LMFDB - Embedded newform 336.6.q.i.193.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,6,Mod(193,336)] 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("336.193"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 4])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,-36,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.3
Root			$13.1471 - 22.7714i$ of defining polynomial
Character	$\chi$	$=$	336.193
Dual form			336.6.q.i.289.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.50000 + 7.79423i) q^{3} +(15.9808 + 27.6796i) q^{5} +(-85.7043 + 97.2716i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(130.442 - 225.932i) q^{11} +769.735 q^{13} -287.654 q^{15} +(776.659 - 1345.21i) q^{17} +(375.024 + 649.561i) q^{19} +(-372.488 - 1105.72i) q^{21} +(377.427 + 653.723i) q^{23} +(1051.73 - 1821.65i) q^{25} +729.000 q^{27} +6008.93 q^{29} +(-3210.02 + 5559.92i) q^{31} +(1173.98 + 2033.39i) q^{33} +(-4062.06 - 817.779i) q^{35} +(2387.86 + 4135.90i) q^{37} +(-3463.81 + 5999.49i) q^{39} -5423.27 q^{41} +11896.4 q^{43} +(1294.44 - 2242.04i) q^{45} +(-8714.00 - 15093.1i) q^{47} +(-2116.54 - 16673.2i) q^{49} +(6989.93 + 12106.9i) q^{51} +(-18825.3 + 32606.4i) q^{53} +8338.26 q^{55} -6750.44 q^{57} +(11039.0 - 19120.2i) q^{59} +(4086.69 + 7078.36i) q^{61} +(10294.4 + 2072.49i) q^{63} +(12301.0 + 21305.9i) q^{65} +(6500.87 - 11259.8i) q^{67} -6793.69 q^{69} +12349.6 q^{71} +(-21800.2 + 37759.0i) q^{73} +(9465.55 + 16394.8i) q^{75} +(10797.3 + 32051.6i) q^{77} +(38374.7 + 66467.0i) q^{79} +(-3280.50 + 5681.99i) q^{81} -21893.6 q^{83} +49646.5 q^{85} +(-27040.2 + 46835.0i) q^{87} +(68483.5 + 118617. i) q^{89} +(-65969.7 + 74873.4i) q^{91} +(-28890.2 - 50039.3i) q^{93} +(-11986.4 + 20761.0i) q^{95} -93050.1 q^{97} -21131.6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 36 q^{3} + 42 q^{7} - 324 q^{9} + 462 q^{11} - 1204 q^{13} + 228 q^{17} - 358 q^{19} + 1404 q^{21} + 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} + 4158 q^{33} - 7692 q^{35}+ \cdots - 74844 q^{99}+O(q^{100}) $ 8 * q - 36 * q^3 + 42 * q^7 - 324 * q^9 + 462 * q^11 - 1204 * q^13 + 228 * q^17 - 358 * q^19 + 1404 * q^21 + 2148 * q^23 - 5454 * q^25 + 5832 * q^27 - 11064 * q^29 - 830 * q^31 + 4158 * q^33 - 7692 * q^35 - 3914 * q^37 + 5418 * q^39 - 16632 * q^41 + 29036 * q^43 - 41700 * q^47 + 41876 * q^49 + 2052 * q^51 + 22164 * q^53 - 7784 * q^55 + 6444 * q^57 - 32886 * q^59 + 83732 * q^61 - 16038 * q^63 - 93192 * q^65 + 80034 * q^67 - 38664 * q^69 - 89544 * q^71 - 22470 * q^73 - 49086 * q^75 + 138732 * q^77 + 75286 * q^79 - 26244 * q^81 + 34836 * q^83 + 278504 * q^85 + 49788 * q^87 + 28944 * q^89 + 12058 * q^91 - 7470 * q^93 + 144120 * q^95 - 433356 * q^97 - 74844 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−4.50000	+	7.79423i	−0.288675	+	0.500000i
$3$	−4.50000	+	7.79423i	−0.288675	+	0.500000i
$4$		0			0
$5$	15.9808	+	27.6796i	0.285873	+	0.495147i	0.972821	−	0.231560i	$-0.0743831\pi$
$5$	15.9808	+	27.6796i	0.285873	+	0.495147i	−0.686947	+	0.726707i	$0.741050\pi$
$6$		0			0
$7$	−85.7043	+	97.2716i	−0.661086	+	0.750311i
$7$	−85.7043	+	97.2716i	−0.661086	+	0.750311i
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	130.442	−	225.932i	0.325039	−	0.562984i	−0.656481	−	0.754342i	$-0.727956\pi$
$11$	130.442	−	225.932i	0.325039	−	0.562984i	0.981520	+	0.191359i	$0.0612893\pi$
$12$		0			0
$13$	769.735			1.26323			0.631616	−	0.775281i	$-0.282392\pi$
$13$	769.735			1.26323			0.631616	+	0.775281i	$0.282392\pi$
$14$		0			0
$15$	−287.654			−0.330098
$16$		0			0
$17$	776.659	−	1345.21i	0.651791	−	1.12893i	−0.330898	−	0.943667i	$-0.607351\pi$
$17$	776.659	−	1345.21i	0.651791	−	1.12893i	0.982688	−	0.185268i	$-0.0593152\pi$
$18$		0			0
$19$	375.024	+	649.561i	0.238328	+	0.412797i	0.960235	−	0.279194i	$-0.0900673\pi$
$19$	375.024	+	649.561i	0.238328	+	0.412797i	−0.721906	+	0.691991i	$0.756734\pi$
$20$		0			0
$21$	−372.488	−	1105.72i	−0.184316	−	0.547139i
$22$		0			0
$23$	377.427	+	653.723i	0.148769	+	0.257676i	0.930773	−	0.365598i	$-0.119135\pi$
$23$	377.427	+	653.723i	0.148769	+	0.257676i	−0.782004	+	0.623274i	$0.785802\pi$
$24$		0			0
$25$	1051.73	−	1821.65i	0.336553	−	0.582927i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	6008.93			1.32679			0.663395	−	0.748269i	$-0.269115\pi$
$29$	6008.93			1.32679			0.663395	+	0.748269i	$0.269115\pi$
$30$		0			0
$31$	−3210.02	+	5559.92i	−0.599934	+	1.03912i	0.392896	+	0.919583i	$0.371473\pi$
$31$	−3210.02	+	5559.92i	−0.599934	+	1.03912i	−0.992830	+	0.119533i	$0.961860\pi$
$32$		0			0
$33$	1173.98	+	2033.39i	0.187661	+	0.325039i
$34$		0			0
$35$	−4062.06	−	817.779i	−0.560501	−	0.112841i
$36$		0			0
$37$	2387.86	+	4135.90i	0.286751	+	0.496668i	0.973032	−	0.230669i	$-0.0740914\pi$
$37$	2387.86	+	4135.90i	0.286751	+	0.496668i	−0.686281	+	0.727336i	$0.740758\pi$
$38$		0			0
$39$	−3463.81	+	5999.49i	−0.364664	+	0.631616i
$40$		0			0
$41$	−5423.27			−0.503850			−0.251925	−	0.967747i	$-0.581064\pi$
$41$	−5423.27			−0.503850			−0.251925	+	0.967747i	$0.581064\pi$
$42$		0			0
$43$	11896.4			0.981171			0.490585	−	0.871393i	$-0.336783\pi$
$43$	11896.4			0.981171			0.490585	+	0.871393i	$0.336783\pi$
$44$		0			0
$45$	1294.44	−	2242.04i	0.0952911	−	0.165049i
$46$		0			0
$47$	−8714.00	−	15093.1i	−0.575404	−	0.996629i	−0.995998	−	0.0893798i	$-0.971512\pi$
$47$	−8714.00	−	15093.1i	−0.575404	−	0.996629i	0.420594	−	0.907249i	$-0.361822\pi$
$48$		0			0
$49$	−2116.54	−	16673.2i	−0.125932	−	0.992039i
$50$		0			0
$51$	6989.93	+	12106.9i	0.376311	+	0.651791i
$52$		0			0
$53$	−18825.3	+	32606.4i	−0.920561	+	1.59446i	−0.122012	+	0.992529i	$0.538935\pi$
$53$	−18825.3	+	32606.4i	−0.920561	+	1.59446i	−0.798549	+	0.601930i	$0.794399\pi$
$54$		0			0
$55$	8338.26			0.371679
$56$		0			0
$57$	−6750.44			−0.275198
$58$		0			0
$59$	11039.0	−	19120.2i	0.412859	−	0.715092i	−0.582342	−	0.812944i	$-0.697864\pi$
$59$	11039.0	−	19120.2i	0.412859	−	0.715092i	0.995201	+	0.0978516i	$0.0311971\pi$
$60$		0			0
$61$	4086.69	+	7078.36i	0.140620	+	0.243561i	0.927730	−	0.373251i	$-0.121757\pi$
$61$	4086.69	+	7078.36i	0.140620	+	0.243561i	−0.787110	+	0.616812i	$0.788424\pi$
$62$		0			0
$63$	10294.4	+	2072.49i	0.326777	+	0.0657872i
$64$		0			0
$65$	12301.0	+	21305.9i	0.361124	+	0.625485i
$66$		0			0
$67$	6500.87	−	11259.8i	0.176923	−	0.306440i	−0.763902	−	0.645332i	$-0.776719\pi$
$67$	6500.87	−	11259.8i	0.176923	−	0.306440i	0.940825	+	0.338892i	$0.110052\pi$
$68$		0			0
$69$	−6793.69			−0.171784
$70$		0			0
$71$	12349.6			0.290742			0.145371	−	0.989377i	$-0.453562\pi$
$71$	12349.6			0.290742			0.145371	+	0.989377i	$0.453562\pi$
$72$		0			0
$73$	−21800.2	+	37759.0i	−0.478798	+	0.829303i	−0.999704	−	0.0243110i	$-0.992261\pi$
