LMFDB - Embedded newform 69.8.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,8,Mod(1,69)]

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, 0]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("69.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	69.a (trivial)

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:	yes
Analytic conductor:	$21.5545667584$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 775x^{5} - 474x^{4} + 167184x^{3} - 33920x^{2} - 9348928x + 28965760 $ x^7 - 775x^5 - 474x^4 + 167184x^3 - 33920x^2 - 9348928*x + 28965760
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$5.40652$ of defining polynomial
Character	$\chi$	$=$	69.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+5.40652 q^{2} -27.0000 q^{3} -98.7695 q^{4} -344.177 q^{5} -145.976 q^{6} +525.740 q^{7} -1226.03 q^{8} +729.000 q^{9} +O(q^{10})$	\(q+5.40652 q^{2} -27.0000 q^{3} -98.7695 q^{4} -344.177 q^{5} -145.976 q^{6} +525.740 q^{7} -1226.03 q^{8} +729.000 q^{9} -1860.80 q^{10} -6690.01 q^{11} +2666.78 q^{12} +3864.31 q^{13} +2842.42 q^{14} +9292.78 q^{15} +6013.92 q^{16} +4992.87 q^{17} +3941.35 q^{18} +52377.6 q^{19} +33994.2 q^{20} -14195.0 q^{21} -36169.7 q^{22} -12167.0 q^{23} +33102.9 q^{24} +40332.9 q^{25} +20892.5 q^{26} -19683.0 q^{27} -51927.1 q^{28} +69355.1 q^{29} +50241.6 q^{30} +74305.9 q^{31} +189447. q^{32} +180630. q^{33} +26994.1 q^{34} -180948. q^{35} -72003.0 q^{36} -345774. q^{37} +283181. q^{38} -104336. q^{39} +421973. q^{40} +304729. q^{41} -76745.4 q^{42} +334242. q^{43} +660769. q^{44} -250905. q^{45} -65781.1 q^{46} -685540. q^{47} -162376. q^{48} -547141. q^{49} +218061. q^{50} -134808. q^{51} -381676. q^{52} +759966. q^{53} -106417. q^{54} +2.30255e6 q^{55} -644575. q^{56} -1.41420e6 q^{57} +374970. q^{58} +893975. q^{59} -917844. q^{60} +1.01834e6 q^{61} +401736. q^{62} +383264. q^{63} +254466. q^{64} -1.33001e6 q^{65} +976581. q^{66} +1.90825e6 q^{67} -493144. q^{68} +328509. q^{69} -978297. q^{70} -3.91381e6 q^{71} -893779. q^{72} +1.34857e6 q^{73} -1.86943e6 q^{74} -1.08899e6 q^{75} -5.17332e6 q^{76} -3.51720e6 q^{77} -564096. q^{78} -5.28649e6 q^{79} -2.06985e6 q^{80} +531441. q^{81} +1.64752e6 q^{82} +4.24345e6 q^{83} +1.40203e6 q^{84} -1.71843e6 q^{85} +1.80708e6 q^{86} -1.87259e6 q^{87} +8.20218e6 q^{88} -4.56295e6 q^{89} -1.35652e6 q^{90} +2.03162e6 q^{91} +1.20173e6 q^{92} -2.00626e6 q^{93} -3.70638e6 q^{94} -1.80272e7 q^{95} -5.11506e6 q^{96} +1.25318e7 q^{97} -2.95813e6 q^{98} -4.87702e6 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) $ 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	5.40652	0.477873	0.238937	−	0.971035i	$-0.423201\pi$
$2$	5.40652	0.477873	0.238937	+	0.971035i	$0.423201\pi$
$3$	−27.0000	−0.577350
$3$	−27.0000	−0.577350
$4$	−98.7695	−0.771637
$5$	−344.177	−1.23137	−0.615683	−	0.787994i	$-0.711120\pi$
$5$	−344.177	−1.23137	−0.615683	+	0.787994i	$0.711120\pi$
$6$	−145.976	−0.275900
$7$	525.740	0.579332	0.289666	−	0.957128i	$-0.406456\pi$
$7$	525.740	0.579332	0.289666	+	0.957128i	$0.406456\pi$
$8$	−1226.03	−0.846618
$9$	729.000	0.333333
$10$	−1860.80	−0.588437
$11$	−6690.01	−1.51549	−0.757744	−	0.652552i	$-0.773698\pi$
$11$	−6690.01	−1.51549	−0.757744	+	0.652552i	$0.773698\pi$
$12$	2666.78	0.445505
$13$	3864.31	0.487831	0.243916	−	0.969796i	$-0.421568\pi$
$13$	3864.31	0.487831	0.243916	+	0.969796i	$0.421568\pi$
$14$	2842.42	0.276847
$15$	9292.78	0.710929
$16$	6013.92	0.367061
$17$	4992.87	0.246479	0.123239	−	0.992377i	$-0.460672\pi$
$17$	4992.87	0.246479	0.123239	+	0.992377i	$0.460672\pi$
$18$	3941.35	0.159291
$19$	52377.6	1.75190	0.875948	−	0.482405i	$-0.160237\pi$
$19$	52377.6	1.75190	0.875948	+	0.482405i	$0.160237\pi$
$20$	33994.2	0.950167
$21$	−14195.0	−0.334478
$22$	−36169.7	−0.724211
$23$	−12167.0	−0.208514
$23$	−12167.0	−0.208514
$24$	33102.9	0.488795
$25$	40332.9	0.516261
$26$	20892.5	0.233122
$27$	−19683.0	−0.192450
$28$	−51927.1	−0.447034
$29$	69355.1	0.528062	0.264031	−	0.964514i	$-0.414948\pi$
$29$	69355.1	0.528062	0.264031	+	0.964514i	$0.414948\pi$
$30$	50241.6	0.339734
$31$	74305.9	0.447979	0.223989	−	0.974592i	$-0.428092\pi$
$31$	74305.9	0.447979	0.223989	+	0.974592i	$0.428092\pi$
$32$	189447.	1.02203
$33$	180630.	0.874967
$34$	26994.1	0.117786
$35$	−180948.	−0.713370
$36$	−72003.0	−0.257212
$37$	−345774.	−1.12224	−0.561121	−	0.827734i	$-0.689630\pi$
$37$	−345774.	−1.12224	−0.561121	+	0.827734i	$0.689630\pi$
$38$	283181.	0.837185
$39$	−104336.	−0.281650
$40$	421973.	1.04250
$41$	304729.	0.690510	0.345255	−	0.938509i	$-0.387792\pi$
$41$	304729.	0.690510	0.345255	+	0.938509i	$0.387792\pi$
$42$	−76745.4	−0.159838
$43$	334242.	0.641093	0.320546	−	0.947233i	$-0.396133\pi$
$43$	334242.	0.641093	0.320546	+	0.947233i	$0.396133\pi$
$44$	660769.	1.16941
$45$	−250905.	−0.410455
$46$	−65781.1	−0.0996435
$47$	−685540.	−0.963141	−0.481571	−	0.876407i	$-0.659934\pi$
$47$	−685540.	−0.963141	−0.481571	+	0.876407i	$0.659934\pi$
$48$	−162376.	−0.211923
$49$	−547141.	−0.664374
$50$	218061.	0.246707
$51$	−134808.	−0.142305
$52$	−381676.	−0.376429
$53$	759966.	0.701179	0.350589	−	0.936529i	$-0.385981\pi$
$53$	759966.	0.701179	0.350589	+	0.936529i	$0.385981\pi$
$54$	−106417.	−0.0919668
$55$	2.30255e6	1.86612
$56$	−644575.	−0.490473
$57$	−1.41420e6	−1.01146
$58$	374970.	0.252347
$59$	893975.	0.566687	0.283343	−	0.959019i	$-0.408556\pi$
$59$	893975.	0.566687	0.283343	+	0.959019i	$0.408556\pi$
$60$	−917844.	−0.548579
$61$	1.01834e6	0.574430	0.287215	−	0.957866i	$-0.407271\pi$
$61$	1.01834e6	0.574430	0.287215	+	0.957866i	$0.407271\pi$
$62$	401736.	0.214077
$63$	383264.	0.193111
$64$	254466.	0.121339
$65$	−1.33001e6	−0.600699
$66$	976581.	0.418123
$67$	1.90825e6	0.775128	0.387564	−	0.921843i	$-0.373317\pi$
$67$	1.90825e6	0.775128	0.387564	+	0.921843i	$0.373317\pi$
$68$	−493144.	−0.190192
$69$	328509.	0.120386
$70$	−978297.	−0.340900
$71$	−3.91381e6	−1.29776	−0.648882	−	0.760889i	$-0.724763\pi$
$71$	−3.91381e6	−1.29776	−0.648882	+	0.760889i	$0.724763\pi$
$72$	−893779.	−0.282206
$73$	1.34857e6	0.405735	0.202868	−	0.979206i	$-0.434974\pi$
$73$	1.34857e6	0.405735	0.202868	+	0.979206i	$0.434974\pi$
$74$	−1.86943e6	−0.536289
$75$	−1.08899e6	−0.298063
$76$	−5.17332e6	−1.35183
$77$	−3.51720e6	−0.877970
$78$	−564096.	−0.134593
$79$	−5.28649e6	−1.20635	−0.603174	−	0.797609i	$-0.706098\pi$
