LMFDB - Embedded newform 87.7.b.a.59.13

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			59.13
Character	$\chi$	$=$	87.59
Dual form			87.7.b.a.59.44

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-9.80663i q^{2} +(-13.8860 - 23.1556i) q^{3} -32.1699 q^{4} -31.8600i q^{5} +(-227.078 + 136.175i) q^{6} -247.832 q^{7} -312.146i q^{8} +(-343.359 + 643.075i) q^{9} -312.439 q^{10} +1235.01i q^{11} +(446.711 + 744.913i) q^{12} +187.470 q^{13} +2430.40i q^{14} +(-737.735 + 442.407i) q^{15} -5119.97 q^{16} +3648.48i q^{17} +(6306.40 + 3367.20i) q^{18} -7863.06 q^{19} +1024.93i q^{20} +(3441.39 + 5738.70i) q^{21} +12111.3 q^{22} -3181.08i q^{23} +(-7227.90 + 4334.45i) q^{24} +14609.9 q^{25} -1838.45i q^{26} +(19658.6 - 979.049i) q^{27} +7972.75 q^{28} -4528.92i q^{29} +(4338.52 + 7234.69i) q^{30} -5020.63 q^{31} +30232.3i q^{32} +(28597.3 - 17149.3i) q^{33} +35779.3 q^{34} +7895.93i q^{35} +(11045.8 - 20687.7i) q^{36} +27281.8 q^{37} +77110.1i q^{38} +(-2603.20 - 4340.97i) q^{39} -9944.94 q^{40} -99324.1i q^{41} +(56277.3 - 33748.5i) q^{42} -28545.6 q^{43} -39730.2i q^{44} +(20488.3 + 10939.4i) q^{45} -31195.6 q^{46} +19529.9i q^{47} +(71095.8 + 118556. i) q^{48} -56228.1 q^{49} -143274. i q^{50} +(84482.5 - 50662.7i) q^{51} -6030.90 q^{52} +291331. i q^{53} +(-9601.17 - 192785. i) q^{54} +39347.3 q^{55} +77359.8i q^{56} +(109186. + 182074. i) q^{57} -44413.5 q^{58} +256441. i q^{59} +(23732.9 - 14232.2i) q^{60} -222205. q^{61} +49235.5i q^{62} +(85095.6 - 159375. i) q^{63} -31200.9 q^{64} -5972.79i q^{65} +(-168177. - 280443. i) q^{66} -366542. q^{67} -117371. i q^{68} +(-73659.6 + 44172.4i) q^{69} +77432.4 q^{70} +270730. i q^{71} +(200733. + 107178. i) q^{72} -253432. q^{73} -267542. i q^{74} +(-202873. - 338301. i) q^{75} +252954. q^{76} -306075. i q^{77} +(-42570.3 + 25528.6i) q^{78} -197960. q^{79} +163122. i q^{80} +(-295650. - 441612. i) q^{81} -974034. q^{82} +397030. i q^{83} +(-110709. - 184614. i) q^{84} +116240. q^{85} +279936. i q^{86} +(-104870. + 62888.5i) q^{87} +385503. q^{88} +147749. i q^{89} +(107279. - 200922. i) q^{90} -46461.1 q^{91} +102335. i q^{92} +(69716.4 + 116256. i) q^{93} +191523. q^{94} +250517. i q^{95} +(700046. - 419805. i) q^{96} -139941. q^{97} +551408. i q^{98} +(-794203. - 424052. 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 $

$2$		−	9.80663i		−	1.22583i	−0.790149	−	0.612914i	$-0.789997\pi$
$2$		−	9.80663i		−	1.22583i	0.790149	−	0.612914i	$-0.210003\pi$
$3$	−13.8860	−	23.1556i	−0.514295	−	0.857613i
$3$	−13.8860	−	23.1556i	−0.514295	−	0.857613i
$4$	−32.1699			−0.502655
$5$		−	31.8600i		−	0.254880i	−0.991846	−	0.127440i	$-0.959324\pi$
$5$		−	31.8600i		−	0.254880i	0.991846	−	0.127440i	$-0.0406760\pi$
$6$	−227.078	+	136.175i	−1.05129	+	0.630438i
$7$	−247.832			−0.722543			−0.361272	−	0.932461i	$-0.617657\pi$
$7$	−247.832			−0.722543			−0.361272	+	0.932461i	$0.617657\pi$
$8$		−	312.146i		−	0.609659i
$9$	−343.359	+	643.075i	−0.471000	+	0.882133i
$10$	−312.439			−0.312439
$11$			1235.01i			0.927881i	0.885866	+	0.463940i	$0.153565\pi$
$11$			1235.01i			0.927881i	−0.885866	+	0.463940i	$0.846435\pi$
$12$	446.711	+	744.913i	0.258513	+	0.431084i
$13$	187.470			0.0853300			0.0426650	−	0.999089i	$-0.486415\pi$
$13$	187.470			0.0853300			0.0426650	+	0.999089i	$0.486415\pi$
$14$			2430.40i			0.885714i
$15$	−737.735	+	442.407i	−0.218588	+	0.131083i
$16$	−5119.97			−1.24999
$17$			3648.48i			0.742617i	0.928509	+	0.371309i	$0.121091\pi$
$17$			3648.48i			0.742617i	−0.928509	+	0.371309i	$0.878909\pi$
$18$	6306.40	+	3367.20i	1.08134	+	0.577366i
$19$	−7863.06			−1.14639			−0.573193	−	0.819420i	$-0.694296\pi$
$19$	−7863.06			−1.14639			−0.573193	+	0.819420i	$0.694296\pi$
$20$			1024.93i			0.128117i
$21$	3441.39	+	5738.70i	0.371601	+	0.619663i
$22$	12111.3			1.13742
$23$		−	3181.08i		−	0.261451i	−0.991419	−	0.130726i	$-0.958269\pi$
$23$		−	3181.08i		−	0.261451i	0.991419	−	0.130726i	$-0.0417307\pi$
$24$	−7227.90	+	4334.45i	−0.522852	+	0.313545i
$25$	14609.9			0.935036
$26$		−	1838.45i		−	0.104600i
$27$	19658.6	−	979.049i	0.998762	−	0.0497409i
$28$	7972.75			0.363190
$29$		−	4528.92i		−	0.185695i
$29$		−	4528.92i		−	0.185695i
$30$	4338.52	+	7234.69i	0.160686	+	0.267952i
$31$	−5020.63			−0.168529			−0.0842643	−	0.996443i	$-0.526854\pi$
$31$	−5020.63			−0.168529			−0.0842643	+	0.996443i	$0.526854\pi$
$32$			30232.3i			0.922618i
$33$	28597.3	−	17149.3i	0.795763	−	0.477205i
$34$	35779.3			0.910321
$35$			7895.93i			0.184162i
$36$	11045.8	−	20687.7i	0.236751	−	0.443409i
$37$	27281.8			0.538601			0.269300	−	0.963056i	$-0.413208\pi$
$37$	27281.8			0.538601			0.269300	+	0.963056i	$0.413208\pi$
$38$			77110.1i			1.40527i
$39$	−2603.20	−	4340.97i	−0.0438848	−	0.0731801i
$40$	−9944.94			−0.155390
$41$		−	99324.1i		−	1.44113i	−0.693388	−	0.720565i	$-0.743883\pi$
$41$		−	99324.1i		−	1.44113i	0.693388	−	0.720565i	$-0.256117\pi$
$42$	56277.3	−	33748.5i	0.759600	−	0.455519i
$43$	−28545.6			−0.359033			−0.179516	−	0.983755i	$-0.557453\pi$
$43$	−28545.6			−0.359033			−0.179516	+	0.983755i	$0.557453\pi$
$44$		−	39730.2i		−	0.466404i
$45$	20488.3	+	10939.4i	0.224838	+	0.120048i
$46$	−31195.6			−0.320494
$47$			19529.9i			0.188108i	0.995567	+	0.0940540i	$0.0299826\pi$
$47$			19529.9i			0.188108i	−0.995567	+	0.0940540i	$0.970017\pi$
$48$	71095.8	+	118556.i	0.642866	+	1.07201i
$49$	−56228.1			−0.477931
$50$		−	143274.i		−	1.14619i
$51$	84482.5	−	50662.7i	0.636878	−	0.381925i
$52$	−6030.90			−0.0428916
$53$			291331.i			1.95685i	0.206594	+	0.978427i	$0.433762\pi$
$53$			291331.i			1.95685i	−0.206594	+	0.978427i	$0.566238\pi$
$54$	−9601.17	−	192785.i	−0.0609738	−	1.22431i
$55$	39347.3			0.236498
$56$			77359.8i			0.440505i
$57$	109186.	+	182074.i	0.589581	+	0.983156i
$58$	−44413.5			−0.227631
$59$			256441.i			1.24862i	0.781176	+	0.624311i	$0.214620\pi$
$59$			256441.i			1.24862i	−0.781176	+	0.624311i	$0.785380\pi$
$60$	23732.9	−	14232.2i	0.109874	−	0.0658898i
$61$	−222205.			−0.978961			−0.489480	−	0.872014i	$-0.662814\pi$
$61$	−222205.			−0.978961			−0.489480	+	0.872014i	$0.662814\pi$
$62$			49235.5i			0.206587i
$63$	85095.6	−	159375.i	0.340318	−	0.637379i
$64$	−31200.9			−0.119022
$65$		−	5972.79i		−	0.0217489i
$66$	−168177.	−	280443.i	−0.584971	−	0.975468i
$67$	−366542.			−1.21871			−0.609353	−	0.792899i	$-0.708571\pi$
$67$	−366542.			−1.21871			−0.609353	+	0.792899i	$0.708571\pi$
$68$		−	117371.i		−	0.373281i
$69$	−73659.6	+	44172.4i	−0.224224	+	0.134463i
$70$	77432.4			0.225751
$71$			270730.i			0.756418i	0.925720	+	0.378209i	$0.123460\pi$
$71$			270730.i			0.756418i	−0.925720	+	0.378209i	$0.876540\pi$
$72$	200733.	+	107178.i	0.537801	+	0.287150i
$73$	−253432.			−0.651468			−0.325734	−	0.945462i	$-0.605611\pi$
$73$	−253432.			−0.651468			−0.325734	+	0.945462i	$0.605611\pi$
$74$		−	267542.i		−	0.660232i
$75$	−202873.	−	338301.i	−0.480885	−	0.801899i
$76$	252954.			0.576237
$77$		−	306075.i		−	0.670434i
$78$	−42570.3	+	25528.6i	−0.0897063	+	0.0537953i
$79$	−197960.			−0.401510			−0.200755	−	0.979641i	$-0.564340\pi$
$79$	−197960.			−0.401510			−0.200755	+	0.979641i	$0.564340\pi$
$80$			163122.i			0.318598i
$81$	−295650.	−	441612.i	−0.556317	−	0.830970i
$82$	−974034.			−1.76658
$83$			397030.i			0.694367i	0.937797	+	0.347184i	$0.112862\pi$
$83$			397030.i			0.694367i	−0.937797	+	0.347184i	$0.887138\pi$
$84$	−110709.	−	184614.i	−0.186787	−	0.311477i
