LMFDB - Embedded newform 162.7.d.d.107.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [162,7,Mod(53,162)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("162.53"); S:= CuspForms(chi, 7); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(162, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 7, names="a")

Level:	$ N $	$=$	$ 162 = 2 \cdot 3^{4} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	162.d (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,64,0,0,968] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$37.2687615464$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			107.2
Root			$-1.22474 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	162.107
Dual form			162.7.d.d.53.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(4.89898 + 2.82843i) q^{2} +(16.0000 + 27.7128i) q^{4} +(-150.644 + 86.9741i) q^{5} +(242.000 - 419.156i) q^{7} +181.019i q^{8} -984.000 q^{10} +(1161.06 + 670.337i) q^{11} +(-1684.00 - 2916.77i) q^{13} +(2371.11 - 1368.96i) q^{14} +(-512.000 + 886.810i) q^{16} -12.7279i q^{17} +5744.00 q^{19} +(-4820.60 - 2783.17i) q^{20} +(3792.00 + 6567.94i) q^{22} +(2924.69 - 1688.57i) q^{23} +(7316.50 - 12672.5i) q^{25} -19052.3i q^{26} +15488.0 q^{28} +(25422.0 + 14677.4i) q^{29} +(19898.0 + 34464.3i) q^{31} +(-5016.55 + 2896.31i) q^{32} +(36.0000 - 62.3538i) q^{34} +84191.0i q^{35} +52526.0 q^{37} +(28139.7 + 16246.5i) q^{38} +(-15744.0 - 27269.4i) q^{40} +(32079.7 - 18521.2i) q^{41} +(-1900.00 + 3290.90i) q^{43} +42901.6i q^{44} +19104.0 q^{46} +(66503.6 + 38395.9i) q^{47} +(-58303.5 - 100985. i) q^{49} +(71686.8 - 41388.4i) q^{50} +(53888.0 - 93336.8i) q^{52} -238738. i q^{53} -233208. q^{55} +(75875.4 + 43806.7i) q^{56} +(83028.0 + 143809. i) q^{58} +(216368. - 124920. i) q^{59} +(-6625.00 + 11474.8i) q^{61} +225120. i q^{62} -32768.0 q^{64} +(507368. + 292929. i) q^{65} +(-84484.0 - 146331. i) q^{67} +(352.727 - 203.647i) q^{68} +(-238128. + 412450. i) q^{70} +531467. i q^{71} +236144. q^{73} +(257324. + 148566. i) q^{74} +(91904.0 + 159182. i) q^{76} +(561952. - 324443. i) q^{77} +(17558.0 - 30411.3i) q^{79} -178123. i q^{80} +209544. q^{82} +(-9508.92 - 5489.98i) q^{83} +(1107.00 + 1917.38i) q^{85} +(-18616.1 + 10748.0i) q^{86} +(-121344. + 210174. i) q^{88} +129328. i q^{89} -1.63011e6 q^{91} +(93590.1 + 54034.3i) q^{92} +(217200. + 376201. i) q^{94} +(-865297. + 499579. i) q^{95} +(160712. - 278361. i) q^{97} -659629. i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 64 q^{4} + 968 q^{7} - 3936 q^{10} - 6736 q^{13} - 2048 q^{16} + 22976 q^{19} + 15168 q^{22} + 29266 q^{25} + 61952 q^{28} + 79592 q^{31} + 144 q^{34} + 210104 q^{37} - 62976 q^{40} - 7600 q^{43}+ \cdots + 642848 q^{97}+O(q^{100}) $ 4 * q + 64 * q^4 + 968 * q^7 - 3936 * q^10 - 6736 * q^13 - 2048 * q^16 + 22976 * q^19 + 15168 * q^22 + 29266 * q^25 + 61952 * q^28 + 79592 * q^31 + 144 * q^34 + 210104 * q^37 - 62976 * q^40 - 7600 * q^43 + 76416 * q^46 - 233214 * q^49 + 215552 * q^52 - 932832 * q^55 + 332112 * q^58 - 26500 * q^61 - 131072 * q^64 - 337936 * q^67 - 952512 * q^70 + 944576 * q^73 + 367616 * q^76 + 70232 * q^79 + 838176 * q^82 + 4428 * q^85 - 485376 * q^88 - 6520448 * q^91 + 868800 * q^94 + 642848 * q^97

Character values

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

$n$	$83$
$\chi(n)$	$e\left(\frac{1}{6}\right)$

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 $

$2$	4.89898	+	2.82843i	0.612372	+	0.353553i
$2$	4.89898	+	2.82843i	0.612372	+	0.353553i
$3$		0			0
$3$		0			0
$4$	16.0000	+	27.7128i	0.250000	+	0.433013i
$5$	−150.644	+	86.9741i	−1.20515	+	0.695793i	−0.961696	−	0.274120i	$-0.911614\pi$
$5$	−150.644	+	86.9741i	−1.20515	+	0.695793i	−0.243453	+	0.969913i	$0.578280\pi$
$6$		0			0
$7$	242.000	−	419.156i	0.705539	−	1.22203i	−0.260957	−	0.965350i	$-0.584038\pi$
$7$	242.000	−	419.156i	0.705539	−	1.22203i	0.966497	−	0.256680i	$-0.0826285\pi$
$8$			181.019i			0.353553i
$9$		0			0
$10$	−984.000			−0.984000
$11$	1161.06	+	670.337i	0.872320	+	0.503634i	0.868119	−	0.496357i	$-0.165329\pi$
$11$	1161.06	+	670.337i	0.872320	+	0.503634i	0.00420160	+	0.999991i	$0.498663\pi$
$12$		0			0
$13$	−1684.00	−	2916.77i	−0.766500	−	1.32762i	−0.939450	−	0.342686i	$-0.888663\pi$
$13$	−1684.00	−	2916.77i	−0.766500	−	1.32762i	0.172950	−	0.984931i	$-0.444670\pi$
$14$	2371.11	−	1368.96i	0.864106	−	0.498892i
$15$		0			0
$16$	−512.000	+	886.810i	−0.125000	+	0.216506i
$17$		−	12.7279i		−	0.00259066i	−0.999999	−	0.00129533i	$-0.999588\pi$
$17$		−	12.7279i		−	0.00259066i	0.999999	−	0.00129533i	$-0.000412317\pi$
$18$		0			0
$19$	5744.00			0.837440			0.418720	−	0.908115i	$-0.362479\pi$
$19$	5744.00			0.837440			0.418720	+	0.908115i	$0.362479\pi$
$20$	−4820.60	−	2783.17i	−0.602574	−	0.347897i
$21$		0			0
$22$	3792.00	+	6567.94i	0.356123	+	0.616824i
$23$	2924.69	−	1688.57i	0.240379	−	0.138783i	−0.374972	−	0.927036i	$-0.622348\pi$
$23$	2924.69	−	1688.57i	0.240379	−	0.138783i	0.615351	+	0.788253i	$0.289014\pi$
$24$		0			0
$25$	7316.50	−	12672.5i	0.468256	−	0.811043i
$26$		−	19052.3i		−	1.08399i
$27$		0			0
$28$	15488.0			0.705539
$29$	25422.0	+	14677.4i	1.04236	+	0.601805i	0.920500	−	0.390743i	$-0.127782\pi$
$29$	25422.0	+	14677.4i	1.04236	+	0.601805i	0.121857	+	0.992548i	$0.461115\pi$
$30$		0			0
$31$	19898.0	+	34464.3i	0.667920	+	1.15687i	0.978485	+	0.206319i	$0.0661484\pi$
$31$	19898.0	+	34464.3i	0.667920	+	1.15687i	−0.310565	+	0.950552i	$0.600518\pi$
$32$	−5016.55	+	2896.31i	−0.153093	+	0.0883883i
$33$		0			0
$34$	36.0000	−	62.3538i	0.000915937	−	0.00158645i
$35$			84191.0i			1.96364i
$36$		0			0
$37$	52526.0			1.03698			0.518489	−	0.855085i	$-0.326495\pi$
$37$	52526.0			1.03698			0.518489	+	0.855085i	$0.326495\pi$
$38$	28139.7	+	16246.5i	0.512825	+	0.296080i
$39$		0			0
$40$	−15744.0	−	27269.4i	−0.246000	−	0.426084i
$41$	32079.7	−	18521.2i	0.465457	−	0.268732i	−0.248879	−	0.968535i	$-0.580062\pi$
$41$	32079.7	−	18521.2i	0.465457	−	0.268732i	0.714336	+	0.699803i	$0.246729\pi$
$42$		0			0
$43$	−1900.00	+	3290.90i	−0.0238973	+	0.0413913i	−0.877727	−	0.479162i	$-0.840941\pi$
$43$	−1900.00	+	3290.90i	−0.0238973	+	0.0413913i	0.853829	+	0.520553i	$0.174274\pi$
$44$			42901.6i			0.503634i
$45$		0			0
$46$	19104.0			0.196269
$47$	66503.6	+	38395.9i	0.640548	+	0.369821i	0.784826	−	0.619717i	$-0.212752\pi$
$47$	66503.6	+	38395.9i	0.640548	+	0.369821i	−0.144277	+	0.989537i	$0.546086\pi$
$48$		0			0
$49$	−58303.5	−	100985.i	−0.495572	−	0.858355i
$50$	71686.8	−	41388.4i	0.573494	−	0.331107i
$51$		0			0
$52$	53888.0	−	93336.8i	0.383250	−	0.663808i
$53$		−	238738.i		−	1.60359i	−0.597599	−	0.801795i	$-0.703879\pi$
$53$		−	238738.i		−	1.60359i	0.597599	−	0.801795i	$-0.296121\pi$
$54$		0			0
$55$	−233208.			−1.40170
$56$	75875.4	+	43806.7i	0.432053	+	0.249446i
$57$		0			0
$58$	83028.0	+	143809.i	0.425540	+	0.737057i
$59$	216368.	−	124920.i	1.05351	−	0.608243i	0.129878	−	0.991530i	$-0.458541\pi$
$59$	216368.	−	124920.i	1.05351	−	0.608243i	0.923629	+	0.383287i	$0.125208\pi$
$60$		0			0
$61$	−6625.00	+	11474.8i	−0.0291875	+	0.0505542i	−0.880250	−	0.474510i	$-0.842625\pi$
$61$	−6625.00	+	11474.8i	−0.0291875	+	0.0505542i	0.851063	+	0.525064i	$0.175959\pi$
$62$			225120.i			0.944581i
$63$		0			0
$64$	−32768.0			−0.125000
$65$	507368.	+	292929.i	1.84749	+	1.06665i
$66$		0			0
$67$	−84484.0	−	146331.i	−0.280899	−	0.486531i	0.690707	−	0.723134i	$-0.257299\pi$
$67$	−84484.0	−	146331.i	−0.280899	−	0.486531i	−0.971606	+	0.236603i	$0.923966\pi$
$68$	352.727	−	203.647i	0.00112179	−	0.000647665i
$69$		0			0
$70$	−238128.	+	412450.i	−0.694251	+	1.20248i
