LMFDB - Embedded newform 29.16.b.a.28.22

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			28.22
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+84.1317i q^{2} -2443.60i q^{3} +25689.9 q^{4} +22406.3 q^{5} +205584. q^{6} +3.47198e6 q^{7} +4.91816e6i q^{8} +8.37775e6 q^{9} +O(q^{10})$	\(q+84.1317i q^{2} -2443.60i q^{3} +25689.9 q^{4} +22406.3 q^{5} +205584. q^{6} +3.47198e6 q^{7} +4.91816e6i q^{8} +8.37775e6 q^{9} +1.88508e6i q^{10} +5.52086e7i q^{11} -6.27756e7i q^{12} -4.72749e7 q^{13} +2.92103e8i q^{14} -5.47519e7i q^{15} +4.28032e8 q^{16} -5.10902e8i q^{17} +7.04834e8i q^{18} +5.33567e9i q^{19} +5.75614e8 q^{20} -8.48411e9i q^{21} -4.64480e9 q^{22} -6.90547e9 q^{23} +1.20180e10 q^{24} -3.00155e10 q^{25} -3.97731e9i q^{26} -5.55348e10i q^{27} +8.91946e10 q^{28} +(7.79779e10 - 5.04840e10i) q^{29} +4.60637e9 q^{30} +8.33256e9i q^{31} +1.97169e11i q^{32} +1.34908e11 q^{33} +4.29831e10 q^{34} +7.77941e10 q^{35} +2.15223e11 q^{36} -6.22321e11i q^{37} -4.48899e11 q^{38} +1.15521e11i q^{39} +1.10198e11i q^{40} +1.08930e12i q^{41} +7.13782e11 q^{42} +1.44300e12i q^{43} +1.41830e12i q^{44} +1.87714e11 q^{45} -5.80969e11i q^{46} +6.00837e12i q^{47} -1.04594e12i q^{48} +7.30706e12 q^{49} -2.52526e12i q^{50} -1.24844e12 q^{51} -1.21448e12 q^{52} +6.81866e12 q^{53} +4.67223e12 q^{54} +1.23702e12i q^{55} +1.70757e13i q^{56} +1.30382e13 q^{57} +(4.24731e12 + 6.56041e12i) q^{58} +2.38753e13 q^{59} -1.40657e12i q^{60} -1.51084e13i q^{61} -7.01033e11 q^{62} +2.90874e13 q^{63} -2.56244e12 q^{64} -1.05925e12 q^{65} +1.13500e13i q^{66} -6.27953e13 q^{67} -1.31250e13i q^{68} +1.68742e13i q^{69} +6.54495e12i q^{70} +7.71181e13 q^{71} +4.12031e13i q^{72} -9.88856e13i q^{73} +5.23569e13 q^{74} +7.33458e13i q^{75} +1.37073e14i q^{76} +1.91683e14i q^{77} -9.71895e12 q^{78} -3.04209e14i q^{79} +9.59061e12 q^{80} -1.54929e13 q^{81} -9.16445e13 q^{82} -1.95561e14 q^{83} -2.17956e14i q^{84} -1.14474e13i q^{85} -1.21402e14 q^{86} +(-1.23363e14 - 1.90546e14i) q^{87} -2.71525e14 q^{88} -3.35259e14i q^{89} +1.57927e13i q^{90} -1.64137e14 q^{91} -1.77400e14 q^{92} +2.03614e13 q^{93} -5.05494e14 q^{94} +1.19553e14i q^{95} +4.81802e14 q^{96} -2.00491e14i q^{97} +6.14756e14i q^{98} +4.62524e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			84.1317i			0.464766i	0.972624	+	0.232383i	$0.0746523\pi$
$2$			84.1317i			0.464766i	−0.972624	+	0.232383i	$0.925348\pi$
$3$		−	2443.60i		−	0.645089i	−0.946554	−	0.322545i	$-0.895462\pi$
$3$		−	2443.60i		−	0.645089i	0.946554	−	0.322545i	$-0.104538\pi$
$4$	25689.9			0.783992
$5$	22406.3			0.128261			0.0641305	−	0.997942i	$-0.479573\pi$
$5$	22406.3			0.128261			0.0641305	+	0.997942i	$0.479573\pi$
$6$	205584.			0.299816
$7$	3.47198e6			1.59346			0.796731	−	0.604334i	$-0.206561\pi$
$7$	3.47198e6			1.59346			0.796731	+	0.604334i	$0.206561\pi$
$8$			4.91816e6i			0.829140i
$9$	8.37775e6			0.583860
$10$			1.88508e6i			0.0596114i
$11$			5.52086e7i			0.854204i	0.904203	+	0.427102i	$0.140466\pi$
$11$			5.52086e7i			0.854204i	−0.904203	+	0.427102i	$0.859534\pi$
$12$		−	6.27756e7i		−	0.505745i
$13$	−4.72749e7			−0.208956			−0.104478	−	0.994527i	$-0.533317\pi$
$13$	−4.72749e7			−0.208956			−0.104478	+	0.994527i	$0.533317\pi$
$14$			2.92103e8i			0.740587i
$15$		−	5.47519e7i		−	0.0827398i
$16$	4.28032e8			0.398636
$17$		−	5.10902e8i		−	0.301975i	−0.988536	−	0.150987i	$-0.951755\pi$
$17$		−	5.10902e8i		−	0.301975i	0.988536	−	0.150987i	$-0.0482453\pi$
$18$			7.04834e8i			0.271358i
$19$			5.33567e9i			1.36942i	0.728815	+	0.684711i	$0.240071\pi$
$19$			5.33567e9i			1.36942i	−0.728815	+	0.684711i	$0.759929\pi$
$20$	5.75614e8			0.100556
$21$		−	8.48411e9i		−	1.02793i
$22$	−4.64480e9			−0.397005
$23$	−6.90547e9			−0.422897			−0.211448	−	0.977389i	$-0.567818\pi$
$23$	−6.90547e9			−0.422897			−0.211448	+	0.977389i	$0.567818\pi$
$24$	1.20180e10			0.534869
$25$	−3.00155e10			−0.983549
$26$		−	3.97731e9i		−	0.0971158i
$27$		−	5.55348e10i		−	1.02173i
$28$	8.91946e10			1.24926
$29$	7.79779e10	−	5.04840e10i	0.839434	−	0.543462i
$29$	7.79779e10	−	5.04840e10i	0.839434	−	0.543462i
$30$	4.60637e9			0.0384547
$31$			8.33256e9i			0.0543959i	0.999630	+	0.0271979i	$0.00865844\pi$
$31$			8.33256e9i			0.0543959i	−0.999630	+	0.0271979i	$0.991342\pi$
$32$			1.97169e11i			1.01441i
$33$	1.34908e11			0.551038
$34$	4.29831e10			0.140348
$35$	7.77941e10			0.204379
$36$	2.15223e11			0.457741
$37$		−	6.22321e11i		−	1.07771i	−0.842399	−	0.538854i	$-0.818857\pi$
$37$		−	6.22321e11i		−	1.07771i	0.842399	−	0.538854i	$-0.181143\pi$
$38$	−4.48899e11			−0.636461
$39$			1.15521e11i			0.134795i
$40$			1.10198e11i			0.106346i
$41$			1.08930e12i			0.873509i	0.899581	+	0.436755i	$0.143872\pi$
$41$			1.08930e12i			0.873509i	−0.899581	+	0.436755i	$0.856128\pi$
$42$	7.13782e11			0.477745
$43$			1.44300e12i			0.809570i	0.914412	+	0.404785i	$0.132654\pi$
$43$			1.44300e12i			0.809570i	−0.914412	+	0.404785i	$0.867346\pi$
$44$			1.41830e12i			0.669689i
$45$	1.87714e11			0.0748865
$46$		−	5.80969e11i		−	0.196548i
$47$			6.00837e12i			1.72991i	0.501853	+	0.864953i	$0.332652\pi$
$47$			6.00837e12i			1.72991i	−0.501853	+	0.864953i	$0.667348\pi$
$48$		−	1.04594e12i		−	0.257156i
$49$	7.30706e12			1.53912
$50$		−	2.52526e12i		−	0.457121i
$51$	−1.24844e12			−0.194801
$52$	−1.21448e12			−0.163820
$53$	6.81866e12			0.797315			0.398657	−	0.917100i	$-0.369476\pi$
$53$	6.81866e12			0.797315			0.398657	+	0.917100i	$0.369476\pi$
$54$	4.67223e12			0.474866
$55$			1.23702e12i			0.109561i
$56$			1.70757e13i			1.32120i
$57$	1.30382e13			0.883399
$58$	4.24731e12	+	6.56041e12i	0.252583	+	0.390141i
$59$	2.38753e13			1.24899			0.624495	−	0.781029i	$-0.285305\pi$
$59$	2.38753e13			1.24899			0.624495	+	0.781029i	$0.285305\pi$
$60$		−	1.40657e12i		−	0.0648674i
$61$		−	1.51084e13i		−	0.615526i	−0.951463	−	0.307763i	$-0.900420\pi$
$61$		−	1.51084e13i		−	0.615526i	0.951463	−	0.307763i	$-0.0995803\pi$
$62$	−7.01033e11			−0.0252814
$63$	2.90874e13			0.930358
$64$	−2.56244e12			−0.0728289
$65$	−1.05925e12			−0.0268009
$66$			1.13500e13i			0.256104i
$67$	−6.27953e13			−1.26580			−0.632901	−	0.774233i	$-0.718136\pi$
$67$	−6.27953e13			−1.26580			−0.632901	+	0.774233i	$0.718136\pi$
$68$		−	1.31250e13i		−	0.236746i
$69$			1.68742e13i			0.272806i
$70$			6.54495e12i			0.0949885i
$71$	7.71181e13			1.00628			0.503140	−	0.864205i	$-0.332178\pi$
$71$	7.71181e13			1.00628			0.503140	+	0.864205i	$0.332178\pi$
$72$			4.12031e13i			0.484101i
$73$		−	9.88856e13i		−	1.04764i	−0.851829	−	0.523820i	$-0.824507\pi$
$73$		−	9.88856e13i		−	1.04764i	0.851829	−	0.523820i	$-0.175493\pi$
$74$	5.23569e13			0.500883
$75$			7.33458e13i			0.634477i
$76$			1.37073e14i			1.07362i
$77$			1.91683e14i			1.36114i
$78$	−9.71895e12			−0.0626484
$79$		−	3.04209e14i		−	1.78225i	−0.453758	−	0.891125i	$-0.649917\pi$
