LMFDB - Embedded newform 33.5.b.a.23.6

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			23.6
Root			$-2.36054i$ of defining polynomial
Character	$\chi$	$=$	33.23
Dual form			33.5.b.a.23.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36054i q^{2} +(7.81446 - 4.46478i) q^{3} +10.4279 q^{4} +9.92884i q^{5} +(-10.5393 - 18.4463i) q^{6} -13.3317 q^{7} -62.3839i q^{8} +(41.1315 - 69.7797i) q^{9} +O(q^{10})$	\(q-2.36054i q^{2} +(7.81446 - 4.46478i) q^{3} +10.4279 q^{4} +9.92884i q^{5} +(-10.5393 - 18.4463i) q^{6} -13.3317 q^{7} -62.3839i q^{8} +(41.1315 - 69.7797i) q^{9} +23.4374 q^{10} +36.4829i q^{11} +(81.4881 - 46.5582i) q^{12} -138.667 q^{13} +31.4699i q^{14} +(44.3301 + 77.5885i) q^{15} +19.5864 q^{16} +445.449i q^{17} +(-164.717 - 97.0923i) q^{18} -151.893 q^{19} +103.537i q^{20} +(-104.180 + 59.5229i) q^{21} +86.1191 q^{22} +661.969i q^{23} +(-278.531 - 487.497i) q^{24} +526.418 q^{25} +327.328i q^{26} +(9.86893 - 728.933i) q^{27} -139.021 q^{28} -130.511i q^{29} +(183.150 - 104.643i) q^{30} -1039.18 q^{31} -1044.38i q^{32} +(162.888 + 285.094i) q^{33} +1051.50 q^{34} -132.368i q^{35} +(428.913 - 727.653i) q^{36} +1110.44 q^{37} +358.549i q^{38} +(-1083.61 + 619.118i) q^{39} +619.400 q^{40} -1278.11i q^{41} +(140.506 + 245.920i) q^{42} -1137.18 q^{43} +380.439i q^{44} +(692.831 + 408.388i) q^{45} +1562.60 q^{46} -915.602i q^{47} +(153.057 - 87.4488i) q^{48} -2223.27 q^{49} -1242.63i q^{50} +(1988.83 + 3480.94i) q^{51} -1446.00 q^{52} +1306.79i q^{53} +(-1720.67 - 23.2960i) q^{54} -362.233 q^{55} +831.681i q^{56} +(-1186.96 + 678.169i) q^{57} -308.077 q^{58} -1578.92i q^{59} +(462.268 + 809.082i) q^{60} -5837.50 q^{61} +2453.03i q^{62} +(-548.351 + 930.279i) q^{63} -2151.91 q^{64} -1376.80i q^{65} +(672.974 - 384.503i) q^{66} +7204.60 q^{67} +4645.08i q^{68} +(2955.55 + 5172.93i) q^{69} -312.459 q^{70} -9590.85i q^{71} +(-4353.13 - 2565.94i) q^{72} +10547.4 q^{73} -2621.24i q^{74} +(4113.67 - 2350.34i) q^{75} -1583.92 q^{76} -486.377i q^{77} +(1461.45 + 2557.89i) q^{78} +1913.70 q^{79} +194.470i q^{80} +(-3177.41 - 5740.28i) q^{81} -3017.02 q^{82} -10896.6i q^{83} +(-1086.37 + 620.697i) q^{84} -4422.79 q^{85} +2684.35i q^{86} +(-582.704 - 1019.87i) q^{87} +2275.95 q^{88} +6029.55i q^{89} +(964.014 - 1635.45i) q^{90} +1848.66 q^{91} +6902.93i q^{92} +(-8120.64 + 4639.72i) q^{93} -2161.31 q^{94} -1508.12i q^{95} +(-4662.92 - 8161.24i) q^{96} -199.952 q^{97} +5248.10i q^{98} +(2545.76 + 1500.59i) 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 $

$2$		−	2.36054i		−	0.590134i	−0.955477	−	0.295067i	$-0.904658\pi$
$2$		−	2.36054i		−	0.590134i	0.955477	−	0.295067i	$-0.0953420\pi$
$3$	7.81446	−	4.46478i	0.868273	−	0.496087i
$3$	7.81446	−	4.46478i	0.868273	−	0.496087i
$4$	10.4279			0.651742
$5$			9.92884i			0.397154i	0.980085	+	0.198577i	$0.0636319\pi$
$5$			9.92884i			0.397154i	−0.980085	+	0.198577i	$0.936368\pi$
$6$	−10.5393	−	18.4463i	−0.292758	−	0.512397i
$7$	−13.3317			−0.272075			−0.136037	−	0.990704i	$-0.543437\pi$
$7$	−13.3317			−0.272075			−0.136037	+	0.990704i	$0.543437\pi$
$8$		−	62.3839i		−	0.974749i
$9$	41.1315	−	69.7797i	0.507796	−	0.861478i
$10$	23.4374			0.234374
$11$			36.4829i			0.301511i
$11$			36.4829i			0.301511i
$12$	81.4881	−	46.5582i	0.565890	−	0.323321i
$13$	−138.667			−0.820515			−0.410257	−	0.911970i	$-0.634561\pi$
$13$	−138.667			−0.820515			−0.410257	+	0.911970i	$0.634561\pi$
$14$			31.4699i			0.160561i
$15$	44.3301	+	77.5885i	0.197023	+	0.344838i
$16$	19.5864			0.0765092
$17$			445.449i			1.54135i	0.637231	+	0.770673i	$0.280080\pi$
$17$			445.449i			1.54135i	−0.637231	+	0.770673i	$0.719920\pi$
$18$	−164.717	−	97.0923i	−0.508387	−	0.299668i
$19$	−151.893			−0.420756			−0.210378	−	0.977620i	$-0.567470\pi$
$19$	−151.893			−0.420756			−0.210378	+	0.977620i	$0.567470\pi$
$20$			103.537i			0.258842i
$21$	−104.180	+	59.5229i	−0.236235	+	0.134973i
$22$	86.1191			0.177932
$23$			661.969i			1.25136i	0.780080	+	0.625680i	$0.215178\pi$
$23$			661.969i			1.25136i	−0.780080	+	0.625680i	$0.784822\pi$
$24$	−278.531	−	487.497i	−0.483560	−	0.846348i
$25$	526.418			0.842269
$26$			327.328i			0.484214i
$27$	9.86893	−	728.933i	0.0135376	−	0.999908i
$28$	−139.021			−0.177322
$29$		−	130.511i		−	0.155186i	−0.996985	−	0.0775929i	$-0.975277\pi$
$29$		−	130.511i		−	0.155186i	0.996985	−	0.0775929i	$-0.0247234\pi$
$30$	183.150	−	104.643i	0.203500	−	0.116270i
$31$	−1039.18			−1.08135			−0.540677	−	0.841230i	$-0.681832\pi$
$31$	−1039.18			−1.08135			−0.540677	+	0.841230i	$0.681832\pi$
$32$		−	1044.38i		−	1.01990i
$33$	162.888	+	285.094i	0.149576	+	0.261794i
$34$	1051.50			0.909601
$35$		−	132.368i		−	0.108055i
$36$	428.913	−	727.653i	0.330952	−	0.561461i
$37$	1110.44			0.811135			0.405568	−	0.914065i	$-0.367074\pi$
$37$	1110.44			0.811135			0.405568	+	0.914065i	$0.367074\pi$
$38$			358.549i			0.248303i
$39$	−1083.61	+	619.118i	−0.712431	+	0.407046i
$40$	619.400			0.387125
$41$		−	1278.11i		−	0.760326i	−0.924919	−	0.380163i	$-0.875868\pi$
$41$		−	1278.11i		−	0.760326i	0.924919	−	0.380163i	$-0.124132\pi$
$42$	140.506	+	245.920i	0.0796520	+	0.139410i
$43$	−1137.18			−0.615023			−0.307511	−	0.951544i	$-0.599496\pi$
$43$	−1137.18			−0.615023			−0.307511	+	0.951544i	$0.599496\pi$
$44$			380.439i			0.196508i
$45$	692.831	+	408.388i	0.342139	+	0.201673i
$46$	1562.60			0.738470
$47$		−	915.602i		−	0.414487i	−0.978289	−	0.207244i	$-0.933551\pi$
$47$		−	915.602i		−	0.414487i	0.978289	−	0.207244i	$-0.0664493\pi$
$48$	153.057	−	87.4488i	0.0664309	−	0.0379552i
$49$	−2223.27			−0.925975
$50$		−	1242.63i		−	0.497052i
$51$	1988.83	+	3480.94i	0.764642	+	1.33831i
$52$	−1446.00			−0.534764
$53$			1306.79i			0.465217i	0.972570	+	0.232609i	$0.0747261\pi$
$53$			1306.79i			0.465217i	−0.972570	+	0.232609i	$0.925274\pi$
$54$	−1720.67	−	23.2960i	−0.590080	−	0.00798902i
$55$	−362.233			−0.119746
$56$			831.681i			0.265205i
$57$	−1186.96	+	678.169i	−0.365331	+	0.208732i
$58$	−308.077			−0.0915804
$59$		−	1578.92i		−	0.453581i	−0.973944	−	0.226791i	$-0.927177\pi$
$59$		−	1578.92i		−	0.453581i	0.973944	−	0.226791i	$-0.0728233\pi$
$60$	462.268	+	809.082i	0.128408	+	0.224745i
$61$	−5837.50			−1.56880			−0.784400	−	0.620255i	$-0.787029\pi$
$61$	−5837.50			−1.56880			−0.784400	+	0.620255i	$0.787029\pi$
$62$			2453.03i			0.638144i
$63$	−548.351	+	930.279i	−0.138158	+	0.234386i
$64$	−2151.91			−0.525368
$65$		−	1376.80i		−	0.325870i
$66$	672.974	−	384.503i	0.154494	−	0.0882698i
$67$	7204.60			1.60494			0.802472	−	0.596689i	$-0.203518\pi$
$67$	7204.60			1.60494			0.802472	+	0.596689i	$0.203518\pi$
$68$			4645.08i			1.00456i
$69$	2955.55	+	5172.93i	0.620783	+	1.08652i
$70$	−312.459			−0.0637672
$71$		−	9590.85i		−	1.90257i	−0.308314	−	0.951285i	$-0.599765\pi$
$71$		−	9590.85i		−	1.90257i	0.308314	−	0.951285i	$-0.400235\pi$
$72$	−4353.13	−	2565.94i	−0.839724	−	0.494973i
$73$	10547.4			1.97924			0.989621	−	0.143704i	$-0.0459013\pi$
$73$	10547.4			1.97924			0.989621	+	0.143704i	$0.0459013\pi$
$74$		−	2621.24i		−	0.478679i
$75$	4113.67	−	2350.34i	0.731319	−	0.417839i
$76$	−1583.92			−0.274225
$77$		−	486.377i		−	0.0820336i
$78$	1461.45	+	2557.89i	0.240212	+	0.420430i
