LMFDB - Embedded newform 16.34.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,34,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 34, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 34);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 34 $
Character orbit:	$[\chi]$	$=$	16.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:	$110.372526210$
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} - x^{2} - 65185566x - 173679864984 $ x^3 - x^2 - 65185566*x - 173679864984
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{3}\cdot 7\cdot 11\cdot 29 $
Twist minimal:	no (minimal twist has level 4)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-6032.20$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+6.39317e7 q^{3} -2.18954e11 q^{5} +4.92542e13 q^{7} -1.47179e15 q^{9} +O(q^{10})$	$q+6.39317e7 q^{3} -2.18954e11 q^{5} +4.92542e13 q^{7} -1.47179e15 q^{9} +2.23089e17 q^{11} +7.57082e17 q^{13} -1.39981e19 q^{15} -6.49754e18 q^{17} +7.92387e20 q^{19} +3.14891e21 q^{21} -2.99746e22 q^{23} -6.84746e22 q^{25} -4.49495e23 q^{27} +1.97632e24 q^{29} -4.54327e24 q^{31} +1.42624e25 q^{33} -1.07844e25 q^{35} +1.28067e26 q^{37} +4.84016e25 q^{39} -3.19347e26 q^{41} +1.35211e27 q^{43} +3.22255e26 q^{45} +3.49210e27 q^{47} -5.30502e27 q^{49} -4.15399e26 q^{51} +3.47590e28 q^{53} -4.88461e28 q^{55} +5.06587e28 q^{57} -1.26922e29 q^{59} -3.88135e29 q^{61} -7.24920e28 q^{63} -1.65766e29 q^{65} +1.28458e30 q^{67} -1.91633e30 q^{69} +2.52414e30 q^{71} +8.98899e30 q^{73} -4.37770e30 q^{75} +1.09881e31 q^{77} -9.32137e30 q^{79} -2.05552e31 q^{81} -1.92342e31 q^{83} +1.42266e30 q^{85} +1.26349e32 q^{87} -6.91532e31 q^{89} +3.72895e31 q^{91} -2.90459e32 q^{93} -1.73496e32 q^{95} +8.57153e32 q^{97} -3.28341e32 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 92491788 q^{3} - 53880683886 q^{5} - 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+O(q^{10}) $ 3 * q - 92491788 * q^3 - 53880683886 * q^5 - 4541009914392 * q^7 + 6032364433690023 * q^9	$ 3 q - 92491788 q^{3} - 53880683886 q^{5} - 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+ \cdots - 22\!\cdots\!00 q^{99}+O(q^{100}) $ 3 * q - 92491788 * q^3 - 53880683886 * q^5 - 4541009914392 * q^7 + 6032364433690023 * q^9 - 227617657302449700 * q^11 + 272970442217358762 * q^13 + 32142473481241105848 * q^15 + 93037188311816716854 * q^17 - 1324887806175081735228 * q^19 - 2289418149258037352352 * q^21 - 11680621362017694733896 * q^23 + 258183723295544823174621 * q^25 - 1166092519364338103061624 * q^27 + 2899576251477878827068378 * q^29 - 15816960746665911900076128 * q^31 + 51019828271446575126952080 * q^33 - 104388880678377160773832848 * q^35 + 34785435467211797617907442 * q^37 - 135170503164419330229696168 * q^39 - 673854719392272123121298562 * q^41 + 1557799443962179540641188220 * q^43 - 8295378544987729028249800086 * q^45 + 7226966919683191159587179376 * q^47 - 4755447639624414087245018709 * q^49 - 24102421034763142313708267736 * q^51 + 54068957931214078433925305634 * q^53 - 72116739028709103312702411480 * q^55 + 276172301449046303471104708848 * q^57 - 201006239114458347216752503476 * q^59 + 59158715139739534320593297466 * q^61 + 1234485289426293666626080288968 * q^63 - 2252651168064734881763466967812 * q^65 + 2904325260543190867603068997428 * q^67 - 1526403599797234628609099310816 * q^69 - 1693346400885181878584585353944 * q^71 + 1318409001451717739930563879902 * q^73 - 20178451712790450065114527827828 * q^75 + 20835723633799560796766684178720 * q^77 - 43046757284148836281181573950704 * q^79 + 60385181780035992188351168542827 * q^81 + 71356083882380783036145089288964 * q^83 - 137360972548514627995666767037884 * q^85 + 253975679679758553189305216266392 * q^87 - 194457621978564019473800095872498 * q^89 + 384667165850137479369424211871408 * q^91 + 554725843432039554369917205602688 * q^93 + 205626706338774416034575296820376 * q^95 + 566218664019390927229117474480422 * q^97 - 2255131944887948953089434155026900 * 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
$3$	6.39317e7	0.857464	0.428732	−	0.903432i	$-0.358960\pi$
$3$	6.39317e7	0.857464	0.428732	+	0.903432i	$0.358960\pi$
$4$	0	0
$5$	−2.18954e11	−0.641723	−0.320861	−	0.947126i	$-0.603972\pi$
$5$	−2.18954e11	−0.641723	−0.320861	+	0.947126i	$0.603972\pi$
$6$	0	0
$7$	4.92542e13	0.560178	0.280089	−	0.959974i	$-0.409636\pi$
$7$	4.92542e13	0.560178	0.280089	+	0.959974i	$0.409636\pi$
$8$	0	0
$9$	−1.47179e15	−0.264756
$10$	0	0
$11$	2.23089e17	1.46386	0.731928	−	0.681382i	$-0.238621\pi$
$11$	2.23089e17	1.46386	0.731928	+	0.681382i	$0.238621\pi$
$12$	0	0
$13$	7.57082e17	0.315557	0.157778	−	0.987475i	$-0.449567\pi$
$13$	7.57082e17	0.315557	0.157778	+	0.987475i	$0.449567\pi$
$14$	0	0
$15$	−1.39981e19	−0.550254
$16$	0	0
$17$	−6.49754e18	−0.0323848	−0.0161924	−	0.999869i	$-0.505154\pi$
$17$	−6.49754e18	−0.0323848	−0.0161924	+	0.999869i	$0.505154\pi$
$18$	0	0
$19$	7.92387e20	0.630236	0.315118	−	0.949053i	$-0.397956\pi$
$19$	7.92387e20	0.630236	0.315118	+	0.949053i	$0.397956\pi$
$20$	0	0
$21$	3.14891e21	0.480332
$22$	0	0
$23$	−2.99746e22	−1.01916	−0.509581	−	0.860423i	$-0.670200\pi$
$23$	−2.99746e22	−1.01916	−0.509581	+	0.860423i	$0.670200\pi$
$24$	0	0
$25$	−6.84746e22	−0.588192
$26$	0	0
$27$	−4.49495e23	−1.08448
$28$	0	0
$29$	1.97632e24	1.46653	0.733263	−	0.679945i	$-0.237996\pi$
