LMFDB - Embedded newform 112.6.i.d.65.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,6,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, 6); 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, 6, names="a")

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	112.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,14] 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:	$17.9629878191$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{79})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 79x^{2} + 6241 $ x^4 + 79*x^2 + 6241
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
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			$4.44410 + 7.69740i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.6.i.d.81.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(12.3882 + 21.4570i) q^{3} +(18.0528 - 31.2683i) q^{5} +(124.435 + 36.3731i) q^{7} +(-185.435 + 321.182i) q^{9} +(77.7174 + 134.610i) q^{11} +1158.87 q^{13} +894.565 q^{15} +(-619.040 - 1072.21i) q^{17} +(-140.140 + 242.729i) q^{19} +(761.065 + 3120.59i) q^{21} +(-1741.20 + 3015.84i) q^{23} +(910.694 + 1577.37i) q^{25} -3168.14 q^{27} -5656.78 q^{29} +(-1157.35 - 2004.59i) q^{31} +(-1925.56 + 3335.16i) q^{33} +(3383.72 - 3234.23i) q^{35} +(1166.59 - 2020.59i) q^{37} +(14356.3 + 24865.8i) q^{39} +3812.61 q^{41} -3925.73 q^{43} +(6695.22 + 11596.5i) q^{45} +(-5558.23 + 9627.13i) q^{47} +(14161.0 + 9052.14i) q^{49} +(15337.6 - 26565.5i) q^{51} +(5593.10 + 9687.54i) q^{53} +5612.06 q^{55} -6944.31 q^{57} +(-3005.18 - 5205.12i) q^{59} +(7419.36 - 12850.7i) q^{61} +(-34756.9 + 33221.4i) q^{63} +(20920.8 - 36235.9i) q^{65} +(-21491.9 - 37225.1i) q^{67} -86281.1 q^{69} +19962.4 q^{71} +(-22775.3 - 39448.0i) q^{73} +(-22563.7 + 39081.5i) q^{75} +(4774.54 + 19577.0i) q^{77} +(54496.0 - 94389.9i) q^{79} +(5813.07 + 10068.5i) q^{81} -55829.0 q^{83} -44701.6 q^{85} +(-70077.3 - 121377. i) q^{87} +(47772.9 - 82745.1i) q^{89} +(144204. + 42151.6i) q^{91} +(28675.0 - 49666.5i) q^{93} +(5059.82 + 8763.86i) q^{95} +15004.9 q^{97} -57646.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{3} - 70 q^{5} - 244 q^{9} + 62 q^{11} + 3640 q^{13} + 4076 q^{15} - 1694 q^{17} + 826 q^{19} + 3542 q^{21} - 2734 q^{23} - 6312 q^{25} - 14308 q^{27} - 5704 q^{29} - 2674 q^{31} - 4858 q^{33}+ \cdots - 139000 q^{99}+O(q^{100}) $ 4 * q + 14 * q^3 - 70 * q^5 - 244 * q^9 + 62 * q^11 + 3640 * q^13 + 4076 * q^15 - 1694 * q^17 + 826 * q^19 + 3542 * q^21 - 2734 * q^23 - 6312 * q^25 - 14308 * q^27 - 5704 * q^29 - 2674 * q^31 - 4858 * q^33 + 13286 * q^35 + 9146 * q^37 + 21588 * q^39 + 12264 * q^41 + 32080 * q^43 + 26852 * q^45 - 25326 * q^47 + 56644 * q^49 + 25762 * q^51 - 14958 * q^53 + 31052 * q^55 - 37732 * q^57 - 1106 * q^59 + 28042 * q^61 - 77308 * q^63 - 28308 * q^65 - 102642 * q^67 - 188692 * q^69 + 22112 * q^71 - 35070 * q^73 - 132776 * q^75 + 27062 * q^77 + 101762 * q^79 + 23750 * q^81 + 89264 * q^83 + 7348 * q^85 - 170380 * q^87 + 75474 * q^89 + 123872 * q^91 + 53478 * q^93 + 127502 * q^95 - 16632 * q^97 - 139000 * 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
$3$	12.3882	+	21.4570i	0.794703	+	1.37647i	0.923028	+	0.384734i	$0.125707\pi$
$3$	12.3882	+	21.4570i	0.794703	+	1.37647i	−0.128325	+	0.991732i	$0.540960\pi$
$4$		0			0
$5$	18.0528	−	31.2683i	0.322938	−	0.559345i	−0.658155	−	0.752883i	$-0.728663\pi$
$5$	18.0528	−	31.2683i	0.322938	−	0.559345i	0.981093	+	0.193538i	$0.0619962\pi$
$6$		0			0
$7$	124.435	+	36.3731i	0.959835	+	0.280566i
$7$	124.435	+	36.3731i	0.959835	+	0.280566i
$8$		0			0
$9$	−185.435	+	321.182i	−0.763106	+	1.32174i
$10$		0			0
$11$	77.7174	+	134.610i	0.193658	+	0.335426i	0.946460	−	0.322821i	$-0.104631\pi$
$11$	77.7174	+	134.610i	0.193658	+	0.335426i	−0.752801	+	0.658248i	$0.771298\pi$
$12$		0			0
$13$	1158.87			1.90185			0.950925	−	0.309422i	$-0.100136\pi$
$13$	1158.87			1.90185			0.950925	+	0.309422i	$0.100136\pi$
$14$		0			0
$15$	894.565			1.02656
$16$		0			0
$17$	−619.040	−	1072.21i	−0.519513	−	0.899823i	−0.999743	−	0.0226804i	$-0.992780\pi$
$17$	−619.040	−	1072.21i	−0.519513	−	0.899823i	0.480230	−	0.877143i	$-0.340553\pi$
$18$		0			0
$19$	−140.140	+	242.729i	−0.0890588	+	0.154254i	−0.907114	−	0.420886i	$-0.861719\pi$
$19$	−140.140	+	242.729i	−0.0890588	+	0.154254i	0.818055	+	0.575140i	$0.195053\pi$
$20$		0			0
$21$	761.065	+	3120.59i	0.376594	+	1.54415i
$22$		0			0
$23$	−1741.20	+	3015.84i	−0.686322	+	1.18874i	0.286698	+	0.958021i	$0.407443\pi$
$23$	−1741.20	+	3015.84i	−0.686322	+	1.18874i	−0.973019	+	0.230723i	$0.925891\pi$
$24$		0			0
$25$	910.694	+	1577.37i	0.291422	+	0.504758i
$26$		0			0
$27$	−3168.14			−0.836364
$28$		0			0
$29$	−5656.78			−1.24903			−0.624517	−	0.781011i	$-0.714704\pi$
$29$	−5656.78			−1.24903			−0.624517	+	0.781011i	$0.714704\pi$
$30$		0			0
$31$	−1157.35	−	2004.59i	−0.216302	−	0.374646i	0.737372	−	0.675486i	$-0.236066\pi$
$31$	−1157.35	−	2004.59i	−0.216302	−	0.374646i	−0.953675	+	0.300840i	$0.902733\pi$
$32$		0			0
$33$	−1925.56	+	3335.16i	−0.307802	+	0.533128i
$34$		0			0
$35$	3383.72	−	3234.23i	0.466900	−	0.446273i
$36$		0			0
$37$	1166.59	−	2020.59i	0.140092	−	0.242646i	−0.787439	−	0.616392i	$-0.788594\pi$
$37$	1166.59	−	2020.59i	0.140092	−	0.242646i	0.927531	+	0.373746i	$0.121927\pi$
$38$		0			0
$39$	14356.3	+	24865.8i	1.51141	+	2.61783i
$40$		0			0
$41$	3812.61			0.354211			0.177106	−	0.984192i	$-0.443327\pi$
$41$	3812.61			0.354211			0.177106	+	0.984192i	$0.443327\pi$
$42$		0			0
$43$	−3925.73			−0.323780			−0.161890	−	0.986809i	$-0.551759\pi$
$43$	−3925.73			−0.323780			−0.161890	+	0.986809i	$0.551759\pi$
$44$		0			0
$45$	6695.22	+	11596.5i	0.492872	+	0.853679i
$46$		0			0
$47$	−5558.23	+	9627.13i	−0.367022	+	0.635700i	−0.989098	−	0.147256i	$-0.952956\pi$
$47$	−5558.23	+	9627.13i	−0.367022	+	0.635700i	0.622077	+	0.782956i	$0.286289\pi$
$48$		0			0
$49$	14161.0	+	9052.14i	0.842566	+	0.538594i
$50$		0			0
$51$	15337.6	−	26565.5i	0.825717	−	1.43018i
$52$		0			0
$53$	5593.10	+	9687.54i	0.273504	+	0.473722i	0.969757	−	0.244074i	$-0.0784841\pi$
$53$	5593.10	+	9687.54i	0.273504	+	0.473722i	−0.696253	+	0.717797i	$0.745151\pi$
$54$		0			0
$55$	5612.06			0.250159
$56$		0			0
$57$	−6944.31			−0.283101
$58$		0			0
$59$	−3005.18	−	5205.12i	−0.112393	−	0.194671i	0.804342	−	0.594167i	$-0.202518\pi$
$59$	−3005.18	−	5205.12i	−0.112393	−	0.194671i	−0.916735	+	0.399497i	$0.869185\pi$
$60$		0			0
$61$	7419.36	−	12850.7i	0.255295	−	0.442183i	−0.709681	−	0.704523i	$-0.751161\pi$
$61$	7419.36	−	12850.7i	0.255295	−	0.442183i	0.964975	+	0.262340i	$0.0844942\pi$
$62$		0			0
$63$	−34756.9	+	33221.4i	−1.10329	+	1.05455i
$64$		0			0
$65$	20920.8	−	36235.9i	0.614179	−	1.06379i
$66$		0			0
$67$	−21491.9	−	37225.1i	−0.584909	−	1.01309i	−0.994887	−	0.100997i	$-0.967797\pi$