$79$	1913.70			0.306634			0.153317	−	0.988177i	$-0.451004\pi$
$79$	1913.70			0.306634			0.153317	+	0.988177i	$0.451004\pi$
$80$			194.470i			0.0303859i
$81$	−3177.41	−	5740.28i	−0.484287	−	0.874909i
$82$	−3017.02			−0.448694
$83$		−	10896.6i		−	1.58173i	−0.611988	−	0.790867i	$-0.709630\pi$
$83$		−	10896.6i		−	1.58173i	0.611988	−	0.790867i	$-0.290370\pi$
$84$	−1086.37	+	620.697i	−0.153964	+	0.0879673i
$85$	−4422.79			−0.612151
$86$			2684.35i			0.362946i
$87$	−582.704	−	1019.87i	−0.0769856	−	0.134744i
$88$	2275.95			0.293898
$89$			6029.55i			0.761211i	0.924737	+	0.380606i	$0.124284\pi$
$89$			6029.55i			0.761211i	−0.924737	+	0.380606i	$0.875716\pi$
$90$	964.014	−	1635.45i	0.119014	−	0.201908i
$91$	1848.66			0.223241
$92$			6902.93i			0.815564i
$93$	−8120.64	+	4639.72i	−0.938911	+	0.536446i
$94$	−2161.31			−0.244603
$95$		−	1508.12i		−	0.167105i
$96$	−4662.92	−	8161.24i	−0.505959	−	0.885551i
$97$	−199.952			−0.0212512			−0.0106256	−	0.999944i	$-0.503382\pi$
$97$	−199.952			−0.0212512			−0.0106256	+	0.999944i	$0.503382\pi$
$98$			5248.10i			0.546450i
$99$	2545.76	+	1500.59i	0.259745	+	0.153106i
$100$	5489.42			0.548942
$101$		−	493.281i		−	0.0483561i	−0.999708	−	0.0241781i	$-0.992303\pi$
$101$		−	493.281i		−	0.0483561i	0.999708	−	0.0241781i	$-0.00769687\pi$
$102$	8216.89	−	4694.71i	0.789782	−	0.451241i
$103$	−1079.74			−0.101775			−0.0508877	−	0.998704i	$-0.516205\pi$
$103$	−1079.74			−0.101775			−0.0508877	+	0.998704i	$0.516205\pi$
$104$			8650.59i			0.799796i
$105$	−590.994	−	1034.38i	−0.0536049	−	0.0938216i
$106$	3084.74			0.274540
$107$			9363.76i			0.817867i	0.912564	+	0.408934i	$0.134099\pi$
$107$			9363.76i			0.817867i	−0.912564	+	0.408934i	$0.865901\pi$
$108$	102.912	−	7601.22i	0.00882304	−	0.651682i
$109$	−10122.9			−0.852027			−0.426014	−	0.904717i	$-0.640082\pi$
$109$	−10122.9			−0.852027			−0.426014	+	0.904717i	$0.640082\pi$
$110$			855.063i			0.0706664i
$111$	8677.52	−	4957.89i	0.704287	−	0.402393i
$112$	−261.119			−0.0208162
$113$			22239.7i			1.74169i	0.491555	+	0.870846i	$0.336429\pi$
$113$			22239.7i			1.74169i	−0.491555	+	0.870846i	$0.663571\pi$
$114$	1600.84	+	2801.87i	0.123180	+	0.215595i
$115$	−6572.59			−0.496982
$116$		−	1360.95i		−	0.101141i
$117$	−5703.57	+	9676.14i	−0.416654	+	0.706855i
$118$	−3727.09			−0.267674
$119$		−	5938.57i		−	0.419361i
$120$	4840.27	−	2765.49i	0.336130	−	0.192048i
$121$	−1331.00			−0.0909091
$122$			13779.6i			0.925802i
$123$	−5706.48	−	9987.72i	−0.377188	−	0.660171i
$124$	−10836.5			−0.704764
$125$			11432.2i			0.731664i
$126$	2195.96	+	1294.40i	0.138319	+	0.0815319i
$127$	−1308.71			−0.0811402			−0.0405701	−	0.999177i	$-0.512917\pi$
$127$	−1308.71			−0.0811402			−0.0405701	+	0.999177i	$0.512917\pi$
$128$		−	11630.4i		−	0.709862i
$129$	−8886.42	+	5077.25i	−0.534007	+	0.305105i
$130$	−3249.99			−0.192307
$131$		−	23314.1i		−	1.35855i	−0.733882	−	0.679277i	$-0.762293\pi$
$131$		−	23314.1i		−	1.35855i	0.733882	−	0.679277i	$-0.237707\pi$
$132$	1698.58	+	2972.92i	0.0974848	+	0.170622i
$133$	2024.99			0.114477
$134$		−	17006.7i		−	0.947132i
$135$	7237.46	+	97.9870i	0.397117	+	0.00537652i
$136$	27788.9			1.50243
$137$		−	19507.7i		−	1.03936i	−0.854361	−	0.519680i	$-0.826051\pi$
$137$		−	19507.7i		−	1.03936i	0.854361	−	0.519680i	$-0.173949\pi$
$138$	12210.9	−	6976.68i	0.641194	−	0.366345i
$139$	18786.5			0.972335			0.486167	−	0.873866i	$-0.338395\pi$
$139$	18786.5			0.972335			0.486167	+	0.873866i	$0.338395\pi$
$140$		−	1380.32i		−	0.0704242i
$141$	−4087.96	−	7154.93i	−0.205622	−	0.359888i
$142$	−22639.6			−1.12277
$143$		−	5058.97i		−	0.247394i
$144$	805.615	−	1366.73i	0.0388510	−	0.0659110i
$145$	1295.83			0.0616326
$146$		−	24897.5i		−	1.16802i
$147$	−17373.6	+	9926.40i	−0.803999	+	0.459364i
$148$	11579.6			0.528651
$149$			27456.7i			1.23673i	0.785890	+	0.618366i	$0.212205\pi$
$149$			27456.7i			1.23673i	−0.785890	+	0.618366i	$0.787795\pi$
$150$	−5548.07	−	9710.47i	−0.246581	−	0.431576i
$151$	32397.6			1.42089			0.710443	−	0.703755i	$-0.248495\pi$
$151$	32397.6			1.42089			0.710443	+	0.703755i	$0.248495\pi$
$152$			9475.69i			0.410132i
$153$	31083.3	+	18322.0i	1.32784	+	0.782689i
$154$	−1148.11			−0.0484108
$155$		−	10317.9i		−	0.429464i
$156$	−11299.7	+	6456.08i	−0.464321	+	0.265289i
$157$	−34439.8			−1.39721			−0.698604	−	0.715509i	$-0.746195\pi$
$157$	−34439.8			−1.39721			−0.698604	+	0.715509i	$0.746195\pi$
$158$		−	4517.37i		−	0.180955i
$159$	5834.55	+	10211.9i	0.230788	+	0.403935i
$160$	10369.5			0.405057
$161$		−	8825.15i		−	0.340463i
$162$	−13550.1	+	7500.38i	−0.516314	+	0.285794i
$163$	5011.87			0.188636			0.0943181	−	0.995542i	$-0.469933\pi$
$163$	5011.87			0.188636			0.0943181	+	0.995542i	$0.469933\pi$
$164$		−	13327.9i		−	0.495536i
$165$	−2830.65	+	1617.29i	−0.103972	+	0.0594046i
$166$	−25721.7			−0.933435
$167$			40070.8i			1.43680i	0.695633	+	0.718398i	$0.255124\pi$
$167$			40070.8i			1.43680i	−0.695633	+	0.718398i	$0.744876\pi$
$168$	3713.28	+	6499.14i	0.131564	+	0.230270i
$169$	−9332.47			−0.326756
$170$			10440.2i			0.361251i
$171$	−6247.58	+	10599.1i	−0.213658	+	0.362472i
$172$	−11858.3			−0.400836
$173$		−	10613.1i		−	0.354609i	−0.984156	−	0.177305i	$-0.943262\pi$
$173$		−	10613.1i		−	0.354609i	0.984156	−	0.177305i	$-0.0567378\pi$
$174$	−2407.45	+	1375.49i	−0.0795168	+	0.0454318i
$175$	−7018.03			−0.229160
$176$			714.567i			0.0230684i
$177$	−7049.51	−	12338.4i	−0.225016	−	0.393832i
$178$	14233.0			0.449217
$179$		−	17343.3i		−	0.541285i	−0.962680	−	0.270643i	$-0.912764\pi$
$179$		−	17343.3i		−	0.541285i	0.962680	−	0.270643i	$-0.0872362\pi$
$180$	7224.75	+	4258.61i	0.222986	+	0.131439i
$181$	27555.1			0.841096			0.420548	−	0.907270i	$-0.361838\pi$
$181$	27555.1			0.841096			0.420548	+	0.907270i	$0.361838\pi$
$182$		−	4363.83i		−	0.131742i
$183$	−45616.9	+	26063.2i	−1.36215	+	0.778261i
$184$	41296.3			1.21976
$185$			11025.4i			0.322145i
$186$	10952.2	+	19169.1i	0.316575	+	0.554083i
$187$	−16251.3			−0.464733
$188$		−	9547.78i		−	0.270139i
$189$	−131.569	+	9717.89i	−0.00368325	+	0.272050i
$190$	−3559.98			−0.0986143
$191$		−	44039.3i		−	1.20719i	−0.797293	−	0.603593i	$-0.793735\pi$
$191$		−	44039.3i		−	1.20719i	0.797293	−	0.603593i	$-0.206265\pi$
$192$	−16816.0	+	9607.80i	−0.456163	+	0.260628i
$193$	−5801.01			−0.155736			−0.0778680	−	0.996964i	$-0.524811\pi$
$193$	−5801.01			−0.155736			−0.0778680	+	0.996964i	$0.524811\pi$
$194$			471.995i			0.0125410i
$195$	−6147.12	−	10759.0i	−0.161660	−	0.282944i
$196$	−23183.9			−0.603497
$197$			63465.5i			1.63533i	0.575694	+	0.817665i	$0.304732\pi$
$197$			63465.5i			1.63533i	−0.575694	+	0.817665i	$0.695268\pi$
$198$	3542.21	−	6009.37i	0.0903532	−	0.153285i
$199$	−71174.9			−1.79730			−0.898650	−	0.438666i	$-0.855451\pi$
$199$	−71174.9			−1.79730			−0.898650	+	0.438666i	$0.855451\pi$
$200$		−	32840.0i		−	0.821001i
$201$	56300.0	−	32166.9i	1.39353	−	0.796192i
$202$	−1164.41			−0.0285366
$203$			1739.93i			0.0422221i
$204$	20739.3	+	36298.8i	0.498349	+	0.872232i
$205$	12690.1			0.301966
$206$			2548.76i			0.0600612i
$207$	46192.0	+	27227.8i	1.07802	+	0.635435i
$208$	−2715.98			−0.0627769
$209$		−	5541.50i		−	0.126863i
