0.265369	−	0.459633i
$122$	8.97926e6	−	1.55525e7i	0.447694	−	0.775429i
$123$		0			0
$124$	5.59654e6	+	9.69350e6i	0.263599	+	0.456567i
$125$	−2.12891e7			−0.974929
$126$		0			0
$127$	−4.98659e6			−0.216018			−0.108009	−	0.994150i	$-0.534448\pi$
$127$	−4.98659e6			−0.216018			−0.108009	+	0.994150i	$0.534448\pi$
$128$	1.04858e6	+	1.81619e6i	0.0441942	+	0.0765466i
$129$		0			0
$130$	3.32574e6	−	5.76035e6i	0.132766	−	0.229957i
$131$	−1.11702e7	+	1.93474e7i	−0.434123	+	0.751923i	−0.997224	−	0.0744655i	$-0.976275\pi$
$131$	−1.11702e7	+	1.93474e7i	−0.434123	+	0.751923i	0.563101	+	0.826388i	$0.309608\pi$
$132$		0			0
$133$	383032.	+	663431.i	0.0141174	+	0.0244520i
$134$	−1.17855e7			−0.423137
$135$		0			0
$136$	−1.63277e6			−0.0556594
$137$	6.43154e6	+	1.11398e7i	0.213694	+	0.370129i	0.952868	−	0.303386i	$-0.0981170\pi$
$137$	6.43154e6	+	1.11398e7i	0.213694	+	0.370129i	−0.739174	+	0.673515i	$0.764784\pi$
$138$		0			0
$139$	−2.20951e7	+	3.82699e7i	−0.697822	+	1.20866i	0.271398	+	0.962467i	$0.412514\pi$
$139$	−2.20951e7	+	3.82699e7i	−0.697822	+	1.20866i	−0.969220	+	0.246196i	$0.920819\pi$
$140$	2.68224e6	−	4.64578e6i	0.0826132	−	0.143090i
$141$		0			0
$142$	2.00276e7	+	3.46888e7i	0.586974	+	1.01667i
$143$	−1.52379e7			−0.435763
$144$		0			0
$145$	1.36397e7			0.371550
$146$	−2.35932e7	−	4.08646e7i	−0.627411	−	1.08671i
$147$		0			0
$148$	1.03542e7	−	1.79340e7i	0.262543	−	0.454738i
$149$	−1.31444e6	+	2.27667e6i	−0.0325527	+	0.0563830i	−0.881843	−	0.471544i	$-0.843697\pi$
$149$	−1.31444e6	+	2.27667e6i	−0.0325527	+	0.0563830i	0.849290	+	0.527927i	$0.177030\pi$
$150$		0			0
$151$	2.76509e7	+	4.78928e7i	0.653568	+	1.13201i	0.982251	+	0.187572i	$0.0600619\pi$
$151$	2.76509e7	+	4.78928e7i	0.653568	+	1.13201i	−0.328683	+	0.944440i	$0.606605\pi$
$152$	−772096.			−0.0178328
$153$		0			0
$154$	−1.22895e7			−0.271152
$155$	1.44286e7	+	2.49911e7i	0.311217	+	0.539043i
$156$		0			0
$157$	1.11855e7	−	1.93739e7i	0.230679	−	0.399548i	−0.727329	−	0.686289i	$-0.759239\pi$
$157$	1.11855e7	−	1.93739e7i	0.230679	−	0.399548i	0.958008	+	0.286741i	$0.0925720\pi$
$158$	2.81151e7	−	4.86967e7i	0.567074	−	0.982200i
$159$		0			0
$160$	2.70336e6	+	4.68236e6i	0.0521776	+	0.0903742i
$161$	−3.84048e7			−0.725262
$162$		0			0
$163$	5.23606e7			0.946995			0.473498	−	0.880795i	$-0.342991\pi$
$163$	5.23606e7			0.946995			0.473498	+	0.880795i	$0.342991\pi$
$164$	−9.85978e6	−	1.70776e7i	−0.174547	−	0.302325i
$165$		0			0
$166$	1.06048e7	−	1.83680e7i	0.179938	−	0.311662i
$167$	−2.73979e7	+	4.74545e7i	−0.455207	+	0.788442i	−0.998700	−	0.0509718i	$-0.983768\pi$
$167$	−2.73979e7	+	4.74545e7i	−0.455207	+	0.788442i	0.543493	+	0.839414i	$0.317101\pi$
$168$		0			0
$169$	1.86785e7	+	3.23521e7i	0.297672	+	0.515584i
$170$	−4.20948e6			−0.0657139
$171$		0			0
$172$	2.15475e7			0.322885
$173$	6.18226e7	+	1.07080e8i	0.907791	+	1.57234i	0.817127	+	0.576458i	$0.195566\pi$
$173$	6.18226e7	+	1.07080e8i	0.907791	+	1.57234i	0.0906639	+	0.995882i	$0.471101\pi$
$174$		0			0
$175$	−1.29286e7	+	2.23930e7i	−0.182355	+	0.315849i
$176$	6.19315e6	−	1.07269e7i	0.0856283	−	0.148313i
$177$		0			0
$178$	2.70836e7	+	4.69102e7i	0.359945	+	0.623443i
$179$	−9.87297e7			−1.28666			−0.643328	−	0.765591i	$-0.722447\pi$
$179$	−9.87297e7			−1.28666			−0.643328	+	0.765591i	$0.722447\pi$
$180$		0			0
$181$	9.19855e7			1.15304			0.576520	−	0.817083i	$-0.304410\pi$
$181$	9.19855e7			1.15304			0.576520	+	0.817083i	$0.304410\pi$
$182$	−1.02392e7	−	1.77349e7i	−0.125898	−	0.218061i
$183$		0			0
$184$	1.93536e7	−	3.35214e7i	0.229034	−	0.396698i
$185$	2.66944e7	−	4.62361e7i	0.309970	−	0.536884i
$186$		0			0
$187$	4.82177e6	+	8.35155e6i	0.0539213	+	0.0933945i
$188$	2.45245e7			0.269184
$189$		0			0
$190$	−1.99056e6			−0.0210541
$191$	6.59893e7	+	1.14297e8i	0.685262	+	1.18691i	0.973354	+	0.229306i	$0.0736457\pi$
$191$	6.59893e7	+	1.14297e8i	0.685262	+	1.18691i	−0.288092	+	0.957603i	$0.593021\pi$
$192$		0			0
$193$	−7.39537e7	+	1.28092e8i	−0.740473	+	1.28254i	0.211807	+	0.977311i	$0.432065\pi$
$193$	−7.39537e7	+	1.28092e8i	−0.740473	+	1.28254i	−0.952280	+	0.305225i	$0.901268\pi$
$194$	6.47055e7	−	1.12073e8i	0.636261	−	1.10204i
$195$		0			0
$196$	1.80953e7	+	3.13420e7i	0.171660	+	0.297325i
$197$	2.69561e7			0.251203			0.125602	−	0.992081i	$-0.459914\pi$
$197$	2.69561e7			0.251203			0.125602	+	0.992081i	$0.459914\pi$
$198$		0			0
$199$	1.35831e8			1.22183			0.610916	−	0.791695i	$-0.290801\pi$
$199$	1.35831e8			1.22183			0.610916	+	0.791695i	$0.290801\pi$
$200$	−1.30304e7	−	2.25693e7i	−0.115174	−	0.199486i
$201$		0			0
$202$	−1.08065e7	+	1.87173e7i	−0.0922473	+	0.159777i
$203$	2.09969e7	−	3.63677e7i	0.176165	−	0.305126i
$204$		0			0
$205$	−2.54197e7	−	4.40283e7i	−0.206078	−	0.356938i
$206$	−1.65513e7			−0.131916
$207$		0			0
$208$	2.06397e7			0.159031
$209$	2.28010e6	+	3.94924e6i	0.0172759	+	0.0299228i
$210$		0			0
