LMFDB - Embedded newform 126.6.g.e.109.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,6,Mod(37,126)] 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("126.37"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	126.g (of order $3$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,-8,0,-32,-42] 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:	$20.2083612964$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{130})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 130x^{2} + 16900 $ x^4 + 130*x^2 + 16900
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3 $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.2
Root			$5.70088 - 9.87421i$ of defining polynomial
Character	$\chi$	$=$	126.109
Dual form			126.6.g.e.37.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-10.5000 - 18.1865i) q^{5} +(126.411 - 28.7642i) q^{7} +64.0000 q^{8} +(-42.0000 + 72.7461i) q^{10} +(-312.937 + 542.023i) q^{11} -206.821 q^{13} +(-352.463 - 380.371i) q^{14} +(-128.000 - 221.703i) q^{16} +(530.732 - 919.254i) q^{17} +(941.989 + 1631.57i) q^{19} +336.000 q^{20} +2503.49 q^{22} +(1858.68 + 3219.34i) q^{23} +(1342.00 - 2324.41i) q^{25} +(413.642 + 716.449i) q^{26} +(-612.716 + 1981.71i) q^{28} +123.747 q^{29} +(4554.63 - 7888.85i) q^{31} +(-512.000 + 886.810i) q^{32} -4245.85 q^{34} +(-1850.43 - 1996.95i) q^{35} +(3014.36 + 5221.03i) q^{37} +(3767.96 - 6526.29i) q^{38} +(-672.000 - 1163.94i) q^{40} +17201.9 q^{41} +5401.98 q^{43} +(-5006.99 - 8672.36i) q^{44} +(7434.74 - 12877.3i) q^{46} +(937.621 + 1624.01i) q^{47} +(15152.2 - 7272.19i) q^{49} -10736.0 q^{50} +(1654.57 - 2865.80i) q^{52} +(9353.62 - 16200.9i) q^{53} +13143.3 q^{55} +(8090.27 - 1840.91i) q^{56} +(-247.495 - 428.673i) q^{58} +(-1267.39 + 2195.18i) q^{59} +(1047.35 + 1814.07i) q^{61} -36437.1 q^{62} +4096.00 q^{64} +(2171.62 + 3761.36i) q^{65} +(-29310.4 + 50767.1i) q^{67} +(8491.71 + 14708.1i) q^{68} +(-3216.76 + 10404.0i) q^{70} +31279.5 q^{71} +(3575.24 - 6192.49i) q^{73} +(12057.5 - 20884.1i) q^{74} -30143.7 q^{76} +(-23967.7 + 77518.7i) q^{77} +(-1489.91 - 2580.59i) q^{79} +(-2688.00 + 4655.75i) q^{80} +(-34403.7 - 59589.0i) q^{82} -45954.6 q^{83} -22290.7 q^{85} +(-10804.0 - 18713.0i) q^{86} +(-20028.0 + 34689.4i) q^{88} +(49520.0 + 85771.1i) q^{89} +(-26144.4 + 5949.04i) q^{91} -59477.9 q^{92} +(3750.48 - 6496.03i) q^{94} +(19781.8 - 34263.0i) q^{95} -115548. q^{97} +(-55496.1 - 37944.5i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} - 32 q^{4} - 42 q^{5} + 232 q^{7} + 256 q^{8} - 168 q^{10} - 294 q^{11} - 280 q^{13} + 232 q^{14} - 512 q^{16} + 1302 q^{17} + 1442 q^{19} + 1344 q^{20} + 2352 q^{22} + 2646 q^{23} + 5368 q^{25}+ \cdots - 63272 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 - 32 * q^4 - 42 * q^5 + 232 * q^7 + 256 * q^8 - 168 * q^10 - 294 * q^11 - 280 * q^13 + 232 * q^14 - 512 * q^16 + 1302 * q^17 + 1442 * q^19 + 1344 * q^20 + 2352 * q^22 + 2646 * q^23 + 5368 * q^25 + 560 * q^26 - 4640 * q^28 - 3336 * q^29 + 14798 * q^31 - 2048 * q^32 - 10416 * q^34 + 1218 * q^35 - 5182 * q^37 + 5768 * q^38 - 2688 * q^40 + 10248 * q^41 - 9040 * q^43 - 4704 * q^44 + 10584 * q^46 - 14994 * q^47 - 2876 * q^49 - 42944 * q^50 + 2240 * q^52 + 24006 * q^53 + 12348 * q^55 + 14848 * q^56 + 6672 * q^58 + 38850 * q^59 + 23618 * q^61 - 118384 * q^62 + 16384 * q^64 + 2940 * q^65 - 32002 * q^67 + 20832 * q^68 - 24360 * q^70 + 178752 * q^71 + 47138 * q^73 - 20728 * q^74 - 46144 * q^76 + 22890 * q^77 + 40970 * q^79 - 10752 * q^80 - 20496 * q^82 - 136752 * q^83 - 54684 * q^85 + 18080 * q^86 - 18816 * q^88 + 123102 * q^89 - 53680 * q^91 - 84672 * q^92 - 59976 * q^94 + 30282 * q^95 - 87304 * q^97 - 63272 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/126\mathbb{Z}\right)^\times$.

$n$	$29$	$73$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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 $

$2$	−2.00000	−	3.46410i	−0.353553	−	0.612372i
$2$	−2.00000	−	3.46410i	−0.353553	−	0.612372i
$3$		0			0
$3$		0			0
$4$	−8.00000	+	13.8564i	−0.250000	+	0.433013i
$5$	−10.5000	−	18.1865i	−0.187830	−	0.325331i	0.756697	−	0.653766i	$-0.226812\pi$
$5$	−10.5000	−	18.1865i	−0.187830	−	0.325331i	−0.944526	+	0.328436i	$0.893479\pi$
$6$		0			0
$7$	126.411	−	28.7642i	0.975075	−	0.221874i
$7$	126.411	−	28.7642i	0.975075	−	0.221874i
$8$	64.0000			0.353553
$9$		0			0
$10$	−42.0000	+	72.7461i	−0.132816	+	0.230043i
$11$	−312.937	+	542.023i	−0.779785	+	1.35063i	0.152280	+	0.988337i	$0.451338\pi$
$11$	−312.937	+	542.023i	−0.779785	+	1.35063i	−0.932065	+	0.362290i	$0.881995\pi$
$12$		0			0
$13$	−206.821			−0.339419			−0.169710	−	0.985494i	$-0.554283\pi$
$13$	−206.821			−0.339419			−0.169710	+	0.985494i	$0.554283\pi$
$14$	−352.463	−	380.371i	−0.480611	−	0.518665i
$15$		0			0
$16$	−128.000	−	221.703i	−0.125000	−	0.216506i
$17$	530.732	−	919.254i	0.445402	−	0.771460i	−0.552678	−	0.833395i	$-0.686394\pi$
$17$	530.732	−	919.254i	0.445402	−	0.771460i	0.998080	+	0.0619354i	$0.0197273\pi$
$18$		0			0
$19$	941.989	+	1631.57i	0.598635	+	1.03687i	0.993023	+	0.117922i	$0.0376233\pi$
$19$	941.989	+	1631.57i	0.598635	+	1.03687i	−0.394388	+	0.918944i	$0.629043\pi$
$20$	336.000			0.187830
$21$		0			0
$22$	2503.49			1.10278
$23$	1858.68	+	3219.34i	0.732632	+	1.26896i	0.955754	+	0.294166i	$0.0950418\pi$
$23$	1858.68	+	3219.34i	0.732632	+	1.26896i	−0.223122	+	0.974790i	$0.571625\pi$
$24$		0			0
$25$	1342.00	−	2324.41i	0.429440	−	0.743812i
$26$	413.642	+	716.449i	0.120003	+	0.207851i
$27$		0			0
$28$	−612.716	+	1981.71i	−0.147694	+	0.477689i
$29$	123.747			0.0273238			0.0136619	−	0.999907i	$-0.495651\pi$
$29$	123.747			0.0273238			0.0136619	+	0.999907i	$0.495651\pi$
$30$		0			0
$31$	4554.63	−	7888.85i	0.851234	−	1.47438i	−0.0288611	−	0.999583i	$-0.509188\pi$
$31$	4554.63	−	7888.85i	0.851234	−	1.47438i	0.880095	−	0.474797i	$-0.157479\pi$
$32$	−512.000	+	886.810i	−0.0883883	+	0.153093i
$33$		0			0
$34$	−4245.85			−0.629894
$35$	−1850.43	−	1996.95i	−0.255331	−	0.275547i
$36$		0			0
$37$	3014.36	+	5221.03i	0.361986	+	0.626977i	0.988288	−	0.152603i	$-0.0487656\pi$
$37$	3014.36	+	5221.03i	0.361986	+	0.626977i	−0.626302	+	0.779581i	$0.715432\pi$
$38$	3767.96	−	6526.29i	0.423299	−	0.733175i
$39$		0			0
$40$	−672.000	−	1163.94i	−0.0664078	−	0.115022i
$41$	17201.9			1.59814			0.799071	−	0.601236i	$-0.205325\pi$
$41$	17201.9			1.59814			0.799071	+	0.601236i	$0.205325\pi$
$42$		0			0
$43$	5401.98			0.445535			0.222767	−	0.974872i	$-0.428491\pi$
$43$	5401.98			0.445535			0.222767	+	0.974872i	$0.428491\pi$
$44$	−5006.99	−	8672.36i	−0.389893	−	0.675314i
$45$		0			0
$46$	7434.74	−	12877.3i	0.518049	−	0.897288i
$47$	937.621	+	1624.01i	0.0619131	+	0.107237i	0.895321	−	0.445422i	$-0.146946\pi$
$47$	937.621	+	1624.01i	0.0619131	+	0.107237i	−0.833407	+	0.552659i	$0.813613\pi$
$48$		0			0
$49$	15152.2	−	7272.19i	0.901544	−	0.432688i
$50$	−10736.0			−0.607320
$51$		0			0
$52$	1654.57	−	2865.80i	0.0848548	−	0.146973i
$53$	9353.62	−	16200.9i	0.457393	−	0.792229i	−0.541429	−	0.840747i	$-0.682116\pi$
$53$	9353.62	−	16200.9i	0.457393	−	0.792229i	0.998822	+	0.0485180i	$0.0154498\pi$
$54$		0			0
$55$	13143.3			0.585867
$56$	8090.27	−	1840.91i	0.344741	−	0.0784444i
$57$		0			0
$58$	−247.495	−	428.673i	−0.00966042	−	0.0167323i
$59$	−1267.39	+	2195.18i	−0.0474002	+	0.0820995i	−0.888752	−	0.458388i	$-0.848427\pi$
$59$	−1267.39	+	2195.18i	−0.0474002	+	0.0820995i	0.841352	+	0.540488i	$0.181760\pi$
$60$		0			0
$61$	1047.35	+	1814.07i	0.0360386	+	0.0624208i	0.883482	−	0.468465i	$-0.155193\pi$
$61$	1047.35	+	1814.07i	0.0360386	+	0.0624208i	−0.847444	+	0.530886i	$0.821859\pi$
