LMFDB - Embedded newform 87.7.b.a.59.19

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [87,7,Mod(59,87)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(87, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 7, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("87.59"); S:= CuspForms(chi, 7); N := Newforms(S);

Level:	$ N $	$=$	$ 87 = 3 \cdot 29 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	87.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$20.0147052749$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			59.19
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.38

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.77572i q^{2} +(6.84673 + 26.1175i) q^{3} +30.6411 q^{4} -163.398i q^{5} +(150.847 - 39.5448i) q^{6} -327.324 q^{7} -546.620i q^{8} +(-635.245 + 357.638i) q^{9} -943.742 q^{10} +366.446i q^{11} +(209.791 + 800.268i) q^{12} -595.662 q^{13} +1890.53i q^{14} +(4267.55 - 1118.74i) q^{15} -1196.10 q^{16} -2829.17i q^{17} +(2065.62 + 3668.99i) q^{18} -11684.7 q^{19} -5006.70i q^{20} +(-2241.10 - 8548.89i) q^{21} +2116.49 q^{22} -5812.35i q^{23} +(14276.3 - 3742.56i) q^{24} -11074.0 q^{25} +3440.38i q^{26} +(-13690.0 - 14142.3i) q^{27} -10029.6 q^{28} -4528.92i q^{29} +(-6461.54 - 24648.2i) q^{30} -4759.27 q^{31} -28075.4i q^{32} +(-9570.65 + 2508.96i) q^{33} -16340.5 q^{34} +53484.2i q^{35} +(-19464.6 + 10958.4i) q^{36} +6824.94 q^{37} +67487.5i q^{38} +(-4078.34 - 15557.2i) q^{39} -89316.8 q^{40} +623.002i q^{41} +(-49376.0 + 12944.0i) q^{42} -99104.4 q^{43} +11228.3i q^{44} +(58437.5 + 103798. i) q^{45} -33570.5 q^{46} +33504.4i q^{47} +(-8189.33 - 31239.0i) q^{48} -10507.7 q^{49} +63960.1i q^{50} +(73890.8 - 19370.6i) q^{51} -18251.7 q^{52} -207466. i q^{53} +(-81682.1 + 79069.3i) q^{54} +59876.6 q^{55} +178922. i q^{56} +(-80001.9 - 305175. i) q^{57} -26157.8 q^{58} -342009. i q^{59} +(130762. - 34279.5i) q^{60} +340446. q^{61} +27488.2i q^{62} +(207931. - 117064. i) q^{63} -238706. q^{64} +97330.1i q^{65} +(14491.0 + 55277.4i) q^{66} -48564.1 q^{67} -86688.8i q^{68} +(151804. - 39795.5i) q^{69} +308910. q^{70} +372179. i q^{71} +(195492. + 347238. i) q^{72} +228755. q^{73} -39418.9i q^{74} +(-75820.4 - 289224. i) q^{75} -358032. q^{76} -119947. i q^{77} +(-89854.0 + 23555.3i) q^{78} +324977. q^{79} +195440. i q^{80} +(275631. - 454376. i) q^{81} +3598.29 q^{82} +280673. i q^{83} +(-68669.7 - 261947. i) q^{84} -462281. q^{85} +572399. i q^{86} +(118284. - 31008.3i) q^{87} +200307. q^{88} +277494. i q^{89} +(599507. - 337518. i) q^{90} +194975. q^{91} -178097. i q^{92} +(-32585.4 - 124300. i) q^{93} +193512. q^{94} +1.90926e6i q^{95} +(733258. - 192224. i) q^{96} +828855. q^{97} +60689.7i q^{98} +(-131055. - 232783. i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) $ 56 * q + 52 * q^3 - 1924 * q^4 - 160 * q^6 + 160 * q^7 - 1060 * q^9 - 3588 * q^10 - 2166 * q^12 - 1400 * q^13 - 6240 * q^15 + 56588 * q^16 - 5978 * q^18 + 25000 * q^19 + 7520 * q^21 + 20970 * q^22 + 1238 * q^24 - 213304 * q^25 + 25492 * q^27 + 7870 * q^28 + 105092 * q^30 - 78656 * q^31 + 18952 * q^33 - 96642 * q^34 - 40224 * q^36 + 52120 * q^37 - 112096 * q^39 + 343440 * q^40 - 341758 * q^42 - 286760 * q^43 + 68312 * q^45 + 255516 * q^46 + 386086 * q^48 + 1051872 * q^49 - 256656 * q^51 + 58606 * q^52 + 450124 * q^54 - 409632 * q^55 - 1141736 * q^57 + 817806 * q^60 + 309976 * q^61 + 567632 * q^63 - 2844730 * q^64 + 958426 * q^66 + 748696 * q^67 - 1323704 * q^69 - 2123748 * q^70 - 1026242 * q^72 - 333728 * q^73 - 331316 * q^75 + 2846728 * q^76 + 292446 * q^78 - 661184 * q^79 + 1614852 * q^81 + 376908 * q^82 - 1670584 * q^84 + 3342000 * q^85 - 927168 * q^88 + 835882 * q^90 + 2248016 * q^91 + 3960776 * q^93 - 1368294 * q^94 - 5811456 * q^96 - 3057296 * q^97 + 4793544 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/87\mathbb{Z}\right)^\times$.

$n$	$31$	$59$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	5.77572i		−	0.721965i	−0.932573	−	0.360982i	$-0.882441\pi$
$2$		−	5.77572i		−	0.721965i	0.932573	−	0.360982i	$-0.117559\pi$
$3$	6.84673	+	26.1175i	0.253582	+	0.967314i
$3$	6.84673	+	26.1175i	0.253582	+	0.967314i
$4$	30.6411			0.478767
$5$		−	163.398i		−	1.30719i	−0.756846	−	0.653593i	$-0.773261\pi$
$5$		−	163.398i		−	1.30719i	0.756846	−	0.653593i	$-0.226739\pi$
$6$	150.847	−	39.5448i	0.698366	−	0.183078i
$7$	−327.324			−0.954299			−0.477149	−	0.878822i	$-0.658330\pi$
$7$	−327.324			−0.954299			−0.477149	+	0.878822i	$0.658330\pi$
$8$		−	546.620i		−	1.06762i
$9$	−635.245	+	357.638i	−0.871392	+	0.490588i
$10$	−943.742			−0.943742
$11$			366.446i			0.275316i	0.990480	+	0.137658i	$0.0439575\pi$
$11$			366.446i			0.275316i	−0.990480	+	0.137658i	$0.956042\pi$
$12$	209.791	+	800.268i	0.121407	+	0.463118i
$13$	−595.662			−0.271125			−0.135563	−	0.990769i	$-0.543284\pi$
$13$	−595.662			−0.271125			−0.135563	+	0.990769i	$0.543284\pi$
$14$			1890.53i			0.688970i
$15$	4267.55	−	1118.74i	1.26446	−	0.331479i
$16$	−1196.10			−0.292015
$17$		−	2829.17i		−	0.575854i	−0.957652	−	0.287927i	$-0.907034\pi$
$17$		−	2829.17i		−	0.575854i	0.957652	−	0.287927i	$-0.0929660\pi$
$18$	2065.62	+	3668.99i	0.354187	+	0.629114i
$19$	−11684.7			−1.70356			−0.851779	−	0.523902i	$-0.824476\pi$
$19$	−11684.7			−1.70356			−0.851779	+	0.523902i	$0.824476\pi$
$20$		−	5006.70i		−	0.625837i
$21$	−2241.10	−	8548.89i	−0.241993	−	0.923106i
$22$	2116.49			0.198769
$23$		−	5812.35i		−	0.477714i	−0.971055	−	0.238857i	$-0.923227\pi$
$23$		−	5812.35i		−	0.477714i	0.971055	−	0.238857i	$-0.0767727\pi$
$24$	14276.3	−	3742.56i	1.03272	−	0.270729i
$25$	−11074.0			−0.708734
$26$			3440.38i			0.195743i
$27$	−13690.0	−	14142.3i	−0.695522	−	0.718505i
$28$	−10029.6			−0.456887
$29$		−	4528.92i		−	0.185695i
$29$		−	4528.92i		−	0.185695i
$30$	−6461.54	−	24648.2i	−0.239316	−	0.912895i
$31$	−4759.27			−0.159755			−0.0798777	−	0.996805i	$-0.525453\pi$
$31$	−4759.27			−0.159755			−0.0798777	+	0.996805i	$0.525453\pi$
$32$		−	28075.4i		−	0.856793i
$33$	−9570.65	+	2508.96i	−0.266317	+	0.0698154i
$34$	−16340.5			−0.415746
$35$			53484.2i			1.24745i
$36$	−19464.6	+	10958.4i	−0.417194	+	0.234877i
$37$	6824.94			0.134739			0.0673695	−	0.997728i	$-0.478539\pi$
$37$	6824.94			0.134739			0.0673695	+	0.997728i	$0.478539\pi$
$38$			67487.5i			1.22991i
$39$	−4078.34	−	15557.2i	−0.0687526	−	0.262263i
$40$	−89316.8			−1.39557
$41$			623.002i			0.00903937i	0.999990	+	0.00451968i	$0.00143867\pi$
$41$			623.002i			0.00903937i	−0.999990	+	0.00451968i	$0.998561\pi$
$42$	−49376.0	+	12944.0i	−0.666450	+	0.174711i
$43$	−99104.4			−1.24649			−0.623244	−	0.782028i	$-0.714185\pi$
$43$	−99104.4			−1.24649			−0.623244	+	0.782028i	$0.714185\pi$
$44$			11228.3i			0.131812i
$45$	58437.5	+	103798.i	0.641289	+	1.13907i
$46$	−33570.5			−0.344893
$47$			33504.4i			0.322707i	0.986897	+	0.161353i	$0.0515859\pi$
$47$			33504.4i			0.322707i	−0.986897	+	0.161353i	$0.948414\pi$
$48$	−8189.33	−	31239.0i	−0.0740500	−	0.282471i
$49$	−10507.7			−0.0893143
$50$			63960.1i			0.511681i
$51$	73890.8	−	19370.6i	0.557031	−	0.146026i
$52$	−18251.7			−0.129806
$53$		−	207466.i		−	1.39354i	−0.717294	−	0.696771i	$-0.754619\pi$
$53$		−	207466.i		−	1.39354i	0.717294	−	0.696771i	$-0.245381\pi$
$54$	−81682.1	+	79069.3i	−0.518735	+	0.502142i
$55$	59876.6			0.359890
$56$			178922.i			1.01883i
$57$	−80001.9	−	305175.i	−0.431992	−	1.64787i
$58$	−26157.8			−0.134065
$59$		−	342009.i		−	1.66526i	−0.553831	−	0.832629i	$-0.686835\pi$
$59$		−	342009.i		−	1.66526i	0.553831	−	0.832629i	$-0.313165\pi$
$60$	130762.	−	34279.5i	0.605381	−	0.158701i
$61$	340446.			1.49989			0.749944	−	0.661501i	$-0.230080\pi$
$61$	340446.			1.49989			0.749944	+	0.661501i	$0.230080\pi$
$62$			27488.2i			0.115338i
$63$	207931.	−	117064.i	0.831568	−	0.468167i
