LMFDB - Embedded newform 21.5.b.a.8.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,5,Mod(8,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, 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("21.8");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	21.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:	$2.17076922476$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 82x^{6} + 2017x^{4} + 13020x^{2} + 756 $ x^8 + 82x^6 + 2017x^4 + 13020*x^2 + 756
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3^{3}\cdot 7^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			8.5
Root			$0.242064i$ of defining polynomial
Character	$\chi$	$=$	21.8
Dual form			21.5.b.a.8.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.242064i q^{2} +(-4.46977 - 7.81161i) q^{3} +15.9414 q^{4} -30.4863i q^{5} +(1.89091 - 1.08197i) q^{6} +18.5203 q^{7} +7.73187i q^{8} +(-41.0424 + 69.8321i) q^{9} +O(q^{10})$	\(q+0.242064i q^{2} +(-4.46977 - 7.81161i) q^{3} +15.9414 q^{4} -30.4863i q^{5} +(1.89091 - 1.08197i) q^{6} +18.5203 q^{7} +7.73187i q^{8} +(-41.0424 + 69.8321i) q^{9} +7.37963 q^{10} +88.0887i q^{11} +(-71.2543 - 124.528i) q^{12} -15.4739 q^{13} +4.48309i q^{14} +(-238.147 + 136.266i) q^{15} +253.191 q^{16} +331.481i q^{17} +(-16.9039 - 9.93490i) q^{18} +298.104 q^{19} -485.994i q^{20} +(-82.7812 - 144.673i) q^{21} -21.3231 q^{22} -815.926i q^{23} +(60.3984 - 34.5597i) q^{24} -304.412 q^{25} -3.74568i q^{26} +(728.951 + 8.47391i) q^{27} +295.239 q^{28} +1565.67i q^{29} +(-32.9852 - 57.6468i) q^{30} -1379.33 q^{31} +184.998i q^{32} +(688.115 - 393.736i) q^{33} -80.2398 q^{34} -564.613i q^{35} +(-654.273 + 1113.22i) q^{36} -556.508 q^{37} +72.1603i q^{38} +(69.1647 + 120.876i) q^{39} +235.716 q^{40} +66.1756i q^{41} +(35.0202 - 20.0384i) q^{42} -1119.70 q^{43} +1404.26i q^{44} +(2128.92 + 1251.23i) q^{45} +197.506 q^{46} -3163.36i q^{47} +(-1131.70 - 1977.83i) q^{48} +343.000 q^{49} -73.6872i q^{50} +(2589.40 - 1481.64i) q^{51} -246.676 q^{52} -212.973i q^{53} +(-2.05123 + 176.453i) q^{54} +2685.50 q^{55} +143.196i q^{56} +(-1332.45 - 2328.67i) q^{57} -378.993 q^{58} +616.765i q^{59} +(-3796.39 + 2172.28i) q^{60} -3132.18 q^{61} -333.886i q^{62} +(-760.116 + 1293.31i) q^{63} +4006.27 q^{64} +471.741i q^{65} +(95.3094 + 166.568i) q^{66} +7028.51 q^{67} +5284.28i q^{68} +(-6373.69 + 3647.00i) q^{69} +136.673 q^{70} -2049.88i q^{71} +(-539.933 - 317.335i) q^{72} -6134.03 q^{73} -134.711i q^{74} +(1360.65 + 2377.94i) q^{75} +4752.19 q^{76} +1631.43i q^{77} +(-29.2598 + 16.7423i) q^{78} -7314.45 q^{79} -7718.84i q^{80} +(-3192.04 - 5732.15i) q^{81} -16.0188 q^{82} +6100.56i q^{83} +(-1319.65 - 2306.29i) q^{84} +10105.6 q^{85} -271.040i q^{86} +(12230.4 - 6998.18i) q^{87} -681.091 q^{88} +6802.07i q^{89} +(-302.878 + 515.335i) q^{90} -286.581 q^{91} -13007.0i q^{92} +(6165.27 + 10774.8i) q^{93} +765.735 q^{94} -9088.06i q^{95} +(1445.14 - 826.900i) q^{96} -9391.65 q^{97} +83.0281i q^{98} +(-6151.42 - 3615.37i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9}+O(q^{10}) $ 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9	$ 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9} - 4 q^{10} + 98 q^{12} + 420 q^{13} + 76 q^{15} - 444 q^{16} - 712 q^{18} - 372 q^{19} + 98 q^{21} - 16 q^{22} + 1146 q^{24} + 1056 q^{25} - 1862 q^{27} + 392 q^{28} + 2348 q^{30} - 2776 q^{31} + 1396 q^{33} + 2928 q^{34} - 3268 q^{36} - 2560 q^{37} - 2540 q^{39} - 1980 q^{40} - 2450 q^{42} + 4720 q^{43} + 9700 q^{45} + 7536 q^{46} - 2962 q^{48} + 2744 q^{49} + 4764 q^{51} - 20252 q^{52} + 4886 q^{54} + 184 q^{55} - 14144 q^{57} - 7504 q^{58} - 13828 q^{60} + 972 q^{61} - 6076 q^{63} + 22772 q^{64} + 36020 q^{66} + 10200 q^{67} - 5760 q^{69} + 10780 q^{70} + 14304 q^{72} - 32008 q^{73} + 2114 q^{75} + 17332 q^{76} - 29668 q^{78} - 23168 q^{79} - 17216 q^{81} - 31976 q^{82} - 14798 q^{84} + 32016 q^{85} + 50764 q^{87} + 29208 q^{88} - 24352 q^{90} + 11956 q^{91} + 31848 q^{93} - 64992 q^{94} + 28630 q^{96} + 28112 q^{97} - 32432 q^{99}+O(q^{100}) $ 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9 - 4 * q^10 + 98 * q^12 + 420 * q^13 + 76 * q^15 - 444 * q^16 - 712 * q^18 - 372 * q^19 + 98 * q^21 - 16 * q^22 + 1146 * q^24 + 1056 * q^25 - 1862 * q^27 + 392 * q^28 + 2348 * q^30 - 2776 * q^31 + 1396 * q^33 + 2928 * q^34 - 3268 * q^36 - 2560 * q^37 - 2540 * q^39 - 1980 * q^40 - 2450 * q^42 + 4720 * q^43 + 9700 * q^45 + 7536 * q^46 - 2962 * q^48 + 2744 * q^49 + 4764 * q^51 - 20252 * q^52 + 4886 * q^54 + 184 * q^55 - 14144 * q^57 - 7504 * q^58 - 13828 * q^60 + 972 * q^61 - 6076 * q^63 + 22772 * q^64 + 36020 * q^66 + 10200 * q^67 - 5760 * q^69 + 10780 * q^70 + 14304 * q^72 - 32008 * q^73 + 2114 * q^75 + 17332 * q^76 - 29668 * q^78 - 23168 * q^79 - 17216 * q^81 - 31976 * q^82 - 14798 * q^84 + 32016 * q^85 + 50764 * q^87 + 29208 * q^88 - 24352 * q^90 + 11956 * q^91 + 31848 * q^93 - 64992 * q^94 + 28630 * q^96 + 28112 * q^97 - 32432 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$.

$n$	$8$	$10$
$\chi(n)$	$-1$	$1$

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 $

$2$			0.242064i			0.0605161i	0.999542	+	0.0302580i	$0.00963290\pi$
$2$			0.242064i			0.0605161i	−0.999542	+	0.0302580i	$0.990367\pi$
$3$	−4.46977	−	7.81161i	−0.496641	−	0.867956i
$3$	−4.46977	−	7.81161i	−0.496641	−	0.867956i
$4$	15.9414			0.996338
$5$		−	30.4863i		−	1.21945i	−0.792613	−	0.609725i	$-0.791280\pi$
$5$		−	30.4863i		−	1.21945i	0.792613	−	0.609725i	$-0.208720\pi$
$6$	1.89091	−	1.08197i	0.0525253	−	0.0300547i
$7$	18.5203			0.377964
$7$	18.5203			0.377964
$8$			7.73187i			0.120811i
$9$	−41.0424	+	69.8321i	−0.506696	+	0.862125i
$10$	7.37963			0.0737963
$11$			88.0887i			0.728006i	0.931398	+	0.364003i	$0.118590\pi$
$11$			88.0887i			0.728006i	−0.931398	+	0.364003i	$0.881410\pi$
$12$	−71.2543	−	124.528i	−0.494822	−	0.864778i
$13$	−15.4739			−0.0915615			−0.0457808	−	0.998952i	$-0.514578\pi$
$13$	−15.4739			−0.0915615			−0.0457808	+	0.998952i	$0.514578\pi$
$14$			4.48309i			0.0228729i
$15$	−238.147	+	136.266i	−1.05843	+	0.605628i
$16$	253.191			0.989027
$17$			331.481i			1.14699i	0.819207	+	0.573497i	$0.194414\pi$
$17$			331.481i			1.14699i	−0.819207	+	0.573497i	$0.805586\pi$
$18$	−16.9039	−	9.93490i	−0.0521724	−	0.0306633i
$19$	298.104			0.825772			0.412886	−	0.910783i	$-0.364521\pi$
$19$	298.104			0.825772			0.412886	+	0.910783i	$0.364521\pi$
$20$		−	485.994i		−	1.21498i
$21$	−82.7812	−	144.673i	−0.187713	−	0.328057i
$22$	−21.3231			−0.0440561
$23$		−	815.926i		−	1.54239i	−0.636598	−	0.771196i	$-0.719659\pi$
$23$		−	815.926i		−	1.54239i	0.636598	−	0.771196i	$-0.280341\pi$
$24$	60.3984	−	34.5597i	0.104858	−	0.0599994i
$25$	−304.412			−0.487058
$26$		−	3.74568i		−	0.00554095i
$27$	728.951	+	8.47391i	0.999932	+	0.0116240i
$28$	295.239			0.376580
$29$			1565.67i			1.86168i	0.365430	+	0.930839i	$0.380922\pi$
$29$			1565.67i			1.86168i	−0.365430	+	0.930839i	$0.619078\pi$
$30$	−32.9852	−	57.6468i	−0.0366503	−	0.0640520i
$31$	−1379.33			−1.43530			−0.717652	−	0.696402i	$-0.754783\pi$
$31$	−1379.33			−1.43530			−0.717652	+	0.696402i	$0.754783\pi$
$32$			184.998i			0.180663i
$33$	688.115	−	393.736i	0.631878	−	0.361557i
$34$	−80.2398			−0.0694116
$35$		−	564.613i		−	0.460909i
$36$	−654.273	+	1113.22i	−0.504841	+	0.858967i
$37$	−556.508			−0.406507			−0.203254	−	0.979126i	$-0.565152\pi$
$37$	−556.508			−0.406507			−0.203254	+	0.979126i	$0.565152\pi$
$38$			72.1603i			0.0499725i
$39$	69.1647	+	120.876i	0.0454732	+	0.0794714i
$40$	235.716			0.147322
$41$			66.1756i			0.0393668i	0.999806	+	0.0196834i	$0.00626583\pi$
$41$			66.1756i			0.0393668i	−0.999806	+	0.0196834i	$0.993734\pi$
$42$	35.0202	−	20.0384i	0.0198527	−	0.0113596i
$43$	−1119.70			−0.605572			−0.302786	−	0.953059i	$-0.597917\pi$
$43$	−1119.70			−0.605572			−0.302786	+	0.953059i	$0.597917\pi$
$44$			1404.26i			0.725340i
