LMFDB - Embedded newform 64.24.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,24,Mod(1,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

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

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	64.a (trivial)

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:	yes
Analytic conductor:	$214.530583901$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2 x^{5} - 376388081 x^{4} + 1624987949956 x^{3} + \cdots + 26\!\cdots\!25 $ x^6 - 2x^5 - 376388081x^4 + 1624987949956x^3 + 32369842380258997x^2 - 236529072927913245962*x + 267804601125256722828825
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{71}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-12175.6$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+95240.8 q^{3} -5.05314e7 q^{5} -3.45287e9 q^{7} -8.50724e10 q^{9} +O(q^{10})$	$q+95240.8 q^{3} -5.05314e7 q^{5} -3.45287e9 q^{7} -8.50724e10 q^{9} -1.58269e12 q^{11} +7.62999e12 q^{13} -4.81265e12 q^{15} +2.45820e14 q^{17} -2.60151e14 q^{19} -3.28854e14 q^{21} -7.85702e15 q^{23} -9.36751e15 q^{25} -1.70686e16 q^{27} -4.59662e16 q^{29} -4.84588e16 q^{31} -1.50737e17 q^{33} +1.74478e17 q^{35} -6.75718e17 q^{37} +7.26686e17 q^{39} +3.34740e18 q^{41} -6.18054e18 q^{43} +4.29883e18 q^{45} -1.60882e19 q^{47} -1.54465e19 q^{49} +2.34120e19 q^{51} -1.07973e20 q^{53} +7.99755e19 q^{55} -2.47770e19 q^{57} +1.50291e20 q^{59} +3.18587e20 q^{61} +2.93743e20 q^{63} -3.85554e20 q^{65} -4.03986e20 q^{67} -7.48309e20 q^{69} -2.40504e21 q^{71} +1.01707e21 q^{73} -8.92168e20 q^{75} +5.46482e21 q^{77} +5.66485e21 q^{79} +6.38335e21 q^{81} +2.07801e22 q^{83} -1.24216e22 q^{85} -4.37785e21 q^{87} -2.24947e22 q^{89} -2.63453e22 q^{91} -4.61526e21 q^{93} +1.31458e22 q^{95} +4.35243e22 q^{97} +1.34643e23 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9}+O(q^{10}) $ 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9	$ 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9} + 926871857520 q^{11} + 3684897167820 q^{13} - 105662245358560 q^{15} + 71064722424780 q^{17} - 453921923982960 q^{19} + 17\!\cdots\!80 q^{21}+ \cdots + 39\!\cdots\!40 q^{99}+O(q^{100}) $ 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9 + 926871857520 * q^11 + 3684897167820 * q^13 - 105662245358560 * q^15 + 71064722424780 * q^17 - 453921923982960 * q^19 + 1712900116737280 * q^21 + 7126581767144736 * q^23 + 21617163289317450 * q^25 + 41851914072529184 * q^27 + 90718382478351852 * q^29 - 53660517982277760 * q^31 + 607667723496123520 * q^33 - 843485446441350720 * q^35 - 1178800099078290660 * q^37 - 387144388087503200 * q^39 + 1672300630976121180 * q^41 + 9707130536958833904 * q^43 - 31072675924013630900 * q^45 + 12637324971625378368 * q^47 + 20589514456941037878 * q^49 - 56125446355277416800 * q^51 + 25955354185131109500 * q^53 - 79415390320029033120 * q^55 + 169187096172727794560 * q^57 + 407713345239006043440 * q^59 + 615884547554394402156 * q^61 + 578320926136902243296 * q^63 - 1095923787012414183240 * q^65 - 523768762996317716400 * q^67 - 596596128814415845632 * q^69 - 948379808687003287200 * q^71 - 4925435605783876580100 * q^73 + 11105120520135150430000 * q^75 - 1628144191735745368320 * q^77 + 6474134783625939495360 * q^79 + 23375719407022646959702 * q^81 + 12413209217367249862032 * q^83 + 5978612007137723165880 * q^85 - 17230630950338425832032 * q^87 + 7892765755425674057628 * q^89 + 117701223538009065704640 * q^91 - 65760567831273830333440 * q^93 - 112886120912929344409440 * q^95 - 12935113600510578486420 * q^97 + 397484583772859161352240 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	95240.8	0.310405	0.155202	−	0.987883i	$-0.450397\pi$
$3$	95240.8	0.310405	0.155202	+	0.987883i	$0.450397\pi$
$4$	0	0
$5$	−5.05314e7	−0.462814	−0.231407	−	0.972857i	$-0.574333\pi$
$5$	−5.05314e7	−0.462814	−0.231407	+	0.972857i	$0.574333\pi$
$6$	0	0
$7$	−3.45287e9	−0.660013	−0.330006	−	0.943979i	$-0.607051\pi$
$7$	−3.45287e9	−0.660013	−0.330006	+	0.943979i	$0.607051\pi$
$8$	0	0
$9$	−8.50724e10	−0.903649
$10$	0	0
$11$	−1.58269e12	−1.67255	−0.836277	−	0.548308i	$-0.815272\pi$
$11$	−1.58269e12	−1.67255	−0.836277	+	0.548308i	$0.815272\pi$
$12$	0	0
$13$	7.62999e12	1.18080	0.590399	−	0.807112i	$-0.298970\pi$
$13$	7.62999e12	1.18080	0.590399	+	0.807112i	$0.298970\pi$
$14$	0	0
$15$	−4.81265e12	−0.143660
$16$	0	0
$17$	2.45820e14	1.73962	0.869809	−	0.493389i	$-0.164242\pi$
$17$	2.45820e14	1.73962	0.869809	+	0.493389i	$0.164242\pi$
$18$	0	0
$19$	−2.60151e14	−0.512342	−0.256171	−	0.966631i	$-0.582461\pi$
$19$	−2.60151e14	−0.512342	−0.256171	+	0.966631i	$0.582461\pi$
$20$	0	0
$21$	−3.28854e14	−0.204871
$22$	0	0
$23$	−7.85702e15	−1.71944	−0.859722	−	0.510763i	$-0.829363\pi$
$23$	−7.85702e15	−1.71944	−0.859722	+	0.510763i	$0.829363\pi$
$24$	0	0
$25$	−9.36751e15	−0.785803
$26$	0	0
$27$	−1.70686e16	−0.590902
$28$	0	0
$29$	−4.59662e16	−0.699619	−0.349809	−	0.936821i	$-0.613754\pi$
$29$	−4.59662e16	−0.699619	−0.349809	+	0.936821i	$0.613754\pi$
$30$	0	0
$31$	−4.84588e16	−0.342542	−0.171271	−	0.985224i	$-0.554787\pi$
$31$	−4.84588e16	−0.342542	−0.171271	+	0.985224i	$0.554787\pi$
$32$	0	0
$33$	−1.50737e17	−0.519169
$34$	0	0
$35$	1.74478e17	0.305463
$36$	0	0
$37$	−6.75718e17	−0.624375	−0.312188	−	0.950020i	$-0.601062\pi$
