LMFDB - Embedded newform 109.8.b.a.108.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [109,8,Mod(108,109)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(109, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("109.108");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 109 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	109.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$34.0499677778$
Analytic rank:	$0$
Dimension:	$62$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			108.9
Character	$\chi$	$=$	109.108
Dual form			109.8.b.a.108.54

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-17.9931i q^{2} +45.7905 q^{3} -195.751 q^{4} +340.028 q^{5} -823.912i q^{6} -1712.93 q^{7} +1219.06i q^{8} -90.2316 q^{9} +O(q^{10})$	\(q-17.9931i q^{2} +45.7905 q^{3} -195.751 q^{4} +340.028 q^{5} -823.912i q^{6} -1712.93 q^{7} +1219.06i q^{8} -90.2316 q^{9} -6118.15i q^{10} +1767.13i q^{11} -8963.55 q^{12} -1945.95i q^{13} +30820.9i q^{14} +15570.0 q^{15} -3121.57 q^{16} -32103.5i q^{17} +1623.55i q^{18} +480.003i q^{19} -66560.9 q^{20} -78435.8 q^{21} +31796.1 q^{22} +75657.6i q^{23} +55821.2i q^{24} +37493.9 q^{25} -35013.6 q^{26} -104276. q^{27} +335308. q^{28} -144421. q^{29} -280153. i q^{30} -192438. q^{31} +212206. i q^{32} +80917.5i q^{33} -577642. q^{34} -582443. q^{35} +17663.0 q^{36} +463049. i q^{37} +8636.74 q^{38} -89105.9i q^{39} +414513. i q^{40} -259362. i q^{41} +1.41130e6i q^{42} +214029. q^{43} -345917. i q^{44} -30681.3 q^{45} +1.36131e6 q^{46} -727507. i q^{47} -142938. q^{48} +2.11058e6 q^{49} -674632. i q^{50} -1.47004e6i q^{51} +380922. i q^{52} -601677. i q^{53} +1.87624e6i q^{54} +600872. i q^{55} -2.08816e6i q^{56} +21979.6i q^{57} +2.59857e6i q^{58} -1.19890e6i q^{59} -3.04786e6 q^{60} +475007. q^{61} +3.46255e6i q^{62} +154560. q^{63} +3.41868e6 q^{64} -661676. i q^{65} +1.45596e6 q^{66} -1.60067e6i q^{67} +6.28431e6i q^{68} +3.46440e6i q^{69} +1.04800e7i q^{70} -5.74148e6 q^{71} -109998. i q^{72} -1.02324e6 q^{73} +8.33168e6 q^{74} +1.71687e6 q^{75} -93961.3i q^{76} -3.02696e6i q^{77} -1.60329e6 q^{78} +6.65950e6i q^{79} -1.06142e6 q^{80} -4.57749e6 q^{81} -4.66672e6 q^{82} +4.32846e6 q^{83} +1.53539e7 q^{84} -1.09161e7i q^{85} -3.85105e6i q^{86} -6.61309e6 q^{87} -2.15423e6 q^{88} +4.93774e6 q^{89} +552051. i q^{90} +3.33327e6i q^{91} -1.48101e7i q^{92} -8.81181e6 q^{93} -1.30901e7 q^{94} +163214. i q^{95} +9.71702e6i q^{96} +553098. q^{97} -3.79758e7i q^{98} -159451. i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9}+O(q^{10}) $ 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9	$ 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9} + 6248 q^{12} - 21376 q^{15} + 227856 q^{16} - 216666 q^{20} + 42914 q^{21} - 101446 q^{22} + 788644 q^{25} - 282610 q^{26} - 235946 q^{27} - 69142 q^{28} - 213678 q^{29} + 300350 q^{31} - 401602 q^{34} - 377134 q^{35} - 2176902 q^{36} - 2421542 q^{38} + 2022550 q^{43} + 2223056 q^{45} + 2665174 q^{46} + 1060208 q^{48} + 7702792 q^{49} + 9894904 q^{60} + 278126 q^{61} - 6010300 q^{63} - 9362182 q^{64} - 23291020 q^{66} + 15373744 q^{71} - 1378934 q^{73} - 19097780 q^{74} - 41528062 q^{75} + 5705654 q^{78} + 42766270 q^{80} + 29291446 q^{81} + 14605464 q^{82} + 1191422 q^{83} - 6959552 q^{84} - 22121732 q^{87} + 6337872 q^{88} + 29157388 q^{89} + 14091936 q^{93} - 17482078 q^{94} - 22763314 q^{97}+O(q^{100}) $ 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9 + 6248 * q^12 - 21376 * q^15 + 227856 * q^16 - 216666 * q^20 + 42914 * q^21 - 101446 * q^22 + 788644 * q^25 - 282610 * q^26 - 235946 * q^27 - 69142 * q^28 - 213678 * q^29 + 300350 * q^31 - 401602 * q^34 - 377134 * q^35 - 2176902 * q^36 - 2421542 * q^38 + 2022550 * q^43 + 2223056 * q^45 + 2665174 * q^46 + 1060208 * q^48 + 7702792 * q^49 + 9894904 * q^60 + 278126 * q^61 - 6010300 * q^63 - 9362182 * q^64 - 23291020 * q^66 + 15373744 * q^71 - 1378934 * q^73 - 19097780 * q^74 - 41528062 * q^75 + 5705654 * q^78 + 42766270 * q^80 + 29291446 * q^81 + 14605464 * q^82 + 1191422 * q^83 - 6959552 * q^84 - 22121732 * q^87 + 6337872 * q^88 + 29157388 * q^89 + 14091936 * q^93 - 17482078 * q^94 - 22763314 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/109\mathbb{Z}\right)^\times$.

$n$	$6$
$\chi(n)$	$-1$

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 $

$2$		−	17.9931i		−	1.59038i	−0.606361	−	0.795190i	$-0.707371\pi$
$2$		−	17.9931i		−	1.59038i	0.606361	−	0.795190i	$-0.292629\pi$
$3$	45.7905			0.979154			0.489577	−	0.871960i	$-0.337151\pi$
$3$	45.7905			0.979154			0.489577	+	0.871960i	$0.337151\pi$
$4$	−195.751			−1.52931
$5$	340.028			1.21652			0.608260	−	0.793738i	$-0.291868\pi$
$5$	340.028			1.21652			0.608260	+	0.793738i	$0.291868\pi$
$6$		−	823.912i		−	1.55723i
$7$	−1712.93			−1.88754			−0.943769	−	0.330604i	$-0.892747\pi$
$7$	−1712.93			−1.88754			−0.943769	+	0.330604i	$0.892747\pi$
$8$			1219.06i			0.841800i
$9$	−90.2316			−0.0412582
$10$		−	6118.15i		−	1.93473i
$11$			1767.13i			0.400307i	0.979765	+	0.200153i	$0.0641441\pi$
$11$			1767.13i			0.400307i	−0.979765	+	0.200153i	$0.935856\pi$
$12$	−8963.55			−1.49743
$13$		−	1945.95i		−	0.245657i	−0.992428	−	0.122829i	$-0.960803\pi$
$13$		−	1945.95i		−	0.245657i	0.992428	−	0.122829i	$-0.0391965\pi$
$14$			30820.9i			3.00190i
$15$	15570.0			1.19116
$16$	−3121.57			−0.190526
$17$		−	32103.5i		−	1.58483i	−0.609985	−	0.792413i	$-0.708824\pi$
$17$		−	32103.5i		−	1.58483i	0.609985	−	0.792413i	$-0.291176\pi$
$18$			1623.55i			0.0656162i
$19$			480.003i			0.0160549i	0.999968	+	0.00802743i	$0.00255524\pi$
$19$			480.003i			0.0160549i	−0.999968	+	0.00802743i	$0.997445\pi$
$20$	−66560.9			−1.86043
$21$	−78435.8			−1.84819
$22$	31796.1			0.636640
$23$			75657.6i			1.29660i	0.761386	+	0.648298i	$0.224519\pi$
$23$			75657.6i			1.29660i	−0.761386	+	0.648298i	$0.775481\pi$
$24$			55821.2i			0.824252i
$25$	37493.9			0.479922
$26$	−35013.6			−0.390688
$27$	−104276.			−1.01955
$28$	335308.			2.88663
$29$	−144421.			−1.09960			−0.549802	−	0.835295i	$-0.685297\pi$
$29$	−144421.			−1.09960			−0.549802	+	0.835295i	$0.685297\pi$
$30$		−	280153.i		−	1.89440i
$31$	−192438.			−1.16018			−0.580088	−	0.814554i	$-0.696982\pi$
$31$	−192438.			−1.16018			−0.580088	+	0.814554i	$0.696982\pi$
$32$			212206.i			1.14481i
$33$			80917.5i			0.391962i
$34$	−577642.			−2.52048
$35$	−582443.			−2.29623
$36$	17663.0			0.0630965
$37$			463049.i			1.50287i	0.659808	+	0.751434i	$0.270638\pi$
$37$			463049.i			1.50287i	−0.659808	+	0.751434i	$0.729362\pi$
$38$	8636.74			0.0255333
$39$		−	89105.9i		−	0.240536i
$40$			414513.i			1.02407i
$41$		−	259362.i		−	0.587710i	−0.955850	−	0.293855i	$-0.905062\pi$
$41$		−	259362.i		−	0.587710i	0.955850	−	0.293855i	$-0.0949382\pi$
$42$			1.41130e6i			2.93932i
$43$	214029.			0.410519			0.205260	−	0.978708i	$-0.434196\pi$
$43$	214029.			0.410519			0.205260	+	0.978708i	$0.434196\pi$
$44$		−	345917.i		−	0.612192i
$45$	−30681.3			−0.0501914
$46$	1.36131e6			2.06208
$47$		−	727507.i		−	1.02210i	−0.859550	−	0.511052i	$-0.829256\pi$