$64$	−238706.			−0.910590
$65$			97330.1i			0.354411i
$66$	14491.0	+	55277.4i	0.0504043	+	0.192272i
$67$	−48564.1			−0.161470			−0.0807349	−	0.996736i	$-0.525727\pi$
$67$	−48564.1			−0.161470			−0.0807349	+	0.996736i	$0.525727\pi$
$68$		−	86688.8i		−	0.275700i
$69$	151804.	−	39795.5i	0.462099	−	0.121140i
$70$	308910.			0.900612
$71$			372179.i			1.03986i	0.854208	+	0.519932i	$0.174043\pi$
$71$			372179.i			1.03986i	−0.854208	+	0.519932i	$0.825957\pi$
$72$	195492.	+	347238.i	0.523760	+	0.930313i
$73$	228755.			0.588034			0.294017	−	0.955800i	$-0.405008\pi$
$73$	228755.			0.588034			0.294017	+	0.955800i	$0.405008\pi$
$74$		−	39418.9i		−	0.0972768i
$75$	−75820.4	−	289224.i	−0.179723	−	0.685568i
$76$	−358032.			−0.815607
$77$		−	119947.i		−	0.262734i
$78$	−89854.0	+	23555.3i	−0.189345	+	0.0496370i
$79$	324977.			0.659130			0.329565	−	0.944133i	$-0.393098\pi$
$79$	324977.			0.659130			0.329565	+	0.944133i	$0.393098\pi$
$80$			195440.i			0.381718i
$81$	275631.	−	454376.i	0.518648	−	0.854988i
$82$	3598.29			0.00652611
$83$			280673.i			0.490869i	0.969413	+	0.245435i	$0.0789307\pi$
$83$			280673.i			0.490869i	−0.969413	+	0.245435i	$0.921069\pi$
$84$	−68669.7	−	261947.i	−0.115858	−	0.441953i
$85$	−462281.			−0.752748
$86$			572399.i			0.899920i
$87$	118284.	−	31008.3i	0.179626	−	0.0470891i
$88$	200307.			0.293933
$89$			277494.i			0.393626i	0.980441	+	0.196813i	$0.0630592\pi$
$89$			277494.i			0.393626i	−0.980441	+	0.196813i	$0.936941\pi$
$90$	599507.	−	337518.i	0.822369	−	0.462988i
$91$	194975.			0.258734
$92$		−	178097.i		−	0.228714i
$93$	−32585.4	−	124300.i	−0.0405112	−	0.154534i
$94$	193512.			0.232983
$95$			1.90926e6i			2.22687i
$96$	733258.	−	192224.i	0.828787	−	0.217268i
$97$	828855.			0.908162			0.454081	−	0.890960i	$-0.349968\pi$
$97$	828855.			0.908162			0.454081	+	0.890960i	$0.349968\pi$
$98$			60689.7i			0.0644817i
$99$	−131055.	−	232783.i	−0.135067	−	0.239908i
$100$	−339318.			−0.339318
$101$			706565.i			0.685785i	0.939375	+	0.342892i	$0.111407\pi$
$101$			706565.i			0.685785i	−0.939375	+	0.342892i	$0.888593\pi$
$102$	−111879.	−	426772.i	−0.105426	−	0.402157i
$103$	1.90844e6			1.74649			0.873247	−	0.487278i	$-0.162010\pi$
$103$	1.90844e6			1.74649			0.873247	+	0.487278i	$0.162010\pi$
$104$			325601.i			0.289458i
$105$	−1.39687e6	+	366192.i	−1.20667	+	0.316330i
$106$	−1.19827e6			−1.00609
$107$			2.14579e6i			1.75161i	0.482668	+	0.875803i	$0.339668\pi$
$107$			2.14579e6i			1.75161i	−0.482668	+	0.875803i	$0.660332\pi$
$108$	−419475.	−	433336.i	−0.332993	−	0.343996i
$109$	−1.19074e6			−0.919467			−0.459733	−	0.888057i	$-0.652055\pi$
$109$	−1.19074e6			−0.919467			−0.459733	+	0.888057i	$0.652055\pi$
$110$		−	345831.i		−	0.259828i
$111$	46728.5	+	178250.i	0.0341675	+	0.130335i
$112$	391511.			0.278670
$113$		−	2.04012e6i		−	1.41390i	−0.707262	−	0.706951i	$-0.750070\pi$
$113$		−	2.04012e6i		−	1.41390i	0.707262	−	0.706951i	$-0.249930\pi$
$114$	−1.76260e6	+	462069.i	−1.18971	+	0.311883i
$115$	−949727.			−0.624461
$116$		−	138771.i		−	0.0889048i
$117$	378391.	−	213032.i	0.236256	−	0.133011i
$118$	−1.97535e6			−1.20226
$119$			926056.i			0.549537i
$120$	−611527.	−	2.33273e6i	−0.353893	−	1.34996i
$121$	1.63728e6			0.924201
$122$		−	1.96632e6i		−	1.08287i
$123$	−16271.2	+	4265.53i	−0.00874391	+	0.00229222i
$124$	−145829.			−0.0764856
$125$		−	743630.i		−	0.380738i
$126$	−676127.	−	1.20095e6i	−0.338000	−	0.600363i
$127$	2.43218e6			1.18737			0.593683	−	0.804699i	$-0.297673\pi$
$127$	2.43218e6			1.18737			0.593683	+	0.804699i	$0.297673\pi$
$128$		−	418128.i		−	0.199379i
$129$	−678541.	−	2.58836e6i	−0.316087	−	1.20574i
$130$	562151.			0.255872
$131$			1.54541e6i			0.687432i	0.939074	+	0.343716i	$0.111686\pi$
$131$			1.54541e6i			0.687432i	−0.939074	+	0.343716i	$0.888314\pi$
$132$	−293255.	+	76877.1i	−0.127504	+	0.0334253i
$133$	3.82469e6			1.62570
$134$			280493.i			0.116576i
$135$	−2.31083e6	+	2.23691e6i	−0.939219	+	0.909176i
$136$	−1.54648e6			−0.614792
$137$		−	1.58473e6i		−	0.616301i	−0.951338	−	0.308150i	$-0.900290\pi$
$137$		−	1.58473e6i		−	0.616301i	0.951338	−	0.308150i	$-0.0997100\pi$
$138$	−229848.	−	876776.i	−0.0874587	−	0.333619i
$139$	−3.19065e6			−1.18805			−0.594025	−	0.804447i	$-0.702462\pi$
$139$	−3.19065e6			−1.18805			−0.594025	+	0.804447i	$0.702462\pi$
$140$			1.63881e6i			0.597235i
$141$	−875050.	+	229395.i	−0.312159	+	0.0818328i
$142$	2.14960e6			0.750745
$143$		−	218278.i		−	0.0746452i
$144$	759813.	−	427769.i	0.254460	−	0.143259i
$145$	−740018.			−0.242738
$146$		−	1.32123e6i		−	0.424540i
$147$	−71943.6	−	274435.i	−0.0226485	−	0.0863949i
$148$	209123.			0.0645086
$149$		−	4.59890e6i		−	1.39026i	−0.718886	−	0.695128i	$-0.755348\pi$
$149$		−	4.59890e6i		−	1.39026i	0.718886	−	0.695128i	$-0.244652\pi$
$150$	−1.67048e6	+	437918.i	−0.494956	+	0.129753i
$151$	−3.62814e6			−1.05379			−0.526894	−	0.849931i	$-0.676644\pi$
$151$	−3.62814e6			−1.05379			−0.526894	+	0.849931i	$0.676644\pi$
$152$			6.38709e6i			1.81875i
$153$	1.01182e6	+	1.79722e6i	0.282507	+	0.501794i
$154$	−692779.			−0.189685
$155$			777657.i			0.208830i
$156$	−124965.	−	476689.i	−0.0329165	−	0.125563i
$157$	695382.			0.179690			0.0898451	−	0.995956i	$-0.471363\pi$
$157$	695382.			0.179690			0.0898451	+	0.995956i	$0.471363\pi$
$158$		−	1.87697e6i		−	0.475869i
$159$	5.41850e6	−	1.42046e6i	1.34799	−	0.353378i
$160$	−4.58747e6			−1.11999
$161$			1.90252e6i			0.455882i
$162$	−2.62435e6	−	1.59197e6i	−0.617271	−	0.374445i
$163$	−7.20308e6			−1.66324			−0.831621	−	0.555343i	$-0.812587\pi$
$163$	−7.20308e6			−1.66324			−0.831621	+	0.555343i	$0.812587\pi$
$164$			19089.5i			0.00432775i
$165$	409959.	+	1.56383e6i	0.0912617	+	0.348126i
$166$	1.62109e6			0.354390
$167$			2.08186e6i			0.446994i	0.974705	+	0.223497i	$0.0717473\pi$
$167$			2.08186e6i			0.446994i	−0.974705	+	0.223497i	$0.928253\pi$
$168$	−4.67299e6	+	1.22503e6i	−0.985524	+	0.258356i
$169$	−4.47200e6			−0.926491
$170$			2.67001e6i			0.543457i
$171$	7.42264e6	−	4.17890e6i	1.48447	−	0.835744i
$172$	−3.03667e6			−0.596777
$173$			5.57736e6i			1.07719i	0.842566	+	0.538593i	$0.181044\pi$
$173$			5.57736e6i			1.07719i	−0.842566	+	0.538593i	$0.818956\pi$
$174$	−179095.	−	683175.i	−0.0339967	−	0.129683i
$175$	3.62478e6			0.676344
$176$		−	438304.i		−	0.0803966i
$177$	8.93241e6	−	2.34164e6i	1.61083	−	0.422280i
$178$	1.60273e6			0.284184
$179$			2.87802e6i			0.501804i	0.968012	+	0.250902i	$0.0807272\pi$
$179$			2.87802e6i			0.501804i	−0.968012	+	0.250902i	$0.919273\pi$
$180$	1.79059e6	+	3.18048e6i	0.307028	+	0.545349i
$181$	7.68611e6			1.29620			0.648099	−	0.761556i	$-0.275564\pi$
$181$	7.68611e6			1.29620			0.648099	+	0.761556i	$0.275564\pi$
$182$		−	1.12612e6i		−	0.186797i
$183$	2.33094e6	+	8.89159e6i	0.380345	+	1.45086i
$184$	−3.17715e6			−0.510016
$185$		−	1.11518e6i		−	0.176129i
$186$	−717923.	+	188204.i	−0.111568	+	0.0292476i
$187$	1.03674e6			0.158542
$188$			1.02661e6i			0.154501i
$189$	4.48106e6	+	4.62913e6i	0.663735	+	0.685668i
$190$	1.10273e7			1.60772
$191$		−	9.31435e6i		−	1.33676i	−0.743821	−	0.668379i	$-0.766989\pi$
$191$		−	9.31435e6i		−	1.33676i	0.743821	−	0.668379i	$-0.233011\pi$
$192$	−1.63435e6	−	6.23439e6i	−0.230910	−	0.880826i
$193$	8.21586e6			1.14283			0.571415	−	0.820662i	$-0.306395\pi$
$193$	8.21586e6			1.14283			0.571415	+	0.820662i	$0.306395\pi$
$194$		−	4.78723e6i		−	0.655661i
$195$	−2.54202e6	+	666393.i	−0.342827	+	0.0898724i
$196$	−321968.			−0.0427607
$197$		−	1.37662e7i		−	1.80059i	−0.435284	−	0.900293i	$-0.643352\pi$
$197$		−	1.37662e7i		−	1.80059i	0.435284	−	0.900293i	$-0.356648\pi$
$198$	−1.34449e6	+	756938.i	−0.173205	+	0.0975135i
$199$	−1.16409e7			−1.47716			−0.738581	−	0.674165i	$-0.764504\pi$
$199$	−1.16409e7			−1.47716			−0.738581	+	0.674165i	$0.764504\pi$
$200$			6.05326e6i			0.756657i
