LMFDB - Embedded newform 21.5.b.a.8.8

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			8.8
Root			$6.02741i$ of defining polynomial
Character	$\chi$	$=$	21.8
Dual form			21.5.b.a.8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.02741i q^{2} +(-8.92230 - 1.18010i) q^{3} -20.3296 q^{4} +15.7540i q^{5} +(7.11292 - 53.7783i) q^{6} -18.5203 q^{7} -26.0965i q^{8} +(78.2147 + 21.0583i) q^{9} +O(q^{10})$	\(q+6.02741i q^{2} +(-8.92230 - 1.18010i) q^{3} -20.3296 q^{4} +15.7540i q^{5} +(7.11292 - 53.7783i) q^{6} -18.5203 q^{7} -26.0965i q^{8} +(78.2147 + 21.0583i) q^{9} -94.9556 q^{10} +196.340i q^{11} +(181.387 + 23.9909i) q^{12} +151.709 q^{13} -111.629i q^{14} +(18.5912 - 140.562i) q^{15} -167.980 q^{16} -421.337i q^{17} +(-126.927 + 471.432i) q^{18} +196.481 q^{19} -320.272i q^{20} +(165.243 + 21.8557i) q^{21} -1183.42 q^{22} +143.456i q^{23} +(-30.7963 + 232.840i) q^{24} +376.813 q^{25} +914.409i q^{26} +(-673.004 - 280.190i) q^{27} +376.510 q^{28} +753.566i q^{29} +(847.222 + 112.057i) q^{30} -635.323 q^{31} -1430.03i q^{32} +(231.700 - 1751.80i) q^{33} +2539.57 q^{34} -291.768i q^{35} +(-1590.08 - 428.108i) q^{36} +1336.52 q^{37} +1184.27i q^{38} +(-1353.59 - 179.031i) q^{39} +411.123 q^{40} +2846.73i q^{41} +(-131.733 + 995.988i) q^{42} +99.0370 q^{43} -3991.52i q^{44} +(-331.752 + 1232.19i) q^{45} -864.666 q^{46} +512.989i q^{47} +(1498.77 + 198.233i) q^{48} +343.000 q^{49} +2271.20i q^{50} +(-497.218 + 3759.29i) q^{51} -3084.18 q^{52} -4252.16i q^{53} +(1688.82 - 4056.47i) q^{54} -3093.13 q^{55} +483.313i q^{56} +(-1753.06 - 231.866i) q^{57} -4542.05 q^{58} -2893.09i q^{59} +(-377.952 + 2857.56i) q^{60} +2661.40 q^{61} -3829.35i q^{62} +(-1448.56 - 390.006i) q^{63} +5931.68 q^{64} +2390.01i q^{65} +(10558.8 + 1396.55i) q^{66} +2567.47 q^{67} +8565.62i q^{68} +(169.292 - 1279.95i) q^{69} +1758.60 q^{70} -1769.31i q^{71} +(549.548 - 2041.13i) q^{72} -3531.14 q^{73} +8055.76i q^{74} +(-3362.03 - 444.675i) q^{75} -3994.38 q^{76} -3636.27i q^{77} +(1079.09 - 8158.63i) q^{78} +2251.95 q^{79} -2646.35i q^{80} +(5674.09 + 3294.15i) q^{81} -17158.4 q^{82} +2063.23i q^{83} +(-3359.33 - 444.318i) q^{84} +6637.72 q^{85} +596.936i q^{86} +(889.280 - 6723.54i) q^{87} +5123.78 q^{88} -3069.01i q^{89} +(-7426.92 - 1999.61i) q^{90} -2809.68 q^{91} -2916.40i q^{92} +(5668.54 + 749.743i) q^{93} -3091.99 q^{94} +3095.35i q^{95} +(-1687.57 + 12759.1i) q^{96} +3416.07 q^{97} +2067.40i q^{98} +(-4134.59 + 15356.7i) 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 $

$2$			6.02741i			1.50685i	0.657533	+	0.753426i	$0.271600\pi$
$2$			6.02741i			1.50685i	−0.657533	+	0.753426i	$0.728400\pi$
$3$	−8.92230	−	1.18010i	−0.991366	−	0.131122i
$3$	−8.92230	−	1.18010i	−0.991366	−	0.131122i
$4$	−20.3296			−1.27060
$5$			15.7540i			0.630159i	0.949065	+	0.315079i	$0.102031\pi$
$5$			15.7540i			0.630159i	−0.949065	+	0.315079i	$0.897969\pi$
$6$	7.11292	−	53.7783i	0.197581	−	1.49384i
$7$	−18.5203			−0.377964
$7$	−18.5203			−0.377964
$8$		−	26.0965i		−	0.407757i
$9$	78.2147	+	21.0583i	0.965614	+	0.259980i
$10$	−94.9556			−0.949556
$11$			196.340i			1.62264i	0.584600	+	0.811322i	$0.301252\pi$
$11$			196.340i			1.62264i	−0.584600	+	0.811322i	$0.698748\pi$
$12$	181.387	+	23.9909i	1.25963	+	0.166604i
$13$	151.709			0.897684			0.448842	−	0.893611i	$-0.351837\pi$
$13$	151.709			0.897684			0.448842	+	0.893611i	$0.351837\pi$
$14$		−	111.629i		−	0.569536i
$15$	18.5912	−	140.562i	0.0826276	−	0.624718i
$16$	−167.980			−0.656173
$17$		−	421.337i		−	1.45791i	−0.684560	−	0.728956i	$-0.740006\pi$
$17$		−	421.337i		−	1.45791i	0.684560	−	0.728956i	$-0.259994\pi$
$18$	−126.927	+	471.432i	−0.391751	+	1.45504i
$19$	196.481			0.544268			0.272134	−	0.962259i	$-0.412271\pi$
$19$	196.481			0.544268			0.272134	+	0.962259i	$0.412271\pi$
$20$		−	320.272i		−	0.800681i
$21$	165.243	+	21.8557i	0.374701	+	0.0495594i
$22$	−1183.42			−2.44508
$23$			143.456i			0.271183i	0.990765	+	0.135591i	$0.0432935\pi$
$23$			143.456i			0.271183i	−0.990765	+	0.135591i	$0.956707\pi$
$24$	−30.7963	+	232.840i	−0.0534659	+	0.404237i
$25$	376.813			0.602900
$26$			914.409i			1.35268i
$27$	−673.004	−	280.190i	−0.923188	−	0.384348i
$28$	376.510			0.480242
$29$			753.566i			0.896035i	0.894025	+	0.448018i	$0.147870\pi$
$29$			753.566i			0.896035i	−0.894025	+	0.448018i	$0.852130\pi$
$30$	847.222	+	112.057i	0.941357	+	0.124507i
$31$	−635.323			−0.661107			−0.330553	−	0.943787i	$-0.607235\pi$
$31$	−635.323			−0.661107			−0.330553	+	0.943787i	$0.607235\pi$
$32$		−	1430.03i		−	1.39651i
$33$	231.700	−	1751.80i	0.212764	−	1.60863i
$34$	2539.57			2.19686
$35$		−	291.768i		−	0.238178i
$36$	−1590.08	−	428.108i	−1.22691	−	0.330331i
$37$	1336.52			0.976276			0.488138	−	0.872767i	$-0.337676\pi$
$37$	1336.52			0.976276			0.488138	+	0.872767i	$0.337676\pi$
$38$			1184.27i			0.820131i
$39$	−1353.59	−	179.031i	−0.889933	−	0.117706i
$40$	411.123			0.256952
$41$			2846.73i			1.69347i	0.532012	+	0.846737i	$0.321436\pi$
$41$			2846.73i			1.69347i	−0.532012	+	0.846737i	$0.678564\pi$
$42$	−131.733	+	995.988i	−0.0746787	+	0.564619i
$43$	99.0370			0.0535624			0.0267812	−	0.999641i	$-0.491474\pi$
$43$	99.0370			0.0535624			0.0267812	+	0.999641i	$0.491474\pi$
$44$		−	3991.52i		−	2.06173i
$45$	−331.752	+	1232.19i	−0.163828	+	0.608490i
$46$	−864.666			−0.408632
$47$			512.989i			0.232227i	0.993236	+	0.116113i	$0.0370436\pi$
$47$			512.989i			0.232227i	−0.993236	+	0.116113i	$0.962956\pi$
$48$	1498.77	+	198.233i	0.650507	+	0.0860386i
$49$	343.000			0.142857
$50$			2271.20i			0.908481i
$51$	−497.218	+	3759.29i	−0.191164	+	1.44532i
$52$	−3084.18			−1.14060
$53$		−	4252.16i		−	1.51376i	−0.653553	−	0.756881i	$-0.726722\pi$
$53$		−	4252.16i		−	1.51376i	0.653553	−	0.756881i	$-0.273278\pi$
$54$	1688.82	−	4056.47i	0.579156	−	1.39111i
$55$	−3093.13			−1.02252
$56$			483.313i			0.154118i
$57$	−1753.06	−	231.866i	−0.539569	−	0.0713654i
$58$	−4542.05			−1.35019
$59$		−	2893.09i		−	0.831110i	−0.909568	−	0.415555i	$-0.863587\pi$
$59$		−	2893.09i		−	0.831110i	0.909568	−	0.415555i	$-0.136413\pi$
$60$	−377.952	+	2857.56i	−0.104987	+	0.793768i
$61$	2661.40			0.715238			0.357619	−	0.933868i	$-0.383589\pi$
$61$	2661.40			0.715238			0.357619	+	0.933868i	$0.383589\pi$
$62$		−	3829.35i		−	0.996190i
$63$	−1448.56	−	390.006i	−0.364968	−	0.0982630i
$64$	5931.68			1.44816
$65$			2390.01i			0.565683i
$66$	10558.8	+	1396.55i	2.42397	+	0.320604i
$67$	2567.47			0.571947			0.285974	−	0.958237i	$-0.407683\pi$
$67$	2567.47			0.571947			0.285974	+	0.958237i	$0.407683\pi$
$68$			8565.62i			1.85243i
$69$	169.292	−	1279.95i	0.0355580	−	0.268842i
$70$	1758.60			0.358898
$71$		−	1769.31i		−	0.350983i	−0.984481	−	0.175492i	$-0.943849\pi$
$71$		−	1769.31i		−	0.350983i	0.984481	−	0.175492i	$-0.0561515\pi$
$72$	549.548	−	2041.13i	0.106009	−	0.393736i
$73$	−3531.14			−0.662628			−0.331314	−	0.943521i	$-0.607492\pi$
$73$	−3531.14			−0.662628			−0.331314	+	0.943521i	$0.607492\pi$
$74$			8055.76i			1.47110i
$75$	−3362.03	−	444.675i	−0.597695	−	0.0790534i
$76$	−3994.38			−0.691548
$77$		−	3636.27i		−	0.613302i
$78$	1079.09	−	8158.63i	0.177365	−	1.34100i
$79$	2251.95			0.360832			0.180416	−	0.983590i	$-0.442256\pi$
$79$	2251.95			0.360832			0.180416	+	0.983590i	$0.442256\pi$