$45$	2128.92	+	1251.23i	1.05132	+	0.617891i
$46$	197.506			0.0933395
$47$		−	3163.36i		−	1.43203i	−0.698085	−	0.716015i	$-0.745964\pi$
$47$		−	3163.36i		−	1.43203i	0.698085	−	0.716015i	$-0.254036\pi$
$48$	−1131.70	−	1977.83i	−0.491191	−	0.858432i
$49$	343.000			0.142857
$50$		−	73.6872i		−	0.0294749i
$51$	2589.40	−	1481.64i	0.995541	−	0.569644i
$52$	−246.676			−0.0912262
$53$		−	212.973i		−	0.0758181i	−0.999281	−	0.0379090i	$-0.987930\pi$
$53$		−	212.973i		−	0.0758181i	0.999281	−	0.0379090i	$-0.0120697\pi$
$54$	−2.05123	+	176.453i	−0.000703440	+	0.0605120i
$55$	2685.50			0.887767
$56$			143.196i			0.0456621i
$57$	−1332.45	−	2328.67i	−0.410112	−	0.716734i
$58$	−378.993			−0.112661
$59$			616.765i			0.177180i	0.996068	+	0.0885902i	$0.0282362\pi$
$59$			616.765i			0.177180i	−0.996068	+	0.0885902i	$0.971764\pi$
$60$	−3796.39	+	2172.28i	−1.05455	+	0.603410i
$61$	−3132.18			−0.841757			−0.420879	−	0.907117i	$-0.638278\pi$
$61$	−3132.18			−0.841757			−0.420879	+	0.907117i	$0.638278\pi$
$62$		−	333.886i		−	0.0868589i
$63$	−760.116	+	1293.31i	−0.191513	+	0.325852i
$64$	4006.27			0.978094
$65$			471.741i			0.111655i
$66$	95.3094	+	166.568i	0.0218800	+	0.0382388i
$67$	7028.51			1.56572			0.782859	−	0.622199i	$-0.213761\pi$
$67$	7028.51			1.56572			0.782859	+	0.622199i	$0.213761\pi$
$68$			5284.28i			1.14279i
$69$	−6373.69	+	3647.00i	−1.33873	+	0.766015i
$70$	136.673			0.0278924
$71$		−	2049.88i		−	0.406642i	−0.979112	−	0.203321i	$-0.934827\pi$
$71$		−	2049.88i		−	0.406642i	0.979112	−	0.203321i	$-0.0651734\pi$
$72$	−539.933	−	317.335i	−0.104154	−	0.0612142i
$73$	−6134.03			−1.15107			−0.575533	−	0.817779i	$-0.695205\pi$
$73$	−6134.03			−1.15107			−0.575533	+	0.817779i	$0.695205\pi$
$74$		−	134.711i		−	0.0246002i
$75$	1360.65	+	2377.94i	0.241893	+	0.422745i
$76$	4752.19			0.822748
$77$			1631.43i			0.275160i
$78$	−29.2598	+	16.7423i	−0.00480930	+	0.00275186i
$79$	−7314.45			−1.17200			−0.586000	−	0.810311i	$-0.699298\pi$
$79$	−7314.45			−1.17200			−0.586000	+	0.810311i	$0.699298\pi$
$80$		−	7718.84i		−	1.20607i
$81$	−3192.04	−	5732.15i	−0.486518	−	0.873671i
$82$	−16.0188			−0.00238233
$83$			6100.56i			0.885551i	0.896633	+	0.442775i	$0.146006\pi$
$83$			6100.56i			0.885551i	−0.896633	+	0.442775i	$0.853994\pi$
$84$	−1319.65	−	2306.29i	−0.187025	−	0.326855i
$85$	10105.6			1.39870
$86$		−	271.040i		−	0.0366468i
$87$	12230.4	−	6998.18i	1.61585	−	0.924585i
$88$	−681.091			−0.0879508
$89$			6802.07i			0.858739i	0.903129	+	0.429369i	$0.141264\pi$
$89$			6802.07i			0.858739i	−0.903129	+	0.429369i	$0.858736\pi$
$90$	−302.878	+	515.335i	−0.0373923	+	0.0636216i
$91$	−286.581			−0.0346070
$92$		−	13007.0i		−	1.53674i
$93$	6165.27	+	10774.8i	0.712830	+	1.24578i
$94$	765.735			0.0866609
$95$		−	9088.06i		−	1.00699i
$96$	1445.14	−	826.900i	0.156807	−	0.0897244i
$97$	−9391.65			−0.998156			−0.499078	−	0.866557i	$-0.666328\pi$
$97$	−9391.65			−0.998156			−0.499078	+	0.866557i	$0.666328\pi$
$98$			83.0281i			0.00864515i
$99$	−6151.42	−	3615.37i	−0.627632	−	0.368878i
$100$	−4852.75			−0.485275
$101$			8054.19i			0.789549i	0.918778	+	0.394774i	$0.129177\pi$
$101$			8054.19i			0.789549i	−0.918778	+	0.394774i	$0.870823\pi$
$102$	358.653	+	626.802i	0.0344726	+	0.0602463i
$103$	−3170.08			−0.298810			−0.149405	−	0.988776i	$-0.547736\pi$
$103$	−3170.08			−0.298810			−0.149405	+	0.988776i	$0.547736\pi$
$104$		−	119.642i		−	0.0110616i
$105$	−4410.54	+	2523.69i	−0.400049	+	0.228906i
$106$	51.5532			0.00458821
$107$		−	3148.48i		−	0.275000i	−0.990502	−	0.137500i	$-0.956093\pi$
$107$		−	3148.48i		−	0.275000i	0.990502	−	0.137500i	$-0.0439067\pi$
$108$	11620.5	+	135.086i	0.996270	+	0.0115814i
$109$	4292.91			0.361326			0.180663	−	0.983545i	$-0.442176\pi$
$109$	4292.91			0.361326			0.180663	+	0.983545i	$0.442176\pi$
$110$			650.063i			0.0537242i
$111$	2487.46	+	4347.22i	0.201888	+	0.352830i
$112$	4689.16			0.373817
$113$			4334.19i			0.339430i	0.985493	+	0.169715i	$0.0542848\pi$
$113$			4334.19i			0.339430i	−0.985493	+	0.169715i	$0.945715\pi$
$114$	563.688	−	322.539i	0.0433739	−	0.0248184i
$115$	−24874.5			−1.88087
$116$			24959.0i			1.85486i
$117$	635.086	−	1080.57i	0.0463939	−	0.0789375i
$118$	−149.297			−0.0107223
$119$			6139.12i			0.433523i
$120$	−1053.59	−	1841.32i	−0.0731663	−	0.127869i
$121$	6881.37			0.470007
$122$		−	758.189i		−	0.0509398i
$123$	516.938	−	295.790i	0.0341687	−	0.0195512i
$124$	−21988.4			−1.43005
$125$		−	9773.54i		−	0.625507i
$126$	−313.064	−	183.997i	−0.0197193	−	0.0115896i
$127$	14744.4			0.914155			0.457078	−	0.889427i	$-0.348896\pi$
$127$	14744.4			0.914155			0.457078	+	0.889427i	$0.348896\pi$
$128$			3929.75i			0.239853i
$129$	5004.81	+	8746.67i	0.300751	+	0.525610i
$130$	−114.192			−0.00675691
$131$		−	20831.8i		−	1.21390i	−0.794740	−	0.606950i	$-0.792393\pi$
$131$		−	20831.8i		−	1.21390i	0.794740	−	0.606950i	$-0.207607\pi$
$132$	10969.5	−	6276.71i	0.629563	−	0.360233i
$133$	5520.96			0.312112
$134$			1701.35i			0.0947511i
$135$	258.338	−	22223.0i	0.0141749	−	1.21937i
$136$	−2562.97			−0.138569
$137$		−	24279.2i		−	1.29358i	−0.762669	−	0.646789i	$-0.776111\pi$
$137$		−	24279.2i		−	1.29358i	0.762669	−	0.646789i	$-0.223889\pi$
$138$	−882.808	−	1542.84i	−0.0463562	−	0.0810146i
$139$	11786.6			0.610042			0.305021	−	0.952346i	$-0.401337\pi$
$139$	11786.6			0.610042			0.305021	+	0.952346i	$0.401337\pi$
$140$		−	9000.73i		−	0.459221i
$141$	−24710.9	+	14139.5i	−1.24294	+	0.711204i
$142$	496.203			0.0246084
$143$		−	1363.08i		−	0.0666574i
$144$	−10391.6	+	17680.8i	−0.501136	+	0.852664i
$145$	47731.4			2.27022
$146$		−	1484.83i		−	0.0696580i
$147$	−1533.13	−	2679.38i	−0.0709487	−	0.123994i
$148$	−8871.52			−0.405018
$149$			6241.91i			0.281154i	0.990070	+	0.140577i	$0.0448958\pi$
$149$			6241.91i			0.281154i	−0.990070	+	0.140577i	$0.955104\pi$
$150$	−575.615	+	329.364i	−0.0255829	+	0.0146384i
$151$	9337.45			0.409519			0.204760	−	0.978812i	$-0.434359\pi$
$151$	9337.45			0.409519			0.204760	+	0.978812i	$0.434359\pi$
$152$			2304.90i			0.0997620i
$153$	−23148.0	−	13604.8i	−0.988852	−	0.581178i
$154$	−394.910			−0.0166516
$155$			42050.5i			1.75028i
$156$	1102.58	+	1926.93i	0.0453066	+	0.0791804i
$157$	8467.95			0.343541			0.171771	−	0.985137i	$-0.445051\pi$
$157$	8467.95			0.343541			0.171771	+	0.985137i	$0.445051\pi$
$158$		−	1770.57i		−	0.0709248i
$159$	−1663.66	+	951.939i	−0.0658068	+	0.0376543i
$160$	5639.91			0.220309
$161$		−	15111.2i		−	0.582970i
$162$	1387.55	−	772.680i	0.0528711	−	0.0294422i
$163$	−7222.49			−0.271839			−0.135919	−	0.990720i	$-0.543399\pi$
$163$	−7222.49			−0.271839			−0.135919	+	0.990720i	$0.543399\pi$
$164$			1054.93i			0.0392227i
$165$	−12003.5	−	20978.0i	−0.440901	−	0.770543i
$166$	−1476.73			−0.0535900
$167$			17419.4i			0.624599i	0.949984	+	0.312300i	$0.101099\pi$
$167$			17419.4i			0.624599i	−0.949984	+	0.312300i	$0.898901\pi$
$168$	1118.59	−	640.054i	0.0396327	−	0.0226776i
$169$	−28321.6			−0.991616
$170$			2446.21i			0.0846440i
$171$	−12234.9	+	20817.2i	−0.418416	+	0.711918i
$172$	−17849.6			−0.603354
$173$		−	15954.1i		−	0.533066i	−0.963826	−	0.266533i	$-0.914122\pi$
$173$		−	15954.1i		−	0.533066i	0.963826	−	0.266533i	$-0.0858781\pi$
$174$	1694.01	+	2960.54i	0.0559522	+	0.0977852i
$175$	−5637.78			−0.184091
$176$			22303.3i			0.720018i
$177$	4817.93	−	2756.80i	0.153785	−	0.0879950i
$178$	−1646.54			−0.0519675
$179$			27975.2i			0.873105i	0.899679	+	0.436553i	$0.143801\pi$
$179$			27975.2i			0.873105i	−0.899679	+	0.436553i	$0.856199\pi$
$180$	33938.0	+	19946.3i	1.04747	+	0.615628i
$181$	25826.2			0.788321			0.394160	−	0.919042i	$-0.371035\pi$
$181$	25826.2			0.788321			0.394160	+	0.919042i	$0.371035\pi$
$182$		−	69.3709i		−	0.00209428i
$183$	14000.1	+	24467.3i	0.418051	+	0.730608i
$184$	6308.63			0.186337
$185$			16965.9i			0.495715i
$186$	−2608.18	+	1492.39i	−0.0753897	+	0.0431377i