$62$	−36437.1			−1.20383
$63$		0			0
$64$	4096.00			0.125000
$65$	2171.62	+	3761.36i	0.0637530	+	0.110423i
$66$		0			0
$67$	−29310.4	+	50767.1i	−0.797691	+	1.38164i	0.123426	+	0.992354i	$0.460612\pi$
$67$	−29310.4	+	50767.1i	−0.797691	+	1.38164i	−0.921117	+	0.389287i	$0.872722\pi$
$68$	8491.71	+	14708.1i	0.222701	+	0.385730i
$69$		0			0
$70$	−3216.76	+	10404.0i	−0.0784645	+	0.253778i
$71$	31279.5			0.736401			0.368201	−	0.929746i	$-0.379974\pi$
$71$	31279.5			0.736401			0.368201	+	0.929746i	$0.379974\pi$
$72$		0			0
$73$	3575.24	−	6192.49i	0.0785231	−	0.136006i	−0.824090	−	0.566459i	$-0.808313\pi$
$73$	3575.24	−	6192.49i	0.0785231	−	0.136006i	0.902613	+	0.430453i	$0.141646\pi$
$74$	12057.5	−	20884.1i	0.255962	−	0.443340i
$75$		0			0
$76$	−30143.7			−0.598635
$77$	−23967.7	+	77518.7i	−0.460680	+	1.48998i
$78$		0			0
$79$	−1489.91	−	2580.59i	−0.0268591	−	0.0465213i	0.852283	−	0.523080i	$-0.175217\pi$
$79$	−1489.91	−	2580.59i	−0.0268591	−	0.0465213i	−0.879143	+	0.476559i	$0.841884\pi$
$80$	−2688.00	+	4655.75i	−0.0469574	+	0.0813327i
$81$		0			0
$82$	−34403.7	−	59589.0i	−0.565029	−	0.978659i
$83$	−45954.6			−0.732207			−0.366103	−	0.930574i	$-0.619308\pi$
$83$	−45954.6			−0.732207			−0.366103	+	0.930574i	$0.619308\pi$
$84$		0			0
$85$	−22290.7			−0.334639
$86$	−10804.0	−	18713.0i	−0.157520	−	0.272833i
$87$		0			0
$88$	−20028.0	+	34689.4i	−0.275696	+	0.477519i
$89$	49520.0	+	85771.1i	0.662682	+	1.14780i	0.979908	+	0.199450i	$0.0639154\pi$
$89$	49520.0	+	85771.1i	0.662682	+	1.14780i	−0.317226	+	0.948350i	$0.602751\pi$
$90$		0			0
$91$	−26144.4	+	5949.04i	−0.330959	+	0.0753084i
$92$	−59477.9			−0.732632
$93$		0			0
$94$	3750.48	−	6496.03i	0.0437792	−	0.0758278i
$95$	19781.8	−	34263.0i	0.224883	−	0.389509i
$96$		0			0
$97$	−115548.			−1.24691			−0.623454	−	0.781860i	$-0.714271\pi$
$97$	−115548.			−1.24691			−0.623454	+	0.781860i	$0.714271\pi$
$98$	−55496.1	−	37944.5i	−0.583710	−	0.399102i
$99$		0			0
$100$	21472.0	+	37190.6i	0.214720	+	0.371906i
$101$	5475.73	−	9484.25i	0.0534120	−	0.0925123i	−0.838083	−	0.545542i	$-0.816324\pi$
$101$	5475.73	−	9484.25i	0.0534120	−	0.0925123i	0.891495	+	0.453030i	$0.149657\pi$
$102$		0			0
$103$	68862.2	+	119273.i	0.639570	+	1.10777i	0.985527	+	0.169517i	$0.0542208\pi$
$103$	68862.2	+	119273.i	0.639570	+	1.10777i	−0.345957	+	0.938250i	$0.612446\pi$
$104$	−13236.5			−0.120003
$105$		0			0
$106$	−74828.9			−0.646852
$107$	37786.5	+	65448.1i	0.319064	+	0.552634i	0.980293	−	0.197550i	$-0.0632983\pi$
$107$	37786.5	+	65448.1i	0.319064	+	0.552634i	−0.661229	+	0.750184i	$0.729965\pi$
$108$		0			0
$109$	−22263.1	+	38560.9i	−0.179482	+	0.310871i	−0.941703	−	0.336445i	$-0.890775\pi$
$109$	−22263.1	+	38560.9i	−0.179482	+	0.310871i	0.762221	+	0.647316i	$0.224109\pi$
$110$	−26286.7	−	45529.9i	−0.207135	−	0.358769i
$111$		0			0
$112$	−22557.6	−	24343.7i	−0.169922	−	0.183376i
$113$	−90456.5			−0.666413			−0.333207	−	0.942854i	$-0.608131\pi$
$113$	−90456.5			−0.666413			−0.333207	+	0.942854i	$0.608131\pi$
$114$		0			0
$115$	39032.4	−	67606.0i	0.275220	−	0.476695i
$116$	−989.979	+	1714.69i	−0.00683095	+	0.0118315i
$117$		0			0
$118$	10139.1			0.0670340
$119$	40648.5	−	131469.i	0.263134	−	0.851055i
$120$		0			0
$121$	−115333.	−	199763.i	−0.716130	−	1.24037i
$122$	4189.41	−	7256.27i	0.0254832	−	0.0441381i
$123$		0			0
$124$	72874.1	+	126222.i	0.425617	+	0.737190i
$125$	−121989.			−0.698306
$126$		0			0
$127$	187707.			1.03269			0.516346	−	0.856380i	$-0.327292\pi$
$127$	187707.			1.03269			0.516346	+	0.856380i	$0.327292\pi$
$128$	−8192.00	−	14189.0i	−0.0441942	−	0.0765466i
$129$		0			0
$130$	8686.48	−	15045.4i	0.0450802	−	0.0780812i
$131$	77206.1	+	133725.i	0.393073	+	0.680823i	0.992853	−	0.119342i	$-0.0380785\pi$
$131$	77206.1	+	133725.i	0.393073	+	0.680823i	−0.599780	+	0.800165i	$0.704745\pi$
$132$		0			0
$133$	166008.	+	179153.i	0.813768	+	0.878201i
$134$	234483.			1.12810
$135$		0			0
$136$	33966.8	−	58832.3i	0.157474	−	0.272752i
$137$	54618.9	−	94602.7i	0.248623	−	0.430628i	−0.714521	−	0.699614i	$-0.753355\pi$
$137$	54618.9	−	94602.7i	0.248623	−	0.430628i	0.963144	+	0.268986i	$0.0866886\pi$
$138$		0			0
$139$	−204695.			−0.898609			−0.449305	−	0.893379i	$-0.648328\pi$
$139$	−204695.			−0.898609			−0.449305	+	0.893379i	$0.648328\pi$
$140$	42473.9	−	9664.77i	0.183148	−	0.0416746i
$141$		0			0
$142$	−62559.1	−	108355.i	−0.260357	−	0.450952i
$143$	64721.9	−	112102.i	0.264674	−	0.458429i
$144$		0			0
$145$	−1299.35	−	2250.54i	−0.00513222	−	0.00888926i
$146$	−28601.9			−0.111048
$147$		0			0
$148$	−96459.6			−0.361986
$149$	−203154.	−	351873.i	−0.749651	−	1.29843i	−0.947990	−	0.318300i	$-0.896888\pi$
$149$	−203154.	−	351873.i	−0.749651	−	1.29843i	0.198339	−	0.980134i	$-0.436445\pi$
$150$		0			0
$151$	208419.	−	360992.i	0.743866	−	1.28841i	−0.206856	−	0.978371i	$-0.566323\pi$
$151$	208419.	−	360992.i	0.743866	−	1.28841i	0.950723	−	0.310043i	$-0.100343\pi$
$152$	60287.3	+	104421.i	0.211649	+	0.366588i
$153$		0			0
$154$	316468.	−	72011.0i	1.07530	−	0.244679i
$155$	−191295.			−0.639548
$156$		0			0
$157$	−59645.3	+	103309.i	−0.193120	+	0.334493i	−0.946283	−	0.323341i	$-0.895194\pi$
$157$	−59645.3	+	103309.i	−0.193120	+	0.334493i	0.753163	+	0.657834i	$0.228527\pi$
$158$	−5959.62	+	10322.4i	−0.0189922	+	0.0328955i
$159$		0			0
$160$	21504.0			0.0664078
$161$	327559.	+	353494.i	0.995920	+	1.07478i
$162$		0			0
$163$	−23686.2	−	41025.7i	−0.0698275	−	0.120945i	0.828998	−	0.559252i	$-0.188912\pi$
$163$	−23686.2	−	41025.7i	−0.0698275	−	0.120945i	−0.898825	+	0.438307i	$0.855578\pi$
$164$	−137615.	+	238356.i	−0.399536	+	0.692016i
$165$		0			0
$166$	91909.2	+	159191.i	0.258874	+	0.448383i
$167$	231669.			0.642802			0.321401	−	0.946943i	$-0.395846\pi$
$167$	231669.			0.642802			0.321401	+	0.946943i	$0.395846\pi$
$168$		0			0
$169$	−328518.			−0.884795
$170$	44581.5	+	77217.3i	0.118313	+	0.204924i
$171$		0			0
$172$	−43215.8	+	74852.0i	−0.111384	+	0.192922i
$173$	67169.6	+	116341.i	0.170631	+	0.295541i	0.938641	−	0.344897i	$-0.112086\pi$
$173$	67169.6	+	116341.i	0.170631	+	0.295541i	−0.768010	+	0.640438i	$0.778753\pi$
$174$		0			0
$175$	102783.	−	332432.i	0.253704	−	0.820554i
$176$	160224.			0.389893
$177$		0			0
$178$	198080.	−	343085.i	0.468587	−	0.811617i
$179$	23292.2	−	40343.2i	0.0543347	−	0.0941105i	−0.837579	−	0.546317i	$-0.816030\pi$
$179$	23292.2	−	40343.2i	0.0543347	−	0.0941105i	0.891913	+	0.452206i	$0.149363\pi$
$180$		0			0
$181$	829210.			1.88134			0.940672	−	0.339317i	$-0.110196\pi$
$181$	829210.			1.88134			0.940672	+	0.339317i	$0.110196\pi$
$182$	72896.8	+	78668.6i	0.163129	+	0.176045i
$183$		0			0
$184$	118956.	+	206037.i	0.259025	+	0.448644i
$185$	63301.6	−	109642.i	0.135983	−	0.235530i
$186$		0			0
$187$	332171.	+	575337.i	0.694636	+	1.20315i
$188$	−30003.9			−0.0619131
$189$		0			0
$190$	−158254.			−0.318032
$191$	−235958.	−	408692.i	−0.468006	−	0.810611i	0.531325	−	0.847168i	$-0.321694\pi$
$191$	−235958.	−	408692.i	−0.468006	−	0.810611i	−0.999332	+	0.0365572i	$0.988361\pi$
$192$		0			0
$193$	−344177.	+	596132.i	−0.665103	+	1.15199i	0.314155	+	0.949372i	$0.398279\pi$
$193$	−344177.	+	596132.i	−0.665103	+	1.15199i	−0.979257	+	0.202620i	$0.935054\pi$
$194$	231097.	+	400271.i	0.440849	+	0.763573i
$195$		0			0
$196$	−20451.5	+	268133.i	−0.0380263	+	0.498552i
$197$	−311915.			−0.572625			−0.286313	−	0.958136i	$-0.592430\pi$
$197$	−311915.			−0.572625			−0.286313	+	0.958136i	$0.592430\pi$
$198$		0			0
$199$	143606.	−	248733.i	0.257063	−	0.445246i	−0.708391	−	0.705820i	$-0.750579\pi$