$29$	1.97632e24	1.46653	0.733263	+	0.679945i	$0.237996\pi$
$30$	0	0
$31$	−4.54327e24	−1.12176	−0.560881	−	0.827897i	$-0.689537\pi$
$31$	−4.54327e24	−1.12176	−0.560881	+	0.827897i	$0.689537\pi$
$32$	0	0
$33$	1.42624e25	1.25520
$34$	0	0
$35$	−1.07844e25	−0.359479
$36$	0	0
$37$	1.28067e26	1.70651	0.853253	−	0.521498i	$-0.174627\pi$
$37$	1.28067e26	1.70651	0.853253	+	0.521498i	$0.174627\pi$
$38$	0	0
$39$	4.84016e25	0.270579
$40$	0	0
$41$	−3.19347e26	−0.782221	−0.391111	−	0.920344i	$-0.627909\pi$
$41$	−3.19347e26	−0.782221	−0.391111	+	0.920344i	$0.627909\pi$
$42$	0	0
$43$	1.35211e27	1.50932	0.754662	−	0.656114i	$-0.227801\pi$
$43$	1.35211e27	1.50932	0.754662	+	0.656114i	$0.227801\pi$
$44$	0	0
$45$	3.22255e26	0.169900
$46$	0	0
$47$	3.49210e27	0.898404	0.449202	−	0.893430i	$-0.351708\pi$
$47$	3.49210e27	0.898404	0.449202	+	0.893430i	$0.351708\pi$
$48$	0	0
$49$	−5.30502e27	−0.686201
$50$	0	0
$51$	−4.15399e26	−0.0277688
$52$	0	0
$53$	3.47590e28	1.23173	0.615864	−	0.787852i	$-0.288807\pi$
$53$	3.47590e28	1.23173	0.615864	+	0.787852i	$0.288807\pi$
$54$	0	0
$55$	−4.88461e28	−0.939389
$56$	0	0
$57$	5.06587e28	0.540404
$58$	0	0
$59$	−1.26922e29	−0.766443	−0.383222	−	0.923656i	$-0.625185\pi$
$59$	−1.26922e29	−0.766443	−0.383222	+	0.923656i	$0.625185\pi$
$60$	0	0
$61$	−3.88135e29	−1.35220	−0.676099	−	0.736811i	$-0.736331\pi$
$61$	−3.88135e29	−1.35220	−0.676099	+	0.736811i	$0.736331\pi$
$62$	0	0
$63$	−7.24920e28	−0.148310
$64$	0	0
$65$	−1.65766e29	−0.202500
$66$	0	0
$67$	1.28458e30	0.951758	0.475879	−	0.879511i	$-0.342130\pi$
$67$	1.28458e30	0.951758	0.475879	+	0.879511i	$0.342130\pi$
$68$	0	0
$69$	−1.91633e30	−0.873895
$70$	0	0
$71$	2.52414e30	0.718373	0.359187	−	0.933266i	$-0.383054\pi$
$71$	2.52414e30	0.718373	0.359187	+	0.933266i	$0.383054\pi$
$72$	0	0
$73$	8.98899e30	1.61765	0.808823	−	0.588052i	$-0.200105\pi$
$73$	8.98899e30	1.61765	0.808823	+	0.588052i	$0.200105\pi$
$74$	0	0
$75$	−4.37770e30	−0.504353
$76$	0	0
$77$	1.09881e31	0.820019
$78$	0	0
$79$	−9.32137e30	−0.455651	−0.227826	−	0.973702i	$-0.573162\pi$
$79$	−9.32137e30	−0.455651	−0.227826	+	0.973702i	$0.573162\pi$
$80$	0	0
$81$	−2.05552e31	−0.665149
$82$	0	0
$83$	−1.92342e31	−0.416185	−0.208093	−	0.978109i	$-0.566726\pi$
$83$	−1.92342e31	−0.416185	−0.208093	+	0.978109i	$0.566726\pi$
$84$	0	0
$85$	1.42266e30	0.0207821
$86$	0	0
$87$	1.26349e32	1.25749
$88$	0	0
$89$	−6.91532e31	−0.473018	−0.236509	−	0.971629i	$-0.576003\pi$
$89$	−6.91532e31	−0.473018	−0.236509	+	0.971629i	$0.576003\pi$
$90$	0	0
$91$	3.72895e31	0.176768
$92$	0	0
$93$	−2.90459e32	−0.961870
$94$	0	0
$95$	−1.73496e32	−0.404436
$96$	0	0
$97$	8.57153e32	1.41685	0.708426	−	0.705785i	$-0.249405\pi$
$97$	8.57153e32	1.41685	0.708426	+	0.705785i	$0.249405\pi$
$98$	0	0
$99$	−3.28341e32	−0.387564
$100$	0	0
$101$	1.67206e33	1.41889	0.709444	−	0.704761i	$-0.248946\pi$
$101$	1.67206e33	1.41889	0.709444	+	0.704761i	$0.248946\pi$
$102$	0	0
$103$	1.55505e33	0.954842	0.477421	−	0.878675i	$-0.341572\pi$
$103$	1.55505e33	0.954842	0.477421	+	0.878675i	$0.341572\pi$
$104$	0	0
$105$	−6.89465e32	−0.308240
$106$	0	0
$107$	3.58097e32	0.117265	0.0586325	−	0.998280i	$-0.481326\pi$
$107$	3.58097e32	0.117265	0.0586325	+	0.998280i	$0.481326\pi$
$108$	0	0
$109$	3.52686e33	0.850847	0.425423	−	0.904995i	$-0.360125\pi$
$109$	3.52686e33	0.850847	0.425423	+	0.904995i	$0.360125\pi$
$110$	0	0
$111$	8.18752e33	1.46327
$112$	0	0
$113$	−4.01423e33	−0.534328	−0.267164	−	0.963651i	$-0.586087\pi$
$113$	−4.01423e33	−0.534328	−0.267164	+	0.963651i	$0.586087\pi$
$114$	0	0
$115$	6.56304e33	0.654020
$116$	0	0
$117$	−1.11427e33	−0.0835455
$118$	0	0
$119$	−3.20031e32	−0.0181413
$120$	0	0
$121$	2.65434e34	1.14287
$122$	0	0
$123$	−2.04164e34	−0.670726
$124$	0	0
$125$	4.04823e34	1.01918
$126$	0	0
$127$	3.61417e34	0.700241	0.350121	−	0.936705i	$-0.386141\pi$
$127$	3.61417e34	0.700241	0.350121	+	0.936705i	$0.386141\pi$
$128$	0	0
$129$	8.64428e34	1.29419
$130$	0	0
$131$	1.50991e35	1.75377	0.876887	−	0.480696i	$-0.159616\pi$
$131$	1.50991e35	1.75377	0.876887	+	0.480696i	$0.159616\pi$
$132$	0	0
$133$	3.90284e34	0.353044
$134$	0	0
$135$	9.84186e34	0.695937
$136$	0	0
$137$	3.42195e34	0.189838	0.0949192	−	0.995485i	$-0.469741\pi$
$137$	3.42195e34	0.189838	0.0949192	+	0.995485i	$0.469741\pi$
$138$	0	0
$139$	3.08655e35	1.34812	0.674060	−	0.738677i	$-0.264549\pi$
$139$	3.08655e35	1.34812	0.674060	+	0.738677i	$0.264549\pi$
$140$	0	0
$141$	2.23256e35	0.770349
$142$	0	0
$143$	1.68896e35	0.461930
$144$	0	0
$145$	−4.32722e35	−0.941103
$146$	0	0
$147$	−3.39159e35	−0.588393
$148$	0	0
$149$	1.12808e36	1.56591	0.782955	−	0.622078i	$-0.213711\pi$
$149$	1.12808e36	1.56591	0.782955	+	0.622078i	$0.213711\pi$
$150$	0	0
$151$	−1.27117e36	−1.41607	−0.708033	−	0.706179i	$-0.750417\pi$
$151$	−1.27117e36	−1.41607	−0.708033	+	0.706179i	$0.750417\pi$
$152$	0	0
$153$	9.56304e33	0.00857407
$154$	0	0
$155$	9.94766e35	0.719860
$156$	0	0
$157$	7.68303e35	0.449975	0.224987	−	0.974362i	$-0.427766\pi$
$157$	7.68303e35	0.449975	0.224987	+	0.974362i	$0.427766\pi$
$158$	0	0
$159$	2.22220e36	1.05616
$160$	0	0
$161$	−1.47637e36	−0.570912
$162$	0	0
$163$	−1.17206e36	−0.369704	−0.184852	−	0.982766i	$-0.559181\pi$
$163$	−1.17206e36	−0.369704	−0.184852	+	0.982766i	$0.559181\pi$
$164$	0	0
$165$	−3.12282e36	−0.805492
$166$	0	0
$167$	−5.02289e35	−0.106202	−0.0531009	−	0.998589i	$-0.516911\pi$
