LMFDB - Embedded newform 29.16.b.a.28.23

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			28.23
Character	$\chi$	$=$	29.28
Dual form			29.16.b.a.28.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+101.334i q^{2} -71.6384i q^{3} +22499.4 q^{4} -295367. q^{5} +7259.42 q^{6} +1.06773e6 q^{7} +5.60048e6i q^{8} +1.43438e7 q^{9} +O(q^{10})$	\(q+101.334i q^{2} -71.6384i q^{3} +22499.4 q^{4} -295367. q^{5} +7259.42 q^{6} +1.06773e6 q^{7} +5.60048e6i q^{8} +1.43438e7 q^{9} -2.99307e7i q^{10} -6.77632e7i q^{11} -1.61182e6i q^{12} -4.89556e7 q^{13} +1.08198e8i q^{14} +2.11596e7i q^{15} +1.69740e8 q^{16} +2.82095e9i q^{17} +1.45351e9i q^{18} -3.07551e9i q^{19} -6.64557e9 q^{20} -7.64907e7i q^{21} +6.86673e9 q^{22} +3.83787e9 q^{23} +4.01209e8 q^{24} +5.67239e10 q^{25} -4.96088e9i q^{26} -2.05550e9i q^{27} +2.40234e10 q^{28} +(-5.63931e10 - 7.38174e10i) q^{29} -2.14419e9 q^{30} +2.68357e11i q^{31} +2.00717e11i q^{32} -4.85445e9 q^{33} -2.85858e11 q^{34} -3.15373e11 q^{35} +3.22726e11 q^{36} +7.50493e11i q^{37} +3.11655e11 q^{38} +3.50710e9i q^{39} -1.65419e12i q^{40} -1.43199e11i q^{41} +7.75113e9 q^{42} +1.06123e12i q^{43} -1.52463e12i q^{44} -4.23667e12 q^{45} +3.88908e11i q^{46} +5.16931e12i q^{47} -1.21599e10i q^{48} -3.60751e12 q^{49} +5.74807e12i q^{50} +2.02088e11 q^{51} -1.10147e12 q^{52} -1.56320e13 q^{53} +2.08292e11 q^{54} +2.00150e13i q^{55} +5.97982e12i q^{56} -2.20325e11 q^{57} +(7.48023e12 - 5.71455e12i) q^{58} -2.07643e12 q^{59} +4.76078e11i q^{60} +9.28687e12i q^{61} -2.71938e13 q^{62} +1.53153e13 q^{63} -1.47774e13 q^{64} +1.44599e13 q^{65} -4.91921e11i q^{66} +7.93908e13 q^{67} +6.34695e13i q^{68} -2.74939e11i q^{69} -3.19581e13i q^{70} -1.42631e14 q^{71} +8.03320e13i q^{72} +1.22324e14i q^{73} -7.60507e13 q^{74} -4.06361e12i q^{75} -6.91971e13i q^{76} -7.23531e13i q^{77} -3.55390e11 q^{78} -1.06705e14i q^{79} -5.01355e13 q^{80} +2.05670e14 q^{81} +1.45109e13 q^{82} +2.36776e14 q^{83} -1.72099e12i q^{84} -8.33213e14i q^{85} -1.07539e14 q^{86} +(-5.28816e12 + 4.03991e12i) q^{87} +3.79506e14 q^{88} +1.89344e14i q^{89} -4.29320e14i q^{90} -5.22716e13 q^{91} +8.63497e13 q^{92} +1.92247e13 q^{93} -5.23828e14 q^{94} +9.08404e14i q^{95} +1.43790e13 q^{96} -5.94650e14i q^{97} -3.65564e14i q^{98} -9.71980e14i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$			101.334i			0.559798i	0.960030	+	0.279899i	$0.0903009\pi$
$2$			101.334i			0.559798i	−0.960030	+	0.279899i	$0.909699\pi$
$3$		−	71.6384i		−	0.0189120i	−0.999955	−	0.00945598i	$-0.996990\pi$
$3$		−	71.6384i		−	0.0189120i	0.999955	−	0.00945598i	$-0.00300998\pi$
$4$	22499.4			0.686627
$5$	−295367.			−1.69078			−0.845389	−	0.534151i	$-0.820631\pi$
$5$	−295367.			−1.69078			−0.845389	+	0.534151i	$0.820631\pi$
$6$	7259.42			0.0105869
$7$	1.06773e6			0.490036			0.245018	−	0.969519i	$-0.421206\pi$
$7$	1.06773e6			0.490036			0.245018	+	0.969519i	$0.421206\pi$
$8$			5.60048e6i			0.944170i
$9$	1.43438e7			0.999642
$10$		−	2.99307e7i		−	0.946493i
$11$		−	6.77632e7i		−	1.04845i	−0.851579	−	0.524226i	$-0.824355\pi$
$11$		−	6.77632e7i		−	1.04845i	0.851579	−	0.524226i	$-0.175645\pi$
$12$		−	1.61182e6i		−	0.0129855i
$13$	−4.89556e7			−0.216385			−0.108193	−	0.994130i	$-0.534506\pi$
$13$	−4.89556e7			−0.216385			−0.108193	+	0.994130i	$0.534506\pi$
$14$			1.08198e8i			0.274321i
$15$			2.11596e7i			0.0319759i
$16$	1.69740e8			0.158083
$17$			2.82095e9i			1.66735i	0.552252	+	0.833677i	$0.313768\pi$
$17$			2.82095e9i			1.66735i	−0.552252	+	0.833677i	$0.686232\pi$
$18$			1.45351e9i			0.559597i
$19$		−	3.07551e9i		−	0.789343i	−0.918822	−	0.394671i	$-0.870858\pi$
$19$		−	3.07551e9i		−	0.789343i	0.918822	−	0.394671i	$-0.129142\pi$
$20$	−6.64557e9			−1.16093
$21$		−	7.64907e7i		−	0.00926753i
$22$	6.86673e9			0.586921
$23$	3.83787e9			0.235034			0.117517	−	0.993071i	$-0.462506\pi$
$23$	3.83787e9			0.235034			0.117517	+	0.993071i	$0.462506\pi$
$24$	4.01209e8			0.0178561
$25$	5.67239e10			1.85873
$26$		−	4.96088e9i		−	0.121132i
$27$		−	2.05550e9i		−	0.0378171i
$28$	2.40234e10			0.336472
$29$	−5.63931e10	−	7.38174e10i	−0.607073	−	0.794646i
$29$	−5.63931e10	−	7.38174e10i	−0.607073	−	0.794646i
$30$	−2.14419e9			−0.0179000
$31$			2.68357e11i			1.75186i	0.482434	+	0.875932i	$0.339753\pi$
$31$			2.68357e11i			1.75186i	−0.482434	+	0.875932i	$0.660247\pi$
$32$			2.00717e11i			1.03266i
$33$	−4.85445e9			−0.0198283
$34$	−2.85858e11			−0.933381
$35$	−3.15373e11			−0.828541
$36$	3.22726e11			0.686381
$37$			7.50493e11i			1.29967i	0.760074	+	0.649836i	$0.225163\pi$
$37$			7.50493e11i			1.29967i	−0.760074	+	0.649836i	$0.774837\pi$
$38$	3.11655e11			0.441872
$39$			3.50710e9i			0.00409227i
$40$		−	1.65419e12i		−	1.59638i
$41$		−	1.43199e11i		−	0.114831i	−0.998350	−	0.0574156i	$-0.981714\pi$
$41$		−	1.43199e11i		−	0.114831i	0.998350	−	0.0574156i	$-0.0182860\pi$
$42$	7.75113e9			0.00518794
$43$			1.06123e12i			0.595385i	0.954662	+	0.297692i	$0.0962170\pi$
$43$			1.06123e12i			0.595385i	−0.954662	+	0.297692i	$0.903783\pi$
$44$		−	1.52463e12i		−	0.719895i
$45$	−4.23667e12			−1.69017
$46$			3.88908e11i			0.131572i
$47$			5.16931e12i			1.48833i	0.667997	+	0.744164i	$0.267152\pi$
$47$			5.16931e12i			1.48833i	−0.667997	+	0.744164i	$0.732848\pi$
$48$		−	1.21599e10i		−	0.00298965i
$49$	−3.60751e12			−0.759865
$50$			5.74807e12i			1.04051i
$51$	2.02088e11			0.0315329
$52$	−1.10147e12			−0.148576
$53$	−1.56320e13			−1.82787			−0.913935	−	0.405860i	$-0.866972\pi$
$53$	−1.56320e13			−1.82787			−0.913935	+	0.405860i	$0.866972\pi$
$54$	2.08292e11			0.0211699
$55$			2.00150e13i			1.77270i
$56$			5.97982e12i			0.462677i
$57$	−2.20325e11			−0.0149280
$58$	7.48023e12	−	5.71455e12i	0.444841	−	0.339838i
$59$	−2.07643e12			−0.108624			−0.0543121	−	0.998524i	$-0.517297\pi$
$59$	−2.07643e12			−0.108624			−0.0543121	+	0.998524i	$0.517297\pi$
$60$			4.76078e11i			0.0219555i
$61$			9.28687e12i			0.378352i	0.981943	+	0.189176i	$0.0605816\pi$
$61$			9.28687e12i			0.378352i	−0.981943	+	0.189176i	$0.939418\pi$
$62$	−2.71938e13			−0.980690
$63$	1.53153e13			0.489860
$64$	−1.47774e13			−0.420000
$65$	1.44599e13			0.365859
$66$		−	4.91921e11i		−	0.0110998i
$67$	7.93908e13			1.60033			0.800165	−	0.599780i	$-0.204745\pi$
$67$	7.93908e13			1.60033			0.800165	+	0.599780i	$0.204745\pi$
$68$			6.34695e13i			1.14485i
$69$		−	2.74939e11i		−	0.00444496i
$70$		−	3.19581e13i		−	0.463816i
$71$	−1.42631e14			−1.86113			−0.930563	−	0.366132i	$-0.880682\pi$
$71$	−1.42631e14			−1.86113			−0.930563	+	0.366132i	$0.880682\pi$
$72$			8.03320e13i			0.943832i
$73$			1.22324e14i			1.29596i	0.761659	+	0.647978i	$0.224385\pi$
$73$			1.22324e14i			1.29596i	−0.761659	+	0.647978i	$0.775615\pi$
$74$	−7.60507e13			−0.727554
$75$		−	4.06361e12i		−	0.0351522i
$76$		−	6.91971e13i		−	0.541984i
$77$		−	7.23531e13i		−	0.513779i
$78$	−3.55390e11			−0.00229084
$79$		−	1.06705e14i		−	0.625143i	−0.949894	−	0.312572i	$-0.898810\pi$
