LMFDB - Embedded newform 8.16.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8,16,Mod(1,8)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("8.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 8 = 2^{3} $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	8.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:	$11.4154804080$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{58}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 58 $ x^2 - 58
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 5 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$7.61577$ of defining polynomial
Character	$\chi$	$=$	8.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2838.09 q^{3} +105337. q^{5} +501765. q^{7} -6.29412e6 q^{9} +O(q^{10})$	$q+2838.09 q^{3} +105337. q^{5} +501765. q^{7} -6.29412e6 q^{9} +9.46973e7 q^{11} +2.61309e8 q^{13} +2.98958e8 q^{15} +2.42709e9 q^{17} +1.15575e9 q^{19} +1.42406e9 q^{21} -2.23730e10 q^{23} -1.94216e10 q^{25} -5.85869e10 q^{27} +1.12679e11 q^{29} -1.67382e11 q^{31} +2.68760e11 q^{33} +5.28546e10 q^{35} +7.53428e11 q^{37} +7.41619e11 q^{39} -1.42319e12 q^{41} -1.58502e12 q^{43} -6.63007e11 q^{45} -1.13188e12 q^{47} -4.49579e12 q^{49} +6.88832e12 q^{51} -5.07146e12 q^{53} +9.97517e12 q^{55} +3.28011e12 q^{57} +1.15328e13 q^{59} +1.56720e13 q^{61} -3.15817e12 q^{63} +2.75256e13 q^{65} -7.82798e13 q^{67} -6.34966e13 q^{69} -6.47540e13 q^{71} +7.33235e13 q^{73} -5.51204e13 q^{75} +4.75157e13 q^{77} +1.87490e14 q^{79} -7.59613e13 q^{81} -4.39668e13 q^{83} +2.55664e14 q^{85} +3.19793e14 q^{87} -5.90336e13 q^{89} +1.31115e14 q^{91} -4.75045e14 q^{93} +1.21743e14 q^{95} -1.06159e15 q^{97} -5.96036e14 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4072 q^{3} - 140260 q^{5} + 126192 q^{7} + 27106378 q^{9}+O(q^{10}) $ 2 * q - 4072 * q^3 - 140260 * q^5 + 126192 * q^7 + 27106378 * q^9	\( 2 q - 4072 q^{3} - 140260 q^{5} + 126192 q^{7} + 27106378 q^{9} + 20682632 q^{11} + 499806476 q^{13} + 1996058960 q^{15} + 3139516900 q^{17} - 474668552 q^{19} + 4019297088 q^{21} - 40776002608 q^{23} + 10378903150 q^{25} - 190235215504 q^{27} + 35253157356 q^{29} - 34389193280 q^{31} + 780208036448 q^{33} + 145094219040 q^{35} + 870228564444 q^{37} - 906424161136 q^{39} + 900085452084 q^{41} + 500707998536 q^{43} - 8866083940340 q^{45} + 1208059119264 q^{47} - 9102300649454 q^{49} + 1965395777200 q^{51} + 1236734202044 q^{53} + 28152970525040 q^{55} + 14546429728672 q^{57} + 14441975905064 q^{59} + 18336303417260 q^{61} - 15702480101712 q^{63} - 31048882497880 q^{65} - 76601421514856 q^{67} + 63670027709888 q^{69} - 145877173886864 q^{71} - 26417269924108 q^{73} - 261044718109400 q^{75} + 75313600116672 q^{77} + 257907833388128 q^{79} + 354480447023698 q^{81} + 255512806582648 q^{83} + 80693958484600 q^{85} + 854812182349584 q^{87} - 719794611712812 q^{89} + 41542153569696 q^{91} - 13\!\cdots\!20 q^{93}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) \) 2 * q - 4072 * q^3 - 140260 * q^5 + 126192 * q^7 + 27106378 * q^9 + 20682632 * q^11 + 499806476 * q^13 + 1996058960 * q^15 + 3139516900 * q^17 - 474668552 * q^19 + 4019297088 * q^21 - 40776002608 * q^23 + 10378903150 * q^25 - 190235215504 * q^27 + 35253157356 * q^29 - 34389193280 * q^31 + 780208036448 * q^33 + 145094219040 * q^35 + 870228564444 * q^37 - 906424161136 * q^39 + 900085452084 * q^41 + 500707998536 * q^43 - 8866083940340 * q^45 + 1208059119264 * q^47 - 9102300649454 * q^49 + 1965395777200 * q^51 + 1236734202044 * q^53 + 28152970525040 * q^55 + 14546429728672 * q^57 + 14441975905064 * q^59 + 18336303417260 * q^61 - 15702480101712 * q^63 - 31048882497880 * q^65 - 76601421514856 * q^67 + 63670027709888 * q^69 - 145877173886864 * q^71 - 26417269924108 * q^73 - 261044718109400 * q^75 + 75313600116672 * q^77 + 257907833388128 * q^79 + 354480447023698 * q^81 + 255512806582648 * q^83 + 80693958484600 * q^85 + 854812182349584 * q^87 - 719794611712812 * q^89 + 41542153569696 * q^91 - 1394036123851520 * q^93 + 522168669596560 * q^95 + 407635590418756 * q^97 - 3068162767300952 * 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
$3$	2838.09	0.749234	0.374617	−	0.927180i	$-0.377774\pi$
$3$	2838.09	0.749234	0.374617	+	0.927180i	$0.377774\pi$
$4$	0	0
$5$	105337.	0.602987	0.301493	−	0.953468i	$-0.402515\pi$
$5$	105337.	0.602987	0.301493	+	0.953468i	$0.402515\pi$
$6$	0	0
$7$	501765.	0.230284	0.115142	−	0.993349i	$-0.463268\pi$
$7$	501765.	0.230284	0.115142	+	0.993349i	$0.463268\pi$
$8$	0	0
$9$	−6.29412e6	−0.438648
$10$	0	0
$11$	9.46973e7	1.46518	0.732592	−	0.680668i	$-0.238310\pi$
$11$	9.46973e7	1.46518	0.732592	+	0.680668i	$0.238310\pi$
$12$	0	0
$13$	2.61309e8	1.15499	0.577495	−	0.816394i	$-0.304030\pi$
$13$	2.61309e8	1.15499	0.577495	+	0.816394i	$0.304030\pi$
$14$	0	0
$15$	2.98958e8	0.451778
$16$	0	0
$17$	2.42709e9	1.43456	0.717281	−	0.696784i	$-0.245386\pi$
$17$	2.42709e9	1.43456	0.717281	+	0.696784i	$0.245386\pi$
$18$	0	0
$19$	1.15575e9	0.296627	0.148313	−	0.988940i	$-0.452616\pi$
$19$	1.15575e9	0.296627	0.148313	+	0.988940i	$0.452616\pi$
$20$	0	0
$21$	1.42406e9	0.172537
$22$	0	0
$23$	−2.23730e10	−1.37014	−0.685070	−	0.728477i	$-0.740228\pi$
$23$	−2.23730e10	−1.37014	−0.685070	+	0.728477i	$0.740228\pi$
$24$	0	0
$25$	−1.94216e10	−0.636407
$26$	0	0
$27$	−5.85869e10	−1.07788
$28$	0	0
$29$	1.12679e11	1.21299	0.606495	−	0.795087i	$-0.292575\pi$
