LMFDB - Embedded newform 169.8.b.f.168.36

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)

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);

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")

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			168.36
Character	$\chi$	$=$	169.168
Dual form			169.8.b.f.168.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+16.7260i q^{2} -13.7721 q^{3} -151.758 q^{4} -0.583029i q^{5} -230.352i q^{6} -274.333i q^{7} -397.369i q^{8} -1997.33 q^{9} +9.75172 q^{10} -1092.42i q^{11} +2090.03 q^{12} +4588.49 q^{14} +8.02956i q^{15} -12778.6 q^{16} -24367.8 q^{17} -33407.2i q^{18} -29441.5i q^{19} +88.4791i q^{20} +3778.16i q^{21} +18271.7 q^{22} +87568.7 q^{23} +5472.63i q^{24} +78124.7 q^{25} +57627.1 q^{27} +41632.2i q^{28} -98544.5 q^{29} -134.302 q^{30} -46345.4i q^{31} -264598. i q^{32} +15044.9i q^{33} -407575. i q^{34} -159.944 q^{35} +303110. q^{36} +217133. i q^{37} +492438. q^{38} -231.678 q^{40} +623135. i q^{41} -63193.3 q^{42} +751529. q^{43} +165783. i q^{44} +1164.50i q^{45} +1.46467e6i q^{46} -1.15731e6i q^{47} +175988. q^{48} +748284. q^{49} +1.30671e6i q^{50} +335597. q^{51} +498300. q^{53} +963869. i q^{54} -636.912 q^{55} -109012. q^{56} +405473. i q^{57} -1.64825e6i q^{58} +1.11248e6i q^{59} -1218.55i q^{60} -624881. q^{61} +775171. q^{62} +547934. i q^{63} +2.78999e6 q^{64} -251641. q^{66} +28848.2i q^{67} +3.69801e6 q^{68} -1.20601e6 q^{69} -2675.22i q^{70} -3.65711e6i q^{71} +793677. i q^{72} +5.20600e6i q^{73} -3.63176e6 q^{74} -1.07594e6 q^{75} +4.46798e6i q^{76} -299687. q^{77} -2.37817e6 q^{79} +7450.29i q^{80} +3.57451e6 q^{81} -1.04225e7 q^{82} -1.77288e6i q^{83} -573364. i q^{84} +14207.2i q^{85} +1.25700e7i q^{86} +1.35717e6 q^{87} -434094. q^{88} -2.72509e6i q^{89} -19477.4 q^{90} -1.32892e7 q^{92} +638275. i q^{93} +1.93570e7 q^{94} -17165.3 q^{95} +3.64407e6i q^{96} +838595. i q^{97} +1.25158e7i q^{98} +2.18192e6i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	16.7260i	1.47838i	0.673497	+	0.739190i	$0.264791\pi$
$2$	16.7260i	1.47838i	−0.673497	+	0.739190i	$0.735209\pi$
$3$	−13.7721	−0.294494	−0.147247	−	0.989100i	$-0.547041\pi$
$3$	−13.7721	−0.294494	−0.147247	+	0.989100i	$0.547041\pi$
$4$	−151.758	−1.18561
$5$	− 0.583029i	− 0.00208591i	−0.999999	−	0.00104295i	$-0.999668\pi$
$5$	− 0.583029i	− 0.00208591i	0.999999	−	0.00104295i	$-0.000331983\pi$
$6$	− 230.352i	− 0.435374i
$7$	− 274.333i	− 0.302298i	−0.988511	−	0.151149i	$-0.951703\pi$
$7$	− 274.333i	− 0.302298i	0.988511	−	0.151149i	$-0.0482974\pi$
$8$	− 397.369i	− 0.274397i
$9$	−1997.33	−0.913273
$10$	9.75172	0.00308377
$11$	− 1092.42i	− 0.247465i	−0.992316	−	0.123733i	$-0.960513\pi$
$11$	− 1092.42i	− 0.247465i	0.992316	−	0.123733i	$-0.0394865\pi$
$12$	2090.03	0.349154
$13$	0	0
$13$	0	0
$14$	4588.49	0.446911
$15$	8.02956i	0 0.000614288i
$16$	−12778.6	−0.779944
$17$	−24367.8	−1.20295	−0.601473	−	0.798893i	$-0.705419\pi$
$17$	−24367.8	−1.20295	−0.601473	+	0.798893i	$0.705419\pi$
$18$	− 33407.2i	− 1.35016i
$19$	− 29441.5i	− 0.984743i	−0.870385	−	0.492372i	$-0.836130\pi$
$19$	− 29441.5i	− 0.984743i	0.870385	−	0.492372i	$-0.163870\pi$
$20$	88.4791i	0.00247307i
$21$	3778.16i	0.0890251i
$22$	18271.7	0.365848
$23$	87568.7	1.50073	0.750363	−	0.661026i	$-0.229879\pi$
$23$	87568.7	1.50073	0.750363	+	0.661026i	$0.229879\pi$
$24$	5472.63i	0.0808084i
$25$	78124.7	0.999996
$26$	0	0
$27$	57627.1	0.563448
$28$	41632.2i	0.358407i
$29$	−98544.5	−0.750308	−0.375154	−	0.926963i	$-0.622410\pi$
$29$	−98544.5	−0.750308	−0.375154	+	0.926963i	$0.622410\pi$
$30$	−134.302	−0.000908151 0
$31$	− 46345.4i	− 0.279409i	−0.990193	−	0.139705i	$-0.955385\pi$
$31$	− 46345.4i	− 0.279409i	0.990193	−	0.139705i	$-0.0446153\pi$
$32$	− 264598.i	− 1.42745i
$33$	15044.9i	0.0728771i
$34$	− 407575.i	− 1.77841i
$35$	−159.944	−0.000630566 0
$36$	303110.	1.08278
$37$	217133.i	0.704727i	0.935863	+	0.352363i	$0.114622\pi$
$37$	217133.i	0.704727i	−0.935863	+	0.352363i	$0.885378\pi$
$38$	492438.	1.45582
$39$	0	0
$40$	−231.678	−0.000572367 0
$41$	623135.i	1.41201i	0.708206	+	0.706006i	$0.249505\pi$
$41$	623135.i	1.41201i	−0.708206	+	0.706006i	$0.750495\pi$
$42$	−63193.3	−0.131613
$43$	751529.	1.44147	0.720736	−	0.693210i	$-0.243804\pi$
$43$	751529.	1.44147	0.720736	+	0.693210i	$0.243804\pi$
$44$	165783.i	0.293397i
$45$	1164.50i	0.00190500i
$46$	1.46467e6i	2.21864i
$47$	− 1.15731e6i	− 1.62594i	−0.582303	−	0.812972i	$-0.697848\pi$
$47$	− 1.15731e6i	− 1.62594i	0.582303	−	0.812972i	$-0.302152\pi$
$48$	175988.	0.229689
$49$	748284.	0.908616
$50$	1.30671e6i	1.47837i
$51$	335597.	0.354260
$52$	0	0
$53$	498300.	0.459753	0.229877	−	0.973220i	$-0.426168\pi$
$53$	498300.	0.459753	0.229877	+	0.973220i	$0.426168\pi$
$54$	963869.i	0.832990i
$55$	−636.912	−0.000516190 0
$56$	−109012.	−0.0829497
$57$	405473.i	0.290001i
$58$	− 1.64825e6i	− 1.10924i
$59$	1.11248e6i	0.705194i	0.935775	+	0.352597i	$0.114701\pi$
$59$	1.11248e6i	0.705194i	−0.935775	+	0.352597i	$0.885299\pi$
$60$	− 1218.55i	0 0.000728304i
$61$	−624881.	−0.352487	−0.176243	−	0.984347i	$-0.556395\pi$
$61$	−624881.	−0.352487	−0.176243	+	0.984347i	$0.556395\pi$
$62$	775171.	0.413073
$63$	547934.i	0.276081i
$64$	2.78999e6	1.33037
$65$	0	0
$66$	−251641.	−0.107740
$67$	28848.2i	0.0117181i	0.999983	+	0.00585904i	$0.00186500\pi$
$67$	28848.2i	0.0117181i	−0.999983	+	0.00585904i	$0.998135\pi$
$68$	3.69801e6	1.42622
$69$	−1.20601e6	−0.441955
$70$	− 2675.22i	0 0.000932217i
$71$	− 3.65711e6i	− 1.21265i	−0.795218	−	0.606323i	$-0.792644\pi$
$71$	− 3.65711e6i	− 1.21265i	0.795218	−	0.606323i	$-0.207356\pi$
$72$	793677.i	0.250600i
$73$	5.20600e6i	1.56630i	0.621835	+	0.783148i	$0.286387\pi$
$73$	5.20600e6i	1.56630i	−0.621835	+	0.783148i	$0.713613\pi$
$74$	−3.63176e6	−1.04185
$75$	−1.07594e6	−0.294493
$76$	4.46798e6i	1.16752i
$77$	−299687.	−0.0748083
$78$	0	0
$79$	−2.37817e6	−0.542685	−0.271343	−	0.962483i	$-0.587468\pi$
$79$	−2.37817e6	−0.542685	−0.271343	+	0.962483i	$0.587468\pi$