$79$		−	1.06705e14i		−	0.625143i	0.949894	−	0.312572i	$-0.101190\pi$
$80$	−5.01355e13			−0.267283
$81$	2.05670e14			0.998927
$82$	1.45109e13			0.0642822
$83$	2.36776e14			0.957750			0.478875	−	0.877883i	$-0.341045\pi$
$83$	2.36776e14			0.957750			0.478875	+	0.877883i	$0.341045\pi$
$84$		−	1.72099e12i		−	0.00636334i
$85$		−	8.33213e14i		−	2.81912i
$86$	−1.07539e14			−0.333295
$87$	−5.28816e12	+	4.03991e12i	−0.0150283	+	0.0114809i
$88$	3.79506e14			0.989917
$89$			1.89344e14i			0.453761i	0.973923	+	0.226881i	$0.0728527\pi$
$89$			1.89344e14i			0.453761i	−0.973923	+	0.226881i	$0.927147\pi$
$90$		−	4.29320e14i		−	0.946155i
$91$	−5.22716e13			−0.106037
$92$	8.63497e13			0.161381
$93$	1.92247e13			0.0331312
$94$	−5.23828e14			−0.833162
$95$			9.08404e14i			1.33460i
$96$	1.43790e13			0.0195297
$97$		−	5.94650e14i		−	0.747263i	−0.927577	−	0.373631i	$-0.878112\pi$
$97$		−	5.94650e14i		−	0.747263i	0.927577	−	0.373631i	$-0.121888\pi$
$98$		−	3.65564e14i		−	0.425371i
$99$		−	9.71980e14i		−	1.04808i
$100$	1.27625e15			1.27625
$101$			1.43997e15i			1.33642i	0.743971	+	0.668212i	$0.232940\pi$
$101$			1.43997e15i			1.33642i	−0.743971	+	0.668212i	$0.767060\pi$
$102$			2.04784e13i			0.0176521i
$103$	7.32317e14			0.586705			0.293353	−	0.956004i	$-0.405229\pi$
$103$	7.32317e14			0.586705			0.293353	+	0.956004i	$0.405229\pi$
$104$		−	2.74175e14i		−	0.204304i
$105$			2.25928e13i			0.0156693i
$106$		−	1.58406e15i		−	1.02324i
$107$	1.34623e15			0.810476			0.405238	−	0.914211i	$-0.367189\pi$
$107$	1.34623e15			0.810476			0.405238	+	0.914211i	$0.367189\pi$
$108$		−	4.62474e13i		−	0.0259663i
$109$	1.40464e15			0.735983			0.367991	−	0.929829i	$-0.380046\pi$
$109$	1.40464e15			0.735983			0.367991	+	0.929829i	$0.380046\pi$
$110$	−2.02820e15			−0.992353
$111$	5.37641e13			0.0245794
$112$	1.81237e14			0.0774662
$113$			2.67019e15i			1.06771i	0.845575	+	0.533857i	$0.179258\pi$
$113$			2.67019e15i			1.06771i	−0.845575	+	0.533857i	$0.820742\pi$
$114$		−	2.23264e13i		−	0.00835667i
$115$	−1.13358e15			−0.397391
$116$	−1.26881e15	−	1.66085e15i	−0.416833	−	0.545625i
$117$	−7.02209e14			−0.216308
$118$		−	2.10413e14i		−	0.0608076i
$119$			3.01202e15i			0.817063i
$120$	−1.18504e14			−0.0301907
$121$	−4.14604e14			−0.0992528
$122$	−9.41078e14			−0.211800
$123$	−1.02585e13			−0.00217168
$124$			6.03787e15i			1.20288i
$125$	−7.74048e15			−1.45192
$126$			1.55197e15i			0.274223i
$127$		−	3.17068e15i		−	0.527989i	−0.964524	−	0.263995i	$-0.914960\pi$
$127$		−	3.17068e15i		−	0.527989i	0.964524	−	0.263995i	$-0.0850401\pi$
$128$			5.07963e15i			0.797549i
$129$	7.60251e13			0.0112599
$130$			1.46528e15i			0.204807i
$131$		−	4.91257e15i		−	0.648297i	−0.946006	−	0.324149i	$-0.894922\pi$
$131$		−	4.91257e15i		−	0.648297i	0.946006	−	0.324149i	$-0.105078\pi$
$132$	−1.09222e14			−0.0136146
$133$		−	3.28383e15i		−	0.386806i
$134$			8.04501e15i			0.895861i
$135$			6.07126e14i			0.0639404i
$136$	−1.57986e16			−1.57426
$137$		−	1.54843e15i		−	0.146046i	−0.997330	−	0.0730228i	$-0.976735\pi$
$137$		−	1.54843e15i		−	0.146046i	0.997330	−	0.0730228i	$-0.0232646\pi$
$138$	2.78607e13			0.00248828
$139$	1.08122e16			0.914750			0.457375	−	0.889274i	$-0.348790\pi$
$139$	1.08122e16			0.914750			0.457375	+	0.889274i	$0.348790\pi$
$140$	−7.09570e15			−0.568899
$141$	3.70321e14			0.0281472
$142$		−	1.44534e16i		−	1.04185i
$143$			3.31739e15i			0.226870i
$144$	2.43471e15			0.158026
$145$	1.66566e16	+	2.18032e16i	1.02643	+	1.34357i
$146$	−1.23956e16			−0.725474
$147$			2.58436e14i			0.0143705i
$148$			1.68856e16i			0.892390i
$149$	−2.53972e15			−0.127611			−0.0638056	−	0.997962i	$-0.520324\pi$
$149$	−2.53972e15			−0.127611			−0.0638056	+	0.997962i	$0.520324\pi$
$150$	4.11783e14			0.0196781
$151$	1.40110e15			0.0637004			0.0318502	−	0.999493i	$-0.489860\pi$
$151$	1.40110e15			0.0637004			0.0318502	+	0.999493i	$0.489860\pi$
$152$	1.72243e16			0.745273
$153$			4.04630e16i			1.66676i
$154$	7.33184e15			0.287612
$155$		−	7.92638e16i		−	2.96201i
$156$			7.89077e13i			0.00280986i
$157$			1.67253e16i			0.567710i	0.958867	+	0.283855i	$0.0916134\pi$
$157$			1.67253e16i			0.567710i	−0.958867	+	0.283855i	$0.908387\pi$
$158$	1.08128e16			0.349954
$159$			1.11985e15i			0.0345686i
$160$		−	5.92851e16i		−	1.74600i
$161$	4.09783e15			0.115175
$162$			2.08414e16i			0.559197i
$163$		−	6.03194e16i		−	1.54543i	−0.634752	−	0.772716i	$-0.718898\pi$
$163$		−	6.03194e16i		−	1.54543i	0.634752	−	0.772716i	$-0.281102\pi$
$164$		−	3.22188e15i		−	0.0788461i
$165$	1.43384e15			0.0335252
$166$			2.39935e16i			0.536146i
$167$	7.09223e16			1.51499			0.757495	−	0.652841i	$-0.226423\pi$
$167$	7.09223e16			1.51499			0.757495	+	0.652841i	$0.226423\pi$
$168$	4.28385e14			0.00875012
$169$	−4.87892e16			−0.953177
$170$	8.44330e16			1.57814
$171$		−	4.41145e16i		−	0.789060i
$172$			2.38771e16i			0.408807i
$173$	−6.54172e16			−1.07237			−0.536187	−	0.844099i	$-0.680136\pi$
$173$	−6.54172e16			−1.07237			−0.536187	+	0.844099i	$0.680136\pi$
$174$	−4.09381e14	−	5.35871e14i	−0.00642700	−	0.00841281i
$175$	6.05660e16			0.910844
$176$		−	1.15021e16i		−	0.165742i
$177$			1.48752e14i			0.00205430i
$178$	−1.91871e16			−0.254014
$179$	−4.00997e16			−0.509030			−0.254515	−	0.967069i	$-0.581916\pi$
$179$	−4.00997e16			−0.509030			−0.254515	+	0.967069i	$0.581916\pi$
$180$	−9.53225e16			−1.16052
$181$	−5.15758e16			−0.602361			−0.301180	−	0.953567i	$-0.597381\pi$
$181$	−5.15758e16			−0.602361			−0.301180	+	0.953567i	$0.597381\pi$
$182$		−	5.29690e15i		−	0.0593590i
$183$	6.65297e14			0.00715537
$184$			2.14939e16i			0.221912i
$185$		−	2.21671e17i		−	2.19746i
$186$			1.94812e15i			0.0185468i
$187$	1.91156e17			1.74814
$188$			1.16306e17i			1.02193i
$189$		−	2.19472e15i		−	0.0185318i
$190$	−9.20524e16			−0.747107
$191$		−	3.99376e16i		−	0.311624i	−0.987787	−	0.155812i	$-0.950201\pi$
$191$		−	3.99376e16i		−	0.311624i	0.987787	−	0.155812i	$-0.0497995\pi$
$192$			1.05863e15i			0.00794302i
$193$			8.28773e16i			0.598075i	0.954241	+	0.299038i	$0.0966656\pi$
$193$			8.28773e16i			0.598075i	−0.954241	+	0.299038i	$0.903334\pi$
$194$	6.02584e16			0.418316
$195$		−	1.03588e15i		−	0.00691912i
$196$	−8.11666e16			−0.521743
$197$	6.00353e16			0.371458			0.185729	−	0.982601i	$-0.440535\pi$
$197$	6.00353e16			0.371458			0.185729	+	0.982601i	$0.440535\pi$
$198$	9.84948e16			0.586711
$199$	4.57678e16			0.262520			0.131260	−	0.991348i	$-0.458098\pi$
$199$	4.57678e16			0.262520			0.131260	+	0.991348i	$0.458098\pi$
$200$			3.17681e17i			1.75496i
$201$		−	5.68743e15i		−	0.0302654i
$202$	−1.45919e17			−0.748127
$203$	−6.02128e16	−	7.88173e16i	−0.297488	−	0.389405i
$204$	4.54685e15			0.0216513
$205$			4.22961e16i			0.194154i
$206$			7.42088e16i			0.328436i
$207$	5.50496e16			0.234950
$208$	−8.30973e15			−0.0342068
$209$	−2.08407e17			−0.827588
$210$	−2.28943e15			−0.00877166