$29$	1.12679e11	1.21299	0.606495	+	0.795087i	$0.292575\pi$
$30$	0	0
$31$	−1.67382e11	−1.09269	−0.546343	−	0.837562i	$-0.683980\pi$
$31$	−1.67382e11	−1.09269	−0.546343	+	0.837562i	$0.683980\pi$
$32$	0	0
$33$	2.68760e11	1.09777
$34$	0	0
$35$	5.28546e10	0.138858
$36$	0	0
$37$	7.53428e11	1.30476	0.652378	−	0.757894i	$-0.273772\pi$
$37$	7.53428e11	1.30476	0.652378	+	0.757894i	$0.273772\pi$
$38$	0	0
$39$	7.41619e11	0.865359
$40$	0	0
$41$	−1.42319e12	−1.14126	−0.570628	−	0.821209i	$-0.693300\pi$
$41$	−1.42319e12	−1.14126	−0.570628	+	0.821209i	$0.693300\pi$
$42$	0	0
$43$	−1.58502e12	−0.889247	−0.444623	−	0.895718i	$-0.646662\pi$
$43$	−1.58502e12	−0.889247	−0.444623	+	0.895718i	$0.646662\pi$
$44$	0	0
$45$	−6.63007e11	−0.264499
$46$	0	0
$47$	−1.13188e12	−0.325885	−0.162943	−	0.986636i	$-0.552099\pi$
$47$	−1.13188e12	−0.325885	−0.162943	+	0.986636i	$0.552099\pi$
$48$	0	0
$49$	−4.49579e12	−0.946969
$50$	0	0
$51$	6.88832e12	1.07482
$52$	0	0
$53$	−5.07146e12	−0.593013	−0.296507	−	0.955031i	$-0.595822\pi$
$53$	−5.07146e12	−0.593013	−0.296507	+	0.955031i	$0.595822\pi$
$54$	0	0
$55$	9.97517e12	0.883487
$56$	0	0
$57$	3.28011e12	0.222243
$58$	0	0
$59$	1.15328e13	0.603317	0.301658	−	0.953416i	$-0.402460\pi$
$59$	1.15328e13	0.603317	0.301658	+	0.953416i	$0.402460\pi$
$60$	0	0
$61$	1.56720e13	0.638485	0.319243	−	0.947673i	$-0.396571\pi$
$61$	1.56720e13	0.638485	0.319243	+	0.947673i	$0.396571\pi$
$62$	0	0
$63$	−3.15817e12	−0.101014
$64$	0	0
$65$	2.75256e13	0.696444
$66$	0	0
$67$	−7.82798e13	−1.57793	−0.788967	−	0.614435i	$-0.789384\pi$
$67$	−7.82798e13	−1.57793	−0.788967	+	0.614435i	$0.789384\pi$
$68$	0	0
$69$	−6.34966e13	−1.02656
$70$	0	0
$71$	−6.47540e13	−0.844946	−0.422473	−	0.906375i	$-0.638838\pi$
$71$	−6.47540e13	−0.844946	−0.422473	+	0.906375i	$0.638838\pi$
$72$	0	0
$73$	7.33235e13	0.776823	0.388411	−	0.921486i	$-0.373024\pi$
$73$	7.33235e13	0.776823	0.388411	+	0.921486i	$0.373024\pi$
$74$	0	0
$75$	−5.51204e13	−0.476818
$76$	0	0
$77$	4.75157e13	0.337409
$78$	0	0
$79$	1.87490e14	1.09843	0.549217	−	0.835680i	$-0.314926\pi$
$79$	1.87490e14	1.09843	0.549217	+	0.835680i	$0.314926\pi$
$80$	0	0
$81$	−7.59613e13	−0.368939
$82$	0	0
$83$	−4.39668e13	−0.177844	−0.0889220	−	0.996039i	$-0.528342\pi$
$83$	−4.39668e13	−0.177844	−0.0889220	+	0.996039i	$0.528342\pi$
$84$	0	0
$85$	2.55664e14	0.865022
$86$	0	0
$87$	3.19793e14	0.908814
$88$	0	0
$89$	−5.90336e13	−0.141473	−0.0707366	−	0.997495i	$-0.522535\pi$
$89$	−5.90336e13	−0.141473	−0.0707366	+	0.997495i	$0.522535\pi$
$90$	0	0
$91$	1.31115e14	0.265976
$92$	0	0
$93$	−4.75045e14	−0.818677
$94$	0	0
$95$	1.21743e14	0.178862
$96$	0	0
$97$	−1.06159e15	−1.33404	−0.667018	−	0.745042i	$-0.732430\pi$
$97$	−1.06159e15	−1.33404	−0.667018	+	0.745042i	$0.732430\pi$
$98$	0	0
$99$	−5.96036e14	−0.642701
$100$	0	0
$101$	−7.55357e14	−0.701039	−0.350520	−	0.936555i	$-0.613995\pi$
$101$	−7.55357e14	−0.701039	−0.350520	+	0.936555i	$0.613995\pi$
$102$	0	0
$103$	1.12617e15	0.902243	0.451121	−	0.892463i	$-0.351024\pi$
$103$	1.12617e15	0.902243	0.451121	+	0.892463i	$0.351024\pi$
$104$	0	0
$105$	1.50006e14	0.104037
$106$	0	0
$107$	−2.50951e15	−1.51081	−0.755407	−	0.655256i	$-0.772561\pi$
$107$	−2.50951e15	−1.51081	−0.755407	+	0.655256i	$0.772561\pi$
$108$	0	0
$109$	3.80630e15	1.99436	0.997182	−	0.0750166i	$-0.0239010\pi$
$109$	3.80630e15	1.99436	0.997182	+	0.0750166i	$0.0239010\pi$
$110$	0	0
$111$	2.13830e15	0.977567
$112$	0	0
$113$	2.30232e15	0.920615	0.460307	−	0.887760i	$-0.347739\pi$
$113$	2.30232e15	0.920615	0.460307	+	0.887760i	$0.347739\pi$
$114$	0	0
$115$	−2.35671e15	−0.826176
$116$	0	0
$117$	−1.64471e15	−0.506635
$118$	0	0
$119$	1.21783e15	0.330357
$120$	0	0
$121$	4.79033e15	1.14677
$122$	0	0
$123$	−4.03914e15	−0.855067
$124$	0	0
$125$	−5.26046e15	−0.986732
$126$	0	0
$127$	8.00128e15	1.33239	0.666195	−	0.745778i	$-0.267922\pi$
$127$	8.00128e15	1.33239	0.666195	+	0.745778i	$0.267922\pi$
$128$	0	0
$129$	−4.49845e15	−0.666254
$130$	0	0
$131$	−1.21422e16	−1.60237	−0.801187	−	0.598414i	$-0.795798\pi$
$131$	−1.21422e16	−1.60237	−0.801187	+	0.598414i	$0.795798\pi$
$132$	0	0
$133$	5.79912e14	0.0683085
$134$	0	0
$135$	−6.17139e15	−0.649950
$136$	0	0
$137$	−9.94937e15	−0.938407	−0.469203	−	0.883090i	$-0.655459\pi$
$137$	−9.94937e15	−0.938407	−0.469203	+	0.883090i	$0.655459\pi$
$138$	0	0
$139$	1.62623e16	1.37585	0.687924	−	0.725783i	$-0.258522\pi$
$139$	1.62623e16	1.37585	0.687924	+	0.725783i	$0.258522\pi$
$140$	0	0
$141$	−3.21237e15	−0.244164
$142$	0	0
$143$	2.47452e16	1.69227
$144$	0	0
$145$	1.18693e16	0.731417
$146$	0	0
$147$	−1.27595e16	−0.709501
$148$	0	0
$149$	7.64837e15	0.384301	0.192151	−	0.981365i	$-0.438454\pi$
$149$	7.64837e15	0.384301	0.192151	+	0.981365i	$0.438454\pi$
$150$	0	0
$151$	3.03865e16	1.38151	0.690754	−	0.723090i	$-0.257279\pi$
$151$	3.03865e16	1.38151	0.690754	+	0.723090i	$0.257279\pi$
$152$	0	0
$153$	−1.52764e16	−0.629268
$154$	0	0
$155$	−1.76316e16	−0.658875
$156$	0	0
$157$	8.13421e15	0.276101	0.138050	−	0.990425i	$-0.455916\pi$
$157$	8.13421e15	0.276101	0.138050	+	0.990425i	$0.455916\pi$
$158$	0	0
$159$	−1.43933e16	−0.444306
$160$	0	0
$161$	−1.12260e16	−0.315522
$162$	0	0
$163$	−1.67285e16	−0.428598	−0.214299	−	0.976768i	$-0.568747\pi$
$163$	−1.67285e16	−0.428598	−0.214299	+	0.976768i	$0.568747\pi$
$164$	0	0
$165$	2.83105e16	0.661938
$166$	0	0
$167$	−8.38919e16	−1.79203	−0.896017	−	0.444019i	$-0.853552\pi$
$167$	−8.38919e16	−1.79203	−0.896017	+	0.444019i	$0.853552\pi$
