LMFDB - Embedded newform 64.24.a.o.1.5

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			1.5
Root			$8019.67$ 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)

$f(q)$	$=$	$q+340627. q^{3} +8.81822e7 q^{5} -8.68945e9 q^{7} +2.18833e10 q^{9} +O(q^{10})$	$q+340627. q^{3} +8.81822e7 q^{5} -8.68945e9 q^{7} +2.18833e10 q^{9} +1.20671e12 q^{11} -1.25514e13 q^{13} +3.00372e13 q^{15} +6.10300e13 q^{17} +2.20847e14 q^{19} -2.95986e15 q^{21} +5.85601e15 q^{23} -4.14484e15 q^{25} -2.46136e16 q^{27} +9.56100e16 q^{29} -7.90042e16 q^{31} +4.11039e17 q^{33} -7.66254e17 q^{35} -2.09565e18 q^{37} -4.27534e18 q^{39} +2.65725e18 q^{41} -1.82334e18 q^{43} +1.92972e18 q^{45} +1.31912e19 q^{47} +4.81378e19 q^{49} +2.07884e19 q^{51} -1.04772e19 q^{53} +1.06411e20 q^{55} +7.52263e19 q^{57} +1.26726e20 q^{59} +2.84167e20 q^{61} -1.90154e20 q^{63} -1.10681e21 q^{65} +9.17044e20 q^{67} +1.99471e21 q^{69} +1.83023e21 q^{71} +1.84172e21 q^{73} -1.41184e21 q^{75} -1.04857e22 q^{77} -4.20132e21 q^{79} -1.04442e22 q^{81} +5.94704e21 q^{83} +5.38176e21 q^{85} +3.25673e22 q^{87} +3.55224e22 q^{89} +1.09065e23 q^{91} -2.69109e22 q^{93} +1.94748e22 q^{95} +2.81006e22 q^{97} +2.64069e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	340627.	1.11016	0.555078	−	0.831798i	$-0.312688\pi$
$3$	340627.	1.11016	0.555078	+	0.831798i	$0.312688\pi$
$4$	0	0
$5$	8.81822e7	0.807655	0.403827	−	0.914835i	$-0.367680\pi$
$5$	8.81822e7	0.807655	0.403827	+	0.914835i	$0.367680\pi$
$6$	0	0
$7$	−8.68945e9	−1.66098	−0.830491	−	0.557032i	$-0.811940\pi$
$7$	−8.68945e9	−1.66098	−0.830491	+	0.557032i	$0.811940\pi$
$8$	0	0
$9$	2.18833e10	0.232447
$10$	0	0
$11$	1.20671e12	1.27523	0.637615	−	0.770355i	$-0.279921\pi$
$11$	1.20671e12	1.27523	0.637615	+	0.770355i	$0.279921\pi$
$12$	0	0
$13$	−1.25514e13	−1.94242	−0.971211	−	0.238219i	$-0.923436\pi$
$13$	−1.25514e13	−1.94242	−0.971211	+	0.238219i	$0.923436\pi$
$14$	0	0
$15$	3.00372e13	0.896623
$16$	0	0
$17$	6.10300e13	0.431897	0.215949	−	0.976405i	$-0.430716\pi$
$17$	6.10300e13	0.431897	0.215949	+	0.976405i	$0.430716\pi$
$18$	0	0
$19$	2.20847e14	0.434936	0.217468	−	0.976067i	$-0.430220\pi$
$19$	2.20847e14	0.434936	0.217468	+	0.976067i	$0.430220\pi$
$20$	0	0
$21$	−2.95986e15	−1.84395
$22$	0	0
$23$	5.85601e15	1.28154	0.640769	−	0.767734i	$-0.278616\pi$
$23$	5.85601e15	1.28154	0.640769	+	0.767734i	$0.278616\pi$
$24$	0	0
$25$	−4.14484e15	−0.347694
$26$	0	0
$27$	−2.46136e16	−0.852104
$28$	0	0
$29$	9.56100e16	1.45521	0.727606	−	0.685995i	$-0.240633\pi$
$29$	9.56100e16	1.45521	0.727606	+	0.685995i	$0.240633\pi$
$30$	0	0
$31$	−7.90042e16	−0.558458	−0.279229	−	0.960224i	$-0.590079\pi$
$31$	−7.90042e16	−0.558458	−0.279229	+	0.960224i	$0.590079\pi$
$32$	0	0
$33$	4.11039e17	1.41570
$34$	0	0
$35$	−7.66254e17	−1.34150
$36$	0	0
$37$	−2.09565e18	−1.93642	−0.968209	−	0.250142i	$-0.919523\pi$
$37$	−2.09565e18	−1.93642	−0.968209	+	0.250142i	$0.919523\pi$
$38$	0	0
$39$	−4.27534e18	−2.15639
$40$	0	0
$41$	2.65725e18	0.754081	0.377041	−	0.926197i	$-0.376942\pi$
$41$	2.65725e18	0.754081	0.377041	+	0.926197i	$0.376942\pi$
$42$	0	0
$43$	−1.82334e18	−0.299213	−0.149606	−	0.988746i	$-0.547801\pi$
$43$	−1.82334e18	−0.299213	−0.149606	+	0.988746i	$0.547801\pi$
$44$	0	0
$45$	1.92972e18	0.187737
$46$	0	0
$47$	1.31912e19	0.778319	0.389159	−	0.921170i	$-0.372766\pi$
$47$	1.31912e19	0.778319	0.389159	+	0.921170i	$0.372766\pi$
$48$	0	0
$49$	4.81378e19	1.75886
$50$	0	0
$51$	2.07884e19	0.479474
$52$	0	0
$53$	−1.04772e19	−0.155265	−0.0776324	−	0.996982i	$-0.524736\pi$
$53$	−1.04772e19	−0.155265	−0.0776324	+	0.996982i	$0.524736\pi$
$54$	0	0
$55$	1.06411e20	1.02995
$56$	0	0
$57$	7.52263e19	0.482846
$58$	0	0
$59$	1.26726e20	0.547099	0.273550	−	0.961858i	$-0.411802\pi$
$59$	1.26726e20	0.547099	0.273550	+	0.961858i	$0.411802\pi$
$60$	0	0
$61$	2.84167e20	0.836143	0.418071	−	0.908414i	$-0.362706\pi$
$61$	2.84167e20	0.836143	0.418071	+	0.908414i	$0.362706\pi$
$62$	0	0
$63$	−1.90154e20	−0.386090
$64$	0	0
$65$	−1.10681e21	−1.56881
$66$	0	0
$67$	9.17044e20	0.917339	0.458670	−	0.888607i	$-0.348326\pi$
$67$	9.17044e20	0.917339	0.458670	+	0.888607i	$0.348326\pi$
$68$	0	0
$69$	1.99471e21	1.42271
$70$	0	0
$71$	1.83023e21	0.939797	0.469899	−	0.882720i	$-0.344291\pi$
$71$	1.83023e21	0.939797	0.469899	+	0.882720i	$0.344291\pi$
$72$	0	0
$73$	1.84172e21	0.687085	0.343543	−	0.939137i	$-0.388373\pi$
$73$	1.84172e21	0.687085	0.343543	+	0.939137i	$0.388373\pi$
$74$	0	0
$75$	−1.41184e21	−0.385995
$76$	0	0
$77$	−1.04857e22	−2.11813
$78$	0	0
$79$	−4.20132e21	−0.631938	−0.315969	−	0.948770i	$-0.602330\pi$
$79$	−4.20132e21	−0.631938	−0.315969	+	0.948770i	$0.602330\pi$
$80$	0	0
$81$	−1.04442e22	−1.17842
$82$	0	0
$83$	5.94704e21	0.506877	0.253438	−	0.967352i	$-0.418438\pi$
