LMFDB - Embedded newform 115.6.b.a.24.45

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [115,6,Mod(24,115)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("115.24"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(115, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 115 = 5 \cdot 23 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	115.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:	$18.4441392785$
Analytic rank:	$0$
Dimension:	$54$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			24.45
Character	$\chi$	$=$	115.24
Dual form			115.6.b.a.24.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+7.77128i q^{2} -16.5107i q^{3} -28.3928 q^{4} +(38.9251 - 40.1228i) q^{5} +128.309 q^{6} +7.31009i q^{7} +28.0326i q^{8} -29.6034 q^{9} +(311.805 + 302.498i) q^{10} -512.382 q^{11} +468.785i q^{12} -603.915i q^{13} -56.8088 q^{14} +(-662.455 - 642.681i) q^{15} -1126.42 q^{16} -1547.97i q^{17} -230.056i q^{18} +2173.79 q^{19} +(-1105.19 + 1139.20i) q^{20} +120.695 q^{21} -3981.86i q^{22} -529.000i q^{23} +462.838 q^{24} +(-94.6706 - 3123.57i) q^{25} +4693.20 q^{26} -3523.33i q^{27} -207.554i q^{28} -2114.61 q^{29} +(4994.45 - 5148.12i) q^{30} -6354.77 q^{31} -7856.67i q^{32} +8459.79i q^{33} +12029.7 q^{34} +(293.301 + 284.546i) q^{35} +840.522 q^{36} +3404.46i q^{37} +16893.1i q^{38} -9971.07 q^{39} +(1124.75 + 1091.17i) q^{40} +8009.77 q^{41} +937.953i q^{42} -14626.7i q^{43} +14548.0 q^{44} +(-1152.31 + 1187.77i) q^{45} +4111.01 q^{46} -2294.90i q^{47} +18598.0i q^{48} +16753.6 q^{49} +(24274.1 - 735.712i) q^{50} -25558.1 q^{51} +17146.8i q^{52} -19770.6i q^{53} +27380.8 q^{54} +(-19944.5 + 20558.2i) q^{55} -204.921 q^{56} -35890.8i q^{57} -16433.2i q^{58} +48436.6 q^{59} +(18808.9 + 18247.5i) q^{60} -2261.14 q^{61} -49384.7i q^{62} -216.403i q^{63} +25011.0 q^{64} +(-24230.8 - 23507.5i) q^{65} -65743.4 q^{66} +47700.6i q^{67} +43951.2i q^{68} -8734.16 q^{69} +(-2211.29 + 2279.32i) q^{70} -59497.4 q^{71} -829.859i q^{72} +12040.2i q^{73} -26457.0 q^{74} +(-51572.3 + 1563.08i) q^{75} -61719.9 q^{76} -3745.56i q^{77} -77488.0i q^{78} -59066.2 q^{79} +(-43846.0 + 45195.0i) q^{80} -65366.3 q^{81} +62246.2i q^{82} -88291.8i q^{83} -3426.86 q^{84} +(-62108.8 - 60254.9i) q^{85} +113668. q^{86} +34913.6i q^{87} -14363.4i q^{88} +61494.9 q^{89} +(-9230.48 - 8954.96i) q^{90} +4414.68 q^{91} +15019.8i q^{92} +104922. i q^{93} +17834.3 q^{94} +(84614.9 - 87218.3i) q^{95} -129719. q^{96} +55907.6i q^{97} +130197. i q^{98} +15168.2 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 54 q - 832 q^{4} + 98 q^{5} - 144 q^{6} - 4270 q^{9} + 942 q^{10} - 380 q^{11} - 3988 q^{14} + 690 q^{15} + 16632 q^{16} - 2440 q^{19} - 2436 q^{20} + 4768 q^{21} - 17940 q^{24} - 9150 q^{25} - 15144 q^{26}+ \cdots - 14088 q^{99}+O(q^{100}) $ 54 * q - 832 * q^4 + 98 * q^5 - 144 * q^6 - 4270 * q^9 + 942 * q^10 - 380 * q^11 - 3988 * q^14 + 690 * q^15 + 16632 * q^16 - 2440 * q^19 - 2436 * q^20 + 4768 * q^21 - 17940 * q^24 - 9150 * q^25 - 15144 * q^26 + 22958 * q^29 + 23034 * q^30 - 33970 * q^31 + 26120 * q^34 - 18694 * q^35 + 71560 * q^36 + 13416 * q^39 - 22230 * q^40 - 33842 * q^41 + 33908 * q^44 + 35260 * q^45 - 8464 * q^46 - 153772 * q^49 - 91728 * q^50 + 96576 * q^51 - 90500 * q^54 + 37344 * q^55 + 201796 * q^56 + 110106 * q^59 - 92484 * q^60 + 132924 * q^61 - 262108 * q^64 + 45090 * q^65 - 123288 * q^66 + 38088 * q^69 + 249664 * q^70 - 310814 * q^71 + 220660 * q^74 + 235776 * q^75 + 42220 * q^76 - 389348 * q^79 - 245138 * q^80 + 364422 * q^81 - 35836 * q^84 - 134118 * q^85 + 111980 * q^86 - 353788 * q^89 - 805800 * q^90 - 226632 * q^91 + 289804 * q^94 - 42676 * q^95 + 294416 * q^96 - 14088 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/115\mathbb{Z}\right)^\times$.

$n$	$47$	$51$
$\chi(n)$	$-1$	$1$

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 $

$2$			7.77128i			1.37378i	0.726761	+	0.686891i	$0.241025\pi$
$2$			7.77128i			1.37378i	−0.726761	+	0.686891i	$0.758975\pi$
$3$		−	16.5107i		−	1.05916i	−0.848259	−	0.529581i	$-0.822349\pi$
$3$		−	16.5107i		−	1.05916i	0.848259	−	0.529581i	$-0.177651\pi$
$4$	−28.3928			−0.887275
$5$	38.9251	−	40.1228i	0.696314	−	0.717738i
$5$	38.9251	−	40.1228i	0.696314	−	0.717738i
$6$	128.309			1.45506
$7$			7.31009i			0.0563868i	0.999602	+	0.0281934i	$0.00897543\pi$
$7$			7.31009i			0.0563868i	−0.999602	+	0.0281934i	$0.991025\pi$
$8$			28.0326i			0.154860i
$9$	−29.6034			−0.121825
$10$	311.805	+	302.498i	0.986014	+	0.956583i
$11$	−512.382			−1.27677			−0.638384	−	0.769718i	$-0.720397\pi$
$11$	−512.382			−1.27677			−0.638384	+	0.769718i	$0.720397\pi$
$12$			468.785i			0.939768i
$13$		−	603.915i		−	0.991101i	−0.868579	−	0.495550i	$-0.834966\pi$
$13$		−	603.915i		−	0.991101i	0.868579	−	0.495550i	$-0.165034\pi$
$14$	−56.8088			−0.0774632
$15$	−662.455	−	642.681i	−0.760201	−	0.737509i
$16$	−1126.42			−1.10002
$17$		−	1547.97i		−	1.29909i	−0.760322	−	0.649546i	$-0.774959\pi$
$17$		−	1547.97i		−	1.29909i	0.760322	−	0.649546i	$-0.225041\pi$
$18$		−	230.056i		−	0.167360i
$19$	2173.79			1.38144			0.690722	−	0.723121i	$-0.257293\pi$
$19$	2173.79			1.38144			0.690722	+	0.723121i	$0.257293\pi$
$20$	−1105.19	+	1139.20i	−0.617822	+	0.636831i
$21$	120.695			0.0597228
$22$		−	3981.86i		−	1.75400i
$23$		−	529.000i		−	0.208514i
$23$		−	529.000i		−	0.208514i
$24$	462.838			0.164022
$25$	−94.6706	−	3123.57i	−0.0302946	−	0.999541i
$26$	4693.20			1.36156
$27$		−	3523.33i		−	0.930130i
$28$		−	207.554i		−	0.0500306i
$29$	−2114.61			−0.466911			−0.233456	−	0.972367i	$-0.575003\pi$
$29$	−2114.61			−0.466911			−0.233456	+	0.972367i	$0.575003\pi$
$30$	4994.45	−	5148.12i	1.01318	−	1.04435i
$31$	−6354.77			−1.18767			−0.593834	−	0.804587i	$-0.702387\pi$
$31$	−6354.77			−1.18767			−0.593834	+	0.804587i	$0.702387\pi$
$32$		−	7856.67i		−	1.35632i
$33$			8459.79i			1.35230i
$34$	12029.7			1.78467
$35$	293.301	+	284.546i	0.0404709	+	0.0392629i
$36$	840.522			0.108092
$37$			3404.46i			0.408831i	0.978884	+	0.204415i	$0.0655293\pi$
$37$			3404.46i			0.408831i	−0.978884	+	0.204415i	$0.934471\pi$
$38$			16893.1i			1.89780i
$39$	−9971.07			−1.04974
$40$	1124.75	+	1091.17i	0.111149	+	0.107831i
$41$	8009.77			0.744150			0.372075	−	0.928203i	$-0.378646\pi$
$41$	8009.77			0.744150			0.372075	+	0.928203i	$0.378646\pi$
$42$			937.953i			0.0820461i
$43$		−	14626.7i		−	1.20635i	−0.797608	−	0.603176i	$-0.793902\pi$
$43$		−	14626.7i		−	1.20635i	0.797608	−	0.603176i	$-0.206098\pi$
$44$	14548.0			1.13284
$45$	−1152.31	+	1187.77i	−0.0848281	+	0.0874381i
$46$	4111.01			0.286453
$47$		−	2294.90i		−	0.151537i	−0.997125	−	0.0757685i	$-0.975859\pi$
$47$		−	2294.90i		−	0.151537i	0.997125	−	0.0757685i	$-0.0241410\pi$
$48$			18598.0i			1.16510i
$49$	16753.6			0.996821
$50$	24274.1	−	735.712i	1.37315	−	0.0416181i
$51$	−25558.1			−1.37595
$52$			17146.8i			0.879379i
$53$		−	19770.6i		−	0.966785i	−0.875404	−	0.483393i	$-0.839404\pi$
$53$		−	19770.6i		−	0.966785i	0.875404	−	0.483393i	$-0.160596\pi$
$54$	27380.8			1.27780
$55$	−19944.5	+	20558.2i	−0.889031	+	0.916385i
$56$	−204.921			−0.00873205
$57$		−	35890.8i		−	1.46317i
$58$		−	16433.2i		−	0.641434i
$59$	48436.6			1.81152			0.905760	−	0.423791i	$-0.139301\pi$
$59$	48436.6			1.81152			0.905760	+	0.423791i	$0.139301\pi$
$60$	18808.9	+	18247.5i	0.674507	+	0.654373i