$80$		−	2646.35i		−	0.413493i
$81$	5674.09	+	3294.15i	0.864821	+	0.502080i
$82$	−17158.4			−2.55181
$83$			2063.23i			0.299496i	0.988724	+	0.149748i	$0.0478463\pi$
$83$			2063.23i			0.299496i	−0.988724	+	0.149748i	$0.952154\pi$
$84$	−3359.33	−	444.318i	−0.476096	−	0.0629703i
$85$	6637.72			0.918716
$86$			596.936i			0.0807107i
$87$	889.280	−	6723.54i	0.117490	−	0.888299i
$88$	5123.78			0.661645
$89$		−	3069.01i		−	0.387453i	−0.981056	−	0.193726i	$-0.937943\pi$
$89$		−	3069.01i		−	0.387453i	0.981056	−	0.193726i	$-0.0620574\pi$
$90$	−7426.92	−	1999.61i	−0.916904	−	0.246865i
$91$	−2809.68			−0.339293
$92$		−	2916.40i		−	0.344566i
$93$	5668.54	+	749.743i	0.655399	+	0.0866855i
$94$	−3091.99			−0.349931
$95$			3095.35i			0.342975i
$96$	−1687.57	+	12759.1i	−0.183113	+	1.38445i
$97$	3416.07			0.363064			0.181532	−	0.983385i	$-0.441894\pi$
$97$	3416.07			0.363064			0.181532	+	0.983385i	$0.441894\pi$
$98$			2067.40i			0.215265i
$99$	−4134.59	+	15356.7i	−0.421854	+	1.56685i
$100$	−7660.46			−0.766046
$101$		−	868.123i		−	0.0851017i	−0.999094	−	0.0425509i	$-0.986452\pi$
$101$		−	868.123i		−	0.0851017i	0.999094	−	0.0425509i	$-0.0135484\pi$
$102$	−22658.8	−	2996.93i	−2.17789	−	0.288056i
$103$	2420.92			0.228195			0.114098	−	0.993470i	$-0.463602\pi$
$103$	2420.92			0.228195			0.114098	+	0.993470i	$0.463602\pi$
$104$		−	3959.06i		−	0.366037i
$105$	−344.314	+	2603.24i	−0.0312303	+	0.236121i
$106$	25629.5			2.28102
$107$			5171.46i			0.451695i	0.974163	+	0.225848i	$0.0725152\pi$
$107$			5171.46i			0.451695i	−0.974163	+	0.225848i	$0.927485\pi$
$108$	13681.9	+	5696.16i	1.17300	+	0.488353i
$109$	18184.1			1.53052			0.765258	−	0.643723i	$-0.222611\pi$
$109$	18184.1			1.53052			0.765258	+	0.643723i	$0.222611\pi$
$110$		−	18643.6i		−	1.54079i
$111$	−11924.8	−	1577.22i	−0.967847	−	0.128011i
$112$	3111.04			0.248010
$113$		−	8272.19i		−	0.647834i	−0.946086	−	0.323917i	$-0.895000\pi$
$113$		−	8272.19i		−	0.647834i	0.946086	−	0.323917i	$-0.105000\pi$
$114$	1397.55	−	10566.4i	0.107537	−	0.813050i
$115$	−2260.00			−0.170888
$116$		−	15319.7i		−	1.13850i
$117$	11865.8	+	3194.73i	0.866816	+	0.233379i
$118$	17437.9			1.25236
$119$			7803.26i			0.551039i
$120$	−3668.16	−	485.164i	−0.254733	−	0.0336920i
$121$	−23908.4			−1.63297
$122$			16041.3i			1.07776i
$123$	3359.42	−	25399.4i	0.222051	−	1.67885i
$124$	12915.9			0.840003
$125$			15782.5i			1.01008i
$126$	2350.72	−	8731.04i	0.148068	−	0.549952i
$127$	−23527.3			−1.45870			−0.729348	−	0.684143i	$-0.760176\pi$
$127$	−23527.3			−1.45870			−0.729348	+	0.684143i	$0.760176\pi$
$128$			12872.2i			0.785656i
$129$	−883.637	−	116.873i	−0.0531000	−	0.00702321i
$130$	−14405.6			−0.852400
$131$		−	19736.1i		−	1.15005i	−0.818134	−	0.575027i	$-0.804992\pi$
$131$		−	19736.1i		−	1.15005i	0.818134	−	0.575027i	$-0.195008\pi$
$132$	−4710.38	+	35613.5i	−0.270339	+	2.04393i
$133$	−3638.87			−0.205714
$134$			15475.2i			0.861840i
$135$	4414.10	−	10602.5i	0.242200	−	0.581755i
$136$	−10995.4			−0.594474
$137$			13653.6i			0.727457i	0.931505	+	0.363728i	$0.118496\pi$
$137$			13653.6i			0.727457i	−0.931505	+	0.363728i	$0.881504\pi$
$138$	7714.81	+	1020.39i	0.405104	+	0.0535806i
$139$	6654.45			0.344416			0.172208	−	0.985061i	$-0.444910\pi$
$139$	6654.45			0.344416			0.172208	+	0.985061i	$0.444910\pi$
$140$			5931.53i			0.302629i
$141$	605.376	−	4577.04i	0.0304500	−	0.230222i
$142$	10664.3			0.528880
$143$			29786.4i			1.45662i
$144$	−13138.5	−	3537.38i	−0.633609	−	0.170591i
$145$	−11871.6			−0.564644
$146$		−	21283.6i		−	0.998482i
$147$	−3060.35	−	404.773i	−0.141624	−	0.0187317i
$148$	−27171.0			−1.24046
$149$		−	25398.8i		−	1.14404i	−0.820239	−	0.572020i	$-0.806160\pi$
$149$		−	25398.8i		−	1.14404i	0.820239	−	0.572020i	$-0.193840\pi$
$150$	2680.24	−	20264.3i	0.119122	−	0.900638i
$151$	−2253.44			−0.0988307			−0.0494154	−	0.998778i	$-0.515736\pi$
$151$	−2253.44			−0.0988307			−0.0494154	+	0.998778i	$0.515736\pi$
$152$		−	5127.45i		−	0.221929i
$153$	8872.65	−	32954.7i	0.379027	−	1.40778i
$154$	21917.3			0.924155
$155$		−	10008.9i		−	0.416602i
$156$	27518.0	+	3639.63i	1.13075	+	0.149557i
$157$	−41471.1			−1.68246			−0.841232	−	0.540674i	$-0.818169\pi$
$157$	−41471.1			−1.68246			−0.841232	+	0.540674i	$0.818169\pi$
$158$			13573.4i			0.543720i
$159$	−5017.96	+	37939.0i	−0.198487	+	1.50069i
$160$	22528.6			0.880024
$161$		−	2656.84i		−	0.102498i
$162$	−19855.2	+	34200.1i	−0.756560	+	1.30316i
$163$	43022.4			1.61927			0.809635	−	0.586934i	$-0.199665\pi$
$163$	43022.4			1.61927			0.809635	+	0.586934i	$0.199665\pi$
$164$		−	57873.0i		−	2.15173i
$165$	27597.8	+	3650.20i	1.01369	+	0.134075i
$166$	−12435.9			−0.451296
$167$		−	13076.2i		−	0.468867i	−0.972132	−	0.234433i	$-0.924677\pi$
$167$		−	13076.2i		−	0.468867i	0.972132	−	0.234433i	$-0.0753234\pi$
$168$	570.356	−	4312.26i	0.0202082	−	0.152787i
$169$	−5545.52			−0.194164
$170$			40008.2i			1.38437i
$171$	15367.7	+	4137.56i	0.525553	+	0.141498i
$172$	−2013.38			−0.0680566
$173$			20091.4i			0.671303i	0.941986	+	0.335652i	$0.108957\pi$
$173$			20091.4i			0.671303i	−0.941986	+	0.335652i	$0.891043\pi$
$174$	40525.5	+	5360.06i	1.33854	+	0.177040i
$175$	−6978.67			−0.227875
$176$		−	32981.2i		−	1.06473i
$177$	−3414.13	+	25813.1i	−0.108977	+	0.823935i
$178$	18498.2			0.583834
$179$			40066.7i			1.25048i	0.780431	+	0.625242i	$0.215000\pi$
$179$			40066.7i			1.25048i	−0.780431	+	0.625242i	$0.785000\pi$
$180$	6744.40	−	25050.0i	0.208161	−	0.773149i
$181$	−63926.2			−1.95129			−0.975645	−	0.219354i	$-0.929605\pi$
$181$	−63926.2			−1.95129			−0.975645	+	0.219354i	$0.929605\pi$
$182$		−	16935.1i		−	0.511264i
$183$	−23745.8	−	3140.71i	−0.709063	−	0.0937834i
$184$	3743.69			0.110577
$185$			21055.5i			0.615209i
$186$	−4519.01	+	34166.6i	−0.130622	+	0.987589i
$187$	82725.2			2.36567
$188$		−	10428.9i		−	0.295068i
$189$	12464.2	+	5189.19i	0.348932	+	0.145270i
$190$	−18656.9			−0.516812
$191$		−	57708.3i		−	1.58187i	−0.611899	−	0.790936i	$-0.709594\pi$
$191$		−	57708.3i		−	1.58187i	0.611899	−	0.790936i	$-0.290406\pi$
$192$	−52924.2	−	6999.95i	−1.43566	−	0.189886i
$193$	−2395.62			−0.0643137			−0.0321568	−	0.999483i	$-0.510238\pi$
$193$	−2395.62			−0.0643137			−0.0321568	+	0.999483i	$0.510238\pi$
$194$			20590.0i			0.547084i
$195$	2820.44	−	21324.4i	0.0741734	−	0.560799i
$196$	−6973.06			−0.181515
$197$		−	2034.61i		−	0.0524263i	−0.999656	−	0.0262132i	$-0.991655\pi$
$197$		−	2034.61i		−	0.0524263i	0.999656	−	0.0262132i	$-0.00834487\pi$
$198$	−92560.9	−	24920.9i	−2.36101	−	0.635672i
$199$	25978.9			0.656016			0.328008	−	0.944675i	$-0.393623\pi$
$199$	25978.9			0.656016			0.328008	+	0.944675i	$0.393623\pi$
$200$		−	9833.47i		−	0.245837i
$201$	−22907.7	−	3029.87i	−0.567009	−	0.0749948i
$202$	5232.53			0.128236
$203$		−	13956.2i		−	0.338670i
$204$	10108.3	−	76425.0i	0.242894	−	1.83643i
$205$	−44847.3			−1.06716
$206$			14591.9i			0.343856i
$207$	−3020.94	+	11220.4i	−0.0705020	+	0.261858i
$208$	−25484.0			−0.589035
$209$			38577.0i			0.883153i
$210$	−15690.8	−	2075.32i	−0.355800	−	0.0470594i
$211$	60437.2			1.35750			0.678749	−	0.734370i	$-0.262522\pi$
$211$	60437.2			1.35750			0.678749	+	0.734370i	$0.262522\pi$
$212$			86444.8i			1.92339i
$213$	−2087.95	+	15786.3i	−0.0460216	+	0.347953i
