LMFDB - Embedded newform 29.16.b.a.28.4

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			28.4
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.33

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-297.362i q^{2} +5034.99i q^{3} -55656.3 q^{4} -122226. q^{5} +1.49722e6 q^{6} +2.97082e6 q^{7} +6.80612e6i q^{8} -1.10022e7 q^{9} +O(q^{10})$	\(q-297.362i q^{2} +5034.99i q^{3} -55656.3 q^{4} -122226. q^{5} +1.49722e6 q^{6} +2.97082e6 q^{7} +6.80612e6i q^{8} -1.10022e7 q^{9} +3.63455e7i q^{10} +6.17092e6i q^{11} -2.80229e8i q^{12} -4.98463e7 q^{13} -8.83410e8i q^{14} -6.15408e8i q^{15} +2.00137e8 q^{16} -2.30401e9i q^{17} +3.27164e9i q^{18} +3.73415e9i q^{19} +6.80266e9 q^{20} +1.49581e10i q^{21} +1.83500e9 q^{22} -2.07415e9 q^{23} -3.42687e10 q^{24} -1.55783e10 q^{25} +1.48224e10i q^{26} +1.68506e10i q^{27} -1.65345e11 q^{28} +(2.92906e10 - 8.81547e10i) q^{29} -1.82999e11 q^{30} +1.24346e11i q^{31} +1.63510e11i q^{32} -3.10705e10 q^{33} -6.85127e11 q^{34} -3.63112e11 q^{35} +6.12342e11 q^{36} +7.98590e11i q^{37} +1.11040e12 q^{38} -2.50975e11i q^{39} -8.31886e11i q^{40} +7.89146e11i q^{41} +4.44796e12 q^{42} +1.01834e12i q^{43} -3.43451e11i q^{44} +1.34476e12 q^{45} +6.16774e11i q^{46} +1.01900e12i q^{47} +1.00769e12i q^{48} +4.07822e12 q^{49} +4.63241e12i q^{50} +1.16007e13 q^{51} +2.77426e12 q^{52} -6.28305e12 q^{53} +5.01075e12 q^{54} -7.54248e11i q^{55} +2.02198e13i q^{56} -1.88014e13 q^{57} +(-2.62139e13 - 8.70991e12i) q^{58} -3.10987e13 q^{59} +3.42513e13i q^{60} +2.04286e13i q^{61} +3.69758e13 q^{62} -3.26856e13 q^{63} +5.51797e13 q^{64} +6.09253e12 q^{65} +9.23920e12i q^{66} +1.18328e12 q^{67} +1.28233e14i q^{68} -1.04433e13i q^{69} +1.07976e14i q^{70} -1.12129e14 q^{71} -7.48823e13i q^{72} -5.76841e13i q^{73} +2.37470e14 q^{74} -7.84367e13i q^{75} -2.07829e14i q^{76} +1.83327e13i q^{77} -7.46306e13 q^{78} -9.00161e13i q^{79} -2.44620e13 q^{80} -2.42712e14 q^{81} +2.34662e14 q^{82} -1.65806e14 q^{83} -8.32510e14i q^{84} +2.81611e14i q^{85} +3.02817e14 q^{86} +(4.43858e14 + 1.47478e14i) q^{87} -4.20000e13 q^{88} +8.14550e14i q^{89} -3.99880e14i q^{90} -1.48084e14 q^{91} +1.15440e14 q^{92} -6.26081e14 q^{93} +3.03012e14 q^{94} -4.56412e14i q^{95} -8.23269e14 q^{96} +1.08661e15i q^{97} -1.21271e15i q^{98} -6.78937e13i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	297.362i		−	1.64271i	−0.570417	−	0.821355i	$-0.693219\pi$
$2$		−	297.362i		−	1.64271i	0.570417	−	0.821355i	$-0.306781\pi$
$3$			5034.99i			1.32920i	0.747201	+	0.664598i	$0.231397\pi$
$3$			5034.99i			1.32920i	−0.747201	+	0.664598i	$0.768603\pi$
$4$	−55656.3			−1.69850
$5$	−122226.			−0.699664			−0.349832	−	0.936812i	$-0.613761\pi$
$5$	−122226.			−0.699664			−0.349832	+	0.936812i	$0.613761\pi$
$6$	1.49722e6			2.18348
$7$	2.97082e6			1.36346			0.681728	−	0.731606i	$-0.261229\pi$
$7$	2.97082e6			1.36346			0.681728	+	0.731606i	$0.261229\pi$
$8$			6.80612e6i			1.14743i
$9$	−1.10022e7			−0.766762
$10$			3.63455e7i			1.14934i
$11$			6.17092e6i			0.0954783i	0.998860	+	0.0477392i	$0.0152016\pi$
$11$			6.17092e6i			0.0954783i	−0.998860	+	0.0477392i	$0.984798\pi$
$12$		−	2.80229e8i		−	2.25763i
$13$	−4.98463e7			−0.220322			−0.110161	−	0.993914i	$-0.535137\pi$
$13$	−4.98463e7			−0.220322			−0.110161	+	0.993914i	$0.535137\pi$
$14$		−	8.83410e8i		−	2.23976i
$15$		−	6.15408e8i		−	0.929990i
$16$	2.00137e8			0.186392
$17$		−	2.30401e9i		−	1.36181i	−0.732369	−	0.680907i	$-0.761586\pi$
$17$		−	2.30401e9i		−	1.36181i	0.732369	−	0.680907i	$-0.238414\pi$
$18$			3.27164e9i			1.25957i
$19$			3.73415e9i			0.958386i	0.877710	+	0.479193i	$0.159071\pi$
$19$			3.73415e9i			0.958386i	−0.877710	+	0.479193i	$0.840929\pi$
$20$	6.80266e9			1.18838
$21$			1.49581e10i			1.81230i
$22$	1.83500e9			0.156843
$23$	−2.07415e9			−0.127023			−0.0635114	−	0.997981i	$-0.520230\pi$
$23$	−2.07415e9			−0.127023			−0.0635114	+	0.997981i	$0.520230\pi$
$24$	−3.42687e10			−1.52515
$25$	−1.55783e10			−0.510470
$26$			1.48224e10i			0.361925i
$27$			1.68506e10i			0.310019i
$28$	−1.65345e11			−2.31583
$29$	2.92906e10	−	8.81547e10i	0.315314	−	0.948987i
$29$	2.92906e10	−	8.81547e10i	0.315314	−	0.948987i
$30$	−1.82999e11			−1.52770
$31$			1.24346e11i			0.811744i	0.913930	+	0.405872i	$0.133032\pi$
$31$			1.24346e11i			0.811744i	−0.913930	+	0.405872i	$0.866968\pi$
$32$			1.63510e11i			0.841238i
$33$	−3.10705e10			−0.126909
$34$	−6.85127e11			−2.23707
$35$	−3.63112e11			−0.953961
$36$	6.12342e11			1.30234
$37$			7.98590e11i			1.38296i	0.722394	+	0.691482i	$0.243042\pi$
$37$			7.98590e11i			1.38296i	−0.722394	+	0.691482i	$0.756958\pi$
$38$	1.11040e12			1.57435
$39$		−	2.50975e11i		−	0.292851i
$40$		−	8.31886e11i		−	0.802813i
$41$			7.89146e11i			0.632818i	0.948623	+	0.316409i	$0.102477\pi$
$41$			7.89146e11i			0.632818i	−0.948623	+	0.316409i	$0.897523\pi$
$42$	4.44796e12			2.97708
$43$			1.01834e12i			0.571322i	0.958331	+	0.285661i	$0.0922132\pi$
$43$			1.01834e12i			0.571322i	−0.958331	+	0.285661i	$0.907787\pi$
$44$		−	3.43451e11i		−	0.162170i
$45$	1.34476e12			0.536476
$46$			6.16774e11i			0.208661i
$47$			1.01900e12i			0.293386i	0.989182	+	0.146693i	$0.0468630\pi$
$47$			1.01900e12i			0.293386i	−0.989182	+	0.146693i	$0.953137\pi$
$48$			1.00769e12i			0.247752i
$49$	4.07822e12			0.859014
$50$			4.63241e12i			0.838555i
$51$	1.16007e13			1.81012
$52$	2.77426e12			0.374216
$53$	−6.28305e12			−0.734686			−0.367343	−	0.930086i	$-0.619732\pi$
$53$	−6.28305e12			−0.734686			−0.367343	+	0.930086i	$0.619732\pi$
$54$	5.01075e12			0.509271
$55$		−	7.54248e11i		−	0.0668027i
$56$			2.02198e13i			1.56447i
$57$	−1.88014e13			−1.27388
$58$	−2.62139e13	−	8.70991e12i	−1.55891	−	0.517969i
$59$	−3.10987e13			−1.62687			−0.813434	−	0.581657i	$-0.802404\pi$
$59$	−3.10987e13			−1.62687			−0.813434	+	0.581657i	$0.802404\pi$
$60$			3.42513e13i			1.57958i
$61$			2.04286e13i			0.832270i	0.909303	+	0.416135i	$0.136616\pi$
$61$			2.04286e13i			0.832270i	−0.909303	+	0.416135i	$0.863384\pi$
$62$	3.69758e13			1.33346
$63$	−3.26856e13			−1.04545
$64$	5.51797e13			1.56830
$65$	6.09253e12			0.154151
$66$			9.23920e12i			0.208475i
$67$	1.18328e12			0.0238521			0.0119261	−	0.999929i	$-0.496204\pi$
$67$	1.18328e12			0.0238521			0.0119261	+	0.999929i	$0.496204\pi$
$68$			1.28233e14i			2.31304i
$69$		−	1.04433e13i		−	0.168838i
$70$			1.07976e14i			1.56708i
$71$	−1.12129e14			−1.46312			−0.731562	−	0.681775i	$-0.761208\pi$
$71$	−1.12129e14			−1.46312			−0.731562	+	0.681775i	$0.761208\pi$
$72$		−	7.48823e13i		−	0.879803i
$73$		−	5.76841e13i		−	0.611131i	−0.952171	−	0.305566i	$-0.901155\pi$
$73$		−	5.76841e13i		−	0.611131i	0.952171	−	0.305566i	$-0.0988455\pi$
$74$	2.37470e14			2.27181
$75$		−	7.84367e13i		−	0.678515i
$76$		−	2.07829e14i		−	1.62781i
$77$			1.83327e13i			0.130181i
$78$	−7.46306e13			−0.481069