$71$			531467.i			1.48491i	0.669894	+	0.742457i	$0.266340\pi$
$71$			531467.i			1.48491i	−0.669894	+	0.742457i	$0.733660\pi$
$72$		0			0
$73$	236144.			0.607027			0.303514	−	0.952827i	$-0.401840\pi$
$73$	236144.			0.607027			0.303514	+	0.952827i	$0.401840\pi$
$74$	257324.	+	148566.i	0.635016	+	0.366627i
$75$		0			0
$76$	91904.0	+	159182.i	0.209360	+	0.362622i
$77$	561952.	−	324443.i	1.23091	−	0.710668i
$78$		0			0
$79$	17558.0	−	30411.3i	0.0356118	−	0.0616814i	−0.847670	−	0.530524i	$-0.821995\pi$
$79$	17558.0	−	30411.3i	0.0356118	−	0.0616814i	0.883282	+	0.468842i	$0.155329\pi$
$80$		−	178123.i		−	0.347897i
$81$		0			0
$82$	209544.			0.380044
$83$	−9508.92	−	5489.98i	−0.0166302	−	0.00960144i	0.491662	−	0.870786i	$-0.336390\pi$
$83$	−9508.92	−	5489.98i	−0.0166302	−	0.00960144i	−0.508292	+	0.861185i	$0.669723\pi$
$84$		0			0
$85$	1107.00	+	1917.38i	0.00180256	+	0.00312213i
$86$	−18616.1	+	10748.0i	−0.0292681	+	0.0168979i
$87$		0			0
$88$	−121344.	+	210174.i	−0.178062	+	0.308412i
$89$			129328.i			0.183453i	0.995784	+	0.0917263i	$0.0292385\pi$
$89$			129328.i			0.183453i	−0.995784	+	0.0917263i	$0.970762\pi$
$90$		0			0
$91$	−1.63011e6			−2.16318
$92$	93590.1	+	54034.3i	0.120189	+	0.0693914i
$93$		0			0
$94$	217200.	+	376201.i	0.261503	+	0.452936i
$95$	−865297.	+	499579.i	−1.00924	+	0.582685i
$96$		0			0
$97$	160712.	−	278361.i	0.176089	−	0.304996i	−0.764448	−	0.644685i	$-0.776989\pi$
$97$	160712.	−	278361.i	0.176089	−	0.304996i	0.940538	+	0.339689i	$0.110322\pi$
$98$		−	659629.i		−	0.700844i
$99$		0			0
$100$	468256.			0.468256
$101$	579181.	+	334390.i	0.562147	+	0.324556i	0.754007	−	0.656867i	$-0.228119\pi$
$101$	579181.	+	334390.i	0.562147	+	0.324556i	−0.191860	+	0.981422i	$0.561452\pi$
$102$		0			0
$103$	−996706.	−	1.72635e6i	−0.912127	−	1.57985i	−0.811054	−	0.584972i	$-0.801106\pi$
$103$	−996706.	−	1.72635e6i	−0.912127	−	1.57985i	−0.101073	−	0.994879i	$-0.532228\pi$
$104$	527992.	−	304837.i	0.469383	−	0.270999i
$105$		0			0
$106$	675252.	−	1.16957e6i	0.566955	−	0.981994i
$107$			260668.i			0.212783i	0.994324	+	0.106391i	$0.0339296\pi$
$107$			260668.i			0.212783i	−0.994324	+	0.106391i	$0.966070\pi$
$108$		0			0
$109$	194456.			0.150156			0.0750779	−	0.997178i	$-0.476079\pi$
$109$	194456.			0.150156			0.0750779	+	0.997178i	$0.476079\pi$
$110$	−1.14248e6	−	659612.i	−0.858363	−	0.495576i
$111$		0			0
$112$	247808.	+	429216.i	0.176385	+	0.305508i
$113$	711784.	−	410949.i	0.493302	−	0.284808i	−0.232641	−	0.972563i	$-0.574737\pi$
$113$	711784.	−	410949.i	0.493302	−	0.284808i	0.725943	+	0.687755i	$0.241404\pi$
$114$		0			0
$115$	−293724.	+	508745.i	−0.193128	+	0.334508i
$116$			939355.i			0.601805i
$117$		0			0
$118$	1.41331e6			0.860185
$119$	−5334.99	−	3080.16i	−0.00316587	−	0.00182781i
$120$		0			0
$121$	12923.5	+	22384.2i	0.00729498	+	0.0126353i
$122$	−64911.5	+	37476.7i	−0.0357472	+	0.0206387i
$123$		0			0
$124$	−636736.	+	1.10286e6i	−0.333960	+	0.578436i
$125$		−	172557.i		−	0.0883490i
$126$		0			0
$127$	3.05721e6			1.49250			0.746250	−	0.665666i	$-0.231852\pi$
$127$	3.05721e6			1.49250			0.746250	+	0.665666i	$0.231852\pi$
$128$	−160530.	−	92681.9i	−0.0765466	−	0.0441942i
$129$		0			0
$130$	1.65706e6	+	2.87011e6i	0.754236	+	1.30637i
$131$	−2.66206e6	+	1.53694e6i	−1.18414	+	0.683664i	−0.956969	−	0.290191i	$-0.906281\pi$
$131$	−2.66206e6	+	1.53694e6i	−1.18414	+	0.683664i	−0.227172	+	0.973855i	$0.572948\pi$
$132$		0			0
$133$	1.39005e6	−	2.40763e6i	0.590847	−	1.02338i
$134$		−	955827.i		−	0.397251i
$135$		0			0
$136$	2304.00			0.000915937
$137$	−3.88336e6	−	2.24206e6i	−1.51024	−	0.871938i	−0.999929	−	0.0119482i	$-0.996197\pi$
$137$	−3.88336e6	−	2.24206e6i	−1.51024	−	0.871938i	−0.510312	−	0.859990i	$-0.670470\pi$
$138$		0			0
$139$	546164.	+	945984.i	0.203366	+	0.352241i	0.949611	−	0.313431i	$-0.101478\pi$
$139$	546164.	+	945984.i	0.203366	+	0.352241i	−0.746245	+	0.665672i	$0.768145\pi$
$140$	−2.33317e6	+	1.34706e6i	−0.850280	+	0.490909i
$141$		0			0
$142$	−1.50322e6	+	2.60365e6i	−0.524996	+	0.909321i
$143$		−	4.51539e6i		−	1.54414i
$144$		0			0
$145$	−5.10622e6			−1.67493
$146$	1.15686e6	+	667916.i	0.371727	+	0.214617i
$147$		0			0
$148$	840416.	+	1.45564e6i	0.259244	+	0.449024i
$149$	1.92333e6	−	1.11044e6i	0.581428	−	0.335687i	−0.180273	−	0.983617i	$-0.557698\pi$
$149$	1.92333e6	−	1.11044e6i	0.581428	−	0.335687i	0.761701	+	0.647929i	$0.224365\pi$
$150$		0			0
$151$	2.03935e6	−	3.53226e6i	0.592327	−	1.02594i	−0.401591	−	0.915819i	$-0.631543\pi$
$151$	2.03935e6	−	3.53226e6i	0.592327	−	1.02594i	0.993918	−	0.110122i	$-0.0351241\pi$
$152$			1.03978e6i			0.296080i
$153$		0			0
$154$	3.67066e6			1.00504
$155$	−5.99501e6	−	3.46122e6i	−1.60989	−	0.929468i
$156$		0			0
$157$	−3.07784e6	−	5.33097e6i	−0.795329	−	1.37755i	−0.922630	−	0.385686i	$-0.873965\pi$
$157$	−3.07784e6	−	5.33097e6i	−0.795329	−	1.37755i	0.127301	−	0.991864i	$-0.459369\pi$
$158$	172033.	−	99323.0i	0.0436154	−	0.0251813i
$159$		0			0
$160$	503808.	−	872621.i	0.123000	−	0.213042i
$161$		−	1.63454e6i		−	0.391667i
$162$		0			0
$163$	800696.			0.184886			0.0924432	−	0.995718i	$-0.470532\pi$
$163$	800696.			0.184886			0.0924432	+	0.995718i	$0.470532\pi$
$164$	1.02655e6	+	592680.i	0.232728	+	0.134366i
$165$		0			0
$166$	−31056.0	−	53790.6i	−0.00678924	−	0.0117593i
$167$	−4.16097e6	+	2.40234e6i	−0.893398	+	0.515804i	−0.875052	−	0.484028i	$-0.839173\pi$
$167$	−4.16097e6	+	2.40234e6i	−0.893398	+	0.515804i	−0.0183455	+	0.999832i	$0.505840\pi$
$168$		0			0
$169$	−3.25831e6	+	5.64355e6i	−0.675044	+	1.16921i
$170$			12524.3i			0.00254921i
$171$		0			0
$172$	−121600.			−0.0238973
$173$	−3.09164e6	−	1.78496e6i	−0.597106	−	0.344739i	0.170796	−	0.985306i	$-0.445366\pi$
$173$	−3.09164e6	−	1.78496e6i	−0.597106	−	0.344739i	−0.767902	+	0.640567i	$0.778699\pi$
$174$		0			0
$175$	−3.54119e6	−	6.13351e6i	−0.660746	−	1.14445i
$176$	−1.18892e6	+	686425.i	−0.218080	+	0.125909i
$177$		0			0
$178$	−365796.	+	633577.i	−0.0648603	+	0.112341i
$179$			7.43698e6i			1.29669i	0.761345	+	0.648347i	$0.224539\pi$
$179$			7.43698e6i			1.29669i	−0.761345	+	0.648347i	$0.775461\pi$
$180$		0			0
$181$	−1.03812e7			−1.75070			−0.875350	−	0.483491i	$-0.839369\pi$
$181$	−1.03812e7			−1.75070			−0.875350	+	0.483491i	$0.839369\pi$
$182$	−7.98589e6	−	4.61065e6i	−1.32467	−	0.764801i
$183$		0			0
$184$	305664.	+	529426.i	0.0490671	+	0.0849868i
$185$	−7.91271e6	+	4.56840e6i	−1.24971	+	0.721521i
$186$		0			0
$187$	8532.00	−	14777.9i	0.00130475	−	0.00225989i
$188$			2.45734e6i			0.369821i
$189$		0			0
$190$	−5.65210e6			−0.824041
$191$	1.12532e7	+	6.49703e6i	1.61501	+	0.932426i	0.988185	+	0.153266i	$0.0489793\pi$
$191$	1.12532e7	+	6.49703e6i	1.61501	+	0.932426i	0.626825	+	0.779160i	$0.284354\pi$
$192$		0			0
$193$	1.96598e6	+	3.40517e6i	0.273468	+	0.473660i	0.969747	−	0.244110i	$-0.0784959\pi$
$193$	1.96598e6	+	3.40517e6i	0.273468	+	0.473660i	−0.696279	+	0.717771i	$0.745163\pi$
$194$	1.57465e6	−	909124.i	0.215665	−	0.124514i
$195$		0			0
$196$	1.86571e6	−	3.23151e6i	0.247786	−	0.429178i
$197$		−	5.37967e6i		−	0.703651i	−0.936066	−	0.351825i	$-0.885561\pi$
$197$		−	5.37967e6i		−	0.703651i	0.936066	−	0.351825i	$-0.114439\pi$
$198$		0			0
$199$	−565900.			−0.0718093			−0.0359046	−	0.999355i	$-0.511431\pi$
$199$	−565900.			−0.0718093			−0.0359046	+	0.999355i	$0.511431\pi$
$200$	2.29398e6	+	1.32443e6i	0.286747	+	0.165553i
$201$		0			0
$202$	1.89160e6	+	3.27634e6i	0.229496	+	0.397498i
$203$	1.23043e7	−	7.10387e6i	1.47085	−	0.849194i
$204$		0			0
$205$	−3.22174e6	+	5.58022e6i	−0.373963	+	0.647723i
$206$		−	1.12764e7i		−	1.28994i
$207$		0			0
