LMFDB - Embedded newform 69.7.d.a.22.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [69,7,Mod(22,69)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("69.22");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 69 = 3 \cdot 23 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	69.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$15.8737317698$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.6
Character	$\chi$	$=$	69.22
Dual form			69.7.d.a.22.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.6305 q^{2} +15.5885 q^{3} +49.0083 q^{4} +66.6032i q^{5} -165.714 q^{6} -306.844i q^{7} +159.370 q^{8} +243.000 q^{9} +O(q^{10})$	\(q-10.6305 q^{2} +15.5885 q^{3} +49.0083 q^{4} +66.6032i q^{5} -165.714 q^{6} -306.844i q^{7} +159.370 q^{8} +243.000 q^{9} -708.028i q^{10} +998.732i q^{11} +763.964 q^{12} -1930.17 q^{13} +3261.92i q^{14} +1038.24i q^{15} -4830.72 q^{16} -9460.64i q^{17} -2583.22 q^{18} +9744.85i q^{19} +3264.11i q^{20} -4783.23i q^{21} -10617.1i q^{22} +(12027.6 + 1836.38i) q^{23} +2484.33 q^{24} +11189.0 q^{25} +20518.7 q^{26} +3788.00 q^{27} -15037.9i q^{28} +44157.2 q^{29} -11037.1i q^{30} +12989.0 q^{31} +41153.4 q^{32} +15568.7i q^{33} +100572. i q^{34} +20436.8 q^{35} +11909.0 q^{36} -93813.5i q^{37} -103593. i q^{38} -30088.4 q^{39} +10614.6i q^{40} +35742.5 q^{41} +50848.3i q^{42} -674.258i q^{43} +48946.1i q^{44} +16184.6i q^{45} +(-127860. - 19521.7i) q^{46} +102972. q^{47} -75303.4 q^{48} +23495.7 q^{49} -118945. q^{50} -147477. i q^{51} -94594.3 q^{52} +185550. i q^{53} -40268.4 q^{54} -66518.8 q^{55} -48901.7i q^{56} +151907. i q^{57} -469414. q^{58} +76235.0 q^{59} +50882.4i q^{60} +405897. i q^{61} -138080. q^{62} -74563.1i q^{63} -128317. q^{64} -128556. i q^{65} -165504. i q^{66} -414252. i q^{67} -463650. i q^{68} +(187492. + 28626.3i) q^{69} -217254. q^{70} +112686. q^{71} +38726.9 q^{72} -120456. q^{73} +997288. i q^{74} +174419. q^{75} +477578. i q^{76} +306455. q^{77} +319856. q^{78} -385722. i q^{79} -321741. i q^{80} +59049.0 q^{81} -379962. q^{82} +130245. i q^{83} -234418. i q^{84} +630109. q^{85} +7167.72i q^{86} +688342. q^{87} +159168. i q^{88} -1.03358e6i q^{89} -172051. i q^{90} +592261. i q^{91} +(589453. + 89997.8i) q^{92} +202478. q^{93} -1.09465e6 q^{94} -649038. q^{95} +641519. q^{96} +27015.5i q^{97} -249772. q^{98} +242692. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	$ 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) $ 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	−10.6305			−1.32882			−0.664408	−	0.747370i	$-0.731316\pi$
$2$	−10.6305			−1.32882			−0.664408	+	0.747370i	$0.731316\pi$
$3$	15.5885			0.577350
$3$	15.5885			0.577350
$4$	49.0083			0.765754
$5$			66.6032i			0.532826i	0.963859	+	0.266413i	$0.0858385\pi$
$5$			66.6032i			0.532826i	−0.963859	+	0.266413i	$0.914162\pi$
$6$	−165.714			−0.767193
$7$		−	306.844i		−	0.894589i	−0.894387	−	0.447295i	$-0.852388\pi$
$7$		−	306.844i		−	0.894589i	0.894387	−	0.447295i	$-0.147612\pi$
$8$	159.370			0.311269
$9$	243.000			0.333333
$10$		−	708.028i		−	0.708028i
$11$			998.732i			0.750362i	0.926952	+	0.375181i	$0.122419\pi$
$11$			998.732i			0.750362i	−0.926952	+	0.375181i	$0.877581\pi$
$12$	763.964			0.442109
$13$	−1930.17			−0.878548			−0.439274	−	0.898353i	$-0.644764\pi$
$13$	−1930.17			−0.878548			−0.439274	+	0.898353i	$0.644764\pi$
$14$			3261.92i			1.18875i
$15$			1038.24i			0.307627i
$16$	−4830.72			−1.17937
$17$		−	9460.64i		−	1.92563i	−0.270155	−	0.962817i	$-0.587075\pi$
$17$		−	9460.64i		−	1.92563i	0.270155	−	0.962817i	$-0.412925\pi$
$18$	−2583.22			−0.442939
$19$			9744.85i			1.42074i	0.703829	+	0.710369i	$0.251472\pi$
$19$			9744.85i			1.42074i	−0.703829	+	0.710369i	$0.748528\pi$
$20$			3264.11i			0.408014i
$21$		−	4783.23i		−	0.516491i
$22$		−	10617.1i		−	0.997094i
$23$	12027.6	+	1836.38i	0.988544	+	0.150931i
$23$	12027.6	+	1836.38i	0.988544	+	0.150931i
$24$	2484.33			0.179712
$25$	11189.0			0.716097
$26$	20518.7			1.16743
$27$	3788.00			0.192450
$28$		−	15037.9i		−	0.685036i
$29$	44157.2			1.81054			0.905268	−	0.424840i	$-0.139670\pi$
$29$	44157.2			1.81054			0.905268	+	0.424840i	$0.139670\pi$
$30$		−	11037.1i		−	0.408780i
$31$	12989.0			0.436003			0.218001	−	0.975948i	$-0.430046\pi$
$31$	12989.0			0.436003			0.218001	+	0.975948i	$0.430046\pi$
$32$	41153.4			1.25590
$33$			15568.7i			0.433222i
$34$			100572.i			2.55881i
$35$	20436.8			0.476660
$36$	11909.0			0.255251
$37$		−	93813.5i		−	1.85208i	−0.377422	−	0.926041i	$-0.623189\pi$
$37$		−	93813.5i		−	1.85208i	0.377422	−	0.926041i	$-0.376811\pi$
$38$		−	103593.i		−	1.88790i
$39$	−30088.4			−0.507230
$40$			10614.6i			0.165852i
$41$	35742.5			0.518601			0.259301	−	0.965797i	$-0.416508\pi$
$41$	35742.5			0.518601			0.259301	+	0.965797i	$0.416508\pi$
$42$			50848.3i			0.686322i
$43$		−	674.258i		−	0.00848048i	−0.999991	−	0.00424024i	$-0.998650\pi$
$43$		−	674.258i		−	0.00848048i	0.999991	−	0.00424024i	$-0.00134971\pi$
$44$			48946.1i			0.574593i
$45$			16184.6i			0.177609i
$46$	−127860.	−	19521.7i	−1.31359	−	0.200560i
$47$	102972.			0.991803			0.495901	−	0.868379i	$-0.334838\pi$
$47$	102972.			0.991803			0.495901	+	0.868379i	$0.334838\pi$
$48$	−75303.4			−0.680912
$49$	23495.7			0.199710
$50$	−118945.			−0.951561
$51$		−	147477.i		−	1.11177i
$52$	−94594.3			−0.672752
$53$			185550.i			1.24633i	0.782089	+	0.623167i	$0.214154\pi$
$53$			185550.i			1.24633i	−0.782089	+	0.623167i	$0.785846\pi$
$54$	−40268.4			−0.255731
$55$	−66518.8			−0.399812
$56$		−	48901.7i		−	0.278458i
$57$			151907.i			0.820264i
$58$	−469414.			−2.40587
$59$	76235.0			0.371192			0.185596	−	0.982626i	$-0.440578\pi$
$59$	76235.0			0.371192			0.185596	+	0.982626i	$0.440578\pi$
$60$			50882.4i			0.235567i
$61$			405897.i			1.78824i	0.447824	+	0.894122i	$0.352199\pi$
$61$			405897.i			1.78824i	−0.447824	+	0.894122i	$0.647801\pi$
$62$	−138080.			−0.579368
$63$		−	74563.1i		−	0.298196i
$64$	−128317.			−0.489491
$65$		−	128556.i		−	0.468113i
$66$		−	165504.i		−	0.575672i
$67$		−	414252.i		−	1.37734i	−0.725076	−	0.688669i	$-0.758195\pi$
$67$		−	414252.i		−	1.37734i	0.725076	−	0.688669i	$-0.241805\pi$
$68$		−	463650.i		−	1.47456i
$69$	187492.	+	28626.3i	0.570736	+	0.0871402i
$70$	−217254.			−0.633394
$71$	112686.			0.314844			0.157422	−	0.987531i	$-0.449682\pi$
$71$	112686.			0.314844			0.157422	+	0.987531i	$0.449682\pi$
$72$	38726.9			0.103756
$73$	−120456.			−0.309642			−0.154821	−	0.987943i	$-0.549480\pi$
$73$	−120456.			−0.309642			−0.154821	+	0.987943i	$0.549480\pi$
$74$			997288.i			2.46108i
$75$	174419.			0.413439
$76$			477578.i			1.08794i
$77$	306455.			0.671266
$78$	319856.			0.674016
$79$		−	385722.i		−	0.782335i	−0.920319	−	0.391168i	$-0.872071\pi$
$79$		−	385722.i		−	0.782335i	0.920319	−	0.391168i	$-0.127929\pi$
$80$		−	321741.i		−	0.628401i
$81$	59049.0			0.111111
$82$	−379962.			−0.689126
