LMFDB - Embedded newform 29.16.b.a.28.16

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			28.16
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-68.3669i q^{2} -2973.41i q^{3} +28094.0 q^{4} +302546. q^{5} -203283. q^{6} -153099. q^{7} -4.16094e6i q^{8} +5.50774e6 q^{9} +O(q^{10})$	\(q-68.3669i q^{2} -2973.41i q^{3} +28094.0 q^{4} +302546. q^{5} -203283. q^{6} -153099. q^{7} -4.16094e6i q^{8} +5.50774e6 q^{9} -2.06841e7i q^{10} -6.80281e7i q^{11} -8.35349e7i q^{12} +2.60817e8 q^{13} +1.04669e7i q^{14} -8.99593e8i q^{15} +6.36113e8 q^{16} +1.34097e9i q^{17} -3.76547e8i q^{18} +7.27524e9i q^{19} +8.49972e9 q^{20} +4.55226e8i q^{21} -4.65087e9 q^{22} -9.77450e9 q^{23} -1.23722e10 q^{24} +6.10165e10 q^{25} -1.78312e10i q^{26} -5.90420e10i q^{27} -4.30116e9 q^{28} +(2.11133e10 + 9.04622e10i) q^{29} -6.15024e10 q^{30} -4.75396e10i q^{31} -1.79835e11i q^{32} -2.02275e11 q^{33} +9.16780e10 q^{34} -4.63195e10 q^{35} +1.54734e11 q^{36} +8.65709e11i q^{37} +4.97385e11 q^{38} -7.75515e11i q^{39} -1.25888e12i q^{40} -7.42121e11i q^{41} +3.11224e10 q^{42} +3.70237e11i q^{43} -1.91118e12i q^{44} +1.66635e12 q^{45} +6.68252e11i q^{46} -7.31985e11i q^{47} -1.89142e12i q^{48} -4.72412e12 q^{49} -4.17151e12i q^{50} +3.98726e12 q^{51} +7.32738e12 q^{52} -1.02252e13 q^{53} -4.03651e12 q^{54} -2.05816e13i q^{55} +6.37036e11i q^{56} +2.16323e13 q^{57} +(6.18462e12 - 1.44345e12i) q^{58} -3.20658e13 q^{59} -2.52732e13i q^{60} -4.06935e13i q^{61} -3.25013e12 q^{62} -8.43230e11 q^{63} +8.54941e12 q^{64} +7.89091e13 q^{65} +1.38289e13i q^{66} +1.43592e13 q^{67} +3.76732e13i q^{68} +2.90636e13i q^{69} +3.16672e12i q^{70} -5.74824e13 q^{71} -2.29174e13i q^{72} +1.16273e13i q^{73} +5.91858e13 q^{74} -1.81427e14i q^{75} +2.04390e14i q^{76} +1.04150e13i q^{77} -5.30195e13 q^{78} -2.57874e14i q^{79} +1.92453e14 q^{80} -9.65259e13 q^{81} -5.07365e13 q^{82} -4.36392e14 q^{83} +1.27891e13i q^{84} +4.05705e14i q^{85} +2.53120e13 q^{86} +(2.68981e14 - 6.27785e13i) q^{87} -2.83061e14 q^{88} +6.21466e14i q^{89} -1.13923e14i q^{90} -3.99308e13 q^{91} -2.74605e14 q^{92} -1.41355e14 q^{93} -5.00435e13 q^{94} +2.20109e15i q^{95} -5.34722e14 q^{96} +3.36361e13i q^{97} +3.22973e14i q^{98} -3.74681e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	68.3669i		−	0.377677i	−0.982008	−	0.188839i	$-0.939528\pi$
$2$		−	68.3669i		−	0.377677i	0.982008	−	0.188839i	$-0.0604723\pi$
$3$		−	2973.41i		−	0.784956i	−0.919761	−	0.392478i	$-0.871618\pi$
$3$		−	2973.41i		−	0.784956i	0.919761	−	0.392478i	$-0.128382\pi$
$4$	28094.0			0.857360
$5$	302546.			1.73187			0.865937	−	0.500153i	$-0.166723\pi$
$5$	302546.			1.73187			0.865937	+	0.500153i	$0.166723\pi$
$6$	−203283.			−0.296460
$7$	−153099.			−0.0702647			−0.0351324	−	0.999383i	$-0.511185\pi$
$7$	−153099.			−0.0702647			−0.0351324	+	0.999383i	$0.511185\pi$
$8$		−	4.16094e6i		−	0.701482i
$9$	5.50774e6			0.383844
$10$		−	2.06841e7i		−	0.654089i
$11$		−	6.80281e7i		−	1.05255i	−0.850314	−	0.526276i	$-0.823588\pi$
$11$		−	6.80281e7i		−	1.05255i	0.850314	−	0.526276i	$-0.176412\pi$
$12$		−	8.35349e7i		−	0.672990i
$13$	2.60817e8			1.15282			0.576409	−	0.817162i	$-0.304454\pi$
$13$	2.60817e8			1.15282			0.576409	+	0.817162i	$0.304454\pi$
$14$			1.04669e7i			0.0265374i
$15$		−	8.99593e8i		−	1.35945i
$16$	6.36113e8			0.592426
$17$			1.34097e9i			0.792597i	0.918122	+	0.396299i	$0.129705\pi$
$17$			1.34097e9i			0.792597i	−0.918122	+	0.396299i	$0.870295\pi$
$18$		−	3.76547e8i		−	0.144969i
$19$			7.27524e9i			1.86722i	0.358293	+	0.933609i	$0.383359\pi$
$19$			7.27524e9i			1.86722i	−0.358293	+	0.933609i	$0.616641\pi$
$20$	8.49972e9			1.48484
$21$			4.55226e8i			0.0551547i
$22$	−4.65087e9			−0.397524
$23$	−9.77450e9			−0.598599			−0.299299	−	0.954159i	$-0.596753\pi$
$23$	−9.77450e9			−0.598599			−0.299299	+	0.954159i	$0.596753\pi$
$24$	−1.23722e10			−0.550633
$25$	6.10165e10			1.99939
$26$		−	1.78312e10i		−	0.435393i
$27$		−	5.90420e10i		−	1.08626i
$28$	−4.30116e9			−0.0602422
$29$	2.11133e10	+	9.04622e10i	0.227285	+	0.973828i
$29$	2.11133e10	+	9.04622e10i	0.227285	+	0.973828i
$30$	−6.15024e10			−0.513431
$31$		−	4.75396e10i		−	0.310343i	−0.987888	−	0.155172i	$-0.950407\pi$
$31$		−	4.75396e10i		−	0.310343i	0.987888	−	0.155172i	$-0.0495931\pi$
$32$		−	1.79835e11i		−	0.925228i
$33$	−2.02275e11			−0.826206
$34$	9.16780e10			0.299346
$35$	−4.63195e10			−0.121690
$36$	1.54734e11			0.329092
$37$			8.65709e11i			1.49920i	0.661893	+	0.749599i	$0.269754\pi$
$37$			8.65709e11i			1.49920i	−0.661893	+	0.749599i	$0.730246\pi$
$38$	4.97385e11			0.705206
$39$		−	7.75515e11i		−	0.904911i
$40$		−	1.25888e12i		−	1.21488i
$41$		−	7.42121e11i		−	0.595108i	−0.954705	−	0.297554i	$-0.903829\pi$
$41$		−	7.42121e11i		−	0.595108i	0.954705	−	0.297554i	$-0.0961708\pi$
$42$	3.11224e10			0.0208307
$43$			3.70237e11i			0.207714i	0.994592	+	0.103857i	$0.0331185\pi$
$43$			3.70237e11i			0.207714i	−0.994592	+	0.103857i	$0.966881\pi$
$44$		−	1.91118e12i		−	0.902415i
$45$	1.66635e12			0.664770
$46$			6.68252e11i			0.226077i
$47$		−	7.31985e11i		−	0.210750i	−0.994433	−	0.105375i	$-0.966396\pi$
$47$		−	7.31985e11i		−	0.210750i	0.994433	−	0.105375i	$-0.0336044\pi$
$48$		−	1.89142e12i		−	0.465029i
$49$	−4.72412e12			−0.995063
$50$		−	4.17151e12i		−	0.755124i
$51$	3.98726e12			0.622154
$52$	7.32738e12			0.988379
$53$	−1.02252e13			−1.19564			−0.597821	−	0.801629i	$-0.703967\pi$
$53$	−1.02252e13			−1.19564			−0.597821	+	0.801629i	$0.703967\pi$
$54$	−4.03651e12			−0.410254
$55$		−	2.05816e13i		−	1.82289i
$56$			6.37036e11i			0.0492895i
$57$	2.16323e13			1.46568
$58$	6.18462e12	−	1.44345e12i	0.367793	−	0.0858404i
$59$	−3.20658e13			−1.67746			−0.838729	−	0.544549i	$-0.816701\pi$
$59$	−3.20658e13			−1.67746			−0.838729	+	0.544549i	$0.816701\pi$
$60$		−	2.52732e13i		−	1.16553i
$61$		−	4.06935e13i		−	1.65787i	−0.559342	−	0.828937i	$-0.688946\pi$
$61$		−	4.06935e13i		−	1.65787i	0.559342	−	0.828937i	$-0.311054\pi$
$62$	−3.25013e12			−0.117210
$63$	−8.43230e11			−0.0269707
$64$	8.54941e12			0.242989
$65$	7.89091e13			1.99653
$66$			1.38289e13i			0.312039i
$67$	1.43592e13			0.289448			0.144724	−	0.989472i	$-0.453771\pi$
$67$	1.43592e13			0.289448			0.144724	+	0.989472i	$0.453771\pi$
$68$			3.76732e13i			0.679541i
$69$			2.90636e13i			0.469874i
$70$			3.16672e12i			0.0459594i
$71$	−5.74824e13			−0.750063			−0.375031	−	0.927012i	$-0.622368\pi$
$71$	−5.74824e13			−0.750063			−0.375031	+	0.927012i	$0.622368\pi$
$72$		−	2.29174e13i		−	0.269260i
$73$			1.16273e13i			0.123185i	0.998101	+	0.0615923i	$0.0196178\pi$
$73$			1.16273e13i			0.123185i	−0.998101	+	0.0615923i	$0.980382\pi$
$74$	5.91858e13			0.566213
$75$		−	1.81427e14i		−	1.56943i
$76$			2.04390e14i			1.60088i
$77$			1.04150e13i			0.0739572i
$78$	−5.30195e13			−0.341764
$79$		−	2.57874e14i		−	1.51079i	−0.655270	−	0.755395i	$-0.727445\pi$
