LMFDB - Embedded newform 69.5.d.a.22.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,5,Mod(22,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(69, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("69.22");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	69.d (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:	$7.13252745279$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 5598 x^{14} + 11369517 x^{12} + 11272666128 x^{10} + 5958872960073 x^{8} + \cdots + 13\!\cdots\!52 $ x^16 + 5598x^14 + 11369517x^12 + 11272666128x^10 + 5958872960073x^8 + 1661250441139200x^6 + 215215956313867368x^4 + 8236967436040727664*x^2 + 13861812915684254052
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{9} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.3
Root			$-1.32778i$ of defining polynomial
Character	$\chi$	$=$	69.22
Dual form			69.5.d.a.22.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.20161 q^{2} -5.19615 q^{3} +11.0568 q^{4} -30.5542i q^{5} +27.0284 q^{6} -6.50343i q^{7} +25.7128 q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-5.20161 q^{2} -5.19615 q^{3} +11.0568 q^{4} -30.5542i q^{5} +27.0284 q^{6} -6.50343i q^{7} +25.7128 q^{8} +27.0000 q^{9} +158.931i q^{10} -127.895i q^{11} -57.4526 q^{12} -221.911 q^{13} +33.8283i q^{14} +158.764i q^{15} -310.656 q^{16} +381.095i q^{17} -140.443 q^{18} +573.894i q^{19} -337.831i q^{20} +33.7928i q^{21} +665.260i q^{22} +(-335.773 - 408.776i) q^{23} -133.608 q^{24} -308.561 q^{25} +1154.29 q^{26} -140.296 q^{27} -71.9068i q^{28} +217.744 q^{29} -825.831i q^{30} -509.950 q^{31} +1204.51 q^{32} +664.562i q^{33} -1982.31i q^{34} -198.707 q^{35} +298.532 q^{36} +76.8550i q^{37} -2985.17i q^{38} +1153.08 q^{39} -785.636i q^{40} +620.166 q^{41} -175.777i q^{42} +3391.21i q^{43} -1414.10i q^{44} -824.964i q^{45} +(1746.56 + 2126.29i) q^{46} -2385.65 q^{47} +1614.22 q^{48} +2358.71 q^{49} +1605.02 q^{50} -1980.23i q^{51} -2453.61 q^{52} +1774.26i q^{53} +729.766 q^{54} -3907.73 q^{55} -167.222i q^{56} -2982.04i q^{57} -1132.62 q^{58} -4491.84 q^{59} +1755.42i q^{60} -368.448i q^{61} +2652.56 q^{62} -175.593i q^{63} -1294.88 q^{64} +6780.31i q^{65} -3456.79i q^{66} +2473.09i q^{67} +4213.67i q^{68} +(1744.73 + 2124.06i) q^{69} +1033.60 q^{70} -2685.71 q^{71} +694.247 q^{72} -5506.59 q^{73} -399.770i q^{74} +1603.33 q^{75} +6345.41i q^{76} -831.756 q^{77} -5997.88 q^{78} -7442.57i q^{79} +9491.86i q^{80} +729.000 q^{81} -3225.86 q^{82} +9961.07i q^{83} +373.639i q^{84} +11644.1 q^{85} -17639.8i q^{86} -1131.43 q^{87} -3288.54i q^{88} -14103.5i q^{89} +4291.14i q^{90} +1443.18i q^{91} +(-3712.56 - 4519.73i) q^{92} +2649.78 q^{93} +12409.2 q^{94} +17534.9 q^{95} -6258.80 q^{96} -3342.83i q^{97} -12269.1 q^{98} -3453.17i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9}+O(q^{10}) $ 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9	$ 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9} + 104 q^{13} + 680 q^{16} + 324 q^{18} - 732 q^{23} - 1764 q^{24} - 2984 q^{25} + 1800 q^{26} - 3528 q^{29} - 400 q^{31} + 5244 q^{32} + 912 q^{35} + 3888 q^{36} + 2016 q^{39} + 1008 q^{41} - 1168 q^{46} - 8664 q^{47} - 2016 q^{48} + 7240 q^{49} - 18852 q^{50} - 20952 q^{52} - 972 q^{54} + 6816 q^{55} - 13352 q^{58} + 20112 q^{59} + 4248 q^{62} - 896 q^{64} - 10044 q^{69} - 10680 q^{70} + 40368 q^{71} + 10044 q^{72} - 9568 q^{73} + 7560 q^{75} + 2952 q^{77} - 6912 q^{78} + 11664 q^{81} + 71800 q^{82} + 42744 q^{85} + 8352 q^{87} - 9876 q^{92} - 10008 q^{93} + 73720 q^{94} + 33312 q^{95} - 24948 q^{96} - 59052 q^{98}+O(q^{100}) $ 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9 + 104 * q^13 + 680 * q^16 + 324 * q^18 - 732 * q^23 - 1764 * q^24 - 2984 * q^25 + 1800 * q^26 - 3528 * q^29 - 400 * q^31 + 5244 * q^32 + 912 * q^35 + 3888 * q^36 + 2016 * q^39 + 1008 * q^41 - 1168 * q^46 - 8664 * q^47 - 2016 * q^48 + 7240 * q^49 - 18852 * q^50 - 20952 * q^52 - 972 * q^54 + 6816 * q^55 - 13352 * q^58 + 20112 * q^59 + 4248 * q^62 - 896 * q^64 - 10044 * q^69 - 10680 * q^70 + 40368 * q^71 + 10044 * q^72 - 9568 * q^73 + 7560 * q^75 + 2952 * q^77 - 6912 * q^78 + 11664 * q^81 + 71800 * q^82 + 42744 * q^85 + 8352 * q^87 - 9876 * q^92 - 10008 * q^93 + 73720 * q^94 + 33312 * q^95 - 24948 * q^96 - 59052 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/69\mathbb{Z}\right)^\times$.

$n$	$28$	$47$
$\chi(n)$	$-1$	$1$

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 $

$2$	−5.20161			−1.30040			−0.650201	−	0.759762i	$-0.725315\pi$
$2$	−5.20161			−1.30040			−0.650201	+	0.759762i	$0.725315\pi$
$3$	−5.19615			−0.577350
$3$	−5.19615			−0.577350
$4$	11.0568			0.691047
$5$		−	30.5542i		−	1.22217i	−0.791565	−	0.611085i	$-0.790733\pi$
$5$		−	30.5542i		−	1.22217i	0.791565	−	0.611085i	$-0.209267\pi$
$6$	27.0284			0.750788
$7$		−	6.50343i		−	0.132723i	−0.997796	−	0.0663615i	$-0.978861\pi$
$7$		−	6.50343i		−	0.132723i	0.997796	−	0.0663615i	$-0.0211391\pi$
$8$	25.7128			0.401763
$9$	27.0000			0.333333
$10$			158.931i			1.58931i
$11$		−	127.895i		−	1.05698i	−0.848938	−	0.528492i	$-0.822758\pi$
$11$		−	127.895i		−	1.05698i	0.848938	−	0.528492i	$-0.177242\pi$
$12$	−57.4526			−0.398976
$13$	−221.911			−1.31308			−0.656540	−	0.754291i	$-0.727981\pi$
$13$	−221.911			−1.31308			−0.656540	+	0.754291i	$0.727981\pi$
$14$			33.8283i			0.172593i
$15$			158.764i			0.705620i
$16$	−310.656			−1.21350
$17$			381.095i			1.31867i	0.751850	+	0.659334i	$0.229162\pi$
$17$			381.095i			1.31867i	−0.751850	+	0.659334i	$0.770838\pi$
$18$	−140.443			−0.433468
$19$			573.894i			1.58973i	0.606783	+	0.794867i	$0.292460\pi$
$19$			573.894i			1.58973i	−0.606783	+	0.794867i	$0.707540\pi$
$20$		−	337.831i		−	0.844577i
$21$			33.7928i			0.0766277i
$22$			665.260i			1.37450i
$23$	−335.773	−	408.776i	−0.634731	−	0.772733i
$23$	−335.773	−	408.776i	−0.634731	−	0.772733i
$24$	−133.608			−0.231958
$25$	−308.561			−0.493698
$26$	1154.29			1.70753
$27$	−140.296			−0.192450
$28$		−	71.9068i		−	0.0917179i
$29$	217.744			0.258911			0.129456	−	0.991585i	$-0.458677\pi$
$29$	217.744			0.258911			0.129456	+	0.991585i	$0.458677\pi$
$30$		−	825.831i		−	0.917590i
$31$	−509.950			−0.530645			−0.265323	−	0.964160i	$-0.585478\pi$
$31$	−509.950			−0.530645			−0.265323	+	0.964160i	$0.585478\pi$
$32$	1204.51			1.17628
$33$			664.562i			0.610250i
$34$		−	1982.31i		−	1.71480i
$35$	−198.707			−0.162210
$36$	298.532			0.230349
$37$			76.8550i			0.0561395i	0.999606	+	0.0280698i	$0.00893606\pi$
$37$			76.8550i			0.0561395i	−0.999606	+	0.0280698i	$0.991064\pi$
$38$		−	2985.17i		−	2.06730i
$39$	1153.08			0.758107
$40$		−	785.636i		−	0.491022i
$41$	620.166			0.368927			0.184463	−	0.982839i	$-0.440945\pi$
$41$	620.166			0.368927			0.184463	+	0.982839i	$0.440945\pi$
$42$		−	175.777i		−	0.0996469i
$43$			3391.21i			1.83408i	0.398797	+	0.917039i	$0.369428\pi$