$83$			130245.i			0.227785i	0.993493	+	0.113893i	$0.0363320\pi$
$83$			130245.i			0.227785i	−0.993493	+	0.113893i	$0.963668\pi$
$84$		−	234418.i		−	0.395505i
$85$	630109.			1.02603
$86$			7167.72i			0.0112690i
$87$	688342.			1.04531
$88$			159168.i			0.233565i
$89$		−	1.03358e6i		−	1.46613i	−0.680156	−	0.733067i	$-0.738088\pi$
$89$		−	1.03358e6i		−	1.46613i	0.680156	−	0.733067i	$-0.261912\pi$
$90$		−	172051.i		−	0.236009i
$91$			592261.i			0.785940i
$92$	589453.	+	89997.8i	0.756982	+	0.115576i
$93$	202478.			0.251726
$94$	−1.09465e6			−1.31792
$95$	−649038.			−0.757006
$96$	641519.			0.725096
$97$			27015.5i			0.0296004i	0.999890	+	0.0148002i	$0.00471122\pi$
$97$			27015.5i			0.0296004i	−0.999890	+	0.0148002i	$0.995289\pi$
$98$	−249772.			−0.265378
$99$			242692.i			0.250121i
$100$	548354.			0.548354
$101$	−662888.			−0.643392			−0.321696	−	0.946843i	$-0.604253\pi$
$101$	−662888.			−0.643392			−0.321696	+	0.946843i	$0.604253\pi$
$102$			1.56776e6i			1.47733i
$103$		−	878837.i		−	0.804260i	−0.915583	−	0.402130i	$-0.868270\pi$
$103$		−	878837.i		−	0.804260i	0.915583	−	0.402130i	$-0.131730\pi$
$104$	−307611.			−0.273465
$105$	318578.			0.275200
$106$		−	1.97250e6i		−	1.65615i
$107$			1.24668e6i			1.01766i	0.860866	+	0.508832i	$0.169922\pi$
$107$			1.24668e6i			1.01766i	−0.860866	+	0.508832i	$0.830078\pi$
$108$	185643.			0.147370
$109$			298907.i			0.230811i	0.993318	+	0.115405i	$0.0368167\pi$
$109$			298907.i			0.230811i	−0.993318	+	0.115405i	$0.963183\pi$
$110$	707130.			0.531278
$111$		−	1.46241e6i		−	1.06930i
$112$			1.48228e6i			1.05506i
$113$			887421.i			0.615027i	0.951544	+	0.307514i	$0.0994970\pi$
$113$			887421.i			0.615027i	−0.951544	+	0.307514i	$0.900503\pi$
$114$		−	1.61485e6i		−	1.08998i
$115$	−122309.	+	801078.i	−0.0804200	+	0.526722i
$116$	2.16407e6			1.38643
$117$	−469031.			−0.292849
$118$	−810419.			−0.493246
$119$	−2.90294e6			−1.72265
$120$			165465.i			0.0957550i
$121$	774095.			0.436957
$122$		−	4.31491e6i		−	2.37625i
$123$	557171.			0.299415
$124$	636567.			0.333871
$125$			1.78590e6i			0.914381i
$126$			792646.i			0.396248i
$127$	−329963.			−0.161085			−0.0805423	−	0.996751i	$-0.525665\pi$
$127$	−329963.			−0.161085			−0.0805423	+	0.996751i	$0.525665\pi$
$128$	−1.26974e6			−0.605459
$129$		−	10510.6i		−	0.00489621i
$130$			1.36661e6i			0.622037i
$131$	3.82865e6			1.70307			0.851533	−	0.524301i	$-0.175673\pi$
$131$	3.82865e6			1.70307			0.851533	+	0.524301i	$0.175673\pi$
$132$			762995.i			0.331742i
$133$	2.99015e6			1.27098
$134$			4.40372e6i			1.83023i
$135$			252293.i			0.102542i
$136$		−	1.50774e6i		−	0.599391i
$137$		−	4.42595e6i		−	1.72125i	−0.509238	−	0.860626i	$-0.670073\pi$
$137$		−	4.42595e6i		−	1.72125i	0.509238	−	0.860626i	$-0.329927\pi$
$138$	−1.99314e6	−	304313.i	−0.758404	−	0.115793i
$139$	1.77040e6			0.659214			0.329607	−	0.944118i	$-0.393084\pi$
$139$	1.77040e6			0.659214			0.329607	+	0.944118i	$0.393084\pi$
$140$	1.00157e6			0.365005
$141$	1.60517e6			0.572618
$142$	−1.19792e6			−0.418371
$143$		−	1.92772e6i		−	0.659229i
$144$	−1.17386e6			−0.393125
$145$			2.94101e6i			0.964701i
$146$	1.28051e6			0.411458
$147$	366262.			0.115303
$148$		−	4.59764e6i		−	1.41824i
$149$			2.44233e6i			0.738323i	0.929365	+	0.369161i	$0.120355\pi$
$149$			2.44233e6i			0.738323i	−0.929365	+	0.369161i	$0.879645\pi$
$150$	−1.85417e6			−0.549384
$151$	−4.93629e6			−1.43374			−0.716869	−	0.697208i	$-0.754425\pi$
$151$	−4.93629e6			−1.43374			−0.716869	+	0.697208i	$0.754425\pi$
$152$			1.55304e6i			0.442233i
$153$		−	2.29894e6i		−	0.641878i
$154$	−3.25778e6			−0.891989
$155$			865107.i			0.232314i
$156$	−1.47458e6			−0.388414
$157$		−	2.26920e6i		−	0.586373i	−0.956055	−	0.293187i	$-0.905284\pi$
$157$		−	2.26920e6i		−	0.586373i	0.956055	−	0.293187i	$-0.0947157\pi$
$158$			4.10043e6i			1.03958i
$159$			2.89244e6i			0.719571i
$160$			2.74095e6i			0.669178i
$161$	563482.	−	3.69060e6i	0.135021	−	0.884341i
$162$	−627722.			−0.147646
$163$	−4.27247e6			−0.986543			−0.493272	−	0.869875i	$-0.664199\pi$
$163$	−4.27247e6			−0.986543			−0.493272	+	0.869875i	$0.664199\pi$
$164$	1.75168e6			0.397121
$165$	−1.03693e6			−0.230832
$166$		−	1.38457e6i		−	0.302685i
$167$	2.94904e6			0.633186			0.316593	−	0.948562i	$-0.397461\pi$
$167$	2.94904e6			0.633186			0.316593	+	0.948562i	$0.397461\pi$
$168$		−	762303.i		−	0.160768i
$169$	−1.10125e6			−0.228153
$170$	−6.69840e6			−1.36340
$171$			2.36800e6i			0.473579i
$172$		−	33044.2i		−	0.00649396i
$173$	−4.97175e6			−0.960221			−0.480110	−	0.877208i	$-0.659403\pi$
$173$	−4.97175e6			−0.960221			−0.480110	+	0.877208i	$0.659403\pi$
$174$	−7.31745e6			−1.38903
$175$		−	3.43328e6i		−	0.640612i
$176$		−	4.82459e6i		−	0.884958i
$177$	1.18839e6			0.214308
$178$			1.09875e7i			1.94822i
$179$	−5.25376e6			−0.916032			−0.458016	−	0.888944i	$-0.651440\pi$
$179$	−5.25376e6			−0.916032			−0.458016	+	0.888944i	$0.651440\pi$
$180$			793179.i			0.136005i
$181$		−	941008.i		−	0.158693i	−0.996847	−	0.0793465i	$-0.974717\pi$
$181$		−	941008.i		−	0.158693i	0.996847	−	0.0793465i	$-0.0252833\pi$
$182$		−	6.29605e6i		−	1.04437i
$183$			6.32731e6i			1.03244i
$184$	1.91684e6	+	292664.i	0.307704	+	0.0469803i
$185$	6.24829e6			0.986838
$186$	−2.15245e6			−0.334498
$187$	9.44864e6			1.44492
$188$	5.04648e6			0.759477
$189$		−	1.16232e6i		−	0.172164i
$190$	6.89962e6			1.00592
$191$			9.08371e6i			1.30366i	0.758367	+	0.651828i	$0.225998\pi$
$191$			9.08371e6i			1.30366i	−0.758367	+	0.651828i	$0.774002\pi$
$192$	−2.00027e6			−0.282608
$193$	−3.73053e6			−0.518918			−0.259459	−	0.965754i	$-0.583544\pi$
$193$	−3.73053e6			−0.518918			−0.259459	+	0.965754i	$0.583544\pi$
$194$		−	287189.i		−	0.0393335i
$195$		−	2.00398e6i		−	0.270265i
$196$	1.15148e6			0.152929
$197$	−7.37802e6			−0.965031			−0.482515	−	0.875887i	$-0.660277\pi$
$197$	−7.37802e6			−0.965031			−0.482515	+	0.875887i	$0.660277\pi$
$198$		−	2.57994e6i		−	0.332365i
$199$		−	797445.i		−	0.101191i	−0.998719	−	0.0505955i	$-0.983888\pi$
$199$		−	797445.i		−	0.101191i	0.998719	−	0.0505955i	$-0.0161119\pi$
$200$	1.78319e6			0.222899
$201$		−	6.45755e6i		−	0.795206i
$202$	7.04685e6			0.854951
$203$		−	1.35494e7i		−	1.61969i
$204$		−	7.22758e6i		−	0.851339i
$205$			2.38057e6i			0.276324i
$206$			9.34251e6i			1.06871i
$207$	2.92271e6	+	446240.i	0.329515	+	0.0503104i
$208$	9.32411e6			1.03614
$209$	−9.73249e6			−1.06607
$210$	−3.38666e6			−0.365690
$211$	8.84501e6			0.941566			0.470783	−	0.882249i	$-0.343971\pi$
$211$	8.84501e6			0.941566			0.470783	+	0.882249i	$0.343971\pi$
$212$			9.09351e6i			0.954385i
$213$	1.75660e6			0.181775
$214$		−	1.32529e7i		−	1.35229i
$215$	44907.7			0.00451862
$216$	603693.			0.0599038
$217$		−	3.98559e6i		−	0.390043i
$218$		−	3.17754e6i		−	0.306705i
$219$	−1.87773e6			−0.178772
$220$	−3.25997e6			−0.306158
$221$			1.82606e7i			1.69176i
