LMFDB - Embedded newform 112.8.i.a.65.2

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 = [4,0,-56] 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:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{2389})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 598x^{2} + 597x + 356409 $ x^4 - x^3 + 598x^2 + 597x + 356409
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.2
Root			$-11.9693 - 20.7315i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.8.i.a.81.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(10.4387 + 18.0804i) q^{3} +(108.377 - 187.715i) q^{5} +(-726.284 + 544.110i) q^{7} +(875.567 - 1516.53i) q^{9} +(1290.93 + 2235.95i) q^{11} -8921.97 q^{13} +4525.28 q^{15} +(5556.44 + 9624.03i) q^{17} +(-4323.40 + 7488.34i) q^{19} +(-17419.1 - 7451.67i) q^{21} +(-33028.6 + 57207.2i) q^{23} +(15571.2 + 26970.1i) q^{25} +82218.0 q^{27} +128836. q^{29} +(-102048. - 176752. i) q^{31} +(-26951.2 + 46680.9i) q^{33} +(23424.9 + 195304. i) q^{35} +(-245856. + 425834. i) q^{37} +(-93133.8 - 161312. i) q^{39} -623293. q^{41} -422919. q^{43} +(-189783. - 328714. i) q^{45} +(-602714. + 1.04393e6i) q^{47} +(231433. - 790356. i) q^{49} +(-116004. + 200925. i) q^{51} +(636483. + 1.10242e6i) q^{53} +559630. q^{55} -180523. q^{57} +(840430. + 1.45567e6i) q^{59} +(-1.06128e6 + 1.83818e6i) q^{61} +(189247. + 1.57783e6i) q^{63} +(-966940. + 1.67479e6i) q^{65} +(-1.67466e6 - 2.90060e6i) q^{67} -1.37910e6 q^{69} -2.49257e6 q^{71} +(1.63676e6 + 2.83496e6i) q^{73} +(-325086. + 563065. i) q^{75} +(-2.15418e6 - 921530. i) q^{77} +(-2.09920e6 + 3.63591e6i) q^{79} +(-1.05662e6 - 1.83011e6i) q^{81} -3.35527e6 q^{83} +2.40877e6 q^{85} +(1.34488e6 + 2.32939e6i) q^{87} +(-1.19979e6 + 2.07810e6i) q^{89} +(6.47988e6 - 4.85453e6i) q^{91} +(2.13049e6 - 3.69011e6i) q^{93} +(937117. + 1.62313e6i) q^{95} +4.51506e6 q^{97} +4.52118e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 56 q^{3} + 238 q^{5} - 168 q^{7} - 1972 q^{9} + 5848 q^{11} + 2632 q^{13} + 5784 q^{15} + 47642 q^{17} - 41048 q^{19} - 169582 q^{21} - 49316 q^{23} + 108816 q^{25} + 400624 q^{27} + 345640 q^{29}+ \cdots - 15278208 q^{99}+O(q^{100}) $ 4 * q - 56 * q^3 + 238 * q^5 - 168 * q^7 - 1972 * q^9 + 5848 * q^11 + 2632 * q^13 + 5784 * q^15 + 47642 * q^17 - 41048 * q^19 - 169582 * q^21 - 49316 * q^23 + 108816 * q^25 + 400624 * q^27 + 345640 * q^29 - 70252 * q^31 + 197190 * q^33 + 36904 * q^35 + 88166 * q^37 - 973336 * q^39 - 2312520 * q^41 - 115088 * q^43 - 300468 * q^45 - 1412292 * q^47 + 465892 * q^49 + 2576256 * q^51 + 2361174 * q^53 + 1258040 * q^55 + 4620796 * q^57 + 1842512 * q^59 - 1278242 * q^61 + 2121448 * q^63 - 1716372 * q^65 - 2121480 * q^67 - 5332236 * q^69 - 7200320 * q^71 + 1634682 * q^73 + 5321176 * q^75 + 1416226 * q^77 - 9192604 * q^79 - 3049498 * q^81 + 56560 * q^83 + 6369676 * q^85 - 691656 * q^87 - 8936550 * q^89 + 26111120 * q^91 + 14550490 * q^93 + 2562604 * q^95 + 11748408 * q^97 - 15278208 * 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
$3$	10.4387	+	18.0804i	0.223214	+	0.386618i	0.955782	−	0.294075i	$-0.0950117\pi$
$3$	10.4387	+	18.0804i	0.223214	+	0.386618i	−0.732568	+	0.680694i	$0.761678\pi$
$4$		0			0
$5$	108.377	−	187.715i	0.387743	−	0.671590i	−0.604403	−	0.796679i	$-0.706588\pi$
$5$	108.377	−	187.715i	0.387743	−	0.671590i	0.992146	+	0.125089i	$0.0399216\pi$
$6$		0			0
$7$	−726.284	+	544.110i	−0.800319	+	0.599575i
$7$	−726.284	+	544.110i	−0.800319	+	0.599575i
$8$		0			0
$9$	875.567	−	1516.53i	0.400351	−	0.693428i
$10$		0			0
$11$	1290.93	+	2235.95i	0.292434	+	0.506511i	0.974385	−	0.224887i	$-0.0722015\pi$
$11$	1290.93	+	2235.95i	0.292434	+	0.506511i	−0.681951	+	0.731398i	$0.738868\pi$
$12$		0			0
$13$	−8921.97			−1.12631			−0.563156	−	0.826350i	$-0.690413\pi$
$13$	−8921.97			−1.12631			−0.563156	+	0.826350i	$0.690413\pi$
$14$		0			0
$15$	4525.28			0.346199
$16$		0			0
$17$	5556.44	+	9624.03i	0.274300	+	0.475101i	0.969958	−	0.243272i	$-0.0782207\pi$
$17$	5556.44	+	9624.03i	0.274300	+	0.475101i	−0.695659	+	0.718373i	$0.744887\pi$
$18$		0			0
$19$	−4323.40	+	7488.34i	−0.144606	+	0.250466i	−0.929226	−	0.369512i	$-0.879525\pi$
$19$	−4323.40	+	7488.34i	−0.144606	+	0.250466i	0.784620	+	0.619977i	$0.212858\pi$
$20$		0			0
$21$	−17419.1	−	7451.67i	−0.410449	−	0.175584i
$22$		0			0
$23$	−33028.6	+	57207.2i	−0.566034	+	0.980399i	0.430919	+	0.902391i	$0.358190\pi$
$23$	−33028.6	+	57207.2i	−0.566034	+	0.980399i	−0.996953	+	0.0780088i	$0.975144\pi$
$24$		0			0
$25$	15571.2	+	26970.1i	0.199311	+	0.345217i
$26$		0			0
$27$	82218.0			0.803885
$28$		0			0
$29$	128836.			0.980941			0.490470	−	0.871458i	$-0.336825\pi$
$29$	128836.			0.980941			0.490470	+	0.871458i	$0.336825\pi$
$30$		0			0
$31$	−102048.	−	176752.i	−0.615229	−	1.06561i	−0.990344	−	0.138630i	$-0.955730\pi$
$31$	−102048.	−	176752.i	−0.615229	−	1.06561i	0.375115	−	0.926978i	$-0.377603\pi$
$32$		0			0
$33$	−26951.2	+	46680.9i	−0.130551	+	0.226121i
$34$		0			0
$35$	23424.9	+	195304.i	0.0923505	+	0.769967i
$36$		0			0
$37$	−245856.	+	425834.i	−0.797947	+	1.38208i	0.123004	+	0.992406i	$0.460747\pi$
$37$	−245856.	+	425834.i	−0.797947	+	1.38208i	−0.920951	+	0.389678i	$0.872586\pi$
$38$		0			0
$39$	−93133.8	−	161312.i	−0.251409	−	0.435453i
$40$		0			0
$41$	−623293.			−1.41237			−0.706185	−	0.708027i	$-0.749585\pi$
$41$	−623293.			−1.41237			−0.706185	+	0.708027i	$0.749585\pi$
$42$		0			0
$43$	−422919.			−0.811181			−0.405591	−	0.914055i	$-0.632934\pi$
$43$	−422919.			−0.811181			−0.405591	+	0.914055i	$0.632934\pi$
$44$		0			0
$45$	−189783.	−	328714.i	−0.310466	−	0.537743i
$46$		0			0
$47$	−602714.	+	1.04393e6i	−0.846777	+	1.46666i	0.0372922	+	0.999304i	$0.488127\pi$
$47$	−602714.	+	1.04393e6i	−0.846777	+	1.46666i	−0.884069	+	0.467356i	$0.845207\pi$
$48$		0			0
$49$	231433.	−	790356.i	0.281021	−	0.959702i
$50$		0			0
$51$	−116004.	+	200925.i	−0.122455	+	0.212099i
$52$		0			0
$53$	636483.	+	1.10242e6i	0.587247	+	1.01714i	0.994591	+	0.103867i	$0.0331217\pi$
$53$	636483.	+	1.10242e6i	0.587247	+	1.01714i	−0.407344	+	0.913275i	$0.633545\pi$
$54$		0			0
$55$	559630.			0.453557
$56$		0			0
$57$	−180523.			−0.129113
$58$		0			0
$59$	840430.	+	1.45567e6i	0.532745	+	0.922741i	0.999269	+	0.0382328i	$0.0121729\pi$
$59$	840430.	+	1.45567e6i	0.532745	+	0.922741i	−0.466524	+	0.884509i	$0.654494\pi$
$60$		0			0
$61$	−1.06128e6	+	1.83818e6i	−0.598651	+	1.03689i	0.394370	+	0.918952i	$0.370963\pi$
$61$	−1.06128e6	+	1.83818e6i	−0.598651	+	1.03689i	−0.993021	+	0.117942i	$0.962370\pi$
$62$		0			0
$63$	189247.	+	1.57783e6i	0.0953534	+	0.795004i
$64$		0			0
$65$	−966940.	+	1.67479e6i	−0.436720	+	0.756421i
$66$		0			0
$67$	−1.67466e6	−	2.90060e6i	−0.680245	−	1.17822i	−0.974906	−	0.222618i	$-0.928540\pi$
$67$	−1.67466e6	−	2.90060e6i	−0.680245	−	1.17822i	0.294660	−	0.955602i	$-0.404793\pi$
$68$		0			0
$69$	−1.37910e6			−0.505387
$70$		0			0