$187$	−29199.8			−0.835019
$188$		−	50428.3i		−	1.42679i
$189$	13500.4	+	156.939i	0.377939	+	0.00439347i
$190$	2199.90			0.0609389
$191$		−	44447.1i		−	1.21836i	−0.793031	−	0.609182i	$-0.791498\pi$
$191$		−	44447.1i		−	1.21836i	0.793031	−	0.609182i	$-0.208502\pi$
$192$	−17907.1	−	31295.4i	−0.485761	−	0.848943i
$193$	17780.1			0.477330			0.238665	−	0.971102i	$-0.423290\pi$
$193$	17780.1			0.477330			0.238665	+	0.971102i	$0.423290\pi$
$194$		−	2273.38i		−	0.0604045i
$195$	3685.06	−	2108.57i	0.0969114	−	0.0554523i
$196$	5467.90			0.142334
$197$		−	51169.6i		−	1.31850i	−0.751924	−	0.659250i	$-0.770874\pi$
$197$		−	51169.6i		−	1.31850i	0.751924	−	0.659250i	$-0.229126\pi$
$198$	875.153	−	1489.04i	0.0223230	−	0.0379818i
$199$	−7247.46			−0.183012			−0.0915061	−	0.995805i	$-0.529168\pi$
$199$	−7247.46			−0.183012			−0.0915061	+	0.995805i	$0.529168\pi$
$200$		−	2353.67i		−	0.0588418i
$201$	−31415.8	−	54903.9i	−0.777599	−	1.35897i
$202$	−1949.63			−0.0477804
$203$			28996.6i			0.703648i
$204$	41278.7	−	23619.5i	0.991895	−	0.567558i
$205$	2017.45			0.0480059
$206$		−	767.363i		−	0.0180828i
$207$	56977.8	+	33487.5i	1.32973	+	0.781524i
$208$	−3917.85			−0.0905568
$209$			26259.6i			0.601167i
$210$	−610.895	−	1067.63i	−0.0138525	−	0.0242094i
$211$	−32606.5			−0.732385			−0.366192	−	0.930539i	$-0.619339\pi$
$211$	−32606.5			−0.732385			−0.366192	+	0.930539i	$0.619339\pi$
$212$		−	3395.09i		−	0.0755404i
$213$	−16012.9	+	9162.48i	−0.352947	+	0.201955i
$214$	762.134			0.0166419
$215$			34135.5i			0.738464i
$216$	−65.5192	+	5636.16i	−0.00140430	+	0.120802i
$217$	−25545.5			−0.542494
$218$			1039.16i			0.0218660i
$219$	27417.7	+	47916.6i	0.571666	+	0.999075i
$220$	42810.6			0.884516
$221$		−	5129.31i		−	0.105021i
$222$	−1052.31	+	602.126i	−0.0213519	+	0.0122175i
$223$	−55353.6			−1.11311			−0.556553	−	0.830812i	$-0.687876\pi$
$223$	−55353.6			−1.11311			−0.556553	+	0.830812i	$0.687876\pi$
$224$			3426.22i			0.0682840i
$225$	12493.8	−	21257.7i	0.246791	−	0.419905i
$226$	−1049.15			−0.0205410
$227$			5840.79i			0.113349i	0.998393	+	0.0566747i	$0.0180498\pi$
$227$			5840.79i			0.113349i	−0.998393	+	0.0566747i	$0.981950\pi$
$228$	−21241.2	−	37122.2i	−0.408610	−	0.714109i
$229$	−2103.16			−0.0401053			−0.0200526	−	0.999799i	$-0.506383\pi$
$229$	−2103.16			−0.0401053			−0.0200526	+	0.999799i	$0.506383\pi$
$230$		−	6021.23i		−	0.113823i
$231$	12744.1	−	7292.09i	0.238827	−	0.136656i
$232$	−12105.6			−0.224910
$233$			61482.6i			1.13251i	0.824232	+	0.566253i	$0.191607\pi$
$233$			61482.6i			1.13251i	−0.824232	+	0.566253i	$0.808393\pi$
$234$	261.569	+	153.732i	0.00477699	+	0.00280758i
$235$	−96438.8			−1.74629
$236$			9832.10i			0.176532i
$237$	32693.9	+	57137.6i	0.582063	+	1.01724i
$238$	−1486.06			−0.0262351
$239$		−	74258.2i		−	1.30001i	−0.759928	−	0.650007i	$-0.774766\pi$
$239$		−	74258.2i		−	1.30001i	0.759928	−	0.650007i	$-0.225234\pi$
$240$	−60296.5	+	34501.4i	−1.04682	+	0.598983i
$241$	111286.			1.91605			0.958026	−	0.286683i	$-0.0925525\pi$
$241$	111286.			1.91605			0.958026	+	0.286683i	$0.0925525\pi$
$242$			1665.73i			0.0284430i
$243$	−30509.6	+	50556.4i	−0.516683	+	0.856177i
$244$	−49931.3			−0.838674
$245$		−	10456.8i		−	0.174207i
$246$	71.6001	+	125.132i	0.00118316	+	0.00206776i
$247$	−4612.83			−0.0756090
$248$		−	10664.8i		−	0.173400i
$249$	47655.2	−	27268.1i	0.768619	−	0.439800i
$250$	2365.83			0.0378532
$251$			41512.0i			0.658910i	0.944171	+	0.329455i	$0.106865\pi$
$251$			41512.0i			0.658910i	−0.944171	+	0.329455i	$0.893135\pi$
$252$	−12117.3	+	20617.2i	−0.190812	+	0.324659i
$253$	71873.9			1.12287
$254$			3569.10i			0.0553211i
$255$	−45169.8	−	78941.2i	−0.694653	−	1.21401i
$256$	63149.1			0.963579
$257$			53152.3i			0.804741i	0.915477	+	0.402370i	$0.131814\pi$
$257$			53152.3i			0.804741i	−0.915477	+	0.402370i	$0.868186\pi$
$258$	−2117.26	+	1211.48i	−0.0318078	+	0.0182003i
$259$	−10306.7			−0.153645
$260$			7520.22i			0.111246i
$261$	−109334.	−	64258.9i	−1.60500	−	0.943305i
$262$	5042.62			0.0734605
$263$		−	64112.0i		−	0.926890i	−0.886126	−	0.463445i	$-0.846613\pi$
$263$		−	64112.0i		−	0.926890i	0.886126	−	0.463445i	$-0.153387\pi$
$264$	3044.32	+	5320.42i	0.0436799	+	0.0763375i
$265$	−6492.75			−0.0924564
$266$			1336.43i			0.0188878i
$267$	53135.1	−	30403.7i	0.745348	−	0.426484i
$268$	112044.			1.55998
$269$		−	115300.i		−	1.59341i	−0.604371	−	0.796703i	$-0.706576\pi$
$269$		−	115300.i		−	1.59341i	0.604371	−	0.796703i	$-0.293424\pi$
$270$	5379.39	+	62.5343i	0.0737914	+	0.000857810i
$271$	108645.			1.47936			0.739678	−	0.672961i	$-0.234978\pi$
$271$	108645.			1.47936			0.739678	+	0.672961i	$0.234978\pi$
$272$			83928.1i			1.13441i
$273$	1280.95	+	2238.66i	0.0171872	+	0.0300374i
$274$	5877.12			0.0782823
$275$		−	26815.2i		−	0.354582i
$276$	−101606.	+	58138.2i	−1.33383	+	0.763209i
$277$	−45058.9			−0.587248			−0.293624	−	0.955921i	$-0.594861\pi$
$277$	−45058.9			−0.587248			−0.293624	+	0.955921i	$0.594861\pi$
$278$			2853.12i			0.0369173i
$279$	56610.9	−	96321.3i	0.727263	−	1.23741i
$280$	4365.52			0.0556826
$281$		−	19372.2i		−	0.245339i	−0.992448	−	0.122670i	$-0.960854\pi$
$281$		−	19372.2i		−	0.245339i	0.992448	−	0.122670i	$-0.0391455\pi$
$282$	−3422.66	−	5981.62i	−0.0430393	−	0.0752178i
$283$	−12147.5			−0.151675			−0.0758375	−	0.997120i	$-0.524163\pi$
$283$	−12147.5			−0.151675			−0.0758375	+	0.997120i	$0.524163\pi$
$284$		−	32678.0i		−	0.405152i
$285$	−70992.4	+	40621.5i	−0.874021	+	0.500111i
$286$	329.952			0.00403384
$287$			1225.59i			0.0148793i
$288$	−12918.8	−	7592.78i	−0.155754	−	0.0915410i
$289$	−26359.0			−0.315597
$290$			11554.1i			0.137385i
$291$	41978.5	+	73363.9i	0.495725	+	0.866356i
$292$	−97785.1			−1.14685
$293$		−	39539.1i		−	0.460566i	−0.973124	−	0.230283i	$-0.926035\pi$
$293$		−	39539.1i		−	0.460566i	0.973124	−	0.230283i	$-0.0739651\pi$
$294$	648.583	−	371.116i	0.00750362	−	0.00429353i
$295$	18802.9			0.216063
$296$		−	4302.85i		−	0.0491103i
$297$	−746.456	+	64212.4i	−0.00846236	+	0.727957i
$298$	−1510.94			−0.0170144
$299$			12625.6i			0.141224i
$300$	21690.6	+	37907.8i	0.241007	+	0.421197i
$301$	−20737.2			−0.228885
$302$			2260.26i			0.0247825i
$303$	62916.2	−	36000.3i	0.685294	−	0.392122i
$304$	75477.1			0.816711
$305$			95488.4i			1.02648i
$306$	3293.23	−	5603.32i	0.0351706	−	0.0598415i
$307$	−51639.4			−0.547904			−0.273952	−	0.961743i	$-0.588331\pi$
$307$	−51639.4			−0.547904			−0.273952	+	0.961743i	$0.588331\pi$
$308$			26007.2i			0.274153i
$309$	14169.5	+	24763.4i	0.148401	+	0.259354i
$310$	−10178.9			−0.105920
$311$			65566.1i			0.677889i	0.940806	+	0.338944i	$0.110070\pi$
$311$			65566.1i			0.677889i	−0.940806	+	0.338944i	$0.889930\pi$
$312$	−934.598	+	534.773i	−0.00960098	+	0.00549364i
$313$	58295.3			0.595038			0.297519	−	0.954716i	$-0.403841\pi$
$313$	58295.3			0.595038			0.297519	+	0.954716i	$0.403841\pi$
$314$			2049.79i			0.0207898i
$315$	39428.1	+	23173.1i	0.397361	+	0.233541i
$316$	−116603.			−1.16771
$317$			125449.i			1.24839i	0.781271	+	0.624193i	$0.214572\pi$
$317$			125449.i			1.24839i	−0.781271	+	0.624193i	$0.785428\pi$
$318$	−230.431	−	402.713i	−0.00227869	−	0.00398237i
$319$	−137918.			−1.35531
$320$		−	122136.i		−	1.19274i
$321$	−24594.6	+	14072.9i	−0.238688	+	0.136576i
$322$	3657.87			0.0352790
$323$			98815.8i			0.947156i
$324$	−50885.7	−	91378.6i	−0.484736	−	0.870471i
$325$	4710.43			0.0445958
$326$		−	1748.31i		−	0.0164506i
$327$	−19188.3	−	33534.5i	−0.179449	−	0.313615i
$328$	−511.662			−0.00475593
$329$		−	58586.2i		−	0.541257i
$330$	5078.03	−	2905.63i	0.0466302	−	0.0266816i
$331$	84438.8			0.770701			0.385351	−	0.922770i	$-0.374080\pi$
$331$	84438.8			0.770701			0.385351	+	0.922770i	$0.374080\pi$
$332$			97251.5i			0.882308i
$333$	22840.4	−	38862.1i	0.205976	−	0.350460i
$334$	−4216.63			−0.0377983
$335$		−	214273.i		−	1.90931i
$336$	−20959.4	−	36629.9i	−0.185653	−	0.324457i
