LMFDB - Embedded newform 29.16.b.a.28.3

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			28.3
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.34

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-317.689i q^{2} -3423.11i q^{3} -68158.0 q^{4} -176108. q^{5} -1.08748e6 q^{6} +98456.3 q^{7} +1.12430e7i q^{8} +2.63125e6 q^{9} +O(q^{10})$	\(q-317.689i q^{2} -3423.11i q^{3} -68158.0 q^{4} -176108. q^{5} -1.08748e6 q^{6} +98456.3 q^{7} +1.12430e7i q^{8} +2.63125e6 q^{9} +5.59476e7i q^{10} +4.71269e7i q^{11} +2.33312e8i q^{12} +3.71888e7 q^{13} -3.12784e7i q^{14} +6.02838e8i q^{15} +1.33837e9 q^{16} -1.54931e7i q^{17} -8.35917e8i q^{18} -2.06467e8i q^{19} +1.20032e10 q^{20} -3.37026e8i q^{21} +1.49717e10 q^{22} -9.21179e8 q^{23} +3.84860e10 q^{24} +4.96573e8 q^{25} -1.18144e10i q^{26} -5.81249e10i q^{27} -6.71059e9 q^{28} +(1.77102e10 + 9.11896e10i) q^{29} +1.91515e11 q^{30} +1.66965e11i q^{31} -5.67748e10i q^{32} +1.61320e11 q^{33} -4.92198e9 q^{34} -1.73390e10 q^{35} -1.79341e11 q^{36} +4.77397e11i q^{37} -6.55922e10 q^{38} -1.27301e11i q^{39} -1.97999e12i q^{40} -1.44307e12i q^{41} -1.07069e11 q^{42} -5.01522e11i q^{43} -3.21208e12i q^{44} -4.63384e11 q^{45} +2.92648e11i q^{46} +1.87213e12i q^{47} -4.58139e12i q^{48} -4.73787e12 q^{49} -1.57756e11i q^{50} -5.30346e10 q^{51} -2.53471e12 q^{52} -4.37851e12 q^{53} -1.84656e13 q^{54} -8.29944e12i q^{55} +1.10694e12i q^{56} -7.06758e11 q^{57} +(2.89699e13 - 5.62632e12i) q^{58} +2.13292e13 q^{59} -4.10882e13i q^{60} -1.53138e13i q^{61} +5.30428e13 q^{62} +2.59063e11 q^{63} +2.58191e13 q^{64} -6.54925e12 q^{65} -5.12497e13i q^{66} -2.28713e13 q^{67} +1.05598e12i q^{68} +3.15329e12i q^{69} +5.50839e12i q^{70} +9.05855e13 q^{71} +2.95831e13i q^{72} +5.94349e13i q^{73} +1.51664e14 q^{74} -1.69982e12i q^{75} +1.40724e13i q^{76} +4.63994e12i q^{77} -4.04421e13 q^{78} +1.36387e14i q^{79} -2.35699e14 q^{80} -1.61212e14 q^{81} -4.58447e14 q^{82} -3.25104e14 q^{83} +2.29711e13i q^{84} +2.72847e12i q^{85} -1.59328e14 q^{86} +(3.12152e14 - 6.06238e13i) q^{87} -5.29848e14 q^{88} +1.75254e14i q^{89} +1.47212e14i q^{90} +3.66147e12 q^{91} +6.27858e13 q^{92} +5.71538e14 q^{93} +5.94754e14 q^{94} +3.63605e13i q^{95} -1.94346e14 q^{96} -6.73699e14i q^{97} +1.50517e15i q^{98} +1.24002e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	317.689i		−	1.75500i	−0.479578	−	0.877499i	$-0.659210\pi$
$2$		−	317.689i		−	1.75500i	0.479578	−	0.877499i	$-0.340790\pi$
$3$		−	3423.11i		−	0.903672i	−0.892101	−	0.451836i	$-0.850769\pi$
$3$		−	3423.11i		−	0.903672i	0.892101	−	0.451836i	$-0.149231\pi$
$4$	−68158.0			−2.08002
$5$	−176108.			−1.00810			−0.504052	−	0.863674i	$-0.668158\pi$
$5$	−176108.			−1.00810			−0.504052	+	0.863674i	$0.668158\pi$
$6$	−1.08748e6			−1.58594
$7$	98456.3			0.0451864			0.0225932	−	0.999745i	$-0.492808\pi$
$7$	98456.3			0.0451864			0.0225932	+	0.999745i	$0.492808\pi$
$8$			1.12430e7i			1.89543i
$9$	2.63125e6			0.183376
$10$			5.59476e7i			1.76922i
$11$			4.71269e7i			0.729162i	0.931172	+	0.364581i	$0.118788\pi$
$11$			4.71269e7i			0.729162i	−0.931172	+	0.364581i	$0.881212\pi$
$12$			2.33312e8i			1.87965i
$13$	3.71888e7			0.164375			0.0821877	−	0.996617i	$-0.473809\pi$
$13$	3.71888e7			0.164375			0.0821877	+	0.996617i	$0.473809\pi$
$14$		−	3.12784e7i		−	0.0793021i
$15$			6.02838e8i			0.910995i
$16$	1.33837e9			1.24646
$17$		−	1.54931e7i		−	0.00915739i	−0.999990	−	0.00457870i	$-0.998543\pi$
$17$		−	1.54931e7i		−	0.00915739i	0.999990	−	0.00457870i	$-0.00145745\pi$
$18$		−	8.35917e8i		−	0.321825i
$19$		−	2.06467e8i		−	0.0529906i	−0.999649	−	0.0264953i	$-0.991565\pi$
$19$		−	2.06467e8i		−	0.0529906i	0.999649	−	0.0264953i	$-0.00843470\pi$
$20$	1.20032e10			2.09687
$21$		−	3.37026e8i		−	0.0408337i
$22$	1.49717e10			1.27968
$23$	−9.21179e8			−0.0564138			−0.0282069	−	0.999602i	$-0.508980\pi$
$23$	−9.21179e8			−0.0564138			−0.0282069	+	0.999602i	$0.508980\pi$
$24$	3.84860e10			1.71285
$25$	4.96573e8			0.0162717
$26$		−	1.18144e10i		−	0.288478i
$27$		−	5.81249e10i		−	1.06938i
$28$	−6.71059e9			−0.0939886
$29$	1.77102e10	+	9.11896e10i	0.190650	+	0.981658i
$29$	1.77102e10	+	9.11896e10i	0.190650	+	0.981658i
$30$	1.91515e11			1.59879
$31$			1.66965e11i			1.08996i	0.838448	+	0.544982i	$0.183464\pi$
$31$			1.66965e11i			1.08996i	−0.838448	+	0.544982i	$0.816536\pi$
$32$		−	5.67748e10i		−	0.292099i
$33$	1.61320e11			0.658923
$34$	−4.92198e9			−0.0160712
$35$	−1.73390e10			−0.0455526
$36$	−1.79341e11			−0.381425
$37$			4.77397e11i			0.826736i	0.910564	+	0.413368i	$0.135648\pi$
$37$			4.77397e11i			0.826736i	−0.910564	+	0.413368i	$0.864352\pi$
$38$	−6.55922e10			−0.0929983
$39$		−	1.27301e11i		−	0.148541i
$40$		−	1.97999e12i		−	1.91079i
$41$		−	1.44307e12i		−	1.15720i	−0.815611	−	0.578601i	$-0.803599\pi$
$41$		−	1.44307e12i		−	1.15720i	0.815611	−	0.578601i	$-0.196401\pi$
$42$	−1.07069e11			−0.0716631
$43$		−	5.01522e11i		−	0.281369i	−0.990054	−	0.140685i	$-0.955070\pi$
$43$		−	5.01522e11i		−	0.281369i	0.990054	−	0.140685i	$-0.0449303\pi$
$44$		−	3.21208e12i		−	1.51667i
$45$	−4.63384e11			−0.184862
$46$			2.92648e11i			0.0990061i
$47$			1.87213e12i			0.539016i	0.962998	+	0.269508i	$0.0868611\pi$
$47$			1.87213e12i			0.539016i	−0.962998	+	0.269508i	$0.913139\pi$
$48$		−	4.58139e12i		−	1.12639i
$49$	−4.73787e12			−0.997958
$50$		−	1.57756e11i		−	0.0285568i
$51$	−5.30346e10			−0.00827528
$52$	−2.53471e12			−0.341904
$53$	−4.37851e12			−0.511985			−0.255992	−	0.966679i	$-0.582402\pi$
$53$	−4.37851e12			−0.511985			−0.255992	+	0.966679i	$0.582402\pi$
$54$	−1.84656e13			−1.87677
$55$		−	8.29944e12i		−	0.735070i
$56$			1.10694e12i			0.0856477i
$57$	−7.06758e11			−0.0478861
$58$	2.89699e13	−	5.62632e12i	1.72281	−	0.334591i
$59$	2.13292e13			1.11579			0.557897	−	0.829910i	$-0.311608\pi$
$59$	2.13292e13			1.11579			0.557897	+	0.829910i	$0.311608\pi$
$60$		−	4.10882e13i		−	1.89489i
$61$		−	1.53138e13i		−	0.623890i	−0.950100	−	0.311945i	$-0.899020\pi$
$61$		−	1.53138e13i		−	0.623890i	0.950100	−	0.311945i	$-0.100980\pi$
$62$	5.30428e13			1.91288
$63$	2.59063e11			0.00828611
$64$	2.58191e13			0.733823
$65$	−6.54925e12			−0.165707
$66$		−	5.12497e13i		−	1.15641i
$67$	−2.28713e13			−0.461030			−0.230515	−	0.973069i	$-0.574041\pi$
$67$	−2.28713e13			−0.461030			−0.230515	+	0.973069i	$0.574041\pi$
$68$			1.05598e12i			0.0190475i
$69$			3.15329e12i			0.0509796i
$70$			5.50839e12i			0.0799447i
$71$	9.05855e13			1.18201			0.591005	−	0.806668i	$-0.298731\pi$
$71$	9.05855e13			1.18201			0.591005	+	0.806668i	$0.298731\pi$
$72$			2.95831e13i			0.347576i
$73$			5.94349e13i			0.629681i	0.949145	+	0.314840i	$0.101951\pi$
$73$			5.94349e13i			0.629681i	−0.949145	+	0.314840i	$0.898049\pi$
$74$	1.51664e14			1.45092
$75$		−	1.69982e12i		−	0.0147043i
$76$			1.40724e13i			0.110221i
$77$			4.63994e12i			0.0329482i