$211$	1.09670e6	−	1.89954e6i	0.00803710	−	0.0139207i	−0.861979	−	0.506944i	$-0.830775\pi$
$211$	1.09670e6	−	1.89954e6i	0.00803710	−	0.0139207i	0.870016	+	0.493024i	$0.164108\pi$
$212$	2.43266e7	−	4.21349e7i	0.175350	−	0.303715i
$213$		0			0
$214$	3.66180e7	+	6.34242e7i	0.255415	+	0.442392i
$215$	5.55522e7			0.381212
$216$		0			0
$217$	8.88451e7			0.590234
$218$	3.47458e7	+	6.01815e7i	0.227146	+	0.393428i
$219$		0			0
$220$	1.59667e7	−	2.76552e7i	0.101097	−	0.175104i
$221$	−8.03469e6	+	1.39165e7i	−0.0500721	+	0.0867274i
$222$		0			0
$223$	−9.97215e6	−	1.72723e7i	−0.0602174	−	0.104300i	0.834345	−	0.551242i	$-0.185846\pi$
$223$	−9.97215e6	−	1.72723e7i	−0.0602174	−	0.104300i	−0.894562	+	0.446943i	$0.852513\pi$
$224$	1.66461e7			0.0989568
$225$		0			0
$226$	−1.99820e8			−1.15149
$227$	−6.29925e7	−	1.09106e8i	−0.357436	−	0.619098i	0.630096	−	0.776518i	$-0.283016\pi$
$227$	−6.29925e7	−	1.09106e8i	−0.357436	−	0.619098i	−0.987532	+	0.157420i	$0.949682\pi$
$228$		0			0
$229$	−1.17523e7	+	2.03556e7i	−0.0646693	+	0.112010i	−0.896547	−	0.442948i	$-0.853933\pi$
$229$	−1.17523e7	+	2.03556e7i	−0.0646693	+	0.112010i	0.831878	+	0.554959i	$0.187266\pi$
$230$	4.98960e7	−	8.64224e7i	0.270407	−	0.468359i
$231$		0			0
$232$	2.11622e7	+	3.66541e7i	0.111264	+	0.192714i
$233$	1.27757e8			0.661665			0.330833	−	0.943689i	$-0.392670\pi$
$233$	1.27757e8			0.661665			0.330833	+	0.943689i	$0.392670\pi$
$234$		0			0
$235$	6.32273e7			0.317810
$236$	−7.12212e7	−	1.23359e8i	−0.352710	−	0.610911i
$237$		0			0
$238$	−6.48005e6	+	1.12238e7i	−0.0311572	+	0.0539659i
$239$	−4.67439e7	+	8.09628e7i	−0.221479	+	0.383613i	−0.955257	−	0.295776i	$-0.904422\pi$
$239$	−4.67439e7	+	8.09628e7i	−0.221479	+	0.383613i	0.733778	+	0.679389i	$0.237755\pi$
$240$		0			0
$241$	6.73901e7	+	1.16723e8i	0.310125	+	0.537152i	0.978389	−	0.206772i	$-0.0662958\pi$
$241$	6.73901e7	+	1.16723e8i	0.310125	+	0.537152i	−0.668264	+	0.743924i	$0.732963\pi$
$242$	8.27408e7			0.375289
$243$		0			0
$244$	1.43668e8			0.633135
$245$	4.66520e7	+	8.08037e7i	0.202670	+	0.351034i
$246$		0			0
$247$	−3.79941e6	+	6.58076e6i	−0.0160427	+	0.0277867i
$248$	−4.47724e7	+	7.75480e7i	−0.186393	+	0.322842i
$249$		0			0
$250$	−8.51565e7	−	1.47495e8i	−0.344689	−	0.597019i
$251$	−1.31470e7			−0.0524771			−0.0262385	−	0.999656i	$-0.508353\pi$
$251$	−1.31470e7			−0.0524771			−0.0262385	+	0.999656i	$0.508353\pi$
$252$		0			0
$253$	−2.28614e8			−0.887527
$254$	−1.99464e7	−	3.45481e7i	−0.0763740	−	0.132284i
$255$		0			0
$256$	−8.38861e6	+	1.45295e7i	−0.0312500	+	0.0541266i
$257$	−6.42355e7	+	1.11259e8i	−0.236053	+	0.408855i	−0.959578	−	0.281443i	$-0.909187\pi$
$257$	−6.42355e7	+	1.11259e8i	−0.236053	+	0.408855i	0.723525	+	0.690298i	$0.242520\pi$
$258$		0			0
$259$	−8.21865e7	−	1.42351e8i	−0.293935	−	0.509111i
$260$	5.32118e7			0.187759
$261$		0			0
$262$	−1.78724e8			−0.613942
$263$	−4.88758e7	−	8.46554e7i	−0.165672	−	0.286952i	0.771222	−	0.636567i	$-0.219646\pi$
$263$	−4.88758e7	−	8.46554e7i	−0.165672	−	0.286952i	−0.936894	+	0.349614i	$0.886313\pi$
$264$		0			0
$265$	6.27170e7	−	1.08629e8i	0.207026	−	0.358579i
$266$	−3.06426e6	+	5.30745e6i	−0.00998250	+	0.0172902i
$267$		0			0
$268$	−4.71420e7	−	8.16524e7i	−0.149602	−	0.259118i
$269$	−3.96851e7			−0.124307			−0.0621534	−	0.998067i	$-0.519797\pi$
$269$	−3.96851e7			−0.124307			−0.0621534	+	0.998067i	$0.519797\pi$
$270$		0			0
$271$	−3.50733e8			−1.07049			−0.535246	−	0.844696i	$-0.679781\pi$
$271$	−3.50733e8			−1.07049			−0.535246	+	0.844696i	$0.679781\pi$
$272$	−6.53107e6	−	1.13121e7i	−0.0196786	−	0.0340843i
$273$		0			0
$274$	−5.14523e7	+	8.91180e7i	−0.151105	+	0.261721i
$275$	−7.69608e7	+	1.33300e8i	−0.223154	+	0.386514i
$276$		0			0
$277$	−2.69801e8	−	4.67309e8i	−0.762719	−	1.32107i	−0.941444	−	0.337169i	$-0.890531\pi$
$277$	−2.69801e8	−	4.67309e8i	−0.762719	−	1.32107i	0.178726	−	0.983899i	$-0.442803\pi$
$278$	−3.53522e8			−0.986869
$279$		0			0
$280$	4.29158e7			0.116833
$281$	−5.04259e7	−	8.73403e7i	−0.135576	−	0.234824i	0.790242	−	0.612796i	$-0.209955\pi$
$281$	−5.04259e7	−	8.73403e7i	−0.135576	−	0.234824i	−0.925817	+	0.377971i	$0.876622\pi$
$282$		0			0
$283$	1.27056e8	−	2.20068e8i	0.333229	−	0.577170i	−0.649914	−	0.760008i	$-0.725195\pi$
$283$	1.27056e8	−	2.20068e8i	0.333229	−	0.577170i	0.983143	+	0.182838i	$0.0585284\pi$
$284$	−1.60221e8	+	2.77510e8i	−0.415053	+	0.718894i
$285$		0			0
$286$	−6.09517e7	−	1.05572e8i	−0.154065	−	0.266849i
$287$	−1.56524e8			−0.390836
$288$		0			0
$289$	−4.00169e8			−0.975216
$290$	5.45589e7	+	9.44988e7i	0.131363	+	0.227527i
$291$		0			0
$292$	1.88746e8	−	3.26917e8i	0.443646	−	0.768418i
$293$	−2.71542e8	+	4.70325e8i	−0.630668	+	1.09235i	0.356748	+	0.934201i	$0.383886\pi$
$293$	−2.71542e8	+	4.70325e8i	−0.630668	+	1.09235i	−0.987415	+	0.158148i	$0.949448\pi$
$294$		0			0
$295$	−1.83617e8	−	3.18034e8i	−0.416425	−	0.721269i