$210$	−2441.70	+	1395.06i	−0.0553673	+	0.0316341i
$211$	44163.4			0.991969			0.495984	−	0.868331i	$-0.334807\pi$
$211$	44163.4			0.991969			0.495984	+	0.868331i	$0.334807\pi$
$212$			13627.1i			0.303201i
$213$	−42821.1	−	74947.3i	−0.943840	−	1.65195i
$214$	22103.5			0.482651
$215$		−	11290.8i		−	0.244258i
$216$	−45473.7	−	615.663i	−0.974660	−	0.0131958i
$217$	13854.0			0.294209
$218$			23895.6i			0.502810i
$219$	82422.0	−	47091.7i	1.71852	−	0.981876i
$220$	−3777.31			−0.0780437
$221$		−	61769.1i		−	1.26470i
$222$	−11703.3	−	20483.6i	−0.237466	−	0.415624i
$223$	−51860.1			−1.04285			−0.521427	−	0.853296i	$-0.674600\pi$
$223$	−51860.1			−1.04285			−0.521427	+	0.853296i	$0.674600\pi$
$224$			13923.3i			0.277489i
$225$	21652.3	−	36733.3i	0.427701	−	0.725596i
$226$	52497.6			1.02783
$227$		−	7471.89i		−	0.145004i	−0.997368	−	0.0725018i	$-0.976902\pi$
$227$		−	7471.89i		−	0.145004i	0.997368	−	0.0725018i	$-0.0230983\pi$
$228$	−12377.5	+	7071.86i	−0.238102	+	0.136039i
$229$	9715.01			0.185256			0.0926280	−	0.995701i	$-0.470473\pi$
$229$	9715.01			0.185256			0.0926280	+	0.995701i	$0.470473\pi$
$230$			15514.8i			0.293286i
$231$	−2171.57	−	3800.77i	−0.0406958	−	0.0712276i
$232$	−8141.81			−0.151267
$233$			46529.7i			0.857075i	0.903524	+	0.428537i	$0.140971\pi$
$233$			46529.7i			0.857075i	−0.903524	+	0.428537i	$0.859029\pi$
$234$	22840.9	+	13463.5i	0.417139	+	0.245882i
$235$	9090.87			0.164615
$236$		−	16464.7i		−	0.295618i
$237$	14954.5	−	8544.27i	0.266242	−	0.152117i
$238$	−14018.2			−0.247479
$239$			35203.7i			0.616301i	0.951338	+	0.308150i	$0.0997100\pi$
$239$			35203.7i			0.616301i	−0.951338	+	0.308150i	$0.900290\pi$
$240$	868.265	+	1519.68i	0.0150740	+	0.0263833i
$241$	39215.8			0.675192			0.337596	−	0.941291i	$-0.390386\pi$
$241$	39215.8			0.675192			0.337596	+	0.941291i	$0.390386\pi$
$242$			3141.87i			0.0536485i
$243$	−50458.8	−	30670.7i	−0.854524	−	0.519412i
$244$	−60872.7			−1.02245
$245$		−	22074.5i		−	0.367754i
$246$	−23576.4	+	13470.3i	−0.389589	+	0.222591i
$247$	21062.6			0.345237
$248$			64828.3i			1.05405i
$249$	−48650.8	−	85150.7i	−0.784677	−	1.37338i
$250$	26986.2			0.431780
$251$			26453.8i			0.419894i	0.977713	+	0.209947i	$0.0673292\pi$
$251$			26453.8i			0.419894i	−0.977713	+	0.209947i	$0.932671\pi$
$252$	−5718.13	+	9700.83i	−0.0900436	+	0.152759i
$253$	−24150.5			−0.377299
$254$			3089.26i			0.0478836i
$255$	−34561.7	+	19746.8i	−0.531514	+	0.303680i
$256$	−61884.5			−0.944282
$257$			51258.5i			0.776068i	0.921645	+	0.388034i	$0.126846\pi$
$257$			51258.5i			0.776068i	−0.921645	+	0.388034i	$0.873154\pi$
$258$	11985.0	+	20976.7i	0.180053	+	0.315136i
$259$	−14804.1			−0.220689
$260$		−	14357.1i		−	0.212383i
$261$	−9107.03	−	5368.12i	−0.133689	−	0.0788027i
$262$	−55033.9			−0.801729
$263$			16153.1i			0.233531i	0.993160	+	0.116765i	$0.0372525\pi$
$263$			16153.1i			0.233531i	−0.993160	+	0.116765i	$0.962747\pi$
$264$	17785.3	−	10161.6i	0.255184	−	0.145799i
$265$	−12975.0			−0.184763
$266$		−	4780.05i		−	0.0675569i
$267$	26920.6	+	47117.7i	0.377627	+	0.660939i
$268$	75128.6			1.04601
$269$			132830.i			1.83565i	0.396984	+	0.917826i	$0.370057\pi$
$269$			132830.i			1.83565i	−0.396984	+	0.917826i	$0.629943\pi$
$270$	231.302	−	17084.3i	0.00317287	−	0.234352i
$271$	52870.0			0.719898			0.359949	−	0.932972i	$-0.382794\pi$
$271$	52870.0			0.719898			0.359949	+	0.932972i	$0.382794\pi$
$272$			8724.72i			0.117927i
$273$	14446.3	−	8253.87i	0.193834	−	0.110747i
$274$	−46048.7			−0.613361
$275$			19205.2i			0.253954i
$276$	30820.1	+	53942.6i	0.404590	+	0.708132i
$277$	113328.			1.47699			0.738495	−	0.674259i	$-0.235537\pi$
$277$	113328.			1.47699			0.738495	+	0.674259i	$0.235537\pi$
$278$		−	44346.2i		−	0.573808i
$279$	−42743.1	+	72513.8i	−0.549107	+	0.931563i
$280$	−8257.63			−0.105327
$281$		−	41838.0i		−	0.529857i	−0.964268	−	0.264928i	$-0.914652\pi$
$281$		−	41838.0i		−	0.529857i	0.964268	−	0.264928i	$-0.0853483\pi$
$282$	−16889.5	+	9649.79i	−0.212382	+	0.121344i
$283$	7318.14			0.0913751			0.0456875	−	0.998956i	$-0.485452\pi$
$283$	7318.14			0.0913751			0.0456875	+	0.998956i	$0.485452\pi$
$284$		−	100012.i		−	1.23998i
$285$	−6733.44	−	11785.2i	−0.0828986	−	0.145093i
$286$	−11941.9			−0.145996
$287$			17039.3i			0.206866i
$288$	−72876.3	−	42956.8i	−0.878621	−	0.517901i
$289$	−114904.			−1.37575
$290$		−	3058.84i		−	0.0363715i
$291$	−1562.52	+	892.743i	−0.0184518	+	0.0105424i
$292$	109987.			1.28995
$293$			41007.6i			0.477671i	0.971060	+	0.238836i	$0.0767657\pi$
$293$			41007.6i			0.477671i	−0.971060	+	0.238836i	$0.923234\pi$
$294$	23431.6	+	41011.1i	0.271086	+	0.474467i
$295$	15676.8			0.180141
$296$		−	69273.9i		−	0.790653i
$297$	26593.6	+	360.047i	0.301484	+	0.00408175i
$298$	64812.5			0.729837
$299$		−	91793.3i		−	1.02676i
$300$	42896.8	−	24509.1i	0.476631	−	0.272323i
$301$	15160.5			0.167332
$302$		−	76475.7i		−	0.838513i
$303$	−2202.39	−	3854.72i	−0.0239888	−	0.0419863i
$304$	−2975.03			−0.0321917
$305$		−	57959.6i		−	0.623054i
$306$	43249.7	−	73373.2i	0.461891	−	0.783601i
$307$	−117657.			−1.24836			−0.624180	−	0.781280i	$-0.714567\pi$
$307$	−117657.			−1.24836			−0.624180	+	0.781280i	$0.714567\pi$
$308$		−	5071.88i		−	0.0534647i
$309$	−8437.55	+	4820.79i	−0.0883689	+	0.0504895i
$310$	−24355.7			−0.253441
$311$			90102.0i			0.931566i	0.884899	+	0.465783i	$0.154227\pi$
$311$			90102.0i			0.931566i	−0.884899	+	0.465783i	$0.845773\pi$
$312$	38623.0	+	67599.7i	0.396768	+	0.694441i
$313$	−47224.3			−0.482033			−0.241016	−	0.970521i	$-0.577481\pi$
$313$	−47224.3			−0.482033			−0.241016	+	0.970521i	$0.577481\pi$
$314$			81296.3i			0.824540i
$315$	−9236.59	−	5444.48i	−0.0930873	−	0.0548701i
$316$	19955.8			0.199846
$317$		−	148932.i		−	1.48208i	−0.671463	−	0.741038i	$-0.734334\pi$
$317$		−	148932.i		−	1.48208i	0.671463	−	0.741038i	$-0.265666\pi$
$318$	24105.5	−	13772.7i	0.238376	−	0.136196i
$319$	4761.43			0.0467903
$320$		−	21366.0i		−	0.208652i
$321$	41807.2	+	73172.7i	0.405733	+	0.710132i
$322$	−20832.1			−0.200919
$323$		−	67660.6i		−	0.648531i
$324$	−33133.6	−	59858.9i	−0.315630	−	0.570215i
$325$	−72996.8			−0.691094
$326$		−	11830.7i		−	0.111321i
$327$	−79105.3	+	45196.7i	−0.739792	+	0.422680i
$328$	−79733.4			−0.741127
$329$			12206.5i			0.112771i
$330$	3817.67	+	6681.85i	0.0350567	+	0.0613577i
$331$	20576.1			0.187805			0.0939027	−	0.995581i	$-0.470066\pi$
$331$	20576.1			0.187805			0.0939027	+	0.995581i	$0.470066\pi$
$332$		−	113628.i		−	1.03088i
$333$	45674.2	−	77486.4i	0.411891	−	0.698775i
$334$	94588.5			0.847902
$335$			71533.3i			0.637409i
$336$	−2040.50	+	1165.84i	−0.0180742	+	0.0103267i
$337$	195633.			1.72259			0.861297	−	0.508102i	$-0.169653\pi$
$337$	195633.			1.72259			0.861297	+	0.508102i	$0.169653\pi$
$338$			22029.6i			0.192830i
$339$	99295.3	+	173791.i	0.864031	+	1.51226i
$340$	−46120.3			−0.398964
$341$		−	37912.3i		−	0.326041i
$342$	25019.4	+	14747.6i	0.213907	+	0.126087i
$343$	61649.1			0.524009
$344$			70941.6i			0.599493i
$345$	−51361.2	+	29345.2i	−0.431516	+	0.246546i
$346$	−25052.6			−0.209267
$347$		−	211691.i		−	1.75810i	−0.476730	−	0.879050i	$-0.658178\pi$
$347$		−	211691.i		−	1.75810i	0.476730	−	0.879050i	$-0.341822\pi$