$211$			3.54151e17i			1.30939i	0.755892	+	0.654696i	$0.227204\pi$
$211$			3.54151e17i			1.30939i	−0.755892	+	0.654696i	$0.772796\pi$
$212$	−3.51710e17			−1.25506
$213$			1.02178e16i			0.0351975i
$214$			1.36419e17i			0.453703i
$215$		−	3.13453e17i		−	1.00666i
$216$	1.15118e16			0.0357058
$217$			2.86534e17i			0.858476i
$218$			1.42338e17i			0.412001i
$219$	8.76310e15			0.0245091
$220$			4.50325e17i			1.21718i
$221$		−	1.38101e17i		−	0.360791i
$222$			5.44815e15i			0.0137595i
$223$	5.33969e16			0.130386			0.0651928	−	0.997873i	$-0.479234\pi$
$223$	5.33969e16			0.130386			0.0651928	+	0.997873i	$0.479234\pi$
$224$			2.14312e17i			0.506042i
$225$	8.13635e17			1.85806
$226$	−2.70582e17			−0.597703
$227$	−6.48085e17			−1.38496			−0.692481	−	0.721436i	$-0.743482\pi$
$227$	−6.48085e17			−1.38496			−0.692481	+	0.721436i	$0.743482\pi$
$228$	−4.95717e15			−0.0102500
$229$		−	3.41476e17i		−	0.683273i	−0.939832	−	0.341637i	$-0.889019\pi$
$229$		−	3.41476e17i		−	0.683273i	0.939832	−	0.341637i	$-0.110981\pi$
$230$		−	1.14870e17i		−	0.222459i
$231$	−5.18326e15			−0.00971657
$232$	4.13412e17	−	3.15828e17i	0.750280	−	0.573180i
$233$	−7.19948e17			−1.26512			−0.632560	−	0.774511i	$-0.717996\pi$
$233$	−7.19948e17			−1.26512			−0.632560	+	0.774511i	$0.717996\pi$
$234$		−	7.11578e16i		−	0.121089i
$235$		−	1.52684e18i		−	2.51643i
$236$	−4.67184e16			−0.0745843
$237$	−7.64414e15			−0.0118227
$238$	−3.05221e17			−0.457390
$239$	7.59385e17			1.10275			0.551376	−	0.834257i	$-0.314103\pi$
$239$	7.59385e17			1.10275			0.551376	+	0.834257i	$0.314103\pi$
$240$			3.59163e15i			0.00505484i
$241$	7.24093e17			0.987794			0.493897	−	0.869520i	$-0.335572\pi$
$241$	7.24093e17			0.987794			0.493897	+	0.869520i	$0.335572\pi$
$242$		−	4.20135e16i		−	0.0555615i
$243$		−	4.42280e16i		−	0.0567088i
$244$			2.08949e17i			0.259786i
$245$	1.06554e18			1.28476
$246$		−	1.03954e15i		−	0.00121570i
$247$			1.50564e17i			0.170802i
$248$	−1.50293e18			−1.65406
$249$		−	1.69623e16i		−	0.0181129i
$250$		−	7.84375e17i		−	0.812781i
$251$			1.55185e18i			1.56062i	0.625393	+	0.780310i	$0.284939\pi$
$251$			1.55185e18i			1.56062i	−0.625393	+	0.780310i	$0.715061\pi$
$252$	3.44586e17			0.336351
$253$		−	2.60066e17i		−	0.246423i
$254$	3.21299e17			0.295567
$255$	−5.96901e16			−0.0533152
$256$	−9.98967e17			−0.866466
$257$	−6.76045e17			−0.569479			−0.284739	−	0.958605i	$-0.591907\pi$
$257$	−6.76045e17			−0.569479			−0.284739	+	0.958605i	$0.591907\pi$
$258$			7.70394e15i			0.00630326i
$259$			8.01327e17i			0.636886i
$260$	3.25338e17			0.251209
$261$	−8.08890e17	−	1.05882e18i	−0.606856	−	0.794362i
$262$	4.97811e17			0.362915
$263$		−	8.44676e17i		−	0.598442i	−0.954184	−	0.299221i	$-0.903273\pi$
$263$		−	8.44676e17i		−	0.598442i	0.954184	−	0.299221i	$-0.0967268\pi$
$264$		−	2.71872e16i		−	0.0187213i
$265$	4.61717e18			3.09052
$266$	3.32764e17			0.216533
$267$	1.35643e16			0.00858151
$268$	1.78624e18			1.09883
$269$		−	2.68401e18i		−	1.60562i	−0.596238	−	0.802808i	$-0.703338\pi$
$269$		−	2.68401e18i		−	1.60562i	0.596238	−	0.802808i	$-0.296662\pi$
$270$	−6.15226e16			−0.0357937
$271$			1.24139e17i			0.0702489i	0.999383	+	0.0351244i	$0.0111828\pi$
$271$			1.24139e17i			0.0702489i	−0.999383	+	0.0351244i	$0.988817\pi$
$272$			4.78827e17i			0.263580i
$273$			3.74465e15i			0.00200536i
$274$	1.56909e17			0.0817560
$275$		−	3.84379e18i		−	1.94879i
$276$		−	6.18595e15i		−	0.00305203i
$277$	−2.63796e18			−1.26669			−0.633345	−	0.773869i	$-0.718319\pi$
$277$	−2.63796e18			−1.26669			−0.633345	+	0.773869i	$0.718319\pi$
$278$			1.09564e18i			0.512075i
$279$			3.84926e18i			1.75124i
$280$		−	1.76624e18i		−	0.782284i
$281$	3.38022e18			1.45763			0.728815	−	0.684710i	$-0.240071\pi$
$281$	3.38022e18			1.45763			0.728815	+	0.684710i	$0.240071\pi$
$282$			3.75262e16i			0.0157567i
$283$	1.80095e18			0.736382			0.368191	−	0.929750i	$-0.379977\pi$
$283$	1.80095e18			0.736382			0.368191	+	0.929750i	$0.379977\pi$
$284$	−3.20911e18			−1.27790
$285$	6.50766e16			0.0252399
$286$	−3.36165e17			−0.127001
$287$		−	1.52898e17i		−	0.0562714i
$288$			2.87904e18i			1.03229i
$289$	−5.09531e18			−1.78007
$290$	−2.20941e18	+	1.68789e18i	−0.752127	+	0.574591i
$291$	−4.25998e16			−0.0141322
$292$			2.75222e18i			0.889838i
$293$			4.16105e18i			1.31128i	0.755073	+	0.655640i	$0.227601\pi$
$293$			4.16105e18i			1.31128i	−0.755073	+	0.655640i	$0.772399\pi$
$294$	−2.61884e16			−0.00804459
$295$	6.13308e17			0.183659
$296$	−4.20312e18			−1.22711
$297$	−1.39287e17			−0.0396495
$298$		−	2.57360e17i		−	0.0714364i
$299$	−1.87885e17			−0.0508580
$300$		−	9.14287e16i		−	0.0241364i
$301$			1.13312e18i			0.291760i
$302$			1.41979e17i			0.0356593i
$303$	1.03157e17			0.0252744
$304$		−	5.22037e17i		−	0.124781i
$305$		−	2.74303e18i		−	0.639709i
$306$	−4.10029e18			−0.933047
$307$		−	1.58095e18i		−	0.351059i	−0.984474	−	0.175530i	$-0.943836\pi$
$307$		−	1.58095e18i		−	0.351059i	0.984474	−	0.175530i	$-0.0561638\pi$
$308$		−	1.62790e18i		−	0.352774i
$309$		−	5.24620e16i		−	0.0110957i
$310$	8.03213e18			1.65813
$311$			6.96779e18i			1.40408i	0.712137	+	0.702040i	$0.247727\pi$
$311$			6.96779e18i			1.40408i	−0.712137	+	0.702040i	$0.752273\pi$
$312$	−1.96415e16			−0.00386380
$313$	−4.05145e18			−0.778087			−0.389044	−	0.921219i	$-0.627194\pi$
$313$	−4.05145e18			−0.778087			−0.389044	+	0.921219i	$0.627194\pi$
$314$	−1.69484e18			−0.317803
$315$	−4.52364e18			−0.828245
$316$		−	2.40079e18i		−	0.429240i
$317$		−	9.38304e18i		−	1.63832i	−0.573564	−	0.819161i	$-0.694440\pi$
$317$		−	9.38304e18i		−	1.63832i	0.573564	−	0.819161i	$-0.305560\pi$
$318$	−1.13479e17			−0.0193514
$319$	−5.00210e18	+	3.82138e18i	−0.833149	+	0.636487i
$320$	4.36476e18			0.710127
$321$		−	9.64416e16i		−	0.0153277i
$322$			4.15250e17i			0.0644749i
$323$	8.67585e18			1.31611
$324$	4.62745e18			0.685890
$325$	−2.77696e18			−0.402202
$326$	6.11242e18			0.865129
$327$		−	1.00626e17i		−	0.0139189i
$328$	8.01980e17			0.108420
$329$			5.51945e18i			0.729334i
$330$			1.45297e17i			0.0187673i
$331$			4.64482e18i			0.586487i	0.956038	+	0.293244i	$0.0947347\pi$
$331$			4.64482e18i			0.586487i	−0.956038	+	0.293244i	$0.905265\pi$
$332$	5.32731e18			0.657616
$333$			1.07649e19i			1.29921i
$334$			7.18686e18i			0.848087i
$335$	−2.34494e19			−2.70580
$336$		−	1.29835e16i		−	0.00146504i
$337$		−	7.42472e17i		−	0.0819324i	−0.999161	−	0.0409662i	$-0.986956\pi$
$337$		−	7.42472e17i		−	0.0819324i	0.999161	−	0.0409662i	$-0.0130436\pi$
$338$		−	4.94402e18i		−	0.533586i
$339$	1.91288e17			0.0201925
$340$		−	1.87468e19i		−	1.93569i
$341$	1.81847e19			1.83675
$342$	4.47030e18			0.441714
$343$	−8.92099e18			−0.862397
$344$	−5.94341e18			−0.562144
$345$			8.12078e16i			0.00751544i
$346$		−	6.62900e18i		−	0.600312i
$347$	−8.17372e18			−0.724351			−0.362175	−	0.932110i	$-0.617966\pi$
$347$	−8.17372e18			−0.724351			−0.362175	+	0.932110i	$0.617966\pi$
