LMFDB - Embedded newform 29.16.b.a.28.17

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			28.17
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.20

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-14.2479i q^{2} +3944.08i q^{3} +32565.0 q^{4} -125301. q^{5} +56194.9 q^{6} -3.40529e6 q^{7} -930860. i q^{8} -1.20682e6 q^{9} +O(q^{10})$	\(q-14.2479i q^{2} +3944.08i q^{3} +32565.0 q^{4} -125301. q^{5} +56194.9 q^{6} -3.40529e6 q^{7} -930860. i q^{8} -1.20682e6 q^{9} +1.78527e6i q^{10} -8.54695e7i q^{11} +1.28439e8i q^{12} +2.96116e8 q^{13} +4.85183e7i q^{14} -4.94195e8i q^{15} +1.05383e9 q^{16} -2.42423e9i q^{17} +1.71947e7i q^{18} +4.03131e9i q^{19} -4.08041e9 q^{20} -1.34307e10i q^{21} -1.21776e9 q^{22} +1.05715e10 q^{23} +3.67138e9 q^{24} -1.48173e10 q^{25} -4.21904e9i q^{26} +5.18334e10i q^{27} -1.10893e11 q^{28} +(9.13600e10 - 1.68088e10i) q^{29} -7.04125e9 q^{30} +2.30071e11i q^{31} -4.55173e10i q^{32} +3.37098e11 q^{33} -3.45402e10 q^{34} +4.26684e11 q^{35} -3.93002e10 q^{36} -1.11994e11i q^{37} +5.74377e10 q^{38} +1.16790e12i q^{39} +1.16637e11i q^{40} -1.39967e12i q^{41} -1.91360e11 q^{42} -1.88668e12i q^{43} -2.78332e12i q^{44} +1.51216e11 q^{45} -1.50622e11i q^{46} +3.19975e12i q^{47} +4.15637e12i q^{48} +6.84841e12 q^{49} +2.11116e11i q^{50} +9.56134e12 q^{51} +9.64303e12 q^{52} +1.02495e13 q^{53} +7.38518e11 q^{54} +1.07094e13i q^{55} +3.16984e12i q^{56} -1.58998e13 q^{57} +(-2.39491e11 - 1.30169e12i) q^{58} +6.18631e12 q^{59} -1.60935e13i q^{60} +3.39020e13i q^{61} +3.27804e12 q^{62} +4.10958e12 q^{63} +3.38833e13 q^{64} -3.71035e13 q^{65} -4.80295e12i q^{66} +7.36893e13 q^{67} -7.89450e13i q^{68} +4.16949e13i q^{69} -6.07937e12i q^{70} +7.54209e13 q^{71} +1.12338e12i q^{72} -1.04221e14i q^{73} -1.59568e12 q^{74} -5.84407e13i q^{75} +1.31279e14i q^{76} +2.91048e14i q^{77} +1.66402e13 q^{78} -2.57827e14i q^{79} -1.32045e14 q^{80} -2.21751e14 q^{81} -1.99424e13 q^{82} -5.96106e13 q^{83} -4.37371e14i q^{84} +3.03757e14i q^{85} -2.68813e13 q^{86} +(6.62953e13 + 3.60331e14i) q^{87} -7.95601e13 q^{88} +2.57647e14i q^{89} -2.15451e12i q^{90} -1.00836e15 q^{91} +3.44262e14 q^{92} -9.07418e14 q^{93} +4.55898e13 q^{94} -5.05125e14i q^{95} +1.79523e14 q^{96} -1.01732e15i q^{97} -9.75756e13i q^{98} +1.03147e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	14.2479i		−	0.0787094i	−0.999225	−	0.0393547i	$-0.987470\pi$
$2$		−	14.2479i		−	0.0787094i	0.999225	−	0.0393547i	$-0.0125302\pi$
$3$			3944.08i			1.04120i	0.853799	+	0.520602i	$0.174292\pi$
$3$			3944.08i			1.04120i	−0.853799	+	0.520602i	$0.825708\pi$
$4$	32565.0			0.993805
$5$	−125301.			−0.717263			−0.358631	−	0.933479i	$-0.616756\pi$
$5$	−125301.			−0.717263			−0.358631	+	0.933479i	$0.616756\pi$
$6$	56194.9			0.0819525
$7$	−3.40529e6			−1.56285			−0.781427	−	0.623997i	$-0.785508\pi$
$7$	−3.40529e6			−1.56285			−0.781427	+	0.623997i	$0.785508\pi$
$8$		−	930860.i		−	0.156931i
$9$	−1.20682e6			−0.0841057
$10$			1.78527e6i			0.0564553i
$11$		−	8.54695e7i		−	1.32241i	−0.750205	−	0.661205i	$-0.770045\pi$
$11$		−	8.54695e7i		−	1.32241i	0.750205	−	0.661205i	$-0.229955\pi$
$12$			1.28439e8i			1.03475i
$13$	2.96116e8			1.30884			0.654421	−	0.756131i	$-0.272912\pi$
$13$	2.96116e8			1.30884			0.654421	+	0.756131i	$0.272912\pi$
$14$			4.85183e7i			0.123011i
$15$		−	4.94195e8i		−	0.746817i
$16$	1.05383e9			0.981453
$17$		−	2.42423e9i		−	1.43287i	−0.697654	−	0.716435i	$-0.745773\pi$
$17$		−	2.42423e9i		−	1.43287i	0.697654	−	0.716435i	$-0.254227\pi$
$18$			1.71947e7i			0.00661991i
$19$			4.03131e9i			1.03465i	0.855789	+	0.517325i	$0.173072\pi$
$19$			4.03131e9i			1.03465i	−0.855789	+	0.517325i	$0.826928\pi$
$20$	−4.08041e9			−0.712819
$21$		−	1.34307e10i		−	1.62725i
$22$	−1.21776e9			−0.104086
$23$	1.05715e10			0.647409			0.323705	−	0.946158i	$-0.395072\pi$
$23$	1.05715e10			0.647409			0.323705	+	0.946158i	$0.395072\pi$
$24$	3.67138e9			0.163397
$25$	−1.48173e10			−0.485534
$26$		−	4.21904e9i		−	0.103018i
$27$			5.18334e10i			0.953633i
$28$	−1.10893e11			−1.55317
$29$	9.13600e10	−	1.68088e10i	0.983493	−	0.180948i
$29$	9.13600e10	−	1.68088e10i	0.983493	−	0.180948i
$30$	−7.04125e9			−0.0587815
$31$			2.30071e11i			1.50193i	0.660342	+	0.750965i	$0.270411\pi$
$31$			2.30071e11i			1.50193i	−0.660342	+	0.750965i	$0.729589\pi$
$32$		−	4.55173e10i		−	0.234181i
$33$	3.37098e11			1.37690
$34$	−3.45402e10			−0.112780
$35$	4.26684e11			1.12098
$36$	−3.93002e10			−0.0835846
$37$		−	1.11994e11i		−	0.193946i	−0.995287	−	0.0969731i	$-0.969084\pi$
$37$		−	1.11994e11i		−	0.193946i	0.995287	−	0.0969731i	$-0.0309161\pi$
$38$	5.74377e10			0.0814367
$39$			1.16790e12i			1.36277i
$40$			1.16637e11i			0.112561i
$41$		−	1.39967e12i		−	1.12240i	−0.827681	−	0.561199i	$-0.810340\pi$
$41$		−	1.39967e12i		−	1.12240i	0.827681	−	0.561199i	$-0.189660\pi$
$42$	−1.91360e11			−0.128080
$43$		−	1.88668e12i		−	1.05848i	−0.848471	−	0.529242i	$-0.822476\pi$
$43$		−	1.88668e12i		−	1.05848i	0.848471	−	0.529242i	$-0.177524\pi$
$44$		−	2.78332e12i		−	1.31422i
$45$	1.51216e11			0.0603259
$46$		−	1.50622e11i		−	0.0509572i
$47$			3.19975e12i			0.921261i	0.887592	+	0.460630i	$0.152377\pi$
$47$			3.19975e12i			0.921261i	−0.887592	+	0.460630i	$0.847623\pi$
$48$			4.15637e12i			1.02189i
$49$	6.84841e12			1.44251
$50$			2.11116e11i			0.0382161i
$51$	9.56134e12			1.49191
$52$	9.64303e12			1.30073
$53$	1.02495e13			1.19849			0.599247	−	0.800564i	$-0.295467\pi$
$53$	1.02495e13			1.19849			0.599247	+	0.800564i	$0.295467\pi$
$54$	7.38518e11			0.0750599
$55$			1.07094e13i			0.948515i
$56$			3.16984e12i			0.245260i
$57$	−1.58998e13			−1.07728
$58$	−2.39491e11	−	1.30169e12i	−0.0142423	−	0.0774101i
$59$	6.18631e12			0.323625			0.161812	−	0.986822i	$-0.448266\pi$
$59$	6.18631e12			0.323625			0.161812	+	0.986822i	$0.448266\pi$
$60$		−	1.60935e13i		−	0.742190i
$61$			3.39020e13i			1.38118i	0.723245	+	0.690592i	$0.242650\pi$
$61$			3.39020e13i			1.38118i	−0.723245	+	0.690592i	$0.757350\pi$
$62$	3.27804e12			0.118216
$63$	4.10958e12			0.131445
$64$	3.38833e13			0.963021
$65$	−3.71035e13			−0.938783
$66$		−	4.80295e12i		−	0.108375i
$67$	7.36893e13			1.48540			0.742700	−	0.669625i	$-0.233545\pi$
$67$	7.36893e13			1.48540			0.742700	+	0.669625i	$0.233545\pi$
$68$		−	7.89450e13i		−	1.42399i
$69$			4.16949e13i			0.674085i
$70$		−	6.07937e12i		−	0.0882314i
$71$	7.54209e13			0.984134			0.492067	−	0.870557i	$-0.336242\pi$
$71$	7.54209e13			0.984134			0.492067	+	0.870557i	$0.336242\pi$
$72$			1.12338e12i			0.0131988i
$73$		−	1.04221e14i		−	1.10416i	−0.833791	−	0.552081i	$-0.813834\pi$
$73$		−	1.04221e14i		−	1.10416i	0.833791	−	0.552081i	$-0.186166\pi$
$74$	−1.59568e12			−0.0152654
$75$		−	5.84407e13i		−	0.505540i
$76$			1.31279e14i			1.02824i
$77$			2.91048e14i			2.06673i
$78$	1.66402e13			0.107263