$79$		−	3.04209e14i		−	1.78225i	0.453758	−	0.891125i	$-0.350083\pi$
$80$	9.59061e12			0.0511295
$81$	−1.54929e13			−0.0752482
$82$	−9.16445e13			−0.405978
$83$	−1.95561e14			−0.791038			−0.395519	−	0.918458i	$-0.629435\pi$
$83$	−1.95561e14			−0.791038			−0.395519	+	0.918458i	$0.629435\pi$
$84$		−	2.17956e14i		−	0.805885i
$85$		−	1.14474e13i		−	0.0387316i
$86$	−1.21402e14			−0.376261
$87$	−1.23363e14	−	1.90546e14i	−0.350581	−	0.541510i
$88$	−2.71525e14			−0.708255
$89$		−	3.35259e14i		−	0.803444i	−0.915762	−	0.401722i	$-0.868412\pi$
$89$		−	3.35259e14i		−	0.803444i	0.915762	−	0.401722i	$-0.131588\pi$
$90$			1.57927e13i			0.0348047i
$91$	−1.64137e14			−0.332963
$92$	−1.77400e14			−0.331548
$93$	2.03614e13			0.0350902
$94$	−5.05494e14			−0.804002
$95$			1.19553e14i			0.175643i
$96$	4.81802e14			0.654387
$97$		−	2.00491e14i		−	0.251946i	−0.992034	−	0.125973i	$-0.959795\pi$
$97$		−	2.00491e14i		−	0.251946i	0.992034	−	0.125973i	$-0.0402052\pi$
$98$			6.14756e14i			0.715331i
$99$			4.62524e14i			0.498735i
$100$	−7.71095e14			−0.771095
$101$		−	1.57147e15i		−	1.45846i	−0.684268	−	0.729231i	$-0.739878\pi$
$101$		−	1.57147e15i		−	1.45846i	0.684268	−	0.729231i	$-0.260122\pi$
$102$		−	1.05033e14i		−	0.0905369i
$103$	−3.50323e14			−0.280666			−0.140333	−	0.990104i	$-0.544817\pi$
$103$	−3.50323e14			−0.280666			−0.140333	+	0.990104i	$0.544817\pi$
$104$		−	2.32505e14i		−	0.173254i
$105$		−	1.90097e14i		−	0.131843i
$106$			5.73665e14i			0.370565i
$107$	−2.24275e14			−0.135021			−0.0675107	−	0.997719i	$-0.521506\pi$
$107$	−2.24275e14			−0.135021			−0.0675107	+	0.997719i	$0.521506\pi$
$108$		−	1.42668e15i		−	0.801029i
$109$	1.47425e15			0.772453			0.386226	−	0.922404i	$-0.373778\pi$
$109$	1.47425e15			0.772453			0.386226	+	0.922404i	$0.373778\pi$
$110$	−1.04073e14			−0.0509203
$111$	−1.52070e15			−0.695218
$112$	1.48612e15			0.635211
$113$			2.03913e15i			0.815375i	0.913122	+	0.407687i	$0.133665\pi$
$113$			2.03913e15i			0.815375i	−0.913122	+	0.407687i	$0.866335\pi$
$114$			1.09693e15i			0.410574i
$115$	−1.54726e14			−0.0542412
$116$	2.00324e15	−	1.29693e15i	0.658110	−	0.426070i
$117$	−3.96057e14			−0.122001
$118$			2.00867e15i			0.580488i
$119$		−	1.77384e15i		−	0.481185i
$120$	2.69279e14			0.0686029
$121$	1.12926e15			0.270335
$122$	1.27110e15			0.286076
$123$	2.66180e15			0.563492
$124$			2.14062e14i			0.0426459i
$125$	−1.35632e15			−0.254412
$126$			2.44717e15i			0.432399i
$127$		−	9.63841e15i		−	1.60501i	−0.596646	−	0.802505i	$-0.703500\pi$
$127$		−	9.63841e15i		−	1.60501i	0.596646	−	0.802505i	$-0.296500\pi$
$128$			6.24526e15i			0.980564i
$129$	3.52612e15			0.522245
$130$		−	8.91168e13i		−	0.0124562i
$131$			9.10129e15i			1.20107i	0.799598	+	0.600535i	$0.205046\pi$
$131$			9.10129e15i			1.20107i	−0.799598	+	0.600535i	$0.794954\pi$
$132$	3.46575e15			0.432010
$133$			1.85253e16i			2.18212i
$134$		−	5.28307e15i		−	0.588303i
$135$		−	1.24433e15i		−	0.131048i
$136$	2.51270e15			0.250379
$137$			7.98143e15i			0.752794i	0.926458	+	0.376397i	$0.122837\pi$
$137$			7.98143e15i			0.752794i	−0.926458	+	0.376397i	$0.877163\pi$
$138$	−1.41965e15			−0.126791
$139$	−2.60257e15			−0.220186			−0.110093	−	0.993921i	$-0.535115\pi$
$139$	−2.60257e15			−0.220186			−0.110093	+	0.993921i	$0.535115\pi$
$140$	1.99852e15			0.160232
$141$	1.46820e16			1.11594
$142$			6.48807e15i			0.467685i
$143$		−	2.60998e15i		−	0.178491i
$144$	3.58594e15			0.232747
$145$	1.74719e15	−	1.13116e15i	0.107667	−	0.0697050i
$146$	8.31942e15			0.486907
$147$		−	1.78555e16i		−	0.992870i
$148$		−	1.59873e16i		−	0.844915i
$149$	−2.13596e14			−0.0107324			−0.00536619	−	0.999986i	$-0.501708\pi$
$149$	−2.13596e14			−0.0107324			−0.00536619	+	0.999986i	$0.501708\pi$
$150$	−6.17071e15			−0.294884
$151$	−2.01487e16			−0.916050			−0.458025	−	0.888939i	$-0.651443\pi$
$151$	−2.01487e16			−0.916050			−0.458025	+	0.888939i	$0.651443\pi$
$152$	−2.62417e16			−1.13544
$153$		−	4.28021e15i		−	0.176311i
$154$	−1.61266e16			−0.632613
$155$			1.86702e14i			0.00697687i
$156$			2.96771e15i			0.105679i
$157$		−	3.09570e16i		−	1.05078i	−0.850862	−	0.525389i	$-0.823920\pi$
$157$		−	3.09570e16i		−	1.05078i	0.850862	−	0.525389i	$-0.176080\pi$
$158$	2.55936e16			0.828330
$159$		−	1.66620e16i		−	0.514339i
$160$			4.41783e15i			0.130110i
$161$	−2.39756e16			−0.673869
$162$		−	1.30345e15i		−	0.0349728i
$163$		−	3.44412e16i		−	0.882411i	−0.897406	−	0.441206i	$-0.854551\pi$
$163$		−	3.44412e16i		−	0.882411i	0.897406	−	0.441206i	$-0.145449\pi$
$164$			2.79839e16i			0.684824i
$165$	3.02278e15			0.0706767
$166$		−	1.64529e16i		−	0.367648i
$167$	1.48846e16			0.317954			0.158977	−	0.987282i	$-0.449180\pi$
$167$	1.48846e16			0.317954			0.158977	+	0.987282i	$0.449180\pi$
$168$	4.17262e16			0.852293
$169$	−4.89510e16			−0.956337
$170$	9.63091e14			0.0180012
$171$			4.47009e16i			0.799550i
$172$			3.70706e16i			0.634696i
$173$	5.35932e15			0.0878544			0.0439272	−	0.999035i	$-0.486013\pi$
$173$	5.35932e15			0.0878544			0.0439272	+	0.999035i	$0.486013\pi$
$174$	1.60310e16	−	1.03787e16i	0.251676	−	0.162938i
$175$	−1.04213e17			−1.56725
$176$			2.36311e16i			0.340516i
$177$		−	5.83416e16i		−	0.805710i
$178$	2.82059e16			0.373414
$179$	−1.29035e17			−1.63799			−0.818994	−	0.573803i	$-0.805468\pi$
$179$	−1.29035e17			−1.63799			−0.818994	+	0.573803i	$0.805468\pi$
$180$	4.82235e15			0.0587104
$181$	−7.03371e16			−0.821477			−0.410738	−	0.911753i	$-0.634729\pi$
$181$	−7.03371e16			−0.821477			−0.410738	+	0.911753i	$0.634729\pi$
$182$		−	1.38091e16i		−	0.154750i
$183$	−3.69189e16			−0.397069
$184$		−	3.39622e16i		−	0.350640i
$185$		−	1.39439e16i		−	0.138228i
$186$			1.71304e15i			0.0163087i
$187$	2.82062e16			0.257948
$188$			1.54354e17i			1.35623i
$189$		−	1.92815e17i		−	1.62809i
$190$	−1.00582e16			−0.0816331
$191$			1.88487e16i			0.147073i	0.997293	+	0.0735363i	$0.0234285\pi$
$191$			1.88487e16i			0.147073i	−0.997293	+	0.0735363i	$0.976572\pi$
$192$			6.26156e15i			0.0469811i
$193$			1.85381e17i			1.33778i	0.743360	+	0.668891i	$0.233231\pi$
$193$			1.85381e17i			1.33778i	−0.743360	+	0.668891i	$0.766769\pi$
$194$	1.68676e16			0.117096
$195$			2.58839e15i			0.0172890i
$196$	1.87717e17			1.20666
$197$	1.97713e16			0.122332			0.0611659	−	0.998128i	$-0.480518\pi$
$197$	1.97713e16			0.122332			0.0611659	+	0.998128i	$0.480518\pi$
$198$	−3.89129e16			−0.231795
$199$	3.33100e17			1.91063			0.955316	−	0.295588i	$-0.0955156\pi$
$199$	3.33100e17			1.91063			0.955316	+	0.295588i	$0.0955156\pi$
$200$		−	1.47621e17i		−	0.815500i
$201$			1.53446e17i			0.816556i
$202$	1.32210e17			0.677844
$203$	2.70737e17	−	1.75279e17i	1.33761	−	0.865985i
$204$	−3.20722e16			−0.152722
$205$			2.44071e16i			0.112037i
$206$		−	2.94733e16i		−	0.130444i
$207$	−5.78523e16			−0.246912
$208$	−2.02352e16			−0.0832974
$209$	−2.94575e17			−1.16977
$210$	1.59932e16			0.0612761
