LMFDB - Embedded newform 64.24.a.k.1.2

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			1.2
Root			$-65.1225$ 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)

$f(q)$	$=$	$q-3705.18 q^{3} +6.91268e7 q^{5} +6.89672e9 q^{7} -9.41295e10 q^{9} +O(q^{10})$	$q-3705.18 q^{3} +6.91268e7 q^{5} +6.89672e9 q^{7} -9.41295e10 q^{9} -4.43283e11 q^{11} +7.05085e11 q^{13} -2.56127e11 q^{15} -2.06386e13 q^{17} +2.50525e13 q^{19} -2.55536e13 q^{21} -3.34652e15 q^{23} -7.14241e15 q^{25} +6.97583e14 q^{27} +8.58309e16 q^{29} -2.39339e17 q^{31} +1.64244e15 q^{33} +4.76749e17 q^{35} +2.03171e18 q^{37} -2.61247e15 q^{39} +2.96442e18 q^{41} +4.96274e18 q^{43} -6.50687e18 q^{45} +1.94347e19 q^{47} +2.01961e19 q^{49} +7.64695e16 q^{51} -9.94325e19 q^{53} -3.06427e19 q^{55} -9.28240e16 q^{57} +2.50005e20 q^{59} -1.62386e20 q^{61} -6.49185e20 q^{63} +4.87403e19 q^{65} +1.42919e21 q^{67} +1.23995e19 q^{69} -6.49160e20 q^{71} +3.52414e21 q^{73} +2.64639e19 q^{75} -3.05720e21 q^{77} -2.14128e21 q^{79} +8.85906e21 q^{81} -1.39600e22 q^{83} -1.42668e21 q^{85} -3.18019e20 q^{87} -3.87945e22 q^{89} +4.86278e21 q^{91} +8.86793e20 q^{93} +1.73180e21 q^{95} -7.34335e22 q^{97} +4.17259e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3705.18	−0.0120758	−0.00603788	−	0.999982i	$-0.501922\pi$
$3$	−3705.18	−0.0120758	−0.00603788	+	0.999982i	$0.501922\pi$
$4$	0	0
$5$	6.91268e7	0.633128	0.316564	−	0.948571i	$-0.397471\pi$
$5$	6.91268e7	0.633128	0.316564	+	0.948571i	$0.397471\pi$
$6$	0	0
$7$	6.89672e9	1.31830	0.659152	−	0.752010i	$-0.270915\pi$
$7$	6.89672e9	1.31830	0.659152	+	0.752010i	$0.270915\pi$
$8$	0	0
$9$	−9.41295e10	−0.999854
$10$	0	0
$11$	−4.43283e11	−0.468452	−0.234226	−	0.972182i	$-0.575256\pi$
$11$	−4.43283e11	−0.468452	−0.234226	+	0.972182i	$0.575256\pi$
$12$	0	0
$13$	7.05085e11	0.109117	0.0545586	−	0.998511i	$-0.482625\pi$
$13$	7.05085e11	0.109117	0.0545586	+	0.998511i	$0.482625\pi$
$14$	0	0
$15$	−2.56127e11	−0.00764550
$16$	0	0
$17$	−2.06386e13	−0.146055	−0.0730276	−	0.997330i	$-0.523266\pi$
$17$	−2.06386e13	−0.146055	−0.0730276	+	0.997330i	$0.523266\pi$
$18$	0	0
$19$	2.50525e13	0.0493384	0.0246692	−	0.999696i	$-0.492147\pi$
$19$	2.50525e13	0.0493384	0.0246692	+	0.999696i	$0.492147\pi$
$20$	0	0
$21$	−2.55536e13	−0.0159195
$22$	0	0
$23$	−3.34652e15	−0.732358	−0.366179	−	0.930544i	$-0.619334\pi$
$23$	−3.34652e15	−0.732358	−0.366179	+	0.930544i	$0.619334\pi$
$24$	0	0
$25$	−7.14241e15	−0.599149
$26$	0	0
$27$	6.97583e14	0.0241498
$28$	0	0
$29$	8.58309e16	1.30637	0.653186	−	0.757198i	$-0.273432\pi$
$29$	8.58309e16	1.30637	0.653186	+	0.757198i	$0.273432\pi$
$30$	0	0
$31$	−2.39339e17	−1.69182	−0.845910	−	0.533326i	$-0.820942\pi$
$31$	−2.39339e17	−1.69182	−0.845910	+	0.533326i	$0.820942\pi$
$32$	0	0
$33$	1.64244e15	0.00565691
$34$	0	0
$35$	4.76749e17	0.834655
$36$	0	0
$37$	2.03171e18	1.87734	0.938670	−	0.344817i	$-0.112059\pi$
$37$	2.03171e18	1.87734	0.938670	+	0.344817i	$0.112059\pi$
$38$	0	0
$39$	−2.61247e15	−0.00131767
$40$	0	0
$41$	2.96442e18	0.841251	0.420626	−	0.907234i	$-0.361811\pi$
$41$	2.96442e18	0.841251	0.420626	+	0.907234i	$0.361811\pi$
$42$	0	0
$43$	4.96274e18	0.814393	0.407197	−	0.913341i	$-0.366506\pi$
$43$	4.96274e18	0.814393	0.407197	+	0.913341i	$0.366506\pi$
$44$	0	0
$45$	−6.50687e18	−0.633036
$46$	0	0
$47$	1.94347e19	1.14671	0.573355	−	0.819307i	$-0.305642\pi$
$47$	1.94347e19	1.14671	0.573355	+	0.819307i	$0.305642\pi$
$48$	0	0
$49$	2.01961e19	0.737924
$50$	0	0
$51$	7.64695e16	0.00176373
$52$	0	0
$53$	−9.94325e19	−1.47352	−0.736759	−	0.676155i	$-0.763645\pi$
$53$	−9.94325e19	−1.47352	−0.736759	+	0.676155i	$0.763645\pi$
$54$	0	0
$55$	−3.06427e19	−0.296590
$56$	0	0
$57$	−9.28240e16	−0.000595799 0
$58$	0	0
$59$	2.50005e20	1.07932	0.539660	−	0.841883i	$-0.318553\pi$
$59$	2.50005e20	1.07932	0.539660	+	0.841883i	$0.318553\pi$
$60$	0	0
$61$	−1.62386e20	−0.477810	−0.238905	−	0.971043i	$-0.576788\pi$
$61$	−1.62386e20	−0.477810	−0.238905	+	0.971043i	$0.576788\pi$
$62$	0	0
$63$	−6.49185e20	−1.31811
$64$	0	0
$65$	4.87403e19	0.0690852
$66$	0	0
$67$	1.42919e21	1.42965	0.714825	−	0.699304i	$-0.246507\pi$
$67$	1.42919e21	1.42965	0.714825	+	0.699304i	$0.246507\pi$
$68$	0	0
$69$	1.23995e19	0.00884378
$70$	0	0
$71$	−6.49160e20	−0.333335	−0.166667	−	0.986013i	$-0.553301\pi$
$71$	−6.49160e20	−0.333335	−0.166667	+	0.986013i	$0.553301\pi$
$72$	0	0
$73$	3.52414e21	1.31474	0.657370	−	0.753568i	$-0.271669\pi$
$73$	3.52414e21	1.31474	0.657370	+	0.753568i	$0.271669\pi$
$74$	0	0
$75$	2.64639e19	0.00723518
$76$	0	0
$77$	−3.05720e21	−0.617562
$78$	0	0
$79$	−2.14128e21	−0.322079	−0.161040	−	0.986948i	$-0.551485\pi$
$79$	−2.14128e21	−0.322079	−0.161040	+	0.986948i	$0.551485\pi$
$80$	0	0
$81$	8.85906e21	0.999563
$82$	0	0
$83$	−1.39600e22	−1.18984	−0.594920	−	0.803785i	$-0.702816\pi$