$80$	7450.29i	0.00162689i
$81$	3.57451e6	0.747341
$82$	−1.04225e7	−2.08749
$83$	− 1.77288e6i	− 0.340335i	−0.985415	−	0.170168i	$-0.945569\pi$
$83$	− 1.77288e6i	− 0.340335i	0.985415	−	0.170168i	$-0.0544309\pi$
$84$	− 573364.i	− 0.105549i
$85$	14207.2i	0.00250923i
$86$	1.25700e7i	2.13104i
$87$	1.35717e6	0.220961
$88$	−434094.	−0.0679038
$89$	− 2.72509e6i	− 0.409746i	−0.978789	−	0.204873i	$-0.934322\pi$
$89$	− 2.72509e6i	− 0.409746i	0.978789	−	0.204873i	$-0.0656782\pi$
$90$	−19477.4	−0.00281632
$91$	0	0
$92$	−1.32892e7	−1.77927
$93$	638275.i	0.0822844i
$94$	1.93570e7	2.40376
$95$	−17165.3	−0.00205408
$96$	3.64407e6i	0.420376i
$97$	838595.i	0.0932934i	0.998911	+	0.0466467i	$0.0148535\pi$
$97$	838595.i	0.0932934i	−0.998911	+	0.0466467i	$0.985147\pi$
$98$	1.25158e7i	1.34328i
$99$	2.18192e6i	0.226003i
$100$	−1.18560e7	−1.18560
$101$	−1.35004e7	−1.30383	−0.651916	−	0.758291i	$-0.726035\pi$
$101$	−1.35004e7	−1.30383	−0.651916	+	0.758291i	$0.726035\pi$
$102$	5.61318e6i	0.523731i
$103$	−1.30297e7	−1.17491	−0.587456	−	0.809256i	$-0.699870\pi$
$103$	−1.30297e7	−1.17491	−0.587456	+	0.809256i	$0.699870\pi$
$104$	0	0
$105$	2202.78	0.000185698 0
$106$	8.33454e6i	0.679690i
$107$	2.04841e7	1.61649	0.808246	−	0.588844i	$-0.200417\pi$
$107$	2.04841e7	1.61649	0.808246	+	0.588844i	$0.200417\pi$
$108$	−8.74536e6	−0.668028
$109$	1.74867e7i	1.29335i	0.762768	+	0.646673i	$0.223840\pi$
$109$	1.74867e7i	1.29335i	−0.762768	+	0.646673i	$0.776160\pi$
$110$	− 10653.0i	0 0.000763125i
$111$	− 2.99039e6i	− 0.207538i
$112$	3.50559e6i	0.235775i
$113$	6.72636e6	0.438536	0.219268	−	0.975665i	$-0.429633\pi$
$113$	6.72636e6	0.438536	0.219268	+	0.975665i	$0.429633\pi$
$114$	−6.78192e6	−0.428732
$115$	− 51055.1i	− 0.00313038i
$116$	1.49549e7	0.889570
$117$	0	0
$118$	−1.86072e7	−1.04254
$119$	6.68491e6i	0.363648i
$120$	3190.70	0.000168559 0
$121$	1.82938e7	0.938761
$122$	− 1.04517e7i	− 0.521109i
$123$	− 8.58190e6i	− 0.415830i
$124$	7.03327e6i	0.331269i
$125$	− 91098.1i	− 0.00417181i
$126$	−9.16472e6	−0.408152
$127$	3.10681e7	1.34587	0.672934	−	0.739703i	$-0.265034\pi$
$127$	3.10681e7	1.34587	0.672934	+	0.739703i	$0.265034\pi$
$128$	1.27967e7i	0.539341i
$129$	−1.03502e7	−0.424505
$130$	0	0
$131$	1.55397e7	0.603940	0.301970	−	0.953317i	$-0.402356\pi$
$131$	1.55397e7	0.603940	0.301970	+	0.953317i	$0.402356\pi$
$132$	− 2.28318e6i	− 0.0864036i
$133$	−8.07680e6	−0.297686
$134$	−482514.	−0.0173238
$135$	− 33598.3i	− 0.00117530i
$136$	9.68304e6i	0.330085i
$137$	1.75618e7i	0.583508i	0.956493	+	0.291754i	$0.0942389\pi$
$137$	1.75618e7i	0.583508i	−0.956493	+	0.291754i	$0.905761\pi$
$138$	− 2.01716e7i	− 0.653378i
$139$	2.32529e7	0.734386	0.367193	−	0.930145i	$-0.380319\pi$
$139$	2.32529e7	0.734386	0.367193	+	0.930145i	$0.380319\pi$
$140$	24272.8	0.000747604 0
$141$	1.59386e7i	0.478831i
$142$	6.11687e7	1.79275
$143$	0	0
$144$	2.55230e7	0.712301
$145$	57454.3i	0.00156507i
$146$	−8.70753e7	−2.31558
$147$	−1.03055e7	−0.267582
$148$	− 3.29517e7i	− 0.835529i
$149$	− 7.31136e7i	− 1.81070i	−0.424666	−	0.905350i	$-0.639609\pi$
$149$	− 7.31136e7i	− 1.81070i	0.424666	−	0.905350i	$-0.360391\pi$
$150$	− 1.79962e7i	− 0.435372i
$151$	− 1.74324e7i	− 0.412039i	−0.978548	−	0.206019i	$-0.933949\pi$
$151$	− 1.74324e7i	− 0.412039i	0.978548	−	0.206019i	$-0.0660510\pi$
$152$	−1.16992e7	−0.270211
$153$	4.86706e7	1.09862
$154$	− 5.01255e6i	− 0.110595i
$155$	−27020.7	−0.000582822 0
$156$	0	0
$157$	8.56670e7	1.76671	0.883354	−	0.468706i	$-0.155280\pi$
$157$	8.56670e7	1.76671	0.883354	+	0.468706i	$0.155280\pi$
$158$	− 3.97772e7i	− 0.802295i
$159$	−6.86265e6	−0.135395
$160$	−154268.	−0.00297753
$161$	− 2.40230e7i	− 0.453667i
$162$	5.97871e7i	1.10485i
$163$	− 4.30758e7i	− 0.779070i	−0.921012	−	0.389535i	$-0.872636\pi$
$163$	− 4.30758e7i	− 0.779070i	0.921012	−	0.389535i	$-0.127364\pi$
$164$	− 9.45655e7i	− 1.67409i
$165$	8771.63	0.000152015 0
$166$	2.96532e7	0.503145
$167$	− 6.57708e7i	− 1.09276i	−0.837537	−	0.546381i	$-0.816005\pi$
$167$	− 6.57708e7i	− 1.09276i	0.837537	−	0.546381i	$-0.183995\pi$
$168$	1.50132e6	0.0244282
$169$	0	0
$170$	−237628.	−0.00370960
$171$	5.88044e7i	0.899340i
$172$	−1.14050e8	−1.70902
$173$	1.02460e8	1.50451	0.752255	−	0.658872i	$-0.228966\pi$
$173$	1.02460e8	1.50451	0.752255	+	0.658872i	$0.228966\pi$
$174$	2.26999e7i	0.326665i
$175$	− 2.14322e7i	− 0.302297i
$176$	1.39596e7i	0.193009i
$177$	− 1.53212e7i	− 0.207675i
$178$	4.55797e7	0.605761
$179$	−6.44115e7	−0.839417	−0.419708	−	0.907659i	$-0.637868\pi$
$179$	−6.44115e7	−0.839417	−0.419708	+	0.907659i	$0.637868\pi$
$180$	− 176722.i	− 0.00225859i
$181$	−9.36975e7	−1.17450	−0.587250	−	0.809406i	$-0.699789\pi$
$181$	−9.36975e7	−1.17450	−0.587250	+	0.809406i	$0.699789\pi$
$182$	0	0
$183$	8.60594e6	0.103805
$184$	− 3.47971e7i	− 0.411795i
$185$	126595.	0.00147000
$186$	−1.06758e7	−0.121648
$187$	2.66199e7i	0.297687i
$188$	1.75630e8i	1.92773i
$189$	− 1.58090e7i	− 0.170329i
$190$	− 287106.i	− 0.00303672i
$191$	1.03934e8	1.07929	0.539647	−	0.841891i	$-0.318558\pi$
$191$	1.03934e8	1.07929	0.539647	+	0.841891i	$0.318558\pi$
$192$	−3.84241e7	−0.391786
$193$	− 4.66141e6i	− 0.0466731i	−0.999728	−	0.0233365i	$-0.992571\pi$
$193$	− 4.66141e6i	− 0.0466731i	0.999728	−	0.0233365i	$-0.00742892\pi$
$194$	−1.40263e7	−0.137923
$195$	0	0
$196$	−1.13558e8	−1.07726
$197$	− 1.01127e8i	− 0.942402i	−0.882026	−	0.471201i	$-0.843821\pi$
$197$	− 1.01127e8i	− 0.942402i	0.882026	−	0.471201i	$-0.156179\pi$
$198$	−3.64947e7	−0.334119
$199$	1.29148e8	1.16172	0.580860	−	0.814003i	$-0.302716\pi$
$199$	1.29148e8	1.16172	0.580860	+	0.814003i	$0.302716\pi$
$200$	− 3.10444e7i	− 0.274396i
$201$	− 397301.i	− 0.00345091i
$202$	− 2.25807e8i	− 1.92756i
$203$	2.70340e7i	0.226817i
$204$	−5.09294e7	−0.420013
$205$	363306.	0.00294533
$206$	− 2.17935e8i	− 1.73696i
$207$	−1.74903e8	−1.37057
$208$	0	0
$209$	−3.21625e7	−0.243690
$210$	36843.5i	0 0.000274532i
$211$	1.43953e8	1.05495	0.527475	−	0.849571i	$-0.323139\pi$
$211$	1.43953e8	1.05495	0.527475	+	0.849571i	$0.323139\pi$