$337$	−185242.			−1.63110			−0.815549	−	0.578689i	$-0.803565\pi$
$337$	−185242.			−1.63110			−0.815549	+	0.578689i	$0.803565\pi$
$338$		−	6855.64i		−	0.0600087i
$339$	33856.9	−	19372.8i	0.294611	−	0.168575i
$340$	161098.			1.39358
$341$		−	121503.i		−	1.04491i
$342$	−5039.10	−	2961.63i	−0.0430825	−	0.0253209i
$343$	6352.45			0.0539949
$344$		−	8657.40i		−	0.0731594i
$345$	111183.	+	194310.i	0.934117	+	1.63251i
$346$	3861.92			0.0322590
$347$			83517.4i			0.693614i	0.937937	+	0.346807i	$0.112734\pi$
$347$			83517.4i			0.693614i	−0.937937	+	0.346807i	$0.887266\pi$
$348$	194970.	−	111561.i	1.60994	−	0.921199i
$349$	100015.			0.821134			0.410567	−	0.911831i	$-0.365331\pi$
$349$	100015.			0.821134			0.410567	+	0.911831i	$0.365331\pi$
$350$		−	1364.71i		−	0.0111405i
$351$	−11279.7	−	131.124i	−0.0915554	−	0.00106431i
$352$	−16296.3			−0.131523
$353$		−	57357.4i		−	0.460299i	−0.973155	−	0.230149i	$-0.926078\pi$
$353$		−	57357.4i		−	0.460299i	0.973155	−	0.230149i	$-0.0739215\pi$
$354$	667.322	+	1166.25i	0.00532511	+	0.00930646i
$355$	−62493.2			−0.495879
$356$			108435.i			0.855594i
$357$	47956.4	−	27440.4i	0.376279	−	0.215305i
$358$	−6771.79			−0.0528369
$359$		−	165789.i		−	1.28637i	−0.765709	−	0.643187i	$-0.777612\pi$
$359$		−	165789.i		−	1.28637i	0.765709	−	0.643187i	$-0.222388\pi$
$360$	−9674.34	+	16460.5i	−0.0746477	+	0.127010i
$361$	−41455.2			−0.318101
$362$			6251.60i			0.0477061i
$363$	−30758.1	−	53754.6i	−0.233425	−	0.407946i
$364$	−4568.50			−0.0344803
$365$			187004.i			1.40367i
$366$	−5922.67	+	3388.93i	−0.0442136	+	0.0252988i
$367$	−78946.7			−0.586141			−0.293070	−	0.956091i	$-0.594677\pi$
$367$	−78946.7			−0.586141			−0.293070	+	0.956091i	$0.594677\pi$
$368$		−	206585.i		−	1.52547i
$369$	−4621.18	−	2716.01i	−0.0339391	−	0.0199470i
$370$	−4106.83			−0.0299987
$371$		−	3944.32i		−	0.0286565i
$372$	98283.0	+	171765.i	0.710219	+	1.24122i
$373$	−236493.			−1.69981			−0.849907	−	0.526932i	$-0.823342\pi$
$373$	−236493.			−1.69981			−0.849907	+	0.526932i	$0.823342\pi$
$374$		−	7068.23i		−	0.0505321i
$375$	−76347.1	+	43685.4i	−0.542912	+	0.310652i
$376$	24458.7			0.173004
$377$		−	24227.0i		−	0.170458i
$378$	−37.9893	+	3267.95i	−0.000265875	+	0.0228714i
$379$	111501.			0.776245			0.388123	−	0.921608i	$-0.373124\pi$
$379$	111501.			0.776245			0.388123	+	0.921608i	$0.373124\pi$
$380$		−	144877.i		−	1.00330i
$381$	−65904.1	−	115178.i	−0.454007	−	0.793447i
$382$	10759.1			0.0737306
$383$			118675.i			0.809024i	0.914533	+	0.404512i	$0.132559\pi$
$383$			118675.i			0.809024i	−0.914533	+	0.404512i	$0.867441\pi$
$384$	30697.7	−	17565.1i	0.208182	−	0.119121i
$385$	49736.1			0.335544
$386$			4303.92i			0.0288862i
$387$	45955.2	−	78191.1i	0.306841	−	0.522078i
$388$	−149716.			−0.994501
$389$			278244.i			1.83877i	0.393361	+	0.919384i	$0.371312\pi$
$389$			278244.i			1.83877i	−0.393361	+	0.919384i	$0.628688\pi$
$390$	510.410	+	892.021i	0.00335575	+	0.00586470i
$391$	270464.			1.76912
$392$			2652.03i			0.0172586i
$393$	−162729.	+	93113.1i	−1.05361	+	0.602873i
$394$	12386.3			0.0797904
$395$			222990.i			1.42919i
$396$	−98062.3	−	57634.1i	−0.625334	−	0.367527i
$397$	108561.			0.688798			0.344399	−	0.938823i	$-0.388083\pi$
$397$	108561.			0.688798			0.344399	+	0.938823i	$0.388083\pi$
$398$		−	1754.35i		−	0.0110752i
$399$	−24677.4	−	43127.5i	−0.155008	−	0.270900i
$400$	−77074.2			−0.481714
$401$		−	15517.8i		−	0.0965029i	−0.998835	−	0.0482514i	$-0.984635\pi$
$401$		−	15517.8i		−	0.0965029i	0.998835	−	0.0482514i	$-0.0153649\pi$
$402$	13290.3	−	7604.64i	0.0822398	−	0.0470572i
$403$	21343.6			0.131419
$404$			128395.i			0.786657i
$405$	−174752.	+	97313.5i	−1.06540	+	0.593284i
$406$	−7019.05			−0.0425820
$407$		−	49022.1i		−	0.295940i
$408$	11455.9	+	20020.9i	0.0688190	+	0.120272i
$409$	−3909.15			−0.0233688			−0.0116844	−	0.999932i	$-0.503719\pi$
$409$	−3909.15			−0.0233688			−0.0116844	+	0.999932i	$0.503719\pi$
$410$			488.352i			0.00290513i
$411$	−189659.	+	108522.i	−1.12277	+	0.642443i
$412$	−50535.5			−0.297716
$413$			11422.7i			0.0669679i
$414$	−8106.14	+	13792.3i	−0.0472948	+	0.0804703i
$415$	185983.			1.07988
$416$		−	2862.65i		−	0.0165417i
$417$	−52683.4	−	92072.4i	−0.302971	−	0.529489i
$418$	−6356.51			−0.0363803
$419$		−	131470.i		−	0.748859i	−0.927255	−	0.374429i	$-0.877839\pi$
$419$		−	131470.i		−	0.748859i	0.927255	−	0.374429i	$-0.122161\pi$
$420$	−70310.2	+	40231.1i	−0.398584	+	0.228068i
$421$	264054.			1.48980			0.744901	−	0.667175i	$-0.232497\pi$
$421$	264054.			1.48980			0.744901	+	0.667175i	$0.232497\pi$
$422$		−	7892.87i		−	0.0443210i
$423$	220904.	+	129832.i	1.23459	+	0.725604i
$424$	1646.68			0.00915962
$425$		−	100907.i		−	0.558653i
$426$	−2217.91	−	3876.14i	−0.0122215	−	0.0213590i
$427$	−58008.8			−0.318154
$428$		−	50191.1i		−	0.273993i
$429$	−10647.8	+	6092.63i	−0.0578557	+	0.0331048i
$430$	−8262.99			−0.0446890
$431$		−	229273.i		−	1.23424i	−0.786870	−	0.617119i	$-0.788300\pi$
$431$		−	229273.i		−	1.23424i	0.786870	−	0.617119i	$-0.211700\pi$
$432$	184564.	+	2145.52i	0.988960	+	0.0114965i
$433$	−110235.			−0.587953			−0.293977	−	0.955813i	$-0.594979\pi$
$433$	−110235.			−0.587953			−0.293977	+	0.955813i	$0.594979\pi$
$434$		−	6183.65i		−	0.0328296i
$435$	−213348.	−	372859.i	−1.12748	−	1.97045i
$436$	68435.0			0.360002
$437$		−	243230.i		−	1.27366i
$438$	−11598.9	+	6636.84i	−0.0604601	+	0.0345950i
$439$	61671.8			0.320005			0.160003	−	0.987117i	$-0.448850\pi$
$439$	61671.8			0.320005			0.160003	+	0.987117i	$0.448850\pi$
$440$			20763.9i			0.107252i
$441$	−14077.5	+	23952.4i	−0.0723852	+	0.123161i
$442$	1241.62			0.00635544
$443$			54702.9i			0.278742i	0.990240	+	0.139371i	$0.0445081\pi$
$443$			54702.9i			0.278742i	−0.990240	+	0.139371i	$0.955492\pi$
$444$	39653.6	+	69300.9i	0.201149	+	0.351538i
$445$	207370.			1.04719
$446$		−	13399.1i		−	0.0673608i
$447$	48759.3	−	27899.9i	0.244030	−	0.139633i
$448$	74197.2			0.369685
$449$		−	110150.i		−	0.546375i	−0.961961	−	0.273188i	$-0.911922\pi$
$449$		−	110150.i		−	0.546375i	0.961961	−	0.273188i	$-0.0880780\pi$
$450$	5145.73	+	3024.30i	0.0254110	+	0.0149348i
$451$	−5829.33			−0.0286593
$452$			69093.0i			0.338187i
$453$	−41736.2	−	72940.5i	−0.203384	−	0.355445i
$454$	−1413.85			−0.00685947
$455$			8736.77i			0.0422015i
$456$	18005.0	−	10302.4i	0.0865890	−	0.0495458i
$457$	−247865.			−1.18681			−0.593407	−	0.804902i	$-0.702218\pi$
$457$	−247865.			−1.18681			−0.593407	+	0.804902i	$0.702218\pi$
$458$		−	509.100i		−	0.00242701i
$459$	−2808.94	+	241634.i	−0.0133327	+	1.14692i
$460$	−396535.			−1.87398
$461$			250553.i			1.17896i	0.807784	+	0.589478i	$0.200667\pi$
$461$			250553.i			1.17896i	−0.807784	+	0.589478i	$0.799333\pi$
$462$	1765.16	+	3084.88i	0.00826988	+	0.0144529i
$463$	192820.			0.899478			0.449739	−	0.893160i	$-0.351517\pi$
$463$	192820.			0.899478			0.449739	+	0.893160i	$0.351517\pi$
$464$			396413.i			1.84125i
$465$	328482.	−	187956.i	1.51917	−	0.869260i
$466$	−14882.7			−0.0685348
$467$		−	86179.2i		−	0.395156i	−0.980287	−	0.197578i	$-0.936692\pi$
$467$		−	86179.2i		−	0.395156i	0.980287	−	0.197578i	$-0.0633076\pi$
$468$	10124.2	−	17225.9i	0.0462240	−	0.0786484i
$469$	130170.			0.591786
$470$		−	23344.4i		−	0.105679i
$471$	−37849.8	−	66148.3i	−0.170617	−	0.298179i
$472$	−4768.75			−0.0214053
$473$		−	98633.1i		−	0.440860i
$474$	−13831.0	+	7914.02i	−0.0615596	+	0.0352241i
$475$	−90746.2			−0.402199
$476$			97866.2i			0.431936i
$477$	14872.4	+	8740.92i	0.0653646	+	0.0384167i
$478$	17975.2			0.0786718
$479$		−	121298.i		−	0.528667i	−0.964431	−	0.264334i	$-0.914848\pi$
$479$		−	121298.i		−	0.528667i	0.964431	−	0.264334i	$-0.0851520\pi$
$480$	−25209.1	−	44056.8i	−0.109414	−	0.191219i
$481$	8611.35			0.0372204
$482$			26938.4i			0.115952i
$483$	−118042.	+	67543.3i	−0.505992	+	0.289526i
$484$	109699.			0.468286
$485$			286316.i			1.21720i
