LMFDB - Embedded newform 169.8.b.f.168.38

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [169,8,Mod(168,169)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(169, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 8, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("169.168"); S:= CuspForms(chi, 8); N := Newforms(S);

Level:	$ N $	$=$	$ 169 = 13^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	169.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$52.7930693068$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			168.38
Character	$\chi$	$=$	169.168
Dual form			169.8.b.f.168.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+19.6237i q^{2} -72.4635 q^{3} -257.091 q^{4} -413.760i q^{5} -1422.00i q^{6} +1005.88i q^{7} -2533.24i q^{8} +3063.96 q^{9} +8119.51 q^{10} +3482.63i q^{11} +18629.7 q^{12} -19739.1 q^{14} +29982.5i q^{15} +16804.1 q^{16} -6157.31 q^{17} +60126.4i q^{18} +41406.0i q^{19} +106374. i q^{20} -72889.6i q^{21} -68342.2 q^{22} +63119.9 q^{23} +183568. i q^{24} -93072.0 q^{25} -63547.8 q^{27} -258602. i q^{28} +75193.0 q^{29} -588368. q^{30} +133868. i q^{31} +5503.36i q^{32} -252364. i q^{33} -120829. i q^{34} +416192. q^{35} -787717. q^{36} +171939. i q^{37} -812541. q^{38} -1.04815e6 q^{40} +747517. i q^{41} +1.43037e6 q^{42} +187227. q^{43} -895352. i q^{44} -1.26774e6i q^{45} +1.23865e6i q^{46} -226649. i q^{47} -1.21768e6 q^{48} -188251. q^{49} -1.82642e6i q^{50} +446180. q^{51} -1.34632e6 q^{53} -1.24705e6i q^{54} +1.44097e6 q^{55} +2.54814e6 q^{56} -3.00043e6i q^{57} +1.47557e6i q^{58} -2.50878e6i q^{59} -7.70822e6i q^{60} -509752. q^{61} -2.62699e6 q^{62} +3.08198e6i q^{63} +2.04292e6 q^{64} +4.95232e6 q^{66} +4.07091e6i q^{67} +1.58299e6 q^{68} -4.57389e6 q^{69} +8.16725e6i q^{70} +256015. i q^{71} -7.76177e6i q^{72} +5.38093e6i q^{73} -3.37408e6 q^{74} +6.74433e6 q^{75} -1.06451e7i q^{76} -3.50311e6 q^{77} +5.18777e6 q^{79} -6.95285e6i q^{80} -2.09599e6 q^{81} -1.46691e7 q^{82} -5.32418e6i q^{83} +1.87392e7i q^{84} +2.54765e6i q^{85} +3.67410e6i q^{86} -5.44875e6 q^{87} +8.82235e6 q^{88} -8.69805e6i q^{89} +2.48779e7 q^{90} -1.62276e7 q^{92} -9.70053e6i q^{93} +4.44770e6 q^{94} +1.71321e7 q^{95} -398793. i q^{96} -3.76799e6i q^{97} -3.69419e6i q^{98} +1.06706e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9} + 10334 q^{10} - 43590 q^{12} - 358 q^{14} + 226410 q^{16} - 90032 q^{17} - 548348 q^{22} + 256210 q^{23} - 733074 q^{25} - 213286 q^{27} - 124658 q^{29} - 19522 q^{30}+ \cdots - 35320062 q^{95}+O(q^{100}) $ 42 * q - 52 * q^3 - 2818 * q^4 + 30930 * q^9 + 10334 * q^10 - 43590 * q^12 - 358 * q^14 + 226410 * q^16 - 90032 * q^17 - 548348 * q^22 + 256210 * q^23 - 733074 * q^25 - 213286 * q^27 - 124658 * q^29 - 19522 * q^30 + 1697720 * q^35 + 548490 * q^36 - 451662 * q^38 - 4577654 * q^40 + 5473672 * q^42 - 1788978 * q^43 + 12604776 * q^48 - 263060 * q^49 + 2368682 * q^51 - 9100054 * q^53 + 11048060 * q^55 - 4668772 * q^56 + 11434024 * q^61 + 27187306 * q^62 - 19535848 * q^64 - 35613954 * q^66 + 24504984 * q^68 - 5544412 * q^69 + 19250156 * q^74 + 86424794 * q^75 - 18892072 * q^77 - 37226562 * q^79 + 82110898 * q^81 - 48380654 * q^82 + 18055788 * q^87 + 151356360 * q^88 - 6671120 * q^90 - 138678780 * q^92 + 57911426 * q^94 - 35320062 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/169\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$	19.6237i	1.73451i	0.497865	+	0.867255i	$0.334118\pi$
$2$	19.6237i	1.73451i	−0.497865	+	0.867255i	$0.665882\pi$
$3$	−72.4635	−1.54951	−0.774756	−	0.632260i	$-0.782127\pi$
$3$	−72.4635	−1.54951	−0.774756	+	0.632260i	$0.782127\pi$
$4$	−257.091	−2.00852
$5$	− 413.760i	− 1.48031i	−0.672436	−	0.740156i	$-0.734752\pi$
$5$	− 413.760i	− 1.48031i	0.672436	−	0.740156i	$-0.265248\pi$
$6$	− 1422.00i	− 2.68764i
$7$	1005.88i	1.10842i	0.832378	+	0.554208i	$0.186979\pi$
$7$	1005.88i	1.10842i	−0.832378	+	0.554208i	$0.813021\pi$
$8$	− 2533.24i	− 1.74929i
$9$	3063.96	1.40099
$10$	8119.51	2.56761
$11$	3482.63i	0.788920i	0.918913	+	0.394460i	$0.129068\pi$
$11$	3482.63i	0.788920i	−0.918913	+	0.394460i	$0.870932\pi$
$12$	18629.7	3.11223
$13$	0	0
$13$	0	0
$14$	−19739.1	−1.92256
$15$	29982.5i	2.29376i
$16$	16804.1	1.02564
$17$	−6157.31	−0.303962	−0.151981	−	0.988383i	$-0.548565\pi$
$17$	−6157.31	−0.303962	−0.151981	+	0.988383i	$0.548565\pi$
$18$	60126.4i	2.43003i
$19$	41406.0i	1.38492i	0.721454	+	0.692462i	$0.243474\pi$
$19$	41406.0i	1.38492i	−0.721454	+	0.692462i	$0.756526\pi$
$20$	106374.i	2.97324i
$21$	− 72889.6i	− 1.71751i
$22$	−68342.2	−1.36839
$23$	63119.9	1.08173	0.540865	−	0.841109i	$-0.318097\pi$
$23$	63119.9	1.08173	0.540865	+	0.841109i	$0.318097\pi$
$24$	183568.i	2.71055i
$25$	−93072.0	−1.19132
$26$	0	0
$27$	−63547.8	−0.621337
$28$	− 258602.i	− 2.22628i
$29$	75193.0	0.572512	0.286256	−	0.958153i	$-0.407589\pi$
$29$	75193.0	0.572512	0.286256	+	0.958153i	$0.407589\pi$
$30$	−588368.	−3.97855
$31$	133868.i	0.807068i	0.914965	+	0.403534i	$0.132218\pi$
$31$	133868.i	0.807068i	−0.914965	+	0.403534i	$0.867782\pi$
$32$	5503.36i	0.0296895i
$33$	− 252364.i	− 1.22244i
$34$	− 120829.i	− 0.527225i
$35$	416192.	1.64080
$36$	−787717.	−2.81392
$37$	171939.i	0.558043i	0.960285	+	0.279022i	$0.0900101\pi$
$37$	171939.i	0.558043i	−0.960285	+	0.279022i	$0.909990\pi$
$38$	−812541.	−2.40216
$39$	0	0
$40$	−1.04815e6	−2.58950
$41$	747517.i	1.69386i	0.531705	+	0.846930i	$0.321552\pi$
$41$	747517.i	1.69386i	−0.531705	+	0.846930i	$0.678448\pi$
$42$	1.43037e6	2.97903
$43$	187227.	0.359112	0.179556	−	0.983748i	$-0.442534\pi$
$43$	187227.	0.359112	0.179556	+	0.983748i	$0.442534\pi$
$44$	− 895352.i	− 1.58456i
$45$	− 1.26774e6i	− 2.07390i
$46$	1.23865e6i	1.87627i
$47$	− 226649.i	− 0.318428i	−0.987244	−	0.159214i	$-0.949104\pi$
$47$	− 226649.i	− 0.318428i	0.987244	−	0.159214i	$-0.0508960\pi$
$48$	−1.21768e6	−1.58924
$49$	−188251.	−0.228587
$50$	− 1.82642e6i	− 2.06636i
$51$	446180.	0.470993
$52$	0	0
$53$	−1.34632e6	−1.24218	−0.621088	−	0.783740i	$-0.713309\pi$
$53$	−1.34632e6	−1.24218	−0.621088	+	0.783740i	$0.713309\pi$
$54$	− 1.24705e6i	− 1.07772i
$55$	1.44097e6	1.16785
$56$	2.54814e6	1.93894
$57$	− 3.00043e6i	− 2.14596i
$58$	1.47557e6i	0.993027i