$73$	−21800.2	+	37759.0i	−0.478798	+	0.829303i	0.520906	+	0.853614i	$0.325594\pi$
$74$		0			0
$75$	9465.55	+	16394.8i	0.194309	+	0.336553i
$76$		0			0
$77$	10797.3	+	32051.6i	0.207534	+	0.616060i
$78$		0			0
$79$	38374.7	+	66467.0i	0.691796	+	1.19823i	0.971249	+	0.238066i	$0.0765134\pi$
$79$	38374.7	+	66467.0i	0.691796	+	1.19823i	−0.279453	+	0.960159i	$0.590153\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−21893.6			−0.348836			−0.174418	−	0.984672i	$-0.555804\pi$
$83$	−21893.6			−0.348836			−0.174418	+	0.984672i	$0.555804\pi$
$84$		0			0
$85$	49646.5			0.745318
$86$		0			0
$87$	−27040.2	+	46835.0i	−0.383011	+	0.663395i
$88$		0			0
$89$	68483.5	+	118617.i	0.916454	+	1.58735i	0.804758	+	0.593603i	$0.202295\pi$
$89$	68483.5	+	118617.i	0.916454	+	1.58735i	0.111696	+	0.993742i	$0.464372\pi$
$90$		0			0
$91$	−65969.7	+	74873.4i	−0.835104	+	0.947816i
$92$		0			0
$93$	−28890.2	−	50039.3i	−0.346372	−	0.599934i
$94$		0			0
$95$	−11986.4	+	20761.0i	−0.136263	+	0.236015i
$96$		0			0
$97$	−93050.1			−1.00412			−0.502062	−	0.864832i	$-0.667425\pi$
$97$	−93050.1			−1.00412			−0.502062	+	0.864832i	$0.667425\pi$
$98$		0			0
$99$	−21131.6			−0.216692
$100$		0			0
$101$	−61150.8	+	105916.i	−0.596484	+	1.03314i	0.396852	+	0.917883i	$0.370103\pi$
$101$	−61150.8	+	105916.i	−0.596484	+	1.03314i	−0.993336	+	0.115258i	$0.963231\pi$
$102$		0			0
$103$	37400.6	+	64779.8i	0.347365	+	0.601654i	0.985781	−	0.168038i	$-0.0537432\pi$
$103$	37400.6	+	64779.8i	0.347365	+	0.601654i	−0.638415	+	0.769692i	$0.720410\pi$
$104$		0			0
$105$	24653.2	−	27980.6i	0.218223	−	0.247676i
$106$		0			0
$107$	35386.2	+	61290.7i	0.298796	+	0.517530i	0.975861	−	0.218394i	$-0.0700817\pi$
$107$	35386.2	+	61290.7i	0.298796	+	0.517530i	−0.677065	+	0.735923i	$0.736748\pi$
$108$		0			0
$109$	71131.0	−	123202.i	0.573446	−	0.993238i	−0.422762	−	0.906241i	$-0.638940\pi$
$109$	71131.0	−	123202.i	0.573446	−	0.993238i	0.996209	−	0.0869972i	$-0.0277271\pi$
$110$		0			0
$111$	−42981.5			−0.331112
$112$		0			0
$113$	198899.			1.46533			0.732667	−	0.680588i	$-0.238275\pi$
$113$	198899.			1.46533			0.732667	+	0.680588i	$0.238275\pi$
$114$		0			0
$115$	−12063.2	+	20894.0i	−0.0850583	+	0.147325i
$116$		0			0
$117$	−31174.3	−	53995.4i	−0.210539	−	0.364664i
$118$		0			0
$119$	64288.0	+	190837.i	0.416162	+	1.23537i
$120$		0			0
$121$	46495.4	+	80532.3i	0.288700	+	0.500042i
$122$		0			0
$123$	24404.7	−	42270.2i	0.145449	−	0.251925i
$124$		0			0
$125$	167110.			0.956592
$126$		0			0
$127$	−219619.			−1.20826			−0.604131	−	0.796885i	$-0.706479\pi$
$127$	−219619.			−1.20826			−0.604131	+	0.796885i	$0.706479\pi$
$128$		0			0
$129$	−53533.8	+	92723.3i	−0.283240	+	0.490585i
$130$		0			0
$131$	27311.1	+	47304.1i	0.139047	+	0.240836i	0.927136	−	0.374725i	$-0.122263\pi$
$131$	27311.1	+	47304.1i	0.139047	+	0.240836i	−0.788089	+	0.615561i	$0.788930\pi$
$132$		0			0
$133$	−95325.1	−	19191.0i	−0.467281	−	0.0940737i
$134$		0			0
$135$	11650.0	+	20178.4i	0.0550163	+	0.0952911i
$136$		0			0
$137$	−36471.4	+	63170.4i	−0.166017	+	0.287549i	−0.937016	−	0.349287i	$-0.886424\pi$
$137$	−36471.4	+	63170.4i	−0.166017	+	0.287549i	0.770999	+	0.636836i	$0.219757\pi$
$138$		0			0
$139$	110668.			0.485831			0.242915	−	0.970047i	$-0.421896\pi$
$139$	110668.			0.485831			0.242915	+	0.970047i	$0.421896\pi$
$140$		0			0
$141$	156852.			0.664419
$142$		0			0
$143$	100406.	−	173908.i	0.410599	−	0.711179i
$144$		0			0
$145$	96027.5	+	166325.i	0.379294	+	0.656956i
$146$		0			0
$147$	139479.	+	58532.6i	0.532373	+	0.223411i
$148$		0			0
$149$	−163513.	−	283212.i	−0.603373	−	1.04507i	−0.992306	−	0.123807i	$-0.960490\pi$
$149$	−163513.	−	283212.i	−0.603373	−	1.04507i	0.388933	−	0.921266i	$-0.372844\pi$
$150$		0			0
$151$	87245.1	−	151113.i	0.311386	−	0.539336i	−0.667277	−	0.744810i	$-0.732540\pi$
$151$	87245.1	−	151113.i	0.311386	−	0.539336i	0.978663	+	0.205474i	$0.0658736\pi$
$152$		0			0
$153$	−125819.			−0.434527
$154$		0			0
$155$	−205195.			−0.686020
$156$		0			0
$157$	−220146.	+	381304.i	−0.712790	+	1.23459i	0.251016	+	0.967983i	$0.419235\pi$
$157$	−220146.	+	381304.i	−0.712790	+	1.23459i	−0.963806	+	0.266606i	$0.914098\pi$
$158$		0			0
$159$	−169428.	−	293458.i	−0.531486	−	0.920561i
$160$		0			0
$161$	−95935.8	−	19313.9i	−0.291686	−	0.0587227i
$162$		0			0
$163$	28246.0	+	48923.5i	0.0832698	+	0.144228i	0.904653	−	0.426150i	$-0.140130\pi$
$163$	28246.0	+	48923.5i	0.0832698	+	0.144228i	−0.821383	+	0.570377i	$0.806797\pi$
$164$		0			0
$165$	−37522.2	+	64990.3i	−0.107295	+	0.185840i
$166$		0			0
$167$	688014.			1.90900			0.954501	−	0.298209i	$-0.0963891\pi$
$167$	688014.			1.90900			0.954501	+	0.298209i	$0.0963891\pi$
$168$		0			0
$169$	221200.			0.595755
$170$		0			0
$171$	30377.0	−	52614.5i	0.0794427	−	0.137599i
$172$		0			0
$173$	−141279.	−	244703.i	−0.358891	−	0.621618i	0.628885	−	0.777499i	$-0.283512\pi$
$173$	−141279.	−	244703.i	−0.358891	−	0.621618i	−0.987776	+	0.155881i	$0.950178\pi$
$174$		0			0
$175$	87056.9	+	258426.i	0.214886	+	0.637884i
$176$		0			0
$177$	99351.4	+	172082.i	0.238364	+	0.412859i
$178$		0			0
$179$	75890.9	−	131447.i	0.177034	−	0.306632i	−0.763829	−	0.645419i	$-0.776683\pi$
$179$	75890.9	−	131447.i	0.177034	−	0.306632i	0.940863	+	0.338786i	$0.110016\pi$
$180$		0			0
$181$	−322258.			−0.731151			−0.365575	−	0.930782i	$-0.619128\pi$
$181$	−322258.			−0.731151			−0.365575	+	0.930782i	$0.619128\pi$
$182$		0			0
$183$	−73560.4			−0.162374
$184$		0			0
$185$	−76319.9	+	132190.i	−0.163949	+	0.283968i
$186$		0			0
$187$	−202618.	−	350944.i	−0.423714	−	0.733895i
$188$		0			0
$189$	−62478.5	+	70911.0i	−0.127226	+	0.144397i
$190$		0			0
$191$	269260.	+	466373.i	0.534059	+	0.925017i	0.999208	+	0.0397847i	$0.0126672\pi$
$191$	269260.	+	466373.i	0.534059	+	0.925017i	−0.465150	+	0.885232i	$0.653999\pi$
$192$		0			0
$193$	−151720.	+	262786.i	−0.293190	+	0.507820i	−0.974562	−	0.224117i	$-0.928050\pi$
$193$	−151720.	+	262786.i	−0.293190	+	0.507820i	0.681372	+	0.731937i	$0.261383\pi$
$194$		0			0
$195$	−221418.			−0.416990
$196$		0			0
$197$	656475.			1.20518			0.602591	−	0.798050i	$-0.294135\pi$
$197$	656475.			1.20518			0.602591	+	0.798050i	$0.294135\pi$
$198$		0			0
$199$	421102.	−	729370.i	0.753797	−	1.30562i	−0.192173	−	0.981361i	$-0.561553\pi$
$199$	421102.	−	729370.i	0.753797	−	1.30562i	0.945970	−	0.324254i	$-0.105113\pi$
$200$		0			0
$201$	58507.8	+	101339.i	0.102147	+	0.176923i
$202$		0			0
$203$	−514991.	+	584498.i	−0.877122	+	0.995505i
$204$		0			0
$205$	−86668.1	−	150114.i	−0.144037	−	0.249480i
$206$		0			0
$207$	30571.6	−	52951.6i	0.0495898	−	0.0858920i
$208$		0			0
$209$	195675.			0.309864
$210$		0			0