$199$	143606.	−	248733.i	0.257063	−	0.445246i	0.965454	+	0.260574i	$0.0839119\pi$
$200$	85888.0	−	148762.i	0.151830	−	0.262977i
$201$		0			0
$202$	−43805.9			−0.0755360
$203$	15643.0	−	3559.49i	0.0266428	−	0.00606245i
$204$		0			0
$205$	−180619.	−	312842.i	−0.300179	−	0.519925i
$206$	275449.	−	477091.i	0.452244	−	0.783310i
$207$		0			0
$208$	26473.1	+	45852.7i	0.0424274	+	0.0734864i
$209$	−1.17913e6			−1.86723
$210$		0			0
$211$	−460493.			−0.712061			−0.356031	−	0.934474i	$-0.615870\pi$
$211$	−460493.			−0.712061			−0.356031	+	0.934474i	$0.615870\pi$
$212$	149658.	+	259215.i	0.228697	+	0.396114i
$213$		0			0
$214$	151146.	−	261793.i	0.225612	−	0.390771i
$215$	−56720.8	−	98243.3i	−0.0836847	−	0.144946i
$216$		0			0
$217$	348837.	−	1.12824e6i	0.502890	−	1.62650i
$218$	178105.			0.253825
$219$		0			0
$220$	−105147.	+	182120.i	−0.146467	+	0.253688i
$221$	−109766.	+	190121.i	−0.151178	+	0.261848i
$222$		0			0
$223$	−1.19776e6			−1.61290			−0.806449	−	0.591304i	$-0.798613\pi$
$223$	−1.19776e6			−1.61290			−0.806449	+	0.591304i	$0.798613\pi$
$224$	−39213.8	+	126829.i	−0.0522179	+	0.168888i
$225$		0			0
$226$	180913.	+	313350.i	0.235613	+	0.408093i
$227$	447281.	−	774713.i	0.576123	−	0.997875i	−0.419795	−	0.907619i	$-0.637898\pi$
$227$	447281.	−	774713.i	0.576123	−	0.997875i	0.995919	−	0.0902559i	$-0.0287685\pi$
$228$		0			0
$229$	129628.	+	224522.i	0.163347	+	0.282925i	0.936067	−	0.351822i	$-0.114438\pi$
$229$	129628.	+	224522.i	0.163347	+	0.282925i	−0.772720	+	0.634747i	$0.781104\pi$
$230$	−312259.			−0.389220
$231$		0			0
$232$	7919.83			0.00966042
$233$	105657.	+	183004.i	0.127500	+	0.220836i	0.922707	−	0.385501i	$-0.125971\pi$
$233$	105657.	+	183004.i	0.127500	+	0.220836i	−0.795207	+	0.606337i	$0.792638\pi$
$234$		0			0
$235$	19690.0	−	34104.2i	0.0232582	−	0.0402845i
$236$	−20278.2	−	35122.9i	−0.0237001	−	0.0410498i
$237$		0			0
$238$	−536720.	+	122129.i	−0.614194	+	0.139757i
$239$	−463.018			−0.000524328			−0.000262164	−	1.00000i	$-0.500083\pi$
$239$	−463.018			−0.000524328			−0.000262164		1.00000i	$0.500083\pi$
$240$		0			0
$241$	−143197.	+	248025.i	−0.158815	+	0.275076i	−0.934442	−	0.356116i	$-0.884101\pi$
$241$	−143197.	+	248025.i	−0.158815	+	0.275076i	0.775626	+	0.631192i	$0.217434\pi$
$242$	−461334.	+	799053.i	−0.506380	+	0.877076i
$243$		0			0
$244$	−33515.3			−0.0360386
$245$	−291355.	−	199209.i	−0.310103	−	0.212028i
$246$		0			0
$247$	−194823.	−	337444.i	−0.203188	−	0.351932i
$248$	291496.	−	504887.i	0.300957	−	0.521272i
$249$		0			0
$250$	243978.	+	422582.i	0.246888	+	0.427623i
$251$	1.37168e6			1.37426			0.687129	−	0.726535i	$-0.258870\pi$
$251$	1.37168e6			1.37426			0.687129	+	0.726535i	$0.258870\pi$
$252$		0			0
$253$	−2.32660e6			−2.28518
$254$	−375414.	−	650236.i	−0.365112	−	0.632392i
$255$		0			0
$256$	−32768.0	+	56755.8i	−0.0312500	+	0.0541266i
$257$	−379612.	−	657507.i	−0.358515	−	0.620965i	0.629198	−	0.777245i	$-0.283383\pi$
$257$	−379612.	−	657507.i	−0.358515	−	0.620965i	−0.987713	+	0.156279i	$0.950050\pi$
$258$		0			0
$259$	531226.	+	573287.i	0.492073	+	0.531035i
$260$	−69491.9			−0.0637530
$261$		0			0
$262$	308825.	−	534900.i	0.277945	−	0.481415i
$263$	−536222.	+	928764.i	−0.478030	+	0.827973i	−0.999683	−	0.0251852i	$-0.991982\pi$
$263$	−536222.	+	928764.i	−0.478030	+	0.827973i	0.521652	+	0.853158i	$0.325316\pi$
$264$		0			0
$265$	−392852.			−0.343648
$266$	288586.	−	933375.i	0.250075	−	0.808820i
$267$		0			0
$268$	−468966.	−	812273.i	−0.398845	−	0.690820i
$269$	−883056.	+	1.52950e6i	−0.744059	+	1.28875i	0.206574	+	0.978431i	$0.433769\pi$
$269$	−883056.	+	1.52950e6i	−0.744059	+	1.28875i	−0.950633	+	0.310317i	$0.899565\pi$
$270$		0			0
$271$	1.02586e6	+	1.77684e6i	0.848522	+	1.46968i	0.882527	+	0.470262i	$0.155841\pi$
$271$	1.02586e6	+	1.77684e6i	0.848522	+	1.46968i	−0.0340046	+	0.999422i	$0.510826\pi$
$272$	−271735.			−0.222701
$273$		0			0
$274$	−436951.			−0.351606
$275$	839922.	+	1.45479e6i	0.669742	+	1.16003i
$276$		0			0
$277$	−435341.	+	754033.i	−0.340903	+	0.590461i	−0.984601	−	0.174819i	$-0.944066\pi$
$277$	−435341.	+	754033.i	−0.340903	+	0.590461i	0.643698	+	0.765280i	$0.277399\pi$
$278$	409391.	+	709085.i	0.317706	+	0.550284i
$279$		0			0
$280$	−118428.	−	127804.i	−0.0902730	−	0.0974207i
$281$	−2.35412e6			−1.77854			−0.889270	−	0.457383i	$-0.848787\pi$
$281$	−2.35412e6			−1.77854			−0.889270	+	0.457383i	$0.848787\pi$
$282$		0			0
$283$	1.09518e6	−	1.89690e6i	0.812863	−	1.40792i	−0.0979887	−	0.995188i	$-0.531241\pi$
$283$	1.09518e6	−	1.89690e6i	0.812863	−	1.40792i	0.910852	−	0.412733i	$-0.135426\pi$
$284$	−250236.	+	433422.i	−0.184100	+	0.318871i
$285$		0			0
$286$	−517775.			−0.374306
$287$	2.17450e6	−	494797.i	1.55831	−	0.354587i
$288$		0			0
$289$	146576.	+	253878.i	0.103233	+	0.178805i
$290$	−5197.39	+	9002.14i	−0.00362903	+	0.00628566i
$291$		0			0
$292$	57203.8	+	99079.9i	0.0392616	+	0.0680030i
$293$	807700.			0.549644			0.274822	−	0.961495i	$-0.411381\pi$
$293$	807700.			0.549644			0.274822	+	0.961495i	$0.411381\pi$
$294$		0			0
$295$	53230.4			0.0356127
$296$	192919.	+	334146.i	0.127981	+	0.221670i
$297$		0			0
$298$	−812615.	+	1.40749e6i	−0.530083	+	0.918131i
$299$	−384415.	−	665826.i	−0.248669	−	0.430708i
$300$		0			0
$301$	682867.	−	155384.i	0.434430	−	0.0988528i
$302$	−1.66735e6			−1.05199
$303$		0			0
$304$	241149.	−	417683.i	0.149659	−	0.259217i
$305$	21994.4	−	38095.4i	0.0135383	−	0.0234489i
$306$		0			0
$307$	211516.			0.128085			0.0640424	−	0.997947i	$-0.479601\pi$
$307$	211516.			0.128085			0.0640424	+	0.997947i	$0.479601\pi$
$308$	−882390.	−	952256.i	−0.530009	−	0.571975i
$309$		0			0
$310$	382589.	+	662664.i	0.226114	+	0.391642i
$311$	−145855.	+	252627.i	−0.0855104	+	0.148108i	−0.905609	−	0.424114i	$-0.860585\pi$
$311$	−145855.	+	252627.i	−0.0855104	+	0.148108i	0.820098	+	0.572223i	$0.193919\pi$
$312$		0			0
$313$	997988.	+	1.72857e6i	0.575791	+	0.997299i	0.995955	+	0.0898507i	$0.0286390\pi$
$313$	997988.	+	1.72857e6i	0.575791	+	0.997299i	−0.420165	+	0.907448i	$0.638028\pi$
$314$	477162.			0.273113
$315$		0			0
$316$	47677.0			0.0268591
$317$	−1.40479e6	−	2.43316e6i	−0.785167	−	1.35995i	−0.928899	−	0.370332i	$-0.879244\pi$
$317$	−1.40479e6	−	2.43316e6i	−0.785167	−	1.35995i	0.143733	−	0.989617i	$-0.454089\pi$
$318$		0			0
$319$	−38725.1	+	67073.9i	−0.0213067	+	0.0369043i
$320$	−43008.0	−	74492.0i	−0.0234787	−	0.0406663i
$321$		0			0
$322$	569423.	−	1.84169e6i	0.306052	−	0.989865i
$323$	1.99977e6			1.06653
$324$		0			0
$325$	−277554.	+	480737.i	−0.145760	+	0.252464i
$326$	−94744.8	+	164103.i	−0.0493755	+	0.0855209i
$327$		0			0
$328$	1.10092e6			0.565029
$329$	165238.	+	178322.i	0.0841630	+	0.0908269i
$330$		0			0
$331$	−690248.	−	1.19554e6i	−0.346286	−	0.599785i	0.639301	−	0.768957i	$-0.279224\pi$
$331$	−690248.	−	1.19554e6i	−0.346286	−	0.599785i	−0.985587	+	0.169172i	$0.945891\pi$
$332$	367637.	−	636766.i	0.183052	−	0.317055i
$333$		0			0
$334$	−463338.	−	802526.i	−0.227265	−	0.393634i
$335$	1.23104e6			0.599320
$336$		0			0
$337$	566429.			0.271688			0.135844	−	0.990730i	$-0.456625\pi$
$337$	566429.			0.271688			0.135844	+	0.990730i	$0.456625\pi$
$338$	657036.	+	1.13802e6i	0.312822	+	0.541824i
$339$		0			0
$340$	178326.	−	308869.i	0.0836598	−	0.144903i
$341$	2.85062e6	+	4.93743e6i	1.32756	+	2.29940i
$342$		0			0
$343$	1.70622e6	−	1.35512e6i	0.783070	−	0.621933i
$344$	345727.			0.157520
$345$		0			0
$346$	268678.	−	465365.i	0.120654	−	0.208979i
$347$	270104.	−	467833.i	0.120422	−	0.208577i	−0.799512	−	0.600650i	$-0.794908\pi$
$347$	270104.	−	467833.i	0.120422	−	0.208577i	0.919934	+	0.392073i	$0.128242\pi$