$486$	−12237.9	−	7385.29i	−0.0518125	−	0.0312676i
$487$	54998.2			0.231895			0.115947	−	0.993255i	$-0.463010\pi$
$487$	54998.2			0.231895			0.115947	+	0.993255i	$0.463010\pi$
$488$		−	24217.6i		−	0.101693i
$489$	32282.8	+	56419.2i	0.135006	+	0.235944i
$490$	2531.21			0.0105423
$491$		−	387549.i		−	1.60755i	−0.594935	−	0.803774i	$-0.702822\pi$
$491$		−	387549.i		−	1.60755i	0.594935	−	0.803774i	$-0.297178\pi$
$492$	8240.72	−	4715.30i	0.0340436	−	0.0194796i
$493$	−518991.			−2.13533
$494$		−	1116.60i		−	0.00457556i
$495$	−110219.	+	187534.i	−0.449828	+	0.765366i
$496$	−349233.			−1.41955
$497$		−	37964.3i		−	0.153696i
$498$	6600.63	+	11535.6i	0.0266150	+	0.0465138i
$499$	−292129.			−1.17320			−0.586602	−	0.809876i	$-0.699535\pi$
$499$	−292129.			−1.17320			−0.586602	+	0.809876i	$0.699535\pi$
$500$		−	155804.i		−	0.623216i
$501$	136074.	−	77860.8i	0.542125	−	0.310201i
$502$	−10048.6			−0.0398747
$503$			456757.i			1.80530i	0.430376	+	0.902650i	$0.358381\pi$
$503$			456757.i			1.80530i	−0.430376	+	0.902650i	$0.641619\pi$
$504$	−9999.70	−	5877.12i	−0.0393664	−	0.0231368i
$505$	245542.			0.962816
$506$			17398.1i			0.0679518i
$507$	126591.	+	221237.i	0.492477	+	0.860680i
$508$	235047.			0.910808
$509$			243339.i			0.939239i	0.882869	+	0.469619i	$0.155609\pi$
$509$			243339.i			0.939239i	−0.882869	+	0.469619i	$0.844391\pi$
$510$	19108.8	−	10934.0i	0.0734673	−	0.0420377i
$511$	−113604.			−0.435062
$512$			78162.2i			0.298165i
$513$	217303.	+	2526.10i	0.825716	+	0.00959879i
$514$	−12866.3			−0.0486998
$515$			96643.8i			0.364384i
$516$	79783.6	+	139434.i	0.299650	+	0.523685i
$517$	278656.			1.04253
$518$		−	2494.88i		−	0.00929801i
$519$	−124627.	+	71311.2i	−0.462678	+	0.264742i
$520$	−3647.44			−0.0134891
$521$		−	83338.8i		−	0.307024i	−0.988147	−	0.153512i	$-0.950942\pi$
$521$		−	83338.8i		−	0.307024i	0.988147	−	0.153512i	$-0.0490583\pi$
$522$	15554.8	−	26465.9i	0.0570851	−	0.0971282i
$523$	5271.56			0.0192724			0.00963620	−	0.999954i	$-0.496933\pi$
$523$	5271.56			0.0192724			0.00963620	+	0.999954i	$0.496933\pi$
$524$		−	332087.i		−	1.20946i
$525$	25199.6	+	44040.1i	0.0914270	+	0.159783i
$526$	15519.2			0.0560917
$527$		−	457221.i		−	1.64629i
$528$	174224.	−	99690.4i	0.624944	−	0.357590i
$529$	−385894.			−1.37897
$530$		−	1571.66i		−	0.00559510i
$531$	−43070.0	−	25313.5i	−0.152752	−	0.0897767i
$532$	88011.8			0.310969
$533$		−	1024.00i		−	0.00360449i
$534$	7359.64	+	12862.1i	0.0258092	+	0.0451055i
$535$	−95985.2			−0.335349
$536$			54343.5i			0.189155i
$537$	218531.	−	125042.i	0.757817	−	0.433620i
$538$	27910.1			0.0964266
$539$			30214.4i			0.104001i
$540$	4118.27	−	354265.i	0.0141230	−	1.21490i
$541$	86693.8			0.296206			0.148103	−	0.988972i	$-0.452683\pi$
$541$	86693.8			0.296206			0.148103	+	0.988972i	$0.452683\pi$
$542$			26299.2i			0.0895248i
$543$	−115437.	−	201744.i	−0.391512	−	0.684228i
$544$	−61323.6			−0.207219
$545$		−	130875.i		−	0.440619i
$546$	−541.899	+	310.072i	−0.00181774	+	0.00104010i
$547$	−43775.8			−0.146305			−0.0731526	−	0.997321i	$-0.523306\pi$
$547$	−43775.8			−0.146305			−0.0731526	+	0.997321i	$0.523306\pi$
$548$		−	387044.i		−	1.28884i
$549$	128552.	−	218727.i	0.426515	−	0.725700i
$550$	6491.01			0.0214579
$551$			466732.i			1.53732i
$552$	−28198.1	−	49280.6i	−0.0925427	−	0.161733i
$553$	−135465.			−0.442974
$554$		−	10907.2i		−	0.0355379i
$555$	132531.	−	75833.4i	0.430259	−	0.246192i
$556$	187895.			0.607807
$557$		−	449830.i		−	1.44990i	−0.688802	−	0.724950i	$-0.741863\pi$
$557$		−	449830.i		−	1.44990i	0.688802	−	0.724950i	$-0.258137\pi$
$558$	23315.9	+	13703.5i	0.0748832	+	0.0440111i
$559$	17326.2			0.0554471
$560$		−	142955.i		−	0.455851i
$561$	130516.	+	228097.i	0.414704	+	0.724760i
$562$	4689.32			0.0148470
$563$			611954.i			1.93064i	0.261066	+	0.965321i	$0.415926\pi$
$563$			611954.i			1.93064i	−0.261066	+	0.965321i	$0.584074\pi$
$564$	−393926.	+	225403.i	−1.23839	+	0.708600i
$565$	132133.			0.413918
$566$		−	2940.47i		−	0.00917877i
$567$	−59117.5	−	106161.i	−0.183886	−	0.330216i
$568$	15849.4			0.0491266
$569$			164656.i			0.508573i	0.967129	+	0.254286i	$0.0818406\pi$
$569$			164656.i			0.508573i	−0.967129	+	0.254286i	$0.918159\pi$
$570$	−9833.02	−	17184.7i	−0.0302648	−	0.0528923i
$571$	275184.			0.844017			0.422009	−	0.906592i	$-0.361325\pi$
$571$	275184.			0.844017			0.422009	+	0.906592i	$0.361325\pi$
$572$		−	21729.4i		−	0.0664133i
$573$	−347203.	+	198668.i	−1.05749	+	0.605089i
$574$	−296.672			−0.000900435
$575$			248377.i			0.751235i
$576$	−164427.	+	279766.i	−0.495596	+	0.843239i
$577$	115231.			0.346112			0.173056	−	0.984912i	$-0.444636\pi$
$577$	115231.			0.346112			0.173056	+	0.984912i	$0.444636\pi$
$578$		−	6380.57i		−	0.0190987i
$579$	−79472.8	−	138891.i	−0.237062	−	0.414302i
$580$	760906.			2.26191
$581$			112984.i			0.334707i
$582$	−17758.8	+	10161.5i	−0.0524285	+	0.0299993i
$583$	18760.5			0.0551960
$584$		−	47427.6i		−	0.139061i
$585$	−32942.7	−	19361.4i	−0.0962603	−	0.0565750i
$586$	9571.00			0.0278716
$587$		−	233476.i		−	0.677588i	−0.940861	−	0.338794i	$-0.889981\pi$
$587$		−	233476.i		−	0.677588i	0.940861	−	0.338794i	$-0.110019\pi$
$588$	−24440.2	−	42713.1i	−0.0706888	−	0.123540i
$589$	−411182.			−1.18523
$590$			4551.50i			0.0130753i
$591$	−399717.	+	228716.i	−1.14440	+	0.654820i
$592$	−140903.			−0.402046
$593$		−	566032.i		−	1.60965i	−0.593511	−	0.804826i	$-0.702259\pi$
$593$		−	566032.i		−	1.60965i	0.593511	−	0.804826i	$-0.297741\pi$
$594$	−15543.5	−	180.690i	−0.0440531	−	0.000512109i
$595$	187159.			0.528660
$596$			99504.8i			0.280125i
$597$	32394.5	+	56614.3i	0.0908912	+	0.158847i
$598$	−3056.20			−0.00854631
$599$			461869.i			1.28726i	0.765338	+	0.643629i	$0.222572\pi$
$599$			461869.i			1.28726i	−0.765338	+	0.643629i	$0.777428\pi$
$600$	−18386.0	+	10520.4i	−0.0510721	+	0.0292232i
$601$	−419840.			−1.16235			−0.581173	−	0.813780i	$-0.697406\pi$
$601$	−419840.			−1.16235			−0.581173	+	0.813780i	$0.697406\pi$
$602$		−	5019.73i		−	0.0138512i
$603$	−288467.	+	490815.i	−0.793343	+	1.34984i
$604$	148852.			0.408020
$605$		−	209787.i		−	0.573150i
$606$	8714.40	+	15229.8i	0.0237297	+	0.0414713i
$607$	26334.2			0.0714732			0.0357366	−	0.999361i	$-0.488622\pi$
$607$	26334.2			0.0714732			0.0357366	+	0.999361i	$0.488622\pi$
$608$			55148.7i			0.149186i
$609$	226510.	−	129608.i	0.610736	−	0.349460i
$610$	−23114.3			−0.0621186
$611$			48949.4i			0.131119i
$612$	−369012.	−	216880.i	−0.985231	−	0.579049i
$613$	587631.			1.56381			0.781905	−	0.623397i	$-0.214248\pi$
$613$	587631.			1.56381			0.781905	+	0.623397i	$0.214248\pi$
$614$		−	12500.1i		−	0.0331570i
$615$	−9017.52	−	15759.5i	−0.0238417	−	0.0416670i
$616$	−12614.0			−0.0332423
$617$			436700.i			1.14713i	0.819159	+	0.573566i	$0.194440\pi$
$617$			436700.i			1.14713i	−0.819159	+	0.573566i	$0.805560\pi$
$618$	−5994.33	+	3429.93i	−0.0156951	+	0.00898066i
$619$	−198892.			−0.519081			−0.259541	−	0.965732i	$-0.583571\pi$
$619$	−198892.			−0.519081			−0.259541	+	0.965732i	$0.583571\pi$
$620$			670344.i			1.74387i
$621$	6914.08	−	594770.i	0.0179288	−	1.54229i
$622$	−15871.2			−0.0410232
$623$			125976.i			0.324573i
$624$	17511.9	+	30604.7i	0.0449742	+	0.0785994i
$625$	−488216.			−1.24983
$626$			14111.2i			0.0360094i
$627$	205129.	−	117374.i	0.521787	−	0.298564i
$628$	134991.			0.342283
$629$		−	184472.i		−	0.466262i
$630$	−5609.38	+	9544.14i	−0.0141330	+	0.0240467i
$631$	−99629.6			−0.250224			−0.125112	−	0.992143i	$-0.539929\pi$
$631$	−99629.6			−0.250224			−0.125112	+	0.992143i	$0.539929\pi$
$632$		−	56554.4i		−	0.141590i
$633$	145743.	+	254709.i	0.363732	+	0.635678i
$634$	−30366.7			−0.0755474
$635$		−	449502.i		−	1.11477i
$636$	−26521.1	+	15175.3i	−0.0655658	+	0.0375164i
$637$	−5307.55			−0.0130802
$638$		−	33385.0i		−	0.0820182i
$639$	143147.	+	84132.0i	0.350576	+	0.206044i
$640$	119803.			0.292489