$211$	−1.01133e6			−1.56382			−0.781908	−	0.623394i	$-0.785753\pi$
$211$	−1.01133e6			−1.56382			−0.781908	+	0.623394i	$0.785753\pi$
$212$		0			0
$213$	−55573.3	+	96255.8i	−0.0839300	+	0.145371i
$214$		0			0
$215$	190114.	+	329287.i	0.280490	+	0.485824i
$216$		0			0
$217$	−265710.	−	788753.i	−0.383052	−	1.13708i
$218$		0			0
$219$	−196201.	−	339831.i	−0.276434	−	0.478798i
$220$		0			0
$221$	597822.	−	1.03546e6i	0.823363	−	1.42611i
$222$		0			0
$223$	−957124.			−1.28886			−0.644431	−	0.764663i	$-0.722906\pi$
$223$	−957124.			−1.28886			−0.644431	+	0.764663i	$0.722906\pi$
$224$		0			0
$225$	−170380.			−0.224369
$226$		0			0
$227$	235740.	−	408313.i	0.303647	−	0.525931i	−0.673312	−	0.739358i	$-0.735129\pi$
$227$	235740.	−	408313.i	0.303647	−	0.525931i	0.976959	+	0.213427i	$0.0684624\pi$
$228$		0			0
$229$	134098.	+	232265.i	0.168980	+	0.292681i	0.938061	−	0.346469i	$-0.112619\pi$
$229$	134098.	+	232265.i	0.168980	+	0.292681i	−0.769082	+	0.639150i	$0.779286\pi$
$230$		0			0
$231$	−298406.	−	60075.5i	−0.367940	−	0.0740742i
$232$		0			0
$233$	−773610.	−	1.33993e6i	−0.933538	−	1.61694i	−0.777220	−	0.629229i	$-0.783371\pi$
$233$	−773610.	−	1.33993e6i	−0.933538	−	1.61694i	−0.156318	−	0.987707i	$-0.549962\pi$
$234$		0			0
$235$	278513.	−	482399.i	0.328985	−	0.569819i
$236$		0			0
$237$	−690745.			−0.798817
$238$		0			0
$239$	937542.			1.06169			0.530843	−	0.847470i	$-0.321875\pi$
$239$	937542.			1.06169			0.530843	+	0.847470i	$0.321875\pi$
$240$		0			0
$241$	642430.	−	1.11272e6i	0.712497	−	1.23408i	−0.251420	−	0.967878i	$-0.580897\pi$
$241$	642430.	−	1.11272e6i	0.712497	−	1.23408i	0.963917	−	0.266203i	$-0.0857692\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	427683.	−	325036.i	0.455204	−	0.345952i
$246$		0			0
$247$	288670.	+	499990.i	0.301064	+	0.521458i
$248$		0			0
$249$	98521.1	−	170644.i	0.100700	−	0.174418i
$250$		0			0
$251$	−709769.			−0.711103			−0.355552	−	0.934657i	$-0.615707\pi$
$251$	−709769.			−0.711103			−0.355552	+	0.934657i	$0.615707\pi$
$252$		0			0
$253$	196929.			0.193423
$254$		0			0
$255$	−223409.	+	386956.i	−0.215155	+	0.372659i
$256$		0			0
$257$	910080.	+	1.57631e6i	0.859502	+	1.48870i	0.872405	+	0.488785i	$0.162560\pi$
$257$	910080.	+	1.57631e6i	0.859502	+	1.48870i	−0.0129024	+	0.999917i	$0.504107\pi$
$258$		0			0
$259$	−606956.	−	122193.i	−0.562222	−	0.113187i
$260$		0			0
$261$	−243362.	−	421515.i	−0.221132	−	0.383011i
$262$		0			0
$263$	179594.	−	311065.i	0.160104	−	0.277308i	−0.774802	−	0.632204i	$-0.782150\pi$
$263$	179594.	−	311065.i	0.160104	−	0.277308i	0.934906	+	0.354896i	$0.115484\pi$
$264$		0			0
$265$	−1.20337e6			−1.05265
$266$		0			0
$267$	−1.23270e6			−1.05823
$268$		0			0
$269$	−578834.	+	1.00257e6i	−0.487723	+	0.844761i	−0.999900	−	0.0141188i	$-0.995506\pi$
$269$	−578834.	+	1.00257e6i	−0.487723	+	0.844761i	0.512177	+	0.858880i	$0.328839\pi$
$270$		0			0
$271$	−390088.	−	675652.i	−0.322655	−	0.558856i	0.658380	−	0.752686i	$-0.271242\pi$
$271$	−390088.	−	675652.i	−0.322655	−	0.558856i	−0.981035	+	0.193830i	$0.937909\pi$
$272$		0			0
$273$	−286717.	−	851113.i	−0.232834	−	0.691163i
$274$		0			0
$275$	−274379.	−	475238.i	−0.218786	−	0.378948i
$276$		0			0
$277$	163165.	−	282611.i	0.127770	−	0.221304i	−0.795042	−	0.606554i	$-0.792551\pi$
$277$	163165.	−	282611.i	0.127770	−	0.221304i	0.922812	+	0.385250i	$0.125885\pi$
$278$		0			0
$279$	520023.			0.399956
$280$		0			0
$281$	364094.			0.275073			0.137536	−	0.990497i	$-0.456082\pi$
$281$	364094.			0.275073			0.137536	+	0.990497i	$0.456082\pi$
$282$		0			0
$283$	−32009.6	+	55442.3i	−0.0237582	+	0.0411505i	−0.877660	−	0.479284i	$-0.840897\pi$
$283$	−32009.6	+	55442.3i	−0.0237582	+	0.0411505i	0.853902	+	0.520434i	$0.174230\pi$
$284$		0			0
$285$	−107877.	−	186849.i	−0.0786717	−	0.136263i
$286$		0			0
$287$	464797.	−	527530.i	0.333088	−	0.378044i
$288$		0			0
$289$	−496470.	−	859911.i	−0.349662	−	0.605632i
$290$		0			0
$291$	418725.	−	725254.i	0.289866	−	0.502062i
$292$		0			0
$293$	398020.			0.270855			0.135427	−	0.990787i	$-0.456759\pi$
$293$	398020.			0.270855			0.135427	+	0.990787i	$0.456759\pi$
$294$		0			0
$295$	705651.			0.472101
$296$		0			0
$297$	95092.1	−	164704.i	0.0625537	−	0.108346i
$298$		0			0
$299$	290519.	+	503194.i	0.187930	+	0.325505i
$300$		0			0
$301$	−1.01957e6	+	1.15718e6i	−0.648638	+	0.736183i
$302$		0			0
$303$	−550357.	−	953247.i	−0.344380	−	0.596484i
$304$		0			0
$305$	−130617.	+	226236.i	−0.0803990	+	0.139255i
$306$		0			0
$307$	638841.			0.386854			0.193427	−	0.981115i	$-0.438040\pi$
$307$	638841.			0.386854			0.193427	+	0.981115i	$0.438040\pi$
$308$		0			0
$309$	−673212.			−0.401103
$310$		0			0
$311$	1.24094e6	−	2.14937e6i	0.727529	−	1.26012i	−0.230395	−	0.973097i	$-0.574002\pi$
$311$	1.24094e6	−	2.14937e6i	0.727529	−	1.26012i	0.957924	−	0.287020i	$-0.0926648\pi$
$312$		0			0
$313$	−738267.	−	1.27872e6i	−0.425944	−	0.737757i	0.570564	−	0.821253i	$-0.306725\pi$
$313$	−738267.	−	1.27872e6i	−0.425944	−	0.737757i	−0.996508	+	0.0834961i	$0.973391\pi$
$314$		0			0
$315$	107148.	+	318066.i	0.0608424	+	0.180609i
$316$		0			0
$317$	1.11563e6	+	1.93233e6i	0.623551	+	1.08002i	0.988819	+	0.149119i	$0.0476438\pi$
$317$	1.11563e6	+	1.93233e6i	0.623551	+	1.08002i	−0.365269	+	0.930902i	$0.619023\pi$
$318$		0			0
$319$	783816.	−	1.35761e6i	0.431258	−	0.746961i
$320$		0			0
$321$	−636952.			−0.345020
$322$		0			0
$323$	1.16506e6			0.621360
$324$		0			0
$325$	809552.	−	1.40219e6i	0.425145	−	0.736372i
$326$		0			0
$327$	640179.	+	1.10882e6i	0.331079	+	0.573446i
$328$		0			0
$329$	2.21496e6	+	445918.i	1.12817	+	0.227125i
$330$		0			0
$331$	−1.20561e6	−	2.08818e6i	−0.604836	−	1.04761i	−0.992077	−	0.125628i	$-0.959905\pi$
$331$	−1.20561e6	−	2.08818e6i	−0.604836	−	1.04761i	0.387242	−	0.921978i	$-0.373428\pi$
$332$		0			0
$333$	193417.	−	335008.i	0.0955837	−	0.165556i
$334$		0			0
$335$	415556.			0.202310
$336$		0			0
$337$	−1.79586e6			−0.861388			−0.430694	−	0.902498i	$-0.641731\pi$
$337$	−1.79586e6			−0.861388			−0.430694	+	0.902498i	$0.641731\pi$
$338$		0			0
$339$	−895045.	+	1.55026e6i	−0.423005	+	0.732667i
$340$		0			0
$341$	837442.	+	1.45049e6i	0.390004	+	0.675506i
$342$		0			0
$343$	1.80323e6	+	1.22309e6i	0.827589	+	0.561335i
$344$		0			0
$345$	−108569.	−	188046.i	−0.0491085	−	0.0850583i
$346$		0			0
$347$	1.10366e6	−	1.91159e6i	0.492051	−	0.852258i	−0.507907	−	0.861412i	$-0.669581\pi$
$347$	1.10366e6	−	1.91159e6i	0.492051	−	0.852258i	0.999958	+	0.00915399i	$0.00291385\pi$
$348$		0			0
$349$	148188.			0.0651252			0.0325626	−	0.999470i	$-0.489633\pi$
$349$	148188.			0.0651252			0.0325626	+	0.999470i	$0.489633\pi$
$350$		0			0
$351$	561137.			0.243109
$352$		0			0
