LMFDB - Embedded newform 16.42.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,42,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, 42, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 42);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 42 $
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:	$170.354672730$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\mathbb{Q}[x]/(x^{2} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1139917559892 $ x^2 - x - 1139917559892
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{7}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.06767e6$ 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)

$f(q)$	$=$	$q-1.05814e10 q^{3} -2.54286e14 q^{5} -2.96016e17 q^{7} +7.54941e19 q^{9} +O(q^{10})$	$q-1.05814e10 q^{3} -2.54286e14 q^{5} -2.96016e17 q^{7} +7.54941e19 q^{9} +1.09620e21 q^{11} +8.09333e22 q^{13} +2.69071e24 q^{15} -5.33128e24 q^{17} +3.50762e25 q^{19} +3.13228e27 q^{21} -6.68932e27 q^{23} +1.91866e28 q^{25} -4.12899e29 q^{27} -1.63586e30 q^{29} -2.77961e30 q^{31} -1.15993e31 q^{33} +7.52726e31 q^{35} +4.65417e31 q^{37} -8.56392e32 q^{39} -8.83679e32 q^{41} +3.69256e32 q^{43} -1.91971e34 q^{45} +1.89619e34 q^{47} +4.30577e34 q^{49} +5.64127e34 q^{51} -2.01460e35 q^{53} -2.78747e35 q^{55} -3.71157e35 q^{57} -5.07603e34 q^{59} -6.57027e36 q^{61} -2.23474e37 q^{63} -2.05802e37 q^{65} -5.28414e37 q^{67} +7.07827e37 q^{69} +1.04768e38 q^{71} -2.83453e38 q^{73} -2.03022e38 q^{75} -3.24491e38 q^{77} +6.59452e38 q^{79} +1.61558e39 q^{81} -1.30773e39 q^{83} +1.35567e39 q^{85} +1.73098e40 q^{87} +2.16819e39 q^{89} -2.39575e40 q^{91} +2.94123e40 q^{93} -8.91937e39 q^{95} -8.40052e39 q^{97} +8.27563e40 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 41\!\cdots\!86 q^{9}+O(q^{10}) $ 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9	$ 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 12\!\cdots\!88 q^{99}+O(q^{100}) $ 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9 - 110594481484939565784 * q^11 + 173293106838448696895068 * q^13 + 3295287287252802285241680 * q^15 - 10287966260741677078429404 * q^17 - 206254882139859790701457000 * q^19 + 3267849131609686416057835584 * q^21 - 14925421378885128304361915568 * q^23 + 97534966241249753807444705150 * q^25 - 533156288863055843864697754320 * q^27 - 1375143011941884388614889894020 * q^29 + 3626496922711160469799126663616 * q^31 - 13672735988327258412044854187424 * q^33 + 103039524095905042910481769779360 * q^35 + 120356679589378915210866994599436 * q^37 - 697708668559917206762862430490352 * q^39 + 1270177139906840106861139346878164 * q^41 + 6045433421160616098770468790853352 * q^43 - 30992658299909156185379775337031460 * q^45 + 25416478966330542868882280891861664 * q^47 + 4716677344275473650585885334355954 * q^49 + 47896619495707425086522465903395056 * q^51 - 301376484284740318156836692232048852 * q^53 - 703399094180619772764269188861842960 * q^55 - 785787658029123941344672503009410400 * q^57 + 3282010250697938085995473868454788040 * q^59 - 4485157745840460017677520841591879236 * q^61 - 24992547810032633572829803615671997008 * q^63 + 11919812053150705903267451825832510120 * q^65 - 55055736076082381526441630307313037256 * q^67 + 56632222305135551516324020091548201152 * q^69 + 48857884738531480102553331099997138416 * q^71 - 485616958845162818165813572349849688812 * q^73 - 68411444542067712502035991381122300600 * q^75 - 419718086944271580508987308621981603648 * q^77 + 905766854341990828970937020702446890080 * q^79 + 2631580786121491108655810678136598411122 * q^81 - 1271755206567258248006583231290595969768 * q^83 - 388512546938141834318136915482004571560 * q^85 + 17757719150026706057484126704955838888080 * q^87 + 2692414558304473820098690391828757436980 * q^89 - 16669534332125295311104493248216990683104 * q^91 + 40418607330508619745600934617627243192576 * q^93 - 93840167647611953736551449995129877215600 * q^95 - 31989468315501178009691057062669320691004 * q^97 + 123209315561064920195916977740234432814088 * 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
$3$	−1.05814e10	−1.75210	−0.876051	−	0.482218i	$-0.839831\pi$
$3$	−1.05814e10	−1.75210	−0.876051	+	0.482218i	$0.839831\pi$
$4$	0	0
$5$	−2.54286e14	−1.19244	−0.596221	−	0.802820i	$-0.703332\pi$
$5$	−2.54286e14	−1.19244	−0.596221	+	0.802820i	$0.703332\pi$
$6$	0	0
$7$	−2.96016e17	−1.40218	−0.701092	−	0.713071i	$-0.747304\pi$
$7$	−2.96016e17	−1.40218	−0.701092	+	0.713071i	$0.747304\pi$
$8$	0	0
$9$	7.54941e19	2.06986
$10$	0	0
$11$	1.09620e21	0.491291	0.245645	−	0.969360i	$-0.421000\pi$
$11$	1.09620e21	0.491291	0.245645	+	0.969360i	$0.421000\pi$
$12$	0	0
$13$	8.09333e22	1.18111	0.590553	−	0.806999i	$-0.298910\pi$
$13$	8.09333e22	1.18111	0.590553	+	0.806999i	$0.298910\pi$
$14$	0	0
$15$	2.69071e24	2.08928
$16$	0	0