$167$	−5.02289e35	−0.106202	−0.0531009	+	0.998589i	$0.516911\pi$
$168$	0	0
$169$	−5.18296e36	−0.900424
$170$	0	0
$171$	−1.16623e36	−0.166859
$172$	0	0
$173$	5.22082e36	0.616564	0.308282	−	0.951295i	$-0.400246\pi$
$173$	5.22082e36	0.616564	0.308282	+	0.951295i	$0.400246\pi$
$174$	0	0
$175$	−3.37266e36	−0.329492
$176$	0	0
$177$	−8.11437e36	−0.657197
$178$	0	0
$179$	−2.47756e37	−1.66705	−0.833527	−	0.552479i	$-0.813682\pi$
$179$	−2.47756e37	−1.66705	−0.833527	+	0.552479i	$0.813682\pi$
$180$	0	0
$181$	8.26307e36	0.462855	0.231427	−	0.972852i	$-0.425660\pi$
$181$	8.26307e36	0.462855	0.231427	+	0.972852i	$0.425660\pi$
$182$	0	0
$183$	−2.48141e37	−1.15946
$184$	0	0
$185$	−2.80407e37	−1.09510
$186$	0	0
$187$	−1.44953e36	−0.0474067
$188$	0	0
$189$	−2.21395e37	−0.607503
$190$	0	0
$191$	2.54554e37	0.587125	0.293562	−	0.955940i	$-0.405159\pi$
$191$	2.54554e37	0.587125	0.293562	+	0.955940i	$0.405159\pi$
$192$	0	0
$193$	−7.64000e37	−1.48388	−0.741938	−	0.670469i	$-0.766093\pi$
$193$	−7.64000e37	−1.48388	−0.741938	+	0.670469i	$0.766093\pi$
$194$	0	0
$195$	−1.05977e37	−0.173636
$196$	0	0
$197$	2.00607e37	0.277749	0.138875	−	0.990310i	$-0.455651\pi$
$197$	2.00607e37	0.277749	0.138875	+	0.990310i	$0.455651\pi$
$198$	0	0
$199$	−2.98496e36	−0.0349835	−0.0174918	−	0.999847i	$-0.505568\pi$
$199$	−2.98496e36	−0.0349835	−0.0174918	+	0.999847i	$0.505568\pi$
$200$	0	0
$201$	8.21253e37	0.816098
$202$	0	0
$203$	9.73419e37	0.821515
$204$	0	0
$205$	6.99223e37	0.501969
$206$	0	0
$207$	4.41164e37	0.269829
$208$	0	0
$209$	1.76773e38	0.922574
$210$	0	0
$211$	−1.29116e38	−0.575863	−0.287931	−	0.957651i	$-0.592968\pi$
$211$	−1.29116e38	−0.575863	−0.287931	+	0.957651i	$0.592968\pi$
$212$	0	0
$213$	1.61372e38	0.615979
$214$	0	0
$215$	−2.96050e38	−0.968568
$216$	0	0
$217$	−2.23775e38	−0.628386
$218$	0	0
$219$	5.74681e38	1.38707
$220$	0	0
$221$	−4.91917e36	−0.0102193
$222$	0	0
$223$	5.28250e38	0.945822	0.472911	−	0.881110i	$-0.343203\pi$
$223$	5.28250e38	0.945822	0.472911	+	0.881110i	$0.343203\pi$
$224$	0	0
$225$	1.00780e38	0.155727
$226$	0	0
$227$	4.73287e38	0.631972	0.315986	−	0.948764i	$-0.397665\pi$
$227$	4.73287e38	0.631972	0.315986	+	0.948764i	$0.397665\pi$
$228$	0	0
$229$	−4.97730e38	−0.575054	−0.287527	−	0.957773i	$-0.592833\pi$
$229$	−4.97730e38	−0.575054	−0.287527	+	0.957773i	$0.592833\pi$
$230$	0	0
$231$	7.02486e38	0.703137
$232$	0	0
$233$	2.20413e39	1.91366	0.956831	−	0.290644i	$-0.0938696\pi$
$233$	2.20413e39	1.91366	0.956831	+	0.290644i	$0.0938696\pi$
$234$	0	0
$235$	−7.64608e38	−0.576526
$236$	0	0
$237$	−5.95931e38	−0.390704
$238$	0	0
$239$	−2.24449e38	−0.128101	−0.0640504	−	0.997947i	$-0.520402\pi$
$239$	−2.24449e38	−0.128101	−0.0640504	+	0.997947i	$0.520402\pi$
$240$	0	0
$241$	2.40422e39	1.19590	0.597949	−	0.801534i	$-0.295983\pi$
$241$	2.40422e39	1.19590	0.597949	+	0.801534i	$0.295983\pi$
$242$	0	0
$243$	1.18464e39	0.514142
$244$	0	0
$245$	1.16155e39	0.440351
$246$	0	0
$247$	5.99902e38	0.198875
$248$	0	0
$249$	−1.22968e39	−0.356864
$250$	0	0
$251$	−7.59890e39	−1.93257	−0.966283	−	0.257481i	$-0.917107\pi$
$251$	−7.59890e39	−1.93257	−0.966283	+	0.257481i	$0.917107\pi$
$252$	0	0
$253$	−6.68698e39	−1.49191
$254$	0	0
$255$	9.09531e37	0.0178199
$256$	0	0
$257$	−7.53622e39	−1.29795	−0.648974	−	0.760811i	$-0.724802\pi$
$257$	−7.53622e39	−1.29795	−0.648974	+	0.760811i	$0.724802\pi$
$258$	0	0
$259$	6.30782e39	0.955946
$260$	0	0
$261$	−2.90873e39	−0.388271
$262$	0	0
$263$	−4.45996e39	−0.524882	−0.262441	−	0.964948i	$-0.584527\pi$
$263$	−4.45996e39	−0.524882	−0.262441	+	0.964948i	$0.584527\pi$
$264$	0	0
$265$	−7.61060e39	−0.790428
$266$	0	0
$267$	−4.42109e39	−0.405596
$268$	0	0
$269$	1.47220e40	1.19414	0.597069	−	0.802190i	$-0.296332\pi$
$269$	1.47220e40	1.19414	0.597069	+	0.802190i	$0.296332\pi$
$270$	0	0
$271$	−1.39791e40	−1.00343	−0.501714	−	0.865034i	$-0.667297\pi$
$271$	−1.39791e40	−1.00343	−0.501714	+	0.865034i	$0.667297\pi$
$272$	0	0
$273$	2.38398e39	0.151572
$274$	0	0
$275$	−1.52759e40	−0.861028
$276$	0	0
$277$	−2.35630e40	−1.17846	−0.589228	−	0.807967i	$-0.700568\pi$
$277$	−2.35630e40	−1.17846	−0.589228	+	0.807967i	$0.700568\pi$
$278$	0	0
$279$	6.68676e39	0.296993
$280$	0	0
$281$	3.71685e40	1.46731	0.733654	−	0.679523i	$-0.237813\pi$
$281$	3.71685e40	1.46731	0.733654	+	0.679523i	$0.237813\pi$
$282$	0	0
$283$	9.67224e39	0.339665	0.169833	−	0.985473i	$-0.445677\pi$
$283$	9.67224e39	0.339665	0.169833	+	0.985473i	$0.445677\pi$
$284$	0	0
$285$	−1.10919e40	−0.346790
$286$	0	0
$287$	−1.57292e40	−0.438183
$288$	0	0
$289$	−4.02123e40	−0.998951
$290$	0	0
$291$	5.47993e40	1.21490
$292$	0	0
$293$	5.46285e40	1.08169	0.540845	−	0.841122i	$-0.318104\pi$
$293$	5.46285e40	1.08169	0.540845	+	0.841122i	$0.318104\pi$
$294$	0	0
$295$	2.77901e40	0.491844
$296$	0	0
$297$	−1.00277e41	−1.58753
$298$	0	0
$299$	−2.26932e40	−0.321604
$300$	0	0
$301$	6.65971e40	0.845490
$302$	0	0
$303$	1.06897e41	1.21665
$304$	0	0
$305$	8.49836e40	0.867736
$306$	0	0
$307$	−1.30064e40	−0.119227	−0.0596135	−	0.998222i	$-0.518987\pi$
$307$	−1.30064e40	−0.119227	−0.0596135	+	0.998222i	$0.518987\pi$
$308$	0	0
$309$	9.94173e40	0.818742
$310$	0	0
$311$	−2.40564e41	−1.78108	−0.890542	−	0.454901i	$-0.849675\pi$
$311$	−2.40564e41	−1.78108	−0.890542	+	0.454901i	$0.849675\pi$
$312$	0	0
$313$	3.63493e40	0.242111	0.121055	−	0.992646i	$-0.461372\pi$
$313$	3.63493e40	0.242111	0.121055	+	0.992646i	$0.461372\pi$