$83$	5.94704e21	0.506877	0.253438	+	0.967352i	$0.418438\pi$
$84$	0	0
$85$	5.38176e21	0.348824
$86$	0	0
$87$	3.25673e22	1.61551
$88$	0	0
$89$	3.55224e22	1.35680	0.678402	−	0.734691i	$-0.262673\pi$
$89$	3.55224e22	1.35680	0.678402	+	0.734691i	$0.262673\pi$
$90$	0	0
$91$	1.09065e23	3.22633
$92$	0	0
$93$	−2.69109e22	−0.619976
$94$	0	0
$95$	1.94748e22	0.351278
$96$	0	0
$97$	2.81006e22	0.398879	0.199439	−	0.979910i	$-0.436088\pi$
$97$	2.81006e22	0.398879	0.199439	+	0.979910i	$0.436088\pi$
$98$	0	0
$99$	2.64069e22	0.296423
$100$	0	0
$101$	7.03798e22	0.627700	0.313850	−	0.949473i	$-0.398381\pi$
$101$	7.03798e22	0.627700	0.313850	+	0.949473i	$0.398381\pi$
$102$	0	0
$103$	7.32709e21	0.0521559	0.0260780	−	0.999660i	$-0.491698\pi$
$103$	7.32709e21	0.0521559	0.0260780	+	0.999660i	$0.491698\pi$
$104$	0	0
$105$	−2.61007e23	−1.48927
$106$	0	0
$107$	−3.97927e22	−0.182764	−0.0913820	−	0.995816i	$-0.529128\pi$
$107$	−3.97927e22	−0.182764	−0.0913820	+	0.995816i	$0.529128\pi$
$108$	0	0
$109$	3.68915e23	1.36937	0.684687	−	0.728838i	$-0.259939\pi$
$109$	3.68915e23	1.36937	0.684687	+	0.728838i	$0.259939\pi$
$110$	0	0
$111$	−7.13834e23	−2.14973
$112$	0	0
$113$	2.27047e23	0.556819	0.278410	−	0.960462i	$-0.410193\pi$
$113$	2.27047e23	0.556819	0.278410	+	0.960462i	$0.410193\pi$
$114$	0	0
$115$	5.16395e23	1.03504
$116$	0	0
$117$	−2.74666e23	−0.451510
$118$	0	0
$119$	−5.30317e23	−0.717374
$120$	0	0
$121$	5.60729e23	0.626211
$122$	0	0
$123$	9.05130e23	0.837148
$124$	0	0
$125$	−1.41671e24	−1.08847
$126$	0	0
$127$	−2.08078e24	−1.33193	−0.665967	−	0.745981i	$-0.731981\pi$
$127$	−2.08078e24	−1.33193	−0.665967	+	0.745981i	$0.731981\pi$
$128$	0	0
$129$	−6.21078e23	−0.332173
$130$	0	0
$131$	3.32142e24	1.48835	0.744173	−	0.667987i	$-0.232844\pi$
$131$	3.32142e24	1.48835	0.744173	+	0.667987i	$0.232844\pi$
$132$	0	0
$133$	−1.91904e24	−0.722420
$134$	0	0
$135$	−2.17048e24	−0.688206
$136$	0	0
$137$	−2.60841e24	−0.698375	−0.349188	−	0.937053i	$-0.613542\pi$
$137$	−2.60841e24	−0.698375	−0.349188	+	0.937053i	$0.613542\pi$
$138$	0	0
$139$	−6.38972e24	−1.44814	−0.724072	−	0.689724i	$-0.757732\pi$
$139$	−6.38972e24	−1.44814	−0.724072	+	0.689724i	$0.757732\pi$
$140$	0	0
$141$	4.49326e24	0.864056
$142$	0	0
$143$	−1.51460e25	−2.47704
$144$	0	0
$145$	8.43109e24	1.17531
$146$	0	0
$147$	1.63970e25	1.95261
$148$	0	0
$149$	1.15043e25	1.17279	0.586393	−	0.810027i	$-0.300547\pi$
$149$	1.15043e25	1.17279	0.586393	+	0.810027i	$0.300547\pi$
$150$	0	0
$151$	−7.39714e24	−0.646888	−0.323444	−	0.946247i	$-0.604841\pi$
$151$	−7.39714e24	−0.646888	−0.323444	+	0.946247i	$0.604841\pi$
$152$	0	0
$153$	1.33554e24	0.100393
$154$	0	0
$155$	−6.96676e24	−0.451042
$156$	0	0
$157$	−9.11900e24	−0.509449	−0.254725	−	0.967014i	$-0.581985\pi$
$157$	−9.11900e24	−0.509449	−0.254725	+	0.967014i	$0.581985\pi$
$158$	0	0
$159$	−3.56882e24	−0.172368
$160$	0	0
$161$	−5.08855e25	−2.12861
$162$	0	0
$163$	4.51803e25	1.63980	0.819901	−	0.572505i	$-0.194028\pi$
$163$	4.51803e25	1.63980	0.819901	+	0.572505i	$0.194028\pi$
$164$	0	0
$165$	3.62463e25	1.14340
$166$	0	0
$167$	6.25311e25	1.71734	0.858670	−	0.512528i	$-0.171291\pi$
$167$	6.25311e25	1.71734	0.858670	+	0.512528i	$0.171291\pi$
$168$	0	0
$169$	1.15784e26	2.77301
$170$	0	0
$171$	4.83286e24	0.101099
$172$	0	0
$173$	−7.30945e25	−1.33769	−0.668844	−	0.743403i	$-0.733210\pi$
$173$	−7.30945e25	−1.33769	−0.668844	+	0.743403i	$0.733210\pi$
$174$	0	0
$175$	3.60163e25	0.577513
$176$	0	0
$177$	4.31661e25	0.607366
$178$	0	0
$179$	−4.39733e25	−0.543725	−0.271863	−	0.962336i	$-0.587640\pi$
$179$	−4.39733e25	−0.543725	−0.271863	+	0.962336i	$0.587640\pi$
$180$	0	0
$181$	−6.96071e25	−0.757443	−0.378722	−	0.925511i	$-0.623636\pi$
$181$	−6.96071e25	−0.757443	−0.378722	+	0.925511i	$0.623636\pi$
$182$	0	0
$183$	9.67948e25	0.928249
$184$	0	0
$185$	−1.84799e26	−1.56396
$186$	0	0
$187$	7.36457e25	0.550769
$188$	0	0
$189$	2.13879e26	1.41533
$190$	0	0
$191$	−2.01749e25	−0.118284	−0.0591422	−	0.998250i	$-0.518837\pi$
$191$	−2.01749e25	−0.118284	−0.0591422	+	0.998250i	$0.518837\pi$
$192$	0	0
$193$	1.73417e25	0.0901949	0.0450974	−	0.998983i	$-0.485640\pi$
$193$	1.73417e25	0.0901949	0.0450974	+	0.998983i	$0.485640\pi$
$194$	0	0
$195$	−3.77009e26	−1.74162
$196$	0	0
$197$	1.91192e26	0.785429	0.392715	−	0.919660i	$-0.371536\pi$
$197$	1.91192e26	0.785429	0.392715	+	0.919660i	$0.371536\pi$
$198$	0	0
$199$	1.57891e26	0.577492	0.288746	−	0.957406i	$-0.406762\pi$
$199$	1.57891e26	0.577492	0.288746	+	0.957406i	$0.406762\pi$
$200$	0	0
$201$	3.12370e26	1.01839
$202$	0	0
$203$	−8.30798e26	−2.41708
$204$	0	0
$205$	2.34322e26	0.609037
$206$	0	0
$207$	1.28149e26	0.297890
$208$	0	0
$209$	2.66499e26	0.554643
$210$	0	0
$211$	−9.03609e25	−0.168551	−0.0842757	−	0.996442i	$-0.526858\pi$