$17$	−5.33128e24	−0.318148	−0.159074	−	0.987267i	$-0.550851\pi$
$17$	−5.33128e24	−0.318148	−0.159074	+	0.987267i	$0.550851\pi$
$18$	0	0
$19$	3.50762e25	0.214074	0.107037	−	0.994255i	$-0.465864\pi$
$19$	3.50762e25	0.214074	0.107037	+	0.994255i	$0.465864\pi$
$20$	0	0
$21$	3.13228e27	2.45677
$22$	0	0
$23$	−6.68932e27	−0.812758	−0.406379	−	0.913705i	$-0.633209\pi$
$23$	−6.68932e27	−0.812758	−0.406379	+	0.913705i	$0.633209\pi$
$24$	0	0
$25$	1.91866e28	0.421917
$26$	0	0
$27$	−4.12899e29	−1.87451
$28$	0	0
$29$	−1.63586e30	−1.71628	−0.858139	−	0.513418i	$-0.828379\pi$
$29$	−1.63586e30	−1.71628	−0.858139	+	0.513418i	$0.828379\pi$
$30$	0	0
$31$	−2.77961e30	−0.743137	−0.371568	−	0.928406i	$-0.621180\pi$
$31$	−2.77961e30	−0.743137	−0.371568	+	0.928406i	$0.621180\pi$
$32$	0	0
$33$	−1.15993e31	−0.860791
$34$	0	0
$35$	7.52726e31	1.67202
$36$	0	0
$37$	4.65417e31	0.330908	0.165454	−	0.986218i	$-0.447091\pi$
$37$	4.65417e31	0.330908	0.165454	+	0.986218i	$0.447091\pi$
$38$	0	0
$39$	−8.56392e32	−2.06942
$40$	0	0
$41$	−8.83679e32	−0.765995	−0.382997	−	0.923749i	$-0.625108\pi$
$41$	−8.83679e32	−0.765995	−0.382997	+	0.923749i	$0.625108\pi$
$42$	0	0
$43$	3.69256e32	0.120566	0.0602829	−	0.998181i	$-0.480800\pi$
$43$	3.69256e32	0.120566	0.0602829	+	0.998181i	$0.480800\pi$
$44$	0	0
$45$	−1.91971e34	−2.46819
$46$	0	0
$47$	1.89619e34	0.999714	0.499857	−	0.866108i	$-0.333386\pi$
$47$	1.89619e34	0.999714	0.499857	+	0.866108i	$0.333386\pi$
$48$	0	0
$49$	4.30577e34	0.966120
$50$	0	0
$51$	5.64127e34	0.557427
$52$	0	0
$53$	−2.01460e35	−0.904761	−0.452380	−	0.891825i	$-0.649425\pi$
$53$	−2.01460e35	−0.904761	−0.452380	+	0.891825i	$0.649425\pi$
$54$	0	0
$55$	−2.78747e35	−0.585835
$56$	0	0
$57$	−3.71157e35	−0.375079
$58$	0	0
$59$	−5.07603e34	−0.0252964	−0.0126482	−	0.999920i	$-0.504026\pi$
$59$	−5.07603e34	−0.0252964	−0.0126482	+	0.999920i	$0.504026\pi$
$60$	0	0
$61$	−6.57027e36	−1.65319	−0.826593	−	0.562801i	$-0.809724\pi$
$61$	−6.57027e36	−1.65319	−0.826593	+	0.562801i	$0.809724\pi$
$62$	0	0
$63$	−2.23474e37	−2.90233
$64$	0	0
$65$	−2.05802e37	−1.40840
$66$	0	0
$67$	−5.28414e37	−1.94286	−0.971429	−	0.237332i	$-0.923727\pi$
$67$	−5.28414e37	−1.94286	−0.971429	+	0.237332i	$0.923727\pi$
$68$	0	0
$69$	7.07827e37	1.42403
$70$	0	0
$71$	1.04768e38	1.17337	0.586684	−	0.809816i	$-0.300433\pi$
$71$	1.04768e38	1.17337	0.586684	+	0.809816i	$0.300433\pi$
$72$	0	0
$73$	−2.83453e38	−1.79623	−0.898117	−	0.439756i	$-0.855065\pi$
$73$	−2.83453e38	−1.79623	−0.898117	+	0.439756i	$0.855065\pi$
$74$	0	0
$75$	−2.03022e38	−0.739242
$76$	0	0
$77$	−3.24491e38	−0.688880
$78$	0	0
$79$	6.59452e38	0.827614	0.413807	−	0.910365i	$-0.364199\pi$
$79$	6.59452e38	0.827614	0.413807	+	0.910365i	$0.364199\pi$
$80$	0	0
$81$	1.61558e39	1.21447
$82$	0	0
$83$	−1.30773e39	−0.596236	−0.298118	−	0.954529i	$-0.596359\pi$
$83$	−1.30773e39	−0.596236	−0.298118	+	0.954529i	$0.596359\pi$
$84$	0	0
$85$	1.35567e39	0.379373
$86$	0	0
$87$	1.73098e40	3.00709
$88$	0	0
$89$	2.16819e39	0.236376	0.118188	−	0.992991i	$-0.462292\pi$
$89$	2.16819e39	0.236376	0.118188	+	0.992991i	$0.462292\pi$
$90$	0	0
$91$	−2.39575e40	−1.65613
$92$	0	0
$93$	2.94123e40	1.30205
$94$	0	0
$95$	−8.91937e39	−0.255270
$96$	0	0
$97$	−8.40052e39	−0.156850	−0.0784252	−	0.996920i	$-0.524989\pi$
$97$	−8.40052e39	−0.156850	−0.0784252	+	0.996920i	$0.524989\pi$
$98$	0	0
$99$	8.27563e40	1.01690
$100$	0	0
$101$	4.81454e40	0.392615	0.196308	−	0.980542i	$-0.437105\pi$
$101$	4.81454e40	0.392615	0.196308	+	0.980542i	$0.437105\pi$
$102$	0	0
$103$	−8.87160e40	−0.483992	−0.241996	−	0.970277i	$-0.577802\pi$
$103$	−8.87160e40	−0.483992	−0.241996	+	0.970277i	$0.577802\pi$
$104$	0	0
$105$	−7.96494e41	−2.92955
$106$	0	0
$107$	−5.55682e41	−1.38822	−0.694111	−	0.719868i	$-0.744203\pi$
$107$	−5.55682e41	−1.38822	−0.694111	+	0.719868i	$0.744203\pi$
$108$	0	0
$109$	−4.77382e41	−0.815873	−0.407936	−	0.913010i	$-0.633752\pi$
$109$	−4.77382e41	−0.815873	−0.407936	+	0.913010i	$0.633752\pi$
$110$	0	0
$111$	−4.92479e41	−0.579785
$112$	0	0
$113$	9.67742e41	0.790045	0.395022	−	0.918672i	$-0.370737\pi$
$113$	9.67742e41	0.790045	0.395022	+	0.918672i	$0.370737\pi$
$114$	0	0
$115$	1.70100e42	0.969167
$116$	0	0
$117$	6.10999e42	2.44473
$118$	0	0
$119$	1.57814e42	0.446102
$120$	0	0
$121$	−3.77687e42	−0.758634
$122$	0	0
$123$	9.35061e42	1.34210
$124$	0	0
$125$	6.68471e42	0.689330
$126$	0	0
$127$	−1.15659e43	−0.861396	−0.430698	−	0.902496i	$-0.641733\pi$
$127$	−1.15659e43	−0.861396	−0.430698	+	0.902496i	$0.641733\pi$
$128$	0	0
$129$	−3.90726e42	−0.211244
$130$	0	0
$131$	−2.42843e43	−0.957776	−0.478888	−	0.877876i	$-0.658960\pi$
$131$	−2.42843e43	−0.957776	−0.478888	+	0.877876i	$0.658960\pi$
$132$	0	0
$133$	−1.03831e43	−0.300171
$134$	0	0
$135$	1.04994e44	2.23524
$136$	0	0
$137$	−1.54500e43	−0.243308	−0.121654	−	0.992573i	$-0.538820\pi$
