LMFDB - Embedded newform 29.16.b.a.28.26

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			28.26
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.11

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+169.905i q^{2} +817.971i q^{3} +3900.43 q^{4} +58678.2 q^{5} -138977. q^{6} -2.72949e6 q^{7} +6.23013e6i q^{8} +1.36798e7 q^{9} +O(q^{10})$	\(q+169.905i q^{2} +817.971i q^{3} +3900.43 q^{4} +58678.2 q^{5} -138977. q^{6} -2.72949e6 q^{7} +6.23013e6i q^{8} +1.36798e7 q^{9} +9.96969e6i q^{10} -4.12308e7i q^{11} +3.19044e6i q^{12} -1.27360e8 q^{13} -4.63753e8i q^{14} +4.79971e7i q^{15} -9.30719e8 q^{16} +1.09993e9i q^{17} +2.32427e9i q^{18} +1.07415e9i q^{19} +2.28870e8 q^{20} -2.23265e9i q^{21} +7.00530e9 q^{22} -2.61528e10 q^{23} -5.09607e9 q^{24} -2.70744e10 q^{25} -2.16390e10i q^{26} +2.29267e10i q^{27} -1.06462e10 q^{28} +(-3.62779e10 + 8.55167e10i) q^{29} -8.15492e9 q^{30} -8.60035e10i q^{31} +4.60156e10i q^{32} +3.37256e10 q^{33} -1.86884e11 q^{34} -1.60162e11 q^{35} +5.33572e10 q^{36} -8.70832e11i q^{37} -1.82503e11 q^{38} -1.04177e11i q^{39} +3.65573e11i q^{40} +2.48384e11i q^{41} +3.79337e11 q^{42} -1.01999e12i q^{43} -1.60818e11i q^{44} +8.02708e11 q^{45} -4.44348e12i q^{46} -1.95008e12i q^{47} -7.61302e11i q^{48} +2.70257e12 q^{49} -4.60007e12i q^{50} -8.99715e11 q^{51} -4.96758e11 q^{52} -5.78513e12 q^{53} -3.89535e12 q^{54} -2.41935e12i q^{55} -1.70051e13i q^{56} -8.78622e11 q^{57} +(-1.45297e13 - 6.16378e12i) q^{58} +1.84391e13 q^{59} +1.87209e11i q^{60} -2.21432e13i q^{61} +1.46124e13 q^{62} -3.73390e13 q^{63} -3.83161e13 q^{64} -7.47325e12 q^{65} +5.73013e12i q^{66} -5.47383e13 q^{67} +4.29022e12i q^{68} -2.13922e13i q^{69} -2.72122e13i q^{70} -3.27160e13 q^{71} +8.52272e13i q^{72} -9.03645e13i q^{73} +1.47958e14 q^{74} -2.21461e13i q^{75} +4.18963e12i q^{76} +1.12539e14i q^{77} +1.77001e13 q^{78} -3.62848e12i q^{79} -5.46129e13 q^{80} +1.77537e14 q^{81} -4.22016e13 q^{82} +9.36521e13 q^{83} -8.70828e12i q^{84} +6.45422e13i q^{85} +1.73301e14 q^{86} +(-6.99502e13 - 2.96743e13i) q^{87} +2.56873e14 q^{88} +3.62287e14i q^{89} +1.36384e14i q^{90} +3.47628e14 q^{91} -1.02007e14 q^{92} +7.03484e13 q^{93} +3.31328e14 q^{94} +6.30290e13i q^{95} -3.76394e13 q^{96} +1.39209e15i q^{97} +4.59178e14i q^{98} -5.64030e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			169.905i			0.938599i	0.883039	+	0.469300i	$0.155494\pi$
$2$			169.905i			0.938599i	−0.883039	+	0.469300i	$0.844506\pi$
$3$			817.971i			0.215938i	0.994154	+	0.107969i	$0.0344347\pi$
$3$			817.971i			0.215938i	−0.994154	+	0.107969i	$0.965565\pi$
$4$	3900.43			0.119032
$5$	58678.2			0.335894			0.167947	−	0.985796i	$-0.446286\pi$
$5$	58678.2			0.335894			0.167947	+	0.985796i	$0.446286\pi$
$6$	−138977.			−0.202679
$7$	−2.72949e6			−1.25270			−0.626349	−	0.779543i	$-0.715452\pi$
$7$	−2.72949e6			−1.25270			−0.626349	+	0.779543i	$0.715452\pi$
$8$			6.23013e6i			1.05032i
$9$	1.36798e7			0.953371
$10$			9.96969e6i			0.315269i
$11$		−	4.12308e7i		−	0.637935i	−0.947766	−	0.318967i	$-0.896664\pi$
$11$		−	4.12308e7i		−	0.637935i	0.947766	−	0.318967i	$-0.103336\pi$
$12$			3.19044e6i			0.0257034i
$13$	−1.27360e8			−0.562934			−0.281467	−	0.959571i	$-0.590821\pi$
$13$	−1.27360e8			−0.562934			−0.281467	+	0.959571i	$0.590821\pi$
$14$		−	4.63753e8i		−	1.17578i
$15$			4.79971e7i			0.0725321i
$16$	−9.30719e8			−0.866800
$17$			1.09993e9i			0.650130i	0.945692	+	0.325065i	$0.105386\pi$
$17$			1.09993e9i			0.650130i	−0.945692	+	0.325065i	$0.894614\pi$
$18$			2.32427e9i			0.894833i
$19$			1.07415e9i			0.275684i	0.990454	+	0.137842i	$0.0440167\pi$
$19$			1.07415e9i			0.275684i	−0.990454	+	0.137842i	$0.955983\pi$
$20$	2.28870e8			0.0399820
$21$		−	2.23265e9i		−	0.270505i
$22$	7.00530e9			0.598765
$23$	−2.61528e10			−1.60162			−0.800810	−	0.598918i	$-0.795597\pi$
$23$	−2.61528e10			−1.60162			−0.800810	+	0.598918i	$0.795597\pi$
$24$	−5.09607e9			−0.226804
$25$	−2.70744e10			−0.887176
$26$		−	2.16390e10i		−	0.528370i
$27$			2.29267e10i			0.421807i
$28$	−1.06462e10			−0.149111
$29$	−3.62779e10	+	8.55167e10i	−0.390533	+	0.920589i
$29$	−3.62779e10	+	8.55167e10i	−0.390533	+	0.920589i
$30$	−8.15492e9			−0.0680786
$31$		−	8.60035e10i		−	0.561440i	−0.959790	−	0.280720i	$-0.909427\pi$
$31$		−	8.60035e10i		−	0.561440i	0.959790	−	0.280720i	$-0.0905732\pi$
$32$			4.60156e10i			0.236745i
$33$	3.37256e10			0.137754
$34$	−1.86884e11			−0.610211
$35$	−1.60162e11			−0.420773
$36$	5.33572e10			0.113481
$37$		−	8.70832e11i		−	1.50807i	−0.656835	−	0.754035i	$-0.728105\pi$
$37$		−	8.70832e11i		−	1.50807i	0.656835	−	0.754035i	$-0.271895\pi$
$38$	−1.82503e11			−0.258757
$39$		−	1.04177e11i		−	0.121559i
$40$			3.65573e11i			0.352796i
$41$			2.48384e11i			0.199180i	0.995029	+	0.0995898i	$0.0317530\pi$
$41$			2.48384e11i			0.199180i	−0.995029	+	0.0995898i	$0.968247\pi$
$42$	3.79337e11			0.253896
$43$		−	1.01999e12i		−	0.572245i	−0.958193	−	0.286123i	$-0.907634\pi$
$43$		−	1.01999e12i		−	0.572245i	0.958193	−	0.286123i	$-0.0923665\pi$
$44$		−	1.60818e11i		−	0.0759344i
$45$	8.02708e11			0.320231
$46$		−	4.44348e12i		−	1.50328i
$47$		−	1.95008e12i		−	0.561460i	−0.959787	−	0.280730i	$-0.909423\pi$
$47$		−	1.95008e12i		−	0.561460i	0.959787	−	0.280730i	$-0.0905766\pi$
$48$		−	7.61302e11i		−	0.187175i
$49$	2.70257e12			0.569254
$50$		−	4.60007e12i		−	0.832702i
$51$	−8.99715e11			−0.140388
$52$	−4.96758e11			−0.0670070
$53$	−5.78513e12			−0.676463			−0.338232	−	0.941063i	$-0.609829\pi$
$53$	−5.78513e12			−0.676463			−0.338232	+	0.941063i	$0.609829\pi$
$54$	−3.89535e12			−0.395907
$55$		−	2.41935e12i		−	0.214278i
$56$		−	1.70051e13i		−	1.31574i
$57$	−8.78622e11			−0.0595306
$58$	−1.45297e13	−	6.16378e12i	−0.864064	−	0.366554i
$59$	1.84391e13			0.964607			0.482304	−	0.876004i	$-0.339800\pi$
$59$	1.84391e13			0.964607			0.482304	+	0.876004i	$0.339800\pi$
$60$			1.87209e11i			0.00863361i
$61$		−	2.21432e13i		−	0.902124i	−0.892493	−	0.451062i	$-0.851045\pi$
$61$		−	2.21432e13i		−	0.902124i	0.892493	−	0.451062i	$-0.148955\pi$
$62$	1.46124e13			0.526967
$63$	−3.73390e13			−1.19429
$64$	−3.83161e13			−1.08901
$65$	−7.47325e12			−0.189086
$66$			5.73013e12i			0.129296i
$67$	−5.47383e13			−1.10339			−0.551697	−	0.834045i	$-0.686019\pi$
$67$	−5.47383e13			−1.10339			−0.551697	+	0.834045i	$0.686019\pi$
$68$			4.29022e12i			0.0773860i
$69$		−	2.13922e13i		−	0.345850i
$70$		−	2.72122e13i		−	0.394937i
$71$	−3.27160e13			−0.426897			−0.213448	−	0.976954i	$-0.568470\pi$
$71$	−3.27160e13			−0.426897			−0.213448	+	0.976954i	$0.568470\pi$
$72$			8.52272e13i			1.00135i
$73$		−	9.03645e13i		−	0.957362i	−0.877989	−	0.478681i	$-0.841115\pi$
$73$		−	9.03645e13i		−	0.957362i	0.877989	−	0.478681i	$-0.158885\pi$
$74$	1.47958e14			1.41547
$75$		−	2.21461e13i		−	0.191575i
$76$			4.18963e12i			0.0328151i
$77$			1.12539e14i			0.799140i
