LMFDB - Embedded newform 23.15.b.b.22.18

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			22.18
Character	$\chi$	$=$	23.22
Dual form			23.15.b.b.22.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+133.181 q^{2} -181.571 q^{3} +1353.30 q^{4} -128567. i q^{5} -24181.9 q^{6} +828509. i q^{7} -2.00181e6 q^{8} -4.75000e6 q^{9} +O(q^{10})$	\(q+133.181 q^{2} -181.571 q^{3} +1353.30 q^{4} -128567. i q^{5} -24181.9 q^{6} +828509. i q^{7} -2.00181e6 q^{8} -4.75000e6 q^{9} -1.71228e7i q^{10} +2.91036e7i q^{11} -245719. q^{12} +1.07801e8 q^{13} +1.10342e8i q^{14} +2.33441e7i q^{15} -2.88776e8 q^{16} +1.43110e8i q^{17} -6.32612e8 q^{18} +6.17346e8i q^{19} -1.73990e8i q^{20} -1.50433e8i q^{21} +3.87606e9i q^{22} +(1.04568e9 + 3.24028e9i) q^{23} +3.63470e8 q^{24} -1.04261e10 q^{25} +1.43570e10 q^{26} +1.73091e9 q^{27} +1.12122e9i q^{28} -4.14188e9 q^{29} +3.10900e9i q^{30} -4.67778e10 q^{31} -5.66199e9 q^{32} -5.28436e9i q^{33} +1.90596e10i q^{34} +1.06519e11 q^{35} -6.42816e9 q^{36} +1.89228e10i q^{37} +8.22190e10i q^{38} -1.95734e10 q^{39} +2.57368e11i q^{40} -3.02755e11 q^{41} -2.00349e10i q^{42} -3.09728e10i q^{43} +3.93857e10i q^{44} +6.10696e11i q^{45} +(1.39266e11 + 4.31545e11i) q^{46} -1.38932e11 q^{47} +5.24334e10 q^{48} -8.20385e9 q^{49} -1.38856e12 q^{50} -2.59845e10i q^{51} +1.45886e11 q^{52} +4.87341e11i q^{53} +2.30525e11 q^{54} +3.74177e12 q^{55} -1.65852e12i q^{56} -1.12092e11i q^{57} -5.51621e11 q^{58} +1.15650e11 q^{59} +3.15915e10i q^{60} -3.87021e12i q^{61} -6.22994e12 q^{62} -3.93542e12i q^{63} +3.97724e12 q^{64} -1.38596e13i q^{65} -7.03778e11i q^{66} +8.82989e12i q^{67} +1.93670e11i q^{68} +(-1.89866e11 - 5.88339e11i) q^{69} +1.41864e13 q^{70} +5.52347e12 q^{71} +9.50860e12 q^{72} -1.52927e13 q^{73} +2.52016e12i q^{74} +1.89307e12 q^{75} +8.35452e11i q^{76} -2.41126e13 q^{77} -2.60682e12 q^{78} -2.57265e13i q^{79} +3.71273e13i q^{80} +2.24048e13 q^{81} -4.03213e13 q^{82} +3.16470e13i q^{83} -2.03580e11i q^{84} +1.83993e13 q^{85} -4.12501e12i q^{86} +7.52044e11 q^{87} -5.82598e13i q^{88} -3.26384e13i q^{89} +8.13333e13i q^{90} +8.93137e13i q^{91} +(1.41512e12 + 4.38505e12i) q^{92} +8.49349e12 q^{93} -1.85032e13 q^{94} +7.93706e13 q^{95} +1.02805e12 q^{96} -2.79045e13i q^{97} -1.09260e12 q^{98} -1.38242e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	133.181			1.04048			0.520240	−	0.854020i	$-0.325842\pi$
$2$	133.181			1.04048			0.520240	+	0.854020i	$0.325842\pi$
$3$	−181.571			−0.0830228			−0.0415114	−	0.999138i	$-0.513217\pi$
$3$	−181.571			−0.0830228			−0.0415114	+	0.999138i	$0.513217\pi$
$4$	1353.30			0.0825986
$5$		−	128567.i		−	1.64566i	−0.568285	−	0.822832i	$-0.692393\pi$
$5$		−	128567.i		−	1.64566i	0.568285	−	0.822832i	$-0.307607\pi$
$6$	−24181.9			−0.0863835
$7$			828509.i			1.00603i	0.864278	+	0.503015i	$0.167776\pi$
$7$			828509.i			1.00603i	−0.864278	+	0.503015i	$0.832224\pi$
$8$	−2.00181e6			−0.954538
$9$	−4.75000e6			−0.993107
$10$		−	1.71228e7i		−	1.71228i
$11$			2.91036e7i			1.49347i	0.665120	+	0.746737i	$0.268380\pi$
$11$			2.91036e7i			1.49347i	−0.665120	+	0.746737i	$0.731620\pi$
$12$	−245719.			−0.00685757
$13$	1.07801e8			1.71798			0.858989	−	0.511994i	$-0.171093\pi$
$13$	1.07801e8			1.71798			0.858989	+	0.511994i	$0.171093\pi$
$14$			1.10342e8i			1.04675i
$15$			2.33441e7i			0.136628i
$16$	−2.88776e8			−1.07578
$17$			1.43110e8i			0.348760i	0.984678	+	0.174380i	$0.0557921\pi$
$17$			1.43110e8i			0.348760i	−0.984678	+	0.174380i	$0.944208\pi$
$18$	−6.32612e8			−1.03331
$19$			6.17346e8i			0.690643i	0.938485	+	0.345321i	$0.112230\pi$
$19$			6.17346e8i			0.690643i	−0.938485	+	0.345321i	$0.887770\pi$
$20$		−	1.73990e8i		−	0.135930i
$21$		−	1.50433e8i		−	0.0835234i
$22$			3.87606e9i			1.55393i
$23$	1.04568e9	+	3.24028e9i	0.307118	+	0.951671i
$23$	1.04568e9	+	3.24028e9i	0.307118	+	0.951671i
$24$	3.63470e8			0.0792484
$25$	−1.04261e10			−1.70821
$26$	1.43570e10			1.78752
$27$	1.73091e9			0.165473
$28$			1.12122e9i			0.0830967i
$29$	−4.14188e9			−0.240111			−0.120055	−	0.992767i	$-0.538307\pi$
$29$	−4.14188e9			−0.240111			−0.120055	+	0.992767i	$0.538307\pi$
$30$			3.10900e9i			0.142158i
$31$	−4.67778e10			−1.70023			−0.850116	−	0.526595i	$-0.823468\pi$
$31$	−4.67778e10			−1.70023			−0.850116	+	0.526595i	$0.823468\pi$
$32$	−5.66199e9			−0.164786
$33$		−	5.28436e9i		−	0.123992i
$34$			1.90596e10i			0.362878i
$35$	1.06519e11			1.65559
$36$	−6.42816e9			−0.0820293
$37$			1.89228e10i			0.199330i	0.995021	+	0.0996651i	$0.0317771\pi$
$37$			1.89228e10i			0.199330i	−0.995021	+	0.0996651i	$0.968223\pi$
$38$			8.22190e10i			0.718600i
$39$	−1.95734e10			−0.142631
$40$			2.57368e11i			1.57085i
$41$	−3.02755e11			−1.55455			−0.777274	−	0.629162i	$-0.783398\pi$
$41$	−3.02755e11			−1.55455			−0.777274	+	0.629162i	$0.783398\pi$
$42$		−	2.00349e10i		−	0.0869044i
$43$		−	3.09728e10i		−	0.113947i	−0.998376	−	0.0569733i	$-0.981855\pi$
$43$		−	3.09728e10i		−	0.113947i	0.998376	−	0.0569733i	$-0.0181450\pi$
$44$			3.93857e10i			0.123359i
$45$			6.10696e11i			1.63432i
$46$	1.39266e11	+	4.31545e11i	0.319550	+	0.990195i
$47$	−1.38932e11			−0.274232			−0.137116	−	0.990555i	$-0.543783\pi$
$47$	−1.38932e11			−0.274232			−0.137116	+	0.990555i	$0.543783\pi$
$48$	5.24334e10			0.0893139
$49$	−8.20385e9			−0.0120961
$50$	−1.38856e12			−1.77736
$51$		−	2.59845e10i		−	0.0289550i
$52$	1.45886e11			0.141903
$53$			4.87341e11i			0.414861i	0.978250	+	0.207430i	$0.0665100\pi$
$53$			4.87341e11i			0.414861i	−0.978250	+	0.207430i	$0.933490\pi$
$54$	2.30525e11			0.172172
$55$	3.74177e12			2.45775
$56$		−	1.65852e12i		−	0.960294i
$57$		−	1.12092e11i		−	0.0573391i
$58$	−5.51621e11			−0.249830
$59$	1.15650e11			0.0464710			0.0232355	−	0.999730i	$-0.492603\pi$
$59$	1.15650e11			0.0464710			0.0232355	+	0.999730i	$0.492603\pi$
$60$			3.15915e10i			0.0112852i
$61$		−	3.87021e12i		−	1.23147i	−0.787952	−	0.615737i	$-0.788858\pi$
$61$		−	3.87021e12i		−	1.23147i	0.787952	−	0.615737i	$-0.211142\pi$
$62$	−6.22994e12			−1.76906
$63$		−	3.93542e12i		−	0.999096i
$64$	3.97724e12			0.904320
$65$		−	1.38596e13i		−	2.82721i
$66$		−	7.03778e11i		−	0.129011i
$67$			8.82989e12i			1.45691i	0.685096	+	0.728453i	$0.259760\pi$
$67$			8.82989e12i			1.45691i	−0.685096	+	0.728453i	$0.740240\pi$
$68$			1.93670e11i			0.0288071i
$69$	−1.89866e11	−	5.88339e11i	−0.0254978	−	0.0790104i
$70$	1.41864e13			1.72261
$71$	5.52347e12			0.607300			0.303650	−	0.952784i	$-0.401795\pi$
$71$	5.52347e12			0.607300			0.303650	+	0.952784i	$0.401795\pi$
$72$	9.50860e12			0.947958
$73$	−1.52927e13			−1.38428			−0.692141	−	0.721763i	$-0.743332\pi$
$73$	−1.52927e13			−1.38428			−0.692141	+	0.721763i	$0.743332\pi$
$74$			2.52016e12i			0.207399i
$75$	1.89307e12			0.141820
$76$			8.35452e11i			0.0570461i
$77$	−2.41126e13			−1.50248
$78$	−2.60682e12			−0.148405
$79$		−	2.57265e13i		−	1.33965i	−0.742520	−	0.669824i	$-0.766369\pi$