$137$	−1.54500e43	−0.243308	−0.121654	+	0.992573i	$0.538820\pi$
$138$	0	0
$139$	−1.40766e44	−1.64700	−0.823501	−	0.567315i	$-0.807982\pi$
$139$	−1.40766e44	−1.64700	−0.823501	+	0.567315i	$0.807982\pi$
$140$	0	0
$141$	−2.00644e44	−1.75160
$142$	0	0
$143$	8.87189e43	0.580266
$144$	0	0
$145$	4.15976e44	2.04656
$146$	0	0
$147$	−4.55613e44	−1.69274
$148$	0	0
$149$	1.57473e44	0.443493	0.221746	−	0.975104i	$-0.428824\pi$
$149$	1.57473e44	0.443493	0.221746	+	0.975104i	$0.428824\pi$
$150$	0	0
$151$	2.82811e44	0.605994	0.302997	−	0.952992i	$-0.402013\pi$
$151$	2.82811e44	0.605994	0.302997	+	0.952992i	$0.402013\pi$
$152$	0	0
$153$	−4.02480e44	−0.658522
$154$	0	0
$155$	7.06815e44	0.886147
$156$	0	0
$157$	2.60906e44	0.251502	0.125751	−	0.992062i	$-0.459866\pi$
$157$	2.60906e44	0.251502	0.125751	+	0.992062i	$0.459866\pi$
$158$	0	0
$159$	2.13174e45	1.58523
$160$	0	0
$161$	1.98014e45	1.13964
$162$	0	0
$163$	1.61270e45	0.720621	0.360310	−	0.932832i	$-0.382671\pi$
$163$	1.61270e45	0.720621	0.360310	+	0.932832i	$0.382671\pi$
$164$	0	0
$165$	2.94955e45	1.02644
$166$	0	0
$167$	1.22668e43	0.00333459	0.00166729	−	0.999999i	$-0.499469\pi$
$167$	1.22668e43	0.00333459	0.00166729	+	0.999999i	$0.499469\pi$
$168$	0	0
$169$	1.85475e45	0.395011
$170$	0	0
$171$	2.64804e45	0.443103
$172$	0	0
$173$	−6.97298e45	−0.919334	−0.459667	−	0.888091i	$-0.652031\pi$
$173$	−6.97298e45	−0.919334	−0.459667	+	0.888091i	$0.652031\pi$
$174$	0	0
$175$	−5.67953e45	−0.591606
$176$	0	0
$177$	5.37118e44	0.0443218
$178$	0	0
$179$	−2.57092e46	−1.68501	−0.842504	−	0.538690i	$-0.818920\pi$
$179$	−2.57092e46	−1.68501	−0.842504	+	0.538690i	$0.818920\pi$
$180$	0	0
$181$	1.71977e46	0.897553	0.448776	−	0.893644i	$-0.351860\pi$
$181$	1.71977e46	0.897553	0.448776	+	0.893644i	$0.351860\pi$
$182$	0	0
$183$	6.95230e46	2.89655
$184$	0	0
$185$	−1.18349e46	−0.394588
$186$	0	0
$187$	−5.84413e45	−0.156303
$188$	0	0
$189$	1.22225e47	2.62840
$190$	0	0
$191$	2.15620e46	0.373684	0.186842	−	0.982390i	$-0.440175\pi$
$191$	2.15620e46	0.373684	0.186842	+	0.982390i	$0.440175\pi$
$192$	0	0
$193$	2.03359e46	0.284668	0.142334	−	0.989819i	$-0.454539\pi$
$193$	2.03359e46	0.284668	0.142334	+	0.989819i	$0.454539\pi$
$194$	0	0
$195$	2.17768e47	2.46766
$196$	0	0
$197$	−1.48334e47	−1.36359	−0.681793	−	0.731545i	$-0.738800\pi$
$197$	−1.48334e47	−1.36359	−0.681793	+	0.731545i	$0.738800\pi$
$198$	0	0
$199$	7.47977e46	0.558985	0.279493	−	0.960148i	$-0.409834\pi$
$199$	7.47977e46	0.558985	0.279493	+	0.960148i	$0.409834\pi$
$200$	0	0
$201$	5.59139e47	3.40408
$202$	0	0
$203$	4.84241e47	2.40654
$204$	0	0
$205$	2.24707e47	0.913404
$206$	0	0
$207$	−5.05004e47	−1.68230
$208$	0	0
$209$	3.84504e46	0.105172
$210$	0	0
$211$	−8.14889e47	−1.83362	−0.916808	−	0.399329i	$-0.869243\pi$
$211$	−8.14889e47	−1.83362	−0.916808	+	0.399329i	$0.869243\pi$
$212$	0	0
$213$	−1.10860e48	−2.05586
$214$	0	0
$215$	−9.38966e46	−0.143768
$216$	0	0
$217$	8.22808e47	1.04201
$218$	0	0
$219$	2.99935e48	3.14719
$220$	0	0
$221$	−4.31479e47	−0.375766
$222$	0	0
$223$	2.94601e45	0.00213297	0.00106648	−	0.999999i	$-0.499661\pi$
$223$	2.94601e45	0.00213297	0.00106648	+	0.999999i	$0.499661\pi$
$224$	0	0
$225$	1.44847e48	0.873311
$226$	0	0
$227$	−1.23961e48	−0.623383	−0.311691	−	0.950183i	$-0.600895\pi$
$227$	−1.23961e48	−0.623383	−0.311691	+	0.950183i	$0.600895\pi$
$228$	0	0
$229$	2.51277e48	1.05567	0.527833	−	0.849348i	$-0.323005\pi$
$229$	2.51277e48	1.05567	0.527833	+	0.849348i	$0.323005\pi$
$230$	0	0
$231$	3.43359e48	1.20699
$232$	0	0
$233$	−3.44195e47	−0.101393	−0.0506966	−	0.998714i	$-0.516144\pi$
$233$	−3.44195e47	−0.101393	−0.0506966	+	0.998714i	$0.516144\pi$
$234$	0	0
$235$	−4.82174e48	−1.19210
$236$	0	0
$237$	−6.97796e48	−1.45006
$238$	0	0
$239$	1.24157e48	0.217177	0.108588	−	0.994087i	$-0.465367\pi$
$239$	1.24157e48	0.217177	0.108588	+	0.994087i	$0.465367\pi$
$240$	0	0
$241$	8.53681e48	1.25877	0.629385	−	0.777094i	$-0.283307\pi$
$241$	8.53681e48	1.25877	0.629385	+	0.777094i	$0.283307\pi$
$242$	0	0
$243$	−2.03549e48	−0.253361
$244$	0	0
$245$	−1.09490e49	−1.15204
$246$	0	0
$247$	2.83883e48	0.252844
$248$	0	0
$249$	1.38377e49	1.04467
$250$	0	0
$251$	−1.82523e49	−1.16952	−0.584759	−	0.811207i	$-0.698811\pi$
$251$	−1.82523e49	−1.16952	−0.584759	+	0.811207i	$0.698811\pi$
$252$	0	0
$253$	−7.33281e48	−0.399300
$254$	0	0
$255$	−1.43450e49	−0.664700
$256$	0	0
$257$	−3.84389e49	−1.51755	−0.758774	−	0.651354i	$-0.774201\pi$
$257$	−3.84389e49	−1.51755	−0.758774	+	0.651354i	$0.774201\pi$
$258$	0	0
$259$	−1.37771e49	−0.463994
$260$	0	0
$261$	−1.23498e50	−3.55246
$262$	0	0
$263$	−5.88587e49	−1.44783	−0.723915	−	0.689889i	$-0.757659\pi$
$263$	−5.88587e49	−1.44783	−0.723915	+	0.689889i	$0.757659\pi$
$264$	0	0
$265$	5.12285e49	1.07887
$266$	0	0
$267$	−2.29426e49	−0.414154
$268$	0	0
$269$	−6.22517e49	−0.964338	−0.482169	−	0.876078i	$-0.660151\pi$