$353$	−985200.	+	1.70642e6i	−0.420811	+	0.728867i	−0.996019	−	0.0891407i	$-0.971588\pi$
$353$	−985200.	+	1.70642e6i	−0.420811	+	0.728867i	0.575208	+	0.818007i	$0.304921\pi$
$354$		0			0
$355$	197357.	+	341832.i	0.0831154	+	0.143960i
$356$		0			0
$357$	−1.77673e6	−	357693.i	−0.737819	−	0.148539i
$358$		0			0
$359$	1.21697e6	+	2.10786e6i	0.498362	+	0.863188i	0.999998	−	0.00189067i	$-0.000601819\pi$
$359$	1.21697e6	+	2.10786e6i	0.498362	+	0.863188i	−0.501636	+	0.865079i	$0.667268\pi$
$360$		0			0
$361$	956763.	−	1.65716e6i	0.386399	−	0.669263i
$362$		0			0
$363$	−836917.			−0.333362
$364$		0			0
$365$	−1.39354e6			−0.547502
$366$		0			0
$367$	−2.15174e6	+	3.72692e6i	−0.833920	+	1.44439i	0.0609873	+	0.998139i	$0.480575\pi$
$367$	−2.15174e6	+	3.72692e6i	−0.833920	+	1.44439i	−0.894907	+	0.446253i	$0.852758\pi$
$368$		0			0
$369$	219642.	+	380432.i	0.0839750	+	0.145449i
$370$		0			0
$371$	−1.55827e6	−	4.62568e6i	−0.587769	−	1.74478i
$372$		0			0
$373$	601526.	+	1.04187e6i	0.223863	+	0.387742i	0.955978	−	0.293439i	$-0.0947999\pi$
$373$	601526.	+	1.04187e6i	0.223863	+	0.387742i	−0.732115	+	0.681181i	$0.761467\pi$
$374$		0			0
$375$	−751994.	+	1.30249e6i	−0.276144	+	0.478296i
$376$		0			0
$377$	4.62529e6			1.67604
$378$		0			0
$379$	−5.11488e6			−1.82910			−0.914550	−	0.404473i	$-0.867455\pi$
$379$	−5.11488e6			−1.82910			−0.914550	+	0.404473i	$0.867455\pi$
$380$		0			0
$381$	988286.	−	1.71176e6i	0.348795	−	0.604131i
$382$		0			0
$383$	2.50528e6	+	4.33926e6i	0.872687	+	1.51154i	0.859206	+	0.511629i	$0.170958\pi$
$383$	2.50528e6	+	4.33926e6i	0.872687	+	1.51154i	0.0134808	+	0.999909i	$0.495709\pi$
$384$		0			0
$385$	−714625.	+	811076.i	−0.245712	+	0.278875i
$386$		0			0
$387$	−481804.	−	834510.i	−0.163528	−	0.283240i
$388$		0			0
$389$	2.04163e6	−	3.53620e6i	0.684073	−	1.18485i	−0.289654	−	0.957131i	$-0.593540\pi$
$389$	2.04163e6	−	3.53620e6i	0.684073	−	1.18485i	0.973727	−	0.227718i	$-0.0731265\pi$
$390$		0			0
$391$	1.17253e6			0.387866
$392$		0			0
$393$	−491599.			−0.160557
$394$		0			0
$395$	−1.22652e6	+	2.12439e6i	−0.395532	+	0.685081i
$396$		0			0
$397$	536498.	+	929242.i	0.170841	+	0.295905i	0.938714	−	0.344697i	$-0.112018\pi$
$397$	536498.	+	929242.i	0.170841	+	0.295905i	−0.767873	+	0.640602i	$0.778685\pi$
$398$		0			0
$399$	578542.	−	656626.i	0.181929	−	0.206484i
$400$		0			0
$401$	949741.	+	1.64500e6i	0.294947	+	0.510863i	0.974973	−	0.222325i	$-0.0713647\pi$
$401$	949741.	+	1.64500e6i	0.294947	+	0.510863i	−0.680026	+	0.733188i	$0.738031\pi$
$402$		0			0
$403$	−2.47087e6	+	4.27967e6i	−0.757856	+	1.31264i
$404$		0			0
$405$	−209700.			−0.0635274
$406$		0			0
$407$	1.24591e6			0.372821
$408$		0			0
$409$	−967887.	+	1.67643e6i	−0.286099	+	0.495538i	−0.972875	−	0.231331i	$-0.925692\pi$
$409$	−967887.	+	1.67643e6i	−0.286099	+	0.495538i	0.686776	+	0.726869i	$0.259025\pi$
$410$		0			0
$411$	−328243.	−	568533.i	−0.0958497	−	0.166017i
$412$		0			0
$413$	913757.	+	2.71247e6i	0.263606	+	0.782509i
$414$		0			0
$415$	−349877.	−	606004.i	−0.0997229	−	0.172725i
$416$		0			0
$417$	−498006.	+	862572.i	−0.140247	+	0.242915i
$418$		0			0
$419$	−3.93448e6			−1.09484			−0.547422	−	0.836856i	$-0.684391\pi$
$419$	−3.93448e6			−1.09484			−0.547422	+	0.836856i	$0.684391\pi$
$420$		0			0
$421$	−5.16927e6			−1.42142			−0.710712	−	0.703483i	$-0.751627\pi$
$421$	−5.16927e6			−1.42142			−0.710712	+	0.703483i	$0.751627\pi$
$422$		0			0
$423$	−705834.	+	1.22254e6i	−0.191801	+	0.332210i
$424$		0			0
$425$	−1.63367e6	−	2.82960e6i	−0.438724	−	0.759893i
$426$		0			0
$427$	−1.03877e6	−	209127.i	−0.275708	−	0.0555060i
$428$		0			0
$429$	903651.	+	1.56517e6i	0.237060	+	0.410599i
$430$		0			0
$431$	−1.06475e6	+	1.84421e6i	−0.276093	+	0.478208i	−0.970410	−	0.241462i	$-0.922373\pi$
$431$	−1.06475e6	+	1.84421e6i	−0.276093	+	0.478208i	0.694317	+	0.719669i	$0.255707\pi$
$432$		0			0
$433$	3.20879e6			0.822473			0.411237	−	0.911529i	$-0.365097\pi$
$433$	3.20879e6			0.822473			0.411237	+	0.911529i	$0.365097\pi$
$434$		0			0
$435$	−1.72849e6			−0.437971
$436$		0			0
$437$	−283089.	+	490324.i	−0.0709119	+	0.122823i
$438$		0			0
$439$	3.49538e6	+	6.05418e6i	0.865632	+	1.49932i	0.866418	+	0.499320i	$0.166417\pi$
$439$	3.49538e6	+	6.05418e6i	0.865632	+	1.49932i	−0.000785585		1.00000i	$0.500250\pi$
$440$		0			0
$441$	−1.08387e6	+	823735.i	−0.265388	+	0.201693i
$442$		0			0
$443$	3.73004e6	+	6.46062e6i	0.903034	+	1.56410i	0.823534	+	0.567266i	$0.191999\pi$
$443$	3.73004e6	+	6.46062e6i	0.903034	+	1.56410i	0.0795000	+	0.996835i	$0.474668\pi$
$444$		0			0
$445$	−2.18884e6	+	3.79118e6i	−0.523979	+	0.907559i
$446$		0			0
$447$	2.94323e6			0.696715
$448$		0			0
$449$	−5.61620e6			−1.31470			−0.657350	−	0.753586i	$-0.728323\pi$
$449$	−5.61620e6			−1.31470			−0.657350	+	0.753586i	$0.728323\pi$
$450$		0			0
$451$	−707421.	+	1.22529e6i	−0.163771	+	0.283659i
$452$		0			0
$453$	785206.	+	1.36002e6i	0.179779	+	0.311386i
$454$		0			0
$455$	−3.12671e6	−	629474.i	−0.708042	−	0.142544i
$456$		0			0
$457$	4.40636e6	+	7.63203e6i	0.986936	+	1.70942i	0.632994	+	0.774156i	$0.281826\pi$
$457$	4.40636e6	+	7.63203e6i	0.986936	+	1.70942i	0.353942	+	0.935267i	$0.384841\pi$
$458$		0			0
$459$	566184.	−	980660.i	0.125437	−	0.217264i
$460$		0			0
$461$	3.24339e6			0.710798			0.355399	−	0.934715i	$-0.384345\pi$
$461$	3.24339e6			0.710798			0.355399	+	0.934715i	$0.384345\pi$
$462$		0			0
$463$	5.70039e6			1.23581			0.617906	−	0.786252i	$-0.287981\pi$
$463$	5.70039e6			1.23581			0.617906	+	0.786252i	$0.287981\pi$
$464$		0			0
$465$	923376.	−	1.59933e6i	0.198037	−	0.343010i
$466$		0			0
$467$	1.09007e6	+	1.88805e6i	0.231292	+	0.400610i	0.958189	−	0.286137i	$-0.0923712\pi$
$467$	1.09007e6	+	1.88805e6i	0.231292	+	0.400610i	−0.726896	+	0.686747i	$0.759038\pi$
$468$		0			0
$469$	538110.	+	1.59737e6i	0.112964	+	0.335330i
$470$		0			0
$471$	−1.98131e6	−	3.43174e6i	−0.411530	−	0.712790i
$472$		0			0
$473$	1.55179e6	−	2.68778e6i	0.318918	−	0.552383i
$474$		0			0
$475$	1.57770e6			0.320840
$476$		0			0
$477$	3.04970e6			0.613707
$478$		0			0
$479$	1.96808e6	−	3.40881e6i	0.391926	−	0.678835i	−0.600778	−	0.799416i	$-0.705142\pi$
$479$	1.96808e6	−	3.40881e6i	0.391926	−	0.678835i	0.992704	+	0.120581i	$0.0384757\pi$
$480$		0			0
$481$	1.83802e6	+	3.18355e6i	0.362233	+	0.627406i
$482$		0			0
$483$	582248.	−	660833.i	0.113564	−	0.128891i
$484$		0			0
$485$	−1.48701e6	−	2.57558e6i	−0.287052	−	0.497189i
$486$		0			0
$487$	3.50088e6	−	6.06370e6i	0.668890	−	1.15855i	−0.309325	−	0.950956i	$-0.600103\pi$
$487$	3.50088e6	−	6.06370e6i	0.668890	−	1.15855i	0.978215	−	0.207595i	$-0.0665636\pi$
$488$		0			0
$489$	−508428.			−0.0961517
$490$		0			0
$491$	9.37114e6			1.75424			0.877119	−	0.480273i	$-0.159463\pi$
