LMFDB - Embedded newform 5.24.b.a.4.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5,24,Mod(4,5)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

N = Newforms(chi, 24, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5.4");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 5 $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	5.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:	$16.7602018673$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 15644845 x^{8} + 79349217360160 x^{6} + \cdots + 34\!\cdots\!24 $ x^10 + 15644845x^8 + 79349217360160x^6 + 150324497245503837440x^4 + 76751558562242644579450880x^2 + 3490504262440921471663078375424
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{12}\cdot 5^{22} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4.1
Root			$-2780.51i$ of defining polynomial
Character	$\chi$	$=$	5.4
Dual form			5.24.b.a.4.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5561.03i q^{2} -421877. i q^{3} -2.25364e7 q^{4} +(7.09941e7 - 8.29504e7i) q^{5} -2.34607e9 q^{6} +7.17617e8i q^{7} +7.86762e10i q^{8} -8.38373e10 q^{9} +O(q^{10})$	\(q-5561.03i q^{2} -421877. i q^{3} -2.25364e7 q^{4} +(7.09941e7 - 8.29504e7i) q^{5} -2.34607e9 q^{6} +7.17617e8i q^{7} +7.86762e10i q^{8} -8.38373e10 q^{9} +(-4.61289e11 - 3.94800e11i) q^{10} -2.81649e11 q^{11} +9.50759e12i q^{12} -8.58779e12i q^{13} +3.99069e12 q^{14} +(-3.49949e13 - 2.99508e13i) q^{15} +2.48471e14 q^{16} -7.16301e13i q^{17} +4.66221e14i q^{18} +3.73965e14 q^{19} +(-1.59995e15 + 1.86940e15i) q^{20} +3.02746e14 q^{21} +1.56626e15i q^{22} +2.05786e15i q^{23} +3.31917e16 q^{24} +(-1.84061e15 - 1.17780e16i) q^{25} -4.77569e16 q^{26} -4.34783e15i q^{27} -1.61725e16i q^{28} +5.28275e16 q^{29} +(-1.66557e17 + 1.94608e17i) q^{30} -6.19487e16 q^{31} -7.21772e17i q^{32} +1.18822e17i q^{33} -3.98337e17 q^{34} +(5.95267e16 + 5.09466e16i) q^{35} +1.88939e18 q^{36} +1.32655e18i q^{37} -2.07963e18i q^{38} -3.62299e18 q^{39} +(6.52622e18 + 5.58554e18i) q^{40} -5.55868e18 q^{41} -1.68358e18i q^{42} +4.87597e18i q^{43} +6.34736e18 q^{44} +(-5.95195e18 + 6.95434e18i) q^{45} +1.14438e19 q^{46} +8.62370e18i q^{47} -1.04824e20i q^{48} +2.68538e19 q^{49} +(-6.54976e19 + 1.02357e19i) q^{50} -3.02191e19 q^{51} +1.93538e20i q^{52} -7.45020e19i q^{53} -2.41784e19 q^{54} +(-1.99954e19 + 2.33629e19i) q^{55} -5.64594e19 q^{56} -1.57767e20i q^{57} -2.93775e20i q^{58} +3.38308e20 q^{59} +(7.88659e20 + 6.74983e20i) q^{60} +8.12197e19 q^{61} +3.44498e20i q^{62} -6.01631e19i q^{63} -1.92946e21 q^{64} +(-7.12360e20 - 6.09682e20i) q^{65} +6.60769e20 q^{66} -2.97339e20i q^{67} +1.61428e21i q^{68} +8.68163e20 q^{69} +(2.83315e20 - 3.31029e20i) q^{70} -1.10318e21 q^{71} -6.59600e21i q^{72} -9.17445e20i q^{73} +7.37700e21 q^{74} +(-4.96886e21 + 7.76513e20i) q^{75} -8.42782e21 q^{76} -2.02117e20i q^{77} +2.01475e22i q^{78} +3.90105e21 q^{79} +(1.76400e22 - 2.06108e22i) q^{80} -9.72696e21 q^{81} +3.09119e22i q^{82} -2.44161e21i q^{83} -6.82281e21 q^{84} +(-5.94174e21 - 5.08531e21i) q^{85} +2.71154e22 q^{86} -2.22867e22i q^{87} -2.21591e22i q^{88} -4.42974e22 q^{89} +(3.86732e22 + 3.30989e22i) q^{90} +6.16274e21 q^{91} -4.63766e22i q^{92} +2.61348e22i q^{93} +4.79566e22 q^{94} +(2.65493e22 - 3.10205e22i) q^{95} -3.04499e23 q^{96} +3.47496e22i q^{97} -1.49335e23i q^{98} +2.36127e22 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 41272680 q^{4} + 124761750 q^{5} - 2262077880 q^{6} - 189250631370 q^{9}+O(q^{10}) $ 10 * q - 41272680 * q^4 + 124761750 * q^5 - 2262077880 * q^6 - 189250631370 * q^9	$ 10 q - 41272680 q^{4} + 124761750 q^{5} - 2262077880 q^{6} - 189250631370 q^{9} - 992748199000 q^{10} - 1448637536280 q^{11} + 20750531044440 q^{14} - 14566613457000 q^{15} + 307971806876960 q^{16} + 887626815301400 q^{19} - 23\!\cdots\!00 q^{20}+ \cdots + 41\!\cdots\!60 q^{99}+O(q^{100}) $ 10 * q - 41272680 * q^4 + 124761750 * q^5 - 2262077880 * q^6 - 189250631370 * q^9 - 992748199000 * q^10 - 1448637536280 * q^11 + 20750531044440 * q^14 - 14566613457000 * q^15 + 307971806876960 * q^16 + 887626815301400 * q^19 - 2363707755099000 * q^20 - 781158183715080 * q^21 + 58473619965621600 * q^24 - 15436708968968750 * q^25 - 49259739315596880 * q^26 + 68538893585329500 * q^29 - 283259561673249000 * q^30 - 303703652281842880 * q^31 - 288299835336191360 * q^34 + 568057466315481000 * q^35 + 3242003939215214760 * q^36 - 8628905972055907440 * q^39 + 9426000384456940000 * q^40 - 6658832954283527580 * q^41 + 14990828914855427040 * q^44 - 8490880963475889750 * q^45 + 16736621075351903320 * q^46 + 51294362631233173870 * q^49 - 96456356624588250000 * q^50 - 117334349391643270080 * q^51 + 6367892697021344400 * q^54 + 8046234911696751000 * q^55 - 93285217288146540000 * q^56 + 461218638014732785800 * q^59 + 1248088814097510276000 * q^60 + 202894074812420701820 * q^61 - 2597882075263561124480 * q^64 + 404224351485216078000 * q^65 - 2067527591150630859360 * q^66 + 1761142984477245282360 * q^69 + 2651929720118465517000 * q^70 - 6362428274364321596880 * q^71 + 21369976437605310645840 * q^74 - 1268180400269994750000 * q^75 - 28533821601830896716000 * q^76 + 8008906997916696975200 * q^79 + 50390556129429851388000 * q^80 - 63620987817532049079390 * q^81 + 26983560053969205971040 * q^84 + 58719411820058643416000 * q^85 - 73642932902287825779480 * q^86 + 17045340818835962686500 * q^89 + 218378490181286718303000 * q^90 - 261754857315780277067280 * q^91 + 163995755653951693894840 * q^94 + 287254967024630517105000 * q^95 - 1029134712278532215134080 * q^96 + 411070875879864759534360 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	5561.03i		−	1.92004i	−0.279934	−	0.960019i	$-0.590313\pi$
$2$		−	5561.03i		−	1.92004i	0.279934	−	0.960019i	$-0.409687\pi$
$3$		−	421877.i		−	1.37497i	−0.726201	−	0.687483i	$-0.758716\pi$
$3$		−	421877.i		−	1.37497i	0.726201	−	0.687483i	$-0.241284\pi$
$4$	−2.25364e7			−2.68655
$5$	7.09941e7	−	8.29504e7i	0.650230	−	0.759737i
$5$	7.09941e7	−	8.29504e7i	0.650230	−	0.759737i
$6$	−2.34607e9			−2.63999
$7$			7.17617e8i			0.137172i	0.997645	+	0.0685860i	$0.0218488\pi$
$7$			7.17617e8i			0.137172i	−0.997645	+	0.0685860i	$0.978151\pi$
$8$			7.86762e10i			3.23824i
$9$	−8.38373e10			−0.890529
$10$	−4.61289e11	−	3.94800e11i	−1.45873	−	1.24847i
$11$	−2.81649e11			−0.297641			−0.148821	−	0.988864i	$-0.547548\pi$
$11$	−2.81649e11			−0.297641			−0.148821	+	0.988864i	$0.547548\pi$
$12$			9.50759e12i			3.69391i
$13$		−	8.58779e12i		−	1.32902i	−0.747278	−	0.664512i	$-0.768640\pi$
$13$		−	8.58779e12i		−	1.32902i	0.747278	−	0.664512i	$-0.231360\pi$
$14$	3.99069e12			0.263376
$15$	−3.49949e13	−	2.99508e13i	−1.04461	−	0.894044i
$16$	2.48471e14			3.53099
$17$		−	7.16301e13i		−	0.506912i	−0.967347	−	0.253456i	$-0.918433\pi$
$17$		−	7.16301e13i		−	0.506912i	0.967347	−	0.253456i	$-0.0815673\pi$
$18$			4.66221e14i			1.70985i
$19$	3.73965e14			0.736486			0.368243	−	0.929730i	$-0.379960\pi$
$19$	3.73965e14			0.736486			0.368243	+	0.929730i	$0.379960\pi$
$20$	−1.59995e15	+	1.86940e15i	−1.74687	+	2.04107i
$21$	3.02746e14			0.188607
$22$			1.56626e15i			0.571483i
$23$			2.05786e15i			0.450344i	0.974319	+	0.225172i	$0.0722945\pi$
$23$			2.05786e15i			0.450344i	−0.974319	+	0.225172i	$0.927706\pi$
$24$	3.31917e16			4.45246
$25$	−1.84061e15	−	1.17780e16i	−0.154402	−	0.988008i
$26$	−4.77569e16			−2.55178
$27$		−	4.34783e15i		−	0.150518i
$28$		−	1.61725e16i		−	0.368519i
$29$	5.28275e16			0.804050			0.402025	−	0.915629i	$-0.368306\pi$
$29$	5.28275e16			0.804050			0.402025	+	0.915629i	$0.368306\pi$
$30$	−1.66557e17	+	1.94608e17i	−1.71660	+	2.00570i
$31$	−6.19487e16			−0.437898			−0.218949	−	0.975736i	$-0.570263\pi$
$31$	−6.19487e16			−0.437898			−0.218949	+	0.975736i	$0.570263\pi$
$32$		−	7.21772e17i		−	3.54140i
$33$			1.18822e17i			0.409246i
$34$	−3.98337e17			−0.973291
$35$	5.95267e16	+	5.09466e16i	0.104215	+	0.0891934i
$36$	1.88939e18			2.39245
$37$			1.32655e18i			1.22576i	0.790176	+	0.612879i	$0.209989\pi$
