LMFDB - Embedded newform 23.15.b.b.22.16

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			22.16
Character	$\chi$	$=$	23.22
Dual form			23.15.b.b.22.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+51.5135 q^{2} -1848.44 q^{3} -13730.4 q^{4} +58352.1i q^{5} -95219.8 q^{6} +395426. i q^{7} -1.55130e6 q^{8} -1.36623e6 q^{9} +O(q^{10})$	\(q+51.5135 q^{2} -1848.44 q^{3} -13730.4 q^{4} +58352.1i q^{5} -95219.8 q^{6} +395426. i q^{7} -1.55130e6 q^{8} -1.36623e6 q^{9} +3.00593e6i q^{10} +2.92117e7i q^{11} +2.53798e7 q^{12} -4.65295e7 q^{13} +2.03698e7i q^{14} -1.07861e8i q^{15} +1.45045e8 q^{16} +5.78305e7i q^{17} -7.03795e7 q^{18} -9.18067e8i q^{19} -8.01196e8i q^{20} -7.30922e8i q^{21} +1.50480e9i q^{22} +(-9.21587e8 - 3.27773e9i) q^{23} +2.86748e9 q^{24} +2.69854e9 q^{25} -2.39690e9 q^{26} +1.13664e10 q^{27} -5.42934e9i q^{28} -2.78579e10 q^{29} -5.55628e9i q^{30} +4.07378e8 q^{31} +3.28883e10 q^{32} -5.39962e10i q^{33} +2.97905e9i q^{34} -2.30739e10 q^{35} +1.87589e10 q^{36} -5.76530e9i q^{37} -4.72929e10i q^{38} +8.60070e10 q^{39} -9.05215e10i q^{40} +1.12103e11 q^{41} -3.76524e10i q^{42} -4.11790e11i q^{43} -4.01088e11i q^{44} -7.97226e10i q^{45} +(-4.74742e10 - 1.68848e11i) q^{46} -1.71340e11 q^{47} -2.68108e11 q^{48} +5.21862e11 q^{49} +1.39012e11 q^{50} -1.06896e11i q^{51} +6.38866e11 q^{52} +1.58400e12i q^{53} +5.85526e11 q^{54} -1.70457e12 q^{55} -6.13423e11i q^{56} +1.69699e12i q^{57} -1.43506e12 q^{58} +2.74875e11 q^{59} +1.48096e12i q^{60} +1.67510e12i q^{61} +2.09855e10 q^{62} -5.40243e11i q^{63} -6.82231e11 q^{64} -2.71509e12i q^{65} -2.78154e12i q^{66} +1.11743e13i q^{67} -7.94033e11i q^{68} +(1.70350e12 + 6.05869e12i) q^{69} -1.18862e12 q^{70} -1.02333e13 q^{71} +2.11943e12 q^{72} +1.41346e13 q^{73} -2.96991e11i q^{74} -4.98810e12 q^{75} +1.26054e13i q^{76} -1.15511e13 q^{77} +4.43053e12 q^{78} +4.77455e12i q^{79} +8.46370e12i q^{80} -1.44756e13 q^{81} +5.77483e12 q^{82} -3.81107e13i q^{83} +1.00358e13i q^{84} -3.37453e12 q^{85} -2.12128e13i q^{86} +5.14937e13 q^{87} -4.53161e13i q^{88} -6.55570e13i q^{89} -4.10679e12i q^{90} -1.83990e13i q^{91} +(1.26537e13 + 4.50044e13i) q^{92} -7.53015e11 q^{93} -8.82632e12 q^{94} +5.35712e13 q^{95} -6.07920e13 q^{96} -1.46266e13i q^{97} +2.68829e13 q^{98} -3.99100e13i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	51.5135			0.402450			0.201225	−	0.979545i	$-0.435508\pi$
$2$	51.5135			0.402450			0.201225	+	0.979545i	$0.435508\pi$
$3$	−1848.44			−0.845195			−0.422598	−	0.906317i	$-0.638882\pi$
$3$	−1848.44			−0.845195			−0.422598	+	0.906317i	$0.638882\pi$
$4$	−13730.4			−0.838034
$5$			58352.1i			0.746907i	0.927649	+	0.373454i	$0.121827\pi$
$5$			58352.1i			0.746907i	−0.927649	+	0.373454i	$0.878173\pi$
$6$	−95219.8			−0.340148
$7$			395426.i			0.480152i	0.970754	+	0.240076i	$0.0771724\pi$
$7$			395426.i			0.480152i	−0.970754	+	0.240076i	$0.922828\pi$
$8$	−1.55130e6			−0.739716
$9$	−1.36623e6			−0.285645
$10$			3.00593e6i			0.300593i
$11$			2.92117e7i			1.49902i	0.661991	+	0.749512i	$0.269712\pi$
$11$			2.92117e7i			1.49902i	−0.661991	+	0.749512i	$0.730288\pi$
$12$	2.53798e7			0.708303
$13$	−4.65295e7			−0.741523			−0.370762	−	0.928728i	$-0.620903\pi$
$13$	−4.65295e7			−0.741523			−0.370762	+	0.928728i	$0.620903\pi$
$14$			2.03698e7i			0.193237i
$15$		−	1.07861e8i		−	0.631283i
$16$	1.45045e8			0.540336
$17$			5.78305e7i			0.140934i	0.997514	+	0.0704668i	$0.0224489\pi$
$17$			5.78305e7i			0.140934i	−0.997514	+	0.0704668i	$0.977551\pi$
$18$	−7.03795e7			−0.114958
$19$		−	9.18067e8i		−	1.02707i	−0.858069	−	0.513534i	$-0.828336\pi$
$19$		−	9.18067e8i		−	1.02707i	0.858069	−	0.513534i	$-0.171664\pi$
$20$		−	8.01196e8i		−	0.625934i
$21$		−	7.30922e8i		−	0.405822i
$22$			1.50480e9i			0.603282i
$23$	−9.21587e8	−	3.27773e9i	−0.270671	−	0.962672i
$23$	−9.21587e8	−	3.27773e9i	−0.270671	−	0.962672i
$24$	2.86748e9			0.625205
$25$	2.69854e9			0.442129
$26$	−2.39690e9			−0.298426
$27$	1.13664e10			1.08662
$28$		−	5.42934e9i		−	0.402384i
$29$	−2.78579e10			−1.61496			−0.807481	−	0.589893i	$-0.799170\pi$
$29$	−2.78579e10			−1.61496			−0.807481	+	0.589893i	$0.799170\pi$
$30$		−	5.55628e9i		−	0.254059i
$31$	4.07378e8			0.0148070			0.00740348	−	0.999973i	$-0.497643\pi$
$31$	4.07378e8			0.0148070			0.00740348	+	0.999973i	$0.497643\pi$
$32$	3.28883e10			0.957174
$33$		−	5.39962e10i		−	1.26697i
$34$			2.97905e9i			0.0567187i
$35$	−2.30739e10			−0.358629
$36$	1.87589e10			0.239380
$37$		−	5.76530e9i		−	0.0607310i	−0.999539	−	0.0303655i	$-0.990333\pi$
$37$		−	5.76530e9i		−	0.0607310i	0.999539	−	0.0303655i	$-0.00966712\pi$
$38$		−	4.72929e10i		−	0.413343i
$39$	8.60070e10			0.626732
$40$		−	9.05215e10i		−	0.552500i
$41$	1.12103e11			0.575613			0.287807	−	0.957689i	$-0.407074\pi$
$41$	1.12103e11			0.575613			0.287807	+	0.957689i	$0.407074\pi$
$42$		−	3.76524e10i		−	0.163323i
$43$		−	4.11790e11i		−	1.51494i	−0.652868	−	0.757472i	$-0.726434\pi$
$43$		−	4.11790e11i		−	1.51494i	0.652868	−	0.757472i	$-0.273566\pi$
$44$		−	4.01088e11i		−	1.25623i
$45$		−	7.97226e10i		−	0.213351i
$46$	−4.74742e10	−	1.68848e11i	−0.108931	−	0.387427i
$47$	−1.71340e11			−0.338200			−0.169100	−	0.985599i	$-0.554086\pi$
$47$	−1.71340e11			−0.338200			−0.169100	+	0.985599i	$0.554086\pi$
$48$	−2.68108e11			−0.456689
$49$	5.21862e11			0.769454
$50$	1.39012e11			0.177935
$51$		−	1.06896e11i		−	0.119116i
$52$	6.38866e11			0.621422
$53$			1.58400e12i			1.34842i	0.738540	+	0.674209i	$0.235515\pi$
$53$			1.58400e12i			1.34842i	−0.738540	+	0.674209i	$0.764485\pi$
$54$	5.85526e11			0.437310
$55$	−1.70457e12			−1.11963
$56$		−	6.13423e11i		−	0.355176i
$57$			1.69699e12i			0.868073i
$58$	−1.43506e12			−0.649941
$59$	2.74875e11			0.110451			0.0552257	−	0.998474i	$-0.482412\pi$
$59$	2.74875e11			0.110451			0.0552257	+	0.998474i	$0.482412\pi$
$60$			1.48096e12i			0.529036i
$61$			1.67510e12i			0.533007i	0.963834	+	0.266504i	$0.0858684\pi$
$61$			1.67510e12i			0.533007i	−0.963834	+	0.266504i	$0.914132\pi$
$62$	2.09855e10			0.00595906
$63$		−	5.40243e11i		−	0.137153i
$64$	−6.82231e11			−0.155121
$65$		−	2.71509e12i		−	0.553849i
$66$		−	2.78154e12i		−	0.509891i
$67$			1.11743e13i			1.84372i	0.387522	+	0.921861i	$0.373331\pi$
$67$			1.11743e13i			1.84372i	−0.387522	+	0.921861i	$0.626669\pi$
$68$		−	7.94033e11i		−	0.118107i
$69$	1.70350e12	+	6.05869e12i	0.228770	+	0.813646i
$70$	−1.18862e12			−0.144330
$71$	−1.02333e13			−1.12514			−0.562568	−	0.826751i	$-0.690187\pi$
$71$	−1.02333e13			−1.12514			−0.562568	+	0.826751i	$0.690187\pi$
$72$	2.11943e12			0.211296
$73$	1.41346e13			1.27945			0.639727	−	0.768603i	$-0.279048\pi$
$73$	1.41346e13			1.27945			0.639727	+	0.768603i	$0.279048\pi$
$74$		−	2.96991e11i		−	0.0244412i
$75$	−4.98810e12			−0.373686
$76$			1.26054e13i			0.860718i
$77$	−1.15511e13			−0.719759
$78$	4.43053e12			0.252228
$79$			4.77455e12i			0.248624i	0.992243	+	0.124312i	$0.0396723\pi$