$79$	−5.28649e6	−1.20635	−0.603174	+	0.797609i	$0.706098\pi$
$80$	−2.06985e6	−0.451986
$81$	531441.	0.111111
$82$	1.64752e6	0.329976
$83$	4.24345e6	0.814603	0.407302	−	0.913294i	$-0.366470\pi$
$83$	4.24345e6	0.814603	0.407302	+	0.913294i	$0.366470\pi$
$84$	1.40203e6	0.258095
$85$	−1.71843e6	−0.303505
$86$	1.80708e6	0.306361
$87$	−1.87259e6	−0.304877
$88$	8.20218e6	1.28304
$89$	−4.56295e6	−0.686089	−0.343045	−	0.939319i	$-0.611458\pi$
$89$	−4.56295e6	−0.686089	−0.343045	+	0.939319i	$0.611458\pi$
$90$	−1.35652e6	−0.196146
$91$	2.03162e6	0.282616
$92$	1.20173e6	0.160897
$93$	−2.00626e6	−0.258641
$94$	−3.70638e6	−0.460260
$95$	−1.80272e7	−2.15722
$96$	−5.11506e6	−0.590067
$97$	1.25318e7	1.39416	0.697081	−	0.716992i	$-0.254482\pi$
$97$	1.25318e7	1.39416	0.697081	+	0.716992i	$0.254482\pi$
$98$	−2.95813e6	−0.317487
$99$	−4.87702e6	−0.505162
$100$	−3.98366e6	−0.398366
$101$	1.24737e7	1.20468	0.602339	−	0.798241i	$-0.294236\pi$
$101$	1.24737e7	1.20468	0.602339	+	0.798241i	$0.294236\pi$
$102$	−728840.	−0.0680035
$103$	4.58949e6	0.413842	0.206921	−	0.978358i	$-0.433656\pi$
$103$	4.58949e6	0.413842	0.206921	+	0.978358i	$0.433656\pi$
$104$	−4.73777e6	−0.413007
$105$	4.88559e6	0.411864
$106$	4.10877e6	0.335075
$107$	−1.62881e7	−1.28537	−0.642684	−	0.766131i	$-0.722179\pi$
$107$	−1.62881e7	−1.28537	−0.642684	+	0.766131i	$0.722179\pi$
$108$	1.94408e6	0.148502
$109$	1.09431e7	0.809370	0.404685	−	0.914456i	$-0.367381\pi$
$109$	1.09431e7	0.809370	0.404685	+	0.914456i	$0.367381\pi$
$110$	1.24488e7	0.891768
$111$	9.33590e6	0.647926
$112$	3.16176e6	0.212650
$113$	2.89680e7	1.88862	0.944308	−	0.329063i	$-0.106733\pi$
$113$	2.89680e7	1.88862	0.944308	+	0.329063i	$0.106733\pi$
$114$	−7.64588e6	−0.483349
$115$	4.18760e6	0.256757
$116$	−6.85017e6	−0.407472
$117$	2.81708e6	0.162610
$118$	4.83329e6	0.270805
$119$	2.62495e6	0.142793
$120$	−1.13933e7	−0.601886
$121$	2.52690e7	1.29670
$122$	5.50566e6	0.274505
$123$	−8.22768e6	−0.398666
$124$	−7.33916e6	−0.345677
$125$	1.30072e7	0.595659
$126$	2.07213e6	0.0922825
$127$	3.95171e7	1.71188	0.855938	−	0.517079i	$-0.172981\pi$
$127$	3.95171e7	1.71188	0.855938	+	0.517079i	$0.172981\pi$
$128$	−2.28734e7	−0.964042
$129$	−9.02452e6	−0.370135
$130$	−7.19070e6	−0.287058
$131$	−261338.	−0.0101567	−0.00507835	−	0.999987i	$-0.501616\pi$
$131$	−261338.	−0.0101567	−0.00507835	+	0.999987i	$0.501616\pi$
$132$	−1.78408e7	−0.675157
$133$	2.75370e7	1.01493
$134$	1.03170e7	0.370413
$135$	6.77444e6	0.236976
$136$	−6.12143e6	−0.208673
$137$	−3.74106e7	−1.24300	−0.621502	−	0.783412i	$-0.713477\pi$
$137$	−3.74106e7	−1.24300	−0.621502	+	0.783412i	$0.713477\pi$
$138$	1.77609e6	0.0575292
$139$	3.90753e7	1.23410	0.617051	−	0.786923i	$-0.288327\pi$
$139$	3.90753e7	1.23410	0.617051	+	0.786923i	$0.288327\pi$
$140$	1.78721e7	0.550462
$141$	1.85096e7	0.556070
$142$	−2.11601e7	−0.620167
$143$	−2.58522e7	−0.739302
$144$	4.38415e6	0.122354
$145$	−2.38704e7	−0.650238
$146$	7.29106e6	0.193890
$147$	1.47728e7	0.383577
$148$	3.41519e7	0.865963
$149$	2.34345e6	0.0580368	0.0290184	−	0.999579i	$-0.490762\pi$
$149$	2.34345e6	0.0580368	0.0290184	+	0.999579i	$0.490762\pi$
$150$	−5.88764e6	−0.142437
$151$	3.81804e7	0.902446	0.451223	−	0.892411i	$-0.350988\pi$
$151$	3.81804e7	0.902446	0.451223	+	0.892411i	$0.350988\pi$
$152$	−6.42168e7	−1.48319
$153$	3.63980e6	0.0821595
$154$	−1.90158e7	−0.419559
$155$	−2.55744e7	−0.551625
$156$	1.03052e7	0.217331
$157$	5.74409e7	1.18460	0.592301	−	0.805717i	$-0.298220\pi$
$157$	5.74409e7	1.18460	0.592301	+	0.805717i	$0.298220\pi$
$158$	−2.85815e7	−0.576482
$159$	−2.05191e7	−0.404826
$160$	−6.52032e7	−1.25849
$161$	−6.39668e6	−0.120799
$162$	2.87325e6	0.0530970
$163$	−3.34219e7	−0.604470	−0.302235	−	0.953233i	$-0.597733\pi$
$163$	−3.34219e7	−0.604470	−0.302235	+	0.953233i	$0.597733\pi$
$164$	−3.00979e7	−0.532823
$165$	−6.21688e7	−1.07740
$166$	2.29423e7	0.389277
$167$	9.67231e7	1.60703	0.803513	−	0.595288i	$-0.202962\pi$
$167$	9.67231e7	1.60703	0.803513	+	0.595288i	$0.202962\pi$
$168$	1.74035e7	0.283175
$169$	−4.78157e7	−0.762020
$170$	−9.29074e6	−0.145037
$171$	3.81833e7	0.583965
$172$	−3.30129e7	−0.494691
$173$	−4.48058e7	−0.657920	−0.328960	−	0.944344i	$-0.606698\pi$
$173$	−4.48058e7	−0.657920	−0.328960	+	0.944344i	$0.606698\pi$
$174$	−1.01242e7	−0.145693
$175$	2.12046e7	0.299087
$176$	−4.02332e7	−0.556276
$177$	−2.41373e7	−0.327177
$178$	−2.46697e7	−0.327864
$179$	7.17236e7	0.934710	0.467355	−	0.884070i	$-0.345207\pi$
$179$	7.17236e7	0.934710	0.467355	+	0.884070i	$0.345207\pi$
$180$	2.47818e7	0.316722
$181$	−1.36040e8	−1.70526	−0.852629	−	0.522516i	$-0.824993\pi$
$181$	−1.36040e8	−1.70526	−0.852629	+	0.522516i	$0.824993\pi$
$182$	1.09840e7	0.135055
$183$	−2.74951e7	−0.331647
$184$	1.49172e7	0.176532
$185$	1.19008e8	1.38189
$186$	−1.08469e7	−0.123597
$187$	−3.34024e7	−0.373535
$188$	6.77104e7	0.743196
$189$	−1.03481e7	−0.111493
$190$	−9.74644e7	−1.03088
$191$	6.40998e7	0.665641	0.332820	−	0.942990i	$-0.392000\pi$
$191$	6.40998e7	0.665641	0.332820	+	0.942990i	$0.392000\pi$
$192$	−6.87058e6	−0.0700549
$193$	−1.11188e8	−1.11329	−0.556645	−	0.830751i	$-0.687912\pi$
$193$	−1.11188e8	−1.11329	−0.556645	+	0.830751i	$0.687912\pi$
$194$	6.77536e7	0.666233
$195$	3.59102e7	0.346814
$196$	5.40408e7	0.512656
$197$	1.37510e8	1.28145	0.640727	−	0.767768i	$-0.278633\pi$
$197$	1.37510e8	1.28145	0.640727	+	0.767768i	$0.278633\pi$
$198$	−2.63677e7	−0.241404
$199$	−1.77736e8	−1.59879	−0.799394	−	0.600808i	$-0.794846\pi$
$199$	−1.77736e8	−1.59879	−0.799394	+	0.600808i	$0.794846\pi$
$200$	−4.94495e7	−0.437076
$201$	−5.15228e7	−0.447521
$202$	6.74393e7	0.575683
$203$	3.64627e7	0.305924
$204$	1.33149e7	0.109807
$205$	−1.04881e8	−0.850271
$206$	2.48132e7	0.197764
$207$	−8.86974e6	−0.0695048
$208$	2.32396e7	0.179064
$209$	−3.50407e8	−2.65498
$210$	2.64140e7	0.196819
$211$	2.06909e8	1.51632	0.758159	−	0.652069i	$-0.226099\pi$
$211$	2.06909e8	1.51632	0.758159	+	0.652069i	$0.226099\pi$
$212$	−7.50615e7	−0.541056
$213$	1.05673e8	0.749265
$214$	−8.80621e7	−0.614244
$215$	−1.15038e8	−0.789419
$216$	2.41320e7	0.162932
$217$	3.90655e7	0.259528
$218$	5.91641e7	0.386777
$219$	−3.64113e7	−0.234251
$220$	−2.27422e8	−1.43997
$221$	1.92940e7	0.120240
$222$	5.04747e7	0.309627
$223$	−1.36885e8	−0.826585	−0.413293	−	0.910598i	$-0.635621\pi$