$348$	−6076.36	−	10635.1i	−0.0501748	−	0.0878181i
$349$	−73946.3			−0.607108			−0.303554	−	0.952814i	$-0.598173\pi$
$349$	−73946.3			−0.607108			−0.303554	+	0.952814i	$0.598173\pi$
$350$			16566.3i			0.135235i
$351$	−1368.49	+	101079.i	−0.0111078	+	0.820439i
$352$	38101.9			0.307511
$353$			124295.i			0.997479i	0.866752	+	0.498740i	$0.166204\pi$
$353$			124295.i			0.997479i	−0.866752	+	0.498740i	$0.833796\pi$
$354$	−29125.2	+	16640.6i	−0.232414	+	0.132789i
$355$	95226.0			0.755612
$356$			62875.4i			0.496113i
$357$	−26514.4	−	46406.7i	−0.208040	−	0.364120i
$358$	−40939.5			−0.319431
$359$			56677.7i			0.439768i	0.975526	+	0.219884i	$0.0705678\pi$
$359$			56677.7i			0.439768i	−0.975526	+	0.219884i	$0.929432\pi$
$360$	25476.8	−	43221.5i	0.196580	−	0.333500i
$361$	−107249.			−0.822964
$362$		−	65044.9i		−	0.496359i
$363$	−10401.0	+	5942.62i	−0.0789339	+	0.0450988i
$364$	19277.6			0.145496
$365$			104723.i			0.786063i
$366$	61523.1	+	107680.i	0.459278	+	0.803849i
$367$	−151562.			−1.12527			−0.562637	−	0.826704i	$-0.690213\pi$
$367$	−151562.			−1.12527			−0.562637	+	0.826704i	$0.690213\pi$
$368$			12965.6i			0.0957406i
$369$	−89186.0	−	52570.5i	−0.655004	−	0.386090i
$370$	26025.9			0.190109
$371$		−	17421.7i		−	0.126574i
$372$	−84681.0	+	48382.4i	−0.611928	+	0.349624i
$373$	43144.4			0.310104			0.155052	−	0.987906i	$-0.450446\pi$
$373$	43144.4			0.310104			0.155052	+	0.987906i	$0.450446\pi$
$374$			38361.7i			0.274255i
$375$	51042.5	+	89336.8i	0.362969	+	0.635284i
$376$	−57118.9			−0.404021
$377$			18097.6i			0.127332i
$378$	22939.4	+	310.574i	0.160546	+	0.00217361i
$379$	−58997.4			−0.410728			−0.205364	−	0.978686i	$-0.565838\pi$
$379$	−58997.4			−0.410728			−0.205364	+	0.978686i	$0.565838\pi$
$380$		−	15726.5i		−	0.108909i
$381$	−10226.9	+	5843.10i	−0.0704518	+	0.0402526i
$382$	−103956.			−0.712401
$383$			76112.7i			0.518871i	0.965760	+	0.259435i	$0.0835365\pi$
$383$			76112.7i			0.518871i	−0.965760	+	0.259435i	$0.916464\pi$
$384$	−51927.1	−	90885.1i	−0.352153	−	0.616354i
$385$	4829.16			0.0325799
$386$			13693.5i			0.0919051i
$387$	−46773.7	+	79351.8i	−0.312306	+	0.529828i
$388$	−2085.08			−0.0138503
$389$		−	148021.i		−	0.978189i	−0.872231	−	0.489094i	$-0.837327\pi$
$389$		−	148021.i		−	0.978189i	0.872231	−	0.489094i	$-0.162673\pi$
$390$	−25396.9	+	14510.5i	−0.166975	+	0.0954010i
$391$	−294874.			−1.92878
$392$			138696.i			0.902594i
$393$	−104093.	−	182187.i	−0.673961	−	1.17960i
$394$	149813.			0.965064
$395$			19000.8i			0.121781i
$396$	26546.9	+	15648.0i	0.169287	+	0.0997857i
$397$	235920.			1.49687			0.748433	−	0.663211i	$-0.230807\pi$
$397$	235920.			1.49687			0.748433	+	0.663211i	$0.230807\pi$
$398$			168011.i			1.06065i
$399$	15824.2	−	9041.12i	0.0993974	−	0.0567906i
$400$	10310.6			0.0644413
$401$		−	161186.i		−	1.00239i	−0.865333	−	0.501197i	$-0.832893\pi$
$401$		−	161186.i		−	1.00239i	0.865333	−	0.501197i	$-0.167107\pi$
$402$	−75931.2	−	132898.i	−0.469860	−	0.822369i
$403$	144100.			0.887267
$404$		−	5143.87i		−	0.0315157i
$405$	56994.3	−	31548.0i	0.347473	−	0.192336i
$406$	4107.17			0.0249167
$407$			40512.2i			0.244566i
$408$	217155.	−	124071.i	1.30452	−	0.745334i
$409$	147844.			0.883808			0.441904	−	0.897062i	$-0.354303\pi$
$409$	147844.			0.883808			0.441904	+	0.897062i	$0.354303\pi$
$410$		−	29955.5i		−	0.178201i
$411$	−87097.8	−	152442.i	−0.515612	−	0.902448i
$412$	−11259.3			−0.0663313
$413$			21049.6i			0.123408i
$414$	64272.1	−	109038.i	0.374992	−	0.636175i
$415$	108190.			0.628191
$416$			144821.i			0.836843i
$417$	146806.	−	83877.5i	0.844252	−	0.482362i
$418$	−13080.9			−0.0748661
$419$		−	154410.i		−	0.879525i	−0.898114	−	0.439762i	$-0.855063\pi$
$419$		−	154410.i		−	0.879525i	0.898114	−	0.439762i	$-0.144937\pi$
$420$	−6162.81	−	10786.4i	−0.0349365	−	0.0611475i
$421$	186090.			1.04993			0.524963	−	0.851125i	$-0.324079\pi$
$421$	186090.			1.04993			0.524963	+	0.851125i	$0.324079\pi$
$422$		−	104249.i		−	0.585395i
$423$	−63890.4	−	37660.1i	−0.357071	−	0.210475i
$424$	81523.0			0.453470
$425$			234492.i			1.29823i
$426$	−176916.	+	101081.i	−0.974872	+	0.556992i
$427$	77823.6			0.426831
$428$			97644.1i			0.533038i
$429$	−22587.2	−	39533.1i	−0.122729	−	0.214806i
$430$	−26652.4			−0.144145
$431$		−	155392.i		−	0.836514i	−0.908329	−	0.418257i	$-0.862641\pi$
$431$		−	155392.i		−	0.836514i	0.908329	−	0.418257i	$-0.137359\pi$
$432$	193.296	−	14277.1i	0.00103575	−	0.0765022i
$433$	33432.8			0.178319			0.0891593	−	0.996017i	$-0.471582\pi$
$433$	33432.8			0.178319			0.0891593	+	0.996017i	$0.471582\pi$
$434$		−	32702.9i		−	0.173623i
$435$	10126.2	−	5785.58i	0.0535139	−	0.0305751i
$436$	−105561.			−0.555302
$437$		−	100549.i		−	0.526518i
$438$	−111162.	−	194560.i	−0.579438	−	1.01416i
$439$	−166902.			−0.866030			−0.433015	−	0.901387i	$-0.642550\pi$
$439$	−166902.			−0.866030			−0.433015	+	0.901387i	$0.642550\pi$
$440$			22597.5i			0.116723i
$441$	−91446.2	+	155139.i	−0.470206	+	0.797707i
$442$	−145808.			−0.746341
$443$		−	25594.0i		−	0.130416i	−0.997872	−	0.0652079i	$-0.979229\pi$
$443$		−	25594.0i		−	0.130416i	0.997872	−	0.0652079i	$-0.0207711\pi$
$444$	90488.0	−	51700.2i	0.459013	−	0.262257i
$445$	−59866.5			−0.302318
$446$			122418.i			0.615424i
$447$	122588.	+	214559.i	0.613526	+	1.07382i
$448$	28688.5			0.142939
$449$			32470.8i			0.161065i	0.996752	+	0.0805323i	$0.0256620\pi$
$449$			32470.8i			0.161065i	−0.996752	+	0.0805323i	$0.974338\pi$
$450$	−86710.3	−	51111.1i	−0.428199	−	0.252401i
$451$	46629.1			0.229247
$452$			231912.i			1.13513i
$453$	253170.	−	144648.i	1.23372	−	0.704882i
$454$	−17637.7			−0.0855715
$455$			18355.1i			0.0886610i
$456$	42306.9	+	74047.4i	0.203461	+	0.356107i
$457$	−234589.			−1.12325			−0.561624	−	0.827392i	$-0.689823\pi$
$457$	−234589.			−1.12325			−0.561624	+	0.827392i	$0.689823\pi$
$458$		−	22932.6i		−	0.109326i
$459$	324703.	+	4396.11i	1.54120	+	0.0208662i
$460$	−68538.1			−0.323904
$461$			96803.3i			0.455500i	0.973720	+	0.227750i	$0.0731369\pi$
$461$			96803.3i			0.455500i	−0.973720	+	0.227750i	$0.926863\pi$
$462$	−8971.86	+	5126.06i	−0.0420338	+	0.0240160i
$463$	−3642.95			−0.0169938			−0.00849691	−	0.999964i	$-0.502705\pi$
$463$	−3642.95			−0.0169938			−0.00849691	+	0.999964i	$0.502705\pi$
$464$		−	2556.24i		−	0.0118731i
$465$	−46067.0	−	80628.5i	−0.213051	−	0.372892i
$466$	109835.			0.505789
$467$		−	182139.i		−	0.835159i	−0.908640	−	0.417579i	$-0.862879\pi$
$467$		−	182139.i		−	0.835159i	0.908640	−	0.417579i	$-0.137121\pi$
$468$	−59476.1	+	100901.i	−0.271551	+	0.460687i
$469$	−96049.2			−0.436665
$470$		−	21459.3i		−	0.0971450i
$471$	−269128.	+	153766.i	−1.21316	+	0.693136i
$472$	−98499.0			−0.442128
$473$		−	41487.5i		−	0.185436i
$474$	−20169.0	−	35300.8i	−0.0897695	−	0.157118i
$475$	−79959.3			−0.354390
$476$		−	61926.7i		−	0.273315i
$477$	91187.7	+	53750.4i	0.400774	+	0.236235i
$478$	83099.6			0.363700
$479$		−	35229.6i		−	0.153545i	−0.997049	−	0.0767727i	$-0.975538\pi$
$479$		−	35229.6i		−	0.153545i	0.997049	−	0.0767727i	$-0.0244616\pi$
$480$	81031.6	−	46297.3i	0.351700	−	0.200943i
$481$	−153982.			−0.665548
$482$		−	92570.4i		−	0.398454i
$483$	−39402.4	−	68963.8i	−0.168899	−	0.295615i