$85$	116240.			0.189278
$86$			279936.i			0.440112i
$87$	−104870.	+	62888.5i	−0.159255	+	0.0955023i
$88$	385503.			0.565691
$89$			147749.i			0.209583i	0.994494	+	0.104791i	$0.0334175\pi$
$89$			147749.i			0.209583i	−0.994494	+	0.104791i	$0.966583\pi$
$90$	107279.	−	200922.i	0.147159	−	0.275613i
$91$	−46461.1			−0.0616546
$92$			102335.i			0.131420i
$93$	69716.4	+	116256.i	0.0866735	+	0.144532i
$94$	191523.			0.230588
$95$			250517.i			0.292191i
$96$	700046.	−	419805.i	0.791249	−	0.474498i
$97$	−139941.			−0.153331			−0.0766656	−	0.997057i	$-0.524427\pi$
$97$	−139941.			−0.153331			−0.0766656	+	0.997057i	$0.524427\pi$
$98$			551408.i			0.585861i
$99$	−794203.	−	424052.i	−0.818514	−	0.437032i
$100$	−470001.			−0.470001
$101$		−	946009.i		−	0.918187i	−0.888388	−	0.459093i	$-0.848174\pi$
$101$		−	946009.i		−	0.918187i	0.888388	−	0.459093i	$-0.151826\pi$
$102$	−496830.	−	828489.i	−0.468174	−	0.780704i
$103$	−1.71704e6			−1.57133			−0.785666	−	0.618651i	$-0.787680\pi$
$103$	−1.71704e6			−1.57133			−0.785666	+	0.618651i	$0.787680\pi$
$104$		−	58517.9i		−	0.0520222i
$105$	182835.	−	109643.i	0.157939	−	0.0947135i
$106$	2.85697e6			2.39877
$107$		−	968294.i		−	0.790416i	−0.918592	−	0.395208i	$-0.870672\pi$
$107$		−	968294.i		−	0.790416i	0.918592	−	0.395208i	$-0.129328\pi$
$108$	−632417.	+	31496.0i	−0.502033	+	0.0250025i
$109$	414004.			0.319687			0.159844	−	0.987142i	$-0.448901\pi$
$109$	414004.			0.319687			0.159844	+	0.987142i	$0.448901\pi$
$110$		−	385865.i		−	0.289906i
$111$	−378834.	−	631724.i	−0.277000	−	0.461911i
$112$	1.26889e6			0.903174
$113$			1.03894e6i			0.720035i	0.932946	+	0.360018i	$0.117229\pi$
$113$			1.03894e6i			0.720035i	−0.932946	+	0.360018i	$0.882771\pi$
$114$	1.78553e6	−	1.07075e6i	1.20518	−	0.722726i
$115$	−101349.			−0.0666386
$116$			145695.i			0.0933407i
$117$	−64369.6	+	120557.i	−0.0401905	+	0.0752724i
$118$	2.51482e6			1.53060
$119$		−	904211.i		−	0.536573i
$120$	138095.	+	230281.i	0.0799162	+	0.133264i
$121$	246313.			0.139038
$122$			2.17909e6i			1.20004i
$123$	−2.29990e6	+	1.37921e6i	−1.23593	+	0.741166i
$124$	161514.			0.0847118
$125$		−	963284.i		−	0.493201i
$126$	−1.56293e6	−	834500.i	−0.781318	−	0.417172i
$127$	490061.			0.239243			0.119621	−	0.992820i	$-0.461832\pi$
$127$	490061.			0.239243			0.119621	+	0.992820i	$0.461832\pi$
$128$			2.24085e6i			1.06852i
$129$	396384.	+	660989.i	0.184649	+	0.307911i
$130$	−58572.9			−0.0266604
$131$			2.48801e6i			1.10672i	0.832942	+	0.553360i	$0.186655\pi$
$131$			2.48801e6i			1.10672i	−0.832942	+	0.553360i	$0.813345\pi$
$132$	−919974.	+	551692.i	−0.399994	+	0.239870i
$133$	1.94872e6			0.828314
$134$			3.59454e6i			1.49392i
$135$	−31192.5	−	626323.i	−0.0126779	−	0.254564i
$136$	1.13886e6			0.452744
$137$			1.35615e6i			0.527407i	0.964604	+	0.263704i	$0.0849441\pi$
$137$			1.35615e6i			0.527407i	−0.964604	+	0.263704i	$0.915056\pi$
$138$	433182.	+	722352.i	0.164829	+	0.274860i
$139$	1.76507e6			0.657232			0.328616	−	0.944464i	$-0.393418\pi$
$139$	1.76507e6			0.657232			0.328616	+	0.944464i	$0.393418\pi$
$140$		−	254012.i		−	0.0925698i
$141$	452226.	−	271192.i	0.161324	−	0.0967430i
$142$	2.65495e6			0.927239
$143$			231527.i			0.0791760i
$144$	1.75799e6	−	3.29253e6i	0.588747	−	1.10266i
$145$	−144291.			−0.0473300
$146$			2.48531e6i			0.798587i
$147$	780782.	+	1.30199e6i	0.245798	+	0.409880i
$148$	−877652.			−0.270731
$149$		−	875793.i		−	0.264754i	−0.991199	−	0.132377i	$-0.957739\pi$
$149$		−	875793.i		−	0.264754i	0.991199	−	0.132377i	$-0.0422610\pi$
$150$	−3.31759e6	+	1.98950e6i	−0.982991	+	0.589482i
$151$	−5.64316e6			−1.63905			−0.819524	−	0.573045i	$-0.805762\pi$
$151$	−5.64316e6			−1.63905			−0.819524	+	0.573045i	$0.805762\pi$
$152$			2.45442e6i			0.698905i
$153$	−2.34625e6	−	1.25274e6i	−0.655087	−	0.349773i
$154$	−3.00157e6			−0.821837
$155$			159957.i			0.0429545i
$156$	83744.9	+	139649.i	0.0220589	+	0.0367844i
$157$	1.94763e6			0.503277			0.251639	−	0.967821i	$-0.419031\pi$
$157$	1.94763e6			0.503277			0.251639	+	0.967821i	$0.419031\pi$
$158$			1.94132e6i			0.492182i
$159$	6.74592e6	−	4.04541e6i	1.67822	−	1.00640i
$160$	963201.			0.235156
$161$			788374.i			0.188910i
$162$	−4.33072e6	+	2.89933e6i	−1.01863	+	0.681949i
$163$	−1.99137e6			−0.459821			−0.229910	−	0.973212i	$-0.573843\pi$
$163$	−1.99137e6			−0.459821			−0.229910	+	0.973212i	$0.573843\pi$
$164$			3.19525e6i			0.724391i
$165$	−546376.	−	911109.i	−0.121630	−	0.202824i
$166$	3.89353e6			0.851175
$167$		−	2.16621e6i		−	0.465105i	−0.972584	−	0.232553i	$-0.925292\pi$
$167$		−	2.16621e6i		−	0.465105i	0.972584	−	0.232553i	$-0.0747078\pi$
$168$	1.79131e6	−	1.07422e6i	0.377783	−	0.226550i
$169$	−4.79166e6			−0.992719
$170$		−	1.13993e6i		−	0.232022i
$171$	2.69986e6	−	5.05654e6i	0.539948	−	1.01127i
$172$	918310.			0.180470
$173$		−	5.92200e6i		−	1.14375i	−0.820341	−	0.571874i	$-0.806217\pi$
$173$		−	5.92200e6i		−	1.14375i	0.820341	−	0.571874i	$-0.193783\pi$
$174$	616724.	+	1.02842e6i	0.117069	+	0.195219i
$175$	−3.62082e6			−0.675604
$176$		−	6.32321e6i		−	1.15984i
$177$	5.93803e6	−	3.56093e6i	1.07083	−	0.642161i
$178$	1.44892e6			0.256912
$179$		−	7.10375e6i		−	1.23859i	−0.785157	−	0.619296i	$-0.787418\pi$
$179$		−	7.10375e6i		−	1.23859i	0.785157	−	0.619296i	$-0.212582\pi$
$180$	−659109.	−	351920.i	−0.113016	−	0.0603430i
$181$	−4.42218e6			−0.745763			−0.372882	−	0.927879i	$-0.621630\pi$
$181$	−4.42218e6			−0.745763			−0.372882	+	0.927879i	$0.621630\pi$
$182$			455627.i			0.0755780i
$183$	3.08554e6	+	5.14529e6i	0.503475	+	0.839570i
$184$	−992959.			−0.159396
$185$		−	869196.i		−	0.137278i
$186$	1.14008e6	−	683683.i	0.177172	−	0.106247i
$187$	−4.50590e6			−0.689060
$188$		−	628277.i		−	0.0945534i
$189$	−4.87205e6	+	242640.i	−0.721649	+	0.0359399i
$190$	2.45673e6			0.358176
$191$		−	7.46641e6i		−	1.07155i	−0.844361	−	0.535774i	$-0.820020\pi$
$191$		−	7.46641e6i		−	1.07155i	0.844361	−	0.535774i	$-0.179980\pi$
$192$	433255.	+	722475.i	0.0612125	+	0.102075i
$193$	1.06116e7			1.47608			0.738041	−	0.674756i	$-0.235751\pi$
$193$	1.06116e7			1.47608			0.738041	+	0.674756i	$0.235751\pi$
$194$			1.37235e6i			0.187958i
$195$	−138303.	+	82938.0i	−0.0186521	+	0.0111853i
$196$	1.80885e6			0.240235
$197$		−	1.02633e7i		−	1.34242i	−0.741266	−	0.671211i	$-0.765774\pi$
$197$		−	1.02633e7i		−	1.34242i	0.741266	−	0.671211i	$-0.234226\pi$
$198$	−4.15852e6	+	7.78846e6i	−0.535726	+	1.00336i
$199$	996251.			0.126418			0.0632091	−	0.998000i	$-0.479867\pi$
$199$	996251.			0.126418			0.0632091	+	0.998000i	$0.479867\pi$
$200$		−	4.56043e6i		−	0.570054i
$201$	5.08979e6	+	8.48747e6i	0.626775	+	1.04518i
$202$	−9.27715e6			−1.12554
$203$			1.12241e6i			0.134173i
$204$	−2.71780e6	+	1.62982e6i	−0.320130	+	0.191976i
$205$	−3.16446e6			−0.367315
$206$			1.68383e7i			1.92618i
$207$	2.04567e6	+	1.09225e6i	0.230635	+	0.123144i
$208$	−959841.			−0.106662
$209$		−	9.71096e6i		−	1.06371i
$210$	−1.07523e6	−	1.79299e6i	−0.116102	−	0.193607i
$211$	−1.71010e7			−1.82043			−0.910213	−	0.414140i	$-0.864082\pi$
$211$	−1.71010e7			−1.82043			−0.910213	+	0.414140i	$0.864082\pi$
$212$		−	9.37208e6i		−	0.983623i
$213$	6.26891e6	−	3.75936e6i	0.648714	−	0.389022i
$214$	−9.49570e6			−0.968915
$215$			909462.i			0.0915101i
$216$	−305606.	−	6.13636e6i	−0.0303250	−	0.608905i
$217$	1.24428e6			0.121769
$218$		−	4.05998e6i		−	0.391882i
$219$	3.51915e6	+	5.86836e6i	0.335047	+	0.558707i
$220$	−1.26580e6			−0.118877