$223$	−1.36885e8	−0.826585	−0.413293	+	0.910598i	$0.635621\pi$
$224$	9.95997e7	0.592093
$225$	2.94027e7	0.172087
$226$	1.56616e8	0.902519
$227$	2.59280e8	1.47123	0.735613	−	0.677402i	$-0.236894\pi$
$227$	2.59280e8	1.47123	0.735613	+	0.677402i	$0.236894\pi$
$228$	1.39680e8	0.780478
$229$	2.74400e8	1.50994	0.754971	−	0.655758i	$-0.227651\pi$
$229$	2.74400e8	1.50994	0.754971	+	0.655758i	$0.227651\pi$
$230$	2.26404e7	0.122698
$231$	9.49645e7	0.506896
$232$	−8.50317e7	−0.447067
$233$	−1.65096e8	−0.855046	−0.427523	−	0.904004i	$-0.640614\pi$
$233$	−1.65096e8	−0.855046	−0.427523	+	0.904004i	$0.640614\pi$
$234$	1.52306e7	0.0777072
$235$	2.35947e8	1.18598
$236$	−8.82975e7	−0.437277
$237$	1.42735e8	0.696486
$238$	1.41919e7	0.0682370
$239$	3.09433e8	1.46613	0.733067	−	0.680157i	$-0.238088\pi$
$239$	3.09433e8	1.46613	0.733067	+	0.680157i	$0.238088\pi$
$240$	5.58861e7	0.260954
$241$	6.69995e6	0.0308327	0.0154164	−	0.999881i	$-0.495093\pi$
$241$	6.69995e6	0.0308327	0.0154164	+	0.999881i	$0.495093\pi$
$242$	1.36618e8	0.619659
$243$	−1.43489e7	−0.0641500
$244$	−1.00581e8	−0.443251
$245$	1.88313e8	0.818087
$246$	−4.44831e7	−0.190512
$247$	2.02403e8	0.854630
$248$	−9.11015e7	−0.379267
$249$	−1.14573e8	−0.470311
$250$	7.03236e7	0.284650
$251$	−3.18373e8	−1.27080	−0.635402	−	0.772182i	$-0.719165\pi$
$251$	−3.18373e8	−1.27080	−0.635402	+	0.772182i	$0.719165\pi$
$252$	−3.78548e7	−0.149011
$253$	8.13973e7	0.316001
$254$	2.13650e8	0.818060
$255$	4.63977e7	0.175229
$256$	−1.56237e8	−0.582029
$257$	−2.14248e8	−0.787320	−0.393660	−	0.919256i	$-0.628791\pi$
$257$	−2.14248e8	−0.787320	−0.393660	+	0.919256i	$0.628791\pi$
$258$	−4.87913e7	−0.176878
$259$	−1.81787e8	−0.650151
$260$	1.31364e8	0.463521
$261$	5.05598e7	0.176021
$262$	−1.41293e6	−0.00485362
$263$	−4.13165e6	−0.0140048	−0.00700242	−	0.999975i	$-0.502229\pi$
$263$	−4.13165e6	−0.0140048	−0.00700242	+	0.999975i	$0.502229\pi$
$264$	−2.21459e8	−0.740763
$265$	−2.61563e8	−0.863407
$266$	1.48879e8	0.485008
$267$	1.23200e8	0.396114
$268$	−1.88477e8	−0.598118
$269$	−2.07795e8	−0.650881	−0.325441	−	0.945562i	$-0.605513\pi$
$269$	−2.07795e8	−0.650881	−0.325441	+	0.945562i	$0.605513\pi$
$270$	3.66261e7	0.113245
$271$	−1.80493e7	−0.0550893	−0.0275447	−	0.999621i	$-0.508769\pi$
$271$	−1.80493e7	−0.0550893	−0.0275447	+	0.999621i	$0.508769\pi$
$272$	3.00268e7	0.0904726
$273$	−5.48537e7	−0.163169
$274$	−2.02261e8	−0.593999
$275$	−2.69827e8	−0.782387
$276$	−3.24467e7	−0.0928942
$277$	−1.54789e8	−0.437583	−0.218791	−	0.975772i	$-0.570211\pi$
$277$	−1.54789e8	−0.437583	−0.218791	+	0.975772i	$0.570211\pi$
$278$	2.11262e8	0.589744
$279$	5.41690e7	0.149326
$280$	2.21848e8	0.603952
$281$	−4.62346e8	−1.24307	−0.621534	−	0.783387i	$-0.713490\pi$
$281$	−4.62346e8	−1.24307	−0.621534	+	0.783387i	$0.713490\pi$
$282$	1.00072e8	0.265731
$283$	6.13374e8	1.60869	0.804346	−	0.594162i	$-0.202516\pi$
$283$	6.13374e8	1.60869	0.804346	+	0.594162i	$0.202516\pi$
$284$	3.86566e8	1.00140
$285$	4.86734e8	1.24547
$286$	−1.39771e8	−0.353293
$287$	1.60208e8	0.400035
$288$	1.38107e8	0.340676
$289$	−3.85410e8	−0.939248
$290$	−1.29056e8	−0.310731
$291$	−3.38359e8	−0.804920
$292$	−1.33197e8	−0.313080
$293$	3.81082e7	0.0885078	0.0442539	−	0.999020i	$-0.485909\pi$
$293$	3.81082e7	0.0885078	0.0442539	+	0.999020i	$0.485909\pi$
$294$	7.98694e7	0.183301
$295$	−3.07686e8	−0.697799
$296$	4.23931e8	0.950110
$297$	1.31679e8	0.291656
$298$	1.26699e7	0.0277342
$299$	−4.70170e7	−0.101720
$300$	1.07559e8	0.229997
$301$	1.75724e8	0.371406
$302$	2.06423e8	0.431255
$303$	−3.36790e8	−0.695521
$304$	3.14995e8	0.643052
$305$	−3.50488e8	−0.707333
$306$	1.96787e7	0.0392619
$307$	−4.87833e8	−0.962247	−0.481124	−	0.876653i	$-0.659771\pi$
$307$	−4.87833e8	−0.962247	−0.481124	+	0.876653i	$0.659771\pi$
$308$	3.47393e8	0.677474
$309$	−1.23916e8	−0.238932
$310$	−1.38268e8	−0.263607
$311$	−8.21826e7	−0.154924	−0.0774619	−	0.996995i	$-0.524682\pi$
$311$	−8.21826e7	−0.154924	−0.0774619	+	0.996995i	$0.524682\pi$
$312$	1.27920e8	0.238450
$313$	9.13866e7	0.168452	0.0842262	−	0.996447i	$-0.473158\pi$
$313$	9.13866e7	0.168452	0.0842262	+	0.996447i	$0.473158\pi$
$314$	3.10556e8	0.566090
$315$	−1.31911e8	−0.237790
$316$	5.22144e8	0.930863
$317$	−6.21150e8	−1.09519	−0.547595	−	0.836744i	$-0.684456\pi$
$317$	−6.21150e8	−1.09519	−0.547595	+	0.836744i	$0.684456\pi$
$318$	−1.10937e8	−0.193455
$319$	−4.63986e8	−0.800272
$320$	−8.75813e7	−0.149412
$321$	4.39779e8	0.742108
$322$	−3.45838e7	−0.0577267
$323$	2.61515e8	0.431805
$324$	−5.24902e7	−0.0857374
$325$	1.55859e8	0.251848
$326$	−1.80696e8	−0.288860
$327$	−2.95463e8	−0.467290
$328$	−3.73608e8	−0.584599
$329$	−3.60415e8	−0.557979
$330$	−3.36117e8	−0.514863
$331$	9.48538e8	1.43766	0.718831	−	0.695185i	$-0.244678\pi$
$331$	9.48538e8	1.43766	0.718831	+	0.695185i	$0.244678\pi$
$332$	−4.19124e8	−0.628578
$333$	−2.52069e8	−0.374081
$334$	5.22936e8	0.767955
$335$	−6.56776e8	−0.954466
$336$	−8.53675e7	−0.122774
$337$	−6.86971e8	−0.977763	−0.488882	−	0.872350i	$-0.662595\pi$
$337$	−6.86971e8	−0.977763	−0.488882	+	0.872350i	$0.662595\pi$
$338$	−2.58516e8	−0.364149
$339$	−7.82135e8	−1.09039
$340$	1.69729e8	0.234196
$341$	−4.97107e8	−0.678906
$342$	2.06439e8	0.279062
$343$	−7.20623e8	−0.964226
$344$	−4.09792e8	−0.542761
$345$	−1.13065e8	−0.148239
$346$	−2.42243e8	−0.314402
$347$	1.20692e8	0.155069	0.0775345	−	0.996990i	$-0.475295\pi$
$347$	1.20692e8	0.155069	0.0775345	+	0.996990i	$0.475295\pi$
$348$	1.84955e8	0.235254
$349$	−9.54223e8	−1.20160	−0.600801	−	0.799398i	$-0.705152\pi$
$349$	−9.54223e8	−1.20160	−0.600801	+	0.799398i	$0.705152\pi$
$350$	1.14643e8	0.142926
$351$	−7.60611e7	−0.0938832
$352$	−1.26740e9	−1.54887
$353$	−3.62096e8	−0.438139	−0.219070	−	0.975709i	$-0.570302\pi$
$353$	−3.62096e8	−0.438139	−0.219070	+	0.975709i	$0.570302\pi$
$354$	−1.30499e8	−0.156349
$355$	1.34704e9	1.59802
$356$	4.50681e8	0.529412
$357$	−7.08737e7	−0.0824416
$358$	3.87775e8	0.446673
$359$	5.77679e8	0.658956	0.329478	−	0.944163i	$-0.393127\pi$
$359$	5.77679e8	0.658956	0.329478	+	0.944163i	$0.393127\pi$
$360$	3.07618e8	0.347499
$361$	1.84955e9	2.06914
$362$	−7.35501e8	−0.814898
$363$	−6.82264e8	−0.748651
$364$	−2.00662e8	−0.218077
$365$	−4.64146e8	−0.499608
$366$	−1.48653e8	−0.158485
$367$	8.05566e8	0.850687	0.425343	−	0.905032i	$-0.360153\pi$
$367$	8.05566e8	0.850687	0.425343	+	0.905032i	$0.360153\pi$