$71$	−2.49257e6			−0.826502			−0.413251	−	0.910617i	$-0.635607\pi$
$71$	−2.49257e6			−0.826502			−0.413251	+	0.910617i	$0.635607\pi$
$72$		0			0
$73$	1.63676e6	+	2.83496e6i	0.492443	+	0.852936i	0.999962	−	0.00870409i	$-0.00277063\pi$
$73$	1.63676e6	+	2.83496e6i	0.492443	+	0.852936i	−0.507519	+	0.861641i	$0.669437\pi$
$74$		0			0
$75$	−325086.	+	563065.i	−0.0889782	+	0.154115i
$76$		0			0
$77$	−2.15418e6	−	921530.i	−0.537731	−	0.230034i
$78$		0			0
$79$	−2.09920e6	+	3.63591e6i	−0.479025	+	0.829695i	−0.999711	−	0.0240529i	$-0.992343\pi$
$79$	−2.09920e6	+	3.63591e6i	−0.479025	+	0.829695i	0.520686	+	0.853748i	$0.325676\pi$
$80$		0			0
$81$	−1.05662e6	−	1.83011e6i	−0.220912	−	0.382631i
$82$		0			0
$83$	−3.35527e6			−0.644102			−0.322051	−	0.946722i	$-0.604372\pi$
$83$	−3.35527e6			−0.644102			−0.322051	+	0.946722i	$0.604372\pi$
$84$		0			0
$85$	2.40877e6			0.425431
$86$		0			0
$87$	1.34488e6	+	2.32939e6i	0.218960	+	0.379250i
$88$		0			0
$89$	−1.19979e6	+	2.07810e6i	−0.180402	+	0.312466i	−0.942018	−	0.335564i	$-0.891073\pi$
$89$	−1.19979e6	+	2.07810e6i	−0.180402	+	0.312466i	0.761615	+	0.648029i	$0.224407\pi$
$90$		0			0
$91$	6.47988e6	−	4.85453e6i	0.901409	−	0.675308i
$92$		0			0
$93$	2.13049e6	−	3.69011e6i	0.274656	−	0.475718i
$94$		0			0
$95$	937117.	+	1.62313e6i	0.112140	+	0.194232i
$96$		0			0
$97$	4.51506e6			0.502299			0.251150	−	0.967948i	$-0.419191\pi$
$97$	4.51506e6			0.502299			0.251150	+	0.967948i	$0.419191\pi$
$98$		0			0
$99$	4.52118e6			0.468305
$100$		0			0
$101$	−3.69997e6	−	6.40853e6i	−0.357333	−	0.618919i	0.630181	−	0.776448i	$-0.282981\pi$
$101$	−3.69997e6	−	6.40853e6i	−0.357333	−	0.618919i	−0.987514	+	0.157529i	$0.949647\pi$
$102$		0			0
$103$	9.22855e6	−	1.59843e7i	0.832153	−	1.44133i	−0.0641744	−	0.997939i	$-0.520441\pi$
$103$	9.22855e6	−	1.59843e7i	0.832153	−	1.44133i	0.896327	−	0.443393i	$-0.146225\pi$
$104$		0			0
$105$	−3.28663e6	+	2.46225e6i	−0.277070	+	0.207572i
$106$		0			0
$107$	−2.49697e6	+	4.32488e6i	−0.197047	+	0.341296i	−0.947570	−	0.319549i	$-0.896469\pi$
$107$	−2.49697e6	+	4.32488e6i	−0.197047	+	0.341296i	0.750523	+	0.660845i	$0.229802\pi$
$108$		0			0
$109$	−9.34351e6	−	1.61834e7i	−0.691062	−	1.19695i	−0.971490	−	0.237080i	$-0.923810\pi$
$109$	−9.34351e6	−	1.61834e7i	−0.691062	−	1.19695i	0.280428	−	0.959875i	$-0.409524\pi$
$110$		0			0
$111$	−1.02656e7			−0.712452
$112$		0			0
$113$	2.49534e7			1.62688			0.813438	−	0.581651i	$-0.197593\pi$
$113$	2.49534e7			1.62688			0.813438	+	0.581651i	$0.197593\pi$
$114$		0			0
$115$	7.15910e6	+	1.23999e7i	0.438951	+	0.760286i
$116$		0			0
$117$	−7.81178e6	+	1.35304e7i	−0.450920	+	0.781017i
$118$		0			0
$119$	−9.27208e6	−	3.96647e6i	−0.504386	−	0.215769i
$120$		0			0
$121$	6.41059e6	−	1.11035e7i	0.328965	−	0.569783i
$122$		0			0
$123$	−6.50636e6	−	1.12694e7i	−0.315261	−	0.546048i
$124$		0			0
$125$	2.36842e7			1.08461
$126$		0			0
$127$	1.72080e7			0.745449			0.372724	−	0.927942i	$-0.378424\pi$
$127$	1.72080e7			0.745449			0.372724	+	0.927942i	$0.378424\pi$
$128$		0			0
$129$	−4.41473e6	−	7.64653e6i	−0.181067	−	0.313618i
$130$		0			0
$131$	1.34633e7	−	2.33191e7i	0.523242	−	0.906281i	−0.476392	−	0.879233i	$-0.658056\pi$
$131$	1.34633e7	−	2.33191e7i	0.523242	−	0.906281i	0.999634	−	0.0270484i	$-0.00861082\pi$
$132$		0			0
$133$	−934467.	−	7.79106e6i	−0.0344416	−	0.287155i
$134$		0			0
$135$	8.91057e6	−	1.54336e7i	0.311700	−	0.539881i
$136$		0			0
$137$	1.36301e7	+	2.36080e7i	0.452873	+	0.784399i	0.998563	−	0.0535877i	$-0.0170657\pi$
$137$	1.36301e7	+	2.36080e7i	0.452873	+	0.784399i	−0.545690	+	0.837987i	$0.683732\pi$
$138$		0			0
$139$	1.36813e7			0.432092			0.216046	−	0.976383i	$-0.430684\pi$
$139$	1.36813e7			0.432092			0.216046	+	0.976383i	$0.430684\pi$
$140$		0			0
$141$	−2.51662e7			−0.756051
$142$		0			0
$143$	−1.15176e7	−	1.99491e7i	−0.329372	−	0.570489i
$144$		0			0
$145$	1.39629e7	−	2.41844e7i	0.380353	−	0.658790i
$146$		0			0
$147$	1.67058e7	−	4.06590e6i	0.433766	−	0.105571i
$148$		0			0
$149$	−1.21944e7	+	2.11214e7i	−0.302002	+	0.523083i	−0.976589	−	0.215112i	$-0.930988\pi$
$149$	−1.21944e7	+	2.11214e7i	−0.302002	+	0.523083i	0.674587	+	0.738195i	$0.264322\pi$
$150$		0			0
$151$	1.69013e7	+	2.92740e7i	0.399486	+	0.691930i	0.993663	−	0.112405i	$-0.0358553\pi$
$151$	1.69013e7	+	2.92740e7i	0.399486	+	0.691930i	−0.594177	+	0.804335i	$0.702522\pi$
$152$		0			0
$153$	1.94601e7			0.439264
$154$		0			0
$155$	−4.42386e7			−0.954202
$156$		0			0
$157$	−3.97833e6	−	6.89067e6i	−0.0820450	−	0.142106i	0.822083	−	0.569367i	$-0.192812\pi$
$157$	−3.97833e6	−	6.89067e6i	−0.0820450	−	0.142106i	−0.904128	+	0.427261i	$0.859478\pi$
$158$		0			0
$159$	−1.32881e7	+	2.30157e7i	−0.262164	+	0.454081i
$160$		0			0
$161$	−7.13886e6	−	5.95198e7i	−0.134815	−	1.12401i
$162$		0			0
$163$	−2.68214e7	+	4.64560e7i	−0.485093	+	0.840205i	−0.999853	−	0.0171289i	$-0.994547\pi$
$163$	−2.68214e7	+	4.64560e7i	−0.485093	+	0.840205i	0.514761	+	0.857334i	$0.327881\pi$
$164$		0			0
$165$	5.84181e6	+	1.01183e7i	0.101240	+	0.175353i
$166$		0			0
$167$	8.20253e7			1.36283			0.681413	−	0.731899i	$-0.261366\pi$
$167$	8.20253e7			1.36283			0.681413	+	0.731899i	$0.261366\pi$
$168$		0			0
$169$	1.68530e7			0.268581
$170$		0			0
$171$	7.57085e6	+	1.31131e7i	0.115787	+	0.200548i
$172$		0			0
$173$	3.39592e7	−	5.88190e7i	0.498650	−	0.863687i	−0.501349	−	0.865245i	$-0.667163\pi$
$173$	3.39592e7	−	5.88190e7i	0.498650	−	0.863687i	0.999999	+	0.00155811i	$0.000495962\pi$
$174$		0			0
$175$	−2.59838e7	−	1.11155e7i	−0.366496	−	0.156782i
$176$		0			0
$177$	−1.75460e7	+	3.03905e7i	−0.237833	+	0.411938i
$178$		0			0
$179$	−3.04898e7	−	5.28098e7i	−0.397346	−	0.688223i	0.596052	−	0.802946i	$-0.296735\pi$
$179$	−3.04898e7	−	5.28098e7i	−0.397346	−	0.688223i	−0.993398	+	0.114723i	$0.963402\pi$
$180$		0			0
$181$	−1.53894e8			−1.92907			−0.964533	−	0.263963i	$-0.914970\pi$
$181$	−1.53894e8			−1.92907			−0.964533	+	0.263963i	$0.914970\pi$
$182$		0			0
$183$	−4.43133e7			−0.534510
$184$		0			0
$185$	5.32904e7	+	9.23016e7i	0.618796	+	1.07179i
$186$		0			0
$187$	−1.43459e7	+	2.48479e7i	−0.160429	+	0.277871i
$188$		0			0
$189$	−5.97136e7	+	4.47356e7i	−0.643364	+	0.481989i
$190$		0			0
$191$	1.12603e7	−	1.95034e7i	0.116932	−	0.202532i	−0.801618	−	0.597836i	$-0.796027\pi$
$191$	1.12603e7	−	1.95034e7i	0.116932	−	0.202532i	0.918550	+	0.395304i	$0.129361\pi$
$192$		0			0
$193$	6.76381e7	+	1.17153e8i	0.677237	+	1.17301i	0.975810	+	0.218622i	$0.0701563\pi$
$193$	6.76381e7	+	1.17153e8i	0.677237	+	1.17301i	−0.298572	+	0.954387i	$0.596510\pi$
$194$		0			0
$195$	−4.03744e7			−0.389928
$196$		0			0
$197$	−8.96233e7			−0.835197			−0.417598	−	0.908632i	$-0.637128\pi$
$197$	−8.96233e7			−0.835197			−0.417598	+	0.908632i	$0.637128\pi$
$198$		0			0
$199$	−2.74707e7	−	4.75806e7i	−0.247106	−	0.428000i	0.715616	−	0.698494i	$-0.246146\pi$
