LMFDB - Embedded newform 23.15.b.b.22.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [23,15,Mod(22,23)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(23, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("23.22");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 23 $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	23.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:	$28.5956626749$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			22.1
Character	$\chi$	$=$	23.22
Dual form			23.15.b.b.22.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-241.960 q^{2} +570.908 q^{3} +42160.6 q^{4} +114189. i q^{5} -138137. q^{6} +270764. i q^{7} -6.23690e6 q^{8} -4.45703e6 q^{9} +O(q^{10})$	\(q-241.960 q^{2} +570.908 q^{3} +42160.6 q^{4} +114189. i q^{5} -138137. q^{6} +270764. i q^{7} -6.23690e6 q^{8} -4.45703e6 q^{9} -2.76291e7i q^{10} +3.28483e7i q^{11} +2.40698e7 q^{12} -1.99048e7 q^{13} -6.55140e7i q^{14} +6.51913e7i q^{15} +8.18321e8 q^{16} +6.85683e8i q^{17} +1.07842e9 q^{18} +9.52590e6i q^{19} +4.81427e9i q^{20} +1.54581e8i q^{21} -7.94797e9i q^{22} +(2.93987e9 - 1.71756e9i) q^{23} -3.56070e9 q^{24} -6.93558e9 q^{25} +4.81616e9 q^{26} -5.27519e9 q^{27} +1.14156e10i q^{28} +1.79234e10 q^{29} -1.57737e10i q^{30} -2.19673e10 q^{31} -9.58155e10 q^{32} +1.87534e10i q^{33} -1.65908e11i q^{34} -3.09182e10 q^{35} -1.87911e11 q^{36} +1.33074e11i q^{37} -2.30489e9i q^{38} -1.13638e10 q^{39} -7.12185e11i q^{40} -9.37366e10 q^{41} -3.74025e10i q^{42} +1.93592e10i q^{43} +1.38490e12i q^{44} -5.08944e11i q^{45} +(-7.11330e11 + 4.15580e11i) q^{46} +6.89754e11 q^{47} +4.67186e11 q^{48} +6.04910e11 q^{49} +1.67813e12 q^{50} +3.91462e11i q^{51} -8.39198e11 q^{52} -1.73630e12i q^{53} +1.27638e12 q^{54} -3.75091e12 q^{55} -1.68873e12i q^{56} +5.43841e9i q^{57} -4.33673e12 q^{58} -3.24962e12 q^{59} +2.74850e12i q^{60} +3.95263e11i q^{61} +5.31519e12 q^{62} -1.20680e12i q^{63} +9.77613e12 q^{64} -2.27291e12i q^{65} -4.53756e12i q^{66} -1.16961e12i q^{67} +2.89088e13i q^{68} +(1.67839e12 - 9.80567e11i) q^{69} +7.48097e12 q^{70} -1.13731e13 q^{71} +2.77981e13 q^{72} -4.90049e12 q^{73} -3.21986e13i q^{74} -3.95958e12 q^{75} +4.01618e11i q^{76} -8.89413e12 q^{77} +2.74959e12 q^{78} -1.94104e13i q^{79} +9.34432e13i q^{80} +1.83062e13 q^{81} +2.26805e13 q^{82} +9.99023e11i q^{83} +6.51724e12i q^{84} -7.82974e13 q^{85} -4.68415e12i q^{86} +1.02326e13 q^{87} -2.04872e14i q^{88} +2.17145e13i q^{89} +1.23144e14i q^{90} -5.38950e12i q^{91} +(1.23947e14 - 7.24133e13i) q^{92} -1.25413e13 q^{93} -1.66893e14 q^{94} -1.08775e12 q^{95} -5.47018e13 q^{96} +5.75389e13i q^{97} -1.46364e14 q^{98} -1.46406e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) $ 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9} - 61408584 q^{12} - 75796032 q^{13} + 61604880 q^{16} + 2078422368 q^{18} + 11061310696 q^{23} - 31030771920 q^{24} - 46421538840 q^{25} - 2664699128 q^{26} + 53616275568 q^{27} + 34752122960 q^{29} - 66504561216 q^{31} - 119385530304 q^{32} + 268767587760 q^{35} - 826304812272 q^{36} - 531040317600 q^{39} - 586388906608 q^{41} - 219014904864 q^{46} + 552854704544 q^{47} + 1314198459696 q^{48} - 2682585958584 q^{49} + 1292374320920 q^{50} - 614830645416 q^{52} - 756511540560 q^{54} - 4886166096240 q^{55} - 4069251029352 q^{58} + 17167207160144 q^{59} + 7479493046896 q^{62} + 18607147469856 q^{64} - 1866422538000 q^{69} + 10676769015360 q^{70} + 15000981757712 q^{71} + 5949703969824 q^{72} + 17535136179168 q^{73} - 79533751619520 q^{75} + 53823905395344 q^{77} + 6433358950512 q^{78} + 149774494087944 q^{81} - 27510435600840 q^{82} - 170228194057680 q^{85} - 286575506293872 q^{87} + 133024510658856 q^{92} + 66347882278032 q^{93} - 205878296932464 q^{94} - 211992774994560 q^{95} + 494703018436320 q^{96} - 537018388090408 q^{98}+O(q^{100}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 - 61408584 * q^12 - 75796032 * q^13 + 61604880 * q^16 + 2078422368 * q^18 + 11061310696 * q^23 - 31030771920 * q^24 - 46421538840 * q^25 - 2664699128 * q^26 + 53616275568 * q^27 + 34752122960 * q^29 - 66504561216 * q^31 - 119385530304 * q^32 + 268767587760 * q^35 - 826304812272 * q^36 - 531040317600 * q^39 - 586388906608 * q^41 - 219014904864 * q^46 + 552854704544 * q^47 + 1314198459696 * q^48 - 2682585958584 * q^49 + 1292374320920 * q^50 - 614830645416 * q^52 - 756511540560 * q^54 - 4886166096240 * q^55 - 4069251029352 * q^58 + 17167207160144 * q^59 + 7479493046896 * q^62 + 18607147469856 * q^64 - 1866422538000 * q^69 + 10676769015360 * q^70 + 15000981757712 * q^71 + 5949703969824 * q^72 + 17535136179168 * q^73 - 79533751619520 * q^75 + 53823905395344 * q^77 + 6433358950512 * q^78 + 149774494087944 * q^81 - 27510435600840 * q^82 - 170228194057680 * q^85 - 286575506293872 * q^87 + 133024510658856 * q^92 + 66347882278032 * q^93 - 205878296932464 * q^94 - 211992774994560 * q^95 + 494703018436320 * q^96 - 537018388090408 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/23\mathbb{Z}\right)^\times$.

$n$	$5$
$\chi(n)$	$-1$

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 $

$2$	−241.960			−1.89031			−0.945156	−	0.326620i	$-0.894090\pi$
$2$	−241.960			−1.89031			−0.945156	+	0.326620i	$0.894090\pi$
$3$	570.908			0.261046			0.130523	−	0.991445i	$-0.458334\pi$
$3$	570.908			0.261046			0.130523	+	0.991445i	$0.458334\pi$
$4$	42160.6			2.57328
$5$			114189.i			1.46162i	0.682582	+	0.730809i	$0.260857\pi$
$5$			114189.i			1.46162i	−0.682582	+	0.730809i	$0.739143\pi$
$6$	−138137.			−0.493459
$7$			270764.i			0.328779i	0.986395	+	0.164390i	$0.0525654\pi$
$7$			270764.i			0.328779i	−0.986395	+	0.164390i	$0.947435\pi$
$8$	−6.23690e6			−2.97399
$9$	−4.45703e6			−0.931855
$10$		−	2.76291e7i		−	2.76291i
$11$			3.28483e7i			1.68564i	0.538198	+	0.842818i	$0.319105\pi$
$11$			3.28483e7i			1.68564i	−0.538198	+	0.842818i	$0.680895\pi$
$12$	2.40698e7			0.671744
$13$	−1.99048e7			−0.317215			−0.158608	−	0.987342i	$-0.550701\pi$
$13$	−1.99048e7			−0.317215			−0.158608	+	0.987342i	$0.550701\pi$
$14$		−	6.55140e7i		−	0.621495i
$15$			6.51913e7i			0.381550i
$16$	8.18321e8			3.04848
$17$			6.85683e8i			1.67102i	0.549477	+	0.835509i	$0.314827\pi$
$17$			6.85683e8i			1.67102i	−0.549477	+	0.835509i	$0.685173\pi$
$18$	1.07842e9			1.76150
$19$			9.52590e6i			0.0106569i	0.999986	+	0.00532845i	$0.00169611\pi$
$19$			9.52590e6i			0.0106569i	−0.999986	+	0.00532845i	$0.998304\pi$
$20$			4.81427e9i			3.76115i
$21$			1.54581e8i			0.0858266i
$22$		−	7.94797e9i		−	3.18638i
$23$	2.93987e9	−	1.71756e9i	0.863442	−	0.504448i
$23$	2.93987e9	−	1.71756e9i	0.863442	−	0.504448i
$24$	−3.56070e9			−0.776348
$25$	−6.93558e9			−1.13633
$26$	4.81616e9			0.599636
$27$	−5.27519e9			−0.504303
$28$			1.14156e10i			0.846041i
$29$	1.79234e10			1.03904			0.519521	−	0.854458i	$-0.326110\pi$
$29$	1.79234e10			1.03904			0.519521	+	0.854458i	$0.326110\pi$
$30$		−	1.57737e10i		−	0.721248i
$31$	−2.19673e10			−0.798443			−0.399221	−	0.916855i	$-0.630720\pi$
$31$	−2.19673e10			−0.798443			−0.399221	+	0.916855i	$0.630720\pi$
$32$	−9.58155e10			−2.78860
$33$			1.87534e10i			0.440029i
$34$		−	1.65908e11i		−	3.15874i
$35$	−3.09182e10			−0.480550
$36$	−1.87911e11			−2.39792
$37$			1.33074e11i			1.40179i	0.713267	+	0.700893i	$0.247215\pi$
$37$			1.33074e11i			1.40179i	−0.713267	+	0.700893i	$0.752785\pi$
$38$		−	2.30489e9i		−	0.0201449i
$39$	−1.13638e10			−0.0828079
$40$		−	7.12185e11i		−	4.34683i