$79$			4.77455e12i			0.248624i	−0.992243	+	0.124312i	$0.960328\pi$
$80$			8.46370e12i			0.403581i
$81$	−1.44756e13			−0.632762
$82$	5.77483e12			0.231655
$83$		−	3.81107e13i		−	1.40443i	−0.711965	−	0.702215i	$-0.752195\pi$
$83$		−	3.81107e13i		−	1.40443i	0.711965	−	0.702215i	$-0.247805\pi$
$84$			1.00358e13i			0.340093i
$85$	−3.37453e12			−0.105264
$86$		−	2.12128e13i		−	0.609689i
$87$	5.14937e13			1.36496
$88$		−	4.53161e13i		−	1.10885i
$89$		−	6.55570e13i		−	1.48214i	−0.671428	−	0.741070i	$-0.734319\pi$
$89$		−	6.55570e13i		−	1.48214i	0.671428	−	0.741070i	$-0.265681\pi$
$90$		−	4.10679e12i		−	0.0858628i
$91$		−	1.83990e13i		−	0.356044i
$92$	1.26537e13	+	4.50044e13i	0.226832	+	0.806752i
$93$	−7.53015e11			−0.0125148
$94$	−8.82632e12			−0.136108
$95$	5.35712e13			0.767124
$96$	−6.07920e13			−0.808999
$97$		−	1.46266e13i		−	0.181026i	−0.995895	−	0.0905130i	$-0.971149\pi$
$97$		−	1.46266e13i		−	0.181026i	0.995895	−	0.0905130i	$-0.0288507\pi$
$98$	2.68829e13			0.309667
$99$		−	3.99100e13i		−	0.428189i
$100$	−3.70520e13			−0.370520
$101$	−4.23916e13			−0.395394			−0.197697	−	0.980263i	$-0.563346\pi$
$101$	−4.23916e13			−0.395394			−0.197697	+	0.980263i	$0.563346\pi$
$102$		−	5.50661e12i		−	0.0479383i
$103$			1.11919e14i			0.910001i	0.890491	+	0.455001i	$0.150361\pi$
$103$			1.11919e14i			0.910001i	−0.890491	+	0.455001i	$0.849639\pi$
$104$	7.21810e13			0.548517
$105$	4.26508e13			0.303112
$106$			8.15976e13i			0.542671i
$107$		−	1.66466e13i		−	0.103667i	−0.998656	−	0.0518333i	$-0.983494\pi$
$107$		−	1.66466e13i		−	0.103667i	0.998656	−	0.0518333i	$-0.0165065\pi$
$108$	−1.56065e14			−0.910626
$109$		−	1.13337e14i		−	0.619991i	−0.950738	−	0.309995i	$-0.899672\pi$
$109$		−	1.13337e14i		−	0.619991i	0.950738	−	0.309995i	$-0.100328\pi$
$110$	−8.78083e13			−0.450596
$111$			1.06568e13i			0.0513295i
$112$			5.73546e13i			0.259443i
$113$		−	1.91598e14i		−	0.814407i	−0.913337	−	0.407204i	$-0.866504\pi$
$113$		−	1.91598e14i		−	0.814407i	0.913337	−	0.407204i	$-0.133496\pi$
$114$			8.74181e13i			0.349355i
$115$	1.91263e14	−	5.37766e13i	0.719027	−	0.202166i
$116$	3.82499e14			1.35339
$117$	6.35701e13			0.211813
$118$	1.41598e13			0.0444511
$119$	−2.28677e13			−0.0676695
$120$			1.67324e14i			0.466970i
$121$	−4.73576e14			−1.24707
$122$			8.62906e13i			0.214508i
$123$	−2.07216e14			−0.486505
$124$	−5.59345e12			−0.0124087
$125$			5.13619e14i			1.07714i
$126$		−	2.78299e13i		−	0.0551972i
$127$	−1.99589e14			−0.374551			−0.187276	−	0.982307i	$-0.559966\pi$
$127$	−1.99589e14			−0.374551			−0.187276	+	0.982307i	$0.559966\pi$
$128$	−5.73985e14			−1.01960
$129$			7.61170e14i			1.28042i
$130$		−	1.39864e14i		−	0.222896i
$131$	1.93285e14			0.291944			0.145972	−	0.989289i	$-0.453369\pi$
$131$	1.93285e14			0.291944			0.145972	+	0.989289i	$0.453369\pi$
$132$			7.41387e14i			1.06176i
$133$	3.63027e14			0.493149
$134$			5.75626e14i			0.742005i
$135$			6.63256e14i			0.811605i
$136$		−	8.97123e13i		−	0.104251i
$137$		−	1.27176e15i		−	1.40398i	−0.712184	−	0.701992i	$-0.752294\pi$
$137$		−	1.27176e15i		−	1.40398i	0.712184	−	0.701992i	$-0.247706\pi$
$138$	8.77534e13	+	3.12105e14i	0.0920683	+	0.327451i
$139$	2.69451e14			0.268767			0.134384	−	0.990929i	$-0.457095\pi$
$139$	2.69451e14			0.268767			0.134384	+	0.990929i	$0.457095\pi$
$140$	3.16813e14			0.300543
$141$	3.16711e14			0.285845
$142$	−5.27151e14			−0.452811
$143$		−	1.35921e15i		−	1.11156i
$144$	−1.98166e14			−0.154344
$145$		−	1.62557e15i		−	1.20623i
$146$	7.28125e14			0.514915
$147$	−9.64631e14			−0.650339
$148$			7.91597e13i			0.0508946i
$149$		−	2.52328e15i		−	1.54761i	−0.633426	−	0.773804i	$-0.718352\pi$
$149$		−	2.52328e15i		−	1.54761i	0.633426	−	0.773804i	$-0.281648\pi$
$150$	−2.56955e14			−0.150390
$151$	2.27828e15			1.27282			0.636411	−	0.771350i	$-0.280418\pi$
$151$	2.27828e15			1.27282			0.636411	+	0.771350i	$0.280418\pi$
$152$			1.42419e15i			0.759739i
$153$		−	7.90099e13i		−	0.0402570i
$154$	−5.95037e14			−0.289667
$155$			2.37714e13i			0.0110594i
$156$	−1.18091e15			−0.525223
$157$		−	3.67037e15i		−	1.56103i	−0.625136	−	0.780516i	$-0.714956\pi$
$157$		−	3.67037e15i		−	1.56103i	0.625136	−	0.780516i	$-0.285044\pi$
$158$			2.45954e14i			0.100058i
$159$		−	2.92794e15i		−	1.13968i
$160$			1.91910e15i			0.714920i
$161$	1.29610e15	−	3.64419e14i	0.462229	−	0.129963i
$162$	−7.45687e14			−0.254655
$163$	1.79033e15			0.585625			0.292813	−	0.956170i	$-0.405409\pi$
$163$	1.79033e15			0.585625			0.292813	+	0.956170i	$0.405409\pi$
$164$	−1.53922e15			−0.482384
$165$	3.15079e15			0.946308
$166$		−	1.96322e15i		−	0.565212i
$167$	2.81573e15			0.777276			0.388638	−	0.921391i	$-0.372946\pi$
$167$	2.81573e15			0.777276			0.388638	+	0.921391i	$0.372946\pi$
$168$			1.13388e15i			0.300193i
$169$	−1.77238e15			−0.450144
$170$	−1.73834e14			−0.0423636
$171$			1.25429e15i			0.293377i
$172$			5.65402e15i			1.26958i
$173$	−7.29859e15			−1.57368			−0.786840	−	0.617157i	$-0.788284\pi$
$173$	−7.29859e15			−1.57368			−0.786840	+	0.617157i	$0.788284\pi$
$174$	2.65262e15			0.549327
$175$			1.06707e15i			0.212289i
$176$			4.23703e15i			0.809977i
$177$	−5.08090e14			−0.0933529
$178$		−	3.37708e15i		−	0.596487i
$179$	−5.39593e15			−0.916421			−0.458210	−	0.888844i	$-0.651509\pi$
$179$	−5.39593e15			−0.916421			−0.458210	+	0.888844i	$0.651509\pi$
$180$			1.09462e15i			0.178795i
$181$			1.42263e15i			0.223534i	0.993734	+	0.111767i	$0.0356510\pi$
$181$			1.42263e15i			0.223534i	−0.993734	+	0.111767i	$0.964349\pi$
$182$		−	9.47795e14i		−	0.143290i
$183$		−	3.09633e15i		−	0.450495i
$184$	1.42966e15	+	5.08473e15i	0.200220	+	0.712104i
$185$	3.36418e14			0.0453604
$186$	−3.87905e13			−0.00503657
$187$	−1.68933e15			−0.211263
$188$	2.35255e15			0.283423
$189$			4.49458e15i			0.521743i
$190$	2.75964e15			0.308729
$191$			1.00023e16i			1.07862i	0.842108	+	0.539309i	$0.181314\pi$
$191$			1.00023e16i			1.07862i	−0.842108	+	0.539309i	$0.818686\pi$
$192$	1.26107e15			0.131108
$193$	5.77968e15			0.579433			0.289716	−	0.957113i	$-0.406439\pi$
$193$	5.77968e15			0.579433			0.289716	+	0.957113i	$0.406439\pi$
$194$		−	7.53468e14i		−	0.0728538i
$195$			5.01869e15i			0.468111i
$196$	−7.16534e15			−0.644829
$197$	−3.72545e15			−0.323530			−0.161765	−	0.986829i	$-0.551719\pi$
$197$	−3.72545e15			−0.323530			−0.161765	+	0.986829i	$0.551719\pi$
$198$		−	2.05591e15i		−	0.172325i
$199$			1.29196e16i			1.04539i	0.852521	+	0.522693i	$0.175073\pi$
$199$			1.29196e16i			1.04539i	−0.852521	+	0.522693i	$0.824927\pi$
$200$	−4.18624e15			−0.327050
$201$		−	2.06550e16i		−	1.55830i
$202$	−2.18374e15			−0.159126
$203$		−	1.10157e16i		−	0.775428i
$204$			1.46772e15i			0.0998236i
$205$			6.54146e15i			0.429930i
$206$			5.76533e15i			0.366230i
$207$	1.25910e15	+	4.47814e15i	0.0773159	+	0.274983i
$208$	−6.74888e15			−0.400671
$209$	2.68183e16			1.53960
$210$	2.19710e15			0.121987