$37$			1.32655e18i			1.22576i	−0.790176	+	0.612879i	$0.790011\pi$
$38$		−	2.07963e18i		−	1.41408i
$39$	−3.62299e18			−1.82736
$40$	6.52622e18	+	5.58554e18i	2.46021	+	2.10560i
$41$	−5.55868e18			−1.57745			−0.788727	−	0.614743i	$-0.789260\pi$
$41$	−5.55868e18			−1.57745			−0.788727	+	0.614743i	$0.789260\pi$
$42$		−	1.68358e18i		−	0.362132i
$43$			4.87597e18i			0.800155i	0.916481	+	0.400077i	$0.131017\pi$
$43$			4.87597e18i			0.800155i	−0.916481	+	0.400077i	$0.868983\pi$
$44$	6.34736e18			0.799627
$45$	−5.95195e18	+	6.95434e18i	−0.579049	+	0.676568i
$46$	1.14438e19			0.864679
$47$			8.62370e18i			0.508825i	0.967096	+	0.254412i	$0.0818820\pi$
$47$			8.62370e18i			0.508825i	−0.967096	+	0.254412i	$0.918118\pi$
$48$		−	1.04824e20i		−	4.85499i
$49$	2.68538e19			0.981184
$50$	−6.54976e19	+	1.02357e19i	−1.89701	+	0.296458i
$51$	−3.02191e19			−0.696987
$52$			1.93538e20i			3.57048i
$53$		−	7.45020e19i		−	1.10407i	−0.833822	−	0.552033i	$-0.813852\pi$
$53$		−	7.45020e19i		−	1.10407i	0.833822	−	0.552033i	$-0.186148\pi$
$54$	−2.41784e19			−0.289001
$55$	−1.99954e19	+	2.33629e19i	−0.193535	+	0.226129i
$56$	−5.64594e19			−0.444195
$57$		−	1.57767e20i		−	1.01264i
$58$		−	2.93775e20i		−	1.54381i
$59$	3.38308e20			1.46054			0.730272	−	0.683157i	$-0.239393\pi$
$59$	3.38308e20			1.46054			0.730272	+	0.683157i	$0.239393\pi$
$60$	7.88659e20	+	6.74983e20i	2.80640	+	2.40189i
$61$	8.12197e19			0.238984			0.119492	−	0.992835i	$-0.461873\pi$
$61$	8.12197e19			0.238984			0.119492	+	0.992835i	$0.461873\pi$
$62$			3.44498e20i			0.840781i
$63$		−	6.01631e19i		−	0.122156i
$64$	−1.92946e21			−3.26863
$65$	−7.12360e20	−	6.09682e20i	−1.00971	−	0.864171i
$66$	6.60769e20			0.785769
$67$		−	2.97339e20i		−	0.297435i	−0.988880	−	0.148717i	$-0.952486\pi$
$67$		−	2.97339e20i		−	0.297435i	0.988880	−	0.148717i	$-0.0475145\pi$
$68$			1.61428e21i			1.36184i
$69$	8.68163e20			0.619208
$70$	2.83315e20	−	3.31029e20i	0.171255	−	0.200096i
$71$	−1.10318e21			−0.566468			−0.283234	−	0.959051i	$-0.591407\pi$
$71$	−1.10318e21			−0.566468			−0.283234	+	0.959051i	$0.591407\pi$
$72$		−	6.59600e21i		−	2.88374i
$73$		−	9.17445e20i		−	0.342269i	−0.985248	−	0.171134i	$-0.945257\pi$
$73$		−	9.17445e20i		−	0.342269i	0.985248	−	0.171134i	$-0.0547432\pi$
$74$	7.37700e21			2.35350
$75$	−4.96886e21	+	7.76513e20i	−1.35848	+	0.212297i
$76$	−8.42782e21			−1.97860
$77$		−	2.02117e20i		−	0.0408280i
$78$			2.01475e22i			3.50860i
$79$	3.90105e21			0.586772			0.293386	−	0.955994i	$-0.405218\pi$
$79$	3.90105e21			0.586772			0.293386	+	0.955994i	$0.405218\pi$
$80$	1.76400e22	−	2.06108e22i	2.29596	−	2.68263i
$81$	−9.72696e21			−1.09749
$82$			3.09119e22i			3.02877i
$83$		−	2.44161e21i		−	0.208103i	−0.994572	−	0.104051i	$-0.966819\pi$
$83$		−	2.44161e21i		−	0.208103i	0.994572	−	0.104051i	$-0.0331806\pi$
$84$	−6.82281e21			−0.506701
$85$	−5.94174e21	−	5.08531e21i	−0.385120	−	0.329609i
$86$	2.71154e22			1.53633
$87$		−	2.22867e22i		−	1.10554i
$88$		−	2.21591e22i		−	0.963832i
$89$	−4.42974e22			−1.69197			−0.845986	−	0.533205i	$-0.820987\pi$
$89$	−4.42974e22			−1.69197			−0.845986	+	0.533205i	$0.820987\pi$
$90$	3.86732e22	+	3.30989e22i	1.29904	+	1.11180i
$91$	6.16274e21			0.182305
$92$		−	4.63766e22i		−	1.20987i
$93$			2.61348e22i			0.602094i
$94$	4.79566e22			0.976963
$95$	2.65493e22	−	3.10205e22i	0.478885	−	0.559536i
$96$	−3.04499e23			−4.86930
$97$			3.47496e22i			0.493259i	0.969110	+	0.246630i	$0.0793231\pi$
$97$			3.47496e22i			0.493259i	−0.969110	+	0.246630i	$0.920677\pi$
$98$		−	1.49335e23i		−	1.88391i
$99$	2.36127e22			0.265058
$100$	4.14808e22	+	2.65433e23i	0.414808	+	2.65433i
$101$	8.49577e22			0.757717			0.378859	−	0.925455i	$-0.376317\pi$
$101$	8.49577e22			0.757717			0.378859	+	0.925455i	$0.376317\pi$
$102$			1.68049e23i			1.33824i
$103$		−	2.06916e23i		−	1.47287i	−0.676506	−	0.736437i	$-0.736507\pi$
$103$		−	2.06916e23i		−	1.47287i	0.676506	−	0.736437i	$-0.263493\pi$
$104$	6.75654e23			4.30369
$105$	2.14932e22	−	2.51129e22i	0.122638	−	0.143292i
$106$	−4.14307e23			−2.11985
$107$		−	2.87531e23i		−	1.32060i	−0.751003	−	0.660299i	$-0.770430\pi$
$107$		−	2.87531e23i		−	1.32060i	0.751003	−	0.660299i	$-0.229570\pi$
$108$			9.79844e22i			0.404375i
$109$	−4.16968e23			−1.54774			−0.773870	−	0.633344i	$-0.781682\pi$
$109$	−4.16968e23			−1.54774			−0.773870	+	0.633344i	$0.781682\pi$
$110$	1.29922e23	+	1.11195e23i	0.434177	+	0.371595i
$111$	5.59643e23			1.68538
$112$			1.78307e23i			0.484353i
$113$			6.61724e23i			1.62284i	0.584466	+	0.811418i	$0.301304\pi$
$113$			6.61724e23i			1.62284i	−0.584466	+	0.811418i	$0.698696\pi$
$114$	−8.77348e23			−1.94431
$115$	1.70700e23	+	1.46096e23i	0.342144	+	0.292827i
$116$	−1.19054e24			−2.16012
$117$			7.19976e23i			1.18353i
$118$		−	1.88134e24i		−	2.80430i
$119$	5.14030e22			0.0695342
$120$	2.35641e24	−	2.75327e24i	2.89512	−	3.38270i
$121$	−8.16104e23			−0.911410
$122$		−	4.51665e23i		−	0.458858i
$123$			2.34508e24i			2.16895i
$124$	1.39610e24			1.17643
$125$	−1.10766e24	−	6.83487e23i	−0.851024	−	0.525128i
$126$	−3.34568e23			−0.234544
$127$		−	1.20320e24i		−	0.770186i	−0.922878	−	0.385093i	$-0.874169\pi$
$127$		−	1.20320e24i		−	0.770186i	0.922878	−	0.385093i	$-0.125831\pi$
$128$			4.67512e24i			2.73450i
$129$	2.05706e24			1.10019
$130$	−3.39046e24	+	3.96145e24i	−1.65924	+	1.93868i
$131$	−4.83455e23			−0.216639			−0.108319	−	0.994116i	$-0.534547\pi$
$131$	−4.83455e23			−0.216639			−0.108319	+	0.994116i	$0.534547\pi$
$132$		−	2.67781e24i		−	1.09946i
$133$			2.68364e23i			0.101025i
$134$	−1.65351e24			−0.571086
$135$	−3.60654e23	−	3.08670e23i	−0.114354	−	0.0978715i
$136$	5.63558e24			1.64150
$137$		−	1.95206e23i		−	0.0522645i	−0.999658	−	0.0261323i	$-0.991681\pi$
$137$		−	1.95206e23i		−	0.0522645i	0.999658	−	0.0261323i	$-0.00831910\pi$
$138$		−	4.82787e24i		−	1.18890i
$139$	8.05971e24			1.82662			0.913312	−	0.407261i	$-0.133516\pi$
$139$	8.05971e24			1.82662			0.913312	+	0.407261i	$0.133516\pi$
$140$	−1.34152e24	−	1.14815e24i	−0.279978	−	0.239622i
$141$	3.63814e24			0.699616
$142$			6.13481e24i			1.08764i
$143$			2.41875e24i			0.395572i
$144$	−2.08312e25			−3.14445
$145$	3.75044e24	−	4.38206e24i	0.522817	−	0.610867i
$146$	−5.10193e24			−0.657169
$147$		−	1.13290e25i		−	1.34909i
$148$		−	2.98957e25i		−	3.29306i
$149$	−6.24452e24			−0.636586			−0.318293	−	0.947992i	$-0.603110\pi$
$149$	−6.24452e24			−0.636586			−0.318293	+	0.947992i	$0.603110\pi$
$150$	4.31821e24	+	2.76320e25i	0.407619	+	2.60833i
$151$	−1.17055e25			−1.02366			−0.511828	−	0.859088i	$-0.671031\pi$
$151$	−1.17055e25			−1.02366			−0.511828	+	0.859088i	$0.671031\pi$
$152$			2.94221e25i			2.38491i
$153$			6.00527e24i			0.451420i
$154$	−1.12398e24			−0.0783914
$155$	−4.39799e24	+	5.13867e24i	−0.284734	+	0.332687i
$156$	8.16492e25			4.90929
$157$		−	1.32088e25i		−	0.737933i	−0.929443	−	0.368966i	$-0.879712\pi$
$157$		−	1.32088e25i		−	0.737933i	0.929443	−	0.368966i	$-0.120288\pi$
$158$		−	2.16938e25i		−	1.12663i
$159$	−3.14307e25			−1.51805
$160$	−5.98712e25	−	5.12415e25i	−2.69053	−	2.30272i
$161$	−1.47675e24			−0.0617747
$162$			5.40919e25i			2.10722i
$163$			2.95975e25i			1.07423i	0.843509	+	0.537115i	$0.180486\pi$
$163$			2.95975e25i			1.07423i	−0.843509	+	0.537115i	$0.819514\pi$
$164$	1.25272e26			4.23791
$165$	9.85629e24	+	8.43562e24i	0.310920	+	0.266104i
$166$	−1.35778e25			−0.399565
$167$		−	9.75764e24i		−	0.267982i	−0.990983	−	0.133991i	$-0.957221\pi$
$167$		−	9.75764e24i		−	0.267982i	0.990983	−	0.133991i	$-0.0427793\pi$
$168$			2.38189e25i			0.610753i
$169$	−3.19962e25			−0.766304
$170$	−2.82795e25	+	3.30422e25i	−0.632863	+	0.739445i
$171$	−3.13522e25			−0.655862
$172$		−	1.09887e26i		−	2.14965i
$173$		−	6.94384e23i		−	0.0127078i	−0.999980	−	0.00635389i	$-0.997977\pi$
$173$		−	6.94384e23i		−	0.0127078i	0.999980	−	0.00635389i	$-0.00202252\pi$
$174$	−1.23937e26			−2.12268
$175$	8.45208e24	−	1.32086e24i	0.135527	−	0.0211796i
$176$	−6.99818e25			−1.05097
$177$		−	1.42725e26i		−	2.00820i
$178$			2.46339e26i			3.24865i
$179$	5.71582e25			0.706754			0.353377	−	0.935481i	$-0.385033\pi$