$79$		−	2.57827e14i		−	1.51051i	−0.655428	−	0.755257i	$-0.727512\pi$
$79$		−	2.57827e14i		−	1.51051i	0.655428	−	0.755257i	$-0.272488\pi$
$80$	−1.32045e14			−0.703959
$81$	−2.21751e14			−1.07703
$82$	−1.99424e13			−0.0883433
$83$	−5.96106e13			−0.241123			−0.120561	−	0.992706i	$-0.538469\pi$
$83$	−5.96106e13			−0.241123			−0.120561	+	0.992706i	$0.538469\pi$
$84$		−	4.37371e14i		−	1.61717i
$85$			3.03757e14i			1.02774i
$86$	−2.68813e13			−0.0833127
$87$	6.62953e13	+	3.60331e14i	0.188403	+	1.02402i
$88$	−7.95601e13			−0.207527
$89$			2.57647e14i			0.617448i	0.951152	+	0.308724i	$0.0999019\pi$
$89$			2.57647e14i			0.617448i	−0.951152	+	0.308724i	$0.900098\pi$
$90$		−	2.15451e12i		−	0.00474821i
$91$	−1.00836e15			−2.04553
$92$	3.44262e14			0.643398
$93$	−9.07418e14			−1.56381
$94$	4.55898e13			0.0725119
$95$		−	5.05125e14i		−	0.742116i
$96$	1.79523e14			0.243830
$97$		−	1.01732e15i		−	1.27841i	−0.769037	−	0.639204i	$-0.779264\pi$
$97$		−	1.01732e15i		−	1.27841i	0.769037	−	0.639204i	$-0.220736\pi$
$98$		−	9.75756e13i		−	0.113539i
$99$			1.03147e14i			0.111222i
$100$	−4.82527e14			−0.482527
$101$		−	7.44684e14i		−	0.691133i	−0.938394	−	0.345567i	$-0.887687\pi$
$101$		−	7.44684e14i		−	0.691133i	0.938394	−	0.345567i	$-0.112313\pi$
$102$		−	1.36229e14i		−	0.117427i
$103$	−6.31407e13			−0.0505860			−0.0252930	−	0.999680i	$-0.508052\pi$
$103$	−6.31407e13			−0.0505860			−0.0252930	+	0.999680i	$0.508052\pi$
$104$		−	2.75643e14i		−	0.205398i
$105$			1.68288e15i			1.16716i
$106$		−	1.46035e14i		−	0.0943327i
$107$	2.06071e15			1.24062			0.620308	−	0.784358i	$-0.287007\pi$
$107$	2.06071e15			1.24062			0.620308	+	0.784358i	$0.287007\pi$
$108$			1.68795e15i			0.947725i
$109$	1.35569e14			0.0710331			0.0355166	−	0.999369i	$-0.488692\pi$
$109$	1.35569e14			0.0710331			0.0355166	+	0.999369i	$0.488692\pi$
$110$	1.52587e14			0.0746571
$111$	4.41712e14			0.201938
$112$	−3.58858e15			−1.53387
$113$		−	4.53053e15i		−	1.81159i	−0.423713	−	0.905797i	$-0.639273\pi$
$113$		−	4.53053e15i		−	1.81159i	0.423713	−	0.905797i	$-0.360727\pi$
$114$			2.26539e14i			0.0847923i
$115$	−1.32462e15			−0.464362
$116$	2.97514e15	−	5.47380e14i	0.977400	−	0.179827i
$117$	−3.57360e14			−0.110081
$118$		−	8.81421e13i		−	0.0254723i
$119$			8.25519e15i			2.23936i
$120$	−4.60026e14			−0.117199
$121$	−3.12780e15			−0.748769
$122$	4.83033e14			0.108712
$123$	5.52041e15			1.16865
$124$			7.49227e15i			1.49262i
$125$	5.68049e15			1.06552
$126$		−	5.85530e13i		−	0.0103459i
$127$		−	6.99510e15i		−	1.16484i	−0.812888	−	0.582420i	$-0.802106\pi$
$127$		−	6.99510e15i		−	1.16484i	0.812888	−	0.582420i	$-0.197894\pi$
$128$		−	1.97428e15i		−	0.309980i
$129$	7.44120e15			1.10210
$130$			5.28649e14i			0.0738911i
$131$			4.47945e15i			0.591139i	0.955321	+	0.295570i	$0.0955094\pi$
$131$			4.47945e15i			0.591139i	−0.955321	+	0.295570i	$0.904491\pi$
$132$	1.09776e16			1.36837
$133$		−	1.37277e16i		−	1.61701i
$134$		−	1.04992e15i		−	0.116915i
$135$		−	6.49475e15i		−	0.684005i
$136$	−2.25662e15			−0.224862
$137$			1.83556e16i			1.73127i	0.500675	+	0.865635i	$0.333085\pi$
$137$			1.83556e16i			1.73127i	−0.500675	+	0.865635i	$0.666915\pi$
$138$	5.94066e14			0.0530568
$139$	5.56954e15			0.471204			0.235602	−	0.971850i	$-0.424294\pi$
$139$	5.56954e15			0.471204			0.235602	+	0.971850i	$0.424294\pi$
$140$	1.38950e16			1.11403
$141$	−1.26201e16			−0.959220
$142$		−	1.07459e15i		−	0.0774606i
$143$		−	2.53089e16i		−	1.73083i
$144$	−1.27178e15			−0.0825458
$145$	−1.14475e16	+	2.10616e15i	−0.705422	+	0.129787i
$146$	−1.48493e15			−0.0869079
$147$			2.70106e16i			1.50195i
$148$		−	3.64708e15i		−	0.192745i
$149$	−3.01494e15			−0.151489			−0.0757446	−	0.997127i	$-0.524133\pi$
$149$	−3.01494e15			−0.151489			−0.0757446	+	0.997127i	$0.524133\pi$
$150$	−8.32659e14			−0.0397908
$151$	−7.50238e15			−0.341092			−0.170546	−	0.985350i	$-0.554553\pi$
$151$	−7.50238e15			−0.341092			−0.170546	+	0.985350i	$0.554553\pi$
$152$	3.75258e15			0.162369
$153$			2.92562e15i			0.120512i
$154$	4.14683e15			0.162671
$155$		−	2.88281e16i		−	1.07728i
$156$			3.80328e16i			1.35433i
$157$		−	2.78855e16i		−	0.946523i	−0.880922	−	0.473261i	$-0.843077\pi$
$157$		−	2.78855e16i		−	0.946523i	0.880922	−	0.473261i	$-0.156923\pi$
$158$	−3.67350e15			−0.118892
$159$			4.04250e16i			1.24788i
$160$			5.70334e15i			0.167969i
$161$	−3.59991e16			−1.01181
$162$			3.15950e15i			0.0847725i
$163$			3.85777e16i			0.988391i	0.869351	+	0.494196i	$0.164537\pi$
$163$			3.85777e16i			0.988391i	−0.869351	+	0.494196i	$0.835463\pi$
$164$		−	4.55803e16i		−	1.11545i
$165$	−4.22386e16			−0.987598
$166$			8.49328e14i			0.0189786i
$167$	−7.31922e16			−1.56348			−0.781738	−	0.623607i	$-0.785666\pi$
$167$	−7.31922e16			−1.56348			−0.781738	+	0.623607i	$0.785666\pi$
$168$	−1.25021e16			−0.255366
$169$	3.64990e16			0.713067
$170$	4.32791e15			0.0808931
$171$		−	4.86508e15i		−	0.0870200i
$172$		−	6.14397e16i		−	1.05193i
$173$	−7.69716e16			−1.26178			−0.630891	−	0.775871i	$-0.717311\pi$
$173$	−7.69716e16			−1.26178			−0.630891	+	0.775871i	$0.717311\pi$
$174$	5.13397e15	−	9.44571e14i	0.0805997	−	0.0148291i
$175$	5.04573e16			0.758819
$176$		−	9.00701e16i		−	1.29788i
$177$			2.43993e16i			0.336959i
$178$	3.67094e15			0.0485989
$179$	7.77385e16			0.986821			0.493410	−	0.869797i	$-0.335750\pi$
$179$	7.77385e16			0.986821			0.493410	+	0.869797i	$0.335750\pi$
$180$	4.92434e15			0.0599521
$181$	1.28466e17			1.50038			0.750189	−	0.661224i	$-0.229963\pi$
$181$	1.28466e17			1.50038			0.750189	+	0.661224i	$0.229963\pi$
$182$			1.43670e16i			0.161002i
$183$	−1.33712e17			−1.43809
$184$		−	9.84060e15i		−	0.101599i
$185$			1.40329e16i			0.139110i
$186$			1.29288e16i			0.123087i
$187$	−2.07198e17			−1.89484
$188$			1.04200e17i			0.915553i
$189$		−	1.76507e17i		−	1.49039i
$190$	−7.19698e15			−0.0584115
$191$		−	4.31031e16i		−	0.336324i	−0.985759	−	0.168162i	$-0.946217\pi$
$191$		−	4.31031e16i		−	0.336324i	0.985759	−	0.168162i	$-0.0537832\pi$
$192$			1.33638e17i			1.00270i
$193$			4.88000e16i			0.352160i	0.984376	+	0.176080i	$0.0563417\pi$
$193$			4.88000e16i			0.352160i	−0.984376	+	0.176080i	$0.943658\pi$
$194$	−1.44947e16			−0.100623
$195$		−	1.46339e17i		−	0.977465i
$196$	2.23018e17			1.43357
$197$	9.28881e16			0.574729			0.287365	−	0.957821i	$-0.407221\pi$
$197$	9.28881e16			0.574729			0.287365	+	0.957821i	$0.407221\pi$
$198$	1.46963e15			0.00875424
$199$	−7.36010e16			−0.422168			−0.211084	−	0.977468i	$-0.567699\pi$
$199$	−7.36010e16			−0.422168			−0.211084	+	0.977468i	$0.567699\pi$
$200$			1.37929e16i			0.0761955i
$201$			2.90636e17i			1.54660i
$202$	−1.06102e16			−0.0543987
$203$	−3.11107e17	+	5.72389e16i	−1.53706	+	0.282795i
$204$	3.11365e17			1.48267
$205$			1.75380e17i			0.805054i
$206$			8.99623e14i			0.00398159i
$207$	−1.27580e16			−0.0544508
$208$	3.12055e17			1.28457
$209$	3.44554e17			1.36823