$41$	−9.37366e10			−0.481307			−0.240654	−	0.970611i	$-0.577362\pi$
$41$	−9.37366e10			−0.481307			−0.240654	+	0.970611i	$0.577362\pi$
$42$		−	3.74025e10i		−	0.162239i
$43$			1.93592e10i			0.0712210i	0.999366	+	0.0356105i	$0.0113376\pi$
$43$			1.93592e10i			0.0712210i	−0.999366	+	0.0356105i	$0.988662\pi$
$44$			1.38490e12i			4.33761i
$45$		−	5.08944e11i		−	1.36202i
$46$	−7.11330e11	+	4.15580e11i	−1.63217	+	0.953564i
$47$	6.89754e11			1.36147			0.680737	−	0.732528i	$-0.261660\pi$
$47$	6.89754e11			1.36147			0.680737	+	0.732528i	$0.261660\pi$
$48$	4.67186e11			0.795795
$49$	6.04910e11			0.891904
$50$	1.67813e12			2.14801
$51$			3.91462e11i			0.436213i
$52$	−8.39198e11			−0.816284
$53$		−	1.73630e12i		−	1.47807i	−0.673668	−	0.739034i	$-0.735282\pi$
$53$		−	1.73630e12i		−	1.47807i	0.673668	−	0.739034i	$-0.264718\pi$
$54$	1.27638e12			0.953290
$55$	−3.75091e12			−2.46376
$56$		−	1.68873e12i		−	0.977785i
$57$			5.43841e9i			0.00278194i
$58$	−4.33673e12			−1.96411
$59$	−3.24962e12			−1.30577			−0.652887	−	0.757455i	$-0.726442\pi$
$59$	−3.24962e12			−1.30577			−0.652887	+	0.757455i	$0.726442\pi$
$60$			2.74850e12i			0.981833i
$61$			3.95263e11i			0.125770i	0.998021	+	0.0628850i	$0.0200301\pi$
$61$			3.95263e11i			0.125770i	−0.998021	+	0.0628850i	$0.979970\pi$
$62$	5.31519e12			1.50931
$63$		−	1.20680e12i		−	0.306375i
$64$	9.77613e12			2.22284
$65$		−	2.27291e12i		−	0.463648i
$66$		−	4.53756e12i		−	0.831792i
$67$		−	1.16961e12i		−	0.192983i	−0.995334	−	0.0964913i	$-0.969238\pi$
$67$		−	1.16961e12i		−	0.192983i	0.995334	−	0.0964913i	$-0.0307620\pi$
$68$			2.89088e13i			4.29999i
$69$	1.67839e12	−	9.80567e11i	0.225398	−	0.131684i
$70$	7.48097e12			0.908388
$71$	−1.13731e13			−1.25046			−0.625230	−	0.780440i	$-0.714995\pi$
$71$	−1.13731e13			−1.25046			−0.625230	+	0.780440i	$0.714995\pi$
$72$	2.77981e13			2.77132
$73$	−4.90049e12			−0.443587			−0.221794	−	0.975094i	$-0.571191\pi$
$73$	−4.90049e12			−0.443587			−0.221794	+	0.975094i	$0.571191\pi$
$74$		−	3.21986e13i		−	2.64981i
$75$	−3.95958e12			−0.296633
$76$			4.01618e11i			0.0274232i
$77$	−8.89413e12			−0.554203
$78$	2.74959e12			0.156533
$79$		−	1.94104e13i		−	1.01075i	−0.862899	−	0.505376i	$-0.831354\pi$
$79$		−	1.94104e13i		−	1.01075i	0.862899	−	0.505376i	$-0.168646\pi$
$80$			9.34432e13i			4.45572i
$81$	1.83062e13			0.800208
$82$	2.26805e13			0.909820
$83$			9.99023e11i			0.0368154i	0.999831	+	0.0184077i	$0.00585968\pi$
$83$			9.99023e11i			0.0368154i	−0.999831	+	0.0184077i	$0.994140\pi$
$84$			6.51724e12i			0.220856i
$85$	−7.82974e13			−2.44239
$86$		−	4.68415e12i		−	0.134630i
$87$	1.02326e13			0.271238
$88$		−	2.04872e14i		−	5.01306i
$89$			2.17145e13i			0.490930i	0.969405	+	0.245465i	$0.0789407\pi$
$89$			2.17145e13i			0.490930i	−0.969405	+	0.245465i	$0.921059\pi$
$90$			1.23144e14i			2.57463i
$91$		−	5.38950e12i		−	0.104294i
$92$	1.23947e14	−	7.24133e13i	2.22188	−	1.29809i
$93$	−1.25413e13			−0.208430
$94$	−1.66893e14			−2.57361
$95$	−1.08775e12			−0.0155763
$96$	−5.47018e13			−0.727953
$97$			5.75389e13i			0.712130i	0.934461	+	0.356065i	$0.115882\pi$
$97$			5.75389e13i			0.712130i	−0.934461	+	0.356065i	$0.884118\pi$
$98$	−1.46364e14			−1.68598
$99$		−	1.46406e14i		−	1.57077i
$100$	−2.92408e14			−2.92408
$101$	5.49921e12			0.0512921			0.0256461	−	0.999671i	$-0.491836\pi$
$101$	5.49921e12			0.0512921			0.0256461	+	0.999671i	$0.491836\pi$
$102$		−	9.47181e13i		−	0.824578i
$103$		−	2.20143e14i		−	1.78997i	−0.446099	−	0.894984i	$-0.647187\pi$
$103$		−	2.20143e14i		−	1.78997i	0.446099	−	0.894984i	$-0.352813\pi$
$104$	1.24144e14			0.943395
$105$	−1.76515e13			−0.125446
$106$			4.20116e14i			2.79401i
$107$		−	9.47322e12i		−	0.0589944i	−0.999565	−	0.0294972i	$-0.990609\pi$
$107$		−	9.47322e12i		−	0.0589944i	0.999565	−	0.0294972i	$-0.00939062\pi$
$108$	−2.22405e14			−1.29771
$109$		−	1.47152e14i		−	0.804972i	−0.915426	−	0.402486i	$-0.868146\pi$
$109$		−	1.47152e14i		−	0.804972i	0.915426	−	0.402486i	$-0.131854\pi$
$110$	9.07570e14			4.65727
$111$			7.59731e13i			0.365931i
$112$			2.21572e14i			1.00228i
$113$			2.88564e14i			1.22657i	0.789861	+	0.613286i	$0.210152\pi$
$113$			2.88564e14i			1.22657i	−0.789861	+	0.613286i	$0.789848\pi$
$114$		−	1.31588e12i		−	0.00525874i
$115$	1.96126e14	+	3.35700e14i	0.737310	+	1.26202i
$116$	7.55659e14			2.67375
$117$	8.87163e13			0.295599
$118$	7.86277e14			2.46832
$119$	−1.85658e14			−0.549396
$120$		−	4.06592e14i		−	1.13472i
$121$	−6.99261e14			−1.84137
$122$		−	9.56377e13i		−	0.237745i
$123$	−5.35150e13			−0.125643
$124$	−9.26152e14			−2.05462
$125$		−	9.50124e13i		−	0.199255i
$126$			2.91998e14i			0.579144i
$127$	4.68567e14			0.879317			0.439659	−	0.898165i	$-0.355099\pi$
$127$	4.68567e14			0.879317			0.439659	+	0.898165i	$0.355099\pi$
$128$	−7.95591e14			−1.41325
$129$			1.10523e13i			0.0185920i
$130$			5.49952e14i			0.876438i
$131$	−1.34620e14			−0.203334			−0.101667	−	0.994818i	$-0.532418\pi$
$131$	−1.34620e14			−0.203334			−0.101667	+	0.994818i	$0.532418\pi$
$132$			7.90652e14i			1.13232i
$133$	−2.57927e12			−0.00350377
$134$			2.82999e14i			0.364797i
$135$		−	6.02368e14i		−	0.737098i
$136$		−	4.27654e15i		−	4.96959i
$137$			6.35747e14i			0.701843i	0.936405	+	0.350922i	$0.114132\pi$
$137$			6.35747e14i			0.701843i	−0.936405	+	0.350922i	$0.885868\pi$
$138$	−4.06104e14	+	2.37258e14i	−0.426073	+	0.248924i
$139$	7.36570e14			0.734700			0.367350	−	0.930083i	$-0.380265\pi$
$139$	7.36570e14			0.734700			0.367350	+	0.930083i	$0.380265\pi$
$140$	−1.30353e15			−1.23659
$141$	3.93786e14			0.355408
$142$	2.75183e15			2.36376
$143$		−	6.53839e14i		−	0.534710i
$144$	−3.64728e15			−2.84074
$145$			2.04665e15i			1.51868i
$146$	1.18572e15			0.838518
$147$	3.45348e14			0.232828
$148$			5.61048e15i			3.60718i
$149$		−	1.71133e15i		−	1.04961i	−0.851222	−	0.524806i	$-0.824138\pi$
$149$		−	1.71133e15i		−	1.04961i	0.851222	−	0.524806i	$-0.175862\pi$
$150$	9.58059e14			0.560729
$151$	−1.30309e15			−0.728008			−0.364004	−	0.931397i	$-0.618590\pi$
$151$	−1.30309e15			−0.728008			−0.364004	+	0.931397i	$0.618590\pi$
$152$		−	5.94121e13i		−	0.0316935i
$153$		−	3.05611e15i		−	1.55715i
$154$	2.15202e15			1.04762
$155$		−	2.50842e15i		−	1.16702i
$156$	−4.79105e14			−0.213088
$157$			4.19111e15i			1.78251i	0.453503	+	0.891255i	$0.350174\pi$
$157$			4.19111e15i			1.78251i	−0.453503	+	0.891255i	$0.649826\pi$
$158$			4.69654e15i			1.91064i
$159$		−	9.91269e14i		−	0.385844i
$160$		−	1.09411e16i		−	4.07586i
$161$	4.65053e14	+	7.96010e14i	0.165852	+	0.283882i
$162$	−4.42937e15			−1.51264
$163$	−3.34171e15			−1.09309			−0.546545	−	0.837430i	$-0.684057\pi$
$163$	−3.34171e15			−1.09309			−0.546545	+	0.837430i	$0.684057\pi$
$164$	−3.95199e15			−1.23854
$165$	−2.14142e15			−0.643154
$166$		−	2.41724e14i		−	0.0695925i
$167$	9.67384e14			0.267044			0.133522	−	0.991046i	$-0.457371\pi$
$167$	9.67384e14			0.267044			0.133522	+	0.991046i	$0.457371\pi$
$168$		−	9.64108e14i		−	0.255247i
$169$	−3.54118e15			−0.899374
$170$	1.89448e16			4.61688
$171$		−	4.24573e13i		−	0.00993068i
$172$			8.16195e14i			0.183271i
$173$	3.82387e15			0.824480			0.412240	−	0.911075i	$-0.364746\pi$
$173$	3.82387e15			0.824480			0.412240	+	0.911075i	$0.364746\pi$
$174$	−2.47588e15			−0.512724
$175$		−	1.87790e15i		−	0.373600i
$176$			2.68805e16i			5.13864i
$177$	−1.85523e15			−0.340867
$178$		−	5.25404e15i		−	0.928012i
$179$	−2.00066e14			−0.0339784			−0.0169892	−	0.999856i	$-0.505408\pi$
$179$	−2.00066e14			−0.0339784			−0.0169892	+	0.999856i	$0.505408\pi$