$79$		−	2.57265e13i		−	1.33965i	0.742520	−	0.669824i	$-0.233631\pi$
$80$			3.71273e13i			1.77037i
$81$	2.24048e13			0.979369
$82$	−4.03213e13			−1.61748
$83$			3.16470e13i			1.16623i	0.812388	+	0.583117i	$0.198167\pi$
$83$			3.16470e13i			1.16623i	−0.812388	+	0.583117i	$0.801833\pi$
$84$		−	2.03580e11i		−	0.00689892i
$85$	1.83993e13			0.573942
$86$		−	4.12501e12i		−	0.118559i
$87$	7.52044e11			0.0199346
$88$		−	5.82598e13i		−	1.42558i
$89$		−	3.26384e13i		−	0.737903i	−0.929449	−	0.368952i	$-0.879717\pi$
$89$		−	3.26384e13i		−	0.737903i	0.929449	−	0.368952i	$-0.120283\pi$
$90$			8.13333e13i			1.70048i
$91$			8.93137e13i			1.72834i
$92$	1.41512e12	+	4.38505e12i	0.0253675	+	0.0786068i
$93$	8.49349e12			0.141158
$94$	−1.85032e13			−0.285333
$95$	7.93706e13			1.13657
$96$	1.02805e12			0.0136810
$97$		−	2.79045e13i		−	0.345360i	−0.984978	−	0.172680i	$-0.944757\pi$
$97$		−	2.79045e13i		−	0.345360i	0.984978	−	0.172680i	$-0.0552427\pi$
$98$	−1.09260e12			−0.0125857
$99$		−	1.38242e14i		−	1.48318i
$100$	−1.41096e13			−0.141096
$101$	−1.21633e14			−1.13449			−0.567247	−	0.823548i	$-0.691992\pi$
$101$	−1.21633e14			−1.13449			−0.567247	+	0.823548i	$0.691992\pi$
$102$		−	3.46066e12i		−	0.0301271i
$103$			1.91267e14i			1.55518i	0.628773	+	0.777589i	$0.283557\pi$
$103$			1.91267e14i			1.55518i	−0.628773	+	0.777589i	$0.716443\pi$
$104$	−2.15796e14			−1.63987
$105$	−1.93408e13			−0.137451
$106$			6.49048e13i			0.431654i
$107$		−	4.78963e13i		−	0.298274i	−0.988817	−	0.149137i	$-0.952350\pi$
$107$		−	4.78963e13i		−	0.298274i	0.988817	−	0.149137i	$-0.0476496\pi$
$108$	2.34243e12			0.0136679
$109$			1.66675e14i			0.911772i	0.890038	+	0.455886i	$0.150678\pi$
$109$			1.66675e14i			0.911772i	−0.890038	+	0.455886i	$0.849322\pi$
$110$	4.98335e14			2.55724
$111$		−	3.43583e12i		−	0.0165489i
$112$		−	2.39254e14i		−	1.08226i
$113$			1.91626e14i			0.814525i	0.913311	+	0.407262i	$0.133517\pi$
$113$			1.91626e14i			0.814525i	−0.913311	+	0.407262i	$0.866483\pi$
$114$		−	1.49286e13i		−	0.0596601i
$115$	4.16594e14	−	1.34441e14i	1.56613	−	0.505413i
$116$	−5.60519e12			−0.0198328
$117$	−5.12053e14			−1.70614
$118$	1.54024e13			0.0483521
$119$	−1.18568e14			−0.350863
$120$		−	4.67305e13i		−	0.130416i
$121$	−4.67268e14			−1.23046
$122$		−	5.15440e14i		−	1.28132i
$123$	5.49715e13			0.129063
$124$	−6.33042e13			−0.140437
$125$			5.55741e14i			1.16547i
$126$		−	5.24125e14i		−	1.03954i
$127$	5.59218e14			1.04943			0.524717	−	0.851277i	$-0.324171\pi$
$127$	5.59218e14			1.04943			0.524717	+	0.851277i	$0.324171\pi$
$128$	6.22461e14			1.10571
$129$			5.62376e12i			0.00946017i
$130$		−	1.84585e15i		−	2.94166i
$131$	3.49656e14			0.528132			0.264066	−	0.964505i	$-0.414936\pi$
$131$	3.49656e14			0.528132			0.264066	+	0.964505i	$0.414936\pi$
$132$		−	7.15130e12i		−	0.0102416i
$133$	−5.11477e14			−0.694807
$134$			1.17598e15i			1.51588i
$135$		−	2.22539e14i		−	0.272313i
$136$		−	2.86479e14i		−	0.332905i
$137$		−	1.25845e15i		−	1.38928i	−0.719356	−	0.694641i	$-0.755563\pi$
$137$		−	1.25845e15i		−	1.38928i	0.719356	−	0.694641i	$-0.244437\pi$
$138$	−2.52866e13	−	7.83559e13i	−0.0265299	−	0.0822087i
$139$	−3.74824e14			−0.373873			−0.186937	−	0.982372i	$-0.559856\pi$
$139$	−3.74824e14			−0.373873			−0.186937	+	0.982372i	$0.559856\pi$
$140$	1.44152e14			0.136749
$141$	2.52261e13			0.0227675
$142$	7.35623e14			0.631883
$143$			3.13738e15i			2.56575i
$144$	1.37169e15			1.06836
$145$			5.32511e14i			0.395141i
$146$	−2.03670e15			−1.44032
$147$	1.48958e12			0.00100425
$148$			2.56081e13i			0.0164644i
$149$			4.34564e14i			0.266532i	0.991080	+	0.133266i	$0.0425465\pi$
$149$			4.34564e14i			0.266532i	−0.991080	+	0.133266i	$0.957453\pi$
$150$	2.52122e14			0.147561
$151$	−1.28472e15			−0.717743			−0.358871	−	0.933387i	$-0.616838\pi$
$151$	−1.28472e15			−0.717743			−0.358871	+	0.933387i	$0.616838\pi$
$152$		−	1.23581e15i		−	0.659245i
$153$		−	6.79771e14i		−	0.346356i
$154$	−3.21135e15			−1.56330
$155$			6.01411e15i			2.79801i
$156$	−2.64886e13			−0.0117811
$157$			4.28217e15i			1.82124i	0.413250	+	0.910618i	$0.364394\pi$
$157$			4.28217e15i			1.82124i	−0.413250	+	0.910618i	$0.635606\pi$
$158$		−	3.42629e15i		−	1.39388i
$159$		−	8.84870e13i		−	0.0344429i
$160$			7.27948e14i			0.271182i
$161$	−2.68460e15	+	8.66358e14i	−0.957410	+	0.308970i
$162$	2.98391e15			1.01901
$163$	4.02398e15			1.31626			0.658131	−	0.752903i	$-0.271347\pi$
$163$	4.02398e15			1.31626			0.658131	+	0.752903i	$0.271347\pi$
$164$	−4.09717e14			−0.128404
$165$	−6.79397e14			−0.204050
$166$			4.21479e15i			1.21344i
$167$	−5.81714e15			−1.60581			−0.802906	−	0.596106i	$-0.796714\pi$
$167$	−5.81714e15			−1.60581			−0.802906	+	0.596106i	$0.796714\pi$
$168$			3.01138e14i			0.0797262i
$169$	7.68358e15			1.95145
$170$	2.45044e15			0.597175
$171$		−	2.93239e15i		−	0.685882i
$172$		−	4.19154e13i		−	0.00941184i
$173$	−1.71482e15			−0.369739			−0.184869	−	0.982763i	$-0.559186\pi$
$173$	−1.71482e15			−0.369739			−0.184869	+	0.982763i	$0.559186\pi$
$174$	1.00158e14			0.0207416
$175$		−	8.63810e15i		−	1.71851i
$176$		−	8.40442e15i		−	1.60664i
$177$	−2.09987e13			−0.00385815
$178$		−	4.34683e15i		−	0.767773i
$179$	−3.80116e15			−0.645573			−0.322786	−	0.946472i	$-0.604620\pi$
$179$	−3.80116e15			−0.645573			−0.322786	+	0.946472i	$0.604620\pi$
$180$			8.26452e14i			0.134993i
$181$		−	8.18032e15i		−	1.28535i	−0.766140	−	0.642673i	$-0.777825\pi$
$181$		−	8.18032e15i		−	1.28535i	0.766140	−	0.642673i	$-0.222175\pi$
$182$			1.18949e16i			1.79830i
$183$			7.02716e14i			0.102240i
$184$	−2.09326e15	−	6.48642e15i	−0.293156	−	0.908406i
$185$	2.43286e15			0.328030
$186$	1.13117e15			0.146872
$187$	−4.16500e15			−0.520864
$188$	−1.88017e14			−0.0226512
$189$			1.43407e15i			0.166471i
$190$	1.05707e16			1.18257
$191$		−	1.02384e15i		−	0.110407i	−0.998475	−	0.0552037i	$-0.982419\pi$
$191$		−	1.02384e15i		−	0.110407i	0.998475	−	0.0552037i	$-0.0175808\pi$
$192$	−7.22151e14			−0.0750791
$193$	4.80418e15			0.481635			0.240817	−	0.970570i	$-0.422585\pi$
$193$	4.80418e15			0.481635			0.240817	+	0.970570i	$0.422585\pi$
$194$		−	3.71636e15i		−	0.359341i
$195$			2.51651e15i			0.234723i
$196$	−1.11022e13			−0.000999121
$197$	3.55152e15			0.308426			0.154213	−	0.988038i	$-0.450716\pi$
$197$	3.55152e15			0.308426			0.154213	+	0.988038i	$0.450716\pi$
$198$		−	1.84113e16i		−	1.54322i
$199$			1.95367e15i			0.158081i	0.996871	+	0.0790406i	$0.0251857\pi$
$199$			1.95367e15i			0.158081i	−0.996871	+	0.0790406i	$0.974814\pi$
$200$	2.08710e16			1.63055
$201$		−	1.60325e15i		−	0.120956i
$202$	−1.61993e16			−1.18042
$203$		−	3.43158e15i		−	0.241558i
$204$		−	3.51648e13i		−	0.00239164i
$205$			3.89244e16i			2.55826i
$206$			2.54732e16i			1.61813i
$207$	−4.96700e15	−	1.53913e16i	−0.305001	−	0.945112i
$208$	−3.11303e16			−1.84816
$209$	−1.79670e16			−1.03146
$210$	−2.57583e15			−0.143015