$179$	5.71582e25			0.706754			0.353377	+	0.935481i	$0.385033\pi$
$180$	1.34135e26	−	1.56726e26i	1.55564	−	1.81763i
$181$	9.84615e25			1.07143			0.535714	−	0.844399i	$-0.320043\pi$
$181$	9.84615e25			1.07143			0.535714	+	0.844399i	$0.320043\pi$
$182$		−	3.42712e25i		−	0.350032i
$183$		−	3.42647e25i		−	0.328594i
$184$	−1.61904e26			−1.45832
$185$	1.10038e26	+	9.41774e25i	0.931255	+	0.797025i
$186$	1.45336e26			1.15604
$187$			2.01746e25i			0.150878i
$188$		−	1.94347e26i		−	1.36698i
$189$	3.12008e24			0.0206469
$190$	−1.72506e26	−	1.47641e26i	−1.07433	−	0.919478i
$191$	−5.82700e24			−0.0341634			−0.0170817	−	0.999854i	$-0.505438\pi$
$191$	−5.82700e24			−0.0341634			−0.0170817	+	0.999854i	$0.505438\pi$
$192$			8.13996e26i			4.49426i
$193$			4.82164e25i			0.250776i	0.992108	+	0.125388i	$0.0400176\pi$
$193$			4.82164e25i			0.250776i	−0.992108	+	0.125388i	$0.959982\pi$
$194$	1.93244e26			0.947076
$195$	−2.57211e26	+	3.00529e26i	−1.18821	+	1.38831i
$196$	−6.05187e26			−2.63600
$197$		−	2.49476e26i		−	1.02486i	−0.858728	−	0.512432i	$-0.828745\pi$
$197$		−	2.49476e26i		−	1.02486i	0.858728	−	0.512432i	$-0.171255\pi$
$198$		−	1.31311e26i		−	0.508922i
$199$	4.45851e26			1.63072			0.815359	−	0.578955i	$-0.196539\pi$
$199$	4.45851e26			1.63072			0.815359	+	0.578955i	$0.196539\pi$
$200$	9.26646e26	−	1.44813e26i	3.19940	−	0.499990i
$201$	−1.25441e26			−0.408963
$202$		−	4.72452e26i		−	1.45485i
$203$			3.79099e25i			0.110293i
$204$	6.81029e26			1.87249
$205$	−3.94633e26	+	4.61094e26i	−1.02571	+	1.19845i
$206$	−1.15066e27			−2.82797
$207$		−	1.72525e26i		−	0.401045i
$208$		−	2.13382e27i		−	4.69277i
$209$	−1.05327e26			−0.219208
$210$	−1.39654e26	−	1.19524e26i	−0.275125	−	0.235469i
$211$	6.03095e26			1.12496			0.562480	−	0.826811i	$-0.309847\pi$
$211$	6.03095e26			1.12496			0.562480	+	0.826811i	$0.309847\pi$
$212$			1.67901e27i			2.96613i
$213$			4.65407e26i			0.778874i
$214$	−1.59896e27			−2.53560
$215$	4.04464e26	+	3.46165e26i	0.607908	+	0.520285i
$216$	3.42071e26			0.487414
$217$		−	4.44555e25i		−	0.0600673i
$218$			2.31877e27i			2.97172i
$219$	−3.87049e26			−0.470607
$220$	4.50625e26	−	5.26516e26i	0.519942	−	0.607507i
$221$	−6.15144e26			−0.673698
$222$		−	3.11219e27i		−	3.23599i
$223$		−	8.71885e26i		−	0.860901i	−0.902614	−	0.430451i	$-0.858355\pi$
$223$		−	8.71885e26i		−	0.860901i	0.902614	−	0.430451i	$-0.141645\pi$
$224$	5.17956e26			0.485781
$225$	1.54312e26	+	9.87433e26i	0.137499	+	0.879850i
$226$	3.67986e27			3.11591
$227$		−	2.04180e27i		−	1.64330i	−0.569994	−	0.821649i	$-0.693054\pi$
$227$		−	2.04180e27i		−	1.64330i	0.569994	−	0.821649i	$-0.306946\pi$
$228$			3.55550e27i			2.72051i
$229$	−1.16297e27			−0.846179			−0.423089	−	0.906088i	$-0.639054\pi$
$229$	−1.16297e27			−0.846179			−0.423089	+	0.906088i	$0.639054\pi$
$230$	8.12441e26	−	9.49267e26i	0.562240	−	0.656929i
$231$	−8.52684e25			−0.0561371
$232$			4.15627e27i			2.60370i
$233$		−	9.61391e26i		−	0.573201i	−0.958050	−	0.286601i	$-0.907475\pi$
$233$		−	9.61391e26i		−	0.573201i	0.958050	−	0.286601i	$-0.0925252\pi$
$234$	4.00381e27			2.27243
$235$	7.15339e26	+	6.12231e26i	0.386573	+	0.330853i
$236$	−7.62425e27			−3.92382
$237$		−	1.64576e27i		−	0.806792i
$238$		−	2.85853e26i		−	0.133508i
$239$	6.13259e26			0.272940			0.136470	−	0.990644i	$-0.456424\pi$
$239$	6.13259e26			0.272940			0.136470	+	0.990644i	$0.456424\pi$
$240$	−8.69523e27	−	7.44191e27i	−3.68852	−	3.15686i
$241$	1.07155e26			0.0433328			0.0216664	−	0.999765i	$-0.493103\pi$
$241$	1.07155e26			0.0433328			0.0216664	+	0.999765i	$0.493103\pi$
$242$			4.53837e27i			1.74994i
$243$			3.69426e27i			1.35849i
$244$	−1.83040e27			−0.642041
$245$	1.90646e27	−	2.22753e27i	0.637995	−	0.745442i
$246$	1.30410e28			4.16446
$247$		−	3.21153e27i		−	0.978807i
$248$		−	4.87389e27i		−	1.41802i
$249$	−1.03006e27			−0.286134
$250$	−3.80089e27	+	6.15973e27i	−1.00827	+	1.63400i
$251$	−5.28884e26			−0.134002			−0.0670012	−	0.997753i	$-0.521343\pi$
$251$	−5.28884e26			−0.134002			−0.0670012	+	0.997753i	$0.521343\pi$
$252$			1.35586e27i			0.328177i
$253$		−	5.79594e26i		−	0.134041i
$254$	−6.69104e27			−1.47879
$255$	−2.14538e27	+	2.50669e27i	−0.453202	+	0.529527i
$256$	9.81299e27			1.98172
$257$		−	4.99426e27i		−	0.964362i	−0.876072	−	0.482181i	$-0.839845\pi$
$257$		−	4.99426e27i		−	0.964362i	0.876072	−	0.482181i	$-0.160155\pi$
$258$		−	1.14394e28i		−	2.11240i
$259$	−9.51958e26			−0.168140
$260$	1.60540e28	+	1.37400e28i	2.71263	+	2.32164i
$261$	−4.42891e27			−0.716030
$262$			2.68850e27i			0.415955i
$263$			4.85892e27i			0.719530i	0.933043	+	0.359765i	$0.117143\pi$
$263$			4.85892e27i			0.719530i	−0.933043	+	0.359765i	$0.882857\pi$
$264$	−9.34842e27			−1.32524
$265$	−6.17997e27	−	5.28920e27i	−0.838801	−	0.717897i
$266$	1.49238e27			0.193972
$267$			1.86881e28i			2.32640i
$268$			6.70095e27i			0.799073i
$269$	1.24045e28			1.41719			0.708596	−	0.705615i	$-0.249329\pi$
$269$	1.24045e28			1.41719			0.708596	+	0.705615i	$0.249329\pi$
$270$	−1.71652e27	+	2.00561e27i	−0.187917	+	0.219565i
$271$	−1.96486e27			−0.206150			−0.103075	−	0.994674i	$-0.532868\pi$
$271$	−1.96486e27			−0.206150			−0.103075	+	0.994674i	$0.532868\pi$
$272$		−	1.77980e28i		−	1.78990i
$273$		−	2.59992e27i		−	0.250663i
$274$	−1.08555e27			−0.100350
$275$	5.18408e26	+	3.31726e27i	0.0459564	+	0.294072i
$276$	−1.95653e28			−1.66353
$277$			2.10364e27i			0.171575i	0.996313	+	0.0857876i	$0.0273406\pi$
$277$			2.10364e27i			0.171575i	−0.996313	+	0.0857876i	$0.972659\pi$
$278$		−	4.48202e28i		−	3.50719i
$279$	5.19361e27			0.389961
$280$	−4.00828e27	+	4.68333e27i	−0.288829	+	0.337472i
$281$	2.56177e27			0.177181			0.0885905	−	0.996068i	$-0.471764\pi$
$281$	2.56177e27			0.177181			0.0885905	+	0.996068i	$0.471764\pi$
$282$		−	2.02318e28i		−	1.34329i
$283$			1.70489e28i			1.08681i	0.839472	+	0.543403i	$0.182864\pi$
$283$			1.70489e28i			1.08681i	−0.839472	+	0.543403i	$0.817136\pi$
$284$	2.48617e28			1.52184
$285$	−1.30869e28	−	1.12005e28i	−0.769342	−	0.658450i
$286$	1.34507e28			0.759514
$287$		−	3.98900e27i		−	0.216383i
$288$			6.05113e28i			3.15372i
$289$	1.48367e28			0.743040
$290$	−2.43688e28	−	2.08563e28i	−1.17289	−	1.00383i
$291$	1.46601e28			0.678214
$292$			2.06759e28i			0.919521i
$293$			3.36186e27i			0.143748i	0.997414	+	0.0718741i	$0.0228980\pi$
$293$			3.36186e27i			0.143748i	−0.997414	+	0.0718741i	$0.977102\pi$
$294$	−6.30008e28			−2.59031
$295$	2.40179e28	−	2.80628e28i	0.949689	−	1.10963i
$296$	−1.04368e29			−3.96930
$297$			1.22456e27i			0.0448005i
$298$			3.47260e28i			1.22227i
$299$	1.76724e28			0.598518
$300$	1.11980e29	−	1.74998e28i	3.64961	−	0.570347i
$301$	−3.49908e27			−0.109759
$302$			6.50944e28i			1.96546i
$303$		−	3.58417e28i		−	1.04183i
$304$	9.29195e28			2.60052
$305$	5.76611e27	−	6.73720e27i	0.155394	−	0.181565i
$306$	3.33954e28			0.866744
$307$			7.14375e28i			1.78581i	0.450245	+	0.892905i	$0.351336\pi$
$307$			7.14375e28i			1.78581i	−0.450245	+	0.892905i	$0.648664\pi$
$308$			4.55498e27i			0.109686i
$309$	−8.72931e28			−2.02515
$310$	2.85763e28	+	2.44573e28i	0.638772	+	0.546701i
$311$	6.08044e28			1.30976			0.654879	−	0.755734i	$-0.272720\pi$
$311$	6.08044e28			1.30976			0.654879	+	0.755734i	$0.272720\pi$
$312$		−	2.85043e29i		−	5.91743i
$313$		−	4.50845e28i		−	0.902126i	−0.892492	−	0.451063i	$-0.851045\pi$
$313$		−	4.50845e28i		−	0.902126i	0.892492	−	0.451063i	$-0.148955\pi$
$314$	−7.34544e28			−1.41686
$315$	−4.99055e27	−	4.27122e27i	−0.0928063	−	0.0794293i
$316$	−8.79155e28			−1.57639
$317$			5.22362e27i			0.0903212i	0.998980	+	0.0451606i	$0.0143800\pi$
$317$			5.22362e27i			0.0903212i	−0.998980	+	0.0451606i	$0.985620\pi$
$318$			1.74787e29i			2.91472i
$319$	−1.48788e28			−0.239318
$320$	−1.36980e29	+	1.60050e29i	−2.12536	+	2.48330i
$321$	−1.21303e29			−1.81578
$322$			8.21226e27i			0.118610i
$323$		−	2.67871e28i		−	0.373334i
$324$	2.19211e29			2.94845
$325$	−1.01147e29	+	1.58068e28i	−1.31309	+	0.205204i
$326$	1.64592e29			2.06256
$327$			1.75909e29i			2.12809i