$43$			3391.21i			1.83408i	−0.398797	+	0.917039i	$0.630572\pi$
$44$		−	1414.10i		−	0.730426i
$45$		−	824.964i		−	0.407390i
$46$	1746.56	+	2126.29i	0.825406	+	1.00486i
$47$	−2385.65			−1.07997			−0.539984	−	0.841675i	$-0.681570\pi$
$47$	−2385.65			−1.07997			−0.539984	+	0.841675i	$0.681570\pi$
$48$	1614.22			0.700615
$49$	2358.71			0.982385
$50$	1605.02			0.642006
$51$		−	1980.23i		−	0.761333i
$52$	−2453.61			−0.907400
$53$			1774.26i			0.631634i	0.948820	+	0.315817i	$0.102279\pi$
$53$			1774.26i			0.631634i	−0.948820	+	0.315817i	$0.897721\pi$
$54$	729.766			0.250263
$55$	−3907.73			−1.29181
$56$		−	167.222i		−	0.0533232i
$57$		−	2982.04i		−	0.917834i
$58$	−1132.62			−0.336689
$59$	−4491.84			−1.29039			−0.645194	−	0.764019i	$-0.723223\pi$
$59$	−4491.84			−1.29039			−0.645194	+	0.764019i	$0.723223\pi$
$60$			1755.42i			0.487617i
$61$		−	368.448i		−	0.0990186i	−0.998774	−	0.0495093i	$-0.984234\pi$
$61$		−	368.448i		−	0.0990186i	0.998774	−	0.0495093i	$-0.0157658\pi$
$62$	2652.56			0.690052
$63$		−	175.593i		−	0.0442410i
$64$	−1294.88			−0.316133
$65$			6780.31i			1.60481i
$66$		−	3456.79i		−	0.793570i
$67$			2473.09i			0.550923i	0.961312	+	0.275461i	$0.0888306\pi$
$67$			2473.09i			0.550923i	−0.961312	+	0.275461i	$0.911169\pi$
$68$			4213.67i			0.911262i
$69$	1744.73	+	2124.06i	0.366462	+	0.446138i
$70$	1033.60			0.210938
$71$	−2685.71			−0.532773			−0.266387	−	0.963866i	$-0.585830\pi$
$71$	−2685.71			−0.532773			−0.266387	+	0.963866i	$0.585830\pi$
$72$	694.247			0.133921
$73$	−5506.59			−1.03332			−0.516662	−	0.856189i	$-0.672826\pi$
$73$	−5506.59			−1.03332			−0.516662	+	0.856189i	$0.672826\pi$
$74$		−	399.770i		−	0.0730040i
$75$	1603.33			0.285037
$76$			6345.41i			1.09858i
$77$	−831.756			−0.140286
$78$	−5997.88			−0.985844
$79$		−	7442.57i		−	1.19253i	−0.802788	−	0.596264i	$-0.796651\pi$
$79$		−	7442.57i		−	1.19253i	0.802788	−	0.596264i	$-0.203349\pi$
$80$			9491.86i			1.48310i
$81$	729.000			0.111111
$82$	−3225.86			−0.479753
$83$			9961.07i			1.44594i	0.690880	+	0.722969i	$0.257223\pi$
$83$			9961.07i			1.44594i	−0.690880	+	0.722969i	$0.742777\pi$
$84$			373.639i			0.0529534i
$85$	11644.1			1.61164
$86$		−	17639.8i		−	2.38504i
$87$	−1131.43			−0.149482
$88$		−	3288.54i		−	0.424657i
$89$		−	14103.5i		−	1.78052i	−0.455448	−	0.890262i	$-0.650521\pi$
$89$		−	14103.5i		−	1.78052i	0.455448	−	0.890262i	$-0.349479\pi$
$90$			4291.14i			0.529771i
$91$			1443.18i			0.174276i
$92$	−3712.56	−	4519.73i	−0.438629	−	0.533995i
$93$	2649.78			0.306368
$94$	12409.2			1.40439
$95$	17534.9			1.94293
$96$	−6258.80			−0.679124
$97$		−	3342.83i		−	0.355280i	−0.984096	−	0.177640i	$-0.943154\pi$
$97$		−	3342.83i		−	0.355280i	0.984096	−	0.177640i	$-0.0568463\pi$
$98$	−12269.1			−1.27750
$99$		−	3453.17i		−	0.352328i
$100$	−3411.69			−0.341169
$101$	−6006.20			−0.588785			−0.294393	−	0.955685i	$-0.595117\pi$
$101$	−6006.20			−0.588785			−0.294393	+	0.955685i	$0.595117\pi$
$102$			10300.4i			0.990040i
$103$		−	5223.23i		−	0.492340i	−0.969227	−	0.246170i	$-0.920828\pi$
$103$		−	5223.23i		−	0.492340i	0.969227	−	0.246170i	$-0.0791721\pi$
$104$	−5705.95			−0.527547
$105$	1032.51			0.0936520
$106$		−	9229.00i		−	0.821378i
$107$			11256.0i			0.983139i	0.870838	+	0.491569i	$0.163577\pi$
$107$			11256.0i			0.983139i	−0.870838	+	0.491569i	$0.836423\pi$
$108$	−1551.22			−0.132992
$109$			2807.96i			0.236341i	0.992993	+	0.118170i	$0.0377029\pi$
$109$			2807.96i			0.236341i	−0.992993	+	0.118170i	$0.962297\pi$
$110$	20326.5			1.67988
$111$		−	399.350i		−	0.0324122i
$112$			2020.33i			0.161060i
$113$		−	5250.11i		−	0.411161i	−0.978640	−	0.205580i	$-0.934092\pi$
$113$		−	5250.11i		−	0.411161i	0.978640	−	0.205580i	$-0.0659082\pi$
$114$			15511.4i			1.19355i
$115$	−12489.8	+	10259.3i	−0.944411	+	0.775749i
$116$	2407.55			0.178920
$117$	−5991.58			−0.437693
$118$	23364.8			1.67802
$119$	2478.42			0.175018
$120$			4082.28i			0.283492i
$121$	−1716.14			−0.117214
$122$			1916.52i			0.128764i
$123$	−3222.48			−0.213000
$124$	−5638.39			−0.366701
$125$		−	9668.54i		−	0.618787i
$126$			913.364i			0.0575311i
$127$	−19316.3			−1.19761			−0.598806	−	0.800894i	$-0.704358\pi$
$127$	−19316.3			−1.19761			−0.598806	+	0.800894i	$0.704358\pi$
$128$	−12536.7			−0.765177
$129$		−	17621.3i		−	1.05891i
$130$		−	35268.5i		−	2.08689i
$131$	8746.45			0.509670			0.254835	−	0.966985i	$-0.417979\pi$
$131$	8746.45			0.509670			0.254835	+	0.966985i	$0.417979\pi$
$132$			7347.90i			0.421711i
$133$	3732.28			0.210994
$134$		−	12864.1i		−	0.716422i
$135$			4286.64i			0.235207i
$136$			9799.03i			0.529792i
$137$		−	15090.9i		−	0.804032i	−0.915633	−	0.402016i	$-0.868310\pi$
$137$		−	15090.9i		−	0.804032i	0.915633	−	0.402016i	$-0.131690\pi$
$138$	−9075.39	−	11048.5i	−0.476548	−	0.580159i
$139$	36495.1			1.88888			0.944440	−	0.328684i	$-0.106605\pi$
$139$	36495.1			1.88888			0.944440	+	0.328684i	$0.106605\pi$
$140$	−2197.06			−0.112095
$141$	12396.2			0.623520
$142$	13970.0			0.692819
$143$			28381.2i			1.38790i
$144$	−8387.72			−0.404500
$145$		−	6653.01i		−	0.316433i
$146$	28643.1			1.34374
$147$	−12256.2			−0.567180
$148$			849.767i			0.0387951i
$149$		−	16685.9i		−	0.751582i	−0.926705	−	0.375791i	$-0.877371\pi$
$149$		−	16685.9i		−	0.751582i	0.926705	−	0.375791i	$-0.122629\pi$
$150$	−8339.91			−0.370662
$151$	−8209.72			−0.360060			−0.180030	−	0.983661i	$-0.557619\pi$
$151$	−8209.72			−0.360060			−0.180030	+	0.983661i	$0.557619\pi$
$152$			14756.4i			0.638697i
$153$			10289.6i			0.439556i
$154$	4326.47			0.182428
$155$			15581.1i			0.648538i
$156$	12749.3			0.523888
$157$			6980.98i			0.283215i	0.989923	+	0.141608i	$0.0452272\pi$
$157$			6980.98i			0.283215i	−0.989923	+	0.141608i	$0.954773\pi$
$158$			38713.3i			1.55077i
$159$		−	9219.32i		−	0.364674i
$160$		−	36802.8i		−	1.43761i
$161$	−2658.44	+	2183.67i	−0.102559	+	0.0842435i
$162$	−3791.97			−0.144489
$163$	−29696.1			−1.11770			−0.558849	−	0.829269i	$-0.688757\pi$
$163$	−29696.1			−1.11770			−0.558849	+	0.829269i	$0.688757\pi$
$164$	6857.02			0.254946
$165$	20305.2			0.745829
$166$		−	51813.6i		−	1.88030i
$167$	−40520.2			−1.45291			−0.726454	−	0.687215i	$-0.758833\pi$
$167$	−40520.2			−1.45291			−0.726454	+	0.687215i	$0.758833\pi$
$168$			868.909i			0.0307862i
$169$	20683.3			0.724179
$170$	−60567.9			−2.09577
$171$			15495.1i			0.529912i
$172$			37495.8i			1.26744i
$173$	−40607.5			−1.35680			−0.678398	−	0.734695i	$-0.737325\pi$
$173$	−40607.5			−1.35680			−0.678398	+	0.734695i	$0.737325\pi$
$174$	5885.27			0.194387
$175$			2006.71i			0.0655251i
$176$			39731.4i			1.28265i
$177$	23340.3			0.745006
$178$			73361.1i			2.31540i
$179$	4124.42			0.128723			0.0643617	−	0.997927i	$-0.479499\pi$
$179$	4124.42			0.128723			0.0643617	+	0.997927i	$0.479499\pi$
$180$		−	9121.43i		−	0.281526i
$181$			30828.9i			0.941025i	0.882393	+	0.470513i	$0.155931\pi$