$78$	1.77001e13			0.114095
$79$		−	3.62848e12i		−	0.0212580i	−0.999944	−	0.0106290i	$-0.996617\pi$
$79$		−	3.62848e12i		−	0.0212580i	0.999944	−	0.0106290i	$-0.00338337\pi$
$80$	−5.46129e13			−0.291152
$81$	1.77537e14			0.862287
$82$	−4.22016e13			−0.186950
$83$	9.36521e13			0.378819			0.189409	−	0.981898i	$-0.439343\pi$
$83$	9.36521e13			0.378819			0.189409	+	0.981898i	$0.439343\pi$
$84$		−	8.70828e12i		−	0.0321986i
$85$			6.45422e13i			0.218374i
$86$	1.73301e14			0.537109
$87$	−6.99502e13	−	2.96743e13i	−0.198790	−	0.0843307i
$88$	2.56873e14			0.670037
$89$			3.62287e14i			0.868215i	0.900861	+	0.434108i	$0.142936\pi$
$89$			3.62287e14i			0.868215i	−0.900861	+	0.434108i	$0.857064\pi$
$90$			1.36384e14i			0.300569i
$91$	3.47628e14			0.705187
$92$	−1.02007e14			−0.190643
$93$	7.03484e13			0.121236
$94$	3.31328e14			0.526986
$95$			6.30290e13i			0.0926005i
$96$	−3.76394e13			−0.0511221
$97$			1.39209e15i			1.74936i	0.484697	+	0.874682i	$0.338930\pi$
$97$			1.39209e15i			1.74936i	−0.484697	+	0.874682i	$0.661070\pi$
$98$			4.59178e14i			0.534301i
$99$		−	5.64030e14i		−	0.608189i
$100$	−1.05602e14			−0.105602
$101$			2.05633e15i			1.90846i	0.299074	+	0.954230i	$0.403322\pi$
$101$			2.05633e15i			1.90846i	−0.299074	+	0.954230i	$0.596678\pi$
$102$		−	1.52866e14i		−	0.131768i
$103$	−1.67213e15			−1.33965			−0.669823	−	0.742521i	$-0.733630\pi$
$103$	−1.67213e15			−1.33965			−0.669823	+	0.742521i	$0.733630\pi$
$104$		−	7.93470e14i		−	0.591263i
$105$		−	1.31008e14i		−	0.0908609i
$106$		−	9.82921e14i		−	0.634928i
$107$	−2.39188e15			−1.43999			−0.719996	−	0.693978i	$-0.755856\pi$
$107$	−2.39188e15			−1.43999			−0.719996	+	0.693978i	$0.755856\pi$
$108$			8.94239e13i			0.0502083i
$109$	3.83622e14			0.201004			0.100502	−	0.994937i	$-0.467955\pi$
$109$	3.83622e14			0.201004			0.100502	+	0.994937i	$0.467955\pi$
$110$	4.11058e14			0.201121
$111$	7.12315e14			0.325649
$112$	2.54039e15			1.08584
$113$		−	1.05666e15i		−	0.422521i	−0.977430	−	0.211261i	$-0.932243\pi$
$113$		−	1.05666e15i		−	0.422521i	0.977430	−	0.211261i	$-0.0677569\pi$
$114$		−	1.49282e14i		−	0.0558754i
$115$	−1.53460e15			−0.537974
$116$	−1.41499e14	+	3.33552e14i	−0.0464857	+	0.109579i
$117$	−1.74226e15			−0.536685
$118$			3.13289e15i			0.905380i
$119$		−	3.00226e15i		−	0.814416i
$120$	−2.99028e14			−0.0761821
$121$	2.47727e15			0.593039
$122$	3.76223e15			0.846733
$123$	−2.03171e14			−0.0430104
$124$		−	3.35450e14i		−	0.0668291i
$125$	−3.37940e15			−0.633890
$126$		−	6.34407e15i		−	1.12096i
$127$		−	1.87562e15i		−	0.312332i	−0.987731	−	0.156166i	$-0.950087\pi$
$127$		−	1.87562e15i		−	0.312332i	0.987731	−	0.156166i	$-0.0499135\pi$
$128$		−	5.00224e15i		−	0.785398i
$129$	8.34322e14			0.123569
$130$		−	1.26974e15i		−	0.177476i
$131$		−	1.26822e16i		−	1.67363i	−0.547487	−	0.836814i	$-0.684416\pi$
$131$		−	1.26822e16i		−	1.67363i	0.547487	−	0.836814i	$-0.315584\pi$
$132$	1.31544e14			0.0163971
$133$		−	2.93188e15i		−	0.345349i
$134$		−	9.30029e15i		−	1.03564i
$135$			1.34530e15i			0.141682i
$136$	−6.85274e15			−0.682846
$137$			1.92530e16i			1.81591i	0.419068	+	0.907955i	$0.362357\pi$
$137$			1.92530e16i			1.81591i	−0.419068	+	0.907955i	$0.637643\pi$
$138$	3.63464e15			0.324615
$139$	1.38925e15			0.117536			0.0587678	−	0.998272i	$-0.481283\pi$
$139$	1.38925e15			0.117536			0.0587678	+	0.998272i	$0.481283\pi$
$140$	−6.24699e14			−0.0500853
$141$	1.59511e15			0.121240
$142$		−	5.55861e15i		−	0.400685i
$143$			5.25115e15i			0.359116i
$144$	−1.27321e16			−0.826382
$145$	−2.12872e15	+	5.01796e15i	−0.131177	+	0.309220i
$146$	1.53533e16			0.898579
$147$			2.21062e15i			0.122923i
$148$		−	3.39662e15i		−	0.179508i
$149$	−1.60485e16			−0.806376			−0.403188	−	0.915117i	$-0.632098\pi$
$149$	−1.60485e16			−0.806376			−0.403188	+	0.915117i	$0.632098\pi$
$150$	3.76273e15			0.179812
$151$	−1.29936e16			−0.590747			−0.295373	−	0.955382i	$-0.595444\pi$
$151$	−1.29936e16			−0.590747			−0.295373	+	0.955382i	$0.595444\pi$
$152$	−6.69208e15			−0.289557
$153$			1.50469e16i			0.619815i
$154$	−1.91209e16			−0.750072
$155$		−	5.04653e15i		−	0.188584i
$156$		−	4.06334e14i		−	0.0144693i
$157$		−	2.18945e14i		−	0.00743168i	−0.999993	−	0.00371584i	$-0.998817\pi$
$157$		−	2.18945e14i		−	0.00743168i	0.999993	−	0.00371584i	$-0.00118279\pi$
$158$	6.16496e14			0.0199527
$159$		−	4.73207e15i		−	0.146074i
$160$			2.70011e15i			0.0795210i
$161$	7.13839e16			2.00635
$162$			3.01644e16i			0.809342i
$163$			5.45445e16i			1.39748i	0.715378	+	0.698738i	$0.246255\pi$
$163$			5.45445e16i			1.39748i	−0.715378	+	0.698738i	$0.753745\pi$
$164$			9.68804e14i			0.0237087i
$165$	1.97896e15			0.0462708
$166$			1.59119e16i			0.355559i
$167$	3.23888e16			0.691866			0.345933	−	0.938259i	$-0.387563\pi$
$167$	3.23888e16			0.691866			0.345933	+	0.938259i	$0.387563\pi$
$168$	1.39097e16			0.284117
$169$	−3.49653e16			−0.683105
$170$	−1.09660e16			−0.204966
$171$			1.46942e16i			0.262829i
$172$		−	3.97839e15i		−	0.0681153i
$173$	−3.75379e16			−0.615353			−0.307676	−	0.951491i	$-0.599551\pi$
$173$	−3.75379e16			−0.615353			−0.307676	+	0.951491i	$0.599551\pi$
$174$	5.04180e15	−	1.18849e16i	0.0791528	−	0.186584i
$175$	7.38995e16			1.11136
$176$			3.83743e16i			0.552962i
$177$			1.50827e16i			0.208295i
$178$	−6.15542e16			−0.814906
$179$	3.83096e16			0.486307			0.243153	−	0.969988i	$-0.421818\pi$
$179$	3.83096e16			0.486307			0.243153	+	0.969988i	$0.421818\pi$
$180$	3.13090e15			0.0381176
$181$	1.26334e17			1.47547			0.737736	−	0.675089i	$-0.235895\pi$
$181$	1.26334e17			1.47547			0.737736	+	0.675089i	$0.235895\pi$
$182$			5.90636e16i			0.661888i
$183$	1.81125e16			0.194803
$184$		−	1.62935e17i		−	1.68222i
$185$		−	5.10988e16i		−	0.506551i
$186$			1.19525e16i			0.113792i
$187$	4.53512e16			0.414740
$188$		−	7.60616e15i		−	0.0668315i
$189$		−	6.25783e16i		−	0.528396i
$190$	−1.07089e16			−0.0869148
$191$			2.04024e17i			1.59195i	0.605327	+	0.795977i	$0.293042\pi$
$191$			2.04024e17i			1.59195i	−0.605327	+	0.795977i	$0.706958\pi$
$192$		−	3.13414e16i		−	0.235158i
$193$			1.31588e17i			0.949591i	0.880096	+	0.474796i	$0.157478\pi$
$193$			1.31588e17i			0.949591i	−0.880096	+	0.474796i	$0.842522\pi$
$194$	−2.36523e17			−1.64195
$195$		−	6.11291e15i		−	0.0408308i
$196$	1.05412e16			0.0677592
$197$	6.11016e16			0.378056			0.189028	−	0.981972i	$-0.439466\pi$
$197$	6.11016e16			0.378056			0.189028	+	0.981972i	$0.439466\pi$
$198$	9.58313e16			0.570845
$199$	−8.70175e16			−0.499124			−0.249562	−	0.968359i	$-0.580287\pi$
$199$	−8.70175e16			−0.499124			−0.249562	+	0.968359i	$0.580287\pi$
$200$		−	1.68677e17i		−	0.931820i
$201$		−	4.47744e16i		−	0.238264i
$202$	−3.49380e17			−1.79128
$203$	9.90203e16	−	2.33417e17i	0.489220	−	1.15322i
$204$	−3.50927e15			−0.0167106
$205$			1.45747e16i			0.0669031i
$206$		−	2.84102e17i		−	1.25739i
$207$	−3.57766e17			−1.52694