$59$	− 2.50878e6i	− 1.59031i	−0.606409	−	0.795153i	$-0.707390\pi$
$59$	− 2.50878e6i	− 1.59031i	0.606409	−	0.795153i	$-0.292610\pi$
$60$	− 7.70822e6i	− 4.60707i
$61$	−509752.	−0.287544	−0.143772	−	0.989611i	$-0.545923\pi$
$61$	−509752.	−0.287544	−0.143772	+	0.989611i	$0.545923\pi$
$62$	−2.62699e6	−1.39987
$63$	3.08198e6i	1.55288i
$64$	2.04292e6	0.974142
$65$	0	0
$66$	4.95232e6	2.12034
$67$	4.07091e6i	1.65360i	0.562498	+	0.826799i	$0.309840\pi$
$67$	4.07091e6i	1.65360i	−0.562498	+	0.826799i	$0.690160\pi$
$68$	1.58299e6	0.610515
$69$	−4.57389e6	−1.67616
$70$	8.16725e6i	2.84599i
$71$	256015.i	0.0848910i	0.999099	+	0.0424455i	$0.0135149\pi$
$71$	256015.i	0.0848910i	−0.999099	+	0.0424455i	$0.986485\pi$
$72$	− 7.76177e6i	− 2.45074i
$73$	5.38093e6i	1.61893i	0.587171	+	0.809463i	$0.300242\pi$
$73$	5.38093e6i	1.61893i	−0.587171	+	0.809463i	$0.699758\pi$
$74$	−3.37408e6	−0.967931
$75$	6.74433e6	1.84597
$76$	− 1.06451e7i	− 2.78165i
$77$	−3.50311e6	−0.874452
$78$	0	0
$79$	5.18777e6	1.18382	0.591911	−	0.806004i	$-0.298374\pi$
$79$	5.18777e6	1.18382	0.591911	+	0.806004i	$0.298374\pi$
$80$	− 6.95285e6i	− 1.51827i
$81$	−2.09599e6	−0.438219
$82$	−1.46691e7	−2.93801
$83$	− 5.32418e6i	− 1.02207i	−0.859561	−	0.511034i	$-0.829263\pi$
$83$	− 5.32418e6i	− 1.02207i	0.859561	−	0.511034i	$-0.170737\pi$
$84$	1.87392e7i	3.44965i
$85$	2.54765e6i	0.449959i
$86$	3.67410e6i	0.622883i
$87$	−5.44875e6	−0.887114
$88$	8.82235e6	1.38005
$89$	− 8.69805e6i	− 1.30785i	−0.756561	−	0.653923i	$-0.773122\pi$
$89$	− 8.69805e6i	− 1.30785i	0.756561	−	0.653923i	$-0.226878\pi$
$90$	2.48779e7	3.59720
$91$	0	0
$92$	−1.62276e7	−2.17268
$93$	− 9.70053e6i	− 1.25056i
$94$	4.44770e6	0.552317
$95$	1.71321e7	2.05012
$96$	− 398793.i	− 0.0460043i
$97$	− 3.76799e6i	− 0.419187i	−0.977789	−	0.209594i	$-0.932786\pi$
$97$	− 3.76799e6i	− 0.419187i	0.977789	−	0.209594i	$-0.0672141\pi$
$98$	− 3.69419e6i	− 0.396486i
$99$	1.06706e7i	1.10527i
$100$	2.39280e7	2.39280
$101$	−1.07132e7	−1.03465	−0.517325	−	0.855789i	$-0.673072\pi$
$101$	−1.07132e7	−1.03465	−0.517325	+	0.855789i	$0.673072\pi$
$102$	8.75572e6i	0.816942i
$103$	8.08002e6	0.728588	0.364294	−	0.931284i	$-0.381310\pi$
$103$	8.08002e6	0.728588	0.364294	+	0.931284i	$0.381310\pi$
$104$	0	0
$105$	−3.01588e7	−2.54244
$106$	− 2.64199e7i	− 2.15457i
$107$	−2.35427e6	−0.185786	−0.0928931	−	0.995676i	$-0.529611\pi$
$107$	−2.35427e6	−0.185786	−0.0928931	+	0.995676i	$0.529611\pi$
$108$	1.63376e7	1.24797
$109$	− 289028.i	− 0.0213770i	−0.999943	−	0.0106885i	$-0.996598\pi$
$109$	− 289028.i	− 0.0213770i	0.999943	−	0.0106885i	$-0.00340232\pi$
$110$	2.82772e7i	2.02564i
$111$	− 1.24593e7i	− 0.864695i
$112$	1.69029e7i	1.13684i
$113$	4.19514e6	0.273509	0.136755	−	0.990605i	$-0.456333\pi$
$113$	4.19514e6	0.273509	0.136755	+	0.990605i	$0.456333\pi$
$114$	5.88796e7	3.72218
$115$	− 2.61165e7i	− 1.60130i
$116$	−1.93314e7	−1.14990
$117$	0	0
$118$	4.92317e7	2.75840
$119$	− 6.19351e6i	− 0.336917i
$120$	7.59529e7	4.01246
$121$	7.35846e6	0.377605
$122$	− 1.00032e7i	− 0.498748i
$123$	− 5.41677e7i	− 2.62466i
$124$	− 3.44162e7i	− 1.62101i
$125$	6.18447e6i	0.283216i
$126$	−6.04799e7	−2.69348
$127$	−3.35328e7	−1.45264	−0.726318	−	0.687358i	$-0.758770\pi$
$127$	−3.35328e7	−1.45264	−0.726318	+	0.687358i	$0.758770\pi$
$128$	4.07942e7i	1.71935i
$129$	−1.35672e7	−0.556449
$130$	0	0
$131$	−2.99754e7	−1.16497	−0.582487	−	0.812840i	$-0.697920\pi$
$131$	−2.99754e7	−1.16497	−0.582487	+	0.812840i	$0.697920\pi$
$132$	6.48804e7i	2.45530i
$133$	−4.16495e7	−1.53507
$134$	−7.98865e7	−2.86818
$135$	2.62935e7i	0.919773i
$136$	1.55980e7i	0.531719i
$137$	− 1.77421e7i	− 0.589501i	−0.955574	−	0.294750i	$-0.904764\pi$
$137$	− 1.77421e7i	− 0.589501i	0.955574	−	0.294750i	$-0.0952365\pi$
$138$	− 8.97569e7i	− 2.90731i
$139$	−2.24539e7	−0.709152	−0.354576	−	0.935027i	$-0.615375\pi$
$139$	−2.24539e7	−0.709152	−0.354576	+	0.935027i	$0.615375\pi$
$140$	−1.06999e8	−3.29559
$141$	1.64238e7i	0.493409i
$142$	−5.02398e6	−0.147244
$143$	0	0
$144$	5.14871e7	1.43691
$145$	− 3.11118e7i	− 0.847495i
$146$	−1.05594e8	−2.80804
$147$	1.36413e7	0.354198
$148$	− 4.42039e7i	− 1.12084i
$149$	1.63242e7i	0.404278i	0.979357	+	0.202139i	$0.0647893\pi$
$149$	1.63242e7i	0.404278i	−0.979357	+	0.202139i	$0.935211\pi$
$150$	1.32349e8i	3.20185i
$151$	− 3.54101e7i	− 0.836965i	−0.908225	−	0.418483i	$-0.862562\pi$
$151$	− 3.54101e7i	− 0.836965i	0.908225	−	0.418483i	$-0.137438\pi$
$152$	1.04892e8	2.42264
$153$	−1.88658e7	−0.425848
$154$	− 6.87440e7i	− 1.51674i
$155$	5.53891e7	1.19471
$156$	0	0
$157$	−1.64297e7	−0.338830	−0.169415	−	0.985545i	$-0.554188\pi$
$157$	−1.64297e7	−0.338830	−0.169415	+	0.985545i	$0.554188\pi$
$158$	1.01803e8i	2.05335i
$159$	9.75593e7	1.92477
$160$	2.27707e6	0.0439497
$161$	6.34911e7i	1.19901i
$162$	− 4.11311e7i	− 0.760095i
$163$	7.24842e7i	1.31095i	0.755216	+	0.655475i	$0.227532\pi$
$163$	7.24842e7i	1.31095i	−0.755216	+	0.655475i	$0.772468\pi$
$164$	− 1.92180e8i	− 3.40215i
$165$	−1.04418e8	−1.80959
$166$	1.04480e8	1.77279
$167$	− 9.79620e7i	− 1.62761i	−0.581139	−	0.813804i	$-0.697393\pi$
$167$	− 9.79620e7i	− 1.62761i	0.581139	−	0.813804i	$-0.302607\pi$
$168$	−1.84647e8	−3.00442
$169$	0	0
$170$	−4.99943e7	−0.780458
$171$	1.26867e8i	1.94026i
$172$	−4.81345e7	−0.721284
$173$	−5.57283e6	−0.0818303	−0.0409151	−	0.999163i	$-0.513027\pi$
$173$	−5.57283e6	−0.0818303	−0.0409151	+	0.999163i	$0.513027\pi$
$174$	− 1.06925e8i	− 1.53871i
$175$	− 9.36193e7i	− 1.32048i
$176$	5.85224e7i	0.809147i
$177$	1.81795e8i	2.46420i
$178$	1.70688e8	2.26847
$179$	−7.61885e7	−0.992896	−0.496448	−	0.868067i	$-0.665363\pi$
$179$	−7.61885e7	−0.992896	−0.496448	+	0.868067i	$0.665363\pi$
$180$	3.25925e8i	4.16547i
$181$	−5.13200e7	−0.643297	−0.321648	−	0.946859i	$-0.604237\pi$
$181$	−5.13200e7	−0.643297	−0.321648	+	0.946859i	$0.604237\pi$
$182$	0	0
$183$	3.69384e7	0.445553
$184$	− 1.59898e8i	− 1.89226i
$185$	7.11413e7	0.826078
$186$	1.90361e8	2.16911
$187$	− 2.14436e7i	− 0.239802i
$188$	5.82694e7i	0.639570i
$189$	− 6.39215e7i	− 0.688701i
$190$	3.36197e8i	3.55595i
$191$	1.19362e7	0.123951	0.0619753	−	0.998078i	$-0.480260\pi$
$191$	1.19362e7	0.123951	0.0619753	+	0.998078i	$0.480260\pi$
$192$	−1.48038e8	−1.50945