$348$	−1.18980e17	+	9.08955e16i	−0.0103188	+	0.00788312i
$349$	−4.24078e18			−0.359961			−0.179981	−	0.983670i	$-0.557603\pi$
$349$	−4.24078e18			−0.359961			−0.179981	+	0.983670i	$0.557603\pi$
$350$			6.13741e18i			0.509888i
$351$			1.00628e17i			0.00818307i
$352$	1.36012e19			1.08270
$353$	−1.33072e16			−0.00103700			−0.000518498	−	1.00000i	$-0.500165\pi$
$353$	−1.33072e16			−0.00103700			−0.000518498		1.00000i	$0.500165\pi$
$354$	−1.50737e16			−0.00114999
$355$	4.21284e19			3.14675
$356$			4.26013e18i			0.311565i
$357$	2.15776e17			0.0154523
$358$		−	4.06347e18i		−	0.284954i
$359$			7.24954e18i			0.497854i	0.968522	+	0.248927i	$0.0800780\pi$
$359$			7.24954e18i			0.497854i	−0.968522	+	0.248927i	$0.919922\pi$
$360$		−	2.37274e19i		−	1.59581i
$361$	5.72235e18			0.376938
$362$		−	5.22639e18i		−	0.337200i
$363$			2.97015e16i			0.00187706i
$364$	−1.17608e18			−0.0728075
$365$		−	3.61305e19i		−	2.19117i
$366$			6.74173e16i			0.00400556i
$367$		−	2.10682e19i		−	1.22640i	−0.789929	−	0.613199i	$-0.789883\pi$
$367$		−	2.10682e19i		−	1.22640i	0.789929	−	0.613199i	$-0.210117\pi$
$368$	6.51440e17			0.0371549
$369$		−	2.05401e18i		−	0.114790i
$370$	2.24628e19			1.23013
$371$	−1.66908e19			−0.895722
$372$	4.32543e17			0.0227488
$373$	−1.91076e19			−0.984894			−0.492447	−	0.870342i	$-0.663897\pi$
$373$	−1.91076e19			−0.984894			−0.492447	+	0.870342i	$0.663897\pi$
$374$			1.93707e19i			0.978605i
$375$			5.54515e17i			0.0274586i
$376$	−2.89506e19			−1.40523
$377$	2.76076e18	+	3.61378e18i	0.131362	+	0.171950i
$378$	2.22401e17			0.0103740
$379$		−	2.60471e19i		−	1.19114i	−0.803302	−	0.595572i	$-0.796925\pi$
$379$		−	2.60471e19i		−	1.19114i	0.803302	−	0.595572i	$-0.203075\pi$
$380$			2.04385e19i			0.916374i
$381$	−2.27143e17			−0.00998531
$382$	4.04704e18			0.174447
$383$	−1.25656e19			−0.531120			−0.265560	−	0.964094i	$-0.585557\pi$
$383$	−1.25656e19			−0.531120			−0.265560	+	0.964094i	$0.585557\pi$
$384$	3.63896e17			0.0150832
$385$			2.13707e19i			0.868686i
$386$	−8.39830e18			−0.334801
$387$			1.52221e19i			0.595172i
$388$		−	1.33793e19i		−	0.513091i
$389$			4.40319e19i			1.65632i	0.560489	+	0.828162i	$0.310613\pi$
$389$			4.40319e19i			1.65632i	−0.560489	+	0.828162i	$0.689387\pi$
$390$	1.04970e17			0.00387330
$391$			1.08264e19i			0.391886i
$392$		−	2.02037e19i		−	0.717441i
$393$	−3.51929e17			−0.0122606
$394$			6.08363e18i			0.207941i
$395$			3.15170e19i			1.05698i
$396$		−	2.18690e19i		−	0.719638i
$397$	1.77903e19			0.574454			0.287227	−	0.957862i	$-0.407267\pi$
$397$	1.77903e19			0.574454			0.287227	+	0.957862i	$0.407267\pi$
$398$			4.63784e18i			0.146958i
$399$	−2.35248e17			−0.00731526
$400$	9.62831e18			0.293833
$401$	−8.91066e18			−0.266887			−0.133443	−	0.991056i	$-0.542603\pi$
$401$	−8.91066e18			−0.266887			−0.133443	+	0.991056i	$0.542603\pi$
$402$	5.76331e17			0.0169425
$403$		−	1.31376e19i		−	0.379078i
$404$			3.23985e19i			0.917625i
$405$	−6.07481e19			−1.68896
$406$	7.98689e18	−	6.10162e18i	0.217988	−	0.166533i
$407$	5.08558e19			1.36265
$408$			1.13179e18i			0.0297724i
$409$		−	7.12780e18i		−	0.184090i	−0.995755	−	0.0920451i	$-0.970660\pi$
$409$		−	7.12780e18i		−	0.184090i	0.995755	−	0.0920451i	$-0.0293404\pi$
$410$	−4.28604e18			−0.108687
$411$	−1.10927e17			−0.00276201
$412$	1.64767e19			0.402847
$413$	−2.21707e18			−0.0532298
$414$			5.57840e18i			0.131525i
$415$	−6.99358e19			−1.61934
$416$		−	9.82622e18i		−	0.223453i
$417$		−	7.74568e17i		−	0.0172997i
$418$		−	2.11187e19i		−	0.463282i
$419$	2.47009e19			0.532240			0.266120	−	0.963940i	$-0.414258\pi$
$419$	2.47009e19			0.532240			0.266120	+	0.963940i	$0.414258\pi$
$420$			5.08324e17i			0.0107590i
$421$			5.94516e18i			0.123608i	0.998088	+	0.0618042i	$0.0196854\pi$
$421$			5.94516e18i			0.123608i	−0.998088	+	0.0618042i	$0.980315\pi$
$422$	−3.58876e19			−0.732995
$423$			7.41474e19i			1.48780i
$424$		−	8.75466e19i		−	1.72582i
$425$			1.60015e20i			3.09916i
$426$	−1.03542e18			−0.0197035
$427$			9.91591e18i			0.185406i
$428$	3.02893e19			0.556495
$429$	2.37653e17			0.00429055
$430$	3.17635e19			0.563528
$431$	5.84639e19			1.01931			0.509657	−	0.860378i	$-0.329772\pi$
$431$	5.84639e19			1.01931			0.509657	+	0.860378i	$0.329772\pi$
$432$		−	3.48900e17i		−	0.00597824i
$433$			4.16093e19i			0.700698i	0.936619	+	0.350349i	$0.113937\pi$
$433$			4.16093e19i			0.700698i	−0.936619	+	0.350349i	$0.886063\pi$
$434$	−2.90357e19			−0.480573
$435$	1.56195e18	−	1.19326e18i	0.0254095	−	0.0194117i
$436$	3.16036e19			0.505345
$437$		−	1.18034e19i		−	0.185523i
$438$			8.88002e17i			0.0137201i
$439$	−6.02481e19			−0.915081			−0.457541	−	0.889189i	$-0.651270\pi$
$439$	−6.02481e19			−0.915081			−0.457541	+	0.889189i	$0.651270\pi$
$440$	−1.12093e20			−1.67373
$441$	−5.17452e19			−0.759593
$442$	1.39944e19			0.201970
$443$		−	8.70545e19i		−	1.23527i	−0.786463	−	0.617637i	$-0.788090\pi$
$443$		−	8.70545e19i		−	1.23527i	0.786463	−	0.617637i	$-0.211910\pi$
$444$	1.20966e18			0.0168768
$445$		−	5.59260e19i		−	0.767209i
$446$			5.41093e18i			0.0729895i
$447$			1.81941e17i			0.00241338i
$448$	−1.57784e19			−0.205815
$449$		−	1.11969e20i		−	1.43632i	−0.695876	−	0.718161i	$-0.744984\pi$
$449$		−	1.11969e20i		−	1.43632i	0.695876	−	0.718161i	$-0.255016\pi$
$450$			8.24490e19i			1.04014i
$451$	−9.70360e18			−0.120395
$452$			6.00777e19i			0.733120i
$453$		−	1.00373e17i		−	0.00120470i
$454$		−	6.56731e19i		−	0.775298i
$455$	1.54393e19			0.179284
$456$		−	1.23392e18i		−	0.0140946i
$457$	5.45514e19			0.612964			0.306482	−	0.951877i	$-0.400848\pi$
$457$	5.45514e19			0.612964			0.306482	+	0.951877i	$0.400848\pi$
$458$	3.46032e19			0.382495
$459$	5.79845e18			0.0630546
$460$	−2.55048e19			−0.272859
$461$		−	3.84912e19i		−	0.405139i	−0.979268	−	0.202569i	$-0.935071\pi$
$461$		−	3.84912e19i		−	0.405139i	0.979268	−	0.202569i	$-0.0649292\pi$
$462$		−	5.25241e17i		−	0.00543931i
$463$	1.31393e20			1.33880			0.669399	−	0.742903i	$-0.266552\pi$
$463$	1.31393e20			1.33880			0.669399	+	0.742903i	$0.266552\pi$
$464$	−9.57217e18	−	1.25298e19i	−0.0959678	−	0.125620i
$465$	−5.67833e18			−0.0560175
$466$		−	7.29553e19i		−	0.708211i
$467$		−	1.20183e20i		−	1.14806i	−0.818833	−	0.574032i	$-0.805378\pi$
$467$		−	1.20183e20i		−	1.14806i	0.818833	−	0.574032i	$-0.194622\pi$
$468$	−1.57993e19			−0.148523
$469$	8.47683e19			0.784219
$470$	1.54721e20			1.40869
$471$	1.19817e18			0.0107365
$472$		−	1.16290e19i		−	0.102560i
$473$	7.19126e19			0.624233
$474$		−	7.74613e17i		−	0.00661831i
$475$		−	1.74455e20i		−	1.46717i
$476$			6.77686e19i			0.561017i
$477$	−2.24222e20			−1.82722
$478$			7.69517e19i			0.617318i
$479$		−	1.60485e20i		−	1.26741i	−0.773574	−	0.633706i	$-0.781533\pi$
$479$		−	1.60485e20i		−	1.26741i	0.773574	−	0.633706i	$-0.218467\pi$
$480$	−4.24709e18			−0.0330204
$481$		−	3.67409e19i		−	0.281230i
$482$			7.33754e19i			0.552965i