$83$	−1.39600e22	−1.18984	−0.594920	+	0.803785i	$0.702816\pi$
$84$	0	0
$85$	−1.42668e21	−0.0924716
$86$	0	0
$87$	−3.18019e20	−0.0157754
$88$	0	0
$89$	−3.87945e22	−1.48179	−0.740893	−	0.671624i	$-0.765597\pi$
$89$	−3.87945e22	−1.48179	−0.740893	+	0.671624i	$0.765597\pi$
$90$	0	0
$91$	4.86278e21	0.143850
$92$	0	0
$93$	8.86793e20	0.0204300
$94$	0	0
$95$	1.73180e21	0.0312375
$96$	0	0
$97$	−7.34335e22	−1.04236	−0.521182	−	0.853446i	$-0.674509\pi$
$97$	−7.34335e22	−1.04236	−0.521182	+	0.853446i	$0.674509\pi$
$98$	0	0
$99$	4.17259e22	0.468383
$100$	0	0
$101$	1.50459e23	1.34191	0.670954	−	0.741499i	$-0.265885\pi$
$101$	1.50459e23	1.34191	0.670954	+	0.741499i	$0.265885\pi$
$102$	0	0
$103$	2.12029e23	1.50927	0.754635	−	0.656145i	$-0.227814\pi$
$103$	2.12029e23	1.50927	0.754635	+	0.656145i	$0.227814\pi$
$104$	0	0
$105$	−1.76644e21	−0.0100791
$106$	0	0
$107$	−1.15563e23	−0.530768	−0.265384	−	0.964143i	$-0.585499\pi$
$107$	−1.15563e23	−0.530768	−0.265384	+	0.964143i	$0.585499\pi$
$108$	0	0
$109$	−2.42738e22	−0.0901018	−0.0450509	−	0.998985i	$-0.514345\pi$
$109$	−2.42738e22	−0.0901018	−0.0450509	+	0.998985i	$0.514345\pi$
$110$	0	0
$111$	−7.52786e21	−0.0226703
$112$	0	0
$113$	6.29941e23	1.54489	0.772445	−	0.635082i	$-0.219034\pi$
$113$	6.29941e23	1.54489	0.772445	+	0.635082i	$0.219034\pi$
$114$	0	0
$115$	−2.31334e23	−0.463676
$116$	0	0
$117$	−6.63693e22	−0.109101
$118$	0	0
$119$	−1.42338e23	−0.192545
$120$	0	0
$121$	−6.98931e23	−0.780553
$122$	0	0
$123$	−1.09837e22	−0.0101587
$124$	0	0
$125$	−1.31779e24	−1.01247
$126$	0	0
$127$	−2.14836e24	−1.37519	−0.687597	−	0.726093i	$-0.741334\pi$
$127$	−2.14836e24	−1.37519	−0.687597	+	0.726093i	$0.741334\pi$
$128$	0	0
$129$	−1.83878e22	−0.00983442
$130$	0	0
$131$	4.11457e24	1.84376	0.921880	−	0.387476i	$-0.126653\pi$
$131$	4.11457e24	1.84376	0.921880	+	0.387476i	$0.126653\pi$
$132$	0	0
$133$	1.72780e23	0.0650430
$134$	0	0
$135$	4.82217e22	0.0152899
$136$	0	0
$137$	6.20749e22	0.0166199	0.00830997	−	0.999965i	$-0.497355\pi$
$137$	6.20749e22	0.0166199	0.00830997	+	0.999965i	$0.497355\pi$
$138$	0	0
$139$	−3.00241e24	−0.680456	−0.340228	−	0.940343i	$-0.610504\pi$
$139$	−3.00241e24	−0.680456	−0.340228	+	0.940343i	$0.610504\pi$
$140$	0	0
$141$	−7.20092e22	−0.0138474
$142$	0	0
$143$	−3.12552e23	−0.0511161
$144$	0	0
$145$	5.93322e24	0.827100
$146$	0	0
$147$	−7.48300e22	−0.00891100
$148$	0	0
$149$	6.49948e23	0.0662577	0.0331288	−	0.999451i	$-0.489453\pi$
$149$	6.49948e23	0.0662577	0.0331288	+	0.999451i	$0.489453\pi$
$150$	0	0
$151$	1.52705e25	1.33542	0.667710	−	0.744422i	$-0.267275\pi$
$151$	1.52705e25	1.33542	0.667710	+	0.744422i	$0.267275\pi$
$152$	0	0
$153$	1.94270e24	0.146034
$154$	0	0
$155$	−1.65447e25	−1.07114
$156$	0	0
$157$	1.22464e25	0.684168	0.342084	−	0.939669i	$-0.388867\pi$
$157$	1.22464e25	0.684168	0.342084	+	0.939669i	$0.388867\pi$
$158$	0	0
$159$	3.68415e23	0.0177939
$160$	0	0
$161$	−2.30800e25	−0.965470
$162$	0	0
$163$	3.19138e25	1.15830	0.579150	−	0.815221i	$-0.303385\pi$
$163$	3.19138e25	1.15830	0.579150	+	0.815221i	$0.303385\pi$
$164$	0	0
$165$	1.13537e23	0.00358155
$166$	0	0
$167$	3.04637e25	0.836649	0.418325	−	0.908298i	$-0.362617\pi$
$167$	3.04637e25	0.836649	0.418325	+	0.908298i	$0.362617\pi$
$168$	0	0
$169$	−4.12568e25	−0.988093
$170$	0	0
$171$	−2.35818e24	−0.0493312
$172$	0	0
$173$	3.57287e25	0.653864	0.326932	−	0.945048i	$-0.393985\pi$
$173$	3.57287e25	0.653864	0.326932	+	0.945048i	$0.393985\pi$
$174$	0	0
$175$	−4.92592e25	−0.789860
$176$	0	0
$177$	−9.26312e23	−0.0130336
$178$	0	0
$179$	6.76601e24	0.0836610	0.0418305	−	0.999125i	$-0.486681\pi$
$179$	6.76601e24	0.0836610	0.0418305	+	0.999125i	$0.486681\pi$
$180$	0	0
$181$	−1.18614e26	−1.29073	−0.645363	−	0.763876i	$-0.723294\pi$
$181$	−1.18614e26	−1.29073	−0.645363	+	0.763876i	$0.723294\pi$
$182$	0	0
$183$	6.01669e23	0.00576992
$184$	0	0
$185$	1.40446e26	1.18860
$186$	0	0
$187$	9.14872e24	0.0684198
$188$	0	0
$189$	4.81104e24	0.0318367
$190$	0	0
$191$	3.58368e25	0.210109	0.105055	−	0.994466i	$-0.466498\pi$
$191$	3.58368e25	0.210109	0.105055	+	0.994466i	$0.466498\pi$
$192$	0	0
$193$	1.73022e26	0.899897	0.449948	−	0.893055i	$-0.351442\pi$
$193$	1.73022e26	0.899897	0.449948	+	0.893055i	$0.351442\pi$
$194$	0	0
$195$	−1.80592e23	−0.000834256 0
$196$	0	0
$197$	−7.62302e25	−0.313159	−0.156580	−	0.987665i	$-0.550047\pi$
$197$	−7.62302e25	−0.313159	−0.156580	+	0.987665i	$0.550047\pi$
$198$	0	0
$199$	3.31684e25	0.121315	0.0606575	−	0.998159i	$-0.480680\pi$
$199$	3.31684e25	0.121315	0.0606575	+	0.998159i	$0.480680\pi$
$200$	0	0
$201$	−5.29540e24	−0.0172641
$202$	0	0
$203$	5.91952e26	1.72219
$204$	0	0
$205$	2.04921e26	0.532620
$206$	0	0
$207$	3.15006e26	0.732251
$208$	0	0
$209$	−1.11053e25	−0.0231127
$210$	0	0
$211$	5.89481e26	1.09957	0.549783	−	0.835307i	$-0.314710\pi$