$211$	2.23343e16			1.19949			0.599744	−	0.800192i	$-0.295269\pi$
$211$	2.23343e16			1.19949			0.599744	+	0.800192i	$0.295269\pi$
$212$			6.59517e14i			0.0342669i
$213$	−1.00290e15			−0.0504197
$214$		−	6.37890e15i		−	0.310348i
$215$	−3.98210e15			−0.187518
$216$	−3.46495e15			−0.157950
$217$		−	3.87558e16i		−	1.71048i
$218$			2.21981e16i			0.948680i
$219$	2.77671e15			0.114927
$220$	5.06372e15			0.203007
$221$			1.54273e16i			0.599162i
$222$		−	4.57588e14i		−	0.0172188i
$223$	−1.07817e16			−0.393144			−0.196572	−	0.980489i	$-0.562981\pi$
$223$	−1.07817e16			−0.393144			−0.196572	+	0.980489i	$0.562981\pi$
$224$		−	4.69101e15i		−	0.165779i
$225$	4.95239e16			1.69643
$226$			2.55210e16i			0.847497i
$227$			1.05375e16i			0.339277i	0.985506	+	0.169639i	$0.0542600\pi$
$227$			1.05375e16i			0.339277i	−0.985506	+	0.169639i	$0.945740\pi$
$228$		−	1.51694e14i		−	0.00473613i
$229$			2.97574e16i			0.901046i	0.892765	+	0.450523i	$0.148762\pi$
$229$			2.97574e16i			0.901046i	−0.892765	+	0.450523i	$0.851238\pi$
$230$	5.54826e16	−	1.79050e16i	1.62953	−	0.525872i
$231$	4.37814e15			0.124740
$232$	8.29125e15			0.229195
$233$	−2.91546e16			−0.782015			−0.391007	−	0.920388i	$-0.627873\pi$
$233$	−2.91546e16			−0.782015			−0.391007	+	0.920388i	$0.627873\pi$
$234$	−6.81959e16			−1.77520
$235$			1.78622e16i			0.451294i
$236$	1.56509e14			0.00383844
$237$			4.67118e15i			0.111221i
$238$	−1.57910e16			−0.365066
$239$	−1.47968e16			−0.332187			−0.166093	−	0.986110i	$-0.553115\pi$
$239$	−1.47968e16			−0.332187			−0.166093	+	0.986110i	$0.553115\pi$
$240$		−	6.74123e15i		−	0.146981i
$241$		−	7.05077e16i		−	1.49320i	−0.665274	−	0.746599i	$-0.731685\pi$
$241$		−	7.05077e16i		−	1.49320i	0.665274	−	0.746599i	$-0.268315\pi$
$242$	−6.22314e16			−1.28027
$243$	−1.23469e16			−0.246783
$244$		−	5.23753e15i		−	0.101718i
$245$			1.05475e15i			0.0199061i
$246$	7.32118e15			0.134287
$247$			6.65502e16i			1.18651i
$248$	9.36404e16			1.62294
$249$		−	5.74617e15i		−	0.0968240i
$250$			7.40144e16i			1.21265i
$251$		−	8.52485e16i		−	1.35822i	−0.734036	−	0.679111i	$-0.762365\pi$
$251$		−	8.52485e16i		−	1.35822i	0.734036	−	0.679111i	$-0.237635\pi$
$252$		−	5.32578e15i		−	0.0825239i
$253$	−9.43036e16	+	3.04331e16i	−1.42130	+	0.458673i
$254$	7.44774e16			1.09191
$255$	−3.34077e15			−0.0476502
$256$	1.77371e16			0.246152
$257$	5.96865e16			0.806017			0.403008	−	0.915196i	$-0.367965\pi$
$257$	5.96865e16			0.806017			0.403008	+	0.915196i	$0.367965\pi$
$258$			7.48980e14i			0.00984311i
$259$	−1.56777e16			−0.200532
$260$		−	1.87562e16i		−	0.233524i
$261$	1.96739e16			0.238456
$262$	4.65677e16			0.549511
$263$			5.25249e15i			0.0603497i	0.999545	+	0.0301749i	$0.00960642\pi$
$263$			5.25249e15i			0.0603497i	−0.999545	+	0.0301749i	$0.990394\pi$
$264$			1.05783e16i			0.118355i
$265$	6.26563e16			0.682721
$266$	−6.81192e16			−0.722933
$267$			5.92619e15i			0.0612628i
$268$			1.19495e16i			0.120338i
$269$	−5.65798e16			−0.555131			−0.277566	−	0.960707i	$-0.589528\pi$
$269$	−5.65798e16			−0.555131			−0.277566	+	0.960707i	$0.589528\pi$
$270$		−	2.96380e16i		−	0.283337i
$271$	−8.93235e16			−0.832110			−0.416055	−	0.909339i	$-0.636588\pi$
$271$	−8.93235e16			−0.832110			−0.416055	+	0.909339i	$0.636588\pi$
$272$		−	4.13267e16i		−	0.375188i
$273$		−	1.62168e16i		−	0.143491i
$274$		−	1.67602e17i		−	1.44552i
$275$		−	3.03436e17i		−	2.55116i
$276$	−2.56944e14	−	7.96197e14i	−0.00210608	−	0.00652615i
$277$	1.22553e17			0.979409			0.489705	−	0.871888i	$-0.337105\pi$
$277$	1.22553e17			0.979409			0.489705	+	0.871888i	$0.337105\pi$
$278$	−4.99197e16			−0.389008
$279$	2.22195e17			1.68851
$280$	−2.13231e17			−1.58032
$281$		−	1.05090e17i		−	0.759658i	−0.925057	−	0.379829i	$-0.875983\pi$
$281$		−	1.05090e17i		−	0.759658i	0.925057	−	0.379829i	$-0.124017\pi$
$282$	3.35964e15			0.0236891
$283$			1.62541e17i			1.11804i	0.829153	+	0.559021i	$0.188823\pi$
$283$			1.62541e17i			1.11804i	−0.829153	+	0.559021i	$0.811177\pi$
$284$	7.47488e15			0.0501621
$285$	−1.44114e16			−0.0943608
$286$			4.17841e17i			2.66961i
$287$		−	2.50835e17i		−	1.56392i
$288$	2.68945e16			0.163650
$289$	1.47897e17			0.878366
$290$			7.09205e16i			0.411137i
$291$			5.06665e15i			0.0286728i
$292$	−2.06956e16			−0.114340
$293$		−	2.27684e17i		−	1.22817i	−0.789240	−	0.614085i	$-0.789525\pi$
$293$		−	2.27684e17i		−	1.22817i	0.789240	−	0.614085i	$-0.210475\pi$
$294$	1.98384e14			0.00104490
$295$		−	1.48688e16i		−	0.0764756i
$296$		−	3.78798e16i		−	0.190268i
$297$			5.03756e16i			0.247130i
$298$			5.78759e16i			0.277322i
$299$	1.12725e17	+	3.49303e17i	0.527622	+	1.63495i
$300$	2.56189e15			0.0117142
$301$	2.56613e16			0.114634
$302$	−1.71100e17			−0.746797
$303$	2.20850e16			0.0941888
$304$		−	1.78275e17i		−	0.742977i
$305$	−4.97583e17			−2.02659
$306$		−	9.05329e16i		−	0.360377i
$307$	3.25958e17			1.26821			0.634107	−	0.773245i	$-0.281368\pi$
$307$	3.25958e17			1.26821			0.634107	+	0.773245i	$0.281368\pi$
$308$	−3.26314e16			−0.124103
$309$		−	3.47285e16i		−	0.129115i
$310$			8.00967e17i			2.91127i
$311$	−1.06840e17			−0.379673			−0.189836	−	0.981816i	$-0.560796\pi$
$311$	−1.06840e17			−0.379673			−0.189836	+	0.981816i	$0.560796\pi$
$312$	3.91823e16			0.136147
$313$			4.91004e17i			1.66830i	0.551536	+	0.834151i	$0.314042\pi$
$313$			4.91004e17i			1.66830i	−0.551536	+	0.834151i	$0.685958\pi$
$314$			5.70305e17i			1.89496i
$315$	−5.05967e17			−1.64418
$316$		−	3.48155e16i		−	0.110653i
$317$	4.22733e17			1.31417			0.657084	−	0.753817i	$-0.271790\pi$
$317$	4.22733e17			1.31417			0.657084	+	0.753817i	$0.271790\pi$
$318$		−	1.17848e16i		−	0.0358371i
$319$		−	1.20543e17i		−	0.358599i
$320$		−	5.11344e17i		−	1.48821i
$321$			8.69657e15i			0.0247635i
$322$	−3.57538e17	+	1.15383e17i	−0.996166	+	0.321477i
$323$	−8.83482e16			−0.240869
$324$	3.03204e16			0.0808945
$325$	−1.12394e18			−2.93466
$326$	5.35919e17			1.36954
$327$		−	3.02634e16i		−	0.0756978i
$328$	6.06058e17			1.48388
$329$		−	1.15107e17i		−	0.275886i
$330$	−9.04830e16			−0.212310
$331$	2.93195e17			0.673536			0.336768	−	0.941588i	$-0.390666\pi$
$331$	2.93195e17			0.673536			0.336768	+	0.941588i	$0.390666\pi$
$332$			4.28278e16i			0.0963294i
$333$		−	8.98833e16i		−	0.197956i
$334$	−7.74736e17			−1.67081
$335$	1.13524e18			2.39758
$336$			4.34415e16i			0.0898525i
$337$			1.89656e17i			0.384200i	0.981375	+	0.192100i	$0.0615298\pi$
$337$			1.89656e17i			0.384200i	−0.981375	+	0.192100i	$0.938470\pi$
$338$	1.02331e18			2.03044
$339$		−	3.47936e16i		−	0.0676241i
$340$	2.48996e16			0.0474068
$341$		−	1.36140e18i		−	2.53925i
$342$		−	3.90540e17i		−	0.713647i
$343$			5.55117e17i			0.993861i
$344$			6.20017e16i			0.108766i
$345$	−7.56413e16	+	2.44105e16i	−0.130025	+	0.0419608i
$346$	−2.28382e17			−0.384706
$347$	−4.06768e17			−0.671491			−0.335746	−	0.941953i	$-0.608988\pi$
$347$	−4.06768e17			−0.671491			−0.335746	+	0.941953i	$0.608988\pi$