$269$	−6.22517e49	−0.964338	−0.482169	+	0.876078i	$0.660151\pi$
$270$	0	0
$271$	8.85718e49	1.17876	0.589380	−	0.807856i	$-0.299372\pi$
$271$	8.85718e49	1.17876	0.589380	+	0.807856i	$0.299372\pi$
$272$	0	0
$273$	2.53506e50	2.90170
$274$	0	0
$275$	2.10323e49	0.207284
$276$	0	0
$277$	−1.94293e50	−1.65053	−0.825264	−	0.564747i	$-0.808974\pi$
$277$	−1.94293e50	−1.65053	−0.825264	+	0.564747i	$0.808974\pi$
$278$	0	0
$279$	−2.09844e50	−1.53819
$280$	0	0
$281$	2.14728e50	1.35960	0.679798	−	0.733400i	$-0.262068\pi$
$281$	2.14728e50	1.35960	0.679798	+	0.733400i	$0.262068\pi$
$282$	0	0
$283$	6.68644e49	0.366078	0.183039	−	0.983106i	$-0.441407\pi$
$283$	6.68644e49	0.366078	0.183039	+	0.983106i	$0.441407\pi$
$284$	0	0
$285$	9.43799e49	0.447260
$286$	0	0
$287$	2.61583e50	1.07407
$288$	0	0
$289$	−2.52383e50	−0.898782
$290$	0	0
$291$	8.88897e49	0.274818
$292$	0	0
$293$	−2.93540e50	−0.788644	−0.394322	−	0.918972i	$-0.629021\pi$
$293$	−2.93540e50	−0.788644	−0.394322	+	0.918972i	$0.629021\pi$
$294$	0	0
$295$	1.29076e49	0.0301645
$296$	0	0
$297$	−4.52619e50	−0.920927
$298$	0	0
$299$	−5.41389e50	−0.959953
$300$	0	0
$301$	−1.09306e50	−0.169055
$302$	0	0
$303$	−5.09449e50	−0.687902
$304$	0	0
$305$	1.67073e51	1.97133
$306$	0	0
$307$	4.35932e50	0.449864	0.224932	−	0.974374i	$-0.427784\pi$
$307$	4.35932e50	0.449864	0.224932	+	0.974374i	$0.427784\pi$
$308$	0	0
$309$	9.38743e50	0.848004
$310$	0	0
$311$	2.38798e50	0.188992	0.0944958	−	0.995525i	$-0.469876\pi$
$311$	2.38798e50	0.188992	0.0944958	+	0.995525i	$0.469876\pi$
$312$	0	0
$313$	−2.03212e51	−1.41023	−0.705117	−	0.709091i	$-0.749106\pi$
$313$	−2.03212e51	−1.41023	−0.705117	+	0.709091i	$0.749106\pi$
$314$	0	0
$315$	5.68264e51	3.46086
$316$	0	0
$317$	2.68276e51	1.43505	0.717525	−	0.696532i	$-0.245275\pi$
$317$	2.68276e51	1.43505	0.717525	+	0.696532i	$0.245275\pi$
$318$	0	0
$319$	−1.79323e51	−0.843191
$320$	0	0
$321$	5.87993e51	2.43231
$322$	0	0
$323$	−1.87001e50	−0.0681071
$324$	0	0
$325$	1.55283e51	0.498329
$326$	0	0
$327$	5.05139e51	1.42949
$328$	0	0
$329$	−5.61302e51	−1.40178
$330$	0	0
$331$	−7.54395e51	−1.66389	−0.831945	−	0.554858i	$-0.812773\pi$
$331$	−7.54395e51	−1.66389	−0.831945	+	0.554858i	$0.812773\pi$
$332$	0	0
$333$	3.51362e51	0.684934
$334$	0	0
$335$	1.34368e52	2.31674
$336$	0	0
$337$	−5.36454e51	−0.818689	−0.409345	−	0.912380i	$-0.634243\pi$
$337$	−5.36454e51	−0.818689	−0.409345	+	0.912380i	$0.634243\pi$
$338$	0	0
$339$	−1.02401e52	−1.38424
$340$	0	0
$341$	−3.04700e51	−0.365096
$342$	0	0
$343$	4.46968e50	0.0475059
$344$	0	0
$345$	−1.79990e52	−1.69808
$346$	0	0
$347$	1.60783e51	0.134737	0.0673684	−	0.997728i	$-0.478540\pi$
$347$	1.60783e51	0.134737	0.0673684	+	0.997728i	$0.478540\pi$
$348$	0	0
$349$	1.51169e52	1.12601	0.563004	−	0.826454i	$-0.309646\pi$
$349$	1.51169e52	1.12601	0.563004	+	0.826454i	$0.309646\pi$
$350$	0	0
$351$	−3.34173e52	−2.21399
$352$	0	0
$353$	8.96540e51	0.528675	0.264337	−	0.964430i	$-0.414847\pi$
$353$	8.96540e51	0.528675	0.264337	+	0.964430i	$0.414847\pi$
$354$	0	0
$355$	−2.66411e52	−1.39917
$356$	0	0
$357$	−1.66990e52	−0.781616
$358$	0	0
$359$	2.57092e52	1.07313	0.536565	−	0.843859i	$-0.319722\pi$
$359$	2.57092e52	1.07313	0.536565	+	0.843859i	$0.319722\pi$
$360$	0	0
$361$	−2.56168e52	−0.954172
$362$	0	0
$363$	3.99648e52	1.32920
$364$	0	0
$365$	7.20782e52	2.14191
$366$	0	0
$367$	−4.37801e52	−1.16311	−0.581557	−	0.813506i	$-0.697556\pi$
$367$	−4.37801e52	−1.16311	−0.581557	+	0.813506i	$0.697556\pi$
$368$	0	0
$369$	−6.67125e52	−1.58550
$370$	0	0
$371$	5.96354e52	1.26864
$372$	0	0
$373$	−3.46418e52	−0.660039	−0.330020	−	0.943974i	$-0.607055\pi$
$373$	−3.46418e52	−0.660039	−0.330020	+	0.943974i	$0.607055\pi$
$374$	0	0
$375$	−7.07339e52	−1.20778
$376$	0	0
$377$	−1.32396e53	−2.02710
$378$	0	0
$379$	−3.98009e52	−0.546750	−0.273375	−	0.961908i	$-0.588140\pi$
$379$	−3.98009e52	−0.546750	−0.273375	+	0.961908i	$0.588140\pi$
$380$	0	0
$381$	1.22384e53	1.50925
$382$	0	0
$383$	−4.24383e52	−0.470092	−0.235046	−	0.971984i	$-0.575524\pi$
$383$	−4.24383e52	−0.470092	−0.235046	+	0.971984i	$0.575524\pi$
$384$	0	0
$385$	8.25136e52	0.821449
$386$	0	0
$387$	2.78766e52	0.249555
$388$	0	0
$389$	−1.66821e53	−1.34364	−0.671821	−	0.740713i	$-0.734488\pi$
$389$	−1.66821e53	−1.34364	−0.671821	+	0.740713i	$0.734488\pi$
$390$	0	0
$391$	3.56627e52	0.258577
$392$	0	0
$393$	2.56963e53	1.67812
$394$	0	0
$395$	−1.67689e53	−0.986882
$396$	0	0
$397$	−2.90239e53	−1.54010	−0.770052	−	0.637981i	$-0.779770\pi$
$397$	−2.90239e53	−1.54010	−0.770052	+	0.637981i	$0.779770\pi$
$398$	0	0
$399$	1.09868e53	0.525930
$400$	0	0
$401$	1.04436e53	0.451222	0.225611	−	0.974217i	$-0.427562\pi$
$401$	1.04436e53	0.451222	0.225611	+	0.974217i	$0.427562\pi$
$402$	0	0
$403$	−2.24963e53	−0.877723
$404$	0	0
$405$	−4.10819e53	−1.44818
$406$	0	0
$407$	5.10189e52	0.162572
$408$	0	0
