LMFDB - Embedded newform 138.5.b.a.91.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [138,5,Mod(91,138)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(138, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("138.91");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 138 = 2 \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	138.b (of order $2$, degree $1$, 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:	$14.2650549056$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 1428 x^{14} - 600 x^{13} + 788282 x^{12} - 529464 x^{11} + 213396724 x^{10} + \cdots + 274129967370817 $ x^16 + 1428x^14 - 600x^13 + 788282x^12 - 529464x^11 + 213396724x^10 - 175079484x^9 + 29130946113x^8 - 26553654912x^7 + 1790717926120x^6 - 1672125302460x^5 + 32715955056348x^4 - 11255739266544x^3 + 140972261417692x^2 - 70961983286748x + 274129967370817
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.8
Root			$-0.707107 - 22.2857i$ of defining polynomial
Character	$\chi$	$=$	138.91
Dual form			138.5.b.a.91.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} +34.5847i q^{5} -14.6969 q^{6} +81.5113i q^{7} -22.6274 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} +34.5847i q^{5} -14.6969 q^{6} +81.5113i q^{7} -22.6274 q^{8} +27.0000 q^{9} -97.8202i q^{10} -67.6260i q^{11} +41.5692 q^{12} -316.869 q^{13} -230.549i q^{14} +179.707i q^{15} +64.0000 q^{16} -44.6757i q^{17} -76.3675 q^{18} -468.905i q^{19} +276.677i q^{20} +423.545i q^{21} +191.275i q^{22} +(491.402 + 195.869i) q^{23} -117.576 q^{24} -571.099 q^{25} +896.240 q^{26} +140.296 q^{27} +652.091i q^{28} -228.778 q^{29} -508.289i q^{30} -1674.25 q^{31} -181.019 q^{32} -351.395i q^{33} +126.362i q^{34} -2819.04 q^{35} +216.000 q^{36} +736.035i q^{37} +1326.26i q^{38} -1646.50 q^{39} -782.562i q^{40} -299.061 q^{41} -1197.97i q^{42} +2677.63i q^{43} -541.008i q^{44} +933.786i q^{45} +(-1389.90 - 554.002i) q^{46} +3461.96 q^{47} +332.554 q^{48} -4243.10 q^{49} +1615.31 q^{50} -232.142i q^{51} -2534.95 q^{52} -142.983i q^{53} -396.817 q^{54} +2338.82 q^{55} -1844.39i q^{56} -2436.50i q^{57} +647.081 q^{58} -1697.68 q^{59} +1437.66i q^{60} +6286.88i q^{61} +4735.50 q^{62} +2200.81i q^{63} +512.000 q^{64} -10958.8i q^{65} +993.896i q^{66} -2566.64i q^{67} -357.406i q^{68} +(2553.40 + 1017.77i) q^{69} +7973.46 q^{70} +5395.67 q^{71} -610.940 q^{72} +1108.66 q^{73} -2081.82i q^{74} -2967.52 q^{75} -3751.24i q^{76} +5512.29 q^{77} +4657.00 q^{78} +4639.58i q^{79} +2213.42i q^{80} +729.000 q^{81} +845.873 q^{82} +12844.1i q^{83} +3388.36i q^{84} +1545.09 q^{85} -7573.49i q^{86} -1188.76 q^{87} +1530.20i q^{88} -3849.23i q^{89} -2641.15i q^{90} -25828.4i q^{91} +(3931.22 + 1566.95i) q^{92} -8699.67 q^{93} -9791.91 q^{94} +16216.9 q^{95} -940.604 q^{96} +940.980i q^{97} +12001.3 q^{98} -1825.90i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) $ 16 * q + 128 * q^4 + 432 * q^9	$ 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) $ 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/138\mathbb{Z}\right)^\times$.

$n$	$47$	$97$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$	−2.82843			−0.707107
$2$	−2.82843			−0.707107
$3$	5.19615			0.577350
$3$	5.19615			0.577350
$4$	8.00000			0.500000
$5$			34.5847i			1.38339i	0.722191	+	0.691693i	$0.243135\pi$
$5$			34.5847i			1.38339i	−0.722191	+	0.691693i	$0.756865\pi$
$6$	−14.6969			−0.408248
$7$			81.5113i			1.66350i	0.555153	+	0.831748i	$0.312660\pi$
$7$			81.5113i			1.66350i	−0.555153	+	0.831748i	$0.687340\pi$
$8$	−22.6274			−0.353553
$9$	27.0000			0.333333
$10$		−	97.8202i		−	0.978202i
$11$		−	67.6260i		−	0.558893i	−0.960161	−	0.279446i	$-0.909849\pi$
$11$		−	67.6260i		−	0.558893i	0.960161	−	0.279446i	$-0.0901509\pi$
$12$	41.5692			0.288675
$13$	−316.869			−1.87496			−0.937482	−	0.348035i	$-0.886849\pi$
$13$	−316.869			−1.87496			−0.937482	+	0.348035i	$0.886849\pi$
$14$		−	230.549i		−	1.17627i
$15$			179.707i			0.798699i
$16$	64.0000			0.250000
$17$		−	44.6757i		−	0.154587i	−0.997008	−	0.0772936i	$-0.975372\pi$
$17$		−	44.6757i		−	0.154587i	0.997008	−	0.0772936i	$-0.0246279\pi$
$18$	−76.3675			−0.235702
$19$		−	468.905i		−	1.29891i	−0.760402	−	0.649453i	$-0.774998\pi$
$19$		−	468.905i		−	1.29891i	0.760402	−	0.649453i	$-0.225002\pi$
$20$			276.677i			0.691693i
$21$			423.545i			0.960420i
$22$			191.275i			0.395197i
$23$	491.402	+	195.869i	0.928927	+	0.370263i
$23$	491.402	+	195.869i	0.928927	+	0.370263i
$24$	−117.576			−0.204124
$25$	−571.099			−0.913758
$26$	896.240			1.32580
$27$	140.296			0.192450
$28$			652.091i			0.831748i
$29$	−228.778			−0.272030			−0.136015	−	0.990707i	$-0.543430\pi$
$29$	−228.778			−0.272030			−0.136015	+	0.990707i	$0.543430\pi$
$30$		−	508.289i		−	0.564765i
$31$	−1674.25			−1.74220			−0.871099	−	0.491108i	$-0.836592\pi$
$31$	−1674.25			−1.74220			−0.871099	+	0.491108i	$0.836592\pi$
$32$	−181.019			−0.176777
$33$		−	351.395i		−	0.322677i
$34$			126.362i			0.109310i
$35$	−2819.04			−2.30126
$36$	216.000			0.166667
$37$			736.035i			0.537644i	0.963190	+	0.268822i	$0.0866344\pi$
$37$			736.035i			0.537644i	−0.963190	+	0.268822i	$0.913366\pi$
$38$			1326.26i			0.918465i
$39$	−1646.50			−1.08251
$40$		−	782.562i		−	0.489101i
$41$	−299.061			−0.177907			−0.0889534	−	0.996036i	$-0.528352\pi$
$41$	−299.061			−0.177907			−0.0889534	+	0.996036i	$0.528352\pi$
$42$		−	1197.97i		−	0.679120i
$43$			2677.63i			1.44815i	0.689721	+	0.724076i	$0.257733\pi$
$43$			2677.63i			1.44815i	−0.689721	+	0.724076i	$0.742267\pi$
$44$		−	541.008i		−	0.279446i
$45$			933.786i			0.461129i
$46$	−1389.90	−	554.002i	−0.656851	−	0.261816i
$47$	3461.96			1.56721			0.783604	−	0.621261i	$-0.213379\pi$
$47$	3461.96			1.56721			0.783604	+	0.621261i	$0.213379\pi$
$48$	332.554			0.144338
$49$	−4243.10			−1.76722
$50$	1615.31			0.646125
$51$		−	232.142i		−	0.0892510i
$52$	−2534.95			−0.937482
$53$		−	142.983i		−	0.0509018i	−0.999676	−	0.0254509i	$-0.991898\pi$
$53$		−	142.983i		−	0.0509018i	0.999676	−	0.0254509i	$-0.00810216\pi$
$54$	−396.817			−0.136083
$55$	2338.82			0.773165
$56$		−	1844.39i		−	0.588135i
$57$		−	2436.50i		−	0.749924i
$58$	647.081			0.192355
$59$	−1697.68			−0.487700			−0.243850	−	0.969813i	$-0.578410\pi$
$59$	−1697.68			−0.487700			−0.243850	+	0.969813i	$0.578410\pi$
$60$			1437.66i			0.399349i
$61$			6286.88i			1.68957i	0.535107	+	0.844784i	$0.320271\pi$
$61$			6286.88i			1.68957i	−0.535107	+	0.844784i	$0.679729\pi$
$62$	4735.50			1.23192
$63$			2200.81i			0.554499i
$64$	512.000			0.125000
$65$		−	10958.8i		−	2.59380i
$66$			993.896i			0.228167i
$67$		−	2566.64i		−	0.571761i	−0.958265	−	0.285881i	$-0.907714\pi$