$67$	−21491.9	−	37225.1i	−0.584909	−	1.01309i	0.409977	−	0.912096i	$-0.365537\pi$
$68$		0			0
$69$	−86281.1			−2.18169
$70$		0			0
$71$	19962.4			0.469967			0.234984	−	0.971999i	$-0.424496\pi$
$71$	19962.4			0.469967			0.234984	+	0.971999i	$0.424496\pi$
$72$		0			0
$73$	−22775.3	−	39448.0i	−0.500215	−	0.866398i	−1.00000		0.000248463i	$-0.999921\pi$
$73$	−22775.3	−	39448.0i	−0.500215	−	0.866398i	0.499785	−	0.866150i	$-0.333412\pi$
$74$		0			0
$75$	−22563.7	+	39081.5i	−0.463188	+	0.802266i
$76$		0			0
$77$	4774.54	+	19577.0i	0.0917709	+	0.376288i
$78$		0			0
$79$	54496.0	−	94389.9i	0.982419	−	1.70160i	0.329534	−	0.944144i	$-0.393108\pi$
$79$	54496.0	−	94389.9i	0.982419	−	1.70160i	0.652885	−	0.757457i	$-0.273558\pi$
$80$		0			0
$81$	5813.07	+	10068.5i	0.0984448	+	0.170511i
$82$		0			0
$83$	−55829.0			−0.889538			−0.444769	−	0.895645i	$-0.646714\pi$
$83$	−55829.0			−0.889538			−0.444769	+	0.895645i	$0.646714\pi$
$84$		0			0
$85$	−44701.6			−0.671082
$86$		0			0
$87$	−70077.3	−	121377.i	−0.992611	−	1.71925i
$88$		0			0
$89$	47772.9	−	82745.1i	0.639303	−	1.10731i	−0.346283	−	0.938130i	$-0.612556\pi$
$89$	47772.9	−	82745.1i	0.639303	−	1.10731i	0.985586	−	0.169175i	$-0.0541103\pi$
$90$		0			0
$91$	144204.	+	42151.6i	1.82546	+	0.533594i
$92$		0			0
$93$	28675.0	−	49666.5i	0.343792	−	0.595465i
$94$		0			0
$95$	5059.82	+	8763.86i	0.0575209	+	0.0996292i
$96$		0			0
$97$	15004.9			0.161922			0.0809609	−	0.996717i	$-0.474201\pi$
$97$	15004.9			0.161922			0.0809609	+	0.996717i	$0.474201\pi$
$98$		0			0
$99$	−57646.0			−0.591127
$100$		0			0
$101$	−13438.1	−	23275.4i	−0.131079	−	0.227036i	0.793014	−	0.609204i	$-0.208511\pi$
$101$	−13438.1	−	23275.4i	−0.131079	−	0.227036i	−0.924093	+	0.382168i	$0.875178\pi$
$102$		0			0
$103$	−25957.6	+	44959.9i	−0.241086	+	0.417573i	−0.961024	−	0.276466i	$-0.910837\pi$
$103$	−25957.6	+	44959.9i	−0.241086	+	0.417573i	0.719938	+	0.694038i	$0.244170\pi$
$104$		0			0
$105$	111315.	+	32538.1i	0.985327	+	0.288017i
$106$		0			0
$107$	−12487.4	+	21628.8i	−0.105442	+	0.182630i	−0.913919	−	0.405898i	$-0.866959\pi$
$107$	−12487.4	+	21628.8i	−0.105442	+	0.182630i	0.808477	+	0.588528i	$0.200292\pi$
$108$		0			0
$109$	1818.07	+	3148.99i	0.0146570	+	0.0253867i	0.873261	−	0.487253i	$-0.162001\pi$
$109$	1818.07	+	3148.99i	0.0146570	+	0.0253867i	−0.858604	+	0.512640i	$0.828668\pi$
$110$		0			0
$111$	57807.7			0.445326
$112$		0			0
$113$	62175.0			0.458057			0.229028	−	0.973420i	$-0.426445\pi$
$113$	62175.0			0.458057			0.229028	+	0.973420i	$0.426445\pi$
$114$		0			0
$115$	62866.8	+	108889.i	0.443279	+	0.767781i
$116$		0			0
$117$	−214895.	+	372208.i	−1.45131	+	2.51375i
$118$		0			0
$119$	−38030.6	−	155936.i	−0.246187	−	1.00944i
$120$		0			0
$121$	68445.5	−	118551.i	0.424993	−	0.736109i
$122$		0			0
$123$	47231.3	+	81807.1i	0.281493	+	0.487560i
$124$		0			0
$125$	178592.			1.02232
$126$		0			0
$127$	−63550.3			−0.349630			−0.174815	−	0.984601i	$-0.555933\pi$
$127$	−63550.3			−0.349630			−0.174815	+	0.984601i	$0.555933\pi$
$128$		0			0
$129$	−48632.7	−	84234.4i	−0.257309	−	0.445672i
$130$		0			0
$131$	68148.5	−	118037.i	0.346959	−	0.600950i	−0.638749	−	0.769415i	$-0.720548\pi$
$131$	68148.5	−	118037.i	0.346959	−	0.600950i	0.985708	+	0.168465i	$0.0538810\pi$
$132$		0			0
$133$	−26267.0	+	25106.6i	−0.128760	+	0.123072i
$134$		0			0
$135$	−57193.8	+	99062.5i	−0.270094	+	0.467816i
$136$		0			0
$137$	−167652.	−	290381.i	−0.763144	−	1.32180i	−0.941222	−	0.337788i	$-0.890321\pi$
$137$	−167652.	−	290381.i	−0.763144	−	1.32180i	0.178078	−	0.984016i	$-0.443012\pi$
$138$		0			0
$139$	−58195.2			−0.255476			−0.127738	−	0.991808i	$-0.540772\pi$
$139$	−58195.2			−0.255476			−0.127738	+	0.991808i	$0.540772\pi$
$140$		0			0
$141$	−275426.			−1.16669
$142$		0			0
$143$	90064.3	+	155996.i	0.368309	+	0.637930i
$144$		0			0
$145$	−102121.	+	176878.i	−0.403360	+	0.698641i
$146$		0			0
$147$	−18802.5	+	415992.i	−0.0717665	+	1.58779i
$148$		0			0
$149$	−141529.	+	245135.i	−0.522251	+	0.904566i	0.477414	+	0.878679i	$0.341574\pi$
$149$	−141529.	+	245135.i	−0.522251	+	0.904566i	−0.999665	+	0.0258870i	$0.991759\pi$
$150$		0			0
$151$	−120779.	−	209195.i	−0.431071	−	0.746637i	0.565895	−	0.824477i	$-0.308531\pi$
$151$	−120779.	−	209195.i	−0.431071	−	0.746637i	−0.996966	+	0.0778407i	$0.975197\pi$
$152$		0			0
$153$	459166.			1.58577
$154$		0			0
$155$	−83573.6			−0.279409
$156$		0			0
$157$	107014.	+	185354.i	0.346492	+	0.600142i	0.985624	−	0.168956i	$-0.0540394\pi$
$157$	107014.	+	185354.i	0.346492	+	0.600142i	−0.639132	+	0.769097i	$0.720706\pi$
$158$		0			0
$159$	−138577.	+	240022.i	−0.434708	+	0.752937i
$160$		0			0
$161$	−326360.	+	311942.i	−0.992277	+	0.948440i
$162$		0			0
$163$	32233.0	−	55829.2i	0.0950237	−	0.164586i	−0.814595	−	0.580030i	$-0.803041\pi$
$163$	32233.0	−	55829.2i	0.0950237	−	0.164586i	0.909619	+	0.415445i	$0.136374\pi$
$164$		0			0
$165$	69523.3	+	120418.i	0.198802	+	0.344335i
$166$		0			0
$167$	442694.			1.22832			0.614162	−	0.789180i	$-0.289494\pi$
$167$	442694.			1.22832			0.614162	+	0.789180i	$0.289494\pi$
$168$		0			0
$169$	971685.			2.61703
$170$		0			0
$171$	−51973.5	−	90020.7i	−0.135923	−	0.235425i
$172$		0			0
$173$	39299.5	−	68068.8i	0.0998325	−	0.172915i	−0.811783	−	0.583960i	$-0.801503\pi$
$173$	39299.5	−	68068.8i	0.0998325	−	0.172915i	0.911615	+	0.411045i	$0.134836\pi$
$174$		0			0
$175$	55948.3	+	229404.i	0.138099	+	0.566247i
$176$		0			0
$177$	74457.4	−	128964.i	0.178638	−	0.309411i
$178$		0			0
$179$	−255214.	−	442043.i	−0.595348	−	1.03117i	−0.993498	−	0.113853i	$-0.963681\pi$
$179$	−255214.	−	442043.i	−0.595348	−	1.03117i	0.398149	−	0.917321i	$-0.369653\pi$
$180$		0			0
$181$	−22051.8			−0.0500319			−0.0250160	−	0.999687i	$-0.507964\pi$
$181$	−22051.8			−0.0500319			−0.0250160	+	0.999687i	$0.507964\pi$
$182$		0			0
$183$	367650.			0.811534
$184$		0			0
$185$	−42120.3	−	72954.5i	−0.0904820	−	0.156719i
$186$		0			0
$187$	96220.4	−	166659.i	0.201216	−	0.348517i
$188$		0			0
$189$	−394227.	−	115235.i	−0.802771	−	0.234655i
$190$		0			0
$191$	−279076.	+	483374.i	−0.553528	+	0.958738i	0.444489	+	0.895784i	$0.353385\pi$
$191$	−279076.	+	483374.i	−0.553528	+	0.958738i	−0.998016	+	0.0629535i	$0.979948\pi$
$192$		0			0
$193$	24658.1	+	42709.1i	0.0476504	+	0.0825330i	0.888867	−	0.458166i	$-0.151493\pi$
$193$	24658.1	+	42709.1i	0.0476504	+	0.0825330i	−0.841216	+	0.540699i	$0.818160\pi$
$194$		0			0
$195$	1.03668e6			1.95236
$196$		0			0
$197$	−941509.			−1.72846			−0.864229	−	0.503099i	$-0.832193\pi$
$197$	−941509.			−1.72846			−0.864229	+	0.503099i	$0.832193\pi$
$198$		0			0
$199$	320006.	+	554267.i	0.572830	+	0.992170i	0.996274	+	0.0862477i	$0.0274876\pi$
$199$	320006.	+	554267.i	0.572830	+	0.992170i	−0.423444	+	0.905922i	$0.639179\pi$
$200$		0			0