$296$	1.65667e8			0.371292
$297$		0			0
$298$	−2.10310e7			−0.0460365
$299$	−1.90474e8	−	3.29911e8i	−0.412085	−	0.713752i
$300$		0			0
$301$	8.55167e7	−	1.48119e8i	0.180746	−	0.313061i
$302$	−2.21208e8	+	3.83143e8i	−0.462142	+	0.800454i
$303$		0			0
$304$	−3.08838e6	−	5.34924e6i	−0.00630484	−	0.0109203i
$305$	3.70394e8			0.747507
$306$		0			0
$307$	−7.29877e8			−1.43968			−0.719839	−	0.694141i	$-0.755784\pi$
$307$	−7.29877e8			−1.43968			−0.719839	+	0.694141i	$0.755784\pi$
$308$	−4.91581e7	−	8.51444e7i	−0.0958667	−	0.166046i
$309$		0			0
$310$	−1.15429e8	+	1.99928e8i	−0.220063	+	0.381161i
$311$	3.19385e8	−	5.53191e8i	0.602078	−	1.04283i	−0.390427	−	0.920634i	$-0.627673\pi$
$311$	3.19385e8	−	5.53191e8i	0.602078	−	1.04283i	0.992506	−	0.122197i	$-0.0389939\pi$
$312$		0			0
$313$	3.18880e8	+	5.52317e8i	0.587790	+	1.01808i	0.994521	+	0.104535i	$0.0333354\pi$
$313$	3.18880e8	+	5.52317e8i	0.587790	+	1.01808i	−0.406731	+	0.913548i	$0.633331\pi$
$314$	1.78968e8			0.326229
$315$		0			0
$316$	4.49841e8			0.801963
$317$	−3.27189e8	−	5.66707e8i	−0.576887	−	0.999198i	−0.995834	−	0.0911869i	$-0.970934\pi$
$317$	−3.27189e8	−	5.66707e8i	−0.576887	−	0.999198i	0.418947	−	0.908011i	$-0.362399\pi$
$318$		0			0
$319$	1.24989e8	−	2.16488e8i	0.215579	−	0.373393i
$320$	−2.16269e7	+	3.74589e7i	−0.0368951	+	0.0639042i
$321$		0			0
$322$	−1.53619e8	−	2.66076e8i	−0.256419	−	0.444130i
$323$	4.80901e6			0.00794049
$324$		0			0
$325$	−2.56485e8			−0.414448
$326$	2.09442e8	+	3.62765e8i	0.334813	+	0.579914i
$327$		0			0
$328$	7.88782e7	−	1.36621e8i	0.123424	−	0.213776i
$329$	9.73318e7	−	1.68584e8i	0.150685	−	0.260993i
$330$		0			0
$331$	−4.03844e8	−	6.99479e8i	−0.612091	−	1.06017i	−0.990887	−	0.134693i	$-0.956995\pi$
$331$	−4.03844e8	−	6.99479e8i	−0.612091	−	1.06017i	0.378796	−	0.925480i	$-0.376338\pi$
$332$	1.69677e8			0.254471
$333$		0			0
$334$	−4.38366e8			−0.643760
$335$	−1.21538e8	−	2.10510e8i	−0.176626	−	0.305926i
$336$		0			0
$337$	9.36402e7	−	1.62190e8i	0.133278	−	0.230844i	−0.791660	−	0.610961i	$-0.790783\pi$
$337$	9.36402e7	−	1.62190e8i	0.133278	−	0.230844i	0.924938	+	0.380117i	$0.124116\pi$
$338$	−1.49428e8	+	2.58817e8i	−0.210486	+	0.364573i
$339$		0			0
$340$	−1.68379e7	−	2.91641e7i	−0.0232334	−	0.0402414i
$341$	5.28873e8			0.722290
$342$		0			0
$343$	7.05623e8			0.944155
$344$	8.61901e7	+	1.49286e8i	0.114157	+	0.197726i
$345$		0			0
$346$	−4.94581e8	+	8.56639e8i	−0.641905	+	1.11181i
$347$	1.41278e8	−	2.44700e8i	0.181518	−	0.314399i	−0.760879	−	0.648893i	$-0.775232\pi$
$347$	1.41278e8	−	2.44700e8i	0.181518	−	0.314399i	0.942398	+	0.334494i	$0.108565\pi$
$348$		0			0
$349$	3.80306e8	+	6.58709e8i	0.478899	+	0.829478i	0.999707	−	0.0241959i	$-0.00770255\pi$
$349$	3.80306e8	+	6.58709e8i	0.478899	+	0.829478i	−0.520808	+	0.853674i	$0.674369\pi$
$350$	−2.06858e8			−0.257889
$351$		0			0
$352$	9.90904e7			0.121097
$353$	−2.34447e8	−	4.06074e8i	−0.283683	−	0.491353i	0.688606	−	0.725135i	$-0.258223\pi$
$353$	−2.34447e8	−	4.06074e8i	−0.283683	−	0.491353i	−0.972289	+	0.233783i	$0.924890\pi$
$354$		0			0
$355$	−4.13069e8	+	7.15456e8i	−0.490030	+	0.848757i
$356$	−2.16669e8	+	3.75281e8i	−0.254520	+	0.440841i
$357$		0			0
$358$	−3.94919e8	−	6.84020e8i	−0.454901	−	0.787912i
$359$	−1.53906e8			−0.175560			−0.0877802	−	0.996140i	$-0.527977\pi$
$359$	−1.53906e8			−0.175560			−0.0877802	+	0.996140i	$0.527977\pi$
$360$		0			0
$361$	−8.91598e8			−0.997456
$362$	3.67942e8	+	6.37294e8i	0.407661	+	0.706090i
$363$		0			0
$364$	8.19140e7	−	1.41879e8i	0.0890232	−	0.154193i
$365$	4.86610e8	−	8.42833e8i	0.523788	−	0.907228i
$366$		0			0
$367$	3.74764e8	+	6.49110e8i	0.395755	+	0.685468i	0.993197	−	0.116444i	$-0.0371496\pi$
$367$	3.74764e8	+	6.49110e8i	0.395755	+	0.685468i	−0.597442	+	0.801912i	$0.703816\pi$
$368$	3.09658e8			0.323903
$369$		0			0
$370$	4.27111e8			0.438364
$371$	−1.93092e8	−	3.34446e8i	−0.196316	−	0.340030i
$372$		0			0
$373$	2.70864e8	−	4.69150e8i	0.270253	−	0.468092i	−0.698674	−	0.715441i	$-0.746226\pi$
$373$	2.70864e8	−	4.69150e8i	0.270253	−	0.468092i	0.968926	+	0.247349i	$0.0795594\pi$
$374$	−3.85741e7	+	6.68124e7i	−0.0381281	+	0.0660399i
$375$		0			0
$376$	9.80982e7	+	1.69911e8i	0.0951707	+	0.164841i
$377$	4.16549e8			0.400379
$378$		0			0
$379$	1.32974e9			1.25467			0.627334	−	0.778750i	$-0.284146\pi$
$379$	1.32974e9			1.25467			0.627334	+	0.778750i	$0.284146\pi$
$380$	−7.96224e6	−	1.37910e7i	−0.00744377	−	0.0128930i
$381$		0			0
$382$	−5.27914e8	+	9.14375e8i	−0.484554	+	0.839271i
$383$	3.59974e8	−	6.23494e8i	0.327398	−	0.567070i	−0.654597	−	0.755978i	$-0.727162\pi$
$383$	3.59974e8	−	6.23494e8i	0.327398	−	0.567070i	0.981995	+	0.188908i	$0.0604949\pi$
$384$		0			0
$385$	−1.26736e8	−	2.19513e8i	−0.113184	−	0.196041i
$386$	−1.18326e9			−1.04719
$387$		0			0
$388$	1.03529e9			0.899809
$389$	1.16034e9	+	2.00976e9i	0.999447	+	1.73109i	0.528517	+	0.848922i	$0.322748\pi$
