LMFDB - Embedded newform 33.5.b.a.23.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [33,5,Mod(23,33)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(33, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("33.23");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 33 = 3 \cdot 11 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	33.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.41120878177$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 162x^{12} + 10041x^{10} + 298396x^{8} + 4418856x^{6} + 32113344x^{4} + 102865552x^{2} + 102193344 $ x^14 + 162x^12 + 10041x^10 + 298396x^8 + 4418856x^6 + 32113344x^4 + 102865552x^2 + 102193344
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{9}\cdot 11^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			23.3
Root			$-6.24203i$ of defining polynomial
Character	$\chi$	$=$	33.23
Dual form			33.5.b.a.23.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-6.24203i q^{2} +(7.09522 + 5.53695i) q^{3} -22.9629 q^{4} -42.0693i q^{5} +(34.5618 - 44.2886i) q^{6} -11.7748 q^{7} +43.4628i q^{8} +(19.6843 + 78.5718i) q^{9} +O(q^{10})$	\(q-6.24203i q^{2} +(7.09522 + 5.53695i) q^{3} -22.9629 q^{4} -42.0693i q^{5} +(34.5618 - 44.2886i) q^{6} -11.7748 q^{7} +43.4628i q^{8} +(19.6843 + 78.5718i) q^{9} -262.598 q^{10} +36.4829i q^{11} +(-162.927 - 127.145i) q^{12} +186.682 q^{13} +73.4985i q^{14} +(232.936 - 298.491i) q^{15} -96.1107 q^{16} +216.992i q^{17} +(490.447 - 122.870i) q^{18} +579.356 q^{19} +966.034i q^{20} +(-83.5446 - 65.1964i) q^{21} +227.727 q^{22} -266.732i q^{23} +(-240.652 + 308.378i) q^{24} -1144.82 q^{25} -1165.27i q^{26} +(-295.384 + 666.475i) q^{27} +270.383 q^{28} +775.677i q^{29} +(-1863.19 - 1453.99i) q^{30} +902.204 q^{31} +1295.33i q^{32} +(-202.004 + 258.854i) q^{33} +1354.47 q^{34} +495.356i q^{35} +(-452.010 - 1804.24i) q^{36} -1100.13 q^{37} -3616.36i q^{38} +(1324.55 + 1033.65i) q^{39} +1828.45 q^{40} +682.715i q^{41} +(-406.958 + 521.488i) q^{42} -1433.48 q^{43} -837.754i q^{44} +(3305.46 - 828.105i) q^{45} -1664.95 q^{46} -3622.90i q^{47} +(-681.927 - 532.161i) q^{48} -2262.35 q^{49} +7146.03i q^{50} +(-1201.47 + 1539.60i) q^{51} -4286.76 q^{52} +987.839i q^{53} +(4160.16 + 1843.79i) q^{54} +1534.81 q^{55} -511.765i q^{56} +(4110.66 + 3207.87i) q^{57} +4841.80 q^{58} +1771.46i q^{59} +(-5348.88 + 6854.22i) q^{60} -2843.76 q^{61} -5631.59i q^{62} +(-231.778 - 925.165i) q^{63} +6547.72 q^{64} -7853.57i q^{65} +(1615.77 + 1260.91i) q^{66} -4896.10 q^{67} -4982.76i q^{68} +(1476.88 - 1892.52i) q^{69} +3092.03 q^{70} -519.758i q^{71} +(-3414.95 + 855.536i) q^{72} +7825.00 q^{73} +6867.04i q^{74} +(-8122.78 - 6338.84i) q^{75} -13303.7 q^{76} -429.578i q^{77} +(6452.07 - 8267.87i) q^{78} +416.507 q^{79} +4043.31i q^{80} +(-5786.06 + 3093.26i) q^{81} +4261.53 q^{82} -2980.79i q^{83} +(1918.43 + 1497.10i) q^{84} +9128.68 q^{85} +8947.81i q^{86} +(-4294.89 + 5503.60i) q^{87} -1585.65 q^{88} -3346.39i q^{89} +(-5169.06 - 20632.8i) q^{90} -2198.14 q^{91} +6124.96i q^{92} +(6401.34 + 4995.46i) q^{93} -22614.3 q^{94} -24373.1i q^{95} +(-7172.19 + 9190.66i) q^{96} -7544.49 q^{97} +14121.7i q^{98} +(-2866.52 + 718.140i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 5 q^{3} - 100 q^{4} - 2 q^{6} + 76 q^{7} - 67 q^{9}+O(q^{10}) $ 14 * q - 5 * q^3 - 100 * q^4 - 2 * q^6 + 76 * q^7 - 67 * q^9	$ 14 q - 5 q^{3} - 100 q^{4} - 2 q^{6} + 76 q^{7} - 67 q^{9} - 156 q^{10} - 100 q^{12} - 104 q^{13} + 151 q^{15} + 356 q^{16} - 34 q^{18} + 1072 q^{19} + 718 q^{21} + 1200 q^{24} - 1060 q^{25} - 1154 q^{27} - 1808 q^{28} - 3026 q^{30} + 3310 q^{31} - 605 q^{33} - 2304 q^{34} + 2644 q^{36} - 362 q^{37} + 4264 q^{39} + 1896 q^{40} - 7364 q^{42} - 6740 q^{43} + 3611 q^{45} - 4068 q^{46} - 2956 q^{48} + 7074 q^{49} - 7046 q^{51} + 13072 q^{52} + 20512 q^{54} + 726 q^{55} + 3876 q^{57} - 7848 q^{58} - 8416 q^{60} - 3560 q^{61} - 17662 q^{63} + 12020 q^{64} + 1210 q^{66} - 16514 q^{67} + 9833 q^{69} + 13320 q^{70} + 8160 q^{72} + 12664 q^{73} - 5386 q^{75} - 43736 q^{76} + 19096 q^{78} + 3052 q^{79} - 11611 q^{81} + 10200 q^{82} - 39184 q^{84} + 34884 q^{85} + 37068 q^{87} - 7260 q^{88} - 26686 q^{90} - 45856 q^{91} + 2719 q^{93} + 6120 q^{94} - 38368 q^{96} - 27854 q^{97} + 4235 q^{99}+O(q^{100}) $ 14 * q - 5 * q^3 - 100 * q^4 - 2 * q^6 + 76 * q^7 - 67 * q^9 - 156 * q^10 - 100 * q^12 - 104 * q^13 + 151 * q^15 + 356 * q^16 - 34 * q^18 + 1072 * q^19 + 718 * q^21 + 1200 * q^24 - 1060 * q^25 - 1154 * q^27 - 1808 * q^28 - 3026 * q^30 + 3310 * q^31 - 605 * q^33 - 2304 * q^34 + 2644 * q^36 - 362 * q^37 + 4264 * q^39 + 1896 * q^40 - 7364 * q^42 - 6740 * q^43 + 3611 * q^45 - 4068 * q^46 - 2956 * q^48 + 7074 * q^49 - 7046 * q^51 + 13072 * q^52 + 20512 * q^54 + 726 * q^55 + 3876 * q^57 - 7848 * q^58 - 8416 * q^60 - 3560 * q^61 - 17662 * q^63 + 12020 * q^64 + 1210 * q^66 - 16514 * q^67 + 9833 * q^69 + 13320 * q^70 + 8160 * q^72 + 12664 * q^73 - 5386 * q^75 - 43736 * q^76 + 19096 * q^78 + 3052 * q^79 - 11611 * q^81 + 10200 * q^82 - 39184 * q^84 + 34884 * q^85 + 37068 * q^87 - 7260 * q^88 - 26686 * q^90 - 45856 * q^91 + 2719 * q^93 + 6120 * q^94 - 38368 * q^96 - 27854 * q^97 + 4235 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$.

$n$	$13$	$23$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	6.24203i		−	1.56051i	−0.625463	−	0.780254i	$-0.715090\pi$
$2$		−	6.24203i		−	1.56051i	0.625463	−	0.780254i	$-0.284910\pi$
$3$	7.09522	+	5.53695i	0.788358	+	0.615217i
$3$	7.09522	+	5.53695i	0.788358	+	0.615217i
$4$	−22.9629			−1.43518
$5$		−	42.0693i		−	1.68277i	−0.540435	−	0.841386i	$-0.681740\pi$
$5$		−	42.0693i		−	1.68277i	0.540435	−	0.841386i	$-0.318260\pi$
$6$	34.5618	−	44.2886i	0.960051	−	1.23024i
$7$	−11.7748			−0.240302			−0.120151	−	0.992756i	$-0.538338\pi$
$7$	−11.7748			−0.240302			−0.120151	+	0.992756i	$0.538338\pi$
$8$			43.4628i			0.679106i
$9$	19.6843	+	78.5718i	0.243016	+	0.970022i
$10$	−262.598			−2.62598
$11$			36.4829i			0.301511i
$11$			36.4829i			0.301511i
$12$	−162.927	−	127.145i	−1.13144	−	0.882949i
$13$	186.682			1.10463			0.552313	−	0.833637i	$-0.313745\pi$
$13$	186.682			1.10463			0.552313	+	0.833637i	$0.313745\pi$
$14$			73.4985i			0.374992i
$15$	232.936	−	298.491i	1.03527	−	1.32663i
$16$	−96.1107			−0.375433
$17$			216.992i			0.750836i	0.926856	+	0.375418i	$0.122501\pi$
$17$			216.992i			0.750836i	−0.926856	+	0.375418i	$0.877499\pi$
$18$	490.447	−	122.870i	1.51373	−	0.379229i
$19$	579.356			1.60487			0.802433	−	0.596743i	$-0.203539\pi$
$19$	579.356			1.60487			0.802433	+	0.596743i	$0.203539\pi$
$20$			966.034i			2.41508i
$21$	−83.5446	−	65.1964i	−0.189444	−	0.147838i
$22$	227.727			0.470511
$23$		−	266.732i		−	0.504220i	−0.967699	−	0.252110i	$-0.918876\pi$
$23$		−	266.732i		−	0.504220i	0.967699	−	0.252110i	$-0.0811245\pi$
$24$	−240.652	+	308.378i	−0.417798	+	0.535379i
$25$	−1144.82			−1.83172
$26$		−	1165.27i		−	1.72378i
$27$	−295.384	+	666.475i	−0.405190	+	0.914232i
$28$	270.383			0.344877
$29$			775.677i			0.922327i	0.887315	+	0.461164i	$0.152568\pi$
$29$			775.677i			0.922327i	−0.887315	+	0.461164i	$0.847432\pi$
$30$	−1863.19	−	1453.99i	−2.07021	−	1.61555i
$31$	902.204			0.938818			0.469409	−	0.882981i	$-0.344467\pi$
$31$	902.204			0.938818			0.469409	+	0.882981i	$0.344467\pi$
$32$			1295.33i			1.26497i
$33$	−202.004	+	258.854i	−0.185495	+	0.237699i
$34$	1354.47			1.17169
$35$			495.356i			0.404373i
$36$	−452.010	−	1804.24i	−0.348773	−	1.39216i
$37$	−1100.13			−0.803600			−0.401800	−	0.915727i	$-0.631615\pi$
$37$	−1100.13			−0.803600			−0.401800	+	0.915727i	$0.631615\pi$
$38$		−	3616.36i		−	2.50440i
$39$	1324.55	+	1033.65i	0.870841	+	0.679585i
$40$	1828.45			1.14278
$41$			682.715i			0.406136i	0.979165	+	0.203068i	$0.0650913\pi$