$214$	−31170.5			−0.680638
$215$			1560.22i			0.0337528i
$216$	−7311.96	+	17563.0i	−0.156721	+	0.376437i
$217$	11766.4			0.249875
$218$			109603.i			2.30626i
$219$	31505.9	+	4167.09i	0.656907	+	0.0868850i
$220$	62882.2			1.29922
$221$		−	63920.4i		−	1.30874i
$222$	9506.58	−	71875.9i	0.192894	−	1.45840i
$223$	11036.5			0.221933			0.110966	−	0.993824i	$-0.464605\pi$
$223$	11036.5			0.221933			0.110966	+	0.993824i	$0.464605\pi$
$224$			26484.5i			0.527832i
$225$	29472.3	+	7935.05i	0.582169	+	0.156742i
$226$	49859.8			0.976189
$227$		−	73080.6i		−	1.41824i	−0.705087	−	0.709121i	$-0.749092\pi$
$227$		−	73080.6i		−	1.41824i	0.705087	−	0.709121i	$-0.250908\pi$
$228$	35639.0	+	4713.75i	0.685577	+	0.0906770i
$229$	15706.6			0.299509			0.149755	−	0.988723i	$-0.452152\pi$
$229$	15706.6			0.299509			0.149755	+	0.988723i	$0.452152\pi$
$230$		−	13621.9i		−	0.257503i
$231$	−4291.15	+	32443.8i	−0.0804173	+	0.608007i
$232$	19665.4			0.365365
$233$		−	58490.4i		−	1.07739i	−0.842501	−	0.538694i	$-0.818918\pi$
$233$		−	58490.4i		−	1.07739i	0.842501	−	0.538694i	$-0.181082\pi$
$234$	−19255.9	+	71520.3i	−0.351668	+	1.30616i
$235$	−8081.61			−0.146340
$236$			58815.6i			1.05601i
$237$	−20092.6	−	2657.52i	−0.357716	−	0.0473129i
$238$	−47033.4			−0.830334
$239$		−	13382.2i		−	0.234277i	−0.993116	−	0.117139i	$-0.962628\pi$
$239$		−	13382.2i		−	0.234277i	0.993116	−	0.117139i	$-0.0373722\pi$
$240$	−3122.95	+	23611.6i	−0.0542179	+	0.409923i
$241$	−310.843			−0.00535189			−0.00267595	−	0.999996i	$-0.500852\pi$
$241$	−310.843			−0.00535189			−0.00267595	+	0.999996i	$0.500852\pi$
$242$		−	144105.i		−	2.46065i
$243$	−46738.5	−	36087.3i	−0.791521	−	0.611142i
$244$	−54105.3			−0.908783
$245$			5403.61i			0.0900227i
$246$	153092.	+	20248.6i	2.52978	+	0.334599i
$247$	29807.8			0.488580
$248$			16579.7i			0.269571i
$249$	2434.81	−	18408.7i	0.0392705	−	0.296910i
$250$	−95127.7			−1.52204
$251$			82720.9i			1.31301i	0.754322	+	0.656505i	$0.227966\pi$
$251$			82720.9i			1.31301i	−0.754322	+	0.656505i	$0.772034\pi$
$252$	29448.6	+	7928.68i	0.463729	+	0.124853i
$253$	−28166.1			−0.440033
$254$		−	141809.i		−	2.19804i
$255$	−59223.7	−	7833.15i	−0.910784	−	0.120464i
$256$	17320.9			0.264297
$257$		−	10849.5i		−	0.164265i	−0.996621	−	0.0821324i	$-0.973827\pi$
$257$		−	10849.5i		−	0.164265i	0.996621	−	0.0821324i	$-0.0261730\pi$
$258$	704.442	−	5326.04i	0.0105829	−	0.0800138i
$259$	−24752.7			−0.368998
$260$		−	48588.0i		−	0.718758i
$261$	−15868.8	+	58940.0i	−0.232951	+	0.865224i
$262$	118957.			1.73296
$263$			40082.3i			0.579484i	0.957105	+	0.289742i	$0.0935695\pi$
$263$			40082.3i			0.579484i	−0.957105	+	0.289742i	$0.906431\pi$
$264$	−45715.8	−	6046.55i	−0.655932	−	0.0867561i
$265$	66988.3			0.953910
$266$		−	21933.0i		−	0.309980i
$267$	−3621.73	+	27382.7i	−0.0508036	+	0.384108i
$268$	−52195.8			−0.726717
$269$			75313.8i			1.04081i	0.853921	+	0.520403i	$0.174218\pi$
$269$			75313.8i			1.04081i	−0.853921	+	0.520403i	$0.825782\pi$
$270$	63905.5	+	26605.6i	0.876618	+	0.364960i
$271$	−113072.			−1.53963			−0.769814	−	0.638268i	$-0.779651\pi$
$271$	−113072.			−1.53963			−0.769814	+	0.638268i	$0.779651\pi$
$272$			70776.2i			0.956642i
$273$	25068.8	+	3315.70i	0.336363	+	0.0444887i
$274$	−82296.0			−1.09617
$275$			73983.4i			0.978292i
$276$	−3441.64	+	26021.0i	−0.0451801	+	0.341591i
$277$	−52989.5			−0.690606			−0.345303	−	0.938491i	$-0.612224\pi$
$277$	−52989.5			−0.690606			−0.345303	+	0.938491i	$0.612224\pi$
$278$			40109.1i			0.518983i
$279$	−49691.7	−	13378.9i	−0.638374	−	0.171874i
$280$	−7614.10			−0.0971186
$281$			48221.1i			0.610695i	0.952241	+	0.305347i	$0.0987726\pi$
$281$			48221.1i			0.610695i	−0.952241	+	0.305347i	$0.901227\pi$
$282$	27587.7	+	3648.85i	0.346910	+	0.0458836i
$283$	63901.6			0.797882			0.398941	−	0.916977i	$-0.369378\pi$
$283$	63901.6			0.797882			0.398941	+	0.916977i	$0.369378\pi$
$284$			35969.4i			0.445960i
$285$	3652.81	−	27617.6i	0.0449715	−	0.340014i
$286$	−179535.			−2.19491
$287$		−	52722.2i		−	0.640073i
$288$	30114.0	−	111849.i	0.363065	−	1.34849i
$289$	−94003.5			−1.12551
$290$		−	71555.2i		−	0.850835i
$291$	−30479.2	−	4031.29i	−0.359930	−	0.0476056i
$292$	71786.9			0.841937
$293$		−	76356.1i		−	0.889423i	−0.895674	−	0.444711i	$-0.853306\pi$
$293$		−	76356.1i		−	0.889423i	0.895674	−	0.444711i	$-0.146694\pi$
$294$	2439.73	−	18446.0i	0.0282259	−	0.213406i
$295$	45577.7			0.523731
$296$		−	34878.5i		−	0.398083i
$297$	55012.4	−	132138.i	0.623660	−	1.49801i
$298$	153089.			1.72390
$299$			21763.5i			0.243436i
$300$	68348.9	+	9040.09i	0.759432	+	0.100445i
$301$	−1834.19			−0.0202447
$302$		−	13582.4i		−	0.148923i
$303$	−1024.47	+	7745.65i	−0.0111587	+	0.0843670i
$304$	−33004.9			−0.357134
$305$			41927.6i			0.450713i
$306$	198632.	+	53479.1i	2.12132	+	0.571138i
$307$	88506.5			0.939071			0.469535	−	0.882914i	$-0.344421\pi$
$307$	88506.5			0.939071			0.469535	+	0.882914i	$0.344421\pi$
$308$			73924.0i			0.779263i
$309$	−21600.2	−	2856.92i	−0.226225	−	0.0299214i
$310$	60327.5			0.627757
$311$			23502.0i			0.242987i	0.992592	+	0.121494i	$0.0387684\pi$
$311$			23502.0i			0.242987i	−0.992592	+	0.121494i	$0.961232\pi$
$312$	−4672.07	+	35323.9i	−0.0479954	+	0.362877i
$313$	158727.			1.62018			0.810089	−	0.586307i	$-0.199419\pi$
$313$	158727.			1.62018			0.810089	+	0.586307i	$0.199419\pi$
$314$		−	249963.i		−	2.53522i
$315$	6144.14	−	22820.5i	0.0619213	−	0.229988i
$316$	−45781.3			−0.458473
$317$			173185.i			1.72342i	0.507403	+	0.861709i	$0.330606\pi$
$317$			173185.i			1.72342i	−0.507403	+	0.861709i	$0.669394\pi$
$318$	−228674.	−	30245.3i	−2.26132	−	0.299091i
$319$	−147955.			−1.45395
$320$			93447.4i			0.912573i
$321$	6102.82	−	46141.3i	0.0592271	−	0.447796i
$322$	16013.8			0.154449
$323$		−	82784.5i		−	0.793494i
$324$	−115352.	−	66968.8i	−1.09884	−	0.637944i
$325$	57165.7			0.541214
$326$			259313.i			2.44000i
$327$	−162244.	−	21459.0i	−1.51730	−	0.200684i
$328$	74289.5			0.690526
$329$		−	9500.68i		−	0.0877734i
$330$	−22001.2	+	166343.i	−0.202031	+	1.52749i
$331$	84106.5			0.767668			0.383834	−	0.923402i	$-0.374603\pi$
$331$	84106.5			0.767668			0.383834	+	0.923402i	$0.374603\pi$
$332$		−	41944.7i		−	0.380540i
$333$	104536.	+	28144.9i	0.942706	+	0.253812i
$334$	78815.7			0.706512
$335$			40447.9i			0.360418i
$336$	−27757.6	−	3671.32i	−0.245869	−	0.0325195i
$337$	3077.59			0.0270989			0.0135494	−	0.999908i	$-0.495687\pi$
$337$	3077.59			0.0270989			0.0135494	+	0.999908i	$0.495687\pi$
$338$		−	33425.1i		−	0.292576i
$339$	−9761.98	+	73806.9i	−0.0849451	+	0.642240i
$340$	−134942.			−1.16732
$341$		−	124739.i		−	1.07274i
$342$	−24938.7	+	92627.3i	−0.213217	+	0.791930i
$343$	−6352.45			−0.0539949
$344$		−	2584.51i		−	0.0218405i
$345$	20164.4	+	2667.02i	0.169413	+	0.0224072i
$346$	−121099.			−1.01155
$347$		−	102480.i		−	0.851101i	−0.904935	−	0.425551i	$-0.860080\pi$
$347$		−	102480.i		−	0.851101i	0.904935	−	0.425551i	$-0.139920\pi$
$348$	−18078.7	+	136687.i	−0.149283	+	1.12867i
$349$	81452.0			0.668730			0.334365	−	0.942444i	$-0.391478\pi$
$349$	81452.0			0.668730			0.334365	+	0.942444i	$0.391478\pi$
$350$		−	42063.3i		−	0.343374i
$351$	−102100.	−	42507.2i	−0.828731	−	0.345023i
$352$	280772.			2.26604