$211$	−3.02530e16			−1.62477			−0.812383	−	0.583124i	$-0.801830\pi$
$211$	−3.02530e16			−1.62477			−0.812383	+	0.583124i	$0.801830\pi$
$212$		−	2.17489e16i		−	1.13002i
$213$	1.89156e16			0.950960
$214$		−	8.57525e14i		−	0.0417206i
$215$	2.40288e16			1.13152
$216$	−1.76327e16			−0.803791
$217$			1.61088e14i			0.00710959i
$218$		−	5.83838e15i		−	0.249515i
$219$	−2.61270e16			−1.08139
$220$	2.34043e16			0.938290
$221$		−	2.69082e15i		−	0.104506i
$222$			5.48971e14i			0.0206575i
$223$	−1.84605e16			−0.673147			−0.336574	−	0.941657i	$-0.609268\pi$
$223$	−1.84605e16			−0.673147			−0.336574	+	0.941657i	$0.609268\pi$
$224$			1.30049e16i			0.459589i
$225$	−3.68684e15			−0.126292
$226$		−	9.86989e15i		−	0.327758i
$227$			1.41561e15i			0.0455789i	0.999740	+	0.0227894i	$0.00725473\pi$
$227$			1.41561e15i			0.0455789i	−0.999740	+	0.0227894i	$0.992745\pi$
$228$		−	2.33003e16i		−	0.727475i
$229$		−	1.76804e16i		−	0.535357i	−0.963508	−	0.267678i	$-0.913744\pi$
$229$		−	1.76804e16i		−	0.535357i	0.963508	−	0.267678i	$-0.0862565\pi$
$230$	9.85261e15	−	2.77022e15i	0.289372	−	0.0813617i
$231$	2.13515e16			0.608337
$232$	4.32159e16			1.19461
$233$	1.43436e16			0.384740			0.192370	−	0.981322i	$-0.438383\pi$
$233$	1.43436e16			0.384740			0.192370	+	0.981322i	$0.438383\pi$
$234$	3.27472e15			0.0852439
$235$		−	9.99804e15i		−	0.252604i
$236$	−3.77413e15			−0.0925620
$237$		−	8.82547e15i		−	0.210136i
$238$	−1.17799e15			−0.0272336
$239$	−3.43196e16			−0.770471			−0.385236	−	0.922818i	$-0.625880\pi$
$239$	−3.43196e16			−0.770471			−0.385236	+	0.922818i	$0.625880\pi$
$240$		−	1.56447e16i		−	0.341105i
$241$		−	7.66060e16i		−	1.62235i	−0.584806	−	0.811173i	$-0.698829\pi$
$241$		−	7.66060e16i		−	1.62235i	0.584806	−	0.811173i	$-0.301171\pi$
$242$	−2.43956e16			−0.501884
$243$	−2.76081e16			−0.551814
$244$		−	2.29998e16i		−	0.446678i
$245$			3.04517e16i			0.574711i
$246$	−1.06744e16			−0.195794
$247$			4.27172e16i			0.761594i
$248$	−6.31965e14			−0.0109530
$249$			7.04454e16i			1.18702i
$250$			2.64583e16i			0.433493i
$251$			2.78858e15i			0.0444290i	0.999753	+	0.0222145i	$0.00707168\pi$
$251$			2.78858e15i			0.0444290i	−0.999753	+	0.0222145i	$0.992928\pi$
$252$			7.41773e15i			0.114939i
$253$	9.57482e16	−	2.69212e16i	1.44307	−	0.405742i
$254$	−1.02816e16			−0.150738
$255$	6.23763e15			0.0889689
$256$	−1.83903e16			−0.255217
$257$	−5.69895e16			−0.769596			−0.384798	−	0.923001i	$-0.625729\pi$
$257$	−5.69895e16			−0.769596			−0.384798	+	0.923001i	$0.625729\pi$
$258$			3.92106e16i			0.515306i
$259$	2.27975e15			0.0291601
$260$			3.72792e16i			0.464145i
$261$	3.80604e16			0.461306
$262$	9.95681e15			0.117493
$263$			5.99737e16i			0.689081i	0.938771	+	0.344541i	$0.111965\pi$
$263$			5.99737e16i			0.689081i	−0.938771	+	0.344541i	$0.888035\pi$
$264$			8.37642e16i			0.937197i
$265$	−9.24299e16			−1.00714
$266$	1.87008e16			0.198467
$267$			1.21178e17i			1.25270i
$268$		−	1.53427e17i		−	1.54510i
$269$	−1.44260e17			−1.41541			−0.707703	−	0.706510i	$-0.750269\pi$
$269$	−1.44260e17			−1.41541			−0.707703	+	0.706510i	$0.750269\pi$
$270$			3.41667e16i			0.326630i
$271$	2.02626e17			1.88761			0.943803	−	0.330510i	$-0.107221\pi$
$271$	2.02626e17			1.88761			0.943803	+	0.330510i	$0.107221\pi$
$272$			8.38804e15i			0.0761515i
$273$			3.40094e16i			0.300926i
$274$		−	6.55131e16i		−	0.565033i
$275$			7.88291e16i			0.662763i
$276$	−2.33897e16	−	8.31880e16i	−0.191717	−	0.681863i
$277$	−1.70502e17			−1.36261			−0.681303	−	0.732002i	$-0.738586\pi$
$277$	−1.70502e17			−1.36261			−0.681303	+	0.732002i	$0.738586\pi$
$278$	1.38804e16			0.108165
$279$	−5.56573e14			−0.00422954
$280$	3.57945e16			0.265284
$281$			1.34474e17i			0.972062i	0.873942	+	0.486031i	$0.161556\pi$
$281$			1.34474e17i			0.972062i	−0.873942	+	0.486031i	$0.838444\pi$
$282$	1.63149e16			0.115038
$283$		−	1.59614e17i		−	1.09791i	−0.835853	−	0.548953i	$-0.815026\pi$
$283$		−	1.59614e17i		−	1.09791i	0.835853	−	0.548953i	$-0.184974\pi$
$284$	1.40506e17			0.942903
$285$	−9.90232e16			−0.648370
$286$		−	7.00176e16i		−	0.447347i
$287$			4.43285e16i			0.276382i
$288$	−4.49330e16			−0.273412
$289$	1.65033e17			0.980138
$290$		−	8.37388e16i		−	0.485446i
$291$			2.70364e16i			0.153002i
$292$	−1.94073e17			−1.07223
$293$		−	6.53828e16i		−	0.352688i	−0.984329	−	0.176344i	$-0.943573\pi$
$293$		−	6.53828e16i		−	0.352688i	0.984329	−	0.176344i	$-0.0564271\pi$
$294$	−4.96915e16			−0.261729
$295$			1.60395e16i			0.0824969i
$296$			8.94370e15i			0.0449237i
$297$			3.32034e17i			1.62887i
$298$		−	1.29983e17i		−	0.622834i
$299$	4.28810e16	+	1.52511e17i	0.200709	+	0.713843i
$300$	6.84884e16			0.313161
$301$	1.62832e17			0.727403
$302$	1.17362e17			0.512247
$303$	7.83584e16			0.334185
$304$		−	1.33161e17i		−	0.554961i
$305$	−9.77459e16			−0.398107
$306$		−	4.07008e15i		−	0.0162014i
$307$	7.63833e16			0.297187			0.148594	−	0.988898i	$-0.452525\pi$
$307$	7.63833e16			0.297187			0.148594	+	0.988898i	$0.452525\pi$
$308$	1.58600e17			0.603183
$309$		−	2.06875e17i		−	0.769128i
$310$			1.22455e15i			0.00445086i
$311$	−5.15827e17			−1.83308			−0.916540	−	0.399944i	$-0.869030\pi$
$311$	−5.15827e17			−1.83308			−0.916540	+	0.399944i	$0.869030\pi$
$312$	−1.33422e17			−0.463604
$313$		−	3.15711e17i		−	1.07270i	−0.843994	−	0.536352i	$-0.819802\pi$
$313$		−	3.15711e17i		−	1.07270i	0.843994	−	0.536352i	$-0.180198\pi$
$314$		−	1.89074e17i		−	0.628237i
$315$	3.15244e16			0.102441
$316$		−	6.55562e16i		−	0.208355i
$317$	1.85530e17			0.576765			0.288383	−	0.957515i	$-0.406883\pi$
$317$	1.85530e17			0.576765			0.288383	+	0.957515i	$0.406883\pi$
$318$		−	1.50828e17i		−	0.458662i
$319$		−	8.13778e17i		−	2.42087i
$320$		−	3.98097e16i		−	0.115861i
$321$			3.07703e16i			0.0876186i
$322$	6.67667e16	−	1.87725e16i	0.186024	−	0.0523036i
$323$	5.30923e16			0.144748
$324$	1.98755e17			0.530276
$325$	−1.25562e17			−0.327849
$326$	9.22263e16			0.235685
$327$			2.09496e17i			0.524013i
$328$	−1.73905e17			−0.425790
$329$		−	6.77521e16i		−	0.162387i
$330$	1.62309e17			0.380841
$331$	7.40625e17			1.70138			0.850692	−	0.525664i	$-0.176183\pi$
$331$	7.40625e17			1.70138			0.850692	+	0.525664i	$0.176183\pi$
$332$			5.23273e17i			1.17696i
$333$			7.87674e15i			0.0173475i
$334$	1.45048e17			0.312814
$335$	−6.52042e17			−1.37709
$336$		−	1.06017e17i		−	0.219280i
$337$			5.06522e16i			0.102610i	0.998683	+	0.0513049i	$0.0163380\pi$
$337$			5.06522e16i			0.102610i	−0.998683	+	0.0513049i	$0.983662\pi$
$338$	−9.13018e16			−0.181160
$339$			3.54158e17i			0.688333i
$340$	4.63335e16			0.0882151
$341$			1.19002e16i			0.0221960i
$342$			6.46131e16i			0.118069i
$343$			4.74544e17i			0.849607i
$344$			6.38809e17i			1.12063i
$345$	−3.53538e17	+	9.94029e16i	−0.607718	+	0.170870i
$346$	−3.75976e17			−0.633327
$347$	2.49740e17			0.412270			0.206135	−	0.978524i	$-0.433911\pi$
$347$	2.49740e17			0.412270			0.206135	+	0.978524i	$0.433911\pi$