$78$	−4.04421e13			−0.260690
$79$			1.36387e14i			0.799043i	0.916724	+	0.399522i	$0.130824\pi$
$79$			1.36387e14i			0.799043i	−0.916724	+	0.399522i	$0.869176\pi$
$80$	−2.35699e14			−1.25656
$81$	−1.61212e14			−0.782997
$82$	−4.58447e14			−2.03089
$83$	−3.25104e14			−1.31503			−0.657517	−	0.753440i	$-0.728393\pi$
$83$	−3.25104e14			−1.31503			−0.657517	+	0.753440i	$0.728393\pi$
$84$			2.29711e13i			0.0849349i
$85$			2.72847e12i			0.00923159i
$86$	−1.59328e14			−0.493802
$87$	3.12152e14	−	6.06238e13i	0.887097	−	0.172286i
$88$	−5.29848e14			−1.38207
$89$			1.75254e14i			0.419994i	0.977702	+	0.209997i	$0.0673454\pi$
$89$			1.75254e14i			0.419994i	−0.977702	+	0.209997i	$0.932655\pi$
$90$			1.47212e14i			0.324432i
$91$	3.66147e12			0.00742753
$92$	6.27858e13			0.117342
$93$	5.71538e14			0.984971
$94$	5.94754e14			0.945973
$95$			3.63605e13i			0.0534199i
$96$	−1.94346e14			−0.263962
$97$		−	6.73699e14i		−	0.846599i	−0.905990	−	0.423299i	$-0.860872\pi$
$97$		−	6.73699e14i		−	0.846599i	0.905990	−	0.423299i	$-0.139128\pi$
$98$			1.50517e15i			1.75141i
$99$			1.24002e14i			0.133711i
$100$	−3.38454e13			−0.0338454
$101$			7.78671e14i			0.722676i	0.932435	+	0.361338i	$0.117680\pi$
$101$			7.78671e14i			0.722676i	−0.932435	+	0.361338i	$0.882320\pi$
$102$			1.68485e13i			0.0145231i
$103$	1.58156e15			1.26709			0.633543	−	0.773708i	$-0.281600\pi$
$103$	1.58156e15			1.26709			0.633543	+	0.773708i	$0.281600\pi$
$104$			4.18113e14i			0.311562i
$105$			5.93532e13i			0.0411646i
$106$			1.39100e15i			0.898532i
$107$	1.34855e15			0.811873			0.405936	−	0.913901i	$-0.366945\pi$
$107$	1.34855e15			0.811873			0.405936	+	0.913901i	$0.366945\pi$
$108$			3.96168e15i			2.22434i
$109$	3.33596e15			1.74792			0.873961	−	0.485996i	$-0.161543\pi$
$109$	3.33596e15			1.74792			0.873961	+	0.485996i	$0.161543\pi$
$110$	−2.63664e15			−1.29005
$111$	1.63418e15			0.747099
$112$	1.31771e14			0.0563229
$113$		−	1.75666e15i		−	0.702425i	−0.936296	−	0.351212i	$-0.885770\pi$
$113$		−	1.75666e15i		−	0.702425i	0.936296	−	0.351212i	$-0.114230\pi$
$114$			2.24529e14i			0.0840400i
$115$	1.62227e14			0.0568709
$116$	−1.20709e15	−	6.21530e15i	−0.396556	−	2.04187i
$117$	9.78528e13			0.0301425
$118$		−	6.77604e15i		−	1.95822i
$119$		−	1.52539e12i		−	0.000413790i
$120$	−6.77771e15			−1.72673
$121$	1.95630e15			0.468323
$122$	−4.86500e15			−1.09493
$123$	−4.93979e15			−1.04573
$124$		−	1.13800e16i		−	2.26714i
$125$	5.28695e15			0.991699
$126$		−	8.23012e13i		−	0.0145421i
$127$		−	1.54425e15i		−	0.257151i	−0.991700	−	0.128576i	$-0.958960\pi$
$127$		−	1.54425e15i		−	0.257151i	0.991700	−	0.128576i	$-0.0410405\pi$
$128$		−	1.00628e16i		−	1.57996i
$129$	−1.71676e15			−0.254266
$130$			2.08062e15i			0.290816i
$131$			8.15145e15i			1.07572i	0.843034	+	0.537861i	$0.180767\pi$
$131$			8.15145e15i			1.07572i	−0.843034	+	0.537861i	$0.819233\pi$
$132$	−1.09953e16			−1.37057
$133$		−	2.03280e13i		−	0.00239445i
$134$			7.26595e15i			0.809107i
$135$			1.02363e16i			1.07805i
$136$	1.74189e14			0.0173572
$137$			1.67168e16i			1.57670i	0.615226	+	0.788351i	$0.289065\pi$
$137$			1.67168e16i			1.57670i	−0.615226	+	0.788351i	$0.710935\pi$
$138$	1.00177e15			0.0894691
$139$	1.20333e16			1.01806			0.509031	−	0.860748i	$-0.330004\pi$
$139$	1.20333e16			1.01806			0.509031	+	0.860748i	$0.330004\pi$
$140$	1.18179e15			0.0947502
$141$	6.40850e15			0.487094
$142$		−	2.87780e16i		−	2.07442i
$143$			1.75259e15i			0.119856i
$144$	3.52159e15			0.228570
$145$	−3.11891e15	−	1.60592e16i	−0.192195	−	0.989612i
$146$	1.88818e16			1.10509
$147$			1.62182e16i			0.901827i
$148$		−	3.25384e16i		−	1.71963i
$149$	3.80367e15			0.191120			0.0955598	−	0.995424i	$-0.469536\pi$
$149$	3.80367e15			0.191120			0.0955598	+	0.995424i	$0.469536\pi$
$150$	−5.40014e14			−0.0258060
$151$	6.20108e15			0.281929			0.140965	−	0.990015i	$-0.454980\pi$
$151$	6.20108e15			0.281929			0.140965	+	0.990015i	$0.454980\pi$
$152$	2.32131e15			0.100440
$153$		−	4.07662e13i		−	0.00167925i
$154$	1.47406e15			0.0578240
$155$		−	2.94039e16i		−	1.09880i
$156$			8.67659e15i			0.308969i
$157$		−	1.54095e16i		−	0.523046i	−0.965197	−	0.261523i	$-0.915775\pi$
$157$		−	1.54095e16i		−	0.523046i	0.965197	−	0.261523i	$-0.0842248\pi$
$158$	4.33286e16			1.40232
$159$			1.49881e16i			0.462666i
$160$			9.99852e15i			0.294466i
$161$	−9.06959e13			−0.00254914
$162$			5.12153e16i			1.37416i
$163$			2.80718e16i			0.719221i	0.933102	+	0.359611i	$0.117090\pi$
$163$			2.80718e16i			0.719221i	−0.933102	+	0.359611i	$0.882910\pi$
$164$			9.83569e16i			2.40700i
$165$	−2.84099e16			−0.664262
$166$			1.03282e17i			2.30788i
$167$	−8.73276e16			−1.86543			−0.932713	−	0.360619i	$-0.882566\pi$
$167$	−8.73276e16			−1.86543			−0.932713	+	0.360619i	$0.882566\pi$
$168$	3.78919e15			0.0773975
$169$	−4.98029e16			−0.972981
$170$	8.66803e14			0.0162014
$171$		−	5.43265e14i		−	0.00971720i
$172$			3.41827e16i			0.585253i
$173$	−2.22166e16			−0.364193			−0.182097	−	0.983281i	$-0.558288\pi$
$173$	−2.22166e16			−0.364193			−0.182097	+	0.983281i	$0.558288\pi$
$174$	−1.92595e16	−	9.91670e16i	−0.302361	−	1.55685i
$175$	4.88907e13			0.000735260
$176$			6.30734e16i			0.908868i
$177$		−	7.30121e16i		−	1.00831i
$178$	5.56762e16			0.737089
$179$	−1.26067e17			−1.60031			−0.800156	−	0.599792i	$-0.795250\pi$
$179$	−1.26067e17			−1.60031			−0.800156	+	0.599792i	$0.795250\pi$
$180$	3.15834e16			0.384516
$181$	1.20109e17			1.40276			0.701382	−	0.712786i	$-0.252567\pi$
$181$	1.20109e17			1.40276			0.701382	+	0.712786i	$0.252567\pi$
$182$		−	1.16321e15i		−	0.0130353i
$183$	−5.24206e16			−0.563792
$184$		−	1.03568e16i		−	0.106928i
$185$		−	8.40736e16i		−	0.833435i
$186$		−	1.81571e17i		−	1.72862i
$187$	7.30142e14			0.00667722
$188$		−	1.27601e17i		−	1.12116i
$189$		−	5.72276e15i		−	0.0483217i
$190$	1.15513e16			0.0937519
$191$			1.92286e17i			1.50036i	0.661231	+	0.750182i	$0.270034\pi$
$191$			1.92286e17i			1.50036i	−0.661231	+	0.750182i	$0.729966\pi$
$192$		−	8.83815e16i		−	0.663135i
$193$			1.75354e17i			1.26542i	0.774389	+	0.632710i	$0.218057\pi$
$193$			1.75354e17i			1.26542i	−0.774389	+	0.632710i	$0.781943\pi$
$194$	−2.14026e17			−1.48578
$195$			2.24188e16i			0.149745i
$196$	3.22924e17			2.07577
$197$	−2.63310e17			−1.62919			−0.814594	−	0.580031i	$-0.803040\pi$
$197$	−2.63310e17			−1.62919			−0.814594	+	0.580031i	$0.803040\pi$
$198$	3.93942e16			0.234662
$199$	4.54547e15			0.0260724			0.0130362	−	0.999915i	$-0.495850\pi$
$199$	4.54547e15			0.0260724			0.0130362	+	0.999915i	$0.495850\pi$
$200$			5.58297e15i			0.0308419i
$201$			7.82909e16i			0.416620i
$202$	2.47375e17			1.26829
$203$	1.74368e15	+	8.97819e15i	0.00861481	+	0.0443576i
$204$	3.61473e15			0.0172127
$205$			2.54137e17i			1.16658i
$206$		−	5.02443e17i		−	2.22373i
$207$	−2.42385e15			−0.0103449
$208$	4.97724e16			0.204887
$209$	9.73015e15			0.0386387