$61$	−2261.14			−0.0778041			−0.0389020	−	0.999243i	$-0.512386\pi$
$61$	−2261.14			−0.0778041			−0.0389020	+	0.999243i	$0.512386\pi$
$62$		−	49384.7i		−	1.63160i
$63$		−	216.403i		−	0.00686930i
$64$	25011.0			0.763275
$65$	−24230.8	−	23507.5i	−0.711350	−	0.690117i
$66$	−65743.4			−1.85777
$67$			47700.6i			1.29819i	0.760709	+	0.649093i	$0.224852\pi$
$67$			47700.6i			1.29819i	−0.760709	+	0.649093i	$0.775148\pi$
$68$			43951.2i			1.15265i
$69$	−8734.16			−0.220851
$70$	−2211.29	+	2279.32i	−0.0539387	+	0.0555982i
$71$	−59497.4			−1.40072			−0.700361	−	0.713789i	$-0.746978\pi$
$71$	−59497.4			−1.40072			−0.700361	+	0.713789i	$0.746978\pi$
$72$		−	829.859i		−	0.0188657i
$73$			12040.2i			0.264441i	0.991220	+	0.132220i	$0.0422106\pi$
$73$			12040.2i			0.264441i	−0.991220	+	0.132220i	$0.957789\pi$
$74$	−26457.0			−0.561644
$75$	−51572.3	+	1563.08i	−1.05868	+	0.0320869i
$76$	−61719.9			−1.22572
$77$		−	3745.56i		−	0.0719929i
$78$		−	77488.0i		−	1.44211i
$79$	−59066.2			−1.06481			−0.532404	−	0.846491i	$-0.678711\pi$
$79$	−59066.2			−1.06481			−0.532404	+	0.846491i	$0.678711\pi$
$80$	−43846.0	+	45195.0i	−0.765958	+	0.789524i
$81$	−65366.3			−1.10698
$82$			62246.2i			1.02230i
$83$		−	88291.8i		−	1.40678i	−0.710806	−	0.703389i	$-0.751669\pi$
$83$		−	88291.8i		−	1.40678i	0.710806	−	0.703389i	$-0.248331\pi$
$84$	−3426.86			−0.0529905
$85$	−62108.8	−	60254.9i	−0.932408	−	0.904576i
$86$	113668.			1.65726
$87$			34913.6i			0.494535i
$88$		−	14363.4i		−	0.197720i
$89$	61494.9			0.822932			0.411466	−	0.911425i	$-0.365017\pi$
$89$	61494.9			0.822932			0.411466	+	0.911425i	$0.365017\pi$
$90$	−9230.48	−	8954.96i	−0.120121	−	0.116535i
$91$	4414.68			0.0558850
$92$			15019.8i			0.185010i
$93$			104922.i			1.25793i
$94$	17834.3			0.208179
$95$	84614.9	−	87218.3i	0.961918	−	0.991514i
$96$	−129719.			−1.43657
$97$			55907.6i			0.603311i	0.953417	+	0.301656i	$0.0975393\pi$
$97$			55907.6i			0.603311i	−0.953417	+	0.301656i	$0.902461\pi$
$98$			130197.i			1.36941i
$99$	15168.2			0.155542
$100$	2687.96	+	88686.8i	0.0268796	+	0.886868i
$101$	−30742.8			−0.299875			−0.149937	−	0.988695i	$-0.547907\pi$
$101$	−30742.8			−0.299875			−0.149937	+	0.988695i	$0.547907\pi$
$102$		−	198619.i		−	1.89025i
$103$			161792.i			1.50267i	0.659919	+	0.751337i	$0.270591\pi$
$103$			161792.i			1.50267i	−0.659919	+	0.751337i	$0.729409\pi$
$104$	16929.3			0.153482
$105$	4698.06	−	4842.61i	0.0415858	−	0.0428653i
$106$	153643.			1.32815
$107$			165359.i			1.39627i	0.715966	+	0.698135i	$0.245986\pi$
$107$			165359.i			1.39627i	−0.715966	+	0.698135i	$0.754014\pi$
$108$			100037.i			0.825281i
$109$	128891.			1.03910			0.519549	−	0.854441i	$-0.326100\pi$
$109$	128891.			1.03910			0.519549	+	0.854441i	$0.326100\pi$
$110$	−159763.	−	154995.i	−1.25891	−	1.22133i
$111$	56210.0			0.433018
$112$		−	8234.22i		−	0.0620265i
$113$			44437.9i			0.327384i	0.986511	+	0.163692i	$0.0523404\pi$
$113$			44437.9i			0.327384i	−0.986511	+	0.163692i	$0.947660\pi$
$114$	278917.			2.01008
$115$	−21224.9	−	20591.4i	−0.149659	−	0.145191i
$116$	60039.5			0.414279
$117$			17877.9i			0.120740i
$118$			376414.i			2.48863i
$119$	11315.8			0.0732517
$120$	18016.0	−	18570.3i	0.114210	−	0.117724i
$121$	101484.			0.630138
$122$		−	17571.9i		−	0.106886i
$123$		−	132247.i		−	0.788175i
$124$	180430.			1.05379
$125$	−129011.	−	117787.i	−0.738503	−	0.674250i
$126$	1681.73			0.00943691
$127$			155427.i			0.855101i	0.903991	+	0.427551i	$0.140623\pi$
$127$			155427.i			0.855101i	−0.903991	+	0.427551i	$0.859377\pi$
$128$		−	57046.0i		−	0.307752i
$129$	−241496.			−1.27772
$130$	182683.	−	188304.i	0.948070	−	0.977240i
$131$	−181481.			−0.923961			−0.461981	−	0.886890i	$-0.652861\pi$
$131$	−181481.			−0.923961			−0.461981	+	0.886890i	$0.652861\pi$
$132$		−	240197.i		−	1.19987i
$133$			15890.6i			0.0778952i
$134$	−370695.			−1.78342
$135$	−141366.	−	137146.i	−0.667589	−	0.647662i
$136$	43393.6			0.201177
$137$			273627.i			1.24554i	0.782406	+	0.622769i	$0.213992\pi$
$137$			273627.i			1.24554i	−0.782406	+	0.622769i	$0.786008\pi$
$138$		−	67875.6i		−	0.303400i
$139$	435154.			1.91032			0.955160	−	0.296090i	$-0.0956829\pi$
$139$	435154.			1.91032			0.955160	+	0.296090i	$0.0956829\pi$
$140$	−8327.63	−	8079.06i	−0.0359089	−	0.0348370i
$141$	−37890.4			−0.160502
$142$		−	462371.i		−	1.92429i
$143$			309435.i			1.26541i
$144$	33345.8			0.134009
$145$	−82311.2	+	84843.8i	−0.325117	+	0.335120i
$146$	−93568.1			−0.363284
$147$		−	276613.i		−	1.05579i
$148$		−	96662.0i		−	0.362745i
$149$	−13438.2			−0.0495877			−0.0247939	−	0.999693i	$-0.507893\pi$
$149$	−13438.2			−0.0495877			−0.0247939	+	0.999693i	$0.507893\pi$
$150$	−12147.1	−	400783.i	−0.0440804	−	1.45439i
$151$	181022.			0.646083			0.323042	−	0.946385i	$-0.395295\pi$
$151$	181022.			0.646083			0.323042	+	0.946385i	$0.395295\pi$
$152$			60936.9i			0.213930i
$153$			45825.1i			0.158261i
$154$	29107.8			0.0989025
$155$	−247360.	+	254971.i	−0.826990	+	0.852435i
$156$	283107.			0.931405
$157$		−	419348.i		−	1.35777i	−0.734245	−	0.678885i	$-0.762464\pi$
$157$		−	419348.i		−	1.35777i	0.734245	−	0.678885i	$-0.237536\pi$
$158$		−	459020.i		−	1.46281i
$159$	−326426.			−1.02398
$160$	−315231.	−	305822.i	−0.973485	−	0.944427i
$161$	3867.04			0.0117575
$162$		−	507980.i		−	1.52075i
$163$			91209.3i			0.268887i	0.990921	+	0.134444i	$0.0429247\pi$
$163$			91209.3i			0.268887i	−0.990921	+	0.134444i	$0.957075\pi$
$164$	−227420.			−0.660265
$165$	339430.	+	329298.i	0.970600	+	0.941628i
$166$	686141.			1.93260
$167$			382602.i			1.06159i	0.847501	+	0.530794i	$0.178106\pi$
$167$			382602.i			1.06159i	−0.847501	+	0.530794i	$0.821894\pi$
$168$			3383.39i			0.00924866i
$169$	6579.14			0.0177195
$170$	468258.	−	482665.i	1.24269	−	1.28092i
$171$	−64351.4			−0.168294
$172$			415292.i			1.07037i
$173$			144391.i			0.366796i	0.983039	+	0.183398i	$0.0587098\pi$
$173$			144391.i			0.366796i	−0.983039	+	0.183398i	$0.941290\pi$
$174$	−271324.			−0.679382
$175$	22833.5	−	692.051i	0.0563609	−	0.00170822i
$176$	577157.			1.40447
$177$		−	799722.i		−	1.91869i
$178$			477894.i			1.13053i
$179$	−521036.			−1.21544			−0.607722	−	0.794150i	$-0.707916\pi$
$179$	−521036.			−1.21544			−0.607722	+	0.794150i	$0.707916\pi$
$180$	32717.4	−	33724.1i	0.0752658	−	0.0775816i
$181$	−103563.			−0.234967			−0.117483	−	0.993075i	$-0.537483\pi$
$181$	−103563.			−0.234967			−0.117483	+	0.993075i	$0.537483\pi$
$182$			34307.7i			0.0767738i
$183$			37333.0i			0.0824071i
$184$	14829.2			0.0322905
$185$	136596.	+	132519.i	0.293433	+	0.284674i
$186$	−815376.			−1.72813
$187$			793152.i			1.65864i
$188$			65158.5i			0.134455i
$189$	25755.9			0.0524471
$190$	677798.	+	657566.i	1.36212	+	1.32146i
$191$	886889.			1.75908			0.879540	−	0.475824i	$-0.157850\pi$
$191$	886889.			1.75908			0.879540	+	0.475824i	$0.157850\pi$
$192$		−	412949.i		−	0.808432i
$193$			175474.i			0.339094i	0.985522	+	0.169547i	$0.0542305\pi$
$193$			175474.i			0.339094i	−0.985522	+	0.169547i	$0.945769\pi$
$194$	−434473.			−0.828817
$195$	−388125.	+	400067.i	−0.730946	+	0.753435i
$196$	−475680.			−0.884454
$197$		−	878098.i		−	1.61205i	−0.591885	−	0.806023i	$-0.701616\pi$
$197$		−	878098.i		−	1.61205i	0.591885	−	0.806023i	$-0.298384\pi$
$198$			117877.i			0.213680i