$211$	−9.03609e25	−0.168551	−0.0842757	+	0.996442i	$0.526858\pi$
$212$	0	0
$213$	6.23424e26	1.04332
$214$	0	0
$215$	−1.60786e26	−0.241661
$216$	0	0
$217$	6.86503e26	0.927589
$218$	0	0
$219$	6.27339e26	0.762772
$220$	0	0
$221$	−7.66012e26	−0.838928
$222$	0	0
$223$	1.81413e27	1.79128	0.895640	−	0.444779i	$-0.146718\pi$
$223$	1.81413e27	1.79128	0.895640	+	0.444779i	$0.146718\pi$
$224$	0	0
$225$	−9.07027e25	−0.0808204
$226$	0	0
$227$	−7.43254e26	−0.598191	−0.299095	−	0.954223i	$-0.596685\pi$
$227$	−7.43254e26	−0.598191	−0.299095	+	0.954223i	$0.596685\pi$
$228$	0	0
$229$	1.72373e27	1.25418	0.627090	−	0.778946i	$-0.284246\pi$
$229$	1.72373e27	1.25418	0.627090	+	0.778946i	$0.284246\pi$
$230$	0	0
$231$	−3.57170e27	−2.35146
$232$	0	0
$233$	3.10007e27	1.84833	0.924164	−	0.381996i	$-0.124763\pi$
$233$	3.10007e27	1.84833	0.924164	+	0.381996i	$0.124763\pi$
$234$	0	0
$235$	1.16322e27	0.628613
$236$	0	0
$237$	−1.43108e27	−0.701550
$238$	0	0
$239$	3.03847e26	0.135232	0.0676160	−	0.997711i	$-0.478461\pi$
$239$	3.03847e26	0.135232	0.0676160	+	0.997711i	$0.478461\pi$
$240$	0	0
$241$	−2.76830e26	−0.111948	−0.0559740	−	0.998432i	$-0.517826\pi$
$241$	−2.76830e26	−0.111948	−0.0559740	+	0.998432i	$0.517826\pi$
$242$	0	0
$243$	−1.24037e27	−0.456121
$244$	0	0
$245$	4.24490e27	1.42055
$246$	0	0
$247$	−2.77194e27	−0.844829
$248$	0	0
$249$	2.02572e27	0.562713
$250$	0	0
$251$	1.12761e27	0.285701	0.142851	−	0.989744i	$-0.454373\pi$
$251$	1.12761e27	0.285701	0.142851	+	0.989744i	$0.454373\pi$
$252$	0	0
$253$	7.06653e27	1.63426
$254$	0	0
$255$	1.83317e27	0.387249
$256$	0	0
$257$	−1.50599e27	−0.290797	−0.145399	−	0.989373i	$-0.546447\pi$
$257$	−1.50599e27	−0.290797	−0.145399	+	0.989373i	$0.546447\pi$
$258$	0	0
$259$	1.82101e28	3.21636
$260$	0	0
$261$	2.09226e27	0.338260
$262$	0	0
$263$	3.82566e27	0.566519	0.283260	−	0.959043i	$-0.408584\pi$
$263$	3.82566e27	0.566519	0.283260	+	0.959043i	$0.408584\pi$
$264$	0	0
$265$	−9.23903e26	−0.125400
$266$	0	0
$267$	1.20999e28	1.50626
$268$	0	0
$269$	1.13837e28	1.30057	0.650285	−	0.759690i	$-0.274649\pi$
$269$	1.13837e28	1.30057	0.650285	+	0.759690i	$0.274649\pi$
$270$	0	0
$271$	−7.76643e27	−0.814844	−0.407422	−	0.913240i	$-0.633572\pi$
$271$	−7.76643e27	−0.814844	−0.407422	+	0.913240i	$0.633572\pi$
$272$	0	0
$273$	3.71504e28	3.58173
$274$	0	0
$275$	−5.00163e27	−0.443390
$276$	0	0
$277$	7.26831e26	0.0592811	0.0296405	−	0.999561i	$-0.490564\pi$
$277$	7.26831e26	0.0592811	0.0296405	+	0.999561i	$0.490564\pi$
$278$	0	0
$279$	−1.72887e27	−0.129812
$280$	0	0
$281$	2.43725e28	1.68569	0.842845	−	0.538157i	$-0.180879\pi$
$281$	2.43725e28	1.68569	0.842845	+	0.538157i	$0.180879\pi$
$282$	0	0
$283$	−2.27674e28	−1.45134	−0.725671	−	0.688042i	$-0.758470\pi$
$283$	−2.27674e28	−1.45134	−0.725671	+	0.688042i	$0.758470\pi$
$284$	0	0
$285$	6.63362e27	0.389973
$286$	0	0
$287$	−2.30901e28	−1.25252
$288$	0	0
$289$	−1.62429e28	−0.813465
$290$	0	0
$291$	9.57182e27	0.442818
$292$	0	0
$293$	9.45044e27	0.404087	0.202043	−	0.979377i	$-0.435242\pi$
$293$	9.45044e27	0.404087	0.202043	+	0.979377i	$0.435242\pi$
$294$	0	0
$295$	1.11749e28	0.441867
$296$	0	0
$297$	−2.97016e28	−1.08663
$298$	0	0
$299$	−7.35011e28	−2.48929
$300$	0	0
$301$	1.58438e28	0.496987
$302$	0	0
$303$	2.39732e28	0.696845
$304$	0	0
$305$	2.50585e28	0.675315
$306$	0	0
$307$	1.05756e28	0.264370	0.132185	−	0.991225i	$-0.457801\pi$
$307$	1.05756e28	0.264370	0.132185	+	0.991225i	$0.457801\pi$
$308$	0	0
$309$	2.49580e27	0.0579012
$310$	0	0
$311$	−6.42553e28	−1.38409	−0.692046	−	0.721854i	$-0.743290\pi$
$311$	−6.42553e28	−1.38409	−0.692046	+	0.721854i	$0.743290\pi$
$312$	0	0
$313$	6.62509e28	1.32566	0.662830	−	0.748770i	$-0.269355\pi$
$313$	6.62509e28	1.32566	0.662830	+	0.748770i	$0.269355\pi$
$314$	0	0
$315$	−1.67682e28	−0.311828
$316$	0	0
$317$	6.35816e28	1.09938	0.549692	−	0.835367i	$-0.314745\pi$
$317$	6.35816e28	1.09938	0.549692	+	0.835367i	$0.314745\pi$
$318$	0	0
$319$	1.15374e29	1.85573
$320$	0	0
$321$	−1.35545e28	−0.202897
$322$	0	0
$323$	1.34783e28	0.187848
$324$	0	0
$325$	5.20235e28	0.675369
$326$	0	0
$327$	1.25662e29	1.52022
$328$	0	0
$329$	−1.14624e29	−1.29277
$330$	0	0
$331$	−1.38442e29	−1.45629	−0.728143	−	0.685425i	$-0.759617\pi$
$331$	−1.38442e29	−1.45629	−0.728143	+	0.685425i	$0.759617\pi$
$332$	0	0
$333$	−4.58598e28	−0.450115
$334$	0	0
$335$	8.08669e28	0.740893
$336$	0	0
$337$	1.53430e29	1.31270	0.656352	−	0.754455i	$-0.272099\pi$
$337$	1.53430e29	1.31270	0.656352	+	0.754455i	$0.272099\pi$
$338$	0	0
$339$	7.73384e28	0.618156
$340$	0	0
$341$	−9.53355e28	−0.712163
$342$	0	0
$343$	−1.80472e29	−1.26045
$344$	0	0
$345$	1.75898e29	1.14906
$346$	0	0
$347$	−1.74317e28	−0.106549	−0.0532747	−	0.998580i	$-0.516966\pi$