$328$		−	4.37335e29i		−	5.10817i
$329$	−6.18851e27			−0.0697965
$330$	4.69107e28	−	5.48111e28i	0.510930	−	0.596978i
$331$	−4.18837e28			−0.440579			−0.220289	−	0.975435i	$-0.570700\pi$
$331$	−4.18837e28			−0.440579			−0.220289	+	0.975435i	$0.570700\pi$
$332$			5.50250e28i			0.559078i
$333$		−	1.11215e29i		−	1.09157i
$334$	−5.42625e28			−0.514536
$335$	−2.46644e28	−	2.11093e28i	−0.225972	−	0.193401i
$336$	7.52238e28			0.665968
$337$		−	3.59677e28i		−	0.307730i	−0.988092	−	0.153865i	$-0.950828\pi$
$337$		−	3.59677e28i		−	0.307730i	0.988092	−	0.153865i	$-0.0491720\pi$
$338$			1.77931e29i			1.47133i
$339$	2.79166e29			2.23134
$340$	1.33905e29	+	1.14605e29i	1.03464	+	0.885511i
$341$	1.74478e28			0.130336
$342$			1.74350e29i			1.25928i
$343$			3.89110e28i			0.271763i
$344$	−3.83623e29			−2.59109
$345$	6.16344e28	−	7.20144e28i	0.402628	−	0.470435i
$346$	−3.86149e27			−0.0243994
$347$		−	1.70931e29i		−	1.04480i	−0.852701	−	0.522400i	$-0.825037\pi$
$347$		−	1.70931e29i		−	1.04480i	0.852701	−	0.522400i	$-0.174963\pi$
$348$			5.02262e29i			2.97009i
$349$	−2.32275e28			−0.132896			−0.0664478	−	0.997790i	$-0.521167\pi$
$349$	−2.32275e28			−0.132896			−0.0664478	+	0.997790i	$0.521167\pi$
$350$	−7.34532e27	−	4.70022e28i	−0.0406657	−	0.260217i
$351$	−3.73383e28			−0.200042
$352$			2.03287e29i			1.05407i
$353$		−	3.45293e29i		−	1.73292i	−0.499248	−	0.866459i	$-0.666390\pi$
$353$		−	3.45293e29i		−	1.73292i	0.499248	−	0.866459i	$-0.333610\pi$
$354$	−7.93695e29			−3.85581
$355$	−7.83193e28	+	9.15093e28i	−0.368335	+	0.430367i
$356$	9.98304e29			4.54556
$357$		−	2.16857e28i		−	0.0956070i
$358$		−	3.17858e29i		−	1.35700i
$359$	4.43799e29			1.83485			0.917423	−	0.397912i	$-0.130265\pi$
$359$	4.43799e29			1.83485			0.917423	+	0.397912i	$0.130265\pi$
$360$	−5.47141e29	−	4.68277e29i	−2.19089	−	1.87510i
$361$	−1.17980e29			−0.457589
$362$		−	5.47547e29i		−	2.05718i
$363$			3.44296e29i			1.25316i
$364$	−1.38886e29			−0.489771
$365$	−7.61024e28	−	6.51331e28i	−0.260034	−	0.222553i
$366$	−1.90547e29			−0.630913
$367$			4.57141e29i			1.46687i	0.679762	+	0.733433i	$0.262083\pi$
$367$			4.57141e29i			1.46687i	−0.679762	+	0.733433i	$0.737917\pi$
$368$			5.11318e29i			1.59016i
$369$	4.66024e29			1.40477
$370$	5.23723e29	−	6.11925e29i	1.53032	−	1.78805i
$371$	5.34639e28			0.151447
$372$		−	5.88983e29i		−	1.61755i
$373$			4.62916e29i			1.23268i	0.787480	+	0.616340i	$0.211385\pi$
$373$			4.62916e29i			1.23268i	−0.787480	+	0.616340i	$0.788615\pi$
$374$	1.12191e29			0.289691
$375$	−2.88347e29	+	4.67297e29i	−0.722032	+	1.17013i
$376$	−6.78480e29			−1.64769
$377$		−	4.53671e29i		−	1.06860i
$378$		−	1.73508e28i		−	0.0396429i
$379$	−3.18271e29			−0.705417			−0.352709	−	0.935733i	$-0.614739\pi$
$379$	−3.18271e29			−0.705417			−0.352709	+	0.935733i	$0.614739\pi$
$380$	−5.98325e29	+	6.99091e29i	−1.28655	+	1.50322i
$381$	−5.07604e29			−1.05898
$382$			3.24041e28i			0.0655951i
$383$			3.21703e29i			0.631931i	0.948771	+	0.315966i	$0.102328\pi$
$383$			3.21703e29i			0.631931i	−0.948771	+	0.315966i	$0.897672\pi$
$384$	1.97233e30			3.75985
$385$	−1.67657e28	−	1.43491e28i	−0.0310186	−	0.0265476i
$386$	2.68133e29			0.481499
$387$		−	4.08788e29i		−	0.712561i
$388$		−	7.83131e29i		−	1.32516i
$389$	−4.88917e29			−0.803183			−0.401592	−	0.915819i	$-0.631543\pi$
$389$	−4.88917e29			−0.803183			−0.401592	+	0.915819i	$0.631543\pi$
$390$	1.67125e30	+	1.43036e30i	2.66562	+	2.28140i
$391$	1.47404e29			0.228285
$392$			2.11275e30i			3.17730i
$393$			2.03959e29i			0.297871i
$394$	−1.38734e30			−1.96778
$395$	2.76951e29	−	3.23593e29i	0.381537	−	0.445793i
$396$	−5.32146e29			−0.712091
$397$		−	6.20956e29i		−	0.807179i	−0.914940	−	0.403590i	$-0.867762\pi$
$397$		−	6.20956e29i		−	0.807179i	0.914940	−	0.403590i	$-0.132238\pi$
$398$		−	2.47939e30i		−	3.13104i
$399$	1.13217e29			0.138906
$400$	−4.57340e29	−	2.92649e30i	−0.545192	−	3.48865i
$401$	1.43035e30			1.65684			0.828422	−	0.560105i	$-0.189239\pi$
$401$	1.43035e30			1.65684			0.828422	+	0.560105i	$0.189239\pi$
$402$			6.97579e29i			0.785224i
$403$			5.32002e29i			0.581976i
$404$	−1.91464e30			−2.03564
$405$	−6.90556e29	+	8.06855e29i	−0.713619	+	0.833802i
$406$	2.10818e29			0.211767
$407$		−	3.73623e29i		−	0.364836i
$408$		−	2.37752e30i		−	2.25701i
$409$	5.87586e29			0.542317			0.271159	−	0.962535i	$-0.412593\pi$
$409$	5.87586e29			0.542317			0.271159	+	0.962535i	$0.412593\pi$
$410$	2.56416e30	+	2.19456e30i	2.30107	+	1.96940i
$411$	−8.23530e28			−0.0718619
$412$			4.66314e30i			3.95695i
$413$			2.42776e29i			0.200346i
$414$	−9.59416e29			−0.770022
$415$	−2.02532e29	−	1.73340e29i	−0.158103	−	0.135315i
$416$	−6.19842e30			−4.70660
$417$		−	3.40021e30i		−	2.51154i
$418$			5.85726e29i			0.420889i
$419$	1.83443e30			1.28245			0.641223	−	0.767355i	$-0.278427\pi$
$419$	1.83443e30			1.28245			0.641223	+	0.767355i	$0.278427\pi$
$420$	−4.84379e29	+	5.65955e29i	−0.329472	+	0.384960i
$421$	−2.14998e30			−1.42296			−0.711478	−	0.702709i	$-0.751974\pi$
$421$	−2.14998e30			−1.42296			−0.711478	+	0.702709i	$0.751974\pi$
$422$		−	3.35382e30i		−	2.15997i
$423$		−	7.22987e29i		−	0.453123i
$424$	5.86153e30			3.57523
$425$	−8.43657e29	+	1.31843e29i	−0.500833	+	0.0782682i
$426$	2.58814e30			1.49547
$427$			5.82846e28i			0.0327819i
$428$			6.47990e30i			3.54785i
$429$	1.02041e30			0.543898
$430$	1.92503e30	−	2.24923e30i	0.998967	−	1.16721i
$431$	−1.32990e30			−0.671941			−0.335970	−	0.941873i	$-0.609064\pi$
$431$	−1.32990e30			−0.671941			−0.335970	+	0.941873i	$0.609064\pi$
$432$		−	1.08031e30i		−	0.531479i
$433$			3.96045e29i			0.189729i	0.995490	+	0.0948645i	$0.0302418\pi$
$433$			3.96045e29i			0.189729i	−0.995490	+	0.0948645i	$0.969758\pi$
$434$	−2.47218e29			−0.115332
$435$	−1.84869e30	−	1.58223e30i	−0.839921	−	0.718856i
$436$	9.39696e30			4.15808
$437$			7.69566e29i			0.331672i
$438$			2.15239e30i			0.903584i
$439$	1.27800e30			0.522622			0.261311	−	0.965255i	$-0.415845\pi$
$439$	1.27800e30			0.522622			0.261311	+	0.965255i	$0.415845\pi$
$440$	−1.83811e30	−	1.57317e30i	−0.732260	−	0.626713i
$441$	−2.25135e30			−0.873773
$442$			3.42083e30i			1.29353i
$443$		−	1.38434e30i		−	0.510035i	−0.966936	−	0.255018i	$-0.917919\pi$
$443$		−	1.38434e30i		−	0.510035i	0.966936	−	0.255018i	$-0.0820813\pi$
$444$	−1.26123e31			−4.52784
$445$	−3.14485e30	+	3.67449e30i	−1.10017	+	1.28545i
$446$	−4.84857e30			−1.65296
$447$			2.63442e30i			0.875284i
$448$		−	1.38461e30i		−	0.448365i
$449$	4.67796e30			1.47647			0.738234	−	0.674545i	$-0.235660\pi$
$449$	4.67796e30			1.47647			0.738234	+	0.674545i	$0.235660\pi$
$450$	5.49114e30	−	8.58133e29i	1.68935	−	0.264004i
$451$	1.56560e30			0.469516
$452$		−	1.49129e31i		−	4.35983i
$453$			4.93827e30i			1.40749i
$454$	−1.13545e31			−3.15520
$455$	4.37518e29	−	5.11202e29i	0.118540	−	0.138504i
$456$	1.24125e31			3.27917
$457$			3.76921e29i			0.0970990i	0.998821	+	0.0485495i	$0.0154599\pi$
$457$			3.76921e29i			0.0970990i	−0.998821	+	0.0485495i	$0.984540\pi$
$458$			6.46733e30i			1.62470i
$459$	−3.11435e29			−0.0762996
$460$	−3.84696e30	−	3.29247e30i	−0.919185	−	0.786695i
$461$	4.12459e29			0.0961214			0.0480607	−	0.998844i	$-0.484696\pi$
$461$	4.12459e29			0.0961214			0.0480607	+	0.998844i	$0.484696\pi$
$462$			4.74180e29i			0.107785i
$463$		−	7.82173e30i		−	1.73429i	−0.498058	−	0.867144i	$-0.665953\pi$
$463$		−	7.82173e30i		−	1.73429i	0.498058	−	0.867144i	$-0.334047\pi$
$464$	1.31261e31			2.83909
$465$	2.16789e30	+	1.85541e30i	0.457434	+	0.391500i
$466$	−5.34632e30			−1.10057
$467$			3.33749e29i			0.0670310i	0.999438	+	0.0335155i	$0.0106703\pi$
$467$			3.33749e29i			0.0670310i	−0.999438	+	0.0335155i	$0.989330\pi$
$468$		−	1.62257e31i		−	3.17962i
$469$	2.13376e29			0.0407997
$470$	3.40463e30	−	3.97802e30i	0.635251	−	0.742235i
$471$	−5.57249e30			−1.01463
$472$			2.66168e31i			4.72958i
$473$		−	1.37332e30i		−	0.238159i
$474$	−9.15213e30			−1.54907
$475$	−6.88325e29	−	4.40455e30i	−0.113715	−	0.727654i
$476$	−1.15844e30			−0.186807
$477$			6.24604e30i			0.983204i
$478$		−	3.41035e30i		−	0.524056i