$211$	5.89481e26	1.09957	0.549783	+	0.835307i	$0.314710\pi$
$212$	0	0
$213$	2.40525e24	0.00402527
$214$	0	0
$215$	3.43058e26	0.515615
$216$	0	0
$217$	−1.65066e27	−2.23033
$218$	0	0
$219$	−1.30576e25	−0.0158765
$220$	0	0
$221$	−1.45520e25	−0.0159371
$222$	0	0
$223$	1.36749e27	1.35026	0.675131	−	0.737698i	$-0.264087\pi$
$223$	1.36749e27	1.35026	0.675131	+	0.737698i	$0.264087\pi$
$224$	0	0
$225$	6.72311e26	0.599062
$226$	0	0
$227$	−1.13000e27	−0.909454	−0.454727	−	0.890631i	$-0.650263\pi$
$227$	−1.13000e27	−0.909454	−0.454727	+	0.890631i	$0.650263\pi$
$228$	0	0
$229$	2.37507e26	0.172810	0.0864050	−	0.996260i	$-0.472462\pi$
$229$	2.37507e26	0.172810	0.0864050	+	0.996260i	$0.472462\pi$
$230$	0	0
$231$	1.13275e25	0.00745753
$232$	0	0
$233$	9.82301e26	0.585668	0.292834	−	0.956163i	$-0.405402\pi$
$233$	9.82301e26	0.585668	0.292834	+	0.956163i	$0.405402\pi$
$234$	0	0
$235$	1.34346e27	0.726014
$236$	0	0
$237$	7.93384e24	0.00388935
$238$	0	0
$239$	3.76754e27	1.67680	0.838402	−	0.545053i	$-0.183490\pi$
$239$	3.76754e27	1.67680	0.838402	+	0.545053i	$0.183490\pi$
$240$	0	0
$241$	6.37252e26	0.257700	0.128850	−	0.991664i	$-0.458871\pi$
$241$	6.37252e26	0.257700	0.128850	+	0.991664i	$0.458871\pi$
$242$	0	0
$243$	−9.84971e25	−0.0362202
$244$	0	0
$245$	1.39609e27	0.467201
$246$	0	0
$247$	1.76642e25	0.00538367
$248$	0	0
$249$	5.17244e25	0.0143682
$250$	0	0
$251$	6.98232e27	1.76910	0.884549	−	0.466448i	$-0.154466\pi$
$251$	6.98232e27	1.76910	0.884549	+	0.466448i	$0.154466\pi$
$252$	0	0
$253$	1.48345e27	0.343074
$254$	0	0
$255$	5.28610e24	0.00111667
$256$	0	0
$257$	1.49876e27	0.289401	0.144700	−	0.989476i	$-0.453778\pi$
$257$	1.49876e27	0.289401	0.144700	+	0.989476i	$0.453778\pi$
$258$	0	0
$259$	1.40122e28	2.47490
$260$	0	0
$261$	−8.07921e27	−1.30618
$262$	0	0
$263$	5.88103e27	0.870889	0.435444	−	0.900216i	$-0.356591\pi$
$263$	5.88103e27	0.870889	0.435444	+	0.900216i	$0.356591\pi$
$264$	0	0
$265$	−6.87345e27	−0.932926
$266$	0	0
$267$	1.43741e26	0.0178937
$268$	0	0
$269$	−3.56788e27	−0.407623	−0.203812	−	0.979010i	$-0.565333\pi$
$269$	−3.56788e27	−0.407623	−0.203812	+	0.979010i	$0.565333\pi$
$270$	0	0
$271$	−2.81760e26	−0.0295619	−0.0147809	−	0.999891i	$-0.504705\pi$
$271$	−2.81760e26	−0.0295619	−0.0147809	+	0.999891i	$0.504705\pi$
$272$	0	0
$273$	−1.80175e25	−0.00173709
$274$	0	0
$275$	3.16611e27	0.280672
$276$	0	0
$277$	3.40753e26	0.0277922	0.0138961	−	0.999903i	$-0.495577\pi$
$277$	3.40753e26	0.0277922	0.0138961	+	0.999903i	$0.495577\pi$
$278$	0	0
$279$	2.25288e28	1.69157
$280$	0	0
$281$	2.49920e28	1.72853	0.864267	−	0.503033i	$-0.167783\pi$
$281$	2.49920e28	1.72853	0.864267	+	0.503033i	$0.167783\pi$
$282$	0	0
$283$	1.18584e28	0.755928	0.377964	−	0.925820i	$-0.376624\pi$
$283$	1.18584e28	0.755928	0.377964	+	0.925820i	$0.376624\pi$
$284$	0	0
$285$	−6.41663e24	−0.000377217 0
$286$	0	0
$287$	2.04448e28	1.10902
$288$	0	0
$289$	−1.95416e28	−0.978668
$290$	0	0
$291$	2.72084e26	0.0125873
$292$	0	0
$293$	1.79941e28	0.769401	0.384700	−	0.923042i	$-0.374305\pi$
$293$	1.79941e28	0.769401	0.384700	+	0.923042i	$0.374305\pi$
$294$	0	0
$295$	1.72820e28	0.683348
$296$	0	0
$297$	−3.09227e26	−0.0113130
$298$	0	0
$299$	−2.35958e27	−0.0799129
$300$	0	0
$301$	3.42266e28	1.07362
$302$	0	0
$303$	−5.57478e26	−0.0162046
$304$	0	0
$305$	−1.12252e28	−0.302515
$306$	0	0
$307$	6.39397e28	1.59838	0.799189	−	0.601080i	$-0.205263\pi$
$307$	6.39397e28	1.59838	0.799189	+	0.601080i	$0.205263\pi$
$308$	0	0
$309$	−7.85605e26	−0.0182256
$310$	0	0
$311$	2.63226e28	0.567001	0.283500	−	0.958972i	$-0.408504\pi$
$311$	2.63226e28	0.567001	0.283500	+	0.958972i	$0.408504\pi$
$312$	0	0
$313$	−5.74270e28	−1.14909	−0.574547	−	0.818471i	$-0.694822\pi$
$313$	−5.74270e28	−1.14909	−0.574547	+	0.818471i	$0.694822\pi$
$314$	0	0
$315$	−4.48761e28	−0.834533
$316$	0	0
$317$	−3.28617e28	−0.568210	−0.284105	−	0.958793i	$-0.591696\pi$
$317$	−3.28617e28	−0.568210	−0.284105	+	0.958793i	$0.591696\pi$
$318$	0	0
$319$	−3.80473e28	−0.611972
$320$	0	0
$321$	4.28181e26	0.00640943
$322$	0	0
$323$	−5.17048e26	−0.00720613
$324$	0	0
$325$	−5.03601e27	−0.0653774
$326$	0	0
$327$	8.99388e25	0.00108805
$328$	0	0
$329$	1.34036e29	1.51171
$330$	0	0
$331$	7.98739e28	0.840201	0.420101	−	0.907478i	$-0.361995\pi$
$331$	7.98739e28	0.840201	0.420101	+	0.907478i	$0.361995\pi$
$332$	0	0
$333$	−1.91244e29	−1.87707
$334$	0	0
$335$	9.87953e28	0.905151
$336$	0	0
$337$	1.41544e28	0.121101	0.0605503	−	0.998165i	$-0.480714\pi$
$337$	1.41544e28	0.121101	0.0605503	+	0.998165i	$0.480714\pi$
$338$	0	0
$339$	−2.33404e27	−0.0186557
$340$	0	0
$341$	1.06095e29	0.792536
$342$	0	0
$343$	−4.94680e28	−0.345495
$344$	0	0
$345$	8.57135e26	0.00559925
$346$	0	0
$347$	7.37917e28	0.451044	0.225522	−	0.974238i	$-0.427591\pi$
$347$	7.37917e28	0.451044	0.225522	+	0.974238i	$0.427591\pi$