$208$	1.18536e17			0.487952
$209$	4.42879e16			0.175869
$210$	2.22588e16			0.0852819
$211$			2.19912e17i			0.813073i	0.913634	+	0.406537i	$0.133264\pi$
$211$			2.19912e17i			0.813073i	−0.913634	+	0.406537i	$0.866736\pi$
$212$	−2.25645e16			−0.0805205
$213$		−	2.67608e16i		−	0.0921832i
$214$		−	4.06391e17i		−	1.35158i
$215$		−	5.98511e16i		−	0.192213i
$216$	−1.42836e17			−0.443033
$217$			2.34746e17i			0.703315i
$218$			6.51792e16i			0.188662i
$219$	7.39155e16			0.206731
$220$		−	9.43649e15i		−	0.0255059i
$221$		−	1.40088e17i		−	0.365980i
$222$			1.21026e17i			0.305654i
$223$	−4.60666e17			−1.12486			−0.562432	−	0.826844i	$-0.690134\pi$
$223$	−4.60666e17			−1.12486			−0.562432	+	0.826844i	$0.690134\pi$
$224$		−	1.25599e17i		−	0.296570i
$225$	−3.70374e17			−0.845807
$226$	1.79532e17			0.396578
$227$	9.18145e17			1.96208			0.981041	−	0.193800i	$-0.0620814\pi$
$227$	9.18145e17			1.96208			0.981041	+	0.193800i	$0.0620814\pi$
$228$	−3.42700e15			−0.00708603
$229$			5.00411e17i			1.00129i	0.865652	+	0.500645i	$0.166904\pi$
$229$			5.00411e17i			1.00129i	−0.865652	+	0.500645i	$0.833096\pi$
$230$		−	2.60735e17i		−	0.504942i
$231$	−9.20537e16			−0.172565
$232$	−5.32780e17	−	2.26016e17i	−0.966915	−	0.410185i
$233$	4.20788e17			0.739425			0.369713	−	0.929146i	$-0.379456\pi$
$233$	4.20788e17			0.739425			0.369713	+	0.929146i	$0.379456\pi$
$234$		−	2.96018e17i		−	0.503732i
$235$		−	1.14427e17i		−	0.188591i
$236$	7.19205e16			0.114819
$237$	2.96799e15			0.00459040
$238$	5.10098e17			0.764411
$239$	9.38375e17			1.36267			0.681337	−	0.731970i	$-0.261399\pi$
$239$	9.38375e17			1.36267			0.681337	+	0.731970i	$0.261399\pi$
$240$		−	4.46718e16i		−	0.0628708i
$241$	−2.28794e17			−0.312117			−0.156058	−	0.987748i	$-0.549879\pi$
$241$	−2.28794e17			−0.312117			−0.156058	+	0.987748i	$0.549879\pi$
$242$			4.20900e17i			0.556626i
$243$			4.74193e17i			0.608007i
$244$		−	8.63679e16i		−	0.107381i
$245$	1.58582e17			0.191209
$246$		−	3.45197e16i		−	0.0403695i
$247$		−	1.36803e17i		−	0.155192i
$248$	5.35813e17			0.589693
$249$			7.66047e16i			0.0818013i
$250$		−	5.74175e17i		−	0.594969i
$251$			6.11302e17i			0.614757i	0.951587	+	0.307378i	$0.0994517\pi$
$251$			6.11302e17i			0.614757i	−0.951587	+	0.307378i	$0.900548\pi$
$252$	−1.45638e17			−0.142158
$253$			1.07830e18i			1.02173i
$254$	3.18676e17			0.293155
$255$	−5.27936e16			−0.0471553
$256$	−4.05638e17			−0.351835
$257$	−1.32364e18			−1.11499			−0.557497	−	0.830179i	$-0.688238\pi$
$257$	−1.32364e18			−1.11499			−0.557497	+	0.830179i	$0.688238\pi$
$258$			1.41755e17i			0.115982i
$259$			2.37693e18i			1.88916i
$260$	−2.91489e16			−0.0225072
$261$	−4.96276e17	+	1.16985e18i	−0.372322	+	0.877663i
$262$	2.15476e18			1.57087
$263$			9.94059e16i			0.0704278i	0.999380	+	0.0352139i	$0.0112113\pi$
$263$			9.94059e16i			0.0704278i	−0.999380	+	0.0352139i	$0.988789\pi$
$264$			2.10115e17i			0.144686i
$265$	−3.39461e17			−0.227220
$266$	4.98139e17			0.324144
$267$	−2.96340e17			−0.187480
$268$	−2.13503e17			−0.131339
$269$		−	1.41430e17i		−	0.0846055i	−0.999105	−	0.0423027i	$-0.986531\pi$
$269$		−	1.41430e17i		−	0.0846055i	0.999105	−	0.0423027i	$-0.0134694\pi$
$270$	−2.28572e17			−0.132983
$271$		−	7.28441e17i		−	0.412216i	−0.978529	−	0.206108i	$-0.933920\pi$
$271$		−	7.28441e17i		−	0.412216i	0.978529	−	0.206108i	$-0.0660798\pi$
$272$		−	1.02373e18i		−	0.563532i
$273$			2.84350e17i			0.152277i
$274$	−3.27117e18			−1.70441
$275$			1.11630e18i			0.565960i
$276$		−	8.34389e16i		−	0.0411671i
$277$	1.63080e18			0.783076			0.391538	−	0.920162i	$-0.371943\pi$
$277$	1.63080e18			0.783076			0.391538	+	0.920162i	$0.371943\pi$
$278$			2.36040e17i			0.110319i
$279$		−	1.17651e18i		−	0.535260i
$280$		−	9.97829e17i		−	0.441948i
$281$	1.70486e17			0.0735174			0.0367587	−	0.999324i	$-0.488297\pi$
$281$	1.70486e17			0.0735174			0.0367587	+	0.999324i	$0.488297\pi$
$282$			2.71017e17i			0.113796i
$283$	−2.08229e18			−0.851417			−0.425709	−	0.904860i	$-0.639975\pi$
$283$	−2.08229e18			−0.851417			−0.425709	+	0.904860i	$0.639975\pi$
$284$	−1.27607e17			−0.0508142
$285$	−5.15559e16			−0.0199960
$286$	−8.92195e17			−0.337066
$287$		−	6.77962e17i		−	0.249512i
$288$			6.29485e17i			0.225705i
$289$	1.65257e18			0.577332
$290$	−8.52575e17	−	3.61680e17i	−0.290234	−	0.123123i
$291$	−1.13869e18			−0.377754
$292$		−	3.52460e17i		−	0.113956i
$293$		−	1.79832e18i		−	0.566710i	−0.959015	−	0.283355i	$-0.908553\pi$
$293$		−	1.79832e18i		−	0.566710i	0.959015	−	0.283355i	$-0.0914475\pi$
$294$	−3.75595e17			−0.115376
$295$	1.08198e18			0.324005
$296$	5.42540e18			1.58396
$297$	9.45286e17			0.269085
$298$		−	2.72671e18i		−	0.756864i
$299$	3.33082e18			0.901607
$300$		−	8.63794e16i		−	0.0228034i
$301$			2.78405e18i			0.716851i
$302$		−	2.20767e18i		−	0.554474i
$303$	−1.68202e18			−0.412109
$304$		−	9.99730e17i		−	0.238963i
$305$		−	1.29932e18i		−	0.303018i
$306$	−2.55654e18			−0.581757
$307$		−	5.96372e18i		−	1.32428i	−0.749381	−	0.662139i	$-0.769649\pi$
$307$		−	5.96372e18i		−	1.32428i	0.749381	−	0.662139i	$-0.230351\pi$
$308$			4.38951e17i			0.0951229i
$309$		−	1.36775e18i		−	0.289280i
$310$	8.57428e17			0.177005
$311$		−	6.42260e18i		−	1.29422i	−0.762397	−	0.647110i	$-0.775978\pi$
$311$		−	6.42260e18i		−	1.29422i	0.762397	−	0.647110i	$-0.224022\pi$
$312$	6.49035e17			0.127676
$313$	−4.65349e18			−0.893710			−0.446855	−	0.894606i	$-0.647456\pi$
$313$	−4.65349e18			−0.893710			−0.446855	+	0.894606i	$0.647456\pi$
$314$	3.71997e16			0.00697537
$315$	−2.19098e18			−0.401153
$316$		−	1.41526e16i		−	0.00253037i
$317$		−	6.90646e18i		−	1.20590i	−0.797779	−	0.602950i	$-0.793992\pi$
$317$		−	6.90646e18i		−	1.20590i	0.797779	−	0.602950i	$-0.206008\pi$
$318$	8.04001e17			0.137105
$319$	3.52592e18	+	1.49577e18i	0.587276	+	0.249134i
$320$	−2.24832e18			−0.365791
$321$		−	1.95649e18i		−	0.310949i
$322$			1.21284e19i			1.88316i
$323$	−1.18149e18			−0.179230
$324$	6.92471e17			0.102639
$325$	3.44820e18			0.499422
$326$	−9.26737e18			−1.31167
$327$			3.13792e17i			0.0434044i
$328$	−1.54747e18			−0.209203
$329$			5.32273e18i			0.703340i
$330$			3.36234e17i			0.0434297i
$331$			7.97779e18i			1.00733i	0.863898	+	0.503666i	$0.168016\pi$
$331$			7.97779e18i			1.00733i	−0.863898	+	0.503666i	$0.831984\pi$
$332$	3.65283e17			0.0450914
$333$		−	1.19128e19i		−	1.43775i
$334$			5.50301e18i			0.649385i
$335$	−3.21195e18			−0.370623
$336$			2.07797e18i			0.234474i
$337$			1.59371e19i			1.75868i	0.476197	+	0.879338i	$0.342015\pi$
$337$			1.59371e19i			1.75868i	−0.476197	+	0.879338i	$0.657985\pi$
$338$		−	5.94077e18i		−	0.641162i
$339$	8.64321e17			0.0912383
$340$			2.51742e17i			0.0259934i
$341$	−3.54599e18			−0.358162
$342$	−2.49660e18			−0.246691
$343$	5.58180e18			0.539595
$344$	6.35467e18			0.601042
$345$		−	1.25526e18i		−	0.116169i
$346$		−	6.37786e18i		−	0.577570i