$193$	1.37269e8i	1.37443i	0.726453	+	0.687216i	$0.241167\pi$
$193$	1.37269e8i	1.37443i	−0.726453	+	0.687216i	$0.758833\pi$
$194$	7.39419e7	0.727084
$195$	0	0
$196$	4.83976e7	0.459122
$197$	9.34188e7i	0.870567i	0.900293	+	0.435284i	$0.143352\pi$
$197$	9.34188e7i	0.870567i	−0.900293	+	0.435284i	$0.856648\pi$
$198$	−2.09398e8	−1.91710
$199$	−1.28639e8	−1.15714	−0.578571	−	0.815632i	$-0.696389\pi$
$199$	−1.28639e8	−1.15714	−0.578571	+	0.815632i	$0.696389\pi$
$200$	2.35774e8i	2.08397i
$201$	− 2.94993e8i	− 2.56227i
$202$	− 2.10232e8i	− 1.79461i
$203$	7.56351e7i	0.634581i
$204$	−1.14709e8	−0.946001
$205$	3.09292e8	2.50744
$206$	1.58560e8i	1.26374i
$207$	1.93397e8	1.51549
$208$	0	0
$209$	−1.44202e8	−1.09259
$210$	− 5.91828e8i	− 4.40989i
$211$	−1.53936e8	−1.12811	−0.564054	−	0.825738i	$-0.690759\pi$
$211$	−1.53936e8	−1.12811	−0.564054	+	0.825738i	$0.690759\pi$
$212$	3.46127e8	2.49494
$213$	− 1.85518e7i	− 0.131540i
$214$	− 4.61996e7i	− 0.322248i
$215$	− 7.74672e7i	− 0.531598i
$216$	1.60982e8i	1.08690i
$217$	−1.34655e8	−0.894568
$218$	5.67181e6	0.0370786
$219$	− 3.89921e8i	− 2.50855i
$220$	−3.70461e8	−2.34565
$221$	0	0
$222$	2.44498e8	1.49982
$223$	1.30054e8i	0.785336i	0.919680	+	0.392668i	$0.128448\pi$
$223$	1.30054e8i	0.785336i	−0.919680	+	0.392668i	$0.871552\pi$
$224$	−5.53572e6	−0.0329083
$225$	−2.85169e8	−1.66903
$226$	8.23244e7i	0.474405i
$227$	3.46925e8i	1.96855i	0.176657	+	0.984273i	$0.443472\pi$
$227$	3.46925e8i	1.96855i	−0.176657	+	0.984273i	$0.556528\pi$
$228$	7.71382e8i	4.31020i
$229$	− 8.27429e7i	− 0.455309i	−0.973742	−	0.227655i	$-0.926894\pi$
$229$	− 8.27429e7i	− 0.455309i	0.973742	−	0.227655i	$-0.0731057\pi$
$230$	5.12503e8	2.77747
$231$	2.53848e8	1.35497
$232$	− 1.90482e8i	− 1.00149i
$233$	1.51866e7	0.0786530	0.0393265	−	0.999226i	$-0.487479\pi$
$233$	1.51866e7	0.0786530	0.0393265	+	0.999226i	$0.487479\pi$
$234$	0	0
$235$	−9.37783e7	−0.471373
$236$	6.44985e8i	3.19417i
$237$	−3.75924e8	−1.83435
$238$	1.21540e8	0.584385
$239$	1.61515e8i	0.765280i	0.923898	+	0.382640i	$0.124985\pi$
$239$	1.61515e8i	0.765280i	−0.923898	+	0.382640i	$0.875015\pi$
$240$	5.03828e8i	2.35257i
$241$	− 1.29574e8i	− 0.596292i	−0.954520	−	0.298146i	$-0.903632\pi$
$241$	− 1.29574e8i	− 0.596292i	0.954520	−	0.298146i	$-0.0963682\pi$
$242$	1.44400e8i	0.654960i
$243$	2.90862e8	1.30036
$244$	1.31053e8	0.577538
$245$	7.78907e7i	0.338380i
$246$	1.06297e9	4.55249
$247$	0	0
$248$	3.39120e8	1.41180
$249$	3.85809e8i	1.58371i
$250$	−1.21362e8	−0.491241
$251$	1.44103e8	0.575196	0.287598	−	0.957751i	$-0.407143\pi$
$251$	1.44103e8	0.575196	0.287598	+	0.957751i	$0.407143\pi$
$252$	− 7.92348e8i	− 3.11899i
$253$	2.19823e8i	0.853399i
$254$	− 6.58039e8i	− 2.51961i
$255$	− 1.84611e8i	− 0.697217i
$256$	−5.39041e8	−2.00808
$257$	−2.48876e8	−0.914572	−0.457286	−	0.889320i	$-0.651178\pi$
$257$	−2.48876e8	−0.914572	−0.457286	+	0.889320i	$0.651178\pi$
$258$	− 2.66238e8i	− 0.965165i
$259$	−1.72950e8	−0.618544
$260$	0	0
$261$	2.30388e8	0.802082
$262$	− 5.88230e8i	− 2.02066i
$263$	1.91513e8	0.649163	0.324582	−	0.945858i	$-0.394777\pi$
$263$	1.91513e8	0.649163	0.324582	+	0.945858i	$0.394777\pi$
$264$	−6.39299e8	−2.13841
$265$	5.57054e8i	1.83881i
$266$	− 8.17318e8i	− 2.66260i
$267$	6.30291e8i	2.02652i
$268$	− 1.04659e9i	− 3.32129i
$269$	−2.90813e8	−0.910922	−0.455461	−	0.890256i	$-0.650526\pi$
$269$	−2.90813e8	−0.910922	−0.455461	+	0.890256i	$0.650526\pi$
$270$	−5.15977e8	−1.59535
$271$	6.34456e7i	0.193646i	0.995302	+	0.0968232i	$0.0308681\pi$
$271$	6.34456e7i	0.193646i	−0.995302	+	0.0968232i	$0.969132\pi$
$272$	−1.03468e8	−0.311756
$273$	0	0
$274$	3.48167e8	1.02249
$275$	− 3.24135e8i	− 0.939858i
$276$	1.17591e9	3.36660
$277$	−2.25716e8	−0.638090	−0.319045	−	0.947739i	$-0.603362\pi$
$277$	−2.25716e8	−0.638090	−0.319045	+	0.947739i	$0.603362\pi$
$278$	− 4.40629e8i	− 1.23003i
$279$	4.10166e8i	1.13069i
$280$	− 1.05432e9i	− 2.87024i
$281$	− 6.67278e6i	− 0.0179405i	−0.999960	−	0.00897025i	$-0.997145\pi$
$281$	− 6.67278e6i	− 0.0179405i	0.999960	−	0.00897025i	$-0.00285536\pi$
$282$	−3.22296e8	−0.855822
$283$	−1.62048e8	−0.425002	−0.212501	−	0.977161i	$-0.568161\pi$
$283$	−1.62048e8	−0.425002	−0.212501	+	0.977161i	$0.568161\pi$
$284$	− 6.58192e7i	− 0.170505i
$285$	−1.24146e9	−3.17669
$286$	0	0
$287$	−7.51912e8	−1.87750
$288$	1.68621e7i	0.0415947i
$289$	−3.72426e8	−0.907607
$290$	6.10530e8	1.46999
$291$	2.73042e8i	0.649536i
$292$	− 1.38339e9i	− 3.25165i
$293$	− 3.53069e8i	− 0.820017i	−0.912082	−	0.410009i	$-0.865526\pi$
$293$	− 3.53069e8i	− 0.820017i	0.912082	−	0.410009i	$-0.134474\pi$
$294$	2.67694e8i	0.614360i
$295$	−1.03803e9	−2.35415
$296$	4.35563e8	0.976180
$297$	− 2.21314e8i	− 0.490186i
$298$	−3.20342e8	−0.701224
$299$	0	0
$300$	−1.73390e9	−3.70767
$301$	1.88328e8i	0.398046i
$302$	6.94878e8	1.45172
$303$	7.76314e8	1.60320
$304$	6.95790e8i	1.42043i
$305$	2.10915e8i	0.425655i
$306$	− 3.70217e8i	− 0.738637i
$307$	− 6.79916e7i	− 0.134113i	−0.997749	−	0.0670565i	$-0.978639\pi$
$307$	− 6.79916e7i	− 0.134113i	0.997749	−	0.0670565i	$-0.0213608\pi$
$308$	9.00617e8	1.75636
$309$	−5.85507e8	−1.12896
$310$	1.08694e9i	2.07224i
$311$	6.98524e8	1.31680	0.658401	−	0.752668i	$-0.271233\pi$
$311$	6.98524e8	1.31680	0.658401	+	0.752668i	$0.271233\pi$
$312$	0	0
$313$	8.41764e8	1.55162	0.775810	−	0.630967i	$-0.217342\pi$
$313$	8.41764e8	1.55162	0.775810	+	0.630967i	$0.217342\pi$
$314$	− 3.22412e8i	− 0.587703i
$315$	1.27520e9	2.29874
$316$	−1.33373e9	−2.37773
$317$	− 9.91402e8i	− 1.74800i	−0.485922	−	0.874002i	$-0.661516\pi$
$317$	− 9.91402e8i	− 1.74800i	0.485922	−	0.874002i	$-0.338484\pi$
$318$	1.91448e9i	3.33853i
$319$	2.61869e8i	0.451666i
$320$	− 8.45280e8i	− 1.44203i
$321$	1.70599e8	0.287878
$322$	−1.24593e9	−2.07969
$323$	− 2.54950e8i	− 0.420965i
$324$	5.38859e8	0.880172
$325$	0	0
$326$	−1.42241e9	−2.27386
$327$	2.09440e7i	0.0331240i
$328$	1.89364e9	2.96305
$329$	2.27982e8	0.352951
$330$	− 2.04907e9i	− 3.13876i
$331$	− 1.46324e8i	− 0.221777i	−0.993833	−	0.110889i	$-0.964630\pi$
$331$	− 1.46324e8i	− 0.221777i	0.993833	−	0.110889i	$-0.0353697\pi$
$332$	1.36880e9i	2.05284i
$333$	5.26814e8i	0.781812i
$334$	1.92238e9	2.82310
$335$	1.68438e9	2.44784