$201$	−332505.	−	1.26837e6i	−0.0409459	−	0.156192i
$202$	4.08092e6			0.495112
$203$			1.48243e6i			0.177209i
$204$	2.26409e6	−	593535.i	0.266688	−	0.0699126i
$205$	101797.			0.0118161
$206$		−	1.10226e7i		−	1.26091i
$207$	2.07872e6	+	3.69226e6i	0.234361	+	0.416276i
$208$	712469.			0.0791728
$209$		−	4.28181e6i		−	0.469017i
$210$	2.11502e6	+	8.06794e6i	0.228379	+	0.871174i
$211$	1.24464e7			1.32494			0.662470	−	0.749088i	$-0.269508\pi$
$211$	1.24464e7			1.32494			0.662470	+	0.749088i	$0.269508\pi$
$212$		−	6.35699e6i		−	0.667182i
$213$	−9.72037e6	+	2.54821e6i	−1.00587	+	0.263691i
$214$	1.23935e7			1.26460
$215$			1.61935e7i			1.62939i
$216$	−7.73049e6	+	7.48321e6i	−0.767089	+	0.742551i
$217$	1.55783e6			0.152454
$218$			6.87735e6i			0.663822i
$219$	1.56622e6	+	5.97451e6i	0.149115	+	0.568813i
$220$	1.83468e6			0.172303
$221$			1.68523e6i			0.156129i
$222$	1.02952e6	−	269890.i	0.0940972	−	0.0246677i
$223$	−1.10197e7			−0.993697			−0.496848	−	0.867837i	$-0.665509\pi$
$223$	−1.10197e7			−0.993697			−0.496848	+	0.867837i	$0.665509\pi$
$224$			9.18976e6i			0.817636i
$225$	7.03468e6	−	3.96048e6i	0.617585	−	0.347696i
$226$	−1.17831e7			−1.02079
$227$		−	6.87026e6i		−	0.587348i	−0.955906	−	0.293674i	$-0.905122\pi$
$227$		−	6.87026e6i		−	0.587348i	0.955906	−	0.293674i	$-0.0948780\pi$
$228$	−2.45135e6	−	9.35088e6i	−0.206824	−	0.788948i
$229$	−1.51626e6			−0.126260			−0.0631301	−	0.998005i	$-0.520108\pi$
$229$	−1.51626e6			−0.126260			−0.0631301	+	0.998005i	$0.520108\pi$
$230$			5.48536e6i			0.450839i
$231$	3.13271e6	−	821242.i	0.254146	−	0.0666247i
$232$	−2.47560e6			−0.198252
$233$			6.67736e6i			0.527883i	0.964539	+	0.263941i	$0.0850225\pi$
$233$			6.67736e6i			0.527883i	−0.964539	+	0.263941i	$0.914977\pi$
$234$	−1.23041e6	−	2.18548e6i	−0.0960290	−	0.170569i
$235$	5.47456e6			0.421838
$236$		−	1.04795e7i		−	0.797271i
$237$	2.22503e6	+	8.48757e6i	0.167144	+	0.637586i
$238$	5.34864e6			0.396746
$239$		−	1.20075e6i		−	0.0879545i	−0.999033	−	0.0439772i	$-0.985997\pi$
$239$		−	1.20075e6i		−	0.0879545i	0.999033	−	0.0439772i	$-0.0140029\pi$
$240$	−5.10439e6	+	1.33812e6i	−0.369241	+	0.0967971i
$241$	−2.04992e7			−1.46449			−0.732244	−	0.681042i	$-0.761527\pi$
$241$	−2.04992e7			−1.46449			−0.732244	+	0.681042i	$0.761527\pi$
$242$		−	9.45646e6i		−	0.667240i
$243$	1.37543e7	+	4.08779e6i	0.958562	+	0.284885i
$244$	1.04316e7			0.718097
$245$			1.71694e6i			0.116750i
$246$	24636.5	+	93978.1i	0.00165491	+	0.00631279i
$247$	6.96013e6			0.461877
$248$			2.60151e6i			0.170558i
$249$	−7.33046e6	+	1.92169e6i	−0.474825	+	0.124476i
$250$	−4.29500e6			−0.274880
$251$		−	2.68741e7i		−	1.69947i	−0.527213	−	0.849733i	$-0.676763\pi$
$251$		−	2.68741e7i		−	1.69947i	0.527213	−	0.849733i	$-0.323237\pi$
$252$	6.37123e6	−	3.58696e6i	0.398127	−	0.224143i
$253$	2.12991e6			0.131523
$254$		−	1.40476e7i		−	0.857236i
$255$	−3.16511e6	−	1.20736e7i	−0.190884	−	0.728143i
$256$	−1.76921e7			−1.05453
$257$		−	1.20305e7i		−	0.708738i	−0.935106	−	0.354369i	$-0.884696\pi$
$257$		−	1.20305e7i		−	0.708738i	0.935106	−	0.354369i	$-0.115304\pi$
$258$	−1.49496e7	+	3.91906e6i	−0.870505	+	0.228204i
$259$	−2.23397e6			−0.128581
$260$			2.98230e6i			0.169680i
$261$	1.61972e6	+	2.87697e6i	0.0910998	+	0.161813i
$262$	8.92585e6			0.496302
$263$		−	6.87462e6i		−	0.377904i	−0.981986	−	0.188952i	$-0.939491\pi$
$263$		−	6.87462e6i		−	0.377904i	0.981986	−	0.188952i	$-0.0605091\pi$
$264$	1.37145e6	+	5.23151e6i	0.0745361	+	0.284325i
$265$	−3.38996e7			−1.82162
$266$		−	2.20903e7i		−	1.17370i
$267$	−7.24745e6	+	1.89993e6i	−0.380760	+	0.0998166i
$268$	−1.48806e6			−0.0773064
$269$			1.83681e6i			0.0943644i	0.998886	+	0.0471822i	$0.0150241\pi$
$269$			1.83681e6i			0.0943644i	−0.998886	+	0.0471822i	$0.984976\pi$
$270$	1.29198e7	+	1.33467e7i	0.656393	+	0.678083i
$271$	−2.91582e7			−1.46505			−0.732527	−	0.680738i	$-0.761659\pi$
$271$	−2.91582e7			−1.46505			−0.732527	+	0.680738i	$0.761659\pi$
$272$			3.38396e6i			0.168158i
$273$	1.33494e6	+	5.09225e6i	0.0656105	+	0.250277i
$274$	−9.15293e6			−0.444947
$275$		−	4.05801e6i		−	0.195126i
$276$	4.65143e6	−	1.21938e6i	0.221238	−	0.0579978i
$277$	−3.19877e6			−0.150503			−0.0752513	−	0.997165i	$-0.523976\pi$
$277$	−3.19877e6			−0.150503			−0.0752513	+	0.997165i	$0.523976\pi$
$278$			1.84283e7i			0.857730i
$279$	3.02330e6	−	1.70210e6i	0.139210	−	0.0783740i
$280$	2.92356e7			1.33179
$281$		−	1.39225e7i		−	0.627475i	−0.949510	−	0.313738i	$-0.898419\pi$
$281$		−	1.39225e7i		−	0.627475i	0.949510	−	0.313738i	$-0.101581\pi$
$282$	1.32492e6	+	5.05404e6i	0.0590804	+	0.225368i
$283$	2.62950e7			1.16015			0.580075	−	0.814563i	$-0.303023\pi$
$283$	2.62950e7			1.16015			0.580075	+	0.814563i	$0.303023\pi$
$284$			1.14040e7i			0.497852i
$285$	−4.98650e7	+	1.30722e7i	−2.15408	+	0.564694i
$286$	−1.26071e6			−0.0538912
$287$		−	203924.i		−	0.00862626i
$288$	1.00408e7	+	1.78347e7i	0.420332	+	0.746602i
$289$	1.61334e7			0.668392
$290$			4.27414e6i			0.175248i
$291$	5.67494e6	+	2.16476e7i	0.230294	+	0.878477i
$292$	7.00931e6			0.281531
$293$			2.99252e7i			1.18969i	0.803840	+	0.594846i	$0.202787\pi$
$293$			2.99252e7i			1.18969i	−0.803840	+	0.594846i	$0.797213\pi$
$294$	−1.58506e6	+	415526.i	−0.0623741	+	0.0163514i
$295$	−5.58837e7			−2.17680
$296$		−	3.73065e6i		−	0.143850i
$297$	5.18240e6	−	5.01663e6i	0.197816	−	0.191489i
$298$	−2.65619e7			−1.00372
$299$			3.46220e6i			0.129520i
$300$	−2.32322e6	−	8.86214e6i	−0.0860452	−	0.328227i
$301$	3.24393e7			1.18952
$302$			2.09551e7i			0.760798i
$303$	−1.84537e7	+	4.83765e6i	−0.663369	+	0.173903i
$304$	1.39760e7			0.497465
$305$		−	5.56283e7i		−	1.96063i
$306$	1.03802e7	−	5.84399e6i	0.362278	−	0.203960i
$307$	−7.60733e6			−0.262916			−0.131458	−	0.991322i	$-0.541966\pi$
$307$	−7.60733e6			−0.262916			−0.131458	+	0.991322i	$0.541966\pi$
$308$		−	3.67530e6i		−	0.125788i
$309$	1.30666e7	+	4.98437e7i	0.442880	+	1.68941i
$310$	4.49153e6			0.150768
$311$		−	5.56350e7i		−	1.84956i	−0.380508	−	0.924778i	$-0.624251\pi$
$311$		−	5.56350e7i		−	1.84956i	0.380508	−	0.924778i	$-0.375749\pi$
$312$	−8.50388e6	+	2.22930e6i	−0.279997	+	0.0734015i
$313$	−1.58519e7			−0.516950			−0.258475	−	0.966018i	$-0.583220\pi$
$313$	−1.58519e7			−0.516950			−0.258475	+	0.966018i	$0.583220\pi$
$314$		−	4.01633e6i		−	0.129730i
$315$	−1.91280e7	−	3.39756e7i	−0.611981	−	1.08701i
$316$	9.95764e6			0.315570
$317$		−	2.67508e7i		−	0.839768i	−0.907578	−	0.419884i	$-0.862071\pi$
$317$		−	2.67508e7i		−	0.839768i	0.907578	−	0.419884i	$-0.137929\pi$
$318$	−8.20420e6	−	3.12957e7i	−0.255126	−	0.973203i
$319$	1.65961e6			0.0511250
$320$			3.90041e7i			1.19031i
$321$	−5.60427e7	+	1.46917e7i	−1.69435	+	0.444177i
$322$	1.09884e7			0.329131
$323$			3.30580e7i			0.981000i
$324$	8.44562e6	−	1.39226e7i	0.248311	−	0.409340i
$325$	6.59635e6			0.192156
$326$			4.16030e7i			1.20080i
$327$	−8.15264e6	−	3.10990e7i	−0.233161	−	0.889413i
$328$	340546.			0.00965059
$329$		−	1.09668e7i		−	0.307959i
$330$	9.03222e6	−	2.36781e6i	0.251335	−	0.0658877i
$331$	6.44078e7			1.77605			0.888023	−	0.459799i	$-0.152079\pi$
$331$	6.44078e7			1.77605			0.888023	+	0.459799i	$0.152079\pi$
$332$			8.60011e6i			0.235012i
$333$	−4.33550e6	+	2.44086e6i	−0.117411	+	0.0661013i
$334$	1.20242e7			0.322714
$335$			7.93529e6i			0.211071i
$336$	2.68057e6	+	1.02253e7i	0.0706658	+	0.269561i
$337$	−2.78171e7			−0.726812			−0.363406	−	0.931631i	$-0.618386\pi$
$337$	−2.78171e7			−0.726812			−0.363406	+	0.931631i	$0.618386\pi$
$338$			2.58290e7i			0.668894i
$339$	5.32827e7	−	1.39681e7i	1.36769	−	0.358541i
$340$	−1.41648e7			−0.360391
$341$		−	1.74402e6i		−	0.0439833i
$342$	−2.41361e7	−	4.28711e7i	−0.603378	−	1.07173i
$343$	4.19488e7			1.03953
$344$			5.41725e7i			1.33077i
$345$	−6.50252e6	−	2.48045e7i	−0.158352	−	0.604050i