$483$		−	2.93562e17i		−	0.00217819i
$484$	−9.32832e18			−0.0681496
$485$			1.75640e20i			1.26346i
$486$	4.48181e18			0.0317455
$487$	1.34429e20			0.937615			0.468807	−	0.883300i	$-0.344684\pi$
$487$	1.34429e20			0.937615			0.468807	+	0.883300i	$0.344684\pi$
$488$	−5.20109e19			−0.357228
$489$	−4.32118e18			−0.0292271
$490$			1.07975e20i			0.719207i
$491$		−	1.27509e20i		−	0.836432i	−0.908347	−	0.418216i	$-0.862655\pi$
$491$		−	1.27509e20i		−	0.836432i	0.908347	−	0.418216i	$-0.137345\pi$
$492$	−2.30810e17			−0.00149113
$493$	2.08235e20	−	1.59082e20i	1.32496	−	1.01221i
$494$	−1.52573e19			−0.0956146
$495$			2.87091e20i			1.77207i
$496$			4.55509e19i			0.276939i
$497$	−1.52292e20			−0.912018
$498$	1.71886e18			0.0101396
$499$	2.37356e20			1.37926			0.689630	−	0.724162i	$-0.257773\pi$
$499$	2.37356e20			1.37926			0.689630	+	0.724162i	$0.257773\pi$
$500$	−1.74156e20			−0.996927
$501$		−	5.08076e18i		−	0.0286514i
$502$	−1.57256e20			−0.873631
$503$			2.30021e20i			1.25894i	0.777023	+	0.629472i	$0.216729\pi$
$503$			2.30021e20i			1.25894i	−0.777023	+	0.629472i	$0.783271\pi$
$504$			8.57732e19i			0.462511i
$505$		−	4.25320e20i		−	2.25960i
$506$	2.63536e19			0.137947
$507$			3.49518e18i			0.0180264i
$508$		−	7.13384e19i		−	0.362531i
$509$	1.65498e20			0.828725			0.414363	−	0.910112i	$-0.364005\pi$
$509$	1.65498e20			0.828725			0.414363	+	0.910112i	$0.364005\pi$
$510$		−	6.04864e18i		−	0.0298457i
$511$			1.30610e20i			0.635065i
$512$			6.52197e19i			0.312503i
$513$	−6.32171e18			−0.0298507
$514$		−	6.85065e19i		−	0.318793i
$515$	−2.16302e20			−0.991988
$516$	1.71052e18			0.00773134
$517$	3.50289e20			1.56044
$518$	−8.12019e19			−0.356527
$519$			4.68638e18i			0.0202807i
$520$			8.09821e19i			0.345433i
$521$	3.33743e20			1.40323			0.701616	−	0.712555i	$-0.252462\pi$
$521$	3.33743e20			1.40323			0.701616	+	0.712555i	$0.252462\pi$
$522$	1.07295e20	−	8.19682e19i	0.444682	−	0.339717i
$523$	−1.78176e20			−0.727924			−0.363962	−	0.931414i	$-0.618576\pi$
$523$	−1.78176e20			−0.727924			−0.363962	+	0.931414i	$0.618576\pi$
$524$		−	1.10530e20i		−	0.445138i
$525$		−	4.33885e18i		−	0.0172258i
$526$	8.55946e19			0.335007
$527$	−7.57021e20			−2.92098
$528$	−8.23994e17			−0.00313451
$529$	−2.51906e20			−0.944759
$530$			4.67877e20i			1.73007i
$531$	−2.97838e19			−0.108585
$532$		−	7.38841e19i		−	0.265591i
$533$			7.01038e18i			0.0248478i
$534$			1.37453e18i			0.00480391i
$535$	−3.97631e20			−1.37033
$536$			4.44626e20i			1.51098i
$537$			2.87268e18i			0.00962675i
$538$	2.71982e20			0.898820
$539$			2.44456e20i			0.796682i
$540$			1.36599e19i			0.0439032i
$541$		−	1.96773e20i		−	0.623715i	−0.950129	−	0.311858i	$-0.899049\pi$
$541$		−	1.96773e20i		−	0.623715i	0.950129	−	0.311858i	$-0.100951\pi$
$542$	−1.25796e19			−0.0393252
$543$			3.69481e18i			0.0113918i
$544$	−5.66211e20			−1.72182
$545$	−4.14885e20			−1.24438
$546$	−3.79462e17			−0.00112259
$547$	2.42812e20			0.708541			0.354271	−	0.935143i	$-0.384729\pi$
$547$	2.42812e20			0.708541			0.354271	+	0.935143i	$0.384729\pi$
$548$		−	3.48388e19i		−	0.100279i
$549$			1.33209e20i			0.378216i
$550$	3.89508e20			1.09093
$551$	−2.27026e20	+	1.73438e20i	−0.627248	+	0.479189i
$552$	1.53979e18			0.00419680
$553$		−	1.13932e20i		−	0.306343i
$554$		−	2.67316e20i		−	0.709090i
$555$	−1.58801e19			−0.0415582
$556$	2.43268e20			0.628092
$557$	5.50489e20			1.40228			0.701141	−	0.713023i	$-0.252675\pi$
$557$	5.50489e20			1.40228			0.701141	+	0.713023i	$0.252675\pi$
$558$	−3.90061e20			−0.980339
$559$		−	5.19534e19i		−	0.128832i
$560$	−5.35314e19			−0.130978
$561$		−	1.36941e19i		−	0.0330608i
$562$			3.42532e20i			0.815978i
$563$			3.15998e20i			0.742800i	0.928473	+	0.371400i	$0.121122\pi$
$563$			3.15998e20i			0.742800i	−0.928473	+	0.371400i	$0.878878\pi$
$564$	8.33199e18			0.0193266
$565$		−	7.88687e20i		−	1.80527i
$566$			1.82498e20i			0.412225i
$567$	2.19601e20			0.489510
$568$		−	7.98801e20i		−	1.75722i
$569$			2.22985e20i			0.484098i	0.970264	+	0.242049i	$0.0778195\pi$
$569$			2.22985e20i			0.484098i	−0.970264	+	0.242049i	$0.922180\pi$
$570$			6.59449e18i			0.0141293i
$571$	4.15776e20			0.879203			0.439601	−	0.898193i	$-0.355120\pi$
$571$	4.15776e20			0.879203			0.439601	+	0.898193i	$0.355120\pi$
$572$			7.46393e19i			0.155775i
$573$	−2.86106e18			−0.00589343
$574$	1.54938e19			0.0315006
$575$	2.17699e20			0.436865
$576$	−2.11964e20			−0.419850
$577$		−	1.30531e20i		−	0.255209i	−0.991825	−	0.127604i	$-0.959271\pi$
$577$		−	1.30531e20i		−	0.255209i	0.991825	−	0.127604i	$-0.0407288\pi$
$578$		−	5.16329e20i		−	0.996478i
$579$	5.93719e18			0.0113108
$580$	3.74764e20	+	4.90558e20i	0.704771	+	0.922531i
$581$	2.52814e20			0.469332
$582$		−	4.31681e18i		−	0.00791117i
$583$			1.05927e21i			1.91644i
$584$	−6.85073e20			−1.22360
$585$	2.07409e20			0.365728
$586$	−4.21656e20			−0.734052
$587$	−5.20537e20			−0.894676			−0.447338	−	0.894365i	$-0.647628\pi$
$587$	−5.20537e20			−0.894676			−0.447338	+	0.894365i	$0.647628\pi$
$588$			5.81465e18i			0.00986719i
$589$	8.25336e20			1.38282
$590$			6.21491e19i			0.102812i
$591$		−	4.30083e18i		−	0.00702500i
$592$			1.27389e20i			0.205456i
$593$	6.47324e19			0.103089			0.0515444	−	0.998671i	$-0.483586\pi$
$593$	6.47324e19			0.103089			0.0515444	+	0.998671i	$0.483586\pi$
$594$		−	1.41145e19i		−	0.0221957i
$595$		−	8.89650e20i		−	1.38147i
$596$	−5.71421e19			−0.0876212
$597$		−	3.27873e18i		−	0.00496476i
$598$		−	1.90392e19i		−	0.0284702i
$599$		−	1.11240e21i		−	1.64271i	−0.570418	−	0.821355i	$-0.693219\pi$
$599$		−	1.11240e21i		−	1.64271i	0.570418	−	0.821355i	$-0.306781\pi$
$600$	2.27581e19			0.0331896
$601$		−	3.14494e20i		−	0.452954i	−0.974017	−	0.226477i	$-0.927279\pi$
$601$		−	3.14494e20i		−	0.452954i	0.974017	−	0.226477i	$-0.0727208\pi$
$602$	−1.14823e20			−0.163326
$603$	1.13876e21			1.59976
$604$	3.15239e19			0.0437384
$605$	1.22460e20			0.167814
$606$			1.04534e19i			0.0141485i
$607$			7.46732e20i			0.998273i	0.866523	+	0.499137i	$0.166349\pi$
$607$			7.46732e20i			0.998273i	−0.866523	+	0.499137i	$0.833651\pi$
$608$	6.17307e20			0.815126
$609$	−5.64635e18	+	4.31355e18i	−0.00736441	+	0.00562607i
$610$	2.77963e20			0.358107
$611$		−	2.53067e20i		−	0.322052i
$612$			9.10393e20i			1.14444i
$613$	−2.01248e20			−0.249907			−0.124953	−	0.992163i	$-0.539878\pi$
$613$	−2.01248e20			−0.249907			−0.124953	+	0.992163i	$0.539878\pi$
$614$	1.60204e20			0.196522
$615$	3.03002e18			0.00367183
$616$	4.05212e20			0.485095
$617$		−	1.45728e21i		−	1.72347i	−0.507355	−	0.861737i	$-0.669377\pi$
$617$		−	1.45728e21i		−	1.72347i	0.507355	−	0.861737i	$-0.330623\pi$
$618$	5.31620e18			0.00621137
$619$		−	4.22640e19i		−	0.0487855i	−0.999702	−	0.0243928i	$-0.992235\pi$
$619$		−	4.22640e19i		−	0.0487855i	0.999702	−	0.0243928i	$-0.00776523\pi$
$620$		−	1.78339e21i		−	2.03380i
$621$		−	7.88873e18i		−	0.00888833i
$622$	−7.06075e20			−0.786001
