LMFDB - Embedded newform 64.24.a.k.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:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 166408x - 10560732 $ x^3 - 166408*x - 10560732
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{3} $
Twist minimal:	no (minimal twist has level 8)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$436.575$ 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)

$f(q)$	$=$	$q+574251. q^{3} -8.41392e7 q^{5} -5.95738e9 q^{7} +2.35621e11 q^{9} +O(q^{10})$	$q+574251. q^{3} -8.41392e7 q^{5} -5.95738e9 q^{7} +2.35621e11 q^{9} -1.24326e12 q^{11} -7.21479e12 q^{13} -4.83170e13 q^{15} -6.50936e13 q^{17} +5.56139e14 q^{19} -3.42103e15 q^{21} -3.08126e15 q^{23} -4.84153e15 q^{25} +8.12435e16 q^{27} +4.69935e16 q^{29} +2.40563e17 q^{31} -7.13941e17 q^{33} +5.01249e17 q^{35} +5.28193e17 q^{37} -4.14310e18 q^{39} +1.59837e18 q^{41} -7.35994e18 q^{43} -1.98249e19 q^{45} -1.63997e19 q^{47} +8.12160e18 q^{49} -3.73800e19 q^{51} +2.64089e19 q^{53} +1.04607e20 q^{55} +3.19363e20 q^{57} +3.24947e20 q^{59} +3.22520e20 q^{61} -1.40368e21 q^{63} +6.07047e20 q^{65} +1.09896e21 q^{67} -1.76941e21 q^{69} -2.30177e20 q^{71} +8.33706e20 q^{73} -2.78025e21 q^{75} +7.40655e21 q^{77} +2.59992e21 q^{79} +2.44721e22 q^{81} +1.05807e22 q^{83} +5.47692e21 q^{85} +2.69861e22 q^{87} +4.07434e22 q^{89} +4.29812e22 q^{91} +1.38143e23 q^{93} -4.67930e22 q^{95} -9.40029e22 q^{97} -2.92937e23 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 213948 q^{3} - 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9}+O(q^{10}) $ 3 * q + 213948 * q^3 - 95628618 * q^5 - 8647912920 * q^7 + 174509823951 * q^9	$ 3 q + 213948 q^{3} - 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9} - 35420906796 q^{11} - 3164858452338 q^{13} - 19825526344392 q^{15} - 30233487828906 q^{17} + 382754784400236 q^{19} - 27788918984928 q^{21} - 37\!\cdots\!20 q^{23}+ \cdots - 19\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q + 213948 * q^3 - 95628618 * q^5 - 8647912920 * q^7 + 174509823951 * q^9 - 35420906796 * q^11 - 3164858452338 * q^13 - 19825526344392 * q^15 - 30233487828906 * q^17 + 382754784400236 * q^19 - 27788918984928 * q^21 - 3754416434163720 * q^23 - 17405885896549371 * q^25 + 103738001159574936 * q^27 + 142892073311612862 * q^29 - 204189369040807008 * q^31 - 1301083012339287408 * q^33 + 1750886716718912208 * q^35 + 900641094521542422 * q^37 - 5338477210392936936 * q^39 - 1818448729363485042 * q^41 - 9021528635440809420 * q^43 - 28993631105890916082 * q^45 + 2644177465669087344 * q^47 + 92864473014347142747 * q^49 - 57094226780716436040 * q^51 - 127243616001408711354 * q^53 - 59143191447597143832 * q^55 + 390032141715811959408 * q^57 + 747107029364442476772 * q^59 + 141236260775173151262 * q^61 - 2369424210278793761592 * q^63 + 386137575836874807036 * q^65 + 3440608449139647701628 * q^67 - 2710328787781605291168 * q^69 - 4012482808005984420504 * q^71 + 4795371647558134022190 * q^73 - 820337660084156915484 * q^75 - 11480348485477300303776 * q^77 + 11708176751738041997328 * q^79 + 22449958139492866882107 * q^81 + 523281952489285038636 * q^83 - 423855162932232129780 * q^85 + 23077946037842683096536 * q^87 - 6011462112251221873026 * q^89 + 15776090363016749995536 * q^91 + 212279712155303705884800 * q^93 - 29064029058982820644968 * q^95 - 233786084347643851364634 * q^97 - 196693345233178755399900 * 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
$3$	574251.	1.87157	0.935787	−	0.352565i	$-0.114691\pi$
$3$	574251.	1.87157	0.935787	+	0.352565i	$0.114691\pi$
$4$	0	0
$5$	−8.41392e7	−0.770625	−0.385312	−	0.922786i	$-0.625906\pi$
$5$	−8.41392e7	−0.770625	−0.385312	+	0.922786i	$0.625906\pi$
$6$	0	0
$7$	−5.95738e9	−1.13875	−0.569374	−	0.822079i	$-0.692814\pi$
$7$	−5.95738e9	−1.13875	−0.569374	+	0.822079i	$0.692814\pi$
$8$	0	0
$9$	2.35621e11	2.50279
$10$	0	0
$11$	−1.24326e12	−1.31385	−0.656924	−	0.753957i	$-0.728143\pi$
$11$	−1.24326e12	−1.31385	−0.656924	+	0.753957i	$0.728143\pi$
$12$	0	0
$13$	−7.21479e12	−1.11654	−0.558271	−	0.829658i	$-0.688535\pi$
$13$	−7.21479e12	−1.11654	−0.558271	+	0.829658i	$0.688535\pi$
$14$	0	0
$15$	−4.83170e13	−1.44228
$16$	0	0
$17$	−6.50936e13	−0.460655	−0.230327	−	0.973113i	$-0.573980\pi$
$17$	−6.50936e13	−0.460655	−0.230327	+	0.973113i	$0.573980\pi$
$18$	0	0
$19$	5.56139e14	1.09526	0.547629	−	0.836721i	$-0.315530\pi$
$19$	5.56139e14	1.09526	0.547629	+	0.836721i	$0.315530\pi$
$20$	0	0
$21$	−3.42103e15	−2.13125
$22$	0	0
$23$	−3.08126e15	−0.674307	−0.337154	−	0.941450i	$-0.609464\pi$
$23$	−3.08126e15	−0.674307	−0.337154	+	0.941450i	$0.609464\pi$
$24$	0	0
$25$	−4.84153e15	−0.406137
$26$	0	0
$27$	8.12435e16	2.81258
$28$	0	0
$29$	4.69935e16	0.715255	0.357628	−	0.933864i	$-0.383586\pi$
$29$	4.69935e16	0.715255	0.357628	+	0.933864i	$0.383586\pi$
$30$	0	0
$31$	2.40563e17	1.70047	0.850234	−	0.526405i	$-0.176460\pi$
$31$	2.40563e17	1.70047	0.850234	+	0.526405i	$0.176460\pi$
$32$	0	0
$33$	−7.13941e17	−2.45896
$34$	0	0
$35$	5.01249e17	0.877548
$36$	0	0
$37$	5.28193e17	0.488059	0.244030	−	0.969768i	$-0.421531\pi$
$37$	5.28193e17	0.488059	0.244030	+	0.969768i	$0.421531\pi$
$38$	0	0
$39$	−4.14310e18	−2.08969
$40$	0	0
$41$	1.59837e18	0.453589	0.226795	−	0.973943i	$-0.427175\pi$