$41$			682.715i			0.406136i	−0.979165	+	0.203068i	$0.934909\pi$
$42$	−406.958	+	521.488i	−0.230702	+	0.295628i
$43$	−1433.48			−0.775272			−0.387636	−	0.921813i	$-0.626708\pi$
$43$	−1433.48			−0.775272			−0.387636	+	0.921813i	$0.626708\pi$
$44$		−	837.754i		−	0.432724i
$45$	3305.46	−	828.105i	1.63233	−	0.408941i
$46$	−1664.95			−0.786839
$47$		−	3622.90i		−	1.64007i	−0.572317	−	0.820033i	$-0.693955\pi$
$47$		−	3622.90i		−	1.64007i	0.572317	−	0.820033i	$-0.306045\pi$
$48$	−681.927	−	532.161i	−0.295975	−	0.230972i
$49$	−2262.35			−0.942255
$50$			7146.03i			2.85841i
$51$	−1201.47	+	1539.60i	−0.461927	+	0.591928i
$52$	−4286.76			−1.58534
$53$			987.839i			0.351669i	0.984420	+	0.175835i	$0.0562624\pi$
$53$			987.839i			0.351669i	−0.984420	+	0.175835i	$0.943738\pi$
$54$	4160.16	+	1843.79i	1.42667	+	0.632302i
$55$	1534.81			0.507375
$56$		−	511.765i		−	0.163190i
$57$	4110.66	+	3207.87i	1.26521	+	0.987340i
$58$	4841.80			1.43930
$59$			1771.46i			0.508895i	0.967087	+	0.254448i	$0.0818936\pi$
$59$			1771.46i			0.508895i	−0.967087	+	0.254448i	$0.918106\pi$
$60$	−5348.88	+	6854.22i	−1.48580	+	1.90395i
$61$	−2843.76			−0.764245			−0.382123	−	0.924112i	$-0.624807\pi$
$61$	−2843.76			−0.764245			−0.382123	+	0.924112i	$0.624807\pi$
$62$		−	5631.59i		−	1.46503i
$63$	−231.778	−	925.165i	−0.0583972	−	0.233098i
$64$	6547.72			1.59857
$65$		−	7853.57i		−	1.85883i
$66$	1615.77	+	1260.91i	0.370931	+	0.289466i
$67$	−4896.10			−1.09069			−0.545344	−	0.838212i	$-0.683601\pi$
$67$	−4896.10			−1.09069			−0.545344	+	0.838212i	$0.683601\pi$
$68$		−	4982.76i		−	1.07759i
$69$	1476.88	−	1892.52i	0.310205	−	0.397506i
$70$	3092.03			0.631026
$71$		−	519.758i		−	0.103106i	−0.998670	−	0.0515531i	$-0.983583\pi$
$71$		−	519.758i		−	0.103106i	0.998670	−	0.0515531i	$-0.0164171\pi$
$72$	−3414.95	+	855.536i	−0.658748	+	0.165034i
$73$	7825.00			1.46838			0.734191	−	0.678943i	$-0.237562\pi$
$73$	7825.00			1.46838			0.734191	+	0.678943i	$0.237562\pi$
$74$			6867.04i			1.25402i
$75$	−8122.78	−	6338.84i	−1.44405	−	1.12690i
$76$	−13303.7			−2.30328
$77$		−	429.578i		−	0.0724536i
$78$	6452.07	−	8267.87i	1.06050	−	1.35895i
$79$	416.507			0.0667372			0.0333686	−	0.999443i	$-0.489376\pi$
$79$	416.507			0.0667372			0.0333686	+	0.999443i	$0.489376\pi$
$80$			4043.31i			0.631767i
$81$	−5786.06	+	3093.26i	−0.881886	+	0.471462i
$82$	4261.53			0.633778
$83$		−	2980.79i		−	0.432688i	−0.976317	−	0.216344i	$-0.930587\pi$
$83$		−	2980.79i		−	0.432688i	0.976317	−	0.216344i	$-0.0694133\pi$
$84$	1918.43	+	1497.10i	0.271886	+	0.212174i
$85$	9128.68			1.26349
$86$			8947.81i			1.20982i
$87$	−4294.89	+	5503.60i	−0.567431	+	0.727124i
$88$	−1585.65			−0.204758
$89$		−	3346.39i		−	0.422470i	−0.977435	−	0.211235i	$-0.932251\pi$
$89$		−	3346.39i		−	0.422470i	0.977435	−	0.211235i	$-0.0677486\pi$
$90$	−5169.06	−	20632.8i	−0.638155	−	2.54726i
$91$	−2198.14			−0.265443
$92$			6124.96i			0.723648i
$93$	6401.34	+	4995.46i	0.740125	+	0.577577i
$94$	−22614.3			−2.55933
$95$		−	24373.1i		−	2.70062i
$96$	−7172.19	+	9190.66i	−0.778232	+	0.997250i
$97$	−7544.49			−0.801837			−0.400919	−	0.916114i	$-0.631309\pi$
$97$	−7544.49			−0.801837			−0.400919	+	0.916114i	$0.631309\pi$
$98$			14121.7i			1.47040i
$99$	−2866.52	+	718.140i	−0.292473	+	0.0732722i
$100$	26288.5			2.62885
$101$			15676.1i			1.53672i	0.640015	+	0.768362i	$0.278928\pi$
$101$			15676.1i			1.53672i	−0.640015	+	0.768362i	$0.721072\pi$
$102$	9610.25	+	7499.63i	0.923707	+	0.720841i
$103$	10923.7			1.02966			0.514830	−	0.857292i	$-0.327855\pi$
$103$	10923.7			1.02966			0.514830	+	0.857292i	$0.327855\pi$
$104$			8113.72i			0.750159i
$105$	−2742.76	+	3514.66i	−0.248777	+	0.318790i
$106$	6166.12			0.548782
$107$			775.638i			0.0677472i	0.999426	+	0.0338736i	$0.0107844\pi$
$107$			775.638i			0.0677472i	−0.999426	+	0.0338736i	$0.989216\pi$
$108$	6782.88	−	15304.2i	0.581522	−	1.31209i
$109$	4458.80			0.375288			0.187644	−	0.982237i	$-0.439915\pi$
$109$	4458.80			0.375288			0.187644	+	0.982237i	$0.439915\pi$
$110$		−	9580.32i		−	0.791762i
$111$	−7805.66	−	6091.36i	−0.633525	−	0.494389i
$112$	1131.68			0.0902170
$113$			15482.2i			1.21248i	0.795281	+	0.606241i	$0.207323\pi$
$113$			15482.2i			1.21248i	−0.795281	+	0.606241i	$0.792677\pi$
$114$	20023.6	−	25658.9i	1.54075	−	1.97437i
$115$	−11221.2			−0.848487
$116$		−	17811.8i		−	1.32371i
$117$	3674.71	+	14667.9i	0.268442	+	1.07151i
$118$	11057.5			0.794134
$119$		−	2555.03i		−	0.180427i
$120$	12973.3	+	10124.0i	0.900920	+	0.703058i
$121$	−1331.00			−0.0909091
$122$			17750.8i			1.19261i
$123$	−3780.16	+	4844.01i	−0.249862	+	0.320181i
$124$	−20717.3			−1.34738
$125$			21868.6i			1.39959i
$126$	−5774.91	+	1446.77i	−0.363751	+	0.0911292i
$127$	3307.23			0.205049			0.102524	−	0.994730i	$-0.467308\pi$
$127$	3307.23			0.205049			0.102524	+	0.994730i	$0.467308\pi$
$128$		−	20145.8i		−	1.22960i
$129$	−10170.8	−	7937.09i	−0.611191	−	0.476960i
$130$	−49022.2			−2.90072
$131$		−	6934.21i		−	0.404068i	−0.979378	−	0.202034i	$-0.935245\pi$
$131$		−	6934.21i		−	0.404068i	0.979378	−	0.202034i	$-0.0647552\pi$
$132$	4638.60	−	5944.05i	0.266219	−	0.341141i
$133$	−6821.79			−0.385652
$134$			30561.6i			1.70203i
$135$	28038.1	+	12426.6i	1.53844	+	0.681843i
$136$	−9431.07			−0.509898
$137$		−	51.4351i		−	0.00274043i	−0.999999	−	0.00137022i	$-0.999564\pi$
$137$		−	51.4351i		−	0.00274043i	0.999999	−	0.00137022i	$-0.000436153\pi$
$138$	−11813.2	−	9218.76i	−0.620311	−	0.484077i
$139$	−10302.8			−0.533246			−0.266623	−	0.963801i	$-0.585908\pi$
$139$	−10302.8			−0.533246			−0.266623	+	0.963801i	$0.585908\pi$
$140$		−	11374.8i		−	0.580349i
$141$	20059.8	−	25705.3i	1.00900	−	1.29296i
$142$	−3244.35			−0.160898
$143$			6810.69i			0.333057i
$144$	−1891.87	−	7551.59i	−0.0912362	−	0.364178i
$145$	32632.2			1.55207
$146$		−	48843.9i		−	2.29142i
$147$	−16051.9	−	12526.5i	−0.742834	−	0.579691i
$148$	25262.2			1.15331
$149$		−	15944.8i		−	0.718203i	−0.933299	−	0.359101i	$-0.883083\pi$
$149$		−	15944.8i		−	0.718203i	0.933299	−	0.359101i	$-0.116917\pi$
$150$	−39567.2	+	50702.6i	−1.75854	+	2.25345i
$151$	−26838.7			−1.17708			−0.588541	−	0.808467i	$-0.700298\pi$
$151$	−26838.7			−1.17708			−0.588541	+	0.808467i	$0.700298\pi$
$152$			25180.5i			1.08987i
$153$	−17049.4	+	4271.33i	−0.728328	+	0.182465i
$154$	−2681.44			−0.113064
$155$		−	37955.1i		−	1.57982i
$156$	−30415.5	−	23735.6i	−1.24982	−	0.975329i
$157$	46406.0			1.88267			0.941336	−	0.337470i	$-0.109571\pi$
$157$	46406.0			1.88267			0.941336	+	0.337470i	$0.109571\pi$
$158$		−	2599.85i		−	0.104144i
$159$	−5469.62	+	7008.94i	−0.216353	+	0.277241i
$160$	54493.6			2.12866
$161$			3140.71i			0.121165i
$162$	19308.2	+	36116.7i	0.735720	+	1.37619i
$163$	23680.7			0.891290			0.445645	−	0.895210i	$-0.352974\pi$
$163$	23680.7			0.891290			0.445645	+	0.895210i	$0.352974\pi$
$164$		−	15677.1i		−	0.582880i
$165$	10889.8	+	8498.16i	0.399993	+	0.312145i
$166$	−18606.2			−0.675213
$167$			5000.00i			0.179282i	0.995974	+	0.0896411i	$0.0285720\pi$
$167$			5000.00i			0.179282i	−0.995974	+	0.0896411i	$0.971428\pi$
$168$	2833.62	−	3631.08i	0.100397	−	0.128652i
$169$	6289.12			0.220200
$170$		−	56981.5i		−	1.97168i
$171$	11404.2	+	45521.1i	0.390008	+	1.55676i
$172$	32916.8			1.11266
$173$			11473.4i			0.383352i	0.981458	+	0.191676i	$0.0613924\pi$
$173$			11473.4i			0.383352i	−0.981458	+	0.191676i	$0.938608\pi$
$174$	34353.6	+	26808.8i	1.13468	+	0.885481i
$175$	13480.0			0.440165
$176$		−	3506.40i		−	0.113197i
$177$	−9808.51	+	12568.9i	−0.313081	+	0.401191i
$178$	−20888.2			−0.659268
$179$			22760.7i			0.710363i	0.934797	+	0.355181i	$0.115581\pi$
$179$			22760.7i			0.710363i	−0.934797	+	0.355181i	$0.884419\pi$
$180$	−75903.0	+	19015.7i	−2.34269	+	0.586905i