$348$	1.01774e15			0.00164657
$349$	−6.34914e17			−1.00679			−0.503393	−	0.864058i	$-0.667915\pi$
$349$	−6.34914e17			−1.00679			−0.503393	+	0.864058i	$0.667915\pi$
$350$		−	1.15043e18i		−	1.78807i
$351$	1.86593e17			0.284279
$352$		−	1.64784e17i		−	0.246103i
$353$	3.41058e17			0.499351			0.249675	−	0.968330i	$-0.419676\pi$
$353$	3.41058e17			0.499351			0.249675	+	0.968330i	$0.419676\pi$
$354$	−2.79663e15			−0.00401433
$355$		−	7.10138e17i		−	0.999411i
$356$		−	4.41695e16i		−	0.0609498i
$357$	2.15284e16			0.0291296
$358$	−5.06244e17			−0.671706
$359$			3.48552e17i			0.453530i	0.973949	+	0.226765i	$0.0728150\pi$
$359$			3.48552e17i			0.453530i	−0.973949	+	0.226765i	$0.927185\pi$
$360$		−	1.22250e18i		−	1.56002i
$361$	4.17891e17			0.523013
$362$		−	1.08947e18i		−	1.33738i
$363$	8.48422e16			0.102156
$364$			1.20868e17i			0.142758i
$365$			1.96615e18i			2.27806i
$366$			9.35888e16i			0.106379i
$367$			5.07101e17i			0.565499i	0.959194	+	0.282750i	$0.0912466\pi$
$367$			5.07101e17i			0.565499i	−0.959194	+	0.282750i	$0.908753\pi$
$368$	−3.01969e17	−	9.35715e17i	−0.330390	−	1.02379i
$369$	1.43809e18			1.54383
$370$	3.24011e17			0.341309
$371$	−4.03767e17			−0.417362
$372$	1.14942e16			0.0116595
$373$			1.79569e18i			1.78760i	0.448468	+	0.893799i	$0.351970\pi$
$373$			1.79569e18i			1.78760i	−0.448468	+	0.893799i	$0.648030\pi$
$374$	−5.54701e17			−0.541948
$375$		−	1.00906e17i		−	0.0967609i
$376$	2.78116e17			0.261765
$377$	−4.46497e17			−0.412504
$378$			1.90992e17i			0.173210i
$379$		−	1.27297e18i		−	1.13329i	−0.823961	−	0.566646i	$-0.808241\pi$
$379$		−	1.27297e18i		−	1.13329i	0.823961	−	0.566646i	$-0.191759\pi$
$380$	1.07412e17			0.0938788
$381$	−1.01538e17			−0.0871269
$382$		−	1.36356e17i		−	0.114877i
$383$			4.21844e17i			0.348948i	0.984662	+	0.174474i	$0.0558224\pi$
$383$			4.21844e17i			0.348948i	−0.984662	+	0.174474i	$0.944178\pi$
$384$	−1.13021e17			−0.0917993
$385$			3.10009e18i			2.47257i
$386$	6.39827e17			0.501131
$387$			1.47121e17i			0.113161i
$388$		−	3.77631e16i		−	0.0285263i
$389$			4.59655e17i			0.341024i	0.985356	+	0.170512i	$0.0545422\pi$
$389$			4.59655e17i			0.341024i	−0.985356	+	0.170512i	$0.945458\pi$
$390$			3.35152e17i			0.244225i
$391$	−4.63715e17	+	1.49647e17i	−0.331905	+	0.107110i
$392$	1.64226e16			0.0115462
$393$	−6.34874e16			−0.0438470
$394$	4.72997e17			0.320911
$395$	−3.30759e18			−2.20461
$396$		−	1.87082e17i		−	0.122509i
$397$	−2.17020e17			−0.139626			−0.0698128	−	0.997560i	$-0.522240\pi$
$397$	−2.17020e17			−0.139626			−0.0698128	+	0.997560i	$0.522240\pi$
$398$			2.60193e17i			0.164480i
$399$	9.28692e16			0.0576848
$400$	3.01081e18			1.83765
$401$		−	8.46130e17i		−	0.507489i	−0.967271	−	0.253745i	$-0.918338\pi$
$401$		−	8.46130e17i		−	0.507489i	0.967271	−	0.253745i	$-0.0816623\pi$
$402$		−	2.13523e17i		−	0.125853i
$403$	−5.04267e18			−2.92096
$404$	−1.64606e17			−0.0937077
$405$		−	2.88053e18i		−	1.61171i
$406$		−	4.57023e17i		−	0.251337i
$407$	−5.50721e17			−0.297694
$408$			5.20161e16i			0.0276387i
$409$	3.35217e18			1.75091			0.875454	−	0.483302i	$-0.160563\pi$
$409$	3.35217e18			1.75091			0.875454	+	0.483302i	$0.160563\pi$
$410$			5.18401e18i			2.66182i
$411$			2.28497e17i			0.115342i
$412$			2.58841e17i			0.128456i
$413$			9.58171e16i			0.0467512i
$414$	−6.61512e17	−	2.04984e18i	−0.317348	−	0.983370i
$415$	4.06878e18			1.91923
$416$	−6.10366e17			−0.283098
$417$	6.80572e16			0.0310400
$418$	−2.39287e18			−1.07321
$419$			2.06872e18i			0.912438i	0.889867	+	0.456219i	$0.150797\pi$
$419$			2.06872e18i			0.912438i	−0.889867	+	0.456219i	$0.849203\pi$
$420$	−2.61738e16			−0.0113533
$421$			1.08195e18i			0.461565i	0.973005	+	0.230783i	$0.0741287\pi$
$421$			1.08195e18i			0.461565i	−0.973005	+	0.230783i	$0.925871\pi$
$422$	2.97452e18			1.24804
$423$	6.59929e17			0.272342
$424$		−	9.75565e17i		−	0.396000i
$425$		−	1.49207e18i		−	0.595755i
$426$	−1.33568e17			−0.0524607
$427$	3.20650e18			1.23890
$428$		−	6.48179e16i		−	0.0246370i
$429$		−	5.69657e17i		−	0.213016i
$430$	−5.30342e17			−0.195109
$431$		−	1.72197e18i		−	0.623281i	−0.950200	−	0.311640i	$-0.899122\pi$
$431$		−	1.72197e18i		−	0.623281i	0.950200	−	0.311640i	$-0.100878\pi$
$432$	−4.99846e17			−0.178012
$433$			3.73791e18i			1.30982i	0.755705	+	0.654912i	$0.227294\pi$
$433$			3.73791e18i			1.30982i	−0.755705	+	0.654912i	$0.772706\pi$
$434$		−	5.16156e18i		−	1.77972i
$435$		−	9.66884e16i		−	0.0328057i
$436$			2.25561e17i			0.0753111i
$437$	−2.00037e18	+	6.45548e17i	−0.657265	+	0.212109i
$438$	3.69806e17			0.119579
$439$	−3.30574e18			−1.05200			−0.526002	−	0.850483i	$-0.676309\pi$
$439$	−3.30574e18			−1.05200			−0.526002	+	0.850483i	$0.676309\pi$
$440$	−7.49032e18			−2.34602
$441$	3.89683e16			0.0120127
$442$			2.05463e18i			0.623416i
$443$	−1.61141e18			−0.481261			−0.240631	−	0.970617i	$-0.577354\pi$
$443$	−1.61141e18			−0.481261			−0.240631	+	0.970617i	$0.577354\pi$
$444$		−	4.64969e15i		−	0.00136692i
$445$	−4.19624e18			−1.21434
$446$	−1.43592e18			−0.409059
$447$		−	7.89042e16i		−	0.0221283i
$448$			3.29518e18i			0.909773i
$449$	4.03873e18			1.09779			0.548897	−	0.835890i	$-0.315048\pi$
$449$	4.03873e18			1.09779			0.548897	+	0.835890i	$0.315048\pi$
$450$	6.59566e18			1.76511
$451$		−	8.81125e18i		−	2.32168i
$452$			2.59326e17i			0.0672786i
$453$	2.33267e17			0.0595890
$454$			1.40339e18i			0.353011i
$455$	1.14828e19			2.84426
$456$			2.24387e17i			0.0547323i
$457$		−	3.93455e18i		−	0.945110i	−0.881301	−	0.472555i	$-0.843332\pi$
$457$		−	3.93455e18i		−	0.945110i	0.881301	−	0.472555i	$-0.156668\pi$
$458$			3.96313e18i			0.937520i
$459$			2.47710e17i			0.0577105i
$460$	5.63775e17	−	1.81938e17i	0.129360	−	0.0417464i
$461$	2.73833e18			0.618842			0.309421	−	0.950925i	$-0.399865\pi$
$461$	2.73833e18			0.618842			0.309421	+	0.950925i	$0.399865\pi$
$462$	5.83087e17			0.129789
$463$	1.72825e18			0.378912			0.189456	−	0.981889i	$-0.439328\pi$
$463$	1.72825e18			0.378912			0.189456	+	0.981889i	$0.439328\pi$
$464$	1.19608e18			0.258305
$465$		−	1.09199e18i		−	0.232299i
$466$	−3.88285e18			−0.813671
$467$		−	4.02736e17i		−	0.0831386i	−0.999136	−	0.0415693i	$-0.986764\pi$
$467$		−	4.02736e17i		−	0.0831386i	0.999136	−	0.0415693i	$-0.0132357\pi$
$468$	−6.92959e17			−0.140924
$469$	−7.31564e18			−1.46569
$470$			2.37891e18i			0.469562i
$471$		−	7.77516e17i		−	0.151204i
$472$	−2.31510e17			−0.0443583
$473$	9.01420e17			0.170176
$474$			6.22114e17i			0.115724i
$475$		−	6.43650e18i		−	1.17976i
$476$	−1.60457e17			−0.0289808
$477$		−	2.31487e18i		−	0.412001i
$478$	−1.97066e18			−0.345634
$479$		−	7.50860e18i		−	1.29781i	−0.760870	−	0.648904i	$-0.775228\pi$
$479$		−	7.50860e18i		−	1.29781i	0.760870	−	0.648904i	$-0.224772\pi$
$480$		−	1.32174e17i		−	0.0225143i
$481$			2.03989e18i			0.342445i
$482$		−	9.39032e18i		−	1.55364i
$483$	4.87444e17	−	1.57305e17i	0.0794868	−	0.0256515i