$67$		−	2566.64i		−	0.571761i	0.958265	−	0.285881i	$-0.0922861\pi$
$68$		−	357.406i		−	0.0772936i
$69$	2553.40	+	1017.77i	0.536316	+	0.213771i
$70$	7973.46			1.62724
$71$	5395.67			1.07036			0.535178	−	0.844739i	$-0.320244\pi$
$71$	5395.67			1.07036			0.535178	+	0.844739i	$0.320244\pi$
$72$	−610.940			−0.117851
$73$	1108.66			0.208042			0.104021	−	0.994575i	$-0.466829\pi$
$73$	1108.66			0.208042			0.104021	+	0.994575i	$0.466829\pi$
$74$		−	2081.82i		−	0.380172i
$75$	−2967.52			−0.527559
$76$		−	3751.24i		−	0.649453i
$77$	5512.29			0.929716
$78$	4657.00			0.765450
$79$			4639.58i			0.743403i	0.928352	+	0.371701i	$0.121225\pi$
$79$			4639.58i			0.743403i	−0.928352	+	0.371701i	$0.878775\pi$
$80$			2213.42i			0.345847i
$81$	729.000			0.111111
$82$	845.873			0.125799
$83$			12844.1i			1.86443i	0.361906	+	0.932214i	$0.382126\pi$
$83$			12844.1i			1.86443i	−0.361906	+	0.932214i	$0.617874\pi$
$84$			3388.36i			0.480210i
$85$	1545.09			0.213854
$86$		−	7573.49i		−	1.02400i
$87$	−1188.76			−0.157057
$88$			1530.20i			0.197598i
$89$		−	3849.23i		−	0.485953i	−0.970032	−	0.242976i	$-0.921876\pi$
$89$		−	3849.23i		−	0.485953i	0.970032	−	0.242976i	$-0.0781237\pi$
$90$		−	2641.15i		−	0.326067i
$91$		−	25828.4i		−	3.11899i
$92$	3931.22	+	1566.95i	0.464463	+	0.185132i
$93$	−8699.67			−1.00586
$94$	−9791.91			−1.10818
$95$	16216.9			1.79689
$96$	−940.604			−0.102062
$97$			940.980i			0.100009i	0.998749	+	0.0500043i	$0.0159235\pi$
$97$			940.980i			0.100009i	−0.998749	+	0.0500043i	$0.984077\pi$
$98$	12001.3			1.24961
$99$		−	1825.90i		−	0.186298i
$100$	−4568.79			−0.456879
$101$	−1542.66			−0.151226			−0.0756132	−	0.997137i	$-0.524091\pi$
$101$	−1542.66			−0.151226			−0.0756132	+	0.997137i	$0.524091\pi$
$102$			656.596i			0.0631100i
$103$		−	12529.7i		−	1.18104i	−0.807022	−	0.590521i	$-0.798922\pi$
$103$		−	12529.7i		−	1.18104i	0.807022	−	0.590521i	$-0.201078\pi$
$104$	7169.92			0.662900
$105$	−14648.2			−1.32863
$106$			404.418i			0.0359930i
$107$			13392.8i			1.16978i	0.811114	+	0.584889i	$0.198862\pi$
$107$			13392.8i			1.16978i	−0.811114	+	0.584889i	$0.801138\pi$
$108$	1122.37			0.0962250
$109$			6720.92i			0.565687i	0.959166	+	0.282843i	$0.0912776\pi$
$109$			6720.92i			0.565687i	−0.959166	+	0.282843i	$0.908722\pi$
$110$	−6615.19			−0.546710
$111$			3824.55i			0.310409i
$112$			5216.73i			0.415874i
$113$			432.194i			0.0338472i	0.999857	+	0.0169236i	$0.00538720\pi$
$113$			432.194i			0.0338472i	−0.999857	+	0.0169236i	$0.994613\pi$
$114$			6891.47i			0.530276i
$115$	−6774.07	+	16995.0i	−0.512217	+	1.28507i
$116$	−1830.22			−0.136015
$117$	−8555.46			−0.624988
$118$	4801.78			0.344856
$119$	3641.58			0.257155
$120$		−	4066.31i		−	0.282383i
$121$	10067.7			0.687639
$122$		−	17782.0i		−	1.19471i
$123$	−1553.97			−0.102715
$124$	−13394.0			−0.871099
$125$			1864.15i			0.119306i
$126$		−	6224.82i		−	0.392090i
$127$	−27147.9			−1.68317			−0.841586	−	0.540123i	$-0.818378\pi$
$127$	−27147.9			−1.68317			−0.841586	+	0.540123i	$0.818378\pi$
$128$	−1448.15			−0.0883883
$129$			13913.4i			0.836090i
$130$			30996.2i			1.83409i
$131$	−24452.9			−1.42491			−0.712457	−	0.701716i	$-0.752417\pi$
$131$	−24452.9			−1.42491			−0.712457	+	0.701716i	$0.752417\pi$
$132$		−	2811.16i		−	0.161338i
$133$	38221.1			2.16073
$134$			7259.54i			0.404296i
$135$			4852.09i			0.266233i
$136$			1010.90i			0.0546549i
$137$		−	24932.8i		−	1.32840i	−0.747553	−	0.664202i	$-0.768771\pi$
$137$		−	24932.8i		−	1.32840i	0.747553	−	0.664202i	$-0.231229\pi$
$138$	−7222.11	−	2878.68i	−0.379233	−	0.151159i
$139$	12421.6			0.642907			0.321454	−	0.946925i	$-0.395829\pi$
$139$	12421.6			0.642907			0.321454	+	0.946925i	$0.395829\pi$
$140$	−22552.3			−1.15063
$141$	17988.9			0.904828
$142$	−15261.2			−0.756856
$143$			21428.6i			1.04790i
$144$	1728.00			0.0833333
$145$		−	7912.20i		−	0.376323i
$146$	−3135.75			−0.147108
$147$	−22047.8			−1.02031
$148$			5888.28i			0.268822i
$149$		−	18427.2i		−	0.830017i	−0.909817	−	0.415009i	$-0.863779\pi$
$149$		−	18427.2i		−	0.830017i	0.909817	−	0.415009i	$-0.136221\pi$
$150$	8393.41			0.373040
$151$	−2685.37			−0.117774			−0.0588871	−	0.998265i	$-0.518755\pi$
$151$	−2685.37			−0.117774			−0.0588871	+	0.998265i	$0.518755\pi$
$152$			10610.1i			0.459233i
$153$		−	1206.24i		−	0.0515291i
$154$	−15591.1			−0.657409
$155$		−	57903.4i		−	2.41013i
$156$	−13172.0			−0.541255
$157$			17500.0i			0.709966i	0.934873	+	0.354983i	$0.115513\pi$
$157$			17500.0i			0.709966i	−0.934873	+	0.354983i	$0.884487\pi$
$158$		−	13122.7i		−	0.525665i
$159$		−	742.963i		−	0.0293882i
$160$		−	6260.49i		−	0.244550i
$161$	−15965.6	+	40054.9i	−0.615931	+	1.54527i
$162$	−2061.92			−0.0785674
$163$	−22049.3			−0.829890			−0.414945	−	0.909847i	$-0.636199\pi$
$163$	−22049.3			−0.829890			−0.414945	+	0.909847i	$0.636199\pi$
$164$	−2392.49			−0.0889534
$165$	12152.9			0.446387
$166$		−	36328.5i		−	1.31835i
$167$	−10720.9			−0.384412			−0.192206	−	0.981355i	$-0.561564\pi$
$167$	−10720.9			−0.384412			−0.192206	+	0.981355i	$0.561564\pi$
$168$		−	9583.74i		−	0.339560i
$169$	71844.8			2.51549
$170$	−4370.19			−0.151218
$171$		−	12660.4i		−	0.432969i
$172$			21421.1i			0.724076i
$173$	45355.3			1.51543			0.757714	−	0.652586i	$-0.226316\pi$
$173$	45355.3			1.51543			0.757714	+	0.652586i	$0.226316\pi$
$174$	3362.33			0.111056
$175$		−	46551.0i		−	1.52003i
$176$		−	4328.07i		−	0.139723i
$177$	−8821.43			−0.281574
$178$			10887.3i			0.343620i
$179$	11298.8			0.352635			0.176318	−	0.984333i	$-0.443581\pi$
$179$	11298.8			0.352635			0.176318	+	0.984333i	$0.443581\pi$
$180$			7470.29i			0.230564i
$181$			31946.7i			0.975143i	0.873083	+	0.487572i	$0.162117\pi$
$181$			31946.7i			0.975143i	−0.873083	+	0.487572i	$0.837883\pi$
$182$			73053.7i			2.20546i
$183$			32667.6i			0.975473i
$184$	−11119.2	−	4432.01i	−0.328425	−	0.130908i
$185$	−25455.5			−0.743770
$186$	24606.4			0.711249
$187$	−3021.24			−0.0863977
$188$	27695.7			0.783604
$189$			11435.7i			0.320140i
$190$	−45868.4			−1.27059
$191$			64795.8i			1.77615i	0.459697	+	0.888076i	$0.347958\pi$
$191$			64795.8i			1.77615i	−0.459697	+	0.888076i	$0.652042\pi$
$192$	2660.43			0.0721688
$193$	20445.5			0.548888			0.274444	−	0.961603i	$-0.411506\pi$
$193$	20445.5			0.548888			0.274444	+	0.961603i	$0.411506\pi$
$194$		−	2661.49i		−	0.0707167i
$195$		−	56943.6i		−	1.49753i
$196$	−33944.8			−0.883611
$197$	−12604.0			−0.324771			−0.162386	−	0.986727i	$-0.551919\pi$
$197$	−12604.0			−0.324771			−0.162386	+	0.986727i	$0.551919\pi$
$198$			5164.43i			0.131732i
$199$			2430.87i			0.0613841i	0.999529	+	0.0306921i	$0.00977112\pi$
$199$			2430.87i			0.0613841i	−0.999529	+	0.0306921i	$0.990229\pi$