$389$	1.16034e9	+	2.00976e9i	0.999447	+	1.73109i	0.470930	+	0.882171i	$0.343919\pi$
$390$		0			0
$391$	−1.20544e8	+	2.08789e8i	−0.101983	+	0.176640i
$392$	−1.44763e8	+	2.50736e8i	−0.121382	+	0.210240i
$393$		0			0
$394$	1.07824e8	+	1.86757e8i	0.0888138	+	0.153830i
$395$	1.15975e9			0.946833
$396$		0			0
$397$	1.18373e9			0.949477			0.474739	−	0.880127i	$-0.342543\pi$
$397$	1.18373e9			0.949477			0.474739	+	0.880127i	$0.342543\pi$
$398$	5.43322e8	+	9.41061e8i	0.431983	+	0.748216i
$399$		0			0
$400$	1.04243e8	−	1.80555e8i	0.0814400	−	0.141058i
$401$	6.33820e8	−	1.09781e9i	0.490863	−	0.850200i	−0.509082	−	0.860718i	$-0.670015\pi$
$401$	6.33820e8	−	1.09781e9i	0.490863	−	0.850200i	0.999945	+	0.0105184i	$0.00334816\pi$
$402$		0			0
$403$	4.40640e8	+	7.63212e8i	0.335364	+	0.580868i
$404$	−1.72903e8			−0.130457
$405$		0			0
$406$	3.35951e8			0.249135
$407$	−4.89236e8	−	8.47382e8i	−0.359698	−	0.623016i
$408$		0			0
$409$	5.09744e8	−	8.82903e8i	0.368401	−	0.638089i	−0.620915	−	0.783878i	$-0.713239\pi$
$409$	5.09744e8	−	8.82903e8i	0.368401	−	0.638089i	0.989316	+	0.145789i	$0.0465721\pi$
$410$	2.03358e8	−	3.52226e8i	0.145719	−	0.252393i
$411$		0			0
$412$	−6.62053e7	−	1.14671e8i	−0.0466394	−	0.0807817i
$413$	−1.13064e9			−0.789765
$414$		0			0
$415$	4.37447e8			0.300440
$416$	8.25590e7	+	1.42996e8i	0.0562261	+	0.0973864i
$417$		0			0
$418$	−1.82408e7	+	3.15939e7i	−0.0122159	+	0.0211586i
$419$	5.20643e8	−	9.01780e8i	0.345773	−	0.598896i	−0.639721	−	0.768607i	$-0.720950\pi$
$419$	5.20643e8	−	9.01780e8i	0.345773	−	0.598896i	0.985494	+	0.169711i	$0.0542834\pi$
$420$		0			0
$421$	9.79224e8	+	1.69607e9i	0.639580	+	1.10779i	0.985525	+	0.169530i	$0.0542251\pi$
$421$	9.79224e8	+	1.69607e9i	0.639580	+	1.10779i	−0.345945	+	0.938255i	$0.612442\pi$
$422$	1.75472e7			0.0113662
$423$		0			0
$424$	3.89225e8			0.247982
$425$	8.11600e7	+	1.40573e8i	0.0512839	+	0.0888263i
$426$		0			0
$427$	5.70183e8	−	9.87586e8i	0.354419	−	0.613871i
$428$	−2.92944e8	+	5.07394e8i	−0.180606	+	0.312818i
$429$		0			0
$430$	2.22209e8	+	3.84877e8i	0.134779	+	0.233444i
$431$	−2.84256e9			−1.71017			−0.855084	−	0.518489i	$-0.826495\pi$
$431$	−2.84256e9			−1.71017			−0.855084	+	0.518489i	$0.826495\pi$
$432$		0			0
$433$	−8.41292e7			−0.0498011			−0.0249006	−	0.999690i	$-0.507927\pi$
$433$	−8.41292e7			−0.0498011			−0.0249006	+	0.999690i	$0.507927\pi$
$434$	3.55381e8	+	6.15537e8i	0.208679	+	0.361443i
$435$		0			0
$436$	−2.77966e8	+	4.81452e8i	−0.160616	+	0.278196i
$437$	−5.70024e7	+	9.87311e7i	−0.0326744	+	0.0565938i
$438$		0			0
$439$	9.35233e8	+	1.61987e9i	0.527587	+	0.913807i	0.999483	+	0.0321530i	$0.0102364\pi$
$439$	9.35233e8	+	1.61987e9i	0.527587	+	0.913807i	−0.471896	+	0.881654i	$0.656430\pi$
$440$	2.55468e8			0.142972
$441$		0			0
$442$	−1.28555e8			−0.0708127
$443$	−5.06663e8	−	8.77566e8i	−0.276889	−	0.479587i	0.693721	−	0.720244i	$-0.255970\pi$
$443$	−5.06663e8	−	8.77566e8i	−0.276889	−	0.479587i	−0.970610	+	0.240658i	$0.922637\pi$
$444$		0			0
$445$	−5.58599e8	+	9.67522e8i	−0.300497	+	0.520476i
$446$	7.97772e7	−	1.38178e8i	0.0425801	−	0.0737509i
$447$		0			0
$448$	6.65846e7	+	1.15328e8i	0.0349865	+	0.0605984i
$449$	1.65756e8			0.0864183			0.0432092	−	0.999066i	$-0.486242\pi$
$449$	1.65756e8			0.0864183			0.0432092	+	0.999066i	$0.486242\pi$
$450$		0			0
$451$	−9.31749e8			−0.478279
$452$	−7.99282e8	−	1.38440e9i	−0.407113	−	0.705141i
$453$		0			0
$454$	5.03940e8	−	8.72850e8i	0.252746	−	0.437768i
$455$	2.11184e8	−	3.65782e8i	0.105105	−	0.182047i
$456$		0			0
$457$	6.90398e8	+	1.19581e9i	0.338371	+	0.586076i	0.984126	−	0.177469i	$-0.0567908\pi$
$457$	6.90398e8	+	1.19581e9i	0.338371	+	0.586076i	−0.645756	+	0.763544i	$0.723457\pi$
$458$	−1.88037e8			−0.0914562
$459$		0			0
$460$	7.98336e8			0.382414
$461$	1.39246e9	+	2.41181e9i	0.661955	+	1.14654i	0.980101	+	0.198498i	$0.0636063\pi$
$461$	1.39246e9	+	2.41181e9i	0.661955	+	1.14654i	−0.318146	+	0.948042i	$0.603060\pi$
$462$		0			0
$463$	−1.39459e9	+	2.41551e9i	−0.653001	+	1.13103i	0.329390	+	0.944194i	$0.393157\pi$
$463$	−1.39459e9	+	2.41551e9i	−0.653001	+	1.13103i	−0.982391	+	0.186837i	$0.940176\pi$
$464$	−1.69298e8	+	2.93233e8i	−0.0786754	+	0.136270i
$465$		0			0
$466$	5.11027e8	+	8.85125e8i	0.233934	+	0.405186i
$467$	−3.97258e9			−1.80494			−0.902471	−	0.430751i	$-0.858249\pi$
$467$	−3.97258e9			−1.80494			−0.902471	+	0.430751i	$0.858249\pi$
$468$		0			0
$469$	−7.48380e8			−0.334979
$470$	2.52909e8	+	4.38052e8i	0.112363	+	0.194618i
$471$		0			0
$472$	5.69770e8	−	9.86871e8i	0.249404	−	0.431980i
$473$	5.09060e8	−	8.81718e8i	0.221185	−	0.383103i
$474$		0			0
$475$	3.83786e7	+	6.64737e7i	0.0164309	+	0.0284592i
$476$	−1.03681e8			−0.0440630
$477$		0			0
$478$	−7.47902e8			−0.313218
$479$	−1.17233e9	−	2.03054e9i	−0.487391	−	0.844186i	0.512504	−	0.858685i	$-0.328718\pi$