$79$		−	2.57874e14i		−	1.51079i	0.655270	−	0.755395i	$-0.272555\pi$
$80$	1.92453e14			1.02601
$81$	−9.65259e13			−0.468820
$82$	−5.07365e13			−0.224759
$83$	−4.36392e14			−1.76519			−0.882593	−	0.470137i	$-0.844204\pi$
$83$	−4.36392e14			−1.76519			−0.882593	+	0.470137i	$0.844204\pi$
$84$			1.27891e13i			0.0472874i
$85$			4.05705e14i			1.37268i
$86$	2.53120e13			0.0784490
$87$	2.68981e14	−	6.27785e13i	0.764412	−	0.178409i
$88$	−2.83061e14			−0.738346
$89$			6.21466e14i			1.48933i	0.667436	+	0.744667i	$0.267392\pi$
$89$			6.21466e14i			1.48933i	−0.667436	+	0.744667i	$0.732608\pi$
$90$		−	1.13923e14i		−	0.251068i
$91$	−3.99308e13			−0.0810024
$92$	−2.74605e14			−0.513215
$93$	−1.41355e14			−0.243606
$94$	−5.00435e13			−0.0795956
$95$			2.20109e15i			3.23379i
$96$	−5.34722e14			−0.726263
$97$			3.36361e13i			0.0422685i	0.999777	+	0.0211343i	$0.00672774\pi$
$97$			3.36361e13i			0.0422685i	−0.999777	+	0.0211343i	$0.993272\pi$
$98$			3.22973e14i			0.375812i
$99$		−	3.74681e14i		−	0.404015i
$100$	1.71420e15			1.71420
$101$		−	2.80524e14i		−	0.260352i	−0.991491	−	0.130176i	$-0.958446\pi$
$101$		−	2.80524e14i		−	0.260352i	0.991491	−	0.130176i	$-0.0415542\pi$
$102$		−	2.72596e14i		−	0.234973i
$103$	1.20538e15			0.965705			0.482852	−	0.875702i	$-0.339601\pi$
$103$	1.20538e15			0.965705			0.482852	+	0.875702i	$0.339601\pi$
$104$		−	1.08524e15i		−	0.808681i
$105$			1.37727e14i			0.0955210i
$106$			6.99062e14i			0.451567i
$107$	1.24421e15			0.749058			0.374529	−	0.927215i	$-0.377804\pi$
$107$	1.24421e15			0.749058			0.374529	+	0.927215i	$0.377804\pi$
$108$		−	1.65872e15i		−	0.931313i
$109$	−2.37546e15			−1.24466			−0.622328	−	0.782757i	$-0.713813\pi$
$109$	−2.37546e15			−1.24466			−0.622328	+	0.782757i	$0.713813\pi$
$110$	−1.40710e15			−0.688462
$111$	2.57411e15			1.17680
$112$	−9.73883e13			−0.0416267
$113$		−	7.00165e13i		−	0.0279971i	−0.999902	−	0.0139985i	$-0.995544\pi$
$113$		−	7.00165e13i		−	0.0279971i	0.999902	−	0.0139985i	$-0.00445602\pi$
$114$		−	1.47893e15i		−	0.553556i
$115$	−2.95724e15			−1.03670
$116$	5.93156e14	+	2.54144e15i	0.194865	+	0.834921i
$117$	1.43651e15			0.442502
$118$			2.19224e15i			0.633537i
$119$		−	2.05301e14i		−	0.0556916i
$120$	−3.74316e15			−0.953627
$121$	−4.50573e14			−0.107864
$122$	−2.78209e15			−0.626141
$123$	−2.20663e15			−0.467134
$124$		−	1.33558e15i		−	0.266076i
$125$	9.22734e15			1.73082
$126$			5.76490e13i			0.0101862i
$127$		−	7.76514e15i		−	1.29307i	−0.762885	−	0.646534i	$-0.776218\pi$
$127$		−	7.76514e15i		−	1.29307i	0.762885	−	0.646534i	$-0.223782\pi$
$128$		−	6.47732e15i		−	1.01700i
$129$	1.10087e15			0.163047
$130$		−	5.39477e15i		−	0.754045i
$131$			3.08228e15i			0.406760i	0.979100	+	0.203380i	$0.0651926\pi$
$131$			3.08228e15i			0.406760i	−0.979100	+	0.203380i	$0.934807\pi$
$132$	−5.68272e15			−0.708356
$133$		−	1.11383e15i		−	0.131200i
$134$		−	9.81697e14i		−	0.109318i
$135$		−	1.78629e16i		−	1.88126i
$136$	5.57970e15			0.555993
$137$		−	4.29836e15i		−	0.405414i	−0.979239	−	0.202707i	$-0.935026\pi$
$137$		−	4.29836e15i		−	0.405414i	0.979239	−	0.202707i	$-0.0649738\pi$
$138$	1.98699e15			0.177461
$139$	1.55185e16			1.31292			0.656462	−	0.754360i	$-0.272052\pi$
$139$	1.55185e16			1.31292			0.656462	+	0.754360i	$0.272052\pi$
$140$	−1.30130e15			−0.104332
$141$	−2.17649e15			−0.165430
$142$			3.92989e15i			0.283281i
$143$		−	1.77429e16i		−	1.21340i
$144$	3.50354e15			0.227399
$145$	6.38774e15	+	2.73690e16i	0.393629	+	1.68655i
$146$	7.94920e14			0.0465240
$147$			1.40468e16i			0.781081i
$148$			2.43212e16i			1.28535i
$149$	−2.28877e16			−1.15002			−0.575008	−	0.818147i	$-0.695001\pi$
$149$	−2.28877e16			−1.15002			−0.575008	+	0.818147i	$0.695001\pi$
$150$	−1.24036e16			−0.592739
$151$	−1.52873e16			−0.695032			−0.347516	−	0.937674i	$-0.612975\pi$
$151$	−1.52873e16			−0.695032			−0.347516	+	0.937674i	$0.612975\pi$
$152$	3.02718e16			1.30982
$153$			7.38572e15i			0.304234i
$154$	7.12043e14			0.0279319
$155$		−	1.43829e16i		−	0.537476i
$156$		−	2.17873e16i		−	0.775834i
$157$			5.43693e16i			1.84547i	0.385439	+	0.922733i	$0.374050\pi$
$157$			5.43693e16i			1.84547i	−0.385439	+	0.922733i	$0.625950\pi$
$158$	−1.76300e16			−0.570591
$159$			3.04036e16i			0.938527i
$160$		−	5.44083e16i		−	1.60238i
$161$	1.49647e15			0.0420604
$162$			6.59917e15i			0.177063i
$163$			1.89053e15i			0.0484370i	0.999707	+	0.0242185i	$0.00770974\pi$
$163$			1.89053e15i			0.0484370i	−0.999707	+	0.0242185i	$0.992290\pi$
$164$		−	2.08491e16i		−	0.510222i
$165$	−6.11976e16			−1.43089
$166$			2.98347e16i			0.666670i
$167$	2.46623e16			0.526817			0.263409	−	0.964684i	$-0.415153\pi$
$167$	2.46623e16			0.526817			0.263409	+	0.964684i	$0.415153\pi$
$168$	1.89417e15			0.0386901
$169$	1.68395e16			0.328988
$170$	2.77368e16			0.518429
$171$			4.00701e16i			0.716721i
$172$			1.04014e16i			0.178086i
$173$	2.06920e16			0.339200			0.169600	−	0.985513i	$-0.445752\pi$
$173$	2.06920e16			0.339200			0.169600	+	0.985513i	$0.445752\pi$
$174$	−4.29197e15	−	1.83894e16i	−0.0673809	−	0.288701i
$175$	−9.34157e15			−0.140487
$176$		−	4.32735e16i		−	0.623559i
$177$			9.53447e16i			1.31673i
$178$	4.24877e16			0.562488
$179$	−4.80751e16			−0.610270			−0.305135	−	0.952309i	$-0.598702\pi$
$179$	−4.80751e16			−0.610270			−0.305135	+	0.952309i	$0.598702\pi$
$180$	4.68143e16			0.569947
$181$	−9.94532e16			−1.16153			−0.580763	−	0.814072i	$-0.697246\pi$
$181$	−9.94532e16			−1.16153			−0.580763	+	0.814072i	$0.697246\pi$
$182$			2.72994e15i			0.0305927i
$183$	−1.20998e17			−1.30136
$184$			4.06711e16i			0.419907i
$185$			2.61917e17i			2.59642i
$186$			9.66397e15i			0.0920044i
$187$	9.12237e16			0.834249
$188$		−	2.05644e16i		−	0.180689i
$189$			9.03927e15i			0.0763255i
$190$	1.50482e17			1.22133
$191$		−	9.25718e16i		−	0.722318i	−0.932504	−	0.361159i	$-0.882381\pi$
$191$		−	9.25718e16i		−	0.722318i	0.932504	−	0.361159i	$-0.117619\pi$
$192$		−	2.54209e16i		−	0.190736i
$193$		−	7.08416e16i		−	0.511221i	−0.966780	−	0.255610i	$-0.917724\pi$
$193$		−	7.08416e16i		−	0.511221i	0.966780	−	0.255610i	$-0.0822764\pi$
$194$	2.29959e15			0.0159638
$195$		−	2.34629e17i		−	1.56719i
$196$	−1.32719e17			−0.853127
$197$	−5.32036e16			−0.329188			−0.164594	−	0.986361i	$-0.552631\pi$
$197$	−5.32036e16			−0.329188			−0.164594	+	0.986361i	$0.552631\pi$
$198$	−2.56158e16			−0.152587
$199$	−7.83455e15			−0.0449382			−0.0224691	−	0.999748i	$-0.507153\pi$
$199$	−7.83455e15			−0.0449382			−0.0224691	+	0.999748i	$0.507153\pi$
$200$		−	2.53886e17i		−	1.40254i
$201$		−	4.26959e16i		−	0.227204i
$202$	−1.91786e16			−0.0983288
$203$	−3.23243e15	−	1.38497e16i	−0.0159701	−	0.0684258i
$204$	1.12018e17			0.533410
$205$		−	2.24526e17i		−	1.03065i
$206$		−	8.24079e16i		−	0.364724i
$207$	−5.38354e16			−0.229769
$208$	1.65909e17			0.682959