$208$	3.44883e6			0.383250
$209$	6.66912e6	+	3.85042e6i	0.730516	+	0.421763i
$210$		0			0
$211$	6.75825e6	+	1.17056e7i	0.719427	+	1.24608i	0.961227	+	0.275759i	$0.0889291\pi$
$211$	6.75825e6	+	1.17056e7i	0.719427	+	1.24608i	−0.241799	+	0.970326i	$0.577738\pi$
$212$	6.61609e6	−	3.81980e6i	0.694375	−	0.400897i
$213$		0			0
$214$	−737280.	+	1.27701e6i	−0.0752300	+	0.130302i
$215$		−	661003.i		−	0.0665102i
$216$		0			0
$217$	1.92613e7			1.88497
$218$	952636.	+	550005.i	0.0919512	+	0.0530881i
$219$		0			0
$220$	−3.73133e6	−	6.46285e6i	−0.350425	−	0.606954i
$221$	−37124.5	+	21433.8i	−0.00343941	+	0.00198574i
$222$		0			0
$223$	2.67742e6	−	4.63742e6i	0.241436	−	0.418179i	−0.719688	−	0.694298i	$-0.755715\pi$
$223$	2.67742e6	−	4.63742e6i	0.241436	−	0.418179i	0.961123	+	0.276119i	$0.0890484\pi$
$224$			2.80363e6i			0.249446i
$225$		0			0
$226$	4.64935e6			0.402779
$227$	−1.17834e7	−	6.80317e6i	−1.00738	−	0.581612i	−0.0969580	−	0.995288i	$-0.530911\pi$
$227$	−1.17834e7	−	6.80317e6i	−1.00738	−	0.581612i	−0.910424	+	0.413676i	$0.864245\pi$
$228$		0			0
$229$	−2.17320e6	−	3.76410e6i	−0.180965	−	0.313440i	0.761245	−	0.648465i	$-0.224589\pi$
$229$	−2.17320e6	−	3.76410e6i	−0.180965	−	0.313440i	−0.942209	+	0.335025i	$0.891255\pi$
$230$	−2.87790e6	+	1.66155e6i	−0.236533	+	0.136562i
$231$		0			0
$232$	−2.65690e6	+	4.60188e6i	−0.212770	+	0.368529i
$233$			2.02333e7i			1.59956i	0.600297	+	0.799778i	$0.295049\pi$
$233$			2.02333e7i			1.59956i	−0.600297	+	0.799778i	$0.704951\pi$
$234$		0			0
$235$	−1.33578e7			−1.02927
$236$	6.92379e6	+	3.99745e6i	0.526754	+	0.304121i
$237$		0			0
$238$	−17424.0	−	30179.3i	−0.00129246	−	0.00223861i
$239$	1.76623e7	−	1.01973e7i	1.29376	−	0.746953i	0.314442	−	0.949277i	$-0.398183\pi$
$239$	1.76623e7	−	1.01973e7i	1.29376	−	0.746953i	0.979319	+	0.202324i	$0.0648494\pi$
$240$		0			0
$241$	1.56046e6	−	2.70280e6i	0.111481	−	0.193092i	−0.804886	−	0.593429i	$-0.797774\pi$
$241$	1.56046e6	−	2.70280e6i	0.111481	−	0.193092i	0.916368	+	0.400337i	$0.131107\pi$
$242$			146213.i			0.0103167i
$243$		0			0
$244$	−424000.			−0.0291875
$245$	1.75661e7	+	1.01418e7i	1.19448	+	0.689631i
$246$		0			0
$247$	−9.67290e6	−	1.67539e7i	−0.641897	−	1.11180i
$248$	−6.23871e6	+	3.60192e6i	−0.409016	+	0.236145i
$249$		0			0
$250$	488064.	−	845352.i	0.0312361	−	0.0541025i
$251$		−	5.09519e6i		−	0.322210i	−0.986937	−	0.161105i	$-0.948494\pi$
$251$		−	5.09519e6i		−	0.322210i	0.986937	−	0.161105i	$-0.0515058\pi$
$252$		0			0
$253$	4.52765e6			0.279583
$254$	1.49772e7	+	8.64710e6i	0.913966	+	0.527678i
$255$		0			0
$256$	−524288.	−	908093.i	−0.0312500	−	0.0541266i
$257$	−1.25031e7	+	7.21869e6i	−0.736580	+	0.425264i	−0.820824	−	0.571181i	$-0.806486\pi$
$257$	−1.25031e7	+	7.21869e6i	−0.736580	+	0.425264i	0.0842448	+	0.996445i	$0.473152\pi$
$258$		0			0
$259$	1.27113e7	−	2.20166e7i	0.731628	−	1.26722i
$260$			1.87474e7i			1.06665i
$261$		0			0
$262$	−1.73885e7			−0.966847
$263$	2.70691e7	+	1.56283e7i	1.48801	+	0.859104i	0.999906	−	0.0136808i	$-0.00435487\pi$
$263$	2.70691e7	+	1.56283e7i	1.48801	+	0.859104i	0.488105	+	0.872785i	$0.337688\pi$
$264$		0			0
$265$	2.07640e7	+	3.59643e7i	1.11577	+	1.93256i
$266$	1.36196e7	−	7.86330e6i	0.723637	−	0.417792i
$267$		0			0
$268$	2.70349e6	−	4.68258e6i	0.140449	−	0.243266i
$269$		−	251338.i		−	0.0129122i	−0.999979	−	0.00645612i	$-0.997945\pi$
$269$		−	251338.i		−	0.0129122i	0.999979	−	0.00645612i	$-0.00205506\pi$
$270$		0			0
$271$	2.96399e7			1.48925			0.744627	−	0.667481i	$-0.232627\pi$
$271$	2.96399e7			1.48925			0.744627	+	0.667481i	$0.232627\pi$
$272$	11287.2	+	6516.70i	0.000560895	+	0.000323833i
$273$		0			0
$274$	−1.26830e7	−	2.19676e7i	−0.616553	−	1.06790i
$275$	1.69898e7	−	9.80904e6i	0.816938	−	0.471660i
$276$		0			0
$277$	−6.61064e6	+	1.14500e7i	−0.311031	+	0.538722i	−0.978586	−	0.205839i	$-0.934008\pi$
$277$	−6.61064e6	+	1.14500e7i	−0.311031	+	0.538722i	0.667555	+	0.744561i	$0.267341\pi$
$278$			6.17914e6i			0.287603i
$279$		0			0
$280$	−1.52402e7			−0.694251
$281$	−5.30320e6	−	3.06180e6i	−0.239011	−	0.137993i	0.375711	−	0.926737i	$-0.377399\pi$
$281$	−5.30320e6	−	3.06180e6i	−0.239011	−	0.137993i	−0.614722	+	0.788744i	$0.710732\pi$
$282$		0			0
$283$	3.37162e6	+	5.83982e6i	0.148758	+	0.257656i	0.930769	−	0.365609i	$-0.119139\pi$
$283$	3.37162e6	+	5.83982e6i	0.148758	+	0.257656i	−0.782011	+	0.623265i	$0.785806\pi$
$284$	−1.47284e7	+	8.50347e6i	−0.642987	+	0.371229i
$285$		0			0
$286$	1.27715e7	−	2.21208e7i	0.545937	−	0.945590i
$287$		−	1.79286e7i		−	0.758403i
$288$		0			0
$289$	2.41374e7			0.999993
$290$	−2.50153e7	−	1.44426e7i	−1.02568	−	0.592176i
$291$		0			0
$292$	3.77830e6	+	6.54421e6i	0.151757	+	0.262851i
$293$	8.68096e6	−	5.01195e6i	0.345116	−	0.199253i	−0.317416	−	0.948286i	$-0.602815\pi$
$293$	8.68096e6	−	5.01195e6i	0.345116	−	0.199253i	0.662532	+	0.749034i	$0.269482\pi$
$294$		0			0
$295$	−2.17297e7	+	3.76369e7i	−0.846422	+	1.46605i
$296$			9.50822e6i			0.366627i
$297$		0			0
$298$	1.25632e7			0.474734
$299$	−9.85036e6	−	5.68711e6i	−0.368501	−	0.212754i
$300$		0			0
$301$	919600.	+	1.59279e6i	0.0337209	+	0.0584064i
$302$	1.99815e7	−	1.15363e7i	0.725450	−	0.418839i
$303$		0			0
$304$	−2.94093e6	+	5.09384e6i	−0.104680	+	0.181311i
$305$		−	2.30481e6i		−	0.0812337i
$306$		0			0
$307$	−5.23060e6			−0.180774			−0.0903871	−	0.995907i	$-0.528810\pi$
$307$	−5.23060e6			−0.180774			−0.0903871	+	0.995907i	$0.528810\pi$
$308$	1.79825e7	+	1.03822e7i	0.615456	+	0.355334i
$309$		0			0
$310$	−1.95796e7	−	3.39129e7i	−0.657233	−	1.13836i
$311$	−2.70392e7	+	1.56111e7i	−0.898901	+	0.518981i	−0.876844	−	0.480776i	$-0.840355\pi$
$311$	−2.70392e7	+	1.56111e7i	−0.898901	+	0.518981i	−0.0220578	+	0.999757i	$0.507022\pi$
$312$		0			0
$313$	−1.12389e7	+	1.94664e7i	−0.366515	+	0.634822i	−0.989018	−	0.147795i	$-0.952782\pi$
$313$	−1.12389e7	+	1.94664e7i	−0.366515	+	0.634822i	0.622503	+	0.782617i	$0.286116\pi$
$314$		−	3.48218e7i		−	1.12477i
$315$		0			0
$316$	1.12371e6			0.0356118
$317$	−2.39206e7	−	1.38106e7i	−0.750921	−	0.433544i	0.0751058	−	0.997176i	$-0.476071\pi$
$317$	−2.39206e7	−	1.38106e7i	−0.750921	−	0.433544i	−0.826026	+	0.563631i	$0.809404\pi$
$318$		0			0
$319$	1.96776e7	+	3.40827e7i	0.606179	+	1.04993i
$320$	4.93629e6	−	2.84997e6i	0.150644	−	0.0869741i
$321$		0			0
$322$	4.62317e6	−	8.00756e6i	0.138475	−	0.239846i
$323$		−	73109.2i		−	0.00216952i
$324$		0			0
$325$	−4.92839e7			−1.43567
$326$	3.92259e6	+	2.26471e6i	0.113219	+	0.0653672i
$327$		0			0
$328$	3.35270e6	+	5.80705e6i	0.0950110	+	0.164564i
$329$	3.21878e7	−	1.85836e7i	0.903864	−	0.521846i
$330$		0			0
$331$	2.88069e6	−	4.98950e6i	0.0794352	−	0.137586i	−0.823571	−	0.567213i	$-0.808022\pi$
$331$	2.88069e6	−	4.98950e6i	0.0794352	−	0.137586i	0.903006	+	0.429627i	$0.141355\pi$
$332$		−	351359.i		−	0.00960144i
$333$		0			0
$334$	−2.71793e7			−0.729456
$335$	2.54540e7	+	1.46958e7i	0.677050	+	0.390895i
$336$		0			0
$337$	2.00526e7	+	3.47321e7i	0.523939	+	0.907489i	0.999612	+	0.0278665i	$0.00887132\pi$
$337$	2.00526e7	+	3.47321e7i	0.523939	+	0.907489i	−0.475673	+	0.879622i	$0.657795\pi$
$338$	−3.19248e7	+	1.84318e7i	−0.826756	+	0.477328i
$339$		0			0
$340$	−35424.0	+	61356.2i	−0.000901282	+	0.00156107i
$341$			5.33535e7i			1.34555i
$342$		0			0
$343$	504328.			0.0124977
$344$	−595716.	−	343937.i	−0.0146340	−	0.00844896i
$345$		0			0
$346$	−1.00973e7	−	1.74890e7i	−0.243767	−	0.422217i
$347$	−5.87275e7	+	3.39064e7i	−1.40557	+	0.811508i	−0.994957	−	0.100301i	$-0.968020\pi$
$347$	−5.87275e7	+	3.39064e7i	−1.40557	+	0.811508i	−0.410616	+	0.911808i	$0.634686\pi$
$348$		0			0
