LMFDB - Embedded newform 68.6.h.a.49.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,6,Mod(9,68)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("68.9");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	68.h (of order $8$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$10.9060997473$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$7$ over $\Q(\zeta_{8})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			49.4
Character	$\chi$	$=$	68.49
Dual form			68.6.h.a.25.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.54560 + 6.14561i) q^{3} +(8.67300 - 3.59247i) q^{5} +(-214.730 - 88.9439i) q^{7} +(140.538 - 140.538i) q^{9} +O(q^{10})$	\(q+(2.54560 + 6.14561i) q^{3} +(8.67300 - 3.59247i) q^{5} +(-214.730 - 88.9439i) q^{7} +(140.538 - 140.538i) q^{9} +(-17.1015 + 41.2866i) q^{11} -190.803i q^{13} +(44.1559 + 44.1559i) q^{15} +(283.339 - 1157.40i) q^{17} +(-1740.56 - 1740.56i) q^{19} -1546.06i q^{21} +(958.085 - 2313.02i) q^{23} +(-2147.39 + 2147.39i) q^{25} +(2714.83 + 1124.52i) q^{27} +(-2026.61 + 839.451i) q^{29} +(-2205.31 - 5324.09i) q^{31} -297.265 q^{33} -2181.88 q^{35} +(-1864.70 - 4501.79i) q^{37} +(1172.60 - 485.707i) q^{39} +(17686.9 + 7326.14i) q^{41} +(-10088.3 + 10088.3i) q^{43} +(714.009 - 1723.77i) q^{45} +22222.4i q^{47} +(26313.4 + 26313.4i) q^{49} +(7834.20 - 1204.98i) q^{51} +(-21151.3 - 21151.3i) q^{53} +419.515i q^{55} +(6266.06 - 15127.6i) q^{57} +(-20226.0 + 20226.0i) q^{59} +(20556.7 + 8514.86i) q^{61} +(-42677.8 + 17677.7i) q^{63} +(-685.454 - 1654.83i) q^{65} +28761.8 q^{67} +16653.8 q^{69} +(34.1214 + 82.3764i) q^{71} +(5758.97 - 2385.44i) q^{73} +(-18663.4 - 7730.65i) q^{75} +(7344.38 - 7344.38i) q^{77} +(13959.4 - 33700.9i) q^{79} -28749.7i q^{81} +(-69193.3 - 69193.3i) q^{83} +(-1700.53 - 11056.0i) q^{85} +(-10317.9 - 10317.9i) q^{87} -47811.7i q^{89} +(-16970.7 + 40971.0i) q^{91} +(27106.0 - 27106.0i) q^{93} +(-21348.8 - 8842.98i) q^{95} +(54218.0 - 22457.8i) q^{97} +(3398.94 + 8205.77i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 44 q^{5} + 44 q^{9}+O(q^{10}) $ 28 * q + 44 * q^5 + 44 * q^9	$ 28 q + 44 q^{5} + 44 q^{9} - 600 q^{11} - 4372 q^{15} + 1212 q^{17} + 284 q^{19} + 1672 q^{23} - 3676 q^{25} - 20796 q^{27} - 11316 q^{29} + 16632 q^{31} + 32888 q^{33} - 6464 q^{35} + 19564 q^{37} - 15880 q^{39} - 14620 q^{41} - 17628 q^{43} + 64172 q^{45} - 64812 q^{49} + 3256 q^{51} + 5396 q^{53} + 35236 q^{57} + 57012 q^{59} + 70364 q^{61} + 93844 q^{63} - 56064 q^{65} + 62528 q^{67} - 159656 q^{69} - 56144 q^{71} + 61828 q^{73} + 92856 q^{75} + 5236 q^{77} - 181064 q^{79} + 129980 q^{83} - 250260 q^{85} - 498484 q^{87} - 20472 q^{91} - 336876 q^{93} - 3044 q^{95} + 317004 q^{97} + 268076 q^{99}+O(q^{100}) $ 28 * q + 44 * q^5 + 44 * q^9 - 600 * q^11 - 4372 * q^15 + 1212 * q^17 + 284 * q^19 + 1672 * q^23 - 3676 * q^25 - 20796 * q^27 - 11316 * q^29 + 16632 * q^31 + 32888 * q^33 - 6464 * q^35 + 19564 * q^37 - 15880 * q^39 - 14620 * q^41 - 17628 * q^43 + 64172 * q^45 - 64812 * q^49 + 3256 * q^51 + 5396 * q^53 + 35236 * q^57 + 57012 * q^59 + 70364 * q^61 + 93844 * q^63 - 56064 * q^65 + 62528 * q^67 - 159656 * q^69 - 56144 * q^71 + 61828 * q^73 + 92856 * q^75 + 5236 * q^77 - 181064 * q^79 + 129980 * q^83 - 250260 * q^85 - 498484 * q^87 - 20472 * q^91 - 336876 * q^93 - 3044 * q^95 + 317004 * q^97 + 268076 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$1$	$e\left(\frac{3}{8}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	2.54560	+	6.14561i	0.163300	+	0.394241i	0.984256	−	0.176751i	$-0.0565587\pi$
$3$	2.54560	+	6.14561i	0.163300	+	0.394241i	−0.820956	+	0.570992i	$0.806559\pi$
$4$		0			0
$5$	8.67300	−	3.59247i	0.155147	−	0.0642641i	−0.303758	−	0.952749i	$-0.598242\pi$
$5$	8.67300	−	3.59247i	0.155147	−	0.0642641i	0.458906	+	0.888485i	$0.348242\pi$
$6$		0			0
$7$	−214.730	−	88.9439i	−1.65633	−	0.686074i	−0.658541	−	0.752545i	$-0.728826\pi$
$7$	−214.730	−	88.9439i	−1.65633	−	0.686074i	−0.997788	+	0.0664711i	$0.978826\pi$
$8$		0			0
$9$	140.538	−	140.538i	0.578348	−	0.578348i
$10$		0			0
$11$	−17.1015	+	41.2866i	−0.0426139	+	0.102879i	−0.943753	−	0.330650i	$-0.892732\pi$
$11$	−17.1015	+	41.2866i	−0.0426139	+	0.102879i	0.901139	+	0.433529i	$0.142732\pi$
$12$		0			0
$13$		−	190.803i		−	0.313131i	−0.987668	−	0.156566i	$-0.949958\pi$
$13$		−	190.803i		−	0.313131i	0.987668	−	0.156566i	$-0.0500423\pi$
$14$		0			0
$15$	44.1559	+	44.1559i	0.0506711	+	0.0506711i
$16$		0			0
$17$	283.339	−	1157.40i	0.237785	−	0.971318i
$17$	283.339	−	1157.40i	0.237785	−	0.971318i
$18$		0			0
$19$	−1740.56	−	1740.56i	−1.10613	−	1.10613i	−0.993655	−	0.112475i	$-0.964122\pi$
$19$	−1740.56	−	1740.56i	−1.10613	−	1.10613i	−0.112475	−	0.993655i	$-0.535878\pi$
$20$		0			0
$21$		−	1546.06i		−	0.765029i
$22$		0			0
$23$	958.085	−	2313.02i	0.377645	−	0.911717i	−0.614761	−	0.788714i	$-0.710747\pi$
$23$	958.085	−	2313.02i	0.377645	−	0.911717i	0.992406	−	0.123003i	$-0.0392526\pi$
$24$		0			0
$25$	−2147.39	+	2147.39i	−0.687166	+	0.687166i
$26$		0			0
$27$	2714.83	+	1124.52i	0.716694	+	0.296864i
$28$		0			0
$29$	−2026.61	+	839.451i	−0.447483	+	0.185353i	−0.595033	−	0.803701i	$-0.702861\pi$
$29$	−2026.61	+	839.451i	−0.447483	+	0.185353i	0.147551	+	0.989055i	$0.452861\pi$
$30$		0			0
$31$	−2205.31	−	5324.09i	−0.412160	−	0.995042i	−0.984557	−	0.175065i	$-0.943986\pi$
$31$	−2205.31	−	5324.09i	−0.412160	−	0.995042i	0.572397	−	0.819977i	$-0.306014\pi$
$32$		0			0
$33$	−297.265			−0.0475180
$34$		0			0
$35$	−2181.88			−0.301065
$36$		0			0
$37$	−1864.70	−	4501.79i	−0.223926	−	0.540606i	0.771490	−	0.636241i	$-0.219512\pi$
$37$	−1864.70	−	4501.79i	−0.223926	−	0.540606i	−0.995416	+	0.0956354i	$0.969512\pi$
$38$		0			0
$39$	1172.60	−	485.707i	0.123449	−	0.0511343i
$40$		0			0
$41$	17686.9	+	7326.14i	1.64320	+	0.680637i	0.996616	−	0.0822028i	$-0.0261955\pi$
$41$	17686.9	+	7326.14i	1.64320	+	0.680637i	0.646587	+	0.762840i	$0.276196\pi$
$42$		0			0
$43$	−10088.3	+	10088.3i	−0.832047	+	0.832047i	−0.987797	−	0.155750i	$-0.950221\pi$
$43$	−10088.3	+	10088.3i	−0.832047	+	0.832047i	0.155750	+	0.987797i	$0.450221\pi$
$44$		0			0
$45$	714.009	−	1723.77i	0.0525621	−	0.126896i
$46$		0			0
$47$			22222.4i			1.46739i	0.679478	+	0.733696i	$0.262206\pi$
$47$			22222.4i			1.46739i	−0.679478	+	0.733696i	$0.737794\pi$
$48$		0			0
$49$	26313.4	+	26313.4i	1.56562	+	1.56562i
$50$		0			0
$51$	7834.20	−	1204.98i	0.421764	−	0.0648717i
$52$		0			0
$53$	−21151.3	−	21151.3i	−1.03430	−	1.03430i	−0.999390	−	0.0349101i	$-0.988885\pi$
$53$	−21151.3	−	21151.3i	−1.03430	−	1.03430i	−0.0349101	−	0.999390i	$-0.511115\pi$
$54$		0			0
$55$			419.515i			0.0187000i
$56$		0			0
$57$	6266.06	−	15127.6i	0.255451	−	0.616713i
$58$		0			0
$59$	−20226.0	+	20226.0i	−0.756449	+	0.756449i	−0.975674	−	0.219226i	$-0.929647\pi$
$59$	−20226.0	+	20226.0i	−0.756449	+	0.756449i	0.219226	+	0.975674i	$0.429647\pi$
$60$		0			0
$61$	20556.7	+	8514.86i	0.707340	+	0.292990i	0.707204	−	0.707010i	$-0.249956\pi$