$181$	−38521.9			−1.17585			−0.587924	−	0.808916i	$-0.700055\pi$
$181$	−38521.9			−1.17585			−0.587924	+	0.808916i	$0.700055\pi$
$182$			13720.8i			0.414226i
$183$	−20177.1	−	15745.7i	−0.602499	−	0.470177i
$184$	11592.9			0.342419
$185$			46281.6i			1.35228i
$186$	31181.8	−	39957.3i	0.901313	−	1.15497i
$187$	−7916.48			−0.226386
$188$			83192.5i			2.35379i
$189$	3478.08	−	7847.60i	0.0973679	−	0.219691i
$190$	−152138.			−4.21434
$191$		−	8633.45i		−	0.236656i	−0.992975	−	0.118328i	$-0.962247\pi$
$191$		−	8633.45i		−	0.236656i	0.992975	−	0.118328i	$-0.0377534\pi$
$192$	46457.5	+	36254.4i	1.26024	+	0.983464i
$193$	−11791.3			−0.316553			−0.158277	−	0.987395i	$-0.550594\pi$
$193$	−11791.3			−0.316553			−0.158277	+	0.987395i	$0.550594\pi$
$194$			47092.9i			1.25127i
$195$	43484.9	−	55722.8i	1.14359	−	1.46543i
$196$	51950.3			1.35231
$197$		−	71709.3i		−	1.84775i	−0.382697	−	0.923874i	$-0.625005\pi$
$197$		−	71709.3i		−	1.84775i	0.382697	−	0.923874i	$-0.374995\pi$
$198$	4482.65	+	17892.9i	0.114342	+	0.456406i
$199$	−47291.0			−1.19419			−0.597093	−	0.802172i	$-0.703678\pi$
$199$	−47291.0			−1.19419			−0.597093	+	0.802172i	$0.703678\pi$
$200$		−	49757.3i		−	1.24393i
$201$	−34738.9	−	27109.5i	−0.859852	−	0.671009i
$202$	97850.8			2.39807
$203$		−	9133.43i		−	0.221637i
$204$	27589.3	−	35353.8i	0.662950	−	0.849524i
$205$	28721.3			0.683434
$206$		−	68185.8i		−	1.60679i
$207$	20957.6	−	5250.44i	0.489105	−	0.122534i
$208$	−17942.1			−0.414713
$209$			21136.6i			0.483885i
$210$	21938.6	+	17120.4i	0.497475	+	0.388218i
$211$	36583.9			0.821722			0.410861	−	0.911698i	$-0.365228\pi$
$211$	36583.9			0.821722			0.410861	+	0.911698i	$0.365228\pi$
$212$		−	22683.7i		−	0.504710i
$213$	2877.88	−	3687.80i	0.0634327	−	0.0812846i
$214$	4841.56			0.105720
$215$			60305.3i			1.30460i
$216$	−28966.9	−	12838.2i	−0.620861	−	0.275167i
$217$	−10623.3			−0.225600
$218$		−	27831.9i		−	0.585640i
$219$	55520.1	+	43326.7i	1.15761	+	0.903373i
$220$	−35243.7			−0.728175
$221$			40508.4i			0.829394i
$222$	−38022.5	+	48723.1i	−0.771497	+	0.988620i
$223$	−49124.9			−0.987852			−0.493926	−	0.869504i	$-0.664439\pi$
$223$	−49124.9			−0.987852			−0.493926	+	0.869504i	$0.664439\pi$
$224$		−	15252.2i		−	0.303975i
$225$	−22535.1	−	89950.9i	−0.445137	−	1.77681i
$226$	96640.2			1.89209
$227$		−	43131.0i		−	0.837024i	−0.908211	−	0.418512i	$-0.862552\pi$
$227$		−	43131.0i		−	0.837024i	0.908211	−	0.418512i	$-0.137448\pi$
$228$	−94392.8	−	73662.1i	−1.81581	−	1.41701i
$229$	−79943.4			−1.52444			−0.762222	−	0.647316i	$-0.775892\pi$
$229$	−79943.4			−1.52444			−0.762222	+	0.647316i	$0.775892\pi$
$230$			70043.3i			1.32407i
$231$	2378.55	−	3047.95i	0.0445747	−	0.0571194i
$232$	−33713.1			−0.626359
$233$		−	48365.2i		−	0.890883i	−0.895311	−	0.445442i	$-0.853047\pi$
$233$		−	48365.2i		−	0.890883i	0.895311	−	0.445442i	$-0.146953\pi$
$234$	91557.7	−	22937.6i	1.67210	−	0.418906i
$235$	−152413.			−2.75985
$236$		−	40678.0i		−	0.730358i
$237$	2955.21	+	2306.18i	0.0526128	+	0.0410579i
$238$	−15948.6			−0.281558
$239$			60448.7i			1.05826i	0.848542	+	0.529128i	$0.177481\pi$
$239$			60448.7i			1.05826i	−0.848542	+	0.529128i	$0.822519\pi$
$240$	−22387.6	+	28688.2i	−0.388674	+	0.498059i
$241$	−67294.4			−1.15863			−0.579315	−	0.815104i	$-0.696680\pi$
$241$	−67294.4			−1.15863			−0.579315	+	0.815104i	$0.696680\pi$
$242$			8308.14i			0.141864i
$243$	−58180.6	−	10089.7i	−0.985294	−	0.170870i
$244$	65301.0			1.09683
$245$			95175.6i			1.58560i
$246$	30236.5	+	23595.9i	0.499644	+	0.389911i
$247$	108155.			1.77278
$248$			39212.3i			0.637558i
$249$	16504.5	−	21149.4i	0.266197	−	0.341113i
$250$	136505.			2.18407
$251$		−	106969.i		−	1.69789i	−0.528483	−	0.848944i	$-0.677239\pi$
$251$		−	106969.i		−	1.69789i	0.528483	−	0.848944i	$-0.322761\pi$
$252$	5322.31	+	21244.5i	0.0838107	+	0.334538i
$253$	9731.16			0.152028
$254$		−	20643.8i		−	0.319980i
$255$	64770.0	+	50545.1i	0.996079	+	0.777318i
$256$	−20987.0			−0.320236
$257$			25616.3i			0.387838i	0.981018	+	0.193919i	$0.0621199\pi$
$257$			25616.3i			0.387838i	−0.981018	+	0.193919i	$0.937880\pi$
$258$	−49543.6	+	63486.7i	−0.744300	+	0.953769i
$259$	12953.8			0.193106
$260$			180341.i			2.66777i
$261$	−60946.4	+	15268.7i	−0.894678	+	0.224141i
$262$	−43283.6			−0.630551
$263$			33290.9i			0.481299i	0.970612	+	0.240649i	$0.0773604\pi$
$263$			33290.9i			0.481299i	−0.970612	+	0.240649i	$0.922640\pi$
$264$	−11250.5	−	8779.66i	−0.161423	−	0.125971i
$265$	41557.7			0.591779
$266$			42581.8i			0.601812i
$267$	18528.8	−	23743.3i	0.259911	−	0.333058i
$268$	112429.			1.56534
$269$			7561.06i			0.104491i	0.998634	+	0.0522454i	$0.0166378\pi$
$269$			7561.06i			0.104491i	−0.998634	+	0.0522454i	$0.983362\pi$
$270$	77567.1	−	175015.i	1.06402	−	2.40075i
$271$	25491.3			0.347100			0.173550	−	0.984825i	$-0.444476\pi$
$271$	25491.3			0.347100			0.173550	+	0.984825i	$0.444476\pi$
$272$		−	20855.2i		−	0.281888i
$273$	−15596.3	−	12171.0i	−0.209264	−	0.163305i
$274$	−321.060			−0.00427646
$275$		−	41766.5i		−	0.552284i
$276$	−33913.6	+	43457.9i	−0.445200	+	0.570494i
$277$	14441.1			0.188209			0.0941045	−	0.995562i	$-0.470001\pi$
$277$	14441.1			0.188209			0.0941045	+	0.995562i	$0.470001\pi$
$278$			64310.7i			0.832134i
$279$	17759.3	+	70887.8i	0.228148	+	0.910675i
$280$	−21529.6			−0.274612
$281$		−	86894.0i		−	1.10047i	−0.835011	−	0.550234i	$-0.814539\pi$
$281$		−	86894.0i		−	1.10047i	0.835011	−	0.550234i	$-0.185461\pi$
$282$	−160453.	−	125214.i	−2.01767	−	1.57455i
$283$	147937.			1.84716			0.923579	−	0.383409i	$-0.125250\pi$
$283$	147937.			1.84716			0.923579	+	0.383409i	$0.125250\pi$
$284$			11935.2i			0.147976i
$285$	134953.	−	172933.i	1.66147	−	2.12906i
$286$	42512.5			0.519739
$287$		−	8038.81i		−	0.0975951i
$288$	−101776.	+	25497.7i	−1.22705	+	0.307409i
$289$	36435.6			0.436245
$290$		−	203691.i		−	2.42201i
$291$	−53529.8	−	41773.5i	−0.632135	−	0.493304i
$292$	−179685.			−2.10740
$293$			98428.0i			1.14653i	0.819372	+	0.573263i	$0.194322\pi$
$293$			98428.0i			1.14653i	−0.819372	+	0.573263i	$0.805678\pi$
$294$	−78191.1	+	100196.i	−0.904613	+	1.15920i
$295$	74524.2			0.856354
$296$		−	47814.7i		−	0.545730i
$297$	−24314.9	−	10776.4i	−0.275651	−	0.122169i
$298$	−99528.1			−1.12076
$299$		−	49794.1i		−	0.556975i
$300$	186523.	+	145558.i	2.07248	+	1.61731i
$301$	16878.9			0.186299
$302$			167528.i			1.83685i
$303$	−86798.0	+	111226.i	−0.945419	+	1.21149i
$304$	−55682.4			−0.602519
$305$			119635.i			1.28605i
$306$	26661.8	+	106423.i	0.284739	+	1.13656i
$307$	−19930.4			−0.211466			−0.105733	−	0.994395i	$-0.533719\pi$
$307$	−19930.4			−0.211466			−0.105733	+	0.994395i	$0.533719\pi$
$308$			9864.36i			0.103984i
$309$	77505.8	+	60483.8i	0.811741	+	0.633464i
$310$	−236917.			−2.46532
$311$		−	95858.8i		−	0.991086i	−0.868584	−	0.495543i	$-0.834969\pi$
$311$		−	95858.8i		−	0.991086i	0.868584	−	0.495543i	$-0.165031\pi$
$312$	−44925.3	+	57568.6i	−0.461510	+	0.591394i
$313$	58734.4			0.599521			0.299760	−	0.954015i	$-0.403093\pi$
$313$	58734.4			0.599521			0.299760	+	0.954015i	$0.403093\pi$
$314$		−	289668.i		−	2.93792i
$315$	−38921.0	+	9750.75i	−0.392250	+	0.0982691i
$316$	−9564.22			−0.0957801
$317$		−	100015.i		−	0.995287i	−0.867382	−	0.497643i	$-0.834199\pi$
$317$		−	100015.i		−	0.995287i	0.867382	−	0.497643i	$-0.165801\pi$
$318$	43750.0	+	34141.5i	0.432637	+	0.337620i
$319$	−28298.9			−0.278092
$320$		−	275458.i		−	2.69002i
$321$	−4294.67	+	5503.32i	−0.0416793	+	0.0534091i
$322$	19604.4			0.189079
$323$			125715.i			1.20499i
$324$	132865.	−	71030.4i	1.26567	−	0.676635i
$325$	−213718.			−2.02337
$326$		−	147816.i		−	1.39087i
$327$	31636.2	+	24688.2i	0.295861	+	0.230884i
$328$	−29672.7			−0.275810
$329$			42658.9i			0.394110i