$168$	0	0
$169$	1.70963e16	0.334004
$170$	0	0
$171$	−7.27441e15	−0.130115
$172$	0	0
$173$	−9.48691e15	−0.155517	−0.0777587	−	0.996972i	$-0.524776\pi$
$173$	−9.48691e15	−0.155517	−0.0777587	+	0.996972i	$0.524776\pi$
$174$	0	0
$175$	−9.74507e15	−0.146555
$176$	0	0
$177$	3.27312e16	0.452025
$178$	0	0
$179$	1.65389e16	0.209947	0.104973	−	0.994475i	$-0.466524\pi$
$179$	1.65389e16	0.209947	0.104973	+	0.994475i	$0.466524\pi$
$180$	0	0
$181$	−4.46776e16	−0.521795	−0.260898	−	0.965366i	$-0.584019\pi$
$181$	−4.46776e16	−0.521795	−0.260898	+	0.965366i	$0.584019\pi$
$182$	0	0
$183$	4.44787e16	0.478375
$184$	0	0
$185$	7.93642e16	0.786750
$186$	0	0
$187$	2.29839e17	2.10190
$188$	0	0
$189$	−2.93968e16	−0.248220
$190$	0	0
$191$	−1.63098e16	−0.127262	−0.0636310	−	0.997973i	$-0.520268\pi$
$191$	−1.63098e16	−0.127262	−0.0636310	+	0.997973i	$0.520268\pi$
$192$	0	0
$193$	−4.40717e16	−0.318039	−0.159019	−	0.987275i	$-0.550833\pi$
$193$	−4.40717e16	−0.318039	−0.159019	+	0.987275i	$0.550833\pi$
$194$	0	0
$195$	7.81202e16	0.521800
$196$	0	0
$197$	1.40145e16	0.0867121	0.0433560	−	0.999060i	$-0.486195\pi$
$197$	1.40145e16	0.0867121	0.0433560	+	0.999060i	$0.486195\pi$
$198$	0	0
$199$	−3.22456e17	−1.84958	−0.924788	−	0.380483i	$-0.875758\pi$
$199$	−3.22456e17	−1.84958	−0.924788	+	0.380483i	$0.875758\pi$
$200$	0	0
$201$	−2.22166e17	−1.18224
$202$	0	0
$203$	5.65383e16	0.279333
$204$	0	0
$205$	−1.49915e17	−0.688162
$206$	0	0
$207$	1.40818e17	0.601010
$208$	0	0
$209$	1.09446e17	0.434613
$210$	0	0
$211$	−2.37005e15	−0.00876272	−0.00438136	−	0.999990i	$-0.501395\pi$
$211$	−2.37005e15	−0.00876272	−0.00438136	+	0.999990i	$0.501395\pi$
$212$	0	0
$213$	−1.83778e17	−0.633062
$214$	0	0
$215$	−1.66962e17	−0.536204
$216$	0	0
$217$	−8.39862e16	−0.251629
$218$	0	0
$219$	2.08099e17	0.582022
$220$	0	0
$221$	6.34220e17	1.65691
$222$	0	0
$223$	−4.18038e17	−1.02077	−0.510387	−	0.859945i	$-0.670498\pi$
$223$	−4.18038e17	−1.02077	−0.510387	+	0.859945i	$0.670498\pi$
$224$	0	0
$225$	1.22242e17	0.279159
$226$	0	0
$227$	1.24320e17	0.265673	0.132837	−	0.991138i	$-0.457591\pi$
$227$	1.24320e17	0.265673	0.132837	+	0.991138i	$0.457591\pi$
$228$	0	0
$229$	3.11336e17	0.622965	0.311482	−	0.950252i	$-0.399175\pi$
$229$	3.11336e17	0.622965	0.311482	+	0.950252i	$0.399175\pi$
$230$	0	0
$231$	1.34854e17	0.252798
$232$	0	0
$233$	7.07921e17	1.24398	0.621992	−	0.783023i	$-0.286323\pi$
$233$	7.07921e17	1.24398	0.621992	+	0.783023i	$0.286323\pi$
$234$	0	0
$235$	−1.19229e17	−0.196504
$236$	0	0
$237$	5.32113e17	0.822984
$238$	0	0
$239$	−7.74058e17	−1.12406	−0.562029	−	0.827117i	$-0.689979\pi$
$239$	−7.74058e17	−1.12406	−0.562029	+	0.827117i	$0.689979\pi$
$240$	0	0
$241$	−1.23307e18	−1.68212	−0.841062	−	0.540938i	$-0.818069\pi$
$241$	−1.23307e18	−1.68212	−0.841062	+	0.540938i	$0.818069\pi$
$242$	0	0
$243$	6.25072e17	0.801463
$244$	0	0
$245$	−4.73575e17	−0.571010
$246$	0	0
$247$	3.02006e17	0.342601
$248$	0	0
$249$	−1.24782e17	−0.133247
$250$	0	0
$251$	−1.40662e18	−1.41456	−0.707282	−	0.706931i	$-0.750079\pi$
$251$	−1.40662e18	−1.41456	−0.707282	+	0.706931i	$0.750079\pi$
$252$	0	0
$253$	−2.11866e18	−2.00751
$254$	0	0
$255$	7.25598e17	0.648104
$256$	0	0
$257$	1.17410e18	0.989026	0.494513	−	0.869170i	$-0.335346\pi$
$257$	1.17410e18	0.989026	0.494513	+	0.869170i	$0.335346\pi$
$258$	0	0
$259$	3.78044e17	0.300465
$260$	0	0
$261$	−7.09215e17	−0.532076
$262$	0	0
$263$	2.08341e18	1.47607	0.738033	−	0.674765i	$-0.235755\pi$
$263$	2.08341e18	1.47607	0.738033	+	0.674765i	$0.235755\pi$
$264$	0	0
$265$	−5.34215e17	−0.357579
$266$	0	0
$267$	−1.67543e17	−0.105997
$268$	0	0
$269$	1.67993e18	1.00496	0.502479	−	0.864589i	$-0.332421\pi$
$269$	1.67993e18	1.00496	0.502479	+	0.864589i	$0.332421\pi$
$270$	0	0
$271$	1.27649e18	0.722348	0.361174	−	0.932498i	$-0.382376\pi$
$271$	1.27649e18	0.722348	0.361174	+	0.932498i	$0.382376\pi$
$272$	0	0
$273$	3.72118e17	0.199279
$274$	0	0
$275$	−1.83917e18	−0.932454
$276$	0	0
$277$	5.35704e17	0.257233	0.128617	−	0.991694i	$-0.458946\pi$
$277$	5.35704e17	0.257233	0.128617	+	0.991694i	$0.458946\pi$
$278$	0	0
$279$	1.05352e18	0.479305
$280$	0	0
$281$	−3.49058e17	−0.150522	−0.0752610	−	0.997164i	$-0.523979\pi$
$281$	−3.49058e17	−0.150522	−0.0752610	+	0.997164i	$0.523979\pi$
$282$	0	0
$283$	2.92413e17	0.119563	0.0597817	−	0.998211i	$-0.480960\pi$
$283$	2.92413e17	0.119563	0.0597817	+	0.998211i	$0.480960\pi$
$284$	0	0
$285$	3.45519e17	0.134009
$286$	0	0
$287$	−7.14104e17	−0.262813
$288$	0	0
$289$	3.02835e18	1.05797
$290$	0	0
$291$	−3.01288e18	−0.999505
$292$	0	0
$293$	−4.39367e16	−0.0138459	−0.00692293	−	0.999976i	$-0.502204\pi$
$293$	−4.39367e16	−0.0138459	−0.00692293	+	0.999976i	$0.502204\pi$
$294$	0	0
$295$	1.21484e18	0.363792
$296$	0	0
$297$	−5.54802e18	−1.57930
$298$	0	0
$299$	−5.84625e18	−1.58250
$300$	0	0
$301$	−7.95309e17	−0.204780
$302$	0	0
$303$	−2.14378e18	−0.525242
$304$	0	0
$305$	1.65085e18	0.384998
$306$	0	0
$307$	2.19940e18	0.488390	0.244195	−	0.969726i	$-0.421476\pi$
$307$	2.19940e18	0.488390	0.244195	+	0.969726i	$0.421476\pi$
$308$	0	0
$309$	3.19617e18	0.675991
$310$	0	0
$311$	−4.13521e18	−0.833286	−0.416643	−	0.909070i	$-0.636794\pi$
$311$	−4.13521e18	−0.833286	−0.416643	+	0.909070i	$0.636794\pi$
$312$	0	0
$313$	3.75720e18	0.721575	0.360788	−	0.932648i	$-0.382508\pi$
$313$	3.75720e18	0.721575	0.360788	+	0.932648i	$0.382508\pi$
$314$	0	0
$315$	−3.32673e17	−0.0609100