$353$		−	198402.i		−	1.59220i	−0.605167	−	0.796099i	$-0.706894\pi$
$353$		−	198402.i		−	1.59220i	0.605167	−	0.796099i	$-0.293106\pi$
$354$	−155586.	−	20578.4i	−1.24155	−	0.164212i
$355$	27873.6			0.221175
$356$			62391.9i			0.492298i
$357$	9208.61	−	69623.0i	0.0722533	−	0.546281i
$358$	−241498.			−1.88429
$359$			20290.7i			0.157437i	0.996897	+	0.0787186i	$0.0250829\pi$
$359$			20290.7i			0.157437i	−0.996897	+	0.0787186i	$0.974917\pi$
$360$	32155.8	+	8657.56i	0.248116	+	0.0668022i
$361$	−91716.4			−0.703773
$362$		−	385309.i		−	2.94031i
$363$	213318.	+	28214.2i	1.61888	+	0.214119i
$364$	57119.8			0.431106
$365$		−	55629.5i		−	0.417561i
$366$	18930.3	−	143126.i	0.141318	−	1.06845i
$367$	−196750.			−1.46077			−0.730385	−	0.683036i	$-0.760659\pi$
$367$	−196750.			−1.46077			−0.730385	+	0.683036i	$0.760659\pi$
$368$		−	24097.7i		−	0.177943i
$369$	−59947.4	+	222656.i	−0.440269	+	1.63524i
$370$	−126910.			−0.927028
$371$			78751.1i			0.572148i
$372$	−115239.	−	15242.0i	−0.832751	−	0.110143i
$373$	−144072.			−1.03553			−0.517765	−	0.855523i	$-0.673236\pi$
$373$	−144072.			−1.03553			−0.517765	+	0.855523i	$0.673236\pi$
$374$			498618.i			3.56472i
$375$	18624.9	−	140816.i	0.132444	−	1.00136i
$376$	13387.2			0.0946921
$377$			114322.i			0.804356i
$378$	−31277.3	+	75126.9i	−0.218900	+	0.525789i
$379$	134879.			0.939004			0.469502	−	0.882931i	$-0.344433\pi$
$379$	134879.			0.939004			0.469502	+	0.882931i	$0.344433\pi$
$380$		−	62927.3i		−	0.435785i
$381$	209918.	+	27764.5i	1.44610	+	0.191267i
$382$	347831.			2.38365
$383$			10393.5i			0.0708540i	0.999372	+	0.0354270i	$0.0112791\pi$
$383$			10393.5i			0.0708540i	−0.999372	+	0.0354270i	$0.988721\pi$
$384$	15190.4	−	114849.i	0.103017	−	0.778873i
$385$	57285.6			0.386477
$386$		−	14439.4i		−	0.0969112i
$387$	7746.15	+	2085.55i	0.0517207	+	0.0139251i
$388$	−69447.5			−0.461310
$389$			22075.3i			0.145884i	0.997336	+	0.0729421i	$0.0232388\pi$
$389$			22075.3i			0.145884i	−0.997336	+	0.0729421i	$0.976761\pi$
$390$	128531.	+	17000.0i	0.845041	+	0.111768i
$391$	60443.2			0.395361
$392$		−	8951.08i		−	0.0582510i
$393$	−23290.5	+	176091.i	−0.150797	+	1.14012i
$394$	12263.4			0.0789987
$395$			35477.1i			0.227381i
$396$	84054.8	−	312196.i	0.536009	−	1.99084i
$397$	39063.7			0.247852			0.123926	−	0.992291i	$-0.460452\pi$
$397$	39063.7			0.247852			0.123926	+	0.992291i	$0.460452\pi$
$398$			156585.i			0.988518i
$399$	32467.1	+	4294.22i	0.203938	+	0.0269736i
$400$	−63297.0			−0.395607
$401$			42017.8i			0.261303i	0.991428	+	0.130652i	$0.0417069\pi$
$401$			42017.8i			0.261303i	−0.991428	+	0.130652i	$0.958293\pi$
$402$	18262.2	−	138074.i	0.113006	−	0.854399i
$403$	−96384.0			−0.593465
$404$			17648.6i			0.108130i
$405$	−51895.9	+	89389.4i	−0.316390	+	0.544975i
$406$	84119.9			0.510325
$407$			262413.i			1.58415i
$408$	98104.1	+	12975.6i	0.589341	+	0.0779485i
$409$	279559.			1.67120			0.835598	−	0.549342i	$-0.185122\pi$
$409$	279559.			1.67120			0.835598	+	0.549342i	$0.185122\pi$
$410$		−	270313.i		−	1.60805i
$411$	16112.6	−	121822.i	0.0953855	−	0.721176i
$412$	−49216.5			−0.289945
$413$			53580.9i			0.314130i
$414$	−67629.6	−	18208.4i	−0.394581	−	0.106236i
$415$	−32504.0			−0.188730
$416$		−	216947.i		−	1.25363i
$417$	−59373.0	−	7852.90i	−0.341442	−	0.0451604i
$418$	−232519.			−1.33078
$419$		−	230293.i		−	1.31175i	−0.754868	−	0.655876i	$-0.772299\pi$
$419$		−	230293.i		−	1.31175i	0.754868	−	0.655876i	$-0.227701\pi$
$420$	6999.77	−	52922.8i	0.0396813	−	0.300016i
$421$	−50251.7			−0.283522			−0.141761	−	0.989901i	$-0.545276\pi$
$421$	−50251.7			−0.283522			−0.141761	+	0.989901i	$0.545276\pi$
$422$			364279.i			2.04555i
$423$	−10802.7	+	40123.3i	−0.0603742	+	0.224241i
$424$	−110966.			−0.617247
$425$		−	158765.i		−	0.878975i
$426$	−95150.3	−	12584.9i	−0.524314	−	0.0693477i
$427$	−49289.8			−0.270335
$428$		−	105134.i		−	0.573925i
$429$	35150.9	−	265763.i	0.190995	−	1.44405i
$430$	−9404.11			−0.0508605
$431$			89839.7i			0.483631i	0.970322	+	0.241815i	$0.0777428\pi$
$431$			89839.7i			0.483631i	−0.970322	+	0.241815i	$0.922257\pi$
$432$	113051.	+	47066.3i	0.605771	+	0.252199i
$433$	−204051.			−1.08833			−0.544167	−	0.838977i	$-0.683154\pi$
$433$	−204051.			−1.08833			−0.544167	+	0.838977i	$0.683154\pi$
$434$			70920.6i			0.376524i
$435$	105922.	+	14009.7i	0.559769	+	0.0740372i
$436$	−369675.			−1.94468
$437$			28186.3i			0.147596i
$438$	−25116.8	+	189899.i	−0.130923	+	0.989862i
$439$	−195110.			−1.01239			−0.506197	−	0.862418i	$-0.668949\pi$
$439$	−195110.			−1.01239			−0.506197	+	0.862418i	$0.668949\pi$
$440$			80719.8i			0.416941i
$441$	26827.7	+	7223.01i	0.137945	+	0.0371399i
$442$	385274.			1.97208
$443$		−	302752.i		−	1.54269i	−0.636416	−	0.771346i	$-0.719584\pi$
$443$		−	302752.i		−	1.54269i	0.636416	−	0.771346i	$-0.280416\pi$
$444$	242428.	+	32064.4i	1.22975	+	0.162651i
$445$	48349.1			0.244157
$446$			66521.5i			0.334420i
$447$	−29973.1	+	226616.i	−0.150009	+	1.13416i
$448$	−109856.			−0.547354
$449$		−	162997.i		−	0.808512i	−0.914646	−	0.404256i	$-0.867530\pi$
$449$		−	162997.i		−	0.808512i	0.914646	−	0.404256i	$-0.132470\pi$
$450$	−47827.8	+	177642.i	−0.236187	+	0.877242i
$451$	−558927.			−2.74790
$452$			168171.i			0.823139i
$453$	20105.9	+	2659.28i	0.0979774	+	0.0129589i
$454$	440486.			2.13708
$455$		−	44263.6i		−	0.213808i
$456$	−6050.88	+	45748.6i	−0.0290997	+	0.220013i
$457$	−156346.			−0.748607			−0.374304	−	0.927306i	$-0.622118\pi$
$457$	−156346.			−0.748607			−0.374304	+	0.927306i	$0.622118\pi$
$458$			94669.9i			0.451316i
$459$	−118054.	+	283561.i	−0.560346	+	1.34593i
$460$	45944.9			0.217131
$461$		−	140167.i		−	0.659543i	−0.944061	−	0.329772i	$-0.893028\pi$
$461$		−	140167.i		−	0.659543i	0.944061	−	0.329772i	$-0.106972\pi$
$462$	−195552.	−	25864.5i	−0.916176	−	0.121177i
$463$	46832.0			0.218464			0.109232	−	0.994016i	$-0.465161\pi$
$463$	46832.0			0.218464			0.109232	+	0.994016i	$0.465161\pi$
$464$		−	126584.i		−	0.587954i
$465$	−11811.4	+	89302.0i	−0.0546256	+	0.413005i
$466$	352545.			1.62347
$467$			364198.i			1.66995i	0.550285	+	0.834977i	$0.314519\pi$
$467$			364198.i			1.66995i	−0.550285	+	0.834977i	$0.685481\pi$
$468$	−241228.	−	64947.7i	−1.10138	−	0.296532i
$469$	−47550.2			−0.216176
$470$		−	48711.1i		−	0.220512i
$471$	370017.	+	48939.9i	1.66794	+	0.220608i
$472$	−75499.5			−0.338891
$473$			19444.9i			0.0869128i
$474$	16018.0	−	121106.i	0.0712936	−	0.539026i
$475$	74036.4			0.328139
$476$		−	158637.i		−	0.700151i
$477$	89543.4	−	332581.i	0.393547	−	1.46171i
$478$	80659.7			0.353021
$479$		−	139602.i		−	0.608445i	−0.952601	−	0.304223i	$-0.901603\pi$
$479$		−	139602.i		−	0.608445i	0.952601	−	0.304223i	$-0.0983967\pi$
$480$	−201007.	−	26585.9i	−0.872426	−	0.115390i
$481$	202762.			0.876387
$482$		−	1873.58i		−	0.00806451i
$483$	−3135.33	+	23705.1i	−0.0134397	+	0.101613i
$484$	486048.			2.07486
$485$			53816.7i			0.228788i
$486$	217513.	−	281712.i	0.920900	−	1.19270i
$487$	−122130.			−0.514950			−0.257475	−	0.966285i	$-0.582891\pi$
$487$	−122130.			−0.514950			−0.257475	+	0.966285i	$0.582891\pi$
$488$		−	69453.1i		−	0.291643i
$489$	−383858.	−	50770.6i	−1.60529	−	0.212322i
$490$	−32569.8			−0.135651
$491$		−	239214.i		−	0.992257i	−0.868249	−	0.496129i	$-0.834754\pi$