$79$		−	9.00161e13i		−	0.527372i	−0.964609	−	0.263686i	$-0.915062\pi$
$79$		−	9.00161e13i		−	0.527372i	0.964609	−	0.263686i	$-0.0849383\pi$
$80$	−2.44620e13			−0.130412
$81$	−2.42712e14			−1.17884
$82$	2.34662e14			1.03954
$83$	−1.65806e14			−0.670680			−0.335340	−	0.942097i	$-0.608851\pi$
$83$	−1.65806e14			−0.670680			−0.335340	+	0.942097i	$0.608851\pi$
$84$		−	8.32510e14i		−	3.07819i
$85$			2.81611e14i			0.952813i
$86$	3.02817e14			0.938516
$87$	4.43858e14	+	1.47478e14i	1.26139	+	0.419114i
$88$	−4.20000e13			−0.109554
$89$			8.14550e14i			1.95206i	0.217642	+	0.976029i	$0.430163\pi$
$89$			8.14550e14i			1.95206i	−0.217642	+	0.976029i	$0.569837\pi$
$90$		−	3.99880e14i		−	0.881274i
$91$	−1.48084e14			−0.300399
$92$	1.15440e14			0.215748
$93$	−6.26081e14			−1.07897
$94$	3.03012e14			0.481948
$95$		−	4.56412e14i		−	0.670548i
$96$	−8.23269e14			−1.11817
$97$			1.08661e15i			1.36548i	0.730663	+	0.682738i	$0.239211\pi$
$97$			1.08661e15i			1.36548i	−0.730663	+	0.682738i	$0.760789\pi$
$98$		−	1.21271e15i		−	1.41111i
$99$		−	6.78937e13i		−	0.0732091i
$100$	8.67032e14			0.867032
$101$		−	2.01082e14i		−	0.186622i	−0.995637	−	0.0933112i	$-0.970255\pi$
$101$		−	2.01082e14i		−	0.186622i	0.995637	−	0.0933112i	$-0.0297452\pi$
$102$		−	3.44960e15i		−	2.97350i
$103$	−1.89595e15			−1.51896			−0.759482	−	0.650528i	$-0.774548\pi$
$103$	−1.89595e15			−1.51896			−0.759482	+	0.650528i	$0.774548\pi$
$104$		−	3.39260e14i		−	0.252803i
$105$		−	1.82827e15i		−	1.26800i
$106$			1.86834e15i			1.20688i
$107$	−3.63216e13			−0.0218669			−0.0109334	−	0.999940i	$-0.503480\pi$
$107$	−3.63216e13			−0.0218669			−0.0109334	+	0.999940i	$0.503480\pi$
$108$		−	9.37845e14i		−	0.526566i
$109$	−3.22700e14			−0.169083			−0.0845415	−	0.996420i	$-0.526943\pi$
$109$	−3.22700e14			−0.169083			−0.0845415	+	0.996420i	$0.526943\pi$
$110$	−2.24285e14			−0.109737
$111$	−4.02089e15			−1.83823
$112$	5.94572e14			0.254138
$113$			2.66539e15i			1.06579i	0.846181	+	0.532895i	$0.178896\pi$
$113$			2.66539e15i			1.06579i	−0.846181	+	0.532895i	$0.821104\pi$
$114$			5.59083e15i			2.09262i
$115$	2.53516e14			0.0888732
$116$	−1.63021e15	+	4.90637e15i	−0.535559	+	1.61185i
$117$	5.48419e14			0.168934
$118$			9.24758e15i			2.67247i
$119$		−	6.84481e15i		−	1.85678i
$120$	4.18854e15			1.06710
$121$	4.13917e15			0.990884
$122$	6.07468e15			1.36718
$123$	−3.97334e15			−0.841139
$124$		−	6.92064e15i		−	1.37874i
$125$	5.63413e15			1.05682
$126$			9.71945e15i			1.71737i
$127$			3.24868e15i			0.540977i	0.962723	+	0.270489i	$0.0871853\pi$
$127$			3.24868e15i			0.540977i	−0.962723	+	0.270489i	$0.912815\pi$
$128$		−	1.10505e16i		−	1.73503i
$129$	−5.12735e15			−0.759399
$130$		−	1.81169e15i		−	0.253226i
$131$		−	1.10002e16i		−	1.45167i	−0.687871	−	0.725833i	$-0.741455\pi$
$131$		−	1.10002e16i		−	1.45167i	0.687871	−	0.725833i	$-0.258545\pi$
$132$	1.72927e15			0.215555
$133$			1.10935e16i			1.30672i
$134$		−	3.51863e14i		−	0.0391821i
$135$		−	2.05959e15i		−	0.216909i
$136$	1.56814e16			1.56258
$137$		−	3.61053e14i		−	0.0340539i	−0.999855	−	0.0170270i	$-0.994580\pi$
$137$		−	3.61053e14i		−	0.0340539i	0.999855	−	0.0170270i	$-0.00542011\pi$
$138$	−3.10545e15			−0.277352
$139$	1.61011e16			1.36221			0.681106	−	0.732185i	$-0.261499\pi$
$139$	1.61011e16			1.36221			0.681106	+	0.732185i	$0.261499\pi$
$140$	2.02095e16			1.62030
$141$	−5.13065e15			−0.389968
$142$			3.33430e16i			2.40349i
$143$		−	3.07597e14i		−	0.0210360i
$144$	−2.20195e15			−0.142918
$145$	−3.58008e15	+	1.07748e16i	−0.220614	+	0.663972i
$146$	−1.71531e16			−1.00391
$147$			2.05338e16i			1.14180i
$148$		−	4.44466e16i		−	2.34896i
$149$	−1.03272e16			−0.518902			−0.259451	−	0.965756i	$-0.583542\pi$
$149$	−1.03272e16			−0.518902			−0.259451	+	0.965756i	$0.583542\pi$
$150$	−2.33241e16			−1.11460
$151$	−2.73660e16			−1.24418			−0.622091	−	0.782945i	$-0.713717\pi$
$151$	−2.73660e16			−1.24418			−0.622091	+	0.782945i	$0.713717\pi$
$152$	−2.54151e16			−1.09968
$153$			2.53492e16i			1.04419i
$154$	5.45145e15			0.213849
$155$		−	1.51984e16i		−	0.567948i
$156$			1.39684e16i			0.497406i
$157$		−	2.10423e16i		−	0.714241i	−0.934058	−	0.357121i	$-0.883758\pi$
$157$		−	2.10423e16i		−	0.714241i	0.934058	−	0.357121i	$-0.116242\pi$
$158$	−2.67674e16			−0.866319
$159$		−	3.16351e16i		−	0.976541i
$160$		−	1.99852e16i		−	0.588584i
$161$	−6.16193e15			−0.173190
$162$			7.21735e16i			1.93649i
$163$		−	3.43458e16i		−	0.879968i	−0.898006	−	0.439984i	$-0.854984\pi$
$163$		−	3.43458e16i		−	0.879968i	0.898006	−	0.439984i	$-0.145016\pi$
$164$		−	4.39210e16i		−	1.07484i
$165$	3.79763e15			0.0887939
$166$			4.93045e16i			1.10173i
$167$	1.73627e16			0.370889			0.185444	−	0.982655i	$-0.440628\pi$
$167$	1.73627e16			0.370889			0.185444	+	0.982655i	$0.440628\pi$
$168$	−1.01806e17			−2.07948
$169$	−4.87012e16			−0.951458
$170$	8.37405e16			1.56519
$171$		−	4.10839e16i		−	0.734854i
$172$		−	5.66773e16i		−	0.970388i
$173$	1.01919e17			1.67074			0.835370	−	0.549688i	$-0.185254\pi$
$173$	1.01919e17			1.67074			0.835370	+	0.549688i	$0.185254\pi$
$174$	4.38543e16	−	1.31987e17i	0.688482	−	2.07210i
$175$	−4.62804e16			−0.696004
$176$			1.23503e15i			0.0177964i
$177$		−	1.56582e17i		−	2.16243i
$178$	2.42216e17			3.20666
$179$	8.90885e16			1.13090			0.565449	−	0.824783i	$-0.308703\pi$
$179$	8.90885e16			1.13090			0.565449	+	0.824783i	$0.308703\pi$
$180$	−7.48442e16			−0.911202
$181$	5.75594e16			0.672244			0.336122	−	0.941818i	$-0.390885\pi$
$181$	5.75594e16			0.672244			0.336122	+	0.941818i	$0.390885\pi$
$182$			4.40347e16i			0.493469i
$183$	−1.02858e17			−1.10625
$184$		−	1.41169e16i		−	0.145749i
$185$		−	9.76086e16i		−	0.967610i
$186$			1.86173e17i			1.77243i
$187$	1.42179e16			0.130024
$188$		−	5.67137e16i		−	0.498315i
$189$			5.00603e16i			0.422697i
$190$	−1.35720e17			−1.10152
$191$			1.50682e17i			1.17574i	0.808955	+	0.587870i	$0.200033\pi$
$191$			1.50682e17i			1.17574i	−0.808955	+	0.587870i	$0.799967\pi$
$192$			2.77829e17i			2.08458i
$193$			9.58914e16i			0.691990i	0.938236	+	0.345995i	$0.112459\pi$
$193$			9.58914e16i			0.691990i	−0.938236	+	0.345995i	$0.887541\pi$
$194$	3.23116e17			2.24308
$195$			3.06758e16i			0.204897i
$196$	−2.26979e17			−1.45903
$197$	2.79115e17			1.72698			0.863488	−	0.504370i	$-0.168275\pi$
$197$	2.79115e17			1.72698			0.863488	+	0.504370i	$0.168275\pi$
$198$	−2.01890e16			−0.120261
$199$	−3.44856e17			−1.97806			−0.989032	−	0.147703i	$-0.952812\pi$
$199$	−3.44856e17			−1.97806			−0.989032	+	0.147703i	$0.952812\pi$
$200$		−	1.06028e17i		−	0.585727i
$201$			5.95781e15i			0.0317042i
$202$	−5.97943e16			−0.306567
$203$	8.70171e16	−	2.61892e17i	0.429917	−	1.29390i
$204$	−6.45651e17			−3.07448
$205$		−	9.64544e16i		−	0.442760i
$206$			5.63784e17i			2.49522i
$207$	2.28202e16			0.0973962
$208$	−9.97609e15			−0.0410663