$336$	− 1.22484e9i	− 1.76154i
$337$	−5.22422e8	−0.743561	−0.371780	−	0.928321i	$-0.621253\pi$
$337$	−5.22422e8	−0.743561	−0.371780	+	0.928321i	$0.621253\pi$
$338$	0	0
$339$	−3.03995e8	−0.423806
$340$	− 6.54976e8i	− 0.903752i
$341$	−4.66212e8	−0.636712
$342$	−2.48960e9	−3.36541
$343$	6.39027e8i	0.855047i
$344$	− 4.74293e8i	− 0.628191i
$345$	1.89249e9i	2.48123i
$346$	− 1.09360e8i	− 0.141935i
$347$	6.74172e7	0.0866199	0.0433100	−	0.999062i	$-0.486210\pi$
$347$	6.74172e7	0.0866199	0.0433100	+	0.999062i	$0.486210\pi$
$348$	1.40082e9	1.78179
$349$	7.24565e8i	0.912406i	0.889876	+	0.456203i	$0.150791\pi$
$349$	7.24565e8i	0.912406i	−0.889876	+	0.456203i	$0.849209\pi$
$350$	1.83716e9	2.29039
$351$	0	0
$352$	−1.91662e7	−0.0234227
$353$	− 4.32002e8i	− 0.522726i	−0.965241	−	0.261363i	$-0.915828\pi$
$353$	− 4.32002e8i	− 0.522726i	0.965241	−	0.261363i	$-0.0841720\pi$
$354$	−3.56750e9	−4.27418
$355$	1.05929e8	0.125665
$356$	2.23619e9i	2.62684i
$357$	4.48804e8i	0.522057i
$358$	− 1.49510e9i	− 1.72219i
$359$	− 1.10242e9i	− 1.25752i	−0.777598	−	0.628761i	$-0.783562\pi$
$359$	− 1.10242e9i	− 1.25752i	0.777598	−	0.628761i	$-0.216438\pi$
$360$	−3.21151e9	−3.62785
$361$	−8.20588e8	−0.918015
$362$	− 1.00709e9i	− 1.11580i
$363$	−5.33220e8	−0.585104
$364$	0	0
$365$	2.22641e9	2.39652
$366$	7.24869e8i	0.772816i
$367$	1.31669e9	1.39044	0.695222	−	0.718796i	$-0.255306\pi$
$367$	1.31669e9	1.39044	0.695222	+	0.718796i	$0.255306\pi$
$368$	1.06067e9	1.10947
$369$	2.29036e9i	2.37308i
$370$	1.39606e9i	1.43284i
$371$	− 1.35424e9i	− 1.37685i
$372$	2.49392e9i	2.51178i
$373$	1.44085e9	1.43760	0.718799	−	0.695218i	$-0.244692\pi$
$373$	1.44085e9	1.43760	0.718799	+	0.695218i	$0.244692\pi$
$374$	4.20804e8	0.415939
$375$	− 4.48149e8i	− 0.438847i
$376$	−5.74158e8	−0.557024
$377$	0	0
$378$	1.25438e9	1.19456
$379$	1.01968e8i	0.0962111i	0.998842	+	0.0481056i	$0.0153184\pi$
$379$	1.01968e8i	0.0962111i	−0.998842	+	0.0481056i	$0.984682\pi$
$380$	−4.40452e9	−4.11771
$381$	2.42991e9	2.25088
$382$	2.34232e8i	0.214993i
$383$	1.94208e9i	1.76633i	0.469065	+	0.883164i	$0.344591\pi$
$383$	1.94208e9i	1.76633i	−0.469065	+	0.883164i	$0.655409\pi$
$384$	− 2.95609e9i	− 2.66415i
$385$	1.44944e9i	1.29446i
$386$	−2.69374e9	−2.38397
$387$	5.73658e8	0.503112
$388$	9.68715e8i	0.841947i
$389$	−1.75657e9	−1.51301	−0.756506	−	0.653987i	$-0.773095\pi$
$389$	−1.75657e9	−1.51301	−0.756506	+	0.653987i	$0.773095\pi$
$390$	0	0
$391$	−3.88649e8	−0.328805
$392$	4.76886e8i	0.399865i
$393$	2.17213e9	1.80514
$394$	−1.83322e9	−1.51001
$395$	− 2.14649e9i	− 1.75242i
$396$	− 2.74333e9i	− 2.21996i
$397$	− 2.22292e9i	− 1.78302i	−0.452998	−	0.891511i	$-0.649646\pi$
$397$	− 2.22292e9i	− 1.78302i	0.452998	−	0.891511i	$-0.350354\pi$
$398$	− 2.52437e9i	− 2.00707i
$399$	3.01807e9	2.37861
$400$	−1.56399e9	−1.22187
$401$	− 1.61957e9i	− 1.25428i	−0.778908	−	0.627139i	$-0.784226\pi$
$401$	− 1.61957e9i	− 1.25428i	0.778908	−	0.627139i	$-0.215774\pi$
$402$	5.78885e9	4.44428
$403$	0	0
$404$	2.75426e9	2.07812
$405$	8.67235e8i	0.648700i
$406$	−1.48424e9	−1.10069
$407$	−5.98799e8	−0.440251
$408$	− 1.13028e9i	− 0.823905i
$409$	− 2.57824e9i	− 1.86334i	−0.363307	−	0.931670i	$-0.618352\pi$
$409$	− 2.57824e9i	− 1.86334i	0.363307	−	0.931670i	$-0.381648\pi$
$410$	6.06947e9i	4.34918i
$411$	1.28566e9i	0.913438i
$412$	−2.07730e9	−1.46339
$413$	2.52353e9	1.76272
$414$	3.79517e9i	2.62864i
$415$	−2.20293e9	−1.51298
$416$	0	0
$417$	1.62709e9	1.09884
$418$	− 2.82978e9i	− 1.89512i
$419$	1.55385e9	1.03196	0.515978	−	0.856602i	$-0.327429\pi$
$419$	1.55385e9	1.03196	0.515978	+	0.856602i	$0.327429\pi$
$420$	7.75354e9	5.10655
$421$	− 1.34533e9i	− 0.878699i	−0.898316	−	0.439350i	$-0.855209\pi$
$421$	− 1.34533e9i	− 0.878699i	0.898316	−	0.439350i	$-0.144791\pi$
$422$	− 3.02079e9i	− 1.95671i
$423$	− 6.94445e8i	− 0.446115i
$424$	3.41056e9i	2.17293i
$425$	5.73073e8	0.362117
$426$	3.64055e8	0.228157
$427$	− 5.12749e8i	− 0.318718i
$428$	6.05262e8	0.373156
$429$	0	0
$430$	1.52019e9	0.922061
$431$	1.60812e9i	0.967495i	0.875208	+	0.483747i	$0.160725\pi$
$431$	1.60812e9i	0.967495i	−0.875208	+	0.483747i	$0.839275\pi$
$432$	−1.06786e9	−0.637268
$433$	1.89392e9	1.12112	0.560562	−	0.828113i	$-0.310585\pi$
$433$	1.89392e9	1.12112	0.560562	+	0.828113i	$0.310585\pi$
$434$	− 2.64243e9i	− 1.55164i
$435$	2.25447e9i	1.31320i
$436$	7.43065e7i	0.0429362i
$437$	2.61355e9i	1.49812i
$438$	7.65171e9	4.35110
$439$	−1.08620e9	−0.612749	−0.306375	−	0.951911i	$-0.599116\pi$
$439$	−1.08620e9	−0.612749	−0.306375	+	0.951911i	$0.599116\pi$
$440$	− 3.65033e9i	− 2.04290i
$441$	−5.76794e8	−0.320248
$442$	0	0
$443$	−2.67935e9	−1.46425	−0.732127	−	0.681168i	$-0.761472\pi$
$443$	−2.67935e9	−1.46425	−0.732127	+	0.681168i	$0.761472\pi$
$444$	3.20317e9i	1.73676i
$445$	−3.59890e9	−1.93602
$446$	−2.55214e9	−1.36217
$447$	− 1.18291e9i	− 0.626434i
$448$	2.05494e9i	1.07976i
$449$	2.77150e9i	1.44495i	0.691398	+	0.722474i	$0.256995\pi$
$449$	2.77150e9i	1.44495i	−0.691398	+	0.722474i	$0.743005\pi$
$450$	− 5.59608e9i	− 2.89495i
$451$	−2.60332e9	−1.33632
$452$	−1.07853e9	−0.549350
$453$	2.56594e9i	1.29689i
$454$	−6.80797e9	−3.41446
$455$	0	0
$456$	−7.60081e9	−3.75390
$457$	1.88973e9i	0.926177i	0.886312	+	0.463088i	$0.153259\pi$
$457$	1.88973e9i	0.926177i	−0.886312	+	0.463088i	$0.846741\pi$
$458$	1.62372e9	0.789738
$459$	3.91284e8	0.188863
$460$	6.71431e9i	3.21624i
$461$	7.18617e8i	0.341621i	0.985304	+	0.170810i	$0.0546386\pi$
$461$	7.18617e8i	0.341621i	−0.985304	+	0.170810i	$0.945361\pi$
$462$	4.98144e9i	2.35022i
$463$	3.35895e8i	0.157279i	0.996903	+	0.0786393i	$0.0250575\pi$
$463$	3.35895e8i	0.157279i	−0.996903	+	0.0786393i	$0.974942\pi$
$464$	1.26355e9	0.587190
$465$	−4.01369e9	−1.85122
$466$	2.98018e8i	0.136424i
$467$	−1.62006e8	−0.0736076	−0.0368038	−	0.999323i	$-0.511718\pi$
$467$	−1.62006e8	−0.0736076	−0.0368038	+	0.999323i	$0.511718\pi$
$468$	0	0
$469$	−4.09485e9	−1.83287
$470$	− 1.84028e9i	− 0.817601i
$471$	1.19056e9	0.525021
$472$	−6.35536e9	−2.78191
$473$	6.52044e8i	0.283311i
$474$	− 7.37704e9i	− 3.18169i
$475$	− 3.85374e9i	− 1.64989i
$476$	1.59230e9i	0.676705i
$477$	−4.12508e9	−1.74028
$478$	−3.16953e9	−1.32738
$479$	− 1.87256e9i	− 0.778505i	−0.921131	−	0.389253i	$-0.872733\pi$