$201$	532493.	−	922304.i	0.929658	−	1.61022i
$202$		0			0
$203$	−703900.	−	205754.i	−1.19887	−	0.350436i
$204$		0			0
$205$	68828.2	−	119214.i	0.114388	−	0.198126i
$206$		0			0
$207$	−645756.	−	1.11848e6i	−1.04747	−	1.81428i
$208$		0			0
$209$	−43565.1			−0.0689879
$210$		0			0
$211$	−921869.			−1.42549			−0.712743	−	0.701425i	$-0.752548\pi$
$211$	−921869.			−1.42549			−0.712743	+	0.701425i	$0.752548\pi$
$212$		0			0
$213$	247298.	+	428333.i	0.373484	+	0.646894i
$214$		0			0
$215$	−70870.4	+	122751.i	−0.104561	+	0.181105i
$216$		0			0
$217$	−71101.5	−	291537.i	−0.102501	−	0.420285i
$218$		0			0
$219$	564290.	−	977378.i	0.795045	−	1.37706i
$220$		0			0
$221$	−717387.	−	1.24255e6i	−0.988036	−	1.71133i
$222$		0			0
$223$	837700.			1.12805			0.564023	−	0.825759i	$-0.309253\pi$
$223$	837700.			1.12805			0.564023	+	0.825759i	$0.309253\pi$
$224$		0			0
$225$	−675497.			−0.889544
$226$		0			0
$227$	−665344.	−	1.15241e6i	−0.857002	−	1.48437i	−0.874775	−	0.484529i	$-0.838991\pi$
$227$	−665344.	−	1.15241e6i	−0.857002	−	1.48437i	0.0177731	−	0.999842i	$-0.494342\pi$
$228$		0			0
$229$	−505317.	+	875234.i	−0.636759	+	1.10290i	0.349381	+	0.936981i	$0.386392\pi$
$229$	−505317.	+	875234.i	−0.636759	+	1.10290i	−0.986140	+	0.165918i	$0.946941\pi$
$230$		0			0
$231$	−360916.	+	344971.i	−0.445017	+	0.425357i
$232$		0			0
$233$	−743050.	+	1.28700e6i	−0.896661	+	1.55306i	−0.0649249	+	0.997890i	$0.520681\pi$
$233$	−743050.	+	1.28700e6i	−0.896661	+	1.55306i	−0.831736	+	0.555172i	$0.812653\pi$
$234$		0			0
$235$	200683.	+	347593.i	0.237050	+	0.410583i
$236$		0			0
$237$	2.70043e6			3.12293
$238$		0			0
$239$	−875637.			−0.991584			−0.495792	−	0.868441i	$-0.665122\pi$
$239$	−875637.			−0.991584			−0.495792	+	0.868441i	$0.665122\pi$
$240$		0			0
$241$	−726472.	−	1.25829e6i	−0.805706	−	1.39552i	−0.915814	−	0.401604i	$-0.868453\pi$
$241$	−726472.	−	1.25829e6i	−0.805706	−	1.39552i	0.110108	−	0.993920i	$-0.464880\pi$
$242$		0			0
$243$	−528956.	+	916179.i	−0.574651	+	0.995324i
$244$		0			0
$245$	538691.	−	279374.i	0.573356	−	0.297352i
$246$		0			0
$247$	−162403.	+	281291.i	−0.169376	+	0.293369i
$248$		0			0
$249$	−691621.	−	1.19792e6i	−0.706919	−	1.22442i
$250$		0			0
$251$	198384.			0.198757			0.0993786	−	0.995050i	$-0.468315\pi$
$251$	198384.			0.198757			0.0993786	+	0.995050i	$0.468315\pi$
$252$		0			0
$253$	−541284.			−0.531648
$254$		0			0
$255$	−553772.	−	959161.i	−0.533311	−	0.923722i
$256$		0			0
$257$	−239786.	+	415321.i	−0.226460	+	0.392240i	−0.956756	−	0.290890i	$-0.906048\pi$
$257$	−239786.	+	415321.i	−0.226460	+	0.392240i	0.730297	+	0.683130i	$0.239382\pi$
$258$		0			0
$259$	218659.	−	208999.i	0.202543	−	0.193595i
$260$		0			0
$261$	1.04896e6	−	1.81686e6i	0.953145	−	1.65090i
$262$		0			0
$263$	227471.	+	393992.i	0.202786	+	0.351235i	0.949425	−	0.313994i	$-0.101667\pi$
$263$	227471.	+	393992.i	0.202786	+	0.351235i	−0.746639	+	0.665229i	$0.768334\pi$
$264$		0			0
$265$	403884.			0.353299
$266$		0			0
$267$	2.36728e6			2.03222
$268$		0			0
$269$	430068.	+	744900.i	0.362374	+	0.627650i	0.988351	−	0.152192i	$-0.0486331\pi$
$269$	430068.	+	744900.i	0.362374	+	0.627650i	−0.625977	+	0.779841i	$0.715300\pi$
$270$		0			0
$271$	652790.	−	1.13067e6i	0.539946	−	0.935214i	−0.458960	−	0.888457i	$-0.651778\pi$
$271$	652790.	−	1.13067e6i	0.539946	−	0.935214i	0.998906	−	0.0467569i	$-0.0148886\pi$
$272$		0			0
$273$	881975.	+	3.61636e6i	0.716226	+	2.93673i
$274$		0			0
$275$	−141554.	+	245178.i	−0.112873	+	0.195501i
$276$		0			0
$277$	271554.	+	470346.i	0.212646	+	0.368314i	0.952542	−	0.304408i	$-0.0984586\pi$
$277$	271554.	+	470346.i	0.212646	+	0.368314i	−0.739896	+	0.672721i	$0.765125\pi$
$278$		0			0
$279$	858452.			0.660246
$280$		0			0
$281$	998089.			0.754055			0.377028	−	0.926202i	$-0.376946\pi$
$281$	998089.			0.754055			0.377028	+	0.926202i	$0.376946\pi$
$282$		0			0
$283$	888660.	+	1.53920e6i	0.659583	+	1.14243i	0.980724	+	0.195400i	$0.0626005\pi$
$283$	888660.	+	1.53920e6i	0.659583	+	1.14243i	−0.321140	+	0.947032i	$0.604066\pi$
$284$		0			0
$285$	−125364.	+	217137.i	−0.0914241	+	0.158351i
$286$		0			0
$287$	474421.	+	138676.i	0.339984	+	0.0993796i
$288$		0			0
$289$	−56493.2	+	97849.1i	−0.0397880	+	0.0689148i
$290$		0			0
$291$	185884.	+	321961.i	0.128680	+	0.222880i
$292$		0			0
$293$	−1.63987e6			−1.11594			−0.557969	−	0.829862i	$-0.688419\pi$
$293$	−1.63987e6			−1.11594			−0.557969	+	0.829862i	$0.688419\pi$
$294$		0			0
$295$	−217007.			−0.145184
$296$		0			0
$297$	−246220.	−	426465.i	−0.161969	−	0.280538i
$298$		0			0
$299$	−2.01782e6	+	3.49496e6i	−1.30528	+	2.26081i
$300$		0			0
$301$	−488498.	−	142791.i	−0.310775	−	0.0908415i
$302$		0			0
$303$	332947.	−	576681.i	0.208338	−	0.360852i
$304$		0			0
$305$	−267880.	−	463982.i	−0.164889	−	0.285595i
$306$		0			0
$307$	−2.38130e6			−1.44201			−0.721006	−	0.692928i	$-0.756320\pi$
$307$	−2.38130e6			−1.44201			−0.721006	+	0.692928i	$0.756320\pi$
$308$		0			0
$309$	−1.28627e6			−0.766366
$310$		0			0
$311$	−622564.	−	1.07831e6i	−0.364992	−	0.632184i	0.623783	−	0.781597i	$-0.285595\pi$
$311$	−622564.	−	1.07831e6i	−0.364992	−	0.632184i	−0.988775	+	0.149413i	$0.952262\pi$
$312$		0			0
$313$	−831495.	+	1.44019e6i	−0.479732	+	0.830921i	−0.999730	−	0.0232470i	$-0.992600\pi$
$313$	−831495.	+	1.44019e6i	−0.479732	+	0.830921i	0.519997	+	0.854168i	$0.325933\pi$
$314$		0			0
$315$	411319.	+	1.68653e6i	0.233562	+	0.957673i
$316$		0			0
$317$	1.31243e6	−	2.27319e6i	0.733547	−	1.27054i	−0.221812	−	0.975090i	$-0.571197\pi$
$317$	1.31243e6	−	2.27319e6i	0.733547	−	1.27054i	0.955358	−	0.295450i	$-0.0954697\pi$
$318$		0			0
$319$	−439630.	−	761462.i	−0.241886	−	0.418959i
$320$		0			0
$321$	−618784.			−0.335179
$322$		0			0
$323$	347008.			0.185069
$324$		0			0
$325$	1.05538e6	+	1.82796e6i	0.554241	+	0.959974i
$326$		0			0
$327$	−45045.3	+	78020.7i	−0.0232959	+	0.0403497i
$328$		0			0
$329$	−1.04180e6	+	995780.i	−0.530636	+	0.507193i
$330$		0			0
$331$	869428.	−	1.50589e6i	0.436178	−	0.755482i	−0.561213	−	0.827671i	$-0.689665\pi$
$331$	869428.	−	1.50589e6i	0.436178	−	0.755482i	0.997391	+	0.0721894i	$0.0229986\pi$
$332$		0			0
$333$	432652.	+	749375.i	0.213810	+	0.370330i
$334$		0			0
$335$	−1.55196e6			−0.755558
$336$		0			0
$337$	853564.			0.409413			0.204706	−	0.978823i	$-0.434376\pi$
$337$	853564.			0.409413			0.204706	+	0.978823i	$0.434376\pi$
$338$		0			0
$339$	770236.	+	1.33409e6i	0.364019	+	0.630500i
$340$		0			0
$341$	179892.	−	311583.i	0.0837774	−	0.145107i
$342$		0			0
$343$	1.43287e6	+	1.64148e6i	0.657613	+	0.753356i
$344$		0			0
$345$	−1.55761e6	+	2.69786e6i	−0.704550	+	1.22032i
$346$		0			0
$347$	−1.49250e6	−	2.58508e6i	−0.665411	−	1.15253i	−0.979174	−	0.203024i	$-0.934923\pi$
$347$	−1.49250e6	−	2.58508e6i	−0.665411	−	1.15253i	0.313763	−	0.949501i	$-0.398410\pi$