$221$			683980.i			0.0633675i
$222$	−6.19508e6	+	3.71508e6i	−0.566224	+	0.339555i
$223$	−1.35473e7			−1.22163			−0.610813	−	0.791775i	$-0.709157\pi$
$223$	−1.35473e7			−1.22163			−0.610813	+	0.791775i	$0.709157\pi$
$224$		−	7.49255e6i		−	0.666631i
$225$	−5.01646e6	+	9.39529e6i	−0.440402	+	0.824826i
$226$	1.01885e7			0.882640
$227$			1.21661e7i			1.04010i	0.854137	+	0.520048i	$0.174086\pi$
$227$			1.21661e7i			1.04010i	−0.854137	+	0.520048i	$0.825914\pi$
$228$	−3.51252e6	−	5.85730e6i	−0.296356	−	0.494189i
$229$	−106942.			−0.00890513			−0.00445257	−	0.999990i	$-0.501417\pi$
$229$	−106942.			−0.00890513			−0.00445257	+	0.999990i	$0.501417\pi$
$230$			993892.i			0.0816875i
$231$	−7.08734e6	+	4.25015e6i	−0.574973	+	0.344801i
$232$	−1.41368e6			−0.113211
$233$			6.22858e6i			0.492403i	0.969219	+	0.246202i	$0.0791826\pi$
$233$			6.22858e6i			0.492403i	−0.969219	+	0.246202i	$0.920817\pi$
$234$	1.18226e6	+	631248.i	0.0922710	+	0.0492666i
$235$	622223.			0.0479449
$236$		−	8.24969e6i		−	0.627627i
$237$	2.74887e6	+	4.58387e6i	0.206495	+	0.344340i
$238$	−8.86726e6			−0.657747
$239$			1.79738e7i			1.31658i	0.752765	+	0.658289i	$0.228720\pi$
$239$			1.79738e7i			1.31658i	−0.752765	+	0.658289i	$0.771280\pi$
$240$	3.77718e6	−	2.26511e6i	0.273234	−	0.163853i
$241$	−1.18320e7			−0.845289			−0.422644	−	0.906296i	$-0.638898\pi$
$241$	−1.18320e7			−0.845289			−0.422644	+	0.906296i	$0.638898\pi$
$242$		−	2.41550e6i		−	0.170436i
$243$	−6.12037e6	+	1.29781e7i	−0.426539	+	0.904469i
$244$	7.14834e6			0.492080
$245$			1.79143e6i			0.121815i
$246$	1.35254e7	+	2.25543e7i	0.908543	+	1.51504i
$247$	−1.47409e6			−0.0978211
$248$			1.56717e6i			0.102745i
$249$	9.19345e6	−	5.51315e6i	0.595498	−	0.357110i
$250$	−9.44657e6			−0.604580
$251$		−	1.53641e7i		−	0.971596i	−0.874071	−	0.485798i	$-0.838529\pi$
$251$		−	1.53641e7i		−	0.971596i	0.874071	−	0.485798i	$-0.161471\pi$
$252$	−2.73752e6	+	5.12708e6i	−0.171063	+	0.320382i
$253$	3.92866e6			0.242596
$254$		−	4.80585e6i		−	0.293271i
$255$	−1.61411e6	−	2.69161e6i	−0.0973448	−	0.162327i
$256$	1.99783e7			1.19080
$257$			2.60637e7i			1.53546i	0.640776	+	0.767728i	$0.278613\pi$
$257$			2.60637e7i			1.53546i	−0.640776	+	0.767728i	$0.721387\pi$
$258$	6.48208e6	−	3.88719e6i	0.377446	−	0.226348i
$259$	−6.76130e6			−0.389163
$260$			192144.i			0.0109322i
$261$	2.91244e6	+	1.55505e6i	0.163808	+	0.0874626i
$262$	2.43990e7			1.35665
$263$			1.23092e6i			0.0676649i	0.999428	+	0.0338324i	$0.0107713\pi$
$263$			1.23092e6i			0.0676649i	−0.999428	+	0.0338324i	$0.989229\pi$
$264$	−5.35308e6	−	8.92653e6i	−0.290932	−	0.485144i
$265$	9.28178e6			0.498762
$266$		−	1.91104e7i		−	1.01537i
$267$	3.42122e6	−	2.05164e6i	0.179741	−	0.107787i
$268$	1.17916e7			0.612589
$269$		−	1.21453e7i		−	0.623952i	−0.950090	−	0.311976i	$-0.899009\pi$
$269$		−	1.21453e7i		−	0.623952i	0.950090	−	0.311976i	$-0.100991\pi$
$270$	−6.14212e6	+	305893.i	−0.312052	+	0.0155410i
$271$	3.72092e7			1.86957			0.934787	−	0.355210i	$-0.115591\pi$
$271$	3.72092e7			1.86957			0.934787	+	0.355210i	$0.115591\pi$
$272$		−	1.86801e7i		−	0.928266i
$273$	645158.	+	1.07583e6i	0.0317087	+	0.0528758i
$274$	1.32993e7			0.646511
$275$			1.80434e7i			0.867602i
$276$	2.36963e6	−	1.42102e6i	0.112707	−	0.0675886i
$277$	3.67120e7			1.72731			0.863653	−	0.504088i	$-0.168171\pi$
$277$	3.67120e7			1.72731			0.863653	+	0.504088i	$0.168171\pi$
$278$		−	1.73094e7i		−	0.805654i
$279$	1.72388e6	−	3.22864e6i	0.0793770	−	0.148665i
$280$	2.46468e6			0.112276
$281$		−	2.53156e7i		−	1.14096i	−0.821313	−	0.570478i	$-0.806758\pi$
$281$		−	2.53156e7i		−	1.14096i	0.821313	−	0.570478i	$-0.193242\pi$
$282$	−2.65948e6	−	4.43481e6i	−0.118590	−	0.197755i
$283$	−2.09587e7			−0.924710			−0.462355	−	0.886695i	$-0.652995\pi$
$283$	−2.09587e7			−0.924710			−0.462355	+	0.886695i	$0.652995\pi$
$284$		−	8.70938e6i		−	0.380218i
$285$	5.80086e6	−	3.47867e6i	0.250587	−	0.150272i
$286$	2.27050e6			0.0970562
$287$			2.46157e7i			1.04128i
$288$	−1.94417e7	−	1.03806e7i	−0.813871	−	0.434553i
$289$	1.08262e7			0.448520
$290$			1.41501e6i			0.0580184i
$291$	1.94322e6	+	3.24042e6i	0.0788576	+	0.131499i
$292$	8.15289e6			0.327464
$293$		−	2.69780e7i		−	1.07252i	−0.844052	−	0.536262i	$-0.819836\pi$
$293$		−	2.69780e7i		−	1.07252i	0.844052	−	0.536262i	$-0.180164\pi$
$294$	1.27682e7	−	7.65684e6i	0.502442	−	0.301306i
$295$	8.17019e6			0.318249
$296$		−	8.51588e6i		−	0.328363i
$297$	1.20913e6	+	2.42786e7i	0.0461536	+	0.926732i
$298$	−8.58858e6			−0.324543
$299$		−	596356.i		−	0.0223096i
$300$	6.52642e6	+	1.08831e7i	0.241719	+	0.403079i
$301$	7.07453e6			0.259417
$302$			5.53404e7i			2.00919i
$303$	−2.19054e7	+	1.31363e7i	−0.787449	+	0.472219i
$304$	4.02587e7			1.43298
$305$			7.07946e6i			0.249517i
$306$	−1.22851e7	+	2.30088e7i	−0.428762	+	0.803025i
$307$	−3.23320e7			−1.11742			−0.558712	−	0.829362i	$-0.688704\pi$
$307$	−3.23320e7			−1.11742			−0.558712	+	0.829362i	$0.688704\pi$
$308$			9.84642e6i			0.336997i
$309$	2.38427e7	+	3.97589e7i	0.808129	+	1.34759i
$310$	1.56864e6			0.0526549
$311$		−	2.29843e7i		−	0.764101i	−0.924141	−	0.382051i	$-0.875218\pi$
$311$		−	2.29843e7i		−	0.764101i	0.924141	−	0.382051i	$-0.124782\pi$
$312$	−1.35501e6	+	812578.i	−0.0446149	+	0.0267548i
$313$	−4.27519e7			−1.39419			−0.697096	−	0.716978i	$-0.745525\pi$
$313$	−4.27519e7			−1.39419			−0.697096	+	0.716978i	$0.745525\pi$
$314$		−	1.90997e7i		−	0.616932i
$315$	−5.07767e6	−	2.71114e6i	−0.162455	−	0.0867402i
$316$	6.36836e6			0.201821
$317$			2.16841e7i			0.680714i	0.940296	+	0.340357i	$0.110548\pi$
$317$			2.16841e7i			0.680714i	−0.940296	+	0.340357i	$0.889452\pi$
$318$	−3.96718e7	−	6.61547e7i	−1.23367	−	2.05721i
$319$	5.59326e6			0.172303
$320$			994060.i			0.0303363i
$321$	−2.24214e7	+	1.34457e7i	−0.677871	+	0.406508i
$322$	7.73129e6			0.231571
$323$		−	2.86882e7i		−	0.851326i
$324$	9.51104e6	+	1.42066e7i	0.279636	+	0.417691i
$325$	2.73893e6			0.0797866
$326$			1.95286e7i			0.563662i
$327$	−5.74885e6	−	9.58649e6i	−0.164414	−	0.274168i
$328$	−3.10036e7			−0.878598
$329$		−	4.84015e6i		−	0.135916i
$330$	−8.93491e6	+	5.35811e6i	−0.248627	+	0.149097i
$331$	−5.08850e7			−1.40316			−0.701578	−	0.712593i	$-0.747521\pi$
$331$	−5.08850e7			−1.40316			−0.701578	+	0.712593i	$0.747521\pi$
$332$		−	1.27724e7i		−	0.349027i
$333$	−9.36744e6	+	1.75442e7i	−0.253681	+	0.475118i
$334$	−2.12432e7			−0.570139
$335$			1.16780e7i			0.310623i
$336$	−1.76198e7	−	2.93820e7i	−0.464498	−	0.774574i
$337$	5.64419e7			1.47473			0.737364	−	0.675495i	$-0.236070\pi$
$337$	5.64419e7			1.47473			0.737364	+	0.675495i	$0.236070\pi$
$338$			4.69901e7i			1.21690i
$339$	2.40572e7	−	1.44266e7i	0.617512	−	0.370311i
$340$	−3.73945e6			−0.0951416
$341$		−	6.20053e6i		−	0.156374i
$342$	−4.95876e7	−	2.64765e7i	−1.23964	−	0.661884i
$343$	4.30924e7			1.06787
$344$			8.91038e6i			0.218888i
$345$	1.40733e6	+	2.34679e6i	0.0342719	+	0.0571501i
$346$	−5.80749e7			−1.40204
$347$		−	2.35217e7i		−	0.562963i	−0.959567	−	0.281482i	$-0.909174\pi$
$347$		−	2.35217e7i		−	0.562963i	0.959567	−	0.281482i	$-0.0908258\pi$
$348$	3.37365e6	−	2.02312e6i	0.0800502	−	0.0480047i
$349$	1.51034e7			0.355302			0.177651	−	0.984094i	$-0.443150\pi$
$349$	1.51034e7			0.355302			0.177651	+	0.984094i	$0.443150\pi$
$350$			3.55080e7i			0.828175i
$351$	3.68540e6	−	183542.i	0.0852244	−	0.00424439i
$352$	−3.73372e7			−0.856079
$353$			3.41559e7i			0.776501i	0.921554	+	0.388251i	$0.126920\pi$