$180$		−	2.14574e16i		−	3.50484i
$181$		−	7.55770e15i		−	1.18752i	−0.804643	−	0.593758i	$-0.797643\pi$
$181$		−	7.55770e15i		−	1.18752i	0.804643	−	0.593758i	$-0.202357\pi$
$182$			1.30404e15i			0.197148i
$183$			2.25659e14i			0.0328318i
$184$	−1.83357e16	+	1.07122e16i	−2.56787	+	1.50022i
$185$	−1.51956e16			−2.04887
$186$	3.03449e15			0.393999
$187$	−2.25235e16			−2.81673
$188$	2.90804e16			3.50345
$189$		−	1.42833e15i		−	0.165804i
$190$	2.63192e14			0.0294441
$191$			4.51226e15i			0.486587i	0.969953	+	0.243294i	$0.0782279\pi$
$191$			4.51226e15i			0.486587i	−0.969953	+	0.243294i	$0.921772\pi$
$192$	5.58127e15			0.580263
$193$	1.68389e16			1.68815			0.844076	−	0.536224i	$-0.180150\pi$
$193$	1.68389e16			1.68815			0.844076	+	0.536224i	$0.180150\pi$
$194$		−	1.39221e16i		−	1.34615i
$195$		−	1.29762e15i		−	0.121033i
$196$	2.55034e16			2.29512
$197$	−5.05981e15			−0.439410			−0.219705	−	0.975566i	$-0.570510\pi$
$197$	−5.05981e15			−0.439410			−0.219705	+	0.975566i	$0.570510\pi$
$198$			3.54244e16i			2.96924i
$199$		−	1.53159e16i		−	1.23928i	−0.784886	−	0.619640i	$-0.787278\pi$
$199$		−	1.53159e16i		−	1.23928i	0.784886	−	0.619640i	$-0.212722\pi$
$200$	4.32565e16			3.37942
$201$		−	6.67741e14i		−	0.0503774i
$202$	−1.33059e15			−0.0969581
$203$			4.85300e15i			0.341616i
$204$			1.65043e16i			1.12250i
$205$		−	1.07037e16i		−	0.703487i
$206$			5.32659e16i			3.38360i
$207$	−1.31031e16	+	7.65521e15i	−0.804603	+	0.470073i
$208$	−1.62885e16			−0.967026
$209$	−3.12910e14			−0.0179637
$210$	4.27094e15			0.237131
$211$	1.00596e16			0.540263			0.270131	−	0.962823i	$-0.412933\pi$
$211$	1.00596e16			0.540263			0.270131	+	0.962823i	$0.412933\pi$
$212$		−	7.32036e16i		−	3.80348i
$213$	−6.49299e15			−0.326428
$214$			2.29214e15i			0.111518i
$215$	−2.21060e15			−0.104098
$216$	3.29008e16			1.49979
$217$		−	5.94794e15i		−	0.262512i
$218$			3.56049e16i			1.52165i
$219$	−2.79773e15			−0.115797
$220$	−1.58141e17			−6.33993
$221$		−	1.36484e16i		−	0.530073i
$222$		−	1.83824e16i		−	0.691723i
$223$	2.30237e16			0.839541			0.419770	−	0.907630i	$-0.362111\pi$
$223$	2.30237e16			0.839541			0.419770	+	0.907630i	$0.362111\pi$
$224$		−	2.59434e16i		−	0.916833i
$225$	3.09121e16			1.05889
$226$		−	6.98209e16i		−	2.31860i
$227$			3.61704e16i			1.16459i	0.812978	+	0.582294i	$0.197845\pi$
$227$			3.61704e16i			1.16459i	−0.812978	+	0.582294i	$0.802155\pi$
$228$			2.29287e14i			0.00715871i
$229$		−	1.98118e16i		−	0.599895i	−0.953956	−	0.299947i	$-0.903031\pi$
$229$		−	1.98118e16i		−	0.599895i	0.953956	−	0.299947i	$-0.0969692\pi$
$230$	−4.74546e16	−	8.12260e16i	−1.39375	−	2.38561i
$231$	−5.07773e15			−0.144672
$232$	−1.11786e17			−3.09010
$233$	4.77952e16			1.28202			0.641008	−	0.767534i	$-0.278517\pi$
$233$	4.77952e16			1.28202			0.641008	+	0.767534i	$0.278517\pi$
$234$	−2.14658e16			−0.558774
$235$			7.87622e16i			1.98995i
$236$	−1.37006e17			−3.36012
$237$		−	1.10816e16i		−	0.263853i
$238$	4.49219e16			1.03853
$239$	3.07410e16			0.690132			0.345066	−	0.938578i	$-0.387856\pi$
$239$	3.07410e16			0.690132			0.345066	+	0.938578i	$0.387856\pi$
$240$			5.33474e16i			1.16315i
$241$		−	3.96101e16i		−	0.838855i	−0.907789	−	0.419427i	$-0.862231\pi$
$241$		−	3.96101e16i		−	0.838855i	0.907789	−	0.419427i	$-0.137769\pi$
$242$	1.69193e17			3.48077
$243$	3.56822e16			0.713195
$244$			1.66645e16i			0.323641i
$245$			6.90740e16i			1.30362i
$246$	1.29485e16			0.237505
$247$		−	1.89611e14i		−	0.00338053i
$248$	1.37008e17			2.37456
$249$			5.70350e14i			0.00961051i
$250$			2.29892e16i			0.376655i
$251$			6.79163e16i			1.08208i	0.840998	+	0.541038i	$0.181969\pi$
$251$			6.79163e16i			1.08208i	−0.840998	+	0.541038i	$0.818031\pi$
$252$		−	5.08796e16i		−	0.788387i
$253$	5.64189e16	+	9.65697e16i	0.850317	+	1.45545i
$254$	−1.13374e17			−1.66218
$255$	−4.47006e16			−0.637576
$256$	3.23290e16			0.448655
$257$	5.57789e16			0.753249			0.376624	−	0.926366i	$-0.377085\pi$
$257$	5.57789e16			0.753249			0.376624	+	0.926366i	$0.377085\pi$
$258$		−	2.67422e15i		−	0.0351446i
$259$	−3.60317e16			−0.460878
$260$		−	9.58271e16i		−	1.19309i
$261$	−7.98850e16			−0.968237
$262$	3.25726e16			0.384364
$263$		−	8.40130e16i		−	0.965286i	−0.875817	−	0.482643i	$-0.839677\pi$
$263$		−	8.40130e16i		−	0.965286i	0.875817	−	0.482643i	$-0.160323\pi$
$264$		−	1.16963e17i		−	1.30864i
$265$	1.98267e17			2.16037
$266$	6.24080e14			0.00662321
$267$			1.23970e16i			0.128155i
$268$		−	4.93115e16i		−	0.496598i
$269$	−1.50760e17			−1.47917			−0.739586	−	0.673062i	$-0.764979\pi$
$269$	−1.50760e17			−1.47917			−0.739586	+	0.673062i	$0.764979\pi$
$270$			1.45749e17i			1.39335i
$271$	1.71798e17			1.60042			0.800208	−	0.599723i	$-0.204723\pi$
$271$	1.71798e17			1.60042			0.800208	+	0.599723i	$0.204723\pi$
$272$			5.61109e17i			5.09407i
$273$		−	3.07691e15i		−	0.0272255i
$274$		−	1.53825e17i		−	1.32670i
$275$		−	2.27822e17i		−	1.91543i
$276$	7.07621e16	−	4.13413e16i	0.580012	−	0.338860i
$277$	6.51041e16			0.520295			0.260148	−	0.965569i	$-0.416229\pi$
$277$	6.51041e16			0.520295			0.260148	+	0.965569i	$0.416229\pi$
$278$	−1.78220e17			−1.38881
$279$	9.79088e16			0.744033
$280$	1.92834e17			1.42915
$281$			2.32763e17i			1.68256i	0.540601	+	0.841279i	$0.318197\pi$
$281$			2.32763e17i			1.68256i	−0.540601	+	0.841279i	$0.681803\pi$
$282$	−9.52805e16			−0.671831
$283$		−	1.81755e16i		−	0.125021i	−0.998044	−	0.0625103i	$-0.980089\pi$
$283$		−	1.81755e16i		−	0.125021i	0.998044	−	0.0625103i	$-0.0199106\pi$
$284$	−4.79496e17			−3.21778
$285$	−6.21006e14			−0.00406614
$286$			1.58203e17i			1.01077i
$287$		−	2.53805e16i		−	0.158244i
$288$	4.27053e17			2.59857
$289$	−3.01784e17			−1.79230
$290$		−	4.95207e17i		−	2.87078i
$291$			3.28494e16i			0.185899i
$292$	−2.06607e17			−1.14147
$293$			2.96008e17i			1.59673i	0.602177	+	0.798363i	$0.294300\pi$
$293$			2.96008e17i			1.59673i	−0.602177	+	0.798363i	$0.705700\pi$
$294$	−8.35603e16			−0.440118
$295$		−	3.71070e17i		−	1.90854i
$296$		−	8.29970e17i		−	4.16889i
$297$		−	1.73281e17i		−	0.850072i
$298$			4.14072e17i			1.98410i
$299$	−5.85175e16	+	3.41876e16i	−0.273897	+	0.160019i
$300$	−1.66938e17			−0.763320
$301$	−5.24177e15			−0.0234160
$302$	3.15296e17			1.37616
$303$	3.13954e15			0.0133896
$304$			7.79525e15i			0.0324874i
$305$	−4.51346e16			−0.183828
$306$			7.39457e17i			2.94349i
$307$	2.89703e17			1.12716			0.563579	−	0.826063i	$-0.309424\pi$
$307$	2.89703e17			1.12716			0.563579	+	0.826063i	$0.309424\pi$
$308$	−3.74982e17			−1.42612
$309$		−	1.25682e17i		−	0.467264i
$310$			6.06936e17i			2.20603i
$311$	−3.59254e17			−1.27667			−0.638336	−	0.769758i	$-0.720377\pi$
$311$	−3.59254e17			−1.27667			−0.638336	+	0.769758i	$0.720377\pi$
$312$	7.08750e16			0.246269
$313$		−	3.44367e16i		−	0.117007i	−0.998287	−	0.0585033i	$-0.981367\pi$
$313$		−	3.44367e16i		−	0.117007i	0.998287	−	0.0585033i	$-0.0186328\pi$
$314$		−	1.01408e18i		−	3.36950i
$315$	1.37804e17			0.447802
$316$		−	8.18354e17i		−	2.60095i
$317$	−4.17730e17			−1.29862			−0.649308	−	0.760525i	$-0.724941\pi$
$317$	−4.17730e17			−1.29862			−0.649308	+	0.760525i	$0.724941\pi$
$318$			2.39847e17i			0.729366i
$319$			5.88752e17i			1.75145i
$320$			1.11633e18i			3.24893i
$321$		−	5.40834e15i		−	0.0154003i
$322$	−1.12524e17	−	1.92603e17i	−0.313512	−	0.536625i
$323$	−6.53175e15			−0.0178079
$324$	7.71800e17			2.05916
$325$	1.38051e17			0.360460
$326$	8.08560e17			2.06628
$327$		−	8.40102e16i		−	0.210135i
$328$	5.84626e17			1.43140