$200$	12922.5			0.323062
$201$		−	13336.6i		−	0.330107i
$202$	4363.30			0.106933
$203$		−	18648.0i		−	0.452522i
$204$		−	1857.13i		−	0.0446255i
$205$		−	10342.9i		−	0.246114i
$206$			35439.3i			0.835123i
$207$	13267.9	+	5288.47i	0.309642	+	0.123421i
$208$	−20279.6			−0.468741
$209$	−31710.2			−0.725949
$210$	41431.3			0.939485
$211$	45269.6			1.01681			0.508407	−	0.861117i	$-0.330234\pi$
$211$	45269.6			1.01681			0.508407	+	0.861117i	$0.330234\pi$
$212$		−	1143.87i		−	0.0254509i
$213$	28036.7			0.617970
$214$		−	37880.5i		−	0.827157i
$215$	−92605.0			−2.00335
$216$	−3174.54			−0.0680414
$217$		−	136471.i		−	2.89814i
$218$		−	19009.6i		−	0.400001i
$219$	5760.74			0.120113
$220$	18710.6			0.386582
$221$			14156.3i			0.289845i
$222$		−	10817.5i		−	0.219492i
$223$	415.977			0.00836488			0.00418244	−	0.999991i	$-0.498669\pi$
$223$	415.977			0.00836488			0.00418244	+	0.999991i	$0.498669\pi$
$224$		−	14755.1i		−	0.294067i
$225$	−15419.7			−0.304586
$226$		−	1222.43i		−	0.0239336i
$227$		−	10839.9i		−	0.210364i	−0.994453	−	0.105182i	$-0.966457\pi$
$227$		−	10839.9i		−	0.210364i	0.994453	−	0.105182i	$-0.0335426\pi$
$228$		−	19492.0i		−	0.374962i
$229$		−	61204.6i		−	1.16711i	−0.812072	−	0.583557i	$-0.801661\pi$
$229$		−	61204.6i		−	1.16711i	0.812072	−	0.583557i	$-0.198339\pi$
$230$	19160.0	−	48069.1i	0.362192	−	0.908678i
$231$	28642.7			0.536772
$232$	5176.65			0.0961773
$233$	71733.3			1.32132			0.660662	−	0.750684i	$-0.270276\pi$
$233$	71733.3			1.32132			0.660662	+	0.750684i	$0.270276\pi$
$234$	24198.5			0.441933
$235$			119731.i			2.16805i
$236$	−13581.5			−0.243850
$237$			24108.0i			0.429204i
$238$	−10299.9			−0.181836
$239$	−42013.5			−0.735517			−0.367759	−	0.929921i	$-0.619875\pi$
$239$	−42013.5			−0.735517			−0.367759	+	0.929921i	$0.619875\pi$
$240$			11501.3i			0.199675i
$241$			33051.7i			0.569062i	0.958667	+	0.284531i	$0.0918379\pi$
$241$			33051.7i			0.569062i	−0.958667	+	0.284531i	$0.908162\pi$
$242$	−28475.8			−0.486234
$243$	3788.00			0.0641500
$244$			50295.1i			0.844784i
$245$		−	146746.i		−	2.44475i
$246$	4395.29			0.0726302
$247$			148581.i			2.43540i
$248$	37884.0			0.615960
$249$			66739.6i			1.07643i
$250$		−	5272.61i		−	0.0843618i
$251$			55703.6i			0.884170i	0.896973	+	0.442085i	$0.145761\pi$
$251$			55703.6i			0.884170i	−0.896973	+	0.442085i	$0.854239\pi$
$252$			17606.4i			0.277249i
$253$	13245.9	−	33231.6i	0.206937	−	0.519171i
$254$	76785.8			1.19018
$255$	8028.55			0.123469
$256$	4096.00			0.0625000
$257$	46826.6			0.708967			0.354484	−	0.935062i	$-0.384657\pi$
$257$	46826.6			0.708967			0.354484	+	0.935062i	$0.384657\pi$
$258$		−	39353.0i		−	0.591205i
$259$	−59995.2			−0.894370
$260$		−	87670.4i		−	1.29690i
$261$	−6177.00			−0.0906768
$262$	69163.3			1.00757
$263$			131665.i			1.90353i	0.306823	+	0.951767i	$0.400734\pi$
$263$			131665.i			1.90353i	−0.306823	+	0.951767i	$0.599266\pi$
$264$			7951.17i			0.114084i
$265$	4945.03			0.0704169
$266$	−108106.			−1.52786
$267$		−	20001.2i		−	0.280565i
$268$		−	20533.1i		−	0.285881i
$269$	16313.5			0.225445			0.112723	−	0.993626i	$-0.464043\pi$
$269$	16313.5			0.225445			0.112723	+	0.993626i	$0.464043\pi$
$270$		−	13723.8i		−	0.188255i
$271$	86263.7			1.17460			0.587299	−	0.809370i	$-0.300191\pi$
$271$	86263.7			1.17460			0.587299	+	0.809370i	$0.300191\pi$
$272$		−	2859.25i		−	0.0386468i
$273$		−	134208.i		−	1.80075i
$274$			70520.7i			0.939324i
$275$			38621.2i			0.510693i
$276$	20427.2	+	8142.13i	0.268158	+	0.106886i
$277$	98843.9			1.28822			0.644110	−	0.764933i	$-0.277228\pi$
$277$	98843.9			1.28822			0.644110	+	0.764933i	$0.277228\pi$
$278$	−35133.6			−0.454604
$279$	−45204.8			−0.580733
$280$	63787.6			0.813618
$281$			7858.32i			0.0995215i	0.998761	+	0.0497608i	$0.0158459\pi$
$281$			7858.32i			0.0995215i	−0.998761	+	0.0497608i	$0.984154\pi$
$282$	−50880.2			−0.639810
$283$		−	100026.i		−	1.24894i	−0.781049	−	0.624469i	$-0.785315\pi$
$283$		−	100026.i		−	1.24894i	0.781049	−	0.624469i	$-0.214685\pi$
$284$	43165.3			0.535178
$285$	84265.6			1.03743
$286$		−	60609.2i		−	0.740980i
$287$		−	24376.9i		−	0.295947i
$288$	−4887.52			−0.0589256
$289$	81525.1			0.976103
$290$			22379.1i			0.266101i
$291$			4889.48i			0.0577400i
$292$	8869.25			0.104021
$293$			44942.9i			0.523511i	0.965134	+	0.261755i	$0.0843014\pi$
$293$			44942.9i			0.523511i	−0.965134	+	0.261755i	$0.915699\pi$
$294$	62360.5			0.721465
$295$		−	58713.8i		−	0.674678i
$296$		−	16654.6i		−	0.190086i
$297$		−	9487.67i		−	0.107559i
$298$			52120.0i			0.586911i
$299$	−155710.	−	62064.8i	−1.74170	−	0.694230i
$300$	−23740.1			−0.263779
$301$	−218257.			−2.40899
$302$	7595.37			0.0832789
$303$	−8015.90			−0.0873106
$304$		−	30009.9i		−	0.324727i
$305$	−217430.			−2.33733
$306$			3411.77i			0.0364366i
$307$	−133001.			−1.41116			−0.705581	−	0.708630i	$-0.749314\pi$
$307$	−133001.			−1.41116			−0.705581	+	0.708630i	$0.749314\pi$
$308$	44098.3			0.464858
$309$		−	65106.1i		−	0.681875i
$310$			163776.i			1.70422i
$311$	−38901.3			−0.402201			−0.201100	−	0.979571i	$-0.564452\pi$
$311$	−38901.3			−0.402201			−0.201100	+	0.979571i	$0.564452\pi$
$312$	37256.0			0.382725
$313$		−	157896.i		−	1.61169i	−0.592126	−	0.805845i	$-0.701711\pi$
$313$		−	157896.i		−	1.61169i	0.592126	−	0.805845i	$-0.298289\pi$
$314$		−	49497.4i		−	0.502022i
$315$	−76114.1			−0.767086
$316$			37116.6i			0.371701i
$317$	38917.3			0.387279			0.193640	−	0.981073i	$-0.437971\pi$
$317$	38917.3			0.387279			0.193640	+	0.981073i	$0.437971\pi$
$318$			2101.42i			0.0207806i
$319$			15471.3i			0.152036i
$320$			17707.3i			0.172923i
$321$			69590.9i			0.675371i
$322$	45157.4	−	113292.i	0.435529	−	1.09267i
$323$	−20948.7			−0.200794
$324$	5832.00			0.0555556
$325$	180963.			1.71326
$326$	62364.9			0.586821
$327$			34922.9i			0.326599i
$328$	6766.99			0.0628996
$329$			282189.i			2.60705i
$330$	−34373.5			−0.315643
$331$	−120375.			−1.09870			−0.549352	−	0.835591i	$-0.685125\pi$
$331$	−120375.			−1.09870			−0.549352	+	0.835591i	$0.685125\pi$
$332$			102752.i			0.932214i
$333$			19873.0i			0.179215i
$334$	30323.2			0.271820
$335$	88766.3			0.790967
$336$			27106.9i			0.240105i
$337$			117125.i			1.03131i	0.856797	+	0.515655i	$0.172451\pi$
$337$			117125.i			1.03131i	−0.856797	+	0.515655i	$0.827549\pi$
$338$	−203208.			−1.77872
$339$			2245.75i			0.0195417i
$340$	12360.8			0.106927
$341$			113223.i			0.973702i
$342$			35809.1i			0.306155i
$343$		−	150152.i		−	1.27627i
$344$		−	60587.9i		−	0.511999i
$345$	−35199.1	+	88308.5i	−0.295729	+	0.741933i
$346$	−128284.			−1.07157
$347$	−39837.5			−0.330851			−0.165426	−	0.986222i	$-0.552900\pi$
$347$	−39837.5			−0.330851			−0.165426	+	0.986222i	$0.552900\pi$