$37$	−6.75718e17	−0.624375	−0.312188	+	0.950020i	$0.601062\pi$
$38$	0	0
$39$	7.26686e17	0.366525
$40$	0	0
$41$	3.34740e18	0.949933	0.474967	−	0.880004i	$-0.342460\pi$
$41$	3.34740e18	0.949933	0.474967	+	0.880004i	$0.342460\pi$
$42$	0	0
$43$	−6.18054e18	−1.01424	−0.507118	−	0.861877i	$-0.669289\pi$
$43$	−6.18054e18	−1.01424	−0.507118	+	0.861877i	$0.669289\pi$
$44$	0	0
$45$	4.29883e18	0.418221
$46$	0	0
$47$	−1.60882e19	−0.949256	−0.474628	−	0.880186i	$-0.657417\pi$
$47$	−1.60882e19	−0.949256	−0.474628	+	0.880186i	$0.657417\pi$
$48$	0	0
$49$	−1.54465e19	−0.564383
$50$	0	0
$51$	2.34120e19	0.539986
$52$	0	0
$53$	−1.07973e20	−1.60008	−0.800041	−	0.599945i	$-0.795189\pi$
$53$	−1.07973e20	−1.60008	−0.800041	+	0.599945i	$0.795189\pi$
$54$	0	0
$55$	7.99755e19	0.774081
$56$	0	0
$57$	−2.47770e19	−0.159033
$58$	0	0
$59$	1.50291e20	0.648837	0.324419	−	0.945914i	$-0.394831\pi$
$59$	1.50291e20	0.648837	0.324419	+	0.945914i	$0.394831\pi$
$60$	0	0
$61$	3.18587e20	0.937422	0.468711	−	0.883352i	$-0.344719\pi$
$61$	3.18587e20	0.937422	0.468711	+	0.883352i	$0.344719\pi$
$62$	0	0
$63$	2.93743e20	0.596420
$64$	0	0
$65$	−3.85554e20	−0.546489
$66$	0	0
$67$	−4.03986e20	−0.404116	−0.202058	−	0.979374i	$-0.564763\pi$
$67$	−4.03986e20	−0.404116	−0.202058	+	0.979374i	$0.564763\pi$
$68$	0	0
$69$	−7.48309e20	−0.533724
$70$	0	0
$71$	−2.40504e21	−1.23495	−0.617477	−	0.786589i	$-0.711845\pi$
$71$	−2.40504e21	−1.23495	−0.617477	+	0.786589i	$0.711845\pi$
$72$	0	0
$73$	1.01707e21	0.379437	0.189719	−	0.981838i	$-0.439242\pi$
$73$	1.01707e21	0.379437	0.189719	+	0.981838i	$0.439242\pi$
$74$	0	0
$75$	−8.92168e20	−0.243917
$76$	0	0
$77$	5.46482e21	1.10391
$78$	0	0
$79$	5.66485e21	0.852073	0.426037	−	0.904706i	$-0.359909\pi$
$79$	5.66485e21	0.852073	0.426037	+	0.904706i	$0.359909\pi$
$80$	0	0
$81$	6.38335e21	0.720230
$82$	0	0
$83$	2.07801e22	1.77112	0.885562	−	0.464522i	$-0.153774\pi$
$83$	2.07801e22	1.77112	0.885562	+	0.464522i	$0.153774\pi$
$84$	0	0
$85$	−1.24216e22	−0.805119
$86$	0	0
$87$	−4.37785e21	−0.217165
$88$	0	0
$89$	−2.24947e22	−0.859201	−0.429600	−	0.903019i	$-0.641346\pi$
$89$	−2.24947e22	−0.859201	−0.429600	+	0.903019i	$0.641346\pi$
$90$	0	0
$91$	−2.63453e22	−0.779341
$92$	0	0
$93$	−4.61526e21	−0.106327
$94$	0	0
$95$	1.31458e22	0.237119
$96$	0	0
$97$	4.35243e22	0.617812	0.308906	−	0.951093i	$-0.400037\pi$
$97$	4.35243e22	0.617812	0.308906	+	0.951093i	$0.400037\pi$
$98$	0	0
$99$	1.34643e23	1.51140
$100$	0	0
$101$	3.79474e22	0.338443	0.169222	−	0.985578i	$-0.445875\pi$
$101$	3.79474e22	0.338443	0.169222	+	0.985578i	$0.445875\pi$
$102$	0	0
$103$	−1.82867e23	−1.30169	−0.650843	−	0.759212i	$-0.725584\pi$
$103$	−1.82867e23	−1.30169	−0.650843	+	0.759212i	$0.725584\pi$
$104$	0	0
$105$	1.66174e22	0.0948172
$106$	0	0
$107$	1.16328e23	0.534281	0.267141	−	0.963658i	$-0.413921\pi$
$107$	1.16328e23	0.534281	0.267141	+	0.963658i	$0.413921\pi$
$108$	0	0
$109$	7.56433e22	0.280780	0.140390	−	0.990096i	$-0.455164\pi$
$109$	7.56433e22	0.280780	0.140390	+	0.990096i	$0.455164\pi$
$110$	0	0
$111$	−6.43559e22	−0.193809
$112$	0	0
$113$	−6.97230e23	−1.70991	−0.854956	−	0.518700i	$-0.826416\pi$
$113$	−6.97230e23	−1.70991	−0.854956	+	0.518700i	$0.826416\pi$
$114$	0	0
$115$	3.97026e23	0.795782
$116$	0	0
$117$	−6.49101e23	−1.06703
$118$	0	0
$119$	−8.48782e23	−1.14817
$120$	0	0
$121$	1.60948e24	1.79743
$122$	0	0
$123$	3.18809e23	0.294864
$124$	0	0
$125$	1.07573e24	0.826494
$126$	0	0
$127$	−2.94049e24	−1.88225	−0.941124	−	0.338063i	$-0.890228\pi$
$127$	−2.94049e24	−1.88225	−0.941124	+	0.338063i	$0.890228\pi$
$128$	0	0
$129$	−5.88639e23	−0.314824
$130$	0	0
$131$	3.34722e24	1.49991	0.749954	−	0.661490i	$-0.230075\pi$
$131$	3.34722e24	1.49991	0.749954	+	0.661490i	$0.230075\pi$
$132$	0	0
$133$	8.98268e23	0.338152
$134$	0	0
$135$	8.62502e23	0.273478
$136$	0	0
$137$	7.64542e23	0.204699	0.102349	−	0.994749i	$-0.467364\pi$
$137$	7.64542e23	0.204699	0.102349	+	0.994749i	$0.467364\pi$
$138$	0	0
$139$	6.21477e24	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$139$	6.21477e24	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$140$	0	0
$141$	−1.53226e24	−0.294654
$142$	0	0
$143$	−1.20759e25	−1.97495
$144$	0	0
$145$	2.32274e24	0.323793
$146$	0	0
$147$	−1.47113e24	−0.175187
$148$	0	0
$149$	−2.29286e24	−0.233742	−0.116871	−	0.993147i	$-0.537286\pi$
$149$	−2.29286e24	−0.233742	−0.116871	+	0.993147i	$0.537286\pi$
$150$	0	0
$151$	−6.62393e24	−0.579270	−0.289635	−	0.957137i	$-0.593534\pi$
$151$	−6.62393e24	−0.579270	−0.289635	+	0.957137i	$0.593534\pi$
$152$	0	0
$153$	−2.09125e25	−1.57200
$154$	0	0
$155$	2.44869e24	0.158533
$156$	0	0
$157$	4.16112e24	0.232469	0.116234	−	0.993222i	$-0.462918\pi$
$157$	4.16112e24	0.232469	0.116234	+	0.993222i	$0.462918\pi$
$158$	0	0
$159$	−1.02834e25	−0.496673
$160$	0	0
$161$	2.71292e25	1.13485
$162$	0	0
$163$	1.99633e25	0.724560	0.362280	−	0.932069i	$-0.381998\pi$
$163$	1.99633e25	0.724560	0.362280	+	0.932069i	$0.381998\pi$
$164$	0	0
$165$	7.61693e24	0.240278
$166$	0	0
$167$	3.16796e25	0.870042	0.435021	−	0.900420i	$-0.356741\pi$
$167$	3.16796e25	0.870042	0.435021	+	0.900420i	$0.356741\pi$
$168$	0	0
$169$	1.64629e25	0.394283
$170$	0	0
$171$	2.21317e25	0.462977
$172$	0	0
$173$	2.31612e25	0.423868	0.211934	−	0.977284i	$-0.432024\pi$