$41$	1.59837e18	0.453589	0.226795	+	0.973943i	$0.427175\pi$
$42$	0	0
$43$	−7.35994e18	−1.20778	−0.603889	−	0.797068i	$-0.706383\pi$
$43$	−7.35994e18	−1.20778	−0.603889	+	0.797068i	$0.706383\pi$
$44$	0	0
$45$	−1.98249e19	−1.92871
$46$	0	0
$47$	−1.63997e19	−0.967631	−0.483815	−	0.875170i	$-0.660749\pi$
$47$	−1.63997e19	−0.967631	−0.483815	+	0.875170i	$0.660749\pi$
$48$	0	0
$49$	8.12160e18	0.296747
$50$	0	0
$51$	−3.73800e19	−0.862149
$52$	0	0
$53$	2.64089e19	0.391361	0.195681	−	0.980668i	$-0.437308\pi$
$53$	2.64089e19	0.391361	0.195681	+	0.980668i	$0.437308\pi$
$54$	0	0
$55$	1.04607e20	1.01248
$56$	0	0
$57$	3.19363e20	2.04986
$58$	0	0
$59$	3.24947e20	1.40286	0.701429	−	0.712739i	$-0.252546\pi$
$59$	3.24947e20	1.40286	0.701429	+	0.712739i	$0.252546\pi$
$60$	0	0
$61$	3.22520e20	0.948995	0.474498	−	0.880257i	$-0.342630\pi$
$61$	3.22520e20	0.948995	0.474498	+	0.880257i	$0.342630\pi$
$62$	0	0
$63$	−1.40368e21	−2.85005
$64$	0	0
$65$	6.07047e20	0.860436
$66$	0	0
$67$	1.09896e21	1.09932	0.549658	−	0.835390i	$-0.314758\pi$
$67$	1.09896e21	1.09932	0.549658	+	0.835390i	$0.314758\pi$
$68$	0	0
$69$	−1.76941e21	−1.26202
$70$	0	0
$71$	−2.30177e20	−0.118193	−0.0590964	−	0.998252i	$-0.518822\pi$
$71$	−2.30177e20	−0.118193	−0.0590964	+	0.998252i	$0.518822\pi$
$72$	0	0
$73$	8.33706e20	0.311028	0.155514	−	0.987834i	$-0.450297\pi$
$73$	8.33706e20	0.311028	0.155514	+	0.987834i	$0.450297\pi$
$74$	0	0
$75$	−2.78025e21	−0.760116
$76$	0	0
$77$	7.40655e21	1.49614
$78$	0	0
$79$	2.59992e21	0.391064	0.195532	−	0.980697i	$-0.437357\pi$
$79$	2.59992e21	0.391064	0.195532	+	0.980697i	$0.437357\pi$
$80$	0	0
$81$	2.44721e22	2.76117
$82$	0	0
$83$	1.05807e22	0.901817	0.450908	−	0.892570i	$-0.351100\pi$
$83$	1.05807e22	0.901817	0.450908	+	0.892570i	$0.351100\pi$
$84$	0	0
$85$	5.47692e21	0.354992
$86$	0	0
$87$	2.69861e22	1.33865
$88$	0	0
$89$	4.07434e22	1.55622	0.778112	−	0.628126i	$-0.216178\pi$
$89$	4.07434e22	1.55622	0.778112	+	0.628126i	$0.216178\pi$
$90$	0	0
$91$	4.29812e22	1.27146
$92$	0	0
$93$	1.38143e23	3.18255
$94$	0	0
$95$	−4.67930e22	−0.844034
$96$	0	0
$97$	−9.40029e22	−1.33434	−0.667169	−	0.744906i	$-0.732494\pi$
$97$	−9.40029e22	−1.33434	−0.667169	+	0.744906i	$0.732494\pi$
$98$	0	0
$99$	−2.92937e23	−3.28828
$100$	0	0
$101$	1.31018e23	1.16851	0.584257	−	0.811568i	$-0.301386\pi$
$101$	1.31018e23	1.16851	0.584257	+	0.811568i	$0.301386\pi$
$102$	0	0
$103$	−5.91184e22	−0.420819	−0.210409	−	0.977613i	$-0.567480\pi$
$103$	−5.91184e22	−0.420819	−0.210409	+	0.977613i	$0.567480\pi$
$104$	0	0
$105$	2.87842e23	1.64240
$106$	0	0
$107$	7.78257e22	0.357445	0.178723	−	0.983899i	$-0.442804\pi$
$107$	7.78257e22	0.357445	0.178723	+	0.983899i	$0.442804\pi$
$108$	0	0
$109$	6.81626e22	0.253012	0.126506	−	0.991966i	$-0.459624\pi$
$109$	6.81626e22	0.253012	0.126506	+	0.991966i	$0.459624\pi$
$110$	0	0
$111$	3.03315e23	0.913439
$112$	0	0
$113$	−4.27400e23	−1.04817	−0.524086	−	0.851665i	$-0.675593\pi$
$113$	−4.27400e23	−1.04817	−0.524086	+	0.851665i	$0.675593\pi$
$114$	0	0
$115$	2.59254e23	0.519638
$116$	0	0
$117$	−1.69995e24	−2.79447
$118$	0	0
$119$	3.87787e23	0.524570
$120$	0	0
$121$	6.50256e23	0.726194
$122$	0	0
$123$	9.17865e23	0.848926
$124$	0	0
$125$	1.41038e24	1.08360
$126$	0	0
$127$	−6.80149e23	−0.435373	−0.217686	−	0.976019i	$-0.569851\pi$
$127$	−6.80149e23	−0.435373	−0.217686	+	0.976019i	$0.569851\pi$
$128$	0	0
$129$	−4.22645e24	−2.26045
$130$	0	0
$131$	−6.38677e22	−0.0286195	−0.0143097	−	0.999898i	$-0.504555\pi$
$131$	−6.38677e22	−0.0286195	−0.0143097	+	0.999898i	$0.504555\pi$
$132$	0	0
$133$	−3.31313e24	−1.24722
$134$	0	0
$135$	−6.83576e24	−2.16745
$136$	0	0
$137$	3.16090e24	0.846300	0.423150	−	0.906060i	$-0.360924\pi$
$137$	3.16090e24	0.846300	0.423150	+	0.906060i	$0.360924\pi$
$138$	0	0
$139$	1.51803e24	0.344040	0.172020	−	0.985093i	$-0.444971\pi$
$139$	1.51803e24	0.344040	0.172020	+	0.985093i	$0.444971\pi$
$140$	0	0
$141$	−9.41752e24	−1.81099
$142$	0	0
$143$	8.96984e24	1.46697
$144$	0	0
$145$	−3.95400e24	−0.551194
$146$	0	0
$147$	4.66384e24	0.555385
$148$	0	0
$149$	5.94136e23	0.0605680	0.0302840	−	0.999541i	$-0.490359\pi$
$149$	5.94136e23	0.0605680	0.0302840	+	0.999541i	$0.490359\pi$
$150$	0	0
$151$	1.03677e25	0.906664	0.453332	−	0.891342i	$-0.350235\pi$
$151$	1.03677e25	0.906664	0.453332	+	0.891342i	$0.350235\pi$
$152$	0	0
$153$	−1.53374e25	−1.15292
$154$	0	0
$155$	−2.02407e25	−1.31042
$156$	0	0
$157$	1.27647e25	0.713125	0.356563	−	0.934271i	$-0.383949\pi$
$157$	1.27647e25	0.713125	0.356563	+	0.934271i	$0.383949\pi$
$158$	0	0
$159$	1.51653e25	0.732461
$160$	0	0
$161$	1.83562e25	0.767866
$162$	0	0
$163$	−4.28936e25	−1.55681	−0.778404	−	0.627764i	$-0.783970\pi$
$163$	−4.28936e25	−1.55681	−0.778404	+	0.627764i	$0.783970\pi$
$164$	0	0
$165$	6.00704e25	1.89494
$166$	0	0
$167$	−6.74074e25	−1.85126	−0.925631	−	0.378427i	$-0.876465\pi$
$167$	−6.74074e25	−1.85126	−0.925631	+	0.378427i	$0.876465\pi$
$168$	0	0
$169$	1.02993e25	0.246668
$170$	0	0
$171$	1.31038e26	2.74120
$172$	0	0
$173$	5.33848e25	0.976984	0.488492	−	0.872569i	$-0.337547\pi$
$173$	5.33848e25	0.976984	0.488492	+	0.872569i	$0.337547\pi$
$174$	0	0
$175$	2.88428e25	0.462488
$176$	0	0
$177$	1.86601e26	2.62555
$178$	0	0
$179$	−5.76759e25	−0.713156	−0.356578	−	0.934266i	$-0.616057\pi$