$210$	2.39775e16			0.0918669
$211$		−	1.15633e17i		−	0.427527i	−0.976885	−	0.213763i	$-0.931428\pi$
$211$		−	1.15633e17i		−	0.427527i	0.976885	−	0.213763i	$-0.0685722\pi$
$212$	3.33776e17			1.19107
$213$			2.97466e17i			1.02468i
$214$		−	2.93608e16i		−	0.0976482i
$215$			2.36402e17i			0.759211i
$216$	4.82496e16			0.149655
$217$		−	7.83458e17i		−	2.34730i
$218$		−	1.93157e15i		−	0.00559098i
$219$	4.11054e17			1.14966
$220$			3.48751e17i			0.942639i
$221$		−	7.17853e17i		−	1.87540i
$222$		−	6.29348e15i		−	0.0158944i
$223$	−4.00565e17			−0.978109			−0.489054	−	0.872253i	$-0.662658\pi$
$223$	−4.00565e17			−0.978109			−0.489054	+	0.872253i	$0.662658\pi$
$224$			1.54999e17i			0.365990i
$225$	1.78819e16			0.0408362
$226$	−6.45507e16			−0.142589
$227$	−1.45220e17			−0.310335			−0.155168	−	0.987888i	$-0.549592\pi$
$227$	−1.45220e17			−0.310335			−0.155168	+	0.987888i	$0.549592\pi$
$228$	−5.17776e17			−1.07061
$229$			2.05484e17i			0.411161i	0.978640	+	0.205581i	$0.0659083\pi$
$229$			2.05484e17i			0.411161i	−0.978640	+	0.205581i	$0.934092\pi$
$230$			1.88731e16i			0.0365497i
$231$	−1.14792e18			−2.15189
$232$	−1.56467e16	−	8.50433e16i	−0.0283963	−	0.154341i
$233$	2.99011e17			0.525434			0.262717	−	0.964873i	$-0.415381\pi$
$233$	2.99011e17			0.525434			0.262717	+	0.964873i	$0.415381\pi$
$234$			5.09164e15i			0.00866441i
$235$		−	4.00931e17i		−	0.660786i
$236$	2.01457e17			0.321620
$237$	1.01689e18			1.57275
$238$	1.17619e17			0.176259
$239$	9.04467e17			1.31343			0.656717	−	0.754137i	$-0.271944\pi$
$239$	9.04467e17			1.31343			0.656717	+	0.754137i	$0.271944\pi$
$240$		−	5.20796e17i		−	0.732965i
$241$	−1.06790e18			−1.45681			−0.728406	−	0.685145i	$-0.759739\pi$
$241$	−1.06790e18			−1.45681			−0.728406	+	0.685145i	$0.759739\pi$
$242$			4.45646e16i			0.0589352i
$243$		−	1.30852e17i		−	0.167777i
$244$			1.10402e18i			1.37263i
$245$	−8.58110e17			−1.03466
$246$		−	7.86544e16i		−	0.0919834i
$247$			1.19374e18i			1.35419i
$248$	2.14164e17			0.235700
$249$		−	2.35109e17i		−	0.251058i
$250$		−	8.09352e16i		−	0.0838663i
$251$			5.78856e17i			0.582126i	0.956704	+	0.291063i	$0.0940090\pi$
$251$			5.78856e17i			0.582126i	−0.956704	+	0.291063i	$0.905991\pi$
$252$	1.33829e17			0.130631
$253$		−	9.03543e17i		−	0.856141i
$254$	−9.96657e16			−0.0916839
$255$	−1.19804e18			−1.07009
$256$	1.08216e18			0.938622
$257$	−1.92799e18			−1.62408			−0.812040	−	0.583602i	$-0.801643\pi$
$257$	−1.92799e18			−1.62408			−0.812040	+	0.583602i	$0.801643\pi$
$258$		−	1.06022e17i		−	0.0867455i
$259$			3.81371e17i			0.303110i
$260$	−1.20828e18			−0.932967
$261$	−1.10256e17	+	2.02853e16i	−0.0827173	+	0.0152187i
$262$	6.38228e16			0.0465282
$263$		−	4.62004e16i		−	0.0327324i	−0.999866	−	0.0163662i	$-0.994790\pi$
$263$		−	4.62004e16i		−	0.0327324i	0.999866	−	0.0163662i	$-0.00520976\pi$
$264$		−	3.13791e17i		−	0.216078i
$265$	−1.28427e18			−0.859634
$266$	−1.95592e17			−0.127274
$267$	−1.01618e18			−0.642889
$268$	2.39969e18			1.47620
$269$			1.41101e17i			0.0844090i	0.999109	+	0.0422045i	$0.0134381\pi$
$269$			1.41101e17i			0.0844090i	−0.999109	+	0.0422045i	$0.986562\pi$
$270$	−9.25367e16			−0.0538376
$271$			6.74057e17i			0.381441i	0.981644	+	0.190720i	$0.0610824\pi$
$271$			6.74057e17i			0.381441i	−0.981644	+	0.190720i	$0.938918\pi$
$272$		−	2.55472e18i		−	1.40629i
$273$		−	3.97705e18i		−	2.12981i
$274$	2.61530e17			0.136267
$275$			1.26643e18i			0.642076i
$276$			1.35779e18i			0.669909i
$277$	−3.66468e18			−1.75970			−0.879849	−	0.475254i	$-0.842356\pi$
$277$	−3.66468e18			−1.75970			−0.879849	+	0.475254i	$0.842356\pi$
$278$		−	7.93544e16i		−	0.0370882i
$279$		−	2.77656e17i		−	0.126321i
$280$		−	3.97183e17i		−	0.175916i
$281$	1.14971e18			0.495781			0.247891	−	0.968788i	$-0.420263\pi$
$281$	1.14971e18			0.495781			0.247891	+	0.968788i	$0.420263\pi$
$282$			1.79810e17i			0.0754997i
$283$	1.19025e18			0.486678			0.243339	−	0.969941i	$-0.421757\pi$
$283$	1.19025e18			0.486678			0.243339	+	0.969941i	$0.421757\pi$
$284$	2.45608e18			0.978037
$285$	1.99225e18			0.772694
$286$	−3.60600e17			−0.136232
$287$			4.76628e18i			1.75414i
$288$			5.49313e16i			0.0196959i
$289$	−3.01446e18			−1.05311
$290$	3.00084e16	+	1.63103e17i	0.0102154	+	0.0555234i
$291$	4.01239e18			1.33108
$292$		−	3.39395e18i		−	1.09732i
$293$		−	5.94617e13i		−	1.87383e-5i	−1.00000		9.36915e-6i	$-0.999997\pi$
$293$		−	5.94617e13i		−	1.87383e-5i	1.00000		9.36915e-6i	$-2.98229e-6\pi$
$294$	3.84846e17			0.118217
$295$	−7.75148e17			−0.232124
$296$	−1.04251e17			−0.0304362
$297$	4.43017e18			1.26109
$298$			4.29566e16i			0.0119236i
$299$	3.13040e18			0.847356
$300$		−	1.90312e18i		−	0.502409i
$301$			6.42468e18i			1.65426i
$302$			1.06893e17i			0.0268472i
$303$	2.93709e18			0.719611
$304$			4.24830e18i			1.01546i
$305$		−	4.24794e18i		−	0.990671i
$306$	4.16840e16			0.00948546
$307$		−	9.93427e17i		−	0.220596i	−0.993899	−	0.110298i	$-0.964819\pi$
$307$		−	9.93427e17i		−	0.220596i	0.993899	−	0.110298i	$-0.0351805\pi$
$308$			9.47799e18i			2.05393i
$309$		−	2.49032e17i		−	0.0526703i
$310$	−4.10740e17			−0.0847919
$311$			9.10828e17i			0.183541i	0.995780	+	0.0917705i	$0.0292526\pi$
$311$			9.10828e17i			0.183541i	−0.995780	+	0.0917705i	$0.970747\pi$
$312$	1.08716e18			0.213861
$313$	1.79762e18			0.345235			0.172617	−	0.984989i	$-0.444778\pi$
$313$	1.79762e18			0.345235			0.172617	+	0.984989i	$0.444778\pi$
$314$	−3.97311e17			−0.0745002
$315$	−5.14933e17			−0.0942805
$316$		−	8.39613e18i		−	1.50116i
$317$		−	3.05214e18i		−	0.532917i	−0.963846	−	0.266459i	$-0.914146\pi$
$317$		−	3.05214e18i		−	0.532917i	0.963846	−	0.266459i	$-0.0858535\pi$
$318$	5.75972e17			0.0982196
$319$	−1.43664e18	−	7.80850e18i	−0.239287	−	1.30058i
$320$	−4.24560e18			−0.690739
$321$			8.12758e18i			1.29174i
$322$			5.12912e17i			0.0796386i
$323$	9.77281e18			1.48252
$324$	−7.22133e18			−1.07036
$325$	−4.38765e18			−0.635488
$326$	5.49651e17			0.0777957
$327$			5.34694e17i			0.0739600i
$328$	−1.30290e18			−0.176139
$329$		−	1.08961e19i		−	1.43980i
$330$			6.01813e17i			0.0777332i
$331$			1.27382e19i			1.60841i	0.594352	+	0.804205i	$0.297409\pi$
$331$			1.27382e19i			1.60841i	−0.594352	+	0.804205i	$0.702591\pi$
$332$	−1.94122e18			−0.239629
$333$			1.35157e17i			0.0163120i
$334$			1.04284e18i			0.123060i
$335$	−9.23331e18			−1.06542
$336$		−	1.41536e19i		−	1.59707i
$337$		−	1.06800e19i		−	1.17855i	−0.807932	−	0.589276i	$-0.799413\pi$
$337$		−	1.06800e19i		−	1.17855i	0.807932	−	0.589276i	$-0.200587\pi$
$338$		−	5.20034e17i		−	0.0561250i
$339$	1.78688e19			1.88624
$340$			9.89185e18i			1.02138i
$341$	1.96641e19			1.98617
$342$	−6.93173e16			−0.00684929
$343$	−7.15399e18			−0.691580
$344$	−1.75623e18			−0.166109
$345$		−	5.22439e18i		−	0.483496i
$346$			1.09668e18i			0.0993141i
$347$	−7.14029e18			−0.632768			−0.316384	−	0.948631i	$-0.602469\pi$
