LMFDB - Embedded newform 29.16.b.a.28.28

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [29,16,Mod(28,29)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(29, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("29.28");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 29 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	29.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:	$41.3811164790$
Analytic rank:	$0$
Dimension:	$36$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			28.28
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+202.124i q^{2} +5797.90i q^{3} -8086.11 q^{4} +172967. q^{5} -1.17189e6 q^{6} +3.88134e6 q^{7} +4.98880e6i q^{8} -1.92667e7 q^{9} +O(q^{10})$	\(q+202.124i q^{2} +5797.90i q^{3} -8086.11 q^{4} +172967. q^{5} -1.17189e6 q^{6} +3.88134e6 q^{7} +4.98880e6i q^{8} -1.92667e7 q^{9} +3.49608e7i q^{10} -1.04599e8i q^{11} -4.68825e7i q^{12} +2.38869e8 q^{13} +7.84512e8i q^{14} +1.00285e9i q^{15} -1.27332e9 q^{16} +8.65555e8i q^{17} -3.89427e9i q^{18} +6.68418e9i q^{19} -1.39863e9 q^{20} +2.25036e10i q^{21} +2.11419e10 q^{22} +7.33746e9 q^{23} -2.89246e10 q^{24} -5.99915e8 q^{25} +4.82812e10i q^{26} -2.85131e10i q^{27} -3.13849e10 q^{28} +(-9.28180e10 + 3.74350e9i) q^{29} -2.02699e11 q^{30} +6.84716e10i q^{31} -9.38959e10i q^{32} +6.06452e11 q^{33} -1.74949e11 q^{34} +6.71345e11 q^{35} +1.55793e11 q^{36} -6.08649e11i q^{37} -1.35103e12 q^{38} +1.38494e12i q^{39} +8.62899e11i q^{40} -6.76659e11i q^{41} -4.54852e12 q^{42} -1.65758e12i q^{43} +8.45795e11i q^{44} -3.33252e12 q^{45} +1.48308e12i q^{46} -2.89060e11i q^{47} -7.38260e12i q^{48} +1.03172e13 q^{49} -1.21257e11i q^{50} -5.01840e12 q^{51} -1.93152e12 q^{52} -6.08957e12 q^{53} +5.76319e12 q^{54} -1.80921e13i q^{55} +1.93632e13i q^{56} -3.87542e13 q^{57} +(-7.56651e11 - 1.87607e13i) q^{58} -1.27876e13 q^{59} -8.10913e12i q^{60} +4.08548e12i q^{61} -1.38398e13 q^{62} -7.47808e13 q^{63} -2.27456e13 q^{64} +4.13165e13 q^{65} +1.22578e14i q^{66} +7.48894e13 q^{67} -6.99898e12i q^{68} +4.25419e13i q^{69} +1.35695e14i q^{70} -1.13931e13 q^{71} -9.61180e13i q^{72} +3.81672e13i q^{73} +1.23022e14 q^{74} -3.47825e12i q^{75} -5.40490e13i q^{76} -4.05983e14i q^{77} -2.79930e14 q^{78} +2.03728e14i q^{79} -2.20243e14 q^{80} -1.11140e14 q^{81} +1.36769e14 q^{82} +3.95848e14 q^{83} -1.81967e14i q^{84} +1.49713e14i q^{85} +3.35037e14 q^{86} +(-2.17044e13 - 5.38149e14i) q^{87} +5.21821e14 q^{88} -3.87504e14i q^{89} -6.73581e14i q^{90} +9.27133e14 q^{91} -5.93315e13 q^{92} -3.96991e14 q^{93} +5.84261e13 q^{94} +1.15614e15i q^{95} +5.44399e14 q^{96} -5.65458e14i q^{97} +2.08536e15i q^{98} +2.01527e15i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) $ 36 * q - 508108 * q^4 + 470082 * q^5 + 1112016 * q^6 - 4820620 * q^7 - 167460710 * q^9	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9} + 133305618 q^{13} + 5626041364 q^{16} - 30737731548 q^{20} - 51638088984 q^{22} - 23459433564 q^{23} - 13473060100 q^{24} + 169887741474 q^{25} + 281303298768 q^{28} - 85550328684 q^{29} - 681215606256 q^{30} + 831111242422 q^{33} - 449988200584 q^{34} + 726838987044 q^{35} + 1809260484664 q^{36} - 2518300733088 q^{38} - 5363921425320 q^{42} - 16561773855556 q^{45} + 29824615981340 q^{49} + 1184881612900 q^{51} + 21527128606228 q^{52} - 40200435711486 q^{53} + 9043904345168 q^{54} + 42099004809572 q^{57} - 3461494533632 q^{58} - 50458797940572 q^{59} - 298531808710416 q^{62} + 159779590145904 q^{63} - 71569159267548 q^{64} + 92095395748902 q^{65} + 130146715692752 q^{67} - 178710878083152 q^{71} - 205323946615296 q^{74} + 13818320315976 q^{78} + 857820862108188 q^{80} + 126746036597568 q^{81} + 249211917251112 q^{82} - 541736282848188 q^{83} + 630538772195064 q^{86} - 633552108095260 q^{87} + 969723837884556 q^{88} - 962583563732444 q^{91} + 22\!\cdots\!64 q^{92}+ \cdots + 40\!\cdots\!64 q^{96}+O(q^{100}) \) 36 * q - 508108 * q^4 + 470082 * q^5 + 1112016 * q^6 - 4820620 * q^7 - 167460710 * q^9 + 133305618 * q^13 + 5626041364 * q^16 - 30737731548 * q^20 - 51638088984 * q^22 - 23459433564 * q^23 - 13473060100 * q^24 + 169887741474 * q^25 + 281303298768 * q^28 - 85550328684 * q^29 - 681215606256 * q^30 + 831111242422 * q^33 - 449988200584 * q^34 + 726838987044 * q^35 + 1809260484664 * q^36 - 2518300733088 * q^38 - 5363921425320 * q^42 - 16561773855556 * q^45 + 29824615981340 * q^49 + 1184881612900 * q^51 + 21527128606228 * q^52 - 40200435711486 * q^53 + 9043904345168 * q^54 + 42099004809572 * q^57 - 3461494533632 * q^58 - 50458797940572 * q^59 - 298531808710416 * q^62 + 159779590145904 * q^63 - 71569159267548 * q^64 + 92095395748902 * q^65 + 130146715692752 * q^67 - 178710878083152 * q^71 - 205323946615296 * q^74 + 13818320315976 * q^78 + 857820862108188 * q^80 + 126746036597568 * q^81 + 249211917251112 * q^82 - 541736282848188 * q^83 + 630538772195064 * q^86 - 633552108095260 * q^87 + 969723837884556 * q^88 - 962583563732444 * q^91 + 2248076163071664 * q^92 - 212103831463414 * q^93 - 5356847193577464 * q^94 + 4021088569901964 * q^96

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/29\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			202.124i			1.11659i	0.829643	+	0.558294i	$0.188544\pi$
$2$			202.124i			1.11659i	−0.829643	+	0.558294i	$0.811456\pi$
$3$			5797.90i			1.53060i	0.643674	+	0.765299i	$0.277409\pi$
$3$			5797.90i			1.53060i	−0.643674	+	0.765299i	$0.722591\pi$
$4$	−8086.11			−0.246769
$5$	172967.			0.990122			0.495061	−	0.868858i	$-0.335146\pi$
$5$	172967.			0.990122			0.495061	+	0.868858i	$0.335146\pi$
$6$	−1.17189e6			−1.70905
$7$	3.88134e6			1.78134			0.890669	−	0.454652i	$-0.150236\pi$
$7$	3.88134e6			1.78134			0.890669	+	0.454652i	$0.150236\pi$
$8$			4.98880e6i			0.841049i
$9$	−1.92667e7			−1.34273
$10$			3.49608e7i			1.10556i
$11$		−	1.04599e8i		−	1.61838i	−0.587547	−	0.809190i	$-0.699906\pi$
$11$		−	1.04599e8i		−	1.61838i	0.587547	−	0.809190i	$-0.300094\pi$
$12$		−	4.68825e7i		−	0.377704i
$13$	2.38869e8			1.05581			0.527904	−	0.849304i	$-0.322978\pi$
$13$	2.38869e8			1.05581			0.527904	+	0.849304i	$0.322978\pi$
$14$			7.84512e8i			1.98902i
$15$			1.00285e9i			1.51548i
$16$	−1.27332e9			−1.18587
$17$			8.65555e8i			0.511597i	0.966730	+	0.255798i	$0.0823383\pi$
$17$			8.65555e8i			0.511597i	−0.966730	+	0.255798i	$0.917662\pi$
$18$		−	3.89427e9i		−	1.49928i
$19$			6.68418e9i			1.71552i	0.514049	+	0.857761i	$0.328145\pi$
$19$			6.68418e9i			1.71552i	−0.514049	+	0.857761i	$0.671855\pi$
$20$	−1.39863e9			−0.244331
$21$			2.25036e10i			2.72651i
$22$	2.11419e10			1.80706
$23$	7.33746e9			0.449352			0.224676	−	0.974433i	$-0.427868\pi$
$23$	7.33746e9			0.449352			0.224676	+	0.974433i	$0.427868\pi$
$24$	−2.89246e10			−1.28731
$25$	−5.99915e8			−0.0196580
$26$			4.82812e10i			1.17890i
$27$		−	2.85131e10i		−	0.524586i
$28$	−3.13849e10			−0.439578
$29$	−9.28180e10	+	3.74350e9i	−0.999188	+	0.0402989i
$29$	−9.28180e10	+	3.74350e9i	−0.999188	+	0.0402989i
$30$	−2.02699e11			−1.69217
$31$			6.84716e10i			0.446990i	0.974705	+	0.223495i	$0.0717466\pi$
$31$			6.84716e10i			0.446990i	−0.974705	+	0.223495i	$0.928253\pi$
$32$		−	9.38959e10i		−	0.483083i
$33$	6.06452e11			2.47709
$34$	−1.74949e11			−0.571243
$35$	6.71345e11			1.76374
$36$	1.55793e11			0.331344
$37$		−	6.08649e11i		−	1.05403i	−0.849855	−	0.527016i	$-0.823311\pi$
$37$		−	6.08649e11i		−	1.05403i	0.849855	−	0.527016i	$-0.176689\pi$
$38$	−1.35103e12			−1.91553
$39$			1.38494e12i			1.61602i