$349$	2.10319e7	−	3.64284e7i	0.494769	−	0.856965i	−0.505213	−	0.862995i	$-0.668586\pi$
$349$	2.10319e7	−	3.64284e7i	0.494769	−	0.856965i	0.999982	+	0.00602956i	$0.00191928\pi$
$350$		−	4.00639e7i		−	0.934436i
$351$		0			0
$352$	−7.76602e6			−0.178062
$353$	−1.52399e7	−	8.79878e6i	−0.346465	−	0.200032i	0.316662	−	0.948538i	$-0.397438\pi$
$353$	−1.52399e7	−	8.79878e6i	−0.346465	−	0.200032i	−0.663127	+	0.748507i	$0.730771\pi$
$354$		0			0
$355$	−4.62239e7	−	8.00621e7i	−1.03319	−	1.78954i
$356$	−3.58405e6	+	2.06925e6i	−0.0794373	+	0.0458632i
$357$		0			0
$358$	−2.10349e7	+	3.64336e7i	−0.458450	+	0.794059i
$359$			1.39920e7i			0.302410i	0.988502	+	0.151205i	$0.0483154\pi$
$359$			1.39920e7i			0.302410i	−0.988502	+	0.151205i	$0.951685\pi$
$360$		0			0
$361$	−1.40523e7			−0.298694
$362$	−5.08572e7	−	2.93624e7i	−1.07208	−	0.618966i
$363$		0			0
$364$	−2.60818e7	−	4.51750e7i	−0.540796	−	0.936686i
$365$	−3.55736e7	+	2.05384e7i	−0.731559	+	0.422365i
$366$		0			0
$367$	1.32927e7	−	2.30237e7i	0.268916	−	0.465776i	−0.699666	−	0.714470i	$-0.746668\pi$
$367$	1.32927e7	−	2.30237e7i	0.268916	−	0.465776i	0.968582	+	0.248694i	$0.0800013\pi$
$368$			3.45819e6i			0.0693914i
$369$		0			0
$370$	−5.16856e7			−1.02039
$371$	−1.00068e8	−	5.77745e7i	−1.95963	−	1.13140i
$372$		0			0
$373$	−8.94146e6	−	1.54871e7i	−0.172299	−	0.298430i	0.766924	−	0.641737i	$-0.221786\pi$
$373$	−8.94146e6	−	1.54871e7i	−0.172299	−	0.298430i	−0.939223	+	0.343307i	$0.888453\pi$
$374$	83596.2	−	48264.3i	0.00159798	−	0.000922595i
$375$		0			0
$376$	−6.95040e6	+	1.20384e7i	−0.130751	+	0.226468i
$377$		−	9.88671e7i		−	1.84513i
$378$		0			0
$379$	7.20978e7			1.32435			0.662177	−	0.749347i	$-0.269633\pi$
$379$	7.20978e7			1.32435			0.662177	+	0.749347i	$0.269633\pi$
$380$	−2.76895e7	−	1.59865e7i	−0.504620	−	0.291342i
$381$		0			0
$382$	3.67527e7	+	6.36576e7i	0.659325	+	1.14198i
$383$	−7.52272e6	+	4.34324e6i	−0.133899	+	0.0773068i	−0.565453	−	0.824780i	$-0.691299\pi$
$383$	−7.52272e6	+	4.34324e6i	−0.133899	+	0.0773068i	0.431554	+	0.902087i	$0.357965\pi$
$384$		0			0
$385$	−5.64363e7	+	9.77506e7i	−0.988955	+	1.71292i
$386$			2.22425e7i			0.386742i
$387$		0			0
$388$	1.02856e7			0.176089
$389$	−4.28173e7	−	2.47206e7i	−0.727395	−	0.419962i	0.0900736	−	0.995935i	$-0.471290\pi$
$389$	−4.28173e7	−	2.47206e7i	−0.727395	−	0.419962i	−0.817468	+	0.575974i	$0.804623\pi$
$390$		0			0
$391$	−21492.0	−	37225.2i	−0.000359539	−	0.000622741i
$392$	1.82802e7	−	1.05541e7i	0.303474	−	0.175211i
$393$		0			0
$394$	1.52160e7	−	2.63549e7i	0.248778	−	0.430896i
$395$			6.10837e6i			0.0991137i
$396$		0			0
$397$	1.56911e7			0.250774			0.125387	−	0.992108i	$-0.459983\pi$
$397$	1.56911e7			0.250774			0.125387	+	0.992108i	$0.459983\pi$
$398$	−2.77233e6	−	1.60061e6i	−0.0439740	−	0.0253884i
$399$		0			0
$400$	7.49210e6	+	1.29767e7i	0.117064	+	0.202761i
$401$	4.10941e7	−	2.37257e7i	0.637304	−	0.367947i	−0.146272	−	0.989244i	$-0.546727\pi$
$401$	4.10941e7	−	2.37257e7i	0.637304	−	0.367947i	0.783575	+	0.621297i	$0.213394\pi$
$402$		0			0
$403$	6.70165e7	−	1.16076e8i	1.02392	−	1.77348i
$404$			2.14010e7i			0.324556i
$405$		0			0
$406$	8.03711e7			1.20094
$407$	6.09857e7	+	3.52101e7i	0.904576	+	0.522257i
$408$		0			0
$409$	5.77558e7	+	1.00036e8i	0.844162	+	1.46213i	0.886347	+	0.463022i	$0.153235\pi$
$409$	5.77558e7	+	1.00036e8i	0.844162	+	1.46213i	−0.0421850	+	0.999110i	$0.513432\pi$
$410$	−3.15665e7	+	1.82249e7i	−0.458009	+	0.264432i
$411$		0			0
$412$	3.18946e7	−	5.52431e7i	0.456064	−	0.789925i
$413$		−	1.20923e8i		−	1.71656i
$414$		0			0
$415$	1.90994e6			0.0267225
$416$	1.68958e7	+	9.75477e6i	0.234692	+	0.135499i
$417$		0			0
$418$	2.17812e7	+	3.77262e7i	0.298232	+	0.516553i
$419$	−1.27040e8	+	7.33466e7i	−1.72702	+	0.997098i	−0.825447	+	0.564480i	$0.809077\pi$
$419$	−1.27040e8	+	7.33466e7i	−1.72702	+	0.997098i	−0.901577	+	0.432618i	$0.857590\pi$
$420$		0			0
$421$	−6.96194e7	+	1.20584e8i	−0.933005	+	1.61601i	−0.154852	+	0.987938i	$0.549490\pi$
$421$	−6.96194e7	+	1.20584e8i	−0.933005	+	1.61601i	−0.778153	+	0.628075i	$0.783843\pi$
$422$			7.64609e7i			1.01742i
$423$		0			0
$424$	4.32161e7			0.566955
$425$	−161295.	−	93123.8i	−0.00210114	−	0.00121309i
$426$		0			0
$427$	3.20650e6	+	5.55382e6i	0.0411858	+	0.0713359i
$428$	−7.22384e6	+	4.17069e6i	−0.0921376	+	0.0531957i
$429$		0			0
$430$	1.86960e6	−	3.23824e6i	0.0235149	−	0.0407290i
$431$			1.00392e8i			1.25391i	0.779056	+	0.626954i	$0.215699\pi$
$431$			1.00392e8i			1.25391i	−0.779056	+	0.626954i	$0.784301\pi$
$432$		0			0
$433$	−4.00631e7			−0.493493			−0.246747	−	0.969080i	$-0.579362\pi$
$433$	−4.00631e7			−0.493493			−0.246747	+	0.969080i	$0.579362\pi$
$434$	9.43605e7	+	5.44791e7i	1.15431	+	0.666439i
$435$		0			0
$436$	3.11130e6	+	5.38892e6i	0.0375389	+	0.0650193i
$437$	1.67994e7	−	9.69915e6i	0.201303	−	0.116222i
$438$		0			0
$439$	6.92959e7	−	1.20024e8i	0.819057	−	1.41865i	−0.0873208	−	0.996180i	$-0.527831\pi$
$439$	6.92959e7	−	1.20024e8i	0.819057	−	1.41865i	0.906378	−	0.422468i	$-0.138836\pi$
$440$		−	4.22152e7i		−	0.495576i
$441$		0			0
$442$	−242496.			−0.00280826
$443$	−9.65121e7	−	5.57213e7i	−1.11012	−	0.640929i	−0.171261	−	0.985226i	$-0.554784\pi$
$443$	−9.65121e7	−	5.57213e7i	−1.11012	−	0.640929i	−0.938861	+	0.344296i	$0.888117\pi$
$444$		0			0
$445$	−1.12482e7	−	1.94825e7i	−0.127645	−	0.221088i
$446$	2.62332e7	−	1.51458e7i	0.295697	−	0.170721i
$447$		0			0
$448$	−7.92986e6	+	1.37349e7i	−0.0881924	+	0.152754i
$449$			6.11166e7i			0.675181i	0.941293	+	0.337591i	$0.109612\pi$
$449$			6.11166e7i			0.675181i	−0.941293	+	0.337591i	$0.890388\pi$
$450$		0			0
$451$	4.96619e7			0.541370
$452$	2.27771e7	+	1.31504e7i	0.246651	+	0.142404i
$453$		0			0
$454$	−3.84845e7	−	6.66572e7i	−0.411262	−	0.712327i
$455$	2.45566e8	−	1.41778e8i	2.60696	−	1.50513i
$456$		0			0
$457$	−1.78332e7	+	3.08881e7i	−0.186845	+	0.323625i	−0.944197	−	0.329382i	$-0.893160\pi$
$457$	−1.78332e7	+	3.08881e7i	−0.186845	+	0.323625i	0.757352	+	0.653007i	$0.226493\pi$
$458$		−	2.45870e7i		−	0.255923i
$459$		0			0
$460$	−1.87983e7			−0.193128
$461$	1.31621e8	+	7.59914e7i	1.34345	+	0.775642i	0.987312	−	0.158791i	$-0.0507595\pi$
$461$	1.31621e8	+	7.59914e7i	1.34345	+	0.775642i	0.356139	+	0.934433i	$0.384093\pi$
$462$		0			0
$463$	−5.74891e7	−	9.95740e7i	−0.579218	−	1.00324i	−0.995569	−	0.0940314i	$-0.970025\pi$
$463$	−5.74891e7	−	9.95740e7i	−0.579218	−	1.00324i	0.416351	−	0.909204i	$-0.363309\pi$
$464$	−2.60322e7	+	1.50297e7i	−0.260589	+	0.150451i
$465$		0			0
$466$	−5.72284e7	+	9.91226e7i	−0.565528	+	0.979523i
$467$		−	8.81705e7i		−	0.865711i	−0.901463	−	0.432855i	$-0.857506\pi$
$467$		−	8.81705e7i		−	0.865711i	0.901463	−	0.432855i	$-0.142494\pi$
$468$		0			0
$469$	−8.17805e7			−0.792741
$470$	−6.54396e7	−	3.77816e7i	−0.630300	−	0.363904i
$471$		0			0
$472$	2.26130e7	+	3.91669e7i	0.215046	+	0.372471i
$473$	−4.41202e6	+	2.54728e6i	−0.0416921	+	0.0240710i
$474$		0			0
$475$	4.20260e7	−	7.27911e7i	0.392136	−	0.679200i
$476$		−	197130.i		−	0.00182781i
$477$		0			0
$478$	1.15370e8			1.05635
$479$	−7.74563e7	−	4.47194e7i	−0.704774	−	0.406902i	0.104349	−	0.994541i	$-0.466724\pi$
$479$	−7.74563e7	−	4.47194e7i	−0.704774	−	0.406902i	−0.809123	+	0.587639i	$0.800057\pi$
$480$		0			0
$481$	−8.84538e7	−	1.53206e8i	−0.794843	−	1.37671i
$482$	1.52894e7	−	8.82732e6i	0.136536	−	0.0788293i
$483$		0			0
$484$	−413552.	+	716293.i	−0.00364749	+	0.00631764i
$485$			5.59111e7i			0.490087i
$486$		0			0
$487$	−7.51688e7			−0.650805			−0.325403	−	0.945576i	$-0.605500\pi$
$487$	−7.51688e7			−0.650805			−0.325403	+	0.945576i	$0.605500\pi$