$346$	3.22133e7			0.777690
$347$			1.66864e7i			0.399368i	0.979860	+	0.199684i	$0.0639916\pi$
$347$			1.66864e7i			0.399368i	−0.979860	+	0.199684i	$0.936008\pi$
$348$	3.62435e6	−	950128.i	0.0859988	−	0.0225447i
$349$	−3.23851e7			−0.761848			−0.380924	−	0.924606i	$-0.624394\pi$
$349$	−3.23851e7			−0.761848			−0.380924	+	0.924606i	$0.624394\pi$
$350$		−	2.09357e7i		−	0.488297i
$351$	8.15459e6	+	8.42405e6i	0.188574	+	0.194805i
$352$	1.02881e7			0.235889
$353$		−	2.00305e7i		−	0.455373i	−0.973734	−	0.227687i	$-0.926884\pi$
$353$		−	2.00305e7i		−	0.455373i	0.973734	−	0.227687i	$-0.0731161\pi$
$354$	−1.35247e7	−	5.15911e7i	−0.304871	−	1.16296i
$355$	6.08133e7			1.35930
$356$			8.50272e6i			0.188455i
$357$	−2.41863e7	+	6.34045e6i	−0.531574	+	0.139353i
$358$	1.66226e7			0.362285
$359$			3.49792e7i			0.756008i	0.925804	+	0.378004i	$0.123389\pi$
$359$			3.49792e7i			0.756008i	−0.925804	+	0.378004i	$0.876611\pi$
$360$	5.67380e7	−	3.19431e7i	1.21609	−	0.684651i
$361$	8.94863e7			1.90211
$362$		−	4.43928e7i		−	0.935809i
$363$	1.12100e7	+	4.27616e7i	0.234361	+	0.893992i
$364$	5.97424e6			0.123873
$365$		−	3.73782e7i		−	0.768670i
$366$	5.13553e7	−	1.34629e7i	1.04747	−	0.274596i
$367$	1.64558e7			0.332906			0.166453	−	0.986049i	$-0.446769\pi$
$367$	1.64558e7			0.332906			0.166453	+	0.986049i	$0.446769\pi$
$368$			6.95212e6i			0.139500i
$369$	−222809.	−	395759.i	−0.00443460	−	0.00787683i
$370$	−6.44098e6			−0.127159
$371$			6.79088e7i			1.32985i
$372$	−998453.	−	3.80869e6i	−0.0193954	−	0.0739856i
$373$	−9.44240e7			−1.81952			−0.909758	−	0.415140i	$-0.863733\pi$
$373$	−9.44240e7			−1.81952			−0.909758	+	0.415140i	$0.863733\pi$
$374$		−	5.98791e6i		−	0.114462i
$375$	1.94217e7	−	5.09143e6i	0.368294	−	0.0965486i
$376$	1.83142e7			0.344528
$377$			2.69771e6i			0.0503467i
$378$	2.67366e7	−	2.58813e7i	0.495028	−	0.479194i
$379$	−2.86638e7			−0.526521			−0.263261	−	0.964725i	$-0.584798\pi$
$379$	−2.86638e7			−0.526521			−0.263261	+	0.964725i	$0.584798\pi$
$380$			5.85017e7i			1.06615i
$381$	1.66525e7	+	6.35224e7i	0.301095	+	1.14855i
$382$	−5.37971e7			−0.965092
$383$			1.57910e7i			0.281070i	0.990076	+	0.140535i	$0.0448823\pi$
$383$			1.57910e7i			0.281070i	−0.990076	+	0.140535i	$0.955118\pi$
$384$	1.09205e7	−	2.86281e6i	0.192862	−	0.0505590i
$385$	−1.95991e7			−0.343442
$386$		−	4.74525e7i		−	0.825082i
$387$	6.29556e7	−	3.54435e7i	1.08618	−	0.611511i
$388$	2.53970e7			0.434798
$389$			2.24187e7i			0.380857i	0.981701	+	0.190428i	$0.0609877\pi$
$389$			2.24187e7i			0.380857i	−0.981701	+	0.190428i	$0.939012\pi$
$390$	3.84890e6	+	1.46820e7i	0.0648847	+	0.247509i
$391$	−1.64441e7			−0.275094
$392$			5.74374e6i			0.0953535i
$393$	−4.03622e7	+	1.05810e7i	−0.664962	+	0.174321i
$394$	−7.95094e7			−1.29996
$395$		−	5.31006e7i		−	0.861605i
$396$	−4.01567e6	−	7.13272e6i	−0.0646655	−	0.114860i
$397$	6.43548e7			1.02851			0.514256	−	0.857637i	$-0.328068\pi$
$397$	6.43548e7			1.02851			0.514256	+	0.857637i	$0.328068\pi$
$398$			6.72347e7i			1.06646i
$399$	2.61866e7	+	9.98912e7i	0.412249	+	1.57256i
$400$	1.32455e7			0.206961
$401$		−	1.12518e8i		−	1.74497i	−0.488637	−	0.872487i	$-0.662506\pi$
$401$		−	1.12518e8i		−	1.74497i	0.488637	−	0.872487i	$-0.337494\pi$
$402$	−7.32576e6	+	1.92046e6i	−0.112765	+	0.0295615i
$403$	2.83492e6			0.0433137
$404$			2.16499e7i			0.328331i
$405$	−7.42442e7	−	4.50376e7i	−1.11763	−	0.677969i
$406$	8.56208e6			0.127939
$407$			2.50097e6i			0.0370959i
$408$	−1.05883e7	−	4.03902e7i	−0.155900	−	0.594697i
$409$	−8.11748e7			−1.18646			−0.593228	−	0.805035i	$-0.702147\pi$
$409$	−8.11748e7			−1.18646			−0.593228	+	0.805035i	$0.702147\pi$
$410$		−	587953.i		−	0.00853083i
$411$	4.13891e7	−	1.08502e7i	0.596156	−	0.156283i
$412$	5.84767e7			0.836164
$413$			1.11948e8i			1.58915i
$414$	2.13255e7	−	1.20061e7i	0.300537	−	0.169200i
$415$	4.58614e7			0.641657
$416$			1.67234e7i			0.232298i
$417$	−2.18455e7	−	8.33317e7i	−0.301269	−	1.14922i
$418$	−2.47305e7			−0.338614
$419$			4.67954e7i			0.636152i	0.948065	+	0.318076i	$0.103037\pi$
$419$			4.67954e7i			0.636152i	−0.948065	+	0.318076i	$0.896963\pi$
$420$	−4.28017e7	+	1.12205e7i	−0.577714	+	0.151448i
$421$	−1.09749e8			−1.47080			−0.735400	−	0.677633i	$-0.763006\pi$
$421$	−1.09749e8			−1.47080			−0.735400	+	0.677633i	$0.763006\pi$
$422$		−	7.18869e7i		−	0.956560i
$423$	−1.19825e7	−	2.12835e7i	−0.158316	−	0.281204i
$424$	−1.13405e8			−1.48777
$425$			3.13302e7i			0.408127i
$426$	1.47177e7	+	5.61421e7i	0.190376	+	0.726206i
$427$	−1.11436e8			−1.43134
$428$			6.57494e7i			0.838611i
$429$	5.70087e6	−	1.49449e6i	0.0722054	−	0.0189287i
$430$	9.35290e7			1.17636
$431$			7.63616e7i			0.953768i	0.878966	+	0.476884i	$0.158234\pi$
$431$			7.63616e7i			0.953768i	−0.878966	+	0.476884i	$0.841766\pi$
$432$	1.63745e7	+	1.69156e7i	0.203103	+	0.209815i
$433$	7.94153e6			0.0978229			0.0489114	−	0.998803i	$-0.484425\pi$
$433$	7.94153e6			0.0978229			0.0489114	+	0.998803i	$0.484425\pi$
$434$		−	8.99757e6i		−	0.110067i
$435$	−5.06670e6	−	1.93274e7i	−0.0615542	−	0.234804i
$436$	−3.64854e7			−0.440210
$437$			6.79155e7i			0.813813i
$438$	3.45071e7	−	9.04607e6i	0.410663	−	0.107656i
$439$	−1.25964e8			−1.48886			−0.744431	−	0.667700i	$-0.767279\pi$
$439$	−1.25964e8			−1.48886			−0.744431	+	0.667700i	$0.767279\pi$
$440$		−	3.27298e7i		−	0.384224i
$441$	6.67498e6	−	3.75797e6i	0.0778277	−	0.0438165i
$442$	9.73341e6			0.112719
$443$		−	9.95345e7i		−	1.14489i	−0.819944	−	0.572443i	$-0.805996\pi$
$443$		−	9.95345e7i		−	1.14489i	0.819944	−	0.572443i	$-0.194004\pi$
$444$	1.43181e6	+	5.46178e6i	0.0163582	+	0.0624000i
$445$	4.53420e7			0.514542
$446$			6.36465e7i			0.717414i
$447$	1.20112e8	−	3.14874e7i	1.34481	−	0.352544i
$448$	7.81342e7			0.868974
$449$			1.26558e8i			1.39813i	0.715056	+	0.699067i	$0.246401\pi$
$449$			1.26558e8i			1.39813i	−0.715056	+	0.699067i	$0.753599\pi$
$450$	−2.28746e7	−	4.06303e7i	−0.251024	−	0.445875i
$451$	−228297.			−0.00248869
$452$		−	6.25113e7i		−	0.676930i
$453$	−2.48409e7	−	9.47579e7i	−0.267222	−	1.01934i
$454$	−3.96807e7			−0.424045
$455$		−	3.18585e7i		−	0.338214i
$456$	−1.66815e8	+	4.37307e7i	−1.75930	+	0.461202i
$457$	7.15421e7			0.749571			0.374786	−	0.927112i	$-0.377716\pi$
$457$	7.15421e7			0.749571			0.374786	+	0.927112i	$0.377716\pi$
$458$			8.75747e6i			0.0911554i
$459$	−4.00111e7	+	3.87312e7i	−0.413754	+	0.400519i
$460$	−2.91007e7			−0.298971
$461$		−	3.62696e7i		−	0.370203i	−0.982719	−	0.185102i	$-0.940739\pi$
$461$		−	3.62696e7i		−	0.370203i	0.982719	−	0.185102i	$-0.0592614\pi$
$462$	−4.74326e6	−	1.80936e7i	−0.0481007	−	0.183485i
$463$	−1.30652e8			−1.31636			−0.658178	−	0.752862i	$-0.728673\pi$
$463$	−1.30652e8			−1.31636			−0.658178	+	0.752862i	$0.728673\pi$
$464$			5.41702e6i			0.0542259i
$465$	−2.03104e7	+	5.32440e6i	−0.202004	+	0.0529556i
$466$	3.85666e7			0.381113
$467$		−	3.88989e7i		−	0.381933i	−0.981597	−	0.190966i	$-0.938838\pi$
$467$		−	3.88989e7i		−	0.381933i	0.981597	−	0.190966i	$-0.0611621\pi$
$468$	1.15943e7	−	6.52752e6i	0.113112	−	0.0636811i
$469$	1.58962e7			0.154090
$470$		−	3.16195e7i		−	0.304552i
$471$	4.76109e6	+	1.81616e7i	0.0455663	+	0.173817i
$472$	−1.86949e8			−1.77786
$473$		−	3.63164e7i		−	0.343178i
$474$	4.90218e7	−	1.28511e7i	0.460314	−	0.120672i
$475$	1.29396e8			1.20737
$476$			2.83754e7i			0.263100i
$477$	7.41979e7	+	1.31792e8i	0.683654	+	1.21432i
$478$	−6.93518e6			−0.0635000
$479$		−	1.23602e8i		−	1.12465i	−0.826916	−	0.562325i	$-0.809907\pi$
$479$		−	1.23602e8i		−	1.12465i	0.826916	−	0.562325i	$-0.190093\pi$
$480$	−3.14091e7	−	1.19813e8i	−0.284009	−	1.08338i
$481$	−4.06536e6			−0.0365312
$482$			1.18398e8i			1.05731i
$483$	−4.96891e7	+	1.30261e7i	−0.440981	+	0.115604i
$484$	5.01680e7			0.442477
$485$		−	1.35433e8i		−	1.18714i
$486$	2.36099e7	−	7.94410e7i	0.205677	−	0.692048i