$347$	−7.14029e18			−0.632768			−0.316384	+	0.948631i	$0.602469\pi$
$348$	2.15891e18	+	1.17342e19i	0.187236	+	1.01767i
$349$	2.51802e18			0.213732			0.106866	−	0.994273i	$-0.465918\pi$
$349$	2.51802e18			0.213732			0.106866	+	0.994273i	$0.465918\pi$
$350$		−	7.18911e17i		−	0.0597262i
$351$			1.53487e19i			1.24815i
$352$	−3.89034e18			−0.309683
$353$	−6.35447e18			−0.495187			−0.247593	−	0.968864i	$-0.579640\pi$
$353$	−6.35447e18			−0.495187			−0.247593	+	0.968864i	$0.579640\pi$
$354$	3.47639e17			0.0265219
$355$	−9.45028e18			−0.705882
$356$			8.39028e18i			0.613623i
$357$	−3.25591e19			−2.33164
$358$		−	1.10761e18i		−	0.0776721i
$359$		−	1.35329e19i		−	0.929358i	−0.885479	−	0.464679i	$-0.846170\pi$
$359$		−	1.35329e19i		−	0.929358i	0.885479	−	0.464679i	$-0.153830\pi$
$360$		−	1.40761e17i		−	0.00946701i
$361$	−1.07030e18			−0.0705021
$362$		−	1.83038e18i		−	0.118094i
$363$		−	1.23363e19i		−	0.779622i
$364$	−3.28373e19			−2.03286
$365$			1.30589e19i			0.791974i
$366$			1.90512e18i			0.113191i
$367$		−	2.21242e19i		−	1.28787i	−0.765080	−	0.643936i	$-0.777300\pi$
$367$		−	2.21242e19i		−	1.28787i	0.765080	−	0.643936i	$-0.222700\pi$
$368$	1.11406e19			0.635402
$369$			1.68916e18i			0.0944001i
$370$	1.99940e17			0.0109493
$371$	−3.49026e19			−1.87307
$372$	−2.95501e19			−1.55413
$373$	2.00591e19			1.03394			0.516969	−	0.856004i	$-0.327060\pi$
$373$	2.00591e19			1.03394			0.516969	+	0.856004i	$0.327060\pi$
$374$			2.95214e18i			0.149142i
$375$			2.24043e19i			1.10942i
$376$	2.97852e18			0.144575
$377$	2.70532e19	−	4.97737e18i	1.28724	−	0.236832i
$378$	−2.51486e18			−0.117308
$379$			3.43988e19i			1.57307i	0.617544	+	0.786536i	$0.288128\pi$
$379$			3.43988e19i			1.57307i	−0.617544	+	0.786536i	$0.711872\pi$
$380$		−	1.64494e19i		−	0.737519i
$381$	2.75892e19			1.21284
$382$	−6.14129e17			−0.0264719
$383$	3.41007e19			1.44136			0.720680	−	0.693268i	$-0.243830\pi$
$383$	3.41007e19			1.44136			0.720680	+	0.693268i	$0.243830\pi$
$384$	7.78669e18			0.322752
$385$		−	3.64685e19i		−	1.48239i
$386$	6.95299e17			0.0277183
$387$			2.27689e18i			0.0890246i
$388$		−	3.31290e19i		−	1.27049i
$389$			8.33942e18i			0.313700i	0.987622	+	0.156850i	$0.0501339\pi$
$389$			8.33942e18i			0.313700i	−0.987622	+	0.156850i	$0.949866\pi$
$390$	−2.08503e18			−0.0769357
$391$		−	2.56278e19i		−	0.927653i
$392$		−	6.37491e18i		−	0.226375i
$393$	−1.76673e19			−0.615497
$394$		−	1.32346e18i		−	0.0452366i
$395$			3.23059e19i			1.08344i
$396$			3.35897e18i			0.110533i
$397$	4.88670e19			1.57793			0.788963	−	0.614441i	$-0.210618\pi$
$397$	4.88670e19			1.57793			0.788963	+	0.614441i	$0.210618\pi$
$398$			1.04866e18i			0.0332286i
$399$	5.41433e19			1.68363
$400$	−1.56149e19			−0.476529
$401$	−9.54145e18			−0.285780			−0.142890	−	0.989739i	$-0.545639\pi$
$401$	−9.54145e18			−0.285780			−0.142890	+	0.989739i	$0.545639\pi$
$402$	4.14096e18			0.121732
$403$			6.81278e19i			1.96579i
$404$		−	2.42506e19i		−	0.686852i
$405$	2.77856e19			0.772515
$406$	8.15536e17	+	4.43263e18i	0.0222586	+	0.120981i
$407$	−9.57207e18			−0.256477
$408$		−	8.90026e18i		−	0.234127i
$409$		−	3.26125e19i		−	0.842285i	−0.906994	−	0.421143i	$-0.861629\pi$
$409$		−	3.26125e19i		−	0.842285i	0.906994	−	0.421143i	$-0.138371\pi$
$410$	2.49880e18			0.0633654
$411$	−7.23960e19			−1.80261
$412$	−2.05618e18			−0.0502726
$413$	−2.10662e19			−0.505778
$414$			1.81775e17i			0.00428579i
$415$	7.46925e18			0.172948
$416$		−	1.34784e19i		−	0.306506i
$417$			2.19667e19i			0.490619i
$418$		−	4.90918e18i		−	0.107693i
$419$	3.92562e19			0.845869			0.422934	−	0.906160i	$-0.361000\pi$
$419$	3.92562e19			0.845869			0.422934	+	0.906160i	$0.361000\pi$
$420$			5.48028e19i			1.15993i
$421$			7.01731e19i			1.45900i	0.683981	+	0.729500i	$0.260247\pi$
$421$			7.01731e19i			1.45900i	−0.683981	+	0.729500i	$0.739753\pi$
$422$	−1.64753e18			−0.0336504
$423$		−	3.86154e18i		−	0.0774833i
$424$		−	9.54089e18i		−	0.188081i
$425$			3.59206e19i			0.695707i
$426$	4.23827e18			0.0806523
$427$		−	1.15446e20i		−	2.15859i
$428$	6.71069e19			1.23293
$429$	9.98203e19			1.80214
$430$	3.36824e18			0.0597571
$431$	1.67187e17			0.00291489			0.00145744	−	0.999999i	$-0.499536\pi$
$431$	1.67187e17			0.00291489			0.00145744	+	0.999999i	$0.499536\pi$
$432$			5.46234e19i			0.935946i
$433$			7.02058e19i			1.18226i	0.806575	+	0.591131i	$0.201318\pi$
$433$			7.02058e19i			1.18226i	−0.806575	+	0.591131i	$0.798682\pi$
$434$	−1.11627e19			−0.184754
$435$	−8.30684e18	−	4.51497e19i	−0.135135	−	0.734489i
$436$	4.41480e18			0.0705931
$437$			4.26170e19i			0.669842i
$438$		−	5.85667e18i		−	0.0904889i
$439$	−2.11244e19			−0.320849			−0.160424	−	0.987048i	$-0.551286\pi$
$439$	−2.11244e19			−0.320849			−0.160424	+	0.987048i	$0.551286\pi$
$440$	9.96893e18			0.148852
$441$	−8.26483e18			−0.121323
$442$	−1.02279e19			−0.147612
$443$		−	4.33681e19i		−	0.615379i	−0.951487	−	0.307689i	$-0.900444\pi$
$443$		−	4.33681e19i		−	0.615379i	0.951487	−	0.307689i	$-0.0995558\pi$
$444$	1.43844e19			0.200687
$445$		−	3.22833e19i		−	0.442872i
$446$			5.70723e18i			0.0769863i
$447$		−	1.18911e19i		−	0.157731i
$448$	−1.15382e20			−1.50506
$449$			2.26273e18i			0.0290258i	0.999895	+	0.0145129i	$0.00461977\pi$
$449$			2.26273e18i			0.0290258i	−0.999895	+	0.0145129i	$0.995380\pi$
$450$		−	2.54780e17i		−	0.00321419i
$451$	−1.19629e20			−1.48427
$452$		−	1.47537e20i		−	1.80037i
$453$		−	2.95900e19i		−	0.355147i
$454$			2.06908e18i			0.0244263i
$455$	1.26348e20			1.46718
$456$			1.48005e19i			0.169059i
$457$	−1.21620e20			−1.36658			−0.683288	−	0.730149i	$-0.739451\pi$
$457$	−1.21620e20			−1.36658			−0.683288	+	0.730149i	$0.739451\pi$
$458$	2.92772e18			0.0323623
$459$	1.25656e20			1.36643
$460$	−4.31362e19			−0.461486
$461$			6.87253e19i			0.723368i	0.932301	+	0.361684i	$0.117798\pi$
$461$			6.87253e19i			0.723368i	−0.932301	+	0.361684i	$0.882202\pi$
$462$			1.63554e19i			0.169374i
$463$	−1.27304e20			−1.29713			−0.648566	−	0.761158i	$-0.724631\pi$
$463$	−1.27304e20			−1.29713			−0.648566	+	0.761158i	$0.724631\pi$
$464$	9.62776e19	−	1.77136e19i	0.965252	−	0.177591i
$465$	1.13700e20			1.12167
$466$		−	4.26029e18i		−	0.0413566i
$467$		−	1.74988e20i		−	1.67159i	−0.549039	−	0.835797i	$-0.685006\pi$
$467$		−	1.74988e20i		−	1.67159i	0.549039	−	0.835797i	$-0.314994\pi$
$468$	−1.16374e19			−0.109399
$469$	−2.50933e20			−2.32146
$470$	−5.71244e18			−0.0520101
$471$	1.09983e20			0.985523
$472$		−	5.75859e18i		−	0.0507868i
$473$	−1.61254e20			−1.39975
$474$		−	1.44886e19i		−	0.123791i
$475$		−	5.97332e19i		−	0.502359i
$476$			2.68830e20i			2.22549i
$477$	−1.23694e19			−0.100800
$478$		−	1.28868e19i		−	0.103380i
$479$			8.83757e19i			0.697937i	0.937134	+	0.348969i	$0.113468\pi$
$479$			8.83757e19i			0.697937i	−0.937134	+	0.348969i	$0.886532\pi$
$480$	−2.24944e19			−0.174890
