LMFDB - Embedded newform 162.8.c.f.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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 54)
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.f.55.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.00000 - 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(156.000 - 270.200i) q^{5} +(-161.500 - 279.726i) q^{7} +512.000 q^{8} -2496.00 q^{10} +(1860.00 + 3221.61i) q^{11} +(7089.50 - 12279.4i) q^{13} +(-1292.00 + 2237.81i) q^{14} +(-2048.00 - 3547.24i) q^{16} -15912.0 q^{17} +22421.0 q^{19} +(9984.00 + 17292.8i) q^{20} +(14880.0 - 25772.9i) q^{22} +(28884.0 - 50028.6i) q^{23} +(-9609.50 - 16644.1i) q^{25} -113432. q^{26} +20672.0 q^{28} +(83328.0 + 144328. i) q^{29} +(-47410.0 + 82116.5i) q^{31} +(-16384.0 + 28377.9i) q^{32} +(63648.0 + 110242. i) q^{34} -100776. q^{35} +453971. q^{37} +(-89684.0 - 155337. i) q^{38} +(79872.0 - 138342. i) q^{40} +(313536. - 543060. i) q^{41} +(21236.0 + 36781.8i) q^{43} -238080. q^{44} -462144. q^{46} +(-617628. - 1.06976e6i) q^{47} +(359607. - 622858. i) q^{49} +(-76876.0 + 133153. i) q^{50} +(453728. + 785880. i) q^{52} -107280. q^{53} +1.16064e6 q^{55} +(-82688.0 - 143220. i) q^{56} +(666624. - 1.15463e6i) q^{58} +(-1.23961e6 + 2.14707e6i) q^{59} +(-1.43719e6 - 2.48929e6i) q^{61} +758560. q^{62} +262144. q^{64} +(-2.21192e6 - 3.83116e6i) q^{65} +(-750548. + 1.29999e6i) q^{67} +(509184. - 881933. i) q^{68} +(403104. + 698197. i) q^{70} -4.73314e6 q^{71} -85111.0 q^{73} +(-1.81588e6 - 3.14520e6i) q^{74} +(-717472. + 1.24270e6i) q^{76} +(600780. - 1.04058e6i) q^{77} +(590410. + 1.02262e6i) q^{79} -1.27795e6 q^{80} -5.01658e6 q^{82} +(-558264. - 966942. i) q^{83} +(-2.48227e6 + 4.29942e6i) q^{85} +(169888. - 294255. i) q^{86} +(952320. + 1.64947e6i) q^{88} -9.36814e6 q^{89} -4.57982e6 q^{91} +(1.84858e6 + 3.20183e6i) q^{92} +(-4.94102e6 + 8.55810e6i) q^{94} +(3.49768e6 - 6.05815e6i) q^{95} +(1.02000e6 + 1.76669e6i) q^{97} -5.75371e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{2} - 64 q^{4} + 312 q^{5} - 323 q^{7} + 1024 q^{8} - 4992 q^{10} + 3720 q^{11} + 14179 q^{13} - 2584 q^{14} - 4096 q^{16} - 31824 q^{17} + 44842 q^{19} + 19968 q^{20} + 29760 q^{22} + 57768 q^{23}+ \cdots - 11507424 q^{98}+O(q^{100}) $ 2 * q - 8 * q^2 - 64 * q^4 + 312 * q^5 - 323 * q^7 + 1024 * q^8 - 4992 * q^10 + 3720 * q^11 + 14179 * q^13 - 2584 * q^14 - 4096 * q^16 - 31824 * q^17 + 44842 * q^19 + 19968 * q^20 + 29760 * q^22 + 57768 * q^23 - 19219 * q^25 - 226864 * q^26 + 41344 * q^28 + 166656 * q^29 - 94820 * q^31 - 32768 * q^32 + 127296 * q^34 - 201552 * q^35 + 907942 * q^37 - 179368 * q^38 + 159744 * q^40 + 627072 * q^41 + 42472 * q^43 - 476160 * q^44 - 924288 * q^46 - 1235256 * q^47 + 719214 * q^49 - 153752 * q^50 + 907456 * q^52 - 214560 * q^53 + 2321280 * q^55 - 165376 * q^56 + 1333248 * q^58 - 2479224 * q^59 - 2874383 * q^61 + 1517120 * q^62 + 524288 * q^64 - 4423848 * q^65 - 1501097 * q^67 + 1018368 * q^68 + 806208 * q^70 - 9466272 * q^71 - 170222 * q^73 - 3631768 * q^74 - 1434944 * q^76 + 1201560 * q^77 + 1180819 * q^79 - 2555904 * q^80 - 10033152 * q^82 - 1116528 * q^83 - 4964544 * q^85 + 339776 * q^86 + 1904640 * q^88 - 18736272 * q^89 - 9159634 * q^91 + 3697152 * q^92 - 9882048 * q^94 + 6995352 * q^95 + 2039995 * q^97 - 11507424 * q^98

Character values

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

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

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
$5$	156.000	−	270.200i	0.558123	−	0.966697i	−0.439531	−	0.898228i	$-0.644855\pi$
$5$	156.000	−	270.200i	0.558123	−	0.966697i	0.997653	−	0.0684691i	$-0.0218114\pi$
$6$		0			0
$7$	−161.500	−	279.726i	−0.177963	−	0.308241i	0.763220	−	0.646139i	$-0.223617\pi$
$7$	−161.500	−	279.726i	−0.177963	−	0.308241i	−0.941183	+	0.337898i	$0.890284\pi$
$8$	512.000			0.353553
$9$		0			0
$10$	−2496.00			−0.789305
$11$	1860.00	+	3221.61i	0.421346	+	0.729792i	0.996071	−	0.0885540i	$-0.0282246\pi$
$11$	1860.00	+	3221.61i	0.421346	+	0.729792i	−0.574726	+	0.818346i	$0.694891\pi$
$12$		0			0
$13$	7089.50	−	12279.4i	0.894981	−	1.55015i	0.0611531	−	0.998128i	$-0.480522\pi$
$13$	7089.50	−	12279.4i	0.894981	−	1.55015i	0.833828	−	0.552024i	$-0.186144\pi$
$14$	−1292.00	+	2237.81i	−0.125839	+	0.217959i
$15$		0			0
$16$	−2048.00	−	3547.24i	−0.125000	−	0.216506i
$17$	−15912.0			−0.785513			−0.392757	−	0.919642i	$-0.628479\pi$
$17$	−15912.0			−0.785513			−0.392757	+	0.919642i	$0.628479\pi$
$18$		0			0
$19$	22421.0			0.749924			0.374962	−	0.927040i	$-0.377656\pi$
$19$	22421.0			0.749924			0.374962	+	0.927040i	$0.377656\pi$
$20$	9984.00	+	17292.8i	0.279061	+	0.483348i
$21$		0			0
$22$	14880.0	−	25772.9i	0.297936	−	0.516041i
$23$	28884.0	−	50028.6i	0.495005	−	0.857374i	−0.504978	−	0.863132i	$-0.668499\pi$
$23$	28884.0	−	50028.6i	0.495005	−	0.857374i	0.999983	+	0.00575773i	$0.00183275\pi$
$24$		0			0
$25$	−9609.50	−	16644.1i	−0.123002	−	0.213045i
$26$	−113432.			−1.26569
$27$		0			0
$28$	20672.0			0.177963
$29$	83328.0	+	144328.i	0.634451	+	1.09890i	0.986631	+	0.162969i	$0.0521070\pi$
$29$	83328.0	+	144328.i	0.634451	+	1.09890i	−0.352180	+	0.935932i	$0.614560\pi$
$30$		0			0
$31$	−47410.0	+	82116.5i	−0.285828	+	0.495068i	−0.972810	−	0.231607i	$-0.925602\pi$
$31$	−47410.0	+	82116.5i	−0.285828	+	0.495068i	0.686982	+	0.726674i	$0.258935\pi$
$32$	−16384.0	+	28377.9i	−0.0883883	+	0.153093i
$33$		0			0
$34$	63648.0	+	110242.i	0.277721	+	0.481027i
$35$	−100776.			−0.397300
$36$		0			0
$37$	453971.			1.47340			0.736702	−	0.676217i	$-0.236382\pi$
$37$	453971.			1.47340			0.736702	+	0.676217i	$0.236382\pi$
$38$	−89684.0	−	155337.i	−0.265138	−	0.459233i
$39$		0			0
$40$	79872.0	−	138342.i	0.197326	−	0.341779i
$41$	313536.	−	543060.i	0.710467	−	1.23056i	−0.254215	−	0.967148i	$-0.581817\pi$
$41$	313536.	−	543060.i	0.710467	−	1.23056i	0.964682	−	0.263417i	$-0.0848496\pi$
$42$		0			0
$43$	21236.0	+	36781.8i	0.0407318	+	0.0705495i	0.885673	−	0.464310i	$-0.153698\pi$
$43$	21236.0	+	36781.8i	0.0407318	+	0.0705495i	−0.844941	+	0.534860i	$0.820364\pi$