$314$	0	0
$315$	1.58724e40	0.0951740
$316$	0	0
$317$	1.26788e41	0.684859	0.342429	−	0.939544i	$-0.388750\pi$
$317$	1.26788e41	0.684859	0.342429	+	0.939544i	$0.388750\pi$
$318$	0	0
$319$	4.40894e41	2.14678
$320$	0	0
$321$	2.28938e40	0.100550
$322$	0	0
$323$	−5.14856e39	−0.0204101
$324$	0	0
$325$	−5.18409e40	−0.185608
$326$	0	0
$327$	2.25478e41	0.729570
$328$	0	0
$329$	1.72001e41	0.503266
$330$	0	0
$331$	2.46365e41	0.652254	0.326127	−	0.945326i	$-0.394256\pi$
$331$	2.46365e41	0.652254	0.326127	+	0.945326i	$0.394256\pi$
$332$	0	0
$333$	−1.88488e41	−0.451807
$334$	0	0
$335$	−2.81263e41	−0.610764
$336$	0	0
$337$	−3.62980e41	−0.714478	−0.357239	−	0.934013i	$-0.616282\pi$
$337$	−3.62980e41	−0.714478	−0.357239	+	0.934013i	$0.616282\pi$
$338$	0	0
$339$	−2.56637e41	−0.458167
$340$	0	0
$341$	−1.01355e42	−1.64210
$342$	0	0
$343$	−6.42078e41	−0.944572
$344$	0	0
$345$	4.19587e41	0.560798
$346$	0	0
$347$	−1.21611e42	−1.47754	−0.738768	−	0.673960i	$-0.764592\pi$
$347$	−1.21611e42	−1.47754	−0.738768	+	0.673960i	$0.764592\pi$
$348$	0	0
$349$	1.93880e41	0.214246	0.107123	−	0.994246i	$-0.465836\pi$
$349$	1.93880e41	0.214246	0.107123	+	0.994246i	$0.465836\pi$
$350$	0	0
$351$	−3.40304e41	−0.342216
$352$	0	0
$353$	2.41824e41	0.221420	0.110710	−	0.993853i	$-0.464688\pi$
$353$	2.41824e41	0.221420	0.110710	+	0.993853i	$0.464688\pi$
$354$	0	0
$355$	−5.52669e41	−0.460996
$356$	0	0
$357$	−2.04601e40	−0.0155555
$358$	0	0
$359$	1.31667e41	0.0912888	0.0456444	−	0.998958i	$-0.485466\pi$
$359$	1.31667e41	0.0912888	0.0456444	+	0.998958i	$0.485466\pi$
$360$	0	0
$361$	−9.52894e41	−0.602803
$362$	0	0
$363$	1.69697e42	0.979973
$364$	0	0
$365$	−1.96817e42	−1.03808
$366$	0	0
$367$	−7.81021e41	−0.376420	−0.188210	−	0.982129i	$-0.560269\pi$
$367$	−7.81021e41	−0.376420	−0.188210	+	0.982129i	$0.560269\pi$
$368$	0	0
$369$	4.70013e41	0.207098
$370$	0	0
$371$	1.71202e42	0.689986
$372$	0	0
$373$	−9.64138e41	−0.355585	−0.177793	−	0.984068i	$-0.556896\pi$
$373$	−9.64138e41	−0.355585	−0.177793	+	0.984068i	$0.556896\pi$
$374$	0	0
$375$	2.58811e42	0.873909
$376$	0	0
$377$	1.49623e42	0.462772
$378$	0	0
$379$	−4.62182e42	−1.30998	−0.654992	−	0.755636i	$-0.727328\pi$
$379$	−4.62182e42	−1.30998	−0.654992	+	0.755636i	$0.727328\pi$
$380$	0	0
$381$	2.31060e42	0.600432
$382$	0	0
$383$	4.15063e41	0.0989313	0.0494657	−	0.998776i	$-0.484248\pi$
$383$	4.15063e41	0.0989313	0.0494657	+	0.998776i	$0.484248\pi$
$384$	0	0
$385$	−2.40588e42	−0.526225
$386$	0	0
$387$	−1.99003e42	−0.399602
$388$	0	0
$389$	−9.10627e41	−0.167947	−0.0839735	−	0.996468i	$-0.526761\pi$
$389$	−9.10627e41	−0.167947	−0.0839735	+	0.996468i	$0.526761\pi$
$390$	0	0
$391$	1.94761e41	0.0330054
$392$	0	0
$393$	9.65309e42	1.50380
$394$	0	0
$395$	2.04095e42	0.292402
$396$	0	0
$397$	1.34295e43	1.77018	0.885088	−	0.465423i	$-0.154098\pi$
$397$	1.34295e43	1.77018	0.885088	+	0.465423i	$0.154098\pi$
$398$	0	0
$399$	2.49515e42	0.302722
$400$	0	0
$401$	−1.37037e43	−1.53093	−0.765465	−	0.643477i	$-0.777491\pi$
$401$	−1.37037e43	−1.53093	−0.765465	+	0.643477i	$0.777491\pi$
$402$	0	0
$403$	−3.43963e42	−0.353980
$404$	0	0
$405$	4.50064e42	0.426841
$406$	0	0
$407$	2.85702e43	2.49808
$408$	0	0
$409$	−5.72611e42	−0.461770	−0.230885	−	0.972981i	$-0.574162\pi$
$409$	−5.72611e42	−0.461770	−0.230885	+	0.972981i	$0.574162\pi$
$410$	0	0
$411$	2.18771e42	0.162780
$412$	0	0
$413$	−6.25146e42	−0.429344
$414$	0	0
$415$	4.21141e42	0.267075
$416$	0	0
$417$	1.97328e43	1.15596
$418$	0	0
$419$	2.34537e42	0.126964	0.0634818	−	0.997983i	$-0.479780\pi$
$419$	2.34537e42	0.126964	0.0634818	+	0.997983i	$0.479780\pi$
$420$	0	0
$421$	5.43051e42	0.271760	0.135880	−	0.990725i	$-0.456614\pi$
$421$	5.43051e42	0.271760	0.135880	+	0.990725i	$0.456614\pi$
$422$	0	0
$423$	−5.13965e42	−0.237858
$424$	0	0
$425$	4.44916e41	0.0190485
$426$	0	0
$427$	−1.91173e43	−0.757471
$428$	0	0
$429$	1.07978e43	0.396088
$430$	0	0
$431$	−2.04863e43	−0.695967	−0.347983	−	0.937501i	$-0.613133\pi$
$431$	−2.04863e43	−0.695967	−0.347983	+	0.937501i	$0.613133\pi$
$432$	0	0
$433$	−2.78864e43	−0.877691	−0.438846	−	0.898562i	$-0.644613\pi$
$433$	−2.78864e43	−0.877691	−0.438846	+	0.898562i	$0.644613\pi$
$434$	0	0
$435$	−2.76647e43	−0.806962
$436$	0	0
$437$	−2.37514e43	−0.642312
$438$	0	0
$439$	−2.25782e43	−0.566272	−0.283136	−	0.959080i	$-0.591375\pi$
$439$	−2.25782e43	−0.566272	−0.283136	+	0.959080i	$0.591375\pi$
$440$	0	0
$441$	7.80789e42	0.181676
$442$	0	0
$443$	−1.19378e43	−0.257789	−0.128894	−	0.991658i	$-0.541143\pi$
$443$	−1.19378e43	−0.257789	−0.128894	+	0.991658i	$0.541143\pi$
$444$	0	0
$445$	1.51414e43	0.303546
$446$	0	0
$447$	7.21202e43	1.34271
$448$	0	0
$449$	−6.42445e43	−1.11115	−0.555573	−	0.831468i	$-0.687501\pi$
$449$	−6.42445e43	−1.11115	−0.555573	+	0.831468i	$0.687501\pi$
$450$	0	0
$451$	−7.12428e43	−1.14506
$452$	0	0
$453$	−8.12680e43	−1.21423
$454$	0	0
$455$	−8.16467e42	−0.113436
$456$	0	0
$457$	−6.72354e43	−0.868922	−0.434461	−	0.900691i	$-0.643061\pi$
$457$	−6.72354e43	−0.868922	−0.434461	+	0.900691i	$0.643061\pi$
$458$	0	0
$459$	2.92061e42	0.0351208
$460$	0	0
$461$	−2.63019e43	−0.294389	−0.147195	−	0.989108i	$-0.547024\pi$
$461$	−2.63019e43	−0.294389	−0.147195	+	0.989108i	$0.547024\pi$
$462$	0	0
$463$	8.76936e43	0.913864	0.456932	−	0.889502i	$-0.348948\pi$
$463$	8.76936e43	0.913864	0.456932	+	0.889502i	$0.348948\pi$
$464$	0	0