$179$	−5.76759e25	−0.713156	−0.356578	+	0.934266i	$0.616057\pi$
$180$	0	0
$181$	1.31592e26	1.43194	0.715971	−	0.698130i	$-0.245984\pi$
$181$	1.31592e26	1.43194	0.715971	+	0.698130i	$0.245984\pi$
$182$	0	0
$183$	1.85208e26	1.77612
$184$	0	0
$185$	−4.44417e25	−0.376111
$186$	0	0
$187$	8.09280e25	0.605230
$188$	0	0
$189$	−4.83999e26	−3.20283
$190$	0	0
$191$	4.07437e25	0.238878	0.119439	−	0.992842i	$-0.461890\pi$
$191$	4.07437e25	0.238878	0.119439	+	0.992842i	$0.461890\pi$
$192$	0	0
$193$	−3.00984e26	−1.56543	−0.782717	−	0.622378i	$-0.786167\pi$
$193$	−3.00984e26	−1.56543	−0.782717	+	0.622378i	$0.786167\pi$
$194$	0	0
$195$	3.48597e26	1.61037
$196$	0	0
$197$	4.17423e25	0.171481	0.0857403	−	0.996318i	$-0.472674\pi$
$197$	4.17423e25	0.171481	0.0857403	+	0.996318i	$0.472674\pi$
$198$	0	0
$199$	−3.13013e25	−0.114486	−0.0572430	−	0.998360i	$-0.518231\pi$
$199$	−3.13013e25	−0.114486	−0.0572430	+	0.998360i	$0.518231\pi$
$200$	0	0
$201$	6.31080e26	2.05745
$202$	0	0
$203$	−2.79958e26	−0.814496
$204$	0	0
$205$	−1.34485e26	−0.349547
$206$	0	0
$207$	−7.26008e26	−1.68765
$208$	0	0
$209$	−6.91423e26	−1.43900
$210$	0	0
$211$	2.37199e26	0.442451	0.221226	−	0.975223i	$-0.428994\pi$
$211$	2.37199e26	0.442451	0.221226	+	0.975223i	$0.428994\pi$
$212$	0	0
$213$	−1.32179e26	−0.221207
$214$	0	0
$215$	6.19259e26	0.930744
$216$	0	0
$217$	−1.43312e27	−1.93640
$218$	0	0
$219$	4.78756e26	0.582113
$220$	0	0
$221$	4.69637e26	0.514341
$222$	0	0
$223$	1.49431e27	1.47549	0.737744	−	0.675080i	$-0.235891\pi$
$223$	1.49431e27	1.47549	0.737744	+	0.675080i	$0.235891\pi$
$224$	0	0
$225$	−1.14076e27	−1.01648
$226$	0	0
$227$	−4.09710e26	−0.329746	−0.164873	−	0.986315i	$-0.552721\pi$
$227$	−4.09710e26	−0.329746	−0.164873	+	0.986315i	$0.552721\pi$
$228$	0	0
$229$	−1.57268e27	−1.14428	−0.572141	−	0.820155i	$-0.693887\pi$
$229$	−1.57268e27	−1.14428	−0.572141	+	0.820155i	$0.693887\pi$
$230$	0	0
$231$	4.25322e27	2.80014
$232$	0	0
$233$	2.46809e27	1.47152	0.735762	−	0.677240i	$-0.236824\pi$
$233$	2.46809e27	1.47152	0.735762	+	0.677240i	$0.236824\pi$
$234$	0	0
$235$	1.37985e27	0.745681
$236$	0	0
$237$	1.49300e27	0.731906
$238$	0	0
$239$	1.15361e27	0.513432	0.256716	−	0.966487i	$-0.417359\pi$
$239$	1.15361e27	0.513432	0.256716	+	0.966487i	$0.417359\pi$
$240$	0	0
$241$	−6.38713e26	−0.258291	−0.129146	−	0.991626i	$-0.541223\pi$
$241$	−6.38713e26	−0.258291	−0.129146	+	0.991626i	$0.541223\pi$
$242$	0	0
$243$	6.40458e27	2.35515
$244$	0	0
$245$	−6.83345e26	−0.228681
$246$	0	0
$247$	−4.01243e27	−1.22290
$248$	0	0
$249$	6.07600e27	1.68782
$250$	0	0
$251$	−5.85944e26	−0.148460	−0.0742298	−	0.997241i	$-0.523650\pi$
$251$	−5.85944e26	−0.148460	−0.0742298	+	0.997241i	$0.523650\pi$
$252$	0	0
$253$	3.83079e27	0.885937
$254$	0	0
$255$	3.14512e27	0.664394
$256$	0	0
$257$	−5.28864e26	−0.102121	−0.0510603	−	0.998696i	$-0.516260\pi$
$257$	−5.28864e26	−0.102121	−0.0510603	+	0.998696i	$0.516260\pi$
$258$	0	0
$259$	−3.14664e27	−0.555777
$260$	0	0
$261$	1.10726e28	1.79013
$262$	0	0
$263$	5.59316e27	0.828259	0.414129	−	0.910218i	$-0.364086\pi$
$263$	5.59316e27	0.828259	0.414129	+	0.910218i	$0.364086\pi$
$264$	0	0
$265$	−2.22202e27	−0.301593
$266$	0	0
$267$	2.33969e28	2.91259
$268$	0	0
$269$	−1.11747e28	−1.27668	−0.638341	−	0.769753i	$-0.720379\pi$
$269$	−1.11747e28	−1.27668	−0.638341	+	0.769753i	$0.720379\pi$
$270$	0	0
$271$	−1.14584e28	−1.20220	−0.601100	−	0.799174i	$-0.705271\pi$
$271$	−1.14584e28	−1.20220	−0.601100	+	0.799174i	$0.705271\pi$
$272$	0	0
$273$	2.46820e28	2.37963
$274$	0	0
$275$	6.01927e27	0.533602
$276$	0	0
$277$	1.04295e28	0.850638	0.425319	−	0.905044i	$-0.360162\pi$
$277$	1.04295e28	0.850638	0.425319	+	0.905044i	$0.360162\pi$
$278$	0	0
$279$	5.66815e28	4.25592
$280$	0	0
$281$	9.50359e25	0.00657302	0.00328651	−	0.999995i	$-0.498954\pi$
$281$	9.50359e25	0.00657302	0.00328651	+	0.999995i	$0.498954\pi$
$282$	0	0
$283$	2.19749e28	1.40082	0.700412	−	0.713738i	$-0.252999\pi$
$283$	2.19749e28	1.40082	0.700412	+	0.713738i	$0.252999\pi$
$284$	0	0
$285$	−2.68709e28	−1.57967
$286$	0	0
$287$	−9.52209e27	−0.516524
$288$	0	0
$289$	−1.57304e28	−0.787797
$290$	0	0
$291$	−5.39812e28	−2.49731
$292$	0	0
$293$	−2.29758e28	−0.982410	−0.491205	−	0.871044i	$-0.663443\pi$
$293$	−2.29758e28	−0.982410	−0.491205	+	0.871044i	$0.663443\pi$
$294$	0	0
$295$	−2.73407e28	−1.08108
$296$	0	0
$297$	−1.01007e29	−3.69531
$298$	0	0
$299$	2.22306e28	0.752893
$300$	0	0
$301$	4.38459e28	1.37535
$302$	0	0
$303$	7.52370e28	2.18696
$304$	0	0
$305$	−2.71366e28	−0.731320
$306$	0	0
$307$	3.40762e28	0.851842	0.425921	−	0.904760i	$-0.359950\pi$
$307$	3.40762e28	0.851842	0.425921	+	0.904760i	$0.359950\pi$
$308$	0	0
$309$	−3.39488e28	−0.787593
$310$	0	0
$311$	5.66460e28	1.22018	0.610091	−	0.792331i	$-0.291133\pi$
$311$	5.66460e28	1.22018	0.610091	+	0.792331i	$0.291133\pi$
$312$	0	0
$313$	3.78832e28	0.758031	0.379016	−	0.925390i	$-0.376263\pi$
$313$	3.78832e28	0.758031	0.379016	+	0.925390i	$0.376263\pi$
$314$	0	0
$315$	1.18105e29	2.19632
$316$	0	0
$317$	−3.72097e28	−0.643390	−0.321695	−	0.946843i	$-0.604253\pi$
$317$	−3.72097e28	−0.643390	−0.321695	+	0.946843i	$0.604253\pi$
$318$	0	0
$319$	−5.84250e28	−0.939736
$320$	0	0
$321$	4.46915e28	0.668985
$322$	0	0
$323$	−3.62010e28	−0.504536