$484$	−13879.5			−0.0592493
$485$		−	1985.29i		−	0.00843998i
$486$	−72399.4	+	119110.i	−0.306522	+	0.504284i
$487$	175239.			0.738880			0.369440	−	0.929255i	$-0.379550\pi$
$487$	175239.			0.738880			0.369440	+	0.929255i	$0.379550\pi$
$488$			364167.i			1.52919i
$489$	39165.1	−	22376.9i	0.163788	−	0.0935799i
$490$	−52107.6			−0.217024
$491$		−	224892.i		−	0.932847i	−0.884562	−	0.466423i	$-0.845542\pi$
$491$		−	224892.i		−	0.932847i	0.884562	−	0.466423i	$-0.154458\pi$
$492$	−59506.4	−	104151.i	−0.245829	−	0.430261i
$493$	58136.1			0.239195
$494$		−	49718.9i		−	0.203736i
$495$	−14899.2	+	25276.5i	−0.0608067	+	0.103159i
$496$	−20353.8			−0.0827336
$497$			127862.i			0.517641i
$498$	−201001.	+	114842.i	−0.810476	+	0.463065i
$499$	−94767.9			−0.380592			−0.190296	−	0.981727i	$-0.560945\pi$
$499$	−94767.9			−0.380592			−0.190296	+	0.981727i	$0.560945\pi$
$500$			119214.i			0.476856i
$501$	178907.	+	313131.i	0.712775	+	1.24753i
$502$	62445.1			0.247794
$503$			233674.i			0.923579i	0.886990	+	0.461789i	$0.152792\pi$
$503$			233674.i			0.923579i	−0.886990	+	0.461789i	$0.847208\pi$
$504$	58034.5	+	34208.3i	0.228468	+	0.134670i
$505$	4897.71			0.0192048
$506$			57008.2i			0.222657i
$507$	−72928.2	+	41667.5i	−0.283713	+	0.162099i
$508$	−13647.1			−0.0528824
$509$		−	152655.i		−	0.589216i	−0.955618	−	0.294608i	$-0.904811\pi$
$509$		−	152655.i		−	0.589216i	0.955618	−	0.294608i	$-0.0951891\pi$
$510$	46613.0	+	81584.2i	0.179212	+	0.313665i
$511$	−140614.			−0.538501
$512$		−	40005.5i		−	0.152609i
$513$	−1499.02	+	110720.i	−0.00569605	+	0.420718i
$514$	120998.			0.457984
$515$		−	10720.5i		−	0.0404205i
$516$	−92666.4	+	52944.9i	−0.348035	+	0.198849i
$517$	33403.8			0.124973
$518$			34945.5i			0.130236i
$519$	−47385.1	−	82935.6i	−0.175917	−	0.307897i
$520$	−85890.3			−0.317642
$521$			282155.i			1.03947i	0.854328	+	0.519735i	$0.173969\pi$
$521$			282155.i			1.03947i	−0.854328	+	0.519735i	$0.826031\pi$
$522$	−12671.6	+	21497.5i	−0.0465042	+	0.0788945i
$523$	403375.			1.47471			0.737354	−	0.675507i	$-0.236075\pi$
$523$	403375.			1.47471			0.737354	+	0.675507i	$0.236075\pi$
$524$		−	243117.i		−	0.885427i
$525$	−54842.1	+	31334.0i	−0.198973	+	0.113683i
$526$	38129.9			0.137814
$527$		−	462903.i		−	1.66674i
$528$	3190.38	+	5583.95i	0.0114439	+	0.0200297i
$529$	−158363.			−0.565902
$530$			30627.8i			0.109035i
$531$	−110176.	−	64943.1i	−0.390750	−	0.230327i
$532$	21116.3			0.0746096
$533$			177231.i			0.623859i
$534$	111223.	−	63547.1i	0.390043	−	0.222850i
$535$	−92971.3			−0.324819
$536$		−	449451.i		−	1.56442i
$537$	−77434.1	−	135529.i	−0.268524	−	0.469983i
$538$	313549.			1.08328
$539$		−	81111.2i		−	0.279192i
$540$	75471.3	+	1021.80i	0.258818	+	0.00350410i
$541$	−299826.			−1.02441			−0.512207	−	0.858862i	$-0.671172\pi$
$541$	−299826.			−1.02441			−0.512207	+	0.858862i	$0.671172\pi$
$542$		−	124802.i		−	0.424836i
$543$	215328.	−	123028.i	0.730301	−	0.417256i
$544$	465217.			1.57202
$545$		−	100509.i		−	0.338386i
$546$	−19483.5	−	34101.0i	−0.0653556	−	0.114388i
$547$	6015.08			0.0201033			0.0100516	−	0.999949i	$-0.496800\pi$
$547$	6015.08			0.0201033			0.0100516	+	0.999949i	$0.496800\pi$
$548$		−	203424.i		−	0.677394i
$549$	−240105.	+	407339.i	−0.796630	+	1.35149i
$550$	45334.7			0.149867
$551$			19823.8i			0.0652954i
$552$	322708.	−	184379.i	1.05909	−	0.605108i
$553$	−25512.8			−0.0834274
$554$		−	267515.i		−	0.871622i
$555$	49226.1	+	86157.7i	0.159812	+	0.279710i
$556$	195903.			0.633711
$557$		−	342983.i		−	1.10551i	−0.833344	−	0.552755i	$-0.813577\pi$
$557$		−	342983.i		−	1.10551i	0.833344	−	0.552755i	$-0.186423\pi$
$558$	171171.	+	100897.i	0.549747	+	0.324047i
$559$	157689.			0.504635
$560$		−	2592.60i		−	0.00826723i
$561$	−126995.	+	72558.3i	−0.403515	+	0.230548i
$562$	−98760.2			−0.312687
$563$		−	451877.i		−	1.42562i	−0.701357	−	0.712810i	$-0.747422\pi$
$563$		−	451877.i		−	1.42562i	0.701357	−	0.712810i	$-0.252578\pi$
$564$	−42628.8	−	74610.7i	−0.134012	−	0.234554i
$565$	−220814.			−0.691719
$566$		−	17274.7i		−	0.0539235i
$567$	42360.1	+	76527.4i	0.131762	+	0.238041i
$568$	−598315.			−1.85453
$569$			476722.i			1.47245i	0.676737	+	0.736225i	$0.263394\pi$
$569$			476722.i			1.47245i	−0.676737	+	0.736225i	$0.736606\pi$
$570$	−27819.3	+	15894.5i	−0.0856241	+	0.0489213i
$571$	484795.			1.48692			0.743458	−	0.668783i	$-0.233184\pi$
$571$	484795.			1.48692			0.743458	+	0.668783i	$0.233184\pi$
$572$		−	52754.3i		−	0.161237i
$573$	−196626.	−	344143.i	−0.598869	−	1.04817i
$574$	40221.9			0.122078
$575$			348473.i			1.05398i
$576$	−88511.1	+	150160.i	−0.266780	+	0.452593i
$577$	−57731.0			−0.173403			−0.0867016	−	0.996234i	$-0.527633\pi$
$577$	−57731.0			−0.173403			−0.0867016	+	0.996234i	$0.527633\pi$
$578$			271235.i			0.811876i
$579$	−45331.7	+	25900.2i	−0.135221	+	0.0772586i
$580$	13512.7			0.0401685
$581$			145269.i			0.430350i
$582$	2107.35	+	3688.38i	0.00622144	+	0.0108890i
$583$	−47675.6			−0.140268
$584$		−	657987.i		−	1.92926i
$585$	−96072.8	−	56629.9i	−0.280730	−	0.165476i
$586$	96799.9			0.281890
$587$		−	197193.i		−	0.572289i	−0.958186	−	0.286144i	$-0.907626\pi$
$587$		−	197193.i		−	0.572289i	0.958186	−	0.286144i	$-0.0923737\pi$
$588$	−181170.	+	103511.i	−0.524000	+	0.299387i
$589$	157845.			0.454987
$590$		−	37005.7i		−	0.106308i
$591$	283360.	+	495949.i	0.811266	+	1.41991i
$592$	21749.6			0.0620593
$593$			298456.i			0.848733i	0.905490	+	0.424367i	$0.139503\pi$
$593$			298456.i			0.848733i	−0.905490	+	0.424367i	$0.860497\pi$
$594$	849.904	−	62775.1i	0.00240878	−	0.177916i
$595$	58963.2			0.166551
$596$			286315.i			0.806030i
$597$	−556193.	+	317780.i	−1.56055	+	0.891617i
$598$	−216681.			−0.605925
$599$			193343.i			0.538859i	0.963020	+	0.269429i	$0.0868351\pi$
$599$			193343.i			0.538859i	−0.963020	+	0.269429i	$0.913165\pi$
$600$	−146624.	−	256627.i	−0.407288	−	0.712853i
$601$	−386484.			−1.07000			−0.534998	−	0.844853i	$-0.679688\pi$
$601$	−386484.			−1.07000			−0.534998	+	0.844853i	$0.679688\pi$
$602$		−	35786.8i		−	0.0987483i
$603$	296335.	−	502734.i	0.814984	−	1.38262i
$604$	337838.			0.926050
$605$		−	13215.3i		−	0.0361049i
$606$	−9099.21	+	5198.82i	−0.0247776	+	0.0141566i
$607$	−191097.			−0.518653			−0.259327	−	0.965790i	$-0.583501\pi$
$607$	−191097.			−0.518653			−0.259327	+	0.965790i	$0.583501\pi$
$608$			158634.i			0.429129i
$609$	7768.41	+	13596.6i	0.0209458	+	0.0366603i
$610$	−136816.			−0.367686
$611$			126964.i			0.340093i
$612$	324132.	+	191059.i	0.865406	+	0.510111i
$613$	−369401.			−0.983053			−0.491526	−	0.870863i	$-0.663561\pi$
$613$	−369401.			−0.983053			−0.491526	+	0.870863i	$0.663561\pi$
$614$			277733.i			0.736700i
$615$	99166.5	−	56658.7i	0.262189	−	0.149802i
$616$	−30342.1			−0.0799622
$617$			429337.i			1.12779i	0.825847	+	0.563895i	$0.190698\pi$
$617$			429337.i			1.12779i	−0.825847	+	0.563895i	$0.809302\pi$
$618$	11379.6	+	19917.1i	0.0297956	+	0.0521495i
$619$	297690.			0.776932			0.388466	−	0.921463i	$-0.373005\pi$
$619$	297690.			0.776932			0.388466	+	0.921463i	$0.373005\pi$
$620$		−	107593.i		−	0.279900i
$621$	482531.	+	6532.93i	1.25125	+	0.0169404i
$622$	212689.			0.549749
$623$		−	80383.9i		−	0.207106i