$181$			30828.9i			0.941025i	−0.882393	+	0.470513i	$0.844069\pi$
$182$		−	7506.86i		−	0.226629i
$183$			1914.51i			0.0571684i
$184$	−8633.67	−	10510.8i	−0.255012	−	0.310456i
$185$	2348.25			0.0686120
$186$	−13783.1			−0.398402
$187$	48740.2			1.39381
$188$	−26377.6			−0.746309
$189$			912.406i			0.0255426i
$190$	−91209.7			−2.52659
$191$		−	29528.0i		−	0.809408i	−0.914448	−	0.404704i	$-0.867375\pi$
$191$		−	29528.0i		−	0.809408i	0.914448	−	0.404704i	$-0.132625\pi$
$192$	6728.39			0.182519
$193$	−42618.9			−1.14416			−0.572081	−	0.820197i	$-0.693864\pi$
$193$	−42618.9			−1.14416			−0.572081	+	0.820197i	$0.693864\pi$
$194$			17388.1i			0.462007i
$195$		−	35231.5i		−	0.926535i
$196$	26079.6			0.678874
$197$	15166.7			0.390803			0.195402	−	0.980723i	$-0.437399\pi$
$197$	15166.7			0.390803			0.195402	+	0.980723i	$0.437399\pi$
$198$			17962.0i			0.458168i
$199$			41253.6i			1.04173i	0.853639	+	0.520865i	$0.174391\pi$
$199$			41253.6i			1.04173i	−0.853639	+	0.520865i	$0.825609\pi$
$200$	−7933.98			−0.198350
$201$		−	12850.6i		−	0.318076i
$202$	31241.9			0.765658
$203$		−	1416.08i		−	0.0343635i
$204$		−	21894.9i		−	0.526117i
$205$		−	18948.7i		−	0.450891i
$206$			27169.2i			0.640240i
$207$	−9065.87	−	11036.9i	−0.211577	−	0.257578i
$208$	68937.9			1.59342
$209$	73398.2			1.68032
$210$	−5370.73			−0.121785
$211$	−20952.4			−0.470619			−0.235309	−	0.971921i	$-0.575610\pi$
$211$	−20952.4			−0.470619			−0.235309	+	0.971921i	$0.575610\pi$
$212$			19617.5i			0.436489i
$213$	13955.4			0.307597
$214$		−	58549.1i		−	1.27848i
$215$	103616.			2.24155
$216$	−3607.41			−0.0773193
$217$			3316.42i			0.0704289i
$218$		−	14605.9i		−	0.307338i
$219$	28613.1			0.596590
$220$	−43206.9			−0.892704
$221$		−	84569.0i		−	1.73152i
$222$			2077.27i			0.0421489i
$223$	−81468.6			−1.63825			−0.819125	−	0.573614i	$-0.805541\pi$
$223$	−81468.6			−1.63825			−0.819125	+	0.573614i	$0.805541\pi$
$224$		−	7833.43i		−	0.156119i
$225$	−8331.15			−0.164566
$226$			27309.0i			0.534674i
$227$		−	98869.9i		−	1.91872i	−0.282178	−	0.959362i	$-0.591057\pi$
$227$		−	98869.9i		−	1.91872i	0.282178	−	0.959362i	$-0.408943\pi$
$228$		−	32971.7i		−	0.634267i
$229$		−	65040.7i		−	1.24027i	−0.784497	−	0.620133i	$-0.787079\pi$
$229$		−	65040.7i		−	1.24027i	0.784497	−	0.620133i	$-0.212921\pi$
$230$	64967.2	−	53364.8i	1.22811	−	1.00879i
$231$	4321.93			0.0809942
$232$	5598.82			0.104021
$233$	86625.0			1.59563			0.797813	−	0.602905i	$-0.205990\pi$
$233$	86625.0			1.59563			0.797813	+	0.602905i	$0.205990\pi$
$234$	31165.9			0.569178
$235$			72891.7i			1.31990i
$236$	−49665.2			−0.891719
$237$			38672.7i			0.688506i
$238$	−12891.8			−0.227593
$239$	−64409.1			−1.12759			−0.563795	−	0.825915i	$-0.690659\pi$
$239$	−64409.1			−1.12759			−0.563795	+	0.825915i	$0.690659\pi$
$240$		−	49321.2i		−	0.856270i
$241$			113191.i			1.94885i	0.224703	+	0.974427i	$0.427859\pi$
$241$			113191.i			1.94885i	−0.224703	+	0.974427i	$0.572141\pi$
$242$	8926.67			0.152426
$243$	−3788.00			−0.0641500
$244$		−	4073.84i		−	0.0684266i
$245$		−	72068.4i		−	1.20064i
$246$	16762.1			0.276986
$247$		−	127353.i		−	2.08745i
$248$	−13112.3			−0.213194
$249$		−	51759.2i		−	0.834813i
$250$			50292.0i			0.804672i
$251$			22920.8i			0.363816i	0.983316	+	0.181908i	$0.0582273\pi$
$251$			22920.8i			0.363816i	−0.983316	+	0.181908i	$0.941773\pi$
$252$		−	1941.48i		−	0.0305726i
$253$	−52280.4	+	42943.7i	−0.816766	+	0.670901i
$254$	100476.			1.55738
$255$	−60504.3			−0.930478
$256$	85928.9			1.31117
$257$	89925.1			1.36149			0.680745	−	0.732520i	$-0.261656\pi$
$257$	89925.1			1.36149			0.680745	+	0.732520i	$0.261656\pi$
$258$			91658.9i			1.37700i
$259$	499.821			0.00745101
$260$			74968.2i			1.10900i
$261$	5879.10			0.0863037
$262$	−45495.6			−0.662776
$263$			42151.5i			0.609399i	0.952448	+	0.304700i	$0.0985561\pi$
$263$			42151.5i			0.609399i	−0.952448	+	0.304700i	$0.901444\pi$
$264$			17087.8i			0.245176i
$265$	54211.1			0.771963
$266$	−19413.9			−0.274378
$267$			73284.1i			1.02799i
$268$			27344.4i			0.380714i
$269$	−123973.			−1.71326			−0.856630	−	0.515931i	$-0.827446\pi$
$269$	−123973.			−1.71326			−0.856630	+	0.515931i	$0.827446\pi$
$270$		−	22297.4i		−	0.305863i
$271$	−82316.6			−1.12085			−0.560427	−	0.828204i	$-0.689363\pi$
$271$	−82316.6			−1.12085			−0.560427	+	0.828204i	$0.689363\pi$
$272$		−	118390.i		−	1.60020i
$273$		−	7498.98i		−	0.100618i
$274$			78496.9i			1.04557i
$275$			39463.4i			0.521831i
$276$	19291.0	+	23485.2i	0.253243	+	0.308302i
$277$	72686.5			0.947315			0.473657	−	0.880709i	$-0.342934\pi$
$277$	72686.5			0.947315			0.473657	+	0.880709i	$0.342934\pi$
$278$	−189833.			−2.45630
$279$	−13768.7			−0.176882
$280$	−5109.33			−0.0651700
$281$		−	76557.0i		−	0.969555i	−0.874638	−	0.484777i	$-0.838901\pi$
$281$		−	76557.0i		−	0.969555i	0.874638	−	0.484777i	$-0.161099\pi$
$282$	−64480.2			−0.810827
$283$			114220.i			1.42616i	0.701083	+	0.713080i	$0.252700\pi$
$283$			114220.i			1.42616i	−0.701083	+	0.713080i	$0.747300\pi$
$284$	−29695.2			−0.368171
$285$	−91114.0			−1.12175
$286$		−	147628.i		−	1.80483i
$287$		−	4033.21i		−	0.0489651i
$288$	32521.7			0.392092
$289$	−61712.4			−0.738885
$290$			34606.4i			0.411491i
$291$			17369.9i			0.205121i
$292$	−60885.0			−0.714076
$293$			152286.i			1.77388i	0.461885	+	0.886940i	$0.347173\pi$
$293$			152286.i			1.77388i	−0.461885	+	0.886940i	$0.652827\pi$
$294$	63751.9			0.737562
$295$			137245.i			1.57707i
$296$			1976.16i			0.0225548i
$297$			17943.2i			0.203417i
$298$			86793.4i			0.977359i
$299$	74511.5	+	90711.6i	0.833453	+	1.01466i
$300$	17727.6			0.196974
$301$	22054.5			0.243425
$302$	42703.8			0.468222
$303$	31209.1			0.339935
$304$		−	178284.i		−	1.92914i
$305$	−11257.7			−0.121018
$306$		−	53522.3i		−	0.571600i
$307$	−44988.8			−0.477340			−0.238670	−	0.971101i	$-0.576711\pi$
$307$	−44988.8			−0.477340			−0.238670	+	0.971101i	$0.576711\pi$
$308$	−9196.53			−0.0969443
$309$			27140.7i			0.284253i
$310$		−	81047.0i		−	0.843361i
$311$	−101400.			−1.04838			−0.524188	−	0.851603i	$-0.675631\pi$
$311$	−101400.			−1.04838			−0.524188	+	0.851603i	$0.675631\pi$
$312$	29649.0			0.304579
$313$		−	59908.7i		−	0.611507i	−0.952111	−	0.305753i	$-0.901092\pi$
$313$		−	59908.7i		−	0.611507i	0.952111	−	0.305753i	$-0.0989083\pi$
$314$		−	36312.3i		−	0.368294i
$315$	−5365.10			−0.0540700
$316$		−	82290.7i		−	0.824093i
$317$	146911.			1.46196			0.730979	−	0.682400i	$-0.239064\pi$
$317$	146911.			1.46196			0.730979	+	0.682400i	$0.239064\pi$
$318$			47955.3i			0.474223i
$319$		−	27848.4i		−	0.273665i
$320$			39564.1i			0.386368i
$321$		−	58487.7i		−	0.567615i
$322$	13828.2	−	11358.6i	0.133369	−	0.109550i
$323$	−218708.			−2.09633
$324$	8060.37			0.0767830
$325$	68473.0			0.648265
$326$	154468.			1.45346
$327$		−	14590.6i		−	0.136451i
$328$	15946.2			0.148221