$222$			1.55462e7i			1.42090i
$223$	1.97755e7			1.78325			0.891626	−	0.452773i	$-0.149565\pi$
$223$	1.97755e7			1.78325			0.891626	+	0.452773i	$0.149565\pi$
$224$		−	1.26277e7i		−	1.12352i
$225$	2.71893e6			0.238699
$226$		−	9.43376e6i		−	0.817259i
$227$			1.08616e7i			0.928570i	0.885686	+	0.464285i	$0.153689\pi$
$227$			1.08616e7i			0.928570i	−0.885686	+	0.464285i	$0.846311\pi$
$228$			7.44471e6i			0.628121i
$229$		−	7.53625e6i		−	0.627551i	−0.949497	−	0.313775i	$-0.898406\pi$
$229$		−	7.53625e6i		−	0.627551i	0.949497	−	0.313775i	$-0.101594\pi$
$230$	1.30021e6	−	8.51589e6i	0.106864	−	0.699917i
$231$	4.77716e6			0.387556
$232$	7.03733e6			0.563565
$233$	1.21256e7			0.958597			0.479299	−	0.877652i	$-0.340891\pi$
$233$	1.21256e7			0.958597			0.479299	+	0.877652i	$0.340891\pi$
$234$	4.98605e6			0.389143
$235$			6.85826e6i			0.528458i
$236$	3.73615e6			0.284242
$237$		−	6.01281e6i		−	0.451681i
$238$	3.08598e7			2.28909
$239$	−1.89545e7			−1.38841			−0.694207	−	0.719776i	$-0.744245\pi$
$239$	−1.89545e7			−1.38841			−0.694207	+	0.719776i	$0.744245\pi$
$240$		−	5.01545e6i		−	0.362808i
$241$			1.43348e7i			1.02409i	0.858957	+	0.512047i	$0.171113\pi$
$241$			1.43348e7i			1.02409i	−0.858957	+	0.512047i	$0.828887\pi$
$242$	−8.22905e6			−0.580635
$243$	920483.			0.0641500
$244$			1.98923e7i			1.36936i
$245$			1.56489e6i			0.106411i
$246$	−5.92303e6			−0.397867
$247$		−	1.88092e7i		−	1.24819i
$248$	2.07005e6			0.135714
$249$			2.03031e6i			0.131512i
$250$		−	1.89851e7i		−	1.21504i
$251$		−	1.74065e7i		−	1.10076i	−0.834916	−	0.550378i	$-0.814484\pi$
$251$		−	1.74065e7i		−	1.10076i	0.834916	−	0.550378i	$-0.185516\pi$
$252$		−	3.65421e6i		−	0.228345i
$253$	−1.83405e6	+	1.20124e7i	−0.113253	+	0.741766i
$254$	3.50768e6			0.214052
$255$	9.82243e6			0.592377
$256$	2.17103e7			1.29404
$257$	3.35748e6			0.197795			0.0988973	−	0.995098i	$-0.468468\pi$
$257$	3.35748e6			0.197795			0.0988973	+	0.995098i	$0.468468\pi$
$258$			111734.i			0.00650616i
$259$	−2.87861e7			−1.65685
$260$		−	6.30029e6i		−	0.358460i
$261$	1.07302e7			0.603512
$262$	−4.07006e7			−2.26306
$263$		−	7.81608e6i		−	0.429657i	−0.976652	−	0.214829i	$-0.931081\pi$
$263$		−	7.81608e6i		−	0.429657i	0.976652	−	0.214829i	$-0.0689193\pi$
$264$			2.48118e6i			0.134849i
$265$	−1.23583e7			−0.664079
$266$	−3.17869e7			−1.68890
$267$		−	1.61119e7i		−	0.846473i
$268$		−	2.03018e7i		−	1.05470i
$269$	−3.72974e6			−0.191611			−0.0958057	−	0.995400i	$-0.530543\pi$
$269$	−3.72974e6			−0.191611			−0.0958057	+	0.995400i	$0.530543\pi$
$270$		−	2.68201e6i		−	0.136260i
$271$	−2.64993e7			−1.33146			−0.665729	−	0.746194i	$-0.731879\pi$
$271$	−2.64993e7			−1.33146			−0.665729	+	0.746194i	$0.731879\pi$
$272$			4.57017e7i			2.27104i
$273$			9.23244e6i			0.453762i
$274$			4.70502e7i			2.28723i
$275$			1.11748e7i			0.537332i
$276$	9.18866e6	+	1.40293e6i	0.437044	+	0.0667280i
$277$	−1.77487e7			−0.835078			−0.417539	−	0.908659i	$-0.637107\pi$
$277$	−1.77487e7			−0.835078			−0.417539	+	0.908659i	$0.637107\pi$
$278$	−1.88203e7			−0.875975
$279$	3.15632e6			0.145334
$280$	3.25701e6			0.148370
$281$		−	2.16898e7i		−	0.977545i	−0.872411	−	0.488773i	$-0.837445\pi$
$281$		−	2.16898e7i		−	0.977545i	0.872411	−	0.488773i	$-0.162555\pi$
$282$	−1.70639e7			−0.760904
$283$			1.79591e7i			0.792367i	0.918171	+	0.396183i	$0.129666\pi$
$283$			1.79591e7i			0.792367i	−0.918171	+	0.396183i	$0.870334\pi$
$284$	5.52256e6			0.241093
$285$	−1.01175e7			−0.437058
$286$			2.04927e7i			0.875995i
$287$		−	1.09674e7i		−	0.463935i
$288$	1.00003e7			0.418634
$289$	−6.53661e7			−2.70807
$290$		−	3.12645e7i		−	1.28191i
$291$			421129.i			0.0170898i
$292$	−5.90335e6			−0.237110
$293$			3.51041e7i			1.39558i	0.716302	+	0.697791i	$0.245833\pi$
$293$			3.51041e7i			1.39558i	−0.716302	+	0.697791i	$0.754167\pi$
$294$	−3.89356e6			−0.153216
$295$			5.07750e6i			0.197781i
$296$		−	1.49511e7i		−	0.576497i
$297$			3.78319e6i			0.144407i
$298$		−	2.59633e7i		−	0.981096i
$299$	−2.32153e7	−	3.54453e6i	−0.868484	−	0.132600i
$300$	8.54799e6			0.316592
$301$	−206892.			−0.00758655
$302$	5.24754e7			1.90517
$303$	−1.03334e7			−0.371463
$304$		−	4.70746e7i		−	1.67558i
$305$	−2.70341e7			−0.952823
$306$			2.44389e7i			0.852938i
$307$	3.31565e7			1.14592			0.572959	−	0.819584i	$-0.305795\pi$
$307$	3.31565e7			1.14592			0.572959	+	0.819584i	$0.305795\pi$
$308$	1.50188e7			0.514025
$309$		−	1.36997e7i		−	0.464340i
$310$		−	9.19655e6i		−	0.308702i
$311$	4.38466e7			1.45766			0.728828	−	0.684697i	$-0.240066\pi$
$311$	4.38466e7			1.45766			0.728828	+	0.684697i	$0.240066\pi$
$312$	−4.79518e6			−0.157885
$313$		−	2.64824e6i		−	0.0863623i	−0.999067	−	0.0431811i	$-0.986251\pi$
$313$		−	2.64824e6i		−	0.0863623i	0.999067	−	0.0431811i	$-0.0137493\pi$
$314$			2.41228e7i			0.779183i
$315$	4.96614e6			0.158887
$316$		−	1.89036e7i		−	0.599077i
$317$	−1.65607e7			−0.519876			−0.259938	−	0.965625i	$-0.583702\pi$
$317$	−1.65607e7			−0.519876			−0.259938	+	0.965625i	$0.583702\pi$
$318$		−	3.07482e7i		−	0.956178i
$319$			4.41012e7i			1.35856i
$320$		−	8.54634e6i		−	0.260814i
$321$			1.94338e7i			0.587548i
$322$	−5.99012e6	+	3.92331e7i	−0.179419	+	1.17513i
$323$	9.21925e7			2.73582
$324$	2.89389e6			0.0850838
$325$	−2.15967e7			−0.629125
$326$	4.54186e7			1.31094
$327$			4.65949e6i			0.133259i
$328$	5.69629e6			0.161425
$329$		−	3.15963e7i		−	0.887256i
$330$	1.10231e7			0.306733
$331$	−4.63044e7			−1.27685			−0.638423	−	0.769686i	$-0.720413\pi$
$331$	−4.63044e7			−1.27685			−0.638423	+	0.769686i	$0.720413\pi$
$332$			6.38307e6i			0.174428i
$333$		−	2.27967e7i		−	0.617361i
$334$	−3.13499e7			−0.841388
$335$	2.75905e7			0.733881
$336$			2.31064e7i			0.609137i
$337$		−	5.12065e7i		−	1.33793i	−0.743292	−	0.668967i	$-0.766737\pi$
$337$		−	5.12065e7i		−	1.33793i	0.743292	−	0.668967i	$-0.233263\pi$
$338$	1.17069e7			0.303174
$339$			1.38335e7i			0.355086i
$340$	3.08806e7			0.785685
$341$			1.29725e7i			0.327160i
$342$		−	2.51731e7i		−	0.629300i
$343$		−	4.33094e7i		−	1.07325i
$344$		−	107456.i		−	0.00263971i
$345$	−1.90661e6	+	1.24876e7i	−0.0464305	+	0.304103i
$346$	5.28524e7			1.27596
$347$	4.88565e6			0.116932			0.0584661	−	0.998289i	$-0.481379\pi$
$347$	4.88565e6			0.116932			0.0584661	+	0.998289i	$0.481379\pi$
$348$	3.37345e7			0.800454
$349$	506526.			0.0119159			0.00595794	−	0.999982i	$-0.498104\pi$
$349$	506526.			0.0119159			0.00595794	+	0.999982i	$0.498104\pi$
$350$			3.64976e7i			0.851256i
$351$	−7.31147e6			−0.169077
$352$			4.11013e7i			0.942382i
$353$	7.37921e7			1.67759			0.838794	−	0.544448i	$-0.183261\pi$
$353$	7.37921e7			1.67759			0.838794	+	0.544448i	$0.183261\pi$
$354$	−1.26332e7			−0.284776
$355$			7.50527e6i			0.167757i
$356$		−	5.06540e7i		−	1.12270i
$357$	−4.52524e7			−0.994573
$358$	5.58502e7			1.21724
$359$			2.09420e7i			0.452621i	0.974055	+	0.226310i	$0.0726663\pi$