$211$			3.63715e17i			1.34475i	0.740209	+	0.672376i	$0.234726\pi$
$211$			3.63715e17i			1.34475i	−0.740209	+	0.672376i	$0.765274\pi$
$212$	1.75170e17			0.625089
$213$		−	1.88445e17i		−	0.649140i
$214$		−	1.88686e16i		−	0.0627534i
$215$			3.23324e16i			0.103836i
$216$	2.73129e17			0.847158
$217$			2.89305e16i			0.0866777i
$218$			1.24031e17i			0.359010i
$219$	−2.41636e17			−0.675821
$220$			3.17789e16i			0.0858951i
$221$			2.41528e16i			0.0630995i
$222$		−	1.27939e17i		−	0.323114i
$223$	6.63989e17			1.62134			0.810671	−	0.585503i	$-0.199103\pi$
$223$	6.63989e17			1.62134			0.810671	+	0.585503i	$0.199103\pi$
$224$			6.84567e17i			1.61643i
$225$	−2.51463e17			−0.574255
$226$	−1.71556e17			−0.378959
$227$	−3.45271e17			−0.737848			−0.368924	−	0.929460i	$-0.620274\pi$
$227$	−3.45271e17			−0.737848			−0.368924	+	0.929460i	$0.620274\pi$
$228$	3.34950e17			0.692578
$229$		−	4.93867e17i		−	0.988198i	−0.869406	−	0.494099i	$-0.835498\pi$
$229$		−	4.93867e17i		−	0.988198i	0.869406	−	0.494099i	$-0.164502\pi$
$230$		−	1.30173e16i		−	0.0252095i
$231$	4.68396e17			0.878058
$232$	2.48288e17	+	3.83508e17i	0.450606	+	0.696008i
$233$	−6.81444e17			−1.19746			−0.598729	−	0.800951i	$-0.704328\pi$
$233$	−6.81444e17			−1.19746			−0.598729	+	0.800951i	$0.704328\pi$
$234$		−	3.33209e16i		−	0.0567020i
$235$			1.34625e17i			0.221880i
$236$	6.13353e17			0.979198
$237$	−7.43363e17			−1.14971
$238$	1.49236e17			0.223639
$239$	−1.34923e18			−1.95930			−0.979650	−	0.200712i	$-0.935675\pi$
$239$	−1.34923e18			−1.95930			−0.979650	+	0.200712i	$0.935675\pi$
$240$		−	2.34356e16i		−	0.0329831i
$241$	−7.41957e17			−1.01216			−0.506082	−	0.862485i	$-0.668907\pi$
$241$	−7.41957e17			−1.01216			−0.506082	+	0.862485i	$0.668907\pi$
$242$			9.50063e16i			0.125643i
$243$		−	7.59005e17i		−	0.973189i
$244$		−	3.88134e17i		−	0.482567i
$245$	1.63724e17			0.197409
$246$			2.23942e17i			0.261892i
$247$		−	2.52243e17i		−	0.286149i
$248$	−4.09809e16			−0.0451018
$249$			4.77873e17i			0.510290i
$250$		−	1.14110e17i		−	0.118242i
$251$		−	4.91672e17i		−	0.494450i	−0.968958	−	0.247225i	$-0.920481\pi$
$251$		−	4.91672e17i		−	0.494450i	0.968958	−	0.247225i	$-0.0795187\pi$
$252$	7.47250e17			0.729393
$253$		−	3.81241e17i		−	0.361240i
$254$	8.10896e17			0.745954
$255$	−2.79729e16			−0.0249853
$256$	−6.09391e17			−0.528562
$257$	1.13101e17			0.0952728			0.0476364	−	0.998865i	$-0.484831\pi$
$257$	1.13101e17			0.0952728			0.0476364	+	0.998865i	$0.484831\pi$
$258$			2.96658e17i			0.242722i
$259$		−	2.16068e18i		−	1.71729i
$260$	−2.72121e16			−0.0210117
$261$	6.53279e17	−	4.22942e17i	0.490112	−	0.317305i
$262$	−7.65707e17			−0.558217
$263$			9.78096e17i			0.692968i	0.938056	+	0.346484i	$0.112625\pi$
$263$			9.78096e17i			0.692968i	−0.938056	+	0.346484i	$0.887375\pi$
$264$			6.63497e17i			0.456888i
$265$	1.52781e17			0.102264
$266$	−1.55857e18			−1.01418
$267$	−8.19238e17			−0.518293
$268$	−1.61320e18			−0.992379
$269$			6.94796e17i			0.415638i	0.978167	+	0.207819i	$0.0666365\pi$
$269$			6.94796e17i			0.415638i	−0.978167	+	0.207819i	$0.933363\pi$
$270$	1.04687e17			0.0609068
$271$		−	2.44024e18i		−	1.38090i	−0.723379	−	0.690451i	$-0.757412\pi$
$271$		−	2.44024e18i		−	1.38090i	0.723379	−	0.690451i	$-0.242588\pi$
$272$		−	2.18682e17i		−	0.120378i
$273$			4.01085e17i			0.214791i
$274$	−6.71491e17			−0.349873
$275$		−	1.65712e18i		−	0.840152i
$276$			4.33495e17i			0.213878i
$277$	−2.54229e18			−1.22075			−0.610375	−	0.792112i	$-0.708981\pi$
$277$	−2.54229e18			−1.22075			−0.610375	+	0.792112i	$0.708981\pi$
$278$		−	2.18958e17i		−	0.102335i
$279$			6.98081e16i			0.0317596i
$280$			3.82604e17i			0.169459i
$281$	2.44170e18			1.05292			0.526460	−	0.850200i	$-0.323519\pi$
$281$	2.44170e18			1.05292			0.526460	+	0.850200i	$0.323519\pi$
$282$			1.23522e18i			0.518653i
$283$	5.47261e17			0.223767			0.111884	−	0.993721i	$-0.464312\pi$
$283$	5.47261e17			0.223767			0.111884	+	0.993721i	$0.464312\pi$
$284$	1.98115e18			0.788915
$285$	2.92138e17			0.113306
$286$	2.19582e17			0.0829567
$287$			3.78202e18i			1.39190i
$288$			1.65183e18i			0.592274i
$289$	2.60140e18			0.908811
$290$	9.51664e16	+	1.46994e17i	0.0323965	+	0.0500399i
$291$	−4.89919e17			−0.162527
$292$		−	2.54036e18i		−	0.821341i
$293$		−	1.86658e17i		−	0.0588218i	−0.999567	−	0.0294109i	$-0.990637\pi$
$293$		−	1.86658e17i		−	0.0588218i	0.999567	−	0.0294109i	$-0.00936313\pi$
$294$	1.50221e18			0.461452
$295$	5.34957e17			0.160197
$296$	3.06067e18			0.893571
$297$	3.06600e18			0.872767
$298$		−	1.79702e16i		−	0.00498805i
$299$	3.26455e17			0.0883668
$300$			1.88424e18i			0.497425i
$301$			5.01008e18i			1.29002i
$302$		−	1.69514e18i		−	0.425749i
$303$	−3.84003e18			−0.940838
$304$			2.28384e18i			0.545900i
$305$		−	3.38524e17i		−	0.0789480i
$306$	3.60101e17			0.0819434
$307$			7.70425e17i			0.171077i	0.996335	+	0.0855387i	$0.0272611\pi$
$307$			7.70425e17i			0.171077i	−0.996335	+	0.0855387i	$0.972739\pi$
$308$			4.92431e18i			1.06712i
$309$			8.56049e17i			0.181055i
$310$	−1.57075e16			−0.00324262
$311$		−	8.90817e18i		−	1.79509i	−0.440926	−	0.897543i	$-0.645350\pi$
$311$		−	8.90817e18i		−	1.79509i	0.440926	−	0.897543i	$-0.354650\pi$
$312$	−5.68149e17			−0.111764
$313$	−2.50899e18			−0.481854			−0.240927	−	0.970543i	$-0.577451\pi$
$313$	−2.50899e18			−0.481854			−0.240927	+	0.970543i	$0.577451\pi$
$314$	2.60446e18			0.488367
$315$	6.51739e17			0.119329
$316$		−	7.81508e18i		−	1.39727i
$317$			1.86186e18i			0.325089i	0.986701	+	0.162544i	$0.0519701\pi$
$317$			1.86186e18i			0.325089i	−0.986701	+	0.162544i	$0.948030\pi$
$318$	1.40181e18			0.239048
$319$	2.78715e18	+	4.30505e18i	0.464227	+	0.717048i
$320$	−5.74147e16			−0.00934111
$321$			5.48038e17i			0.0871009i
$322$		−	2.01711e18i		−	0.313192i
$323$	2.72601e18			0.413531
$324$	−3.98011e17			−0.0589940
$325$	1.41898e18			0.205519
$326$	2.89760e18			0.410115
$327$		−	3.60247e18i		−	0.498301i
$328$	−5.35734e18			−0.724261
$329$			2.08609e19i			2.75654i
$330$			2.54311e17i			0.0328482i
$331$			5.16038e18i			0.651586i	0.945441	+	0.325793i	$0.105631\pi$
$331$			5.16038e18i			0.651586i	−0.945441	+	0.325793i	$0.894369\pi$
$332$	−5.02394e18			−0.620168
$333$		−	5.21365e18i		−	0.629230i
$334$			1.25227e18i			0.147775i
$335$	−1.40701e18			−0.162353
$336$		−	3.63147e18i		−	0.409768i
$337$		−	1.20372e19i		−	1.32832i	−0.747590	−	0.664160i	$-0.768789\pi$
$337$		−	1.20372e19i		−	1.32832i	0.747590	−	0.664160i	$-0.231211\pi$
$338$		−	4.11833e18i		−	0.444473i
$339$	4.98282e18			0.525989
$340$		−	2.94082e17i		−	0.0303653i
$341$	−4.60029e17			−0.0464652
$342$	−3.76076e18			−0.371604
$343$	8.88654e18			0.859066
$344$	−7.09693e18			−0.671246
$345$			3.78087e17i			0.0349904i
$346$			4.50888e17i			0.0408318i
$347$	−1.89464e19			−1.67902			−0.839508	−	0.543348i	$-0.817157\pi$
$347$	−1.89464e19			−1.67902			−0.839508	+	0.543348i	$0.817157\pi$