$199$	621709.			1.11290			0.556448	−	0.830882i	$-0.312164\pi$
$199$	621709.			1.11290			0.556448	+	0.830882i	$0.312164\pi$
$200$	87561.7	−	2653.86i	0.154789	−	0.00469141i
$201$	787571.			1.37499
$202$		−	238911.i		−	0.411963i
$203$		−	15458.0i		−	0.0263276i
$204$	725665.			1.22085
$205$	311781.	−	321374.i	0.518162	−	0.534104i
$206$	−1.25733e6			−2.06434
$207$			15660.2i			0.0254022i
$208$			680262.i			1.09023i
$209$	−1.11381e6			−1.76378
$210$	37633.2	+	36509.9i	0.0588875	+	0.0571298i
$211$	531532.			0.821908			0.410954	−	0.911656i	$-0.365196\pi$
$211$	531532.			0.821908			0.410954	+	0.911656i	$0.365196\pi$
$212$			561342.i			0.857804i
$213$			982344.i			1.48359i
$214$	−1.28505e6			−1.91817
$215$	−586862.	−	569344.i	−0.865844	−	0.839999i
$216$	98768.1			0.144040
$217$		−	46453.9i		−	0.0669689i
$218$			1.00165e6i			1.42749i
$219$	198793.			0.280086
$220$	566281.	−	583704.i	0.788815	−	0.813085i
$221$	−934843.			−1.28753
$222$			436823.i			0.594872i
$223$			148971.i			0.200605i	0.994957	+	0.100302i	$0.0319810\pi$
$223$			148971.i			0.200605i	−0.994957	+	0.100302i	$0.968019\pi$
$224$	57433.0			0.0764788
$225$	2802.57	+	92468.0i	0.00369062	+	0.121769i
$226$	−345340.			−0.449754
$227$			912828.i			1.17577i	0.808943	+	0.587887i	$0.200040\pi$
$227$			912828.i			1.17577i	−0.808943	+	0.587887i	$0.799960\pi$
$228$			1.01904e6i			1.29824i
$229$	602691.			0.759462			0.379731	−	0.925097i	$-0.376017\pi$
$229$	602691.			0.759462			0.379731	+	0.925097i	$0.376017\pi$
$230$	160021.	−	164945.i	0.199461	−	0.205598i
$231$	−61841.8			−0.0762522
$232$		−	59277.9i		−	0.0723057i
$233$		−	1.08901e6i		−	1.31414i	−0.753829	−	0.657070i	$-0.771795\pi$
$233$		−	1.08901e6i		−	1.31414i	0.753829	−	0.657070i	$-0.228205\pi$
$234$	−138934.			−0.165871
$235$	−92077.6	−	89329.1i	−0.108764	−	0.105517i
$236$	−1.37525e6			−1.60732
$237$			975224.i			1.12780i
$238$			87938.2i			0.100632i
$239$	234332.			0.265361			0.132680	−	0.991159i	$-0.457642\pi$
$239$	234332.			0.265361			0.132680	+	0.991159i	$0.457642\pi$
$240$	746202.	+	723928.i	0.836234	+	0.811273i
$241$	−179624.			−0.199215			−0.0996076	−	0.995027i	$-0.531759\pi$
$241$	−179624.			−0.199215			−0.0996076	+	0.995027i	$0.531759\pi$
$242$			788663.i			0.865672i
$243$			223074.i			0.242345i
$244$	64200.0			0.0690336
$245$	652134.	−	672199.i	0.694100	−	0.715456i
$246$	1.02773e6			1.08278
$247$		−	1.31278e6i		−	1.36915i
$248$		−	178141.i		−	0.183922i
$249$	−1.45776e6			−1.49001
$250$	915354.	−	1.00258e6i	0.926273	−	1.01454i
$251$	−526227.			−0.527217			−0.263608	−	0.964630i	$-0.584913\pi$
$251$	−526227.			−0.527217			−0.263608	+	0.964630i	$0.584913\pi$
$252$			6144.29i			0.00609496i
$253$			271050.i			0.266225i
$254$	−1.20787e6			−1.17472
$255$	−994851.	+	1.02546e6i	−0.958093	+	0.987571i
$256$	1.24367e6			1.18606
$257$		−	1.81654e6i		−	1.71558i	−0.513997	−	0.857792i	$-0.671836\pi$
$257$		−	1.81654e6i		−	1.71558i	0.513997	−	0.857792i	$-0.328164\pi$
$258$		−	1.87674e6i		−	1.75531i
$259$	−24886.9			−0.0230527
$260$	687979.	+	667443.i	0.631163	+	0.612323i
$261$	62599.4			0.0568812
$262$		−	1.41034e6i		−	1.26932i
$263$			153539.i			0.136877i	0.997655	+	0.0684385i	$0.0218017\pi$
$263$			153539.i			0.136877i	−0.997655	+	0.0684385i	$0.978198\pi$
$264$	−237150.			−0.209418
$265$	−793250.	−	769572.i	−0.693898	−	0.673186i
$266$	−123490.			−0.107011
$267$		−	1.01532e6i		−	0.871619i
$268$		−	1.35435e6i		−	1.15185i
$269$	733721.			0.618230			0.309115	−	0.951025i	$-0.399967\pi$
$269$	733721.			0.618230			0.309115	+	0.951025i	$0.399967\pi$
$270$	1.06580e6	−	1.09859e6i	0.889746	−	0.917122i
$271$	595559.			0.492608			0.246304	−	0.969193i	$-0.420784\pi$
$271$	595559.			0.492608			0.246304	+	0.969193i	$0.420784\pi$
$272$			1.74366e6i			1.42903i
$273$		−	72889.4i		−	0.0591913i
$274$	−2.12643e6			−1.71110
$275$	48507.5	+	1.60046e6i	0.0386792	+	1.27618i
$276$	247987.			0.195955
$277$			1.32205e6i			1.03526i	0.855605	+	0.517629i	$0.173185\pi$
$277$			1.32205e6i			1.03526i	−0.855605	+	0.517629i	$0.826815\pi$
$278$			3.38170e6i			2.62436i
$279$	188122.			0.144687
$280$	−7976.57	+	8221.99i	−0.00608024	+	0.00626732i
$281$	−1.85236e6			−1.39946			−0.699729	−	0.714408i	$-0.746696\pi$
$281$	−1.85236e6			−1.39946			−0.699729	+	0.714408i	$0.746696\pi$
$282$		−	294457.i		−	0.220495i
$283$			840403.i			0.623766i	0.950121	+	0.311883i	$0.100960\pi$
$283$			840403.i			0.623766i	−0.950121	+	0.311883i	$0.899040\pi$
$284$	1.68930e6			1.24283
$285$	−1.44004e6	−	1.39705e6i	−1.05017	−	1.01883i
$286$	−2.40471e6			−1.73839
$287$			58552.1i			0.0419602i
$288$			232584.i			0.165234i
$289$	−976352.			−0.687641
$290$	−659345.	−	639664.i	−0.460381	−	0.446639i
$291$	923073.			0.639004
$292$		−	341856.i		−	0.234632i
$293$		−	130541.i		−	0.0888336i	−0.999013	−	0.0444168i	$-0.985857\pi$
$293$		−	130541.i		−	0.0888336i	0.999013	−	0.0444168i	$-0.0141430\pi$
$294$	2.14964e6			1.45043
$295$	1.88540e6	−	1.94341e6i	1.26139	−	1.30020i
$296$	−95435.8			−0.0633114
$297$			1.80529e6i			1.18756i
$298$		−	104432.i		−	0.0681227i
$299$	−319471.			−0.206659
$300$	1.46428e6	−	44380.2i	0.939337	−	0.0284699i
$301$	106922.			0.0680224
$302$			1.40677e6i			0.887577i
$303$			507585.i			0.317616i
$304$	−2.44859e6			−1.51961
$305$	−88015.0	+	90723.0i	−0.0541760	+	0.0558429i
$306$	−356120.			−0.217416
$307$		−	371862.i		−	0.225183i	−0.993641	−	0.112591i	$-0.964085\pi$
$307$		−	371862.i		−	0.225183i	0.993641	−	0.112591i	$-0.0359151\pi$
$308$			106347.i			0.0638775i
$309$	2.67130e6			1.59157
$310$	−1.98145e6	−	1.92230e6i	−1.17106	−	1.13610i
$311$	−1.29561e6			−0.759581			−0.379791	−	0.925073i	$-0.624004\pi$
$311$	−1.29561e6			−0.759581			−0.379791	+	0.925073i	$0.624004\pi$
$312$		−	279515.i		−	0.162562i
$313$			305809.i			0.176437i	0.996101	+	0.0882183i	$0.0281173\pi$
$313$			305809.i			0.176437i	−0.996101	+	0.0882183i	$0.971883\pi$
$314$	3.25887e6			1.86528
$315$	−8682.69	−	8423.52i	−0.00493035	−	0.00478319i
$316$	1.67705e6			0.944777
$317$		−	2.57648e6i		−	1.44005i	−0.693946	−	0.720027i	$-0.744129\pi$
$317$		−	2.57648e6i		−	1.44005i	0.693946	−	0.720027i	$-0.255871\pi$
$318$		−	2.53675e6i		−	1.40673i
$319$	1.08349e6			0.596138
$320$	973556.	−	1.00351e6i	0.531479	−	0.547831i
$321$	2.73020e6			1.47888
$322$			30051.8i			0.0161522i
$323$		−	3.36496e6i		−	1.79462i
$324$	1.85593e6			0.982198
$325$	−1.88637e6	+	57173.0i	−0.990646	+	0.0300250i
$326$	−708813.			−0.369392
$327$		−	2.12808e6i		−	1.10057i
$328$			224535.i			0.115239i
$329$	16775.9			0.00854469
$330$	−2.55907e6	+	2.63781e6i	−1.29359	+	1.33339i
$331$	2.22853e6			1.11801			0.559007	−	0.829163i	$-0.311182\pi$
$331$	2.22853e6			1.11801			0.559007	+	0.829163i	$0.311182\pi$
$332$			2.50685e6i			1.24820i
$333$		−	100783.i		−	0.0498056i
$334$	−2.97330e6			−1.45839
$335$	1.91388e6	+	1.85675e6i	0.931758	+	0.903945i
$336$	−135953.			−0.0656962
$337$		−	2.26386e6i		−	1.08586i	−0.839777	−	0.542931i	$-0.817314\pi$
$337$		−	2.26386e6i		−	1.08586i	0.839777	−	0.542931i	$-0.182686\pi$
$338$			51128.3i			0.0243428i
$339$	733702.			0.346753
$340$	1.76344e6	+	1.71080e6i	0.827302	+	0.802607i
$341$	3.25607e6			1.51638
$342$		−	500093.i		−	0.231199i
$343$			245331.i			0.112594i
$344$	410023.			0.186815
$345$	−339978.	+	350439.i	−0.153781	+	0.158513i
$346$	−1.12210e6			−0.503898
$347$			2.65136e6i			1.18207i	0.806645	+	0.591037i	$0.201281\pi$