$409$	2.12912e53	0.613581	0.306790	−	0.951777i	$-0.400745\pi$
$409$	2.12912e53	0.613581	0.306790	+	0.951777i	$0.400745\pi$
$410$	0	0
$411$	1.63483e53	0.426301
$412$	0	0
$413$	1.50259e52	0.0354702
$414$	0	0
$415$	3.32538e53	0.710976
$416$	0	0
$417$	1.48951e54	2.88571
$418$	0	0
$419$	8.36673e53	1.46950	0.734748	−	0.678340i	$-0.237300\pi$
$419$	8.36673e53	1.46950	0.734748	+	0.678340i	$0.237300\pi$
$420$	0	0
$421$	1.14983e54	1.83169	0.915845	−	0.401533i	$-0.131522\pi$
$421$	1.14983e54	1.83169	0.915845	+	0.401533i	$0.131522\pi$
$422$	0	0
$423$	1.43151e54	2.06927
$424$	0	0
$425$	−1.02289e53	−0.134232
$426$	0	0
$427$	1.94490e54	2.31807
$428$	0	0
$429$	−9.38774e53	−1.01669
$430$	0	0
$431$	−1.32392e54	−1.30340	−0.651701	−	0.758476i	$-0.725944\pi$
$431$	−1.32392e54	−1.30340	−0.651701	+	0.758476i	$0.725944\pi$
$432$	0	0
$433$	−1.08006e54	−0.967046	−0.483523	−	0.875332i	$-0.660643\pi$
$433$	−1.08006e54	−0.967046	−0.483523	+	0.875332i	$0.660643\pi$
$434$	0	0
$435$	−4.40163e54	−3.58578
$436$	0	0
$437$	−2.34636e53	−0.173990
$438$	0	0
$439$	−1.14481e53	−0.0773059	−0.0386529	−	0.999253i	$-0.512307\pi$
$439$	−1.14481e53	−0.0773059	−0.0386529	+	0.999253i	$0.512307\pi$
$440$	0	0
$441$	3.25060e54	1.99973
$442$	0	0
$443$	9.71784e53	0.544870	0.272435	−	0.962174i	$-0.412171\pi$
$443$	9.71784e53	0.544870	0.272435	+	0.962174i	$0.412171\pi$
$444$	0	0
$445$	−5.51341e53	−0.281864
$446$	0	0
$447$	−1.66629e54	−0.777045
$448$	0	0
$449$	−3.40657e54	−1.44966	−0.724832	−	0.688926i	$-0.758083\pi$
$449$	−3.40657e54	−1.44966	−0.724832	+	0.688926i	$0.758083\pi$
$450$	0	0
$451$	−9.68686e53	−0.376326
$452$	0	0
$453$	−2.99255e54	−1.06176
$454$	0	0
$455$	6.09207e54	1.97484
$456$	0	0
$457$	1.58378e54	0.469258	0.234629	−	0.972085i	$-0.424612\pi$
$457$	1.58378e54	0.469258	0.234629	+	0.972085i	$0.424612\pi$
$458$	0	0
$459$	2.20128e54	0.596370
$460$	0	0
$461$	−2.93741e53	−0.0727939	−0.0363970	−	0.999337i	$-0.511588\pi$
$461$	−2.93741e53	−0.0727939	−0.0363970	+	0.999337i	$0.511588\pi$
$462$	0	0
$463$	−7.57194e54	−1.71711	−0.858553	−	0.512725i	$-0.828636\pi$
$463$	−7.57194e54	−1.71711	−0.858553	+	0.512725i	$0.828636\pi$
$464$	0	0
$465$	−7.47913e54	−1.55262
$466$	0	0
$467$	2.70909e54	0.515024	0.257512	−	0.966275i	$-0.417097\pi$
$467$	2.70909e54	0.515024	0.257512	+	0.966275i	$0.417097\pi$
$468$	0	0
$469$	1.56419e55	2.72424
$470$	0	0
$471$	−2.76077e54	−0.440657
$472$	0	0
$473$	4.04777e53	0.0592328
$474$	0	0
$475$	6.72992e53	0.0903214
$476$	0	0
$477$	−1.52090e55	−1.87273
$478$	0	0
$479$	7.24145e53	0.0818366	0.0409183	−	0.999162i	$-0.486972\pi$
$479$	7.24145e53	0.0818366	0.0409183	+	0.999162i	$0.486972\pi$
$480$	0	0
$481$	3.76678e54	0.390837
$482$	0	0
$483$	−2.09528e55	−1.99676
$484$	0	0
$485$	2.13613e54	0.187035
$486$	0	0
$487$	−3.33837e54	−0.268653	−0.134326	−	0.990937i	$-0.542887\pi$
$487$	−3.33837e54	−0.268653	−0.134326	+	0.990937i	$0.542887\pi$
$488$	0	0
$489$	−1.70647e55	−1.26260
$490$	0	0
$491$	1.88223e55	1.28086	0.640430	−	0.768016i	$-0.278756\pi$
$491$	1.88223e55	1.28086	0.640430	+	0.768016i	$0.278756\pi$
$492$	0	0
$493$	8.72124e54	0.546030
$494$	0	0
$495$	−2.10438e55	−1.21260
$496$	0	0
$497$	−3.10131e55	−1.64528
$498$	0	0
$499$	6.93745e54	0.338953	0.169476	−	0.985534i	$-0.445792\pi$
$499$	6.93745e54	0.338953	0.169476	+	0.985534i	$0.445792\pi$
$500$	0	0
$501$	−1.29800e53	−0.00584254
$502$	0	0
$503$	−2.98925e55	−1.23999	−0.619995	−	0.784606i	$-0.712865\pi$
$503$	−2.98925e55	−1.23999	−0.619995	+	0.784606i	$0.712865\pi$
$504$	0	0
$505$	−1.22427e55	−0.468171
$506$	0	0
$507$	−1.96260e55	−0.692099
$508$	0	0
$509$	5.65842e55	1.84069	0.920345	−	0.391108i	$-0.127908\pi$
$509$	5.65842e55	1.84069	0.920345	+	0.391108i	$0.127908\pi$
$510$	0	0
$511$	8.39067e55	2.51865
$512$	0	0
$513$	−1.44829e55	−0.401282
$514$	0	0
$515$	2.25592e55	0.577133
$516$	0	0
$517$	2.07860e55	0.491150
$518$	0	0
$519$	7.37843e55	1.61077
$520$	0	0
$521$	3.31572e55	0.668967	0.334483	−	0.942402i	$-0.391438\pi$
$521$	3.31572e55	0.668967	0.334483	+	0.942402i	$0.391438\pi$
$522$	0	0
$523$	−7.77683e55	−1.45050	−0.725250	−	0.688486i	$-0.758276\pi$
$523$	−7.77683e55	−1.45050	−0.725250	+	0.688486i	$0.758276\pi$
$524$	0	0
$525$	6.00977e55	1.03655
$526$	0	0
$527$	1.48189e55	0.236427
$528$	0	0
$529$	−2.29924e55	−0.339424
$530$	0	0
$531$	−3.83210e54	−0.0523600
$532$	0	0
$533$	−7.15191e55	−0.904721
$534$	0	0
$535$	1.41302e56	1.65538
$536$	0	0
$537$	2.72041e56	2.95231
$538$	0	0
$539$	4.71997e55	0.474646
$540$	0	0
$541$	8.91733e55	0.831172	0.415586	−	0.909554i	$-0.363577\pi$
$541$	8.91733e55	0.831172	0.415586	+	0.909554i	$0.363577\pi$
$542$	0	0
$543$	−1.81977e56	−1.57260
$544$	0	0
$545$	1.21391e56	0.972881
$546$	0	0
$547$	8.97461e55	0.667230	0.333615	−	0.942709i	$-0.391732\pi$
$547$	8.97461e55	0.667230	0.333615	+	0.942709i	$0.391732\pi$
$548$	0	0
$549$	−4.96016e56	−3.42186
$550$	0	0
$551$	−5.73797e55	−0.367410
$552$	0	0