$209$	4.94921e17			1.96534
$210$	9.41596e15			0.0360761
$211$		−	4.27631e16i		−	0.158107i	−0.996870	−	0.0790535i	$-0.974810\pi$
$211$		−	4.27631e16i		−	0.158107i	0.996870	−	0.0790535i	$-0.0251898\pi$
$212$	−2.87266e17			−1.02510
$213$			1.70919e17i			0.588766i
$214$		−	8.50627e16i		−	0.282902i
$215$			1.12014e17i			0.359735i
$216$	−2.45670e17			−0.761990
$217$			7.27826e15i			0.0218062i
$218$			1.62403e17i			0.470078i
$219$	3.45726e16			0.0966945
$220$		−	5.78220e17i		−	1.56287i
$221$			3.49748e17i			0.913720i
$222$		−	1.75984e17i		−	0.444452i
$223$	−2.26392e17			−0.552808			−0.276404	−	0.961042i	$-0.589143\pi$
$223$	−2.26392e17			−0.552808			−0.276404	+	0.961042i	$0.589143\pi$
$224$			2.75325e16i			0.0650109i
$225$	3.36063e17			0.767454
$226$	−4.78681e15			−0.0105738
$227$	3.67175e15			0.00784655			0.00392328	−	0.999992i	$-0.498751\pi$
$227$	3.67175e15			0.00784655			0.00392328	+	0.999992i	$0.498751\pi$
$228$	6.07736e17			1.25662
$229$			4.37863e17i			0.876137i	0.898942	+	0.438069i	$0.144337\pi$
$229$			4.37863e17i			0.876137i	−0.898942	+	0.438069i	$0.855663\pi$
$230$			2.02177e17i			0.391537i
$231$	3.09682e16			0.0580532
$232$	3.76408e17	−	8.78512e16i	0.683123	−	0.159437i
$233$	4.12061e17			0.724090			0.362045	−	0.932161i	$-0.382079\pi$
$233$	4.12061e17			0.724090			0.362045	+	0.932161i	$0.382079\pi$
$234$		−	9.82098e16i		−	0.167123i
$235$		−	2.21459e17i		−	0.364993i
$236$	−9.00855e17			−1.43819
$237$	−7.66764e17			−1.18590
$238$	−1.40358e16			−0.0210334
$239$	−9.39170e17			−1.36383			−0.681914	−	0.731432i	$-0.738852\pi$
$239$	−9.39170e17			−1.36383			−0.681914	+	0.731432i	$0.738852\pi$
$240$		−	5.72243e17i		−	0.805371i
$241$	1.21580e18			1.65857			0.829287	−	0.558823i	$-0.188747\pi$
$241$	1.21580e18			1.65857			0.829287	+	0.558823i	$0.188747\pi$
$242$			3.08043e16i			0.0407376i
$243$		−	5.60177e17i		−	0.718254i
$244$		−	1.14324e18i		−	1.42139i
$245$	−1.42926e18			−1.72332
$246$			1.50860e17i			0.176426i
$247$			1.89750e18i			2.15256i
$248$	−1.97809e17			−0.217700
$249$			1.29757e18i			1.38559i
$250$		−	6.30844e17i		−	0.653690i
$251$			1.60426e18i			1.61332i	0.591014	+	0.806662i	$0.298728\pi$
$251$			1.60426e18i			1.61332i	−0.591014	+	0.806662i	$0.701272\pi$
$252$	−2.36897e16			−0.0231236
$253$			6.64941e17i			0.630056i
$254$	−5.30878e17			−0.488362
$255$	1.20633e18			1.07749
$256$	−1.62687e17			−0.141109
$257$	−8.32744e17			−0.701476			−0.350738	−	0.936474i	$-0.614069\pi$
$257$	−8.32744e17			−0.701476			−0.350738	+	0.936474i	$0.614069\pi$
$258$		−	7.52628e16i		−	0.0615790i
$259$		−	1.32539e17i		−	0.105341i
$260$	2.21687e18			1.71175
$261$	1.16287e17	+	4.98243e17i	0.0872420	+	0.373798i
$262$	2.10726e17			0.153624
$263$			1.11748e18i			0.791720i	0.918311	+	0.395860i	$0.129553\pi$
$263$			1.11748e18i			0.791720i	−0.918311	+	0.395860i	$0.870447\pi$
$264$			8.41656e17i			0.579569i
$265$	−3.09358e18			−2.07070
$266$	−7.61492e16			−0.0495511
$267$	1.84787e18			1.16906
$268$	4.03408e17			0.248161
$269$		−	1.39416e18i		−	0.834009i	−0.908904	−	0.417005i	$-0.863080\pi$
$269$		−	1.39416e18i		−	0.834009i	0.908904	−	0.417005i	$-0.136920\pi$
$270$	−1.22123e18			−0.710509
$271$			2.40874e18i			1.36308i	0.731782	+	0.681539i	$0.238689\pi$
$271$			2.40874e18i			1.36308i	−0.731782	+	0.681539i	$0.761311\pi$
$272$			8.53009e17i			0.469555i
$273$			1.18731e17i			0.0635833i
$274$	−2.93865e17			−0.153115
$275$		−	4.15084e18i		−	2.10446i
$276$			8.16512e17i			0.402851i
$277$	3.61373e18			1.73523			0.867616	−	0.497235i	$-0.165651\pi$
$277$	3.61373e18			1.73523			0.867616	+	0.497235i	$0.165651\pi$
$278$		−	1.06095e18i		−	0.495861i
$279$		−	2.61836e17i		−	0.119123i
$280$			1.92733e17i			0.0853632i
$281$	7.95062e17			0.342849			0.171425	−	0.985197i	$-0.445163\pi$
$281$	7.95062e17			0.342849			0.171425	+	0.985197i	$0.445163\pi$
$282$			1.48800e17i			0.0624791i
$283$	−3.01991e18			−1.23480			−0.617399	−	0.786650i	$-0.711814\pi$
$283$	−3.01991e18			−1.23480			−0.617399	+	0.786650i	$0.711814\pi$
$284$	−1.61491e18			−0.643074
$285$	6.54476e18			2.53838
$286$	−1.21302e18			−0.458273
$287$			1.13618e17i			0.0418151i
$288$		−	9.90483e17i		−	0.355143i
$289$	1.06422e18			0.371790
$290$	1.87113e18	−	4.36710e17i	0.636971	−	0.148665i
$291$	1.00014e17			0.0331789
$292$			3.26656e17i			0.105613i
$293$		−	2.22130e18i		−	0.700004i	−0.936749	−	0.350002i	$-0.886181\pi$
$293$		−	2.22130e18i		−	0.700004i	0.936749	−	0.350002i	$-0.113819\pi$
$294$	9.60332e17			0.294996
$295$	−9.70137e18			−2.90515
$296$	3.60216e18			1.05166
$297$	−4.01651e18			−1.14334
$298$			1.56476e18i			0.434335i
$299$	−2.54935e18			−0.690075
$300$		−	5.09701e18i		−	1.34557i
$301$		−	5.66830e16i		−	0.0145950i
$302$			1.04515e18i			0.262498i
$303$	−8.34114e17			−0.204365
$304$			4.62787e18i			1.10619i
$305$		−	1.23117e19i		−	2.87123i
$306$	5.04939e17			0.114902
$307$			5.96633e18i			1.32486i	0.749124	+	0.662429i	$0.230475\pi$
$307$			5.96633e18i			1.32486i	−0.749124	+	0.662429i	$0.769525\pi$
$308$			2.92600e17i			0.0634079i
$309$		−	3.58408e18i		−	0.758036i
$310$	−9.83314e17			−0.202992
$311$		−	3.84496e18i		−	0.774798i	−0.921912	−	0.387399i	$-0.873374\pi$
$311$		−	3.84496e18i		−	0.774798i	0.921912	−	0.387399i	$-0.126626\pi$
$312$	−3.22687e18			−0.634779
$313$	4.92058e18			0.945004			0.472502	−	0.881330i	$-0.343351\pi$
$313$	4.92058e18			0.945004			0.472502	+	0.881330i	$0.343351\pi$
$314$	3.71706e18			0.696990
$315$	−2.55116e17			−0.0467098
$316$		−	7.24470e18i		−	1.29529i
$317$			2.63329e18i			0.459785i	0.973216	+	0.229892i	$0.0738374\pi$
$317$			2.63329e18i			0.459785i	−0.973216	+	0.229892i	$0.926163\pi$
$318$	2.07860e18			0.354460
$319$	6.15397e18	−	1.43630e18i	1.02500	−	0.239229i
$320$	2.58659e18			0.420826
$321$		−	3.69955e18i		−	0.587978i
$322$		−	1.02309e17i		−	0.0158852i
$323$	−9.75588e18			−1.47995
$324$	−2.71179e18			−0.401947
$325$	1.59141e19			2.30493
$326$	1.29250e17			0.0182935
$327$			7.06322e18i			0.977000i
$328$	−3.08792e18			−0.417458
$329$			1.12066e17i			0.0148083i
$330$			4.18389e18i			0.540413i
$331$			4.78966e18i			0.604776i	0.953185	+	0.302388i	$0.0977838\pi$
$331$			4.78966e18i			0.604776i	−0.953185	+	0.302388i	$0.902216\pi$
$332$	−1.22600e19			−1.51340
$333$			4.76810e18i			0.575458i
$334$		−	1.68608e18i		−	0.198967i
$335$	4.34433e18			0.501288
$336$			2.89575e17i			0.0326751i
$337$			1.13607e18i			0.125366i	0.998033	+	0.0626830i	$0.0199657\pi$
$337$			1.13607e18i			0.125366i	−0.998033	+	0.0626830i	$0.980034\pi$
$338$		−	1.15127e18i		−	0.124251i
$339$	−2.08188e17			−0.0219765
$340$			1.13979e19i			1.17688i
$341$	−3.23403e18			−0.326652
$342$	2.73947e18			0.270689
$343$	1.45011e18			0.140183
$344$	1.54054e18			0.145708
$345$			8.79308e18i			0.813763i
$346$		−	1.41465e18i		−	0.128108i
$347$	1.65645e19			1.46794			0.733969	−	0.679183i	$-0.237666\pi$