$488$	−2.07717e6	−	1.19925e6i	−0.0178736	−	0.0103193i
$489$		0			0
$490$	5.73706e7	+	9.93689e7i	0.487642	+	0.844621i
$491$	−3.90424e7	+	2.25411e7i	−0.329831	+	0.190428i	−0.655766	−	0.754964i	$-0.727654\pi$
$491$	−3.90424e7	+	2.25411e7i	−0.329831	+	0.190428i	0.325935	+	0.945392i	$0.394321\pi$
$492$		0			0
$493$	186813.	−	323570.i	0.00155907	−	0.00270039i
$494$		−	1.09436e8i		−	0.907780i
$495$		0			0
$496$	−4.07511e7			−0.333960
$497$	2.22768e8	+	1.28615e8i	1.81461	+	1.04767i
$498$		0			0
$499$	−4.57729e7	−	7.92810e7i	−0.368389	−	0.638069i	0.620925	−	0.783870i	$-0.286757\pi$
$499$	−4.57729e7	−	7.92810e7i	−0.368389	−	0.638069i	−0.989314	+	0.145801i	$0.953424\pi$
$500$	4.78203e6	−	2.76091e6i	0.0382562	−	0.0220873i
$501$		0			0
$502$	1.44114e7	−	2.49612e7i	0.113919	−	0.197313i
$503$			1.61043e8i			1.26543i	0.774386	+	0.632713i	$0.218059\pi$
$503$			1.61043e8i			1.26543i	−0.774386	+	0.632713i	$0.781941\pi$
$504$		0			0
$505$	−1.16333e8			−0.903295
$506$	2.21809e7	+	1.28061e7i	0.171209	+	0.0988476i
$507$		0			0
$508$	4.89154e7	+	8.47239e7i	0.373125	+	0.646272i
$509$	−2.07841e7	+	1.19997e7i	−0.157608	+	0.0909951i	−0.576730	−	0.816935i	$-0.695671\pi$
$509$	−2.07841e7	+	1.19997e7i	−0.157608	+	0.0909951i	0.419122	+	0.907930i	$0.362338\pi$
$510$		0			0
$511$	5.71468e7	−	9.89812e7i	0.428282	−	0.741806i
$512$		−	5.93164e6i		−	0.0441942i
$513$		0			0
$514$	−8.16702e7			−0.601415
$515$	3.00295e8	+	1.73375e8i	2.19850	+	1.26930i
$516$		0			0
$517$	5.14764e7	+	8.91597e7i	0.372509	+	0.645204i
$518$	1.24545e8	−	7.19059e7i	0.896058	−	0.517339i
$519$		0			0
$520$	−5.30258e7	+	9.18434e7i	−0.377118	+	0.653187i
$521$			9.00897e7i			0.637033i	0.947917	+	0.318517i	$0.103185\pi$
$521$			9.00897e7i			0.637033i	−0.947917	+	0.318517i	$0.896815\pi$
$522$		0			0
$523$	−3.77691e7			−0.264016			−0.132008	−	0.991249i	$-0.542143\pi$
$523$	−3.77691e7			−0.264016			−0.132008	+	0.991249i	$0.542143\pi$
$524$	−8.51858e7	−	4.91820e7i	−0.592070	−	0.341832i
$525$		0			0
$526$	8.84073e7	+	1.53126e8i	0.607478	+	1.05218i
$527$	438660.	−	253260.i	0.00299706	−	0.00173035i
$528$		0			0
$529$	−6.83154e7	+	1.18326e8i	−0.461479	+	0.799304i
$530$			2.34918e8i			1.57793i
$531$		0			0
$532$	8.89631e7			0.590847
$533$	−1.08045e8	−	6.23796e7i	−0.713545	−	0.411965i
$534$		0			0
$535$	−2.26714e7	−	3.92679e7i	−0.148053	−	0.256435i
$536$	2.64887e7	−	1.52932e7i	0.172015	−	0.0993128i
$537$		0			0
$538$	710892.	−	1.23130e6i	0.00456517	−	0.00790710i
$539$		−	1.56332e8i		−	0.998347i
$540$		0			0
$541$	2.54800e7			0.160919			0.0804595	−	0.996758i	$-0.474361\pi$
$541$	2.54800e7			0.160919			0.0804595	+	0.996758i	$0.474361\pi$
$542$	1.45205e8	+	8.38343e7i	0.911978	+	0.526531i
$543$		0			0
$544$	36864.0	+	63850.3i	0.000228984	+	0.000396612i
$545$	−2.92936e7	+	1.69126e7i	−0.180960	+	0.104477i
$546$		0			0
$547$	−1.02608e8	+	1.77722e8i	−0.626930	+	1.08587i	0.361234	+	0.932475i	$0.382355\pi$
$547$	−1.02608e8	+	1.77722e8i	−0.626930	+	1.08587i	−0.988164	+	0.153399i	$0.950978\pi$
$548$		−	1.43492e8i		−	0.871938i
$549$		0			0
$550$	1.10977e8			0.667027
$551$	1.46024e8	+	8.43071e7i	0.872911	+	0.503975i
$552$		0			0
$553$	−8.49807e6	−	1.47191e7i	−0.0502510	−	0.0870373i
$554$	−6.47707e7	+	3.73954e7i	−0.380934	+	0.219932i
$555$		0			0
$556$	−1.74772e7	+	3.02715e7i	−0.101683	+	0.176120i
$557$			2.41143e8i			1.39543i	0.716375	+	0.697715i	$0.245800\pi$
$557$			2.41143e8i			1.39543i	−0.716375	+	0.697715i	$0.754200\pi$
$558$		0			0
$559$	1.27984e7			0.0732690
$560$	−7.46614e7	−	4.31058e7i	−0.425140	−	0.245455i
$561$		0			0
$562$	−1.73202e7	−	2.99994e7i	−0.0975760	−	0.169007i
$563$	1.46252e8	−	8.44386e7i	0.819552	−	0.473168i	−0.0307102	−	0.999528i	$-0.509777\pi$
$563$	1.46252e8	−	8.44386e7i	0.819552	−	0.473168i	0.850262	+	0.526360i	$0.176444\pi$
$564$		0			0
$565$	−7.14838e7	+	1.23814e8i	−0.396335	+	0.686472i
$566$			3.81456e7i			0.210375i
$567$		0			0
$568$	−9.62058e7			−0.524996
$569$	−2.11306e8	−	1.21998e8i	−1.14703	−	0.662238i	−0.198869	−	0.980026i	$-0.563727\pi$
$569$	−2.11306e8	−	1.21998e8i	−1.14703	−	0.662238i	−0.948162	+	0.317788i	$0.897060\pi$
$570$		0			0
$571$	−1.20751e8	−	2.09147e8i	−0.648608	−	1.12342i	−0.983456	−	0.181149i	$-0.942018\pi$
$571$	−1.20751e8	−	2.09147e8i	−0.648608	−	1.12342i	0.334848	−	0.942272i	$-0.391315\pi$
$572$	1.25134e8	−	7.22463e7i	0.668633	−	0.386036i
$573$		0			0
$574$	5.07096e7	−	8.78317e7i	0.268136	−	0.464425i
$575$		−	4.94177e7i		−	0.259944i
$576$		0			0
$577$	−4.93979e7			−0.257147			−0.128573	−	0.991700i	$-0.541040\pi$
$577$	−4.93979e7			−0.257147			−0.128573	+	0.991700i	$0.541040\pi$
$578$	1.18249e8	+	6.82709e7i	0.612368	+	0.353551i
$579$		0			0
$580$	−8.16996e7	−	1.41508e8i	−0.418732	−	0.725264i
$581$	−4.60232e6	+	2.65715e6i	−0.0234665	+	0.0135484i
$582$		0			0
$583$	1.60035e8	−	2.77188e8i	0.807623	−	1.39884i
$584$			4.27466e7i			0.214617i
$585$		0			0
$586$	5.67038e7			0.281786
$587$	−1.49001e8	−	8.60260e7i	−0.736675	−	0.425320i	0.0841840	−	0.996450i	$-0.473172\pi$
$587$	−1.49001e8	−	8.60260e7i	−0.736675	−	0.425320i	−0.820859	+	0.571131i	$0.806505\pi$
$588$		0			0
$589$	1.14294e8	+	1.97963e8i	0.559343	+	0.968810i
$590$	−2.12906e8	+	1.22922e8i	−1.03665	+	0.598511i
$591$		0			0
$592$	−2.68933e7	+	4.65806e7i	−0.129622	+	0.224512i
$593$		−	2.70643e8i		−	1.29788i	−0.760841	−	0.648938i	$-0.775213\pi$
$593$		−	2.70643e8i		−	1.29788i	0.760841	−	0.648938i	$-0.224787\pi$
$594$		0			0
$595$	1.07158e6			0.00508712
$596$	6.15467e7	+	3.55340e7i	0.290714	+	0.167844i
$597$		0			0
$598$	−3.21711e7	−	5.57220e7i	−0.150440	−	0.260569i
$599$	1.50082e8	−	8.66497e7i	0.698308	−	0.403169i	−0.108409	−	0.994106i	$-0.534575\pi$
$599$	1.50082e8	−	8.66497e7i	0.698308	−	0.403169i	0.806717	+	0.590938i	$0.201242\pi$
$600$		0			0
$601$	2.15545e8	−	3.73335e8i	0.992921	−	1.71979i	0.393598	−	0.919282i	$-0.371230\pi$
$601$	2.15545e8	−	3.73335e8i	0.992921	−	1.71979i	0.599323	−	0.800507i	$-0.295437\pi$
$602$			1.04041e7i			0.0476886i
$603$		0			0
$604$	1.30519e8			0.592327
$605$	−3.89369e6	−	2.24802e6i	−0.0175831	−	0.0101516i
$606$		0			0
$607$	−8.34953e6	−	1.44618e7i	−0.0373332	−	0.0646631i	0.846755	−	0.531983i	$-0.178553\pi$
$607$	−8.34953e6	−	1.44618e7i	−0.0373332	−	0.0646631i	−0.884088	+	0.467320i	$0.845220\pi$
$608$	−2.88151e7	+	1.66364e7i	−0.128206	+	0.0740199i
$609$		0			0
$610$	6.51900e6	−	1.12912e7i	0.0287205	−	0.0497453i
$611$		−	2.58635e8i		−	1.13387i
$612$		0			0
$613$	−1.92321e8			−0.834920			−0.417460	−	0.908695i	$-0.637080\pi$
$613$	−1.92321e8			−0.834920			−0.417460	+	0.908695i	$0.637080\pi$
$614$	−2.56246e7	−	1.47944e7i	−0.110701	−	0.0639133i
$615$		0			0
$616$	5.87305e7	+	1.01724e8i	0.251259	+	0.435193i
$617$	1.61967e8	−	9.35117e7i	0.689558	−	0.398117i	−0.113888	−	0.993494i	$-0.536331\pi$
$617$	1.61967e8	−	9.35117e7i	0.689558	−	0.398117i	0.803447	+	0.595377i	$0.202997\pi$
$618$		0			0
$619$	−1.27437e8	+	2.20727e8i	−0.537307	+	0.930643i	0.461741	+	0.887015i	$0.347225\pi$
$619$	−1.27437e8	+	2.20727e8i	−0.537307	+	0.930643i	−0.999048	+	0.0436278i	$0.986108\pi$
$620$		−	2.21518e8i		−	0.929468i
$621$		0			0
$622$	−1.76619e8			−0.733950
$623$	5.42088e7	+	3.12975e7i	0.224185	+	0.129433i
$624$		0			0
$625$	1.29328e8	+	2.24003e8i	0.529729	+	0.917517i
$626$	−1.10118e8	+	6.35769e7i	−0.448887	+	0.259165i
$627$		0			0
$628$	9.84908e7	−	1.70591e8i	0.397665	−	0.688775i
$629$		−	668547.i		−	0.00268646i
$630$		0			0
$631$	9.23602e7			0.367618			0.183809	−	0.982962i	$-0.441157\pi$
$631$	9.23602e7			0.367618			0.183809	+	0.982962i	$0.441157\pi$
$632$	5.50504e6	+	3.17834e6i	0.0218077	+	0.0125907i
$633$		0			0