$347$			2.65136e6i			1.18207i	−0.806645	+	0.591037i	$0.798719\pi$
$348$		−	991295.i		−	0.438788i
$349$	3.79224e6			1.66660			0.833301	−	0.552819i	$-0.186448\pi$
$349$	3.79224e6			1.66660			0.833301	+	0.552819i	$0.186448\pi$
$350$	5378.12	+	177446.i	0.00234672	+	0.0774276i
$351$	−2.12779e6			−0.921853
$352$			4.02562e6i			1.73171i
$353$		−	2.17078e6i		−	0.927213i	−0.886041	−	0.463607i	$-0.846555\pi$
$353$		−	2.17078e6i		−	0.927213i	0.886041	−	0.463607i	$-0.153445\pi$
$354$	6.21486e6			2.63587
$355$	−2.31594e6	+	2.38720e6i	−0.975342	+	1.00535i
$356$	−1.74601e6			−0.730167
$357$		−	186832.i		−	0.0775854i
$358$		−	4.04911e6i		−	1.66975i
$359$	−1.75257e6			−0.717695			−0.358847	−	0.933396i	$-0.616830\pi$
$359$	−1.75257e6			−0.717695			−0.358847	+	0.933396i	$0.616830\pi$
$360$	−33296.2	−	32302.4i	−0.0135406	−	0.0131365i
$361$	2.24925e6			0.908386
$362$		−	804814.i		−	0.322793i
$363$		−	1.67558e6i		−	0.667418i
$364$	−125345.			−0.0495854
$365$	483088.	+	468668.i	0.189799	+	0.184134i
$366$	−290125.			−0.113209
$367$		−	2.76408e6i		−	1.07124i	−0.844460	−	0.535618i	$-0.820079\pi$
$367$		−	2.76408e6i		−	1.07124i	0.844460	−	0.535618i	$-0.179921\pi$
$368$			595875.i			0.229370i
$369$	−237116.			−0.0906557
$370$	−1.02984e6	+	1.06153e6i	−0.391080	+	0.403113i
$371$	144525.			0.0545139
$372$		−	2.97902e6i		−	1.11613i
$373$		−	1.89476e6i		−	0.705151i	−0.935783	−	0.352576i	$-0.885306\pi$
$373$		−	1.89476e6i		−	0.705151i	0.935783	−	0.352576i	$-0.114694\pi$
$374$	−6.16380e6			−2.27861
$375$	−1.94474e6	+	2.13006e6i	−0.714141	+	0.782194i
$376$	64331.9			0.0234670
$377$			1.27704e6i			0.462756i
$378$			200156.i			0.0720508i
$379$	−209091.			−0.0747716			−0.0373858	−	0.999301i	$-0.511903\pi$
$379$	−209091.			−0.0747716			−0.0373858	+	0.999301i	$0.511903\pi$
$380$	−2.40245e6	+	2.47637e6i	−0.853486	+	0.879745i
$381$	2.56621e6			0.905691
$382$			6.89226e6i			2.41659i
$383$		−	3.11090e6i		−	1.08365i	−0.840491	−	0.541825i	$-0.817733\pi$
$383$		−	3.11090e6i		−	1.08365i	0.840491	−	0.541825i	$-0.182267\pi$
$384$	−941870.			−0.325959
$385$	−150282.	−	145796.i	−0.0516720	−	0.0501297i
$386$	−1.36366e6			−0.465842
$387$			432998.i			0.146963i
$388$		−	1.58737e6i		−	0.535303i
$389$	−1.50348e6			−0.503762			−0.251881	−	0.967758i	$-0.581049\pi$
$389$	−1.50348e6			−0.503762			−0.251881	+	0.967758i	$0.581049\pi$
$390$	−3.10903e6	−	3.01623e6i	−1.03506	−	1.00416i
$391$	−818876.			−0.270880
$392$			469646.i			0.154367i
$393$			2.99639e6i			0.978625i
$394$	6.82394e6			2.21460
$395$	−2.29916e6	+	2.36990e6i	−0.741440	+	0.764252i
$396$	−430668.			−0.138008
$397$		−	640908.i		−	0.204089i	−0.994780	−	0.102044i	$-0.967462\pi$
$397$		−	640908.i		−	0.204089i	0.994780	−	0.102044i	$-0.0325384\pi$
$398$			4.83148e6i			1.52888i
$399$	262365.			0.0825037
$400$	106639.	+	3.51844e6i	0.0333246	+	1.09951i
$401$	5.49966e6			1.70795			0.853975	−	0.520314i	$-0.174185\pi$
$401$	5.49966e6			1.70795			0.853975	+	0.520314i	$0.174185\pi$
$402$			6.12044e6i			1.88894i
$403$			3.83774e6i			1.17710i
$404$	872874.			0.266071
$405$	−2.54439e6	+	2.62267e6i	−0.770808	+	0.794524i
$406$	120128.			0.0361684
$407$		−	1.74438e6i		−	0.521982i
$408$		−	716459.i		−	0.213079i
$409$	−1.53891e6			−0.454888			−0.227444	−	0.973791i	$-0.573037\pi$
$409$	−1.53891e6			−0.454888			−0.227444	+	0.973791i	$0.573037\pi$
$410$	2.49749e6	+	2.42294e6i	0.733742	+	0.711841i
$411$	4.51777e6			1.31923
$412$		−	4.59373e6i		−	1.33328i
$413$			354076.i			0.102146i
$414$	−121700.			−0.0348970
$415$	−3.54251e6	−	3.43677e6i	−1.00970	−	0.979558i
$416$	−4.74477e6			−1.34425
$417$		−	7.18470e6i		−	2.02334i
$418$		−	8.65573e6i		−	2.42305i
$419$	−1.43328e6			−0.398837			−0.199419	−	0.979914i	$-0.563905\pi$
$419$	−1.43328e6			−0.398837			−0.199419	+	0.979914i	$0.563905\pi$
$420$	−133391.	+	137495.i	−0.0368980	+	0.0380333i
$421$	−4.41511e6			−1.21405			−0.607024	−	0.794683i	$-0.707637\pi$
$421$	−4.41511e6			−1.21405			−0.607024	+	0.794683i	$0.707637\pi$
$422$			4.13068e6i			1.12912i
$423$			67936.6i			0.0184609i
$424$	554221.			0.149716
$425$	−4.83518e6	+	146547.i	−1.29850	+	0.0393555i
$426$	−7.63407e6			−2.03813
$427$		−	16529.1i		−	0.00438712i
$428$		−	4.69501e6i		−	1.23888i
$429$	5.10900e6			1.34027
$430$	4.42453e6	−	4.56067e6i	1.15398	−	1.18948i
$431$	−323287.			−0.0838291			−0.0419146	−	0.999121i	$-0.513346\pi$
$431$	−323287.			−0.0838291			−0.0419146	+	0.999121i	$0.513346\pi$
$432$			3.96874e6i			1.02316i
$433$			408265.i			0.104646i	0.998630	+	0.0523230i	$0.0166625\pi$
$433$			408265.i			0.104646i	−0.998630	+	0.0523230i	$0.983337\pi$
$434$	361006.			0.0920006
$435$	1.40083e6	+	1.35902e6i	0.354946	+	0.344351i
$436$	−3.65958e6			−0.921966
$437$		−	1.14993e6i		−	0.288051i
$438$			1.54488e6i			0.384776i
$439$	−3.88258e6			−0.961521			−0.480761	−	0.876852i	$-0.659639\pi$
$439$	−3.88258e6			−0.961521			−0.480761	+	0.876852i	$0.659639\pi$
$440$	−576299.	−	559097.i	−0.141911	−	0.137675i
$441$	−495962.			−0.121437
$442$		−	7.26492e6i		−	1.76879i
$443$			2.35832e6i			0.570943i	0.958387	+	0.285471i	$0.0921502\pi$
$443$			2.35832e6i			0.570943i	−0.958387	+	0.285471i	$0.907850\pi$
$444$	−1.59596e6			−0.384206
$445$	2.39370e6	−	2.46734e6i	0.573019	−	0.590649i
$446$	−1.15770e6			−0.275587
$447$			221874.i			0.0525214i
$448$			182833.i			0.0430387i
$449$	6.47326e6			1.51533			0.757665	−	0.652644i	$-0.226340\pi$
$449$	6.47326e6			1.51533			0.757665	+	0.652644i	$0.226340\pi$
$450$	−718595.	+	21779.5i	−0.167283	+	0.00507011i
$451$	−4.10406e6			−0.950107
$452$		−	1.26172e6i		−	0.290480i
$453$		−	2.98880e6i		−	0.684307i
$454$	−7.09384e6			−1.61526
$455$	171842.	−	177129.i	0.0389135	−	0.0401108i
$456$	1.00611e6			0.226587
$457$			2.07856e6i			0.465557i	0.972530	+	0.232779i	$0.0747817\pi$
$457$			2.07856e6i			0.465557i	−0.972530	+	0.232779i	$0.925218\pi$
$458$			4.68368e6i			1.04333i
$459$	−5.45401e6			−1.20833
$460$	602635.	+	584647.i	0.132788	+	0.128825i
$461$	4.43220e6			0.971332			0.485666	−	0.874145i	$-0.338577\pi$
$461$	4.43220e6			0.971332			0.485666	+	0.874145i	$0.338577\pi$
$462$		−	480590.i		−	0.104754i
$463$			2.25296e6i			0.488428i	0.969721	+	0.244214i	$0.0785299\pi$
$463$			2.25296e6i			0.488428i	−0.969721	+	0.244214i	$0.921470\pi$
$464$	2.38193e6			0.513611
$465$	4.20975e6	+	4.08409e6i	0.902867	+	0.875917i
$466$	8.46300e6			1.80534
$467$			1.37928e6i			0.292658i	0.989236	+	0.146329i	$0.0467457\pi$
$467$			1.37928e6i			0.292658i	−0.989236	+	0.146329i	$0.953254\pi$
$468$		−	507604.i		−	0.107130i
$469$	−348696.			−0.0732006
$470$	694202.	−	715561.i	0.144958	−	0.149418i
$471$	−6.92374e6			−1.43810
$472$			1.35780e6i			0.280532i
$473$			7.49444e6i			1.54023i
$474$	−7.57874e6			−1.54936
$475$	−205794.	−	6.78997e6i	−0.0418503	−	1.38081i
$476$	−321287.			−0.0649944
$477$			585276.i			0.117778i
$478$			1.82106e6i			0.364548i
$479$	−9.86038e6			−1.96361			−0.981805	−	0.189894i	$-0.939185\pi$
$479$	−9.86038e6			−1.96361			−0.981805	+	0.189894i	$0.939185\pi$
$480$	−5.04933e6	+	5.20469e6i	−1.00030	+	1.03108i
$481$	2.05600e6			0.405192
$482$		−	1.39591e6i		−	0.273678i
$483$		−	63847.5i		−	0.0124531i
$484$	−2.88142e6			−0.559106
$485$	2.24317e6	+	2.17621e6i	0.433019	+	0.420094i
$486$	−1.73357e6			−0.332928
$487$			5.47539e6i			1.04615i	0.852288	+	0.523073i	$0.175214\pi$