$209$	−2.30432e16			−0.0915050
$210$	−5.43657e17			−2.08296
$211$			5.93836e16i			0.219557i	0.993956	+	0.109779i	$0.0350142\pi$
$211$			5.93836e16i			0.219557i	−0.993956	+	0.109779i	$0.964986\pi$
$212$	3.49691e17			1.24786
$213$		−	5.64569e17i		−	1.94478i
$214$			1.08007e16i			0.0359210i
$215$		−	1.24468e17i		−	0.399733i
$216$	−1.14688e17			−0.355724
$217$			3.69410e17i			1.10678i
$218$			9.59587e16i			0.277754i
$219$	2.90439e17			0.812313
$220$			4.19787e16i			0.113464i
$221$			1.14847e17i			0.300038i
$222$			1.19566e18i			3.01968i
$223$	3.05094e17			0.744985			0.372493	−	0.928035i	$-0.378503\pi$
$223$	3.05094e17			0.744985			0.372493	+	0.928035i	$0.378503\pi$
$224$			4.85758e17i			1.14699i
$225$	1.71396e17			0.391409
$226$	7.92585e17			1.75078
$227$	2.43727e17			0.520847			0.260423	−	0.965495i	$-0.416138\pi$
$227$	2.43727e17			0.520847			0.260423	+	0.965495i	$0.416138\pi$
$228$	1.04642e18			2.16368
$229$		−	8.97125e17i		−	1.79509i	−0.440921	−	0.897546i	$-0.645348\pi$
$229$		−	8.97125e17i		−	1.79509i	0.440921	−	0.897546i	$-0.354652\pi$
$230$		−	7.53860e16i		−	0.145993i
$231$	−9.23049e16			−0.173035
$232$	5.99991e17	+	1.99355e17i	1.08889	+	0.361799i
$233$	−5.29542e17			−0.930532			−0.465266	−	0.885171i	$-0.654041\pi$
$233$	−5.29542e17			−0.930532			−0.465266	+	0.885171i	$0.654041\pi$
$234$		−	1.63079e17i		−	0.277510i
$235$		−	1.24548e17i		−	0.205272i
$236$	1.73084e18			2.76323
$237$	4.53230e17			0.700981
$238$	−2.03539e18			−3.05014
$239$	−1.12330e18			−1.63121			−0.815607	−	0.578606i	$-0.803597\pi$
$239$	−1.12330e18			−1.63121			−0.815607	+	0.578606i	$0.803597\pi$
$240$		−	1.23166e17i		−	0.173343i
$241$	−7.25800e17			−0.990124			−0.495062	−	0.868858i	$-0.664855\pi$
$241$	−7.25800e17			−0.990124			−0.495062	+	0.868858i	$0.664855\pi$
$242$		−	1.23083e18i		−	1.62773i
$243$		−	9.80265e17i		−	1.25689i
$244$		−	1.13698e18i		−	1.41361i
$245$	−4.98465e17			−0.601021
$246$			1.18152e18i			1.38175i
$247$		−	1.86134e17i		−	0.211153i
$248$	−8.46314e17			−0.931417
$249$		−	8.34832e17i		−	0.891465i
$250$		−	1.67538e18i		−	1.73605i
$251$		−	5.56424e17i		−	0.559568i	−0.960063	−	0.279784i	$-0.909737\pi$
$251$		−	5.56424e17i		−	0.559568i	0.960063	−	0.279784i	$-0.0902628\pi$
$252$	1.81916e18			1.77569
$253$		−	1.27994e16i		−	0.0121279i
$254$	9.66035e17			0.888669
$255$	−1.41791e18			−1.26647
$256$	−1.47787e18			−1.28184
$257$	1.53202e18			1.29052			0.645261	−	0.763962i	$-0.276749\pi$
$257$	1.53202e18			1.29052			0.645261	+	0.763962i	$0.276749\pi$
$258$			1.52468e18i			1.24747i
$259$			2.37247e18i			1.88561i
$260$	−3.39087e17			−0.261825
$261$	−3.22261e17	+	9.69895e17i	−0.241771	+	0.727648i
$262$	−3.27105e18			−2.38467
$263$		−	6.26844e17i		−	0.444111i	−0.975034	−	0.222055i	$-0.928723\pi$
$263$		−	6.26844e17i		−	0.444111i	0.975034	−	0.222055i	$-0.0712766\pi$
$264$		−	2.11470e17i		−	0.145619i
$265$	7.67954e17			0.514033
$266$	3.29879e18			2.14656
$267$	−4.10125e18			−2.59467
$268$	−6.58571e16			−0.0405127
$269$			4.64212e17i			0.277699i	0.990314	+	0.138849i	$0.0443404\pi$
$269$			4.64212e17i			0.277699i	−0.990314	+	0.138849i	$0.955660\pi$
$270$	−6.12445e17			−0.356319
$271$			2.11667e18i			1.19780i	0.800825	+	0.598898i	$0.204395\pi$
$271$			2.11667e18i			1.19780i	−0.800825	+	0.598898i	$0.795605\pi$
$272$		−	4.61119e17i		−	0.253832i
$273$		−	7.45603e17i		−	0.399290i
$274$	−1.07364e17			−0.0559407
$275$		−	9.61326e16i		−	0.0487389i
$276$			5.81237e17i			0.286771i
$277$	1.00573e18			0.482930			0.241465	−	0.970410i	$-0.422372\pi$
$277$	1.00573e18			0.482930			0.241465	+	0.970410i	$0.422372\pi$
$278$		−	4.78786e18i		−	2.23772i
$279$		−	1.36808e18i		−	0.622415i
$280$		−	2.47139e18i		−	1.09460i
$281$	5.32904e17			0.229801			0.114900	−	0.993377i	$-0.463345\pi$
$281$	5.32904e17			0.229801			0.114900	+	0.993377i	$0.463345\pi$
$282$			1.52566e18i			0.640604i
$283$	−4.05420e18			−1.65770			−0.828851	−	0.559469i	$-0.811005\pi$
$283$	−4.05420e18			−1.65770			−0.828851	+	0.559469i	$0.811005\pi$
$284$	6.24069e18			2.48511
$285$	2.29803e18			0.891289
$286$	−9.14679e16			−0.0345560
$287$			2.34441e18i			0.862819i
$288$		−	1.79897e18i		−	0.645029i
$289$	−2.44606e18			−0.854540
$290$	3.20402e18	+	1.06458e18i	1.09071	+	0.362404i
$291$	−5.47105e18			−1.81498
$292$			3.21048e18i			1.03800i
$293$		−	2.69877e18i		−	0.850471i	−0.905083	−	0.425235i	$-0.860191\pi$
$293$		−	2.69877e18i		−	0.850471i	0.905083	−	0.425235i	$-0.139809\pi$
$294$	6.10597e18			1.87564
$295$	3.80108e18			1.13826
$296$	−5.43530e18			−1.58685
$297$	−1.03984e17			−0.0296001
$298$			3.07092e18i			0.852406i
$299$	1.03389e17			0.0279859
$300$			4.36550e18i			1.15246i
$301$			3.02532e18i			0.778973i
$302$			8.13761e18i			2.04383i
$303$	1.01245e18			0.248058
$304$			7.47343e17i			0.178636i
$305$		−	2.49691e18i		−	0.582309i
$306$	7.53790e18			1.71530
$307$			6.13712e18i			1.36278i	0.731919	+	0.681391i	$0.238625\pi$
$307$			6.13712e18i			1.36278i	−0.731919	+	0.681391i	$0.761375\pi$
$308$		−	1.02033e18i		−	0.221111i
$309$		−	9.54608e18i		−	2.01900i
$310$	−4.51942e18			−0.932974
$311$		−	9.45807e18i		−	1.90590i	−0.303130	−	0.952949i	$-0.598032\pi$
$311$		−	9.45807e18i		−	1.90590i	0.303130	−	0.952949i	$-0.401968\pi$
$312$	1.70817e18			0.336025
$313$	9.31364e18			1.78870			0.894349	−	0.447370i	$-0.147639\pi$
$313$	9.31364e18			1.78870			0.894349	+	0.447370i	$0.147639\pi$
$314$	−6.25717e18			−1.17329
$315$	3.99503e18			0.731461
$316$			5.00996e18i			0.895739i
$317$			6.12252e18i			1.06902i	0.845162	+	0.534510i	$0.179504\pi$
$317$			6.12252e18i			1.06902i	−0.845162	+	0.534510i	$0.820496\pi$
$318$	−9.40708e18			−1.60417
$319$	5.43996e17	+	1.80750e17i	0.0906077	+	0.0301056i
$320$	−6.74441e18			−1.09728
$321$		−	1.82879e17i		−	0.0290654i
$322$			1.83233e18i			0.284501i
$323$	8.60354e18			1.30514
$324$	1.35085e19			2.00225
$325$	7.76522e17			0.112468
$326$	−1.02131e19			−1.44553
$327$		−	1.62479e18i		−	0.224744i
$328$	−5.37102e18			−0.726111
$329$			3.02726e18i			0.400019i
$330$		−	1.12927e18i		−	0.145863i
$331$			7.42217e18i			0.937175i	0.883417	+	0.468587i	$0.155237\pi$
$331$			7.42217e18i			0.937175i	−0.883417	+	0.468587i	$0.844763\pi$
$332$	9.22816e18			1.13915
$333$		−	8.78624e18i		−	1.06040i
$334$		−	5.16301e18i		−	0.609262i
$335$	−1.44628e17			−0.0166885
$336$			2.99366e18i			0.337799i
$337$			1.09693e19i			1.21047i	0.796045	+	0.605237i	$0.206922\pi$
$337$			1.09693e19i			1.21047i	−0.796045	+	0.605237i	$0.793078\pi$
$338$			1.44819e19i			1.56297i
$339$	−1.34202e19			−1.41664
$340$		−	1.56734e19i		−	1.61835i
$341$	−7.67329e17			−0.0775040
$342$	−1.22168e19			−1.20715
$343$	−1.98850e18			−0.192229
$344$	−6.93097e18			−0.655550
$345$			1.27645e18i			0.118130i
$346$		−	3.03068e19i		−	2.74454i
$347$	6.58748e18			0.583779			0.291889	−	0.956452i	$-0.405716\pi$