$329$			15514.9i			0.143337i
$330$	−105620.			−0.969878
$331$	44707.2			0.408058			0.204029	−	0.978965i	$-0.434596\pi$
$331$	44707.2			0.408058			0.204029	+	0.978965i	$0.434596\pi$
$332$			110137.i			0.999212i
$333$			2075.09i			0.0187132i
$334$	210770.			1.88937
$335$	75563.5			0.673321
$336$		−	10497.9i		−	0.0929878i
$337$		−	115710.i		−	1.01885i	−0.860515	−	0.509425i	$-0.829858\pi$
$337$		−	115710.i		−	1.01885i	0.860515	−	0.509425i	$-0.170142\pi$
$338$	−107586.			−0.941724
$339$			27280.4i			0.237384i
$340$	128746.			1.11372
$341$			65220.1i			0.560883i
$342$		−	80599.7i		−	0.689099i
$343$		−	30954.4i		−	0.263108i
$344$			87197.7i			0.736865i
$345$	64899.1	−	53308.8i	0.545256	−	0.447879i
$346$	211225.			1.76438
$347$	−100868.			−0.837712			−0.418856	−	0.908053i	$-0.637569\pi$
$347$	−100868.			−0.837712			−0.418856	+	0.908053i	$0.637569\pi$
$348$	−12510.0			−0.103299
$349$	−116200.			−0.954014			−0.477007	−	0.878900i	$-0.658278\pi$
$349$	−116200.			−0.954014			−0.477007	+	0.878900i	$0.658278\pi$
$350$		−	10438.1i		−	0.0852090i
$351$	31133.2			0.252702
$352$		−	154051.i		−	1.24331i
$353$	−73359.8			−0.588720			−0.294360	−	0.955695i	$-0.595106\pi$
$353$	−73359.8			−0.588720			−0.294360	+	0.955695i	$0.595106\pi$
$354$	−121407.			−0.968807
$355$			82059.8i			0.651139i
$356$		−	155939.i		−	1.23043i
$357$	−12878.3			−0.101046
$358$	−21453.6			−0.167392
$359$			32378.7i			0.251229i	0.992079	+	0.125615i	$0.0400903\pi$
$359$			32378.7i			0.251229i	−0.992079	+	0.125615i	$0.959910\pi$
$360$		−	21212.2i		−	0.163674i
$361$	−199034.			−1.52726
$362$		−	160360.i		−	1.22371i
$363$	8917.30			0.0676738
$364$			15956.9i			0.120433i
$365$			168250.i			1.26290i
$366$		−	9958.56i		−	0.0743420i
$367$		−	43984.9i		−	0.326566i	−0.986579	−	0.163283i	$-0.947792\pi$
$367$		−	43984.9i		−	0.326566i	0.986579	−	0.163283i	$-0.0522084\pi$
$368$	104310.	+	126989.i	0.770247	+	0.937712i
$369$	16744.5			0.122976
$370$	−12214.7			−0.0892233
$371$	11538.8			0.0838323
$372$	29297.9			0.211715
$373$		−	54773.8i		−	0.393691i	−0.980435	−	0.196845i	$-0.936930\pi$
$373$		−	54773.8i		−	0.393691i	0.980435	−	0.196845i	$-0.0630697\pi$
$374$	−253527.			−1.81251
$375$			50239.2i			0.357257i
$376$	−61341.8			−0.433891
$377$	−48319.8			−0.339971
$378$		−	4745.98i		−	0.0332156i
$379$		−	135366.i		−	0.942392i	−0.882028	−	0.471196i	$-0.843822\pi$
$379$		−	135366.i		−	0.942392i	0.882028	−	0.471196i	$-0.156178\pi$
$380$	193879.			1.34265
$381$	100370.			0.691442
$382$			153593.i			1.05256i
$383$			84643.1i			0.577024i	0.957476	+	0.288512i	$0.0931606\pi$
$383$			84643.1i			0.577024i	−0.957476	+	0.288512i	$0.906839\pi$
$384$	65142.4			0.441775
$385$			25413.7i			0.171453i
$386$	221687.			1.48787
$387$			91562.7i			0.611360i
$388$		−	36960.8i		−	0.245515i
$389$		−	192639.i		−	1.27305i	−0.771257	−	0.636524i	$-0.780372\pi$
$389$		−	192639.i		−	1.27305i	0.771257	−	0.636524i	$-0.219628\pi$
$390$			183261.i			1.20487i
$391$	155782.	−	127961.i	1.01898	−	0.837000i
$392$	60649.0			0.394686
$393$	−45447.9			−0.294258
$394$	−78891.2			−0.508202
$395$	−227402.			−1.45747
$396$		−	38180.8i		−	0.243475i
$397$	227103.			1.44092			0.720462	−	0.693495i	$-0.243930\pi$
$397$	227103.			1.44092			0.720462	+	0.693495i	$0.243930\pi$
$398$		−	214585.i		−	1.35467i
$399$	−19393.5			−0.121818
$400$	95856.5			0.599103
$401$		−	26544.2i		−	0.165075i	−0.996588	−	0.0825374i	$-0.973698\pi$
$401$		−	26544.2i		−	0.165075i	0.996588	−	0.0825374i	$-0.0263024\pi$
$402$			66843.7i			0.413626i
$403$	113163.			0.696780
$404$	−66409.1			−0.406878
$405$		−	22274.0i		−	0.135797i
$406$			7365.92i			0.0446864i
$407$	9829.38			0.0593386
$408$		−	50917.3i		−	0.305876i
$409$	104060.			0.622065			0.311032	−	0.950399i	$-0.399325\pi$
$409$	104060.			0.622065			0.311032	+	0.950399i	$0.399325\pi$
$410$			98563.7i			0.586340i
$411$			78414.5i			0.464208i
$412$		−	57752.0i		−	0.340230i
$413$			29212.4i			0.171264i
$414$	47157.1	+	57409.9i	0.275135	+	0.334955i
$415$	304353.			1.76718
$416$	−267293.			−1.54455
$417$	−189634.			−1.09055
$418$	−381789.			−2.18510
$419$			36614.0i			0.208554i	0.994548	+	0.104277i	$0.0332529\pi$
$419$			36614.0i			0.208554i	−0.994548	+	0.104277i	$0.966747\pi$
$420$	11416.2			0.0647180
$421$			161260.i			0.909832i	0.890534	+	0.454916i	$0.150331\pi$
$421$			161260.i			0.909832i	−0.890534	+	0.454916i	$0.849669\pi$
$422$	108986.			0.611994
$423$	−64412.6			−0.359990
$424$			45621.2i			0.253767i
$425$		−	117591.i		−	0.651024i
$426$	−72590.3			−0.400000
$427$	−2396.18			−0.0131421
$428$			124454.i			0.679395i
$429$		−	147473.i		−	0.801307i
$430$	−538969.			−2.91492
$431$		−	95486.0i		−	0.514026i	−0.966408	−	0.257013i	$-0.917262\pi$
$431$		−	95486.0i		−	0.514026i	0.966408	−	0.257013i	$-0.0827383\pi$
$432$	43583.9			0.233538
$433$		−	318487.i		−	1.69870i	−0.527832	−	0.849349i	$-0.676995\pi$
$433$		−	318487.i		−	1.69870i	0.527832	−	0.849349i	$-0.323005\pi$
$434$		−	17250.7i		−	0.0915859i
$435$			34570.1i			0.182693i
$436$			31047.0i			0.163323i
$437$	234594.	−	192698.i	1.22844	−	1.00905i
$438$	−148834.			−0.775807
$439$	285294.			1.48035			0.740175	−	0.672414i	$-0.234743\pi$
$439$	285294.			1.48035			0.740175	+	0.672414i	$0.234743\pi$
$440$	−100479.			−0.519003
$441$	63685.0			0.327462
$442$			439895.i			2.25167i
$443$	43384.6			0.221069			0.110534	−	0.993872i	$-0.464744\pi$
$443$	43384.6			0.221069			0.110534	+	0.993872i	$0.464744\pi$
$444$		−	4415.52i		−	0.0223983i
$445$	−430923.			−2.17610
$446$	423768.			2.13039
$447$			86702.3i			0.433926i
$448$			8421.16i			0.0419581i
$449$	73572.5			0.364941			0.182471	−	0.983211i	$-0.441591\pi$
$449$	73572.5			0.364941			0.182471	+	0.983211i	$0.441591\pi$
$450$	43335.4			0.214002
$451$		−	79316.1i		−	0.389950i
$452$		−	58049.2i		−	0.284131i
$453$	42658.9			0.207880
$454$			514283.i			2.49511i
$455$	44095.2			0.212995
$456$		−	76676.8i		−	0.368752i
$457$		−	39705.0i		−	0.190113i	−0.995472	−	0.0950567i	$-0.969697\pi$
$457$		−	39705.0i		−	0.190113i	0.995472	−	0.0950567i	$-0.0303032\pi$
$458$			338317.i			1.61284i
$459$		−	53466.1i		−	0.253778i
$460$	−138097.	+	113434.i	−0.652632	+	0.536079i
$461$	−115961.			−0.545644			−0.272822	−	0.962065i	$-0.587957\pi$
$461$	−115961.			−0.545644			−0.272822	+	0.962065i	$0.587957\pi$
$462$	−22481.0			−0.105325
$463$	331421.			1.54603			0.773015	−	0.634388i	$-0.218748\pi$
$463$	331421.			1.54603			0.773015	+	0.634388i	$0.218748\pi$
$464$	−67643.6			−0.314189
$465$		−	80961.9i		−	0.374434i
$466$	−450589.			−2.07496
$467$			70162.0i			0.321713i	0.986978	+	0.160856i	$0.0514256\pi$
$467$			70162.0i			0.321713i	−0.986978	+	0.160856i	$0.948574\pi$
$468$	−66247.5			−0.302467
$469$	16083.6			0.0731202
$470$		−	379154.i		−	1.71641i
$471$		−	36274.2i		−	0.163515i
$472$	−115498.			−0.518430
$473$	433719.			1.93859
$474$		−	201160.i		−	0.895336i