$487$			5.47539e6i			1.04615i	−0.852288	+	0.523073i	$0.824786\pi$
$488$		−	63385.5i		−	0.0120487i
$489$	1.50593e6			0.284795
$490$	5.22385e6	+	5.06792e6i	0.982879	+	0.953541i
$491$	−5.62532e6			−1.05304			−0.526519	−	0.850164i	$-0.676503\pi$
$491$	−5.62532e6			−1.05304			−0.526519	+	0.850164i	$0.676503\pi$
$492$			3.75486e6i			0.699328i
$493$			3.27334e6i			0.606561i
$494$	1.02020e7			1.88091
$495$	590425.	−	608591.i	0.108306	−	0.111638i
$496$	7.15813e6			1.30646
$497$		−	434931.i		−	0.0789823i
$498$		−	1.13287e7i		−	2.04694i
$499$	9.22947e6			1.65930			0.829651	−	0.558282i	$-0.188539\pi$
$499$	9.22947e6			1.65930			0.829651	+	0.558282i	$0.188539\pi$
$500$	3.66299e6	+	3.34429e6i	0.655255	+	0.598245i
$501$	6.31702e6			1.12439
$502$		−	4.08946e6i		−	0.724280i
$503$		−	965448.i		−	0.170141i	−0.996375	−	0.0850705i	$-0.972888\pi$
$503$		−	965448.i		−	0.170141i	0.996375	−	0.0850705i	$-0.0271116\pi$
$504$	6066.35			0.00106378
$505$	−1.19667e6	+	1.23349e6i	−0.208807	+	0.215231i
$506$	−2.10641e6			−0.365734
$507$		−	108626.i		−	0.0187679i
$508$		−	4.41301e6i		−	0.758710i
$509$	3.27345e6			0.560030			0.280015	−	0.959996i	$-0.409661\pi$
$509$	3.27345e6			0.560030			0.280015	+	0.959996i	$0.409661\pi$
$510$	−7.96914e6	−	7.73126e6i	−1.35671	−	1.31621i
$511$	−88015.3			−0.0149110
$512$			7.83945e6i			1.32163i
$513$		−	7.65897e6i		−	1.28492i
$514$	1.41168e7			2.35684
$515$	6.49155e6	+	6.29778e6i	1.07853	+	1.04633i
$516$	6.85676e6			1.13369
$517$			1.17586e6i			0.193478i
$518$		−	193403.i		−	0.0316693i
$519$	2.38400e6			0.388497
$520$	658976.	−	679251.i	0.106871	−	0.110159i
$521$	−592762.			−0.0956723			−0.0478361	−	0.998855i	$-0.515233\pi$
$521$	−592762.			−0.0956723			−0.0478361	+	0.998855i	$0.515233\pi$
$522$			486478.i			0.0781424i
$523$		−	822001.i		−	0.131407i	−0.997839	−	0.0657034i	$-0.979071\pi$
$523$		−	822001.i		−	0.131407i	0.997839	−	0.0657034i	$-0.0209291\pi$
$524$	5.15276e6			0.819808
$525$	−11426.2	−	376998.i	−0.00180928	−	0.0596954i
$526$	−1.19320e6			−0.188039
$527$			9.83699e6i			1.54289i
$528$		−	9.52926e6i		−	1.48756i
$529$	−279841.			−0.0434783
$530$	5.98056e6	−	6.16457e6i	0.924810	−	0.953264i
$531$	−1.43388e6			−0.220688
$532$		−	451178.i		−	0.0691145i
$533$		−	4.83722e6i		−	0.737527i
$534$	7.89037e6			1.19741
$535$	6.63467e6	+	6.43663e6i	1.00216	+	0.972242i
$536$	−1.33717e6			−0.201037
$537$			8.60266e6i			1.28735i
$538$			5.70195e6i			0.849313i
$539$	−8.58422e6			−1.27271
$540$	4.01377e6	+	3.89396e6i	0.592335	+	0.574655i
$541$	6.54620e6			0.961604			0.480802	−	0.876829i	$-0.340346\pi$
$541$	6.54620e6			0.961604			0.480802	+	0.876829i	$0.340346\pi$
$542$			4.62826e6i			0.676736i
$543$			1.70989e6i			0.248868i
$544$	−1.21619e7			−1.76199
$545$	5.01710e6	−	5.17146e6i	0.723538	−	0.745800i
$546$	566444.			0.0813159
$547$		−	5.44281e6i		−	0.777776i	−0.921285	−	0.388888i	$-0.872859\pi$
$547$		−	5.44281e6i		−	0.777776i	0.921285	−	0.388888i	$-0.127141\pi$
$548$		−	7.76903e6i		−	1.10513i
$549$	66937.2			0.00947844
$550$	−1.24376e7	+	376966.i	−1.75320	+	0.0531367i
$551$	−4.59670e6			−0.645011
$552$		−	244841.i		−	0.0342009i
$553$		−	431779.i		−	0.0600411i
$554$	−1.02740e7			−1.42222
$555$	2.18798e6	−	2.25530e6i	0.301516	−	0.310793i
$556$	−1.23552e7			−1.69498
$557$			1.44274e6i			0.197037i	0.995135	+	0.0985187i	$0.0314104\pi$
$557$			1.44274e6i			0.197037i	−0.995135	+	0.0985187i	$0.968590\pi$
$558$			1.46195e6i			0.198769i
$559$	−8.83327e6			−1.19562
$560$	−330380.	−	320518.i	−0.0445188	−	0.0431899i
$561$	1.30955e7			1.75677
$562$		−	1.43952e7i		−	1.92255i
$563$		−	1.81146e6i		−	0.240857i	−0.992722	−	0.120428i	$-0.961573\pi$
$563$		−	1.81146e6i		−	0.240857i	0.992722	−	0.120428i	$-0.0384268\pi$
$564$	1.07581e6			0.142410
$565$	1.78297e6	+	1.72975e6i	0.234976	+	0.227962i
$566$	−6.53101e6			−0.856918
$567$		−	477833.i		−	0.0624193i
$568$		−	1.66787e6i		−	0.216915i
$569$	−9.21951e6			−1.19379			−0.596894	−	0.802320i	$-0.703599\pi$
$569$	−9.21951e6			−1.19379			−0.596894	+	0.802320i	$0.703599\pi$
$570$	1.08569e7	−	1.11909e7i	1.39965	−	1.44271i
$571$	1.07294e7			1.37716			0.688582	−	0.725158i	$-0.258234\pi$
$571$	1.07294e7			1.37716			0.688582	+	0.725158i	$0.258234\pi$
$572$		−	8.78574e6i		−	1.12276i
$573$		−	1.46432e7i		−	1.86315i
$574$	−455025.			−0.0576442
$575$	−1.65237e6	+	50080.8i	−0.208419	+	0.00631686i
$576$	−740410.			−0.0929856
$577$			9.83068e6i			1.22926i	0.788815	+	0.614630i	$0.210695\pi$
$577$			9.83068e6i			1.22926i	−0.788815	+	0.614630i	$0.789305\pi$
$578$		−	7.58751e6i		−	0.944669i
$579$	2.89721e6			0.359156
$580$	2.33705e6	−	2.40895e6i	0.288468	−	0.297343i
$581$	645421.			0.0793237
$582$			7.17346e6i			0.877852i
$583$			1.01301e7i			1.23436i
$584$	−337519.			−0.0409512
$585$	717312.	+	695900.i	0.0866599	+	0.0840732i
$586$	1.01447e6			0.122038
$587$		−	1.12085e7i		−	1.34262i	−0.741177	−	0.671310i	$-0.765732\pi$
$587$		−	1.12085e7i		−	1.34262i	0.741177	−	0.671310i	$-0.234268\pi$
$588$			7.85382e6i			0.936780i
$589$	−1.38139e7			−1.64070
$590$	1.51028e7	+	1.46520e7i	1.78619	+	1.73287i
$591$	−1.44980e7			−1.70742
$592$		−	3.83484e6i		−	0.449721i
$593$		−	1.38972e7i		−	1.62290i	−0.584423	−	0.811449i	$-0.698679\pi$
$593$		−	1.38972e7i		−	1.62290i	0.584423	−	0.811449i	$-0.301321\pi$
$594$	−1.40294e7			−1.63145
$595$	440469.	−	454021.i	0.0510062	−	0.0525755i
$596$	381547.			0.0439979
$597$		−	1.02649e7i		−	1.17874i
$598$		−	2.48270e6i		−	0.283904i
$599$	−5.10108e6			−0.580891			−0.290446	−	0.956892i	$-0.593804\pi$
$599$	−5.10108e6			−0.580891			−0.290446	+	0.956892i	$0.593804\pi$
$600$	−43817.2	−	1.44570e6i	−0.00496897	−	0.163946i
$601$	9.41211e6			1.06292			0.531461	−	0.847083i	$-0.321643\pi$
$601$	9.41211e6			1.06292			0.531461	+	0.847083i	$0.321643\pi$
$602$			830922.i			0.0934478i
$603$		−	1.41210e6i		−	0.158151i
$604$	−5.13972e6			−0.573254
$605$	3.95029e6	−	4.07183e6i	0.438774	−	0.452274i
$606$	−3.94459e6			−0.436335
$607$		−	1.61401e6i		−	0.177801i	−0.996041	−	0.0889005i	$-0.971665\pi$
$607$		−	1.61401e6i		−	0.177801i	0.996041	−	0.0889005i	$-0.0283353\pi$
$608$		−	1.70787e7i		−	1.87369i
$609$	−255222.			−0.0278852
$610$	−705034.	−	683989.i	−0.0767159	−	0.0744260i
$611$	−1.38592e6			−0.150188
$612$		−	1.30110e6i		−	0.140421i
$613$			1.39952e7i			1.50427i	0.659007	+	0.752137i	$0.270977\pi$
$613$			1.39952e7i			1.50427i	−0.659007	+	0.752137i	$0.729023\pi$
$614$	2.88984e6			0.309352
$615$	−5.30611e6	−	5.14773e6i	−0.565703	−	0.548817i
$616$	104998.			0.0111488
$617$		−	1.34927e7i		−	1.42687i	−0.700720	−	0.713437i	$-0.747138\pi$
$617$		−	1.34927e7i		−	1.42687i	0.700720	−	0.713437i	$-0.252862\pi$
$618$			2.07594e7i			2.18648i
$619$	−3.97835e6			−0.417327			−0.208664	−	0.977987i	$-0.566911\pi$
$619$	−3.97835e6			−0.417327			−0.208664	+	0.977987i	$0.566911\pi$
$620$	7.02324e6	−	7.23933e6i	0.733768	−	0.756344i
$621$	−1.86384e6			−0.193946
$622$		−	1.00686e7i		−	1.04350i
$623$			449533.i			0.0464025i
$624$	1.12316e7			1.15473
$625$	−9.74770e6	+	591420.i	−0.998164	+	0.0605614i
$626$	−2.37652e6			−0.242385
$627$			1.83898e7i			1.86813i
$628$			1.19065e7i			1.20471i
$629$	5.26999e6			0.531109
$630$	65461.6	−	67475.7i	0.00657105	−	0.00677323i
$631$	−1.37318e7			−1.37295			−0.686474	−	0.727155i	$-0.740842\pi$
$631$	−1.37318e7			−1.37295			−0.686474	+	0.727155i	$0.740842\pi$