$40$			8.62899e11i			0.832741i
$41$		−	6.76659e11i		−	0.542614i	−0.962493	−	0.271307i	$-0.912544\pi$
$41$		−	6.76659e11i		−	0.542614i	0.962493	−	0.271307i	$-0.0874558\pi$
$42$	−4.54852e12			−3.04439
$43$		−	1.65758e12i		−	0.929953i	−0.885323	−	0.464977i	$-0.846063\pi$
$43$		−	1.65758e12i		−	0.929953i	0.885323	−	0.464977i	$-0.153937\pi$
$44$			8.45795e11i			0.399365i
$45$	−3.33252e12			−1.32947
$46$			1.48308e12i			0.501741i
$47$		−	2.89060e11i		−	0.0832252i	−0.999134	−	0.0416126i	$-0.986750\pi$
$47$		−	2.89060e11i		−	0.0832252i	0.999134	−	0.0416126i	$-0.0132495\pi$
$48$		−	7.38260e12i		−	1.81510i
$49$	1.03172e13			2.17317
$50$		−	1.21257e11i		−	0.0219499i
$51$	−5.01840e12			−0.783050
$52$	−1.93152e12			−0.260540
$53$	−6.08957e12			−0.712062			−0.356031	−	0.934474i	$-0.615870\pi$
$53$	−6.08957e12			−0.712062			−0.356031	+	0.934474i	$0.615870\pi$
$54$	5.76319e12			0.585746
$55$		−	1.80921e13i		−	1.60239i
$56$			1.93632e13i			1.49819i
$57$	−3.87542e13			−2.62578
$58$	−7.56651e11	−	1.87607e13i	−0.0449972	−	1.11568i
$59$	−1.27876e13			−0.668956			−0.334478	−	0.942403i	$-0.608560\pi$
$59$	−1.27876e13			−0.668956			−0.334478	+	0.942403i	$0.608560\pi$
$60$		−	8.10913e12i		−	0.373973i
$61$			4.08548e12i			0.166444i	0.996531	+	0.0832221i	$0.0265211\pi$
$61$			4.08548e12i			0.166444i	−0.996531	+	0.0832221i	$0.973479\pi$
$62$	−1.38398e13			−0.499103
$63$	−7.47808e13			−2.39186
$64$	−2.27456e13			−0.646469
$65$	4.13165e13			1.04538
$66$			1.22578e14i			2.76589i
$67$	7.48894e13			1.50959			0.754796	−	0.655960i	$-0.227736\pi$
$67$	7.48894e13			1.50959			0.754796	+	0.655960i	$0.227736\pi$
$68$		−	6.99898e12i		−	0.126246i
$69$			4.25419e13i			0.687778i
$70$			1.35695e14i			1.96937i
$71$	−1.13931e13			−0.148663			−0.0743316	−	0.997234i	$-0.523682\pi$
$71$	−1.13931e13			−0.148663			−0.0743316	+	0.997234i	$0.523682\pi$
$72$		−	9.61180e13i		−	1.12930i
$73$			3.81672e13i			0.404361i	0.979348	+	0.202180i	$0.0648027\pi$
$73$			3.81672e13i			0.404361i	−0.979348	+	0.202180i	$0.935197\pi$
$74$	1.23022e14			1.17692
$75$		−	3.47825e12i		−	0.0300885i
$76$		−	5.40490e13i		−	0.423337i
$77$		−	4.05983e14i		−	2.88288i
$78$	−2.79930e14			−1.80443
$79$			2.03728e14i			1.19357i	0.802402	+	0.596785i	$0.203555\pi$
$79$			2.03728e14i			1.19357i	−0.802402	+	0.596785i	$0.796445\pi$
$80$	−2.20243e14			−1.17416
$81$	−1.11140e14			−0.539802
$82$	1.36769e14			0.605876
$83$	3.95848e14			1.60119			0.800596	−	0.599205i	$-0.204517\pi$
$83$	3.95848e14			1.60119			0.800596	+	0.599205i	$0.204517\pi$
$84$		−	1.81967e14i		−	0.672818i
$85$			1.49713e14i			0.506543i
$86$	3.35037e14			1.03837
$87$	−2.17044e13	−	5.38149e14i	−0.0616814	−	1.52936i
$88$	5.21821e14			1.36114
$89$		−	3.87504e14i		−	0.928648i	−0.885665	−	0.464324i	$-0.846297\pi$
$89$		−	3.87504e14i		−	0.928648i	0.885665	−	0.464324i	$-0.153703\pi$
$90$		−	6.73581e14i		−	1.48447i
$91$	9.27133e14			1.88075
$92$	−5.93315e13			−0.110886
$93$	−3.96991e14			−0.684162
$94$	5.84261e13			0.0929282
$95$			1.15614e15i			1.69858i
$96$	5.44399e14			0.739407
$97$		−	5.65458e14i		−	0.710579i	−0.934756	−	0.355290i	$-0.884382\pi$
$97$		−	5.65458e14i		−	0.710579i	0.934756	−	0.355290i	$-0.115618\pi$
$98$			2.08536e15i			2.42653i
$99$			2.01527e15i			2.17305i
$100$	4.85098e12			0.00485098
$101$		−	1.82022e15i		−	1.68932i	−0.535302	−	0.844661i	$-0.679802\pi$
$101$		−	1.82022e15i		−	1.68932i	0.535302	−	0.844661i	$-0.320198\pi$
$102$		−	1.01434e15i		−	0.874344i
$103$	−2.00823e15			−1.60892			−0.804459	−	0.594008i	$-0.797545\pi$
$103$	−2.00823e15			−1.60892			−0.804459	+	0.594008i	$0.797545\pi$
$104$			1.19167e15i			0.887986i
$105$			3.89239e15i			2.69958i
$106$		−	1.23085e15i		−	0.795079i
$107$	2.54294e15			1.53094			0.765468	−	0.643474i	$-0.222508\pi$
$107$	2.54294e15			1.53094			0.765468	+	0.643474i	$0.222508\pi$
$108$			2.30560e14i			0.129451i
$109$	−8.85669e14			−0.464058			−0.232029	−	0.972709i	$-0.574537\pi$
$109$	−8.85669e14			−0.464058			−0.232029	+	0.972709i	$0.574537\pi$
$110$	3.65685e15			1.78921
$111$	3.52888e15			1.61330
$112$	−4.94220e15			−2.11244
$113$			2.87444e15i			1.14938i	0.818369	+	0.574692i	$0.194878\pi$
$113$			2.87444e15i			1.14938i	−0.818369	+	0.574692i	$0.805122\pi$
$114$		−	7.83316e15i		−	2.93191i
$115$	1.26914e15			0.444914
$116$	7.50536e14	−	3.02704e13i	0.246568	−	0.00994449i
$117$	−4.60223e15			−1.41767
$118$		−	2.58467e15i		−	0.746949i
$119$			3.35951e15i			0.911327i
$120$	−5.00300e15			−1.27459
$121$	−6.76360e15			−1.61915
$122$	−8.25773e14			−0.185850
$123$	3.92320e15			0.830524
$124$		−	5.53669e14i		−	0.110303i
$125$	−5.38231e15			−1.00959
$126$		−	1.51150e16i		−	2.67072i
$127$		−	3.27759e15i		−	0.545791i	−0.962044	−	0.272896i	$-0.912019\pi$
$127$		−	3.27759e15i		−	0.545791i	0.962044	−	0.272896i	$-0.0879814\pi$
$128$		−	7.67421e15i		−	1.20492i
$129$	9.61049e15			1.42339
$130$			8.35106e15i			1.16726i
$131$		−	9.76379e15i		−	1.28850i	−0.764816	−	0.644249i	$-0.777170\pi$
$131$		−	9.76379e15i		−	1.28850i	0.764816	−	0.644249i	$-0.222830\pi$
$132$	−4.90384e15			−0.611268
$133$			2.59436e16i			3.05592i
$134$			1.51369e16i			1.68559i
$135$		−	4.93184e15i		−	0.519404i
$136$	−4.31808e15			−0.430278
$137$		−	2.25933e15i		−	0.213096i	−0.994308	−	0.106548i	$-0.966020\pi$
$137$		−	2.25933e15i		−	0.213096i	0.994308	−	0.106548i	$-0.0339798\pi$
$138$	−8.59873e15			−0.767965
$139$	−1.32717e16			−1.12283			−0.561415	−	0.827534i	$-0.689743\pi$
$139$	−1.32717e16			−1.12283			−0.561415	+	0.827534i	$0.689743\pi$
$140$	−5.42857e15			−0.435236
$141$	1.67594e15			0.127384
$142$		−	2.30281e15i		−	0.165995i
$143$		−	2.49854e16i		−	1.70870i
$144$	2.45328e16			1.59231
$145$	−1.60545e16	+	6.47503e14i	−0.989318	+	0.0399008i
$146$	−7.71451e15			−0.451504
$147$			5.98184e16i			3.32625i
$148$			4.92160e15i			0.260102i
$149$	1.31906e16			0.662779			0.331389	−	0.943494i	$-0.392483\pi$
$149$	1.31906e16			0.662779			0.331389	+	0.943494i	$0.392483\pi$
$150$	7.03037e14			0.0335965
$151$	8.90893e15			0.405041			0.202520	−	0.979278i	$-0.435087\pi$
$151$	8.90893e15			0.405041			0.202520	+	0.979278i	$0.435087\pi$
$152$	−3.33461e16			−1.44284
$153$		−	1.66764e16i		−	0.686938i
$154$	8.20588e16			3.21899
$155$			1.18433e16i			0.442575i
$156$		−	1.11988e16i		−	0.398782i
$157$		−	6.50750e15i		−	0.220885i	−0.993882	−	0.110443i	$-0.964773\pi$
$157$		−	6.50750e15i		−	0.220885i	0.993882	−	0.110443i	$-0.0352269\pi$
$158$	−4.11783e16			−1.33272
$159$		−	3.53067e16i		−	1.08988i
$160$		−	1.62409e16i		−	0.478311i
$161$	2.84792e16			0.800449
$162$		−	2.24641e16i		−	0.602736i
$163$			1.26200e16i			0.323335i	0.986845	+	0.161668i	$0.0516872\pi$
$163$			1.26200e16i			0.323335i	−0.986845	+	0.161668i	$0.948313\pi$
$164$			5.47154e15i			0.133900i
$165$	1.04896e17			2.45262
$166$			8.00105e16i			1.78787i
$167$	3.53108e15			0.0754283			0.0377141	−	0.999289i	$-0.487992\pi$
$167$	3.53108e15			0.0754283			0.0377141	+	0.999289i	$0.487992\pi$
$168$	−1.12266e17			−2.29313
$169$	5.87259e15			0.114731
$170$	−3.02605e16			−0.565600
$171$		−	1.28782e17i		−	2.30349i
$172$			1.34034e16i			0.229483i
$173$	8.75395e16			1.43502			0.717511	−	0.696547i	$-0.245281\pi$
$173$	8.75395e16			1.43502			0.717511	+	0.696547i	$0.245281\pi$
$174$	1.08773e17	−	4.38699e15i	1.70766	−	0.0688727i
$175$	−2.32847e15			−0.0350176
$176$			1.33188e17i			1.91919i
$177$		−	7.41410e16i		−	1.02390i
$178$	7.83239e16			1.03692
$179$	−1.02651e17			−1.30307			−0.651534	−	0.758620i	$-0.725874\pi$