$481$		−	3.31632e19i		−	0.253845i
$482$			1.52154e19i			0.114665i
$483$		−	1.41983e20i		−	1.05350i
$484$	−1.01857e20			−0.744131
$485$			1.27471e20i			0.916955i
$486$	−1.86437e18			−0.0132056
$487$	−1.51917e20			−1.05959			−0.529797	−	0.848125i	$-0.677732\pi$
$487$	−1.51917e20			−1.05959			−0.529797	+	0.848125i	$0.677732\pi$
$488$	3.15580e19			0.216751
$489$	−1.52153e20			−1.02912
$490$			1.22263e19i			0.0814374i
$491$			1.82720e20i			1.19860i	0.800523	+	0.599302i	$0.204555\pi$
$491$			1.82720e20i			1.19860i	−0.800523	+	0.599302i	$0.795445\pi$
$492$	1.79772e20			1.16141
$493$	−4.07485e19	−	2.21478e20i	−0.259274	−	1.40922i
$494$	1.70082e19			0.106588
$495$		−	1.29244e19i		−	0.0797755i
$496$			2.42455e20i			1.47407i
$497$	−2.56830e20			−1.53806
$498$	−3.34981e18			−0.0197606
$499$	9.74050e19			0.566014			0.283007	−	0.959118i	$-0.408668\pi$
$499$	9.74050e19			0.566014			0.283007	+	0.959118i	$0.408668\pi$
$500$	1.84985e20			1.05892
$501$		−	2.88675e20i		−	1.62790i
$502$	8.24749e18			0.0458188
$503$		−	3.36436e19i		−	0.184138i	−0.995753	−	0.0920688i	$-0.970652\pi$
$503$		−	3.36436e19i		−	0.184138i	0.995753	−	0.0920688i	$-0.0293480\pi$
$504$		−	3.82544e18i		−	0.0206278i
$505$			9.33094e19i			0.495724i
$506$	−1.28736e19			−0.0673863
$507$			1.43955e20i			0.742448i
$508$		−	2.27796e20i		−	1.15762i
$509$	3.26198e20			1.63342			0.816710	−	0.577049i	$-0.195796\pi$
$509$	3.26198e20			1.63342			0.816710	+	0.577049i	$0.195796\pi$
$510$			1.70696e19i			0.0842262i
$511$			3.54901e20i			1.72564i
$512$		−	8.01116e19i		−	0.383858i
$513$	−2.08956e20			−0.986677
$514$			2.74699e19i			0.127830i
$515$	7.91156e18			0.0362834
$516$	2.42323e20			1.09527
$517$	2.73481e20			1.21828
$518$	5.43375e18			0.0238576
$519$		−	3.03582e20i		−	1.31377i
$520$			3.45382e19i			0.147324i
$521$	−3.13876e20			−1.31970			−0.659849	−	0.751398i	$-0.729380\pi$
$521$	−3.13876e20			−1.31970			−0.659849	+	0.751398i	$0.729380\pi$
$522$	2.89024e17	+	1.57091e18i	0.00119786	+	0.00651063i
$523$	1.55746e20			0.636288			0.318144	−	0.948042i	$-0.396941\pi$
$523$	1.55746e20			0.636288			0.318144	+	0.948042i	$0.396941\pi$
$524$			1.45873e20i			0.587477i
$525$			1.99007e20i			0.790086i
$526$	−6.58260e17			−0.00257635
$527$	5.57745e20			2.15207
$528$	3.55243e20			1.35136
$529$	−1.54878e20			−0.580861
$530$			1.82982e19i			0.0676613i
$531$	−7.46579e18			−0.0272187
$532$		−	4.47044e20i		−	1.60699i
$533$		−	4.14465e20i		−	1.46904i
$534$			1.44785e19i			0.0506014i
$535$	−2.58208e20			−0.889848
$536$		−	6.85943e19i		−	0.233105i
$537$			3.06606e20i			1.02748i
$538$	2.01040e18			0.00664378
$539$		−	5.85331e20i		−	1.90759i
$540$		−	2.11502e20i		−	0.679768i
$541$		−	8.53677e19i		−	0.270591i	−0.990805	−	0.135296i	$-0.956802\pi$
$541$		−	8.53677e19i		−	0.270591i	0.990805	−	0.135296i	$-0.0431984\pi$
$542$	9.60392e18			0.0300230
$543$			5.06682e20i			1.56220i
$544$	−1.10344e20			−0.335550
$545$	−1.69869e19			−0.0509494
$546$	−5.66647e19			−0.167636
$547$	−1.20909e20			−0.352822			−0.176411	−	0.984317i	$-0.556449\pi$
$547$	−1.20909e20			−0.352822			−0.176411	+	0.984317i	$0.556449\pi$
$548$			5.97751e20i			1.72055i
$549$		−	4.09137e19i		−	0.116165i
$550$	1.80440e19			0.0505374
$551$	6.77616e19	+	3.68300e20i	0.187218	+	1.01757i
$552$	3.88121e19			0.105785
$553$			8.77974e20i			2.36071i
$554$			5.22141e19i			0.138505i
$555$	−5.53468e19			−0.144842
$556$	1.81372e20			0.468284
$557$	2.80542e20			0.714634			0.357317	−	0.933983i	$-0.383692\pi$
$557$	2.80542e20			0.714634			0.357317	+	0.933983i	$0.383692\pi$
$558$	−3.95602e18			−0.00994264
$559$		−	5.58676e20i		−	1.38539i
$560$	4.49652e20			1.10019
$561$		−	8.17203e20i		−	1.97292i
$562$		−	1.63810e19i		−	0.0390227i
$563$			1.73524e20i			0.407893i	0.978982	+	0.203946i	$0.0653768\pi$
$563$			1.73524e20i			0.407893i	−0.978982	+	0.203946i	$0.934623\pi$
$564$	−4.10973e20			−0.953278
$565$			5.67678e20i			1.29939i
$566$		−	1.69586e19i		−	0.0383061i
$567$	7.55127e20			1.68324
$568$		−	7.02063e19i		−	0.154441i
$569$			4.80192e20i			1.04249i	0.853406	+	0.521246i	$0.174533\pi$
$569$			4.80192e20i			1.04249i	−0.853406	+	0.521246i	$0.825467\pi$
$570$		−	2.83854e19i		−	0.0608183i
$571$	−4.96467e20			−1.04983			−0.524915	−	0.851154i	$-0.675903\pi$
$571$	−4.96467e20			−1.04983			−0.524915	+	0.851154i	$0.675903\pi$
$572$		−	8.24185e20i		−	1.72010i
$573$	1.70002e20			0.350182
$574$	6.79096e19			0.138068
$575$	−1.56642e20			−0.314339
$576$	−4.08912e19			−0.0809955
$577$			6.85429e20i			1.34012i	0.742306	+	0.670061i	$0.233732\pi$
$577$			6.85429e20i			1.34012i	−0.742306	+	0.670061i	$0.766268\pi$
$578$			4.29498e19i			0.0828900i
$579$	−1.92471e20			−0.366670
$580$	−3.72787e20	+	6.85870e19i	−0.701052	+	0.128983i
$581$	2.02991e20			0.376839
$582$		−	5.71682e19i		−	0.104769i
$583$		−	8.76024e20i		−	1.58490i
$584$	−9.70149e19			−0.173277
$585$	4.47775e19			0.0789570
$586$	−8.47206e14			−1.47488e−6
$587$	−5.31893e20			−0.914194			−0.457097	−	0.889417i	$-0.651111\pi$
$587$	−5.31893e20			−0.914194			−0.457097	+	0.889417i	$0.651111\pi$
$588$			8.79602e20i			1.49264i
$589$	−9.27487e20			−1.55397
$590$			1.10443e19i			0.0182703i
$591$			3.66358e20i			0.598410i
$592$		−	1.18022e20i		−	0.190349i
$593$	−2.71133e20			−0.431790			−0.215895	−	0.976417i	$-0.569267\pi$
$593$	−2.71133e20			−0.431790			−0.215895	+	0.976417i	$0.569267\pi$
$594$		−	6.31208e19i		−	0.0992599i
$595$		−	1.03438e21i		−	1.60621i
$596$	−9.81815e19			−0.150551
$597$		−	2.90288e20i		−	0.439563i
$598$		−	4.46017e19i		−	0.0666949i
$599$		−	1.06998e20i		−	0.158006i	−0.996874	−	0.0790028i	$-0.974826\pi$
$599$		−	1.06998e20i		−	0.158006i	0.996874	−	0.0790028i	$-0.0251736\pi$
$600$	−5.44001e19			−0.0793351
$601$		−	3.67790e20i		−	0.529714i	−0.964288	−	0.264857i	$-0.914675\pi$
$601$		−	3.67790e20i		−	0.529714i	0.964288	−	0.264857i	$-0.0853247\pi$
$602$	9.15384e19			0.130206
$603$	−8.89300e19			−0.124931
$604$	−2.44315e20			−0.338979
$605$	3.91915e20			0.537064
$606$		−	4.18474e19i		−	0.0566401i
$607$		−	2.20939e20i		−	0.295363i	−0.989035	−	0.147682i	$-0.952819\pi$
$607$		−	2.20939e20i		−	0.295363i	0.989035	−	0.147682i	$-0.0471811\pi$
$608$	1.83494e20			0.242295
$609$	−2.25755e20	−	1.22703e21i	−0.294447	−	1.60039i
$610$	−6.05243e19			−0.0779751
$611$			9.47499e20i			1.20578i
$612$			9.52728e19i			0.119766i
$613$	−4.12845e20			−0.512665			−0.256333	−	0.966589i	$-0.582514\pi$
$613$	−4.12845e20			−0.512665			−0.256333	+	0.966589i	$0.582514\pi$
$614$	−1.41543e19			−0.0173630
$615$	−6.91711e20			−0.838226
$616$	2.70925e20			0.324335
$617$		−	4.45244e20i		−	0.526574i	−0.964718	−	0.263287i	$-0.915193\pi$
$617$		−	4.45244e20i		−	0.526574i	0.964718	−	0.263287i	$-0.0848066\pi$
$618$	−3.54818e18			−0.00414565
$619$			6.95431e20i			0.802739i	0.915916	+	0.401369i	$0.131466\pi$
$619$			6.95431e20i			0.802739i	−0.915916	+	0.401369i	$0.868534\pi$