$632$		−	1.65578e6i		−	0.164896i
$633$		−	8.77597e6i		−	0.870534i
$634$	2.00226e7			1.97832
$635$	6.23617e6	+	6.05002e6i	0.613738	+	0.595419i
$636$	9.26816e6			0.908554
$637$		−	1.01177e7i		−	0.987949i
$638$			8.42007e6i			0.818963i
$639$	1.76132e6			0.170642
$640$	−2.28884e6	−	2.22052e6i	−0.220885	−	0.214292i
$641$	−1.70708e7			−1.64100			−0.820499	−	0.571649i	$-0.806304\pi$
$641$	−1.70708e7			−1.64100			−0.820499	+	0.571649i	$0.806304\pi$
$642$			2.12171e7i			2.03165i
$643$			1.32344e7i			1.26234i	0.775645	+	0.631169i	$0.217424\pi$
$643$			1.32344e7i			1.26234i	−0.775645	+	0.631169i	$0.782576\pi$
$644$	−109796.			−0.0104321
$645$	−9.40028e6	+	9.68950e6i	−0.889695	+	0.917069i
$646$	2.61500e7			2.46542
$647$		−	6.34727e6i		−	0.596110i	−0.954549	−	0.298055i	$-0.903662\pi$
$647$		−	6.34727e6i		−	0.596110i	0.954549	−	0.298055i	$-0.0963378\pi$
$648$		−	1.83239e6i		−	0.171427i
$649$	−2.48180e7			−2.31289
$650$	−444308.	−	1.46595e7i	−0.0412478	−	1.36093i
$651$	−766987.			−0.0709309
$652$		−	2.58969e6i		−	0.238577i
$653$			1.60903e7i			1.47666i	0.674437	+	0.738332i	$0.264386\pi$
$653$			1.60903e7i			1.47666i	−0.674437	+	0.738332i	$0.735614\pi$
$654$	1.65379e7			1.51195
$655$	−7.06418e6	+	7.28153e6i	−0.643367	+	0.663162i
$656$	−9.02235e6			−0.818578
$657$		−	356432.i		−	0.0322154i
$658$			130370.i			0.0117385i
$659$	−2.11630e6			−0.189830			−0.0949148	−	0.995485i	$-0.530258\pi$
$659$	−2.11630e6			−0.189830			−0.0949148	+	0.995485i	$0.530258\pi$
$660$	−9.63737e6	−	9.34970e6i	−0.861189	−	0.835483i
$661$	5.77601e6			0.514191			0.257095	−	0.966386i	$-0.417235\pi$
$661$	5.77601e6			0.514191			0.257095	+	0.966386i	$0.417235\pi$
$662$			1.73185e7i			1.53591i
$663$			1.54349e7i			1.36370i
$664$	2.47505e6			0.217853
$665$	637574.	+	618543.i	0.0559083	+	0.0542395i
$666$	783216.			0.0684220
$667$			1.11863e6i			0.0973577i
$668$		−	1.08631e7i		−	0.941919i
$669$	2.45962e6			0.212473
$670$	−1.44293e7	+	1.48733e7i	−1.24182	+	1.28003i
$671$	1.15857e6			0.0993378
$672$		−	948259.i		−	0.0810035i
$673$			8.72533e6i			0.742582i	0.928517	+	0.371291i	$0.121085\pi$
$673$			8.72533e6i			0.742582i	−0.928517	+	0.371291i	$0.878915\pi$
$674$	1.75931e7			1.49174
$675$	−1.10053e7	+	333556.i	−0.929703	+	0.0281779i
$676$	−186800.			−0.0157221
$677$			2.64813e6i			0.222058i	0.993817	+	0.111029i	$0.0354147\pi$
$677$			2.64813e6i			0.222058i	−0.993817	+	0.111029i	$0.964585\pi$
$678$			5.70180e6i			0.476363i
$679$	−408689.			−0.0340188
$680$	1.68910e6	−	1.74107e6i	0.140082	−	0.144392i
$681$	1.50714e7			1.24534
$682$			2.53038e7i			2.08317i
$683$		−	2.03196e6i		−	0.166673i	−0.996521	−	0.0833363i	$-0.973442\pi$
$683$		−	2.03196e6i		−	0.166673i	0.996521	−	0.0833363i	$-0.0265576\pi$
$684$	1.82712e6			0.149323
$685$	1.09787e7	+	1.06510e7i	0.893970	+	0.867285i
$686$	−1.90653e6			−0.154680
$687$		−	9.95085e6i		−	0.804393i
$688$			1.64757e7i			1.32701i
$689$	−1.19398e7			−0.958181
$690$	−2.72336e6	−	2.64207e6i	−0.217762	−	0.211262i
$691$	1.68677e7			1.34388			0.671942	−	0.740604i	$-0.265460\pi$
$691$	1.68677e7			1.34388			0.671942	+	0.740604i	$0.265460\pi$
$692$		−	4.09967e6i		−	0.325449i
$693$			110881.i			0.00877051i
$694$	−2.06044e7			−1.62391
$695$	1.69384e7	−	1.74596e7i	1.33018	−	1.37111i
$696$	−978720.			−0.0765835
$697$		−	1.23989e7i		−	0.966719i
$698$			2.94705e7i			2.28955i
$699$	−1.79803e7			−1.39189
$700$	−648308.	+	19649.3i	−0.0500077	+	0.00151566i
$701$	−9.68545e6			−0.744431			−0.372216	−	0.928146i	$-0.621402\pi$
$701$	−9.68545e6			−0.744431			−0.372216	+	0.928146i	$0.621402\pi$
$702$		−	1.65357e7i		−	1.26642i
$703$			7.40057e6i			0.564776i
$704$	−1.28152e7			−0.974526
$705$	−1.47489e6	+	1.52027e6i	−0.111760	+	0.115198i
$706$	1.68698e7			1.27379
$707$		−	224733.i		−	0.0169090i
$708$			2.27063e7i			1.70241i
$709$	−6.35593e6			−0.474858			−0.237429	−	0.971405i	$-0.576305\pi$
$709$	−6.35593e6			−0.474858			−0.237429	+	0.971405i	$0.576305\pi$
$710$	−1.85516e7	−	1.79978e7i	−1.38113	−	1.33991i
$711$	1.74856e6			0.129720
$712$			1.72386e6i			0.127439i
$713$			3.36167e6i			0.247646i
$714$	1.45192e6			0.106585
$715$	1.24154e7	+	1.20448e7i	0.908230	+	0.881120i
$716$	1.47937e7			1.07843
$717$		−	3.86898e6i		−	0.281060i
$718$		−	1.36197e7i		−	0.985955i
$719$	2.63364e7			1.89991			0.949956	−	0.312384i	$-0.101128\pi$
$719$	2.63364e7			1.89991			0.949956	+	0.312384i	$0.101128\pi$
$720$	1.29799e6	−	1.33792e6i	0.0933124	−	0.0961835i
$721$	−1.18272e6			−0.0847310
$722$			1.74796e7i			1.24792i
$723$			2.96572e6i			0.211001i
$724$	2.94043e6			0.208480
$725$	200191.	+	6.60511e6i	0.0141449	+	0.466697i
$726$	1.30214e7			0.916887
$727$		−	1.98997e7i		−	1.39640i	−0.715902	−	0.698200i	$-0.753984\pi$
$727$		−	1.98997e7i		−	1.39640i	0.715902	−	0.698200i	$-0.246016\pi$
$728$			123755.i			0.00865434i
$729$	−1.22009e7			−0.850301
$730$	−3.64215e6	+	3.75421e6i	−0.252959	+	0.260742i
$731$	−2.26416e7			−1.56716
$732$		−	1.05999e6i		−	0.0731178i
$733$			1.47917e7i			1.01685i	0.861106	+	0.508425i	$0.169772\pi$
$733$			1.47917e7i			1.01685i	−0.861106	+	0.508425i	$0.830228\pi$
$734$	2.14804e7			1.47164
$735$	−1.10985e7	−	1.07672e7i	−0.757784	−	0.735164i
$736$	−4.15618e6			−0.282813
$737$		−	2.44410e7i		−	1.65748i
$738$		−	1.84270e6i		−	0.124541i
$739$	2.93771e7			1.97878			0.989391	−	0.145278i	$-0.0464075\pi$
$739$	2.93771e7			1.97878			0.989391	+	0.145278i	$0.0464075\pi$
$740$	−3.87835e6	−	3.76258e6i	−0.260356	−	0.252584i
$741$	−2.16750e7			−1.45015
$742$			1.12314e6i			0.0748902i
$743$			8.89487e6i			0.591109i	0.955326	+	0.295555i	$0.0955044\pi$
$743$			8.89487e6i			0.591109i	−0.955326	+	0.295555i	$0.904496\pi$
$744$	−2.94123e6			−0.194803
$745$	−523082.	+	539176.i	−0.0345286	+	0.0355910i
$746$	1.47247e7			0.968723
$747$			2.61374e6i			0.171380i
$748$		−	2.25198e7i		−	1.47167i
$749$	−1.20879e6			−0.0787312
$750$	−1.65533e7	−	1.51131e7i	−1.07456	−	0.981073i
$751$	−2.45791e7			−1.59025			−0.795125	−	0.606445i	$-0.792595\pi$
$751$	−2.45791e7			−1.59025			−0.795125	+	0.606445i	$0.792595\pi$
$752$			2.58501e6i			0.166693i
$753$			8.68838e6i			0.558408i
$754$	−9.92426e6			−0.635725
$755$	7.04630e6	−	7.26310e6i	0.449877	−	0.463718i
$756$	−731281.			−0.0465350
$757$		−	2.57709e7i		−	1.63452i	−0.576268	−	0.817261i	$-0.695492\pi$
$757$		−	2.57709e7i		−	1.63452i	0.576268	−	0.817261i	$-0.304508\pi$
$758$		−	1.62490e6i		−	0.102720i
$759$	4.47523e6			0.281975
$760$	2.44496e6	+	2.37198e6i	0.153546	+	0.148962i
$761$	1.91790e7			1.20051			0.600253	−	0.799810i	$-0.295067\pi$
$761$	1.91790e7			1.20051			0.600253	+	0.799810i	$0.295067\pi$
$762$			1.99428e7i			1.24422i
$763$			942205.i			0.0585915i
$764$	−2.51813e7			−1.56079
$765$	1.83863e6	+	1.78375e6i	0.113590	+	0.110200i
$766$	2.41757e7			1.48870
$767$		−	2.92516e7i		−	1.79540i
$768$		−	2.05339e7i		−	1.25623i
$769$	1.95840e7			1.19422			0.597111	−	0.802159i	$-0.296315\pi$
$769$	1.95840e7			1.19422			0.597111	+	0.802159i	$0.296315\pi$
$770$	1.13302e6	−	1.16788e6i	0.0688672	−	0.0709861i
$771$	−2.99923e7			−1.81708
$772$		−	4.98221e6i		−	0.300870i
$773$		−	1.93830e7i		−	1.16674i	−0.812208	−	0.583368i	$-0.801734\pi$
$773$		−	1.93830e7i		−	1.16674i	0.812208	−	0.583368i	$-0.198266\pi$
$774$	−3.36495e6			−0.201895
$775$	601610.	+	1.98495e7i	0.0359799	+	1.18712i
$776$	−1.56723e6			−0.0934286
$777$			410900.i			0.0244165i