$179$	−1.02651e17			−1.30307			−0.651534	+	0.758620i	$0.725874\pi$
$180$	2.69471e16			0.328071
$181$	2.24341e16			0.262011			0.131006	−	0.991382i	$-0.458179\pi$
$181$	2.24341e16			0.262011			0.131006	+	0.991382i	$0.458179\pi$
$182$			1.87396e17i			2.10002i
$183$	−2.36872e16			−0.254759
$184$			3.66051e16i			0.377927i
$185$		−	1.05276e17i		−	1.04362i
$186$		−	8.02415e16i		−	0.763927i
$187$	9.05358e16			0.827958
$188$			2.33737e15i			0.0205374i
$189$		−	1.10669e17i		−	0.934465i
$190$	−2.33685e17			−1.89661
$191$		−	1.61101e17i		−	1.25703i	−0.777796	−	0.628517i	$-0.783662\pi$
$191$		−	1.61101e17i		−	1.25703i	0.777796	−	0.628517i	$-0.216338\pi$
$192$		−	1.31877e17i		−	0.989485i
$193$			2.19085e16i			0.158100i	0.996871	+	0.0790502i	$0.0251887\pi$
$193$			2.19085e16i			0.158100i	−0.996871	+	0.0790502i	$0.974811\pi$
$194$	1.14293e17			0.793424
$195$			2.39549e17i			1.60006i
$196$	−8.34264e16			−0.536269
$197$	−9.55276e15			−0.0591061			−0.0295531	−	0.999563i	$-0.509408\pi$
$197$	−9.55276e15			−0.0591061			−0.0295531	+	0.999563i	$0.509408\pi$
$198$	−4.07335e17			−2.42640
$199$	1.83870e17			1.05466			0.527331	−	0.849660i	$-0.323193\pi$
$199$	1.83870e17			1.05466			0.527331	+	0.849660i	$0.323193\pi$
$200$		−	2.99286e15i		−	0.0165334i
$201$			4.34201e17i			2.31058i
$202$	3.67909e17			1.88628
$203$	−3.60258e17	+	1.45298e16i	−1.77989	+	0.0717859i
$204$	4.05794e16			0.193232
$205$		−	1.17040e17i		−	0.537254i
$206$		−	4.05911e17i		−	1.79650i
$207$	−1.41369e17			−0.603360
$208$	−3.04157e17			−1.25206
$209$	6.99156e17			2.77637
$210$	−7.86745e17			−3.01432
$211$		−	4.61787e16i		−	0.170735i	−0.996350	−	0.0853676i	$-0.972794\pi$
$211$		−	4.61787e16i		−	0.170735i	0.996350	−	0.0853676i	$-0.0272065\pi$
$212$	4.92409e16			0.175714
$213$		−	6.60559e16i		−	0.227544i
$214$			5.13988e17i			1.70942i
$215$		−	2.86707e17i		−	0.920767i
$216$	1.42246e17			0.441203
$217$			2.65762e17i			0.796240i
$218$		−	1.79015e17i		−	0.518162i
$219$	−2.21290e17			−0.618914
$220$			1.46295e17i			0.395420i
$221$			2.06754e17i			0.540148i
$222$			7.13272e17i			1.80139i
$223$	−4.51345e17			−1.10210			−0.551052	−	0.834471i	$-0.685773\pi$
$223$	−4.51345e17			−1.10210			−0.551052	+	0.834471i	$0.685773\pi$
$224$		−	3.64442e17i		−	0.860535i
$225$	1.15584e16			0.0263955
$226$	−5.80994e17			−1.28339
$227$	−1.19365e17			−0.255084			−0.127542	−	0.991833i	$-0.540709\pi$
$227$	−1.19365e17			−0.255084			−0.127542	+	0.991833i	$0.540709\pi$
$228$	3.13371e17			0.647959
$229$			5.86018e17i			1.17259i	0.810099	+	0.586293i	$0.199413\pi$
$229$			5.86018e17i			1.17259i	−0.810099	+	0.586293i	$0.800587\pi$
$230$			2.56524e17i			0.496785i
$231$	2.35385e18			4.41254
$232$	−1.86756e16	−	4.63050e17i	−0.0338933	−	0.840366i
$233$	−7.44566e17			−1.30838			−0.654189	−	0.756331i	$-0.726990\pi$
$233$	−7.44566e17			−1.30838			−0.654189	+	0.756331i	$0.726990\pi$
$234$		−	9.30221e17i		−	1.58295i
$235$		−	4.99980e16i		−	0.0824031i
$236$	1.03402e17			0.165077
$237$	−1.18119e18			−1.82688
$238$	−6.79039e17			−1.01758
$239$	7.41572e16			0.107688			0.0538442	−	0.998549i	$-0.482853\pi$
$239$	7.41572e16			0.107688			0.0538442	+	0.998549i	$0.482853\pi$
$240$		−	1.27695e18i		−	1.79717i
$241$	5.04459e17			0.688173			0.344087	−	0.938938i	$-0.388189\pi$
$241$	5.04459e17			0.688173			0.344087	+	0.938938i	$0.388189\pi$
$242$		−	1.36709e18i		−	1.80793i
$243$		−	1.05351e18i		−	1.35081i
$244$		−	3.30356e16i		−	0.0410732i
$245$	1.78455e18			2.15170
$246$			7.92973e17i			0.927353i
$247$			1.59664e18i			1.81126i
$248$	−3.41591e17			−0.375940
$249$			2.29509e18i			2.45078i
$250$		−	1.08789e18i		−	1.12729i
$251$		−	9.87778e17i		−	0.993359i	−0.867934	−	0.496680i	$-0.834552\pi$
$251$		−	9.87778e17i		−	0.993359i	0.867934	−	0.496680i	$-0.165448\pi$
$252$	6.04686e17			0.590236
$253$		−	7.67488e17i		−	0.727223i
$254$	6.62480e17			0.609424
$255$	−8.68019e17			−0.775315
$256$	8.05815e17			0.698933
$257$	1.69417e18			1.42711			0.713555	−	0.700599i	$-0.247084\pi$
$257$	1.69417e18			1.42711			0.713555	+	0.700599i	$0.247084\pi$
$258$			1.94251e18i			1.58933i
$259$		−	2.36237e18i		−	1.87759i
$260$	−3.34090e17			−0.257967
$261$	1.78830e18	−	7.21251e16i	1.34164	−	0.0541106i
$262$	1.97350e18			1.43872
$263$		−	1.26434e18i		−	0.895765i	−0.894092	−	0.447883i	$-0.852178\pi$
$263$		−	1.26434e18i		−	0.895765i	0.894092	−	0.447883i	$-0.147822\pi$
$264$			3.02547e18i			2.08335i
$265$	−1.05330e18			−0.705028
$266$	−5.24382e18			−3.41221
$267$	2.24671e18			1.42139
$268$	−6.05564e17			−0.372520
$269$			1.50934e18i			0.902914i	0.892293	+	0.451457i	$0.149096\pi$
$269$			1.50934e18i			0.902914i	−0.892293	+	0.451457i	$0.850904\pi$
$270$	9.96843e17			0.579961
$271$			2.06639e18i			1.16934i	0.811270	+	0.584672i	$0.198777\pi$
$271$			2.06639e18i			1.16934i	−0.811270	+	0.584672i	$0.801223\pi$
$272$		−	1.10213e18i		−	0.606689i
$273$			5.37542e18i			2.87868i
$274$	4.56665e17			0.237941
$275$			6.27502e16i			0.0318141i
$276$		−	3.43998e17i		−	0.169722i
$277$	1.40692e18			0.675570			0.337785	−	0.941223i	$-0.390322\pi$
$277$	1.40692e18			0.675570			0.337785	+	0.941223i	$0.390322\pi$
$278$		−	2.68252e18i		−	1.25374i
$279$		−	1.31922e18i		−	0.600188i
$280$			3.34921e18i			1.48339i
$281$	−3.39166e18			−1.46257			−0.731283	−	0.682074i	$-0.761078\pi$
$281$	−3.39166e18			−1.46257			−0.731283	+	0.682074i	$0.761078\pi$
$282$			3.38748e17i			0.142236i
$283$	−8.03806e16			−0.0328665			−0.0164332	−	0.999865i	$-0.505231\pi$
$283$	−8.03806e16			−0.0328665			−0.0164332	+	0.999865i	$0.505231\pi$
$284$	9.21257e16			0.0366854
$285$	−6.70321e18			−2.59984
$286$	5.05014e18			1.90791
$287$		−	2.62634e18i		−	0.966579i
$288$			1.80907e18i			0.648652i
$289$	2.11324e18			0.738269
$290$	−1.30876e17	−	3.24499e18i	−0.0445527	−	1.10466i
$291$	3.27847e18			1.08761
$292$		−	3.08624e17i		−	0.0997834i
$293$		−	3.63655e18i		−	1.14599i	−0.819558	−	0.572997i	$-0.805781\pi$
$293$		−	3.63655e18i		−	1.14599i	0.819558	−	0.572997i	$-0.194219\pi$
$294$	−1.20907e19			−3.71405
$295$	−2.21183e18			−0.662349
$296$	3.03643e18			0.886492
$297$	−2.98243e18			−0.848979
$298$			2.66614e18i			0.740051i
$299$	1.75269e18			0.474430
$300$			2.81255e16i			0.00742490i
$301$		−	6.43363e18i		−	1.65656i
$302$			1.80071e18i			0.452263i
$303$	1.05534e19			2.58567
$304$		−	8.51112e18i		−	2.03439i
$305$			7.06653e17i			0.164800i
$306$	3.37071e18			0.767026
$307$		−	1.47469e17i		−	0.0327462i	−0.999866	−	0.0163731i	$-0.994788\pi$
$307$		−	1.47469e17i		−	0.0327462i	0.999866	−	0.0163731i	$-0.00521196\pi$
$308$			3.28282e18i			0.711405i
$309$		−	1.16435e19i		−	2.46261i
$310$	−2.39382e18			−0.494173
$311$		−	3.83423e18i		−	0.772636i	−0.922366	−	0.386318i	$-0.873747\pi$
$311$		−	3.83423e18i		−	0.772636i	0.922366	−	0.386318i	$-0.126253\pi$
$312$	−6.90919e18			−1.35915
$313$	−8.06608e18			−1.54910			−0.774551	−	0.632512i	$-0.782024\pi$
$313$	−8.06608e18			−1.54910			−0.774551	+	0.632512i	$0.782024\pi$
$314$	1.31532e18			0.246638
$315$	−1.29346e19			−2.36823
$316$		−	1.64737e18i		−	0.294535i
$317$			2.75412e18i			0.480883i	0.970664	+	0.240441i	$0.0772922\pi$
$317$			2.75412e18i			0.480883i	−0.970664	+	0.240441i	$0.922708\pi$
$318$	7.13633e18			1.21695
$319$	3.91565e17	+	9.70862e18i	0.0652189	+	1.61707i
$320$	−3.93424e18			−0.640083
$321$			1.47437e19i			2.34325i
$322$			5.75633e18i			0.893771i
$323$	−5.78553e18			−0.877656
$324$	8.98694e17			0.133206
$325$	−1.43301e17			−0.0207551
$326$	−2.55081e18			−0.361032
$327$		−	5.13502e18i		−	0.710287i