$44$	−238080.			−0.421346
$45$		0			0
$46$	−462144.			−0.700043
$47$	−617628.	−	1.06976e6i	−0.867730	−	1.50295i	−0.864311	−	0.502958i	$-0.832245\pi$
$47$	−617628.	−	1.06976e6i	−0.867730	−	1.50295i	−0.00341876	−	0.999994i	$-0.501088\pi$
$48$		0			0
$49$	359607.	−	622858.i	0.436658	−	0.756315i
$50$	−76876.0	+	133153.i	−0.0869753	+	0.150646i
$51$		0			0
$52$	453728.	+	785880.i	0.447491	+	0.775076i
$53$	−107280.			−0.0989813			−0.0494907	−	0.998775i	$-0.515760\pi$
$53$	−107280.			−0.0989813			−0.0494907	+	0.998775i	$0.515760\pi$
$54$		0			0
$55$	1.16064e6			0.940650
$56$	−82688.0	−	143220.i	−0.0629194	−	0.108980i
$57$		0			0
$58$	666624.	−	1.15463e6i	0.448624	−	0.777040i
$59$	−1.23961e6	+	2.14707e6i	−0.785785	+	1.36102i	0.142744	+	0.989760i	$0.454407\pi$
$59$	−1.23961e6	+	2.14707e6i	−0.785785	+	1.36102i	−0.928529	+	0.371260i	$0.878926\pi$
$60$		0			0
$61$	−1.43719e6	−	2.48929e6i	−0.810700	−	1.40417i	−0.912375	−	0.409356i	$-0.865753\pi$
$61$	−1.43719e6	−	2.48929e6i	−0.810700	−	1.40417i	0.101675	−	0.994818i	$-0.467580\pi$
$62$	758560.			0.404221
$63$		0			0
$64$	262144.			0.125000
$65$	−2.21192e6	−	3.83116e6i	−0.999018	−	1.73035i
$66$		0			0
$67$	−750548.	+	1.29999e6i	−0.304872	+	0.528053i	−0.977233	−	0.212170i	$-0.931947\pi$
$67$	−750548.	+	1.29999e6i	−0.304872	+	0.528053i	0.672361	+	0.740223i	$0.265280\pi$
$68$	509184.	−	881933.i	0.196378	−	0.340137i
$69$		0			0
$70$	403104.	+	698197.i	0.140467	+	0.243296i
$71$	−4.73314e6			−1.56944			−0.784720	−	0.619850i	$-0.787193\pi$
$71$	−4.73314e6			−1.56944			−0.784720	+	0.619850i	$0.787193\pi$
$72$		0			0
$73$	−85111.0			−0.0256068			−0.0128034	−	0.999918i	$-0.504076\pi$
$73$	−85111.0			−0.0256068			−0.0128034	+	0.999918i	$0.504076\pi$
$74$	−1.81588e6	−	3.14520e6i	−0.520927	−	0.902272i
$75$		0			0
$76$	−717472.	+	1.24270e6i	−0.187481	+	0.324727i
$77$	600780.	−	1.04058e6i	0.149968	−	0.259752i
$78$		0			0
$79$	590410.	+	1.02262e6i	0.134728	+	0.233356i	0.925494	−	0.378763i	$-0.123651\pi$
$79$	590410.	+	1.02262e6i	0.134728	+	0.233356i	−0.790765	+	0.612119i	$0.790317\pi$
$80$	−1.27795e6			−0.279061
$81$		0			0
$82$	−5.01658e6			−1.00475
$83$	−558264.	−	966942.i	−0.107168	−	0.185621i	0.807454	−	0.589931i	$-0.200845\pi$
$83$	−558264.	−	966942.i	−0.107168	−	0.185621i	−0.914622	+	0.404310i	$0.867512\pi$
$84$		0			0
$85$	−2.48227e6	+	4.29942e6i	−0.438413	+	0.759353i
$86$	169888.	−	294255.i	0.0288017	−	0.0498860i
$87$		0			0
$88$	952320.	+	1.64947e6i	0.148968	+	0.258020i
$89$	−9.36814e6			−1.40860			−0.704301	−	0.709902i	$-0.748739\pi$
$89$	−9.36814e6			−1.40860			−0.704301	+	0.709902i	$0.748739\pi$
$90$		0			0
$91$	−4.57982e6			−0.637094
$92$	1.84858e6	+	3.20183e6i	0.247503	+	0.428687i
$93$		0			0
$94$	−4.94102e6	+	8.55810e6i	−0.613578	+	1.06275i
$95$	3.49768e6	−	6.05815e6i	0.418550	−	0.724949i
$96$		0			0
$97$	1.02000e6	+	1.76669e6i	0.113474	+	0.196543i	0.917169	−	0.398499i	$-0.130469\pi$
$97$	1.02000e6	+	1.76669e6i	0.113474	+	0.196543i	−0.803694	+	0.595042i	$0.797135\pi$
$98$	−5.75371e6			−0.617528
$99$		0			0
$100$	1.23002e6			0.123002
$101$	−7.62874e6	−	1.32134e7i	−0.736763	−	1.27611i	−0.953946	−	0.299980i	$-0.903020\pi$
$101$	−7.62874e6	−	1.32134e7i	−0.736763	−	1.27611i	0.217183	−	0.976131i	$-0.430313\pi$
$102$		0			0
$103$	9.62167e6	−	1.66652e7i	0.867601	−	1.50273i	0.00316080	−	0.999995i	$-0.498994\pi$
$103$	9.62167e6	−	1.66652e7i	0.867601	−	1.50273i	0.864441	−	0.502735i	$-0.167673\pi$
$104$	3.62982e6	−	6.28704e6i	0.316424	−	0.548062i
$105$		0			0
$106$	429120.	+	743258.i	0.0349952	+	0.0606134i
$107$	−1.28571e7			−1.01461			−0.507306	−	0.861766i	$-0.669359\pi$
$107$	−1.28571e7			−1.01461			−0.507306	+	0.861766i	$0.669359\pi$
$108$		0			0
$109$	−1.02835e7			−0.760589			−0.380294	−	0.924865i	$-0.624177\pi$
$109$	−1.02835e7			−0.760589			−0.380294	+	0.924865i	$0.624177\pi$
$110$	−4.64256e6	−	8.04115e6i	−0.332570	−	0.576028i
$111$		0			0
$112$	−661504.	+	1.14576e6i	−0.0444907	+	0.0770602i
$113$	9.33863e6	−	1.61750e7i	0.608847	−	1.05455i	−0.382583	−	0.923921i	$-0.624965\pi$
$113$	9.33863e6	−	1.61750e7i	0.608847	−	1.05455i	0.991431	−	0.130634i	$-0.0417012\pi$
$114$		0			0
$115$	−9.01181e6	−	1.56089e7i	−0.552547	−	0.957040i
$116$	−1.06660e7			−0.634451
$117$		0			0
$118$	1.98338e7			1.11127
$119$	2.56979e6	+	4.45100e6i	0.139792	+	0.242127i
$120$		0			0
$121$	2.82439e6	−	4.89198e6i	0.144936	−	0.251036i
$122$	−1.14975e7	+	1.99143e7i	−0.573252	+	0.992901i
$123$		0			0
$124$	−3.03424e6	−	5.25546e6i	−0.142914	−	0.247534i
$125$	1.83787e7			0.841645
$126$		0			0
$127$	3.53659e6			0.153204			0.0766022	−	0.997062i	$-0.475593\pi$
$127$	3.53659e6			0.153204			0.0766022	+	0.997062i	$0.475593\pi$
$128$	−1.04858e6	−	1.81619e6i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	−1.76954e7	+	3.06493e7i	−0.706413	+	1.22354i
$131$	−1.50503e7	+	2.60678e7i	−0.584917	+	1.01311i	0.409969	+	0.912100i	$0.365540\pi$
$131$	−1.50503e7	+	2.60678e7i	−0.584917	+	1.01311i	−0.994886	+	0.101007i	$0.967794\pi$
$132$		0			0
$133$	−3.62099e6	−	6.27174e6i	−0.133459	−	0.231157i
$134$	1.20088e7			0.431154
$135$		0			0
$136$	−8.14694e6			−0.277721
$137$	−3.68663e6	−	6.38543e6i	−0.122492	−	0.212162i	0.798258	−	0.602316i	$-0.205755\pi$
$137$	−3.68663e6	−	6.38543e6i	−0.122492	−	0.212162i	−0.920750	+	0.390154i	$0.872422\pi$
$138$		0			0
$139$	2.96261e7	−	5.13139e7i	0.935670	−	1.62063i	0.162235	−	0.986752i	$-0.448130\pi$
$139$	2.96261e7	−	5.13139e7i	0.935670	−	1.62063i	0.773435	−	0.633876i	$-0.218537\pi$
$140$	3.22483e6	−	5.58557e6i	0.0993251	−	0.172036i
$141$		0			0
$142$	1.89325e7	+	3.27921e7i	0.554881	+	0.961082i
$143$	5.27459e7			1.50839
$144$		0			0
$145$	5.19967e7			1.41641
$146$	340444.	+	589666.i	0.00905338	+	0.0156809i
$147$		0			0
$148$	−1.45271e7	+	2.51616e7i	−0.368351	+	0.638003i
$149$	−1.16316e7	+	2.01466e7i	−0.288064	+	0.498941i	−0.973347	−	0.229335i	$-0.926345\pi$
$149$	−1.16316e7	+	2.01466e7i	−0.288064	+	0.498941i	0.685284	+	0.728276i	$0.259678\pi$
$150$		0			0
$151$	2.91230e7	+	5.04425e7i	0.688361	+	1.19228i	0.972368	+	0.233455i	$0.0750030\pi$