$465$	6.35971e43	0.617254
$466$	0	0
$467$	−1.35357e44	−1.22392	−0.611958	−	0.790890i	$-0.709618\pi$
$467$	−1.35357e44	−1.22392	−0.611958	+	0.790890i	$0.709618\pi$
$468$	0	0
$469$	6.32709e43	0.533153
$470$	0	0
$471$	4.91190e43	0.385837
$472$	0	0
$473$	3.01641e44	2.20943
$474$	0	0
$475$	−5.42584e43	−0.370700
$476$	0	0
$477$	−5.11580e43	−0.326107
$478$	0	0
$479$	7.85163e43	0.467115	0.233558	−	0.972343i	$-0.424963\pi$
$479$	7.85163e43	0.467115	0.233558	+	0.972343i	$0.424963\pi$
$480$	0	0
$481$	9.69569e43	0.538499
$482$	0	0
$483$	−9.43871e43	−0.489536
$484$	0	0
$485$	−1.87677e44	−0.909226
$486$	0	0
$487$	3.82489e44	1.73138	0.865688	−	0.500584i	$-0.166881\pi$
$487$	3.82489e44	1.73138	0.865688	+	0.500584i	$0.166881\pi$
$488$	0	0
$489$	−7.49317e43	−0.317007
$490$	0	0
$491$	−4.02288e44	−1.59108	−0.795541	−	0.605900i	$-0.792813\pi$
$491$	−4.02288e44	−1.59108	−0.795541	+	0.605900i	$0.792813\pi$
$492$	0	0
$493$	−1.28412e43	−0.0474932
$494$	0	0
$495$	7.18914e43	0.248709
$496$	0	0
$497$	1.24324e44	0.402416
$498$	0	0
$499$	4.25677e44	1.28950	0.644750	−	0.764393i	$-0.276961\pi$
$499$	4.25677e44	1.28950	0.644750	+	0.764393i	$0.276961\pi$
$500$	0	0
$501$	−3.21122e43	−0.0910643
$502$	0	0
$503$	5.24439e44	1.39259	0.696295	−	0.717756i	$-0.254830\pi$
$503$	5.24439e44	1.39259	0.696295	+	0.717756i	$0.254830\pi$
$504$	0	0
$505$	−3.66103e44	−0.910533
$506$	0	0
$507$	−3.31355e44	−0.772081
$508$	0	0
$509$	8.39856e44	1.83384	0.916920	−	0.399071i	$-0.130667\pi$
$509$	8.39856e44	1.83384	0.916920	+	0.399071i	$0.130667\pi$
$510$	0	0
$511$	4.42745e44	0.906169
$512$	0	0
$513$	−3.56174e44	−0.683479
$514$	0	0
$515$	−3.40485e44	−0.612744
$516$	0	0
$517$	7.79048e44	1.31513
$518$	0	0
$519$	3.33776e44	0.528681
$520$	0	0
$521$	8.55665e44	1.27198	0.635991	−	0.771696i	$-0.280591\pi$
$521$	8.55665e44	1.27198	0.635991	+	0.771696i	$0.280591\pi$
$522$	0	0
$523$	−8.81639e44	−1.23030	−0.615152	−	0.788408i	$-0.710906\pi$
$523$	−8.81639e44	−1.23030	−0.615152	+	0.788408i	$0.710906\pi$
$524$	0	0
$525$	−2.15620e44	−0.282527
$526$	0	0
$527$	2.95201e43	0.0363281
$528$	0	0
$529$	3.34689e43	0.0386921
$530$	0	0
$531$	1.86804e44	0.202920
$532$	0	0
$533$	−2.41772e44	−0.246835
$534$	0	0
$535$	−7.84067e43	−0.0752516
$536$	0	0
$537$	−1.58395e45	−1.42944
$538$	0	0
$539$	−1.18349e45	−1.00450
$540$	0	0
$541$	−2.37260e45	−1.89439	−0.947197	−	0.320651i	$-0.896098\pi$
$541$	−2.37260e45	−1.89439	−0.947197	+	0.320651i	$0.896098\pi$
$542$	0	0
$543$	5.28272e44	0.396881
$544$	0	0
$545$	−7.72220e44	−0.546007
$546$	0	0
$547$	−1.48406e45	−0.987778	−0.493889	−	0.869525i	$-0.664425\pi$
$547$	−1.48406e45	−0.987778	−0.493889	+	0.869525i	$0.664425\pi$
$548$	0	0
$549$	5.71254e44	0.358002
$550$	0	0
$551$	1.56601e45	0.924257
$552$	0	0
$553$	−4.59116e44	−0.255246
$554$	0	0
$555$	−1.79269e45	−0.939011
$556$	0	0
$557$	3.92126e44	0.193560	0.0967799	−	0.995306i	$-0.469146\pi$
$557$	3.92126e44	0.193560	0.0967799	+	0.995306i	$0.469146\pi$
$558$	0	0
$559$	1.02366e45	0.476278
$560$	0	0
$561$	−9.26708e43	−0.0406495
$562$	0	0
$563$	−1.13519e45	−0.469548	−0.234774	−	0.972050i	$-0.575435\pi$
$563$	−1.13519e45	−0.469548	−0.234774	+	0.972050i	$0.575435\pi$
$564$	0	0
$565$	8.78930e44	0.342890
$566$	0	0
$567$	−1.01243e45	−0.372601
$568$	0	0
$569$	−1.83884e45	−0.638544	−0.319272	−	0.947663i	$-0.603438\pi$
$569$	−1.83884e45	−0.638544	−0.319272	+	0.947663i	$0.603438\pi$
$570$	0	0
$571$	−3.16473e45	−1.03715	−0.518574	−	0.855033i	$-0.673537\pi$
$571$	−3.16473e45	−1.03715	−0.518574	+	0.855033i	$0.673537\pi$
$572$	0	0
$573$	1.62741e45	0.503438
$574$	0	0
$575$	2.05249e45	0.599463
$576$	0	0
$577$	6.15172e45	1.69666	0.848332	−	0.529464i	$-0.177607\pi$
$577$	6.15172e45	1.69666	0.848332	+	0.529464i	$0.177607\pi$
$578$	0	0
$579$	−4.88438e45	−1.27237
$580$	0	0
$581$	−9.47367e44	−0.233137
$582$	0	0
$583$	7.75433e45	1.80307
$584$	0	0
$585$	2.43973e44	0.0536130
$586$	0	0
$587$	6.28489e45	1.30548	0.652738	−	0.757584i	$-0.273620\pi$
$587$	6.28489e45	1.30548	0.652738	+	0.757584i	$0.273620\pi$
$588$	0	0
$589$	−3.60003e45	−0.706974
$590$	0	0
$591$	1.28251e45	0.238160
$592$	0	0
$593$	−3.26071e45	−0.572677	−0.286338	−	0.958129i	$-0.592438\pi$
$593$	−3.26071e45	−0.572677	−0.286338	+	0.958129i	$0.592438\pi$
$594$	0	0
$595$	7.00720e43	0.0116417
$596$	0	0
$597$	−1.90834e44	−0.0299971
$598$	0	0
$599$	6.21811e45	0.924944	0.462472	−	0.886634i	$-0.346963\pi$
$599$	6.21811e45	0.924944	0.462472	+	0.886634i	$0.346963\pi$
$600$	0	0
$601$	8.29634e45	1.16804	0.584020	−	0.811739i	$-0.301479\pi$
$601$	8.29634e45	1.16804	0.584020	+	0.811739i	$0.301479\pi$
$602$	0	0
$603$	−1.89063e45	−0.251983
$604$	0	0
$605$	−5.81178e45	−0.733408
$606$	0	0
$607$	−6.16593e45	−0.736861	−0.368430	−	0.929655i	$-0.620105\pi$
$607$	−6.16593e45	−0.736861	−0.368430	+	0.929655i	$0.620105\pi$
$608$	0	0
$609$	6.22324e45	0.704419
$610$	0	0
$611$	2.64380e45	0.283498
$612$	0	0
$613$	5.34569e45	0.543133	0.271566	−	0.962420i	$-0.412458\pi$
$613$	5.34569e45	0.543133	0.271566	+	0.962420i	$0.412458\pi$
$614$	0	0
$615$	4.47025e45	0.430420
$616$	0	0
$617$	2.09959e45	0.191615	0.0958077	−	0.995400i	$-0.469457\pi$
$617$	2.09959e45	0.191615	0.0958077	+	0.995400i	$0.469457\pi$
$618$	0	0
$619$	−6.78373e45	−0.586912	−0.293456	−	0.955973i	$-0.594805\pi$
$619$	−6.78373e45	−0.586912	−0.293456	+	0.955973i	$0.594805\pi$
$620$	0	0
$621$	1.34734e46	1.10526
$622$	0	0
$623$	−3.40609e45	−0.264974
