LMFDB - Embedded newform 112.8.i.d.65.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,8,Mod(65,112)] 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("112.65"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(112, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,-27] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$34.9871228542$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 $ x^10 - x^9 + 342x^8 + 2165x^7 + 113605x^6 + 319380x^5 + 1438128x^4 + 1705752x^3 + 7544016x^2 + 8573040x + 23619600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{4}\cdot 7^{5} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.4
Root			$-8.10827 + 14.0439i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.8.i.d.81.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(9.22936 + 15.9857i) q^{3} +(-88.1161 + 152.621i) q^{5} +(-836.746 - 351.283i) q^{7} +(923.138 - 1598.92i) q^{9} +(-1357.32 - 2350.94i) q^{11} +13223.2 q^{13} -3253.02 q^{15} +(11131.2 + 19279.8i) q^{17} +(-4462.71 + 7729.64i) q^{19} +(-2107.12 - 16618.1i) q^{21} +(-11763.5 + 20375.0i) q^{23} +(23533.6 + 40761.4i) q^{25} +74449.1 q^{27} -186244. q^{29} +(67236.0 + 116456. i) q^{31} +(25054.4 - 43395.4i) q^{33} +(127344. - 96751.7i) q^{35} +(-99319.8 + 172027. i) q^{37} +(122041. + 211382. i) q^{39} +263027. q^{41} -235255. q^{43} +(162687. + 281781. i) q^{45} +(-390528. + 676414. i) q^{47} +(576744. + 587869. i) q^{49} +(-205468. + 355881. i) q^{51} +(549100. + 951068. i) q^{53} +478406. q^{55} -164752. q^{57} +(-422267. - 731387. i) q^{59} +(455474. - 788904. i) q^{61} +(-1.33411e6 + 1.01361e6i) q^{63} +(-1.16517e6 + 2.01814e6i) q^{65} +(2.30285e6 + 3.98865e6i) q^{67} -434279. q^{69} +5.88754e6 q^{71} +(1.39756e6 + 2.42065e6i) q^{73} +(-434401. + 752404. i) q^{75} +(309884. + 2.44394e6i) q^{77} +(-469333. + 812909. i) q^{79} +(-1.33178e6 - 2.30672e6i) q^{81} +4.16545e6 q^{83} -3.92336e6 q^{85} +(-1.71891e6 - 2.97724e6i) q^{87} +(4.85226e6 - 8.40436e6i) q^{89} +(-1.10644e7 - 4.64507e6i) q^{91} +(-1.24109e6 + 2.14963e6i) q^{93} +(-786473. - 1.36221e6i) q^{95} +3.19719e6 q^{97} -5.01197e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) $ 10 * q - 27 * q^3 + 249 * q^5 - 332 * q^7 - 5702 * q^9 - 6399 * q^11 - 26988 * q^13 - 19294 * q^15 + 3609 * q^17 + 12403 * q^19 + 16099 * q^21 + 13959 * q^23 - 162364 * q^25 - 161550 * q^27 + 26148 * q^29 + 20181 * q^31 + 264239 * q^33 - 791715 * q^35 - 54763 * q^37 - 4750 * q^39 - 1824468 * q^41 + 1938424 * q^43 + 1936430 * q^45 - 1217253 * q^47 + 1722058 * q^49 + 1353871 * q^51 - 2349159 * q^53 - 3123494 * q^55 - 9823194 * q^57 - 2188611 * q^59 + 6107445 * q^61 - 10275990 * q^63 - 3022014 * q^65 + 3171581 * q^67 - 11255938 * q^69 + 18938976 * q^71 + 8034853 * q^73 - 29173588 * q^75 + 24100191 * q^77 + 7589559 * q^79 - 9846485 * q^81 + 13452408 * q^83 - 48717670 * q^85 - 30033142 * q^87 + 29864757 * q^89 - 34267512 * q^91 - 20217029 * q^93 + 25812009 * q^95 + 285580 * q^97 + 119277812 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	9.22936	+	15.9857i	0.197355	+	0.341828i	0.947670	−	0.319252i	$-0.103432\pi$
$3$	9.22936	+	15.9857i	0.197355	+	0.341828i	−0.750315	+	0.661080i	$0.770098\pi$
$4$		0			0
$5$	−88.1161	+	152.621i	−0.315254	+	0.546035i	−0.979491	−	0.201486i	$-0.935423\pi$
$5$	−88.1161	+	152.621i	−0.315254	+	0.546035i	0.664238	+	0.747521i	$0.268756\pi$
$6$		0			0
$7$	−836.746	−	351.283i	−0.922041	−	0.387092i
$7$	−836.746	−	351.283i	−0.922041	−	0.387092i
$8$		0			0
$9$	923.138	−	1598.92i	0.422102	−	0.731103i
$10$		0			0
$11$	−1357.32	−	2350.94i	−0.307473	−	0.532559i	0.670336	−	0.742058i	$-0.266150\pi$
$11$	−1357.32	−	2350.94i	−0.307473	−	0.532559i	−0.977809	+	0.209499i	$0.932817\pi$
$12$		0			0
$13$	13223.2			1.66930			0.834649	−	0.550783i	$-0.185671\pi$
$13$	13223.2			1.66930			0.834649	+	0.550783i	$0.185671\pi$
$14$		0			0
$15$	−3253.02			−0.248867
$16$		0			0
$17$	11131.2	+	19279.8i	0.549505	+	0.951771i	0.998308	+	0.0581402i	$0.0185170\pi$
$17$	11131.2	+	19279.8i	0.549505	+	0.951771i	−0.448803	+	0.893631i	$0.648150\pi$
$18$		0			0
$19$	−4462.71	+	7729.64i	−0.149266	+	0.258536i	−0.930956	−	0.365130i	$-0.881024\pi$
$19$	−4462.71	+	7729.64i	−0.149266	+	0.258536i	0.781690	+	0.623667i	$0.214358\pi$
$20$		0			0
$21$	−2107.12	−	16618.1i	−0.0496502	−	0.391574i
$22$		0			0
$23$	−11763.5	+	20375.0i	−0.201600	+	0.349181i	−0.949044	−	0.315144i	$-0.897947\pi$
$23$	−11763.5	+	20375.0i	−0.201600	+	0.349181i	0.747444	+	0.664324i	$0.231281\pi$
$24$		0			0
$25$	23533.6	+	40761.4i	0.301230	+	0.521746i
$26$		0			0
$27$	74449.1			0.727924
$28$		0			0
$29$	−186244.			−1.41804			−0.709020	−	0.705189i	$-0.750862\pi$
$29$	−186244.			−1.41804			−0.709020	+	0.705189i	$0.750862\pi$
$30$		0			0
$31$	67236.0	+	116456.i	0.405355	+	0.702096i	0.994363	−	0.106032i	$-0.0338145\pi$
$31$	67236.0	+	116456.i	0.405355	+	0.702096i	−0.589008	+	0.808127i	$0.700481\pi$
$32$		0			0
$33$	25054.4	−	43395.4i	0.121362	−	0.210206i
$34$		0			0
$35$	127344.	−	96751.7i	0.502043	−	0.381435i
$36$		0			0
$37$	−99319.8	+	172027.i	−0.322351	+	0.558329i	−0.980973	−	0.194145i	$-0.937807\pi$
$37$	−99319.8	+	172027.i	−0.322351	+	0.558329i	0.658621	+	0.752475i	$0.271140\pi$
$38$		0			0
$39$	122041.	+	211382.i	0.329443	+	0.570613i
$40$		0			0
$41$	263027.			0.596015			0.298007	−	0.954564i	$-0.403678\pi$
$41$	263027.			0.596015			0.298007	+	0.954564i	$0.403678\pi$
$42$		0			0
$43$	−235255.			−0.451231			−0.225615	−	0.974216i	$-0.572439\pi$
$43$	−235255.			−0.451231			−0.225615	+	0.974216i	$0.572439\pi$
$44$		0			0
$45$	162687.	+	281781.i	0.266139	+	0.460966i
$46$		0			0
$47$	−390528.	+	676414.i	−0.548668	+	0.950320i	0.449698	+	0.893180i	$0.351531\pi$
$47$	−390528.	+	676414.i	−0.548668	+	0.950320i	−0.998366	+	0.0571400i	$0.981802\pi$
$48$		0			0
$49$	576744.	+	587869.i	0.700320	+	0.713829i
$50$		0			0
$51$	−205468.	+	355881.i	−0.216895	+	0.375673i
$52$		0			0
$53$	549100.	+	951068.i	0.506624	+	0.877498i	0.999971	+	0.00766543i	$0.00244001\pi$
$53$	549100.	+	951068.i	0.506624	+	0.877498i	−0.493347	+	0.869833i	$0.664227\pi$
$54$		0			0
$55$	478406.			0.387728
$56$		0			0
$57$	−164752.			−0.117833
$58$		0			0
$59$	−422267.	−	731387.i	−0.267673	−	0.463623i	0.700587	−	0.713567i	$-0.252921\pi$
$59$	−422267.	−	731387.i	−0.267673	−	0.463623i	−0.968260	+	0.249943i	$0.919588\pi$
$60$		0			0
$61$	455474.	−	788904.i	0.256927	−	0.445010i	−0.708490	−	0.705721i	$-0.750623\pi$
$61$	455474.	−	788904.i	0.256927	−	0.445010i	0.965417	+	0.260710i	$0.0839568\pi$
$62$		0			0
$63$	−1.33411e6	+	1.01361e6i	−0.672200	+	0.510715i
$64$		0			0
$65$	−1.16517e6	+	2.01814e6i	−0.526252	+	0.911495i
$66$		0			0
$67$	2.30285e6	+	3.98865e6i	0.935414	+	1.62018i	0.773894	+	0.633315i	$0.218306\pi$
$67$	2.30285e6	+	3.98865e6i	0.935414	+	1.62018i	0.161520	+	0.986869i	$0.448360\pi$
$68$		0			0
$69$	−434279.			−0.159146