$329$			1.86761e17i			0.447624i
$330$	5.18139e17			1.21576
$331$	−5.29530e17			−1.21645			−0.608226	−	0.793764i	$-0.708118\pi$
$331$	−5.29530e17			−1.21645			−0.608226	+	0.793764i	$0.708118\pi$
$332$			4.21194e16i			0.0947362i
$333$		−	5.93116e17i		−	1.30626i
$334$	−2.34068e17			−0.504797
$335$	1.33557e17			0.282067
$336$			1.26497e17i			0.261641i
$337$		−	2.58685e17i		−	0.524038i	−0.965063	−	0.262019i	$-0.915612\pi$
$337$		−	2.58685e17i		−	0.524038i	0.965063	−	0.262019i	$-0.0843883\pi$
$338$	8.56822e17			1.70010
$339$			1.64743e17i			0.320192i
$340$	−3.30106e18			−6.28495
$341$		−	7.21587e17i		−	1.34588i
$342$			1.02730e16i			0.0187721i
$343$			3.47426e17i			0.622019i
$344$		−	1.20741e17i		−	0.211810i
$345$	1.11970e17	+	1.91654e17i	0.192472	+	0.329446i
$346$	−9.25223e17			−1.55852
$347$	3.92500e17			0.647938			0.323969	−	0.946068i	$-0.394983\pi$
$347$	3.92500e17			0.647938			0.323969	+	0.946068i	$0.394983\pi$
$348$	4.31412e17			0.697971
$349$	−8.95638e17			−1.42022			−0.710108	−	0.704093i	$-0.751354\pi$
$349$	−8.95638e17			−1.42022			−0.710108	+	0.704093i	$0.751354\pi$
$350$			4.54378e17i			0.706221i
$351$	1.05002e17			0.159973
$352$		−	3.14738e18i		−	4.70056i
$353$	−3.34945e17			−0.490400			−0.245200	−	0.969473i	$-0.578854\pi$
$353$	−3.34945e17			−0.490400			−0.245200	+	0.969473i	$0.578854\pi$
$354$	4.48892e17			0.644345
$355$		−	1.29868e18i		−	1.82769i
$356$			9.15497e17i			1.26330i
$357$	−1.05994e17			−0.143418
$358$	4.84080e16			0.0642297
$359$			1.23019e18i			1.60071i	0.599527	+	0.800354i	$0.295355\pi$
$359$			1.23019e18i			1.60071i	−0.599527	+	0.800354i	$0.704645\pi$
$360$			3.17423e18i			4.05062i
$361$	7.98916e17			0.999886
$362$			1.82866e18i			2.24478i
$363$	−3.99213e17			−0.480683
$364$		−	2.27225e17i		−	0.268377i
$365$		−	5.59581e17i		−	0.648355i
$366$		−	5.46003e16i		−	0.0620623i
$367$			1.02504e18i			1.14308i	0.820574	+	0.571540i	$0.193654\pi$
$367$			1.02504e18i			1.14308i	−0.820574	+	0.571540i	$0.806346\pi$
$368$	2.40576e18	−	1.40551e18i	2.63219	−	1.53780i
$369$	4.17787e17			0.448508
$370$	3.67672e18			3.87301
$371$	4.70128e17			0.485958
$372$	−5.28748e17			−0.536350
$373$			1.19199e18i			1.18662i	0.804973	+	0.593311i	$0.202180\pi$
$373$			1.19199e18i			1.18662i	−0.804973	+	0.593311i	$0.797820\pi$
$374$	5.44979e18			5.32450
$375$		−	5.42433e16i		−	0.0520149i
$376$	−4.30193e18			−4.04901
$377$	−3.56761e17			−0.329600
$378$			3.45599e17i			0.313422i
$379$		−	7.17083e17i		−	0.638403i	−0.947687	−	0.319202i	$-0.896585\pi$
$379$		−	7.17083e17i		−	0.638403i	0.947687	−	0.319202i	$-0.103415\pi$
$380$	−4.58603e16			−0.0400822
$381$	2.67509e17			0.229542
$382$		−	1.09179e18i		−	0.919802i
$383$		−	2.03531e18i		−	1.68360i	−0.539786	−	0.841802i	$-0.681495\pi$
$383$		−	2.03531e18i		−	1.68360i	0.539786	−	0.841802i	$-0.318505\pi$
$384$	−4.54209e17			−0.368925
$385$		−	1.01561e18i		−	0.810032i
$386$	−4.07433e18			−3.19113
$387$		−	8.62846e16i		−	0.0663676i
$388$			2.42588e18i			1.83251i
$389$			7.29931e17i			0.541545i	0.962643	+	0.270773i	$0.0872791\pi$
$389$			7.29931e17i			0.541545i	−0.962643	+	0.270773i	$0.912721\pi$
$390$			3.13972e17i			0.228791i
$391$	1.17770e18	+	2.01582e18i	0.842942	+	1.44283i
$392$	−3.77276e18			−2.65251
$393$	−7.68555e16			−0.0530795
$394$	1.22427e18			0.830623
$395$	2.21645e18			1.47733
$396$		−	6.17256e18i		−	4.04203i
$397$	−7.75333e17			−0.498833			−0.249416	−	0.968396i	$-0.580239\pi$
$397$	−7.75333e17			−0.498833			−0.249416	+	0.968396i	$0.580239\pi$
$398$			3.70582e18i			2.34263i
$399$	−1.47253e15			−0.000914645
$400$	−5.67553e18			−3.46407
$401$		−	1.28256e18i		−	0.769249i	−0.923073	−	0.384625i	$-0.874331\pi$
$401$		−	1.28256e18i		−	0.769249i	0.923073	−	0.384625i	$-0.125669\pi$
$402$			1.61566e17i			0.0952289i
$403$	4.37254e17			0.253278
$404$	2.31850e17			0.131989
$405$			2.09036e18i			1.16960i
$406$		−	1.17423e18i		−	0.645760i
$407$	−4.37126e18			−2.36290
$408$		−	2.44151e18i		−	1.29729i
$409$	8.49090e16			0.0443497			0.0221748	−	0.999754i	$-0.492941\pi$
$409$	8.49090e16			0.0443497			0.0221748	+	0.999754i	$0.492941\pi$
$410$			2.58986e18i			1.32981i
$411$			3.62953e17i			0.183213i
$412$		−	9.28138e18i		−	4.60608i
$413$		−	8.79879e17i		−	0.429311i
$414$	3.17042e18	−	1.85225e18i	1.52095	−	0.888584i
$415$	−1.14077e17			−0.0538100
$416$	1.90719e18			0.884586
$417$	4.20514e17			0.191791
$418$	7.57116e16			0.0339569
$419$			1.97821e18i			0.872519i	0.899821	+	0.436260i	$0.143697\pi$
$419$			1.97821e18i			0.872519i	−0.899821	+	0.436260i	$0.856303\pi$
$420$	−7.44196e17			−0.322806
$421$			1.37871e18i			0.588165i	0.955780	+	0.294082i	$0.0950140\pi$
$421$			1.37871e18i			0.588165i	−0.955780	+	0.294082i	$0.904986\pi$
$422$	−2.43403e18			−1.02126
$423$	−3.07426e18			−1.26870
$424$			1.08292e19i			4.39576i
$425$		−	4.75561e18i		−	1.89882i
$426$	1.57104e18			0.617050
$427$	−1.07023e17			−0.0413506
$428$		−	3.99397e17i		−	0.151809i
$429$		−	3.73282e17i		−	0.139584i
$430$	5.34878e17			0.196777
$431$			2.36907e18i			0.857507i	0.903422	+	0.428753i	$0.141047\pi$
$431$			2.36907e18i			0.857507i	−0.903422	+	0.428753i	$0.858953\pi$
$432$	−4.31680e18			−1.53736
$433$		−	2.17841e18i		−	0.763353i	−0.924296	−	0.381676i	$-0.875347\pi$
$433$		−	2.17841e18i		−	0.763353i	0.924296	−	0.381676i	$-0.124653\pi$
$434$			1.43916e18i			0.496229i
$435$			1.16845e18i			0.396446i
$436$		−	6.20401e18i		−	2.07142i
$437$	1.63613e16	+	2.80049e16i	0.00537585	+	0.00920161i
$438$	6.76937e17			0.218892
$439$	1.33727e18			0.425568			0.212784	−	0.977099i	$-0.431747\pi$
$439$	1.33727e18			0.425568			0.212784	+	0.977099i	$0.431747\pi$
$440$	2.33941e19			7.32718
$441$	−2.69610e18			−0.831125
$442$			3.30236e18i			1.00200i
$443$	1.65087e18			0.493045			0.246523	−	0.969137i	$-0.420712\pi$
$443$	1.65087e18			0.493045			0.246523	+	0.969137i	$0.420712\pi$
$444$			3.20307e18i			0.941641i
$445$	−2.47955e18			−0.717552
$446$	−5.57082e18			−1.58699
$447$		−	9.77010e17i		−	0.273997i
$448$			2.64702e18i			0.730822i
$449$	−6.19143e17			−0.168293			−0.0841466	−	0.996453i	$-0.526816\pi$
$449$	−6.19143e17			−0.168293			−0.0841466	+	0.996453i	$0.526816\pi$
$450$	−7.47949e18			−2.00163
$451$		−	3.07909e18i		−	0.811309i
$452$			1.21660e19i			3.15631i
$453$	−7.43945e17			−0.190044
$454$		−	8.75179e18i		−	2.20143i
$455$	6.15421e17			0.152438
$456$		−	3.39188e16i		−	0.00827346i
$457$		−	3.26785e18i		−	0.784963i	−0.919760	−	0.392482i	$-0.871617\pi$
$457$		−	3.26785e18i		−	0.784963i	0.919760	−	0.392482i	$-0.128383\pi$
$458$			4.79366e18i			1.13399i
$459$		−	3.61711e18i		−	0.842700i
$460$	8.26879e18	+	1.41533e19i	1.89730	+	3.24753i
$461$	4.86417e17			0.109926			0.0549632	−	0.998488i	$-0.482496\pi$
$461$	4.86417e17			0.109926			0.0549632	+	0.998488i	$0.482496\pi$
$462$	1.22861e18			0.273476
$463$	7.86519e18			1.72442			0.862209	−	0.506553i	$-0.169081\pi$
$463$	7.86519e18			1.72442			0.862209	+	0.506553i	$0.169081\pi$
$464$	1.46671e19			3.16750
$465$		−	1.43207e18i		−	0.304646i
$466$	−1.15645e19			−2.42341
$467$			1.75637e18i			0.362576i	0.983430	+	0.181288i	$0.0580266\pi$
$467$			1.75637e18i			0.362576i	−0.983430	+	0.181288i	$0.941973\pi$
$468$	3.74033e18			0.760658
$469$	3.16689e17			0.0634487
$470$		−	1.90573e19i		−	3.76163i
$471$			2.39274e18i			0.465317i
$472$	2.02675e19			3.88335
$473$	−6.35917e17			−0.120053
$474$			2.68129e18i			0.498765i
$475$		−	6.60676e16i		−	0.0121097i
$476$	−7.82746e18			−1.41375
$477$			7.73876e18i			1.37735i
$478$	−7.43809e18			−1.30457