$475$		−	177082.i		−	0.784849i
$476$	27403.3			0.120945
$477$			47905.0i			0.210545i
$478$	335031.			1.46632
$479$			288365.i			1.25681i	0.777884	+	0.628407i	$0.216293\pi$
$479$			288365.i			1.25681i	−0.777884	+	0.628407i	$0.783707\pi$
$480$			191233.i			0.830004i
$481$		−	17054.9i		−	0.0737157i
$482$		−	588778.i		−	2.53430i
$483$	13813.7	−	11346.7i	0.0592127	−	0.0486380i
$484$	−18974.9			−0.0810007
$485$	−102138.			−0.434212
$486$	19703.7			0.0834209
$487$	−365404.			−1.54069			−0.770345	−	0.637627i	$-0.779916\pi$
$487$	−365404.			−1.54069			−0.770345	+	0.637627i	$0.779916\pi$
$488$		−	9473.85i		−	0.0397820i
$489$	154306.			0.645303
$490$			374872.i			1.56132i
$491$	−194346.			−0.806145			−0.403073	−	0.915168i	$-0.632058\pi$
$491$	−194346.			−0.806145			−0.403073	+	0.915168i	$0.632058\pi$
$492$	−35630.1			−0.147193
$493$			82981.3i			0.341418i
$494$			662442.i			2.71452i
$495$	−105509.			−0.430604
$496$	158419.			0.643938
$497$			17466.3i			0.0707113i
$498$			269231.i			1.08559i
$499$	−270272.			−1.08542			−0.542712	−	0.839919i	$-0.682602\pi$
$499$	−270272.			−1.08542			−0.542712	+	0.839919i	$0.682602\pi$
$500$		−	106903.i		−	0.427611i
$501$	210549.			0.838837
$502$		−	119225.i		−	0.473107i
$503$		−	291983.i		−	1.15404i	−0.816730	−	0.577020i	$-0.804215\pi$
$503$		−	291983.i		−	1.15404i	0.816730	−	0.577020i	$-0.195785\pi$
$504$		−	4514.98i		−	0.0177744i
$505$			183515.i			0.719595i
$506$	271942.	−	223376.i	1.06212	−	0.872441i
$507$	−107473.			−0.418105
$508$	−213575.			−0.827607
$509$	−200051.			−0.772157			−0.386078	−	0.922466i	$-0.626171\pi$
$509$	−200051.			−0.772157			−0.386078	+	0.922466i	$0.626171\pi$
$510$	314720.			1.21000
$511$			35811.7i			0.137146i
$512$	−246382.			−0.939873
$513$		−	80515.1i		−	0.305945i
$514$	−467755.			−1.77049
$515$	−159592.			−0.601723
$516$		−	194834.i		−	0.731754i
$517$			305113.i			1.14151i
$518$	−2599.88			−0.00968932
$519$	211003.			0.783347
$520$			174341.i			0.644752i
$521$			472823.i			1.74190i	0.491372	+	0.870950i	$0.336496\pi$
$521$			472823.i			1.74190i	−0.491372	+	0.870950i	$0.663504\pi$
$522$	−30580.8			−0.112230
$523$		−	417272.i		−	1.52551i	−0.646686	−	0.762756i	$-0.723846\pi$
$523$		−	417272.i		−	1.52551i	0.646686	−	0.762756i	$-0.276154\pi$
$524$	96707.3			0.352206
$525$		−	10427.2i		−	0.0378309i
$526$		−	219256.i		−	0.792465i
$527$		−	194339.i		−	0.699745i
$528$		−	206450.i		−	0.740539i
$529$	−54354.2	+	274512.i	−0.194233	+	0.980956i
$530$	−281985.			−1.00386
$531$	−121280.			−0.430129
$532$	41266.9			0.145807
$533$	−137621.			−0.484430
$534$		−	381195.i		−	1.33680i
$535$	343917.			1.20156
$536$			63590.2i			0.221340i
$537$	−21431.1			−0.0743184
$538$	644860.			2.22793
$539$		−	301667.i		−	1.03836i
$540$			47396.3i			0.162539i
$541$	366173.			1.25110			0.625550	−	0.780184i	$-0.284875\pi$
$541$	366173.			1.25110			0.625550	+	0.780184i	$0.284875\pi$
$542$	428179.			1.45756
$543$		−	160192.i		−	0.543301i
$544$			459032.i			1.55112i
$545$	85795.2			0.288848
$546$			39006.8i			0.130844i
$547$	−280150.			−0.936303			−0.468152	−	0.883648i	$-0.655080\pi$
$547$	−280150.			−0.936303			−0.468152	+	0.883648i	$0.655080\pi$
$548$		−	166856.i		−	0.555624i
$549$		−	9948.11i		−	0.0330062i
$550$		−	205273.i		−	0.678590i
$551$			124962.i			0.411600i
$552$	44861.9	+	54615.6i	0.147231	+	0.179242i
$553$	−48402.2			−0.158276
$554$	−378087.			−1.23189
$555$	−12201.8			−0.0396132
$556$	403517.			1.30531
$557$			114961.i			0.370544i	0.982687	+	0.185272i	$0.0593166\pi$
$557$			114961.i			0.370544i	−0.982687	+	0.185272i	$0.940683\pi$
$558$	71619.2			0.230017
$559$		−	752546.i		−	2.40829i
$560$	61729.7			0.196842
$561$	−253261.			−0.804717
$562$			398220.i			1.26081i
$563$			351660.i			1.10945i	0.832035	+	0.554723i	$0.187176\pi$
$563$			351660.i			1.10945i	−0.832035	+	0.554723i	$0.812824\pi$
$564$	137062.			0.430882
$565$	−160413.			−0.502508
$566$		−	594127.i		−	1.85458i
$567$		−	4741.00i		−	0.0147470i
$568$	−69057.2			−0.214049
$569$			247322.i			0.763904i	0.924182	+	0.381952i	$0.124748\pi$
$569$			247322.i			0.763904i	−0.924182	+	0.381952i	$0.875252\pi$
$570$	473940.			1.45872
$571$		−	507858.i		−	1.55765i	−0.627240	−	0.778826i	$-0.715816\pi$
$571$		−	507858.i		−	1.55765i	0.627240	−	0.778826i	$-0.284184\pi$
$572$			313805.i			0.959107i
$573$			153432.i			0.467312i
$574$			20979.2i			0.0636743i
$575$	103606.	+	126132.i	0.313366	+	0.381497i
$576$	−34961.7			−0.105378
$577$	466836.			1.40221			0.701104	−	0.713059i	$-0.252691\pi$
$577$	466836.			1.40221			0.701104	+	0.713059i	$0.252691\pi$
$578$	321004.			0.960848
$579$	221454.			0.660583
$580$		−	73560.7i		−	0.218670i
$581$	64781.1			0.191909
$582$		−	90351.2i		−	0.266740i
$583$	226919.			0.667626
$584$	−141590.			−0.415152
$585$			183068.i			0.534935i
$586$		−	792132.i		−	2.30676i
$587$	332078.			0.963750			0.481875	−	0.876240i	$-0.339956\pi$
$587$	332078.			0.963750			0.481875	+	0.876240i	$0.339956\pi$
$588$	−135514.			−0.391948
$589$		−	292657.i		−	0.843585i
$590$		−	713894.i		−	2.05083i
$591$	−78808.4			−0.225630
$592$		−	23875.5i		−	0.0681254i
$593$	−605840.			−1.72285			−0.861427	−	0.507882i	$-0.830429\pi$
$593$	−605840.			−1.72285			−0.861427	+	0.507882i	$0.830429\pi$
$594$		−	93333.4i		−	0.264523i
$595$		−	75726.4i		−	0.213901i
$596$		−	184492.i		−	0.519378i
$597$		−	214360.i		−	0.601443i
$598$	−387580.	−	471847.i	−1.08382	−	1.31947i
$599$	−118309.			−0.329734			−0.164867	−	0.986316i	$-0.552720\pi$
$599$	−118309.			−0.329734			−0.164867	+	0.986316i	$0.552720\pi$
$600$	41226.2			0.114517
$601$	−143467.			−0.397194			−0.198597	−	0.980081i	$-0.563638\pi$
$601$	−143467.			−0.397194			−0.198597	+	0.980081i	$0.563638\pi$
$602$	−114719.			−0.316550
$603$			66773.5i			0.183641i
$604$	−90772.8			−0.248818
$605$			52435.2i			0.143256i
$606$	−162338.			−0.442053
$607$	221803.			0.601990			0.300995	−	0.953626i	$-0.402681\pi$
$607$	221803.			0.601990			0.300995	+	0.953626i	$0.402681\pi$
$608$			691260.i			1.86997i
$609$			7358.19i			0.0198398i
$610$	58558.0			0.157372
$611$	529401.			1.41809
$612$			113769.i			0.303754i
$613$			489220.i			1.30192i	0.759114	+	0.650958i	$0.225633\pi$
$613$			489220.i			1.30192i	−0.759114	+	0.650958i	$0.774367\pi$
$614$	234014.			0.620734
$615$			98460.3i			0.260322i
$616$	−21386.8			−0.0563618
$617$		−	219023.i		−	0.575334i	−0.957730	−	0.287667i	$-0.907120\pi$
$617$		−	219023.i		−	0.575334i	0.957730	−	0.287667i	$-0.0928797\pi$
$618$		−	141175.i		−	0.369643i
$619$			520987.i			1.35971i	0.733347	+	0.679854i	$0.237957\pi$
$619$			520987.i			1.35971i	−0.733347	+	0.679854i	$0.762043\pi$
$620$			172277.i			0.448171i
$621$	47107.6	+	57349.6i	0.122154	+	0.148713i
$622$	527443.			1.36331
$623$	−91721.4			−0.236317
$624$	−358212.			−0.919964
$625$	−488266.			−1.24996
$626$			311622.i			0.795205i
$627$	−381388.			−0.970135
$628$			77187.0i			0.195715i