$348$	−9510.11			−0.0785284
$349$	−61183.9			−0.502327			−0.251163	−	0.967945i	$-0.580813\pi$
$349$	−61183.9			−0.502327			−0.251163	+	0.967945i	$0.580813\pi$
$350$			131666.i			1.07483i
$351$	−44455.5			−0.360837
$352$			12241.6i			0.0987992i
$353$	14940.3			0.119898			0.0599488	−	0.998201i	$-0.480906\pi$
$353$	14940.3			0.119898			0.0599488	+	0.998201i	$0.480906\pi$
$354$	24950.8			0.199103
$355$			186607.i			1.48072i
$356$		−	30793.8i		−	0.242976i
$357$	18922.2			0.148469
$358$	−31957.8			−0.249351
$359$		−	187105.i		−	1.45177i	−0.687817	−	0.725884i	$-0.741431\pi$
$359$		−	187105.i		−	1.45177i	0.687817	−	0.725884i	$-0.258569\pi$
$360$		−	21129.2i		−	0.163034i
$361$	−89551.0			−0.687157
$362$		−	90358.8i		−	0.689530i
$363$	52313.4			0.397008
$364$		−	206627.i		−	1.55950i
$365$			38342.5i			0.287802i
$366$		−	92397.9i		−	0.689763i
$367$		−	15335.7i		−	0.113860i	−0.998378	−	0.0569299i	$-0.981869\pi$
$367$		−	15335.7i		−	0.113860i	0.998378	−	0.0569299i	$-0.0181311\pi$
$368$	31449.8	+	12535.6i	0.232232	+	0.0925658i
$369$	−8074.66			−0.0593023
$370$	71999.1			0.525925
$371$	11654.8			0.0846751
$372$	−69597.4			−0.502929
$373$		−	89174.0i		−	0.640945i	−0.947258	−	0.320472i	$-0.896158\pi$
$373$		−	89174.0i		−	0.640945i	0.947258	−	0.320472i	$-0.103842\pi$
$374$	8545.36			0.0610924
$375$			9686.41i			0.0688811i
$376$	−78335.3			−0.554092
$377$	72492.5			0.510047
$378$		−	32345.1i		−	0.226373i
$379$			208424.i			1.45100i	0.688221	+	0.725501i	$0.258392\pi$
$379$			208424.i			1.45100i	−0.688221	+	0.725501i	$0.741608\pi$
$380$	129735.			0.898445
$381$	−141065.			−0.971780
$382$		−	183270.i		−	1.25593i
$383$		−	119229.i		−	0.812804i	−0.913694	−	0.406402i	$-0.866783\pi$
$383$		−	119229.i		−	0.812804i	0.913694	−	0.406402i	$-0.133217\pi$
$384$	−7524.83			−0.0510310
$385$			190641.i			1.28616i
$386$	−57828.7			−0.388122
$387$			72296.0i			0.482717i
$388$			7527.84i			0.0500043i
$389$		−	208451.i		−	1.37754i	−0.724978	−	0.688772i	$-0.758150\pi$
$389$		−	208451.i		−	1.37754i	0.724978	−	0.688772i	$-0.241850\pi$
$390$			161061.i			1.05891i
$391$	8750.60	−	21953.8i	0.0572380	−	0.143600i
$392$	96010.3			0.624807
$393$	−127061.			−0.822674
$394$	35649.6			0.229648
$395$	−160458.			−1.02841
$396$		−	14607.2i		−	0.0931488i
$397$	1940.42			0.0123116			0.00615581	−	0.999981i	$-0.498041\pi$
$397$	1940.42			0.0123116			0.00615581	+	0.999981i	$0.498041\pi$
$398$		−	6875.55i		−	0.0434051i
$399$	198603.			1.24750
$400$	−36550.3			−0.228440
$401$			221336.i			1.37646i	0.725493	+	0.688230i	$0.241612\pi$
$401$			221336.i			1.37646i	−0.725493	+	0.688230i	$0.758388\pi$
$402$			37721.7i			0.233421i
$403$	530518.			3.26656
$404$	−12341.3			−0.0756132
$405$			25212.2i			0.153710i
$406$			52744.4i			0.319981i
$407$	49775.1			0.300486
$408$			5252.77i			0.0315550i
$409$	153371.			0.916845			0.458423	−	0.888734i	$-0.348415\pi$
$409$	153371.			0.916845			0.458423	+	0.888734i	$0.348415\pi$
$410$			29254.2i			0.174029i
$411$		−	129555.i		−	0.766954i
$412$		−	100237.i		−	0.590521i
$413$		−	138381.i		−	0.811288i
$414$	−37527.2	−	14958.0i	−0.218950	−	0.0872718i
$415$	−444207.			−2.57923
$416$	57359.4			0.331450
$417$	64544.6			0.371183
$418$	89690.0			0.513324
$419$		−	255314.i		−	1.45427i	−0.686492	−	0.727137i	$-0.740850\pi$
$419$		−	255314.i		−	1.45427i	0.686492	−	0.727137i	$-0.259150\pi$
$420$	−117185.			−0.664316
$421$			86647.5i			0.488868i	0.969666	+	0.244434i	$0.0786022\pi$
$421$			86647.5i			0.488868i	−0.969666	+	0.244434i	$0.921398\pi$
$422$	−128042.			−0.718997
$423$	93473.0			0.522403
$424$			3235.34i			0.0179965i
$425$			25514.3i			0.141255i
$426$	−79299.8			−0.436971
$427$	−512452.			−2.81059
$428$			107142.i			0.584889i
$429$			111346.i			0.605007i
$430$	261926.			1.41658
$431$		−	331914.i		−	1.78678i	−0.449282	−	0.893390i	$-0.648320\pi$
$431$		−	331914.i		−	1.78678i	0.449282	−	0.893390i	$-0.351680\pi$
$432$	8978.95			0.0481125
$433$			14258.0i			0.0760470i	0.999277	+	0.0380235i	$0.0121062\pi$
$433$			14258.0i			0.0760470i	−0.999277	+	0.0380235i	$0.987894\pi$
$434$			385997.i			2.04929i
$435$		−	41113.0i		−	0.217270i
$436$			53767.4i			0.282843i
$437$	91844.1	−	230421.i	0.480937	−	1.20659i
$438$	−16293.8			−0.0849328
$439$	−65318.9			−0.338930			−0.169465	−	0.985536i	$-0.554204\pi$
$439$	−65318.9			−0.338930			−0.169465	+	0.985536i	$0.554204\pi$
$440$	−52921.5			−0.273355
$441$	−114564.			−0.589074
$442$		−	40040.2i		−	0.204952i
$443$	303676.			1.54740			0.773700	−	0.633553i	$-0.218404\pi$
$443$	303676.			1.54740			0.773700	+	0.633553i	$0.218404\pi$
$444$			30596.4i			0.155205i
$445$	133124.			0.672260
$446$	−1176.56			−0.00591486
$447$		−	95750.6i		−	0.479211i
$448$			41733.8i			0.207937i
$449$	296049.			1.46849			0.734246	−	0.678884i	$-0.237536\pi$
$449$	296049.			1.46849			0.734246	+	0.678884i	$0.237536\pi$
$450$	43613.4			0.215375
$451$			20224.3i			0.0994309i
$452$			3457.55i			0.0169236i
$453$	−13953.6			−0.0679970
$454$			30659.8i			0.148750i
$455$	893266.			4.31478
$456$			55131.8i			0.265138i
$457$		−	114510.i		−	0.548289i	−0.961689	−	0.274144i	$-0.911605\pi$
$457$		−	114510.i		−	0.548289i	0.961689	−	0.274144i	$-0.0883946\pi$
$458$			173113.i			0.825274i
$459$		−	6267.83i		−	0.0297503i
$460$	−54192.6	+	135960.i	−0.256108	+	0.642533i
$461$	69910.9			0.328960			0.164480	−	0.986380i	$-0.447405\pi$
$461$	69910.9			0.328960			0.164480	+	0.986380i	$0.447405\pi$
$462$	−81013.8			−0.379555
$463$	−133673.			−0.623564			−0.311782	−	0.950154i	$-0.600926\pi$
$463$	−133673.			−0.623564			−0.311782	+	0.950154i	$0.600926\pi$
$464$	−14641.8			−0.0680076
$465$		−	300875.i		−	1.39149i
$466$	−202893.			−0.934317
$467$			19129.4i			0.0877139i	0.999038	+	0.0438570i	$0.0139646\pi$
$467$			19129.4i			0.0877139i	−0.999038	+	0.0438570i	$0.986035\pi$
$468$	−68443.7			−0.312494
$469$	209210.			0.951123
$470$		−	338650.i		−	1.53305i
$471$			90932.5i			0.409899i
$472$	38414.2			0.172428
$473$	181078.			0.809361
$474$		−	68187.6i		−	0.303493i
$475$			267791.i			1.18689i
$476$	29132.6			0.128578
$477$		−	3860.55i		−	0.0169673i
$478$	118832.			0.520089
$479$		−	193492.i		−	0.843318i	−0.906754	−	0.421659i	$-0.861448\pi$
$479$		−	193492.i		−	0.843318i	0.906754	−	0.421659i	$-0.138552\pi$
$480$		−	32530.5i		−	0.141191i
$481$		−	233227.i		−	1.00806i
$482$		−	93484.3i		−	0.402388i
$483$	−82959.5	+	208131.i	−0.355608	+	0.892160i
$484$	80541.8			0.343819
$485$	−32543.5			−0.138350
$486$	−10714.1			−0.0453609
$487$	330551.			1.39374			0.696868	−	0.717200i	$-0.254576\pi$
$487$	330551.			1.39374			0.696868	+	0.717200i	$0.254576\pi$
$488$		−	142256.i		−	0.597353i