$359$			2.09420e7i			0.452621i	−0.974055	+	0.226310i	$0.927334\pi$
$360$			2.57934e6i			0.0552841i
$361$	−4.79161e7			−1.01850
$362$			1.00034e7i			0.210874i
$363$	1.20669e7			0.252277
$364$			2.90257e7i			0.601837i
$365$		−	8.02277e6i		−	0.164985i
$366$		−	6.72627e7i		−	1.37193i
$367$			4.05559e6i			0.0820458i	0.999158	+	0.0410229i	$0.0130617\pi$
$367$			4.05559e6i			0.0820458i	−0.999158	+	0.0410229i	$0.986938\pi$
$368$	−5.81020e7	−	8.87103e6i	−1.16586	−	0.178004i
$369$	8.68544e6			0.172867
$370$	−6.64226e7			−1.31133
$371$	5.69350e7			1.11496
$372$	9.92309e6			0.192761
$373$			7.87774e7i			1.51801i	0.651084	+	0.759006i	$0.274315\pi$
$373$			7.87774e7i			1.51801i	−0.651084	+	0.759006i	$0.725685\pi$
$374$	−1.00444e8			−1.92004
$375$			2.78394e7i			0.527918i
$376$	1.64106e7			0.308718
$377$	−8.52309e7			−1.59064
$378$			1.23561e7i			0.228774i
$379$			4.61194e7i			0.847160i	0.905859	+	0.423580i	$0.139227\pi$
$379$			4.61194e7i			0.847160i	−0.905859	+	0.423580i	$0.860773\pi$
$380$	−3.18082e7			−0.579681
$381$	−5.14361e6			−0.0930022
$382$		−	9.65647e7i		−	1.73232i
$383$		−	8.48875e7i		−	1.51094i	−0.655182	−	0.755471i	$-0.727408\pi$
$383$		−	8.48875e7i		−	1.51094i	0.655182	−	0.755471i	$-0.272592\pi$
$384$	−1.97933e7			−0.349562
$385$			2.04109e7i			0.357668i
$386$	3.96575e7			0.689547
$387$		−	163845.i		−	0.00282683i
$388$			1.32398e6i			0.0226666i
$389$			9.12890e6i			0.155085i	0.996989	+	0.0775425i	$0.0247073\pi$
$389$			9.12890e6i			0.155085i	−0.996989	+	0.0775425i	$0.975293\pi$
$390$			2.13034e7i			0.359133i
$391$	1.73733e7	−	1.13789e8i	0.290638	−	1.90357i
$392$	3.74451e6			0.0621637
$393$	5.96827e7			0.983266
$394$	7.84323e7			1.28235
$395$	2.56903e7			0.416848
$396$			1.18939e7i			0.191531i
$397$	−1.10456e8			−1.76529			−0.882647	−	0.470036i	$-0.844241\pi$
$397$	−1.10456e8			−1.76529			−0.882647	+	0.470036i	$0.844241\pi$
$398$			8.47727e6i			0.134464i
$399$	4.66118e7			0.733799
$400$	−5.40510e7			−0.844546
$401$			6.83433e7i			1.05990i	0.848030	+	0.529948i	$0.177788\pi$
$401$			6.83433e7i			1.05990i	−0.848030	+	0.529948i	$0.822212\pi$
$402$			6.86472e7i			1.05668i
$403$	−2.50709e7			−0.383049
$404$	−3.24870e7			−0.492681
$405$			3.93285e6i			0.0592029i
$406$			1.44037e8i			2.15227i
$407$	9.36946e7			1.38973
$408$		−	2.35034e7i		−	0.346059i
$409$	4.14001e7			0.605105			0.302553	−	0.953133i	$-0.402161\pi$
$409$	4.14001e7			0.605105			0.302553	+	0.953133i	$0.402161\pi$
$410$		−	2.53067e7i		−	0.367184i
$411$		−	6.89937e7i		−	0.993765i
$412$		−	4.30703e7i		−	0.615866i
$413$		−	2.33923e7i		−	0.332064i
$414$	−3.10700e7	−	4.74377e6i	−0.437865	−	0.0668533i
$415$	−8.67472e6			−0.121370
$416$	−7.94331e7			−1.10337
$417$	2.75978e7			0.380598
$418$	1.03462e8			1.41661
$419$			2.59180e7i			0.352337i	0.984360	+	0.176169i	$0.0563704\pi$
$419$			2.59180e7i			0.352337i	−0.984360	+	0.176169i	$0.943630\pi$
$420$	1.56130e7			0.210736
$421$		−	6.48650e7i		−	0.869289i	−0.900602	−	0.434645i	$-0.856874\pi$
$421$		−	6.48650e7i		−	0.869289i	0.900602	−	0.434645i	$-0.143126\pi$
$422$	−9.40272e7			−1.25117
$423$	2.50222e7			0.330601
$424$			2.95712e7i			0.387946i
$425$		−	1.05855e8i		−	1.37894i
$426$	−1.86737e7			−0.241546
$427$	1.24547e8			1.59974
$428$			6.10977e7i			0.779280i
$429$		−	3.00502e7i		−	0.380606i
$430$	−477393.			−0.00600442
$431$		−	4.89259e6i		−	0.0611092i	−0.999533	−	0.0305546i	$-0.990273\pi$
$431$		−	4.89259e6i		−	0.0611092i	0.999533	−	0.0305546i	$-0.00972735\pi$
$432$	−1.82987e7			−0.226971
$433$			1.91942e7i			0.236432i	0.992988	+	0.118216i	$0.0377176\pi$
$433$			1.91942e7i			0.236432i	−0.992988	+	0.118216i	$0.962282\pi$
$434$			4.23689e7i			0.518296i
$435$			4.58458e7i			0.556970i
$436$			1.46489e7i			0.176744i
$437$	−1.78952e7	+	1.17207e8i	−0.214434	+	1.40446i
$438$	1.99612e7			0.237555
$439$	4.92575e7			0.582208			0.291104	−	0.956691i	$-0.405977\pi$
$439$	4.92575e7			0.582208			0.291104	+	0.956691i	$0.405977\pi$
$440$	−1.06011e7			−0.124449
$441$	5.70946e6			0.0665701
$442$		−	1.94120e8i		−	2.24804i
$443$	2.46468e6			0.0283498			0.0141749	−	0.999900i	$-0.495488\pi$
$443$	2.46468e6			0.0283498			0.0141749	+	0.999900i	$0.495488\pi$
$444$		−	7.16701e7i		−	0.818822i
$445$	6.88398e7			0.781195
$446$	−2.10224e8			−2.36962
$447$			3.80722e7i			0.426271i
$448$			3.93734e7i			0.437893i
$449$	4.30400e7			0.475481			0.237741	−	0.971329i	$-0.423593\pi$
$449$	4.30400e7			0.475481			0.237741	+	0.971329i	$0.423593\pi$
$450$	−2.89037e7			−0.317187
$451$			3.56972e7i			0.389139i
$452$			4.34910e7i			0.470960i
$453$	−7.69491e7			−0.827769
$454$		−	1.15464e8i		−	1.23390i
$455$	−3.94465e7			−0.418769
$456$			2.42094e7i			0.255323i
$457$			2.02402e7i			0.212063i	0.994363	+	0.106032i	$0.0338145\pi$
$457$			2.02402e7i			0.212063i	−0.994363	+	0.106032i	$0.966186\pi$
$458$			8.01144e7i			0.833900i
$459$		−	3.58369e7i		−	0.370588i
$460$	−5.99415e6	+	3.92595e7i	−0.0615820	+	0.403340i
$461$	1.33765e8			1.36534			0.682668	−	0.730729i	$-0.260820\pi$
$461$	1.33765e8			1.36534			0.682668	+	0.730729i	$0.260820\pi$
$462$	−5.07838e7			−0.514990
$463$	1.37084e8			1.38116			0.690578	−	0.723258i	$-0.257356\pi$
$463$	1.37084e8			1.38116			0.690578	+	0.723258i	$0.257356\pi$
$464$	−2.13311e8			−2.13530
$465$			1.34857e7i			0.134126i
$466$	−1.28902e8			−1.27380
$467$			3.01091e7i			0.295629i	0.989015	+	0.147815i	$0.0472239\pi$
$467$			3.01091e7i			0.295629i	−0.989015	+	0.147815i	$0.952776\pi$
$468$	−2.29864e7			−0.224251
$469$	−1.27111e8			−1.23215
$470$		−	7.29070e7i		−	0.702224i
$471$		−	3.53734e7i		−	0.338543i
$472$	1.21496e7			0.115541
$473$	673403.			0.00636343
$474$			6.39194e7i			0.600202i
$475$			1.09035e8i			1.01739i
$476$	−1.42268e8			−1.31913
$477$			4.50887e7i			0.415444i
$478$	2.01497e8			1.84495
$479$		−	1.64681e8i		−	1.49843i	−0.662326	−	0.749216i	$-0.730430\pi$
$479$		−	1.64681e8i		−	1.49843i	0.662326	−	0.749216i	$-0.269570\pi$
$480$			4.27272e7i			0.386350i
$481$			1.81076e8i			1.62714i
$482$		−	1.52386e8i		−	1.36083i
$483$	8.78382e6	−	5.75308e7i	0.0779546	−	0.510575i
$484$	3.79371e7			0.334601
$485$	−1.79932e6			−0.0157718
$486$	−9.78523e6			−0.0852436
$487$	3.34119e7			0.289277			0.144638	−	0.989485i	$-0.453798\pi$
$487$	3.34119e7			0.289277			0.144638	+	0.989485i	$0.453798\pi$
$488$			6.46879e7i			0.556626i
$489$	−6.66012e7			−0.569581
$490$		−	1.66356e7i		−	0.141400i
$491$	−5.10915e7			−0.431622			−0.215811	−	0.976435i	$-0.569240\pi$
$491$	−5.10915e7			−0.431622			−0.215811	+	0.976435i	$0.569240\pi$
$492$	2.73060e7			0.229278
$493$		−	4.17755e8i		−	3.48643i
$494$			1.99952e8i			1.65861i
$495$	−1.61641e7			−0.133271
$496$	−6.27460e7			−0.514211
$497$		−	3.45771e7i		−	0.281656i
$498$		−	2.15833e7i		−	0.174755i
$499$	4.60487e7			0.370609			0.185305	−	0.982681i	$-0.440673\pi$
$499$	4.60487e7			0.370609			0.185305	+	0.982681i	$0.440673\pi$
$500$			8.75239e7i			0.700191i
$501$	4.59710e7			0.365570