$151$	2.91230e7	+	5.04425e7i	0.688361	+	1.19228i	−0.284006	+	0.958822i	$0.591664\pi$
$152$	1.14796e7			0.265138
$153$		0			0
$154$	−9.61248e6			−0.212086
$155$	1.47919e7	+	2.56204e7i	0.319054	+	0.552617i
$156$		0			0
$157$	1.46788e7	−	2.54244e7i	0.302721	−	0.524328i	−0.674031	−	0.738703i	$-0.735438\pi$
$157$	1.46788e7	−	2.54244e7i	0.302721	−	0.524328i	0.976751	+	0.214376i	$0.0687717\pi$
$158$	4.72328e6	−	8.18095e6i	0.0952672	−	0.165008i
$159$		0			0
$160$	5.11181e6	+	8.85391e6i	0.0986631	+	0.170889i
$161$	−1.86591e7			−0.352370
$162$		0			0
$163$	−3.50196e7			−0.633366			−0.316683	−	0.948531i	$-0.602569\pi$
$163$	−3.50196e7			−0.633366			−0.316683	+	0.948531i	$0.602569\pi$
$164$	2.00663e7	+	3.47559e7i	0.355234	+	0.615282i
$165$		0			0
$166$	−4.46611e6	+	7.73553e6i	−0.0757794	+	0.131254i
$167$	−3.02274e7	+	5.23554e7i	−0.502219	+	0.869868i	0.497778	+	0.867305i	$0.334149\pi$
$167$	−3.02274e7	+	5.23554e7i	−0.502219	+	0.869868i	−0.999997	+	0.00256395i	$0.999184\pi$
$168$		0			0
$169$	−6.91478e7	−	1.19767e8i	−1.10198	−	1.90869i
$170$	3.97164e7			0.620009
$171$		0			0
$172$	−2.71821e6			−0.0407318
$173$	3.21740e7	+	5.57270e7i	0.472437	+	0.818285i	0.999502	−	0.0315398i	$-0.0100411\pi$
$173$	3.21740e7	+	5.57270e7i	0.472437	+	0.818285i	−0.527065	+	0.849825i	$0.676708\pi$
$174$		0			0
$175$	−3.10387e6	+	5.37606e6i	−0.0437794	+	0.0758282i
$176$	7.61856e6	−	1.31957e7i	0.105336	−	0.182448i
$177$		0			0
$178$	3.74725e7	+	6.49044e7i	0.498016	+	0.862589i
$179$	−1.05862e7			−0.137960			−0.0689799	−	0.997618i	$-0.521974\pi$
$179$	−1.05862e7			−0.137960			−0.0689799	+	0.997618i	$0.521974\pi$
$180$		0			0
$181$	6.41578e7			0.804219			0.402109	−	0.915592i	$-0.368277\pi$
$181$	6.41578e7			0.804219			0.402109	+	0.915592i	$0.368277\pi$
$182$	1.83193e7	+	3.17299e7i	0.225247	+	0.390139i
$183$		0			0
$184$	1.47886e7	−	2.56146e7i	0.175011	−	0.303128i
$185$	7.08195e7	−	1.22663e8i	0.822340	−	1.42434i
$186$		0			0
$187$	−2.95963e7	−	5.12623e7i	−0.330973	−	0.573261i
$188$	7.90564e7			0.867730
$189$		0			0
$190$	−5.59628e7			−0.591919
$191$	7.81664e7	+	1.35388e8i	0.811715	+	1.40593i	0.911663	+	0.410939i	$0.134799\pi$
$191$	7.81664e7	+	1.35388e8i	0.811715	+	1.40593i	−0.0999482	+	0.994993i	$0.531868\pi$
$192$		0			0
$193$	1.76630e7	−	3.05931e7i	0.176853	−	0.306319i	−0.763948	−	0.645278i	$-0.776742\pi$
$193$	1.76630e7	−	3.05931e7i	0.176853	−	0.306319i	0.940801	+	0.338959i	$0.110075\pi$
$194$	8.15998e6	−	1.41335e7i	0.0802385	−	0.138977i
$195$		0			0
$196$	2.30148e7	+	3.98629e7i	0.218329	+	0.378157i
$197$	1.06306e8			0.990666			0.495333	−	0.868703i	$-0.335046\pi$
$197$	1.06306e8			0.990666			0.495333	+	0.868703i	$0.335046\pi$
$198$		0			0
$199$	1.59628e7			0.143590			0.0717949	−	0.997419i	$-0.477127\pi$
$199$	1.59628e7			0.143590			0.0717949	+	0.997419i	$0.477127\pi$
$200$	−4.92006e6	−	8.52180e6i	−0.0434876	−	0.0753228i
$201$		0			0
$202$	−6.10299e7	+	1.05707e8i	−0.520970	+	0.902347i
$203$	2.69149e7	−	4.66181e7i	0.225817	−	0.391127i
$204$		0			0
$205$	−9.78232e7	−	1.69435e8i	−0.793055	−	1.37361i
$206$	−1.53947e8			−1.22697
$207$		0			0
$208$	−5.80772e7			−0.447491
$209$	4.17031e7	+	7.22318e7i	0.315977	+	0.547289i
$210$		0			0
$211$	1.32890e7	−	2.30172e7i	0.0973876	−	0.168680i	−0.813215	−	0.581963i	$-0.802285\pi$
$211$	1.32890e7	−	2.30172e7i	0.0973876	−	0.168680i	0.910603	+	0.413283i	$0.135618\pi$
$212$	3.43296e6	−	5.94606e6i	0.0247453	−	0.0428602i
$213$		0			0
$214$	5.14284e7	+	8.90767e7i	0.358720	+	0.621321i
$215$	1.32513e7			0.0909332
$216$		0			0
$217$	3.06269e7			0.203467
$218$	4.11342e7	+	7.12465e7i	0.268909	+	0.465764i
$219$		0			0
$220$	−3.71405e7	+	6.43292e7i	−0.235163	+	0.407313i
$221$	−1.12808e8	+	1.95389e8i	−0.703020	+	1.21767i
$222$		0			0
$223$	8.56114e7	+	1.48283e8i	0.516969	+	0.895416i	0.999806	+	0.0197061i	$0.00627304\pi$
$223$	8.56114e7	+	1.48283e8i	0.516969	+	0.895416i	−0.482837	+	0.875710i	$0.660394\pi$
$224$	1.05841e7			0.0629194
$225$		0			0
$226$	−1.49418e8			−0.861040
$227$	−2.16949e7	−	3.75766e7i	−0.123102	−	0.213220i	0.797887	−	0.602807i	$-0.205951\pi$
$227$	−2.16949e7	−	3.75766e7i	−0.123102	−	0.213220i	−0.920990	+	0.389587i	$0.872618\pi$
$228$		0			0
$229$	−1.32199e8	+	2.28975e8i	−0.727451	+	1.25998i	0.230506	+	0.973071i	$0.425962\pi$
$229$	−1.32199e8	+	2.28975e8i	−0.727451	+	1.25998i	−0.957957	+	0.286912i	$0.907371\pi$
$230$	−7.20945e7	+	1.24871e8i	−0.390710	+	0.676730i
$231$		0			0
$232$	4.26639e7	+	7.38961e7i	0.224312	+	0.388520i
$233$	1.69022e8			0.875380			0.437690	−	0.899126i	$-0.355797\pi$
$233$	1.69022e8			0.875380			0.437690	+	0.899126i	$0.355797\pi$
$234$		0			0
$235$	−3.85400e8			−1.93720
$236$	−7.93352e7	−	1.37413e8i	−0.392892	−	0.680510i
$237$		0			0
$238$	2.05583e7	−	3.56080e7i	0.0988480	−	0.171210i
$239$	3.00453e7	−	5.20400e7i	0.142359	−	0.246572i	−0.786026	−	0.618194i	$-0.787865\pi$
$239$	3.00453e7	−	5.20400e7i	0.142359	−	0.246572i	0.928384	+	0.371621i	$0.121198\pi$
$240$		0			0
$241$	2.42028e7	+	4.19205e7i	0.111380	+	0.192915i	0.916327	−	0.400431i	$-0.131140\pi$
$241$	2.42028e7	+	4.19205e7i	0.111380	+	0.192915i	−0.804947	+	0.593347i	$0.797806\pi$
$242$	−4.51902e7			−0.204970
$243$		0			0
$244$	1.83961e8			0.810700
$245$	−1.12197e8	−	1.94332e8i	−0.487418	−	0.844232i
$246$		0			0
$247$	1.58954e8	−	2.75316e8i	0.671168	−	1.16250i
$248$	−2.42739e7	+	4.20437e7i	−0.101055	+	0.175033i
$249$		0			0
$250$	−7.35147e7	−	1.27331e8i	−0.297567	−	0.515400i
$251$	−2.72664e7			−0.108835			−0.0544176	−	0.998518i	$-0.517330\pi$
$251$	−2.72664e7			−0.108835			−0.0544176	+	0.998518i	$0.517330\pi$
$252$		0			0
$253$	2.14897e8			0.834273
$254$	−1.41464e7	−	2.45022e7i	−0.0541660	−	0.0938182i
$255$		0			0
$256$	−8.38861e6	+	1.45295e7i	−0.0312500	+	0.0541266i
$257$	−1.34388e7	+	2.32767e7i	−0.0493850	+	0.0855374i	−0.889661	−	0.456621i	$-0.849059\pi$
$257$	−1.34388e7	+	2.32767e7i	−0.0493850	+	0.0855374i	0.840276	+	0.542159i	$0.182393\pi$
$258$		0			0
$259$	−7.33163e7	−	1.26988e8i	−0.262211	−	0.454163i
$260$	2.83126e8			0.999018
$261$		0			0