$553$	−1.95208e56	−1.16047
$554$	0	0
$555$	1.25230e56	0.691359
$556$	0	0
$557$	3.47843e55	0.178382	0.0891911	−	0.996015i	$-0.471572\pi$
$557$	3.47843e55	0.178382	0.0891911	+	0.996015i	$0.471572\pi$
$558$	0	0
$559$	2.98851e55	0.142401
$560$	0	0
$561$	6.18394e55	0.273859
$562$	0	0
$563$	−2.29948e55	−0.0946688	−0.0473344	−	0.998879i	$-0.515073\pi$
$563$	−2.29948e55	−0.0946688	−0.0473344	+	0.998879i	$0.515073\pi$
$564$	0	0
$565$	−2.46083e56	−0.942082
$566$	0	0
$567$	−4.78237e56	−1.70290
$568$	0	0
$569$	3.18790e56	1.05610	0.528048	−	0.849214i	$-0.322924\pi$
$569$	3.18790e56	1.05610	0.528048	+	0.849214i	$0.322924\pi$
$570$	0	0
$571$	−1.16154e54	−0.00358091	−0.00179046	−	0.999998i	$-0.500570\pi$
$571$	−1.16154e54	−0.00358091	−0.00179046	+	0.999998i	$0.500570\pi$
$572$	0	0
$573$	−2.28157e56	−0.654733
$574$	0	0
$575$	−1.28345e56	−0.342917
$576$	0	0
$577$	6.45847e56	1.60703	0.803517	−	0.595281i	$-0.202959\pi$
$577$	6.45847e56	1.60703	0.803517	+	0.595281i	$0.202959\pi$
$578$	0	0
$579$	−2.15183e56	−0.498767
$580$	0	0
$581$	3.87109e56	0.836032
$582$	0	0
$583$	−2.20840e56	−0.444500
$584$	0	0
$585$	−1.55368e57	−2.91519
$586$	0	0
$587$	1.53676e56	0.268859	0.134430	−	0.990923i	$-0.457080\pi$
$587$	1.53676e56	0.268859	0.134430	+	0.990923i	$0.457080\pi$
$588$	0	0
$589$	−9.74980e55	−0.159086
$590$	0	0
$591$	1.56959e57	2.38914
$592$	0	0
$593$	7.79761e56	1.10749	0.553746	−	0.832686i	$-0.313198\pi$
$593$	7.79761e56	1.10749	0.553746	+	0.832686i	$0.313198\pi$
$594$	0	0
$595$	−4.01300e56	−0.531950
$596$	0	0
$597$	−7.91468e56	−0.979399
$598$	0	0
$599$	9.18440e56	1.06121	0.530605	−	0.847619i	$-0.321965\pi$
$599$	9.18440e56	1.06121	0.530605	+	0.847619i	$0.321965\pi$
$600$	0	0
$601$	1.34315e57	1.44943	0.724714	−	0.689050i	$-0.241972\pi$
$601$	1.34315e57	1.44943	0.724714	+	0.689050i	$0.241972\pi$
$602$	0	0
$603$	−3.98921e57	−4.02145
$604$	0	0
$605$	9.60405e56	0.904626
$606$	0	0
$607$	1.37250e56	0.120821	0.0604105	−	0.998174i	$-0.480759\pi$
$607$	1.37250e56	0.120821	0.0604105	+	0.998174i	$0.480759\pi$
$608$	0	0
$609$	−5.12397e57	−4.21650
$610$	0	0
$611$	1.53465e57	1.18077
$612$	0	0
$613$	−4.58982e56	−0.330260	−0.165130	−	0.986272i	$-0.552804\pi$
$613$	−4.58982e56	−0.330260	−0.165130	+	0.986272i	$0.552804\pi$
$614$	0	0
$615$	−2.37773e57	−1.60038
$616$	0	0
$617$	−4.69500e56	−0.295658	−0.147829	−	0.989013i	$-0.547229\pi$
$617$	−4.69500e56	−0.295658	−0.147829	+	0.989013i	$0.547229\pi$
$618$	0	0
$619$	8.01009e56	0.472040	0.236020	−	0.971748i	$-0.424157\pi$
$619$	8.01009e56	0.472040	0.236020	+	0.971748i	$0.424157\pi$
$620$	0	0
$621$	2.76202e57	1.52352
$622$	0	0
$623$	−6.41819e56	−0.331442
$624$	0	0
$625$	−2.57233e57	−1.24390
$626$	0	0
$627$	−4.06861e56	−0.184273
$628$	0	0
$629$	−2.48127e56	−0.105278
$630$	0	0
$631$	1.26000e57	0.500920	0.250460	−	0.968127i	$-0.419418\pi$
$631$	1.26000e57	0.500920	0.250460	+	0.968127i	$0.419418\pi$
$632$	0	0
$633$	8.62271e57	3.21268
$634$	0	0
$635$	2.94104e57	1.02716
$636$	0	0
$637$	3.48480e57	1.14109
$638$	0	0
$639$	7.90939e57	2.42871
$640$	0	0
$641$	4.29334e57	1.23653	0.618267	−	0.785968i	$-0.287835\pi$
$641$	4.29334e57	1.23653	0.618267	+	0.785968i	$0.287835\pi$
$642$	0	0
$643$	3.66649e57	0.990662	0.495331	−	0.868704i	$-0.335047\pi$
$643$	3.66649e57	0.990662	0.495331	+	0.868704i	$0.335047\pi$
$644$	0	0
$645$	9.93562e56	0.251896
$646$	0	0
$647$	−7.19666e56	−0.171235	−0.0856177	−	0.996328i	$-0.527286\pi$
$647$	−7.19666e56	−0.171235	−0.0856177	+	0.996328i	$0.527286\pi$
$648$	0	0
$649$	−5.56433e55	−0.0124279
$650$	0	0
$651$	−8.70650e57	−1.82572
$652$	0	0
$653$	1.77671e56	0.0349862	0.0174931	−	0.999847i	$-0.494431\pi$
$653$	1.77671e56	0.0349862	0.0174931	+	0.999847i	$0.494431\pi$
$654$	0	0
$655$	6.17515e57	1.14209
$656$	0	0
$657$	−2.13990e58	−3.71796
$658$	0	0
$659$	−5.94098e57	−0.969855	−0.484928	−	0.874554i	$-0.661154\pi$
$659$	−5.94098e57	−0.969855	−0.484928	+	0.874554i	$0.661154\pi$
$660$	0	0
$661$	8.22733e57	1.26220	0.631101	−	0.775700i	$-0.282603\pi$
$661$	8.22733e57	1.26220	0.631101	+	0.775700i	$0.282603\pi$
$662$	0	0
$663$	4.56567e57	0.658381
$664$	0	0
$665$	2.64027e57	0.357936
$666$	0	0
$667$	1.09428e58	1.39492
$668$	0	0
$669$	−3.11730e55	−0.00373717
$670$	0	0
$671$	−7.20231e57	−0.812194
$672$	0	0
$673$	3.40645e57	0.361404	0.180702	−	0.983538i	$-0.442163\pi$
$673$	3.40645e57	0.361404	0.180702	+	0.983538i	$0.442163\pi$
$674$	0	0
$675$	−7.92213e57	−0.790887
$676$	0	0
$677$	−3.42769e57	−0.322058	−0.161029	−	0.986950i	$-0.551481\pi$
$677$	−3.42769e57	−0.322058	−0.161029	+	0.986950i	$0.551481\pi$
$678$	0	0
$679$	2.48669e57	0.219933
$680$	0	0
$681$	1.31169e58	1.09223
$682$	0	0
$683$	−8.22982e57	−0.645306	−0.322653	−	0.946517i	$-0.604575\pi$
$683$	−8.22982e57	−0.645306	−0.322653	+	0.946517i	$0.604575\pi$
$684$	0	0
$685$	3.92872e57	0.290131
$686$	0	0
$687$	−2.65888e58	−1.84963
$688$	0	0