$491$	9.37114e6			1.75424			0.877119	+	0.480273i	$0.159463\pi$
$492$		0			0
$493$	4.66689e6	−	8.08329e6i	0.864789	−	1.49786i
$494$		0			0
$495$	−337699.	−	584912.i	−0.0619466	−	0.107295i
$496$		0			0
$497$	−1.05842e6	+	1.20127e6i	−0.192205	+	0.218147i
$498$		0			0
$499$	−4.87966e6	−	8.45182e6i	−0.877280	−	1.51949i	−0.854314	−	0.519758i	$-0.826022\pi$
$499$	−4.87966e6	−	8.45182e6i	−0.877280	−	1.51949i	−0.0229668	−	0.999736i	$-0.507311\pi$
$500$		0			0
$501$	−3.09606e6	+	5.36254e6i	−0.551081	+	0.954501i
$502$		0			0
$503$	1.69667e6			0.299004			0.149502	−	0.988761i	$-0.452233\pi$
$503$	1.69667e6			0.299004			0.149502	+	0.988761i	$0.452233\pi$
$504$		0			0
$505$	−3.90895e6			−0.682075
$506$		0			0
$507$	−995398.	+	1.72408e6i	−0.171980	+	0.297877i
$508$		0			0
$509$	1.86783e6	+	3.23518e6i	0.319553	+	0.553482i	0.980395	−	0.197043i	$-0.0631339\pi$
$509$	1.86783e6	+	3.23518e6i	0.319553	+	0.553482i	−0.660842	+	0.750525i	$0.729801\pi$
$510$		0			0
$511$	−1.80451e6	−	5.35665e6i	−0.305708	−	0.907488i
$512$		0			0
$513$	273393.	+	473530.i	0.0458663	+	0.0794427i
$514$		0			0
$515$	−1.19538e6	+	2.07047e6i	−0.198605	+	0.343993i
$516$		0			0
$517$	−4.54668e6			−0.748114
$518$		0			0
$519$	2.54302e6			0.414412
$520$		0			0
$521$	−3.66359e6	+	6.34552e6i	−0.591306	+	1.02417i	0.402750	+	0.915310i	$0.368054\pi$
$521$	−3.66359e6	+	6.34552e6i	−0.591306	+	1.02417i	−0.994057	+	0.108863i	$0.965279\pi$
$522$		0			0
$523$	−4795.71	−	8306.42i	−0.000766653	−	0.00132788i	0.865642	−	0.500664i	$-0.166911\pi$
$523$	−4795.71	−	8306.42i	−0.000766653	−	0.00132788i	−0.866408	+	0.499336i	$0.833577\pi$
$524$		0			0
$525$	−2.40599e6	−	484377.i	−0.380974	−	0.0766983i
$526$		0			0
$527$	4.98618e6	+	8.63632e6i	0.782063	+	1.35457i
$528$		0			0
$529$	2.93327e6	−	5.08057e6i	0.455735	−	0.789357i
$530$		0			0
$531$	−1.78832e6			−0.275239
$532$		0			0
$533$	−4.17448e6			−0.636479
$534$		0			0
$535$	−1.13100e6	+	1.95895e6i	−0.170836	+	0.295896i
$536$		0			0
$537$	683018.	+	1.18302e6i	0.102211	+	0.177034i
$538$		0			0
$539$	−4.04309e6	−	1.69669e6i	−0.599434	−	0.251554i
$540$		0			0
$541$	1.87846e6	+	3.25359e6i	0.275936	+	0.477935i	0.970371	−	0.241620i	$-0.0776788\pi$
$541$	1.87846e6	+	3.25359e6i	0.275936	+	0.477935i	−0.694435	+	0.719556i	$0.744346\pi$
$542$		0			0
$543$	1.45016e6	−	2.51175e6i	0.211065	−	0.365575i
$544$		0			0
$545$	4.54692e6			0.655731
$546$		0			0
$547$	−1.03391e7			−1.47746			−0.738729	−	0.674002i	$-0.764574\pi$
$547$	−1.03391e7			−1.47746			−0.738729	+	0.674002i	$0.764574\pi$
$548$		0			0
$549$	331022.	−	573347.i	0.0468734	−	0.0811870i
$550$		0			0
$551$	2.25350e6	+	3.90317e6i	0.316212	+	0.547694i
$552$		0			0
$553$	−9.75424e6	−	1.96374e6i	−1.35638	−	0.273068i
$554$		0			0
$555$	−686879.	−	1.18971e6i	−0.0946560	−	0.163949i
$556$		0			0
$557$	5.02378e6	−	8.70144e6i	0.686108	−	1.18837i	−0.286979	−	0.957937i	$-0.592651\pi$
$557$	5.02378e6	−	8.70144e6i	0.686108	−	1.18837i	0.973087	−	0.230437i	$-0.0740156\pi$
$558$		0			0
$559$	9.15708e6			1.23945
$560$		0			0
$561$	3.64712e6			0.489263
$562$		0			0
$563$	−4.24690e6	+	7.35585e6i	−0.564678	+	0.978052i	0.432401	+	0.901681i	$0.357666\pi$
$563$	−4.24690e6	+	7.35585e6i	−0.564678	+	0.978052i	−0.997080	+	0.0763704i	$0.975667\pi$
$564$		0			0
$565$	3.17856e6	+	5.50543e6i	0.418899	+	0.725555i
$566$		0			0
$567$	−271544.	−	806071.i	−0.0354717	−	0.105297i
$568$		0			0
$569$	−1.90765e6	−	3.30415e6i	−0.247012	−	0.427838i	0.715683	−	0.698425i	$-0.246115\pi$
$569$	−1.90765e6	−	3.30415e6i	−0.247012	−	0.427838i	−0.962695	+	0.270587i	$0.912782\pi$
$570$		0			0
$571$	6.10664e6	−	1.05770e7i	0.783812	−	1.35760i	−0.145894	−	0.989300i	$-0.546606\pi$
$571$	6.10664e6	−	1.05770e7i	0.783812	−	1.35760i	0.929706	−	0.368302i	$-0.120061\pi$
$572$		0			0
$573$	−4.84669e6			−0.616678
$574$		0			0
$575$	1.58780e6			0.200275
$576$		0			0
$577$	1.14735e6	−	1.98727e6i	0.143469	−	0.248495i	−0.785332	−	0.619075i	$-0.787508\pi$
$577$	1.14735e6	−	1.98727e6i	0.143469	−	0.248495i	0.928801	+	0.370580i	$0.120841\pi$
$578$		0			0
$579$	−1.36548e6	−	2.36508e6i	−0.169273	−	0.293190i
$580$		0			0
$581$	1.87637e6	−	2.12962e6i	0.230611	−	0.261735i
$582$		0			0
$583$	4.91122e6	+	8.50648e6i	0.598436	+	1.03652i
$584$		0			0
$585$	996380.	−	1.72578e6i	0.120375	−	0.208495i
$586$		0			0
$587$	−7.53391e6			−0.902454			−0.451227	−	0.892409i	$-0.649014\pi$
$587$	−7.53391e6			−0.902454			−0.451227	+	0.892409i	$0.649014\pi$
$588$		0			0
$589$	−4.81534e6			−0.571925
$590$		0			0
$591$	−2.95414e6	+	5.11671e6i	−0.347906	+	0.602591i
$592$		0			0
$593$	−3.61760e6	−	6.26588e6i	−0.422459	−	0.731720i	0.573721	−	0.819051i	$-0.305500\pi$
$593$	−3.61760e6	−	6.26588e6i	−0.422459	−	0.731720i	−0.996179	+	0.0873310i	$0.972166\pi$
$594$		0			0
$595$	−4.25492e6	+	4.82920e6i	−0.492719	+	0.559220i
$596$		0			0
$597$	3.78992e6	+	6.56433e6i	0.435205	+	0.753797i
$598$		0			0
$599$	1.66928e6	−	2.89128e6i	0.190091	−	0.329247i	−0.755189	−	0.655507i	$-0.772455\pi$
$599$	1.66928e6	−	2.89128e6i	0.190091	−	0.329247i	0.945280	+	0.326259i	$0.105788\pi$
$600$		0			0
$601$	1.48297e7			1.67474			0.837370	−	0.546636i	$-0.184092\pi$
$601$	1.48297e7			1.67474			0.837370	+	0.546636i	$0.184092\pi$
$602$		0			0
$603$	−1.05314e6			−0.117949
$604$		0			0
$605$	−1.48607e6	+	2.57394e6i	−0.165063	+	0.285897i
$606$		0			0
$607$	−3.18737e6	−	5.52068e6i	−0.351124	−	0.608164i	0.635323	−	0.772247i	$-0.280867\pi$
$607$	−3.18737e6	−	5.52068e6i	−0.351124	−	0.608164i	−0.986447	+	0.164083i	$0.947534\pi$
$608$		0			0
$609$	−2.23825e6	−	6.64420e6i	−0.244549	−	0.725938i
$610$		0			0
$611$	−6.70747e6	−	1.16177e7i	−0.726869	−	1.25897i
$612$		0			0
$613$	7.74344e6	−	1.34120e7i	0.832306	−	1.44160i	−0.0638996	−	0.997956i	$-0.520354\pi$
$613$	7.74344e6	−	1.34120e7i	0.832306	−	1.44160i	0.896205	−	0.443639i	$-0.146313\pi$
$614$		0			0
$615$	1.56003e6			0.166320
$616$		0			0
$617$	−4.55324e6			−0.481513			−0.240756	−	0.970586i	$-0.577396\pi$
$617$	−4.55324e6			−0.481513			−0.240756	+	0.970586i	$0.577396\pi$
$618$		0			0
$619$	−5.82108e6	+	1.00824e7i	−0.610628	+	1.05764i	0.380507	+	0.924778i	$0.375750\pi$
$619$	−5.82108e6	+	1.00824e7i	−0.610628	+	1.05764i	−0.991135	+	0.132861i	$0.957584\pi$
$620$		0			0
$621$	275144.	+	476564.i	0.0286307	+	0.0495898i
$622$		0			0
$623$	−1.74074e7	−	3.50448e6i	−1.79686	−	0.361746i
$624$		0			0
$625$	−616103.	−	1.06712e6i	−0.0630890	−	0.109273i
$626$		0			0
$627$	−880539.	+	1.52514e6i	−0.0894499	+	0.154932i
$628$		0			0
$629$	7.41822e6			0.747607
$630$		0			0
$631$	−1.07436e7			−1.07417			−0.537087	−	0.843527i	$-0.680475\pi$
$631$	−1.07436e7			−1.07417			−0.537087	+	0.843527i	$0.680475\pi$
$632$		0			0