$348$	−3.16917e18	−	4.89511e18i	−0.274853	−	0.424540i
$349$	1.59629e19			1.35494			0.677471	−	0.735549i	$-0.263076\pi$
$349$	1.59629e19			1.35494			0.677471	+	0.735549i	$0.263076\pi$
$350$		−	8.76764e18i		−	0.728404i
$351$			2.62540e18i			0.213497i
$352$	−1.08854e19			−0.866515
$353$	−3.11803e18			−0.242980			−0.121490	−	0.992593i	$-0.538767\pi$
$353$	−3.11803e18			−0.242980			−0.121490	+	0.992593i	$0.538767\pi$
$354$	4.90838e18			0.374467
$355$	1.72793e18			0.129066
$356$		−	8.61276e18i		−	0.629894i
$357$	−4.33455e18			−0.310408
$358$		−	1.08560e19i		−	0.761282i
$359$			4.24591e18i			0.291583i	0.989315	+	0.145792i	$0.0465729\pi$
$359$			4.24591e18i			0.291583i	−0.989315	+	0.145792i	$0.953427\pi$
$360$			9.23208e17i			0.0620913i
$361$	−1.32883e19			−0.875314
$362$		−	5.91758e18i		−	0.381795i
$363$		−	2.75945e18i		−	0.174390i
$364$	−4.21666e18			−0.261041
$365$		−	2.21566e18i		−	0.134371i
$366$		−	3.10605e18i		−	0.184544i
$367$			1.16940e18i			0.0680718i	0.999421	+	0.0340359i	$0.0108361\pi$
$367$			1.16940e18i			0.0680718i	−0.999421	+	0.0340359i	$0.989164\pi$
$368$	−2.95576e18			−0.168582
$369$			9.12586e18i			0.510007i
$370$	1.17312e18			0.0642437
$371$	2.36742e19			1.27049
$372$	5.23082e17			0.0275104
$373$	−3.30534e19			−1.70373			−0.851864	−	0.523763i	$-0.824528\pi$
$373$	−3.30534e19			−1.70373			−0.851864	+	0.523763i	$0.824528\pi$
$374$			2.37304e18i			0.119886i
$375$			3.31430e18i			0.164119i
$376$	−2.95501e19			−1.43433
$377$	−3.68639e18	+	2.38662e18i	−0.175405	+	0.113560i
$378$	1.62219e19			0.756681
$379$			1.60910e18i			0.0735847i	0.999323	+	0.0367924i	$0.0117140\pi$
$379$			1.60910e18i			0.0735847i	−0.999323	+	0.0367924i	$0.988286\pi$
$380$			3.07129e18i			0.137703i
$381$	−2.35524e19			−1.03537
$382$	−1.58578e18			−0.0683544
$383$	3.13759e19			1.32619			0.663094	−	0.748536i	$-0.269243\pi$
$383$	3.13759e19			1.32619			0.663094	+	0.748536i	$0.269243\pi$
$384$	1.52609e19			0.632551
$385$			4.29490e18i			0.174581i
$386$	−1.55964e19			−0.621756
$387$			1.20891e19i			0.472675i
$388$		−	5.15058e18i		−	0.197523i
$389$		−	2.77845e19i		−	1.04515i	−0.852592	−	0.522577i	$-0.824971\pi$
$389$		−	2.77845e19i		−	1.04515i	0.852592	−	0.522577i	$-0.175029\pi$
$390$	−2.17765e17			−0.00803534
$391$			3.52802e18i			0.127704i
$392$			3.59373e19i			1.27615i
$393$	2.22399e19			0.774797
$394$			1.66340e18i			0.0568557i
$395$		−	6.81619e18i		−	0.228593i
$396$			1.18822e19i			0.391005i
$397$	−1.89100e19			−0.610608			−0.305304	−	0.952255i	$-0.598758\pi$
$397$	−1.89100e19			−0.610608			−0.305304	+	0.952255i	$0.598758\pi$
$398$			2.80243e19i			0.887997i
$399$	4.52684e19			1.40766
$400$	−1.28476e19			−0.392078
$401$	3.25218e19			0.974073			0.487037	−	0.873381i	$-0.338078\pi$
$401$	3.25218e19			0.974073			0.487037	+	0.873381i	$0.338078\pi$
$402$	−1.29097e19			−0.379508
$403$		−	3.93921e17i		−	0.0113664i
$404$		−	4.03708e19i		−	1.14342i
$405$	−3.47139e17			−0.00965141
$406$	1.47466e19	+	2.27776e19i	0.402481	+	0.621674i
$407$	3.43575e19			0.920583
$408$		−	6.14002e18i		−	0.161517i
$409$			4.18594e19i			1.08111i	0.841310	+	0.540553i	$0.181785\pi$
$409$			4.18594e19i			1.08111i	−0.841310	+	0.540553i	$0.818215\pi$
$410$	−2.05341e18			−0.0520711
$411$	1.95034e19			0.485619
$412$	−8.99976e18			−0.220040
$413$	8.28945e19			1.99022
$414$		−	4.86721e18i		−	0.114757i
$415$	−4.38180e18			−0.101459
$416$		−	9.32115e18i		−	0.211968i
$417$			6.35962e18i			0.142040i
$418$		−	2.47831e19i		−	0.543668i
$419$	−3.86256e19			−0.832282			−0.416141	−	0.909300i	$-0.636618\pi$
$419$	−3.86256e19			−0.832282			−0.416141	+	0.909300i	$0.636618\pi$
$420$		−	4.88357e18i		−	0.103364i
$421$		−	5.27714e19i		−	1.09719i	−0.836087	−	0.548597i	$-0.815162\pi$
$421$		−	5.27714e19i		−	1.09719i	0.836087	−	0.548597i	$-0.184838\pi$
$422$	−3.05999e19			−0.624996
$423$			5.03366e19i			1.01002i
$424$			3.35352e19i			0.661085i
$425$			1.53350e19i			0.297007i
$426$	1.58542e19			0.301698
$427$		−	5.24562e19i		−	0.980816i
$428$	−5.76160e18			−0.105856
$429$	−6.37773e18			−0.115143
$430$	−2.72018e18			−0.0482596
$431$	4.07343e19			0.710201			0.355100	−	0.934828i	$-0.384447\pi$
$431$	4.07343e19			0.710201			0.355100	+	0.934828i	$0.384447\pi$
$432$		−	2.37707e19i		−	0.407299i
$433$		−	2.78381e19i		−	0.468793i	−0.972141	−	0.234396i	$-0.924689\pi$
$433$		−	2.78381e19i		−	0.468793i	0.972141	−	0.234396i	$-0.0753113\pi$
$434$	−2.43397e18			−0.0402849
$435$	−2.76410e18	−	4.26944e18i	−0.0449659	−	0.0694546i
$436$	3.78732e19			0.605597
$437$		−	3.68453e19i		−	0.579124i
$438$		−	2.03293e19i		−	0.314099i
$439$	−5.13885e19			−0.780517			−0.390258	−	0.920705i	$-0.627614\pi$
$439$	−5.13885e19			−0.780517			−0.390258	+	0.920705i	$0.627614\pi$
$440$	−6.08386e18			−0.0908415
$441$	6.12167e19			0.898630
$442$	−2.03202e18			−0.0293265
$443$			1.35880e20i			1.92809i	0.265739	+	0.964045i	$0.414384\pi$
$443$			1.35880e20i			1.92809i	−0.265739	+	0.964045i	$0.585616\pi$
$444$	−3.90666e19			−0.545046
$445$		−	7.51191e18i		−	0.103051i
$446$			5.58625e19i			0.753545i
$447$			5.21942e17i			0.00692335i
$448$	−8.89673e18			−0.116050
$449$			4.14674e19i			0.531935i	0.963982	+	0.265968i	$0.0856914\pi$
$449$			4.14674e19i			0.531935i	−0.963982	+	0.265968i	$0.914309\pi$
$450$		−	2.11560e19i		−	0.266894i
$451$	−6.01386e19			−0.746155
$452$			5.23850e19i			0.639247i
$453$			4.92352e19i			0.590934i
$454$		−	2.90483e19i		−	0.342927i
$455$	−3.67770e18			−0.0427062
$456$			6.41240e19i			0.732461i
$457$	3.00206e19			0.337324			0.168662	−	0.985674i	$-0.446055\pi$
$457$	3.00206e19			0.337324			0.168662	+	0.985674i	$0.446055\pi$
$458$	4.15499e19			0.459281
$459$	−2.83728e19			−0.308537
$460$	−3.97488e18			−0.0425246
$461$		−	1.33446e20i		−	1.40458i	−0.711890	−	0.702291i	$-0.752160\pi$
$461$		−	1.33446e20i		−	1.40458i	0.711890	−	0.702291i	$-0.247840\pi$
$462$			3.94069e19i			0.408092i
$463$	−1.20749e20			−1.23034			−0.615172	−	0.788393i	$-0.710913\pi$
$463$	−1.20749e20			−1.23034			−0.615172	+	0.788393i	$0.710913\pi$
$464$	3.33770e19	−	2.16088e19i	0.334628	−	0.216643i
$465$	4.56224e17			0.00450071
$466$		−	5.73310e19i		−	0.556539i
$467$		−	6.15440e19i		−	0.587908i	−0.955820	−	0.293954i	$-0.905029\pi$
$467$		−	6.15440e19i		−	0.587908i	0.955820	−	0.293954i	$-0.0949712\pi$
$468$	−1.01746e19			−0.0956479
$469$	−2.18024e20			−2.01701
$470$	−1.13262e19			−0.103122
$471$	−7.56463e19			−0.677846
$472$			1.17423e20i			1.03559i
$473$	−7.96663e19			−0.691538
$474$		−	6.25404e19i		−	0.534347i
$475$		−	1.60153e20i		−	1.34689i
$476$		−	4.55697e19i		−	0.377246i
$477$	5.71250e19			0.465520
$478$		−	1.13513e20i		−	0.910617i
$479$		−	6.33265e19i		−	0.500114i	−0.968231	−	0.250057i	$-0.919551\pi$
$479$		−	6.33265e19i		−	0.500114i	0.968231	−	0.250057i	$-0.0804494\pi$
$480$	1.07954e19			0.0839323
$481$			2.94201e19i			0.225194i