$61$	20556.7	+	8514.86i	0.707340	+	0.292990i	0.000136863		1.00000i	$0.499956\pi$
$62$		0			0
$63$	−42677.8	+	17677.7i	−1.35472	+	0.561145i
$64$		0			0
$65$	−685.454	−	1654.83i	−0.0201231	−	0.0485815i
$66$		0			0
$67$	28761.8			0.782760			0.391380	−	0.920229i	$-0.371998\pi$
$67$	28761.8			0.782760			0.391380	+	0.920229i	$0.371998\pi$
$68$		0			0
$69$	16653.8			0.421106
$70$		0			0
$71$	34.1214	+	82.3764i	0.000803306	+	0.00193935i	0.924281	−	0.381714i	$-0.124666\pi$
$71$	34.1214	+	82.3764i	0.000803306	+	0.00193935i	−0.923477	+	0.383653i	$0.874666\pi$
$72$		0			0
$73$	5758.97	−	2385.44i	0.126485	−	0.0523916i	−0.318544	−	0.947908i	$-0.603194\pi$
$73$	5758.97	−	2385.44i	0.126485	−	0.0523916i	0.445028	+	0.895517i	$0.353194\pi$
$74$		0			0
$75$	−18663.4	−	7730.65i	−0.383123	−	0.158695i
$76$		0			0
$77$	7344.38	−	7344.38i	0.141165	−	0.141165i
$78$		0			0
$79$	13959.4	−	33700.9i	0.251651	−	0.607539i	−0.746687	−	0.665176i	$-0.768357\pi$
$79$	13959.4	−	33700.9i	0.251651	−	0.607539i	0.998338	+	0.0576369i	$0.0183566\pi$
$80$		0			0
$81$		−	28749.7i		−	0.486879i
$82$		0			0
$83$	−69193.3	−	69193.3i	−1.10248	−	1.10248i	−0.994111	−	0.108365i	$-0.965439\pi$
$83$	−69193.3	−	69193.3i	−1.10248	−	1.10248i	−0.108365	−	0.994111i	$-0.534561\pi$
$84$		0			0
$85$	−1700.53	−	11056.0i	−0.0255292	−	0.165978i
$86$		0			0
$87$	−10317.9	−	10317.9i	−0.146148	−	0.146148i
$88$		0			0
$89$		−	47811.7i		−	0.639822i	−0.947448	−	0.319911i	$-0.896347\pi$
$89$		−	47811.7i		−	0.639822i	0.947448	−	0.319911i	$-0.103653\pi$
$90$		0			0
$91$	−16970.7	+	40971.0i	−0.214831	+	0.518648i
$92$		0			0
$93$	27106.0	−	27106.0i	0.324981	−	0.324981i
$94$		0			0
$95$	−21348.8	−	8842.98i	−0.242698	−	0.100529i
$96$		0			0
$97$	54218.0	−	22457.8i	0.585078	−	0.242347i	−0.0704535	−	0.997515i	$-0.522445\pi$
$97$	54218.0	−	22457.8i	0.585078	−	0.242347i	0.655532	+	0.755168i	$0.272445\pi$
$98$		0			0
$99$	3398.94	+	8205.77i	0.0348542	+	0.0841455i
$100$		0			0
$101$	75591.4			0.737342			0.368671	−	0.929560i	$-0.379813\pi$
$101$	75591.4			0.737342			0.368671	+	0.929560i	$0.379813\pi$
$102$		0			0
$103$	−110126.			−1.02281			−0.511405	−	0.859340i	$-0.670875\pi$
$103$	−110126.			−1.02281			−0.511405	+	0.859340i	$0.670875\pi$
$104$		0			0
$105$	−5554.18	−	13409.0i	−0.0491639	−	0.118692i
$106$		0			0
$107$	−33845.3	+	14019.2i	−0.285785	+	0.118376i	−0.520971	−	0.853574i	$-0.674430\pi$
$107$	−33845.3	+	14019.2i	−0.285785	+	0.118376i	0.235186	+	0.971950i	$0.424430\pi$
$108$		0			0
$109$	−50187.8	−	20788.5i	−0.404606	−	0.167593i	0.171093	−	0.985255i	$-0.445270\pi$
$109$	−50187.8	−	20788.5i	−0.404606	−	0.167593i	−0.575699	+	0.817662i	$0.695270\pi$
$110$		0			0
$111$	22919.5	−	22919.5i	0.176562	−	0.176562i
$112$		0			0
$113$	28118.6	−	67884.4i	0.207156	−	0.500120i	−0.785817	−	0.618459i	$-0.787757\pi$
$113$	28118.6	−	67884.4i	0.207156	−	0.500120i	0.992973	+	0.118340i	$0.0377572\pi$
$114$		0			0
$115$		−	23502.7i		−	0.165719i
$116$		0			0
$117$	−26815.1	−	26815.1i	−0.181099	−	0.181099i
$118$		0			0
$119$	−163785.	+	223327.i	−1.06025	+	1.44568i
$120$		0			0
$121$	112468.	+	112468.i	0.698339	+	0.698339i
$122$		0			0
$123$			127346.i			0.758966i
$124$		0			0
$125$	−22136.4	+	53441.9i	−0.126716	+	0.305919i
$126$		0			0
$127$	154693.	−	154693.i	0.851061	−	0.851061i	−0.139203	−	0.990264i	$-0.544454\pi$
$127$	154693.	−	154693.i	0.851061	−	0.851061i	0.990264	+	0.139203i	$0.0444540\pi$
$128$		0			0
$129$	−87679.7	−	36318.1i	−0.463900	−	0.192154i
$130$		0			0
$131$	217179.	−	89958.4i	1.10570	−	0.457998i	0.246248	−	0.969207i	$-0.420802\pi$
$131$	217179.	−	89958.4i	1.10570	−	0.457998i	0.859456	+	0.511209i	$0.170802\pi$
$132$		0			0
$133$	218938.	+	528563.i	1.07323	+	2.59100i
$134$		0			0
$135$	27585.6			0.130271
$136$		0			0
$137$	85863.1			0.390845			0.195423	−	0.980719i	$-0.437392\pi$
$137$	85863.1			0.390845			0.195423	+	0.980719i	$0.437392\pi$
$138$		0			0
$139$	41930.2	+	101229.i	0.184073	+	0.444392i	0.988799	−	0.149255i	$-0.0476876\pi$
$139$	41930.2	+	101229.i	0.184073	+	0.444392i	−0.804726	+	0.593647i	$0.797688\pi$
$140$		0			0
$141$	−136570.	+	56569.2i	−0.578506	+	0.239625i
$142$		0			0
$143$	7877.59	+	3263.01i	0.0322147	+	0.0133437i
$144$		0			0
$145$	−14561.1	+	14561.1i	−0.0575141	+	0.0575141i
$146$		0			0
$147$	−94728.7	+	228695.i	−0.361566	+	0.872899i
$148$		0			0
$149$		−	224743.i		−	0.829316i	−0.909977	−	0.414658i	$-0.863901\pi$
$149$		−	224743.i		−	0.829316i	0.909977	−	0.414658i	$-0.136099\pi$
$150$		0			0
$151$	−83213.2	−	83213.2i	−0.296996	−	0.296996i	0.542840	−	0.839836i	$-0.317349\pi$
$151$	−83213.2	−	83213.2i	−0.296996	−	0.296996i	−0.839836	+	0.542840i	$0.817349\pi$
$152$		0			0
$153$	−122839.	−	202479.i	−0.424237	−	0.699282i
$154$		0			0
$155$	−38253.3	−	38253.3i	−0.127891	−	0.127891i
$156$		0			0
$157$			63051.8i			0.204150i	0.994777	+	0.102075i	$0.0325481\pi$
$157$			63051.8i			0.204150i	−0.994777	+	0.102075i	$0.967452\pi$
$158$		0			0
$159$	76144.9	−	183830.i	0.238863	−	0.576665i
$160$		0			0
$161$	−411458.	+	411458.i	−1.25101	+	1.25101i
$162$		0			0
$163$	482612.	+	199904.i	1.42275	+	0.589323i	0.955550	−	0.294828i	$-0.0952624\pi$
$163$	482612.	+	199904.i	1.42275	+	0.589323i	0.467201	+	0.884151i	$0.345262\pi$
$164$		0			0
$165$	−2578.18	+	1067.92i	−0.00737230	+	0.00305370i
$166$		0			0
$167$	−134104.	−	323756.i	−0.372093	−	0.898311i	−0.993396	−	0.114740i	$-0.963396\pi$
$167$	−134104.	−	323756.i	−0.372093	−	0.898311i	0.621303	−	0.783570i	$-0.286604\pi$
$168$		0			0
$169$	334887.			0.901949
$170$		0			0
$171$	−489232.			−1.27945
$172$		0			0
$173$	−125196.	−	302249.i	−0.318035	−	0.767803i	−0.999358	−	0.0358210i	$-0.988595\pi$
$173$	−125196.	−	302249.i	−0.318035	−	0.767803i	0.681324	−	0.731982i	$-0.261405\pi$
$174$		0			0
$175$	652106.	−	270111.i	1.60962	−	0.666726i
$176$		0			0
$177$	−175788.	−	72813.8i	−0.421751	−	0.174695i
$178$		0			0
$179$	−290334.	+	290334.i	−0.677276	+	0.677276i	−0.959383	−	0.282107i	$-0.908967\pi$
$179$	−290334.	+	290334.i	−0.677276	+	0.677276i	0.282107	+	0.959383i	$0.408967\pi$
$180$		0			0
$181$	83929.9	−	202625.i	0.190423	−	0.459723i	−0.799616	−	0.600511i	$-0.794964\pi$
$181$	83929.9	−	202625.i	0.190423	−	0.459723i	0.990040	+	0.140788i	$0.0449637\pi$
$182$		0			0
$183$			148009.i			0.326708i
$184$		0			0
$185$	−32345.1	−	32345.1i	−0.0694831	−	0.0694831i
$186$		0			0
$187$	42939.6	+	31491.3i	0.0897954	+	0.0658547i
$188$		0			0
$189$	−482935.	−	482935.i	−0.983410	−	0.983410i
$190$		0			0
$191$		−	464279.i		−	0.920864i	−0.887695	−	0.460432i	$-0.847694\pi$
$191$		−	464279.i		−	0.920864i	0.887695	−	0.460432i	$-0.152306\pi$
$192$		0			0
$193$	229518.	−	554106.i	0.443531	−	1.07078i	−0.531170	−	0.847265i	$-0.678247\pi$
$193$	229518.	−	554106.i	0.443531	−	1.07078i	0.974701	−	0.223513i	$-0.0717526\pi$
$194$		0			0
$195$	8425.07	−	8425.07i	0.0158667	−	0.0158667i
$196$		0			0
$197$	−730667.	−	302652.i	−1.34139	−	0.555620i	−0.407504	−	0.913203i	$-0.633601\pi$