$624$	−21223.9	+	12126.3i	−0.0545075	+	0.0311428i
$625$	215502.			0.551686
$626$			111475.i			0.284464i
$627$	−24741.6	−	43303.8i	−0.0629350	−	0.110152i
$628$	−359133.			−0.910618
$629$			494646.i			1.25024i
$630$	−12851.9	+	21803.3i	−0.0323807	+	0.0549340i
$631$	−238825.			−0.599821			−0.299911	−	0.953967i	$-0.596957\pi$
$631$	−238825.			−0.599821			−0.299911	+	0.953967i	$0.596957\pi$
$632$		−	119384.i		−	0.298891i
$633$	345113.	−	197180.i	0.861300	−	0.492103i
$634$	−351560.			−0.874623
$635$		−	12994.0i		−	0.0322251i
$636$	60841.9	+	106488.i	0.150414	+	0.263262i
$637$	308294.			0.759776
$638$		−	11239.5i		−	0.0276125i
$639$	−669247.	−	394486.i	−1.63902	−	0.966117i
$640$	115476.			0.281924
$641$			88451.4i			0.215273i	0.994190	+	0.107636i	$0.0343282\pi$
$641$			88451.4i			0.215273i	−0.994190	+	0.107636i	$0.965672\pi$
$642$	172727.	−	98687.3i	0.419073	−	0.239437i
$643$	−107915.			−0.261012			−0.130506	−	0.991448i	$-0.541660\pi$
$643$	−107915.			−0.261012			−0.130506	+	0.991448i	$0.541660\pi$
$644$		−	92027.5i		−	0.221894i
$645$	−50411.2	−	88231.8i	−0.121173	−	0.212083i
$646$	−159715.			−0.382720
$647$			293113.i			0.700207i	0.936711	+	0.350103i	$0.113854\pi$
$647$			293113.i			0.700207i	−0.936711	+	0.350103i	$0.886146\pi$
$648$	−358101.	+	198219.i	−0.852817	+	0.472058i
$649$	57603.4			0.136760
$650$			172312.i			0.407838i
$651$	108262.	−	61855.2i	0.255454	−	0.145953i
$652$	52263.2			0.122942
$653$			167595.i			0.393038i	0.980500	+	0.196519i	$0.0629637\pi$
$653$			167595.i			0.393038i	−0.980500	+	0.196519i	$0.937036\pi$
$654$	106688.	+	186731.i	0.249438	+	0.436577i
$655$	231482.			0.539555
$656$		−	25033.5i		−	0.0581720i
$657$	433829.	−	735993.i	1.00505	−	1.70507i
$658$	28813.9			0.0665503
$659$			322012.i			0.741482i	0.928736	+	0.370741i	$0.120896\pi$
$659$			322012.i			0.741482i	−0.928736	+	0.370741i	$0.879104\pi$
$660$	−29517.7	+	16864.9i	−0.0677632	+	0.0387164i
$661$	−166204.			−0.380399			−0.190200	−	0.981745i	$-0.560913\pi$
$661$	−166204.			−0.380399			−0.190200	+	0.981745i	$0.560913\pi$
$662$		−	48570.7i		−	0.110830i
$663$	−275785.	−	482692.i	−0.627400	−	1.09810i
$664$	−679770.			−1.54179
$665$			20105.8i			0.0454650i
$666$	−182910.	−	107816.i	−0.412371	−	0.243071i
$667$	86394.5			0.194193
$668$			417853.i			0.936420i
$669$	−405258.	+	231544.i	−0.905482	+	0.517346i
$670$	168857.			0.376157
$671$		−	212969.i		−	0.473011i
$672$	62164.4	+	108803.i	0.137659	+	0.240936i
$673$	354661.			0.783040			0.391520	−	0.920170i	$-0.371949\pi$
$673$	354661.			0.783040			0.391520	+	0.920170i	$0.371949\pi$
$674$		−	461799.i		−	1.01656i
$675$	5195.18	−	383724.i	0.0114023	−	0.842192i
$676$	−97317.8			−0.212960
$677$		−	222358.i		−	0.485150i	−0.970133	−	0.242575i	$-0.922008\pi$
$677$		−	222358.i		−	0.485150i	0.970133	−	0.242575i	$-0.0779921\pi$
$678$	410240.	−	234390.i	0.892439	−	0.509894i
$679$	2665.70			0.00578191
$680$			275911.i			0.596694i
$681$	−33360.4	−	58388.8i	−0.0719344	−	0.125903i
$682$	−89493.4			−0.192408
$683$		−	438076.i		−	0.939092i	−0.882908	−	0.469546i	$-0.844418\pi$
$683$		−	438076.i		−	0.939092i	0.882908	−	0.469546i	$-0.155582\pi$
$684$	−65149.0	+	110526.i	−0.139250	+	0.236238i
$685$	193689.			0.412785
$686$		−	145525.i		−	0.309236i
$687$	75917.6	−	43375.4i	0.160853	−	0.0919031i
$688$	−22273.1			−0.0470549
$689$		−	181209.i		−	0.381717i
$690$	69270.3	+	121240.i	0.145495	+	0.254652i
$691$	−47579.0			−0.0996458			−0.0498229	−	0.998758i	$-0.515866\pi$
$691$	−47579.0			−0.0996458			−0.0498229	+	0.998758i	$0.515866\pi$
$692$		−	110672.i		−	0.231114i
$693$	−33939.2	−	20005.4i	−0.0706701	−	0.0416563i
$694$	−499704.			−1.03751
$695$			186528.i			0.386166i
$696$	−63623.8	+	36351.4i	−0.131341	+	0.0750417i
$697$	569332.			1.17193
$698$			174553.i			0.358275i
$699$	207745.	+	363605.i	0.425184	+	0.744175i
$700$	−73183.1			−0.149353
$701$		−	316760.i		−	0.644607i	−0.946636	−	0.322303i	$-0.895543\pi$
$701$		−	316760.i		−	0.644607i	0.946636	−	0.322303i	$-0.104457\pi$
$702$	238601.	+	3230.38i	0.484169	+	0.00655510i
$703$	−168669.			−0.341290
$704$		−	78507.8i		−	0.158405i
$705$	71040.2	−	40588.7i	0.142931	−	0.0816634i
$706$	293403.			0.588647
$707$			6576.25i			0.0131565i
$708$	−73511.4	−	128663.i	−0.146652	−	0.256677i
$709$	455144.			0.905433			0.452716	−	0.891655i	$-0.350455\pi$
$709$	455144.			0.905433			0.452716	+	0.891655i	$0.350455\pi$
$710$		−	224784.i		−	0.445912i
$711$	78713.4	−	133538.i	0.155707	−	0.264158i
$712$	376147.			0.741990
$713$		−	687907.i		−	1.35316i
$714$	−109545.	+	62588.3i	−0.214880	+	0.122771i
$715$	50229.7			0.0982536
$716$		−	180854.i		−	0.352778i
$717$	157177.	+	275098.i	0.305739	+	0.535117i
$718$	133790.			0.259522
$719$			496408.i			0.960243i	0.877202	+	0.480121i	$0.159407\pi$
$719$			496408.i			0.960243i	−0.877202	+	0.480121i	$0.840593\pi$
$720$	13570.0	+	7998.82i	0.0261768	+	0.0154298i
$721$	14394.7			0.0276905
$722$			253166.i			0.485659i
$723$	306450.	−	175090.i	0.586251	−	0.334954i
$724$	287341.			0.548177
$725$		−	68703.5i		−	0.130708i
$726$	14027.8	+	24552.0i	0.0266143	+	0.0465816i
$727$	684607.			1.29531			0.647654	−	0.761935i	$-0.275750\pi$
$727$	684607.			1.29531			0.647654	+	0.761935i	$0.275750\pi$
$728$		−	115327.i		−	0.217604i
$729$	−531246.	−	14387.6i	−0.999633	−	0.0270728i
$730$	247203.			0.463882
$731$		−	506554.i		−	0.947963i
$732$	−475687.	+	271783.i	−0.887768	+	0.507225i
$733$	−645792.			−1.20194			−0.600972	−	0.799270i	$-0.705220\pi$
$733$	−645792.			−1.20194			−0.600972	+	0.799270i	$0.705220\pi$
$734$			357768.i			0.664062i
$735$	−98557.6	−	172500.i	−0.182438	−	0.319311i
$736$	691346.			1.27626
$737$			262844.i			0.483909i
$738$	−124094.	+	210527.i	−0.227845	+	0.386540i
$739$	−963549.			−1.76435			−0.882176	−	0.470920i	$-0.843922\pi$
$739$	−963549.			−1.76435			−0.882176	+	0.470920i	$0.843922\pi$
$740$			114972.i			0.209956i
$741$	164592.	−	94039.7i	0.299760	−	0.171267i
$742$	−41124.6			−0.0746955
$743$			618795.i			1.12091i	0.828186	+	0.560453i	$0.189373\pi$
$743$			618795.i			1.12091i	−0.828186	+	0.560453i	$0.810627\pi$
$744$	289444.	+	506598.i	0.522900	+	0.915203i
$745$	−272613.			−0.491172
$746$		−	101844.i		−	0.183003i
$747$	−760359.	−	448191.i	−1.36263	−	0.803198i
$748$	−169466.			−0.302886
$749$		−	124835.i		−	0.222521i
$750$	210883.	−	120488.i	0.374903	−	0.214200i
$751$	261213.			0.463143			0.231571	−	0.972818i	$-0.425613\pi$
$751$	261213.			0.463143			0.231571	+	0.972818i	$0.425613\pi$
$752$		−	17933.3i		−	0.0317121i
$753$	118110.	+	206722.i	0.208304	+	0.364583i
$754$	42720.0			0.0751431
$755$			321671.i			0.564310i
$756$	−1371.99	+	101337.i	−0.00240053	+	0.177306i
$757$	−536053.			−0.935439			−0.467720	−	0.883877i	$-0.654924\pi$
$757$	−536053.			−0.935439			−0.467720	+	0.883877i	$0.654924\pi$
$758$			139265.i			0.242385i
$759$	−188723.	+	107827.i	−0.327599	+	0.187173i
$760$	−94082.6			−0.162885
$761$		−	565879.i		−	0.977135i	−0.872526	−	0.488568i	$-0.837520\pi$
$761$		−	565879.i		−	0.977135i	0.872526	−	0.488568i	$-0.162480\pi$
$762$	13792.9	+	24140.9i	0.0237544	+	0.0415760i
$763$	134956.			0.231815
$764$		−	459236.i		−	0.786773i
$765$	−181916.	+	308621.i	−0.310848	+	0.527354i