$70$		0			0
$71$	5.88754e6			1.95223			0.976113	−	0.217265i	$-0.0697135\pi$
$71$	5.88754e6			1.95223			0.976113	+	0.217265i	$0.0697135\pi$
$72$		0			0
$73$	1.39756e6	+	2.42065e6i	0.420476	+	0.728286i	0.995986	−	0.0895085i	$-0.0285296\pi$
$73$	1.39756e6	+	2.42065e6i	0.420476	+	0.728286i	−0.575510	+	0.817795i	$0.695196\pi$
$74$		0			0
$75$	−434401.	+	752404.i	−0.118898	+	0.205938i
$76$		0			0
$77$	309884.	+	2.44394e6i	0.0773537	+	0.610062i
$78$		0			0
$79$	−469333.	+	812909.i	−0.107099	+	0.185501i	−0.914594	−	0.404373i	$-0.867490\pi$
$79$	−469333.	+	812909.i	−0.107099	+	0.185501i	0.807495	+	0.589875i	$0.200823\pi$
$80$		0			0
$81$	−1.33178e6	−	2.30672e6i	−0.278443	−	0.482278i
$82$		0			0
$83$	4.16545e6			0.799630			0.399815	−	0.916596i	$-0.369074\pi$
$83$	4.16545e6			0.799630			0.399815	+	0.916596i	$0.369074\pi$
$84$		0			0
$85$	−3.92336e6			−0.692934
$86$		0			0
$87$	−1.71891e6	−	2.97724e6i	−0.279857	−	0.484726i
$88$		0			0
$89$	4.85226e6	−	8.40436e6i	0.729590	−	1.26369i	−0.227467	−	0.973786i	$-0.573044\pi$
$89$	4.85226e6	−	8.40436e6i	0.729590	−	1.26369i	0.957057	−	0.289901i	$-0.0936223\pi$
$90$		0			0
$91$	−1.10644e7	−	4.64507e6i	−1.53916	−	0.646171i
$92$		0			0
$93$	−1.24109e6	+	2.14963e6i	−0.159997	+	0.277124i
$94$		0			0
$95$	−786473.	−	1.36221e6i	−0.0941133	−	0.163009i
$96$		0			0
$97$	3.19719e6			0.355686			0.177843	−	0.984059i	$-0.443088\pi$
$97$	3.19719e6			0.355686			0.177843	+	0.984059i	$0.443088\pi$
$98$		0			0
$99$	−5.01197e6			−0.519141
$100$		0			0
$101$	5.17419e6	+	8.96196e6i	0.499709	+	0.865522i	1.00000		0.000335479i	$-0.000106786\pi$
$101$	5.17419e6	+	8.96196e6i	0.499709	+	0.865522i	−0.500291	+	0.865858i	$0.666773\pi$
$102$		0			0
$103$	−8.88935e6	+	1.53968e7i	−0.801567	+	1.38835i	0.117018	+	0.993130i	$0.462667\pi$
$103$	−8.88935e6	+	1.53968e7i	−0.801567	+	1.38835i	−0.918585	+	0.395225i	$0.870667\pi$
$104$		0			0
$105$	2.72195e6	+	1.14273e6i	0.229466	+	0.0963343i
$106$		0			0
$107$	9.18742e6	−	1.59131e7i	0.725020	−	1.25577i	−0.233945	−	0.972250i	$-0.575164\pi$
$107$	9.18742e6	−	1.59131e7i	0.725020	−	1.25577i	0.958966	−	0.283522i	$-0.0915030\pi$
$108$		0			0
$109$	−2.95020e6	−	5.10989e6i	−0.218202	−	0.377937i	0.736056	−	0.676920i	$-0.236686\pi$
$109$	−2.95020e6	−	5.10989e6i	−0.218202	−	0.377937i	−0.954258	+	0.298983i	$0.903352\pi$
$110$		0			0
$111$	−3.66663e6			−0.254470
$112$		0			0
$113$	−2.25373e7			−1.46935			−0.734677	−	0.678417i	$-0.762666\pi$
$113$	−2.25373e7			−1.46935			−0.734677	+	0.678417i	$0.762666\pi$
$114$		0			0
$115$	−2.07311e6	−	3.59073e6i	−0.127110	−	0.220161i
$116$		0			0
$117$	1.22068e7	−	2.11428e7i	0.704614	−	1.22043i
$118$		0			0
$119$	−2.54132e6	−	2.00425e7i	−0.138244	−	1.09028i
$120$		0			0
$121$	6.05896e6	−	1.04944e7i	0.310920	−	0.538530i
$122$		0			0
$123$	2.42757e6	+	4.20468e6i	0.117626	+	0.203735i
$124$		0			0
$125$	−2.20629e7			−1.01036
$126$		0			0
$127$	3.43100e7			1.48630			0.743152	−	0.669122i	$-0.233330\pi$
$127$	3.43100e7			1.48630			0.743152	+	0.669122i	$0.233330\pi$
$128$		0			0
$129$	−2.17125e6	−	3.76072e6i	−0.0890525	−	0.154243i
$130$		0			0
$131$	1.02849e7	−	1.78139e7i	0.399713	−	0.692324i	−0.593977	−	0.804482i	$-0.702443\pi$
$131$	1.02849e7	−	1.78139e7i	0.399713	−	0.692324i	0.993690	+	0.112158i	$0.0357764\pi$
$132$		0			0
$133$	6.44944e6	−	4.90007e6i	0.237707	−	0.180602i
$134$		0			0
$135$	−6.56016e6	+	1.13625e7i	−0.229481	+	0.397472i
$136$		0			0
$137$	−1.36313e7	−	2.36101e7i	−0.452913	−	0.784469i	0.545652	−	0.838012i	$-0.316282\pi$
$137$	−1.36313e7	−	2.36101e7i	−0.452913	−	0.784469i	−0.998566	+	0.0535428i	$0.982949\pi$
$138$		0			0
$139$	3.94701e6			0.124657			0.0623285	−	0.998056i	$-0.480147\pi$
$139$	3.94701e6			0.124657			0.0623285	+	0.998056i	$0.480147\pi$
$140$		0			0
$141$	−1.44173e7			−0.433128
$142$		0			0
$143$	−1.79480e7	−	3.10869e7i	−0.513264	−	0.889000i
$144$		0			0
$145$	1.64111e7	−	2.84248e7i	0.447042	−	0.774300i
$146$		0			0
$147$	−4.07453e6	+	1.46453e7i	−0.105795	+	0.380267i
$148$		0			0
$149$	−3.13115e7	+	5.42331e7i	−0.775446	+	1.34311i	0.159097	+	0.987263i	$0.449142\pi$
$149$	−3.13115e7	+	5.42331e7i	−0.775446	+	1.34311i	−0.934543	+	0.355850i	$0.884192\pi$
$150$		0			0
$151$	−1.89202e7	−	3.27707e7i	−0.447205	−	0.774581i	0.550998	−	0.834506i	$-0.314247\pi$
$151$	−1.89202e7	−	3.27707e7i	−0.447205	−	0.774581i	−0.998203	+	0.0599252i	$0.980914\pi$
$152$		0			0
$153$	4.11026e7			0.927789
$154$		0			0
$155$	−2.36983e7			−0.511159
$156$		0			0
$157$	−462647.	−	801328.i	−0.00954115	−	0.0165258i	0.861215	−	0.508240i	$-0.169704\pi$
$157$	−462647.	−	801328.i	−0.00954115	−	0.0165258i	−0.870757	+	0.491714i	$0.836370\pi$
$158$		0			0
$159$	−1.01357e7	+	1.75555e7i	−0.199969	+	0.346357i
$160$		0			0
$161$	1.70005e7	−	1.29164e7i	0.321048	−	0.243922i
$162$		0			0
$163$	−3.05628e7	+	5.29364e7i	−0.552761	+	0.957409i	0.445313	+	0.895375i	$0.353092\pi$
$163$	−3.05628e7	+	5.29364e7i	−0.552761	+	0.957409i	−0.998074	+	0.0620347i	$0.980241\pi$
$164$		0			0
$165$	4.41538e6	+	7.64767e6i	0.0765199	+	0.132536i
$166$		0			0
$167$	1.21239e6			0.0201434			0.0100717	−	0.999949i	$-0.496794\pi$
$167$	1.21239e6			0.0201434			0.0100717	+	0.999949i	$0.496794\pi$
$168$		0			0
$169$	1.12104e8			1.78655
$170$		0			0
$171$	8.23939e6	+	1.42710e7i	0.126011	+	0.218258i
$172$		0			0
$173$	−4.93351e7	+	8.54510e7i	−0.724428	+	1.25475i	0.234781	+	0.972048i	$0.424563\pi$
$173$	−4.93351e7	+	8.54510e7i	−0.724428	+	1.25475i	−0.959209	+	0.282697i	$0.908771\pi$
$174$		0			0
$175$	−5.37286e6	−	4.23739e7i	−0.0757832	−	0.597675i
$176$		0			0
$177$	7.79450e6	−	1.35005e7i	0.105653	−	0.182996i
$178$		0			0
$179$	2.40837e7	+	4.17142e7i	0.313861	+	0.543623i	0.979195	−	0.202923i	$-0.0650441\pi$
$179$	2.40837e7	+	4.17142e7i	0.313861	+	0.543623i	−0.665334	+	0.746546i	$0.731711\pi$
$180$		0			0
$181$	−6.53009e7			−0.818548			−0.409274	−	0.912411i	$-0.634218\pi$
$181$	−6.53009e7			−0.818548			−0.409274	+	0.912411i	$0.634218\pi$
$182$		0			0
$183$	1.68149e7			0.202823
$184$		0			0
$185$	−1.75033e7	−	3.03167e7i	−0.203245	−	0.352031i
$186$		0			0
$187$	3.02172e7	−	5.23378e7i	0.337916	−	0.585288i
$188$		0			0
$189$	−6.22950e7	−	2.61527e7i	−0.671176	−	0.281774i
$190$		0			0
$191$	1.33404e6	−	2.31063e6i	0.0138533	−	0.0239946i	−0.859016	−	0.511949i	$-0.828924\pi$
$191$	1.33404e6	−	2.31063e6i	0.0138533	−	0.0239946i	0.872869	+	0.487955i	$0.162257\pi$
$192$		0			0
$193$	1.70478e7	+	2.95277e7i	0.170694	+	0.295651i	0.938663	−	0.344837i	$-0.112066\pi$
$193$	1.70478e7	+	2.95277e7i	0.170694	+	0.295651i	−0.767969	+	0.640487i	$0.778732\pi$
$194$		0			0
$195$	−4.30152e7			−0.415433
$196$		0			0
$197$	2.65031e7			0.246982			0.123491	−	0.992346i	$-0.460591\pi$
$197$	2.65031e7			0.246982			0.123491	+	0.992346i	$0.460591\pi$
$198$		0			0
$199$	1.06544e8	+	1.84539e8i	0.958388	+	1.65998i	0.726417	+	0.687254i	$0.241184\pi$