$348$		0			0
$349$	1.73807e6			0.763841			0.381921	−	0.924195i	$-0.375263\pi$
$349$	1.73807e6			0.763841			0.381921	+	0.924195i	$0.375263\pi$
$350$	−1.35714e6	+	308812.i	−0.592183	+	0.134749i
$351$		0			0
$352$	−320447.	−	555031.i	−0.137848	−	0.238759i
$353$	537109.	−	930300.i	0.229417	−	0.397362i	−0.728219	−	0.685345i	$-0.759651\pi$
$353$	537109.	−	930300.i	0.229417	−	0.397362i	0.957635	+	0.287983i	$0.0929847\pi$
$354$		0			0
$355$	−328435.	−	568866.i	−0.138318	−	0.239574i
$356$	−1.58464e6			−0.662682
$357$		0			0
$358$	−186337.			−0.0768409
$359$	60601.9	+	104966.i	0.0248170	+	0.0429844i	0.878167	−	0.478354i	$-0.158766\pi$
$359$	60601.9	+	104966.i	0.0248170	+	0.0429844i	−0.853350	+	0.521338i	$0.825433\pi$
$360$		0			0
$361$	−536639.	+	929486.i	−0.216728	+	0.375383i
$362$	−1.65842e6	−	2.87247e6i	−0.665156	−	1.15208i
$363$		0			0
$364$	126723.	−	409859.i	0.0501303	−	0.162137i
$365$	−150160.			−0.0589959
$366$		0			0
$367$	276914.	−	479630.i	0.107320	−	0.185884i	−0.807364	−	0.590054i	$-0.799106\pi$
$367$	276914.	−	479630.i	0.107320	−	0.185884i	0.914684	+	0.404171i	$0.132440\pi$
$368$	475823.	−	824150.i	0.183158	−	0.317239i
$369$		0			0
$370$	−506413.			−0.192309
$371$	716389.	−	2.31702e6i	0.270218	−	0.873966i
$372$		0			0
$373$	250739.	+	434293.i	0.0933146	+	0.161626i	0.908904	−	0.417006i	$-0.136920\pi$
$373$	250739.	+	434293.i	0.0933146	+	0.161626i	−0.815589	+	0.578631i	$0.803587\pi$
$374$	1.32868e6	−	2.30135e6i	0.491182	−	0.850752i
$375$		0			0
$376$	60007.7	+	103936.i	0.0218896	+	0.0379139i
$377$	−25593.6			−0.00927422
$378$		0			0
$379$	999004.			0.357248			0.178624	−	0.983917i	$-0.442835\pi$
$379$	999004.			0.357248			0.178624	+	0.983917i	$0.442835\pi$
$380$	316508.	+	548209.i	0.112441	+	0.194754i
$381$		0			0
$382$	−943833.	+	1.63477e6i	−0.330930	+	0.573188i
$383$	326232.	+	565050.i	0.113640	+	0.196829i	0.917235	−	0.398346i	$-0.130416\pi$
$383$	326232.	+	565050.i	0.113640	+	0.196829i	−0.803596	+	0.595176i	$0.797082\pi$
$384$		0			0
$385$	1.66146e6	−	378058.i	0.571265	−	0.129989i
$386$	2.75342e6			0.940597
$387$		0			0
$388$	924387.	−	1.60109e6i	0.311727	−	0.539927i
$389$	−39626.5	+	68635.1i	−0.0132774	+	0.0229971i	−0.872588	−	0.488457i	$-0.837560\pi$
$389$	−39626.5	+	68635.1i	−0.0132774	+	0.0229971i	0.859310	+	0.511454i	$0.170893\pi$
$390$		0			0
$391$	3.94585e6			1.30526
$392$	969743.	−	465420.i	0.318744	−	0.152978i
$393$		0			0
$394$	623830.	+	1.08050e6i	0.202454	+	0.350660i
$395$	−31288.0	+	54192.4i	−0.0100899	+	0.0174762i
$396$		0			0
$397$	−2.00443e6	−	3.47177e6i	−0.638284	−	1.10554i	−0.985809	−	0.167870i	$-0.946311\pi$
$397$	−2.00443e6	−	3.47177e6i	−0.638284	−	1.10554i	0.347525	−	0.937671i	$-0.387022\pi$
$398$	−1.14885e6			−0.363542
$399$		0			0
$400$	−687104.			−0.214720
$401$	337209.	+	584063.i	0.104722	+	0.181384i	0.913625	−	0.406559i	$-0.133271\pi$
$401$	337209.	+	584063.i	0.104722	+	0.181384i	−0.808903	+	0.587943i	$0.799938\pi$
$402$		0			0
$403$	−941994.	+	1.63158e6i	−0.288925	+	0.500433i
$404$	87611.7	+	151748.i	0.0267060	+	0.0462561i
$405$		0			0
$406$	−43616.4	−	47069.8i	−0.0131321	−	0.0141719i
$407$	−3.77322e6			−1.12908
$408$		0			0
$409$	1.42706e6	−	2.47175e6i	0.421828	−	0.730627i	−0.574291	−	0.818652i	$-0.694722\pi$
$409$	1.42706e6	−	2.47175e6i	0.421828	−	0.730627i	0.996118	+	0.0880244i	$0.0280553\pi$
$410$	−722478.	+	1.25137e6i	−0.212258	+	0.367642i
$411$		0			0
$412$	−2.20359e6			−0.639570
$413$	−97068.7	+	313950.i	−0.0280030	+	0.0905701i
$414$		0			0
$415$	482523.	+	835755.i	0.137530	+	0.238209i
$416$	105892.	−	183411.i	0.0300007	−	0.0519627i
$417$		0			0
$418$	2.35827e6	+	4.08464e6i	0.660164	+	1.14344i
$419$	4.66553e6			1.29827			0.649136	−	0.760672i	$-0.275130\pi$
$419$	4.66553e6			1.29827			0.649136	+	0.760672i	$0.275130\pi$
$420$		0			0
$421$	−3.73317e6			−1.02653			−0.513266	−	0.858229i	$-0.671565\pi$
$421$	−3.73317e6			−1.02653			−0.513266	+	0.858229i	$0.671565\pi$
$422$	920987.	+	1.59520e6i	0.251752	+	0.436047i
$423$		0			0
$424$	598631.	−	1.03686e6i	0.161713	−	0.280095i
$425$	−1.42448e6	−	2.46728e6i	−0.382547	−	0.662591i
$426$		0			0
$427$	184577.	+	199191.i	0.0489899	+	0.0528689i
$428$	−1.20917e6			−0.319064
$429$		0			0
$430$	−226883.	+	392973.i	−0.0591740	+	0.102492i
$431$	482335.	−	835429.i	0.125071	−	0.216629i	−0.796690	−	0.604388i	$-0.793418\pi$
$431$	482335.	−	835429.i	0.125071	−	0.216629i	0.921761	+	0.387760i	$0.126751\pi$
$432$		0			0
$433$	−6.18096e6			−1.58429			−0.792147	−	0.610330i	$-0.791037\pi$
$433$	−6.18096e6			−1.58429			−0.792147	+	0.610330i	$0.791037\pi$
$434$	−4.60603e6	+	1.04808e6i	−1.17382	+	0.267098i
$435$		0			0
$436$	−356210.	−	616974.i	−0.0897409	−	0.155436i
$437$	−3.50172e6	+	6.06516e6i	−0.877158	+	1.51928i
$438$		0			0
$439$	107567.	+	186312.i	0.0266391	+	0.0461402i	0.879038	−	0.476752i	$-0.158186\pi$
$439$	107567.	+	186312.i	0.0266391	+	0.0461402i	−0.852399	+	0.522893i	$0.824853\pi$
$440$	841174.			0.207135
$441$		0			0
$442$	878132.			0.213798
$443$	−3.59934e6	−	6.23423e6i	−0.871391	−	1.50929i	−0.860558	−	0.509353i	$-0.829885\pi$
$443$	−3.59934e6	−	6.23423e6i	−0.871391	−	1.50929i	−0.0108335	−	0.999941i	$-0.503448\pi$
$444$		0			0
$445$	1.03992e6	−	1.80119e6i	0.248943	−	0.431182i
$446$	2.39551e6	+	4.14915e6i	0.570245	+	0.987694i
$447$		0			0
$448$	517778.	−	117818.i	0.121884	−	0.0277343i
$449$	−3.92153e6			−0.917994			−0.458997	−	0.888438i	$-0.651791\pi$
$449$	−3.92153e6			−0.917994			−0.458997	+	0.888438i	$0.651791\pi$
$450$		0			0
$451$	−5.38309e6	+	9.32379e6i	−1.24621	+	2.15850i
$452$	723652.	−	1.25340e6i	0.166603	−	0.288565i
$453$		0			0
$454$	−3.57824e6			−0.814761
$455$	382708.	+	413010.i	0.0866641	+	0.0935260i
$456$		0			0
$457$	−660839.	−	1.14461e6i	−0.148015	−	0.256369i	0.782479	−	0.622677i	$-0.213955\pi$
$457$	−660839.	−	1.14461e6i	−0.148015	−	0.256369i	−0.930494	+	0.366308i	$0.880622\pi$
$458$	518512.	−	898090.i	0.115504	−	0.200058i
$459$		0			0
$460$	624518.	+	1.08170e6i	0.137610	+	0.238348i
$461$	75459.1			0.0165371			0.00826855	−	0.999966i	$-0.497368\pi$
$461$	75459.1			0.0165371			0.00826855	+	0.999966i	$0.497368\pi$
$462$		0			0
$463$	−3.28757e6			−0.712727			−0.356363	−	0.934347i	$-0.615984\pi$
$463$	−3.28757e6			−0.712727			−0.356363	+	0.934347i	$0.615984\pi$
$464$	−15839.7	−	27435.1i	−0.00341547	−	0.00591577i
$465$		0			0
$466$	422630.	−	732016.i	0.0901561	−	0.156155i
$467$	−321691.	−	557185.i	−0.0682569	−	0.118224i	0.829877	−	0.557946i	$-0.188410\pi$
$467$	−321691.	−	557185.i	−0.0682569	−	0.118224i	−0.898134	+	0.439722i	$0.855077\pi$
$468$		0			0
$469$	−2.24487e6	+	7.26058e6i	−0.471258	+	1.52419i
$470$	−157520.			−0.0328921
$471$		0			0
$472$	−81112.9	+	140492.i	−0.0167585	+	0.0290266i
$473$	−1.69048e6	+	2.92799e6i	−0.347422	+	0.601752i
$474$		0			0
$475$	5.05660e6			1.02831
$476$	1.49651e6	+	1.61500e6i	0.302734	+	0.326704i
$477$		0			0
$478$	926.036	+	1603.94i	0.000185378	+	0.000321084i
$479$	−3.71324e6	+	6.43152e6i	−0.739459	+	1.28078i	0.213280	+	0.976991i	$0.431585\pi$
$479$	−3.71324e6	+	6.43152e6i	−0.739459	+	1.28078i	−0.952739	+	0.303790i	$0.901748\pi$
$480$		0			0
$481$	−623434.	−	1.07982e6i	−0.122865	−	0.212808i
$482$	1.14558e6			0.224599
$483$		0			0
$484$	3.69067e6			0.716130
$485$	1.21326e6	+	2.10143e6i	0.234207	+	0.405658i
$486$		0			0
$487$	2.08542e6	−	3.61206e6i	0.398448	−	0.690132i	−0.595087	−	0.803662i	$-0.702882\pi$
$487$	2.08542e6	−	3.61206e6i	0.398448	−	0.690132i	0.993535	+	0.113529i	$0.0362156\pi$
$488$	67030.6	+	116100.i	0.0127416	+	0.0220691i
$489$		0			0
$490$	−107370.	+	1.40770e6i	−0.0202019	+	0.264862i