$482$		−	6.24221e19i		−	0.470420i
$483$			5.85867e19i			0.434706i
$484$	2.90105e19			0.211941
$485$		−	4.49226e18i		−	0.0323148i
$486$	6.38563e19			0.452306
$487$	−1.52913e20			−1.06654			−0.533272	−	0.845944i	$-0.679038\pi$
$487$	−1.52913e20			−1.06654			−0.533272	+	0.845944i	$0.679038\pi$
$488$	7.43058e19			0.510357
$489$	−8.41603e19			−0.569234
$490$			1.37744e19i			0.0917491i
$491$		−	3.33386e19i		−	0.218694i	−0.994004	−	0.109347i	$-0.965124\pi$
$491$		−	3.33386e19i		−	0.218694i	0.994004	−	0.109347i	$-0.0348760\pi$
$492$	6.83813e19			0.441773
$493$	−2.57924e19	−	3.98391e19i	−0.164112	−	0.253488i
$494$	2.12216e19			0.132992
$495$			1.03634e19i			0.0639683i
$496$			3.56660e18i			0.0216841i
$497$	2.67752e20			1.60347
$498$	−4.02043e19			−0.237166
$499$	4.17248e19			0.242460			0.121230	−	0.992624i	$-0.461316\pi$
$499$	4.17248e19			0.242460			0.121230	+	0.992624i	$0.461316\pi$
$500$	−3.48437e19			−0.199457
$501$		−	3.63721e19i		−	0.205109i
$502$	4.13652e19			0.229804
$503$			2.15598e20i			1.18001i	0.807400	+	0.590004i	$0.200874\pi$
$503$			2.15598e20i			1.18001i	−0.807400	+	0.590004i	$0.799126\pi$
$504$			1.43056e20i			0.771397i
$505$		−	3.52107e19i		−	0.187064i
$506$	3.20745e19			0.167892
$507$			1.19616e20i			0.616923i
$508$		−	2.47609e20i		−	1.25831i
$509$	−2.20372e20			−1.10350			−0.551751	−	0.834009i	$-0.686040\pi$
$509$	−2.20372e20			−1.10350			−0.551751	+	0.834009i	$0.686040\pi$
$510$		−	2.35340e18i		−	0.0116124i
$511$		−	3.43329e20i		−	1.66937i
$512$			1.53376e20i			0.734906i
$513$	2.96315e20			1.39918
$514$			9.51540e18i			0.0442796i
$515$	−7.84944e18			−0.0359985
$516$	9.05855e19			0.409436
$517$	−3.31714e20			−1.47769
$518$	1.81782e20			0.798137
$519$		−	1.30960e19i		−	0.0566739i
$520$		−	5.20958e18i		−	0.0222217i
$521$	4.23741e20			1.78163			0.890816	−	0.454364i	$-0.150134\pi$
$521$	4.23741e20			1.78163			0.890816	+	0.454364i	$0.150134\pi$
$522$	3.55829e19	+	5.49615e19i	0.147473	+	0.227787i
$523$	−2.15725e20			−0.881329			−0.440665	−	0.897672i	$-0.645257\pi$
$523$	−2.15725e20			−0.881329			−0.440665	+	0.897672i	$0.645257\pi$
$524$			2.33811e20i			0.941629i
$525$			2.54655e20i			1.01101i
$526$	−8.22889e19			−0.322068
$527$	4.25712e18			0.0164262
$528$	5.77447e19			0.219664
$529$	−2.18950e20			−0.821158
$530$			1.28537e19i			0.0475291i
$531$	2.00021e20			0.729235
$532$			4.75913e20i			1.71076i
$533$		−	5.14964e19i		−	0.182525i
$534$		−	6.89239e19i		−	0.240885i
$535$	−5.02517e18			−0.0173180
$536$		−	3.08837e20i		−	1.04953i
$537$			3.15310e20i			1.05665i
$538$	−5.84544e19			−0.193175
$539$			4.03413e20i			1.31472i
$540$		−	3.19666e19i		−	0.102741i
$541$		−	5.89461e19i		−	0.186842i	−0.995627	−	0.0934212i	$-0.970220\pi$
$541$		−	5.89461e19i		−	0.186842i	0.995627	−	0.0934212i	$-0.0297803\pi$
$542$	2.05301e20			0.641797
$543$			1.71876e20i			0.529926i
$544$	1.00734e20			0.306327
$545$	3.30324e19			0.0990756
$546$	−3.37440e19			−0.0998277
$547$	1.74365e20			0.508809			0.254405	−	0.967098i	$-0.418121\pi$
$547$	1.74365e20			0.508809			0.254405	+	0.967098i	$0.418121\pi$
$548$			2.05042e20i			0.590185i
$549$		−	1.26575e20i		−	0.359381i
$550$	1.39416e20			0.390474
$551$	2.69366e20	+	4.16064e20i	0.744228	+	1.14954i
$552$	−8.29898e19			−0.226194
$553$		−	1.05621e21i		−	2.83995i
$554$		−	2.13887e20i		−	0.567364i
$555$	−3.40732e19			−0.0891694
$556$	−6.68595e19			−0.172624
$557$	−5.26304e20			−1.34067			−0.670336	−	0.742058i	$-0.733850\pi$
$557$	−5.26304e20			−1.34067			−0.670336	+	0.742058i	$0.733850\pi$
$558$	−5.87308e18			−0.0147608
$559$		−	6.82178e19i		−	0.169165i
$560$	3.32984e19			0.0814728
$561$		−	6.89245e19i		−	0.166400i
$562$			2.05425e20i			0.489362i
$563$		−	6.45323e20i		−	1.51693i	−0.651716	−	0.758463i	$-0.725951\pi$
$563$		−	6.45323e20i		−	1.51693i	0.651716	−	0.758463i	$-0.274049\pi$
$564$	3.77179e20			0.874891
$565$			4.56894e19i			0.104581i
$566$			4.60420e19i			0.103999i
$567$	−5.37911e19			−0.119905
$568$			3.79279e20i			0.834346i
$569$			2.38864e20i			0.518571i	0.965801	+	0.259285i	$0.0834870\pi$
$569$			2.38864e20i			0.518571i	−0.965801	+	0.259285i	$0.916513\pi$
$570$			2.45781e19i			0.0526607i
$571$	1.15964e20			0.245218			0.122609	−	0.992455i	$-0.460874\pi$
$571$	1.15964e20			0.245218			0.122609	+	0.992455i	$0.460874\pi$
$572$		−	6.70500e19i		−	0.139936i
$573$	4.60587e19			0.0948750
$574$	−3.18187e20			−0.646910
$575$	2.07271e20			0.415940
$576$	−2.14675e19			−0.0425218
$577$		−	2.68838e20i		−	0.525619i	−0.964848	−	0.262810i	$-0.915351\pi$
$577$		−	2.68838e20i		−	0.525619i	0.964848	−	0.262810i	$-0.0846491\pi$
$578$			2.18860e20i			0.422385i
$579$	4.52996e20			0.862989
$580$	4.48852e19	−	2.90593e19i	0.0844098	−	0.0546482i
$581$	−6.78984e20			−1.26049
$582$		−	4.12177e19i		−	0.0755373i
$583$			3.76449e20i			0.681070i
$584$	4.86335e20			0.868639
$585$	−8.87416e18			−0.0156480
$586$	1.57038e19			0.0273384
$587$	1.22242e20			0.210105			0.105052	−	0.994467i	$-0.466499\pi$
$587$	1.22242e20			0.210105			0.105052	+	0.994467i	$0.466499\pi$
$588$		−	4.58705e20i		−	0.778402i
$589$	−4.44598e19			−0.0744909
$590$			4.50068e19i			0.0744540i
$591$		−	4.83131e19i		−	0.0789149i
$592$		−	2.66373e20i		−	0.429613i
$593$	−5.95219e20			−0.947909			−0.473955	−	0.880549i	$-0.657174\pi$
$593$	−5.95219e20			−0.947909			−0.473955	+	0.880549i	$0.657174\pi$
$594$			2.57948e20i			0.405633i
$595$		−	3.97452e19i		−	0.0617173i
$596$	−5.48725e18			−0.00841411
$597$		−	8.13962e20i		−	1.23253i
$598$			2.74652e19i			0.0410699i
$599$			4.80867e20i			0.710106i	0.934846	+	0.355053i	$0.115537\pi$
$599$			4.80867e20i			0.710106i	−0.934846	+	0.355053i	$0.884463\pi$
$600$	−3.60726e20			−0.526070
$601$		−	5.27133e20i		−	0.759210i	−0.925149	−	0.379605i	$-0.876060\pi$
$601$		−	5.27133e20i		−	0.759210i	0.925149	−	0.379605i	$-0.123940\pi$
$602$	−4.21506e20			−0.599557
$603$	−5.26083e20			−0.739051
$604$	−5.17616e20			−0.718176
$605$	2.53025e19			0.0346735
$606$		−	3.23068e20i		−	0.437270i
$607$		−	4.44369e20i		−	0.594057i	−0.954869	−	0.297029i	$-0.904004\pi$
$607$		−	4.44369e20i		−	0.594057i	0.954869	−	0.297029i	$-0.0959957\pi$
$608$	−1.05203e21			−1.38916
$609$	−4.28312e20	−	6.61573e20i	−0.558638	−	0.862875i
$610$	2.84806e19			0.0366924
$611$		−	2.84045e20i		−	0.361474i
$612$		−	1.09958e20i		−	0.138226i
$613$	−9.86041e19			−0.122445			−0.0612225	−	0.998124i	$-0.519500\pi$
$613$	−9.86041e19			−0.122445			−0.0612225	+	0.998124i	$0.519500\pi$
$614$	−6.48172e19			−0.0795110
$615$	5.96411e19			0.0722740
$616$	−9.42728e20			−1.12858
$617$			9.58926e20i			1.13409i	0.823688	+	0.567043i	$0.191913\pi$
$617$			9.58926e20i			1.13409i	−0.823688	+	0.567043i	$0.808087\pi$
$618$	−7.20208e19			−0.0841481
$619$			1.35236e21i			1.56104i	0.625132	+	0.780519i	$0.285045\pi$
$619$			1.35236e21i			1.56104i	−0.625132	+	0.780519i	$0.714955\pi$
$620$			4.79634e18i			0.00546981i