$330$	53045.8	−	67974.5i	0.487105	−	0.624192i
$331$	64359.9			0.587434			0.293717	−	0.955892i	$-0.405108\pi$
$331$	64359.9			0.587434			0.293717	+	0.955892i	$0.405108\pi$
$332$			68447.6i			0.620987i
$333$	−21655.3	−	86439.1i	−0.195288	−	0.779510i
$334$	31210.1			0.279771
$335$			205975.i			1.83538i
$336$	8029.54	+	6266.07i	0.0711233	+	0.0555030i
$337$	59843.0			0.526931			0.263466	−	0.964669i	$-0.415134\pi$
$337$	59843.0			0.526931			0.263466	+	0.964669i	$0.415134\pi$
$338$		−	39256.9i		−	0.343623i
$339$	−85724.1	+	109849.i	−0.745939	+	0.955869i
$340$	−209621.			−1.81333
$341$			32915.0i			0.283064i
$342$	284144.	−	71185.6i	2.42933	−	0.608611i
$343$	54910.0			0.466727
$344$		−	62302.9i		−	0.526492i
$345$	−79617.2	−	62131.5i	−0.668911	−	0.522003i
$346$	71617.0			0.598224
$347$		−	12845.8i		−	0.106685i	−0.998576	−	0.0533425i	$-0.983012\pi$
$347$		−	12845.8i		−	0.106685i	0.998576	−	0.0533425i	$-0.0169875\pi$
$348$	98623.2	−	126379.i	0.814368	−	1.04356i
$349$	−130947.			−1.07509			−0.537546	−	0.843234i	$-0.680649\pi$
$349$	−130947.			−1.07509			−0.537546	+	0.843234i	$0.680649\pi$
$350$		−	84142.9i		−	0.686881i
$351$	−55142.8	+	124419.i	−0.447584	+	1.00989i
$352$	−47257.4			−0.381403
$353$		−	15375.0i		−	0.123386i	−0.998095	−	0.0616928i	$-0.980350\pi$
$353$		−	15375.0i		−	0.123386i	0.998095	−	0.0616928i	$-0.0196499\pi$
$354$	78455.6	+	61225.0i	0.626062	+	0.488565i
$355$	−21865.9			−0.173504
$356$			76842.8i			0.606322i
$357$	14147.1	−	18128.5i	0.111002	−	0.142241i
$358$	142073.			1.10853
$359$			55146.4i			0.427886i	0.976846	+	0.213943i	$0.0686307\pi$
$359$			55146.4i			0.427886i	−0.976846	+	0.213943i	$0.931369\pi$
$360$	35991.8	+	143665.i	0.277714	+	1.10852i
$361$	205333.			1.57559
$362$			240455.i			1.83492i
$363$	−9443.74	−	7369.68i	−0.0716689	−	0.0559288i
$364$	50475.7			0.380960
$365$		−	329192.i		−	2.47095i
$366$	−98285.4	+	125946.i	−0.733714	+	0.940204i
$367$	160740.			1.19341			0.596706	−	0.802460i	$-0.296476\pi$
$367$	160740.			1.19341			0.596706	+	0.802460i	$0.296476\pi$
$368$			25635.8i			0.189301i
$369$	−53642.1	+	13438.8i	−0.393961	+	0.0986977i
$370$	288891.			2.11024
$371$		−	11631.6i		−	0.0845067i
$372$	−146994.	−	114710.i	−1.06221	−	0.828929i
$373$	−45525.6			−0.327219			−0.163609	−	0.986525i	$-0.552314\pi$
$373$	−45525.6			−0.327219			−0.163609	+	0.986525i	$0.552314\pi$
$374$			49414.9i			0.353276i
$375$	−121086.	+	155163.i	−0.861053	+	1.10338i
$376$	157462.			1.11378
$377$			144805.i			1.01883i
$378$	−48984.9	−	21710.3i	−0.342830	−	0.151943i
$379$	−201699.			−1.40419			−0.702093	−	0.712085i	$-0.747751\pi$
$379$	−201699.			−1.40419			−0.702093	+	0.712085i	$0.747751\pi$
$380$			559678.i			3.87589i
$381$	23465.5	+	18312.0i	0.161652	+	0.126149i
$382$	−53890.2			−0.369303
$383$		−	226958.i		−	1.54721i	−0.633669	−	0.773604i	$-0.718452\pi$
$383$		−	226958.i		−	1.54721i	0.633669	−	0.773604i	$-0.281548\pi$
$384$	111546.	−	142939.i	0.756471	−	0.969365i
$385$	−18072.0			−0.121923
$386$			73601.6i			0.493984i
$387$	−28217.0	−	112631.i	−0.188404	−	0.752031i
$388$	173244.			1.15078
$389$		−	9157.77i		−	0.0605188i	−0.999542	−	0.0302594i	$-0.990367\pi$
$389$		−	9157.77i		−	0.0605188i	0.999542	−	0.0302594i	$-0.00963334\pi$
$390$	−347824.	−	271434.i	−2.28681	−	1.78457i
$391$	57878.7			0.378587
$392$		−	98328.3i		−	0.639892i
$393$	38394.4	−	49199.8i	0.248590	−	0.318550i
$394$	−447611.			−2.88342
$395$		−	17522.2i		−	0.112303i
$396$	65823.8	−	16490.6i	0.419752	−	0.105159i
$397$	−45936.2			−0.291457			−0.145728	−	0.989325i	$-0.546553\pi$
$397$	−45936.2			−0.291457			−0.145728	+	0.989325i	$0.546553\pi$
$398$			295192.i			1.86354i
$399$	−48402.1	−	37771.9i	−0.304032	−	0.237259i
$400$	110030.			0.687687
$401$			87213.0i			0.542366i	0.962528	+	0.271183i	$0.0874148\pi$
$401$			87213.0i			0.542366i	−0.962528	+	0.271183i	$0.912585\pi$
$402$	−169218.	+	216841.i	−1.04712	+	1.34181i
$403$	168425.			1.03704
$404$		−	359970.i		−	2.20548i
$405$	130131.	+	243415.i	0.793363	+	1.48401i
$406$	−57011.1			−0.345866
$407$		−	40135.9i		−	0.242295i
$408$	−66915.5	−	52219.4i	−0.401982	−	0.313698i
$409$	191495.			1.14475			0.572377	−	0.819991i	$-0.306022\pi$
$409$	191495.			1.14475			0.572377	+	0.819991i	$0.306022\pi$
$410$		−	179279.i		−	1.06650i
$411$	284.794	−	364.944i	0.00168596	−	0.00216044i
$412$	−250839.			−1.47775
$413$		−	20858.6i		−	0.122288i
$414$	−32773.4	−	130818.i	−0.191215	−	0.763251i
$415$	−125400.			−0.728115
$416$			241815.i			1.39732i
$417$	−73101.0	−	57046.4i	−0.420389	−	0.328062i
$418$	131935.			0.755106
$419$			19793.3i			0.112743i	0.998410	+	0.0563716i	$0.0179531\pi$
$419$			19793.3i			0.112743i	−0.998410	+	0.0563716i	$0.982047\pi$
$420$	62981.9	−	80707.0i	0.357040	−	0.457522i
$421$	−215445.			−1.21555			−0.607775	−	0.794109i	$-0.707938\pi$
$421$	−215445.			−1.21555			−0.607775	+	0.794109i	$0.707938\pi$
$422$		−	228358.i		−	1.28230i
$423$	284658.	−	71314.4i	1.59090	−	0.398562i
$424$	−42934.3			−0.238821
$425$		−	248417.i		−	1.37532i
$426$	−23019.4	−	17963.8i	−0.126845	−	0.0989871i
$427$	33484.6			0.183649
$428$		−	17810.9i		−	0.0972297i
$429$	−37710.5	+	48323.4i	−0.204903	+	0.262568i
$430$	376428.			2.03584
$431$			69765.5i			0.375566i	0.982211	+	0.187783i	$0.0601302\pi$
$431$			69765.5i			0.375566i	−0.982211	+	0.187783i	$0.939870\pi$
$432$	28389.5	−	64055.4i	0.152122	−	0.343233i
$433$	−290735.			−1.55068			−0.775340	−	0.631544i	$-0.782421\pi$
$433$	−290735.			−1.55068			−0.775340	+	0.631544i	$0.782421\pi$
$434$			66310.7i			0.352050i
$435$	231533.	+	180683.i	1.22358	+	0.954857i
$436$	−102387.			−0.538607
$437$		−	154533.i		−	0.809205i
$438$	270446.	−	346558.i	1.40972	−	1.80646i
$439$	252609.			1.31075			0.655376	−	0.755303i	$-0.272510\pi$
$439$	252609.			1.31075			0.655376	+	0.755303i	$0.272510\pi$
$440$			66707.1i			0.344561i
$441$	−44532.9	−	177757.i	−0.228983	−	0.914008i
$442$	252855.			1.29427
$443$		−	383213.i		−	1.95269i	−0.216225	−	0.976343i	$-0.569375\pi$
$443$		−	383213.i		−	1.95269i	0.216225	−	0.976343i	$-0.430625\pi$
$444$	179241.	+	139876.i	0.909224	+	0.709538i
$445$	−140780.			−0.710921
$446$			306639.i			1.54155i
$447$	88285.7	−	113132.i	0.441851	−	0.566201i
$448$	−77098.0			−0.384138
$449$			297348.i			1.47493i	0.675383	+	0.737467i	$0.263978\pi$
$449$			297348.i			1.47493i	−0.675383	+	0.737467i	$0.736022\pi$
$450$	−561476.	+	140665.i	−2.77272	+	0.694640i
$451$	−24907.4			−0.122455
$452$		−	355516.i		−	1.74013i
$453$	−190426.	−	148604.i	−0.927962	−	0.724161i
$454$	−269225.			−1.30618
$455$			92474.1i			0.446681i
$456$	−139423.	+	178661.i	−0.670509	+	0.859211i
$457$	−109053.			−0.522161			−0.261080	−	0.965317i	$-0.584079\pi$
$457$	−109053.			−0.522161			−0.261080	+	0.965317i	$0.584079\pi$
$458$			499009.i			2.37891i
$459$	−144620.	−	64095.8i	−0.686439	−	0.304232i
$460$	257672.			1.21773
$461$			98599.7i			0.463953i	0.972721	+	0.231976i	$0.0745192\pi$
$461$			98599.7i			0.463953i	−0.972721	+	0.231976i	$0.925481\pi$
$462$	−19025.4	−	14847.0i	−0.0891352	−	0.0695592i
$463$	155562.			0.725674			0.362837	−	0.931853i	$-0.381808\pi$
$463$	155562.			0.725674			0.362837	+	0.931853i	$0.381808\pi$
$464$		−	74550.9i		−	0.346272i
$465$	210156.	−	269300.i	0.971930	−	1.24546i
$466$	−301897.			−1.39023
$467$		−	131455.i		−	0.602757i	−0.953505	−	0.301378i	$-0.902553\pi$
$467$		−	131455.i		−	0.602757i	0.953505	−	0.301378i	$-0.0974467\pi$
$468$	−84382.0	−	336819.i	−0.385264	−	1.53782i
$469$	57650.4			0.262094
$470$			951366.i			4.30677i
$471$	329261.	+	256948.i	1.48422	+	1.15825i
$472$	−76992.8			−0.345594
$473$		−	52297.4i		−	0.233753i
$474$	14395.2	−	18446.5i	0.0640711	−	0.0821027i
$475$	−663261.			−2.93966
$476$			58670.9i			0.258946i
$477$	−77616.3	+	19444.9i	−0.341127	+	0.0854613i
$478$	377322.			1.65142