$47$		−	727507.i		−	1.02210i	0.859550	−	0.511052i	$-0.170744\pi$
$48$	−142938.			−0.186554
$49$	2.11058e6			2.56280
$50$		−	674632.i		−	0.763259i
$51$		−	1.47004e6i		−	1.55179i
$52$			380922.i			0.375685i
$53$		−	601677.i		−	0.555134i	−0.960706	−	0.277567i	$-0.910472\pi$
$53$		−	601677.i		−	0.555134i	0.960706	−	0.277567i	$-0.0895281\pi$
$54$			1.87624e6i			1.62147i
$55$			600872.i			0.486982i
$56$		−	2.08816e6i		−	1.58893i
$57$			21979.6i			0.0157202i
$58$			2.59857e6i			1.74879i
$59$		−	1.19890e6i		−	0.759977i	−0.924991	−	0.379989i	$-0.875928\pi$
$59$		−	1.19890e6i		−	0.759977i	0.924991	−	0.379989i	$-0.124072\pi$
$60$	−3.04786e6			−1.82165
$61$	475007.			0.267945			0.133972	−	0.990985i	$-0.457227\pi$
$61$	475007.			0.267945			0.133972	+	0.990985i	$0.457227\pi$
$62$			3.46255e6i			1.84512i
$63$	154560.			0.0778764
$64$	3.41868e6			1.63015
$65$		−	661676.i		−	0.298847i
$66$	1.45596e6			0.623368
$67$		−	1.60067e6i		−	0.650191i	−0.945681	−	0.325096i	$-0.894604\pi$
$67$		−	1.60067e6i		−	0.650191i	0.945681	−	0.325096i	$-0.105396\pi$
$68$			6.28431e6i			2.42369i
$69$			3.46440e6i			1.26957i
$70$			1.04800e7i			3.65188i
$71$	−5.74148e6			−1.90379			−0.951897	−	0.306420i	$-0.900869\pi$
$71$	−5.74148e6			−1.90379			−0.951897	+	0.306420i	$0.900869\pi$
$72$		−	109998.i		−	0.0347311i
$73$	−1.02324e6			−0.307855			−0.153928	−	0.988082i	$-0.549192\pi$
$73$	−1.02324e6			−0.307855			−0.153928	+	0.988082i	$0.549192\pi$
$74$	8.33168e6			2.39013
$75$	1.71687e6			0.469918
$76$		−	93961.3i		−	0.0245528i
$77$		−	3.02696e6i		−	0.755595i
$78$	−1.60329e6			−0.382544
$79$			6.65950e6i			1.51966i	0.650121	+	0.759831i	$0.274718\pi$
$79$			6.65950e6i			1.51966i	−0.650121	+	0.759831i	$0.725282\pi$
$80$	−1.06142e6			−0.231778
$81$	−4.57749e6			−0.957040
$82$	−4.66672e6			−0.934682
$83$	4.32846e6			0.830923			0.415461	−	0.909611i	$-0.363620\pi$
$83$	4.32846e6			0.830923			0.415461	+	0.909611i	$0.363620\pi$
$84$	1.53539e7			2.82645
$85$		−	1.09161e7i		−	1.92797i
$86$		−	3.85105e6i		−	0.652882i
$87$	−6.61309e6			−1.07668
$88$	−2.15423e6			−0.336978
$89$	4.93774e6			0.742443			0.371221	−	0.928544i	$-0.378939\pi$
$89$	4.93774e6			0.742443			0.371221	+	0.928544i	$0.378939\pi$
$90$			552051.i			0.0798234i
$91$			3.33327e6i			0.463687i
$92$		−	1.48101e7i		−	1.98290i
$93$	−8.81181e6			−1.13599
$94$	−1.30901e7			−1.62553
$95$			163214.i			0.0195311i
$96$			9.71702e6i			1.12094i
$97$	553098.			0.0615320			0.0307660	−	0.999527i	$-0.490205\pi$
$97$	553098.			0.0615320			0.0307660	+	0.999527i	$0.490205\pi$
$98$		−	3.79758e7i		−	4.07583i
$99$		−	159451.i		−	0.0165159i
$100$	−7.33949e6			−0.733949
$101$		−	1.89960e7i		−	1.83458i	−0.398215	−	0.917292i	$-0.630370\pi$
$101$		−	1.89960e7i		−	1.83458i	0.398215	−	0.917292i	$-0.369630\pi$
$102$	−2.64505e7			−2.46793
$103$			2.83884e6i			0.255983i	0.991775	+	0.127992i	$0.0408530\pi$
$103$			2.83884e6i			0.255983i	−0.991775	+	0.127992i	$0.959147\pi$
$104$	2.37222e6			0.206794
$105$	−2.66704e7			−2.24836
$106$	−1.08260e7			−0.882874
$107$		−	1.90746e7i		−	1.50526i	−0.658444	−	0.752630i	$-0.728785\pi$
$107$		−	1.90746e7i		−	1.50526i	0.658444	−	0.752630i	$-0.271215\pi$
$108$	2.04121e7			1.55921
$109$	−737491.	−	1.35004e7i	−0.0545461	−	0.998511i
$109$	−737491.	−	1.35004e7i	−0.0545461	−	0.998511i
$110$	1.08115e7			0.774486
$111$			2.12032e7i			1.47154i
$112$	5.34702e6			0.359624
$113$	1.58711e7			1.03475			0.517373	−	0.855760i	$-0.326910\pi$
$113$	1.58711e7			1.03475			0.517373	+	0.855760i	$0.326910\pi$
$114$	395481.			0.0250010
$115$			2.57257e7i			1.57734i
$116$	2.82705e7			1.68163
$117$			175586.i			0.0101354i
$118$	−2.15719e7			−1.20865
$119$			5.49910e7i			2.99142i
$120$			1.89808e7i			1.00272i
$121$	1.63644e7			0.839754
$122$		−	8.54684e6i		−	0.426134i
$123$		−	1.18763e7i		−	0.575458i
$124$	3.76699e7			1.77427
$125$	−1.38157e7			−0.632685
$126$		−	2.78102e6i		−	0.123853i
$127$		−	1.14842e7i		−	0.497493i	−0.968569	−	0.248747i	$-0.919981\pi$
$127$		−	1.14842e7i		−	0.497493i	0.968569	−	0.248747i	$-0.0800187\pi$
$128$		−	3.43503e7i		−	1.44776i
$129$	9.80051e6			0.401962
$130$	−1.19056e7			−0.475280
$131$	−4.34179e7			−1.68740			−0.843702	−	0.536812i	$-0.819629\pi$
$131$	−4.34179e7			−1.68740			−0.843702	+	0.536812i	$0.819629\pi$
$132$		−	1.58397e7i		−	0.599430i
$133$		−	822211.i		−	0.0303042i
$134$	−2.88011e7			−1.03405
$135$	−3.54566e7			−1.24031
$136$	3.91360e7			1.33411
$137$	−2.39322e7			−0.795170			−0.397585	−	0.917565i	$-0.630152\pi$
$137$	−2.39322e7			−0.795170			−0.397585	+	0.917565i	$0.630152\pi$
$138$	6.23352e7			2.01909
$139$		−	1.00897e7i		−	0.318658i	−0.987226	−	0.159329i	$-0.949067\pi$
$139$		−	1.00897e7i		−	0.318658i	0.987226	−	0.159329i	$-0.0509331\pi$
$140$	1.14014e8			3.51164
$141$		−	3.33129e7i		−	1.00080i
$142$			1.03307e8i			3.02775i
$143$	3.43873e6			0.0983382
$144$	281664.			0.00786074
$145$	−4.91070e7			−1.33769
$146$			1.84112e7i			0.489607i
$147$	9.66444e7			2.50938
$148$		−	9.06425e7i		−	2.29835i
$149$		−	3.83742e7i		−	0.950357i	−0.879889	−	0.475179i	$-0.842383\pi$
$149$		−	3.83742e7i		−	0.950357i	0.879889	−	0.475179i	$-0.157617\pi$
$150$		−	3.08917e7i		−	0.747348i
$151$		−	3.57774e7i		−	0.845647i	−0.906212	−	0.422823i	$-0.861039\pi$
$151$		−	3.57774e7i		−	0.845647i	0.906212	−	0.422823i	$-0.138961\pi$
$152$	−585151.			−0.0135150
$153$			2.89675e6i			0.0653870i
$154$	−5.44643e7			−1.20168
$155$	−6.54341e7			−1.41138
$156$			1.74426e7i			0.367854i
$157$	1.00383e7			0.207020			0.103510	−	0.994628i	$-0.466993\pi$
$157$	1.00383e7			0.207020			0.103510	+	0.994628i	$0.466993\pi$
$158$	1.19825e8			2.41684
$159$		−	2.75511e7i		−	0.543562i
$160$			7.21560e7i			1.39268i
$161$		−	1.29596e8i		−	2.44738i
$162$			8.23632e7i			1.52206i
$163$		−	7.82315e6i		−	0.141490i	−0.997494	−	0.0707449i	$-0.977462\pi$
$163$		−	7.82315e6i		−	0.141490i	0.997494	−	0.0707449i	$-0.0225376\pi$
$164$			5.07705e7i			0.898789i
$165$			2.75142e7i			0.476830i
$166$		−	7.78825e7i		−	1.32148i
$167$			4.37412e7i			0.726747i	0.931644	+	0.363373i	$0.118375\pi$
$167$			4.37412e7i			0.726747i	−0.931644	+	0.363373i	$0.881625\pi$
$168$		−	9.56177e7i		−	1.55581i
$169$	5.89618e7			0.939653
$170$	−1.96414e8			−3.06621
$171$		−	43311.5i		−	0.000662394i
$172$	−4.18965e7			−0.627811
$173$	−4.44065e7			−0.652057			−0.326028	−	0.945360i	$-0.605711\pi$
$173$	−4.44065e7			−0.652057			−0.326028	+	0.945360i	$0.605711\pi$
$174$			1.18990e8i			1.71233i
$175$	−6.42244e7			−0.905872
$176$		−	5.51621e6i		−	0.0762687i
$177$		−	5.48982e7i		−	0.744134i
$178$		−	8.88452e7i		−	1.18077i
$179$			9.01705e6i			0.117511i	0.998272	+	0.0587556i	$0.0187133\pi$
$179$			9.01705e6i			0.117511i	−0.998272	+	0.0587556i	$0.981287\pi$
$180$	6.00590e6			0.0767581
$181$		−	1.88632e7i		−	0.236450i	−0.992987	−	0.118225i	$-0.962280\pi$
$181$		−	1.88632e7i		−	0.236450i	0.992987	−	0.118225i	$-0.0377204\pi$
$182$	5.99758e7			0.737439
$183$	2.17508e7			0.262359
$184$	−9.22309e7			−1.09148
$185$			1.57450e8i			1.82827i
$186$			1.58552e8i			1.80666i
$187$	5.67310e7			0.634417
$188$			1.42411e8i			1.56311i