$316$	0	0
$317$	6.34412e18	1.10771	0.553856	−	0.832612i	$-0.313156\pi$
$317$	6.34412e18	1.10771	0.553856	+	0.832612i	$0.313156\pi$
$318$	0	0
$319$	1.06704e19	1.77726
$320$	0	0
$321$	−7.12223e18	−1.13195
$322$	0	0
$323$	2.80510e18	0.425529
$324$	0	0
$325$	−5.07503e18	−0.735045
$326$	0	0
$327$	1.08026e19	1.49425
$328$	0	0
$329$	−5.67935e17	−0.0750463
$330$	0	0
$331$	2.37002e18	0.299255	0.149628	−	0.988742i	$-0.452193\pi$
$331$	2.37002e18	0.299255	0.149628	+	0.988742i	$0.452193\pi$
$332$	0	0
$333$	−4.74217e18	−0.572329
$334$	0	0
$335$	−8.24580e18	−0.951473
$336$	0	0
$337$	7.13045e18	0.786851	0.393426	−	0.919356i	$-0.371290\pi$
$337$	7.13045e18	0.786851	0.393426	+	0.919356i	$0.371290\pi$
$338$	0	0
$339$	6.53421e18	0.689756
$340$	0	0
$341$	−1.58506e19	−1.60099
$342$	0	0
$343$	−4.63799e18	−0.448357
$344$	0	0
$345$	−6.68857e18	−0.618999
$346$	0	0
$347$	4.94119e18	0.437886	0.218943	−	0.975738i	$-0.429739\pi$
$347$	4.94119e18	0.437886	0.218943	+	0.975738i	$0.429739\pi$
$348$	0	0
$349$	5.12359e18	0.434895	0.217447	−	0.976072i	$-0.430227\pi$
$349$	5.12359e18	0.434895	0.217447	+	0.976072i	$0.430227\pi$
$350$	0	0
$351$	−1.53093e19	−1.24495
$352$	0	0
$353$	1.08442e19	0.845061	0.422531	−	0.906349i	$-0.361142\pi$
$353$	1.08442e19	0.845061	0.422531	+	0.906349i	$0.361142\pi$
$354$	0	0
$355$	−6.82102e18	−0.509491
$356$	0	0
$357$	3.45631e18	0.247515
$358$	0	0
$359$	−2.05131e19	−1.40871	−0.704357	−	0.709846i	$-0.748765\pi$
$359$	−2.05131e19	−1.40871	−0.704357	+	0.709846i	$0.748765\pi$
$360$	0	0
$361$	−1.38454e19	−0.912013
$362$	0	0
$363$	1.35954e19	0.859196
$364$	0	0
$365$	7.72371e18	0.468414
$366$	0	0
$367$	1.07085e19	0.623351	0.311675	−	0.950189i	$-0.399110\pi$
$367$	1.07085e19	0.623351	0.311675	+	0.950189i	$0.399110\pi$
$368$	0	0
$369$	8.95771e18	0.500610
$370$	0	0
$371$	−2.54468e18	−0.136562
$372$	0	0
$373$	2.96204e18	0.152678	0.0763388	−	0.997082i	$-0.475677\pi$
$373$	2.96204e18	0.152678	0.0763388	+	0.997082i	$0.475677\pi$
$374$	0	0
$375$	−1.49297e19	−0.739293
$376$	0	0
$377$	2.94440e19	1.40099
$378$	0	0
$379$	2.57574e19	1.17790	0.588949	−	0.808170i	$-0.299542\pi$
$379$	2.57574e19	1.17790	0.588949	+	0.808170i	$0.299542\pi$
$380$	0	0
$381$	2.27084e19	0.998272
$382$	0	0
$383$	6.85656e18	0.289812	0.144906	−	0.989445i	$-0.453712\pi$
$383$	6.85656e18	0.289812	0.144906	+	0.989445i	$0.453712\pi$
$384$	0	0
$385$	5.00518e18	0.203453
$386$	0	0
$387$	9.97634e18	0.390067
$388$	0	0
$389$	−2.27546e19	−0.855946	−0.427973	−	0.903791i	$-0.640772\pi$
$389$	−2.27546e19	−0.855946	−0.427973	+	0.903791i	$0.640772\pi$
$390$	0	0
$391$	−5.43013e19	−1.96555
$392$	0	0
$393$	−3.44608e19	−1.20055
$394$	0	0
$395$	1.97497e19	0.662341
$396$	0	0
$397$	2.05219e19	0.662655	0.331328	−	0.943516i	$-0.392503\pi$
$397$	2.05219e19	0.662655	0.331328	+	0.943516i	$0.392503\pi$
$398$	0	0
$399$	1.64585e18	0.0511791
$400$	0	0
$401$	−3.06518e19	−0.918064	−0.459032	−	0.888420i	$-0.651804\pi$
$401$	−3.06518e19	−0.918064	−0.459032	+	0.888420i	$0.651804\pi$
$402$	0	0
$403$	−4.37383e19	−1.26204
$404$	0	0
$405$	−8.00157e18	−0.222465
$406$	0	0
$407$	7.13476e19	1.91171
$408$	0	0
$409$	−4.69448e19	−1.21245	−0.606223	−	0.795295i	$-0.707316\pi$
$409$	−4.69448e19	−1.21245	−0.606223	+	0.795295i	$0.707316\pi$
$410$	0	0
$411$	−2.82372e19	−0.703086
$412$	0	0
$413$	5.78676e18	0.138934
$414$	0	0
$415$	−4.63135e18	−0.107238
$416$	0	0
$417$	4.61539e19	1.03083
$418$	0	0
$419$	5.68679e19	1.22536	0.612678	−	0.790333i	$-0.290092\pi$
$419$	5.68679e19	1.22536	0.612678	+	0.790333i	$0.290092\pi$
$420$	0	0
$421$	−9.32398e19	−1.93859	−0.969294	−	0.245903i	$-0.920916\pi$
$421$	−9.32398e19	−1.93859	−0.969294	+	0.245903i	$0.920916\pi$
$422$	0	0
$423$	7.12416e18	0.142949
$424$	0	0
$425$	−4.71380e19	−0.912966
$426$	0	0
$427$	7.86366e18	0.147033
$428$	0	0
$429$	7.02293e19	1.26791
$430$	0	0
$431$	9.67848e17	0.0168744	0.00843718	−	0.999964i	$-0.497314\pi$
$431$	9.67848e17	0.0168744	0.00843718	+	0.999964i	$0.497314\pi$
$432$	0	0
$433$	1.04924e20	1.76691	0.883455	−	0.468516i	$-0.155211\pi$
$433$	1.04924e20	1.76691	0.883455	+	0.468516i	$0.155211\pi$
$434$	0	0
$435$	3.36862e19	0.548003
$436$	0	0
$437$	−2.58575e19	−0.406420
$438$	0	0
$439$	1.78155e19	0.270592	0.135296	−	0.990805i	$-0.456801\pi$
$439$	1.78155e19	0.270592	0.135296	+	0.990805i	$0.456801\pi$
$440$	0	0
$441$	2.82971e19	0.415386
$442$	0	0
$443$	−4.08199e19	−0.579221	−0.289611	−	0.957145i	$-0.593526\pi$
$443$	−4.08199e19	−0.579221	−0.289611	+	0.957145i	$0.593526\pi$
$444$	0	0
$445$	−6.21845e18	−0.0853064
$446$	0	0
$447$	2.17068e19	0.287932
$448$	0	0
$449$	8.52239e19	1.09324	0.546618	−	0.837382i	$-0.315915\pi$
$449$	8.52239e19	1.09324	0.546618	+	0.837382i	$0.315915\pi$
$450$	0	0
$451$	−1.34772e20	−1.67215
$452$	0	0
$453$	8.62397e19	1.03507
$454$	0	0
$455$	1.38114e19	0.160380
$456$	0	0
$457$	−6.30917e19	−0.708926	−0.354463	−	0.935070i	$-0.615336\pi$
$457$	−6.30917e19	−0.708926	−0.354463	+	0.935070i	$0.615336\pi$
$458$	0	0
$459$	−1.42196e20	−1.54629
$460$	0	0
$461$	−3.72713e19	−0.392299	−0.196150	−	0.980574i	$-0.562844\pi$
$461$	−3.72713e19	−0.392299	−0.196150	+	0.980574i	$0.562844\pi$
$462$	0	0
$463$	−1.93933e19	−0.197603	−0.0988015	−	0.995107i	$-0.531501\pi$
$463$	−1.93933e19	−0.197603	−0.0988015	+	0.995107i	$0.531501\pi$
$464$	0	0
$465$	−5.00400e19	−0.493651
$466$	0	0
$467$	−1.53169e19	−0.146317	−0.0731586	−	0.997320i	$-0.523308\pi$
$467$	−1.53169e19	−0.146317	−0.0731586	+	0.997320i	$0.523308\pi$