$347$	6.58748e18			0.583779			0.291889	+	0.956452i	$0.405716\pi$
$348$	−2.47035e19	−	8.20806e18i	−2.14247	−	0.711863i
$349$	−1.99310e18			−0.169176			−0.0845880	−	0.996416i	$-0.526957\pi$
$349$	−1.99310e18			−0.169176			−0.0845880	+	0.996416i	$0.526957\pi$
$350$			1.37620e19i			1.14333i
$351$		−	8.39942e17i		−	0.0683040i
$352$	−1.00901e18			−0.0803200
$353$	2.13299e19			1.66218			0.831089	−	0.556139i	$-0.187718\pi$
$353$	2.13299e19			1.66218			0.831089	+	0.556139i	$0.187718\pi$
$354$	−4.65615e19			−3.55224
$355$	1.37051e19			1.02369
$356$		−	4.53348e19i		−	3.31556i
$357$	3.44636e19			2.46802
$358$		−	2.64915e19i		−	1.85774i
$359$			2.07330e19i			1.42381i	0.702275	+	0.711906i	$0.252168\pi$
$359$			2.07330e19i			1.42381i	−0.702275	+	0.711906i	$0.747832\pi$
$360$			9.15258e18i			0.615566i
$361$	1.23721e18			0.0814969
$362$		−	1.71160e19i		−	1.10430i
$363$			2.08407e19i			1.31708i
$364$	8.24183e18			0.510227
$365$			7.05051e18i			0.427586i
$366$			3.05860e19i			1.81725i
$367$		−	1.88384e19i		−	1.09660i	−0.836282	−	0.548300i	$-0.815275\pi$
$367$		−	1.88384e19i		−	1.09660i	0.836282	−	0.548300i	$-0.184725\pi$
$368$	−4.15114e17			−0.0236760
$369$		−	8.68234e18i		−	0.485220i
$370$	−2.90251e19			−1.58950
$371$	−1.86658e19			−1.00171
$372$	3.48454e19			1.83262
$373$	2.54096e18			0.130973			0.0654864	−	0.997853i	$-0.479140\pi$
$373$	2.54096e18			0.130973			0.0654864	+	0.997853i	$0.479140\pi$
$374$		−	4.22786e18i		−	0.213591i
$375$			2.83678e19i			1.40472i
$376$	−6.93543e18			−0.336639
$377$	−1.46003e18	+	4.39419e18i	−0.0694705	+	0.209083i
$378$	1.48860e19			0.694369
$379$		−	1.82425e19i		−	0.834237i	−0.908852	−	0.417118i	$-0.863040\pi$
$379$		−	1.82425e19i		−	0.834237i	0.908852	−	0.417118i	$-0.136960\pi$
$380$			2.54022e19i			1.13892i
$381$	−1.63571e19			−0.719065
$382$	4.48072e19			1.93140
$383$	−8.61526e18			−0.364148			−0.182074	−	0.983285i	$-0.558281\pi$
$383$	−8.61526e18			−0.364148			−0.182074	+	0.983285i	$0.558281\pi$
$384$	5.56390e19			2.30619
$385$		−	2.24074e18i		−	0.0910826i
$386$	2.85145e19			1.13674
$387$		−	1.12040e19i		−	0.438068i
$388$		−	6.04765e19i		−	2.31925i
$389$		−	1.60495e19i		−	0.603724i	−0.953352	−	0.301862i	$-0.902392\pi$
$389$		−	1.60495e19i		−	0.603724i	0.953352	−	0.301862i	$-0.0976082\pi$
$390$	9.12182e18			0.336587
$391$			4.77887e18i			0.172981i
$392$			2.77569e19i			0.985654i
$393$	5.53860e19			1.92955
$394$		−	8.29982e19i		−	2.83692i
$395$			1.10023e19i			0.368983i
$396$			3.77871e18i			0.124345i
$397$	−5.04705e19			−1.62971			−0.814853	−	0.579668i	$-0.803182\pi$
$397$	−5.04705e19			−1.62971			−0.814853	+	0.579668i	$0.803182\pi$
$398$			1.02547e20i			3.24938i
$399$	−5.58557e19			−1.73688
$400$	−3.11780e18			−0.0951477
$401$	−6.65554e19			−1.99343			−0.996714	−	0.0810000i	$-0.974189\pi$
$401$	−6.65554e19			−1.99343			−0.996714	+	0.0810000i	$0.974189\pi$
$402$	1.77163e18			0.0520807
$403$		−	6.19819e18i		−	0.178845i
$404$			1.11915e19i			0.316977i
$405$	2.96658e19			0.824790
$406$	−7.78768e19	−	2.58756e19i	−2.12551	−	0.706228i
$407$	−4.92803e18			−0.132043
$408$			7.89556e19i			2.07698i
$409$		−	3.98314e19i		−	1.02873i	−0.857572	−	0.514364i	$-0.828028\pi$
$409$		−	3.98314e19i		−	1.02873i	0.857572	−	0.514364i	$-0.171972\pi$
$410$	−2.86819e19			−0.727326
$411$	1.81790e18			0.0452644
$412$	1.05522e20			2.57995
$413$	−9.23887e19			−2.21816
$414$		−	6.78587e18i		−	0.159994i
$415$	2.02659e19			0.469250
$416$		−	8.15035e18i		−	0.185343i
$417$			8.10689e19i			1.81065i
$418$			6.85217e18i			0.150316i
$419$	8.58991e19			1.85090			0.925451	−	0.378868i	$-0.123686\pi$
$419$	8.58991e19			1.85090			0.925451	+	0.378868i	$0.123686\pi$
$420$			1.01755e20i			2.15370i
$421$			5.75677e19i			1.19692i	0.801155	+	0.598458i	$0.204219\pi$
$421$			5.75677e19i			1.19692i	−0.801155	+	0.598458i	$0.795781\pi$
$422$	1.76584e19			0.360669
$423$		−	1.12112e19i		−	0.224957i
$424$		−	4.27632e19i		−	0.842998i
$425$			3.58927e19i			0.695166i
$426$	−1.67881e20			−3.19471
$427$			6.06896e19i			1.13476i
$428$	2.02153e18			0.0371408
$429$	1.54875e18			0.0279609
$430$	−3.70122e19			−0.656646
$431$	−6.67146e19			−1.16316			−0.581582	−	0.813488i	$-0.697566\pi$
$431$	−6.67146e19			−1.16316			−0.581582	+	0.813488i	$0.697566\pi$
$432$			3.37244e18i			0.0577851i
$433$			9.41582e19i			1.58562i	0.609469	+	0.792810i	$0.291383\pi$
$433$			9.41582e19i			1.58562i	−0.609469	+	0.792810i	$0.708617\pi$
$434$	1.09849e20			1.81812
$435$	−5.42511e19	−	1.80256e19i	−0.882549	−	0.293239i
$436$	1.79603e19			0.287187
$437$		−	7.74520e18i		−	0.121737i
$438$		−	8.63655e19i		−	1.33439i
$439$	5.58859e19			0.848826			0.424413	−	0.905469i	$-0.360481\pi$
$439$	5.58859e19			0.848826			0.424413	+	0.905469i	$0.360481\pi$
$440$	5.13350e18			0.0766512
$441$	−4.48694e19			−0.658659
$442$	3.41510e19			0.492875
$443$			8.30529e19i			1.17849i	0.807953	+	0.589247i	$0.200575\pi$
$443$			8.30529e19i			1.17849i	−0.807953	+	0.589247i	$0.799425\pi$
$444$	2.23788e20			3.12223
$445$		−	9.95594e19i		−	1.36578i
$446$		−	9.07236e19i		−	1.22380i
$447$		−	5.19973e19i		−	0.689723i
$448$	1.63929e20			2.13831
$449$			1.26828e19i			0.162693i	0.996686	+	0.0813465i	$0.0259220\pi$
$449$			1.26828e19i			0.162693i	−0.996686	+	0.0813465i	$0.974078\pi$
$450$		−	5.09666e19i		−	0.642972i
$451$	−4.86976e18			−0.0604204
$452$		−	1.48346e20i		−	1.81024i
$453$		−	1.37787e20i		−	1.65376i
$454$		−	7.24753e19i		−	0.855600i
$455$	1.80998e19			0.210179
$456$		−	1.27965e20i		−	1.46169i
$457$	−2.80247e19			−0.314898			−0.157449	−	0.987527i	$-0.550327\pi$
$457$	−2.80247e19			−0.314898			−0.157449	+	0.987527i	$0.550327\pi$
$458$	−2.66771e20			−2.94881
$459$	3.88241e19			0.422189
$460$	−1.41097e19			−0.150951
$461$			8.17933e19i			0.860916i	0.902611	+	0.430458i	$0.141648\pi$
$461$			8.17933e19i			0.860916i	−0.902611	+	0.430458i	$0.858352\pi$
$462$			2.74480e19i			0.284247i
$463$	9.52790e19			0.970822			0.485411	−	0.874286i	$-0.338670\pi$
$463$	9.52790e19			0.970822			0.485411	+	0.874286i	$0.338670\pi$
$464$	5.86213e18	−	1.76430e19i	0.0587720	−	0.176884i
$465$	7.65235e19			0.754915
$466$			1.57466e20i			1.52859i
$467$		−	7.80766e19i		−	0.745838i	−0.927864	−	0.372919i	$-0.878357\pi$
$467$		−	7.80766e19i		−	0.745838i	0.927864	−	0.372919i	$-0.121643\pi$
$468$	−3.05230e19			−0.286935
$469$	3.51532e18			0.0325213
$470$	−3.70360e19			−0.337202
$471$	1.05948e20			0.949367
$472$		−	2.11662e20i		−	1.86671i
$473$	−6.28412e18			−0.0545489
$474$		−	1.34774e20i		−	1.15151i
$475$		−	5.81719e19i		−	0.489228i
$476$			3.80957e20i			3.15373i
$477$	6.91274e19			0.563329
$478$			3.34027e20i			2.67961i
$479$		−	1.39440e20i		−	1.10122i	−0.834764	−	0.550608i	$-0.814396\pi$
$479$		−	1.39440e20i		−	1.10122i	0.834764	−	0.550608i	$-0.185604\pi$
$480$	1.00625e20			0.782343
$481$		−	3.98067e19i		−	0.304697i