$197$	−730667.	−	302652.i	−1.34139	−	0.555620i	−0.933881	+	0.357583i	$0.883601\pi$
$198$		0			0
$199$	−50067.7	+	20738.7i	−0.0896241	+	0.0371235i	−0.427045	−	0.904230i	$-0.640445\pi$
$199$	−50067.7	+	20738.7i	−0.0896241	+	0.0371235i	0.337421	+	0.941354i	$0.390445\pi$
$200$		0			0
$201$	73215.8	+	176759.i	0.127825	+	0.308596i
$202$		0			0
$203$	509838.			0.868344
$204$		0			0
$205$	179717.			0.298679
$206$		0			0
$207$	−190421.	−	459716.i	−0.308879	−	0.745700i
$208$		0			0
$209$	101628.	−	42095.8i	0.160934	−	0.0666611i
$210$		0			0
$211$	667462.	+	276472.i	1.03210	+	0.427509i	0.833469	−	0.552566i	$-0.186351\pi$
$211$	667462.	+	276472.i	1.03210	+	0.427509i	0.198628	+	0.980075i	$0.436351\pi$
$212$		0			0
$213$	−419.394	+	419.394i	−0.000633393	+	0.000633393i
$214$		0			0
$215$	−51254.0	+	123738.i	−0.0756191	+	0.182561i
$216$		0			0
$217$			1.33939e6i			1.93089i
$218$		0			0
$219$	29320.0	+	29320.0i	0.0413099	+	0.0413099i
$220$		0			0
$221$	−220835.	−	54061.9i	−0.304150	−	0.0744578i
$222$		0			0
$223$	−253145.	−	253145.i	−0.340884	−	0.340884i	0.515815	−	0.856700i	$-0.327489\pi$
$223$	−253145.	−	253145.i	−0.340884	−	0.340884i	−0.856700	+	0.515815i	$0.827489\pi$
$224$		0			0
$225$			603583.i			0.794842i
$226$		0			0
$227$	528338.	−	1.27552e6i	0.680529	−	1.64294i	−0.0825094	−	0.996590i	$-0.526293\pi$
$227$	528338.	−	1.27552e6i	0.680529	−	1.64294i	0.763039	−	0.646353i	$-0.223707\pi$
$228$		0			0
$229$	−808869.	+	808869.i	−1.01927	+	1.01927i	−0.0194596	+	0.999811i	$0.506195\pi$
$229$	−808869.	+	808869.i	−1.01927	+	1.01927i	−0.999811	+	0.0194596i	$0.993805\pi$
$230$		0			0
$231$	63831.5	+	26439.9i	0.0787055	+	0.0326009i
$232$		0			0
$233$	−936310.	+	387832.i	−1.12987	+	0.468009i	−0.867738	−	0.497022i	$-0.834427\pi$
$233$	−936310.	+	387832.i	−1.12987	+	0.468009i	−0.262135	+	0.965031i	$0.584427\pi$
$234$		0			0
$235$	79833.3	+	192735.i	0.0943006	+	0.227662i
$236$		0			0
$237$	242648.			0.280611
$238$		0			0
$239$	197846.			0.224044			0.112022	−	0.993706i	$-0.464267\pi$
$239$	197846.			0.224044			0.112022	+	0.993706i	$0.464267\pi$
$240$		0			0
$241$	−265397.	−	640726.i	−0.294343	−	0.710607i	−0.999998	−	0.00205148i	$-0.999347\pi$
$241$	−265397.	−	640726.i	−0.294343	−	0.710607i	0.705655	−	0.708556i	$-0.250653\pi$
$242$		0			0
$243$	836389.	−	346444.i	0.908641	−	0.376372i
$244$		0			0
$245$	322746.	+	133686.i	0.343515	+	0.142289i
$246$		0			0
$247$	−332105.	+	332105.i	−0.346364	+	0.346364i
$248$		0			0
$249$	249097.	−	601374.i	0.254607	−	0.614676i
$250$		0			0
$251$			1.91454e6i			1.91814i	0.283172	+	0.959069i	$0.408613\pi$
$251$			1.91454e6i			1.91814i	−0.283172	+	0.959069i	$0.591387\pi$
$252$		0			0
$253$	79112.1	+	79112.1i	0.0777037	+	0.0777037i
$254$		0			0
$255$	63617.1	−	38595.0i	0.0612666	−	0.0371689i
$256$		0			0
$257$	461989.	+	461989.i	0.436314	+	0.436314i	0.890769	−	0.454456i	$-0.150166\pi$
$257$	461989.	+	461989.i	0.436314	+	0.436314i	−0.454456	+	0.890769i	$0.650166\pi$
$258$		0			0
$259$			1.13252e6i			1.04905i
$260$		0			0
$261$	−166842.	+	402792.i	−0.151602	+	0.365999i
$262$		0			0
$263$	−122604.	+	122604.i	−0.109299	+	0.109299i	−0.759641	−	0.650342i	$-0.774625\pi$
$263$	−122604.	+	122604.i	−0.109299	+	0.109299i	0.650342	+	0.759641i	$0.274625\pi$
$264$		0			0
$265$	−259430.	−	107460.i	−0.226937	−	0.0940005i
$266$		0			0
$267$	293832.	−	121709.i	0.252244	−	0.104483i
$268$		0			0
$269$	861511.	+	2.07987e6i	0.725906	+	1.75249i	0.655779	+	0.754953i	$0.272340\pi$
$269$	861511.	+	2.07987e6i	0.725906	+	1.75249i	0.0701268	+	0.997538i	$0.477660\pi$
$270$		0			0
$271$	2.01390e6			1.66577			0.832884	−	0.553448i	$-0.186688\pi$
$271$	2.01390e6			1.66577			0.832884	+	0.553448i	$0.186688\pi$
$272$		0			0
$273$	−294992.			−0.239554
$274$		0			0
$275$	−51935.0	−	125382.i	−0.0414122	−	0.0999779i
$276$		0			0
$277$	1.38965e6	−	575613.i	1.08820	−	0.450745i	0.234819	−	0.972039i	$-0.424550\pi$
$277$	1.38965e6	−	575613.i	1.08820	−	0.450745i	0.853377	+	0.521294i	$0.174550\pi$
$278$		0			0
$279$	−1.05817e6	−	438309.i	−0.813851	−	0.337108i
$280$		0			0
$281$	260071.	−	260071.i	0.196483	−	0.196483i	−0.602007	−	0.798491i	$-0.705632\pi$
$281$	260071.	−	260071.i	0.196483	−	0.196483i	0.798491	+	0.602007i	$0.205632\pi$
$282$		0			0
$283$	−623879.	+	1.50618e6i	−0.463057	+	1.11792i	0.504079	+	0.863658i	$0.331832\pi$
$283$	−623879.	+	1.50618e6i	−0.463057	+	1.11792i	−0.967136	+	0.254261i	$0.918168\pi$
$284$		0			0
$285$		−	153712.i		−	0.112098i
$286$		0			0
$287$	−3.14628e6	−	3.14628e6i	−2.25472	−	2.25472i
$288$		0			0
$289$	−1.25929e6	−	655873.i	−0.886917	−	0.461929i
$290$		0			0
$291$	276034.	+	276034.i	0.191087	+	0.191087i
$292$		0			0
$293$		−	1.03184e6i		−	0.702174i	−0.936343	−	0.351087i	$-0.885812\pi$
$293$		−	1.03184e6i		−	0.702174i	0.936343	−	0.351087i	$-0.114188\pi$
$294$		0			0
$295$	−102759.	+	248081.i	−0.0687485	+	0.165974i
$296$		0			0
$297$	−92855.2	+	92855.2i	−0.0610823	+	0.0610823i
$298$		0			0
$299$	−441331.	−	182805.i	−0.285487	−	0.118253i
$300$		0			0
$301$	3.06355e6	−	1.26897e6i	1.94899	−	0.807298i
$302$		0			0
$303$	192425.	+	464556.i	0.120408	+	0.290691i
$304$		0			0
$305$	208878.			0.128571
$306$		0			0
$307$	−546859.			−0.331154			−0.165577	−	0.986197i	$-0.552949\pi$
$307$	−546859.			−0.331154			−0.165577	+	0.986197i	$0.552949\pi$
$308$		0			0
$309$	−280335.	−	676789.i	−0.167025	−	0.403234i
$310$		0			0
$311$	1.37692e6	−	570339.i	0.807249	−	0.334373i	0.0593932	−	0.998235i	$-0.481083\pi$
$311$	1.37692e6	−	570339.i	0.807249	−	0.334373i	0.747856	+	0.663861i	$0.231083\pi$
$312$		0			0
$313$	−2.15748e6	−	893659.i	−1.24476	−	0.515598i	−0.339563	−	0.940583i	$-0.610279\pi$
$313$	−2.15748e6	−	893659.i	−1.24476	−	0.515598i	−0.905200	+	0.424986i	$0.860279\pi$
$314$		0			0
$315$	−306638.	+	306638.i	−0.174120	+	0.174120i
$316$		0			0
$317$	−818333.	+	1.97563e6i	−0.457385	+	1.10423i	0.512067	+	0.858945i	$0.328880\pi$
$317$	−818333.	+	1.97563e6i	−0.457385	+	1.10423i	−0.969452	+	0.245280i	$0.921120\pi$
$318$		0			0
$319$		−	98027.8i		−	0.0539352i
$320$		0			0
$321$	−172313.	−	172313.i	−0.0933374	−	0.0933374i
$322$		0			0
$323$	−2.50770e6	+	1.52136e6i	−1.33742	+	0.811383i
$324$		0			0
$325$	409729.	+	409729.i	0.215173	+	0.215173i
$326$		0			0
$327$		−	361354.i		−	0.186880i
$328$		0			0
$329$	1.97655e6	−	4.77180e6i	1.00674	−	2.43048i
$330$		0			0
$331$	694449.	−	694449.i	0.348394	−	0.348394i	−0.511117	−	0.859511i	$-0.670768\pi$
$331$	694449.	−	694449.i	0.348394	−	0.348394i	0.859511	+	0.511117i	$0.170768\pi$
$332$		0			0
$333$	−894737.	−	370612.i	−0.442165	−	0.183151i
$334$		0			0
$335$	249451.	−	103326.i	0.121443	−	0.0503034i
$336$		0			0
$337$	349928.	+	844801.i	0.167843	+	0.405209i	0.985312	−	0.170763i	$-0.0546234\pi$
$337$	349928.	+	844801.i	0.167843	+	0.405209i	−0.817469	+	0.575973i	$0.804623\pi$
$338$		0			0
$339$	488770.			0.230996
$340$		0			0
$341$	257528.			0.119933
$342$		0			0