$210$	1.88558e16			0.0722438
$211$			4.76214e17i			1.76069i	0.474332	+	0.880346i	$0.342690\pi$
$211$			4.76214e17i			1.76069i	−0.474332	+	0.880346i	$0.657310\pi$
$212$	2.98430e17			1.06494
$213$		−	3.10084e17i		−	1.06815i
$214$		−	4.28418e17i		−	1.42484i
$215$			8.83222e16i			0.283649i
$216$	6.53498e17			2.02694
$217$			1.64387e16i			0.0492516i
$218$		−	1.05980e18i		−	3.06760i
$219$	2.03452e17			0.569025
$220$			5.65674e17i			1.52896i
$221$		−	5.76170e14i		−	0.00150525i
$222$		−	5.19161e17i		−	1.31116i
$223$	3.51478e16			0.0858247			0.0429123	−	0.999079i	$-0.486336\pi$
$223$	3.51478e16			0.0858247			0.0429123	+	0.999079i	$0.486336\pi$
$224$		−	5.58984e15i		−	0.0131989i
$225$	1.30661e15			0.00298384
$226$	−5.58071e17			−1.23275
$227$	−4.37279e17			−0.934470			−0.467235	−	0.884133i	$-0.654750\pi$
$227$	−4.37279e17			−0.934470			−0.467235	+	0.884133i	$0.654750\pi$
$228$	4.81713e16			0.0996040
$229$			3.24227e17i			0.648758i	0.945927	+	0.324379i	$0.105155\pi$
$229$			3.24227e17i			0.648758i	−0.945927	+	0.324379i	$0.894845\pi$
$230$		−	5.15378e16i		−	0.0998084i
$231$	1.58830e16			0.0297744
$232$	−1.02525e18	+	1.99116e17i	−1.86066	+	0.361364i
$233$	3.01861e17			0.530443			0.265221	−	0.964188i	$-0.414555\pi$
$233$	3.01861e17			0.530443			0.265221	+	0.964188i	$0.414555\pi$
$234$		−	3.10867e16i		−	0.0529000i
$235$		−	3.29698e17i		−	0.543384i
$236$	−1.45376e18			−2.32087
$237$	4.66868e17			0.722073
$238$	−4.84600e14			−0.000726200
$239$	−3.18843e17			−0.463012			−0.231506	−	0.972833i	$-0.574365\pi$
$239$	−3.18843e17			−0.463012			−0.231506	+	0.972833i	$0.574365\pi$
$240$			8.06821e17i			1.13552i
$241$	8.46057e17			1.15417			0.577087	−	0.816682i	$-0.304189\pi$
$241$	8.46057e17			1.15417			0.577087	+	0.816682i	$0.304189\pi$
$242$		−	6.21495e17i		−	0.821907i
$243$		−	2.82182e17i		−	0.361811i
$244$			1.04376e18i			1.29770i
$245$	8.34378e17			1.00604
$246$			1.56931e18i			1.83526i
$247$		−	7.67825e15i		−	0.00871034i
$248$	−1.87719e18			−2.06595
$249$			1.11287e18i			1.18836i
$250$		−	1.67960e18i		−	1.74043i
$251$			7.34384e17i			0.738533i	0.929323	+	0.369267i	$0.120391\pi$
$251$			7.34384e17i			0.738533i	−0.929323	+	0.369267i	$0.879609\pi$
$252$	−1.76572e16			−0.0172353
$253$		−	4.34123e16i		−	0.0411348i
$254$	−4.90589e17			−0.451300
$255$	9.33983e15			0.00834234
$256$	−2.35080e18			−2.03900
$257$	−2.71509e17			−0.228710			−0.114355	−	0.993440i	$-0.536480\pi$
$257$	−2.71509e17			−0.228710			−0.114355	+	0.993440i	$0.536480\pi$
$258$			5.45396e17i			0.446236i
$259$			4.70027e16i			0.0373572i
$260$	4.46384e17			0.344674
$261$	4.65998e16	+	2.39942e17i	0.0349607	+	0.180013i
$262$	2.58962e18			1.88789
$263$		−	5.93521e17i		−	0.420502i	−0.977647	−	0.210251i	$-0.932572\pi$
$263$		−	5.93521e17i		−	0.420502i	0.977647	−	0.210251i	$-0.0674281\pi$
$264$			1.81373e18i			1.24894i
$265$	7.71091e17			0.516133
$266$	−6.45796e15			−0.00420226
$267$	5.99913e17			0.379537
$268$	1.55886e18			0.958951
$269$			1.70153e18i			1.01788i	0.860802	+	0.508940i	$0.169962\pi$
$269$			1.70153e18i			1.01788i	−0.860802	+	0.508940i	$0.830038\pi$
$270$	3.25195e18			1.89197
$271$			5.57804e17i			0.315655i	0.987467	+	0.157827i	$0.0504489\pi$
$271$			5.57804e17i			0.315655i	−0.987467	+	0.157827i	$0.949551\pi$
$272$		−	2.07356e16i		−	0.0114143i
$273$		−	1.25336e16i		−	0.00671206i
$274$	5.31074e18			2.76711
$275$			2.34020e16i			0.0118647i
$276$		−	2.14922e17i		−	0.106038i
$277$	2.16955e18			1.04177			0.520885	−	0.853627i	$-0.325602\pi$
$277$	2.16955e18			1.04177			0.520885	+	0.853627i	$0.325602\pi$
$278$		−	3.82284e18i		−	1.78670i
$279$			4.39325e17i			0.199873i
$280$		−	1.94942e17i		−	0.0863417i
$281$	−4.19297e18			−1.80811			−0.904055	−	0.427417i	$-0.859424\pi$
$281$	−4.19297e18			−1.80811			−0.904055	+	0.427417i	$0.859424\pi$
$282$		−	2.03591e18i		−	0.854849i
$283$	−1.16121e18			−0.474803			−0.237402	−	0.971412i	$-0.576296\pi$
$283$	−1.16121e18			−0.474803			−0.237402	+	0.971412i	$0.576296\pi$
$284$	−6.17413e18			−2.45860
$285$	1.24466e17			0.0482741
$286$	5.56778e17			0.210347
$287$		−	1.42079e17i		−	0.0522898i
$288$		−	1.49388e17i		−	0.0535641i
$289$	2.86218e18			0.999916
$290$	−5.10184e18	+	9.90842e17i	−1.73677	+	0.337302i
$291$	−2.30614e18			−0.765048
$292$		−	4.05097e18i		−	1.30975i
$293$			5.62865e18i			1.77377i	0.461990	+	0.886885i	$0.347135\pi$
$293$			5.62865e18i			1.77377i	−0.461990	+	0.886885i	$0.652865\pi$
$294$	5.15235e18			1.58271
$295$	−3.75625e18			−1.12484
$296$	−5.36738e18			−1.56702
$297$	2.73925e18			0.779754
$298$		−	1.20838e18i		−	0.335415i
$299$	−3.42575e16			−0.00927304
$300$			1.15857e17i			0.0305852i
$301$		−	4.93780e16i		−	0.0127141i
$302$		−	1.97001e18i		−	0.494785i
$303$	2.66547e18			0.653062
$304$		−	2.76330e17i		−	0.0660504i
$305$			2.69688e18i			0.628945i
$306$	−1.29510e16			−0.00294707
$307$		−	5.90607e18i		−	1.31148i	−0.754988	−	0.655738i	$-0.772357\pi$
$307$		−	5.90607e18i		−	1.31148i	0.754988	−	0.655738i	$-0.227643\pi$
$308$		−	3.16249e17i		−	0.0685329i
$309$		−	5.41384e18i		−	1.14503i
$310$	−9.34128e18			−1.92839
$311$			2.20195e17i			0.0443716i	0.999754	+	0.0221858i	$0.00706253\pi$
$311$			2.20195e17i			0.0443716i	−0.999754	+	0.0221858i	$0.992937\pi$
$312$	1.43125e18			0.281550
$313$	−3.72724e18			−0.715821			−0.357911	−	0.933756i	$-0.616511\pi$
$313$	−3.72724e18			−0.715821			−0.357911	+	0.933756i	$0.616511\pi$
$314$	−4.89541e18			−0.917945
$315$	−4.56231e16			−0.00835325
$316$		−	9.29588e18i		−	1.66202i
$317$		−	4.56969e18i		−	0.797889i	−0.916975	−	0.398945i	$-0.869377\pi$
$317$		−	4.56969e18i		−	0.797889i	0.916975	−	0.398945i	$-0.130623\pi$
$318$	4.76155e18			0.811979
$319$	−4.29748e18	+	8.34626e17i	−0.715787	+	0.139015i
$320$	−4.54696e18			−0.739769
$321$		−	4.61622e18i		−	0.733667i
$322$			2.88130e16i			0.00447373i
$323$	−3.19882e15			−0.000485255
$324$	1.09879e19			1.62865
$325$	1.84669e16			0.00267467
$326$	8.91807e18			1.26223
$327$		−	1.14193e19i		−	1.57955i
$328$	1.62245e19			2.19339
$329$			1.84323e17i			0.0243562i
$330$			9.02549e18i			1.16578i
$331$		−	1.18185e18i		−	0.149229i	−0.997212	−	0.0746143i	$-0.976227\pi$
$331$		−	1.18185e18i		−	0.149229i	0.997212	−	0.0746143i	$-0.0237726\pi$
$332$	2.21585e19			2.73529
$333$			1.25615e18i			0.151604i
$334$			2.77430e19i			3.27382i
$335$	4.02782e18			0.464766
$336$		−	4.51067e17i		−	0.0508975i
$337$			5.63607e18i			0.621945i	0.950419	+	0.310972i	$0.100655\pi$
$337$			5.63607e18i			0.621945i	−0.950419	+	0.310972i	$0.899345\pi$
$338$			1.58218e19i			1.70758i
$339$	−6.01324e18			−0.634762
$340$		−	1.85967e17i		−	0.0192019i
$341$	−7.86853e18			−0.794760
$342$	−1.72589e17			−0.0170537
$343$	−9.33900e17			−0.0902806
$344$	5.63861e18			0.533315
$345$		−	5.55322e17i		−	0.0513927i
$346$			7.05796e18i			0.639158i
$347$	6.33473e18			0.561381			0.280690	−	0.959798i	$-0.409437\pi$