$491$	3.73674e6			0.699502			0.349751	−	0.936843i	$-0.386266\pi$
$491$	3.73674e6			0.699502			0.349751	+	0.936843i	$0.386266\pi$
$492$		0			0
$493$	65676.6	−	113755.i	0.0121701	−	0.0210792i
$494$	−779293.	+	1.34978e6i	−0.143676	+	0.248854i
$495$		0			0
$496$	−2.33197e6			−0.425617
$497$	3.95406e6	−	899731.i	0.718047	−	0.163389i
$498$		0			0
$499$	−4.35431e6	−	7.54188e6i	−0.782831	−	1.35590i	−0.930286	−	0.366834i	$-0.880442\pi$
$499$	−4.35431e6	−	7.54188e6i	−0.782831	−	1.35590i	0.147456	−	0.989069i	$-0.452892\pi$
$500$	975912.	−	1.69033e6i	0.174576	−	0.302375i
$501$		0			0
$502$	−2.74336e6	−	4.75164e6i	−0.485874	−	0.841558i
$503$	−3.28384e6			−0.578711			−0.289355	−	0.957222i	$-0.593441\pi$
$503$	−3.28384e6			−0.578711			−0.289355	+	0.957222i	$0.593441\pi$
$504$		0			0
$505$	−229981.			−0.0401294
$506$	4.65321e6	+	8.05959e6i	0.807934	+	1.39938i
$507$		0			0
$508$	−1.50166e6	+	2.60094e6i	−0.258173	+	0.447169i
$509$	−4.71851e6	−	8.17271e6i	−0.807255	−	1.39821i	−0.914758	−	0.404002i	$-0.867619\pi$
$509$	−4.71851e6	−	8.17271e6i	−0.807255	−	1.39821i	0.107503	−	0.994205i	$-0.465714\pi$
$510$		0			0
$511$	273826.	−	885635.i	0.0463897	−	0.150038i
$512$	262144.			0.0441942
$513$		0			0
$514$	−1.51845e6	+	2.63003e6i	−0.253508	+	0.439089i
$515$	1.44611e6	−	2.50473e6i	0.240260	−	0.416143i
$516$		0			0
$517$	−1.17366e6			−0.193116
$518$	923474.	−	2.98680e6i	0.151217	−	0.489081i
$519$		0			0
$520$	138984.	+	240727.i	0.0225401	+	0.0390406i
$521$	2.95363e6	−	5.11584e6i	0.476719	−	0.825701i	−0.522926	−	0.852378i	$-0.675159\pi$
$521$	2.95363e6	−	5.11584e6i	0.476719	−	0.825701i	0.999644	+	0.0266776i	$0.00849276\pi$
$522$		0			0
$523$	−327168.	−	566672.i	−0.0523019	−	0.0905895i	0.838689	−	0.544610i	$-0.183322\pi$
$523$	−327168.	−	566672.i	−0.0523019	−	0.0905895i	−0.890991	+	0.454021i	$0.849989\pi$
$524$	−2.47060e6			−0.393073
$525$		0			0
$526$	4.28978e6			0.676037
$527$	−4.83457e6	−	8.37373e6i	−0.758284	−	1.31339i
$528$		0			0
$529$	−3.69124e6	+	6.39342e6i	−0.573500	+	0.993331i
$530$	785704.	+	1.36088e6i	0.121498	+	0.210441i
$531$		0			0
$532$	−3.81048e6	+	867058.i	−0.583714	+	0.132822i
$533$	−3.55771e6			−0.542440
$534$		0			0
$535$	793516.	−	1.37441e6i	0.119859	−	0.207602i
$536$	−1.87586e6	+	3.24909e6i	−0.282026	+	0.488484i
$537$		0			0
$538$	7.06445e6			1.05226
$539$	−800002.	+	1.04886e7i	−0.118609	+	1.55505i
$540$		0			0
$541$	431558.	+	747480.i	0.0633936	+	0.109801i	0.895980	−	0.444094i	$-0.146474\pi$
$541$	431558.	+	747480.i	0.0633936	+	0.109801i	−0.832587	+	0.553895i	$0.813141\pi$
$542$	4.10342e6	−	7.10734e6i	0.599996	−	1.03922i
$543$		0			0
$544$	543469.	+	941316.i	0.0787368	+	0.136376i
$545$	935052.			0.134848
$546$		0			0
$547$	−5.45692e6			−0.779794			−0.389897	−	0.920859i	$-0.627489\pi$
$547$	−5.45692e6			−0.779794			−0.389897	+	0.920859i	$0.627489\pi$
$548$	873903.	+	1.51364e6i	0.124312	+	0.215314i
$549$		0			0
$550$	3.35969e6	−	5.81915e6i	0.473579	−	0.820263i
$551$	116569.	+	201903.i	0.0163570	+	0.0283311i
$552$		0			0
$553$	−262568.	−	283358.i	−0.0365115	−	0.0394024i
$554$	3.48273e6			0.482109
$555$		0			0
$556$	1.63756e6	−	2.83634e6i	0.224652	−	0.389109i
$557$	5.58881e6	−	9.68010e6i	0.763275	−	1.32203i	−0.177879	−	0.984052i	$-0.556923\pi$
$557$	5.58881e6	−	9.68010e6i	0.763275	−	1.32203i	0.941154	−	0.337979i	$-0.109743\pi$
$558$		0			0
$559$	−1.11724e6			−0.151223
$560$	−205873.	+	665854.i	−0.0277414	+	0.0897241i
$561$		0			0
$562$	4.70825e6	+	8.15493e6i	0.628809	+	1.08913i
$563$	440132.	−	762332.i	0.0585211	−	0.101361i	−0.835281	−	0.549824i	$-0.814695\pi$
$563$	440132.	−	762332.i	0.0585211	−	0.101361i	0.893802	+	0.448462i	$0.148028\pi$
$564$		0			0
$565$	949793.	+	1.64509e6i	0.125172	+	0.216805i
$566$	−8.76140e6			−1.14956
$567$		0			0
$568$	2.00189e6			0.260357
$569$	−774858.	−	1.34209e6i	−0.100332	−	0.173781i	0.811489	−	0.584367i	$-0.198657\pi$
$569$	−774858.	−	1.34209e6i	−0.100332	−	0.173781i	−0.911822	+	0.410587i	$0.865324\pi$
$570$		0			0
$571$	5.48507e6	−	9.50043e6i	0.704032	−	1.21942i	−0.263008	−	0.964794i	$-0.584715\pi$
$571$	5.48507e6	−	9.50043e6i	0.704032	−	1.21942i	0.967040	−	0.254625i	$-0.0819521\pi$
$572$	1.03555e6	+	1.79363e6i	0.132337	+	0.229214i
$573$		0			0
$574$	−6.06302e6	−	6.54308e6i	−0.768085	−	0.828900i
$575$	9.97742e6			1.25849
$576$		0			0
$577$	6.29341e6	−	1.09005e7i	0.786948	−	1.36303i	−0.140880	−	0.990027i	$-0.544993\pi$
$577$	6.29341e6	−	1.09005e7i	0.786948	−	1.36303i	0.927828	−	0.373008i	$-0.121674\pi$
$578$	586306.	−	1.01551e6i	0.0729969	−	0.126434i
$579$		0			0
$580$	41579.1			0.00513222
$581$	−5.80915e6	+	1.32185e6i	−0.713957	+	0.162458i
$582$		0			0
$583$	5.85418e6	+	1.01397e7i	0.713337	+	1.23554i
$584$	228815.	−	396319.i	0.0277621	−	0.0480854i
$585$		0			0
$586$	−1.61540e6	−	2.79796e6i	−0.194328	−	0.336587i
$587$	−1.19962e7			−1.43697			−0.718484	−	0.695544i	$-0.755164\pi$
$587$	−1.19962e7			−1.43697			−0.718484	+	0.695544i	$0.755164\pi$
$588$		0			0
$589$	1.71617e7			2.03831
$590$	−106461.	−	184395.i	−0.0125910	−	0.0218082i
$591$		0			0
$592$	771677.	−	1.33658e6i	0.0904964	−	0.156744i
$593$	2.66154e6	+	4.60993e6i	0.310811	+	0.538341i	0.978538	−	0.206065i	$-0.0660659\pi$
$593$	2.66154e6	+	4.60993e6i	0.310811	+	0.538341i	−0.667727	+	0.744406i	$0.732733\pi$
$594$		0			0
$595$	−2.81778e6	+	641175.i	−0.326298	+	0.0742479i
$596$	6.50092e6			0.749651
$597$		0			0
$598$	−1.53766e6	+	2.66331e6i	−0.175836	+	0.304557i
$599$	3.77757e6	−	6.54293e6i	0.430175	−	0.745085i	−0.566713	−	0.823915i	$-0.691785\pi$
$599$	3.77757e6	−	6.54293e6i	0.430175	−	0.745085i	0.996888	+	0.0788305i	$0.0251186\pi$
$600$		0			0
$601$	4.44758e6			0.502270			0.251135	−	0.967952i	$-0.419196\pi$
$601$	4.44758e6			0.502270			0.251135	+	0.967952i	$0.419196\pi$
$602$	−1.90400e6	−	2.05475e6i	−0.214129	−	0.231083i
$603$		0			0
$604$	3.33470e6	+	5.77588e6i	0.371933	+	0.644207i
$605$	−2.42200e6	+	4.19503e6i	−0.269021	+	0.465958i
$606$		0			0
$607$	−2.78660e6	−	4.82653e6i	−0.306975	−	0.531696i	0.670724	−	0.741707i	$-0.265983\pi$
$607$	−2.78660e6	−	4.82653e6i	−0.306975	−	0.531696i	−0.977699	+	0.210011i	$0.932650\pi$
$608$	−1.92919e6			−0.211649
$609$		0			0
$610$	−175955.			−0.0191460
$611$	−193920.	−	335879.i	−0.0210145	−	0.0363982i
$612$		0			0
$613$	5.56323e6	−	9.63580e6i	0.597965	−	1.03571i	−0.395156	−	0.918614i	$-0.629309\pi$
$613$	5.56323e6	−	9.63580e6i	0.597965	−	1.03571i	0.993121	−	0.117092i	$-0.0373573\pi$
$614$	−423032.	−	732713.i	−0.0452848	−	0.0784356i
$615$		0			0
$616$	−1.53393e6	+	4.96120e6i	−0.162875	+	0.526787i
$617$	1.07454e7			1.13634			0.568170	−	0.822911i	$-0.307651\pi$
$617$	1.07454e7			1.13634			0.568170	+	0.822911i	$0.307651\pi$
$618$		0			0
$619$	−6.78781e6	+	1.17568e7i	−0.712038	+	1.23329i	0.252053	+	0.967713i	$0.418894\pi$
$619$	−6.78781e6	+	1.17568e7i	−0.712038	+	1.23329i	−0.964091	+	0.265572i	$0.914439\pi$
$620$	1.53036e6	−	2.65065e6i	0.159887	−	0.276933i
$621$		0			0
$622$	1.16684e6			0.120930
$623$	8.72698e6	+	9.41797e6i	0.900833	+	0.972159i
$624$		0			0
$625$	−2.91287e6	−	5.04523e6i	−0.298277	−	0.516632i
$626$	3.99195e6	−	6.91427e6i	0.407145	−	0.705197i
$627$		0			0
$628$	−954324.	−	1.65294e6i	−0.0965599	−	0.167247i
$629$	6.39927e6			0.644917
$630$		0			0
$631$	1.27986e7			1.27964			0.639820	−	0.768525i	$-0.279009\pi$
$631$	1.27986e7			1.27964			0.639820	+	0.768525i	$0.279009\pi$
$632$	−95353.9	−	165158.i	−0.00949611	−	0.0164478i
$633$		0			0
$634$	−5.61914e6	+	9.73264e6i	−0.555197	+	0.961629i
$635$	−1.97092e6	−	3.41374e6i	−0.193970	−	0.335966i
$636$		0			0
$637$	−3.13380e6	+	1.50404e6i	−0.306001	+	0.146863i