$620$		−	9.38786e20i		−	1.07060i
$621$			5.47958e20i			0.617391i
$622$	1.29774e19			0.0144464
$623$		−	8.77362e20i		−	0.964980i
$624$			1.23077e21i			1.33750i
$625$	−2.59580e20			−0.278722
$626$		−	2.56123e19i		−	0.0271732i
$627$			1.35895e21i			1.42461i
$628$		−	9.08091e20i		−	0.940659i
$629$	−2.71499e20			−0.277900
$630$			7.33673e18i			0.00742076i
$631$	−1.87196e21			−1.87101			−0.935504	−	0.353317i	$-0.885054\pi$
$631$	−1.87196e21			−1.87101			−0.935504	+	0.353317i	$0.885054\pi$
$632$	−2.40001e20			−0.237047
$633$	4.56065e20			0.445143
$634$	−4.34866e19			−0.0419456
$635$			8.76491e20i			0.835496i
$636$			1.31644e21i			1.24015i
$637$	2.02793e21			1.88802
$638$	−1.11255e20	+	2.04692e19i	−0.102368	+	0.0188341i
$639$	−9.10198e19			−0.0827712
$640$			2.47378e20i			0.222337i
$641$		−	1.46571e21i		−	1.30201i	−0.759074	−	0.651004i	$-0.774348\pi$
$641$		−	1.46571e21i		−	1.30201i	0.759074	−	0.651004i	$-0.225652\pi$
$642$	1.15801e20			0.101672
$643$	5.41636e20			0.470029			0.235015	−	0.971992i	$-0.424486\pi$
$643$	5.41636e20			0.470029			0.235015	+	0.971992i	$0.424486\pi$
$644$	−1.17231e21			−1.00554
$645$	−9.32387e20			−0.790494
$646$		−	1.39242e20i		−	0.116688i
$647$	1.91487e21			1.58620			0.793098	−	0.609094i	$-0.208467\pi$
$647$	1.91487e21			1.58620			0.793098	+	0.609094i	$0.208467\pi$
$648$			2.06419e20i			0.169020i
$649$		−	5.28741e20i		−	0.427965i
$650$			6.25150e19i			0.0500189i
$651$	3.09002e21			2.44401
$652$			1.25628e21i			0.982268i
$653$			6.80771e20i			0.526202i	0.964768	+	0.263101i	$0.0847452\pi$
$653$			6.80771e20i			0.526202i	−0.964768	+	0.263101i	$0.915255\pi$
$654$	7.61827e18			0.00582135
$655$		−	5.61278e20i		−	0.424002i
$656$		−	1.47501e21i		−	1.10158i
$657$			1.25776e20i			0.0928663i
$658$	−1.55246e20			−0.113325
$659$			7.73275e20i			0.558076i	0.960280	+	0.279038i	$0.0900155\pi$
$659$			7.73275e20i			0.558076i	−0.960280	+	0.279038i	$0.909984\pi$
$660$	−1.37550e21			−0.981480
$661$	4.90627e20			0.346131			0.173065	−	0.984910i	$-0.444633\pi$
$661$	4.90627e20			0.346131			0.173065	+	0.984910i	$0.444633\pi$
$662$	1.81492e20			0.126597
$663$	2.83127e21			1.95267
$664$			5.54891e19i			0.0378397i
$665$			1.72010e21i			1.15982i
$666$	1.92571e18			0.00128391
$667$	9.65815e20	−	1.77695e20i	0.636722	−	0.117147i
$668$	−2.38350e21			−1.55379
$669$		−	1.57986e21i		−	1.01841i
$670$			1.31555e20i			0.0838587i
$671$	2.89759e21			1.82649
$672$	−6.11329e20			−0.381070
$673$	−1.54038e21			−0.949541			−0.474771	−	0.880110i	$-0.657469\pi$
$673$	−1.54038e21			−0.949541			−0.474771	+	0.880110i	$0.657469\pi$
$674$	−1.52168e20			−0.0927631
$675$		−	7.68032e20i		−	0.463022i
$676$	1.18859e21			0.708649
$677$		−	9.28869e20i		−	0.547696i	−0.961773	−	0.273848i	$-0.911704\pi$
$677$		−	9.28869e20i		−	0.547696i	0.961773	−	0.273848i	$-0.0882965\pi$
$678$		−	2.54593e20i		−	0.148465i
$679$			3.46427e21i			1.99797i
$680$	2.82755e20			0.161285
$681$		−	5.72757e20i		−	0.323122i
$682$		−	2.80172e20i		−	0.156330i
$683$	1.14912e21			0.634179			0.317089	−	0.948396i	$-0.397294\pi$
$683$	1.14912e21			0.634179			0.317089	+	0.948396i	$0.397294\pi$
$684$		−	1.58431e20i		−	0.0864809i
$685$		−	2.29997e21i		−	1.24178i
$686$			1.01930e20i			0.0544339i
$687$	−8.10445e20			−0.428103
$688$		−	1.98823e21i		−	1.03885i
$689$	3.03506e21			1.56864
$690$	−7.44368e19			−0.0380557
$691$	−1.26071e21			−0.637575			−0.318788	−	0.947826i	$-0.603276\pi$
$691$	−1.26071e21			−0.637575			−0.318788	+	0.947826i	$0.603276\pi$
$692$	−2.50658e21			−1.25397
$693$		−	3.51244e20i		−	0.173824i
$694$			1.01734e20i			0.0498048i
$695$	−6.97867e20			−0.337977
$696$	3.35417e20	−	6.17116e19i	0.160700	−	0.0295664i
$697$	−3.39312e21			−1.60825
$698$		−	3.58766e19i		−	0.0168227i
$699$			1.17932e21i			0.547084i
$700$	1.64314e21			0.754118
$701$	−1.26828e21			−0.575876			−0.287938	−	0.957649i	$-0.592970\pi$
$701$	−1.26828e21			−0.575876			−0.287938	+	0.957649i	$0.592970\pi$
$702$	2.18687e20			0.0982415
$703$	4.51482e20			0.200667
$704$		−	2.89599e21i		−	1.27351i
$705$	1.58130e21			0.688013
$706$			9.05380e19i			0.0389759i
$707$			2.53586e21i			1.08014i
$708$			7.94562e20i			0.334872i
$709$	2.83708e21			1.18311			0.591555	−	0.806265i	$-0.298514\pi$
$709$	2.83708e21			1.18311			0.591555	+	0.806265i	$0.298514\pi$
$710$			1.34647e20i			0.0555596i
$711$			3.11152e20i			0.127043i
$712$	2.39833e20			0.0968968
$713$			2.43220e21i			0.972363i
$714$			4.63899e20i			0.183522i
$715$			3.17122e21i			1.24146i
$716$	2.53155e21			0.980707
$717$			3.56729e21i			1.36755i
$718$	−1.92816e20			−0.0731492
$719$	9.38368e20			0.352295			0.176148	−	0.984364i	$-0.443636\pi$
$719$	9.38368e20			0.352295			0.176148	+	0.984364i	$0.443636\pi$
$720$	1.59355e20			0.0592070
$721$	2.15012e20			0.0790584
$722$			1.52496e19i			0.00554918i
$723$		−	4.21189e21i		−	1.51684i
$724$	4.18351e21			1.49108
$725$	−1.35371e21	+	2.49062e20i	−0.477520	+	0.0878563i
$726$	−1.75766e20			−0.0613635
$727$			1.75310e21i			0.605756i	0.953029	+	0.302878i	$0.0979475\pi$
$727$			1.75310e21i			0.605756i	−0.953029	+	0.302878i	$0.902053\pi$
$728$			9.38642e20i			0.321007i
$729$	−2.66580e21			−0.902342
$730$	1.86063e20			0.0623358
$731$	−4.57374e21			−1.51667
$732$	−4.35433e21			−1.42918
$733$		−	2.19731e21i		−	0.713859i	−0.934131	−	0.356930i	$-0.883824\pi$
$733$		−	2.19731e21i		−	0.713859i	0.934131	−	0.356930i	$-0.116176\pi$
$734$	−3.15224e20			−0.101368
$735$		−	3.38445e21i		−	1.07729i
$736$		−	4.81187e20i		−	0.151611i
$737$		−	6.29819e21i		−	1.96431i
$738$	2.40670e19			0.00743018
$739$		−	7.30410e20i		−	0.223220i	−0.993752	−	0.111610i	$-0.964399\pi$
$739$		−	7.30410e20i		−	0.223220i	0.993752	−	0.111610i	$-0.0356008\pi$
$740$			4.56981e20i			0.138249i
$741$	−4.70818e21			−1.40999
$742$			4.97290e20i			0.147428i
$743$			2.48383e21i			0.728965i	0.931210	+	0.364482i	$0.118754\pi$
$743$			2.48383e21i			0.728965i	−0.931210	+	0.364482i	$0.881246\pi$
$744$			8.44679e20i			0.245411i
$745$	3.77774e20			0.108658
$746$		−	2.85800e20i		−	0.0813807i
$747$	7.19396e19			0.0202798
$748$	−6.74739e21			−1.88310
$749$	−7.01729e21			−1.93890
$750$	3.19215e20			0.0873219
$751$			5.14643e20i			0.139382i	0.997569	+	0.0696910i	$0.0222013\pi$
$751$			5.14643e20i			0.139382i	−0.997569	+	0.0696910i	$0.977799\pi$
$752$			3.37199e21i			0.904174i
$753$	−2.28305e21			−0.606112
$754$	−7.09172e19	−	3.85452e20i	−0.0186409	−	0.101318i
$755$	9.40053e20			0.244653
$756$		−	5.74796e21i		−	1.48116i
$757$			2.75164e21i			0.702057i	0.936365	+	0.351029i	$0.114168\pi$
$757$			2.75164e21i			0.702057i	−0.936365	+	0.351029i	$0.885832\pi$
$758$	4.90111e20			0.123816
$759$	3.56364e21			0.891417
$760$	−4.70201e20			−0.116461
$761$	1.24448e21			0.305213			0.152607	−	0.988287i	$-0.451233\pi$
$761$	1.24448e21			0.305213			0.152607	+	0.988287i	$0.451233\pi$
$762$		−	3.93089e20i		−	0.0954616i