$347$	6.33473e18			0.561381			0.280690	+	0.959798i	$0.409437\pi$
$348$	−2.12756e19	+	4.13200e18i	−1.84518	+	0.358357i
$349$	−8.25316e18			−0.700534			−0.350267	−	0.936650i	$-0.613909\pi$
$349$	−8.25316e18			−0.700534			−0.350267	+	0.936650i	$0.613909\pi$
$350$		−	1.55320e16i		−	0.00129038i
$351$		−	2.16159e18i		−	0.175780i
$352$	2.67562e18			0.212988
$353$	2.19722e19			1.71223			0.856116	−	0.516784i	$-0.172871\pi$
$353$	2.19722e19			1.71223			0.856116	+	0.516784i	$0.172871\pi$
$354$	−2.31951e19			−1.76959
$355$	−1.59529e19			−1.19159
$356$		−	1.19450e19i		−	0.873595i
$357$	−5.22159e15			−0.000373931
$358$			4.00501e19i			2.80854i
$359$			1.81219e19i			1.24450i	0.782817	+	0.622252i	$0.213782\pi$
$359$			1.81219e19i			1.24450i	−0.782817	+	0.622252i	$0.786218\pi$
$360$		−	5.20983e18i		−	0.350393i
$361$	1.51385e19			0.997192
$362$		−	3.81571e19i		−	2.46185i
$363$		−	6.69664e18i		−	0.423211i
$364$	−2.49558e17			−0.0154494
$365$		−	1.04670e19i		−	0.634783i
$366$			1.66534e19i			0.989454i
$367$		−	1.93663e19i		−	1.12733i	−0.826003	−	0.563666i	$-0.809390\pi$
$367$		−	1.93663e19i		−	1.12733i	0.826003	−	0.563666i	$-0.190610\pi$
$368$	−1.23288e18			−0.0703174
$369$		−	3.79708e18i		−	0.212203i
$370$	−2.67092e19			−1.46268
$371$	−4.31091e17			−0.0231348
$372$	−3.89549e19			−2.04876
$373$	−1.16110e18			−0.0598484			−0.0299242	−	0.999552i	$-0.509527\pi$
$373$	−1.16110e18			−0.0598484			−0.0299242	+	0.999552i	$0.509527\pi$
$374$		−	2.31958e17i		−	0.0117185i
$375$		−	1.80978e19i		−	0.896172i
$376$	−2.10484e19			−1.02167
$377$	6.58619e17	+	3.39123e18i	0.0313382	+	0.161360i
$378$	−1.81806e18			−0.0848044
$379$			2.61049e19i			1.19379i	0.802320	+	0.596895i	$0.203599\pi$
$379$			2.61049e19i			1.19379i	−0.802320	+	0.596895i	$0.796401\pi$
$380$		−	2.47826e18i		−	0.111114i
$381$	−5.28612e18			−0.232380
$382$	6.10870e19			2.63314
$383$	−3.51851e19			−1.48720			−0.743598	−	0.668627i	$-0.766882\pi$
$383$	−3.51851e19			−1.48720			−0.743598	+	0.668627i	$0.766882\pi$
$384$	−3.44461e19			−1.42776
$385$		−	8.17132e17i		−	0.0332152i
$386$	5.57078e19			2.22081
$387$		−	1.31963e18i		−	0.0515964i
$388$			4.59180e19i			1.76094i
$389$		−	2.18901e18i		−	0.0823429i	−0.999152	−	0.0411714i	$-0.986891\pi$
$389$		−	2.18901e18i		−	0.0823429i	0.999152	−	0.0411714i	$-0.0131090\pi$
$390$	7.12219e18			0.262802
$391$			1.42719e16i			0.000516603i
$392$		−	5.32679e19i		−	1.89156i
$393$	2.79033e19			0.972100
$394$			8.36507e19i			2.85922i
$395$		−	2.40189e19i		−	0.805518i
$396$		−	8.45176e18i		−	0.278121i
$397$	−2.66254e19			−0.859739			−0.429870	−	0.902891i	$-0.641440\pi$
$397$	−2.66254e19			−0.859739			−0.429870	+	0.902891i	$0.641440\pi$
$398$		−	1.44404e18i		−	0.0457569i
$399$	−6.95848e16			−0.00216380
$400$	6.64600e17			0.0202820
$401$	4.86962e18			0.145852			0.0729260	−	0.997337i	$-0.476766\pi$
$401$	4.86962e18			0.145852			0.0729260	+	0.997337i	$0.476766\pi$
$402$	2.48721e19			0.731168
$403$			6.20921e18i			0.179163i
$404$		−	5.30727e19i		−	1.50318i
$405$	2.83908e19			0.789342
$406$	2.85227e18	−	5.53946e17i	0.0778475	−	0.0151190i
$407$	−2.24982e19			−0.602824
$408$		−	5.96268e17i		−	0.0156852i
$409$		−	4.17175e19i		−	1.07744i	−0.842484	−	0.538721i	$-0.818908\pi$
$409$		−	4.17175e19i		−	1.07744i	0.842484	−	0.538721i	$-0.181092\pi$
$410$	8.07364e19			2.04734
$411$	5.72235e19			1.42482
$412$	−1.07796e20			−2.63556
$413$	2.09999e18			0.0504188
$414$			7.70029e17i			0.0181553i
$415$	5.72536e19			1.32569
$416$		−	2.11138e18i		−	0.0480139i
$417$		−	4.11913e19i		−	0.919994i
$418$		−	3.09116e18i		−	0.0678108i
$419$	1.95325e19			0.420874			0.210437	−	0.977607i	$-0.432511\pi$
$419$	1.95325e19			0.420874			0.210437	+	0.977607i	$0.432511\pi$
$420$		−	4.04539e18i		−	0.0856231i
$421$		−	1.98401e19i		−	0.412505i	−0.978499	−	0.206252i	$-0.933873\pi$
$421$		−	1.98401e19i		−	0.412505i	0.978499	−	0.206252i	$-0.0661268\pi$
$422$	1.51288e20			3.09001
$423$			4.92603e18i			0.0988427i
$424$		−	4.92276e19i		−	0.970431i
$425$		−	7.69346e15i		−	0.000149006i
$426$	−9.85101e19			−1.87460
$427$		−	1.50773e18i		−	0.0281914i
$428$	−9.19144e19			−1.68871
$429$	5.99931e18			0.108311
$430$	2.80590e19			0.497804
$431$	4.69287e19			0.818200			0.409100	−	0.912490i	$-0.365843\pi$
$431$	4.69287e19			0.818200			0.409100	+	0.912490i	$0.365843\pi$
$432$		−	7.77927e19i		−	1.33294i
$433$			2.74687e19i			0.462571i	0.972886	+	0.231286i	$0.0742932\pi$
$433$			2.74687e19i			0.462571i	−0.972886	+	0.231286i	$0.925707\pi$
$434$	5.22240e18			0.0864364
$435$	−5.49725e19	+	1.06764e19i	−0.894285	+	0.173682i
$436$	−2.27372e20			−3.63571
$437$			1.90193e17i			0.00298940i
$438$		−	6.46344e19i		−	0.998638i
$439$	5.63371e19			0.855679			0.427839	−	0.903855i	$-0.359275\pi$
$439$	5.63371e19			0.855679			0.427839	+	0.903855i	$0.359275\pi$
$440$	9.33107e19			1.39327
$441$	−1.24665e19			−0.183002
$442$	−1.83043e17			−0.00264171
$443$			1.18590e20i			1.68275i	0.540452	+	0.841375i	$0.318253\pi$
$443$			1.18590e20i			1.68275i	−0.540452	+	0.841375i	$0.681747\pi$
$444$	−1.11383e20			−1.55398
$445$		−	3.08637e19i		−	0.423397i
$446$		−	1.11661e19i		−	0.150622i
$447$		−	1.30204e19i		−	0.172710i
$448$	2.54205e18			0.0331588
$449$			8.27629e19i			1.06167i	0.847476	+	0.530833i	$0.178121\pi$
$449$			8.27629e19i			1.06167i	−0.847476	+	0.530833i	$0.821879\pi$
$450$		−	4.15094e17i		−	0.00523664i
$451$	6.80075e19			0.843787
$452$			1.19731e20i			1.46106i
$453$		−	2.12269e19i		−	0.254772i
$454$			1.38919e20i			1.63999i
$455$	−6.44815e17			−0.00748772
$456$		−	7.94609e18i		−	0.0907647i
$457$	−1.04561e20			−1.17489			−0.587447	−	0.809262i	$-0.699867\pi$
$457$	−1.04561e20			−1.17489			−0.587447	+	0.809262i	$0.699867\pi$
$458$	1.03003e20			1.13857
$459$	−9.00535e17			−0.00979277
$460$	−1.10571e19			−0.118293
$461$		−	1.20828e19i		−	0.127178i	−0.997976	−	0.0635888i	$-0.979745\pi$
$461$		−	1.20828e19i		−	0.127178i	0.997976	−	0.0635888i	$-0.0202546\pi$
$462$		−	5.04585e18i		−	0.0522540i
$463$	−3.74198e19			−0.381281			−0.190640	−	0.981660i	$-0.561056\pi$
$463$	−3.74198e19			−0.381281			−0.190640	+	0.981660i	$0.561056\pi$
$464$	2.37028e19	+	1.22046e20i	0.237637	+	1.22359i
$465$	−1.00653e20			−0.992952
$466$		−	9.58979e19i		−	0.930926i
$467$			1.85673e19i			0.177366i	0.996060	+	0.0886832i	$0.0282659\pi$
$467$			1.85673e19i			0.177366i	−0.996060	+	0.0886832i	$0.971734\pi$
$468$	−6.66945e18			−0.0626969
$469$	−2.25182e18			−0.0208323
$470$	−1.04741e20			−0.953638
$471$	−5.27482e19			−0.472663
$472$			2.39804e20i			2.11491i
$473$	2.36352e19			0.205164
$474$		−	1.48319e20i		−	1.26724i
$475$		−	1.02526e17i		−	0.000862247i
$476$			1.03968e17i			0.000860690i
$477$	−1.15209e19			−0.0938857
$478$			1.01293e20i			0.812586i
$479$			2.79797e19i			0.220967i	0.993878	+	0.110483i	$0.0352399\pi$
$479$			2.79797e19i			0.220967i	−0.993878	+	0.110483i	$0.964760\pi$
$480$	3.42260e19			0.266101