$623$			2.02169e20i			0.222359i
$624$			5.95296e17i			0.000646917i
$625$	5.55203e20			0.596144
$626$		−	4.10551e20i		−	0.435571i
$627$			1.49299e19i			0.0156513i
$628$			3.76309e20i			0.389805i
$629$	−2.11710e21			−2.16701
$630$		−	4.58399e20i		−	0.463650i
$631$	3.19825e19			0.0319663			0.0159831	−	0.999872i	$-0.494912\pi$
$631$	3.19825e19			0.0319663			0.0159831	+	0.999872i	$0.494912\pi$
$632$	5.97596e20			0.590241
$633$	2.53708e19			0.0247632
$634$	9.50823e20			0.917128
$635$			9.36514e20i			0.892712i
$636$			2.51960e19i			0.0237357i
$637$	1.76608e20			0.164424
$638$	−3.87236e20	−	5.06884e20i	−0.356304	−	0.466395i
$639$	−2.04586e21			−1.86046
$640$		−	1.50035e21i		−	1.34848i
$641$			1.46413e21i			1.30061i	0.759675	+	0.650303i	$0.225358\pi$
$641$			1.46413e21i			1.30061i	−0.759675	+	0.650303i	$0.774642\pi$
$642$	9.77283e18			0.00858040
$643$	5.22158e20			0.453127			0.226564	−	0.973996i	$-0.427251\pi$
$643$	5.22158e20			0.453127			0.226564	+	0.973996i	$0.427251\pi$
$644$	9.21985e19			0.0790824
$645$	−2.24553e19			−0.0190380
$646$			8.79161e20i			0.736757i
$647$	1.84336e19			0.0152696			0.00763482	−	0.999971i	$-0.497570\pi$
$647$	1.84336e19			0.0152696			0.00763482	+	0.999971i	$0.497570\pi$
$648$			1.15185e21i			0.943157i
$649$			1.40705e20i			0.113887i
$650$		−	2.81401e20i		−	0.225151i
$651$	2.05268e19			0.0162355
$652$		−	1.35715e21i		−	1.06113i
$653$			6.15583e20i			0.475814i	0.971288	+	0.237907i	$0.0764614\pi$
$653$			6.15583e20i			0.475814i	−0.971288	+	0.237907i	$0.923539\pi$
$654$	1.01969e19			0.00779175
$655$			1.45101e21i			1.09613i
$656$		−	2.43065e19i		−	0.0181528i
$657$			1.75459e21i			1.29549i
$658$	−5.59309e20			−0.408279
$659$			3.79046e20i			0.273559i	0.990601	+	0.136780i	$0.0436753\pi$
$659$			3.79046e20i			0.273559i	−0.990601	+	0.136780i	$0.956325\pi$
$660$	3.22606e19			0.0230193
$661$	−1.58987e21			−1.12163			−0.560817	−	0.827940i	$-0.689513\pi$
$661$	−1.58987e21			−1.12163			−0.560817	+	0.827940i	$0.689513\pi$
$662$	−4.70679e20			−0.328314
$663$	−9.89335e18			−0.00682326
$664$			1.32606e21i			0.904278i
$665$			9.69934e20i			0.654003i
$666$	−1.09085e21			−0.727293
$667$	−2.16429e20	−	2.83302e20i	−0.142683	−	0.186769i
$668$	1.59571e21			1.04023
$669$		−	3.82527e18i		−	0.00246584i
$670$		−	2.37623e21i		−	1.51470i
$671$	6.29308e20			0.396684
$672$	1.53530e19			0.00957025
$673$	−2.52753e21			−1.55806			−0.779030	−	0.626987i	$-0.784288\pi$
$673$	−2.52753e21			−1.55806			−0.779030	+	0.626987i	$0.784288\pi$
$674$	7.52378e19			0.0458656
$675$		−	1.16596e20i		−	0.0702918i
$676$	−1.09773e21			−0.654477
$677$			2.43644e21i			1.43662i	0.695724	+	0.718309i	$0.255084\pi$
$677$			2.43644e21i			1.43662i	−0.695724	+	0.718309i	$0.744916\pi$
$678$			1.93841e19i			0.0113037i
$679$		−	6.34928e20i		−	0.366186i
$680$	4.66639e21			2.66173
$681$			4.64277e19i			0.0261923i
$682$			1.84274e21i			1.02821i
$683$	1.07027e21			0.590663			0.295332	−	0.955395i	$-0.404570\pi$
$683$	1.07027e21			0.590663			0.295332	+	0.955395i	$0.404570\pi$
$684$		−	9.92548e20i		−	0.541790i
$685$			4.57356e20i			0.246931i
$686$		−	9.04001e20i		−	0.482768i
$687$	−2.44628e19			−0.0129220
$688$			1.80134e20i			0.0941200i
$689$	7.65275e20			0.395524
$690$	−8.22913e18			−0.00420713
$691$	−1.51562e21			−0.766489			−0.383245	−	0.923647i	$-0.625193\pi$
$691$	−1.51562e21			−0.766489			−0.383245	+	0.923647i	$0.625193\pi$
$692$	−1.47185e21			−0.736320
$693$		−	1.03782e21i		−	0.513595i
$694$		−	8.28278e20i		−	0.405490i
$695$	−3.19356e21			−1.54664
$696$	−2.26254e19	−	2.96162e19i	−0.0108400	−	0.0141893i
$697$	4.03955e20			0.191464
$698$		−	4.29737e20i		−	0.201505i
$699$			5.15759e19i			0.0239259i
$700$	1.36270e21			0.625409
$701$	2.17910e21			0.989448			0.494724	−	0.869050i	$-0.335269\pi$
$701$	2.17910e21			0.989448			0.494724	+	0.869050i	$0.335269\pi$
$702$	−1.01971e19			−0.00458086
$703$	2.30815e21			1.02589
$704$			1.00137e21i			0.440350i
$705$	−1.09381e20			−0.0475906
$706$		−	1.34848e18i		−	0.000580507i
$707$			1.53751e21i			0.654896i
$708$			3.34683e18i			0.00141054i
$709$	1.09213e21			0.455438			0.227719	−	0.973727i	$-0.426873\pi$
$709$	1.09213e21			0.455438			0.227719	+	0.973727i	$0.426873\pi$
$710$			4.26905e21i			1.76154i
$711$		−	1.53055e21i		−	0.624920i
$712$	−1.06042e21			−0.428428
$713$			1.02992e21i			0.411749i
$714$			2.18655e19i			0.00865014i
$715$		−	9.79847e20i		−	0.383586i
$716$	−9.02218e20			−0.349513
$717$		−	5.44011e19i		−	0.0208552i
$718$	−7.34626e20			−0.278698
$719$	−4.45097e21			−1.67105			−0.835523	−	0.549456i	$-0.814835\pi$
$719$	−4.45097e21			−1.67105			−0.835523	+	0.549456i	$0.814835\pi$
$720$	−7.19133e20			−0.267187
$721$	7.81920e20			0.287507
$722$			5.79870e20i			0.211009i
$723$		−	5.18729e19i		−	0.0186811i
$724$	−1.16042e21			−0.413597
$725$	−3.19884e21	−	4.18721e21i	−1.12838	−	1.47703i
$726$	−3.00978e18			−0.00105078
$727$			1.84962e21i			0.639107i	0.947568	+	0.319553i	$0.103533\pi$
$727$			1.84962e21i			0.639107i	−0.947568	+	0.319553i	$0.896467\pi$
$728$		−	2.92746e20i		−	0.100116i
$729$	2.94797e21			0.997855
$730$	3.66125e21			1.22661
$731$	−2.99368e21			−0.992717
$732$	1.49688e19			0.00491307
$733$		−	1.52763e20i		−	0.0496294i	−0.999692	−	0.0248147i	$-0.992100\pi$
$733$		−	1.52763e20i		−	0.0496294i	0.999692	−	0.0248147i	$-0.00789957\pi$
$734$	2.13493e21			0.686534
$735$		−	7.63334e19i		−	0.0242974i
$736$			7.70325e20i			0.242712i
$737$		−	5.37978e21i		−	1.67787i
$738$	2.08141e20			0.0642592
$739$		−	5.26034e21i		−	1.60761i	−0.594892	−	0.803805i	$-0.702805\pi$
$739$		−	5.26034e21i		−	1.60761i	0.594892	−	0.803805i	$-0.297195\pi$
$740$		−	4.98745e21i		−	1.50883i
$741$	1.07861e19			0.00323020
$742$		−	1.69135e21i		−	0.501423i
$743$			3.85678e21i			1.13190i	0.824439	+	0.565950i	$0.191491\pi$
$743$			3.85678e21i			1.13190i	−0.824439	+	0.565950i	$0.808509\pi$
$744$			1.07667e20i			0.0312815i
$745$	7.50149e20			0.215762
$746$		−	1.93625e21i		−	0.551341i
$747$	3.39626e21			0.957407
$748$	4.30090e21			1.20032
$749$	1.43741e21			0.397162
$750$	−5.61914e19			−0.0153713
$751$			2.08180e21i			0.563819i	0.959441	+	0.281910i	$0.0909679\pi$
$751$			2.08180e21i			0.563819i	−0.959441	+	0.281910i	$0.909032\pi$
$752$			8.77438e20i			0.235279i
$753$	1.11172e20			0.0295144
$754$	−3.66199e20	+	2.79760e20i	−0.0962570	+	0.0735360i
$755$	−4.13838e20			−0.107703
$756$		−	4.93799e19i		−	0.0127244i
$757$		−	2.45794e21i		−	0.627121i	−0.949568	−	0.313561i	$-0.898478\pi$
$757$		−	2.45794e21i		−	0.627121i	0.949568	−	0.313561i	$-0.101522\pi$
$758$	2.63946e21			0.666800
$759$	−1.86307e19			−0.00466033
$760$	−5.08749e21			−1.26009
$761$	6.51724e21			1.59838			0.799188	−	0.601081i	$-0.205263\pi$
$761$	6.51724e21			1.59838			0.799188	+	0.601081i	$0.205263\pi$
$762$		−	2.30173e19i		−	0.00558975i
$763$	1.49979e21			0.360658
$764$		−	8.98571e20i		−	0.213970i
$765$		−	1.19514e22i		−	2.81812i
$766$		−	1.27333e21i		−	0.297320i