$347$	8.41833e18			0.746028			0.373014	−	0.927826i	$-0.378324\pi$
$347$	8.41833e18			0.746028			0.373014	+	0.927826i	$0.378324\pi$
$348$	−2.72836e17	−	1.15742e17i	−0.0236623	−	0.0100380i
$349$	−6.29262e18			−0.534122			−0.267061	−	0.963680i	$-0.586053\pi$
$349$	−6.29262e18			−0.534122			−0.267061	+	0.963680i	$0.586053\pi$
$350$			1.25559e19i			1.04312i
$351$		−	2.91994e18i		−	0.237449i
$352$	1.89726e18			0.151028
$353$	−1.33832e19			−1.04292			−0.521458	−	0.853277i	$-0.674612\pi$
$353$	−1.33832e19			−1.04292			−0.521458	+	0.853277i	$0.674612\pi$
$354$	−2.56262e18			−0.195506
$355$	−1.91972e18			−0.143392
$356$			1.41307e18i			0.103345i
$357$	2.45576e18			0.175863
$358$			6.50898e18i			0.456447i
$359$			9.78455e17i			0.0671943i	0.999435	+	0.0335972i	$0.0106963\pi$
$359$			9.78455e17i			0.0671943i	−0.999435	+	0.0335972i	$0.989304\pi$
$360$			5.00098e18i			0.336346i
$361$	1.40273e19			0.923998
$362$			2.14647e19i			1.38488i
$363$			2.02634e18i			0.128060i
$364$	1.35590e18			0.0839396
$365$		−	5.30242e18i		−	0.321572i
$366$			3.07739e18i			0.182842i
$367$		−	4.45939e18i		−	0.259585i	−0.991541	−	0.129793i	$-0.958569\pi$
$367$		−	4.45939e18i		−	0.259585i	0.991541	−	0.129793i	$-0.0414311\pi$
$368$	2.43409e19			1.38828
$369$			3.39785e18i			0.189892i
$370$	8.68193e18			0.475448
$371$	1.57905e19			0.847405
$372$	2.74389e17			0.0144309
$373$	1.47777e19			0.761714			0.380857	−	0.924634i	$-0.375629\pi$
$373$	1.47777e19			0.761714			0.380857	+	0.924634i	$0.375629\pi$
$374$			7.70537e18i			0.389275i
$375$		−	2.76425e18i		−	0.136881i
$376$	1.21493e19			0.589714
$377$	4.62035e18	−	1.08914e19i	0.219844	−	0.518231i
$378$	1.06323e19			0.495952
$379$		−	6.35200e18i		−	0.290480i	−0.989396	−	0.145240i	$-0.953605\pi$
$379$		−	6.35200e18i		−	0.290480i	0.989396	−	0.145240i	$-0.0463954\pi$
$380$			2.45840e17i			0.0110224i
$381$	1.53420e18			0.0674443
$382$	−3.46646e19			−1.49421
$383$	−2.93493e19			−1.24053			−0.620264	−	0.784393i	$-0.712975\pi$
$383$	−2.93493e19			−1.24053			−0.620264	+	0.784393i	$0.712975\pi$
$384$	4.09169e18			0.169597
$385$			6.60359e18i			0.268426i
$386$	−2.23574e19			−0.891286
$387$		−	1.39533e19i		−	0.545562i
$388$			5.42976e18i			0.208230i
$389$		−	2.44544e19i		−	0.919888i	−0.887948	−	0.459944i	$-0.847869\pi$
$389$		−	2.44544e19i		−	0.919888i	0.887948	−	0.459944i	$-0.152131\pi$
$390$	1.03861e18			0.0383238
$391$		−	2.87664e19i		−	1.04126i
$392$			1.68374e19i			0.597900i
$393$	1.03737e19			0.361400
$394$			1.03814e19i			0.354843i
$395$		−	2.12913e17i		−	0.00714041i
$396$		−	2.19996e18i		−	0.0723937i
$397$	−2.77548e19			−0.896208			−0.448104	−	0.893981i	$-0.647901\pi$
$397$	−2.77548e19			−0.896208			−0.448104	+	0.893981i	$0.647901\pi$
$398$		−	1.47847e19i		−	0.468478i
$399$	2.39819e18			0.0745739
$400$	2.51987e19			0.769004
$401$	−4.14308e19			−1.24091			−0.620455	−	0.784242i	$-0.713052\pi$
$401$	−4.14308e19			−1.24091			−0.620455	+	0.784242i	$0.713052\pi$
$402$	7.60737e18			0.223635
$403$			1.09534e19i			0.316054i
$404$			8.02058e18i			0.227167i
$405$	1.04176e19			0.289637
$406$	3.96586e19	+	1.68240e19i	1.08241	+	0.459181i
$407$	−3.59051e19			−0.962050
$408$		−	5.60534e18i		−	0.147452i
$409$		−	1.58705e19i		−	0.409889i	−0.978774	−	0.204945i	$-0.934299\pi$
$409$		−	1.58705e19i		−	0.409889i	0.978774	−	0.204945i	$-0.0657014\pi$
$410$	−2.47631e18			−0.0627952
$411$	−1.57484e19			−0.392123
$412$	−6.52201e18			−0.159460
$413$	−5.03295e19			−1.20836
$414$		−	6.07861e19i		−	1.43318i
$415$	5.49533e18			0.127243
$416$		−	5.86054e18i		−	0.133272i
$417$			1.13637e18i			0.0253804i
$418$			7.52472e18i			0.165070i
$419$	−2.08514e19			−0.449294			−0.224647	−	0.974440i	$-0.572123\pi$
$419$	−2.08514e19			−0.449294			−0.224647	+	0.974440i	$0.572123\pi$
$420$		−	5.10986e17i		−	0.0108153i
$421$		−	5.01088e19i		−	1.04183i	−0.853607	−	0.520917i	$-0.825590\pi$
$421$		−	5.01088e19i		−	1.04183i	0.853607	−	0.520917i	$-0.174410\pi$
$422$	−3.73640e19			−0.763150
$423$		−	2.66768e19i		−	0.535280i
$424$		−	3.60422e19i		−	0.710504i
$425$		−	2.97801e19i		−	0.576779i
$426$	4.54678e18			0.0865231
$427$			6.04396e19i			1.13009i
$428$	−9.32934e18			−0.171405
$429$	−4.29529e18			−0.0775466
$430$	1.01690e19			0.180411
$431$	5.53406e19			0.964859			0.482430	−	0.875935i	$-0.339754\pi$
$431$	5.53406e19			0.964859			0.482430	+	0.875935i	$0.339754\pi$
$432$		−	2.13383e19i		−	0.365622i
$433$			1.09845e20i			1.84979i	0.380228	+	0.924893i	$0.375846\pi$
$433$			1.09845e20i			1.84979i	−0.380228	+	0.924893i	$0.624154\pi$
$434$	−3.98844e19			−0.660131
$435$	−4.10455e18	−	1.74123e18i	−0.0667723	−	0.0283261i
$436$	1.49629e18			0.0239259
$437$		−	2.80920e19i		−	0.441541i
$438$			1.25586e19i			0.194037i
$439$	−1.78239e19			−0.270720			−0.135360	−	0.990797i	$-0.543219\pi$
$439$	−1.78239e19			−0.270720			−0.135360	+	0.990797i	$0.543219\pi$
$440$	1.50729e19			0.225061
$441$	3.69707e19			0.542710
$442$	2.38015e19			0.343509
$443$			1.92591e19i			0.273280i	0.990621	+	0.136640i	$0.0436303\pi$
$443$			1.92591e19i			0.273280i	−0.990621	+	0.136640i	$0.956370\pi$
$444$	2.77833e18			0.0387625
$445$			2.12583e19i			0.291628i
$446$		−	7.82693e19i		−	1.05580i
$447$		−	1.31272e19i		−	0.174127i
$448$	1.04583e20			1.36420
$449$			1.89732e19i			0.243384i	0.992568	+	0.121692i	$0.0388320\pi$
$449$			1.89732e19i			0.243384i	−0.992568	+	0.121692i	$0.961168\pi$
$450$		−	6.29282e19i		−	0.793874i
$451$	1.02411e19			0.127064
$452$		−	4.12144e18i		−	0.0502934i
$453$		−	1.06284e19i		−	0.127564i
$454$			1.55997e20i			1.84161i
$455$	2.03982e19			0.236868
$456$		−	5.47393e18i		−	0.0625263i
$457$	1.36342e20			1.53200			0.766000	−	0.642841i	$-0.222244\pi$
$457$	1.36342e20			1.53200			0.766000	+	0.642841i	$0.222244\pi$
$458$	−8.50220e19			−0.939811
$459$	−2.52179e19			−0.274229
$460$	−5.98559e18			−0.0640359
$461$			8.47067e19i			0.891581i	0.895137	+	0.445790i	$0.147077\pi$
$461$			8.47067e19i			0.891581i	−0.895137	+	0.445790i	$0.852923\pi$
$462$		−	1.56404e19i		−	0.161969i
$463$	−7.84555e19			−0.799403			−0.399702	−	0.916645i	$-0.630886\pi$
$463$	−7.84555e19			−0.799403			−0.399702	+	0.916645i	$0.630886\pi$
$464$	3.37645e19	−	7.95920e19i	0.338514	−	0.797967i
$465$	4.12791e18			0.0407224
$466$			7.14939e19i			0.694024i
$467$		−	7.96602e19i		−	0.760966i	−0.924788	−	0.380483i	$-0.875758\pi$
$467$		−	7.96602e19i		−	0.760966i	0.924788	−	0.380483i	$-0.124242\pi$
$468$	−6.79557e18			−0.0638825
$469$	1.49408e20			1.38222
$470$	1.94417e19			0.177011
$471$	1.79091e17			0.00160478
$472$			1.14878e20i			1.01315i
$473$	−4.20549e19			−0.365055
$474$			5.04276e17i			0.00430854i
$475$		−	2.90819e19i		−	0.244580i
$476$		−	1.17101e19i		−	0.0969413i
$477$	−7.91396e19			−0.644920
$478$			1.59434e20i			1.27900i
$479$			3.57275e19i			0.282154i	0.989999	+	0.141077i	$0.0450565\pi$
$479$			3.57275e19i			0.282154i	−0.989999	+	0.141077i	$0.954943\pi$