$324$	0	0
$325$	3.49306e28	0.453469
$326$	0	0
$327$	3.91424e28	0.473531
$328$	0	0
$329$	9.76990e28	1.10189
$330$	0	0
$331$	5.62405e28	0.591599	0.295799	−	0.955250i	$-0.404414\pi$
$331$	5.62405e28	0.591599	0.295799	+	0.955250i	$0.404414\pi$
$332$	0	0
$333$	1.24453e29	1.22151
$334$	0	0
$335$	−9.24657e28	−0.847160
$336$	0	0
$337$	9.79219e28	0.837792	0.418896	−	0.908034i	$-0.362417\pi$
$337$	9.79219e28	0.837792	0.418896	+	0.908034i	$0.362417\pi$
$338$	0	0
$339$	−2.45435e29	−1.96173
$340$	0	0
$341$	−2.99081e29	−2.23416
$342$	0	0
$343$	1.14663e29	0.800828
$344$	0	0
$345$	1.48877e29	0.972541
$346$	0	0
$347$	−7.68930e28	−0.470000	−0.235000	−	0.971995i	$-0.575509\pi$
$347$	−7.68930e28	−0.470000	−0.235000	+	0.971995i	$0.575509\pi$
$348$	0	0
$349$	3.15740e29	1.80650	0.903249	−	0.429116i	$-0.141175\pi$
$349$	3.15740e29	1.80650	0.903249	+	0.429116i	$0.141175\pi$
$350$	0	0
$351$	−5.86155e29	−3.14037
$352$	0	0
$353$	−2.41820e29	−1.21362	−0.606809	−	0.794847i	$-0.707551\pi$
$353$	−2.41820e29	−1.21362	−0.606809	+	0.794847i	$0.707551\pi$
$354$	0	0
$355$	1.93669e28	0.0910824
$356$	0	0
$357$	2.22687e29	0.981771
$358$	0	0
$359$	2.43466e29	1.00659	0.503294	−	0.864115i	$-0.332121\pi$
$359$	2.43466e29	1.00659	0.503294	+	0.864115i	$0.332121\pi$
$360$	0	0
$361$	5.14606e28	0.199592
$362$	0	0
$363$	3.73410e29	1.35913
$364$	0	0
$365$	−7.01473e28	−0.239686
$366$	0	0
$367$	−2.79274e29	−0.896128	−0.448064	−	0.894001i	$-0.647886\pi$
$367$	−2.79274e29	−0.896128	−0.448064	+	0.894001i	$0.647886\pi$
$368$	0	0
$369$	3.76609e29	1.13524
$370$	0	0
$371$	−1.57328e29	−0.445662
$372$	0	0
$373$	1.37422e29	0.365937	0.182968	−	0.983119i	$-0.441429\pi$
$373$	1.37422e29	0.365937	0.182968	+	0.983119i	$0.441429\pi$
$374$	0	0
$375$	8.09911e29	2.02805
$376$	0	0
$377$	−3.39049e29	−0.798613
$378$	0	0
$379$	−1.24160e29	−0.275189	−0.137594	−	0.990489i	$-0.543937\pi$
$379$	−1.24160e29	−0.275189	−0.137594	+	0.990489i	$0.543937\pi$
$380$	0	0
$381$	−3.90576e29	−0.814832
$382$	0	0
$383$	−3.88270e29	−0.762690	−0.381345	−	0.924433i	$-0.624539\pi$
$383$	−3.88270e29	−0.762690	−0.381345	+	0.924433i	$0.624539\pi$
$384$	0	0
$385$	−6.23181e29	−1.15296
$386$	0	0
$387$	−1.73415e30	−3.02282
$388$	0	0
$389$	1.17387e30	1.92841	0.964203	−	0.265165i	$-0.0854265\pi$
$389$	1.17387e30	1.92841	0.964203	+	0.265165i	$0.0854265\pi$
$390$	0	0
$391$	2.00570e29	0.310623
$392$	0	0
$393$	−3.66761e28	−0.0535634
$394$	0	0
$395$	−2.18755e29	−0.301364
$396$	0	0
$397$	−1.47771e30	−1.92087	−0.960433	−	0.278511i	$-0.910159\pi$
$397$	−1.47771e30	−1.92087	−0.960433	+	0.278511i	$0.910159\pi$
$398$	0	0
$399$	−1.90257e30	−2.33427
$400$	0	0
$401$	−4.26558e29	−0.494103	−0.247052	−	0.969002i	$-0.579462\pi$
$401$	−4.26558e29	−0.494103	−0.247052	+	0.969002i	$0.579462\pi$
$402$	0	0
$403$	−1.73561e30	−1.89865
$404$	0	0
$405$	−2.05906e30	−2.12783
$406$	0	0
$407$	−6.56679e29	−0.641235
$408$	0	0
$409$	−1.42830e30	−1.31826	−0.659129	−	0.752030i	$-0.729075\pi$
$409$	−1.42830e30	−1.31826	−0.659129	+	0.752030i	$0.729075\pi$
$410$	0	0
$411$	1.81515e30	1.58391
$412$	0	0
$413$	−1.93583e30	−1.59750
$414$	0	0
$415$	−8.90255e29	−0.694963
$416$	0	0
$417$	8.71728e29	0.643897
$418$	0	0
$419$	1.76509e30	1.23397	0.616985	−	0.786975i	$-0.288354\pi$
$419$	1.76509e30	1.23397	0.616985	+	0.786975i	$0.288354\pi$
$420$	0	0
$421$	2.02002e30	1.33694	0.668471	−	0.743739i	$-0.266949\pi$
$421$	2.02002e30	1.33694	0.668471	+	0.743739i	$0.266949\pi$
$422$	0	0
$423$	−3.86410e30	−2.42178
$424$	0	0
$425$	3.15153e29	0.187089
$426$	0	0
$427$	−1.92138e30	−1.08067
$428$	0	0
$429$	5.15094e30	2.74554
$430$	0	0
$431$	2.37148e30	1.19820	0.599102	−	0.800673i	$-0.295524\pi$
$431$	2.37148e30	1.19820	0.599102	+	0.800673i	$0.295524\pi$
$432$	0	0
$433$	2.23333e30	1.06990	0.534949	−	0.844884i	$-0.320331\pi$
$433$	2.23333e30	1.06990	0.534949	+	0.844884i	$0.320331\pi$
$434$	0	0
$435$	−2.27059e30	−1.03160
$436$	0	0
$437$	−1.71361e30	−0.738541
$438$	0	0
$439$	3.90666e30	1.59758	0.798792	−	0.601607i	$-0.205473\pi$
$439$	3.90666e30	1.59758	0.798792	+	0.601607i	$0.205473\pi$
$440$	0	0
$441$	1.91362e30	0.742697
$442$	0	0
$443$	3.27739e30	1.20749	0.603747	−	0.797176i	$-0.293674\pi$
$443$	3.27739e30	1.20749	0.603747	+	0.797176i	$0.293674\pi$
$444$	0	0
$445$	−3.42811e30	−1.19927
$446$	0	0
$447$	3.41183e29	0.113358
$448$	0	0
$449$	8.52929e29	0.269203	0.134602	−	0.990900i	$-0.457025\pi$
$449$	8.52929e29	0.269203	0.134602	+	0.990900i	$0.457025\pi$
$450$	0	0
$451$	−1.98718e30	−0.595947
$452$	0	0
$453$	5.95364e30	1.69689
$454$	0	0
$455$	−3.61641e30	−0.979820
$456$	0	0
$457$	7.07801e30	1.82337	0.911685	−	0.410890i	$-0.134782\pi$
$457$	7.07801e30	1.82337	0.911685	+	0.410890i	$0.134782\pi$
$458$	0	0
$459$	−5.28843e30	−1.29563
$460$	0	0
$461$	5.36478e30	1.25023	0.625117	−	0.780531i	$-0.285051\pi$
$461$	5.36478e30	1.25023	0.625117	+	0.780531i	$0.285051\pi$
$462$	0	0
$463$	−2.58238e30	−0.572584	−0.286292	−	0.958142i	$-0.592423\pi$
$463$	−2.58238e30	−0.572584	−0.286292	+	0.958142i	$0.592423\pi$
$464$	0	0
$465$	−1.16233e31	−2.45255
$466$	0	0
$467$	3.99016e30	0.801394	0.400697	−	0.916211i	$-0.368768\pi$
$467$	3.99016e30	0.801394	0.400697	+	0.916211i	$0.368768\pi$
$468$	0	0
$469$	−6.54693e30	−1.25184
$470$	0	0
$471$	7.33016e30	1.33467
$472$	0	0