$641$		−	568044.i		−	1.38250i	−0.722615	−	0.691251i	$-0.757060\pi$
$641$		−	568044.i		−	1.38250i	0.722615	−	0.691251i	$-0.242940\pi$
$642$	−3406.56	−	5953.49i	−0.00826505	−	0.0144445i
$643$	547371.			1.32391			0.661957	−	0.749542i	$-0.269726\pi$
$643$	547371.			1.32391			0.661957	+	0.749542i	$0.269726\pi$
$644$		−	240893.i		−	0.580835i
$645$	266653.	−	152578.i	0.640955	−	0.366751i
$646$	−23919.8			−0.0573182
$647$			208631.i			0.498390i	0.968453	+	0.249195i	$0.0801661\pi$
$647$			208631.i			0.498390i	−0.968453	+	0.249195i	$0.919834\pi$
$648$	44320.3	−	24680.5i	0.105549	−	0.0587765i
$649$	−54330.1			−0.128988
$650$			1140.23i			0.00269876i
$651$	114182.	+	199551.i	0.269424	+	0.470861i
$652$	−115137.			−0.270843
$653$		−	44613.2i		−	0.104625i	−0.998631	−	0.0523127i	$-0.983341\pi$
$653$		−	44613.2i		−	0.104625i	0.998631	−	0.0523127i	$-0.0166592\pi$
$654$	8117.51	−	4644.80i	0.0189787	−	0.0108596i
$655$	−635082.			−1.48029
$656$			16755.1i			0.0389349i
$657$	251755.	−	428352.i	0.583241	−	0.992362i
$658$	14181.6			0.0327547
$659$			33171.4i			0.0763823i	0.999270	+	0.0381912i	$0.0121596\pi$
$659$			33171.4i			0.0763823i	−0.999270	+	0.0381912i	$0.987840\pi$
$660$	−191353.	−	334419.i	−0.439287	−	0.767721i
$661$	456359.			1.04449			0.522244	−	0.852796i	$-0.325095\pi$
$661$	456359.			1.04449			0.522244	+	0.852796i	$0.325095\pi$
$662$			20439.6i			0.0466398i
$663$	−40068.2	+	22926.8i	−0.0911533	+	0.0521575i
$664$	−47168.7			−0.106984
$665$		−	168313.i		−	0.380606i
$666$	9407.14	+	5528.85i	0.0212085	+	0.0124648i
$667$	1.27747e6			2.87144
$668$			277690.i			0.622312i
$669$	247418.	+	432401.i	0.552813	+	0.966127i
$670$	51867.8			0.115544
$671$		−	275910.i		−	0.612804i
$672$	26764.3	−	15314.4i	0.0592676	−	0.0339126i
$673$	−555138.			−1.22566			−0.612831	−	0.790214i	$-0.709969\pi$
$673$	−555138.			−1.22566			−0.612831	+	0.790214i	$0.709969\pi$
$674$		−	44840.5i		−	0.0987076i
$675$	−221901.	−	2579.56i	−0.487026	−	0.00566158i
$676$	−451485.			−0.987985
$677$		−	290742.i		−	0.634353i	−0.948367	−	0.317176i	$-0.897265\pi$
$677$		−	290742.i		−	0.634353i	0.948367	−	0.317176i	$-0.102735\pi$
$678$	4689.46	+	8195.56i	0.0102015	+	0.0178287i
$679$	−173936.			−0.377268
$680$			78135.4i			0.168978i
$681$	45625.9	−	26106.9i	0.0983824	−	0.0562940i
$682$	29411.6			0.0632338
$683$		−	179246.i		−	0.384244i	−0.981371	−	0.192122i	$-0.938463\pi$
$683$		−	179246.i		−	0.384244i	0.981371	−	0.192122i	$-0.0615370\pi$
$684$	−195041.	+	331856.i	−0.416883	+	0.709311i
$685$	−740181.			−1.57745
$686$			1537.70i			0.00326756i
$687$	9400.64	+	16429.1i	0.0199179	+	0.0348096i
$688$	−283498.			−0.598927
$689$			3295.52i			0.00694202i
$690$	−47035.5	+	26913.5i	−0.0987933	+	0.0565291i
$691$	260140.			0.544818			0.272409	−	0.962182i	$-0.412180\pi$
$691$	260140.			0.544818			0.272409	+	0.962182i	$0.412180\pi$
$692$		−	254331.i		−	0.531114i
$693$	−113926.	−	66957.6i	−0.237223	−	0.139423i
$694$	−20216.6			−0.0419748
$695$		−	359330.i		−	0.743915i
$696$	54109.1	+	94563.9i	0.111700	+	0.195212i
$697$	−21936.0			−0.0451536
$698$			24210.0i			0.0496918i
$699$	480278.	−	274813.i	0.982966	−	0.562448i
$700$	−89874.1			−0.183417
$701$			167363.i			0.340584i	0.985394	+	0.170292i	$0.0544710\pi$
$701$			167363.i			0.340584i	−0.985394	+	0.170292i	$0.945529\pi$
$702$	31.7405	−	2730.42i	6.44080e−5	−	0.00554057i
$703$	−165897.			−0.335682
$704$			352908.i			0.712058i
$705$	431059.	+	753342.i	0.867278	+	1.51570i
$706$	13884.2			0.0278555
$707$			149166.i			0.298421i
$708$	76804.5	−	43947.2i	0.153222	−	0.0876728i
$709$	−112915.			−0.224626			−0.112313	−	0.993673i	$-0.535826\pi$
$709$	−112915.			−0.224626			−0.112313	+	0.993673i	$0.535826\pi$
$710$		−	15127.4i		−	0.0300087i
$711$	300202.	−	510783.i	0.593848	−	1.01041i
$712$	−52592.7			−0.103745
$713$			1.12543e6i			2.21380i
$714$	6642.35	+	11608.5i	0.0130294	+	0.0227709i
$715$	−41555.1			−0.0812853
$716$			445963.i			0.869908i
$717$	−580075.	+	331917.i	−1.12836	+	0.645640i
$718$	40131.6			0.0778463
$719$			757578.i			1.46545i	0.680527	+	0.732723i	$0.261751\pi$
$719$			757578.i			1.46545i	−0.680527	+	0.732723i	$0.738249\pi$
$720$	539023.	+	316800.i	1.03978	+	0.611110i
$721$	−58710.6			−0.112940
$722$		−	10034.8i		−	0.0192502i
$723$	−497423.	−	869324.i	−0.951589	−	1.66305i
$724$	411706.			0.785434
$725$		−	476608.i		−	0.906746i
$726$	13012.1	−	7445.44i	0.0246873	−	0.0141259i
$727$	−939062.			−1.77675			−0.888373	−	0.459123i	$-0.848164\pi$
$727$	−939062.			−1.77675			−0.888373	+	0.459123i	$0.848164\pi$
$728$		−	2215.81i		−	0.00418089i
$729$	531297.	+	12354.1i	0.999730	+	0.0232465i
$730$	−45266.9			−0.0849444
$731$		−	371160.i		−	0.694588i
$732$	223181.	+	390044.i	0.416520	+	0.727933i
$733$	−24447.0			−0.0455007			−0.0227504	−	0.999741i	$-0.507242\pi$
$733$	−24447.0			−0.0455007			−0.0227504	+	0.999741i	$0.507242\pi$
$734$		−	19110.2i		−	0.0354709i
$735$	−81684.3	+	46739.4i	−0.151204	+	0.0865183i
$736$	150945.			0.278653
$737$			619132.i			1.13985i
$738$	657.448	−	1118.62i	0.00120712	−	0.00205386i
$739$	−605915.			−1.10949			−0.554744	−	0.832021i	$-0.687184\pi$
$739$	−605915.			−1.10949			−0.554744	+	0.832021i	$0.687184\pi$
$740$			270460.i			0.493900i
$741$	20618.3	+	36033.6i	0.0375505	+	0.0656253i
$742$	954.778			0.00173418
$743$		−	207400.i		−	0.375690i	−0.982199	−	0.187845i	$-0.939850\pi$
$743$		−	207400.i		−	0.375690i	0.982199	−	0.187845i	$-0.0601503\pi$
$744$	−83309.1	+	47669.1i	−0.150503	+	0.0861174i
$745$	190292.			0.342854
$746$		−	57246.6i		−	0.102866i
$747$	−426015.	−	250381.i	−0.763455	−	0.448705i
$748$	−465486.			−0.831961
$749$		−	58310.6i		−	0.103940i
$750$	−10574.7	−	18480.9i	−0.0187994	−	0.0328549i
$751$	−6227.67			−0.0110419			−0.00552097	−	0.999985i	$-0.501757\pi$
$751$	−6227.67			−0.0110419			−0.00552097	+	0.999985i	$0.501757\pi$
$752$		−	800933.i		−	1.41632i
$753$	324275.	−	185549.i	0.571905	−	0.327242i
$754$	5864.50			0.0103155
$755$		−	284664.i		−	0.499388i
$756$	215215.	+	2501.83i	0.376555	+	0.00437738i
$757$	212971.			0.371645			0.185823	−	0.982583i	$-0.440505\pi$
$757$	212971.			0.371645			0.185823	+	0.982583i	$0.440505\pi$
$758$			26990.3i			0.0469753i
$759$	−321259.	−	561450.i	−0.557663	−	0.974603i
$760$	70267.8			0.121655
$761$		−	41126.2i		−	0.0710149i	−0.999369	−	0.0355075i	$-0.988695\pi$
$761$		−	41126.2i		−	0.0710149i	0.999369	−	0.0355075i	$-0.0113048\pi$
$762$	27880.4	−	15953.0i	0.0480163	−	0.0274747i
$763$	79505.8			0.136568
$764$		−	708550.i		−	1.21390i
$765$	−414759.	+	705697.i	−0.708717	+	1.20586i
$766$	−28727.0			−0.0489590
$767$		−	9543.76i		−	0.0162229i
$768$	−282262.	−	493296.i	−0.478552	−	0.836344i
$769$	1.15006e6			1.94477			0.972386	−	0.233380i	$-0.0749785\pi$
$769$	1.15006e6			1.94477			0.972386	+	0.233380i	$0.0749785\pi$
$770$			12039.3i			0.0203058i
$771$	415205.	−	237578.i	0.698480	−	0.399667i
$772$	283439.			0.475582
$773$		−	134095.i		−	0.224416i	−0.993685	−	0.112208i	$-0.964208\pi$
$773$		−	134095.i		−	0.224416i	0.993685	−	0.112208i	$-0.0357924\pi$
$774$	18927.3	+	11124.1i	0.0315941	+	0.0185688i
$775$	419883.			0.699077
$776$		−	72615.1i		−	0.120588i
$777$	46068.4	+	80511.7i	0.0763065	+	0.133357i
$778$	−67353.0			−0.111275
$779$			19727.2i			0.0325080i
$780$	58745.0	−	33613.6i	0.0965565	−	0.0552492i
$781$	180571.			0.296038
$782$			65469.7i			0.107060i
$783$	−13267.3	+	1.14130e6i	−0.0216402	+	1.86155i
$784$	86844.5			0.141290
$785$		−	258156.i		−	0.418932i
$786$	−22539.3	−	39391.0i	−0.0364835	−	0.0637605i
$787$	922036.			1.48867			0.744335	−	0.667807i	$-0.232767\pi$
$787$	922036.			1.48867			0.744335	+	0.667807i	$0.232767\pi$
$788$		−	815716.i		−	1.31367i
$789$	−500818.	+	286566.i	−0.804500	+	0.460331i
$790$	−53978.0			−0.0864893
$791$			80270.2i			0.128293i
$792$	27953.6	−	47562.0i	0.0445643	−	0.0758246i
$793$	48467.0			0.0770726
$794$			26278.7i			0.0416833i