$479$	−1.17233e9	−	2.03054e9i	−0.487391	−	0.844186i	−0.999895	+	0.0144991i	$0.995385\pi$
$480$		0			0
$481$	8.15232e8	−	1.41202e9i	0.334021	−	0.578541i
$482$	−5.39121e8	+	9.33785e8i	−0.219291	+	0.379824i
$483$		0			0
$484$	3.30963e8	+	5.73245e8i	0.132685	+	0.229817i
$485$	2.66910e9			1.06235
$486$		0			0
$487$	3.08381e9			1.20986			0.604931	−	0.796278i	$-0.293201\pi$
$487$	3.08381e9			1.20986			0.604931	+	0.796278i	$0.293201\pi$
$488$	5.74673e8	+	9.95362e8i	0.223847	+	0.387714i
$489$		0			0
$490$	−3.73216e8	+	6.46429e8i	−0.143309	+	0.248219i
$491$	−1.85925e9	+	3.22032e9i	−0.708848	+	1.22776i	0.256436	+	0.966561i	$0.417452\pi$
$491$	−1.85925e9	+	3.22032e9i	−0.708848	+	1.22776i	−0.965285	+	0.261200i	$0.915882\pi$
$492$		0			0
$493$	−1.31809e8	−	2.28300e8i	−0.0495430	−	0.0858109i
$494$	−6.07905e7			−0.0226877
$495$		0			0
$496$	−7.16358e8			−0.263599
$497$	1.27175e9	+	2.20274e9i	0.464681	+	0.804850i
$498$		0			0
$499$	−2.18213e8	+	3.77956e8i	−0.0786192	+	0.136172i	−0.902654	−	0.430366i	$-0.858384\pi$
$499$	−2.18213e8	+	3.77956e8i	−0.0786192	+	0.136172i	0.824035	+	0.566539i	$0.191718\pi$
$500$	6.81252e8	−	1.17996e9i	0.243732	−	0.422156i
$501$		0			0
$502$	−5.25881e7	−	9.10853e7i	−0.0185535	−	0.0321355i
$503$	5.04603e8			0.176792			0.0883958	−	0.996085i	$-0.471826\pi$
$503$	5.04603e8			0.176792			0.0883958	+	0.996085i	$0.471826\pi$
$504$		0			0
$505$	−4.45766e8			−0.154024
$506$	−9.14458e8	−	1.58389e9i	−0.313788	−	0.543497i
$507$		0			0
$508$	1.59571e8	−	2.76385e8i	0.0540046	−	0.0935387i
$509$	−2.30043e9	+	3.98446e9i	−0.773208	+	1.33923i	0.162589	+	0.986694i	$0.448016\pi$
$509$	−2.30043e9	+	3.98446e9i	−0.773208	+	1.33923i	−0.935796	+	0.352541i	$0.885318\pi$
$510$		0			0
$511$	−1.49817e9	−	2.59490e9i	−0.496692	−	0.860296i
$512$	−1.34218e8			−0.0441942
$513$		0			0
$514$	−1.02777e9			−0.333829
$515$	−1.70686e8	−	2.95636e8i	−0.0550645	−	0.0953745i
$516$		0			0
$517$	5.79392e8	−	1.00354e9i	0.184398	−	0.319386i
$518$	6.57492e8	−	1.13881e9i	0.207844	−	0.359996i
$519$		0			0
$520$	2.12847e8	+	3.68662e8i	0.0663830	+	0.114979i
$521$	−7.14937e7			−0.0221481			−0.0110740	−	0.999939i	$-0.503525\pi$
$521$	−7.14937e7			−0.0221481			−0.0110740	+	0.999939i	$0.503525\pi$
$522$		0			0
$523$	−2.12182e9			−0.648563			−0.324281	−	0.945961i	$-0.605122\pi$
$523$	−2.12182e9			−0.648563			−0.324281	+	0.945961i	$0.605122\pi$
$524$	−7.14895e8	−	1.23823e9i	−0.217061	−	0.375961i
$525$		0			0
$526$	3.91007e8	−	6.77243e8i	0.117148	−	0.202906i
$527$	2.78865e8	−	4.83009e8i	0.0829960	−	0.143753i
$528$		0			0
$529$	−1.15527e9	−	2.00098e9i	−0.339303	−	0.587690i
$530$	1.00347e9			0.292779
$531$		0			0
$532$	−4.90281e7			−0.0141174
$533$	−7.76303e8	−	1.34460e9i	−0.222068	−	0.384633i
$534$		0			0
$535$	−7.55246e8	+	1.30812e9i	−0.213231	+	0.369327i
$536$	3.77136e8	−	6.53219e8i	0.105784	−	0.183224i
$537$		0			0
$538$	−1.58740e8	−	2.74946e8i	−0.0439491	−	0.0761220i
$539$	1.71001e9			0.470368
$540$		0			0
$541$	−2.11351e9			−0.573870			−0.286935	−	0.957950i	$-0.592636\pi$
$541$	−2.11351e9			−0.573870			−0.286935	+	0.957950i	$0.592636\pi$
$542$	−1.40293e9	−	2.42995e9i	−0.378476	−	0.655540i
$543$		0			0
$544$	5.22486e7	−	9.04972e7i	0.0139148	−	0.0241012i
$545$	−7.16632e8	+	1.24124e9i	−0.189631	+	0.328450i
$546$		0			0
$547$	−3.12777e9	−	5.41745e9i	−0.817107	−	1.41527i	−0.907805	−	0.419392i	$-0.862243\pi$
$547$	−3.12777e9	−	5.41745e9i	−0.817107	−	1.41527i	0.0906987	−	0.995878i	$-0.471090\pi$
$548$	−8.23237e8			−0.213694
$549$		0			0
$550$	−1.23137e9			−0.315588
$551$	−6.23294e7	−	1.07958e8i	−0.0158731	−	0.0274931i
$552$		0			0
$553$	1.78531e9	−	3.09224e9i	0.448926	−	0.777563i
$554$	2.15841e9	−	3.73847e9i	0.539324	−	0.934136i
$555$		0			0
$556$	−1.41409e9	−	2.44927e9i	−0.348911	−	0.604331i
$557$	6.50846e9			1.59583			0.797913	−	0.602773i	$-0.205938\pi$
$557$	6.50846e9			1.59583			0.797913	+	0.602773i	$0.205938\pi$
$558$		0			0
$559$	1.69653e9			0.410790
$560$	1.71663e8	+	2.97330e8i	0.0413066	+	0.0715451i
$561$		0			0
$562$	4.03408e8	−	6.98722e8i	0.0958665	−	0.166046i
$563$	−8.66244e7	+	1.50038e8i	−0.0204579	+	0.0354341i	−0.876073	−	0.482178i	$-0.839846\pi$
$563$	−8.66244e7	+	1.50038e8i	−0.0204579	+	0.0354341i	0.855615	+	0.517612i	$0.173179\pi$
$564$		0			0
$565$	−2.06065e9	−	3.56915e9i	−0.480656	−	0.832520i
$566$	2.03290e9			0.471257
$567$		0			0
$568$	−2.56353e9			−0.586974
$569$	3.64441e9	+	6.31230e9i	0.829343	+	1.43646i	0.898555	+	0.438862i	$0.144618\pi$
$569$	3.64441e9	+	6.31230e9i	0.829343	+	1.43646i	−0.0692121	+	0.997602i	$0.522049\pi$
$570$		0			0
$571$	−2.14518e9	+	3.71556e9i	−0.482212	+	0.835215i	−0.999791	−	0.0204199i	$-0.993500\pi$
$571$	−2.14518e9	+	3.71556e9i	−0.482212	+	0.835215i	0.517580	+	0.855635i	$0.326833\pi$
$572$	4.87614e8	−	8.44572e8i	0.108941	−	0.188691i
$573$		0			0
$574$	−6.26096e8	−	1.08443e9i	−0.138181	−	0.239337i
$575$	−3.84804e9			−0.844116
$576$		0			0