$343$	−1.81497e6	−	4.38172e6i	−0.832978	−	2.01099i
$344$		0			0
$345$	144439.	−	59828.4i	0.0653334	−	0.0270620i
$346$		0			0
$347$	136580.	+	56573.5i	0.0608926	+	0.0252226i	0.412922	−	0.910766i	$-0.364508\pi$
$347$	136580.	+	56573.5i	0.0608926	+	0.0252226i	−0.352029	+	0.935989i	$0.614508\pi$
$348$		0			0
$349$	−260158.	+	260158.i	−0.114334	+	0.114334i	−0.761959	−	0.647625i	$-0.775762\pi$
$349$	−260158.	+	260158.i	−0.114334	+	0.114334i	0.647625	+	0.761959i	$0.275762\pi$
$350$		0			0
$351$	214562.	−	517998.i	0.0929575	−	0.224419i
$352$		0			0
$353$		−	564379.i		−	0.241065i	−0.992709	−	0.120532i	$-0.961540\pi$
$353$		−	564379.i		−	0.241065i	0.992709	−	0.120532i	$-0.0384602\pi$
$354$		0			0
$355$	591.870	+	591.870i	0.000249262	+	0.000249262i
$356$		0			0
$357$	−1.78941e6	−	438059.i	−0.743086	−	0.181912i
$358$		0			0
$359$	342104.	+	342104.i	0.140095	+	0.140095i	0.773676	−	0.633581i	$-0.218416\pi$
$359$	342104.	+	342104.i	0.140095	+	0.140095i	−0.633581	+	0.773676i	$0.718416\pi$
$360$		0			0
$361$			3.58303e6i			1.44705i
$362$		0			0
$363$	−404887.	+	977484.i	−0.161275	+	0.389353i
$364$		0			0
$365$	41377.9	−	41377.9i	0.0162568	−	0.0162568i
$366$		0			0
$367$	−1.99880e6	−	827930.i	−0.774647	−	0.320869i	−0.0398942	−	0.999204i	$-0.512702\pi$
$367$	−1.99880e6	−	827930.i	−0.774647	−	0.320869i	−0.734753	+	0.678334i	$0.762702\pi$
$368$		0			0
$369$	3.51529e6	−	1.45608e6i	1.34399	−	0.556698i
$370$		0			0
$371$	2.66053e6	+	6.42308e6i	1.00354	+	2.42275i
$372$		0			0
$373$	3.42012e6			1.27283			0.636413	−	0.771348i	$-0.280417\pi$
$373$	3.42012e6			1.27283			0.636413	+	0.771348i	$0.280417\pi$
$374$		0			0
$375$	−384784.			−0.141299
$376$		0			0
$377$	160170.	+	386684.i	0.0580399	+	0.140121i
$378$		0			0
$379$	300356.	−	124412.i	0.107408	−	0.0444900i	−0.328332	−	0.944562i	$-0.606487\pi$
$379$	300356.	−	124412.i	0.107408	−	0.0444900i	0.435741	+	0.900072i	$0.356487\pi$
$380$		0			0
$381$	1.34447e6	+	556896.i	0.474502	+	0.196545i
$382$		0			0
$383$	−1.84299e6	+	1.84299e6i	−0.641986	+	0.641986i	−0.951043	−	0.309058i	$-0.899986\pi$
$383$	−1.84299e6	+	1.84299e6i	−0.641986	+	0.641986i	0.309058	+	0.951043i	$0.399986\pi$
$384$		0			0
$385$	37313.3	−	90082.2i	0.0128296	−	0.0309733i
$386$		0			0
$387$			2.83559e6i			0.962424i
$388$		0			0
$389$	−2.45342e6	−	2.45342e6i	−0.822049	−	0.822049i	0.164353	−	0.986402i	$-0.447447\pi$
$389$	−2.45342e6	−	2.45342e6i	−0.822049	−	0.822049i	−0.986402	+	0.164353i	$0.947447\pi$
$390$		0			0
$391$	−2.40563e6	−	1.76426e6i	−0.795769	−	0.583606i
$392$		0			0
$393$	1.10570e6	+	1.10570e6i	0.361123	+	0.361123i
$394$		0			0
$395$		−	342437.i		−	0.110430i
$396$		0			0
$397$	658987.	−	1.59093e6i	0.209846	−	0.506613i	−0.783553	−	0.621325i	$-0.786595\pi$
$397$	658987.	−	1.59093e6i	0.209846	−	0.506613i	0.993399	+	0.114712i	$0.0365947\pi$
$398$		0			0
$399$	−2.69102e6	+	2.69102e6i	−0.846221	+	0.846221i
$400$		0			0
$401$	4.50770e6	+	1.86715e6i	1.39989	+	0.579854i	0.949723	−	0.313090i	$-0.101364\pi$
$401$	4.50770e6	+	1.86715e6i	1.39989	+	0.579854i	0.450168	+	0.892944i	$0.351364\pi$
$402$		0			0
$403$	−1.01585e6	+	420779.i	−0.311579	+	0.129060i
$404$		0			0
$405$	−103283.	−	249346.i	−0.0312888	−	0.0755380i
$406$		0			0
$407$	217753.			0.0651594
$408$		0			0
$409$	−6.33036e6			−1.87120			−0.935600	−	0.353063i	$-0.885140\pi$
$409$	−6.33036e6			−1.87120			−0.935600	+	0.353063i	$0.885140\pi$
$410$		0			0
$411$	218573.	+	527681.i	0.0638251	+	0.154087i
$412$		0			0
$413$	6.14209e6	−	2.54414e6i	1.77191	−	0.733948i
$414$		0			0
$415$	−848689.	−	351539.i	−0.241896	−	0.100197i
$416$		0			0
$417$	−515374.	+	515374.i	−0.145138	+	0.145138i
$418$		0			0
$419$	−731626.	+	1.76630e6i	−0.203589	+	0.491507i	−0.992389	−	0.123143i	$-0.960703\pi$
$419$	−731626.	+	1.76630e6i	−0.203589	+	0.491507i	0.788800	+	0.614650i	$0.210703\pi$
$420$		0			0
$421$			6.16638e6i			1.69561i	0.530310	+	0.847804i	$0.322075\pi$
$421$			6.16638e6i			1.69561i	−0.530310	+	0.847804i	$0.677925\pi$
$422$		0			0
$423$	3.12310e6	+	3.12310e6i	0.848662	+	0.848662i
$424$		0			0
$425$	1.87695e6	+	3.09383e6i	0.504059	+	0.830854i
$426$		0			0
$427$	−3.65678e6	−	3.65678e6i	−0.970576	−	0.970576i
$428$		0			0
$429$			56718.9i			0.0148794i
$430$		0			0
$431$	2.14864e6	−	5.18727e6i	0.557148	−	1.34507i	−0.354868	−	0.934917i	$-0.615474\pi$
$431$	2.14864e6	−	5.18727e6i	0.557148	−	1.34507i	0.912015	−	0.410157i	$-0.134526\pi$
$432$		0			0
$433$	−2.98579e6	+	2.98579e6i	−0.765314	+	0.765314i	−0.977278	−	0.211964i	$-0.932014\pi$
$433$	−2.98579e6	+	2.98579e6i	−0.765314	+	0.765314i	0.211964	+	0.977278i	$0.432014\pi$
$434$		0			0
$435$	−126554.	−	52420.3i	−0.0320665	−	0.0132824i
$436$		0			0
$437$	−5.69357e6	+	2.35835e6i	−1.42620	+	0.590752i
$438$		0			0
$439$	−108478.	−	261888.i	−0.0268645	−	0.0648567i	0.909878	−	0.414876i	$-0.136175\pi$
$439$	−108478.	−	261888.i	−0.0268645	−	0.0648567i	−0.936742	+	0.350019i	$0.886175\pi$
$440$		0			0
$441$	7.39609e6			1.81095
$442$		0			0
$443$	7.20519e6			1.74436			0.872181	−	0.489184i	$-0.162705\pi$
$443$	7.20519e6			1.74436			0.872181	+	0.489184i	$0.162705\pi$
$444$		0			0
$445$	−171762.	−	414671.i	−0.0411176	−	0.0992667i
$446$		0			0
$447$	1.38118e6	−	572104.i	0.326951	−	0.135427i
$448$		0			0
$449$	5.30031e6	+	2.19546e6i	1.24075	+	0.513937i	0.903950	−	0.427639i	$-0.140654\pi$
$449$	5.30031e6	+	2.19546e6i	1.24075	+	0.513937i	0.336803	+	0.941575i	$0.390654\pi$
$450$		0			0
$451$	−604942.	+	604942.i	−0.140047	+	0.140047i
$452$		0			0
$453$	299569.	−	723223.i	0.0685885	−	0.165587i
$454$		0			0
$455$			416308.i			0.0942728i
$456$		0			0
$457$	−5.42313e6	−	5.42313e6i	−1.21467	−	1.21467i	−0.969473	−	0.245200i	$-0.921146\pi$
$457$	−5.42313e6	−	5.42313e6i	−1.21467	−	1.21467i	−0.245200	−	0.969473i	$-0.578854\pi$
$458$		0			0
$459$	2.07074e6	−	2.82353e6i	0.458768	−	0.625548i
$460$		0			0
$461$	−5.70178e6	−	5.70178e6i	−1.24956	−	1.24956i	−0.955913	−	0.293651i	$-0.905130\pi$
$461$	−5.70178e6	−	5.70178e6i	−1.24956	−	1.24956i	−0.293651	−	0.955913i	$-0.594870\pi$
$462$		0			0
$463$		−	2.95066e6i		−	0.639687i	−0.947470	−	0.319843i	$-0.896370\pi$
$463$		−	2.95066e6i		−	0.639687i	0.947470	−	0.319843i	$-0.103630\pi$
$464$		0			0
$465$	137713.	−	332467.i	0.0295353	−	0.0713045i
$466$		0			0
$467$	−3.86660e6	+	3.86660e6i	−0.820422	+	0.820422i	−0.986168	−	0.165747i	$-0.946997\pi$
$467$	−3.86660e6	+	3.86660e6i	−0.820422	+	0.820422i	0.165747	+	0.986168i	$0.446997\pi$
$468$		0			0
$469$	−6.17600e6	−	2.55818e6i	−1.29651	−	0.537031i
$470$		0			0
$471$	−387492.	+	160504.i	−0.0804841	+	0.0333376i
$472$		0			0
$473$	−243987.	−	589037.i	−0.0501434	−	0.121057i
$474$		0			0
$475$	7.47535e6			1.52019
$476$		0			0
$477$	−5.94513e6			−1.19637
$478$		0			0
$479$	−966797.	−	2.33405e6i	−0.192529	−	0.464806i	0.797907	−	0.602781i	$-0.205941\pi$
$479$	−966797.	−	2.33405e6i	−0.192529	−	0.464806i	−0.990436	+	0.137975i	$0.955941\pi$
$480$		0			0
$481$	−858954.	+	355790.i	−0.169281	+	0.0701183i
$482$		0			0