$482$			2.15826e20i			1.62649i
$483$		−	3.10252e19i		−	0.230203i
$484$	−2.30371e20			−1.68301
$485$		−	1.32812e20i		−	0.955374i
$486$	−2.91494e20			−2.06470
$487$	−5.95438e19			−0.415307			−0.207654	−	0.978202i	$-0.566583\pi$
$487$	−5.95438e19			−0.415307			−0.207654	+	0.978202i	$0.566583\pi$
$488$	−1.39039e20			−0.954968
$489$	1.72931e20			1.16965
$490$			1.48225e20i			0.987303i
$491$			2.86451e19i			0.187905i	0.995577	+	0.0939527i	$0.0299503\pi$
$491$			2.86451e19i			0.187905i	−0.995577	+	0.0939527i	$0.970050\pi$
$492$	2.21142e20			1.42867
$493$	−2.03110e20	−	6.74859e19i	−1.29235	−	0.429399i
$494$	−5.53492e19			−0.346864
$495$			8.29839e18i			0.0512218i
$496$			2.48863e19i			0.151303i
$497$	−3.33116e20			−1.99490
$498$	−2.48248e20			−1.46442
$499$	4.73096e19			0.274913			0.137456	−	0.990508i	$-0.456107\pi$
$499$	4.73096e19			0.274913			0.137456	+	0.990508i	$0.456107\pi$
$500$	−3.13575e20			−1.79501
$501$			8.74209e19i			0.492984i
$502$	−1.65459e20			−0.919208
$503$		−	2.63859e20i		−	1.44415i	−0.691815	−	0.722075i	$-0.743189\pi$
$503$		−	2.63859e20i		−	1.44415i	0.691815	−	0.722075i	$-0.256811\pi$
$504$		−	2.22462e20i		−	1.19957i
$505$			2.45776e19i			0.130573i
$506$	−3.80606e18			−0.0199226
$507$		−	2.45210e20i		−	1.26467i
$508$		−	1.80810e20i		−	0.918848i
$509$	1.26490e20			0.633391			0.316696	−	0.948527i	$-0.397427\pi$
$509$	1.26490e20			0.633391			0.316696	+	0.948527i	$0.397427\pi$
$510$			4.21632e20i			2.08045i
$511$		−	1.71369e20i		−	0.833251i
$512$			7.73595e19i			0.370671i
$513$	−6.29229e19			−0.297118
$514$		−	4.55564e20i		−	2.11995i
$515$	2.31735e20			1.06276
$516$	2.85369e20			1.28984
$517$	−6.28816e18			−0.0280120
$518$	7.05482e20			3.09751
$519$			5.13160e20i			2.22074i
$520$			4.14665e19i			0.176877i
$521$	1.46111e20			0.614328			0.307164	−	0.951657i	$-0.400620\pi$
$521$	1.46111e20			0.614328			0.307164	+	0.951657i	$0.400620\pi$
$522$	2.88410e20	+	9.58282e19i	1.19531	+	0.397159i
$523$	−1.94612e20			−0.795072			−0.397536	−	0.917586i	$-0.630135\pi$
$523$	−1.94612e20			−0.795072			−0.397536	+	0.917586i	$0.630135\pi$
$524$			6.12232e20i			2.46565i
$525$		−	2.33021e20i		−	0.925126i
$526$	−1.86400e20			−0.729545
$527$	2.86495e20			1.10545
$528$	−6.21836e18			−0.0236549
$529$	−2.62333e20			−0.983865
$530$		−	2.28360e20i		−	0.844407i
$531$	3.42154e20			1.24742
$532$		−	6.17424e20i		−	2.21945i
$533$		−	3.93360e19i		−	0.139424i
$534$			1.21956e21i			4.26228i
$535$	4.43946e18			0.0152995
$536$			8.05356e18i			0.0273686i
$537$			4.48559e20i			1.50319i
$538$	1.38039e20			0.456179
$539$			2.51664e19i			0.0820172i
$540$			1.14629e20i			0.368419i
$541$		−	7.57764e19i		−	0.240190i	−0.992762	−	0.120095i	$-0.961680\pi$
$541$		−	7.57764e19i		−	0.240190i	0.992762	−	0.120095i	$-0.0383199\pi$
$542$	6.29417e20			1.96763
$543$			2.89811e20i			0.893544i
$544$	3.76729e20			1.14561
$545$	3.94424e19			0.118301
$546$	−2.21714e20			−0.655917
$547$	−2.45363e20			−0.715985			−0.357993	−	0.933724i	$-0.616539\pi$
$547$	−2.45363e20			−0.715985			−0.357993	+	0.933724i	$0.616539\pi$
$548$			2.00949e19i			0.0578405i
$549$		−	2.24759e20i		−	0.638153i
$550$	−2.85862e19			−0.0800638
$551$	3.29183e20	+	1.09376e20i	0.909496	+	0.302192i
$552$	7.10785e19			0.193729
$553$		−	2.67422e20i		−	0.719049i
$554$		−	2.99067e20i		−	0.793313i
$555$	4.91458e20			1.28614
$556$	−8.96128e20			−2.31371
$557$	1.08528e20			0.276457			0.138228	−	0.990400i	$-0.455859\pi$
$557$	1.08528e20			0.276457			0.138228	+	0.990400i	$0.455859\pi$
$558$	−4.06815e20			−1.02245
$559$		−	5.07607e19i		−	0.125875i
$560$	−7.26723e19			−0.177811
$561$			7.15869e19i			0.172827i
$562$		−	1.58466e20i		−	0.377496i
$563$		−	1.85001e20i		−	0.434871i	−0.976075	−	0.217436i	$-0.930231\pi$
$563$		−	1.85001e20i		−	0.434871i	0.976075	−	0.217436i	$-0.0697692\pi$
$564$	2.85553e20			0.662359
$565$		−	3.25780e20i		−	0.745695i
$566$			1.20557e21i			2.72313i
$567$	−7.21055e20			−1.60729
$568$		−	7.63164e20i		−	1.67883i
$569$			1.39744e20i			0.303383i	0.988428	+	0.151691i	$0.0484720\pi$
$569$			1.39744e20i			0.303383i	−0.988428	+	0.151691i	$0.951528\pi$
$570$		−	6.83347e20i		−	1.46413i
$571$	−3.26334e20			−0.690067			−0.345033	−	0.938590i	$-0.612132\pi$
$571$	−3.26334e20			−0.690067			−0.345033	+	0.938590i	$0.612132\pi$
$572$			1.71197e19i			0.0357295i
$573$	−7.58683e20			−1.56279
$574$	6.97140e20			1.41736
$575$	3.23118e19			0.0648413
$576$	−6.07098e20			−1.20251
$577$		−	4.06759e20i		−	0.795277i	−0.917542	−	0.397638i	$-0.869830\pi$
$577$		−	4.06759e20i		−	0.795277i	0.917542	−	0.397638i	$-0.130170\pi$
$578$			7.27364e20i			1.40376i
$579$	−4.82812e20			−0.919790
$580$	1.99254e20	−	5.99687e20i	0.374711	−	1.12775i
$581$	−4.92581e20			−0.914443
$582$			1.62688e21i			2.98149i
$583$		−	3.87722e19i		−	0.0701466i
$584$	3.92605e20			0.701228
$585$	−6.70312e19			−0.118197
$586$	−8.02513e20			−1.39708
$587$	1.61383e20			0.277377			0.138689	−	0.990336i	$-0.455711\pi$
$587$	1.61383e20			0.277377			0.138689	+	0.990336i	$0.455711\pi$
$588$		−	1.14283e21i		−	1.93934i
$589$	−4.64327e20			−0.777964
$590$		−	1.13030e21i		−	1.86983i
$591$			1.40534e21i			2.29549i
$592$			1.59827e20i			0.257774i
$593$	−6.20547e20			−0.988245			−0.494122	−	0.869392i	$-0.664511\pi$
$593$	−6.20547e20			−0.988245			−0.494122	+	0.869392i	$0.664511\pi$
$594$			3.09209e19i			0.0486244i
$595$			8.36616e20i			1.29912i
$596$	5.74774e20			0.881353
$597$		−	1.73635e21i		−	2.62923i
$598$		−	3.07439e19i		−	0.0459727i
$599$		−	4.69018e20i		−	0.692609i	−0.938122	−	0.346305i	$-0.887436\pi$
$599$		−	4.69018e20i		−	0.692609i	0.938122	−	0.346305i	$-0.112564\pi$
$600$	5.33849e20			0.778546
$601$			7.10153e20i			1.02281i	0.859341	+	0.511403i	$0.170874\pi$
$601$			7.10153e20i			1.02281i	−0.859341	+	0.511403i	$0.829126\pi$
$602$	8.99615e20			1.27963
$603$	−1.30187e19			−0.0182889
$604$	1.52309e21			2.11324
$605$	−5.05915e20			−0.693286
$606$		−	3.01064e20i		−	0.407487i
$607$		−	1.01882e21i		−	1.36202i	−0.732276	−	0.681008i	$-0.761542\pi$
$607$		−	1.01882e21i		−	1.36202i	0.732276	−	0.681008i	$-0.238458\pi$
$608$	−6.10571e20			−0.806230
$609$	1.31862e21	+	4.38130e20i	1.71985	+	0.571443i
$610$	−7.42486e20			−0.956565
$611$		−	5.07933e19i		−	0.0646394i
$612$		−	1.41084e21i		−	1.77355i
$613$	2.63752e20			0.327524			0.163762	−	0.986500i	$-0.447637\pi$
$613$	2.63752e20			0.327524			0.163762	+	0.986500i	$0.447637\pi$
$614$	1.82495e21			2.23866
$615$	4.85647e20			0.588514
$616$	−1.24775e20			−0.149373
$617$		−	5.29424e20i		−	0.626130i	−0.949732	−	0.313065i	$-0.898644\pi$
$617$		−	5.29424e20i		−	0.626130i	0.949732	−	0.313065i	$-0.101356\pi$
$618$	−2.83864e21			−3.31663
$619$		−	8.80376e19i		−	0.101622i	−0.998708	−	0.0508111i	$-0.983819\pi$
$619$		−	8.80376e19i		−	0.101622i	0.998708	−	0.0508111i	$-0.0161806\pi$
$620$			8.45884e20i			0.964658i