$348$	−7.07027e17			−1.14388
$349$	−8.23125e16			−0.130523			−0.0652616	−	0.997868i	$-0.520788\pi$
$349$	−8.23125e16			−0.130523			−0.0652616	+	0.997868i	$0.520788\pi$
$350$			5.49687e16i			0.0854357i
$351$	−5.28874e17			−0.805755
$352$			9.60723e17i			1.43483i
$353$	6.18195e17			0.905112			0.452556	−	0.891736i	$-0.350512\pi$
$353$	6.18195e17			0.905112			0.452556	+	0.891736i	$0.350512\pi$
$354$	−2.61735e16			−0.0375699
$355$		−	5.97132e17i		−	0.840373i
$356$			9.00121e17i			1.24208i
$357$	4.22696e16			0.0571940
$358$	−2.77963e17			−0.368813
$359$			4.17733e17i			0.543547i	0.962361	+	0.271774i	$0.0876102\pi$
$359$			4.17733e17i			0.543547i	−0.962361	+	0.271774i	$0.912390\pi$
$360$			1.23673e17i			0.157819i
$361$	−4.38398e16			−0.0548679
$362$			7.32849e16i			0.0899611i
$363$	8.75378e17			1.05402
$364$			2.52624e17i			0.298377i
$365$			8.24786e17i			0.955633i
$366$		−	1.59503e17i		−	0.181302i
$367$		−	1.24506e18i		−	1.38844i	−0.719761	−	0.694222i	$-0.755749\pi$
$367$		−	1.24506e18i		−	1.38844i	0.719761	−	0.694222i	$-0.244251\pi$
$368$	−1.33672e17	−	4.75419e17i	−0.146253	−	0.520166i
$369$	−1.53159e17			−0.164421
$370$	1.73301e16			0.0182553
$371$	−6.26355e17			−0.647446
$372$	1.03392e16			0.0104878
$373$		−	1.68422e17i		−	0.167663i	−0.996480	−	0.0838313i	$-0.973284\pi$
$373$		−	1.68422e17i		−	0.167663i	0.996480	−	0.0838313i	$-0.0267157\pi$
$374$	−8.70234e16			−0.0850227
$375$		−	9.49395e17i		−	0.910391i
$376$	2.65799e17			0.250172
$377$	1.29621e18			1.19753
$378$			2.31532e17i			0.209975i
$379$		−	1.57540e18i		−	1.40255i	−0.712892	−	0.701274i	$-0.752615\pi$
$379$		−	1.57540e18i		−	1.40255i	0.712892	−	0.701274i	$-0.247385\pi$
$380$	−7.35551e17			−0.642877
$381$	3.68929e17			0.316569
$382$			5.15255e17i			0.434089i
$383$			1.84741e17i			0.152817i	0.997077	+	0.0764084i	$0.0243453\pi$
$383$			1.84741e17i			0.152817i	−0.997077	+	0.0764084i	$0.975655\pi$
$384$	1.06098e18			0.861763
$385$		−	6.74030e17i		−	0.537594i
$386$	2.97732e17			0.233192
$387$			5.62601e17i			0.432737i
$388$			2.00828e17i			0.151706i
$389$		−	1.83941e18i		−	1.36468i	−0.731033	−	0.682342i	$-0.760962\pi$
$389$		−	1.83941e18i		−	1.36468i	0.731033	−	0.682342i	$-0.239038\pi$
$390$			2.58531e17i			0.188391i
$391$	1.89553e17	−	5.32959e16i	0.135673	−	0.0381466i
$392$	−8.09562e17			−0.569178
$393$	−3.57277e17			−0.246750
$394$	−1.91911e17			−0.130205
$395$	−2.78605e17			−0.185699
$396$			5.47979e17i			0.358837i
$397$	−1.28972e18			−0.829779			−0.414890	−	0.909872i	$-0.636180\pi$
$397$	−1.28972e18			−0.829779			−0.414890	+	0.909872i	$0.636180\pi$
$398$			6.65533e17i			0.420715i
$399$	−6.71035e17			−0.416807
$400$	3.91411e17			0.238898
$401$			6.44589e17i			0.386609i	0.981139	+	0.193305i	$0.0619205\pi$
$401$			6.44589e17i			0.386609i	−0.981139	+	0.193305i	$0.938079\pi$
$402$		−	1.06401e18i		−	0.627139i
$403$	−1.89551e16			−0.0109797
$404$	5.82052e17			0.331354
$405$		−	8.44680e17i		−	0.472614i
$406$		−	5.67460e17i		−	0.312071i
$407$	1.68415e17			0.0910372
$408$			1.65828e17i			0.0881123i
$409$	1.98719e18			1.03795			0.518974	−	0.854790i	$-0.326314\pi$
$409$	1.98719e18			1.03795			0.518974	+	0.854790i	$0.326314\pi$
$410$			3.36974e17i			0.173025i
$411$			2.35078e18i			1.18664i
$412$		−	1.53668e18i		−	0.762612i
$413$			1.08693e17i			0.0530334i
$414$	6.48608e16	+	2.30685e17i	0.0311157	+	0.110667i
$415$	2.22384e18			1.04898
$416$	−1.53027e18			−0.709767
$417$	−4.98064e17			−0.227161
$418$	1.38151e18			0.619611
$419$		−	2.64598e18i		−	1.16705i	−0.812097	−	0.583523i	$-0.801674\pi$
$419$		−	2.64598e18i		−	1.16705i	0.812097	−	0.583523i	$-0.198326\pi$
$420$	−5.85611e17			−0.254018
$421$		−	2.51645e18i		−	1.07353i	−0.843732	−	0.536765i	$-0.819646\pi$
$421$		−	2.51645e18i		−	1.07353i	0.843732	−	0.536765i	$-0.180354\pi$
$422$	−1.55844e18			−0.653886
$423$	2.34090e17			0.0966051
$424$		−	2.45726e18i		−	0.997447i
$425$			1.56058e17i			0.0623109i
$426$	9.74408e17			0.382713
$427$	−6.62379e17			−0.255924
$428$			2.28564e17i			0.0868762i
$429$			2.51241e18i			0.939486i
$430$	1.23781e18			0.455381
$431$		−	4.45616e17i		−	0.161295i	−0.996743	−	0.0806474i	$-0.974301\pi$
$431$		−	4.45616e17i		−	0.161295i	0.996743	−	0.0806474i	$-0.0256988\pi$
$432$	1.64865e18			0.587140
$433$		−	2.34977e18i		−	0.823399i	−0.911320	−	0.411700i	$-0.864935\pi$
$433$		−	2.34977e18i		−	0.823399i	0.911320	−	0.411700i	$-0.135065\pi$
$434$			8.29821e15i			0.00286125i
$435$			3.00477e18i			1.01950i
$436$			1.55615e18i			0.519574i
$437$	−3.00917e18	+	8.46079e17i	−0.988729	+	0.277997i
$438$	−1.34590e18			−0.435204
$439$	−4.07452e18			−1.29666			−0.648328	−	0.761361i	$-0.724532\pi$
$439$	−4.07452e18			−1.29666			−0.648328	+	0.761361i	$0.724532\pi$
$440$	2.64429e18			0.828210
$441$	−7.12984e17			−0.219791
$442$		−	1.38614e17i		−	0.0420582i
$443$	6.53099e16			0.0195053			0.00975265	−	0.999952i	$-0.496896\pi$
$443$	6.53099e16			0.0195053			0.00975265	+	0.999952i	$0.496896\pi$
$444$		−	1.46322e17i		−	0.0430159i
$445$	3.82539e18			1.10702
$446$	−9.50967e17			−0.270908
$447$			4.66413e18i			1.30803i
$448$		−	2.69772e17i		−	0.0744819i
$449$	6.25349e17			0.169980			0.0849900	−	0.996382i	$-0.472914\pi$
$449$	6.25349e17			0.169980			0.0849900	+	0.996382i	$0.472914\pi$
$450$	−1.89922e17			−0.0508262
$451$			3.27473e18i			0.862858i
$452$			2.63071e18i			0.682501i
$453$	−4.21126e18			−1.07578
$454$			7.29233e16i			0.0183432i
$455$	1.07362e18			0.265932
$456$		−	2.63254e18i		−	0.642127i
$457$		−	1.14549e18i		−	0.275156i	−0.990491	−	0.137578i	$-0.956068\pi$
$457$		−	1.14549e18i		−	0.275156i	0.990491	−	0.137578i	$-0.0439318\pi$
$458$		−	9.10780e17i		−	0.215454i
$459$			6.57327e17i			0.153141i
$460$	−2.62610e18	+	7.38372e17i	−0.602569	+	0.169422i
$461$	3.70961e18			0.838342			0.419171	−	0.907907i	$-0.362321\pi$
$461$	3.70961e18			0.838342			0.419171	+	0.907907i	$0.362321\pi$
$462$	1.09989e18			0.244825
$463$	−2.31156e16			−0.00506802			−0.00253401	−	0.999997i	$-0.500807\pi$
$463$	−2.31156e16			−0.00506802			−0.00253401	+	0.999997i	$0.500807\pi$
$464$	−4.04066e18			−0.872622
$465$		−	4.39400e16i		−	0.00934738i
$466$	7.38892e17			0.154839
$467$			8.85768e18i			1.82853i	0.405119	+	0.914264i	$0.367230\pi$
$467$			8.85768e18i			1.82853i	−0.405119	+	0.914264i	$0.632770\pi$
$468$	−8.72839e17			−0.177506
$469$	−4.41859e18			−0.885266
$470$		−	5.15034e17i		−	0.101660i
$471$			6.78446e18i			1.31938i
$472$	−4.26413e17			−0.0817027
$473$	1.20291e19			2.27094
$474$		−	4.54631e17i		−	0.0845690i
$475$		−	2.47744e18i		−	0.454097i
$476$	3.13981e17			0.0567094
$477$		−	2.16411e18i		−	0.385169i
$478$	−1.76792e18			−0.310076
$479$		−	1.70202e16i		−	0.00294182i	−0.999999	−	0.00147091i	$-0.999532\pi$
$479$		−	1.70202e16i		−	0.00294182i	0.999999	−	0.00147091i	$-0.000468205\pi$
$480$		−	3.54734e18i		−	0.604247i
$481$			2.68257e17i			0.0450334i
$482$		−	3.94625e18i		−	0.652913i