$484$	−6.32352e17			−0.101634
$485$	−3.58761e18			−0.568347
$486$	−1.64438e18			−0.256773
$487$	2.13159e18			0.328096			0.164048	−	0.986452i	$-0.447545\pi$
$487$	2.13159e18			0.328096			0.164048	+	0.986452i	$0.447545\pi$
$488$			7.74742e18i			1.17549i
$489$	−7.30637e17			−0.109280
$490$			1.40473e17i			0.0207119i
$491$	1.30649e19			1.89904			0.949520	−	0.313707i	$-0.101571\pi$
$491$	1.30649e19			1.89904			0.949520	+	0.313707i	$0.101571\pi$
$492$	7.43926e16			0.0106604
$493$		−	5.92743e17i		−	0.0837409i
$494$			8.86325e18i			1.23454i
$495$	−1.77734e19			−2.44081
$496$	1.35083e19			1.82907
$497$			4.57624e18i			0.610962i
$498$		−	7.65283e17i		−	0.100743i
$499$	6.85221e17			0.0889461			0.0444731	−	0.999011i	$-0.485839\pi$
$499$	6.85221e17			0.0889461			0.0444731	+	0.999011i	$0.485839\pi$
$500$			7.52082e17i			0.0962665i
$501$	1.05622e18			0.133319
$502$		−	1.13535e19i		−	1.41320i
$503$			3.84675e18i			0.472192i	0.971730	+	0.236096i	$0.0758680\pi$
$503$			3.84675e18i			0.472192i	−0.971730	+	0.236096i	$0.924132\pi$
$504$			7.87796e18i			0.953674i
$505$			1.56381e19i			1.86700i
$506$	−1.25595e19	+	4.05313e18i	−1.47883	+	0.477240i
$507$	−1.39511e18			−0.162014
$508$	7.56787e17			0.0866818
$509$	−1.08906e19			−1.23035			−0.615175	−	0.788391i	$-0.710915\pi$
$509$	−1.08906e19			−1.23035			−0.615175	+	0.788391i	$0.710915\pi$
$510$	−4.44928e17			−0.0495791
$511$		−	1.26701e19i		−	1.39263i
$512$	−7.83614e18			−0.849596
$513$			1.06857e18i			0.114283i
$514$	7.94913e18			0.838644
$515$	2.45907e19			2.55930
$516$			7.61061e15i			0.000781397i
$517$		−	4.04343e18i		−	0.409558i
$518$	−2.08798e18			−0.208650
$519$	3.11361e17			0.0306967
$520$			2.77444e19i			2.69868i
$521$			4.87339e18i			0.467700i	0.972273	+	0.233850i	$0.0751325\pi$
$521$			4.87339e18i			0.467700i	−0.972273	+	0.233850i	$0.924868\pi$
$522$	2.62020e18			0.248108
$523$			1.19814e19i			1.11943i	0.828685	+	0.559715i	$0.189089\pi$
$523$			1.19814e19i			1.11943i	−0.828685	+	0.559715i	$0.810911\pi$
$524$	4.73189e17			0.0436230
$525$			1.56843e18i			0.142675i
$526$			6.99535e17i			0.0627927i
$527$		−	6.69436e18i		−	0.592973i
$528$			1.52600e18i			0.133388i
$529$	−9.40593e18	+	6.77660e18i	−0.811357	+	0.584551i
$530$	8.34465e18			0.710358
$531$	−5.49338e17			−0.0461507
$532$	−6.92179e17			−0.0573901
$533$	−3.26371e19			−2.67068
$534$			7.89258e17i			0.0637427i
$535$	−6.15791e18			−0.490859
$536$		−	1.76758e19i		−	1.39067i
$537$	6.90180e17			0.0535973
$538$	−7.53538e18			−0.577603
$539$		−	2.38761e17i		−	0.0180652i
$540$		−	3.01161e17i		−	0.0224927i
$541$	5.75772e17			0.0424492			0.0212246	−	0.999775i	$-0.493243\pi$
$541$	5.75772e17			0.0424492			0.0212246	+	0.999775i	$0.493243\pi$
$542$	−1.18962e19			−0.865794
$543$			1.48531e18i			0.106713i
$544$		−	8.10286e17i		−	0.0574707i
$545$	2.14290e19			1.50047
$546$		−	2.15977e18i		−	0.149300i
$547$	9.92681e18			0.677483			0.338742	−	0.940879i	$-0.389999\pi$
$547$	9.92681e18			0.677483			0.338742	+	0.940879i	$0.389999\pi$
$548$		−	1.70305e18i		−	0.114753i
$549$			1.83835e19i			1.22299i
$550$		−	4.04121e19i		−	2.65444i
$551$		−	2.55697e18i		−	0.165831i
$552$	3.80075e17	+	1.17774e18i	0.0243386	+	0.0754184i
$553$	2.13146e19			1.34773
$554$	1.63217e19			1.01906
$555$	−4.41735e17			−0.0272340
$556$	−5.07248e17			−0.0308814
$557$		−	2.22177e19i		−	1.33571i	−0.744290	−	0.667856i	$-0.767212\pi$
$557$		−	2.22177e19i		−	1.33571i	0.744290	−	0.667856i	$-0.232788\pi$
$558$	2.95922e19			1.75686
$559$		−	3.33889e18i		−	0.195758i
$560$	−3.07603e19			−1.78104
$561$	7.56243e17			0.0432435
$562$		−	1.39961e19i		−	0.790409i
$563$		−	1.71897e18i		−	0.0958762i	−0.998850	−	0.0479381i	$-0.984735\pi$
$563$		−	1.71897e18i		−	0.0958762i	0.998850	−	0.0479381i	$-0.0152650\pi$
$564$	3.41383e16			0.00188057
$565$	2.46368e19			1.34043
$566$			2.16475e19i			1.16330i
$567$			1.85626e19i			0.985275i
$568$	−1.10569e19			−0.579691
$569$		−	2.88669e19i		−	1.49491i	−0.664314	−	0.747454i	$-0.731276\pi$
$569$		−	2.88669e19i		−	1.49491i	0.664314	−	0.747454i	$-0.268724\pi$
$570$	−1.91933e18			−0.0981805
$571$		−	9.58358e18i		−	0.484256i	−0.970244	−	0.242128i	$-0.922155\pi$
$571$		−	9.58358e18i		−	0.484256i	0.970244	−	0.242128i	$-0.0778454\pi$
$572$			4.24580e18i			0.211928i
$573$			1.85899e17i			0.00916633i
$574$		−	3.34066e19i		−	1.62723i
$575$	−1.09024e19	−	3.37834e19i	−0.524622	−	1.62565i
$576$	−1.88919e19			−0.898087
$577$	3.00915e18			0.141323			0.0706614	−	0.997500i	$-0.477489\pi$
$577$	3.00915e18			0.141323			0.0706614	+	0.997500i	$0.477489\pi$
$578$	1.96972e19			0.913923
$579$	−8.72298e17			−0.0399866
$580$			7.20645e17i			0.0326381i
$581$	−2.62198e19			−1.17327
$582$			6.74783e17i			0.0298334i
$583$	−1.41834e19			−0.619583
$584$	3.06131e19			1.32135
$585$			6.58333e19i			2.80773i
$586$		−	3.03232e19i		−	1.27789i
$587$	1.69874e19			0.707393			0.353697	−	0.935360i	$-0.384925\pi$
$587$	1.69874e19			0.707393			0.353697	+	0.935360i	$0.384925\pi$
$588$	2.01584e15			8.29498e−5
$589$		−	2.88781e19i		−	1.17425i
$590$		−	1.98025e18i		−	0.0795713i
$591$	−6.44853e17			−0.0256064
$592$		−	5.46446e18i		−	0.214435i
$593$	8.45941e18			0.328063			0.164031	−	0.986455i	$-0.447550\pi$
$593$	8.45941e18			0.328063			0.164031	+	0.986455i	$0.447550\pi$
$594$			6.70910e18i			0.257134i
$595$			1.52439e19i			0.577402i
$596$			5.88094e17i			0.0220152i
$597$		−	3.54730e17i		−	0.0131243i
$598$	1.50129e19	+	4.65207e19i	0.548980	+	1.70113i
$599$	−4.14535e19			−1.49821			−0.749106	−	0.662450i	$-0.769517\pi$
$599$	−4.14535e19			−1.49821			−0.749106	+	0.662450i	$0.769517\pi$
$600$	−3.78957e18			−0.135373
$601$	−1.41256e19			−0.498754			−0.249377	−	0.968407i	$-0.580226\pi$
$601$	−1.41256e19			−0.498754			−0.249377	+	0.968407i	$0.580226\pi$
$602$	3.41760e18			0.119274
$603$		−	4.19420e19i		−	1.44686i
$604$	−1.73860e18			−0.0592846
$605$			6.00754e19i			2.02493i
$606$	2.94132e18			0.0980016
$607$	−2.16960e18			−0.0714592			−0.0357296	−	0.999361i	$-0.511376\pi$
$607$	−2.16960e18			−0.0714592			−0.0357296	+	0.999361i	$0.511376\pi$
$608$		−	3.49541e18i		−	0.113808i
$609$			6.23075e17i			0.0200548i
$610$	−6.62688e19			−2.10863
$611$	−1.49770e19			−0.471125
$612$		−	9.19932e17i		−	0.0286085i
$613$			6.08379e19i			1.87047i	0.354027	+	0.935235i	$0.384812\pi$
$613$			6.08379e19i			1.87047i	−0.354027	+	0.935235i	$0.615188\pi$
$614$	4.34115e19			1.31955
$615$		−	7.06754e18i		−	0.212394i
$616$	4.82688e19			1.43417
$617$			6.08140e19i			1.78652i	0.449543	+	0.893258i	$0.351587\pi$
$617$			6.08140e19i			1.78652i	−0.449543	+	0.893258i	$0.648413\pi$
$618$		−	4.62520e18i		−	0.134342i
$619$		−	1.27565e19i		−	0.366351i	−0.983080	−	0.183176i	$-0.941362\pi$
$619$		−	1.27565e19i		−	0.366351i	0.983080	−	0.183176i	$-0.0586377\pi$
$620$			8.13887e18i			0.231112i
$621$	1.80998e18	+	5.60862e18i	0.0508198	+	0.157476i
$622$	−1.42291e19			−0.395042