$347$	1.65645e19			1.46794			0.733969	+	0.679183i	$0.237666\pi$
$348$	7.55675e18	−	1.76370e18i	0.655377	−	0.152961i
$349$	6.83177e18			0.579886			0.289943	−	0.957044i	$-0.406364\pi$
$349$	6.83177e18			0.579886			0.289943	+	0.957044i	$0.406364\pi$
$350$			6.38654e17i			0.0530585i
$351$		−	1.53991e19i		−	1.25226i
$352$	−1.22338e19			−0.973850
$353$	−1.05879e19			−0.825083			−0.412542	−	0.910939i	$-0.635359\pi$
$353$	−1.05879e19			−0.825083			−0.412542	+	0.910939i	$0.635359\pi$
$354$	6.51841e18			0.497299
$355$	−1.73911e19			−1.29901
$356$			1.74595e19i			1.27690i
$357$	−6.10445e17			−0.0437155
$358$			3.28674e18i			0.230485i
$359$		−	3.57518e18i		−	0.245522i	−0.992436	−	0.122761i	$-0.960825\pi$
$359$		−	3.57518e18i		−	0.245522i	0.992436	−	0.122761i	$-0.0391748\pi$
$360$		−	6.93357e18i		−	0.466324i
$361$	−3.77480e19			−2.48651
$362$			6.79930e18i			0.438682i
$363$			1.33974e18i			0.0846683i
$364$	−1.12182e18			−0.0694482
$365$			3.51778e18i			0.213340i
$366$			8.27229e18i			0.491493i
$367$			9.32273e18i			0.542685i	0.962483	+	0.271342i	$0.0874675\pi$
$367$			9.32273e18i			0.542685i	−0.962483	+	0.271342i	$0.912532\pi$
$368$	−6.21769e18			−0.354626
$369$		−	4.08741e18i		−	0.228429i
$370$	1.79064e19			0.980609
$371$	1.56546e18			0.0840115
$372$	−3.97121e18			−0.208858
$373$	1.98409e19			1.02269			0.511346	−	0.859375i	$-0.329147\pi$
$373$	1.98409e19			1.02269			0.511346	+	0.859375i	$0.329147\pi$
$374$		−	6.23668e18i		−	0.315077i
$375$		−	2.74366e19i		−	1.35862i
$376$	−3.04575e18			−0.147838
$377$	5.50670e18	+	2.35941e19i	0.262018	+	1.12265i
$378$	6.17986e17			0.0288264
$379$			4.91039e17i			0.0224554i	0.999937	+	0.0112277i	$0.00357397\pi$
$379$			4.91039e17i			0.0224554i	−0.999937	+	0.0112277i	$0.996426\pi$
$380$			6.18375e19i			2.77252i
$381$	−2.30889e19			−1.01500
$382$	−6.32884e18			−0.272803
$383$	2.79702e19			1.18224			0.591120	−	0.806584i	$-0.298686\pi$
$383$	2.79702e19			1.18224			0.591120	+	0.806584i	$0.298686\pi$
$384$	−1.92597e19			−0.798300
$385$			3.15103e18i			0.128085i
$386$	−4.84322e18			−0.193076
$387$			2.03917e18i			0.0797299i
$388$			9.44970e17i			0.0362393i
$389$		−	2.96552e19i		−	1.11552i	−0.830001	−	0.557762i	$-0.811660\pi$
$389$		−	2.96552e19i		−	1.11552i	0.830001	−	0.557762i	$-0.188340\pi$
$390$	−1.60409e19			−0.591893
$391$		−	1.31073e19i		−	0.474448i
$392$			1.96568e19i			0.698019i
$393$	9.16489e18			0.319288
$394$			3.63736e18i			0.124327i
$395$		−	7.80187e19i		−	2.61650i
$396$		−	1.05263e19i		−	0.346387i
$397$	4.81669e19			1.55532			0.777661	−	0.628684i	$-0.216406\pi$
$397$	4.81669e19			1.55532			0.777661	+	0.628684i	$0.216406\pi$
$398$			5.35624e17i			0.0169721i
$399$	−3.31188e18			−0.102986
$400$	3.88134e19			1.18449
$401$	2.53878e19			0.760399			0.380200	−	0.924904i	$-0.375855\pi$
$401$	2.53878e19			0.760399			0.380200	+	0.924904i	$0.375855\pi$
$402$	−2.91899e18			−0.0858098
$403$		−	1.23991e19i		−	0.357769i
$404$		−	7.88104e18i		−	0.223215i
$405$	−2.92035e19			−0.811937
$406$	−9.46860e17	+	2.20991e17i	−0.0258428	+	0.00603155i
$407$	5.88925e19			1.57798
$408$		−	1.65907e19i		−	0.436430i
$409$		−	4.28813e19i		−	1.10750i	−0.832683	−	0.553749i	$-0.813197\pi$
$409$		−	4.28813e19i		−	1.10750i	0.832683	−	0.553749i	$-0.186803\pi$
$410$	−1.53501e19			−0.389254
$411$	−1.27808e19			−0.318232
$412$	3.38639e19			0.827957
$413$	4.90924e18			0.117866
$414$			3.68056e18i			0.0867783i
$415$	−1.32029e20			−3.05708
$416$		−	4.69039e19i		−	1.06662i
$417$		−	4.61429e19i		−	1.03059i
$418$		−	3.38362e19i		−	0.742265i
$419$	3.06755e19			0.660978			0.330489	−	0.943810i	$-0.392786\pi$
$419$	3.06755e19			0.660978			0.330489	+	0.943810i	$0.392786\pi$
$420$			3.86930e18i			0.0818959i
$421$			5.02163e19i			1.04407i	0.852924	+	0.522035i	$0.174827\pi$
$421$			5.02163e19i			1.04407i	−0.852924	+	0.522035i	$0.825173\pi$
$422$	−2.92358e18			−0.0597134
$423$		−	4.03159e18i		−	0.0808953i
$424$			4.25463e19i			0.838722i
$425$			8.18214e19i			1.58471i
$426$	1.16852e19			0.222363
$427$			6.23014e18i			0.116490i
$428$	3.49548e19			0.642212
$429$	−5.27568e19			−0.952465
$430$	7.65803e18			0.135864
$431$	−2.55511e19			−0.445482			−0.222741	−	0.974878i	$-0.571500\pi$
$431$	−2.55511e19			−0.445482			−0.222741	+	0.974878i	$0.571500\pi$
$432$		−	3.75573e19i		−	0.643527i
$433$			6.33837e19i			1.06738i	0.845681	+	0.533689i	$0.179195\pi$
$433$			6.33837e19i			1.06738i	−0.845681	+	0.533689i	$0.820805\pi$
$434$	4.97592e17			0.00823570
$435$	8.13792e19	−	1.89934e19i	1.32387	−	0.308982i
$436$	−6.67362e19			−1.06712
$437$		−	7.11118e19i		−	1.11772i
$438$		−	2.36362e18i		−	0.0365193i
$439$	6.64443e19			1.00919			0.504596	−	0.863355i	$-0.331641\pi$
$439$	6.64443e19			1.00919			0.504596	+	0.863355i	$0.331641\pi$
$440$	−8.56390e19			−1.27872
$441$	−2.60192e19			−0.381949
$442$	2.39112e19			0.345091
$443$		−	9.35217e19i		−	1.32704i	−0.748158	−	0.663521i	$-0.769061\pi$
$443$		−	9.35217e19i		−	1.32704i	0.748158	−	0.663521i	$-0.230939\pi$
$444$	7.23169e19			1.00894
$445$			1.88022e20i			2.57934i
$446$			1.54777e19i			0.208783i
$447$			6.80544e19i			0.902713i
$448$	−1.30891e18			−0.0170735
$449$		−	1.13610e20i		−	1.45737i	−0.684847	−	0.728687i	$-0.740131\pi$
$449$		−	1.13610e20i		−	1.45737i	0.684847	−	0.728687i	$-0.259869\pi$
$450$		−	2.29756e19i		−	0.289850i
$451$	−5.04851e19			−0.626382
$452$		−	1.96704e18i		−	0.0240036i
$453$			4.54555e19i			0.545570i
$454$		−	2.51026e17i		−	0.00296346i
$455$	−1.20809e19			−0.140286
$456$		−	9.00106e19i		−	1.02815i
$457$	3.58252e19			0.402547			0.201274	−	0.979535i	$-0.435492\pi$
$457$	3.58252e19			0.402547			0.201274	+	0.979535i	$0.435492\pi$
$458$	2.99353e19			0.330897
$459$	7.91736e19			0.860964
$460$	−8.30805e19			−0.888824
$461$		−	1.38542e20i		−	1.45822i	−0.684395	−	0.729112i	$-0.739933\pi$
$461$		−	1.38542e20i		−	1.45822i	0.684395	−	0.729112i	$-0.260067\pi$
$462$		−	2.11720e18i		−	0.0219253i
$463$	−9.51639e19			−0.969649			−0.484825	−	0.874611i	$-0.661117\pi$
$463$	−9.51639e19			−0.969649			−0.484825	+	0.874611i	$0.661117\pi$
$464$	1.34304e19	+	5.75442e19i	0.134650	+	0.576921i
$465$	−4.27663e19			−0.421895
$466$		−	2.81713e19i		−	0.273472i
$467$			1.84027e19i			0.175794i	0.996130	+	0.0878971i	$0.0280147\pi$
$467$			1.84027e19i			0.175794i	−0.996130	+	0.0878971i	$0.971985\pi$
$468$	4.03573e19			0.379383
$469$	−2.19839e18			−0.0203380
$470$	−1.51405e19			−0.137850
$471$	1.61662e20			1.44861
$472$			1.33424e20i			1.17671i
$473$	2.51865e19			0.218630
$474$			5.24213e19i			0.447889i
$475$			4.43910e20i			3.73330i
$476$		−	5.76773e18i		−	0.0477478i
$477$	−5.63176e19			−0.458940
$478$			6.42081e19i			0.515087i
$479$		−	1.15809e19i		−	0.0914587i	−0.998954	−	0.0457294i	$-0.985439\pi$
$479$		−	1.15809e19i		−	0.0914587i	0.998954	−	0.0457294i	$-0.0145612\pi$
$480$	−1.61778e20			−1.25780
$481$			2.25791e20i			1.72830i