$173$	2.31612e25	0.423868	0.211934	+	0.977284i	$0.432024\pi$
$174$	0	0
$175$	3.23447e25	0.518640
$176$	0	0
$177$	1.43139e25	0.201402
$178$	0	0
$179$	2.45902e25	0.304055	0.152028	−	0.988376i	$-0.451420\pi$
$179$	2.45902e25	0.304055	0.152028	+	0.988376i	$0.451420\pi$
$180$	0	0
$181$	−3.47537e25	−0.378179	−0.189090	−	0.981960i	$-0.560554\pi$
$181$	−3.47537e25	−0.378179	−0.189090	+	0.981960i	$0.560554\pi$
$182$	0	0
$183$	3.03425e25	0.290980
$184$	0	0
$185$	3.41450e25	0.288970
$186$	0	0
$187$	−3.89056e26	−2.90960
$188$	0	0
$189$	5.89357e25	0.390003
$190$	0	0
$191$	1.72211e26	1.00966	0.504832	−	0.863218i	$-0.331555\pi$
$191$	1.72211e26	1.00966	0.504832	+	0.863218i	$0.331555\pi$
$192$	0	0
$193$	5.20728e25	0.270833	0.135417	−	0.990789i	$-0.456763\pi$
$193$	5.20728e25	0.270833	0.135417	+	0.990789i	$0.456763\pi$
$194$	0	0
$195$	−3.67205e25	−0.169633
$196$	0	0
$197$	3.52872e26	1.44962	0.724812	−	0.688947i	$-0.241927\pi$
$197$	3.52872e26	1.44962	0.724812	+	0.688947i	$0.241927\pi$
$198$	0	0
$199$	−7.23679e25	−0.264689	−0.132344	−	0.991204i	$-0.542250\pi$
$199$	−7.23679e25	−0.264689	−0.132344	+	0.991204i	$0.542250\pi$
$200$	0	0
$201$	−3.84759e25	−0.125440
$202$	0	0
$203$	1.58715e26	0.461757
$204$	0	0
$205$	−1.69149e26	−0.439642
$206$	0	0
$207$	6.68416e26	1.55377
$208$	0	0
$209$	4.11739e26	0.856919
$210$	0	0
$211$	−8.62974e26	−1.60972	−0.804859	−	0.593466i	$-0.797759\pi$
$211$	−8.62974e26	−1.60972	−0.804859	+	0.593466i	$0.797759\pi$
$212$	0	0
$213$	−2.29058e26	−0.383336
$214$	0	0
$215$	3.12311e26	0.469402
$216$	0	0
$217$	1.67322e26	0.226082
$218$	0	0
$219$	9.68670e25	0.117779
$220$	0	0
$221$	1.87560e27	2.05414
$222$	0	0
$223$	−9.55444e26	−0.943408	−0.471704	−	0.881757i	$-0.656361\pi$
$223$	−9.55444e26	−0.943408	−0.471704	+	0.881757i	$0.656361\pi$
$224$	0	0
$225$	7.96916e26	0.710090
$226$	0	0
$227$	−6.03916e26	−0.486048	−0.243024	−	0.970020i	$-0.578139\pi$
$227$	−6.03916e26	−0.486048	−0.243024	+	0.970020i	$0.578139\pi$
$228$	0	0
$229$	−9.01699e25	−0.0656075	−0.0328038	−	0.999462i	$-0.510444\pi$
$229$	−9.01699e25	−0.0656075	−0.0328038	+	0.999462i	$0.510444\pi$
$230$	0	0
$231$	5.20473e26	0.342658
$232$	0	0
$233$	1.61120e27	0.960630	0.480315	−	0.877096i	$-0.340522\pi$
$233$	1.61120e27	0.960630	0.480315	+	0.877096i	$0.340522\pi$
$234$	0	0
$235$	8.12962e26	0.439329
$236$	0	0
$237$	5.39525e26	0.264488
$238$	0	0
$239$	−1.13961e27	−0.507200	−0.253600	−	0.967309i	$-0.581615\pi$
$239$	−1.13961e27	−0.507200	−0.253600	+	0.967309i	$0.581615\pi$
$240$	0	0
$241$	1.27258e27	0.514621	0.257311	−	0.966329i	$-0.417164\pi$
$241$	1.27258e27	0.514621	0.257311	+	0.966329i	$0.417164\pi$
$242$	0	0
$243$	2.21485e27	0.814465
$244$	0	0
$245$	7.80532e26	0.261204
$246$	0	0
$247$	−1.98495e27	−0.604972
$248$	0	0
$249$	1.97911e27	0.549765
$250$	0	0
$251$	5.02954e27	1.27433	0.637163	−	0.770729i	$-0.280108\pi$
$251$	5.02954e27	1.27433	0.637163	+	0.770729i	$0.280108\pi$
$252$	0	0
$253$	1.24352e28	2.87586
$254$	0	0
$255$	−1.18304e27	−0.249913
$256$	0	0
$257$	6.15155e27	1.18783	0.593914	−	0.804528i	$-0.297582\pi$
$257$	6.15155e27	1.18783	0.593914	+	0.804528i	$0.297582\pi$
$258$	0	0
$259$	2.33316e27	0.412096
$260$	0	0
$261$	3.91045e27	0.632210
$262$	0	0
$263$	−1.04885e28	−1.55318	−0.776588	−	0.630008i	$-0.783051\pi$
$263$	−1.04885e28	−1.55318	−0.776588	+	0.630008i	$0.783051\pi$
$264$	0	0
$265$	5.45602e27	0.740540
$266$	0	0
$267$	−2.14241e27	−0.266700
$268$	0	0
$269$	−1.32175e28	−1.51007	−0.755034	−	0.655686i	$-0.772380\pi$
$269$	−1.32175e28	−1.51007	−0.755034	+	0.655686i	$0.772380\pi$
$270$	0	0
$271$	5.86632e27	0.615487	0.307743	−	0.951469i	$-0.400426\pi$
$271$	5.86632e27	0.615487	0.307743	+	0.951469i	$0.400426\pi$
$272$	0	0
$273$	−2.50915e27	−0.241911
$274$	0	0
$275$	1.48259e28	1.31430
$276$	0	0
$277$	1.68075e28	1.37084	0.685419	−	0.728149i	$-0.259619\pi$
$277$	1.68075e28	1.37084	0.685419	+	0.728149i	$0.259619\pi$
$278$	0	0
$279$	4.12251e27	0.309537
$280$	0	0
$281$	3.15261e27	0.218046	0.109023	−	0.994039i	$-0.465228\pi$
$281$	3.15261e27	0.218046	0.109023	+	0.994039i	$0.465228\pi$
$282$	0	0
$283$	4.01138e26	0.0255711	0.0127856	−	0.999918i	$-0.495930\pi$
$283$	4.01138e26	0.0255711	0.0127856	+	0.999918i	$0.495930\pi$
$284$	0	0
$285$	1.25202e27	0.0736029
$286$	0	0
$287$	−1.15581e28	−0.626968
$288$	0	0
$289$	4.04597e28	2.02627
$290$	0	0
$291$	4.14528e27	0.191772
$292$	0	0
$293$	4.48822e28	1.91910	0.959548	−	0.281544i	$-0.0908466\pi$
$293$	4.48822e28	1.91910	0.959548	+	0.281544i	$0.0908466\pi$
$294$	0	0
$295$	−7.59443e27	−0.300291
$296$	0	0
$297$	2.70143e28	0.988315
$298$	0	0
$299$	−5.99490e28	−2.03031
$300$	0	0
$301$	2.13406e28	0.669408
$302$	0	0
$303$	3.61414e27	0.105054
$304$	0	0
$305$	−1.60987e28	−0.433852
$306$	0	0
$307$	−1.73640e28	−0.434069	−0.217034	−	0.976164i	$-0.569638\pi$
$307$	−1.73640e28	−0.434069	−0.217034	+	0.976164i	$0.569638\pi$
$308$	0	0
$309$	−1.74163e28	−0.404049
$310$	0	0
$311$	5.68774e28	1.22517	0.612584	−	0.790406i	$-0.290130\pi$
$311$	5.68774e28	1.22517	0.612584	+	0.790406i	$0.290130\pi$
$312$	0	0
$313$	2.01685e27	0.0403566	0.0201783	−	0.999796i	$-0.493577\pi$
$313$	2.01685e27	0.0403566	0.0201783	+	0.999796i	$0.493577\pi$
$314$	0	0
$315$	−1.48433e28	−0.276031
$316$	0	0