$629$	−29289.1			−0.0740294
$630$	27907.1			0.0703128
$631$		−	51494.1i		−	0.129330i	−0.997907	−	0.0646650i	$-0.979402\pi$
$631$		−	51494.1i		−	0.129330i	0.997907	−	0.0646650i	$-0.0205979\pi$
$632$		−	191370.i		−	0.479114i
$633$	108872.			0.271712
$634$	−764173.			−1.90114
$635$			590194.i			1.46368i
$636$		−	101936.i		−	0.252007i
$637$	−523422.			−1.28995
$638$			144857.i			0.355875i
$639$	−72514.1			−0.177591
$640$			383048.i			0.935176i
$641$			412825.i			1.00473i	0.864655	+	0.502366i	$0.167537\pi$
$641$			412825.i			1.00473i	−0.864655	+	0.502366i	$0.832463\pi$
$642$			304230.i			0.738129i
$643$			383967.i			0.928693i	0.885654	+	0.464346i	$0.153711\pi$
$643$			383967.i			0.928693i	−0.885654	+	0.464346i	$0.846289\pi$
$644$	−29393.8	+	24144.4i	−0.0708734	+	0.0582162i
$645$	−538404.			−1.29416
$646$	1.13764e6			2.72608
$647$	388563.			0.928224			0.464112	−	0.885777i	$-0.346374\pi$
$647$	388563.			0.928224			0.464112	+	0.885777i	$0.346374\pi$
$648$	18744.7			0.0446403
$649$			574484.i			1.36392i
$650$	−356170.			−0.843005
$651$		−	17232.6i		−	0.0406621i
$652$	−328343.			−0.772382
$653$	−701971.			−1.64624			−0.823119	−	0.567869i	$-0.807768\pi$
$653$	−701971.			−1.64624			−0.823119	+	0.567869i	$0.807768\pi$
$654$			75894.6i			0.177442i
$655$		−	267241.i		−	0.622903i
$656$	−192658.			−0.447693
$657$	−148678.			−0.344441
$658$		−	80702.5i		−	0.186395i
$659$		−	64055.8i		−	0.147499i	−0.997277	−	0.0737493i	$-0.976504\pi$
$659$		−	64055.8i		−	0.147499i	0.997277	−	0.0737493i	$-0.0234965\pi$
$660$	224509.			0.515403
$661$			555941.i			1.27241i	0.771522	+	0.636203i	$0.219496\pi$
$661$			555941.i			1.27241i	−0.771522	+	0.636203i	$0.780504\pi$
$662$	−232550.			−0.530640
$663$			439433.i			0.999691i
$664$			256127.i			0.580925i
$665$		−	114037.i		−	0.257871i
$666$		−	10793.8i		−	0.0243347i
$667$	−73112.6	−	89008.6i	−0.164339	−	0.200069i
$668$	−448022.			−1.00403
$669$	423323.			0.945845
$670$	−393052.			−0.875589
$671$	−47122.7			−0.104661
$672$			40703.7i			0.0901354i
$673$	−5360.56			−0.0118353			−0.00591766	−	0.999982i	$-0.501884\pi$
$673$	−5360.56			−0.0118353			−0.00591766	+	0.999982i	$0.501884\pi$
$674$			601877.i			1.32491i
$675$	43289.9			0.0950122
$676$	228690.			0.500442
$677$		−	567618.i		−	1.23845i	−0.785213	−	0.619226i	$-0.787446\pi$
$677$		−	567618.i		−	1.23845i	0.785213	−	0.619226i	$-0.212554\pi$
$678$		−	141902.i		−	0.308694i
$679$	−21739.9			−0.0471538
$680$	299402.			0.647496
$681$			513743.i			1.10778i
$682$		−	339249.i		−	0.729374i
$683$	131682.			0.282283			0.141142	−	0.989989i	$-0.454923\pi$
$683$	131682.			0.282283			0.141142	+	0.989989i	$0.454923\pi$
$684$			171326.i			0.366194i
$685$	−461090.			−0.982663
$686$			161013.i			0.342147i
$687$			337962.i			0.716067i
$688$		−	1.05350e6i		−	2.22566i
$689$		−	393727.i		−	0.829385i
$690$	−337580.	+	277292.i	−0.709052	+	0.582423i
$691$	89639.0			0.187733			0.0938665	−	0.995585i	$-0.470077\pi$
$691$	89639.0			0.187733			0.0938665	+	0.995585i	$0.470077\pi$
$692$	−448988.			−0.937610
$693$	−22457.4			−0.0467620
$694$	524677.			1.08936
$695$		−	1.11508e6i		−	2.30853i
$696$	−29092.3			−0.0600565
$697$			236342.i			0.486492i
$698$	604426.			1.24060
$699$	−450117.			−0.921235
$700$			22187.7i			0.0452809i
$701$		−	521133.i		−	1.06050i	−0.847840	−	0.530252i	$-0.822097\pi$
$701$		−	521133.i		−	1.06050i	0.847840	−	0.530252i	$-0.177903\pi$
$702$	−161943.			−0.328615
$703$	−44106.7			−0.0892470
$704$			165609.i			0.334147i
$705$		−	378757.i		−	0.762047i
$706$	381589.			0.765573
$707$			39060.9i			0.0781454i
$708$	258068.			0.514834
$709$			183547.i			0.365137i	0.983193	+	0.182568i	$0.0584411\pi$
$709$			183547.i			0.365137i	−0.983193	+	0.182568i	$0.941559\pi$
$710$		−	426843.i		−	0.846743i
$711$		−	200949.i		−	0.397509i
$712$		−	362642.i		−	0.715349i
$713$	171227.	+	208455.i	0.336817	+	0.410047i
$714$	66987.8			0.131401
$715$	867167.			1.69625
$716$	45602.7			0.0889539
$717$	334679.			0.651015
$718$		−	168421.i		−	0.326699i
$719$	516750.			0.999592			0.499796	−	0.866143i	$-0.333408\pi$
$719$	516750.			0.999592			0.499796	+	0.866143i	$0.333408\pi$
$720$			256280.i			0.494368i
$721$	−33968.9			−0.0653448
$722$	1.03530e6			1.98605
$723$		−	588160.i		−	1.12517i
$724$			340868.i			0.650293i
$725$	−67187.5			−0.127824
$726$	−46384.3			−0.0880031
$727$		−	34702.6i		−	0.0656588i	−0.999461	−	0.0328294i	$-0.989548\pi$
$727$		−	34702.6i		−	0.0656588i	0.999461	−	0.0328294i	$-0.0104518\pi$
$728$			37108.2i			0.0700176i
$729$	19683.0			0.0370370
$730$		−	875169.i		−	1.64228i
$731$	−1.29237e6			−2.41854
$732$			21168.3i			0.0395061i
$733$		−	544453.i		−	1.01333i	−0.862142	−	0.506667i	$-0.830877\pi$
$733$		−	544453.i		−	1.01333i	0.862142	−	0.506667i	$-0.169123\pi$
$734$			228792.i			0.424668i
$735$			374479.i			0.693190i
$736$	−404441.	−	492373.i	−0.746620	−	0.908948i
$737$	316296.			0.582317
$738$	−87098.3			−0.159918
$739$	607468.			1.11233			0.556166	−	0.831071i	$-0.312272\pi$
$739$	607468.			1.11233			0.556166	+	0.831071i	$0.312272\pi$
$740$	25964.0			0.0474141
$741$			661747.i			1.20519i
$742$	−60020.2			−0.109016
$743$		−	954490.i		−	1.72900i	−0.502637	−	0.864498i	$-0.667637\pi$
$743$		−	954490.i		−	1.72900i	0.502637	−	0.864498i	$-0.332363\pi$
$744$	68133.3			0.123087
$745$	−509824.			−0.918560
$746$			284912.i			0.511957i
$747$			268949.i			0.481980i
$748$	538908.			0.963189
$749$	73202.3			0.130485
$750$		−	261325.i		−	0.464578i
$751$		−	112016.i		−	0.198609i	−0.995057	−	0.0993046i	$-0.968338\pi$
$751$		−	112016.i		−	0.198609i	0.995057	−	0.0993046i	$-0.0316618\pi$
$752$	741117.			1.31054
$753$		−	119100.i		−	0.210049i
$754$	251341.			0.442099
$755$			250842.i			0.440054i
$756$			10088.2i			0.0176511i
$757$			832742.i			1.45318i	0.687072	+	0.726589i	$0.258896\pi$
$757$			832742.i			1.45318i	−0.687072	+	0.726589i	$0.741104\pi$
$758$			704122.i			1.22549i
$759$	271657.	−	223142.i	0.471560	−	0.387345i
$760$	450872.			0.780596
$761$	−391690.			−0.676353			−0.338176	−	0.941083i	$-0.609810\pi$
$761$	−391690.			−0.676353			−0.338176	+	0.941083i	$0.609810\pi$
$762$	−522088.			−0.899153
$763$	18261.4			0.0313679
$764$		−	326484.i		−	0.559339i
$765$	314390.			0.537212
$766$		−	440281.i		−	0.750364i
$767$	996786.			1.69438
$768$	−446500.			−0.757005
$769$		−	508594.i		−	0.860040i	−0.902819	−	0.430020i	$-0.858507\pi$
$769$		−	508594.i		−	0.860040i	0.902819	−	0.430020i	$-0.141493\pi$
$770$		−	132192.i		−	0.222958i
$771$	−467265.			−0.786057
$772$	−471227.			−0.790670
$773$			67467.2i			0.112910i	0.998405	+	0.0564551i	$0.0179798\pi$
$773$			67467.2i			0.112910i	−0.998405	+	0.0564551i	$0.982020\pi$
$774$		−	476274.i		−	0.795014i
$775$	157351.			0.261978
$776$		−	85953.6i		−	0.142738i
$777$	−2597.15			−0.00430184
$778$			1.00203e6i			1.65547i
$779$			355910.i			0.586496i
$780$		−	389546.i		−	0.640280i