$479$		−	3.20322e16i		−	0.00553654i	−0.999996	−	0.00276827i	$-0.999119\pi$
$479$		−	3.20322e16i		−	0.00553654i	0.999996	−	0.00276827i	$-0.000881169\pi$
$480$		−	6.24634e18i		−	1.06399i
$481$		−	2.64881e18i		−	0.444668i
$482$			9.58406e18i			1.58570i
$483$	2.65502e17	+	4.54449e17i	0.0432951	+	0.0741063i
$484$	−2.94812e19			−4.73836
$485$	−6.57030e18			−1.04086
$486$	−8.63367e18			−1.34816
$487$	3.18280e18			0.489899			0.244950	−	0.969536i	$-0.421229\pi$
$487$	3.18280e18			0.489899			0.244950	+	0.969536i	$0.421229\pi$
$488$		−	2.46522e18i		−	0.374038i
$489$	−1.90781e18			−0.285347
$490$		−	1.67131e19i		−	2.46425i
$491$	−6.17285e18			−0.897253			−0.448627	−	0.893719i	$-0.648087\pi$
$491$	−6.17285e18			−0.897253			−0.448627	+	0.893719i	$0.648087\pi$
$492$	−2.25622e18			−0.323315
$493$			1.22897e19i			1.73626i
$494$			4.58783e16i			0.00639026i
$495$	1.67179e19			2.29586
$496$	−1.79763e19			−2.43404
$497$		−	3.07942e18i		−	0.411126i
$498$		−	1.38002e17i		−	0.0181669i
$499$	−6.29414e18			−0.817020			−0.408510	−	0.912754i	$-0.633951\pi$
$499$	−6.29414e18			−0.817020			−0.408510	+	0.912754i	$0.633951\pi$
$500$		−	4.00578e18i		−	0.512740i
$501$	5.52287e17			0.0697109
$502$		−	1.64330e19i		−	2.04546i
$503$			6.55633e17i			0.0804794i	0.999190	+	0.0402397i	$0.0128122\pi$
$503$			6.55633e17i			0.0804794i	−0.999190	+	0.0402397i	$0.987188\pi$
$504$			7.52672e18i			0.911154i
$505$			6.27949e17i			0.0749695i
$506$	−1.36511e19	−	2.33660e19i	−1.60736	−	2.75125i
$507$	−2.02168e18			−0.234778
$508$	1.97551e19			2.26273
$509$	−6.27979e18			−0.709448			−0.354724	−	0.934971i	$-0.615425\pi$
$509$	−6.27979e18			−0.709448			−0.354724	+	0.934971i	$0.615425\pi$
$510$	1.08158e19			1.20522
$511$		−	1.32687e18i		−	0.145842i
$512$	5.21264e18			0.565155
$513$		−	5.02509e16i		−	0.00537431i
$514$	−1.34963e19			−1.42387
$515$	2.51379e19			2.61625
$516$			4.65972e17i			0.0478423i
$517$			2.26573e19i			2.29495i
$518$	8.71822e18			0.871203
$519$	2.18308e18			0.215227
$520$			1.41759e19i			1.37888i
$521$		−	3.95790e18i		−	0.379840i	−0.981800	−	0.189920i	$-0.939177\pi$
$521$		−	3.95790e18i		−	0.379840i	0.981800	−	0.189920i	$-0.0608229\pi$
$522$	1.93290e19			1.83027
$523$			1.10110e19i			1.02877i	0.857561	+	0.514383i	$0.171979\pi$
$523$			1.10110e19i			1.02877i	−0.857561	+	0.514383i	$0.828021\pi$
$524$	−5.67565e18			−0.523235
$525$		−	1.07211e18i		−	0.0975269i
$526$			2.03278e19i			1.82469i
$527$		−	1.50626e19i		−	1.33421i
$528$			1.53463e19i			1.34142i
$529$	5.69283e18	−	1.00988e19i	0.491064	−	0.871123i
$530$	−4.79725e19			−4.08377
$531$	1.44836e19			1.21679
$532$	−1.08744e17			−0.00901617
$533$	1.86581e18			0.152678
$534$		−	2.99957e18i		−	0.242254i
$535$	1.08174e18			0.0862273
$536$			7.29475e18i			0.573928i
$537$	−1.14219e17			−0.00886992
$538$	3.64778e19			2.79610
$539$			1.98703e19i			1.50343i
$540$		−	2.53962e19i		−	1.89676i
$541$	7.38748e18			0.544647			0.272324	−	0.962206i	$-0.412208\pi$
$541$	7.38748e18			0.544647			0.272324	+	0.962206i	$0.412208\pi$
$542$	−4.15682e19			−3.02528
$543$		−	4.31475e18i		−	0.309997i
$544$		−	6.56991e19i		−	4.65980i
$545$	1.68031e19			1.17656
$546$			7.44488e17i			0.0514647i
$547$	−2.04008e18			−0.139231			−0.0696156	−	0.997574i	$-0.522177\pi$
$547$	−2.04008e18			−0.139231			−0.0696156	+	0.997574i	$0.522177\pi$
$548$			2.68035e19i			1.80604i
$549$		−	1.76170e18i		−	0.117199i
$550$			5.51238e19i			3.62076i
$551$			1.70736e17i			0.0110730i
$552$	−1.04680e19	+	6.11570e18i	−0.670331	+	0.391627i
$553$	5.25564e18			0.332315
$554$	−1.57526e19			−0.983520
$555$	−8.67528e18			−0.534851
$556$	3.10542e19			1.89059
$557$		−	1.85043e19i		−	1.11247i	−0.831027	−	0.556233i	$-0.812246\pi$
$557$		−	1.85043e19i		−	1.11247i	0.831027	−	0.556233i	$-0.187754\pi$
$558$	−2.36900e19			−1.40645
$559$		−	3.85341e17i		−	0.0225924i
$560$	−2.53010e19			−1.46495
$561$	−1.28589e19			−0.735296
$562$		−	5.63194e19i		−	3.18056i
$563$			1.58386e19i			0.883403i	0.897162	+	0.441702i	$0.145625\pi$
$563$			1.58386e19i			0.883403i	−0.897162	+	0.441702i	$0.854375\pi$
$564$	1.66023e19			0.914563
$565$	−3.29508e19			−1.79278
$566$			4.39774e18i			0.236328i
$567$			4.95666e18i			0.263092i
$568$	7.09328e19			3.71885
$569$		−	3.33123e19i		−	1.72512i	−0.505958	−	0.862558i	$-0.668861\pi$
$569$		−	3.33123e19i		−	1.72512i	0.505958	−	0.862558i	$-0.331139\pi$
$570$	1.50259e17			0.00768626
$571$			7.16610e18i			0.362101i	0.983474	+	0.181051i	$0.0579498\pi$
$571$			7.16610e18i			0.362101i	−0.983474	+	0.181051i	$0.942050\pi$
$572$		−	2.75662e19i		−	1.37596i
$573$			2.57609e18i			0.127022i
$574$			6.14106e18i			0.299130i
$575$	−2.03897e19	+	1.19123e19i	−0.981151	+	0.573217i
$576$	−4.35726e19			−2.07136
$577$	−1.35379e19			−0.635800			−0.317900	−	0.948124i	$-0.602978\pi$
$577$	−1.35379e19			−0.635800			−0.317900	+	0.948124i	$0.602978\pi$
$578$	7.30196e19			3.38801
$579$	9.61344e18			0.440685
$580$			8.62879e19i			3.90799i
$581$	−2.70499e17			−0.0121041
$582$		−	7.94824e18i		−	0.351407i
$583$	5.70346e19			2.49149
$584$	3.05638e19			1.31922
$585$			1.01304e19i			0.432052i
$586$		−	7.16221e19i		−	3.01831i
$587$	−3.21334e19			−1.33810			−0.669051	−	0.743216i	$-0.733299\pi$
$587$	−3.21334e19			−1.33810			−0.669051	+	0.743216i	$0.733299\pi$
$588$	1.45601e19			0.599132
$589$		−	2.09258e17i		−	0.00850893i
$590$			8.97840e19i			3.60774i
$591$	−2.88868e18			−0.114706
$592$			1.08897e20i			4.27332i
$593$	−3.00743e19			−1.16631			−0.583154	−	0.812362i	$-0.698181\pi$
$593$	−3.00743e19			−1.16631			−0.583154	+	0.812362i	$0.698181\pi$
$594$			4.19271e19i			1.60690i
$595$		−	2.12001e19i		−	0.803007i
$596$		−	7.21506e19i		−	2.70095i
$597$		−	8.74394e18i		−	0.323509i
$598$	1.41589e19	−	8.27204e18i	0.517751	−	0.302485i
$599$	−7.49746e18			−0.270973			−0.135487	−	0.990779i	$-0.543260\pi$
$599$	−7.49746e18			−0.270973			−0.135487	+	0.990779i	$0.543260\pi$
$600$	2.46955e19			0.882184
$601$	1.86252e18			0.0657627			0.0328813	−	0.999459i	$-0.489532\pi$
$601$	1.86252e18			0.0657627			0.0328813	+	0.999459i	$0.489532\pi$
$602$	1.26830e18			0.0442635
$603$			5.21300e18i			0.179832i
$604$	−5.49391e19			−1.87337
$605$		−	7.98478e19i		−	2.69138i
$606$	−7.59644e17			−0.0253105
$607$	−2.18355e18			−0.0719187			−0.0359593	−	0.999353i	$-0.511449\pi$
$607$	−2.18355e18			−0.0719187			−0.0359593	+	0.999353i	$0.511449\pi$
$608$		−	9.12729e17i		−	0.0297178i
$609$			2.77061e18i			0.0891774i
$610$	1.09208e19			0.347492
$611$	−1.37294e19			−0.431881
$612$		−	1.28848e20i		−	4.00697i
$613$		−	1.24347e19i		−	0.382308i	−0.981560	−	0.191154i	$-0.938777\pi$
$613$		−	1.24347e19i		−	0.382308i	0.981560	−	0.191154i	$-0.0612230\pi$
$614$	−7.00965e19			−2.13068
$615$		−	6.11081e18i		−	0.183642i
$616$	5.54718e19			1.64819
$617$			2.87934e19i			0.845856i	0.906163	+	0.422928i	$0.138998\pi$
$617$			2.87934e19i			0.845856i	−0.906163	+	0.422928i	$0.861002\pi$
$618$			3.04099e19i			0.883275i
$619$			2.18492e19i			0.627482i	0.949509	+	0.313741i	$0.101582\pi$
$619$			2.18492e19i			0.627482i	−0.949509	+	0.313741i	$0.898418\pi$
$620$		−	1.05756e20i		−	3.00306i
$621$	−1.55084e19	+	9.06045e18i	−0.435437	+	0.254395i
$622$	8.69252e19			2.41331
$623$	−5.87950e18			−0.161408
$624$	−9.29924e18			−0.252438
$625$	−3.14821e19			−0.845090
$626$			8.33229e18i			0.221179i
$627$	−1.78643e17			−0.00468934
$628$			1.76700e20i			4.58689i
$629$	−9.12467e19			−2.34241
$630$	−3.33429e19			−0.846486
$631$			4.78636e19i			1.20171i	0.799358	+	0.600855i	$0.205173\pi$