$199$	−2.74707e7	−	4.75806e7i	−0.247106	−	0.428000i	−0.962722	+	0.270494i	$0.912813\pi$
$200$		0			0
$201$	3.49626e7	−	6.05570e7i	0.303681	−	0.525991i
$202$		0			0
$203$	−9.35712e7	+	7.01007e7i	−0.785066	+	0.588147i
$204$		0			0
$205$	−6.75508e7	+	1.17001e8i	−0.547636	+	0.948534i
$206$		0			0
$207$	5.78375e7	+	1.00177e8i	0.453224	+	0.785007i
$208$		0			0
$209$	−2.23248e7			−0.169151
$210$		0			0
$211$	1.24539e8			0.912677			0.456338	−	0.889806i	$-0.349161\pi$
$211$	1.24539e8			0.912677			0.456338	+	0.889806i	$0.349161\pi$
$212$		0			0
$213$	−2.60192e7	−	4.50666e7i	−0.184487	−	0.319541i
$214$		0			0
$215$	−4.58349e7	+	7.93884e7i	−0.314530	+	0.544781i
$216$		0			0
$217$	1.70288e8	+	7.28467e7i	1.13129	+	0.483950i
$218$		0			0
$219$	−3.41714e7	+	5.91866e7i	−0.219841	+	0.380775i
$220$		0			0
$221$	−4.95744e7	−	8.58653e7i	−0.308947	−	0.535112i
$222$		0			0
$223$	−2.85464e8			−1.72379			−0.861895	−	0.507087i	$-0.830722\pi$
$223$	−2.85464e8			−1.72379			−0.861895	+	0.507087i	$0.830722\pi$
$224$		0			0
$225$	5.45345e7			0.319177
$226$		0			0
$227$	3.37141e7	+	5.83946e7i	0.191303	+	0.331346i	0.945682	−	0.325092i	$-0.105395\pi$
$227$	3.37141e7	+	5.83946e7i	0.191303	+	0.331346i	−0.754379	+	0.656439i	$0.772062\pi$
$228$		0			0
$229$	−3.58459e7	+	6.20869e7i	−0.197249	+	0.341646i	−0.947636	−	0.319354i	$-0.896534\pi$
$229$	−3.58459e7	+	6.20869e7i	−0.197249	+	0.341646i	0.750386	+	0.660999i	$0.229867\pi$
$230$		0			0
$231$	−5.82529e6	−	4.85680e7i	−0.0310939	−	0.259244i
$232$		0			0
$233$	−1.08990e6	+	1.88777e6i	−0.00564472	+	0.00977693i	−0.868834	−	0.495104i	$-0.835130\pi$
$233$	−1.08990e6	+	1.88777e6i	−0.00564472	+	0.00977693i	0.863189	+	0.504881i	$0.168463\pi$
$234$		0			0
$235$	1.30641e8	+	2.26277e8i	0.656663	+	1.13737i
$236$		0			0
$237$	−8.76515e7			−0.427701
$238$		0			0
$239$	1.87131e8			0.886654			0.443327	−	0.896360i	$-0.353798\pi$
$239$	1.87131e8			0.886654			0.443327	+	0.896360i	$0.353798\pi$
$240$		0			0
$241$	6.26650e7	+	1.08539e8i	0.288380	+	0.499489i	0.973423	−	0.229014i	$-0.0735500\pi$
$241$	6.26650e7	+	1.08539e8i	0.288380	+	0.499489i	−0.685043	+	0.728502i	$0.740217\pi$
$242$		0			0
$243$	1.11965e8	−	1.93929e8i	0.500564	−	0.867002i
$244$		0			0
$245$	−1.23280e8	−	1.29100e8i	−0.535562	−	0.560848i
$246$		0			0
$247$	3.85732e7	−	6.68108e7i	0.162872	−	0.282103i
$248$		0			0
$249$	−3.50247e7	−	6.06645e7i	−0.143773	−	0.249022i
$250$		0			0
$251$	−3.48978e7			−0.139296			−0.0696482	−	0.997572i	$-0.522188\pi$
$251$	−3.48978e7			−0.139296			−0.0696482	+	0.997572i	$0.522188\pi$
$252$		0			0
$253$	−1.70550e8			−0.662110
$254$		0			0
$255$	2.51444e7	+	4.35514e7i	0.0949622	+	0.164479i
$256$		0			0
$257$	2.96784e7	−	5.14046e7i	0.109062	−	0.188902i	−0.806328	−	0.591468i	$-0.798548\pi$
$257$	2.96784e7	−	5.14046e7i	0.109062	−	0.188902i	0.915391	+	0.402567i	$0.131882\pi$
$258$		0			0
$259$	−5.31396e7	−	4.43049e8i	−0.190051	−	1.58454i
$260$		0			0
$261$	1.12804e8	−	1.95383e8i	0.392720	−	0.680212i
$262$		0			0
$263$	−1.06373e7	−	1.84243e7i	−0.0360567	−	0.0624521i	0.847434	−	0.530901i	$-0.178146\pi$
$263$	−1.06373e7	−	1.84243e7i	−0.0360567	−	0.0624521i	−0.883491	+	0.468449i	$0.844813\pi$
$264$		0			0
$265$	2.75921e8			0.910804
$266$		0			0
$267$	−5.00972e7			−0.161073
$268$		0			0
$269$	−5.85529e7	−	1.01417e8i	−0.183407	−	0.317670i	0.759632	−	0.650353i	$-0.225379\pi$
$269$	−5.85529e7	−	1.01417e8i	−0.183407	−	0.317670i	−0.943038	+	0.332684i	$0.892046\pi$
$270$		0			0
$271$	−1.01385e8	+	1.75603e8i	−0.309442	+	0.535970i	−0.978240	−	0.207474i	$-0.933476\pi$
$271$	−1.01385e8	+	1.75603e8i	−0.309442	+	0.535970i	0.668798	+	0.743444i	$0.266809\pi$
$272$		0			0
$273$	1.55413e8	+	6.64836e7i	0.462294	+	0.197763i
$274$		0			0
$275$	−4.02026e7	+	6.96329e7i	−0.116571	+	0.201906i
$276$		0			0
$277$	−5.05757e7	−	8.75996e7i	−0.142976	−	0.247641i	0.785640	−	0.618684i	$-0.212334\pi$
$277$	−5.05757e7	−	8.75996e7i	−0.142976	−	0.247641i	−0.928616	+	0.371042i	$0.879000\pi$
$278$		0			0
$279$	−3.57398e8			−0.985230
$280$		0			0
$281$	3.69631e8			0.993793			0.496897	−	0.867810i	$-0.334473\pi$
$281$	3.69631e8			0.993793			0.496897	+	0.867810i	$0.334473\pi$
$282$		0			0
$283$	−1.09831e8	−	1.90232e8i	−0.288052	−	0.498921i	0.685293	−	0.728268i	$-0.259674\pi$
$283$	−1.09831e8	−	1.90232e8i	−0.288052	−	0.498921i	−0.973345	+	0.229347i	$0.926341\pi$
$284$		0			0
$285$	−1.95646e7	+	3.38868e7i	−0.0500626	+	0.0867109i
$286$		0			0
$287$	4.52687e8	−	3.39140e8i	1.13035	−	0.846821i
$288$		0			0
$289$	1.43421e8	−	2.48413e8i	0.349519	−	0.605385i
$290$		0			0
$291$	4.71314e7	+	8.16339e7i	0.112120	+	0.194198i
$292$		0			0
$293$	−7.22230e8			−1.67741			−0.838704	−	0.544587i	$-0.816686\pi$
$293$	−7.22230e8			−1.67741			−0.838704	+	0.544587i	$0.816686\pi$
$294$		0			0
$295$	3.64334e8			0.826272
$296$		0			0
$297$	1.06138e8	+	1.83836e8i	0.235083	+	0.407176i
$298$		0			0
$299$	2.94680e8	−	5.10401e8i	0.637531	−	1.10424i
$300$		0			0
$301$	3.07159e8	−	2.30114e8i	0.649204	−	0.486364i
$302$		0			0
$303$	7.72457e7	−	1.33794e8i	0.159524	−	0.276303i
$304$		0			0
$305$	2.30036e8	+	3.98435e8i	0.464245	+	0.804096i
$306$		0			0
$307$	6.39545e7			0.126150			0.0630750	−	0.998009i	$-0.479909\pi$
$307$	6.39545e7			0.126150			0.0630750	+	0.998009i	$0.479909\pi$
$308$		0			0
$309$	3.85336e8			0.742994
$310$		0			0
$311$	7.73216e7	+	1.33925e8i	0.145760	+	0.252464i	0.929656	−	0.368428i	$-0.120104\pi$
$311$	7.73216e7	+	1.33925e8i	0.145760	+	0.252464i	−0.783896	+	0.620892i	$0.786771\pi$
$312$		0			0
$313$	2.46992e8	−	4.27803e8i	0.455279	−	0.788567i	−0.543425	−	0.839458i	$-0.682873\pi$
$313$	2.46992e8	−	4.27803e8i	0.455279	−	0.788567i	0.998704	+	0.0508911i	$0.0162062\pi$
$314$		0			0
$315$	3.16693e8	+	1.35477e8i	0.570889	+	0.244218i
$316$		0			0
$317$	−2.14969e8	+	3.72338e8i	−0.379026	+	0.656492i	−0.990921	−	0.134447i	$-0.957074\pi$
$317$	−2.14969e8	+	3.72338e8i	−0.379026	+	0.656492i	0.611895	+	0.790939i	$0.290408\pi$
$318$		0			0
$319$	1.66318e8	+	2.88071e8i	0.286861	+	0.496857i
$320$		0			0
$321$	−1.04260e8			−0.175935
$322$		0			0
$323$	−9.60907e7			−0.158662
$324$		0			0
$325$	−1.38926e8	−	2.40626e8i	−0.224487	−	0.388822i
$326$		0			0
$327$	1.95068e8	−	3.37868e8i	0.308510	−	0.534355i
$328$		0			0
$329$	−1.30272e8	−	1.08613e9i	−0.201681	−	1.68150i
$330$		0			0
$331$	−1.64085e8	+	2.84203e8i	−0.248697	+	0.430755i	−0.963164	−	0.268913i	$-0.913336\pi$
$331$	−1.64085e8	+	2.84203e8i	−0.248697	+	0.430755i	0.714468	+	0.699668i	$0.246669\pi$
$332$		0			0
$333$	4.30526e8	+	7.45693e8i	0.638917	+	1.10664i
$334$		0			0
$335$	−7.25983e8			−1.05504
$336$		0			0
$337$	−1.14409e8			−0.162838			−0.0814189	−	0.996680i	$-0.525945\pi$
$337$	−1.14409e8			−0.162838			−0.0814189	+	0.996680i	$0.525945\pi$
$338$		0			0
$339$	2.60481e8	+	4.51166e8i	0.363142	+	0.628981i
$340$		0			0
$341$	2.63472e8	−	4.56348e8i	0.359828	−	0.623240i
$342$		0			0
$343$	2.61954e8	+	6.99947e8i	0.350507	+	0.936560i
$344$		0			0