$348$	0	0
$349$	5.91695e28	0.338537	0.169268	−	0.985570i	$-0.445860\pi$
$349$	5.91695e28	0.338537	0.169268	+	0.985570i	$0.445860\pi$
$350$	0	0
$351$	4.91856e26	0.00263515
$352$	0	0
$353$	1.85300e29	0.929964	0.464982	−	0.885320i	$-0.346061\pi$
$353$	1.85300e29	0.929964	0.464982	+	0.885320i	$0.346061\pi$
$354$	0	0
$355$	−4.48744e28	−0.211044
$356$	0	0
$357$	5.27389e26	0.00232513
$358$	0	0
$359$	−4.04102e29	−1.67072	−0.835361	−	0.549701i	$-0.814742\pi$
$359$	−4.04102e29	−1.67072	−0.835361	+	0.549701i	$0.814742\pi$
$360$	0	0
$361$	−2.57202e29	−0.997566
$362$	0	0
$363$	2.58966e27	0.00942577
$364$	0	0
$365$	2.43613e29	0.832399
$366$	0	0
$367$	−1.25397e29	−0.402373	−0.201186	−	0.979553i	$-0.564480\pi$
$367$	−1.25397e29	−0.402373	−0.201186	+	0.979553i	$0.564480\pi$
$368$	0	0
$369$	−2.79039e29	−0.841128
$370$	0	0
$371$	−6.85758e29	−1.94255
$372$	0	0
$373$	2.52057e28	0.0671192	0.0335596	−	0.999437i	$-0.489316\pi$
$373$	2.52057e28	0.0671192	0.0335596	+	0.999437i	$0.489316\pi$
$374$	0	0
$375$	4.88264e27	0.0122263
$376$	0	0
$377$	6.05181e28	0.142548
$378$	0	0
$379$	−5.58331e29	−1.23749	−0.618744	−	0.785593i	$-0.712358\pi$
$379$	−5.58331e29	−1.23749	−0.618744	+	0.785593i	$0.712358\pi$
$380$	0	0
$381$	7.96005e27	0.0166065
$382$	0	0
$383$	−2.07862e29	−0.408309	−0.204155	−	0.978939i	$-0.565445\pi$
$383$	−2.07862e29	−0.408309	−0.204155	+	0.978939i	$0.565445\pi$
$384$	0	0
$385$	−2.11334e29	−0.390996
$386$	0	0
$387$	−4.67140e29	−0.814274
$388$	0	0
$389$	−5.76005e29	−0.946249	−0.473125	−	0.880996i	$-0.656874\pi$
$389$	−5.76005e29	−0.946249	−0.473125	+	0.880996i	$0.656874\pi$
$390$	0	0
$391$	6.90674e28	0.106965
$392$	0	0
$393$	−1.52452e28	−0.0222648
$394$	0	0
$395$	−1.48020e29	−0.203917
$396$	0	0
$397$	−3.01173e29	−0.391494	−0.195747	−	0.980654i	$-0.562713\pi$
$397$	−3.01173e29	−0.391494	−0.195747	+	0.980654i	$0.562713\pi$
$398$	0	0
$399$	−6.40182e26	−0.000785444 0
$400$	0	0
$401$	4.50852e29	0.522245	0.261122	−	0.965306i	$-0.415907\pi$
$401$	4.50852e29	0.522245	0.261122	+	0.965306i	$0.415907\pi$
$402$	0	0
$403$	−1.68754e29	−0.184607
$404$	0	0
$405$	6.12399e29	0.632851
$406$	0	0
$407$	−9.00624e29	−0.879443
$408$	0	0
$409$	4.57914e29	0.422636	0.211318	−	0.977417i	$-0.432224\pi$
$409$	4.57914e29	0.422636	0.211318	+	0.977417i	$0.432224\pi$
$410$	0	0
$411$	−2.29999e26	−0.000200699 0
$412$	0	0
$413$	1.72421e30	1.42287
$414$	0	0
$415$	−9.65014e29	−0.753322
$416$	0	0
$417$	1.11245e28	0.00821702
$418$	0	0
$419$	1.18732e30	0.830055	0.415027	−	0.909809i	$-0.363772\pi$
$419$	1.18732e30	0.830055	0.415027	+	0.909809i	$0.363772\pi$
$420$	0	0
$421$	−1.90661e30	−1.26188	−0.630940	−	0.775831i	$-0.717331\pi$
$421$	−1.90661e30	−1.26188	−0.630940	+	0.775831i	$0.717331\pi$
$422$	0	0
$423$	−1.82938e30	−1.14654
$424$	0	0
$425$	1.47409e29	0.0875088
$426$	0	0
$427$	−1.11993e30	−0.629899
$428$	0	0
$429$	1.15806e27	0.000617266 0
$430$	0	0
$431$	−3.07278e30	−1.55254	−0.776270	−	0.630401i	$-0.782891\pi$
$431$	−3.07278e30	−1.55254	−0.776270	+	0.630401i	$0.782891\pi$
$432$	0	0
$433$	1.32819e30	0.636282	0.318141	−	0.948043i	$-0.396942\pi$
$433$	1.32819e30	0.636282	0.318141	+	0.948043i	$0.396942\pi$
$434$	0	0
$435$	−2.19836e28	−0.00998786
$436$	0	0
$437$	−8.38388e28	−0.0361334
$438$	0	0
$439$	3.14284e30	1.28523	0.642613	−	0.766191i	$-0.277850\pi$
$439$	3.14284e30	1.28523	0.642613	+	0.766191i	$0.277850\pi$
$440$	0	0
$441$	−1.90104e30	−0.737817
$442$	0	0
$443$	−3.18342e30	−1.17287	−0.586437	−	0.809995i	$-0.699470\pi$
$443$	−3.18342e30	−1.17287	−0.586437	+	0.809995i	$0.699470\pi$
$444$	0	0
$445$	−2.68174e30	−0.938160
$446$	0	0
$447$	−2.40817e27	−0.000800112 0
$448$	0	0
$449$	−6.17935e29	−0.195034	−0.0975169	−	0.995234i	$-0.531090\pi$
$449$	−6.17935e29	−0.195034	−0.0975169	+	0.995234i	$0.531090\pi$
$450$	0	0
$451$	−1.31408e30	−0.394085
$452$	0	0
$453$	−5.65798e28	−0.0161262
$454$	0	0
$455$	3.36149e29	0.0910752
$456$	0	0
$457$	−4.23877e29	−0.109195	−0.0545976	−	0.998508i	$-0.517388\pi$
$457$	−4.23877e29	−0.109195	−0.0545976	+	0.998508i	$0.517388\pi$
$458$	0	0
$459$	−1.43971e28	−0.00352720
$460$	0	0
$461$	−2.62733e30	−0.612286	−0.306143	−	0.951985i	$-0.599039\pi$
$461$	−2.62733e30	−0.612286	−0.306143	+	0.951985i	$0.599039\pi$
$462$	0	0
$463$	5.83993e30	1.29487	0.647435	−	0.762121i	$-0.275842\pi$
$463$	5.83993e30	1.29487	0.647435	+	0.762121i	$0.275842\pi$
$464$	0	0
$465$	6.13012e28	0.0129348
$466$	0	0
$467$	2.69186e30	0.540640	0.270320	−	0.962771i	$-0.412870\pi$
$467$	2.69186e30	0.540640	0.270320	+	0.962771i	$0.412870\pi$
$468$	0	0
$469$	9.85672e30	1.88471
$470$	0	0
$471$	−4.53751e28	−0.00826185
$472$	0	0
$473$	−2.19990e30	−0.381504
$474$	0	0
$475$	−1.78935e29	−0.0295610
$476$	0	0
$477$	9.35952e30	1.47330
$478$	0	0
$479$	−3.66675e30	−0.550077	−0.275038	−	0.961433i	$-0.588691\pi$