$480$	−2.20861e18			−0.0171716
$481$			1.10909e20i			0.848944i
$482$		−	3.88732e19i		−	0.292953i
$483$			5.83900e19i			0.433246i
$484$	9.66242e18			0.0705904
$485$			8.16855e19i			0.587600i
$486$	−8.05676e19			−0.570675
$487$	5.79748e19			0.404364			0.202182	−	0.979348i	$-0.435197\pi$
$487$	5.79748e19			0.404364			0.202182	+	0.979348i	$0.435197\pi$
$488$	1.37955e20			0.947521
$489$	−4.46159e19			−0.301768
$490$			2.69438e19i			0.179468i
$491$			4.54721e19i			0.298287i	0.988816	+	0.149143i	$0.0476516\pi$
$491$			4.54721e19i			0.298287i	−0.988816	+	0.149143i	$0.952348\pi$
$492$	−7.92454e17			−0.00511960
$493$	−9.40627e19	−	3.99033e19i	−0.598502	−	0.253897i
$494$	2.32435e19			0.145663
$495$		−	3.30963e19i		−	0.204287i
$496$			8.00451e19i			0.486656i
$497$	8.92982e19			0.534773
$498$	−1.30155e19			−0.0767786
$499$	−1.09676e19			−0.0637321			−0.0318660	−	0.999492i	$-0.510145\pi$
$499$	−1.09676e19			−0.0637321			−0.0318660	+	0.999492i	$0.510145\pi$
$500$	−1.31811e19			−0.0754530
$501$			2.64931e19i			0.149400i
$502$	−1.03863e20			−0.577010
$503$		−	9.41369e19i		−	0.515229i	−0.966248	−	0.257614i	$-0.917064\pi$
$503$		−	9.41369e19i		−	0.515229i	0.966248	−	0.257614i	$-0.0829364\pi$
$504$		−	2.32627e20i		−	1.25439i
$505$			1.20662e20i			0.641039i
$506$	−1.83208e20			−0.958994
$507$		−	2.86006e19i		−	0.147508i
$508$		−	7.31571e18i		−	0.0371774i
$509$	1.10594e20			0.553795			0.276898	−	0.960899i	$-0.410694\pi$
$509$	1.10594e20			0.553795			0.276898	+	0.960899i	$0.410694\pi$
$510$		−	8.96988e18i		−	0.0442599i
$511$			2.46649e20i			1.19929i
$512$		−	2.32833e20i		−	1.11563i
$513$	−2.46267e19			−0.116285
$514$		−	2.24893e20i		−	1.04653i
$515$	−9.81174e19			−0.449978
$516$	3.25421e18			0.0147087
$517$	−8.04034e19			−0.358175
$518$	−4.03851e20			−1.77316
$519$		−	3.07049e19i		−	0.132878i
$520$		−	4.65594e19i		−	0.198601i
$521$	7.41673e19			0.311839			0.155919	−	0.987770i	$-0.450166\pi$
$521$	7.41673e19			0.311839			0.155919	+	0.987770i	$0.450166\pi$
$522$	−1.98764e20	−	8.43195e19i	−0.823774	−	0.349461i
$523$	2.13671e20			0.872938			0.436469	−	0.899719i	$-0.356229\pi$
$523$	2.13671e20			0.872938			0.436469	+	0.899719i	$0.356229\pi$
$524$		−	4.94659e19i		−	0.199215i
$525$			6.04477e19i			0.239985i
$526$	−1.68895e19			−0.0661035
$527$	9.45982e19			0.365009
$528$	−3.13891e19			−0.119405
$529$	4.17334e20			1.56519
$530$		−	5.76760e19i		−	0.213268i
$531$	2.52244e20			0.919629
$532$		−	1.14356e19i		−	0.0411075i
$533$		−	3.16342e19i		−	0.112125i
$534$		−	5.03495e19i		−	0.175969i
$535$	−1.40351e20			−0.483684
$536$		−	3.41027e20i		−	1.15892i
$537$			3.13362e19i			0.105012i
$538$	2.40296e19			0.0794106
$539$		−	1.11429e20i		−	0.363147i
$540$			5.24723e18i			0.0168646i
$541$		−	2.53970e20i		−	0.805014i	−0.915417	−	0.402507i	$-0.868139\pi$
$541$		−	2.53970e20i		−	0.805014i	0.915417	−	0.402507i	$-0.131861\pi$
$542$	1.23766e20			0.386906
$543$			1.03338e20i			0.318610i
$544$	−5.06141e19			−0.153915
$545$	2.25103e19			0.0675160
$546$	−4.83123e19			−0.142927
$547$	7.69794e19			0.224631			0.112315	−	0.993673i	$-0.464173\pi$
$547$	7.69794e19			0.224631			0.112315	+	0.993673i	$0.464173\pi$
$548$			7.50950e19i			0.216151i
$549$		−	3.02915e20i		−	0.860059i
$550$	−1.89665e20			−0.531210
$551$	−9.18575e19	−	3.89678e19i	−0.253792	−	0.107664i
$552$	1.33277e20			0.363254
$553$			9.90391e18i			0.0266298i
$554$			2.77081e20i			0.734994i
$555$	4.17974e19			0.109383
$556$	5.41867e18			0.0139904
$557$	−9.96556e19			−0.253856			−0.126928	−	0.991912i	$-0.540512\pi$
$557$	−9.96556e19			−0.253856			−0.126928	+	0.991912i	$0.540512\pi$
$558$	1.99895e20			0.502395
$559$			1.29906e20i			0.322137i
$560$	1.49066e20			0.364726
$561$			3.70959e19i			0.0895581i
$562$			2.89663e19i			0.0690034i
$563$		−	6.62490e20i		−	1.55728i	−0.627472	−	0.778639i	$-0.715910\pi$
$563$		−	6.62490e20i		−	1.55728i	0.627472	−	0.778639i	$-0.284090\pi$
$564$	6.22162e18			0.0144315
$565$		−	6.20032e19i		−	0.141922i
$566$		−	3.53790e20i		−	0.799140i
$567$	−4.84586e20			−1.08019
$568$		−	2.03825e20i		−	0.448379i
$569$			4.66723e20i			1.01325i	0.862166	+	0.506626i	$0.169108\pi$
$569$			4.66723e20i			1.01325i	−0.862166	+	0.506626i	$0.830892\pi$
$570$		−	8.75959e18i		−	0.0187682i
$571$	−6.11423e20			−1.29292			−0.646459	−	0.762948i	$-0.723751\pi$
$571$	−6.11423e20			−1.29292			−0.646459	+	0.762948i	$0.723751\pi$
$572$			2.04817e19i			0.0427461i
$573$	−1.66886e20			−0.343763
$574$	1.15189e20			0.234192
$575$	7.08073e20			1.42092
$576$	−5.24157e20			−1.03823
$577$			8.17579e20i			1.59850i	0.601002	+	0.799248i	$0.294768\pi$
$577$			8.17579e20i			1.59850i	−0.601002	+	0.799248i	$0.705232\pi$
$578$			2.80779e20i			0.541883i
$579$	−1.07635e20			−0.205053
$580$	−8.30293e18	+	1.95722e19i	−0.0156143	+	0.0368069i
$581$	−2.55623e20			−0.474546
$582$		−	1.93469e20i		−	0.354559i
$583$			2.38525e20i			0.431540i
$584$	5.62983e20			1.00554
$585$	−1.02233e20			−0.180269
$586$	3.05544e20			0.531914
$587$	2.72842e20			0.468950			0.234475	−	0.972122i	$-0.424663\pi$
$587$	2.72842e20			0.468950			0.234475	+	0.972122i	$0.424663\pi$
$588$			8.62237e18i			0.0146318i
$589$	9.23804e19			0.154780
$590$			1.83833e20i			0.304111i
$591$			4.99794e19i			0.0816365i
$592$			8.10500e20i			1.30719i
$593$	6.63290e19			0.105631			0.0528157	−	0.998604i	$-0.483180\pi$
$593$	6.63290e19			0.105631			0.0528157	+	0.998604i	$0.483180\pi$
$594$			1.60608e20i			0.252563i
$595$		−	1.76167e20i		−	0.273557i
$596$	−6.25960e19			−0.0959842
$597$		−	7.11778e19i		−	0.107780i
$598$			5.65922e20i			0.846248i
$599$		−	1.80408e20i		−	0.266413i	−0.991088	−	0.133206i	$-0.957473\pi$
$599$		−	1.80408e20i		−	0.266413i	0.991088	−	0.133206i	$-0.0425273\pi$
$600$	1.37973e20			0.201215
$601$			8.26199e20i			1.18994i	0.803747	+	0.594972i	$0.202837\pi$
$601$			8.26199e20i			1.18994i	−0.803747	+	0.594972i	$0.797163\pi$
$602$	−4.73023e20			−0.672836
$603$	−7.48811e20			−1.05194
$604$	−5.06805e19			−0.0703175
$605$	1.45362e20			0.199198
$606$		−	2.85783e20i		−	0.386805i
$607$			1.12257e21i			1.50072i	0.661030	+	0.750360i	$0.270120\pi$
$607$			1.12257e21i			1.50072i	−0.661030	+	0.750360i	$0.729880\pi$
$608$	−4.94275e19			−0.0652667
$609$	1.90928e20	+	8.09957e19i	0.249024	+	0.105641i
$610$	2.20761e20			0.284412
$611$			2.48362e20i			0.316065i
$612$			5.86894e19i			0.0737775i
$613$	−1.03406e21			−1.28409			−0.642043	−	0.766669i	$-0.721913\pi$
$613$	−1.03406e21			−1.28409			−0.642043	+	0.766669i	$0.721913\pi$
$614$	1.01326e21			1.24297
$615$	−1.19217e19			−0.0144469
$616$	−7.01134e20			−0.839355
$617$		−	1.35553e21i		−	1.60313i	−0.597907	−	0.801566i	$-0.704001\pi$
$617$		−	1.35553e21i		−	1.60313i	0.597907	−	0.801566i	$-0.295999\pi$
$618$	2.32387e20			0.271518
$619$			1.14614e21i			1.32299i	0.749948	+	0.661497i	$0.230078\pi$
$619$			1.14614e21i			1.32299i	−0.749948	+	0.661497i	$0.769922\pi$