$621$			3.83493e20i			0.432087i
$622$	7.49459e20			0.834296
$623$		−	1.16401e21i		−	1.28026i
$624$			4.94465e19i			0.0537343i
$625$	8.85611e20			0.950918
$626$		−	2.11085e20i		−	0.223950i
$627$			7.19822e20i			0.754603i
$628$		−	7.95280e20i		−	0.823802i
$629$	−3.17945e20			−0.325441
$630$			5.48319e19i			0.0554600i
$631$	−6.94354e20			−0.694002			−0.347001	−	0.937865i	$-0.612800\pi$
$631$	−6.94354e20			−0.694002			−0.347001	+	0.937865i	$0.612800\pi$
$632$	1.49615e21			1.47773
$633$	8.88772e20			0.867486
$634$	−1.56641e20			−0.151090
$635$		−	2.15961e20i		−	0.205860i
$636$		−	4.28045e20i		−	0.403238i
$637$	−3.45440e20			−0.321608
$638$	−3.62191e20	+	2.34488e20i	−0.333260	+	0.215757i
$639$	6.46076e20			0.587526
$640$			1.39933e20i			0.125768i
$641$			1.85594e21i			1.64865i	0.566116	+	0.824326i	$0.308446\pi$
$641$			1.85594e21i			1.64865i	−0.566116	+	0.824326i	$0.691554\pi$
$642$	−4.61073e19			−0.0404816
$643$	1.26804e21			1.10040			0.550199	−	0.835033i	$-0.314552\pi$
$643$	1.26804e21			1.10040			0.550199	+	0.835033i	$0.314552\pi$
$644$	−6.15930e20			−0.528308
$645$	7.90072e19			0.0669837
$646$			2.29344e20i			0.192195i
$647$	1.27709e21			1.05789			0.528944	−	0.848656i	$-0.322588\pi$
$647$	1.27709e21			1.05789			0.528944	+	0.848656i	$0.322588\pi$
$648$		−	7.61967e19i		−	0.0623913i
$649$			1.31812e21i			1.06689i
$650$			1.19381e20i			0.0955181i
$651$	7.06944e19			0.0559149
$652$		−	8.84789e20i		−	0.691803i
$653$		−	7.85614e20i		−	0.607240i	−0.952793	−	0.303620i	$-0.901805\pi$
$653$		−	7.85614e20i		−	0.607240i	0.952793	−	0.303620i	$-0.0981954\pi$
$654$	3.03082e20			0.231594
$655$			2.03926e20i			0.154050i
$656$			4.66254e20i			0.348212i
$657$		−	8.28439e20i		−	0.611674i
$658$	−1.75506e21			−1.28115
$659$			2.33164e20i			0.168275i	0.996454	+	0.0841377i	$0.0268135\pi$
$659$			2.33164e20i			0.168275i	−0.996454	+	0.0841377i	$0.973186\pi$
$660$	7.76547e19			0.0554100
$661$	3.90919e20			0.275789			0.137894	−	0.990447i	$-0.455967\pi$
$661$	3.90919e20			0.275789			0.137894	+	0.990447i	$0.455967\pi$
$662$	−4.34151e20			−0.302835
$663$	5.90197e19			0.0407048
$664$		−	9.61802e20i		−	0.655881i
$665$			4.15084e20i			0.279881i
$666$	4.38633e20			0.292445
$667$	−5.38474e20	+	3.48616e20i	−0.354994	+	0.229828i
$668$	3.82384e20			0.249274
$669$		−	1.62252e21i		−	1.04591i
$670$		−	1.18374e20i		−	0.0754563i
$671$	8.34117e20			0.525785
$672$	1.67281e21			1.04274
$673$	−7.22789e20			−0.445552			−0.222776	−	0.974870i	$-0.571512\pi$
$673$	−7.22789e20			−0.445552			−0.222776	+	0.974870i	$0.571512\pi$
$674$	1.01271e21			0.617359
$675$			1.66691e21i			1.00492i
$676$	−1.25754e21			−0.749761
$677$			8.88659e20i			0.523987i	0.965070	+	0.261993i	$0.0843798\pi$
$677$			8.88659e20i			0.523987i	−0.965070	+	0.261993i	$0.915620\pi$
$678$			4.19213e20i			0.244462i
$679$		−	6.96100e20i		−	0.401466i
$680$	5.63002e19			0.0321139
$681$			8.43704e20i			0.475978i
$682$		−	3.87031e19i		−	0.0215955i
$683$	2.08949e21			1.15315			0.576575	−	0.817044i	$-0.304389\pi$
$683$	2.08949e21			1.15315			0.576575	+	0.817044i	$0.304389\pi$
$684$			1.14836e21i			0.626841i
$685$			1.78834e20i			0.0965542i
$686$			7.47639e20i			0.399265i
$687$	−1.20681e21			−0.637476
$688$			6.17652e20i			0.322724i
$689$	−3.22351e20			−0.166604
$690$	−3.18091e19			−0.0162624
$691$	1.52780e21			0.772648			0.386324	−	0.922363i	$-0.373745\pi$
$691$	1.52780e21			0.772648			0.386324	+	0.922363i	$0.373745\pi$
$692$	1.37680e20			0.0688772
$693$			1.60587e21i			0.794716i
$694$		−	1.59399e21i		−	0.780350i
$695$	−5.83138e19			−0.0282413
$696$	9.37138e20	−	6.06717e20i	0.448987	−	0.290681i
$697$	5.56524e20			0.263778
$698$			1.34299e21i			0.629732i
$699$			1.66517e21i			0.772468i
$700$	−2.67722e21			−1.22871
$701$	2.89059e21			1.31251			0.656254	−	0.754540i	$-0.272140\pi$
$701$	2.89059e21			1.31251			0.656254	+	0.754540i	$0.272140\pi$
$702$	−2.20879e20			−0.0992262
$703$	3.32050e21			1.47584
$704$		−	1.41469e20i		−	0.0622107i
$705$	3.28969e20			0.143132
$706$		−	2.62325e20i		−	0.112929i
$707$		−	5.45610e21i		−	2.32400i
$708$		−	1.49879e21i		−	0.631670i
$709$	−1.34642e21			−0.561478			−0.280739	−	0.959784i	$-0.590579\pi$
$709$	−1.34642e21			−0.561478			−0.280739	+	0.959784i	$0.590579\pi$
$710$			1.45374e20i			0.0599857i
$711$		−	2.54858e21i		−	1.04058i
$712$	1.64886e21			0.666167
$713$		−	5.75402e19i		−	0.0230038i
$714$		−	3.64673e20i		−	0.144267i
$715$		−	5.84799e19i		−	0.0228935i
$716$	−3.31490e21			−1.28417
$717$			3.29697e21i			1.26392i
$718$	−3.57216e20			−0.135518
$719$	−5.24783e20			−0.197021			−0.0985106	−	0.995136i	$-0.531408\pi$
$719$	−5.24783e20			−0.197021			−0.0985106	+	0.995136i	$0.531408\pi$
$720$	8.03477e19			0.0298524
$721$	−1.21631e21			−0.447231
$722$		−	1.11796e21i		−	0.406817i
$723$			1.81304e21i			0.652936i
$724$	−1.80695e21			−0.644031
$725$	−2.34055e21	+	1.51531e21i	−0.825624	+	0.534521i
$726$	2.32157e20			0.0810508
$727$			2.54031e21i			0.877767i	0.898544	+	0.438883i	$0.144626\pi$
$727$			2.54031e21i			0.877767i	−0.898544	+	0.438883i	$0.855374\pi$
$728$		−	8.07253e20i		−	0.276073i
$729$	−2.07701e21			−0.703042
$730$	1.86407e20			0.0624513
$731$	7.37234e20			0.244470
$732$	−9.48442e20			−0.311299
$733$		−	2.92969e21i		−	0.951791i	−0.879502	−	0.475895i	$-0.842124\pi$
$733$		−	2.92969e21i		−	0.951791i	0.879502	−	0.475895i	$-0.157876\pi$
$734$	−9.83835e19			−0.0316375
$735$		−	4.00076e20i		−	0.127346i
$736$		−	1.36155e21i		−	0.428991i
$737$		−	3.46684e21i		−	1.08125i
$738$	−7.67774e20			−0.237034
$739$			1.76805e21i			0.540331i	0.962814	+	0.270166i	$0.0870785\pi$
$739$			1.76805e21i			0.540331i	−0.962814	+	0.270166i	$0.912922\pi$
$740$		−	3.58217e20i		−	0.108370i
$741$	−6.16380e20			−0.184592
$742$			1.99175e21i			0.590481i
$743$		−	1.47827e21i		−	0.433849i	−0.976188	−	0.216924i	$-0.930397\pi$
$743$		−	1.47827e21i		−	0.433849i	0.976188	−	0.216924i	$-0.0696025\pi$
$744$			1.00141e20i			0.0290947i
$745$	−4.78589e18			−0.00137655
$746$		−	2.78084e21i		−	0.791836i
$747$	−1.63836e21			−0.461855
$748$	7.24613e20			0.202229
$749$	−7.78678e20			−0.215151
$750$	−2.78838e20			−0.0762768
$751$			6.63139e21i			1.79600i	0.440000	+	0.897998i	$0.354978\pi$
$751$			6.63139e21i			1.79600i	−0.440000	+	0.897998i	$0.645022\pi$
$752$			2.57177e21i			0.689603i
$753$	−1.20145e21			−0.318965
$754$	−2.00791e20	−	3.10143e20i	−0.0527787	−	0.0815223i
$755$	−4.51456e20			−0.117493
$756$		−	4.95340e21i		−	1.27641i
$757$		−	3.55007e21i		−	0.905771i	−0.891569	−	0.452885i	$-0.850395\pi$
$757$		−	3.55007e21i		−	0.905771i	0.891569	−	0.452885i	$-0.149605\pi$
$758$	−1.35376e20			−0.0341997
$759$	−9.31599e20			−0.233032
$760$	−5.87978e20			−0.145633
$761$	−5.32461e21			−1.30588			−0.652939	−	0.757411i	$-0.726464\pi$
$761$	−5.32461e21			−1.30588			−0.652939	+	0.757411i	$0.726464\pi$
$762$		−	1.98150e21i		−	0.481207i