$199$	1.06544e8	+	1.84539e8i	0.958388	+	1.65998i	0.231971	+	0.972723i	$0.425483\pi$
$200$		0			0
$201$	−4.25077e7	+	7.36254e7i	−0.369216	+	0.639502i
$202$		0			0
$203$	1.55839e8	+	6.54242e7i	1.30749	+	0.548911i
$204$		0			0
$205$	−2.31769e7	+	4.01436e7i	−0.187896	+	0.325445i
$206$		0			0
$207$	2.17187e7	+	3.76179e7i	0.170191	+	0.294780i
$208$		0			0
$209$	2.42293e7			0.183581
$210$		0			0
$211$	−3.67400e7			−0.269247			−0.134624	−	0.990897i	$-0.542983\pi$
$211$	−3.67400e7			−0.269247			−0.134624	+	0.990897i	$0.542983\pi$
$212$		0			0
$213$	5.43383e7	+	9.41166e7i	0.385281	+	0.667326i
$214$		0			0
$215$	2.07297e7	−	3.59049e7i	0.142252	−	0.246388i
$216$		0			0
$217$	−1.53504e7	−	1.21063e8i	−0.101979	−	0.804271i
$218$		0			0
$219$	−2.57972e7	+	4.46821e7i	−0.165966	+	0.287461i
$220$		0			0
$221$	1.47190e8	+	2.54941e8i	0.917287	+	1.58879i
$222$		0			0
$223$	−3.23173e7			−0.195150			−0.0975749	−	0.995228i	$-0.531109\pi$
$223$	−3.23173e7			−0.195150			−0.0975749	+	0.995228i	$0.531109\pi$
$224$		0			0
$225$	8.68991e7			0.508600
$226$		0			0
$227$	6.22771e7	+	1.07867e8i	0.353376	+	0.612066i	0.986839	−	0.161708i	$-0.0517001\pi$
$227$	6.22771e7	+	1.07867e8i	0.353376	+	0.612066i	−0.633462	+	0.773774i	$0.718367\pi$
$228$		0			0
$229$	−8.82568e7	+	1.52865e8i	−0.485651	+	0.841171i	−0.999864	−	0.0164909i	$-0.994751\pi$
$229$	−8.82568e7	+	1.52865e8i	−0.485651	+	0.841171i	0.514214	+	0.857662i	$0.328084\pi$
$230$		0			0
$231$	−3.62082e7	+	2.75098e7i	−0.193270	+	0.146840i
$232$		0			0
$233$	1.57549e8	−	2.72882e8i	0.815960	−	1.41328i	−0.0926758	−	0.995696i	$-0.529542\pi$
$233$	1.57549e8	−	2.72882e8i	0.815960	−	1.41328i	0.908636	−	0.417589i	$-0.137125\pi$
$234$		0			0
$235$	−6.88235e7	−	1.19206e8i	−0.345939	−	0.599184i
$236$		0			0
$237$	−1.73266e7			−0.0845461
$238$		0			0
$239$	−2.34353e8			−1.11040			−0.555198	−	0.831718i	$-0.687358\pi$
$239$	−2.34353e8			−1.11040			−0.555198	+	0.831718i	$0.687358\pi$
$240$		0			0
$241$	5.54267e6	+	9.60019e6i	0.0255070	+	0.0441794i	0.878497	−	0.477748i	$-0.158547\pi$
$241$	5.54267e6	+	9.60019e6i	0.0255070	+	0.0441794i	−0.852990	+	0.521927i	$0.825213\pi$
$242$		0			0
$243$	1.05993e8	−	1.83586e8i	0.473866	−	0.820760i
$244$		0			0
$245$	−1.40542e8	+	3.62228e7i	−0.610554	+	0.157362i
$246$		0			0
$247$	−5.90111e7	+	1.02210e8i	−0.249169	+	0.431574i
$248$		0			0
$249$	3.84445e7	+	6.65878e7i	0.157811	+	0.273336i
$250$		0			0
$251$	−4.57338e8			−1.82549			−0.912744	−	0.408531i	$-0.866041\pi$
$251$	−4.57338e8			−1.82549			−0.912744	+	0.408531i	$0.866041\pi$
$252$		0			0
$253$	6.38673e7			0.247946
$254$		0			0
$255$	−3.62101e7	−	6.27177e7i	−0.136754	−	0.236864i
$256$		0			0
$257$	2.06665e8	−	3.57954e8i	0.759453	−	1.31541i	−0.183677	−	0.982987i	$-0.558800\pi$
$257$	2.06665e8	−	3.57954e8i	0.759453	−	1.31541i	0.943130	−	0.332424i	$-0.107867\pi$
$258$		0			0
$259$	1.43535e8	−	1.09053e8i	0.513346	−	0.390023i
$260$		0			0
$261$	−1.71929e8	+	2.97789e8i	−0.598558	+	1.03673i
$262$		0			0
$263$	1.35557e8	+	2.34792e8i	0.459492	+	0.795864i	0.998934	−	0.0461589i	$-0.0146980\pi$
$263$	1.35557e8	+	2.34792e8i	0.459492	+	0.795864i	−0.539442	+	0.842023i	$0.681365\pi$
$264$		0			0
$265$	−1.93538e8			−0.638860
$266$		0			0
$267$	1.79133e8			0.575952
$268$		0			0
$269$	−2.29263e8	−	3.97095e8i	−0.718126	−	1.24383i	−0.961741	−	0.273959i	$-0.911667\pi$
$269$	−2.29263e8	−	3.97095e8i	−0.718126	−	1.24383i	0.243615	−	0.969872i	$-0.421667\pi$
$270$		0			0
$271$	2.70728e8	−	4.68915e8i	0.826306	−	1.43120i	−0.0746117	−	0.997213i	$-0.523772\pi$
$271$	2.70728e8	−	4.68915e8i	0.826306	−	1.43120i	0.900917	−	0.433991i	$-0.142895\pi$
$272$		0			0
$273$	−2.78628e7	−	2.19744e8i	−0.0828810	−	0.653653i
$274$		0			0
$275$	6.38852e7	−	1.10652e8i	0.185240	−	0.320846i
$276$		0			0
$277$	−6.53840e7	−	1.13248e8i	−0.184839	−	0.320150i	0.758684	−	0.651459i	$-0.225843\pi$
$277$	−6.53840e7	−	1.13248e8i	−0.184839	−	0.320150i	−0.943522	+	0.331310i	$0.892510\pi$
$278$		0			0
$279$	2.48272e8			0.684405
$280$		0			0
$281$	1.71135e8			0.460114			0.230057	−	0.973177i	$-0.426109\pi$
$281$	1.71135e8			0.460114			0.230057	+	0.973177i	$0.426109\pi$
$282$		0			0
$283$	−5.47927e7	−	9.49037e7i	−0.143704	−	0.248903i	0.785184	−	0.619262i	$-0.212568\pi$
$283$	−5.47927e7	−	9.49037e7i	−0.143704	−	0.248903i	−0.928889	+	0.370359i	$0.879235\pi$
$284$		0			0
$285$	1.45173e7	−	2.51447e7i	0.0371474	−	0.0643412i
$286$		0			0
$287$	−2.20087e8	−	9.23969e7i	−0.549550	−	0.230712i
$288$		0			0
$289$	−4.26390e7	+	7.38528e7i	−0.103912	+	0.179980i
$290$		0			0
$291$	2.95080e7	+	5.11094e7i	0.0701964	+	0.121584i
$292$		0			0
$293$	−7.45719e8			−1.73196			−0.865981	−	0.500077i	$-0.833305\pi$
$293$	−7.45719e8			−1.73196			−0.865981	+	0.500077i	$0.833305\pi$
$294$		0			0
$295$	1.48834e8			0.337540
$296$		0			0
$297$	−1.01051e8	−	1.75026e8i	−0.223817	−	0.387663i
$298$		0			0
$299$	−1.55551e8	+	2.69422e8i	−0.336530	+	0.582887i
$300$		0			0
$301$	1.96848e8	+	8.26410e7i	0.416053	+	0.174668i
$302$		0			0
$303$	−9.55089e7	+	1.65426e8i	−0.197240	+	0.341630i
$304$		0			0
$305$	8.02692e7	+	1.39030e8i	0.161994	+	0.280582i
$306$		0			0
$307$	−3.32575e8			−0.656002			−0.328001	−	0.944677i	$-0.606375\pi$
$307$	−3.32575e8			−0.656002			−0.328001	+	0.944677i	$0.606375\pi$
$308$		0			0
$309$	−3.28172e8			−0.632772
$310$		0			0
$311$	−4.04524e8	−	7.00656e8i	−0.762575	−	1.32082i	−0.941519	−	0.336960i	$-0.890601\pi$
$311$	−4.04524e8	−	7.00656e8i	−0.762575	−	1.32082i	0.178944	−	0.983859i	$-0.442732\pi$
$312$		0			0
$313$	3.26018e8	−	5.64680e8i	0.600948	−	1.04087i	−0.391730	−	0.920080i	$-0.628123\pi$
$313$	3.26018e8	−	5.64680e8i	0.600948	−	1.04087i	0.992678	−	0.120792i	$-0.0385433\pi$
$314$		0			0
$315$	−3.71423e7	−	2.92928e8i	−0.0669548	−	0.528049i
$316$		0			0
$317$	1.16084e8	−	2.01063e8i	0.204675	−	0.354508i	−0.745354	−	0.666669i	$-0.767720\pi$
$317$	1.16084e8	−	2.01063e8i	0.204675	−	0.354508i	0.950029	+	0.312161i	$0.101053\pi$
$318$		0			0
$319$	2.52792e8	+	4.37848e8i	0.436009	+	0.755190i
$320$		0			0
$321$	3.39176e8			0.572344
$322$		0			0
$323$	−1.98702e8			−0.328090
$324$		0			0
$325$	3.11189e8	+	5.38995e8i	0.502843	+	0.870950i
$326$		0			0
$327$	5.44569e7	−	9.43221e7i	0.0861263	−	0.149175i
$328$		0			0
$329$	5.64385e8	−	4.28801e8i	0.873755	−	0.663850i
$330$		0			0
$331$	3.98968e8	−	6.91033e8i	0.604700	−	1.04737i	−0.387399	−	0.921912i	$-0.626626\pi$
$331$	3.98968e8	−	6.91033e8i	0.604700	−	1.04737i	0.992099	−	0.125459i	$-0.0400403\pi$
$332$		0			0
$333$	1.83372e8	+	3.17609e8i	0.272131	+	0.471344i
$334$		0			0
$335$	−8.11672e8			−1.17957
$336$		0			0
$337$	−4.79255e8			−0.682122			−0.341061	−	0.940041i	$-0.610786\pi$
$337$	−4.79255e8			−0.682122			−0.341061	+	0.940041i	$0.610786\pi$
$338$		0			0
$339$	−2.08004e8	−	3.60274e8i	−0.289984	−	0.502267i
$340$		0			0
$341$	1.82521e8	−	3.16136e8i	0.249272	−	0.431751i
$342$		0			0
$343$	−2.76080e8	−	6.94497e8i	−0.369407	−	0.929268i