$502$			1.85041e8i			1.46270i
$503$		−	2.04567e7i		−	0.160743i	−0.996765	−	0.0803715i	$-0.974389\pi$
$503$		−	2.04567e7i		−	0.160743i	0.996765	−	0.0803715i	$-0.0256107\pi$
$504$		−	1.18831e7i		−	0.0928194i
$505$		−	4.41505e7i		−	0.342816i
$506$	1.94969e7	−	1.27698e8i	0.150493	−	0.985672i
$507$	−1.71668e7			−0.131724
$508$	−1.61709e7			−0.123351
$509$	−6.63534e7			−0.503165			−0.251582	−	0.967836i	$-0.580951\pi$
$509$	−6.63534e7			−0.503165			−0.251582	+	0.967836i	$0.580951\pi$
$510$	−1.04418e8			−0.787161
$511$			3.69613e7i			0.277003i
$512$	−1.49529e8			−1.11408
$513$			3.69134e7i			0.273421i
$514$	−3.56918e7			−0.262833
$515$	5.85334e7			0.428531
$516$		−	515108.i		−	0.00374929i
$517$			1.02841e8i			0.744211i
$518$	3.06012e8			2.20165
$519$	−7.75019e7			−0.554384
$520$		−	2.04879e7i		−	0.145709i
$521$			1.38712e8i			0.980849i	0.871484	+	0.490424i	$0.163158\pi$
$521$			1.38712e8i			0.980849i	−0.871484	+	0.490424i	$0.836842\pi$
$522$	−1.14068e8			−0.801957
$523$			1.00444e8i			0.702133i	0.936350	+	0.351067i	$0.114181\pi$
$523$			1.00444e8i			0.702133i	−0.936350	+	0.351067i	$0.885819\pi$
$524$	1.87635e8			1.30413
$525$		−	5.35196e7i		−	0.369858i
$526$			8.30892e7i			0.570936i
$527$		−	1.22884e8i		−	0.839582i
$528$		−	7.52080e7i		−	0.510931i
$529$	1.41291e8	+	4.41745e7i	0.954440	+	0.298404i
$530$	1.31375e8			0.882439
$531$	1.85251e7			0.123731
$532$	1.46542e8			0.973256
$533$	−6.89892e7			−0.455616
$534$			1.71278e8i			1.12481i
$535$	−8.30330e7			−0.542237
$536$		−	6.60194e7i		−	0.428723i
$537$	−8.18980e7			−0.528872
$538$	3.96491e7			0.254617
$539$			2.34659e7i			0.149855i
$540$			1.23644e7i			0.0785223i
$541$	−1.94692e8			−1.22958			−0.614790	−	0.788691i	$-0.710759\pi$
$541$	−1.94692e8			−1.22958			−0.614790	+	0.788691i	$0.710759\pi$
$542$	2.81702e8			1.76926
$543$		−	1.46689e7i		−	0.0916214i
$544$		−	3.89338e8i		−	2.41841i
$545$	−1.99081e7			−0.122982
$546$		−	9.81458e7i		−	0.602967i
$547$	1.93151e8			1.18015			0.590073	−	0.807350i	$-0.299099\pi$
$547$	1.93151e8			1.18015			0.590073	+	0.807350i	$0.299099\pi$
$548$		−	2.16908e8i		−	1.31806i
$549$			9.86331e7i			0.596081i
$550$		−	1.18794e8i		−	0.714016i
$551$			4.30305e8i			2.57230i
$552$	2.98806e7	+	4.56218e6i	0.177653	+	0.0271241i
$553$	−1.18356e8			−0.699869
$554$	1.88678e8			1.10967
$555$	9.74011e7			0.569751
$556$	8.67642e7			0.504796
$557$		−	2.74291e8i		−	1.58725i	−0.608407	−	0.793625i	$-0.708191\pi$
$557$		−	2.74291e8i		−	1.58725i	0.608407	−	0.793625i	$-0.291809\pi$
$558$	−3.35533e7			−0.193123
$559$			1.30143e6i			0.00745051i
$560$	−9.87245e7			−0.562161
$561$	1.47290e8			0.834227
$562$			2.30574e8i			1.29898i
$563$			1.27056e8i			0.711982i	0.934489	+	0.355991i	$0.115857\pi$
$563$			1.27056e8i			0.711982i	−0.934489	+	0.355991i	$0.884143\pi$
$564$	7.86668e7			0.438484
$565$	−5.91051e7			−0.327702
$566$		−	1.90915e8i		−	1.05291i
$567$		−	1.81188e7i		−	0.0993988i
$568$	1.79588e7			0.0980014
$569$		−	9.70750e7i		−	0.526952i	−0.964666	−	0.263476i	$-0.915131\pi$
$569$		−	9.70750e7i		−	0.526952i	0.964666	−	0.263476i	$-0.0848689\pi$
$570$	1.07554e8			0.580770
$571$		−	2.66556e8i		−	1.43179i	−0.698208	−	0.715895i	$-0.746019\pi$
$571$		−	2.66556e8i		−	1.43179i	0.698208	−	0.715895i	$-0.253981\pi$
$572$		−	9.44744e7i		−	0.504808i
$573$			1.41601e8i			0.752666i
$574$			1.16589e8i			0.616485i
$575$	1.34577e8	+	2.05473e7i	0.707893	+	0.108081i
$576$	−3.11811e7			−0.163164
$577$	−5.10701e7			−0.265851			−0.132926	−	0.991126i	$-0.542437\pi$
$577$	−5.10701e7			−0.265851			−0.132926	+	0.991126i	$0.542437\pi$
$578$	6.94877e8			3.59852
$579$	−5.81532e7			−0.299597
$580$			1.44134e8i			0.738724i
$581$	3.99648e7			0.203774
$582$		−	4.47683e6i		−	0.0227092i
$583$	−1.85315e8			−0.935201
$584$	−1.91971e7			−0.0963822
$585$		−	3.12390e7i		−	0.156038i
$586$		−	3.73175e8i		−	1.85447i
$587$	8.57660e7			0.424034			0.212017	−	0.977266i	$-0.431997\pi$
$587$	8.57660e7			0.424034			0.212017	+	0.977266i	$0.431997\pi$
$588$	1.79499e7			0.0882936
$589$			1.26575e8i			0.619446i
$590$		−	5.39765e7i		−	0.262814i
$591$	−1.15012e8			−0.557161
$592$			4.53187e8i			2.18430i
$593$	−1.07927e8			−0.517565			−0.258783	−	0.965936i	$-0.583321\pi$
$593$	−1.07927e8			−0.517565			−0.258783	+	0.965936i	$0.583321\pi$
$594$		−	4.02174e7i		−	0.191891i
$595$		−	1.93345e8i		−	0.917873i
$596$			1.19695e8i			0.565374i
$597$		−	1.24309e7i		−	0.0584226i
$598$	2.46792e8	+	3.76802e7i	1.15406	+	0.176202i
$599$	1.07396e8			0.499698			0.249849	−	0.968285i	$-0.419619\pi$
$599$	1.07396e8			0.499698			0.249849	+	0.968285i	$0.419619\pi$
$600$	2.77972e7			0.128691
$601$	−2.63192e8			−1.21241			−0.606204	−	0.795309i	$-0.707308\pi$
$601$	−2.63192e8			−1.21241			−0.606204	+	0.795309i	$0.707308\pi$
$602$	2.19937e6			0.0100811
$603$		−	1.00663e8i		−	0.459113i
$604$	−2.41919e8			−1.09789
$605$			5.15572e7i			0.232822i
$606$	1.09850e8			0.493606
$607$	−2.19680e8			−0.982255			−0.491128	−	0.871088i	$-0.663415\pi$
$607$	−2.19680e8			−0.982255			−0.491128	+	0.871088i	$0.663415\pi$
$608$			4.01034e8i			1.78431i
$609$		−	2.11214e8i		−	0.935126i
$610$	2.87387e8			1.26613
$611$	−1.98753e8			−0.871346
$612$		−	1.12667e8i		−	0.491521i
$613$			2.33622e8i			1.01422i	0.861881	+	0.507110i	$0.169286\pi$
$613$			2.33622e8i			1.01422i	−0.861881	+	0.507110i	$0.830714\pi$
$614$	−3.52471e8			−1.52272
$615$			3.71094e7i			0.159536i
$616$	4.88397e7			0.208945
$617$			2.56113e8i			1.09038i	0.838313	+	0.545189i	$0.183542\pi$
$617$			2.56113e8i			1.09038i	−0.838313	+	0.545189i	$0.816458\pi$
$618$			1.45635e8i			0.617023i
$619$		−	2.28763e8i		−	0.964524i	−0.876027	−	0.482262i	$-0.839815\pi$
$619$		−	2.28763e8i		−	0.964524i	0.876027	−	0.482262i	$-0.160185\pi$
$620$			4.23974e7i			0.177895i
$621$	4.55606e7	+	6.95620e6i	0.190245	+	0.0290467i
$622$	−4.66113e8			−1.93696
$623$	−3.17148e8			−1.31159
$624$	1.45348e8			0.598214
$625$	5.58816e7			0.228891
$626$			2.81522e7i			0.114760i
$627$	−1.51715e8			−0.615495
$628$		−	1.11210e8i		−	0.449018i
$629$	−8.87536e8			−3.56643
$630$	−5.27928e7			−0.211131
$631$			2.85720e8i			1.13724i	0.822600	+	0.568621i	$0.192523\pi$
$631$			2.85720e8i			1.13724i	−0.822600	+	0.568621i	$0.807477\pi$
$632$		−	6.14725e7i		−	0.243517i
$633$	1.37880e8			0.543613
$634$	1.76049e8			0.690820
$635$		−	2.19766e7i		−	0.0858300i
$636$			1.41754e8i			0.551015i
$637$	−4.53507e7			−0.175455
$638$		−	4.68819e8i		−	1.80528i
$639$	2.73828e7			0.104948
$640$		−	8.45688e7i		−	0.322604i
$641$		−	8.85056e7i		−	0.336044i	−0.985783	−	0.168022i	$-0.946262\pi$
$641$		−	8.85056e7i		−	0.336044i	0.985783	−	0.168022i	$-0.0537380\pi$
$642$		−	2.06592e8i		−	0.780744i
$643$		−	2.30481e8i		−	0.866966i	−0.901162	−	0.433483i	$-0.857284\pi$
$643$		−	2.30481e8i		−	0.866966i	0.901162	−	0.433483i	$-0.142716\pi$
$644$	2.76153e7	−	1.80870e8i	0.103393	−	0.677188i
$645$	700042.			0.00260883
$646$	−9.80055e8			−3.63541