$483$	−2.39576e18	+	6.73608e17i	−0.390674	+	0.109844i
$484$	6.50237e18			1.04509
$485$	8.53493e17			0.135210
$486$	−1.42219e18			−0.222077
$487$	−8.56402e18			−1.31818			−0.659090	−	0.752064i	$-0.729059\pi$
$487$	−8.56402e18			−1.31818			−0.659090	+	0.752064i	$0.729059\pi$
$488$		−	2.59858e18i		−	0.394274i
$489$	−3.30932e18			−0.494968
$490$			1.56868e18i			0.231292i
$491$	−9.31161e18			−1.35349			−0.676744	−	0.736219i	$-0.736609\pi$
$491$	−9.31161e18			−1.35349			−0.676744	+	0.736219i	$0.736609\pi$
$492$	2.84515e18			0.407708
$493$		−	1.61104e18i		−	0.227602i
$494$			2.20051e18i			0.306503i
$495$	2.32884e18			0.319818
$496$	5.90883e16			0.00800073
$497$		−	4.04649e18i		−	0.540236i
$498$			3.62889e18i			0.477715i
$499$	1.28185e19			1.66392			0.831960	−	0.554836i	$-0.187219\pi$
$499$	1.28185e19			1.66392			0.831960	+	0.554836i	$0.187219\pi$
$500$		−	7.05217e18i		−	0.902678i
$501$	−5.20471e18			−0.656950
$502$			1.43650e17i			0.0178804i
$503$		−	5.07533e18i		−	0.623001i	−0.950246	−	0.311500i	$-0.899168\pi$
$503$		−	5.07533e18i		−	0.623001i	0.950246	−	0.311500i	$-0.100832\pi$
$504$			8.38078e17i			0.101454i
$505$		−	2.47364e18i		−	0.295323i
$506$	4.93233e18	−	1.38681e18i	0.580762	−	0.163291i
$507$	3.27615e18			0.380459
$508$	2.74043e18			0.313887
$509$	−2.07786e18			−0.234742			−0.117371	−	0.993088i	$-0.537447\pi$
$509$	−2.07786e18			−0.234742			−0.117371	+	0.993088i	$0.537447\pi$
$510$	3.21322e17			0.0358055
$511$			5.58920e18i			0.614332i
$512$	8.45682e18			0.916891
$513$		−	1.04352e19i		−	1.11603i
$514$	−2.93573e18			−0.309724
$515$	−6.53069e18			−0.679687
$516$		−	1.04511e19i		−	1.07304i
$517$		−	5.00513e18i		−	0.506969i
$518$	1.17438e17			0.0117355
$519$	1.34910e19			1.33007
$520$			4.21192e18i			0.409691i
$521$		−	1.53803e19i		−	1.47605i	−0.674773	−	0.738025i	$-0.735759\pi$
$521$		−	1.53803e19i		−	1.47605i	0.674773	−	0.738025i	$-0.264241\pi$
$522$	1.96062e18			0.185653
$523$			1.48511e19i			1.38755i	0.720194	+	0.693773i	$0.244053\pi$
$523$			1.48511e19i			1.38755i	−0.720194	+	0.693773i	$0.755947\pi$
$524$	−2.65388e18			−0.244659
$525$		−	1.97242e18i		−	0.179426i
$526$			3.08946e18i			0.277320i
$527$			2.35589e16i			0.00208680i
$528$		−	7.83190e18i		−	0.684588i
$529$	−9.89419e18	+	6.04143e18i	−0.853474	+	0.521135i
$530$	−4.76139e18			−0.405325
$531$	−3.75543e17			−0.0315499
$532$	−4.98449e18			−0.413275
$533$	−5.21610e18			−0.426830
$534$			6.24233e18i			0.504148i
$535$	9.71365e17			0.0774294
$536$		−	1.73346e19i		−	1.36383i
$537$	9.97406e18			0.774554
$538$	−7.43136e18			−0.569630
$539$			1.52445e19i			1.15343i
$540$		−	9.10674e18i		−	0.680153i
$541$	−1.82698e18			−0.134695			−0.0673476	−	0.997730i	$-0.521454\pi$
$541$	−1.82698e18			−0.134695			−0.0673476	+	0.997730i	$0.521454\pi$
$542$	1.04380e19			0.759666
$543$		−	2.62966e18i		−	0.188930i
$544$			1.90194e18i			0.134898i
$545$	6.61344e18			0.463076
$546$			1.75194e18i			0.121108i
$547$	−4.46574e18			−0.304777			−0.152388	−	0.988321i	$-0.548696\pi$
$547$	−4.46574e18			−0.304777			−0.152388	+	0.988321i	$0.548696\pi$
$548$			1.74618e19i			1.17659i
$549$		−	2.28858e18i		−	0.152251i
$550$			4.06077e18i			0.266729i
$551$			2.55754e19i			1.65868i
$552$	−2.64264e18	−	9.39883e18i	−0.169225	−	0.601867i
$553$	−1.88798e18			−0.119377
$554$	−8.78314e18			−0.548380
$555$	−6.21849e17			−0.0383384
$556$	−3.69966e18			−0.225236
$557$		−	3.63887e18i		−	0.218766i	−0.994000	−	0.109383i	$-0.965112\pi$
$557$		−	3.63887e18i		−	0.218766i	0.994000	−	0.109383i	$-0.0348876\pi$
$558$	−2.86711e16			−0.00170218
$559$			1.91604e19i			1.12337i
$560$	−3.34677e18			−0.193780
$561$	3.12263e18			0.178558
$562$			6.92723e18i			0.391206i
$563$		−	1.12816e18i		−	0.0629236i	−0.999505	−	0.0314618i	$-0.989984\pi$
$563$		−	1.12816e18i		−	0.0629236i	0.999505	−	0.0314618i	$-0.0100162\pi$
$564$	−4.34856e18			−0.239548
$565$	1.11801e19			0.608287
$566$		−	8.22227e18i		−	0.441852i
$567$		−	5.72401e18i		−	0.303822i
$568$	1.58748e19			0.832282
$569$			1.31420e19i			0.680575i	0.940321	+	0.340287i	$0.110524\pi$
$569$			1.31420e19i			0.680575i	−0.940321	+	0.340287i	$0.889476\pi$
$570$	−5.10103e18			−0.260936
$571$			8.27571e18i			0.418170i	0.977898	+	0.209085i	$0.0670485\pi$
$571$			8.27571e18i			0.418170i	−0.977898	+	0.209085i	$0.932952\pi$
$572$			1.86624e19i			0.931526i
$573$		−	1.84887e19i		−	0.911642i
$574$			2.28352e18i			0.111230i
$575$	−2.48694e18	−	8.84510e18i	−0.119672	−	0.425625i
$576$	9.32087e17			0.0443097
$577$	1.86518e19			0.875971			0.437986	−	0.898982i	$-0.355692\pi$
$577$	1.86518e19			0.875971			0.437986	+	0.898982i	$0.355692\pi$
$578$	8.50146e18			0.394456
$579$	−1.06834e19			−0.489734
$580$			2.23196e19i			1.01086i
$581$	1.50699e19			0.674340
$582$			1.39274e18i			0.0615757i
$583$	−4.62715e19			−2.02131
$584$	−2.19270e19			−0.946432
$585$			3.70945e18i			0.158204i
$586$		−	3.36810e18i		−	0.141939i
$587$	−4.55238e19			−1.89571			−0.947856	−	0.318700i	$-0.896754\pi$
$587$	−4.55238e19			−1.89571			−0.947856	+	0.318700i	$0.896754\pi$
$588$	1.32447e19			0.545006
$589$		−	3.74000e17i		−	0.0152078i
$590$			8.26254e17i			0.0332009i
$591$	6.88628e18			0.273446
$592$		−	8.36230e17i		−	0.0328151i
$593$	−4.23905e18			−0.164394			−0.0821970	−	0.996616i	$-0.526194\pi$
$593$	−4.23905e18			−0.164394			−0.0821970	+	0.996616i	$0.526194\pi$
$594$			1.71042e19i			0.655539i
$595$		−	1.33438e18i		−	0.0505429i
$596$			3.46455e19i			1.29695i
$597$		−	2.38811e19i		−	0.883555i
$598$	2.20895e18	+	7.85638e18i	0.0807752	+	0.287286i
$599$	2.40551e19			0.869400			0.434700	−	0.900575i	$-0.356854\pi$
$599$	2.40551e19			0.869400			0.434700	+	0.900575i	$0.356854\pi$
$600$	7.73803e18			0.276421
$601$	2.83341e19			1.00043			0.500216	−	0.865900i	$-0.333254\pi$
$601$	2.83341e19			1.00043			0.500216	+	0.865900i	$0.333254\pi$
$602$	8.38807e18			0.292743
$603$		−	1.52666e19i		−	0.526650i
$604$	−3.12815e19			−1.06667
$605$		−	2.76342e19i		−	0.931449i
$606$	4.03652e18			0.134493
$607$	3.53417e19			1.16404			0.582018	−	0.813176i	$-0.302263\pi$
$607$	3.53417e19			1.16404			0.582018	+	0.813176i	$0.302263\pi$
$608$		−	3.01936e19i		−	0.983083i
$609$			2.03619e19i			0.655388i
$610$	−5.03524e18			−0.160218
$611$	7.97235e18			0.250783
$612$			1.08483e18i			0.0337368i
$613$		−	1.97679e19i		−	0.607767i	−0.952709	−	0.303884i	$-0.901717\pi$
$613$		−	1.97679e19i		−	0.607767i	0.952709	−	0.303884i	$-0.0982834\pi$
$614$	3.93478e18			0.119603
$615$		−	1.20915e19i		−	0.363374i
$616$	1.79192e19			0.532418
$617$		−	3.18510e19i		−	0.935680i	−0.883813	−	0.467840i	$-0.845032\pi$
$617$		−	3.18510e19i		−	0.935680i	0.883813	−	0.467840i	$-0.154968\pi$
$618$		−	1.06569e19i		−	0.309535i
$619$			2.42805e18i			0.0697306i	0.999392	+	0.0348653i	$0.0111002\pi$
$619$			2.42805e18i			0.0697306i	−0.999392	+	0.0348653i	$0.988900\pi$
$620$		−	3.26390e17i		−	0.00926818i
$621$	−1.04752e19	−	3.72561e19i	−0.294117	−	1.04606i