$481$			1.77538e19i			0.135895i
$482$		−	2.68782e20i		−	2.02557i
$483$			3.10462e17i			0.00230359i
$484$	−1.33338e20			−0.974121
$485$			1.18644e20i			0.853459i
$486$	−8.96460e19			−0.634978
$487$	2.12938e20			1.48520			0.742602	−	0.669733i	$-0.233591\pi$
$487$	2.12938e20			1.48520			0.742602	+	0.669733i	$0.233591\pi$
$488$	1.72173e20			1.18254
$489$	9.60926e19			0.649940
$490$		−	2.65072e20i		−	1.76561i
$491$		−	1.10705e20i		−	0.726200i	−0.931750	−	0.363100i	$-0.881718\pi$
$491$		−	1.10705e20i		−	0.726200i	0.931750	−	0.363100i	$-0.118282\pi$
$492$	3.36686e20			2.17514
$493$	1.41281e18	−	2.74386e17i	0.00898943	−	0.00174586i
$494$	−2.43929e18			−0.0152866
$495$		−	2.18379e19i		−	0.134794i
$496$			2.23461e20i			1.35859i
$497$	8.91871e18			0.0534108
$498$	3.53545e20			2.08557
$499$	−1.61785e20			−0.940122			−0.470061	−	0.882634i	$-0.655768\pi$
$499$	−1.61785e20			−0.940122			−0.470061	+	0.882634i	$0.655768\pi$
$500$	−3.60348e20			−2.06275
$501$			2.98932e20i			1.68573i
$502$	2.33305e20			1.29612
$503$		−	1.33609e20i		−	0.731266i	−0.930759	−	0.365633i	$-0.880853\pi$
$503$		−	1.33609e20i		−	0.731266i	0.930759	−	0.365633i	$-0.119147\pi$
$504$			2.91264e18i			0.0157057i
$505$		−	1.37130e20i		−	0.728532i
$506$	−1.37916e19			−0.0721914
$507$			1.70481e20i			0.879256i
$508$			1.05253e20i			0.534879i
$509$	−5.29378e19			−0.265083			−0.132542	−	0.991177i	$-0.542314\pi$
$509$	−5.29378e19			−0.265083			−0.132542	+	0.991177i	$0.542314\pi$
$510$		−	2.96716e18i		−	0.0146408i
$511$			5.85174e18i			0.0284530i
$512$			4.17085e20i			1.99848i
$513$	−1.20009e19			−0.0566673
$514$			8.62554e19i			0.401386i
$515$	−2.78526e20			−1.27735
$516$	1.17011e20			0.528877
$517$	−8.82277e19			−0.393030
$518$	1.49322e19			0.0655619
$519$			7.60498e19i			0.329111i
$520$		−	7.36333e19i		−	0.314086i
$521$	−2.15144e20			−0.904580			−0.452290	−	0.891871i	$-0.649393\pi$
$521$	−2.15144e20			−0.904580			−0.452290	+	0.891871i	$0.649393\pi$
$522$	7.62269e19	−	1.48042e19i	0.315922	−	0.0613560i
$523$	3.04817e20			1.24531			0.622653	−	0.782498i	$-0.286055\pi$
$523$	3.04817e20			1.24531			0.622653	+	0.782498i	$0.286055\pi$
$524$		−	5.55586e20i		−	2.23752i
$525$		−	1.67358e17i		−	0.000664435i
$526$	−1.88555e20			−0.737980
$527$	2.58680e18			0.00998123
$528$	2.15907e20			0.821319
$529$	−2.65787e20			−0.996817
$530$		−	2.44967e20i		−	0.905813i
$531$	5.61223e19			0.204610
$532$			1.38551e18i			0.00498051i
$533$		−	5.36661e19i		−	0.190215i
$534$		−	1.90586e20i		−	0.666087i
$535$	−2.37491e20			−0.818452
$536$		−	2.57142e20i		−	0.873850i
$537$			4.31542e20i			1.44616i
$538$	5.40555e20			1.78638
$539$		−	2.23281e20i		−	0.727673i
$540$		−	6.97684e20i		−	2.24236i
$541$			1.43403e20i			0.454546i	0.973831	+	0.227273i	$0.0729810\pi$
$541$			1.43403e20i			0.454546i	−0.973831	+	0.227273i	$0.927019\pi$
$542$	1.77208e20			0.553973
$543$		−	4.11144e20i		−	1.26764i
$544$	−8.79618e17			−0.00267487
$545$	−5.87490e20			−1.76209
$546$	−3.98178e18			−0.0117796
$547$	1.82908e19			0.0533739			0.0266869	−	0.999644i	$-0.491504\pi$
$547$	1.82908e19			0.0533739			0.0266869	+	0.999644i	$0.491504\pi$
$548$		−	1.13939e21i		−	3.27957i
$549$		−	4.02942e19i		−	0.114406i
$550$	7.43453e18			0.0208225
$551$	1.88276e19	−	3.65656e18i	0.0520186	−	0.0101027i
$552$	−3.54525e19			−0.0966282
$553$			1.34282e19i			0.0361059i
$554$		−	6.89242e20i		−	1.82830i
$555$	−2.87793e20			−0.753152
$556$	−8.20166e20			−2.11759
$557$	−3.14290e19			−0.0800601			−0.0400300	−	0.999198i	$-0.512745\pi$
$557$	−3.14290e19			−0.0800601			−0.0400300	+	0.999198i	$0.512745\pi$
$558$	1.39569e20			0.350777
$559$		−	1.86510e19i		−	0.0462501i
$560$	−2.32060e19			−0.0567793
$561$		−	2.49936e18i		−	0.00603402i
$562$			1.33206e21i			3.17323i
$563$			3.44279e20i			0.809277i	0.914477	+	0.404638i	$0.132603\pi$
$563$			3.44279e20i			0.809277i	−0.914477	+	0.404638i	$0.867397\pi$
$564$	−4.36791e20			−1.01316
$565$			3.09363e20i			0.708116i
$566$			3.68904e20i			0.833279i
$567$	−1.58723e19			−0.0353809
$568$			1.01845e21i			2.24042i
$569$			1.51855e20i			0.329675i	0.986321	+	0.164838i	$0.0527100\pi$
$569$			1.51855e20i			0.329675i	−0.986321	+	0.164838i	$0.947290\pi$
$570$		−	3.95414e19i		−	0.0847210i
$571$	1.47904e19			0.0312759			0.0156379	−	0.999878i	$-0.495022\pi$
$571$	1.47904e19			0.0312759			0.0156379	+	0.999878i	$0.495022\pi$
$572$		−	1.19453e20i		−	0.249303i
$573$	6.58215e20			1.35584
$574$	−4.51370e19			−0.0917685
$575$	−4.57433e17			−0.000917949
$576$	6.79364e19			0.134565
$577$			4.25411e20i			0.831744i	0.909423	+	0.415872i	$0.136524\pi$
$577$			4.25411e20i			0.831744i	−0.909423	+	0.415872i	$0.863476\pi$
$578$		−	9.09283e20i		−	1.75485i
$579$	6.00254e20			1.14353
$580$	2.12579e20	+	1.09457e21i	0.399770	+	2.05841i
$581$	−3.20085e19			−0.0594217
$582$			7.32635e20i			1.34266i
$583$		−	2.06345e20i		−	0.373319i
$584$	−6.68227e20			−1.19352
$585$	−1.72327e19			−0.0303867
$586$	1.78816e21			3.11296
$587$	8.73618e20			1.50154			0.750768	−	0.660566i	$-0.229684\pi$
$587$	8.73618e20			1.50154			0.750768	+	0.660566i	$0.229684\pi$
$588$		−	1.10540e21i		−	1.87582i
$589$	3.44727e19			0.0577578
$590$			1.19332e21i			1.97409i
$591$			9.01340e20i			1.47225i
$592$			6.38935e20i			1.03049i
$593$	5.28498e20			0.841653			0.420826	−	0.907141i	$-0.361740\pi$
$593$	5.28498e20			0.841653			0.420826	+	0.907141i	$0.361740\pi$
$594$		−	8.70227e20i		−	1.36847i
$595$			2.68635e17i			0.000417143i
$596$	−2.59250e20			−0.397532
$597$		−	1.55596e19i		−	0.0235609i
$598$			1.08832e19i			0.0162742i
$599$		−	1.27821e21i		−	1.88755i	−0.330582	−	0.943777i	$-0.607245\pi$
$599$		−	1.27821e21i		−	1.88755i	0.330582	−	0.943777i	$-0.392755\pi$
$600$	1.91111e19			0.0278709
$601$		−	3.88099e20i		−	0.558965i	−0.960151	−	0.279482i	$-0.909837\pi$
$601$		−	3.88099e20i		−	0.558965i	0.960151	−	0.279482i	$-0.0901629\pi$
$602$	−1.56868e19			−0.0223132
$603$	−6.01800e19			−0.0845419
$604$	−4.22653e20			−0.586418
$605$	−3.44521e20			−0.472118
$606$		−	8.46791e20i		−	1.14612i
$607$		−	9.56868e20i		−	1.27920i	−0.768710	−	0.639598i	$-0.779101\pi$
$607$		−	9.56868e20i		−	1.27920i	0.768710	−	0.639598i	$-0.220899\pi$
$608$	−1.17221e19			−0.0154785
$609$	3.07333e19	−	5.96879e18i	0.0400848	−	0.00778497i
$610$	8.56768e20			1.10380
$611$			6.96222e19i			0.0886010i
$612$			2.77854e18i			0.00349286i
$613$	4.65061e20			0.577506			0.288753	−	0.957404i	$-0.406759\pi$
$613$	4.65061e20			0.577506			0.288753	+	0.957404i	$0.406759\pi$
$614$	−1.87629e21			−2.30164
$615$	8.69938e20			1.05420
$616$	−5.21669e19			−0.0624510
$617$			3.83707e19i			0.0453796i	0.999743	+	0.0226898i	$0.00722301\pi$
$617$			3.83707e19i			0.0453796i	−0.999743	+	0.0226898i	$0.992777\pi$
$618$	−1.71992e21			−2.00953
$619$			8.90641e20i			1.02807i	0.857769	+	0.514035i	$0.171850\pi$
$619$			8.90641e20i			1.02807i	−0.857769	+	0.514035i	$0.828150\pi$