$344$		0			0
$345$	3.82669e7	−	6.62803e7i	0.0501715	−	0.0868996i
$346$		0			0
$347$	6.57809e8	+	1.13936e9i	0.845175	+	1.46389i	0.885469	+	0.464698i	$0.153837\pi$
$347$	6.57809e8	+	1.13936e9i	0.845175	+	1.46389i	−0.0402946	+	0.999188i	$0.512830\pi$
$348$		0			0
$349$	−2.67516e8			−0.336869			−0.168434	−	0.985713i	$-0.553871\pi$
$349$	−2.67516e8			−0.336869			−0.168434	+	0.985713i	$0.553871\pi$
$350$		0			0
$351$	9.84453e8			1.21512
$352$		0			0
$353$	3.21078e8	+	5.56124e8i	0.388507	+	0.672915i	0.992249	−	0.124266i	$-0.0396575\pi$
$353$	3.21078e8	+	5.56124e8i	0.388507	+	0.672915i	−0.603742	+	0.797180i	$0.706324\pi$
$354$		0			0
$355$	−5.18787e8	+	8.98566e8i	−0.615446	+	1.06598i
$356$		0			0
$357$	2.96940e8	−	2.25605e8i	0.345406	−	0.262428i
$358$		0			0
$359$	−3.14068e8	+	5.43981e8i	−0.358255	+	0.620517i	−0.987669	−	0.156553i	$-0.949962\pi$
$359$	−3.14068e8	+	5.43981e8i	−0.358255	+	0.620517i	0.629414	+	0.777070i	$0.283295\pi$
$360$		0			0
$361$	4.07104e8	+	7.05125e8i	0.455439	+	0.788844i
$362$		0			0
$363$	2.23681e8			0.245446
$364$		0			0
$365$	−4.92591e8			−0.530227
$366$		0			0
$367$	7.84137e8	+	1.35816e9i	0.828058	+	1.43424i	0.899560	+	0.436798i	$0.143887\pi$
$367$	7.84137e8	+	1.35816e9i	0.828058	+	1.43424i	−0.0715019	+	0.997440i	$0.522779\pi$
$368$		0			0
$369$	2.42810e8	−	4.20560e8i	0.251579	−	0.435748i
$370$		0			0
$371$	−1.25363e8	−	9.88692e8i	−0.127456	−	1.00520i
$372$		0			0
$373$	−6.05018e8	+	1.04792e9i	−0.603653	+	1.04556i	0.388610	+	0.921402i	$0.372955\pi$
$373$	−6.05018e8	+	1.04792e9i	−0.603653	+	1.04556i	−0.992263	+	0.124155i	$0.960378\pi$
$374$		0			0
$375$	−2.03626e8	−	3.52691e8i	−0.199400	−	0.345371i
$376$		0			0
$377$	−2.46273e9			−2.36713
$378$		0			0
$379$	1.19750e9			1.12990			0.564949	−	0.825126i	$-0.308896\pi$
$379$	1.19750e9			1.12990			0.564949	+	0.825126i	$0.308896\pi$
$380$		0			0
$381$	3.16659e8	+	5.48470e8i	0.293329	+	0.508061i
$382$		0			0
$383$	−1.20905e8	+	2.09413e8i	−0.109963	+	0.190462i	−0.915755	−	0.401737i	$-0.868407\pi$
$383$	−1.20905e8	+	2.09413e8i	−0.109963	+	0.190462i	0.805792	+	0.592199i	$0.201740\pi$
$384$		0			0
$385$	−4.00304e8	−	1.68056e8i	−0.357501	−	0.150086i
$386$		0			0
$387$	−2.17173e8	+	3.76154e8i	−0.190466	+	0.329896i
$388$		0			0
$389$	−1.97619e8	−	3.42287e8i	−0.170218	−	0.294827i	0.768278	−	0.640116i	$-0.221114\pi$
$389$	−1.97619e8	−	3.42287e8i	−0.170218	−	0.294827i	−0.938496	+	0.345290i	$0.887781\pi$
$390$		0			0
$391$	−5.23769e8			−0.443120
$392$		0			0
$393$	3.79690e8			0.315541
$394$		0			0
$395$	−8.27116e7	−	1.43261e8i	−0.0675269	−	0.116960i
$396$		0			0
$397$	9.37843e8	−	1.62439e9i	0.752252	−	1.30294i	−0.194477	−	0.980907i	$-0.562301\pi$
$397$	9.37843e8	−	1.62439e9i	0.752252	−	1.30294i	0.946729	−	0.322032i	$-0.104366\pi$
$398$		0			0
$399$	1.37855e8	+	5.78745e7i	0.108647	+	0.0456123i
$400$		0			0
$401$	−9.31917e8	+	1.61413e9i	−0.721725	+	1.25006i	0.238583	+	0.971122i	$0.423317\pi$
$401$	−9.31917e8	+	1.61413e9i	−0.721725	+	1.25006i	−0.960308	+	0.278942i	$0.910016\pi$
$402$		0			0
$403$	8.89072e8	+	1.53992e9i	0.676658	+	1.17201i
$404$		0			0
$405$	4.69406e8			0.351121
$406$		0			0
$407$	5.39234e8			0.396458
$408$		0			0
$409$	7.54934e8	+	1.30758e9i	0.545604	+	0.945014i	0.998569	+	0.0534853i	$0.0170330\pi$
$409$	7.54934e8	+	1.30758e9i	0.545604	+	0.945014i	−0.452965	+	0.891528i	$0.649634\pi$
$410$		0			0
$411$	2.51616e8	−	4.35812e8i	0.178769	−	0.309637i
$412$		0			0
$413$	9.64059e7	+	7.60320e8i	0.0673408	+	0.531094i
$414$		0			0
$415$	−3.67043e8	+	6.35738e8i	−0.252086	+	0.436626i
$416$		0			0
$417$	3.64284e7	+	6.30958e7i	0.0246016	+	0.0426112i
$418$		0			0
$419$	−1.60700e9			−1.06725			−0.533625	−	0.845721i	$-0.679171\pi$
$419$	−1.60700e9			−1.06725			−0.533625	+	0.845721i	$0.679171\pi$
$420$		0			0
$421$	2.88949e9			1.88727			0.943636	−	0.330985i	$-0.107381\pi$
$421$	2.88949e9			1.88727			0.943636	+	0.330985i	$0.107381\pi$
$422$		0			0
$423$	7.21022e8	+	1.24885e9i	0.463188	+	0.802265i
$424$		0			0
$425$	−5.23916e8	+	9.07449e8i	−0.331055	+	0.573404i
$426$		0			0
$427$	−6.58245e8	+	5.00112e8i	−0.409157	+	0.310863i
$428$		0			0
$429$	3.31298e8	−	5.73825e8i	0.202590	−	0.350896i
$430$		0			0
$431$	−8.83704e8	−	1.53062e9i	−0.531663	−	0.920867i	−0.999317	−	0.0369554i	$-0.988234\pi$
$431$	−8.83704e8	−	1.53062e9i	−0.531663	−	0.920867i	0.467654	−	0.883912i	$-0.345099\pi$
$432$		0			0
$433$	4.77962e8			0.282935			0.141467	−	0.989943i	$-0.454818\pi$
$433$	4.77962e8			0.282935			0.141467	+	0.989943i	$0.454818\pi$
$434$		0			0
$435$	6.05854e8			0.352903
$436$		0			0
$437$	−1.04994e8	−	1.81855e8i	−0.0601840	−	0.104242i
$438$		0			0
$439$	5.84043e8	−	1.01159e9i	0.329472	−	0.570663i	−0.652935	−	0.757414i	$-0.726463\pi$
$439$	5.84043e8	−	1.01159e9i	0.329472	−	0.570663i	0.982407	+	0.186751i	$0.0597958\pi$
$440$		0			0
$441$	1.47237e9	−	3.79484e8i	0.817489	−	0.210697i
$442$		0			0
$443$	−8.46483e8	+	1.46615e9i	−0.462600	+	0.801246i	−0.999090	−	0.0426605i	$-0.986417\pi$
$443$	−8.46483e8	+	1.46615e9i	−0.462600	+	0.801246i	0.536490	+	0.843907i	$0.319750\pi$
$444$		0			0
$445$	8.55123e8	+	1.48112e9i	0.460012	+	0.796763i
$446$		0			0
$447$	−1.15594e9			−0.612152
$448$		0			0
$449$	−8.50779e8			−0.443562			−0.221781	−	0.975096i	$-0.571187\pi$
$449$	−8.50779e8			−0.443562			−0.221781	+	0.975096i	$0.571187\pi$
$450$		0			0
$451$	−3.57011e8	−	6.18362e8i	−0.183259	−	0.317413i
$452$		0			0
$453$	3.49243e8	−	6.04906e8i	0.176516	−	0.305734i
$454$		0			0
$455$	1.68389e9	−	1.27936e9i	0.838058	−	0.636728i
$456$		0			0
$457$	9.22058e7	−	1.59705e8i	0.0451910	−	0.0782730i	−0.842545	−	0.538626i	$-0.818944\pi$
$457$	9.22058e7	−	1.59705e8i	0.0451910	−	0.0782730i	0.887736	+	0.460353i	$0.152277\pi$
$458$		0			0
$459$	8.28710e8	+	1.43537e9i	0.399998	+	0.692817i
$460$		0			0
$461$	−2.86503e9			−1.36199			−0.680997	−	0.732286i	$-0.738454\pi$
$461$	−2.86503e9			−1.36199			−0.680997	+	0.732286i	$0.738454\pi$
$462$		0			0
$463$	−7.30600e8			−0.342095			−0.171047	−	0.985263i	$-0.554715\pi$
$463$	−7.30600e8			−0.342095			−0.171047	+	0.985263i	$0.554715\pi$
$464$		0			0
$465$	−2.18720e8	−	3.78834e8i	−0.100879	−	0.174728i
$466$		0			0
$467$	1.51899e9	−	2.63096e9i	0.690153	−	1.19538i	−0.281635	−	0.959522i	$-0.590877\pi$
$467$	1.51899e9	−	2.63096e9i	0.690153	−	1.19538i	0.971788	−	0.235858i	$-0.0757899\pi$
$468$		0			0
$469$	−5.25754e8	−	4.14644e9i	−0.235330	−	1.85597i
$470$		0			0
$471$	8.53987e6	−	1.47915e7i	0.00376598	−	0.00652287i
$472$		0			0
$473$	3.19315e8	+	5.53071e8i	0.138741	+	0.240307i
$474$		0			0
$475$	−4.20095e8			−0.179854
$476$		0			0
$477$	2.02758e9			0.855388
$478$		0			0
$479$	−5.30387e8	−	9.18658e8i	−0.220505	−	0.381926i	0.734456	−	0.678656i	$-0.237437\pi$
$479$	−5.30387e8	−	9.18658e8i	−0.220505	−	0.381926i	−0.954961	+	0.296730i	$0.904104\pi$
$480$		0			0
$481$	−1.31332e9	+	2.27474e9i	−0.538100	+	0.932017i