$479$	−3.66675e30	−0.550077	−0.275038	+	0.961433i	$0.588691\pi$
$480$	0	0
$481$	1.43253e30	0.204850
$482$	0	0
$483$	8.55156e28	0.0116588
$484$	0	0
$485$	−5.07622e30	−0.659950
$486$	0	0
$487$	7.51702e30	0.932101	0.466050	−	0.884758i	$-0.345677\pi$
$487$	7.51702e30	0.932101	0.466050	+	0.884758i	$0.345677\pi$
$488$	0	0
$489$	−1.18246e29	−0.0139874
$490$	0	0
$491$	1.69672e31	1.91501	0.957507	−	0.288409i	$-0.0931263\pi$
$491$	1.69672e31	1.91501	0.957507	+	0.288409i	$0.0931263\pi$
$492$	0	0
$493$	−1.77143e30	−0.190802
$494$	0	0
$495$	2.88438e30	0.296547
$496$	0	0
$497$	−4.47708e30	−0.439437
$498$	0	0
$499$	7.24838e29	0.0679336	0.0339668	−	0.999423i	$-0.489186\pi$
$499$	7.24838e29	0.0679336	0.0339668	+	0.999423i	$0.489186\pi$
$500$	0	0
$501$	−1.12873e29	−0.0101032
$502$	0	0
$503$	−1.42943e30	−0.122217	−0.0611086	−	0.998131i	$-0.519464\pi$
$503$	−1.42943e30	−0.122217	−0.0611086	+	0.998131i	$0.519464\pi$
$504$	0	0
$505$	1.04008e31	0.849600
$506$	0	0
$507$	1.52864e29	0.0119320
$508$	0	0
$509$	−2.42629e31	−1.81004	−0.905021	−	0.425367i	$-0.860145\pi$
$509$	−2.42629e31	−1.81004	−0.905021	+	0.425367i	$0.860145\pi$
$510$	0	0
$511$	2.43050e31	1.73323
$512$	0	0
$513$	1.74762e28	0.00119151
$514$	0	0
$515$	1.46569e31	0.955561
$516$	0	0
$517$	−8.61508e30	−0.537178
$518$	0	0
$519$	−1.32381e29	−0.00789590
$520$	0	0
$521$	1.27172e31	0.725703	0.362851	−	0.931847i	$-0.381803\pi$
$521$	1.27172e31	0.725703	0.362851	+	0.931847i	$0.381803\pi$
$522$	0	0
$523$	3.28752e31	1.79514	0.897572	−	0.440867i	$-0.145329\pi$
$523$	3.28752e31	1.79514	0.897572	+	0.440867i	$0.145329\pi$
$524$	0	0
$525$	1.82514e29	0.00953816
$526$	0	0
$527$	4.93961e30	0.247099
$528$	0	0
$529$	−9.68126e30	−0.463652
$530$	0	0
$531$	−2.35328e31	−1.07916
$532$	0	0
$533$	2.09017e30	0.0917950
$534$	0	0
$535$	−7.98850e30	−0.336044
$536$	0	0
$537$	−2.50693e28	−0.00101027
$538$	0	0
$539$	−8.95256e30	−0.345682
$540$	0	0
$541$	4.23264e31	1.56618	0.783091	−	0.621907i	$-0.213642\pi$
$541$	4.23264e31	1.56618	0.783091	+	0.621907i	$0.213642\pi$
$542$	0	0
$543$	4.39488e29	0.0155865
$544$	0	0
$545$	−1.67797e30	−0.0570460
$546$	0	0
$547$	4.02195e31	1.31094	0.655470	−	0.755221i	$-0.272471\pi$
$547$	4.02195e31	1.31094	0.655470	+	0.755221i	$0.272471\pi$
$548$	0	0
$549$	1.52853e31	0.477740
$550$	0	0
$551$	2.15028e30	0.0644543
$552$	0	0
$553$	−1.47679e31	−0.424598
$554$	0	0
$555$	−5.20377e29	−0.0143532
$556$	0	0
$557$	−1.77757e31	−0.470427	−0.235214	−	0.971944i	$-0.575579\pi$
$557$	−1.77757e31	−0.470427	−0.235214	+	0.971944i	$0.575579\pi$
$558$	0	0
$559$	3.49915e30	0.0888643
$560$	0	0
$561$	−3.38976e28	−0.000826221 0
$562$	0	0
$563$	−2.99411e30	−0.0700521	−0.0350260	−	0.999386i	$-0.511151\pi$
$563$	−2.99411e30	−0.0700521	−0.0350260	+	0.999386i	$0.511151\pi$
$564$	0	0
$565$	4.35458e31	0.978113
$566$	0	0
$567$	6.10985e31	1.31773
$568$	0	0
$569$	−6.71175e30	−0.139010	−0.0695048	−	0.997582i	$-0.522142\pi$
$569$	−6.71175e30	−0.139010	−0.0695048	+	0.997582i	$0.522142\pi$
$570$	0	0
$571$	−6.56935e31	−1.30679	−0.653397	−	0.757016i	$-0.726657\pi$
$571$	−6.56935e31	−1.30679	−0.653397	+	0.757016i	$0.726657\pi$
$572$	0	0
$573$	−1.32782e29	−0.00253723
$574$	0	0
$575$	2.39022e31	0.438792
$576$	0	0
$577$	−3.58953e31	−0.633163	−0.316581	−	0.948565i	$-0.602535\pi$
$577$	−3.58953e31	−0.633163	−0.316581	+	0.948565i	$0.602535\pi$
$578$	0	0
$579$	−6.41078e29	−0.0108669
$580$	0	0
$581$	−9.62786e31	−1.56857
$582$	0	0
$583$	4.40767e31	0.690272
$584$	0	0
$585$	−4.58790e30	−0.0690751
$586$	0	0
$587$	8.28891e31	1.19994	0.599970	−	0.800023i	$-0.295179\pi$
$587$	8.28891e31	1.19994	0.599970	+	0.800023i	$0.295179\pi$
$588$	0	0
$589$	−5.99605e30	−0.0834717
$590$	0	0
$591$	2.82446e29	0.00378164
$592$	0	0
$593$	1.37258e32	1.76770	0.883848	−	0.467775i	$-0.154944\pi$
$593$	1.37258e32	1.76770	0.883848	+	0.467775i	$0.154944\pi$
$594$	0	0
$595$	−9.83941e30	−0.121906
$596$	0	0
$597$	−1.22895e29	−0.00146497
$598$	0	0
$599$	−1.28400e32	−1.47285	−0.736424	−	0.676520i	$-0.763487\pi$
$599$	−1.28400e32	−1.47285	−0.736424	+	0.676520i	$0.763487\pi$
$600$	0	0
$601$	−1.11025e32	−1.22564	−0.612822	−	0.790221i	$-0.709966\pi$
$601$	−1.11025e32	−1.22564	−0.612822	+	0.790221i	$0.709966\pi$
$602$	0	0
$603$	−1.34529e32	−1.42944
$604$	0	0
$605$	−4.83149e31	−0.494190
$606$	0	0
$607$	−1.72788e32	−1.70155	−0.850773	−	0.525533i	$-0.823866\pi$
$607$	−1.72788e32	−1.70155	−0.850773	+	0.525533i	$0.823866\pi$
$608$	0	0
$609$	−2.19329e30	−0.0207968
$610$	0	0
$611$	1.37032e31	0.125126
$612$	0	0
$613$	1.24240e32	1.09261	0.546306	−	0.837586i	$-0.316034\pi$
$613$	1.24240e32	1.09261	0.546306	+	0.837586i	$0.316034\pi$
$614$	0	0
$615$	−7.59269e29	−0.00643179
$616$	0	0
$617$	7.67716e31	0.626500	0.313250	−	0.949671i	$-0.398582\pi$
$617$	7.67716e31	0.626500	0.313250	+	0.949671i	$0.398582\pi$
$618$	0	0