$262$	2.40804e8			0.827198
$263$	−2.01854e8	−	3.49621e8i	−0.684214	−	1.18509i	−0.973683	−	0.227907i	$-0.926812\pi$
$263$	−2.01854e8	−	3.49621e8i	−0.684214	−	1.18509i	0.289469	−	0.957188i	$-0.406521\pi$
$264$		0			0
$265$	−1.67357e7	+	2.89870e7i	−0.0552437	+	0.0956849i
$266$	−2.89679e7	+	5.01739e7i	−0.0943695	+	0.163453i
$267$		0			0
$268$	−4.80351e7	−	8.31992e7i	−0.152436	−	0.264027i
$269$	−3.44677e8			−1.07964			−0.539820	−	0.841780i	$-0.681508\pi$
$269$	−3.44677e8			−1.07964			−0.539820	+	0.841780i	$0.681508\pi$
$270$		0			0
$271$	3.41243e8			1.04153			0.520765	−	0.853700i	$-0.325647\pi$
$271$	3.41243e8			1.04153			0.520765	+	0.853700i	$0.325647\pi$
$272$	3.25878e7	+	5.64437e7i	0.0981892	+	0.170069i
$273$		0			0
$274$	−2.94930e7	+	5.10834e7i	−0.0866148	+	0.150021i
$275$	3.57473e7	−	6.19162e7i	0.103652	−	0.179531i
$276$		0			0
$277$	2.70810e7	+	4.69056e7i	0.0765570	+	0.132601i	0.901762	−	0.432232i	$-0.142274\pi$
$277$	2.70810e7	+	4.69056e7i	0.0765570	+	0.132601i	−0.825205	+	0.564833i	$0.808941\pi$
$278$	−4.74018e8			−1.32324
$279$		0			0
$280$	−5.15973e7			−0.140467
$281$	−2.74858e6	−	4.76067e6i	−0.00738985	−	0.0127996i	0.862307	−	0.506386i	$-0.169019\pi$
$281$	−2.74858e6	−	4.76067e6i	−0.00738985	−	0.0127996i	−0.869697	+	0.493587i	$0.835686\pi$
$282$		0			0
$283$	2.79386e8	−	4.83912e8i	0.732745	−	1.26915i	−0.222961	−	0.974827i	$-0.571572\pi$
$283$	2.79386e8	−	4.83912e8i	0.732745	−	1.26915i	0.955706	−	0.294324i	$-0.0950945\pi$
$284$	1.51460e8	−	2.62337e8i	0.392360	−	0.679588i
$285$		0			0
$286$	−2.10984e8	−	3.65434e8i	−0.533295	−	0.923694i
$287$	−2.02544e8			−0.505747
$288$		0			0
$289$	−1.57147e8			−0.382969
$290$	−2.07987e8	−	3.60244e8i	−0.500775	−	0.867368i
$291$		0			0
$292$	2.72355e6	−	4.71733e6i	0.00640171	−	0.0110881i
$293$	−6.23684e6	+	1.08025e7i	−0.0144853	+	0.0250893i	−0.873177	−	0.487403i	$-0.837944\pi$
$293$	−6.23684e6	+	1.08025e7i	−0.0144853	+	0.0250893i	0.858692	+	0.512492i	$0.171278\pi$
$294$		0			0
$295$	3.86759e8	+	6.69886e8i	0.877129	+	1.51923i
$296$	2.32433e8			0.520927
$297$		0			0
$298$	1.86106e8			0.407383
$299$	−4.09546e8	−	7.09355e8i	−0.886041	−	1.53467i
$300$		0			0
$301$	6.85923e6	−	1.18805e7i	0.0144975	−	0.0251104i
$302$	2.32984e8	−	4.03540e8i	0.486745	−	0.843067i
$303$		0			0
$304$	−4.59182e7	−	7.95327e7i	−0.0937405	−	0.162363i
$305$	−8.96807e8			−1.80988
$306$		0			0
$307$	−1.80343e7			−0.0355725			−0.0177863	−	0.999842i	$-0.505662\pi$
$307$	−1.80343e7			−0.0355725			−0.0177863	+	0.999842i	$0.505662\pi$
$308$	3.84499e7	+	6.65972e7i	0.0749839	+	0.129876i
$309$		0			0
$310$	1.18335e8	−	2.04963e8i	0.225605	−	0.390759i
$311$	2.97327e8	−	5.14986e8i	0.560497	−	0.970809i	−0.436956	−	0.899483i	$-0.643944\pi$
$311$	2.97327e8	−	5.14986e8i	0.560497	−	0.970809i	0.997453	−	0.0713260i	$-0.0227231\pi$
$312$		0			0
$313$	−1.86935e8	−	3.23782e8i	−0.344577	−	0.596825i	0.640700	−	0.767791i	$-0.278644\pi$
$313$	−1.86935e8	−	3.23782e8i	−0.344577	−	0.596825i	−0.985277	+	0.170967i	$0.945311\pi$
$314$	−2.34861e8			−0.428112
$315$		0			0
$316$	−7.55724e7			−0.134728
$317$	2.94682e8	+	5.10404e8i	0.519572	+	0.899925i	0.999741	+	0.0227491i	$0.00724188\pi$
$317$	2.94682e8	+	5.10404e8i	0.519572	+	0.899925i	−0.480169	+	0.877176i	$0.659425\pi$
$318$		0			0
$319$	−3.09980e8	+	5.36901e8i	−0.534646	+	0.926034i
$320$	4.08945e7	−	7.08313e7i	0.0697653	−	0.120837i
$321$		0			0
$322$	7.46363e7	+	1.29274e8i	0.124582	+	0.215782i
$323$	−3.56763e8			−0.589075
$324$		0			0
$325$	−2.72506e8			−0.440336
$326$	1.40079e8	+	2.42623e8i	0.223929	+	0.387856i
$327$		0			0
$328$	1.60530e8	−	2.78047e8i	0.251188	−	0.435070i
$329$	−1.99494e8	+	3.45533e8i	−0.308847	+	0.534939i
$330$		0			0
$331$	5.78287e8	+	1.00162e9i	0.876487	+	1.51812i	0.855170	+	0.518347i	$0.173453\pi$
$331$	5.78287e8	+	1.00162e9i	0.876487	+	1.51812i	0.0213168	+	0.999773i	$0.493214\pi$
$332$	7.14578e7			0.107168
$333$		0			0
$334$	4.83638e8			0.710245
$335$	2.34171e8	+	4.05596e8i	0.340311	+	0.589437i
$336$		0			0
$337$	−2.89409e8	+	5.01271e8i	−0.411915	+	0.713458i	−0.995099	−	0.0988824i	$-0.968473\pi$
$337$	−2.89409e8	+	5.01271e8i	−0.411915	+	0.713458i	0.583184	+	0.812340i	$0.301807\pi$
$338$	−5.53182e8	+	9.58139e8i	−0.779219	+	1.34965i
$339$		0			0
$340$	−1.58865e8	−	2.75163e8i	−0.219206	−	0.379677i
$341$	−3.52730e8			−0.481729
$342$		0			0
$343$	−4.98311e8			−0.666762
$344$	1.08728e7	+	1.88323e7i	0.0144009	+	0.0249430i
$345$		0			0
$346$	2.57392e8	−	4.45816e8i	0.334063	−	0.578615i
$347$	1.62500e8	−	2.81458e8i	0.208785	−	0.361627i	−0.742547	−	0.669794i	$-0.766382\pi$
$347$	1.62500e8	−	2.81458e8i	0.208785	−	0.361627i	0.951332	+	0.308167i	$0.0997157\pi$
$348$		0			0
$349$	−1.25383e8	−	2.17169e8i	−0.157888	−	0.273470i	0.776219	−	0.630463i	$-0.217135\pi$
$349$	−1.25383e8	−	2.17169e8i	−0.157888	−	0.273470i	−0.934107	+	0.356994i	$0.883802\pi$
$350$	4.96619e7			0.0619135
$351$		0			0
$352$	−1.21897e8			−0.148968
$353$	−5.20373e8	−	9.01312e8i	−0.629656	−	1.09060i	−0.987621	−	0.156861i	$-0.949863\pi$
$353$	−5.20373e8	−	9.01312e8i	−0.629656	−	1.09060i	0.357965	−	0.933735i	$-0.383471\pi$
$354$		0			0
$355$	−7.38369e8	+	1.27889e9i	−0.875940	+	1.51717i
$356$	2.99780e8	−	5.19235e8i	0.352150	−	0.609942i
$357$		0			0
$358$	4.23446e7	+	7.33431e7i	0.0487762	+	0.0844828i
$359$	1.02411e9			1.16820			0.584101	−	0.811681i	$-0.301447\pi$
$359$	1.02411e9			1.16820			0.584101	+	0.811681i	$0.301447\pi$
$360$		0			0
$361$	−3.91170e8			−0.437614
$362$	−2.56631e8	−	4.44498e8i	−0.284334	−	0.492481i
$363$		0			0
$364$	1.46554e8	−	2.53839e8i	0.159273	−	0.275870i
$365$	−1.32773e7	+	2.29970e7i	−0.0142917	+	0.0247540i
$366$		0			0
$367$	−4.99967e8	−	8.65967e8i	−0.527971	−	0.914472i	−0.999468	−	0.0326046i	$-0.989620\pi$
$367$	−4.99967e8	−	8.65967e8i	−0.527971	−	0.914472i	0.471498	−	0.881867i	$-0.343714\pi$
$368$	−2.36618e8			−0.247503
$369$		0			0
$370$	−1.13311e9			−1.16297
$371$	1.73257e7	+	3.00090e7i	0.0176150	+	0.0305101i
$372$		0			0
$373$	−9.62905e7	+	1.66780e8i	−0.0960732	+	0.166404i	−0.910056	−	0.414485i	$-0.863962\pi$