$468$	0	0
$469$	−3.92780e19	−0.363374
$470$	0	0
$471$	2.30856e19	0.206864
$472$	0	0
$473$	−1.50097e20	−1.30291
$474$	0	0
$475$	−2.24464e19	−0.188775
$476$	0	0
$477$	3.19204e19	0.260124
$478$	0	0
$479$	1.66176e20	1.31236	0.656178	−	0.754606i	$-0.272172\pi$
$479$	1.66176e20	1.31236	0.656178	+	0.754606i	$0.272172\pi$
$480$	0	0
$481$	1.96877e20	1.50698
$482$	0	0
$483$	−3.18604e19	−0.236400
$484$	0	0
$485$	−1.11825e20	−0.804406
$486$	0	0
$487$	1.23288e20	0.859914	0.429957	−	0.902849i	$-0.358529\pi$
$487$	1.23288e20	0.859914	0.429957	+	0.902849i	$0.358529\pi$
$488$	0	0
$489$	−4.74771e19	−0.321120
$490$	0	0
$491$	2.12623e19	0.139476	0.0697378	−	0.997565i	$-0.477784\pi$
$491$	2.12623e19	0.139476	0.0697378	+	0.997565i	$0.477784\pi$
$492$	0	0
$493$	2.73482e20	1.74011
$494$	0	0
$495$	−6.27849e19	−0.387540
$496$	0	0
$497$	−3.24913e19	−0.194578
$498$	0	0
$499$	1.08110e20	0.628219	0.314110	−	0.949387i	$-0.398294\pi$
$499$	1.08110e20	0.628219	0.314110	+	0.949387i	$0.398294\pi$
$500$	0	0
$501$	−2.38093e20	−1.34265
$502$	0	0
$503$	1.09452e20	0.599053	0.299527	−	0.954088i	$-0.403171\pi$
$503$	1.09452e20	0.599053	0.299527	+	0.954088i	$0.403171\pi$
$504$	0	0
$505$	−7.95674e19	−0.422717
$506$	0	0
$507$	4.85209e19	0.250247
$508$	0	0
$509$	2.36773e20	1.18563	0.592815	−	0.805339i	$-0.298017\pi$
$509$	2.36773e20	1.18563	0.592815	+	0.805339i	$0.298017\pi$
$510$	0	0
$511$	3.67911e19	0.178890
$512$	0	0
$513$	−6.77115e19	−0.319729
$514$	0	0
$515$	1.18627e20	0.544040
$516$	0	0
$517$	−1.07186e20	−0.477482
$518$	0	0
$519$	−2.69248e19	−0.116519
$520$	0	0
$521$	−2.76006e20	−1.16048	−0.580238	−	0.814447i	$-0.697041\pi$
$521$	−2.76006e20	−1.16048	−0.580238	+	0.814447i	$0.697041\pi$
$522$	0	0
$523$	9.11560e19	0.372411	0.186206	−	0.982511i	$-0.440381\pi$
$523$	9.11560e19	0.372411	0.186206	+	0.982511i	$0.440381\pi$
$524$	0	0
$525$	−2.76574e19	−0.109804
$526$	0	0
$527$	−4.06251e20	−1.56753
$528$	0	0
$529$	2.33915e20	0.877284
$530$	0	0
$531$	−7.25890e19	−0.264644
$532$	0	0
$533$	−3.71891e20	−1.31814
$534$	0	0
$535$	−2.64345e20	−0.911000
$536$	0	0
$537$	4.69390e19	0.157299
$538$	0	0
$539$	−4.25739e20	−1.38748
$540$	0	0
$541$	3.65644e20	1.15899	0.579495	−	0.814976i	$-0.303250\pi$
$541$	3.65644e20	1.15899	0.579495	+	0.814976i	$0.303250\pi$
$542$	0	0
$543$	−1.26799e20	−0.390947
$544$	0	0
$545$	4.00946e20	1.20257
$546$	0	0
$547$	4.06995e20	1.18764	0.593818	−	0.804599i	$-0.297620\pi$
$547$	4.06995e20	1.18764	0.593818	+	0.804599i	$0.297620\pi$
$548$	0	0
$549$	−9.86416e19	−0.280071
$550$	0	0
$551$	1.30228e20	0.359805
$552$	0	0
$553$	9.40756e19	0.252952
$554$	0	0
$555$	2.25243e20	0.589460
$556$	0	0
$557$	6.37675e19	0.162437	0.0812187	−	0.996696i	$-0.474119\pi$
$557$	6.37675e19	0.162437	0.0812187	+	0.996696i	$0.474119\pi$
$558$	0	0
$559$	−4.14180e20	−1.02707
$560$	0	0
$561$	6.52305e20	1.57481
$562$	0	0
$563$	4.05240e18	0.00952576	0.00476288	−	0.999989i	$-0.498484\pi$
$563$	4.05240e18	0.00952576	0.00476288	+	0.999989i	$0.498484\pi$
$564$	0	0
$565$	2.42521e20	0.555118
$566$	0	0
$567$	−3.81147e19	−0.0849610
$568$	0	0
$569$	−2.04215e20	−0.443349	−0.221674	−	0.975121i	$-0.571152\pi$
$569$	−2.04215e20	−0.443349	−0.221674	+	0.975121i	$0.571152\pi$
$570$	0	0
$571$	5.85266e20	1.23761	0.618803	−	0.785546i	$-0.287618\pi$
$571$	5.85266e20	1.23761	0.618803	+	0.785546i	$0.287618\pi$
$572$	0	0
$573$	−4.62888e19	−0.0953490
$574$	0	0
$575$	4.34519e20	0.871967
$576$	0	0
$577$	−9.55684e20	−1.86851	−0.934255	−	0.356605i	$-0.883934\pi$
$577$	−9.55684e20	−1.86851	−0.934255	+	0.356605i	$0.883934\pi$
$578$	0	0
$579$	−1.25080e20	−0.238286
$580$	0	0
$581$	−2.20610e19	−0.0409547
$582$	0	0
$583$	−4.80254e20	−0.868874
$584$	0	0
$585$	−1.73249e20	−0.305494
$586$	0	0
$587$	4.14089e20	0.711718	0.355859	−	0.934540i	$-0.384188\pi$
$587$	4.14089e20	0.711718	0.355859	+	0.934540i	$0.384188\pi$
$588$	0	0
$589$	−1.93451e20	−0.324120
$590$	0	0
$591$	3.97744e19	0.0649676
$592$	0	0
$593$	3.67145e20	0.584693	0.292346	−	0.956312i	$-0.405564\pi$
$593$	3.67145e20	0.584693	0.292346	+	0.956312i	$0.405564\pi$
$594$	0	0
$595$	1.28283e20	0.199201
$596$	0	0
$597$	−9.15160e20	−1.38577
$598$	0	0
$599$	−9.75975e20	−1.44124	−0.720622	−	0.693328i	$-0.756144\pi$
$599$	−9.75975e20	−1.44124	−0.720622	+	0.693328i	$0.756144\pi$
$600$	0	0
$601$	−7.76470e20	−1.11832	−0.559160	−	0.829060i	$-0.688876\pi$
$601$	−7.76470e20	−1.11832	−0.559160	+	0.829060i	$0.688876\pi$
$602$	0	0
$603$	4.92703e20	0.692158
$604$	0	0
$605$	5.04601e20	0.691484
$606$	0	0
$607$	8.90456e20	1.19041	0.595206	−	0.803573i	$-0.297070\pi$
$607$	8.90456e20	1.19041	0.595206	+	0.803573i	$0.297070\pi$
$608$	0	0
$609$	1.60461e20	0.209286
$610$	0	0
$611$	−2.95769e20	−0.376394
$612$	0	0
$613$	−5.79932e20	−0.720151	−0.360075	−	0.932923i	$-0.617249\pi$
$613$	−5.79932e20	−0.720151	−0.360075	+	0.932923i	$0.617249\pi$
$614$	0	0
$615$	−4.25472e20	−0.515594
$616$	0	0
$617$	−4.20982e20	−0.497880	−0.248940	−	0.968519i	$-0.580082\pi$
$617$	−4.20982e20	−0.497880	−0.248940	+	0.968519i	$0.580082\pi$
$618$	0	0
$619$	−8.78332e20	−1.01386	−0.506931	−	0.861987i	$-0.669220\pi$
$619$	−8.78332e20	−1.01386	−0.506931	+	0.861987i	$0.669220\pi$
$620$	0	0
$621$	1.31076e21	1.47685
$622$	0	0
$623$	−2.96210e19	−0.0325791
$624$	0	0
$625$	3.85767e19	0.0414214
$626$	0	0
$627$	3.10618e20	0.325627
$628$	0	0
$629$	1.82864e21	1.87175
$630$	0	0
$631$	1.63256e21	1.63174	0.815869	−	0.578237i	$-0.196259\pi$