$633$	4.55097e6	−	7.88252e6i	0.451435	−	0.781908i
$634$		0			0
$635$	−3.50969e6	−	6.07896e6i	−0.345409	−	0.598267i
$636$		0			0
$637$	−1.62917e6	−	1.28339e7i	−0.159081	−	1.25318i
$638$		0			0
$639$	−500160.	−	866302.i	−0.0484570	−	0.0839300i
$640$		0			0
$641$	−6.15689e6	+	1.06641e7i	−0.591857	+	1.02513i	0.402126	+	0.915585i	$0.368272\pi$
$641$	−6.15689e6	+	1.06641e7i	−0.591857	+	1.02513i	−0.993982	+	0.109541i	$0.965062\pi$
$642$		0			0
$643$	1.67944e7			1.60191			0.800955	−	0.598724i	$-0.204325\pi$
$643$	1.67944e7			1.60191			0.800955	+	0.598724i	$0.204325\pi$
$644$		0			0
$645$	−3.42205e6			−0.323882
$646$		0			0
$647$	9.95539e6	−	1.72432e7i	0.934970	−	1.61942i	0.160283	−	0.987071i	$-0.448759\pi$
$647$	9.95539e6	−	1.72432e7i	0.934970	−	1.61942i	0.774687	−	0.632345i	$-0.217907\pi$
$648$		0			0
$649$	−2.87990e6	−	4.98814e6i	−0.268390	−	0.464865i
$650$		0			0
$651$	7.34341e6	+	1.47839e6i	0.679118	+	0.136721i
$652$		0			0
$653$	−1.05495e7	−	1.82723e7i	−0.968163	−	1.67691i	−0.700864	−	0.713295i	$-0.747202\pi$
$653$	−1.05495e7	−	1.82723e7i	−0.968163	−	1.67691i	−0.267299	−	0.963614i	$-0.586131\pi$
$654$		0			0
$655$	−872905.	+	1.51192e6i	−0.0794994	+	0.137697i
$656$		0			0
$657$	3.53163e6			0.319199
$658$		0			0
$659$	−1.75070e7			−1.57035			−0.785177	−	0.619271i	$-0.787428\pi$
$659$	−1.75070e7			−1.57035			−0.785177	+	0.619271i	$0.787428\pi$
$660$		0			0
$661$	−9.00844e6	+	1.56031e7i	−0.801947	+	1.38901i	0.116385	+	0.993204i	$0.462869\pi$
$661$	−9.00844e6	+	1.56031e7i	−0.801947	+	1.38901i	−0.918333	+	0.395810i	$0.870464\pi$
$662$		0			0
$663$	5.38040e6	+	9.31912e6i	0.475369	+	0.823363i
$664$		0			0
$665$	−992173.	−	2.94524e6i	−0.0870028	−	0.258266i
$666$		0			0
$667$	2.26793e6	+	3.92818e6i	0.197386	+	0.341882i
$668$		0			0
$669$	4.30706e6	−	7.46004e6i	0.372062	−	0.644431i
$670$		0			0
$671$	2.13230e6			0.182828
$672$		0			0
$673$	1.48593e7			1.26462			0.632312	−	0.774714i	$-0.282106\pi$
$673$	1.48593e7			1.26462			0.632312	+	0.774714i	$0.282106\pi$
$674$		0			0
$675$	766710.	−	1.32798e6i	0.0647697	−	0.112184i
$676$		0			0
$677$	−3.59160e6	−	6.22084e6i	−0.301173	−	0.521648i	0.675229	−	0.737609i	$-0.264045\pi$
$677$	−3.59160e6	−	6.22084e6i	−0.301173	−	0.521648i	−0.976402	+	0.215961i	$0.930712\pi$
$678$		0			0
$679$	7.97480e6	−	9.05113e6i	0.663812	−	0.753405i
$680$		0			0
$681$	2.12166e6	+	3.67482e6i	0.175310	+	0.303647i
$682$		0			0
$683$	825461.	−	1.42974e6i	0.0677087	−	0.117275i	−0.830184	−	0.557490i	$-0.811764\pi$
$683$	825461.	−	1.42974e6i	0.0677087	−	0.117275i	0.897892	+	0.440215i	$0.145098\pi$
$684$		0			0
$685$	−2.33137e6			−0.189839
$686$		0			0
$687$	−2.41377e6			−0.195121
$688$		0			0
$689$	−1.44905e7	+	2.50983e7i	−1.16288	+	2.01417i
$690$		0			0
$691$	388722.	+	673286.i	0.0309702	+	0.0536419i	0.881095	−	0.472939i	$-0.156807\pi$
$691$	388722.	+	673286.i	0.0309702	+	0.0536419i	−0.850125	+	0.526581i	$0.823474\pi$
$692$		0			0
$693$	1.81107e6	−	2.05550e6i	0.143252	−	0.162587i
$694$		0			0
$695$	1.76856e6	+	3.06324e6i	0.138886	+	0.240558i
$696$		0			0
$697$	−4.21203e6	+	7.29545e6i	−0.328405	+	0.568814i
$698$		0			0
$699$	1.39250e7			1.07796
$700$		0			0
$701$	1.65671e6			0.127336			0.0636681	−	0.997971i	$-0.479720\pi$
$701$	1.65671e6			0.127336			0.0636681	+	0.997971i	$0.479720\pi$
$702$		0			0
$703$	−1.79101e6	+	3.10213e6i	−0.136682	+	0.236740i
$704$		0			0
$705$	2.50662e6	+	4.34159e6i	0.189940	+	0.328985i
$706$		0			0
$707$	−5.06176e6	−	1.50257e7i	−0.380849	−	1.13054i
$708$		0			0
$709$	−2.49950e6	−	4.32927e6i	−0.186740	−	0.323444i	0.757421	−	0.652927i	$-0.226459\pi$
$709$	−2.49950e6	−	4.32927e6i	−0.186740	−	0.323444i	−0.944162	+	0.329483i	$0.893126\pi$
$710$		0			0
$711$	3.10835e6	−	5.38383e6i	0.230599	−	0.399408i
$712$		0			0
$713$	−4.84619e6			−0.357007
$714$		0			0
$715$	6.41825e6			0.469517
$716$		0			0
$717$	−4.21894e6	+	7.30742e6i	−0.306482	+	0.530843i
$718$		0			0
$719$	−9.06918e6	−	1.57083e7i	−0.654253	−	1.13320i	−0.982081	−	0.188462i	$-0.939650\pi$
$719$	−9.06918e6	−	1.57083e7i	−0.654253	−	1.13320i	0.327828	−	0.944738i	$-0.393683\pi$
$720$		0			0
$721$	−9.50664e6	−	1.91389e6i	−0.681065	−	0.137113i
$722$		0			0
$723$	5.78187e6	+	1.00145e7i	0.411360	+	0.712497i
$724$		0			0
$725$	6.31976e6	−	1.09461e7i	0.446535	−	0.773422i
$726$		0			0
$727$	−1.17918e7			−0.827454			−0.413727	−	0.910401i	$-0.635773\pi$
$727$	−1.17918e7			−0.827454			−0.413727	+	0.910401i	$0.635773\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	9.23945e6	−	1.60032e7i	0.639518	−	1.10768i
$732$		0			0
$733$	2.48235e6	+	4.29955e6i	0.170648	+	0.295572i	0.938647	−	0.344880i	$-0.112080\pi$
$733$	2.48235e6	+	4.29955e6i	0.170648	+	0.295572i	−0.767998	+	0.640452i	$0.778747\pi$
$734$		0			0
$735$	608831.	+	4.79612e6i	0.0415698	+	0.327470i
$736$		0			0
$737$	−1.69597e6	−	2.93751e6i	−0.115014	−	0.199210i
$738$		0			0
$739$	9.31461e6	−	1.61334e7i	0.627413	−	1.08671i	−0.360656	−	0.932699i	$-0.617447\pi$
$739$	9.31461e6	−	1.61334e7i	0.627413	−	1.08671i	0.988069	−	0.154013i	$-0.0492196\pi$
$740$		0			0
$741$	−5.19605e6			−0.347639
$742$		0			0
$743$	−1.77727e7			−1.18109			−0.590544	−	0.807006i	$-0.701087\pi$
$743$	−1.77727e7			−1.18109			−0.590544	+	0.807006i	$0.701087\pi$
$744$		0			0
$745$	5.22613e6	−	9.05192e6i	0.344976	−	0.597516i
$746$		0			0
$747$	886690.	+	1.53579e6i	0.0581394	+	0.100700i
$748$		0			0
$749$	−8.99460e6	−	1.81081e6i	−0.585838	−	0.117942i
$750$		0			0
$751$	2.33030e6	+	4.03619e6i	0.150769	+	0.261139i	0.931510	−	0.363715i	$-0.118492\pi$
$751$	2.33030e6	+	4.03619e6i	0.150769	+	0.261139i	−0.780742	+	0.624854i	$0.785158\pi$
$752$		0			0
$753$	3.19396e6	−	5.53210e6i	0.205278	−	0.355552i
$754$		0			0
$755$	5.57699e6			0.356067
$756$		0			0
$757$	−2.36520e6			−0.150013			−0.0750064	−	0.997183i	$-0.523898\pi$
$757$	−2.36520e6			−0.150013			−0.0750064	+	0.997183i	$0.523898\pi$
$758$		0			0
$759$	−886181.	+	1.53491e6i	−0.0558365	+	0.0967116i
$760$		0			0
$761$	8.84728e6	+	1.53239e7i	0.553794	+	0.959199i	0.997996	+	0.0632726i	$0.0201538\pi$
$761$	8.84728e6	+	1.53239e7i	0.553794	+	0.959199i	−0.444202	+	0.895926i	$0.646513\pi$
$762$		0			0
$763$	5.88787e6	+	1.74780e7i	0.366140	+	1.08688i
$764$		0			0
$765$	−2.01068e6	−	3.48261e6i	−0.124220	−	0.215155i
$766$		0			0
$767$	8.49714e6	−	1.47175e7i	0.521536	−	0.903327i
$768$		0			0
$769$	−7.90324e6			−0.481936			−0.240968	−	0.970533i	$-0.577465\pi$
$769$	−7.90324e6			−0.481936			−0.240968	+	0.970533i	$0.577465\pi$
$770$		0			0
$771$	−1.63814e7			−0.992468
$772$		0			0
$773$	−1.11230e7	+	1.92656e7i	−0.669534	+	1.15967i	0.308501	+	0.951224i	$0.400173\pi$
$773$	−1.11230e7	+	1.92656e7i	−0.669534	+	1.15967i	−0.978035	+	0.208442i	$0.933161\pi$
$774$		0			0