$347$	−1.74317e28	−0.106549	−0.0532747	+	0.998580i	$0.516966\pi$
$348$	0	0
$349$	−7.65514e28	−0.437987	−0.218993	−	0.975726i	$-0.570277\pi$
$349$	−7.65514e28	−0.437987	−0.218993	+	0.975726i	$0.570277\pi$
$350$	0	0
$351$	3.08936e29	1.65515
$352$	0	0
$353$	−1.58464e29	−0.795281	−0.397640	−	0.917541i	$-0.630171\pi$
$353$	−1.58464e29	−0.795281	−0.397640	+	0.917541i	$0.630171\pi$
$354$	0	0
$355$	1.61393e29	0.759031
$356$	0	0
$357$	−1.80640e29	−0.796397
$358$	0	0
$359$	−3.89889e29	−1.61196	−0.805981	−	0.591941i	$-0.798362\pi$
$359$	−3.89889e29	−1.61196	−0.805981	+	0.591941i	$0.798362\pi$
$360$	0	0
$361$	−2.09056e29	−0.810831
$362$	0	0
$363$	1.90999e29	0.695193
$364$	0	0
$365$	1.62407e29	0.554928
$366$	0	0
$367$	1.14694e29	0.368028	0.184014	−	0.982924i	$-0.441091\pi$
$367$	1.14694e29	0.368028	0.184014	+	0.982924i	$0.441091\pi$
$368$	0	0
$369$	5.81494e28	0.175284
$370$	0	0
$371$	9.10412e28	0.257892
$372$	0	0
$373$	1.31351e29	0.349769	0.174884	−	0.984589i	$-0.444045\pi$
$373$	1.31351e29	0.349769	0.174884	+	0.984589i	$0.444045\pi$
$374$	0	0
$375$	−4.82570e29	−1.20837
$376$	0	0
$377$	−1.20004e30	−2.82664
$378$	0	0
$379$	4.48519e29	0.994099	0.497050	−	0.867722i	$-0.334417\pi$
$379$	4.48519e29	0.994099	0.497050	+	0.867722i	$0.334417\pi$
$380$	0	0
$381$	−7.08768e29	−1.47865
$382$	0	0
$383$	−2.39567e29	−0.470588	−0.235294	−	0.971924i	$-0.575605\pi$
$383$	−2.39567e29	−0.470588	−0.235294	+	0.971924i	$0.575605\pi$
$384$	0	0
$385$	−9.24650e29	−1.71072
$386$	0	0
$387$	−3.99007e28	−0.0695511
$388$	0	0
$389$	−7.24038e29	−1.18944	−0.594718	−	0.803934i	$-0.702736\pi$
$389$	−7.24038e29	−1.18944	−0.594718	+	0.803934i	$0.702736\pi$
$390$	0	0
$391$	3.57392e29	0.553493
$392$	0	0
$393$	1.13136e30	1.65230
$394$	0	0
$395$	−3.70482e29	−0.510388
$396$	0	0
$397$	2.84086e29	0.369282	0.184641	−	0.982806i	$-0.440888\pi$
$397$	2.84086e29	0.369282	0.184641	+	0.982806i	$0.440888\pi$
$398$	0	0
$399$	−6.53676e29	−0.801999
$400$	0	0
$401$	3.09276e29	0.358250	0.179125	−	0.983826i	$-0.442673\pi$
$401$	3.09276e29	0.358250	0.179125	+	0.983826i	$0.442673\pi$
$402$	0	0
$403$	9.91614e29	1.08476
$404$	0	0
$405$	−9.20994e29	−0.951753
$406$	0	0
$407$	−2.52885e30	−2.46938
$408$	0	0
$409$	3.80891e29	0.351546	0.175773	−	0.984431i	$-0.443757\pi$
$409$	3.80891e29	0.351546	0.175773	+	0.984431i	$0.443757\pi$
$410$	0	0
$411$	−8.88493e29	−0.775306
$412$	0	0
$413$	−1.10118e30	−0.908722
$414$	0	0
$415$	5.24422e29	0.409381
$416$	0	0
$417$	−2.17651e30	−1.60767
$418$	0	0
$419$	−9.25744e29	−0.647187	−0.323593	−	0.946196i	$-0.604891\pi$
$419$	−9.25744e29	−0.647187	−0.323593	+	0.946196i	$0.604891\pi$
$420$	0	0
$421$	2.34765e30	1.55378	0.776891	−	0.629635i	$-0.216796\pi$
$421$	2.34765e30	1.55378	0.776891	+	0.629635i	$0.216796\pi$
$422$	0	0
$423$	2.88666e29	0.180918
$424$	0	0
$425$	−2.52959e29	−0.150168
$426$	0	0
$427$	−2.46925e30	−1.38882
$428$	0	0
$429$	−5.15912e30	−2.74990
$430$	0	0
$431$	1.70610e30	0.862017	0.431008	−	0.902348i	$-0.358158\pi$
$431$	1.70610e30	0.862017	0.431008	+	0.902348i	$0.358158\pi$
$432$	0	0
$433$	−2.19791e30	−1.05293	−0.526465	−	0.850197i	$-0.676483\pi$
$433$	−2.19791e30	−1.05293	−0.526465	+	0.850197i	$0.676483\pi$
$434$	0	0
$435$	2.87185e30	1.30478
$436$	0	0
$437$	1.29328e30	0.557386
$438$	0	0
$439$	3.62941e30	1.48421	0.742103	−	0.670285i	$-0.233828\pi$
$439$	3.62941e30	1.48421	0.742103	+	0.670285i	$0.233828\pi$
$440$	0	0
$441$	1.05341e30	0.408842
$442$	0	0
$443$	−1.28577e30	−0.473719	−0.236860	−	0.971544i	$-0.576118\pi$
$443$	−1.28577e30	−0.473719	−0.236860	+	0.971544i	$0.576118\pi$
$444$	0	0
$445$	3.13244e30	1.09583
$446$	0	0
$447$	3.91868e30	1.30198
$448$	0	0
$449$	2.19913e30	0.694093	0.347047	−	0.937848i	$-0.387185\pi$
$449$	2.19913e30	0.694093	0.347047	+	0.937848i	$0.387185\pi$
$450$	0	0
$451$	3.20654e30	0.961627
$452$	0	0
$453$	−2.51966e30	−0.718147
$454$	0	0
$455$	9.61757e30	2.60576
$456$	0	0
$457$	−6.35571e30	−1.63730	−0.818650	−	0.574293i	$-0.805277\pi$
$457$	−6.35571e30	−1.63730	−0.818650	+	0.574293i	$0.805277\pi$
$458$	0	0
$459$	−1.50217e30	−0.368021
$460$	0	0
$461$	6.33133e30	1.47548	0.737742	−	0.675083i	$-0.235892\pi$
$461$	6.33133e30	1.47548	0.737742	+	0.675083i	$0.235892\pi$
$462$	0	0
$463$	−7.01448e30	−1.55530	−0.777650	−	0.628698i	$-0.783588\pi$
$463$	−7.01448e30	−1.55530	−0.777650	+	0.628698i	$0.783588\pi$
$464$	0	0
$465$	−2.37307e30	−0.500727
$466$	0	0
$467$	−2.07602e30	−0.416954	−0.208477	−	0.978027i	$-0.566851\pi$
$467$	−2.07602e30	−0.416954	−0.208477	+	0.978027i	$0.566851\pi$
$468$	0	0
$469$	−7.96861e30	−1.52368
$470$	0	0
$471$	−3.10617e30	−0.565569
$472$	0	0
$473$	−2.20025e30	−0.381565
$474$	0	0
$475$	−9.15374e29	−0.151225
$476$	0	0
$477$	−2.29276e29	−0.0360909
$478$	0	0
$479$	−1.74232e30	−0.261378	−0.130689	−	0.991423i	$-0.541719\pi$