$317$	4.73005e28	0.817870	0.408935	−	0.912564i	$-0.365900\pi$
$317$	4.73005e28	0.817870	0.408935	+	0.912564i	$0.365900\pi$
$318$	0	0
$319$	7.27502e28	1.17015
$320$	0	0
$321$	1.10791e28	0.165844
$322$	0	0
$323$	−6.39503e28	−0.891279
$324$	0	0
$325$	−7.14740e28	−0.927875
$326$	0	0
$327$	7.20432e27	0.0871554
$328$	0	0
$329$	5.55506e28	0.626521
$330$	0	0
$331$	1.10623e29	1.16365	0.581825	−	0.813314i	$-0.302339\pi$
$331$	1.10623e29	1.16365	0.581825	+	0.813314i	$0.302339\pi$
$332$	0	0
$333$	5.74850e28	0.564216
$334$	0	0
$335$	2.04140e28	0.187031
$336$	0	0
$337$	−1.17852e29	−1.00831	−0.504153	−	0.863614i	$-0.668195\pi$
$337$	−1.17852e29	−1.00831	−0.504153	+	0.863614i	$0.668195\pi$
$338$	0	0
$339$	−6.64047e28	−0.530765
$340$	0	0
$341$	7.66953e28	0.572919
$342$	0	0
$343$	1.47835e29	1.03251
$344$	0	0
$345$	3.78131e28	0.247015
$346$	0	0
$347$	−2.07676e29	−1.26939	−0.634697	−	0.772761i	$-0.718875\pi$
$347$	−2.07676e29	−1.26939	−0.634697	+	0.772761i	$0.718875\pi$
$348$	0	0
$349$	−2.66957e29	−1.52739	−0.763693	−	0.645580i	$-0.776616\pi$
$349$	−2.66957e29	−1.52739	−0.763693	+	0.645580i	$0.776616\pi$
$350$	0	0
$351$	−1.30233e29	−0.697735
$352$	0	0
$353$	9.20874e28	0.462159	0.231079	−	0.972935i	$-0.425774\pi$
$353$	9.20874e28	0.462159	0.231079	+	0.972935i	$0.425774\pi$
$354$	0	0
$355$	1.21530e29	0.571554
$356$	0	0
$357$	−8.08386e28	−0.356397
$358$	0	0
$359$	−4.77074e29	−1.97242	−0.986210	−	0.165497i	$-0.947077\pi$
$359$	−4.77074e29	−1.97242	−0.986210	+	0.165497i	$0.947077\pi$
$360$	0	0
$361$	−1.90151e29	−0.737506
$362$	0	0
$363$	1.53288e29	0.557932
$364$	0	0
$365$	−5.13942e28	−0.175609
$366$	0	0
$367$	3.82679e29	1.22793	0.613967	−	0.789332i	$-0.289573\pi$
$367$	3.82679e29	1.22793	0.613967	+	0.789332i	$0.289573\pi$
$368$	0	0
$369$	−2.84771e29	−0.858406
$370$	0	0
$371$	3.72816e29	1.05607
$372$	0	0
$373$	2.01388e29	0.536267	0.268134	−	0.963382i	$-0.413593\pi$
$373$	2.01388e29	0.536267	0.268134	+	0.963382i	$0.413593\pi$
$374$	0	0
$375$	1.02454e29	0.256548
$376$	0	0
$377$	−3.50721e29	−0.826108
$378$	0	0
$379$	3.02112e29	0.669603	0.334802	−	0.942289i	$-0.391331\pi$
$379$	3.02112e29	0.669603	0.334802	+	0.942289i	$0.391331\pi$
$380$	0	0
$381$	−2.80054e29	−0.584259
$382$	0	0
$383$	−1.71878e29	−0.337626	−0.168813	−	0.985648i	$-0.553993\pi$
$383$	−1.71878e29	−0.337626	−0.168813	+	0.985648i	$0.553993\pi$
$384$	0	0
$385$	−2.76145e29	−0.510903
$386$	0	0
$387$	5.25793e29	0.916513
$388$	0	0
$389$	−9.90623e29	−1.62738	−0.813688	−	0.581302i	$-0.802544\pi$
$389$	−9.90623e29	−1.62738	−0.813688	+	0.581302i	$0.802544\pi$
$390$	0	0
$391$	−1.93141e30	−2.99117
$392$	0	0
$393$	3.18792e29	0.465579
$394$	0	0
$395$	−2.86253e29	−0.394351
$396$	0	0
$397$	1.49973e28	0.0194950	0.00974749	−	0.999952i	$-0.496897\pi$
$397$	1.49973e28	0.0194950	0.00974749	+	0.999952i	$0.496897\pi$
$398$	0	0
$399$	8.55517e28	0.104964
$400$	0	0
$401$	−3.65211e29	−0.423042	−0.211521	−	0.977373i	$-0.567842\pi$
$401$	−3.65211e29	−0.423042	−0.211521	+	0.977373i	$0.567842\pi$
$402$	0	0
$403$	−3.69740e29	−0.404472
$404$	0	0
$405$	−3.22560e29	−0.333332
$406$	0	0
$407$	1.06945e30	1.04430
$408$	0	0
$409$	−1.82691e30	−1.68616	−0.843082	−	0.537785i	$-0.819261\pi$
$409$	−1.82691e30	−1.68616	−0.843082	+	0.537785i	$0.819261\pi$
$410$	0	0
$411$	7.28155e28	0.0635394
$412$	0	0
$413$	−5.18936e29	−0.428241
$414$	0	0
$415$	−1.05005e30	−0.819700
$416$	0	0
$417$	5.91900e29	0.437203
$418$	0	0
$419$	1.37022e29	0.0957917	0.0478959	−	0.998852i	$-0.484748\pi$
$419$	1.37022e29	0.0957917	0.0478959	+	0.998852i	$0.484748\pi$
$420$	0	0
$421$	−2.02197e30	−1.33823	−0.669115	−	0.743159i	$-0.733327\pi$
$421$	−2.02197e30	−1.33823	−0.669115	+	0.743159i	$0.733327\pi$
$422$	0	0
$423$	1.36867e30	0.857794
$424$	0	0
$425$	−2.30272e30	−1.36700
$426$	0	0
$427$	−1.10004e30	−0.618710
$428$	0	0
$429$	−1.15012e30	−0.613033
$430$	0	0
$431$	1.76499e29	0.0891773	0.0445886	−	0.999005i	$-0.485802\pi$
$431$	1.76499e29	0.0891773	0.0445886	+	0.999005i	$0.485802\pi$
$432$	0	0
$433$	7.09811e29	0.340041	0.170021	−	0.985440i	$-0.445617\pi$
$433$	7.09811e29	0.340041	0.170021	+	0.985440i	$0.445617\pi$
$434$	0	0
$435$	2.21219e29	0.100507
$436$	0	0
$437$	2.04402e30	0.880943
$438$	0	0
$439$	3.12218e30	1.27678	0.638391	−	0.769712i	$-0.279600\pi$
$439$	3.12218e30	1.27678	0.638391	+	0.769712i	$0.279600\pi$
$440$	0	0
$441$	1.31407e30	0.510004
$442$	0	0
$443$	1.18121e30	0.435196	0.217598	−	0.976038i	$-0.430178\pi$
$443$	1.18121e30	0.435196	0.217598	+	0.976038i	$0.430178\pi$
$444$	0	0
$445$	1.13669e30	0.397650
$446$	0	0
$447$	−2.18374e29	−0.0725545
$448$	0	0
$449$	−2.39338e30	−0.755404	−0.377702	−	0.925927i	$-0.623286\pi$
$449$	−2.39338e30	−0.755404	−0.377702	+	0.925927i	$0.623286\pi$
$450$	0	0
$451$	−5.29789e30	−1.58881
$452$	0	0
$453$	−6.30868e29	−0.179808
$454$	0	0
$455$	1.33127e30	0.360690
$456$	0	0
$457$	6.58631e29	0.169670	0.0848351	−	0.996395i	$-0.472964\pi$
$457$	6.58631e29	0.169670	0.0848351	+	0.996395i	$0.472964\pi$
$458$	0	0
$459$	−4.19580e30	−1.02794
$460$	0	0
$461$	−9.71777e29	−0.226468	−0.113234	−	0.993568i	$-0.536121\pi$
$461$	−9.71777e29	−0.226468	−0.113234	+	0.993568i	$0.536121\pi$
$462$	0	0
$463$	5.72854e30	1.27017	0.635086	−	0.772441i	$-0.280965\pi$
$463$	5.72854e30	1.27017	0.635086	+	0.772441i	$0.280965\pi$
$464$	0	0
$465$	2.33215e29	0.0492094