$483$	−3.57607e6	−	1.48126e6i	−0.697490	−	0.288910i
$484$		0			0
$485$	389553.	−	389553.i	0.0751991	−	0.0751991i
$486$		0			0
$487$	3.17242e6	−	7.65889e6i	0.606133	−	1.46333i	−0.261040	−	0.965328i	$-0.584065\pi$
$487$	3.17242e6	−	7.65889e6i	0.606133	−	1.46333i	0.867173	−	0.498007i	$-0.165935\pi$
$488$		0			0
$489$			3.47482e6i			0.657144i
$490$		0			0
$491$	−530314.	−	530314.i	−0.0992727	−	0.0992727i	0.655726	−	0.754999i	$-0.272363\pi$
$491$	−530314.	−	530314.i	−0.0992727	−	0.0992727i	−0.754999	+	0.655726i	$0.772363\pi$
$492$		0			0
$493$	397362.	+	2.58345e6i	0.0736325	+	0.478722i
$494$		0			0
$495$	58958.0	+	58958.0i	0.0108151	+	0.0108151i
$496$		0			0
$497$		−	20723.5i		−	0.00376333i
$498$		0			0
$499$	1.08891e6	−	2.62885e6i	0.195767	−	0.472623i	−0.795263	−	0.606265i	$-0.792667\pi$
$499$	1.08891e6	−	2.62885e6i	0.195767	−	0.472623i	0.991030	+	0.133642i	$0.0426671\pi$
$500$		0			0
$501$	1.64830e6	−	1.64830e6i	0.293388	−	0.293388i
$502$		0			0
$503$	2.49895e6	+	1.03510e6i	0.440390	+	0.182416i	0.591851	−	0.806048i	$-0.298398\pi$
$503$	2.49895e6	+	1.03510e6i	0.440390	+	0.182416i	−0.151460	+	0.988463i	$0.548398\pi$
$504$		0			0
$505$	655605.	−	271560.i	0.114397	−	0.0473847i
$506$		0			0
$507$	852488.	+	2.05809e6i	0.147288	+	0.355585i
$508$		0			0
$509$	−3.37337e6			−0.577125			−0.288562	−	0.957461i	$-0.593177\pi$
$509$	−3.37337e6			−0.577125			−0.288562	+	0.957461i	$0.593177\pi$
$510$		0			0
$511$	−1.44879e6			−0.245445
$512$		0			0
$513$	−2.76804e6	−	6.68264e6i	−0.464386	−	1.12113i
$514$		0			0
$515$	−955119.	+	395623.i	−0.158686	+	0.0657300i
$516$		0			0
$517$	−917486.	−	380035.i	−0.150964	−	0.0625313i
$518$		0			0
$519$	1.53881e6	−	1.53881e6i	0.250765	−	0.250765i
$520$		0			0
$521$	1.47410e6	−	3.55879e6i	0.237921	−	0.574392i	−0.759147	−	0.650920i	$-0.774383\pi$
$521$	1.47410e6	−	3.55879e6i	0.237921	−	0.574392i	0.997067	+	0.0765280i	$0.0243835\pi$
$522$		0			0
$523$		−	2.57424e6i		−	0.411524i	−0.978602	−	0.205762i	$-0.934033\pi$
$523$		−	2.57424e6i		−	0.411524i	0.978602	−	0.205762i	$-0.0659672\pi$
$524$		0			0
$525$	3.32000e6	+	3.32000e6i	0.525702	+	0.525702i
$526$		0			0
$527$	−6.78696e6	+	1.04390e6i	−1.06451	+	0.163732i
$528$		0			0
$529$	119042.	+	119042.i	0.0184953	+	0.0184953i
$530$		0			0
$531$			5.68506e6i			0.874981i
$532$		0			0
$533$	1.39785e6	−	3.37470e6i	0.213129	−	0.514538i
$534$		0			0
$535$	−243177.	+	243177.i	−0.0367314	+	0.0367314i
$536$		0			0
$537$	−2.52335e6	−	1.04521e6i	−0.377609	−	0.156411i
$538$		0			0
$539$	−1.53639e6	+	636393.i	−0.227787	+	0.0943525i
$540$		0			0
$541$	978368.	+	2.36199e6i	0.143717	+	0.346964i	0.979304	−	0.202394i	$-0.0648722\pi$
$541$	978368.	+	2.36199e6i	0.143717	+	0.346964i	−0.835587	+	0.549358i	$0.814872\pi$
$542$		0			0
$543$	1.45891e6			0.212338
$544$		0			0
$545$	−509961.			−0.0735437
$546$		0			0
$547$	−2.39970e6	−	5.79339e6i	−0.342917	−	0.827874i	−0.997418	−	0.0718132i	$-0.977121\pi$
$547$	−2.39970e6	−	5.79339e6i	−0.342917	−	0.827874i	0.654501	−	0.756061i	$-0.272879\pi$
$548$		0			0
$549$	4.08567e6	−	1.69234e6i	0.578539	−	0.239639i
$550$		0			0
$551$	4.98857e6	+	2.06633e6i	0.699999	+	0.289949i
$552$		0			0
$553$	−5.99498e6	+	5.99498e6i	−0.833633	+	0.833633i
$554$		0			0
$555$	116443.	−	281118.i	0.0160465	−	0.0387397i
$556$		0			0
$557$		−	1.67037e6i		−	0.228126i	−0.993474	−	0.114063i	$-0.963613\pi$
$557$		−	1.67037e6i		−	0.228126i	0.993474	−	0.114063i	$-0.0363866\pi$
$558$		0			0
$559$	1.92488e6	+	1.92488e6i	0.260540	+	0.260540i
$560$		0			0
$561$	−84226.7	+	344054.i	−0.0112991	+	0.0461551i
$562$		0			0
$563$	5.11742e6	+	5.11742e6i	0.680425	+	0.680425i	0.960096	−	0.279671i	$-0.0902254\pi$
$563$	5.11742e6	+	5.11742e6i	0.680425	+	0.680425i	−0.279671	+	0.960096i	$0.590225\pi$
$564$		0			0
$565$		−	689777.i		−	0.0909050i
$566$		0			0
$567$	−2.55711e6	+	6.17341e6i	−0.334035	+	0.806432i
$568$		0			0
$569$	2.14187e6	−	2.14187e6i	0.277340	−	0.277340i	−0.554706	−	0.832046i	$-0.687169\pi$
$569$	2.14187e6	−	2.14187e6i	0.277340	−	0.277340i	0.832046	+	0.554706i	$0.187169\pi$
$570$		0			0
$571$	8.19710e6	+	3.39535e6i	1.05213	+	0.435807i	0.840652	−	0.541575i	$-0.182172\pi$
$571$	8.19710e6	+	3.39535e6i	1.05213	+	0.435807i	0.211479	+	0.977382i	$0.432172\pi$
$572$		0			0
$573$	2.85328e6	−	1.18187e6i	0.363043	−	0.150377i
$574$		0			0
$575$	2.90958e6	+	7.02435e6i	0.366996	+	0.886006i
$576$		0			0
$577$	−5.13461e6			−0.642049			−0.321024	−	0.947071i	$-0.604027\pi$
$577$	−5.13461e6			−0.642049			−0.321024	+	0.947071i	$0.604027\pi$
$578$		0			0
$579$	3.98958e6			0.494574
$580$		0			0
$581$	8.70353e6	+	2.10122e7i	1.06968	+	2.58244i
$582$		0			0
$583$	1.23498e6	−	511546.i	0.150484	−	0.0623323i
$584$		0			0
$585$	−328900.	−	136235.i	−0.0397351	−	0.0164588i
$586$		0			0
$587$	1.53446e6	−	1.53446e6i	0.183807	−	0.183807i	−0.609206	−	0.793012i	$-0.708512\pi$
$587$	1.53446e6	−	1.53446e6i	0.183807	−	0.183807i	0.793012	+	0.609206i	$0.208512\pi$
$588$		0			0
$589$	−5.42844e6	+	1.31054e7i	−0.644743	+	1.55655i
$590$		0			0
$591$		−	5.26082e6i		−	0.619562i
$592$		0			0
$593$	−8.54665e6	−	8.54665e6i	−0.998066	−	0.998066i	0.00193176	−	0.999998i	$-0.499385\pi$
$593$	−8.54665e6	−	8.54665e6i	−0.998066	−	0.998066i	−0.999998	+	0.00193176i	$0.999385\pi$
$594$		0			0
$595$	−618211.	+	2.52531e6i	−0.0715887	+	0.292430i
$596$		0			0
$597$	−254904.	−	254904.i	−0.0292712	−	0.0292712i
$598$		0			0
$599$		−	5.85648e6i		−	0.666914i	−0.942765	−	0.333457i	$-0.891785\pi$
$599$		−	5.85648e6i		−	0.666914i	0.942765	−	0.333457i	$-0.108215\pi$
$600$		0			0
$601$	−3.75181e6	+	9.05767e6i	−0.423696	+	1.02289i	0.557551	+	0.830142i	$0.311741\pi$
$601$	−3.75181e6	+	9.05767e6i	−0.423696	+	1.02289i	−0.981248	+	0.192751i	$0.938259\pi$
$602$		0			0
$603$	4.04213e6	−	4.04213e6i	0.452707	−	0.452707i
$604$		0			0
$605$	1.37947e6	+	571397.i	0.153223	+	0.0634672i
$606$		0			0
$607$	9.20074e6	−	3.81107e6i	1.01356	−	0.419832i	0.186810	−	0.982396i	$-0.440185\pi$
$607$	9.20074e6	−	3.81107e6i	1.01356	−	0.419832i	0.826753	+	0.562564i	$0.190185\pi$
$608$		0			0
$609$	1.29784e6	+	3.13327e6i	0.141801	+	0.342337i
$610$		0			0
$611$	4.24009e6			0.459486
$612$		0			0
$613$	−9.44978e6			−1.01571			−0.507856	−	0.861442i	$-0.669562\pi$
$613$	−9.44978e6			−1.01571			−0.507856	+	0.861442i	$0.669562\pi$
$614$		0			0
$615$	457487.	+	1.10447e6i	0.0487743	+	0.117752i
$616$		0			0
$617$	−1.26142e7	+	5.22499e6i	−1.33398	+	0.552551i	−0.931787	−	0.363006i	$-0.881751\pi$
$617$	−1.26142e7	+	5.22499e6i	−1.33398	+	0.552551i	−0.402189	+	0.915557i	$0.631751\pi$
$618$		0			0
$619$	−1.45548e7	−	6.02880e6i	−1.52679	−	0.632418i	−0.547854	−	0.836574i	$-0.684555\pi$
$619$	−1.45548e7	−	6.02880e6i	−1.52679	−	0.632418i	−0.978938	+	0.204156i	$0.934555\pi$
$620$		0			0
$621$	5.20208e6	−	5.20208e6i	0.541312	−	0.541312i
$622$		0			0
$623$	−4.25256e6	+	1.02666e7i	−0.438965	+	1.05976i
$624$		0			0
$625$		−	8.94720e6i		−	0.916194i
$626$		0			0
$627$	517408.	+	517408.i	0.0525611	+	0.0525611i