$353$			3.41559e7i			0.776501i	−0.921554	+	0.388251i	$0.873080\pi$
$354$	−3.49207e7	−	5.82320e7i	−0.787179	−	1.31266i
$355$	8.62546e6			0.192796
$356$		−	4.75309e6i		−	0.105348i
$357$	−2.09375e7	+	1.25559e7i	−0.460172	+	0.275957i
$358$	−6.96638e7			−1.51830
$359$		−	7.94032e6i		−	0.171615i	−0.996312	−	0.0858074i	$-0.972653\pi$
$359$		−	7.94032e6i		−	0.171615i	0.996312	−	0.0858074i	$-0.0273470\pi$
$360$	3.41469e6	−	6.39534e6i	0.0731886	−	0.137074i
$361$	1.47819e7			0.314202
$362$			4.33667e7i			0.914178i
$363$	−3.42030e6	−	5.70353e6i	−0.0715064	−	0.119240i
$364$	1.49465e6			0.0309910
$365$			8.07433e6i			0.166046i
$366$	5.04580e7	−	3.02587e7i	1.02917	−	0.617174i
$367$	−7.30410e6			−0.147764			−0.0738820	−	0.997267i	$-0.523539\pi$
$367$	−7.30410e6			−0.147764			−0.0738820	+	0.997267i	$0.523539\pi$
$368$			1.62870e7i			0.326812i
$369$	6.38728e7	+	3.41038e7i	1.27127	+	0.678773i
$370$	−8.52388e6			−0.168280
$371$		−	7.22011e7i		−	1.41391i
$372$	−2.24277e6	−	3.73993e6i	−0.0435669	−	0.0726499i
$373$	3.98886e7			0.768638			0.384319	−	0.923200i	$-0.374436\pi$
$373$	3.98886e7			0.768638			0.384319	+	0.923200i	$0.374436\pi$
$374$			4.41877e7i			0.844670i
$375$	−2.23054e7	+	1.33761e7i	−0.422976	+	0.253651i
$376$	6.09618e6			0.114682
$377$		−	849037.i		−	0.0158454i
$378$	2.37948e6	+	4.77783e7i	0.0440562	+	0.884618i
$379$	9.74833e7			1.79066			0.895329	−	0.445405i	$-0.146940\pi$
$379$	9.74833e7			1.79066			0.895329	+	0.445405i	$0.146940\pi$
$380$		−	8.05911e6i		−	0.146871i
$381$	−6.80498e6	−	1.13476e7i	−0.123042	−	0.205178i
$382$	−7.32203e7			−1.31353
$383$		−	6.25259e7i		−	1.11292i	−0.830875	−	0.556459i	$-0.812160\pi$
$383$		−	6.25259e7i		−	1.11292i	0.830875	−	0.556459i	$-0.187840\pi$
$384$	5.18880e7	−	3.11163e7i	0.916375	−	0.549534i
$385$	−9.75155e6			−0.170880
$386$		−	1.04064e8i		−	1.80942i
$387$	9.80140e6	−	1.83570e7i	0.169105	−	0.316715i
$388$	4.50190e6			0.0770728
$389$			1.44314e7i			0.245166i	0.992458	+	0.122583i	$0.0391178\pi$
$389$			1.44314e7i			0.245166i	−0.992458	+	0.122583i	$0.960882\pi$
$390$	813342.	+	1.35629e6i	0.0137113	+	0.0228643i
$391$	1.16061e7			0.194158
$392$			1.75514e7i			0.291375i
$393$	5.76112e7	−	3.45484e7i	0.949138	−	0.569182i
$394$	−1.00649e8			−1.64558
$395$			6.30700e6i			0.102337i
$396$	2.55495e7	+	1.36417e7i	0.411430	+	0.219677i
$397$	−6.06096e7			−0.968658			−0.484329	−	0.874886i	$-0.660936\pi$
$397$	−6.06096e7			−0.968658			−0.484329	+	0.874886i	$0.660936\pi$
$398$		−	9.76986e6i		−	0.154967i
$399$	−2.70599e7	−	4.51237e7i	−0.425998	−	0.710373i
$400$	−7.48025e7			−1.16879
$401$			7.19363e6i			0.111562i	0.998443	+	0.0557809i	$0.0177648\pi$
$401$			7.19363e6i			0.111562i	−0.998443	+	0.0557809i	$0.982235\pi$
$402$	8.32335e7	−	4.99137e7i	1.28121	−	0.768318i
$403$	−941218.			−0.0143805
$404$			3.04330e7i			0.461531i
$405$	−1.40697e7	+	9.41939e6i	−0.211797	+	0.141794i
$406$	1.10071e7			0.164473
$407$			3.36932e7i			0.499757i
$408$	−1.58141e7	−	2.63709e7i	−0.232844	−	0.388279i
$409$	−6.65153e7			−0.972191			−0.486096	−	0.873906i	$-0.661579\pi$
$409$	−6.65153e7			−0.972191			−0.486096	+	0.873906i	$0.661579\pi$
$410$			3.10327e7i			0.450265i
$411$	3.14024e7	−	1.88315e7i	0.452311	−	0.271243i
$412$	5.52370e7			0.789838
$413$		−	6.35543e7i		−	0.902184i
$414$	1.07113e7	−	2.00611e7i	0.150953	−	0.282719i
$415$	1.26494e7			0.176980
$416$			5.66765e6i			0.0787269i
$417$	−2.45098e7	−	4.08713e7i	−0.338011	−	0.563651i
$418$	−9.52317e7			−1.30393
$419$		−	8.77437e7i		−	1.19282i	−0.802681	−	0.596408i	$-0.796594\pi$
$419$		−	8.77437e7i		−	1.19282i	0.802681	−	0.596408i	$-0.203406\pi$
$420$	−5.88178e6	+	3.52720e6i	−0.0793891	+	0.0476082i
$421$	9.26942e6			0.124224			0.0621121	−	0.998069i	$-0.480216\pi$
$421$	9.26942e6			0.124224			0.0621121	+	0.998069i	$0.480216\pi$
$422$			1.67703e8i			2.23153i
$423$	−1.25592e7	−	6.70578e6i	−0.165936	−	0.0885989i
$424$	9.09375e7			1.19301
$425$			5.33041e7i			0.694374i
$426$	−3.68666e7	−	6.14769e7i	−0.476875	−	0.795212i
$427$	5.50697e7			0.707342
$428$			3.11500e7i			0.397307i
$429$	5.36114e6	−	3.21498e6i	0.0679024	−	0.0407199i
$430$	8.91875e6			0.112176
$431$			8.61114e7i			1.07555i	0.843090	+	0.537773i	$0.180734\pi$
$431$			8.61114e7i			1.07555i	−0.843090	+	0.537773i	$0.819266\pi$
$432$	−1.00652e8	+	5.01270e6i	−1.24845	+	0.0621757i
$433$	6.72358e7			0.828204			0.414102	−	0.910231i	$-0.364096\pi$
$433$	6.72358e7			0.828204			0.414102	+	0.910231i	$0.364096\pi$
$434$		−	1.22022e7i		−	0.149268i
$435$	2.00363e6	+	3.34115e6i	0.0243416	+	0.0405908i
$436$	−1.33185e7			−0.160692
$437$			2.50130e7i			0.299724i
$438$	5.75488e7	−	3.45110e7i	0.684879	−	0.410710i
$439$	−2.84352e7			−0.336096			−0.168048	−	0.985779i	$-0.553746\pi$
$439$	−2.84352e7			−0.336096			−0.168048	+	0.985779i	$0.553746\pi$
$440$		−	1.22821e7i		−	0.144183i
$441$	1.93064e7	−	3.61589e7i	0.225106	−	0.421599i
$442$	6.70754e6			0.0776777
$443$		−	1.31152e8i		−	1.50857i	−0.656550	−	0.754283i	$-0.727985\pi$
$443$		−	1.31152e8i		−	1.50857i	0.656550	−	0.754283i	$-0.272015\pi$
$444$	1.21871e7	+	2.03225e7i	0.139236	+	0.232182i
$445$	4.70729e6			0.0534184
$446$			1.32853e8i			1.49750i
$447$	−2.02795e7	+	1.21612e7i	−0.227057	+	0.136162i
$448$	7.73260e6			0.0859986
$449$			1.20424e8i			1.33037i	0.746678	+	0.665186i	$0.231648\pi$
$449$			1.20424e8i			1.33037i	−0.746678	+	0.665186i	$0.768352\pi$
$450$	9.21361e7	+	4.91946e7i	1.01110	+	0.539858i
$451$	1.22666e8			1.33720
$452$		−	3.34225e7i		−	0.361929i
$453$	7.83608e7	+	1.30670e8i	0.842955	+	1.40567i
$454$	1.19308e8			1.27498
$455$			1.48025e6i			0.0157145i
$456$	5.68335e7	−	3.40820e7i	0.599390	−	0.359444i
$457$	−1.22301e8			−1.28139			−0.640696	−	0.767794i	$-0.721354\pi$
$457$	−1.22301e8			−1.28139			−0.640696	+	0.767794i	$0.721354\pi$
$458$			1.04874e6i			0.0109162i
$459$	3.57204e6	+	7.17241e7i	0.0369384	+	0.741698i
$460$	3.26039e6			0.0334963
$461$			3.21147e7i			0.327794i	0.986477	+	0.163897i	$0.0524065\pi$
$461$			3.21147e7i			0.327794i	−0.986477	+	0.163897i	$0.947594\pi$
$462$	4.16797e7	+	6.95029e7i	0.422667	+	0.704818i
$463$	−1.20156e8			−1.21061			−0.605304	−	0.795995i	$-0.706948\pi$
$463$	−1.20156e8			−1.21061			−0.605304	+	0.795995i	$0.706948\pi$
$464$			2.31880e7i			0.232118i
$465$	3.70390e6	−	2.22116e6i	0.0368384	−	0.0220913i
$466$	6.10813e7			0.603602
$467$			1.87217e8i			1.83821i	0.394015	+	0.919104i	$0.371086\pi$
$467$			1.87217e8i			1.83821i	−0.394015	+	0.919104i	$0.628914\pi$
$468$	2.07076e6	−	3.87832e6i	0.0202019	−	0.0378361i
$469$	9.08409e7			0.880568
$470$		−	6.10191e6i		−	0.0587722i
$471$	−2.70447e7	−	4.50985e7i	−0.258833	−	0.431617i
$472$	8.00469e7			0.761234
$473$		−	3.52541e7i		−	0.333139i
$474$	4.49523e7	−	2.69571e7i	0.422102	−	0.253127i
$475$	−1.14879e8			−1.07191
$476$			2.90884e7i			0.269711i
$477$	−1.87347e8	−	1.00031e8i	−1.72621	−	0.921679i
$478$	1.76263e8			1.61390
$479$		−	1.72673e8i		−	1.57115i	−0.618764	−	0.785577i	$-0.712366\pi$
$479$		−	1.72673e8i		−	1.57115i	0.618764	−	0.785577i	$-0.287634\pi$
$480$	−1.33750e7	−	2.23035e7i	−0.120940	−	0.201673i
$481$	5.11451e6			0.0459588
$482$			1.16032e8i			1.03618i
$483$	1.82552e7	−	1.09473e7i	0.162012	−	0.0971555i
$484$	−7.92389e6			−0.0698880
$485$			4.45852e6i			0.0390810i
$486$	1.27272e8	+	6.00202e7i	1.10872	+	0.522864i
$487$	−1.89868e8			−1.64386			−0.821932	−	0.569586i	$-0.807104\pi$
$487$	−1.89868e8			−1.64386			−0.821932	+	0.569586i	$0.807104\pi$