$491$		−	239214.i		−	0.992257i	0.868249	−	0.496129i	$-0.165246\pi$
$492$	−68295.7	+	516360.i	−0.282139	+	2.13315i
$493$	317505.			1.30634
$494$			179664.i			0.736218i
$495$	−241929.	−	65136.2i	−0.987363	−	0.265835i
$496$	106722.			0.433800
$497$			32768.0i			0.132659i
$498$	110957.	+	14675.6i	0.447400	+	0.0591748i
$499$	28578.8			0.114774			0.0573869	−	0.998352i	$-0.481723\pi$
$499$	28578.8			0.114774			0.0573869	+	0.998352i	$0.481723\pi$
$500$		−	320853.i		−	1.28341i
$501$	−15431.2	+	116670.i	−0.0614787	+	0.464818i
$502$	−498593.			−1.97851
$503$		−	394825.i		−	1.56052i	−0.625457	−	0.780258i	$-0.715088\pi$
$503$		−	394825.i		−	1.56052i	0.625457	−	0.780258i	$-0.284912\pi$
$504$	−10177.8	+	37802.2i	−0.0400675	+	0.148818i
$505$	13676.4			0.0536276
$506$		−	169769.i		−	0.663065i
$507$	49478.8	+	6544.25i	0.192488	+	0.0254591i
$508$	478301.			1.85342
$509$			370203.i			1.42891i	0.699681	+	0.714455i	$0.253325\pi$
$509$			370203.i			1.42891i	−0.699681	+	0.714455i	$0.746675\pi$
$510$	47213.6	−	356965.i	0.181521	−	1.37242i
$511$	65397.7			0.250450
$512$			310355.i			1.18391i
$513$	−132232.	−	55051.9i	−0.502462	−	0.209188i
$514$	65394.5			0.247523
$515$			38139.1i			0.143799i
$516$	17964.0	+	2375.99i	0.0674690	+	0.00892370i
$517$	−100720.			−0.376821
$518$		−	149195.i		−	0.556025i
$519$	23709.8	−	179262.i	0.0880226	−	0.665508i
$520$	62370.8			0.230661
$521$			319934.i			1.17865i	0.807896	+	0.589325i	$0.200606\pi$
$521$			319934.i			1.17865i	−0.807896	+	0.589325i	$0.799394\pi$
$522$	−355255.	−	95648.0i	−1.30376	−	0.351022i
$523$	365694.			1.33695			0.668474	−	0.743736i	$-0.266948\pi$
$523$	365694.			1.33695			0.668474	+	0.743736i	$0.266948\pi$
$524$			401227.i			1.46126i
$525$	62265.7	+	8235.50i	0.225907	+	0.0298794i
$526$	−241592.			−0.873196
$527$			267685.i			0.963835i
$528$	−38921.0	+	294268.i	−0.139610	+	1.05554i
$529$	259261.			0.926460
$530$			403766.i			1.43740i
$531$	60923.8	−	226283.i	0.216072	−	0.802532i
$532$	73976.9			0.261380
$533$			431873.i			1.52020i
$534$	−165046.	−	21829.7i	−0.578793	−	0.0765534i
$535$	−81471.0			−0.284640
$536$		−	67001.9i		−	0.233216i
$537$	47282.6	−	357487.i	0.163966	−	1.23969i
$538$	−453947.			−1.56834
$539$			67344.6i			0.231806i
$540$	−89737.0	+	215545.i	−0.307740	+	0.739179i
$541$	−398244.			−1.36068			−0.680339	−	0.732898i	$-0.738167\pi$
$541$	−398244.			−1.36068			−0.680339	+	0.732898i	$0.738167\pi$
$542$		−	681530.i		−	2.31999i
$543$	570369.	+	75439.1i	1.93444	+	0.255857i
$544$	−602523.			−2.03599
$545$			286471.i			0.964468i
$546$	−19985.0	+	151100.i	−0.0670378	+	0.506849i
$547$	310182.			1.03667			0.518336	−	0.855177i	$-0.326552\pi$
$547$	310182.			1.03667			0.518336	+	0.855177i	$0.326552\pi$
$548$		−	277573.i		−	0.924308i
$549$	208161.	+	56044.7i	0.690644	+	0.185947i
$550$	−445928.			−1.47414
$551$			148061.i			0.487683i
$552$	−33402.3	−	4417.91i	−0.109622	−	0.0144990i
$553$	−41706.7			−0.136382
$554$		−	319389.i		−	1.04064i
$555$	24847.5	−	187864.i	0.0806673	−	0.609897i
$556$	−135283.			−0.437615
$557$		−	446126.i		−	1.43796i	−0.695030	−	0.718980i	$-0.744609\pi$
$557$		−	446126.i		−	1.43796i	0.695030	−	0.718980i	$-0.255391\pi$
$558$	80639.8	−	299512.i	0.258989	−	0.961935i
$559$	15024.8			0.0480821
$560$			49011.2i			0.156286i
$561$	−738099.	−	97623.7i	−2.34525	−	0.310191i
$562$	−290648.			−0.920226
$563$			489563.i			1.54451i	0.635310	+	0.772257i	$0.280872\pi$
$563$			489563.i			1.54451i	−0.635310	+	0.772257i	$0.719128\pi$
$564$	−12307.1	+	93049.5i	−0.0386898	+	0.292520i
$565$	130320.			0.408238
$566$			385161.i			1.20229i
$567$	−105086.	−	61008.4i	−0.326872	−	0.189768i
$568$	−46172.6			−0.143116
$569$		−	444748.i		−	1.37369i	−0.726803	−	0.686846i	$-0.758995\pi$
$569$		−	444748.i		−	1.37369i	0.726803	−	0.686846i	$-0.241005\pi$
$570$	166463.	+	22017.0i	0.512350	+	0.0677654i
$571$	48894.3			0.149964			0.0749818	−	0.997185i	$-0.476110\pi$
$571$	48894.3			0.149964			0.0749818	+	0.997185i	$0.476110\pi$
$572$		−	605547.i		−	1.85079i
$573$	−68101.4	+	514890.i	−0.207418	+	1.56821i
$574$	317778.			0.964495
$575$			54055.9i			0.163496i
$576$	463945.	+	124911.i	1.39837	+	0.376493i
$577$	−80474.7			−0.241717			−0.120859	−	0.992670i	$-0.538565\pi$
$577$	−80474.7			−0.241717			−0.120859	+	0.992670i	$0.538565\pi$
$578$		−	566598.i		−	1.69597i
$579$	21374.4	+	2827.06i	0.0637584	+	0.00843293i
$580$	241346.			0.717438
$581$		−	38211.5i		−	0.113199i
$582$	24298.2	−	183710.i	0.0717347	−	0.542360i
$583$	834868.			2.45630
$584$			92150.4i			0.270191i
$585$	−50329.7	+	186934.i	−0.147066	+	0.546232i
$586$	460229.			1.34023
$587$		−	54845.0i		−	0.159170i	−0.996828	−	0.0795849i	$-0.974641\pi$
$587$		−	54845.0i		−	0.159170i	0.996828	−	0.0795849i	$-0.0253595\pi$
$588$	62215.7	+	8228.89i	0.179947	+	0.0238005i
$589$	−124829.			−0.359819
$590$			274715.i			0.789185i
$591$	−2401.04	+	18153.4i	−0.00687424	+	0.0519737i
$592$	−224509.			−0.640605
$593$		−	192797.i		−	0.548265i	−0.961692	−	0.274133i	$-0.911609\pi$
$593$		−	192797.i		−	0.548265i	0.961692	−	0.274133i	$-0.0883907\pi$
$594$	796447.	+	331582.i	2.25727	+	0.939763i
$595$	−122932.			−0.347242
$596$			516349.i			1.45362i
$597$	−231791.	−	30657.6i	−0.650352	−	0.0860180i
$598$	−131177.			−0.366823
$599$		−	466565.i		−	1.30035i	−0.759786	−	0.650173i	$-0.774696\pi$
$599$		−	466565.i		−	1.30035i	0.759786	−	0.650173i	$-0.225304\pi$
$600$	−11604.4	+	87737.2i	−0.0322346	+	0.243714i
$601$	125070.			0.346261			0.173130	−	0.984899i	$-0.444612\pi$
$601$	125070.			0.346261			0.173130	+	0.984899i	$0.444612\pi$
$602$		−	11055.4i		−	0.0305058i
$603$	200814.	+	54066.7i	0.552280	+	0.148695i
$604$	45811.6			0.125575
$605$		−	376652.i		−	1.02903i
$606$	−46686.2	−	6174.89i	−0.127129	−	0.0168145i
$607$	−385152.			−1.04533			−0.522667	−	0.852537i	$-0.675063\pi$
$607$	−385152.			−1.04533			−0.522667	+	0.852537i	$0.675063\pi$
$608$		−	280973.i		−	0.760076i
$609$	−16469.7	+	124522.i	−0.0444070	+	0.335746i
$610$	−252715.			−0.679158
$611$			77824.8i			0.208466i
$612$	−180378.	+	669958.i	−0.481593	+	1.78873i
$613$	−135252.			−0.359933			−0.179967	−	0.983673i	$-0.557599\pi$
$613$	−135252.			−0.359933			−0.179967	+	0.983673i	$0.557599\pi$
$614$			533464.i			1.41504i
$615$	400141.	+	52924.1i	1.05794	+	0.139928i
$616$	−94893.7			−0.250078
$617$			378025.i			0.993003i	0.868036	+	0.496502i	$0.165382\pi$
$617$			378025.i			0.993003i	−0.868036	+	0.496502i	$0.834618\pi$
$618$	17219.8	−	130193.i	0.0450871	−	0.340888i
$619$	−286244.			−0.747059			−0.373529	−	0.927618i	$-0.621853\pi$
$619$	−286244.			−0.747059			−0.373529	+	0.927618i	$0.621853\pi$
$620$			203476.i			0.529335i
$621$	40194.8	−	96546.3i	0.104229	−	0.250353i
$622$	−141656.			−0.366146
$623$			56838.9i			0.146443i
$624$	227376.	+	30073.6i	0.583950	+	0.0772354i
$625$	−13129.4			−0.0336112
$626$			956713.i			2.44137i
$627$	45524.6	−	344195.i	0.115801	−	0.875528i
$628$	843091.			2.13774
$629$		−	563125.i		−	1.42332i
$630$	137549.	+	37033.2i	0.346557	+	0.0933062i
$631$	−104005.			−0.261214			−0.130607	−	0.991434i	$-0.541693\pi$
$631$	−104005.			−0.261214			−0.130607	+	0.991434i	$0.541693\pi$
$632$		−	58767.9i		−	0.147132i
$633$	−539238.	−	71321.7i	−1.34578	−	0.177998i
$634$	−1.04385e6			−2.59694
$635$		−	370648.i		−	0.919210i