$368$	−7.31714e7	−0.0765374
$369$	2.22147e8	0.230170
$370$	6.43417e8	0.660368
$371$	3.99545e8	0.406215
$372$	1.98157e8	0.199577
$373$	1.53686e9	1.53340	0.766698	−	0.642008i	$-0.221898\pi$
$373$	1.53686e9	1.53340	0.766698	+	0.642008i	$0.221898\pi$
$374$	−1.80591e8	−0.178503
$375$	−3.51194e8	−0.343904
$376$	8.40495e8	0.815413
$377$	2.68009e8	0.257605
$378$	−5.59474e7	−0.0532793
$379$	1.48201e9	1.39835	0.699173	−	0.714953i	$-0.253552\pi$
$379$	1.48201e9	1.39835	0.699173	+	0.714953i	$0.253552\pi$
$380$	1.78054e9	1.66459
$381$	−1.06696e9	−0.988352
$382$	3.46557e8	0.318092
$383$	−8.66673e8	−0.788242	−0.394121	−	0.919059i	$-0.628951\pi$
$383$	−8.66673e8	−0.788242	−0.394121	+	0.919059i	$0.628951\pi$
$384$	6.17582e8	0.556590
$385$	1.21054e9	1.08110
$386$	−6.01141e8	−0.532011
$387$	2.43662e8	0.213698
$388$	−1.23776e9	−1.07579
$389$	−1.38485e9	−1.19283	−0.596417	−	0.802674i	$-0.703410\pi$
$389$	−1.38485e9	−1.19283	−0.596417	+	0.802674i	$0.703410\pi$
$390$	1.94149e8	0.165733
$391$	−6.07483e7	−0.0513944
$392$	6.70813e8	0.562471
$393$	7.05612e6	0.00586397
$394$	7.43452e8	0.612373
$395$	1.81949e9	1.48546
$396$	4.81701e8	0.389802
$397$	−9.97021e8	−0.799719	−0.399859	−	0.916576i	$-0.630941\pi$
$397$	−9.97021e8	−0.799719	−0.399859	+	0.916576i	$0.630941\pi$
$398$	−9.60936e8	−0.764018
$399$	−7.43499e8	−0.585970
$400$	2.42559e8	0.189499
$401$	2.19367e9	1.69889	0.849445	−	0.527677i	$-0.176937\pi$
$401$	2.19367e9	1.69889	0.849445	+	0.527677i	$0.176937\pi$
$402$	−2.78559e8	−0.213858
$403$	2.87141e8	0.218538
$404$	−1.23202e9	−0.929573
$405$	−1.82910e8	−0.136818
$406$	1.97136e8	0.146193
$407$	2.31323e9	1.70074
$408$	1.65279e8	0.120478
$409$	3.18875e8	0.230456	0.115228	−	0.993339i	$-0.463240\pi$
$409$	3.18875e8	0.230456	0.115228	+	0.993339i	$0.463240\pi$
$410$	−5.67040e8	−0.406322
$411$	1.01009e9	0.717649
$412$	−4.53302e8	−0.319336
$413$	4.69998e8	0.328300
$414$	−4.79544e7	−0.0332145
$415$	−1.46050e9	−1.00307
$416$	7.32080e8	0.498577
$417$	−1.05503e9	−0.712509
$418$	−1.89448e9	−1.26874
$419$	−7.98056e8	−0.530010	−0.265005	−	0.964247i	$-0.585374\pi$
$419$	−7.98056e8	−0.530010	−0.265005	+	0.964247i	$0.585374\pi$
$420$	−4.82547e8	−0.317810
$421$	2.70591e9	1.76737	0.883684	−	0.468085i	$-0.155056\pi$
$421$	2.70591e9	1.76737	0.883684	+	0.468085i	$0.155056\pi$
$422$	1.11866e9	0.724608
$423$	−4.99758e8	−0.321047
$424$	−9.31745e8	−0.593631
$425$	2.01377e8	0.127247
$426$	5.71323e8	0.358054
$427$	5.35380e8	0.332786
$428$	1.60877e9	0.991838
$429$	6.98010e8	0.426836
$430$	−6.21957e8	−0.377243
$431$	4.37854e8	0.263426	0.131713	−	0.991288i	$-0.457952\pi$
$431$	4.37854e8	0.263426	0.131713	+	0.991288i	$0.457952\pi$
$432$	−1.18372e8	−0.0706409
$433$	−8.64340e8	−0.511655	−0.255827	−	0.966722i	$-0.582348\pi$
$433$	−8.64340e8	−0.511655	−0.255827	+	0.966722i	$0.582348\pi$
$434$	2.11209e8	0.124022
$435$	6.44502e8	0.375415
$436$	−1.08084e9	−0.624540
$437$	−6.37279e8	−0.365296
$438$	−1.96858e8	−0.111942
$439$	1.19683e8	0.0675160	0.0337580	−	0.999430i	$-0.489252\pi$
$439$	1.19683e8	0.0675160	0.0337580	+	0.999430i	$0.489252\pi$
$440$	−2.82300e9	−1.57989
$441$	−3.98866e8	−0.221458
$442$	1.04313e8	0.0574595
$443$	2.73826e9	1.49645	0.748225	−	0.663445i	$-0.230906\pi$
$443$	2.73826e9	1.49645	0.748225	+	0.663445i	$0.230906\pi$
$444$	−9.22102e8	−0.499964
$445$	1.57046e9	0.844827
$446$	−7.40069e8	−0.395003
$447$	−6.32731e7	−0.0335076
$448$	1.33783e8	0.0702954
$449$	−1.10118e9	−0.574111	−0.287055	−	0.957914i	$-0.592676\pi$
$449$	−1.10118e9	−0.574111	−0.287055	+	0.957914i	$0.592676\pi$
$450$	1.58966e8	0.0822358
$451$	−2.03864e9	−1.04646
$452$	−2.86115e9	−1.45733
$453$	−1.03087e9	−0.521028
$454$	1.40181e9	0.703060
$455$	−6.99237e8	−0.348004
$456$	1.73385e9	0.856319
$457$	1.98372e9	0.972239	0.486119	−	0.873892i	$-0.338412\pi$
$457$	1.98372e9	0.972239	0.486119	+	0.873892i	$0.338412\pi$
$458$	1.48355e9	0.721561
$459$	−9.82747e7	−0.0474348
$460$	−4.13608e8	−0.198124
$461$	−7.95089e8	−0.377975	−0.188987	−	0.981980i	$-0.560520\pi$
$461$	−7.95089e8	−0.377975	−0.188987	+	0.981980i	$0.560520\pi$
$462$	5.13427e8	0.242232
$463$	−7.69252e8	−0.360193	−0.180097	−	0.983649i	$-0.557641\pi$
$463$	−7.69252e8	−0.360193	−0.180097	+	0.983649i	$0.557641\pi$
$464$	4.17096e8	0.193831
$465$	6.90508e8	0.318481
$466$	−8.92592e8	−0.408604
$467$	−2.36312e9	−1.07368	−0.536841	−	0.843683i	$-0.680383\pi$
$467$	−2.36312e9	−1.07368	−0.536841	+	0.843683i	$0.680383\pi$
$468$	−2.78242e8	−0.125476
$469$	1.00324e9	0.449057
$470$	1.27565e9	0.566748
$471$	−1.55091e9	−0.683931
$472$	−1.09604e9	−0.479767
$473$	−2.23608e9	−0.971568
$474$	7.71701e8	0.332832
$475$	2.11254e9	0.904436
$476$	−2.59265e8	−0.110184
$477$	5.54015e8	0.233726
$478$	1.67295e9	0.700626
$479$	−3.11846e9	−1.29648	−0.648240	−	0.761436i	$-0.724495\pi$
$479$	−3.11846e9	−1.29648	−0.648240	+	0.761436i	$0.724495\pi$
$480$	1.76049e9	0.726589
$481$	−1.33618e9	−0.547465
$482$	3.62234e7	0.0147341
$483$	1.72710e8	0.0697434
$484$	−2.49581e9	−1.00058
$485$	−4.31317e9	−1.71672
$486$	−7.75777e7	−0.0306556
$487$	−3.77527e9	−1.48114	−0.740570	−	0.671979i	$-0.765444\pi$
$487$	−3.77527e9	−1.48114	−0.740570	+	0.671979i	$0.765444\pi$
$488$	−1.24852e9	−0.486323
$489$	9.02391e8	0.348991
$490$	1.01812e9	0.390942
$491$	1.21830e9	0.464482	0.232241	−	0.972658i	$-0.425394\pi$
$491$	1.21830e9	0.464482	0.232241	+	0.972658i	$0.425394\pi$
$492$	8.12644e8	0.307626
$493$	3.46281e8	0.130156
$494$	1.09430e9	0.408405
$495$	1.67856e9	0.622039
$496$	4.46870e8	0.164435
$497$	−2.05765e9	−0.751837
$498$	−6.19443e8	−0.224749
$499$	−4.76230e9	−1.71579	−0.857897	−	0.513822i	$-0.828229\pi$
$499$	−4.76230e9	−1.71579	−0.857897	+	0.513822i	$0.828229\pi$
$500$	−1.28471e9	−0.459633
$501$	−2.61152e9	−0.927817
$502$	−1.72129e9	−0.607283
$503$	4.21755e9	1.47765	0.738826	−	0.673896i	$-0.235380\pi$
$503$	4.21755e9	1.47765	0.738826	+	0.673896i	$0.235380\pi$
$504$	−4.69895e8	−0.163491
$505$	−4.29316e9	−1.48340
$506$	4.40076e8	0.151008
$507$	1.29102e9	0.439953
$508$	−3.90309e9	−1.32095
$509$	1.94818e9	0.654813	0.327407	−	0.944884i	$-0.393825\pi$
$509$	1.94818e9	0.654813	0.327407	+	0.944884i	$0.393825\pi$
$510$	2.50850e8	0.0837372
$511$	7.08995e8	0.235055
$512$	2.08310e9	0.685906
$513$	−1.03095e9	−0.337153
$514$	−1.15834e9	−0.376239
$515$	−1.57960e9	−0.509591
$516$	8.91348e8	0.285610
$517$	4.58627e9	1.45963