$624$	0	0
$625$	−8.92271e44	−0.0658380
$626$	0	0
$627$	1.13014e46	0.791074
$628$	0	0
$629$	−8.32118e44	−0.0552649
$630$	0	0
$631$	−6.16884e45	−0.388794	−0.194397	−	0.980923i	$-0.562275\pi$
$631$	−6.16884e45	−0.388794	−0.194397	+	0.980923i	$0.562275\pi$
$632$	0	0
$633$	−8.25460e45	−0.493782
$634$	0	0
$635$	−7.91337e45	−0.449361
$636$	0	0
$637$	−4.01633e45	−0.216535
$638$	0	0
$639$	−3.71501e45	−0.190193
$640$	0	0
$641$	8.65757e45	0.420958	0.210479	−	0.977598i	$-0.432498\pi$
$641$	8.65757e45	0.420958	0.210479	+	0.977598i	$0.432498\pi$
$642$	0	0
$643$	3.14944e46	1.45463	0.727314	−	0.686305i	$-0.240769\pi$
$643$	3.14944e46	1.45463	0.727314	+	0.686305i	$0.240769\pi$
$644$	0	0
$645$	−1.89270e46	−0.830512
$646$	0	0
$647$	1.65528e46	0.690163	0.345081	−	0.938573i	$-0.387851\pi$
$647$	1.65528e46	0.690163	0.345081	+	0.938573i	$0.387851\pi$
$648$	0	0
$649$	−2.83149e46	−1.12196
$650$	0	0
$651$	−1.43063e46	−0.538818
$652$	0	0
$653$	−1.91669e46	−0.686252	−0.343126	−	0.939289i	$-0.611486\pi$
$653$	−1.91669e46	−0.686252	−0.343126	+	0.939289i	$0.611486\pi$
$654$	0	0
$655$	−3.30600e46	−1.12544
$656$	0	0
$657$	−1.32299e46	−0.428281
$658$	0	0
$659$	−1.47214e46	−0.453251	−0.226626	−	0.973982i	$-0.572769\pi$
$659$	−1.47214e46	−0.453251	−0.226626	+	0.973982i	$0.572769\pi$
$660$	0	0
$661$	2.36391e46	0.692319	0.346160	−	0.938176i	$-0.387486\pi$
$661$	2.36391e46	0.692319	0.346160	+	0.938176i	$0.387486\pi$
$662$	0	0
$663$	−3.14491e44	−0.00876264
$664$	0	0
$665$	−8.54541e45	−0.226556
$666$	0	0
$667$	−5.92392e46	−1.49463
$668$	0	0
$669$	3.37719e46	0.811008
$670$	0	0
$671$	−8.65885e46	−1.97942
$672$	0	0
$673$	−6.32698e46	−1.37704	−0.688522	−	0.725216i	$-0.741740\pi$
$673$	−6.32698e46	−1.37704	−0.688522	+	0.725216i	$0.741740\pi$
$674$	0	0
$675$	3.07790e46	0.637884
$676$	0	0
$677$	7.67075e44	0.0151400	0.00756998	−	0.999971i	$-0.497590\pi$
$677$	7.67075e44	0.0151400	0.00756998	+	0.999971i	$0.497590\pi$
$678$	0	0
$679$	4.22184e46	0.793689
$680$	0	0
$681$	3.02581e46	0.541893
$682$	0	0
$683$	1.85282e46	0.316148	0.158074	−	0.987427i	$-0.449472\pi$
$683$	1.85282e46	0.316148	0.158074	+	0.987427i	$0.449472\pi$
$684$	0	0
$685$	−7.49248e45	−0.121824
$686$	0	0
$687$	−3.18208e46	−0.493088
$688$	0	0
$689$	2.63154e46	0.388680
$690$	0	0
$691$	1.06135e47	1.49441	0.747207	−	0.664591i	$-0.231394\pi$
$691$	1.06135e47	1.49441	0.747207	+	0.664591i	$0.231394\pi$
$692$	0	0
$693$	−1.61722e46	−0.217105
$694$	0	0
$695$	−6.75811e46	−0.865119
$696$	0	0
$697$	2.07497e45	0.0253321
$698$	0	0
$699$	1.40914e47	1.64090
$700$	0	0
$701$	8.20778e46	0.911756	0.455878	−	0.890042i	$-0.349325\pi$
$701$	8.20778e46	0.911756	0.455878	+	0.890042i	$0.349325\pi$
$702$	0	0
$703$	1.01478e47	1.07550
$704$	0	0
$705$	−4.88827e46	−0.494351
$706$	0	0
$707$	8.23558e46	0.794830
$708$	0	0
$709$	5.53905e45	0.0510238	0.0255119	−	0.999675i	$-0.491878\pi$
$709$	5.53905e45	0.0510238	0.0255119	+	0.999675i	$0.491878\pi$
$710$	0	0
$711$	1.37191e46	0.120636
$712$	0	0
$713$	1.36182e47	1.14326
$714$	0	0
$715$	−3.69805e46	−0.296431
$716$	0	0
$717$	−1.43494e46	−0.109842
$718$	0	0
$719$	−2.50176e47	−1.82902	−0.914512	−	0.404559i	$-0.867425\pi$
$719$	−2.50176e47	−1.82902	−0.914512	+	0.404559i	$0.867425\pi$
$720$	0	0
$721$	7.65930e46	0.534881
$722$	0	0
$723$	1.53706e47	1.02544
$724$	0	0
$725$	−1.35327e47	−0.862599
$726$	0	0
$727$	−3.23979e47	−1.97333	−0.986664	−	0.162769i	$-0.947958\pi$
$727$	−3.23979e47	−1.97333	−0.986664	+	0.162769i	$0.947958\pi$
$728$	0	0
$729$	1.90004e47	1.10601
$730$	0	0
$731$	−8.78539e45	−0.0488792
$732$	0	0
$733$	−1.15065e47	−0.611968	−0.305984	−	0.952037i	$-0.598985\pi$
$733$	−1.15065e47	−0.611968	−0.305984	+	0.952037i	$0.598985\pi$
$734$	0	0
$735$	7.42601e46	0.377585
$736$	0	0
$737$	2.86575e47	1.39324
$738$	0	0
$739$	2.15723e47	1.00291	0.501456	−	0.865183i	$-0.332798\pi$
$739$	2.15723e47	1.00291	0.501456	+	0.865183i	$0.332798\pi$
$740$	0	0
$741$	3.83528e46	0.170528
$742$	0	0
$743$	4.13812e47	1.75990	0.879949	−	0.475069i	$-0.157577\pi$
$743$	4.13812e47	1.75990	0.879949	+	0.475069i	$0.157577\pi$
$744$	0	0
$745$	−2.46998e47	−1.00488
$746$	0	0
$747$	2.83088e46	0.110187
$748$	0	0
$749$	1.76378e46	0.0656892
$750$	0	0
$751$	−3.22086e47	−1.14792	−0.573962	−	0.818882i	$-0.694594\pi$
$751$	−3.22086e47	−1.14792	−0.573962	+	0.818882i	$0.694594\pi$
$752$	0	0
$753$	−4.85811e47	−1.65711
$754$	0	0
$755$	2.78327e47	0.908722
$756$	0	0
$757$	2.08055e47	0.650274	0.325137	−	0.945667i	$-0.394590\pi$
$757$	2.08055e47	0.650274	0.325137	+	0.945667i	$0.394590\pi$
$758$	0	0
$759$	−4.27511e47	−1.27926
$760$	0	0
$761$	−9.02741e46	−0.258652	−0.129326	−	0.991602i	$-0.541281\pi$
$761$	−9.02741e46	−0.258652	−0.129326	+	0.991602i	$0.541281\pi$
$762$	0	0
$763$	1.73713e47	0.476625
$764$	0	0
$765$	−2.09386e45	−0.00550218
$766$	0	0
$767$	−9.60906e46	−0.241856
$768$	0	0
$769$	−1.05116e47	−0.253444	−0.126722	−	0.991938i	$-0.540446\pi$
$769$	−1.05116e47	−0.253444	−0.126722	+	0.991938i	$0.540446\pi$
$770$	0	0
$771$	−4.81804e47	−1.11294
$772$	0	0
$773$	−6.02385e47	−1.33325	−0.666627	−	0.745392i	$-0.732262\pi$
$773$	−6.02385e47	−1.33325	−0.666627	+	0.745392i	$0.732262\pi$
$774$	0	0
$775$	3.11098e47	0.659811
$776$	0	0
$777$	4.03270e47	0.819689
$778$	0	0
$779$	−2.53047e47	−0.492984
$780$	0	0
$781$	5.63106e47	1.05159
$782$	0	0
$783$	−8.88344e47	−1.59042
$784$	0	0
$785$	−1.68223e47	−0.288759
$786$	0	0