$487$	2.15452e8			1.86537			0.932683	−	0.360696i	$-0.117461\pi$
$487$	2.15452e8			1.86537			0.932683	+	0.360696i	$0.117461\pi$
$488$		−	1.86095e8i		−	1.60131i
$489$	−4.93175e7	−	1.88126e8i	−0.421769	−	1.60888i
$490$	9.91659e6			0.0842896
$491$			4.93099e6i			0.0416572i	0.999783	+	0.0208286i	$0.00663042\pi$
$491$			4.93099e6i			0.0416572i	−0.999783	+	0.0208286i	$0.993370\pi$
$492$	−498569.	+	130700.i	−0.00418629	+	0.00109744i
$493$	−1.28131e7			−0.106933
$494$		−	4.01998e7i		−	0.333459i
$495$	−3.80363e7	+	2.14142e7i	−0.313605	+	0.176557i
$496$	5.69254e6			0.0466510
$497$		−	1.21823e8i		−	0.992341i
$498$	1.10991e7	+	4.23387e7i	0.0898671	+	0.342807i
$499$	−1.22156e8			−0.983139			−0.491570	−	0.870838i	$-0.663577\pi$
$499$	−1.22156e8			−0.983139			−0.491570	+	0.870838i	$0.663577\pi$
$500$		−	2.27856e7i		−	0.182285i
$501$	−5.43729e7	+	1.42539e7i	−0.432384	+	0.113350i
$502$	−1.55217e8			−1.22696
$503$			1.55994e8i			1.22576i	0.790177	+	0.612879i	$0.209989\pi$
$503$			1.55994e8i			1.22576i	−0.790177	+	0.612879i	$0.790011\pi$
$504$	−6.39894e7	−	1.13659e8i	−0.499823	−	0.887797i
$505$	1.15451e8			0.896448
$506$		−	1.23018e7i		−	0.0949546i
$507$	−3.06185e7	−	1.16797e8i	−0.234942	−	0.896208i
$508$	7.45246e7			0.568471
$509$			1.31192e8i			0.994840i	0.867510	+	0.497420i	$0.165719\pi$
$509$			1.31192e8i			0.994840i	−0.867510	+	0.497420i	$0.834281\pi$
$510$	−6.97338e7	+	1.82808e7i	−0.525694	+	0.137811i
$511$	−7.48772e7			−0.561160
$512$			7.54247e7i			0.561957i
$513$	1.59963e8	+	1.65249e8i	1.18486	+	1.22401i
$514$	−6.94850e7			−0.511684
$515$		−	3.11836e8i		−	2.28299i
$516$	−2.07912e7	−	7.93101e7i	−0.151332	−	0.577270i
$517$	−1.22776e7			−0.0888465
$518$			1.29028e7i			0.0928311i
$519$	−1.45667e8	+	3.81867e7i	−1.04198	+	0.273155i
$520$	5.32026e7			0.378375
$521$			1.67968e8i			1.18772i	0.804568	+	0.593860i	$0.202397\pi$
$521$			1.67968e8i			1.18772i	−0.804568	+	0.593860i	$0.797603\pi$
$522$	1.66166e7	−	9.35503e6i	0.116824	−	0.0657709i
$523$	2.03638e8			1.42349			0.711745	−	0.702438i	$-0.247905\pi$
$523$	2.03638e8			1.42349			0.711745	+	0.702438i	$0.247905\pi$
$524$			4.73530e7i			0.329120i
$525$	2.48179e7	+	9.46701e7i	0.171509	+	0.654237i
$526$	−3.97059e7			−0.272833
$527$			1.34648e7i			0.0919958i
$528$	1.14474e7	−	3.00095e6i	0.0777688	−	0.0203872i
$529$	1.14253e8			0.771789
$530$			1.95795e8i			1.31514i
$531$	1.22316e8	+	2.17259e8i	0.816955	+	1.45109i
$532$	1.17193e8			0.778332
$533$		−	371099.i		−	0.00245080i
$534$	1.09734e7	+	4.18592e7i	0.0720641	+	0.274895i
$535$	3.50619e8			2.28967
$536$			2.65461e7i			0.172388i
$537$	−7.51666e7	+	1.97050e7i	−0.485402	+	0.127249i
$538$	1.06089e7			0.0681278
$539$		−	3.85052e6i		−	0.0245897i
$540$	−7.08064e7	+	6.85415e7i	−0.449667	+	0.435283i
$541$	−1.46567e8			−0.925644			−0.462822	−	0.886451i	$-0.653163\pi$
$541$	−1.46567e8			−0.925644			−0.462822	+	0.886451i	$0.653163\pi$
$542$			1.68410e8i			1.05772i
$543$	5.26247e7	+	2.00742e8i	0.328693	+	1.25383i
$544$	−7.94300e7			−0.493387
$545$			1.94564e8i			1.20191i
$546$	2.94114e7	−	7.71023e6i	0.180691	−	0.0473685i
$547$	1.23166e8			0.752541			0.376271	−	0.926510i	$-0.377206\pi$
$547$	1.23166e8			0.752541			0.376271	+	0.926510i	$0.377206\pi$
$548$		−	4.85577e7i		−	0.295064i
$549$	−2.16267e8	+	1.21757e8i	−1.30699	+	0.735827i
$550$	−2.34379e7			−0.140874
$551$			5.29191e7i			0.316343i
$552$	−2.17530e7	−	8.29790e7i	−0.129331	−	0.493345i
$553$	−1.06373e8			−0.629007
$554$			1.84752e7i			0.108658i
$555$	2.91257e7	−	7.63535e6i	0.170372	−	0.0446632i
$556$	−9.77649e7			−0.568799
$557$			3.73880e7i			0.216355i	0.994132	+	0.108177i	$0.0345014\pi$
$557$			3.73880e7i			0.216355i	−0.994132	+	0.108177i	$0.965499\pi$
$558$	−9.83084e6	−	1.74617e7i	−0.0565833	−	0.100504i
$559$	5.90328e7			0.337954
$560$		−	6.39722e7i		−	0.364273i
$561$	7.09826e6	+	2.70770e7i	0.0402035	+	0.153360i
$562$	−8.04122e7			−0.453015
$563$		−	1.05723e8i		−	0.592439i	−0.955120	−	0.296220i	$-0.904274\pi$
$563$		−	1.05723e8i		−	0.592439i	0.955120	−	0.296220i	$-0.0957260\pi$
$564$	−2.68125e7	+	7.02892e6i	−0.149451	+	0.0391788i
$565$	−3.33351e8			−1.84823
$566$		−	1.51873e8i		−	0.837588i
$567$	−9.02206e7	+	1.48728e8i	−0.494945	+	0.815914i
$568$	2.03440e8			1.11018
$569$		−	3.07047e8i		−	1.66674i	−0.552716	−	0.833370i	$-0.686409\pi$
$569$		−	3.07047e8i		−	1.66674i	0.552716	−	0.833370i	$-0.313591\pi$
$570$	7.55012e7	+	2.88006e8i	0.407689	+	1.55517i
$571$	2.83737e8			1.52408			0.762039	−	0.647532i	$-0.224199\pi$
$571$	2.83737e8			1.52408			0.762039	+	0.647532i	$0.224199\pi$
$572$		−	6.68828e6i		−	0.0357377i
$573$	2.43267e8	−	6.37728e7i	1.29306	−	0.338978i
$574$	−1.17781e6			−0.00622785
$575$			6.43658e7i			0.338572i
$576$	1.51636e8	−	8.53703e7i	0.793480	−	0.446724i
$577$	5.34014e7			0.277987			0.138994	−	0.990293i	$-0.455613\pi$
$577$	5.34014e7			0.277987			0.138994	+	0.990293i	$0.455613\pi$
$578$		−	9.31818e7i		−	0.482556i
$579$	5.62518e7	+	2.14578e8i	0.289801	+	1.10547i
$580$	−2.26749e7			−0.116215
$581$		−	9.18710e7i		−	0.468436i
$582$	1.25030e8	−	3.27769e7i	0.634230	−	0.166264i
$583$	7.60252e7			0.383665
$584$		−	1.25042e8i		−	0.627795i
$585$	−3.48090e7	−	6.18285e7i	−0.173870	−	0.308831i
$586$	1.72840e8			0.858916
$587$		−	6.07355e7i		−	0.300281i	−0.988665	−	0.150141i	$-0.952027\pi$
$587$		−	6.07355e7i		−	0.300281i	0.988665	−	0.150141i	$-0.0479727\pi$
$588$	−2.20443e6	−	8.40900e6i	−0.0108434	−	0.0413630i
$589$	5.56107e7			0.272152
$590$			3.22768e8i			1.57157i
$591$	3.59537e8	−	9.42531e7i	1.74173	−	0.456597i
$592$	−8.16327e6			−0.0393459
$593$		−	4.94899e7i		−	0.237330i	−0.992934	−	0.118665i	$-0.962139\pi$
$593$		−	4.94899e7i		−	0.237330i	0.992934	−	0.118665i	$-0.0378615\pi$
$594$	−2.89746e7	−	2.99321e7i	−0.138248	−	0.142816i
$595$	1.51316e8			0.718346
$596$		−	1.40915e8i		−	0.665609i
$597$	−7.97022e7	−	3.04032e8i	−0.374582	−	1.42888i
$598$	1.99967e7			0.0935091
$599$		−	2.30866e8i		−	1.07418i	−0.843524	−	0.537092i	$-0.819523\pi$
$599$		−	2.30866e8i		−	1.07418i	0.843524	−	0.537092i	$-0.180477\pi$
$600$	−1.58096e8	+	4.14450e7i	−0.731925	+	0.191875i
$601$	3.25174e6			0.0149793			0.00748967	−	0.999972i	$-0.497616\pi$
$601$	3.25174e6			0.0149793			0.00748967	+	0.999972i	$0.497616\pi$
$602$		−	1.87360e8i		−	0.858792i
$603$	3.08501e7	−	1.73684e7i	0.140703	−	0.0792151i
$604$	−1.11170e8			−0.504519
$605$		−	2.67528e8i		−	1.20810i
$606$	2.79409e7	+	1.06583e8i	0.125552	+	0.478929i
$607$	3.72624e8			1.66612			0.833058	−	0.553186i	$-0.186588\pi$
$607$	3.72624e8			1.66612			0.833058	+	0.553186i	$0.186588\pi$
$608$			3.28052e8i			1.45960i
$609$	−3.87173e7	+	1.01498e7i	−0.171417	+	0.0449370i
$610$	−3.21293e8			−1.41551
$611$		−	1.99573e7i		−	0.0874940i
$612$	3.10032e7	+	5.50686e7i	0.135255	+	0.240243i
$613$	1.40087e7			0.0608159			0.0304080	−	0.999538i	$-0.490319\pi$
$613$	1.40087e7			0.0608159			0.0304080	+	0.999538i	$0.490319\pi$
$614$			4.39378e7i			0.189816i
$615$	696979.	+	2.65869e6i	0.00299636	+	0.0114299i
$616$	−6.55653e7			−0.280499
$617$		−	8.97046e7i		−	0.381909i	−0.981599	−	0.190954i	$-0.938842\pi$
$617$		−	8.97046e7i		−	0.381909i	0.981599	−	0.190954i	$-0.0611582\pi$
$618$	2.87883e8	−	7.54688e7i	1.21969	−	0.319744i
$619$	−3.52389e8			−1.48577			−0.742883	−	0.669421i	$-0.766542\pi$
$619$	−3.52389e8			−1.48577			−0.742883	+	0.669421i	$0.766542\pi$
$620$			2.38282e7i			0.0999809i
$621$	−8.22002e7	+	7.95708e7i	−0.343240	+	0.332261i
$622$	−3.21332e8			−1.33531
$623$		−	9.08306e7i		−	0.375637i
$624$	4.87808e6	+	1.86079e7i	0.0200768	+	0.0765849i
$625$	−2.94539e8			−1.20643
$626$			9.15562e7i			0.373220i
$627$	1.11830e8	−	2.93164e7i	0.453687	−	0.118935i
$628$	2.13073e7			0.0860298
$629$		−	1.93089e7i		−	0.0775900i
$630$	−1.96233e8	+	1.10478e8i	−0.784786	+	0.441829i
$631$	8.18274e7			0.325695			0.162847	−	0.986651i	$-0.447932\pi$