$620$		−	1.96836e19i		−	0.0224475i
$621$		−	5.99598e20i		−	0.675574i
$622$	1.09123e21			1.21475
$623$		−	9.88859e20i		−	1.08761i
$624$			9.69594e19i			0.105367i
$625$	6.27950e20			0.674256
$626$		−	7.90650e20i		−	0.838835i
$627$			3.62263e19i			0.0379767i
$628$		−	8.53978e17i		−	0.000884605i
$629$	9.57858e20			0.980441
$630$		−	3.72258e20i		−	0.376522i
$631$	1.52711e21			1.52633			0.763167	−	0.646201i	$-0.223643\pi$
$631$	1.52711e21			1.52633			0.763167	+	0.646201i	$0.223643\pi$
$632$	2.26059e19			0.0223277
$633$	−1.79881e20			−0.175573
$634$	1.17344e21			1.13186
$635$		−	1.10058e20i		−	0.104910i
$636$		−	1.84571e19i		−	0.0173874i
$637$	−3.44199e20			−0.320452
$638$	−2.54138e20	+	5.99070e20i	−0.233837	+	0.551217i
$639$	−4.47550e20			−0.406991
$640$		−	2.93522e20i		−	0.263810i
$641$			1.61008e21i			1.43025i	0.698994	+	0.715127i	$0.253631\pi$
$641$			1.61008e21i			1.43025i	−0.698994	+	0.715127i	$0.746369\pi$
$642$	3.32416e20			0.291856
$643$	−1.27769e21			−1.10878			−0.554389	−	0.832257i	$-0.687048\pi$
$643$	−1.27769e21			−1.10878			−0.554389	+	0.832257i	$0.687048\pi$
$644$	2.78428e20			0.238819
$645$	4.89565e19			0.0415061
$646$		−	2.00741e20i		−	0.168226i
$647$	−9.94209e20			−0.823560			−0.411780	−	0.911283i	$-0.635093\pi$
$647$	−9.94209e20			−0.823560			−0.411780	+	0.911283i	$0.635093\pi$
$648$			1.10608e21i			0.905679i
$649$		−	7.60260e20i		−	0.615357i
$650$			5.85865e20i			0.468757i
$651$	−1.92015e20			−0.151872
$652$			2.12747e20i			0.166344i
$653$		−	1.13584e21i		−	0.877946i	−0.898500	−	0.438973i	$-0.855342\pi$
$653$		−	1.13584e21i		−	0.877946i	0.898500	−	0.438973i	$-0.144658\pi$
$654$	−5.33147e19			−0.0407394
$655$		−	7.44167e20i		−	0.562161i
$656$		−	2.31176e20i		−	0.172649i
$657$		−	1.23617e21i		−	0.912721i
$658$	−9.04357e20			−0.660155
$659$		−	1.26018e21i		−	0.909477i	−0.890625	−	0.454739i	$-0.849733\pi$
$659$		−	1.26018e21i		−	0.909477i	0.890625	−	0.454739i	$-0.150267\pi$
$660$	7.71878e18			0.00550768
$661$	3.08463e20			0.217617			0.108808	−	0.994063i	$-0.465296\pi$
$661$	3.08463e20			0.217617			0.108808	+	0.994063i	$0.465296\pi$
$662$	−1.35546e21			−0.945481
$663$	1.14588e20			0.0790290
$664$			5.83465e20i			0.397882i
$665$		−	1.72037e20i		−	0.116001i
$666$	2.02404e21			1.34947
$667$	9.48769e20	−	2.23650e21i	0.625485	−	1.47443i
$668$	1.26330e20			0.0823539
$669$		−	3.76812e20i		−	0.242901i
$670$		−	5.45724e20i		−	0.347866i
$671$	−9.12980e20			−0.575496
$672$	1.02736e20			0.0640406
$673$	−2.43341e21			−1.50004			−0.750020	−	0.661416i	$-0.769956\pi$
$673$	−2.43341e21			−1.50004			−0.750020	+	0.661416i	$0.769956\pi$
$674$	−2.70779e21			−1.65069
$675$		−	6.20728e20i		−	0.374216i
$676$	−1.36380e20			−0.0813111
$677$			2.69247e21i			1.58758i	0.608192	+	0.793790i	$0.291895\pi$
$677$			2.69247e21i			1.58758i	−0.608192	+	0.793790i	$0.708105\pi$
$678$			1.46852e20i			0.0856362i
$679$		−	3.79971e21i		−	2.19143i
$680$	−4.02106e20			−0.229363
$681$			7.51016e20i			0.423688i
$682$		−	6.02480e20i		−	0.336171i
$683$	1.37390e21			0.758226			0.379113	−	0.925350i	$-0.376229\pi$
$683$	1.37390e21			0.758226			0.379113	+	0.925350i	$0.376229\pi$
$684$			5.73135e19i			0.0312850i
$685$			1.12973e21i			0.609952i
$686$			9.48373e20i			0.506464i
$687$	−4.09321e20			−0.216217
$688$			9.49324e20i			0.496022i
$689$	7.36794e20			0.380805
$690$	2.13274e20			0.109036
$691$	−2.99176e21			−1.51301			−0.756505	−	0.653988i	$-0.773095\pi$
$691$	−2.99176e21			−1.51301			−0.756505	+	0.653988i	$0.773095\pi$
$692$	−1.46414e20			−0.0732464
$693$			1.53952e21i			0.761877i
$694$			1.43031e21i			0.700221i
$695$	8.15186e19			0.0394794
$696$	1.84875e20	−	4.35799e20i	0.0885744	−	0.208793i
$697$	−2.73206e20			−0.129493
$698$		−	1.06914e21i		−	0.501327i
$699$			3.44193e20i			0.159670i
$700$	2.88240e20			0.132287
$701$	−2.72814e21			−1.23874			−0.619371	−	0.785099i	$-0.712612\pi$
$701$	−2.72814e21			−1.23874			−0.619371	+	0.785099i	$0.712612\pi$
$702$	4.96112e20			0.222870
$703$	9.35402e20			0.415751
$704$			1.57980e21i			0.694716i
$705$	9.35982e19			0.0407239
$706$		−	2.27387e21i		−	0.978881i
$707$		−	5.61274e21i		−	2.39073i
$708$			5.88289e19i			0.0247937i
$709$	2.48395e21			1.03585			0.517923	−	0.855427i	$-0.326705\pi$
$709$	2.48395e21			1.03585			0.517923	+	0.855427i	$0.326705\pi$
$710$		−	3.26169e20i		−	0.134588i
$711$		−	4.96370e19i		−	0.0202667i
$712$	−2.25710e21			−0.911906
$713$			2.24923e21i			0.899213i
$714$			4.17246e20i			0.165065i
$715$			3.08128e20i			0.120625i
$716$	1.49424e20			0.0578859
$717$			7.67564e20i			0.294253i
$718$	−1.66244e20			−0.0630685
$719$	1.09077e21			0.409511			0.204755	−	0.978813i	$-0.434360\pi$
$719$	1.09077e21			0.409511			0.204755	+	0.978813i	$0.434360\pi$
$720$	−7.47095e20			−0.277576
$721$	4.56406e21			1.67817
$722$			2.38331e21i			0.867264i
$723$		−	1.87147e20i		−	0.0673978i
$724$	4.92757e20			0.175628
$725$	9.82204e20	−	2.31532e21i	0.346471	−	0.816724i
$726$	−3.44284e20			−0.120197
$727$		−	3.48814e21i		−	1.20528i	−0.798015	−	0.602638i	$-0.794116\pi$
$727$		−	3.48814e21i		−	1.20528i	0.798015	−	0.602638i	$-0.205884\pi$
$728$			2.16577e21i			0.740674i
$729$	2.15959e21			0.730995
$730$	9.00906e20			0.301827
$731$	1.12192e21			0.372034
$732$	7.06464e19			0.0231877
$733$			4.63529e21i			1.50590i	0.658075	+	0.752952i	$0.271371\pi$
$733$			4.63529e21i			1.50590i	−0.658075	+	0.752952i	$0.728629\pi$
$734$	7.57670e20			0.243646
$735$			1.29715e20i			0.0412892i
$736$		−	1.20344e21i		−	0.379175i
$737$			2.25690e21i			0.703893i
$738$	−5.77311e20			−0.178232
$739$		−	4.21811e21i		−	1.28909i	−0.764565	−	0.644547i	$-0.777046\pi$
$739$		−	4.21811e21i		−	1.28909i	0.764565	−	0.644547i	$-0.222954\pi$
$740$		−	1.99307e20i		−	0.0602956i
$741$	1.11901e20			0.0335118
$742$			2.68287e21i			0.795373i
$743$			4.10826e21i			1.20571i	0.797852	+	0.602854i	$0.205970\pi$
$743$			4.10826e21i			1.20571i	−0.797852	+	0.602854i	$0.794030\pi$
$744$			4.38280e20i			0.127337i
$745$	−9.41697e20			−0.270856
$746$			2.51081e21i			0.714944i
$747$	1.28114e21			0.361155
$748$	1.76889e20			0.0493672
$749$	6.52861e21			1.80388
$750$	4.69658e20			0.128476
$751$			2.91770e21i			0.790208i	0.918636	+	0.395104i	$0.129291\pi$
$751$			2.91770e21i			0.790208i	−0.918636	+	0.395104i	$0.870709\pi$
$752$			1.81498e21i			0.486674i
$753$	−5.00028e20			−0.132749
$754$	1.85050e21	+	7.85019e20i	0.486412	+	0.206346i
$755$	−7.62439e20			−0.198428
$756$		−	2.44082e20i		−	0.0628959i
$757$			2.71701e21i			0.693222i	0.938009	+	0.346611i	$0.112668\pi$
$757$			2.71701e21i			0.693222i	−0.938009	+	0.346611i	$0.887332\pi$
$758$	1.07923e21			0.272644
$759$	−8.82019e20			−0.220630
$760$	−3.92679e20			−0.0972604
$761$	−4.81957e21			−1.18202			−0.591008	−	0.806666i	$-0.701270\pi$
$761$	−4.81957e21			−1.18202			−0.591008	+	0.806666i	$0.701270\pi$
$762$			2.60668e20i			0.0633031i
$763$	−1.04709e21			−0.251798