$345$	−1.49463e8	+	2.58878e8i	−0.195960	+	0.339413i
$346$		0			0
$347$	−3.28339e8	−	5.68699e8i	−0.421860	−	0.730684i	0.574261	−	0.818672i	$-0.305289\pi$
$347$	−3.28339e8	−	5.68699e8i	−0.421860	−	0.730684i	−0.996121	+	0.0879887i	$0.971956\pi$
$348$		0			0
$349$	−8.85994e8			−1.11569			−0.557843	−	0.829947i	$-0.688371\pi$
$349$	−8.85994e8			−1.11569			−0.557843	+	0.829947i	$0.688371\pi$
$350$		0			0
$351$	−7.33546e8			−0.905426
$352$		0			0
$353$	−3.43700e8	−	5.95305e8i	−0.415880	−	0.720324i	0.579641	−	0.814872i	$-0.303193\pi$
$353$	−3.43700e8	−	5.95305e8i	−0.415880	−	0.720324i	−0.995520	+	0.0945476i	$0.969860\pi$
$354$		0			0
$355$	−2.70139e8	+	4.67894e8i	−0.320470	+	0.555071i
$356$		0			0
$357$	−2.50733e7	−	2.09047e8i	−0.0291657	−	0.243168i
$358$		0			0
$359$	1.61222e7	−	2.79244e7i	0.0183905	−	0.0318533i	−0.856684	−	0.515842i	$-0.827479\pi$
$359$	1.61222e7	−	2.79244e7i	0.0183905	−	0.0318533i	0.875074	+	0.483989i	$0.160812\pi$
$360$		0			0
$361$	4.09552e8	+	7.09365e8i	0.458178	+	0.793588i
$362$		0			0
$363$	2.67673e8			0.293718
$364$		0			0
$365$	7.09553e8			0.763765
$366$		0			0
$367$	−3.58275e8	−	6.20551e8i	−0.378343	−	0.655309i	0.612479	−	0.790487i	$-0.290173\pi$
$367$	−3.58275e8	−	6.20551e8i	−0.378343	−	0.655309i	−0.990821	+	0.135178i	$0.956839\pi$
$368$		0			0
$369$	−5.45735e8	+	9.45240e8i	−0.565444	+	0.979377i
$370$		0			0
$371$	−1.06210e9	−	4.54353e8i	−1.07984	−	0.461940i
$372$		0			0
$373$	−6.11134e8	+	1.05852e9i	−0.609755	+	1.05613i	0.381525	+	0.924358i	$0.375399\pi$
$373$	−6.11134e8	+	1.05852e9i	−0.609755	+	1.05613i	−0.991281	+	0.131769i	$0.957934\pi$
$374$		0			0
$375$	2.47232e8	+	4.28219e8i	0.242101	+	0.419331i
$376$		0			0
$377$	−1.14947e9			−1.10485
$378$		0			0
$379$	6.83298e8			0.644723			0.322361	−	0.946617i	$-0.395523\pi$
$379$	6.83298e8			0.644723			0.322361	+	0.946617i	$0.395523\pi$
$380$		0			0
$381$	1.79629e8	+	3.11127e8i	0.166395	+	0.288204i
$382$		0			0
$383$	1.01294e9	−	1.75447e9i	0.921273	−	1.59569i	0.123826	−	0.992304i	$-0.460484\pi$
$383$	1.01294e9	−	1.75447e9i	0.921273	−	1.59569i	0.797447	−	0.603388i	$-0.206183\pi$
$384$		0			0
$385$	−4.06450e8	+	3.04500e8i	−0.362990	+	0.271941i
$386$		0			0
$387$	−3.70294e8	+	6.41368e8i	−0.324757	+	0.562496i
$388$		0			0
$389$	3.64506e8	+	6.31342e8i	0.313965	+	0.543803i	0.979217	−	0.202816i	$-0.0650093\pi$
$389$	3.64506e8	+	6.31342e8i	0.313965	+	0.543803i	−0.665252	+	0.746619i	$0.731676\pi$
$390$		0			0
$391$	−7.34085e8			−0.621052
$392$		0			0
$393$	5.62158e8			0.467180
$394$		0			0
$395$	4.55011e8	+	7.88102e8i	0.371477	+	0.643417i
$396$		0			0
$397$	6.63836e8	−	1.14980e9i	0.532469	−	0.922263i	−0.466812	−	0.884356i	$-0.654598\pi$
$397$	6.63836e8	−	1.14980e9i	0.532469	−	0.922263i	0.999281	−	0.0379069i	$-0.0120690\pi$
$398$		0			0
$399$	1.31111e8	−	9.82240e7i	0.103331	−	0.0774128i
$400$		0			0
$401$	4.28975e8	−	7.43007e8i	0.332221	−	0.575423i	−0.650726	−	0.759312i	$-0.725535\pi$
$401$	4.28975e8	−	7.43007e8i	0.332221	−	0.575423i	0.982947	+	0.183889i	$0.0588688\pi$
$402$		0			0
$403$	9.10465e8	+	1.57697e9i	0.692940	+	1.20021i
$404$		0			0
$405$	−4.58053e8			−0.342629
$406$		0			0
$407$	−1.26953e9			−0.933387
$408$		0			0
$409$	1.05133e9	+	1.82096e9i	0.759814	+	1.31604i	0.942945	+	0.332948i	$0.108044\pi$
$409$	1.05133e9	+	1.82096e9i	0.759814	+	1.31604i	−0.183131	+	0.983089i	$0.558623\pi$
$410$		0			0
$411$	−2.84561e8	+	4.92874e8i	−0.202176	+	0.350178i
$412$		0			0
$413$	−1.40243e9	−	5.99941e8i	−0.979618	−	0.419067i
$414$		0			0
$415$	−3.63636e8	+	6.29835e8i	−0.249746	+	0.432572i
$416$		0			0
$417$	1.42815e8	+	2.47363e8i	0.0964490	+	0.167055i
$418$		0			0
$419$	−2.47991e9			−1.64697			−0.823486	−	0.567336i	$-0.807974\pi$
$419$	−2.47991e9			−1.64697			−0.823486	+	0.567336i	$0.807974\pi$
$420$		0			0
$421$	−5.51855e7			−0.0360444			−0.0180222	−	0.999838i	$-0.505737\pi$
$421$	−5.51855e7			−0.0360444			−0.0180222	+	0.999838i	$0.505737\pi$
$422$		0			0
$423$	1.05543e9	+	1.82806e9i	0.678016	+	1.17436i
$424$		0			0
$425$	−1.73041e8	+	2.99715e8i	−0.109342	+	0.189386i
$426$		0			0
$427$	−2.29386e8	−	1.91249e9i	−0.142583	−	1.18878i
$428$		0			0
$429$	2.40458e8	−	4.16486e8i	0.147041	−	0.254683i
$430$		0			0
$431$	8.42361e8	+	1.45901e9i	0.506790	+	0.877786i	0.999969	+	0.00785824i	$0.00250138\pi$
$431$	8.42361e8	+	1.45901e9i	0.506790	+	0.877786i	−0.493179	+	0.869928i	$0.664165\pi$
$432$		0			0
$433$	−1.29692e9			−0.767724			−0.383862	−	0.923390i	$-0.625406\pi$
$433$	−1.29692e9			−0.767724			−0.383862	+	0.923390i	$0.625406\pi$
$434$		0			0
$435$	5.83017e8			0.339601
$436$		0			0
$437$	−2.85591e8	−	4.94659e8i	−0.163704	−	0.283544i
$438$		0			0
$439$	−7.03191e7	+	1.21796e8i	−0.0396686	+	0.0687081i	−0.885178	−	0.465252i	$-0.845964\pi$
$439$	−7.03191e7	+	1.21796e8i	−0.0396686	+	0.0687081i	0.845509	+	0.533960i	$0.179297\pi$
$440$		0			0
$441$	−9.95961e8	−	1.04298e9i	−0.552977	−	0.579085i
$442$		0			0
$443$	−9.24666e8	+	1.60157e9i	−0.505326	+	0.875251i	0.494655	+	0.869090i	$0.335295\pi$
$443$	−9.24666e8	+	1.60157e9i	−0.505326	+	0.875251i	−0.999981	+	0.00616141i	$0.998039\pi$
$444$		0			0
$445$	2.60061e8	+	4.50439e8i	0.139899	+	0.242313i
$446$		0			0
$447$	−5.09176e8			−0.269645
$448$		0			0
$449$	−2.91794e9			−1.52130			−0.760648	−	0.649165i	$-0.775118\pi$
$449$	−2.91794e9			−1.52130			−0.760648	+	0.649165i	$0.775118\pi$
$450$		0			0
$451$	−8.04627e8	−	1.39365e9i	−0.413025	−	0.715381i
$452$		0			0
$453$	−3.52856e8	+	6.11164e8i	−0.178342	+	0.308897i
$454$		0			0
$455$	−2.08996e8	−	1.74249e9i	−0.104016	−	0.867224i
$456$		0			0
$457$	−1.42040e9	+	2.46021e9i	−0.696153	+	1.20577i	0.273638	+	0.961833i	$0.411773\pi$
$457$	−1.42040e9	+	2.46021e9i	−0.696153	+	1.20577i	−0.969791	+	0.243939i	$0.921560\pi$
$458$		0			0
$459$	4.56839e8	+	7.91269e8i	0.220505	+	0.381926i
$460$		0			0
$461$	3.96057e9			1.88280			0.941400	−	0.337291i	$-0.109511\pi$
$461$	3.96057e9			1.88280			0.941400	+	0.337291i	$0.109511\pi$
$462$		0			0
$463$	2.19815e9			1.02926			0.514629	−	0.857413i	$-0.327930\pi$
$463$	2.19815e9			1.02926			0.514629	+	0.857413i	$0.327930\pi$
$464$		0			0
$465$	−4.61793e8	−	7.99850e8i	−0.212992	−	0.368912i
$466$		0			0
$467$	1.56120e9	−	2.70408e9i	0.709332	−	1.22860i	−0.255773	−	0.966737i	$-0.582330\pi$
$467$	1.56120e9	−	2.70408e9i	0.709332	−	1.22860i	0.965105	−	0.261862i	$-0.0843366\pi$
$468$		0			0
$469$	2.79453e9	+	1.19546e9i	1.25084	+	0.535094i
$470$		0			0
$471$	8.30572e7	−	1.43859e8i	0.0366272	−	0.0634403i
$472$		0			0
$473$	−5.45959e8	−	9.45629e8i	−0.237217	−	0.410872i
$474$		0			0
$475$	−2.69282e8			−0.115287
$476$		0			0
$477$	2.22913e9			0.940420
$478$		0			0
$479$	−6.79522e8	−	1.17697e9i	−0.282507	−	0.489317i	0.689494	−	0.724291i	$-0.257833\pi$
$479$	−6.79522e8	−	1.17697e9i	−0.282507	−	0.489317i	−0.972002	+	0.234974i	$0.924499\pi$
$480$		0			0
$481$	2.19352e9	−	3.79928e9i	0.898738	−	1.55666i
$482$		0			0