$689$	−1.63048e58	−1.06862
$690$	0	0
$691$	−1.60618e58	−0.991960	−0.495980	−	0.868334i	$-0.665191\pi$
$691$	−1.60618e58	−0.991960	−0.495980	+	0.868334i	$0.665191\pi$
$692$	0	0
$693$	−2.44972e58	−1.42589
$694$	0	0
$695$	3.57948e58	1.96395
$696$	0	0
$697$	4.71115e57	0.243699
$698$	0	0
$699$	3.64208e57	0.177651
$700$	0	0
$701$	−3.61228e58	−1.66174	−0.830869	−	0.556468i	$-0.812156\pi$
$701$	−3.61228e58	−1.66174	−0.830869	+	0.556468i	$0.812156\pi$
$702$	0	0
$703$	1.63251e57	0.0708387
$704$	0	0
$705$	5.10210e58	2.08868
$706$	0	0
$707$	−1.42518e58	−0.550519
$708$	0	0
$709$	−2.71836e58	−0.990965	−0.495483	−	0.868618i	$-0.665009\pi$
$709$	−2.71836e58	−0.990965	−0.495483	+	0.868618i	$0.665009\pi$
$710$	0	0
$711$	4.97847e58	1.71305
$712$	0	0
$713$	1.85937e58	0.603990
$714$	0	0
$715$	−2.25600e58	−0.691934
$716$	0	0
$717$	−1.31376e58	−0.380516
$718$	0	0
$719$	1.95054e58	0.533595	0.266797	−	0.963753i	$-0.414035\pi$
$719$	1.95054e58	0.533595	0.266797	+	0.963753i	$0.414035\pi$
$720$	0	0
$721$	2.62613e58	0.678647
$722$	0	0
$723$	−9.03318e58	−2.20549
$724$	0	0
$725$	−3.13866e58	−0.724127
$726$	0	0
$727$	−6.41681e58	−1.39915	−0.699575	−	0.714560i	$-0.746627\pi$
$727$	−6.41681e58	−1.39915	−0.699575	+	0.714560i	$0.746627\pi$
$728$	0	0
$729$	−3.73866e58	−0.770552
$730$	0	0
$731$	−1.96861e57	−0.0383577
$732$	0	0
$733$	1.98495e58	0.365695	0.182847	−	0.983141i	$-0.441469\pi$
$733$	1.98495e58	0.365695	0.182847	+	0.983141i	$0.441469\pi$
$734$	0	0
$735$	1.15856e59	2.01850
$736$	0	0
$737$	−5.79246e58	−0.954507
$738$	0	0
$739$	−9.09593e58	−1.41787	−0.708934	−	0.705275i	$-0.750824\pi$
$739$	−9.09593e58	−1.41787	−0.708934	+	0.705275i	$0.750824\pi$
$740$	0	0
$741$	−3.00389e58	−0.443008
$742$	0	0
$743$	1.22229e59	1.70570	0.852849	−	0.522158i	$-0.174873\pi$
$743$	1.22229e59	1.70570	0.852849	+	0.522158i	$0.174873\pi$
$744$	0	0
$745$	−4.00431e58	−0.528839
$746$	0	0
$747$	−9.87260e58	−1.23413
$748$	0	0
$749$	1.64491e59	1.94654
$750$	0	0
$751$	3.17979e57	0.0356270	0.0178135	−	0.999841i	$-0.494329\pi$
$751$	3.17979e57	0.0356270	0.0178135	+	0.999841i	$0.494329\pi$
$752$	0	0
$753$	1.93136e59	2.04912
$754$	0	0
$755$	−7.19147e58	−0.722612
$756$	0	0
$757$	−5.74234e58	−0.546542	−0.273271	−	0.961937i	$-0.588106\pi$
$757$	−5.74234e58	−0.546542	−0.273271	+	0.961937i	$0.588106\pi$
$758$	0	0
$759$	7.75917e58	0.699615
$760$	0	0
$761$	−1.47303e59	−1.25842	−0.629209	−	0.777236i	$-0.716621\pi$
$761$	−1.47303e59	−1.25842	−0.629209	+	0.777236i	$0.716621\pi$
$762$	0	0
$763$	1.41313e59	1.14400
$764$	0	0
$765$	1.02345e59	0.785249
$766$	0	0
$767$	−4.10820e57	−0.0298777
$768$	0	0
$769$	−1.33564e59	−0.920873	−0.460437	−	0.887693i	$-0.652307\pi$
$769$	−1.33564e59	−0.920873	−0.460437	+	0.887693i	$0.652307\pi$
$770$	0	0
$771$	4.06740e59	2.65890
$772$	0	0
$773$	2.25801e59	1.39974	0.699868	−	0.714272i	$-0.253242\pi$
$773$	2.25801e59	1.39974	0.699868	+	0.714272i	$0.253242\pi$
$774$	0	0
$775$	−5.33312e58	−0.313542
$776$	0	0
$777$	1.45782e59	0.812965
$778$	0	0
$779$	−3.09961e58	−0.163979
$780$	0	0
$781$	1.14847e59	0.576464
$782$	0	0
$783$	6.75446e59	3.21717
$784$	0	0
$785$	−6.63448e58	−0.299902
$786$	0	0
$787$	−3.77733e59	−1.62070	−0.810351	−	0.585945i	$-0.800723\pi$
$787$	−3.77733e59	−1.62070	−0.810351	+	0.585945i	$0.800723\pi$
$788$	0	0
$789$	6.22810e59	2.53675
$790$	0	0
$791$	−2.86467e59	−1.10779
$792$	0	0
$793$	−5.31754e59	−1.95259
$794$	0	0
$795$	−5.42071e59	−1.89030
$796$	0	0
$797$	−1.32373e59	−0.438433	−0.219217	−	0.975676i	$-0.570350\pi$
$797$	−1.32373e59	−0.438433	−0.219217	+	0.975676i	$0.570350\pi$
$798$	0	0
$799$	−1.01091e59	−0.318057
$800$	0	0
$801$	1.63686e59	0.489265
$802$	0	0
$803$	−3.10721e59	−0.882473
$804$	0	0
$805$	−5.03523e59	−1.35895
$806$	0	0
$807$	6.58713e59	1.68962
$808$	0	0
$809$	8.51729e58	0.207662	0.103831	−	0.994595i	$-0.466890\pi$
$809$	8.51729e58	0.207662	0.103831	+	0.994595i	$0.466890\pi$
$810$	0	0
$811$	1.20156e59	0.278496	0.139248	−	0.990258i	$-0.455531\pi$
$811$	1.20156e59	0.278496	0.139248	+	0.990258i	$0.455531\pi$
$812$	0	0
$813$	−9.37218e59	−2.06531
$814$	0	0
$815$	−4.10086e59	−0.859298
$816$	0	0
$817$	1.29521e58	0.0258100
$818$	0	0
$819$	−1.80865e60	−3.42796
$820$	0	0
$821$	−1.35794e59	−0.244820	−0.122410	−	0.992480i	$-0.539062\pi$
$821$	−1.35794e59	−0.244820	−0.122410	+	0.992480i	$0.539062\pi$
$822$	0	0
$823$	8.83563e59	1.51545	0.757724	−	0.652575i	$-0.226311\pi$
$823$	8.83563e59	1.51545	0.757724	+	0.652575i	$0.226311\pi$
$824$	0	0
$825$	−2.22552e59	−0.363183
$826$	0	0
$827$	2.24391e59	0.348451	0.174226	−	0.984706i	$-0.444258\pi$
$827$	2.24391e59	0.348451	0.174226	+	0.984706i	$0.444258\pi$
$828$	0	0
$829$	−8.39406e58	−0.124052	−0.0620260	−	0.998075i	$-0.519756\pi$
$829$	−8.39406e58	−0.124052	−0.0620260	+	0.998075i	$0.519756\pi$
$830$	0	0
$831$	2.05591e60	2.89189
$832$	0	0
$833$	−2.29553e59	−0.307369
$834$	0	0
$835$	−3.11926e57	−0.00397630