$638$	309801.			0.0301322
$639$		0			0
$640$	−172032.	+	297968.i	−0.0166020	+	0.0287554i
$641$	−442417.	+	766288.i	−0.0425291	+	0.0736626i	−0.886506	−	0.462716i	$-0.846875\pi$
$641$	−442417.	+	766288.i	−0.0425291	+	0.0736626i	0.843977	+	0.536379i	$0.180208\pi$
$642$		0			0
$643$	6.66271e6			0.635511			0.317756	−	0.948173i	$-0.397071\pi$
$643$	6.66271e6			0.635511			0.317756	+	0.948173i	$0.397071\pi$
$644$	−7.51863e6	+	1.71083e6i	−0.714372	+	0.162552i
$645$		0			0
$646$	−3.99955e6	−	6.92742e6i	−0.377077	−	0.653116i
$647$	−8.45887e6	+	1.46512e7i	−0.794423	+	1.37598i	0.128782	+	0.991673i	$0.458893\pi$
$647$	−8.45887e6	+	1.46512e7i	−0.794423	+	1.37598i	−0.923205	+	0.384308i	$0.874440\pi$
$648$		0			0
$649$	−793226.	−	1.37391e6i	−0.0739239	−	0.128040i
$650$	2.22043e6			0.206136
$651$		0			0
$652$	757959.			0.0698275
$653$	4.74654e6	+	8.22124e6i	0.435606	+	0.754492i	0.997345	−	0.0728230i	$-0.0232008\pi$
$653$	4.74654e6	+	8.22124e6i	0.435606	+	0.754492i	−0.561739	+	0.827315i	$0.689867\pi$
$654$		0			0
$655$	1.62133e6	−	2.80822e6i	0.147662	−	0.255758i
$656$	−2.20184e6	−	3.81369e6i	−0.199768	−	0.346008i
$657$		0			0
$658$	287248.	−	929046.i	0.0258638	−	0.0836513i
$659$	4.97563e6			0.446308			0.223154	−	0.974783i	$-0.428365\pi$
$659$	4.97563e6			0.446308			0.223154	+	0.974783i	$0.428365\pi$
$660$		0			0
$661$	−1.05765e7	+	1.83191e7i	−0.941543	+	1.63080i	−0.179013	+	0.983847i	$0.557290\pi$
$661$	−1.05765e7	+	1.83191e7i	−0.941543	+	1.63080i	−0.762530	+	0.646953i	$0.776043\pi$
$662$	−2.76099e6	+	4.78218e6i	−0.244861	+	0.424112i
$663$		0			0
$664$	−2.94110e6			−0.258874
$665$	1.51508e6	−	4.90022e6i	0.132856	−	0.429696i
$666$		0			0
$667$	230007.	+	398384.i	0.0200183	+	0.0346727i
$668$	−1.85335e6	+	3.21010e6i	−0.160700	+	0.278341i
$669$		0			0
$670$	−2.46207e6	−	4.26443e6i	−0.211892	−	0.367007i
$671$	−1.31102e6			−0.112410
$672$		0			0
$673$	417573.			0.0355382			0.0177691	−	0.999842i	$-0.494344\pi$
$673$	417573.			0.0355382			0.0177691	+	0.999842i	$0.494344\pi$
$674$	−1.13286e6	−	1.96217e6i	−0.0960562	−	0.166374i
$675$		0			0
$676$	2.62814e6	−	4.55208e6i	0.221199	−	0.383127i
$677$	1.31234e6	+	2.27304e6i	0.110046	+	0.190605i	0.915789	−	0.401661i	$-0.131567\pi$
$677$	1.31234e6	+	2.27304e6i	0.110046	+	0.190605i	−0.805743	+	0.592266i	$0.798233\pi$
$678$		0			0
$679$	−1.46065e7	+	3.32366e6i	−1.21583	+	0.276657i
$680$	−1.42661e6			−0.118313
$681$		0			0
$682$	1.14025e7	−	1.97497e7i	0.938726	−	1.62592i
$683$	−4.37227e6	+	7.57300e6i	−0.358637	+	0.621178i	−0.987733	−	0.156149i	$-0.950092\pi$
$683$	−4.37227e6	+	7.57300e6i	−0.358637	+	0.621178i	0.629096	+	0.777328i	$0.283425\pi$
$684$		0			0
$685$	−2.29399e6			−0.186795
$686$	−8.10674e6	−	3.20029e6i	−0.657712	−	0.259644i
$687$		0			0
$688$	−691453.	−	1.19763e6i	−0.0556919	−	0.0964611i
$689$	−1.93452e6	+	3.35069e6i	−0.155248	+	0.268898i
$690$		0			0
$691$	−2.19834e6	−	3.80763e6i	−0.175146	−	0.303361i	0.765066	−	0.643952i	$-0.222706\pi$
$691$	−2.19834e6	−	3.80763e6i	−0.175146	−	0.303361i	−0.940212	+	0.340591i	$0.889373\pi$
$692$	−2.14943e6			−0.170631
$693$		0			0
$694$	−2.16083e6			−0.170303
$695$	2.14930e6	+	3.72270e6i	0.168786	+	0.292345i
$696$		0			0
$697$	9.12957e6	−	1.58129e7i	0.711817	−	1.23290i
$698$	−3.47614e6	−	6.02084e6i	−0.270059	−	0.467755i
$699$		0			0
$700$	3.78404e6	+	4.08366e6i	0.291885	+	0.314995i
$701$	−6.51339e6			−0.500624			−0.250312	−	0.968165i	$-0.580533\pi$
$701$	−6.51339e6			−0.500624			−0.250312	+	0.968165i	$0.580533\pi$
$702$		0			0
$703$	−5.67900e6	+	9.83631e6i	−0.433394	+	0.750661i
$704$	−1.28179e6	+	2.22012e6i	−0.0974731	+	0.168828i
$705$		0			0
$706$	−4.29687e6			−0.324445
$707$	419383.	−	1.35641e6i	0.0315546	−	0.102057i
$708$		0			0
$709$	5.23256e6	+	9.06305e6i	0.390929	+	0.677110i	0.992572	−	0.121655i	$-0.0388203\pi$
$709$	5.23256e6	+	9.06305e6i	0.390929	+	0.677110i	−0.601643	+	0.798765i	$0.705487\pi$
$710$	−1.31374e6	+	2.27547e6i	−0.0978056	+	0.169404i
$711$		0			0
$712$	3.16928e6	+	5.48935e6i	0.234294	+	0.405808i
$713$	3.38625e7			2.49457
$714$		0			0
$715$	−2.71832e6			−0.198855
$716$	372675.	+	645492.i	0.0271674	+	0.0470553i
$717$		0			0
$718$	242408.	−	419862.i	0.0175483	−	0.0303946i
$719$	−8.40746e6	−	1.45621e7i	−0.606516	−	1.05052i	−0.991810	−	0.127723i	$-0.959233\pi$
$719$	−8.40746e6	−	1.45621e7i	−0.606516	−	1.05052i	0.385294	−	0.922794i	$-0.374100\pi$
$720$		0			0
$721$	1.21357e7	+	1.30966e7i	0.869414	+	0.938252i
$722$	4.29311e6			0.306499
$723$		0			0
$724$	−6.63368e6	+	1.14899e7i	−0.470336	+	0.814646i
$725$	166069.	−	287640.i	0.0117339	−	0.0203238i
$726$		0			0
$727$	−1.71928e7			−1.20646			−0.603228	−	0.797569i	$-0.706119\pi$
$727$	−1.71928e7			−1.20646			−0.603228	+	0.797569i	$0.706119\pi$
$728$	−1.67324e6	+	380739.i	−0.117012	+	0.0266255i
$729$		0			0
$730$	300320.	+	520169.i	0.0208582	+	0.0361275i
$731$	2.86700e6	−	4.96579e6i	0.198442	−	0.343712i
$732$		0			0
$733$	−9.86827e6	−	1.70923e7i	−0.678393	−	1.17501i	−0.975465	−	0.220155i	$-0.929344\pi$
$733$	−9.86827e6	−	1.70923e7i	−0.678393	−	1.17501i	0.297072	−	0.954855i	$-0.403990\pi$
$734$	−2.21531e6			−0.151773
$735$		0			0
$736$	−3.80659e6			−0.259025
$737$	−1.83446e7	−	3.17738e7i	−1.24405	−	2.15477i
$738$		0			0
$739$	−6.43593e6	+	1.11474e7i	−0.433511	+	0.750863i	−0.997173	−	0.0751426i	$-0.976059\pi$
$739$	−6.43593e6	+	1.11474e7i	−0.433511	+	0.750863i	0.563662	+	0.826006i	$0.309392\pi$
$740$	1.01283e6	+	1.75427e6i	0.0679916	+	0.117765i
$741$		0			0
$742$	−9.45916e6	+	2.15239e6i	−0.630729	+	0.143520i
$743$	−2.45606e7			−1.63218			−0.816089	−	0.577927i	$-0.803862\pi$
$743$	−2.45606e7			−1.63218			−0.816089	+	0.577927i	$0.803862\pi$
$744$		0			0
$745$	−4.26623e6	+	7.38933e6i	−0.281614	+	0.487769i
$746$	1.00296e6	−	1.73717e6i	0.0659834	−	0.114287i
$747$		0			0
$748$	−1.06295e7			−0.694636
$749$	6.65917e6	+	7.18643e6i	0.433726	+	0.468068i
$750$		0			0
$751$	162983.	+	282294.i	0.0105449	+	0.0182643i	0.871250	−	0.490840i	$-0.163310\pi$
$751$	162983.	+	282294.i	0.0105449	+	0.0182643i	−0.860705	+	0.509104i	$0.829977\pi$
$752$	240031.	−	415746.i	0.0154783	−	0.0268092i
$753$		0			0
$754$	51187.1	+	88658.7i	0.00327893	+	0.00567928i
$755$	−8.75360e6			−0.558881
$756$		0			0
$757$	1.84659e7			1.17120			0.585599	−	0.810601i	$-0.300859\pi$
$757$	1.84659e7			1.17120			0.585599	+	0.810601i	$0.300859\pi$
$758$	−1.99801e6	−	3.46065e6i	−0.126306	−	0.218769i
$759$		0			0
$760$	1.26603e6	−	2.19283e6i	0.0795081	−	0.137712i
$761$	2.56313e6	+	4.43947e6i	0.160439	+	0.277888i	0.935026	−	0.354579i	$-0.115376\pi$
$761$	2.56313e6	+	4.43947e6i	0.160439	+	0.277888i	−0.774587	+	0.632467i	$0.782042\pi$
$762$		0			0
$763$	−1.70512e6	+	5.51488e6i	−0.106034	+	0.342945i
$764$	7.55066e6			0.468006
$765$		0			0
$766$	1.30493e6	−	2.26020e6i	0.0803553	−	0.139179i
$767$	262123.	−	454010.i	0.0160885	−	0.0278662i
$768$		0			0
$769$	1.63432e6			0.0996602			0.0498301	−	0.998758i	$-0.484132\pi$
$769$	1.63432e6			0.0996602			0.0498301	+	0.998758i	$0.484132\pi$
$770$	−4.63255e6	−	4.99934e6i	−0.281574	−	0.303869i
$771$		0			0
$772$	−5.50683e6	−	9.53811e6i	−0.332551	−	0.575996i
$773$	−2.14128e6	+	3.70881e6i	−0.128892	+	0.223247i	−0.923248	−	0.384206i	$-0.874475\pi$
$773$	−2.14128e6	+	3.70881e6i	−0.128892	+	0.223247i	0.794356	+	0.607453i	$0.207809\pi$
$774$		0			0
$775$	−1.22246e7	−	2.11737e7i	−0.731108	−	1.26632i
$776$	−7.39510e6			−0.440849
$777$		0			0
$778$	317012.			0.0187770
$779$	1.62040e7	+	2.80661e7i	0.956704	+	1.65706i
$780$		0			0
$781$	−9.78852e6	+	1.69542e7i	−0.574235	+	0.994604i
$782$	−7.89170e6	−	1.36688e7i	−0.461481	−	0.799308i
$783$		0			0