$488$			6.93605e7i			0.596833i
$489$	2.76521e7	+	4.61112e7i	0.236484	+	0.394348i
$490$	1.75678e7			0.149324
$491$		−	4.55181e7i		−	0.384539i	−0.981342	−	0.192269i	$-0.938415\pi$
$491$		−	4.55181e7i		−	0.384539i	0.981342	−	0.192269i	$-0.0615847\pi$
$492$	7.39878e7	−	4.43692e7i	0.621248	−	0.372551i
$493$	1.65237e7			0.137901
$494$			1.44558e7i			0.119912i
$495$	−1.35103e7	+	2.53033e7i	−0.111391	+	0.208623i
$496$	2.57055e7			0.210660
$497$		−	6.70958e7i		−	0.546545i
$498$	−5.40654e7	−	9.01567e7i	−0.437755	−	0.729979i
$499$	−6.85454e7			−0.551666			−0.275833	−	0.961206i	$-0.588954\pi$
$499$	−6.85454e7			−0.551666			−0.275833	+	0.961206i	$0.588954\pi$
$500$			3.09888e7i			0.247910i
$501$	−5.01598e7	+	3.00799e7i	−0.398880	+	0.239202i
$502$	−1.50670e8			−1.19101
$503$		−	4.84361e6i		−	0.0380597i	−0.999819	−	0.0190298i	$-0.993942\pi$
$503$		−	4.84361e6i		−	0.0380597i	0.999819	−	0.0190298i	$-0.00605775\pi$
$504$	−4.97481e7	−	2.65622e7i	−0.388584	−	0.207478i
$505$	−3.01398e7			−0.234027
$506$		−	3.85269e7i		−	0.297381i
$507$	6.65369e7	+	1.10954e8i	0.510551	+	0.851369i
$508$	−1.57652e7			−0.120257
$509$		−	5.06611e7i		−	0.384168i	−0.981378	−	0.192084i	$-0.938475\pi$
$509$		−	5.06611e7i		−	0.384168i	0.981378	−	0.192084i	$-0.0615246\pi$
$510$	−2.63956e7	+	1.58290e7i	−0.198985	+	0.119328i
$511$	6.28086e7			0.470714
$512$		−	5.25054e7i		−	0.391196i
$513$	−1.54577e8	+	7.69833e6i	−1.14497	+	0.0570223i
$514$	2.55597e8			1.88221
$515$			5.47047e7i			0.400501i
$516$	−1.27516e7	−	2.12640e7i	−0.0928147	−	0.154773i
$517$	−2.41196e7			−0.174542
$518$			6.63056e7i			0.477047i
$519$	−1.37127e8	+	8.22328e7i	−0.980893	+	0.588225i
$520$	−1.86438e6			−0.0132594
$521$		−	1.13166e8i		−	0.800207i	−0.916470	−	0.400104i	$-0.868974\pi$
$521$		−	1.13166e8i		−	0.800207i	0.916470	−	0.400104i	$-0.131026\pi$
$522$	1.52498e7	−	2.85612e7i	0.107214	−	0.200800i
$523$	6.70587e7			0.468760			0.234380	−	0.972145i	$-0.424694\pi$
$523$	6.70587e7			0.468760			0.234380	+	0.972145i	$0.424694\pi$
$524$		−	8.00391e7i		−	0.556299i
$525$	5.02786e7	+	8.38420e7i	0.347460	+	0.579407i
$526$	1.20712e7			0.0829456
$527$		−	1.83177e7i		−	0.125152i
$528$	−1.46417e8	+	8.78040e7i	−0.994698	+	0.596503i
$529$	1.37917e8			0.931643
$530$		−	9.10229e7i		−	0.611397i
$531$	−1.64911e8	−	8.80513e7i	−1.10145	−	0.588102i
$532$	−6.26903e7			−0.416356
$533$		−	1.86203e7i		−	0.122972i
$534$	−2.01197e7	−	3.35506e7i	−0.132129	−	0.220331i
$535$	−3.08498e7			−0.201461
$536$			1.14414e8i			0.742995i
$537$	−1.64491e8	+	9.86425e7i	−1.06223	+	0.637003i
$538$	−1.19104e8			−0.764858
$539$		−	6.94422e7i		−	0.443463i
$540$	1.00346e6	+	2.01488e7i	0.00637263	+	0.127958i
$541$	1.82515e8			1.15268			0.576338	−	0.817212i	$-0.304481\pi$
$541$	1.82515e8			1.15268			0.576338	+	0.817212i	$0.304481\pi$
$542$		−	3.64897e8i		−	2.29178i
$543$	6.14063e7	+	1.02398e8i	0.383543	+	0.639576i
$544$	−1.10302e8			−0.685152
$545$		−	1.31902e7i		−	0.0814818i
$546$	1.05503e7	−	6.32683e6i	0.0648167	−	0.0388694i
$547$	1.37327e8			0.839062			0.419531	−	0.907741i	$-0.362195\pi$
$547$	1.37327e8			0.839062			0.419531	+	0.907741i	$0.362195\pi$
$548$		−	4.36273e7i		−	0.265104i
$549$	7.62963e7	−	1.42895e8i	0.461091	−	0.863574i
$550$	1.76945e8			1.06353
$551$			3.56112e7i			0.212879i
$552$	1.37882e7	+	2.29925e7i	0.0819767	+	0.136700i
$553$	4.90609e7			0.290108
$554$		−	3.60021e8i		−	2.11738i
$555$	−2.01267e7	+	1.20696e7i	−0.117732	+	0.0706017i
$556$	−5.67823e7			−0.330361
$557$			2.73001e8i			1.57979i	0.613244	+	0.789894i	$0.289864\pi$
$557$			2.73001e8i			1.57979i	−0.613244	+	0.789894i	$0.710136\pi$
$558$	−3.16621e7	−	1.69055e7i	−0.182237	−	0.0973026i
$559$	−5.35144e6			−0.0306362
$560$		−	4.04269e7i		−	0.230201i
$561$	6.25689e7	+	1.04337e8i	0.354381	+	0.590947i
$562$	−2.48261e8			−1.39862
$563$		−	2.80300e8i		−	1.57072i	−0.619042	−	0.785358i	$-0.712479\pi$
$563$		−	2.80300e8i		−	1.57072i	0.619042	−	0.785358i	$-0.287521\pi$
$564$	−1.45481e7	+	8.72423e6i	−0.0810903	+	0.0486284i
$565$	3.31005e7			0.183522
$566$			2.05534e8i			1.13354i
$567$	7.32716e7	+	1.09446e8i	0.401963	+	0.600412i
$568$	8.45073e7			0.461157
$569$		−	2.41862e7i		−	0.131290i	−0.997843	−	0.0656448i	$-0.979090\pi$
$569$		−	2.41862e7i		−	0.131290i	0.997843	−	0.0656448i	$-0.0209104\pi$
$570$	−3.41140e7	−	5.68869e7i	−0.184208	−	0.307176i
$571$	−3.11828e8			−1.67497			−0.837485	−	0.546461i	$-0.815975\pi$
$571$	−3.11828e8			−1.67497			−0.837485	+	0.546461i	$0.815975\pi$
$572$		−	7.44821e6i		−	0.0397983i
$573$	−1.72889e8	+	1.03678e8i	−0.918974	+	0.551092i
$574$	2.41397e8			1.27643
$575$		−	4.64754e7i		−	0.244466i
$576$	1.07131e7	−	2.00645e7i	0.0560594	−	0.104993i
$577$	2.20391e8			1.14727			0.573637	−	0.819110i	$-0.305532\pi$
$577$	2.20391e8			1.14727			0.573637	+	0.819110i	$0.305532\pi$
$578$		−	1.06168e8i		−	0.549808i
$579$	−1.47353e8	−	2.45718e8i	−0.759142	−	1.26591i
$580$	4.64184e6			0.0237907
$581$		−	9.83969e7i		−	0.501710i
$582$	3.17776e7	−	1.90565e7i	0.161195	−	0.0966659i
$583$	−3.59796e8			−1.81573
$584$			7.91077e7i			0.397173i
$585$	3.84095e6	+	2.05081e6i	0.0191854	+	0.0102437i
$586$	−2.64563e8			−1.31473
$587$		−	2.32721e8i		−	1.15059i	−0.817945	−	0.575296i	$-0.804887\pi$
$587$		−	2.32721e8i		−	1.15059i	0.817945	−	0.575296i	$-0.195113\pi$
$588$	−2.51177e7	−	4.18850e7i	−0.123552	−	0.206028i
$589$	3.94776e7			0.193199
$590$		−	8.01221e7i		−	0.390118i
$591$	−2.37653e8	+	1.42516e8i	−1.15128	+	0.690402i
$592$	−1.39682e8			−0.673247
$593$		−	1.41260e8i		−	0.677414i	−0.940892	−	0.338707i	$-0.890011\pi$
$593$		−	1.41260e8i		−	0.677414i	0.940892	−	0.338707i	$-0.109989\pi$
$594$	2.38091e8	−	1.18575e7i	1.13601	−	0.0565764i
$595$	−2.88081e7			−0.136762
$596$			2.81742e7i			0.133080i
$597$	−1.38339e7	−	2.30687e7i	−0.0650163	−	0.108418i
$598$	−5.84825e6			−0.0273478
$599$			1.05860e8i			0.492551i	0.969200	+	0.246275i	$0.0792068\pi$
$599$			1.05860e8i			0.492551i	−0.969200	+	0.246275i	$0.920793\pi$
$600$	−1.05599e8	+	6.33260e7i	−0.488885	+	0.293176i
$601$	−1.45836e8			−0.671800			−0.335900	−	0.941898i	$-0.609040\pi$
$601$	−1.45836e8			−0.671800			−0.335900	+	0.941898i	$0.609040\pi$
$602$		−	6.93772e7i		−	0.318000i
$603$	1.25855e8	−	2.35714e8i	0.574011	−	1.07506i
$604$	1.81540e8			0.823876
$605$		−	7.84754e6i		−	0.0354378i
$606$	1.28822e8	+	2.14818e8i	0.578860	+	0.965277i
$607$	−1.23952e8			−0.554229			−0.277114	−	0.960837i	$-0.589378\pi$
$607$	−1.23952e8			−0.554229			−0.277114	+	0.960837i	$0.589378\pi$
$608$		−	2.37719e8i		−	1.05768i
$609$	2.59901e7	−	1.55858e7i	0.115068	−	0.0690045i
$610$	6.94256e7			0.305865
$611$			3.66128e6i			0.0160512i
$612$	7.54786e7	+	4.03005e7i	0.329283	+	0.175815i
$613$	4.23699e8			1.83940			0.919700	−	0.392621i	$-0.128432\pi$
$613$	4.23699e8			1.83940			0.919700	+	0.392621i	$0.128432\pi$
$614$			3.17068e8i			1.36977i
$615$	4.39416e7	+	7.32749e7i	0.188908	+	0.315014i
$616$	−9.55400e7			−0.408736
$617$			2.95729e8i			1.25904i	0.776986	+	0.629518i	$0.216748\pi$
$617$			2.95729e8i			1.25904i	−0.776986	+	0.629518i	$0.783252\pi$
$618$	3.89901e8	−	2.33817e8i	1.65192	−	0.990627i
$619$	8.50387e7			0.358546			0.179273	−	0.983799i	$-0.442625\pi$
$619$	8.50387e7			0.358546			0.179273	+	0.983799i	$0.442625\pi$
$620$		−	5.14581e6i		−	0.0215913i
$621$	−3.11443e6	−	6.25356e7i	−0.0130048	−	0.261128i
$622$	−2.25399e8			−0.936657
$623$		−	3.66171e7i		−	0.151433i
$624$	1.33283e7	+	2.22256e7i	0.0548557	+	0.0914746i