$212$	−7.56208e7	−0.545087
$213$	5.03663e7i	0.357118i
$214$	3.42616e8i	2.38979i
$215$	− 438163.i	− 0.00300678i
$216$	− 2.28993e7i	− 0.154608i
$217$	−1.27141e7	−0.0844649
$218$	−2.92481e8	−1.91206
$219$	− 7.16977e7i	− 0.461265i
$220$	96656.2	0.000611998 0
$221$	0	0
$222$	5.00171e7	0.306820
$223$	− 9.50963e7i	− 0.574244i	−0.957894	−	0.287122i	$-0.907301\pi$
$223$	− 9.50963e7i	− 0.574244i	0.957894	−	0.287122i	$-0.0926986\pi$
$224$	−7.25879e7	−0.431515
$225$	−1.56041e8	−0.913269
$226$	1.12505e8i	0.648323i
$227$	− 2.36462e8i	− 1.34175i	−0.741571	−	0.670874i	$-0.765919\pi$
$227$	− 2.36462e8i	− 1.34175i	0.741571	−	0.670874i	$-0.234081\pi$
$228$	− 6.15336e7i	− 0.343827i
$229$	2.40501e8i	1.32340i	0.749767	+	0.661702i	$0.230166\pi$
$229$	2.40501e8i	1.32340i	−0.749767	+	0.661702i	$0.769834\pi$
$230$	853946.	0.00462789
$231$	4.12732e6	0.0220306
$232$	3.91586e7i	0.205882i
$233$	−2.23639e8	−1.15825	−0.579123	−	0.815240i	$-0.696605\pi$
$233$	−2.23639e8	−1.15825	−0.579123	+	0.815240i	$0.696605\pi$
$234$	0	0
$235$	−674743.	−0.00339157
$236$	− 1.68827e8i	− 0.836082i
$237$	3.27525e7	0.159818
$238$	−1.11812e8	−0.537610
$239$	− 2.60257e8i	− 1.23313i	−0.787302	−	0.616567i	$-0.788523\pi$
$239$	− 2.60257e8i	− 1.23313i	0.787302	−	0.616567i	$-0.211477\pi$
$240$	− 102606.i	0 0.000479110i
$241$	− 2.92081e8i	− 1.34414i	−0.740489	−	0.672069i	$-0.765406\pi$
$241$	− 2.92081e8i	− 1.34414i	0.740489	−	0.672069i	$-0.234594\pi$
$242$	3.05981e8i	1.38785i
$243$	−1.75259e8	−0.783535
$244$	9.48305e7	0.417911
$245$	− 436272.i	− 0.00189529i
$246$	1.43540e8	0.614754
$247$	0	0
$248$	−1.84162e7	−0.0766691
$249$	2.44164e7i	0.100227i
$250$	1.52370e6	0.00616752
$251$	−1.00598e8	−0.401542	−0.200771	−	0.979638i	$-0.564345\pi$
$251$	−1.00598e8	−0.401542	−0.200771	+	0.979638i	$0.564345\pi$
$252$	− 8.31532e7i	− 0.327323i
$253$	− 9.56616e7i	− 0.371378i
$254$	5.19644e8i	1.98970i
$255$	− 195663.i	0 0.000738955i
$256$	1.43081e8	0.533018
$257$	1.67536e7	0.0615662	0.0307831	−	0.999526i	$-0.490200\pi$
$257$	1.67536e7	0.0615662	0.0307831	+	0.999526i	$0.490200\pi$
$258$	− 1.73116e8i	− 0.627580i
$259$	5.95669e7	0.213038
$260$	0	0
$261$	1.96826e8	0.685236
$262$	2.59917e8i	0.892853i
$263$	−2.50229e7	−0.0848187	−0.0424094	−	0.999100i	$-0.513503\pi$
$263$	−2.50229e7	−0.0848187	−0.0424094	+	0.999100i	$0.513503\pi$
$264$	5.97839e6	0.0199973
$265$	− 290523.i	0 0.000959004i
$266$	− 1.35092e8i	− 0.440093i
$267$	3.75302e7i	0.120668i
$268$	− 4.37793e6i	− 0.0138930i
$269$	2.47248e7	0.0774462	0.0387231	−	0.999250i	$-0.487671\pi$
$269$	2.47248e7	0.0774462	0.0387231	+	0.999250i	$0.487671\pi$
$270$	561964.	0.00173754
$271$	5.81639e8i	1.77526i	0.460561	+	0.887628i	$0.347648\pi$
$271$	5.81639e8i	1.77526i	−0.460561	+	0.887628i	$0.652352\pi$
$272$	3.11387e8	0.938229
$273$	0	0
$274$	−2.93738e8	−0.862647
$275$	− 8.53448e7i	− 0.247464i
$276$	1.83021e8	0.523985
$277$	2.85853e8	0.808097	0.404048	−	0.914738i	$-0.367603\pi$
$277$	2.85853e8	0.808097	0.404048	+	0.914738i	$0.367603\pi$
$278$	3.88926e8i	1.08570i
$279$	9.25670e7i	0.255177i
$280$	63557.0i	0 0.000173026i
$281$	− 5.88850e8i	− 1.58319i	−0.611048	−	0.791593i	$-0.709252\pi$
$281$	− 5.88850e8i	− 1.58319i	0.611048	−	0.791593i	$-0.290748\pi$
$282$	−2.66588e8	−0.707894
$283$	4.95887e8	1.30056	0.650279	−	0.759695i	$-0.274652\pi$
$283$	4.95887e8	1.30056	0.650279	+	0.759695i	$0.274652\pi$
$284$	5.54995e8i	1.43772i
$285$	236403.	0.000604916 0
$286$	0	0
$287$	1.70947e8	0.426849
$288$	5.28488e8i	1.30365i
$289$	1.83453e8	0.447077
$290$	−960979.	−0.00231377
$291$	− 1.15492e7i	− 0.0274744i
$292$	− 7.90050e8i	− 1.85701i
$293$	5.41027e8i	1.25656i	0.777988	+	0.628279i	$0.216240\pi$
$293$	5.41027e8i	1.25656i	−0.777988	+	0.628279i	$0.783760\pi$
$294$	− 1.72369e8i	− 0.395588i
$295$	648606.	0.00147097
$296$	8.62822e7	0.193375
$297$	− 6.29529e7i	− 0.139434i
$298$	1.22290e9	2.67690
$299$	0	0
$300$	1.63283e8	0.349153
$301$	− 2.06169e8i	− 0.435754i
$302$	2.91574e8	0.609149
$303$	1.85929e8	0.383971
$304$	3.76222e8i	0.768044i
$305$	364324.i	0 0.000735255i
$306$	8.14062e8i	1.62417i
$307$	− 4.24834e8i	− 0.837982i	−0.907990	−	0.418991i	$-0.862384\pi$
$307$	− 4.24834e8i	− 0.837982i	0.907990	−	0.418991i	$-0.137616\pi$
$308$	4.54797e7	0.0886932
$309$	1.79447e8	0.346005
$310$	− 451947.i	0 0.000861632i
$311$	−2.89109e8	−0.545005	−0.272503	−	0.962155i	$-0.587851\pi$
$311$	−2.89109e8	−0.545005	−0.272503	+	0.962155i	$0.587851\pi$
$312$	0	0
$313$	−9.46652e8	−1.74496	−0.872479	−	0.488651i	$-0.837489\pi$
$313$	−9.46652e8	−1.74496	−0.872479	+	0.488651i	$0.837489\pi$
$314$	1.43286e9i	2.61187i
$315$	319461.	0.000575879 0
$316$	3.60906e8	0.643412
$317$	− 4.18587e8i	− 0.738038i	−0.929422	−	0.369019i	$-0.879694\pi$
$317$	− 4.18587e8i	− 0.738038i	0.929422	−	0.369019i	$-0.120306\pi$
$318$	− 1.14784e8i	− 0.200165i
$319$	1.07652e8i	0.185675i
$320$	− 1.62664e6i	− 0.00277503i
$321$	−2.82110e8	−0.476048
$322$	4.01808e8	0.670692
$323$	7.17427e8i	1.18459i
$324$	−5.42459e8	−0.886052
$325$	0	0
$326$	7.20483e8	1.15176
$327$	− 2.40829e8i	− 0.380883i
$328$	2.47615e8	0.387452
$329$	−3.17488e8	−0.491520
$330$	146714.i	0 0.000224736i
$331$	− 4.12347e8i	− 0.624978i	−0.949921	−	0.312489i	$-0.898837\pi$
$331$	− 4.12347e8i	− 0.624978i	0.949921	−	0.312489i	$-0.101163\pi$
$332$	2.69049e8i	0.403504i
$333$	− 4.33687e8i	− 0.643608i
$334$	1.10008e9	1.61552
$335$	16819.3	2.44429e−5 0
$336$	− 4.82795e7i	− 0.0694345i
$337$	5.74808e8	0.818123	0.409061	−	0.912507i	$-0.365856\pi$
$337$	5.74808e8	0.818123	0.409061	+	0.912507i	$0.365856\pi$
$338$	0	0
$339$	−9.26363e7	−0.129146
$340$	− 2.15605e6i	− 0.00297496i
$341$	−5.06285e7	−0.0691441
$342$	−9.83561e8	−1.32957
$343$	− 4.31205e8i	− 0.576971i
$344$	− 2.98635e8i	− 0.395536i
$345$	703138.i	0 0.000921878i
$346$	1.71375e9i	2.22424i
$347$	−1.46161e9	−1.87792	−0.938962	−	0.344020i	$-0.888211\pi$
$347$	−1.46161e9	−1.87792	−0.938962	+	0.344020i	$0.888211\pi$
$348$	−2.05961e8	−0.261973
$349$	4.00590e8i	0.504441i	0.967670	+	0.252221i	$0.0811609\pi$