$373$	−9.62905e7	+	1.66780e8i	−0.0960732	+	0.166404i	0.813983	+	0.580889i	$0.197295\pi$
$374$	−2.36771e8	+	4.10099e8i	−0.234033	+	0.405357i
$375$		0			0
$376$	−3.16226e8	−	5.47719e8i	−0.306789	−	0.531374i
$377$	2.36302e9			2.27129
$378$		0			0
$379$	1.91512e9			1.80700			0.903501	−	0.428585i	$-0.140988\pi$
$379$	1.91512e9			1.80700			0.903501	+	0.428585i	$0.140988\pi$
$380$	2.23851e8	+	3.87722e8i	0.209275	+	0.362475i
$381$		0			0
$382$	6.25331e8	−	1.08311e9i	0.573969	−	0.994144i
$383$	−7.55549e8	+	1.30865e9i	−0.687175	+	1.19022i	0.285573	+	0.958357i	$0.407816\pi$
$383$	−7.55549e8	+	1.30865e9i	−0.687175	+	1.19022i	−0.972748	+	0.231865i	$0.925517\pi$
$384$		0			0
$385$	−1.87443e8	−	3.24661e8i	−0.167401	−	0.289947i
$386$	−2.82607e8			−0.250108
$387$		0			0
$388$	−1.30560e8			−0.113474
$389$	3.68584e8	+	6.38406e8i	0.317477	+	0.549887i	0.979961	−	0.199189i	$-0.0638309\pi$
$389$	3.68584e8	+	6.38406e8i	0.317477	+	0.549887i	−0.662484	+	0.749076i	$0.730498\pi$
$390$		0			0
$391$	−4.59602e8	+	7.96054e8i	−0.388833	+	0.673479i
$392$	1.84119e8	−	3.18903e8i	0.154382	−	0.267398i
$393$		0			0
$394$	−4.25225e8	−	7.36512e8i	−0.350253	−	0.606657i
$395$	3.68416e8			0.300779
$396$		0			0
$397$	−8.55916e8			−0.686538			−0.343269	−	0.939237i	$-0.611534\pi$
$397$	−8.55916e8			−0.686538			−0.343269	+	0.939237i	$0.611534\pi$
$398$	−6.38512e7	−	1.10594e8i	−0.0507666	−	0.0879304i
$399$		0			0
$400$	−3.93605e7	+	6.81744e7i	−0.0307504	+	0.0532613i
$401$	1.19476e8	−	2.06939e8i	0.0925288	−	0.160265i	−0.816046	−	0.577987i	$-0.803838\pi$
$401$	1.19476e8	−	2.06939e8i	0.0925288	−	0.160265i	0.908575	+	0.417723i	$0.137172\pi$
$402$		0			0
$403$	6.72226e8	+	1.16433e9i	0.511620	+	0.886153i
$404$	9.76478e8			0.736763
$405$		0			0
$406$	−4.30639e8			−0.319354
$407$	8.44386e8	+	1.46252e9i	0.620813	+	1.07528i
$408$		0			0
$409$	4.76770e8	−	8.25789e8i	0.344570	−	0.596812i	−0.640706	−	0.767786i	$-0.721358\pi$
$409$	4.76770e8	−	8.25789e8i	0.344570	−	0.596812i	0.985275	+	0.170974i	$0.0546915\pi$
$410$	−7.82586e8	+	1.35548e9i	−0.560775	+	0.971290i
$411$		0			0
$412$	6.15787e8	+	1.06657e9i	0.433801	+	0.751365i
$413$	8.00789e8			0.559362
$414$		0			0
$415$	−3.48357e8			−0.239252
$416$	2.32309e8	+	4.02371e8i	0.158212	+	0.274031i
$417$		0			0
$418$	3.33624e8	−	5.77855e8i	0.223430	−	0.386992i
$419$	2.72334e8	−	4.71696e8i	0.180864	−	0.313266i	−0.761311	−	0.648387i	$-0.775444\pi$
$419$	2.72334e8	−	4.71696e8i	0.180864	−	0.313266i	0.942175	+	0.335121i	$0.108777\pi$
$420$		0			0
$421$	3.30635e8	+	5.72677e8i	0.215954	+	0.374044i	0.953567	−	0.301180i	$-0.0973805\pi$
$421$	3.30635e8	+	5.72677e8i	0.215954	+	0.374044i	−0.737613	+	0.675224i	$0.764047\pi$
$422$	−2.12624e8			−0.137727
$423$		0			0
$424$	−5.49274e7			−0.0349952
$425$	1.52906e8	+	2.64842e8i	0.0966194	+	0.167350i
$426$		0			0
$427$	−4.64213e8	+	8.04040e8i	−0.288549	+	0.499782i
$428$	4.11428e8	−	7.12613e8i	0.253653	−	0.439340i
$429$		0			0
$430$	−5.30051e7	−	9.18075e7i	−0.0321498	−	0.0556850i
$431$	−2.10789e8			−0.126817			−0.0634086	−	0.997988i	$-0.520197\pi$
$431$	−2.10789e8			−0.126817			−0.0634086	+	0.997988i	$0.520197\pi$
$432$		0			0
$433$	−1.72888e9			−1.02343			−0.511714	−	0.859156i	$-0.670989\pi$
$433$	−1.72888e9			−1.02343			−0.511714	+	0.859156i	$0.670989\pi$
$434$	−1.22507e8	−	2.12189e8i	−0.0719364	−	0.124597i
$435$		0			0
$436$	3.29073e8	−	5.69972e8i	0.190147	−	0.329345i
$437$	6.47608e8	−	1.12169e9i	0.371217	−	0.642966i
$438$		0			0
$439$	−6.97147e7	−	1.20749e8i	−0.0393277	−	0.0681175i	0.845692	−	0.533672i	$-0.179188\pi$
$439$	−6.97147e7	−	1.20749e8i	−0.0393277	−	0.0681175i	−0.885019	+	0.465554i	$0.845855\pi$
$440$	5.94248e8			0.332570
$441$		0			0
$442$	1.80493e9			0.994220
$443$	7.00603e8	+	1.21348e9i	0.382877	+	0.663162i	0.991472	−	0.130319i	$-0.0416001\pi$
$443$	7.00603e8	+	1.21348e9i	0.382877	+	0.663162i	−0.608595	+	0.793481i	$0.708267\pi$
$444$		0			0
$445$	−1.46143e9	+	2.53127e9i	−0.786172	+	1.36169i
$446$	6.84891e8	−	1.18627e9i	0.365552	−	0.633155i
$447$		0			0
$448$	−4.23363e7	−	7.33285e7i	−0.0222454	−	0.0385301i
$449$	1.78420e7			0.00930209			0.00465104	−	0.999989i	$-0.498520\pi$
$449$	1.78420e7			0.00930209			0.00465104	+	0.999989i	$0.498520\pi$
$450$		0			0
$451$	2.33271e9			1.19741
$452$	5.97672e8	+	1.03520e9i	0.304424	+	0.527277i
$453$		0			0
$454$	−1.73559e8	+	3.00613e8i	−0.0870466	+	0.150769i
$455$	−7.14451e8	+	1.23747e9i	−0.355576	+	0.615876i
$456$		0			0
$457$	8.72066e8	+	1.51046e9i	0.427408	+	0.740292i	0.996642	−	0.0818833i	$-0.0260935\pi$
$457$	8.72066e8	+	1.51046e9i	0.427408	+	0.740292i	−0.569234	+	0.822176i	$0.692760\pi$
$458$	2.11518e9			1.02877
$459$		0			0
$460$	1.15351e9			0.552547
$461$	6.79956e8	+	1.17772e9i	0.323242	+	0.559871i	0.981155	−	0.193223i	$-0.0618940\pi$
$461$	6.79956e8	+	1.17772e9i	0.323242	+	0.559871i	−0.657913	+	0.753094i	$0.728561\pi$
$462$		0			0
$463$	9.88793e8	−	1.71264e9i	0.462991	−	0.801923i	−0.536118	−	0.844143i	$-0.680110\pi$
$463$	9.88793e8	−	1.71264e9i	0.462991	−	0.801923i	0.999108	+	0.0422200i	$0.0134431\pi$
$464$	3.41311e8	−	5.91169e8i	0.158613	−	0.274725i
$465$		0			0
$466$	−6.76087e8	−	1.17102e9i	−0.309494	−	0.536059i
$467$	−3.46852e9			−1.57592			−0.787961	−	0.615726i	$-0.788863\pi$
$467$	−3.46852e9			−1.57592			−0.787961	+	0.615726i	$0.788863\pi$
$468$		0			0
$469$	4.84854e8			0.217023
$470$	1.54160e9	+	2.67013e9i	0.684903	+	1.18629i
$471$		0			0
$472$	−6.34681e8	+	1.09930e9i	−0.277817	+	0.481193i
$473$	−7.89979e7	+	1.36828e8i	−0.0343243	+	0.0594514i
$474$		0			0
$475$	−2.15455e8	−	3.73178e8i	−0.0922419	−	0.159768i
$476$	−3.28933e8			−0.139792
$477$		0			0
$478$	−4.80725e8			−0.201325
$479$	1.09021e9	+	1.88830e9i	0.453247	+	0.785047i	0.998586	−	0.0531688i	$-0.0169321\pi$
$479$	1.09021e9	+	1.88830e9i	0.453247	+	0.785047i	−0.545338	+	0.838216i	$0.683599\pi$
$480$		0			0
$481$	3.21843e9	−	5.57448e9i	1.31867	−	2.28400i
$482$	1.93622e8	−	3.35364e8i	0.0787573	−	0.136412i
$483$		0			0
$484$	1.80761e8	+	3.13087e8i	0.0724678	+	0.125518i
$485$	6.36478e8			0.253331
$486$		0			0
$487$	3.00745e9			1.17990			0.589952	−	0.807439i	$-0.299147\pi$