$631$			4.78636e19i			1.20171i	−0.799358	+	0.600855i	$0.794827\pi$
$632$			1.21061e20i			3.00597i
$633$	5.74312e18			0.141033
$634$	1.01074e20			2.45479
$635$			5.35051e19i			1.28523i
$636$		−	4.17925e19i		−	0.992884i
$637$	−1.20406e19			−0.282926
$638$		−	1.42454e20i		−	3.31078i
$639$	5.06902e19			1.16525
$640$		−	9.08477e19i		−	2.06564i
$641$		−	4.62860e19i		−	1.04098i	−0.853867	−	0.520491i	$-0.825749\pi$
$641$		−	4.62860e19i		−	1.04098i	0.853867	−	0.520491i	$-0.174251\pi$
$642$			1.30860e18i			0.0291113i
$643$			1.95775e19i			0.430805i	0.976525	+	0.215402i	$0.0691063\pi$
$643$			1.95775e19i			0.430805i	−0.976525	+	0.215402i	$0.930894\pi$
$644$	1.96069e19	+	3.35603e19i	0.426784	+	0.730507i
$645$	−1.26205e18			−0.0271743
$646$	1.58042e18			0.0336624
$647$	1.24196e19			0.261683			0.130842	−	0.991403i	$-0.458232\pi$
$647$	1.24196e19			0.261683			0.130842	+	0.991403i	$0.458232\pi$
$648$	−1.14174e20			−2.37981
$649$		−	1.06744e20i		−	2.20106i
$650$	−3.34029e19			−0.681382
$651$		−	3.39573e18i		−	0.0685276i
$652$	−1.40889e20			−2.81282
$653$	−4.69031e19			−0.926424			−0.463212	−	0.886248i	$-0.653303\pi$
$653$	−4.69031e19			−0.926424			−0.463212	+	0.886248i	$0.653303\pi$
$654$			2.03271e19i			0.397220i
$655$		−	1.53721e19i		−	0.297196i
$656$	−7.67066e19			−1.46726
$657$	2.18416e19			0.413359
$658$		−	4.51886e19i		−	0.846150i
$659$		−	8.30878e19i		−	1.53936i	−0.638432	−	0.769678i	$-0.720417\pi$
$659$		−	8.30878e19i		−	1.53936i	0.638432	−	0.769678i	$-0.279583\pi$
$660$	−9.02837e19			−1.65501
$661$		−	4.94192e18i		−	0.0896366i	−0.998995	−	0.0448183i	$-0.985729\pi$
$661$		−	4.94192e18i		−	0.0896366i	0.998995	−	0.0448183i	$-0.0142709\pi$
$662$	1.28125e20			2.29947
$663$		−	7.79197e18i		−	0.138373i
$664$		−	6.23081e18i		−	0.109488i
$665$		−	2.94524e17i		−	0.00512117i
$666$			1.43510e20i			2.46924i
$667$	5.26923e19	−	3.07844e19i	0.897153	−	0.524143i
$668$	4.07855e19			0.687180
$669$	1.31444e19			0.219159
$670$	−3.23153e19			−0.533194
$671$	−1.29837e19			−0.212003
$672$		−	1.48113e19i		−	0.239336i
$673$	−2.04557e19			−0.327122			−0.163561	−	0.986533i	$-0.552298\pi$
$673$	−2.04557e19			−0.327122			−0.163561	+	0.986533i	$0.552298\pi$
$674$			6.25915e19i			0.990595i
$675$	3.65865e19			0.573053
$676$	−1.49298e20			−2.31434
$677$			2.85065e19i			0.437345i	0.975798	+	0.218672i	$0.0701726\pi$
$677$			2.85065e19i			0.437345i	−0.975798	+	0.218672i	$0.929827\pi$
$678$		−	3.98613e19i		−	0.605262i
$679$	−1.55795e19			−0.234134
$680$	4.88333e20			7.26363
$681$			2.06500e19i			0.304011i
$682$			1.74595e20i			2.54414i
$683$	−5.41268e18			−0.0780670			−0.0390335	−	0.999238i	$-0.512428\pi$
$683$	−5.41268e18			−0.0780670			−0.0390335	+	0.999238i	$0.512428\pi$
$684$		−	1.79002e18i		−	0.0255544i
$685$	−7.25952e19			−1.02583
$686$		−	8.40632e19i		−	1.17581i
$687$		−	1.13107e19i		−	0.156600i
$688$			1.58420e19i			0.217116i
$689$			3.45608e19i			0.468866i
$690$	−2.70922e19	−	4.63726e19i	−0.363832	−	0.622755i
$691$	1.12379e20			1.49396			0.746982	−	0.664844i	$-0.231502\pi$
$691$	1.12379e20			1.49396			0.746982	+	0.664844i	$0.231502\pi$
$692$	1.61217e20			2.12162
$693$	3.96414e19			0.516436
$694$	−9.49693e19			−1.22480
$695$			8.41081e19i			1.07385i
$696$	−6.38196e19			−0.806658
$697$		−	6.42736e19i		−	0.804273i
$698$	2.16708e20			2.68465
$699$	2.72867e19			0.334665
$700$		−	7.91736e19i		−	0.961377i
$701$			1.21041e20i			1.45514i	0.686033	+	0.727571i	$0.259351\pi$
$701$			1.21041e20i			1.45514i	−0.686033	+	0.727571i	$0.740649\pi$
$702$	−2.54062e19			−0.302398
$703$	−1.26765e18			−0.0149387
$704$			3.21129e20i			3.74689i
$705$			4.49660e19i			0.519470i
$706$	8.10432e19			0.927009
$707$			1.48899e18i			0.0168638i
$708$	−7.82177e19			−0.877146
$709$			3.56441e19i			0.395789i	0.980223	+	0.197895i	$0.0634104\pi$
$709$			3.56441e19i			0.395789i	−0.980223	+	0.197895i	$0.936590\pi$
$710$			3.14228e20i			3.45491i
$711$			8.65128e19i			0.941875i
$712$		−	1.35431e20i		−	1.46002i
$713$	−6.45809e19	+	3.77300e19i	−0.689409	+	0.402773i
$714$	2.56462e19			0.271104
$715$	7.46611e19			0.781542
$716$	−8.43491e18			−0.0874358
$717$	1.75503e19			0.180156
$718$		−	2.97657e20i		−	3.02584i
$719$	1.70625e20			1.71767			0.858836	−	0.512250i	$-0.171188\pi$
$719$	1.70625e20			1.71767			0.858836	+	0.512250i	$0.171188\pi$
$720$		−	4.16479e20i		−	4.15208i
$721$	5.96069e19			0.588504
$722$	−1.93306e20			−1.89010
$723$		−	2.26137e19i		−	0.218980i
$724$		−	3.18637e20i		−	3.05581i
$725$	−1.24309e20			−1.18069
$726$	9.65936e19			0.908641
$727$			1.29724e20i			1.20859i	0.796761	+	0.604295i	$0.206545\pi$
$727$			1.29724e20i			1.20859i	−0.796761	+	0.604295i	$0.793455\pi$
$728$			3.36138e19i			0.310169i
$729$	−6.71867e19			−0.614032
$730$			1.35396e20i			1.22559i
$731$	−1.32743e19			−0.119012
$732$			9.51390e18i			0.0844853i
$733$		−	8.39509e19i		−	0.738410i	−0.929348	−	0.369205i	$-0.879630\pi$
$733$		−	8.39509e19i		−	0.738410i	0.929348	−	0.369205i	$-0.120370\pi$
$734$		−	2.48018e20i		−	2.16078i
$735$			3.94349e19i			0.340306i
$736$	−2.81685e20	+	1.64569e20i	−2.40779	+	1.40670i
$737$	3.84198e19			0.325299
$738$	−1.01088e20			−0.847821
$739$	−2.47586e19			−0.205691			−0.102845	−	0.994697i	$-0.532795\pi$
$739$	−2.47586e19			−0.205691			−0.102845	+	0.994697i	$0.532795\pi$
$740$	−6.40655e20			−5.27232
$741$		−	1.08250e17i		−	0.000882475i
$742$	−1.13752e20			−0.918613
$743$		−	5.13007e19i		−	0.410394i	−0.978721	−	0.205197i	$-0.934216\pi$
$743$		−	5.13007e19i		−	0.410394i	0.978721	−	0.205197i	$-0.0657835\pi$
$744$	7.82187e19			0.619869
$745$	1.95414e20			1.53413
$746$		−	2.88415e20i		−	2.24309i
$747$		−	4.45268e18i		−	0.0343066i
$748$	−9.49605e20			−7.24823
$749$	2.56501e18			0.0193962
$750$			1.31247e19i			0.0983243i
$751$		−	8.91181e19i		−	0.661434i	−0.943730	−	0.330717i	$-0.892710\pi$
$751$		−	8.91181e19i		−	0.661434i	0.943730	−	0.330717i	$-0.107290\pi$
$752$	5.64440e20			4.15043
$753$			3.87740e19i			0.282472i
$754$	8.63218e19			0.623047
$755$		−	1.48798e20i		−	1.06407i
$756$		−	6.02193e19i		−	0.426661i
$757$			2.66897e18i			0.0187358i	0.999956	+	0.00936792i	$0.00298194\pi$
$757$			2.66897e18i			0.0187358i	−0.999956	+	0.00936792i	$0.997018\pi$
$758$			1.73505e20i			1.20678i
$759$	3.22100e19	+	5.51324e19i	0.221972	+	0.379940i
$760$	6.78420e18			0.0463237
$761$	−1.11694e19			−0.0755680			−0.0377840	−	0.999286i	$-0.512030\pi$
$761$	−1.11694e19			−0.0755680			−0.0377840	+	0.999286i	$0.512030\pi$
$762$	−6.47264e19			−0.433907
$763$	3.98434e19			0.264658
$764$			1.90240e20i			1.25212i
$765$	3.48974e20			2.27595
$766$			4.92464e20i			3.18254i
$767$	6.46830e19			0.414212
$768$	1.84569e19			0.117120
$769$			2.10960e20i			1.32652i	0.748387	+	0.663262i	$0.230828\pi$
$769$			2.10960e20i			1.32652i	−0.748387	+	0.663262i	$0.769172\pi$
$770$			2.45737e20i			1.53121i
$771$	3.18446e19			0.196633
$772$	7.09936e20			4.34408
$773$		−	4.03101e19i		−	0.244431i	−0.992504	−	0.122216i	$-0.961000\pi$
$773$		−	4.03101e19i		−	0.244431i	0.992504	−	0.122216i	$-0.0390000\pi$
$774$			2.08774e19i			0.125456i
$775$	1.52356e20			0.907291
$776$		−	3.58865e20i		−	2.11787i
$777$	−2.05708e19			−0.120310
$778$		−	1.76614e20i		−	1.02369i
$779$		−	8.92926e17i		−	0.00512924i
$780$		−	5.47084e19i		−	0.311453i
$781$		−	3.73587e20i		−	2.10782i
$782$	−2.84956e20	−	4.87747e20i	−1.59342	−	2.72739i
$783$	−9.45491e19			−0.523992
$784$	4.95011e20			2.71896
$785$	−4.78579e20			−2.60535
$786$	1.85959e19			0.100337
$787$			3.07854e20i			1.64635i	0.567788	+	0.823175i	$0.307799\pi$