$518$	−9.82836e8	−0.310690
$519$	1.20976e9	0.379850
$520$	1.63063e9	0.508563
$521$	4.07821e9	1.26339	0.631695	−	0.775217i	$-0.282359\pi$
$521$	4.07821e9	1.26339	0.631695	+	0.775217i	$0.282359\pi$
$522$	2.73353e8	0.0841157
$523$	6.06342e9	1.85337	0.926684	−	0.375841i	$-0.122646\pi$
$523$	6.06342e9	1.85337	0.926684	+	0.375841i	$0.122646\pi$
$524$	2.58122e7	0.00783729
$525$	−5.72524e8	−0.172678
$526$	−2.23378e7	−0.00669254
$527$	3.71000e8	0.110417
$528$	1.08630e9	0.321166
$529$	1.48036e8	0.0434783
$530$	−1.41415e9	−0.412599
$531$	6.51707e8	0.188896
$532$	−2.71982e9	−0.783158
$533$	1.17757e9	0.336853
$534$	6.66081e8	0.189292
$535$	5.60600e9	1.58276
$536$	−2.33958e9	−0.656238
$537$	−1.93654e9	−0.539655
$538$	−1.12345e9	−0.311039
$539$	3.66038e9	1.00685
$540$	−6.69108e8	−0.182860
$541$	−6.00472e9	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$541$	−6.00472e9	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$542$	−9.75838e7	−0.0263257
$543$	3.67307e9	0.984532
$544$	9.45884e8	0.251908
$545$	−3.76636e9	−0.996631
$546$	−2.96568e8	−0.0779740
$547$	5.57969e9	1.45765	0.728827	−	0.684698i	$-0.240066\pi$
$547$	5.57969e9	1.45765	0.728827	+	0.684698i	$0.240066\pi$
$548$	3.69503e9	0.959148
$549$	7.42368e8	0.191477
$550$	−1.45883e9	−0.373882
$551$	3.63266e9	0.925111
$552$	−4.02763e8	−0.101921
$553$	−2.77932e9	−0.698877
$554$	−8.36868e8	−0.209109
$555$	−3.21320e9	−0.797834
$556$	−3.85945e9	−0.952278
$557$	3.59539e9	0.881562	0.440781	−	0.897615i	$-0.354702\pi$
$557$	3.59539e9	0.881562	0.440781	+	0.897615i	$0.354702\pi$
$558$	2.92866e8	0.0713590
$559$	1.29161e9	0.312745
$560$	−1.08820e9	−0.261850
$561$	9.01864e8	0.215661
$562$	−2.49968e9	−0.594029
$563$	−9.98324e8	−0.235772	−0.117886	−	0.993027i	$-0.537612\pi$
$563$	−9.98324e8	−0.235772	−0.117886	+	0.993027i	$0.537612\pi$
$564$	−1.82818e9	−0.429084
$565$	−9.97012e9	−2.32558
$566$	3.31622e9	0.768751
$567$	2.79400e8	0.0643702
$568$	4.79847e9	1.09871
$569$	3.60988e9	0.821486	0.410743	−	0.911751i	$-0.365269\pi$
$569$	3.60988e9	0.821486	0.410743	+	0.911751i	$0.365269\pi$
$570$	2.63154e9	0.595179
$571$	3.06436e9	0.688832	0.344416	−	0.938817i	$-0.388077\pi$
$571$	3.06436e9	0.688832	0.344416	+	0.938817i	$0.388077\pi$
$572$	2.55341e9	0.570473
$573$	−1.73069e9	−0.384308
$574$	8.66168e8	0.191166
$575$	−4.90730e8	−0.107648
$576$	1.85506e8	0.0404462
$577$	−1.25398e9	−0.271753	−0.135876	−	0.990726i	$-0.543385\pi$
$577$	−1.25398e9	−0.271753	−0.135876	+	0.990726i	$0.543385\pi$
$578$	−2.08373e9	−0.448842
$579$	3.00208e9	0.642758
$580$	2.35767e9	0.501748
$581$	2.23095e9	0.471926
$582$	−1.82935e9	−0.384650
$583$	−5.08418e9	−1.06263
$584$	−1.65339e9	−0.343503
$585$	−9.69574e8	−0.200233
$586$	2.06033e8	0.0422955
$587$	6.31504e9	1.28867	0.644336	−	0.764742i	$-0.277134\pi$
$587$	6.31504e9	1.28867	0.644336	+	0.764742i	$0.277134\pi$
$588$	−1.45910e9	−0.295982
$589$	3.89197e9	0.784812
$590$	−1.66351e9	−0.333459
$591$	−3.71278e9	−0.739848
$592$	−2.07946e9	−0.411931
$593$	4.12267e9	0.811872	0.405936	−	0.913902i	$-0.366946\pi$
$593$	4.12267e9	0.811872	0.405936	+	0.913902i	$0.366946\pi$
$594$	7.11927e8	0.139374
$595$	−9.03448e8	−0.175830
$596$	−2.31461e8	−0.0447833
$597$	4.79888e9	0.923060
$598$	−2.54198e8	−0.0486092
$599$	8.48783e9	1.61363	0.806813	−	0.590808i	$-0.201191\pi$
$599$	8.48783e9	1.61363	0.806813	+	0.590808i	$0.201191\pi$
$600$	1.33514e9	0.252346
$601$	−1.68246e9	−0.316144	−0.158072	−	0.987428i	$-0.550528\pi$
$601$	−1.68246e9	−0.316144	−0.158072	+	0.987428i	$0.550528\pi$
$602$	9.50056e8	0.177485
$603$	1.39111e9	0.258376
$604$	−3.77106e9	−0.696361
$605$	−8.69702e9	−1.59671
$606$	−1.82086e9	−0.332371
$607$	−2.99741e9	−0.543984	−0.271992	−	0.962299i	$-0.587682\pi$
$607$	−2.99741e9	−0.543984	−0.271992	+	0.962299i	$0.587682\pi$
$608$	9.92278e9	1.79048
$609$	−9.84493e8	−0.176625
$610$	−1.89492e9	−0.338016
$611$	−2.64914e9	−0.469851
$612$	−3.59502e8	−0.0633973
$613$	3.01310e9	0.528325	0.264163	−	0.964478i	$-0.414904\pi$
$613$	3.01310e9	0.528325	0.264163	+	0.964478i	$0.414904\pi$
$614$	−2.63748e9	−0.459832
$615$	2.83178e9	0.490904
$616$	4.31221e9	0.743306
$617$	−1.05608e9	−0.181008	−0.0905040	−	0.995896i	$-0.528848\pi$
$617$	−1.05608e9	−0.181008	−0.0905040	+	0.995896i	$0.528848\pi$
$618$	−6.69956e8	−0.114179
$619$	5.70924e9	0.967522	0.483761	−	0.875200i	$-0.339270\pi$
$619$	5.70924e9	0.967522	0.483761	+	0.875200i	$0.339270\pi$
$620$	2.52597e9	0.425654
$621$	2.39483e8	0.0401286
$622$	−4.44322e8	−0.0740340
$623$	−2.39892e9	−0.397474
$624$	−6.27470e8	−0.103382
$625$	−7.62778e9	−1.24974
$626$	4.94083e8	0.0804989
$627$	9.46099e9	1.53285
$628$	−5.67341e9	−0.914083
$629$	−1.72641e9	−0.276609
$630$	−7.13178e8	−0.113633
$631$	1.21714e10	1.92858	0.964288	−	0.264856i	$-0.0853244\pi$
$631$	1.21714e10	1.92858	0.964288	+	0.264856i	$0.0853244\pi$
$632$	6.48142e9	1.02132
$633$	−5.58654e9	−0.875447
$634$	−3.35826e9	−0.523362
$635$	−1.36009e10	−2.10794
$636$	2.02666e9	0.312379
$637$	−2.11432e9	−0.324103
$638$	−2.50855e9	−0.382429
$639$	−2.85317e9	−0.432588
$640$	7.87250e9	1.18709
$641$	−3.46116e9	−0.519062	−0.259531	−	0.965735i	$-0.583568\pi$
$641$	−3.46116e9	−0.519062	−0.259531	+	0.965735i	$0.583568\pi$
$642$	2.37768e9	0.354634
$643$	2.11373e9	0.313553	0.156777	−	0.987634i	$-0.449890\pi$
$643$	2.11373e9	0.313553	0.156777	+	0.987634i	$0.449890\pi$
$644$	6.31797e8	0.0932131
$645$	3.10603e9	0.455772
$646$	1.41389e9	0.206348
$647$	8.83366e9	1.28226	0.641130	−	0.767433i	$-0.278466\pi$
$647$	8.83366e9	1.28226	0.641130	+	0.767433i	$0.278466\pi$
$648$	−6.51565e8	−0.0940687
$649$	−5.98070e9	−0.858807
$650$	8.42653e8	0.120352
$651$	−1.05477e9	−0.149839
$652$	3.30107e9	0.466431
$653$	−5.02955e9	−0.706859	−0.353430	−	0.935461i	$-0.614985\pi$
$653$	−5.02955e9	−0.706859	−0.353430	+	0.935461i	$0.614985\pi$
$654$	−1.59743e9	−0.223306
$655$	8.99465e7	0.0125066
$656$	1.83262e9	0.253459
$657$	9.83105e8	0.135245
$658$	−1.94859e9	−0.266643
$659$	3.65728e9	0.497805	0.248902	−	0.968529i	$-0.419930\pi$
$659$	3.65728e9	0.497805	0.248902	+	0.968529i	$0.419930\pi$
$660$	6.14038e9	0.831365
$661$	8.02439e9	1.08070	0.540352	−	0.841439i	$-0.318291\pi$
$661$	8.02439e9	1.08070	0.540352	+	0.841439i	$0.318291\pi$
$662$	5.12829e9	0.687020
$663$	−5.20938e8	−0.0694206
$664$	−5.20262e9	−0.689658
$665$	−9.47761e9	−1.24975
$666$	−1.36282e9	−0.178763
$667$	−8.43843e8	−0.110109