$348$		0			0
$349$	2.87467e6			1.26335			0.631676	−	0.775233i	$-0.282367\pi$
$349$	2.87467e6			1.26335			0.631676	+	0.775233i	$0.282367\pi$
$350$		0			0
$351$	−3.67146e6			−1.59064
$352$		0			0
$353$	−735546.	−	1.27400e6i	−0.314176	−	0.544169i	0.665086	−	0.746767i	$-0.268395\pi$
$353$	−735546.	−	1.27400e6i	−0.314176	−	0.544169i	−0.979262	+	0.202598i	$0.935062\pi$
$354$		0			0
$355$	360377.	−	624192.i	0.151770	−	0.262874i
$356$		0			0
$357$	2.87480e6	−	2.74779e6i	1.19381	−	1.14107i
$358$		0			0
$359$	1.53900e6	−	2.66562e6i	0.630233	−	1.09160i	−0.357270	−	0.934001i	$-0.616293\pi$
$359$	1.53900e6	−	2.66562e6i	0.630233	−	1.09160i	0.987504	−	0.157595i	$-0.0503741\pi$
$360$		0			0
$361$	1.19877e6	+	2.07633e6i	0.484137	+	0.838550i
$362$		0			0
$363$	3.39167e6			1.35097
$364$		0			0
$365$	−1.64463e6			−0.646154
$366$		0			0
$367$	808349.	+	1.40010e6i	0.313281	+	0.542618i	0.979071	−	0.203521i	$-0.0652386\pi$
$367$	808349.	+	1.40010e6i	0.313281	+	0.542618i	−0.665790	+	0.746139i	$0.731905\pi$
$368$		0			0
$369$	−706990.	+	1.22454e6i	−0.270301	+	0.468175i
$370$		0			0
$371$	343611.	+	1.40890e6i	0.129608	+	0.531431i
$372$		0			0
$373$	2.38059e6	−	4.12331e6i	0.885958	−	1.53452i	0.0413462	−	0.999145i	$-0.486835\pi$
$373$	2.38059e6	−	4.12331e6i	0.885958	−	1.53452i	0.844612	−	0.535379i	$-0.179831\pi$
$374$		0			0
$375$	2.21243e6	+	3.83205e6i	0.812442	+	1.40719i
$376$		0			0
$377$	−6.55547e6			−2.37548
$378$		0			0
$379$	−1.00193e6			−0.358294			−0.179147	−	0.983822i	$-0.557334\pi$
$379$	−1.00193e6			−0.358294			−0.179147	+	0.983822i	$0.557334\pi$
$380$		0			0
$381$	−787273.	−	1.36360e6i	−0.277852	−	0.481253i
$382$		0			0
$383$	−1.41076e6	+	2.44351e6i	−0.491424	+	0.851172i	−0.999951	−	0.00987425i	$-0.996857\pi$
$383$	−1.41076e6	+	2.44351e6i	−0.491424	+	0.851172i	0.508527	+	0.861046i	$0.330190\pi$
$384$		0			0
$385$	698335.	+	204128.i	0.240111	+	0.0701859i
$386$		0			0
$387$	727967.	−	1.26088e6i	0.247078	−	0.427952i
$388$		0			0
$389$	621198.	+	1.07595e6i	0.208140	+	0.360510i	0.951129	−	0.308795i	$-0.0999256\pi$
$389$	621198.	+	1.07595e6i	0.208140	+	0.360510i	−0.742988	+	0.669304i	$0.766592\pi$
$390$		0			0
$391$	4.31148e6			1.42621
$392$		0			0
$393$	3.37695e6			1.10292
$394$		0			0
$395$	−1.96761e6	−	3.40800e6i	−0.634521	−	1.09902i
$396$		0			0
$397$	−1.37281e6	+	2.37778e6i	−0.437154	+	0.757173i	−0.997469	−	0.0711068i	$-0.977347\pi$
$397$	−1.37281e6	+	2.37778e6i	−0.437154	+	0.757173i	0.560315	+	0.828280i	$0.310680\pi$
$398$		0			0
$399$	−864113.	−	252586.i	−0.271730	−	0.0794285i
$400$		0			0
$401$	1.85668e6	−	3.21586e6i	0.576601	−	0.998702i	−0.419265	−	0.907864i	$-0.637712\pi$
$401$	1.85668e6	−	3.21586e6i	0.576601	−	0.998702i	0.995866	−	0.0908378i	$-0.0289545\pi$
$402$		0			0
$403$	−1.34122e6	−	2.32306e6i	−0.411374	−	0.712521i
$404$		0			0
$405$	419768.			0.127166
$406$		0			0
$407$	362656.			0.108520
$408$		0			0
$409$	−1.87970e6	−	3.25574e6i	−0.555623	−	0.962367i	−0.997855	−	0.0654664i	$-0.979146\pi$
$409$	−1.87970e6	−	3.25574e6i	−0.555623	−	0.962367i	0.442232	−	0.896901i	$-0.354187\pi$
$410$		0			0
$411$	4.15380e6	−	7.19460e6i	1.21295	−	2.10088i
$412$		0			0
$413$	−184622.	−	757005.i	−0.0532609	−	0.218385i
$414$		0			0
$415$	−1.00787e6	+	1.74568e6i	−0.287266	+	0.497559i
$416$		0			0
$417$	−720934.	−	1.24869e6i	−0.203028	−	0.351654i
$418$		0			0
$419$	−4.60027e6			−1.28011			−0.640056	−	0.768328i	$-0.721089\pi$
$419$	−4.60027e6			−1.28011			−0.640056	+	0.768328i	$0.721089\pi$
$420$		0			0
$421$	−404864.			−0.111328			−0.0556639	−	0.998450i	$-0.517728\pi$
$421$	−404864.			−0.111328			−0.0556639	+	0.998450i	$0.517728\pi$
$422$		0			0
$423$	−2.06138e6	−	3.57041e6i	−0.560153	−	0.970213i
$424$		0			0
$425$	1.12751e6	−	1.95291e6i	0.302795	−	0.524457i
$426$		0			0
$427$	1.39065e6	−	1.32921e6i	0.369102	−	0.352796i
$428$		0			0
$429$	−2.23147e6	+	3.86502e6i	−0.585393	+	1.01393i
$430$		0			0
$431$	47379.2	+	82063.2i	0.0122856	+	0.0212792i	0.872103	−	0.489323i	$-0.162756\pi$
$431$	47379.2	+	82063.2i	0.0122856	+	0.0212792i	−0.859817	+	0.510602i	$0.829423\pi$
$432$		0			0
$433$	4.31727e6			1.10660			0.553298	−	0.832983i	$-0.313369\pi$
$433$	4.31727e6			1.10660			0.553298	+	0.832983i	$0.313369\pi$
$434$		0			0
$435$	−5.06036e6			−1.28221
$436$		0			0
$437$	−488021.	−	845277.i	−0.122246	−	0.211736i
$438$		0			0
$439$	−1.11184e6	+	1.92576e6i	−0.275346	+	0.476914i	−0.970222	−	0.242216i	$-0.922126\pi$
$439$	−1.11184e6	+	1.92576e6i	−0.275346	+	0.476914i	0.694876	+	0.719130i	$0.255459\pi$
$440$		0			0
$441$	−5.53333e6	+	2.86968e6i	−1.35485	+	0.702647i
$442$		0			0
$443$	−2.34213e6	+	4.05668e6i	−0.567024	+	0.982114i	0.429835	+	0.902908i	$0.358572\pi$
$443$	−2.34213e6	+	4.05668e6i	−0.567024	+	0.982114i	−0.996858	+	0.0792062i	$0.974761\pi$
$444$		0			0
$445$	−1.72487e6	−	2.98756e6i	−0.412910	−	0.715182i
$446$		0			0
$447$	−7.01315e6			−1.66014
$448$		0			0
$449$	897932.			0.210198			0.105099	−	0.994462i	$-0.466484\pi$
$449$	897932.			0.210198			0.105099	+	0.994462i	$0.466484\pi$
$450$		0			0
$451$	296306.	+	513217.i	0.0685960	+	0.118812i
$452$		0			0
$453$	2.99246e6	−	5.18310e6i	0.685147	−	1.18671i
$454$		0			0
$455$	3.92129e6	−	3.74805e6i	0.887974	−	0.848745i
$456$		0			0
$457$	−2.99091e6	+	5.18040e6i	−0.669903	+	1.16031i	0.308027	+	0.951377i	$0.400331\pi$
$457$	−2.99091e6	+	5.18040e6i	−0.669903	+	1.16031i	−0.977931	+	0.208929i	$0.933002\pi$
$458$		0			0
$459$	1.96121e6	+	3.39691e6i	0.434502	+	0.752580i
$460$		0			0
$461$	1.62507e6			0.356138			0.178069	−	0.984018i	$-0.443015\pi$
$461$	1.62507e6			0.356138			0.178069	+	0.984018i	$0.443015\pi$
$462$		0			0
$463$	8.46295e6			1.83472			0.917359	−	0.398060i	$-0.130317\pi$
$463$	8.46295e6			1.83472			0.917359	+	0.398060i	$0.130317\pi$
$464$		0			0
$465$	−1.03533e6	−	1.79324e6i	−0.222047	−	0.384596i
$466$		0			0
$467$	3.83297e6	−	6.63890e6i	0.813286	−	1.40865i	−0.0972658	−	0.995258i	$-0.531010\pi$
$467$	3.83297e6	−	6.63890e6i	0.813286	−	1.40865i	0.910552	−	0.413395i	$-0.135657\pi$
$468$		0			0
$469$	−1.32035e6	−	5.41383e6i	−0.277177	−	1.13651i
$470$		0			0
$471$	−2.65143e6	+	4.59241e6i	−0.550717	+	0.953869i
$472$		0			0
$473$	−305098.	−	528445.i	−0.0627027	−	0.108604i
$474$		0			0
$475$	−510497.			−0.103815
$476$		0			0
$477$	−4.14862e6			−0.834849
$478$		0			0
$479$	−1.20146e6	−	2.08099e6i	−0.239261	−	0.414412i	0.721242	−	0.692683i	$-0.243572\pi$
$479$	−1.20146e6	−	2.08099e6i	−0.239261	−	0.414412i	−0.960502	+	0.278272i	$0.910238\pi$
$480$		0			0
$481$	1.35192e6	−	2.34160e6i	0.266434	−	0.461477i
$482$		0			0
$483$	−1.07364e7	−	3.13831e6i	−2.09406	−	0.612107i
$484$		0			0
$485$	270881.	−	469180.i	0.0522907	−	0.0905701i
$486$		0			0
$487$	−85150.8	−	147486.i	−0.0162692	−	0.0281791i	0.857776	−	0.514023i	$-0.171846\pi$