$487$	3.00745e9			1.17990			0.589952	+	0.807439i	$0.299147\pi$
$488$	−7.35842e8	−	1.27452e9i	−0.286626	−	0.496450i
$489$		0			0
$490$	−8.97579e8	+	1.55465e9i	−0.344656	+	0.596963i
$491$	−1.31485e9	+	2.27739e9i	−0.501292	+	0.868263i	0.498707	+	0.866771i	$0.333808\pi$
$491$	−1.31485e9	+	2.27739e9i	−0.501292	+	0.868263i	−0.999999	+	0.00149237i	$0.999525\pi$
$492$		0			0
$493$	−1.32592e9	−	2.29655e9i	−0.498370	−	0.863201i
$494$	−2.54326e9			−0.949175
$495$		0			0
$496$	3.88383e8			0.142914
$497$	7.64401e8	+	1.32398e9i	0.279302	+	0.483765i
$498$		0			0
$499$	1.38495e8	−	2.39880e8i	0.0498978	−	0.0864256i	−0.839998	−	0.542590i	$-0.817444\pi$
$499$	1.38495e8	−	2.39880e8i	0.0498978	−	0.0864256i	0.889896	+	0.456164i	$0.150777\pi$
$500$	−5.88118e8	+	1.01865e9i	−0.210411	+	0.364443i
$501$		0			0
$502$	1.09066e8	+	1.88907e8i	0.0384791	+	0.0666477i
$503$	4.59216e9			1.60890			0.804451	−	0.594019i	$-0.202460\pi$
$503$	4.59216e9			1.60890			0.804451	+	0.594019i	$0.202460\pi$
$504$		0			0
$505$	−4.76033e9			−1.64482
$506$	−8.59588e8	−	1.48885e9i	−0.294960	−	0.510886i
$507$		0			0
$508$	−1.13171e8	+	1.96018e8i	−0.0383011	+	0.0663395i
$509$	1.62733e9	−	2.81862e9i	0.546970	−	0.947379i	−0.451510	−	0.892266i	$-0.649115\pi$
$509$	1.62733e9	−	2.81862e9i	0.546970	−	0.947379i	0.998480	−	0.0551135i	$-0.0175521\pi$
$510$		0			0
$511$	1.37454e7	+	2.38078e7i	0.00455706	+	0.00789307i
$512$	1.34218e8			0.0441942
$513$		0			0
$514$	2.15021e8			0.0698410
$515$	−3.00196e9	−	5.19955e9i	−0.968456	−	1.67741i
$516$		0			0
$517$	2.29758e9	−	3.97952e9i	0.731228	−	1.26652i
$518$	−5.86531e8	+	1.01590e9i	−0.185411	+	0.321142i
$519$		0			0
$520$	−1.13251e9	−	1.96156e9i	−0.353206	−	0.611771i
$521$	−5.57306e9			−1.72648			−0.863241	−	0.504793i	$-0.831569\pi$
$521$	−5.57306e9			−1.72648			−0.863241	+	0.504793i	$0.831569\pi$
$522$		0			0
$523$	2.33783e9			0.714590			0.357295	−	0.933992i	$-0.383699\pi$
$523$	2.33783e9			0.714590			0.357295	+	0.933992i	$0.383699\pi$
$524$	−9.63216e8	−	1.66834e9i	−0.292459	−	0.506553i
$525$		0			0
$526$	−1.61483e9	+	2.79697e9i	−0.483813	+	0.837988i
$527$	7.54388e8	−	1.30664e9i	0.224521	−	0.388882i
$528$		0			0
$529$	3.38418e7	+	5.86157e7i	0.00993937	+	0.0172155i
$530$	2.67771e8			0.0781264
$531$		0			0
$532$	4.63487e8			0.133459
$533$	−4.44563e9	−	7.70005e9i	−1.27171	−	2.20266i
$534$		0			0
$535$	−2.00571e9	+	3.47399e9i	−0.566278	+	0.980822i
$536$	−3.84281e8	+	6.65594e8i	−0.107788	+	0.186695i
$537$		0			0
$538$	1.37871e9	+	2.38799e9i	0.381710	+	0.661142i
$539$	2.67548e9			0.735937
$540$		0			0
$541$	−5.60874e9			−1.52291			−0.761456	−	0.648217i	$-0.775515\pi$
$541$	−5.60874e9			−1.52291			−0.761456	+	0.648217i	$0.775515\pi$
$542$	−1.36497e9	−	2.36420e9i	−0.368236	−	0.637804i
$543$		0			0
$544$	2.60702e8	−	4.51549e8i	0.0694302	−	0.120257i
$545$	−1.60423e9	+	2.77861e9i	−0.424502	+	0.735259i
$546$		0			0
$547$	2.25517e9	+	3.90606e9i	0.589146	+	1.02043i	0.994345	+	0.106202i	$0.0338690\pi$
$547$	2.25517e9	+	3.90606e9i	0.589146	+	1.02043i	−0.405199	+	0.914229i	$0.632798\pi$
$548$	4.71888e8			0.122492
$549$		0			0
$550$	−5.71957e8			−0.146587
$551$	1.86830e9	+	3.23599e9i	0.475790	+	0.824093i
$552$		0			0
$553$	1.90702e8	−	3.30306e8i	0.0479532	−	0.0830574i
$554$	2.16648e8	−	3.75245e8i	0.0541340	−	0.0937628i
$555$		0			0
$556$	1.89607e9	+	3.28409e9i	0.467835	+	0.810314i
$557$	−3.30015e8			−0.0809171			−0.0404585	−	0.999181i	$-0.512882\pi$
$557$	−3.30015e8			−0.0809171			−0.0404585	+	0.999181i	$0.512882\pi$
$558$		0			0
$559$	6.02210e8			0.145817
$560$	2.06389e8	+	3.57477e8i	0.0496625	+	0.0860181i
$561$		0			0
$562$	−2.19886e7	+	3.80854e7i	−0.00522541	+	0.00905068i
$563$	1.28786e9	−	2.23064e9i	0.304151	−	0.526805i	−0.672921	−	0.739715i	$-0.734961\pi$
$563$	1.28786e9	−	2.23064e9i	0.304151	−	0.526805i	0.977072	+	0.212909i	$0.0682938\pi$
$564$		0			0
$565$	−2.91365e9	−	5.04659e9i	−0.679623	−	1.17714i
$566$	−4.47018e9			−1.03626
$567$		0			0
$568$	−2.42337e9			−0.554881
$569$	−1.11446e9	−	1.93030e9i	−0.253612	−	0.439269i	0.710905	−	0.703288i	$-0.248285\pi$
$569$	−1.11446e9	−	1.93030e9i	−0.253612	−	0.439269i	−0.964518	+	0.264018i	$0.914952\pi$
$570$		0			0
$571$	−1.41716e9	+	2.45460e9i	−0.318562	+	0.551765i	−0.980188	−	0.198069i	$-0.936533\pi$
$571$	−1.41716e9	+	2.45460e9i	−0.318562	+	0.551765i	0.661626	+	0.749834i	$0.269866\pi$
$572$	−1.68787e9	+	2.92347e9i	−0.377096	+	0.653150i
$573$		0			0
$574$	8.10177e8	+	1.40327e9i	0.178809	+	0.309706i
$575$	−1.11024e9			−0.243546
$576$		0			0
$577$	1.24427e7			0.00269649			0.00134825	−	0.999999i	$-0.499571\pi$
$577$	1.24427e7			0.00269649			0.00134825	+	0.999999i	$0.499571\pi$
$578$	6.28588e8	+	1.08875e9i	0.135400	+	0.234520i
$579$		0			0
$580$	−1.66389e9	+	2.88195e9i	−0.354101	+	0.613321i
$581$	−1.80319e8	+	3.12322e8i	−0.0381440	+	0.0660673i
$582$		0			0
$583$	−1.99541e8	−	3.45615e8i	−0.0417053	−	0.0722358i
$584$	−4.35768e7			−0.00905338
$585$		0			0
$586$	9.97895e7			0.0204853
$587$	−4.30705e9	−	7.46003e9i	−0.878914	−	1.52232i	−0.852534	−	0.522672i	$-0.824935\pi$
$587$	−4.30705e9	−	7.46003e9i	−0.878914	−	1.52232i	−0.0263805	−	0.999652i	$-0.508398\pi$
$588$		0			0
$589$	−1.06298e9	+	1.84113e9i	−0.214349	+	0.371263i
$590$	3.09407e9	−	5.35909e9i	0.620224	−	1.07426i
$591$		0			0
$592$	−9.29733e8	−	1.61034e9i	−0.184176	−	0.319001i
$593$	4.55490e9			0.896989			0.448495	−	0.893786i	$-0.351960\pi$
$593$	4.55490e9			0.896989			0.448495	+	0.893786i	$0.351960\pi$
$594$		0			0
$595$	1.60355e9			0.312085
$596$	−7.44424e8	−	1.28938e9i	−0.144032	−	0.249470i
$597$		0			0
$598$	−3.27637e9	+	5.67484e9i	−0.626526	+	1.08517i
$599$	3.28583e8	−	5.69123e8i	0.0624672	−	0.108196i	−0.833100	−	0.553122i	$-0.813436\pi$
$599$	3.28583e8	−	5.69123e8i	0.0624672	−	0.108196i	0.895568	+	0.444925i	$0.146770\pi$
$600$		0			0
$601$	−4.19703e9	−	7.26947e9i	−0.788645	−	1.36597i	−0.926797	−	0.375562i	$-0.877450\pi$
$601$	−4.19703e9	−	7.26947e9i	−0.788645	−	1.36597i	0.138152	−	0.990411i	$-0.455884\pi$
$602$	−1.09748e8			−0.0205025
$603$		0			0
$604$	−3.72774e9			−0.688361