$479$	− 1.87256e9i	− 0.778505i	0.921131	−	0.389253i	$-0.127267\pi$
$480$	−1.65004e8	−0.0681007
$481$	0	0
$482$	2.54273e9	1.03427
$483$	− 4.60079e9i	− 1.85788i
$484$	−1.89179e9	−0.758428
$485$	−1.55904e9	−0.620528
$486$	5.70779e9i	2.25549i
$487$	2.79396e9i	1.09615i	0.836431	+	0.548073i	$0.184638\pi$
$487$	2.79396e9i	1.09615i	−0.836431	+	0.548073i	$0.815362\pi$
$488$	1.29133e9i	0.502998i
$489$	− 5.25246e9i	− 2.03133i
$490$	−1.52851e9	−0.586923
$491$	1.87416e8	0.0714532	0.0357266	−	0.999362i	$-0.488625\pi$
$491$	1.87416e8	0.0714532	0.0357266	+	0.999362i	$0.488625\pi$
$492$	1.39260e10i	5.27168i
$493$	−4.62986e8	−0.174022
$494$	0	0
$495$	4.41508e9	1.63614
$496$	2.24952e9i	0.827761i
$497$	−2.57521e8	−0.0940946
$498$	−7.57101e9	−2.74695
$499$	− 3.05663e9i	− 1.10126i	−0.834749	−	0.550631i	$-0.814387\pi$
$499$	− 3.05663e9i	− 1.10126i	0.834749	−	0.550631i	$-0.185613\pi$
$500$	− 1.58997e9i	− 0.568845i
$501$	7.09867e9i	2.52200i
$502$	2.82784e9i	0.997682i
$503$	−1.10435e9	−0.386916	−0.193458	−	0.981109i	$-0.561970\pi$
$503$	−1.10435e9	−0.386916	−0.193458	+	0.981109i	$0.561970\pi$
$504$	7.80740e9	2.71644
$505$	4.43268e9i	1.53160i
$506$	−4.31376e9	−1.48023
$507$	0	0
$508$	8.62098e9	2.91765
$509$	− 2.84435e8i	− 0.0956029i	−0.998857	−	0.0478014i	$-0.984779\pi$
$509$	− 2.84435e8i	− 0.0956029i	0.998857	−	0.0478014i	$-0.0152215\pi$
$510$	3.62276e9	1.20933
$511$	−5.41257e9	−1.79444
$512$	− 5.35633e9i	− 1.76369i
$513$	− 2.63126e9i	− 0.860505i
$514$	− 4.88388e9i	− 1.58633i
$515$	− 3.34318e9i	− 1.07854i
$516$	3.48799e9	1.11764
$517$	7.89335e8	0.251214
$518$	− 3.39392e9i	− 1.07287i
$519$	4.03827e8	0.126797
$520$	0	0
$521$	−6.21830e8	−0.192637	−0.0963184	−	0.995351i	$-0.530707\pi$
$521$	−6.21830e8	−0.192637	−0.0963184	+	0.995351i	$0.530707\pi$
$522$	4.52108e9i	1.39122i
$523$	−1.32430e9	−0.404790	−0.202395	−	0.979304i	$-0.564873\pi$
$523$	−1.32430e9	−0.404790	−0.202395	+	0.979304i	$0.564873\pi$
$524$	7.70641e9	2.33988
$525$	6.78398e9i	2.04610i
$526$	3.75821e9i	1.12598i
$527$	− 8.24266e8i	− 0.245318i
$528$	− 4.24074e9i	− 1.25378i
$529$	5.79302e8	0.170141
$530$	−1.09315e10	−3.18943
$531$	− 7.68681e9i	− 2.22800i
$532$	1.07077e10	3.08323
$533$	0	0
$534$	−1.23687e10	−3.51503
$535$	9.74103e8i	0.275022i
$536$	1.03126e10	2.89262
$537$	5.52089e9	1.53850
$538$	− 5.70684e9i	− 1.58000i
$539$	− 6.55609e8i	− 0.180337i
$540$	− 6.75982e9i	− 1.84738i
$541$	2.63957e9i	0.716708i	0.933586	+	0.358354i	$0.116662\pi$
$541$	2.63957e9i	0.716708i	−0.933586	+	0.358354i	$0.883338\pi$
$542$	−1.24504e9	−0.335881
$543$	3.71883e9	0.996797
$544$	− 3.38859e7i	− 0.00902449i
$545$	−1.19588e8	−0.0316447
$546$	0	0
$547$	−3.91983e9	−1.02403	−0.512013	−	0.858977i	$-0.671100\pi$
$547$	−3.91983e9	−1.02403	−0.512013	+	0.858977i	$0.671100\pi$
$548$	4.56134e9i	1.18402i
$549$	−1.56186e9	−0.402846
$550$	6.36075e9	1.63019
$551$	3.11344e9i	0.792885i
$552$	1.15868e10i	2.93208i
$553$	5.21828e9i	1.31217i
$554$	− 4.42938e9i	− 1.10677i
$555$	−5.15515e9	−1.28002
$556$	5.77269e9	1.42435
$557$	4.28372e9i	1.05034i	0.850999	+	0.525168i	$0.175997\pi$
$557$	4.28372e9i	1.05034i	−0.850999	+	0.525168i	$0.824003\pi$
$558$	−8.04899e9	−1.96120
$559$	0	0
$560$	6.99373e9	1.68287
$561$	1.55388e9i	0.371576i
$562$	1.30945e8	0.0311180
$563$	7.15933e9	1.69080	0.845402	−	0.534131i	$-0.179361\pi$
$563$	7.15933e9	1.69080	0.845402	+	0.534131i	$0.179361\pi$
$564$	− 4.22241e9i	− 0.991022i
$565$	− 1.73578e9i	− 0.404879i
$566$	− 3.17999e9i	− 0.737171i
$567$	− 2.10831e9i	− 0.485729i
$568$	6.48549e8	0.148499
$569$	5.31555e9	1.20964	0.604818	−	0.796364i	$-0.293246\pi$
$569$	5.31555e9	1.20964	0.604818	+	0.796364i	$0.293246\pi$
$570$	− 2.43620e10i	− 5.50999i
$571$	1.41625e9	0.318356	0.159178	−	0.987250i	$-0.449116\pi$
$571$	1.41625e9	0.318356	0.159178	+	0.987250i	$0.449116\pi$
$572$	0	0
$573$	−8.64938e8	−0.192063
$574$	− 1.47553e10i	− 3.25654i
$575$	−5.87470e9	−1.28869
$576$	6.25945e9	1.36476
$577$	− 3.09413e9i	− 0.670538i	−0.942123	−	0.335269i	$-0.891173\pi$
$577$	− 3.09413e9i	− 0.670538i	0.942123	−	0.335269i	$-0.108827\pi$
$578$	− 7.30839e9i	− 1.57425i
$579$	− 9.94703e9i	− 2.12970i
$580$	7.99856e9i	1.70221i
$581$	5.35549e9	1.13288
$582$	−5.35809e9	−1.12663
$583$	− 4.68874e9i	− 0.979978i
$584$	1.36312e10	2.83197
$585$	0	0
$586$	6.92853e9	1.42233
$587$	− 5.10437e8i	− 0.104162i	−0.998643	−	0.0520809i	$-0.983415\pi$
$587$	− 5.10437e8i	− 0.104162i	0.998643	−	0.0520809i	$-0.0165854\pi$
$588$	−3.50706e9	−0.711415
$589$	−5.54293e9	−1.11773
$590$	− 2.03701e10i	− 4.08329i
$591$	− 6.76945e9i	− 1.34895i
$592$	2.88927e9i	0.572351i
$593$	6.99937e9i	1.37838i	0.724583	+	0.689188i	$0.242032\pi$
$593$	6.99937e9i	1.37838i	−0.724583	+	0.689188i	$0.757968\pi$
$594$	4.34300e9	0.850231
$595$	−2.56263e9	−0.498742
$596$	− 4.19680e9i	− 0.812002i
$597$	9.32162e9	1.79300
$598$	0	0
$599$	1.10847e9	0.210733	0.105366	−	0.994433i	$-0.466398\pi$
$599$	1.10847e9	0.210733	0.105366	+	0.994433i	$0.466398\pi$
$600$	− 1.70850e10i	− 3.22914i
$601$	−3.46969e9	−0.651973	−0.325987	−	0.945374i	$-0.605696\pi$
$601$	−3.46969e9	−0.651973	−0.325987	+	0.945374i	$0.605696\pi$
$602$	−3.69570e9	−0.690414
$603$	1.24731e10i	2.31667i
$604$	9.10361e9i	1.68106i
$605$	− 3.04463e9i	− 0.558973i
$606$	1.52342e10i	2.78077i
$607$	−8.78548e9	−1.59443	−0.797215	−	0.603696i	$-0.793694\pi$
$607$	−8.78548e9	−1.59443	−0.797215	+	0.603696i	$0.793694\pi$
$608$	−2.27872e8	−0.0411177
$609$	− 5.48079e9i	− 0.983291i
$610$	−4.13893e9	−0.738302
$611$	0	0
$612$	4.85022e9	0.855325
$613$	1.72321e9i	0.302152i	0.988522	+	0.151076i	$0.0482739\pi$
$613$	1.72321e9i	0.302152i	−0.988522	+	0.151076i	$0.951726\pi$
$614$	1.33425e9	0.232620
$615$	−2.24124e10	−3.88531
$616$	8.87423e9i	1.52967i
$617$	3.95049e9i	0.677101i	0.940948	+	0.338550i	$0.109937\pi$
$617$	3.95049e9i	0.677101i	−0.940948	+	0.338550i	$0.890063\pi$
$618$	− 1.14898e10i	− 1.95818i
$619$	− 3.95281e9i	− 0.669866i	−0.942242	−	0.334933i	$-0.891286\pi$
$619$	− 3.95281e9i	− 0.669866i	0.942242	−	0.334933i	$-0.108714\pi$
$620$	−1.42400e10	−2.39961
$621$	−4.01114e9	−0.672120
$622$	1.37077e10i	2.28400i
$623$	8.74919e9	1.44964
$624$	0	0
$625$	−4.71237e9	−0.772074
$626$	1.65186e10i	2.69130i
$627$	1.04494e10	1.69299
$628$	4.22393e9	0.680547