$349$	4.00590e8i	0.504441i	−0.967670	+	0.252221i	$0.918839\pi$
$350$	3.58474e8	0.446910
$351$	0	0
$352$	−2.89051e8	−0.353244
$353$	− 5.97796e8i	− 0.723338i	−0.932307	−	0.361669i	$-0.882207\pi$
$353$	− 5.97796e8i	− 0.723338i	0.932307	−	0.361669i	$-0.117793\pi$
$354$	2.56261e8	0.307023
$355$	−2.13220e6	−0.00252947
$356$	4.13553e8i	0.485798i
$357$	− 9.20655e7i	− 0.107092i
$358$	− 1.07734e9i	− 1.24098i
$359$	6.48576e8i	0.739827i	0.929066	+	0.369913i	$0.120613\pi$
$359$	6.48576e8i	0.739827i	−0.929066	+	0.369913i	$0.879387\pi$
$360$	462737.	0.000522728 0
$361$	2.70670e7	0.0302806
$362$	− 1.56718e9i	− 1.73636i
$363$	−2.51945e8	−0.276460
$364$	0	0
$365$	3.03525e6	0.00326715
$366$	1.43943e8i	0.153464i
$367$	6.24121e8	0.659080	0.329540	−	0.944142i	$-0.393106\pi$
$367$	6.24121e8	0.659080	0.329540	+	0.944142i	$0.393106\pi$
$368$	−1.11901e9	−1.17048
$369$	− 1.24461e9i	− 1.28955i
$370$	2.11742e6i	0.00217321i
$371$	− 1.36700e8i	− 0.138983i
$372$	− 9.68631e7i	− 0.0975569i
$373$	−1.14420e8	−0.114162	−0.0570810	−	0.998370i	$-0.518179\pi$
$373$	−1.14420e8	−0.114162	−0.0570810	+	0.998370i	$0.518179\pi$
$374$	−4.45243e8	−0.440095
$375$	1.25462e6i	0.00122857i
$376$	−4.59878e8	−0.446154
$377$	0	0
$378$	2.64421e8	0.251811
$379$	− 1.79026e9i	− 1.68919i	−0.535407	−	0.844594i	$-0.679842\pi$
$379$	− 1.79026e9i	− 1.68919i	0.535407	−	0.844594i	$-0.320158\pi$
$380$	2.60496e6	0.00243534
$381$	−4.27874e8	−0.396350
$382$	1.73839e9i	1.59561i
$383$	1.71291e9i	1.55789i	0.627090	+	0.778947i	$0.284246\pi$
$383$	1.71291e9i	1.55789i	−0.627090	+	0.778947i	$0.715754\pi$
$384$	− 1.76238e8i	− 0.158833i
$385$	174726.i	0 0.000156043i
$386$	7.79665e7	0.0690005
$387$	−1.50105e9	−1.31646
$388$	− 1.27263e8i	− 0.110609i
$389$	−1.68291e9	−1.44957	−0.724783	−	0.688978i	$-0.758060\pi$
$389$	−1.68291e9	−1.44957	−0.724783	+	0.688978i	$0.758060\pi$
$390$	0	0
$391$	−2.13386e9	−1.80529
$392$	− 2.97345e8i	− 0.249322i
$393$	−2.14015e8	−0.177857
$394$	1.69145e9	1.39323
$395$	1.38654e6i	0.00113199i
$396$	− 3.31123e8i	− 0.267951i
$397$	− 2.12886e9i	− 1.70758i	−0.520619	−	0.853789i	$-0.674299\pi$
$397$	− 2.12886e9i	− 1.70758i	0.520619	−	0.853789i	$-0.325701\pi$
$398$	2.16012e9i	1.71746i
$399$	1.11235e8	0.0876668
$400$	−9.98323e8	−0.779940
$401$	9.05120e8i	0.700972i	0.936568	+	0.350486i	$0.113984\pi$
$401$	9.05120e8i	0.700972i	−0.936568	+	0.350486i	$0.886016\pi$
$402$	6.64524e6	0.00510175
$403$	0	0
$404$	2.04879e9	1.54583
$405$	− 2.08404e6i	− 0.00155889i
$406$	−4.52170e8	−0.335321
$407$	2.37200e8	0.174395
$408$	− 1.33356e8i	− 0.0972080i
$409$	1.60956e9i	1.16326i	0.813454	+	0.581629i	$0.197585\pi$
$409$	1.60956e9i	1.16326i	−0.813454	+	0.581629i	$0.802415\pi$
$410$	6.07664e6i	0.00435432i
$411$	− 2.41863e8i	− 0.171840i
$412$	1.97736e9	1.39298
$413$	3.05189e8	0.213179
$414$	− 2.92543e9i	− 2.02623i
$415$	−1.03364e6	−0.000709908 0
$416$	0	0
$417$	−3.20241e8	−0.216273
$418$	− 5.37948e8i	− 0.360266i
$419$	−2.87655e8	−0.191039	−0.0955196	−	0.995428i	$-0.530451\pi$
$419$	−2.87655e8	−0.191039	−0.0955196	+	0.995428i	$0.530451\pi$
$420$	−334288.	−0.000220165 0
$421$	− 1.52458e7i	− 0.00995781i	−0.999988	−	0.00497891i	$-0.998415\pi$
$421$	− 1.52458e7i	− 0.00995781i	0.999988	−	0.00497891i	$-0.00158484\pi$
$422$	2.40775e9i	1.55962i
$423$	2.31152e9i	1.48493i
$424$	− 1.98009e8i	− 0.126155i
$425$	−1.90373e9	−1.20294
$426$	−8.42424e8	−0.527955
$427$	1.71426e8i	0.106556i
$428$	−3.10862e9	−1.91652
$429$	0	0
$430$	7.32870e6	0.00444516
$431$	9.22436e8i	0.554965i	0.960731	+	0.277483i	$0.0895001\pi$
$431$	9.22436e8i	0.554965i	−0.960731	+	0.277483i	$0.910500\pi$
$432$	−7.36394e8	−0.439458
$433$	1.88411e9	1.11532	0.557660	−	0.830069i	$-0.311699\pi$
$433$	1.88411e9	1.11532	0.557660	+	0.830069i	$0.311699\pi$
$434$	− 2.12655e8i	− 0.124871i
$435$	− 791269.i	0 0.000460905i
$436$	− 2.65374e9i	− 1.53340i
$437$	− 2.57816e9i	− 1.47783i
$438$	1.19921e9	0.681925
$439$	8.65096e8	0.488021	0.244010	−	0.969773i	$-0.421537\pi$
$439$	8.65096e8	0.488021	0.244010	+	0.969773i	$0.421537\pi$
$440$	253089.i	0 0.000141641i
$441$	−1.49457e9	−0.829814
$442$	0	0
$443$	9.40807e8	0.514147	0.257074	−	0.966392i	$-0.417242\pi$
$443$	9.40807e8	0.514147	0.257074	+	0.966392i	$0.417242\pi$
$444$	4.53815e8i	0.246058i
$445$	−1.58880e6	−0.000854694 0
$446$	1.59058e9	0.848951
$447$	1.00693e9i	0.533241i
$448$	− 7.65386e8i	− 0.402168i
$449$	− 8.79790e7i	− 0.0458687i	−0.999737	−	0.0229344i	$-0.992699\pi$
$449$	− 8.79790e7i	− 0.0458687i	0.999737	−	0.0229344i	$-0.00730087\pi$
$450$	− 2.60993e9i	− 1.35016i
$451$	6.80724e8	0.349424
$452$	−1.02078e9	−0.519931
$453$	2.40081e8i	0.121343i
$454$	3.95506e9	1.98361
$455$	0	0
$456$	1.61123e8	0.0795755
$457$	− 1.95139e9i	− 0.956396i	−0.878252	−	0.478198i	$-0.841290\pi$
$457$	− 1.95139e9i	− 0.956396i	0.878252	−	0.478198i	$-0.158710\pi$
$458$	−4.02261e9	−1.95649
$459$	−1.40425e9	−0.677797
$460$	7.74801e6i	0.00371140i
$461$	1.15818e8i	0.0550585i	0.999621	+	0.0275293i	$0.00876394\pi$
$461$	1.15818e8i	0.0550585i	−0.999621	+	0.0275293i	$0.991236\pi$
$462$	6.90335e7i	0.0325696i
$463$	− 1.46488e9i	− 0.685914i	−0.939351	−	0.342957i	$-0.888571\pi$
$463$	− 1.46488e9i	− 0.685914i	0.939351	−	0.342957i	$-0.111429\pi$
$464$	1.25926e9	0.585198
$465$	372133.	0.000171638 0
$466$	− 3.74057e9i	− 1.71233i
$467$	6.23541e8	0.283306	0.141653	−	0.989916i	$-0.454758\pi$
$467$	6.23541e8	0.283306	0.141653	+	0.989916i	$0.454758\pi$
$468$	0	0
$469$	7.91402e6	0.00354236
$470$	− 1.12857e7i	− 0.00501403i
$471$	−1.17982e9	−0.520286
$472$	4.42064e8	0.193503
$473$	− 8.20984e8i	− 0.356714i
$474$	5.47817e8i	0.236271i
$475$	− 2.30011e9i	− 0.984739i
$476$	− 1.01449e9i	− 0.431144i
$477$	−9.95268e8	−0.419880
$478$	4.35305e9	1.82304
$479$	2.10997e9i	0.877207i	0.898681	+	0.438604i	$0.144527\pi$
$479$	2.10997e9i	0.877207i	−0.898681	+	0.438604i	$0.855473\pi$
$480$	2.12460e6	0.000876865 0
$481$	0	0
$482$	4.88534e9	1.98715
$483$	3.30848e8i	0.133602i
$484$	−2.77622e9	−1.11300
$485$	488925.	0.000194602 0
$486$	− 2.93138e9i	− 1.15836i