$634$	−7.81243e7	−	1.35315e8i	−0.306562	−	0.530981i
$635$	−4.60549e8	+	2.65898e8i	−1.79869	+	1.03847i
$636$		0			0
$637$	−1.96366e8	+	3.40116e8i	−0.759711	+	1.31586i
$638$			2.22627e8i			0.857267i
$639$		0			0
$640$	3.22437e7			0.123000
$641$	−3.67771e8	−	2.12333e8i	−1.39638	−	0.806200i	−0.402368	−	0.915478i	$-0.631813\pi$
$641$	−3.67771e8	−	2.12333e8i	−1.39638	−	0.806200i	−0.994011	+	0.109278i	$0.965146\pi$
$642$		0			0
$643$	−1.87973e8	−	3.25579e8i	−0.707071	−	1.22468i	−0.965939	−	0.258770i	$-0.916683\pi$
$643$	−1.87973e8	−	3.25579e8i	−0.707071	−	1.22468i	0.258868	−	0.965913i	$-0.416650\pi$
$644$	4.52976e7	−	2.61526e7i	0.169597	−	0.0979168i
$645$		0			0
$646$	206784.	−	358160.i	0.000767042	−	0.00132856i
$647$		−	2.63747e7i		−	0.0973813i	−0.998814	−	0.0486906i	$-0.984495\pi$
$647$		−	2.63747e7i		−	0.0973813i	0.998814	−	0.0486906i	$-0.0155048\pi$
$648$		0			0
$649$	3.34955e8			1.22533
$650$	−2.41441e8	−	1.39396e8i	−0.879166	−	0.507587i
$651$		0			0
$652$	1.28111e7	+	2.21895e7i	0.0462216	+	0.0800581i
$653$	−2.24090e8	+	1.29378e8i	−0.804790	+	0.464645i	−0.845143	−	0.534540i	$-0.820485\pi$
$653$	−2.24090e8	+	1.29378e8i	−0.804790	+	0.464645i	0.0403536	+	0.999185i	$0.487152\pi$
$654$		0			0
$655$	2.67348e8	−	4.63060e8i	0.951377	−	1.64783i
$656$			3.79315e7i			0.134366i
$657$		0			0
$658$	2.10250e8			0.738002
$659$	−1.20676e8	−	6.96725e7i	−0.421663	−	0.243447i	0.274126	−	0.961694i	$-0.411612\pi$
$659$	−1.20676e8	−	6.96725e7i	−0.421663	−	0.243447i	−0.695789	+	0.718247i	$0.744945\pi$
$660$		0			0
$661$	2.36272e8	+	4.09236e8i	0.818104	+	1.41700i	0.907077	+	0.420964i	$0.138308\pi$
$661$	2.36272e8	+	4.09236e8i	0.818104	+	1.41700i	−0.0889731	+	0.996034i	$0.528359\pi$
$662$	2.82249e7	−	1.62957e7i	0.0972878	−	0.0561691i
$663$		0			0
$664$	993792.	−	1.72130e6i	0.00339462	−	0.00587966i
$665$			4.83593e8i			1.64443i
$666$		0			0
$667$	9.91354e7			0.334081
$668$	−1.33151e8	−	7.68747e7i	−0.446699	−	0.257902i
$669$		0			0
$670$	8.31323e7	+	1.43989e8i	0.276405	+	0.478747i
$671$	−1.53840e7	+	8.88197e6i	−0.0509216	+	0.0293996i
$672$		0			0
$673$	−2.74417e8	+	4.75304e8i	−0.900254	+	1.55929i	−0.0730906	+	0.997325i	$0.523286\pi$
$673$	−2.74417e8	+	4.75304e8i	−0.900254	+	1.55929i	−0.827164	+	0.561961i	$0.810047\pi$
$674$			2.26869e8i			0.740961i
$675$		0			0
$676$	−2.08532e8			−0.675044
$677$	−8.72609e7	−	5.03801e7i	−0.281225	−	0.162365i	0.352753	−	0.935717i	$-0.385246\pi$
$677$	−8.72609e7	−	5.03801e7i	−0.281225	−	0.162365i	−0.633978	+	0.773351i	$0.718579\pi$
$678$		0			0
$679$	−7.77846e7	−	1.34727e8i	−0.248476	−	0.430373i
$680$	−347083.	+	200388.i	−0.00110384	+	0.000637303i
$681$		0			0
$682$	−1.50906e8	+	2.61378e8i	−0.475724	+	0.823977i
$683$			313056.i			0.000982562i	1.00000		0.000491281i	$0.000156380\pi$
$683$			313056.i			0.000982562i	−1.00000		0.000491281i	$0.999844\pi$
$684$		0			0
$685$	7.80005e8			2.42675
$686$	2.47069e6	+	1.42645e6i	0.00765326	+	0.00441861i
$687$		0			0
$688$	−1.94560e6	−	3.36988e6i	−0.00597432	−	0.0103478i
$689$	−6.96344e8	+	4.02034e8i	−2.12895	+	1.22915i
$690$		0			0
$691$	1.86406e8	−	3.22865e8i	0.564971	−	0.978558i	−0.432082	−	0.901834i	$-0.642221\pi$
$691$	1.86406e8	−	3.22865e8i	0.564971	−	0.978558i	0.997052	−	0.0767236i	$-0.0244459\pi$
$692$		−	1.14238e8i		−	0.344739i
$693$		0			0
$694$	−3.83607e8			−1.14765
$695$	−1.64552e8	−	9.50043e7i	−0.490173	−	0.283002i
$696$		0			0
$697$	−235737.	−	408308.i	−0.000696193	−	0.00120584i
$698$	2.06070e8	−	1.18975e8i	0.605966	−	0.349855i
$699$		0			0
$700$	1.13318e8	−	1.96272e8i	0.330373	−	0.572223i
$701$		−	6.21170e8i		−	1.80325i	−0.432517	−	0.901626i	$-0.642374\pi$
$701$		−	6.21170e8i		−	1.80325i	0.432517	−	0.901626i	$-0.357626\pi$
$702$		0			0
$703$	3.01709e8			0.868406
$704$	−3.80456e7	−	2.19656e7i	−0.109040	−	0.0629543i
$705$		0			0
$706$	−4.97734e7	−	8.62101e7i	−0.141444	−	0.244988i
$707$	2.80323e8	−	1.61845e8i	0.793234	−	0.457974i
$708$		0			0
$709$	1.23255e8	−	2.13484e8i	0.345833	−	0.599000i	−0.639672	−	0.768648i	$-0.720930\pi$
$709$	1.23255e8	−	2.13484e8i	0.345833	−	0.599000i	0.985505	+	0.169648i	$0.0542631\pi$
$710$		−	5.22964e8i		−	1.46116i
$711$		0			0
$712$	−2.34109e7			−0.0648603
$713$	1.16391e8	+	6.71984e7i	0.321108	+	0.185392i
$714$		0			0
$715$	3.92722e8	+	6.80215e8i	1.07440	+	1.86092i
$716$	−2.06100e8	+	1.18992e8i	−0.561485	+	0.324173i
$717$		0			0
$718$	−3.95754e7	+	6.85466e7i	−0.106918	+	0.185188i
$719$			9.60389e7i			0.258381i	0.991620	+	0.129191i	$0.0412379\pi$
$719$			9.60389e7i			0.258381i	−0.991620	+	0.129191i	$0.958762\pi$
$720$		0			0
$721$	−9.64811e8			−2.57417
$722$	−6.88421e7	−	3.97460e7i	−0.182912	−	0.105604i
$723$		0			0
$724$	−1.66099e8	−	2.87692e8i	−0.437675	−	0.758075i
$725$	3.72001e8	−	2.14775e8i	0.976179	−	0.563597i
$726$		0			0
$727$	−1.95685e8	+	3.38937e8i	−0.509278	+	0.882096i	0.490664	+	0.871349i	$0.336754\pi$
$727$	−1.95685e8	+	3.38937e8i	−0.509278	+	0.882096i	−0.999942	+	0.0107471i	$0.996579\pi$
$728$		−	2.95082e8i		−	0.764801i
$729$		0			0
$730$	−2.32366e8			−0.597315
$731$	41886.3	+	24183.1i	0.000107231	+	6.19097e-5i
$732$		0			0
$733$	−1.74539e7	−	3.02311e7i	−0.0443181	−	0.0767611i	0.843015	−	0.537889i	$-0.180778\pi$
$733$	−1.74539e7	−	3.02311e7i	−0.0443181	−	0.0767611i	−0.887334	+	0.461128i	$0.847445\pi$
$734$	1.30242e8	−	7.51951e7i	0.329353	−	0.190152i
$735$		0			0
$736$	−9.78125e6	+	1.69416e7i	−0.0245336	+	0.0424934i
$737$		−	2.26531e8i		−	0.565881i
$738$		0			0
$739$	−3.02999e8			−0.750773			−0.375386	−	0.926868i	$-0.622490\pi$
$739$	−3.02999e8			−0.750773			−0.375386	+	0.926868i	$0.622490\pi$
$740$	−2.53207e8	−	1.46189e8i	−0.624856	−	0.360761i
$741$		0			0
$742$	−3.26822e8	−	5.66072e8i	−0.800018	−	1.38567i
$743$	−2.12720e8	+	1.22814e8i	−0.518612	+	0.299421i	−0.736367	−	0.676583i	$-0.763460\pi$
$743$	−2.12720e8	+	1.22814e8i	−0.518612	+	0.299421i	0.217754	+	0.976004i	$0.430127\pi$
$744$		0			0
$745$	−1.93159e8	+	3.34560e8i	−0.467138	+	0.809107i
$746$		−	1.01161e8i		−	0.243667i
$747$		0			0
$748$	546048.			0.00130475
$749$	1.09261e8	+	6.30816e7i	0.260027	+	0.150127i
$750$		0			0
$751$	4.11635e7	+	7.12973e7i	0.0971835	+	0.168327i	0.910518	−	0.413470i	$-0.135683\pi$
$751$	4.11635e7	+	7.12973e7i	0.0971835	+	0.168327i	−0.813334	+	0.581797i	$0.802350\pi$
$752$	−6.80997e7	+	3.93174e7i	−0.160137	+	0.0924552i
$753$		0			0
$754$	2.79638e8	−	4.84348e8i	0.652353	−	1.12991i
$755$			7.09484e8i			1.64855i
$756$		0			0
$757$	−6.03579e8			−1.39138			−0.695691	−	0.718341i	$-0.744902\pi$
$757$	−6.03579e8			−1.39138			−0.695691	+	0.718341i	$0.744902\pi$
$758$	3.53205e8	+	2.03923e8i	0.810998	+	0.468230i
$759$		0			0
$760$	−9.04335e7	−	1.56635e8i	−0.206010	−	0.356820i
$761$	2.01769e8	−	1.16491e8i	0.457825	−	0.264325i	−0.253304	−	0.967387i	$-0.581517\pi$
$761$	2.01769e8	−	1.16491e8i	0.457825	−	0.264325i	0.711129	+	0.703061i	$0.248184\pi$
$762$		0			0
$763$	4.70584e7	−	8.15075e7i	0.105941	−	0.183495i
$764$			4.15810e8i			0.932426i
$765$		0			0
$766$	−4.91382e7			−0.109328
$767$	−7.28729e8	−	4.20732e8i	−1.61503	−	0.932436i
$768$		0			0
$769$	−4.07898e8	−	7.06500e8i	−0.896958	−	1.55358i	−0.831362	−	0.555731i	$-0.812438\pi$
$769$	−4.07898e8	−	7.06500e8i	−0.896958	−	1.55358i	−0.0655965	−	0.997846i	$-0.520895\pi$
$770$	−5.52961e8	+	3.19252e8i	−1.21122	+	0.699297i
$771$		0			0
$772$	−6.29113e7	+	1.08966e8i	−0.136734	+	0.236830i
$773$			3.66587e8i			0.793667i	0.917891	+	0.396833i	$0.129891\pi$
$773$			3.66587e8i			0.793667i	−0.917891	+	0.396833i	$0.870109\pi$
$774$		0			0
$775$	5.82335e8			1.25103
$776$	5.03888e7	+	2.90920e7i	0.107832	+	0.0622570i
$777$		0			0
$778$	−1.39841e8	−	2.42211e8i	−0.296958	−	0.514346i