$619$	1.50459e31	0.118297	0.0591487	−	0.998249i	$-0.481161\pi$
$619$	1.50459e31	0.118297	0.0591487	+	0.998249i	$0.481161\pi$
$620$	0	0
$621$	−2.33448e30	−0.0176863
$622$	0	0
$623$	−2.67555e32	−1.95344
$624$	0	0
$625$	−5.95033e30	−0.0418718
$626$	0	0
$627$	4.11473e28	0.000279103 0
$628$	0	0
$629$	−4.19317e31	−0.274195
$630$	0	0
$631$	−6.09550e31	−0.384301	−0.192151	−	0.981365i	$-0.561546\pi$
$631$	−6.09550e31	−0.384301	−0.192151	+	0.981365i	$0.561546\pi$
$632$	0	0
$633$	−2.18413e30	−0.0132781
$634$	0	0
$635$	−1.48509e32	−0.870674
$636$	0	0
$637$	1.42400e31	0.0805202
$638$	0	0
$639$	6.11051e31	0.333286
$640$	0	0
$641$	2.74644e32	1.44512	0.722559	−	0.691309i	$-0.242966\pi$
$641$	2.74644e32	1.44512	0.722559	+	0.691309i	$0.242966\pi$
$642$	0	0
$643$	−1.88032e32	−0.954563	−0.477281	−	0.878750i	$-0.658378\pi$
$643$	−1.88032e32	−0.954563	−0.477281	+	0.878750i	$0.658378\pi$
$644$	0	0
$645$	−1.27109e30	−0.00622645
$646$	0	0
$647$	−4.29086e31	−0.202836	−0.101418	−	0.994844i	$-0.532338\pi$
$647$	−4.29086e31	−0.202836	−0.101418	+	0.994844i	$0.532338\pi$
$648$	0	0
$649$	−1.10823e32	−0.505609
$650$	0	0
$651$	6.11597e30	0.0269330
$652$	0	0
$653$	1.34844e32	0.573233	0.286616	−	0.958045i	$-0.407470\pi$
$653$	1.34844e32	0.573233	0.286616	+	0.958045i	$0.407470\pi$
$654$	0	0
$655$	2.84427e32	1.16734
$656$	0	0
$657$	−3.31725e32	−1.31455
$658$	0	0
$659$	−2.44361e32	−0.935083	−0.467541	−	0.883971i	$-0.654860\pi$
$659$	−2.44361e32	−0.935083	−0.467541	+	0.883971i	$0.654860\pi$
$660$	0	0
$661$	−4.07278e32	−1.50513	−0.752565	−	0.658517i	$-0.771184\pi$
$661$	−4.07278e32	−1.50513	−0.752565	+	0.658517i	$0.771184\pi$
$662$	0	0
$663$	5.39176e28	0.000192453 0
$664$	0	0
$665$	1.19438e31	0.0411805
$666$	0	0
$667$	−2.87235e32	−0.956731
$668$	0	0
$669$	−5.06679e30	−0.0163054
$670$	0	0
$671$	7.19828e31	0.223831
$672$	0	0
$673$	−2.01182e32	−0.604528	−0.302264	−	0.953224i	$-0.597742\pi$
$673$	−2.01182e32	−0.604528	−0.302264	+	0.953224i	$0.597742\pi$
$674$	0	0
$675$	−4.98243e30	−0.0144693
$676$	0	0
$677$	4.58560e32	1.28714	0.643571	−	0.765386i	$-0.277452\pi$
$677$	4.58560e32	1.28714	0.643571	+	0.765386i	$0.277452\pi$
$678$	0	0
$679$	−5.06451e32	−1.37415
$680$	0	0
$681$	4.18685e30	0.0109823
$682$	0	0
$683$	5.31237e32	1.34726	0.673629	−	0.739070i	$-0.264735\pi$
$683$	5.31237e32	1.34726	0.673629	+	0.739070i	$0.264735\pi$
$684$	0	0
$685$	4.29104e30	0.0105226
$686$	0	0
$687$	−8.80006e29	−0.00208681
$688$	0	0
$689$	−7.01084e31	−0.160786
$690$	0	0
$691$	−6.16076e32	−1.36659	−0.683293	−	0.730145i	$-0.739453\pi$
$691$	−6.16076e32	−1.36659	−0.683293	+	0.730145i	$0.739453\pi$
$692$	0	0
$693$	2.87772e32	0.617472
$694$	0	0
$695$	−2.07547e32	−0.430816
$696$	0	0
$697$	−6.11814e31	−0.122869
$698$	0	0
$699$	−3.63960e30	−0.00707238
$700$	0	0
$701$	−2.55445e32	−0.480330	−0.240165	−	0.970732i	$-0.577201\pi$
$701$	−2.55445e32	−0.480330	−0.240165	+	0.970732i	$0.577201\pi$
$702$	0	0
$703$	5.08996e31	0.0926249
$704$	0	0
$705$	−4.97776e30	−0.00876717
$706$	0	0
$707$	1.03768e33	1.76904
$708$	0	0
$709$	2.65623e32	0.438363	0.219182	−	0.975684i	$-0.429661\pi$
$709$	2.65623e32	0.438363	0.219182	+	0.975684i	$0.429661\pi$
$710$	0	0
$711$	2.01558e32	0.322032
$712$	0	0
$713$	8.00953e32	1.23902
$714$	0	0
$715$	−2.16057e31	−0.0323631
$716$	0	0
$717$	−1.39594e31	−0.0202487
$718$	0	0
$719$	−3.26668e32	−0.458907	−0.229453	−	0.973320i	$-0.573694\pi$
$719$	−3.26668e32	−0.458907	−0.229453	+	0.973320i	$0.573694\pi$
$720$	0	0
$721$	1.46230e33	1.98968
$722$	0	0
$723$	−2.36113e30	−0.00311193
$724$	0	0
$725$	−6.13039e32	−0.782711
$726$	0	0
$727$	7.70663e32	0.953277	0.476638	−	0.879100i	$-0.341855\pi$
$727$	7.70663e32	0.953277	0.476638	+	0.879100i	$0.341855\pi$
$728$	0	0
$729$	−8.33655e32	−0.999125
$730$	0	0
$731$	−1.02424e32	−0.118946
$732$	0	0
$733$	−1.14925e33	−1.29336	−0.646680	−	0.762762i	$-0.723843\pi$
$733$	−1.14925e33	−1.29336	−0.646680	+	0.762762i	$0.723843\pi$
$734$	0	0
$735$	−5.17276e30	−0.00564180
$736$	0	0
$737$	−6.33535e32	−0.669722
$738$	0	0
$739$	−2.14242e31	−0.0219530	−0.0109765	−	0.999940i	$-0.503494\pi$
$739$	−2.14242e31	−0.0219530	−0.0109765	+	0.999940i	$0.503494\pi$
$740$	0	0
$741$	−6.54489e28	−6.50119e−5 0
$742$	0	0
$743$	3.03724e32	0.292488	0.146244	−	0.989249i	$-0.453282\pi$
$743$	3.03724e32	0.292488	0.146244	+	0.989249i	$0.453282\pi$
$744$	0	0
$745$	4.49288e31	0.0419496
$746$	0	0
$747$	1.31405e33	1.18967
$748$	0	0
$749$	−7.97005e32	−0.699714
$750$	0	0
$751$	1.08183e33	0.921084	0.460542	−	0.887638i	$-0.347655\pi$
$751$	1.08183e33	0.921084	0.460542	+	0.887638i	$0.347655\pi$
$752$	0	0
$753$	−2.58707e31	−0.0213632
$754$	0	0
$755$	1.05560e33	0.845492
$756$	0	0
$757$	1.54813e33	1.20283	0.601415	−	0.798937i	$-0.294604\pi$
$757$	1.54813e33	1.20283	0.601415	+	0.798937i	$0.294604\pi$
$758$	0	0
$759$	−5.49646e30	−0.00414288