$620$			2.00411e21i			2.28552i
$621$			5.35434e19i			0.0603280i
$622$	6.99535e19			0.0778720
$623$			1.72549e19i			0.0189780i
$624$		−	1.70376e20i		−	0.185150i
$625$	−9.46230e20			−1.01601
$626$			1.18410e21i			1.25626i
$627$		−	3.33073e19i		−	0.0349167i
$628$			1.05028e21i			1.08795i
$629$	7.39637e18			0.00757074
$630$			1.44939e19i			0.0146599i
$631$	1.00257e21			1.00206			0.501030	−	0.865430i	$-0.332955\pi$
$631$	1.00257e21			1.00206			0.501030	+	0.865430i	$0.332955\pi$
$632$	−1.53340e21			−1.51453
$633$	1.63013e21			1.59109
$634$	−1.45174e21			−1.40029
$635$			2.71954e20i			0.259235i
$636$		−	1.02156e21i		−	0.962354i
$637$	−1.76195e20			−0.164040
$638$	2.65151e20	+	1.36526e21i	0.243971	+	1.25621i
$639$	2.38353e20			0.216752
$640$			1.77215e21i			1.59276i
$641$		−	1.88090e21i		−	1.67082i	−0.549627	−	0.835410i	$-0.685230\pi$
$641$		−	1.88090e21i		−	1.67082i	0.549627	−	0.835410i	$-0.314770\pi$
$642$	−1.46652e21			−1.28758
$643$	−2.03224e21			−1.76357			−0.881784	−	0.471653i	$-0.843657\pi$
$643$	−2.03224e21			−1.76357			−0.881784	+	0.471653i	$0.843657\pi$
$644$	6.18165e18			0.00530225
$645$	3.02336e20			0.256326
$646$			1.01623e18i			0.000851622i
$647$	−1.66185e21			−1.37660			−0.688302	−	0.725424i	$-0.741644\pi$
$647$	−1.66185e21			−1.37660			−0.688302	+	0.725424i	$0.741644\pi$
$648$		−	1.81251e21i		−	1.48412i
$649$			1.00518e21i			0.813595i
$650$		−	5.86673e18i		−	0.00469403i
$651$	5.62715e19			0.0445073
$652$		−	1.91332e21i		−	1.49599i
$653$			1.74054e21i			1.34535i	0.739938	+	0.672675i	$0.234855\pi$
$653$			1.74054e21i			1.34535i	−0.739938	+	0.672675i	$0.765145\pi$
$654$	−3.62780e21			−2.77211
$655$		−	1.43554e21i		−	1.08444i
$656$		−	1.93137e21i		−	1.44240i
$657$			1.56388e20i			0.115468i
$658$	5.85573e19			0.0427451
$659$		−	1.79982e21i		−	1.29894i	−0.760389	−	0.649468i	$-0.774992\pi$
$659$		−	1.79982e21i		−	1.29894i	0.760389	−	0.649468i	$-0.225008\pi$
$660$	1.93636e21			1.38168
$661$	2.26994e21			1.60141			0.800706	−	0.599058i	$-0.204458\pi$
$661$	2.26994e21			1.60141			0.800706	+	0.599058i	$0.204458\pi$
$662$	−3.75460e20			−0.261896
$663$	−1.97229e18			−0.00136025
$664$		−	3.65515e21i		−	2.49255i
$665$			3.57992e18i			0.00241386i
$666$	3.99064e20			0.266064
$667$	−1.63142e19	−	8.40019e19i	−0.0107553	−	0.0553791i
$668$	5.95208e21			3.88012
$669$		−	1.20315e20i		−	0.0775574i
$670$		−	1.27959e21i		−	0.815664i
$671$	7.21690e20			0.454916
$672$	−1.91346e19			−0.0119275
$673$	1.71100e21			1.05472			0.527361	−	0.849641i	$-0.323182\pi$
$673$	1.71100e21			1.05472			0.527361	+	0.849641i	$0.323182\pi$
$674$	1.79051e21			1.09151
$675$		−	2.88632e19i		−	0.0174007i
$676$	3.39447e21			2.02382
$677$		−	9.61071e20i		−	0.566683i	−0.959019	−	0.283342i	$-0.908557\pi$
$677$		−	9.61071e20i		−	0.566683i	0.959019	−	0.283342i	$-0.0914431\pi$
$678$			1.91034e21i			1.11401i
$679$		−	6.63298e19i		−	0.0382548i
$680$	−3.06762e19			−0.0174978
$681$			1.49685e21i			0.844454i
$682$			2.49974e21i			1.39480i
$683$	−1.32976e21			−0.733869			−0.366935	−	0.930247i	$-0.619593\pi$
$683$	−1.32976e21			−0.733869			−0.366935	+	0.930247i	$0.619593\pi$
$684$			3.70279e19i			0.0202119i
$685$		−	2.94397e21i		−	1.58948i
$686$			2.96689e20i			0.158442i
$687$	1.10986e21			0.586265
$688$		−	6.71223e20i		−	0.350714i
$689$	−1.62831e20			−0.0841576
$690$	−1.76419e20			−0.0901941
$691$	−7.46842e20			−0.377697			−0.188848	−	0.982006i	$-0.560475\pi$
$691$	−7.46842e20			−0.377697			−0.188848	+	0.982006i	$0.560475\pi$
$692$	1.51424e21			0.757528
$693$			1.22088e19i			0.00604191i
$694$		−	2.01247e21i		−	0.985222i
$695$	−2.11917e21			−1.02631
$696$	6.81594e20	+	3.50952e21i	0.326555	+	1.68143i
$697$	−2.23577e19			−0.0105969
$698$			2.62193e21i			1.22944i
$699$		−	1.03330e21i		−	0.479346i
$700$	−3.33230e18			−0.00152935
$701$	−1.44201e21			−0.654761			−0.327380	−	0.944893i	$-0.606166\pi$
$701$	−1.44201e21			−0.654761			−0.327380	+	0.944893i	$0.606166\pi$
$702$	−6.86713e20			−0.308494
$703$	9.85667e19			0.0438092
$704$			1.21677e21i			0.535075i
$705$	−1.12859e21			−0.491041
$706$		−	6.98030e21i		−	3.00496i
$707$			7.66650e19i			0.0326552i
$708$			4.97636e21i			2.09731i
$709$	−3.24521e21			−1.35331			−0.676653	−	0.736302i	$-0.736570\pi$
$709$	−3.24521e21			−1.35331			−0.676653	+	0.736302i	$0.736570\pi$
$710$			5.06804e21i			2.09123i
$711$			3.58868e20i			0.146525i
$712$	−1.97038e21			−0.796069
$713$		−	1.53805e20i		−	0.0614890i
$714$			1.65884e18i			0.000656247i
$715$		−	3.08646e20i		−	0.120827i
$716$	8.59250e21			3.32868
$717$			1.09143e21i			0.418412i
$718$	5.75714e21			2.18410
$719$	−3.67140e21			−1.37837			−0.689184	−	0.724586i	$-0.742031\pi$
$719$	−3.67140e21			−1.37837			−0.689184	+	0.724586i	$0.742031\pi$
$720$	−6.20181e20			−0.230422
$721$	1.55714e20			0.0572551
$722$		−	4.80933e21i		−	1.75007i
$723$		−	2.89614e21i		−	1.04300i
$724$	−8.18636e21			−2.91777
$725$	8.79439e18	+	4.52823e19i	0.00310221	+	0.0159733i
$726$	−2.12744e21			−0.742734
$727$		−	2.21986e21i		−	0.767038i	−0.923533	−	0.383519i	$-0.874712\pi$
$727$		−	2.21986e21i		−	0.767038i	0.923533	−	0.383519i	$-0.125288\pi$
$728$			4.11659e19i			0.0140784i
$729$	−3.27916e21			−1.10996
$730$	−3.32524e21			−1.11404
$731$	−7.77014e18			−0.00257661
$732$	3.57289e21			1.17270
$733$			3.93637e21i			1.27884i	0.768858	+	0.639420i	$0.220825\pi$
$733$			3.93637e21i			1.27884i	−0.768858	+	0.639420i	$0.779175\pi$
$734$	−6.15246e21			−1.97847
$735$		−	2.85617e21i		−	0.909135i
$736$			5.22998e19i			0.0164784i
$737$		−	1.07785e21i		−	0.336166i
$738$	−1.20629e21			−0.372416
$739$			5.68428e21i			1.73717i	0.495541	+	0.868585i	$0.334970\pi$
$739$			5.68428e21i			1.73717i	−0.495541	+	0.868585i	$0.665030\pi$
$740$			5.73029e21i			1.73356i
$741$	−2.62835e19			−0.00787129
$742$			1.36953e20i			0.0406015i
$743$		−	4.33316e21i		−	1.27171i	−0.771808	−	0.635856i	$-0.780647\pi$
$743$		−	4.33316e21i		−	1.27171i	0.771808	−	0.635856i	$-0.219353\pi$
$744$			6.42581e21i			1.86694i
$745$	−6.69857e20			−0.192668
$746$			3.68868e20i			0.105034i
$747$	−8.55429e20			−0.241146
$748$	−4.97651e19			−0.0138887
$749$	1.32773e20			0.0366856
$750$	−5.74946e21			−1.57278
$751$		−	3.43843e21i		−	0.931238i	−0.884985	−	0.465619i	$-0.845832\pi$
$751$		−	3.43843e21i		−	0.931238i	0.884985	−	0.465619i	$-0.154168\pi$
$752$			2.50561e21i			0.671860i
$753$	2.51387e21			0.667392
$754$	1.07735e21	−	2.09236e20i	0.283187	−	0.0549985i
$755$	−1.09206e21			−0.284214
$756$			3.90052e20i			0.100510i
$757$		−	4.39960e21i		−	1.12252i	−0.827640	−	0.561259i	$-0.810317\pi$
$757$		−	4.39960e21i		−	1.12252i	0.827640	−	0.561259i	$-0.189683\pi$
$758$	8.29322e21			2.09510
$759$	−1.48605e20			−0.0371724
$760$	−4.08802e20			−0.101254
$761$	3.88612e21			0.953085			0.476543	−	0.879151i	$-0.341890\pi$
$761$	3.88612e21			0.953085			0.476543	+	0.879151i	$0.341890\pi$
$762$			1.67934e21i			0.407827i