$631$	8.18274e7			0.325695			0.162847	+	0.986651i	$0.447932\pi$
$632$		−	1.77639e8i		−	0.703699i
$633$	8.52171e7	+	3.25068e8i	0.335982	+	1.28163i
$634$	−1.54505e8			−0.606283
$635$		−	3.97414e8i		−	1.55211i
$636$	1.66029e8	−	4.35246e7i	0.645374	−	0.169186i
$637$	6.25906e6			0.0242154
$638$		−	9.58542e6i		−	0.0369104i
$639$	−1.33105e8	−	2.36425e8i	−0.510144	−	0.906129i
$640$	−6.83214e7			−0.260626
$641$		−	1.50979e8i		−	0.573246i	−0.958043	−	0.286623i	$-0.907467\pi$
$641$		−	1.50979e8i		−	0.573246i	0.958043	−	0.286623i	$-0.0925328\pi$
$642$	8.48549e7	+	3.23687e8i	0.320680	+	1.22326i
$643$	3.48554e8			1.31111			0.655553	−	0.755150i	$-0.272436\pi$
$643$	3.48554e8			1.31111			0.655553	+	0.755150i	$0.272436\pi$
$644$			5.82954e7i			0.218261i
$645$	−4.22933e8	+	1.10872e8i	−1.57613	+	0.413185i
$646$	1.90934e8			0.708247
$647$			1.97403e8i			0.728856i	0.931232	+	0.364428i	$0.118735\pi$
$647$			1.97403e8i			0.728856i	−0.931232	+	0.364428i	$0.881265\pi$
$648$	−2.48371e8	−	1.50665e8i	−0.912800	−	0.553717i
$649$	1.25328e8			0.458473
$650$		−	3.80986e7i		−	0.138730i
$651$	1.06660e7	+	4.06865e7i	0.0386597	+	0.147471i
$652$	−2.20710e8			−0.796305
$653$			2.27833e8i			0.818232i	0.912482	+	0.409116i	$0.134163\pi$
$653$			2.27833e8i			0.818232i	−0.912482	+	0.409116i	$0.865837\pi$
$654$	−1.79619e8	+	4.70874e7i	−0.642125	+	0.168334i
$655$	2.52517e8			0.898601
$656$		−	745170.i		−	0.00263963i
$657$	−1.45316e8	+	8.18116e7i	−0.512408	+	0.288482i
$658$	−6.33412e7			−0.222335
$659$			4.93079e8i			1.72290i	0.507842	+	0.861450i	$0.330443\pi$
$659$			4.93079e8i			1.72290i	−0.507842	+	0.861450i	$0.669557\pi$
$660$	1.25616e7	+	4.79173e7i	0.0436931	+	0.166671i
$661$	2.08745e8			0.722790			0.361395	−	0.932413i	$-0.382301\pi$
$661$	2.08745e8			0.722790			0.361395	+	0.932413i	$0.382301\pi$
$662$		−	3.72001e8i		−	1.28224i
$663$	−4.40139e7	+	1.15383e7i	−0.151025	+	0.0395915i
$664$	1.53421e8			0.524061
$665$		−	6.24947e8i		−	2.12509i
$666$	1.40977e7	+	2.50407e7i	0.0477228	+	0.0847663i
$667$	−2.63237e7			−0.0887093
$668$			6.37904e7i			0.214006i
$669$	−7.54486e7	−	2.87806e8i	−0.251984	−	0.961216i
$670$	4.58320e7			0.152386
$671$			1.24755e8i			0.412944i
$672$	−2.40013e8	+	6.29198e7i	−0.790911	+	0.207338i
$673$	−1.32382e8			−0.434295			−0.217148	−	0.976139i	$-0.569675\pi$
$673$	−1.32382e8			−0.434295			−0.217148	+	0.976139i	$0.569675\pi$
$674$			1.60664e8i			0.524733i
$675$	1.51602e8	+	1.56612e8i	0.492940	+	0.509229i
$676$	−1.37027e8			−0.443573
$677$			1.96331e8i			0.632735i	0.948637	+	0.316368i	$0.102463\pi$
$677$			1.96331e8i			0.632735i	−0.948637	+	0.316368i	$0.897537\pi$
$678$	−8.06759e7	−	3.07746e8i	−0.258854	−	0.987422i
$679$	−2.71304e8			−0.866657
$680$			2.52692e8i			0.803647i
$681$	1.79434e8	−	4.70388e7i	0.568150	−	0.148941i
$682$	−1.00730e7			−0.0317544
$683$		−	2.95106e8i		−	0.926225i	−0.886300	−	0.463113i	$-0.846733\pi$
$683$		−	2.95106e8i		−	0.926225i	0.886300	−	0.463113i	$-0.153267\pi$
$684$	2.27438e8	−	1.28046e8i	0.710713	−	0.400126i
$685$	−2.58941e8			−0.805619
$686$		−	2.42285e8i		−	0.750505i
$687$	−1.03814e7	−	3.96008e7i	−0.0320174	−	0.122133i
$688$	1.18538e8			0.363993
$689$			1.23580e8i			0.377824i
$690$	−1.43264e8	+	3.75567e7i	−0.436103	+	0.114325i
$691$	−5.66600e8			−1.71729			−0.858643	−	0.512574i	$-0.828692\pi$
$691$	−5.66600e8			−1.71729			−0.858643	+	0.512574i	$0.828692\pi$
$692$			1.70896e8i			0.515721i
$693$	4.28976e7	+	7.61955e7i	0.128894	+	0.228944i
$694$	9.63757e7			0.288330
$695$			5.21346e8i			1.55300i
$696$	−1.69498e7	−	6.46564e7i	−0.0502731	−	0.191772i
$697$	1.76258e6			0.00520536
$698$			1.87047e8i			0.550027i
$699$	−1.74396e8	+	4.57181e7i	−0.510628	+	0.133862i
$700$	1.11067e8			0.323811
$701$		−	2.59989e8i		−	0.754745i	−0.926061	−	0.377373i	$-0.876828\pi$
$701$		−	2.59989e8i		−	0.754745i	0.926061	−	0.377373i	$-0.123172\pi$
$702$	4.86550e7	−	4.70986e7i	0.140642	−	0.136143i
$703$	−7.97473e7			−0.229536
$704$		−	8.74727e7i		−	0.250700i
$705$	3.74828e7	+	1.42982e8i	0.106971	+	0.408050i
$706$	−1.15690e8			−0.328763
$707$		−	2.31276e8i		−	0.654443i
$708$	2.73699e8	−	7.17505e7i	0.771211	−	0.202174i
$709$	−6.05427e8			−1.69872			−0.849362	−	0.527810i	$-0.823013\pi$
$709$	−6.05427e8			−1.69872			−0.849362	+	0.527810i	$0.823013\pi$
$710$		−	3.51241e8i		−	0.981363i
$711$	−2.06440e8	+	1.16224e8i	−0.574361	+	0.323361i
$712$	1.51684e8			0.420242
$713$			2.76625e7i			0.0763174i
$714$	3.66207e7	+	1.39693e8i	0.100608	+	0.383778i
$715$	−3.56662e7			−0.0975752
$716$			8.81856e7i			0.240247i
$717$	3.13605e7	−	8.22119e6i	0.0850796	−	0.0223037i
$718$	2.02030e8			0.545811
$719$			6.06595e8i			1.63197i	0.578072	+	0.815986i	$0.303805\pi$
$719$			6.06595e8i			1.63197i	−0.578072	+	0.815986i	$0.696195\pi$
$720$	−6.98968e7	−	1.24152e8i	−0.187266	−	0.332626i
$721$	−6.24679e8			−1.66668
$722$		−	5.16848e8i		−	1.37325i
$723$	−1.40352e8	−	5.35387e8i	−0.371368	−	1.41662i
$724$	2.35511e8			0.620576
$725$			5.01532e7i			0.131609i
$726$	2.46979e8	−	6.47458e7i	0.645431	−	0.169200i
$727$	−3.32454e8			−0.865222			−0.432611	−	0.901581i	$-0.642408\pi$
$727$	−3.32454e8			−0.865222			−0.432611	+	0.901581i	$0.642408\pi$
$728$		−	1.06577e8i		−	0.276229i
$729$	−1.25908e7	+	3.87216e8i	−0.0324990	+	0.999472i
$730$	−2.15886e8			−0.554952
$731$			2.80383e8i			0.717794i
$732$	7.14226e7	+	2.72448e8i	0.182097	+	0.694625i
$733$	4.11192e8			1.04408			0.522039	−	0.852922i	$-0.325171\pi$
$733$	4.11192e8			1.04408			0.522039	+	0.852922i	$0.325171\pi$
$734$		−	9.50443e7i		−	0.240347i
$735$	−4.48423e7	+	1.17554e7i	−0.112934	+	0.0296058i
$736$	−1.63184e8			−0.409302
$737$		−	1.77961e7i		−	0.0444553i
$738$	−2.28579e6	+	1.28688e6i	−0.00568680	+	0.00320163i
$739$	1.51845e8			0.376243			0.188121	−	0.982146i	$-0.439760\pi$
$739$	1.51845e8			0.376243			0.188121	+	0.982146i	$0.439760\pi$
$740$		−	3.41704e7i		−	0.0843247i
$741$	4.76541e7	+	1.81781e8i	0.117124	+	0.446780i
$742$	3.92222e8			0.960108
$743$		−	1.27236e8i		−	0.310200i	−0.987899	−	0.155100i	$-0.950430\pi$
$743$		−	1.27236e8i		−	0.310200i	0.987899	−	0.155100i	$-0.0495700\pi$
$744$	−6.79450e7	+	1.78119e7i	−0.164983	+	0.0432504i
$745$	−7.51451e8			−1.81732
$746$			5.45366e8i			1.31363i
$747$	−1.00379e8	−	1.78296e8i	−0.240814	−	0.427739i
$748$	3.17668e7			0.0759047
$749$		−	7.02371e8i		−	1.67156i
$750$	−2.94067e7	−	1.12174e8i	−0.0697047	−	0.265895i
$751$	−1.39159e8			−0.328543			−0.164271	−	0.986415i	$-0.552527\pi$
$751$	−1.39159e8			−0.328543			−0.164271	+	0.986415i	$0.552527\pi$
$752$		−	4.00745e7i		−	0.0942354i
$753$	7.01883e8	−	1.84000e8i	1.64392	−	0.430955i
$754$	1.55812e7			0.0363485
$755$			5.92832e8i			1.37750i
$756$	1.37304e8	+	1.41842e8i	0.317775	+	0.328275i
$757$	−3.27923e8			−0.755935			−0.377967	−	0.925819i	$-0.623377\pi$
$757$	−3.27923e8			−0.755935			−0.377967	+	0.925819i	$0.623377\pi$
$758$			1.65554e8i			0.380130i
$759$	1.45829e7	+	5.56279e7i	0.0333518	+	0.127224i
$760$	1.04364e9			2.37744
$761$			6.66847e8i			1.51312i	0.653926	+	0.756558i	$0.273121\pi$
$761$			6.66847e8i			1.51312i	−0.653926	+	0.756558i	$0.726879\pi$
$762$	3.66887e8	−	9.61799e7i	0.829216	−	0.217380i
$763$	3.89757e8			0.877446
$764$		−	2.85402e8i		−	0.639995i
$765$	2.93662e8	−	1.65330e8i	0.655938	−	0.369289i
$766$	9.12046e7			0.202923
$767$			2.03722e8i			0.451494i
$768$	−1.21133e8	−	4.62074e8i	−0.267411	−	1.02007i
$769$	1.35476e8			0.297909			0.148955	−	0.988844i	$-0.452409\pi$
$769$	1.35476e8			0.297909			0.148955	+	0.988844i	$0.452409\pi$
$770$			1.13199e8i			0.247953i
$771$	3.14207e8	−	8.23698e7i	0.685572	−	0.179724i
$772$	2.51743e8			0.547149
$773$			4.07346e8i			0.881911i	0.897529	+	0.440955i	$0.145360\pi$
$773$			4.07346e8i			0.881911i	−0.897529	+	0.440955i	$0.854640\pi$
$774$	−2.04712e8	−	3.63614e8i	−0.441489	−	0.784183i