$483$	1.00162e9	−	7.50382e8i	0.404471	−	0.303017i
$484$		0			0
$485$	4.89330e8	−	8.47545e8i	0.194763	−	0.337339i
$486$		0			0
$487$	1.19733e9	+	2.07383e9i	0.469745	+	0.813622i	0.999402	−	0.0345902i	$-0.0110126\pi$
$487$	1.19733e9	+	2.07383e9i	0.469745	+	0.813622i	−0.529657	+	0.848212i	$0.677679\pi$
$488$		0			0
$489$	−1.11992e9			−0.433118
$490$		0			0
$491$	1.95738e9			0.746261			0.373130	−	0.927779i	$-0.378284\pi$
$491$	1.95738e9			0.746261			0.373130	+	0.927779i	$0.378284\pi$
$492$		0			0
$493$	7.15867e8	+	1.23992e9i	0.269072	+	0.466046i
$494$		0			0
$495$	4.89994e8	−	8.48694e8i	0.181582	−	0.314509i
$496$		0			0
$497$	1.81032e9	−	1.35623e9i	0.661465	−	0.495550i
$498$		0			0
$499$	1.47577e8	−	2.55611e8i	0.0531699	−	0.0920931i	−0.838215	−	0.545339i	$-0.816401\pi$
$499$	1.47577e8	−	2.55611e8i	0.0531699	−	0.0920931i	0.891385	+	0.453246i	$0.149734\pi$
$500$		0			0
$501$	8.56238e8	+	1.48305e9i	0.304202	+	0.526893i
$502$		0			0
$503$	−2.51431e9			−0.880909			−0.440454	−	0.897775i	$-0.645183\pi$
$503$	−2.51431e9			−0.880909			−0.440454	+	0.897775i	$0.645183\pi$
$504$		0			0
$505$	−1.60397e9			−0.554213
$506$		0			0
$507$	1.75924e8	+	3.04709e8i	0.0599510	+	0.103838i
$508$		0			0
$509$	1.11783e9	−	1.93615e9i	0.375721	−	0.650767i	−0.614714	−	0.788750i	$-0.710729\pi$
$509$	1.11783e9	−	1.93615e9i	0.375721	−	0.650767i	0.990435	+	0.137983i	$0.0440619\pi$
$510$		0			0
$511$	−2.73128e9	−	1.16840e9i	−0.905511	−	0.387365i
$512$		0			0
$513$	−3.55461e8	+	6.15676e8i	−0.116247	+	0.201345i
$514$		0			0
$515$	−2.00033e9	−	3.46468e9i	−0.645323	−	1.11773i
$516$		0			0
$517$	−3.11225e9			−0.990506
$518$		0			0
$519$	1.41796e9			0.445223
$520$		0			0
$521$	−1.59740e9	−	2.76677e9i	−0.494858	−	0.857120i	0.505124	−	0.863047i	$-0.331447\pi$
$521$	−1.59740e9	−	2.76677e9i	−0.494858	−	0.857120i	−0.999982	+	0.00592705i	$0.998113\pi$
$522$		0			0
$523$	−4.70741e8	+	8.15348e8i	−0.143889	+	0.249222i	−0.928958	−	0.370186i	$-0.879294\pi$
$523$	−4.70741e8	+	8.15348e8i	−0.143889	+	0.249222i	0.785069	+	0.619408i	$0.212627\pi$
$524$		0			0
$525$	−7.02646e7	−	5.85827e8i	−0.0211923	−	0.176690i
$526$		0			0
$527$	1.13404e9	−	1.96422e9i	0.337514	−	0.584592i
$528$		0			0
$529$	−4.79361e8	−	8.30278e8i	−0.140789	−	0.243853i
$530$		0			0
$531$	2.94341e9			0.853140
$532$		0			0
$533$	5.56100e9			1.59077
$534$		0			0
$535$	5.41230e8	+	9.37438e8i	0.152807	+	0.264670i
$536$		0			0
$537$	6.36547e8	−	1.10253e9i	0.177387	−	0.307242i
$538$		0			0
$539$	2.06596e9	−	5.02820e8i	0.568279	−	0.138309i
$540$		0			0
$541$	−1.93667e9	+	3.35441e9i	−0.525854	+	0.910806i	0.473692	+	0.880691i	$0.342921\pi$
$541$	−1.93667e9	+	3.35441e9i	−0.525854	+	0.910806i	−0.999546	+	0.0301159i	$0.990412\pi$
$542$		0			0
$543$	−1.60645e9	−	2.78246e9i	−0.430595	−	0.745813i
$544$		0			0
$545$	−4.05050e9			−1.07182
$546$		0			0
$547$	1.92865e9			0.503845			0.251922	−	0.967747i	$-0.418937\pi$
$547$	1.92865e9			0.503845			0.251922	+	0.967747i	$0.418937\pi$
$548$		0			0
$549$	1.85844e9	+	3.21890e9i	0.479341	+	0.830242i
$550$		0			0
$551$	−5.57007e8	+	9.64765e8i	−0.141850	+	0.245692i
$552$		0			0
$553$	−4.53724e8	−	3.78290e9i	−0.114092	−	0.951232i
$554$		0			0
$555$	−1.11256e9	+	1.92702e9i	−0.276248	+	0.478476i
$556$		0			0
$557$	−1.65798e9	−	2.87170e9i	−0.406524	−	0.704120i	0.587974	−	0.808880i	$-0.299926\pi$
$557$	−1.65798e9	−	2.87170e9i	−0.406524	−	0.704120i	−0.994498	+	0.104760i	$0.966593\pi$
$558$		0			0
$559$	3.77327e9			0.913644
$560$		0			0
$561$	−5.99012e8			−0.143240
$562$		0			0
$563$	−9.36100e8	−	1.62137e9i	−0.221077	−	0.382916i	0.734059	−	0.679086i	$-0.237624\pi$
$563$	−9.36100e8	−	1.62137e9i	−0.221077	−	0.382916i	−0.955135	+	0.296170i	$0.904290\pi$
$564$		0			0
$565$	2.70438e9	−	4.68413e9i	0.630810	−	1.09259i
$566$		0			0
$567$	1.76319e9	+	7.54266e8i	0.406216	+	0.173774i
$568$		0			0
$569$	3.17886e8	−	5.50595e8i	0.0723400	−	0.125297i	−0.827586	−	0.561338i	$-0.810287\pi$
$569$	3.17886e8	−	5.50595e8i	0.0723400	−	0.125297i	0.899926	+	0.436042i	$0.143620\pi$
$570$		0			0
$571$	2.85689e9	+	4.94828e9i	0.642196	+	1.11232i	0.984942	+	0.172887i	$0.0553097\pi$
$571$	2.85689e9	+	4.94828e9i	0.642196	+	1.11232i	−0.342746	+	0.939428i	$0.611357\pi$
$572$		0			0
$573$	4.70171e8			0.104403
$574$		0			0
$575$	−2.05718e9			−0.451267
$576$		0			0
$577$	3.17926e9	+	5.50664e9i	0.688986	+	1.19336i	0.972166	+	0.234292i	$0.0752773\pi$
$577$	3.17926e9	+	5.50664e9i	0.688986	+	1.19336i	−0.283180	+	0.959067i	$0.591389\pi$
$578$		0			0
$579$	−1.41211e9	+	2.44584e9i	−0.302338	+	0.523665i
$580$		0			0
$581$	2.43688e9	−	1.82564e9i	0.515487	−	0.386187i
$582$		0			0
$583$	−1.64331e9	+	2.84629e9i	−0.343462	+	0.594894i
$584$		0			0
$585$	1.69324e9	+	2.93278e9i	0.349682	+	0.605667i
$586$		0			0
$587$	−7.71077e8			−0.157349			−0.0786745	−	0.996900i	$-0.525069\pi$
$587$	−7.71077e8			−0.157349			−0.0786745	+	0.996900i	$0.525069\pi$
$588$		0			0
$589$	1.76477e9			0.355864
$590$		0			0
$591$	−9.35550e8	−	1.62042e9i	−0.186428	−	0.322903i
$592$		0			0
$593$	−3.20093e9	+	5.54417e9i	−0.630354	+	1.09181i	0.357125	+	0.934057i	$0.383757\pi$
$593$	−3.20093e9	+	5.54417e9i	−0.630354	+	1.09181i	−0.987479	+	0.157749i	$0.949576\pi$
$594$		0			0
$595$	−1.74945e9	+	1.31063e9i	−0.340480	+	0.255078i
$596$		0			0
$597$	5.73516e8	−	9.93359e8i	0.110315	−	0.191072i
$598$		0			0
$599$	−6.60575e8	−	1.14415e9i	−0.125582	−	0.217515i	0.796378	−	0.604799i	$-0.206747\pi$
$599$	−6.60575e8	−	1.14415e9i	−0.125582	−	0.217515i	−0.921960	+	0.387284i	$0.873413\pi$
$600$		0			0
$601$	−2.93204e9			−0.550947			−0.275474	−	0.961309i	$-0.588835\pi$
$601$	−2.93204e9			−0.550947			−0.275474	+	0.961309i	$0.588835\pi$
$602$		0			0
$603$	−5.86512e9			−1.08935
$604$		0			0
$605$	−1.38953e9	−	2.40673e9i	−0.255107	−	0.441859i
$606$		0			0
$607$	−2.76216e9	+	4.78420e9i	−0.501289	+	0.868258i	0.498710	+	0.866769i	$0.333807\pi$
$607$	−2.76216e9	+	4.78420e9i	−0.501289	+	0.868258i	−0.999999	+	0.00148902i	$0.999526\pi$
$608$		0			0
$609$	−2.24421e9	−	9.60040e8i	−0.402626	−	0.172238i
$610$		0			0
$611$	5.37740e9	−	9.31393e9i	0.953736	−	1.65192i
$612$		0			0
$613$	3.14956e9	+	5.45520e9i	0.552253	+	0.956530i	0.998112	+	0.0614270i	$0.0195651\pi$
$613$	3.14956e9	+	5.45520e9i	0.552253	+	0.956530i	−0.445858	+	0.895103i	$0.647102\pi$
$614$		0			0
$615$	−2.82057e9			−0.488961
$616$		0			0
$617$	−4.76835e9			−0.817278			−0.408639	−	0.912696i	$-0.633996\pi$
$617$	−4.76835e9			−0.817278			−0.408639	+	0.912696i	$0.633996\pi$
$618$		0			0
$619$	3.93891e8	+	6.82239e8i	0.0667511	+	0.115616i	0.897469	−	0.441077i	$-0.145403\pi$
$619$	3.93891e8	+	6.82239e8i	0.0667511	+	0.115616i	−0.830718	+	0.556693i	$0.812070\pi$
$620$		0			0
$621$	−2.71554e9	+	4.70346e9i	−0.455026	+	0.788128i
$622$		0			0
$623$	−2.59326e8	−	2.16211e9i	−0.0429672	−	0.358237i
$624$		0			0
$625$	1.35034e9	−	2.33885e9i	0.221239	−	0.383197i
$626$		0			0
$627$	−2.33042e8	−	4.03640e8i	−0.0377570	−	0.0653970i