$763$	3.28446e20			0.0789824
$764$		−	1.31058e22i		−	3.12079i
$765$			7.17927e18i			0.00169285i
$766$			1.11779e22i			2.61002i
$767$	7.93206e20			0.183409
$768$			8.04706e21i			1.84259i
$769$			7.22878e21i			1.63915i	0.572975	+	0.819573i	$0.305789\pi$
$769$			7.22878e21i			1.63915i	−0.572975	+	0.819573i	$0.694211\pi$
$770$	−2.59594e20			−0.0582926
$771$			9.29405e20i			0.206679i
$772$		−	1.19517e22i		−	2.63210i
$773$			8.53536e21i			1.86156i	0.365586	+	0.930778i	$0.380869\pi$
$773$			8.53536e21i			1.86156i	−0.365586	+	0.930778i	$0.619131\pi$
$774$	−4.19231e20			−0.0905515
$775$			8.29102e19i			0.0177356i
$776$	7.57440e21			1.60467
$777$	1.60895e20			0.0337587
$778$	−6.95424e20			−0.144512
$779$	−2.97947e20			−0.0613208
$780$		−	1.52802e21i		−	0.311472i
$781$			4.26901e21i			0.861876i
$782$	4.53403e18			0.000906638
$783$	5.30038e21	−	1.02940e21i	1.04977	−	0.203879i
$784$	−6.34103e21			−1.24391
$785$			2.71374e21i			0.527285i
$786$		−	8.86455e21i		−	1.70603i
$787$	7.35084e21			1.40128			0.700642	−	0.713513i	$-0.252897\pi$
$787$	7.35084e21			1.40128			0.700642	+	0.713513i	$0.252897\pi$
$788$	1.79467e22			3.38874
$789$	−2.03169e21			−0.379996
$790$	−7.63054e21			−1.41368
$791$		−	1.72954e20i		−	0.0317401i
$792$	−1.39416e21			−0.253439
$793$		−	5.69499e20i		−	0.102552i
$794$			8.45857e21i			1.50884i
$795$		−	2.63953e21i		−	0.466415i
$796$	−3.09810e20			−0.0542310
$797$			2.43455e20i			0.0422164i	0.999777	+	0.0211082i	$0.00671944\pi$
$797$			2.43455e20i			0.0422164i	−0.999777	+	0.0211082i	$0.993281\pi$
$798$			2.21063e19i			0.00379747i
$799$	2.90051e19			0.00493598
$800$		−	2.81928e19i		−	0.00475296i
$801$			4.61137e20i			0.0770168i
$802$		−	1.54702e21i		−	0.255970i
$803$	−2.80098e21			−0.459139
$804$		−	5.33615e21i		−	0.866578i
$805$	1.59723e19			0.00256979
$806$	1.97260e21			0.314431
$807$	5.82451e21			0.919830
$808$	−8.75460e21			−1.36978
$809$			7.88988e21i			1.22308i	0.791212	+	0.611542i	$0.209450\pi$
$809$			7.88988e21i			1.22308i	−0.791212	+	0.611542i	$0.790550\pi$
$810$		−	9.01944e21i		−	1.38529i
$811$	−1.18587e21			−0.180461			−0.0902303	−	0.995921i	$-0.528760\pi$
$811$	−1.18587e21			−0.180461			−0.0902303	+	0.995921i	$0.528760\pi$
$812$	−1.18846e20	−	6.11935e20i	−0.0179190	−	0.0922647i
$813$	1.90942e21			0.285248
$814$			7.14744e21i			1.05795i
$815$		−	4.94367e21i		−	0.725049i
$816$	−7.09800e19			−0.0103148
$817$	−1.03548e20			−0.0149099
$818$	−1.32532e22			−1.89091
$819$	9.63422e18			0.00136203
$820$		−	1.73215e22i		−	2.42650i
$821$	−3.19326e21			−0.443262			−0.221631	−	0.975131i	$-0.571138\pi$
$821$	−3.19326e21			−0.443262			−0.221631	+	0.975131i	$0.571138\pi$
$822$		−	1.81792e22i		−	2.50056i
$823$		−	1.95717e21i		−	0.266766i	−0.991065	−	0.133383i	$-0.957416\pi$
$823$		−	1.95717e21i		−	0.266766i	0.991065	−	0.133383i	$-0.0425840\pi$
$824$			1.77815e22i			2.40167i
$825$	8.01074e19			0.0107218
$826$		−	6.67144e20i		−	0.0884849i
$827$		−	2.70634e21i		−	0.355706i	−0.984057	−	0.177853i	$-0.943085\pi$
$827$		−	2.70634e21i		−	0.355706i	0.984057	−	0.177853i	$-0.0569152\pi$
$828$	1.65205e20			0.0215177
$829$			5.92529e21i			0.764805i	0.923996	+	0.382403i	$0.124903\pi$
$829$			5.92529e21i			0.764805i	−0.923996	+	0.382403i	$0.875097\pi$
$830$		−	1.81888e22i		−	2.32658i
$831$		−	7.42661e21i		−	0.941419i
$832$	9.60180e20			0.120622
$833$			7.34043e19i			0.00913869i
$834$	−1.30860e22			−1.61459
$835$	1.53791e22			1.88054
$836$	−6.63188e20			−0.0803691
$837$	9.70481e21			1.16559
$838$		−	6.20524e21i		−	0.738632i
$839$		−	7.60765e21i		−	0.897503i	−0.893657	−	0.448751i	$-0.851869\pi$
$839$		−	7.60765e21i		−	0.897503i	0.893657	−	0.448751i	$-0.148131\pi$
$840$	−6.67308e20			−0.0780246
$841$	−8.00189e21	+	3.22997e21i	−0.927305	+	0.374307i
$842$	−6.30298e21			−0.723945
$843$			1.43530e22i			1.63394i
$844$		−	3.24578e22i		−	3.66227i
$845$	8.77070e21			0.980865
$846$	1.56494e21			0.173469
$847$	1.92610e20			0.0211619
$848$	−5.86007e21			−0.638167
$849$			3.97496e21i			0.429067i
$850$	−2.44413e18			−0.000261506
$851$		−	4.39768e20i		−	0.0466393i
$852$			2.11347e22i			2.22177i
$853$			1.22235e22i			1.27373i	0.770976	+	0.636864i	$0.219769\pi$
$853$			1.22235e22i			1.27373i	−0.770976	+	0.636864i	$0.780231\pi$
$854$	−4.78990e20			−0.0494758
$855$			9.56735e19i			0.00979594i
$856$			1.51617e22i			1.53885i
$857$	−1.19829e22			−1.20561			−0.602803	−	0.797890i	$-0.705950\pi$
$857$	−1.19829e22			−1.20561			−0.602803	+	0.797890i	$0.705950\pi$
$858$		−	1.90591e21i		−	0.190085i
$859$			3.18645e21i			0.315035i	0.987516	+	0.157517i	$0.0503491\pi$
$859$			3.18645e21i			0.315035i	−0.987516	+	0.157517i	$0.949651\pi$
$860$		−	6.01987e21i		−	0.589995i
$861$	−4.86353e20			−0.0472529
$862$		−	1.49087e22i		−	1.43594i
$863$	−1.01668e22			−0.970739			−0.485369	−	0.874309i	$-0.661315\pi$
$863$	−1.01668e22			−0.970739			−0.485369	+	0.874309i	$0.661315\pi$
$864$	−3.30003e21			−0.312367
$865$	3.91253e21			0.367144
$866$	8.72648e21			0.811811
$867$		−	9.79756e21i		−	0.903597i
$868$		−	1.12043e21i		−	0.102444i
$869$	−6.42750e21			−0.582632
$870$	3.39176e21	+	1.74641e22i	0.304811	+	1.56947i
$871$	−8.50555e20			−0.0757820
$872$			3.75062e22i			3.31306i
$873$		−	1.77267e21i		−	0.155246i
$874$	6.04222e19			0.00524639
$875$	5.20533e20			0.0448114
$876$	−1.38669e22			−1.18358
$877$	−7.41138e21			−0.627194			−0.313597	−	0.949556i	$-0.601534\pi$
$877$	−7.41138e21			−0.627194			−0.313597	+	0.949556i	$0.601534\pi$
$878$		−	1.78977e22i		−	1.50171i
$879$	1.92675e22			1.60291
$880$		−	1.11077e22i		−	0.916233i
$881$			2.15552e22i			1.76292i	0.472255	+	0.881462i	$0.343440\pi$
$881$			2.15552e22i			1.76292i	−0.472255	+	0.881462i	$0.656560\pi$
$882$			3.96046e21i			0.321167i
$883$	−8.21932e21			−0.660892			−0.330446	−	0.943825i	$-0.607199\pi$
$883$	−8.21932e21			−0.660892			−0.330446	+	0.943825i	$0.607199\pi$
$884$			3.92706e19i			0.00313094i
$885$			1.28580e22i			1.01648i
$886$	3.76746e22			2.95322
$887$		−	4.21157e21i		−	0.327353i	−0.986514	−	0.163677i	$-0.947665\pi$
$887$		−	4.21157e21i		−	0.327353i	0.986514	−	0.163677i	$-0.0523353\pi$
$888$			1.83731e22i			1.41607i
$889$		−	1.52041e20i		−	0.0116197i
$890$	−9.80505e21			−0.743061
$891$		−	7.59743e21i		−	0.570931i
$892$	−2.39561e21			−0.178517
$893$	3.86533e20			0.0285628
$894$	−4.13642e21			−0.303105
$895$	2.22015e22			1.61328
$896$		−	9.90748e20i		−	0.0713926i
$897$			1.17267e20i			0.00837979i
$898$	2.62928e22			1.86322
$899$	−1.52254e22	+	2.95698e21i	−1.06997	+	0.207802i
$900$	−8.90557e19			−0.00620644
$901$			6.78367e19i			0.00468844i
$902$		−	2.16052e22i		−	1.48084i
$903$	−1.69026e20			−0.0114894
$904$	1.97502e22			1.33140
$905$	−2.11521e22			−1.41413
$906$	−6.74356e21			−0.447124
$907$		−	1.89617e22i		−	1.24688i	−0.781873	−	0.623438i	$-0.785735\pi$
$907$		−	1.89617e22i		−	1.24688i	0.781873	−	0.623438i	$-0.214265\pi$
$908$	2.98041e22			1.94371