$781$			343489.i			0.563132i
$782$	−810319.	+	665605.i	−1.32508	+	1.08844i
$783$	−30548.7			−0.0498275
$784$	−732747.			−1.19212
$785$	213298.			0.346137
$786$	236402.			0.382654
$787$		−	1.08119e6i		−	1.74563i	−0.488051	−	0.872815i	$-0.662292\pi$
$787$		−	1.08119e6i		−	1.74563i	0.488051	−	0.872815i	$-0.337708\pi$
$788$	167694.			0.270064
$789$		−	219026.i		−	0.351837i
$790$	1.18286e6			1.89530
$791$	−34143.7			−0.0545705
$792$		−	88790.7i		−	0.141552i
$793$			81762.6i			0.130019i
$794$	−1.18130e6			−1.87378
$795$	−281689.			−0.445693
$796$			456131.i			0.719885i
$797$		−	348275.i		−	0.548284i	−0.961689	−	0.274142i	$-0.911606\pi$
$797$		−	348275.i		−	0.548284i	0.961689	−	0.274142i	$-0.0883937\pi$
$798$	100877.			0.158412
$799$		−	909159.i		−	1.42412i
$800$	−371664.			−0.580726
$801$		−	380795.i		−	0.593508i
$802$			138072.i			0.214664i
$803$			704265.i			1.09221i
$804$		−	142086.i		−	0.219805i
$805$	66720.5	+	81226.7i	0.102960	+	0.125345i
$806$	−588631.			−0.906094
$807$	644184.			0.989151
$808$	−154436.			−0.236552
$809$	949142.			1.45022			0.725110	−	0.688633i	$-0.241789\pi$
$809$	949142.			1.45022			0.725110	+	0.688633i	$0.241789\pi$
$810$			115861.i			0.176590i
$811$	1541.50			0.00234370			0.00117185	−	0.999999i	$-0.499627\pi$
$811$	1541.50			0.00234370			0.00117185	+	0.999999i	$0.499627\pi$
$812$		−	15657.3i		−	0.0237468i
$813$	427730.			0.647125
$814$	−51128.6			−0.0771640
$815$			907343.i			1.36602i
$816$			615170.i			0.923879i
$817$	−1.94620e6			−2.91570
$818$	−541277.			−0.808935
$819$			38965.8i			0.0580920i
$820$		−	209511.i		−	0.311587i
$821$	686475.			1.01845			0.509224	−	0.860634i	$-0.329933\pi$
$821$	686475.			1.01845			0.509224	+	0.860634i	$0.329933\pi$
$822$		−	407882.i		−	0.603657i
$823$	319752.			0.472078			0.236039	−	0.971744i	$-0.424151\pi$
$823$	319752.			0.472078			0.236039	+	0.971744i	$0.424151\pi$
$824$		−	134304.i		−	0.197804i
$825$		−	205058.i		−	0.301279i
$826$		−	151951.i		−	0.222712i
$827$		−	58086.5i		−	0.0849306i	−0.999098	−	0.0424653i	$-0.986479\pi$
$827$		−	58086.5i		−	0.0849306i	0.999098	−	0.0424653i	$-0.0135212\pi$
$828$	−100239.	−	122033.i	−0.146210	−	0.177998i
$829$	159231.			0.231697			0.115848	−	0.993267i	$-0.463041\pi$
$829$	159231.			0.231697			0.115848	+	0.993267i	$0.463041\pi$
$830$	−1.58313e6			−2.29805
$831$	−377690.			−0.546932
$832$	287347.			0.415107
$833$			898891.i			1.29544i
$834$	986401.			1.41815
$835$			1.23806e6i			1.77570i
$836$	811546.			1.16118
$837$	71544.0			0.102123
$838$		−	190452.i		−	0.271205i
$839$			27961.6i			0.0397227i	0.999803	+	0.0198613i	$0.00632248\pi$
$839$			27961.6i			0.0397227i	−0.999803	+	0.0198613i	$0.993678\pi$
$840$	26548.8			0.0376259
$841$	−659868.			−0.932965
$842$		−	838809.i		−	1.18315i
$843$			397802.i			0.559773i
$844$	−231666.			−0.325220
$845$		−	631962.i		−	0.885069i
$846$	335049.			0.468131
$847$			11160.8i			0.0155571i
$848$		−	551185.i		−	0.766488i
$849$		−	593503.i		−	0.823394i
$850$			611663.i			0.846593i
$851$	31416.5	−	25805.8i	0.0433809	−	0.0356335i
$852$	154301.			0.212564
$853$	720507.			0.990239			0.495120	−	0.868825i	$-0.335124\pi$
$853$	720507.			0.990239			0.495120	+	0.868825i	$0.335124\pi$
$854$	12464.0			0.0170900
$855$	473442.			0.647642
$856$			289423.i			0.394989i
$857$	−230073.			−0.313260			−0.156630	−	0.987657i	$-0.550063\pi$
$857$	−230073.			−0.313260			−0.156630	+	0.987657i	$0.550063\pi$
$858$			767099.i			1.04202i
$859$	−166530.			−0.225687			−0.112844	−	0.993613i	$-0.535996\pi$
$859$	−166530.			−0.225687			−0.112844	+	0.993613i	$0.535996\pi$
$860$	1.14566e6			1.54902
$861$			20957.2i			0.0282700i
$862$			496681.i			0.668441i
$863$	−87423.6			−0.117384			−0.0586918	−	0.998276i	$-0.518693\pi$
$863$	−87423.6			−0.117384			−0.0586918	+	0.998276i	$0.518693\pi$
$864$	−168988.			−0.226375
$865$			1.24073e6i			1.65823i
$866$			1.65665e6i			2.20899i
$867$	320667.			0.426595
$868$			36668.9i			0.0486697i
$869$	−951867.			−1.26048
$870$		−	179820.i		−	0.237574i
$871$		−	548805.i		−	0.723406i
$872$			72200.7i			0.0949529i
$873$		−	90256.4i		−	0.118427i
$874$	−1.22027e6	+	1.00234e6i	−1.59747	+	1.31218i
$875$	−62878.7			−0.0821273
$876$	316368.			0.412272
$877$	275442.			0.358123			0.179061	−	0.983838i	$-0.442694\pi$
$877$	275442.			0.358123			0.179061	+	0.983838i	$0.442694\pi$
$878$	−1.48399e6			−1.92505
$879$		−	791300.i		−	1.02415i
$880$	1.21396e6			1.56762
$881$			124394.i			0.160268i	0.996784	+	0.0801342i	$0.0255349\pi$
$881$			124394.i			0.160268i	−0.996784	+	0.0801342i	$0.974465\pi$
$882$	−331265.			−0.425832
$883$	−655207.			−0.840345			−0.420172	−	0.907444i	$-0.638030\pi$
$883$	−655207.			−0.840345			−0.420172	+	0.907444i	$0.638030\pi$
$884$		−	935059.i		−	1.19656i
$885$		−	713144.i		−	0.910523i
$886$	−225670.			−0.287479
$887$	−789097.			−1.00296			−0.501479	−	0.865170i	$-0.667211\pi$
$887$	−789097.			−1.00296			−0.501479	+	0.865170i	$0.667211\pi$
$888$		−	10268.4i		−	0.0130220i
$889$			125622.i			0.158951i
$890$	2.24149e6			2.82981
$891$		−	93235.5i		−	0.117443i
$892$	−900778.			−1.13211
$893$		−	1.36911e6i		−	1.71686i
$894$		−	450992.i		−	0.564278i
$895$		−	126019.i		−	0.157322i
$896$			81531.3i			0.101557i
$897$	−387173.	−	471351.i	−0.481194	−	0.585814i
$898$	−382696.			−0.474571
$899$	−111039.			−0.137390
$900$	−92115.5			−0.113723
$901$	−676161.			−0.832915
$902$			412572.i			0.507091i
$903$	−114599.			−0.140541
$904$		−	134995.i		−	0.165189i
$905$	941954.			1.15009
$906$	−221895.			−0.270328
$907$			962496.i			1.17000i	0.811035	+	0.584998i	$0.198905\pi$
$907$			962496.i			1.17000i	−0.811035	+	0.584998i	$0.801095\pi$
$908$		−	1.09318e6i		−	1.32593i
$909$	−162167.			−0.196262
$910$	−229366.			−0.276979
$911$		−	272436.i		−	0.328267i	−0.986438	−	0.164133i	$-0.947517\pi$
$911$		−	272436.i		−	0.328267i	0.986438	−	0.164133i	$-0.0524828\pi$
$912$			926390.i			1.11379i
$913$	1.27397e6			1.52833
$914$			206530.i			0.247224i
$915$	58496.5			0.0698695
$916$		−	719140.i		−	0.857082i
$917$		−	56881.9i		−	0.0676450i
$918$			278110.i			0.330013i
$919$		−	1.31885e6i		−	1.56158i	−0.624790	−	0.780792i	$-0.714816\pi$
$919$		−	1.31885e6i		−	1.56158i	0.624790	−	0.780792i	$-0.285184\pi$
$920$	−321149.	+	263795.i	−0.379429	+	0.311667i
$921$	233769.			0.275592
$922$	603183.			0.709556
$923$	595987.			0.699574
$924$	47786.5			0.0559708
$925$		−	23714.5i		−	0.0277160i
$926$	−1.72392e6			−2.01046
$927$		−	141027.i		−	0.164113i
$928$	262275.			0.304551
$929$	412883.			0.478405			0.239203	−	0.970970i	$-0.423114\pi$
$929$	412883.			0.478405			0.239203	+	0.970970i	$0.423114\pi$
$930$			421133.i			0.486915i
$931$			1.35365e6i			1.56173i
$932$	957791.			1.10265
$933$	526889.			0.605280
$934$		−	364955.i		−	0.418356i
$935$		−	1.48922e6i		−	1.70347i
$936$	−154061.			−0.175849
$937$			1.38279e6i			1.57499i	0.616320	+	0.787495i	$0.288623\pi$