$482$		0			0
$483$	3.63381e8	+	1.52555e8i	0.146740	+	0.0616043i
$484$		0			0
$485$	−2.81724e8	+	4.87960e8i	−0.112131	+	0.194217i
$486$		0			0
$487$	9.70344e8	+	1.68069e9i	0.380693	+	0.659379i	0.991161	−	0.132661i	$-0.0423523\pi$
$487$	9.70344e8	+	1.68069e9i	0.380693	+	0.659379i	−0.610469	+	0.792040i	$0.709019\pi$
$488$		0			0
$489$	−1.12830e9			−0.436359
$490$		0			0
$491$	−1.92020e9			−0.732086			−0.366043	−	0.930598i	$-0.619288\pi$
$491$	−1.92020e9			−0.732086			−0.366043	+	0.930598i	$0.619288\pi$
$492$		0			0
$493$	−2.07312e9	−	3.59075e9i	−0.779220	−	1.34965i
$494$		0			0
$495$	4.41635e8	−	7.64934e8i	0.163661	−	0.283469i
$496$		0			0
$497$	−4.92638e9	−	2.06819e9i	−1.80003	−	0.755690i
$498$		0			0
$499$	2.24206e8	−	3.88337e8i	0.0807785	−	0.139912i	−0.822806	−	0.568322i	$-0.807593\pi$
$499$	2.24206e8	−	3.88337e8i	0.0807785	−	0.139912i	0.903585	+	0.428410i	$0.140926\pi$
$500$		0			0
$501$	1.11896e7	+	1.93809e7i	0.00397540	+	0.00688559i
$502$		0			0
$503$	5.32401e8			0.186531			0.0932654	−	0.995641i	$-0.470269\pi$
$503$	5.32401e8			0.186531			0.0932654	+	0.995641i	$0.470269\pi$
$504$		0			0
$505$	−1.82372e9			−0.630141
$506$		0			0
$507$	1.03464e9	+	1.79206e9i	0.352585	+	0.610694i
$508$		0			0
$509$	4.70740e8	−	8.15346e8i	0.158223	−	0.274050i	−0.776005	−	0.630727i	$-0.782757\pi$
$509$	4.70740e8	−	8.15346e8i	0.158223	−	0.274050i	0.934228	+	0.356677i	$0.116090\pi$
$510$		0			0
$511$	−3.19072e8	−	2.51641e9i	−0.105783	−	0.834273i
$512$		0			0
$513$	−3.32245e8	+	5.75465e8i	−0.108654	+	0.188195i
$514$		0			0
$515$	−1.56659e9	−	2.71341e9i	−0.505394	−	0.875367i
$516$		0			0
$517$	2.12028e9			0.674802
$518$		0			0
$519$	−1.82133e9			−0.571877
$520$		0			0
$521$	−2.99067e8	−	5.17999e8i	−0.0926481	−	0.160471i	0.815977	−	0.578085i	$-0.196200\pi$
$521$	−2.99067e8	−	5.17999e8i	−0.0926481	−	0.160471i	−0.908625	+	0.417614i	$0.862866\pi$
$522$		0			0
$523$	2.21813e9	−	3.84191e9i	0.678001	−	1.17433i	−0.297580	−	0.954697i	$-0.596180\pi$
$523$	2.21813e9	−	3.84191e9i	0.678001	−	1.17433i	0.975582	−	0.219636i	$-0.0704871\pi$
$524$		0			0
$525$	6.27789e8	−	4.76973e8i	0.189346	−	0.143859i
$526$		0			0
$527$	−1.49684e9	+	2.59260e9i	−0.445489	+	0.771610i
$528$		0			0
$529$	1.42565e9	+	2.46930e9i	0.418715	+	0.725236i
$530$		0			0
$531$	−1.55924e9			−0.451942
$532$		0			0
$533$	3.47805e9			0.994926
$534$		0			0
$535$	1.61912e9	+	2.80439e9i	0.457131	+	0.791773i
$536$		0			0
$537$	−4.44554e8	+	7.69990e8i	−0.123884	+	0.214573i
$538$		0			0
$539$	5.99222e8	−	2.15382e9i	0.164827	−	0.592445i
$540$		0			0
$541$	1.81228e9	−	3.13896e9i	0.492078	−	0.852305i	−0.507880	−	0.861428i	$-0.669571\pi$
$541$	1.81228e9	−	3.13896e9i	0.492078	−	0.852305i	0.999958	+	0.00912313i	$0.00290402\pi$
$542$		0			0
$543$	−6.02686e8	−	1.04388e9i	−0.161544	−	0.279803i
$544$		0			0
$545$	1.03984e9			0.275156
$546$		0			0
$547$	−3.39025e9			−0.885677			−0.442839	−	0.896601i	$-0.646029\pi$
$547$	−3.39025e9			−0.885677			−0.442839	+	0.896601i	$0.646029\pi$
$548$		0			0
$549$	−8.40931e8	−	1.45653e9i	−0.216899	−	0.375680i
$550$		0			0
$551$	8.31151e8	−	1.43960e9i	0.211665	−	0.366615i
$552$		0			0
$553$	6.78274e8	−	5.15329e8i	0.170556	−	0.129583i
$554$		0			0
$555$	3.23089e8	−	5.59607e8i	0.0802226	−	0.138950i
$556$		0			0
$557$	−3.08886e8	−	5.35006e8i	−0.0757364	−	0.131179i	0.825670	−	0.564154i	$-0.190797\pi$
$557$	−3.08886e8	−	5.35006e8i	−0.0757364	−	0.131179i	−0.901406	+	0.432974i	$0.857464\pi$
$558$		0			0
$559$	−3.11081e9			−0.753238
$560$		0			0
$561$	1.11554e9			0.266757
$562$		0			0
$563$	−1.96047e9	−	3.39563e9i	−0.462999	−	0.801938i	0.536109	−	0.844149i	$-0.319893\pi$
$563$	−1.96047e9	−	3.39563e9i	−0.462999	−	0.801938i	−0.999109	+	0.0422101i	$0.986560\pi$
$564$		0			0
$565$	1.98589e9	−	3.43967e9i	0.463219	−	0.802319i
$566$		0			0
$567$	3.04054e8	+	2.39797e9i	0.0700504	+	0.552463i
$568$		0			0
$569$	−3.12972e8	+	5.42083e8i	−0.0712216	+	0.123359i	−0.899437	−	0.437051i	$-0.856023\pi$
$569$	−3.12972e8	+	5.42083e8i	−0.0712216	+	0.123359i	0.828215	+	0.560410i	$0.189356\pi$
$570$		0			0
$571$	−6.65326e8	−	1.15238e9i	−0.149558	−	0.259041i	0.781506	−	0.623897i	$-0.214452\pi$
$571$	−6.65326e8	−	1.15238e9i	−0.149558	−	0.259041i	−0.931064	+	0.364856i	$0.881118\pi$
$572$		0			0
$573$	4.92494e7			0.0109360
$574$		0			0
$575$	−1.10735e9			−0.242912
$576$		0			0
$577$	1.03775e9	+	1.79744e9i	0.224895	+	0.389529i	0.956288	−	0.292427i	$-0.0944629\pi$
$577$	1.03775e9	+	1.79744e9i	0.224895	+	0.389529i	−0.731393	+	0.681956i	$0.761130\pi$
$578$		0			0
$579$	−3.14681e8	+	5.45044e8i	−0.0673745	+	0.116696i
$580$		0			0
$581$	−3.48542e9	−	1.46325e9i	−0.737291	−	0.309530i
$582$		0			0
$583$	1.49061e9	−	2.58180e9i	0.311546	−	0.539614i
$584$		0			0
$585$	2.15123e9	+	3.72604e9i	0.444264	+	0.769489i
$586$		0			0
$587$	1.08864e9			0.222152			0.111076	−	0.993812i	$-0.464570\pi$
$587$	1.08864e9			0.222152			0.111076	+	0.993812i	$0.464570\pi$
$588$		0			0
$589$	−1.20022e9			−0.242023
$590$		0			0
$591$	2.44607e8	+	4.23672e8i	0.0487430	+	0.0844254i
$592$		0			0
$593$	3.22909e8	−	5.59295e8i	0.0635900	−	0.110141i	−0.832478	−	0.554059i	$-0.813078\pi$
$593$	3.22909e8	−	5.59295e8i	0.0635900	−	0.110141i	0.896068	+	0.443917i	$0.146412\pi$
$594$		0			0
$595$	3.28285e9	+	1.37821e9i	0.638914	+	0.268229i
$596$		0			0
$597$	−1.96666e9	+	3.40635e9i	−0.378285	+	0.655208i
$598$		0			0
$599$	3.94789e9	+	6.83794e9i	0.750535	+	1.29996i	0.947564	+	0.319567i	$0.103537\pi$
$599$	3.94789e9	+	6.83794e9i	0.750535	+	1.29996i	−0.197029	+	0.980398i	$0.563129\pi$
$600$		0			0
$601$	−5.19643e9			−0.976437			−0.488219	−	0.872721i	$-0.662353\pi$
$601$	−5.19643e9			−0.976437			−0.488219	+	0.872721i	$0.662353\pi$
$602$		0			0
$603$	8.50339e9			1.57936
$604$		0			0
$605$	1.06778e9	+	1.84946e9i	0.196038	+	0.339547i
$606$		0			0
$607$	1.52131e9	−	2.63499e9i	0.276094	−	0.478209i	−0.694316	−	0.719670i	$-0.744293\pi$
$607$	1.52131e9	−	2.63499e9i	0.276094	−	0.478209i	0.970411	+	0.241461i	$0.0776265\pi$
$608$		0			0
$609$	3.92437e8	+	3.09501e9i	0.0704060	+	0.555267i
$610$		0			0
$611$	−5.16401e9	+	8.94433e9i	−0.915890	+	1.58637i
$612$		0			0
$613$	1.76390e9	+	3.05516e9i	0.309287	+	0.535701i	0.978207	−	0.207634i	$-0.0665762\pi$
$613$	1.76390e9	+	3.05516e9i	0.309287	+	0.535701i	−0.668919	+	0.743335i	$0.733243\pi$
$614$		0			0
$615$	−8.55632e8			−0.148328
$616$		0			0
$617$	−1.68479e9			−0.288768			−0.144384	−	0.989522i	$-0.546120\pi$
$617$	−1.68479e9			−0.288768			−0.144384	+	0.989522i	$0.546120\pi$
$618$		0			0
$619$	3.22124e9	+	5.57935e9i	0.545891	+	0.945511i	0.998550	+	0.0538276i	$0.0171421\pi$
$619$	3.22124e9	+	5.57935e9i	0.545891	+	0.945511i	−0.452659	+	0.891684i	$0.649525\pi$
$620$		0			0
$621$	−8.75783e8	+	1.51690e9i	−0.146749	+	0.254177i
$622$		0			0
$623$	−7.01241e9	+	5.32779e9i	−1.16187	+	0.882753i
$624$		0			0
$625$	1.05531e8	−	1.82786e8i	0.0172903	−	0.0299476i