$487$	− 3.08900e9i	− 1.21190i	−0.795503	−	0.605949i	$-0.792793\pi$
$487$	− 3.08900e9i	− 1.21190i	0.795503	−	0.605949i	$-0.207207\pi$
$488$	2.48309e8i	0.0967214i
$489$	5.93245e8i	0.229431i
$490$	7.29706e6	0.00280196
$491$	3.63222e9	1.38480	0.692399	−	0.721515i	$-0.256554\pi$
$491$	3.63222e9	1.38480	0.692399	+	0.721515i	$0.256554\pi$
$492$	1.30237e9i	0.493010i
$493$	2.40132e9	0.902579
$494$	0	0
$495$	1.27212e6	0.000471423 0
$496$	5.92229e8i	0.217923i
$497$	−1.00327e9	−0.366581
$498$	−4.08388e8	−0.148173
$499$	− 4.71935e9i	− 1.70032i	−0.526525	−	0.850160i	$-0.676505\pi$
$499$	− 4.71935e9i	− 1.70032i	0.526525	−	0.850160i	$-0.323495\pi$
$500$	1.38248e7i	0.00494612i
$501$	9.05805e8i	0.321812i
$502$	− 1.68260e9i	− 0.593632i
$503$	7.89607e8	0.276645	0.138323	−	0.990387i	$-0.455829\pi$
$503$	7.89607e8	0.276645	0.138323	+	0.990387i	$0.455829\pi$
$504$	2.17732e8	0.0757558
$505$	7.87113e6i	0.00271968i
$506$	1.60003e9	0.549037
$507$	0	0
$508$	−4.71483e9	−1.59567
$509$	− 9.69489e7i	− 0.0325860i	−0.999867	−	0.0162930i	$-0.994814\pi$
$509$	− 9.69489e7i	− 0.0325860i	0.999867	−	0.0162930i	$-0.00518645\pi$
$510$	3.27265e6	0.00109246
$511$	1.42818e9	0.473489
$512$	4.03115e9i	1.32734i
$513$	− 1.69663e9i	− 0.554852i
$514$	2.80220e8i	0.0910182i
$515$	7.59671e6i	0.00245076i
$516$	1.57072e9	0.503296
$517$	−1.26426e9	−0.402365
$518$	9.96314e8i	0.314950i
$519$	−1.41110e9	−0.443069
$520$	0	0
$521$	4.10985e9	1.27319	0.636596	−	0.771197i	$-0.280342\pi$
$521$	4.10985e9	1.27319	0.636596	+	0.771197i	$0.280342\pi$
$522$	3.29210e9i	1.01304i
$523$	2.38880e8	0.0730169	0.0365084	−	0.999333i	$-0.488376\pi$
$523$	2.38880e8	0.0730169	0.0365084	+	0.999333i	$0.488376\pi$
$524$	−2.35827e9	−0.716036
$525$	2.95167e8i	0.0890247i
$526$	− 4.18531e8i	− 0.125394i
$527$	1.12934e9i	0.336114i
$528$	− 1.92253e8i	− 0.0568400i
$529$	4.26345e9	1.25218
$530$	4.85928e6	0.00141777
$531$	− 2.22198e9i	− 0.644034i
$532$	1.22572e9	0.352939
$533$	0	0
$534$	−6.27729e8	−0.178393
$535$	− 1.19428e7i	− 0.00337186i
$536$	1.14634e7	0.00321541
$537$	8.87083e8	0.247203
$538$	4.13546e8i	0.114495i
$539$	− 8.17439e8i	− 0.224851i
$540$	5.09880e6i	0.00139344i
$541$	5.59019e8i	0.151788i	0.997116	+	0.0758938i	$0.0241810\pi$
$541$	5.59019e8i	0.151788i	−0.997116	+	0.0758938i	$0.975819\pi$
$542$	−9.72847e9	−2.62450
$543$	1.29041e9	0.345883
$544$	6.44767e9i	1.71714i
$545$	1.01952e7	0.00269780
$546$	0	0
$547$	−7.01923e8	−0.183372	−0.0916861	−	0.995788i	$-0.529226\pi$
$547$	−7.01923e8	−0.183372	−0.0916861	+	0.995788i	$0.529226\pi$
$548$	− 2.66514e9i	− 0.691812i
$549$	1.24809e9	0.321917
$550$	1.42747e9	0.365846
$551$	2.90130e9i	0.738860i
$552$	4.79231e8i	0.121271i
$553$	6.52412e8i	0.164053i
$554$	4.78116e9i	1.19467i
$555$	−1.74348e6	−0.000432905 0
$556$	−3.52880e9	−0.870693
$557$	− 3.51904e9i	− 0.862840i	−0.902151	−	0.431420i	$-0.858013\pi$
$557$	− 3.51904e9i	− 0.862840i	0.902151	−	0.431420i	$-0.141987\pi$
$558$	−1.54827e9	−0.377248
$559$	0	0
$560$	2.04386e6	0.000491806 0
$561$	− 3.66612e8i	− 0.0876672i
$562$	9.84907e9	2.34055
$563$	3.27161e8	0.0772648	0.0386324	−	0.999253i	$-0.487700\pi$
$563$	3.27161e8	0.0772648	0.0386324	+	0.999253i	$0.487700\pi$
$564$	− 2.41880e9i	− 0.567705i
$565$	− 3.92166e6i	0 0.000914746i
$566$	8.29418e9i	1.92272i
$567$	− 9.80607e8i	− 0.225920i
$568$	−1.45323e9	−0.332747
$569$	−5.92681e9	−1.34874	−0.674369	−	0.738394i	$-0.735584\pi$
$569$	−5.92681e9	−1.34874	−0.674369	+	0.738394i	$0.735584\pi$
$570$	3.95406e6i	0 0.000894296i
$571$	−3.50991e9	−0.788988	−0.394494	−	0.918899i	$-0.629080\pi$
$571$	−3.50991e9	−0.788988	−0.394494	+	0.918899i	$0.629080\pi$
$572$	0	0
$573$	−1.43139e9	−0.317846
$574$	2.85925e9i	0.631045i
$575$	6.84128e9	1.50072
$576$	−5.57252e9	−1.21499
$577$	9.14479e9i	1.98179i	0.134621	+	0.990897i	$0.457018\pi$
$577$	9.14479e9i	1.98179i	−0.134621	+	0.990897i	$0.542982\pi$
$578$	3.06843e9i	0.660950i
$579$	6.41975e7i	0.0137450i
$580$	− 8.71913e6i	− 0.00185556i
$581$	−4.86361e8	−0.102883
$582$	1.93172e8	0.0406176
$583$	− 5.44351e8i	− 0.113773i
$584$	2.06871e9	0.429787
$585$	0	0
$586$	−9.04920e9	−1.85767
$587$	1.65538e9i	0.337804i	0.985633	+	0.168902i	$0.0540221\pi$
$587$	1.65538e9i	0.337804i	−0.985633	+	0.168902i	$0.945978\pi$
$588$	1.56393e9	0.317247
$589$	−1.36448e9	−0.275146
$590$	1.08486e7i	0.00217465i
$591$	1.39274e9i	0.277532i
$592$	− 2.77466e9i	− 0.549647i
$593$	7.99836e9i	1.57510i	0.616248	+	0.787552i	$0.288652\pi$
$593$	7.99836e9i	1.57510i	−0.616248	+	0.787552i	$0.711348\pi$
$594$	1.05295e9	0.206136
$595$	3.89750e6	0.000758537 0
$596$	1.10956e10i	2.14678i
$597$	−1.77864e9	−0.342120
$598$	0	0
$599$	−8.80866e8	−0.167462	−0.0837309	−	0.996488i	$-0.526684\pi$
$599$	−8.80866e8	−0.167462	−0.0837309	+	0.996488i	$0.526684\pi$
$600$	4.27547e8i	0.0808080i
$601$	2.97804e9	0.559589	0.279795	−	0.960060i	$-0.409734\pi$
$601$	2.97804e9	0.559589	0.279795	+	0.960060i	$0.409734\pi$
$602$	3.44838e9	0.644210
$603$	− 5.76193e7i	− 0.0107018i
$604$	2.64550e9i	0.488516i
$605$	− 1.06658e7i	− 0.00195817i
$606$	3.10985e9i	0.567655i
$607$	−3.31888e8	−0.0602325	−0.0301163	−	0.999546i	$-0.509588\pi$
$607$	−3.31888e8	−0.0602325	−0.0301163	+	0.999546i	$0.509588\pi$
$608$	−7.79016e9	−1.40567
$609$	− 3.72316e8i	− 0.0667962i
$610$	−6.09367e6	−0.00108699
$611$	0	0
$612$	−7.38613e9	−1.30253
$613$	3.93427e9i	0.689846i	0.938631	+	0.344923i	$0.112095\pi$
$613$	3.93427e9i	0.689846i	−0.938631	+	0.344923i	$0.887905\pi$
$614$	7.10575e9	1.23886
$615$	−5.00350e6	−0.000867383 0
$616$	1.19086e8i	0.0205272i
$617$	− 7.24044e9i	− 1.24099i	−0.784212	−	0.620493i	$-0.786933\pi$
$617$	− 7.24044e9i	− 1.24099i	0.784212	−	0.620493i	$-0.213067\pi$
$618$	3.00142e9i	0.511526i
$619$	1.16885e10i	1.98080i	0.138220	+	0.990401i	$0.455862\pi$
$619$	1.16885e10i	1.98080i	−0.138220	+	0.990401i	$0.544138\pi$
$620$	4.10060e6	0.000690998 0
$621$	5.04633e9	0.845581
$622$	− 4.83563e9i	− 0.805725i
$623$	−7.47582e8	−0.123866
$624$	0	0
$625$	6.10344e9	0.999987
$626$	− 1.58337e10i	− 2.57971i
$627$	4.42946e8	0.0717652
$628$	−1.30006e10	−2.09462