$328$	3.37572e18			0.456365
$329$		−	1.12194e18i		−	0.148252i
$330$			2.12021e19i			2.73857i
$331$		−	1.19799e19i		−	1.51267i	−0.654184	−	0.756336i	$-0.726988\pi$
$331$		−	1.19799e19i		−	1.51267i	0.654184	−	0.756336i	$-0.273012\pi$
$332$	−3.20087e18			−0.395124
$333$			1.17267e19i			1.41528i
$334$			7.13716e17i			0.0842223i
$335$	1.29534e19			1.49468
$336$		−	2.86544e19i		−	3.23330i
$337$			1.66222e19i			1.83427i	0.398574	+	0.917136i	$0.369505\pi$
$337$			1.66222e19i			1.83427i	−0.398574	+	0.917136i	$0.630495\pi$
$338$			1.18699e18i			0.128107i
$339$	−1.66657e19			−1.75925
$340$		−	1.21059e18i		−	0.124999i
$341$	7.16203e18			0.723399
$342$	2.60300e19			2.57205
$343$	2.16178e19			2.08981
$344$	8.26934e18			0.782136
$345$			7.35835e18i			0.680984i
$346$			1.76938e19i			1.60233i
$347$	−5.94255e18			−0.526625			−0.263313	−	0.964711i	$-0.584815\pi$
$347$	−5.94255e18			−0.526625			−0.263313	+	0.964711i	$0.584815\pi$
$348$	1.75504e17	+	4.35154e18i	0.0152210	+	0.377397i
$349$	1.67526e19			1.42198			0.710988	−	0.703204i	$-0.248248\pi$
$349$	1.67526e19			1.42198			0.710988	+	0.703204i	$0.248248\pi$
$350$		−	4.70640e17i		−	0.0391002i
$351$		−	6.81091e18i		−	0.553862i
$352$	−9.82138e18			−0.781812
$353$	4.08263e18			0.318148			0.159074	−	0.987267i	$-0.449149\pi$
$353$	4.08263e18			0.318148			0.159074	+	0.987267i	$0.449149\pi$
$354$	1.49857e19			1.14328
$355$	−1.97063e18			−0.147195
$356$			3.13340e18i			0.229161i
$357$	−1.94781e19			−1.39488
$358$		−	2.07483e19i		−	1.45499i
$359$		−	9.79779e18i		−	0.672852i	−0.941710	−	0.336426i	$-0.890782\pi$
$359$		−	9.79779e18i		−	0.672852i	0.941710	−	0.336426i	$-0.109218\pi$
$360$		−	1.66253e19i		−	1.11815i
$361$	−2.94972e19			−1.94301
$362$			4.53448e18i			0.292559i
$363$		−	3.92147e19i		−	2.47827i
$364$	−7.49690e18			−0.464110
$365$			6.60167e18i			0.400366i
$366$		−	4.78775e18i		−	0.284461i
$367$		−	8.09702e18i		−	0.471335i	−0.971834	−	0.235668i	$-0.924272\pi$
$367$		−	8.09702e18i		−	0.471335i	0.971834	−	0.235668i	$-0.0757276\pi$
$368$	−9.34295e18			−0.532875
$369$			1.30370e19i			0.728585i
$370$	2.12789e19			1.16529
$371$	−2.36357e19			−1.26842
$372$	3.21012e18			0.168830
$373$	2.63174e19			1.35652			0.678262	−	0.734820i	$-0.262734\pi$
$373$	2.63174e19			1.35652			0.678262	+	0.734820i	$0.262734\pi$
$374$			1.82995e19i			0.924488i
$375$		−	3.12061e19i		−	1.54527i
$376$	1.44207e18			0.0699965
$377$	−2.21714e19	+	8.94207e17i	−1.05495	+	0.0425479i
$378$	2.23689e19			1.04341
$379$			3.92978e19i			1.79711i	0.438862	+	0.898555i	$0.355382\pi$
$379$			3.92978e19i			1.79711i	−0.438862	+	0.898555i	$0.644618\pi$
$380$		−	9.34871e18i		−	0.419155i
$381$	1.90031e19			0.835388
$382$	3.25623e19			1.40359
$383$	2.48046e19			1.04844			0.524218	−	0.851584i	$-0.324358\pi$
$383$	2.48046e19			1.04844			0.524218	+	0.851584i	$0.324358\pi$
$384$	4.44943e19			1.84425
$385$		−	7.02217e19i		−	2.85441i
$386$	−4.42823e18			−0.176533
$387$			3.19362e19i			1.24868i
$388$			4.57236e18i			0.175349i
$389$			1.32961e19i			0.500153i	0.968226	+	0.250077i	$0.0804558\pi$
$389$			1.32961e19i			0.500153i	−0.968226	+	0.250077i	$0.919544\pi$
$390$	−4.84186e19			−1.78660
$391$			6.35098e18i			0.229887i
$392$			5.14707e19i			1.82774i
$393$	5.66095e19			1.97217
$394$		−	1.93084e18i		−	0.0659972i
$395$			3.52383e19i			1.18178i
$396$		−	1.62957e19i		−	0.536241i
$397$	−1.23143e19			−0.397630			−0.198815	−	0.980037i	$-0.563709\pi$
$397$	−1.23143e19			−0.397630			−0.198815	+	0.980037i	$0.563709\pi$
$398$			3.71646e19i			1.17762i
$399$	−1.50418e20			−4.67739
$400$	7.63885e17			0.0233119
$401$	1.38520e19			0.414886			0.207443	−	0.978247i	$-0.433486\pi$
$401$	1.38520e19			0.414886			0.207443	+	0.978247i	$0.433486\pi$
$402$	−8.77625e19			−2.57996
$403$			1.63558e19i			0.471935i
$404$			1.47185e19i			0.416871i
$405$	−1.92236e19			−0.534470
$406$	−2.93682e18	−	7.28168e19i	−0.0801553	−	1.98741i
$407$	−6.36637e19			−1.70582
$408$		−	2.50358e19i		−	0.658583i
$409$			2.94970e19i			0.761821i	0.924612	+	0.380911i	$0.124389\pi$
$409$			2.94970e19i			0.761821i	−0.924612	+	0.380911i	$0.875611\pi$
$410$	2.36566e19			0.599891
$411$	1.30994e19			0.326165
$412$	1.62388e19			0.397030
$413$	−4.96329e19			−1.19164
$414$		−	2.85741e19i		−	0.673704i
$415$	6.84688e19			1.58538
$416$		−	2.24288e19i		−	0.510043i
$417$		−	7.69478e19i		−	1.71860i
$418$			1.41316e20i			3.10006i
$419$	−3.59914e19			−0.775521			−0.387761	−	0.921760i	$-0.626751\pi$
$419$	−3.59914e19			−0.775521			−0.387761	+	0.921760i	$0.626751\pi$
$420$		−	3.14743e19i		−	0.666172i
$421$			3.42241e19i			0.711568i	0.934568	+	0.355784i	$0.115786\pi$
$421$			3.42241e19i			0.711568i	−0.934568	+	0.355784i	$0.884214\pi$
$422$	9.33382e18			0.190641
$423$			5.56925e18i			0.111749i
$424$		−	3.03797e19i		−	0.598879i
$425$		−	5.19259e17i		−	0.0100570i
$426$	1.33515e19			0.254072
$427$			1.58571e19i			0.296494i
$428$	−2.05625e19			−0.377787
$429$	1.44863e20			2.61533
$430$	5.79504e19			1.02812
$431$	−2.40632e19			−0.419540			−0.209770	−	0.977751i	$-0.567272\pi$
$431$	−2.40632e19			−0.419540			−0.209770	+	0.977751i	$0.567272\pi$
$432$			3.63064e19i			0.622093i
$433$		−	5.16099e19i		−	0.869108i	−0.900646	−	0.434554i	$-0.856906\pi$
$433$		−	5.16099e19i		−	0.869108i	0.900646	−	0.434554i	$-0.143094\pi$
$434$	−5.37168e19			−0.889072
$435$	−3.75416e18	−	9.30822e19i	−0.0610721	−	1.51425i
$436$	7.16162e18			0.114515
$437$			4.90449e19i			0.770874i
$438$		−	4.47279e19i		−	0.691072i
$439$	5.46716e19			0.830382			0.415191	−	0.909734i	$-0.363715\pi$
$439$	5.46716e19			0.830382			0.415191	+	0.909734i	$0.363715\pi$
$440$	9.02580e19			1.34769
$441$	−1.98780e20			−2.91798
$442$	−4.17900e19			−0.603123
$443$			6.17014e19i			0.875522i	0.899091	+	0.437761i	$0.144228\pi$
$443$			6.17014e19i			0.875522i	−0.899091	+	0.437761i	$0.855772\pi$
$444$	−2.85349e19			−0.398112
$445$		−	6.70255e19i		−	0.919475i
$446$		−	9.12277e19i		−	1.23059i
$447$			7.64780e19i			1.01445i
$448$	−8.82834e19			−1.15158
$449$			4.68380e19i			0.600829i	0.953809	+	0.300415i	$0.0971250\pi$
$449$			4.68380e19i			0.600829i	−0.953809	+	0.300415i	$0.902875\pi$
$450$			2.33623e18i			0.0294728i
$451$	−7.07775e19			−0.878155
$452$		−	2.32431e19i		−	0.283632i
$453$			5.16531e19i			0.619955i
$454$		−	2.41265e19i		−	0.284824i
$455$	1.60364e20			1.86217
$456$		−	1.93337e20i		−	2.20841i
$457$	1.54257e20			1.73330			0.866650	−	0.498917i	$-0.166269\pi$
$457$	1.54257e20			1.73330			0.866650	+	0.498917i	$0.166269\pi$
$458$	−1.18448e20			−1.30930
$459$	2.46797e19			0.268377
$460$	−1.02624e19			−0.109791
$461$			5.76215e19i			0.606495i	0.952912	+	0.303248i	$0.0980710\pi$
$461$			5.76215e19i			0.606495i	−0.952912	+	0.303248i	$0.901929\pi$
$462$			4.75769e20i			4.92698i
$463$	2.29865e19			0.234216			0.117108	−	0.993119i	$-0.462638\pi$
$463$	2.29865e19			0.234216			0.117108	+	0.993119i	$0.462638\pi$
$464$	1.18187e20	−	4.76668e18i	1.18491	−	0.0477894i
$465$	−6.86665e19			−0.677404
$466$		−	1.50495e20i		−	1.46092i
$467$		−	1.72688e19i		−	0.164962i	−0.996593	−	0.0824812i	$-0.973716\pi$
$467$		−	1.72688e19i		−	0.164962i	0.996593	−	0.0824812i	$-0.0262844\pi$
$468$	3.72142e19			0.349836
$469$	2.90671e20			2.68909
$470$	1.01058e19			0.0920103
$471$	3.77298e19			0.338087
$472$		−	6.37946e19i		−	0.562625i
$473$	−1.73380e20			−1.50502
$474$		−	2.38748e20i		−	2.03987i
$475$		−	4.00994e18i		−	0.0337237i
$476$		−	2.71654e19i		−	0.224887i
$477$	1.17326e20			0.956108
$478$			1.49889e19i			0.120244i