$628$		0			0
$629$	−5.73871e6	+	882674.i	−0.578346	+	0.0889557i
$630$		0			0
$631$	5.72900e6	+	5.72900e6i	0.572804	+	0.572804i	0.932911	−	0.360107i	$-0.117260\pi$
$631$	5.72900e6	+	5.72900e6i	0.572804	+	0.572804i	−0.360107	+	0.932911i	$0.617260\pi$
$632$		0			0
$633$			4.80575e6i			0.476707i
$634$		0			0
$635$	785921.	−	1.89738e6i	0.0773471	−	0.186733i
$636$		0			0
$637$	5.02067e6	−	5.02067e6i	0.490245	−	0.490245i
$638$		0			0
$639$	16372.4	+	6781.68i	0.00158621	+	0.000657030i
$640$		0			0
$641$	1.54690e7	−	6.40746e6i	1.48702	−	0.615944i	0.516354	−	0.856375i	$-0.327289\pi$
$641$	1.54690e7	−	6.40746e6i	1.48702	−	0.615944i	0.970666	+	0.240432i	$0.0772890\pi$
$642$		0			0
$643$	−2.00926e6	−	4.85077e6i	−0.191650	−	0.462683i	0.798622	−	0.601833i	$-0.205563\pi$
$643$	−2.00926e6	−	4.85077e6i	−0.191650	−	0.462683i	−0.990271	+	0.139150i	$0.955563\pi$
$644$		0			0
$645$	−890918.			−0.0843215
$646$		0			0
$647$	5.89611e6			0.553739			0.276869	−	0.960908i	$-0.410703\pi$
$647$	5.89611e6			0.553739			0.276869	+	0.960908i	$0.410703\pi$
$648$		0			0
$649$	−489168.	−	1.18096e6i	−0.0455875	−	0.110058i
$650$		0			0
$651$	−8.23136e6	+	3.40954e6i	−0.761236	+	0.315314i
$652$		0			0
$653$	1.63410e6	+	676868.i	0.149967	+	0.0621185i	0.456405	−	0.889772i	$-0.349137\pi$
$653$	1.63410e6	+	676868.i	0.149967	+	0.0621185i	−0.306437	+	0.951891i	$0.599137\pi$
$654$		0			0
$655$	1.56042e6	−	1.56042e6i	0.142114	−	0.142114i
$656$		0			0
$657$	474110.	−	1.14460e6i	0.0428515	−	0.103453i
$658$		0			0
$659$		−	5.33113e6i		−	0.478196i	−0.970995	−	0.239098i	$-0.923148\pi$
$659$		−	5.33113e6i		−	0.478196i	0.970995	−	0.239098i	$-0.0768517\pi$
$660$		0			0
$661$	6.97391e6	+	6.97391e6i	0.620830	+	0.620830i	0.945744	−	0.324913i	$-0.105335\pi$
$661$	6.97391e6	+	6.97391e6i	0.620830	+	0.620830i	−0.324913	+	0.945744i	$0.605335\pi$
$662$		0			0
$663$	−229914.	−	1.49479e6i	−0.0203133	−	0.132067i
$664$		0			0
$665$	3.79770e6	+	3.79770e6i	0.333017	+	0.333017i
$666$		0			0
$667$			5.49187e6i			0.477975i
$668$		0			0
$669$	911325.	−	2.20013e6i	0.0787242	−	0.190057i
$670$		0			0
$671$	−703099.	+	703099.i	−0.0602851	+	0.0602851i
$672$		0			0
$673$	2.04105e6	+	845431.i	0.173707	+	0.0719516i	0.467842	−	0.883812i	$-0.345032\pi$
$673$	2.04105e6	+	845431.i	0.173707	+	0.0719516i	−0.294135	+	0.955764i	$0.595032\pi$
$674$		0			0
$675$	−8.24460e6	+	3.41503e6i	−0.696483	+	0.288493i
$676$		0			0
$677$	−5.19409e6	−	1.25396e7i	−0.435550	−	1.05151i	−0.977469	−	0.211079i	$-0.932302\pi$
$677$	−5.19409e6	−	1.25396e7i	−0.435550	−	1.05151i	0.541919	−	0.840431i	$-0.317698\pi$
$678$		0			0
$679$	−1.36397e7			−1.13535
$680$		0			0
$681$	9.18378e6			0.758846
$682$		0			0
$683$	−4.63905e6	−	1.11997e7i	−0.380520	−	0.918656i	−0.991865	−	0.127292i	$-0.959371\pi$
$683$	−4.63905e6	−	1.11997e7i	−0.380520	−	0.918656i	0.611345	−	0.791364i	$-0.290629\pi$
$684$		0			0
$685$	744690.	−	308461.i	0.0606386	−	0.0251173i
$686$		0			0
$687$	−7.03005e6	−	2.91194e6i	−0.568285	−	0.235391i
$688$		0			0
$689$	−4.03572e6	+	4.03572e6i	−0.323872	+	0.323872i
$690$		0			0
$691$	6.90580e6	−	1.66721e7i	0.550198	−	1.32830i	−0.367133	−	0.930169i	$-0.619660\pi$
$691$	6.90580e6	−	1.66721e7i	0.550198	−	1.32830i	0.917330	−	0.398127i	$-0.130340\pi$
$692$		0			0
$693$		−	2.06433e6i		−	0.163285i
$694$		0			0
$695$	727322.	+	727322.i	0.0571169	+	0.0571169i
$696$		0			0
$697$	1.34907e7	−	1.83950e7i	1.05184	−	1.43423i
$698$		0			0
$699$	−4.76693e6	−	4.76693e6i	−0.369017	−	0.369017i
$700$		0			0
$701$		−	2.44145e7i		−	1.87652i	−0.345929	−	0.938261i	$-0.612436\pi$
$701$		−	2.44145e7i		−	1.87652i	0.345929	−	0.938261i	$-0.387564\pi$
$702$		0			0
$703$	−4.59002e6	+	1.10813e7i	−0.350289	+	0.845672i
$704$		0			0
$705$	−981249.	+	981249.i	−0.0743544	+	0.0743544i
$706$		0			0
$707$	−1.62317e7	−	6.72340e6i	−1.22128	−	0.505871i
$708$		0			0
$709$	−7.47753e6	+	3.09730e6i	−0.558654	+	0.231402i	−0.644101	−	0.764941i	$-0.722768\pi$
$709$	−7.47753e6	+	3.09730e6i	−0.558654	+	0.231402i	0.0854469	+	0.996343i	$0.472768\pi$
$710$		0			0
$711$	−2.77445e6	−	6.69811e6i	−0.205827	−	0.496910i
$712$		0			0
$713$	−1.44276e7			−1.06285
$714$		0			0
$715$	80044.6			0.00585554
$716$		0			0
$717$	503636.	+	1.21589e6i	0.0365864	+	0.0883273i
$718$		0			0
$719$	6.53953e6	−	2.70876e6i	0.471763	−	0.195411i	−0.134119	−	0.990965i	$-0.542820\pi$
$719$	6.53953e6	−	2.70876e6i	0.471763	−	0.195411i	0.605882	+	0.795555i	$0.292820\pi$
$720$		0			0
$721$	2.36472e7	+	9.79499e6i	1.69411	+	0.701723i
$722$		0			0
$723$	3.26206e6	−	3.26206e6i	0.232084	−	0.232084i
$724$		0			0
$725$	2.54931e6	−	6.15457e6i	0.180126	−	0.434863i
$726$		0			0
$727$			1.25841e7i			0.883049i	0.897249	+	0.441525i	$0.145562\pi$
$727$			1.25841e7i			0.883049i	−0.897249	+	0.441525i	$0.854438\pi$
$728$		0			0
$729$	−681759.	−	681759.i	−0.0475130	−	0.0475130i
$730$		0			0
$731$	8.81781e6	+	1.45346e7i	0.610334	+	1.00603i
$732$		0			0
$733$	−962306.	−	962306.i	−0.0661536	−	0.0661536i	0.673256	−	0.739410i	$-0.264895\pi$
$733$	−962306.	−	962306.i	−0.0661536	−	0.0661536i	−0.739410	+	0.673256i	$0.764895\pi$
$734$		0			0
$735$			2.32378e6i			0.158664i
$736$		0			0
$737$	−491868.	+	1.18747e6i	−0.0333565	+	0.0805296i
$738$		0			0
$739$	1.26071e7	−	1.26071e7i	0.849185	−	0.849185i	−0.140846	−	0.990032i	$-0.544982\pi$
$739$	1.26071e7	−	1.26071e7i	0.849185	−	0.849185i	0.990032	+	0.140846i	$0.0449823\pi$
$740$		0			0
$741$	−2.88639e6	−	1.19558e6i	−0.193112	−	0.0799896i
$742$		0			0
$743$	−2.72968e7	+	1.13067e7i	−1.81401	+	0.751389i	−0.834202	+	0.551458i	$0.814072\pi$
$743$	−2.72968e7	+	1.13067e7i	−1.81401	+	0.751389i	−0.979810	+	0.199930i	$0.935928\pi$
$744$		0			0
$745$	−807383.	−	1.94919e6i	−0.0532953	−	0.128666i
$746$		0			0
$747$	−1.94487e7			−1.27523
$748$		0			0
$749$	8.51451e6			0.554568
$750$		0			0
$751$	7.14488e6	+	1.72493e7i	0.462269	+	1.11602i	0.967463	+	0.253011i	$0.0814210\pi$
$751$	7.14488e6	+	1.72493e7i	0.462269	+	1.11602i	−0.505194	+	0.863006i	$0.668579\pi$
$752$		0			0
$753$	−1.17660e7	+	4.87364e6i	−0.756209	+	0.313232i
$754$		0			0
$755$	−1.02065e6	−	422767.i	−0.0651642	−	0.0269919i
$756$		0			0
$757$	1.63302e7	−	1.63302e7i	1.03574	−	1.03574i	0.0364068	−	0.999337i	$-0.488409\pi$
$757$	1.63302e7	−	1.63302e7i	1.03574	−	1.03574i	0.999337	−	0.0364068i	$-0.0115912\pi$
$758$		0			0
$759$	−284805.	+	687579.i	−0.0179450	+	0.0433230i
$760$		0			0
$761$			1.58860e6i			0.0994383i	0.998763	+	0.0497192i	$0.0158326\pi$
$761$			1.58860e6i			0.0994383i	−0.998763	+	0.0497192i	$0.984167\pi$
$762$		0			0
$763$	8.92780e6	+	8.92780e6i	0.555179	+	0.555179i
$764$		0			0
$765$	−1.79279e6	−	1.31481e6i	−0.110758	−	0.0812284i
$766$		0			0
$767$	3.85917e6	+	3.85917e6i	0.236868	+	0.236868i
$768$		0			0
$769$			2.19957e7i			1.34129i	0.741780	+	0.670644i	$0.233982\pi$
$769$			2.19957e7i			1.34129i	−0.741780	+	0.670644i	$0.766018\pi$
$770$		0			0
$771$	−1.66317e6	+	4.01524e6i	−0.100763	+	0.243263i
$772$		0			0