$577$	6.28378e8			0.136178			0.0680888	−	0.997679i	$-0.478310\pi$
$577$	6.28378e8			0.136178			0.0680888	+	0.997679i	$0.478310\pi$
$578$	−1.60068e9	−	2.77245e9i	−0.344791	−	0.597196i
$579$		0			0
$580$	−4.36471e8	+	7.55990e8i	−0.0928876	+	0.160886i
$581$	6.73404e8	−	1.16637e9i	0.142449	−	0.246729i
$582$		0			0
$583$	−1.14943e9	−	1.99087e9i	−0.240239	−	0.416106i
$584$	3.01993e9			0.627411
$585$		0			0
$586$	−4.34467e9			−0.891899
$587$	−4.48401e9	−	7.76654e9i	−0.915026	−	1.58487i	−0.806863	−	0.590738i	$-0.798837\pi$
$587$	−4.48401e9	−	7.76654e9i	−0.915026	−	1.58487i	−0.108163	−	0.994133i	$-0.534497\pi$
$588$		0			0
$589$	1.31869e8	−	2.28403e8i	0.0265912	−	0.0460573i
$590$	1.46894e9	−	2.54428e9i	0.294457	−	0.510014i
$591$		0			0
$592$	6.62669e8	+	1.14778e9i	0.131272	+	0.227369i
$593$	7.57484e9			1.49170			0.745851	−	0.666112i	$-0.232043\pi$
$593$	7.57484e9			1.49170			0.745851	+	0.666112i	$0.232043\pi$
$594$		0			0
$595$	−2.67302e8			−0.0520227
$596$	−8.41239e7	−	1.45707e8i	−0.0162764	−	0.0281915i
$597$		0			0
$598$	1.52379e9	−	2.63929e9i	0.291388	−	0.504699i
$599$	−3.02973e9	+	5.24765e9i	−0.575984	+	0.997633i	0.419950	+	0.907547i	$0.362047\pi$
$599$	−3.02973e9	+	5.24765e9i	−0.575984	+	0.997633i	−0.995934	+	0.0900859i	$0.971286\pi$
$600$		0			0
$601$	3.27899e8	+	5.67937e8i	0.0616139	+	0.106718i	0.895187	−	0.445691i	$-0.147042\pi$
$601$	3.27899e8	+	5.67937e8i	0.0616139	+	0.106718i	−0.833573	+	0.552409i	$0.813709\pi$
$602$	1.36827e9			0.255613
$603$		0			0
$604$	−3.53932e9			−0.653568
$605$	8.53264e8	+	1.47790e9i	0.156653	+	0.271332i
$606$		0			0
$607$	−1.92806e9	+	3.33949e9i	−0.349913	+	0.606066i	−0.986234	−	0.165358i	$-0.947122\pi$
$607$	−1.92806e9	+	3.33949e9i	−0.349913	+	0.606066i	0.636321	+	0.771424i	$0.280455\pi$
$608$	2.47071e7	−	4.27939e7i	0.00445819	−	0.00772181i
$609$		0			0
$610$	1.48158e9	+	2.56617e9i	0.264284	+	0.457752i
$611$	1.93092e9			0.342469
$612$		0			0
$613$	−4.89889e9			−0.858986			−0.429493	−	0.903070i	$-0.641308\pi$
$613$	−4.89889e9			−0.858986			−0.429493	+	0.903070i	$0.641308\pi$
$614$	−2.91951e9	−	5.05674e9i	−0.509003	−	0.881619i
$615$		0			0
$616$	3.93265e8	−	6.81155e8i	0.0677880	−	0.117412i
$617$	−3.32728e9	+	5.76302e9i	−0.570284	+	0.987761i	0.426253	+	0.904604i	$0.359833\pi$
$617$	−3.32728e9	+	5.76302e9i	−0.570284	+	0.987761i	−0.996537	+	0.0831565i	$0.973500\pi$
$618$		0			0
$619$	−5.27657e9	−	9.13929e9i	−0.894199	−	1.54880i	−0.834792	−	0.550565i	$-0.814412\pi$
$619$	−5.27657e9	−	9.13929e9i	−0.894199	−	1.54880i	−0.0594070	−	0.998234i	$-0.518921\pi$
$620$	−1.84686e9			−0.311217
$621$		0			0
$622$	5.11016e9			0.851468
$623$	1.71981e9	+	2.97880e9i	0.284952	+	0.493552i
$624$		0			0
$625$	−2.31928e8	+	4.01712e8i	−0.0379992	+	0.0658165i
$626$	−2.55104e9	+	4.41853e9i	−0.415631	+	0.719893i
$627$		0			0
$628$	7.15874e8	+	1.23993e9i	0.115339	+	0.199774i
$629$	−1.03186e9			−0.165327
$630$		0			0
$631$	6.37775e9			1.01057			0.505283	−	0.862954i	$-0.331388\pi$
$631$	6.37775e9			1.01057			0.505283	+	0.862954i	$0.331388\pi$
$632$	1.79936e9	+	3.11659e9i	0.283537	+	0.491100i
$633$		0			0
$634$	2.61751e9	−	4.53366e9i	0.407921	−	0.706540i
$635$	4.11394e8	−	7.12555e8i	0.0637602	−	0.110436i
$636$		0			0
$637$	1.42472e9	+	2.46769e9i	0.218395	+	0.378271i
$638$	1.99983e9			0.304874
$639$		0			0
$640$	−3.46030e8			−0.0521776
$641$	3.34351e8	+	5.79113e8i	0.0501418	+	0.0868481i	0.890007	−	0.455947i	$-0.150699\pi$
$641$	3.34351e8	+	5.79113e8i	0.0501418	+	0.0868481i	−0.839865	+	0.542795i	$0.817366\pi$
$642$		0			0
$643$	−1.62316e9	+	2.81140e9i	−0.240782	+	0.417046i	−0.960937	−	0.276767i	$-0.910737\pi$
$643$	−1.62316e9	+	2.81140e9i	−0.240782	+	0.417046i	0.720155	+	0.693813i	$0.244070\pi$
$644$	1.22895e9	−	2.12861e9i	0.181316	−	0.314048i
$645$		0			0
$646$	1.92360e7	+	3.33178e7i	0.00280739	+	0.00486253i
$647$	−7.46321e9			−1.08333			−0.541665	−	0.840595i	$-0.682206\pi$
$647$	−7.46321e9			−1.08333			−0.541665	+	0.840595i	$0.682206\pi$
$648$		0			0
$649$	−6.73041e9			−0.966462
$650$	−1.02594e9	−	1.77698e9i	−0.146530	−	0.253797i
$651$		0			0
$652$	−1.67554e9	+	2.90212e9i	−0.236749	+	0.410061i
$653$	−5.08150e9	+	8.80141e9i	−0.714160	+	1.23696i	0.249122	+	0.968472i	$0.419858\pi$
$653$	−5.08150e9	+	8.80141e9i	−0.714160	+	1.23696i	−0.963283	+	0.268490i	$0.913475\pi$
$654$		0			0
$655$	−1.84309e9	−	3.19232e9i	−0.256272	−	0.443876i
$656$	1.26205e9			0.174547
$657$		0			0
$658$	1.55731e9			0.213100
$659$	−1.41534e9	−	2.45144e9i	−0.192647	−	0.333674i	0.753480	−	0.657471i	$-0.228374\pi$
$659$	−1.41534e9	−	2.45144e9i	−0.192647	−	0.333674i	−0.946127	+	0.323797i	$0.895041\pi$
$660$		0			0
$661$	−1.98137e9	+	3.43184e9i	−0.266846	+	0.462192i	−0.968046	−	0.250774i	$-0.919315\pi$
$661$	−1.98137e9	+	3.43184e9i	−0.266846	+	0.462192i	0.701199	+	0.712965i	$0.252648\pi$
$662$	3.23075e9	−	5.59583e9i	0.432814	−	0.749655i
$663$		0			0
$664$	6.78706e8	+	1.17555e9i	0.0899691	+	0.155831i