$766$	179667.			0.306203
$767$			218943.i			0.372170i
$768$	−483593.	+	276301.i	−0.819895	+	0.468446i
$769$	−171758.			−0.290445			−0.145223	−	0.989399i	$-0.546390\pi$
$769$	−171758.			−0.290445			−0.145223	+	0.989399i	$0.546390\pi$
$770$		−	11399.4i		−	0.0192265i
$771$	228858.	+	400558.i	0.384997	+	0.673839i
$772$	−60492.2			−0.101500
$773$		−	148705.i		−	0.248867i	−0.992228	−	0.124433i	$-0.960289\pi$
$773$		−	148705.i		−	0.248867i	0.992228	−	0.124433i	$-0.0397113\pi$
$774$	187313.	+	110411.i	0.312670	+	0.184302i
$775$	−547044.			−0.910792
$776$			12473.8i			0.0207146i
$777$	−115686.	+	66096.9i	−0.191619	+	0.109481i
$778$	−349408.			−0.577263
$779$			194136.i			0.319912i
$780$	−64101.4	−	112193.i	−0.105361	−	0.184407i
$781$	349902.			0.573646
$782$			696060.i			1.13824i
$783$	−95134.0	−	1288.01i	−0.155172	−	0.00210085i
$784$	−43545.7			−0.0708456
$785$		−	341947.i		−	0.554906i
$786$	−430060.	+	245714.i	−0.696120	+	0.397727i
$787$	−781097.			−1.26112			−0.630559	−	0.776141i	$-0.717174\pi$
$787$	−781097.			−1.26112			−0.630559	+	0.776141i	$0.717174\pi$
$788$			661810.i			1.06581i
$789$	72119.9	+	126227.i	0.115851	+	0.202768i
$790$	44852.2			0.0718670
$791$		−	296492.i		−	0.473870i
$792$	93612.9	−	158815.i	0.149240	−	0.253186i
$793$	809469.			1.28722
$794$		−	556897.i		−	0.883351i
$795$	−101392.	+	57930.3i	−0.160424	+	0.0916583i
$796$	−742202.			−1.17138
$797$			69274.9i			0.109058i	0.998512	+	0.0545292i	$0.0173658\pi$
$797$			69274.9i			0.109058i	−0.998512	+	0.0545292i	$0.982634\pi$
$798$	−21341.9	−	37353.5i	−0.0335141	−	0.0586578i
$799$	407854.			0.638868
$800$		−	549779.i		−	0.859030i
$801$	420740.	+	248004.i	0.655766	+	0.386540i
$802$	−380486.			−0.591547
$803$			384799.i			0.596764i
$804$	587089.	−	335433.i	0.908222	−	0.518911i
$805$	87623.5			0.135216
$806$		−	340154.i		−	0.523607i
$807$	593055.	+	1.03799e6i	0.910642	+	1.59385i
$808$	−30772.8			−0.0471351
$809$			497706.i			0.760459i	0.924892	+	0.380230i	$0.124155\pi$
$809$			497706.i			0.760459i	−0.924892	+	0.380230i	$0.875845\pi$
$810$	−74470.1	−	134537.i	−0.113504	−	0.205056i
$811$	−346897.			−0.527422			−0.263711	−	0.964602i	$-0.584947\pi$
$811$	−346897.			−0.527422			−0.263711	+	0.964602i	$0.584947\pi$
$812$			18143.8i			0.0275179i
$813$	413151.	−	236053.i	0.625068	−	0.357132i
$814$	95630.5			0.144327
$815$			49762.1i			0.0749175i
$816$	38954.0	+	68179.0i	0.0585021	+	0.102393i
$817$	172729.			0.258775
$818$		−	348992.i		−	0.521565i
$819$	76038.1	−	128999.i	0.113361	−	0.192317i
$820$	132331.			0.196804
$821$		−	921375.i		−	1.36694i	−0.729977	−	0.683471i	$-0.760469\pi$
$821$		−	921375.i		−	1.36694i	0.729977	−	0.683471i	$-0.239531\pi$
$822$	−359846.	+	205597.i	−0.532565	+	0.304280i
$823$	−329225.			−0.486064			−0.243032	−	0.970018i	$-0.578142\pi$
$823$	−329225.			−0.486064			−0.243032	+	0.970018i	$0.578142\pi$
$824$			67358.2i			0.0992055i
$825$	85747.2	+	150079.i	0.125983	+	0.220501i
$826$	49688.3			0.0728272
$827$			350069.i			0.511850i	0.966697	+	0.255925i	$0.0823799\pi$
$827$			350069.i			0.511850i	−0.966697	+	0.255925i	$0.917620\pi$
$828$	481684.	+	283928.i	0.702590	+	0.414140i
$829$	707766.			1.02987			0.514933	−	0.857231i	$-0.327817\pi$
$829$	707766.			1.02987			0.514933	+	0.857231i	$0.327817\pi$
$830$		−	255387.i		−	0.370717i
$831$	885597.	−	505985.i	1.28243	−	0.732715i
$832$	298399.			0.431072
$833$		−	990352.i		−	1.42725i
$834$	−197996.	−	346541.i	−0.284658	−	0.498222i
$835$	−397856.			−0.570628
$836$		−	57786.0i		−	0.0826818i
$837$	−10255.6	+	757494.i	−0.0146390	+	1.08126i
$838$	−364491.			−0.519038
$839$		−	260427.i		−	0.369966i	−0.982742	−	0.184983i	$-0.940777\pi$
$839$		−	260427.i		−	0.369966i	0.982742	−	0.184983i	$-0.0592231\pi$
$840$	−64528.9	+	36868.5i	−0.0914525	+	0.0522513i
$841$	690248.			0.975917
$842$		−	439272.i		−	0.619597i
$843$	−186798.	−	326941.i	−0.262855	−	0.460060i
$844$	460531.			0.646508
$845$		−	92660.6i		−	0.129772i
$846$	−88897.9	+	150816.i	−0.124208	+	0.210720i
$847$	17744.4			0.0247341
$848$			25595.3i			0.0355934i
$849$	57187.3	−	32673.9i	0.0793385	−	0.0453300i
$850$	553528.			0.766129
$851$			735080.i			1.01502i
$852$	−446532.	−	781541.i	−0.615140	−	1.07664i
$853$	−851668.			−1.17050			−0.585251	−	0.810852i	$-0.699004\pi$
$853$	−851668.			−1.17050			−0.585251	+	0.810852i	$0.699004\pi$
$854$		−	183705.i		−	0.251887i
$855$	−105236.	−	62031.3i	−0.143957	−	0.0848552i
$856$	584149.			0.797216
$857$		−	1.34625e6i		−	1.83301i	−0.400022	−	0.916505i	$-0.630998\pi$
$857$		−	1.34625e6i		−	1.83301i	0.400022	−	0.916505i	$-0.369002\pi$
$858$	−93319.3	+	53317.9i	−0.126764	+	0.0724266i
$859$	901853.			1.22222			0.611110	−	0.791546i	$-0.290723\pi$
$859$	901853.			1.22222			0.611110	+	0.791546i	$0.290723\pi$
$860$		−	117739.i		−	0.159193i
$861$	76076.8	+	133153.i	0.102623	+	0.179616i
$862$	−366808.			−0.493655
$863$			593241.i			0.796543i	0.917268	+	0.398272i	$0.130390\pi$
$863$			593241.i			0.796543i	−0.917268	+	0.398272i	$0.869610\pi$
$864$	−761281.	−	10306.9i	−1.01981	−	0.0138070i
$865$	105376.			0.140834
$866$		−	78919.3i		−	0.105232i
$867$	−897911.	+	513021.i	−1.19452	+	0.682491i
$868$	144468.			0.191748
$869$			69817.4i			0.0924536i
$870$	−13657.1	−	23903.2i	−0.0180434	−	0.0315804i
$871$	−999039.			−1.31688
$872$			631509.i			0.830513i
$873$	−8224.33	+	13952.6i	−0.0107913	+	0.0183074i
$874$	−237349.			−0.310716
$875$		−	152411.i		−	0.199067i
$876$	859486.	−	491066.i	1.12003	−	0.639929i
$877$	−745074.			−0.968724			−0.484362	−	0.874868i	$-0.660948\pi$
$877$	−745074.			−0.968724			−0.484362	+	0.874868i	$0.660948\pi$
$878$			393979.i			0.511074i
$879$	183090.	+	320452.i	0.236966	+	0.414749i
$880$	−7094.82			−0.00916169
$881$		−	95207.0i		−	0.122664i	−0.998117	−	0.0613320i	$-0.980465\pi$
$881$		−	95207.0i		−	0.122664i	0.998117	−	0.0613320i	$-0.0195348\pi$
$882$	366211.	+	215862.i	0.470754	+	0.277485i
$883$	789087.			1.01205			0.506027	−	0.862518i	$-0.331114\pi$
$883$	789087.			1.01205			0.506027	+	0.862518i	$0.331114\pi$
$884$		−	644120.i		−	0.824256i
$885$	122506.	−	69993.5i	0.156412	−	0.0893657i
$886$	−60415.5			−0.0769628
$887$			1.06364e6i			1.35190i	0.736946	+	0.675951i	$0.236267\pi$
$887$			1.06364e6i			1.35190i	−0.736946	+	0.675951i	$0.763733\pi$
$888$	−309293.	−	541338.i	−0.392233	−	0.686503i
$889$	17447.3			0.0220762
$890$			141317.i			0.178408i
$891$	209422.	−	115921.i	0.263795	−	0.146018i
$892$	−540790.			−0.679672
$893$			139074.i			0.174398i
$894$	506474.	−	289374.i	0.633698	−	0.362063i
$895$	172199.			0.214973
$896$			155052.i			0.193135i
$897$	−409837.	−	717315.i	−0.509362	−	0.891507i
$898$	76648.5			0.0950497
$899$			135625.i			0.167811i
$900$	225788.	−	383050.i	0.278750	−	0.472901i
$901$	−582110.			−0.717061
$902$		−	110070.i		−	0.135286i
$903$	118471.	−	67688.1i	0.145290	−	0.0830112i
$904$	1.38740e6			1.69771
$905$			273591.i			0.334044i
$906$	−341447.	−	597616.i	−0.415975	−	0.728058i
$907$	−1.32897e6			−1.61548			−0.807740	−	0.589539i	$-0.799310\pi$
$907$	−1.32897e6			−1.61548			−0.807740	+	0.589539i	$0.799310\pi$
$908$		−	77915.9i		−	0.0945049i
$909$	−34421.0	−	20289.4i	−0.0416577	−	0.0245550i
$910$	43327.8			0.0523219