$622$	−2.65721e19			−0.737722
$623$	2.59229e19			0.711652
$624$	1.24749e19			0.338646
$625$	−1.35002e19			−0.362392
$626$		−	1.62634e19i		−	0.431709i
$627$	−4.95721e19			−1.30126
$628$			5.03954e19i			1.30820i
$629$	3.33410e17			0.00855903
$630$	1.62393e18			0.0412272
$631$		−	4.97679e19i		−	1.24952i	−0.780816	−	0.624761i	$-0.785196\pi$
$631$		−	4.97679e19i		−	1.24952i	0.780816	−	0.624761i	$-0.214804\pi$
$632$		−	7.40674e18i		−	0.183911i
$633$	5.59208e19			1.37324
$634$	9.55730e18			0.232119
$635$		−	1.16465e19i		−	0.279755i
$636$			4.02016e19i			0.955088i
$637$	−2.42819e19			−0.570568
$638$		−	4.19206e19i		−	0.974278i
$639$	1.39810e19			0.321390
$640$		−	3.34933e19i		−	0.761549i
$641$		−	6.63434e19i		−	1.49208i	−0.665902	−	0.746039i	$-0.731953\pi$
$641$		−	6.63434e19i		−	1.49208i	0.665902	−	0.746039i	$-0.268047\pi$
$642$			1.58509e18i			0.0352621i
$643$			3.77286e19i			0.830222i	0.909771	+	0.415111i	$0.136257\pi$
$643$			3.77286e19i			0.830222i	−0.909771	+	0.415111i	$0.863743\pi$
$644$	−1.77959e19	+	5.00361e18i	−0.387364	+	0.108914i
$645$	−4.44159e19			−0.956358
$646$	2.73497e18			0.0582539
$647$	5.34038e19			1.12523			0.562616	−	0.826719i	$-0.309795\pi$
$647$	5.34038e19			1.12523			0.562616	+	0.826719i	$0.309795\pi$
$648$	2.24559e19			0.468064
$649$			8.02958e18i			0.165569i
$650$	−6.46813e18			−0.131943
$651$		−	2.97762e17i		−	0.00600899i
$652$	−2.45819e19			−0.490774
$653$	1.58434e19			0.312936			0.156468	−	0.987683i	$-0.449989\pi$
$653$	1.58434e19			0.312936			0.156468	+	0.987683i	$0.449989\pi$
$654$			1.07919e19i			0.210889i
$655$			1.12786e19i			0.218055i
$656$	1.62600e19			0.311024
$657$	−1.93112e19			−0.365470
$658$		−	3.49015e18i		−	0.0653527i
$659$			3.26048e19i			0.604064i	0.953298	+	0.302032i	$0.0976650\pi$
$659$			3.26048e19i			0.604064i	−0.953298	+	0.302032i	$0.902335\pi$
$660$	−4.32615e19			−0.793038
$661$			2.68765e19i			0.487487i	0.969840	+	0.243743i	$0.0783755\pi$
$661$			2.68765e19i			0.487487i	−0.969840	+	0.243743i	$0.921625\pi$
$662$	3.81522e19			0.684722
$663$			4.97383e18i			0.0883275i
$664$			5.91210e19i			1.03888i
$665$			2.11834e19i			0.368336i
$666$			4.05759e17i			0.00698150i
$667$	2.56735e19	+	9.13107e19i	0.437124	+	1.55468i
$668$	−3.86609e19			−0.651384
$669$	3.41232e19			0.568941
$670$	−3.35890e19			−0.554209
$671$	−4.89327e19			−0.798991
$672$		−	2.40387e19i		−	0.388442i
$673$	−7.87482e19			−1.25932			−0.629659	−	0.776872i	$-0.716805\pi$
$673$	−7.87482e19			−1.25932			−0.629659	+	0.776872i	$0.716805\pi$
$674$			2.60927e18i			0.0412953i
$675$	3.06728e19			0.480427
$676$	2.43355e19			0.377236
$677$			7.16898e19i			1.09986i	0.835211	+	0.549929i	$0.185345\pi$
$677$			7.16898e19i			1.09986i	−0.835211	+	0.549929i	$0.814655\pi$
$678$			1.82439e19i			0.277019i
$679$	5.78373e18			0.0869200
$680$	5.23490e18			0.0778657
$681$		−	2.61668e18i		−	0.0385231i
$682$			6.13023e17i			0.00893277i
$683$	2.59362e19			0.374078			0.187039	−	0.982353i	$-0.440111\pi$
$683$	2.59362e19			0.374078			0.187039	+	0.982353i	$0.440111\pi$
$684$		−	1.72219e19i		−	0.245860i
$685$	7.42101e19			1.04865
$686$			2.44455e19i			0.341924i
$687$			3.26812e19i			0.452481i
$688$		−	5.97282e19i		−	0.818579i
$689$		−	7.37028e19i		−	0.999883i
$690$	−1.82120e19	+	5.12060e18i	−0.244576	+	0.0687665i
$691$	−1.33717e20			−1.77762			−0.888811	−	0.458273i	$-0.848468\pi$
$691$	−1.33717e20			−1.77762			−0.888811	+	0.458273i	$0.848468\pi$
$692$	1.00212e20			1.31880
$693$	1.57815e19			0.205596
$694$	1.28650e19			0.165918
$695$			1.57230e19i			0.200744i
$696$	−7.98821e19			−1.00968
$697$			6.48298e18i			0.0811232i
$698$	−4.24021e18			−0.0525290
$699$	−2.65134e19			−0.325181
$700$		−	1.46513e19i		−	0.177906i
$701$		−	4.72278e19i		−	0.567769i	−0.958858	−	0.283885i	$-0.908377\pi$
$701$		−	4.72278e19i		−	0.567769i	0.958858	−	0.283885i	$-0.0916233\pi$
$702$	−2.72442e19			−0.324276
$703$	−5.29293e18			−0.0623748
$704$		−	1.99292e19i		−	0.232531i
$705$			1.84808e19i			0.213499i
$706$	3.18454e19			0.364262
$707$		−	1.67627e19i		−	0.189849i
$708$	6.97626e18			0.0782330
$709$		−	8.79330e19i		−	0.976401i	−0.872732	−	0.488201i	$-0.837653\pi$
$709$		−	8.79330e19i		−	0.976401i	0.872732	−	0.488201i	$-0.162347\pi$
$710$		−	3.07604e19i		−	0.338208i
$711$		−	6.52314e18i		−	0.0710182i
$712$			1.01698e20i			1.09636i
$713$	−3.75435e17	−	1.33528e18i	−0.00400782	−	0.0142542i
$714$	2.17745e18			0.0230177
$715$	7.93126e19			0.830233
$716$	7.40880e19			0.767992
$717$	6.34377e19			0.651199
$718$			2.15189e19i			0.218750i
$719$	−1.34080e20			−1.34978			−0.674888	−	0.737920i	$-0.735808\pi$
$719$	−1.34080e20			−1.34978			−0.674888	+	0.737920i	$0.735808\pi$
$720$		−	1.15634e19i		−	0.115281i
$721$	−4.42555e19			−0.436939
$722$	−2.25835e18			−0.0220816
$723$			1.41602e20i			1.37120i
$724$		−	1.95333e19i		−	0.187329i
$725$	−7.51758e19			−0.714022
$726$	4.50938e19			0.424190
$727$			1.71611e20i			1.59884i	0.600772	+	0.799420i	$0.294860\pi$
$727$			1.71611e20i			1.59884i	−0.600772	+	0.799420i	$0.705140\pi$
$728$			2.85422e19i			0.263371i
$729$	1.20268e20			1.09915
$730$			4.24876e19i			0.384594i
$731$	2.38140e19			0.213507
$732$			4.25137e19i			0.377530i
$733$			1.52675e20i			1.34289i	0.741055	+	0.671444i	$0.234326\pi$
$733$			1.52675e20i			1.34289i	−0.741055	+	0.671444i	$0.765674\pi$
$734$		−	6.41375e19i		−	0.558779i
$735$		−	5.62883e19i		−	0.485743i
$736$	−3.03094e19	−	1.07799e20i	−0.259079	−	0.921445i
$737$	−3.26420e20			−2.76378
$738$	−7.88976e18			−0.0661712
$739$	1.42895e20			1.18715			0.593577	−	0.804777i	$-0.297715\pi$
$739$	1.42895e20			1.18715			0.593577	+	0.804777i	$0.297715\pi$
$740$	−4.61914e18			−0.0380136
$741$		−	7.89602e19i		−	0.643696i
$742$	−3.22658e19			−0.260564
$743$		−	8.70274e19i		−	0.696201i	−0.937457	−	0.348100i	$-0.886827\pi$
$743$		−	8.70274e19i		−	0.696201i	0.937457	−	0.348100i	$-0.113173\pi$
$744$	1.16815e18			0.00925738
$745$	1.47239e20			1.15592
$746$		−	8.67600e18i		−	0.0674757i
$747$			5.20680e19i			0.401169i
$748$	2.31951e19			0.177046
$749$	6.58249e18			0.0497757
$750$		−	4.89067e19i		−	0.366387i
$751$		−	2.43081e20i		−	1.80415i	−0.431581	−	0.902074i	$-0.642044\pi$
$751$		−	2.43081e20i		−	1.80415i	0.431581	−	0.902074i	$-0.357956\pi$
$752$	−2.48520e19			−0.182741
$753$		−	5.15453e18i		−	0.0375512i
$754$	6.67726e19			0.481946
$755$			1.32942e20i			0.950680i
$756$		−	6.17122e19i		−	0.437239i
$757$		−	1.62867e20i		−	1.14330i	−0.820497	−	0.571651i	$-0.806303\pi$
$757$		−	1.62867e20i		−	1.14330i	0.820497	−	0.571651i	$-0.193697\pi$
$758$		−	8.11547e19i		−	0.564455i
$759$	−1.76985e20	+	4.97622e19i	−1.21967	+	0.342931i
$760$	−8.31048e19			−0.567454
$761$	2.47437e20			1.67406			0.837031	−	0.547156i	$-0.184290\pi$
$761$	2.47437e20			1.67406			0.837031	+	0.547156i	$0.184290\pi$
$762$	1.90048e19			0.127403
$763$	4.48163e19			0.297690
$764$		−	1.37335e20i		−	0.903918i
$765$	4.61040e18			0.0300683