$466$	0	0
$467$	1.35172e30	0.271484	0.135742	−	0.990744i	$-0.456658\pi$
$467$	1.35172e30	0.271484	0.135742	+	0.990744i	$0.456658\pi$
$468$	0	0
$469$	1.39491e30	0.266722
$470$	0	0
$471$	3.96308e29	0.0721593
$472$	0	0
$473$	9.78187e30	1.69636
$474$	0	0
$475$	2.43697e30	0.402600
$476$	0	0
$477$	9.18551e30	1.44591
$478$	0	0
$479$	1.63527e30	0.245318	0.122659	−	0.992449i	$-0.460858\pi$
$479$	1.63527e30	0.245318	0.122659	+	0.992449i	$0.460858\pi$
$480$	0	0
$481$	−5.15572e30	−0.737261
$482$	0	0
$483$	2.58381e30	0.352264
$484$	0	0
$485$	−2.19934e30	−0.285932
$486$	0	0
$487$	−6.62345e30	−0.821299	−0.410650	−	0.911793i	$-0.634698\pi$
$487$	−6.62345e30	−0.821299	−0.410650	+	0.911793i	$0.634698\pi$
$488$	0	0
$489$	1.90132e30	0.224907
$490$	0	0
$491$	−6.45947e30	−0.729054	−0.364527	−	0.931193i	$-0.618769\pi$
$491$	−6.45947e30	−0.729054	−0.364527	+	0.931193i	$0.618769\pi$
$492$	0	0
$493$	−1.12994e31	−1.21707
$494$	0	0
$495$	−6.80371e30	−0.699497
$496$	0	0
$497$	8.30427e30	0.815085
$498$	0	0
$499$	3.55880e30	0.333539	0.166770	−	0.985996i	$-0.446666\pi$
$499$	3.55880e30	0.333539	0.166770	+	0.985996i	$0.446666\pi$
$500$	0	0
$501$	3.01719e30	0.270065
$502$	0	0
$503$	−9.63472e30	−0.823772	−0.411886	−	0.911235i	$-0.635130\pi$
$503$	−9.63472e30	−0.823772	−0.411886	+	0.911235i	$0.635130\pi$
$504$	0	0
$505$	−1.91754e30	−0.156636
$506$	0	0
$507$	1.56793e30	0.122387
$508$	0	0
$509$	2.15087e31	1.60457	0.802286	−	0.596939i	$-0.203617\pi$
$509$	2.15087e31	1.60457	0.802286	+	0.596939i	$0.203617\pi$
$510$	0	0
$511$	−3.51182e30	−0.250433
$512$	0	0
$513$	4.44043e30	0.302744
$514$	0	0
$515$	9.24050e30	0.602438
$516$	0	0
$517$	2.54627e31	1.58768
$518$	0	0
$519$	2.20589e30	0.131571
$520$	0	0
$521$	9.58671e30	0.547061	0.273530	−	0.961863i	$-0.411809\pi$
$521$	9.58671e30	0.547061	0.273530	+	0.961863i	$0.411809\pi$
$522$	0	0
$523$	3.36312e31	1.83643	0.918213	−	0.396087i	$-0.129632\pi$
$523$	3.36312e31	1.83643	0.918213	+	0.396087i	$0.129632\pi$
$524$	0	0
$525$	3.08054e30	0.160988
$526$	0	0
$527$	−1.19121e31	−0.595892
$528$	0	0
$529$	4.08523e31	1.95649
$530$	0	0
$531$	−1.27856e31	−0.586321
$532$	0	0
$533$	2.55406e31	1.12168
$534$	0	0
$535$	−5.87821e30	−0.247273
$536$	0	0
$537$	2.34199e30	0.0943802
$538$	0	0
$539$	2.44470e31	0.943961
$540$	0	0
$541$	−2.35726e31	−0.872246	−0.436123	−	0.899887i	$-0.643649\pi$
$541$	−2.35726e31	−0.872246	−0.436123	+	0.899887i	$0.643649\pi$
$542$	0	0
$543$	−3.30997e30	−0.117389
$544$	0	0
$545$	−3.82236e30	−0.129949
$546$	0	0
$547$	−2.53269e31	−0.825520	−0.412760	−	0.910840i	$-0.635435\pi$
$547$	−2.53269e31	−0.825520	−0.412760	+	0.910840i	$0.635435\pi$
$548$	0	0
$549$	−2.71030e31	−0.847100
$550$	0	0
$551$	1.19582e31	0.358444
$552$	0	0
$553$	−1.95600e31	−0.562379
$554$	0	0
$555$	3.25200e30	0.0896975
$556$	0	0
$557$	2.33287e31	0.617386	0.308693	−	0.951162i	$-0.400108\pi$
$557$	2.33287e31	0.617386	0.308693	+	0.951162i	$0.400108\pi$
$558$	0	0
$559$	−4.71574e31	−1.19761
$560$	0	0
$561$	−3.70540e31	−0.903155
$562$	0	0
$563$	2.74247e30	0.0641646	0.0320823	−	0.999485i	$-0.489786\pi$
$563$	2.74247e30	0.0641646	0.0320823	+	0.999485i	$0.489786\pi$
$564$	0	0
$565$	3.52320e31	0.791371
$566$	0	0
$567$	−2.20409e31	−0.475361
$568$	0	0
$569$	7.41380e31	1.53550	0.767750	−	0.640749i	$-0.221376\pi$
$569$	7.41380e31	1.53550	0.767750	+	0.640749i	$0.221376\pi$
$570$	0	0
$571$	−1.88279e31	−0.374530	−0.187265	−	0.982309i	$-0.559962\pi$
$571$	−1.88279e31	−0.374530	−0.187265	+	0.982309i	$0.559962\pi$
$572$	0	0
$573$	1.64015e31	0.313404
$574$	0	0
$575$	7.36007e31	1.35114
$576$	0	0
$577$	3.30037e31	0.582158	0.291079	−	0.956699i	$-0.405986\pi$
$577$	3.30037e31	0.582158	0.291079	+	0.956699i	$0.405986\pi$
$578$	0	0
$579$	4.95945e30	0.0840679
$580$	0	0
$581$	−7.17508e31	−1.16896
$582$	0	0
$583$	1.70888e32	2.67622
$584$	0	0
$585$	3.28000e31	0.493835
$586$	0	0
$587$	−1.03331e32	−1.49586	−0.747931	−	0.663777i	$-0.768952\pi$
$587$	−1.03331e32	−1.49586	−0.747931	+	0.663777i	$0.768952\pi$
$588$	0	0
$589$	1.26066e31	0.175498
$590$	0	0
$591$	3.36078e31	0.449970
$592$	0	0
$593$	−1.42855e32	−1.83978	−0.919888	−	0.392181i	$-0.871721\pi$
$593$	−1.42855e32	−1.83978	−0.919888	+	0.392181i	$0.871721\pi$
$594$	0	0
$595$	4.28901e31	0.531389
$596$	0	0
$597$	−6.89237e30	−0.0821607
$598$	0	0
$599$	−3.11386e31	−0.357182	−0.178591	−	0.983923i	$-0.557154\pi$
$599$	−3.11386e31	−0.357182	−0.178591	+	0.983923i	$0.557154\pi$
$600$	0	0
$601$	−1.43698e32	−1.58633	−0.793167	−	0.609004i	$-0.791569\pi$
$601$	−1.43698e32	−1.58633	−0.793167	+	0.609004i	$0.791569\pi$
$602$	0	0
$603$	3.43681e31	0.365179
$604$	0	0
$605$	−8.13291e31	−0.831877
$606$	0	0
$607$	−6.18855e31	−0.609423	−0.304712	−	0.952445i	$-0.598560\pi$
$607$	−6.18855e31	−0.609423	−0.304712	+	0.952445i	$0.598560\pi$
$608$	0	0
$609$	1.51161e31	0.143332
$610$	0	0
$611$	−1.22753e32	−1.12088
$612$	0	0
$613$	−8.50847e31	−0.748266	−0.374133	−	0.927375i	$-0.622060\pi$
$613$	−8.50847e31	−0.748266	−0.374133	+	0.927375i	$0.622060\pi$
$614$	0	0
$615$	−1.61099e31	−0.136467
$616$	0	0
$617$	−2.66734e30	−0.0217670	−0.0108835	−	0.999941i	$-0.503464\pi$
$617$	−2.66734e30	−0.0217670	−0.0108835	+	0.999941i	$0.503464\pi$
$618$	0	0
$619$	−2.36189e32	−1.85702	−0.928512	−	0.371304i	$-0.878911\pi$
$619$	−2.36189e32	−1.85702	−0.928512	+	0.371304i	$0.878911\pi$