$787$			3.07854e20i			1.64635i	−0.567788	+	0.823175i	$0.692201\pi$
$788$	−2.13324e20			−1.13073
$789$		−	4.79637e19i		−	0.251984i
$790$	−5.36293e20			−2.79262
$791$	−7.81326e19			−0.403271
$792$			9.13120e20i			4.67145i
$793$		−	7.86763e18i		−	0.0398962i
$794$	1.87600e20			0.942949
$795$	1.13192e20			0.563956
$796$		−	6.45725e20i		−	3.18901i
$797$			8.51416e18i			0.0416805i	0.999783	+	0.0208403i	$0.00663415\pi$
$797$			8.51416e18i			0.0416805i	−0.999783	+	0.0208403i	$0.993366\pi$
$798$	3.56292e17			0.00172896
$799$			4.72953e20i			2.27505i
$800$	6.64536e20			3.16875
$801$		−	9.67823e19i		−	0.457476i
$802$			3.10328e20i			1.45412i
$803$		−	1.60973e20i		−	0.747727i
$804$		−	2.81523e19i		−	0.129635i
$805$	−9.08955e19	+	5.31038e19i	−0.414927	+	0.242412i
$806$	−1.05798e20			−0.478775
$807$	−8.60698e19			−0.386132
$808$	−3.42980e19			−0.152542
$809$	−2.03286e20			−0.896331			−0.448165	−	0.893951i	$-0.647922\pi$
$809$	−2.03286e20			−0.896331			−0.448165	+	0.893951i	$0.647922\pi$
$810$		−	5.05784e20i		−	2.21091i
$811$	−2.76778e20			−1.19946			−0.599730	−	0.800202i	$-0.704726\pi$
$811$	−2.76778e20			−1.19946			−0.599730	+	0.800202i	$0.704726\pi$
$812$			2.04605e20i			0.879072i
$813$	9.80808e19			0.417782
$814$	1.05767e21			4.46662
$815$		−	3.81586e20i		−	1.59768i
$816$			3.20342e20i			1.32979i
$817$	−1.84414e17			−0.000758995
$818$	−2.05446e19			−0.0838347
$819$			2.40212e19i			0.0971868i
$820$		−	4.51273e20i		−	1.81027i
$821$	−2.24259e20			−0.891965			−0.445983	−	0.895042i	$-0.647146\pi$
$821$	−2.24259e20			−0.891965			−0.445983	+	0.895042i	$0.647146\pi$
$822$		−	8.78200e19i		−	0.346331i
$823$	−4.26047e18			−0.0166594			−0.00832970	−	0.999965i	$-0.502651\pi$
$823$	−4.26047e18			−0.0166594			−0.00832970	+	0.999965i	$0.502651\pi$
$824$			1.37301e21i			5.32334i
$825$		−	1.30065e20i		−	0.500016i
$826$			2.12895e20i			0.811532i
$827$		−	4.75362e20i		−	1.79674i	−0.439238	−	0.898371i	$-0.644751\pi$
$827$		−	4.75362e20i		−	1.79674i	0.439238	−	0.898371i	$-0.355249\pi$
$828$	−5.52434e20	+	3.22748e20i	−2.07047	+	1.20963i
$829$	7.00253e19			0.260240			0.130120	−	0.991498i	$-0.458464\pi$
$829$	7.00253e19			0.260240			0.130120	+	0.991498i	$0.458464\pi$
$830$	2.76021e19			0.101718
$831$	3.71684e19			0.135821
$832$	−1.94592e20			−0.705118
$833$			4.14777e20i			1.49039i
$834$	−1.01747e20			−0.362544
$835$			1.10464e20i			0.390317i
$836$	−1.31925e19			−0.0462255
$837$	1.15881e20			0.402657
$838$		−	4.78648e20i		−	1.64933i
$839$		−	2.75750e20i		−	0.942283i	−0.882058	−	0.471142i	$-0.843842\pi$
$839$		−	2.75750e20i		−	0.942283i	0.882058	−	0.471142i	$-0.156158\pi$
$840$	1.10090e20			0.373074
$841$	2.36884e19			0.0796092
$842$		−	3.33593e20i		−	1.11181i
$843$			1.32886e20i			0.439225i
$844$	4.24120e20			1.39025
$845$		−	4.04363e20i		−	1.31454i
$846$	7.43847e20			2.39823
$847$		−	1.89335e20i		−	0.605405i
$848$		−	1.42085e21i		−	4.50587i
$849$		−	1.03765e19i		−	0.0326361i
$850$			1.15067e21i			3.58936i
$851$	2.28563e20	+	3.91220e20i	0.707128	+	1.21036i
$852$	−2.73748e20			−0.839990
$853$	−3.45416e20			−1.05123			−0.525617	−	0.850722i	$-0.676165\pi$
$853$	−3.45416e20			−1.05123			−0.525617	+	0.850722i	$0.676165\pi$
$854$	2.58952e19			0.0781655
$855$	4.84815e18			0.0145149
$856$			5.90835e19i			0.175449i
$857$	−4.80756e20			−1.41598			−0.707992	−	0.706220i	$-0.750399\pi$
$857$	−4.80756e20			−1.41598			−0.707992	+	0.706220i	$0.750399\pi$
$858$			9.03192e19i			0.263857i
$859$	−6.11224e20			−1.77112			−0.885560	−	0.464525i	$-0.846225\pi$
$859$	−6.11224e20			−1.77112			−0.885560	+	0.464525i	$0.846225\pi$
$860$	−9.32004e19			−0.267873
$861$		−	1.44899e19i		−	0.0413089i
$862$		−	5.73221e20i		−	1.62096i
$863$	4.00999e20			1.12478			0.562390	−	0.826872i	$-0.309882\pi$
$863$	4.00999e20			1.12478			0.562390	+	0.826872i	$0.309882\pi$
$864$	5.05445e20			1.40630
$865$			4.36643e20i			1.20507i
$866$			5.27089e20i			1.44297i
$867$	−1.72291e20			−0.467873
$868$		−	2.50769e20i		−	0.675515i
$869$	6.37599e20			1.70376
$870$		−	2.82717e20i		−	0.749407i
$871$			2.32809e19i			0.0612171i
$872$			9.17772e20i			2.39398i
$873$		−	2.56453e20i		−	0.663602i
$874$	−3.95878e18	−	6.77606e18i	−0.0101620	−	0.0173939i
$875$	2.57259e19			0.0655110
$876$	−1.17954e20			−0.297977
$877$	3.10594e20			0.778387			0.389194	−	0.921156i	$-0.372754\pi$
$877$	3.10594e20			0.778387			0.389194	+	0.921156i	$0.372754\pi$
$878$	−3.23566e20			−0.804455
$879$			1.68993e20i			0.416819i
$880$	−3.06945e21			−7.51072
$881$		−	3.84524e18i		−	0.00933452i	−0.999989	−	0.00466726i	$-0.998514\pi$
$881$		−	3.84524e18i		−	0.00933452i	0.999989	−	0.00466726i	$-0.00148564\pi$
$882$	6.52349e20			1.57109
$883$	−4.54855e19			−0.108680			−0.0543398	−	0.998523i	$-0.517305\pi$
$883$	−4.54855e19			−0.108680			−0.0543398	+	0.998523i	$0.517305\pi$
$884$		−	5.75424e20i		−	1.36402i
$885$		−	2.11847e20i		−	0.498217i
$886$	−3.99444e20			−0.932009
$887$	−4.61644e20			−1.06867			−0.534333	−	0.845274i	$-0.679437\pi$
$887$	−4.61644e20			−1.06867			−0.534333	+	0.845274i	$0.679437\pi$
$888$		−	4.73837e20i		−	1.08827i
$889$			1.26871e20i			0.289101i
$890$	5.99953e20			1.35640
$891$			6.01328e20i			1.34886i
$892$	9.70694e20			2.16037
$893$			6.57053e18i			0.0145091i
$894$			2.36397e20i			0.517940i
$895$		−	2.28453e19i		−	0.0496634i
$896$		−	2.15417e20i		−	0.464649i
$897$	−3.34081e19	+	1.95180e19i	−0.0714998	+	0.0417723i
$898$	1.49808e20			0.318127
$899$	−3.93727e20			−0.829616
$900$	1.30327e21			2.72482
$901$	1.19055e21			2.46988
$902$			7.45016e20i			1.53363i
$903$	−2.99257e18			−0.00611265
$904$		−	1.79974e21i		−	3.64781i
$905$	8.63005e20			1.73570
$906$	1.80005e20			0.359242
$907$		−	4.37917e20i		−	0.867242i	−0.901095	−	0.433621i	$-0.857236\pi$
$907$		−	4.37917e20i		−	0.867242i	0.901095	−	0.433621i	$-0.142764\pi$
$908$			1.52497e21i			2.99681i
$909$	−2.45102e19			−0.0477968
$910$	−1.48907e20			−0.288155
$911$			3.85629e20i			0.740528i	0.928927	+	0.370264i	$0.120733\pi$
$911$			3.85629e20i			0.740528i	−0.928927	+	0.370264i	$0.879267\pi$
$912$			4.45037e18i			0.00848071i
$913$	−3.28162e19			−0.0620573
$914$			7.90690e20i			1.48383i
$915$	−2.57677e19			−0.0479875
$916$		−	8.35277e20i		−	1.54370i
$917$		−	3.64502e19i		−	0.0668520i
$918$			8.75195e20i			1.59297i
$919$			4.10807e20i			0.742044i	0.928624	+	0.371022i	$0.120993\pi$
$919$			4.10807e20i			0.742044i	−0.928624	+	0.371022i	$0.879007\pi$
$920$	−1.22322e21	−	2.09373e21i	−2.19275	−	3.75324i
$921$	1.65394e20			0.294240
$922$	−1.17693e20			−0.207795
$923$	2.26379e20			0.396665
$924$	−2.14080e20			−0.372282
$925$		−	9.22946e20i		−	1.59288i
$926$	−1.90306e21			−3.25969
$927$			9.81187e20i			1.66799i
$928$	−1.71733e21			−2.89747
$929$	−7.46618e20			−1.25023			−0.625113	−	0.780535i	$-0.714947\pi$
$929$	−7.46618e20			−1.25023			−0.625113	+	0.780535i	$0.714947\pi$
$930$			3.46505e20i			0.575875i
$931$			5.76231e18i			0.00950493i
$932$	2.01508e21			3.29898
$933$	−2.05101e20			−0.333270
$934$		−	4.24972e20i		−	0.685381i
$935$		−	2.57194e21i		−	4.11698i
$936$	−5.53315e20			−0.879107
$937$		−	4.57298e20i		−	0.721144i	−0.932731	−	0.360572i	$-0.882581\pi$
$937$		−	4.57298e20i		−	0.721144i	0.932731	−	0.360572i	$-0.117419\pi$
$938$	−7.66259e19			−0.119938
$939$		−	1.96602e19i		−	0.0305441i
$940$			3.32066e21i			5.12071i
$941$		−	4.92785e20i		−	0.754275i	−0.926157	−	0.377138i	$-0.876908\pi$
$941$		−	4.92785e20i		−	0.754275i	0.926157	−	0.377138i	$-0.123092\pi$
$942$		−	5.78947e20i		−	0.879595i