$775$	5.27041e7			0.113224
$776$		−	4.53069e8i		−	0.969569i
$777$	−1.52954e7	−	5.83456e7i	−0.0326059	−	0.124378i
$778$	1.29484e8			0.274965
$779$		−	7.27959e6i		−	0.0153991i
$780$	−7.78901e7	+	2.04190e7i	−0.164134	+	0.0430279i
$781$	−1.36383e8			−0.286292
$782$			9.49766e7i			0.198608i
$783$	−6.40496e7	+	6.20008e7i	−0.133423	+	0.129155i
$784$	1.25682e7			0.0260811
$785$		−	1.13624e8i		−	0.234889i
$786$	6.11128e7	+	2.33121e8i	0.125853	+	0.480079i
$787$	−3.34146e8			−0.685508			−0.342754	−	0.939425i	$-0.611360\pi$
$787$	−3.34146e8			−0.685508			−0.342754	+	0.939425i	$0.611360\pi$
$788$		−	4.21810e8i		−	0.862061i
$789$	1.79548e8	−	4.70686e7i	0.365552	−	0.0958298i
$790$	−3.06694e8			−0.622049
$791$			6.67780e8i			1.34929i
$792$	−1.27244e8	+	7.16374e7i	−0.256130	+	0.144200i
$793$	−2.02791e8			−0.406658
$794$		−	3.71695e8i		−	0.742550i
$795$	−2.32101e8	−	8.85372e8i	−0.461930	−	1.76208i
$796$	−3.56690e8			−0.707216
$797$			8.03163e8i			1.58646i	0.608923	+	0.793229i	$0.291602\pi$
$797$			8.03163e8i			1.58646i	−0.608923	+	0.793229i	$0.708398\pi$
$798$	5.76943e8	−	1.51246e8i	1.13534	−	0.297630i
$799$	9.47897e7			0.185832
$800$			3.10906e8i			0.607238i
$801$	−9.92425e7	−	1.76277e8i	−0.193108	−	0.343003i
$802$	−6.49872e8			−1.25981
$803$			8.38265e7i			0.161895i
$804$	−1.01883e7	−	3.88643e7i	−0.0196035	−	0.0747795i
$805$	3.10869e8			0.595922
$806$		−	1.63737e7i		−	0.0312710i
$807$	−4.79729e7	+	1.25762e7i	−0.0912800	+	0.0239292i
$808$	3.86223e8			0.732156
$809$		−	9.26866e8i		−	1.75054i	−0.483637	−	0.875269i	$-0.660685\pi$
$809$		−	9.26866e8i		−	1.75054i	0.483637	−	0.875269i	$-0.339315\pi$
$810$	−2.60124e8	+	4.28813e8i	−0.489470	+	0.806888i
$811$	4.65619e8			0.872907			0.436454	−	0.899727i	$-0.356234\pi$
$811$	4.65619e8			0.872907			0.436454	+	0.899727i	$0.356234\pi$
$812$			4.54232e7i			0.0848417i
$813$	−1.99638e8	−	7.61539e8i	−0.371512	−	1.41717i
$814$	1.44449e7			0.0267819
$815$			1.17697e9i			2.17417i
$816$	−8.83804e7	+	2.31690e7i	−0.162662	+	0.0426420i
$817$	1.15801e9			2.12346
$818$			4.68843e8i			0.856579i
$819$	−1.23857e8	+	6.97305e7i	−0.225459	+	0.126932i
$820$	3.11918e6			0.00565717
$821$			2.75443e8i			0.497739i	0.968537	+	0.248870i	$0.0800591\pi$
$821$			2.75443e8i			0.497739i	−0.968537	+	0.248870i	$0.919941\pi$
$822$	−6.26676e7	−	2.39052e8i	−0.112831	−	0.430404i
$823$	3.31587e8			0.594837			0.297418	−	0.954747i	$-0.403874\pi$
$823$	3.31587e8			0.594837			0.297418	+	0.954747i	$0.403874\pi$
$824$		−	1.04319e9i		−	1.86459i
$825$	1.05985e8	−	2.77841e7i	0.188748	−	0.0494806i
$826$	6.46580e8			1.14731
$827$		−	4.69560e8i		−	0.830184i	−0.909779	−	0.415092i	$-0.863749\pi$
$827$		−	4.69560e8i		−	0.830184i	0.909779	−	0.415092i	$-0.136251\pi$
$828$	6.36942e7	+	1.13135e8i	0.112204	+	0.199299i
$829$	1.66978e8			0.293086			0.146543	−	0.989204i	$-0.453185\pi$
$829$	1.66978e8			0.293086			0.146543	+	0.989204i	$0.453185\pi$
$830$		−	2.64883e8i		−	0.463254i
$831$	−2.19011e7	−	8.35439e7i	−0.0381648	−	0.145583i
$832$	1.42188e8			0.246884
$833$			2.97282e7i			0.0514320i
$834$	−4.81300e8	+	1.26173e8i	−0.829694	+	0.217505i
$835$	3.40172e8			0.584304
$836$		−	1.31199e8i		−	0.224550i
$837$	6.51542e7	+	6.73072e7i	0.111113	+	0.114785i
$838$	2.70277e8			0.459279
$839$		−	3.49915e8i		−	0.592485i	−0.955113	−	0.296242i	$-0.904266\pi$
$839$		−	3.49915e8i		−	0.592485i	0.955113	−	0.296242i	$-0.0957336\pi$
$840$	2.00168e8	+	7.63559e8i	0.337720	+	1.28826i
$841$	−2.05111e7			−0.0344828
$842$			6.33879e8i			1.06187i
$843$	3.63619e8	−	9.53232e7i	0.606966	−	0.159117i
$844$	3.81371e8			0.634338
$845$			7.30716e8i			1.21110i
$846$	−1.22927e8	+	6.92073e7i	−0.203020	+	0.114299i
$847$	−5.35921e8			−0.881964
$848$			2.48149e8i			0.406936i
$849$	1.80035e8	+	6.86760e8i	0.294194	+	1.12223i
$850$	1.80954e8			0.294654
$851$		−	3.96689e7i		−	0.0643667i
$852$	−2.97843e8	+	7.80798e7i	−0.481580	+	0.126247i
$853$	−4.25660e8			−0.685829			−0.342914	−	0.939367i	$-0.611414\pi$
$853$	−4.25660e8			−0.685829			−0.342914	+	0.939367i	$0.611414\pi$
$854$			6.43625e8i			1.03338i
$855$	−6.82824e8	−	1.21285e9i	−1.09247	−	1.94047i
$856$	1.17293e9			1.87005
$857$		−	9.69039e8i		−	1.53957i	−0.638305	−	0.769784i	$-0.720364\pi$
$857$		−	9.69039e8i		−	1.53957i	0.638305	−	0.769784i	$-0.279636\pi$
$858$	−8.63175e6	−	3.29266e7i	−0.0136659	−	0.0521297i
$859$	−2.73280e8			−0.431149			−0.215575	−	0.976487i	$-0.569162\pi$
$859$	−2.73280e8			−0.431149			−0.215575	+	0.976487i	$0.569162\pi$
$860$			4.96186e8i			0.780098i
$861$	5.32598e6	−	1.39621e6i	0.00834430	−	0.00218747i
$862$	4.41043e8			0.688587
$863$		−	9.71130e8i		−	1.51093i	−0.655188	−	0.755466i	$-0.727410\pi$
$863$		−	9.71130e8i		−	1.51093i	0.655188	−	0.755466i	$-0.272590\pi$
$864$	−3.97051e8	+	3.84351e8i	−0.615610	+	0.595918i
$865$	9.11331e8			1.40808
$866$		−	4.58680e7i		−	0.0706247i
$867$	1.10461e8	+	4.21363e8i	0.169493	+	0.646545i
$868$	4.77335e7			0.0729901
$869$			1.19087e8i			0.181469i
$870$	−1.11630e8	+	2.92638e7i	−0.169520	+	0.0444399i
$871$	2.89278e7			0.0437785
$872$			6.50880e8i			0.981639i
$873$	−5.26526e8	+	2.96430e8i	−0.791365	+	0.445533i
$874$	3.92261e8			0.587544
$875$			2.43408e8i			0.363338i
$876$	4.79908e7	+	1.83065e8i	0.0713914	+	0.272329i
$877$	−2.97277e8			−0.440720			−0.220360	−	0.975419i	$-0.570723\pi$
$877$	−2.97277e8			−0.440720			−0.220360	+	0.975419i	$0.570723\pi$
$878$			7.27535e8i			1.07491i
$879$	−7.81571e8	+	2.04890e8i	−1.15081	+	0.301685i
$880$	−7.16181e7			−0.105093
$881$		−	4.91287e8i		−	0.718468i	−0.933247	−	0.359234i	$-0.883038\pi$
$881$		−	4.91287e8i		−	0.718468i	0.933247	−	0.359234i	$-0.116962\pi$
$882$	−2.17050e7	−	3.85528e7i	−0.0316339	−	0.0561889i
$883$	−6.91819e8			−1.00487			−0.502436	−	0.864614i	$-0.667563\pi$
$883$	−6.91819e8			−1.00487			−0.502436	+	0.864614i	$0.667563\pi$
$884$			5.16373e7i			0.0747492i
$885$	−3.82620e8	−	1.45954e9i	−0.551999	−	2.10565i
$886$	−5.74883e8			−0.826568
$887$			1.01434e9i			1.45350i	0.686903	+	0.726749i	$0.258970\pi$
$887$			1.01434e9i			1.45350i	−0.686903	+	0.726749i	$0.741030\pi$
$888$	9.74351e7	−	2.55427e7i	0.139148	−	0.0364778i
$889$	−7.96111e8			−1.13310
$890$		−	2.61883e8i		−	0.371481i
$891$	1.66504e8	+	1.01004e8i	0.235392	+	0.142792i
$892$	−3.37654e8			−0.475749
$893$		−	3.91489e8i		−	0.549750i
$894$	−1.81862e8	−	6.93730e8i	−0.254525	−	0.970908i
$895$	4.70263e8			0.655951
$896$			1.36864e8i			0.190267i
$897$	−9.04238e7	+	2.37047e7i	−0.125287	+	0.0328441i
$898$	7.30961e8			1.00940
$899$			2.15544e7i			0.0296658i
$900$	2.15550e8	−	1.21353e8i	0.295679	−	0.166465i
$901$	−5.86957e8			−0.802476
$902$			1.31858e6i			0.00179674i
$903$	2.22103e8	+	8.47233e8i	0.301642	+	1.15064i
$904$	−1.11517e9			−1.50951
$905$		−	1.25590e9i		−	1.69437i
$906$	−5.47295e8	+	1.43474e8i	−0.735931	+	0.192925i
$907$	1.55138e8			0.207921			0.103960	−	0.994581i	$-0.466849\pi$
$907$	1.55138e8			0.207921			0.103960	+	0.994581i	$0.466849\pi$
$908$		−	2.10512e8i		−	0.281203i
$909$	−2.52695e8	−	4.48842e8i	−0.336437	−	0.597587i
$910$	−1.84006e8			−0.244179
$911$		−	5.58763e8i		−	0.739048i	−0.929221	−	0.369524i	$-0.879521\pi$
$911$		−	5.58763e8i		−	0.739048i	0.929221	−	0.369524i	$-0.120479\pi$
$912$	9.56899e7	+	3.65018e8i	0.126148	+	0.481205i
$913$	−1.02851e8			−0.135144
$914$		−	4.13207e8i		−	0.541164i
$915$	1.45287e9	−	3.80872e8i	1.89655	−	0.497182i
$916$	−4.64598e7			−0.0604492
$917$		−	5.05850e8i		−	0.656015i
$918$	2.23701e8	+	2.31093e8i	0.289161	+	0.298716i
$919$	−4.44174e8			−0.572278			−0.286139	−	0.958188i	$-0.592372\pi$
$919$	−4.44174e8			−0.572278			−0.286139	+	0.958188i	$0.592372\pi$
$920$			5.19140e8i			0.666686i
$921$	−5.20853e7	−	1.98684e8i	−0.0666709	−	0.254322i
$922$	−2.09483e8			−0.267274
$923$		−	2.21693e8i		−	0.281933i
$924$	9.59895e7	−	2.51638e7i	0.121677	−	0.0318977i
$925$	−7.55792e7			−0.0954942