$479$		−	3.28572e19i		−	0.259486i	−0.991548	−	0.129743i	$-0.958585\pi$
$479$		−	3.28572e19i		−	0.259486i	0.991548	−	0.129743i	$-0.0414152\pi$
$480$	9.41632e19			0.732103
$481$		−	1.45387e20i		−	1.11286i
$482$			1.01963e20i			0.768406i
$483$			1.65119e20i			1.22517i
$484$	5.46913e19			0.399556
$485$		−	9.78058e19i		−	0.703561i
$486$	2.12940e20			1.50829
$487$	1.29693e20			0.904587			0.452293	−	0.891869i	$-0.350606\pi$
$487$	1.29693e20			0.904587			0.452293	+	0.891869i	$0.350606\pi$
$488$	−2.03816e19			−0.139988
$489$	−7.31696e19			−0.494897
$490$			3.60699e20i			2.40256i
$491$		−	2.53589e19i		−	0.166349i	−0.996535	−	0.0831744i	$-0.973494\pi$
$491$		−	2.53589e19i		−	0.166349i	0.996535	−	0.0831744i	$-0.0265059\pi$
$492$	−3.17234e19			−0.204947
$493$	−3.24021e18	−	8.03391e19i	−0.0206168	−	0.511181i
$494$	−3.22720e20			−2.02243
$495$			3.48576e20i			2.15159i
$496$		−	8.71864e19i		−	0.530074i
$497$	−4.42204e19			−0.264819
$498$	−4.63893e20			−2.73651
$499$	−1.25981e19			−0.0732066			−0.0366033	−	0.999330i	$-0.511654\pi$
$499$	−1.25981e19			−0.0732066			−0.0366033	+	0.999330i	$0.511654\pi$
$500$	4.35219e19			0.249134
$501$			2.04729e19i			0.115450i
$502$	1.99654e20			1.10917
$503$		−	7.88684e19i		−	0.431661i	−0.976431	−	0.215831i	$-0.930754\pi$
$503$		−	7.88684e19i		−	0.431661i	0.976431	−	0.215831i	$-0.0692459\pi$
$504$		−	3.73067e20i		−	2.01167i
$505$		−	3.14838e20i		−	1.67264i
$506$	1.55128e20			0.812008
$507$			3.40487e19i			0.175606i
$508$			2.65030e19i			0.134684i
$509$	−1.02949e20			−0.515513			−0.257756	−	0.966210i	$-0.582983\pi$
$509$	−1.02949e20			−0.515513			−0.257756	+	0.966210i	$0.582983\pi$
$510$		−	1.75448e20i		−	0.865707i
$511$			1.48140e20i			0.720303i
$512$		−	8.85941e19i		−	0.424502i
$513$	1.90587e20			0.899939
$514$			3.42432e20i			1.59349i
$515$	−3.47358e20			−1.59303
$516$	−7.77114e19			−0.351247
$517$	−3.02353e19			−0.134690
$518$	4.77492e20			2.09649
$519$			5.07546e20i			2.19644i
$520$			2.06120e20i			0.879215i
$521$	−8.92247e19			−0.375147			−0.187574	−	0.982251i	$-0.560062\pi$
$521$	−8.92247e19			−0.375147			−0.187574	+	0.982251i	$0.560062\pi$
$522$	1.45782e19	+	3.61458e20i	0.0604192	+	1.49806i
$523$	−4.86317e19			−0.198681			−0.0993405	−	0.995053i	$-0.531673\pi$
$523$	−4.86317e19			−0.198681			−0.0993405	+	0.995053i	$0.531673\pi$
$524$			7.89511e19i			0.317961i
$525$		−	1.35003e19i		−	0.0535978i
$526$	2.55552e20			1.00020
$527$	−5.92659e19			−0.228679
$528$	−7.72209e20			−2.93752
$529$	−2.12797e20			−0.798082
$530$		−	2.12896e20i		−	0.787226i
$531$	2.46375e20			0.898230
$532$		−	2.09783e20i		−	0.754106i
$533$		−	1.61633e20i		−	0.572896i
$534$			4.54114e20i			1.58710i
$535$	4.39845e20			1.51581
$536$			3.73608e20i			1.26964i
$537$		−	5.95162e20i		−	1.99447i
$538$	−3.05075e20			−1.00818
$539$		−	1.07917e21i		−	3.51701i
$540$			3.98794e19i			0.128173i
$541$		−	5.37144e20i		−	1.70259i	−0.524685	−	0.851297i	$-0.675817\pi$
$541$		−	5.37144e20i		−	1.70259i	0.524685	−	0.851297i	$-0.324183\pi$
$542$	−4.17667e20			−1.30568
$543$			1.30071e20i			0.401034i
$544$	8.12721e19			0.247144
$545$	−1.53192e20			−0.459475
$546$	−1.08650e21			−3.21429
$547$	4.82635e20			1.40836			0.704180	−	0.710021i	$-0.251315\pi$
$547$	4.82635e20			1.40836			0.704180	+	0.710021i	$0.251315\pi$
$548$			1.82692e19i			0.0525854i
$549$		−	7.87138e19i		−	0.223490i
$550$	−1.26833e19			−0.0355233
$551$	−2.50222e19	−	6.20412e20i	−0.0691336	−	1.71413i
$552$	−2.12233e20			−0.578455
$553$			7.90738e20i			2.12615i
$554$			2.84372e20i			0.754333i
$555$	6.10381e20			1.59736
$556$	1.07316e20			0.277079
$557$	1.26345e20			0.321842			0.160921	−	0.986967i	$-0.448553\pi$
$557$	1.26345e20			0.321842			0.160921	+	0.986967i	$0.448553\pi$
$558$	2.66647e20			0.670162
$559$		−	3.95945e20i		−	0.981852i
$560$	−8.54838e20			−2.09158
$561$			5.24918e20i			1.26727i
$562$		−	6.85537e20i		−	1.63308i
$563$			3.27567e20i			0.769994i	0.922918	+	0.384997i	$0.125798\pi$
$563$			3.27567e20i			0.769994i	−0.922918	+	0.384997i	$0.874202\pi$
$564$	−1.35519e19			−0.0314344
$565$			4.97185e20i			1.13803i
$566$		−	1.62468e19i		−	0.0366983i
$567$	−4.31374e20			−0.961570
$568$		−	5.68378e19i		−	0.125033i
$569$			3.27748e20i			0.711537i	0.934574	+	0.355768i	$0.115781\pi$
$569$			3.27748e20i			0.711537i	−0.934574	+	0.355768i	$0.884219\pi$
$570$		−	1.35488e21i		−	2.90295i
$571$	−1.58317e20			−0.334778			−0.167389	−	0.985891i	$-0.553534\pi$
$571$	−1.58317e20			−0.334778			−0.167389	+	0.985891i	$0.553534\pi$
$572$			2.02034e20i			0.421653i
$573$	9.34046e20			1.92402
$574$	5.30847e20			1.07927
$575$	−4.40185e18			−0.00883337
$576$	4.38234e20			0.868035
$577$		−	4.85705e20i		−	0.949629i	−0.880086	−	0.474814i	$-0.842515\pi$
$577$		−	4.85705e20i		−	0.949629i	0.880086	−	0.474814i	$-0.157485\pi$
$578$			4.27136e20i			0.824342i
$579$	−1.27023e20			−0.241988
$580$	1.29818e20	−	5.23578e18i	0.244133	−	0.00984626i
$581$	1.53642e21			2.85226
$582$			6.62658e20i			1.21441i
$583$			6.36960e20i			1.15239i
$584$	−1.90409e20			−0.340087
$585$	−7.96035e20			−1.40366
$586$	7.35033e20			1.27960
$587$	−1.10884e21			−1.90582			−0.952911	−	0.303251i	$-0.901928\pi$
$587$	−1.10884e21			−1.90582			−0.952911	+	0.303251i	$0.901928\pi$
$588$		−	4.83698e20i		−	0.820813i
$589$	−4.57676e20			−0.766821
$590$		−	4.47064e20i		−	0.739570i
$591$		−	5.53860e19i		−	0.0904677i
$592$			7.75006e20i			1.24995i
$593$	1.02947e21			1.63948			0.819738	−	0.572738i	$-0.194119\pi$
$593$	1.02947e21			1.63948			0.819738	+	0.572738i	$0.194119\pi$
$594$		−	6.02821e20i		−	0.947960i
$595$			5.81086e20i			0.902325i
$596$	−1.06661e20			−0.163553
$597$			1.06606e21i			1.61426i
$598$			3.54261e20i			0.529743i
$599$			4.13646e20i			0.610841i	0.952218	+	0.305420i	$0.0987970\pi$
$599$			4.13646e20i			0.610841i	−0.952218	+	0.305420i	$0.901203\pi$
$600$	1.73523e19			0.0253059
$601$			1.11267e21i			1.60254i	0.598302	+	0.801271i	$0.295842\pi$
$601$			1.11267e21i			1.60254i	−0.598302	+	0.801271i	$0.704158\pi$
$602$	1.30039e21			1.84970
$603$	−1.44288e21			−2.02698
$604$	−7.20386e19			−0.0999513
$605$	−1.16988e21			−1.60316
$606$			2.13310e21i			2.88713i
$607$		−	9.71925e19i		−	0.129932i	−0.997887	−	0.0649662i	$-0.979306\pi$
$607$		−	9.71925e19i		−	0.129932i	0.997887	−	0.0649662i	$-0.0206940\pi$
$608$	6.27617e20			0.828740
$609$	−8.42423e19	−	2.08874e21i	−0.109875	−	2.72430i
$610$	−1.42832e20			−0.184014
$611$		−	6.90476e19i		−	0.0878698i
$612$			1.34847e20i			0.169515i
$613$	−4.21274e20			−0.523131			−0.261566	−	0.965186i	$-0.584239\pi$
$613$	−4.21274e20			−0.523131			−0.261566	+	0.965186i	$0.584239\pi$
$614$	2.98069e19			0.0365641
$615$	6.78585e20			0.822320
$616$	2.02537e21			2.42465
$617$		−	8.67640e20i		−	1.02613i	−0.858351	−	0.513063i	$-0.828511\pi$
$617$		−	8.67640e20i		−	1.02613i	0.858351	−	0.513063i	$-0.171489\pi$
$618$	2.35343e21			2.74972
$619$			6.26656e19i			0.0723351i	0.999346	+	0.0361676i	$0.0115150\pi$
$619$			6.26656e19i			0.0723351i	−0.999346	+	0.0361676i	$0.988485\pi$
$620$		−	9.57666e19i		−	0.109213i
$621$		−	2.09214e20i		−	0.235724i
$622$	7.74989e20			0.862716
$623$		−	1.50404e21i		−	1.65424i
$624$		−	1.76347e21i		−	1.91639i
$625$	−9.12655e20			−0.979956
$626$		−	1.63035e21i		−	1.72971i
$627$			4.05363e21i			4.24950i
$628$			5.26204e19i			0.0545075i
$629$	5.26819e20			0.539239
$630$		−	2.61440e21i		−	2.64434i
$631$	−1.58765e21			−1.58684			−0.793422	−	0.608672i	$-0.791702\pi$
$631$	−1.58765e21			−1.58684			−0.793422	+	0.608672i	$0.791702\pi$