$482$		−	8.31205e19i		−	0.626405i
$483$		−	4.44961e18i		−	0.0330156i
$484$	−1.26584e19			−0.0924780
$485$			1.01765e19i			0.0732038i
$486$	−3.82975e19			−0.271268
$487$	6.03790e19			0.421133			0.210566	−	0.977580i	$-0.432469\pi$
$487$	6.03790e19			0.421133			0.210566	+	0.977580i	$0.432469\pi$
$488$	−1.69323e20			−1.16297
$489$	5.62132e18			0.0380209
$490$			9.77143e19i			0.650860i
$491$		−	1.60218e20i		−	1.05099i	−0.850796	−	0.525497i	$-0.823880\pi$
$491$		−	1.60218e20i		−	1.05099i	0.850796	−	0.525497i	$-0.176120\pi$
$492$	−6.19930e19			−0.400502
$493$	−1.21307e20	+	2.83123e19i	−0.771853	+	0.180146i
$494$	1.29726e20			0.812973
$495$		−	1.13358e20i		−	0.699704i
$496$		−	3.02405e19i		−	0.183856i
$497$	8.80051e18			0.0527029
$498$	8.87108e19			0.523307
$499$	−2.83240e20			−1.64589			−0.822946	−	0.568120i	$-0.807671\pi$
$499$	−2.83240e20			−1.64589			−0.822946	+	0.568120i	$0.807671\pi$
$500$	2.59233e20			1.48393
$501$		−	7.33311e19i		−	0.413528i
$502$	1.09678e20			0.609315
$503$			1.09281e20i			0.598115i	0.954235	+	0.299057i	$0.0966722\pi$
$503$			1.09281e20i			0.598115i	−0.954235	+	0.299057i	$0.903328\pi$
$504$			3.50863e18i			0.0189195i
$505$		−	8.48715e19i		−	0.450896i
$506$	4.54599e19			0.237958
$507$		−	5.00708e19i		−	0.258241i
$508$		−	2.18154e20i		−	1.10862i
$509$	−2.46728e20			−1.23548			−0.617738	−	0.786384i	$-0.711951\pi$
$509$	−2.46728e20			−1.23548			−0.617738	+	0.786384i	$0.711951\pi$
$510$		−	8.24729e19i		−	0.406944i
$511$		−	1.78012e18i		−	0.00865553i
$512$		−	2.01126e20i		−	0.963706i
$513$	4.29544e20			2.02828
$514$			5.69321e19i			0.264932i
$515$	3.64683e20			1.67248
$516$	3.09277e19			0.139790
$517$	−4.97956e19			−0.221826
$518$	−9.06129e18			−0.0397848
$519$		−	6.15258e19i		−	0.266257i
$520$		−	3.28336e20i		−	1.40053i
$521$	−1.42182e20			−0.597809			−0.298905	−	0.954283i	$-0.596621\pi$
$521$	−1.42182e20			−0.597809			−0.298905	+	0.954283i	$0.596621\pi$
$522$	3.40633e19	−	7.95015e18i	0.141175	−	0.0329493i
$523$	−1.64568e19			−0.0672332			−0.0336166	−	0.999435i	$-0.510703\pi$
$523$	−1.64568e19			−0.0672332			−0.0336166	+	0.999435i	$0.510703\pi$
$524$			8.65936e19i			0.348739i
$525$			2.77763e19i			0.110276i
$526$	7.63986e19			0.299014
$527$	6.37492e19			0.245977
$528$	−1.28670e20			−0.489466
$529$	−1.71094e20			−0.641679
$530$			2.11499e20i			0.782057i
$531$	−1.76610e20			−0.643882
$532$		−	3.12920e19i		−	0.112485i
$533$		−	1.93558e20i		−	0.686051i
$534$		−	1.26333e20i		−	0.441528i
$535$	3.76431e20			1.29727
$536$		−	5.97480e19i		−	0.203043i
$537$			1.42947e20i			0.479035i
$538$	−9.53144e19			−0.314986
$539$			3.21373e20i			1.04735i
$540$		−	5.01840e20i		−	1.61292i
$541$			2.56282e20i			0.812341i	0.913797	+	0.406171i	$0.133136\pi$
$541$			2.56282e20i			0.812341i	−0.913797	+	0.406171i	$0.866864\pi$
$542$	1.64678e20			0.514803
$543$			2.95715e20i			0.911748i
$544$	2.41153e20			0.733333
$545$	−7.18687e20			−2.15559
$546$	8.11724e18			0.0240140
$547$	−1.64682e20			−0.480554			−0.240277	−	0.970704i	$-0.577238\pi$
$547$	−1.64682e20			−0.480554			−0.240277	+	0.970704i	$0.577238\pi$
$548$		−	1.20758e20i		−	0.347585i
$549$		−	2.24129e20i		−	0.636365i
$550$	−2.83780e20			−0.794806
$551$	−6.58134e20	+	1.53604e20i	−1.81835	+	0.424391i
$552$	1.20932e20			0.329608
$553$			3.94802e19i			0.106155i
$554$		−	2.47059e20i		−	0.655357i
$555$	7.78786e20			2.03808
$556$	4.35977e20			1.12565
$557$	2.06253e20			0.525394			0.262697	−	0.964878i	$-0.415388\pi$
$557$	2.06253e20			0.525394			0.262697	+	0.964878i	$0.415388\pi$
$558$	−1.79009e19			−0.0449902
$559$			9.65641e19i			0.239457i
$560$	−2.94644e19			−0.0720922
$561$		−	2.71245e20i		−	0.654849i
$562$		−	5.43559e19i		−	0.129486i
$563$		−	8.01248e20i		−	1.88345i	−0.336385	−	0.941724i	$-0.609204\pi$
$563$		−	8.01248e20i		−	1.88345i	0.336385	−	0.941724i	$-0.390796\pi$
$564$	−6.11463e19			−0.141833
$565$		−	2.11832e19i		−	0.0484874i
$566$			2.06462e20i			0.466355i
$567$	1.47780e19			0.0329415
$568$			2.39181e20i			0.526156i
$569$			3.45486e20i			0.750046i	0.927015	+	0.375023i	$0.122365\pi$
$569$			3.45486e20i			0.750046i	−0.927015	+	0.375023i	$0.877635\pi$
$570$		−	4.47444e20i		−	0.958689i
$571$	−1.85022e20			−0.391249			−0.195625	−	0.980679i	$-0.562673\pi$
$571$	−1.85022e20			−0.391249			−0.195625	+	0.980679i	$0.562673\pi$
$572$		−	4.98468e20i		−	1.04032i
$573$	−2.75254e20			−0.566988
$574$	7.76771e18			0.0157926
$575$	−5.96406e20			−1.19683
$576$	4.70879e19			0.0932698
$577$		−	8.22899e20i		−	1.60890i	−0.594023	−	0.804448i	$-0.702461\pi$
$577$		−	8.22899e20i		−	1.60890i	0.594023	−	0.804448i	$-0.297539\pi$
$578$		−	7.27573e19i		−	0.140416i
$579$	−2.10641e20			−0.401286
$580$	1.79457e20	+	7.68904e20i	0.337482	+	1.44598i
$581$	6.68111e19			0.124030
$582$		−	6.83763e18i		−	0.0125309i
$583$			6.95599e20i			1.25847i
$584$	4.83804e19			0.0864118
$585$	4.34611e20			0.766358
$586$	−1.51863e20			−0.264376
$587$	5.81309e20			0.999129			0.499564	−	0.866277i	$-0.333493\pi$
$587$	5.81309e20			0.999129			0.499564	+	0.866277i	$0.333493\pi$
$588$			3.94629e20i			0.669667i
$589$	3.45862e20			0.579479
$590$			6.63252e20i			1.09721i
$591$			1.58196e20i			0.258398i
$592$			5.50688e20i			0.888164i
$593$	8.22728e20			1.31023			0.655113	−	0.755531i	$-0.272621\pi$
$593$	8.22728e20			1.31023			0.655113	+	0.755531i	$0.272621\pi$
$594$			2.74596e20i			0.431814i
$595$		−	6.21131e19i		−	0.0964509i
$596$	−6.43005e20			−0.985979
$597$			2.32953e19i			0.0352745i
$598$			1.74291e20i			0.260626i
$599$			1.04863e21i			1.54854i	0.632854	+	0.774271i	$0.281883\pi$
$599$			1.04863e21i			1.54854i	−0.632854	+	0.774271i	$0.718117\pi$
$600$	−7.54908e20			−1.10093
$601$			4.96797e20i			0.715517i	0.933814	+	0.357759i	$0.116459\pi$
$601$			4.96797e20i			0.715517i	−0.933814	+	0.357759i	$0.883541\pi$
$602$	−3.87524e18			−0.00551220
$603$	7.90870e19			0.111103
$604$	−4.29482e20			−0.595893
$605$	−1.36319e20			−0.186806
$606$			5.70257e19i			0.0771838i
$607$			5.43059e19i			0.0725993i	0.999341	+	0.0362996i	$0.0115571\pi$
$607$			5.43059e19i			0.0725993i	−0.999341	+	0.0362996i	$0.988443\pi$
$608$	1.30834e21			1.72760
$609$	−4.11808e19	+	9.61133e18i	−0.0537112	+	0.0125358i
$610$	−8.41710e20			−1.08440
$611$		−	1.90914e20i		−	0.242957i
$612$			2.07494e20i			0.260838i
$613$	5.92676e20			0.735976			0.367988	−	0.929831i	$-0.380047\pi$
$613$	5.92676e20			0.735976			0.367988	+	0.929831i	$0.380047\pi$
$614$	4.07899e20			0.500369
$615$	−6.67607e20			−0.809017
$616$	4.33364e19			0.0518797
$617$			1.57405e21i			1.86158i	0.365559	+	0.930788i	$0.380878\pi$
$617$			1.57405e21i			1.86158i	−0.365559	+	0.930788i	$0.619122\pi$
$618$	−2.45033e20			−0.286293
$619$		−	1.01032e21i		−	1.16622i	−0.812394	−	0.583108i	$-0.801836\pi$
$619$		−	1.01032e21i		−	1.16622i	0.812394	−	0.583108i	$-0.198164\pi$
$620$		−	4.04073e20i		−	0.460810i