$775$	6.75214e6	+	1.16950e7i	0.403819	+	0.699436i
$776$		0			0
$777$	3.68370e6	−	4.18088e6i	0.218893	−	0.248437i
$778$		0			0
$779$	−2.03386e6	−	3.52274e6i	−0.120082	−	0.207988i
$780$		0			0
$781$	1.61091e6	−	2.79017e6i	0.0945024	−	0.163683i
$782$		0			0
$783$	4.38051e6			0.255341
$784$		0			0
$785$	−1.40724e7			−0.815070
$786$		0			0
$787$	1.24949e7	−	2.16417e7i	0.719108	−	1.24553i	−0.242245	−	0.970215i	$-0.577884\pi$
$787$	1.24949e7	−	2.16417e7i	0.719108	−	1.24553i	0.961353	−	0.275317i	$-0.0887828\pi$
$788$		0			0
$789$	1.61634e6	+	2.79959e6i	0.0924360	+	0.160104i
$790$		0			0
$791$	−1.70465e7	+	1.93472e7i	−0.968711	+	1.09945i
$792$		0			0
$793$	3.14567e6	+	5.44846e6i	0.177636	+	0.307674i
$794$		0			0
$795$	5.41518e6	−	9.37937e6i	0.303875	−	0.526327i
$796$		0			0
$797$	1.40897e7			0.785696			0.392848	−	0.919603i	$-0.371490\pi$
$797$	1.40897e7			0.785696			0.392848	+	0.919603i	$0.371490\pi$
$798$		0			0
$799$	−2.70712e7			−1.50017
$800$		0			0
$801$	5.54716e6	−	9.60797e6i	0.305485	−	0.529115i
$802$		0			0
$803$	5.68730e6	+	9.85070e6i	0.311256	+	0.539111i
$804$		0			0
$805$	−998530.	−	2.96411e6i	−0.0543089	−	0.161215i
$806$		0			0
$807$	−5.20950e6	−	9.02313e6i	−0.281587	−	0.487723i
$808$		0			0
$809$	1.27153e7	−	2.20235e7i	0.683053	−	1.18308i	−0.290992	−	0.956726i	$-0.593985\pi$
$809$	1.27153e7	−	2.20235e7i	0.683053	−	1.18308i	0.974045	−	0.226356i	$-0.0726814\pi$
$810$		0			0
$811$	−1.97155e6			−0.105258			−0.0526291	−	0.998614i	$-0.516760\pi$
$811$	−1.97155e6			−0.105258			−0.0526291	+	0.998614i	$0.516760\pi$
$812$		0			0
$813$	7.02158e6			0.372570
$814$		0			0
$815$	−902786.	+	1.56367e6i	−0.0476092	+	0.0824616i
$816$		0			0
$817$	4.46144e6	+	7.72744e6i	0.233841	+	0.405024i
$818$		0			0
$819$	7.92399e6	+	1.59527e6i	0.412795	+	0.0831045i
$820$		0			0
$821$	557255.	+	965194.i	0.0288534	+	0.0499755i	0.880092	−	0.474804i	$-0.157481\pi$
$821$	557255.	+	965194.i	0.0288534	+	0.0499755i	−0.851238	+	0.524780i	$0.824148\pi$
$822$		0			0
$823$	1.39919e6	−	2.42347e6i	0.0720073	−	0.124720i	−0.827774	−	0.561062i	$-0.810393\pi$
$823$	1.39919e6	−	2.42347e6i	0.0720073	−	0.124720i	0.899781	+	0.436342i	$0.143726\pi$
$824$		0			0
$825$	4.93882e6			0.252632
$826$		0			0
$827$	−1.56547e6			−0.0795940			−0.0397970	−	0.999208i	$-0.512671\pi$
$827$	−1.56547e6			−0.0795940			−0.0397970	+	0.999208i	$0.512671\pi$
$828$		0			0
$829$	6.68603e6	−	1.15806e7i	0.337895	−	0.585252i	−0.646141	−	0.763218i	$-0.723618\pi$
$829$	6.68603e6	−	1.15806e7i	0.337895	−	0.585252i	0.984037	+	0.177966i	$0.0569517\pi$
$830$		0			0
$831$	1.46849e6	+	2.54350e6i	0.0737680	+	0.127770i
$832$		0			0
$833$	−2.40728e7	−	1.01022e7i	−1.20203	−	0.504433i
$834$		0			0
$835$	1.09950e7	+	1.90439e7i	0.545732	+	0.945236i
$836$		0			0
$837$	−2.34010e6	+	4.05318e6i	−0.115457	+	0.199978i
$838$		0			0
$839$	−4.37187e6			−0.214418			−0.107209	−	0.994236i	$-0.534191\pi$
$839$	−4.37187e6			−0.214418			−0.107209	+	0.994236i	$0.534191\pi$
$840$		0			0
$841$	1.55961e7			0.760372
$842$		0			0
$843$	−1.63842e6	+	2.83783e6i	−0.0794067	+	0.137536i
$844$		0			0
$845$	3.53494e6	+	6.12270e6i	0.170310	+	0.294986i
$846$		0			0
$847$	−1.18184e7	−	2.37929e6i	−0.566042	−	0.113956i
$848$		0			0
$849$	−288086.	−	498980.i	−0.0137168	−	0.0237582i
$850$		0			0
$851$	−1.80249e6	+	3.12200e6i	−0.0853196	+	0.147778i
$852$		0			0
$853$	−1.45910e7			−0.686612			−0.343306	−	0.939224i	$-0.611547\pi$
$853$	−1.45910e7			−0.686612			−0.343306	+	0.939224i	$0.611547\pi$
$854$		0			0
$855$	1.94179e6			0.0908422
$856$		0			0
$857$	6.56378e6	−	1.13688e7i	0.305283	−	0.528765i	−0.672042	−	0.740513i	$-0.734582\pi$
$857$	6.56378e6	−	1.13688e7i	0.305283	−	0.528765i	0.977324	+	0.211749i	$0.0679158\pi$
$858$		0			0
$859$	7.10653e6	+	1.23089e7i	0.328605	+	0.569161i	0.982235	−	0.187653i	$-0.0600881\pi$
$859$	7.10653e6	+	1.23089e7i	0.328605	+	0.569161i	−0.653630	+	0.756814i	$0.726755\pi$
$860$		0			0
$861$	2.02010e6	+	5.99662e6i	0.0928678	+	0.275676i
$862$		0			0
$863$	1.33085e7	+	2.30510e7i	0.608278	+	1.05357i	0.991524	+	0.129922i	$0.0414726\pi$
$863$	1.33085e7	+	2.30510e7i	0.608278	+	1.05357i	−0.383247	+	0.923646i	$0.625194\pi$
$864$		0			0
$865$	4.51551e6	−	7.82109e6i	0.205195	−	0.355408i
$866$		0			0
$867$	8.93646e6			0.403755
$868$		0			0
$869$	2.00227e7			0.899441
$870$		0			0
$871$	5.00395e6	−	8.66710e6i	0.223495	−	0.387104i
$872$		0			0
$873$	3.76853e6	+	6.52728e6i	0.167354	+	0.289866i
$874$		0			0
$875$	−1.43220e7	+	1.62550e7i	−0.632389	+	0.717741i
$876$		0			0
$877$	7.11916e6	+	1.23307e7i	0.312557	+	0.541365i	0.978915	−	0.204267i	$-0.0654811\pi$
$877$	7.11916e6	+	1.23307e7i	0.312557	+	0.541365i	−0.666358	+	0.745632i	$0.732148\pi$
$878$		0			0
$879$	−1.79109e6	+	3.10226e6i	−0.0781890	+	0.135427i
$880$		0			0
$881$	−1.51921e7			−0.659445			−0.329722	−	0.944078i	$-0.606955\pi$
$881$	−1.51921e7			−0.659445			−0.329722	+	0.944078i	$0.606955\pi$
$882$		0			0
$883$	−1.03997e7			−0.448868			−0.224434	−	0.974489i	$-0.572053\pi$
$883$	−1.03997e7			−0.448868			−0.224434	+	0.974489i	$0.572053\pi$
$884$		0			0
$885$	−3.17543e6	+	5.50000e6i	−0.136284	+	0.236050i
$886$		0			0
$887$	1.01064e7	+	1.75048e7i	0.431309	+	0.747049i	0.996986	−	0.0775776i	$-0.0247185\pi$
$887$	1.01064e7	+	1.75048e7i	0.431309	+	0.747049i	−0.565677	+	0.824627i	$0.691385\pi$
$888$		0			0
$889$	1.88223e7	−	2.13627e7i	0.798764	−	0.906571i
$890$		0			0
$891$	855829.	+	1.48234e6i	0.0361154	+	0.0625537i
$892$		0			0
$893$	6.53592e6	−	1.13206e7i	0.274270	−	0.475050i
$894$		0			0
$895$	4.85119e6			0.202437
$896$		0			0
$897$	−5.22934e6			−0.217003
$898$		0			0
$899$	−1.92888e7	+	3.34092e7i	−0.795987	+	1.37869i
$900$		0			0
$901$	2.92417e7	+	5.06481e7i	1.20003	+	2.07851i
$902$		0			0
$903$	−4.43126e6	−	1.31541e7i	−0.180846	−	0.536836i
$904$		0			0
$905$	−5.14994e6	−	8.91995e6i	−0.209016	−	0.362027i
$906$		0			0
$907$	−2.17821e6	+	3.77278e6i	−0.0879189	+	0.152280i	−0.906631	−	0.421924i	$-0.861355\pi$
$907$	−2.17821e6	+	3.77278e6i	−0.0879189	+	0.152280i	0.818712	+	0.574204i	$0.194688\pi$
$908$		0			0
$909$	9.90643e6			0.397656
$910$		0			0
$911$	−1.99500e7			−0.796427			−0.398214	−	0.917293i	$-0.630370\pi$
$911$	−1.99500e7			−0.796427			−0.398214	+	0.917293i	$0.630370\pi$
$912$		0			0
$913$	−2.85584e6	+	4.94646e6i	−0.113385	+	0.196389i
$914$		0			0
$915$	−1.17555e6	−	2.03612e6i	−0.0464184	−	0.0803990i
$916$		0			0
$917$	−6.94203e6	−	1.39758e6i	−0.272623	−	0.0548849i
$918$		0			0
$919$	−4.77586e6	−	8.27204e6i	−0.186536	−	0.323090i	0.757557	−	0.652769i	$-0.226393\pi$
$919$	−4.77586e6	−	8.27204e6i	−0.186536	−	0.323090i	−0.944093	+	0.329679i	$0.893059\pi$
$920$		0			0
$921$	−2.87478e6	+	4.97927e6i	−0.111675	+	0.193427i