$909$			2.04887e21i			0.132521i
$910$			2.04850e20i			0.0131409i
$911$			1.03115e22i			0.656046i	0.944670	+	0.328023i	$0.106382\pi$
$911$			1.03115e22i			0.656046i	−0.944670	+	0.328023i	$0.893618\pi$
$912$	−9.45906e20			−0.0596880
$913$		−	1.53212e22i		−	0.958872i
$914$			3.32179e22i			2.06194i
$915$	9.23171e21			0.568360
$916$		−	2.20987e22i		−	1.34943i
$917$			8.02561e20i			0.0486080i
$918$			2.86090e20i			0.0171863i
$919$	−2.39044e22			−1.42434			−0.712168	−	0.702009i	$-0.752287\pi$
$919$	−2.39044e22			−1.42434			−0.712168	+	0.702009i	$0.752287\pi$
$920$			1.82392e21i			0.107795i
$921$	−2.02171e22			−1.18515
$922$	−3.83857e21			−0.223197
$923$	3.36876e21			0.194293
$924$	−1.08255e21			−0.0619313
$925$			2.37063e20i			0.0134524i
$926$			1.18879e22i			0.669147i
$927$	4.16147e21			0.232353
$928$	5.17727e21	−	1.00549e21i	0.286742	−	0.0556889i
$929$	−1.83870e22			−1.01017			−0.505085	−	0.863070i	$-0.668539\pi$
$929$	−1.83870e22			−1.01017			−0.505085	+	0.863070i	$0.668539\pi$
$930$			3.19762e22i			1.74263i
$931$			9.78213e20i			0.0528824i
$932$	−2.05743e22			−1.10333
$933$	7.53751e20			0.0400974
$934$	5.89861e21			0.311278
$935$	−1.28584e20			−0.00673132
$936$			1.10016e21i			0.0571330i
$937$	1.65997e21			0.0855173			0.0427587	−	0.999085i	$-0.486385\pi$
$937$	1.65997e21			0.0855173			0.0427587	+	0.999085i	$0.486385\pi$
$938$			7.15378e20i			0.0365607i
$939$			1.27587e22i			0.646868i
$940$			2.24715e22i			1.13025i
$941$	2.09842e22			1.04706			0.523528	−	0.852009i	$-0.324616\pi$
$941$	2.09842e22			1.04706			0.523528	+	0.852009i	$0.324616\pi$
$942$			1.67575e22i			0.829522i
$943$			1.32933e21i			0.0652821i
$944$	2.85464e22			1.39079
$945$			1.00783e21i			0.0487132i
$946$		−	7.50863e21i		−	0.360062i
$947$		−	1.23214e19i		−	0.000586186i	−1.00000		0.000293093i	$-0.999907\pi$
$947$		−	1.23214e19i		−	0.000586186i	1.00000		0.000293093i	$-9.32944e-5\pi$
$948$	−3.18208e22			−1.50193
$949$			2.21031e21i			0.103504i
$950$	−3.25713e19			−0.00151324
$951$	−1.56425e22			−0.721030
$952$	1.71500e19			0.000784309
$953$	−1.79546e22			−0.814665			−0.407333	−	0.913280i	$-0.633541\pi$
$953$	−1.79546e22			−0.814665			−0.407333	+	0.913280i	$0.633541\pi$
$954$			3.66007e21i			0.164769i
$955$		−	3.38631e22i		−	1.51252i
$956$	2.17317e22			0.963074
$957$	2.85701e21	+	1.47107e22i	0.125624	+	0.646837i
$958$	8.88884e21			0.387796
$959$			1.64588e21i			0.0712455i
$960$			1.55647e22i			0.668509i
$961$	−4.41198e21			−0.188022
$962$	5.64018e21			0.238495
$963$	3.54836e21			0.148878
$964$	−5.76655e22			−2.40070
$965$		−	3.08812e22i		−	1.27567i
$966$	9.86301e19			0.00404279
$967$		−	3.62443e22i		−	1.47415i	−0.675812	−	0.737074i	$-0.736207\pi$
$967$		−	3.62443e22i		−	1.47415i	0.675812	−	0.737074i	$-0.263793\pi$
$968$			2.19947e22i			0.887674i
$969$			1.09499e19i			0.000438512i
$970$	3.76918e22			1.49782
$971$			4.19127e21i			0.165273i	0.996580	+	0.0826364i	$0.0263340\pi$
$971$			4.19127e21i			0.165273i	−0.996580	+	0.0826364i	$0.973666\pi$
$972$			1.92330e22i			0.752574i
$973$	1.18475e21			0.0460026
$974$		−	6.76480e22i		−	2.60653i
$975$		−	6.32143e19i		−	0.00241702i
$976$		−	2.04955e22i		−	0.777651i
$977$	−4.80319e22			−1.80851			−0.904255	−	0.426993i	$-0.859573\pi$
$977$	−4.80319e22			−1.80851			−0.904255	+	0.426993i	$0.859573\pi$
$978$		−	3.05275e22i		−	1.14064i
$979$	−8.25918e21			−0.306243
$980$	−5.68696e22			−2.09259
$981$	8.77773e21			0.320527
$982$	−3.51697e22			−1.27448
$983$		−	4.66031e22i		−	1.67596i	−0.545702	−	0.837979i	$-0.683737\pi$
$983$		−	4.66031e22i		−	1.67596i	0.545702	−	0.837979i	$-0.316263\pi$
$984$		−	5.55381e22i		−	1.98211i
$985$	4.63712e22			1.64239
$986$	−8.71692e19	−	4.48834e20i	−0.00306398	−	0.0157764i
$987$	6.30957e20			0.0220101
$988$			5.23334e20i			0.0181177i
$989$			4.61992e20i			0.0158731i
$990$	−6.93764e21			−0.236564
$991$	1.89001e22			0.639605			0.319802	−	0.947484i	$-0.396384\pi$
$991$	1.89001e22			0.639605			0.319802	+	0.947484i	$0.396384\pi$
$992$	9.47939e21			0.318378
$993$	−4.04560e21			−0.134854
$994$		−	2.83337e21i		−	0.0937358i
$995$	−8.00495e20			−0.0262836
$996$		−	7.58508e22i		−	2.47181i
$997$			2.26175e22i			0.731527i	0.930708	+	0.365763i	$0.119192\pi$
$997$			2.26175e22i			0.731527i	−0.930708	+	0.365763i	$0.880808\pi$
$998$			5.13973e22i			1.64991i
$999$	2.77486e22			0.884098

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.3	✓	36
29.28	even	2	inner	29.16.b.a.28.34	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.3	✓	36	1.1	even	1	trivial
29.16.b.a.28.34	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)

\(f(q)\)	\(=\)	\(q-317.689i q^{2} -3423.11i q^{3} -68158.0 q^{4} -176108. q^{5} -1.08748e6 q^{6} +98456.3 q^{7} +1.12430e7i q^{8} +2.63125e6 q^{9} +O(q^{10})\)	\(q-317.689i q^{2} -3423.11i q^{3} -68158.0 q^{4} -176108. q^{5} -1.08748e6 q^{6} +98456.3 q^{7} +1.12430e7i q^{8} +2.63125e6 q^{9} +5.59476e7i q^{10} +4.71269e7i q^{11} +2.33312e8i q^{12} +3.71888e7 q^{13} -3.12784e7i q^{14} +6.02838e8i q^{15} +1.33837e9 q^{16} -1.54931e7i q^{17} -8.35917e8i q^{18} -2.06467e8i q^{19} +1.20032e10 q^{20} -3.37026e8i q^{21} +1.49717e10 q^{22} -9.21179e8 q^{23} +3.84860e10 q^{24} +4.96573e8 q^{25} -1.18144e10i q^{26} -5.81249e10i q^{27} -6.71059e9 q^{28} +(1.77102e10 + 9.11896e10i) q^{29} +1.91515e11 q^{30} +1.66965e11i q^{31} -5.67748e10i q^{32} +1.61320e11 q^{33} -4.92198e9 q^{34} -1.73390e10 q^{35} -1.79341e11 q^{36} +4.77397e11i q^{37} -6.55922e10 q^{38} -1.27301e11i q^{39} -1.97999e12i q^{40} -1.44307e12i q^{41} -1.07069e11 q^{42} -5.01522e11i q^{43} -3.21208e12i q^{44} -4.63384e11 q^{45} +2.92648e11i q^{46} +1.87213e12i q^{47} -4.58139e12i q^{48} -4.73787e12 q^{49} -1.57756e11i q^{50} -5.30346e10 q^{51} -2.53471e12 q^{52} -4.37851e12 q^{53} -1.84656e13 q^{54} -8.29944e12i q^{55} +1.10694e12i q^{56} -7.06758e11 q^{57} +(2.89699e13 - 5.62632e12i) q^{58} +2.13292e13 q^{59} -4.10882e13i q^{60} -1.53138e13i q^{61} +5.30428e13 q^{62} +2.59063e11 q^{63} +2.58191e13 q^{64} -6.54925e12 q^{65} -5.12497e13i q^{66} -2.28713e13 q^{67} +1.05598e12i q^{68} +3.15329e12i q^{69} +5.50839e12i q^{70} +9.05855e13 q^{71} +2.95831e13i q^{72} +5.94349e13i q^{73} +1.51664e14 q^{74} -1.69982e12i q^{75} +1.40724e13i q^{76} +4.63994e12i q^{77} -4.04421e13 q^{78} +1.36387e14i q^{79} -2.35699e14 q^{80} -1.61212e14 q^{81} -4.58447e14 q^{82} -3.25104e14 q^{83} +2.29711e13i q^{84} +2.72847e12i q^{85} -1.59328e14 q^{86} +(3.12152e14 - 6.06238e13i) q^{87} -5.29848e14 q^{88} +1.75254e14i q^{89} +1.47212e14i q^{90} +3.66147e12 q^{91} +6.27858e13 q^{92} +5.71538e14 q^{93} +5.94754e14 q^{94} +3.63605e13i q^{95} -1.94346e14 q^{96} -6.73699e14i q^{97} +1.50517e15i q^{98} +1.24002e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 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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