$629$	− 1.05868e9i	− 0.169624i
$630$	2.50241e10i	3.98719i
$631$	− 6.84671e9i	− 1.08487i	−0.840097	−	0.542437i	$-0.817502\pi$
$631$	− 6.84671e9i	− 1.08487i	0.840097	−	0.542437i	$-0.182498\pi$
$632$	− 1.31419e10i	− 2.07085i
$633$	1.11547e10	1.74802
$634$	1.94550e10	3.03193
$635$	1.38745e10i	2.15035i
$636$	−2.50816e10	−3.86594
$637$	0	0
$638$	−5.13885e9	−0.783419
$639$	7.84421e8i	0.118931i
$640$	1.68790e10	2.54517
$641$	−3.12250e9	−0.468273	−0.234136	−	0.972204i	$-0.575226\pi$
$641$	−3.12250e9	−0.468273	−0.234136	+	0.972204i	$0.575226\pi$
$642$	3.34779e9i	0.499327i
$643$	− 8.57775e9i	− 1.27243i	−0.771511	−	0.636216i	$-0.780499\pi$
$643$	− 8.57775e9i	− 1.27243i	0.771511	−	0.636216i	$-0.219501\pi$
$644$	− 1.63230e10i	− 2.40823i
$645$	5.61354e9i	0.823717i
$646$	5.00307e9	0.730167
$647$	−8.70898e9	−1.26416	−0.632080	−	0.774903i	$-0.717799\pi$
$647$	−8.70898e9	−1.26416	−0.632080	+	0.774903i	$0.717799\pi$
$648$	5.30965e9i	0.766572i
$649$	8.73716e9	1.25462
$650$	0	0
$651$	9.75757e9	1.38614
$652$	− 1.86350e10i	− 2.63307i
$653$	−9.71192e9	−1.36493	−0.682463	−	0.730920i	$-0.739091\pi$
$653$	−9.71192e9	−1.36493	−0.682463	+	0.730920i	$0.739091\pi$
$654$	−4.10999e8	−0.0574538
$655$	1.24026e10i	1.72452i
$656$	1.25613e10i	1.73729i
$657$	1.64870e10i	2.26810i
$658$	4.47385e9i	0.612197i
$659$	3.34780e9	0.455680	0.227840	−	0.973699i	$-0.426834\pi$
$659$	3.34780e9	0.455680	0.227840	+	0.973699i	$0.426834\pi$
$660$	2.68449e10	3.63461
$661$	− 8.98798e9i	− 1.21048i	−0.796044	−	0.605239i	$-0.793078\pi$
$661$	− 8.98798e9i	− 1.21048i	0.796044	−	0.605239i	$-0.206922\pi$
$662$	2.87142e9	0.384675
$663$	0	0
$664$	−1.34874e10	−1.78789
$665$	1.72329e10i	2.27239i
$666$	−1.03381e10	−1.35606
$667$	4.74618e9	0.619303
$668$	2.51851e10i	3.26909i
$669$	− 9.42414e9i	− 1.21689i
$670$	3.30538e10i	4.24580i
$671$	− 1.77528e9i	− 0.226849i
$672$	4.01138e8	0.0509919
$673$	1.90573e9	0.240995	0.120498	−	0.992714i	$-0.461551\pi$
$673$	1.90573e9	0.240995	0.120498	+	0.992714i	$0.461551\pi$
$674$	− 1.02519e10i	− 1.28971i
$675$	5.91452e9	0.740213
$676$	0	0
$677$	1.10831e10	1.37278	0.686388	−	0.727236i	$-0.259195\pi$
$677$	1.10831e10	1.37278	0.686388	+	0.727236i	$0.259195\pi$
$678$	− 5.96552e9i	− 0.735096i
$679$	3.79014e9	0.464634
$680$	6.45381e9	0.787109
$681$	− 2.51394e10i	− 3.05029i
$682$	− 9.14882e9i	− 1.10438i
$683$	5.53250e9i	0.664429i	0.943204	+	0.332215i	$0.107796\pi$
$683$	5.53250e9i	0.664429i	−0.943204	+	0.332215i	$0.892204\pi$
$684$	− 3.26162e10i	− 3.89706i
$685$	−7.34098e9	−0.872644
$686$	−1.25401e10	−1.48309
$687$	5.99584e9i	0.705507i
$688$	3.14618e9	0.368319
$689$	0	0
$690$	−3.71378e10	−4.30372
$691$	− 9.96854e9i	− 1.14937i	−0.818376	−	0.574683i	$-0.805125\pi$
$691$	− 9.96854e9i	− 1.14937i	0.818376	−	0.574683i	$-0.194875\pi$
$692$	1.43272e9	0.164358
$693$	−1.07334e10	−1.22510
$694$	1.32298e9i	0.150243i
$695$	9.29051e9i	1.04977i
$696$	1.38030e10i	1.55182i
$697$	− 4.60269e9i	− 0.514869i
$698$	−1.42187e10	−1.58258
$699$	−1.10048e9	−0.121874
$700$	2.40687e10i	2.65221i
$701$	−1.56464e9	−0.171554	−0.0857769	−	0.996314i	$-0.527337\pi$
$701$	−1.56464e9	−0.171554	−0.0857769	+	0.996314i	$0.527337\pi$
$702$	0	0
$703$	−7.11930e9	−0.772847
$704$	7.11475e9i	0.768520i
$705$	6.79550e9	0.730398
$706$	8.47749e9	0.906673
$707$	− 1.07762e10i	− 1.14682i
$708$	− 4.67379e10i	− 4.94940i
$709$	− 4.49879e9i	− 0.474061i	−0.971502	−	0.237031i	$-0.923826\pi$
$709$	− 4.49879e9i	− 0.474061i	0.971502	−	0.237031i	$-0.0761741\pi$
$710$	2.07872e9i	0.217967i
$711$	1.58951e10	1.65852
$712$	−2.20343e10	−2.28780
$713$	8.44973e9i	0.873030i
$714$	−8.80721e9	−0.905512
$715$	0	0
$716$	1.95874e10	1.99425
$717$	− 1.17039e10i	− 1.18581i
$718$	2.16336e10	2.18118
$719$	1.28814e9	0.129245	0.0646223	−	0.997910i	$-0.479416\pi$
$719$	1.28814e9	0.129245	0.0646223	+	0.997910i	$0.479416\pi$
$720$	− 2.13033e10i	− 2.12707i
$721$	8.12753e9i	0.807579i
$722$	− 1.61030e10i	− 1.59231i
$723$	9.38940e9i	0.923962i
$724$	1.31939e10	1.29208
$725$	−6.99836e9	−0.682046
$726$	− 1.04638e10i	− 1.01487i
$727$	−1.64110e9	−0.158403	−0.0792017	−	0.996859i	$-0.525237\pi$
$727$	−1.64110e9	−0.158403	−0.0792017	+	0.996859i	$0.525237\pi$
$728$	0	0
$729$	−1.64929e10	−1.57671
$730$	4.36905e10i	4.15678i
$731$	−1.15282e9	−0.109157
$732$	−9.49653e9	−0.894903
$733$	− 6.09340e9i	− 0.571473i	−0.958308	−	0.285737i	$-0.907762\pi$
$733$	− 6.09340e9i	− 0.571473i	0.958308	−	0.285737i	$-0.0922382\pi$
$734$	2.58384e10i	2.41174i
$735$	− 5.64424e9i	− 0.524324i
$736$	3.47372e8i	0.0321161i
$737$	−1.41775e10	−1.30456
$738$	−4.49455e10	−4.11613
$739$	− 1.10220e10i	− 1.00462i	−0.864687	−	0.502311i	$-0.832483\pi$
$739$	− 1.10220e10i	− 1.00462i	0.864687	−	0.502311i	$-0.167517\pi$
$740$	−1.82898e10	−1.65919
$741$	0	0
$742$	2.65752e10	2.38816
$743$	− 1.44250e9i	− 0.129020i	−0.997917	−	0.0645098i	$-0.979452\pi$
$743$	− 1.44250e9i	− 0.129020i	0.997917	−	0.0645098i	$-0.0205484\pi$
$744$	−2.45738e10	−2.18760
$745$	6.75430e9	0.598457
$746$	2.82748e10i	2.49353i
$747$	− 1.63131e10i	− 1.43190i
$748$	5.51296e9i	0.481648i
$749$	− 2.36812e9i	− 0.205929i
$750$	8.79435e9	0.761183
$751$	4.23430e9	0.364789	0.182395	−	0.983225i	$-0.441615\pi$
$751$	4.23430e9	0.364789	0.182395	+	0.983225i	$0.441615\pi$
$752$	− 3.80863e9i	− 0.326593i
$753$	−1.04422e10	−0.891273
$754$	0	0
$755$	−1.46513e10	−1.23897
$756$	1.64336e10i	1.38327i
$757$	−4.85119e9	−0.406456	−0.203228	−	0.979131i	$-0.565143\pi$
$757$	−4.85119e9	−0.406456	−0.203228	+	0.979131i	$0.565143\pi$
$758$	−2.00099e9	−0.166879
$759$	− 1.59292e10i	− 1.32235i
$760$	− 4.33999e10i	− 3.58625i
$761$	− 1.68175e10i	− 1.38329i	−0.722236	−	0.691647i	$-0.756885\pi$
$761$	− 1.68175e10i	− 1.38329i	0.722236	−	0.691647i	$-0.243115\pi$
$762$	4.76838e10i	3.90417i
$763$	2.90727e8	0.0236946
$764$	−3.06868e9	−0.248958
$765$	7.80589e9i	0.630387i
$766$	−3.81108e10	−3.06371
$767$	0	0
$768$	3.90608e10	3.11155
$769$	1.62454e9i	0.128821i	0.997923	+	0.0644107i	$0.0205168\pi$
$769$	1.62454e9i	0.128821i	−0.997923	+	0.0644107i	$0.979483\pi$
$770$	−2.84435e10	−2.24525
$771$	1.80345e10	1.41714
$772$	− 3.52907e10i	− 2.76058i
$773$	− 2.06113e10i	− 1.60501i	−0.596648	−	0.802503i	$-0.703501\pi$
$773$	− 2.06113e10i	− 1.60501i	0.596648	−	0.802503i	$-0.296499\pi$