$479$	−1.74232e30	−0.261378	−0.130689	+	0.991423i	$0.541719\pi$
$480$	0	0
$481$	2.63034e31	3.76134
$482$	0	0
$483$	−1.73329e31	−2.36309
$484$	0	0
$485$	2.47797e30	0.322156
$486$	0	0
$487$	−6.17353e30	−0.765510	−0.382755	−	0.923850i	$-0.625025\pi$
$487$	−6.17353e30	−0.765510	−0.382755	+	0.923850i	$0.625025\pi$
$488$	0	0
$489$	1.53896e31	1.82044
$490$	0	0
$491$	8.81707e30	0.995147	0.497574	−	0.867422i	$-0.334224\pi$
$491$	8.81707e30	0.995147	0.497574	+	0.867422i	$0.334224\pi$
$492$	0	0
$493$	5.83507e30	0.628502
$494$	0	0
$495$	2.32862e30	0.239408
$496$	0	0
$497$	−1.59037e31	−1.56099
$498$	0	0
$499$	−7.56335e30	−0.708856	−0.354428	−	0.935083i	$-0.615324\pi$
$499$	−7.56335e30	−0.708856	−0.354428	+	0.935083i	$0.615324\pi$
$500$	0	0
$501$	2.12997e31	1.90652
$502$	0	0
$503$	5.37249e30	0.459350	0.229675	−	0.973267i	$-0.426234\pi$
$503$	5.37249e30	0.459350	0.229675	+	0.973267i	$0.426234\pi$
$504$	0	0
$505$	6.20624e30	0.506965
$506$	0	0
$507$	3.94391e31	3.07847
$508$	0	0
$509$	−4.61117e27	−0.000343999 0	−0.000171999	−	1.00000i	$-0.500055\pi$
$509$	−4.61117e27	−0.000343999 0	−0.000171999		1.00000i	$0.500055\pi$
$510$	0	0
$511$	−1.60035e31	−1.14124
$512$	0	0
$513$	−5.43585e30	−0.370610
$514$	0	0
$515$	6.46119e29	0.0421240
$516$	0	0
$517$	1.59180e31	0.992535
$518$	0	0
$519$	−2.48979e31	−1.48504
$520$	0	0
$521$	1.51435e30	0.0864156	0.0432078	−	0.999066i	$-0.486242\pi$
$521$	1.51435e30	0.0864156	0.0432078	+	0.999066i	$0.486242\pi$
$522$	0	0
$523$	−2.06888e31	−1.12971	−0.564854	−	0.825191i	$-0.691068\pi$
$523$	−2.06888e31	−1.12971	−0.564854	+	0.825191i	$0.691068\pi$
$524$	0	0
$525$	1.22681e31	0.641130
$526$	0	0
$527$	−4.82163e30	−0.241197
$528$	0	0
$529$	1.34123e31	0.642339
$530$	0	0
$531$	2.77317e30	0.127172
$532$	0	0
$533$	−3.33522e31	−1.46475
$534$	0	0
$535$	−3.50901e30	−0.147610
$536$	0	0
$537$	−1.49785e31	−0.603620
$538$	0	0
$539$	5.80886e31	2.24295
$540$	0	0
$541$	2.64799e31	0.979824	0.489912	−	0.871772i	$-0.337029\pi$
$541$	2.64799e31	0.979824	0.489912	+	0.871772i	$0.337029\pi$
$542$	0	0
$543$	−2.37100e31	−0.840881
$544$	0	0
$545$	3.25317e31	1.10598
$546$	0	0
$547$	−1.01796e31	−0.331800	−0.165900	−	0.986143i	$-0.553053\pi$
$547$	−1.01796e31	−0.331800	−0.165900	+	0.986143i	$0.553053\pi$
$548$	0	0
$549$	6.21851e30	0.194359
$550$	0	0
$551$	2.11152e31	0.632923
$552$	0	0
$553$	3.65072e31	1.04964
$554$	0	0
$555$	−6.29475e31	−1.73624
$556$	0	0
$557$	7.11797e29	0.0188374	0.00941871	−	0.999956i	$-0.497002\pi$
$557$	7.11797e29	0.0188374	0.00941871	+	0.999956i	$0.497002\pi$
$558$	0	0
$559$	2.28855e31	0.581198
$560$	0	0
$561$	2.50857e31	0.611439
$562$	0	0
$563$	2.46264e31	0.576174	0.288087	−	0.957604i	$-0.406981\pi$
$563$	2.46264e31	0.576174	0.288087	+	0.957604i	$0.406981\pi$
$564$	0	0
$565$	2.00215e31	0.449718
$566$	0	0
$567$	9.07546e31	1.95733
$568$	0	0
$569$	−4.12510e30	−0.0854364	−0.0427182	−	0.999087i	$-0.513602\pi$
$569$	−4.12510e30	−0.0854364	−0.0427182	+	0.999087i	$0.513602\pi$
$570$	0	0
$571$	6.44326e31	1.28171	0.640856	−	0.767661i	$-0.278580\pi$
$571$	6.44326e31	1.28171	0.640856	+	0.767661i	$0.278580\pi$
$572$	0	0
$573$	−6.87211e30	−0.131314
$574$	0	0
$575$	−2.42722e31	−0.445583
$576$	0	0
$577$	−5.81944e31	−1.02650	−0.513250	−	0.858239i	$-0.671559\pi$
$577$	−5.81944e31	−1.02650	−0.513250	+	0.858239i	$0.671559\pi$
$578$	0	0
$579$	5.90703e30	0.100130
$580$	0	0
$581$	−5.16765e31	−0.841913
$582$	0	0
$583$	−1.26430e31	−0.197998
$584$	0	0
$585$	−2.42207e31	−0.364664
$586$	0	0
$587$	−7.80377e31	−1.12971	−0.564854	−	0.825191i	$-0.691067\pi$
$587$	−7.80377e31	−1.12971	−0.564854	+	0.825191i	$0.691067\pi$
$588$	0	0
$589$	−1.74478e31	−0.242893
$590$	0	0
$591$	6.51249e31	0.871949
$592$	0	0
$593$	7.88203e31	1.01510	0.507550	−	0.861622i	$-0.330551\pi$
$593$	7.88203e31	1.01510	0.507550	+	0.861622i	$0.330551\pi$
$594$	0	0
$595$	−4.67645e31	−0.579390
$596$	0	0
$597$	5.37817e31	0.641106
$598$	0	0
$599$	9.02537e31	1.03528	0.517638	−	0.855599i	$-0.326811\pi$
$599$	9.02537e31	1.03528	0.517638	+	0.855599i	$0.326811\pi$
$600$	0	0
$601$	−6.99470e30	−0.0772170	−0.0386085	−	0.999254i	$-0.512293\pi$
$601$	−6.99470e30	−0.0772170	−0.0386085	+	0.999254i	$0.512293\pi$
$602$	0	0
$603$	2.00680e31	0.213233
$604$	0	0
$605$	4.94463e31	0.505763
$606$	0	0
$607$	1.21642e31	0.119788	0.0598940	−	0.998205i	$-0.480924\pi$
$607$	1.21642e31	0.119788	0.0598940	+	0.998205i	$0.480924\pi$
$608$	0	0
$609$	−2.82992e32	−2.68334
$610$	0	0
$611$	−1.65568e32	−1.51182
$612$	0	0
$613$	2.32749e31	0.204688	0.102344	−	0.994749i	$-0.467366\pi$
$613$	2.32749e31	0.204688	0.102344	+	0.994749i	$0.467366\pi$
$614$	0	0
$615$	7.98164e31	0.676127
$616$	0	0
$617$	−1.02201e31	−0.0834020	−0.0417010	−	0.999130i	$-0.513278\pi$
$617$	−1.02201e31	−0.0834020	−0.0417010	+	0.999130i	$0.513278\pi$