$479$	−2.16416e30			−0.324661			−0.162331	−	0.986736i	$-0.551901\pi$
$479$	−2.16416e30			−0.324661			−0.162331	+	0.986736i	$0.551901\pi$
$480$	−2.16176e31	+	2.52583e31i	−3.16617	+	3.69939i
$481$	1.13922e31			1.62906
$482$		−	5.95892e29i		−	0.0832006i
$483$			6.23009e29i			0.0849380i
$484$	1.83920e31			2.44855
$485$	2.88250e30	+	2.46702e30i	0.374747	+	0.320732i
$486$	2.05439e31			2.60835
$487$			1.12464e31i			1.39454i	0.716807	+	0.697271i	$0.245603\pi$
$487$			1.12464e31i			1.39454i	−0.716807	+	0.697271i	$0.754397\pi$
$488$			6.39005e30i			0.773885i
$489$	1.24865e31			1.47703
$490$	−1.23874e31	−	1.06019e31i	−1.43128	−	1.22498i
$491$	−3.55690e30			−0.401452			−0.200726	−	0.979647i	$-0.564330\pi$
$491$	−3.55690e30			−0.401452			−0.200726	+	0.979647i	$0.564330\pi$
$492$		−	5.28496e31i		−	5.82698i
$493$		−	3.78404e30i		−	0.407583i
$494$	−1.78594e31			−1.87935
$495$	1.67636e30	−	1.95868e30i	0.172349	−	0.201375i
$496$	−1.53925e31			−1.54621
$497$		−	7.91661e29i		−	0.0777036i
$498$			5.72818e30i			0.549388i
$499$	−6.76832e30			−0.634344			−0.317172	−	0.948368i	$-0.602733\pi$
$499$	−6.76832e30			−0.634344			−0.317172	+	0.948368i	$0.602733\pi$
$500$	2.49627e31	+	1.54033e31i	2.28632	+	1.41078i
$501$	−4.11653e30			−0.368466
$502$			2.94114e30i			0.257290i
$503$		−	1.84387e31i		−	1.57652i	−0.615344	−	0.788259i	$-0.710983\pi$
$503$		−	1.84387e31i		−	1.57652i	0.615344	−	0.788259i	$-0.289017\pi$
$504$	4.73340e30			0.395569
$505$	6.03149e30	−	7.04728e30i	0.492690	−	0.575666i
$506$	−3.22314e30			−0.257364
$507$			1.34985e31i			1.05364i
$508$			2.71158e31i			2.06914i
$509$	−9.19455e30			−0.685924			−0.342962	−	0.939349i	$-0.611430\pi$
$509$	−9.19455e30			−0.685924			−0.342962	+	0.939349i	$0.611430\pi$
$510$	1.39397e31	+	1.19305e31i	1.01671	+	0.870164i
$511$	6.58374e29			0.0469497
$512$		−	1.53525e31i		−	1.07047i
$513$		−	1.62594e30i		−	0.110855i
$514$	−2.77732e31			−1.85161
$515$	−1.71638e31	−	1.46898e31i	−1.11900	−	0.957707i
$516$	−4.63588e31			−2.95570
$517$		−	2.42886e30i		−	0.151447i
$518$			5.29386e30i			0.322835i
$519$	−2.92945e29			−0.0174728
$520$	4.79674e31	−	5.60458e31i	2.79839	−	3.26968i
$521$	7.42230e30			0.423550			0.211775	−	0.977318i	$-0.432076\pi$
$521$	7.42230e30			0.423550			0.211775	+	0.977318i	$0.432076\pi$
$522$			2.46293e31i			1.37481i
$523$			1.95954e31i			1.07000i	0.844852	+	0.535001i	$0.179689\pi$
$523$			1.95954e31i			1.07000i	−0.844852	+	0.535001i	$0.820311\pi$
$524$	1.08953e31			0.582010
$525$	−5.57239e29	−	3.56574e30i	−0.0291212	−	0.186345i
$526$	2.70206e31			1.38152
$527$			4.43739e30i			0.221976i
$528$			2.95237e31i			1.44504i
$529$	1.66457e31			0.797190
$530$	−2.94134e31	+	3.43670e31i	−1.37839	+	1.61053i
$531$	−2.83628e31			−1.30066
$532$		−	6.04795e30i		−	0.271409i
$533$			4.77367e31i			2.09647i
$534$	1.03925e32			4.46678
$535$	−2.38508e31	−	2.04130e31i	−1.00331	−	0.858692i
$536$	2.33935e31			0.963164
$537$		−	2.41137e31i		−	0.971763i
$538$		−	6.89818e31i		−	2.72106i
$539$	−7.56335e30			−0.292041
$540$	8.12785e30	+	6.95631e30i	0.307219	+	0.262937i
$541$	−3.23562e30			−0.119726			−0.0598630	−	0.998207i	$-0.519066\pi$
$541$	−3.23562e30			−0.119726			−0.0598630	+	0.998207i	$0.519066\pi$
$542$			1.09266e31i			0.395816i
$543$		−	4.15387e31i		−	1.47318i
$544$	−5.17005e31			−1.79518
$545$	−2.96023e31	+	3.45877e31i	−1.00639	+	1.17588i
$546$	−1.44582e31			−0.481282
$547$			7.06020e30i			0.230124i	0.993358	+	0.115062i	$0.0367067\pi$
$547$			7.06020e30i			0.230124i	−0.993358	+	0.115062i	$0.963293\pi$
$548$			4.39924e30i			0.140411i
$549$	−6.80923e30			−0.212822
$550$	1.84474e31	−	2.88288e30i	0.564629	−	0.0882380i
$551$	1.97556e31			0.592171
$552$			6.83037e31i			2.00514i
$553$			2.79946e30i			0.0804887i
$554$	1.16984e31			0.329431
$555$	3.97313e31	−	4.64226e31i	1.09588	−	1.28044i
$556$	−1.81637e32			−4.90731
$557$		−	1.57411e31i		−	0.416581i	−0.978067	−	0.208290i	$-0.933210\pi$
$557$		−	1.57411e31i		−	0.416581i	0.978067	−	0.208290i	$-0.0667899\pi$
$558$		−	2.88818e31i		−	0.748740i
$559$	4.18738e31			1.06342
$560$	1.47907e31	+	1.26588e31i	0.367981	+	0.314941i
$561$	8.51119e30			0.207452
$562$		−	1.42461e31i		−	0.340194i
$563$		−	2.77197e30i		−	0.0648548i	−0.999474	−	0.0324274i	$-0.989676\pi$
$563$		−	2.77197e30i		−	0.0648548i	0.999474	−	0.0324274i	$-0.0103238\pi$
$564$	−8.19906e31			−1.87955
$565$	5.48903e31	+	4.69785e31i	1.23293	+	1.05522i
$566$	9.48093e31			2.08671
$567$		−	6.98023e30i		−	0.150544i
$568$		−	8.67940e31i		−	1.83436i
$569$	−8.98034e31			−1.85995			−0.929976	−	0.367621i	$-0.880172\pi$
$569$	−8.98034e31			−1.85995			−0.929976	+	0.367621i	$0.880172\pi$
$570$	−6.22865e31	+	7.27764e31i	−1.26425	+	1.47717i
$571$	8.25741e31			1.64259			0.821294	−	0.570505i	$-0.193252\pi$
$571$	8.25741e31			1.64259			0.821294	+	0.570505i	$0.193252\pi$
$572$		−	5.45098e31i		−	1.06272i
$573$			2.45828e30i			0.0469735i
$574$	−2.21829e31			−0.415463
$575$	2.42374e31	−	3.78772e30i	0.444944	−	0.0695340i
$576$	1.61761e32			2.91081
$577$		−	4.50714e31i		−	0.795022i	−0.917597	−	0.397511i	$-0.869874\pi$
$577$		−	4.50714e31i		−	0.795022i	0.917597	−	0.397511i	$-0.130126\pi$
$578$		−	8.25073e31i		−	1.42667i
$579$	2.03414e31			0.344808
$580$	−8.45214e31	+	9.87559e31i	−1.40457	+	1.64112i
$581$	1.75214e30			0.0285459
$582$		−	8.15250e31i		−	1.30220i
$583$			2.09834e31i			0.328616i
$584$	7.21811e31			1.10835
$585$	5.97224e31	+	5.11141e31i	0.899175	+	0.769570i
$586$	1.86954e31			0.276002
$587$			5.23555e31i			0.757921i	0.925413	+	0.378961i	$0.123718\pi$
$587$			5.23555e31i			0.757921i	−0.925413	+	0.378961i	$0.876282\pi$
$588$			2.55315e32i			3.62440i
$589$	−2.31666e31			−0.322505
$590$	−1.56058e32	−	1.33564e32i	−2.13053	−	1.82344i
$591$	−1.05248e32			−1.40915
$592$			3.29611e32i			4.32814i
$593$			1.26536e32i			1.62961i	0.579735	+	0.814805i	$0.303156\pi$
$593$			1.26536e32i			1.62961i	−0.579735	+	0.814805i	$0.696844\pi$
$594$	6.80983e30			0.0860186
$595$	3.64931e30	−	4.26390e30i	0.0452132	−	0.0528277i
$596$	1.40729e32			1.71022
$597$		−	1.88094e32i		−	2.24218i
$598$		−	9.82768e31i		−	1.14918i
$599$	−1.54403e32			−1.77112			−0.885560	−	0.464525i	$-0.846225\pi$
$599$	−1.54403e32			−1.77112			−0.885560	+	0.464525i	$0.846225\pi$
$600$	−6.10931e31	−	3.90931e32i	−0.687469	−	4.39907i
$601$	−7.90359e30			−0.0872505			−0.0436253	−	0.999048i	$-0.513891\pi$
$601$	−7.90359e30			−0.0872505			−0.0436253	+	0.999048i	$0.513891\pi$
$602$			1.94585e31i			0.210741i
$603$			2.49281e31i			0.264875i
$604$	2.63799e32			2.75010
$605$	−5.79385e31	+	6.76962e31i	−0.592626	+	0.692432i
$606$	−1.99317e32			−2.00036
$607$			1.59589e32i			1.57157i	0.618501	+	0.785784i	$0.287740\pi$
$607$			1.59589e32i			1.57157i	−0.618501	+	0.785784i	$0.712260\pi$
$608$		−	2.69917e32i		−	2.60819i
$609$	1.59933e31			0.151649
$610$	−3.74658e31	−	3.20655e31i	−0.348611	−	0.298363i
$611$	7.40585e31			0.676240
$612$		−	1.35337e32i		−	1.21276i
$613$		−	8.29203e31i		−	0.729232i	−0.931158	−	0.364616i	$-0.881200\pi$
$613$		−	8.29203e31i		−	0.729232i	0.931158	−	0.364616i	$-0.118800\pi$
$614$	3.97266e32			3.42882
$615$	1.94525e32	+	1.66487e32i	1.64783	+	1.41031i
$616$	1.59018e31			0.132211
$617$		−	1.24950e32i		−	1.01966i	−0.860275	−	0.509830i	$-0.829708\pi$
$617$		−	1.24950e32i		−	1.01966i	0.860275	−	0.509830i	$-0.170292\pi$
$618$			4.85439e32i			3.88837i
$619$	−1.34368e32			−1.05646			−0.528230	−	0.849101i	$-0.677144\pi$
$619$	−1.34368e32			−1.05646			−0.528230	+	0.849101i	$0.677144\pi$
$620$	9.91148e31	−	1.15807e32i	0.764952	−	0.893780i
$621$	8.94721e30			0.0677851
$622$		−	3.38135e32i		−	2.51478i
$623$		−	3.17886e31i		−	0.232091i
$624$	−9.00210e32			−6.45239
$625$	−1.35333e32	+	4.33574e31i	−0.952320	+	0.305101i
$626$	−2.50716e32			−1.73212
$627$			4.44351e31i			0.301404i
$628$			2.97678e32i			1.98249i
$629$	9.50211e31			0.621352
$630$	−2.37524e31	+	2.77526e31i	−0.152507	+	0.178192i
$631$	1.21648e32			0.766951			0.383476	−	0.923551i	$-0.374727\pi$
$631$	1.21648e32			0.766951			0.383476	+	0.923551i	$0.374727\pi$