$626$		0			0
$627$	2.23621e8	+	3.87322e8i	0.0362306	+	0.0627532i
$628$		0			0
$629$	−4.42220e9			−0.708535
$630$		0			0
$631$	9.82780e9			1.55723			0.778617	−	0.627500i	$-0.215922\pi$
$631$	9.82780e9			1.55723			0.778617	+	0.627500i	$0.215922\pi$
$632$		0			0
$633$	−3.39087e8	−	5.87316e8i	−0.0531372	−	0.0920362i
$634$		0			0
$635$	−3.02326e9	+	5.23645e9i	−0.468563	+	0.811575i
$636$		0			0
$637$	7.62638e9	+	7.77349e9i	1.16904	+	1.19159i
$638$		0			0
$639$	5.43501e9	−	9.41372e9i	0.824039	−	1.42728i
$640$		0			0
$641$	−5.29277e9	−	9.16734e9i	−0.793743	−	1.37480i	−0.923634	−	0.383275i	$-0.874796\pi$
$641$	−5.29277e9	−	9.16734e9i	−0.793743	−	1.37480i	0.129892	−	0.991528i	$-0.458537\pi$
$642$		0			0
$643$	2.05978e9			0.305551			0.152775	−	0.988261i	$-0.451179\pi$
$643$	2.05978e9			0.305551			0.152775	+	0.988261i	$0.451179\pi$
$644$		0			0
$645$	7.65288e8			0.112296
$646$		0			0
$647$	−4.88958e9	−	8.46900e9i	−0.709752	−	1.22933i	−0.964949	−	0.262437i	$-0.915474\pi$
$647$	−4.88958e9	−	8.46900e9i	−0.709752	−	1.22933i	0.255197	−	0.966889i	$-0.417860\pi$
$648$		0			0
$649$	−1.14630e9	+	1.98545e9i	−0.164605	+	0.285103i
$650$		0			0
$651$	1.79360e9	−	1.36272e9i	0.254796	−	0.193586i
$652$		0			0
$653$	1.79886e9	−	3.11571e9i	0.252814	−	0.437886i	−0.711486	−	0.702701i	$-0.751977\pi$
$653$	1.79886e9	−	3.11571e9i	0.252814	−	0.437886i	0.964299	+	0.264814i	$0.0853107\pi$
$654$		0			0
$655$	1.81252e9	+	3.13938e9i	0.252022	+	0.436515i
$656$		0			0
$657$	5.16058e9			0.709936
$658$		0			0
$659$	−1.04261e10			−1.41913			−0.709567	−	0.704638i	$-0.751109\pi$
$659$	−1.04261e10			−1.41913			−0.709567	+	0.704638i	$0.751109\pi$
$660$		0			0
$661$	2.58339e9	+	4.47456e9i	0.347924	+	0.602622i	0.985881	−	0.167450i	$-0.0535534\pi$
$661$	2.58339e9	+	4.47456e9i	0.347924	+	0.602622i	−0.637957	+	0.770072i	$0.720220\pi$
$662$		0			0
$663$	−2.71694e9	+	4.70588e9i	−0.362062	+	0.627109i
$664$		0			0
$665$	1.79556e8	+	1.41610e9i	0.0236769	+	0.186732i
$666$		0			0
$667$	2.19088e9	−	3.79471e9i	0.285876	−	0.495152i
$668$		0			0
$669$	−2.98268e8	−	5.16616e8i	−0.0385137	−	0.0667077i
$670$		0			0
$671$	−2.47289e9			−0.315992
$672$		0			0
$673$	−4.78512e9			−0.605118			−0.302559	−	0.953131i	$-0.597841\pi$
$673$	−4.78512e9			−0.605118			−0.302559	+	0.953131i	$0.597841\pi$
$674$		0			0
$675$	1.75206e9	+	3.03465e9i	0.219273	+	0.379792i
$676$		0			0
$677$	3.41978e9	−	5.92323e9i	0.423582	−	0.733666i	−0.572704	−	0.819762i	$-0.694106\pi$
$677$	3.41978e9	−	5.92323e9i	0.423582	−	0.733666i	0.996287	+	0.0860956i	$0.0274391\pi$
$678$		0			0
$679$	−2.67523e9	−	1.12312e9i	−0.327958	−	0.137683i
$680$		0			0
$681$	−1.14955e9	+	1.99109e9i	−0.139481	+	0.241588i
$682$		0			0
$683$	−7.98324e7	−	1.38274e8i	−0.00958753	−	0.0166061i	0.861192	−	0.508280i	$-0.169719\pi$
$683$	−7.98324e7	−	1.38274e8i	−0.00958753	−	0.0166061i	−0.870779	+	0.491674i	$0.836385\pi$
$684$		0			0
$685$	4.80454e9			0.571130
$686$		0			0
$687$	−3.25821e9			−0.383381
$688$		0			0
$689$	7.26083e9	+	1.25761e10i	0.845706	+	1.46481i
$690$		0			0
$691$	−7.02899e9	+	1.21746e10i	−0.810437	+	1.40372i	0.102121	+	0.994772i	$0.467437\pi$
$691$	−7.02899e9	+	1.21746e10i	−0.810437	+	1.40372i	−0.912558	+	0.408947i	$0.865896\pi$
$692$		0			0
$693$	4.19374e9	+	1.76062e9i	0.478669	+	0.200955i
$694$		0			0
$695$	−3.47795e8	+	6.02399e8i	−0.0392985	+	0.0680671i
$696$		0			0
$697$	2.92781e9	+	5.07112e9i	0.327513	+	0.567269i
$698$		0			0
$699$	5.81629e9			0.644134
$700$		0			0
$701$	−1.44650e9			−0.158600			−0.0793002	−	0.996851i	$-0.525269\pi$
$701$	−1.44650e9			−0.158600			−0.0793002	+	0.996851i	$0.525269\pi$
$702$		0			0
$703$	−8.86471e8	−	1.53541e9i	−0.0962323	−	0.166679i
$704$		0			0
$705$	1.27039e9	−	2.20039e9i	0.136545	−	0.236503i
$706$		0			0
$707$	−1.18130e9	−	9.31649e9i	−0.125716	−	0.991480i
$708$		0			0
$709$	−4.20057e9	+	7.27561e9i	−0.442636	+	0.766668i	−0.997884	−	0.0650166i	$-0.979290\pi$
$709$	−4.20057e9	+	7.27561e9i	−0.442636	+	0.766668i	0.555248	+	0.831685i	$0.312623\pi$
$710$		0			0
$711$	8.66519e8	+	1.50085e9i	0.0904137	+	0.156601i
$712$		0			0
$713$	−3.16372e9			−0.326878
$714$		0			0
$715$	6.32604e9			0.647234
$716$		0			0
$717$	−2.16293e9	−	3.74630e9i	−0.219142	−	0.379565i
$718$		0			0
$719$	−7.19176e9	+	1.24565e10i	−0.721579	+	1.24981i	0.238787	+	0.971072i	$0.423250\pi$
$719$	−7.19176e9	+	1.24565e10i	−0.721579	+	1.24981i	−0.960367	+	0.278740i	$0.910083\pi$
$720$		0			0
$721$	1.28468e10	−	9.76053e9i	1.27650	−	0.969840i
$722$		0			0
$723$	−1.02311e8	+	1.77207e8i	−0.0100679	+	0.0174380i
$724$		0			0
$725$	−4.38299e9	−	7.59155e9i	−0.427157	−	0.739857i
$726$		0			0
$727$	9.99667e9			0.964906			0.482453	−	0.875922i	$-0.339746\pi$
$727$	9.99667e9			0.964906			0.482453	+	0.875922i	$0.339746\pi$
$728$		0			0
$729$	−1.91223e9			−0.182807
$730$		0			0
$731$	−2.61867e9	−	4.53567e9i	−0.247954	−	0.429468i
$732$		0			0
$733$	−7.02053e9	+	1.21599e10i	−0.658425	+	1.14043i	0.322599	+	0.946536i	$0.395444\pi$
$733$	−7.02053e9	+	1.21599e10i	−0.658425	+	1.14043i	−0.981023	+	0.193889i	$0.937890\pi$
$734$		0			0
$735$	−1.87616e9	−	1.91235e9i	−0.174287	−	0.177648i
$736$		0			0
$737$	6.25140e9	−	1.08277e10i	0.575229	−	0.996327i
$738$		0			0
$739$	−2.69376e9	−	4.66574e9i	−0.245530	−	0.425270i	0.716751	−	0.697330i	$-0.245629\pi$
$739$	−2.69376e9	−	4.66574e9i	−0.245530	−	0.425270i	−0.962280	+	0.272060i	$0.912295\pi$
$740$		0			0
$741$	−2.17854e9			−0.196699
$742$		0			0
$743$	1.07384e10			0.960458			0.480229	−	0.877143i	$-0.340553\pi$
$743$	1.07384e10			0.960458			0.480229	+	0.877143i	$0.340553\pi$
$744$		0			0
$745$	−5.51809e9	−	9.55761e9i	−0.488925	−	0.846842i
$746$		0			0
$747$	3.84529e9	−	6.66023e9i	0.337526	−	0.584611i
$748$		0			0
$749$	−1.32775e10	+	1.00878e10i	−1.15460	+	0.877224i
$750$		0			0
$751$	4.22337e9	−	7.31509e9i	0.363847	−	0.630202i	−0.624743	−	0.780830i	$-0.714796\pi$
$751$	4.22337e9	−	7.31509e9i	0.363847	−	0.630202i	0.988590	+	0.150628i	$0.0481297\pi$
$752$		0			0
$753$	−4.22094e9	−	7.31087e9i	−0.360269	−	0.624003i
$754$		0			0
$755$	6.66869e9			0.563932
$756$		0			0
$757$	−2.05285e10			−1.71998			−0.859988	−	0.510314i	$-0.829529\pi$
$757$	−2.05285e10			−1.71998			−0.859988	+	0.510314i	$0.829529\pi$
$758$		0			0
$759$	5.89455e8	+	1.02097e9i	0.0489333	+	0.0847549i
$760$		0			0
$761$	6.29140e9	−	1.08970e10i	0.517489	−	0.896317i	−0.482305	−	0.876004i	$-0.660200\pi$
$761$	6.29140e9	−	1.08970e10i	0.517489	−	0.896317i	0.999794	−	0.0203138i	$-0.00646652\pi$
$762$		0			0
$763$	6.73548e8	+	5.31204e9i	0.0548950	+	0.432937i
$764$		0			0
$765$	−3.62180e9	+	6.27314e9i	−0.292489	+	0.506606i
$766$		0			0
$767$	−5.58370e9	−	9.67125e9i	−0.446826	−	0.773925i
$768$		0			0
$769$	1.73875e10			1.37878			0.689391	−	0.724390i	$-0.257878\pi$
$769$	1.73875e10			1.37878			0.689391	+	0.724390i	$0.257878\pi$
$770$		0			0
$771$	7.62954e9			0.599526
$772$		0			0