$618$	0	0
$619$	−1.95465e32	−1.53683	−0.768415	−	0.639952i	$-0.778954\pi$
$619$	−1.95465e32	−1.53683	−0.768415	+	0.639952i	$0.778954\pi$
$620$	0	0
$621$	−1.44138e32	−1.09200
$622$	0	0
$623$	−3.08670e32	−2.25363
$624$	0	0
$625$	−7.55186e31	−0.531415
$626$	0	0
$627$	9.07767e31	0.615740
$628$	0	0
$629$	−1.27898e32	−0.836334
$630$	0	0
$631$	−2.63924e31	−0.166396	−0.0831978	−	0.996533i	$-0.526513\pi$
$631$	−2.63924e31	−0.166396	−0.0831978	+	0.996533i	$0.526513\pi$
$632$	0	0
$633$	−3.07793e31	−0.187118
$634$	0	0
$635$	−1.83487e32	−1.07574
$636$	0	0
$637$	−6.04197e32	−3.41645
$638$	0	0
$639$	4.00514e31	0.218453
$640$	0	0
$641$	−2.72987e31	−0.143640	−0.0718198	−	0.997418i	$-0.522881\pi$
$641$	−2.72987e31	−0.143640	−0.0718198	+	0.997418i	$0.522881\pi$
$642$	0	0
$643$	6.39170e31	0.324482	0.162241	−	0.986751i	$-0.448128\pi$
$643$	6.39170e31	0.324482	0.162241	+	0.986751i	$0.448128\pi$
$644$	0	0
$645$	−5.47680e31	−0.268281
$646$	0	0
$647$	−3.40848e32	−1.61124	−0.805621	−	0.592431i	$-0.798168\pi$
$647$	−3.40848e32	−1.61124	−0.805621	+	0.592431i	$0.798168\pi$
$648$	0	0
$649$	1.52922e32	0.697678
$650$	0	0
$651$	2.33841e32	1.02977
$652$	0	0
$653$	2.54363e32	1.08132	0.540658	−	0.841242i	$-0.318175\pi$
$653$	2.54363e32	1.08132	0.540658	+	0.841242i	$0.318175\pi$
$654$	0	0
$655$	2.92890e32	1.20207
$656$	0	0
$657$	4.03029e31	0.159711
$658$	0	0
$659$	3.31885e32	1.27001	0.635003	−	0.772510i	$-0.280999\pi$
$659$	3.31885e32	1.27001	0.635003	+	0.772510i	$0.280999\pi$
$660$	0	0
$661$	−1.25983e31	−0.0465582	−0.0232791	−	0.999729i	$-0.507411\pi$
$661$	−1.25983e31	−0.0465582	−0.0232791	+	0.999729i	$0.507411\pi$
$662$	0	0
$663$	−2.60924e32	−0.931341
$664$	0	0
$665$	−1.69225e32	−0.583466
$666$	0	0
$667$	5.59893e32	1.86491
$668$	0	0
$669$	6.17942e32	1.98860
$670$	0	0
$671$	3.42908e32	1.06627
$672$	0	0
$673$	−4.29458e32	−1.29047	−0.645234	−	0.763985i	$-0.723240\pi$
$673$	−4.29458e32	−1.29047	−0.645234	+	0.763985i	$0.723240\pi$
$674$	0	0
$675$	1.02019e32	0.296271
$676$	0	0
$677$	−3.30342e32	−0.927245	−0.463622	−	0.886033i	$-0.653451\pi$
$677$	−3.30342e32	−0.927245	−0.463622	+	0.886033i	$0.653451\pi$
$678$	0	0
$679$	−2.44179e32	−0.662531
$680$	0	0
$681$	−2.53172e32	−0.664085
$682$	0	0
$683$	−6.63921e32	−1.68375	−0.841876	−	0.539671i	$-0.818548\pi$
$683$	−6.63921e32	−1.68375	−0.841876	+	0.539671i	$0.818548\pi$
$684$	0	0
$685$	−2.30015e32	−0.564046
$686$	0	0
$687$	5.87147e32	1.39234
$688$	0	0
$689$	1.31504e32	0.301590
$690$	0	0
$691$	7.78066e32	1.72591	0.862956	−	0.505279i	$-0.168610\pi$
$691$	7.78066e32	1.72591	0.862956	+	0.505279i	$0.168610\pi$
$692$	0	0
$693$	−2.29461e32	−0.492354
$694$	0	0
$695$	−5.63459e32	−1.16960
$696$	0	0
$697$	1.62172e32	0.325686
$698$	0	0
$699$	1.05597e33	2.05193
$700$	0	0
$701$	2.42065e32	0.455171	0.227585	−	0.973758i	$-0.426917\pi$
$701$	2.42065e32	0.455171	0.227585	+	0.973758i	$0.426917\pi$
$702$	0	0
$703$	−4.62818e32	−0.842217
$704$	0	0
$705$	3.96225e32	0.697858
$706$	0	0
$707$	−6.11562e32	−1.04260
$708$	0	0
$709$	1.37942e32	0.227648	0.113824	−	0.993501i	$-0.463690\pi$
$709$	1.37942e32	0.227648	0.113824	+	0.993501i	$0.463690\pi$
$710$	0	0
$711$	−9.19388e31	−0.146892
$712$	0	0
$713$	−4.62649e32	−0.715686
$714$	0	0
$715$	−1.33560e33	−2.00059
$716$	0	0
$717$	1.03498e32	0.150129
$718$	0	0
$719$	1.65919e31	0.0233085	0.0116543	−	0.999932i	$-0.496290\pi$
$719$	1.65919e31	0.0233085	0.0116543	+	0.999932i	$0.496290\pi$
$720$	0	0
$721$	−6.36684e31	−0.0866300
$722$	0	0
$723$	−9.42955e31	−0.124280
$724$	0	0
$725$	−3.96288e32	−0.505968
$726$	0	0
$727$	−9.86135e32	−1.21981	−0.609903	−	0.792476i	$-0.708792\pi$
$727$	−9.86135e32	−1.21981	−0.609903	+	0.792476i	$0.708792\pi$
$728$	0	0
$729$	5.60748e32	0.672049
$730$	0	0
$731$	−1.11278e32	−0.129229
$732$	0	0
$733$	8.48526e32	0.954925	0.477463	−	0.878652i	$-0.341557\pi$
$733$	8.48526e32	0.954925	0.477463	+	0.878652i	$0.341557\pi$
$734$	0	0
$735$	1.44592e33	1.57703
$736$	0	0
$737$	1.10661e33	1.16982
$738$	0	0
$739$	7.18742e31	0.0736482	0.0368241	−	0.999322i	$-0.488276\pi$
$739$	7.18742e31	0.0736482	0.0368241	+	0.999322i	$0.488276\pi$
$740$	0	0
$741$	−9.44196e32	−0.937892
$742$	0	0
$743$	1.86324e33	1.79431	0.897153	−	0.441720i	$-0.145631\pi$
$743$	1.86324e33	1.79431	0.897153	+	0.441720i	$0.145631\pi$
$744$	0	0
$745$	1.01448e33	0.947206
$746$	0	0
$747$	1.30141e32	0.117822
$748$	0	0
$749$	3.45777e32	0.303568
$750$	0	0
$751$	−6.95114e32	−0.591829	−0.295915	−	0.955214i	$-0.595624\pi$
$751$	−6.95114e32	−0.591829	−0.295915	+	0.955214i	$0.595624\pi$
$752$	0	0
$753$	3.84095e32	0.317173
$754$	0	0
$755$	−6.52296e32	−0.522462
$756$	0	0
$757$	1.49058e33	1.15812	0.579059	−	0.815286i	$-0.303420\pi$
$757$	1.49058e33	1.15812	0.579059	+	0.815286i	$0.303420\pi$