$605$	−8.81208e8	−	1.52630e9i	−0.161784	−	0.280218i
$606$		0			0
$607$	5.24763e8	−	9.08915e8i	0.0952363	−	0.164954i	−0.814471	−	0.580204i	$-0.802973\pi$
$607$	5.24763e8	−	9.08915e8i	0.0952363	−	0.164954i	0.909707	+	0.415250i	$0.136306\pi$
$608$	−3.67346e8	+	6.36261e8i	−0.0662846	+	0.114808i
$609$		0			0
$610$	3.58723e9	+	6.21326e9i	0.639889	+	1.10832i
$611$	−1.75147e10			−3.10641
$612$		0			0
$613$	4.22848e9			0.741433			0.370717	−	0.928746i	$-0.379112\pi$
$613$	4.22848e9			0.741433			0.370717	+	0.928746i	$0.379112\pi$
$614$	7.21372e7	+	1.24945e8i	0.0125768	+	0.0217836i
$615$		0			0
$616$	3.07599e8	−	5.32778e8i	0.0530216	−	0.0918361i
$617$	2.42730e9	−	4.20421e9i	0.416031	−	0.720586i	−0.579505	−	0.814968i	$-0.696754\pi$
$617$	2.42730e9	−	4.20421e9i	0.416031	−	0.720586i	0.995536	+	0.0943822i	$0.0300876\pi$
$618$		0			0
$619$	−1.16137e9	−	2.01156e9i	−0.196813	−	0.340891i	0.750680	−	0.660666i	$-0.229726\pi$
$619$	−1.16137e9	−	2.01156e9i	−0.196813	−	0.340891i	−0.947493	+	0.319775i	$0.896393\pi$
$620$	−1.89337e9			−0.319054
$621$		0			0
$622$	−4.75723e9			−0.792662
$623$	1.51295e9	+	2.62051e9i	0.250679	+	0.434188i
$624$		0			0
$625$	3.61782e9	−	6.26624e9i	0.592743	−	1.02666i
$626$	−1.49548e9	+	2.59025e9i	−0.243653	+	0.422019i
$627$		0			0
$628$	9.39444e8	+	1.62716e9i	0.151360	+	0.262164i
$629$	−7.22359e9			−1.15738
$630$		0			0
$631$	5.84987e9			0.926923			0.463461	−	0.886117i	$-0.346607\pi$
$631$	5.84987e9			0.926923			0.463461	+	0.886117i	$0.346607\pi$
$632$	3.02290e8	+	5.23581e8i	0.0476336	+	0.0825038i
$633$		0			0
$634$	2.35745e9	−	4.08323e9i	0.367393	−	0.636343i
$635$	5.51708e8	−	9.55586e8i	0.0855069	−	0.148102i
$636$		0			0
$637$	−5.09887e9	−	8.83150e9i	−0.781602	−	1.35377i
$638$	4.95968e9			0.756104
$639$		0			0
$640$	−6.54311e8			−0.0986631
$641$	4.90859e9	+	8.50193e9i	0.736129	+	1.27501i	0.954226	+	0.299086i	$0.0966817\pi$
$641$	4.90859e9	+	8.50193e9i	0.736129	+	1.27501i	−0.218097	+	0.975927i	$0.569985\pi$
$642$		0			0
$643$	2.36522e9	−	4.09668e9i	0.350859	−	0.607706i	−0.635541	−	0.772067i	$-0.719223\pi$
$643$	2.36522e9	−	4.09668e9i	0.350859	−	0.607706i	0.986400	+	0.164361i	$0.0525562\pi$
$644$	5.97090e8	−	1.03419e9i	0.0880926	−	0.152581i
$645$		0			0
$646$	1.42705e9	+	2.47173e9i	0.208270	+	0.360734i
$647$	4.28999e9			0.622718			0.311359	−	0.950292i	$-0.399216\pi$
$647$	4.28999e9			0.622718			0.311359	+	0.950292i	$0.399216\pi$
$648$		0			0
$649$	−9.22271e9			−1.32435
$650$	1.09002e9	+	1.88798e9i	0.155682	+	0.269650i
$651$		0			0
$652$	1.12063e9	−	1.94099e9i	0.158342	−	0.274256i
$653$	−6.60538e9	+	1.14409e10i	−0.928328	+	1.60791i	−0.142209	+	0.989837i	$0.545421\pi$
$653$	−6.60538e9	+	1.14409e10i	−0.928328	+	1.60791i	−0.786119	+	0.618075i	$0.787913\pi$
$654$		0			0
$655$	4.69568e9	+	8.13316e9i	0.652911	+	1.13087i
$656$	−2.56849e9			−0.355234
$657$		0			0
$658$	3.19190e9			0.436776
$659$	2.80273e9	+	4.85448e9i	0.381490	+	0.660760i	0.991275	−	0.131807i	$-0.0420778\pi$
$659$	2.80273e9	+	4.85448e9i	0.381490	+	0.660760i	−0.609786	+	0.792566i	$0.708744\pi$
$660$		0			0
$661$	8.85352e8	−	1.53347e9i	0.119237	−	0.206524i	−0.800229	−	0.599695i	$-0.795289\pi$
$661$	8.85352e8	−	1.53347e9i	0.119237	−	0.206524i	0.919465	+	0.393171i	$0.128622\pi$
$662$	4.62630e9	−	8.01298e9i	0.619770	−	1.07347i
$663$		0			0
$664$	−2.85831e8	−	4.95074e8i	−0.0378897	−	0.0656269i
$665$	−2.25950e9			−0.297945
$666$		0			0
$667$	9.62738e9			1.25623
$668$	−1.93455e9	−	3.35074e9i	−0.251109	−	0.434934i
$669$		0			0
$670$	1.87337e9	−	3.24477e9i	0.240637	−	0.416795i
$671$	5.34635e9	−	9.26015e9i	0.683170	−	1.18329i
$672$		0			0
$673$	3.68018e8	+	6.37426e8i	0.0465389	+	0.0806077i	0.888356	−	0.459154i	$-0.151848\pi$
$673$	3.68018e8	+	6.37426e8i	0.0465389	+	0.0806077i	−0.841818	+	0.539762i	$0.818514\pi$
$674$	4.63055e9			0.582536
$675$		0			0
$676$	8.85091e9			1.10198
$677$	3.11281e9	+	5.39154e9i	0.385560	+	0.667810i	0.991847	−	0.127437i	$-0.0406749\pi$
$677$	3.11281e9	+	5.39154e9i	0.385560	+	0.667810i	−0.606287	+	0.795246i	$0.707342\pi$
$678$		0			0
$679$	3.29459e8	−	5.70640e8i	0.0403885	−	0.0699549i
$680$	−1.27092e9	+	2.20130e9i	−0.155002	+	0.268472i
$681$		0			0
$682$	1.41092e9	+	2.44379e9i	0.170317	+	0.294997i
$683$	6.78088e9			0.814355			0.407178	−	0.913349i	$-0.366513\pi$
$683$	6.78088e9			0.814355			0.407178	+	0.913349i	$0.366513\pi$
$684$		0			0
$685$	−2.30046e9			−0.273462
$686$	1.99324e9	+	3.45240e9i	0.235736	+	0.408306i
$687$		0			0
$688$	8.69827e7	−	1.50658e8i	0.0101829	−	0.0176374i
$689$	−7.60562e8	+	1.31733e9i	−0.0885864	+	0.153436i
$690$		0			0
$691$	2.07498e9	+	3.59398e9i	0.239244	+	0.414383i	0.960498	−	0.278288i	$-0.0897670\pi$
$691$	2.07498e9	+	3.59398e9i	0.239244	+	0.414383i	−0.721253	+	0.692671i	$0.756434\pi$
$692$	−4.11827e9			−0.472437
$693$		0			0
$694$	−2.60000e9			−0.295267
$695$	−9.24334e9	−	1.60099e10i	−1.04444	−	1.80902i
$696$		0			0
$697$	−4.98898e9	+	8.64118e9i	−0.558081	+	0.966625i
$698$	−1.00306e9	+	1.73735e9i	−0.111644	+	0.193372i
$699$		0			0
$700$	−1.98648e8	−	3.44068e8i	−0.0218897	−	0.0379141i
$701$	1.61210e10			1.76758			0.883791	−	0.467881i	$-0.154983\pi$
$701$	1.61210e10			1.76758			0.883791	+	0.467881i	$0.154983\pi$
$702$		0			0
$703$	1.01785e10			1.10494
$704$	4.87588e8	+	8.44527e8i	0.0526682	+	0.0912240i
$705$		0			0
$706$	−4.16298e9	+	7.21050e9i	−0.445234	+	0.771168i
$707$	−2.46408e9	+	4.26791e9i	−0.262233	+	0.454201i
$708$		0			0
$709$	−5.66442e9	−	9.81106e9i	−0.596889	−	1.03384i	−0.993277	−	0.115759i	$-0.963070\pi$
$709$	−5.66442e9	−	9.81106e9i	−0.596889	−	1.03384i	0.396388	−	0.918083i	$-0.370263\pi$
$710$	1.18139e10			1.23877
$711$		0			0
$712$	−4.79649e9			−0.498016
$713$	2.73878e9	+	4.74371e9i	0.282972	+	0.490122i
$714$		0			0
$715$	8.22836e9	−	1.42519e10i	0.841864	−	1.45815i
$716$	3.38757e8	−	5.86745e8i	0.0344900	−	0.0597384i
$717$		0			0
$718$	−4.09646e9	−	7.09528e9i	−0.413022	−	0.715375i
$719$	8.97759e9			0.900760			0.450380	−	0.892837i	$-0.351289\pi$
$719$	8.97759e9			0.900760			0.450380	+	0.892837i	$0.351289\pi$
$720$		0			0
$721$	−6.21560e9			−0.617603