$629$	− 5.29107e9i	− 0.847747i
$630$	5.34330e6i	0 0.000851368i
$631$	− 6.38796e8i	− 0.101218i	−0.998719	−	0.0506092i	$-0.983884\pi$
$631$	− 6.38796e8i	− 0.101218i	0.998719	−	0.0506092i	$-0.0161163\pi$
$632$	9.45013e8i	0.148911i
$633$	−1.98254e9	−0.310677
$634$	7.00127e9	1.09110
$635$	− 1.81136e7i	− 0.00280736i
$636$	1.04146e9	0.160525
$637$	0	0
$638$	−1.80058e9	−0.274498
$639$	7.30446e9i	1.10748i
$640$	7.46086e6	0.00112502
$641$	−6.15856e8	−0.0923584	−0.0461792	−	0.998933i	$-0.514705\pi$
$641$	−6.15856e8	−0.0923584	−0.0461792	+	0.998933i	$0.514705\pi$
$642$	− 4.71856e9i	− 0.703779i
$643$	− 4.86092e9i	− 0.721075i	−0.932745	−	0.360537i	$-0.882593\pi$
$643$	− 4.86092e9i	− 0.721075i	0.932745	−	0.360537i	$-0.117407\pi$
$644$	3.64568e9i	0.537870i
$645$	6.03444e6i	0 0.000885479i
$646$	−1.19997e10	−1.75128
$647$	−7.67484e9	−1.11405	−0.557025	−	0.830496i	$-0.688057\pi$
$647$	−7.67484e9	−1.11405	−0.557025	+	0.830496i	$0.688057\pi$
$648$	− 1.42040e9i	− 0.205068i
$649$	1.21529e9	0.174511
$650$	0	0
$651$	1.75100e8	0.0248744
$652$	6.53707e9i	0.923670i
$653$	−5.86264e9	−0.823942	−0.411971	−	0.911197i	$-0.635160\pi$
$653$	−5.86264e9	−0.823942	−0.411971	+	0.911197i	$0.635160\pi$
$654$	4.02809e9	0.563089
$655$	− 9.06012e6i	− 0.00125976i
$656$	− 7.96279e9i	− 1.10129i
$657$	− 1.03981e10i	− 1.43046i
$658$	− 5.31028e9i	− 0.726653i
$659$	1.17798e10	1.60338	0.801692	−	0.597737i	$-0.203933\pi$
$659$	1.17798e10	1.60338	0.801692	+	0.597737i	$0.203933\pi$
$660$	−1.33116e6	−0.000180230 0
$661$	− 3.65943e9i	− 0.492843i	−0.969163	−	0.246421i	$-0.920745\pi$
$661$	− 3.65943e9i	− 0.492843i	0.969163	−	0.246421i	$-0.0792547\pi$
$662$	6.89690e9	0.923955
$663$	0	0
$664$	−7.04490e8	−0.0933870
$665$	4.70901e6i	0 0.000620946i
$666$	7.25383e9	0.951497
$667$	−8.62942e9	−1.12601
$668$	9.98123e9i	1.29559i
$669$	1.30968e9i	0.169112i
$670$	281319.i	0 3.61358e-5i
$671$	6.82631e8i	0.0872283i
$672$	9.99691e8	0.127079
$673$	8.09730e9	1.02397	0.511985	−	0.858994i	$-0.328910\pi$
$673$	8.09730e9	1.02397	0.511985	+	0.858994i	$0.328910\pi$
$674$	9.61422e9i	1.20950i
$675$	4.50210e9	0.563445
$676$	0	0
$677$	6.04454e9	0.748691	0.374346	−	0.927289i	$-0.377867\pi$
$677$	6.04454e9	0.748691	0.374346	+	0.927289i	$0.377867\pi$
$678$	− 1.54943e9i	− 0.190927i
$679$	2.30055e8	0.0282024
$680$	5.64549e6	0.000688527 0
$681$	3.25659e9i	0.395137i
$682$	− 8.46811e8i	− 0.102221i
$683$	6.73839e9i	0.809252i	0.914482	+	0.404626i	$0.132598\pi$
$683$	6.73839e9i	0.809252i	−0.914482	+	0.404626i	$0.867402\pi$
$684$	− 8.92402e9i	− 1.06626i
$685$	1.02390e7	0.00121715
$686$	7.21231e9	0.852982
$687$	− 3.31221e9i	− 0.389735i
$688$	−9.60348e9	−1.12427
$689$	0	0
$690$	−1.17607e7	−0.00136289
$691$	3.86203e9i	0.445289i	0.974900	+	0.222645i	$0.0714690\pi$
$691$	3.86203e9i	0.445289i	−0.974900	+	0.222645i	$0.928531\pi$
$692$	−1.55492e10	−1.78376
$693$	5.98573e8	0.0683204
$694$	− 2.44468e10i	− 2.77629i
$695$	− 1.35571e7i	− 0.00153186i
$696$	− 5.39297e8i	− 0.0606311i
$697$	− 1.51845e10i	− 1.69857i
$698$	−6.70024e9	−0.745756
$699$	3.07998e9	0.341097
$700$	3.25250e9i	0.358405i
$701$	6.81601e9	0.747338	0.373669	−	0.927562i	$-0.378100\pi$
$701$	6.81601e9	0.747338	0.373669	+	0.927562i	$0.378100\pi$
$702$	0	0
$703$	6.39274e9	0.693975
$704$	− 3.04783e9i	− 0.329220i
$705$	9.29265e6	0.000998798 0
$706$	9.99871e9	1.06937
$707$	3.70361e9i	0.394146i
$708$	2.32510e9i	0.246221i
$709$	7.39468e9i	0.779215i	0.920981	+	0.389608i	$0.127389\pi$
$709$	7.39468e9i	0.779215i	−0.920981	+	0.389608i	$0.872611\pi$
$710$	− 3.56632e7i	− 0.00373952i
$711$	4.74999e9	0.495620
$712$	−1.08287e9	−0.112433
$713$	− 4.05841e9i	− 0.419317i
$714$	1.53988e9	0.158323
$715$	0	0
$716$	9.77493e9	0.995218
$717$	3.58430e9i	0.363151i
$718$	−1.08481e10	−1.09375
$719$	1.85081e10	1.85700	0.928500	−	0.371333i	$-0.121099\pi$
$719$	1.85081e10	1.85700	0.928500	+	0.371333i	$0.121099\pi$
$720$	− 1.48807e7i	− 0.00148580i
$721$	3.57449e9i	0.355173i
$722$	4.52721e8i	0.0447663i
$723$	4.02258e9i	0.395841i
$724$	1.42193e10	1.39249
$725$	−7.69876e9	−0.750304
$726$	− 4.21401e9i	− 0.408712i
$727$	4.04178e9	0.390124	0.195062	−	0.980791i	$-0.437509\pi$
$727$	4.04178e9	0.390124	0.195062	+	0.980791i	$0.437509\pi$
$728$	0	0
$729$	−5.40376e9	−0.516594
$730$	5.07675e7i	0.00483009i
$731$	−1.83131e10	−1.73401
$732$	−1.30602e9	−0.123072
$733$	1.54392e10i	1.44797i	0.689816	+	0.723985i	$0.257692\pi$
$733$	1.54392e10i	1.44797i	−0.689816	+	0.723985i	$0.742308\pi$
$734$	1.04390e10i	0.974370i
$735$	6.00839e6i	0 0.000558152i
$736$	− 2.31705e10i	− 2.14221i
$737$	3.15143e7	0.00289982
$738$	2.08172e10	1.90645
$739$	6.64104e8i	0.0605313i	0.999542	+	0.0302657i	$0.00963533\pi$
$739$	6.64104e8i	0.0605313i	−0.999542	+	0.0302657i	$0.990365\pi$
$740$	−1.92118e7	−0.00174284
$741$	0	0
$742$	2.28644e9	0.205469
$743$	− 1.54094e9i	− 0.137824i	−0.997623	−	0.0689118i	$-0.978047\pi$
$743$	− 1.54094e9i	− 0.137824i	0.997623	−	0.0689118i	$-0.0219527\pi$
$744$	2.53631e8	0.0225786
$745$	−4.26274e7	−0.00377695
$746$	− 1.91379e9i	− 0.168775i
$747$	3.54103e9i	0.310819i
$748$	− 4.03977e9i	− 0.352940i
$749$	− 5.61947e9i	− 0.488663i
$750$	−2.09846e7	−0.00181630
$751$	−3.60098e9	−0.310228	−0.155114	−	0.987897i	$-0.549574\pi$
$751$	−3.60098e9	−0.310228	−0.155114	+	0.987897i	$0.549574\pi$
$752$	1.47887e10i	1.26814i
$753$	1.38545e9	0.118252
$754$	0	0
$755$	−1.01636e7	−0.000859475 0
$756$	2.39914e9i	0.201944i
$757$	3.39784e8	0.0284686	0.0142343	−	0.999899i	$-0.495469\pi$
$757$	3.39784e8	0.0284686	0.0142343	+	0.999899i	$0.495469\pi$
$758$	2.99438e10	2.49726
$759$	1.31746e9i	0.109369i
$760$	6.82096e6i	0 0.000563635i
$761$	− 1.96039e9i	− 0.161249i	−0.996745	−	0.0806243i	$-0.974309\pi$
$761$	− 1.96039e9i	− 0.161249i	0.996745	−	0.0806243i	$-0.0256914\pi$
$762$	− 7.15661e9i	− 0.585956i
$763$	4.79718e9	0.390976
$764$	−1.57727e10	−1.27962
$765$	− 2.83764e7i	− 0.00229162i
$766$	−2.86500e10	−2.30316
$767$	0	0
$768$	−1.97053e9	−0.156971
$769$	1.11615e10i	0.885077i	0.896750	+	0.442538i	$0.145922\pi$
$769$	1.11615e10i	0.885077i	−0.896750	+	0.442538i	$0.854078\pi$