$760$	0	0
$761$	−1.32317e33	−0.967590	−0.483795	−	0.875181i	$-0.660742\pi$
$761$	−1.32317e33	−0.967590	−0.483795	+	0.875181i	$0.660742\pi$
$762$	0	0
$763$	−1.67410e32	−0.118781
$764$	0	0
$765$	1.34292e32	0.0924581
$766$	0	0
$767$	1.76275e32	0.117772
$768$	0	0
$769$	−7.55866e32	−0.490108	−0.245054	−	0.969509i	$-0.578806\pi$
$769$	−7.55866e32	−0.490108	−0.245054	+	0.969509i	$0.578806\pi$
$770$	0	0
$771$	−5.55315e30	−0.00349474
$772$	0	0
$773$	2.44363e33	1.49270	0.746348	−	0.665556i	$-0.231805\pi$
$773$	2.44363e33	1.49270	0.746348	+	0.665556i	$0.231805\pi$
$774$	0	0
$775$	1.70946e33	1.01365
$776$	0	0
$777$	−5.19176e31	−0.0298863
$778$	0	0
$779$	7.42662e31	0.0415060
$780$	0	0
$781$	2.87761e32	0.156151
$782$	0	0
$783$	5.98742e31	0.0315486
$784$	0	0
$785$	8.46555e32	0.433166
$786$	0	0
$787$	3.66922e33	1.82333	0.911664	−	0.410937i	$-0.134798\pi$
$787$	3.66922e33	1.82333	0.911664	+	0.410937i	$0.134798\pi$
$788$	0	0
$789$	−2.17903e31	−0.0105166
$790$	0	0
$791$	4.34453e33	2.03663
$792$	0	0
$793$	−1.14496e32	−0.0521373
$794$	0	0
$795$	2.54673e31	0.0112658
$796$	0	0
$797$	6.91587e32	0.297219	0.148609	−	0.988896i	$-0.452520\pi$
$797$	6.91587e32	0.297219	0.148609	+	0.988896i	$0.452520\pi$
$798$	0	0
$799$	−4.01105e32	−0.167483
$800$	0	0
$801$	3.65171e33	1.48157
$802$	0	0
$803$	−1.56219e33	−0.615893
$804$	0	0
$805$	−1.59545e33	−0.611266
$806$	0	0
$807$	1.32196e31	0.00492236
$808$	0	0
$809$	−2.54447e33	−0.920850	−0.460425	−	0.887699i	$-0.652303\pi$
$809$	−2.54447e33	−0.920850	−0.460425	+	0.887699i	$0.652303\pi$
$810$	0	0
$811$	−3.26107e33	−1.14715	−0.573575	−	0.819153i	$-0.694444\pi$
$811$	−3.26107e33	−1.14715	−0.573575	+	0.819153i	$0.694444\pi$
$812$	0	0
$813$	1.04397e30	0.000356982 0
$814$	0	0
$815$	2.20610e33	0.733352
$816$	0	0
$817$	1.24329e32	0.0401809
$818$	0	0
$819$	−4.57731e32	−0.143829
$820$	0	0
$821$	−3.75926e33	−1.14857	−0.574283	−	0.818657i	$-0.694719\pi$
$821$	−3.75926e33	−1.14857	−0.574283	+	0.818657i	$0.694719\pi$
$822$	0	0
$823$	−5.33635e33	−1.58543	−0.792714	−	0.609594i	$-0.791332\pi$
$823$	−5.33635e33	−1.58543	−0.792714	+	0.609594i	$0.791332\pi$
$824$	0	0
$825$	−1.17310e31	−0.00338933
$826$	0	0
$827$	3.60699e33	1.01352	0.506759	−	0.862088i	$-0.330843\pi$
$827$	3.60699e33	1.01352	0.506759	+	0.862088i	$0.330843\pi$
$828$	0	0
$829$	−1.84212e33	−0.503432	−0.251716	−	0.967801i	$-0.580995\pi$
$829$	−1.84212e33	−0.503432	−0.251716	+	0.967801i	$0.580995\pi$
$830$	0	0
$831$	−1.26255e30	−0.000335612 0
$832$	0	0
$833$	−4.16818e32	−0.107778
$834$	0	0
$835$	2.10586e33	0.529706
$836$	0	0
$837$	−1.66959e32	−0.0408570
$838$	0	0
$839$	−5.31109e33	−1.26451	−0.632253	−	0.774762i	$-0.717870\pi$
$839$	−5.31109e33	−1.26451	−0.632253	+	0.774762i	$0.717870\pi$
$840$	0	0
$841$	3.05022e33	0.706605
$842$	0	0
$843$	−9.25997e31	−0.0208734
$844$	0	0
$845$	−2.85195e33	−0.625590
$846$	0	0
$847$	−4.82033e33	−1.02901
$848$	0	0
$849$	−4.39373e31	−0.00912841
$850$	0	0
$851$	−6.79918e33	−1.37488
$852$	0	0
$853$	7.87613e33	1.55024	0.775120	−	0.631814i	$-0.217689\pi$
$853$	7.87613e33	1.55024	0.775120	+	0.631814i	$0.217689\pi$
$854$	0	0
$855$	−1.63014e32	−0.0312330
$856$	0	0
$857$	−9.22373e33	−1.72039	−0.860195	−	0.509964i	$-0.829659\pi$
$857$	−9.22373e33	−1.72039	−0.860195	+	0.509964i	$0.829659\pi$
$858$	0	0
$859$	9.31982e33	1.69233	0.846167	−	0.532918i	$-0.178904\pi$
$859$	9.31982e33	1.69233	0.846167	+	0.532918i	$0.178904\pi$
$860$	0	0
$861$	−7.57516e31	−0.0133923
$862$	0	0
$863$	3.57093e33	0.614691	0.307345	−	0.951598i	$-0.400559\pi$
$863$	3.57093e33	0.614691	0.307345	+	0.951598i	$0.400559\pi$
$864$	0	0
$865$	2.46981e33	0.413980
$866$	0	0
$867$	7.24052e31	0.0118182
$868$	0	0
$869$	9.49194e32	0.150879
$870$	0	0
$871$	1.00770e33	0.155999
$872$	0	0
$873$	6.91225e33	1.04221
$874$	0	0
$875$	−9.08842e33	−1.33474
$876$	0	0
$877$	5.08782e32	0.0727841	0.0363920	−	0.999338i	$-0.488413\pi$
$877$	5.08782e32	0.0727841	0.0363920	+	0.999338i	$0.488413\pi$
$878$	0	0
$879$	−6.66713e31	−0.00929110
$880$	0	0
$881$	1.28302e34	1.74184	0.870922	−	0.491422i	$-0.163523\pi$
$881$	1.28302e34	1.74184	0.870922	+	0.491422i	$0.163523\pi$
$882$	0	0
$883$	1.12890e32	0.0149316	0.00746580	−	0.999972i	$-0.497624\pi$
$883$	1.12890e32	0.0149316	0.00746580	+	0.999972i	$0.497624\pi$
$884$	0	0
$885$	−6.40330e31	−0.00825194
$886$	0	0
$887$	−8.52749e32	−0.107078	−0.0535389	−	0.998566i	$-0.517050\pi$
$887$	−8.52749e32	−0.107078	−0.0535389	+	0.998566i	$0.517050\pi$
$888$	0	0
$889$	−1.48166e34	−1.81292
$890$	0	0
$891$	−3.92707e33	−0.468247
$892$	0	0
$893$	4.86889e32	0.0565768
$894$	0	0
$895$	4.67713e32	0.0529681
$896$	0	0
$897$	8.74267e30	0.000965009 0
$898$	0	0
$899$	−2.05427e34	−2.21014
$900$	0	0
$901$	2.05214e33	0.215215
$902$	0	0
$903$	−1.26816e32	−0.0129647
$904$	0	0
$905$	−8.19944e33	−0.817195
$906$	0	0