$763$	5.11856e21			1.23087
$764$			4.84221e20i			0.115304i
$765$		−	9.59036e19i		−	0.0226138i
$766$			2.63971e21i			0.616368i
$767$	−1.12870e21			−0.260984
$768$			1.48910e21i			0.340970i
$769$		−	5.73085e21i		−	1.29949i	−0.760154	−	0.649743i	$-0.774877\pi$
$769$		−	5.73085e21i		−	1.29949i	0.760154	−	0.649743i	$-0.225123\pi$
$770$	−3.61338e20			−0.0811396
$771$		−	2.76374e20i		−	0.0614594i
$772$			4.76241e21i			1.04881i
$773$		−	1.56757e21i		−	0.341886i	−0.985281	−	0.170943i	$-0.945319\pi$
$773$		−	1.56757e21i		−	0.341886i	0.985281	−	0.170943i	$-0.0546814\pi$
$774$	−1.01708e21			−0.219684
$775$		−	2.50106e20i		−	0.0535010i
$776$	9.86046e20			0.208898
$777$	−5.27984e21			−1.10780
$778$	2.33756e21			0.485752
$779$	−5.81213e21			−1.19620
$780$			6.64953e19i			0.0135544i
$781$			4.25758e21i			0.859568i
$782$	−2.96818e20			−0.0593526
$783$	−2.80362e21	−	4.33048e21i	−0.555272	−	0.857676i
$784$	3.12766e21			0.613548
$785$		−	6.93631e20i		−	0.134774i
$786$			1.87108e21i			0.360100i
$787$	7.14540e21			1.36212			0.681061	−	0.732226i	$-0.261519\pi$
$787$	7.14540e21			1.36212			0.681061	+	0.732226i	$0.261519\pi$
$788$	5.07922e20			0.0959071
$789$	2.39007e21			0.447027
$790$	5.73457e20			0.106242
$791$			7.07982e21i			1.29927i
$792$	−2.27477e21			−0.413521
$793$			7.14250e20i			0.128618i
$794$		−	1.59093e21i		−	0.283790i
$795$		−	3.73334e20i		−	0.0659697i
$796$	8.55730e21			1.49792
$797$		−	1.05469e22i		−	1.82888i	−0.404719	−	0.914441i	$-0.632631\pi$
$797$		−	1.05469e22i		−	1.82888i	0.404719	−	0.914441i	$-0.367369\pi$
$798$			3.80851e21i			0.654234i
$799$	3.06969e21			0.522388
$800$		−	5.91814e21i		−	0.997724i
$801$		−	2.80872e21i		−	0.469099i
$802$			2.73611e21i			0.452717i
$803$	5.45934e21			0.894898
$804$			3.94201e21i			0.640173i
$805$	−5.37204e20			−0.0864312
$806$	3.31412e19			0.00528270
$807$	1.69780e21			0.268124
$808$	7.72873e21			1.20927
$809$			5.17099e21i			0.801603i	0.916165	+	0.400802i	$0.131268\pi$
$809$			5.17099e21i			0.801603i	−0.916165	+	0.400802i	$0.868732\pi$
$810$		−	2.92054e19i		−	0.00448565i
$811$	1.06051e22			1.61383			0.806914	−	0.590669i	$-0.201136\pi$
$811$	1.06051e22			1.61383			0.806914	+	0.590669i	$0.201136\pi$
$812$	6.95521e21	−	4.50290e21i	1.04867	−	0.678926i
$813$	−5.96296e21			−0.890805
$814$			2.89055e21i			0.427856i
$815$		−	7.71699e20i		−	0.113179i
$816$	−5.34371e20			−0.0776546
$817$	−7.69940e21			−1.10864
$818$	−3.52170e21			−0.502462
$819$	−1.37510e21			−0.194404
$820$			6.27015e20i			0.0878363i
$821$	1.66870e21			0.231635			0.115817	−	0.993271i	$-0.463051\pi$
$821$	1.66870e21			0.231635			0.115817	+	0.993271i	$0.463051\pi$
$822$			1.64085e21i			0.225700i
$823$		−	5.74273e20i		−	0.0782744i	−0.999234	−	0.0391372i	$-0.987539\pi$
$823$		−	5.74273e20i		−	0.0782744i	0.999234	−	0.0391372i	$-0.0124609\pi$
$824$		−	1.72295e21i		−	0.232711i
$825$	−4.04932e21			−0.541973
$826$			6.97406e21i			0.924986i
$827$			2.46914e21i			0.324529i	0.986747	+	0.162265i	$0.0518798\pi$
$827$			2.46914e21i			0.324529i	−0.986747	+	0.162265i	$0.948120\pi$
$828$	−1.48622e21			−0.193577
$829$			2.13800e21i			0.275962i	0.990435	+	0.137981i	$0.0440612\pi$
$829$			2.13800e21i			0.275962i	−0.990435	+	0.137981i	$0.955939\pi$
$830$		−	3.68649e20i		−	0.0471549i
$831$			6.21233e21i			0.787493i
$832$	1.21139e20			0.0152180
$833$		−	3.73319e21i		−	0.464775i
$834$	−5.35045e20			−0.0660154
$835$	3.33510e20			0.0407812
$836$	−7.56759e21			−0.917087
$837$	4.62747e20			0.0555780
$838$		−	3.24964e21i		−	0.386817i
$839$			7.81756e21i			0.922266i	0.887331	+	0.461133i	$0.152557\pi$
$839$			7.81756e21i			0.922266i	−0.887331	+	0.461133i	$0.847443\pi$
$840$	9.34929e20			0.109316
$841$	3.53191e21	−	7.87328e21i	0.409299	−	0.912400i
$842$	4.43975e21			0.509939
$843$		−	5.96654e21i		−	0.679228i
$844$			9.34378e21i			1.05428i
$845$	−1.09681e21			−0.122661
$846$	−4.23490e21			−0.469424
$847$	3.92075e21			0.430769
$848$	2.91860e21			0.317838
$849$		−	1.33728e21i		−	0.144350i
$850$	−1.29016e21			−0.138039
$851$			4.29741e21i			0.455759i
$852$		−	4.84113e21i		−	0.508921i
$853$			7.83526e21i			0.816462i	0.912879	+	0.408231i	$0.133854\pi$
$853$			7.83526e21i			0.816462i	−0.912879	+	0.408231i	$0.866146\pi$
$854$	4.41323e21			0.455850
$855$			1.00158e21i			0.102551i
$856$		−	1.10302e21i		−	0.111952i
$857$	−1.83538e22			−1.84659			−0.923293	−	0.384097i	$-0.874513\pi$
$857$	−1.83538e22			−1.84659			−0.923293	+	0.384097i	$0.874513\pi$
$858$		−	5.36570e20i		−	0.0535145i
$859$			1.94892e22i			1.92684i	0.267991	+	0.963421i	$0.413640\pi$
$859$			1.94892e22i			1.92684i	−0.267991	+	0.963421i	$0.586360\pi$
$860$			8.30614e20i			0.0814068i
$861$	9.24172e21			0.897902
$862$			3.42705e21i			0.330077i
$863$	5.81540e21			0.555263			0.277631	−	0.960688i	$-0.410451\pi$
$863$	5.81540e21			0.555263			0.277631	+	0.960688i	$0.410451\pi$
$864$	1.09497e22			1.03646
$865$	1.20082e20			0.0112683
$866$	2.34207e21			0.217879
$867$		−	6.35677e21i		−	0.586264i
$868$			7.43220e20i			0.0679547i
$869$	1.67949e22			1.52241
$870$	3.59195e20	−	2.32548e20i	0.0322802	−	0.0208987i
$871$	2.96864e21			0.264497
$872$			7.25059e21i			0.640471i
$873$		−	1.67966e21i		−	0.147101i
$874$	3.09986e21			0.269157
$875$	−4.70912e21			−0.405396
$876$	−6.20761e21			−0.529838
$877$	−1.20835e22			−1.02257			−0.511286	−	0.859411i	$-0.670831\pi$
$877$	−1.20835e22			−1.02257			−0.511286	+	0.859411i	$0.670831\pi$
$878$		−	4.32340e21i		−	0.362758i
$879$	−4.56116e20			−0.0379453
$880$			5.29484e20i			0.0436750i
$881$		−	1.53380e22i		−	1.25444i	−0.778844	−	0.627218i	$-0.784194\pi$
$881$		−	1.53380e22i		−	1.25444i	0.778844	−	0.627218i	$-0.215806\pi$
$882$			5.15027e21i			0.417653i
$883$	−1.76527e22			−1.41941			−0.709703	−	0.704501i	$-0.751171\pi$
$883$	−1.76527e22			−1.41941			−0.709703	+	0.704501i	$0.751171\pi$
$884$			6.20482e20i			0.0494695i
$885$		−	1.30722e21i		−	0.103341i
$886$	−1.14318e22			−0.896111
$887$		−	8.54180e21i		−	0.663930i	−0.943292	−	0.331965i	$-0.892288\pi$
$887$		−	8.54180e21i		−	0.663930i	0.943292	−	0.331965i	$-0.107712\pi$
$888$		−	7.47904e21i		−	0.576433i
$889$		−	3.34643e22i		−	2.55752i
$890$	6.31990e20			0.0478945
$891$		−	8.55343e20i		−	0.0642773i
$892$	1.70578e22			1.27112
$893$	−3.20587e22			−2.36897
$894$	−4.39119e19			−0.00321774
$895$	−2.89120e21			−0.210090
$896$			2.16834e22i			1.56249i
$897$		−	7.97724e20i		−	0.0570045i
$898$	−3.48872e21			−0.247226
$899$	4.20661e20	+	6.49756e20i	0.0295621	+	0.0456617i
$900$	−6.46004e21			−0.450211
$901$		−	3.48367e21i		−	0.240769i
$902$		−	5.05956e21i		−	0.346788i
$903$	1.22426e22			0.832177
$904$	−1.00288e22			−0.676059
$905$	−1.57599e21			−0.105363
$906$	−4.14224e21			−0.274646
$907$		−	2.86825e22i		−	1.88609i	−0.332669	−	0.943044i	$-0.607949\pi$
$907$		−	2.86825e22i		−	1.88609i	0.332669	−	0.943044i	$-0.392051\pi$
$908$	−8.86997e21			−0.578467