$473$	9.15029e30	1.58684
$474$	0	0
$475$	−2.69256e30	−0.444825
$476$	0	0
$477$	6.22248e30	0.979495
$478$	0	0
$479$	5.56629e30	0.835041	0.417520	−	0.908668i	$-0.362899\pi$
$479$	5.56629e30	0.835041	0.417520	+	0.908668i	$0.362899\pi$
$480$	0	0
$481$	−3.81080e30	−0.544939
$482$	0	0
$483$	1.05411e31	1.43712
$484$	0	0
$485$	7.90932e30	1.02827
$486$	0	0
$487$	−3.57505e30	−0.443301	−0.221651	−	0.975126i	$-0.571144\pi$
$487$	−3.57505e30	−0.443301	−0.221651	+	0.975126i	$0.571144\pi$
$488$	0	0
$489$	−2.46317e31	−2.91368
$490$	0	0
$491$	6.34262e30	0.715865	0.357933	−	0.933747i	$-0.383482\pi$
$491$	6.34262e30	0.715865	0.357933	+	0.933747i	$0.383482\pi$
$492$	0	0
$493$	−3.05898e30	−0.329486
$494$	0	0
$495$	2.46475e31	2.53403
$496$	0	0
$497$	1.37125e30	0.134592
$498$	0	0
$499$	3.30165e30	0.309439	0.154720	−	0.987958i	$-0.450553\pi$
$499$	3.30165e30	0.309439	0.154720	+	0.987958i	$0.450553\pi$
$500$	0	0
$501$	−3.87087e31	−3.46477
$502$	0	0
$503$	−2.32758e31	−1.99009	−0.995044	−	0.0994338i	$-0.968297\pi$
$503$	−2.32758e31	−1.99009	−0.995044	+	0.0994338i	$0.968297\pi$
$504$	0	0
$505$	−1.10237e31	−0.900487
$506$	0	0
$507$	5.91440e30	0.461657
$508$	0	0
$509$	−2.41190e31	−1.79930	−0.899652	−	0.436607i	$-0.856180\pi$
$509$	−2.41190e31	−1.79930	−0.899652	+	0.436607i	$0.856180\pi$
$510$	0	0
$511$	−4.96670e30	−0.354183
$512$	0	0
$513$	4.51827e31	3.08051
$514$	0	0
$515$	4.97418e30	0.324293
$516$	0	0
$517$	2.03890e31	1.27132
$518$	0	0
$519$	3.06562e31	1.82850
$520$	0	0
$521$	6.94775e30	0.396470	0.198235	−	0.980155i	$-0.436479\pi$
$521$	6.94775e30	0.396470	0.198235	+	0.980155i	$0.436479\pi$
$522$	0	0
$523$	−2.69454e31	−1.47135	−0.735673	−	0.677337i	$-0.763134\pi$
$523$	−2.69454e31	−1.47135	−0.735673	+	0.677337i	$0.763134\pi$
$524$	0	0
$525$	1.65630e31	0.865580
$526$	0	0
$527$	−1.56591e31	−0.783328
$528$	0	0
$529$	−1.13863e31	−0.545310
$530$	0	0
$531$	7.65641e31	3.51106
$532$	0	0
$533$	−1.15319e31	−0.506452
$534$	0	0
$535$	−6.54819e30	−0.275456
$536$	0	0
$537$	−3.31204e31	−1.33472
$538$	0	0
$539$	−1.00972e31	−0.389881
$540$	0	0
$541$	−7.63627e30	−0.282561	−0.141281	−	0.989970i	$-0.545122\pi$
$541$	−7.63627e30	−0.282561	−0.141281	+	0.989970i	$0.545122\pi$
$542$	0	0
$543$	7.55667e31	2.67999
$544$	0	0
$545$	−5.73515e30	−0.194978
$546$	0	0
$547$	2.26514e31	0.738312	0.369156	−	0.929367i	$-0.379647\pi$
$547$	2.26514e31	0.738312	0.369156	+	0.929367i	$0.379647\pi$
$548$	0	0
$549$	7.59925e31	2.37514
$550$	0	0
$551$	2.61349e31	0.783390
$552$	0	0
$553$	−1.54887e31	−0.445324
$554$	0	0
$555$	−2.55207e31	−0.703919
$556$	0	0
$557$	−1.58983e31	−0.420741	−0.210371	−	0.977622i	$-0.567467\pi$
$557$	−1.58983e31	−0.420741	−0.210371	+	0.977622i	$0.567467\pi$
$558$	0	0
$559$	5.31005e31	1.34854
$560$	0	0
$561$	4.64730e31	1.13273
$562$	0	0
$563$	7.95857e31	1.86204	0.931019	−	0.364972i	$-0.118921\pi$
$563$	7.95857e31	1.86204	0.931019	+	0.364972i	$0.118921\pi$
$564$	0	0
$565$	3.59611e31	0.807748
$566$	0	0
$567$	−1.45789e32	−3.14428
$568$	0	0
$569$	7.26713e31	1.50512	0.752561	−	0.658522i	$-0.228818\pi$
$569$	7.26713e31	1.50512	0.752561	+	0.658522i	$0.228818\pi$
$570$	0	0
$571$	−3.78373e31	−0.752670	−0.376335	−	0.926484i	$-0.622816\pi$
$571$	−3.78373e31	−0.752670	−0.376335	+	0.926484i	$0.622816\pi$
$572$	0	0
$573$	2.33971e31	0.447078
$574$	0	0
$575$	1.49180e31	0.273861
$576$	0	0
$577$	−6.81473e31	−1.20206	−0.601031	−	0.799225i	$-0.705243\pi$
$577$	−6.81473e31	−1.20206	−0.601031	+	0.799225i	$0.705243\pi$
$578$	0	0
$579$	−1.72840e32	−2.92982
$580$	0	0
$581$	−6.30335e31	−1.02694
$582$	0	0
$583$	−3.28330e31	−0.514189
$584$	0	0
$585$	1.43033e32	2.15349
$586$	0	0
$587$	−1.10842e32	−1.60459	−0.802296	−	0.596927i	$-0.796388\pi$
$587$	−1.10842e32	−1.60459	−0.802296	+	0.596927i	$0.796388\pi$
$588$	0	0
$589$	1.33786e32	1.86245
$590$	0	0
$591$	2.39706e31	0.320939
$592$	0	0
$593$	1.12848e31	0.145333	0.0726666	−	0.997356i	$-0.476849\pi$
$593$	1.12848e31	0.145333	0.0726666	+	0.997356i	$0.476849\pi$
$594$	0	0
$595$	−3.26281e31	−0.404246
$596$	0	0
$597$	−1.79748e31	−0.214269
$598$	0	0
$599$	7.82635e31	0.897741	0.448870	−	0.893597i	$-0.351826\pi$
$599$	7.82635e31	0.897741	0.448870	+	0.893597i	$0.351826\pi$
$600$	0	0
$601$	−8.73347e31	−0.964119	−0.482059	−	0.876139i	$-0.660111\pi$
$601$	−8.73347e31	−0.964119	−0.482059	+	0.876139i	$0.660111\pi$
$602$	0	0
$603$	2.58938e32	2.75136
$604$	0	0
$605$	−5.47120e31	−0.559623
$606$	0	0
$607$	−9.31517e31	−0.917320	−0.458660	−	0.888612i	$-0.651670\pi$
$607$	−9.31517e31	−0.917320	−0.458660	+	0.888612i	$0.651670\pi$
$608$	0	0
$609$	−1.60766e32	−1.52439
$610$	0	0
$611$	1.18320e32	1.08040
$612$	0	0
$613$	−1.93566e32	−1.70229	−0.851145	−	0.524930i	$-0.824091\pi$
$613$	−1.93566e32	−1.70229	−0.851145	+	0.524930i	$0.824091\pi$
$614$	0	0
$615$	−7.72284e31	−0.654204
$616$	0	0
$617$	−7.76155e31	−0.633387	−0.316694	−	0.948528i	$-0.602573\pi$
$617$	−7.76155e31	−0.633387	−0.316694	+	0.948528i	$0.602573\pi$
$618$	0	0
$619$	1.48903e32	1.17074	0.585372	−	0.810765i	$-0.300949\pi$
$619$	1.48903e32	1.17074	0.585372	+	0.810765i	$0.300949\pi$
$620$	0	0
$621$	−2.50332e32	−1.89655
$622$	0	0
$623$	−2.42724e32	−1.77215
$624$	0	0
$625$	−6.09526e31	−0.428916
$626$	0	0
$627$	−3.97050e32	−2.69320
$628$	0	0
$629$	−3.43819e31	−0.224827
$630$	0	0