$784$	−3.55175e6	−	2.42845e6i	−0.206373	−	0.141104i
$785$	2.50510e6			0.145095
$786$		0			0
$787$	3.91482e6	−	6.78066e6i	0.225307	−	0.390243i	−0.731105	−	0.682265i	$-0.760995\pi$
$787$	3.91482e6	−	6.78066e6i	0.225307	−	0.390243i	0.956411	+	0.292022i	$0.0943282\pi$
$788$	2.49532e6	−	4.32202e6i	0.143156	−	0.247954i
$789$		0			0
$790$	250304.			0.0142692
$791$	−1.14346e7	+	2.60191e6i	−0.649803	+	0.147860i
$792$		0			0
$793$	−216615.	−	375187.i	−0.0122322	−	0.0211868i
$794$	−8.01771e6	+	1.38871e7i	−0.451335	+	0.781736i
$795$		0			0
$796$	2.29769e6	+	3.97972e6i	0.128531	+	0.222623i
$797$	3.68238e6			0.205344			0.102672	−	0.994715i	$-0.467261\pi$
$797$	3.68238e6			0.205344			0.102672	+	0.994715i	$0.467261\pi$
$798$		0			0
$799$	1.99050e6			0.110305
$800$	1.37421e6	+	2.38020e6i	0.0759150	+	0.131489i
$801$		0			0
$802$	1.34884e6	−	2.33625e6i	0.0740497	−	0.128258i
$803$	2.23765e6	+	3.87572e6i	0.122462	+	0.212111i
$804$		0			0
$805$	2.98947e6	−	9.66885e6i	0.162594	−	0.525878i
$806$	7.53595e6			0.408602
$807$		0			0
$808$	350447.	−	606992.i	0.0188840	−	0.0327080i
$809$	1.04091e7	−	1.80290e7i	0.559165	−	0.968502i	−0.438402	−	0.898779i	$-0.644455\pi$
$809$	1.04091e7	−	1.80290e7i	0.559165	−	0.968502i	0.997566	−	0.0697227i	$-0.0222114\pi$
$810$		0			0
$811$	3.03542e7			1.62057			0.810283	−	0.586039i	$-0.199313\pi$
$811$	3.03542e7			1.62057			0.810283	+	0.586039i	$0.199313\pi$
$812$	−75822.0	+	245231.i	−0.00403557	+	0.0130523i
$813$		0			0
$814$	7.54644e6	+	1.30708e7i	0.399191	+	0.691420i
$815$	−497410.	+	861540.i	−0.0262314	+	0.0454341i
$816$		0			0
$817$	5.08861e6	+	8.81373e6i	0.266713	+	0.461960i
$818$	−1.14165e7			−0.596555
$819$		0			0
$820$	5.77982e6			0.300179
$821$	1.37193e7	+	2.37626e7i	0.710355	+	1.23037i	0.964724	+	0.263264i	$0.0847990\pi$
$821$	1.37193e7	+	2.37626e7i	0.710355	+	1.23037i	−0.254369	+	0.967107i	$0.581868\pi$
$822$		0			0
$823$	−7.95629e6	+	1.37807e7i	−0.409459	+	0.709204i	−0.994829	−	0.101562i	$-0.967616\pi$
$823$	−7.95629e6	+	1.37807e7i	−0.409459	+	0.709204i	0.585370	+	0.810766i	$0.300949\pi$
$824$	4.40718e6	+	7.63346e6i	0.226122	+	0.391655i
$825$		0			0
$826$	1.28169e6	−	291643.i	0.0653632	−	0.0148731i
$827$	−1.40824e7			−0.716001			−0.358001	−	0.933721i	$-0.616541\pi$
$827$	−1.40824e7			−0.716001			−0.358001	+	0.933721i	$0.616541\pi$
$828$		0			0
$829$	−1.09441e7	+	1.89558e7i	−0.553089	+	0.957978i	0.444960	+	0.895550i	$0.353218\pi$
$829$	−1.09441e7	+	1.89558e7i	−0.553089	+	0.957978i	−0.998049	+	0.0624281i	$0.980116\pi$
$830$	1.93009e6	−	3.34302e6i	0.0972486	−	0.168439i
$831$		0			0
$832$	−847139.			−0.0424274
$833$	1.35678e6	−	1.77883e7i	0.0677480	−	0.888225i
$834$		0			0
$835$	−2.43253e6	−	4.21326e6i	−0.120737	−	0.209123i
$836$	9.43306e6	−	1.63385e7i	0.466807	−	0.808533i
$837$		0			0
$838$	−9.33106e6	−	1.61619e7i	−0.459009	−	0.795026i
$839$	1.98669e7			0.974372			0.487186	−	0.873298i	$-0.338023\pi$
$839$	1.98669e7			0.974372			0.487186	+	0.873298i	$0.338023\pi$
$840$		0			0
$841$	−2.04958e7			−0.999253
$842$	7.46635e6	+	1.29321e7i	0.362934	+	0.628620i
$843$		0			0
$844$	3.68395e6	−	6.38078e6i	0.178015	−	0.308332i
$845$	3.44944e6	+	5.97460e6i	0.166191	+	0.287851i
$846$		0			0
$847$	−2.03254e7	−	2.19347e7i	−0.973488	−	1.05057i
$848$	−4.78905e6			−0.228697
$849$		0			0
$850$	−5.69793e6	+	9.86911e6i	−0.270502	+	0.468523i
$851$	−1.12055e7	+	1.94085e7i	−0.530405	+	0.918688i
$852$		0			0
$853$	8.75258e6			0.411873			0.205937	−	0.978565i	$-0.433976\pi$
$853$	8.75258e6			0.411873			0.205937	+	0.978565i	$0.433976\pi$
$854$	320865.	−	1.03777e6i	0.0150549	−	0.0486921i
$855$		0			0
$856$	2.41834e6	+	4.18868e6i	0.112806	+	0.195386i
$857$	−977384.	+	1.69288e6i	−0.0454583	+	0.0787361i	−0.887859	−	0.460115i	$-0.847808\pi$
$857$	−977384.	+	1.69288e6i	−0.0454583	+	0.0787361i	0.842401	+	0.538851i	$0.181141\pi$
$858$		0			0
$859$	−2.49225e6	−	4.31670e6i	−0.115241	−	0.199604i	0.802635	−	0.596471i	$-0.203431\pi$
$859$	−2.49225e6	−	4.31670e6i	−0.115241	−	0.199604i	−0.917876	+	0.396867i	$0.870098\pi$
$860$	1.81506e6			0.0836847
$861$		0			0
$862$	−3.85868e6			−0.176877
$863$	−8.49654e6	−	1.47164e7i	−0.388343	−	0.672629i	0.603884	−	0.797072i	$-0.293619\pi$
$863$	−8.49654e6	−	1.47164e7i	−0.388343	−	0.672629i	−0.992227	+	0.124443i	$0.960286\pi$
$864$		0			0
$865$	1.41056e6	−	2.44316e6i	0.0640991	−	0.111023i
$866$	1.23619e7	+	2.14115e7i	0.560133	+	0.970179i
$867$		0			0
$868$	1.28427e7	+	1.38596e7i	0.578572	+	0.624383i
$869$	1.86498e6			0.0837772
$870$		0			0
$871$	6.06200e6	−	1.04997e7i	0.270751	−	0.468955i
$872$	−1.42484e6	+	2.46790e6i	−0.0634564	+	0.109910i
$873$		0			0
$874$	2.80138e7			1.24049
$875$	−1.54207e7	+	3.50892e6i	−0.680901	+	0.154936i
$876$		0			0
$877$	−9.09797e6	−	1.57581e7i	−0.399434	−	0.691840i	0.594222	−	0.804301i	$-0.297460\pi$
$877$	−9.09797e6	−	1.57581e7i	−0.399434	−	0.691840i	−0.993656	+	0.112461i	$0.964127\pi$
$878$	430269.	−	745249.i	0.0188367	−	0.0326261i
$879$		0			0
$880$	−1.68235e6	−	2.91391e6i	−0.0732334	−	0.126844i
$881$	−1.77637e7			−0.771068			−0.385534	−	0.922694i	$-0.625983\pi$
$881$	−1.77637e7			−0.771068			−0.385534	+	0.922694i	$0.625983\pi$
$882$		0			0
$883$	1.71479e6			0.0740131			0.0370065	−	0.999315i	$-0.488218\pi$
$883$	1.71479e6			0.0740131			0.0370065	+	0.999315i	$0.488218\pi$
$884$	−1.75626e6	−	3.04194e6i	−0.0755891	−	0.130924i
$885$		0			0
$886$	−1.43973e7	+	2.49369e7i	−0.616167	+	1.06723i
$887$	−1.21218e7	−	2.09955e7i	−0.517318	−	0.896021i	−0.999798	−	0.0201139i	$-0.993597\pi$
$887$	−1.21218e7	−	2.09955e7i	−0.517318	−	0.896021i	0.482480	−	0.875907i	$-0.339736\pi$
$888$		0			0
$889$	2.37281e7	−	5.39924e6i	1.00695	−	0.229128i
$890$	−8.31936e6			−0.352058
$891$		0			0
$892$	9.58206e6	−	1.65966e7i	0.403224	−	0.698405i
$893$	−1.76646e6	+	3.05960e6i	−0.0741267	+	0.128391i
$894$		0			0
$895$	−978272.			−0.0408227
$896$	−1.44369e6	−	1.55800e6i	−0.0600764	−	0.0648331i
$897$		0			0
$898$	7.84306e6	+	1.35846e7i	0.324560	+	0.562154i
$899$	563624.	−	976225.i	0.0232589	−	0.0402857i
$900$		0			0
$901$	−9.92852e6	−	1.71967e7i	−0.407448	−	0.705721i
$902$	4.30647e7			1.76240
$903$		0			0
$904$	−5.78921e6			−0.235613
$905$	−8.70671e6	−	1.50805e7i	−0.353372	−	0.612059i
$906$		0			0
$907$	9.52161e6	−	1.64919e7i	0.384319	−	0.665661i	−0.607355	−	0.794430i	$-0.707769\pi$
$907$	9.52161e6	−	1.64919e7i	0.384319	−	0.665661i	0.991675	+	0.128770i	$0.0411028\pi$
$908$	7.15649e6	+	1.23954e7i	0.288062	+	0.498937i
$909$		0			0
$910$	665293.	−	2.15176e6i	0.0266324	−	0.0861371i
$911$	3.18731e7			1.27242			0.636208	−	0.771518i	$-0.280502\pi$
$911$	3.18731e7			1.27242			0.636208	+	0.771518i	$0.280502\pi$
$912$		0			0
$913$	1.43809e7	−	2.49084e7i	0.570964	−	0.988939i
$914$	−2.64336e6	+	4.57843e6i	−0.104662	+	0.181280i
$915$		0			0
$916$	−4.14810e6			−0.163347
$917$	1.36062e7	+	1.46835e7i	0.534333	+	0.576641i
$918$		0			0
$919$	1.47429e7	+	2.55355e7i	0.575830	+	0.997367i	0.995951	+	0.0898988i	$0.0286544\pi$
$919$	1.47429e7	+	2.55355e7i	0.575830	+	0.997367i	−0.420121	+	0.907468i	$0.638012\pi$
$920$	2.49807e6	−	4.32679e6i	0.0973050	−	0.168537i
$921$		0			0
$922$	−150918.	−	261398.i	−0.00584675	−	0.0101269i
$923$	−6.46927e6			−0.249949
$924$		0			0
$925$	1.61811e7			0.621804
$926$	6.57515e6	+	1.13885e7i	0.251987	+	0.436454i
$927$		0			0
$928$	−63358.6	+	109740.i	−0.00241510	+	0.00418308i
$929$	−4.86935e6	−	8.43397e6i	−0.185111	−	0.320621i	0.758503	−	0.651670i	$-0.225931\pi$
$929$	−4.86935e6	−	8.43397e6i	−0.185111	−	0.320621i	−0.943614	+	0.331048i	$0.892598\pi$
$930$		0			0
$931$	2.61384e7	+	1.78717e7i	0.988335	+	0.675757i
$932$	−3.38104e6			−0.127500