$636$	102013.	−	771286.i	0.252198	−	1.90678i
$637$	52036.0			0.128241
$638$		−	891785.i		−	2.19088i
$639$	37258.7	−	138386.i	0.0912485	−	0.338914i
$640$	−202788.			−0.495088
$641$			313789.i			0.763699i	0.924224	+	0.381850i	$0.124713\pi$
$641$			313789.i			0.763699i	−0.924224	+	0.381850i	$0.875287\pi$
$642$	278112.	+	36784.2i	0.674761	+	0.0892465i
$643$	−17439.4			−0.0421803			−0.0210902	−	0.999778i	$-0.506714\pi$
$643$	−17439.4			−0.0421803			−0.0210902	+	0.999778i	$0.506714\pi$
$644$			54012.5i			0.130234i
$645$	1841.22	−	13920.8i	0.00442573	−	0.0334614i
$646$	498976.			1.19568
$647$		−	470170.i		−	1.12317i	−0.827419	−	0.561586i	$-0.810192\pi$
$647$		−	470170.i		−	1.12317i	0.827419	−	0.561586i	$-0.189808\pi$
$648$	85965.5	−	148074.i	0.204727	−	0.352637i
$649$	568030.			1.34860
$650$			344561.i			0.815529i
$651$	−104983.	−	13885.4i	−0.247717	−	0.0327641i
$652$	−874629.			−2.05745
$653$			426505.i			1.00022i	0.865961	+	0.500112i	$0.166708\pi$
$653$			426505.i			1.00022i	−0.865961	+	0.500112i	$0.833292\pi$
$654$	129342.	−	977908.i	0.302401	−	2.28635i
$655$	310921.			0.724716
$656$		−	478194.i		−	1.11121i
$657$	−276188.	−	74360.1i	−0.639843	−	0.172270i
$658$	57264.5			0.132262
$659$			201868.i			0.464833i	0.972616	+	0.232416i	$0.0746632\pi$
$659$			201868.i			0.464833i	−0.972616	+	0.232416i	$0.925337\pi$
$660$	−561054.	−	74207.1i	−1.28800	−	0.170356i
$661$	−64778.0			−0.148260			−0.0741301	−	0.997249i	$-0.523618\pi$
$661$	−64778.0			−0.148260			−0.0741301	+	0.997249i	$0.523618\pi$
$662$			506944.i			1.15676i
$663$	−75432.2	+	570316.i	−0.171605	+	1.29744i
$664$	53843.0			0.122122
$665$		−	57326.7i		−	0.129632i
$666$	−169641.	+	630079.i	−0.382457	+	1.42052i
$667$	−108103.			−0.242989
$668$			265835.i			0.595743i
$669$	−98470.9	−	13024.1i	−0.220017	−	0.0291002i
$670$	−243796.			−0.543096
$671$			522539.i			1.16058i
$672$	31254.3	−	236302.i	0.0692103	−	0.523275i
$673$	−417567.			−0.921925			−0.460963	−	0.887420i	$-0.652496\pi$
$673$	−417567.			−0.921925			−0.460963	+	0.887420i	$0.652496\pi$
$674$			18549.9i			0.0408340i
$675$	−253596.	−	105579.i	−0.556590	−	0.231724i
$676$	112738.			0.246705
$677$			291796.i			0.636652i	0.947981	+	0.318326i	$0.103121\pi$
$677$			291796.i			0.636652i	−0.947981	+	0.318326i	$0.896879\pi$
$678$	−444864.	−	58839.4i	−0.967761	−	0.128000i
$679$	−63266.5			−0.137225
$680$		−	173221.i		−	0.374613i
$681$	−86242.2	+	652047.i	−0.185963	+	1.40600i
$682$	751855.			1.61646
$683$			493655.i			1.05823i	0.848549	+	0.529117i	$0.177477\pi$
$683$			493655.i			1.05823i	−0.848549	+	0.529117i	$0.822523\pi$
$684$	−312419.	−	84115.0i	−0.667768	−	0.179788i
$685$	−215099.			−0.458413
$686$		−	38288.8i		−	0.0813623i
$687$	−140139.	−	18535.3i	−0.296923	−	0.0392722i
$688$	−16636.2			−0.0351462
$689$		−	645089.i		−	1.35888i
$690$	−16075.2	+	121539.i	−0.0337643	+	0.255280i
$691$	−56687.4			−0.118722			−0.0593609	−	0.998237i	$-0.518906\pi$
$691$	−56687.4			−0.118722			−0.0593609	+	0.998237i	$0.518906\pi$
$692$		−	408452.i		−	0.852959i
$693$	76573.8	−	284410.i	0.159446	−	0.592213i
$694$	617690.			1.28248
$695$			104834.i			0.217036i
$696$	−175460.	−	23207.1i	−0.362210	−	0.0479073i
$697$	1.19943e6			2.46894
$698$			490944.i			1.00768i
$699$	−69024.3	+	521868.i	−0.141269	+	1.06809i
$700$	141874.			0.289538
$701$		−	441765.i		−	0.898992i	−0.893282	−	0.449496i	$-0.851604\pi$
$701$		−	441765.i		−	0.898992i	0.893282	−	0.449496i	$-0.148396\pi$
$702$	256208.	−	615401.i	0.519899	−	1.24877i
$703$	262601.			0.531355
$704$			1.16463e6i			2.34985i
$705$	72106.5	+	9537.08i	0.145076	+	0.0191883i
$706$	1.19585e6			2.39921
$707$			16077.9i			0.0321654i
$708$	69408.0	−	524770.i	0.138466	−	1.04689i
$709$	−590139.			−1.17398			−0.586992	−	0.809593i	$-0.699688\pi$
$709$	−590139.			−1.17398			−0.586992	+	0.809593i	$0.699688\pi$
$710$			168005.i			0.333278i
$711$	176136.	+	47422.4i	0.348424	+	0.0938089i
$712$	−80090.4			−0.157987
$713$		−	91140.8i		−	0.179281i
$714$	419646.	+	55504.0i	0.823165	+	0.108875i
$715$	−469255.			−0.917902
$716$		−	814542.i		−	1.58887i
$717$	−15792.2	+	119400.i	−0.0307189	+	0.232255i
$718$	−122300.			−0.237234
$719$		−	918202.i		−	1.77615i	−0.459695	−	0.888077i	$-0.652041\pi$
$719$		−	918202.i		−	1.77615i	0.459695	−	0.888077i	$-0.347959\pi$
$720$	55727.8	−	206984.i	0.107500	−	0.399274i
$721$	−44836.1			−0.0862497
$722$		−	552812.i		−	1.06048i
$723$	2773.44	+	366.825i	0.00530569	+	0.000701750i
$724$	1.29960e6			2.47931
$725$			283953.i			0.540220i
$726$	−170058.	+	1.28575e6i	−0.322645	+	2.43940i
$727$	−242969.			−0.459708			−0.229854	−	0.973225i	$-0.573825\pi$
$727$	−242969.			−0.459708			−0.229854	+	0.973225i	$0.573825\pi$
$728$			73322.7i			0.138349i
$729$	374428.	+	377138.i	0.704553	+	0.709651i
$730$	335302.			0.629202
$731$		−	41727.9i		−	0.0780893i
$732$	482744.	+	63849.5i	0.900937	+	0.119161i
$733$	−145455.			−0.270721			−0.135361	−	0.990796i	$-0.543219\pi$
$733$	−145455.			−0.270721			−0.135361	+	0.990796i	$0.543219\pi$
$734$		−	1.18589e6i		−	2.20116i
$735$	6376.78	−	48212.6i	0.0118039	−	0.0892454i
$736$	205146.			0.378710
$737$			504097.i			0.928067i
$738$	−1.34204e6	−	361327.i	−2.46407	−	0.663419i
$739$	−971847.			−1.77955			−0.889773	−	0.456404i	$-0.849137\pi$
$739$	−971847.			−1.77955			−0.889773	+	0.456404i	$0.849137\pi$
$740$		−	428051.i		−	0.781685i
$741$	−265954.	−	35176.1i	−0.484362	−	0.0640636i
$742$	−474665.			−0.862143
$743$			26014.8i			0.0471241i	0.999722	+	0.0235620i	$0.00750072\pi$
$743$			26014.8i			0.0471241i	−0.999722	+	0.0235620i	$0.992499\pi$
$744$	19565.6	−	147929.i	0.0353466	−	0.267244i
$745$	400133.			0.720927
$746$		−	868381.i		−	1.56039i
$747$	−43448.2	+	161375.i	−0.0778629	+	0.289198i
$748$	−1.68177e6			−3.00583
$749$		−	95776.8i		−	0.170725i
$750$	848757.	+	112260.i	1.50890	+	0.199573i
$751$	424520.			0.752694			0.376347	−	0.926479i	$-0.377180\pi$
$751$	424520.			0.752694			0.376347	+	0.926479i	$0.377180\pi$
$752$		−	86171.9i		−	0.152381i
$753$	97618.7	−	738061.i	0.172164	−	1.30167i
$754$	−689067.			−1.21205
$755$		−	35500.6i		−	0.0622790i
$756$	−253393.	−	105494.i	−0.443354	−	0.184580i
$757$	241891.			0.422112			0.211056	−	0.977474i	$-0.432310\pi$
$757$	241891.			0.422112			0.211056	+	0.977474i	$0.432310\pi$
$758$			812974.i			1.41494i
$759$	251306.	+	33238.7i	0.436234	+	0.0576980i
$760$	80777.6			0.139850
$761$			57363.8i			0.0990531i	0.998773	+	0.0495266i	$0.0157712\pi$
$761$			57363.8i			0.0990531i	−0.998773	+	0.0495266i	$0.984229\pi$
$762$	−167348.	+	1.26526e6i	−0.288211	+	2.17906i
$763$	−336774.			−0.578481
$764$			1.17319e6i			2.00993i
$765$	519168.	+	139779.i	0.887125	+	0.238847i
$766$	−62645.8			−0.106766
$767$		−	438907.i		−	0.746074i
$768$	−154543.	−	20440.4i	−0.262015	−	0.0346551i
$769$	919966.			1.55568			0.777838	−	0.628465i	$-0.216316\pi$
$769$	919966.			1.55568			0.777838	+	0.628465i	$0.216316\pi$
$770$			345284.i			0.582364i
$771$	−12803.5	+	96802.7i	−0.0215387	+	0.162847i
$772$	48702.1			0.0817171
$773$			620899.i			1.03911i	0.854437	+	0.519556i	$0.173903\pi$
$773$			620899.i			1.03911i	−0.854437	+	0.519556i	$0.826097\pi$
$774$	−12570.5	+	46689.2i	−0.0209831	+	0.0779354i
$775$	−239398.			−0.398581
$776$		−	89147.3i		−	0.148042i