$795$	29021.1	+	50718.8i	0.0459176	+	0.0802481i
$796$	−115535.			−0.182342
$797$			737363.i			1.16082i	0.814325	+	0.580409i	$0.197107\pi$
$797$			737363.i			1.16082i	−0.814325	+	0.580409i	$0.802893\pi$
$798$	10439.6	−	5973.51i	0.0163938	−	0.00938046i
$799$	1.04859e6			1.64253
$800$		−	56315.7i		−	0.0879932i
$801$	−475003.	−	279173.i	−0.740340	−	0.435120i
$802$	3756.30			0.00583997
$803$		−	540339.i		−	0.837983i
$804$	−500812.	−	875246.i	−0.774751	−	1.35400i
$805$	−460682.			−0.710902
$806$			5166.51i			0.00795294i
$807$	−900681.	+	515366.i	−1.38301	+	0.791350i
$808$	−62274.0			−0.0953858
$809$			1.05292e6i			1.60879i	0.594095	+	0.804395i	$0.297510\pi$
$809$			1.05292e6i			1.60879i	−0.594095	+	0.804395i	$0.702490\pi$
$810$	−23556.1	−	42301.2i	−0.0359032	−	0.0644737i
$811$	−1.10643e6			−1.68221			−0.841105	−	0.540871i	$-0.818095\pi$
$811$	−1.10643e6			−1.68221			−0.841105	+	0.540871i	$0.818095\pi$
$812$			462247.i			0.701071i
$813$	−485619.	−	848695.i	−0.734708	−	1.28402i
$814$	11866.5			0.0179091
$815$			220187.i			0.331494i
$816$	655613.	−	375139.i	0.984617	−	0.563393i
$817$	−333787.			−0.500064
$818$		−	946.265i		−	0.00141419i
$819$	11762.0	−	20012.5i	0.0175352	−	0.0298356i
$820$	32160.9			0.0478301
$821$		−	123241.i		−	0.182840i	−0.995812	−	0.0914198i	$-0.970859\pi$
$821$		−	123241.i		−	0.182840i	0.995812	−	0.0914198i	$-0.0291405\pi$
$822$	−26269.3	−	45909.7i	−0.0388781	−	0.0679456i
$823$	792158.			1.16953			0.584766	−	0.811202i	$-0.301186\pi$
$823$	792158.			1.16953			0.584766	+	0.811202i	$0.301186\pi$
$824$		−	24510.6i		−	0.0360994i
$825$	−209470.	+	119858.i	−0.307761	+	0.176100i
$826$	−2765.02			−0.00405264
$827$		−	699053.i		−	1.02211i	−0.859547	−	0.511057i	$-0.829254\pi$
$827$		−	699053.i		−	1.02211i	0.859547	−	0.511057i	$-0.170746\pi$
$828$	908306.	+	533838.i	1.32486	+	0.778662i
$829$	1.22153e6			1.77744			0.888721	−	0.458449i	$-0.151595\pi$
$829$	1.22153e6			1.77744			0.888721	+	0.458449i	$0.151595\pi$
$830$			45019.9i			0.0653504i
$831$	201403.	+	351983.i	0.291651	+	0.509705i
$832$	−61992.7			−0.0895558
$833$			113698.i			0.163856i
$834$	22287.4	−	12752.8i	0.0320426	−	0.0183346i
$835$	531054.			0.761667
$836$			418615.i			0.598965i
$837$	−1.00546e6	−	11688.3i	−1.43521	−	0.0166840i
$838$	31824.3			0.0453180
$839$		−	682312.i		−	0.969303i	−0.874707	−	0.484651i	$-0.838947\pi$
$839$		−	682312.i		−	0.969303i	0.874707	−	0.484651i	$-0.161053\pi$
$840$	−19512.8	−	34101.7i	−0.0276543	−	0.0483301i
$841$	−1.74404e6			−2.46584
$842$			63918.1i			0.0901570i
$843$	−151328.	+	86589.3i	−0.212944	+	0.121845i
$844$	−519793.			−0.729702
$845$			863418.i			1.20923i
$846$	−31427.6	+	53472.9i	−0.0439107	+	0.0747125i
$847$	127445.			0.177646
$848$		−	53922.8i		−	0.0749861i
$849$	54296.4	+	94891.4i	0.0753279	+	0.131647i
$850$	24425.9			0.0338075
$851$			454069.i			0.626994i
$852$	−255267.	+	146063.i	−0.351655	+	0.201215i
$853$	765325.			1.05184			0.525918	−	0.850535i	$-0.323722\pi$
$853$	765325.			1.05184			0.525918	+	0.850535i	$0.323722\pi$
$854$		−	14041.8i		−	0.0192534i
$855$	634639.	+	372996.i	0.868149	+	0.510237i
$856$	24343.6			0.0332229
$857$			1.36272e6i			1.85543i	0.373289	+	0.927715i	$0.378230\pi$
$857$			1.36272e6i			1.85543i	−0.373289	+	0.927715i	$0.621770\pi$
$858$	−1474.81	−	2577.46i	−0.00200337	−	0.00350120i
$859$	−1.03986e6			−1.40926			−0.704628	−	0.709577i	$-0.748886\pi$
$859$	−1.03986e6			−1.40926			−0.704628	+	0.709577i	$0.748886\pi$
$860$			544168.i			0.735760i
$861$	9573.83	−	5478.10i	0.0129146	−	0.00738965i
$862$	55498.9			0.0746913
$863$		−	563532.i		−	0.756653i	−0.925672	−	0.378327i	$-0.876500\pi$
$863$		−	563532.i		−	0.756653i	0.925672	−	0.378327i	$-0.123500\pi$
$864$	−1567.66	+	134855.i	−0.00210002	+	0.180650i
$865$	−486381.			−0.650047
$866$		−	26683.9i		−	0.0355806i
$867$	117818.	+	205906.i	0.156738	+	0.273924i
$868$	−407231.			−0.540507
$869$		−	644321.i		−	0.853223i
$870$	90255.9	−	51644.0i	0.119244	−	0.0682310i
$871$	−108758.			−0.143360
$872$			33192.2i			0.0436520i
$873$	385456.	−	655839.i	0.505762	−	0.860535i
$874$	58877.4			0.0770772
$875$		−	181009.i		−	0.236419i
$876$	437076.	+	763858.i	0.569573	+	0.995416i
$877$	−646978.			−0.841183			−0.420592	−	0.907250i	$-0.638177\pi$
$877$	−646978.			−0.841183			−0.420592	+	0.907250i	$0.638177\pi$
$878$			14928.5i			0.0193655i
$879$	−308864.	+	176730.i	−0.399751	+	0.228736i
$880$	679943.			0.878026
$881$		−	530305.i		−	0.683242i	−0.939838	−	0.341621i	$-0.889024\pi$
$881$		−	530305.i		−	0.683242i	0.939838	−	0.341621i	$-0.110976\pi$
$882$	−5798.02	−	3407.67i	−0.00745320	−	0.00438047i
$883$	485702.			0.622943			0.311471	−	0.950256i	$-0.399178\pi$
$883$	485702.			0.622943			0.311471	+	0.950256i	$0.399178\pi$
$884$		−	81768.4i		−	0.104636i
$885$	−84044.4	−	146881.i	−0.107306	−	0.187533i
$886$	−13241.6			−0.0168684
$887$			670457.i			0.852165i	0.904684	+	0.426082i	$0.140107\pi$
$887$			670457.i			0.852165i	−0.904684	+	0.426082i	$0.859893\pi$
$888$	−33612.2	+	19232.7i	−0.0426256	+	0.0243902i
$889$	273070.			0.345518
$890$			50196.8i			0.0633718i
$891$	504938.	−	281183.i	0.636038	−	0.354188i
$892$	−882415.			−1.10903
$893$		−	943008.i		−	1.18253i
$894$	6753.56	+	11802.9i	0.00845003	+	0.0147677i
$895$	852858.			1.06471
$896$			72780.0i			0.0906559i
$897$	98625.8	−	56433.3i	0.122576	−	0.0701375i
$898$	26663.3			0.0330645
$899$		−	2.15957e6i		−	2.67207i
$900$	199168.	−	338878.i	0.245887	−	0.418367i
$901$	70596.6			0.0869630
$902$		−	1411.07i		−	0.00173435i
$903$	92690.3	+	161991.i	0.113673	+	0.198662i
$904$	−33511.4			−0.0410068
$905$		−	787343.i		−	0.961318i
$906$	17656.3	−	10102.8i	0.0215101	−	0.0123080i
$907$	916171.			1.11368			0.556842	−	0.830618i	$-0.312013\pi$
$907$	916171.			1.11368			0.556842	+	0.830618i	$0.312013\pi$
$908$			93110.3i			0.112934i
$909$	−562441.	−	330563.i	−0.680690	−	0.400061i
$910$	−2114.86			−0.00255387
$911$			1.30288e6i			1.56989i	0.619567	+	0.784944i	$0.287308\pi$
$911$			1.30288e6i			1.56989i	−0.619567	+	0.784944i	$0.712692\pi$
$912$	−337365.	−	589598.i	−0.405612	−	0.708869i
$913$	−537390.			−0.644686
$914$		−	59999.3i		−	0.0718214i
$915$	745918.	−	426811.i	0.890940	−	0.509792i
$916$	−33527.3			−0.0399584
$917$		−	385809.i		−	0.458811i
$918$	−58490.9	−	679.945i	−0.0694069	−	0.000806842i
$919$	−307397.			−0.363972			−0.181986	−	0.983301i	$-0.558253\pi$
$919$	−307397.			−0.363972			−0.181986	+	0.983301i	$0.558253\pi$
$920$		−	192327.i		−	0.227229i
$921$	230816.	+	403387.i	0.272111	+	0.475557i
$922$	−60649.9			−0.0713458
$923$			31719.6i			0.0372327i
$924$	203158.	−	116246.i	0.237953	−	0.136155i
$925$	169408.			0.197993
$926$			46674.9i			0.0544329i
$927$	130108.	−	221373.i	0.151406	−	0.257612i
$928$	−289647.			−0.336335
$929$		−	955189.i		−	1.10677i	−0.832925	−	0.553385i	$-0.813336\pi$
$929$		−	955189.i		−	1.10677i	0.832925	−	0.553385i	$-0.186664\pi$
$930$	45497.4	+	79513.7i	0.0526042	+	0.0919340i
$931$	102250.			0.117967
$932$			980119.i			1.12836i
$933$	512176.	−	293065.i	0.588378	−	0.336667i
$934$	20860.9			0.0239133
$935$			890192.i			1.01826i
$936$	8354.87	+	4910.40i	0.00953648	+	0.00560487i
$937$	93998.6			0.107064			0.0535318	−	0.998566i	$-0.482952\pi$
$937$	93998.6			0.107064			0.0535318	+	0.998566i	$0.482952\pi$
$938$			31509.5i			0.0358126i
$939$	−260566.	−	455380.i	−0.295520	−	0.516467i
$940$	−1.53737e6			−1.73989
$941$			904009.i			1.02092i	0.859900	+	0.510462i	$0.170526\pi$
$941$			904009.i			1.02092i	−0.859900	+	0.510462i	$0.829474\pi$
$942$	16012.1	−	9162.08i	0.0180446	−	0.0103251i
$943$	53994.4			0.0607191
$944$			156159.i			0.175236i
$945$	4784.48	−	411575.i	0.00535761	−	0.460878i
$946$	23875.6			0.0266791
$947$			1.03257e6i			1.15138i	0.817669	+	0.575689i	$0.195266\pi$
$947$			1.03257e6i			1.15138i	−0.817669	+	0.575689i	$0.804734\pi$
$948$	521186.	+	910854.i	0.579931	+	1.01352i
$949$	94917.4			0.105393
$950$		−	21966.4i		−	0.0243395i