$632$	−1.01636e21			−1.00385
$633$	2.67739e20			0.261327
$634$	−5.56675e20			−0.536948
$635$		−	5.66916e20i		−	0.540400i
$636$			2.85494e20i			0.268948i
$637$	2.46447e21			2.29445
$638$	−1.96235e21	+	7.91446e19i	−1.80560	+	0.0728226i
$639$	2.19507e20			0.199615
$640$		−	1.32739e21i		−	1.19302i
$641$		−	5.56122e20i		−	0.494009i	−0.969014	−	0.247005i	$-0.920554\pi$
$641$		−	5.56122e20i		−	0.494009i	0.969014	−	0.247005i	$-0.0794463\pi$
$642$	−2.98005e21			−2.61644
$643$	−4.80994e20			−0.417404			−0.208702	−	0.977979i	$-0.566924\pi$
$643$	−4.80994e20			−0.417404			−0.208702	+	0.977979i	$0.566924\pi$
$644$	−2.30286e20			−0.197526
$645$	1.66230e21			1.40933
$646$		−	1.16939e21i		−	0.979980i
$647$	1.04535e20			0.0865921			0.0432961	−	0.999062i	$-0.486214\pi$
$647$	1.04535e20			0.0865921			0.0432961	+	0.999062i	$0.486214\pi$
$648$		−	5.54457e20i		−	0.454000i
$649$			1.33756e21i			1.08263i
$650$		−	2.89646e19i		−	0.0231749i
$651$	−1.54086e21			−1.21872
$652$		−	1.02047e20i		−	0.0797890i
$653$			1.54623e21i			1.19516i	0.801809	+	0.597580i	$0.203871\pi$
$653$			1.54623e21i			1.19516i	−0.801809	+	0.597580i	$0.796129\pi$
$654$	1.03791e21			0.793098
$655$		−	1.68882e21i		−	1.27577i
$656$			8.61605e20i			0.643471i
$657$		−	7.35358e20i		−	0.542948i
$658$	2.26771e20			0.165537
$659$		−	4.69139e20i		−	0.338580i	−0.985566	−	0.169290i	$-0.945853\pi$
$659$		−	4.69139e20i		−	0.338580i	0.985566	−	0.169290i	$-0.0541474\pi$
$660$	−8.48203e20			−0.605230
$661$	−4.47179e20			−0.315479			−0.157740	−	0.987481i	$-0.550421\pi$
$661$	−4.47179e20			−0.315479			−0.157740	+	0.987481i	$0.550421\pi$
$662$	2.42143e21			1.68903
$663$	−1.19874e21			−0.826750
$664$			1.97481e21i			1.34668i
$665$			4.48739e21i			3.02574i
$666$	−2.37024e21			−1.58029
$667$	−6.81048e20	+	2.74678e19i	−0.448987	+	0.0181084i
$668$	−2.85527e19			−0.0186133
$669$		−	2.61685e21i		−	1.68688i
$670$			2.61820e21i			1.66894i
$671$	4.27335e20			0.269370
$672$	2.11300e21			1.31713
$673$	2.28024e21			1.40562			0.702809	−	0.711379i	$-0.251929\pi$
$673$	2.28024e21			1.40562			0.702809	+	0.711379i	$0.251929\pi$
$674$	−3.35974e21			−2.04813
$675$			1.71055e19i			0.0103123i
$676$	−4.74864e19			−0.0283119
$677$			4.44766e20i			0.262251i	0.991366	+	0.131125i	$0.0418590\pi$
$677$			4.44766e20i			0.262251i	−0.991366	+	0.131125i	$0.958141\pi$
$678$		−	3.36855e21i		−	1.96435i
$679$		−	2.19474e21i		−	1.26578i
$680$	−7.46887e20			−0.426028
$681$		−	6.92066e20i		−	0.390431i
$682$			1.44762e21i			0.807739i
$683$	2.09377e21			1.15551			0.577756	−	0.816210i	$-0.303929\pi$
$683$	2.09377e21			1.15551			0.577756	+	0.816210i	$0.303929\pi$
$684$			1.04135e21i			0.568428i
$685$		−	3.90790e20i		−	0.210991i
$686$			4.36948e21i			2.33345i
$687$	−3.39767e21			−1.79476
$688$			2.11063e21i			1.10281i
$689$	−1.45461e21			−0.751800
$690$	−1.48730e21			−0.760379
$691$	−3.09625e21			−1.56585			−0.782925	−	0.622116i	$-0.786273\pi$
$691$	−3.09625e21			−1.56585			−0.782925	+	0.622116i	$0.786273\pi$
$692$	−7.07854e20			−0.354118
$693$			7.82196e21i			3.87094i
$694$		−	1.20113e21i		−	0.588023i
$695$	−2.29556e21			−1.11174
$696$	2.68472e21	−	1.08279e20i	1.28626	−	0.0518771i
$697$	5.85686e20			0.277600
$698$			3.38611e21i			1.58776i
$699$		−	4.31692e21i		−	2.00260i
$700$	1.88283e19			0.00864123
$701$	−2.54235e21			−1.15438			−0.577191	−	0.816609i	$-0.695851\pi$
$701$	−2.54235e21			−1.15438			−0.577191	+	0.816609i	$0.695851\pi$
$702$	1.37665e21			0.618436
$703$	4.06832e21			1.80821
$704$			2.37916e21i			1.04623i
$705$	2.89883e20			0.126126
$706$			8.25197e20i			0.355240i
$707$		−	7.06488e21i		−	3.00925i
$708$			5.99513e20i			0.252667i
$709$	−2.97451e21			−1.24042			−0.620209	−	0.784437i	$-0.712952\pi$
$709$	−2.97451e21			−1.24042			−0.620209	+	0.784437i	$0.712952\pi$
$710$		−	3.98311e20i		−	0.164356i
$711$		−	3.92518e21i		−	1.60264i
$712$	1.93318e21			0.781039
$713$			5.02408e20i			0.200856i
$714$		−	3.93700e21i		−	1.55750i
$715$		−	4.32165e21i		−	1.69182i
$716$	8.30050e20			0.321556
$717$			4.29956e20i			0.164828i
$718$	1.98037e21			0.751298
$719$	3.65604e21			1.37260			0.686301	−	0.727317i	$-0.259233\pi$
$719$	3.65604e21			1.37260			0.686301	+	0.727317i	$0.259233\pi$
$720$	4.24337e21			1.57658
$721$	−7.79462e21			−2.86603
$722$		−	5.96208e21i		−	2.16955i
$723$			2.92480e21i			1.05332i
$724$	−1.81405e20			−0.0646562
$725$	5.56829e19	−	2.24578e18i	0.0196420	−	0.000792195i
$726$	7.92623e21			2.76721
$727$		−	3.55779e21i		−	1.22934i	−0.788783	−	0.614671i	$-0.789289\pi$
$727$		−	3.55779e21i		−	1.22934i	0.788783	−	0.614671i	$-0.210711\pi$
$728$			4.62528e21i			1.58180i
$729$	4.51342e21			1.52774
$730$	−1.33436e21			−0.447044
$731$	1.43473e21			0.475761
$732$	1.91537e20			0.0628666
$733$		−	1.28932e21i		−	0.418872i	−0.977822	−	0.209436i	$-0.932837\pi$
$733$		−	1.28932e21i		−	0.418872i	0.977822	−	0.209436i	$-0.0671627\pi$
$734$	1.63660e21			0.526287
$735$			1.03466e22i			3.29339i
$736$		−	6.88958e20i		−	0.217075i
$737$		−	7.83332e21i		−	2.44309i
$738$	−2.63509e21			−0.813529
$739$			7.11265e20i			0.217369i	0.994076	+	0.108685i	$0.0346639\pi$
$739$			7.11265e20i			0.217369i	−0.994076	+	0.108685i	$0.965336\pi$
$740$			8.51275e20i			0.257533i
$741$	−9.25719e21			−2.77231
$742$		−	4.77734e21i		−	1.41631i
$743$		−	1.73488e21i		−	0.509159i	−0.967052	−	0.254580i	$-0.918063\pi$
$743$		−	1.73488e21i		−	0.509159i	0.967052	−	0.254580i	$-0.0819371\pi$
$744$		−	1.98051e21i		−	0.575414i
$745$	2.28155e21			0.656232
$746$			5.31939e21i			1.51468i
$747$	−7.62671e21			−2.14997
$748$	−7.32083e20			−0.204314
$749$	9.87000e21			2.72712
$750$	6.30750e21			1.72543
$751$			1.57861e20i			0.0427537i	0.999771	+	0.0213769i	$0.00680499\pi$
$751$			1.57861e20i			0.0427537i	−0.999771	+	0.0213769i	$0.993195\pi$
$752$			3.68067e20i			0.0986946i
$753$	5.72704e21			1.52043
$754$	−1.80741e20	−	4.48136e21i	−0.0475084	−	1.17794i
$755$	1.54095e21			0.401040
$756$			8.94883e20i			0.230597i
$757$			4.27417e21i			1.09052i	0.838268	+	0.545258i	$0.183568\pi$
$757$			4.27417e21i			1.09052i	−0.838268	+	0.545258i	$0.816432\pi$
$758$	−7.94303e21			−2.00663
$759$	4.44982e21			1.11309
$760$	−5.76778e21			−1.42859
$761$	−6.57891e20			−0.161350			−0.0806750	−	0.996740i	$-0.525708\pi$
$761$	−6.57891e20			−0.161350			−0.0806750	+	0.996740i	$0.525708\pi$
$762$			3.84099e21i			0.932784i
$763$	−3.43758e21			−0.826645
$764$			1.30268e21i			0.310196i
$765$		−	2.88448e21i		−	0.680152i
$766$			5.01361e21i			1.17067i
$767$	−3.05456e21			−0.706290
$768$			4.67203e21i			1.06979i
$769$		−	5.27770e21i		−	1.19673i	−0.801223	−	0.598366i	$-0.795817\pi$
$769$		−	5.27770e21i		−	1.19673i	0.801223	−	0.598366i	$-0.204183\pi$
$770$	1.41935e22			3.18719
$771$			9.82261e21i			2.18433i
$772$		−	1.77155e20i		−	0.0390142i
$773$		−	3.02437e21i		−	0.659613i	−0.944049	−	0.329806i	$-0.893017\pi$
$773$		−	3.02437e21i		−	0.659613i	0.944049	−	0.329806i	$-0.106983\pi$
$774$	−6.45507e21			−1.39426
$775$		−	4.10771e19i		−	0.00878693i
$776$	2.82096e21			0.597632
$777$	1.36968e22			2.87383
$778$	−2.68747e21			−0.558465
$779$	4.52291e21			0.930866
$780$		−	1.93702e21i		−	0.394843i
$781$			1.19170e21i			0.240593i
$782$	−1.28368e21			−0.256689
$783$	1.06739e20	+	2.64653e21i	0.0211402	+	0.524160i
$784$	−1.31372e22			−2.57710
$785$		−	1.12558e21i		−	0.218703i
$786$			1.14421e22i			2.20211i
$787$	1.82321e21			0.347557			0.173778	−	0.984785i	$-0.444402\pi$
$787$	1.82321e21			0.347557			0.173778	+	0.984785i	$0.444402\pi$