$774$	1.12573e10i	0.872652i
$775$	− 1.24593e10i	− 0.961478i
$776$	−9.54523e9	−0.733281
$777$	1.25325e10	0.958442
$778$	− 3.44705e10i	− 2.62433i
$779$	−3.09517e10	−2.34587
$780$	0	0
$781$	−8.91607e8	−0.0669722
$782$	− 7.62674e9i	− 0.570316i
$783$	−4.77835e9	−0.355723
$784$	−3.16339e9	−0.234448
$785$	6.79795e9i	0.501573i
$786$	4.26252e10i	3.13103i
$787$	7.97469e9i	0.583179i	0.956543	+	0.291590i	$0.0941842\pi$
$787$	7.97469e9i	0.583179i	−0.956543	+	0.291590i	$0.905816\pi$
$788$	− 2.40171e10i	− 1.74855i
$789$	−1.38777e10	−1.00589
$790$	4.21222e10	3.03960
$791$	4.21981e9i	0.303162i
$792$	2.70314e10	1.93344
$793$	0	0
$794$	4.36220e10	3.09267
$795$	− 4.03661e10i	− 2.84926i
$796$	3.30719e10	2.32414
$797$	2.05018e10	1.43446	0.717228	−	0.696839i	$-0.245411\pi$
$797$	2.05018e10	1.43446	0.717228	+	0.696839i	$0.245411\pi$
$798$	5.92258e10i	4.12573i
$799$	1.39555e9i	0.0967902i
$800$	− 5.12209e8i	− 0.0353698i
$801$	− 2.66505e10i	− 1.83228i
$802$	3.17819e10	2.17556
$803$	−1.87398e10	−1.27720
$804$	7.58399e10i	5.14638i
$805$	2.62700e10	1.77491
$806$	0	0
$807$	2.10734e10	1.41149
$808$	2.71391e10i	1.80990i
$809$	−1.78384e10	−1.18450	−0.592252	−	0.805753i	$-0.701761\pi$
$809$	−1.78384e10	−1.18450	−0.592252	+	0.805753i	$0.701761\pi$
$810$	−1.70184e10	−1.12518
$811$	1.20810e10i	0.795297i	0.917538	+	0.397649i	$0.130174\pi$
$811$	1.20810e10i	0.795297i	−0.917538	+	0.397649i	$0.869826\pi$
$812$	− 1.94451e10i	− 1.27457i
$813$	− 4.59750e9i	− 0.300057i
$814$	− 1.17507e10i	− 0.763620i
$815$	2.99910e10	1.94062
$816$	7.49765e9	0.483069
$817$	7.75235e9i	0.497343i
$818$	5.05947e10	3.23198
$819$	0	0
$820$	−7.95162e10	−5.03625
$821$	− 2.10707e10i	− 1.32886i	−0.747352	−	0.664428i	$-0.768675\pi$
$821$	− 2.10707e10i	− 1.32886i	0.747352	−	0.664428i	$-0.231325\pi$
$822$	−2.52294e10	−1.58437
$823$	−4.97330e9	−0.310989	−0.155494	−	0.987837i	$-0.549697\pi$
$823$	−4.97330e9	−0.310989	−0.155494	+	0.987837i	$0.549697\pi$
$824$	− 2.04687e10i	− 1.27451i
$825$	2.34880e10i	1.45632i
$826$	4.95211e10i	3.05746i
$827$	− 4.23151e9i	− 0.260151i	−0.991504	−	0.130076i	$-0.958478\pi$
$827$	− 4.23151e9i	− 0.260151i	0.991504	−	0.130076i	$-0.0415220\pi$
$828$	−4.97206e10	−3.04390
$829$	−1.94458e10	−1.18546	−0.592729	−	0.805402i	$-0.701949\pi$
$829$	−1.94458e10	−1.18546	−0.592729	+	0.805402i	$0.701949\pi$
$830$	− 4.32297e10i	− 2.62427i
$831$	1.63561e10	0.988729
$832$	0	0
$833$	1.15912e9	0.0694818
$834$	3.19295e10i	1.90595i
$835$	−4.05327e10	−2.40937
$836$	3.70730e10	2.19450
$837$	− 8.50701e9i	− 0.501462i
$838$	3.04924e10i	1.78994i
$839$	− 1.48186e9i	− 0.0866244i	−0.999062	−	0.0433122i	$-0.986209\pi$
$839$	− 1.48186e9i	− 0.0866244i	0.999062	−	0.0433122i	$-0.0137910\pi$
$840$	7.63995e10i	4.44747i
$841$	−1.15959e10	−0.672231
$842$	2.64003e10	1.52411
$843$	4.83533e8i	0.0277990i
$844$	3.95754e10	2.26583
$845$	0	0
$846$	1.36276e10	0.773790
$847$	7.40172e9i	0.418544i
$848$	−2.26237e10	−1.27403
$849$	1.17426e10	0.658547
$850$	1.12458e10i	0.628095i
$851$	1.08528e10i	0.603652i
$852$	4.76949e9i	0.264200i
$853$	5.95360e9i	0.328442i	0.986424	+	0.164221i	$0.0525109\pi$
$853$	5.95360e9i	0.328442i	−0.986424	+	0.164221i	$0.947489\pi$
$854$	1.00620e10	0.552820
$855$	5.24923e10	2.87219
$856$	5.96395e9i	0.324994i
$857$	8.37353e9	0.454439	0.227220	−	0.973844i	$-0.427037\pi$
$857$	8.37353e9	0.454439	0.227220	+	0.973844i	$0.427037\pi$
$858$	0	0
$859$	−2.14894e10	−1.15677	−0.578386	−	0.815763i	$-0.696317\pi$
$859$	−2.14894e10	−1.15677	−0.578386	+	0.815763i	$0.696317\pi$
$860$	1.99161e10i	1.06773i
$861$	5.44862e10	2.90921
$862$	−3.15574e10	−1.67813
$863$	− 9.93195e9i	− 0.526013i	−0.964794	−	0.263007i	$-0.915286\pi$
$863$	− 9.93195e9i	− 0.526013i	0.964794	−	0.263007i	$-0.0847141\pi$
$864$	− 3.49727e8i	− 0.0184472i
$865$	2.30581e9i	0.121134i
$866$	3.71657e10i	1.94460i
$867$	2.69873e10	1.40635
$868$	3.46185e10	1.79676
$869$	1.80671e10i	0.933940i
$870$	−4.42412e10	−2.27777
$871$	0	0
$872$	−7.32179e8	−0.0373946
$873$	− 1.15450e10i	− 0.587277i
$874$	−5.12875e10	−2.59849
$875$	−6.22084e9	−0.313921
$876$	1.00245e11i	5.03847i
$877$	3.23745e10i	1.62071i	0.585940	+	0.810355i	$0.300725\pi$
$877$	3.23745e10i	1.62071i	−0.585940	+	0.810355i	$0.699275\pi$
$878$	− 2.13152e10i	− 1.06282i
$879$	2.55846e10i	1.27063i
$880$	2.42142e10	1.19779
$881$	−6.00548e9	−0.295891	−0.147946	−	0.988995i	$-0.547266\pi$
$881$	−6.00548e9	−0.295891	−0.147946	+	0.988995i	$0.547266\pi$
$882$	− 1.13189e10i	− 0.555473i
$883$	2.76852e9	0.135327	0.0676636	−	0.997708i	$-0.478446\pi$
$883$	2.76852e9	0.135327	0.0676636	+	0.997708i	$0.478446\pi$
$884$	0	0
$885$	7.52195e10	3.64778
$886$	− 5.25788e10i	− 2.53976i
$887$	3.08999e10	1.48670	0.743351	−	0.668902i	$-0.233235\pi$
$887$	3.08999e10	1.48670	0.743351	+	0.668902i	$0.233235\pi$
$888$	−3.15624e10	−1.51260
$889$	− 3.37300e10i	− 1.61013i
$890$	− 7.06239e10i	− 3.35804i
$891$	− 7.29955e9i	− 0.345720i
$892$	− 3.34356e10i	− 1.57736i
$893$	9.38464e9	0.440999
$894$	2.32131e10	1.08656
$895$	3.15237e10i	1.46980i
$896$	−4.10341e10	−1.90575
$897$	0	0
$898$	−5.43871e10	−2.50627
$899$	1.00659e10i	0.462056i
$900$	7.33144e10	3.35228
$901$	8.28972e9	0.377575
$902$	− 5.10869e10i	− 2.31786i
$903$	− 1.36469e10i	− 0.616777i
$904$	− 1.06273e10i	− 0.478448i
$905$	2.12341e10i	0.952280i
$906$	−5.03533e10	−2.24946
$907$	−2.63801e10	−1.17395	−0.586977	−	0.809604i	$-0.699682\pi$
$907$	−2.63801e10	−1.17395	−0.586977	+	0.809604i	$0.699682\pi$
$908$	− 8.91913e10i	− 3.95387i
$909$	−3.28247e10	−1.44953
$910$	0	0
$911$	9.42968e9	0.413221	0.206611	−	0.978423i	$-0.433757\pi$
$911$	9.42968e9	0.413221	0.206611	+	0.978423i	$0.433757\pi$
$912$	− 5.04194e10i	− 2.20098i
$913$	1.85422e10	0.806329
$914$	−3.70836e10	−1.60646
$915$	− 1.52836e10i	− 0.659557i
$916$	2.12724e10i	0.914498i
$917$	− 3.01517e10i	− 1.29128i
$918$	7.67844e9i	0.327585i
$919$	3.27533e10	1.39204	0.696019	−	0.718023i	$-0.254953\pi$
$919$	3.27533e10	1.39204	0.696019	+	0.718023i	$0.254953\pi$
$920$	−6.61594e10	−2.80114
$921$	4.92691e9i	0.207810i
$922$	−1.41019e10	−0.592544
$923$	0	0
$924$	−6.52619e10	−2.72150
$925$	− 1.60027e10i	− 0.664809i
$926$	−6.59151e9	−0.272801
$927$	2.47569e10	1.02074
$928$	4.13814e8i	0.0169976i