$668$	−9.55330e9	−1.24004
$669$	3.69588e9	0.477229
$670$	−3.55087e9	−0.456114
$671$	−6.81268e9	−0.870541
$672$	−2.68919e9	−0.341845
$673$	−6.32514e9	−0.799867	−0.399933	−	0.916544i	$-0.630967\pi$
$673$	−6.32514e9	−0.799867	−0.399933	+	0.916544i	$0.630967\pi$
$674$	−3.71412e9	−0.467247
$675$	−7.93872e8	−0.0993545
$676$	4.72273e9	0.588003
$677$	−7.59623e9	−0.940888	−0.470444	−	0.882430i	$-0.655906\pi$
$677$	−7.59623e9	−0.940888	−0.470444	+	0.882430i	$0.655906\pi$
$678$	−4.22863e9	−0.521070
$679$	6.58848e9	0.807683
$680$	2.10686e9	0.256953
$681$	−7.00057e9	−0.849412
$682$	−2.68762e9	−0.324431
$683$	−1.36892e10	−1.64402	−0.822008	−	0.569476i	$-0.807146\pi$
$683$	−1.36892e10	−1.64402	−0.822008	+	0.569476i	$0.807146\pi$
$684$	−3.77135e9	−0.450609
$685$	1.28759e10	1.53059
$686$	−3.89606e9	−0.460778
$687$	−7.40881e9	−0.871766
$688$	2.01010e9	0.235320
$689$	2.93674e9	0.342057
$690$	−6.11290e8	−0.0708395
$691$	7.98642e9	0.920829	0.460414	−	0.887704i	$-0.347701\pi$
$691$	7.98642e9	0.920829	0.460414	+	0.887704i	$0.347701\pi$
$692$	4.42545e9	0.507675
$693$	−2.56404e9	−0.292657
$694$	6.52523e8	0.0741033
$695$	−1.34488e10	−1.51963
$696$	2.29586e9	0.258114
$697$	1.52147e9	0.170196
$698$	−5.15903e9	−0.574214
$699$	4.45758e9	0.493661
$700$	−2.09437e9	−0.230786
$701$	1.29422e10	1.41904	0.709520	−	0.704685i	$-0.248912\pi$
$701$	1.29422e10	1.41904	0.709520	+	0.704685i	$0.248912\pi$
$702$	−4.11226e8	−0.0448643
$703$	−1.81108e10	−1.96605
$704$	−1.70238e9	−0.183887
$705$	−6.37057e9	−0.684725
$706$	−1.95768e9	−0.209375
$707$	6.55792e9	0.697908
$708$	2.38403e9	0.252462
$709$	−1.36942e9	−0.144303	−0.0721513	−	0.997394i	$-0.522986\pi$
$709$	−1.36942e9	−0.144303	−0.0721513	+	0.997394i	$0.522986\pi$
$710$	7.28283e9	0.763652
$711$	−3.85385e9	−0.402116
$712$	5.59433e9	0.580856
$713$	−9.04080e8	−0.0934100
$714$	−3.83180e8	−0.0393966
$715$	8.89775e9	0.910351
$716$	−7.08411e9	−0.721257
$717$	−8.35468e9	−0.846473
$718$	3.12323e9	0.314897
$719$	1.52783e10	1.53294	0.766470	−	0.642280i	$-0.222012\pi$
$719$	1.52783e10	1.53294	0.766470	+	0.642280i	$0.222012\pi$
$720$	−1.50892e9	−0.150662
$721$	2.41288e9	0.239752
$722$	9.99961e9	0.988787
$723$	−1.80899e8	−0.0178013
$724$	1.34366e10	1.31584
$725$	2.79729e9	0.272618
$726$	−3.68867e9	−0.357760
$727$	4.60649e9	0.444631	0.222315	−	0.974975i	$-0.428639\pi$
$727$	4.60649e9	0.444631	0.222315	+	0.974975i	$0.428639\pi$
$728$	−2.49083e9	−0.239268
$729$	3.87420e8	0.0370370
$730$	−2.50941e9	−0.238749
$731$	1.66883e9	0.158016
$732$	2.71568e9	0.255911
$733$	−1.57570e10	−1.47778	−0.738888	−	0.673829i	$-0.764649\pi$
$733$	−1.57570e10	−1.47778	−0.738888	+	0.673829i	$0.764649\pi$
$734$	4.35531e9	0.406521
$735$	−5.08446e9	−0.472323
$736$	−2.30500e9	−0.213107
$737$	−1.27662e10	−1.17470
$738$	1.20104e9	0.109992
$739$	−1.35450e10	−1.23459	−0.617296	−	0.786731i	$-0.711772\pi$
$739$	−1.35450e10	−1.23459	−0.617296	+	0.786731i	$0.711772\pi$
$740$	−1.17543e10	−1.06632
$741$	−5.46489e9	−0.493421
$742$	2.16015e9	0.194120
$743$	−1.32438e10	−1.18455	−0.592273	−	0.805738i	$-0.701769\pi$
$743$	−1.32438e10	−1.18455	−0.592273	+	0.805738i	$0.701769\pi$
$744$	2.45974e9	0.218970
$745$	−8.06561e8	−0.0714645
$746$	8.30908e9	0.732769
$747$	3.09348e9	0.271534
$748$	3.29914e9	0.288234
$749$	−8.56331e9	−0.744656
$750$	−1.89874e9	−0.164343
$751$	1.68901e10	1.45510	0.727548	−	0.686057i	$-0.240660\pi$
$751$	1.68901e10	1.45510	0.727548	+	0.686057i	$0.240660\pi$
$752$	−4.12278e9	−0.353531
$753$	8.59607e9	0.733698
$754$	1.44900e9	0.123103
$755$	−1.31408e10	−1.11124
$756$	1.02208e9	0.0860318
$757$	−1.85508e10	−1.55427	−0.777134	−	0.629335i	$-0.783327\pi$
$757$	−1.85508e10	−1.55427	−0.777134	+	0.629335i	$0.783327\pi$
$758$	8.01253e9	0.668232
$759$	−2.19773e9	−0.182443
$760$	2.21019e10	1.82635
$761$	7.79417e9	0.641097	0.320548	−	0.947232i	$-0.396133\pi$
$761$	7.79417e9	0.641097	0.320548	+	0.947232i	$0.396133\pi$
$762$	−5.76855e9	−0.472307
$763$	5.75322e9	0.468894
$764$	−6.33111e9	−0.513633
$765$	−1.25274e9	−0.101168
$766$	−4.68569e9	−0.376680
$767$	3.45459e9	0.276448
$768$	4.21840e9	0.336035
$769$	−3.68753e9	−0.292411	−0.146205	−	0.989254i	$-0.546706\pi$
$769$	−3.68753e9	−0.292411	−0.146205	+	0.989254i	$0.546706\pi$
$770$	6.54481e9	0.516630
$771$	5.78470e9	0.454559
$772$	1.09820e10	0.859055
$773$	−9.90277e8	−0.0771132	−0.0385566	−	0.999256i	$-0.512276\pi$
$773$	−9.90277e8	−0.0771132	−0.0385566	+	0.999256i	$0.512276\pi$
$774$	1.31736e9	0.102120
$775$	2.99697e9	0.231274
$776$	−1.53644e10	−1.18032
$777$	4.90825e9	0.375365
$778$	−7.48724e9	−0.570024
$779$	1.59610e10	1.20970
$780$	−3.54683e9	−0.267614
$781$	2.61834e10	1.96675
$782$	−3.28437e8	−0.0245600
$783$	−1.36512e9	−0.101626
$784$	−3.29046e9	−0.243866
$785$	−1.97699e10	−1.45868
$786$	3.81491e7	0.00280224
$787$	−3.85492e9	−0.281905	−0.140953	−	0.990016i	$-0.545017\pi$
$787$	−3.85492e9	−0.281905	−0.140953	+	0.990016i	$0.545017\pi$
$788$	−1.35818e10	−0.988818
$789$	1.11555e8	0.00808570
$790$	9.83711e9	0.709860
$791$	1.52296e10	1.09414
$792$	5.97939e9	0.427680
$793$	3.93517e9	0.280225
$794$	−5.39041e9	−0.382164
$795$	7.06220e9	0.498489
$796$	1.75549e10	1.23368
$797$	2.68172e8	0.0187633	0.00938164	−	0.999956i	$-0.497014\pi$
$797$	2.68172e8	0.0187633	0.00938164	+	0.999956i	$0.497014\pi$
$798$	−4.01974e9	−0.280020
$799$	−3.42281e9	−0.237394
$800$	7.64094e9	0.527633
$801$	−3.32639e9	−0.228696
$802$	1.18601e10	0.811854
$803$	−9.02192e9	−0.614886
$804$	5.08888e9	0.345323
$805$	2.20159e9	0.148748
$806$	1.55243e9	0.104434
$807$	5.61046e9	0.375786
$808$	−1.52932e10	−1.01990
$809$	1.50724e10	1.00083	0.500416	−	0.865785i	$-0.333180\pi$
$809$	1.50724e10	1.00083	0.500416	+	0.865785i	$0.333180\pi$
$810$	−9.88906e8	−0.0653819
$811$	−2.54700e10	−1.67670	−0.838352	−	0.545130i	$-0.816480\pi$
$811$	−2.54700e10	−1.67670	−0.838352	+	0.545130i	$0.816480\pi$
$812$	−3.60141e9	−0.236062
$813$	4.87331e8	0.0318058
$814$	1.25065e10	0.812740
$815$	1.15031e10	0.744323
$816$	−8.10722e8	−0.0522344
$817$	1.75068e10	1.12313
$818$	1.72400e9	0.110129
$819$	1.48105e9	0.0942055
$820$	1.03590e10	0.656100
$821$	−8.54784e9	−0.539083	−0.269541	−	0.962989i	$-0.586872\pi$
$821$	−8.54784e9	−0.539083	−0.269541	+	0.962989i	$0.586872\pi$
$822$	5.46105e9	0.342945
$823$	9.89326e8	0.0618643	0.0309321	−	0.999521i	$-0.490152\pi$
$823$	9.89326e8	0.0618643	0.0309321	+	0.999521i	$0.490152\pi$