$621$			5.77106e20i			0.650232i
$622$	−2.62868e20			−0.292623
$623$		−	9.51459e19i		−	0.104648i
$624$		−	4.93315e20i		−	0.536093i
$625$	9.29617e20			0.998169
$626$		−	3.36405e20i		−	0.356906i
$627$		−	1.47160e21i		−	1.54271i
$628$			1.52745e21i			1.58223i
$629$	−1.16089e21			−1.18826
$630$			1.74415e19i			0.0176412i
$631$	7.94742e20			0.794339			0.397169	−	0.917745i	$-0.369993\pi$
$631$	7.94742e20			0.794339			0.397169	+	0.917745i	$0.369993\pi$
$632$	−1.07300e21			−1.05979
$633$	−1.27152e20			−0.124107
$634$	1.80030e20			0.173650
$635$		−	2.34931e21i		−	2.23943i
$636$			8.54158e20i			0.804655i
$637$	−1.23213e21			−1.14713
$638$	−9.81951e19	−	4.20728e20i	−0.0903514	−	0.387120i
$639$	−3.16598e20			−0.287907
$640$		−	1.95969e21i		−	1.76132i
$641$		−	1.27620e21i		−	1.13366i	−0.823833	−	0.566832i	$-0.808169\pi$
$641$		−	1.27620e21i		−	1.13366i	0.823833	−	0.566832i	$-0.191831\pi$
$642$	−2.52926e20			−0.222066
$643$	−1.18315e21			−1.02673			−0.513366	−	0.858170i	$-0.671602\pi$
$643$	−1.18315e21			−1.02673			−0.513366	+	0.858170i	$0.671602\pi$
$644$	4.20417e19			0.0360609
$645$	3.33063e20			0.282376
$646$			6.66979e20i			0.558944i
$647$	1.75015e21			1.44975			0.724873	−	0.688882i	$-0.241898\pi$
$647$	1.75015e21			1.44975			0.724873	+	0.688882i	$0.241898\pi$
$648$			4.01638e20i			0.328869i
$649$			2.18137e21i			1.76561i
$650$		−	1.08800e21i		−	0.870519i
$651$	2.16413e19			0.0171169
$652$			5.31125e19i			0.0415279i
$653$			1.04450e21i			0.807346i	0.914903	+	0.403673i	$0.132267\pi$
$653$			1.04450e21i			0.807346i	−0.914903	+	0.403673i	$0.867733\pi$
$654$	4.82890e20			0.368991
$655$			9.32532e20i			0.704456i
$656$		−	4.72073e20i		−	0.352558i
$657$			6.40400e19i			0.0472836i
$658$	7.66162e18			0.00559276
$659$		−	1.59642e21i		−	1.15214i	−0.817399	−	0.576072i	$-0.804585\pi$
$659$		−	1.59642e21i		−	1.15214i	0.817399	−	0.576072i	$-0.195415\pi$
$660$	−1.71928e21			−1.22678
$661$	2.39380e21			1.68879			0.844396	−	0.535719i	$-0.179959\pi$
$661$	2.39380e21			1.68879			0.844396	+	0.535719i	$0.179959\pi$
$662$	3.27454e20			0.228410
$663$	1.03994e21			0.717230
$664$			1.81580e21i			1.23825i
$665$		−	3.36986e20i		−	0.227221i
$666$	3.25980e20			0.217337
$667$	−2.06372e20	−	8.84223e20i	−0.136053	−	0.582933i
$668$	6.92862e20			0.451672
$669$			6.73156e20i			0.433930i
$670$		−	2.97008e20i		−	0.189325i
$671$	−2.76830e21			−1.74500
$672$	8.18655e19			0.0510307
$673$	−7.85050e20			−0.483932			−0.241966	−	0.970285i	$-0.577792\pi$
$673$	−7.85050e20			−0.483932			−0.241966	+	0.970285i	$0.577792\pi$
$674$	7.76693e19			0.0473478
$675$		−	3.60253e21i		−	2.17185i
$676$	4.73089e20			0.282061
$677$		−	6.90395e20i		−	0.407083i	−0.979066	−	0.203541i	$-0.934755\pi$
$677$		−	6.90395e20i		−	0.407083i	0.979066	−	0.203541i	$-0.0652452\pi$
$678$			1.42332e19i			0.00830000i
$679$		−	5.14965e18i		−	0.00296999i
$680$	1.68812e21			0.962910
$681$		−	1.09176e19i		−	0.00615920i
$682$			2.21100e20i			0.123369i
$683$	8.84618e20			0.488203			0.244102	−	0.969750i	$-0.421507\pi$
$683$	8.84618e20			0.488203			0.244102	+	0.969750i	$0.421507\pi$
$684$			1.12573e21i			0.614488i
$685$		−	1.30045e21i		−	0.702126i
$686$		−	9.91392e19i		−	0.0529437i
$687$	1.30195e21			0.687729
$688$			2.35513e20i			0.123055i
$689$	−2.66690e21			−1.37836
$690$	6.01155e20			0.307339
$691$	3.09280e21			1.56410			0.782052	−	0.623213i	$-0.214173\pi$
$691$	3.09280e21			1.56410			0.782052	+	0.623213i	$0.214173\pi$
$692$	5.81320e20			0.290817
$693$			5.73633e19i			0.0283880i
$694$		−	1.13246e21i		−	0.554406i
$695$	4.69507e21			2.27382
$696$	−2.61218e20	−	1.11922e21i	−0.125151	−	0.536222i
$697$	9.95163e20			0.471681
$698$		−	4.67067e20i		−	0.219010i
$699$		−	1.22523e21i		−	0.568379i
$700$	−2.62442e20			−0.120448
$701$	−1.33684e20			−0.0607008			−0.0303504	−	0.999539i	$-0.509662\pi$
$701$	−1.33684e20			−0.0607008			−0.0303504	+	0.999539i	$0.509662\pi$
$702$	−1.05279e21			−0.472948
$703$	−6.29824e21			−2.79933
$704$		−	5.81600e20i		−	0.255758i
$705$	−6.58489e20			−0.286504
$706$			7.23858e20i			0.311615i
$707$			4.29480e19i			0.0182935i
$708$			2.67861e21i			1.12891i
$709$	−4.35517e20			−0.181618			−0.0908090	−	0.995868i	$-0.528945\pi$
$709$	−4.35517e20			−0.181618			−0.0908090	+	0.995868i	$0.528945\pi$
$710$			1.18897e21i			0.490608i
$711$		−	1.42030e21i		−	0.579907i
$712$	2.58588e21			1.04474
$713$			4.64676e20i			0.185771i
$714$			4.17342e19i			0.0165103i
$715$		−	5.36804e21i		−	2.10145i
$716$	−1.35062e21			−0.523221
$717$			2.79254e21i			1.07055i
$718$	−2.44424e20			−0.0927279
$719$	−1.96532e21			−0.737849			−0.368925	−	0.929459i	$-0.620274\pi$
$719$	−1.96532e21			−0.737849			−0.368925	+	0.929459i	$0.620274\pi$
$720$	1.05998e21			0.393827
$721$	−1.84542e20			−0.0678550
$722$			2.58071e21i			0.939096i
$723$		−	3.61507e21i		−	1.30191i
$724$	−2.79403e21			−0.995847
$725$	1.28826e21	+	5.51969e21i	0.454432	+	1.94706i
$726$	9.15938e19			0.0319773
$727$			8.77278e20i			0.303130i	0.988447	+	0.151565i	$0.0484313\pi$
$727$			8.77278e20i			0.303130i	−0.988447	+	0.151565i	$0.951569\pi$
$728$			1.66150e20i			0.0568217i
$729$	−3.05067e21			−1.03262
$730$	2.40500e20			0.0805737
$731$	−4.96478e20			−0.164634
$732$	−3.39933e21			−1.11573
$733$		−	1.09775e21i		−	0.356634i	−0.983973	−	0.178317i	$-0.942935\pi$
$733$		−	1.09775e21i		−	0.356634i	0.983973	−	0.178317i	$-0.0570653\pi$
$734$	6.37366e20			0.204960
$735$			4.24979e21i			1.35273i
$736$			1.75780e21i			0.553841i
$737$		−	9.76832e20i		−	0.304659i
$738$	−2.79443e20			−0.0862722
$739$		−	3.05524e20i		−	0.0933709i	−0.998910	−	0.0466854i	$-0.985134\pi$
$739$		−	3.05524e20i		−	0.0933709i	0.998910	−	0.0466854i	$-0.0148658\pi$
$740$			7.35828e21i			2.22607i
$741$	5.64206e21			1.68967
$742$		−	1.07026e20i		−	0.0317292i
$743$			1.86048e21i			0.546020i	0.962011	+	0.273010i	$0.0880193\pi$
$743$			1.86048e21i			0.546020i	−0.962011	+	0.273010i	$0.911981\pi$
$744$			5.88168e20i			0.170885i
$745$	−6.92457e21			−1.99169
$746$		−	1.35646e21i		−	0.386248i
$747$	−2.40353e21			−0.677556
$748$	2.56284e21			0.715252
$749$	−1.90487e20			−0.0526323
$750$	−1.87576e21			−0.513118
$751$		−	1.08566e21i		−	0.294032i	−0.989134	−	0.147016i	$-0.953033\pi$
$751$		−	1.08566e21i		−	0.294032i	0.989134	−	0.147016i	$-0.0469669\pi$
$752$		−	4.65625e20i		−	0.124854i
$753$	4.77012e21			1.26639
$754$	1.61305e21	−	3.76476e20i	0.423998	−	0.0989583i
$755$	−4.62512e21			−1.20371
$756$			2.53949e20i			0.0654384i
$757$		−	1.43564e21i		−	0.366292i	−0.983086	−	0.183146i	$-0.941372\pi$
$757$		−	1.43564e21i		−	0.366292i	0.983086	−	0.183146i	$-0.0586281\pi$
$758$	3.35708e19			0.00848091
$759$	1.97714e21			0.494566
$760$	9.15863e21			2.26845
$761$	4.23492e21			1.03863			0.519315	−	0.854583i	$-0.326187\pi$
$761$	4.23492e21			1.03863			0.519315	+	0.854583i	$0.326187\pi$
$762$			1.57852e21i			0.383343i
$763$	3.63681e20			0.0874554