$632$			3.06920e32i			1.90011i
$633$		−	2.54432e32i		−	1.54678i
$634$	2.90487e31			0.173420
$635$	−9.98061e31	−	8.54202e31i	−0.585139	−	0.500798i
$636$	7.08335e32			4.07832
$637$		−	2.30614e32i		−	1.30402i
$638$			8.27416e31i			0.459501i
$639$	9.24876e31			0.504457
$640$	3.87803e32	+	3.31906e32i	2.07750	+	1.77806i
$641$	−1.17828e32			−0.619987			−0.309994	−	0.950739i	$-0.600327\pi$
$641$	−1.17828e32			−0.619987			−0.309994	+	0.950739i	$0.600327\pi$
$642$			6.74567e32i			3.48636i
$643$		−	2.42059e31i		−	0.122884i	−0.998111	−	0.0614420i	$-0.980430\pi$
$643$		−	2.42059e31i		−	0.122884i	0.998111	−	0.0614420i	$-0.0195699\pi$
$644$	3.32807e31			0.165961
$645$	1.46039e32	−	1.70634e32i	0.715373	−	0.835852i
$646$	−1.48964e32			−0.716815
$647$		−	3.42544e31i		−	0.161926i	−0.996717	−	0.0809631i	$-0.974200\pi$
$647$		−	3.42544e31i		−	0.161926i	0.996717	−	0.0809631i	$-0.0257996\pi$
$648$		−	7.65280e32i		−	3.55392i
$649$	−9.52843e31			−0.434718
$650$	8.79020e31	+	5.62480e32i	0.393999	+	2.52118i
$651$	−1.87548e31			−0.0825905
$652$		−	6.67021e32i		−	2.88597i
$653$		−	2.57719e32i		−	1.09558i	−0.836615	−	0.547791i	$-0.815469\pi$
$653$		−	2.57719e32i		−	1.09558i	0.836615	−	0.547791i	$-0.184531\pi$
$654$	9.78236e32			4.08601
$655$	−3.43224e31	+	4.01028e31i	−0.140865	+	0.164589i
$656$	−1.38117e33			−5.56998
$657$			7.69161e31i			0.304800i
$658$			3.44145e31i			0.134012i
$659$	9.86405e31			0.377462			0.188731	−	0.982029i	$-0.439563\pi$
$659$	9.86405e31			0.377462			0.188731	+	0.982029i	$0.439563\pi$
$660$	−2.22125e32	−	1.90108e32i	−0.835301	−	0.714902i
$661$	3.09535e32			1.14391			0.571957	−	0.820284i	$-0.306184\pi$
$661$	3.09535e32			1.14391			0.571957	+	0.820284i	$0.306184\pi$
$662$			2.32917e32i			0.845928i
$663$			2.59515e32i			0.926312i
$664$	1.92096e32			0.673885
$665$	2.22609e31	+	1.90522e31i	0.0767526	+	0.0656896i
$666$	−6.18467e32			−2.09586
$667$			1.08711e32i			0.362099i
$668$			2.19902e32i			0.719946i
$669$	−3.67828e32			−1.18371
$670$	−1.17389e32	+	1.37159e32i	−0.371338	+	0.433876i
$671$	−2.28755e31			−0.0711314
$672$		−	2.18514e32i		−	0.667932i
$673$			5.47386e32i			1.64483i	0.568889	+	0.822414i	$0.307373\pi$
$673$			5.47386e32i			1.64483i	−0.568889	+	0.822414i	$0.692627\pi$
$674$	−2.00018e32			−0.590853
$675$	−5.12087e31	+	8.00268e30i	−0.148713	+	0.0232403i
$676$	7.21078e32			2.05871
$677$			2.30892e32i			0.648095i	0.946041	+	0.324047i	$0.105044\pi$
$677$			2.30892e32i			0.648095i	−0.946041	+	0.324047i	$0.894956\pi$
$678$		−	1.55245e33i		−	4.28426i
$679$	−2.49369e31			−0.0676613
$680$	4.00093e32	−	4.67474e32i	1.06735	−	1.24711i
$681$	−8.61390e32			−2.25948
$682$		−	9.70278e31i		−	0.250251i
$683$		−	3.50225e32i		−	0.888196i	−0.895978	−	0.444098i	$-0.853524\pi$
$683$		−	3.50225e32i		−	0.888196i	0.895978	−	0.444098i	$-0.146476\pi$
$684$	7.06565e32			1.76200
$685$	−1.61924e31	−	1.38585e31i	−0.0397073	−	0.0339840i
$686$	2.16385e32			0.521795
$687$			4.90632e32i			1.16347i
$688$			1.21154e33i			2.82534i
$689$	−6.39807e32			−1.46733
$690$	−4.00474e32	−	3.42750e32i	−0.903254	−	0.773060i
$691$	7.28404e32			1.61575			0.807876	−	0.589353i	$-0.200617\pi$
$691$	7.28404e32			1.61575			0.807876	+	0.589353i	$0.200617\pi$
$692$			1.56489e31i			0.0341401i
$693$			1.69449e31i			0.0363586i
$694$	−9.50554e32			−2.00606
$695$	5.72191e32	−	6.68556e32i	1.18773	−	1.38775i
$696$	1.75343e33			3.58000
$697$			3.98168e32i			0.799631i
$698$			1.29169e32i			0.255164i
$699$	−4.05589e32			−0.788132
$700$	−1.90479e32	+	2.97673e31i	−0.364100	+	0.0569001i
$701$	8.26850e32			1.55478			0.777391	−	0.629017i	$-0.216542\pi$
$701$	8.26850e32			1.55478			0.777391	+	0.629017i	$0.216542\pi$
$702$			2.07639e32i			0.384089i
$703$			4.96084e32i			0.902754i
$704$	5.43432e32			0.972880
$705$	2.58286e32	−	3.01785e32i	0.454911	−	0.531525i
$706$	−1.92018e33			−3.32727
$707$			6.09671e31i			0.103938i
$708$			3.21650e33i			5.39512i
$709$	1.92680e32			0.317983			0.158992	−	0.987280i	$-0.449176\pi$
$709$	1.92680e32			0.317983			0.158992	+	0.987280i	$0.449176\pi$
$710$	5.08885e32	+	4.35535e32i	0.826321	+	0.707217i
$711$	−3.27053e32			−0.522538
$712$		−	3.48515e33i		−	5.47900i
$713$		−	1.27481e32i		−	0.197205i
$714$	−1.20595e32			−0.183569
$715$	2.00636e32	+	1.71717e32i	0.300531	+	0.257213i
$716$	−1.28814e33			−1.89873
$717$		−	2.58720e32i		−	0.375284i
$718$		−	2.46798e33i		−	3.52298i
$719$	9.68172e32			1.36010			0.680049	−	0.733167i	$-0.261959\pi$
$719$	9.68172e32			1.36010			0.680049	+	0.733167i	$0.261959\pi$
$720$	−1.47889e33	+	1.72795e33i	−2.04462	+	2.38896i
$721$	1.48486e32			0.202037
$722$			6.56090e32i			0.878588i
$723$		−	4.52063e31i		−	0.0595811i
$724$	−2.21897e33			−2.87844
$725$	−9.72351e31	−	6.22201e32i	−0.124147	−	0.794408i
$726$	1.91464e33			2.40611
$727$			4.49904e32i			0.556511i	0.960507	+	0.278256i	$0.0897562\pi$
$727$			4.49904e32i			0.556511i	−0.960507	+	0.278256i	$0.910244\pi$
$728$			4.84861e32i			0.590346i
$729$	6.42799e32			0.770387
$730$	−3.62207e32	+	4.23208e32i	−0.427311	+	0.499276i
$731$	3.49266e32			0.405608
$732$			7.72203e32i			0.882784i
$733$			5.61083e31i			0.0631440i	0.999501	+	0.0315720i	$0.0100513\pi$
$733$			5.61083e31i			0.0631440i	−0.999501	+	0.0315720i	$0.989949\pi$
$734$	2.54217e33			2.81644
$735$	−9.39745e32	−	8.04292e32i	−1.02496	−	0.877221i
$736$	1.48530e33			1.59485
$737$			8.37454e31i			0.0885289i
$738$		−	2.59157e33i		−	2.69721i
$739$	8.23472e32			0.843797			0.421898	−	0.906643i	$-0.361364\pi$
$739$	8.23472e32			0.843797			0.421898	+	0.906643i	$0.361364\pi$
$740$	−2.47986e33	−	2.12242e33i	−2.50186	−	2.14125i
$741$	−1.35487e33			−1.34583
$742$		−	2.97314e32i		−	0.290784i
$743$		−	7.56295e32i		−	0.728316i	−0.931337	−	0.364158i	$-0.881357\pi$
$743$		−	7.56295e32i		−	0.728316i	0.931337	−	0.364158i	$-0.118643\pi$
$744$	−2.05618e33			−1.94972
$745$	−4.43324e32	+	5.17986e32i	−0.413927	+	0.483638i
$746$	2.57429e33			2.36679
$747$			2.04698e32i			0.185322i
$748$		−	4.54662e32i		−	0.405341i
$749$	2.06337e32			0.181149
$750$	2.59865e33	+	1.60351e33i	2.24669	+	1.38633i
$751$	−1.71152e33			−1.45721			−0.728605	−	0.684934i	$-0.759831\pi$
$751$	−1.71152e33			−1.45721			−0.728605	+	0.684934i	$0.759831\pi$
$752$			2.14274e33i			1.79665i
$753$			2.23124e32i			0.184249i
$754$	−2.52288e33			−2.05176
$755$	−8.31019e32	+	9.70973e32i	−0.665612	+	0.777709i
$756$	−7.03153e31			−0.0554689
$757$			1.05505e33i			0.819726i	0.912147	+	0.409863i	$0.134423\pi$
$757$			1.05505e33i			0.819726i	−0.912147	+	0.409863i	$0.865577\pi$
$758$			1.76991e33i			1.35443i
$759$	−2.44518e32			−0.184302
$760$	2.44058e33	+	2.08880e33i	1.81191	+	1.55074i
$761$	1.01410e33			0.741579			0.370789	−	0.928717i	$-0.379087\pi$
$761$	1.01410e33			0.741579			0.370789	+	0.928717i	$0.379087\pi$
$762$			2.82280e33i			2.03328i
$763$		−	2.99224e32i		−	0.212307i
$764$	1.31320e32			0.0917816
$765$	4.98139e32	+	4.26338e32i	0.342961	+	0.293527i
$766$	1.78900e33			1.21333
$767$		−	2.90532e33i		−	1.94110i
$768$		−	4.13988e33i		−	2.72479i
$769$	1.10268e33			0.714984			0.357492	−	0.933916i	$-0.383632\pi$
$769$	1.10268e33			0.714984			0.357492	+	0.933916i	$0.383632\pi$
$770$	−7.97956e31	+	9.32342e31i	−0.0509724	+	0.0595569i
$771$	−2.10696e33			−1.32596
$772$		−	1.08662e33i		−	0.673721i
$773$		−	9.27895e32i		−	0.566807i	−0.959001	−	0.283403i	$-0.908537\pi$
$773$		−	9.27895e32i		−	0.566807i	0.959001	−	0.283403i	$-0.0914635\pi$
$774$	−2.27328e33			−1.36815
$775$	1.14024e32	+	7.29630e32i	0.0676123	+	0.432647i
$776$	−2.73397e33			−1.59729
$777$			4.01609e32i			0.231186i
$778$			2.71888e33i			1.54214i
$779$	−2.07875e33			−1.16177
$780$	5.79661e33	−	6.77283e33i	3.19217	−	3.72977i
$781$	3.10710e32			0.168604
$782$		−	8.19719e32i		−	0.438316i
$783$		−	2.29685e32i		−	0.121024i
$784$	6.67239e33			3.46455
$785$	−1.09567e33	−	9.37745e32i	−0.560635	−	0.479826i
$786$	1.13422e33			0.571923
$787$		−	1.36537e33i		−	0.678488i	−0.940698	−	0.339244i	$-0.889829\pi$
$787$		−	1.36537e33i		−	0.678488i	0.940698	−	0.339244i	$-0.110171\pi$
$788$			5.62228e33i			2.75335i