$824$	−5.62688e9	−0.350366
$825$	7.28534e9	0.451711
$826$	2.54105e9	0.156886
$827$	2.72623e10	1.67608	0.838038	−	0.545612i	$-0.183703\pi$
$827$	2.72623e10	1.67608	0.838038	+	0.545612i	$0.183703\pi$
$828$	8.76060e8	0.0536325
$829$	−1.87927e10	−1.14564	−0.572821	−	0.819681i	$-0.694151\pi$
$829$	−1.87927e10	−1.14564	−0.572821	+	0.819681i	$0.694151\pi$
$830$	−7.89622e9	−0.479342
$831$	4.17929e9	0.252638
$832$	9.83334e8	0.0591928
$833$	−2.73180e9	−0.163754
$834$	−5.70406e9	−0.340489
$835$	−3.32899e10	−1.97884
$836$	3.46095e10	2.04868
$837$	−1.46256e9	−0.0862135
$838$	−4.31471e9	−0.253278
$839$	−1.42627e10	−0.833749	−0.416874	−	0.908964i	$-0.636875\pi$
$839$	−1.42627e10	−0.833749	−0.416874	+	0.908964i	$0.636875\pi$
$840$	−5.98989e9	−0.348692
$841$	−1.24398e10	−0.721150
$842$	1.46296e10	0.844578
$843$	1.24833e10	0.717686
$844$	−2.04363e10	−1.17005
$845$	1.64571e10	0.938326
$846$	−2.70195e9	−0.153420
$847$	1.32849e10	0.751221
$848$	4.57038e9	0.257375
$849$	−1.65611e10	−0.928778
$850$	1.08875e9	0.0608081
$851$	4.20703e9	0.234004
$852$	−1.04373e10	−0.578160
$853$	1.24913e10	0.689108	0.344554	−	0.938766i	$-0.388030\pi$
$853$	1.24913e10	0.689108	0.344554	+	0.938766i	$0.388030\pi$
$854$	2.89454e9	0.159029
$855$	−1.31418e10	−0.719075
$856$	1.99698e10	1.08822
$857$	−1.94250e10	−1.05421	−0.527106	−	0.849800i	$-0.676723\pi$
$857$	−1.94250e10	−1.05421	−0.527106	+	0.849800i	$0.676723\pi$
$858$	3.77381e9	0.203974
$859$	−3.37547e10	−1.81701	−0.908506	−	0.417873i	$-0.862776\pi$
$859$	−3.37547e10	−1.81701	−0.908506	+	0.417873i	$0.862776\pi$
$860$	1.13623e10	0.609145
$861$	−4.32562e9	−0.230960
$862$	2.36727e9	0.125884
$863$	−1.20894e10	−0.640278	−0.320139	−	0.947371i	$-0.603730\pi$
$863$	−1.20894e10	−0.640278	−0.320139	+	0.947371i	$0.603730\pi$
$864$	−3.72888e9	−0.196689
$865$	1.54211e10	0.810140
$866$	−4.67307e9	−0.244506
$867$	1.04061e10	0.542275
$868$	−3.85849e9	−0.200262
$869$	3.53667e10	1.82821
$870$	3.48451e9	0.179401
$871$	7.37407e9	0.378132
$872$	−1.34166e10	−0.685228
$873$	9.13570e9	0.464721
$874$	−3.44546e9	−0.174565
$875$	6.83839e9	0.345085
$876$	3.59633e9	0.180757
$877$	−3.99637e9	−0.200063	−0.100032	−	0.994984i	$-0.531894\pi$
$877$	−3.99637e9	−0.200063	−0.100032	+	0.994984i	$0.531894\pi$
$878$	6.47069e8	0.0322641
$879$	−1.02892e9	−0.0511000
$880$	1.38473e10	0.684979
$881$	1.00727e10	0.496286	0.248143	−	0.968723i	$-0.420180\pi$
$881$	1.00727e10	0.496286	0.248143	+	0.968723i	$0.420180\pi$
$882$	−2.15647e9	−0.105829
$883$	−1.22925e10	−0.600868	−0.300434	−	0.953803i	$-0.597131\pi$
$883$	−1.22925e10	−0.600868	−0.300434	+	0.953803i	$0.597131\pi$
$884$	−1.90566e9	−0.0927817
$885$	8.30751e9	0.402874
$886$	1.48045e10	0.715114
$887$	−2.39436e10	−1.15201	−0.576005	−	0.817446i	$-0.695389\pi$
$887$	−2.39436e10	−1.15201	−0.576005	+	0.817446i	$0.695389\pi$
$888$	−1.14461e10	−0.548546
$889$	2.07757e10	0.991744
$890$	8.49074e9	0.403720
$891$	−3.55534e9	−0.168387
$892$	1.35200e10	0.637824
$893$	−3.59070e10	−1.68732
$894$	−3.42087e8	−0.0160124
$895$	−2.46856e10	−1.15097
$896$	−1.20255e10	−0.558501
$897$	1.26946e9	0.0587280
$898$	−5.95354e9	−0.274352
$899$	5.15349e9	0.236561
$900$	−2.90409e9	−0.132789
$901$	3.79442e9	0.172826
$902$	−1.10219e10	−0.500075
$903$	−4.74455e9	−0.214431
$904$	−3.55157e10	−1.59894
$905$	4.68217e10	2.09980
$906$	−5.57343e9	−0.248985
$907$	3.93573e10	1.75146	0.875729	−	0.482804i	$-0.160382\pi$
$907$	3.93573e10	1.75146	0.875729	+	0.482804i	$0.160382\pi$
$908$	−2.56090e10	−1.13525
$909$	9.09333e9	0.401559
$910$	−3.78044e9	−0.166302
$911$	1.01487e10	0.444730	0.222365	−	0.974963i	$-0.428622\pi$
$911$	1.01487e10	0.444730	0.222365	+	0.974963i	$0.428622\pi$
$912$	−8.50487e9	−0.371266
$913$	−2.83887e10	−1.23452
$914$	1.07250e10	0.464607
$915$	9.46319e9	0.408379
$916$	−2.71024e10	−1.16513
$917$	−1.37396e8	−0.00588410
$918$	−5.31324e8	−0.0226678
$919$	7.75386e9	0.329544	0.164772	−	0.986332i	$-0.447311\pi$
$919$	7.75386e9	0.329544	0.164772	+	0.986332i	$0.447311\pi$
$920$	−5.13414e9	−0.217376
$921$	1.31715e10	0.555554
$922$	−4.29867e9	−0.180624
$923$	−1.51242e10	−0.633090
$924$	−9.37960e9	−0.391140
$925$	−1.39461e10	−0.579370
$926$	−4.15898e9	−0.172127
$927$	3.34574e9	0.137947
$928$	1.31391e10	0.539694
$929$	8.63007e9	0.353150	0.176575	−	0.984287i	$-0.443498\pi$
$929$	8.63007e9	0.353150	0.176575	+	0.984287i	$0.443498\pi$
$930$	3.73325e9	0.152194
$931$	−2.86579e10	−1.16391
$932$	1.63064e10	0.659785
$933$	2.21893e9	0.0894454
$934$	−1.27762e10	−0.513084
$935$	1.14963e10	0.459958
$936$	−3.45384e9	−0.137669
$937$	−1.83344e10	−0.728076	−0.364038	−	0.931384i	$-0.618602\pi$
$937$	−1.83344e10	−0.728076	−0.364038	+	0.931384i	$0.618602\pi$
$938$	5.42406e9	0.214592
$939$	−2.46744e9	−0.0972561
$940$	−2.33044e10	−0.915145
$941$	−1.16186e10	−0.454559	−0.227279	−	0.973830i	$-0.572983\pi$
$941$	−1.16186e10	−0.454559	−0.227279	+	0.973830i	$0.572983\pi$
$942$	−8.38500e9	−0.326832
$943$	−3.70764e9	−0.143981
$944$	5.37629e9	0.208008
$945$	3.56159e9	0.137288
$946$	−1.20894e10	−0.464286
$947$	−3.46540e10	−1.32595	−0.662977	−	0.748640i	$-0.730707\pi$
$947$	−3.46540e10	−1.32595	−0.662977	+	0.748640i	$0.730707\pi$
$948$	−1.40979e10	−0.537434
$949$	5.21128e9	0.197930
$950$	1.14215e10	0.432206
$951$	1.67711e10	0.632308
$952$	−3.21828e9	−0.120891
$953$	2.12304e10	0.794571	0.397285	−	0.917695i	$-0.369952\pi$
$953$	2.12304e10	0.794571	0.397285	+	0.917695i	$0.369952\pi$
$954$	2.99530e9	0.111692
$955$	−2.20617e10	−0.819647
$956$	−3.05625e10	−1.13132
$957$	1.25276e10	0.462037
$958$	−1.68600e10	−0.619554
$959$	−1.96682e10	−0.720113
$960$	2.36469e9	0.0862632
$961$	−2.19913e10	−0.799315
$962$	−7.22407e9	−0.261619
$963$	−1.18740e10	−0.428456
$964$	−6.61751e8	−0.0237917
$965$	3.82684e10	1.37087
$966$	9.33761e8	0.0333285
$967$	−7.36471e9	−0.261917	−0.130958	−	0.991388i	$-0.541805\pi$
$967$	−7.36471e9	−0.261917	−0.130958	+	0.991388i	$0.541805\pi$
$968$	−3.09807e10	−1.09781
$969$	−7.06090e9	−0.249303
$970$	−2.33192e10	−0.820376
$971$	2.52233e10	0.884167	0.442084	−	0.896974i	$-0.354239\pi$
$971$	2.52233e10	0.884167	0.442084	+	0.896974i	$0.354239\pi$
$972$	1.41723e9	0.0495005
$973$	2.05435e10	0.714955
$974$	−2.04111e10	−0.707798
$975$	−4.20818e9	−0.145405
$976$	6.12420e9	0.210851
$977$	−1.66661e10	−0.571745	−0.285872	−	0.958268i	$-0.592283\pi$
$977$	−1.66661e10	−0.571745	−0.285872	+	0.958268i	$0.592283\pi$