$189$	1.78616e8			1.92444
$190$	2.93673e6			0.0310618
$191$	2.20764e7			0.229251			0.114625	−	0.993409i	$-0.463433\pi$
$191$	2.20764e7			0.229251			0.114625	+	0.993409i	$0.463433\pi$
$192$	1.56543e8			1.59617
$193$	8.36333e7			0.837391			0.418696	−	0.908127i	$-0.362487\pi$
$193$	8.36333e7			0.837391			0.418696	+	0.908127i	$0.362487\pi$
$194$		−	9.95194e6i		−	0.0978592i
$195$		−	3.02985e7i		−	0.292617i
$196$	−4.13149e8			−3.91931
$197$	−1.41175e8			−1.31561			−0.657804	−	0.753189i	$-0.728515\pi$
$197$	−1.41175e8			−1.31561			−0.657804	+	0.753189i	$0.728515\pi$
$198$	−2.86901e6			−0.0262666
$199$			2.00179e8i			1.80067i	0.435201	+	0.900333i	$0.356677\pi$
$199$			2.00179e8i			1.80067i	−0.435201	+	0.900333i	$0.643323\pi$
$200$			4.57072e7i			0.403999i
$201$		−	7.32956e7i		−	0.636637i
$202$	−3.41797e8			−2.91769
$203$	2.47382e8			2.07555
$204$			2.87762e8i			2.37316i
$205$		−	8.81903e7i		−	0.714961i
$206$	5.10796e7			0.407110
$207$		−	6.82671e6i		−	0.0534952i
$208$			6.07441e6i			0.0468040i
$209$	−848226.			−0.00642687
$210$			4.79882e8i			3.57575i
$211$	−1.46193e8			−1.07137			−0.535684	−	0.844418i	$-0.679946\pi$
$211$	−1.46193e8			−1.07137			−0.535684	+	0.844418i	$0.679946\pi$
$212$			1.17779e8i			0.848971i
$213$	−2.62905e8			−1.86411
$214$	−3.43210e8			−2.39393
$215$	7.27759e7			0.499405
$216$		−	1.27118e8i		−	0.858259i
$217$	3.29632e8			2.18988
$218$	−2.42913e8	+	1.32697e7i	−1.58801	+	0.0867490i
$219$	−4.68546e7			−0.301438
$220$		−	1.17622e8i		−	0.744745i
$221$	−6.24718e7			−0.389324
$222$	3.81512e8			2.34031
$223$	−6.77639e7			−0.409196			−0.204598	−	0.978846i	$-0.565589\pi$
$223$	−6.77639e7			−0.409196			−0.204598	+	0.978846i	$0.565589\pi$
$224$		−	3.63494e8i		−	2.16087i
$225$	−3.38314e6			−0.0198007
$226$		−	2.85571e8i		−	1.64564i
$227$	5.23416e7			0.297000			0.148500	−	0.988912i	$-0.452556\pi$
$227$	5.23416e7			0.297000			0.148500	+	0.988912i	$0.452556\pi$
$228$		−	4.30253e6i		−	0.0240410i
$229$		−	1.47965e8i		−	0.814205i	−0.913382	−	0.407103i	$-0.866539\pi$
$229$		−	1.47965e8i		−	0.814205i	0.913382	−	0.407103i	$-0.133461\pi$
$230$	4.62885e8			2.50856
$231$		−	1.38606e8i		−	0.739843i
$232$		−	1.76057e8i		−	0.925647i
$233$	−2.94709e8			−1.52633			−0.763163	−	0.646206i	$-0.776355\pi$
$233$	−2.94709e8			−1.52633			−0.763163	+	0.646206i	$0.776355\pi$
$234$	3.15934e6			0.0161191
$235$		−	2.47373e8i		−	1.24341i
$236$			2.34686e8i			1.16224i
$237$			3.04942e8i			1.48798i
$238$	9.89459e8			4.75750
$239$	−8.47882e7			−0.401738			−0.200869	−	0.979618i	$-0.564377\pi$
$239$	−8.47882e7			−0.401738			−0.200869	+	0.979618i	$0.564377\pi$
$240$	−4.86030e7			−0.226946
$241$			3.23027e8i			1.48655i	0.668986	+	0.743275i	$0.266729\pi$
$241$			3.23027e8i			1.48655i	−0.668986	+	0.743275i	$0.733271\pi$
$242$		−	2.94447e8i		−	1.33553i
$243$	1.84451e7			0.0824629
$244$	−9.29832e7			−0.409770
$245$	7.17655e8			3.11770
$246$	−2.13692e8			−0.915197
$247$	934061.			0.00394399
$248$		−	2.34592e8i		−	0.976637i
$249$	1.98202e8			0.813601
$250$			2.48587e8i			1.00621i
$251$			2.39647e8i			0.956564i	0.878206	+	0.478282i	$0.158740\pi$
$251$			2.39647e8i			0.956564i	−0.878206	+	0.478282i	$0.841260\pi$
$252$	−3.02554e7			−0.119097
$253$	−1.33696e8			−0.519037
$254$	−2.06636e8			−0.791203
$255$		−	4.99853e8i		−	1.88778i
$256$	−1.80477e8			−0.672328
$257$		−	1.55900e8i		−	0.572903i	−0.958095	−	0.286452i	$-0.907524\pi$
$257$		−	1.55900e8i		−	0.572903i	0.958095	−	0.286452i	$-0.0924758\pi$
$258$		−	1.76341e8i		−	0.639272i
$259$		−	7.93170e8i		−	2.83672i
$260$			1.29524e8i			0.457029i
$261$	1.30313e7			0.0453676
$262$			7.81222e8i			2.68361i
$263$	2.51505e8			0.852515			0.426258	−	0.904602i	$-0.359832\pi$
$263$	2.51505e8			0.852515			0.426258	+	0.904602i	$0.359832\pi$
$264$	−9.86431e7			−0.329954
$265$		−	2.04587e8i		−	0.675332i
$266$	−1.47941e7			−0.0481951
$267$	2.26101e8			0.726966
$268$			3.13334e8i			0.994343i
$269$			9.21674e7i			0.288698i	0.989527	+	0.144349i	$0.0461088\pi$
$269$			9.21674e7i			0.288698i	−0.989527	+	0.144349i	$0.953891\pi$
$270$			6.37974e8i			1.97256i
$271$		−	1.36943e8i		−	0.417971i	−0.977919	−	0.208986i	$-0.932984\pi$
$271$		−	1.36943e8i		−	0.417971i	0.977919	−	0.208986i	$-0.0670162\pi$
$272$			1.00213e8i			0.301950i
$273$			1.52632e8i			0.454021i
$274$			4.30614e8i			1.26462i
$275$			6.62565e7i			0.192116i
$276$		−	6.78161e8i		−	1.94156i
$277$		−	1.20225e8i		−	0.339872i	−0.985455	−	0.169936i	$-0.945644\pi$
$277$		−	1.20225e8i		−	0.339872i	0.985455	−	0.169936i	$-0.0543560\pi$
$278$	−1.81544e8			−0.506788
$279$	1.73640e7			0.0478668
$280$		−	7.10031e8i		−	1.93297i
$281$	−3.18915e8			−0.857437			−0.428719	−	0.903438i	$-0.641035\pi$
$281$	−3.18915e8			−0.857437			−0.428719	+	0.903438i	$0.641035\pi$
$282$	−5.99402e8			−1.59165
$283$		−	1.10218e8i		−	0.289069i	−0.989500	−	0.144535i	$-0.953832\pi$
$283$		−	1.10218e8i		−	0.289069i	0.989500	−	0.144535i	$-0.0461685\pi$
$284$	1.12390e9			2.91149
$285$			7.47367e6i			0.0191239i
$286$		−	6.18735e7i		−	0.156395i
$287$			4.44268e8i			1.10932i
$288$		−	1.91477e7i		−	0.0472327i
$289$	−6.20298e8			−1.51167
$290$			8.83587e8i			2.12744i
$291$	2.53266e7			0.0602493
$292$	2.00300e8			0.470806
$293$	7.22136e8			1.67719			0.838595	−	0.544755i	$-0.183377\pi$
$293$	7.22136e8			1.67719			0.838595	+	0.544755i	$0.183377\pi$
$294$		−	1.73893e9i		−	3.99086i
$295$		−	4.07659e8i		−	0.924528i
$296$	−5.64483e8			−1.26512
$297$		−	1.84268e8i		−	0.408134i
$298$	−6.90470e8			−1.51143
$299$	1.47226e8			0.318518
$300$	−3.36079e8			−0.718649
$301$	−3.66617e8			−0.774871
$302$	−6.43746e8			−1.34490
$303$		−	8.69836e8i		−	1.79634i
$304$		−	1.49836e6i		−	0.00305886i
$305$	1.61515e8			0.325960
$306$	5.21216e7			0.103990
$307$	−4.10483e8			−0.809676			−0.404838	−	0.914388i	$-0.632672\pi$
$307$	−4.10483e8			−0.809676			−0.404838	+	0.914388i	$0.632672\pi$
$308$			5.92531e8i			1.15554i
$309$			1.29992e8i			0.250647i
$310$			1.17736e9i			2.24463i
$311$	4.53139e8			0.854220			0.427110	−	0.904200i	$-0.359532\pi$
$311$	4.53139e8			0.854220			0.427110	+	0.904200i	$0.359532\pi$
$312$	1.08625e8			0.202483
$313$		−	9.14577e8i		−	1.68584i	−0.538043	−	0.842918i	$-0.680836\pi$
$313$		−	9.14577e8i		−	1.68584i	0.538043	−	0.842918i	$-0.319164\pi$
$314$		−	1.80620e8i		−	0.329240i
$315$	5.25548e7			0.0947383
$316$		−	1.30361e9i		−	2.32403i
$317$			9.46943e8i			1.66961i	0.550542	+	0.834807i	$0.314421\pi$
$317$			9.46943e8i			1.66961i	−0.550542	+	0.834807i	$0.685579\pi$
$318$	−4.95729e8			−0.864470
$319$		−	2.55209e8i		−	0.440179i
$320$	1.16245e9			1.98312
$321$		−	8.73433e8i		−	1.47388i
$322$	−2.33183e9			−3.89226
$323$	1.54098e7			0.0254442
$324$	8.96050e8			1.46361
$325$		−	7.29612e7i		−	0.117896i
$326$	−1.40763e8			−0.225022
$327$	−3.37701e7	−	6.18189e8i	−0.0534090	−	0.977696i
$328$	3.16177e8			0.494734
$329$			1.24617e9i			1.92926i
$330$	4.95066e8			0.758340
$331$	3.08751e8			0.467962			0.233981	−	0.972241i	$-0.424825\pi$
$331$	3.08751e8			0.467962			0.233981	+	0.972241i	$0.424825\pi$
$332$	−8.47303e8			−1.27074
$333$		−	4.17817e7i		−	0.0620056i