$911$		−	1.12314e6i		−	1.35331i	−0.736302	−	0.676653i	$-0.763430\pi$
$911$		−	1.12314e6i		−	1.35331i	0.736302	−	0.676653i	$-0.236570\pi$
$912$	−23248.3	+	13282.9i	−0.0279512	+	0.0159699i
$913$	397538.			0.476911
$914$			553757.i			0.662867i
$915$	−258777.	−	452923.i	−0.309089	−	0.540981i
$916$	101307.			0.120739
$917$			310816.i			0.369628i
$918$	10377.2	−	766472.i	0.0123138	−	0.909518i
$919$	−154862.			−0.183364			−0.0916821	−	0.995788i	$-0.529224\pi$
$919$	−154862.			−0.183364			−0.0916821	+	0.995788i	$0.529224\pi$
$920$			410024.i			0.484433i
$921$	−919424.	+	525312.i	−1.08392	+	0.619295i
$922$	228508.			0.268806
$923$			1.32993e6i			1.56109i
$924$	−22644.8	−	39634.0i	−0.0265231	−	0.0464220i
$925$	584558.			0.683194
$926$			8599.31i			0.0100286i
$927$	−44411.1	+	75343.6i	−0.0516811	+	0.0876773i
$928$	−136303.			−0.158274
$929$			992052.i			1.14948i	0.818335	+	0.574742i	$0.194898\pi$
$929$			992052.i			1.14948i	−0.818335	+	0.574742i	$0.805102\pi$
$930$	−190327.	+	108743.i	−0.220056	+	0.125729i
$931$	337699.			0.389610
$932$			485206.i			0.558592i
$933$	402286.	+	704098.i	0.462138	+	0.808853i
$934$	−429945.			−0.492856
$935$		−	161356.i		−	0.184571i
$936$	603635.	+	355811.i	0.689006	+	0.406133i
$937$	294602.			0.335550			0.167775	−	0.985825i	$-0.446342\pi$
$937$	294602.			0.335550			0.167775	+	0.985825i	$0.446342\pi$
$938$			226728.i			0.257691i
$939$	−369032.	+	210846.i	−0.418536	+	0.239130i
$940$	94798.4			0.107287
$941$		−	1.13953e6i		−	1.28690i	−0.765488	−	0.643451i	$-0.777502\pi$
$941$		−	1.13953e6i		−	1.28690i	0.765488	−	0.643451i	$-0.222498\pi$
$942$	362970.	+	635286.i	0.409043	+	0.715925i
$943$	846069.			0.951442
$944$		−	30925.2i		−	0.0347031i
$945$	−96487.4	−	1306.33i	−0.108046	−	0.00146281i
$946$	−97932.7			−0.109432
$947$			1.04095e6i			1.16073i	0.814356	+	0.580366i	$0.197090\pi$
$947$			1.04095e6i			1.16073i	−0.814356	+	0.580366i	$0.802910\pi$
$948$	155944.	−	89098.5i	0.173521	−	0.0991411i
$949$	−1.46257e6			−1.62400
$950$			188747.i			0.209138i
$951$	−664950.	−	1.16382e6i	−0.735238	−	1.28685i
$952$	−370472.			−0.408772
$953$			1.41370e6i			1.55658i	0.627905	+	0.778290i	$0.283913\pi$
$953$			1.41370e6i			1.55658i	−0.627905	+	0.778290i	$0.716087\pi$
$954$	126880.	−	215252.i	0.139410	−	0.236510i
$955$	437259.			0.479438
$956$			367100.i			0.401669i
$957$	37208.0	−	21258.7i	0.0406267	−	0.0232120i
$958$	−83160.7			−0.0906123
$959$			260070.i			0.282783i
$960$	−95394.3	−	166963.i	−0.103509	−	0.181167i
$961$	156378.			0.169328
$962$			363480.i			0.392763i
$963$	653400.	+	385145.i	0.704574	+	0.415310i
$964$	408938.			0.440051
$965$		−	57597.3i		−	0.0618511i
$966$	−162791.	+	93010.7i	−0.174453	+	0.0996733i
$967$	−1.01698e6			−1.08758			−0.543788	−	0.839222i	$-0.683011\pi$
$967$	−1.01698e6			−1.08758			−0.543788	+	0.839222i	$0.683011\pi$
$968$			83033.0i			0.0886136i
$969$	−302090.	−	528731.i	−0.321728	−	0.563102i
$970$	−4686.36			−0.00498072
$971$			1.67536e6i			1.77693i	0.458944	+	0.888465i	$0.348228\pi$
$971$			1.67536e6i			1.77693i	−0.458944	+	0.888465i	$0.651772\pi$
$972$	−526178.	−	319830.i	−0.556929	−	0.338522i
$973$	−250455.			−0.264548
$974$		−	413659.i		−	0.436038i
$975$	−570430.	+	325915.i	−0.600058	+	0.342843i
$976$	−114335.			−0.120028
$977$		−	468877.i		−	0.491213i	−0.969370	−	0.245607i	$-0.921013\pi$
$977$		−	468877.i		−	0.491213i	0.969370	−	0.245607i	$-0.0789871\pi$
$978$	−52821.5	−	92450.5i	−0.0552247	−	0.0966567i
$979$	−219975.			−0.229514
$980$		−	230190.i		−	0.239681i
$981$	−416371.	+	706375.i	−0.432656	+	0.734003i
$982$	−530865.			−0.550505
$983$		−	1.72528e6i		−	1.78547i	−0.450580	−	0.892736i	$-0.648783\pi$
$983$		−	1.72528e6i		−	1.78547i	0.450580	−	0.892736i	$-0.351217\pi$
$984$	−623074.	+	355992.i	−0.643501	+	0.367664i
$985$	−630139.			−0.649477
$986$		−	137232.i		−	0.141157i
$987$	54499.3	+	95387.1i	0.0559444	+	0.0979164i
$988$	219638.			0.225005
$989$		−	752776.i		−	0.769615i
$990$	59666.0	+	35170.0i	0.0608775	+	0.0358841i
$991$	724559.			0.737779			0.368889	−	0.929473i	$-0.379738\pi$
$991$	724559.			0.737779			0.368889	+	0.929473i	$0.379738\pi$
$992$			1.08530e6i			1.10287i
$993$	160791.	−	91868.0i	0.163066	−	0.0931678i
$994$	301823.			0.305478
$995$		−	706684.i		−	0.713804i
$996$	−507324.	−	887940.i	−0.511407	−	0.895087i
$997$	1.67803e6			1.68814			0.844071	−	0.536232i	$-0.180153\pi$
$997$	1.67803e6			1.68814			0.844071	+	0.536232i	$0.180153\pi$
$998$			223703.i			0.224601i
$999$	10958.9	−	809440.i	0.0109808	−	0.811061i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	33.5.b.a.23.6	✓	14
3.2	odd	2	inner	33.5.b.a.23.9	yes	14
4.3	odd	2		528.5.i.d.353.2		14
12.11	even	2		528.5.i.d.353.1		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.b.a.23.6	✓	14	1.1	even	1	trivial
33.5.b.a.23.9	yes	14	3.2	odd	2	inner
528.5.i.d.353.1		14	12.11	even	2
528.5.i.d.353.2		14	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-2.36054i q^{2} +(7.81446 - 4.46478i) q^{3} +10.4279 q^{4} +9.92884i q^{5} +(-10.5393 - 18.4463i) q^{6} -13.3317 q^{7} -62.3839i q^{8} +(41.1315 - 69.7797i) q^{9} +O(q^{10})\)	\(q-2.36054i q^{2} +(7.81446 - 4.46478i) q^{3} +10.4279 q^{4} +9.92884i q^{5} +(-10.5393 - 18.4463i) q^{6} -13.3317 q^{7} -62.3839i q^{8} +(41.1315 - 69.7797i) q^{9} +23.4374 q^{10} +36.4829i q^{11} +(81.4881 - 46.5582i) q^{12} -138.667 q^{13} +31.4699i q^{14} +(44.3301 + 77.5885i) q^{15} +19.5864 q^{16} +445.449i q^{17} +(-164.717 - 97.0923i) q^{18} -151.893 q^{19} +103.537i q^{20} +(-104.180 + 59.5229i) q^{21} +86.1191 q^{22} +661.969i q^{23} +(-278.531 - 487.497i) q^{24} +526.418 q^{25} +327.328i q^{26} +(9.86893 - 728.933i) q^{27} -139.021 q^{28} -130.511i q^{29} +(183.150 - 104.643i) q^{30} -1039.18 q^{31} -1044.38i q^{32} +(162.888 + 285.094i) q^{33} +1051.50 q^{34} -132.368i q^{35} +(428.913 - 727.653i) q^{36} +1110.44 q^{37} +358.549i q^{38} +(-1083.61 + 619.118i) q^{39} +619.400 q^{40} -1278.11i q^{41} +(140.506 + 245.920i) q^{42} -1137.18 q^{43} +380.439i q^{44} +(692.831 + 408.388i) q^{45} +1562.60 q^{46} -915.602i q^{47} +(153.057 - 87.4488i) q^{48} -2223.27 q^{49} -1242.63i q^{50} +(1988.83 + 3480.94i) q^{51} -1446.00 q^{52} +1306.79i q^{53} +(-1720.67 - 23.2960i) q^{54} -362.233 q^{55} +831.681i q^{56} +(-1186.96 + 678.169i) q^{57} -308.077 q^{58} -1578.92i q^{59} +(462.268 + 809.082i) q^{60} -5837.50 q^{61} +2453.03i q^{62} +(-548.351 + 930.279i) q^{63} -2151.91 q^{64} -1376.80i q^{65} +(672.974 - 384.503i) q^{66} +7204.60 q^{67} +4645.08i q^{68} +(2955.55 + 5172.93i) q^{69} -312.459 q^{70} -9590.85i q^{71} +(-4353.13 - 2565.94i) q^{72} +10547.4 q^{73} -2621.24i q^{74} +(4113.67 - 2350.34i) q^{75} -1583.92 q^{76} -486.377i q^{77} +(1461.45 + 2557.89i) q^{78} +1913.70 q^{79} +194.470i q^{80} +(-3177.41 - 5740.28i) q^{81} -3017.02 q^{82} -10896.6i q^{83} +(-1086.37 + 620.697i) q^{84} -4422.79 q^{85} +2684.35i q^{86} +(-582.704 - 1019.87i) q^{87} +2275.95 q^{88} +6029.55i q^{89} +(964.014 - 1635.45i) q^{90} +1848.66 q^{91} +6902.93i q^{92} +(-8120.64 + 4639.72i) q^{93} -2161.31 q^{94} -1508.12i q^{95} +(-4662.92 - 8161.24i) q^{96} -199.952 q^{97} +5248.10i q^{98} +(2545.76 + 1500.59i) 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