$766$			9.51664e18i			0.0615010i
$767$	−1.27898e19			−0.0819022
$768$	3.39935e19			0.215708
$769$		−	1.94776e20i		−	1.22476i	−0.790563	−	0.612380i	$-0.790212\pi$
$769$		−	1.94776e20i		−	1.22476i	0.790563	−	0.612380i	$-0.209788\pi$
$770$		−	3.47217e19i		−	0.216354i
$771$	1.05342e20			0.650459
$772$	−7.93571e19			−0.485584
$773$			1.88179e20i			1.14108i	0.821271	+	0.570538i	$0.193265\pi$
$773$			1.88179e20i			1.14108i	−0.821271	+	0.570538i	$0.806735\pi$
$774$			2.89816e19i			0.174155i
$775$	1.09933e18			0.00654659
$776$			2.26902e19i			0.133908i
$777$	−4.21398e18			−0.0246460
$778$		−	9.47547e19i		−	0.549216i
$779$		−	1.02918e20i		−	0.591194i
$780$		−	6.89084e19i		−	0.392293i
$781$		−	2.98931e20i		−	1.68661i
$782$	9.76454e18	−	2.74546e18i	0.0546015	−	0.0153521i
$783$	−3.16645e20			−1.75485
$784$	7.56936e19			0.415764
$785$	2.14174e20			1.16595
$786$	−1.84046e19			−0.0993043
$787$			2.31347e20i			1.23720i	0.785705	+	0.618602i	$0.212301\pi$
$787$			2.31347e20i			1.23720i	−0.785705	+	0.618602i	$0.787699\pi$
$788$	5.11517e19			0.271130
$789$		−	1.10858e20i		−	0.582408i
$790$	−1.43519e19			−0.0747344
$791$	7.57627e19			0.391039
$792$			6.19123e19i			0.316738i
$793$		−	7.79417e19i		−	0.395237i
$794$	−6.64381e19			−0.333944
$795$	1.70851e20			0.851233
$796$		−	1.77390e20i		−	0.876069i
$797$			3.78069e20i			1.85081i	0.378974	+	0.925407i	$0.376277\pi$
$797$			3.78069e20i			1.85081i	−0.378974	+	0.925407i	$0.623723\pi$
$798$	−3.45674e19			−0.167744
$799$		−	9.90866e18i		−	0.0476637i
$800$	8.87504e19			0.423195
$801$			8.95661e19i			0.423366i
$802$			3.32051e19i			0.155591i
$803$			4.12897e20i			1.91793i
$804$			2.83600e20i			1.30591i
$805$	2.12647e19	+	7.56301e19i	0.0970705	+	0.345242i
$806$	−9.76444e17			−0.00441878
$807$	2.66657e20			1.19629
$808$	6.57620e19			0.292479
$809$	−3.83404e20			−1.69051			−0.845253	−	0.534366i	$-0.820550\pi$
$809$	−3.83404e20			−1.69051			−0.845253	+	0.534366i	$0.820550\pi$
$810$		−	4.35124e19i		−	0.190203i
$811$	−3.14403e20			−1.36251			−0.681256	−	0.732045i	$-0.738566\pi$
$811$	−3.14403e20			−1.36251			−0.681256	+	0.732045i	$0.738566\pi$
$812$			1.51250e20i			0.649835i
$813$	−3.74543e20			−1.59539
$814$	8.67563e18			0.0366379
$815$			1.04470e20i			0.437408i
$816$		−	1.55048e19i		−	0.0643628i
$817$	−3.78051e20			−1.55595
$818$	1.02367e20			0.417722
$819$			2.51372e19i			0.101702i
$820$		−	8.98165e19i		−	0.360296i
$821$	1.96362e20			0.781009			0.390505	−	0.920601i	$-0.372301\pi$
$821$	1.96362e20			0.781009			0.390505	+	0.920601i	$0.372301\pi$
$822$			1.21097e20i			0.477563i
$823$	−2.67560e20			−1.04622			−0.523108	−	0.852266i	$-0.675228\pi$
$823$	−2.67560e20			−1.04622			−0.523108	+	0.852266i	$0.675228\pi$
$824$		−	1.73619e20i		−	0.673143i
$825$		−	1.45711e20i		−	0.560164i
$826$			5.59914e18i			0.0213433i
$827$			3.95204e20i			1.49377i	0.664955	+	0.746884i	$0.268451\pi$
$827$			3.95204e20i			1.49377i	−0.664955	+	0.746884i	$0.731549\pi$
$828$	−1.72879e19	−	6.14865e19i	−0.0647933	−	0.230445i
$829$	1.54288e20			0.573390			0.286695	−	0.958022i	$-0.407443\pi$
$829$	1.54288e20			0.573390			0.286695	+	0.958022i	$0.407443\pi$
$830$	1.14558e20			0.422161
$831$	3.15162e20			1.15167
$832$	3.17439e19			0.115026
$833$			3.01795e19i			0.108442i
$834$	−2.56571e19			−0.0914207
$835$			1.64304e20i			0.580553i
$836$	−3.68225e20			−1.29024
$837$	4.63044e18			0.0160896
$838$		−	1.36304e20i		−	0.469677i
$839$			3.39198e20i			1.15910i	0.814938	+	0.579548i	$0.196771\pi$
$839$			3.39198e20i			1.15910i	−0.814938	+	0.579548i	$0.803229\pi$
$840$	−6.61641e19			−0.224217
$841$	4.78505e20			1.60810
$842$		−	1.29631e20i		−	0.432042i
$843$		−	2.48567e20i		−	0.821582i
$844$	4.15384e20			1.36161
$845$		−	1.03422e20i		−	0.336216i
$846$	1.20588e19			0.0388787
$847$		−	1.87264e20i		−	0.598785i
$848$			2.29752e20i			0.728599i
$849$			2.95037e20i			0.927945i
$850$			8.03911e18i			0.0250770i
$851$	−1.88971e19	+	5.31323e18i	−0.0584640	+	0.0164381i
$852$	−2.59717e20			−0.796937
$853$	3.67399e20			1.11814			0.559068	−	0.829122i	$-0.311159\pi$
$853$	3.67399e20			1.11814			0.559068	+	0.829122i	$0.311159\pi$
$854$	−3.41215e19			−0.102997
$855$	−7.31906e19			−0.219125
$856$			2.58238e19i			0.0766839i
$857$	−4.80669e20			−1.41573			−0.707865	−	0.706347i	$-0.750342\pi$
$857$	−4.80669e20			−1.41573			−0.707865	+	0.706347i	$0.750342\pi$
$858$			1.29423e20i			0.378096i
$859$	−2.52686e20			−0.732200			−0.366100	−	0.930576i	$-0.619307\pi$
$859$	−2.52686e20			−0.732200			−0.366100	+	0.930576i	$0.619307\pi$
$860$	−3.29924e20			−0.948255
$861$		−	8.19386e19i		−	0.233597i
$862$		−	2.29553e19i		−	0.0649130i
$863$	4.44726e20			1.24743			0.623716	−	0.781651i	$-0.285622\pi$
$863$	4.44726e20			1.24743			0.623716	+	0.781651i	$0.285622\pi$
$864$	3.73822e20			1.04009
$865$		−	4.25889e20i		−	1.17539i
$866$		−	1.21045e20i		−	0.331377i
$867$	−3.05055e20			−0.828408
$868$		−	2.21179e18i		−	0.00595808i
$869$	−1.39473e20			−0.372693
$870$			1.54786e20i			0.410296i
$871$		−	5.19932e20i		−	1.36716i
$872$			1.75819e20i			0.458617i
$873$			1.99833e19i			0.0517092i
$874$	−1.55013e20	+	4.35845e19i	−0.397914	+	0.111880i
$875$	−2.03098e20			−0.517189
$876$	3.58733e20			0.906240
$877$	5.88100e20			1.47385			0.736927	−	0.675973i	$-0.236276\pi$
$877$	5.88100e20			1.47385			0.736927	+	0.675973i	$0.236276\pi$
$878$	−2.09893e20			−0.521839
$879$			1.20856e20i			0.298090i
$880$	−2.47240e20			−0.604978
$881$		−	4.05132e20i		−	0.983478i	−0.870743	−	0.491739i	$-0.836361\pi$
$881$		−	4.05132e20i		−	0.983478i	0.870743	−	0.491739i	$-0.163639\pi$
$882$	−3.67283e19			−0.0884548
$883$	−5.07118e20			−1.21167			−0.605835	−	0.795591i	$-0.707161\pi$
$883$	−5.07118e20			−1.21167			−0.605835	+	0.795591i	$0.707161\pi$
$884$			3.69459e19i			0.0875792i
$885$		−	2.96482e19i		−	0.0697260i
$886$	3.36434e18			0.00784990
$887$	7.53773e20			1.74492			0.872460	−	0.488686i	$-0.162524\pi$
$887$	7.53773e20			1.74492			0.872460	+	0.488686i	$0.162524\pi$
$888$		−	1.65319e19i		−	0.0379693i
$889$		−	7.89227e19i		−	0.179841i
$890$	1.97060e20			0.445520
$891$		−	4.22856e20i		−	0.948525i
$892$	2.53470e20			0.564121
$893$			1.57301e20i			0.347354i
$894$			2.40266e20i			0.526416i
$895$		−	3.14864e20i		−	0.684481i
$896$		−	2.26969e20i		−	0.489564i
$897$	−7.92630e19	−	2.81908e20i	−0.169638	−	0.603337i
$898$	3.22139e19			0.0684084
$899$	−1.13487e19			−0.0239127
$900$	5.06216e19			0.105837
$901$	−9.16036e19			−0.190037
$902$			1.68693e20i			0.347257i
$903$	−3.00986e20			−0.614798
$904$			2.97225e20i			0.602430i
$905$	−8.30137e19			−0.166959
$906$	−2.16937e20			−0.432949
$907$		−	5.65794e20i		−	1.12049i	−0.828328	−	0.560244i	$-0.810707\pi$
$907$		−	5.65794e20i		−	1.12049i	0.828328	−	0.560244i	$-0.189293\pi$
$908$		−	1.94369e19i		−	0.0381967i
$909$	5.79168e19			0.112942
$910$	5.53059e19			0.107024
$911$		−	5.80856e20i		−	1.11542i	−0.830035	−	0.557711i	$-0.811680\pi$