$489$	−114572.			−0.479137
$490$			415061.i			1.72870i
$491$	−4576.09			−0.0189816			−0.00949078	−	0.999955i	$-0.503021\pi$
$491$	−4576.09			−0.0189816			−0.00949078	+	0.999955i	$0.503021\pi$
$492$	−12431.8			−0.0513573
$493$			10220.8i			0.0420524i
$494$		−	420252.i		−	1.72209i
$495$	63148.2			0.257722
$496$	−107152.			−0.435549
$497$			439808.i			1.78053i
$498$		−	188768.i		−	0.761150i
$499$	−256326.			−1.02942			−0.514708	−	0.857366i	$-0.672100\pi$
$499$	−256326.			−1.02942			−0.514708	+	0.857366i	$0.672100\pi$
$500$			14913.2i			0.0596528i
$501$	−55707.2			−0.221940
$502$		−	157553.i		−	0.625202i
$503$			376177.i			1.48681i	0.668840	+	0.743407i	$0.266791\pi$
$503$			376177.i			1.48681i	−0.668840	+	0.743407i	$0.733209\pi$
$504$		−	49798.6i		−	0.196045i
$505$		−	53352.4i		−	0.209205i
$506$	−37464.9	+	93993.1i	−0.146327	+	0.367109i
$507$	373317.			1.45232
$508$	−217183.			−0.841586
$509$	−183323.			−0.707588			−0.353794	−	0.935323i	$-0.615109\pi$
$509$	−183323.			−0.707588			−0.353794	+	0.935323i	$0.615109\pi$
$510$	−22708.2			−0.0873055
$511$			90368.0i			0.346077i
$512$	−11585.2			−0.0441942
$513$		−	65785.6i		−	0.249975i
$514$	−132446.			−0.501316
$515$	433335.			1.63384
$516$			111307.i			0.418045i
$517$		−	234119.i		−	0.875901i
$518$	169692.			0.632415
$519$	235673.			0.874933
$520$			247969.i			0.917046i
$521$		−	366081.i		−	1.34866i	−0.738430	−	0.674330i	$-0.764433\pi$
$521$		−	366081.i		−	1.34866i	0.738430	−	0.674330i	$-0.235567\pi$
$522$	17471.2			0.0641182
$523$		−	236643.i		−	0.865148i	−0.901598	−	0.432574i	$-0.857605\pi$
$523$		−	236643.i		−	0.865148i	0.901598	−	0.432574i	$-0.142395\pi$
$524$	−195623.			−0.712457
$525$		−	241886.i		−	0.877592i
$526$		−	372406.i		−	1.34600i
$527$			74798.4i			0.269322i
$528$		−	22489.3i		−	0.0806692i
$529$	203112.	+	192501.i	0.725810	+	0.687895i
$530$	−13986.7			−0.0497923
$531$	−45837.5			−0.162567
$532$	305769.			1.08036
$533$	94763.2			0.333569
$534$			56571.9i			0.198389i
$535$	−463185.			−1.61825
$536$			58076.4i			0.202148i
$537$	58710.3			0.203594
$538$	−46141.4			−0.159414
$539$			286944.i			0.987687i
$540$			38816.8i			0.133116i
$541$	−386905.			−1.32194			−0.660968	−	0.750414i	$-0.729854\pi$
$541$	−386905.			−1.32194			−0.660968	+	0.750414i	$0.729854\pi$
$542$	−243991.			−0.830567
$543$			166000.i			0.562999i
$544$			8087.17i			0.0273274i
$545$	−232441.			−0.782563
$546$			379598.i			1.27332i
$547$	5255.98			0.0175663			0.00878313	−	0.999961i	$-0.497204\pi$
$547$	5255.98			0.0175663			0.00878313	+	0.999961i	$0.497204\pi$
$548$		−	199463.i		−	0.664202i
$549$			169746.i			0.563189i
$550$		−	109237.i		−	0.361114i
$551$			107275.i			0.353342i
$552$	−57776.9	−	23029.4i	−0.189616	−	0.0755796i
$553$	−378178.			−1.23665
$554$	−279573.			−0.910910
$555$	−132271.			−0.429416
$556$	99372.9			0.321454
$557$		−	148246.i		−	0.477830i	−0.971040	−	0.238915i	$-0.923208\pi$
$557$		−	148246.i		−	0.477830i	0.971040	−	0.238915i	$-0.0767918\pi$
$558$	127858.			0.410640
$559$		−	848458.i		−	2.71523i
$560$	−180419.			−0.575315
$561$	−15698.8			−0.0498817
$562$		−	22226.7i		−	0.0703723i
$563$			468399.i			1.47774i	0.673846	+	0.738872i	$0.264641\pi$
$563$			468399.i			1.47774i	−0.673846	+	0.738872i	$0.735359\pi$
$564$	143911.			0.452414
$565$	−14947.3			−0.0468237
$566$			282917.i			0.883133i
$567$			59421.8i			0.184833i
$568$	−122090.			−0.378428
$569$		−	287885.i		−	0.889191i	−0.895731	−	0.444595i	$-0.853347\pi$
$569$		−	287885.i		−	0.889191i	0.895731	−	0.444595i	$-0.146653\pi$
$570$	−238339.			−0.733577
$571$			563564.i			1.72851i	0.503056	+	0.864254i	$0.332209\pi$
$571$			563564.i			1.72851i	−0.503056	+	0.864254i	$0.667791\pi$
$572$			171429.i			0.523952i
$573$			336689.i			1.02546i
$574$			68948.3i			0.209266i
$575$	−280639.	−	111861.i	−0.848815	−	0.338331i
$576$	13824.0			0.0416667
$577$	−385391.			−1.15758			−0.578788	−	0.815478i	$-0.696474\pi$
$577$	−385391.			−1.15758			−0.578788	+	0.815478i	$0.696474\pi$
$578$	−230588.			−0.690209
$579$	106238.			0.316901
$580$		−	63297.6i		−	0.188162i
$581$	−1.04694e6			−3.10147
$582$		−	13829.5i		−	0.0408283i
$583$	−9669.39			−0.0284487
$584$	−25086.0			−0.0735539
$585$		−	295888.i		−	0.864600i
$586$		−	127118.i		−	0.370178i
$587$	473689.			1.37473			0.687365	−	0.726312i	$-0.258767\pi$
$587$	473689.			1.37473			0.687365	+	0.726312i	$0.258767\pi$
$588$	−176382.			−0.510153
$589$			785065.i			2.26295i
$590$			166068.i			0.477069i
$591$	−65492.5			−0.187507
$592$			47106.3i			0.134411i
$593$	−41779.3			−0.118810			−0.0594048	−	0.998234i	$-0.518920\pi$
$593$	−41779.3			−0.118810			−0.0594048	+	0.998234i	$0.518920\pi$
$594$			26835.2i			0.0760557i
$595$			125943.i			0.355745i
$596$		−	147418.i		−	0.415009i
$597$			12631.2i			0.0354402i
$598$	440415.	+	175546.i	1.23157	+	0.490894i
$599$	84704.2			0.236076			0.118038	−	0.993009i	$-0.462340\pi$
$599$	84704.2			0.236076			0.118038	+	0.993009i	$0.462340\pi$
$600$	67147.2			0.186520
$601$	−555826.			−1.53883			−0.769414	−	0.638750i	$-0.779452\pi$
$601$	−555826.			−1.53883			−0.769414	+	0.638750i	$0.779452\pi$
$602$	617325.			1.70342
$603$		−	69299.2i		−	0.190587i
$604$	−21483.0			−0.0588871
$605$			348189.i			0.951270i
$606$	22672.4			0.0617379
$607$	212157.			0.575810			0.287905	−	0.957659i	$-0.407041\pi$
$607$	212157.			0.575810			0.287905	+	0.957659i	$0.407041\pi$
$608$			84880.9i			0.229616i
$609$		−	96897.7i		−	0.261264i
$610$	614984.			1.65274
$611$	−1.09699e6			−2.93846
$612$		−	9649.96i		−	0.0257645i
$613$			94405.1i			0.251232i	0.992079	+	0.125616i	$0.0400907\pi$
$613$			94405.1i			0.251232i	−0.992079	+	0.125616i	$0.959909\pi$
$614$	376182.			0.997842
$615$		−	53743.5i		−	0.142094i
$616$	−124729.			−0.328704
$617$		−	174142.i		−	0.457438i	−0.973493	−	0.228719i	$-0.926546\pi$
$617$		−	174142.i		−	0.457438i	0.973493	−	0.228719i	$-0.0734537\pi$
$618$			184148.i			0.482159i
$619$		−	136471.i		−	0.356171i	−0.984015	−	0.178086i	$-0.943010\pi$
$619$		−	136471.i		−	0.356171i	0.984015	−	0.178086i	$-0.0569904\pi$
$620$		−	463228.i		−	1.20507i
$621$	68941.8	+	27479.7i	0.178772	+	0.0712572i
$622$	110029.			0.284399
$623$	313756.			0.808381
$624$	−105376.			−0.270628
$625$	−421408.			−1.07880
$626$			446596.i			1.13964i
$627$	−164771.			−0.419127
$628$			140000.i			0.354983i
$629$	32882.9			0.0831130
$630$	215283.			0.542412
$631$			159317.i			0.400132i	0.979782	+	0.200066i	$0.0641156\pi$
$631$			159317.i			0.400132i	−0.979782	+	0.200066i	$0.935884\pi$
$632$		−	104982.i		−	0.262833i
$633$	235228.			0.587058
$634$	−110075.			−0.273848
$635$		−	938901.i		−	2.32848i
$636$		−	5943.70i		−	0.0146941i
$637$	1.34451e6			3.31347
$638$		−	43759.5i		−	0.107506i