$778$		−	1.16840e7i		−	0.692058i
$779$	1.74115e7			1.02800
$780$	1.10200e7	−	1.13590e7i	0.648550	−	0.668504i
$781$	3.04854e7			1.78840
$782$		−	6.36371e6i		−	0.372129i
$783$			7.45045e6i			0.434288i
$784$	−1.88715e7			−1.09652
$785$	−1.68254e7	−	1.63232e7i	−0.974522	−	0.945433i
$786$	−2.32858e7			−1.34442
$787$			2.91524e7i			1.67779i	0.544294	+	0.838895i	$0.316798\pi$
$787$			2.91524e7i			1.67779i	−0.544294	+	0.838895i	$0.683202\pi$
$788$			2.49316e7i			1.43033i
$789$	2.53504e6			0.144975
$790$	−1.84171e7	−	1.78674e7i	−1.04992	−	1.01858i
$791$	−324845.			−0.0184602
$792$			425205.i			0.0240871i
$793$			1.36554e6i			0.0771116i
$794$	4.98067e6			0.280373
$795$	−1.27062e7	+	1.30971e7i	−0.713013	+	0.734950i
$796$	−1.76521e7			−0.987445
$797$		−	1.26902e7i		−	0.707659i	−0.935310	−	0.353829i	$-0.884879\pi$
$797$		−	1.26902e7i		−	0.707659i	0.935310	−	0.353829i	$-0.115121\pi$
$798$			2.03891e6i			0.113342i
$799$	−3.55243e6			−0.196860
$800$	−2.45408e7	+	743796.i	−1.35570	+	0.0410893i
$801$	−1.82046e6			−0.100253
$802$			4.27394e7i			2.34635i
$803$		−	6.16921e6i		−	0.337630i
$804$	−2.23613e7			−1.21999
$805$	150525.	−	155156.i	0.00818688	−	0.00843878i
$806$	−2.98242e7			−1.61708
$807$		−	1.21143e7i		−	0.654806i
$808$		−	861801.i		−	0.0464385i
$809$	2.51445e7			1.35074			0.675369	−	0.737480i	$-0.263985\pi$
$809$	2.51445e7			1.35074			0.675369	+	0.737480i	$0.263985\pi$
$810$	−2.03815e7	−	1.97732e7i	−1.09150	−	1.05892i
$811$	−2.62629e7			−1.40214			−0.701069	−	0.713094i	$-0.747293\pi$
$811$	−2.62629e7			−1.40214			−0.701069	+	0.713094i	$0.747293\pi$
$812$			438895.i			0.0233599i
$813$		−	9.83310e6i		−	0.521752i
$814$	1.35561e7			0.717089
$815$	3.65957e6	+	3.55033e6i	0.192990	+	0.187230i
$816$	2.87891e7			1.51357
$817$		−	3.17953e7i		−	1.66651i
$818$		−	1.19593e7i		−	0.624917i
$819$	−130689.			−0.00680817
$820$	−8.85234e6	+	9.12470e6i	−0.459752	+	0.473897i
$821$	−3.02297e7			−1.56522			−0.782612	−	0.622510i	$-0.786113\pi$
$821$	−3.02297e7			−1.56522			−0.782612	+	0.622510i	$0.786113\pi$
$822$			3.51089e7i			1.81233i
$823$		−	5.41689e6i		−	0.278773i	−0.990238	−	0.139386i	$-0.955487\pi$
$823$		−	5.41689e6i		−	0.278773i	0.990238	−	0.139386i	$-0.0445130\pi$
$824$	−4.53546e6			−0.232704
$825$	2.64247e7	−	800893.i	1.35168	−	0.0409675i
$826$	−2.75162e6			−0.140326
$827$			1.84505e7i			0.938088i	0.883175	+	0.469044i	$0.155401\pi$
$827$			1.84505e7i			0.938088i	−0.883175	+	0.469044i	$0.844599\pi$
$828$		−	444636.i		−	0.0225387i
$829$	2.08062e7			1.05149			0.525747	−	0.850641i	$-0.323786\pi$
$829$	2.08062e7			1.05149			0.525747	+	0.850641i	$0.323786\pi$
$830$	2.67081e7	−	2.75298e7i	1.34570	−	1.38710i
$831$	2.18280e7			1.09651
$832$		−	1.51045e7i		−	0.756482i
$833$		−	2.59340e7i		−	1.29496i
$834$	5.58343e7			2.77962
$835$	1.53510e7	+	1.48928e7i	0.761941	+	0.739198i
$836$	3.16242e7			1.56496
$837$			2.23899e7i			1.10469i
$838$		−	1.11384e7i		−	0.547915i
$839$	3.38768e7			1.66149			0.830744	−	0.556655i	$-0.187915\pi$
$839$	3.38768e7			1.66149			0.830744	+	0.556655i	$0.187915\pi$
$840$	135751.	+	131699.i	0.00663811	+	0.00643997i
$841$	−1.60396e7			−0.781994
$842$		−	3.43110e7i		−	1.66784i
$843$			3.05838e7i			1.48225i
$844$	−1.50917e7			−0.729258
$845$	256094.	−	263973.i	0.0123384	−	0.0127180i
$846$	−527955.			−0.0253613
$847$			741860.i			0.0355315i
$848$			2.22700e7i			1.06348i
$849$	1.38756e7			0.660669
$850$	−1.13886e6	−	3.75756e7i	−0.0540658	−	1.78385i
$851$	1.80096e6			0.0852471
$852$		−	2.78915e7i		−	1.31635i
$853$		−	1.79175e7i		−	0.843150i	−0.906793	−	0.421575i	$-0.861477\pi$
$853$		−	1.79175e7i		−	0.843150i	0.906793	−	0.421575i	$-0.138523\pi$
$854$	128452.			0.00602695
$855$	−2.50489e6	+	2.58196e6i	−0.117185	+	0.120791i
$856$	−4.63545e6			−0.216226
$857$		−	730568.i		−	0.0339788i	−0.999856	−	0.0169894i	$-0.994592\pi$
$857$		−	730568.i		−	0.0339788i	0.999856	−	0.0169894i	$-0.00540816\pi$
$858$			3.97034e7i			1.84124i
$859$	2.57647e7			1.19136			0.595679	−	0.803223i	$-0.296883\pi$
$859$	2.57647e7			1.19136			0.595679	+	0.803223i	$0.296883\pi$
$860$	1.66626e7	+	1.61653e7i	0.768242	+	0.745310i
$861$	966737.			0.0444427
$862$		−	2.51235e6i		−	0.115163i
$863$		−	1.02389e7i		−	0.467980i	−0.972239	−	0.233990i	$-0.924822\pi$
$863$		−	1.02389e7i		−	0.467980i	0.972239	−	0.233990i	$-0.0751782\pi$
$864$	−2.76816e7			−1.26156
$865$	5.79337e6	+	5.62044e6i	0.263264	+	0.255405i
$866$	−3.17274e6			−0.143761
$867$			1.61203e7i			0.728324i
$868$			1.31896e6i			0.0594198i
$869$	3.02644e7			1.35951
$870$	−1.05613e7	+	1.08862e7i	−0.473063	+	0.487618i
$871$	2.88072e7			1.28663
$872$			3.61315e6i			0.160914i
$873$		−	1.65505e6i		−	0.0734981i
$874$	8.93646e6			0.395719
$875$	861032.	−	943083.i	0.0380188	−	0.0416418i
$876$	−5.64429e6			−0.248513
$877$			3.81971e7i			1.67699i	0.544907	+	0.838497i	$0.316565\pi$
$877$			3.81971e7i			1.67699i	−0.544907	+	0.838497i	$0.683435\pi$
$878$		−	3.01726e7i		−	1.32092i
$879$	−2.15532e6			−0.0940891
$880$	2.24659e7	−	2.31571e7i	0.977951	−	1.00804i
$881$	3.54313e7			1.53797			0.768983	−	0.639269i	$-0.220763\pi$
$881$	3.54313e7			1.53797			0.768983	+	0.639269i	$0.220763\pi$
$882$		−	3.85426e6i		−	0.166828i
$883$		−	1.38612e7i		−	0.598270i	−0.954211	−	0.299135i	$-0.903302\pi$
$883$		−	1.38612e7i		−	0.598270i	0.954211	−	0.299135i	$-0.0966982\pi$
$884$	2.65428e7			1.14239
$885$	−3.20870e7	−	3.11293e7i	−1.37712	−	1.33601i
$886$	−1.83271e7			−0.784351
$887$		−	1.95520e7i		−	0.834417i	−0.908811	−	0.417208i	$-0.863009\pi$
$887$		−	1.95520e7i		−	0.834417i	0.908811	−	0.417208i	$-0.136991\pi$
$888$			1.57571e6i			0.0670570i
$889$	−1.13619e6			−0.0482165
$890$	1.91744e7	+	1.86021e7i	0.811423	+	0.787202i
$891$	3.34925e7			1.41336
$892$		−	4.22971e6i		−	0.177991i
$893$		−	4.98862e6i		−	0.209340i
$894$	−1.72424e6			−0.0721530
$895$	−2.02814e7	+	2.09054e7i	−0.846330	+	0.872370i
$896$	417012.			0.0173531
$897$			5.27470e6i			0.218885i
$898$			5.03055e7i			2.08173i
$899$	1.34378e7			0.554536
$900$	−79572.7	−	2.62543e6i	−0.00327460	−	0.108042i
$901$	−3.06043e7			−1.25594
$902$		−	3.18938e7i		−	1.30524i
$903$		−	1.76536e6i		−	0.0720467i
$904$	−1.24571e6			−0.0506986
$905$	−4.03119e6	+	4.15522e6i	−0.163611	+	0.168645i
$906$	2.32268e7			0.940088
$907$			1.43934e7i			0.580960i	0.956881	+	0.290480i	$0.0938149\pi$
$907$			1.43934e7i			0.580960i	−0.956881	+	0.290480i	$0.906185\pi$
$908$		−	2.59177e7i		−	1.04324i
$909$	910091.			0.0365321
$910$	1.37652e6	+	1.33543e6i	0.0551034	+	0.0534586i
$911$	−1.75068e7			−0.698893			−0.349446	−	0.936956i	$-0.613630\pi$
$911$	−1.75068e7			−0.698893			−0.349446	+	0.936956i	$0.613630\pi$
$912$			4.04280e7i			1.60952i
$913$			4.52392e7i			1.79613i
$914$	−1.61531e7			−0.639573
$915$	1.49790e6	+	1.45319e6i	0.0591467	+	0.0573812i
$916$	−1.71121e7			−0.673851
$917$		−	1.32665e6i		−	0.0520993i
$918$		−	4.23846e7i		−	1.65997i
$919$	−3.61554e7			−1.41216			−0.706081	−	0.708131i	$-0.749539\pi$
$919$	−3.61554e7			−1.41216			−0.706081	+	0.708131i	$0.749539\pi$
$920$	577230.	−	594990.i	0.0224843	−	0.0231761i
$921$	−6.13970e6			−0.238505
$922$			3.44439e7i			1.33440i
$923$			3.59314e7i			1.38826i
$924$	1.75586e6			0.0676566
$925$	1.06340e7	−	322302.i	0.408643	−	0.0123854i
$926$	−1.75084e7			−0.670993
$927$		−	4.78959e6i		−	0.183062i
$928$			1.66138e7i			0.633283i