$647$	−1.65118e6			−0.00609652			−0.00304826	−	0.999995i	$-0.500970\pi$
$647$	−1.65118e6			−0.00609652			−0.00304826	+	0.999995i	$0.500970\pi$
$648$	9.41064e6			0.0345855
$649$			7.61384e7i			0.278528i
$650$	2.29584e8			0.835992
$651$		−	6.21291e7i		−	0.225192i
$652$	−2.09386e8			−0.755450
$653$	−1.94390e8			−0.698128			−0.349064	−	0.937099i	$-0.613500\pi$
$653$	−1.94390e8			−0.698128			−0.349064	+	0.937099i	$0.613500\pi$
$654$		−	4.95329e7i		−	0.177076i
$655$			2.55000e8i			0.907438i
$656$	−1.72662e8			−0.611625
$657$	−2.92708e7			−0.103214
$658$			3.35886e8i			1.17900i
$659$		−	1.16673e8i		−	0.407677i	−0.979005	−	0.203838i	$-0.934658\pi$
$659$		−	1.16673e8i		−	0.407677i	0.979005	−	0.203838i	$-0.0653417\pi$
$660$	−5.08179e7			−0.176760
$661$		−	1.78775e8i		−	0.619018i	−0.950896	−	0.309509i	$-0.899835\pi$
$661$		−	1.78775e8i		−	0.619018i	0.950896	−	0.309509i	$-0.100165\pi$
$662$	4.92240e8			1.69669
$663$			2.84655e8i			0.976739i
$664$			2.07571e7i			0.0709026i
$665$			1.99154e8i			0.677210i
$666$			2.42341e8i			0.820360i
$667$	5.31106e8	+	8.10894e7i	1.78980	+	0.273266i
$668$	1.44527e8			0.484865
$669$	3.08269e8			1.02956
$670$	−2.93302e8			−0.975194
$671$	−4.05383e8			−1.34183
$672$		−	1.96846e8i		−	0.648663i
$673$	2.82390e8			0.926413			0.463206	−	0.886251i	$-0.346699\pi$
$673$	2.82390e8			0.926413			0.463206	+	0.886251i	$0.346699\pi$
$674$			5.44352e8i			1.77787i
$675$	4.23839e7			0.137813
$676$	−5.39705e7			−0.174709
$677$		−	4.01139e8i		−	1.29279i	−0.763002	−	0.646396i	$-0.776275\pi$
$677$		−	4.01139e8i		−	1.29279i	0.763002	−	0.646396i	$-0.223725\pi$
$678$		−	1.47058e8i		−	0.471844i
$679$	8.28953e6			0.0264802
$680$	1.00420e8			0.319371
$681$			1.69315e8i			0.536110i
$682$		−	1.37905e8i		−	0.434736i
$683$	−4.05428e8			−1.27248			−0.636241	−	0.771490i	$-0.719512\pi$
$683$	−4.05428e8			−1.27248			−0.636241	+	0.771490i	$0.719512\pi$
$684$			1.16051e8i			0.362646i
$685$	2.94782e8			0.917127
$686$			4.60402e8i			1.42615i
$687$		−	1.17479e8i		−	0.362317i
$688$			3.25715e6i			0.0100017i
$689$		−	3.58144e8i		−	1.09496i
$690$	2.02682e7	−	1.32750e8i	0.0616977	−	0.404097i
$691$	−1.27788e8			−0.387306			−0.193653	−	0.981070i	$-0.562034\pi$
$691$	−1.27788e8			−0.387306			−0.193653	+	0.981070i	$0.562034\pi$
$692$	−2.43657e8			−0.735293
$693$	7.44686e7			0.223755
$694$	−5.19371e7			−0.155381
$695$			1.17914e8i			0.351247i
$696$	1.09701e8			0.325374
$697$		−	3.38147e8i		−	0.998637i
$698$	−5.38465e6			−0.0158340
$699$	1.89020e8			0.553446
$700$		−	1.68259e8i		−	0.490552i
$701$			1.37425e8i			0.398943i	0.979904	+	0.199472i	$0.0639226\pi$
$701$			1.37425e8i			0.398943i	−0.979904	+	0.199472i	$0.936077\pi$
$702$	7.77249e7			0.224672
$703$	9.14198e8			2.63133
$704$		−	1.28154e8i		−	0.367296i
$705$			1.06910e8i			0.305105i
$706$	−7.84449e8			−2.22921
$707$			2.03403e8i			0.575572i
$708$	5.82408e7			0.164107
$709$			2.82804e8i			0.793501i	0.917926	+	0.396751i	$0.129862\pi$
$709$			2.82804e8i			0.793501i	−0.917926	+	0.396751i	$0.870138\pi$
$710$		−	7.97850e7i		−	0.222919i
$711$		−	9.37304e7i		−	0.260778i
$712$		−	1.64722e8i		−	0.456363i
$713$	1.56226e8	+	2.38527e7i	0.431008	+	0.0658064i
$714$	4.81057e8			1.32161
$715$	1.28393e8			0.351254
$716$	−2.57478e8			−0.701456
$717$	−2.95472e8			−0.801601
$718$		−	2.22624e8i		−	0.601450i
$719$	1.49055e8			0.401013			0.200507	−	0.979692i	$-0.435741\pi$
$719$	1.49055e8			0.401013			0.200507	+	0.979692i	$0.435741\pi$
$720$		−	7.81832e7i		−	0.209467i
$721$	−2.69666e8			−0.719483
$722$	5.09374e8			1.35340
$723$			2.23457e8i			0.591261i
$724$		−	4.61172e7i		−	0.121520i
$725$	4.94075e8			1.29652
$726$	−1.28278e8			−0.335230
$727$		−	2.60476e8i		−	0.677899i	−0.940805	−	0.338949i	$-0.889928\pi$
$727$		−	2.60476e8i		−	0.677899i	0.940805	−	0.338949i	$-0.110072\pi$
$728$			9.43887e7i			0.244639i
$729$	1.43489e7			0.0370370
$730$			8.52863e7i			0.219236i
$731$	−6.37891e6			−0.0163303
$732$			3.10091e8i			0.790598i
$733$			4.18481e8i			1.06258i	0.847189	+	0.531292i	$0.178293\pi$
$733$			4.18481e8i			1.06258i	−0.847189	+	0.531292i	$0.821707\pi$
$734$		−	4.31131e7i		−	0.109024i
$735$			2.43942e7i			0.0614363i
$736$	4.94978e8	+	7.55733e7i	1.24152	+	0.189555i
$737$	4.13727e8			1.03350
$738$	−9.23308e7			−0.229709
$739$	1.61438e7			0.0400012			0.0200006	−	0.999800i	$-0.493633\pi$
$739$	1.61438e7			0.0400012			0.0200006	+	0.999800i	$0.493633\pi$
$740$	3.06218e8			0.755675
$741$		−	2.93207e8i		−	0.720641i
$742$	−6.05250e8			−1.48157
$743$		−	1.27402e8i		−	0.310607i	−0.987867	−	0.155303i	$-0.950364\pi$
$743$		−	1.27402e8i		−	0.310607i	0.987867	−	0.155303i	$-0.0496355\pi$
$744$	3.22689e7			0.0783547
$745$	−1.62667e8			−0.393397
$746$		−	8.37446e8i		−	2.01716i
$747$			3.16495e7i			0.0759285i
$748$	4.63062e8			1.10646
$749$	3.82537e8			0.910391
$750$		−	2.95948e8i		−	0.701506i
$751$		−	4.55203e8i		−	1.07470i	−0.843361	−	0.537348i	$-0.819426\pi$
$751$		−	4.55203e8i		−	1.07470i	0.843361	−	0.537348i	$-0.180574\pi$
$752$	−4.97428e8			−1.16971
$753$		−	2.71341e8i		−	0.635522i
$754$	9.06050e8			2.11367
$755$		−	3.28773e8i		−	0.763932i
$756$		−	5.69635e7i		−	0.131835i
$757$			8.11546e7i			0.187079i	0.995616	+	0.0935396i	$0.0298182\pi$
$757$			8.11546e7i			0.187079i	−0.995616	+	0.0935396i	$0.970182\pi$
$758$		−	4.90274e8i		−	1.12572i
$759$	−2.85900e7	+	1.87254e8i	−0.0653867	+	0.428259i
$760$	−1.03437e8			−0.235633
$761$	1.18408e8			0.268675			0.134337	−	0.990936i	$-0.457109\pi$
$761$	1.18408e8			0.268675			0.134337	+	0.990936i	$0.457109\pi$
$762$	5.46794e7			0.123583
$763$	9.17177e7			0.206481
$764$			4.45177e8i			0.998280i
$765$	1.53117e8			0.342009
$766$			9.02400e8i			2.00776i
$767$	−1.47147e8			−0.326110
$768$	3.38430e8			0.747112
$769$			6.04390e8i			1.32904i	0.747270	+	0.664521i	$0.231364\pi$
$769$			6.04390e8i			1.32904i	−0.747270	+	0.664521i	$0.768636\pi$
$770$		−	2.16979e8i		−	0.475275i
$771$	5.23380e7			0.114197
$772$	−1.82827e8			−0.397363
$773$			2.25811e8i			0.488886i	0.969664	+	0.244443i	$0.0786051\pi$
$773$			2.25811e8i			0.488886i	−0.969664	+	0.244443i	$0.921395\pi$
$774$			1.74176e6i			0.00375634i
$775$	1.45334e8			0.312220
$776$			4.30545e6i			0.00921369i
$777$	−4.48731e8			−0.956585
$778$		−	9.70451e7i		−	0.206080i
$779$			3.48305e8i			0.736797i
$780$		−	9.82118e7i		−	0.206957i
$781$			1.12543e8i			0.236247i
$782$	−1.84688e8	+	1.20964e9i	−0.386205	+	2.52950i
$783$	1.67267e8			0.348438
$784$	−1.13501e8			−0.235533
$785$	1.51136e8			0.312435
$786$	−6.34459e8			−1.30658
$787$			5.31648e8i			1.09069i	0.838213	+	0.545343i	$0.183601\pi$
$787$			5.31648e8i			1.09069i	−0.838213	+	0.545343i	$0.816399\pi$
$788$	−3.61584e8			−0.738977
$789$		−	1.21841e8i		−	0.248063i
$790$	−2.73102e8			−0.553915
$791$	2.72300e8			0.550197
$792$			3.86778e7i			0.0778550i
$793$		−	7.83451e8i		−	1.57106i
$794$	1.17420e9			2.34575
$795$	−1.92646e8			−0.383406