$625$	1.97590e8			0.809329
$626$			4.19252e8i			1.70904i
$627$	−2.24863e8	+	1.34846e8i	−0.912251	+	0.547061i
$628$	−6.26551e7			−0.252975
$629$			9.95369e7i			0.399974i
$630$	−2.65871e7	+	4.97949e7i	−0.106329	+	0.199142i
$631$	−3.80935e8			−1.51622			−0.758111	−	0.652126i	$-0.773877\pi$
$631$	−3.80935e8			−1.51622			−0.758111	+	0.652126i	$0.773877\pi$
$632$			6.17923e7i			0.244784i
$633$	2.37463e8	+	3.95982e8i	0.936237	+	1.56122i
$634$	2.12648e8			0.834438
$635$		−	1.56133e7i		−	0.0609781i
$636$	−2.17016e8	+	1.30141e8i	−0.843568	+	0.505873i
$637$	−1.05411e7			−0.0407818
$638$		−	5.48510e7i		−	0.211214i
$639$	−1.74100e8	−	9.29578e7i	−0.667262	−	0.356273i
$640$	7.13932e7			0.272344
$641$		−	2.77323e8i		−	1.05296i	−0.850188	−	0.526479i	$-0.823512\pi$
$641$		−	2.77323e8i		−	1.05296i	0.850188	−	0.526479i	$-0.176488\pi$
$642$	1.31857e8	+	2.19878e8i	0.498308	+	0.830954i
$643$	3.47519e8			1.30721			0.653605	−	0.756836i	$-0.273256\pi$
$643$	3.47519e8			1.30721			0.653605	+	0.756836i	$0.273256\pi$
$644$		−	2.53619e7i		−	0.0949566i
$645$	2.10591e7	−	1.26288e7i	0.0784803	−	0.0470632i
$646$	−2.81335e8			−1.04358
$647$		−	2.64335e8i		−	0.975981i	−0.872849	−	0.487991i	$-0.837730\pi$
$647$		−	2.64335e8i		−	0.975981i	0.872849	−	0.487991i	$-0.162270\pi$
$648$	−1.37847e8	+	9.22858e7i	−0.506609	+	0.339164i
$649$	−3.16707e8			−1.15857
$650$		−	2.68596e7i		−	0.0978047i
$651$	−1.72780e7	−	2.88119e7i	−0.0626254	−	0.104431i
$652$	6.40622e7			0.231131
$653$			9.01727e7i			0.323844i	0.986804	+	0.161922i	$0.0517693\pi$
$653$			9.01727e7i			0.323844i	−0.986804	+	0.161922i	$0.948231\pi$
$654$	−9.40112e7	+	5.63768e7i	−0.336083	+	0.201543i
$655$	7.92679e7			0.282081
$656$			5.08536e8i			1.80140i
$657$	8.70182e7	−	1.62976e8i	0.306842	−	0.574681i
$658$	−4.74655e7			−0.166610
$659$			4.08939e7i			0.142890i	0.997445	+	0.0714451i	$0.0227611\pi$
$659$			4.08939e7i			0.142890i	−0.997445	+	0.0714451i	$0.977239\pi$
$660$	1.75769e7	+	2.93103e7i	0.0611379	+	0.101950i
$661$	−1.12172e7			−0.0388401			−0.0194200	−	0.999811i	$-0.506182\pi$
$661$	−1.12172e7			−0.0388401			−0.0194200	+	0.999811i	$0.506182\pi$
$662$			4.99011e8i			1.72003i
$663$	1.58379e7	−	9.49773e6i	0.0543448	−	0.0325896i
$664$	1.23931e8			0.423327
$665$		−	6.20862e7i		−	0.211120i
$666$	1.72050e8	+	9.18630e7i	0.582413	+	0.310970i
$667$	−1.44069e7			−0.0485503
$668$			6.96869e7i			0.233788i
$669$	1.88118e8	+	3.13695e8i	0.628277	+	1.04768i
$670$	1.14522e8			0.380771
$671$		−	2.74426e8i		−	0.908359i
$672$	−1.73494e8	+	1.04041e8i	−0.571712	+	0.342845i
$673$	−3.83326e8			−1.25754			−0.628771	−	0.777590i	$-0.716442\pi$
$673$	−3.83326e8			−1.25754			−0.628771	+	0.777590i	$0.716442\pi$
$674$		−	5.53505e8i		−	1.80776i
$675$	2.87212e8	−	1.43039e7i	0.933879	−	0.0465095i
$676$	1.54148e8			0.498995
$677$		−	3.42935e7i		−	0.110521i	−0.998472	−	0.0552607i	$-0.982401\pi$
$677$		−	3.42935e7i		−	0.110521i	0.998472	−	0.0552607i	$-0.0175990\pi$
$678$	−1.41477e8	−	2.35920e8i	−0.453937	−	0.756963i
$679$	3.46820e7			0.110788
$680$		−	3.62839e7i		−	0.115395i
$681$	2.81713e8	−	1.68938e8i	0.892000	−	0.534917i
$682$	−6.08063e7			−0.191688
$683$			7.40065e7i			0.232278i	0.993233	+	0.116139i	$0.0370518\pi$
$683$			7.40065e7i			0.232278i	−0.993233	+	0.116139i	$0.962948\pi$
$684$	−8.68542e7	+	1.62669e8i	−0.271408	+	0.508318i
$685$	4.32069e7			0.134425
$686$		−	4.22591e8i		−	1.30902i
$687$	1.48499e6	+	2.47629e6i	0.00457987	+	0.00763716i
$688$	1.46153e8			0.448788
$689$			5.46157e7i			0.166978i
$690$	2.30141e7	−	1.38012e7i	0.0700563	−	0.0420115i
$691$	−5.50557e8			−1.66866			−0.834330	−	0.551265i	$-0.814145\pi$
$691$	−5.50557e8			−1.66866			−0.834330	+	0.551265i	$0.814145\pi$
$692$			1.90511e8i			0.574911i
$693$	1.96829e8	+	1.05094e8i	0.591412	+	0.315775i
$694$	−2.30668e8			−0.690096
$695$		−	5.62352e7i		−	0.167515i
$696$	1.96304e7	+	3.27346e7i	0.0582238	+	0.0970911i
$697$	3.62382e8			1.07021
$698$		−	1.48113e8i		−	0.435540i
$699$	1.44226e8	−	8.64899e7i	0.422292	−	0.253241i
$700$	1.16481e8			0.339596
$701$			2.31653e8i			0.672487i	0.941775	+	0.336243i	$0.109157\pi$
$701$			2.31653e8i			0.672487i	−0.941775	+	0.336243i	$0.890843\pi$
$702$	−1.79993e6	−	3.61414e7i	−0.00520289	−	0.104470i
$703$	−2.14518e8			−0.617445
$704$		−	3.85334e7i		−	0.110438i
$705$	−8.64017e6	−	1.44079e7i	−0.0246578	−	0.0411182i
$706$	3.34955e8			0.951857
$707$			2.34452e8i			0.663430i
$708$	−1.91026e8	+	1.14555e8i	−0.538261	+	0.322786i
$709$	3.37520e8			0.947023			0.473512	−	0.880788i	$-0.342986\pi$
$709$	3.37520e8			0.947023			0.473512	+	0.880788i	$0.342986\pi$
$710$		−	8.45867e7i		−	0.236334i
$711$	6.79714e7	−	1.27303e8i	0.189111	−	0.354185i
$712$	4.61193e7			0.127774
$713$			1.59710e7i			0.0440620i
$714$	1.23131e8	+	2.05326e8i	0.338276	+	0.564092i
$715$	7.37645e6			0.0201804
$716$			2.28527e8i			0.622585i
$717$	4.16194e8	−	2.49584e8i	1.12911	−	0.677110i
$718$	−7.78678e7			−0.210370
$719$			2.68649e8i			0.722766i	0.932417	+	0.361383i	$0.117695\pi$
$719$			2.68649e8i			0.722766i	−0.932417	+	0.361383i	$0.882305\pi$
$720$	−1.04900e8	−	5.60095e7i	−0.281046	−	0.150060i
$721$	4.25537e8			1.13536
$722$		−	1.44961e8i		−	0.385158i
$723$	1.64298e8	+	2.73975e8i	0.434728	+	0.724931i
$724$	1.42261e8			0.374862
$725$		−	6.61673e7i		−	0.173632i
$726$	−5.59323e7	+	3.35416e7i	−0.146168	+	0.0876545i
$727$	5.84091e8			1.52012			0.760059	−	0.649855i	$-0.225170\pi$
$727$	5.84091e8			1.52012			0.760059	+	0.649855i	$0.225170\pi$
$728$			1.45026e7i			0.0375883i
$729$	3.85503e8	−	3.84935e7i	0.995052	−	0.0993586i
$730$	7.91820e7			0.203544
$731$		−	1.04148e8i		−	0.266624i
$732$	−9.92616e7	−	1.65524e8i	−0.253074	−	0.422014i
$733$	−1.05493e8			−0.267862			−0.133931	−	0.990991i	$-0.542760\pi$
$733$	−1.05493e8			−0.267862			−0.133931	+	0.990991i	$0.542760\pi$
$734$			7.16286e7i			0.181133i
$735$	4.14814e7	−	2.48757e7i	0.104470	−	0.0626489i
$736$	9.61714e7			0.241220
$737$		−	4.52682e8i		−	1.13081i
$738$	3.34444e8	−	6.26377e8i	0.832059	−	1.55836i
$739$	−1.77720e8			−0.440355			−0.220177	−	0.975460i	$-0.570664\pi$
$739$	−1.77720e8			−0.440355			−0.220177	+	0.975460i	$0.570664\pi$
$740$			2.79620e7i			0.0690037i
$741$	2.04692e7	+	3.41333e7i	0.0503090	+	0.0838927i
$742$	−7.08050e8			−1.73321
$743$			2.65973e8i			0.648441i	0.945982	+	0.324220i	$0.105102\pi$
$743$			2.65973e8i			0.648441i	−0.945982	+	0.324220i	$0.894898\pi$
$744$	3.62887e7	−	2.17617e7i	0.0881155	−	0.0528413i
$745$	−2.79027e7			−0.0674804
$746$		−	3.91172e8i		−	0.942218i
$747$	−2.55320e8	−	1.36324e8i	−0.612524	−	0.327047i
$748$	1.44955e8			0.346360
$749$			2.39975e8i			0.571110i
$750$	1.31175e8	+	2.18741e8i	0.310933	+	0.518496i
$751$	−5.29864e8			−1.25096			−0.625482	−	0.780239i	$-0.715098\pi$
$751$	−5.29864e8			−1.25096			−0.625482	+	0.780239i	$0.715098\pi$
$752$		−	9.99927e7i		−	0.235134i
$753$	−3.55764e8	+	2.13345e8i	−0.833253	+	0.499687i
$754$	−8.32619e6			−0.0194237
$755$			1.79791e8i			0.417760i
$756$	1.56733e8	−	7.80572e6i	0.362741	−	0.0180654i
$757$	2.80306e8			0.646166			0.323083	−	0.946371i	$-0.395281\pi$
$757$	2.80306e8			0.646166			0.323083	+	0.946371i	$0.395281\pi$
$758$		−	9.55982e8i		−	2.19504i
$759$	−5.45533e7	−	9.09703e7i	−0.124766	−	0.208053i
$760$	7.81977e7			0.178137
$761$		−	4.05753e8i		−	0.920677i	−0.887743	−	0.460339i	$-0.847728\pi$
$761$		−	4.05753e8i		−	0.920677i	0.887743	−	0.460339i	$-0.152272\pi$
$762$	−1.11282e8	+	6.67339e7i	−0.251513	+	0.150828i