$789$	2.04987e33			0.989329
$790$	−1.79951e33	−	1.54013e33i	−0.855940	−	0.732566i
$791$	−4.74864e32			−0.222608
$792$			1.85776e33i			0.858321i
$793$		−	6.97497e32i		−	0.317615i
$794$	−3.45315e33			−1.54982
$795$	−2.23139e33	+	2.60719e33i	−0.987084	+	1.15332i
$796$	−1.00479e34			−4.38100
$797$			2.78580e33i			1.19724i	0.801035	+	0.598618i	$0.204283\pi$
$797$			2.78580e33i			1.19724i	−0.801035	+	0.598618i	$0.795717\pi$
$798$		−	6.29600e32i		−	0.266705i
$799$	6.17716e32			0.257929
$800$	−8.50101e33	+	1.32850e33i	−3.49893	+	0.546799i
$801$	3.71377e33			1.50675
$802$		−	7.95420e33i		−	3.18120i
$803$			2.58398e32i			0.101873i
$804$	2.82698e33			1.09870
$805$	−1.04841e32	+	1.22497e32i	−0.0401677	+	0.0469325i
$806$	2.95848e33			1.11742
$807$		−	5.23318e33i		−	1.94859i
$808$			6.68415e33i			2.45367i
$809$	−6.23586e31			−0.0225678			−0.0112839	−	0.999936i	$-0.503592\pi$
$809$	−6.23586e31			−0.0225678			−0.0112839	+	0.999936i	$0.503592\pi$
$810$	4.48694e33	+	3.84020e33i	1.60093	+	1.37018i
$811$	2.57245e33			0.904912			0.452456	−	0.891787i	$-0.350548\pi$
$811$	2.57245e33			0.904912			0.452456	+	0.891787i	$0.350548\pi$
$812$		−	8.54353e32i		−	0.296308i
$813$			8.28928e32i			0.283449i
$814$	−2.07773e33			−0.700500
$815$	2.45513e33	+	2.10125e33i	0.816133	+	0.698497i
$816$	−7.50858e33			−2.46105
$817$			1.82344e33i			0.589303i
$818$		−	3.26758e33i		−	1.04127i
$819$	−5.16668e32			−0.162348
$820$	8.89360e33	−	1.03914e34i	2.75561	−	3.21970i
$821$	−1.78200e33			−0.544454			−0.272227	−	0.962233i	$-0.587760\pi$
$821$	−1.78200e33			−0.544454			−0.272227	+	0.962233i	$0.587760\pi$
$822$			4.57967e32i			0.137978i
$823$			3.21458e33i			0.955051i	0.878618	+	0.477526i	$0.158466\pi$
$823$			3.21458e33i			0.955051i	−0.878618	+	0.477526i	$0.841534\pi$
$824$	1.62793e34			4.76951
$825$	1.39948e33	−	2.18705e32i	0.404339	−	0.0631884i
$826$	1.35008e33			0.384671
$827$		−	4.79466e33i		−	1.34724i	−0.739078	−	0.673620i	$-0.764739\pi$
$827$		−	4.79466e33i		−	1.34724i	0.739078	−	0.673620i	$-0.235261\pi$
$828$			3.88809e33i			1.07743i
$829$	5.32701e32			0.145581			0.0727907	−	0.997347i	$-0.476809\pi$
$829$	5.32701e32			0.145581			0.0727907	+	0.997347i	$0.476809\pi$
$830$	−9.63946e32	+	1.12629e33i	−0.259809	+	0.303565i
$831$	8.87479e32			0.235910
$832$			1.65698e34i			4.34409i
$833$		−	1.92354e33i		−	0.497374i
$834$	−1.89086e34			−4.82226
$835$	−8.09401e32	−	6.92735e32i	−0.203596	−	0.174250i
$836$	2.37369e33			0.588914
$837$			2.69343e32i			0.0659117i
$838$		−	1.02013e34i		−	2.46234i
$839$	−1.26191e30			−0.000300445			−0.000150223	−	1.00000i	$-0.500048\pi$
$839$	−1.26191e30			−0.000300445			−0.000150223		1.00000i	$0.500048\pi$
$840$	1.97579e33	+	1.69100e33i	0.464012	+	0.397130i
$841$	−1.52598e33			−0.353503
$842$			1.19561e34i			2.73213i
$843$		−	1.08075e33i		−	0.243618i
$844$	−1.35916e34			−3.02226
$845$	−2.27154e33	+	2.65410e33i	−0.498274	+	0.582190i
$846$	−4.02055e33			−0.870014
$847$		−	5.85650e32i		−	0.125020i
$848$		−	1.85116e34i		−	3.89845i
$849$	7.19254e33			1.49432
$850$	7.33184e32	+	4.69160e33i	0.150278	+	0.961619i
$851$	−2.72986e33			−0.552014
$852$		−	1.04886e34i		−	2.09248i
$853$			7.20246e33i			1.41764i	0.705388	+	0.708822i	$0.250773\pi$
$853$			7.20246e33i			1.41764i	−0.705388	+	0.708822i	$0.749227\pi$
$854$	3.24122e32			0.0629424
$855$	−2.22582e33	+	2.60068e33i	−0.426461	+	0.498283i
$856$	2.26218e34			4.27641
$857$			1.61829e33i			0.301841i	0.988546	+	0.150920i	$0.0482237\pi$
$857$			1.61829e33i			0.301841i	−0.988546	+	0.150920i	$0.951776\pi$
$858$		−	5.67455e33i		−	1.04430i
$859$	5.54266e33			1.00646			0.503230	−	0.864152i	$-0.332145\pi$
$859$	5.54266e33			1.00646			0.503230	+	0.864152i	$0.332145\pi$
$860$	−9.11516e33	−	7.80131e33i	−1.63317	−	1.39777i
$861$	−1.68287e33			−0.297519
$862$			7.39562e33i			1.29015i
$863$			6.97005e33i			1.19981i	0.800072	+	0.599904i	$0.204795\pi$
$863$			6.97005e33i			1.19981i	−0.800072	+	0.599904i	$0.795205\pi$
$864$	−3.13814e33			−0.533046
$865$	−5.75995e31	−	4.92972e31i	−0.00965458	−	0.00826298i
$866$	2.20242e33			0.364287
$867$		−	6.25927e33i		−	1.02165i
$868$			1.00187e33i			0.161374i
$869$	−1.09873e33			−0.174648
$870$	−8.79879e33	+	1.02806e34i	−1.38023	+	1.61268i
$871$	−2.55349e33			−0.395298
$872$		−	3.28055e34i		−	5.01195i
$873$		−	2.91331e33i		−	0.439262i
$874$	4.27957e33			0.636823
$875$	4.90482e32	−	7.94877e32i	0.0720328	−	0.116737i
$876$	8.72269e33			1.26431
$877$			9.85482e33i			1.40979i	0.709313	+	0.704893i	$0.249005\pi$
$877$			9.85482e33i			1.40979i	−0.709313	+	0.704893i	$0.750995\pi$
$878$		−	7.10697e33i		−	1.00345i
$879$	1.41829e33			0.197649
$880$	−4.96829e33	+	5.80502e33i	−0.683371	+	0.798460i
$881$	−7.41172e33			−1.00623			−0.503113	−	0.864220i	$-0.667812\pi$
$881$	−7.41172e33			−1.00623			−0.503113	+	0.864220i	$0.667812\pi$
$882$			1.25198e34i			1.67768i
$883$		−	9.60835e33i		−	1.27087i	−0.772155	−	0.635434i	$-0.780821\pi$
$883$		−	9.60835e33i		−	1.27087i	0.772155	−	0.635434i	$-0.219179\pi$
$884$	1.38631e34			1.80992
$885$	−1.18391e34	−	1.01326e34i	−1.52570	−	1.30579i
$886$	−7.69836e33			−0.979287
$887$			3.98373e33i			0.500229i	0.968216	+	0.250114i	$0.0804682\pi$
$887$			3.98373e33i			0.500229i	−0.968216	+	0.250114i	$0.919532\pi$
$888$			4.40306e34i			5.45764i
$889$	8.63439e32			0.105648
$890$	2.04339e34	+	1.74886e34i	2.46812	+	2.11237i
$891$	2.73959e33			0.326657
$892$			1.96491e34i			2.31285i
$893$			3.22496e33i			0.374742i
$894$	1.46501e34			1.68058
$895$	4.05789e33	−	4.74129e33i	0.459553	−	0.536948i
$896$	−3.35495e33			−0.375097
$897$		−	7.45559e33i		−	0.822942i
$898$		−	2.60143e34i		−	2.83487i
$899$	−3.27260e33			−0.352092
$900$	−3.47764e33	−	2.22532e34i	−0.369399	−	2.36376i
$901$	−5.33658e33			−0.559665
$902$		−	8.70633e33i		−	0.901488i
$903$			1.47618e33i			0.150915i
$904$	−5.20619e34			−5.25513
$905$	6.99019e33	−	8.16743e33i	0.696675	−	0.814004i
$906$	2.74618e34			2.70244
$907$		−	3.15842e33i		−	0.306892i	−0.988157	−	0.153446i	$-0.950963\pi$
$907$		−	3.15842e33i		−	0.306892i	0.988157	−	0.153446i	$-0.0490372\pi$
$908$			4.60149e34i			4.41480i
$909$	−7.12262e33			−0.674769
$910$	−2.84281e33	−	2.43305e33i	−0.265933	−	0.227601i
$911$	−4.61804e33			−0.426576			−0.213288	−	0.976989i	$-0.568417\pi$
$911$	−4.61804e33			−0.426576			−0.213288	+	0.976989i	$0.568417\pi$
$912$		−	3.92006e34i		−	3.57563i
$913$			6.87677e32i			0.0619399i
$914$	2.09607e33			0.186434
$915$	−2.84227e33	−	2.43259e33i	−0.249645	−	0.213662i
$916$	2.62092e34			2.27330
$917$		−	3.46936e32i		−	0.0297168i
$918$			1.73190e33i			0.146498i
$919$	1.02790e34			0.858666			0.429333	−	0.903146i	$-0.358749\pi$
$919$	1.02790e34			0.858666			0.429333	+	0.903146i	$0.358749\pi$
$920$	−1.14942e34	+	1.34300e34i	−0.948244	+	1.10794i
$921$	3.01379e34			2.45543
$922$		−	2.29369e33i		−	0.184557i
$923$			9.47388e33i			0.752850i
$924$	1.92164e33			0.150815
$925$	1.56241e34	−	2.44167e33i	1.21106	−	0.189260i
$926$	−4.34968e34			−3.32990
$927$			1.73473e34i			1.31164i
$928$		−	3.81294e34i		−	2.84746i
$929$	−1.66556e34			−1.22851			−0.614257	−	0.789106i	$-0.710544\pi$
$929$	−1.66556e34			−1.22851			−0.614257	+	0.789106i	$0.710544\pi$
$930$	1.03180e34	−	1.20557e34i	0.751694	−	0.878290i
$931$	1.00424e34			0.722628
$932$			2.16663e34i			1.53993i
$933$		−	2.56520e34i		−	1.80087i
$934$	1.85599e33			0.128702
$935$	1.67349e33	+	1.43227e33i	0.114628	+	0.0981054i
$936$	−5.66450e34			−3.83256
$937$		−	2.95665e33i		−	0.197603i	−0.995107	−	0.0988017i	$-0.968499\pi$
$937$		−	2.95665e33i		−	0.197603i	0.995107	−	0.0988017i	$-0.0315009\pi$
$938$		−	1.18659e33i		−	0.0783371i
$939$	−1.90201e34			−1.24039
$940$	−1.61212e34	−	1.37975e34i	−1.03855	−	0.888852i
$941$	−9.22670e33			−0.587173			−0.293586	−	0.955933i	$-0.594849\pi$
$941$	−9.22670e33			−0.587173			−0.293586	+	0.955933i	$0.594849\pi$
$942$			3.09887e34i			1.94813i
$943$		−	1.14390e34i		−	0.710398i
$944$	8.40599e34			5.15716
$945$	2.21507e32	−	2.58812e32i	0.0134252	−	0.0156862i
$946$	−7.63704e33			−0.457275
$947$			2.32205e34i			1.37356i	0.726867	+	0.686778i	$0.240976\pi$