$933$		0			0
$934$	−1.28676e6	+	2.22874e6i	−0.0482649	+	0.0835973i
$935$	6.97559e6	−	1.20821e7i	0.260947	−	0.451973i
$936$		0			0
$937$	2.66734e7			0.992498			0.496249	−	0.868180i	$-0.334710\pi$
$937$	2.66734e7			0.992498			0.496249	+	0.868180i	$0.334710\pi$
$938$	2.96411e7	−	6.74472e6i	1.09999	−	0.250298i
$939$		0			0
$940$	315041.	+	545666.i	0.0116291	+	0.0201422i
$941$	1.61432e6	−	2.79609e6i	0.0594315	−	0.102938i	−0.834779	−	0.550585i	$-0.814405\pi$
$941$	1.61432e6	−	2.79609e6i	0.0594315	−	0.102938i	0.894210	+	0.447647i	$0.147738\pi$
$942$		0			0
$943$	3.19728e7	+	5.53785e7i	1.17085	+	2.02797i
$944$	648903.			0.0237001
$945$		0			0
$946$	1.35238e7			0.491328
$947$	−2.29622e6	−	3.97717e6i	−0.0832029	−	0.144112i	0.821421	−	0.570322i	$-0.193182\pi$
$947$	−2.29622e6	−	3.97717e6i	−0.0832029	−	0.144112i	−0.904624	+	0.426210i	$0.859848\pi$
$948$		0			0
$949$	−739434.	+	1.28074e6i	−0.0266523	+	0.0461631i
$950$	−1.01132e7	−	1.75166e7i	−0.363563	−	0.629709i
$951$		0			0
$952$	2.60150e6	−	8.41405e6i	0.0930318	−	0.300893i
$953$	−2.97125e7			−1.05976			−0.529880	−	0.848073i	$-0.677763\pi$
$953$	−2.97125e7			−1.05976			−0.529880	+	0.848073i	$0.677763\pi$
$954$		0			0
$955$	−4.95512e6	+	8.58253e6i	−0.175811	+	0.304514i
$956$	3704.14	−	6415.77i	0.000131082	−	0.000227041i
$957$		0			0
$958$	2.97059e7			1.04575
$959$	4.18323e6	−	1.35299e7i	0.146881	−	0.475058i
$960$		0			0
$961$	−2.71748e7	−	4.70681e7i	−0.949199	−	1.64406i
$962$	−2.49373e6	+	4.31928e6i	−0.0868786	+	0.150478i
$963$		0			0
$964$	−2.29116e6	−	3.96840e6i	−0.0794077	−	0.137538i
$965$	1.44554e7			0.499704
$966$		0			0
$967$	7.64435e6			0.262890			0.131445	−	0.991323i	$-0.458038\pi$
$967$	7.64435e6			0.262890			0.131445	+	0.991323i	$0.458038\pi$
$968$	−7.38134e6	−	1.27849e7i	−0.253190	−	0.438538i
$969$		0			0
$970$	4.85303e6	−	8.40570e6i	0.165609	−	0.286843i
$971$	−2.49810e7	−	4.32684e7i	−0.850281	−	1.47273i	−0.880955	−	0.473200i	$-0.843099\pi$
$971$	−2.49810e7	−	4.32684e7i	−0.850281	−	1.47273i	0.0306743	−	0.999529i	$-0.490235\pi$
$972$		0			0
$973$	−2.58756e7	+	5.88790e6i	−0.876212	+	0.199378i
$974$	−1.66834e7			−0.563491
$975$		0			0
$976$	268122.	−	464401.i	0.00900966	−	0.0156052i
$977$	2.02697e7	−	3.51082e7i	0.679378	−	1.17672i	−0.295791	−	0.955253i	$-0.595583\pi$
$977$	2.02697e7	−	3.51082e7i	0.679378	−	1.17672i	0.975169	−	0.221464i	$-0.0710836\pi$
$978$		0			0
$979$	−6.19865e7			−2.06700
$980$	5.09115e6	−	2.44346e6i	0.169337	−	0.0812717i
$981$		0			0
$982$	−7.47348e6	−	1.29445e7i	−0.247311	−	0.428356i
$983$	−2.06528e7	+	3.57717e7i	−0.681703	+	1.18074i	0.292757	+	0.956187i	$0.405427\pi$
$983$	−2.06528e7	+	3.57717e7i	−0.681703	+	1.18074i	−0.974461	+	0.224558i	$0.927906\pi$
$984$		0			0
$985$	3.27511e6	+	5.67265e6i	0.107556	+	0.186292i
$986$	−525413.			−0.0172111
$987$		0			0
$988$	6.23434e6			0.203188
$989$	1.00406e7	+	1.73908e7i	0.326413	+	0.565364i
$990$		0			0
$991$	9.20419e6	−	1.59421e7i	0.297715	−	0.515658i	−0.677897	−	0.735156i	$-0.737109\pi$
$991$	9.20419e6	−	1.59421e7i	0.297715	−	0.515658i	0.975613	+	0.219498i	$0.0704420\pi$
$992$	4.66394e6	+	8.07819e6i	0.150478	+	0.260636i
$993$		0			0
$994$	−1.10249e7	−	1.18978e7i	−0.353922	−	0.381945i
$995$	−6.03144e6			−0.193136
$996$		0			0
$997$	1.69854e7	−	2.94196e7i	0.541175	−	0.937343i	−0.457662	−	0.889126i	$-0.651313\pi$
$997$	1.69854e7	−	2.94196e7i	0.541175	−	0.937343i	0.998837	−	0.0482164i	$-0.0153537\pi$
$998$	−1.74172e7	+	3.01675e7i	−0.553545	+	0.958768i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.6.g.e.109.2		4
3.2	odd	2		14.6.c.b.11.1	yes	4
7.2	even	3	inner	126.6.g.e.37.2		4
7.3	odd	6		882.6.a.bl.1.2		2
7.4	even	3		882.6.a.bt.1.2		2
12.11	even	2		112.6.i.b.81.2		4
21.2	odd	6		14.6.c.b.9.1	✓	4
21.5	even	6		98.6.c.f.79.2		4
21.11	odd	6		98.6.a.c.1.2		2
21.17	even	6		98.6.a.f.1.1		2
21.20	even	2		98.6.c.f.67.2		4
84.11	even	6		784.6.a.bc.1.1		2
84.23	even	6		112.6.i.b.65.2		4
84.59	odd	6		784.6.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.6.c.b.9.1	✓	4	21.2	odd	6
14.6.c.b.11.1	yes	4	3.2	odd	2
98.6.a.c.1.2		2	21.11	odd	6
98.6.a.f.1.1		2	21.17	even	6
98.6.c.f.67.2		4	21.20	even	2
98.6.c.f.79.2		4	21.5	even	6
112.6.i.b.65.2		4	84.23	even	6
112.6.i.b.81.2		4	12.11	even	2
126.6.g.e.37.2		4	7.2	even	3	inner
126.6.g.e.109.2		4	1.1	even	1	trivial
784.6.a.r.1.2		2	84.59	odd	6
784.6.a.bc.1.1		2	84.11	even	6
882.6.a.bl.1.2		2	7.3	odd	6
882.6.a.bt.1.2		2	7.4	even	3

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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.g (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.00000 - 3.46410i) q^{2} +(-8.00000 + 13.8564i) q^{4} +(-10.5000 - 18.1865i) q^{5} +(126.411 - 28.7642i) q^{7} +64.0000 q^{8} +(-42.0000 + 72.7461i) q^{10} +(-312.937 + 542.023i) q^{11} -206.821 q^{13} +(-352.463 - 380.371i) q^{14} +(-128.000 - 221.703i) q^{16} +(530.732 - 919.254i) q^{17} +(941.989 + 1631.57i) q^{19} +336.000 q^{20} +2503.49 q^{22} +(1858.68 + 3219.34i) q^{23} +(1342.00 - 2324.41i) q^{25} +(413.642 + 716.449i) q^{26} +(-612.716 + 1981.71i) q^{28} +123.747 q^{29} +(4554.63 - 7888.85i) q^{31} +(-512.000 + 886.810i) q^{32} -4245.85 q^{34} +(-1850.43 - 1996.95i) q^{35} +(3014.36 + 5221.03i) q^{37} +(3767.96 - 6526.29i) q^{38} +(-672.000 - 1163.94i) q^{40} +17201.9 q^{41} +5401.98 q^{43} +(-5006.99 - 8672.36i) q^{44} +(7434.74 - 12877.3i) q^{46} +(937.621 + 1624.01i) q^{47} +(15152.2 - 7272.19i) q^{49} -10736.0 q^{50} +(1654.57 - 2865.80i) q^{52} +(9353.62 - 16200.9i) q^{53} +13143.3 q^{55} +(8090.27 - 1840.91i) q^{56} +(-247.495 - 428.673i) q^{58} +(-1267.39 + 2195.18i) q^{59} +(1047.35 + 1814.07i) q^{61} -36437.1 q^{62} +4096.00 q^{64} +(2171.62 + 3761.36i) q^{65} +(-29310.4 + 50767.1i) q^{67} +(8491.71 + 14708.1i) q^{68} +(-3216.76 + 10404.0i) q^{70} +31279.5 q^{71} +(3575.24 - 6192.49i) q^{73} +(12057.5 - 20884.1i) q^{74} -30143.7 q^{76} +(-23967.7 + 77518.7i) q^{77} +(-1489.91 - 2580.59i) q^{79} +(-2688.00 + 4655.75i) q^{80} +(-34403.7 - 59589.0i) q^{82} -45954.6 q^{83} -22290.7 q^{85} +(-10804.0 - 18713.0i) q^{86} +(-20028.0 + 34689.4i) q^{88} +(49520.0 + 85771.1i) q^{89} +(-26144.4 + 5949.04i) q^{91} -59477.9 q^{92} +(3750.48 - 6496.03i) q^{94} +(19781.8 - 34263.0i) q^{95} -115548. q^{97} +(-55496.1 - 37944.5i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} - 32 q^{4} - 42 q^{5} + 232 q^{7} + 256 q^{8} - 168 q^{10} - 294 q^{11} - 280 q^{13} + 232 q^{14} - 512 q^{16} + 1302 q^{17} + 1442 q^{19} + 1344 q^{20} + 2352 q^{22} + 2646 q^{23} + 5368 q^{25}+ \cdots - 63272 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 - 32 * q^4 - 42 * q^5 + 232 * q^7 + 256 * q^8 - 168 * q^10 - 294 * q^11 - 280 * q^13 + 232 * q^14 - 512 * q^16 + 1302 * q^17 + 1442 * q^19 + 1344 * q^20 + 2352 * q^22 + 2646 * q^23 + 5368 * q^25 + 560 * q^26 - 4640 * q^28 - 3336 * q^29 + 14798 * q^31 - 2048 * q^32 - 10416 * q^34 + 1218 * q^35 - 5182 * q^37 + 5768 * q^38 - 2688 * q^40 + 10248 * q^41 - 9040 * q^43 - 4704 * q^44 + 10584 * q^46 - 14994 * q^47 - 2876 * q^49 - 42944 * q^50 + 2240 * q^52 + 24006 * q^53 + 12348 * q^55 + 14848 * q^56 + 6672 * q^58 + 38850 * q^59 + 23618 * q^61 - 118384 * q^62 + 16384 * q^64 + 2940 * q^65 - 32002 * q^67 + 20832 * q^68 - 24360 * q^70 + 178752 * q^71 + 47138 * q^73 - 20728 * q^74 - 46144 * q^76 + 22890 * q^77 + 40970 * q^79 - 10752 * q^80 - 20496 * q^82 - 136752 * q^83 - 54684 * q^85 + 18080 * q^86 - 18816 * q^88 + 123102 * q^89 - 53680 * q^91 - 84672 * q^92 - 59976 * q^94 + 30282 * q^95 - 87304 * q^97 - 63272 * q^98

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