$620$	0	0
$621$	1.34109e32	1.01602
$622$	0	0
$623$	7.76711e31	0.567083
$624$	0	0
$625$	5.73110e31	0.403290
$626$	0	0
$627$	3.92143e31	0.265992
$628$	0	0
$629$	−1.66105e32	−1.08617
$630$	0	0
$631$	−2.01959e32	−1.27329	−0.636643	−	0.771159i	$-0.719677\pi$
$631$	−2.01959e32	−1.27329	−0.636643	+	0.771159i	$0.719677\pi$
$632$	0	0
$633$	−8.21903e31	−0.499664
$634$	0	0
$635$	1.48587e32	0.871130
$636$	0	0
$637$	−1.17856e32	−0.666422
$638$	0	0
$639$	2.04602e32	1.11597
$640$	0	0
$641$	1.73563e30	0.00913252	0.00456626	−	0.999990i	$-0.498547\pi$
$641$	1.73563e30	0.00913252	0.00456626	+	0.999990i	$0.498547\pi$
$642$	0	0
$643$	5.73750e31	0.291270	0.145635	−	0.989338i	$-0.453477\pi$
$643$	5.73750e31	0.291270	0.145635	+	0.989338i	$0.453477\pi$
$644$	0	0
$645$	2.97448e31	0.145705
$646$	0	0
$647$	−4.44014e31	−0.209893	−0.104946	−	0.994478i	$-0.533467\pi$
$647$	−4.44014e31	−0.209893	−0.104946	+	0.994478i	$0.533467\pi$
$648$	0	0
$649$	−2.37864e32	−1.08521
$650$	0	0
$651$	1.59359e31	0.0701769
$652$	0	0
$653$	−1.61980e32	−0.688587	−0.344293	−	0.938862i	$-0.611881\pi$
$653$	−1.61980e32	−0.688587	−0.344293	+	0.938862i	$0.611881\pi$
$654$	0	0
$655$	−1.69140e32	−0.694178
$656$	0	0
$657$	−8.65250e31	−0.342878
$658$	0	0
$659$	1.80882e32	0.692172	0.346086	−	0.938203i	$-0.387511\pi$
$659$	1.80882e32	0.692172	0.346086	+	0.938203i	$0.387511\pi$
$660$	0	0
$661$	−2.87349e32	−1.06192	−0.530961	−	0.847396i	$-0.678169\pi$
$661$	−2.87349e32	−1.06192	−0.530961	+	0.847396i	$0.678169\pi$
$662$	0	0
$663$	1.78634e32	0.637614
$664$	0	0
$665$	−4.53908e31	−0.156501
$666$	0	0
$667$	3.61157e32	1.20295
$668$	0	0
$669$	−9.09972e31	−0.292838
$670$	0	0
$671$	−5.04225e32	−1.56789
$672$	0	0
$673$	−9.79432e31	−0.294307	−0.147154	−	0.989114i	$-0.547011\pi$
$673$	−9.79432e31	−0.294307	−0.147154	+	0.989114i	$0.547011\pi$
$674$	0	0
$675$	1.59890e32	0.464333
$676$	0	0
$677$	3.42960e32	0.962663	0.481332	−	0.876539i	$-0.340153\pi$
$677$	3.42960e32	0.962663	0.481332	+	0.876539i	$0.340153\pi$
$678$	0	0
$679$	−1.50283e32	−0.407764
$680$	0	0
$681$	−5.75174e31	−0.150872
$682$	0	0
$683$	1.58252e32	0.401339	0.200669	−	0.979659i	$-0.435688\pi$
$683$	1.58252e32	0.401339	0.200669	+	0.979659i	$0.435688\pi$
$684$	0	0
$685$	−3.86334e31	−0.0947373
$686$	0	0
$687$	−8.58785e30	−0.0203649
$688$	0	0
$689$	−8.23832e32	−1.88937
$690$	0	0
$691$	−2.46301e32	−0.546347	−0.273174	−	0.961965i	$-0.588073\pi$
$691$	−2.46301e32	−0.546347	−0.273174	+	0.961965i	$0.588073\pi$
$692$	0	0
$693$	−4.64905e32	−0.997543
$694$	0	0
$695$	−3.14041e32	−0.651870
$696$	0	0
$697$	8.22856e32	1.65252
$698$	0	0
$699$	1.53452e32	0.298184
$700$	0	0
$701$	4.85823e31	0.0913526	0.0456763	−	0.998956i	$-0.485456\pi$
$701$	4.85823e31	0.0913526	0.0456763	+	0.998956i	$0.485456\pi$
$702$	0	0
$703$	1.75789e32	0.319894
$704$	0	0
$705$	7.74271e31	0.136370
$706$	0	0
$707$	−1.31027e32	−0.223377
$708$	0	0
$709$	5.56379e32	0.918203	0.459102	−	0.888384i	$-0.348171\pi$
$709$	5.56379e32	0.918203	0.459102	+	0.888384i	$0.348171\pi$
$710$	0	0
$711$	−4.81922e32	−0.769975
$712$	0	0
$713$	3.80742e32	0.588981
$714$	0	0
$715$	6.10213e32	0.914033
$716$	0	0
$717$	−1.08537e32	−0.157437
$718$	0	0
$719$	7.56246e32	1.06238	0.531191	−	0.847252i	$-0.321745\pi$
$719$	7.56246e32	1.06238	0.531191	+	0.847252i	$0.321745\pi$
$720$	0	0
$721$	6.31414e32	0.859129
$722$	0	0
$723$	1.21201e32	0.159741
$724$	0	0
$725$	4.30588e32	0.549763
$726$	0	0
$727$	1.18063e33	1.46038	0.730192	−	0.683242i	$-0.239431\pi$
$727$	1.18063e33	1.46038	0.730192	+	0.683242i	$0.239431\pi$
$728$	0	0
$729$	−3.90005e32	−0.467416
$730$	0	0
$731$	−1.51930e33	−1.76438
$732$	0	0
$733$	1.34202e33	1.51030	0.755149	−	0.655553i	$-0.227564\pi$
$733$	1.34202e33	1.51030	0.755149	+	0.655553i	$0.227564\pi$
$734$	0	0
$735$	7.43384e31	0.0810791
$736$	0	0
$737$	6.39385e32	0.675906
$738$	0	0
$739$	6.77535e31	0.0694258	0.0347129	−	0.999397i	$-0.488948\pi$
$739$	6.77535e31	0.0694258	0.0347129	+	0.999397i	$0.488948\pi$
$740$	0	0
$741$	−1.89048e32	−0.187786
$742$	0	0
$743$	−1.24750e33	−1.20135	−0.600676	−	0.799493i	$-0.705102\pi$
$743$	−1.24750e33	−1.20135	−0.600676	+	0.799493i	$0.705102\pi$
$744$	0	0
$745$	1.15862e32	0.108179
$746$	0	0
$747$	−1.76781e33	−1.60047
$748$	0	0
$749$	−4.01664e32	−0.352632
$750$	0	0
$751$	1.14148e32	0.0971874	0.0485937	−	0.998819i	$-0.484526\pi$
$751$	1.14148e32	0.0971874	0.0485937	+	0.998819i	$0.484526\pi$
$752$	0	0
$753$	4.79018e32	0.395557
$754$	0	0
$755$	3.34716e32	0.268094
$756$	0	0
$757$	−6.41998e32	−0.498805	−0.249403	−	0.968400i	$-0.580234\pi$
$757$	−6.41998e32	−0.498805	−0.249403	+	0.968400i	$0.580234\pi$
$758$	0	0
$759$	1.18434e33	0.892681
$760$	0	0
$761$	2.39413e33	1.75075	0.875377	−	0.483441i	$-0.160613\pi$
$761$	2.39413e33	1.75075	0.875377	+	0.483441i	$0.160613\pi$
$762$	0	0
$763$	−2.61186e32	−0.185318
$764$	0	0
$765$	1.05674e33	0.727545
$766$	0	0
$767$	1.14672e33	0.766145
$768$	0	0
$769$	−2.35012e33	−1.52383	−0.761917	−	0.647675i	$-0.775742\pi$
$769$	−2.35012e33	−1.52383	−0.761917	+	0.647675i	$0.775742\pi$
$770$	0	0
$771$	5.85879e32	0.368708
$772$	0	0
$773$	6.66378e32	0.407058	0.203529	−	0.979069i	$-0.434759\pi$
$773$	6.66378e32	0.407058	0.203529	+	0.979069i	$0.434759\pi$
$774$	0	0
$775$	4.53938e32	0.269170
$776$	0	0