$937$			1.38279e6i			1.57499i	−0.616320	+	0.787495i	$0.711377\pi$
$938$	−83660.6			−0.0950857
$939$			311295.i			0.353054i
$940$			805946.i			0.912116i
$941$			683087.i			0.771430i	0.922618	+	0.385715i	$0.126045\pi$
$941$			683087.i			0.771430i	−0.922618	+	0.385715i	$0.873955\pi$
$942$			188684.i			0.212635i
$943$	−208235.	−	253509.i	−0.234169	−	0.285082i
$944$	1.39542e6			1.56589
$945$	27877.9			0.0312173
$946$	−2.25604e6			−2.52095
$947$	−924645.			−1.03104			−0.515520	−	0.856878i	$-0.672401\pi$
$947$	−924645.			−1.03104			−0.515520	+	0.856878i	$0.672401\pi$
$948$			427595.i			0.475790i
$949$	1.22197e6			1.35684
$950$			921109.i			1.02062i
$951$	−763371.			−0.844062
$952$	63727.3			0.0703156
$953$			29574.2i			0.0325632i	0.999867	+	0.0162816i	$0.00518282\pi$
$953$			29574.2i			0.0325632i	−0.999867	+	0.0162816i	$0.994817\pi$
$954$		−	249183.i		−	0.273793i
$955$	−902206.			−0.989233
$956$	−712155.			−0.779218
$957$			144705.i			0.158001i
$958$		−	1.49996e6i		−	1.63437i
$959$	−98142.4			−0.106714
$960$		−	205581.i		−	0.223069i
$961$	−663472.			−0.718416
$962$			88713.2i			0.0958601i
$963$			303911.i			0.327713i
$964$			1.25153e6i			1.34675i
$965$			1.30219e6i			1.39836i
$966$	−71853.4	+	59021.2i	−0.0770004	+	0.0632490i
$967$	404908.			0.433015			0.216508	−	0.976281i	$-0.430533\pi$
$967$	404908.			0.433015			0.216508	+	0.976281i	$0.430533\pi$
$968$	−44126.7			−0.0470924
$969$	1.13644e6			1.21032
$970$	531280.			0.564651
$971$			617547.i			0.654985i	0.944854	+	0.327493i	$0.106204\pi$
$971$			617547.i			0.654985i	−0.944854	+	0.327493i	$0.893796\pi$
$972$	−41882.9			−0.0443307
$973$		−	237343.i		−	0.250698i
$974$	1.90069e6			2.00352
$975$	−355796.			−0.374276
$976$			114461.i			0.120159i
$977$		−	1.28409e6i		−	1.34526i	−0.739980	−	0.672629i	$-0.765165\pi$
$977$		−	1.28409e6i		−	1.34526i	0.739980	−	0.672629i	$-0.234835\pi$
$978$	−802638.			−0.839154
$979$	−1.80377e6			−1.88199
$980$		−	796843.i		−	0.829699i
$981$			75815.0i			0.0787802i
$982$	1.01091e6			1.04831
$983$			1.51010e6i			1.56278i	0.624043	+	0.781390i	$0.285489\pi$
$983$			1.51010e6i			1.56278i	−0.624043	+	0.781390i	$0.714511\pi$
$984$	−82859.0			−0.0855755
$985$		−	463407.i		−	0.477628i
$986$		−	431636.i		−	0.443981i
$987$		−	80617.8i		−	0.0827555i
$988$		−	1.40811e6i		−	1.44253i
$989$	1.38625e6	−	1.13868e6i	1.41725	−	1.16415i
$990$	548816.			0.559959
$991$	−26657.3			−0.0271437			−0.0135719	−	0.999908i	$-0.504320\pi$
$991$	−26657.3			−0.0271437			−0.0135719	+	0.999908i	$0.504320\pi$
$992$	−614239.			−0.624186
$993$	−232306.			−0.235592
$994$		−	90853.0i		−	0.0919531i
$995$	1.26047e6			1.27317
$996$		−	572289.i		−	0.576895i
$997$	33629.9			0.0338326			0.0169163	−	0.999857i	$-0.494615\pi$
$997$	33629.9			0.0338326			0.0169163	+	0.999857i	$0.494615\pi$
$998$	1.40585e6			1.41149
$999$		−	10782.5i		−	0.0108041i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.5.d.a.22.3	✓	16
3.2	odd	2		207.5.d.c.91.14		16
4.3	odd	2		1104.5.c.c.1057.10		16
23.22	odd	2	inner	69.5.d.a.22.4	yes	16
69.68	even	2		207.5.d.c.91.13		16
92.91	even	2		1104.5.c.c.1057.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.5.d.a.22.3	✓	16	1.1	even	1	trivial
69.5.d.a.22.4	yes	16	23.22	odd	2	inner
207.5.d.c.91.13		16	69.68	even	2
207.5.d.c.91.14		16	3.2	odd	2
1104.5.c.c.1057.10		16	4.3	odd	2
1104.5.c.c.1057.15		16	92.91	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5.20161 q^{2} -5.19615 q^{3} +11.0568 q^{4} -30.5542i q^{5} +27.0284 q^{6} -6.50343i q^{7} +25.7128 q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-5.20161 q^{2} -5.19615 q^{3} +11.0568 q^{4} -30.5542i q^{5} +27.0284 q^{6} -6.50343i q^{7} +25.7128 q^{8} +27.0000 q^{9} +158.931i q^{10} -127.895i q^{11} -57.4526 q^{12} -221.911 q^{13} +33.8283i q^{14} +158.764i q^{15} -310.656 q^{16} +381.095i q^{17} -140.443 q^{18} +573.894i q^{19} -337.831i q^{20} +33.7928i q^{21} +665.260i q^{22} +(-335.773 - 408.776i) q^{23} -133.608 q^{24} -308.561 q^{25} +1154.29 q^{26} -140.296 q^{27} -71.9068i q^{28} +217.744 q^{29} -825.831i q^{30} -509.950 q^{31} +1204.51 q^{32} +664.562i q^{33} -1982.31i q^{34} -198.707 q^{35} +298.532 q^{36} +76.8550i q^{37} -2985.17i q^{38} +1153.08 q^{39} -785.636i q^{40} +620.166 q^{41} -175.777i q^{42} +3391.21i q^{43} -1414.10i q^{44} -824.964i q^{45} +(1746.56 + 2126.29i) q^{46} -2385.65 q^{47} +1614.22 q^{48} +2358.71 q^{49} +1605.02 q^{50} -1980.23i q^{51} -2453.61 q^{52} +1774.26i q^{53} +729.766 q^{54} -3907.73 q^{55} -167.222i q^{56} -2982.04i q^{57} -1132.62 q^{58} -4491.84 q^{59} +1755.42i q^{60} -368.448i q^{61} +2652.56 q^{62} -175.593i q^{63} -1294.88 q^{64} +6780.31i q^{65} -3456.79i q^{66} +2473.09i q^{67} +4213.67i q^{68} +(1744.73 + 2124.06i) q^{69} +1033.60 q^{70} -2685.71 q^{71} +694.247 q^{72} -5506.59 q^{73} -399.770i q^{74} +1603.33 q^{75} +6345.41i q^{76} -831.756 q^{77} -5997.88 q^{78} -7442.57i q^{79} +9491.86i q^{80} +729.000 q^{81} -3225.86 q^{82} +9961.07i q^{83} +373.639i q^{84} +11644.1 q^{85} -17639.8i q^{86} -1131.43 q^{87} -3288.54i q^{88} -14103.5i q^{89} +4291.14i q^{90} +1443.18i q^{91} +(-3712.56 - 4519.73i) q^{92} +2649.78 q^{93} +12409.2 q^{94} +17534.9 q^{95} -6258.80 q^{96} -3342.83i q^{97} -12269.1 q^{98} -3453.17i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9}+O(q^{10}) \) 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9	\( 16 q + 12 q^{2} + 144 q^{4} - 36 q^{6} + 372 q^{8} + 432 q^{9} + 104 q^{13} + 680 q^{16} + 324 q^{18} - 732 q^{23} - 1764 q^{24} - 2984 q^{25} + 1800 q^{26} - 3528 q^{29} - 400 q^{31} + 5244 q^{32} + 912 q^{35} + 3888 q^{36} + 2016 q^{39} + 1008 q^{41} - 1168 q^{46} - 8664 q^{47} - 2016 q^{48} + 7240 q^{49} - 18852 q^{50} - 20952 q^{52} - 972 q^{54} + 6816 q^{55} - 13352 q^{58} + 20112 q^{59} + 4248 q^{62} - 896 q^{64} - 10044 q^{69} - 10680 q^{70} + 40368 q^{71} + 10044 q^{72} - 9568 q^{73} + 7560 q^{75} + 2952 q^{77} - 6912 q^{78} + 11664 q^{81} + 71800 q^{82} + 42744 q^{85} + 8352 q^{87} - 9876 q^{92} - 10008 q^{93} + 73720 q^{94} + 33312 q^{95} - 24948 q^{96} - 59052 q^{98}+O(q^{100}) \) 16 * q + 12 * q^2 + 144 * q^4 - 36 * q^6 + 372 * q^8 + 432 * q^9 + 104 * q^13 + 680 * q^16 + 324 * q^18 - 732 * q^23 - 1764 * q^24 - 2984 * q^25 + 1800 * q^26 - 3528 * q^29 - 400 * q^31 + 5244 * q^32 + 912 * q^35 + 3888 * q^36 + 2016 * q^39 + 1008 * q^41 - 1168 * q^46 - 8664 * q^47 - 2016 * q^48 + 7240 * q^49 - 18852 * q^50 - 20952 * q^52 - 972 * q^54 + 6816 * q^55 - 13352 * q^58 + 20112 * q^59 + 4248 * q^62 - 896 * q^64 - 10044 * q^69 - 10680 * q^70 + 40368 * q^71 + 10044 * q^72 - 9568 * q^73 + 7560 * q^75 + 2952 * q^77 - 6912 * q^78 + 11664 * q^81 + 71800 * q^82 + 42744 * q^85 + 8352 * q^87 - 9876 * q^92 - 10008 * q^93 + 73720 * q^94 + 33312 * q^95 - 24948 * q^96 - 59052 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more