$764$			7.95780e20i			0.189493i
$765$			8.82926e20i			0.208192i
$766$		−	4.98658e21i		−	1.16436i
$767$	−2.34841e21			−0.543011
$768$		−	3.31800e20i		−	0.0759744i
$769$		−	7.71359e21i		−	1.74908i	−0.484955	−	0.874539i	$-0.661164\pi$
$769$		−	7.71359e21i		−	1.74908i	0.484955	−	0.874539i	$-0.338836\pi$
$770$	−1.12198e21			−0.251944
$771$		−	1.08270e21i		−	0.240769i
$772$			5.13250e20i			0.113031i
$773$			1.42172e21i			0.310075i	0.987909	+	0.155038i	$0.0495499\pi$
$773$			1.42172e21i			0.310075i	−0.987909	+	0.155038i	$0.950450\pi$
$774$	2.37073e21			0.512064
$775$			2.32850e21i			0.498096i
$776$	−8.67293e21			−1.83740
$777$	−1.94426e21			−0.407940
$778$	4.15492e21			0.863406
$779$	−2.66801e20			−0.0549107
$780$		−	2.38429e19i		−	0.00486016i
$781$			1.34891e21i			0.272332i
$782$	4.88754e21			0.977326
$783$	−1.96062e21	−	8.31733e20i	−0.388311	−	0.164729i
$784$	−2.51533e21			−0.493429
$785$		−	1.28473e19i		−	0.00249625i
$786$			1.76253e21i			0.339209i
$787$	−2.73464e21			−0.521302			−0.260651	−	0.965433i	$-0.583937\pi$
$787$	−2.73464e21			−0.521302			−0.260651	+	0.965433i	$0.583937\pi$
$788$	2.38322e20			0.0450006
$789$	−8.13112e19			−0.0152080
$790$	3.61748e19			0.00670199
$791$			2.88416e21i			0.529292i
$792$	3.51398e21			0.638794
$793$			2.82015e21i			0.507837i
$794$		−	4.71566e21i		−	0.841180i
$795$		−	2.77669e20i		−	0.0490653i
$796$	−3.39406e20			−0.0594116
$797$			4.59646e21i			0.797050i	0.917157	+	0.398525i	$0.130478\pi$
$797$			4.59646e21i			0.797050i	−0.917157	+	0.398525i	$0.869522\pi$
$798$			4.07464e20i			0.0699950i
$799$	2.14496e21			0.365022
$800$		−	1.24585e21i		−	0.210034i
$801$			4.95602e21i			0.827731i
$802$		−	7.03928e21i		−	1.16472i
$803$	−3.72580e21			−0.610735
$804$		−	1.74639e20i		−	0.0283610i
$805$	4.18868e21			0.673919
$806$	−1.86103e21			−0.296648
$807$	1.15685e20			0.0182695
$808$	−1.28112e22			−2.00450
$809$		−	6.01459e20i		−	0.0932379i	−0.998913	−	0.0466189i	$-0.985155\pi$
$809$		−	6.01459e20i		−	0.0932379i	0.998913	−	0.0466189i	$-0.0148446\pi$
$810$			1.76999e21i			0.271853i
$811$	−4.20103e21			−0.639293			−0.319646	−	0.947537i	$-0.603564\pi$
$811$	−4.20103e21			−0.639293			−0.319646	+	0.947537i	$0.603564\pi$
$812$	3.86221e20	−	9.10427e20i	0.0582326	−	0.137270i
$813$	5.95844e20			0.0890130
$814$		−	6.10044e21i		−	0.902979i
$815$			3.20057e21i			0.469403i
$816$	8.37382e20			0.121688
$817$	1.09562e21			0.157759
$818$	2.69648e21			0.384722
$819$	4.75549e21			0.672305
$820$			5.68477e19i			0.00796359i
$821$	−3.46756e21			−0.481338			−0.240669	−	0.970607i	$-0.577367\pi$
$821$	−3.46756e21			−0.481338			−0.240669	+	0.970607i	$0.577367\pi$
$822$		−	2.67573e21i		−	0.368047i
$823$		−	1.15438e22i		−	1.57343i	−0.617313	−	0.786717i	$-0.711779\pi$
$823$		−	1.15438e22i		−	1.57343i	0.617313	−	0.786717i	$-0.288221\pi$
$824$		−	1.04176e22i		−	1.40706i
$825$	−9.13102e20			−0.122212
$826$		−	8.55121e21i		−	1.13417i
$827$		−	1.12694e22i		−	1.48119i	−0.671953	−	0.740594i	$-0.734544\pi$
$827$		−	1.12694e22i		−	1.48119i	0.671953	−	0.740594i	$-0.265456\pi$
$828$	−1.39544e21			−0.181754
$829$			3.88650e21i			0.501649i	0.968033	+	0.250824i	$0.0807017\pi$
$829$			3.88650e21i			0.501649i	−0.968033	+	0.250824i	$0.919298\pi$
$830$			9.33682e20i			0.119430i
$831$			1.33395e21i			0.169096i
$832$	4.87993e21			0.613040
$833$			2.97265e21i			0.370089i
$834$	−1.93074e20			−0.0238220
$835$	1.90052e21			0.232393
$836$	1.72742e20			0.0209339
$837$	1.97178e21			0.236819
$838$		−	3.54276e21i		−	0.421707i
$839$			5.65290e21i			0.666893i	0.942769	+	0.333447i	$0.108212\pi$
$839$			5.65290e21i			0.666893i	−0.942769	+	0.333447i	$0.891788\pi$
$840$	8.16195e20			0.0954332
$841$	−5.99702e21	−	6.20473e21i	−0.694969	−	0.719040i
$842$	8.51372e21			0.977864
$843$			1.39452e20i			0.0158752i
$844$			8.57749e20i			0.0967814i
$845$	−2.05170e21			−0.229450
$846$	4.53251e21			0.502413
$847$	−6.76169e21			−0.742899
$848$	5.38433e21			0.586358
$849$		−	1.70325e21i		−	0.183853i
$850$	5.05978e21			0.541364
$851$			2.27747e22i			2.41535i
$852$		−	1.04378e20i		−	0.0109727i
$853$		−	7.49535e21i		−	0.781042i	−0.920594	−	0.390521i	$-0.872295\pi$
$853$		−	7.49535e21i		−	0.781042i	0.920594	−	0.390521i	$-0.127705\pi$
$854$	−1.02690e22			−1.06070
$855$			8.62226e20i			0.0882827i
$856$		−	1.49017e22i		−	1.51246i
$857$	1.12200e22			1.12885			0.564424	−	0.825485i	$-0.309098\pi$
$857$	1.12200e22			1.12885			0.564424	+	0.825485i	$0.309098\pi$
$858$		−	7.29790e20i		−	0.0727852i
$859$		−	5.00293e21i		−	0.494624i	−0.968936	−	0.247312i	$-0.920453\pi$
$859$		−	5.00293e21i		−	0.494624i	0.968936	−	0.247312i	$-0.0795473\pi$
$860$		−	2.33445e20i		−	0.0228795i
$861$	5.54554e20			0.0538791
$862$			9.40261e21i			0.905616i
$863$	4.42987e21			0.422971			0.211485	−	0.977381i	$-0.432170\pi$
$863$	4.42987e21			0.422971			0.211485	+	0.977381i	$0.432170\pi$
$864$	−1.05499e21			−0.0998604
$865$	−2.20266e21			−0.206693
$866$	−1.86632e22			−1.73621
$867$			1.35175e21i			0.124668i
$868$			9.15609e20i			0.0837167i
$869$	−1.49605e20			−0.0135612
$870$	2.95844e20	−	6.97382e20i	0.0265869	−	0.0626724i
$871$	6.97147e21			0.621138
$872$			2.39002e21i			0.211119i
$873$			1.90436e22i			1.66779i
$874$	4.77295e21			0.414430
$875$	9.22403e21			0.794073
$876$	2.88302e20			0.0246075
$877$	−2.47786e21			−0.209691			−0.104846	−	0.994489i	$-0.533435\pi$
$877$	−2.47786e21			−0.209691			−0.104846	+	0.994489i	$0.533435\pi$
$878$		−	3.02837e21i		−	0.254097i
$879$	1.47098e21			0.122374
$880$			2.25173e21i			0.185736i
$881$			1.90543e21i			0.155838i	0.996960	+	0.0779191i	$0.0248276\pi$
$881$			1.90543e21i			0.155838i	−0.996960	+	0.0779191i	$0.975172\pi$
$882$			6.28148e21i			0.509387i
$883$	−1.62994e22			−1.31059			−0.655296	−	0.755373i	$-0.727456\pi$
$883$	−1.62994e22			−1.31059			−0.655296	+	0.755373i	$0.727456\pi$
$884$		−	5.46402e20i		−	0.0435632i
$885$			8.85025e20i			0.0699650i
$886$	−3.27221e21			−0.256500
$887$		−	1.32560e22i		−	1.03035i	−0.857085	−	0.515175i	$-0.827727\pi$
$887$		−	1.32560e22i		−	1.03035i	0.857085	−	0.515175i	$-0.172273\pi$
$888$			4.43782e21i			0.342037i
$889$			5.11949e21i			0.391258i
$890$	−3.61189e21			−0.273722
$891$		−	7.32000e21i		−	0.550083i
$892$	−1.79680e21			−0.133894
$893$	2.09468e21			0.154786
$894$	2.23037e21			0.163435
$895$	2.24794e21			0.163347
$896$			1.36536e22i			0.983866i
$897$			2.72452e21i			0.194691i
$898$	−3.22363e21			−0.228440
$899$	7.35473e21	+	3.12003e21i	0.516855	+	0.219261i
$900$	−1.44462e21			−0.100678
$901$		−	6.36327e21i		−	0.439789i
$902$			1.74000e21i			0.119262i
$903$	−2.27728e21			−0.154795
$904$	6.58316e21			0.443784
$905$	7.41306e21			0.495602
$906$	1.80581e21			0.119732
$907$		−	2.10302e21i		−	0.138289i	−0.997607	−	0.0691447i	$-0.977973\pi$
$907$		−	2.10302e21i		−	0.138289i	0.997607	−	0.0691447i	$-0.0220270\pi$
$908$	3.58116e21			0.233550