$779$	1.84266e8	−	1.06386e8i	0.389792	−	0.225047i
$780$		0			0
$781$	−3.56262e8	+	6.17064e8i	−0.747854	+	1.29532i
$782$		−	243154.i		−	0.000508466i
$783$		0			0
$784$	1.19406e8			0.247786
$785$	9.27314e8	+	5.35385e8i	1.91698	+	1.10677i
$786$		0			0
$787$	2.01231e8	+	3.48542e8i	0.412830	+	0.715042i	0.995198	−	0.0978832i	$-0.0312072\pi$
$787$	2.01231e8	+	3.48542e8i	0.412830	+	0.715042i	−0.582368	+	0.812925i	$0.697874\pi$
$788$	1.49086e8	−	8.60748e7i	0.304690	−	0.175913i
$789$		0			0
$790$	−1.72771e7	+	2.99248e7i	−0.0350420	+	0.0606945i
$791$		−	3.97798e8i		−	0.803773i
$792$		0			0
$793$	4.46260e7			0.0894887
$794$	7.68704e7	+	4.43811e7i	0.153567	+	0.0886619i
$795$		0			0
$796$	−9.05440e6	−	1.56827e7i	−0.0179523	−	0.0310943i
$797$	4.49415e8	−	2.59470e8i	0.887713	−	0.512521i	0.0145190	−	0.999895i	$-0.495378\pi$
$797$	4.49415e8	−	2.59470e8i	0.887713	−	0.512521i	0.873194	+	0.487373i	$0.162045\pi$
$798$		0			0
$799$	488700.	−	846453.i	0.000958081	−	0.00165944i
$800$			8.47634e7i			0.165553i
$801$		0			0
$802$	2.68426e8			0.520356
$803$	2.74177e8	+	1.58296e8i	0.529522	+	0.305720i
$804$		0			0
$805$	1.42162e8	+	2.46233e8i	0.272519	+	0.472017i
$806$	6.56625e8	−	3.79102e8i	1.25404	−	0.724021i
$807$		0			0
$808$	−6.05311e7	+	1.04843e8i	−0.114748	+	0.198749i
$809$			3.04036e6i			0.00574221i	0.999996	+	0.00287110i	$0.000913902\pi$
$809$			3.04036e6i			0.00574221i	−0.999996	+	0.00287110i	$0.999086\pi$
$810$		0			0
$811$	2.25521e8			0.422790			0.211395	−	0.977401i	$-0.432199\pi$
$811$	2.25521e8			0.422790			0.211395	+	0.977401i	$0.432199\pi$
$812$	3.93736e8	+	2.27324e8i	0.735423	+	0.424597i
$813$		0			0
$814$	1.99179e8	+	3.44987e8i	0.369292	+	0.639632i
$815$	−1.20620e8	+	6.96398e7i	−0.222816	+	0.128643i
$816$		0			0
$817$	−1.09136e7	+	1.89029e7i	−0.0200125	+	0.0346627i
$818$			6.53432e8i			1.19383i
$819$		0			0
$820$	−2.06191e8			−0.373963
$821$	2.39920e8	+	1.38518e8i	0.433547	+	0.250309i	0.700857	−	0.713302i	$-0.252801\pi$
$821$	2.39920e8	+	1.38518e8i	0.433547	+	0.250309i	−0.267309	+	0.963611i	$0.586135\pi$
$822$		0			0
$823$	3.53668e8	+	6.12571e8i	0.634448	+	1.09890i	0.986632	+	0.162966i	$0.0521059\pi$
$823$	3.53668e8	+	6.12571e8i	0.634448	+	1.09890i	−0.352184	+	0.935931i	$0.614561\pi$
$824$	3.12502e8	−	1.80423e8i	0.558562	−	0.322486i
$825$		0			0
$826$	3.42022e8	−	5.92399e8i	0.606895	−	1.05117i
$827$			2.66346e8i			0.470900i	0.971886	+	0.235450i	$0.0756564\pi$
$827$			2.66346e8i			0.470900i	−0.971886	+	0.235450i	$0.924344\pi$
$828$		0			0
$829$	5.03826e8			0.884336			0.442168	−	0.896932i	$-0.354209\pi$
$829$	5.03826e8			0.884336			0.442168	+	0.896932i	$0.354209\pi$
$830$	9.35678e6	+	5.40214e6i	0.0163641	+	0.00944781i
$831$		0			0
$832$	5.51813e7	+	9.55768e7i	0.0958125	+	0.165952i
$833$	−1.28532e6	+	742082.i	−0.00222371	+	0.00128386i
$834$		0			0
$835$	4.17882e8	−	7.23793e8i	0.717785	−	1.24324i
$836$			2.46427e8i			0.421763i
$837$		0			0
$838$	−8.29822e8			−1.41011
$839$	6.61093e8	+	3.81682e8i	1.11938	+	0.646273i	0.941242	−	0.337733i	$-0.109660\pi$
$839$	6.61093e8	+	3.81682e8i	1.11938	+	0.646273i	0.178136	+	0.984006i	$0.442993\pi$
$840$		0			0
$841$	1.33441e8	+	2.31127e8i	0.224338	+	0.388565i
$842$	−6.82128e8	+	3.93827e8i	−1.14269	+	0.659734i
$843$		0			0
$844$	−2.16264e8	+	3.74580e8i	−0.359714	+	0.623042i
$845$		−	1.13355e9i		−	1.87876i
$846$		0			0
$847$	1.25099e7			0.0205876
$848$	2.11715e8	+	1.22234e8i	0.347187	+	0.200449i
$849$		0			0
$850$	−526788.	−	912424.i	−0.000857786	−	0.00148573i
$851$	1.53622e8	−	8.86939e7i	0.249267	−	0.143915i
$852$		0			0
$853$	9.39926e6	−	1.62800e7i	0.0151442	−	0.0262305i	−0.858354	−	0.513058i	$-0.828513\pi$
$853$	9.39926e6	−	1.62800e7i	0.0151442	−	0.0262305i	0.873498	+	0.486827i	$0.161846\pi$
$854$			3.62774e7i			0.0582455i
$855$		0			0
$856$	−4.71859e7			−0.0752300
$857$	−5.94463e8	−	3.43214e8i	−0.944458	−	0.545283i	−0.0531031	−	0.998589i	$-0.516911\pi$
$857$	−5.94463e8	−	3.43214e8i	−0.944458	−	0.545283i	−0.891355	+	0.453306i	$0.850245\pi$
$858$		0			0
$859$	−2.75966e8	−	4.77987e8i	−0.435387	−	0.754113i	0.561940	−	0.827178i	$-0.310055\pi$
$859$	−2.75966e8	−	4.77987e8i	−0.435387	−	0.754113i	−0.997327	+	0.0730651i	$0.976722\pi$
$860$	1.83183e7	−	1.05761e7i	0.0287998	−	0.0166276i
$861$		0			0
$862$	−2.83950e8	+	4.91816e8i	−0.443323	+	0.767858i
$863$		−	3.65665e8i		−	0.568920i	−0.958688	−	0.284460i	$-0.908186\pi$
$863$		−	3.65665e8i		−	0.568920i	0.958688	−	0.284460i	$-0.0918142\pi$
$864$		0			0
$865$	6.20982e8			0.959468
$866$	−1.96268e8	−	1.13316e8i	−0.302202	−	0.174476i
$867$		0			0
$868$	3.08180e8	+	5.33784e8i	0.471244	+	0.816218i
$869$	4.07717e7	−	2.35396e7i	0.0621298	−	0.0358706i
$870$		0			0
$871$	−2.84542e8	+	4.92841e8i	−0.430618	+	0.745852i
$872$			3.52003e7i			0.0530881i
$873$		0			0
$874$	1.09733e8			0.164363
$875$	−7.23282e7	−	4.17587e7i	−0.107965	−	0.0623337i
$876$		0			0
$877$	2.92694e8	+	5.06960e8i	0.433925	+	0.751580i	0.997207	−	0.0746848i	$-0.0237951\pi$
$877$	2.92694e8	+	5.06960e8i	0.433925	+	0.751580i	−0.563283	+	0.826264i	$0.690462\pi$
$878$	6.78959e8	−	3.91997e8i	1.00314	−	0.579161i
$879$		0			0
$880$	1.19402e8	−	2.06811e8i	0.175213	−	0.303477i
$881$			4.29761e8i			0.628491i	0.949342	+	0.314246i	$0.101752\pi$
$881$			4.29761e8i			0.628491i	−0.949342	+	0.314246i	$0.898248\pi$
$882$		0			0
$883$	2.20085e8			0.319675			0.159837	−	0.987143i	$-0.448903\pi$
$883$	2.20085e8			0.319675			0.159837	+	0.987143i	$0.448903\pi$
$884$	−1.18798e6	−	685882.i	−0.00171970	−	0.000992871i
$885$		0			0
$886$	−3.15207e8	−	5.45955e8i	−0.453205	−	0.784975i
$887$	−1.01495e9	+	5.85981e8i	−1.45437	+	0.839678i	−0.998725	−	0.0504855i	$-0.983923\pi$
$887$	−1.01495e9	+	5.85981e8i	−1.45437	+	0.839678i	−0.455641	+	0.890164i	$0.650590\pi$
$888$		0			0
$889$	7.39845e8	−	1.28145e9i	1.05302	−	1.82388i
$890$		−	1.27259e8i		−	0.180517i
$891$		0			0
$892$	1.71355e8			0.241436
$893$	3.81997e8	+	2.20546e8i	0.536421	+	0.309703i
$894$		0			0
$895$	−6.46825e8	−	1.12033e9i	−0.902230	−	1.56271i
$896$	−7.76964e7	+	4.48580e7i	−0.108013	+	0.0623615i
$897$		0			0
$898$	−1.72864e8	+	2.99409e8i	−0.238713	+	0.413462i
$899$			1.16820e9i			1.60783i
$900$		0			0
$901$	−3.03863e6			−0.00415436
$902$	2.43293e8	+	1.40465e8i	0.331520	+	0.191403i
$903$		0			0
$904$	7.43896e7	+	1.28847e8i	0.100695	+	0.174409i
$905$	1.56386e9	−	9.02895e8i	2.10985	−	1.21812i
$906$		0			0
$907$	−3.65807e8	+	6.33597e8i	−0.490264	+	0.849163i	−0.999937	−	0.0112055i	$-0.996433\pi$
$907$	−3.65807e8	+	6.33597e8i	−0.490264	+	0.849163i	0.509673	+	0.860368i	$0.329766\pi$
$908$		−	4.35403e8i		−	0.581612i
$909$		0			0
$910$	1.60403e9			2.12857
$911$	7.95527e8	+	4.59298e8i	1.05220	+	0.607490i	0.923265	−	0.384163i	$-0.125510\pi$
$911$	7.95527e8	+	4.59298e8i	1.05220	+	0.607490i	0.128938	+	0.991653i	$0.458843\pi$
$912$		0			0
$913$	−7.36027e6	−	1.27484e7i	−0.00967123	−	0.0167511i
$914$	−1.74729e8	+	1.00880e8i	−0.228838	+	0.132119i
$915$		0			0
$916$	6.95425e7	−	1.20451e8i	0.0904824	−	0.156720i
$917$			1.48776e9i			1.92941i
$918$		0			0
$919$	−2.15987e8			−0.278279			−0.139139	−	0.990273i	$-0.544434\pi$
$919$	−2.15987e8			−0.278279			−0.139139	+	0.990273i	$0.544434\pi$
$920$	−9.20927e7	−	5.31697e7i	−0.118266	−	0.0682812i
$921$		0			0
$922$	4.29872e8	+	7.44560e8i	0.548462	+	0.949964i
$923$	1.55017e9	−	8.94991e8i	1.97140	−	1.13819i
$924$		0			0
$925$	3.84306e8	−	6.65638e8i	0.485571	−	0.841033i
$926$		−	6.50414e8i		−	0.819138i
$927$		0			0
$928$	−1.70041e8			−0.212770
$929$	2.68575e8	+	1.55062e8i	0.334980	+	0.193401i	0.658050	−	0.752974i	$-0.271382\pi$