$631$	2.00939e31	0.126685	0.0633426	−	0.997992i	$-0.479824\pi$
$631$	2.00939e31	0.126685	0.0633426	+	0.997992i	$0.479824\pi$
$632$	0	0
$633$	1.36212e32	0.828080
$634$	0	0
$635$	5.72271e31	0.335509
$636$	0	0
$637$	−5.85957e31	−0.331331
$638$	0	0
$639$	−5.42345e31	−0.295812
$640$	0	0
$641$	2.52315e31	0.132762	0.0663812	−	0.997794i	$-0.478855\pi$
$641$	2.52315e31	0.132762	0.0663812	+	0.997794i	$0.478855\pi$
$642$	0	0
$643$	−7.18001e31	−0.364501	−0.182251	−	0.983252i	$-0.558338\pi$
$643$	−7.18001e31	−0.364501	−0.182251	+	0.983252i	$0.558338\pi$
$644$	0	0
$645$	3.55610e32	1.74196
$646$	0	0
$647$	−1.17766e32	−0.556700	−0.278350	−	0.960480i	$-0.589788\pi$
$647$	−1.17766e32	−0.556700	−0.278350	+	0.960480i	$0.589788\pi$
$648$	0	0
$649$	−4.03992e32	−1.84314
$650$	0	0
$651$	−8.22971e32	−3.62413
$652$	0	0
$653$	−2.88510e32	−1.22648	−0.613238	−	0.789898i	$-0.710133\pi$
$653$	−2.88510e32	−1.22648	−0.613238	+	0.789898i	$0.710133\pi$
$654$	0	0
$655$	5.37377e30	0.0220549
$656$	0	0
$657$	1.96438e32	0.778439
$658$	0	0
$659$	1.55020e32	0.593208	0.296604	−	0.955001i	$-0.404146\pi$
$659$	1.55020e32	0.593208	0.296604	+	0.955001i	$0.404146\pi$
$660$	0	0
$661$	2.65192e32	0.980039	0.490020	−	0.871711i	$-0.336990\pi$
$661$	2.65192e32	0.980039	0.490020	+	0.871711i	$0.336990\pi$
$662$	0	0
$663$	2.69689e32	0.962627
$664$	0	0
$665$	2.78764e32	0.961142
$666$	0	0
$667$	−1.44799e32	−0.482302
$668$	0	0
$669$	8.58111e32	2.76149
$670$	0	0
$671$	−4.00976e32	−1.24683
$672$	0	0
$673$	5.25147e32	1.57800	0.789001	−	0.614392i	$-0.210599\pi$
$673$	5.25147e32	1.57800	0.789001	+	0.614392i	$0.210599\pi$
$674$	0	0
$675$	−3.93343e32	−1.14229
$676$	0	0
$677$	−4.13293e32	−1.16008	−0.580041	−	0.814587i	$-0.696963\pi$
$677$	−4.13293e32	−1.16008	−0.580041	+	0.814587i	$0.696963\pi$
$678$	0	0
$679$	5.60011e32	1.51948
$680$	0	0
$681$	−2.35276e32	−0.617143
$682$	0	0
$683$	5.82971e32	1.47846	0.739229	−	0.673454i	$-0.235190\pi$
$683$	5.82971e32	1.47846	0.739229	+	0.673454i	$0.235190\pi$
$684$	0	0
$685$	−2.65956e32	−0.652180
$686$	0	0
$687$	−9.03114e32	−2.14161
$688$	0	0
$689$	−1.90535e32	−0.436971
$690$	0	0
$691$	7.25689e30	0.0160973	0.00804865	−	0.999968i	$-0.497438\pi$
$691$	7.25689e30	0.0160973	0.00804865	+	0.999968i	$0.497438\pi$
$692$	0	0
$693$	1.74514e33	3.74453
$694$	0	0
$695$	−1.27725e32	−0.265126
$696$	0	0
$697$	−1.04044e32	−0.208948
$698$	0	0
$699$	1.41730e33	2.75407
$700$	0	0
$701$	−8.94327e32	−1.68166	−0.840832	−	0.541297i	$-0.817934\pi$
$701$	−8.94327e32	−1.68166	−0.840832	+	0.541297i	$0.817934\pi$
$702$	0	0
$703$	2.93748e32	0.534551
$704$	0	0
$705$	7.92382e32	1.39560
$706$	0	0
$707$	−7.80522e32	−1.33064
$708$	0	0
$709$	−1.06809e33	−1.76269	−0.881346	−	0.472472i	$-0.843362\pi$
$709$	−1.06809e33	−1.76269	−0.881346	+	0.472472i	$0.843362\pi$
$710$	0	0
$711$	6.12594e32	0.978752
$712$	0	0
$713$	−7.41235e32	−1.14664
$714$	0	0
$715$	−7.54715e32	−1.13048
$716$	0	0
$717$	6.62461e32	0.960927
$718$	0	0
$719$	−1.23737e32	−0.173827	−0.0869133	−	0.996216i	$-0.527700\pi$
$719$	−1.23737e32	−0.173827	−0.0869133	+	0.996216i	$0.527700\pi$
$720$	0	0
$721$	3.52191e32	0.479206
$722$	0	0
$723$	−3.66781e32	−0.483411
$724$	0	0
$725$	−2.27521e32	−0.290492
$726$	0	0
$727$	1.92923e32	0.238637	0.119319	−	0.992856i	$-0.461929\pi$
$727$	1.92923e32	0.238637	0.119319	+	0.992856i	$0.461929\pi$
$728$	0	0
$729$	1.37396e33	1.64667
$730$	0	0
$731$	4.79085e32	0.556368
$732$	0	0
$733$	−5.14954e31	−0.0579526	−0.0289763	−	0.999580i	$-0.509225\pi$
$733$	−5.14954e31	−0.0579526	−0.0289763	+	0.999580i	$0.509225\pi$
$734$	0	0
$735$	−3.92411e32	−0.427993
$736$	0	0
$737$	−1.36629e33	−1.44433
$738$	0	0
$739$	1.78518e33	1.82924	0.914621	−	0.404312i	$-0.132489\pi$
$739$	1.78518e33	1.82924	0.914621	+	0.404312i	$0.132489\pi$
$740$	0	0
$741$	−2.30414e33	−2.28875
$742$	0	0
$743$	1.03301e32	0.0994792	0.0497396	−	0.998762i	$-0.484161\pi$
$743$	1.03301e32	0.0994792	0.0497396	+	0.998762i	$0.484161\pi$
$744$	0	0
$745$	−4.99901e31	−0.0466752
$746$	0	0
$747$	2.49304e33	2.25706
$748$	0	0
$749$	−4.63637e32	−0.407040
$750$	0	0
$751$	2.00354e33	1.70584	0.852922	−	0.522038i	$-0.174828\pi$
$751$	2.00354e33	1.70584	0.852922	+	0.522038i	$0.174828\pi$
$752$	0	0
$753$	−3.36479e32	−0.277853
$754$	0	0
$755$	−8.72327e32	−0.698698
$756$	0	0
$757$	−2.90156e32	−0.225439	−0.112719	−	0.993627i	$-0.535956\pi$
$757$	−2.90156e32	−0.225439	−0.112719	+	0.993627i	$0.535956\pi$
$758$	0	0
$759$	2.19984e33	1.65810
$760$	0	0
$761$	−1.59845e33	−1.16889	−0.584446	−	0.811433i	$-0.698688\pi$
$761$	−1.59845e33	−1.16889	−0.584446	+	0.811433i	$0.698688\pi$
$762$	0	0
$763$	−4.06071e32	−0.288117
$764$	0	0
$765$	1.29047e33	0.888471
$766$	0	0
$767$	−2.34442e33	−1.56635
$768$	0	0
$769$	−8.79322e32	−0.570158	−0.285079	−	0.958504i	$-0.592020\pi$
$769$	−8.79322e32	−0.570158	−0.285079	+	0.958504i	$0.592020\pi$
$770$	0	0
$771$	−3.03701e32	−0.191126
$772$	0	0
$773$	8.24155e32	0.503437	0.251718	−	0.967801i	$-0.419004\pi$
$773$	8.24155e32	0.503437	0.251718	+	0.967801i	$0.419004\pi$
$774$	0	0
$775$	−1.16469e33	−0.690623
$776$	0	0
$777$	−1.80696e33	−1.04018
$778$	0	0
$779$	8.88915e32	0.496798
$780$	0	0
$781$	2.86169e32	0.155287
$782$	0	0
$783$	3.81792e33	2.01172
$784$	0	0
$785$	−1.07401e33	−0.549552
$786$	0	0
$787$	1.89467e31	0.00941509	0.00470755	−	0.999989i	$-0.498502\pi$