$334$	7.87040e8			1.15580
$335$		−	5.44274e8i		−	0.790971i
$336$	2.44843e8			0.352127
$337$			6.23289e8i			0.887125i	0.896243	+	0.443562i	$0.146286\pi$
$337$			6.23289e8i			0.887125i	−0.896243	+	0.443562i	$0.853714\pi$
$338$		−	1.06091e9i		−	1.49440i
$339$	7.26747e8			1.01317
$340$			2.13684e9i			2.94847i
$341$		−	3.40061e8i		−	0.464427i
$342$	−779307.			−0.00105346
$343$	−2.20460e9			−2.94985
$344$			2.60914e8i			0.345575i
$345$			1.17799e9i			1.54446i
$346$			7.99011e8i			1.03702i
$347$	6.52819e8			0.838763			0.419382	−	0.907810i	$-0.362247\pi$
$347$	6.52819e8			0.838763			0.419382	+	0.907810i	$0.362247\pi$
$348$	1.29452e9			1.64658
$349$	4.35376e8			0.548246			0.274123	−	0.961695i	$-0.411612\pi$
$349$	4.35376e8			0.548246			0.274123	+	0.961695i	$0.411612\pi$
$350$			1.15560e9i			1.44068i
$351$			2.02915e8i			0.250460i
$352$	−3.74995e8			−0.458275
$353$	−9.90304e8			−1.19828			−0.599138	−	0.800646i	$-0.704490\pi$
$353$	−9.90304e8			−1.19828			−0.599138	+	0.800646i	$0.704490\pi$
$354$	−9.87788e8			−1.18346
$355$	−1.95226e9			−2.31600
$356$	−9.66569e8			−1.13542
$357$			2.51807e9i			2.92906i
$358$	1.62245e8			0.186887
$359$		−	1.38154e9i		−	1.57592i	−0.615727	−	0.787960i	$-0.711137\pi$
$359$		−	1.38154e9i		−	1.57592i	0.615727	−	0.787960i	$-0.288863\pi$
$360$		−	3.74022e7i		−	0.0422512i
$361$	8.93641e8			0.999742
$362$	−3.39407e8			−0.376046
$363$	7.49336e8			0.822249
$364$		−	6.52492e8i		−	0.709121i
$365$	−3.47930e8			−0.374513
$366$		−	3.91364e8i		−	0.417251i
$367$		−	7.26877e8i		−	0.767590i	−0.923418	−	0.383795i	$-0.874617\pi$
$367$		−	7.26877e8i		−	0.767590i	0.923418	−	0.383795i	$-0.125383\pi$
$368$		−	2.36170e8i		−	0.247035i
$369$			2.34027e7i			0.0242478i
$370$	2.83300e9			2.90764
$371$			1.03063e9i			1.04784i
$372$	1.72492e9			1.73728
$373$	8.14962e8			0.813124			0.406562	−	0.913623i	$-0.366728\pi$
$373$	8.14962e8			0.813124			0.406562	+	0.913623i	$0.366728\pi$
$374$		−	1.02077e9i		−	1.00896i
$375$	−6.32627e8			−0.619496
$376$	8.86873e8			0.860407
$377$			2.81035e8i			0.270126i
$378$		−	3.21386e9i		−	3.06060i
$379$			1.72344e9i			1.62615i	0.582162	+	0.813073i	$0.302207\pi$
$379$			1.72344e9i			1.62615i	−0.582162	+	0.813073i	$0.697793\pi$
$380$		−	3.19495e7i		−	0.0298690i
$381$		−	5.25867e8i		−	0.487122i
$382$		−	3.97222e8i		−	0.364596i
$383$			1.55826e9i			1.41724i	0.705589	+	0.708621i	$0.250683\pi$
$383$			1.55826e9i			1.41724i	−0.705589	+	0.708621i	$0.749317\pi$
$384$		−	1.57292e9i		−	1.41758i
$385$		−	1.02925e9i		−	0.919197i
$386$		−	1.50482e9i		−	1.33177i
$387$	−1.93122e7			−0.0169373
$388$	−1.08270e8			−0.0941014
$389$		−	8.28871e8i		−	0.713943i	−0.934115	−	0.356972i	$-0.883809\pi$
$389$		−	8.28871e8i		−	0.713943i	0.934115	−	0.356972i	$-0.116191\pi$
$390$	−5.45163e8			−0.465372
$391$	2.42888e9			2.05488
$392$			2.57292e9i			2.15737i
$393$	−1.98813e9			−1.65223
$394$			2.54018e9i			2.09232i
$395$			2.26442e9i			1.84870i
$396$			3.12127e7i			0.0252579i
$397$			1.16670e9i			0.935822i	0.883775	+	0.467911i	$0.154993\pi$
$397$			1.16670e9i			0.935822i	−0.883775	+	0.467911i	$0.845007\pi$
$398$	3.60184e9			2.86374
$399$		−	3.76494e7i		−	0.0296724i
$400$	−1.17040e8			−0.0914375
$401$	−2.01439e8			−0.156005			−0.0780024	−	0.996953i	$-0.524854\pi$
$401$	−2.01439e8			−0.156005			−0.0780024	+	0.996953i	$0.524854\pi$
$402$	−1.31882e9			−1.01249
$403$			3.74473e8i			0.285006i
$404$			3.71850e9i			2.80564i
$405$	−1.55647e9			−1.16426
$406$		−	4.45117e9i		−	3.30090i
$407$	−8.18266e8			−0.601609
$408$	1.79206e9			1.30630
$409$	1.68571e9			1.21829			0.609145	−	0.793059i	$-0.291513\pi$
$409$	1.68571e9			1.21829			0.609145	+	0.793059i	$0.291513\pi$
$410$	−1.58682e9			−1.13706
$411$	−1.09587e9			−0.778594
$412$		−	5.55708e8i		−	0.391477i
$413$			2.05363e9i			1.43449i
$414$	−1.22834e8			−0.0850777
$415$	1.47180e9			1.01083
$416$	4.12942e8			0.281230
$417$		−	4.62011e8i		−	0.312015i
$418$			1.52622e7i			0.0102212i
$419$		−	1.51671e9i		−	1.00729i	−0.863912	−	0.503643i	$-0.831993\pi$
$419$		−	1.51671e9i		−	1.00729i	0.863912	−	0.503643i	$-0.168007\pi$
$420$	5.22076e9			3.43844
$421$	−9.08273e8			−0.593238			−0.296619	−	0.954996i	$-0.595859\pi$
$421$	−9.08273e8			−0.593238			−0.296619	+	0.954996i	$0.595859\pi$
$422$			2.63047e9i			1.70388i
$423$			6.56441e7i			0.0421701i
$424$	7.33479e8			0.467312
$425$		−	1.20369e9i		−	0.760593i
$426$			4.73048e9i			2.96464i
$427$	−8.13652e8			−0.505756
$428$			3.73387e9i			2.30200i
$429$	1.57461e8			0.0962882
$430$		−	1.30946e9i		−	0.794244i
$431$	1.18690e9			0.714074			0.357037	−	0.934090i	$-0.383787\pi$
$431$	1.18690e9			0.714074			0.357037	+	0.934090i	$0.383787\pi$
$432$	3.25503e8			0.194251
$433$	−2.41268e9			−1.42821			−0.714105	−	0.700039i	$-0.753166\pi$
$433$	−2.41268e9			−1.42821			−0.714105	+	0.700039i	$0.753166\pi$
$434$		−	5.93109e9i		−	3.48274i
$435$	−2.24863e9			−1.30980
$436$	1.44365e8	+	2.64272e9i	0.0834178	+	1.52703i
$437$	−3.63159e7			−0.0208167
$438$			8.43059e8i			0.479401i
$439$	−1.78601e9			−1.00753			−0.503764	−	0.863841i	$-0.668052\pi$
$439$	−1.78601e9			−1.00753			−0.503764	+	0.863841i	$0.668052\pi$
$440$	−7.32497e8			−0.409941
$441$	−1.90441e8			−0.105737
$442$			1.12406e9i			0.619173i
$443$	−2.79697e9			−1.52853			−0.764265	−	0.644902i	$-0.776898\pi$
$443$	−2.79697e9			−1.52853			−0.764265	+	0.644902i	$0.776898\pi$
$444$		−	4.15056e9i		−	2.25044i
$445$	1.67897e9			0.903197
$446$			1.21928e9i			0.650777i
$447$		−	1.75717e9i		−	0.930546i
$448$	−5.85595e9			−3.07698
$449$		−	3.45347e9i		−	1.80050i	−0.435373	−	0.900250i	$-0.643384\pi$
$449$		−	3.45347e9i		−	1.80050i	0.435373	−	0.900250i	$-0.356616\pi$
$450$			6.08731e7i			0.0314907i
$451$	4.58325e8			0.235264
$452$	−3.10680e9			−1.58244
$453$		−	1.63826e9i		−	0.828018i
$454$		−	9.41787e8i		−	0.472343i
$455$			1.13340e9i			0.564085i
$456$	−2.67944e7			−0.0132332
$457$	1.38673e9			0.679651			0.339825	−	0.940489i	$-0.389632\pi$
$457$	1.38673e9			0.679651			0.339825	+	0.940489i	$0.389632\pi$
$458$	−2.66234e9			−1.29490
$459$			3.34761e9i			1.61581i
$460$		−	5.03584e9i		−	2.41223i
$461$	−2.09283e9			−0.994901			−0.497450	−	0.867492i	$-0.665730\pi$
$461$	−2.09283e9			−0.994901			−0.497450	+	0.867492i	$0.665730\pi$
$462$	−2.49395e9			−1.17663
$463$	3.79605e8			0.177746			0.0888728	−	0.996043i	$-0.471674\pi$
$463$	3.79605e8			0.177746			0.0888728	+	0.996043i	$0.471674\pi$
$464$	4.50819e8			0.209503
$465$	−2.99626e9			−1.38196
$466$			5.30272e9i			2.42744i
$467$	−1.06276e8			−0.0482863			−0.0241432	−	0.999709i	$-0.507686\pi$
$467$	−1.06276e8			−0.0482863			−0.0241432	+	0.999709i	$0.507686\pi$
$468$		−	3.43712e7i		−	0.0155001i
$469$			2.74184e9i			1.22726i
$470$	−4.45100e9			−1.97749
$471$	4.59658e8			0.202704
$472$	1.46153e9			0.639749
$473$			3.78217e8i			0.164334i
$474$	5.48685e9			2.36646
$475$			1.79972e7i			0.00770509i
$476$		−	1.07646e10i		−	4.57480i
$477$			5.42903e7i			0.0229038i
$478$			1.52560e9i			0.638916i
$479$	4.37587e9			1.81924			0.909620	−	0.415440i	$-0.136373\pi$
$479$	4.37587e9			1.81924			0.909620	+	0.415440i	$0.136373\pi$