$763$	−4.61651e20			−0.111014
$764$		−	1.40365e21i		−	0.334240i
$765$		−	3.66582e20i		−	0.0864391i
$766$		−	4.85864e20i		−	0.113449i
$767$	1.83187e21			0.423573
$768$			4.26811e21i			0.977297i
$769$			2.71662e20i			0.0616001i	0.999526	+	0.0308001i	$0.00980552\pi$
$769$			2.71662e20i			0.0616001i	−0.999526	+	0.0308001i	$0.990194\pi$
$770$	−5.19601e20			−0.116678
$771$		−	7.60415e21i		−	1.69100i
$772$			1.58917e21i			0.349978i
$773$		−	2.14952e21i		−	0.468809i	−0.972139	−	0.234404i	$-0.924686\pi$
$773$		−	2.14952e21i		−	0.468809i	0.972139	−	0.234404i	$-0.0753140\pi$
$774$	3.24410e19			0.00700707
$775$		−	3.40904e21i		−	0.729239i
$776$	−9.46982e20			−0.200622
$777$	−1.50416e21			−0.315599
$778$	1.18819e20			0.0246911
$779$	5.64250e21			1.16129
$780$		−	4.76554e21i		−	0.971409i
$781$		−	6.44619e21i		−	1.30143i
$782$	−3.65143e20			−0.0730150
$783$	8.71259e20	+	4.73550e21i	0.172558	+	0.937891i
$784$	7.21704e21			1.41576
$785$			3.49407e21i			0.678905i
$786$			2.51722e20i			0.0484454i
$787$	8.58522e20			0.163659			0.0818297	−	0.996646i	$-0.473924\pi$
$787$	8.58522e20			0.163659			0.0818297	+	0.996646i	$0.473924\pi$
$788$	3.02490e21			0.571169
$789$	1.82218e20			0.0340811
$790$	4.60291e20			0.0852766
$791$			1.54278e22i			2.83126i
$792$	9.60151e19			0.0174542
$793$			1.00389e22i			1.80775i
$794$		−	6.96253e20i		−	0.124198i
$795$		−	5.06528e21i		−	0.895055i
$796$	−2.39682e21			−0.419553
$797$		−	3.13571e21i		−	0.543749i	−0.962333	−	0.271875i	$-0.912356\pi$
$797$		−	3.13571e21i		−	0.543749i	0.962333	−	0.271875i	$-0.0876436\pi$
$798$		−	7.71429e20i		−	0.132518i
$799$	7.75693e21			1.32005
$800$			6.74444e20i			0.113703i
$801$		−	3.10935e20i		−	0.0519309i
$802$			1.35946e20i			0.0224936i
$803$	−8.90770e21			−1.46015
$804$			9.46456e21i			1.53702i
$805$	4.51070e21			0.725730
$806$	9.70680e20			0.154726
$807$	−5.56514e20			−0.0878869
$808$	−6.93196e20			−0.108460
$809$		−	5.58644e21i		−	0.866007i	−0.901392	−	0.433003i	$-0.857454\pi$
$809$		−	5.58644e21i		−	0.866007i	0.901392	−	0.433003i	$-0.142546\pi$
$810$		−	3.95887e20i		−	0.0608042i
$811$	−2.60607e21			−0.396579			−0.198290	−	0.980143i	$-0.563539\pi$
$811$	−2.60607e21			−0.396579			−0.198290	+	0.980143i	$0.563539\pi$
$812$	−1.01312e22	+	1.86398e21i	−1.52753	+	0.281043i
$813$	−2.65853e21			−0.397158
$814$			1.36382e20i			0.0201871i
$815$		−	4.83380e21i		−	0.708936i
$816$	1.00760e22			1.46424
$817$	7.60578e21			1.09516
$818$	−4.64660e20			−0.0662958
$819$	1.21691e21			0.172041
$820$			5.71124e21i			0.800067i
$821$	−4.65689e21			−0.646431			−0.323216	−	0.946325i	$-0.604764\pi$
$821$	−4.65689e21			−0.646431			−0.323216	+	0.946325i	$0.604764\pi$
$822$			1.03149e21i			0.141882i
$823$			5.24824e21i			0.715344i	0.933847	+	0.357672i	$0.116429\pi$
$823$			5.24824e21i			0.715344i	−0.933847	+	0.357672i	$0.883571\pi$
$824$			5.87751e19i			0.00793851i
$825$	−4.99490e21			−0.668532
$826$			3.00149e20i			0.0398095i
$827$		−	2.00975e21i		−	0.264150i	−0.991240	−	0.132075i	$-0.957836\pi$
$827$		−	2.00975e21i		−	0.264150i	0.991240	−	0.132075i	$-0.0421640\pi$
$828$	−4.15463e20			−0.0541135
$829$			1.21086e22i			1.56291i	0.623961	+	0.781456i	$0.285522\pi$
$829$			1.21086e22i			1.56291i	−0.623961	+	0.781456i	$0.714478\pi$
$830$		−	1.06421e20i		−	0.0136127i
$831$		−	1.44538e22i		−	1.83220i
$832$	1.00334e22			1.26044
$833$		−	1.66021e22i		−	2.06693i
$834$	3.12980e20			0.0386163
$835$	9.17102e21			1.12142
$836$	1.12204e22			1.35976
$837$	−1.19254e22			−1.43229
$838$		−	5.59319e20i		−	0.0665778i
$839$			3.15940e21i			0.372726i	0.982481	+	0.186363i	$0.0596701\pi$
$839$			3.15940e21i			0.372726i	−0.982481	+	0.186363i	$0.940330\pi$
$840$	1.56652e21			0.183165
$841$	8.06411e21	−	3.07131e21i	0.934516	−	0.355921i
$842$	9.99822e20			0.114837
$843$			4.53454e21i			0.516210i
$844$		−	3.76559e21i		−	0.424878i
$845$	−4.57334e21			−0.511456
$846$	−5.50190e19			−0.00609866
$847$	1.06510e22			1.17022
$848$	1.08012e22			1.17626
$849$			4.69445e21i			0.506731i
$850$	5.11794e20			0.0547587
$851$		−	1.18395e21i		−	0.125563i
$852$			9.68697e21i			1.01834i
$853$			1.01957e22i			1.06242i	0.847239	+	0.531212i	$0.178263\pi$
$853$			1.01957e22i			1.06242i	−0.847239	+	0.531212i	$0.821737\pi$
$854$	−1.64486e21			−0.169901
$855$			6.09597e20i			0.0624162i
$856$		−	1.91823e21i		−	0.194691i
$857$	−1.60369e22			−1.61348			−0.806740	−	0.590907i	$-0.798770\pi$
$857$	−1.60369e22			−1.61348			−0.806740	+	0.590907i	$0.798770\pi$
$858$		−	1.42223e21i		−	0.141846i
$859$		−	1.33194e22i		−	1.31685i	−0.752646	−	0.658425i	$-0.771223\pi$
$859$		−	1.33194e22i		−	1.31685i	0.752646	−	0.658425i	$-0.228777\pi$
$860$			7.69843e21i			0.754508i
$861$	−1.87986e22			−1.82642
$862$		−	2.38206e18i		−	0.000229429i
$863$	1.27526e22			1.21763			0.608816	−	0.793311i	$-0.291645\pi$
$863$	1.27526e22			1.21763			0.608816	+	0.793311i	$0.291645\pi$
$864$	2.35931e21			0.223322
$865$	9.64458e21			0.905029
$866$	1.00029e21			0.0930552
$867$		−	1.18893e22i		−	1.09651i
$868$		−	2.55133e22i		−	2.33275i
$869$	−2.20363e22			−1.99752
$870$	−6.43289e20	+	1.18355e20i	−0.0578112	+	0.0106364i
$871$	2.18206e22			1.94415
$872$		−	1.26196e20i		−	0.0111473i
$873$			1.22773e21i			0.107521i
$874$	6.07204e20			0.0527229
$875$	−1.93437e22			−1.66525
$876$	1.33860e22			1.14254
$877$	4.54242e21			0.384406			0.192203	−	0.981355i	$-0.438437\pi$
$877$	4.54242e21			0.384406			0.192203	+	0.981355i	$0.438437\pi$
$878$			3.00979e20i			0.0252538i
$879$	2.34521e17			1.95104e−5
$880$			1.12858e22i			0.930923i
$881$		−	1.34058e21i		−	0.109641i	−0.998496	−	0.0548207i	$-0.982541\pi$
$881$		−	1.34058e21i		−	0.109641i	0.998496	−	0.0548207i	$-0.0174587\pi$
$882$			1.17757e20i			0.00954929i
$883$	−6.09854e21			−0.490366			−0.245183	−	0.969477i	$-0.578848\pi$
$883$	−6.09854e21			−0.490366			−0.245183	+	0.969477i	$0.578848\pi$
$884$		−	2.33769e22i		−	1.86378i
$885$		−	3.05724e21i		−	0.241688i
$886$	−6.17906e20			−0.0484361
$887$			2.00026e22i			1.55475i	0.629038	+	0.777375i	$0.283449\pi$
$887$			2.00026e22i			1.55475i	−0.629038	+	0.777375i	$0.716551\pi$
$888$		−	4.11172e20i		−	0.0316903i
$889$			2.38203e22i			1.82047i
$890$	−4.59971e20			−0.0348582
$891$			1.89530e22i			1.42428i
$892$	−1.30444e22			−0.972049
$893$	−1.28992e22			−0.953183
$894$	−1.69424e20			−0.0124149
$895$	−9.74068e21			−0.707809
$896$			6.72297e21i			0.484453i
$897$			1.23465e22i			0.882271i
$898$	3.22392e19			0.00228461
$899$	3.86723e21	+	2.10193e22i	0.271770	+	1.47714i
$900$	5.82325e20			0.0405832
$901$		−	2.48472e22i		−	1.71728i
$902$			1.70447e21i			0.116826i
$903$	−2.53394e22			−1.72242
$904$	−4.21729e21			−0.284296
$905$	−1.60969e22			−1.07616
$906$	−4.21596e20			−0.0279534
$907$		−	1.89642e22i		−	1.24704i	−0.781807	−	0.623521i	$-0.785702\pi$
$907$		−	1.89642e22i		−	1.24704i	0.781807	−	0.623521i	$-0.214298\pi$
$908$	−4.72907e21			−0.308413