$788$	7.72447e19			0.0145855
$789$	7.33049e21			1.37106
$790$	−7.12250e21			−1.31956
$791$			1.11567e22i			2.04744i
$792$	−1.00538e22			−1.82764
$793$			9.75894e20i			0.175733i
$794$		−	2.48901e21i		−	0.443989i
$795$		−	6.10690e21i		−	1.07911i
$796$	−1.48680e21			−0.260257
$797$			2.23773e21i			0.388035i	0.980998	+	0.194018i	$0.0621519\pi$
$797$			2.23773e21i			0.388035i	−0.980998	+	0.194018i	$0.937848\pi$
$798$		−	3.04032e22i		−	5.22272i
$799$	2.50198e20			0.0425777
$800$			5.63296e19i			0.00949645i
$801$			7.46594e21i			1.24693i
$802$			2.79982e21i			0.463257i
$803$	3.99223e21			0.654409
$804$		−	3.51100e21i		−	0.570178i
$805$	4.92597e21			0.792542
$806$	−3.30589e21			−0.526957
$807$	−8.75103e21			−1.38200
$808$	9.08069e21			1.42080
$809$			4.56072e21i			0.707001i	0.935434	+	0.353500i	$0.115009\pi$
$809$			4.56072e21i			0.707001i	−0.935434	+	0.353500i	$0.884991\pi$
$810$		−	3.88556e21i		−	0.596782i
$811$	−1.91938e21			−0.292082			−0.146041	−	0.989279i	$-0.546653\pi$
$811$	−1.91938e21			−0.292082			−0.146041	+	0.989279i	$0.546653\pi$
$812$	2.91309e21	−	1.17490e20i	0.439221	−	0.0177145i
$813$	−1.19807e22			−1.78980
$814$		−	1.28680e22i		−	1.90470i
$815$			2.18285e21i			0.320141i
$816$	6.39005e21			0.928598
$817$	1.10796e22			1.59536
$818$	−5.96205e21			−0.850640
$819$	−1.78628e22			−2.52535
$820$			9.46397e20i			0.132577i
$821$	−7.39062e21			−1.02591			−0.512953	−	0.858417i	$-0.671448\pi$
$821$	−7.39062e21			−1.02591			−0.512953	+	0.858417i	$0.671448\pi$
$822$			2.64770e21i			0.364192i
$823$			9.89173e21i			1.34826i	0.738613	+	0.674129i	$0.235481\pi$
$823$			9.89173e21i			1.34826i	−0.738613	+	0.674129i	$0.764519\pi$
$824$		−	1.00187e22i		−	1.35318i
$825$	−3.63819e20			−0.0486947
$826$		−	1.00320e22i		−	1.33057i
$827$			2.99392e21i			0.393504i	0.980453	+	0.196752i	$0.0630393\pi$
$827$			2.99392e21i			0.393504i	−0.980453	+	0.196752i	$0.936961\pi$
$828$	1.14313e21			0.148890
$829$		−	4.85482e21i		−	0.626634i	−0.949649	−	0.313317i	$-0.898560\pi$
$829$		−	4.85482e21i		−	0.626634i	0.949649	−	0.313317i	$-0.101440\pi$
$830$			1.38392e22i			1.77021i
$831$			8.15717e21i			1.03403i
$832$	−5.43322e21			−0.682547
$833$			8.93014e21i			1.11179i
$834$	1.55530e22			1.91897
$835$	6.10761e20			0.0746832
$836$	−5.65345e21			−0.685120
$837$	1.95234e21			0.234485
$838$		−	7.27473e21i		−	0.865938i
$839$		−	1.12402e22i		−	1.32605i	−0.748599	−	0.663023i	$-0.769273\pi$
$839$		−	1.12402e22i		−	1.32605i	0.748599	−	0.663023i	$-0.230727\pi$
$840$	−1.94184e22			−2.27048
$841$	8.60116e21	−	6.94928e20i	0.996752	−	0.0805323i
$842$	−6.91751e21			−0.794528
$843$		−	1.96645e22i		−	2.23860i
$844$			3.73406e20i			0.0421321i
$845$	1.01576e21			0.113597
$846$	−1.12568e21			−0.124778
$847$	−2.62519e22			−2.88426
$848$	7.75398e21			0.844415
$849$		−	4.66039e20i		−	0.0503054i
$850$	1.04955e20			0.0112295
$851$		−	4.46593e21i		−	0.473632i
$852$			5.34135e20i			0.0561506i
$853$		−	9.35030e19i		−	0.00974334i	−0.999988	−	0.00487167i	$-0.998449\pi$
$853$		−	9.35030e19i		−	0.00974334i	0.999988	−	0.00487167i	$-0.00155071\pi$
$854$	−3.20511e21			−0.331061
$855$		−	2.22751e22i		−	2.28073i
$856$			1.26862e22i			1.28759i
$857$	−1.12839e21			−0.113529			−0.0567643	−	0.998388i	$-0.518078\pi$
$857$	−1.12839e21			−0.113529			−0.0567643	+	0.998388i	$0.518078\pi$
$858$			2.92802e22i			2.92025i
$859$			1.46937e22i			1.45272i	0.687312	+	0.726362i	$0.258790\pi$
$859$			1.46937e22i			1.45272i	−0.687312	+	0.726362i	$0.741210\pi$
$860$			2.31834e21i			0.227216i
$861$	1.52273e22			1.47944
$862$		−	4.86375e21i		−	0.468453i
$863$	−1.10631e22			−1.05632			−0.528160	−	0.849145i	$-0.677118\pi$
$863$	−1.10631e22			−1.05632			−0.528160	+	0.849145i	$0.677118\pi$
$864$	−2.67727e21			−0.253419
$865$	1.51415e22			1.42085
$866$	1.04316e22			0.970435
$867$			1.22523e22i			1.12999i
$868$		−	2.14898e21i		−	0.196487i
$869$	2.13096e22			1.93165
$870$	1.88141e22	−	7.58805e20i	1.69079	−	0.0681924i
$871$	1.78888e22			1.59384
$872$		−	4.41843e21i		−	0.390296i
$873$			1.08945e22i			0.954118i
$874$	−9.91315e21			−0.860748
$875$	−2.08906e22			−1.79841
$876$	1.78937e21			0.152728
$877$	−9.09691e21			−0.769834			−0.384917	−	0.922951i	$-0.625770\pi$
$877$	−9.09691e21			−0.769834			−0.384917	+	0.922951i	$0.625770\pi$
$878$			1.10504e22i			0.927195i
$879$	2.10843e22			1.75406
$880$			2.30371e22i			1.90024i
$881$		−	1.40325e22i		−	1.14767i	−0.818972	−	0.573833i	$-0.805456\pi$
$881$		−	1.40325e22i		−	1.14767i	0.818972	−	0.573833i	$-0.194544\pi$
$882$		−	4.01781e22i		−	3.25818i
$883$	−9.10087e20			−0.0731775			−0.0365887	−	0.999330i	$-0.511649\pi$
$883$	−9.10087e20			−0.0731775			−0.0365887	+	0.999330i	$0.511649\pi$
$884$		−	1.67184e21i		−	0.133292i
$885$		−	1.28240e22i		−	1.01379i
$886$	−1.24713e22			−0.977598
$887$			1.04844e22i			0.814922i	0.913223	+	0.407461i	$0.133586\pi$
$887$			1.04844e22i			0.814922i	−0.913223	+	0.407461i	$0.866414\pi$
$888$			1.76049e22i			1.35686i
$889$		−	1.27214e22i		−	0.972239i
$890$	1.35475e22			1.02667
$891$			1.16251e22i			0.873604i
$892$	3.64963e21			0.271964
$893$	1.93213e21			0.142775
$894$	−1.54580e22			−1.13272
$895$	−1.77553e22			−1.29020
$896$		−	2.97862e22i		−	2.14638i
$897$			1.01619e22i			0.726162i
$898$	−9.46709e21			−0.670879
$899$	−2.56323e20	−	6.35539e21i	−0.0180132	−	0.446627i
$900$	−9.34626e19			−0.00651357
$901$		−	5.27086e21i		−	0.364288i
$902$		−	1.43058e22i		−	0.980537i
$903$	3.73016e22			2.53553
$904$	−1.43400e22			−0.966689
$905$	3.88037e21			0.259423
$906$	−1.04403e22			−0.692234
$907$			1.33039e22i			0.874829i	0.899260	+	0.437415i	$0.144106\pi$
$907$			1.33039e22i			0.874829i	−0.899260	+	0.437415i	$0.855894\pi$
$908$	9.65199e20			0.0629467
$909$			3.50696e22i			2.26831i
$910$			3.24133e22i			2.07928i
$911$		−	1.34471e22i		−	0.855540i	−0.903888	−	0.427770i	$-0.859299\pi$
$911$		−	1.34471e22i		−	0.855540i	0.903888	−	0.427770i	$-0.140701\pi$
$912$	4.93466e22			3.11384
$913$		−	4.14052e22i		−	2.59134i
$914$			3.11791e22i			1.93538i
$915$	−4.09711e21			−0.252243
$916$		−	4.73861e21i		−	0.289357i
$917$		−	3.78966e22i		−	2.29525i
$918$			4.98836e21i			0.299666i
$919$	−1.44526e22			−0.861149			−0.430574	−	0.902555i	$-0.641689\pi$
$919$	−1.44526e22			−0.861149			−0.430574	+	0.902555i	$0.641689\pi$
$920$			6.33149e21i			0.374194i
$921$	8.55008e20			0.0501214
$922$	−1.16467e22			−0.677206
$923$	−2.72145e21			−0.156960
$924$	−1.90335e22			−1.08887
$925$			3.65137e20i			0.0207202i
$926$			4.64613e21i			0.261522i
$927$	3.86920e22			2.16035
$928$	3.51499e20	+	8.71523e21i	0.0194677	+	0.482691i
$929$	−1.57480e22			−0.865182			−0.432591	−	0.901590i	$-0.642401\pi$
$929$	−1.57480e22			−0.865182			−0.432591	+	0.901590i	$0.642401\pi$
$930$		−	1.38791e22i		−	0.756381i
$931$			6.89623e22i			3.72811i
$932$	6.02064e21			0.322867
$933$	2.22305e22			1.18260
$934$	3.49043e21			0.184195
$935$	1.56597e22			0.819780
$936$		−	2.29596e22i		−	1.19233i
$937$	2.71282e22			1.39757			0.698785	−	0.715332i	$-0.253724\pi$
$937$	2.71282e22			1.39757			0.698785	+	0.715332i	$0.253724\pi$
$938$			5.87516e22i			3.00261i
$939$		−	4.67663e22i		−	2.37105i
$940$			4.04289e20i			0.0203345i
$941$	−1.79526e22			−0.895789			−0.447895	−	0.894086i	$-0.647826\pi$
$941$	−1.79526e22			−0.895789			−0.447895	+	0.894086i	$0.647826\pi$
$942$			7.62611e21i			0.377504i
$943$		−	4.96496e21i		−	0.243825i
$944$	1.62827e22			0.793298
$945$		−	1.91421e22i		−	0.925235i
$946$		−	3.50443e22i		−	1.68048i