$758$	0	0
$759$	2.40705e33	1.81428
$760$	0	0
$761$	−8.32636e32	−0.608880	−0.304440	−	0.952531i	$-0.598469\pi$
$761$	−8.32636e32	−0.608880	−0.304440	+	0.952531i	$0.598469\pi$
$762$	0	0
$763$	−3.20567e33	−2.27450
$764$	0	0
$765$	1.17771e32	0.0810831
$766$	0	0
$767$	−1.59058e33	−1.06270
$768$	0	0
$769$	−1.64117e33	−1.06414	−0.532072	−	0.846699i	$-0.678586\pi$
$769$	−1.64117e33	−1.06414	−0.532072	+	0.846699i	$0.678586\pi$
$770$	0	0
$771$	−5.12980e32	−0.322831
$772$	0	0
$773$	3.05094e33	1.86367	0.931836	−	0.362881i	$-0.118207\pi$
$773$	3.05094e33	1.86367	0.931836	+	0.362881i	$0.118207\pi$
$774$	0	0
$775$	3.27460e32	0.194173
$776$	0	0
$777$	6.20283e33	3.57066
$778$	0	0
$779$	5.86846e32	0.327977
$780$	0	0
$781$	2.20856e33	1.19846
$782$	0	0
$783$	−2.35331e33	−1.23999
$784$	0	0
$785$	−8.04133e32	−0.411459
$786$	0	0
$787$	−7.22068e32	−0.358814	−0.179407	−	0.983775i	$-0.557418\pi$
$787$	−7.22068e32	−0.358814	−0.179407	+	0.983775i	$0.557418\pi$
$788$	0	0
$789$	1.30312e33	0.628925
$790$	0	0
$791$	−1.97292e33	−0.924867
$792$	0	0
$793$	−3.56669e33	−1.62414
$794$	0	0
$795$	−3.14706e32	−0.139214
$796$	0	0
$797$	1.89134e33	0.812828	0.406414	−	0.913689i	$-0.366779\pi$
$797$	1.89134e33	0.812828	0.406414	+	0.913689i	$0.366779\pi$
$798$	0	0
$799$	8.05056e32	0.336154
$800$	0	0
$801$	7.77347e32	0.315385
$802$	0	0
$803$	2.22243e33	0.876192
$804$	0	0
$805$	−4.48719e33	−1.71918
$806$	0	0
$807$	3.87760e33	1.44384
$808$	0	0
$809$	3.49570e33	1.26510	0.632552	−	0.774518i	$-0.282007\pi$
$809$	3.49570e33	1.26510	0.632552	+	0.774518i	$0.282007\pi$
$810$	0	0
$811$	8.88899e32	0.312689	0.156344	−	0.987703i	$-0.450029\pi$
$811$	8.88899e32	0.312689	0.156344	+	0.987703i	$0.450029\pi$
$812$	0	0
$813$	−2.64545e33	−0.904605
$814$	0	0
$815$	3.98409e33	1.32439
$816$	0	0
$817$	−4.02679e32	−0.130138
$818$	0	0
$819$	2.38670e33	0.749951
$820$	0	0
$821$	−6.23273e33	−1.90428	−0.952142	−	0.305655i	$-0.901125\pi$
$821$	−6.23273e33	−1.90428	−0.952142	+	0.305655i	$0.901125\pi$
$822$	0	0
$823$	−2.75064e33	−0.817214	−0.408607	−	0.912710i	$-0.633985\pi$
$823$	−2.75064e33	−0.817214	−0.408607	+	0.912710i	$0.633985\pi$
$824$	0	0
$825$	−1.70369e33	−0.492232
$826$	0	0
$827$	4.03024e33	1.13245	0.566224	−	0.824252i	$-0.308404\pi$
$827$	4.03024e33	1.13245	0.566224	+	0.824252i	$0.308404\pi$
$828$	0	0
$829$	6.75700e32	0.184662	0.0923308	−	0.995728i	$-0.470568\pi$
$829$	6.75700e32	0.184662	0.0923308	+	0.995728i	$0.470568\pi$
$830$	0	0
$831$	2.47578e32	0.0658112
$832$	0	0
$833$	2.93785e33	0.759647
$834$	0	0
$835$	5.51412e33	1.38702
$836$	0	0
$837$	1.94458e33	0.475865
$838$	0	0
$839$	−9.11612e32	−0.217044	−0.108522	−	0.994094i	$-0.534612\pi$
$839$	−9.11612e32	−0.217044	−0.108522	+	0.994094i	$0.534612\pi$
$840$	0	0
$841$	4.82454e33	1.11764
$842$	0	0
$843$	8.30192e33	1.87138
$844$	0	0
$845$	1.02101e34	2.23963
$846$	0	0
$847$	−4.87242e33	−1.04013
$848$	0	0
$849$	−7.75519e33	−1.61122
$850$	0	0
$851$	−1.22721e34	−2.48159
$852$	0	0
$853$	1.72683e33	0.339889	0.169944	−	0.985454i	$-0.445641\pi$
$853$	1.72683e33	0.339889	0.169944	+	0.985454i	$0.445641\pi$
$854$	0	0
$855$	4.26172e32	0.0816535
$856$	0	0
$857$	−4.84834e33	−0.904302	−0.452151	−	0.891941i	$-0.649343\pi$
$857$	−4.84834e33	−0.904302	−0.452151	+	0.891941i	$0.649343\pi$
$858$	0	0
$859$	3.04672e33	0.553236	0.276618	−	0.960980i	$-0.410786\pi$
$859$	3.04672e33	0.553236	0.276618	+	0.960980i	$0.410786\pi$
$860$	0	0
$861$	−7.86509e33	−1.39049
$862$	0	0
$863$	−8.56648e33	−1.47461	−0.737307	−	0.675558i	$-0.763903\pi$
$863$	−8.56648e33	−1.47461	−0.737307	+	0.675558i	$0.763903\pi$
$864$	0	0
$865$	−6.44563e33	−1.08039
$866$	0	0
$867$	−5.53277e33	−0.903073
$868$	0	0
$869$	−5.06980e33	−0.805866
$870$	0	0
$871$	−1.15102e34	−1.78186
$872$	0	0
$873$	6.14935e32	0.0927182
$874$	0	0
$875$	1.23105e34	1.80793
$876$	0	0
$877$	−7.54302e33	−1.07907	−0.539535	−	0.841963i	$-0.681400\pi$
$877$	−7.54302e33	−1.07907	−0.539535	+	0.841963i	$0.681400\pi$
$878$	0	0
$879$	3.21907e33	0.448599
$880$	0	0
$881$	−4.20317e33	−0.570629	−0.285315	−	0.958434i	$-0.592098\pi$
$881$	−4.20317e33	−0.570629	−0.285315	+	0.958434i	$0.592098\pi$
$882$	0	0
$883$	−7.59687e33	−1.00482	−0.502408	−	0.864630i	$-0.667553\pi$
$883$	−7.59687e33	−1.00482	−0.502408	+	0.864630i	$0.667553\pi$
$884$	0	0
$885$	3.80648e33	0.490542
$886$	0	0
$887$	8.87296e33	1.11416	0.557079	−	0.830459i	$-0.311922\pi$
$887$	8.87296e33	1.11416	0.557079	+	0.830459i	$0.311922\pi$
$888$	0	0
$889$	1.80808e34	2.21232
$890$	0	0
$891$	−1.26032e34	−1.50275
$892$	0	0
$893$	2.91323e33	0.338519
$894$	0	0
$895$	−3.87766e33	−0.439142
$896$	0	0
$897$	−2.50364e34	−2.76350
$898$	0	0
$899$	−7.55359e33	−0.812675
$900$	0	0
$901$	−6.39424e32	−0.0670585
$902$	0	0
$903$	5.39683e33	0.551733
$904$	0	0