$777$	2.22212e32	0.127916
$778$	0	0
$779$	−8.70831e32	−0.486691
$780$	0	0
$781$	3.80643e33	2.06553
$782$	0	0
$783$	7.84579e32	0.413406
$784$	0	0
$785$	−2.10267e32	−0.107590
$786$	0	0
$787$	−2.01139e33	−0.999509	−0.499755	−	0.866167i	$-0.666576\pi$
$787$	−2.01139e33	−0.999509	−0.499755	+	0.866167i	$0.666576\pi$
$788$	0	0
$789$	−9.98930e32	−0.482114
$790$	0	0
$791$	2.40744e33	1.12856
$792$	0	0
$793$	2.43082e33	1.10691
$794$	0	0
$795$	5.19636e32	0.229867
$796$	0	0
$797$	5.38154e32	0.231279	0.115639	−	0.993291i	$-0.463108\pi$
$797$	5.38154e32	0.231279	0.115639	+	0.993291i	$0.463108\pi$
$798$	0	0
$799$	−3.95481e33	−1.65134
$800$	0	0
$801$	1.91368e33	0.776416
$802$	0	0
$803$	−1.60971e33	−0.634629
$804$	0	0
$805$	−1.37088e33	−0.525226
$806$	0	0
$807$	−1.25884e33	−0.468732
$808$	0	0
$809$	7.38405e31	0.0267231	0.0133615	−	0.999911i	$-0.495747\pi$
$809$	7.38405e31	0.0267231	0.0133615	+	0.999911i	$0.495747\pi$
$810$	0	0
$811$	4.41113e33	1.55171	0.775854	−	0.630913i	$-0.217320\pi$
$811$	4.41113e33	1.55171	0.775854	+	0.630913i	$0.217320\pi$
$812$	0	0
$813$	5.58712e32	0.191050
$814$	0	0
$815$	−1.00877e33	−0.335337
$816$	0	0
$817$	1.60788e33	0.519636
$818$	0	0
$819$	2.24126e33	0.704251
$820$	0	0
$821$	2.88258e33	0.880714	0.440357	−	0.897823i	$-0.354852\pi$
$821$	2.88258e33	0.880714	0.440357	+	0.897823i	$0.354852\pi$
$822$	0	0
$823$	−3.09630e33	−0.919909	−0.459955	−	0.887942i	$-0.652134\pi$
$823$	−3.09630e33	−0.919909	−0.459955	+	0.887942i	$0.652134\pi$
$824$	0	0
$825$	1.41203e33	0.407964
$826$	0	0
$827$	3.57006e33	1.00314	0.501571	−	0.865117i	$-0.332756\pi$
$827$	3.57006e33	1.00314	0.501571	+	0.865117i	$0.332756\pi$
$828$	0	0
$829$	−2.69087e33	−0.735385	−0.367693	−	0.929947i	$-0.619852\pi$
$829$	−2.69087e33	−0.735385	−0.367693	+	0.929947i	$0.619852\pi$
$830$	0	0
$831$	1.60076e33	0.425515
$832$	0	0
$833$	−3.79704e33	−0.981811
$834$	0	0
$835$	−1.60082e33	−0.402668
$836$	0	0
$837$	8.27126e32	0.202409
$838$	0	0
$839$	3.84140e33	0.914592	0.457296	−	0.889315i	$-0.348818\pi$
$839$	3.84140e33	0.914592	0.457296	+	0.889315i	$0.348818\pi$
$840$	0	0
$841$	−2.20383e33	−0.510534
$842$	0	0
$843$	3.00257e32	0.0676825
$844$	0	0
$845$	−8.31891e32	−0.182480
$846$	0	0
$847$	−5.55731e33	−1.18633
$848$	0	0
$849$	3.82047e31	0.00793740
$850$	0	0
$851$	5.30913e33	1.07358
$852$	0	0
$853$	−4.26346e33	−0.839168	−0.419584	−	0.907717i	$-0.637824\pi$
$853$	−4.26346e33	−0.839168	−0.419584	+	0.907717i	$0.637824\pi$
$854$	0	0
$855$	−1.11835e33	−0.214272
$856$	0	0
$857$	−2.64302e33	−0.492971	−0.246485	−	0.969147i	$-0.579276\pi$
$857$	−2.64302e33	−0.492971	−0.246485	+	0.969147i	$0.579276\pi$
$858$	0	0
$859$	3.76153e33	0.683035	0.341517	−	0.939875i	$-0.389059\pi$
$859$	3.76153e33	0.683035	0.341517	+	0.939875i	$0.389059\pi$
$860$	0	0
$861$	−1.10080e33	−0.194614
$862$	0	0
$863$	−1.72146e33	−0.296328	−0.148164	−	0.988963i	$-0.547336\pi$
$863$	−1.72146e33	−0.296328	−0.148164	+	0.988963i	$0.547336\pi$
$864$	0	0
$865$	−1.17037e33	−0.196172
$866$	0	0
$867$	3.85341e33	0.628964
$868$	0	0
$869$	−8.96570e33	−1.42514
$870$	0	0
$871$	−3.08241e33	−0.477179
$872$	0	0
$873$	−3.70271e33	−0.558285
$874$	0	0
$875$	−3.71437e33	−0.545497
$876$	0	0
$877$	1.04746e34	1.49846	0.749228	−	0.662312i	$-0.230425\pi$
$877$	1.04746e34	1.49846	0.749228	+	0.662312i	$0.230425\pi$
$878$	0	0
$879$	4.27462e33	0.595697
$880$	0	0
$881$	−7.56118e33	−1.02652	−0.513259	−	0.858234i	$-0.671562\pi$
$881$	−7.56118e33	−1.02652	−0.513259	+	0.858234i	$0.671562\pi$
$882$	0	0
$883$	−1.20422e34	−1.59279	−0.796394	−	0.604778i	$-0.793262\pi$
$883$	−1.20422e34	−1.59279	−0.796394	+	0.604778i	$0.793262\pi$
$884$	0	0
$885$	−7.23299e32	−0.0932117
$886$	0	0
$887$	−1.27059e34	−1.59545	−0.797727	−	0.603019i	$-0.793964\pi$
$887$	−1.27059e34	−1.59545	−0.797727	+	0.603019i	$0.793964\pi$
$888$	0	0
$889$	1.01531e34	1.24231
$890$	0	0
$891$	−1.01029e34	−1.20462
$892$	0	0
$893$	4.18538e33	0.486344
$894$	0	0
$895$	−1.24258e33	−0.140721
$896$	0	0
$897$	−5.70959e33	−0.630219
$898$	0	0
$899$	2.22747e33	0.239649
$900$	0	0
$901$	−2.65418e34	−2.78353
$902$	0	0
$903$	2.03249e33	0.207788
$904$	0	0
$905$	1.75616e33	0.175027
$906$	0	0
$907$	1.49263e34	1.45034	0.725169	−	0.688570i	$-0.241761\pi$
$907$	1.49263e34	1.45034	0.725169	+	0.688570i	$0.241761\pi$
$908$	0	0
$909$	−3.22827e33	−0.305834
$910$	0	0
$911$	1.60175e34	1.47956	0.739779	−	0.672849i	$-0.234930\pi$
$911$	1.60175e34	1.47956	0.739779	+	0.672849i	$0.234930\pi$
$912$	0	0
$913$	−3.28884e34	−2.96230
$914$	0	0
$915$	−1.53325e33	−0.134670
$916$	0	0
$917$	−1.15575e34	−0.989959
$918$	0	0
$919$	9.89056e33	0.826214	0.413107	−	0.910682i	$-0.364444\pi$
$919$	9.89056e33	0.826214	0.413107	+	0.910682i	$0.364444\pi$
$920$	0	0
$921$	−1.65376e33	−0.134737
$922$	0	0
$923$	−1.83504e34	−1.45823
$924$	0	0
$925$	6.32979e33	0.490636
$926$	0	0
$927$	1.55569e34	1.17627
$928$	0	0
$929$	8.43420e33	0.622104	0.311052	−	0.950393i	$-0.399319\pi$
$929$	8.43420e33	0.622104	0.311052	+	0.950393i	$0.399319\pi$
$930$	0	0
$931$	4.01842e33	0.289157
$932$	0	0
$933$	5.41705e33	0.380298
$934$	0	0
$935$	1.96595e34	1.34660
$936$	0	0
$937$	1.33822e34	0.894377	0.447189	−	0.894440i	$-0.352425\pi$
$937$	1.33822e34	0.894377	0.447189	+	0.894440i	$0.352425\pi$
$938$	0	0
$939$	1.92087e32	0.0125269
$940$	0	0