$479$		−	123234.i		−	0.537104i	−0.963265	−	0.268552i	$-0.913455\pi$
$479$		−	123234.i		−	0.537104i	0.963265	−	0.268552i	$-0.0865451\pi$
$480$	386644.	+	301729.i	1.67814	+	1.30959i
$481$	−205374.			−0.887678
$482$			420053.i			1.80805i
$483$	−17390.0	+	22284.1i	−0.0745427	+	0.0955213i
$484$	30563.7			0.130471
$485$			317391.i			1.34931i
$486$	−62980.3	+	363165.i	−0.266644	+	1.53756i
$487$	191384.			0.806952			0.403476	−	0.914990i	$-0.367802\pi$
$487$	191384.			0.806952			0.403476	+	0.914990i	$0.367802\pi$
$488$		−	123598.i		−	0.519004i
$489$	168020.	+	131119.i	0.702656	+	0.548337i
$490$	594089.			2.47434
$491$		−	163093.i		−	0.676509i	−0.941055	−	0.338254i	$-0.890164\pi$
$491$		−	163093.i		−	0.676509i	0.941055	−	0.338254i	$-0.109836\pi$
$492$	86803.5	−	111233.i	0.358597	−	0.459518i
$493$	−168316.			−0.692517
$494$		−	675109.i		−	2.76643i
$495$	30211.6	+	120593.i	0.123300	+	0.492165i
$496$	−86711.5			−0.352463
$497$			6120.04i			0.0247766i
$498$	−132015.	−	103021.i	−0.532309	−	0.415402i
$499$	−158644.			−0.637120			−0.318560	−	0.947903i	$-0.603199\pi$
$499$	−158644.			−0.637120			−0.318560	+	0.947903i	$0.603199\pi$
$500$		−	502168.i		−	2.00867i
$501$	−27684.8	+	35476.1i	−0.110297	+	0.141338i
$502$	−667701.			−2.64957
$503$		−	386864.i		−	1.52905i	−0.644592	−	0.764526i	$-0.722973\pi$
$503$		−	386864.i		−	1.52905i	0.644592	−	0.764526i	$-0.277027\pi$
$504$	40210.3	−	10073.7i	0.158298	−	0.0396579i
$505$	659483.			2.58596
$506$		−	60742.2i		−	0.237241i
$507$	44622.7	+	34822.6i	0.173596	+	0.135471i
$508$	−75943.7			−0.294283
$509$			398158.i			1.53681i	0.639963	+	0.768405i	$0.278949\pi$
$509$			398158.i			1.53681i	−0.639963	+	0.768405i	$0.721051\pi$
$510$	315504.	−	404296.i	1.21301	−	1.55439i
$511$	−92137.7			−0.352854
$512$		−	191331.i		−	0.729870i
$513$	−171132.	+	386127.i	−0.650276	+	1.46722i
$514$	159898.			0.605224
$515$		−	459551.i		−	1.73268i
$516$	233552.	+	182259.i	0.877172	+	0.684525i
$517$	132174.			0.494498
$518$		−	80857.8i		−	0.301344i
$519$	−63527.4	+	81406.0i	−0.235845	+	0.302219i
$520$	341338.			1.26235
$521$			50104.4i			0.184587i	0.995732	+	0.0922933i	$0.0294197\pi$
$521$			50104.4i			0.184587i	−0.995732	+	0.0922933i	$0.970580\pi$
$522$	95307.5	+	380429.i	0.349773	+	1.39615i
$523$	−537778.			−1.96607			−0.983036	−	0.183415i	$-0.941285\pi$
$523$	−537778.			−1.96607			−0.983036	+	0.183415i	$0.941285\pi$
$524$			159230.i			0.579912i
$525$	95643.9	+	74638.4i	0.347007	+	0.270797i
$526$	207803.			0.751070
$527$			195771.i			0.704899i
$528$	19414.7	−	24878.7i	0.0696408	−	0.0892399i
$529$	208695.			0.745762
$530$		−	259404.i		−	0.923475i
$531$	−139187.	+	34870.0i	−0.493639	+	0.123670i
$532$	156648.			0.553481
$533$			127450.i			0.448629i
$534$	−148207.	−	115657.i	−0.519739	−	0.405593i
$535$	32630.5			0.114003
$536$		−	212798.i		−	0.740693i
$537$	−126025.	+	161492.i	−0.437027	+	0.560020i
$538$	47196.3			0.163059
$539$		−	82537.2i		−	0.284101i
$540$	−643838.	−	285351.i	−2.20795	−	0.978569i
$541$	277152.			0.946942			0.473471	−	0.880809i	$-0.343001\pi$
$541$	277152.			0.946942			0.473471	+	0.880809i	$0.343001\pi$
$542$		−	159118.i		−	0.541651i
$543$	−273322.	−	213294.i	−0.926989	−	0.723401i
$544$	−281076.			−0.949786
$545$		−	187578.i		−	0.631524i
$546$	−75971.6	+	97352.4i	−0.254839	+	0.326559i
$547$	212103.			0.708879			0.354440	−	0.935079i	$-0.384672\pi$
$547$	212103.			0.708879			0.354440	+	0.935079i	$0.384672\pi$
$548$			1181.10i			0.00393302i
$549$	−55977.4	−	223439.i	−0.185724	−	0.741335i
$550$	−260708.			−0.861843
$551$			449394.i			1.48021i
$552$	82254.5	+	64189.5i	0.269949	+	0.210662i
$553$	−4904.28			−0.0160371
$554$		−	90141.7i		−	0.293701i
$555$	−256259.	+	328378.i	−0.831943	+	1.06608i
$556$	236584.			0.765306
$557$		−	438459.i		−	1.41325i	−0.707588	−	0.706625i	$-0.750217\pi$
$557$		−	438459.i		−	1.41325i	0.707588	−	0.706625i	$-0.249783\pi$
$558$	442484.	−	110854.i	1.42111	−	0.356027i
$559$	−267604.			−0.856385
$560$		−	47609.1i		−	0.151815i
$561$	−56169.2	−	43833.2i	−0.178473	−	0.139276i
$562$	−542395.			−1.71729
$563$			390786.i			1.23288i	0.787401	+	0.616442i	$0.211426\pi$
$563$			390786.i			1.23288i	−0.787401	+	0.616442i	$0.788574\pi$
$564$	−460633.	+	590269.i	−1.44809	+	1.85563i
$565$	651324.			2.04033
$566$		−	923427.i		−	2.88250i
$567$	68129.5	−	36422.5i	0.211919	−	0.113293i
$568$	22590.2			0.0700201
$569$			520018.i			1.60618i	0.595859	+	0.803089i	$0.296812\pi$
$569$			520018.i			1.60618i	−0.595859	+	0.803089i	$0.703188\pi$
$570$	−1.07945e6	−	842379.i	−3.32241	−	2.59273i
$571$	−513581.			−1.57520			−0.787602	−	0.616185i	$-0.788677\pi$
$571$	−513581.			−1.57520			−0.787602	+	0.616185i	$0.788677\pi$
$572$		−	156393.i		−	0.477998i
$573$	47803.0	−	61256.2i	0.145595	−	0.186570i
$574$	−50178.5			−0.152298
$575$			305362.i			0.923589i
$576$	128887.	+	514466.i	0.388477	+	1.55064i
$577$	151067.			0.453752			0.226876	−	0.973924i	$-0.427149\pi$
$577$	151067.			0.453752			0.226876	+	0.973924i	$0.427149\pi$
$578$		−	227432.i		−	0.680764i
$579$	−83661.8	−	65287.8i	−0.249557	−	0.194749i
$580$	−749331.			−2.22750
$581$			35098.1i			0.103976i
$582$	−260751.	+	334135.i	−0.769804	+	0.986451i
$583$	−36039.2			−0.106032
$584$			340097.i			0.997187i
$585$	617069.	−	154592.i	1.80311	−	0.451727i
$586$	614391.			1.78916
$587$			266575.i			0.773648i	0.922154	+	0.386824i	$0.126428\pi$
$587$			266575.i			0.773648i	−0.922154	+	0.386824i	$0.873572\pi$
$588$	368599.	+	287646.i	1.06610	+	0.831963i
$589$	522698.			1.50668
$590$		−	465182.i		−	1.33635i
$591$	397051.	−	508793.i	1.13677	−	1.45669i
$592$	105734.			0.301698
$593$			583555.i			1.65948i	0.558149	+	0.829741i	$0.311512\pi$
$593$			583555.i			1.65948i	−0.558149	+	0.829741i	$0.688488\pi$
$594$	−67266.9	+	151775.i	−0.190646	+	0.430156i
$595$	−107488.			−0.303618
$596$			366140.i			1.03075i
$597$	−335540.	−	261848.i	−0.941446	−	0.734684i
$598$	−310816.			−0.869163
$599$		−	3287.16i		−	0.00916151i	−0.999990	−	0.00458075i	$-0.998542\pi$
$599$		−	3287.16i		−	0.00916151i	0.999990	−	0.00458075i	$-0.00145810\pi$
$600$	275504.	−	353039.i	0.765288	−	0.980664i
$601$	−393802.			−1.09026			−0.545128	−	0.838353i	$-0.683519\pi$
$601$	−393802.			−1.09026			−0.545128	+	0.838353i	$0.683519\pi$
$602$		−	105358.i		−	0.290721i
$603$	−96376.3	−	384695.i	−0.265055	−	1.05799i
$604$	616294.			1.68933
$605$			55994.2i			0.152979i
$606$	694273.	+	541795.i	1.89054	+	1.47533i
$607$	287851.			0.781251			0.390626	−	0.920550i	$-0.372259\pi$
$607$	287851.			0.781251			0.390626	+	0.920550i	$0.372259\pi$
$608$			750458.i			2.03011i
$609$	50571.4	−	64803.7i	0.136355	−	0.174729i
$610$	746764.			2.00689
$611$		−	676331.i		−	1.81166i
$612$	391505.	−	98082.3i	1.04528	−	0.261871i
$613$	343049.			0.912925			0.456463	−	0.889743i	$-0.349116\pi$
$613$	343049.			0.912925			0.456463	+	0.889743i	$0.349116\pi$
$614$			124406.i			0.329994i
$615$	203784.	+	159029.i	0.538791	+	0.420460i
$616$	18670.7			0.0492037
$617$		−	569755.i		−	1.49664i	−0.663338	−	0.748320i	$-0.730861\pi$
$617$		−	569755.i		−	1.49664i	0.663338	−	0.748320i	$-0.269139\pi$
$618$	377542.	−	483794.i	0.988526	−	1.26673i
$619$	347551.			0.907062			0.453531	−	0.891241i	$-0.350164\pi$
$619$	347551.			0.907062			0.453531	+	0.891241i	$0.350164\pi$
$620$			871560.i			2.26733i
$621$	177771.	+	78788.4i	0.460974	+	0.204305i
$622$	−598353.			−1.54660
$623$			39402.9i			0.101520i
$624$	−127303.	−	99344.7i	−0.326942	−	0.255138i
$625$	204482.			0.523475
$626$		−	366622.i		−	0.935556i
$627$	−117032.	+	149969.i	−0.297694	+	0.381475i
$628$	−1.06562e6			−2.70198
$629$		−	238719.i		−	0.603372i
$630$	60864.5	+	242946.i	0.153350	+	0.612110i
$631$	−275097.			−0.690918			−0.345459	−	0.938434i	$-0.612277\pi$
$631$	−275097.			−0.690918			−0.345459	+	0.938434i	$0.612277\pi$