$631$	1.63256e21	1.63174	0.815869	+	0.578237i	$0.196259\pi$
$632$	0	0
$633$	−6.72642e18	−0.00656533
$634$	0	0
$635$	8.42834e20	0.803413
$636$	0	0
$637$	−1.17479e21	−1.09374
$638$	0	0
$639$	4.07570e20	0.370634
$640$	0	0
$641$	1.40859e21	1.25127	0.625634	−	0.780116i	$-0.284840\pi$
$641$	1.40859e21	1.25127	0.625634	+	0.780116i	$0.284840\pi$
$642$	0	0
$643$	−1.39010e21	−1.20632	−0.603161	−	0.797619i	$-0.706092\pi$
$643$	−1.39010e21	−1.20632	−0.603161	+	0.797619i	$0.706092\pi$
$644$	0	0
$645$	−4.73855e20	−0.401742
$646$	0	0
$647$	−5.53660e20	−0.458629	−0.229314	−	0.973352i	$-0.573648\pi$
$647$	−5.53660e20	−0.458629	−0.229314	+	0.973352i	$0.573648\pi$
$648$	0	0
$649$	1.09213e21	0.883970
$650$	0	0
$651$	−2.38361e20	−0.188529
$652$	0	0
$653$	3.48743e19	0.0269561	0.0134780	−	0.999909i	$-0.495710\pi$
$653$	3.48743e19	0.0269561	0.0134780	+	0.999909i	$0.495710\pi$
$654$	0	0
$655$	−1.27903e21	−0.966210
$656$	0	0
$657$	−4.61507e20	−0.340752
$658$	0	0
$659$	−2.63820e21	−1.90400	−0.951999	−	0.306101i	$-0.900975\pi$
$659$	−2.63820e21	−1.90400	−0.951999	+	0.306101i	$0.900975\pi$
$660$	0	0
$661$	−9.99388e19	−0.0705055	−0.0352528	−	0.999378i	$-0.511224\pi$
$661$	−9.99388e19	−0.0705055	−0.0352528	+	0.999378i	$0.511224\pi$
$662$	0	0
$663$	1.79998e21	1.24141
$664$	0	0
$665$	6.10864e19	0.0411891
$666$	0	0
$667$	−2.52096e21	−1.66197
$668$	0	0
$669$	−1.18643e21	−0.764798
$670$	0	0
$671$	1.48410e21	0.935499
$672$	0	0
$673$	3.80991e20	0.234856	0.117428	−	0.993081i	$-0.462535\pi$
$673$	3.80991e20	0.234856	0.117428	+	0.993081i	$0.462535\pi$
$674$	0	0
$675$	1.13785e21	0.685973
$676$	0	0
$677$	−1.10899e20	−0.0653900	−0.0326950	−	0.999465i	$-0.510409\pi$
$677$	−1.10899e20	−0.0653900	−0.0326950	+	0.999465i	$0.510409\pi$
$678$	0	0
$679$	−5.32667e20	−0.307208
$680$	0	0
$681$	3.52833e20	0.199051
$682$	0	0
$683$	−1.63079e21	−0.899998	−0.449999	−	0.893029i	$-0.648576\pi$
$683$	−1.63079e21	−0.899998	−0.449999	+	0.893029i	$0.648576\pi$
$684$	0	0
$685$	−1.04804e21	−0.565847
$686$	0	0
$687$	8.83602e20	0.466746
$688$	0	0
$689$	−1.32522e21	−0.684925
$690$	0	0
$691$	3.25588e21	1.64658	0.823290	−	0.567621i	$-0.192136\pi$
$691$	3.25588e21	1.64658	0.823290	+	0.567621i	$0.192136\pi$
$692$	0	0
$693$	−2.99070e20	−0.148004
$694$	0	0
$695$	1.71303e21	0.829617
$696$	0	0
$697$	−3.45420e21	−1.63720
$698$	0	0
$699$	2.00915e21	0.932036
$700$	0	0
$701$	1.82945e21	0.830683	0.415341	−	0.909666i	$-0.363662\pi$
$701$	1.82945e21	0.830683	0.415341	+	0.909666i	$0.363662\pi$
$702$	0	0
$703$	8.70771e20	0.387025
$704$	0	0
$705$	−3.38383e20	−0.147228
$706$	0	0
$707$	−3.79012e20	−0.161438
$708$	0	0
$709$	2.19318e21	0.914592	0.457296	−	0.889314i	$-0.348818\pi$
$709$	2.19318e21	0.914592	0.457296	+	0.889314i	$0.348818\pi$
$710$	0	0
$711$	−1.18008e21	−0.481826
$712$	0	0
$713$	3.74483e21	1.49713
$714$	0	0
$715$	2.60660e21	1.02042
$716$	0	0
$717$	−2.19685e21	−0.842183
$718$	0	0
$719$	2.51347e20	0.0943642	0.0471821	−	0.998886i	$-0.484976\pi$
$719$	2.51347e20	0.0943642	0.0471821	+	0.998886i	$0.484976\pi$
$720$	0	0
$721$	5.65070e20	0.207772
$722$	0	0
$723$	−3.49956e21	−1.26030
$724$	0	0
$725$	−2.18840e21	−0.771956
$726$	0	0
$727$	3.28520e21	1.13515	0.567575	−	0.823322i	$-0.307882\pi$
$727$	3.28520e21	1.13515	0.567575	+	0.823322i	$0.307882\pi$
$728$	0	0
$729$	2.86398e21	0.969422
$730$	0	0
$731$	−3.84700e21	−1.27568
$732$	0	0
$733$	3.05120e21	0.991267	0.495634	−	0.868532i	$-0.334936\pi$
$733$	3.05120e21	0.991267	0.495634	+	0.868532i	$0.334936\pi$
$734$	0	0
$735$	−1.34405e21	−0.427820
$736$	0	0
$737$	−7.41289e21	−2.31197
$738$	0	0
$739$	−1.12412e21	−0.343542	−0.171771	−	0.985137i	$-0.554949\pi$
$739$	−1.12412e21	−0.343542	−0.171771	+	0.985137i	$0.554949\pi$
$740$	0	0
$741$	8.57122e20	0.256688
$742$	0	0
$743$	−2.31463e21	−0.679306	−0.339653	−	0.940551i	$-0.610310\pi$
$743$	−2.31463e21	−0.679306	−0.339653	+	0.940551i	$0.610310\pi$
$744$	0	0
$745$	8.05660e20	0.231729
$746$	0	0
$747$	2.76733e20	0.0780110
$748$	0	0
$749$	−1.25918e21	−0.347917
$750$	0	0
$751$	−5.64662e21	−1.52929	−0.764644	−	0.644453i	$-0.777085\pi$
$751$	−5.64662e21	−1.52929	−0.764644	+	0.644453i	$0.777085\pi$
$752$	0	0
$753$	−3.99211e21	−1.05984
$754$	0	0
$755$	3.20083e21	0.833031
$756$	0	0
$757$	4.91904e21	1.25505	0.627526	−	0.778596i	$-0.284068\pi$
$757$	4.91904e21	1.25505	0.627526	+	0.778596i	$0.284068\pi$
$758$	0	0
$759$	−6.01296e21	−1.50409
$760$	0	0
$761$	4.57420e21	1.12184	0.560919	−	0.827871i	$-0.310448\pi$
$761$	4.57420e21	1.12184	0.560919	+	0.827871i	$0.310448\pi$
$762$	0	0
$763$	1.90987e21	0.459271
$764$	0	0
$765$	−1.60918e21	−0.379440
$766$	0	0
$767$	3.01362e21	0.696825
$768$	0	0
$769$	3.76756e21	0.854304	0.427152	−	0.904180i	$-0.359517\pi$
$769$	3.76756e21	0.854304	0.427152	+	0.904180i	$0.359517\pi$
$770$	0	0
$771$	3.33221e21	0.741012
$772$	0	0
$773$	2.33428e21	0.509104	0.254552	−	0.967059i	$-0.418072\pi$
$773$	2.33428e21	0.509104	0.254552	+	0.967059i	$0.418072\pi$
$774$	0	0
$775$	3.25082e21	0.695393
$776$	0	0
$777$	1.07292e21	0.225119
$778$	0	0
$779$	−1.64484e21	−0.338527
$780$	0	0
$781$	−6.13203e21	−1.23800
$782$	0	0
$783$	−6.60150e21	−1.30746
$784$	0	0
$785$	8.56836e20	0.166485
$786$	0	0
$787$	8.45397e21	1.61157	0.805787	−	0.592206i	$-0.201743\pi$
$787$	8.45397e21	1.61157	0.805787	+	0.592206i	$0.201743\pi$
$788$	0	0
$789$	5.91290e21	1.10592
$790$	0	0
$791$	1.15522e21	0.212003
$792$	0	0
$793$	4.09523e21	0.737445
$794$	0	0
$795$	−1.51615e21	−0.267910