$787$	1.89467e31	0.00941509	0.00470755	+	0.999989i	$0.498502\pi$
$788$	0	0
$789$	3.21187e33	1.55015
$790$	0	0
$791$	2.54619e33	1.19360
$792$	0	0
$793$	−2.32692e33	−1.05959
$794$	0	0
$795$	−1.27600e33	−0.564453
$796$	0	0
$797$	5.78084e32	0.248439	0.124220	−	0.992255i	$-0.460357\pi$
$797$	5.78084e32	0.248439	0.124220	+	0.992255i	$0.460357\pi$
$798$	0	0
$799$	1.06751e33	0.445744
$800$	0	0
$801$	9.59998e33	3.89490
$802$	0	0
$803$	−1.03651e33	−0.408644
$804$	0	0
$805$	−1.54448e33	−0.591737
$806$	0	0
$807$	−6.41706e33	−2.38941
$808$	0	0
$809$	4.46172e33	1.61471	0.807355	−	0.590066i	$-0.200898\pi$
$809$	4.46172e33	1.61471	0.807355	+	0.590066i	$0.200898\pi$
$810$	0	0
$811$	−2.69737e33	−0.948857	−0.474429	−	0.880294i	$-0.657345\pi$
$811$	−2.69737e33	−0.948857	−0.474429	+	0.880294i	$0.657345\pi$
$812$	0	0
$813$	−6.57999e33	−2.25001
$814$	0	0
$815$	3.60903e33	1.19971
$816$	0	0
$817$	−4.09315e33	−1.32283
$818$	0	0
$819$	1.01273e34	3.18220
$820$	0	0
$821$	3.28224e33	1.00282	0.501412	−	0.865209i	$-0.332814\pi$
$821$	3.28224e33	1.00282	0.501412	+	0.865209i	$0.332814\pi$
$822$	0	0
$823$	−2.64872e33	−0.786933	−0.393466	−	0.919339i	$-0.628724\pi$
$823$	−2.64872e33	−0.786933	−0.393466	+	0.919339i	$0.628724\pi$
$824$	0	0
$825$	3.45657e33	0.998676
$826$	0	0
$827$	−1.65312e33	−0.464505	−0.232253	−	0.972655i	$-0.574610\pi$
$827$	−1.65312e33	−0.464505	−0.232253	+	0.972655i	$0.574610\pi$
$828$	0	0
$829$	−3.48612e33	−0.952719	−0.476360	−	0.879251i	$-0.658044\pi$
$829$	−3.48612e33	−0.952719	−0.476360	+	0.879251i	$0.658044\pi$
$830$	0	0
$831$	5.98913e33	1.59203
$832$	0	0
$833$	−5.28664e32	−0.136698
$834$	0	0
$835$	5.67160e33	1.42663
$836$	0	0
$837$	1.95442e34	4.78271
$838$	0	0
$839$	4.22678e33	1.00634	0.503172	−	0.864186i	$-0.332166\pi$
$839$	4.22678e33	1.00634	0.503172	+	0.864186i	$0.332166\pi$
$840$	0	0
$841$	−2.10833e33	−0.488410
$842$	0	0
$843$	5.45745e31	0.0123019
$844$	0	0
$845$	−8.66577e32	−0.190088
$846$	0	0
$847$	−3.87382e33	−0.826952
$848$	0	0
$849$	1.26191e34	2.62175
$850$	0	0
$851$	−1.62750e33	−0.329102
$852$	0	0
$853$	−4.60865e33	−0.907111	−0.453555	−	0.891228i	$-0.649845\pi$
$853$	−4.60865e33	−0.907111	−0.453555	+	0.891228i	$0.649845\pi$
$854$	0	0
$855$	−1.10254e34	−2.11244
$856$	0	0
$857$	−1.09714e33	−0.204637	−0.102318	−	0.994752i	$-0.532626\pi$
$857$	−1.09714e33	−0.204637	−0.102318	+	0.994752i	$0.532626\pi$
$858$	0	0
$859$	−2.95181e33	−0.536003	−0.268001	−	0.963419i	$-0.586363\pi$
$859$	−2.95181e33	−0.536003	−0.268001	+	0.963419i	$0.586363\pi$
$860$	0	0
$861$	−5.46807e33	−0.966713
$862$	0	0
$863$	−1.04670e33	−0.180177	−0.0900884	−	0.995934i	$-0.528715\pi$
$863$	−1.04670e33	−0.180177	−0.0900884	+	0.995934i	$0.528715\pi$
$864$	0	0
$865$	−4.49175e33	−0.752888
$866$	0	0
$867$	−9.03319e33	−1.47442
$868$	0	0
$869$	−3.23237e33	−0.513799
$870$	0	0
$871$	−7.92878e33	−1.22743
$872$	0	0
$873$	−2.21490e34	−3.33957
$874$	0	0
$875$	−8.40216e33	−1.23395
$876$	0	0
$877$	−3.86117e33	−0.552362	−0.276181	−	0.961106i	$-0.589069\pi$
$877$	−3.86117e33	−0.552362	−0.276181	+	0.961106i	$0.589069\pi$
$878$	0	0
$879$	−1.31939e34	−1.83865
$880$	0	0
$881$	−7.78654e33	−1.05711	−0.528557	−	0.848898i	$-0.677267\pi$
$881$	−7.78654e33	−1.05711	−0.528557	+	0.848898i	$0.677267\pi$
$882$	0	0
$883$	4.55914e33	0.603024	0.301512	−	0.953462i	$-0.402509\pi$
$883$	4.55914e33	0.603024	0.301512	+	0.953462i	$0.402509\pi$
$884$	0	0
$885$	−1.57004e34	−2.02332
$886$	0	0
$887$	3.58889e32	0.0450649	0.0225325	−	0.999746i	$-0.492827\pi$
$887$	3.58889e32	0.0450649	0.0225325	+	0.999746i	$0.492827\pi$
$888$	0	0
$889$	4.05190e33	0.495780
$890$	0	0
$891$	−3.04251e34	−3.62776
$892$	0	0
$893$	−9.12049e33	−1.05981
$894$	0	0
$895$	4.85280e33	0.549576
$896$	0	0
$897$	1.27660e34	1.40909
$898$	0	0
$899$	1.13049e34	1.21627
$900$	0	0
$901$	−1.71905e33	−0.180282
$902$	0	0
$903$	2.51786e34	2.57408
$904$	0	0
$905$	−1.10720e34	−1.10349
$906$	0	0
$907$	1.09184e34	1.06090	0.530450	−	0.847716i	$-0.322023\pi$
$907$	1.09184e34	1.06090	0.530450	+	0.847716i	$0.322023\pi$
$908$	0	0
$909$	3.08705e34	2.92455
$910$	0	0
$911$	1.42933e34	1.32029	0.660146	−	0.751137i	$-0.270494\pi$
$911$	1.42933e34	1.32029	0.660146	+	0.751137i	$0.270494\pi$
$912$	0	0
$913$	−1.31546e34	−1.18485
$914$	0	0
$915$	−1.55832e34	−1.36872
$916$	0	0
$917$	3.80484e32	0.0325904
$918$	0	0
$919$	−2.15002e34	−1.79604	−0.898018	−	0.439959i	$-0.854993\pi$
$919$	−2.15002e34	−1.79604	−0.898018	+	0.439959i	$0.854993\pi$
$920$	0	0
$921$	1.95683e34	1.59429
$922$	0	0
$923$	1.66068e33	0.131967
$924$	0	0
$925$	−2.55726e33	−0.198219
$926$	0	0
$927$	−1.39295e34	−1.05322
$928$	0	0
$929$	−1.53709e34	−1.13376	−0.566878	−	0.823801i	$-0.691849\pi$
$929$	−1.53709e34	−1.13376	−0.566878	+	0.823801i	$0.691849\pi$
$930$	0	0
$931$	4.51674e33	0.325015
$932$	0	0
$933$	3.25290e34	2.28366
$934$	0	0
$935$	−6.80921e33	−0.466405
$936$	0	0
$937$	−3.02985e33	−0.202496	−0.101248	−	0.994861i	$-0.532284\pi$
$937$	−3.02985e33	−0.202496	−0.101248	+	0.994861i	$0.532284\pi$
$938$	0	0
$939$	2.17545e34	1.41871
$940$	0	0
$941$	−1.40737e34	−0.895631	−0.447815	−	0.894126i	$-0.647798\pi$
$941$	−1.40737e34	−0.895631	−0.447815	+	0.894126i	$0.647798\pi$
$942$	0	0
$943$	−4.92499e33	−0.305859
$944$	0	0
$945$	4.07232e34	2.46818