$777$	220851.	+	29210.6i	0.365812	+	0.0483837i
$778$	−133057.			−0.219826
$779$			559327.i			0.921703i
$780$	−57338.6	+	433517.i	−0.0942449	+	0.712552i
$781$	347386.			0.569521
$782$			364316.i			0.595750i
$783$	211141.	−	507153.i	0.344389	−	0.827209i
$784$	−57617.2			−0.0937389
$785$		−	653334.i		−	1.06022i
$786$	−1.06137e6	−	140381.i	−1.71800	−	0.227229i
$787$	−559334.			−0.903071			−0.451536	−	0.892253i	$-0.649124\pi$
$787$	−559334.			−0.903071			−0.451536	+	0.892253i	$0.649124\pi$
$788$			41363.0i			0.0666130i
$789$	47301.0	−	357626.i	0.0759830	−	0.574481i
$790$	−213835.			−0.342630
$791$			153203.i			0.244858i
$792$	400755.	+	107898.i	0.638893	+	0.172014i
$793$	403757.			0.642058
$794$			235453.i			0.373476i
$795$	−597690.	−	79052.7i	−0.945674	−	0.125078i
$796$	−528141.			−0.833535
$797$			280663.i			0.441843i	0.975292	+	0.220922i	$0.0709065\pi$
$797$			280663.i			0.441843i	−0.975292	+	0.220922i	$0.929093\pi$
$798$	−25883.0	+	195692.i	−0.0406452	+	0.307304i
$799$	216141.			0.338566
$800$		−	538853.i		−	0.841957i
$801$	64628.4	−	240042.i	0.100730	−	0.374130i
$802$	−253258.			−0.393745
$803$		−	693305.i		−	1.07521i
$804$	465706.	+	61596.0i	0.720443	+	0.0952885i
$805$	41855.7			0.0645897
$806$		−	580946.i		−	0.894263i
$807$	88877.5	−	671972.i	0.136472	−	1.03182i
$808$	−22654.9			−0.0347008
$809$		−	572672.i		−	0.875002i	−0.899218	−	0.437501i	$-0.855864\pi$
$809$		−	572672.i		−	0.875002i	0.899218	−	0.437501i	$-0.144136\pi$
$810$	−538787.	−	312797.i	−0.821196	−	0.476753i
$811$	−980201.			−1.49030			−0.745149	−	0.666898i	$-0.767622\pi$
$811$	−980201.			−1.49030			−0.745149	+	0.666898i	$0.767622\pi$
$812$			283725.i			0.430314i
$813$	1.00886e6	+	133436.i	1.52634	+	0.201879i
$814$	−1.58167e6			−2.38708
$815$			677773.i			1.02040i
$816$	83522.8	−	631486.i	0.125437	−	0.948383i
$817$	19458.8			0.0291523
$818$			1.68502e6i			2.51824i
$819$	−219759.	−	59167.2i	−0.327626	−	0.0882091i
$820$	911728.			1.35593
$821$		−	263704.i		−	0.391228i	−0.980681	−	0.195614i	$-0.937330\pi$
$821$		−	263704.i		−	0.391228i	0.980681	−	0.195614i	$-0.0626700\pi$
$822$	734270.	+	97117.3i	1.08671	+	0.143732i
$823$	−148647.			−0.219461			−0.109730	−	0.993961i	$-0.534999\pi$
$823$	−148647.			−0.219461			−0.109730	+	0.993961i	$0.534999\pi$
$824$		−	63177.5i		−	0.0930482i
$825$	87307.5	−	660101.i	0.128276	−	0.969846i
$826$	−322954.			−0.473348
$827$			30591.1i			0.0447285i	0.999750	+	0.0223642i	$0.00711935\pi$
$827$			30591.1i			0.0447285i	−0.999750	+	0.0223642i	$0.992881\pi$
$828$	61414.6	−	228106.i	0.0895800	−	0.332717i
$829$	285462.			0.415374			0.207687	−	0.978195i	$-0.433407\pi$
$829$	285462.			0.415374			0.207687	+	0.978195i	$0.433407\pi$
$830$		−	195915.i		−	0.284388i
$831$	472788.	+	62532.7i	0.684643	+	0.0905535i
$832$	899886.			1.29999
$833$		−	144518.i		−	0.208273i
$834$	47332.6	−	357865.i	0.0680500	−	0.514502i
$835$	206002.			0.295460
$836$		−	784256.i		−	1.12214i
$837$	427575.	+	178011.i	0.610326	+	0.254095i
$838$	1.38807e6			1.97662
$839$			1.03015e6i			1.46345i	0.681602	+	0.731723i	$0.261284\pi$
$839$			1.03015e6i			1.46345i	−0.681602	+	0.731723i	$0.738716\pi$
$840$	67935.2	+	8985.37i	0.0962801	+	0.0127344i
$841$	139420.			0.197121
$842$		−	302887.i		−	0.427225i
$843$	56905.5	−	430243.i	0.0800754	−	0.605422i
$844$	−1.22867e6			−1.72484
$845$		−	87363.9i		−	0.122354i
$846$	−241839.	−	65112.2i	−0.337898	−	0.0909750i
$847$	442789.			0.617206
$848$			714278.i			0.993289i
$849$	−570149.	−	75410.1i	−0.790994	−	0.104620i
$850$	956941.			1.32449
$851$			191732.i			0.264749i
$852$	42447.3	−	320929.i	0.0584751	−	0.442110i
$853$	−1.02297e6			−1.40593			−0.702967	−	0.711223i	$-0.748142\pi$
$853$	−1.02297e6			−1.40593			−0.702967	+	0.711223i	$0.748142\pi$
$854$		−	297090.i		−	0.407354i
$855$	−65182.9	+	242102.i	−0.0891665	+	0.331181i
$856$	134957.			0.184182
$857$			838435.i			1.14158i	0.821095	+	0.570792i	$0.193364\pi$
$857$			838435.i			1.14158i	−0.821095	+	0.570792i	$0.806636\pi$
$858$	1.60186e6	+	211869.i	2.17596	+	0.287801i
$859$	272504.			0.369306			0.184653	−	0.982804i	$-0.440884\pi$
$859$	272504.			0.369306			0.184653	+	0.982804i	$0.440884\pi$
$860$		−	31718.8i		−	0.0428864i
$861$	−62217.3	+	470403.i	−0.0839275	+	0.634547i
$862$	−541501.			−0.728760
$863$		−	600703.i		−	0.806563i	−0.915076	−	0.403281i	$-0.867870\pi$
$863$		−	600703.i		−	0.806563i	0.915076	−	0.403281i	$-0.132130\pi$
$864$	−400679.	+	962415.i	−0.536747	+	1.28924i
$865$	−316520.			−0.423028
$866$		−	1.22990e6i		−	1.63996i
$867$	838727.	+	110933.i	1.11579	+	0.147579i
$868$	−239206.			−0.317491
$869$			442148.i			0.585501i
$870$	−84442.1	+	638437.i	−0.111563	+	0.843489i
$871$	389507.			0.513428
$872$		−	474540.i		−	0.624079i
$873$	267187.	+	71936.8i	0.350580	+	0.0943893i
$874$	−169890.			−0.222405
$875$		−	292296.i		−	0.381775i
$876$	−640504.	−	84715.5i	−0.834667	−	0.110396i
$877$	−446820.			−0.580943			−0.290472	−	0.956884i	$-0.593812\pi$
$877$	−446820.			−0.580943			−0.290472	+	0.956884i	$0.593812\pi$
$878$		−	1.17600e6i		−	1.52553i
$879$	−90107.5	+	681271.i	−0.116623	+	0.881744i
$880$	519585.			0.670952
$881$			437687.i			0.563913i	0.959427	+	0.281956i	$0.0909833\pi$
$881$			437687.i			0.563913i	−0.959427	+	0.281956i	$0.909017\pi$
$882$	−43536.0	+	161701.i	−0.0559644	+	0.207862i
$883$	−80288.7			−0.102975			−0.0514876	−	0.998674i	$-0.516396\pi$
$883$	−80288.7			−0.102975			−0.0514876	+	0.998674i	$0.516396\pi$
$884$			1.29948e6i			1.66289i
$885$	−406658.	−	53786.1i	−0.519210	−	0.0686726i
$886$	1.82481e6			2.32461
$887$			1.44826e6i			1.84076i	0.391022	+	0.920381i	$0.372122\pi$
$887$			1.44826e6i			1.84076i	−0.391022	+	0.920381i	$0.627878\pi$
$888$	−41160.0	+	311196.i	−0.0521974	+	0.394646i
$889$	435732.			0.551335
$890$			291420.i			0.367908i
$891$	−646772.	+	1.11405e6i	−0.814697	+	1.40330i
$892$	−224368.			−0.281988
$893$			100792.i			0.126393i
$894$	−1.36591e6	−	180660.i	−1.70902	−	0.226041i
$895$	−631210.			−0.788003
$896$		−	238396.i		−	0.296950i
$897$	25683.0	−	194180.i	0.0319198	−	0.241335i
$898$	982448.			1.21831
$899$		−	478758.i		−	0.592375i
$900$	−599161.	−	161317.i	−0.739705	−	0.199156i
$901$	−1.79159e6			−2.20693
$902$		−	3.36888e6i		−	4.14069i
$903$	16365.2	+	2164.52i	0.0200699	+	0.00265452i
$904$	−215875.			−0.264159
$905$		−	1.00709e6i		−	1.22962i
$906$	−16028.5	+	121186.i	−0.0195271	+	0.147637i
$907$	−903376.			−1.09813			−0.549065	−	0.835779i	$-0.685016\pi$
$907$	−903376.			−1.09813			−0.549065	+	0.835779i	$0.685016\pi$
$908$			1.48570e6i			1.80202i
$909$	18281.2	−	67900.0i	0.0221247	−	0.0821754i
$910$	266795.			0.322177
$911$		−	1.15959e6i		−	1.39723i	−0.715499	−	0.698614i	$-0.753800\pi$
$911$		−	1.15959e6i		−	1.39723i	0.715499	−	0.698614i	$-0.246200\pi$
$912$	294479.	+	38948.9i	0.354050	+	0.0468280i
$913$	−405094.			−0.485976
$914$		−	942360.i		−	1.12804i
$915$	49478.6	−	374091.i	0.0590984	−	0.446822i
$916$	−319309.			−0.380557
$917$			365517.i			0.434680i
$918$	−1.70914e6	−	711561.i	−2.02811	−	0.844358i
$919$	1.39808e6			1.65539			0.827697	−	0.561176i	$-0.189651\pi$
$919$	1.39808e6			1.65539			0.827697	+	0.561176i	$0.189651\pi$
$920$			58977.9i			0.0696809i
$921$	−789681.	−	104446.i	−0.930963	−	0.123133i
$922$	844842.			0.993834
$923$		−	268419.i		−	0.315072i