$487$	−85150.8	−	147486.i	−0.0162692	−	0.0281791i	−0.874045	+	0.485844i	$0.838512\pi$
$488$		0			0
$489$	1.59724e6			0.302063
$490$		0			0
$491$	5.77628e6			1.08130			0.540648	−	0.841249i	$-0.318179\pi$
$491$	5.77628e6			1.08130			0.540648	+	0.841249i	$0.318179\pi$
$492$		0			0
$493$	3.50177e6	+	6.06525e6i	0.648890	+	1.12391i
$494$		0			0
$495$	−1.04067e6	+	1.80249e6i	−0.190897	+	0.330644i
$496$		0			0
$497$	2.48402e6	+	726095.i	0.451091	+	0.131857i
$498$		0			0
$499$	271801.	−	470773.i	0.0488652	−	0.0846369i	−0.840558	−	0.541721i	$-0.817773\pi$
$499$	271801.	−	470773.i	0.0488652	−	0.0846369i	0.889423	+	0.457084i	$0.151106\pi$
$500$		0			0
$501$	5.48418e6	+	9.49888e6i	0.976152	+	1.69075i
$502$		0			0
$503$	5.40086e6			0.951794			0.475897	−	0.879501i	$-0.342124\pi$
$503$	5.40086e6			0.951794			0.475897	+	0.879501i	$0.342124\pi$
$504$		0			0
$505$	−970378.			−0.169322
$506$		0			0
$507$	1.20374e7	+	2.08494e7i	2.07976	+	3.60225i
$508$		0			0
$509$	−4.17209e6	+	7.22628e6i	−0.713772	+	1.23629i	0.249659	+	0.968334i	$0.419681\pi$
$509$	−4.17209e6	+	7.22628e6i	−0.713772	+	1.23629i	−0.963431	+	0.267955i	$0.913652\pi$
$510$		0			0
$511$	−1.39919e6	−	5.73710e6i	−0.237042	−	0.971942i
$512$		0			0
$513$	443982.	−	769000.i	0.0744856	−	0.129013i
$514$		0			0
$515$	937213.	+	1.62330e6i	0.155711	+	0.269700i
$516$		0			0
$517$	−1.72788e6			−0.284307
$518$		0			0
$519$	1.94740e6			0.317349
$520$		0			0
$521$	5.73919e6	+	9.94057e6i	0.926310	+	1.60442i	0.789441	+	0.613827i	$0.210371\pi$
$521$	5.73919e6	+	9.94057e6i	0.926310	+	1.60442i	0.136870	+	0.990589i	$0.456296\pi$
$522$		0			0
$523$	−2.19367e6	+	3.79955e6i	−0.350685	+	0.607404i	−0.986370	−	0.164545i	$-0.947384\pi$
$523$	−2.19367e6	+	3.79955e6i	−0.350685	+	0.607404i	0.635685	+	0.771949i	$0.280718\pi$
$524$		0			0
$525$	−4.22922e6	+	4.04238e6i	−0.669673	+	0.640088i
$526$		0			0
$527$	−1.43289e6	+	2.48184e6i	−0.224744	+	0.389267i
$528$		0			0
$529$	−2.84535e6	−	4.92829e6i	−0.442075	−	0.765697i
$530$		0			0
$531$	2.22906e6			0.343071
$532$		0			0
$533$	4.41832e6			0.673657
$534$		0			0
$535$	450864.	+	780919.i	0.0681022	+	0.117956i
$536$		0			0
$537$	6.32327e6	−	1.09522e7i	0.946250	−	1.63895i
$538$		0			0
$539$	−117957.	+	2.60973e6i	−0.0174885	+	0.386922i
$540$		0			0
$541$	−3.61153e6	+	6.25535e6i	−0.530515	+	0.918879i	0.468851	+	0.883277i	$0.344668\pi$
$541$	−3.61153e6	+	6.25535e6i	−0.530515	+	0.918879i	−0.999366	+	0.0356020i	$0.988665\pi$
$542$		0			0
$543$	−273182.	−	473165.i	−0.0397605	−	0.0688673i
$544$		0			0
$545$	131285.			0.0189332
$546$		0			0
$547$	−3.54529e6			−0.506622			−0.253311	−	0.967385i	$-0.581520\pi$
$547$	−3.54529e6			−0.506622			−0.253311	+	0.967385i	$0.581520\pi$
$548$		0			0
$549$	2.75161e6	+	4.76593e6i	0.389634	+	0.674865i
$550$		0			0
$551$	792739.	−	1.37306e6i	0.111237	−	0.192669i
$552$		0			0
$553$	1.02144e7	−	9.76319e6i	1.42037	−	1.35762i
$554$		0			0
$555$	1.04359e6	−	1.80755e6i	0.143813	−	0.249091i
$556$		0			0
$557$	−5.18314e6	−	8.97747e6i	−0.707873	−	1.22607i	−0.965645	−	0.259866i	$-0.916322\pi$
$557$	−5.18314e6	−	8.97747e6i	−0.707873	−	1.22607i	0.257772	−	0.966206i	$-0.417012\pi$
$558$		0			0
$559$	−4.54941e6			−0.615780
$560$		0			0
$561$	4.76799e6			0.639628
$562$		0			0
$563$	−2.24141e6	−	3.88223e6i	−0.298023	−	0.516191i	0.677660	−	0.735375i	$-0.262994\pi$
$563$	−2.24141e6	−	3.88223e6i	−0.298023	−	0.516191i	−0.975684	+	0.219184i	$0.929661\pi$
$564$		0			0
$565$	1.12243e6	−	1.94411e6i	0.147924	−	0.256212i
$566$		0			0
$567$	357124.	+	1.46431e6i	0.0466511	+	0.191283i
$568$		0			0
$569$	−3.24668e6	+	5.62341e6i	−0.420396	+	0.728147i	−0.995978	−	0.0895969i	$-0.971442\pi$
$569$	−3.24668e6	+	5.62341e6i	−0.420396	+	0.728147i	0.575582	+	0.817744i	$0.304775\pi$
$570$		0			0
$571$	−248754.	−	430854.i	−0.0319286	−	0.0553019i	0.849620	−	0.527396i	$-0.176832\pi$
$571$	−248754.	−	430854.i	−0.0319286	−	0.0553019i	−0.881548	+	0.472094i	$0.843498\pi$
$572$		0			0
$573$	−1.38290e7			−1.75956
$574$		0			0
$575$	−6.34279e6			−0.800038
$576$		0			0
$577$	3.27561e6	+	5.67353e6i	0.409594	+	0.709437i	0.994844	−	0.101416i	$-0.0323372\pi$
$577$	3.27561e6	+	5.67353e6i	0.409594	+	0.709437i	−0.585250	+	0.810853i	$0.699004\pi$
$578$		0			0
$579$	−610939.	+	1.05818e6i	−0.0757359	+	0.131178i
$580$		0			0
$581$	−6.94707e6	−	2.03067e6i	−0.853810	−	0.249574i
$582$		0			0
$583$	−869363.	+	1.50578e6i	−0.105933	+	0.183481i
$584$		0			0
$585$	7.75889e6	+	1.34388e7i	0.937368	+	1.62357i
$586$		0			0
$587$	30793.5			0.00368862			0.00184431	−	0.999998i	$-0.499413\pi$
$587$	30793.5			0.00368862			0.00184431	+	0.999998i	$0.499413\pi$
$588$		0			0
$589$	648763.			0.0770544
$590$		0			0
$591$	−1.16636e7	−	2.02019e7i	−1.37361	−	2.37916i
$592$		0			0
$593$	−285468.	+	494446.i	−0.0333366	+	0.0577407i	−0.882212	−	0.470852i	$-0.843947\pi$
$593$	−285468.	+	494446.i	−0.0333366	+	0.0577407i	0.848876	+	0.528592i	$0.177280\pi$
$594$		0			0
$595$	−5.56243e6	−	1.62593e6i	−0.644128	−	0.188283i
$596$		0			0
$597$	−7.92859e6	+	1.37327e7i	−0.910459	+	1.57696i
$598$		0			0
$599$	4.53555e6	+	7.85581e6i	0.516492	+	0.894590i	0.999817	+	0.0191488i	$0.00609561\pi$
$599$	4.53555e6	+	7.85581e6i	0.516492	+	0.894590i	−0.483325	+	0.875441i	$0.660571\pi$
$600$		0			0
$601$	−1.36309e7			−1.53936			−0.769679	−	0.638431i	$-0.779584\pi$
$601$	−1.36309e7			−1.53936			−0.769679	+	0.638431i	$0.779584\pi$
$602$		0			0
$603$	1.59414e7			1.78539
$604$		0			0
$605$	−2.47126e6	−	4.28035e6i	−0.274493	−	0.475435i
$606$		0			0
$607$	−1.97825e6	+	3.42644e6i	−0.217927	+	0.377460i	−0.954174	−	0.299253i	$-0.903263\pi$
$607$	−1.97825e6	+	3.42644e6i	−0.217927	+	0.377460i	0.736247	+	0.676713i	$0.236596\pi$
$608$		0			0
$609$	−4.30518e6	−	1.76525e7i	−0.470379	−	1.92869i
$610$		0			0
$611$	−6.44126e6	+	1.11566e7i	−0.698020	+	1.20901i
$612$		0			0
$613$	−661552.	−	1.14584e6i	−0.0711070	−	0.123161i	0.828280	−	0.560315i	$-0.189320\pi$
$613$	−661552.	−	1.14584e6i	−0.0711070	−	0.123161i	−0.899387	+	0.437154i	$0.855987\pi$
$614$		0			0
$615$	3.41063e6			0.363619
$616$		0			0
$617$	9.26370e6			0.979651			0.489826	−	0.871820i	$-0.337060\pi$
$617$	9.26370e6			0.979651			0.489826	+	0.871820i	$0.337060\pi$
$618$		0			0
$619$	334693.	+	579705.i	0.0351091	+	0.0608107i	0.883046	−	0.469286i	$-0.155489\pi$
$619$	334693.	+	579705.i	0.0351091	+	0.0608107i	−0.847937	+	0.530097i	$0.822155\pi$
$620$		0			0
$621$	5.51636e6	−	9.55461e6i	0.574015	−	0.994223i
$622$		0			0
$623$	8.95430e6	−	8.55872e6i	0.924297	−	0.883463i
$624$		0			0
$625$	378164.	−	654999.i	0.0387240	−	0.0670719i
$626$		0			0
$627$	−539693.	−	934776.i	−0.0548249	−	0.0949595i
$628$		0			0
$629$	−2.88866e6			−0.291118
$630$		0			0
$631$	666246.			0.0666133			0.0333067	−	0.999445i	$-0.489396\pi$
$631$	666246.			0.0666133			0.0333067	+	0.999445i	$0.489396\pi$