$943$	−2.75573e20	+	1.60998e20i	−0.415581	+	0.242794i
$944$	−2.65923e21			−3.98063
$945$	1.63099e20			0.242343
$946$	1.53866e20			0.226937
$947$	−1.78353e20			−0.261114			−0.130557	−	0.991441i	$-0.541677\pi$
$947$	−1.78353e20			−0.261114			−0.130557	+	0.991441i	$0.541677\pi$
$948$		−	4.67205e20i		−	0.678968i
$949$	9.75432e19			0.140713
$950$			1.59857e19i			0.0228911i
$951$	−2.38485e20			−0.338999
$952$	1.15793e21			1.63390
$953$			8.92980e20i			1.25081i	0.780300	+	0.625405i	$0.215066\pi$
$953$			8.92980e20i			1.25081i	−0.780300	+	0.625405i	$0.784934\pi$
$954$		−	1.87247e21i		−	2.60361i
$955$	−5.15250e20			−0.711204
$956$	1.29606e21			1.77590
$957$			3.36123e20i			0.457209i
$958$			7.75051e18i			0.0104658i
$959$	−1.72137e20			−0.230751
$960$			6.37319e20i			0.848122i
$961$	−2.74384e20			−0.362489
$962$			6.40907e20i			0.840561i
$963$			4.22224e19i			0.0549743i
$964$		−	1.66999e21i		−	2.15861i
$965$			1.92281e21i			2.46743i
$966$	−6.42409e19	−	1.09958e20i	−0.0818411	−	0.140084i
$967$	7.91089e20			1.00055			0.500276	−	0.865866i	$-0.333232\pi$
$967$	7.91089e20			1.00055			0.500276	+	0.865866i	$0.333232\pi$
$968$	4.36122e21			5.47622
$969$	−3.72903e18			−0.00464868
$970$	1.58975e21			1.96755
$971$			3.10524e20i			0.381558i	0.981633	+	0.190779i	$0.0611013\pi$
$971$			3.10524e20i			0.381558i	−0.981633	+	0.190779i	$0.938899\pi$
$972$	1.50438e21			1.83525
$973$			1.99436e20i			0.241554i
$974$	−7.70110e20			−0.926062
$975$	7.88146e19			0.0940967
$976$			3.23452e20i			0.383408i
$977$			1.32074e21i			1.55438i	0.629269	+	0.777188i	$0.283354\pi$
$977$			1.32074e21i			1.55438i	−0.629269	+	0.777188i	$0.716646\pi$
$978$	4.61613e20			0.539394
$979$	−7.13285e20			−0.827530
$980$			2.91220e21i			3.35458i
$981$			6.55861e20i			0.750117i
$982$	1.49358e21			1.69609
$983$		−	5.26009e20i		−	0.593087i	−0.955019	−	0.296544i	$-0.904166\pi$
$983$		−	5.26009e20i		−	0.593087i	0.955019	−	0.296544i	$-0.0958340\pi$
$984$	3.33768e20			0.373662
$985$		−	5.77773e20i		−	0.642250i
$986$		−	2.97363e21i		−	3.28207i
$987$			1.06623e20i			0.116851i
$988$		−	7.99412e18i		−	0.00869905i
$989$	3.32505e19	+	5.69135e19i	0.0359273	+	0.0614952i
$990$	−4.04507e21			−4.33990
$991$	−7.41619e20			−0.790069			−0.395035	−	0.918666i	$-0.629267\pi$
$991$	−7.41619e20			−0.790069			−0.395035	+	0.918666i	$0.629267\pi$
$992$	2.10480e21			2.22654
$993$	−3.02313e20			−0.317550
$994$			7.45097e20i			0.777155i
$995$	1.74890e21			1.81135
$996$			2.40463e19i			0.0247305i
$997$	−1.12392e21			−1.14780			−0.573902	−	0.818924i	$-0.694571\pi$
$997$	−1.12392e21			−1.14780			−0.573902	+	0.818924i	$0.694571\pi$
$998$	1.52293e21			1.54442
$999$		−	7.01991e20i		−	0.706925i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.b.b.22.1	✓	24
23.22	odd	2	inner	23.15.b.b.22.2	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.b.b.22.1	✓	24	1.1	even	1	trivial
23.15.b.b.22.2	yes	24	23.22	odd	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:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-241.960 q^{2} +570.908 q^{3} +42160.6 q^{4} +114189. i q^{5} -138137. q^{6} +270764. i q^{7} -6.23690e6 q^{8} -4.45703e6 q^{9} +O(q^{10})\)	\(q-241.960 q^{2} +570.908 q^{3} +42160.6 q^{4} +114189. i q^{5} -138137. q^{6} +270764. i q^{7} -6.23690e6 q^{8} -4.45703e6 q^{9} -2.76291e7i q^{10} +3.28483e7i q^{11} +2.40698e7 q^{12} -1.99048e7 q^{13} -6.55140e7i q^{14} +6.51913e7i q^{15} +8.18321e8 q^{16} +6.85683e8i q^{17} +1.07842e9 q^{18} +9.52590e6i q^{19} +4.81427e9i q^{20} +1.54581e8i q^{21} -7.94797e9i q^{22} +(2.93987e9 - 1.71756e9i) q^{23} -3.56070e9 q^{24} -6.93558e9 q^{25} +4.81616e9 q^{26} -5.27519e9 q^{27} +1.14156e10i q^{28} +1.79234e10 q^{29} -1.57737e10i q^{30} -2.19673e10 q^{31} -9.58155e10 q^{32} +1.87534e10i q^{33} -1.65908e11i q^{34} -3.09182e10 q^{35} -1.87911e11 q^{36} +1.33074e11i q^{37} -2.30489e9i q^{38} -1.13638e10 q^{39} -7.12185e11i q^{40} -9.37366e10 q^{41} -3.74025e10i q^{42} +1.93592e10i q^{43} +1.38490e12i q^{44} -5.08944e11i q^{45} +(-7.11330e11 + 4.15580e11i) q^{46} +6.89754e11 q^{47} +4.67186e11 q^{48} +6.04910e11 q^{49} +1.67813e12 q^{50} +3.91462e11i q^{51} -8.39198e11 q^{52} -1.73630e12i q^{53} +1.27638e12 q^{54} -3.75091e12 q^{55} -1.68873e12i q^{56} +5.43841e9i q^{57} -4.33673e12 q^{58} -3.24962e12 q^{59} +2.74850e12i q^{60} +3.95263e11i q^{61} +5.31519e12 q^{62} -1.20680e12i q^{63} +9.77613e12 q^{64} -2.27291e12i q^{65} -4.53756e12i q^{66} -1.16961e12i q^{67} +2.89088e13i q^{68} +(1.67839e12 - 9.80567e11i) q^{69} +7.48097e12 q^{70} -1.13731e13 q^{71} +2.77981e13 q^{72} -4.90049e12 q^{73} -3.21986e13i q^{74} -3.95958e12 q^{75} +4.01618e11i q^{76} -8.89413e12 q^{77} +2.74959e12 q^{78} -1.94104e13i q^{79} +9.34432e13i q^{80} +1.83062e13 q^{81} +2.26805e13 q^{82} +9.99023e11i q^{83} +6.51724e12i q^{84} -7.82974e13 q^{85} -4.68415e12i q^{86} +1.02326e13 q^{87} -2.04872e14i q^{88} +2.17145e13i q^{89} +1.23144e14i q^{90} -5.38950e12i q^{91} +(1.23947e14 - 7.24133e13i) q^{92} -1.25413e13 q^{93} -1.66893e14 q^{94} -1.08775e12 q^{95} -5.47018e13 q^{96} +5.75389e13i q^{97} -1.46364e14 q^{98} -1.46406e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9} - 61408584 q^{12} - 75796032 q^{13} + 61604880 q^{16} + 2078422368 q^{18} + 11061310696 q^{23} - 31030771920 q^{24} - 46421538840 q^{25} - 2664699128 q^{26} + 53616275568 q^{27} + 34752122960 q^{29} - 66504561216 q^{31} - 119385530304 q^{32} + 268767587760 q^{35} - 826304812272 q^{36} - 531040317600 q^{39} - 586388906608 q^{41} - 219014904864 q^{46} + 552854704544 q^{47} + 1314198459696 q^{48} - 2682585958584 q^{49} + 1292374320920 q^{50} - 614830645416 q^{52} - 756511540560 q^{54} - 4886166096240 q^{55} - 4069251029352 q^{58} + 17167207160144 q^{59} + 7479493046896 q^{62} + 18607147469856 q^{64} - 1866422538000 q^{69} + 10676769015360 q^{70} + 15000981757712 q^{71} + 5949703969824 q^{72} + 17535136179168 q^{73} - 79533751619520 q^{75} + 53823905395344 q^{77} + 6433358950512 q^{78} + 149774494087944 q^{81} - 27510435600840 q^{82} - 170228194057680 q^{85} - 286575506293872 q^{87} + 133024510658856 q^{92} + 66347882278032 q^{93} - 205878296932464 q^{94} - 211992774994560 q^{95} + 494703018436320 q^{96} - 537018388090408 q^{98}+O(q^{100}) \) 24 * q - 184 * q^2 - 2160 * q^3 + 134952 * q^4 - 1666320 * q^6 - 6887840 * q^8 + 24850248 * q^9 - 61408584 * q^12 - 75796032 * q^13 + 61604880 * q^16 + 2078422368 * q^18 + 11061310696 * q^23 - 31030771920 * q^24 - 46421538840 * q^25 - 2664699128 * q^26 + 53616275568 * q^27 + 34752122960 * q^29 - 66504561216 * q^31 - 119385530304 * q^32 + 268767587760 * q^35 - 826304812272 * q^36 - 531040317600 * q^39 - 586388906608 * q^41 - 219014904864 * q^46 + 552854704544 * q^47 + 1314198459696 * q^48 - 2682585958584 * q^49 + 1292374320920 * q^50 - 614830645416 * q^52 - 756511540560 * q^54 - 4886166096240 * q^55 - 4069251029352 * q^58 + 17167207160144 * q^59 + 7479493046896 * q^62 + 18607147469856 * q^64 - 1866422538000 * q^69 + 10676769015360 * q^70 + 15000981757712 * q^71 + 5949703969824 * q^72 + 17535136179168 * q^73 - 79533751619520 * q^75 + 53823905395344 * q^77 + 6433358950512 * q^78 + 149774494087944 * q^81 - 27510435600840 * q^82 - 170228194057680 * q^85 - 286575506293872 * q^87 + 133024510658856 * q^92 + 66347882278032 * q^93 - 205878296932464 * q^94 - 211992774994560 * q^95 + 494703018436320 * q^96 - 537018388090408 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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