$628$		0			0
$629$	−5.46432e9			−0.875506
$630$		0			0
$631$	9.52407e9			1.50911			0.754553	−	0.656239i	$-0.227854\pi$
$631$	9.52407e9			1.50911			0.754553	+	0.656239i	$0.227854\pi$
$632$		0			0
$633$	1.30003e9	+	2.25171e9i	0.203723	+	0.352858i
$634$		0			0
$635$	1.86496e9	−	3.23021e9i	0.289042	−	0.500636i
$636$		0			0
$637$	−2.06484e9	+	7.05153e9i	−0.316517	+	1.08092i
$638$		0			0
$639$	−2.18242e9	+	3.78006e9i	−0.330891	+	0.573120i
$640$		0			0
$641$	1.32160e9	+	2.28909e9i	0.198198	+	0.343288i	0.947944	−	0.318437i	$-0.103158\pi$
$641$	1.32160e9	+	2.28909e9i	0.198198	+	0.343288i	−0.749746	+	0.661725i	$0.769825\pi$
$642$		0			0
$643$	−7.78643e9			−1.15505			−0.577524	−	0.816374i	$-0.695981\pi$
$643$	−7.78643e9			−1.15505			−0.577524	+	0.816374i	$0.695981\pi$
$644$		0			0
$645$	−1.91383e9			−0.280830
$646$		0			0
$647$	4.77250e9	+	8.26622e9i	0.692757	+	1.19989i	0.970931	+	0.239360i	$0.0769377\pi$
$647$	4.77250e9	+	8.26622e9i	0.692757	+	1.19989i	−0.278173	+	0.960531i	$0.589729\pi$
$648$		0			0
$649$	−2.16987e9	+	3.75833e9i	−0.311586	+	0.539682i
$650$		0			0
$651$	4.60487e8	+	3.83929e9i	0.0654161	+	0.545403i
$652$		0			0
$653$	5.15813e8	−	8.93414e8i	0.0724930	−	0.125562i	−0.827500	−	0.561465i	$-0.810238\pi$
$653$	5.15813e8	−	8.93414e8i	0.0724930	−	0.125562i	0.899993	+	0.435904i	$0.143571\pi$
$654$		0			0
$655$	−2.91824e9	−	5.05453e9i	−0.405766	−	0.702808i
$656$		0			0
$657$	5.73239e9			0.788600
$658$		0			0
$659$	−9.97213e9			−1.35734			−0.678671	−	0.734443i	$-0.737444\pi$
$659$	−9.97213e9			−1.35734			−0.678671	+	0.734443i	$0.737444\pi$
$660$		0			0
$661$	−4.39852e9	−	7.61845e9i	−0.592381	−	1.02603i	−0.993911	−	0.110188i	$-0.964855\pi$
$661$	−4.39852e9	−	7.61845e9i	−0.592381	−	1.02603i	0.401530	−	0.915846i	$-0.368479\pi$
$662$		0			0
$663$	1.03498e9	−	1.79264e9i	0.137923	−	0.238889i
$664$		0			0
$665$	−1.56378e9	−	6.68961e8i	−0.206205	−	0.0882115i
$666$		0			0
$667$	−4.25526e9	+	7.37032e9i	−0.555246	+	0.961714i
$668$		0			0
$669$	−2.97987e9	−	5.16129e9i	−0.384774	−	0.666449i
$670$		0			0
$671$	−5.48012e9			−0.700264
$672$		0			0
$673$	7.42386e9			0.938808			0.469404	−	0.882983i	$-0.344469\pi$
$673$	7.42386e9			0.938808			0.469404	+	0.882983i	$0.344469\pi$
$674$		0			0
$675$	1.28023e9	+	2.21743e9i	0.160223	+	0.277515i
$676$		0			0
$677$	5.72981e8	−	9.92432e8i	0.0709708	−	0.122925i	−0.828356	−	0.560202i	$-0.810724\pi$
$677$	5.72981e8	−	9.92432e8i	0.0709708	−	0.122925i	0.899327	+	0.437277i	$0.144057\pi$
$678$		0			0
$679$	−3.27921e9	+	2.45669e9i	−0.402000	+	0.301166i
$680$		0			0
$681$	−7.03863e8	+	1.21913e9i	−0.0854030	+	0.147922i
$682$		0			0
$683$	5.29365e9	+	9.16887e9i	0.635745	+	1.10114i	0.986357	+	0.164621i	$0.0526402\pi$
$683$	5.29365e9	+	9.16887e9i	0.635745	+	1.10114i	−0.350612	+	0.936521i	$0.614026\pi$
$684$		0			0
$685$	5.90877e9			0.702393
$686$		0			0
$687$	−1.49674e9			−0.176115
$688$		0			0
$689$	−5.67868e9	−	9.83576e9i	−0.661424	−	1.14562i
$690$		0			0
$691$	3.78065e9	−	6.54827e9i	0.435906	−	0.755011i	−0.561463	−	0.827502i	$-0.689761\pi$
$691$	3.78065e9	−	6.54827e9i	0.435906	−	0.755011i	0.997369	+	0.0724904i	$0.0230947\pi$
$692$		0			0
$693$	−3.28366e9	+	2.46002e9i	−0.374793	+	0.280784i
$694$		0			0
$695$	1.48275e9	−	2.56819e9i	0.167540	−	0.290189i
$696$		0			0
$697$	−3.46329e9	−	5.99859e9i	−0.387413	−	0.671018i
$698$		0			0
$699$	−4.55087e7			−0.00503992
$700$		0			0
$701$	−1.31100e10			−1.43744			−0.718719	−	0.695301i	$-0.755271\pi$
$701$	−1.31100e10			−1.43744			−0.718719	+	0.695301i	$0.755271\pi$
$702$		0			0
$703$	−2.12586e9	−	3.68210e9i	−0.230776	−	0.399716i
$704$		0			0
$705$	−2.72745e9	+	4.72408e9i	−0.293153	+	0.507756i
$706$		0			0
$707$	6.17417e9	+	2.64122e9i	0.657069	+	0.281085i
$708$		0			0
$709$	−1.55398e8	+	2.69157e8i	−0.0163750	+	0.0283624i	−0.874097	−	0.485752i	$-0.838546\pi$
$709$	−1.55398e8	+	2.69157e8i	−0.0163750	+	0.0283624i	0.857722	+	0.514114i	$0.171879\pi$
$710$		0			0
$711$	3.67597e9	+	6.36697e9i	0.383556	+	0.664338i
$712$		0			0
$713$	1.34819e10			1.39296
$714$		0			0
$715$	−4.99300e9			−0.510847
$716$		0			0
$717$	1.95341e9	+	3.38340e9i	0.197914	+	0.342797i
$718$		0			0
$719$	2.09243e9	−	3.62420e9i	0.209943	−	0.363631i	−0.741754	−	0.670673i	$-0.766006\pi$
$719$	2.09243e9	−	3.62420e9i	0.209943	−	0.363631i	0.951696	+	0.307041i	$0.0993389\pi$
$720$		0			0
$721$	1.99468e9	+	1.66305e10i	0.198198	+	1.65246i
$722$		0			0
$723$	−1.30828e9	+	2.26601e9i	−0.128741	+	0.222986i
$724$		0			0
$725$	2.00612e9	+	3.47471e9i	0.195512	+	0.338637i
$726$		0			0
$727$	9.54370e9			0.921184			0.460592	−	0.887612i	$-0.347637\pi$
$727$	9.54370e9			0.921184			0.460592	+	0.887612i	$0.347637\pi$
$728$		0			0
$729$	5.34261e7			0.00510749
$730$		0			0
$731$	−2.34993e9	−	4.07019e9i	−0.222507	−	0.385393i
$732$		0			0
$733$	−5.03140e9	+	8.71464e9i	−0.471873	+	0.817307i	−0.999482	−	0.0321796i	$-0.989755\pi$
$733$	−5.03140e9	+	8.71464e9i	−0.471873	+	0.817307i	0.527609	+	0.849487i	$0.323088\pi$
$734$		0			0
$735$	1.04730e9	−	3.57658e9i	0.0972891	−	0.332248i
$736$		0			0
$737$	4.32374e9	−	7.48894e9i	0.397854	−	0.689103i
$738$		0			0
$739$	−3.54880e9	−	6.14670e9i	−0.323464	−	0.560256i	0.657736	−	0.753248i	$-0.271514\pi$
$739$	−3.54880e9	−	6.14670e9i	−0.323464	−	0.560256i	−0.981200	+	0.192992i	$0.938181\pi$
$740$		0			0
$741$	1.61062e9			0.145421
$742$		0			0
$743$	−3.16400e8			−0.0282993			−0.0141497	−	0.999900i	$-0.504504\pi$
$743$	−3.16400e8			−0.0282993			−0.0141497	+	0.999900i	$0.504504\pi$
$744$		0			0
$745$	2.64320e9	+	4.57816e9i	0.234198	+	0.405643i
$746$		0			0
$747$	−2.93777e9	+	5.08836e9i	−0.257867	+	0.446638i
$748$		0			0
$749$	−5.39699e8	−	4.49971e9i	−0.0469316	−	0.391290i
$750$		0			0
$751$	3.22967e9	−	5.59396e9i	0.278240	−	0.481925i	−0.692708	−	0.721218i	$-0.743582\pi$
$751$	3.22967e9	−	5.59396e9i	0.278240	−	0.481925i	0.970947	+	0.239293i	$0.0769158\pi$
$752$		0			0
$753$	−3.64287e8	−	6.30964e8i	−0.0310929	−	0.0538545i
$754$		0			0
$755$	7.32689e9			0.619591
$756$		0			0
$757$	1.07167e10			0.897892			0.448946	−	0.893559i	$-0.351800\pi$
$757$	1.07167e10			0.897892			0.448946	+	0.893559i	$0.351800\pi$
$758$		0			0
$759$	−1.78032e9	−	3.08361e9i	−0.147793	−	0.255984i
$760$		0			0
$761$	4.51415e9	−	7.81874e9i	0.371304	−	0.643117i	−0.618462	−	0.785814i	$-0.712244\pi$
$761$	4.51415e9	−	7.81874e9i	0.371304	−	0.643117i	0.989766	+	0.142697i	$0.0455774\pi$
$762$		0			0
$763$	1.55916e10	+	6.66987e9i	1.27073	+	0.543602i
$764$		0			0
$765$	2.10904e9	−	3.65296e9i	0.170322	−	0.295006i
$766$		0			0
$767$	−7.49829e9	−	1.29874e10i	−0.600038	−	1.03930i
$768$		0			0
$769$	8.30919e9			0.658895			0.329448	−	0.944174i	$-0.393137\pi$
$769$	8.30919e9			0.658895			0.329448	+	0.944174i	$0.393137\pi$
$770$		0			0
$771$	1.23922e9			0.0973771
$772$		0			0
$773$	−5.63091e9	−	9.75302e9i	−0.438480	−	0.759470i	0.559092	−	0.829106i	$-0.311150\pi$