$623$	2.70412e19			0.742352
$624$	5.65235e18			0.153439
$625$	7.81451e18			0.209769
$626$			6.53926e19i			1.73583i
$627$	3.26228e18			0.0856344
$628$			5.79504e18i			0.150432i
$629$	−2.70804e18			−0.0695184
$630$	−6.73854e19			−1.71073
$631$		−	4.23586e19i		−	1.06350i	−0.846902	−	0.531748i	$-0.821535\pi$
$631$		−	4.23586e19i		−	1.06350i	0.846902	−	0.531748i	$-0.178465\pi$
$632$			5.14996e19i			1.27874i
$633$	−4.05526e18			−0.0995848
$634$	5.63001e19			1.36737
$635$		−	7.18972e19i		−	1.72702i
$636$		−	1.19749e17i		−	0.00284493i
$637$	−8.84380e17			−0.0207808
$638$		−	1.60541e19i		−	0.373115i
$639$	−2.62365e19			−0.603114
$640$		−	8.00282e19i		−	1.81963i
$641$			1.34691e16i			0.000302923i	1.00000		0.000151461i	$4.82117e-5\pi$
$641$			1.34691e16i			0.000302923i	−1.00000		0.000151461i	$0.999952\pi$
$642$			1.15822e18i			0.0257660i
$643$			2.16660e19i			0.476761i	0.971172	+	0.238381i	$0.0766166\pi$
$643$			2.16660e19i			0.476761i	−0.971172	+	0.238381i	$0.923383\pi$
$644$	−3.63305e18	+	1.17244e18i	−0.0790807	+	0.0255205i
$645$	7.23033e17			0.0155683
$646$	−1.17663e19			−0.250619
$647$	1.53611e19			0.323663			0.161831	−	0.986818i	$-0.448260\pi$
$647$	1.53611e19			0.323663			0.161831	+	0.986818i	$0.448260\pi$
$648$	−4.48502e19			−0.934845
$649$			3.36583e18i			0.0694031i
$650$	−1.49688e20			−3.05346
$651$			7.03693e18i			0.142009i
$652$	5.44563e18			0.108721
$653$	−3.02518e19			−0.597529			−0.298765	−	0.954327i	$-0.596575\pi$
$653$	−3.02518e19			−0.597529			−0.298765	+	0.954327i	$0.596575\pi$
$654$		−	4.03052e18i		−	0.0787621i
$655$		−	4.49544e19i		−	0.869127i
$656$	8.74285e19			1.67235
$657$	7.26404e19			1.37474
$658$		−	1.53301e19i		−	0.287054i
$659$			3.03163e19i			0.561667i	0.959757	+	0.280833i	$0.0906108\pi$
$659$			3.03163e19i			0.561667i	−0.959757	+	0.280833i	$0.909389\pi$
$660$	−9.19425e17			−0.0168542
$661$			6.67580e19i			1.21086i	0.795899	+	0.605429i	$0.206998\pi$
$661$			6.67580e19i			1.21086i	−0.795899	+	0.605429i	$0.793002\pi$
$662$	3.90481e19			0.700801
$663$		−	2.80115e18i		−	0.0497441i
$664$		−	6.33513e19i		−	1.11322i
$665$			6.57593e19i			1.14342i
$666$		−	1.19708e19i		−	0.205970i
$667$	−4.33109e18	−	1.34208e19i	−0.0737423	−	0.228506i
$668$	−7.87232e18			−0.132638
$669$	1.95764e18			0.0326399
$670$	1.51192e20			2.49463
$671$	1.12637e20			1.83917
$672$			8.51751e17i			0.0137635i
$673$	3.80537e18			0.0608544			0.0304272	−	0.999537i	$-0.490313\pi$
$673$	3.80537e18			0.0608544			0.0304272	+	0.999537i	$0.490313\pi$
$674$			2.52587e19i			0.399752i
$675$	−1.80466e19			−0.282663
$676$	1.03982e19			0.161187
$677$		−	1.12901e20i		−	1.73211i	−0.499946	−	0.866057i	$-0.666647\pi$
$677$		−	1.12901e20i		−	1.73211i	0.499946	−	0.866057i	$-0.333353\pi$
$678$		−	4.63386e18i		−	0.0703615i
$679$	2.31191e19			0.347443
$680$	−3.68318e19			−0.547849
$681$		−	1.91329e18i		−	0.0281677i
$682$		−	1.81313e20i		−	2.64204i
$683$	−5.40040e19			−0.778899			−0.389449	−	0.921048i	$-0.627335\pi$
$683$	−5.40040e19			−0.778899			−0.389449	+	0.921048i	$0.627335\pi$
$684$		−	3.96840e18i		−	0.0566529i
$685$	−1.61795e20			−2.28629
$686$			7.39313e19i			1.03409i
$687$		−	5.40308e18i		−	0.0748073i
$688$			8.94422e18i			0.122581i
$689$			5.25357e19i			0.712721i
$690$	−1.00740e19	+	3.25103e18i	−0.135288	+	0.0436594i
$691$	−1.18915e20			−1.58085			−0.790425	−	0.612558i	$-0.790140\pi$
$691$	−1.18915e20			−1.58085			−0.790425	+	0.612558i	$0.790140\pi$
$692$	−2.32066e18			−0.0305399
$693$	1.14535e20			1.49212
$694$	−5.41739e19			−0.698673
$695$			4.81902e19i			0.615269i
$696$	−1.50545e18			−0.0190284
$697$		−	4.33272e19i		−	0.542164i
$698$	−8.45587e19			−1.04754
$699$	5.29361e18			0.0649250
$700$		−	1.16899e19i		−	0.141947i
$701$			7.98930e19i			0.960467i	0.877141	+	0.480234i	$0.159448\pi$
$701$			7.98930e19i			0.960467i	−0.877141	+	0.480234i	$0.840552\pi$
$702$	2.48507e19			0.295787
$703$	−1.16819e19			−0.137666
$704$			1.15752e20i			1.35058i
$705$		−	3.24325e18i		−	0.0374677i
$706$	4.54226e19			0.519564
$707$		−	1.00774e20i		−	1.14134i
$708$	−2.84174e16			−0.000318678
$709$		−	4.16109e19i		−	0.462044i	−0.972949	−	0.231022i	$-0.925793\pi$
$709$		−	4.16109e19i		−	0.462044i	0.972949	−	0.231022i	$-0.0742069\pi$
$710$		−	9.45772e19i		−	1.03987i
$711$			1.22201e20i			1.33041i
$712$			6.53360e19i			0.704356i
$713$	−4.89148e19	−	1.51573e20i	−0.522172	−	1.61806i
$714$	2.86719e18			0.0303088
$715$	4.03365e20			4.22237
$716$	−5.14410e18			−0.0533234
$717$	2.68667e18			0.0275791
$718$			4.64206e19i			0.471889i
$719$	1.83111e19			0.184337			0.0921684	−	0.995743i	$-0.470620\pi$
$719$	1.83111e19			0.184337			0.0921684	+	0.995743i	$0.470620\pi$
$720$		−	1.76355e20i		−	1.75816i
$721$	−1.58467e20			−1.56456
$722$	5.56553e19			0.544184
$723$			1.28021e19i			0.123969i
$724$		−	1.10704e19i		−	0.106168i
$725$	4.31835e19			0.410159
$726$	1.12994e19			0.106292
$727$		−	6.19839e19i		−	0.577481i	−0.957407	−	0.288741i	$-0.906763\pi$
$727$		−	6.19839e19i		−	0.577481i	0.957407	−	0.288741i	$-0.0932366\pi$
$728$		−	1.78789e20i		−	1.64976i
$729$	−1.04920e20			−0.958881
$730$			2.61854e20i			2.37028i
$731$	4.43251e18			0.0397400
$732$			9.50983e17i			0.00844491i
$733$			1.61738e20i			1.42260i	0.702887	+	0.711302i	$0.251894\pi$
$733$			1.61738e20i			1.42260i	−0.702887	+	0.711302i	$0.748106\pi$
$734$			6.75365e19i			0.588391i
$735$		−	1.91512e17i		−	0.00165266i
$736$	−5.92065e18	−	1.83464e19i	−0.0506087	−	0.156822i
$737$	−2.56981e20			−2.17585
$738$	1.91526e20			1.60633
$739$	9.94832e18			0.0826492			0.0413246	−	0.999146i	$-0.486842\pi$
$739$	9.94832e18			0.0826492			0.0413246	+	0.999146i	$0.486842\pi$
$740$	3.29237e18			0.0270949
$741$		−	1.20836e19i		−	0.0985072i
$742$	−5.37742e19			−0.434257
$743$		−	1.21082e20i		−	0.968633i	−0.874893	−	0.484316i	$-0.839068\pi$
$743$		−	1.21082e20i		−	0.968633i	0.874893	−	0.484316i	$-0.160932\pi$
$744$	−1.70024e19			−0.134741
$745$	5.58708e19			0.438623
$746$			2.39153e20i			1.85996i
$747$		−	1.50323e20i		−	1.15820i
$748$	−5.63648e18			−0.0430226
$749$	3.96825e19			0.300073
$750$		−	1.34389e19i		−	0.100678i
$751$		−	1.73180e19i		−	0.128534i	−0.997933	−	0.0642669i	$-0.979529\pi$
$751$		−	1.73180e19i		−	0.128534i	0.997933	−	0.0642669i	$-0.0204709\pi$
$752$	4.01204e19			0.295012
$753$			1.54786e19i			0.112763i
$754$	−5.94651e19			−0.429203
$755$			1.65173e20i			1.18116i
$756$			1.94073e18i			0.0137503i
$757$			2.95045e17i			0.00207118i	0.999999	+	0.00103559i	$0.000329638\pi$
$757$			2.95045e17i			0.00207118i	−0.999999	+	0.00103559i	$0.999670\pi$
$758$		−	1.69535e20i		−	1.17917i
$759$	1.71228e19	−	5.52576e18i	0.118000	−	0.0380803i
$760$	−1.58885e20			−1.08489
$761$	−1.54129e20			−1.04278			−0.521390	−	0.853318i	$-0.674586\pi$
$761$	−1.54129e20			−1.04278			−0.521390	+	0.853318i	$0.674586\pi$
$762$	−1.35229e19			−0.0906538
$763$	−1.38092e20			−0.917270
$764$		−	1.38556e18i		−	0.00911950i