$639$	145683.			0.356785
$640$		−	50083.9i		−	0.122275i
$641$		−	278560.i		−	0.677958i	−0.940794	−	0.338979i	$-0.889918\pi$
$641$		−	278560.i		−	0.677958i	0.940794	−	0.338979i	$-0.110082\pi$
$642$		−	196833.i		−	0.477559i
$643$			611844.i			1.47985i	0.672687	+	0.739927i	$0.265140\pi$
$643$			611844.i			1.47985i	−0.672687	+	0.739927i	$0.734860\pi$
$644$	−127724.	+	320439.i	−0.307966	+	0.772633i
$645$	−481190.			−1.15664
$646$	59251.8			0.141983
$647$	−549943.			−1.31374			−0.656870	−	0.754004i	$-0.728120\pi$
$647$	−549943.			−1.31374			−0.656870	+	0.754004i	$0.728120\pi$
$648$	−16495.4			−0.0392837
$649$			114808.i			0.272572i
$650$	−511842.			−1.21146
$651$		−	709122.i		−	1.67324i
$652$	−176395.			−0.414945
$653$	−192358.			−0.451111			−0.225556	−	0.974230i	$-0.572420\pi$
$653$	−192358.			−0.451111			−0.225556	+	0.974230i	$0.572420\pi$
$654$		−	98777.0i		−	0.230941i
$655$		−	845697.i		−	1.97121i
$656$	−19139.9			−0.0444767
$657$	29933.7			0.0693473
$658$		−	798151.i		−	1.84346i
$659$			494216.i			1.13801i	0.822334	+	0.569005i	$0.192672\pi$
$659$			494216.i			1.13801i	−0.822334	+	0.569005i	$0.807328\pi$
$660$	97223.1			0.223193
$661$		−	117269.i		−	0.268399i	−0.990954	−	0.134199i	$-0.957154\pi$
$661$		−	117269.i		−	0.268399i	0.990954	−	0.134199i	$-0.0428463\pi$
$662$	340472.			0.776901
$663$			73558.5i			0.167342i
$664$		−	290628.i		−	0.659175i
$665$			1.32186e6i			2.98912i
$666$		−	56209.2i		−	0.126724i
$667$	−112422.	−	44810.5i	−0.252696	−	0.100723i
$668$	−85766.9			−0.192206
$669$	2161.48			0.00482946
$670$	−251069.			−0.559298
$671$	425157.			0.944288
$672$		−	76669.9i		−	0.169780i
$673$	22739.7			0.0502059			0.0251029	−	0.999685i	$-0.492009\pi$
$673$	22739.7			0.0502059			0.0251029	+	0.999685i	$0.492009\pi$
$674$		−	331279.i		−	0.729246i
$675$	−80123.0			−0.175853
$676$	574759.			1.25774
$677$		−	384761.i		−	0.839488i	−0.907643	−	0.419744i	$-0.862120\pi$
$677$		−	384761.i		−	0.839488i	0.907643	−	0.419744i	$-0.137880\pi$
$678$		−	6351.93i		−	0.0138180i
$679$	−76700.6			−0.166364
$680$	−34961.5			−0.0756088
$681$		−	56325.6i		−	0.121454i
$682$		−	320243.i		−	0.688511i
$683$	−197744.			−0.423898			−0.211949	−	0.977281i	$-0.567981\pi$
$683$	−197744.			−0.423898			−0.211949	+	0.977281i	$0.567981\pi$
$684$		−	101284.i		−	0.216484i
$685$	862293.			1.83770
$686$			424694.i			0.902459i
$687$		−	318028.i		−	0.673833i
$688$			171368.i			0.362038i
$689$			45306.9i			0.0954391i
$690$	99558.1	−	249774.i	0.209112	−	0.524626i
$691$	515275.			1.07915			0.539576	−	0.841937i	$-0.318585\pi$
$691$	515275.			1.07915			0.539576	+	0.841937i	$0.318585\pi$
$692$	362842.			0.757714
$693$	148832.			0.309905
$694$	112677.			0.233947
$695$			429597.i			0.889390i
$696$	26898.6			0.0555280
$697$			13360.8i			0.0275021i
$698$	173054.			0.355199
$699$	372737.			0.762867
$700$		−	372408.i		−	0.760017i
$701$			359121.i			0.730811i	0.930848	+	0.365406i	$0.119070\pi$
$701$			359121.i			0.730811i	−0.930848	+	0.365406i	$0.880930\pi$
$702$	125739.			0.255150
$703$	345131.			0.698350
$704$		−	34624.5i		−	0.0698616i
$705$			622139.i			1.25173i
$706$	−42257.6			−0.0847804
$707$		−	125744.i		−	0.251565i
$708$	−70571.4			−0.140787
$709$			25288.2i			0.0503066i	0.999684	+	0.0251533i	$0.00800739\pi$
$709$			25288.2i			0.0503066i	−0.999684	+	0.0251533i	$0.991993\pi$
$710$		−	527805.i		−	1.04702i
$711$			125269.i			0.247801i
$712$			87098.2i			0.171810i
$713$	−822731.	−	327934.i	−1.61837	−	0.645072i
$714$	−53520.0			−0.104983
$715$	−741100.			−1.44966
$716$	90390.3			0.176318
$717$	−218309.			−0.424651
$718$			529214.i			1.02655i
$719$	401185.			0.776045			0.388022	−	0.921650i	$-0.373158\pi$
$719$	401185.			0.776045			0.388022	+	0.921650i	$0.373158\pi$
$720$			59762.3i			0.115282i
$721$	1.02131e6			1.96466
$722$	253288.			0.485893
$723$			171742.i			0.328548i
$724$			255573.i			0.487572i
$725$	130655.			0.248570
$726$	−147965.			−0.280727
$727$			20759.7i			0.0392782i	0.999807	+	0.0196391i	$0.00625172\pi$
$727$			20759.7i			0.0392782i	−0.999807	+	0.0196391i	$0.993748\pi$
$728$			584430.i			1.10273i
$729$	19683.0			0.0370370
$730$		−	108449.i		−	0.203507i
$731$	119625.			0.223866
$732$			261341.i			0.487736i
$733$		−	70627.4i		−	0.131451i	−0.997838	−	0.0657257i	$-0.979064\pi$
$733$		−	70627.4i		−	0.131451i	0.997838	−	0.0657257i	$-0.0209362\pi$
$734$			43375.8i			0.0805110i
$735$		−	762515.i		−	1.41148i
$736$	−88953.3	−	35456.1i	−0.164213	−	0.0654539i
$737$	−173571.			−0.319553
$738$	22838.6			0.0419330
$739$	−194006.			−0.355243			−0.177621	−	0.984099i	$-0.556840\pi$
$739$	−194006.			−0.355243			−0.177621	+	0.984099i	$0.556840\pi$
$740$	−203644.			−0.371885
$741$			772052.i			1.40608i
$742$	−32964.6			−0.0598743
$743$		−	679406.i		−	1.23070i	−0.788254	−	0.615350i	$-0.789015\pi$
$743$		−	679406.i		−	1.23070i	0.788254	−	0.615350i	$-0.210985\pi$
$744$	196851.			0.355625
$745$	637299.			1.14823
$746$			252222.i			0.453216i
$747$			346789.i			0.621476i
$748$	−24169.9			−0.0431989
$749$	−1.09166e6			−1.94592
$750$		−	27397.3i		−	0.0487063i
$751$		−	169607.i		−	0.300722i	−0.988631	−	0.150361i	$-0.951956\pi$
$751$		−	169607.i		−	0.300722i	0.988631	−	0.150361i	$-0.0480435\pi$
$752$	221566.			0.391802
$753$			289444.i			0.510475i
$754$	−205040.			−0.360658
$755$		−	92872.6i		−	0.162927i
$756$			91485.8i			0.160070i
$757$			952503.i			1.66217i	0.556147	+	0.831084i	$0.312279\pi$
$757$			952503.i			1.66217i	−0.556147	+	0.831084i	$0.687721\pi$
$758$		−	589511.i		−	1.02601i
$759$	68827.5	−	172676.i	0.119475	−	0.299743i
$760$	−366947.			−0.635296
$761$	−332609.			−0.574335			−0.287167	−	0.957880i	$-0.592714\pi$
$761$	−332609.			−0.574335			−0.287167	+	0.957880i	$0.592714\pi$
$762$	398991.			0.687152
$763$	−547831.			−0.941018
$764$			518366.i			0.888076i
$765$	41717.6			0.0712846
$766$			337232.i			0.574739i
$767$	537943.			0.914420
$768$	21283.4			0.0360844
$769$		−	481757.i		−	0.814658i	−0.913281	−	0.407329i	$-0.866460\pi$
$769$		−	481757.i		−	0.814658i	0.913281	−	0.407329i	$-0.133540\pi$
$770$		−	539213.i		−	0.909450i
$771$	243318.			0.409322
$772$	163564.			0.274444
$773$		−	516049.i		−	0.863638i	−0.901960	−	0.431819i	$-0.857872\pi$
$773$		−	516049.i		−	0.863638i	0.901960	−	0.431819i	$-0.142128\pi$
$774$		−	204484.i		−	0.341332i
$775$	956164.			1.59195
$776$		−	21292.0i		−	0.0353584i
$777$	−311744.			−0.516365
$778$			589589.i			0.974070i
$779$			140231.i			0.231084i
$780$		−	455549.i		−	0.748765i
$781$		−	364888.i		−	0.598215i
$782$	−24750.4	+	62094.6i	−0.0404733	+	0.101541i
$783$	−32096.6			−0.0523523
$784$	−271558.			−0.441805
$785$	−605230.			−0.982158
$786$	359383.			0.581718