$480$			3.30406e9i			1.36365i
$481$	9.01069e8			0.369190
$482$	5.81226e9			2.36418
$483$		−	5.93426e9i		−	2.39636i
$484$	−3.20336e9			−1.28424
$485$	1.88069e8			0.0748549
$486$		−	3.31884e8i		−	0.131147i
$487$			4.20391e9i			1.64931i	0.565637	+	0.824654i	$0.308630\pi$
$487$			4.20391e9i			1.64931i	−0.565637	+	0.824654i	$0.691370\pi$
$488$			5.79060e8i			0.225556i
$489$		−	3.58226e8i		−	0.138540i
$490$		−	1.29128e10i		−	4.95833i
$491$		−	2.86235e9i		−	1.09128i	−0.838018	−	0.545642i	$-0.816286\pi$
$491$		−	2.86235e9i		−	1.09128i	0.838018	−	0.545642i	$-0.183714\pi$
$492$			2.32480e9i			0.880053i
$493$			4.63641e9i			1.74268i
$494$		−	1.68066e7i		−	0.00627244i
$495$		−	5.42177e7i		−	0.0200920i
$496$	6.00707e8			0.221043
$497$	9.83474e9			3.59348
$498$		−	3.56628e9i		−	1.29393i
$499$	−7.59608e7			−0.0273677			−0.0136838	−	0.999906i	$-0.504356\pi$
$499$	−7.59608e7			−0.0273677			−0.0136838	+	0.999906i	$0.504356\pi$
$500$	2.70444e9			0.967570
$501$			2.00293e9i			0.711597i
$502$	4.31199e9			1.52130
$503$			4.32061e9i			1.51376i	0.653553	+	0.756880i	$0.273278\pi$
$503$			4.32061e9i			1.51376i	−0.653553	+	0.756880i	$0.726722\pi$
$504$			1.88418e8i			0.0655564i
$505$		−	6.45917e9i		−	2.23181i
$506$			2.40561e9i			0.825465i
$507$	2.69989e9			0.920064
$508$			2.24805e9i			0.760821i
$509$	−3.71104e9			−1.24733			−0.623667	−	0.781690i	$-0.714358\pi$
$509$	−3.71104e9			−1.24733			−0.623667	+	0.781690i	$0.714358\pi$
$510$	−8.99391e9			−3.00229
$511$	1.75273e9			0.581089
$512$		−	1.14950e9i		−	0.378500i
$513$		−	5.00526e7i		−	0.0163688i
$514$	−2.80513e9			−0.911134
$515$			9.65286e8i			0.311409i
$516$	−1.91846e9			−0.614723
$517$	1.28560e9			0.409155
$518$	−1.42716e10			−4.51147
$519$	−2.03340e9			−0.638464
$520$	8.06621e8			0.251569
$521$		−	7.82349e8i		−	0.242364i	−0.992630	−	0.121182i	$-0.961331\pi$
$521$		−	7.82349e8i		−	0.242364i	0.992630	−	0.121182i	$-0.0386685\pi$
$522$		−	2.34474e8i		−	0.0721518i
$523$	−1.86669e9			−0.570579			−0.285289	−	0.958441i	$-0.592090\pi$
$523$	−1.86669e9			−0.570579			−0.285289	+	0.958441i	$0.592090\pi$
$524$	8.49911e9			2.58056
$525$	−2.94087e9			−0.886988
$526$		−	4.52536e9i		−	1.35582i
$527$			6.17793e9i			1.83868i
$528$		−	2.52590e8i		−	0.0746787i
$529$	−2.31924e9			−0.681164
$530$	−3.68115e9			−1.07403
$531$			1.08179e8i			0.0313553i
$532$			1.60949e8i			0.0463444i
$533$	−5.04705e8			−0.144375
$534$		−	4.06826e9i		−	1.15615i
$535$		−	6.48588e9i		−	1.83118i
$536$	1.95131e9			0.547331
$537$			4.12895e8i			0.115061i
$538$	1.65838e9			0.459140
$539$			3.72966e9i			1.02591i
$540$	6.94068e9			1.89681
$541$	6.24714e9			1.69625			0.848126	−	0.529794i	$-0.177731\pi$
$541$	6.24714e9			1.69625			0.848126	+	0.529794i	$0.177731\pi$
$542$	−2.46402e9			−0.664733
$543$		−	8.63754e8i		−	0.231521i
$544$	6.81256e9			1.81432
$545$	−2.50767e8	−	4.59050e9i	−0.0663565	−	1.21471i
$546$	2.74632e9			0.722066
$547$			1.73392e9i			0.452974i	0.974014	+	0.226487i	$0.0727240\pi$
$547$			1.73392e9i			0.452974i	−0.974014	+	0.226487i	$0.927276\pi$
$548$	4.68476e9			1.21606
$549$	−4.28606e7			−0.0110549
$550$	1.19216e9			0.305538
$551$		−	6.93224e7i		−	0.0176540i
$552$	−4.22330e9			−1.06872
$553$		−	1.14072e10i		−	2.86842i
$554$	−2.16322e9			−0.540525
$555$			7.20969e9i			1.79016i
$556$			1.97507e9i			0.487327i
$557$	−2.41341e9			−0.591749			−0.295875	−	0.955227i	$-0.595611\pi$
$557$	−2.41341e9			−0.591749			−0.295875	+	0.955227i	$0.595611\pi$
$558$		−	3.12431e8i		−	0.0761263i
$559$		−	4.16490e8i		−	0.100847i
$560$	1.81814e9			0.437490
$561$	2.59774e9			0.621192
$562$			5.73826e9i			1.36365i
$563$			5.93493e9i			1.40164i	0.713338	+	0.700820i	$0.247182\pi$
$563$			5.93493e9i			1.40164i	−0.713338	+	0.700820i	$0.752818\pi$
$564$			6.52105e9i			1.53052i
$565$	5.39663e9			1.25879
$566$	−1.98317e9			−0.459730
$567$	7.84091e9			1.80645
$568$		−	6.99919e9i		−	1.60261i
$569$		−	5.28834e9i		−	1.20344i	−0.798705	−	0.601722i	$-0.794481\pi$
$569$		−	5.28834e9i		−	1.20344i	0.798705	−	0.601722i	$-0.205519\pi$
$570$	1.34474e8			0.0304143
$571$	3.86278e9			0.868308			0.434154	−	0.900839i	$-0.357047\pi$
$571$	3.86278e9			0.868308			0.434154	+	0.900839i	$0.357047\pi$
$572$	−6.73137e8			−0.150389
$573$	1.01089e9			0.224472
$574$	7.99376e9			1.76425
$575$			2.83670e9i			0.622266i
$576$	−3.08473e8			−0.0672572
$577$		−	3.13269e9i		−	0.678895i	−0.940625	−	0.339447i	$-0.889760\pi$
$577$		−	3.13269e9i		−	0.678895i	0.940625	−	0.339447i	$-0.110240\pi$
$578$			1.11611e10i			2.40414i
$579$	3.82961e9			0.819935
$580$	9.61277e9			2.04574
$581$	−7.41435e9			−1.56840
$582$		−	4.55704e8i		−	0.0958192i
$583$	1.06324e9			0.222224
$584$		−	1.24739e9i		−	0.259153i
$585$			5.97041e7i			0.0123299i
$586$		−	1.29935e10i		−	2.66737i
$587$		−	1.60374e9i		−	0.327265i	−0.986521	−	0.163633i	$-0.947679\pi$
$587$		−	1.60374e9i		−	0.327265i	0.986521	−	0.163633i	$-0.0523212\pi$
$588$	−1.89183e10			−3.83761
$589$		−	9.23707e7i		−	0.0186265i
$590$	−7.33505e9			−1.47035
$591$	−6.46448e9			−1.28818
$592$		−	1.44544e9i		−	0.286335i
$593$	8.18680e8			0.161221			0.0806107	−	0.996746i	$-0.474313\pi$
$593$	8.18680e8			0.161221			0.0806107	+	0.996746i	$0.474313\pi$
$594$	−3.31555e9			−0.649087
$595$			1.86985e10i			3.63913i
$596$			7.51180e9i			1.45339i
$597$			9.16631e9i			1.76313i
$598$		−	2.64905e9i		−	0.506565i
$599$			1.62405e9i			0.308750i	0.988012	+	0.154375i	$0.0493363\pi$
$599$			1.62405e9i			0.308750i	−0.988012	+	0.154375i	$0.950664\pi$
$600$			2.09296e9i			0.395577i
$601$			1.09418e9i			0.205603i	0.994702	+	0.102802i	$0.0327807\pi$
$601$			1.09418e9i			0.205603i	−0.994702	+	0.102802i	$0.967219\pi$
$602$			6.59657e9i			1.23234i
$603$			1.44431e8i			0.0268257i
$604$			7.00347e9i			1.29325i
$605$	5.56436e9			1.02158
$606$	−1.56510e10			−2.85686
$607$			4.23756e9i			0.769052i	0.923114	+	0.384526i	$0.125635\pi$
$607$			4.23756e9i			0.769052i	−0.923114	+	0.384526i	$0.874365\pi$
$608$	−1.01860e8			−0.0183797
$609$	1.13277e10			2.03228
$610$		−	2.90616e9i		−	0.518401i
$611$	−1.41569e9			−0.251087
$612$		−	5.67044e8i		−	0.0999969i
$613$			4.52109e9i			0.792741i	0.918091	+	0.396370i	$0.129730\pi$
$613$			4.52109e9i			0.792741i	−0.918091	+	0.396370i	$0.870270\pi$
$614$			7.38587e9i			1.28769i
$615$		−	4.03828e9i		−	0.700057i
$616$	3.69004e9			0.636060
$617$		−	7.20205e9i		−	1.23441i	−0.786804	−	0.617203i	$-0.788266\pi$
$617$		−	7.20205e9i		−	1.23441i	0.786804	−	0.617203i	$-0.211734\pi$
$618$	2.33896e9			0.398624
$619$	−7.57007e9			−1.28287			−0.641435	−	0.767177i	$-0.721661\pi$
$619$	−7.57007e9			−1.28287			−0.641435	+	0.767177i	$0.721661\pi$
$620$	1.28088e10			2.15843
$621$		−	7.88923e9i		−	1.32195i
$622$		−	8.15337e9i		−	1.35853i
$623$	−8.45799e9			−1.40139
$624$			2.78150e8i			0.0458283i
$625$	−7.62693e9			−1.24960
$626$	−1.64561e10			−2.68112
$627$	−3.88407e7			−0.00629289
$628$	−1.96501e9			−0.316597
$629$	1.48655e10			2.38179
$630$		−	9.45623e8i		−	0.150670i
$631$			4.45488e9i			0.705884i	0.935645	+	0.352942i	$0.114819\pi$
$631$			4.45488e9i			0.705884i	−0.935645	+	0.352942i	$0.885181\pi$