$621$		−	3.49508e19i		−	0.0393795i
$622$	−2.81247e21			−3.13084
$623$			2.41988e21i			2.66155i
$624$		−	5.02295e19i		−	0.0545852i
$625$	−2.13226e20			−0.228949
$626$		−	2.76952e21i		−	2.93831i
$627$		−	1.16022e20i		−	0.121628i
$628$			1.17113e21i			1.21314i
$629$	1.83996e21			1.88334
$630$		−	1.18797e21i		−	1.20158i
$631$	5.18869e20			0.518606			0.259303	−	0.965796i	$-0.416507\pi$
$631$	5.18869e20			0.518606			0.259303	+	0.965796i	$0.416507\pi$
$632$	6.12660e20			0.605120
$633$	−2.98996e20			−0.291835
$634$	1.82061e21			1.75609
$635$		−	3.97074e20i		−	0.378502i
$636$			1.76069e21i			1.65865i
$637$	−2.03284e20			−0.189260
$638$	5.37482e19	−	1.61764e20i	0.0494548	−	0.148842i
$639$	1.23367e21			1.12187
$640$			1.35066e21i			1.21394i
$641$		−	7.08839e20i		−	0.629669i	−0.949147	−	0.314835i	$-0.898051\pi$
$641$		−	7.08839e20i		−	0.629669i	0.949147	−	0.314835i	$-0.101949\pi$
$642$	−5.43813e19			−0.0477460
$643$	−4.84714e20			−0.420633			−0.210317	−	0.977633i	$-0.567449\pi$
$643$	−4.84714e20			−0.420633			−0.210317	+	0.977633i	$0.567449\pi$
$644$	3.42950e20			0.294162
$645$	6.26697e20			0.531324
$646$		−	2.55837e21i		−	2.14397i
$647$	−1.17500e21			−0.973322			−0.486661	−	0.873591i	$-0.661785\pi$
$647$	−1.17500e21			−0.973322			−0.486661	+	0.873591i	$0.661785\pi$
$648$		−	1.65193e21i		−	1.35263i
$649$		−	1.91908e20i		−	0.155331i
$650$		−	2.30908e20i		−	0.184752i
$651$	−1.85997e21			−1.47113
$652$			1.91156e21i			1.49462i
$653$			1.30413e21i			1.00802i	0.863696	+	0.504012i	$0.168143\pi$
$653$			1.30413e21i			1.00802i	−0.863696	+	0.504012i	$0.831857\pi$
$654$	−4.83151e20			−0.369190
$655$			1.34452e21i			1.01568i
$656$			1.57937e20i			0.117952i
$657$			6.34651e20i			0.468592i
$658$	9.00194e20			0.657116
$659$			2.25239e20i			0.162556i	0.996691	+	0.0812782i	$0.0259002\pi$
$659$			2.25239e20i			0.162556i	−0.996691	+	0.0812782i	$0.974100\pi$
$660$	−2.11362e20			−0.150816
$661$	1.05754e21			0.746083			0.373042	−	0.927815i	$-0.378315\pi$
$661$	1.05754e21			0.746083			0.373042	+	0.927815i	$0.378315\pi$
$662$	2.20707e21			1.53951
$663$	−5.78251e20			−0.398809
$664$		−	1.12850e21i		−	0.769555i
$665$		−	1.35592e21i		−	0.914263i
$666$	−2.61270e21			−1.74194
$667$	−6.07530e19	+	1.82846e20i	−0.0400520	+	0.120543i
$668$	−9.66343e20			−0.629953
$669$			1.53615e21i			0.990232i
$670$			4.30069e19i			0.0274143i
$671$	−1.26063e20			−0.0794637
$672$	−2.44579e21			−1.52458
$673$	7.74771e20			0.477596			0.238798	−	0.971069i	$-0.423247\pi$
$673$	7.74771e20			0.477596			0.238798	+	0.971069i	$0.423247\pi$
$674$	3.26186e21			1.98846
$675$		−	2.62505e20i		−	0.158256i
$676$	2.71053e21			1.61605
$677$			2.46945e20i			0.145608i	0.997346	+	0.0728039i	$0.0231947\pi$
$677$			2.46945e20i			0.145608i	−0.997346	+	0.0728039i	$0.976805\pi$
$678$			3.99066e21i			2.32714i
$679$			3.22811e21i			1.86177i
$680$	−1.91668e21			−1.09328
$681$			1.22716e21i			0.692307i
$682$			2.28175e20i			0.127317i
$683$	−1.33057e21			−0.734313			−0.367156	−	0.930159i	$-0.619669\pi$
$683$	−1.33057e21			−0.734313			−0.367156	+	0.930159i	$0.619669\pi$
$684$			2.28658e21i			1.24815i
$685$			4.41302e19i			0.0238263i
$686$			5.91303e20i			0.315776i
$687$	4.51701e21			2.38603
$688$			2.03808e20i			0.106490i
$689$	3.13187e20			0.161867
$690$	3.79567e20			0.194053
$691$	2.17084e20			0.109785			0.0548925	−	0.998492i	$-0.482518\pi$
$691$	2.17084e20			0.109785			0.0548925	+	0.998492i	$0.482518\pi$
$692$	−5.67243e21			−2.83774
$693$		−	2.01700e20i		−	0.0998175i
$694$		−	1.95887e21i		−	0.958979i
$695$	−1.96798e21			−0.953091
$696$	−1.00375e21	+	3.02095e21i	−0.480902	+	1.44735i
$697$	1.81820e21			0.861780
$698$			5.92673e20i			0.277907i
$699$		−	2.66624e21i		−	1.23686i
$700$	2.57580e21			1.18216
$701$	1.92643e21			0.874716			0.437358	−	0.899287i	$-0.355914\pi$
$701$	1.92643e21			0.874716			0.437358	+	0.899287i	$0.355914\pi$
$702$	−2.49767e20			−0.112204
$703$	−2.98206e21			−1.32541
$704$			3.40510e20i			0.149739i
$705$	6.27100e20			0.272846
$706$		−	6.34270e21i		−	2.73048i
$707$		−	5.97380e20i		−	0.254452i
$708$			8.71476e21i			3.67287i
$709$	1.14897e21			0.479139			0.239569	−	0.970879i	$-0.422994\pi$
$709$	1.14897e21			0.479139			0.239569	+	0.970879i	$0.422994\pi$
$710$		−	4.07539e21i		−	1.68163i
$711$			9.90375e20i			0.404369i
$712$	−5.54392e21			−2.23984
$713$		−	2.57912e20i		−	0.103110i
$714$		−	1.02482e22i		−	4.05424i
$715$			3.75965e19i			0.0147181i
$716$	−4.95834e21			−1.92083
$717$		−	5.65580e21i		−	2.16820i
$718$	6.16520e21			2.33891
$719$	−3.82614e20			−0.143646			−0.0718230	−	0.997417i	$-0.522882\pi$
$719$	−3.82614e20			−0.143646			−0.0718230	+	0.997417i	$0.522882\pi$
$720$	2.69136e20			0.0999949
$721$	−5.63253e21			−2.07104
$722$		−	3.67901e20i		−	0.133876i
$723$		−	3.65440e21i		−	1.31607i
$724$	−3.20354e21			−1.14180
$725$	−4.56298e20	+	1.37330e21i	−0.160958	+	0.484430i
$726$	6.19723e21			2.16358
$727$			4.07925e21i			1.40952i	0.709444	+	0.704762i	$0.248946\pi$
$727$			4.07925e21i			1.40952i	−0.709444	+	0.704762i	$0.751054\pi$
$728$		−	1.00788e21i		−	0.344686i
$729$	1.45297e21			0.491812
$730$	2.09655e21			0.702400
$731$	2.34628e21			0.778035
$732$	5.72467e21			1.87896
$733$			3.38238e21i			1.09886i	0.835540	+	0.549429i	$0.185155\pi$
$733$			3.38238e21i			1.09886i	−0.835540	+	0.549429i	$0.814845\pi$
$734$	−5.60182e21			−1.80139
$735$		−	2.50977e21i		−	0.798874i
$736$		−	3.39144e20i		−	0.106856i
$737$			7.30194e18i			0.00227736i
$738$	−2.58180e21			−0.797076
$739$			3.65664e21i			1.11750i	0.829335	+	0.558752i	$0.188719\pi$
$739$			3.65664e21i			1.11750i	−0.829335	+	0.558752i	$0.811281\pi$
$740$			5.43254e21i			1.64348i
$741$	9.37181e20			0.280664
$742$			5.55051e21i			1.64552i
$743$			4.12300e20i			0.121003i	0.998168	+	0.0605017i	$0.0192701\pi$
$743$			4.12300e20i			0.121003i	−0.998168	+	0.0605017i	$0.980730\pi$
$744$		−	4.26118e21i		−	1.23804i
$745$	1.26225e21			0.363057
$746$		−	7.55584e20i		−	0.215150i
$747$	1.82423e21			0.514252
$748$	−7.91315e20			−0.220845
$749$	−1.07905e20			−0.0298146
$750$	8.43550e21			2.30755
$751$		−	1.04438e21i		−	0.282853i	−0.989949	−	0.141426i	$-0.954831\pi$
$751$		−	1.04438e21i		−	0.282853i	0.989949	−	0.141426i	$-0.0451689\pi$
$752$			2.03939e20i			0.0546849i
$753$	2.80159e21			0.743775
$754$	1.30666e21	+	4.34157e20i	0.343462	+	0.114120i
$755$	3.34484e21			0.870509
$756$		−	2.78617e21i		−	0.717950i
$757$		−	5.95050e21i		−	1.51822i	−0.650964	−	0.759109i	$-0.725635\pi$
$757$		−	5.95050e21i		−	1.51822i	0.650964	−	0.759109i	$-0.274365\pi$
$758$	−5.42462e21			−1.37041
$759$	6.44449e19			0.0161204
$760$	3.10639e21			0.769404
$761$	1.43903e21			0.352926			0.176463	−	0.984307i	$-0.443534\pi$
$761$	1.43903e21			0.352926			0.176463	+	0.984307i	$0.443534\pi$
$762$			4.86397e21i			1.18121i
$763$	−9.58683e20			−0.230537
$764$		−	8.38641e21i		−	1.99699i