$773$	−2.34632e6	+	2.34632e6i	−0.141234	+	0.141234i	−0.774189	−	0.632955i	$-0.781842\pi$
$773$	−2.34632e6	+	2.34632e6i	−0.141234	+	0.141234i	0.632955	+	0.774189i	$0.281842\pi$
$774$		0			0
$775$	1.61686e7	+	6.69725e6i	0.966981	+	0.400537i
$776$		0			0
$777$	−6.96003e6	+	2.88294e6i	−0.413579	+	0.171310i
$778$		0			0
$779$	−1.80335e7	−	4.35367e7i	−1.06472	−	2.57047i
$780$		0			0
$781$	−3984.57			−0.000233751
$782$		0			0
$783$	−6.44590e6			−0.375733
$784$		0			0
$785$	226512.	+	546848.i	0.0131195	+	0.0316733i
$786$		0			0
$787$	1.36897e7	−	5.67044e6i	0.787872	−	0.326347i	0.0477845	−	0.998858i	$-0.484784\pi$
$787$	1.36897e7	−	5.67044e6i	0.787872	−	0.326347i	0.740088	+	0.672510i	$0.234784\pi$
$788$		0			0
$789$	−1.06558e6	−	441377.i	−0.0609386	−	0.0252416i
$790$		0			0
$791$	−1.20758e7	+	1.20758e7i	−0.686238	+	0.686238i
$792$		0			0
$793$	1.62466e6	−	3.92227e6i	0.0917443	−	0.221490i
$794$		0			0
$795$		−	1.86791e6i		−	0.104818i
$796$		0			0
$797$	1.97709e7	+	1.97709e7i	1.10250	+	1.10250i	0.994108	+	0.108395i	$0.0345710\pi$
$797$	1.97709e7	+	1.97709e7i	1.10250	+	1.10250i	0.108395	+	0.994108i	$0.465429\pi$
$798$		0			0
$799$	2.57202e7	+	6.29647e6i	1.42530	+	0.348923i
$800$		0			0
$801$	−6.71939e6	−	6.71939e6i	−0.370040	−	0.370040i
$802$		0			0
$803$			278563.i			0.0152452i
$804$		0			0
$805$	−2.09042e6	+	5.04673e6i	−0.113696	+	0.274486i
$806$		0			0
$807$	−1.05890e7	+	1.05890e7i	−0.572364	+	0.572364i
$808$		0			0
$809$	4.63354e6	+	1.91928e6i	0.248910	+	0.103102i	0.503650	−	0.863908i	$-0.331990\pi$
$809$	4.63354e6	+	1.91928e6i	0.248910	+	0.103102i	−0.254741	+	0.967009i	$0.581990\pi$
$810$		0			0
$811$	−1.29253e7	+	5.35384e6i	−0.690064	+	0.285834i	−0.700027	−	0.714117i	$-0.746829\pi$
$811$	−1.29253e7	+	5.35384e6i	−0.690064	+	0.285834i	0.00996307	+	0.999950i	$0.496829\pi$
$812$		0			0
$813$	5.12657e6	+	1.23766e7i	0.272020	+	0.656714i
$814$		0			0
$815$	4.90384e6			0.258608
$816$		0			0
$817$	3.51187e7			1.84070
$818$		0			0
$819$	3.37296e6	+	8.14304e6i	0.175712	+	0.424206i
$820$		0			0
$821$	−1.34106e7	+	5.55484e6i	−0.694367	+	0.287616i	−0.701818	−	0.712356i	$-0.747628\pi$
$821$	−1.34106e7	+	5.55484e6i	−0.694367	+	0.287616i	0.00745128	+	0.999972i	$0.497628\pi$
$822$		0			0
$823$	1.54410e7	+	6.39588e6i	0.794651	+	0.329155i	0.742812	−	0.669500i	$-0.233492\pi$
$823$	1.54410e7	+	6.39588e6i	0.794651	+	0.329155i	0.0518389	+	0.998655i	$0.483492\pi$
$824$		0			0
$825$	638344.	−	638344.i	0.0326528	−	0.0326528i
$826$		0			0
$827$	−1.46207e7	+	3.52975e7i	−0.743370	+	1.79465i	−0.151779	+	0.988414i	$0.548500\pi$
$827$	−1.46207e7	+	3.52975e7i	−0.743370	+	1.79465i	−0.591591	+	0.806238i	$0.701500\pi$
$828$		0			0
$829$			1.74260e7i			0.880665i	0.897835	+	0.440332i	$0.145139\pi$
$829$			1.74260e7i			0.880665i	−0.897835	+	0.440332i	$0.854861\pi$
$830$		0			0
$831$	7.07499e6	+	7.07499e6i	0.355405	+	0.355405i
$832$		0			0
$833$	3.79108e7	−	2.29995e7i	1.89300	−	1.14844i
$834$		0			0
$835$	−2.32617e6	−	2.32617e6i	−0.115458	−	0.115458i
$836$		0			0
$837$		−	1.69339e7i		−	0.835496i
$838$		0			0
$839$	−1.25378e7	+	3.02689e7i	−0.614917	+	1.48454i	0.242621	+	0.970121i	$0.421993\pi$
$839$	−1.25378e7	+	3.02689e7i	−0.614917	+	1.48454i	−0.857538	+	0.514420i	$0.828007\pi$
$840$		0			0
$841$	−1.11011e7	+	1.11011e7i	−0.541222	+	0.541222i
$842$		0			0
$843$	2.26033e6	+	936259.i	0.109548	+	0.0453761i
$844$		0			0
$845$	2.90448e6	−	1.20307e6i	0.139935	−	0.0579630i
$846$		0			0
$847$	−1.41469e7	−	3.41536e7i	−0.677567	−	1.63579i
$848$		0			0
$849$	−1.08445e7			−0.516347
$850$		0			0
$851$	−1.21993e7			−0.577444
$852$		0			0
$853$	1.15034e7	+	2.77718e7i	0.541321	+	1.30687i	0.923791	+	0.382897i	$0.125074\pi$
$853$	1.15034e7	+	2.77718e7i	0.541321	+	1.30687i	−0.382469	+	0.923968i	$0.624926\pi$
$854$		0			0
$855$	−4.24311e6	+	1.75755e6i	−0.198504	+	0.0822231i
$856$		0			0
$857$	−1.60000e7	−	6.62743e6i	−0.744164	−	0.308243i	−0.0218061	−	0.999762i	$-0.506942\pi$
$857$	−1.60000e7	−	6.62743e6i	−0.744164	−	0.308243i	−0.722358	+	0.691519i	$0.756942\pi$
$858$		0			0
$859$	−8.06042e6	+	8.06042e6i	−0.372713	+	0.372713i	−0.868464	−	0.495751i	$-0.834893\pi$
$859$	−8.06042e6	+	8.06042e6i	−0.372713	+	0.372713i	0.495751	+	0.868464i	$0.334893\pi$
$860$		0			0
$861$	1.13266e7	−	2.73449e7i	0.520707	−	1.25710i
$862$		0			0
$863$		−	2.37355e7i		−	1.08485i	−0.840103	−	0.542426i	$-0.817506\pi$
$863$		−	2.37355e7i		−	1.08485i	0.840103	−	0.542426i	$-0.182494\pi$
$864$		0			0
$865$	−2.17164e6	−	2.17164e6i	−0.0986844	−	0.0986844i
$866$		0			0
$867$	825088.	−	9.40873e6i	0.0372780	−	0.425092i
$868$		0			0
$869$	1.15267e6	+	1.15267e6i	0.0517792	+	0.0517792i
$870$		0			0
$871$		−	5.48782e6i		−	0.245107i
$872$		0			0
$873$	4.46352e6	−	1.07759e7i	0.198218	−	0.478539i
$874$		0			0
$875$	9.50666e6	−	9.50666e6i	0.419767	−	0.419767i
$876$		0			0
$877$	−1.74460e7	−	7.22636e6i	−0.765942	−	0.317264i	−0.0347147	−	0.999397i	$-0.511052\pi$
$877$	−1.74460e7	−	7.22636e6i	−0.765942	−	0.317264i	−0.731228	+	0.682134i	$0.761052\pi$
$878$		0			0
$879$	6.34131e6	−	2.62666e6i	0.276826	−	0.114665i
$880$		0			0
$881$	−2.08546e6	−	5.03475e6i	−0.0905238	−	0.218544i	0.872133	−	0.489269i	$-0.162736\pi$
$881$	−2.08546e6	−	5.03475e6i	−0.0905238	−	0.218544i	−0.962657	+	0.270725i	$0.912736\pi$
$882$		0			0
$883$	−2.14840e7			−0.927285			−0.463643	−	0.886022i	$-0.653458\pi$
$883$	−2.14840e7			−0.927285			−0.463643	+	0.886022i	$0.653458\pi$
$884$		0			0
$885$	−1.78619e6			−0.0766602
$886$		0			0
$887$	−177367.	−	428203.i	−0.00756946	−	0.0182743i	0.920050	−	0.391802i	$-0.128148\pi$
$887$	−177367.	−	428203.i	−0.00756946	−	0.0182743i	−0.927619	+	0.373528i	$0.878148\pi$
$888$		0			0
$889$	−4.69761e7	+	1.94581e7i	−1.99353	+	0.825746i
$890$		0			0
$891$	1.18698e6	+	491662.i	0.0500897	+	0.0207478i
$892$		0			0
$893$	3.86795e7	−	3.86795e7i	1.62313	−	1.62313i
$894$		0			0
$895$	−1.47505e6	+	3.56108e6i	−0.0615530	+	0.148602i
$896$		0			0
$897$		−	3.17760e6i		−	0.131861i
$898$		0			0
$899$	8.93863e6	+	8.93863e6i	0.368869	+	0.368869i
$900$		0			0
$901$	−3.04735e7	+	1.84875e7i	−1.25058	+	0.758694i
$902$		0			0
$903$	1.55971e7	+	1.55971e7i	0.636540	+	0.636540i
$904$		0			0
$905$		−	2.05888e6i		−	0.0835622i
$906$		0			0
$907$	4.52800e6	−	1.09316e7i	0.182763	−	0.441229i	−0.805771	−	0.592227i	$-0.798249\pi$
$907$	4.52800e6	−	1.09316e7i	0.182763	−	0.441229i	0.988534	+	0.150998i	$0.0482488\pi$
$908$		0			0
$909$	1.06235e7	−	1.06235e7i	0.426440	−	0.426440i
$910$		0			0
$911$	1.82858e7	+	7.57424e6i	0.729994	+	0.302373i	0.716549	−	0.697536i	$-0.245720\pi$
$911$	1.82858e7	+	7.57424e6i	0.729994	+	0.302373i	0.0134442	+	0.999910i	$0.495720\pi$
$912$		0			0
$913$	4.04006e6	−	1.67345e6i	0.160403	−	0.0664409i
$914$		0			0
$915$	531718.	+	1.28368e6i	0.0209956	+	0.0506879i
$916$		0			0
$917$	−5.46359e7			−2.14563
$918$		0			0
$919$	2.05183e7			0.801405			0.400703	−	0.916208i	$-0.368766\pi$
$919$	2.05183e7			0.801405			0.400703	+	0.916208i	$0.368766\pi$
$920$		0			0