$909$			8.98703e20i			0.0581282i
$910$		−	1.80020e21i		−	0.115481i
$911$		−	3.11962e22i		−	1.98479i	−0.123101	−	0.992394i	$-0.539284\pi$
$911$		−	3.11962e22i		−	1.98479i	0.123101	−	0.992394i	$-0.460716\pi$
$912$	−1.67556e22			−1.05730
$913$			5.09489e21i			0.318863i
$914$			1.73283e21i			0.107562i
$915$	1.67542e22			1.03149
$916$			6.69159e21i			0.408614i
$917$		−	1.52538e22i		−	0.923864i
$918$		−	1.79034e21i		−	0.107551i
$919$	−5.07645e21			−0.302478			−0.151239	−	0.988497i	$-0.548326\pi$
$919$	−5.07645e21			−0.302478			−0.151239	+	0.988497i	$0.548326\pi$
$920$			1.23303e21i			0.0728729i
$921$	3.91815e21			0.229686
$922$	9.79192e20			0.0569359
$923$	2.23334e22			1.28808
$924$	−3.73819e22			−2.13856
$925$			1.65945e21i			0.0941676i
$926$			1.81382e21i			0.102096i
$927$	7.61997e19			0.00425457
$928$	−7.65092e20	−	4.15846e21i	−0.0423744	−	0.230315i
$929$	1.24434e21			0.0683632			0.0341816	−	0.999416i	$-0.489118\pi$
$929$	1.24434e21			0.0683632			0.0341816	+	0.999416i	$0.489118\pi$
$930$		−	1.61999e21i		−	0.0882857i
$931$			2.76080e22i			1.49250i
$932$	9.73730e21			0.522179
$933$	−3.59237e21			−0.191104
$934$	−2.49321e21			−0.131570
$935$	2.59620e22			1.35910
$936$			3.32652e20i			0.0172751i
$937$	−1.25146e22			−0.644718			−0.322359	−	0.946618i	$-0.604476\pi$
$937$	−1.25146e22			−0.644718			−0.322359	+	0.946618i	$0.604476\pi$
$938$			3.57527e21i			0.182721i
$939$			7.08994e21i			0.359460i
$940$		−	1.30563e22i		−	0.656692i
$941$	−8.53000e21			−0.425625			−0.212812	−	0.977093i	$-0.568262\pi$
$941$	−8.53000e21			−0.425625			−0.212812	+	0.977093i	$0.568262\pi$
$942$		−	1.56702e21i		−	0.0775699i
$943$		−	1.47967e22i		−	0.726651i
$944$	6.51930e21			0.317622
$945$			2.21165e22i			1.06900i
$946$			2.29753e21i			0.110174i
$947$			7.04573e21i			0.335198i	0.985855	+	0.167599i	$0.0536013\pi$
$947$			7.04573e21i			0.335198i	−0.985855	+	0.167599i	$0.946399\pi$
$948$	3.31150e22			1.56301
$949$		−	3.08615e22i		−	1.44517i
$950$	−8.51074e20			−0.0395403
$951$	1.20379e22			0.554875
$952$	7.68442e21			0.351426
$953$	1.60123e22			0.726537			0.363269	−	0.931684i	$-0.381661\pi$
$953$	1.60123e22			0.726537			0.363269	+	0.931684i	$0.381661\pi$
$954$			1.76238e20i			0.00793392i
$955$			5.40084e21i			0.241233i
$956$	2.94540e22			1.30530
$957$	3.07973e22	−	5.66623e21i	1.35417	−	0.249147i
$958$	1.25917e21			0.0549342
$959$		−	6.25062e22i		−	2.70572i
$960$		−	1.67449e22i		−	0.719200i
$961$	−2.94675e22			−1.25579
$962$	−4.72507e20			−0.0199800
$963$	−2.48691e21			−0.104343
$964$	−3.47762e22			−1.44779
$965$		−	6.11467e21i		−	0.252591i
$966$	−2.02296e21			−0.0829200
$967$			1.53890e22i			0.625912i	0.949768	+	0.312956i	$0.101319\pi$
$967$			1.53890e22i			0.625912i	−0.949768	+	0.312956i	$0.898681\pi$
$968$			2.91154e21i			0.117505i
$969$			3.85447e22i			1.54360i
$970$	1.81620e21			0.0721730
$971$			3.06084e22i			1.20697i	0.797374	+	0.603485i	$0.206222\pi$
$971$			3.06084e22i			1.20697i	−0.797374	+	0.603485i	$0.793778\pi$
$972$		−	4.26119e21i		−	0.166738i
$973$	−1.89659e22			−0.736422
$974$			2.16450e21i			0.0834000i
$975$		−	1.73052e22i		−	0.661672i
$976$			3.57268e22i			1.35557i
$977$	1.14158e22			0.429829			0.214914	−	0.976633i	$-0.431053\pi$
$977$	1.14158e22			0.429829			0.214914	+	0.976633i	$0.431053\pi$
$978$			2.16787e21i			0.0810012i
$979$	2.20210e22			0.816519
$980$	−2.79443e22			−1.02825
$981$	−1.63608e20			−0.00597429
$982$	2.60339e21			0.0943415
$983$		−	2.72845e22i		−	0.981215i	−0.871381	−	0.490608i	$-0.836775\pi$
$983$		−	2.72845e22i		−	0.981215i	0.871381	−	0.490608i	$-0.163225\pi$
$984$		−	5.13873e21i		−	0.183397i
$985$	−1.16389e22			−0.412232
$986$	−3.15559e21	+	5.80581e20i	−0.110919	+	0.0204073i
$987$	4.29749e22			1.49912
$988$			3.88740e22i			1.34580i
$989$		−	1.99451e22i		−	0.685273i
$990$	−1.84145e20			−0.00627909
$991$	1.32105e22			0.447061			0.223531	−	0.974697i	$-0.428242\pi$
$991$	1.32105e22			0.447061			0.223531	+	0.974697i	$0.428242\pi$
$992$	1.04722e22			0.351723
$993$	−5.02403e22			−1.67468
$994$			3.65929e21i			0.121060i
$995$	9.22225e21			0.302806
$996$		−	7.65632e21i		−	0.249503i
$997$			1.31992e22i			0.426907i	0.976953	+	0.213454i	$0.0684712\pi$
$997$			1.31992e22i			0.426907i	−0.976953	+	0.213454i	$0.931529\pi$
$998$		−	1.38782e21i		−	0.0445506i
$999$	5.80502e21			0.184953

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.17	✓	36
29.28	even	2	inner	29.16.b.a.28.20	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.17	✓	36	1.1	even	1	trivial
29.16.b.a.28.20	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)

\(f(q)\)	\(=\)	\(q-14.2479i q^{2} +3944.08i q^{3} +32565.0 q^{4} -125301. q^{5} +56194.9 q^{6} -3.40529e6 q^{7} -930860. i q^{8} -1.20682e6 q^{9} +O(q^{10})\)	\(q-14.2479i q^{2} +3944.08i q^{3} +32565.0 q^{4} -125301. q^{5} +56194.9 q^{6} -3.40529e6 q^{7} -930860. i q^{8} -1.20682e6 q^{9} +1.78527e6i q^{10} -8.54695e7i q^{11} +1.28439e8i q^{12} +2.96116e8 q^{13} +4.85183e7i q^{14} -4.94195e8i q^{15} +1.05383e9 q^{16} -2.42423e9i q^{17} +1.71947e7i q^{18} +4.03131e9i q^{19} -4.08041e9 q^{20} -1.34307e10i q^{21} -1.21776e9 q^{22} +1.05715e10 q^{23} +3.67138e9 q^{24} -1.48173e10 q^{25} -4.21904e9i q^{26} +5.18334e10i q^{27} -1.10893e11 q^{28} +(9.13600e10 - 1.68088e10i) q^{29} -7.04125e9 q^{30} +2.30071e11i q^{31} -4.55173e10i q^{32} +3.37098e11 q^{33} -3.45402e10 q^{34} +4.26684e11 q^{35} -3.93002e10 q^{36} -1.11994e11i q^{37} +5.74377e10 q^{38} +1.16790e12i q^{39} +1.16637e11i q^{40} -1.39967e12i q^{41} -1.91360e11 q^{42} -1.88668e12i q^{43} -2.78332e12i q^{44} +1.51216e11 q^{45} -1.50622e11i q^{46} +3.19975e12i q^{47} +4.15637e12i q^{48} +6.84841e12 q^{49} +2.11116e11i q^{50} +9.56134e12 q^{51} +9.64303e12 q^{52} +1.02495e13 q^{53} +7.38518e11 q^{54} +1.07094e13i q^{55} +3.16984e12i q^{56} -1.58998e13 q^{57} +(-2.39491e11 - 1.30169e12i) q^{58} +6.18631e12 q^{59} -1.60935e13i q^{60} +3.39020e13i q^{61} +3.27804e12 q^{62} +4.10958e12 q^{63} +3.38833e13 q^{64} -3.71035e13 q^{65} -4.80295e12i q^{66} +7.36893e13 q^{67} -7.89450e13i q^{68} +4.16949e13i q^{69} -6.07937e12i q^{70} +7.54209e13 q^{71} +1.12338e12i q^{72} -1.04221e14i q^{73} -1.59568e12 q^{74} -5.84407e13i q^{75} +1.31279e14i q^{76} +2.91048e14i q^{77} +1.66402e13 q^{78} -2.57827e14i q^{79} -1.32045e14 q^{80} -2.21751e14 q^{81} -1.99424e13 q^{82} -5.96106e13 q^{83} -4.37371e14i q^{84} +3.03757e14i q^{85} -2.68813e13 q^{86} +(6.62953e13 + 3.60331e14i) q^{87} -7.95601e13 q^{88} +2.57647e14i q^{89} -2.15451e12i q^{90} -1.00836e15 q^{91} +3.44262e14 q^{92} -9.07418e14 q^{93} +4.55898e13 q^{94} -5.05125e14i q^{95} +1.79523e14 q^{96} -1.01732e15i q^{97} -9.75756e13i q^{98} +1.03147e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 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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