$909$			2.81303e22i			1.81947i
$910$			3.46575e21i			0.222324i
$911$		−	1.97297e22i		−	1.25526i	−0.778514	−	0.627628i	$-0.784026\pi$
$911$		−	1.97297e22i		−	1.25526i	0.778514	−	0.627628i	$-0.215974\pi$
$912$	8.17750e20			0.0516011
$913$		−	3.86135e21i		−	0.241662i
$914$			2.31652e22i			1.43793i
$915$	1.06281e21			0.0654329
$916$			1.95182e21i			0.119185i
$917$			3.46159e22i			2.09655i
$918$		−	4.28463e21i		−	0.257391i
$919$	−2.24253e22			−1.33620			−0.668100	−	0.744071i	$-0.732892\pi$
$919$	−2.24253e22			−1.33620			−0.668100	+	0.744071i	$0.732892\pi$
$920$		−	9.56076e21i		−	0.565046i
$921$	4.87815e21			0.285962
$922$	−1.43921e22			−0.836837
$923$	4.16671e21			0.240315
$924$	−3.59049e20			−0.0205406
$925$			2.35773e22i			1.33792i
$926$		−	1.33299e22i		−	0.750319i
$927$	−2.28744e22			−1.27718
$928$	−3.93510e21	−	1.66935e21i	−0.217944	−	0.0924564i
$929$	1.96502e22			1.07957			0.539783	−	0.841804i	$-0.318506\pi$
$929$	1.96502e22			1.07957			0.539783	+	0.841804i	$0.318506\pi$
$930$			7.01352e20i			0.0382220i
$931$			2.90295e21i			0.156934i
$932$	1.64126e21			0.0880150
$933$	5.25350e21			0.279471
$934$	1.35346e22			0.714242
$935$	2.66112e21			0.139309
$936$		−	1.08545e22i		−	0.563692i
$937$	−1.43415e22			−0.738833			−0.369417	−	0.929264i	$-0.620442\pi$
$937$	−1.43415e22			−0.738833			−0.369417	+	0.929264i	$0.620442\pi$
$938$			2.53851e22i			1.29735i
$939$		−	3.80642e21i		−	0.192986i
$940$		−	4.46315e20i		−	0.0224483i
$941$	3.79021e22			1.89122			0.945608	−	0.325308i	$-0.105468\pi$
$941$	3.79021e22			1.89122			0.945608	+	0.325308i	$0.105468\pi$
$942$			3.04283e19i			0.00150625i
$943$		−	6.49594e21i		−	0.319010i
$944$	−1.71617e22			−0.836122
$945$		−	3.67198e21i		−	0.177485i
$946$		−	7.14533e21i		−	0.342641i
$947$		−	3.06750e22i		−	1.45935i	−0.683794	−	0.729675i	$-0.739671\pi$
$947$		−	3.06750e22i		−	1.45935i	0.683794	−	0.729675i	$-0.260329\pi$
$948$	1.15764e19			0.000546403
$949$			1.15088e22i			0.538932i
$950$	4.94116e21			0.229563
$951$	5.64929e21			0.260399
$952$	1.87045e22			0.855400
$953$	3.72333e22			1.68941			0.844705	−	0.535232i	$-0.179776\pi$
$953$	3.72333e22			1.68941			0.844705	+	0.535232i	$0.179776\pi$
$954$		−	1.34462e22i		−	0.605322i
$955$			1.19717e22i			0.534727i
$956$	3.66006e21			0.162201
$957$	−1.22349e21	+	2.88410e21i	−0.0537975	+	0.126815i
$958$	−6.07027e21			−0.264830
$959$		−	5.25509e22i		−	2.27479i
$960$		−	1.83906e21i		−	0.0789880i
$961$	1.60687e22			0.684785
$962$	−1.88440e22			−0.796818
$963$	−3.27204e22			−1.37285
$964$	−8.92395e20			−0.0371518
$965$			7.72135e21i			0.318962i
$966$	−9.92072e21			−0.406644
$967$		−	3.17334e22i		−	1.29068i	−0.763896	−	0.645339i	$-0.776716\pi$
$967$		−	3.17334e22i		−	1.29068i	0.763896	−	0.645339i	$-0.223284\pi$
$968$			1.54337e22i			0.622882i
$969$		−	9.66426e20i		−	0.0387026i
$970$	−1.38787e22			−0.551521
$971$		−	4.52472e22i		−	1.78422i	−0.451820	−	0.892109i	$-0.649225\pi$
$971$		−	4.52472e22i		−	1.78422i	0.451820	−	0.892109i	$-0.350775\pi$
$972$			1.84956e21i			0.0723720i
$973$	−3.79195e21			−0.147237
$974$			9.85019e21i			0.379535i
$975$			2.82053e21i			0.107844i
$976$			2.06091e22i			0.781961i
$977$	−2.29301e22			−0.863370			−0.431685	−	0.902025i	$-0.642081\pi$
$977$	−2.29301e22			−0.863370			−0.431685	+	0.902025i	$0.642081\pi$
$978$		−	7.58044e21i		−	0.283239i
$979$	1.49374e22			0.553865
$980$	6.18537e20			0.0227599
$981$	5.24789e21			0.191632
$982$	−7.72593e21			−0.279972
$983$		−	3.39173e20i		−	0.0121975i	−0.999981	−	0.00609874i	$-0.998059\pi$
$983$		−	3.39173e20i		−	0.0121975i	0.999981	−	0.00609874i	$-0.00194130\pi$
$984$		−	1.26578e21i		−	0.0451748i
$985$	3.58533e21			0.126987
$986$	6.77976e21	−	1.59817e22i	0.238307	−	0.561754i
$987$	−4.35384e21			−0.151878
$988$		−	5.33592e20i		−	0.0184728i
$989$			2.66756e22i			0.916519i
$990$	5.62321e21			0.191743
$991$	−1.16590e22			−0.394555			−0.197278	−	0.980348i	$-0.563210\pi$
$991$	−1.16590e22			−0.394555			−0.197278	+	0.980348i	$0.563210\pi$
$992$	3.95750e21			0.132918
$993$	−6.52560e21			−0.217521
$994$			1.51722e22i			0.501938i
$995$	−5.10603e21			−0.167653
$996$			2.98791e20i			0.00973694i
$997$		−	5.33541e22i		−	1.72565i	−0.505499	−	0.862827i	$-0.668691\pi$
$997$		−	5.33541e22i		−	1.72565i	0.505499	−	0.862827i	$-0.331309\pi$
$998$		−	1.86345e21i		−	0.0598189i
$999$	1.99653e22			0.636114

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.26	yes	36
29.28	even	2	inner	29.16.b.a.28.11	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.11	✓	36	29.28	even	2	inner
29.16.b.a.28.26	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)

\(f(q)\)	\(=\)	\(q+169.905i q^{2} +817.971i q^{3} +3900.43 q^{4} +58678.2 q^{5} -138977. q^{6} -2.72949e6 q^{7} +6.23013e6i q^{8} +1.36798e7 q^{9} +O(q^{10})\)	\(q+169.905i q^{2} +817.971i q^{3} +3900.43 q^{4} +58678.2 q^{5} -138977. q^{6} -2.72949e6 q^{7} +6.23013e6i q^{8} +1.36798e7 q^{9} +9.96969e6i q^{10} -4.12308e7i q^{11} +3.19044e6i q^{12} -1.27360e8 q^{13} -4.63753e8i q^{14} +4.79971e7i q^{15} -9.30719e8 q^{16} +1.09993e9i q^{17} +2.32427e9i q^{18} +1.07415e9i q^{19} +2.28870e8 q^{20} -2.23265e9i q^{21} +7.00530e9 q^{22} -2.61528e10 q^{23} -5.09607e9 q^{24} -2.70744e10 q^{25} -2.16390e10i q^{26} +2.29267e10i q^{27} -1.06462e10 q^{28} +(-3.62779e10 + 8.55167e10i) q^{29} -8.15492e9 q^{30} -8.60035e10i q^{31} +4.60156e10i q^{32} +3.37256e10 q^{33} -1.86884e11 q^{34} -1.60162e11 q^{35} +5.33572e10 q^{36} -8.70832e11i q^{37} -1.82503e11 q^{38} -1.04177e11i q^{39} +3.65573e11i q^{40} +2.48384e11i q^{41} +3.79337e11 q^{42} -1.01999e12i q^{43} -1.60818e11i q^{44} +8.02708e11 q^{45} -4.44348e12i q^{46} -1.95008e12i q^{47} -7.61302e11i q^{48} +2.70257e12 q^{49} -4.60007e12i q^{50} -8.99715e11 q^{51} -4.96758e11 q^{52} -5.78513e12 q^{53} -3.89535e12 q^{54} -2.41935e12i q^{55} -1.70051e13i q^{56} -8.78622e11 q^{57} +(-1.45297e13 - 6.16378e12i) q^{58} +1.84391e13 q^{59} +1.87209e11i q^{60} -2.21432e13i q^{61} +1.46124e13 q^{62} -3.73390e13 q^{63} -3.83161e13 q^{64} -7.47325e12 q^{65} +5.73013e12i q^{66} -5.47383e13 q^{67} +4.29022e12i q^{68} -2.13922e13i q^{69} -2.72122e13i q^{70} -3.27160e13 q^{71} +8.52272e13i q^{72} -9.03645e13i q^{73} +1.47958e14 q^{74} -2.21461e13i q^{75} +4.18963e12i q^{76} +1.12539e14i q^{77} +1.77001e13 q^{78} -3.62848e12i q^{79} -5.46129e13 q^{80} +1.77537e14 q^{81} -4.22016e13 q^{82} +9.36521e13 q^{83} -8.70828e12i q^{84} +6.45422e13i q^{85} +1.73301e14 q^{86} +(-6.99502e13 - 2.96743e13i) q^{87} +2.56873e14 q^{88} +3.62287e14i q^{89} +1.36384e14i q^{90} +3.47628e14 q^{91} -1.02007e14 q^{92} +7.03484e13 q^{93} +3.31328e14 q^{94} +6.30290e13i q^{95} -3.76394e13 q^{96} +1.39209e15i q^{97} +4.59178e14i q^{98} -5.64030e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 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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