$765$		−	3.09834e21i		−	0.730581i
$766$			2.56185e21i			0.598189i
$767$	1.55016e21			0.358435
$768$		−	7.44104e21i		−	1.70382i
$769$		−	3.40407e21i		−	0.771881i	−0.922524	−	0.385941i	$-0.873877\pi$
$769$		−	3.40407e21i		−	0.771881i	0.922524	−	0.385941i	$-0.126123\pi$
$770$	−6.66311e20			−0.149622
$771$			7.71369e21i			1.71536i
$772$		−	5.33696e21i		−	1.17534i
$773$		−	1.00918e21i		−	0.220102i	−0.993926	−	0.110051i	$-0.964899\pi$
$773$		−	1.00918e21i		−	0.220102i	0.993926	−	0.110051i	$-0.0351014\pi$
$774$	−3.33165e21			−0.719619
$775$		−	1.93710e21i		−	0.414372i
$776$	−7.39557e21			−1.56678
$777$	−1.19453e22			−2.50635
$778$	−4.77250e21			−0.991743
$779$	−2.94679e21			−0.606483
$780$		−	1.70730e21i		−	0.348017i
$781$		−	6.91940e20i		−	0.139697i
$782$	1.42106e21			0.284158
$783$	1.48546e21	+	4.93565e20i	0.294204	+	0.0977532i
$784$	8.16203e20			0.160113
$785$			2.57192e21i			0.499729i
$786$		−	1.64697e22i		−	3.16969i
$787$	−7.31680e21			−1.39480			−0.697398	−	0.716684i	$-0.745659\pi$
$787$	−7.31680e21			−1.39480			−0.697398	+	0.716684i	$0.745659\pi$
$788$	−1.55345e22			−2.93326
$789$	3.15615e21			0.590310
$790$	3.27168e21			0.606132
$791$			7.91839e21i			1.45316i
$792$	4.62092e20			0.0840021
$793$		−	1.01829e21i		−	0.183367i
$794$			1.50080e22i			2.67713i
$795$			3.86664e21i			0.683251i
$796$	1.91934e22			3.35973
$797$			3.32503e21i			0.576577i	0.957544	+	0.288289i	$0.0930863\pi$
$797$			3.32503e21i			0.576577i	−0.957544	+	0.288289i	$0.906914\pi$
$798$			1.66094e22i			2.85320i
$799$	2.34779e21			0.399538
$800$		−	2.54721e21i		−	0.429427i
$801$		−	8.96184e21i		−	1.49676i
$802$			1.97911e22i			3.27462i
$803$	3.55964e20			0.0583498
$804$		−	3.31590e20i		−	0.0538494i
$805$	7.53150e20			0.121175
$806$	−1.84311e21			−0.293791
$807$	−2.33730e21			−0.369116
$808$	1.36859e21			0.214135
$809$		−	9.32580e21i		−	1.44568i	−0.691016	−	0.722840i	$-0.742836\pi$
$809$		−	9.32580e21i		−	1.44568i	0.691016	−	0.722840i	$-0.257164\pi$
$810$		−	8.82149e21i		−	1.35489i
$811$	5.03881e21			0.766781			0.383390	−	0.923586i	$-0.374756\pi$
$811$	5.03881e21			0.766781			0.383390	+	0.923586i	$0.374756\pi$
$812$	−4.84305e21	+	1.45759e22i	−0.730211	+	2.19769i
$813$	−1.06574e22			−1.59211
$814$			1.46541e21i			0.216908i
$815$			4.19796e21i			0.615682i
$816$	2.32173e21			0.337392
$817$	−3.80265e21			−0.547547
$818$	−1.18444e22			−1.68990
$819$	1.62925e21			0.230335
$820$			5.36829e21i			0.752025i
$821$	1.17383e22			1.62941			0.814705	−	0.579875i	$-0.196899\pi$
$821$	1.17383e22			1.62941			0.814705	+	0.579875i	$0.196899\pi$
$822$		−	5.40575e20i		−	0.0743562i
$823$			8.93525e21i			1.21789i	0.793212	+	0.608945i	$0.208407\pi$
$823$			8.93525e21i			1.21789i	−0.793212	+	0.608945i	$0.791593\pi$
$824$		−	1.29041e22i		−	1.74290i
$825$	4.84026e20			0.0647835
$826$			2.74729e22i			3.64380i
$827$			2.21452e21i			0.291064i	0.989354	+	0.145532i	$0.0464893\pi$
$827$			2.21452e21i			0.291064i	−0.989354	+	0.145532i	$0.953511\pi$
$828$	−1.27009e21			−0.165427
$829$		−	5.00286e21i		−	0.645743i	−0.946443	−	0.322871i	$-0.895352\pi$
$829$		−	5.00286e21i		−	0.645743i	0.946443	−	0.322871i	$-0.104648\pi$
$830$		−	6.02631e21i		−	0.770842i
$831$			5.06385e21i			0.641908i
$832$	−2.75050e21			−0.345531
$833$		−	9.39627e21i		−	1.16982i
$834$	2.41068e22			2.97437
$835$	−2.12218e21			−0.259497
$836$	1.28250e21			0.155421
$837$	−2.09531e21			−0.251656
$838$		−	2.55431e22i		−	3.04049i
$839$		−	3.33668e21i		−	0.393640i	−0.980440	−	0.196820i	$-0.936939\pi$
$839$		−	3.33668e21i		−	0.393640i	0.980440	−	0.196820i	$-0.0630615\pi$
$840$	1.24434e22			1.45494
$841$	−6.91331e21	−	5.16420e21i	−0.801155	−	0.598458i
$842$	1.71185e22			1.96618
$843$			2.68317e21i			0.305450i
$844$		−	3.30507e21i		−	0.372917i
$845$	5.95257e21			0.665701
$846$	−3.33380e21			−0.369540
$847$	1.22967e22			1.35103
$848$	−1.25747e21			−0.136940
$849$		−	2.04128e22i		−	2.20341i
$850$	1.06731e22			1.14196
$851$		−	1.65639e21i		−	0.175668i
$852$			3.14218e22i			3.30320i
$853$		−	8.57058e21i		−	0.893085i	−0.894763	−	0.446542i	$-0.852655\pi$
$853$		−	8.57058e21i		−	0.893085i	0.894763	−	0.446542i	$-0.147345\pi$
$854$	1.80468e22			1.86409
$855$			5.02153e21i			0.514151i
$856$		−	2.47209e20i		−	0.0250906i
$857$	−1.64569e22			−1.65574			−0.827869	−	0.560921i	$-0.810447\pi$
$857$	−1.64569e22			−1.65574			−0.827869	+	0.560921i	$0.810447\pi$
$858$		−	4.60540e20i		−	0.0459317i
$859$			4.50872e21i			0.445764i	0.974845	+	0.222882i	$0.0715464\pi$
$859$			4.50872e21i			0.445764i	−0.974845	+	0.222882i	$0.928454\pi$
$860$			6.92745e21i			0.678946i
$861$	−1.18041e22			−1.14686
$862$			1.98384e22i			1.91074i
$863$	−4.93480e21			−0.471182			−0.235591	−	0.971852i	$-0.575703\pi$
$863$	−4.93480e21			−0.471182			−0.235591	+	0.971852i	$0.575703\pi$
$864$	−2.75524e21			−0.260800
$865$	−1.24572e22			−1.16896
$866$	2.79991e22			2.60471
$867$		−	1.23159e22i		−	1.13585i
$868$		−	2.05600e22i		−	1.87986i
$869$	5.55482e20			0.0503526
$870$	−5.36015e21	+	1.61322e22i	−0.481706	+	1.44977i
$871$	−5.89822e19			−0.00525515
$872$		−	2.19633e21i		−	0.194010i
$873$		−	1.19551e22i		−	1.04699i
$874$	−2.30313e21			−0.199978
$875$	1.67380e22			1.44093
$876$	−1.61647e22			−1.37971
$877$	−2.29448e22			−1.94172			−0.970861	−	0.239645i	$-0.922969\pi$
$877$	−2.29448e22			−1.94172			−0.970861	+	0.239645i	$0.922969\pi$
$878$		−	1.66184e22i		−	1.39437i
$879$	1.35883e22			1.13044
$880$		−	1.50953e20i		−	0.0124515i
$881$			1.24532e22i			1.01850i	0.860617	+	0.509252i	$0.170078\pi$
$881$			1.24532e22i			1.01850i	−0.860617	+	0.509252i	$0.829922\pi$
$882$			1.33425e22i			1.08199i
$883$	−3.20868e21			−0.258001			−0.129000	−	0.991645i	$-0.541177\pi$
$883$	−3.20868e21			−0.258001			−0.129000	+	0.991645i	$0.541177\pi$
$884$		−	6.39193e21i		−	0.509613i
$885$			1.91384e22i			1.51297i
$886$	2.46968e22			1.93592
$887$		−	1.38065e22i		−	1.07314i	−0.843855	−	0.536571i	$-0.819719\pi$
$887$		−	1.38065e22i		−	1.07314i	0.843855	−	0.536571i	$-0.180281\pi$
$888$		−	2.73667e22i		−	2.10923i
$889$			9.65125e21i			0.737599i
$890$	−2.96052e22			−2.24359
$891$		−	1.49776e21i		−	0.112553i
$892$	−1.69804e22			−1.26535
$893$	−3.80510e21			−0.281177
$894$	−1.54620e22			−1.13301
$895$	−1.08889e22			−0.791249
$896$		−	3.28290e22i		−	2.36563i
$897$			5.20561e20i			0.0371987i
$898$	3.77140e21			0.267257
$899$	1.09617e22	+	3.64217e21i	0.770335	+	0.255954i
$900$	−9.53926e21			−0.664807
$901$			1.44762e22i			1.00051i
$902$			1.44808e21i			0.0992531i
$903$	−1.52324e22			−1.03541
$904$	−1.81409e22			−1.22292
$905$	−7.03527e21			−0.470345
$906$	−4.09728e22			−2.71665
$907$			5.68432e21i			0.373787i	0.982380	+	0.186893i	$0.0598419\pi$
$907$			5.68432e21i			0.373787i	−0.982380	+	0.186893i	$0.940158\pi$
$908$	−1.35650e22			−0.884656
$909$			2.21235e21i			0.143095i
$910$		−	5.38220e21i		−	0.345262i