$722$	1.56468e9	+	2.71011e9i	0.154720	+	0.267983i
$723$		0			0
$724$	−2.05305e9	+	3.55598e9i	−0.201055	+	0.348237i
$725$	1.60148e9	−	2.77385e9i	0.156077	−	0.270333i
$726$		0			0
$727$	9.90246e9	+	1.71516e10i	0.955813	+	1.65552i	0.732497	+	0.680770i	$0.238355\pi$
$727$	9.90246e9	+	1.71516e10i	0.955813	+	1.65552i	0.223315	+	0.974746i	$0.428312\pi$
$728$	−2.34487e9			−0.225247
$729$		0			0
$730$	2.12437e8			0.0202116
$731$	−3.37907e8	−	5.85272e8i	−0.0319953	−	0.0554175i
$732$		0			0
$733$	−9.43023e9	+	1.63336e10i	−0.884420	+	1.53186i	−0.0380432	+	0.999276i	$0.512112\pi$
$733$	−9.43023e9	+	1.63336e10i	−0.884420	+	1.53186i	−0.846377	+	0.532584i	$0.821221\pi$
$734$	−3.99973e9	+	6.92774e9i	−0.373332	+	0.646629i
$735$		0			0
$736$	9.46471e8	+	1.63934e9i	0.0875054	+	0.151564i
$737$	−5.58408e9			−0.513825
$738$		0			0
$739$	−1.38397e10			−1.26145			−0.630725	−	0.776006i	$-0.717243\pi$
$739$	−1.38397e10			−1.26145			−0.630725	+	0.776006i	$0.717243\pi$
$740$	4.53245e9	+	7.85043e9i	0.411170	+	0.712168i
$741$		0			0
$742$	1.38606e8	−	2.40072e8i	0.0124557	−	0.0215739i
$743$	−9.04418e9	+	1.56650e10i	−0.808926	+	1.40110i	0.104683	+	0.994506i	$0.466617\pi$
$743$	−9.04418e9	+	1.56650e10i	−0.808926	+	1.40110i	−0.913609	+	0.406595i	$0.866716\pi$
$744$		0			0
$745$	3.62907e9	+	6.28573e9i	0.321550	+	0.556940i
$746$	1.54065e9			0.135868
$747$		0			0
$748$	3.78833e9			0.330973
$749$	2.07642e9	+	3.59647e9i	0.180563	+	0.312745i
$750$		0			0
$751$	1.11013e9	−	1.92281e9i	0.0956390	−	0.165652i	−0.814236	−	0.580534i	$-0.802844\pi$
$751$	1.11013e9	−	1.92281e9i	0.0956390	−	0.165652i	0.909875	+	0.414882i	$0.136177\pi$
$752$	−2.52980e9	+	4.38175e9i	−0.216932	+	0.375738i
$753$		0			0
$754$	−9.45206e9	−	1.63715e10i	−0.803021	−	1.39087i
$755$	1.81727e10			1.53676
$756$		0			0
$757$	2.21883e10			1.85904			0.929521	−	0.368770i	$-0.120221\pi$
$757$	2.21883e10			1.85904			0.929521	+	0.368770i	$0.120221\pi$
$758$	−7.66048e9	−	1.32683e10i	−0.638872	−	1.10656i
$759$		0			0
$760$	1.79081e9	−	3.10177e9i	0.147980	−	0.256308i
$761$	1.51911e9	−	2.63117e9i	0.124952	−	0.216423i	−0.796762	−	0.604293i	$-0.793456\pi$
$761$	1.51911e9	−	2.63117e9i	0.124952	−	0.216423i	0.921714	+	0.387870i	$0.126789\pi$
$762$		0			0
$763$	1.66079e9	+	2.87658e9i	0.135357	+	0.234444i
$764$	−1.00053e10			−0.811715
$765$		0			0
$766$	1.20888e10			0.971812
$767$	1.75765e10	+	3.04433e10i	1.40653	+	2.43617i
$768$		0			0
$769$	9.12045e9	−	1.57971e10i	0.723226	−	1.25266i	−0.236474	−	0.971638i	$-0.575992\pi$
$769$	9.12045e9	−	1.57971e10i	0.723226	−	1.25266i	0.959700	−	0.281027i	$-0.0906750\pi$
$770$	−1.49955e9	+	2.59729e9i	−0.118370	+	0.205023i
$771$		0			0
$772$	1.13043e9	+	1.95796e9i	0.0884266	+	0.153159i
$773$	1.20659e10			0.939576			0.469788	−	0.882779i	$-0.344330\pi$
$773$	1.20659e10			0.939576			0.469788	+	0.882779i	$0.344330\pi$

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)

\(f(q)\)	\(=\)	\(q+(-4.00000 - 6.92820i) q^{2} +(-32.0000 + 55.4256i) q^{4} +(156.000 - 270.200i) q^{5} +(-161.500 - 279.726i) q^{7} +512.000 q^{8} -2496.00 q^{10} +(1860.00 + 3221.61i) q^{11} +(7089.50 - 12279.4i) q^{13} +(-1292.00 + 2237.81i) q^{14} +(-2048.00 - 3547.24i) q^{16} -15912.0 q^{17} +22421.0 q^{19} +(9984.00 + 17292.8i) q^{20} +(14880.0 - 25772.9i) q^{22} +(28884.0 - 50028.6i) q^{23} +(-9609.50 - 16644.1i) q^{25} -113432. q^{26} +20672.0 q^{28} +(83328.0 + 144328. i) q^{29} +(-47410.0 + 82116.5i) q^{31} +(-16384.0 + 28377.9i) q^{32} +(63648.0 + 110242. i) q^{34} -100776. q^{35} +453971. q^{37} +(-89684.0 - 155337. i) q^{38} +(79872.0 - 138342. i) q^{40} +(313536. - 543060. i) q^{41} +(21236.0 + 36781.8i) q^{43} -238080. q^{44} -462144. q^{46} +(-617628. - 1.06976e6i) q^{47} +(359607. - 622858. i) q^{49} +(-76876.0 + 133153. i) q^{50} +(453728. + 785880. i) q^{52} -107280. q^{53} +1.16064e6 q^{55} +(-82688.0 - 143220. i) q^{56} +(666624. - 1.15463e6i) q^{58} +(-1.23961e6 + 2.14707e6i) q^{59} +(-1.43719e6 - 2.48929e6i) q^{61} +758560. q^{62} +262144. q^{64} +(-2.21192e6 - 3.83116e6i) q^{65} +(-750548. + 1.29999e6i) q^{67} +(509184. - 881933. i) q^{68} +(403104. + 698197. i) q^{70} -4.73314e6 q^{71} -85111.0 q^{73} +(-1.81588e6 - 3.14520e6i) q^{74} +(-717472. + 1.24270e6i) q^{76} +(600780. - 1.04058e6i) q^{77} +(590410. + 1.02262e6i) q^{79} -1.27795e6 q^{80} -5.01658e6 q^{82} +(-558264. - 966942. i) q^{83} +(-2.48227e6 + 4.29942e6i) q^{85} +(169888. - 294255. i) q^{86} +(952320. + 1.64947e6i) q^{88} -9.36814e6 q^{89} -4.57982e6 q^{91} +(1.84858e6 + 3.20183e6i) q^{92} +(-4.94102e6 + 8.55810e6i) q^{94} +(3.49768e6 - 6.05815e6i) q^{95} +(1.02000e6 + 1.76669e6i) q^{97} -5.75371e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{2} - 64 q^{4} + 312 q^{5} - 323 q^{7} + 1024 q^{8} - 4992 q^{10} + 3720 q^{11} + 14179 q^{13} - 2584 q^{14} - 4096 q^{16} - 31824 q^{17} + 44842 q^{19} + 19968 q^{20} + 29760 q^{22} + 57768 q^{23}+ \cdots - 11507424 q^{98}+O(q^{100}) \) 2 * q - 8 * q^2 - 64 * q^4 + 312 * q^5 - 323 * q^7 + 1024 * q^8 - 4992 * q^10 + 3720 * q^11 + 14179 * q^13 - 2584 * q^14 - 4096 * q^16 - 31824 * q^17 + 44842 * q^19 + 19968 * q^20 + 29760 * q^22 + 57768 * q^23 - 19219 * q^25 - 226864 * q^26 + 41344 * q^28 + 166656 * q^29 - 94820 * q^31 - 32768 * q^32 + 127296 * q^34 - 201552 * q^35 + 907942 * q^37 - 179368 * q^38 + 159744 * q^40 + 627072 * q^41 + 42472 * q^43 - 476160 * q^44 - 924288 * q^46 - 1235256 * q^47 + 719214 * q^49 - 153752 * q^50 + 907456 * q^52 - 214560 * q^53 + 2321280 * q^55 - 165376 * q^56 + 1333248 * q^58 - 2479224 * q^59 - 2874383 * q^61 + 1517120 * q^62 + 524288 * q^64 - 4423848 * q^65 - 1501097 * q^67 + 1018368 * q^68 + 806208 * q^70 - 9466272 * q^71 - 170222 * q^73 - 3631768 * q^74 - 1434944 * q^76 + 1201560 * q^77 + 1180819 * q^79 - 2555904 * q^80 - 10033152 * q^82 - 1116528 * q^83 - 4964544 * q^85 + 339776 * q^86 + 1904640 * q^88 - 18736272 * q^89 - 9159634 * q^91 + 3697152 * q^92 - 9882048 * q^94 + 6995352 * q^95 + 2039995 * q^97 - 11507424 * q^98

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