$632$			18102.6i			0.0453217i
$633$	259571.	+	202563.i	0.647811	+	0.505537i
$634$	−624299.			−1.55315
$635$		−	139133.i		−	0.345050i
$636$	125598.	−	160946.i	0.310506	−	0.397892i
$637$	−422341.			−1.04084
$638$			176643.i			0.433965i
$639$	40838.3	−	10231.1i	0.100015	−	0.0250565i
$640$	−847518.			−2.06914
$641$		−	16424.6i		−	0.0399740i	−0.999800	−	0.0199870i	$-0.993638\pi$
$641$		−	16424.6i		−	0.0399740i	0.999800	−	0.0199870i	$-0.00636249\pi$
$642$	34351.9	+	26807.5i	0.0833453	+	0.0650408i
$643$	85020.1			0.205636			0.102818	−	0.994700i	$-0.467214\pi$
$643$	85020.1			0.205636			0.102818	+	0.994700i	$0.467214\pi$
$644$		−	72120.0i		−	0.173894i
$645$	−333908.	+	427880.i	−0.802615	+	1.02850i
$646$	784720.			1.88040
$647$		−	4502.05i		−	0.0107548i	−0.999986	−	0.00537740i	$-0.998288\pi$
$647$		−	4502.05i		−	0.0107548i	0.999986	−	0.00537740i	$-0.00171169\pi$
$648$	−134442.	−	251478.i	−0.320173	−	0.598895i
$649$	−64628.1			−0.153438
$650$			1.33403e6i			3.15748i
$651$	−75374.3	−	58820.5i	−0.177853	−	0.138793i
$652$	−543778.			−1.27916
$653$			181202.i			0.424950i	0.977167	+	0.212475i	$0.0681524\pi$
$653$			181202.i			0.424950i	−0.977167	+	0.212475i	$0.931848\pi$
$654$	154104.	−	197474.i	0.360296	−	0.461694i
$655$	−291717.			−0.679954
$656$		−	65616.2i		−	0.152477i
$657$	154030.	+	614825.i	0.356840	+	1.42436i
$658$	266278.			0.615012
$659$		−	584133.i		−	1.34506i	−0.740071	−	0.672529i	$-0.765208\pi$
$659$		−	584133.i		−	1.34506i	0.740071	−	0.672529i	$-0.234792\pi$
$660$	−250062.	−	195143.i	−0.574063	−	0.447986i
$661$	85750.2			0.196260			0.0981301	−	0.995174i	$-0.468714\pi$
$661$	85750.2			0.196260			0.0981301	+	0.995174i	$0.468714\pi$
$662$		−	401736.i		−	0.916695i
$663$	−224293.	+	287416.i	−0.510257	+	0.653859i
$664$	129553.			0.293841
$665$			286988.i			0.648963i
$666$	−539556.	+	135173.i	−1.21643	+	0.304748i
$667$	206898.			0.465056
$668$		−	114815.i		−	0.257303i
$669$	−348552.	−	272002.i	−0.778781	−	0.607744i
$670$	1.28570e6			2.86412
$671$		−	103748.i		−	0.230429i
$672$	84450.9	−	108218.i	0.187010	−	0.239641i
$673$	−764933.			−1.68886			−0.844429	−	0.535667i	$-0.820060\pi$
$673$	−764933.			−1.68886			−0.844429	+	0.535667i	$0.820060\pi$
$674$		−	373542.i		−	0.822280i
$675$	338162.	−	762997.i	0.742195	−	1.67462i
$676$	−144417.			−0.316027
$677$			394004.i			0.859654i	0.902911	+	0.429827i	$0.141425\pi$
$677$			394004.i			0.859654i	−0.902911	+	0.429827i	$0.858575\pi$
$678$	685684.	+	535092.i	1.49164	+	1.16404i
$679$	88834.6			0.192683
$680$			396758.i			0.858041i
$681$	238814.	−	306024.i	0.514951	−	0.659875i
$682$	205456.			0.441724
$683$			85034.7i			0.182287i	0.995838	+	0.0911433i	$0.0290521\pi$
$683$			85034.7i			0.182287i	−0.995838	+	0.0911433i	$0.970948\pi$
$684$	−261875.	−	1.04530e6i	−0.559733	−	2.23423i
$685$	−2163.84			−0.00461152
$686$		−	342750.i		−	0.728331i
$687$	−567216.	−	442643.i	−1.20181	−	0.937864i
$688$	137773.			0.291062
$689$			184412.i			0.388463i
$690$	−387826.	+	496973.i	−0.814590	+	1.04384i
$691$	362399.			0.758981			0.379491	−	0.925196i	$-0.376099\pi$
$691$	362399.			0.758981			0.379491	+	0.925196i	$0.376099\pi$
$692$		−	263462.i		−	0.550181i
$693$	33752.7	−	8455.94i	0.0702816	−	0.0176074i
$694$	−80184.1			−0.166483
$695$			433433.i			0.897331i
$696$	−239202.	−	186668.i	−0.493795	−	0.385346i
$697$	−148143.			−0.304942
$698$			817377.i			1.67769i
$699$	267796.	−	343162.i	0.548087	−	0.702335i
$700$	−309541.			−0.631717
$701$			431063.i			0.877212i	0.898680	+	0.438606i	$0.144528\pi$
$701$			431063.i			0.877212i	−0.898680	+	0.438606i	$0.855472\pi$
$702$	776626.	+	344203.i	1.57593	+	0.698458i
$703$	−637367.			−1.28967
$704$			238880.i			0.481985i
$705$	−1.08140e6	−	843903.i	−2.17575	−	1.69791i
$706$	−95971.0			−0.192544
$707$		−	184583.i		−	0.369277i
$708$	225232.	−	288619.i	0.449328	−	0.575783i
$709$	409134.			0.813905			0.406952	−	0.913449i	$-0.366592\pi$
$709$	409134.			0.813905			0.406952	+	0.913449i	$0.366592\pi$
$710$			136487.i			0.270754i
$711$	8198.66	+	32725.7i	0.0162182	+	0.0647366i
$712$	145443.			0.286902
$713$		−	240647.i		−	0.473371i
$714$	−113159.	−	88306.4i	−0.221968	−	0.173219i
$715$	286521.			0.560459
$716$		−	522653.i		−	1.01950i
$717$	−334701.	+	428897.i	−0.651057	+	0.834285i
$718$	344226.			0.667720
$719$			631414.i			1.22140i	0.791863	+	0.610698i	$0.209111\pi$
$719$			631414.i			1.22140i	−0.791863	+	0.610698i	$0.790889\pi$
$720$	−317690.	+	79589.8i	−0.612828	+	0.153530i
$721$	−128624.			−0.247429
$722$		−	1.28169e6i		−	2.45872i
$723$	−477468.	−	372606.i	−0.913415	−	0.712808i
$724$	884577.			1.68756
$725$		−	888014.i		−	1.68944i
$726$	−46001.8	+	58948.1i	−0.0872773	+	0.111840i
$727$	711266.			1.34575			0.672873	−	0.739758i	$-0.265060\pi$
$727$	711266.			1.34575			0.672873	+	0.739758i	$0.265060\pi$
$728$		−	95537.2i		−	0.180264i
$729$	−356938.	−	393732.i	−0.671642	−	0.740876i
$730$	−2.05483e6			−3.85593
$731$		−	311053.i		−	0.582102i
$732$	463325.	+	361568.i	0.864696	+	0.674789i
$733$	−36344.9			−0.0676449			−0.0338225	−	0.999428i	$-0.510768\pi$
$733$	−36344.9			−0.0676449			−0.0338225	+	0.999428i	$0.510768\pi$
$734$		−	1.00334e6i		−	1.86233i
$735$	−526983.	+	675292.i	−0.975488	+	1.25002i
$736$	345507.			0.637824
$737$		−	178624.i		−	0.328855i
$738$	83885.2	+	334836.i	0.154018	+	0.614779i
$739$	800264.			1.46536			0.732680	−	0.680574i	$-0.238269\pi$
$739$	800264.			1.46536			0.732680	+	0.680574i	$0.238269\pi$
$740$		−	1.06276e6i		−	1.94076i
$741$	767386.	+	598851.i	1.39758	+	1.09064i
$742$	−72604.7			−0.131873
$743$		−	41609.1i		−	0.0753722i	−0.999290	−	0.0376861i	$-0.988001\pi$
$743$		−	41609.1i		−	0.0753722i	0.999290	−	0.0376861i	$-0.0119987\pi$
$744$	−217117.	+	278220.i	−0.392236	+	0.502624i
$745$	−670787.			−1.20857
$746$			284172.i			0.510627i
$747$	234206.	−	58674.8i	0.419717	−	0.105150i
$748$	181786.			0.324905
$749$		−	9132.97i		−	0.0162798i
$750$	968530.	+	755820.i	1.72183	+	1.34368i
$751$	864924.			1.53355			0.766776	−	0.641915i	$-0.221860\pi$
$751$	864924.			1.53355			0.766776	+	0.641915i	$0.221860\pi$
$752$			348200.i			0.615734i
$753$	592280.	−	758966.i	1.04457	−	1.33854i
$754$	903877.			1.58989
$755$			1.12908e6i			1.98076i
$756$	−79866.9	+	180204.i	−0.139741	+	0.315297i
$757$	−151740.			−0.264794			−0.132397	−	0.991197i	$-0.542267\pi$
$757$	−151740.			−0.264794			−0.132397	+	0.991197i	$0.542267\pi$
$758$			1.25901e6i			2.19124i
$759$	69044.7	+	53881.0i	0.119852	+	0.0935302i
$760$	1.05932e6			1.83401
$761$			1.05603e6i			1.82351i	0.410735	+	0.911755i	$0.365272\pi$
$761$			1.05603e6i			1.82351i	−0.410735	+	0.911755i	$0.634728\pi$
$762$	114304.	−	146473.i	0.196857	−	0.252259i
$763$	−52501.4			−0.0901823
$764$			198249.i			0.339645i
$765$	179692.	+	717257.i	0.307047	+	1.22561i
$766$	−1.41668e6			−2.41443
$767$			330700.i			0.562139i
$768$	−148907.	−	116204.i	−0.252461	−	0.197015i
$769$	−848741.			−1.43523			−0.717617	−	0.696438i	$-0.754767\pi$
$769$	−848741.			−1.43523			−0.717617	+	0.696438i	$0.754767\pi$
$770$			112806.i			0.190262i
$771$	−141836.	+	181754.i	−0.238605	+	0.305755i
$772$	270763.			0.454312
$773$			296519.i			0.496242i	0.968729	+	0.248121i	$0.0798130\pi$
$773$			296519.i			0.496242i	−0.968729	+	0.248121i	$0.920187\pi$
$774$	−703045.	+	176131.i	−1.17355	+	0.294005i
$775$	−1.03287e6			−1.71965
$776$		−	327905.i		−	0.544533i
$777$	91909.9	+	71724.4i	0.152237	+	0.118802i
$778$	−57163.0			−0.0944400
$779$			395535.i			0.651794i
$780$	−998540.	+	1.27956e6i	−1.64126	+	2.10315i
$781$	18962.3			0.0310877
$782$		−	361280.i		−	0.590787i
$783$	−516970.	−	229123.i	−0.843222	−	0.373718i
$784$	217437.			0.353753
$785$		−	1.95227e6i		−	3.16811i
$786$	−307106.	−	239659.i	−0.497100	−	0.387926i