$787$		−	48857.7i		−	0.0788831i	−0.999222	−	0.0394415i	$-0.987442\pi$
$787$		−	48857.7i		−	0.0788831i	0.999222	−	0.0394415i	$-0.0125579\pi$
$788$	−100832.			−0.162386
$789$			684154.i			1.09901i
$790$	453844.			0.727198
$791$	−35228.7			−0.0563046
$792$			41315.5i			0.0658662i
$793$		−	1.99212e6i		−	3.16788i
$794$	−5488.35			−0.00870564
$795$	25695.1			0.0406552
$796$			19447.0i			0.0306921i
$797$		−	67546.8i		−	0.106338i	−0.998586	−	0.0531690i	$-0.983068\pi$
$797$		−	67546.8i		−	0.106338i	0.998586	−	0.0531690i	$-0.0169322\pi$
$798$	−561733.			−0.882113
$799$		−	154666.i		−	0.242270i
$800$	103380.			0.161531
$801$		−	103929.i		−	0.161984i
$802$		−	626033.i		−	0.973304i
$803$		−	74974.0i		−	0.116273i
$804$		−	106693.i		−	0.165053i
$805$	−1.38528e6	−	552163.i	−2.13770	−	0.852071i
$806$	−1.50053e6			−2.30980
$807$	84767.2			0.130161
$808$	34906.4			0.0534666
$809$	853381.			1.30391			0.651953	−	0.758260i	$-0.273950\pi$
$809$	853381.			1.30391			0.651953	+	0.758260i	$0.273950\pi$
$810$		−	71310.9i		−	0.108689i
$811$	−584283.			−0.888344			−0.444172	−	0.895942i	$-0.646502\pi$
$811$	−584283.			−0.888344			−0.444172	+	0.895942i	$0.646502\pi$
$812$		−	149184.i		−	0.226261i
$813$	448239.			0.678155
$814$	−140785.			−0.212475
$815$		−	762569.i		−	1.14806i
$816$		−	14857.1i		−	0.0223127i
$817$	1.25556e6			1.88101
$818$	−433798.			−0.648307
$819$		−	697367.i		−	1.03966i
$820$		−	82743.5i		−	0.123057i
$821$	−681669.			−1.01132			−0.505658	−	0.862734i	$-0.668750\pi$
$821$	−681669.			−1.01132			−0.505658	+	0.862734i	$0.668750\pi$
$822$			366436.i			0.542319i
$823$	830671.			1.22639			0.613196	−	0.789931i	$-0.289884\pi$
$823$	830671.			1.22639			0.613196	+	0.789931i	$0.289884\pi$
$824$			283514.i			0.417562i
$825$			200681.i			0.294849i
$826$			391399.i			0.573667i
$827$			164064.i			0.239885i	0.992781	+	0.119942i	$0.0382710\pi$
$827$			164064.i			0.239885i	−0.992781	+	0.119942i	$0.961729\pi$
$828$	106143.	+	42307.7i	0.154821	+	0.0617105i
$829$	−11061.6			−0.0160956			−0.00804780	−	0.999968i	$-0.502562\pi$
$829$	−11061.6			−0.0160956			−0.00804780	+	0.999968i	$0.502562\pi$
$830$	1.25641e6			1.82379
$831$	513608.			0.743755
$832$	−162237.			−0.234370
$833$			189563.i			0.273190i
$834$	−182560.			−0.262466
$835$		−	370777.i		−	0.531790i
$836$	−253682.			−0.362975
$837$	−234891.			−0.335286
$838$			722137.i			1.02833i
$839$			1.01500e6i			1.44192i	0.692978	+	0.720959i	$0.256298\pi$
$839$			1.01500e6i			1.44192i	−0.692978	+	0.720959i	$0.743702\pi$
$840$	331450.			0.469742
$841$	−654942.			−0.925999
$842$		−	245076.i		−	0.345682i
$843$			40833.0i			0.0574588i
$844$	362157.			0.508407
$845$			2.48473e6i			3.47989i
$846$	−264382.			−0.369394
$847$			820633.i			1.14388i
$848$		−	9150.93i		−	0.0127255i
$849$		−	519752.i		−	0.721075i
$850$		−	72165.2i		−	0.0998826i
$851$	−144167.	+	361689.i	−0.199070	+	0.499432i
$852$	224294.			0.308985
$853$	−694837.			−0.954960			−0.477480	−	0.878643i	$-0.658450\pi$
$853$	−694837.			−0.954960			−0.477480	+	0.878643i	$0.658450\pi$
$854$	1.44943e6			1.98739
$855$	437857.			0.598963
$856$		−	303044.i		−	0.413579i
$857$	816252.			1.11138			0.555690	−	0.831389i	$-0.312454\pi$
$857$	816252.			1.11138			0.555690	+	0.831389i	$0.312454\pi$
$858$		−	314935.i		−	0.427805i
$859$	−568625.			−0.770619			−0.385309	−	0.922787i	$-0.625905\pi$
$859$	−568625.			−0.770619			−0.385309	+	0.922787i	$0.625905\pi$
$860$	−740840.			−1.00168
$861$		−	126666.i		−	0.170865i
$862$			938795.i			1.26344i
$863$	1.05589e6			1.41774			0.708870	−	0.705339i	$-0.249205\pi$
$863$	1.05589e6			1.41774			0.708870	+	0.705339i	$0.249205\pi$
$864$	−25396.3			−0.0340207
$865$			1.56860e6i			2.09642i
$866$		−	40327.6i		−	0.0537734i
$867$	423617.			0.563553
$868$		−	1.09176e6i		−	1.44907i
$869$	313756.			0.415483
$870$			116285.i			0.153633i
$871$			813287.i			1.07203i
$872$		−	152077.i		−	0.200000i
$873$			25406.5i			0.0333362i
$874$	−259774.	+	651729.i	−0.340074	+	0.853187i
$875$	−151949.			−0.198464
$876$	46086.0			0.0600565
$877$	194231.			0.252533			0.126267	−	0.991996i	$-0.459701\pi$
$877$	194231.			0.252533			0.126267	+	0.991996i	$0.459701\pi$
$878$	184750.			0.239660
$879$			233530.i			0.302249i
$880$	149685.			0.193291
$881$			529510.i			0.682217i	0.940024	+	0.341109i	$0.110802\pi$
$881$			529510.i			0.682217i	−0.940024	+	0.341109i	$0.889198\pi$
$882$	324035.			0.416538
$883$	892558.			1.14476			0.572381	−	0.819988i	$-0.306020\pi$
$883$	892558.			1.14476			0.572381	+	0.819988i	$0.306020\pi$
$884$			113251.i			0.144923i
$885$		−	305086.i		−	0.389525i
$886$	−858924.			−1.09418
$887$	938546.			1.19291			0.596456	−	0.802646i	$-0.296575\pi$
$887$	938546.			1.19291			0.596456	+	0.802646i	$0.296575\pi$
$888$		−	86539.7i		−	0.109746i
$889$		−	2.21286e6i		−	2.79995i
$890$	−376533.			−0.475360
$891$		−	49299.4i		−	0.0620992i
$892$	3327.82			0.00418244
$893$		−	1.62333e6i		−	2.03566i
$894$			270824.i			0.338853i
$895$			390765.i			0.487831i
$896$		−	118041.i		−	0.147034i
$897$	−809093.	−	322498.i	−1.00557	−	0.400814i
$898$	−837354.			−1.03838
$899$	383031.			0.473931
$900$	−123357.			−0.152293
$901$	−6387.88			−0.00786878
$902$		−	57203.1i		−	0.0703082i
$903$	−1.13410e6			−1.39083
$904$		−	9779.44i		−	0.0119668i
$905$	−1.10486e6			−1.34900
$906$	39466.7			0.0480811
$907$		−	854017.i		−	1.03813i	−0.854735	−	0.519065i	$-0.826280\pi$
$907$		−	854017.i		−	1.03813i	0.854735	−	0.519065i	$-0.173720\pi$
$908$		−	86718.9i		−	0.105182i
$909$	−41651.8			−0.0504088
$910$	−2.52654e6			−3.05101
$911$			739645.i			0.891224i	0.895226	+	0.445612i	$0.147014\pi$
$911$			739645.i			0.891224i	−0.895226	+	0.445612i	$0.852986\pi$
$912$		−	155936.i		−	0.187481i
$913$	868592.			1.04202
$914$			323882.i			0.387699i
$915$	−1.12980e6			−1.34946
$916$		−	489637.i		−	0.583557i
$917$		−	1.99319e6i		−	2.37034i
$918$			17728.1i			0.0210367i
$919$			777357.i			0.920427i	0.887808	+	0.460213i	$0.152227\pi$
$919$			777357.i			0.920427i	−0.887808	+	0.460213i	$0.847773\pi$
$920$	153280.	−	384553.i	0.181096	−	0.454339i
$921$	−691091.			−0.814734
$922$	−197738.			−0.232610
$923$	−1.70972e6			−2.00688
$924$	229142.			0.268386
$925$		−	420349.i		−	0.491277i
$926$	378084.			0.440926
$927$		−	338301.i		−	0.393681i
$928$	41413.2			0.0480886
$929$	1.28666e6			1.49084			0.745422	−	0.666592i	$-0.232248\pi$
$929$	1.28666e6			1.49084			0.745422	+	0.666592i	$0.232248\pi$
$930$			851003.i			0.983933i
$931$			1.98961e6i			2.29545i
$932$	573867.			0.660662
$933$	−202137.			−0.232211
$934$		−	54106.2i		−	0.0620231i
$935$		−	104489.i		−	0.119521i
$936$	193588.			0.220967
$937$			713777.i			0.812987i	0.913654	+	0.406493i	$0.133249\pi$