$665$	−1.26401e8			−0.0166676
$666$		0			0
$667$	6.24947e9			0.815461
$668$	−1.75346e9	−	3.03709e9i	−0.227604	−	0.394221i
$669$		0			0
$670$	9.72304e8	−	1.68408e9i	0.124894	−	0.216322i
$671$	3.39416e9	−	5.87886e9i	0.433714	−	0.751215i
$672$		0			0
$673$	5.93077e9	+	1.02724e10i	0.749995	+	1.29903i	0.947824	+	0.318794i	$0.103278\pi$
$673$	5.93077e9	+	1.02724e10i	0.749995	+	1.29903i	−0.197829	+	0.980237i	$0.563389\pi$
$674$	1.49824e9			0.188483
$675$		0			0
$676$	−2.39085e9			−0.297672
$677$	4.93418e9	+	8.54625e9i	0.611159	+	1.05856i	0.991045	+	0.133526i	$0.0426299\pi$
$677$	4.93418e9	+	8.54625e9i	0.611159	+	1.05856i	−0.379886	+	0.925033i	$0.624037\pi$
$678$		0			0
$679$	4.10880e9	−	7.11665e9i	0.503699	−	0.872432i
$680$	1.34703e8	−	2.33313e8i	0.0164285	−	0.0284550i
$681$		0			0
$682$	2.11549e9	+	3.66414e9i	0.255368	+	0.442310i
$683$	−1.36287e10			−1.63674			−0.818372	−	0.574689i	$-0.805123\pi$
$683$	−1.36287e10			−1.63674			−0.818372	+	0.574689i	$0.805123\pi$
$684$		0			0
$685$	−2.12241e9			−0.252297
$686$	2.82249e9	+	4.88870e9i	0.333809	+	0.578175i
$687$		0			0
$688$	−6.89521e8	+	1.19428e9i	−0.0807212	+	0.139813i
$689$	1.91534e9	−	3.31746e9i	0.223089	−	0.386402i
$690$		0			0
$691$	1.08135e9	+	1.87295e9i	0.124679	+	0.215950i	0.921607	−	0.388124i	$-0.126877\pi$
$691$	1.08135e9	+	1.87295e9i	0.124679	+	0.215950i	−0.796929	+	0.604074i	$0.793543\pi$
$692$	−7.91329e9			−0.907791
$693$		0			0
$694$	2.26045e9			0.256706
$695$	−3.64569e9	−	6.31453e9i	−0.411940	−	0.713500i
$696$		0			0
$697$	−4.91294e8	+	8.50946e8i	−0.0549575	+	0.0951892i
$698$	−3.04245e9	+	5.26968e9i	−0.338633	+	0.586530i
$699$		0			0
$700$	−8.27430e8	−	1.43315e9i	−0.0911776	−	0.157924i
$701$	−2.93914e9			−0.322260			−0.161130	−	0.986933i	$-0.551514\pi$
$701$	−2.93914e9			−0.322260			−0.161130	+	0.986933i	$0.551514\pi$
$702$		0			0
$703$	−4.87942e8			−0.0529693
$704$	3.96362e8	+	6.86519e8i	0.0428142	+	0.0741563i
$705$		0			0
$706$	1.87557e9	−	3.24859e9i	0.200594	−	0.347439i
$707$	−6.86210e8	+	1.18855e9i	−0.0730279	+	0.126488i
$708$		0			0
$709$	6.54572e9	+	1.13375e10i	0.689756	+	1.19469i	0.971917	+	0.235326i	$0.0756157\pi$
$709$	6.54572e9	+	1.13375e10i	0.689756	+	1.19469i	−0.282160	+	0.959367i	$0.591051\pi$
$710$	−6.60910e9			−0.693007
$711$		0			0
$712$	−3.46670e9			−0.359945
$713$	6.61092e9	+	1.14504e10i	0.683044	+	1.18307i
$714$		0			0
$715$	1.25713e9	−	2.17741e9i	0.128620	−	0.222777i
$716$	3.15935e9	−	5.47216e9i	0.321664	−	0.557138i
$717$		0			0
$718$	−6.15626e8	−	1.06630e9i	−0.0620699	−	0.107508i
$719$	3.03015e9			0.304027			0.152014	−	0.988378i	$-0.451424\pi$
$719$	3.03015e9			0.304027			0.152014	+	0.988378i	$0.451424\pi$
$720$		0			0
$721$	−1.05101e9			−0.104432
$722$	−3.56639e9	−	6.17717e9i	−0.352654	−	0.610815i
$723$		0			0
$724$	−2.94354e9	+	5.09836e9i	−0.288260	+	0.499281i
$725$	2.10382e9	−	3.64393e9i	0.205034	−	0.355130i
$726$		0			0
$727$	5.34238e9	+	9.25328e9i	0.515661	+	0.893151i	0.999835	+	0.0181794i	$0.00578701\pi$
$727$	5.34238e9	+	9.25328e9i	0.515661	+	0.893151i	−0.484174	+	0.874972i	$0.660880\pi$
$728$	1.31062e9			0.125898
$729$		0			0
$730$	7.78576e9			0.740749
$731$	−5.36836e8	−	9.29828e8i	−0.0508313	−	0.0880424i
$732$		0			0
$733$	8.38524e9	−	1.45237e10i	0.786414	−	1.36211i	−0.141736	−	0.989904i	$-0.545268\pi$
$733$	8.38524e9	−	1.45237e10i	0.786414	−	1.36211i	0.928150	−	0.372205i	$-0.121398\pi$
$734$	−2.99811e9	+	5.19288e9i	−0.279841	+	0.484699i
$735$		0			0
$736$	1.23863e9	+	2.14537e9i	0.114517	+	0.198349i
$737$	−4.45492e9			−0.409924
$738$		0			0
$739$	1.68052e10			1.53175			0.765877	−	0.642987i	$-0.222305\pi$
$739$	1.68052e10			1.53175			0.765877	+	0.642987i	$0.222305\pi$
$740$	1.70844e9	+	2.95911e9i	0.154985	+	0.268442i
$741$		0			0
$742$	1.54474e9	−	2.67557e9i	0.138817	−	0.240437i
$743$	1.03917e10	−	1.79989e10i	0.929448	−	1.60985i	0.145203	−	0.989402i	$-0.453617\pi$
$743$	1.03917e10	−	1.79989e10i	0.929448	−	1.60985i	0.784246	−	0.620450i	$-0.213050\pi$
$744$		0			0
$745$	−2.16882e8	−	3.75651e8i	−0.0192166	−	0.0332841i
$746$	4.33382e9			0.382195
$747$		0			0
$748$	−6.17186e8			−0.0539213
$749$	2.32524e9	+	4.02744e9i	0.202200	+	0.350221i
$750$		0			0
$751$	−5.03006e9	+	8.71232e9i	−0.433344	+	0.750575i	−0.997159	−	0.0753269i	$-0.976000\pi$
$751$	−5.03006e9	+	8.71232e9i	−0.433344	+	0.750575i	0.563814	+	0.825901i	$0.309333\pi$
$752$	−7.84785e8	+	1.35929e9i	−0.0672959	+	0.116560i
$753$		0			0
$754$	1.66620e9	+	2.88594e9i	0.141555	+	0.245181i
$755$	−9.12481e9			−0.771631
$756$		0			0
$757$	6.59893e9			0.552889			0.276445	−	0.961030i	$-0.410844\pi$
$757$	6.59893e9			0.552889			0.276445	+	0.961030i	$0.410844\pi$
$758$	5.31896e9	+	9.21270e9i	0.443592	+	0.768325i
$759$		0			0
$760$	6.36979e7	−	1.10328e8i	0.00526354	−	0.00911671i
$761$	−3.03962e9	+	5.26478e9i	−0.250019	+	0.433046i	−0.963531	−	0.267597i	$-0.913770\pi$