$773$	−5.63091e9	−	9.75302e9i	−0.438480	−	0.759470i	−0.997573	+	0.0696352i	$0.977816\pi$
$774$		0			0
$775$	3.17800e9	−	5.50446e9i	0.245244	−	0.424775i
$776$		0			0
$777$	7.45577e9	−	5.58564e9i	0.570189	−	0.427168i
$778$		0			0
$779$	2.69474e9	−	4.66743e9i	0.204238	−	0.353750i
$780$		0			0
$781$	−3.21774e9	−	5.57328e9i	−0.241697	−	0.418632i
$782$		0			0
$783$	1.05926e10			0.788563
$784$		0			0
$785$	−1.72465e9			−0.127249
$786$		0			0
$787$	−1.11102e10	−	1.92434e10i	−0.812475	−	1.40725i	−0.911127	−	0.412127i	$-0.864786\pi$
$787$	−1.11102e10	−	1.92434e10i	−0.812475	−	1.40725i	0.0986511	−	0.995122i	$-0.468547\pi$
$788$		0			0
$789$	2.22079e8	−	3.84652e8i	0.0160968	−	0.0278804i
$790$		0			0
$791$	−1.81232e10	+	1.35774e10i	−1.30202	+	0.975434i
$792$		0			0
$793$	9.46866e9	−	1.64002e10i	0.674268	−	1.16787i
$794$		0			0
$795$	2.88026e9	+	4.98876e9i	0.203304	+	0.352133i
$796$		0			0
$797$	−6.49195e9			−0.454225			−0.227113	−	0.973868i	$-0.572929\pi$
$797$	−6.49195e9			−0.454225			−0.227113	+	0.973868i	$0.572929\pi$
$798$		0			0
$799$	−1.33958e10			−0.929082
$800$		0			0
$801$	2.10100e9	+	3.63904e9i	0.144448	+	0.250192i
$802$		0			0
$803$	−4.22589e9	+	7.31946e9i	−0.288014	+	0.498855i
$804$		0			0
$805$	−1.19465e10	−	5.11053e9i	−0.807149	−	0.345287i
$806$		0			0
$807$	1.22243e9	−	2.11731e9i	0.0818780	−	0.141817i
$808$		0			0
$809$	5.80562e9	+	1.00556e10i	0.385504	+	0.667712i	0.991839	−	0.127497i	$-0.0406944\pi$
$809$	5.80562e9	+	1.00556e10i	0.385504	+	0.667712i	−0.606335	+	0.795209i	$0.707361\pi$
$810$		0			0
$811$	1.72427e10			1.13510			0.567548	−	0.823340i	$-0.307892\pi$
$811$	1.72427e10			1.13510			0.567548	+	0.823340i	$0.307892\pi$
$812$		0			0
$813$	−4.23329e9			−0.276288
$814$		0			0
$815$	5.81366e9	+	1.00696e10i	0.376182	+	0.651567i
$816$		0			0
$817$	1.82845e9	−	3.16696e9i	0.117302	−	0.203173i
$818$		0			0
$819$	−1.68845e9	−	1.40774e10i	−0.107398	−	0.895423i
$820$		0			0
$821$	−7.15770e9	+	1.23975e10i	−0.451412	+	0.781868i	−0.998474	−	0.0552240i	$-0.982413\pi$
$821$	−7.15770e9	+	1.23975e10i	−0.451412	+	0.781868i	0.547062	+	0.837092i	$0.315746\pi$
$822$		0			0
$823$	−2.33905e9	−	4.05135e9i	−0.146265	−	0.253338i	0.783579	−	0.621292i	$-0.213392\pi$
$823$	−2.33905e9	−	4.05135e9i	−0.146265	−	0.253338i	−0.929844	+	0.367954i	$0.880059\pi$
$824$		0			0
$825$	−1.67865e9			−0.104081
$826$		0			0
$827$	1.01540e10			0.624264			0.312132	−	0.950039i	$-0.398957\pi$
$827$	1.01540e10			0.624264			0.312132	+	0.950039i	$0.398957\pi$
$828$		0			0
$829$	−5.51455e9	−	9.55148e9i	−0.336178	−	0.582278i	0.647532	−	0.762038i	$-0.275801\pi$
$829$	−5.51455e9	−	9.55148e9i	−0.336178	−	0.582278i	−0.983710	+	0.179760i	$0.942468\pi$
$830$		0			0
$831$	1.05589e9	−	1.82885e9i	0.0638284	−	0.110554i
$832$		0			0
$833$	8.89235e9	−	2.16425e9i	0.533039	−	0.129733i
$834$		0			0
$835$	8.88969e9	−	1.53974e10i	0.528426	−	0.915260i
$836$		0			0
$837$	−8.39015e9	−	1.45322e10i	−0.494573	−	0.856626i
$838$		0			0
$839$	1.71068e10			1.00001			0.500003	−	0.866023i	$-0.333332\pi$
$839$	1.71068e10			1.00001			0.500003	+	0.866023i	$0.333332\pi$
$840$		0			0
$841$	−6.51269e8			−0.0377550
$842$		0			0
$843$	3.85847e9	+	6.68306e9i	0.221829	+	0.384219i
$844$		0			0
$845$	1.82649e9	−	3.16357e9i	0.104140	−	0.180376i
$846$		0			0
$847$	1.38560e9	+	1.15523e10i	0.0783510	+	0.653247i
$848$		0			0
$849$	2.29298e9	−	3.97156e9i	0.128595	−	0.222733i
$850$		0			0
$851$	−1.62405e10	−	2.81294e10i	−0.903330	−	1.56461i
$852$		0			0
$853$	−2.88309e9			−0.159051			−0.0795256	−	0.996833i	$-0.525341\pi$
$853$	−2.88309e9			−0.159051			−0.0795256	+	0.996833i	$0.525341\pi$
$854$		0			0
$855$	3.28203e9			0.179582
$856$		0			0
$857$	1.20159e10	+	2.08122e10i	0.652116	+	1.12950i	0.982608	+	0.185690i	$0.0594520\pi$
$857$	1.20159e10	+	2.08122e10i	0.652116	+	1.12950i	−0.330492	+	0.943809i	$0.607215\pi$
$858$		0			0
$859$	−1.70872e10	+	2.95958e10i	−0.919801	+	1.59314i	−0.120084	+	0.992764i	$0.538317\pi$
$859$	−1.70872e10	+	2.95958e10i	−0.919801	+	1.59314i	−0.799716	+	0.600378i	$0.795017\pi$
$860$		0			0
$861$	1.08572e10	+	4.64457e9i	0.579706	+	0.247990i
$862$		0			0
$863$	1.48357e10	−	2.56962e10i	0.785724	−	1.36091i	−0.142841	−	0.989746i	$-0.545624\pi$
$863$	1.48357e10	−	2.56962e10i	0.785724	−	1.36091i	0.928566	−	0.371168i	$-0.121043\pi$
$864$		0			0
$865$	−7.36082e9	−	1.27493e10i	−0.386696	−	0.669777i
$866$		0			0
$867$	5.98853e9			0.312071
$868$		0			0
$869$	−1.08397e10			−0.560333
$870$		0			0
$871$	1.49413e10	+	2.58791e10i	0.766169	+	1.32704i
$872$		0			0
$873$	3.95324e9	−	6.84721e9i	0.201096	−	0.348308i
$874$		0			0
$875$	−1.72015e10	+	1.28868e10i	−0.868035	+	0.650305i
$876$		0			0
$877$	3.70786e9	−	6.42220e9i	0.185620	−	0.321503i	−0.758165	−	0.652062i	$-0.773904\pi$
$877$	3.70786e9	−	6.42220e9i	0.185620	−	0.321503i	0.943785	+	0.330559i	$0.107237\pi$
$878$		0			0
$879$	−7.53914e9	−	1.30582e10i	−0.374422	−	0.648517i
$880$		0			0
$881$	3.89699e10			1.92005			0.960027	−	0.279907i	$-0.0903037\pi$
$881$	3.89699e10			1.92005			0.960027	+	0.279907i	$0.0903037\pi$
$882$		0			0
$883$	6.37811e9			0.311767			0.155883	−	0.987775i	$-0.450178\pi$
$883$	6.37811e9			0.311767			0.155883	+	0.987775i	$0.450178\pi$
$884$		0			0
$885$	3.80318e9	+	6.58730e9i	0.184436	+	0.319452i
$886$		0			0
$887$	−1.84996e10	+	3.20422e10i	−0.890079	+	1.54166i	−0.0503004	+	0.998734i	$0.516018\pi$
$887$	−1.84996e10	+	3.20422e10i	−0.890079	+	1.54166i	−0.839779	+	0.542928i	$0.817315\pi$
$888$		0			0
$889$	−1.24979e10	+	9.36305e9i	−0.596597	+	0.446952i
$890$		0			0
$891$	2.72803e9	−	4.72509e9i	0.129205	−	0.223789i
$892$		0			0
$893$	−5.21155e9	−	9.02666e9i	−0.244899	−	0.424177i
$894$		0			0
$895$	−1.32176e10			−0.616272
$896$		0			0
$897$	1.23043e10			0.569224
$898$		0			0
$899$	−1.31474e10	−	2.27719e10i	−0.603503	−	1.04530i
$900$		0			0
$901$	−7.07315e9	+	1.22511e10i	−0.322163	+	0.558003i
$902$		0			0
$903$	7.36690e9	+	3.15146e9i	0.332949	+	0.142431i
$904$		0			0
$905$	−1.66786e10	+	2.88883e10i	−0.747981	+	1.29554i
$906$		0			0
$907$	−3.20212e9	−	5.54623e9i	−0.142499	−	0.246816i	0.785938	−	0.618305i	$-0.212180\pi$
$907$	−3.20212e9	−	5.54623e9i	−0.142499	−	0.246816i	−0.928437	+	0.371490i	$0.878847\pi$
$908$		0			0
$909$	−1.29583e10			−0.572234
$910$		0			0
$911$	2.59678e9			0.113794			0.0568971	−	0.998380i	$-0.481879\pi$
$911$	2.59678e9			0.113794			0.0568971	+	0.998380i	$0.481879\pi$
$912$		0			0
$913$	−4.33142e9	−	7.50224e9i	−0.188357	−	0.326244i
$914$		0			0
$915$	−4.80256e9	+	8.31828e9i	−0.207252	+	0.358971i
$916$		0			0
$917$	2.90998e9	+	2.42618e10i	0.124623	+	1.03904i
$918$		0			0
$919$	−1.87721e10	+	3.25142e10i	−0.797825	+	1.38187i	0.123205	+	0.992381i	$0.460683\pi$
$919$	−1.87721e10	+	3.25142e10i	−0.797825	+	1.38187i	−0.921030	+	0.389492i	$0.872651\pi$
$920$		0			0
$921$	6.67602e8	+	1.15632e9i	0.0281585	+	0.0487719i
$922$		0			0
$923$	2.22387e10			0.930900
$924$		0			0