$909$		−	1.31654e22i		−	0.851537i
$910$		−	3.09412e20i		−	0.0198484i
$911$		−	1.45472e22i		−	0.925533i	−0.886480	−	0.462767i	$-0.846857\pi$
$911$		−	1.45472e22i		−	0.925533i	0.886480	−	0.462767i	$-0.153143\pi$
$912$	5.58078e21			0.352155
$913$		−	1.07967e22i		−	0.675708i
$914$			2.52568e21i			0.156777i
$915$	−8.27216e20			−0.0509285
$916$		−	1.26874e22i		−	0.774740i
$917$			3.15995e22i			1.91386i
$918$		−	2.38705e21i		−	0.143398i
$919$	1.19884e22			0.714323			0.357161	−	0.934043i	$-0.383745\pi$
$919$	1.19884e22			0.714323			0.357161	+	0.934043i	$0.383745\pi$
$920$		−	7.60966e20i		−	0.0449735i
$921$	1.88261e21			0.110360
$922$	1.12270e22			0.652803
$923$	−3.64574e21			−0.210268
$924$	1.20330e22			0.688391
$925$			1.86793e22i			1.05998i
$926$		−	1.01588e22i		−	0.571822i
$927$	−2.93492e21			−0.163870
$928$	9.95390e21	+	1.53748e22i	0.551294	+	0.851532i
$929$	−1.01298e22			−0.556522			−0.278261	−	0.960505i	$-0.589758\pi$
$929$	−1.01298e22			−0.556522			−0.278261	+	0.960505i	$0.589758\pi$
$930$			3.83829e19i			0.00209178i
$931$			3.89881e22i			2.10770i
$932$	−1.75062e22			−0.938798
$933$	−2.17680e22			−1.15799
$934$	5.17780e21			0.273240
$935$	6.31996e20			0.0330847
$936$		−	1.94787e21i		−	0.101156i
$937$	−4.87862e21			−0.251333			−0.125667	−	0.992073i	$-0.540107\pi$
$937$	−4.87862e21			−0.251333			−0.125667	+	0.992073i	$0.540107\pi$
$938$		−	1.83427e22i		−	0.937437i
$939$			6.13095e21i			0.310839i
$940$			3.45850e21i			0.173952i
$941$	1.30363e22			0.650478			0.325239	−	0.945632i	$-0.394555\pi$
$941$	1.30363e22			0.650478			0.325239	+	0.945632i	$0.394555\pi$
$942$		−	6.36426e21i		−	0.315040i
$943$		−	7.52211e21i		−	0.369404i
$944$	1.02194e22			0.497892
$945$		−	4.32028e21i		−	0.208820i
$946$		−	6.70246e21i		−	0.321404i
$947$			7.80350e21i			0.371249i	0.982621	+	0.185624i	$0.0594307\pi$
$947$			7.80350e21i			0.371249i	−0.982621	+	0.185624i	$0.940569\pi$
$948$	−1.90969e22			−0.901364
$949$			4.67480e21i			0.218911i
$950$	1.34739e22			0.625991
$951$	4.54963e21			0.209711
$952$	8.72403e21			0.398970
$953$	−1.40127e22			−0.635806			−0.317903	−	0.948123i	$-0.602979\pi$
$953$	−1.40127e22			−0.635806			−0.317903	+	0.948123i	$0.602979\pi$
$954$			4.80602e21i			0.216358i
$955$			4.22330e20i			0.0188637i
$956$	−3.46615e22			−1.53608
$957$	1.05198e22	−	6.81067e21i	0.462560	−	0.299468i
$958$	5.32777e21			0.232436
$959$			2.77113e22i			1.19955i
$960$			1.40298e20i			0.00602585i
$961$	2.33958e22			0.997041
$962$	−2.47516e21			−0.104662
$963$	−1.87892e21			−0.0788336
$964$	−1.90608e22			−0.793529
$965$			4.15370e21i			0.171585i
$966$	−4.92900e21			−0.202037
$967$			4.72581e22i			1.92211i	0.276359	+	0.961055i	$0.410872\pi$
$967$			4.72581e22i			1.92211i	−0.276359	+	0.961055i	$0.589128\pi$
$968$			5.55387e21i			0.224146i
$969$		−	6.66125e21i		−	0.266764i
$970$	3.77941e20			0.0150188
$971$			2.87113e22i			1.13216i	0.824350	+	0.566081i	$0.191541\pi$
$971$			2.87113e22i			1.13216i	−0.824350	+	0.566081i	$0.808459\pi$
$972$		−	1.94987e22i		−	0.762973i
$973$	−9.03605e21			−0.350859
$974$		−	1.28649e22i		−	0.495693i
$975$		−	3.46741e21i		−	0.132578i
$976$		−	6.46690e21i		−	0.245371i
$977$	1.85141e22			0.697099			0.348549	−	0.937290i	$-0.386674\pi$
$977$	1.85141e22			0.697099			0.348549	+	0.937290i	$0.386674\pi$
$978$		−	7.08055e21i		−	0.264561i
$979$	1.85092e22			0.686305
$980$	4.20605e21			0.154767
$981$	1.23509e22			0.451004
$982$	2.80484e21			0.101642
$983$		−	1.17554e22i		−	0.422752i	−0.977405	−	0.211376i	$-0.932206\pi$
$983$		−	1.17554e22i		−	0.422752i	0.977405	−	0.211376i	$-0.0677944\pi$
$984$			1.30912e22i			0.467213i
$985$	4.43002e20			0.0156904
$986$	3.35173e21	−	2.16996e21i	0.117813	−	0.0762736i
$987$	5.09756e22			1.77821
$988$		−	6.48009e21i		−	0.224338i
$989$		−	9.96462e21i		−	0.342364i
$990$	−8.71894e20			−0.0297303
$991$	5.12666e22			1.73493			0.867465	−	0.497498i	$-0.165748\pi$
$991$	5.12666e22			1.73493			0.867465	+	0.497498i	$0.165748\pi$
$992$	−1.64293e21			−0.0551798
$993$	1.26099e22			0.420331
$994$			2.25264e22i			0.745238i
$995$	7.46354e21			0.245060
$996$			1.22765e22i			0.400063i
$997$		−	2.73703e22i		−	0.885250i	−0.896707	−	0.442625i	$-0.854047\pi$
$997$		−	2.73703e22i		−	0.885250i	0.896707	−	0.442625i	$-0.145953\pi$
$998$			3.51038e21i			0.112687i
$999$	−3.45604e22			−1.10113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.22	yes	36
29.28	even	2	inner	29.16.b.a.28.15	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.15	✓	36	29.28	even	2	inner
29.16.b.a.28.22	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)

\(f(q)\)	\(=\)	\(q+84.1317i q^{2} -2443.60i q^{3} +25689.9 q^{4} +22406.3 q^{5} +205584. q^{6} +3.47198e6 q^{7} +4.91816e6i q^{8} +8.37775e6 q^{9} +O(q^{10})\)	\(q+84.1317i q^{2} -2443.60i q^{3} +25689.9 q^{4} +22406.3 q^{5} +205584. q^{6} +3.47198e6 q^{7} +4.91816e6i q^{8} +8.37775e6 q^{9} +1.88508e6i q^{10} +5.52086e7i q^{11} -6.27756e7i q^{12} -4.72749e7 q^{13} +2.92103e8i q^{14} -5.47519e7i q^{15} +4.28032e8 q^{16} -5.10902e8i q^{17} +7.04834e8i q^{18} +5.33567e9i q^{19} +5.75614e8 q^{20} -8.48411e9i q^{21} -4.64480e9 q^{22} -6.90547e9 q^{23} +1.20180e10 q^{24} -3.00155e10 q^{25} -3.97731e9i q^{26} -5.55348e10i q^{27} +8.91946e10 q^{28} +(7.79779e10 - 5.04840e10i) q^{29} +4.60637e9 q^{30} +8.33256e9i q^{31} +1.97169e11i q^{32} +1.34908e11 q^{33} +4.29831e10 q^{34} +7.77941e10 q^{35} +2.15223e11 q^{36} -6.22321e11i q^{37} -4.48899e11 q^{38} +1.15521e11i q^{39} +1.10198e11i q^{40} +1.08930e12i q^{41} +7.13782e11 q^{42} +1.44300e12i q^{43} +1.41830e12i q^{44} +1.87714e11 q^{45} -5.80969e11i q^{46} +6.00837e12i q^{47} -1.04594e12i q^{48} +7.30706e12 q^{49} -2.52526e12i q^{50} -1.24844e12 q^{51} -1.21448e12 q^{52} +6.81866e12 q^{53} +4.67223e12 q^{54} +1.23702e12i q^{55} +1.70757e13i q^{56} +1.30382e13 q^{57} +(4.24731e12 + 6.56041e12i) q^{58} +2.38753e13 q^{59} -1.40657e12i q^{60} -1.51084e13i q^{61} -7.01033e11 q^{62} +2.90874e13 q^{63} -2.56244e12 q^{64} -1.05925e12 q^{65} +1.13500e13i q^{66} -6.27953e13 q^{67} -1.31250e13i q^{68} +1.68742e13i q^{69} +6.54495e12i q^{70} +7.71181e13 q^{71} +4.12031e13i q^{72} -9.88856e13i q^{73} +5.23569e13 q^{74} +7.33458e13i q^{75} +1.37073e14i q^{76} +1.91683e14i q^{77} -9.71895e12 q^{78} -3.04209e14i q^{79} +9.59061e12 q^{80} -1.54929e13 q^{81} -9.16445e13 q^{82} -1.95561e14 q^{83} -2.17956e14i q^{84} -1.14474e13i q^{85} -1.21402e14 q^{86} +(-1.23363e14 - 1.90546e14i) q^{87} -2.71525e14 q^{88} -3.35259e14i q^{89} +1.57927e13i q^{90} -1.64137e14 q^{91} -1.77400e14 q^{92} +2.03614e13 q^{93} -5.05494e14 q^{94} +1.19553e14i q^{95} +4.81802e14 q^{96} -2.00491e14i q^{97} +6.14756e14i q^{98} +4.62524e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 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

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