$761$	−3.03962e9	+	5.26478e9i	−0.250019	+	0.433046i	0.713511	+	0.700644i	$0.247104\pi$
$762$		0			0
$763$	2.20636e9	+	3.82153e9i	0.179821	+	0.311459i
$764$	−8.44663e9			−0.685262
$765$		0			0
$766$	5.75959e9			0.463011
$767$	−5.60756e9	−	9.71258e9i	−0.448735	−	0.777232i
$768$		0			0
$769$	−2.42525e9	+	4.20065e9i	−0.192315	+	0.333100i	−0.946017	−	0.324117i	$-0.894933\pi$
$769$	−2.42525e9	+	4.20065e9i	−0.192315	+	0.333100i	0.753702	+	0.657216i	$0.228266\pi$
$770$	1.01389e9	−	1.75610e9i	0.0800335	−	0.138622i
$771$		0			0
$772$	−4.73304e9

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)

\(f(q)\)	\(=\)	\(q+(4.00000 + 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(-82.5000 + 142.894i) q^{5} +(254.000 + 439.941i) q^{7} -512.000 q^{8} +O(q^{10})\)	\(q+(4.00000 + 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(-82.5000 + 142.894i) q^{5} +(254.000 + 439.941i) q^{7} -512.000 q^{8} -1320.00 q^{10} +(1512.00 + 2618.86i) q^{11} +(-2519.50 + 4363.90i) q^{13} +(-2032.00 + 3519.53i) q^{14} +(-2048.00 - 3547.24i) q^{16} +3189.00 q^{17} +1508.00 q^{19} +(-5280.00 - 9145.23i) q^{20} +(-12096.0 + 20950.9i) q^{22} +(-37800.0 + 65471.5i) q^{23} +(25450.0 + 44080.7i) q^{25} -40312.0 q^{26} -32512.0 q^{28} +(-41332.5 - 71590.0i) q^{29} +(87446.0 - 151461. i) q^{31} +(16384.0 - 28377.9i) q^{32} +(12756.0 + 22094.0i) q^{34} -83820.0 q^{35} -323569. q^{37} +(6032.00 + 10447.7i) q^{38} +(42240.0 - 73161.8i) q^{40} +(-154059. + 266838. i) q^{41} +(-168340. - 291573. i) q^{43} -193536. q^{44} -604800. q^{46} +(-191598. - 331857. i) q^{47} +(282740. - 489719. i) q^{49} +(-203600. + 352646. i) q^{50} +(-161248. - 279290. i) q^{52} -760206. q^{53} -498960. q^{55} +(-130048. - 225250. i) q^{56} +(330660. - 572720. i) q^{58} +(-1.11283e6 + 1.92748e6i) q^{59} +(-1.12241e6 - 1.94407e6i) q^{61} +1.39914e6 q^{62} +262144. q^{64} +(-415718. - 720044. i) q^{65} +(-736594. + 1.27582e6i) q^{67} +(-102048. + 176752. i) q^{68} +(-335280. - 580722. i) q^{70} +5.00689e6 q^{71} -5.89830e6 q^{73} +(-1.29428e6 - 2.24175e6i) q^{74} +(-48256.0 + 83581.8i) q^{76} +(-768096. + 1.33038e6i) q^{77} +(-3.51438e6 - 6.08709e6i) q^{79} +675840. q^{80} -2.46494e6 q^{82} +(-1.32560e6 - 2.29600e6i) q^{83} +(-263092. + 455690. i) q^{85} +(1.34672e6 - 2.33259e6i) q^{86} +(-774144. - 1.34086e6i) q^{88} +6.77090e6 q^{89} -2.55981e6 q^{91} +(-2.41920e6 - 4.19018e6i) q^{92} +(1.53278e6 - 2.65486e6i) q^{94} +(-124410. + 215484. i) q^{95} +(-8.08819e6 - 1.40092e7i) q^{97} +4.52383e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 8 q^{2} - 64 q^{4} - 165 q^{5} + 508 q^{7} - 1024 q^{8}+O(q^{10}) \) 2 * q + 8 * q^2 - 64 * q^4 - 165 * q^5 + 508 * q^7 - 1024 * q^8	\( 2 q + 8 q^{2} - 64 q^{4} - 165 q^{5} + 508 q^{7} - 1024 q^{8} - 2640 q^{10} + 3024 q^{11} - 5039 q^{13} - 4064 q^{14} - 4096 q^{16} + 6378 q^{17} + 3016 q^{19} - 10560 q^{20} - 24192 q^{22} - 75600 q^{23} + 50900 q^{25} - 80624 q^{26} - 65024 q^{28} - 82665 q^{29} + 174892 q^{31} + 32768 q^{32} + 25512 q^{34} - 167640 q^{35} - 647138 q^{37} + 12064 q^{38} + 84480 q^{40} - 308118 q^{41} - 336680 q^{43} - 387072 q^{44} - 1209600 q^{46} - 383196 q^{47} + 565479 q^{49} - 407200 q^{50} - 322496 q^{52} - 1520412 q^{53} - 997920 q^{55} - 260096 q^{56} + 661320 q^{58} - 2225664 q^{59} - 2244815 q^{61} + 2798272 q^{62} + 524288 q^{64} - 831435 q^{65} - 1473188 q^{67} - 204096 q^{68} - 670560 q^{70} + 10013784 q^{71} - 11796602 q^{73} - 2588552 q^{74} - 96512 q^{76} - 1536192 q^{77} - 7028768 q^{79} + 1351680 q^{80} - 4929888 q^{82} - 2651196 q^{83} - 526185 q^{85} + 2693440 q^{86} - 1548288 q^{88} + 13541802 q^{89} - 5119624 q^{91} - 4838400 q^{92} + 3065568 q^{94} - 248820 q^{95} - 16176386 q^{97} + 9047664 q^{98}+O(q^{100}) \) 2 * q + 8 * q^2 - 64 * q^4 - 165 * q^5 + 508 * q^7 - 1024 * q^8 - 2640 * q^10 + 3024 * q^11 - 5039 * q^13 - 4064 * q^14 - 4096 * q^16 + 6378 * q^17 + 3016 * q^19 - 10560 * q^20 - 24192 * q^22 - 75600 * q^23 + 50900 * q^25 - 80624 * q^26 - 65024 * q^28 - 82665 * q^29 + 174892 * q^31 + 32768 * q^32 + 25512 * q^34 - 167640 * q^35 - 647138 * q^37 + 12064 * q^38 + 84480 * q^40 - 308118 * q^41 - 336680 * q^43 - 387072 * q^44 - 1209600 * q^46 - 383196 * q^47 + 565479 * q^49 - 407200 * q^50 - 322496 * q^52 - 1520412 * q^53 - 997920 * q^55 - 260096 * q^56 + 661320 * q^58 - 2225664 * q^59 - 2244815 * q^61 + 2798272 * q^62 + 524288 * q^64 - 831435 * q^65 - 1473188 * q^67 - 204096 * q^68 - 670560 * q^70 + 10013784 * q^71 - 11796602 * q^73 - 2588552 * q^74 - 96512 * q^76 - 1536192 * q^77 - 7028768 * q^79 + 1351680 * q^80 - 4929888 * q^82 - 2651196 * q^83 - 526185 * q^85 + 2693440 * q^86 - 1548288 * q^88 + 13541802 * q^89 - 5119624 * q^91 - 4838400 * q^92 + 3065568 * q^94 - 248820 * q^95 - 16176386 * q^97 + 9047664 * q^98

Introduction

L-functions

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