$632$	−8.11831e9			−1.27925
$633$	−6.69426e9			−1.04903
$634$	1.70384e10			2.65532
$635$		−	3.90494e9i		−	0.605211i
$636$			5.39316e9i			0.831273i
$637$		−	4.10708e9i		−	0.629571i
$638$	−4.59201e9			−0.700052
$639$	5.18063e8			0.0785470
$640$		−	1.16801e10i		−	1.76123i
$641$			3.57170e8i			0.0535638i	0.999641	+	0.0267819i	$0.00852596\pi$
$641$			3.57170e8i			0.0535638i	−0.999641	+	0.0267819i	$0.991474\pi$
$642$	−1.57158e10			−2.34403
$643$		−	7.03379e9i		−	1.04340i	−0.853129	−	0.521700i	$-0.825298\pi$
$643$		−	7.03379e9i		−	1.04340i	0.853129	−	0.521700i	$-0.174702\pi$
$644$			2.53686e10i			3.74279i
$645$	3.33245e9			0.488995
$646$		−	2.77270e8i		−	0.0404659i
$647$	−8.70270e9			−1.26325			−0.631625	−	0.775274i	$-0.717612\pi$
$647$	−8.70270e9			−1.26325			−0.631625	+	0.775274i	$0.717612\pi$
$648$		−	5.58022e9i		−	0.805636i
$649$	2.11860e9			0.304224
$650$	−1.31280e9			−0.187500
$651$	1.50940e10			2.14423
$652$			1.53139e9i			0.216381i
$653$	−9.24396e9			−1.29916			−0.649579	−	0.760294i	$-0.725055\pi$
$653$	−9.24396e9			−1.29916			−0.649579	+	0.760294i	$0.725055\pi$
$654$	−1.11231e10	+	6.07628e8i	−1.55491	+	0.0849406i
$655$	−1.47633e10			−2.05276
$656$			8.09617e8i			0.111974i
$657$	9.23285e7			0.0127016
$658$	2.24224e10			3.06825
$659$	7.00325e9			0.953236			0.476618	−	0.879110i	$-0.341862\pi$
$659$	7.00325e9			0.953236			0.476618	+	0.879110i	$0.341862\pi$
$660$		−	5.38595e9i		−	0.729219i
$661$	−6.26024e9			−0.843113			−0.421557	−	0.906802i	$-0.638516\pi$
$661$	−6.26024e9			−0.843113			−0.421557	+	0.906802i	$0.638516\pi$
$662$		−	5.55539e9i		−	0.744237i
$663$	−2.86061e9			−0.381208
$664$			5.27665e9i			0.699471i
$665$		−	2.79575e8i		−	0.0368656i
$666$	−7.51782e8			−0.0986125
$667$		−	1.09265e10i		−	1.42574i
$668$		−	8.56240e9i		−	1.11142i
$669$	−3.10294e9			−0.400666
$670$	−9.79317e9			−1.25794
$671$			8.39396e8i			0.107260i
$672$		−	1.66445e10i		−	2.11582i
$673$			1.53507e10i			1.94123i	0.240644	+	0.970614i	$0.422642\pi$
$673$			1.53507e10i			1.94123i	−0.240644	+	0.970614i	$0.577358\pi$
$674$	1.12149e10			1.41087
$675$	−3.90970e9			−0.489306
$676$	−1.15419e10			−1.43702
$677$			8.48297e9i			1.05072i	0.850880	+	0.525361i	$0.176070\pi$
$677$			8.48297e9i			1.05072i	−0.850880	+	0.525361i	$0.823930\pi$
$678$		−	1.30764e10i		−	1.61133i
$679$	−9.47417e8			−0.116144
$680$	1.33073e10			1.62297
$681$	2.39675e9			0.290809
$682$	−6.11876e9			−0.738615
$683$	1.01783e10			1.22237			0.611185	−	0.791488i	$-0.290693\pi$
$683$	1.01783e10			1.22237			0.611185	+	0.791488i	$0.290693\pi$
$684$			8.47828e6i			0.00101300i
$685$	−8.13761e9			−0.967341
$686$			3.96676e10i			4.69138i
$687$		−	6.77538e9i		−	0.797232i
$688$	−6.68108e8			−0.0782144
$689$	−1.17083e9			−0.136373
$690$	2.11957e10			2.45627
$691$			1.77242e9i			0.204359i	0.994766	+	0.102179i	$0.0325816\pi$
$691$			1.77242e9i			0.204359i	−0.994766	+	0.102179i	$0.967418\pi$
$692$	8.69264e9			0.997195
$693$			2.73127e8i			0.0311745i
$694$		−	1.17462e10i		−	1.33395i
$695$		−	3.43077e9i		−	0.387654i
$696$		−	8.06174e9i		−	0.906351i
$697$	−8.32644e9			−0.931418
$698$		−	7.83376e9i		−	0.871920i
$699$	−1.34949e10			−1.49451
$700$	1.25720e10			1.38536
$701$		−	5.69892e9i		−	0.624856i	−0.949941	−	0.312428i	$-0.898858\pi$
$701$		−	5.69892e9i		−	0.624856i	0.949941	−	0.312428i	$-0.101142\pi$
$702$	3.65106e9			0.398327
$703$	−2.22265e8			−0.0241283
$704$			6.04124e9i			0.652562i
$705$		−	1.13273e10i		−	1.21749i
$706$			1.78186e10i			1.90571i
$707$			3.25388e10i			3.46285i
$708$			1.07464e10i			1.13801i
$709$		−	1.84560e9i		−	0.194481i	−0.995261	−	0.0972404i	$-0.968998\pi$
$709$		−	1.84560e9i		−	0.194481i	0.995261	−	0.0972404i	$-0.0310016\pi$
$710$			3.51273e10i			3.68333i
$711$		−	6.00898e8i		−	0.0626985i
$712$			6.01939e9i			0.624989i
$713$		−	1.45594e10i		−	1.50428i
$714$	4.53078e10			4.65832
$715$	1.16927e9			0.119630
$716$		−	1.76510e9i		−	0.179711i
$717$	−3.88249e9			−0.393363
$718$	−2.48582e10			−2.50631
$719$		−	1.80315e10i		−	1.80918i	−0.426285	−	0.904589i	$-0.640178\pi$
$719$		−	1.80315e10i		−	1.80918i	0.426285	−	0.904589i	$-0.359822\pi$
$720$	9.57737e7			0.00956275
$721$		−	4.86273e9i		−	0.483178i
$722$		−	1.60794e10i		−	1.58997i
$723$			1.47916e10i			1.45556i
$724$			3.69249e9i			0.361605i
$725$	−5.41490e9			−0.527724
$726$		−	1.34829e10i		−	1.30769i
$727$	1.01100e10			0.975846			0.487923	−	0.872887i	$-0.337755\pi$
$727$	1.01100e10			0.975846			0.487923	+	0.872887i	$0.337755\pi$
$728$	−4.06344e9			−0.390332
$729$	1.08556e10			1.03778
$730$			6.26033e9i			0.595617i
$731$		−	6.87110e9i		−	0.650602i
$732$	−4.25775e9			−0.401228
$733$		−	1.71878e10i		−	1.61197i	−0.591937	−	0.805984i	$-0.701637\pi$
$733$		−	1.71878e10i		−	1.61197i	0.591937	−	0.805984i	$-0.298363\pi$
$734$	−1.30788e10			−1.22076
$735$	3.28618e10			3.05271
$736$	−1.60550e10			−1.48435
$737$	2.82859e9			0.260276
$738$	4.21086e8			0.0385633
$739$			1.82666e10i			1.66496i	0.554058	+	0.832478i	$0.313079\pi$
$739$			1.82666e10i			1.66496i	−0.554058	+	0.832478i	$0.686921\pi$
$740$		−	3.08210e10i		−	2.79599i
$741$	4.27711e7			0.00386177
$742$	1.85442e10			1.66646
$743$	1.41512e10			1.26570			0.632850	−	0.774274i	$-0.281885\pi$
$743$	1.41512e10			1.26570			0.632850	+	0.774274i	$0.281885\pi$
$744$		−	1.07421e10i		−	0.956277i
$745$		−	1.30483e10i		−	1.15613i
$746$		−	1.46637e10i		−	1.29318i
$747$	−3.90564e8			−0.0342823
$748$	−1.11052e10			−0.970219
$749$			3.26733e10i			2.84123i
$750$			1.13829e10i			0.985234i
$751$	8.57515e9			0.738757			0.369379	−	0.929279i	$-0.379571\pi$
$751$	8.57515e9			0.738757			0.369379	+	0.929279i	$0.379571\pi$
$752$			2.27096e9i			0.194737i
$753$			1.09736e10i			0.936623i
$754$	5.05669e9			0.429602
$755$		−	1.21653e10i		−	1.02875i
$756$	−3.49644e10			−2.94307
$757$		−	4.18820e9i		−	0.350907i	−0.984488	−	0.175454i	$-0.943861\pi$
$757$		−	4.18820e9i		−	0.350907i	0.984488	−	0.175454i	$-0.0561392\pi$
$758$	3.10101e10			2.58619
$759$	−6.12203e9			−0.508217
$760$	−1.98968e8			−0.0164413
$761$		−	7.93710e9i		−	0.652853i	−0.945223	−	0.326427i	$-0.894155\pi$
$761$		−	7.93710e9i		−	0.652853i	0.945223	−	0.326427i	$-0.105845\pi$
$762$	−9.46197e9			−0.774710
$763$	1.26327e9	+	2.31252e10i	0.102958	+	1.88473i
$764$	−4.32148e9			−0.350595
$765$			9.84977e8i			0.0795447i
$766$	2.80379e10			2.25395
$767$	−2.33299e9			−0.186694
$768$	−8.26411e9			−0.658312
$769$			2.01924e10i			1.60120i	0.599197	+	0.800602i	$0.295487\pi$
$769$			2.01924e10i			1.60120i	−0.599197	+	0.800602i	$0.704513\pi$
$770$	−1.85194e10			−1.46187
$771$		−	7.13876e9i		−	0.560960i
$772$	−1.63713e10			−1.28063
$773$		−	8.68532e9i		−	0.676329i	−0.941087	−	0.338164i	$-0.890194\pi$
$773$		−	8.68532e9i		−	0.676329i	0.941087	−	0.338164i	$-0.109806\pi$
$774$			3.47487e8i			0.0269367i
$775$	−7.21524e9			−0.556795
$776$			6.74258e8i			0.0517977i
$777$		−	3.63196e10i		−	2.77759i
$778$	−1.49140e10			−1.13544
$779$	1.24495e8			0.00943560
$780$			5.93097e9i			0.447502i
$781$		−	1.01459e10i		−	0.762101i
$782$		−	4.37030e10i		−	3.26804i
$783$	1.50595e10			1.12110
$784$	−6.58832e9			−0.488279