$929$	2.68575e8	+	1.55062e8i	0.334980	+	0.193401i	−0.323070	+	0.946375i	$0.604715\pi$
$930$		0			0
$931$	−3.34895e8	−	5.80056e8i	−0.415011	−	0.718821i
$932$	−5.60722e8	+	3.23733e8i	−0.692628	+	0.399889i
$933$		0			0
$934$	2.49384e8	−	4.31946e8i	0.306075	−	0.530137i
$935$			2.96825e6i			0.00363133i
$936$		0			0
$937$	−7.42448e8			−0.902501			−0.451250	−	0.892397i	$-0.649022\pi$
$937$	−7.42448e8			−0.902501			−0.451250	+	0.892397i	$0.649022\pi$
$938$	−4.00641e8	−	2.31310e8i	−0.485453	−	0.280276i
$939$		0			0
$940$	−2.13725e8	−	3.70182e8i	−0.257319	−	0.445689i
$941$	−1.57414e8	+	9.08828e7i	−0.188918	+	0.109072i	−0.591476	−	0.806323i	$-0.701455\pi$
$941$	−1.57414e8	+	9.08828e7i	−0.188918	+	0.109072i	0.402558	+	0.915395i	$0.368121\pi$
$942$		0			0
$943$	6.25489e7	−	1.08338e8i	0.0745907	−	0.129195i
$944$			2.55837e8i			0.304121i
$945$		0			0
$946$	−2.88192e7			−0.0340415
$947$	7.44080e8	+	4.29595e8i	0.876132	+	0.505835i	0.869381	−	0.494142i	$-0.164518\pi$
$947$	7.44080e8	+	4.29595e8i	0.876132	+	0.505835i	0.00675089	+	0.999977i	$0.497851\pi$
$948$		0			0
$949$	−3.97666e8	−	6.88779e8i	−0.465286	−	0.805900i
$950$	4.11769e8	−	2.37735e8i	0.480267	−	0.277282i
$951$		0			0
$952$	557568.	−	965736.i	0.000646230	−	0.00111930i
$953$		−	6.86819e8i		−	0.793530i	−0.917920	−	0.396765i	$-0.870133\pi$
$953$		−	6.86819e8i		−	0.793530i	0.917920	−	0.396765i	$-0.129867\pi$
$954$		0			0
$955$	−2.26029e9			−2.59510
$956$	5.65194e8	+	3.26315e8i	0.646880	+	0.373476i
$957$		0			0
$958$	−2.52971e8	−	4.38159e8i	−0.287723	−	0.498351i
$959$	−1.87955e9	+	1.08516e9i	−2.13107	+	1.23037i
$960$		0			0
$961$	−3.48109e8	+	6.02942e8i	−0.392234	+	0.679369i
$962$		−	1.00074e9i		−	1.12408i
$963$		0			0
$964$	9.98697e7			0.111481
$965$	−5.92324e8	−	3.41978e8i	−0.659139	−	0.380554i
$966$		0			0
$967$	5.47055e8	+	9.47527e8i	0.604995	+	1.04788i	0.992052	+	0.125826i	$0.0401581\pi$
$967$	5.47055e8	+	9.47527e8i	0.604995	+	1.04788i	−0.387058	+	0.922055i	$0.626509\pi$
$968$	−4.05197e6	+	2.33940e6i	−0.00446724	+	0.00257916i
$969$		0			0
$970$	−1.58141e8	+	2.73908e8i	−0.173272	+	0.300116i
$971$		−	4.43115e8i		−	0.484014i	−0.970274	−	0.242007i	$-0.922194\pi$
$971$		−	4.43115e8i		−	0.484014i	0.970274	−	0.242007i	$-0.0778058\pi$
$972$		0			0
$973$	5.28687e8			0.573931
$974$	−3.68251e8	−	2.12610e8i	−0.398535	−	0.230094i
$975$		0			0
$976$	−6.78400e6	−	1.17502e7i	−0.00729687	−	0.0126385i
$977$	−1.03061e9	+	5.95022e8i	−1.10512	+	0.638042i	−0.937561	−	0.347820i	$-0.886922\pi$
$977$	−1.03061e9	+	5.95022e8i	−1.10512	+	0.638042i	−0.167560	+	0.985862i	$0.553589\pi$
$978$		0			0
$979$	−8.66937e7	+	1.50158e8i	−0.0923930	+	0.160029i
$980$			6.49075e8i			0.689631i
$981$		0			0
$982$	−2.55024e8			−0.269306
$983$	−1.02353e9	−	5.90933e8i	−1.07755	−	0.622125i	−0.147317	−	0.989089i	$-0.547064\pi$
$983$	−1.02353e9	−	5.90933e8i	−1.07755	−	0.622125i	−0.930235	+	0.366965i	$0.880397\pi$
$984$		0			0
$985$	4.67892e8	+	8.10413e8i	0.489595	+	0.848004i
$986$	1.83039e6	−	1.05677e6i	0.00190947	−	0.00110243i
$987$		0			0
$988$	3.09533e8	−	5.36126e8i	0.320949	−	0.555900i
$989$			1.28331e7i			0.0132661i
$990$		0			0
$991$	5.09602e8			0.523613			0.261806	−	0.965120i	$-0.415682\pi$
$991$	5.09602e8			0.523613			0.261806	+	0.965120i	$0.415682\pi$
$992$	−1.99639e8	−	1.15262e8i	−0.204508	−	0.118073i
$993$		0			0
$994$	7.27557e8	+	1.26016e9i	0.740811	+	1.28312i
$995$	8.52492e7	−	4.92187e7i	0.0865409	−	0.0499644i
$996$		0			0
$997$	4.95390e8	−	8.58041e8i	0.499875	−	0.865810i	−0.500125	−	0.865953i	$-0.666712\pi$
$997$	4.95390e8	−	8.58041e8i	0.499875	−	0.865810i	1.00000		0.000143848i	$4.57883e-5\pi$
$998$		−	5.17861e8i		−	0.520981i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	162.7.d.d.107.2		4
3.2	odd	2	inner	162.7.d.d.107.1		4
9.2	odd	6		18.7.b.a.17.2	yes	2
9.4	even	3	inner	162.7.d.d.53.1		4
9.5	odd	6	inner	162.7.d.d.53.2		4
9.7	even	3		18.7.b.a.17.1	✓	2
36.7	odd	6		144.7.e.d.17.1		2
36.11	even	6		144.7.e.d.17.2		2
45.2	even	12		450.7.b.a.449.1		4
45.7	odd	12		450.7.b.a.449.3		4
45.29	odd	6		450.7.d.a.251.1		2
45.34	even	6		450.7.d.a.251.2		2
45.38	even	12		450.7.b.a.449.4		4
45.43	odd	12		450.7.b.a.449.2		4
72.11	even	6		576.7.e.k.449.1		2
72.29	odd	6		576.7.e.b.449.1		2
72.43	odd	6		576.7.e.k.449.2		2
72.61	even	6		576.7.e.b.449.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.7.b.a.17.1	✓	2	9.7	even	3
18.7.b.a.17.2	yes	2	9.2	odd	6
144.7.e.d.17.1		2	36.7	odd	6
144.7.e.d.17.2		2	36.11	even	6
162.7.d.d.53.1		4	9.4	even	3	inner
162.7.d.d.53.2		4	9.5	odd	6	inner
162.7.d.d.107.1		4	3.2	odd	2	inner
162.7.d.d.107.2		4	1.1	even	1	trivial
450.7.b.a.449.1		4	45.2	even	12
450.7.b.a.449.2		4	45.43	odd	12
450.7.b.a.449.3		4	45.7	odd	12
450.7.b.a.449.4		4	45.38	even	12
450.7.d.a.251.1		2	45.29	odd	6
450.7.d.a.251.2		2	45.34	even	6
576.7.e.b.449.1		2	72.29	odd	6
576.7.e.b.449.2		2	72.61	even	6
576.7.e.k.449.1		2	72.11	even	6
576.7.e.k.449.2		2	72.43	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 162 = 2 \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	162.d (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(4.89898 + 2.82843i) q^{2} +(16.0000 + 27.7128i) q^{4} +(-150.644 + 86.9741i) q^{5} +(242.000 - 419.156i) q^{7} +181.019i q^{8} -984.000 q^{10} +(1161.06 + 670.337i) q^{11} +(-1684.00 - 2916.77i) q^{13} +(2371.11 - 1368.96i) q^{14} +(-512.000 + 886.810i) q^{16} -12.7279i q^{17} +5744.00 q^{19} +(-4820.60 - 2783.17i) q^{20} +(3792.00 + 6567.94i) q^{22} +(2924.69 - 1688.57i) q^{23} +(7316.50 - 12672.5i) q^{25} -19052.3i q^{26} +15488.0 q^{28} +(25422.0 + 14677.4i) q^{29} +(19898.0 + 34464.3i) q^{31} +(-5016.55 + 2896.31i) q^{32} +(36.0000 - 62.3538i) q^{34} +84191.0i q^{35} +52526.0 q^{37} +(28139.7 + 16246.5i) q^{38} +(-15744.0 - 27269.4i) q^{40} +(32079.7 - 18521.2i) q^{41} +(-1900.00 + 3290.90i) q^{43} +42901.6i q^{44} +19104.0 q^{46} +(66503.6 + 38395.9i) q^{47} +(-58303.5 - 100985. i) q^{49} +(71686.8 - 41388.4i) q^{50} +(53888.0 - 93336.8i) q^{52} -238738. i q^{53} -233208. q^{55} +(75875.4 + 43806.7i) q^{56} +(83028.0 + 143809. i) q^{58} +(216368. - 124920. i) q^{59} +(-6625.00 + 11474.8i) q^{61} +225120. i q^{62} -32768.0 q^{64} +(507368. + 292929. i) q^{65} +(-84484.0 - 146331. i) q^{67} +(352.727 - 203.647i) q^{68} +(-238128. + 412450. i) q^{70} +531467. i q^{71} +236144. q^{73} +(257324. + 148566. i) q^{74} +(91904.0 + 159182. i) q^{76} +(561952. - 324443. i) q^{77} +(17558.0 - 30411.3i) q^{79} -178123. i q^{80} +209544. q^{82} +(-9508.92 - 5489.98i) q^{83} +(1107.00 + 1917.38i) q^{85} +(-18616.1 + 10748.0i) q^{86} +(-121344. + 210174. i) q^{88} +129328. i q^{89} -1.63011e6 q^{91} +(93590.1 + 54034.3i) q^{92} +(217200. + 376201. i) q^{94} +(-865297. + 499579. i) q^{95} +(160712. - 278361. i) q^{97} -659629. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 64 q^{4} + 968 q^{7} - 3936 q^{10} - 6736 q^{13} - 2048 q^{16} + 22976 q^{19} + 15168 q^{22} + 29266 q^{25} + 61952 q^{28} + 79592 q^{31} + 144 q^{34} + 210104 q^{37} - 62976 q^{40} - 7600 q^{43}+ \cdots + 642848 q^{97}+O(q^{100}) \) 4 * q + 64 * q^4 + 968 * q^7 - 3936 * q^10 - 6736 * q^13 - 2048 * q^16 + 22976 * q^19 + 15168 * q^22 + 29266 * q^25 + 61952 * q^28 + 79592 * q^31 + 144 * q^34 + 210104 * q^37 - 62976 * q^40 - 7600 * q^43 + 76416 * q^46 - 233214 * q^49 + 215552 * q^52 - 932832 * q^55 + 332112 * q^58 - 26500 * q^61 - 131072 * q^64 - 337936 * q^67 - 952512 * q^70 + 944576 * q^73 + 367616 * q^76 + 70232 * q^79 + 838176 * q^82 + 4428 * q^85 - 485376 * q^88 - 6520448 * q^91 + 868800 * q^94 + 642848 * q^97

Introduction

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