$796$	0	0
$797$	−9.41594e21	−1.63278	−0.816388	−	0.577504i	$-0.804027\pi$
$797$	−9.41594e21	−1.63278	−0.816388	+	0.577504i	$0.804027\pi$
$798$	0	0
$799$	−2.74717e21	−0.467503
$800$	0	0
$801$	3.71565e20	0.0620570
$802$	0	0
$803$	6.94354e21	1.13819
$804$	0	0
$805$	−1.18251e21	−0.190256
$806$	0	0
$807$	4.76779e21	0.752949
$808$	0	0
$809$	−4.69655e21	−0.728056	−0.364028	−	0.931388i	$-0.618599\pi$
$809$	−4.69655e21	−0.728056	−0.364028	+	0.931388i	$0.618599\pi$
$810$	0	0
$811$	2.33661e21	0.355573	0.177787	−	0.984069i	$-0.443106\pi$
$811$	2.33661e21	0.355573	0.177787	+	0.984069i	$0.443106\pi$
$812$	0	0
$813$	3.62279e21	0.541208
$814$	0	0
$815$	−1.76214e21	−0.258439
$816$	0	0
$817$	−1.83188e21	−0.263774
$818$	0	0
$819$	−8.25257e20	−0.116670
$820$	0	0
$821$	−1.02936e22	−1.42887	−0.714433	−	0.699704i	$-0.753315\pi$
$821$	−1.02936e22	−1.42887	−0.714433	+	0.699704i	$0.753315\pi$
$822$	0	0
$823$	−1.25719e22	−1.71358	−0.856788	−	0.515670i	$-0.827543\pi$
$823$	−1.25719e22	−1.71358	−0.856788	+	0.515670i	$0.827543\pi$
$824$	0	0
$825$	−5.21975e21	−0.698626
$826$	0	0
$827$	8.69184e21	1.14241	0.571203	−	0.820809i	$-0.306477\pi$
$827$	8.69184e21	1.14241	0.571203	+	0.820809i	$0.306477\pi$
$828$	0	0
$829$	−5.63874e21	−0.727819	−0.363909	−	0.931434i	$-0.618558\pi$
$829$	−5.63874e21	−0.727819	−0.363909	+	0.931434i	$0.618558\pi$
$830$	0	0
$831$	1.52038e21	0.192728
$832$	0	0
$833$	−1.09117e22	−1.35849
$834$	0	0
$835$	−8.83695e21	−1.08057
$836$	0	0
$837$	9.80637e21	1.17779
$838$	0	0
$839$	8.28411e21	0.977307	0.488654	−	0.872478i	$-0.337488\pi$
$839$	8.28411e21	0.977307	0.488654	+	0.872478i	$0.337488\pi$
$840$	0	0
$841$	4.06734e21	0.471346
$842$	0	0
$843$	−9.90660e20	−0.112776
$844$	0	0
$845$	1.80088e21	0.201400
$846$	0	0
$847$	2.40362e21	0.264082
$848$	0	0
$849$	8.29896e20	0.0895810
$850$	0	0
$851$	−1.68564e22	−1.78770
$852$	0	0
$853$	1.28659e22	1.34068	0.670338	−	0.742056i	$-0.266149\pi$
$853$	1.28659e22	1.34068	0.670338	+	0.742056i	$0.266149\pi$
$854$	0	0
$855$	−7.66267e20	−0.0784575
$856$	0	0
$857$	1.65564e22	1.66575	0.832874	−	0.553462i	$-0.186694\pi$
$857$	1.65564e22	1.66575	0.832874	+	0.553462i	$0.186694\pi$
$858$	0	0
$859$	1.81629e21	0.179571	0.0897853	−	0.995961i	$-0.471382\pi$
$859$	1.81629e21	0.179571	0.0897853	+	0.995961i	$0.471382\pi$
$860$	0	0
$861$	−2.02670e21	−0.196909
$862$	0	0
$863$	1.38764e22	1.32494	0.662468	−	0.749091i	$-0.269509\pi$
$863$	1.38764e22	1.32494	0.662468	+	0.749091i	$0.269509\pi$
$864$	0	0
$865$	−9.99327e20	−0.0937749
$866$	0	0
$867$	8.59475e21	0.792666
$868$	0	0
$869$	1.77547e22	1.60941
$870$	0	0
$871$	−2.04552e22	−1.82250
$872$	0	0
$873$	6.68176e21	0.585173
$874$	0	0
$875$	−2.63951e21	−0.227229
$876$	0	0
$877$	−9.64407e21	−0.816137	−0.408069	−	0.912951i	$-0.633798\pi$
$877$	−9.64407e21	−0.816137	−0.408069	+	0.912951i	$0.633798\pi$
$878$	0	0
$879$	−1.24696e20	−0.0103738
$880$	0	0
$881$	−7.69910e21	−0.629681	−0.314841	−	0.949145i	$-0.601951\pi$
$881$	−7.69910e21	−0.629681	−0.314841	+	0.949145i	$0.601951\pi$
$882$	0	0
$883$	−7.13934e21	−0.574054	−0.287027	−	0.957922i	$-0.592667\pi$
$883$	−7.13934e21	−0.574054	−0.287027	+	0.957922i	$0.592667\pi$
$884$	0	0
$885$	3.44782e21	0.272565
$886$	0	0
$887$	2.10141e22	1.63336	0.816682	−	0.577088i	$-0.195811\pi$
$887$	2.10141e22	1.63336	0.816682	+	0.577088i	$0.195811\pi$
$888$	0	0
$889$	4.01476e21	0.306829
$890$	0	0
$891$	−7.19333e21	−0.540564
$892$	0	0
$893$	−1.30816e21	−0.0966662
$894$	0	0
$895$	1.74217e21	0.126595
$896$	0	0
$897$	−1.65922e22	−1.18566
$898$	0	0
$899$	−1.88604e22	−1.32542
$900$	0	0
$901$	−1.23089e22	−0.850714
$902$	0	0
$903$	−2.25716e21	−0.153428
$904$	0	0
$905$	−4.70622e21	−0.314636
$906$	0	0
$907$	7.53615e21	0.495558	0.247779	−	0.968817i	$-0.420299\pi$
$907$	7.53615e21	0.495558	0.247779	+	0.968817i	$0.420299\pi$
$908$	0	0
$909$	4.75431e21	0.307510
$910$	0	0
$911$	1.45464e22	0.925481	0.462741	−	0.886494i	$-0.346866\pi$
$911$	1.45464e22	0.925481	0.462741	+	0.886494i	$0.346866\pi$
$912$	0	0
$913$	−4.16354e21	−0.260574
$914$	0	0
$915$	4.68527e21	0.288454
$916$	0	0
$917$	−6.09254e21	−0.369002
$918$	0	0
$919$	−1.01562e22	−0.605155	−0.302577	−	0.953125i	$-0.597847\pi$
$919$	−1.01562e22	−0.605155	−0.302577	+	0.953125i	$0.597847\pi$
$920$	0	0
$921$	6.24211e21	0.365918
$922$	0	0
$923$	−1.69208e22	−0.975905
$924$	0	0
$925$	−1.46328e22	−0.830356
$926$	0	0
$927$	−7.08823e21	−0.395767
$928$	0	0
$929$	2.60392e22	1.43057	0.715287	−	0.698831i	$-0.246296\pi$
$929$	2.60392e22	1.43057	0.715287	+	0.698831i	$0.246296\pi$
$930$	0	0
$931$	−5.19599e21	−0.280896
$932$	0	0
$933$	−1.17361e22	−0.624327
$934$	0	0
$935$	2.42106e22	1.26742
$936$	0	0
$937$	−1.29295e22	−0.666094	−0.333047	−	0.942910i	$-0.608077\pi$
$937$	−1.29295e22	−0.666094	−0.333047	+	0.942910i	$0.608077\pi$
$938$	0	0
$939$	1.06633e22	0.540629
$940$	0	0
$941$	−2.30229e22	−1.14878	−0.574391	−	0.818581i	$-0.694761\pi$
$941$	−2.30229e22	−1.14878	−0.574391	+	0.818581i	$0.694761\pi$
$942$	0	0
$943$	3.18409e22	1.56368
$944$	0	0
$945$	−3.09658e21	−0.149673
$946$	0	0
$947$	7.53643e21	0.358543	0.179271	−	0.983800i	$-0.442626\pi$
$947$	7.53643e21	0.358543	0.179271	+	0.983800i	$0.442626\pi$
$948$	0	0
$949$	1.91601e22	0.897223
$950$	0	0
$951$	1.80052e22	0.829936
$952$	0	0
$953$	8.83861e21	0.401039	0.200520	−	0.979690i	$-0.435737\pi$
$953$	8.83861e21	0.401039	0.200520	+	0.979690i	$0.435737\pi$
$954$	0	0
$955$	−1.71803e21	−0.0767373
$956$	0	0
$957$	3.02836e22	1.33158