$787$	789809.			1.27518			0.637591	−	0.770375i	$-0.279931\pi$
$787$	789809.			1.27518			0.637591	+	0.770375i	$0.279931\pi$
$788$			1.64665e6i			2.65186i
$789$	−184330.	+	236207.i	−0.296103	+	0.379436i
$790$	−109374.			−0.175250
$791$		−	182299.i		−	0.291361i
$792$	−31212.4	−	124587.i	−0.0497596	−	0.198620i
$793$	−530878.			−0.844205
$794$			286735.i			0.454821i
$795$	294861.	+	230103.i	0.466534	+	0.364072i
$796$	1.08594e6			1.71388
$797$		−	1.22354e6i		−	1.92620i	−0.269147	−	0.963099i	$-0.586742\pi$
$797$		−	1.22354e6i		−	1.92620i	0.269147	−	0.963099i	$-0.413258\pi$
$798$	−235774.	+	302127.i	−0.370245	+	0.474443i
$799$	786140.			1.23142
$800$		−	1.48293e6i		−	2.31707i
$801$	262932.	−	65871.3i	0.409805	−	0.102667i
$802$	544386.			0.846366
$803$			285479.i			0.442734i
$804$	797707.	+	622512.i	1.23405	+	0.963021i
$805$	132128.			0.203893
$806$		−	1.05132e6i		−	1.61831i
$807$	−41865.2	+	53647.4i	−0.0642845	+	0.0823761i
$808$	−681328.			−1.04360
$809$			672652.i			1.02776i	0.857861	+	0.513882i	$0.171793\pi$
$809$			672652.i			1.02776i	−0.857861	+	0.513882i	$0.828207\pi$
$810$	1.51940e6	−	812284.i	2.31581	−	1.23805i
$811$	−617539.			−0.938907			−0.469453	−	0.882957i	$-0.655549\pi$
$811$	−617539.			−0.938907			−0.469453	+	0.882957i	$0.655549\pi$
$812$			209730.i			0.318089i
$813$	180867.	+	141144.i	0.273639	+	0.213542i
$814$	−250529.			−0.378103
$815$		−	996230.i		−	1.49984i
$816$	115474.	−	147972.i	0.173422	−	0.222229i
$817$	−830494.			−1.24421
$818$		−	1.19532e6i		−	1.78640i
$819$	−43268.8	−	172712.i	−0.0645071	−	0.257486i
$820$	−659526.			−0.980853
$821$		−	991868.i		−	1.47153i	−0.677240	−	0.735763i	$-0.736824\pi$
$821$		−	991868.i		−	1.47153i	0.677240	−	0.735763i	$-0.263176\pi$
$822$	−2277.99	−	1777.69i	−0.00337138	−	0.00263095i
$823$	−989527.			−1.46092			−0.730462	−	0.682953i	$-0.760695\pi$
$823$	−989527.			−1.46092			−0.730462	+	0.682953i	$0.760695\pi$
$824$			474773.i			0.699249i
$825$	231259.	−	296342.i	0.339774	−	0.435397i
$826$	−130200.			−0.190832
$827$			445972.i			0.652074i	0.945357	+	0.326037i	$0.105713\pi$
$827$			445972.i			0.652074i	−0.945357	+	0.326037i	$0.894287\pi$
$828$	−481249.	+	120566.i	−0.701955	+	0.175858i
$829$	728510.			1.06005			0.530025	−	0.847982i	$-0.322183\pi$
$829$	728510.			1.06005			0.530025	+	0.847982i	$0.322183\pi$
$830$			782748.i			1.13623i
$831$	102463.	+	79959.6i	0.148376	+	0.115789i
$832$	1.22234e6			1.76582
$833$		−	490912.i		−	0.707479i
$834$	−356085.	+	456298.i	−0.511943	+	0.656020i
$835$	210346.			0.301691
$836$		−	485358.i		−	0.694464i
$837$	−266497.	+	601297.i	−0.380400	+	0.858298i
$838$	123550.			0.175937
$839$			370123.i			0.525801i	0.964823	+	0.262901i	$0.0846791\pi$
$839$			370123.i			0.525801i	−0.964823	+	0.262901i	$0.915321\pi$
$840$	−152757.	−	119208.i	−0.216493	−	0.168946i
$841$	105606.			0.149312
$842$			1.34482e6i			1.89687i
$843$	481128.	−	616532.i	0.677026	−	0.867562i
$844$	−840073.			−1.17932
$845$		−	264579.i		−	0.370546i
$846$	−445146.	−	1.77684e6i	−0.621960	−	2.48261i
$847$	15672.2			0.0218456
$848$		−	94941.9i		−	0.132028i
$849$	1.04965e6	+	819120.i	1.45622	+	1.13640i
$850$	−1.55063e6			−2.14620
$851$			293440.i			0.405191i
$852$	−66084.5	+	84682.7i	−0.0910375	+	0.116658i
$853$	237569.			0.326506			0.163253	−	0.986584i	$-0.447801\pi$
$853$	237569.			0.326506			0.163253	+	0.986584i	$0.447801\pi$
$854$		−	209012.i		−	0.286586i
$855$	1.91504e6	−	479768.i	2.61966	−	0.656295i
$856$	−33711.4			−0.0460076
$857$			1.20972e6i			1.64711i	0.567238	+	0.823554i	$0.308012\pi$
$857$			1.20972e6i			1.64711i	−0.567238	+	0.823554i	$0.691988\pi$
$858$	301636.	+	235390.i	0.409740	+	0.319752i
$859$	31520.0			0.0427169			0.0213584	−	0.999772i	$-0.493201\pi$
$859$	31520.0			0.0427169			0.0213584	+	0.999772i	$0.493201\pi$
$860$		−	1.38479e6i		−	1.87235i
$861$	44510.5	−	57037.2i	0.0600422	−	0.0769399i
$862$	435478.			0.586073
$863$			965213.i			1.29599i	0.761645	+	0.647995i	$0.224392\pi$
$863$			965213.i			1.29599i	−0.761645	+	0.647995i	$0.775608\pi$
$864$	−863306.	−	382620.i	−1.15648	−	0.512554i
$865$	482676.			0.645094
$866$			1.81478e6i			2.41985i
$867$	258519.	+	201742.i	0.343917	+	0.268385i
$868$	243941.			0.323777
$869$			15195.4i			0.0201220i
$870$	1.12783e6	−	1.44523e6i	1.49006	−	1.90941i
$871$	−914012.			−1.20480
$872$			193792.i			0.254861i
$873$	−148508.	−	592784.i	−0.194859	−	0.777800i
$874$	−964600.			−1.26277
$875$		−	257498.i		−	0.336324i
$876$	−1.27490e6	−	994907.i	−1.66138	−	1.29651i
$877$	−1.33894e6			−1.74085			−0.870424	−	0.492304i	$-0.836155\pi$
$877$	−1.33894e6			−1.74085			−0.870424	+	0.492304i	$0.836155\pi$
$878$		−	1.57679e6i		−	2.04544i
$879$	−544991.	+	698369.i	−0.705362	+	0.903872i
$880$	−147512.			−0.190485
$881$		−	879725.i		−	1.13343i	−0.823913	−	0.566716i	$-0.808214\pi$
$881$		−	879725.i		−	1.13343i	0.823913	−	0.566716i	$-0.191786\pi$
$882$	−1.10957e6	+	277976.i	−1.42632	+	0.357330i
$883$	277527.			0.355946			0.177973	−	0.984035i	$-0.443046\pi$
$883$	277527.			0.355946			0.177973	+	0.984035i	$0.443046\pi$
$884$		−	930192.i		−	1.19033i
$885$	528766.	+	412637.i	0.675113	+	0.526843i
$886$	−2.39203e6			−3.04718
$887$			632003.i			0.803289i	0.915796	+	0.401645i	$0.131561\pi$
$887$			632003.i			0.803289i	−0.915796	+	0.401645i	$0.868439\pi$
$888$	264748.	−	339256.i	0.335742	−	0.430231i
$889$	−38941.9			−0.0492735
$890$			878753.i			1.10940i
$891$	−112851.	−	211092.i	−0.142151	−	0.265899i
$892$	1.12805e6			1.41775
$893$		−	2.09895e6i		−	2.63208i
$894$	−706174.	−	551082.i	−0.883561	−	0.689511i
$895$	957528.			1.19538
$896$			237212.i			0.295475i
$897$	275708.	−	353300.i	0.342660	−	0.439095i
$898$	1.85606e6			2.30164
$899$			699820.i			0.865898i
$900$	517471.	+	2.06554e6i	0.638854	+	2.55004i
$901$	−214353.			−0.264046
$902$			155473.i			0.191091i
$903$	119759.	+	93457.5i	0.146870	+	0.114614i
$904$	−672899.			−0.823404
$905$			1.62059e6i			1.97868i
$906$	−927593.	+	1.18865e6i	−1.13006	+	1.44809i
$907$	−461433.			−0.560911			−0.280455	−	0.959867i	$-0.590485\pi$
$907$	−461433.			−0.560911			−0.280455	+	0.959867i	$0.590485\pi$
$908$			990415.i			1.20128i
$909$	−1.23170e6	+	308574.i	−1.49066	+	0.373449i
$910$	577226.			0.697048
$911$		−	108286.i		−	0.130477i	−0.997870	−	0.0652387i	$-0.979219\pi$
$911$		−	108286.i		−	0.130477i	0.997870	−	0.0652387i	$-0.0207809\pi$
$912$	−395079.	−	308311.i	−0.475000	−	0.370680i
$913$	108748.			0.130460
$914$			680710.i			0.814835i
$915$	−662412.	+	848835.i	−0.791200	+	1.01387i
$916$	1.83573e6			2.18786
$917$			81648.8i			0.0970982i
$918$	−400088.	+	902720.i	−0.474756	+	1.07119i
$919$	1.12294e6			1.32962			0.664809	−	0.747014i	$-0.268513\pi$
$919$	1.12294e6			1.32962			0.664809	+	0.747014i	$0.268513\pi$
$920$		−	487707.i		−	0.576213i
$921$	−141411.	−	110354.i	−0.166711	−	0.130097i
$922$	615462.			0.724001
$923$		−	97029.4i		−	0.113894i
$924$	−54618.5	+	69989.8i	−0.0639729	+	0.0819768i
$925$	1.25945e6			1.47197
$926$		−	971023.i		−	1.13242i
$927$	215025.	+	858292.i	0.250224	+	0.998793i
$928$	−1.00476e6			−1.16672
$929$		−	748563.i		−	0.867355i	−0.901068	−	0.433678i	$-0.857216\pi$
$929$		−	748563.i		−	0.867355i	0.901068	−	0.433678i	$-0.142784\pi$
$930$	−1.68098e6	−	1.31180e6i	−1.94355	−	1.51670i
$931$	−1.31071e6			−1.51219
$932$			1.11061e6i			1.27858i
$933$	530766.	−	680139.i	0.609733	−	0.781330i
$934$	−820544.			−0.940606
$935$			333041.i			0.380955i
$936$	−637510.	+	159713.i	−0.727671	+	0.182301i
$937$	812345.			0.925255			0.462627	−	0.886553i	$-0.346907\pi$
$937$	812345.			0.925255			0.462627	+	0.886553i	$0.346907\pi$
$938$		−	359856.i		−	0.408999i
$939$	416734.	+	325210.i	0.472637	+	0.368835i
$940$	3.49985e6			3.96090
$941$		−	781678.i		−	0.882772i	−0.897317	−	0.441386i	$-0.854487\pi$
$941$		−	781678.i		−	0.882772i	0.897317	−	0.441386i	$-0.145513\pi$