$922$		0			0
$923$	9.50594e6			0.367275
$924$		0			0
$925$	1.00455e7			0.386028
$926$		0			0
$927$	3.02945e6	−	5.24717e6i	0.115788	−	0.200551i
$928$		0			0
$929$	6.30124e6	+	1.09141e7i	0.239545	+	0.414904i	0.960584	−	0.277991i	$-0.0896685\pi$
$929$	6.30124e6	+	1.09141e7i	0.239545	+	0.414904i	−0.721039	+	0.692895i	$0.756335\pi$
$930$		0			0
$931$	1.00365e7	−	7.62768e6i	0.379497	−	0.288415i
$932$		0			0
$933$	1.11685e7	+	1.93444e7i	0.420039	+	0.727529i
$934$		0			0
$935$	6.47598e6	−	1.12167e7i	0.242257	−	0.419602i
$936$		0			0
$937$	1.00216e7			0.372897			0.186448	−	0.982465i	$-0.440302\pi$
$937$	1.00216e7			0.372897			0.186448	+	0.982465i	$0.440302\pi$
$938$		0			0
$939$	1.32888e7			0.491838
$940$		0			0
$941$	770565.	−	1.33466e6i	0.0283684	−	0.0491356i	−0.851493	−	0.524367i	$-0.824302\pi$
$941$	770565.	−	1.33466e6i	0.0283684	−	0.0491356i	0.879861	+	0.475231i	$0.157635\pi$
$942$		0			0
$943$	−2.04689e6	−	3.54531e6i	−0.0749574	−	0.129830i
$944$		0			0
$945$	−2.96124e6	−	596161.i	−0.107868	−	0.0217162i
$946$		0			0
$947$	1.53523e7	+	2.65910e7i	0.556287	+	0.963517i	0.997802	+	0.0662633i	$0.0211077\pi$
$947$	1.53523e7	+	2.65910e7i	0.556287	+	0.963517i	−0.441515	+	0.897254i	$0.645559\pi$
$948$		0			0
$949$	−1.67804e7	+	2.90644e7i	−0.604833	+	1.04760i
$950$		0			0
$951$	−2.00813e7			−0.720014
$952$		0			0
$953$	−4.39769e7			−1.56853			−0.784263	−	0.620428i	$-0.786959\pi$
$953$	−4.39769e7			−1.56853			−0.784263	+	0.620428i	$0.786959\pi$
$954$		0			0
$955$	−8.60599e6	+	1.49060e7i	−0.305346	+	0.528875i
$956$		0			0
$957$	7.05434e6	+	1.22185e7i	0.248987	+	0.431258i
$958$		0			0
$959$	−3.01892e6	−	8.96161e6i	−0.106000	−	0.314658i
$960$		0			0
$961$	−6.29389e6	−	1.09013e7i	−0.219842	−	0.380777i
$962$		0			0
$963$	2.86628e6	−	4.96455e6i	0.0995987	−	0.172510i
$964$		0			0
$965$	−9.69841e6			−0.335261
$966$		0			0
$967$	4.20732e7			1.44690			0.723452	−	0.690375i	$-0.242555\pi$
$967$	4.20732e7			1.44690			0.723452	+	0.690375i	$0.242555\pi$
$968$		0			0
$969$	−5.24279e6	+	9.08078e6i	−0.179371	+	0.310680i
$970$		0			0
$971$	1.04003e7	+	1.80139e7i	0.353996	+	0.613139i	0.986946	−	0.161054i	$-0.0514893\pi$
$971$	1.04003e7	+	1.80139e7i	0.353996	+	0.613139i	−0.632950	+	0.774193i	$0.718156\pi$
$972$		0			0
$973$	−9.48473e6	+	1.07649e7i	−0.321176	+	0.364524i
$974$		0			0
$975$	7.28597e6	+	1.26197e7i	0.245457	+	0.425145i
$976$		0			0
$977$	−1.87169e7	+	3.24186e7i	−0.627332	+	1.08657i	0.360753	+	0.932661i	$0.382520\pi$
$977$	−1.87169e7	+	3.24186e7i	−0.627332	+	1.08657i	−0.988085	+	0.153909i	$0.950814\pi$
$978$		0			0
$979$	3.57324e7			1.19153
$980$		0			0
$981$	−1.15232e7			−0.382297
$982$		0			0
$983$	−1.20981e7	+	2.09546e7i	−0.399332	+	0.691663i	−0.993644	−	0.112571i	$-0.964091\pi$
$983$	−1.20981e7	+	2.09546e7i	−0.399332	+	0.691663i	0.594312	+	0.804235i	$0.297425\pi$
$984$		0			0
$985$	1.04910e7	+	1.81709e7i	0.344529	+	0.596742i
$986$		0			0
$987$	−1.34429e7	+	1.52572e7i	−0.439238	+	0.498521i
$988$		0			0
$989$	4.49002e6	+	7.77695e6i	0.145968	+	0.252824i
$990$		0			0
$991$	−1.26040e7	+	2.18308e7i	−0.407684	+	0.706130i	−0.994630	−	0.103496i	$-0.966997\pi$
$991$	−1.26040e7	+	2.18308e7i	−0.407684	+	0.706130i	0.586945	+	0.809627i	$0.300330\pi$
$992$		0			0
$993$	2.17010e7			0.698404
$994$		0			0
$995$	2.69182e7			0.861962
$996$		0			0
$997$	−2.47755e7	+	4.29125e7i	−0.789379	+	1.36724i	0.136969	+	0.990575i	$0.456264\pi$
$997$	−2.47755e7	+	4.29125e7i	−0.789379	+	1.36724i	−0.926348	+	0.376669i	$0.877070\pi$
$998$		0			0
$999$	1.74075e6	+	3.01507e6i	0.0551853	+	0.0955837i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.q.i.193.3		8
4.3	odd	2		84.6.i.c.25.3	✓	8
7.2	even	3	inner	336.6.q.i.289.3		8
12.11	even	2		252.6.k.f.109.2		8
28.3	even	6		588.6.a.p.1.3		4
28.11	odd	6		588.6.a.n.1.2		4
28.19	even	6		588.6.i.o.373.2		8
28.23	odd	6		84.6.i.c.37.3	yes	8
28.27	even	2		588.6.i.o.361.2		8
84.23	even	6		252.6.k.f.37.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.3	✓	8	4.3	odd	2
84.6.i.c.37.3	yes	8	28.23	odd	6
252.6.k.f.37.2		8	84.23	even	6
252.6.k.f.109.2		8	12.11	even	2
336.6.q.i.193.3		8	1.1	even	1	trivial
336.6.q.i.289.3		8	7.2	even	3	inner
588.6.a.n.1.2		4	28.11	odd	6
588.6.a.p.1.3		4	28.3	even	6
588.6.i.o.361.2		8	28.27	even	2
588.6.i.o.373.2		8	28.19	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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} +(15.9808 + 27.6796i) q^{5} +(-85.7043 + 97.2716i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(130.442 - 225.932i) q^{11} +769.735 q^{13} -287.654 q^{15} +(776.659 - 1345.21i) q^{17} +(375.024 + 649.561i) q^{19} +(-372.488 - 1105.72i) q^{21} +(377.427 + 653.723i) q^{23} +(1051.73 - 1821.65i) q^{25} +729.000 q^{27} +6008.93 q^{29} +(-3210.02 + 5559.92i) q^{31} +(1173.98 + 2033.39i) q^{33} +(-4062.06 - 817.779i) q^{35} +(2387.86 + 4135.90i) q^{37} +(-3463.81 + 5999.49i) q^{39} -5423.27 q^{41} +11896.4 q^{43} +(1294.44 - 2242.04i) q^{45} +(-8714.00 - 15093.1i) q^{47} +(-2116.54 - 16673.2i) q^{49} +(6989.93 + 12106.9i) q^{51} +(-18825.3 + 32606.4i) q^{53} +8338.26 q^{55} -6750.44 q^{57} +(11039.0 - 19120.2i) q^{59} +(4086.69 + 7078.36i) q^{61} +(10294.4 + 2072.49i) q^{63} +(12301.0 + 21305.9i) q^{65} +(6500.87 - 11259.8i) q^{67} -6793.69 q^{69} +12349.6 q^{71} +(-21800.2 + 37759.0i) q^{73} +(9465.55 + 16394.8i) q^{75} +(10797.3 + 32051.6i) q^{77} +(38374.7 + 66467.0i) q^{79} +(-3280.50 + 5681.99i) q^{81} -21893.6 q^{83} +49646.5 q^{85} +(-27040.2 + 46835.0i) q^{87} +(68483.5 + 118617. i) q^{89} +(-65969.7 + 74873.4i) q^{91} +(-28890.2 - 50039.3i) q^{93} +(-11986.4 + 20761.0i) q^{95} -93050.1 q^{97} -21131.6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 36 q^{3} + 42 q^{7} - 324 q^{9} + 462 q^{11} - 1204 q^{13} + 228 q^{17} - 358 q^{19} + 1404 q^{21} + 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} + 4158 q^{33} - 7692 q^{35}+ \cdots - 74844 q^{99}+O(q^{100}) \) 8 * q - 36 * q^3 + 42 * q^7 - 324 * q^9 + 462 * q^11 - 1204 * q^13 + 228 * q^17 - 358 * q^19 + 1404 * q^21 + 2148 * q^23 - 5454 * q^25 + 5832 * q^27 - 11064 * q^29 - 830 * q^31 + 4158 * q^33 - 7692 * q^35 - 3914 * q^37 + 5418 * q^39 - 16632 * q^41 + 29036 * q^43 - 41700 * q^47 + 41876 * q^49 + 2052 * q^51 + 22164 * q^53 - 7784 * q^55 + 6444 * q^57 - 32886 * q^59 + 83732 * q^61 - 16038 * q^63 - 93192 * q^65 + 80034 * q^67 - 38664 * q^69 - 89544 * q^71 - 22470 * q^73 - 49086 * q^75 + 138732 * q^77 + 75286 * q^79 - 26244 * q^81 + 34836 * q^83 + 278504 * q^85 + 49788 * q^87 + 28944 * q^89 + 12058 * q^91 - 7470 * q^93 + 144120 * q^95 - 433356 * q^97 - 74844 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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