$924$	87237.4	−	659571.i	0.102178	−	0.772535i
$925$	503618.			0.588597
$926$			282276.i			0.329194i
$927$	189352.	+	50980.7i	0.220349	+	0.0593261i
$928$	1.07762e6			1.25132
$929$			1.23771e6i			1.43413i	0.697006	+	0.717066i	$0.254515\pi$
$929$			1.23771e6i			1.43413i	−0.697006	+	0.717066i	$0.745485\pi$
$930$	−538260.	−	71192.3i	−0.622338	−	0.0823127i
$931$	67392.9			0.0777525
$932$			1.18909e6i			1.36893i
$933$	27734.6	−	209691.i	0.0318609	−	0.240889i
$934$	−2.19517e6			−2.51637
$935$			1.30325e6i			1.49075i
$936$	83371.2	−	309656.i	0.0951621	−	0.353450i
$937$	1.41105e6			1.60718			0.803589	−	0.595185i	$-0.202921\pi$
$937$	1.41105e6			1.60718			0.803589	+	0.595185i	$0.202921\pi$
$938$		−	286605.i		−	0.325745i
$939$	−1.41621e6	−	187313.i	−1.60619	−	0.212441i
$940$	164296.			0.185939
$941$		−	1.68557e6i		−	1.90356i	−0.306775	−	0.951782i	$-0.599250\pi$
$941$		−	1.68557e6i		−	1.90356i	0.306775	−	0.951782i	$-0.400750\pi$
$942$	−294980.	+	2.23024e6i	−0.332423	+	2.51334i
$943$	−408380.			−0.459241
$944$			485983.i			0.545352i
$945$	−81750.3	+	196361.i	−0.0915431	+	0.219883i
$946$	−117202.			−0.130965
$947$		−	1.28636e6i		−	1.43437i	−0.696883	−	0.717185i	$-0.745430\pi$
$947$		−	1.28636e6i		−	1.43437i	0.696883	−	0.717185i	$-0.254570\pi$
$948$	408475.	+	54026.4i	0.454515	+	0.0601159i
$949$	−535705.			−0.594830
$950$			446247.i			0.494457i
$951$	204375.	−	1.54520e6i	0.225978	−	1.70854i
$952$	203637.			0.224690
$953$			261159.i			0.287553i	0.989610	+	0.143777i	$0.0459247\pi$
$953$			261159.i			0.287553i	−0.989610	+	0.143777i	$0.954075\pi$
$954$	2.00460e6	+	539715.i	2.20258	+	0.593017i
$955$	909134.			0.996830
$956$			272054.i			0.297673i
$957$	1.32010e6	+	174601.i	1.44139	+	0.190644i
$958$	841440.			0.916837
$959$		−	252869.i		−	0.274953i
$960$	110277.	−	833766.i	0.119658	−	0.904694i
$961$	−519885.			−0.562938
$962$			1.22213e6i			1.32059i
$963$	−108902.	+	404484.i	−0.117432	+	0.436163i
$964$	6319.33			0.00680013
$965$		−	37740.5i		−	0.0405278i
$966$	−142880.	−	18897.9i	−0.153115	−	0.0202516i
$967$	−522045.			−0.558284			−0.279142	−	0.960250i	$-0.590050\pi$
$967$	−522045.			−0.558284			−0.279142	+	0.960250i	$0.590050\pi$
$968$			623924.i			0.665857i
$969$	−97693.7	+	738628.i	−0.104044	+	0.786644i
$970$	−324375.			−0.344750
$971$			527907.i			0.559911i	0.960013	+	0.279956i	$0.0903198\pi$
$971$			527907.i			0.559911i	−0.960013	+	0.279956i	$0.909680\pi$
$972$	950177.	+	733642.i	1.00571	+	0.776518i
$973$	−123242.			−0.130177
$974$		−	736129.i		−	0.775954i
$975$	−510049.	−	67461.1i	−0.536541	−	0.0709649i
$976$	−447063.			−0.469320
$977$			730919.i			0.765737i	0.923803	+	0.382869i	$0.125064\pi$
$977$			730919.i			0.765737i	−0.923803	+	0.382869i	$0.874936\pi$
$978$	306015.	−	2.31367e6i	0.319937	−	2.41893i
$979$	602570.			0.628698
$980$		−	109853.i		−	0.114383i
$981$	1.42226e6	+	382926.i	1.47789	+	0.397903i
$982$	1.44184e6			1.49518
$983$			163325.i			0.169023i	0.996423	+	0.0845115i	$0.0269330\pi$
$983$			163325.i			0.169023i	−0.996423	+	0.0845115i	$0.973067\pi$
$984$	−662833.	−	87668.8i	−0.684564	−	0.0905430i
$985$	32053.2			0.0330369
$986$			1.91373e6i			1.96846i
$987$	−11211.7	+	84767.9i	−0.0115090	+	0.0870156i
$988$	−605981.			−0.620791
$989$			14207.4i			0.0145252i
$990$	392603.	−	1.45820e6i	0.400574	−	1.48781i
$991$	1.24036e6			1.26299			0.631494	−	0.775381i	$-0.282442\pi$
$991$	1.24036e6			1.26299			0.631494	+	0.775381i	$0.282442\pi$
$992$			908530.i			0.923243i
$993$	−750423.	−	99253.8i	−0.761040	−	0.100658i
$994$	−197506.			−0.199898
$995$			409270.i			0.413394i
$996$	−49498.8	+	374243.i	−0.0498972	+	0.377255i
$997$	−881732.			−0.887046			−0.443523	−	0.896263i	$-0.646272\pi$
$997$	−881732.			−0.887046			−0.443523	+	0.896263i	$0.646272\pi$
$998$			172256.i			0.172947i
$999$	−899485.	−	374480.i	−0.901286	−	0.375230i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.5.b.a.8.8	yes	8
3.2	odd	2	inner	21.5.b.a.8.1	✓	8
4.3	odd	2		336.5.d.b.113.8		8
7.2	even	3		147.5.h.e.116.8		16
7.3	odd	6		147.5.h.c.128.1		16
7.4	even	3		147.5.h.e.128.1		16
7.5	odd	6		147.5.h.c.116.8		16
7.6	odd	2		147.5.b.e.50.8		8
12.11	even	2		336.5.d.b.113.7		8
21.2	odd	6		147.5.h.e.116.1		16
21.5	even	6		147.5.h.c.116.1		16
21.11	odd	6		147.5.h.e.128.8		16
21.17	even	6		147.5.h.c.128.8		16
21.20	even	2		147.5.b.e.50.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.5.b.a.8.1	✓	8	3.2	odd	2	inner
21.5.b.a.8.8	yes	8	1.1	even	1	trivial
147.5.b.e.50.1		8	21.20	even	2
147.5.b.e.50.8		8	7.6	odd	2
147.5.h.c.116.1		16	21.5	even	6
147.5.h.c.116.8		16	7.5	odd	6
147.5.h.c.128.1		16	7.3	odd	6
147.5.h.c.128.8		16	21.17	even	6
147.5.h.e.116.1		16	21.2	odd	6
147.5.h.e.116.8		16	7.2	even	3
147.5.h.e.128.1		16	7.4	even	3
147.5.h.e.128.8		16	21.11	odd	6
336.5.d.b.113.7		8	12.11	even	2
336.5.d.b.113.8		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)

\(f(q)\)	\(=\)	\(q+6.02741i q^{2} +(-8.92230 - 1.18010i) q^{3} -20.3296 q^{4} +15.7540i q^{5} +(7.11292 - 53.7783i) q^{6} -18.5203 q^{7} -26.0965i q^{8} +(78.2147 + 21.0583i) q^{9} +O(q^{10})\)	\(q+6.02741i q^{2} +(-8.92230 - 1.18010i) q^{3} -20.3296 q^{4} +15.7540i q^{5} +(7.11292 - 53.7783i) q^{6} -18.5203 q^{7} -26.0965i q^{8} +(78.2147 + 21.0583i) q^{9} -94.9556 q^{10} +196.340i q^{11} +(181.387 + 23.9909i) q^{12} +151.709 q^{13} -111.629i q^{14} +(18.5912 - 140.562i) q^{15} -167.980 q^{16} -421.337i q^{17} +(-126.927 + 471.432i) q^{18} +196.481 q^{19} -320.272i q^{20} +(165.243 + 21.8557i) q^{21} -1183.42 q^{22} +143.456i q^{23} +(-30.7963 + 232.840i) q^{24} +376.813 q^{25} +914.409i q^{26} +(-673.004 - 280.190i) q^{27} +376.510 q^{28} +753.566i q^{29} +(847.222 + 112.057i) q^{30} -635.323 q^{31} -1430.03i q^{32} +(231.700 - 1751.80i) q^{33} +2539.57 q^{34} -291.768i q^{35} +(-1590.08 - 428.108i) q^{36} +1336.52 q^{37} +1184.27i q^{38} +(-1353.59 - 179.031i) q^{39} +411.123 q^{40} +2846.73i q^{41} +(-131.733 + 995.988i) q^{42} +99.0370 q^{43} -3991.52i q^{44} +(-331.752 + 1232.19i) q^{45} -864.666 q^{46} +512.989i q^{47} +(1498.77 + 198.233i) q^{48} +343.000 q^{49} +2271.20i q^{50} +(-497.218 + 3759.29i) q^{51} -3084.18 q^{52} -4252.16i q^{53} +(1688.82 - 4056.47i) q^{54} -3093.13 q^{55} +483.313i q^{56} +(-1753.06 - 231.866i) q^{57} -4542.05 q^{58} -2893.09i q^{59} +(-377.952 + 2857.56i) q^{60} +2661.40 q^{61} -3829.35i q^{62} +(-1448.56 - 390.006i) q^{63} +5931.68 q^{64} +2390.01i q^{65} +(10558.8 + 1396.55i) q^{66} +2567.47 q^{67} +8565.62i q^{68} +(169.292 - 1279.95i) q^{69} +1758.60 q^{70} -1769.31i q^{71} +(549.548 - 2041.13i) q^{72} -3531.14 q^{73} +8055.76i q^{74} +(-3362.03 - 444.675i) q^{75} -3994.38 q^{76} -3636.27i q^{77} +(1079.09 - 8158.63i) q^{78} +2251.95 q^{79} -2646.35i q^{80} +(5674.09 + 3294.15i) q^{81} -17158.4 q^{82} +2063.23i q^{83} +(-3359.33 - 444.318i) q^{84} +6637.72 q^{85} +596.936i q^{86} +(889.280 - 6723.54i) q^{87} +5123.78 q^{88} -3069.01i q^{89} +(-7426.92 - 1999.61i) q^{90} -2809.68 q^{91} -2916.40i q^{92} +(5668.54 + 749.743i) q^{93} -3091.99 q^{94} +3095.35i q^{95} +(-1687.57 + 12759.1i) q^{96} +3416.07 q^{97} +2067.40i q^{98} +(-4134.59 + 15356.7i) 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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more