$951$	979958.	−	560728.i	1.08354	−	0.619999i
$952$	−47466.9			−0.0523742
$953$			1.24014e6i			1.36548i	0.730664	+	0.682738i	$0.239211\pi$
$953$			1.24014e6i			1.36548i	−0.730664	+	0.682738i	$0.760789\pi$
$954$	−2115.87	+	3600.07i	−0.00232483	+	0.00395561i
$955$	−1.35503e6			−1.48573
$956$		−	1.18378e6i		−	1.29525i
$957$	616461.	+	1.07736e6i	0.673103	+	1.17635i
$958$	29361.9			0.0319929
$959$		−	449656.i		−	0.488926i
$960$	−954080.	+	545920.i	−1.03524	+	0.592361i
$961$	979020.			1.06010
$962$			2084.50i			0.00225243i
$963$	219865.	+	129221.i	0.237084	+	0.139341i
$964$	1.77406e6			1.90903
$965$		−	542048.i		−	0.582081i
$966$	−16349.8	−	28573.8i	−0.0175210	−	0.0306207i
$967$	−1.40510e6			−1.50264			−0.751319	−	0.659940i	$-0.770582\pi$
$967$	−1.40510e6			−1.50264			−0.751319	+	0.659940i	$0.770582\pi$
$968$			53205.9i			0.0567818i
$969$	771911.	−	441684.i	0.822090	−	0.470396i
$970$	−69306.9			−0.0736603
$971$		−	1.48138e6i		−	1.57119i	−0.618742	−	0.785594i	$-0.712357\pi$
$971$		−	1.48138e6i		−	1.57119i	0.618742	−	0.785594i	$-0.287643\pi$
$972$	−486366.	+	805940.i	−0.514791	+	0.853041i
$973$	218291.			0.230574
$974$			13313.1i			0.0140334i
$975$	−21054.5	−	36796.1i	−0.0221481	−	0.0387072i
$976$	−793039.			−0.832520
$977$			94526.4i			0.0990293i	0.998773	+	0.0495147i	$0.0157675\pi$
$977$			94526.4i			0.0990293i	−0.998773	+	0.0495147i	$0.984233\pi$
$978$	−13657.1	+	7814.52i	−0.0142784	+	0.00817005i
$979$	−599186.			−0.625167
$980$		−	166696.i		−	0.173569i
$981$	−176191.	+	299783.i	−0.183082	+	0.311508i
$982$	93811.9			0.0972825
$983$			338813.i			0.350633i	0.984512	+	0.175316i	$0.0560949\pi$
$983$			338813.i			0.350633i	−0.984512	+	0.175316i	$0.943905\pi$
$984$	2287.01	+	3996.90i	0.00236199	+	0.00412794i
$985$	−1.55997e6			−1.60784
$986$		−	125629.i		−	0.129222i
$987$	−457652.	+	261866.i	−0.469787	+	0.268810i
$988$	−73534.9			−0.0753321
$989$			913594.i			0.934029i
$990$	−45395.2	−	26680.1i	−0.0463169	−	0.0272218i
$991$	−1.43200e6			−1.45813			−0.729066	−	0.684443i	$-0.760045\pi$
$991$	−1.43200e6			−1.45813			−0.729066	+	0.684443i	$0.760045\pi$
$992$		−	255173.i		−	0.259306i
$993$	−377422.	−	659603.i	−0.382762	−	0.668935i
$994$	9189.81			0.00930108
$995$			220948.i			0.223174i
$996$	759690.	−	434691.i	0.765804	−	0.438190i
$997$	1.20016e6			1.20739			0.603697	−	0.797214i	$-0.293694\pi$
$997$	1.20016e6			1.20739			0.603697	+	0.797214i	$0.293694\pi$
$998$		−	70714.0i		−	0.0709977i
$999$	−405667.	−	4715.80i	−0.406480	−	0.00472525i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.b.a.8.5	yes	8
3.2	odd	2	inner	21.5.b.a.8.4	✓	8
4.3	odd	2		336.5.d.b.113.6		8
7.2	even	3		147.5.h.e.116.5		16
7.3	odd	6		147.5.h.c.128.4		16
7.4	even	3		147.5.h.e.128.4		16
7.5	odd	6		147.5.h.c.116.5		16
7.6	odd	2		147.5.b.e.50.5		8
12.11	even	2		336.5.d.b.113.5		8
21.2	odd	6		147.5.h.e.116.4		16
21.5	even	6		147.5.h.c.116.4		16
21.11	odd	6		147.5.h.e.128.5		16
21.17	even	6		147.5.h.c.128.5		16
21.20	even	2		147.5.b.e.50.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.b.a.8.4	✓	8	3.2	odd	2	inner
21.5.b.a.8.5	yes	8	1.1	even	1	trivial
147.5.b.e.50.4		8	21.20	even	2
147.5.b.e.50.5		8	7.6	odd	2
147.5.h.c.116.4		16	21.5	even	6
147.5.h.c.116.5		16	7.5	odd	6
147.5.h.c.128.4		16	7.3	odd	6
147.5.h.c.128.5		16	21.17	even	6
147.5.h.e.116.4		16	21.2	odd	6
147.5.h.e.116.5		16	7.2	even	3
147.5.h.e.128.4		16	7.4	even	3
147.5.h.e.128.5		16	21.11	odd	6
336.5.d.b.113.5		8	12.11	even	2
336.5.d.b.113.6		8	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	21.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.242064i q^{2} +(-4.46977 - 7.81161i) q^{3} +15.9414 q^{4} -30.4863i q^{5} +(1.89091 - 1.08197i) q^{6} +18.5203 q^{7} +7.73187i q^{8} +(-41.0424 + 69.8321i) q^{9} +O(q^{10})\)	\(q+0.242064i q^{2} +(-4.46977 - 7.81161i) q^{3} +15.9414 q^{4} -30.4863i q^{5} +(1.89091 - 1.08197i) q^{6} +18.5203 q^{7} +7.73187i q^{8} +(-41.0424 + 69.8321i) q^{9} +7.37963 q^{10} +88.0887i q^{11} +(-71.2543 - 124.528i) q^{12} -15.4739 q^{13} +4.48309i q^{14} +(-238.147 + 136.266i) q^{15} +253.191 q^{16} +331.481i q^{17} +(-16.9039 - 9.93490i) q^{18} +298.104 q^{19} -485.994i q^{20} +(-82.7812 - 144.673i) q^{21} -21.3231 q^{22} -815.926i q^{23} +(60.3984 - 34.5597i) q^{24} -304.412 q^{25} -3.74568i q^{26} +(728.951 + 8.47391i) q^{27} +295.239 q^{28} +1565.67i q^{29} +(-32.9852 - 57.6468i) q^{30} -1379.33 q^{31} +184.998i q^{32} +(688.115 - 393.736i) q^{33} -80.2398 q^{34} -564.613i q^{35} +(-654.273 + 1113.22i) q^{36} -556.508 q^{37} +72.1603i q^{38} +(69.1647 + 120.876i) q^{39} +235.716 q^{40} +66.1756i q^{41} +(35.0202 - 20.0384i) q^{42} -1119.70 q^{43} +1404.26i q^{44} +(2128.92 + 1251.23i) q^{45} +197.506 q^{46} -3163.36i q^{47} +(-1131.70 - 1977.83i) q^{48} +343.000 q^{49} -73.6872i q^{50} +(2589.40 - 1481.64i) q^{51} -246.676 q^{52} -212.973i q^{53} +(-2.05123 + 176.453i) q^{54} +2685.50 q^{55} +143.196i q^{56} +(-1332.45 - 2328.67i) q^{57} -378.993 q^{58} +616.765i q^{59} +(-3796.39 + 2172.28i) q^{60} -3132.18 q^{61} -333.886i q^{62} +(-760.116 + 1293.31i) q^{63} +4006.27 q^{64} +471.741i q^{65} +(95.3094 + 166.568i) q^{66} +7028.51 q^{67} +5284.28i q^{68} +(-6373.69 + 3647.00i) q^{69} +136.673 q^{70} -2049.88i q^{71} +(-539.933 - 317.335i) q^{72} -6134.03 q^{73} -134.711i q^{74} +(1360.65 + 2377.94i) q^{75} +4752.19 q^{76} +1631.43i q^{77} +(-29.2598 + 16.7423i) q^{78} -7314.45 q^{79} -7718.84i q^{80} +(-3192.04 - 5732.15i) q^{81} -16.0188 q^{82} +6100.56i q^{83} +(-1319.65 - 2306.29i) q^{84} +10105.6 q^{85} -271.040i q^{86} +(12230.4 - 6998.18i) q^{87} -681.091 q^{88} +6802.07i q^{89} +(-302.878 + 515.335i) q^{90} -286.581 q^{91} -13007.0i q^{92} +(6165.27 + 10774.8i) q^{93} +765.735 q^{94} -9088.06i q^{95} +(1445.14 - 826.900i) q^{96} -9391.65 q^{97} +83.0281i q^{98} +(-6151.42 - 3615.37i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9}+O(q^{10}) \) 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9	\( 8 q - 2 q^{3} - 36 q^{4} - 34 q^{6} + 64 q^{9} - 4 q^{10} + 98 q^{12} + 420 q^{13} + 76 q^{15} - 444 q^{16} - 712 q^{18} - 372 q^{19} + 98 q^{21} - 16 q^{22} + 1146 q^{24} + 1056 q^{25} - 1862 q^{27} + 392 q^{28} + 2348 q^{30} - 2776 q^{31} + 1396 q^{33} + 2928 q^{34} - 3268 q^{36} - 2560 q^{37} - 2540 q^{39} - 1980 q^{40} - 2450 q^{42} + 4720 q^{43} + 9700 q^{45} + 7536 q^{46} - 2962 q^{48} + 2744 q^{49} + 4764 q^{51} - 20252 q^{52} + 4886 q^{54} + 184 q^{55} - 14144 q^{57} - 7504 q^{58} - 13828 q^{60} + 972 q^{61} - 6076 q^{63} + 22772 q^{64} + 36020 q^{66} + 10200 q^{67} - 5760 q^{69} + 10780 q^{70} + 14304 q^{72} - 32008 q^{73} + 2114 q^{75} + 17332 q^{76} - 29668 q^{78} - 23168 q^{79} - 17216 q^{81} - 31976 q^{82} - 14798 q^{84} + 32016 q^{85} + 50764 q^{87} + 29208 q^{88} - 24352 q^{90} + 11956 q^{91} + 31848 q^{93} - 64992 q^{94} + 28630 q^{96} + 28112 q^{97} - 32432 q^{99}+O(q^{100}) \) 8 * q - 2 * q^3 - 36 * q^4 - 34 * q^6 + 64 * q^9 - 4 * q^10 + 98 * q^12 + 420 * q^13 + 76 * q^15 - 444 * q^16 - 712 * q^18 - 372 * q^19 + 98 * q^21 - 16 * q^22 + 1146 * q^24 + 1056 * q^25 - 1862 * q^27 + 392 * q^28 + 2348 * q^30 - 2776 * q^31 + 1396 * q^33 + 2928 * q^34 - 3268 * q^36 - 2560 * q^37 - 2540 * q^39 - 1980 * q^40 - 2450 * q^42 + 4720 * q^43 + 9700 * q^45 + 7536 * q^46 - 2962 * q^48 + 2744 * q^49 + 4764 * q^51 - 20252 * q^52 + 4886 * q^54 + 184 * q^55 - 14144 * q^57 - 7504 * q^58 - 13828 * q^60 + 972 * q^61 - 6076 * q^63 + 22772 * q^64 + 36020 * q^66 + 10200 * q^67 - 5760 * q^69 + 10780 * q^70 + 14304 * q^72 - 32008 * q^73 + 2114 * q^75 + 17332 * q^76 - 29668 * q^78 - 23168 * q^79 - 17216 * q^81 - 31976 * q^82 - 14798 * q^84 + 32016 * q^85 + 50764 * q^87 + 29208 * q^88 - 24352 * q^90 + 11956 * q^91 + 31848 * q^93 - 64992 * q^94 + 28630 * q^96 + 28112 * q^97 - 32432 * q^99

Introduction

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