$773$	6.00100e8	+	1.03940e9i	0.0467300	+	0.0809387i	0.888444	−	0.458984i	$-0.151787\pi$
$773$	6.00100e8	+	1.03940e9i	0.0467300	+	0.0809387i	−0.841714	+	0.539923i	$0.818453\pi$
$774$		0			0
$775$	−3.16461e9	+	5.48127e9i	−0.244210	+	0.422985i
$776$		0			0
$777$	3.06804e9	+	1.28802e9i	0.234632	+	0.0985033i
$778$		0			0
$779$	−1.17381e9	+	2.03311e9i	−0.0889648	+	0.154092i
$780$		0			0
$781$	−7.99127e9	−	1.38413e10i	−0.600257	−	1.03968i
$782$		0			0
$783$	−1.38657e10			−1.03223
$784$		0			0
$785$	1.63066e8			0.0120315
$786$		0			0
$787$	−1.06710e10	−	1.84827e10i	−0.780359	−	1.35162i	−0.931733	−	0.363144i	$-0.881703\pi$
$787$	−1.06710e10	−	1.84827e10i	−0.780359	−	1.35162i	0.151374	−	0.988477i	$-0.451630\pi$
$788$		0			0
$789$	−2.50222e9	+	4.33397e9i	−0.181366	+	0.314135i
$790$		0			0
$791$	1.88579e10	+	7.91695e9i	1.35480	+	0.568775i
$792$		0			0
$793$	6.02281e9	−	1.04318e10i	0.428887	−	0.742854i
$794$		0			0
$795$	−1.78623e9	−	3.09384e9i	−0.126082	−	0.218380i
$796$		0			0
$797$	−2.52753e9			−0.176845			−0.0884225	−	0.996083i	$-0.528183\pi$
$797$	−2.52753e9			−0.176845			−0.0884225	+	0.996083i	$0.528183\pi$
$798$		0			0
$799$	−1.73882e10			−1.20598
$800$		0			0
$801$	−8.95860e9	−	1.55168e10i	−0.615923	−	1.06681i
$802$		0			0
$803$	3.79388e9	−	6.57119e9i	0.258570	−	0.447857i
$804$		0			0
$805$	4.73303e8	+	3.73278e9i	0.0319782	+	0.252201i
$806$		0			0
$807$	4.23190e9	−	7.32986e9i	0.283451	−	0.490951i
$808$		0			0
$809$	4.28924e8	+	7.42918e8i	0.0284813	+	0.0493311i	0.879915	−	0.475132i	$-0.157600\pi$
$809$	4.28924e8	+	7.42918e8i	0.0284813	+	0.0493311i	−0.851433	+	0.524463i	$0.824266\pi$
$810$		0			0
$811$	7.71014e9			0.507562			0.253781	−	0.967262i	$-0.418326\pi$
$811$	7.71014e9			0.507562			0.253781	+	0.967262i	$0.418326\pi$
$812$		0			0
$813$	9.99459e9			0.652301
$814$		0			0
$815$	−5.38615e9	−	9.32909e9i	−0.348520	−	0.603654i
$816$		0			0
$817$	1.04987e9	−	1.81843e9i	0.0673534	−	0.116660i
$818$		0			0
$819$	−1.76411e10	+	1.34031e10i	−1.12210	+	0.852534i
$820$		0			0
$821$	7.96146e9	−	1.37896e10i	0.502101	−	0.869665i	−0.497896	−	0.867237i	$-0.665894\pi$
$821$	7.96146e9	−	1.37896e10i	0.502101	−	0.869665i	0.999997	−	0.00242818i	$-0.000772915\pi$
$822$		0			0
$823$	−1.25789e10	−	2.17873e10i	−0.786579	−	1.36240i	−0.928051	−	0.372453i	$-0.878517\pi$
$823$	−1.25789e10	−	2.17873e10i	−0.786579	−	1.36240i	0.141472	−	0.989942i	$-0.454817\pi$
$824$		0			0
$825$	2.35848e9			0.146232
$826$		0			0
$827$	−1.68241e10			−1.03434			−0.517168	−	0.855884i	$-0.673014\pi$
$827$	−1.68241e10			−1.03434			−0.517168	+	0.855884i	$0.673014\pi$
$828$		0			0
$829$	−4.45917e9	−	7.72350e9i	−0.271840	−	0.470840i	0.697493	−	0.716591i	$-0.254299\pi$
$829$	−4.45917e9	−	7.72350e9i	−0.271840	−	0.470840i	−0.969333	+	0.245751i	$0.920965\pi$
$830$		0			0
$831$	1.20691e9	−	2.09042e9i	0.0729575	−	0.126366i
$832$		0			0
$833$	−4.91416e9	+	1.76632e10i	−0.294572	+	1.05880i
$834$		0			0
$835$	−1.06831e8	+	1.85036e8i	−0.00635029	+	0.0109990i
$836$		0			0
$837$	5.00566e9	+	8.67005e9i	0.295068	+	0.511073i
$838$		0			0
$839$	1.16315e10			0.679936			0.339968	−	0.940437i	$-0.389584\pi$
$839$	1.16315e10			0.679936			0.339968	+	0.940437i	$0.389584\pi$
$840$		0			0
$841$	1.74368e10			1.01084
$842$		0			0
$843$	1.57946e9	+	2.73571e9i	0.0908056	+	0.157280i
$844$		0			0
$845$	−9.87813e9	+	1.71094e10i	−0.563217	+	0.975521i
$846$		0			0
$847$	−8.75632e9	+	6.65276e9i	−0.495142	+	0.376192i
$848$		0			0
$849$	1.01140e9	−	1.75180e9i	0.0567214	−	0.0982444i
$850$		0			0
$851$	−2.33670e9	−	4.04728e9i	−0.129972	−	0.225118i
$852$		0			0
$853$	1.23761e10			0.682748			0.341374	−	0.939928i	$-0.389108\pi$
$853$	1.23761e10			0.682748			0.341374	+	0.939928i	$0.389108\pi$
$854$		0			0
$855$	−2.90409e9			−0.158902
$856$		0			0
$857$	4.27039e9	+	7.39653e9i	0.231758	+	0.401416i	0.958326	−	0.285679i	$-0.0922190\pi$
$857$	4.27039e9	+	7.39653e9i	0.231758	+	0.401416i	−0.726568	+	0.687095i	$0.758886\pi$
$858$		0			0
$859$	2.68950e9	−	4.65836e9i	0.144776	−	0.250759i	−0.784513	−	0.620112i	$-0.787087\pi$
$859$	2.68950e9	−	4.65836e9i	0.144776	−	0.250759i	0.929289	+	0.369353i	$0.120421\pi$
$860$		0			0
$861$	−5.54229e8	−	4.37101e9i	−0.0295923	−	0.233384i
$862$		0			0
$863$	−2.91171e8	+	5.04323e8i	−0.0154209	+	0.0267098i	−0.873633	−	0.486586i	$-0.838242\pi$
$863$	−2.91171e8	+	5.04323e8i	−0.0154209	+	0.0267098i	0.858212	+	0.513295i	$0.171575\pi$
$864$		0			0
$865$	−8.69444e9	−	1.50592e10i	−0.456757	−	0.791126i
$866$		0			0
$867$	−1.57412e9			−0.0820297
$868$		0			0
$869$	2.54814e9			0.131721
$870$		0			0
$871$	3.04510e10	+	5.27426e10i	1.56148	+	2.70457i
$872$		0			0
$873$	2.95145e9	−	5.11206e9i	0.150136	−	0.260043i
$874$		0			0
$875$	1.84610e10	+	7.75032e9i	0.931596	+	0.391103i
$876$		0			0
$877$	−9.35669e9	+	1.62063e10i	−0.468407	+	0.811306i	−0.999348	−	0.0361035i	$-0.988505\pi$
$877$	−9.35669e9	+	1.62063e10i	−0.468407	+	0.811306i	0.530941	+	0.847409i	$0.321839\pi$
$878$		0			0
$879$	−6.88251e9	−	1.19209e10i	−0.341811	−	0.592034i
$880$		0			0
$881$	1.01779e8			0.00501469			0.00250734	−	0.999997i	$-0.499202\pi$
$881$	1.01779e8			0.00501469			0.00250734	+	0.999997i	$0.499202\pi$
$882$		0			0
$883$	−2.40782e10			−1.17696			−0.588480	−	0.808512i	$-0.700274\pi$
$883$	−2.40782e10			−1.17696			−0.588480	+	0.808512i	$0.700274\pi$
$884$		0			0
$885$	1.37364e9	+	2.37922e9i	0.0666150	+	0.115381i
$886$		0			0
$887$	1.46914e10	−	2.54463e10i	0.706856	−	1.22431i	−0.259161	−	0.965834i	$-0.583446\pi$
$887$	1.46914e10	−	2.54463e10i	0.706856	−	1.22431i	0.966017	−	0.258477i	$-0.0832207\pi$
$888$		0			0
$889$	−2.87088e10	−	1.20525e10i	−1.37043	−	0.575336i
$890$		0			0
$891$	−3.61531e9	+	6.26190e9i	−0.171228	+	0.296575i
$892$		0			0
$893$	−3.48562e9	−	6.03728e9i	−0.163795	−	0.283701i
$894$		0			0
$895$	−8.48864e9			−0.395783
$896$		0			0
$897$	−5.74254e9			−0.265663
$898$		0			0
$899$	−1.25223e10	−	2.16892e10i	−0.574810	−	0.995599i
$900$		0			0
$901$	−1.22243e10	+	2.11731e10i	−0.556785	+	0.964379i
$902$		0			0
$903$	4.95709e8	+	3.90949e9i	0.0224037	+	0.176690i
$904$		0			0
$905$	5.75406e9	−	9.96632e9i	0.258050	−	0.446956i
$906$		0			0
$907$	1.23474e10	+	2.13864e10i	0.549479	+	0.951725i	0.998310	+	0.0581085i	$0.0185069\pi$
$907$	1.23474e10	+	2.13864e10i	0.549479	+	0.951725i	−0.448832	+	0.893616i	$0.648160\pi$
$908$		0			0
$909$	1.91060e10			0.843714
$910$		0			0
$911$	−6.77617e9			−0.296941			−0.148470	−	0.988917i	$-0.547435\pi$
$911$	−6.77617e9			−0.296941			−0.148470	+	0.988917i	$0.547435\pi$
$912$		0			0
$913$	−5.65384e9	−	9.79274e9i	−0.245865	−	0.425850i
$914$		0			0
$915$	−1.48167e9	+	2.56632e9i	−0.0639406	+	0.110748i
$916$		0			0
$917$	−1.48635e10	+	1.12928e10i	−0.636545	+	0.483625i
$918$		0			0
$919$	6.14974e9	−	1.06517e10i	0.261368	−	0.452702i	−0.705238	−	0.708971i	$-0.749160\pi$
$919$	6.14974e9	−	1.06517e10i	0.261368	−	0.452702i	0.966606	+	0.256268i	$0.0824931\pi$