$764$		−	2.60071e21i		−	0.619286i
$765$			2.23452e21i			0.526894i
$766$		−	1.91224e21i		−	0.446505i
$767$	−8.36329e21			−1.93380
$768$			4.83736e20i			0.110764i
$769$		−	3.24518e21i		−	0.735853i	−0.929855	−	0.367927i	$-0.880068\pi$
$769$		−	3.24518e21i		−	0.735853i	0.929855	−	0.367927i	$-0.119932\pi$
$770$	2.15426e20			0.0483746
$771$			2.47609e21i			0.550628i
$772$		−	1.99022e21i		−	0.438300i
$773$			7.38078e21i			1.60974i	0.593450	+	0.804871i	$0.297765\pi$
$773$			7.38078e21i			1.60974i	−0.593450	+	0.804871i	$0.702235\pi$
$774$	1.39412e20			0.0301122
$775$		−	2.90070e21i		−	0.620497i
$776$	1.39958e20			0.0296506
$777$	−3.94093e20			−0.0826878
$778$	−2.02743e21			−0.421308
$779$	5.39911e21			1.11120
$780$		−	6.59166e21i		−	1.34365i
$781$			3.91042e21i			0.789479i
$782$	−8.96107e20			−0.179188
$783$	5.34107e21	−	1.24657e21i	1.05783	−	0.246890i
$784$	−3.00507e21			−0.589501
$785$			1.64492e22i			3.19612i
$786$		−	6.26575e20i		−	0.120588i
$787$	−3.02606e20			−0.0576855			−0.0288428	−	0.999584i	$-0.509182\pi$
$787$	−3.02606e20			−0.0576855			−0.0288428	+	0.999584i	$0.509182\pi$
$788$	−1.49470e21			−0.282233
$789$	3.32273e21			0.621465
$790$	−5.33389e21			−0.988191
$791$			1.07195e19i			0.00196721i
$792$	−1.55903e21			−0.283410
$793$		−	1.06136e22i		−	1.91123i
$794$		−	3.29302e21i		−	0.587409i
$795$			9.19849e21i			1.62541i
$796$	−2.20104e20			−0.0385282
$797$		−	6.90843e21i		−	1.19796i	−0.800764	−	0.598979i	$-0.795573\pi$
$797$		−	6.90843e21i		−	1.19796i	0.800764	−	0.598979i	$-0.204427\pi$
$798$			2.26423e20i			0.0388954i
$799$	9.81571e20			0.167040
$800$		−	1.09729e22i		−	1.84989i
$801$			3.42287e21i			0.571672i
$802$		−	1.73568e21i		−	0.287185i
$803$	7.90981e20			0.129658
$804$		−	1.19950e21i		−	0.194796i
$805$	4.52750e20			0.0728433
$806$	−8.47689e20			−0.135121
$807$	−4.14541e21			−0.654661
$808$	−1.16725e21			−0.182632
$809$			2.52921e21i			0.392077i	0.980596	+	0.196038i	$0.0628077\pi$
$809$			2.52921e21i			0.392077i	−0.980596	+	0.196038i	$0.937192\pi$
$810$			1.99655e21i			0.306650i
$811$	9.91129e21			1.50825			0.754126	−	0.656730i	$-0.228061\pi$
$811$	9.91129e21			1.50825			0.754126	+	0.656730i	$0.228061\pi$
$812$	−9.08117e19	−	3.89093e20i	−0.0136921	−	0.0586655i
$813$	7.16218e21			1.06996
$814$		−	4.02630e21i		−	0.595968i
$815$			5.71973e20i			0.0838867i
$816$	2.53635e21			0.368580
$817$	−2.69356e21			−0.387848
$818$	−2.93166e21			−0.418277
$819$	−2.19929e20			−0.0310923
$820$		−	6.30782e21i		−	0.883640i
$821$	−6.66434e21			−0.925090			−0.462545	−	0.886596i	$-0.653064\pi$
$821$	−6.66434e21			−0.925090			−0.462545	+	0.886596i	$0.653064\pi$
$822$			8.73782e20i			0.120189i
$823$			2.80072e21i			0.381743i	0.981615	+	0.190871i	$0.0611313\pi$
$823$			2.80072e21i			0.381743i	−0.981615	+	0.190871i	$0.938869\pi$
$824$		−	5.01551e21i		−	0.677425i
$825$	−1.23421e22			−1.65191
$826$		−	3.35629e20i		−	0.0445153i
$827$		−	2.08247e21i		−	0.273708i	−0.990591	−	0.136854i	$-0.956301\pi$
$827$		−	2.08247e21i		−	0.273708i	0.990591	−	0.136854i	$-0.0436991\pi$
$828$	−1.51245e21			−0.196994
$829$		−	1.46294e22i		−	1.88828i	−0.329546	−	0.944139i	$-0.606896\pi$
$829$		−	1.46294e22i		−	1.88828i	0.329546	−	0.944139i	$-0.393104\pi$
$830$			9.02637e21i			1.15459i
$831$		−	1.07451e22i		−	1.36208i
$832$	2.22983e21			0.280122
$833$		−	6.33491e21i		−	0.788684i
$834$	−3.15465e21			−0.389229
$835$	7.46148e21			0.912381
$836$	1.39043e22			1.68501
$837$	−2.80683e21			−0.337113
$838$		−	2.09719e21i		−	0.249636i
$839$		−	1.19108e21i		−	0.140516i	−0.997529	−	0.0702579i	$-0.977618\pi$
$839$		−	1.19108e21i		−	0.140516i	0.997529	−	0.0702579i	$-0.0223822\pi$
$840$	5.73074e20			0.0670063
$841$	−7.73765e21	+	3.81991e21i	−0.896683	+	0.442673i
$842$	3.43313e21			0.394321
$843$		−	2.36404e21i		−	0.269122i
$844$		−	1.20139e21i		−	0.135555i
$845$	5.09473e21			0.569765
$846$	−2.75627e20			−0.0305523
$847$	6.89824e19			0.00757901
$848$	−6.50436e21			−0.708330
$849$			8.97943e21i			0.969262i
$850$	5.59387e21			0.598509
$851$		−	8.46187e21i		−	0.897418i
$852$			4.80179e21i			0.504785i
$853$			1.27722e22i			1.33091i	0.746438	+	0.665455i	$0.231762\pi$
$853$			1.27722e22i			1.33091i	−0.746438	+	0.665455i	$0.768238\pi$
$854$	4.25935e20			0.0439956
$855$			1.21231e22i			1.24127i
$856$		−	5.17709e21i		−	0.525451i
$857$	2.21569e21			0.222922			0.111461	−	0.993769i	$-0.464447\pi$
$857$	2.21569e21			0.222922			0.111461	+	0.993769i	$0.464447\pi$
$858$			3.60682e21i			0.359724i
$859$		−	7.80461e21i		−	0.771618i	−0.922579	−	0.385809i	$-0.873922\pi$
$859$		−	7.80461e21i		−	0.771618i	0.922579	−	0.385809i	$-0.126078\pi$
$860$			3.14691e21i			0.308423i
$861$	3.37833e20			0.0328230
$862$			1.74685e21i			0.168248i
$863$	1.19161e22			1.13777			0.568885	−	0.822417i	$-0.307375\pi$
$863$	1.19161e22			1.13777			0.568885	+	0.822417i	$0.307375\pi$
$864$	−1.06178e22			−1.00504
$865$	6.26028e21			0.587453
$866$	4.33335e21			0.403124
$867$		−	3.16436e21i		−	0.291839i
$868$			2.04475e20i			0.0186958i
$869$	−1.75427e22			−1.59018
$870$	−1.29852e21	−	5.56364e21i	−0.116695	−	0.499994i
$871$	3.74513e21			0.333681
$872$			9.88416e21i			0.873104i
$873$			1.85259e20i			0.0162245i
$874$	−4.86169e21			−0.422135
$875$	−1.41270e21			−0.121615
$876$	9.71283e20			0.0829020
$877$	1.96955e21			0.166675			0.0833373	−	0.996521i	$-0.473442\pi$
$877$	1.96955e21			0.166675			0.0833373	+	0.996521i	$0.473442\pi$
$878$		−	4.54259e21i		−	0.381149i
$879$	−6.60484e21			−0.549472
$880$		−	1.30922e22i		−	1.07993i
$881$		−	1.20129e22i		−	0.982488i	−0.871022	−	0.491244i	$-0.836542\pi$
$881$		−	1.20129e22i		−	0.982488i	0.871022	−	0.491244i	$-0.163458\pi$
$882$			1.77885e21i			0.144253i
$883$	1.22366e22			0.983912			0.491956	−	0.870620i	$-0.336282\pi$
$883$	1.22366e22			0.983912			0.491956	+	0.870620i	$0.336282\pi$
$884$			9.82581e21i			0.783387i
$885$			2.88462e22i			2.28041i
$886$	−6.39379e21			−0.501193
$887$			5.13950e21i			0.399479i	0.979849	+	0.199739i	$0.0640096\pi$
$887$			5.13950e21i			0.399479i	−0.979849	+	0.199739i	$0.935990\pi$
$888$		−	1.07107e22i		−	0.825507i
$889$			1.18884e21i			0.0908571i
$890$	1.28545e22			0.974158
$891$			6.56647e21i			0.493457i
$892$	−6.36025e21			−0.473955
$893$	5.32537e21			0.393517
$894$	4.65266e21			0.340934
$895$	−1.45449e22			−1.05691
$896$			9.91672e20i			0.0714592i
$897$			7.58028e21i			0.541679i
$898$	−7.76719e21			−0.550417
$899$	4.30054e21	−	1.00372e21i	0.302221	−	0.0705364i
$900$	9.44135e21			0.657984
$901$		−	1.37117e22i		−	0.947663i
$902$			3.45151e21i			0.236570i
$903$	−1.68542e20			−0.0114564
$904$	−2.91335e20			−0.0196394
$905$	−3.00892e22			−2.01162
$906$	3.10765e21			0.206049
$907$		−	9.26742e21i		−	0.609402i	−0.952448	−	0.304701i	$-0.901443\pi$
$907$		−	9.26742e21i		−	0.609402i	0.952448	−	0.304701i	$-0.0985566\pi$
$908$	1.03154e20			0.00672732