$785$	3.41330e9			0.251843
$786$			3.57725e10i			2.62767i
$787$		−	2.41242e9i		−	0.176418i	−0.996102	−	0.0882088i	$-0.971886\pi$
$787$		−	2.41242e9i		−	0.176418i	0.996102	−	0.0882088i	$-0.0281143\pi$
$788$	2.76353e10			2.01197
$789$	1.15166e10			0.834744
$790$	4.07439e10			2.94014
$791$	−2.71861e10			−1.95312
$792$	1.94379e8			0.0139031
$793$		−	9.24338e8i		−	0.0658225i
$794$	2.09926e10			1.48831
$795$		−	9.36814e9i		−	0.661254i
$796$		−	3.91854e10i		−	2.75377i
$797$	−1.79800e10			−1.25801			−0.629007	−	0.777399i	$-0.716538\pi$
$797$	−1.79800e10			−1.25801			−0.629007	+	0.777399i	$0.716538\pi$
$798$	−6.77430e8			−0.0471904
$799$	−2.33555e10			−1.61986
$800$			7.95644e9i			0.549419i
$801$	−4.45540e8			−0.0306318
$802$			3.62451e9i			0.248107i
$803$		−	1.80819e9i		−	0.123237i
$804$			1.43477e10i			0.973614i
$805$		−	4.40662e10i		−	2.97728i
$806$	6.73794e9			0.453267
$807$			4.22039e9i			0.282680i
$808$	2.31572e10			1.54435
$809$	−1.89775e10			−1.26014			−0.630069	−	0.776539i	$-0.716974\pi$
$809$	−1.89775e10			−1.26014			−0.630069	+	0.776539i	$0.716974\pi$
$810$			2.80058e10i			1.85161i
$811$	−1.60954e10			−1.05956			−0.529782	−	0.848134i	$-0.677726\pi$
$811$	−1.60954e10			−1.05956			−0.529782	+	0.848134i	$0.677726\pi$
$812$	−4.84254e10			−3.17415
$813$		−	6.27068e9i		−	0.409258i
$814$			1.47231e10i			0.956786i
$815$		−	2.66009e9i		−	0.172125i
$816$			4.58882e9i			0.295655i
$817$			1.02735e8i			0.00659083i
$818$		−	3.03311e10i		−	1.93754i
$819$		−	3.00766e8i		−	0.0191309i
$820$			1.72634e10i			1.09340i
$821$		−	2.02942e10i		−	1.27989i	−0.768422	−	0.639944i	$-0.778958\pi$
$821$		−	2.02942e10i		−	1.27989i	0.768422	−	0.639944i	$-0.221042\pi$
$822$			1.97180e10i			1.23826i
$823$	−1.76119e10			−1.10130			−0.550652	−	0.834735i	$-0.685621\pi$
$823$	−1.76119e10			−1.10130			−0.550652	+	0.834735i	$0.685621\pi$
$824$	−3.46071e9			−0.215487
$825$			3.03392e9i			0.188111i
$826$	3.69511e10			2.28138
$827$	2.06148e10			1.26739			0.633694	−	0.773584i	$-0.281538\pi$
$827$	2.06148e10			1.26739			0.633694	+	0.773584i	$0.281538\pi$
$828$			1.33634e9i			0.0818107i
$829$	−7.97922e9			−0.486429			−0.243214	−	0.969973i	$-0.578202\pi$
$829$	−7.97922e9			−0.486429			−0.243214	+	0.969973i	$0.578202\pi$
$830$		−	2.64822e10i		−	1.60761i
$831$		−	5.50515e9i		−	0.332787i
$832$		−	6.65258e9i		−	0.400459i
$833$		−	6.77570e10i		−	4.06160i
$834$	−8.31301e9			−0.496223
$835$			1.48732e10i			0.884102i
$836$	1.66041e8			0.00982866
$837$	2.00665e10			1.18286
$838$	−2.72903e10			−1.60197
$839$			2.95185e10i			1.72555i	0.505589	+	0.862775i	$0.331275\pi$
$839$			2.95185e10i			1.72555i	−0.505589	+	0.862775i	$0.668725\pi$
$840$		−	3.25127e10i		−	1.89267i
$841$	3.60744e9			0.209129
$842$			1.63426e10i			0.943474i
$843$	−1.46033e10			−0.839563
$844$	2.86175e10			1.63845
$845$	2.00487e10			1.14311
$846$	1.18114e9			0.0670665
$847$	−2.80311e10			−1.58507
$848$			1.87818e9i			0.105767i
$849$		−	5.04695e9i		−	0.283043i
$850$	−2.16581e10			−1.20963
$851$	−3.50332e10			−1.94861
$852$	5.14641e10			2.85079
$853$			4.25073e9i			0.234499i	0.993102	+	0.117250i	$0.0374078\pi$
$853$			4.25073e9i			0.234499i	−0.993102	+	0.117250i	$0.962592\pi$
$854$			1.46401e10i			0.804344i
$855$		−	1.47271e7i		−	0.000805816i
$856$	2.32530e10			1.26713
$857$	−1.20682e10			−0.654951			−0.327476	−	0.944860i	$-0.606198\pi$
$857$	−1.20682e10			−0.654951			−0.327476	+	0.944860i	$0.606198\pi$
$858$		−	2.83322e9i		−	0.153135i
$859$			1.73259e10i			0.932652i	0.884613	+	0.466326i	$0.154423\pi$
$859$			1.73259e10i			0.932652i	−0.884613	+	0.466326i	$0.845577\pi$
$860$	−1.42460e10			−0.763744
$861$			2.03433e10i			1.08620i
$862$		−	2.13560e10i		−	1.13565i
$863$	−5.65804e9			−0.299660			−0.149830	−	0.988712i	$-0.547873\pi$
$863$	−5.65804e9			−0.299660			−0.149830	+	0.988712i	$0.547873\pi$
$864$		−	2.21279e10i		−	1.16719i
$865$	−1.50995e10			−0.793240
$866$			4.34116e10i			2.27140i
$867$	−2.84038e10			−1.48016
$868$	−6.45259e10			−3.34900
$869$	−1.17682e10			−0.608331
$870$			4.04599e10i			2.08309i
$871$	−3.11483e9			−0.159724
$872$	1.64577e10	−	8.99043e8i	0.840547	−	0.0459169i
$873$	−4.99069e7			−0.00253870
$874$			6.53435e8i			0.0331064i
$875$	2.36653e10			1.19422
$876$	9.17185e9			0.460991
$877$	−1.36550e10			−0.683588			−0.341794	−	0.939775i	$-0.611035\pi$
$877$	−1.36550e10			−0.683588			−0.341794	+	0.939775i	$0.611035\pi$
$878$			3.21358e10i			1.60235i
$879$	3.30670e10			1.64223
$880$		−	1.87566e9i		−	0.0927824i
$881$	−1.62214e9			−0.0799231			−0.0399615	−	0.999201i	$-0.512724\pi$
$881$	−1.62214e9			−0.0799231			−0.0399615	+	0.999201i	$0.512724\pi$
$882$			3.42662e9i			0.168161i
$883$			1.91163e10i			0.934418i	0.884147	+	0.467209i	$0.154740\pi$
$883$			1.91163e10i			0.934418i	−0.884147	+	0.467209i	$0.845260\pi$
$884$	1.22289e10			0.595396
$885$		−	1.86669e10i		−	0.905255i
$886$			5.03261e10i			2.43094i
$887$	−3.53540e9			−0.170101			−0.0850503	−	0.996377i	$-0.527105\pi$
$887$	−3.53540e9			−0.170101			−0.0850503	+	0.996377i	$0.527105\pi$
$888$	−2.58480e10			−1.23874
$889$			1.96716e10i			0.939038i
$890$		−	3.02098e10i		−	1.43643i
$891$		−	8.08900e9i		−	0.383110i
$892$	1.32649e10			0.625787
$893$	3.49206e8			0.0164097
$894$	−3.16169e10			−1.47992
$895$			3.06605e9i			0.142955i
$896$			5.88396e10i			2.73270i
$897$	6.74154e9			0.311878
$898$	−6.21386e10			−2.86348
$899$	2.77920e10			1.27573
$900$	6.62254e8			0.0302814
$901$	−1.93160e10			−0.879791
$902$		−	8.24669e9i		−	0.374159i
$903$	−1.67876e10			−0.758718
$904$			1.93478e10i			0.871049i
$905$		−	6.41400e9i		−	0.287646i
$906$	−2.94774e10			−1.31686
$907$	2.06661e10			0.919674			0.459837	−	0.888003i	$-0.347908\pi$
$907$	2.06661e10			0.919674			0.459837	+	0.888003i	$0.347908\pi$
$908$	−1.02459e10			−0.454204
$909$			1.71404e9i			0.0756916i
$910$	2.03934e10			0.897110
$911$		−	2.66567e9i		−	0.116813i	−0.998293	−	0.0584066i	$-0.981398\pi$
$911$		−	2.66567e9i		−	0.116813i	0.998293	−	0.0584066i	$-0.0186020\pi$
$912$		−	6.86108e7i		−	0.00299509i
$913$			7.64894e9i			0.332624i
$914$		−	2.49516e10i		−	1.08090i
$915$	7.39587e9			0.319165
$916$			2.89643e10i			1.24517i
$917$	7.43717e10			3.18504
$918$	6.02339e10			2.56976
$919$		−	1.12679e10i		−	0.478892i	−0.970910	−	0.239446i	$-0.923034\pi$
$919$		−	1.12679e10i		−	0.478892i	0.970910	−	0.239446i	$-0.0769659\pi$
$920$	−3.13611e10			−1.32780
$921$	−1.87962e10			−0.792797
$922$			3.76564e10i			1.58227i
$923$			1.11726e10i			0.467680i
$924$			2.71323e10i			1.13145i
$925$			1.73615e10i			0.721260i
$926$		−	6.83027e9i		−	0.282683i
$927$		−	2.56154e8i		−	0.0105614i
$928$		−	3.06469e10i		−	1.25884i
$929$			1.30335e10i			0.533342i	0.963788	+	0.266671i	$0.0859237\pi$
$929$			1.30335e10i			0.533342i	−0.963788	+	0.266671i	$0.914076\pi$
$930$			5.39120e10i			2.19784i
$931$			1.01308e9i			0.0411454i
$932$	5.76896e10			2.33422
$933$	2.07494e10			0.836413
$934$			1.91223e9i			0.0767936i
$935$	1.92901e10			0.771781
$936$	−2.14049e8			−0.00853195
$937$			1.87697e10i			0.745365i	0.927959	+	0.372682i	$0.121562\pi$