$796$		−	3.90814e7i		−	0.0774874i
$797$			3.18375e8i			0.628874i	0.949278	+	0.314437i	$0.101816\pi$
$797$			3.18375e8i			0.628874i	−0.949278	+	0.314437i	$0.898184\pi$
$798$	−4.95508e8			−0.975085
$799$		−	9.74180e8i		−	1.90985i
$800$	4.60466e8			0.899348
$801$		−	2.51160e8i		−	0.488712i
$802$		−	7.26526e8i		−	1.40841i
$803$		−	1.20303e8i		−	0.232344i
$804$		−	3.16474e8i		−	0.608933i
$805$	2.45806e8	+	3.75297e7i	0.471200	+	0.0719429i
$806$	2.66517e8			0.509003
$807$	−5.81409e7			−0.110627
$808$	−1.05644e8			−0.200268
$809$	6.18466e6			0.0116807			0.00584037	−	0.999983i	$-0.498141\pi$
$809$	6.18466e6			0.0116807			0.00584037	+	0.999983i	$0.498141\pi$
$810$		−	4.18083e7i		−	0.0786698i
$811$	−9.26477e7			−0.173689			−0.0868445	−	0.996222i	$-0.527678\pi$
$811$	−9.26477e7			−0.173689			−0.0868445	+	0.996222i	$0.527678\pi$
$812$		−	6.64031e8i		−	1.24028i
$813$	−4.13084e8			−0.768717
$814$	−9.96024e8			−1.84670
$815$		−	2.84560e8i		−	0.525656i
$816$			7.12419e8i			1.31119i
$817$	6.57054e6			0.0120485
$818$	−4.40105e8			−0.804074
$819$			1.43919e8i			0.261980i
$820$			1.16668e8i			0.211597i
$821$	−9.05632e8			−1.63652			−0.818262	−	0.574846i	$-0.805062\pi$
$821$	−9.05632e8			−1.63652			−0.818262	+	0.574846i	$0.805062\pi$
$822$			7.33440e8i			1.32053i
$823$	−1.89363e8			−0.339700			−0.169850	−	0.985470i	$-0.554328\pi$
$823$	−1.89363e8			−0.339700			−0.169850	+	0.985470i	$0.554328\pi$
$824$		−	1.40060e8i		−	0.250342i
$825$			1.74198e8i			0.310229i
$826$			2.48672e8i			0.441253i
$827$			4.82304e8i			0.852716i	0.904555	+	0.426358i	$0.140204\pi$
$827$			4.82304e8i			0.852716i	−0.904555	+	0.426358i	$0.859796\pi$
$828$	1.43237e8	+	2.18695e7i	0.252327	+	0.0385254i
$829$	2.77778e8			0.487567			0.243784	−	0.969830i	$-0.421611\pi$
$829$	2.77778e8			0.487567			0.243784	+	0.969830i	$0.421611\pi$
$830$	9.22169e7			0.161278
$831$	−2.76675e8			−0.482133
$832$	2.47674e8			0.430041
$833$		−	2.22284e8i		−	0.384569i
$834$	−2.93379e8			−0.505745
$835$			1.96416e8i			0.337378i
$836$	−4.76973e8			−0.816347
$837$	4.92021e7			0.0839088
$838$		−	2.75522e8i		−	0.468192i
$839$			3.89000e8i			0.658663i	0.944214	+	0.329332i	$0.106823\pi$
$839$			3.89000e8i			0.658663i	−0.944214	+	0.329332i	$0.893177\pi$
$840$	5.07718e7			0.0856613
$841$	1.35503e9			2.27804
$842$			6.89550e8i			1.15513i
$843$		−	3.38111e8i		−	0.564386i
$844$	4.33479e8			0.721008
$845$		−	7.33470e7i		−	0.121566i
$846$	−2.65999e8			−0.439308
$847$		−	2.37527e8i		−	0.390897i
$848$		−	8.96342e8i		−	1.46989i
$849$			2.79955e8i			0.457473i
$850$			1.12530e9i			1.83236i
$851$	1.72277e8	−	1.12835e9i	0.279537	−	1.83087i
$852$	8.60882e7			0.139195
$853$	−1.11450e9			−1.79569			−0.897846	−	0.440310i	$-0.854869\pi$
$853$	−1.11450e9			−1.79569			−0.897846	+	0.440310i	$0.854869\pi$
$854$	−1.32400e9			−2.12577
$855$	−1.57716e8			−0.252335
$856$			1.98684e8i			0.316768i
$857$	2.94792e8			0.468353			0.234176	−	0.972194i	$-0.424761\pi$
$857$	2.94792e8			0.468353			0.234176	+	0.972194i	$0.424761\pi$
$858$			3.19450e8i			0.505756i
$859$	6.01262e8			0.948602			0.474301	−	0.880363i	$-0.342701\pi$
$859$	6.01262e8			0.948602			0.474301	+	0.880363i	$0.342701\pi$
$860$	2.20085e6			0.00346015
$861$		−	1.70965e8i		−	0.267853i
$862$			5.20108e7i			0.0812029i
$863$	−5.54362e8			−0.862503			−0.431252	−	0.902232i	$-0.641928\pi$
$863$	−5.54362e8			−0.862503			−0.431252	+	0.902232i	$0.641928\pi$
$864$	1.55889e8			0.241699
$865$		−	3.31135e8i		−	0.511630i
$866$		−	2.04045e8i		−	0.314175i
$867$	−1.01896e9			−1.56350
$868$		−	1.95327e8i		−	0.298677i
$869$	3.85233e8			0.587035
$870$		−	4.87366e8i		−	0.740112i
$871$			7.99577e8i			1.21006i
$872$			4.76367e7i			0.0718444i
$873$			6.56476e6i			0.00986679i
$874$	1.90236e8	−	1.24598e9i	0.284943	−	1.86627i
$875$	5.47993e8			0.817995
$876$	−9.20241e7			−0.136896
$877$	−8.19418e8			−1.21481			−0.607403	−	0.794394i	$-0.707789\pi$
$877$	−8.19418e8			−1.21481			−0.607403	+	0.794394i	$0.707789\pi$
$878$	−5.23633e8			−0.773648
$879$			5.47219e8i			0.805739i
$880$	3.21334e8			0.471529
$881$			3.16544e8i			0.462920i	0.972844	+	0.231460i	$0.0743503\pi$
$881$			3.16544e8i			0.462920i	−0.972844	+	0.231460i	$0.925650\pi$
$882$	−6.06946e7			−0.0884594
$883$	1.46132e7			0.0212257			0.0106129	−	0.999944i	$-0.496622\pi$
$883$	1.46132e7			0.0212257			0.0106129	+	0.999944i	$0.496622\pi$
$884$			8.94923e8i			1.29547i
$885$			7.91504e7i			0.114189i
$886$	−2.62009e7			−0.0376717
$887$	−3.75892e8			−0.538633			−0.269316	−	0.963052i	$-0.586798\pi$
$887$	−3.75892e8			−0.538633			−0.269316	+	0.963052i	$0.586798\pi$
$888$		−	2.33064e8i		−	0.332841i
$889$			1.01247e8i			0.144105i
$890$	−7.31803e8			−1.03806
$891$			5.89741e7i			0.0833736i
$892$	9.69163e8			1.36553
$893$			1.00345e9i			1.40909i
$894$		−	4.04728e8i		−	0.566436i
$895$		−	3.49917e8i		−	0.488086i
$896$			3.89612e8i			0.541637i
$897$	−3.61891e8	−	5.52537e7i	−0.501419	−	0.0765568i
$898$	−4.57538e8			−0.631828
$899$	5.73556e8			0.789399
$900$	1.33250e8			0.182785
$901$	1.75543e9			2.39998
$902$		−	3.79481e8i		−	0.517094i
$903$	−3.22513e6			−0.00438009
$904$			1.41428e8i			0.191439i
$905$	6.26742e7			0.0845557
$906$	8.18010e8			1.09995
$907$			7.34089e8i			0.983846i	0.870639	+	0.491923i	$0.163706\pi$
$907$			7.34089e8i			0.983846i	−0.870639	+	0.491923i	$0.836294\pi$
$908$			5.32306e8i			0.711057i
$909$	−1.61082e8			−0.214464
$910$	4.19338e8			0.556467
$911$		−	2.02554e8i		−	0.267908i	−0.990988	−	0.133954i	$-0.957233\pi$
$911$		−	2.02554e8i		−	0.267908i	0.990988	−	0.133954i	$-0.0427674\pi$
$912$		−	7.33820e8i		−	0.967398i
$913$	−1.30080e8			−0.170922
$914$		−	2.15164e8i		−	0.281793i
$915$	−4.21420e8			−0.550112
$916$		−	3.69339e8i		−	0.480550i
$917$		−	1.17480e9i		−	1.52354i
$918$			3.80965e8i			0.492444i
$919$		−	3.90694e7i		−	0.0503373i	−0.999683	−	0.0251686i	$-0.991988\pi$
$919$		−	3.90694e7i		−	0.0503373i	0.999683	−	0.0251686i	$-0.00801227\pi$
$920$	−1.94924e7	+	1.27668e8i	−0.0250323	+	0.163952i
$921$	5.16859e8			0.661596
$922$	−1.42199e9			−1.81428
$923$	−2.17504e8			−0.276606
$924$	2.34120e8			0.296772
$925$		−	1.04968e9i		−	1.32627i
$926$	−1.45727e9			−1.83530
$927$		−	2.13557e8i		−	0.268087i
$928$	1.81722e9			2.27386
$929$	1.46032e8			0.182138			0.0910690	−	0.995845i	$-0.470972\pi$
$929$	1.46032e8			0.182138			0.0910690	+	0.995845i	$0.470972\pi$
$930$		−	1.43360e8i		−	0.178229i
$931$			2.28962e8i			0.283736i
$932$	5.94256e8			0.734050
$933$	6.83501e8			0.841578
$934$		−	3.20076e8i		−	0.392837i
$935$			6.29310e8i			0.769892i
$936$	−7.47495e7			−0.0911551
$937$			9.27597e8i			1.12756i	0.825924	+	0.563781i	$0.190654\pi$
$937$			9.27597e8i			1.12756i	−0.825924	+	0.563781i	$0.809346\pi$
$938$	1.35126e9			1.63730
$939$		−	4.12819e7i		−	0.0498613i
$940$			3.36112e8i			0.404669i
$941$		−	5.99047e8i		−	0.718939i	−0.933157	−	0.359470i	$-0.882958\pi$