$765$	−8.73965e19			−0.569986
$766$			5.61817e19i			0.363073i
$767$	1.24671e19			0.0798361
$768$	−3.22054e18			−0.0204362
$769$		−	1.20311e20i		−	0.756520i	−0.925699	−	0.378260i	$-0.876523\pi$
$769$		−	1.20311e20i		−	0.756520i	0.925699	−	0.378260i	$-0.123477\pi$
$770$			4.12875e20i			2.57266i
$771$	−1.08373e19			−0.0669177
$772$	6.50147e18			0.0397824
$773$			1.13473e20i			0.688077i	0.938956	+	0.344038i	$0.111795\pi$
$773$			1.13473e20i			0.688077i	−0.938956	+	0.344038i	$0.888205\pi$
$774$			1.95938e19i			0.117742i
$775$	4.87709e20			2.90435
$776$			5.58596e19i			0.329659i
$777$	2.84661e18			0.0166487
$778$			6.12176e19i			0.354829i
$779$		−	1.86905e20i		−	1.07364i
$780$			3.40558e18i			0.0193878i
$781$			1.60753e20i			0.906986i
$782$	−6.17582e19	+	1.99303e19i	−0.345340	+	0.111446i
$783$	−7.16921e18			−0.0397319
$784$	2.36908e18			0.0130127
$785$	5.50547e20			2.99714
$786$	−8.45534e18			−0.0456219
$787$			2.46348e20i			1.31742i	0.752395	+	0.658712i	$0.228898\pi$
$787$			2.46348e20i			1.31742i	−0.752395	+	0.658712i	$0.771102\pi$
$788$	4.80626e18			0.0254756
$789$		−	9.53699e17i		−	0.00501040i
$790$	−4.40509e20			−2.29385
$791$	−1.58763e20			−0.819436
$792$			2.76734e20i			1.41575i
$793$		−	4.17210e20i		−	2.11564i
$794$	−2.89030e19			−0.145278
$795$	−1.13765e19			−0.0566814
$796$			2.64390e18i			0.0130573i
$797$			1.86159e20i			0.911330i	0.890151	+	0.455665i	$0.150598\pi$
$797$			1.86159e20i			0.911330i	−0.890151	+	0.455665i	$0.849402\pi$
$798$	1.23685e19			0.0600199
$799$		−	1.98826e19i		−	0.0956412i
$800$	5.90324e19			0.281488
$801$			1.55033e20i			0.732817i
$802$		−	1.12689e20i		−	0.528032i
$803$		−	4.45072e20i		−	2.06739i
$804$		−	2.16967e18i		−	0.00999083i
$805$	1.11385e20	+	3.45152e20i	0.508461	+	1.57557i
$806$	−6.71591e20			−3.03920
$807$	1.02732e19			0.0460885
$808$	2.43487e20			1.08292
$809$	9.11032e19			0.401693			0.200846	−	0.979623i	$-0.435631\pi$
$809$	9.11032e19			0.401693			0.200846	+	0.979623i	$0.435631\pi$
$810$		−	3.83633e20i		−	1.67695i
$811$	−1.94003e20			−0.840743			−0.420371	−	0.907352i	$-0.638100\pi$
$811$	−1.94003e20			−0.840743			−0.420371	+	0.907352i	$0.638100\pi$
$812$		−	4.64395e18i		−	0.0199524i
$813$	1.62185e19			0.0690841
$814$	−7.33458e19			−0.309745
$815$		−	5.17353e20i		−	2.16612i
$816$			7.50372e18i			0.0311491i
$817$	1.91209e19			0.0786964
$818$	4.46447e20			1.82178
$819$		−	4.24240e20i		−	1.71642i
$820$			5.26763e19i			0.211309i
$821$	3.98230e20			1.58392			0.791958	−	0.610575i	$-0.209062\pi$
$821$	3.98230e20			1.58392			0.791958	+	0.610575i	$0.209062\pi$
$822$			3.04316e19i			0.120011i
$823$	−4.41145e20			−1.72498			−0.862488	−	0.506078i	$-0.831095\pi$
$823$	−4.41145e20			−1.72498			−0.862488	+	0.506078i	$0.831095\pi$
$824$		−	3.82881e20i		−	1.48448i
$825$			5.50951e19i			0.211805i
$826$			1.27611e19i			0.0486437i
$827$		−	4.03979e20i		−	1.52693i	−0.645848	−	0.763466i	$-0.723496\pi$
$827$		−	4.03979e20i		−	1.52693i	0.645848	−	0.763466i	$-0.276504\pi$
$828$	−6.72182e18	−	2.08290e19i	−0.0251927	−	0.0780649i
$829$	−2.15075e20			−0.799296			−0.399648	−	0.916669i	$-0.630868\pi$
$829$	−2.15075e20			−0.799296			−0.399648	+	0.916669i	$0.630868\pi$
$830$	5.41885e20			1.99692
$831$	−2.22520e19			−0.0813133
$832$	4.28749e20			1.55360
$833$		−	1.17405e18i		−	0.00421863i
$834$	9.06395e18			0.0322965
$835$			7.47896e20i			2.64263i
$836$	−2.43146e19			−0.0851969
$837$	−8.09682e19			−0.281343
$838$			2.75515e20i			0.949374i
$839$			2.31692e20i			0.791730i	0.918309	+	0.395865i	$0.129555\pi$
$839$			2.31692e20i			0.791730i	−0.918309	+	0.395865i	$0.870445\pi$
$840$	3.87166e19			0.131203
$841$	−2.80403e20			−0.942347
$842$			1.44096e20i			0.480250i
$843$			1.90813e19i			0.0630689i
$844$	3.02250e19			0.0990760
$845$		−	9.87858e20i		−	3.21142i
$846$	8.78903e19			0.283366
$847$		−	3.87136e20i		−	1.23788i
$848$		−	1.40733e20i		−	0.446297i
$849$		−	2.95127e19i		−	0.0928230i
$850$		−	1.98716e20i		−	0.619871i
$851$	−6.13150e19	+	1.97872e19i	−0.189697	+	0.0612179i
$852$	−1.35722e18			−0.00416460
$853$	5.09647e20			1.55105			0.775525	−	0.631316i	$-0.217485\pi$
$853$	5.09647e20			1.55105			0.775525	+	0.631316i	$0.217485\pi$
$854$	4.27046e20			1.28905
$855$	−3.77010e20			−1.12873
$856$			9.58793e19i			0.284714i
$857$	−3.66446e20			−1.07931			−0.539653	−	0.841888i	$-0.681444\pi$
$857$	−3.66446e20			−1.07931			−0.539653	+	0.841888i	$0.681444\pi$
$858$		−	7.58677e19i		−	0.221639i
$859$	−3.94719e20			−1.14376			−0.571881	−	0.820337i	$-0.693786\pi$
$859$	−3.94719e20			−1.14376			−0.571881	+	0.820337i	$0.693786\pi$
$860$	−5.38896e18			−0.0154887
$861$			4.55443e19i			0.129841i
$862$		−	2.29334e20i		−	0.648511i
$863$	1.70493e20			0.478223			0.239112	−	0.970992i	$-0.423144\pi$
$863$	1.70493e20			0.478223			0.239112	+	0.970992i	$0.423144\pi$
$864$	−9.80040e18			−0.0272676
$865$			2.20470e20i			0.608466i
$866$			4.97820e20i			1.36285i
$867$	−2.68539e19			−0.0729244
$868$		−	5.24481e19i		−	0.141284i
$869$	7.48732e20			2.00073
$870$		−	1.28771e19i		−	0.0341337i
$871$			9.51867e20i			2.50293i
$872$		−	3.33653e20i		−	0.870321i
$873$			1.32547e20i			0.342980i
$874$	−2.66412e20	+	8.59751e19i	−0.683871	+	0.220695i
$875$	−4.60437e20			−1.17250
$876$	3.75771e18			0.00949280
$877$	2.51722e20			0.630846			0.315423	−	0.948951i	$-0.397854\pi$
$877$	2.51722e20			0.630846			0.315423	+	0.948951i	$0.397854\pi$
$878$	−4.40263e20			−1.09459
$879$			4.13407e19i			0.101966i
$880$	−1.08054e21			−2.64399
$881$			5.81902e20i			1.41260i	0.707913	+	0.706299i	$0.249637\pi$
$881$			5.81902e20i			1.41260i	−0.707913	+	0.706299i	$0.750363\pi$
$882$	5.18986e18			0.0124990
$883$	2.55346e20			0.610104			0.305052	−	0.952336i	$-0.401326\pi$
$883$	2.55346e20			0.610104			0.305052	+	0.952336i	$0.401326\pi$
$884$			2.08777e19i			0.0494899i
$885$			2.69975e18i			0.00634921i
$886$	−2.14610e20			−0.500742
$887$	4.73816e20			1.09684			0.548422	−	0.836202i	$-0.315229\pi$
$887$	4.73816e20			1.09684			0.548422	+	0.836202i	$0.315229\pi$
$888$			6.87787e18i			0.0157966i
$889$			4.63317e20i			1.05576i
$890$	−5.58861e20			−1.26350
$891$			6.52060e20i			1.46266i
$892$	−1.45908e19			−0.0324732
$893$		−	8.57693e19i		−	0.189396i
$894$		−	1.05086e19i		−	0.0230240i
$895$			4.88706e20i			1.06240i
$896$			5.15714e20i			1.11238i
$897$	−2.04676e19	−	6.34233e19i	−0.0438046	−	0.135738i
$898$	5.37884e20			1.14223
$899$	1.93748e20			0.408244
$900$	6.70205e19			0.140123
$901$	−6.97433e19			−0.144687
$902$		−	1.17349e21i		−	2.41566i
$903$	−4.65933e18			−0.00951721
$904$		−	3.83598e20i		−	0.777495i
$905$	−1.05172e21			−2.11525
$906$	3.10668e19			0.0620011
$907$			8.51038e19i			0.168538i	0.996443	+	0.0842690i	$0.0268555\pi$
$907$			8.51038e19i			0.168538i	−0.996443	+	0.0842690i	$0.973144\pi$
$908$			1.42603e19i			0.0280238i
$909$	5.77758e20			1.12667
$910$	1.52930e21			2.95940
$911$			6.74813e20i			1.29585i	0.761704	+	0.647925i	$0.224363\pi$