$632$		0			0
$633$	−1.14203e7	−	1.97805e7i	−1.13284	−	1.96213i
$634$		0			0
$635$	−1.14726e6	+	1.98711e6i	−0.112909	+	0.195564i
$636$		0			0
$637$	1.64108e7	+	1.04903e7i	1.60243	+	1.02432i
$638$		0			0
$639$	−3.70173e6	+	6.41158e6i	−0.358635	+	0.621173i
$640$		0			0
$641$	1.05439e6	+	1.82625e6i	0.101357	+	0.175556i	0.912244	−	0.409647i	$-0.134348\pi$
$641$	1.05439e6	+	1.82625e6i	0.101357	+	0.175556i	−0.810887	+	0.585203i	$0.801015\pi$
$642$		0			0
$643$	−1.25977e7			−1.20161			−0.600806	−	0.799395i	$-0.705154\pi$
$643$	−1.25977e7			−1.20161			−0.600806	+	0.799395i	$0.705154\pi$
$644$		0			0
$645$	−3.51182e6			−0.332379
$646$		0			0
$647$	8.56496e6	+	1.48349e7i	0.804386	+	1.39324i	0.916705	+	0.399565i	$0.130839\pi$
$647$	8.56496e6	+	1.48349e7i	0.804386	+	1.39324i	−0.112319	+	0.993672i	$0.535828\pi$
$648$		0			0
$649$	467109.	−	809056.i	0.0435318	−	0.0753992i
$650$		0			0
$651$	5.37469e6	−	5.13724e6i	0.497051	−	0.475092i
$652$		0			0
$653$	4.12762e6	−	7.14925e6i	0.378806	−	0.656112i	−0.612083	−	0.790794i	$-0.709668\pi$
$653$	4.12762e6	−	7.14925e6i	0.378806	−	0.656112i	0.990889	+	0.134682i	$0.0430013\pi$
$654$		0			0
$655$	−2.46054e6	−	4.26178e6i	−0.224092	−	0.388139i
$656$		0			0
$657$	1.68933e7			1.52687
$658$		0			0
$659$	1.05417e7			0.945581			0.472790	−	0.881175i	$-0.343247\pi$
$659$	1.05417e7			0.945581			0.472790	+	0.881175i	$0.343247\pi$
$660$		0			0
$661$	−2.23820e6	−	3.87668e6i	−0.199249	−	0.345109i	0.749036	−	0.662529i	$-0.230517\pi$
$661$	−2.23820e6	−	3.87668e6i	−0.199249	−	0.345109i	−0.948285	+	0.317420i	$0.897184\pi$
$662$		0			0
$663$	1.77743e7	−	3.07859e7i	1.57039	−	2.72000i
$664$		0			0
$665$	310848.	+	1.27457e6i	0.0272580	+	0.111766i
$666$		0			0
$667$	9.84956e6	−	1.70599e7i	0.857240	−	1.48478i
$668$		0			0
$669$	1.03776e7	+	1.79745e7i	0.896461	+	1.55272i
$670$		0			0
$671$	2.30645e6			0.197760
$672$		0			0
$673$	−1.87165e7			−1.59289			−0.796447	−	0.604708i	$-0.793290\pi$
$673$	−1.87165e7			−1.59289			−0.796447	+	0.604708i	$0.793290\pi$
$674$		0			0
$675$	−2.88521e6	−	4.99733e6i	−0.243735	−	0.422162i
$676$		0			0
$677$	−2.08116e6	+	3.60467e6i	−0.174515	+	0.302270i	−0.939993	−	0.341192i	$-0.889169\pi$
$677$	−2.08116e6	+	3.60467e6i	−0.174515	+	0.302270i	0.765478	+	0.643462i	$0.222503\pi$
$678$		0			0
$679$	1.86714e6	+	545776.i	0.155418	+	0.0454297i
$680$		0			0
$681$	1.64848e7	−	2.85526e7i	1.36212	−	2.35927i
$682$		0			0
$683$	−6.43455e6	−	1.11450e7i	−0.527796	−	0.914170i	−0.999475	−	0.0323992i	$-0.989685\pi$
$683$	−6.43455e6	−	1.11450e7i	−0.527796	−	0.914170i	0.471679	−	0.881770i	$-0.343648\pi$
$684$		0			0
$685$	−1.21063e7			−0.985793
$686$		0			0
$687$	−2.50398e7			−2.02414
$688$		0			0
$689$	6.48168e6	+	1.12266e7i	0.520163	+	0.900949i
$690$		0			0
$691$	−4.64518e6	+	8.04569e6i	−0.370090	+	0.641015i	−0.989579	−	0.143990i	$-0.954007\pi$
$691$	−4.64518e6	+	8.04569e6i	−0.370090	+	0.641015i	0.619489	+	0.785006i	$0.287340\pi$
$692$		0			0
$693$	−7.17316e6	−	2.09676e6i	−0.567385	−	0.165850i
$694$		0			0
$695$	−1.05059e6	+	1.81967e6i	−0.0825029	+	0.142899i
$696$		0			0
$697$	−2.36016e6	−	4.08791e6i	−0.184018	−	0.318728i
$698$		0			0
$699$	−3.68202e7			−2.85032
$700$		0			0
$701$	2.50809e7			1.92774			0.963870	−	0.266375i	$-0.0858258\pi$
$701$	2.50809e7			1.92774			0.963870	+	0.266375i	$0.0858258\pi$
$702$		0			0
$703$	326970.	+	566329.i	0.0249528	+	0.0432196i
$704$		0			0
$705$	−4.97220e6	+	8.61210e6i	−0.376769	+	0.652584i
$706$		0			0
$707$	−825564.	−	3.38505e6i	−0.0621158	−	0.254693i
$708$		0			0
$709$	−4.96890e6	+	8.60639e6i	−0.371231	+	0.642991i	−0.989755	−	0.142774i	$-0.954398\pi$
$709$	−4.96890e6	+	8.60639e6i	−0.371231	+	0.642991i	0.618524	+	0.785766i	$0.287731\pi$
$710$		0			0
$711$	2.02109e7	+	3.50063e7i	1.49938	+	2.59700i
$712$		0			0
$713$	8.06069e6			0.593811
$714$		0			0
$715$	6.50364e6			0.475764
$716$		0			0
$717$	−1.08476e7	−	1.87885e7i	−0.788015	−	1.36488i
$718$		0			0
$719$	−7.00047e6	+	1.21252e7i	−0.505016	+	0.874713i	0.494968	+	0.868911i	$0.335180\pi$
$719$	−7.00047e6	+	1.21252e7i	−0.505016	+	0.874713i	−0.999983	+	0.00580117i	$0.998153\pi$
$720$		0			0
$721$	−4.86536e6	+	4.65041e6i	−0.348559	+	0.333160i
$722$		0			0
$723$	1.79994e7	−	3.11758e7i	1.28059	−	2.21805i
$724$		0			0
$725$	−5.15160e6	−	8.92283e6i	−0.363996	−	0.630460i
$726$		0			0
$727$	−8.27315e6			−0.580544			−0.290272	−	0.956944i	$-0.593746\pi$
$727$	−8.27315e6			−0.580544			−0.290272	+	0.956944i	$0.593746\pi$
$728$		0			0
$729$	−2.33861e7			−1.62982
$730$		0			0
$731$	2.43019e6	+	4.20921e6i	0.168208	+	0.291345i
$732$		0			0
$733$	1.64295e6	−	2.84568e6i	0.112945	−	0.195626i	−0.804012	−	0.594614i	$-0.797305\pi$
$733$	1.64295e6	−	2.84568e6i	0.112945	−	0.195626i	0.916956	+	0.398988i	$0.130638\pi$
$734$		0			0
$735$	1.26679e7	+	8.09773e6i	0.864943	+	0.552898i
$736$		0			0
$737$	3.34059e6	−	5.78608e6i	0.226545	−	0.392388i
$738$		0			0
$739$	5.04564e6	+	8.73930e6i	0.339864	+	0.588662i	0.984407	−	0.175907i	$-0.0562857\pi$
$739$	5.04564e6	+	8.73930e6i	0.339864	+	0.588662i	−0.644543	+	0.764568i	$0.722952\pi$
$740$		0			0
$741$	−8.04754e6			−0.538416
$742$		0			0
$743$	−1.73443e7			−1.15261			−0.576307	−	0.817233i	$-0.695507\pi$
$743$	−1.73443e7			−1.15261			−0.576307	+	0.817233i	$0.695507\pi$
$744$		0			0
$745$	5.10998e6	+	8.85074e6i	0.337309	+	0.584237i
$746$		0			0
$747$	1.03526e7	−	1.79313e7i	0.678812	−	1.17574i
$748$		0			0
$749$	−2.34057e6	+	2.23717e6i	−0.152446	+	0.145711i
$750$		0			0
$751$	−8.79857e6	+	1.52396e7i	−0.569262	+	0.985990i	0.427377	+	0.904073i	$0.359438\pi$
$751$	−8.79857e6	+	1.52396e7i	−0.569262	+	0.985990i	−0.996639	+	0.0819170i	$0.973896\pi$
$752$		0			0
$753$	2.45762e6	+	4.25673e6i	0.157953	+	0.273583i
$754$		0			0
$755$	−8.72158e6			−0.556836
$756$		0			0
$757$	−4.66480e6			−0.295865			−0.147932	−	0.988997i	$-0.547262\pi$
$757$	−4.66480e6			−0.295865			−0.147932	+	0.988997i	$0.547262\pi$
$758$		0			0
$759$	−6.70554e6	−	1.16143e7i	−0.422502	−	0.731795i
$760$		0			0
$761$	−9.13585e6	+	1.58238e7i	−0.571857	+	0.990485i	0.424518	+	0.905419i	$0.360443\pi$
$761$	−9.13585e6	+	1.58238e7i	−0.571857	+	0.990485i	−0.996375	+	0.0850658i	$0.972890\pi$
$762$		0			0
$763$	111693.	+	457973.i	0.00694566	+	0.0284792i
$764$		0			0
$765$	8.28923e6	−	1.43574e7i	0.512107	−	0.886995i
$766$		0			0
$767$	−3.48261e6	−	6.03205e6i	−0.213755	−	0.370234i
$768$		0			0
$769$	−2.31895e7			−1.41409			−0.707043	−	0.707171i	$-0.749971\pi$
$769$	−2.31895e7			−1.41409			−0.707043	+	0.707171i	$0.749971\pi$
$770$		0			0
$771$	−1.18821e7			−0.719873
$772$		0			0
$773$	−1.01074e6	−	1.75066e6i	−0.0608405	−	0.105379i	0.834001	−	0.551763i	$-0.186045\pi$
$773$	−1.01074e6	−	1.75066e6i	−0.0608405	−	0.105379i	−0.894841	+	0.446384i	$0.852711\pi$
$774$		0			0