$911$			5.92329e21i			0.376856i	0.982087	+	0.188428i	$0.0603392\pi$
$911$			5.92329e21i			0.376856i	−0.982087	+	0.188428i	$0.939661\pi$
$912$	−3.76286e21			−0.237442
$913$		−	1.02318e21i		−	0.0640354i
$914$			8.33349e21i			0.517286i
$915$	1.25719e22			0.774003
$916$			4.99306e22i			3.04896i
$917$		−	3.26797e22i		−	1.97928i
$918$		−	1.15448e22i		−	0.693533i
$919$	2.21430e22			1.31938			0.659691	−	0.751537i	$-0.270687\pi$
$919$	2.21430e22			1.31938			0.659691	+	0.751537i	$0.270687\pi$
$920$			1.72546e21i			0.101975i
$921$	−3.09003e22			−1.81141
$922$	2.43223e22			1.41424
$923$	5.58922e21			0.322358
$924$	5.13735e21			0.293900
$925$		−	1.24407e22i		−	0.705962i
$926$		−	2.83324e22i		−	1.59478i
$927$	2.08596e22			1.16468
$928$	1.44141e22	+	4.78929e21i	0.798324	+	0.265254i
$929$	−9.44688e21			−0.519004			−0.259502	−	0.965743i	$-0.583558\pi$
$929$	−9.44688e21			−0.519004			−0.259502	+	0.965743i	$0.583558\pi$
$930$		−	2.27552e22i		−	1.24011i
$931$			1.52287e22i			0.823266i
$932$	2.94724e22			1.58050
$933$	4.76213e22			2.53331
$934$	−2.32170e22			−1.22520
$935$	−1.73780e21			−0.0909730
$936$			3.73260e21i			0.193840i
$937$	1.09040e22			0.561746			0.280873	−	0.959745i	$-0.409376\pi$
$937$	1.09040e22			0.561746			0.280873	+	0.959745i	$0.409376\pi$
$938$		−	1.04532e21i		−	0.0534231i
$939$			4.68941e22i			2.37753i
$940$			6.93190e21i			0.348653i
$941$	2.03205e22			1.01394			0.506971	−	0.861963i	$-0.330765\pi$
$941$	2.03205e22			1.01394			0.506971	+	0.861963i	$0.330765\pi$
$942$		−	3.15048e22i		−	1.55953i
$943$		−	1.63681e21i		−	0.0803822i
$944$	−6.22401e21			−0.303236
$945$		−	6.11868e21i		−	0.295746i
$946$			1.86866e21i			0.0896080i
$947$		−	1.67922e22i		−	0.798885i	−0.916758	−	0.399442i	$-0.869204\pi$
$947$		−	1.67922e22i		−	0.798885i	0.916758	−	0.399442i	$-0.130796\pi$
$948$	−2.52251e22			−1.19061
$949$			2.87534e21i			0.134646i
$950$	−1.72981e22			−0.803659
$951$	−3.08268e22			−1.42094
$952$	4.65866e22			2.13051
$953$	4.39325e21			0.199338			0.0996688	−	0.995021i	$-0.468222\pi$
$953$	4.39325e21			0.199338			0.0996688	+	0.995021i	$0.468222\pi$
$954$		−	2.05559e22i		−	0.925386i
$955$		−	1.84173e22i		−	0.822623i
$956$	6.25187e22			2.77061
$957$	−9.10073e20	+	2.73901e21i	−0.0400163	+	0.120435i
$958$	−4.14643e22			−1.80898
$959$		−	1.07263e21i		−	0.0464311i
$960$		−	3.39580e22i		−	1.45851i
$961$	8.00332e21			0.341071
$962$	−1.18370e22			−0.500529
$963$	3.99618e20			0.0167667
$964$	4.03954e22			1.68172
$965$		−	1.17204e22i		−	0.484160i
$966$	−9.22574e21			−0.378157
$967$		−	3.09574e22i		−	1.25912i	−0.776953	−	0.629559i	$-0.783236\pi$
$967$		−	3.09574e22i		−	1.25912i	0.776953	−	0.629559i	$-0.216764\pi$
$968$			2.81717e22i			1.13697i
$969$			4.33187e22i			1.73479i
$970$	−3.94932e22			−1.56940
$971$		−	4.12664e22i		−	1.62724i	−0.581394	−	0.813622i	$-0.697493\pi$
$971$		−	4.12664e22i		−	1.62724i	0.581394	−	0.813622i	$-0.302507\pi$
$972$			5.45579e22i			2.13482i
$973$	4.78335e22			1.85732
$974$			1.77061e22i			0.682229i
$975$			3.90978e21i			0.149492i
$976$			4.08851e21i			0.155129i
$977$	4.91341e22			1.85001			0.925005	−	0.379956i	$-0.124061\pi$
$977$	4.91341e22			1.85001			0.925005	+	0.379956i	$0.124061\pi$
$978$		−	5.14231e22i		−	1.92140i
$979$	−5.02652e21			−0.186379
$980$	2.77427e22			1.02083
$981$	3.55041e21			0.129646
$982$	8.51798e21			0.308674
$983$			3.36183e22i			1.20899i	0.796607	+	0.604497i	$0.206626\pi$
$983$			3.36183e22i			1.20899i	−0.796607	+	0.604497i	$0.793374\pi$
$984$		−	2.70430e22i		−	0.965144i
$985$	−3.41152e22			−1.20830
$986$	−2.00678e22	+	6.03971e22i	−0.705378	+	2.12295i
$987$	−1.52422e22			−0.531704
$988$			1.03595e22i			0.358643i
$989$		−	2.11220e21i		−	0.0725709i
$990$	2.46763e21			0.0841425
$991$	5.12032e22			1.73279			0.866393	−	0.499362i	$-0.166432\pi$
$991$	5.12032e22			1.73279			0.866393	+	0.499362i	$0.166432\pi$
$992$	−2.03318e22			−0.682870
$993$	−3.73705e22			−1.24569
$994$			9.90560e22i			3.27705i
$995$	4.21505e22			1.38398
$996$			4.64637e22i			1.51415i
$997$		−	4.66495e21i		−	0.150881i	−0.997150	−	0.0754403i	$-0.975964\pi$
$997$		−	4.66495e21i		−	0.150881i	0.997150	−	0.0754403i	$-0.0240362\pi$
$998$		−	1.40681e22i		−	0.451602i
$999$	−1.34568e22			−0.428745

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.4	✓	36
29.28	even	2	inner	29.16.b.a.28.33	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.4	✓	36	1.1	even	1	trivial
29.16.b.a.28.33	yes	36	29.28	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 29 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	29.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-297.362i q^{2} +5034.99i q^{3} -55656.3 q^{4} -122226. q^{5} +1.49722e6 q^{6} +2.97082e6 q^{7} +6.80612e6i q^{8} -1.10022e7 q^{9} +O(q^{10})\)	\(q-297.362i q^{2} +5034.99i q^{3} -55656.3 q^{4} -122226. q^{5} +1.49722e6 q^{6} +2.97082e6 q^{7} +6.80612e6i q^{8} -1.10022e7 q^{9} +3.63455e7i q^{10} +6.17092e6i q^{11} -2.80229e8i q^{12} -4.98463e7 q^{13} -8.83410e8i q^{14} -6.15408e8i q^{15} +2.00137e8 q^{16} -2.30401e9i q^{17} +3.27164e9i q^{18} +3.73415e9i q^{19} +6.80266e9 q^{20} +1.49581e10i q^{21} +1.83500e9 q^{22} -2.07415e9 q^{23} -3.42687e10 q^{24} -1.55783e10 q^{25} +1.48224e10i q^{26} +1.68506e10i q^{27} -1.65345e11 q^{28} +(2.92906e10 - 8.81547e10i) q^{29} -1.82999e11 q^{30} +1.24346e11i q^{31} +1.63510e11i q^{32} -3.10705e10 q^{33} -6.85127e11 q^{34} -3.63112e11 q^{35} +6.12342e11 q^{36} +7.98590e11i q^{37} +1.11040e12 q^{38} -2.50975e11i q^{39} -8.31886e11i q^{40} +7.89146e11i q^{41} +4.44796e12 q^{42} +1.01834e12i q^{43} -3.43451e11i q^{44} +1.34476e12 q^{45} +6.16774e11i q^{46} +1.01900e12i q^{47} +1.00769e12i q^{48} +4.07822e12 q^{49} +4.63241e12i q^{50} +1.16007e13 q^{51} +2.77426e12 q^{52} -6.28305e12 q^{53} +5.01075e12 q^{54} -7.54248e11i q^{55} +2.02198e13i q^{56} -1.88014e13 q^{57} +(-2.62139e13 - 8.70991e12i) q^{58} -3.10987e13 q^{59} +3.42513e13i q^{60} +2.04286e13i q^{61} +3.69758e13 q^{62} -3.26856e13 q^{63} +5.51797e13 q^{64} +6.09253e12 q^{65} +9.23920e12i q^{66} +1.18328e12 q^{67} +1.28233e14i q^{68} -1.04433e13i q^{69} +1.07976e14i q^{70} -1.12129e14 q^{71} -7.48823e13i q^{72} -5.76841e13i q^{73} +2.37470e14 q^{74} -7.84367e13i q^{75} -2.07829e14i q^{76} +1.83327e13i q^{77} -7.46306e13 q^{78} -9.00161e13i q^{79} -2.44620e13 q^{80} -2.42712e14 q^{81} +2.34662e14 q^{82} -1.65806e14 q^{83} -8.32510e14i q^{84} +2.81611e14i q^{85} +3.02817e14 q^{86} +(4.43858e14 + 1.47478e14i) q^{87} -4.20000e13 q^{88} +8.14550e14i q^{89} -3.99880e14i q^{90} -1.48084e14 q^{91} +1.15440e14 q^{92} -6.26081e14 q^{93} +3.03012e14 q^{94} -4.56412e14i q^{95} -8.23269e14 q^{96} +1.08661e15i q^{97} -1.21271e15i q^{98} -6.78937e13i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 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