$958$	0	0
$959$	−4.99224e21	−0.216100
$960$	0	0
$961$	4.55138e21	0.193962
$962$	0	0
$963$	1.57952e22	0.662716
$964$	0	0
$965$	−4.64240e21	−0.191773
$966$	0	0
$967$	−1.02519e19	−0.000416969 0	−0.000208485	−	1.00000i	$-0.500066\pi$
$967$	−1.02519e19	−0.000416969 0	−0.000208485		1.00000i	$0.500066\pi$
$968$	0	0
$969$	7.96114e21	0.318821
$970$	0	0
$971$	−2.99934e22	−1.18272	−0.591359	−	0.806408i	$-0.701409\pi$
$971$	−2.99934e22	−1.18272	−0.591359	+	0.806408i	$0.701409\pi$
$972$	0	0
$973$	8.15983e21	0.316836
$974$	0	0
$975$	−1.44034e22	−0.550720
$976$	0	0
$977$	−3.21653e22	−1.21110	−0.605549	−	0.795808i	$-0.707046\pi$
$977$	−3.21653e22	−1.21110	−0.605549	+	0.795808i	$0.707046\pi$
$978$	0	0
$979$	−5.59032e21	−0.207284
$980$	0	0
$981$	−2.39573e22	−0.874825
$982$	0	0
$983$	−1.85327e22	−0.666482	−0.333241	−	0.942842i	$-0.608142\pi$
$983$	−1.85327e22	−0.666482	−0.333241	+	0.942842i	$0.608142\pi$
$984$	0	0
$985$	1.47625e21	0.0522862
$986$	0	0
$987$	−1.61185e21	−0.0562272
$988$	0	0
$989$	3.54617e22	1.21839
$990$	0	0
$991$	−2.67234e22	−0.904354	−0.452177	−	0.891928i	$-0.649353\pi$
$991$	−2.67234e22	−0.904354	−0.452177	+	0.891928i	$0.649353\pi$
$992$	0	0
$993$	6.72634e21	0.224212
$994$	0	0
$995$	−3.39667e22	−1.11527
$996$	0	0
$997$	2.12518e22	0.687356	0.343678	−	0.939088i	$-0.388327\pi$
$997$	2.12518e22	0.687356	0.343678	+	0.939088i	$0.388327\pi$
$998$	0	0
$999$	−4.41410e22	−1.40638

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8.16.a.c.1.2	✓	2
3.2	odd	2		72.16.a.g.1.1		2
4.3	odd	2		16.16.a.f.1.1		2
8.3	odd	2		64.16.a.l.1.2		2
8.5	even	2		64.16.a.n.1.1		2
12.11	even	2		144.16.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.16.a.c.1.2	✓	2	1.1	even	1	trivial
16.16.a.f.1.1		2	4.3	odd	2
64.16.a.l.1.2		2	8.3	odd	2
64.16.a.n.1.1		2	8.5	even	2
72.16.a.g.1.1		2	3.2	odd	2
144.16.a.v.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)

\(f(q)\)	\(=\)	\(q+2838.09 q^{3} +105337. q^{5} +501765. q^{7} -6.29412e6 q^{9} +O(q^{10})\)	\(q+2838.09 q^{3} +105337. q^{5} +501765. q^{7} -6.29412e6 q^{9} +9.46973e7 q^{11} +2.61309e8 q^{13} +2.98958e8 q^{15} +2.42709e9 q^{17} +1.15575e9 q^{19} +1.42406e9 q^{21} -2.23730e10 q^{23} -1.94216e10 q^{25} -5.85869e10 q^{27} +1.12679e11 q^{29} -1.67382e11 q^{31} +2.68760e11 q^{33} +5.28546e10 q^{35} +7.53428e11 q^{37} +7.41619e11 q^{39} -1.42319e12 q^{41} -1.58502e12 q^{43} -6.63007e11 q^{45} -1.13188e12 q^{47} -4.49579e12 q^{49} +6.88832e12 q^{51} -5.07146e12 q^{53} +9.97517e12 q^{55} +3.28011e12 q^{57} +1.15328e13 q^{59} +1.56720e13 q^{61} -3.15817e12 q^{63} +2.75256e13 q^{65} -7.82798e13 q^{67} -6.34966e13 q^{69} -6.47540e13 q^{71} +7.33235e13 q^{73} -5.51204e13 q^{75} +4.75157e13 q^{77} +1.87490e14 q^{79} -7.59613e13 q^{81} -4.39668e13 q^{83} +2.55664e14 q^{85} +3.19793e14 q^{87} -5.90336e13 q^{89} +1.31115e14 q^{91} -4.75045e14 q^{93} +1.21743e14 q^{95} -1.06159e15 q^{97} -5.96036e14 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4072 q^{3} - 140260 q^{5} + 126192 q^{7} + 27106378 q^{9}+O(q^{10}) \) 2 * q - 4072 * q^3 - 140260 * q^5 + 126192 * q^7 + 27106378 * q^9	\( 2 q - 4072 q^{3} - 140260 q^{5} + 126192 q^{7} + 27106378 q^{9} + 20682632 q^{11} + 499806476 q^{13} + 1996058960 q^{15} + 3139516900 q^{17} - 474668552 q^{19} + 4019297088 q^{21} - 40776002608 q^{23} + 10378903150 q^{25} - 190235215504 q^{27} + 35253157356 q^{29} - 34389193280 q^{31} + 780208036448 q^{33} + 145094219040 q^{35} + 870228564444 q^{37} - 906424161136 q^{39} + 900085452084 q^{41} + 500707998536 q^{43} - 8866083940340 q^{45} + 1208059119264 q^{47} - 9102300649454 q^{49} + 1965395777200 q^{51} + 1236734202044 q^{53} + 28152970525040 q^{55} + 14546429728672 q^{57} + 14441975905064 q^{59} + 18336303417260 q^{61} - 15702480101712 q^{63} - 31048882497880 q^{65} - 76601421514856 q^{67} + 63670027709888 q^{69} - 145877173886864 q^{71} - 26417269924108 q^{73} - 261044718109400 q^{75} + 75313600116672 q^{77} + 257907833388128 q^{79} + 354480447023698 q^{81} + 255512806582648 q^{83} + 80693958484600 q^{85} + 854812182349584 q^{87} - 719794611712812 q^{89} + 41542153569696 q^{91} - 13\!\cdots\!20 q^{93}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) \) 2 * q - 4072 * q^3 - 140260 * q^5 + 126192 * q^7 + 27106378 * q^9 + 20682632 * q^11 + 499806476 * q^13 + 1996058960 * q^15 + 3139516900 * q^17 - 474668552 * q^19 + 4019297088 * q^21 - 40776002608 * q^23 + 10378903150 * q^25 - 190235215504 * q^27 + 35253157356 * q^29 - 34389193280 * q^31 + 780208036448 * q^33 + 145094219040 * q^35 + 870228564444 * q^37 - 906424161136 * q^39 + 900085452084 * q^41 + 500707998536 * q^43 - 8866083940340 * q^45 + 1208059119264 * q^47 - 9102300649454 * q^49 + 1965395777200 * q^51 + 1236734202044 * q^53 + 28152970525040 * q^55 + 14546429728672 * q^57 + 14441975905064 * q^59 + 18336303417260 * q^61 - 15702480101712 * q^63 - 31048882497880 * q^65 - 76601421514856 * q^67 + 63670027709888 * q^69 - 145877173886864 * q^71 - 26417269924108 * q^73 - 261044718109400 * q^75 + 75313600116672 * q^77 + 257907833388128 * q^79 + 354480447023698 * q^81 + 255512806582648 * q^83 + 80693958484600 * q^85 + 854812182349584 * q^87 - 719794611712812 * q^89 + 41542153569696 * q^91 - 1394036123851520 * q^93 + 522168669596560 * q^95 + 407635590418756 * q^97 - 3068162767300952 * q^99

Introduction

L-functions

Modular forms

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