$941$		−	5.99047e8i		−	0.718939i	0.933157	−	0.359470i	$-0.117042\pi$
$942$			3.76038e8i			0.449861i
$943$	4.29898e8	+	6.56369e7i	0.512661	+	0.0782731i
$944$	−3.68270e8			−0.437774
$945$	7.74145e7			0.0917333
$946$	−7.15863e6			−0.00845584
$947$	3.43415e8			0.404361			0.202180	−	0.979348i	$-0.435197\pi$
$947$	3.43415e8			0.404361			0.202180	+	0.979348i	$0.435197\pi$
$948$		−	2.94677e8i		−	0.345877i
$949$	2.32501e8			0.272036
$950$		−	1.15910e9i		−	1.35192i
$951$	−2.58155e8			−0.300151
$952$	−4.62642e8			−0.536209
$953$			1.55314e9i			1.79445i	0.441575	+	0.897224i	$0.354420\pi$
$953$			1.55314e9i			1.79445i	−0.441575	+	0.897224i	$0.645580\pi$
$954$		−	4.79317e8i		−	0.552050i
$955$	−6.05004e8			−0.694622
$956$	−9.28928e8			−1.06318
$957$			6.87470e8i			0.784364i
$958$			1.75065e9i			1.99114i
$959$	−1.35808e9			−1.53981
$960$		−	1.33224e8i		−	0.150581i
$961$	−7.18791e8			−0.809902
$962$		−	1.92494e9i		−	2.16218i
$963$			3.02944e8i			0.339221i
$964$			7.02523e8i			0.784205i
$965$		−	2.48465e8i		−	0.276493i
$966$	−9.33767e7	+	6.11583e8i	−0.103587	+	0.678460i
$967$	−1.21371e9			−1.34226			−0.671129	−	0.741341i	$-0.734190\pi$
$967$	−1.21371e9			−1.34226			−0.671129	+	0.741341i	$0.734190\pi$
$968$	1.23368e8			0.136011
$969$	1.43714e9			1.57953
$970$	1.91277e7			0.0209579
$971$		−	1.12801e9i		−	1.23213i	−0.787695	−	0.616065i	$-0.788726\pi$
$971$		−	1.12801e9i		−	1.23213i	0.787695	−	0.616065i	$-0.211274\pi$
$972$	4.51113e7			0.0491232
$973$		−	5.43236e8i		−	0.589726i
$974$	−3.55186e8			−0.384396
$975$	−3.36659e8			−0.363226
$976$		−	1.96078e9i		−	2.10901i
$977$		−	7.68649e8i		−	0.824222i	−0.911134	−	0.412111i	$-0.864792\pi$
$977$		−	7.68649e8i		−	0.824222i	0.911134	−	0.412111i	$-0.135208\pi$
$978$	7.08006e8			0.756869
$979$	1.03227e9			1.10013
$980$			7.66926e7i			0.0814845i
$981$			7.26343e7i			0.0769369i
$982$	5.43130e8			0.573547
$983$			1.63647e9i			1.72285i	0.507885	+	0.861425i	$0.330427\pi$
$983$			1.63647e9i			1.72285i	−0.507885	+	0.861425i	$0.669573\pi$
$984$	8.87963e7			0.0931987
$985$		−	4.91400e8i		−	0.514193i
$986$			4.44096e9i			4.63283i
$987$		−	4.92538e8i		−	0.512257i
$988$		−	9.21807e8i		−	0.955805i
$989$	1.23819e6	−	8.10971e6i	0.00127997	−	0.00838333i
$990$	1.71833e8			0.177093
$991$	−6.22107e8			−0.639211			−0.319605	−	0.947551i	$-0.603550\pi$
$991$	−6.22107e8			−0.639211			−0.319605	+	0.947551i	$0.603550\pi$
$992$	5.34540e8			0.547577
$993$	−7.21814e8			−0.737187
$994$			3.67573e8i			0.374270i
$995$	5.31124e7			0.0539171
$996$			9.95022e7i			0.100706i
$997$	−1.06328e9			−1.07291			−0.536455	−	0.843929i	$-0.680237\pi$
$997$	−1.06328e9			−1.07291			−0.536455	+	0.843929i	$0.680237\pi$
$998$	−4.89523e8			−0.492472
$999$		−	3.55365e8i		−	0.356433i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	69.7.d.a.22.6	yes	24
3.2	odd	2		207.7.d.e.91.19		24
23.22	odd	2	inner	69.7.d.a.22.5	✓	24
69.68	even	2		207.7.d.e.91.20		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
69.7.d.a.22.5	✓	24	23.22	odd	2	inner
69.7.d.a.22.6	yes	24	1.1	even	1	trivial
207.7.d.e.91.19		24	3.2	odd	2
207.7.d.e.91.20		24	69.68	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 69 = 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	69.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-10.6305 q^{2} +15.5885 q^{3} +49.0083 q^{4} +66.6032i q^{5} -165.714 q^{6} -306.844i q^{7} +159.370 q^{8} +243.000 q^{9} +O(q^{10})\)	\(q-10.6305 q^{2} +15.5885 q^{3} +49.0083 q^{4} +66.6032i q^{5} -165.714 q^{6} -306.844i q^{7} +159.370 q^{8} +243.000 q^{9} -708.028i q^{10} +998.732i q^{11} +763.964 q^{12} -1930.17 q^{13} +3261.92i q^{14} +1038.24i q^{15} -4830.72 q^{16} -9460.64i q^{17} -2583.22 q^{18} +9744.85i q^{19} +3264.11i q^{20} -4783.23i q^{21} -10617.1i q^{22} +(12027.6 + 1836.38i) q^{23} +2484.33 q^{24} +11189.0 q^{25} +20518.7 q^{26} +3788.00 q^{27} -15037.9i q^{28} +44157.2 q^{29} -11037.1i q^{30} +12989.0 q^{31} +41153.4 q^{32} +15568.7i q^{33} +100572. i q^{34} +20436.8 q^{35} +11909.0 q^{36} -93813.5i q^{37} -103593. i q^{38} -30088.4 q^{39} +10614.6i q^{40} +35742.5 q^{41} +50848.3i q^{42} -674.258i q^{43} +48946.1i q^{44} +16184.6i q^{45} +(-127860. - 19521.7i) q^{46} +102972. q^{47} -75303.4 q^{48} +23495.7 q^{49} -118945. q^{50} -147477. i q^{51} -94594.3 q^{52} +185550. i q^{53} -40268.4 q^{54} -66518.8 q^{55} -48901.7i q^{56} +151907. i q^{57} -469414. q^{58} +76235.0 q^{59} +50882.4i q^{60} +405897. i q^{61} -138080. q^{62} -74563.1i q^{63} -128317. q^{64} -128556. i q^{65} -165504. i q^{66} -414252. i q^{67} -463650. i q^{68} +(187492. + 28626.3i) q^{69} -217254. q^{70} +112686. q^{71} +38726.9 q^{72} -120456. q^{73} +997288. i q^{74} +174419. q^{75} +477578. i q^{76} +306455. q^{77} +319856. q^{78} -385722. i q^{79} -321741. i q^{80} +59049.0 q^{81} -379962. q^{82} +130245. i q^{83} -234418. i q^{84} +630109. q^{85} +7167.72i q^{86} +688342. q^{87} +159168. i q^{88} -1.03358e6i q^{89} -172051. i q^{90} +592261. i q^{91} +(589453. + 89997.8i) q^{92} +202478. q^{93} -1.09465e6 q^{94} -649038. q^{95} +641519. q^{96} +27015.5i q^{97} -249772. q^{98} +242692. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9}+O(q^{10}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9	\( 24 q - 20 q^{2} + 816 q^{4} - 324 q^{6} - 940 q^{8} + 5832 q^{9} + 384 q^{13} + 29544 q^{16} - 4860 q^{18} + 29336 q^{23} - 39204 q^{24} - 61272 q^{25} + 10088 q^{26} + 64672 q^{29} + 9696 q^{31} - 319620 q^{32} - 225744 q^{35} + 198288 q^{36} - 11664 q^{39} + 135280 q^{41} + 233232 q^{46} - 74336 q^{47} + 552096 q^{48} - 722136 q^{49} + 619324 q^{50} + 1059720 q^{52} - 78732 q^{54} - 1019328 q^{55} - 694344 q^{58} + 1057648 q^{59} - 488776 q^{62} - 273888 q^{64} - 23328 q^{69} + 2785512 q^{70} - 255392 q^{71} - 228420 q^{72} - 322560 q^{73} - 365472 q^{75} - 1002960 q^{77} - 171072 q^{78} + 1417176 q^{81} - 5732712 q^{82} - 2704704 q^{85} + 611712 q^{87} - 1611444 q^{92} + 2484432 q^{93} - 147720 q^{94} - 1672656 q^{95} - 1818612 q^{96} + 9104212 q^{98}+O(q^{100}) \) 24 * q - 20 * q^2 + 816 * q^4 - 324 * q^6 - 940 * q^8 + 5832 * q^9 + 384 * q^13 + 29544 * q^16 - 4860 * q^18 + 29336 * q^23 - 39204 * q^24 - 61272 * q^25 + 10088 * q^26 + 64672 * q^29 + 9696 * q^31 - 319620 * q^32 - 225744 * q^35 + 198288 * q^36 - 11664 * q^39 + 135280 * q^41 + 233232 * q^46 - 74336 * q^47 + 552096 * q^48 - 722136 * q^49 + 619324 * q^50 + 1059720 * q^52 - 78732 * q^54 - 1019328 * q^55 - 694344 * q^58 + 1057648 * q^59 - 488776 * q^62 - 273888 * q^64 - 23328 * q^69 + 2785512 * q^70 - 255392 * q^71 - 228420 * q^72 - 322560 * q^73 - 365472 * q^75 - 1002960 * q^77 - 171072 * q^78 + 1417176 * q^81 - 5732712 * q^82 - 2704704 * q^85 + 611712 * q^87 - 1611444 * q^92 + 2484432 * q^93 - 147720 * q^94 - 1672656 * q^95 - 1818612 * q^96 + 9104212 * q^98

Introduction

L-functions

Modular forms

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