$929$	2.49414e10i	1.02062i	0.859989	+	0.510312i	$0.170470\pi$
$929$	2.49414e10i	1.02062i	−0.859989	+	0.510312i	$0.829530\pi$
$930$	− 7.87636e10i	− 3.21096i
$931$	− 7.79473e9i	− 0.316576i
$932$	−3.90434e9	−0.157976
$933$	−5.06175e10	−2.04040
$934$	− 3.17917e9i	− 0.127673i
$935$	−8.87251e9	−0.354982
$936$	0	0
$937$	−1.59590e10	−0.633750	−0.316875	−	0.948467i	$-0.602634\pi$
$937$	−1.59590e10	−0.633750	−0.316875	+	0.948467i	$0.602634\pi$
$938$	− 8.03562e10i	− 3.17914i
$939$	−6.09972e10	−2.40425
$940$	2.41095e10	0.946763
$941$	2.91329e8i	0.0113978i	0.999984	+	0.00569888i	$0.00181402\pi$
$941$	2.91329e8i	0.0113978i	−0.999984	+	0.00569888i	$0.998186\pi$
$942$	2.33631e10i	0.910653i
$943$	4.71832e10i	1.83230i
$944$	− 4.21577e10i	− 1.63108i
$945$	−2.64481e10	−1.01949
$946$	−1.27955e10	−0.491405
$947$	− 1.58421e10i	− 0.606160i	−0.952965	−	0.303080i	$-0.901985\pi$
$947$	− 1.58421e10i	− 0.606160i	0.952965	−	0.303080i	$-0.0980149\pi$
$948$	9.66467e10	3.68432
$949$	0	0
$950$	7.56248e10	2.86175
$951$	7.18405e10i	2.70856i
$952$	−1.56897e10	−0.589366
$953$	−4.30011e10	−1.60937	−0.804683	−	0.593705i	$-0.797665\pi$
$953$	−4.30011e10	−1.60937	−0.804683	+	0.593705i	$0.797665\pi$
$954$	− 8.09495e10i	− 3.01853i
$955$	− 4.93871e9i	− 0.183486i
$956$	− 4.15240e10i	− 1.53708i
$957$	− 1.89760e10i	− 0.699862i
$958$	3.67466e10	1.35032
$959$	1.78465e10	0.653412
$960$	6.12519e10i	2.23445i
$961$	9.59202e9	0.348641
$962$	0	0
$963$	−7.21341e9	−0.260285
$964$	3.33123e10i	1.19767i
$965$	5.67966e10	2.03459
$966$	9.02846e10	3.22251
$967$	4.06842e10i	1.44688i	0.690386	+	0.723441i	$0.257441\pi$
$967$	4.06842e10i	1.44688i	−0.690386	+	0.723441i	$0.742559\pi$
$968$	− 1.86408e10i	− 0.660541i
$969$	1.84746e10i	0.652290i
$970$	− 3.05942e10i	− 1.07631i
$971$	5.95290e9	0.208671	0.104335	−	0.994542i	$-0.466729\pi$
$971$	5.95290e9	0.208671	0.104335	+	0.994542i	$0.466729\pi$
$972$	−7.47779e10	−2.61181
$973$	− 2.25859e10i	− 0.786036i
$974$	−5.48279e10	−1.90128
$975$	0	0
$976$	−8.56590e9	−0.294916
$977$	− 8.40433e9i	− 0.288318i	−0.989555	−	0.144159i	$-0.953952\pi$
$977$	− 8.40433e9i	− 0.288318i	0.989555	−	0.144159i	$-0.0460477\pi$
$978$	1.03073e11	3.52337
$979$	3.02921e10	1.03179
$980$	− 2.00250e10i	− 0.679643i
$981$	− 8.85571e8i	− 0.0299490i
$982$	3.67780e9i	0.123936i
$983$	− 3.76311e10i	− 1.26360i	−0.775132	−	0.631799i	$-0.782317\pi$
$983$	− 3.76311e10i	− 1.26360i	0.775132	−	0.631799i	$-0.217683\pi$
$984$	−1.37220e11	−4.59129
$985$	3.86529e10	1.28871
$986$	− 9.08552e9i	− 0.301843i
$987$	−1.65204e10	−0.546902
$988$	0	0
$989$	1.18178e10	0.388462
$990$	8.66404e10i	2.83790i
$991$	4.38732e10	1.43199	0.715997	−	0.698103i	$-0.245972\pi$
$991$	4.38732e10	1.43199	0.715997	+	0.698103i	$0.245972\pi$
$992$	−7.36723e8	−0.0239615
$993$	1.06031e10i	0.343647i
$994$	− 5.05352e9i	− 0.163208i
$995$	5.32255e10i	1.71293i
$996$	− 9.91879e10i	− 3.18091i
$997$	2.15416e10	0.688408	0.344204	−	0.938895i	$-0.388149\pi$
$997$	2.15416e10	0.688408	0.344204	+	0.938895i	$0.388149\pi$
$998$	5.99824e10	1.91015
$999$	− 1.09263e10i	− 0.346733i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.f.168.38		42
13.5	odd	4		169.8.a.i.1.19	yes	21
13.8	odd	4		169.8.a.h.1.3	✓	21
13.12	even	2	inner	169.8.b.f.168.5		42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.3	✓	21	13.8	odd	4
169.8.a.i.1.19	yes	21	13.5	odd	4
169.8.b.f.168.5		42	13.12	even	2	inner
169.8.b.f.168.38		42	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	169.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+19.6237i q^{2} -72.4635 q^{3} -257.091 q^{4} -413.760i q^{5} -1422.00i q^{6} +1005.88i q^{7} -2533.24i q^{8} +3063.96 q^{9} +8119.51 q^{10} +3482.63i q^{11} +18629.7 q^{12} -19739.1 q^{14} +29982.5i q^{15} +16804.1 q^{16} -6157.31 q^{17} +60126.4i q^{18} +41406.0i q^{19} +106374. i q^{20} -72889.6i q^{21} -68342.2 q^{22} +63119.9 q^{23} +183568. i q^{24} -93072.0 q^{25} -63547.8 q^{27} -258602. i q^{28} +75193.0 q^{29} -588368. q^{30} +133868. i q^{31} +5503.36i q^{32} -252364. i q^{33} -120829. i q^{34} +416192. q^{35} -787717. q^{36} +171939. i q^{37} -812541. q^{38} -1.04815e6 q^{40} +747517. i q^{41} +1.43037e6 q^{42} +187227. q^{43} -895352. i q^{44} -1.26774e6i q^{45} +1.23865e6i q^{46} -226649. i q^{47} -1.21768e6 q^{48} -188251. q^{49} -1.82642e6i q^{50} +446180. q^{51} -1.34632e6 q^{53} -1.24705e6i q^{54} +1.44097e6 q^{55} +2.54814e6 q^{56} -3.00043e6i q^{57} +1.47557e6i q^{58} -2.50878e6i q^{59} -7.70822e6i q^{60} -509752. q^{61} -2.62699e6 q^{62} +3.08198e6i q^{63} +2.04292e6 q^{64} +4.95232e6 q^{66} +4.07091e6i q^{67} +1.58299e6 q^{68} -4.57389e6 q^{69} +8.16725e6i q^{70} +256015. i q^{71} -7.76177e6i q^{72} +5.38093e6i q^{73} -3.37408e6 q^{74} +6.74433e6 q^{75} -1.06451e7i q^{76} -3.50311e6 q^{77} +5.18777e6 q^{79} -6.95285e6i q^{80} -2.09599e6 q^{81} -1.46691e7 q^{82} -5.32418e6i q^{83} +1.87392e7i q^{84} +2.54765e6i q^{85} +3.67410e6i q^{86} -5.44875e6 q^{87} +8.82235e6 q^{88} -8.69805e6i q^{89} +2.48779e7 q^{90} -1.62276e7 q^{92} -9.70053e6i q^{93} +4.44770e6 q^{94} +1.71321e7 q^{95} -398793. i q^{96} -3.76799e6i q^{97} -3.69419e6i q^{98} +1.06706e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9} + 10334 q^{10} - 43590 q^{12} - 358 q^{14} + 226410 q^{16} - 90032 q^{17} - 548348 q^{22} + 256210 q^{23} - 733074 q^{25} - 213286 q^{27} - 124658 q^{29} - 19522 q^{30}+ \cdots - 35320062 q^{95}+O(q^{100}) \) 42 * q - 52 * q^3 - 2818 * q^4 + 30930 * q^9 + 10334 * q^10 - 43590 * q^12 - 358 * q^14 + 226410 * q^16 - 90032 * q^17 - 548348 * q^22 + 256210 * q^23 - 733074 * q^25 - 213286 * q^27 - 124658 * q^29 - 19522 * q^30 + 1697720 * q^35 + 548490 * q^36 - 451662 * q^38 - 4577654 * q^40 + 5473672 * q^42 - 1788978 * q^43 + 12604776 * q^48 - 263060 * q^49 + 2368682 * q^51 - 9100054 * q^53 + 11048060 * q^55 - 4668772 * q^56 + 11434024 * q^61 + 27187306 * q^62 - 19535848 * q^64 - 35613954 * q^66 + 24504984 * q^68 - 5544412 * q^69 + 19250156 * q^74 + 86424794 * q^75 - 18892072 * q^77 - 37226562 * q^79 + 82110898 * q^81 - 48380654 * q^82 + 18055788 * q^87 + 151356360 * q^88 - 6671120 * q^90 - 138678780 * q^92 + 57911426 * q^94 - 35320062 * q^95

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