$787$	−3.16244e47	−0.520523	−0.260261	−	0.965538i	$-0.583809\pi$
$787$	−3.16244e47	−0.520523	−0.260261	+	0.965538i	$0.583809\pi$
$788$	0	0
$789$	−2.85133e47	−0.450067
$790$	0	0
$791$	−1.97718e47	−0.299318
$792$	0	0
$793$	−2.93850e47	−0.426695
$794$	0	0
$795$	−4.86559e47	−0.677763
$796$	0	0
$797$	−4.41105e47	−0.589495	−0.294747	−	0.955575i	$-0.595235\pi$
$797$	−4.41105e47	−0.589495	−0.294747	+	0.955575i	$0.595235\pi$
$798$	0	0
$799$	−2.26900e46	−0.0290947
$800$	0	0
$801$	1.01779e47	0.125234
$802$	0	0
$803$	2.00534e48	2.36800
$804$	0	0
$805$	3.23257e47	0.366367
$806$	0	0
$807$	9.41204e47	1.02393
$808$	0	0
$809$	−1.10768e48	−1.15681	−0.578406	−	0.815749i	$-0.696325\pi$
$809$	−1.10768e48	−1.15681	−0.578406	+	0.815749i	$0.696325\pi$
$810$	0	0
$811$	−9.46114e47	−0.948635	−0.474318	−	0.880354i	$-0.657305\pi$
$811$	−9.46114e47	−0.948635	−0.474318	+	0.880354i	$0.657305\pi$
$812$	0	0
$813$	−8.93709e47	−0.860403
$814$	0	0
$815$	2.56627e47	0.237247
$816$	0	0
$817$	1.07139e48	0.951230
$818$	0	0
$819$	−5.48824e46	−0.0468003
$820$	0	0
$821$	2.23829e48	1.83338	0.916692	−	0.399594i	$-0.130849\pi$
$821$	2.23829e48	1.83338	0.916692	+	0.399594i	$0.130849\pi$
$822$	0	0
$823$	1.31671e48	1.03608	0.518040	−	0.855356i	$-0.326662\pi$
$823$	1.31671e48	1.03608	0.518040	+	0.855356i	$0.326662\pi$
$824$	0	0
$825$	−9.76615e47	−0.738301
$826$	0	0
$827$	5.96543e47	0.433312	0.216656	−	0.976248i	$-0.430485\pi$
$827$	5.96543e47	0.433312	0.216656	+	0.976248i	$0.430485\pi$
$828$	0	0
$829$	1.00487e48	0.701392	0.350696	−	0.936489i	$-0.385945\pi$
$829$	1.00487e48	0.701392	0.350696	+	0.936489i	$0.385945\pi$
$830$	0	0
$831$	−1.50642e48	−1.01048
$832$	0	0
$833$	3.44695e46	0.0222225
$834$	0	0
$835$	1.09978e47	0.0681521
$836$	0	0
$837$	2.04218e48	1.21653
$838$	0	0
$839$	4.96184e47	0.284165	0.142083	−	0.989855i	$-0.454620\pi$
$839$	4.96184e47	0.284165	0.142083	+	0.989855i	$0.454620\pi$
$840$	0	0
$841$	2.08975e48	1.15070
$842$	0	0
$843$	2.37625e48	1.25816
$844$	0	0
$845$	1.13483e48	0.577822
$846$	0	0
$847$	1.30737e48	0.640212
$848$	0	0
$849$	6.18363e47	0.291251
$850$	0	0
$851$	−3.83874e48	−1.73921
$852$	0	0
$853$	−3.70417e48	−1.61448	−0.807238	−	0.590226i	$-0.799039\pi$
$853$	−3.70417e48	−1.61448	−0.807238	+	0.590226i	$0.799039\pi$
$854$	0	0
$855$	2.55350e47	0.107077
$856$	0	0
$857$	4.82096e47	0.194514	0.0972569	−	0.995259i	$-0.468993\pi$
$857$	4.82096e47	0.194514	0.0972569	+	0.995259i	$0.468993\pi$
$858$	0	0
$859$	−2.42649e48	−0.942087	−0.471043	−	0.882110i	$-0.656122\pi$
$859$	−2.42649e48	−0.942087	−0.471043	+	0.882110i	$0.656122\pi$
$860$	0	0
$861$	−1.00559e48	−0.375726
$862$	0	0
$863$	4.96761e48	1.78636	0.893181	−	0.449698i	$-0.148468\pi$
$863$	4.96761e48	1.78636	0.893181	+	0.449698i	$0.148468\pi$
$864$	0	0
$865$	−1.14312e48	−0.395663
$866$	0	0
$867$	−2.57084e48	−0.856565
$868$	0	0
$869$	−2.07949e48	−0.667008
$870$	0	0
$871$	9.72531e47	0.300334
$872$	0	0
$873$	−1.26155e48	−0.375120
$874$	0	0
$875$	1.99393e48	0.570921
$876$	0	0
$877$	−5.66923e48	−1.56326	−0.781629	−	0.623743i	$-0.785611\pi$
$877$	−5.66923e48	−1.56326	−0.781629	+	0.623743i	$0.785611\pi$
$878$	0	0
$879$	3.49249e48	0.927511
$880$	0	0
$881$	4.15832e48	1.06369	0.531845	−	0.846842i	$-0.321499\pi$
$881$	4.15832e48	1.06369	0.531845	+	0.846842i	$0.321499\pi$
$882$	0	0
$883$	−2.01243e48	−0.495870	−0.247935	−	0.968777i	$-0.579752\pi$
$883$	−2.01243e48	−0.495870	−0.247935	+	0.968777i	$0.579752\pi$
$884$	0	0
$885$	1.77667e48	0.421738
$886$	0	0
$887$	−7.27157e48	−1.66299	−0.831493	−	0.555535i	$-0.812513\pi$
$887$	−7.27157e48	−1.66299	−0.831493	+	0.555535i	$0.812513\pi$
$888$	0	0
$889$	1.78013e48	0.392259
$890$	0	0
$891$	−4.58563e48	−0.973681
$892$	0	0
$893$	2.76709e48	0.566206
$894$	0	0
$895$	5.42472e48	1.06979
$896$	0	0
$897$	−1.45082e48	−0.275763
$898$	0	0
$899$	−8.97894e48	−1.64509
$900$	0	0
$901$	−2.25848e47	−0.0398893
$902$	0	0
$903$	4.25767e48	0.724977
$904$	0	0
$905$	−1.80923e48	−0.297024
$906$	0	0
$907$	−7.77548e48	−1.23085	−0.615427	−	0.788194i	$-0.711017\pi$
$907$	−7.77548e48	−1.23085	−0.615427	+	0.788194i	$0.711017\pi$
$908$	0	0
$909$	−2.46092e48	−0.375659
$910$	0	0
$911$	2.43513e48	0.358484	0.179242	−	0.983805i	$-0.442636\pi$
$911$	2.43513e48	0.358484	0.179242	+	0.983805i	$0.442636\pi$
$912$	0	0
$913$	−4.29094e48	−0.609235
$914$	0	0
$915$	5.43315e48	0.744052
$916$	0	0
$917$	7.43692e48	0.982425
$918$	0	0
$919$	−2.51449e48	−0.320438	−0.160219	−	0.987081i	$-0.551220\pi$
$919$	−2.51449e48	−0.320438	−0.160219	+	0.987081i	$0.551220\pi$
$920$	0	0
$921$	−8.31524e47	−0.102233
$922$	0	0
$923$	1.91098e48	0.226687
$924$	0	0
$925$	−8.76931e48	−1.00375
$926$	0	0
$927$	−2.28872e48	−0.252800
$928$	0	0
$929$	−3.90806e48	−0.416584	−0.208292	−	0.978067i	$-0.566790\pi$
$929$	−3.90806e48	−0.416584	−0.208292	+	0.978067i	$0.566790\pi$
$930$	0	0
$931$	−4.20363e48	−0.432468
$932$	0	0
$933$	−1.53797e49	−1.52721
$934$	0	0
$935$	3.17379e47	0.0304220
$936$	0	0
$937$	−6.90657e48	−0.639086	−0.319543	−	0.947572i	$-0.603529\pi$
$937$	−6.90657e48	−0.639086	−0.319543	+	0.947572i	$0.603529\pi$
$938$	0	0
$939$	2.32387e48	0.207601
$940$	0	0
$941$	1.15202e49	0.993646	0.496823	−	0.867852i	$-0.334500\pi$
$941$	1.15202e49	0.993646	0.496823	+	0.867852i	$0.334500\pi$
$942$	0	0
$943$	9.57229e48	0.797210
$944$	0	0
$945$	4.84753e48	0.389848
$946$	0	0
$947$	2.25510e48	0.175142	0.0875711	−	0.996158i	$-0.472090\pi$