$942$	1.60388e6	−	2.05526e6i	1.80746	−	2.31614i
$943$	182102.			0.204782
$944$		−	170257.i		−	0.191056i
$945$	−330143.	−	146320.i	−0.369690	−	0.163848i
$946$	−326442.			−0.364774
$947$			135861.i			0.151494i	0.997127	+	0.0757468i	$0.0241341\pi$
$947$			135861.i			0.151494i	−0.997127	+	0.0757468i	$0.975866\pi$
$948$	−67860.3	−	52956.6i	−0.0755090	−	0.0589256i
$949$	1.46079e6			1.62201
$950$			4.14010e6i			4.58736i
$951$	553780.	−	709631.i	0.612317	−	0.784642i
$952$	111049.			0.122529
$953$			622966.i			0.685928i	0.939348	+	0.342964i	$0.111431\pi$
$953$			622966.i			0.685928i	−0.939348	+	0.342964i	$0.888569\pi$
$954$	121376.	+	484483.i	0.133363	+	0.532331i
$955$	−363203.			−0.398238
$956$		−	1.38808e6i		−	1.51879i
$957$	−200787.	−	156690.i	−0.219236	−	0.171087i
$958$	−769228.			−0.838155
$959$			605.637i			0.000658530i
$960$	1.52520e6	−	1.95443e6i	1.65495	−	2.12070i
$961$	−109548.			−0.118620
$962$			1.28195e6i			1.38523i
$963$	−60943.3	+	15267.9i	−0.0657163	+	0.0164637i
$964$	1.54528e6			1.66285
$965$			496051.i			0.532687i
$966$	139098.	+	108549.i	0.149062	+	0.116324i
$967$	−98316.7			−0.105142			−0.0525708	−	0.998617i	$-0.516742\pi$
$967$	−98316.7			−0.105142			−0.0525708	+	0.998617i	$0.516742\pi$
$968$		−	57849.0i		−	0.0617369i
$969$	−696081.	+	891979.i	−0.741331	+	0.949964i
$970$	1.98116e6			2.10561
$971$		−	681994.i		−	0.723339i	−0.932306	−	0.361670i	$-0.882207\pi$
$971$		−	681994.i		−	0.723339i	0.932306	−	0.361670i	$-0.117793\pi$
$972$	1.33600e6	+	231690.i	1.41408	+	0.245230i
$973$	121314.			0.128140
$974$		−	1.19462e6i		−	1.25925i
$975$	−1.51638e6	−	1.18335e6i	−1.59514	−	1.24481i
$976$	273315.			0.286922
$977$			781651.i			0.818887i	0.912335	+	0.409443i	$0.134277\pi$
$977$			781651.i			0.818887i	−0.912335	+	0.409443i	$0.865723\pi$
$978$	818448.	−	1.04878e6i	0.855684	−	1.09650i
$979$	122086.			0.127380
$980$		−	2.18551e6i		−	2.27563i
$981$	87768.4	+	350336.i	0.0912011	+	0.364038i
$982$	−1.01803e6			−1.05570
$983$		−	1.82458e6i		−	1.88823i	−0.329615	−	0.944116i	$-0.606919\pi$
$983$		−	1.82458e6i		−	1.88823i	0.329615	−	0.944116i	$-0.393081\pi$
$984$	−210534.	−	164296.i	−0.217437	−	0.169683i
$985$	−3.01676e6			−3.10934
$986$			1.05063e6i			1.08068i
$987$	−236200.	+	302674.i	−0.242463	+	0.310700i
$988$	−2.48356e6			−2.54426
$989$			382355.i			0.390907i
$990$	752743.	−	188582.i	0.768027	−	0.192411i
$991$	847057.			0.862513			0.431256	−	0.902229i	$-0.358070\pi$
$991$	847057.			0.862513			0.431256	+	0.902229i	$0.358070\pi$
$992$			1.16865e6i			1.18758i
$993$	456647.	+	356358.i	0.463108	+	0.361399i
$994$	38201.5			0.0386640
$995$			1.98950e6i			2.00954i
$996$	−378991.	+	485651.i	−0.382041	+	0.489560i
$997$	−1.49155e6			−1.50054			−0.750271	−	0.661130i	$-0.770077\pi$
$997$	−1.49155e6			−1.50054			−0.750271	+	0.661130i	$0.770077\pi$
$998$			990258.i			0.994231i
$999$	324960.	−	733209.i	0.325611	−	0.734678i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.b.a.23.3	✓	14
3.2	odd	2	inner	33.5.b.a.23.12	yes	14
4.3	odd	2		528.5.i.d.353.3		14
12.11	even	2		528.5.i.d.353.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.b.a.23.3	✓	14	1.1	even	1	trivial
33.5.b.a.23.12	yes	14	3.2	odd	2	inner
528.5.i.d.353.3		14	4.3	odd	2
528.5.i.d.353.4		14	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 33 = 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	33.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-6.24203i q^{2} +(7.09522 + 5.53695i) q^{3} -22.9629 q^{4} -42.0693i q^{5} +(34.5618 - 44.2886i) q^{6} -11.7748 q^{7} +43.4628i q^{8} +(19.6843 + 78.5718i) q^{9} +O(q^{10})\)	\(q-6.24203i q^{2} +(7.09522 + 5.53695i) q^{3} -22.9629 q^{4} -42.0693i q^{5} +(34.5618 - 44.2886i) q^{6} -11.7748 q^{7} +43.4628i q^{8} +(19.6843 + 78.5718i) q^{9} -262.598 q^{10} +36.4829i q^{11} +(-162.927 - 127.145i) q^{12} +186.682 q^{13} +73.4985i q^{14} +(232.936 - 298.491i) q^{15} -96.1107 q^{16} +216.992i q^{17} +(490.447 - 122.870i) q^{18} +579.356 q^{19} +966.034i q^{20} +(-83.5446 - 65.1964i) q^{21} +227.727 q^{22} -266.732i q^{23} +(-240.652 + 308.378i) q^{24} -1144.82 q^{25} -1165.27i q^{26} +(-295.384 + 666.475i) q^{27} +270.383 q^{28} +775.677i q^{29} +(-1863.19 - 1453.99i) q^{30} +902.204 q^{31} +1295.33i q^{32} +(-202.004 + 258.854i) q^{33} +1354.47 q^{34} +495.356i q^{35} +(-452.010 - 1804.24i) q^{36} -1100.13 q^{37} -3616.36i q^{38} +(1324.55 + 1033.65i) q^{39} +1828.45 q^{40} +682.715i q^{41} +(-406.958 + 521.488i) q^{42} -1433.48 q^{43} -837.754i q^{44} +(3305.46 - 828.105i) q^{45} -1664.95 q^{46} -3622.90i q^{47} +(-681.927 - 532.161i) q^{48} -2262.35 q^{49} +7146.03i q^{50} +(-1201.47 + 1539.60i) q^{51} -4286.76 q^{52} +987.839i q^{53} +(4160.16 + 1843.79i) q^{54} +1534.81 q^{55} -511.765i q^{56} +(4110.66 + 3207.87i) q^{57} +4841.80 q^{58} +1771.46i q^{59} +(-5348.88 + 6854.22i) q^{60} -2843.76 q^{61} -5631.59i q^{62} +(-231.778 - 925.165i) q^{63} +6547.72 q^{64} -7853.57i q^{65} +(1615.77 + 1260.91i) q^{66} -4896.10 q^{67} -4982.76i q^{68} +(1476.88 - 1892.52i) q^{69} +3092.03 q^{70} -519.758i q^{71} +(-3414.95 + 855.536i) q^{72} +7825.00 q^{73} +6867.04i q^{74} +(-8122.78 - 6338.84i) q^{75} -13303.7 q^{76} -429.578i q^{77} +(6452.07 - 8267.87i) q^{78} +416.507 q^{79} +4043.31i q^{80} +(-5786.06 + 3093.26i) q^{81} +4261.53 q^{82} -2980.79i q^{83} +(1918.43 + 1497.10i) q^{84} +9128.68 q^{85} +8947.81i q^{86} +(-4294.89 + 5503.60i) q^{87} -1585.65 q^{88} -3346.39i q^{89} +(-5169.06 - 20632.8i) q^{90} -2198.14 q^{91} +6124.96i q^{92} +(6401.34 + 4995.46i) q^{93} -22614.3 q^{94} -24373.1i q^{95} +(-7172.19 + 9190.66i) q^{96} -7544.49 q^{97} +14121.7i q^{98} +(-2866.52 + 718.140i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 5 q^{3} - 100 q^{4} - 2 q^{6} + 76 q^{7} - 67 q^{9}+O(q^{10}) \) 14 * q - 5 * q^3 - 100 * q^4 - 2 * q^6 + 76 * q^7 - 67 * q^9	\( 14 q - 5 q^{3} - 100 q^{4} - 2 q^{6} + 76 q^{7} - 67 q^{9} - 156 q^{10} - 100 q^{12} - 104 q^{13} + 151 q^{15} + 356 q^{16} - 34 q^{18} + 1072 q^{19} + 718 q^{21} + 1200 q^{24} - 1060 q^{25} - 1154 q^{27} - 1808 q^{28} - 3026 q^{30} + 3310 q^{31} - 605 q^{33} - 2304 q^{34} + 2644 q^{36} - 362 q^{37} + 4264 q^{39} + 1896 q^{40} - 7364 q^{42} - 6740 q^{43} + 3611 q^{45} - 4068 q^{46} - 2956 q^{48} + 7074 q^{49} - 7046 q^{51} + 13072 q^{52} + 20512 q^{54} + 726 q^{55} + 3876 q^{57} - 7848 q^{58} - 8416 q^{60} - 3560 q^{61} - 17662 q^{63} + 12020 q^{64} + 1210 q^{66} - 16514 q^{67} + 9833 q^{69} + 13320 q^{70} + 8160 q^{72} + 12664 q^{73} - 5386 q^{75} - 43736 q^{76} + 19096 q^{78} + 3052 q^{79} - 11611 q^{81} + 10200 q^{82} - 39184 q^{84} + 34884 q^{85} + 37068 q^{87} - 7260 q^{88} - 26686 q^{90} - 45856 q^{91} + 2719 q^{93} + 6120 q^{94} - 38368 q^{96} - 27854 q^{97} + 4235 q^{99}+O(q^{100}) \) 14 * q - 5 * q^3 - 100 * q^4 - 2 * q^6 + 76 * q^7 - 67 * q^9 - 156 * q^10 - 100 * q^12 - 104 * q^13 + 151 * q^15 + 356 * q^16 - 34 * q^18 + 1072 * q^19 + 718 * q^21 + 1200 * q^24 - 1060 * q^25 - 1154 * q^27 - 1808 * q^28 - 3026 * q^30 + 3310 * q^31 - 605 * q^33 - 2304 * q^34 + 2644 * q^36 - 362 * q^37 + 4264 * q^39 + 1896 * q^40 - 7364 * q^42 - 6740 * q^43 + 3611 * q^45 - 4068 * q^46 - 2956 * q^48 + 7074 * q^49 - 7046 * q^51 + 13072 * q^52 + 20512 * q^54 + 726 * q^55 + 3876 * q^57 - 7848 * q^58 - 8416 * q^60 - 3560 * q^61 - 17662 * q^63 + 12020 * q^64 + 1210 * q^66 - 16514 * q^67 + 9833 * q^69 + 13320 * q^70 + 8160 * q^72 + 12664 * q^73 - 5386 * q^75 - 43736 * q^76 + 19096 * q^78 + 3052 * q^79 - 11611 * q^81 + 10200 * q^82 - 39184 * q^84 + 34884 * q^85 + 37068 * q^87 - 7260 * q^88 - 26686 * q^90 - 45856 * q^91 + 2719 * q^93 + 6120 * q^94 - 38368 * q^96 - 27854 * q^97 + 4235 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more