$775$	2.10799e6	−	3.65114e6i	0.126070	−	0.218360i
$776$		0			0
$777$	7.19328e6	+	2.10264e6i	0.427439	+	0.124943i
$778$		0			0
$779$	−534297.	+	925430.i	−0.0315456	+	0.0546387i
$780$		0			0
$781$	1.55143e6	+	2.68715e6i	0.0910131	+	0.157639i
$782$		0			0
$783$	1.79215e7			1.04465
$784$		0			0
$785$	7.72763e6			0.447582
$786$		0			0
$787$	7.88722e6	+	1.36611e7i	0.453928	+	0.786227i	0.998626	−	0.0524059i	$-0.0166889\pi$
$787$	7.88722e6	+	1.36611e7i	0.453928	+	0.786227i	−0.544698	+	0.838632i	$0.683356\pi$
$788$		0			0
$789$	−5.63592e6	+	9.76170e6i	−0.322309	+	0.558255i
$790$		0			0
$791$	7.73672e6	+	2.26149e6i	0.439659	+	0.128515i
$792$		0			0
$793$	8.59807e6	−	1.48923e7i	0.485532	−	0.840966i
$794$		0			0
$795$	5.00340e6	+	8.66614e6i	0.280768	+	0.486304i
$796$		0			0
$797$	1.22436e6			0.0682750			0.0341375	−	0.999417i	$-0.489132\pi$
$797$	1.22436e6			0.0682750			0.0341375	+	0.999417i	$0.489132\pi$
$798$		0			0
$799$	1.37631e7			0.762690
$800$		0			0
$801$	1.77175e7	+	3.06876e7i	0.975712	+	1.68998i
$802$		0			0
$803$	3.54007e6	−	6.13158e6i	0.193742	−	0.335571i
$804$		0			0
$805$	3.86221e6	+	1.58362e7i	0.210061	+	0.861312i
$806$		0			0
$807$	−1.06555e7	+	1.84559e7i	−0.575959	+	0.997590i
$808$		0			0
$809$	1.38662e7	+	2.40170e7i	0.744882	+	1.29017i	0.950250	+	0.311489i	$0.100828\pi$
$809$	1.38662e7	+	2.40170e7i	0.744882	+	1.29017i	−0.205368	+	0.978685i	$0.565839\pi$
$810$		0			0
$811$	1.19246e7			0.636636			0.318318	−	0.947984i	$-0.396882\pi$
$811$	1.19246e7			0.636636			0.318318	+	0.947984i	$0.396882\pi$
$812$		0			0
$813$	3.23476e7			1.71639
$814$		0			0
$815$	−1.16379e6	−	2.01575e6i	−0.0613735	−	0.106302i
$816$		0			0
$817$	550151.	−	952889.i	0.0288354	−	0.0499444i
$818$		0			0
$819$	−4.02787e7	+	3.84993e7i	−2.09829	+	2.00559i
$820$		0			0
$821$	−8.02312e6	+	1.38965e7i	−0.415418	+	0.719525i	−0.995472	−	0.0950526i	$-0.969698\pi$
$821$	−8.02312e6	+	1.38965e7i	−0.415418	+	0.719525i	0.580054	+	0.814578i	$0.303031\pi$
$822$		0			0
$823$	−5.61533e6	−	9.72603e6i	−0.288985	−	0.500537i	0.684583	−	0.728935i	$-0.259984\pi$
$823$	−5.61533e6	−	9.72603e6i	−0.288985	−	0.500537i	−0.973568	+	0.228398i	$0.926651\pi$
$824$		0			0
$825$	−7.01437e6			−0.358801
$826$		0			0
$827$	2.68427e7			1.36478			0.682389	−	0.730989i	$-0.260941\pi$
$827$	2.68427e7			1.36478			0.682389	+	0.730989i	$0.260941\pi$
$828$		0			0
$829$	−1.40210e7	−	2.42850e7i	−0.708584	−	1.22730i	−0.965382	−	0.260839i	$-0.916001\pi$
$829$	−1.40210e7	−	2.42850e7i	−0.708584	−	1.22730i	0.256798	−	0.966465i	$-0.417332\pi$
$830$		0			0
$831$	−6.72813e6	+	1.16535e7i	−0.337981	+	0.585400i
$832$		0			0
$833$	939564.	−	2.07872e7i	0.0469152	−	1.03797i
$834$		0			0
$835$	7.99186e6	−	1.38423e7i	0.396672	−	0.687056i
$836$		0			0
$837$	3.66665e6	+	6.35083e6i	0.180907	+	0.313341i
$838$		0			0
$839$	3.30603e7			1.62145			0.810723	−	0.585430i	$-0.199074\pi$
$839$	3.30603e7			1.62145			0.810723	+	0.585430i	$0.199074\pi$
$840$		0			0
$841$	1.14880e7			0.560086
$842$		0			0
$843$	1.23645e7	+	2.14160e7i	0.599250	+	1.03793i
$844$		0			0
$845$	1.75416e7	−	3.03830e7i	0.845139	−	1.46382i
$846$		0			0
$847$	1.28291e7	−	1.22623e7i	0.614450	−	0.587305i
$848$		0			0
$849$	−2.20178e7	+	3.81359e7i	−1.04835	+	1.81579i
$850$		0			0
$851$	4.06251e6	+	7.03648e6i	0.192296	+	0.333067i
$852$		0			0
$853$	2.73897e7			1.28889			0.644444	−	0.764652i	$-0.277089\pi$
$853$	2.73897e7			1.28889			0.644444	+	0.764652i	$0.277089\pi$
$854$		0			0
$855$	−3.75306e6			−0.175578
$856$		0			0
$857$	1.48360e6	+	2.56968e6i	0.0690027	+	0.119516i	0.898463	−	0.439050i	$-0.144685\pi$
$857$	1.48360e6	+	2.56968e6i	0.0690027	+	0.119516i	−0.829460	+	0.558566i	$0.811352\pi$
$858$		0			0
$859$	−1.98980e7	+	3.44643e7i	−0.920081	+	1.59363i	−0.120794	+	0.992678i	$0.538544\pi$
$859$	−1.98980e7	+	3.44643e7i	−0.920081	+	1.59363i	−0.799287	+	0.600949i	$0.794789\pi$
$860$		0			0
$861$	2.90164e6	+	1.18976e7i	0.133394	+	0.546954i
$862$		0			0
$863$	3.59799e6	−	6.23190e6i	0.164450	−	0.284835i	−0.772010	−	0.635610i	$-0.780749\pi$
$863$	3.59799e6	−	6.23190e6i	0.164450	−	0.284835i	0.936460	+	0.350775i	$0.114082\pi$
$864$		0			0
$865$	−1.41893e6	−	2.45766e6i	−0.0644794	−	0.111682i
$866$		0			0
$867$	−2.79940e6			−0.126478
$868$		0			0
$869$	1.69411e7			0.761015
$870$		0			0
$871$	−2.49063e7	−	4.31391e7i	−1.11241	−	1.92675i
$872$		0			0
$873$	−2.78244e6	+	4.81932e6i	−0.123563	+	0.214018i
$874$		0			0
$875$	2.22231e7	+	6.49594e6i	0.981259	+	0.286828i
$876$		0			0
$877$	−1.25621e7	+	2.17582e7i	−0.551523	+	0.955266i	0.446642	+	0.894713i	$0.352620\pi$
$877$	−1.25621e7	+	2.17582e7i	−0.551523	+	0.955266i	−0.998165	+	0.0605532i	$0.980714\pi$
$878$		0			0
$879$	−2.03150e7	−	3.51866e7i	−0.886839	−	1.53605i
$880$		0			0
$881$	120262.			0.00522022			0.00261011	−	0.999997i	$-0.499169\pi$
$881$	120262.			0.00522022			0.00261011	+	0.999997i	$0.499169\pi$
$882$		0			0
$883$	−2.43497e7			−1.05097			−0.525487	−	0.850802i	$-0.676117\pi$
$883$	−2.43497e7			−1.05097			−0.525487	+	0.850802i	$0.676117\pi$
$884$		0			0
$885$	−2.68833e6	−	4.65632e6i	−0.115378	−	0.199841i
$886$		0			0
$887$	−1.41383e7	+	2.44883e7i	−0.603377	+	1.04508i	0.388929	+	0.921268i	$0.372845\pi$
$887$	−1.41383e7	+	2.44883e7i	−0.603377	+	1.04508i	−0.992306	+	0.123812i	$0.960488\pi$
$888$		0			0
$889$	−7.90786e6	−	2.31152e6i	−0.335587	−	0.0980941i
$890$		0			0
$891$	−903552.	+	1.56500e6i	−0.0381293	+	0.0660419i
$892$		0			0
$893$	−1.55786e6	−	2.69828e6i	−0.0653730	−	0.113229i
$894$		0			0
$895$	−1.84293e7			−0.769042
$896$		0			0
$897$	−9.99885e7			−4.14924
$898$		0			0
$899$	6.54688e6	+	1.13395e7i	0.270169	+	0.467946i
$900$		0			0
$901$	6.92471e6	−	1.19940e7i	0.284178	−	0.492210i
$902$		0			0
$903$	−2.98774e6	−	1.22506e7i	−0.121934	−	0.499963i
$904$		0			0
$905$	−398096.	+	689523.i	−0.0161572	+	0.0279851i
$906$		0			0
$907$	−1.21584e7	−	2.10590e7i	−0.490748	−	0.850000i	0.509195	−	0.860651i	$-0.329943\pi$
$907$	−1.21584e7	−	2.10590e7i	−0.490748	−	0.850000i	−0.999943	+	0.0106507i	$0.996610\pi$
$908$		0			0
$909$	9.96754e6			0.400109
$910$		0			0
$911$	3.36224e7			1.34225			0.671123	−	0.741346i	$-0.265812\pi$
$911$	3.36224e7			1.34225			0.671123	+	0.741346i	$0.265812\pi$
$912$		0			0
$913$	−4.33888e6	−	7.51517e6i	−0.172267	−	0.298374i
$914$		0			0
$915$	6.63710e6	−	1.14958e7i	0.262075	−	0.453927i
$916$		0			0
$917$	1.27734e7	−	1.22091e7i	0.501629	−	0.479468i
$918$		0			0
$919$	−4.06874e6	+	7.04727e6i	−0.158917	+	0.275253i	−0.934479	−	0.356019i	$-0.884134\pi$
$919$	−4.06874e6	+	7.04727e6i	−0.158917	+	0.275253i	0.775561	+	0.631272i	$0.217467\pi$
$920$		0			0
$921$	−2.95001e7	−	5.10956e7i	−1.14597	−	1.98488i
$922$		0			0
$923$	2.31338e7			0.893807
$924$		0			0
$925$	4.24962e6			0.163304
$926$		0			0
$927$	−9.62688e6	−	1.66742e7i	−0.367948	−	0.637304i
$928$		0			0