$909$		−	1.54506e21i		−	0.0999344i
$910$			8.25934e20i			0.0529828i
$911$			1.19360e22i			0.759398i	0.925110	+	0.379699i	$0.123972\pi$
$911$			1.19360e22i			0.759398i	−0.925110	+	0.379699i	$0.876028\pi$
$912$	1.37606e22			0.868310
$913$			2.96869e22i			1.85795i
$914$		−	2.44925e21i		−	0.152033i
$915$	−3.66076e22			−2.25379
$916$			1.23013e22i			0.751165i
$917$		−	4.71895e20i		−	0.0285808i
$918$		−	5.41285e21i		−	0.325166i
$919$	1.35886e22			0.809668			0.404834	−	0.914390i	$-0.367329\pi$
$919$	1.35886e22			0.809668			0.404834	+	0.914390i	$0.367329\pi$
$920$			1.23049e22i			0.727225i
$921$	1.77403e22			1.03996
$922$	−9.47167e21			−0.550738
$923$	−1.49924e22			−0.864685
$924$	8.70019e20			0.0497725
$925$			5.28225e22i			2.99748i
$926$			6.50605e21i			0.366214i
$927$	6.63891e21			0.370680
$928$	1.62683e22	−	3.79690e21i	0.901013	−	0.210291i
$929$	1.17069e21			0.0643165			0.0321582	−	0.999483i	$-0.489762\pi$
$929$	1.17069e21			0.0643165			0.0321582	+	0.999483i	$0.489762\pi$
$930$			2.92380e21i			0.159340i
$931$		−	3.43691e22i		−	1.85800i
$932$	1.15764e22			0.620806
$933$	−1.14326e22			−0.608182
$934$	1.25813e21			0.0663935
$935$	2.75994e22			1.44481
$936$		−	5.97724e21i		−	0.310407i
$937$	1.05906e22			0.545598			0.272799	−	0.962071i	$-0.412051\pi$
$937$	1.05906e22			0.545598			0.272799	+	0.962071i	$0.412051\pi$
$938$			1.50297e20i			0.00768119i
$939$		−	1.46309e22i		−	0.741787i
$940$		−	6.22167e21i		−	0.312931i
$941$	−1.39302e22			−0.695080			−0.347540	−	0.937665i	$-0.612983\pi$
$941$	−1.39302e22			−0.695080			−0.347540	+	0.937665i	$0.612983\pi$
$942$		−	1.10523e22i		−	0.547107i
$943$			7.25386e21i			0.356231i
$944$	−2.03974e22			−0.993770
$945$			2.73479e21i			0.132186i
$946$		−	1.72192e21i		−	0.0825716i
$947$			3.07125e22i			1.46113i	0.682841	+	0.730567i	$0.260744\pi$
$947$			3.07125e22i			1.46113i	−0.682841	+	0.730567i	$0.739256\pi$
$948$	−2.15415e22			−1.01675
$949$			3.03259e21i			0.142009i
$950$	3.03487e22			1.40998
$951$	7.82986e21			0.360911
$952$	−8.54247e20			−0.0390667
$953$	−3.23518e22			−1.46792			−0.733960	−	0.679193i	$-0.762330\pi$
$953$	−3.23518e22			−1.46792			−0.733960	+	0.679193i	$0.762330\pi$
$954$			3.85026e21i			0.173331i
$955$		−	2.80072e22i		−	1.25096i
$956$	−2.63850e22			−1.16929
$957$	−4.27070e21	−	1.82983e22i	−0.187784	−	0.804583i
$958$	−7.91748e20			−0.0345419
$959$			6.58075e20i			0.0284863i
$960$		−	7.69099e21i		−	0.330330i
$961$	2.12053e22			0.903687
$962$	1.54366e22			0.652740
$963$	6.85279e21			0.287521
$964$	3.41567e22			1.42199
$965$		−	2.14328e22i		−	0.885370i
$966$	−3.04206e20			−0.0124692
$967$		−	3.08748e22i		−	1.25576i	−0.778311	−	0.627879i	$-0.783923\pi$
$967$		−	3.08748e22i		−	1.25576i	0.778311	−	0.627879i	$-0.216077\pi$
$968$			1.87481e21i			0.0756645i
$969$			2.90082e22i			1.16170i
$970$	6.95732e20			0.0276474
$971$		−	4.19273e21i		−	0.165330i	−0.996577	−	0.0826652i	$-0.973657\pi$
$971$		−	4.19273e21i		−	0.165330i	0.996577	−	0.0826652i	$-0.0263432\pi$
$972$		−	1.57376e22i		−	0.615802i
$973$	−2.37587e21			−0.0922522
$974$		−	4.12792e21i		−	0.159052i
$975$		−	4.73192e22i		−	1.80927i
$976$		−	2.58857e22i		−	0.982168i
$977$	9.61266e21			0.361938			0.180969	−	0.983489i	$-0.442077\pi$
$977$	9.61266e21			0.361938			0.180969	+	0.983489i	$0.442077\pi$
$978$		−	3.84312e20i		−	0.0143596i
$979$	4.22772e22			1.56760
$980$	−4.01537e22			−1.47751
$981$	−1.30834e22			−0.477754
$982$	−1.09536e22			−0.396936
$983$		−	3.65898e22i		−	1.31586i	−0.753081	−	0.657928i	$-0.771433\pi$
$983$		−	3.65898e22i		−	1.31586i	0.753081	−	0.657928i	$-0.228567\pi$
$984$			9.18166e21i			0.327686i
$985$	−1.60965e22			−0.570113
$986$	1.93562e21	+	8.29340e21i	0.0680368	+	0.291511i
$987$	3.33219e20			0.0116239
$988$			5.33084e22i			1.84552i
$989$		−	3.61889e21i		−	0.124338i
$990$	−7.74995e21			−0.264262
$991$	−3.35883e22			−1.13667			−0.568336	−	0.822797i	$-0.692413\pi$
$991$	−3.35883e22			−1.13667			−0.568336	+	0.822797i	$0.692413\pi$
$992$	−8.54927e21			−0.287138
$993$	1.42416e22			0.474723
$994$		−	6.01663e20i		−	0.0199047i
$995$	−2.37031e21			−0.0778274
$996$			3.64539e22i			1.18795i
$997$			3.11834e22i			1.00858i	0.863535	+	0.504289i	$0.168245\pi$
$997$			3.11834e22i			1.00858i	−0.863535	+	0.504289i	$0.831755\pi$
$998$			1.93643e22i			0.621615i
$999$	5.11131e22			1.62851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.16	✓	36
29.28	even	2	inner	29.16.b.a.28.21	yes	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.16	✓	36	1.1	even	1	trivial
29.16.b.a.28.21	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)

\(f(q)\)	\(=\)	\(q-68.3669i q^{2} -2973.41i q^{3} +28094.0 q^{4} +302546. q^{5} -203283. q^{6} -153099. q^{7} -4.16094e6i q^{8} +5.50774e6 q^{9} +O(q^{10})\)	\(q-68.3669i q^{2} -2973.41i q^{3} +28094.0 q^{4} +302546. q^{5} -203283. q^{6} -153099. q^{7} -4.16094e6i q^{8} +5.50774e6 q^{9} -2.06841e7i q^{10} -6.80281e7i q^{11} -8.35349e7i q^{12} +2.60817e8 q^{13} +1.04669e7i q^{14} -8.99593e8i q^{15} +6.36113e8 q^{16} +1.34097e9i q^{17} -3.76547e8i q^{18} +7.27524e9i q^{19} +8.49972e9 q^{20} +4.55226e8i q^{21} -4.65087e9 q^{22} -9.77450e9 q^{23} -1.23722e10 q^{24} +6.10165e10 q^{25} -1.78312e10i q^{26} -5.90420e10i q^{27} -4.30116e9 q^{28} +(2.11133e10 + 9.04622e10i) q^{29} -6.15024e10 q^{30} -4.75396e10i q^{31} -1.79835e11i q^{32} -2.02275e11 q^{33} +9.16780e10 q^{34} -4.63195e10 q^{35} +1.54734e11 q^{36} +8.65709e11i q^{37} +4.97385e11 q^{38} -7.75515e11i q^{39} -1.25888e12i q^{40} -7.42121e11i q^{41} +3.11224e10 q^{42} +3.70237e11i q^{43} -1.91118e12i q^{44} +1.66635e12 q^{45} +6.68252e11i q^{46} -7.31985e11i q^{47} -1.89142e12i q^{48} -4.72412e12 q^{49} -4.17151e12i q^{50} +3.98726e12 q^{51} +7.32738e12 q^{52} -1.02252e13 q^{53} -4.03651e12 q^{54} -2.05816e13i q^{55} +6.37036e11i q^{56} +2.16323e13 q^{57} +(6.18462e12 - 1.44345e12i) q^{58} -3.20658e13 q^{59} -2.52732e13i q^{60} -4.06935e13i q^{61} -3.25013e12 q^{62} -8.43230e11 q^{63} +8.54941e12 q^{64} +7.89091e13 q^{65} +1.38289e13i q^{66} +1.43592e13 q^{67} +3.76732e13i q^{68} +2.90636e13i q^{69} +3.16672e12i q^{70} -5.74824e13 q^{71} -2.29174e13i q^{72} +1.16273e13i q^{73} +5.91858e13 q^{74} -1.81427e14i q^{75} +2.04390e14i q^{76} +1.04150e13i q^{77} -5.30195e13 q^{78} -2.57874e14i q^{79} +1.92453e14 q^{80} -9.65259e13 q^{81} -5.07365e13 q^{82} -4.36392e14 q^{83} +1.27891e13i q^{84} +4.05705e14i q^{85} +2.53120e13 q^{86} +(2.68981e14 - 6.27785e13i) q^{87} -2.83061e14 q^{88} +6.21466e14i q^{89} -1.13923e14i q^{90} -3.99308e13 q^{91} -2.74605e14 q^{92} -1.41355e14 q^{93} -5.00435e13 q^{94} +2.20109e15i q^{95} -5.34722e14 q^{96} +3.36361e13i q^{97} +3.22973e14i q^{98} -3.74681e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 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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