$926$			7.54610e8i			0.950363i
$927$	−1.21233e9	+	6.82532e8i	−1.52188	+	0.856808i
$928$	−1.27151e8			−0.159102
$929$			8.07835e8i			1.00757i	0.863829	+	0.503785i	$0.168060\pi$
$929$			8.07835e8i			1.00757i	−0.863829	+	0.503785i	$0.831940\pi$
$930$	3.07522e7	+	1.17307e8i	0.0382321	+	0.145840i
$931$	1.22780e8			0.152152
$932$			2.04602e8i			0.252733i
$933$	1.45305e9	−	3.80918e8i	1.78910	−	0.469015i
$934$	−2.24669e8			−0.275742
$935$		−	1.69401e8i		−	0.207244i
$936$	−1.16447e8	−	2.06836e8i	−0.142005	−	0.252231i
$937$	7.46226e8			0.907093			0.453546	−	0.891233i	$-0.350159\pi$
$937$	7.46226e8			0.907093			0.453546	+	0.891233i	$0.350159\pi$
$938$		−	9.18121e7i		−	0.111248i
$939$	−1.08534e8	−	4.14012e8i	−0.131089	−	0.500053i
$940$	1.67746e8			0.201962
$941$			1.30543e9i			1.56669i	0.621585	+	0.783347i	$0.286489\pi$
$941$			1.30543e9i			1.56669i	−0.621585	+	0.783347i	$0.713511\pi$
$942$	1.04896e8	−	2.74987e7i	0.125490	−	0.0328973i
$943$	3.62111e6			0.00431823
$944$			4.09075e8i			0.486281i
$945$	7.56392e8	−	7.32197e8i	0.896296	−	0.867625i
$946$	−2.09754e8			−0.247763
$947$		−	1.91366e8i		−	0.225328i	−0.993633	−	0.112664i	$-0.964062\pi$
$947$		−	1.91366e8i		−	0.225328i	0.993633	−	0.112664i	$-0.0359383\pi$
$948$	6.81772e7	+	2.60068e8i	0.0800229	+	0.305255i
$949$	−1.36261e8			−0.159431
$950$		−	7.47355e8i		−	0.871678i
$951$	6.98664e8	−	1.83156e8i	0.812319	−	0.212950i
$952$	5.06201e8			0.586695
$953$			1.09696e9i			1.26739i	0.773581	+	0.633697i	$0.218464\pi$
$953$			1.09696e9i			1.26739i	−0.773581	+	0.633697i	$0.781536\pi$
$954$	7.61193e8	−	4.28546e8i	0.876697	−	0.493574i
$955$	−1.52195e9			−1.74739
$956$		−	3.67922e7i		−	0.0421097i
$957$	1.13629e7	+	4.33447e7i	0.0129644	+	0.0494539i
$958$	−7.13888e8			−0.811958
$959$			5.18720e8i			0.588135i
$960$	−1.01869e9	+	2.67050e8i	−1.15140	+	0.301842i
$961$	−8.64853e8			−0.974478
$962$			2.34804e7i			0.0263742i
$963$	−7.67418e8	−	1.36310e9i	−0.859316	−	1.52634i
$964$	−6.28118e8			−0.701148
$965$		−	1.34246e9i		−	1.49389i
$966$	7.52348e7	+	2.86990e8i	0.0834617	+	0.318373i
$967$	1.28508e9			1.42119			0.710594	−	0.703602i	$-0.248426\pi$
$967$	1.28508e9			1.42119			0.710594	+	0.703602i	$0.248426\pi$
$968$		−	8.94969e8i		−	0.986693i
$969$	−8.63391e8	+	2.26339e8i	−0.948935	+	0.248764i
$970$	−7.82225e8			−0.857070
$971$		−	3.79540e8i		−	0.414572i	−0.978280	−	0.207286i	$-0.933537\pi$
$971$		−	3.79540e8i		−	0.414572i	0.978280	−	0.207286i	$-0.0664631\pi$
$972$	4.21447e8	+	1.25254e8i	0.458928	+	0.136394i
$973$	1.04438e9			1.13375
$974$		−	1.24439e9i		−	1.34673i
$975$	4.51634e7	+	1.72280e8i	0.0487273	+	0.185875i
$976$	−4.07206e8			−0.437991
$977$			1.61992e8i			0.173704i	0.996221	+	0.0868521i	$0.0276808\pi$
$977$			1.61992e8i			0.173704i	−0.996221	+	0.0868521i	$0.972319\pi$
$978$	−1.08656e9	+	2.84844e8i	−1.16155	+	0.304502i
$979$	−1.01687e8			−0.108372
$980$			5.26090e7i			0.0558962i
$981$	7.56409e8	−	4.25853e8i	0.801216	−	0.451079i
$982$	2.84800e7			0.0300750
$983$			1.65767e8i			0.174516i	0.996186	+	0.0872582i	$0.0278105\pi$
$983$			1.65767e8i			0.174516i	−0.996186	+	0.0872582i	$0.972189\pi$
$984$	2.33162e6	+	8.89419e6i	0.00244722	+	0.00933515i
$985$	−2.24937e9			−2.35370
$986$			7.40048e7i			0.0772021i
$987$	2.86425e8	−	7.50867e7i	0.297893	−	0.0780929i
$988$	2.13266e8			0.221132
$989$			5.76029e8i			0.595464i
$990$	1.23682e8	+	2.19687e8i	0.127468	+	0.226412i
$991$	2.67257e8			0.274605			0.137302	−	0.990529i	$-0.456157\pi$
$991$	2.67257e8			0.274605			0.137302	+	0.990529i	$0.456157\pi$
$992$			1.33618e8i			0.136877i
$993$	4.40982e8	+	1.68217e9i	0.450374	+	1.71799i
$994$	−7.03616e8			−0.716435
$995$			1.90211e9i			1.93093i
$996$	−2.24613e8	+	5.88826e7i	−0.227330	+	0.0595949i
$997$	1.87948e7			0.0189650			0.00948250	−	0.999955i	$-0.496982\pi$
$997$	1.87948e7			0.0189650			0.00948250	+	0.999955i	$0.496982\pi$
$998$			7.05541e8i			0.709792i
$999$	−9.34331e7	−	9.65205e7i	−0.0937139	−	0.0968107i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.19	✓	56
3.2	odd	2	inner	87.7.b.a.59.38	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.19	✓	56	1.1	even	1	trivial
87.7.b.a.59.38	yes	56	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 87 = 3 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	87.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.77572i q^{2} +(6.84673 + 26.1175i) q^{3} +30.6411 q^{4} -163.398i q^{5} +(150.847 - 39.5448i) q^{6} -327.324 q^{7} -546.620i q^{8} +(-635.245 + 357.638i) q^{9} -943.742 q^{10} +366.446i q^{11} +(209.791 + 800.268i) q^{12} -595.662 q^{13} +1890.53i q^{14} +(4267.55 - 1118.74i) q^{15} -1196.10 q^{16} -2829.17i q^{17} +(2065.62 + 3668.99i) q^{18} -11684.7 q^{19} -5006.70i q^{20} +(-2241.10 - 8548.89i) q^{21} +2116.49 q^{22} -5812.35i q^{23} +(14276.3 - 3742.56i) q^{24} -11074.0 q^{25} +3440.38i q^{26} +(-13690.0 - 14142.3i) q^{27} -10029.6 q^{28} -4528.92i q^{29} +(-6461.54 - 24648.2i) q^{30} -4759.27 q^{31} -28075.4i q^{32} +(-9570.65 + 2508.96i) q^{33} -16340.5 q^{34} +53484.2i q^{35} +(-19464.6 + 10958.4i) q^{36} +6824.94 q^{37} +67487.5i q^{38} +(-4078.34 - 15557.2i) q^{39} -89316.8 q^{40} +623.002i q^{41} +(-49376.0 + 12944.0i) q^{42} -99104.4 q^{43} +11228.3i q^{44} +(58437.5 + 103798. i) q^{45} -33570.5 q^{46} +33504.4i q^{47} +(-8189.33 - 31239.0i) q^{48} -10507.7 q^{49} +63960.1i q^{50} +(73890.8 - 19370.6i) q^{51} -18251.7 q^{52} -207466. i q^{53} +(-81682.1 + 79069.3i) q^{54} +59876.6 q^{55} +178922. i q^{56} +(-80001.9 - 305175. i) q^{57} -26157.8 q^{58} -342009. i q^{59} +(130762. - 34279.5i) q^{60} +340446. q^{61} +27488.2i q^{62} +(207931. - 117064. i) q^{63} -238706. q^{64} +97330.1i q^{65} +(14491.0 + 55277.4i) q^{66} -48564.1 q^{67} -86688.8i q^{68} +(151804. - 39795.5i) q^{69} +308910. q^{70} +372179. i q^{71} +(195492. + 347238. i) q^{72} +228755. q^{73} -39418.9i q^{74} +(-75820.4 - 289224. i) q^{75} -358032. q^{76} -119947. i q^{77} +(-89854.0 + 23555.3i) q^{78} +324977. q^{79} +195440. i q^{80} +(275631. - 454376. i) q^{81} +3598.29 q^{82} +280673. i q^{83} +(-68669.7 - 261947. i) q^{84} -462281. q^{85} +572399. i q^{86} +(118284. - 31008.3i) q^{87} +200307. q^{88} +277494. i q^{89} +(599507. - 337518. i) q^{90} +194975. q^{91} -178097. i q^{92} +(-32585.4 - 124300. i) q^{93} +193512. q^{94} +1.90926e6i q^{95} +(733258. - 192224. i) q^{96} +828855. q^{97} +60689.7i q^{98} +(-131055. - 232783. i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q + 52 q^{3} - 1924 q^{4} - 160 q^{6} + 160 q^{7} - 1060 q^{9} - 3588 q^{10} - 2166 q^{12} - 1400 q^{13} - 6240 q^{15} + 56588 q^{16} - 5978 q^{18} + 25000 q^{19} + 7520 q^{21} + 20970 q^{22} + 1238 q^{24}+ \cdots + 4793544 q^{99}+O(q^{100}) \) 56 * q + 52 * q^3 - 1924 * q^4 - 160 * q^6 + 160 * q^7 - 1060 * q^9 - 3588 * q^10 - 2166 * q^12 - 1400 * q^13 - 6240 * q^15 + 56588 * q^16 - 5978 * q^18 + 25000 * q^19 + 7520 * q^21 + 20970 * q^22 + 1238 * q^24 - 213304 * q^25 + 25492 * q^27 + 7870 * q^28 + 105092 * q^30 - 78656 * q^31 + 18952 * q^33 - 96642 * q^34 - 40224 * q^36 + 52120 * q^37 - 112096 * q^39 + 343440 * q^40 - 341758 * q^42 - 286760 * q^43 + 68312 * q^45 + 255516 * q^46 + 386086 * q^48 + 1051872 * q^49 - 256656 * q^51 + 58606 * q^52 + 450124 * q^54 - 409632 * q^55 - 1141736 * q^57 + 817806 * q^60 + 309976 * q^61 + 567632 * q^63 - 2844730 * q^64 + 958426 * q^66 + 748696 * q^67 - 1323704 * q^69 - 2123748 * q^70 - 1026242 * q^72 - 333728 * q^73 - 331316 * q^75 + 2846728 * q^76 + 292446 * q^78 - 661184 * q^79 + 1614852 * q^81 + 376908 * q^82 - 1670584 * q^84 + 3342000 * q^85 - 927168 * q^88 + 835882 * q^90 + 2248016 * q^91 + 3960776 * q^93 - 1368294 * q^94 - 5811456 * q^96 - 3057296 * q^97 + 4793544 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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