$947$			2.77747e21i			0.132137i	0.997815	+	0.0660686i	$0.0210456\pi$
$947$			2.77747e21i			0.132137i	−0.997815	+	0.0660686i	$0.978954\pi$
$948$	9.55127e21			0.450815
$949$			9.11697e21i			0.426927i
$950$	8.10505e20			0.0376555
$951$	−1.59681e22			−0.736038
$952$	−1.67600e22			−0.766471
$953$	3.23832e22			1.46934			0.734672	−	0.678422i	$-0.237336\pi$
$953$	3.23832e22			1.46934			0.734672	+	0.678422i	$0.237336\pi$
$954$			2.37144e22i			1.06758i
$955$		−	2.78651e22i		−	1.24462i
$956$	−5.99643e20			−0.0265741
$957$	−5.62896e22	+	2.27025e21i	−2.47508	+	0.0998239i
$958$	6.64122e21			0.289739
$959$		−	8.76923e21i		−	0.379596i
$960$		−	2.28104e22i		−	0.979711i
$961$	1.87769e22			0.800200
$962$	2.93863e22			1.24260
$963$	−4.89941e22			−2.05564
$964$	−4.07911e21			−0.169820
$965$			3.78945e21i			0.156539i
$966$	−3.33746e22			−1.36801
$967$			4.51942e22i			1.83816i	0.394068	+	0.919081i	$0.371067\pi$
$967$			4.51942e22i			1.83816i	−0.394068	+	0.919081i	$0.628933\pi$
$968$		−	3.37423e22i		−	1.36179i
$969$		−	3.35439e22i		−	1.34334i
$970$	1.97689e22			0.785587
$971$			1.81863e22i			0.717135i	0.933504	+	0.358567i	$0.116735\pi$
$971$			1.81863e22i			0.717135i	−0.933504	+	0.358567i	$0.883265\pi$
$972$			8.51883e21i			0.333336i
$973$	−5.15118e22			−2.00014
$974$			2.62141e22i			1.01005i
$975$		−	8.30846e20i		−	0.0317677i
$976$		−	5.20213e21i		−	0.197382i
$977$	−4.66096e22			−1.75496			−0.877478	−	0.479617i	$-0.840776\pi$
$977$	−4.66096e22			−1.75496			−0.877478	+	0.479617i	$0.840776\pi$
$978$		−	1.47893e22i		−	0.552595i
$979$	−4.05324e22			−1.50291
$980$	−1.44300e22			−0.530972
$981$	1.70640e22			0.623106
$982$	5.12565e21			0.185743
$983$		−	2.07808e22i		−	0.747327i	−0.927564	−	0.373663i	$-0.878102\pi$
$983$		−	2.07808e22i		−	0.747327i	0.927564	−	0.373663i	$-0.121898\pi$
$984$			1.95721e22i			0.698511i
$985$	−1.65232e21			−0.0585223
$986$	1.62385e22	−	6.54923e20i	0.570779	−	0.0230204i
$987$	6.50491e21			0.226915
$988$		−	1.29106e22i		−	0.446962i
$989$		−	1.21624e22i		−	0.417877i
$990$	−7.04556e22			−2.40244
$991$	−4.66660e22			−1.57924			−0.789619	−	0.613597i	$-0.789722\pi$
$991$	−4.66660e22			−1.57924			−0.789619	+	0.613597i	$0.789722\pi$
$992$	6.42920e21			0.215933
$993$	6.94585e22			2.31529
$994$		−	8.93800e21i		−	0.295694i
$995$	3.18035e22			1.04424
$996$		−	1.85584e22i		−	0.604776i
$997$			6.13783e22i			1.98519i	0.121484	+	0.992593i	$0.461235\pi$
$997$			6.13783e22i			1.98519i	−0.121484	+	0.992593i	$0.538765\pi$
$998$		−	2.54638e21i		−	0.0817417i
$999$	−1.73545e22			−0.552930

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.28	yes	36
29.28	even	2	inner	29.16.b.a.28.9	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.9	✓	36	29.28	even	2	inner
29.16.b.a.28.28	yes	36	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:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	29.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+202.124i q^{2} +5797.90i q^{3} -8086.11 q^{4} +172967. q^{5} -1.17189e6 q^{6} +3.88134e6 q^{7} +4.98880e6i q^{8} -1.92667e7 q^{9} +O(q^{10})\)	\(q+202.124i q^{2} +5797.90i q^{3} -8086.11 q^{4} +172967. q^{5} -1.17189e6 q^{6} +3.88134e6 q^{7} +4.98880e6i q^{8} -1.92667e7 q^{9} +3.49608e7i q^{10} -1.04599e8i q^{11} -4.68825e7i q^{12} +2.38869e8 q^{13} +7.84512e8i q^{14} +1.00285e9i q^{15} -1.27332e9 q^{16} +8.65555e8i q^{17} -3.89427e9i q^{18} +6.68418e9i q^{19} -1.39863e9 q^{20} +2.25036e10i q^{21} +2.11419e10 q^{22} +7.33746e9 q^{23} -2.89246e10 q^{24} -5.99915e8 q^{25} +4.82812e10i q^{26} -2.85131e10i q^{27} -3.13849e10 q^{28} +(-9.28180e10 + 3.74350e9i) q^{29} -2.02699e11 q^{30} +6.84716e10i q^{31} -9.38959e10i q^{32} +6.06452e11 q^{33} -1.74949e11 q^{34} +6.71345e11 q^{35} +1.55793e11 q^{36} -6.08649e11i q^{37} -1.35103e12 q^{38} +1.38494e12i q^{39} +8.62899e11i q^{40} -6.76659e11i q^{41} -4.54852e12 q^{42} -1.65758e12i q^{43} +8.45795e11i q^{44} -3.33252e12 q^{45} +1.48308e12i q^{46} -2.89060e11i q^{47} -7.38260e12i q^{48} +1.03172e13 q^{49} -1.21257e11i q^{50} -5.01840e12 q^{51} -1.93152e12 q^{52} -6.08957e12 q^{53} +5.76319e12 q^{54} -1.80921e13i q^{55} +1.93632e13i q^{56} -3.87542e13 q^{57} +(-7.56651e11 - 1.87607e13i) q^{58} -1.27876e13 q^{59} -8.10913e12i q^{60} +4.08548e12i q^{61} -1.38398e13 q^{62} -7.47808e13 q^{63} -2.27456e13 q^{64} +4.13165e13 q^{65} +1.22578e14i q^{66} +7.48894e13 q^{67} -6.99898e12i q^{68} +4.25419e13i q^{69} +1.35695e14i q^{70} -1.13931e13 q^{71} -9.61180e13i q^{72} +3.81672e13i q^{73} +1.23022e14 q^{74} -3.47825e12i q^{75} -5.40490e13i q^{76} -4.05983e14i q^{77} -2.79930e14 q^{78} +2.03728e14i q^{79} -2.20243e14 q^{80} -1.11140e14 q^{81} +1.36769e14 q^{82} +3.95848e14 q^{83} -1.81967e14i q^{84} +1.49713e14i q^{85} +3.35037e14 q^{86} +(-2.17044e13 - 5.38149e14i) q^{87} +5.21821e14 q^{88} -3.87504e14i q^{89} -6.73581e14i q^{90} +9.27133e14 q^{91} -5.93315e13 q^{92} -3.96991e14 q^{93} +5.84261e13 q^{94} +1.15614e15i q^{95} +5.44399e14 q^{96} -5.65458e14i q^{97} +2.08536e15i q^{98} +2.01527e15i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 36 * q - 508108 * q^4 + 470082 * q^5 + 1112016 * q^6 - 4820620 * q^7 - 167460710 * q^9	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9} + 133305618 q^{13} + 5626041364 q^{16} - 30737731548 q^{20} - 51638088984 q^{22} - 23459433564 q^{23} - 13473060100 q^{24} + 169887741474 q^{25} + 281303298768 q^{28} - 85550328684 q^{29} - 681215606256 q^{30} + 831111242422 q^{33} - 449988200584 q^{34} + 726838987044 q^{35} + 1809260484664 q^{36} - 2518300733088 q^{38} - 5363921425320 q^{42} - 16561773855556 q^{45} + 29824615981340 q^{49} + 1184881612900 q^{51} + 21527128606228 q^{52} - 40200435711486 q^{53} + 9043904345168 q^{54} + 42099004809572 q^{57} - 3461494533632 q^{58} - 50458797940572 q^{59} - 298531808710416 q^{62} + 159779590145904 q^{63} - 71569159267548 q^{64} + 92095395748902 q^{65} + 130146715692752 q^{67} - 178710878083152 q^{71} - 205323946615296 q^{74} + 13818320315976 q^{78} + 857820862108188 q^{80} + 126746036597568 q^{81} + 249211917251112 q^{82} - 541736282848188 q^{83} + 630538772195064 q^{86} - 633552108095260 q^{87} + 969723837884556 q^{88} - 962583563732444 q^{91} + 22\!\cdots\!64 q^{92}+ \cdots + 40\!\cdots\!64 q^{96}+O(q^{100}) \) 36 * q - 508108 * q^4 + 470082 * q^5 + 1112016 * q^6 - 4820620 * q^7 - 167460710 * q^9 + 133305618 * q^13 + 5626041364 * q^16 - 30737731548 * q^20 - 51638088984 * q^22 - 23459433564 * q^23 - 13473060100 * q^24 + 169887741474 * q^25 + 281303298768 * q^28 - 85550328684 * q^29 - 681215606256 * q^30 + 831111242422 * q^33 - 449988200584 * q^34 + 726838987044 * q^35 + 1809260484664 * q^36 - 2518300733088 * q^38 - 5363921425320 * q^42 - 16561773855556 * q^45 + 29824615981340 * q^49 + 1184881612900 * q^51 + 21527128606228 * q^52 - 40200435711486 * q^53 + 9043904345168 * q^54 + 42099004809572 * q^57 - 3461494533632 * q^58 - 50458797940572 * q^59 - 298531808710416 * q^62 + 159779590145904 * q^63 - 71569159267548 * q^64 + 92095395748902 * q^65 + 130146715692752 * q^67 - 178710878083152 * q^71 - 205323946615296 * q^74 + 13818320315976 * q^78 + 857820862108188 * q^80 + 126746036597568 * q^81 + 249211917251112 * q^82 - 541736282848188 * q^83 + 630538772195064 * q^86 - 633552108095260 * q^87 + 969723837884556 * q^88 - 962583563732444 * q^91 + 2248076163071664 * q^92 - 212103831463414 * q^93 - 5356847193577464 * q^94 + 4021088569901964 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more