$767$	1.01653e20			0.0235047
$768$			7.15644e19i			0.0163866i
$769$		−	3.81755e21i		−	0.865640i	−0.901480	−	0.432820i	$-0.857519\pi$
$769$		−	3.81755e21i		−	0.865640i	0.901480	−	0.432820i	$-0.142481\pi$
$770$	−2.16558e21			−0.486289
$771$			4.84308e19i			0.0107700i
$772$			1.86469e21i			0.410654i
$773$			1.93933e21i			0.422966i	0.977382	+	0.211483i	$0.0678293\pi$
$773$			1.93933e21i			0.422966i	−0.977382	+	0.211483i	$0.932171\pi$
$774$	−1.54252e21			−0.333176
$775$			1.52223e22i			3.25624i
$776$	3.33032e21			0.705543
$777$	5.74058e19			0.0120448
$778$	−4.46194e21			−0.927206
$779$	−4.40409e20			−0.0906411
$780$		−	2.33067e19i		−	0.00475085i
$781$			9.66512e21i			1.95130i
$782$	−1.09709e21			−0.219377
$783$	−1.51731e20	+	1.15916e20i	−0.0300512	+	0.0229578i
$784$	−6.12338e20			−0.120121
$785$		−	4.94009e21i		−	0.959871i
$786$		−	3.56624e19i		−	0.00686344i
$787$	−1.30378e21			−0.248540			−0.124270	−	0.992248i	$-0.539659\pi$
$787$	−1.30378e21			−0.248540			−0.124270	+	0.992248i	$0.539659\pi$
$788$	1.35076e21			0.255053
$789$	−6.05113e19			−0.0113177
$790$	−3.19375e21			−0.591694
$791$			2.85106e21i			0.523218i
$792$	5.44355e21			0.989563
$793$		−	4.54645e20i		−	0.0818697i
$794$			1.80277e21i			0.321578i
$795$		−	3.30767e20i		−	0.0584478i
$796$	1.02975e21			0.180253
$797$		−	5.47364e21i		−	0.949159i	−0.880213	−	0.474580i	$-0.842600\pi$
$797$		−	5.47364e21i		−	0.949159i	0.880213	−	0.474580i	$-0.157400\pi$
$798$		−	2.38387e19i		−	0.00409506i
$799$	−1.45823e22			−2.48157
$800$			1.13854e22i			1.91944i
$801$			2.71591e21i			0.453599i
$802$		−	9.02955e20i		−	0.149403i
$803$	8.28907e21			1.35875
$804$		−	1.27964e20i		−	0.0207810i
$805$	−1.21036e21			−0.194736
$806$	1.33129e21			0.212207
$807$	−1.92278e20			−0.0303653
$808$	−8.06454e21			−1.26181
$809$		−	2.52442e21i		−	0.391333i	−0.980670	−	0.195667i	$-0.937313\pi$
$809$		−	2.52442e21i		−	0.391333i	0.980670	−	0.195667i	$-0.0626871\pi$
$810$		−	6.15586e21i		−	0.945478i
$811$	6.21866e21			0.946325			0.473162	−	0.880975i	$-0.343112\pi$
$811$	6.21866e21			0.946325			0.473162	+	0.880975i	$0.343112\pi$
$812$	−1.35475e21	−	1.77334e21i	−0.204263	−	0.267376i
$813$	8.89314e18			0.00132854
$814$			5.15344e21i			0.762806i
$815$			1.78163e22i			2.61298i
$816$	3.43024e19			0.00498481
$817$	3.26384e21			0.469963
$818$	7.22290e20			0.103053
$819$	−7.49772e20			−0.105999
$820$			9.51636e20i			0.133311i
$821$	5.35907e21			0.743902			0.371951	−	0.928252i	$-0.378689\pi$
$821$	5.35907e21			0.743902			0.371951	+	0.928252i	$0.378689\pi$
$822$		−	1.12407e19i		−	0.00154617i
$823$		−	9.41625e20i		−	0.128345i	−0.997939	−	0.0641725i	$-0.979559\pi$
$823$		−	9.41625e20i		−	0.128345i	0.997939	−	0.0641725i	$-0.0204408\pi$
$824$			4.10132e21i			0.553949i
$825$	−2.75363e20			−0.0368554
$826$		−	2.24665e20i		−	0.0297979i
$827$		−	1.11536e22i		−	1.46596i	−0.680248	−	0.732982i	$-0.738128\pi$
$827$		−	1.11536e22i		−	1.46596i	0.680248	−	0.732982i	$-0.261872\pi$
$828$	1.23858e21			0.161323
$829$		−	5.48564e21i		−	0.708057i	−0.935235	−	0.354029i	$-0.884812\pi$
$829$		−	5.48564e21i		−	0.708057i	0.935235	−	0.354029i	$-0.115188\pi$
$830$		−	7.08688e21i		−	0.906504i
$831$			1.88979e20i			0.0239556i
$832$	7.23439e20			0.0908818
$833$		−	1.01766e22i		−	1.26696i
$834$	7.84902e19			0.00968434
$835$	−2.09481e22			−2.56151
$836$	−4.68902e21			−0.568244
$837$	5.51608e20			0.0662505
$838$			2.50305e21i			0.297947i
$839$		−	7.14822e21i		−	0.843302i	−0.906758	−	0.421651i	$-0.861451\pi$
$839$		−	7.14822e21i		−	0.843302i	0.906758	−	0.421651i	$-0.138549\pi$
$840$	−1.26531e20			−0.0147945
$841$	−2.26882e21	+	8.32558e21i	−0.262924	+	0.964817i
$842$	−6.02448e20			−0.0691957
$843$		−	2.42154e20i		−	0.0275666i
$844$			7.96818e21i			0.899064i
$845$	1.44107e22			1.61161
$846$	−7.51367e21			−0.832864
$847$	−4.42686e20			−0.0486374
$848$	−2.65337e21			−0.288955
$849$		−	1.29017e20i		−	0.0139264i
$850$	−1.62150e22			−1.73490
$851$			2.88030e21i			0.305468i
$852$			2.29895e20i			0.0241676i
$853$		−	2.57881e21i		−	0.268721i	−0.990933	−	0.134360i	$-0.957102\pi$
$853$		−	2.57881e21i		−	0.268721i	0.990933	−	0.134360i	$-0.0428979\pi$
$854$	−1.00482e21			−0.103790
$855$			1.30299e22i			1.33413i
$856$			7.53952e21i			0.765227i
$857$	3.40922e21			0.343004			0.171502	−	0.985184i	$-0.445138\pi$
$857$	3.40922e21			0.343004			0.171502	+	0.985184i	$0.445138\pi$
$858$			2.40823e19i			0.00240184i
$859$			1.28285e22i			1.26832i	0.773203	+	0.634159i	$0.218653\pi$
$859$			1.28285e22i			1.26832i	−0.773203	+	0.634159i	$0.781347\pi$
$860$		−	7.05250e21i		−	0.691202i
$861$	−1.09534e19			−0.00106420
$862$			5.92439e21i			0.570610i
$863$	−2.58418e20			−0.0246741			−0.0123371	−	0.999924i	$-0.503927\pi$
$863$	−2.58418e20			−0.0246741			−0.0123371	+	0.999924i	$0.503927\pi$
$864$	4.12573e20			0.0390524
$865$	1.93221e22			1.81315
$866$	−4.21644e21			−0.392249
$867$			3.65020e20i			0.0336646i
$868$			6.44684e21i			0.589453i
$869$	−7.23064e21			−0.655433
$870$	1.20918e20	+	1.58279e20i	0.0108666	+	0.0142242i
$871$	−3.88663e21			−0.346288
$872$			7.86667e21i			0.694892i
$873$		−	8.52953e21i		−	0.746996i
$874$	1.19609e21			0.103855
$875$	−8.26477e21			−0.711492
$876$	1.97164e20			0.0168286
$877$	4.39862e21			0.372237			0.186119	−	0.982527i	$-0.440409\pi$
$877$	4.39862e21			0.372237			0.186119	+	0.982527i	$0.440409\pi$
$878$		−	6.10520e21i		−	0.512260i
$879$	2.98091e20			0.0247989
$880$			3.39734e21i			0.280233i
$881$		−	1.37936e22i		−	1.12813i	−0.825731	−	0.564064i	$-0.809237\pi$
$881$		−	1.37936e22i		−	1.12813i	0.825731	−	0.564064i	$-0.190763\pi$
$882$		−	5.24356e21i		−	0.425218i
$883$	−4.49686e21			−0.361580			−0.180790	−	0.983522i	$-0.557865\pi$
$883$	−4.49686e21			−0.361580			−0.180790	+	0.983522i	$0.557865\pi$
$884$		−	3.10719e21i		−	0.247729i
$885$		−	4.39364e19i		−	0.00347336i
$886$	8.82160e21			0.691504
$887$		−	1.83607e22i		−	1.42712i	−0.700592	−	0.713562i	$-0.747081\pi$
$887$		−	1.83607e22i		−	1.42712i	0.700592	−	0.713562i	$-0.252919\pi$
$888$			3.01105e20i			0.0232071i
$889$		−	3.38545e21i		−	0.258734i
$890$	5.66722e21			0.429482
$891$		−	1.39369e22i		−	1.04733i
$892$	1.20140e21			0.0895262
$893$	1.58983e22			1.17480
$894$	−1.84369e19			−0.00135100
$895$	1.18441e22			0.860656
$896$			5.42369e21i			0.390827i
$897$			1.34598e19i			0.000961824i
$898$	1.13463e22			0.804050
$899$	1.98094e22	−	1.51335e22i	1.39211	−	1.06351i
$900$	1.83063e22			1.27580
$901$		−	4.40970e22i		−	3.04771i
$902$		−	9.83306e20i		−	0.0673969i
$903$	8.11746e19			0.00551775
$904$	−1.49544e22			−1.00810
$905$	1.52338e22			1.01846
$906$	1.01712e19			0.000674387
$907$			1.97080e22i			1.29595i	0.761661	+	0.647976i	$0.224384\pi$
$907$			1.97080e22i			1.29595i	−0.761661	+	0.647976i	$0.775616\pi$
$908$	−1.45815e22			−0.950952
$909$			2.06547e22i			1.33595i
$910$			1.56453e21i			0.100363i
$911$			1.26838e22i			0.806976i	0.914985	+	0.403488i	$0.132202\pi$