$929$	4.18440e6			0.159072			0.0795361	−	0.996832i	$-0.474656\pi$
$929$	4.18440e6			0.159072			0.0795361	+	0.996832i	$0.474656\pi$
$930$	−3.17386e7	+	3.27151e7i	−1.20332	+	1.24034i
$931$	3.64187e7			1.37705
$932$			3.09200e7i			1.16600i
$933$			2.13915e7i			0.804520i
$934$	−1.07188e7			−0.402047
$935$	3.18234e7	+	3.08735e7i	1.19047	+	1.15493i
$936$	−501165.			−0.0186978
$937$		−	2.01116e7i		−	0.748337i	−0.927361	−	0.374168i	$-0.877928\pi$
$937$		−	2.01116e7i		−	0.748337i	0.927361	−	0.374168i	$-0.122072\pi$
$938$		−	2.70981e6i		−	0.100562i
$939$	5.04911e6			0.186875
$940$	2.61434e6	+	2.53630e6i	0.0965033	+	0.0936228i
$941$	−2.16937e7			−0.798654			−0.399327	−	0.916808i	$-0.630756\pi$
$941$	−2.16937e7			−0.798654			−0.399327	+	0.916808i	$0.630756\pi$
$942$		−	5.38063e7i		−	1.97563i
$943$		−	4.23717e6i		−	0.155166i
$944$	−5.45598e7			−1.99271
$945$	1.00255e6	−	1.03340e6i	0.0365196	−	0.0376433i
$946$	−5.82414e7			−2.11594
$947$			3.88024e7i			1.40600i	0.711192	+	0.702998i	$0.248156\pi$
$947$			3.88024e7i			1.40600i	−0.711192	+	0.702998i	$0.751844\pi$
$948$		−	2.76893e7i		−	1.00067i
$949$	7.27129e6			0.262087
$950$	5.27667e7	−	1.59928e6i	1.89693	−	0.0574931i
$951$	−4.25395e7			−1.52525
$952$			317211.i			0.0113437i
$953$			4.09847e7i			1.46181i	0.682481	+	0.730903i	$0.260901\pi$
$953$			4.09847e7i			1.46181i	−0.682481	+	0.730903i	$0.739099\pi$
$954$	−4.54834e6			−0.161801
$955$	3.45222e7	−	3.55844e7i	1.22487	−	1.26256i
$956$	−6.65334e6			−0.235448
$957$		−	1.78891e7i		−	0.631406i
$958$		−	7.66278e7i		−	2.69757i
$959$	−2.00024e6			−0.0702320
$960$	−1.65687e7	−	1.60741e7i	−0.580242	−	0.562922i
$961$	1.17539e7			0.410558
$962$			1.59778e7i			0.556646i
$963$		−	4.89519e6i		−	0.170100i
$964$	5.10004e6			0.176759
$965$	7.04052e6	+	6.83036e6i	0.243381	+	0.236116i
$966$	496177.			0.0171078
$967$		−	3.06485e7i		−	1.05401i	−0.849864	−	0.527003i	$-0.823316\pi$
$967$		−	3.06485e7i		−	1.05401i	0.849864	−	0.527003i	$-0.176684\pi$
$968$			2.84487e6i			0.0975830i
$969$	−5.55578e7			−1.90080
$970$	−1.69119e7	+	1.74323e7i	−0.577117	+	0.594873i
$971$	−4.41972e7			−1.50434			−0.752172	−	0.658967i	$-0.770994\pi$
$971$	−4.41972e7			−1.50434			−0.752172	+	0.658967i	$0.770994\pi$
$972$		−	6.33370e6i		−	0.215026i
$973$			3.18102e6i			0.107717i
$974$	−4.25508e7			−1.43718
$975$	943967.	+	3.11453e7i	0.0318013	+	1.04925i
$976$	2.54699e6			0.0855859
$977$			2.11557e7i			0.709073i	0.935042	+	0.354537i	$0.115361\pi$
$977$			2.11557e7i			0.709073i	−0.935042	+	0.354537i	$0.884639\pi$
$978$			1.17030e7i			0.391246i
$979$	−3.15089e7			−1.05069
$980$	−1.85159e7	+	1.90856e7i	−0.615857	+	0.634806i
$981$	−3.81561e6			−0.126588
$982$		−	4.37160e7i		−	1.44664i
$983$		−	2.27966e7i		−	0.752466i	−0.926525	−	0.376233i	$-0.877219\pi$
$983$		−	2.27966e7i		−	0.752466i	0.926525	−	0.376233i	$-0.122781\pi$
$984$	3.70723e6			0.122057
$985$	−3.52317e7	−	3.41801e7i	−1.15703	−	1.12249i
$986$	−2.54381e7			−0.833282
$987$		−	276982.i		−	0.00905021i
$988$			3.72736e7i			1.21481i
$989$	−7.73750e6			−0.251542
$990$	4.72953e6	+	4.58836e6i	0.153366	+	0.148789i
$991$	3.93077e7			1.27143			0.635716	−	0.771923i	$-0.280705\pi$
$991$	3.93077e7			1.27143			0.635716	+	0.771923i	$0.280705\pi$
$992$			4.99273e7i			1.61086i
$993$		−	3.67945e7i		−	1.18416i
$994$	3.37997e6			0.108504
$995$	2.42001e7	−	2.49447e7i	0.774925	−	0.798767i
$996$	4.13899e7			1.32204
$997$		−	2.02220e7i		−	0.644298i	−0.946689	−	0.322149i	$-0.895595\pi$
$997$		−	2.02220e7i		−	0.644298i	0.946689	−	0.322149i	$-0.104405\pi$
$998$			7.17248e7i			2.27952i
$999$	1.19950e7			0.380266

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	115.6.b.a.24.45	yes	54
5.2	odd	4		575.6.a.m.1.6		27
5.3	odd	4		575.6.a.l.1.22		27
5.4	even	2	inner	115.6.b.a.24.10	✓	54

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
115.6.b.a.24.10	✓	54	5.4	even	2	inner
115.6.b.a.24.45	yes	54	1.1	even	1	trivial
575.6.a.l.1.22		27	5.3	odd	4
575.6.a.m.1.6		27	5.2	odd	4

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:	\( N \)	\(=\)	\( 115 = 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	115.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+7.77128i q^{2} -16.5107i q^{3} -28.3928 q^{4} +(38.9251 - 40.1228i) q^{5} +128.309 q^{6} +7.31009i q^{7} +28.0326i q^{8} -29.6034 q^{9} +(311.805 + 302.498i) q^{10} -512.382 q^{11} +468.785i q^{12} -603.915i q^{13} -56.8088 q^{14} +(-662.455 - 642.681i) q^{15} -1126.42 q^{16} -1547.97i q^{17} -230.056i q^{18} +2173.79 q^{19} +(-1105.19 + 1139.20i) q^{20} +120.695 q^{21} -3981.86i q^{22} -529.000i q^{23} +462.838 q^{24} +(-94.6706 - 3123.57i) q^{25} +4693.20 q^{26} -3523.33i q^{27} -207.554i q^{28} -2114.61 q^{29} +(4994.45 - 5148.12i) q^{30} -6354.77 q^{31} -7856.67i q^{32} +8459.79i q^{33} +12029.7 q^{34} +(293.301 + 284.546i) q^{35} +840.522 q^{36} +3404.46i q^{37} +16893.1i q^{38} -9971.07 q^{39} +(1124.75 + 1091.17i) q^{40} +8009.77 q^{41} +937.953i q^{42} -14626.7i q^{43} +14548.0 q^{44} +(-1152.31 + 1187.77i) q^{45} +4111.01 q^{46} -2294.90i q^{47} +18598.0i q^{48} +16753.6 q^{49} +(24274.1 - 735.712i) q^{50} -25558.1 q^{51} +17146.8i q^{52} -19770.6i q^{53} +27380.8 q^{54} +(-19944.5 + 20558.2i) q^{55} -204.921 q^{56} -35890.8i q^{57} -16433.2i q^{58} +48436.6 q^{59} +(18808.9 + 18247.5i) q^{60} -2261.14 q^{61} -49384.7i q^{62} -216.403i q^{63} +25011.0 q^{64} +(-24230.8 - 23507.5i) q^{65} -65743.4 q^{66} +47700.6i q^{67} +43951.2i q^{68} -8734.16 q^{69} +(-2211.29 + 2279.32i) q^{70} -59497.4 q^{71} -829.859i q^{72} +12040.2i q^{73} -26457.0 q^{74} +(-51572.3 + 1563.08i) q^{75} -61719.9 q^{76} -3745.56i q^{77} -77488.0i q^{78} -59066.2 q^{79} +(-43846.0 + 45195.0i) q^{80} -65366.3 q^{81} +62246.2i q^{82} -88291.8i q^{83} -3426.86 q^{84} +(-62108.8 - 60254.9i) q^{85} +113668. q^{86} +34913.6i q^{87} -14363.4i q^{88} +61494.9 q^{89} +(-9230.48 - 8954.96i) q^{90} +4414.68 q^{91} +15019.8i q^{92} +104922. i q^{93} +17834.3 q^{94} +(84614.9 - 87218.3i) q^{95} -129719. q^{96} +55907.6i q^{97} +130197. i q^{98} +15168.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 54 q - 832 q^{4} + 98 q^{5} - 144 q^{6} - 4270 q^{9} + 942 q^{10} - 380 q^{11} - 3988 q^{14} + 690 q^{15} + 16632 q^{16} - 2440 q^{19} - 2436 q^{20} + 4768 q^{21} - 17940 q^{24} - 9150 q^{25} - 15144 q^{26}+ \cdots - 14088 q^{99}+O(q^{100}) \) 54 * q - 832 * q^4 + 98 * q^5 - 144 * q^6 - 4270 * q^9 + 942 * q^10 - 380 * q^11 - 3988 * q^14 + 690 * q^15 + 16632 * q^16 - 2440 * q^19 - 2436 * q^20 + 4768 * q^21 - 17940 * q^24 - 9150 * q^25 - 15144 * q^26 + 22958 * q^29 + 23034 * q^30 - 33970 * q^31 + 26120 * q^34 - 18694 * q^35 + 71560 * q^36 + 13416 * q^39 - 22230 * q^40 - 33842 * q^41 + 33908 * q^44 + 35260 * q^45 - 8464 * q^46 - 153772 * q^49 - 91728 * q^50 + 96576 * q^51 - 90500 * q^54 + 37344 * q^55 + 201796 * q^56 + 110106 * q^59 - 92484 * q^60 + 132924 * q^61 - 262108 * q^64 + 45090 * q^65 - 123288 * q^66 + 38088 * q^69 + 249664 * q^70 - 310814 * q^71 + 220660 * q^74 + 235776 * q^75 + 42220 * q^76 - 389348 * q^79 - 245138 * q^80 + 364422 * q^81 - 35836 * q^84 - 134118 * q^85 + 111980 * q^86 - 353788 * q^89 - 805800 * q^90 - 226632 * q^91 + 289804 * q^94 - 42676 * q^95 + 294416 * q^96 - 14088 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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