$920$		0			0
$921$	−3.06945e9	−	5.31645e9i	−0.129465	−	0.224240i
$922$		0			0
$923$	7.78520e10			3.25884
$924$		0			0
$925$	−9.34941e9			−0.388408
$926$		0			0
$927$	1.64122e10	+	2.84267e10i	0.676686	+	1.17206i
$928$		0			0
$929$	−2.23527e10	+	3.87160e10i	−0.914691	+	1.58429i	−0.107337	+	0.994223i	$0.534233\pi$
$929$	−2.23527e10	+	3.87160e10i	−0.914691	+	1.58429i	−0.807353	+	0.590068i	$0.799101\pi$
$930$		0			0
$931$	−7.11785e9	+	1.83453e9i	−0.289085	+	0.0745078i
$932$		0			0
$933$	7.46699e9	−	1.29332e10i	0.300995	−	0.521339i
$934$		0			0
$935$	5.32525e9	+	9.22360e9i	0.213059	+	0.369028i
$936$		0			0
$937$	4.82318e10			1.91534			0.957668	−	0.287874i	$-0.0929486\pi$
$937$	4.82318e10			1.91534			0.957668	+	0.287874i	$0.0929486\pi$
$938$		0			0
$939$	1.20358e10			0.474399
$940$		0			0
$941$	8.39260e9	+	1.45364e10i	0.328347	+	0.568714i	0.982184	−	0.187922i	$-0.0601751\pi$
$941$	8.39260e9	+	1.45364e10i	0.328347	+	0.568714i	−0.653837	+	0.756635i	$0.726842\pi$
$942$		0			0
$943$	−3.09412e9	+	5.35918e9i	−0.120156	+	0.208117i
$944$		0			0
$945$	9.48065e9	−	7.20308e9i	0.365449	−	0.277656i
$946$		0			0
$947$	−1.08057e10	+	1.87160e10i	−0.413455	+	0.716125i	−0.995265	−	0.0972000i	$-0.969011\pi$
$947$	−1.08057e10	+	1.87160e10i	−0.413455	+	0.716125i	0.581810	+	0.813325i	$0.302345\pi$
$948$		0			0
$949$	1.84802e10	+	3.20087e10i	0.701900	+	1.21573i
$950$		0			0
$951$	4.28553e9			0.161574
$952$		0			0
$953$	2.72269e10			1.01900			0.509498	−	0.860472i	$-0.329831\pi$
$953$	2.72269e10			1.01900			0.509498	+	0.860472i	$0.329831\pi$
$954$		0			0
$955$	2.35101e8	+	4.07207e8i	0.00873459	+	0.0151287i
$956$		0			0
$957$	−4.66621e9	+	8.08212e9i	−0.172097	+	0.298080i
$958$		0			0
$959$	3.11211e9	+	2.45441e10i	0.113943	+	0.898632i
$960$		0			0
$961$	4.71496e9	−	8.16655e9i	0.171374	−	0.296829i
$962$		0			0
$963$	−1.69625e10	−	2.93799e10i	−0.612066	−	1.06013i
$964$		0			0
$965$	−6.00875e9			−0.215248
$966$		0			0
$967$	−2.88074e9			−0.102450			−0.0512250	−	0.998687i	$-0.516313\pi$
$967$	−2.88074e9			−0.102450			−0.0512250	+	0.998687i	$0.516313\pi$
$968$		0			0
$969$	−1.83389e9	−	3.17639e9i	−0.0647500	−	0.112150i
$970$		0			0
$971$	−1.33908e10	+	2.31936e10i	−0.469396	+	0.813019i	−0.999388	−	0.0349845i	$-0.988862\pi$
$971$	−1.33908e10	+	2.31936e10i	−0.469396	+	0.813019i	0.529991	+	0.848003i	$0.322195\pi$
$972$		0			0
$973$	−3.30264e9	−	1.38652e9i	−0.114939	−	0.0482537i
$974$		0			0
$975$	−5.74415e9	+	9.94916e9i	−0.198477	+	0.343772i
$976$		0			0
$977$	1.03023e10	+	1.78441e10i	0.353430	+	0.612159i	0.986848	−	0.161651i	$-0.0516819\pi$
$977$	1.03023e10	+	1.78441e10i	0.353430	+	0.612159i	−0.633418	+	0.773810i	$0.718349\pi$
$978$		0			0
$979$	−2.63442e10			−0.897317
$980$		0			0
$981$	−1.08938e10			−0.368414
$982$		0			0
$983$	−7.94195e9	−	1.37559e10i	−0.266679	−	0.461902i	0.701323	−	0.712844i	$-0.252593\pi$
$983$	−7.94195e9	−	1.37559e10i	−0.266679	−	0.461902i	−0.968002	+	0.250941i	$0.919260\pi$
$984$		0			0
$985$	−2.33535e9	+	4.04495e9i	−0.0778620	+	0.134861i
$986$		0			0
$987$	1.20636e10	+	5.06455e9i	0.399362	+	0.167660i
$988$		0			0
$989$	2.76742e9	−	4.79332e9i	0.0909680	−	0.157561i
$990$		0			0
$991$	−1.79584e10	−	3.11048e10i	−0.586150	−	1.01524i	−0.994731	−	0.102519i	$-0.967310\pi$
$991$	−1.79584e10	−	3.11048e10i	−0.586150	−	1.01524i	0.408581	−	0.912722i	$-0.366024\pi$
$992$		0			0
$993$	1.47289e10			0.477361
$994$		0			0
$995$	−3.75528e10			−1.20854
$996$		0			0
$997$	−4.39087e9	−	7.60521e9i	−0.140319	−	0.243040i	0.787297	−	0.616573i	$-0.211480\pi$
$997$	−4.39087e9	−	7.60521e9i	−0.140319	−	0.243040i	−0.927617	+	0.373533i	$0.878146\pi$
$998$		0			0
$999$	−7.39427e9	+	1.28072e10i	−0.234648	+	0.406421i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.8.i.d.65.4		10
4.3	odd	2		28.8.e.a.9.2	✓	10
7.4	even	3	inner	112.8.i.d.81.4		10
12.11	even	2		252.8.k.c.37.4		10
28.3	even	6		196.8.e.f.165.4		10
28.11	odd	6		28.8.e.a.25.2	yes	10
28.19	even	6		196.8.a.e.1.2		5
28.23	odd	6		196.8.a.d.1.4		5
28.27	even	2		196.8.e.f.177.4		10
84.11	even	6		252.8.k.c.109.4		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.e.a.9.2	✓	10	4.3	odd	2
28.8.e.a.25.2	yes	10	28.11	odd	6
112.8.i.d.65.4		10	1.1	even	1	trivial
112.8.i.d.81.4		10	7.4	even	3	inner
196.8.a.d.1.4		5	28.23	odd	6
196.8.a.e.1.2		5	28.19	even	6
196.8.e.f.165.4		10	28.3	even	6
196.8.e.f.177.4		10	28.27	even	2
252.8.k.c.37.4		10	12.11	even	2
252.8.k.c.109.4		10	84.11	even	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:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(9.22936 + 15.9857i) q^{3} +(-88.1161 + 152.621i) q^{5} +(-836.746 - 351.283i) q^{7} +(923.138 - 1598.92i) q^{9} +(-1357.32 - 2350.94i) q^{11} +13223.2 q^{13} -3253.02 q^{15} +(11131.2 + 19279.8i) q^{17} +(-4462.71 + 7729.64i) q^{19} +(-2107.12 - 16618.1i) q^{21} +(-11763.5 + 20375.0i) q^{23} +(23533.6 + 40761.4i) q^{25} +74449.1 q^{27} -186244. q^{29} +(67236.0 + 116456. i) q^{31} +(25054.4 - 43395.4i) q^{33} +(127344. - 96751.7i) q^{35} +(-99319.8 + 172027. i) q^{37} +(122041. + 211382. i) q^{39} +263027. q^{41} -235255. q^{43} +(162687. + 281781. i) q^{45} +(-390528. + 676414. i) q^{47} +(576744. + 587869. i) q^{49} +(-205468. + 355881. i) q^{51} +(549100. + 951068. i) q^{53} +478406. q^{55} -164752. q^{57} +(-422267. - 731387. i) q^{59} +(455474. - 788904. i) q^{61} +(-1.33411e6 + 1.01361e6i) q^{63} +(-1.16517e6 + 2.01814e6i) q^{65} +(2.30285e6 + 3.98865e6i) q^{67} -434279. q^{69} +5.88754e6 q^{71} +(1.39756e6 + 2.42065e6i) q^{73} +(-434401. + 752404. i) q^{75} +(309884. + 2.44394e6i) q^{77} +(-469333. + 812909. i) q^{79} +(-1.33178e6 - 2.30672e6i) q^{81} +4.16545e6 q^{83} -3.92336e6 q^{85} +(-1.71891e6 - 2.97724e6i) q^{87} +(4.85226e6 - 8.40436e6i) q^{89} +(-1.10644e7 - 4.64507e6i) q^{91} +(-1.24109e6 + 2.14963e6i) q^{93} +(-786473. - 1.36221e6i) q^{95} +3.19719e6 q^{97} -5.01197e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) \) 10 * q - 27 * q^3 + 249 * q^5 - 332 * q^7 - 5702 * q^9 - 6399 * q^11 - 26988 * q^13 - 19294 * q^15 + 3609 * q^17 + 12403 * q^19 + 16099 * q^21 + 13959 * q^23 - 162364 * q^25 - 161550 * q^27 + 26148 * q^29 + 20181 * q^31 + 264239 * q^33 - 791715 * q^35 - 54763 * q^37 - 4750 * q^39 - 1824468 * q^41 + 1938424 * q^43 + 1936430 * q^45 - 1217253 * q^47 + 1722058 * q^49 + 1353871 * q^51 - 2349159 * q^53 - 3123494 * q^55 - 9823194 * q^57 - 2188611 * q^59 + 6107445 * q^61 - 10275990 * q^63 - 3022014 * q^65 + 3171581 * q^67 - 11255938 * q^69 + 18938976 * q^71 + 8034853 * q^73 - 29173588 * q^75 + 24100191 * q^77 + 7589559 * q^79 - 9846485 * q^81 + 13452408 * q^83 - 48717670 * q^85 - 30033142 * q^87 + 29864757 * q^89 - 34267512 * q^91 - 20217029 * q^93 + 25812009 * q^95 + 285580 * q^97 + 119277812 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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