$929$	−1.79027e7	+	3.10084e7i	−0.680580	+	1.17880i	0.294224	+	0.955736i	$0.404939\pi$
$929$	−1.79027e7	+	3.10084e7i	−0.680580	+	1.17880i	−0.974804	+	0.223063i	$0.928395\pi$
$930$		0			0
$931$	−4.18173e6	+	2.16872e6i	−0.158118	+	0.0820029i
$932$		0			0
$933$	1.54249e7	−	2.67167e7i	0.580120	−	1.00480i
$934$		0			0
$935$	−3.47409e6	−	6.01730e6i	−0.129961	−	0.225098i
$936$		0			0
$937$	4.65849e7			1.73339			0.866694	−	0.498840i	$-0.166241\pi$
$937$	4.65849e7			1.73339			0.866694	+	0.498840i	$0.166241\pi$
$938$		0			0
$939$	−4.12029e7			−1.52498
$940$		0			0
$941$	1.03362e6	+	1.79029e6i	0.0380529	+	0.0659096i	0.884425	−	0.466683i	$-0.154551\pi$
$941$	1.03362e6	+	1.79029e6i	0.0380529	+	0.0659096i	−0.846372	+	0.532593i	$0.821218\pi$
$942$		0			0
$943$	−6.63850e6	+	1.14982e7i	−0.243103	+	0.421067i
$944$		0			0
$945$	−1.07201e7	+	1.02465e7i	−0.390498	+	0.373247i
$946$		0			0
$947$	1.18386e7	−	2.05051e7i	0.428970	−	0.742998i	−0.567812	−	0.823158i	$-0.692210\pi$
$947$	1.18386e7	−	2.05051e7i	0.428970	−	0.742998i	0.996782	+	0.0801602i	$0.0255432\pi$
$948$		0			0
$949$	−2.63936e7	−	4.57150e7i	−0.951334	−	1.64776i
$950$		0			0
$951$	6.50345e7			2.33181
$952$		0			0
$953$	−3.40521e6			−0.121454			−0.0607270	−	0.998154i	$-0.519342\pi$
$953$	−3.40521e6			−0.121454			−0.0607270	+	0.998154i	$0.519342\pi$
$954$		0			0
$955$	1.00762e7	+	1.74525e7i	0.357510	+	0.619226i
$956$		0			0
$957$	1.08924e7	−	1.88663e7i	0.384455	−	0.665896i
$958$		0			0
$959$	−1.02996e7	−	4.22315e7i	−0.361639	−	1.48283i
$960$		0			0
$961$	1.16357e7	−	2.01535e7i	0.406427	−	0.703952i
$962$		0			0
$963$	−4.63119e6	−	8.02145e6i	−0.160926	−	0.278732i
$964$		0			0
$965$	1.78059e6			0.0615525
$966$		0			0
$967$	−1.44238e6			−0.0496036			−0.0248018	−	0.999692i	$-0.507895\pi$
$967$	−1.44238e6			−0.0496036			−0.0248018	+	0.999692i	$0.507895\pi$
$968$		0			0
$969$	4.29880e6	+	7.44575e6i	0.147075	+	0.254741i
$970$		0			0
$971$	−2.57977e6	+	4.46830e6i	−0.0878078	+	0.152088i	−0.906584	−	0.422025i	$-0.861319\pi$
$971$	−2.57977e6	+	4.46830e6i	−0.0878078	+	0.152088i	0.818776	+	0.574113i	$0.194653\pi$
$972$		0			0
$973$	−7.24151e6	−	2.11674e6i	−0.245215	−	0.0716779i
$974$		0			0
$975$	−2.61484e7	+	4.52904e7i	−0.880914	+	1.52579i
$976$		0			0
$977$	3.48825e6	+	6.04183e6i	0.116915	+	0.202503i	0.918544	−	0.395319i	$-0.129366\pi$
$977$	3.48825e6	+	6.04183e6i	0.116915	+	0.202503i	−0.801628	+	0.597823i	$0.796033\pi$
$978$		0			0
$979$	1.48511e7			0.495225
$980$		0			0
$981$	−1.34853e6			−0.0447394
$982$		0			0
$983$	−1.40784e7	−	2.43845e7i	−0.464696	−	0.804878i	0.534491	−	0.845174i	$-0.320503\pi$
$983$	−1.40784e7	−	2.43845e7i	−0.464696	−	0.804878i	−0.999188	+	0.0402961i	$0.987170\pi$
$984$		0			0
$985$	−1.69968e7	+	2.94394e7i	−0.558184	+	0.966804i
$986$		0			0
$987$	−3.42725e7	−	1.00181e7i	−1.11983	−	0.327334i
$988$		0			0
$989$	6.83547e6	−	1.18394e7i	0.222217	−	0.384891i
$990$		0			0
$991$	−1.17022e7	−	2.02687e7i	−0.378514	−	0.655606i	0.612332	−	0.790601i	$-0.290232\pi$
$991$	−1.17022e7	−	2.02687e7i	−0.378514	−	0.655606i	−0.990846	+	0.134995i	$0.956898\pi$
$992$		0			0
$993$	4.30825e7			1.38653
$994$		0			0
$995$	2.31080e7			0.739954
$996$		0			0
$997$	2.15142e7	+	3.72637e7i	0.685468	+	1.18727i	0.973290	+	0.229581i	$0.0737356\pi$
$997$	2.15142e7	+	3.72637e7i	0.685468	+	1.18727i	−0.287822	+	0.957684i	$0.592931\pi$
$998$		0			0
$999$	−3.69592e6	+	6.40151e6i	−0.117168	+	0.202941i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.6.i.d.65.2		4
4.3	odd	2		14.6.c.a.9.1	✓	4
7.2	even	3		784.6.a.s.1.1		2
7.4	even	3	inner	112.6.i.d.81.2		4
7.5	odd	6		784.6.a.bb.1.2		2
12.11	even	2		126.6.g.j.37.1		4
28.3	even	6		98.6.c.e.67.2		4
28.11	odd	6		14.6.c.a.11.1	yes	4
28.19	even	6		98.6.a.g.1.1		2
28.23	odd	6		98.6.a.h.1.2		2
28.27	even	2		98.6.c.e.79.2		4
84.11	even	6		126.6.g.j.109.1		4
84.23	even	6		882.6.a.ba.1.2		2
84.47	odd	6		882.6.a.bi.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.a.9.1	✓	4	4.3	odd	2
14.6.c.a.11.1	yes	4	28.11	odd	6
98.6.a.g.1.1		2	28.19	even	6
98.6.a.h.1.2		2	28.23	odd	6
98.6.c.e.67.2		4	28.3	even	6
98.6.c.e.79.2		4	28.27	even	2
112.6.i.d.65.2		4	1.1	even	1	trivial
112.6.i.d.81.2		4	7.4	even	3	inner
126.6.g.j.37.1		4	12.11	even	2
126.6.g.j.109.1		4	84.11	even	6
784.6.a.s.1.1		2	7.2	even	3
784.6.a.bb.1.2		2	7.5	odd	6
882.6.a.ba.1.2		2	84.23	even	6
882.6.a.bi.1.1		2	84.47	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(12.3882 + 21.4570i) q^{3} +(18.0528 - 31.2683i) q^{5} +(124.435 + 36.3731i) q^{7} +(-185.435 + 321.182i) q^{9} +(77.7174 + 134.610i) q^{11} +1158.87 q^{13} +894.565 q^{15} +(-619.040 - 1072.21i) q^{17} +(-140.140 + 242.729i) q^{19} +(761.065 + 3120.59i) q^{21} +(-1741.20 + 3015.84i) q^{23} +(910.694 + 1577.37i) q^{25} -3168.14 q^{27} -5656.78 q^{29} +(-1157.35 - 2004.59i) q^{31} +(-1925.56 + 3335.16i) q^{33} +(3383.72 - 3234.23i) q^{35} +(1166.59 - 2020.59i) q^{37} +(14356.3 + 24865.8i) q^{39} +3812.61 q^{41} -3925.73 q^{43} +(6695.22 + 11596.5i) q^{45} +(-5558.23 + 9627.13i) q^{47} +(14161.0 + 9052.14i) q^{49} +(15337.6 - 26565.5i) q^{51} +(5593.10 + 9687.54i) q^{53} +5612.06 q^{55} -6944.31 q^{57} +(-3005.18 - 5205.12i) q^{59} +(7419.36 - 12850.7i) q^{61} +(-34756.9 + 33221.4i) q^{63} +(20920.8 - 36235.9i) q^{65} +(-21491.9 - 37225.1i) q^{67} -86281.1 q^{69} +19962.4 q^{71} +(-22775.3 - 39448.0i) q^{73} +(-22563.7 + 39081.5i) q^{75} +(4774.54 + 19577.0i) q^{77} +(54496.0 - 94389.9i) q^{79} +(5813.07 + 10068.5i) q^{81} -55829.0 q^{83} -44701.6 q^{85} +(-70077.3 - 121377. i) q^{87} +(47772.9 - 82745.1i) q^{89} +(144204. + 42151.6i) q^{91} +(28675.0 - 49666.5i) q^{93} +(5059.82 + 8763.86i) q^{95} +15004.9 q^{97} -57646.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{3} - 70 q^{5} - 244 q^{9} + 62 q^{11} + 3640 q^{13} + 4076 q^{15} - 1694 q^{17} + 826 q^{19} + 3542 q^{21} - 2734 q^{23} - 6312 q^{25} - 14308 q^{27} - 5704 q^{29} - 2674 q^{31} - 4858 q^{33}+ \cdots - 139000 q^{99}+O(q^{100}) \) 4 * q + 14 * q^3 - 70 * q^5 - 244 * q^9 + 62 * q^11 + 3640 * q^13 + 4076 * q^15 - 1694 * q^17 + 826 * q^19 + 3542 * q^21 - 2734 * q^23 - 6312 * q^25 - 14308 * q^27 - 5704 * q^29 - 2674 * q^31 - 4858 * q^33 + 13286 * q^35 + 9146 * q^37 + 21588 * q^39 + 12264 * q^41 + 32080 * q^43 + 26852 * q^45 - 25326 * q^47 + 56644 * q^49 + 25762 * q^51 - 14958 * q^53 + 31052 * q^55 - 37732 * q^57 - 1106 * q^59 + 28042 * q^61 - 77308 * q^63 - 28308 * q^65 - 102642 * q^67 - 188692 * q^69 + 22112 * q^71 - 35070 * q^73 - 132776 * q^75 + 27062 * q^77 + 101762 * q^79 + 23750 * q^81 + 89264 * q^83 + 7348 * q^85 - 170380 * q^87 + 75474 * q^89 + 123872 * q^91 + 53478 * q^93 + 127502 * q^95 - 16632 * q^97 - 139000 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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