$770$	−2.92246e6	−0.000230691 0
$771$	−2.30733e8	−0.0181309
$772$	7.07404e8i	0.0553359i
$773$	− 2.85420e9i	− 0.222258i	−0.993806	−	0.111129i	$-0.964553\pi$
$773$	− 2.85420e9i	− 0.222258i	0.993806	−	0.111129i	$-0.0354466\pi$
$774$	− 2.51065e10i	− 1.94622i
$775$	− 3.62072e9i	− 0.279408i
$776$	3.33232e8	0.0255994
$777$	−8.20364e8	−0.0627383
$778$	− 2.81483e10i	− 2.14301i
$779$	1.83461e10	1.39047
$780$	0	0
$781$	−3.99510e9	−0.300088
$782$	− 3.56909e10i	− 2.66891i
$783$	−5.67884e9	−0.422759
$784$	−9.56202e9	−0.708669
$785$	− 4.99464e7i	− 0.00368519i
$786$	− 3.57961e9i	− 0.262940i
$787$	1.33429e10i	0.975749i	0.872914	+	0.487874i	$0.162228\pi$
$787$	1.33429e10i	0.975749i	−0.872914	+	0.487874i	$0.837772\pi$
$788$	1.53468e10i	1.11732i
$789$	3.44618e8	0.0249786
$790$	−2.31913e7	−0.00167351
$791$	− 1.84526e9i	− 0.132569i
$792$	8.67027e8	0.0620147
$793$	0	0
$794$	3.56073e10	2.52445
$795$	4.00112e6i	0 0.000282421i
$796$	−1.95992e10	−1.37734
$797$	−8.42860e9	−0.589727	−0.294864	−	0.955539i	$-0.595274\pi$
$797$	−8.42860e9	−0.589727	−0.294864	+	0.955539i	$0.595274\pi$
$798$	1.86051e9i	0.129605i
$799$	2.82010e10i	1.95592i
$800$	− 2.06716e10i	− 1.42744i
$801$	5.44289e9i	0.374210i
$802$	−1.51390e10	−1.03630
$803$	5.68713e9	0.387604
$804$	6.02935e7i	0.00409142i
$805$	−1.40061e7	−0.000946308 0
$806$	0	0
$807$	−3.40513e8	−0.0228075
$808$	5.36465e9i	0.357768i
$809$	9.44279e9	0.627019	0.313509	−	0.949585i	$-0.398495\pi$
$809$	9.44279e9	0.627019	0.313509	+	0.949585i	$0.398495\pi$
$810$	3.48576e7	0.00230462
$811$	− 1.75509e10i	− 1.15539i	−0.816254	−	0.577693i	$-0.803953\pi$
$811$	− 1.75509e10i	− 1.15539i	0.816254	−	0.577693i	$-0.196047\pi$
$812$	− 4.10262e9i	− 0.268915i
$813$	− 8.01041e9i	− 0.522803i
$814$	3.96740e9i	0.257823i
$815$	−2.51144e7	−0.00162507
$816$	−4.28846e9	−0.276303
$817$	− 2.21262e10i	− 1.41948i
$818$	−2.69215e10	−1.71974
$819$	0	0
$820$	−5.51344e7	−0.00349200
$821$	− 1.65515e10i	− 1.04384i	−0.852993	−	0.521922i	$-0.825215\pi$
$821$	− 1.65515e10i	− 1.04384i	0.852993	−	0.521922i	$-0.174785\pi$
$822$	4.04540e9	0.254045
$823$	−2.59526e10	−1.62286	−0.811430	−	0.584449i	$-0.801311\pi$
$823$	−2.59526e10	−1.62286	−0.811430	+	0.584449i	$0.801311\pi$
$824$	5.17762e9i	0.322392i
$825$	1.17538e9i	0.0728768i
$826$	5.10458e9i	0.315159i
$827$	2.53210e10i	1.55672i	0.627816	+	0.778362i	$0.283949\pi$
$827$	2.53210e10i	1.55672i	−0.627816	+	0.778362i	$0.716051\pi$
$828$	2.65429e10	1.62496
$829$	−2.70822e10	−1.65099	−0.825494	−	0.564411i	$-0.809103\pi$
$829$	−2.70822e10	−1.65099	−0.825494	+	0.564411i	$0.809103\pi$
$830$	− 1.72887e7i	− 0.00104951i
$831$	−3.93680e9	−0.237980
$832$	0	0
$833$	−1.82341e10	−1.09301
$834$	− 5.35634e9i	− 0.319733i
$835$	−3.83463e7	−0.00227940
$836$	4.88090e9	0.288920
$837$	− 2.67075e9i	− 0.157432i
$838$	− 4.81130e9i	− 0.282428i
$839$	− 1.26152e10i	− 0.737442i	−0.929540	−	0.368721i	$-0.879796\pi$
$839$	− 1.26152e10i	− 0.737442i	0.929540	−	0.368721i	$-0.120204\pi$
$840$	− 875316.i	0 5.09550e-5i
$841$	−7.53886e9	−0.437038
$842$	2.55001e8	0.0147214
$843$	8.10971e9i	0.466239i
$844$	−2.18460e10	−1.25076
$845$	0	0
$846$	−3.86624e10	−2.19529
$847$	− 5.01860e9i	− 0.283786i
$848$	−6.36757e9	−0.358582
$849$	−6.82942e9	−0.383007
$850$	− 3.18417e10i	− 1.77840i
$851$	1.90141e10i	1.05760i
$852$	− 7.64347e9i	− 0.423401i
$853$	− 1.11666e9i	− 0.0616027i	−0.999526	−	0.0308013i	$-0.990194\pi$
$853$	− 1.11666e9i	− 0.0616027i	0.999526	−	0.0308013i	$-0.00980592\pi$
$854$	−2.86726e9	−0.157530
$855$	3.42847e7	0.00187594
$856$	− 8.13976e9i	− 0.443561i
$857$	1.48481e10	0.805821	0.402911	−	0.915239i	$-0.367999\pi$
$857$	1.48481e10	0.805821	0.402911	+	0.915239i	$0.367999\pi$
$858$	0	0
$859$	1.26319e10	0.679974	0.339987	−	0.940430i	$-0.389577\pi$
$859$	1.26319e10	0.679974	0.339987	+	0.940430i	$0.389577\pi$
$860$	6.64946e7i	0.00356486i
$861$	−2.35430e9	−0.125705
$862$	−1.54286e10	−0.820449
$863$	2.29823e10i	1.21718i	0.793484	+	0.608591i	$0.208265\pi$
$863$	2.29823e10i	1.21718i	−0.793484	+	0.608591i	$0.791735\pi$
$864$	− 1.52480e10i	− 0.804294i
$865$	− 5.97374e7i	− 0.00313827i
$866$	3.15136e10i	1.64887i
$867$	−2.52654e9	−0.131662
$868$	1.92946e9	0.100142
$869$	2.59796e9i	0.134296i
$870$	1.32347e7	0.000681393 0
$871$	0	0
$872$	6.94867e9	0.354890
$873$	− 1.67495e9i	− 0.0852024i
$874$	4.31222e10	2.18479
$875$	−2.49913e7	−0.00126113
$876$	1.08807e10i	0.546879i
$877$	1.63120e10i	0.816599i	0.912848	+	0.408299i	$0.133878\pi$
$877$	1.63120e10i	0.816599i	−0.912848	+	0.408299i	$0.866122\pi$
$878$	1.44696e10i	0.721480i
$879$	− 7.45110e9i	− 0.370049i
$880$	8.13883e6	0.000402599 0
$881$	−2.07580e9	−0.102275	−0.0511374	−	0.998692i	$-0.516285\pi$
$881$	−2.07580e9	−0.102275	−0.0511374	+	0.998692i	$0.516285\pi$
$882$	− 2.49981e10i	− 1.22678i
$883$	−1.70041e10	−0.831175	−0.415587	−	0.909553i	$-0.636424\pi$
$883$	−1.70041e10	−0.831175	−0.415587	+	0.909553i	$0.636424\pi$
$884$	0	0
$885$	−8.93268e6	−0.000433192 0
$886$	1.57359e10i	0.760105i
$887$	−1.27005e10	−0.611066	−0.305533	−	0.952182i	$-0.598835\pi$
$887$	−1.27005e10	−0.611066	−0.305533	+	0.952182i	$0.598835\pi$
$888$	−1.18829e9	−0.0569478
$889$	− 8.52303e9i	− 0.406853i
$890$	− 2.65743e7i	− 0.00126356i
$891$	− 3.90486e9i	− 0.184941i
$892$	1.44316e10i	0.680828i
$893$	−3.40729e10	−1.60114
$894$	−1.68419e10	−0.788332
$895$	3.75538e7i	0.00175095i
$896$	3.51057e9	0.163042
$897$	0	0
$898$	1.47153e9	0.0678114
$899$	4.56708e9i	0.209643i
$900$	2.36804e10	1.08278
$901$	−1.21425e10	−0.553058
$902$	1.13858e10i	0.516582i
$903$	2.83939e9i	0.128327i
$904$	− 2.67285e9i	− 0.120333i
$905$	5.46284e7i	0.00244990i
$906$	−4.01559e9	−0.179391
$907$	5.79320e9	0.257806	0.128903	−	0.991657i	$-0.458854\pi$
$907$	5.79320e9	0.257806	0.128903	+	0.991657i	$0.458854\pi$
$908$	3.58849e10i	1.59079i
$909$	2.69647e10	1.19076
$910$	0	0
$911$	−2.92730e10	−1.28278	−0.641390	−	0.767215i	$-0.721642\pi$
$911$	−2.92730e10	−1.28278	−0.641390	+	0.767215i	$0.721642\pi$
$912$	− 5.18137e9i	− 0.226185i
$913$	−1.93673e9	−0.0842212