$925$	−1.53130e10			−0.636159
$926$		0			0
$927$	−1.61604e10	−	2.79907e10i	−0.666306	−	1.15408i
$928$		0			0
$929$	−1.26185e10	+	2.18559e10i	−0.516361	+	0.894363i	0.483459	+	0.875367i	$0.339380\pi$
$929$	−1.26185e10	+	2.18559e10i	−0.516361	+	0.894363i	−0.999820	+	0.0189960i	$0.993953\pi$
$930$		0			0
$931$	4.91788e9	+	5.15007e9i	0.199735	+	0.209165i
$932$		0			0
$933$	−1.61427e9	+	2.79600e9i	−0.0650716	+	0.112707i
$934$		0			0
$935$	3.10955e9	+	5.38590e9i	0.124410	+	0.215485i
$936$		0			0
$937$	−6.75432e9			−0.268221			−0.134111	−	0.990966i	$-0.542818\pi$
$937$	−6.75432e9			−0.268221			−0.134111	+	0.990966i	$0.542818\pi$
$938$		0			0
$939$	1.03131e10			0.406499
$940$		0			0
$941$	4.73345e9	+	8.19857e9i	0.185188	+	0.320756i	0.943640	−	0.330974i	$-0.107377\pi$
$941$	4.73345e9	+	8.19857e9i	0.185188	+	0.320756i	−0.758452	+	0.651729i	$0.774044\pi$
$942$		0			0
$943$	2.05865e10	−	3.56568e10i	0.799449	−	1.38469i
$944$		0			0
$945$	1.92595e9	+	1.60575e10i	0.0742392	+	0.618965i
$946$		0			0
$947$	−2.51579e10	+	4.35748e10i	−0.962608	+	1.66729i	−0.246701	+	0.969092i	$0.579346\pi$
$947$	−2.51579e10	+	4.35748e10i	−0.962608	+	1.66729i	−0.715908	+	0.698195i	$0.753987\pi$
$948$		0			0
$949$	−1.46032e10	−	2.52934e10i	−0.554645	−	0.960673i
$950$		0			0
$951$	−8.97599e9			−0.338416
$952$		0			0
$953$	−2.56329e9			−0.0959342			−0.0479671	−	0.998849i	$-0.515274\pi$
$953$	−2.56329e9			−0.0959342			−0.0479671	+	0.998849i	$0.515274\pi$
$954$		0			0
$955$	−2.44072e9	−	4.22745e9i	−0.0906790	−	0.157061i
$956$		0			0
$957$	−3.47228e9	+	6.01416e9i	−0.128063	+	0.221811i
$958$		0			0
$959$	−2.27446e10	−	9.72984e9i	−0.832749	−	0.356238i
$960$		0			0
$961$	−7.07111e9	+	1.22475e10i	−0.257013	+	0.445160i
$962$		0			0
$963$	4.37253e9	+	7.57344e9i	0.157776	+	0.273276i
$964$		0			0
$965$	2.93218e10			1.05038
$966$		0			0
$967$	−1.17322e10			−0.417241			−0.208621	−	0.977997i	$-0.566897\pi$
$967$	−1.17322e10			−0.417241			−0.208621	+	0.977997i	$0.566897\pi$
$968$		0			0
$969$	−1.00306e9	−	1.73735e9i	−0.0354156	−	0.0613416i
$970$		0			0
$971$	−8.01185e9	+	1.38769e10i	−0.280844	+	0.486437i	−0.971593	−	0.236658i	$-0.923948\pi$
$971$	−8.01185e9	+	1.38769e10i	−0.280844	+	0.486437i	0.690749	+	0.723095i	$0.257281\pi$
$972$		0			0
$973$	−9.93651e9	+	7.44413e9i	−0.345811	+	0.259071i
$974$		0			0
$975$	2.90040e9	−	5.02365e9i	0.100217	−	0.173581i
$976$		0			0
$977$	−2.08302e10	−	3.60790e10i	−0.714601	−	1.23772i	−0.963113	−	0.269096i	$-0.913275\pi$
$977$	−2.08302e10	−	3.60790e10i	−0.714601	−	1.23772i	0.248513	−	0.968629i	$-0.420058\pi$
$978$		0			0
$979$	−6.19540e9			−0.211023
$980$		0			0
$981$	−3.27235e10			−1.10667
$982$		0			0
$983$	2.36038e10	+	4.08830e10i	0.792584	+	1.37280i	0.924362	+	0.381516i	$0.124598\pi$
$983$	2.36038e10	+	4.08830e10i	0.792584	+	1.37280i	−0.131778	+	0.991279i	$0.542069\pi$
$984$		0			0
$985$	−9.71314e9	+	1.68236e10i	−0.323842	+	0.560910i
$986$		0			0
$987$	1.82778e10	−	1.36932e10i	0.605082	−	0.453309i
$988$		0			0
$989$	1.39684e10	−	2.41940e10i	0.459156	−	0.795282i
$990$		0			0
$991$	−3.20329e9	−	5.54826e9i	−0.104553	−	0.181092i	0.809002	−	0.587806i	$-0.200008\pi$
$991$	−3.20329e9	−	5.54826e9i	−0.104553	−	0.181092i	−0.913556	+	0.406714i	$0.866675\pi$
$992$		0			0
$993$	−6.85132e9			−0.222051
$994$		0			0
$995$	−1.19088e10			−0.383254
$996$		0			0
$997$	6.01597e9	+	1.04200e10i	0.192253	+	0.332991i	0.945996	−	0.324177i	$-0.105087\pi$
$997$	6.01597e9	+	1.04200e10i	0.192253	+	0.332991i	−0.753744	+	0.657168i	$0.771754\pi$
$998$		0			0
$999$	−2.02137e10	+	3.50112e10i	−0.641457	+	1.11104i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.8.i.a.65.2		4
4.3	odd	2		14.8.c.a.9.1	✓	4
7.4	even	3	inner	112.8.i.a.81.2		4
12.11	even	2		126.8.g.e.37.1		4
28.3	even	6		98.8.c.h.67.2		4
28.11	odd	6		14.8.c.a.11.1	yes	4
28.19	even	6		98.8.a.k.1.1		2
28.23	odd	6		98.8.a.h.1.2		2
28.27	even	2		98.8.c.h.79.2		4
84.11	even	6		126.8.g.e.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.c.a.9.1	✓	4	4.3	odd	2
14.8.c.a.11.1	yes	4	28.11	odd	6
98.8.a.h.1.2		2	28.23	odd	6
98.8.a.k.1.1		2	28.19	even	6
98.8.c.h.67.2		4	28.3	even	6
98.8.c.h.79.2		4	28.27	even	2
112.8.i.a.65.2		4	1.1	even	1	trivial
112.8.i.a.81.2		4	7.4	even	3	inner
126.8.g.e.37.1		4	12.11	even	2
126.8.g.e.109.1		4	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)

\(f(q)\)	\(=\)	\(q+(10.4387 + 18.0804i) q^{3} +(108.377 - 187.715i) q^{5} +(-726.284 + 544.110i) q^{7} +(875.567 - 1516.53i) q^{9} +(1290.93 + 2235.95i) q^{11} -8921.97 q^{13} +4525.28 q^{15} +(5556.44 + 9624.03i) q^{17} +(-4323.40 + 7488.34i) q^{19} +(-17419.1 - 7451.67i) q^{21} +(-33028.6 + 57207.2i) q^{23} +(15571.2 + 26970.1i) q^{25} +82218.0 q^{27} +128836. q^{29} +(-102048. - 176752. i) q^{31} +(-26951.2 + 46680.9i) q^{33} +(23424.9 + 195304. i) q^{35} +(-245856. + 425834. i) q^{37} +(-93133.8 - 161312. i) q^{39} -623293. q^{41} -422919. q^{43} +(-189783. - 328714. i) q^{45} +(-602714. + 1.04393e6i) q^{47} +(231433. - 790356. i) q^{49} +(-116004. + 200925. i) q^{51} +(636483. + 1.10242e6i) q^{53} +559630. q^{55} -180523. q^{57} +(840430. + 1.45567e6i) q^{59} +(-1.06128e6 + 1.83818e6i) q^{61} +(189247. + 1.57783e6i) q^{63} +(-966940. + 1.67479e6i) q^{65} +(-1.67466e6 - 2.90060e6i) q^{67} -1.37910e6 q^{69} -2.49257e6 q^{71} +(1.63676e6 + 2.83496e6i) q^{73} +(-325086. + 563065. i) q^{75} +(-2.15418e6 - 921530. i) q^{77} +(-2.09920e6 + 3.63591e6i) q^{79} +(-1.05662e6 - 1.83011e6i) q^{81} -3.35527e6 q^{83} +2.40877e6 q^{85} +(1.34488e6 + 2.32939e6i) q^{87} +(-1.19979e6 + 2.07810e6i) q^{89} +(6.47988e6 - 4.85453e6i) q^{91} +(2.13049e6 - 3.69011e6i) q^{93} +(937117. + 1.62313e6i) q^{95} +4.51506e6 q^{97} +4.52118e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 56 q^{3} + 238 q^{5} - 168 q^{7} - 1972 q^{9} + 5848 q^{11} + 2632 q^{13} + 5784 q^{15} + 47642 q^{17} - 41048 q^{19} - 169582 q^{21} - 49316 q^{23} + 108816 q^{25} + 400624 q^{27} + 345640 q^{29}+ \cdots - 15278208 q^{99}+O(q^{100}) \) 4 * q - 56 * q^3 + 238 * q^5 - 168 * q^7 - 1972 * q^9 + 5848 * q^11 + 2632 * q^13 + 5784 * q^15 + 47642 * q^17 - 41048 * q^19 - 169582 * q^21 - 49316 * q^23 + 108816 * q^25 + 400624 * q^27 + 345640 * q^29 - 70252 * q^31 + 197190 * q^33 + 36904 * q^35 + 88166 * q^37 - 973336 * q^39 - 2312520 * q^41 - 115088 * q^43 - 300468 * q^45 - 1412292 * q^47 + 465892 * q^49 + 2576256 * q^51 + 2361174 * q^53 + 1258040 * q^55 + 4620796 * q^57 + 1842512 * q^59 - 1278242 * q^61 + 2121448 * q^63 - 1716372 * q^65 - 2121480 * q^67 - 5332236 * q^69 - 7200320 * q^71 + 1634682 * q^73 + 5321176 * q^75 + 1416226 * q^77 - 9192604 * q^79 - 3049498 * q^81 + 56560 * q^83 + 6369676 * q^85 - 691656 * q^87 - 8936550 * q^89 + 26111120 * q^91 + 14550490 * q^93 + 2562604 * q^95 + 11748408 * q^97 - 15278208 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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