$911$			6.74813e20i			1.29585i	−0.761704	+	0.647925i	$0.775637\pi$
$912$			3.23695e19i			0.0616840i
$913$	−9.21041e20			−1.74174
$914$		−	5.24010e20i		−	0.983368i
$915$	9.03465e19			0.168253
$916$			4.02706e19i			0.0744251i
$917$			2.89693e20i			0.531316i
$918$			3.29904e19i			0.0600466i
$919$			5.39537e20i			0.974570i	0.873243	+	0.487285i	$0.162013\pi$
$919$			5.39537e20i			0.974570i	−0.873243	+	0.487285i	$0.837987\pi$
$920$	−8.33942e20	+	2.69125e20i	−1.49493	+	0.482436i
$921$	−5.91844e19			−0.105291
$922$	3.64695e20			0.643893
$923$	5.95433e20			1.04333
$924$	5.92491e18			0.0103033
$925$		−	1.97291e20i		−	0.340498i
$926$	2.30170e20			0.394251
$927$		−	9.08520e20i		−	1.54446i
$928$	2.34513e19			0.0395668
$929$	−1.08207e20			−0.181194			−0.0905971	−	0.995888i	$-0.528878\pi$
$929$	−1.08207e20			−0.181194			−0.0905971	+	0.995888i	$0.528878\pi$
$930$		−	1.45432e20i		−	0.241702i
$931$		−	5.06461e18i		−	0.00835408i
$932$	−3.94547e19			−0.0645933
$933$	1.93990e19			0.0315215
$934$		−	5.36370e19i		−	0.0865040i
$935$			5.35484e20i			0.857167i
$936$	1.02503e21			1.62857
$937$			4.83695e20i			0.762772i	0.924416	+	0.381386i	$0.124553\pi$
$937$			4.83695e20i			0.762772i	−0.924416	+	0.381386i	$0.875447\pi$
$938$	−9.74307e20			−1.52502
$939$		−	8.91520e19i		−	0.138507i
$940$			2.41728e19i			0.0372763i
$941$		−	6.96586e20i		−	1.06622i	−0.846045	−	0.533111i	$-0.821023\pi$
$941$		−	6.96586e20i		−	1.06622i	0.846045	−	0.533111i	$-0.178977\pi$
$942$		−	1.03551e20i		−	0.157325i
$943$	−3.16586e20	−	9.81009e20i	−0.477430	−	1.47942i
$944$	−3.33970e19			−0.0499924
$945$	1.84375e20			0.273955
$946$	1.20052e20			0.177065
$947$	−3.16102e20			−0.462783			−0.231391	−	0.972861i	$-0.574328\pi$
$947$	−3.16102e20			−0.462783			−0.231391	+	0.972861i	$0.574328\pi$
$948$			6.32149e18i			0.00918673i
$949$	−1.64856e21			−2.37816
$950$		−	8.57222e20i		−	1.22752i
$951$	−7.67559e19			−0.109106
$952$	2.37350e20			0.334912
$953$		−	7.22634e20i		−	1.01220i	−0.862474	−	0.506102i	$-0.831086\pi$
$953$		−	7.22634e20i		−	1.01220i	0.862474	−	0.506102i	$-0.168914\pi$
$954$		−	3.08298e20i		−	0.428679i
$955$	−1.31632e20			−0.181694
$956$	−2.00244e19			−0.0274382
$957$			2.18872e19i			0.0297719i
$958$		−	1.00001e21i		−	1.35034i
$959$	1.04263e21			1.39766
$960$			9.28451e19i			0.123555i
$961$	1.43122e21			1.89079
$962$			2.71675e20i			0.356307i
$963$			2.27507e20i			0.296218i
$964$		−	9.54178e19i		−	0.123336i
$965$		−	6.17661e20i		−	0.792609i
$966$	6.49185e19	−	2.09501e19i	0.0827044	−	0.0266899i
$967$	−8.98896e20			−1.13690			−0.568452	−	0.822716i	$-0.692458\pi$
$967$	−8.98896e20			−1.13690			−0.568452	+	0.822716i	$0.692458\pi$
$968$	9.35382e20			1.17452
$969$	1.60415e19			0.0199976
$970$	−4.77804e20			−0.591354
$971$			9.55081e20i			1.17356i	0.809746	+	0.586781i	$0.199605\pi$
$971$			9.55081e20i			1.17356i	−0.809746	+	0.586781i	$0.800395\pi$
$972$	−1.67091e19			−0.0203840
$973$		−	3.10545e20i		−	0.376128i
$974$	2.83888e20			0.341377
$975$	2.04074e20			0.243644
$976$			1.11762e21i			1.32479i
$977$			7.70929e20i			0.907303i	0.891179	+	0.453651i	$0.149879\pi$
$977$			7.70929e20i			0.907303i	−0.891179	+	0.453651i	$0.850121\pi$
$978$	−9.73073e19			−0.113703
$979$	9.49895e20			1.10204
$980$			1.42739e18i			0.00164422i
$981$		−	7.91709e20i		−	0.905487i
$982$	1.74000e21			1.97591
$983$			2.43846e20i			0.274942i	0.990506	+	0.137471i	$0.0438973\pi$
$983$			2.43846e20i			0.274942i	−0.990506	+	0.137471i	$0.956103\pi$
$984$	−1.10042e20			−0.123195
$985$		−	4.56610e20i		−	0.507565i
$986$		−	7.89423e19i		−	0.0871308i
$987$			2.09000e19i			0.0229048i
$988$			9.00621e19i			0.0980040i
$989$	1.00360e20	−	3.23878e19i	0.108440	−	0.0349951i
$990$	−2.36709e21			−2.53962
$991$	−2.48884e20			−0.265143			−0.132572	−	0.991173i	$-0.542323\pi$
$991$	−2.48884e20			−0.265143			−0.132572	+	0.991173i	$0.542323\pi$
$992$	2.64856e20			0.280174
$993$	−5.32357e19			−0.0559188
$994$			6.09470e20i			0.635694i
$995$	2.51179e20			0.260148
$996$		−	7.77627e18i		−	0.00799753i
$997$	6.98572e20			0.713420			0.356710	−	0.934215i	$-0.383898\pi$
$997$	6.98572e20			0.713420			0.356710	+	0.934215i	$0.383898\pi$
$998$	9.12587e19			0.0925466
$999$			3.27536e19i			0.0329838i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.b.b.22.18	yes	24
23.22	odd	2	inner	23.15.b.b.22.17	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.b.b.22.17	✓	24	23.22	odd	2	inner
23.15.b.b.22.18	yes	24	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 23 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	23.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+133.181 q^{2} -181.571 q^{3} +1353.30 q^{4} -128567. i q^{5} -24181.9 q^{6} +828509. i q^{7} -2.00181e6 q^{8} -4.75000e6 q^{9} +O(q^{10})\)	\(q+133.181 q^{2} -181.571 q^{3} +1353.30 q^{4} -128567. i q^{5} -24181.9 q^{6} +828509. i q^{7} -2.00181e6 q^{8} -4.75000e6 q^{9} -1.71228e7i q^{10} +2.91036e7i q^{11} -245719. q^{12} +1.07801e8 q^{13} +1.10342e8i q^{14} +2.33441e7i q^{15} -2.88776e8 q^{16} +1.43110e8i q^{17} -6.32612e8 q^{18} +6.17346e8i q^{19} -1.73990e8i q^{20} -1.50433e8i q^{21} +3.87606e9i q^{22} +(1.04568e9 + 3.24028e9i) q^{23} +3.63470e8 q^{24} -1.04261e10 q^{25} +1.43570e10 q^{26} +1.73091e9 q^{27} +1.12122e9i q^{28} -4.14188e9 q^{29} +3.10900e9i q^{30} -4.67778e10 q^{31} -5.66199e9 q^{32} -5.28436e9i q^{33} +1.90596e10i q^{34} +1.06519e11 q^{35} -6.42816e9 q^{36} +1.89228e10i q^{37} +8.22190e10i q^{38} -1.95734e10 q^{39} +2.57368e11i q^{40} -3.02755e11 q^{41} -2.00349e10i q^{42} -3.09728e10i q^{43} +3.93857e10i q^{44} +6.10696e11i q^{45} +(1.39266e11 + 4.31545e11i) q^{46} -1.38932e11 q^{47} +5.24334e10 q^{48} -8.20385e9 q^{49} -1.38856e12 q^{50} -2.59845e10i q^{51} +1.45886e11 q^{52} +4.87341e11i q^{53} +2.30525e11 q^{54} +3.74177e12 q^{55} -1.65852e12i q^{56} -1.12092e11i q^{57} -5.51621e11 q^{58} +1.15650e11 q^{59} +3.15915e10i q^{60} -3.87021e12i q^{61} -6.22994e12 q^{62} -3.93542e12i q^{63} +3.97724e12 q^{64} -1.38596e13i q^{65} -7.03778e11i q^{66} +8.82989e12i q^{67} +1.93670e11i q^{68} +(-1.89866e11 - 5.88339e11i) q^{69} +1.41864e13 q^{70} +5.52347e12 q^{71} +9.50860e12 q^{72} -1.52927e13 q^{73} +2.52016e12i q^{74} +1.89307e12 q^{75} +8.35452e11i q^{76} -2.41126e13 q^{77} -2.60682e12 q^{78} -2.57265e13i q^{79} +3.71273e13i q^{80} +2.24048e13 q^{81} -4.03213e13 q^{82} +3.16470e13i q^{83} -2.03580e11i q^{84} +1.83993e13 q^{85} -4.12501e12i q^{86} +7.52044e11 q^{87} -5.82598e13i q^{88} -3.26384e13i q^{89} +8.13333e13i q^{90} +8.93137e13i q^{91} +(1.41512e12 + 4.38505e12i) q^{92} +8.49349e12 q^{93} -1.85032e13 q^{94} +7.93706e13 q^{95} +1.02805e12 q^{96} -2.79045e13i q^{97} -1.09260e12 q^{98} -1.38242e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) \) 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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