$763$	−1.02604e8			−0.230988
$764$			2.40194e8i			0.538619i
$765$	−3.99122e7	+	7.47513e7i	−0.0891500	+	0.166968i
$766$	−6.13168e8			−1.36425
$767$			4.80750e7i			0.106545i
$768$	−2.77418e8	−	4.62608e8i	−0.612422	−	1.02124i
$769$	−5.95981e7			−0.131055			−0.0655275	−	0.997851i	$-0.520873\pi$
$769$	−5.95981e7			−0.131055			−0.0655275	+	0.997851i	$0.520873\pi$
$770$			9.56298e7i			0.209470i
$771$	6.03520e8	−	3.61920e8i	1.31683	−	0.789678i
$772$	−3.41376e8			−0.741960
$773$		−	4.90915e8i		−	1.06284i	−0.847109	−	0.531420i	$-0.821659\pi$
$773$		−	4.90915e8i		−	1.06284i	0.847109	−	0.531420i	$-0.178341\pi$
$774$	−1.80020e8	−	9.61187e7i	−0.388238	−	0.207293i
$775$	−7.33512e7			−0.157580
$776$			4.36821e7i			0.0934798i
$777$	9.38873e7	+	1.56562e8i	0.200145	+	0.333751i
$778$	1.41524e8			0.300532
$779$			7.80992e8i			1.65209i
$780$	4.44920e6	−	2.66811e6i	0.00937559	−	0.00562238i
$781$	−3.34355e8			−0.701866
$782$		−	1.13817e8i		−	0.238005i
$783$	−4.43404e6	−	8.90325e7i	−0.00923665	−	0.185465i
$784$	2.87886e8			0.597410
$785$		−	6.20514e7i		−	0.128275i
$786$	−3.38804e8	−	5.64972e8i	−0.697719	−	1.16348i
$787$	1.56894e8			0.321870			0.160935	−	0.986965i	$-0.448549\pi$
$787$	1.56894e8			0.321870			0.160935	+	0.986965i	$0.448549\pi$
$788$			3.30170e8i			0.674776i
$789$	2.85027e7	−	1.70926e7i	0.0580303	−	0.0347997i
$790$	6.18504e7			0.125447
$791$		−	2.57482e8i		−	0.520257i
$792$	−1.32366e8	+	2.47907e8i	−0.266441	+	0.499015i
$793$	−4.16569e7			−0.0835347
$794$			5.94376e8i			1.18741i
$795$	−1.28887e8	−	2.14925e8i	−0.256511	−	0.427745i
$796$	−3.20493e7			−0.0635448
$797$			9.14070e8i			1.80553i	0.430135	+	0.902765i	$0.358466\pi$
$797$			9.14070e8i			1.80553i	−0.430135	+	0.902765i	$0.641534\pi$
$798$	−4.42512e8	+	2.65366e8i	−0.870795	+	0.522201i
$799$	−7.12545e7			−0.139692
$800$			4.41693e8i			0.862681i
$801$	−9.50139e7	−	5.07311e7i	−0.184880	−	0.0987135i
$802$	7.05453e7			0.136756
$803$		−	3.12991e8i		−	0.604484i
$804$	−1.63738e8	−	2.73042e8i	−0.315052	−	0.525364i
$805$	2.51176e7			0.0481493
$806$			9.23018e6i			0.0176281i
$807$	−2.81231e8	+	1.68649e8i	−0.535109	+	0.320896i
$808$	−2.95292e8			−0.559781
$809$			8.62430e8i			1.62884i	0.580276	+	0.814420i	$0.302945\pi$
$809$			8.62430e8i			1.62884i	−0.580276	+	0.814420i	$0.697055\pi$
$810$	9.23725e7	+	1.37977e8i	0.173815	+	0.259627i
$811$	1.97242e8			0.369775			0.184888	−	0.982760i	$-0.440808\pi$
$811$	1.97242e8			0.369775			0.184888	+	0.982760i	$0.440808\pi$
$812$		−	3.61080e7i		−	0.0674427i
$813$	−5.16686e8	−	8.61600e8i	−0.961513	−	1.60337i
$814$	3.30417e8			0.612617
$815$			6.34449e7i			0.117199i
$816$	−4.32548e8	+	2.59392e8i	−0.796093	+	0.477403i
$817$	2.24456e8			0.411590
$818$			6.52291e8i			1.19174i
$819$	1.59529e7	−	2.98780e7i	0.0290394	−	0.0543876i
$820$	1.01801e8			0.184633
$821$			5.31778e8i			0.960950i	0.877008	+	0.480475i	$0.159536\pi$
$821$			5.31778e8i			0.960950i	−0.877008	+	0.480475i	$0.840464\pi$
$822$	−1.84673e8	−	3.07952e8i	−0.332498	−	0.554456i
$823$	−8.38836e8			−1.50480			−0.752398	−	0.658709i	$-0.771103\pi$
$823$	−8.38836e8			−1.50480			−0.752398	+	0.658709i	$0.771103\pi$
$824$			5.35965e8i			0.957977i
$825$	4.17805e8	−	2.50550e8i	0.744067	−	0.446204i
$826$	−6.23254e8			−1.10592
$827$			7.12797e8i			1.26023i	0.776502	+	0.630114i	$0.216992\pi$
$827$			7.12797e8i			1.26023i	−0.776502	+	0.630114i	$0.783008\pi$
$828$	−6.58091e7	−	3.51377e7i	−0.115930	−	0.0618988i
$829$	1.24432e8			0.218408			0.109204	−	0.994019i	$-0.465170\pi$
$829$	1.24432e8			0.218408			0.109204	+	0.994019i	$0.465170\pi$
$830$		−	1.24048e8i		−	0.216947i
$831$	−5.09782e8	−	8.50087e8i	−0.888345	−	1.48136i
$832$	−5.84924e6			−0.0101562
$833$		−	2.05147e8i		−	0.354920i
$834$	−4.00809e8	+	2.40358e8i	−0.690939	+	0.414344i
$835$	−6.90154e7			−0.118546
$836$			3.12401e8i			0.534679i
$837$	−9.86988e7	+	4.91545e6i	−0.168320	+	0.00838276i
$838$	−8.60469e8			−1.46219
$839$			1.80792e8i			0.306122i	0.988217	+	0.153061i	$0.0489130\pi$
$839$			1.80792e8i			0.306122i	−0.988217	+	0.153061i	$0.951087\pi$
$840$	−3.42245e7	−	5.70710e7i	−0.0577430	−	0.0962892i
$841$	−2.05111e7			−0.0344828
$842$		−	9.09017e7i		−	0.152277i
$843$	−5.86197e8	+	3.51532e8i	−0.978500	+	0.586789i
$844$	5.50137e8			0.915047
$845$			1.52662e8i			0.253024i
$846$	−6.57611e7	+	1.23163e8i	−0.108607	+	0.203409i
$847$	−6.10445e7			−0.100461
$848$		−	1.49160e9i		−	2.44605i
$849$	2.91032e8	+	4.85311e8i	0.475574	+	0.793043i
$850$	5.22733e8			0.851184
$851$		−	8.67854e7i		−	0.140818i
$852$	−2.01671e8	+	1.20938e8i	−0.326080	+	0.195544i
$853$	4.34651e7			0.0700315			0.0350158	−	0.999387i	$-0.488852\pi$
$853$	4.34651e7			0.0700315			0.0350158	+	0.999387i	$0.488852\pi$
$854$		−	5.40048e8i		−	0.867079i
$855$	−1.61101e8	−	8.60173e7i	−0.257751	−	0.137622i
$856$	−3.02249e8			−0.481885
$857$		−	7.59544e8i		−	1.20673i	−0.797465	−	0.603366i	$-0.793826\pi$
$857$		−	7.59544e8i		−	1.20673i	0.797465	−	0.603366i	$-0.206174\pi$
$858$	−3.15281e7	−	5.25747e7i	−0.0499156	−	0.0832367i
$859$	1.13631e9			1.79274			0.896370	−	0.443307i	$-0.146195\pi$
$859$	1.13631e9			1.79274			0.896370	+	0.443307i	$0.146195\pi$
$860$		−	2.92573e7i		−	0.0459980i
$861$	5.69991e8	−	3.41813e8i	0.893014	−	0.535525i
$862$	8.44462e8			1.31843
$863$			7.27656e8i			1.13212i	0.824363	+	0.566062i	$0.191534\pi$
$863$			7.27656e8i			1.13212i	−0.824363	+	0.566062i	$0.808466\pi$
$864$	2.95989e7	+	5.94326e8i	0.0458918	+	0.921476i
$865$	−1.88675e8			−0.291518
$866$		−	6.59357e8i		−	1.01524i
$867$	−1.50332e8	−	2.50686e8i	−0.230672	−	0.384656i
$868$	−4.00283e7			−0.0612079
$869$		−	2.44482e8i		−	0.372553i
$870$	3.27654e7	−	1.96488e7i	0.0497574	−	0.0298386i
$871$	−6.87156e7			−0.103992
$872$		−	1.29230e8i		−	0.194900i
$873$	4.80502e7	−	8.99928e7i	0.0722191	−	0.135259i
$874$	2.45293e8			0.367410
$875$			2.38733e8i			0.356359i
$876$	−1.13211e8	−	1.88785e8i	−0.168413	−	0.280837i
$877$	2.08164e8			0.308608			0.154304	−	0.988023i	$-0.450686\pi$
$877$	2.08164e8			0.308608			0.154304	+	0.988023i	$0.450686\pi$
$878$			2.78854e8i			0.411996i
$879$	−6.24690e8	+	3.74616e8i	−0.919810	+	0.551594i
$880$	−2.01457e8			−0.295621
$881$		−	8.68090e8i		−	1.26951i	−0.772712	−	0.634757i	$-0.781100\pi$
$881$		−	8.68090e8i		−	1.26951i	0.772712	−	0.634757i	$-0.218900\pi$
$882$	−3.54597e8	−	1.89331e8i	−0.516808	−	0.275941i
$883$	2.93853e8			0.426824			0.213412	−	0.976962i	$-0.431542\pi$
$883$	2.93853e8			0.426824			0.213412	+	0.976962i	$0.431542\pi$
$884$		−	2.20036e7i		−	0.0318520i
$885$	−1.13451e8	−	1.89185e8i	−0.163674	−	0.272934i
$886$	−1.28616e9			−1.84924
$887$			6.47260e8i			0.927487i	0.885970	+	0.463743i	$0.153494\pi$
$887$			6.47260e8i			0.927487i	−0.885970	+	0.463743i	$0.846506\pi$
$888$	−1.97190e8	+	1.18251e8i	−0.281608	+	0.168876i
$889$	−1.21453e8			−0.172863
$890$		−	4.61626e7i		−	0.0654817i
$891$	5.45394e8	−	3.65130e8i	0.771041	−	0.516196i
$892$	4.35816e8			0.614057
$893$		−	1.53565e8i		−	0.215644i
$894$	1.19261e8	+	1.98873e8i	0.166911	+	0.278332i
$895$	−2.26325e8			−0.315692
$896$		−	5.55354e8i		−	0.772051i
$897$	−1.38090e7	+	8.28099e6i	−0.0191330	+	0.0114737i
$898$	1.18095e9			1.63081
$899$			2.27381e7i			0.0312950i
$900$	1.61379e8	−	3.02246e8i	0.221371	−	0.414603i
$901$	−1.06291e9			−1.45319
$902$		−	1.20294e9i		−	1.63917i
$903$	−9.82367e7	−	1.63815e8i	−0.133417	−	0.222479i
$904$	3.24299e8			0.438976
$905$			1.40891e8i			0.190080i
$906$	1.28144e9	−	7.68455e8i	1.72311	−	1.03332i