$911$		−	5.80856e20i		−	1.11542i	0.830035	−	0.557711i	$-0.188320\pi$
$912$			2.46141e20i			0.469051i
$913$	1.11328e21			2.10527
$914$		−	5.90084e19i		−	0.110736i
$915$	1.80678e20			0.336478
$916$			2.42758e20i			0.448647i
$917$			7.64300e19i			0.140178i
$918$			3.38612e19i			0.0616317i
$919$		−	4.37341e20i		−	0.789971i	−0.918687	−	0.394986i	$-0.870750\pi$
$919$		−	4.37341e20i		−	0.789971i	0.918687	−	0.394986i	$-0.129250\pi$
$920$	−2.96705e20	+	8.34235e19i	−0.531876	+	0.149546i
$921$	−1.41190e20			−0.251181
$922$	1.91095e20			0.337390
$923$	4.76148e20			0.834315
$924$	−2.93164e20			−0.509807
$925$		−	1.55579e19i		−	0.0268509i
$926$	−1.19077e18			−0.00203962
$927$		−	1.52907e20i		−	0.259937i
$928$	−9.16198e20			−1.54580
$929$	−2.52232e20			−0.422367			−0.211184	−	0.977446i	$-0.567732\pi$
$929$	−2.52232e20			−0.422367			−0.211184	+	0.977446i	$0.567732\pi$
$930$		−	2.26351e18i		−	0.00376185i
$931$		−	4.79104e20i		−	0.790281i
$932$	−1.96943e20			−0.322426
$933$	9.53476e20			1.54931
$934$			4.56290e20i			0.735890i
$935$		−	9.85760e19i		−	0.157794i
$936$	−9.86161e19			−0.156681
$937$			9.32411e20i			1.47038i	0.677860	+	0.735191i	$0.262908\pi$
$937$			9.32411e20i			1.47038i	−0.677860	+	0.735191i	$0.737092\pi$
$938$	−2.27617e20			−0.356275
$939$			5.83574e20i			0.906644i
$940$			1.37277e20i			0.211691i
$941$		−	1.13782e21i		−	1.74159i	−0.491650	−	0.870793i	$-0.663606\pi$
$941$		−	1.13782e21i		−	1.74159i	0.491650	−	0.870793i	$-0.336394\pi$
$942$			3.49491e20i			0.530983i
$943$	−1.03313e20	−	3.67444e20i	−0.155802	−	0.554127i
$944$	3.98693e19			0.0596808
$945$	−2.62269e20			−0.389694
$946$	6.19662e20			0.913938
$947$	−3.31289e20			−0.485017			−0.242509	−	0.970149i	$-0.577970\pi$
$947$	−3.31289e20			−0.485017			−0.242509	+	0.970149i	$0.577970\pi$
$948$			1.21177e20i			0.176101i
$949$	−6.57677e20			−0.948744
$950$		−	1.27622e20i		−	0.182751i
$951$	−3.42941e20			−0.487479
$952$	3.54746e19			0.0500563
$953$			7.58897e20i			1.06300i	0.847059	+	0.531499i	$0.178371\pi$
$953$			7.58897e20i			1.06300i	−0.847059	+	0.531499i	$0.821629\pi$
$954$		−	1.11481e20i		−	0.155011i
$955$	−5.83657e20			−0.805627
$956$	4.71220e20			0.645681
$957$			1.50422e21i			2.04611i
$958$		−	8.76771e17i		−	0.00118393i
$959$	5.02888e20			0.674126
$960$			7.35858e19i			0.0979255i
$961$	−7.56778e20			−0.999781
$962$			1.38188e19i			0.0181237i
$963$			2.27431e19i			0.0296119i
$964$			1.05183e21i			1.35958i
$965$			3.37257e20i			0.432783i
$966$	−1.23414e20	+	3.46999e19i	−0.157226	+	0.0442068i
$967$	−9.30556e19			−0.117695			−0.0588474	−	0.998267i	$-0.518743\pi$
$967$	−9.30556e19			−0.117695			−0.0588474	+	0.998267i	$0.518743\pi$
$968$	7.34657e20			0.922481
$969$	−9.81379e19			−0.122341
$970$	4.39664e19			0.0544151
$971$		−	4.17834e20i		−	0.513416i	−0.966489	−	0.256708i	$-0.917362\pi$
$971$		−	4.17834e20i		−	0.513416i	0.966489	−	0.256708i	$-0.0826379\pi$
$972$	3.79069e20			0.462439
$973$			1.06548e20i			0.129049i
$974$	−4.41163e20			−0.530501
$975$	2.32094e20			0.277096
$976$			2.42966e20i			0.288003i
$977$			3.89012e20i			0.457827i	0.973447	+	0.228913i	$0.0735172\pi$
$977$			3.89012e20i			0.457827i	−0.973447	+	0.228913i	$0.926483\pi$
$978$	−1.70475e20			−0.199200
$979$	1.91504e21			2.22176
$980$		−	4.18113e20i		−	0.481628i
$981$			1.54844e20i			0.177097i
$982$	−4.79674e20			−0.544710
$983$			5.11392e20i			0.576606i	0.957539	+	0.288303i	$0.0930910\pi$
$983$			5.11392e20i			0.576606i	−0.957539	+	0.288303i	$0.906909\pi$
$984$	3.21454e20			0.359876
$985$		−	2.17388e20i		−	0.241647i
$986$		−	8.29902e19i		−	0.0915985i
$987$			1.25236e20i			0.137249i
$988$		−	5.86522e20i		−	0.638242i
$989$	−1.34974e21	+	3.79500e20i	−1.45839	+	0.410051i
$990$	1.19967e20			0.128710
$991$	3.20012e20			0.340918			0.170459	−	0.985365i	$-0.445475\pi$
$991$	3.20012e20			0.340918			0.170459	+	0.985365i	$0.445475\pi$
$992$	1.33980e19			0.0141728
$993$	−1.36900e21			−1.43800
$994$		−	2.08449e20i		−	0.217418i
$995$	−7.53885e20			−0.780806
$996$		−	9.67240e20i		−	0.994761i
$997$	−1.11604e21			−1.13976			−0.569878	−	0.821729i	$-0.693009\pi$
$997$	−1.11604e21			−1.13976			−0.569878	+	0.821729i	$0.693009\pi$
$998$	6.60325e20			0.669644
$999$		−	6.55310e19i		−	0.0659915i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	23.15.b.b.22.16	yes	24
23.22	odd	2	inner	23.15.b.b.22.15	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
23.15.b.b.22.15	✓	24	23.22	odd	2	inner
23.15.b.b.22.16	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)

\(f(q)\)	\(=\)	\(q+51.5135 q^{2} -1848.44 q^{3} -13730.4 q^{4} +58352.1i q^{5} -95219.8 q^{6} +395426. i q^{7} -1.55130e6 q^{8} -1.36623e6 q^{9} +O(q^{10})\)	\(q+51.5135 q^{2} -1848.44 q^{3} -13730.4 q^{4} +58352.1i q^{5} -95219.8 q^{6} +395426. i q^{7} -1.55130e6 q^{8} -1.36623e6 q^{9} +3.00593e6i q^{10} +2.92117e7i q^{11} +2.53798e7 q^{12} -4.65295e7 q^{13} +2.03698e7i q^{14} -1.07861e8i q^{15} +1.45045e8 q^{16} +5.78305e7i q^{17} -7.03795e7 q^{18} -9.18067e8i q^{19} -8.01196e8i q^{20} -7.30922e8i q^{21} +1.50480e9i q^{22} +(-9.21587e8 - 3.27773e9i) q^{23} +2.86748e9 q^{24} +2.69854e9 q^{25} -2.39690e9 q^{26} +1.13664e10 q^{27} -5.42934e9i q^{28} -2.78579e10 q^{29} -5.55628e9i q^{30} +4.07378e8 q^{31} +3.28883e10 q^{32} -5.39962e10i q^{33} +2.97905e9i q^{34} -2.30739e10 q^{35} +1.87589e10 q^{36} -5.76530e9i q^{37} -4.72929e10i q^{38} +8.60070e10 q^{39} -9.05215e10i q^{40} +1.12103e11 q^{41} -3.76524e10i q^{42} -4.11790e11i q^{43} -4.01088e11i q^{44} -7.97226e10i q^{45} +(-4.74742e10 - 1.68848e11i) q^{46} -1.71340e11 q^{47} -2.68108e11 q^{48} +5.21862e11 q^{49} +1.39012e11 q^{50} -1.06896e11i q^{51} +6.38866e11 q^{52} +1.58400e12i q^{53} +5.85526e11 q^{54} -1.70457e12 q^{55} -6.13423e11i q^{56} +1.69699e12i q^{57} -1.43506e12 q^{58} +2.74875e11 q^{59} +1.48096e12i q^{60} +1.67510e12i q^{61} +2.09855e10 q^{62} -5.40243e11i q^{63} -6.82231e11 q^{64} -2.71509e12i q^{65} -2.78154e12i q^{66} +1.11743e13i q^{67} -7.94033e11i q^{68} +(1.70350e12 + 6.05869e12i) q^{69} -1.18862e12 q^{70} -1.02333e13 q^{71} +2.11943e12 q^{72} +1.41346e13 q^{73} -2.96991e11i q^{74} -4.98810e12 q^{75} +1.26054e13i q^{76} -1.15511e13 q^{77} +4.43053e12 q^{78} +4.77455e12i q^{79} +8.46370e12i q^{80} -1.44756e13 q^{81} +5.77483e12 q^{82} -3.81107e13i q^{83} +1.00358e13i q^{84} -3.37453e12 q^{85} -2.12128e13i q^{86} +5.14937e13 q^{87} -4.53161e13i q^{88} -6.55570e13i q^{89} -4.10679e12i q^{90} -1.83990e13i q^{91} +(1.26537e13 + 4.50044e13i) q^{92} -7.53015e11 q^{93} -8.82632e12 q^{94} +5.35712e13 q^{95} -6.07920e13 q^{96} -1.46266e13i q^{97} +2.68829e13 q^{98} -3.99100e13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 184 q^{2} - 2160 q^{3} + 134952 q^{4} - 1666320 q^{6} - 6887840 q^{8} + 24850248 q^{9}+O(q^{10}) \) 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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