$914$	3.26389e10	1.41392
$915$	− 5.01752e6i	0 0.000216528i
$916$	− 3.64978e10i	− 1.56904i
$917$	− 4.26307e9i	− 0.182570i
$918$	− 2.34874e10i	− 1.00204i
$919$	1.74294e10	0.740759	0.370380	−	0.928880i	$-0.379228\pi$
$919$	1.74294e10	0.740759	0.370380	+	0.928880i	$0.379228\pi$
$920$	−2.02877e7	−0.000858967 0
$921$	5.85087e9i	0.246781i
$922$	−1.93717e9	−0.0813974
$923$	0	0
$924$	−6.26353e8	−0.0261196
$925$	1.69635e10i	0.704723i
$926$	2.45016e10	1.01404
$927$	2.60246e10	1.07301
$928$	2.60746e10i	1.07103i
$929$	1.64663e10i	0.673814i	0.941538	+	0.336907i	$0.109381\pi$
$929$	1.64663e10i	0.673814i	−0.941538	+	0.336907i	$0.890619\pi$
$930$	6.22428e6i	0 0.000253746i
$931$	− 2.20306e10i	− 0.894753i
$932$	3.39389e10	1.37322
$933$	3.98165e9	0.160501
$934$	1.04293e10i	0.418834i
$935$	1.55202e7	0.000620948 0
$936$	0	0
$937$	−1.22607e10	−0.486885	−0.243442	−	0.969915i	$-0.578277\pi$
$937$	−1.22607e10	−0.486885	−0.243442	+	0.969915i	$0.578277\pi$
$938$	1.32370e8i	0.00523695i
$939$	1.30374e10	0.513880
$940$	1.02397e8	0.00402107
$941$	3.58066e10i	1.40088i	0.713713	+	0.700438i	$0.247012\pi$
$941$	3.58066e10i	1.40088i	−0.713713	+	0.700438i	$0.752988\pi$
$942$	− 1.97336e10i	− 0.769180i
$943$	5.45671e10i	2.11904i
$944$	− 1.42159e10i	− 0.550011i
$945$	−9.21713e6	−0.000355291 0
$946$	1.37317e10	0.527359
$947$	3.11279e10i	1.19104i	0.803342	+	0.595518i	$0.203053\pi$
$947$	3.11279e10i	1.19104i	−0.803342	+	0.595518i	$0.796947\pi$
$948$	−4.97044e9	−0.189481
$949$	0	0
$950$	3.84716e10	1.45582
$951$	5.76484e9i	0.217348i
$952$	2.65638e9	0.0997840
$953$	−6.95907e9	−0.260451	−0.130225	−	0.991484i	$-0.541570\pi$
$953$	−6.95907e9	−0.260451	−0.130225	+	0.991484i	$0.541570\pi$
$954$	− 1.66468e10i	− 0.620743i
$955$	− 6.05964e7i	− 0.00225131i
$956$	3.94961e10i	1.46201i
$957$	− 1.48259e9i	− 0.0546803i
$958$	−3.52913e10	−1.29685
$959$	4.81779e9	0.176394
$960$	2.24024e7i	0 0.000817230i
$961$	2.53647e10	0.921931
$962$	0	0
$963$	−4.09135e10	−1.47630
$964$	4.43255e10i	1.59362i
$965$	−2.71774e6	−9.73558e−5 0
$966$	−5.53375e9	−0.197515
$967$	− 3.75275e10i	− 1.33462i	−0.744780	−	0.667310i	$-0.767446\pi$
$967$	− 3.75275e10i	− 1.33462i	0.744780	−	0.667310i	$-0.232554\pi$
$968$	− 7.26940e9i	− 0.257593i
$969$	− 9.88050e9i	− 0.348856i
$970$	8.17774e6i	0 0.000287695i
$971$	1.78987e9	0.0627414	0.0313707	−	0.999508i	$-0.490013\pi$
$971$	1.78987e9	0.0627414	0.0313707	+	0.999508i	$0.490013\pi$
$972$	2.65969e10	0.928965
$973$	− 6.37904e9i	− 0.222004i
$974$	5.16665e10	1.79165
$975$	0	0
$976$	7.98510e9	0.274920
$977$	1.98276e10i	0.680203i	0.940389	+	0.340101i	$0.110461\pi$
$977$	1.98276e10i	0.680203i	−0.940389	+	0.340101i	$0.889539\pi$
$978$	−9.92259e9	−0.339187
$979$	−2.97693e9	−0.101398
$980$	6.62075e7i	0.00224707i
$981$	− 3.49266e10i	− 1.18118i
$982$	6.07523e10i	2.04726i
$983$	3.24213e10i	1.08866i	0.838870	+	0.544332i	$0.183217\pi$
$983$	3.24213e10i	1.08866i	−0.838870	+	0.544332i	$0.816783\pi$
$984$	−3.41018e9	−0.114102
$985$	−5.89601e7	−0.00196576
$986$	4.01643e10i	1.33435i
$987$	4.37248e9	0.144750
$988$	0	0
$989$	6.58104e10	2.16326
$990$	2.12775e7i	0 0.000696942i
$991$	−2.45428e10	−0.801063	−0.400532	−	0.916283i	$-0.631175\pi$
$991$	−2.45428e10	−0.801063	−0.400532	+	0.916283i	$0.631175\pi$
$992$	−1.22629e10	−0.398843
$993$	5.67890e9i	0.184052i
$994$	− 1.67806e10i	− 0.541946i
$995$	− 7.52970e7i	− 0.00242324i
$996$	− 3.70537e9i	− 0.118830i
$997$	−1.08000e10	−0.345137	−0.172568	−	0.984998i	$-0.555207\pi$
$997$	−1.08000e10	−0.345137	−0.172568	+	0.984998i	$0.555207\pi$
$998$	7.89357e10	2.51372
$999$	1.25128e10i	0.397077i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	169.8.b.f.168.36		42
13.5	odd	4		169.8.a.h.1.19	✓	21
13.8	odd	4		169.8.a.i.1.3	yes	21
13.12	even	2	inner	169.8.b.f.168.7		42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
169.8.a.h.1.19	✓	21	13.5	odd	4
169.8.a.i.1.3	yes	21	13.8	odd	4
169.8.b.f.168.7		42	13.12	even	2	inner
169.8.b.f.168.36		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)

\(f(q)\)	\(=\)	\(q+16.7260i q^{2} -13.7721 q^{3} -151.758 q^{4} -0.583029i q^{5} -230.352i q^{6} -274.333i q^{7} -397.369i q^{8} -1997.33 q^{9} +9.75172 q^{10} -1092.42i q^{11} +2090.03 q^{12} +4588.49 q^{14} +8.02956i q^{15} -12778.6 q^{16} -24367.8 q^{17} -33407.2i q^{18} -29441.5i q^{19} +88.4791i q^{20} +3778.16i q^{21} +18271.7 q^{22} +87568.7 q^{23} +5472.63i q^{24} +78124.7 q^{25} +57627.1 q^{27} +41632.2i q^{28} -98544.5 q^{29} -134.302 q^{30} -46345.4i q^{31} -264598. i q^{32} +15044.9i q^{33} -407575. i q^{34} -159.944 q^{35} +303110. q^{36} +217133. i q^{37} +492438. q^{38} -231.678 q^{40} +623135. i q^{41} -63193.3 q^{42} +751529. q^{43} +165783. i q^{44} +1164.50i q^{45} +1.46467e6i q^{46} -1.15731e6i q^{47} +175988. q^{48} +748284. q^{49} +1.30671e6i q^{50} +335597. q^{51} +498300. q^{53} +963869. i q^{54} -636.912 q^{55} -109012. q^{56} +405473. i q^{57} -1.64825e6i q^{58} +1.11248e6i q^{59} -1218.55i q^{60} -624881. q^{61} +775171. q^{62} +547934. i q^{63} +2.78999e6 q^{64} -251641. q^{66} +28848.2i q^{67} +3.69801e6 q^{68} -1.20601e6 q^{69} -2675.22i q^{70} -3.65711e6i q^{71} +793677. i q^{72} +5.20600e6i q^{73} -3.63176e6 q^{74} -1.07594e6 q^{75} +4.46798e6i q^{76} -299687. q^{77} -2.37817e6 q^{79} +7450.29i q^{80} +3.57451e6 q^{81} -1.04225e7 q^{82} -1.77288e6i q^{83} -573364. i q^{84} +14207.2i q^{85} +1.25700e7i q^{86} +1.35717e6 q^{87} -434094. q^{88} -2.72509e6i q^{89} -19477.4 q^{90} -1.32892e7 q^{92} +638275. i q^{93} +1.93570e7 q^{94} -17165.3 q^{95} +3.64407e6i q^{96} +838595. i q^{97} +1.25158e7i q^{98} +2.18192e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 52 q^{3} - 2818 q^{4} + 30930 q^{9} + 10334 q^{10} - 43590 q^{12} - 358 q^{14} + 226410 q^{16} - 90032 q^{17} - 548348 q^{22} + 256210 q^{23} - 733074 q^{25} - 213286 q^{27} - 124658 q^{29} - 19522 q^{30}+ \cdots - 35320062 q^{95}+O(q^{100}) \) 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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