$911$			1.26838e22i			0.806976i	−0.914985	+	0.403488i	$0.867798\pi$
$912$	−3.73979e19			−0.00235986
$913$		−	1.60447e22i		−	1.00416i
$914$			5.52793e21i			0.343136i
$915$	−1.96506e20			−0.0120981
$916$		−	7.68301e21i		−	0.469154i
$917$		−	5.24532e21i		−	0.317689i
$918$			5.87581e20i			0.0352978i
$919$	−9.70665e21			−0.578366			−0.289183	−	0.957274i	$-0.593384\pi$
$919$	−9.70665e21			−0.578366			−0.289183	+	0.957274i	$0.593384\pi$
$920$		−	6.34858e21i		−	0.375205i
$921$	−1.13257e20			−0.00663922
$922$	3.90047e21			0.226796
$923$	6.98258e21			0.402720
$924$	−1.16620e20			−0.00667166
$925$			4.25709e22i			2.41574i
$926$			1.33146e22i			0.749456i
$927$	1.05042e22			0.586495
$928$	1.48164e22	−	1.13190e22i	0.820602	−	0.626903i
$929$	−3.17674e21			−0.174528			−0.0872639	−	0.996185i	$-0.527812\pi$
$929$	−3.17674e21			−0.174528			−0.0872639	+	0.996185i	$0.527812\pi$
$930$		−	5.75409e20i		−	0.0313584i
$931$			1.10949e22i			0.599794i
$932$	−1.61984e22			−0.868665
$933$	4.99161e20			0.0265539
$934$	1.21786e22			0.642684
$935$	−5.64612e22			−2.95572
$936$		−	3.93270e21i		−	0.204231i
$937$	1.50064e22			0.773091			0.386546	−	0.922270i	$-0.373668\pi$
$937$	1.50064e22			0.773091			0.386546	+	0.922270i	$0.373668\pi$
$938$			8.58993e21i			0.439004i
$939$			2.90240e20i			0.0147151i
$940$		−	3.43530e22i		−	1.72785i
$941$	−3.56770e22			−1.78019			−0.890095	−	0.455775i	$-0.849362\pi$
$941$	−3.56770e22			−1.78019			−0.890095	+	0.455775i	$0.849362\pi$
$942$			1.21416e20i			0.00601027i
$943$		−	5.49578e20i		−	0.0269893i
$944$	−3.52453e20			−0.0171716
$945$			6.48249e20i			0.0313331i
$946$			7.28721e21i			0.349444i
$947$		−	7.65493e21i		−	0.364180i	−0.983282	−	0.182090i	$-0.941714\pi$
$947$		−	7.65493e21i		−	0.364180i	0.983282	−	0.182090i	$-0.0582862\pi$
$948$	−1.71988e20			−0.00811777
$949$		−	5.98845e21i		−	0.280426i
$950$	1.76783e22			0.821320
$951$	−6.72186e20			−0.0309839
$952$	−1.68687e22			−0.771446
$953$	−3.50901e21			−0.159217			−0.0796083	−	0.996826i	$-0.525367\pi$
$953$	−3.50901e21			−0.159217			−0.0796083	+	0.996826i	$0.525367\pi$
$954$		−	2.27213e22i		−	1.02287i
$955$			1.17962e22i			0.526888i
$956$	1.70857e22			0.757179
$957$	2.73757e20	+	3.58343e20i	0.0120372	+	0.0157565i
$958$	1.62626e22			0.709495
$959$		−	1.65332e21i		−	0.0715676i
$960$		−	3.12685e20i		−	0.0134299i
$961$	−4.85503e22			−2.06903
$962$	3.72311e21			0.157432
$963$	1.93100e22			0.810186
$964$	1.62916e22			0.678246
$965$		−	2.44792e22i		−	1.01121i
$966$	2.97478e19			0.00121935
$967$			2.38189e22i			0.968775i	0.874853	+	0.484388i	$0.160958\pi$
$967$			2.38189e22i			0.968775i	−0.874853	+	0.484388i	$0.839042\pi$
$968$		−	2.32198e21i		−	0.0937115i
$969$		−	6.21524e20i		−	0.0248903i
$970$	−1.77983e22			−0.707279
$971$			1.90090e22i			0.749576i	0.927110	+	0.374788i	$0.122285\pi$
$971$			1.90090e22i			0.749576i	−0.927110	+	0.374788i	$0.877715\pi$
$972$		−	9.95103e20i		−	0.0389378i
$973$	1.15445e22			0.448260
$974$			1.36222e22i			0.524874i
$975$			1.98937e20i			0.00760642i
$976$			1.57635e21i			0.0598108i
$977$	−2.43637e22			−0.917349			−0.458675	−	0.888604i	$-0.651676\pi$
$977$	−2.43637e22			−0.917349			−0.458675	+	0.888604i	$0.651676\pi$
$978$		−	4.37884e20i		−	0.0163613i
$979$	1.28306e22			0.475747
$980$	2.39739e22			0.882152
$981$	2.01479e22			0.735719
$982$	1.29211e22			0.468233
$983$		−	4.57417e21i		−	0.164498i	−0.996612	−	0.0822490i	$-0.973790\pi$
$983$		−	4.57417e21i		−	0.164498i	0.996612	−	0.0822490i	$-0.0262103\pi$
$984$		−	5.74526e19i		−	0.00205044i
$985$	−1.77324e22			−0.628053
$986$	1.61204e22	+	2.11013e22i	0.566630	+	0.741707i
$987$	3.95404e20			0.0137931
$988$			3.38759e21i			0.117277i
$989$			4.07288e21i			0.139936i
$990$	−2.90921e22			−0.991998
$991$	−4.76444e22			−1.61235			−0.806176	−	0.591676i	$-0.798467\pi$
$991$	−4.76444e22			−1.61235			−0.806176	+	0.591676i	$0.798467\pi$
$992$	−5.38638e22			−1.80909
$993$	3.32747e20			0.0110916
$994$		−	1.54324e22i		−	0.510546i
$995$	−1.35183e22			−0.443863
$996$		−	3.81640e20i		−	0.0124368i
$997$			4.39317e22i			1.42090i	0.703747	+	0.710451i	$0.251509\pi$
$997$			4.39317e22i			1.42090i	−0.703747	+	0.710451i	$0.748491\pi$
$998$			2.40523e22i			0.772106i
$999$	1.54264e21			0.0491499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	29.16.b.a.28.23	yes	36
29.28	even	2	inner	29.16.b.a.28.14	✓	36

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
29.16.b.a.28.14	✓	36	29.28	even	2	inner
29.16.b.a.28.23	yes	36	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+101.334i q^{2} -71.6384i q^{3} +22499.4 q^{4} -295367. q^{5} +7259.42 q^{6} +1.06773e6 q^{7} +5.60048e6i q^{8} +1.43438e7 q^{9} +O(q^{10})\)	\(q+101.334i q^{2} -71.6384i q^{3} +22499.4 q^{4} -295367. q^{5} +7259.42 q^{6} +1.06773e6 q^{7} +5.60048e6i q^{8} +1.43438e7 q^{9} -2.99307e7i q^{10} -6.77632e7i q^{11} -1.61182e6i q^{12} -4.89556e7 q^{13} +1.08198e8i q^{14} +2.11596e7i q^{15} +1.69740e8 q^{16} +2.82095e9i q^{17} +1.45351e9i q^{18} -3.07551e9i q^{19} -6.64557e9 q^{20} -7.64907e7i q^{21} +6.86673e9 q^{22} +3.83787e9 q^{23} +4.01209e8 q^{24} +5.67239e10 q^{25} -4.96088e9i q^{26} -2.05550e9i q^{27} +2.40234e10 q^{28} +(-5.63931e10 - 7.38174e10i) q^{29} -2.14419e9 q^{30} +2.68357e11i q^{31} +2.00717e11i q^{32} -4.85445e9 q^{33} -2.85858e11 q^{34} -3.15373e11 q^{35} +3.22726e11 q^{36} +7.50493e11i q^{37} +3.11655e11 q^{38} +3.50710e9i q^{39} -1.65419e12i q^{40} -1.43199e11i q^{41} +7.75113e9 q^{42} +1.06123e12i q^{43} -1.52463e12i q^{44} -4.23667e12 q^{45} +3.88908e11i q^{46} +5.16931e12i q^{47} -1.21599e10i q^{48} -3.60751e12 q^{49} +5.74807e12i q^{50} +2.02088e11 q^{51} -1.10147e12 q^{52} -1.56320e13 q^{53} +2.08292e11 q^{54} +2.00150e13i q^{55} +5.97982e12i q^{56} -2.20325e11 q^{57} +(7.48023e12 - 5.71455e12i) q^{58} -2.07643e12 q^{59} +4.76078e11i q^{60} +9.28687e12i q^{61} -2.71938e13 q^{62} +1.53153e13 q^{63} -1.47774e13 q^{64} +1.44599e13 q^{65} -4.91921e11i q^{66} +7.93908e13 q^{67} +6.34695e13i q^{68} -2.74939e11i q^{69} -3.19581e13i q^{70} -1.42631e14 q^{71} +8.03320e13i q^{72} +1.22324e14i q^{73} -7.60507e13 q^{74} -4.06361e12i q^{75} -6.91971e13i q^{76} -7.23531e13i q^{77} -3.55390e11 q^{78} -1.06705e14i q^{79} -5.01355e13 q^{80} +2.05670e14 q^{81} +1.45109e13 q^{82} +2.36776e14 q^{83} -1.72099e12i q^{84} -8.33213e14i q^{85} -1.07539e14 q^{86} +(-5.28816e12 + 4.03991e12i) q^{87} +3.79506e14 q^{88} +1.89344e14i q^{89} -4.29320e14i q^{90} -5.22716e13 q^{91} +8.63497e13 q^{92} +1.92247e13 q^{93} -5.23828e14 q^{94} +9.08404e14i q^{95} +1.43790e13 q^{96} -5.94650e14i q^{97} -3.65564e14i q^{98} -9.71980e14i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 36 q - 508108 q^{4} + 470082 q^{5} + 1112016 q^{6} - 4820620 q^{7} - 167460710 q^{9}+O(q^{10}) \) 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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