$946$	0	0
$947$	−2.84567e34	−1.68329	−0.841647	−	0.540028i	$-0.818414\pi$
$947$	−2.84567e34	−1.68329	−0.841647	+	0.540028i	$0.818414\pi$
$948$	0	0
$949$	−6.01502e33	−0.347276
$950$	0	0
$951$	−2.13677e34	−1.20415
$952$	0	0
$953$	−3.91513e33	−0.215366	−0.107683	−	0.994185i	$-0.534343\pi$
$953$	−3.91513e33	−0.215366	−0.107683	+	0.994185i	$0.534343\pi$
$954$	0	0
$955$	−3.42814e33	−0.184085
$956$	0	0
$957$	−3.35506e34	−1.75879
$958$	0	0
$959$	−1.88307e34	−0.963723
$960$	0	0
$961$	3.78570e34	1.89159
$962$	0	0
$963$	1.83373e34	0.894611
$964$	0	0
$965$	2.53245e34	1.20636
$966$	0	0
$967$	−1.18860e34	−0.552882	−0.276441	−	0.961031i	$-0.589155\pi$
$967$	−1.18860e34	−0.552882	−0.276441	+	0.961031i	$0.589155\pi$
$968$	0	0
$969$	−2.07885e34	−0.944277
$970$	0	0
$971$	2.66114e33	0.118045	0.0590224	−	0.998257i	$-0.481202\pi$
$971$	2.66114e33	0.118045	0.0590224	+	0.998257i	$0.481202\pi$
$972$	0	0
$973$	−9.04346e33	−0.391775
$974$	0	0
$975$	2.00589e34	0.848702
$976$	0	0
$977$	1.53142e34	0.632860	0.316430	−	0.948616i	$-0.397516\pi$
$977$	1.53142e34	0.632860	0.316430	+	0.948616i	$0.397516\pi$
$978$	0	0
$979$	−5.06545e34	−2.04464
$980$	0	0
$981$	1.60605e34	0.633237
$982$	0	0
$983$	1.82163e34	0.701610	0.350805	−	0.936449i	$-0.385908\pi$
$983$	1.82163e34	0.701610	0.350805	+	0.936449i	$0.385908\pi$
$984$	0	0
$985$	−3.51217e33	−0.132147
$986$	0	0
$987$	5.61037e34	2.06227
$988$	0	0
$989$	2.26779e34	0.814413
$990$	0	0
$991$	4.71132e34	1.65309	0.826543	−	0.562874i	$-0.190304\pi$
$991$	4.71132e34	1.65309	0.826543	+	0.562874i	$0.190304\pi$
$992$	0	0
$993$	3.22961e34	1.10722
$994$	0	0
$995$	2.63367e33	0.0882257
$996$	0	0
$997$	2.34856e34	0.768791	0.384395	−	0.923169i	$-0.374410\pi$
$997$	2.34856e34	0.768791	0.384395	+	0.923169i	$0.374410\pi$
$998$	0	0
$999$	4.29122e34	1.37271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.k.1.3		3
4.3	odd	2		64.24.a.h.1.1		3
8.3	odd	2		16.24.a.e.1.3		3
8.5	even	2		8.24.a.a.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.24.a.a.1.1	✓	3	8.5	even	2
16.24.a.e.1.3		3	8.3	odd	2
64.24.a.h.1.1		3	4.3	odd	2
64.24.a.k.1.3		3	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)

\(f(q)\)	\(=\)	\(q+574251. q^{3} -8.41392e7 q^{5} -5.95738e9 q^{7} +2.35621e11 q^{9} +O(q^{10})\)	\(q+574251. q^{3} -8.41392e7 q^{5} -5.95738e9 q^{7} +2.35621e11 q^{9} -1.24326e12 q^{11} -7.21479e12 q^{13} -4.83170e13 q^{15} -6.50936e13 q^{17} +5.56139e14 q^{19} -3.42103e15 q^{21} -3.08126e15 q^{23} -4.84153e15 q^{25} +8.12435e16 q^{27} +4.69935e16 q^{29} +2.40563e17 q^{31} -7.13941e17 q^{33} +5.01249e17 q^{35} +5.28193e17 q^{37} -4.14310e18 q^{39} +1.59837e18 q^{41} -7.35994e18 q^{43} -1.98249e19 q^{45} -1.63997e19 q^{47} +8.12160e18 q^{49} -3.73800e19 q^{51} +2.64089e19 q^{53} +1.04607e20 q^{55} +3.19363e20 q^{57} +3.24947e20 q^{59} +3.22520e20 q^{61} -1.40368e21 q^{63} +6.07047e20 q^{65} +1.09896e21 q^{67} -1.76941e21 q^{69} -2.30177e20 q^{71} +8.33706e20 q^{73} -2.78025e21 q^{75} +7.40655e21 q^{77} +2.59992e21 q^{79} +2.44721e22 q^{81} +1.05807e22 q^{83} +5.47692e21 q^{85} +2.69861e22 q^{87} +4.07434e22 q^{89} +4.29812e22 q^{91} +1.38143e23 q^{93} -4.67930e22 q^{95} -9.40029e22 q^{97} -2.92937e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 213948 q^{3} - 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9}+O(q^{10}) \) 3 * q + 213948 * q^3 - 95628618 * q^5 - 8647912920 * q^7 + 174509823951 * q^9	\( 3 q + 213948 q^{3} - 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9} - 35420906796 q^{11} - 3164858452338 q^{13} - 19825526344392 q^{15} - 30233487828906 q^{17} + 382754784400236 q^{19} - 27788918984928 q^{21} - 37\!\cdots\!20 q^{23}+ \cdots - 19\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q + 213948 * q^3 - 95628618 * q^5 - 8647912920 * q^7 + 174509823951 * q^9 - 35420906796 * q^11 - 3164858452338 * q^13 - 19825526344392 * q^15 - 30233487828906 * q^17 + 382754784400236 * q^19 - 27788918984928 * q^21 - 3754416434163720 * q^23 - 17405885896549371 * q^25 + 103738001159574936 * q^27 + 142892073311612862 * q^29 - 204189369040807008 * q^31 - 1301083012339287408 * q^33 + 1750886716718912208 * q^35 + 900641094521542422 * q^37 - 5338477210392936936 * q^39 - 1818448729363485042 * q^41 - 9021528635440809420 * q^43 - 28993631105890916082 * q^45 + 2644177465669087344 * q^47 + 92864473014347142747 * q^49 - 57094226780716436040 * q^51 - 127243616001408711354 * q^53 - 59143191447597143832 * q^55 + 390032141715811959408 * q^57 + 747107029364442476772 * q^59 + 141236260775173151262 * q^61 - 2369424210278793761592 * q^63 + 386137575836874807036 * q^65 + 3440608449139647701628 * q^67 - 2710328787781605291168 * q^69 - 4012482808005984420504 * q^71 + 4795371647558134022190 * q^73 - 820337660084156915484 * q^75 - 11480348485477300303776 * q^77 + 11708176751738041997328 * q^79 + 22449958139492866882107 * q^81 + 523281952489285038636 * q^83 - 423855162932232129780 * q^85 + 23077946037842683096536 * q^87 - 6011462112251221873026 * q^89 + 15776090363016749995536 * q^91 + 212279712155303705884800 * q^93 - 29064029058982820644968 * q^95 - 233786084347643851364634 * q^97 - 196693345233178755399900 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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