$921$	−1.39208e6	−	3.36078e6i	−0.0540774	−	0.130554i
$922$		0			0
$923$	15717.6	−	6510.46i	0.000607272	−	0.000251540i
$924$		0			0
$925$	1.36714e7	+	5.66286e6i	0.525360	+	0.217611i
$926$		0			0
$927$	−1.54769e7	+	1.54769e7i	−0.591540	+	0.591540i
$928$		0			0
$929$	1.03317e7	−	2.49430e7i	0.392766	−	0.948221i	−0.596569	−	0.802562i	$-0.703470\pi$
$929$	1.03317e7	−	2.49430e7i	0.392766	−	0.948221i	0.989335	−	0.145659i	$-0.0465302\pi$
$930$		0			0
$931$		−	9.16003e7i		−	3.46356i
$932$		0			0
$933$	7.01016e6	+	7.01016e6i	0.263648	+	0.263648i
$934$		0			0
$935$	485547.	+	118865.i	0.0181636	+	0.00444657i
$936$		0			0
$937$	2.52215e7	+	2.52215e7i	0.938474	+	0.938474i	0.998214	−	0.0597401i	$-0.0190272\pi$
$937$	2.52215e7	+	2.52215e7i	0.938474	+	0.938474i	−0.0597401	+	0.998214i	$0.519027\pi$
$938$		0			0
$939$		−	1.55339e7i		−	0.574934i
$940$		0			0
$941$	−1.78091e7	+	4.29950e7i	−0.655644	+	1.58286i	0.148822	+	0.988864i	$0.452452\pi$
$941$	−1.78091e7	+	4.29950e7i	−0.655644	+	1.58286i	−0.804465	+	0.594000i	$0.797548\pi$
$942$		0			0
$943$	3.38910e7	−	3.38910e7i	1.24110	−	1.24110i
$944$		0			0
$945$	−5.92343e6	−	2.45357e6i	−0.215771	−	0.0893754i
$946$		0			0
$947$	3.24382e7	−	1.34363e7i	1.17539	−	0.486863i	0.292419	−	0.956290i	$-0.405540\pi$
$947$	3.24382e7	−	1.34363e7i	1.17539	−	0.486863i	0.882971	+	0.469428i	$0.155540\pi$
$948$		0			0
$949$	−455149.	−	1.09883e6i	−0.0164054	−	0.0396063i
$950$		0			0
$951$	−1.42246e7			−0.510022
$952$		0			0
$953$	−5.56065e7			−1.98332			−0.991661	−	0.128872i	$-0.958864\pi$
$953$	−5.56065e7			−1.98332			−0.991661	+	0.128872i	$0.958864\pi$
$954$		0			0
$955$	−1.66791e6	−	4.02669e6i	−0.0591785	−	0.142870i
$956$		0			0
$957$	602441.	−	249539.i	0.0212635	−	0.00880763i
$958$		0			0
$959$	−1.84373e7	−	7.63699e6i	−0.647369	−	0.268149i
$960$		0			0
$961$	−3.23868e6	+	3.23868e6i	−0.113125	+	0.113125i
$962$		0			0
$963$	−2.78633e6	+	6.72681e6i	−0.0968205	+	0.233745i
$964$		0			0
$965$		−	5.63030e6i		−	0.194632i
$966$		0			0
$967$	1.07364e6	+	1.07364e6i	0.0369226	+	0.0369226i	0.725327	−	0.688404i	$-0.241689\pi$
$967$	1.07364e6	+	1.07364e6i	0.0369226	+	0.0369226i	−0.688404	+	0.725327i	$0.741689\pi$
$968$		0			0
$969$	−1.57333e7	−	1.15386e7i	−0.538282	−	0.394769i
$970$		0			0
$971$	−2.79828e7	−	2.79828e7i	−0.952450	−	0.952450i	0.0464693	−	0.998920i	$-0.485203\pi$
$971$	−2.79828e7	−	2.79828e7i	−0.952450	−	0.952450i	−0.998920	+	0.0464693i	$0.985203\pi$
$972$		0			0
$973$		−	2.54662e7i		−	0.862347i
$974$		0			0
$975$	−1.47503e6	+	3.56104e6i	−0.0496923	+	0.119968i
$976$		0			0
$977$	3.11425e7	−	3.11425e7i	1.04380	−	1.04380i	0.0448020	−	0.998996i	$-0.485734\pi$
$977$	3.11425e7	−	3.11425e7i	1.04380	−	1.04380i	0.998996	−	0.0448020i	$-0.0142657\pi$
$978$		0			0
$979$	1.97398e6	+	817650.i	0.0658244	+	0.0272653i
$980$		0			0
$981$	−9.97490e6	+	4.13174e6i	−0.330930	+	0.137076i
$982$		0			0
$983$	1.61420e6	+	3.89702e6i	0.0532811	+	0.128632i	0.948279	−	0.317439i	$-0.102823\pi$
$983$	1.61420e6	+	3.89702e6i	0.0532811	+	0.128632i	−0.894998	+	0.446071i	$0.852823\pi$
$984$		0			0
$985$	−7.42434e6			−0.243819
$986$		0			0
$987$	3.43571e7			1.12260
$988$		0			0
$989$	1.36690e7	+	3.30000e7i	0.444372	+	1.07281i
$990$		0			0
$991$	−1.64943e6	+	683216.i	−0.0533518	+	0.0220990i	−0.409200	−	0.912445i	$-0.634192\pi$
$991$	−1.64943e6	+	683216.i	−0.0533518	+	0.0220990i	0.355848	+	0.934544i	$0.384192\pi$
$992$		0			0
$993$	6.03560e6	+	2.50003e6i	0.194244	+	0.0804585i
$994$		0			0
$995$	−359734.	+	359734.i	−0.0115192	+	0.0115192i
$996$		0			0
$997$	−2.06878e7	+	4.99448e7i	−0.659138	+	1.59130i	0.139999	+	0.990152i	$0.455290\pi$
$997$	−2.06878e7	+	4.99448e7i	−0.659138	+	1.59130i	−0.799137	+	0.601149i	$0.794710\pi$
$998$		0			0
$999$		−	1.43185e7i		−	0.453925i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.6.h.a.49.4	yes	28
17.8	even	8	inner	68.6.h.a.25.4	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.6.h.a.25.4	✓	28	17.8	even	8	inner
68.6.h.a.49.4	yes	28	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	68.h (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(2.54560 + 6.14561i) q^{3} +(8.67300 - 3.59247i) q^{5} +(-214.730 - 88.9439i) q^{7} +(140.538 - 140.538i) q^{9} +O(q^{10})\)	\(q+(2.54560 + 6.14561i) q^{3} +(8.67300 - 3.59247i) q^{5} +(-214.730 - 88.9439i) q^{7} +(140.538 - 140.538i) q^{9} +(-17.1015 + 41.2866i) q^{11} -190.803i q^{13} +(44.1559 + 44.1559i) q^{15} +(283.339 - 1157.40i) q^{17} +(-1740.56 - 1740.56i) q^{19} -1546.06i q^{21} +(958.085 - 2313.02i) q^{23} +(-2147.39 + 2147.39i) q^{25} +(2714.83 + 1124.52i) q^{27} +(-2026.61 + 839.451i) q^{29} +(-2205.31 - 5324.09i) q^{31} -297.265 q^{33} -2181.88 q^{35} +(-1864.70 - 4501.79i) q^{37} +(1172.60 - 485.707i) q^{39} +(17686.9 + 7326.14i) q^{41} +(-10088.3 + 10088.3i) q^{43} +(714.009 - 1723.77i) q^{45} +22222.4i q^{47} +(26313.4 + 26313.4i) q^{49} +(7834.20 - 1204.98i) q^{51} +(-21151.3 - 21151.3i) q^{53} +419.515i q^{55} +(6266.06 - 15127.6i) q^{57} +(-20226.0 + 20226.0i) q^{59} +(20556.7 + 8514.86i) q^{61} +(-42677.8 + 17677.7i) q^{63} +(-685.454 - 1654.83i) q^{65} +28761.8 q^{67} +16653.8 q^{69} +(34.1214 + 82.3764i) q^{71} +(5758.97 - 2385.44i) q^{73} +(-18663.4 - 7730.65i) q^{75} +(7344.38 - 7344.38i) q^{77} +(13959.4 - 33700.9i) q^{79} -28749.7i q^{81} +(-69193.3 - 69193.3i) q^{83} +(-1700.53 - 11056.0i) q^{85} +(-10317.9 - 10317.9i) q^{87} -47811.7i q^{89} +(-16970.7 + 40971.0i) q^{91} +(27106.0 - 27106.0i) q^{93} +(-21348.8 - 8842.98i) q^{95} +(54218.0 - 22457.8i) q^{97} +(3398.94 + 8205.77i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 44 q^{5} + 44 q^{9}+O(q^{10}) \) 28 * q + 44 * q^5 + 44 * q^9	\( 28 q + 44 q^{5} + 44 q^{9} - 600 q^{11} - 4372 q^{15} + 1212 q^{17} + 284 q^{19} + 1672 q^{23} - 3676 q^{25} - 20796 q^{27} - 11316 q^{29} + 16632 q^{31} + 32888 q^{33} - 6464 q^{35} + 19564 q^{37} - 15880 q^{39} - 14620 q^{41} - 17628 q^{43} + 64172 q^{45} - 64812 q^{49} + 3256 q^{51} + 5396 q^{53} + 35236 q^{57} + 57012 q^{59} + 70364 q^{61} + 93844 q^{63} - 56064 q^{65} + 62528 q^{67} - 159656 q^{69} - 56144 q^{71} + 61828 q^{73} + 92856 q^{75} + 5236 q^{77} - 181064 q^{79} + 129980 q^{83} - 250260 q^{85} - 498484 q^{87} - 20472 q^{91} - 336876 q^{93} - 3044 q^{95} + 317004 q^{97} + 268076 q^{99}+O(q^{100}) \) 28 * q + 44 * q^5 + 44 * q^9 - 600 * q^11 - 4372 * q^15 + 1212 * q^17 + 284 * q^19 + 1672 * q^23 - 3676 * q^25 - 20796 * q^27 - 11316 * q^29 + 16632 * q^31 + 32888 * q^33 - 6464 * q^35 + 19564 * q^37 - 15880 * q^39 - 14620 * q^41 - 17628 * q^43 + 64172 * q^45 - 64812 * q^49 + 3256 * q^51 + 5396 * q^53 + 35236 * q^57 + 57012 * q^59 + 70364 * q^61 + 93844 * q^63 - 56064 * q^65 + 62528 * q^67 - 159656 * q^69 - 56144 * q^71 + 61828 * q^73 + 92856 * q^75 + 5236 * q^77 - 181064 * q^79 + 129980 * q^83 - 250260 * q^85 - 498484 * q^87 - 20472 * q^91 - 336876 * q^93 - 3044 * q^95 + 317004 * q^97 + 268076 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more