$941$	−3.43499e33	−0.218597	−0.109299	−	0.994009i	$-0.534861\pi$
$941$	−3.43499e33	−0.218597	−0.109299	+	0.994009i	$0.534861\pi$
$942$	0	0
$943$	−2.63006e34	−1.63336
$944$	0	0
$945$	−2.97810e33	−0.180499
$946$	0	0
$947$	−1.92854e34	−1.14078	−0.570392	−	0.821372i	$-0.693209\pi$
$947$	−1.92854e34	−1.14078	−0.570392	+	0.821372i	$0.693209\pi$
$948$	0	0
$949$	7.76027e33	0.448039
$950$	0	0
$951$	4.50494e33	0.253871
$952$	0	0
$953$	2.73918e34	1.50679	0.753395	−	0.657569i	$-0.228415\pi$
$953$	2.73918e34	1.50679	0.753395	+	0.657569i	$0.228415\pi$
$954$	0	0
$955$	−8.70205e33	−0.467286
$956$	0	0
$957$	6.92878e33	0.363220
$958$	0	0
$959$	−2.63986e33	−0.135104
$960$	0	0
$961$	−1.76651e34	−0.882665
$962$	0	0
$963$	−9.89628e33	−0.482803
$964$	0	0
$965$	−2.63131e33	−0.125345
$966$	0	0
$967$	2.97112e34	1.38202	0.691011	−	0.722844i	$-0.257166\pi$
$967$	2.97112e34	1.38202	0.691011	+	0.722844i	$0.257166\pi$
$968$	0	0
$969$	−6.09068e33	−0.276657
$970$	0	0
$971$	−4.23034e33	−0.187653	−0.0938263	−	0.995589i	$-0.529910\pi$
$971$	−4.23034e33	−0.187653	−0.0938263	+	0.995589i	$0.529910\pi$
$972$	0	0
$973$	−2.14588e34	−0.929624
$974$	0	0
$975$	−6.80724e33	−0.288017
$976$	0	0
$977$	−3.54564e34	−1.46523	−0.732617	−	0.680641i	$-0.761701\pi$
$977$	−3.54564e34	−1.46523	−0.732617	+	0.680641i	$0.761701\pi$
$978$	0	0
$979$	3.56021e34	1.43706
$980$	0	0
$981$	−6.43515e33	−0.253726
$982$	0	0
$983$	−4.16127e34	−1.60273	−0.801365	−	0.598176i	$-0.795892\pi$
$983$	−4.16127e34	−1.60273	−0.801365	+	0.598176i	$0.795892\pi$
$984$	0	0
$985$	−1.78311e34	−0.670906
$986$	0	0
$987$	5.29068e33	0.194475
$988$	0	0
$989$	4.85606e34	1.74392
$990$	0	0
$991$	−1.22239e34	−0.428908	−0.214454	−	0.976734i	$-0.568797\pi$
$991$	−1.22239e34	−0.428908	−0.214454	+	0.976734i	$0.568797\pi$
$992$	0	0
$993$	1.05358e34	0.361202
$994$	0	0
$995$	3.65685e33	0.122502
$996$	0	0
$997$	−1.92647e34	−0.630620	−0.315310	−	0.948989i	$-0.602108\pi$
$997$	−1.92647e34	−0.630620	−0.315310	+	0.948989i	$0.602108\pi$
$998$	0	0
$999$	1.15336e34	0.368945

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.o.1.3		6
4.3	odd	2		64.24.a.m.1.4		6
8.3	odd	2		32.24.a.e.1.3	yes	6
8.5	even	2		32.24.a.c.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.c.1.4	✓	6	8.5	even	2
32.24.a.e.1.3	yes	6	8.3	odd	2
64.24.a.m.1.4		6	4.3	odd	2
64.24.a.o.1.3		6	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q+95240.8 q^{3} -5.05314e7 q^{5} -3.45287e9 q^{7} -8.50724e10 q^{9} +O(q^{10})\)	\(q+95240.8 q^{3} -5.05314e7 q^{5} -3.45287e9 q^{7} -8.50724e10 q^{9} -1.58269e12 q^{11} +7.62999e12 q^{13} -4.81265e12 q^{15} +2.45820e14 q^{17} -2.60151e14 q^{19} -3.28854e14 q^{21} -7.85702e15 q^{23} -9.36751e15 q^{25} -1.70686e16 q^{27} -4.59662e16 q^{29} -4.84588e16 q^{31} -1.50737e17 q^{33} +1.74478e17 q^{35} -6.75718e17 q^{37} +7.26686e17 q^{39} +3.34740e18 q^{41} -6.18054e18 q^{43} +4.29883e18 q^{45} -1.60882e19 q^{47} -1.54465e19 q^{49} +2.34120e19 q^{51} -1.07973e20 q^{53} +7.99755e19 q^{55} -2.47770e19 q^{57} +1.50291e20 q^{59} +3.18587e20 q^{61} +2.93743e20 q^{63} -3.85554e20 q^{65} -4.03986e20 q^{67} -7.48309e20 q^{69} -2.40504e21 q^{71} +1.01707e21 q^{73} -8.92168e20 q^{75} +5.46482e21 q^{77} +5.66485e21 q^{79} +6.38335e21 q^{81} +2.07801e22 q^{83} -1.24216e22 q^{85} -4.37785e21 q^{87} -2.24947e22 q^{89} -2.63453e22 q^{91} -4.61526e21 q^{93} +1.31458e22 q^{95} +4.35243e22 q^{97} +1.34643e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9}+O(q^{10}) \) 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9	\( 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9} + 926871857520 q^{11} + 3684897167820 q^{13} - 105662245358560 q^{15} + 71064722424780 q^{17} - 453921923982960 q^{19} + 17\!\cdots\!80 q^{21}+ \cdots + 39\!\cdots\!40 q^{99}+O(q^{100}) \) 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9 + 926871857520 * q^11 + 3684897167820 * q^13 - 105662245358560 * q^15 + 71064722424780 * q^17 - 453921923982960 * q^19 + 1712900116737280 * q^21 + 7126581767144736 * q^23 + 21617163289317450 * q^25 + 41851914072529184 * q^27 + 90718382478351852 * q^29 - 53660517982277760 * q^31 + 607667723496123520 * q^33 - 843485446441350720 * q^35 - 1178800099078290660 * q^37 - 387144388087503200 * q^39 + 1672300630976121180 * q^41 + 9707130536958833904 * q^43 - 31072675924013630900 * q^45 + 12637324971625378368 * q^47 + 20589514456941037878 * q^49 - 56125446355277416800 * q^51 + 25955354185131109500 * q^53 - 79415390320029033120 * q^55 + 169187096172727794560 * q^57 + 407713345239006043440 * q^59 + 615884547554394402156 * q^61 + 578320926136902243296 * q^63 - 1095923787012414183240 * q^65 - 523768762996317716400 * q^67 - 596596128814415845632 * q^69 - 948379808687003287200 * q^71 - 4925435605783876580100 * q^73 + 11105120520135150430000 * q^75 - 1628144191735745368320 * q^77 + 6474134783625939495360 * q^79 + 23375719407022646959702 * q^81 + 12413209217367249862032 * q^83 + 5978612007137723165880 * q^85 - 17230630950338425832032 * q^87 + 7892765755425674057628 * q^89 + 117701223538009065704640 * q^91 - 65760567831273830333440 * q^93 - 112886120912929344409440 * q^95 - 12935113600510578486420 * q^97 + 397484583772859161352240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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