$907$	1.29705e9			1.73834			0.869168	−	0.494516i	$-0.164655\pi$
$907$	1.29705e9			1.73834			0.869168	+	0.494516i	$0.164655\pi$
$908$		−	3.91383e8i		−	0.522810i
$909$	6.08354e8	+	3.24821e8i	0.809963	+	0.432466i
$910$	1.45163e7			0.0192633
$911$			9.76953e8i			1.29217i	0.763267	+	0.646084i	$0.223594\pi$
$911$			9.76953e8i			1.29217i	−0.763267	+	0.646084i	$0.776406\pi$
$912$	−5.59031e8	−	9.32212e8i	−0.736973	−	1.22894i
$913$	−4.90336e8			−0.644290
$914$			1.19936e9i			1.57077i
$915$	1.63929e8	−	9.83052e7i	0.213989	−	0.128326i
$916$	3.44031e6			0.00447621
$917$		−	6.16609e8i		−	0.799654i
$918$	7.03372e8	−	3.50297e7i	0.909195	−	0.0452802i
$919$	1.08791e9			1.40167			0.700837	−	0.713322i	$-0.252810\pi$
$919$	1.08791e9			1.40167			0.700837	+	0.713322i	$0.252810\pi$
$920$			3.16356e7i			0.0406268i
$921$	4.48962e8	+	7.48666e8i	0.574686	+	0.958317i
$922$	3.14937e8			0.401819
$923$			5.07538e7i			0.0645452i
$924$	2.27999e8	−	1.36727e8i	0.289013	−	0.173316i
$925$	3.98585e8			0.503612
$926$			1.17833e9i			1.48400i
$927$	5.89561e8	−	1.10418e9i	0.740098	−	1.38612i
$928$	1.36920e8			0.171326
$929$			1.28342e8i			0.160074i	0.996792	+	0.0800370i	$0.0255039\pi$
$929$			1.28342e8i			0.160074i	−0.996792	+	0.0800370i	$0.974496\pi$
$930$	−2.17821e7	−	3.63227e7i	−0.0270802	−	0.0451575i
$931$	4.42125e8			0.547894
$932$		−	2.00373e8i		−	0.247509i
$933$	−5.32215e8	+	3.19160e8i	−0.655303	+	0.392974i
$934$	1.83597e9			2.25333
$935$			1.43558e8i			0.175627i
$936$	3.76314e7	+	2.00927e7i	0.0458905	+	0.0245025i
$937$	4.77275e8			0.580163			0.290082	−	0.957002i	$-0.406318\pi$
$937$	4.77275e8			0.580163			0.290082	+	0.957002i	$0.406318\pi$
$938$		−	8.90843e8i		−	1.07943i
$939$	5.93652e8	+	9.89944e8i	0.717026	+	1.19568i
$940$	−2.00169e7			−0.0240998
$941$			8.34145e8i			1.00109i	0.865711	+	0.500545i	$0.166867\pi$
$941$			8.34145e8i			1.00109i	−0.865711	+	0.500545i	$0.833133\pi$
$942$	−4.42264e8	+	2.65218e8i	−0.529089	+	0.317285i
$943$	−3.15958e8			−0.376785
$944$		−	1.31297e9i		−	1.56077i
$945$	7.73051e6	+	1.55223e8i	0.00916036	+	0.183934i
$946$	−3.45724e8			−0.408372
$947$		−	5.49043e8i		−	0.646481i	−0.946317	−	0.323241i	$-0.895228\pi$
$947$		−	5.49043e8i		−	0.646481i	0.946317	−	0.323241i	$-0.104772\pi$
$948$	−8.84309e7	−	1.47463e8i	−0.103796	−	0.173084i
$949$	−4.75109e7			−0.0555897
$950$			1.12657e9i			1.31398i
$951$	5.02108e8	−	3.01106e8i	0.583789	−	0.350088i
$952$	−2.82246e8			−0.327127
$953$			1.36926e9i			1.58200i	0.611817	+	0.791000i	$0.290439\pi$
$953$			1.36926e9i			1.58200i	−0.611817	+	0.791000i	$0.709561\pi$
$954$	−9.80967e8	+	1.83725e9i	−1.12982	+	2.11603i
$955$	−2.37880e8			−0.273116
$956$		−	5.78217e8i		−	0.661785i
$957$	−7.76679e7	−	1.29515e8i	−0.0886147	−	0.147769i
$958$	−1.69334e9			−1.92597
$959$		−	3.36098e8i		−	0.381075i
$960$	2.30180e7	−	1.38035e7i	0.0260168	−	0.0156018i
$961$	−8.62297e8			−0.971598
$962$		−	5.01561e7i		−	0.0563376i
$963$	6.22686e8	+	3.32473e8i	0.697252	+	0.372286i
$964$	3.80633e8			0.424889
$965$		−	3.38086e8i		−	0.376223i
$966$	−1.07357e8	−	1.79022e8i	−0.119096	−	0.198598i
$967$	5.12824e8			0.567138			0.283569	−	0.958952i	$-0.408481\pi$
$967$	5.12824e8			0.567138			0.283569	+	0.958952i	$0.408481\pi$
$968$		−	7.68857e7i		−	0.0847655i
$969$	−6.64292e8	+	3.98364e8i	−0.730109	+	0.437833i
$970$	4.37231e7			0.0479066
$971$			3.05458e8i			0.333653i	0.985986	+	0.166826i	$0.0533519\pi$
$971$			3.05458e8i			0.333653i	−0.985986	+	0.166826i	$0.946648\pi$
$972$	1.96892e8	−	4.17506e8i	0.214402	−	0.454636i
$973$	−4.37443e8			−0.474879
$974$			1.86197e9i			2.01510i
$975$	−3.80327e7	−	6.34213e7i	−0.0410339	−	0.0684261i
$976$	1.13769e9			1.22369
$977$			4.41991e8i			0.473947i	0.971516	+	0.236974i	$0.0761555\pi$
$977$			4.41991e8i			0.473947i	−0.971516	+	0.236974i	$0.923845\pi$
$978$	4.52196e8	−	2.71174e8i	0.483404	−	0.289889i
$979$	−1.82472e8			−0.194468
$980$		−	5.76300e7i		−	0.0612309i
$981$	−1.42152e8	+	2.66236e8i	−0.150573	+	0.282007i
$982$	−4.46379e8			−0.471378
$983$		−	1.94251e7i		−	0.0204504i	−0.999948	−	0.0102252i	$-0.996745\pi$
$983$		−	1.94251e7i		−	0.0204504i	0.999948	−	0.0102252i	$-0.00325484\pi$
$984$	4.30515e8	+	7.17905e8i	0.451859	+	0.753497i
$985$	−3.26989e8			−0.342156
$986$		−	1.62042e8i		−	0.169042i
$987$	−1.12076e8	+	6.72102e7i	−0.116563	+	0.0699010i
$988$	4.74213e7			0.0491703
$989$			9.08058e7i			0.0938695i
$990$	2.48140e8	+	1.32490e8i	0.255736	+	0.136546i
$991$	−1.41258e9			−1.45141			−0.725707	−	0.688004i	$-0.758487\pi$
$991$	−1.41258e9			−1.45141			−0.725707	+	0.688004i	$0.758487\pi$
$992$		−	1.51786e8i		−	0.155487i
$993$	7.06588e8	+	1.17827e9i	0.721637	+	1.20337i
$994$	−6.57983e8			−0.669970
$995$		−	3.17405e7i		−	0.0322214i
$996$	−2.95753e8	+	1.77358e8i	−0.299330	+	0.179503i
$997$	1.44017e9			1.45321			0.726606	−	0.687054i	$-0.241097\pi$
$997$	1.44017e9			1.45321			0.726606	+	0.687054i	$0.241097\pi$
$998$			6.72199e8i			0.676248i
$999$	5.36322e8	−	2.67102e7i	0.537934	−	0.0267905i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	87.7.b.a.59.13	✓	56
3.2	odd	2	inner	87.7.b.a.59.44	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.7.b.a.59.13	✓	56	1.1	even	1	trivial
87.7.b.a.59.44	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-9.80663i q^{2} +(-13.8860 - 23.1556i) q^{3} -32.1699 q^{4} -31.8600i q^{5} +(-227.078 + 136.175i) q^{6} -247.832 q^{7} -312.146i q^{8} +(-343.359 + 643.075i) q^{9} -312.439 q^{10} +1235.01i q^{11} +(446.711 + 744.913i) q^{12} +187.470 q^{13} +2430.40i q^{14} +(-737.735 + 442.407i) q^{15} -5119.97 q^{16} +3648.48i q^{17} +(6306.40 + 3367.20i) q^{18} -7863.06 q^{19} +1024.93i q^{20} +(3441.39 + 5738.70i) q^{21} +12111.3 q^{22} -3181.08i q^{23} +(-7227.90 + 4334.45i) q^{24} +14609.9 q^{25} -1838.45i q^{26} +(19658.6 - 979.049i) q^{27} +7972.75 q^{28} -4528.92i q^{29} +(4338.52 + 7234.69i) q^{30} -5020.63 q^{31} +30232.3i q^{32} +(28597.3 - 17149.3i) q^{33} +35779.3 q^{34} +7895.93i q^{35} +(11045.8 - 20687.7i) q^{36} +27281.8 q^{37} +77110.1i q^{38} +(-2603.20 - 4340.97i) q^{39} -9944.94 q^{40} -99324.1i q^{41} +(56277.3 - 33748.5i) q^{42} -28545.6 q^{43} -39730.2i q^{44} +(20488.3 + 10939.4i) q^{45} -31195.6 q^{46} +19529.9i q^{47} +(71095.8 + 118556. i) q^{48} -56228.1 q^{49} -143274. i q^{50} +(84482.5 - 50662.7i) q^{51} -6030.90 q^{52} +291331. i q^{53} +(-9601.17 - 192785. i) q^{54} +39347.3 q^{55} +77359.8i q^{56} +(109186. + 182074. i) q^{57} -44413.5 q^{58} +256441. i q^{59} +(23732.9 - 14232.2i) q^{60} -222205. q^{61} +49235.5i q^{62} +(85095.6 - 159375. i) q^{63} -31200.9 q^{64} -5972.79i q^{65} +(-168177. - 280443. i) q^{66} -366542. q^{67} -117371. i q^{68} +(-73659.6 + 44172.4i) q^{69} +77432.4 q^{70} +270730. i q^{71} +(200733. + 107178. i) q^{72} -253432. q^{73} -267542. i q^{74} +(-202873. - 338301. i) q^{75} +252954. q^{76} -306075. i q^{77} +(-42570.3 + 25528.6i) q^{78} -197960. q^{79} +163122. i q^{80} +(-295650. - 441612. i) q^{81} -974034. q^{82} +397030. i q^{83} +(-110709. - 184614. i) q^{84} +116240. q^{85} +279936. i q^{86} +(-104870. + 62888.5i) q^{87} +385503. q^{88} +147749. i q^{89} +(107279. - 200922. i) q^{90} -46461.1 q^{91} +102335. i q^{92} +(69716.4 + 116256. i) q^{93} +191523. q^{94} +250517. i q^{95} +(700046. - 419805. i) q^{96} -139941. q^{97} +551408. i q^{98} +(-794203. - 424052. 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

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