$937$			713777.i			0.812987i	−0.913654	+	0.406493i	$0.866751\pi$
$938$	−591735.			−0.672546
$939$		−	820450.i		−	0.930510i
$940$			957846.i			1.08403i
$941$		−	1.38256e6i		−	1.56137i	−0.624927	−	0.780683i	$-0.714871\pi$
$941$		−	1.38256e6i		−	1.56137i	0.624927	−	0.780683i	$-0.285129\pi$
$942$		−	257196.i		−	0.289843i
$943$	−146959.	−	58576.9i	−0.165262	−	0.0658723i
$944$	−108652.			−0.121925
$945$	−395501.			−0.442877
$946$	−512165.			−0.572305
$947$	−610787.			−0.681067			−0.340533	−	0.940232i	$-0.610608\pi$
$947$	−610787.			−0.681067			−0.340533	+	0.940232i	$0.610608\pi$
$948$			192864.i			0.214602i
$949$	−351298.			−0.390071
$950$		−	757428.i		−	0.839255i
$951$	202220.			0.223596
$952$	−82399.5			−0.0909182
$953$			746406.i			0.821843i	0.911671	+	0.410922i	$0.134793\pi$
$953$			746406.i			0.821843i	−0.911671	+	0.410922i	$0.865207\pi$
$954$			10919.3i			0.0119977i
$955$	−2.24094e6			−2.45710
$956$	−336108.			−0.367759
$957$			80391.4i			0.0877780i
$958$			547277.i			0.596316i
$959$	2.03231e6			2.20980
$960$			92010.1i			0.0998373i
$961$	1.87960e6			2.03525
$962$			659664.i			0.712809i
$963$			361605.i			0.389926i
$964$			264414.i			0.284531i
$965$			707102.i			0.759324i
$966$	234645.	−	588684.i	0.251453	−	0.630853i
$967$	110775.			0.118464			0.0592322	−	0.998244i	$-0.481135\pi$
$967$	110775.			0.118464			0.0592322	+	0.998244i	$0.481135\pi$
$968$	−227806.			−0.243117
$969$	−108852.			−0.115929
$970$	92046.9			0.0978286
$971$		−	1.13722e6i		−	1.20616i	−0.797681	−	0.603080i	$-0.793940\pi$
$971$		−	1.13722e6i		−	1.20616i	0.797681	−	0.603080i	$-0.206060\pi$
$972$	30304.0			0.0320750
$973$			1.01250e6i			1.06947i
$974$	−934939.			−0.985520
$975$	940313.			0.989153
$976$			402361.i			0.422392i
$977$		−	343214.i		−	0.359564i	−0.983706	−	0.179782i	$-0.942461\pi$
$977$		−	343214.i		−	0.359564i	0.983706	−	0.179782i	$-0.0575392\pi$
$978$	324058.			0.338801
$979$	−260308.			−0.271595
$980$		−	1.17397e6i		−	1.22237i
$981$			181465.i			0.188562i
$982$	12943.1			0.0134220
$983$		−	1.41908e6i		−	1.46859i	−0.678832	−	0.734293i	$-0.737514\pi$
$983$		−	1.41908e6i		−	1.46859i	0.678832	−	0.734293i	$-0.262486\pi$
$984$	35162.3			0.0363151
$985$		−	435906.i		−	0.449284i
$986$		−	28908.8i		−	0.0297356i
$987$			1.46630e6i			1.50518i
$988$			1.18865e6i			1.21770i
$989$	−524465.	+	1.31579e6i	−0.536197	+	1.34523i
$990$	−178610.			−0.182237
$991$	147325.			0.150013			0.0750065	−	0.997183i	$-0.476102\pi$
$991$	147325.			0.150013			0.0750065	+	0.997183i	$0.476102\pi$
$992$	303072.			0.307980
$993$	−625487.			−0.634337
$994$		−	1.24396e6i		−	1.25903i
$995$	−84070.9			−0.0849180
$996$			533917.i			0.538214i
$997$	−289452.			−0.291196			−0.145598	−	0.989344i	$-0.546511\pi$
$997$	−289452.			−0.291196			−0.145598	+	0.989344i	$0.546511\pi$
$998$	724998.			0.727907
$999$			103263.i			0.103470i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	138.5.b.a.91.8	yes	16
3.2	odd	2		414.5.b.b.91.9		16
4.3	odd	2		1104.5.c.a.1057.8		16
23.22	odd	2	inner	138.5.b.a.91.5	✓	16
69.68	even	2		414.5.b.b.91.16		16
92.91	even	2		1104.5.c.a.1057.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
138.5.b.a.91.5	✓	16	23.22	odd	2	inner
138.5.b.a.91.8	yes	16	1.1	even	1	trivial
414.5.b.b.91.9		16	3.2	odd	2
414.5.b.b.91.16		16	69.68	even	2
1104.5.c.a.1057.1		16	92.91	even	2
1104.5.c.a.1057.8		16	4.3	odd	2

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:	\( N \)	\(=\)	\( 138 = 2 \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	138.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} +34.5847i q^{5} -14.6969 q^{6} +81.5113i q^{7} -22.6274 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{2} +5.19615 q^{3} +8.00000 q^{4} +34.5847i q^{5} -14.6969 q^{6} +81.5113i q^{7} -22.6274 q^{8} +27.0000 q^{9} -97.8202i q^{10} -67.6260i q^{11} +41.5692 q^{12} -316.869 q^{13} -230.549i q^{14} +179.707i q^{15} +64.0000 q^{16} -44.6757i q^{17} -76.3675 q^{18} -468.905i q^{19} +276.677i q^{20} +423.545i q^{21} +191.275i q^{22} +(491.402 + 195.869i) q^{23} -117.576 q^{24} -571.099 q^{25} +896.240 q^{26} +140.296 q^{27} +652.091i q^{28} -228.778 q^{29} -508.289i q^{30} -1674.25 q^{31} -181.019 q^{32} -351.395i q^{33} +126.362i q^{34} -2819.04 q^{35} +216.000 q^{36} +736.035i q^{37} +1326.26i q^{38} -1646.50 q^{39} -782.562i q^{40} -299.061 q^{41} -1197.97i q^{42} +2677.63i q^{43} -541.008i q^{44} +933.786i q^{45} +(-1389.90 - 554.002i) q^{46} +3461.96 q^{47} +332.554 q^{48} -4243.10 q^{49} +1615.31 q^{50} -232.142i q^{51} -2534.95 q^{52} -142.983i q^{53} -396.817 q^{54} +2338.82 q^{55} -1844.39i q^{56} -2436.50i q^{57} +647.081 q^{58} -1697.68 q^{59} +1437.66i q^{60} +6286.88i q^{61} +4735.50 q^{62} +2200.81i q^{63} +512.000 q^{64} -10958.8i q^{65} +993.896i q^{66} -2566.64i q^{67} -357.406i q^{68} +(2553.40 + 1017.77i) q^{69} +7973.46 q^{70} +5395.67 q^{71} -610.940 q^{72} +1108.66 q^{73} -2081.82i q^{74} -2967.52 q^{75} -3751.24i q^{76} +5512.29 q^{77} +4657.00 q^{78} +4639.58i q^{79} +2213.42i q^{80} +729.000 q^{81} +845.873 q^{82} +12844.1i q^{83} +3388.36i q^{84} +1545.09 q^{85} -7573.49i q^{86} -1188.76 q^{87} +1530.20i q^{88} -3849.23i q^{89} -2641.15i q^{90} -25828.4i q^{91} +(3931.22 + 1566.95i) q^{92} -8699.67 q^{93} -9791.91 q^{94} +16216.9 q^{95} -940.604 q^{96} +940.980i q^{97} +12001.3 q^{98} -1825.90i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 128 q^{4} + 432 q^{9}+O(q^{10}) \) 16 * q + 128 * q^4 + 432 * q^9	\( 16 q + 128 q^{4} + 432 q^{9} - 208 q^{13} + 1024 q^{16} + 840 q^{23} + 1056 q^{25} + 1920 q^{26} + 3600 q^{29} + 224 q^{31} - 3264 q^{35} + 3456 q^{36} - 2016 q^{39} - 6144 q^{41} + 1280 q^{46} + 8880 q^{47} - 13888 q^{49} + 7296 q^{50} - 1664 q^{52} + 832 q^{55} + 2944 q^{58} - 18240 q^{59} + 8192 q^{64} + 10584 q^{69} + 19584 q^{70} - 30048 q^{71} + 9536 q^{73} - 4176 q^{75} + 14160 q^{77} + 6912 q^{78} + 11664 q^{81} - 19584 q^{82} - 32496 q^{85} - 8064 q^{87} + 6720 q^{92} - 11952 q^{93} - 21248 q^{94} - 20064 q^{95} + 21504 q^{98}+O(q^{100}) \) 16 * q + 128 * q^4 + 432 * q^9 - 208 * q^13 + 1024 * q^16 + 840 * q^23 + 1056 * q^25 + 1920 * q^26 + 3600 * q^29 + 224 * q^31 - 3264 * q^35 + 3456 * q^36 - 2016 * q^39 - 6144 * q^41 + 1280 * q^46 + 8880 * q^47 - 13888 * q^49 + 7296 * q^50 - 1664 * q^52 + 832 * q^55 + 2944 * q^58 - 18240 * q^59 + 8192 * q^64 + 10584 * q^69 + 19584 * q^70 - 30048 * q^71 + 9536 * q^73 - 4176 * q^75 + 14160 * q^77 + 6912 * q^78 + 11664 * q^81 - 19584 * q^82 - 32496 * q^85 - 8064 * q^87 + 6720 * q^92 - 11952 * q^93 - 21248 * q^94 - 20064 * q^95 + 21504 * q^98

Introduction

L-functions

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