$937$			1.87697e10i			0.745365i	−0.927959	+	0.372682i	$0.878438\pi$
$938$	4.93342e10			1.95181
$939$		−	4.18789e10i		−	1.65069i
$940$			4.84235e10i			1.90156i
$941$			2.44437e10i			0.956319i	0.878273	+	0.478160i	$0.158696\pi$
$941$			2.44437e10i			0.956319i	−0.878273	+	0.478160i	$0.841304\pi$
$942$		−	8.27068e9i		−	0.322376i
$943$	1.96227e10			0.762023
$944$			3.74245e9i			0.144795i
$945$	6.07346e10			2.34113
$946$	6.80529e9			0.261353
$947$	−2.22665e10			−0.851975			−0.425988	−	0.904729i	$-0.640073\pi$
$947$	−2.22665e10			−0.851975			−0.425988	+	0.904729i	$0.640073\pi$
$948$		−	5.96928e10i		−	2.27558i
$949$			1.99117e9i			0.0756269i
$950$	3.23825e8			0.0122540
$951$			4.33610e10i			1.63481i
$952$	−6.70372e10			−2.51818
$953$	−9.37541e7			−0.00350885			−0.00175443	−	0.999998i	$-0.500558\pi$
$953$	−9.37541e7			−0.00350885			−0.00175443	+	0.999998i	$0.500558\pi$
$954$	9.76851e8			0.0364258
$955$	7.50658e9			0.278888
$956$	1.65974e10			0.614381
$957$		−	1.16862e10i		−	0.431003i
$958$		−	7.87354e10i		−	2.89328i
$959$	4.09941e10			1.50092
$960$	5.32290e10			1.94178
$961$	9.51961e9			0.346009
$962$		−	1.62130e10i		−	0.587153i
$963$			1.72113e9i			0.0621042i
$964$		−	6.32331e10i		−	2.27339i
$965$	2.84376e10			1.01870
$966$	−1.06776e11			−3.81112
$967$		−	3.79386e10i		−	1.34924i	−0.738166	−	0.674619i	$-0.764308\pi$
$967$		−	3.79386e10i		−	1.34924i	0.738166	−	0.674619i	$-0.235692\pi$
$968$			1.99492e10i			0.706906i
$969$	7.05622e8			0.0249137
$970$		−	3.38394e9i		−	0.119048i
$971$		−	4.00926e10i		−	1.40539i	−0.711492	−	0.702694i	$-0.751980\pi$
$971$		−	4.00926e10i		−	1.40539i	0.711492	−	0.702694i	$-0.248020\pi$
$972$	−3.61065e9			−0.126111
$973$			1.72829e10i			0.601480i
$974$	7.56413e10			2.62303
$975$		−	3.34093e9i		−	0.115439i
$976$	−1.48277e9			−0.0510503
$977$	2.52845e10			0.867407			0.433703	−	0.901056i	$-0.357207\pi$
$977$	2.52845e10			0.867407			0.433703	+	0.901056i	$0.357207\pi$
$978$	−6.44559e9			−0.220331
$979$			8.72561e9i			0.297205i
$980$	−1.40482e11			−4.76793
$981$	6.65450e7	+	1.21816e9i	0.00225047	+	0.0411968i
$982$	−5.15026e10			−1.73556
$983$		−	2.63118e10i		−	0.883514i	−0.897135	−	0.441757i	$-0.854355\pi$
$983$		−	2.63118e10i		−	0.883514i	0.897135	−	0.441757i	$-0.145645\pi$
$984$	1.44779e10			0.484421
$985$	−4.80035e10			−1.60047
$986$	8.34234e10			2.77152
$987$			5.70626e10i			1.88904i
$988$	−1.82844e8			−0.00603158
$989$			1.61929e10i			0.532278i
$990$	−9.75543e8			−0.0319539
$991$		−	2.09090e9i		−	0.0682457i	−0.999418	−	0.0341228i	$-0.989136\pi$
$991$		−	2.09090e9i		−	0.0682457i	0.999418	−	0.0341228i	$-0.0108637\pi$
$992$		−	4.08364e10i		−	1.32818i
$993$	1.41379e10			0.458207
$994$		−	1.76957e11i		−	5.71500i
$995$			6.80665e10i			2.19055i
$996$	−3.87984e10			−1.24425
$997$	−3.23088e10			−1.03249			−0.516247	−	0.856440i	$-0.672671\pi$
$997$	−3.23088e10			−1.03249			−0.516247	+	0.856440i	$0.672671\pi$
$998$			1.36677e9i			0.0435250i
$999$		−	4.82847e10i		−	1.53225i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	109.8.b.a.108.9	✓	62
109.108	even	2	inner	109.8.b.a.108.54	yes	62

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
109.8.b.a.108.9	✓	62	1.1	even	1	trivial
109.8.b.a.108.54	yes	62	109.108	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 109 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	109.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-17.9931i q^{2} +45.7905 q^{3} -195.751 q^{4} +340.028 q^{5} -823.912i q^{6} -1712.93 q^{7} +1219.06i q^{8} -90.2316 q^{9} +O(q^{10})\)	\(q-17.9931i q^{2} +45.7905 q^{3} -195.751 q^{4} +340.028 q^{5} -823.912i q^{6} -1712.93 q^{7} +1219.06i q^{8} -90.2316 q^{9} -6118.15i q^{10} +1767.13i q^{11} -8963.55 q^{12} -1945.95i q^{13} +30820.9i q^{14} +15570.0 q^{15} -3121.57 q^{16} -32103.5i q^{17} +1623.55i q^{18} +480.003i q^{19} -66560.9 q^{20} -78435.8 q^{21} +31796.1 q^{22} +75657.6i q^{23} +55821.2i q^{24} +37493.9 q^{25} -35013.6 q^{26} -104276. q^{27} +335308. q^{28} -144421. q^{29} -280153. i q^{30} -192438. q^{31} +212206. i q^{32} +80917.5i q^{33} -577642. q^{34} -582443. q^{35} +17663.0 q^{36} +463049. i q^{37} +8636.74 q^{38} -89105.9i q^{39} +414513. i q^{40} -259362. i q^{41} +1.41130e6i q^{42} +214029. q^{43} -345917. i q^{44} -30681.3 q^{45} +1.36131e6 q^{46} -727507. i q^{47} -142938. q^{48} +2.11058e6 q^{49} -674632. i q^{50} -1.47004e6i q^{51} +380922. i q^{52} -601677. i q^{53} +1.87624e6i q^{54} +600872. i q^{55} -2.08816e6i q^{56} +21979.6i q^{57} +2.59857e6i q^{58} -1.19890e6i q^{59} -3.04786e6 q^{60} +475007. q^{61} +3.46255e6i q^{62} +154560. q^{63} +3.41868e6 q^{64} -661676. i q^{65} +1.45596e6 q^{66} -1.60067e6i q^{67} +6.28431e6i q^{68} +3.46440e6i q^{69} +1.04800e7i q^{70} -5.74148e6 q^{71} -109998. i q^{72} -1.02324e6 q^{73} +8.33168e6 q^{74} +1.71687e6 q^{75} -93961.3i q^{76} -3.02696e6i q^{77} -1.60329e6 q^{78} +6.65950e6i q^{79} -1.06142e6 q^{80} -4.57749e6 q^{81} -4.66672e6 q^{82} +4.32846e6 q^{83} +1.53539e7 q^{84} -1.09161e7i q^{85} -3.85105e6i q^{86} -6.61309e6 q^{87} -2.15423e6 q^{88} +4.93774e6 q^{89} +552051. i q^{90} +3.33327e6i q^{91} -1.48101e7i q^{92} -8.81181e6 q^{93} -1.30901e7 q^{94} +163214. i q^{95} +9.71702e6i q^{96} +553098. q^{97} -3.79758e7i q^{98} -159451. i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9}+O(q^{10}) \) 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9	\( 62 q - 56 q^{3} - 3584 q^{4} + 194 q^{5} + 370 q^{7} + 40462 q^{9} + 6248 q^{12} - 21376 q^{15} + 227856 q^{16} - 216666 q^{20} + 42914 q^{21} - 101446 q^{22} + 788644 q^{25} - 282610 q^{26} - 235946 q^{27} - 69142 q^{28} - 213678 q^{29} + 300350 q^{31} - 401602 q^{34} - 377134 q^{35} - 2176902 q^{36} - 2421542 q^{38} + 2022550 q^{43} + 2223056 q^{45} + 2665174 q^{46} + 1060208 q^{48} + 7702792 q^{49} + 9894904 q^{60} + 278126 q^{61} - 6010300 q^{63} - 9362182 q^{64} - 23291020 q^{66} + 15373744 q^{71} - 1378934 q^{73} - 19097780 q^{74} - 41528062 q^{75} + 5705654 q^{78} + 42766270 q^{80} + 29291446 q^{81} + 14605464 q^{82} + 1191422 q^{83} - 6959552 q^{84} - 22121732 q^{87} + 6337872 q^{88} + 29157388 q^{89} + 14091936 q^{93} - 17482078 q^{94} - 22763314 q^{97}+O(q^{100}) \) 62 * q - 56 * q^3 - 3584 * q^4 + 194 * q^5 + 370 * q^7 + 40462 * q^9 + 6248 * q^12 - 21376 * q^15 + 227856 * q^16 - 216666 * q^20 + 42914 * q^21 - 101446 * q^22 + 788644 * q^25 - 282610 * q^26 - 235946 * q^27 - 69142 * q^28 - 213678 * q^29 + 300350 * q^31 - 401602 * q^34 - 377134 * q^35 - 2176902 * q^36 - 2421542 * q^38 + 2022550 * q^43 + 2223056 * q^45 + 2665174 * q^46 + 1060208 * q^48 + 7702792 * q^49 + 9894904 * q^60 + 278126 * q^61 - 6010300 * q^63 - 9362182 * q^64 - 23291020 * q^66 + 15373744 * q^71 - 1378934 * q^73 - 19097780 * q^74 - 41528062 * q^75 + 5705654 * q^78 + 42766270 * q^80 + 29291446 * q^81 + 14605464 * q^82 + 1191422 * q^83 - 6959552 * q^84 - 22121732 * q^87 + 6337872 * q^88 + 29157388 * q^89 + 14091936 * q^93 - 17482078 * q^94 - 22763314 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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