$907$	−1.04711e34	−1.01744	−0.508719	−	0.860932i	$-0.669881\pi$
$907$	−1.04711e34	−1.01744	−0.508719	+	0.860932i	$0.669881\pi$
$908$	0	0
$909$	−1.41626e34	−1.34171
$910$	0	0
$911$	−7.18890e32	−0.0664051	−0.0332025	−	0.999449i	$-0.510571\pi$
$911$	−7.18890e32	−0.0664051	−0.0332025	+	0.999449i	$0.510571\pi$
$912$	0	0
$913$	6.18825e33	0.557383
$914$	0	0
$915$	4.15914e31	0.00365310
$916$	0	0
$917$	2.83770e34	2.43063
$918$	0	0
$919$	−1.34022e34	−1.11956	−0.559780	−	0.828641i	$-0.689115\pi$
$919$	−1.34022e34	−1.11956	−0.559780	+	0.828641i	$0.689115\pi$
$920$	0	0
$921$	−2.36908e32	−0.0193016
$922$	0	0
$923$	−4.57713e32	−0.0363726
$924$	0	0
$925$	−1.45113e34	−1.12481
$926$	0	0
$927$	−1.99582e34	−1.50905
$928$	0	0
$929$	−1.11463e34	−0.822150	−0.411075	−	0.911602i	$-0.634847\pi$
$929$	−1.11463e34	−0.822150	−0.411075	+	0.911602i	$0.634847\pi$
$930$	0	0
$931$	5.05962e32	0.0364080
$932$	0	0
$933$	−9.75297e31	−0.00684697
$934$	0	0
$935$	6.32422e32	0.0433185
$936$	0	0
$937$	−7.01654e33	−0.468940	−0.234470	−	0.972123i	$-0.575336\pi$
$937$	−7.01654e33	−0.468940	−0.234470	+	0.972123i	$0.575336\pi$
$938$	0	0
$939$	2.12777e32	0.0138762
$940$	0	0
$941$	−1.03649e34	−0.659603	−0.329801	−	0.944050i	$-0.606982\pi$
$941$	−1.03649e34	−0.659603	−0.329801	+	0.944050i	$0.606982\pi$
$942$	0	0
$943$	−9.92050e33	−0.616097
$944$	0	0
$945$	3.32572e32	0.0201567
$946$	0	0
$947$	1.35194e34	0.799708	0.399854	−	0.916579i	$-0.369061\pi$
$947$	1.35194e34	0.799708	0.399854	+	0.916579i	$0.369061\pi$
$948$	0	0
$949$	2.48482e33	0.143461
$950$	0	0
$951$	1.21759e32	0.00686157
$952$	0	0
$953$	−3.88752e32	−0.0213848	−0.0106924	−	0.999943i	$-0.503404\pi$
$953$	−3.88752e32	−0.0213848	−0.0106924	+	0.999943i	$0.503404\pi$
$954$	0	0
$955$	2.47728e33	0.133026
$956$	0	0
$957$	1.40972e32	0.00739003
$958$	0	0
$959$	4.28114e32	0.0219101
$960$	0	0
$961$	3.72698e34	1.86225
$962$	0	0
$963$	1.08779e34	0.530691
$964$	0	0
$965$	1.19605e34	0.569750
$966$	0	0
$967$	1.19396e34	0.555373	0.277687	−	0.960672i	$-0.410432\pi$
$967$	1.19396e34	0.555373	0.277687	+	0.960672i	$0.410432\pi$
$968$	0	0
$969$	1.91575e30	8.70195e−5 0
$970$	0	0
$971$	−1.75730e34	−0.779517	−0.389758	−	0.920917i	$-0.627441\pi$
$971$	−1.75730e34	−0.779517	−0.389758	+	0.920917i	$0.627441\pi$
$972$	0	0
$973$	−2.07068e34	−0.897047
$974$	0	0
$975$	1.86593e31	0.000789482 0
$976$	0	0
$977$	2.46767e34	1.01976	0.509881	−	0.860245i	$-0.329689\pi$
$977$	2.46767e34	1.01976	0.509881	+	0.860245i	$0.329689\pi$
$978$	0	0
$979$	1.71969e34	0.694145
$980$	0	0
$981$	2.28488e33	0.0900886
$982$	0	0
$983$	4.26700e33	0.164345	0.0821727	−	0.996618i	$-0.473814\pi$
$983$	4.26700e33	0.164345	0.0821727	+	0.996618i	$0.473814\pi$
$984$	0	0
$985$	−5.26955e33	−0.198270
$986$	0	0
$987$	−4.96627e32	−0.0182551
$988$	0	0
$989$	−1.66079e34	−0.596427
$990$	0	0
$991$	2.56312e34	0.899336	0.449668	−	0.893196i	$-0.351542\pi$
$991$	2.56312e34	0.899336	0.449668	+	0.893196i	$0.351542\pi$
$992$	0	0
$993$	−2.95947e32	−0.0101461
$994$	0	0
$995$	2.29283e33	0.0768079
$996$	0	0
$997$	2.59336e33	0.0848924	0.0424462	−	0.999099i	$-0.486485\pi$
$997$	2.59336e33	0.0848924	0.0424462	+	0.999099i	$0.486485\pi$
$998$	0	0
$999$	1.41729e33	0.0453373

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.k.1.2		3
4.3	odd	2		64.24.a.h.1.2		3
8.3	odd	2		16.24.a.e.1.2		3
8.5	even	2		8.24.a.a.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.24.a.a.1.2	✓	3	8.5	even	2
16.24.a.e.1.2		3	8.3	odd	2
64.24.a.h.1.2		3	4.3	odd	2
64.24.a.k.1.2		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)

\(f(q)\)	\(=\)	\(q-3705.18 q^{3} +6.91268e7 q^{5} +6.89672e9 q^{7} -9.41295e10 q^{9} +O(q^{10})\)	\(q-3705.18 q^{3} +6.91268e7 q^{5} +6.89672e9 q^{7} -9.41295e10 q^{9} -4.43283e11 q^{11} +7.05085e11 q^{13} -2.56127e11 q^{15} -2.06386e13 q^{17} +2.50525e13 q^{19} -2.55536e13 q^{21} -3.34652e15 q^{23} -7.14241e15 q^{25} +6.97583e14 q^{27} +8.58309e16 q^{29} -2.39339e17 q^{31} +1.64244e15 q^{33} +4.76749e17 q^{35} +2.03171e18 q^{37} -2.61247e15 q^{39} +2.96442e18 q^{41} +4.96274e18 q^{43} -6.50687e18 q^{45} +1.94347e19 q^{47} +2.01961e19 q^{49} +7.64695e16 q^{51} -9.94325e19 q^{53} -3.06427e19 q^{55} -9.28240e16 q^{57} +2.50005e20 q^{59} -1.62386e20 q^{61} -6.49185e20 q^{63} +4.87403e19 q^{65} +1.42919e21 q^{67} +1.23995e19 q^{69} -6.49160e20 q^{71} +3.52414e21 q^{73} +2.64639e19 q^{75} -3.05720e21 q^{77} -2.14128e21 q^{79} +8.85906e21 q^{81} -1.39600e22 q^{83} -1.42668e21 q^{85} -3.18019e20 q^{87} -3.87945e22 q^{89} +4.86278e21 q^{91} +8.86793e20 q^{93} +1.73180e21 q^{95} -7.34335e22 q^{97} +4.17259e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 213948 q^{3} - 95628618 q^{5} - 8647912920 q^{7} + 174509823951 q^{9}+O(q^{10}) \) 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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