$905$	−6.13811e33	−0.611753
$906$	0	0
$907$	−2.93488e33	−0.285172	−0.142586	−	0.989782i	$-0.545542\pi$
$907$	−2.93488e33	−0.285172	−0.142586	+	0.989782i	$0.545542\pi$
$908$	0	0
$909$	1.54014e33	0.145907
$910$	0	0
$911$	−2.22675e33	−0.205689	−0.102844	−	0.994697i	$-0.532794\pi$
$911$	−2.22675e33	−0.205689	−0.102844	+	0.994697i	$0.532794\pi$
$912$	0	0
$913$	7.17637e33	0.646385
$914$	0	0
$915$	8.53558e33	0.749705
$916$	0	0
$917$	−2.88613e34	−2.47211
$918$	0	0
$919$	1.21822e34	1.01764	0.508822	−	0.860872i	$-0.330081\pi$
$919$	1.21822e34	1.01764	0.508822	+	0.860872i	$0.330081\pi$
$920$	0	0
$921$	3.60231e33	0.293492
$922$	0	0
$923$	−2.29719e34	−1.82548
$924$	0	0
$925$	8.68613e33	0.673281
$926$	0	0
$927$	1.60341e32	0.0121235
$928$	0	0
$929$	−7.28510e33	−0.537347	−0.268673	−	0.963231i	$-0.586585\pi$
$929$	−7.28510e33	−0.537347	−0.268673	+	0.963231i	$0.586585\pi$
$930$	0	0
$931$	1.06311e34	0.764991
$932$	0	0
$933$	−2.18871e34	−1.53656
$934$	0	0
$935$	6.49424e33	0.444831
$936$	0	0
$937$	−1.00574e34	−0.672173	−0.336087	−	0.941831i	$-0.609104\pi$
$937$	−1.00574e34	−0.672173	−0.336087	+	0.941831i	$0.609104\pi$
$938$	0	0
$939$	2.25668e34	1.47169
$940$	0	0
$941$	−1.25828e34	−0.800750	−0.400375	−	0.916351i	$-0.631120\pi$
$941$	−1.25828e34	−0.800750	−0.400375	+	0.916351i	$0.631120\pi$
$942$	0	0
$943$	1.55609e34	0.966384
$944$	0	0
$945$	1.88603e34	1.14310
$946$	0	0
$947$	−1.38583e34	−0.819754	−0.409877	−	0.912141i	$-0.634428\pi$
$947$	−1.38583e34	−0.819754	−0.409877	+	0.912141i	$0.634428\pi$
$948$	0	0
$949$	−2.31162e34	−1.33461
$950$	0	0
$951$	2.16576e34	1.22049
$952$	0	0
$953$	−2.56485e34	−1.41089	−0.705447	−	0.708763i	$-0.749254\pi$
$953$	−2.56485e34	−1.41089	−0.705447	+	0.708763i	$0.749254\pi$
$954$	0	0
$955$	−1.77907e33	−0.0955330
$956$	0	0
$957$	3.92994e34	2.06015
$958$	0	0
$959$	2.26656e34	1.15999
$960$	0	0
$961$	−1.37716e34	−0.688124
$962$	0	0
$963$	−8.70797e32	−0.0424829
$964$	0	0
$965$	1.52923e33	0.0728463
$966$	0	0
$967$	−3.46204e34	−1.61038	−0.805189	−	0.593018i	$-0.797936\pi$
$967$	−3.46204e34	−1.61038	−0.805189	+	0.593018i	$0.797936\pi$
$968$	0	0
$969$	4.59106e33	0.208540
$970$	0	0
$971$	−2.98018e34	−1.32197	−0.660986	−	0.750399i	$-0.729862\pi$
$971$	−2.98018e34	−1.32197	−0.660986	+	0.750399i	$0.729862\pi$
$972$	0	0
$973$	5.55232e34	2.40534
$974$	0	0
$975$	1.77206e34	0.749765
$976$	0	0
$977$	3.27297e33	0.135255	0.0676277	−	0.997711i	$-0.478457\pi$
$977$	3.27297e33	0.135255	0.0676277	+	0.997711i	$0.478457\pi$
$978$	0	0
$979$	4.28653e34	1.73024
$980$	0	0
$981$	8.07308e33	0.318307
$982$	0	0
$983$	−1.99566e34	−0.768635	−0.384317	−	0.923201i	$-0.625563\pi$
$983$	−1.99566e34	−0.768635	−0.384317	+	0.923201i	$0.625563\pi$
$984$	0	0
$985$	1.68597e34	0.634355
$986$	0	0
$987$	−3.90440e34	−1.43518
$988$	0	0
$989$	−1.06775e34	−0.383453
$990$	0	0
$991$	−4.07384e34	−1.42941	−0.714705	−	0.699426i	$-0.753439\pi$
$991$	−4.07384e34	−1.42941	−0.714705	+	0.699426i	$0.753439\pi$
$992$	0	0
$993$	−4.71571e34	−1.61671
$994$	0	0
$995$	1.39231e34	0.466414
$996$	0	0
$997$	−7.51486e33	−0.245995	−0.122998	−	0.992407i	$-0.539251\pi$
$997$	−7.51486e33	−0.245995	−0.122998	+	0.992407i	$0.539251\pi$
$998$	0	0
$999$	5.15816e34	1.65003

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.o.1.5		6
4.3	odd	2		64.24.a.m.1.2		6
8.3	odd	2		32.24.a.e.1.5	yes	6
8.5	even	2		32.24.a.c.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.c.1.2	✓	6	8.5	even	2
32.24.a.e.1.5	yes	6	8.3	odd	2
64.24.a.m.1.2		6	4.3	odd	2
64.24.a.o.1.5		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)

\(f(q)\)	\(=\)	\(q+340627. q^{3} +8.81822e7 q^{5} -8.68945e9 q^{7} +2.18833e10 q^{9} +O(q^{10})\)	\(q+340627. q^{3} +8.81822e7 q^{5} -8.68945e9 q^{7} +2.18833e10 q^{9} +1.20671e12 q^{11} -1.25514e13 q^{13} +3.00372e13 q^{15} +6.10300e13 q^{17} +2.20847e14 q^{19} -2.95986e15 q^{21} +5.85601e15 q^{23} -4.14484e15 q^{25} -2.46136e16 q^{27} +9.56100e16 q^{29} -7.90042e16 q^{31} +4.11039e17 q^{33} -7.66254e17 q^{35} -2.09565e18 q^{37} -4.27534e18 q^{39} +2.65725e18 q^{41} -1.82334e18 q^{43} +1.92972e18 q^{45} +1.31912e19 q^{47} +4.81378e19 q^{49} +2.07884e19 q^{51} -1.04772e19 q^{53} +1.06411e20 q^{55} +7.52263e19 q^{57} +1.26726e20 q^{59} +2.84167e20 q^{61} -1.90154e20 q^{63} -1.10681e21 q^{65} +9.17044e20 q^{67} +1.99471e21 q^{69} +1.83023e21 q^{71} +1.84172e21 q^{73} -1.41184e21 q^{75} -1.04857e22 q^{77} -4.20132e21 q^{79} -1.04442e22 q^{81} +5.94704e21 q^{83} +5.38176e21 q^{85} +3.25673e22 q^{87} +3.55224e22 q^{89} +1.09065e23 q^{91} -2.69109e22 q^{93} +1.94748e22 q^{95} +2.81006e22 q^{97} +2.64069e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9}+O(q^{10}) \) 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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