$978$	4.87880e9	0.166773
$979$	3.05262e10	1.03976
$980$	−1.85996e10	−0.631267
$981$	7.97751e9	0.269790
$982$	6.58677e9	0.221964
$983$	1.86831e10	0.627354	0.313677	−	0.949530i	$-0.398439\pi$
$983$	1.86831e10	0.627354	0.313677	+	0.949530i	$0.398439\pi$
$984$	1.00874e10	0.337518
$985$	−4.73279e10	−1.57794
$986$	1.87218e9	0.0621981
$987$	9.73122e9	0.322149
$988$	−1.99913e10	−0.659464
$989$	−4.06672e9	−0.133677
$990$	9.07515e9	0.297256
$991$	3.89119e10	1.27006	0.635031	−	0.772487i	$-0.280987\pi$
$991$	3.89119e10	1.27006	0.635031	+	0.772487i	$0.280987\pi$
$992$	1.40770e10	0.457846
$993$	−2.56105e10	−0.830034
$994$	−1.11247e10	−0.359283
$995$	6.11728e10	1.96869
$996$	1.13163e10	0.362910
$997$	−4.49037e10	−1.43499	−0.717496	−	0.696563i	$-0.754712\pi$
$997$	−4.49037e10	−1.43499	−0.717496	+	0.696563i	$0.754712\pi$
$998$	−2.57475e10	−0.819932
$999$	6.80587e9	0.215975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.8.a.c.1.5	✓	7
3.2	odd	2		207.8.a.d.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.8.a.c.1.5	✓	7	1.1	even	1	trivial
207.8.a.d.1.3		7	3.2	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:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	69.a (trivial)

\(f(q)\)	\(=\)	\(q+5.40652 q^{2} -27.0000 q^{3} -98.7695 q^{4} -344.177 q^{5} -145.976 q^{6} +525.740 q^{7} -1226.03 q^{8} +729.000 q^{9} +O(q^{10})\)	\(q+5.40652 q^{2} -27.0000 q^{3} -98.7695 q^{4} -344.177 q^{5} -145.976 q^{6} +525.740 q^{7} -1226.03 q^{8} +729.000 q^{9} -1860.80 q^{10} -6690.01 q^{11} +2666.78 q^{12} +3864.31 q^{13} +2842.42 q^{14} +9292.78 q^{15} +6013.92 q^{16} +4992.87 q^{17} +3941.35 q^{18} +52377.6 q^{19} +33994.2 q^{20} -14195.0 q^{21} -36169.7 q^{22} -12167.0 q^{23} +33102.9 q^{24} +40332.9 q^{25} +20892.5 q^{26} -19683.0 q^{27} -51927.1 q^{28} +69355.1 q^{29} +50241.6 q^{30} +74305.9 q^{31} +189447. q^{32} +180630. q^{33} +26994.1 q^{34} -180948. q^{35} -72003.0 q^{36} -345774. q^{37} +283181. q^{38} -104336. q^{39} +421973. q^{40} +304729. q^{41} -76745.4 q^{42} +334242. q^{43} +660769. q^{44} -250905. q^{45} -65781.1 q^{46} -685540. q^{47} -162376. q^{48} -547141. q^{49} +218061. q^{50} -134808. q^{51} -381676. q^{52} +759966. q^{53} -106417. q^{54} +2.30255e6 q^{55} -644575. q^{56} -1.41420e6 q^{57} +374970. q^{58} +893975. q^{59} -917844. q^{60} +1.01834e6 q^{61} +401736. q^{62} +383264. q^{63} +254466. q^{64} -1.33001e6 q^{65} +976581. q^{66} +1.90825e6 q^{67} -493144. q^{68} +328509. q^{69} -978297. q^{70} -3.91381e6 q^{71} -893779. q^{72} +1.34857e6 q^{73} -1.86943e6 q^{74} -1.08899e6 q^{75} -5.17332e6 q^{76} -3.51720e6 q^{77} -564096. q^{78} -5.28649e6 q^{79} -2.06985e6 q^{80} +531441. q^{81} +1.64752e6 q^{82} +4.24345e6 q^{83} +1.40203e6 q^{84} -1.71843e6 q^{85} +1.80708e6 q^{86} -1.87259e6 q^{87} +8.20218e6 q^{88} -4.56295e6 q^{89} -1.35652e6 q^{90} +2.03162e6 q^{91} +1.20173e6 q^{92} -2.00626e6 q^{93} -3.70638e6 q^{94} -1.80272e7 q^{95} -5.11506e6 q^{96} +1.25318e7 q^{97} -2.95813e6 q^{98} -4.87702e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9}+O(q^{10}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9	\( 7 q - 189 q^{3} + 654 q^{4} - 516 q^{5} + 1018 q^{7} + 1422 q^{8} + 5103 q^{9} - 15310 q^{10} + 9040 q^{11} - 17658 q^{12} + 3774 q^{13} + 4536 q^{14} + 13932 q^{15} + 52002 q^{16} - 40760 q^{17} + 81598 q^{19} - 88946 q^{20} - 27486 q^{21} + 245034 q^{22} - 85169 q^{23} - 38394 q^{24} + 321325 q^{25} + 412748 q^{26} - 137781 q^{27} + 965948 q^{28} + 154126 q^{29} + 413370 q^{30} + 243132 q^{31} + 1278286 q^{32} - 244080 q^{33} + 984836 q^{34} - 130296 q^{35} + 476766 q^{36} + 582114 q^{37} + 772558 q^{38} - 101898 q^{39} - 132618 q^{40} + 113062 q^{41} - 122472 q^{42} - 659778 q^{43} + 659390 q^{44} - 376164 q^{45} - 591032 q^{47} - 1404054 q^{48} + 3263235 q^{49} - 702684 q^{50} + 1100520 q^{51} + 1793280 q^{52} + 207128 q^{53} + 184664 q^{55} + 5390508 q^{56} - 2203146 q^{57} - 1142916 q^{58} + 447148 q^{59} + 2401542 q^{60} + 2248970 q^{61} - 5729060 q^{62} + 742122 q^{63} + 7212922 q^{64} - 827096 q^{65} - 6615918 q^{66} + 4467570 q^{67} - 5477620 q^{68} + 2299563 q^{69} - 12744284 q^{70} - 5154608 q^{71} + 1036638 q^{72} - 13239250 q^{73} - 2827426 q^{74} - 8675775 q^{75} - 527434 q^{76} - 18415912 q^{77} - 11144196 q^{78} + 9594446 q^{79} - 55932394 q^{80} + 3720087 q^{81} - 20889952 q^{82} - 573720 q^{83} - 26080596 q^{84} + 7477272 q^{85} - 28416910 q^{86} - 4161402 q^{87} + 26555702 q^{88} - 3810540 q^{89} - 11160990 q^{90} + 36092068 q^{91} - 7957218 q^{92} - 6564564 q^{93} + 33545768 q^{94} + 10497320 q^{95} - 34513722 q^{96} + 49497978 q^{97} - 1023376 q^{98} + 6590160 q^{99}+O(q^{100}) \) 7 * q - 189 * q^3 + 654 * q^4 - 516 * q^5 + 1018 * q^7 + 1422 * q^8 + 5103 * q^9 - 15310 * q^10 + 9040 * q^11 - 17658 * q^12 + 3774 * q^13 + 4536 * q^14 + 13932 * q^15 + 52002 * q^16 - 40760 * q^17 + 81598 * q^19 - 88946 * q^20 - 27486 * q^21 + 245034 * q^22 - 85169 * q^23 - 38394 * q^24 + 321325 * q^25 + 412748 * q^26 - 137781 * q^27 + 965948 * q^28 + 154126 * q^29 + 413370 * q^30 + 243132 * q^31 + 1278286 * q^32 - 244080 * q^33 + 984836 * q^34 - 130296 * q^35 + 476766 * q^36 + 582114 * q^37 + 772558 * q^38 - 101898 * q^39 - 132618 * q^40 + 113062 * q^41 - 122472 * q^42 - 659778 * q^43 + 659390 * q^44 - 376164 * q^45 - 591032 * q^47 - 1404054 * q^48 + 3263235 * q^49 - 702684 * q^50 + 1100520 * q^51 + 1793280 * q^52 + 207128 * q^53 + 184664 * q^55 + 5390508 * q^56 - 2203146 * q^57 - 1142916 * q^58 + 447148 * q^59 + 2401542 * q^60 + 2248970 * q^61 - 5729060 * q^62 + 742122 * q^63 + 7212922 * q^64 - 827096 * q^65 - 6615918 * q^66 + 4467570 * q^67 - 5477620 * q^68 + 2299563 * q^69 - 12744284 * q^70 - 5154608 * q^71 + 1036638 * q^72 - 13239250 * q^73 - 2827426 * q^74 - 8675775 * q^75 - 527434 * q^76 - 18415912 * q^77 - 11144196 * q^78 + 9594446 * q^79 - 55932394 * q^80 + 3720087 * q^81 - 20889952 * q^82 - 573720 * q^83 - 26080596 * q^84 + 7477272 * q^85 - 28416910 * q^86 - 4161402 * q^87 + 26555702 * q^88 - 3810540 * q^89 - 11160990 * q^90 + 36092068 * q^91 - 7957218 * q^92 - 6564564 * q^93 + 33545768 * q^94 + 10497320 * q^95 - 34513722 * q^96 + 49497978 * q^97 - 1023376 * q^98 + 6590160 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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