$947$	2.25510e48	0.175142	0.0875711	+	0.996158i	$0.472090\pi$
$948$	0	0
$949$	6.80540e48	0.510459
$950$	0	0
$951$	8.10579e48	0.587242
$952$	0	0
$953$	1.18218e49	0.827277	0.413638	−	0.910441i	$-0.364258\pi$
$953$	1.18218e49	0.827277	0.413638	+	0.910441i	$0.364258\pi$
$954$	0	0
$955$	−5.57357e48	−0.376771
$956$	0	0
$957$	2.81871e49	1.84079
$958$	0	0
$959$	1.68545e48	0.106343
$960$	0	0
$961$	4.23783e48	0.258349
$962$	0	0
$963$	−5.27045e47	−0.0310466
$964$	0	0
$965$	1.67281e49	0.952237
$966$	0	0
$967$	−3.25640e48	−0.179144	−0.0895718	−	0.995980i	$-0.528550\pi$
$967$	−3.25640e48	−0.179144	−0.0895718	+	0.995980i	$0.528550\pi$
$968$	0	0
$969$	−3.29157e47	−0.0175009
$970$	0	0
$971$	2.66132e49	1.36767	0.683833	−	0.729639i	$-0.260312\pi$
$971$	2.66132e49	1.36767	0.683833	+	0.729639i	$0.260312\pi$
$972$	0	0
$973$	1.52025e49	0.755187
$974$	0	0
$975$	−3.31428e48	−0.159152
$976$	0	0
$977$	−1.77551e49	−0.824256	−0.412128	−	0.911126i	$-0.635214\pi$
$977$	−1.77551e49	−0.824256	−0.412128	+	0.911126i	$0.635214\pi$
$978$	0	0
$979$	−1.54273e49	−0.692430
$980$	0	0
$981$	−5.19081e48	−0.225267
$982$	0	0
$983$	2.83065e49	1.18782	0.593912	−	0.804530i	$-0.297583\pi$
$983$	2.83065e49	1.18782	0.593912	+	0.804530i	$0.297583\pi$
$984$	0	0
$985$	−4.39236e48	−0.178238
$986$	0	0
$987$	1.09963e49	0.431532
$988$	0	0
$989$	−4.05289e49	−1.53825
$990$	0	0
$991$	1.61657e49	0.593442	0.296721	−	0.954964i	$-0.404107\pi$
$991$	1.61657e49	0.593442	0.296721	+	0.954964i	$0.404107\pi$
$992$	0	0
$993$	1.57505e49	0.559284
$994$	0	0
$995$	6.53569e47	0.0224497
$996$	0	0
$997$	−8.74014e48	−0.290434	−0.145217	−	0.989400i	$-0.546388\pi$
$997$	−8.74014e48	−0.290434	−0.145217	+	0.989400i	$0.546388\pi$
$998$	0	0
$999$	−5.75653e49	−1.85067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.34.a.d.1.3		3
4.3	odd	2		4.34.a.a.1.1	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4.34.a.a.1.1	✓	3	4.3	odd	2
16.34.a.d.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 34 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q+6.39317e7 q^{3} -2.18954e11 q^{5} +4.92542e13 q^{7} -1.47179e15 q^{9} +O(q^{10})\)	\(q+6.39317e7 q^{3} -2.18954e11 q^{5} +4.92542e13 q^{7} -1.47179e15 q^{9} +2.23089e17 q^{11} +7.57082e17 q^{13} -1.39981e19 q^{15} -6.49754e18 q^{17} +7.92387e20 q^{19} +3.14891e21 q^{21} -2.99746e22 q^{23} -6.84746e22 q^{25} -4.49495e23 q^{27} +1.97632e24 q^{29} -4.54327e24 q^{31} +1.42624e25 q^{33} -1.07844e25 q^{35} +1.28067e26 q^{37} +4.84016e25 q^{39} -3.19347e26 q^{41} +1.35211e27 q^{43} +3.22255e26 q^{45} +3.49210e27 q^{47} -5.30502e27 q^{49} -4.15399e26 q^{51} +3.47590e28 q^{53} -4.88461e28 q^{55} +5.06587e28 q^{57} -1.26922e29 q^{59} -3.88135e29 q^{61} -7.24920e28 q^{63} -1.65766e29 q^{65} +1.28458e30 q^{67} -1.91633e30 q^{69} +2.52414e30 q^{71} +8.98899e30 q^{73} -4.37770e30 q^{75} +1.09881e31 q^{77} -9.32137e30 q^{79} -2.05552e31 q^{81} -1.92342e31 q^{83} +1.42266e30 q^{85} +1.26349e32 q^{87} -6.91532e31 q^{89} +3.72895e31 q^{91} -2.90459e32 q^{93} -1.73496e32 q^{95} +8.57153e32 q^{97} -3.28341e32 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 92491788 q^{3} - 53880683886 q^{5} - 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+O(q^{10}) \) 3 * q - 92491788 * q^3 - 53880683886 * q^5 - 4541009914392 * q^7 + 6032364433690023 * q^9	\( 3 q - 92491788 q^{3} - 53880683886 q^{5} - 4541009914392 q^{7} + 60\!\cdots\!23 q^{9}+ \cdots - 22\!\cdots\!00 q^{99}+O(q^{100}) \) 3 * q - 92491788 * q^3 - 53880683886 * q^5 - 4541009914392 * q^7 + 6032364433690023 * q^9 - 227617657302449700 * q^11 + 272970442217358762 * q^13 + 32142473481241105848 * q^15 + 93037188311816716854 * q^17 - 1324887806175081735228 * q^19 - 2289418149258037352352 * q^21 - 11680621362017694733896 * q^23 + 258183723295544823174621 * q^25 - 1166092519364338103061624 * q^27 + 2899576251477878827068378 * q^29 - 15816960746665911900076128 * q^31 + 51019828271446575126952080 * q^33 - 104388880678377160773832848 * q^35 + 34785435467211797617907442 * q^37 - 135170503164419330229696168 * q^39 - 673854719392272123121298562 * q^41 + 1557799443962179540641188220 * q^43 - 8295378544987729028249800086 * q^45 + 7226966919683191159587179376 * q^47 - 4755447639624414087245018709 * q^49 - 24102421034763142313708267736 * q^51 + 54068957931214078433925305634 * q^53 - 72116739028709103312702411480 * q^55 + 276172301449046303471104708848 * q^57 - 201006239114458347216752503476 * q^59 + 59158715139739534320593297466 * q^61 + 1234485289426293666626080288968 * q^63 - 2252651168064734881763466967812 * q^65 + 2904325260543190867603068997428 * q^67 - 1526403599797234628609099310816 * q^69 - 1693346400885181878584585353944 * q^71 + 1318409001451717739930563879902 * q^73 - 20178451712790450065114527827828 * q^75 + 20835723633799560796766684178720 * q^77 - 43046757284148836281181573950704 * q^79 + 60385181780035992188351168542827 * q^81 + 71356083882380783036145089288964 * q^83 - 137360972548514627995666767037884 * q^85 + 253975679679758553189305216266392 * q^87 - 194457621978564019473800095872498 * q^89 + 384667165850137479369424211871408 * q^91 + 554725843432039554369917205602688 * q^93 + 205626706338774416034575296820376 * q^95 + 566218664019390927229117474480422 * q^97 - 2255131944887948953089434155026900 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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