$947$			2.32205e34i			1.37356i	−0.726867	+	0.686778i	$0.759024\pi$
$948$			3.70896e34i			2.16748i
$949$	−7.87882e33			−0.454883
$950$	−2.44938e34	+	3.82779e33i	−1.39712	+	0.218337i
$951$	2.20373e33			0.124189
$952$			4.04419e33i			0.225168i
$953$		−	2.23030e34i		−	1.22686i	−0.789748	−	0.613431i	$-0.789789\pi$
$953$		−	2.23030e34i		−	1.22686i	0.789748	−	0.613431i	$-0.210211\pi$
$954$	3.47344e34			1.88779
$955$	−4.13682e32	+	4.83352e32i	−0.0222141	+	0.0259552i
$956$	−1.38206e34			−0.733267
$957$			6.27704e33i			0.329055i
$958$			1.20349e34i			0.623362i
$959$	1.40083e32			0.00716923
$960$	6.75213e34	+	5.77889e34i	3.41446	+	2.92230i
$961$	−1.61757e34			−0.808246
$962$		−	6.33521e34i		−	3.12786i
$963$			2.41058e34i			1.17603i
$964$	−2.41489e33			−0.116416
$965$	3.99957e33	+	3.42308e33i	0.190524	+	0.163062i
$966$	3.46457e33			0.163084
$967$			2.43134e34i			1.13094i	0.824767	+	0.565472i	$0.191306\pi$
$967$			2.43134e34i			1.13094i	−0.824767	+	0.565472i	$0.808694\pi$
$968$		−	6.42079e34i		−	2.95136i
$969$	−1.13009e34			−0.513321
$970$	1.37191e34	−	1.60296e34i	0.615817	−	0.719529i
$971$	−1.26565e34			−0.561427			−0.280714	−	0.959792i	$-0.590571\pi$
$971$	−1.26565e34			−0.561427			−0.280714	+	0.959792i	$0.590571\pi$
$972$		−	8.32554e34i		−	3.64964i
$973$			5.78379e33i			0.250562i
$974$	6.25417e34			2.67758
$975$	6.66853e33	+	4.26715e34i	0.282148	+	1.80545i
$976$	2.01808e34			0.843849
$977$			2.22019e34i			0.917492i	0.888567	+	0.458746i	$0.151701\pi$
$977$			2.22019e34i			0.917492i	−0.888567	+	0.458746i	$0.848299\pi$
$978$		−	6.94378e34i		−	2.83595i
$979$	1.24763e34			0.503601
$980$	−4.29647e34	+	5.02005e34i	−1.71400	+	2.00267i
$981$	3.49575e34			1.37831
$982$			1.97800e34i			0.770804i
$983$			2.55841e33i			0.0985380i	0.998786	+	0.0492690i	$0.0156892\pi$
$983$			2.55841e33i			0.0985380i	−0.998786	+	0.0492690i	$0.984311\pi$
$984$	−1.84502e35			−7.02356
$985$	−2.06941e34	−	1.77113e34i	−0.778628	−	0.666397i
$986$	−2.10431e34			−0.782575
$987$			2.61079e33i			0.0959678i
$988$			7.23763e34i			2.62961i
$989$	−1.00340e34			−0.360345
$990$	−1.08923e34	−	9.32230e33i	−0.386647	−	0.330916i
$991$	2.73946e34			0.961208			0.480604	−	0.876938i	$-0.340417\pi$
$991$	2.73946e34			0.961208			0.480604	+	0.876938i	$0.340417\pi$
$992$			4.47128e34i			1.55077i
$993$			1.76698e34i			0.605780i
$994$	−4.40245e33			−0.149194
$995$	3.16527e34	−	3.69835e34i	1.06034	−	1.23892i
$996$	2.32138e34			0.768712
$997$		−	2.50955e34i		−	0.821488i	−0.911751	−	0.410744i	$-0.865269\pi$
$997$		−	2.50955e34i		−	0.821488i	0.911751	−	0.410744i	$-0.134731\pi$
$998$			3.76388e34i			1.21796i
$999$	5.76763e33			0.184499

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5.24.b.a.4.1	✓	10
3.2	odd	2		45.24.b.b.19.10		10
5.2	odd	4		25.24.a.f.1.10		10
5.3	odd	4		25.24.a.f.1.1		10
5.4	even	2	inner	5.24.b.a.4.10	yes	10
15.14	odd	2		45.24.b.b.19.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.24.b.a.4.1	✓	10	1.1	even	1	trivial
5.24.b.a.4.10	yes	10	5.4	even	2	inner
25.24.a.f.1.1		10	5.3	odd	4
25.24.a.f.1.10		10	5.2	odd	4
45.24.b.b.19.1		10	15.14	odd	2
45.24.b.b.19.10		10	3.2	odd	2

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:	\( N \)	\(=\)	\( 5 \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	5.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-5561.03i q^{2} -421877. i q^{3} -2.25364e7 q^{4} +(7.09941e7 - 8.29504e7i) q^{5} -2.34607e9 q^{6} +7.17617e8i q^{7} +7.86762e10i q^{8} -8.38373e10 q^{9} +O(q^{10})\)	\(q-5561.03i q^{2} -421877. i q^{3} -2.25364e7 q^{4} +(7.09941e7 - 8.29504e7i) q^{5} -2.34607e9 q^{6} +7.17617e8i q^{7} +7.86762e10i q^{8} -8.38373e10 q^{9} +(-4.61289e11 - 3.94800e11i) q^{10} -2.81649e11 q^{11} +9.50759e12i q^{12} -8.58779e12i q^{13} +3.99069e12 q^{14} +(-3.49949e13 - 2.99508e13i) q^{15} +2.48471e14 q^{16} -7.16301e13i q^{17} +4.66221e14i q^{18} +3.73965e14 q^{19} +(-1.59995e15 + 1.86940e15i) q^{20} +3.02746e14 q^{21} +1.56626e15i q^{22} +2.05786e15i q^{23} +3.31917e16 q^{24} +(-1.84061e15 - 1.17780e16i) q^{25} -4.77569e16 q^{26} -4.34783e15i q^{27} -1.61725e16i q^{28} +5.28275e16 q^{29} +(-1.66557e17 + 1.94608e17i) q^{30} -6.19487e16 q^{31} -7.21772e17i q^{32} +1.18822e17i q^{33} -3.98337e17 q^{34} +(5.95267e16 + 5.09466e16i) q^{35} +1.88939e18 q^{36} +1.32655e18i q^{37} -2.07963e18i q^{38} -3.62299e18 q^{39} +(6.52622e18 + 5.58554e18i) q^{40} -5.55868e18 q^{41} -1.68358e18i q^{42} +4.87597e18i q^{43} +6.34736e18 q^{44} +(-5.95195e18 + 6.95434e18i) q^{45} +1.14438e19 q^{46} +8.62370e18i q^{47} -1.04824e20i q^{48} +2.68538e19 q^{49} +(-6.54976e19 + 1.02357e19i) q^{50} -3.02191e19 q^{51} +1.93538e20i q^{52} -7.45020e19i q^{53} -2.41784e19 q^{54} +(-1.99954e19 + 2.33629e19i) q^{55} -5.64594e19 q^{56} -1.57767e20i q^{57} -2.93775e20i q^{58} +3.38308e20 q^{59} +(7.88659e20 + 6.74983e20i) q^{60} +8.12197e19 q^{61} +3.44498e20i q^{62} -6.01631e19i q^{63} -1.92946e21 q^{64} +(-7.12360e20 - 6.09682e20i) q^{65} +6.60769e20 q^{66} -2.97339e20i q^{67} +1.61428e21i q^{68} +8.68163e20 q^{69} +(2.83315e20 - 3.31029e20i) q^{70} -1.10318e21 q^{71} -6.59600e21i q^{72} -9.17445e20i q^{73} +7.37700e21 q^{74} +(-4.96886e21 + 7.76513e20i) q^{75} -8.42782e21 q^{76} -2.02117e20i q^{77} +2.01475e22i q^{78} +3.90105e21 q^{79} +(1.76400e22 - 2.06108e22i) q^{80} -9.72696e21 q^{81} +3.09119e22i q^{82} -2.44161e21i q^{83} -6.82281e21 q^{84} +(-5.94174e21 - 5.08531e21i) q^{85} +2.71154e22 q^{86} -2.22867e22i q^{87} -2.21591e22i q^{88} -4.42974e22 q^{89} +(3.86732e22 + 3.30989e22i) q^{90} +6.16274e21 q^{91} -4.63766e22i q^{92} +2.61348e22i q^{93} +4.79566e22 q^{94} +(2.65493e22 - 3.10205e22i) q^{95} -3.04499e23 q^{96} +3.47496e22i q^{97} -1.49335e23i q^{98} +2.36127e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 41272680 q^{4} + 124761750 q^{5} - 2262077880 q^{6} - 189250631370 q^{9}+O(q^{10}) \) 10 * q - 41272680 * q^4 + 124761750 * q^5 - 2262077880 * q^6 - 189250631370 * q^9	\( 10 q - 41272680 q^{4} + 124761750 q^{5} - 2262077880 q^{6} - 189250631370 q^{9} - 992748199000 q^{10} - 1448637536280 q^{11} + 20750531044440 q^{14} - 14566613457000 q^{15} + 307971806876960 q^{16} + 887626815301400 q^{19} - 23\!\cdots\!00 q^{20}+ \cdots + 41\!\cdots\!60 q^{99}+O(q^{100}) \) 10 * q - 41272680 * q^4 + 124761750 * q^5 - 2262077880 * q^6 - 189250631370 * q^9 - 992748199000 * q^10 - 1448637536280 * q^11 + 20750531044440 * q^14 - 14566613457000 * q^15 + 307971806876960 * q^16 + 887626815301400 * q^19 - 2363707755099000 * q^20 - 781158183715080 * q^21 + 58473619965621600 * q^24 - 15436708968968750 * q^25 - 49259739315596880 * q^26 + 68538893585329500 * q^29 - 283259561673249000 * q^30 - 303703652281842880 * q^31 - 288299835336191360 * q^34 + 568057466315481000 * q^35 + 3242003939215214760 * q^36 - 8628905972055907440 * q^39 + 9426000384456940000 * q^40 - 6658832954283527580 * q^41 + 14990828914855427040 * q^44 - 8490880963475889750 * q^45 + 16736621075351903320 * q^46 + 51294362631233173870 * q^49 - 96456356624588250000 * q^50 - 117334349391643270080 * q^51 + 6367892697021344400 * q^54 + 8046234911696751000 * q^55 - 93285217288146540000 * q^56 + 461218638014732785800 * q^59 + 1248088814097510276000 * q^60 + 202894074812420701820 * q^61 - 2597882075263561124480 * q^64 + 404224351485216078000 * q^65 - 2067527591150630859360 * q^66 + 1761142984477245282360 * q^69 + 2651929720118465517000 * q^70 - 6362428274364321596880 * q^71 + 21369976437605310645840 * q^74 - 1268180400269994750000 * q^75 - 28533821601830896716000 * q^76 + 8008906997916696975200 * q^79 + 50390556129429851388000 * q^80 - 63620987817532049079390 * q^81 + 26983560053969205971040 * q^84 + 58719411820058643416000 * q^85 - 73642932902287825779480 * q^86 + 17045340818835962686500 * q^89 + 218378490181286718303000 * q^90 - 261754857315780277067280 * q^91 + 163995755653951693894840 * q^94 + 287254967024630517105000 * q^95 - 1029134712278532215134080 * q^96 + 411070875879864759534360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more