$836$	0	0
$837$	1.14770e60	1.39301
$838$	0	0
$839$	−7.27294e59	−0.840600	−0.420300	−	0.907385i	$-0.638075\pi$
$839$	−7.27294e59	−0.840600	−0.420300	+	0.907385i	$0.638075\pi$
$840$	0	0
$841$	1.76756e60	1.94561
$842$	0	0
$843$	−2.27214e60	−2.38215
$844$	0	0
$845$	−4.71638e59	−0.471027
$846$	0	0
$847$	1.11801e60	1.06374
$848$	0	0
$849$	−7.07522e59	−0.641405
$850$	0	0
$851$	−3.11332e59	−0.268948
$852$	0	0
$853$	9.33512e58	0.0768537	0.0384269	−	0.999261i	$-0.487765\pi$
$853$	9.33512e58	0.0768537	0.0384269	+	0.999261i	$0.487765\pi$
$854$	0	0
$855$	−6.73360e59	−0.528374
$856$	0	0
$857$	−1.10562e60	−0.826990	−0.413495	−	0.910506i	$-0.635692\pi$
$857$	−1.10562e60	−0.826990	−0.413495	+	0.910506i	$0.635692\pi$
$858$	0	0
$859$	6.75499e59	0.481687	0.240844	−	0.970564i	$-0.422576\pi$
$859$	6.75499e59	0.481687	0.240844	+	0.970564i	$0.422576\pi$
$860$	0	0
$861$	−2.76793e60	−1.88187
$862$	0	0
$863$	−5.90285e59	−0.382684	−0.191342	−	0.981523i	$-0.561284\pi$
$863$	−5.90285e59	−0.382684	−0.191342	+	0.981523i	$0.561284\pi$
$864$	0	0
$865$	1.77313e60	1.09625
$866$	0	0
$867$	2.67058e60	1.57476
$868$	0	0
$869$	7.22889e59	0.406599
$870$	0	0
$871$	−4.27663e60	−2.29472
$872$	0	0
$873$	−6.34190e59	−0.324659
$874$	0	0
$875$	−1.97878e60	−0.966567
$876$	0	0
$877$	−2.63402e60	−1.22780	−0.613900	−	0.789384i	$-0.710400\pi$
$877$	−2.63402e60	−1.22780	−0.613900	+	0.789384i	$0.710400\pi$
$878$	0	0
$879$	3.10608e60	1.38179
$880$	0	0
$881$	1.68258e60	0.714445	0.357223	−	0.934019i	$-0.383724\pi$
$881$	1.68258e60	0.714445	0.357223	+	0.934019i	$0.383724\pi$
$882$	0	0
$883$	−2.43606e60	−0.987400	−0.493700	−	0.869632i	$-0.664356\pi$
$883$	−2.43606e60	−0.987400	−0.493700	+	0.869632i	$0.664356\pi$
$884$	0	0
$885$	−1.36582e59	−0.0528512
$886$	0	0
$887$	2.73192e60	1.00933	0.504666	−	0.863315i	$-0.331616\pi$
$887$	2.73192e60	1.00933	0.504666	+	0.863315i	$0.331616\pi$
$888$	0	0
$889$	3.42369e60	1.20784
$890$	0	0
$891$	1.77099e60	0.596655
$892$	0	0
$893$	6.65111e59	0.214012
$894$	0	0
$895$	6.53750e60	2.00927
$896$	0	0
$897$	5.72868e60	1.68194
$898$	0	0
$899$	4.54705e60	1.27543
$900$	0	0
$901$	1.07404e60	0.287848
$902$	0	0
$903$	1.15661e60	0.296202
$904$	0	0
$905$	−4.37314e60	−1.07028
$906$	0	0
$907$	9.66132e59	0.225989	0.112994	−	0.993596i	$-0.463956\pi$
$907$	9.66132e59	0.225989	0.112994	+	0.993596i	$0.463956\pi$
$908$	0	0
$909$	3.63469e60	0.812659
$910$	0	0
$911$	3.27643e60	0.700284	0.350142	−	0.936697i	$-0.386133\pi$
$911$	3.27643e60	0.700284	0.350142	+	0.936697i	$0.386133\pi$
$912$	0	0
$913$	−1.43353e60	−0.292925
$914$	0	0
$915$	−1.76787e61	−3.45397
$916$	0	0
$917$	7.18853e60	1.34298
$918$	0	0
$919$	−2.29076e60	−0.409271	−0.204636	−	0.978838i	$-0.565601\pi$
$919$	−2.29076e60	−0.409271	−0.204636	+	0.978838i	$0.565601\pi$
$920$	0	0
$921$	−4.61279e60	−0.788208
$922$	0	0
$923$	8.47926e60	1.38587
$924$	0	0
$925$	8.92977e59	0.139616
$926$	0	0
$927$	−6.69753e60	−1.00180
$928$	0	0
$929$	6.62994e60	0.948828	0.474414	−	0.880302i	$-0.342660\pi$
$929$	6.62994e60	0.948828	0.474414	+	0.880302i	$0.342660\pi$
$930$	0	0
$931$	1.51030e60	0.206821
$932$	0	0
$933$	−2.52683e60	−0.331133
$934$	0	0
$935$	1.48608e60	0.186382
$936$	0	0
$937$	−2.63986e60	−0.316898	−0.158449	−	0.987367i	$-0.550649\pi$
$937$	−2.63986e60	−0.316898	−0.158449	+	0.987367i	$0.550649\pi$
$938$	0	0
$939$	2.15028e61	2.47088
$940$	0	0
$941$	1.40617e61	1.54685	0.773427	−	0.633885i	$-0.218541\pi$
$941$	1.40617e61	1.54685	0.773427	+	0.633885i	$0.218541\pi$
$942$	0	0
$943$	5.91121e60	0.622568
$944$	0	0
$945$	−3.10800e61	−3.13422
$946$	0	0
$947$	−1.18800e61	−1.14721	−0.573606	−	0.819131i	$-0.694456\pi$
$947$	−1.18800e61	−1.14721	−0.573606	+	0.819131i	$0.694456\pi$
$948$	0	0
$949$	−2.29408e61	−2.12154
$950$	0	0
$951$	−2.83875e61	−2.51436
$952$	0	0
$953$	−1.12712e61	−0.956235	−0.478117	−	0.878296i	$-0.658681\pi$
$953$	−1.12712e61	−0.956235	−0.478117	+	0.878296i	$0.658681\pi$
$954$	0	0
$955$	−5.48291e60	−0.445597
$956$	0	0
$957$	1.89749e61	1.47736
$958$	0	0
$959$	4.57345e60	0.341163
$960$	0	0
$961$	−6.26416e60	−0.447748
$962$	0	0
$963$	−4.19507e61	−2.87343
$964$	0	0
$965$	−5.17113e60	−0.339450
$966$	0	0
$967$	6.94733e60	0.437095	0.218547	−	0.975826i	$-0.429868\pi$
$967$	6.94733e60	0.437095	0.218547	+	0.975826i	$0.429868\pi$
$968$	0	0
$969$	1.97874e60	0.119331
$970$	0	0
$971$	2.12424e61	1.22803	0.614016	−	0.789294i	$-0.289553\pi$
$971$	2.12424e61	1.22803	0.614016	+	0.789294i	$0.289553\pi$
$972$	0	0
$973$	4.16689e61	2.30940
$974$	0	0
$975$	−1.64312e61	−0.873124
$976$	0	0
$977$	−2.72440e61	−1.38814	−0.694068	−	0.719909i	$-0.744183\pi$
$977$	−2.72440e61	−1.38814	−0.694068	+	0.719909i	$0.744183\pi$
$978$	0	0
$979$	2.37676e60	0.116129
$980$	0	0
$981$	−3.60395e61	−1.68874
$982$	0	0
$983$	6.88350e60	0.309359	0.154679	−	0.987965i	$-0.450566\pi$
$983$	6.88350e60	0.309359	0.154679	+	0.987965i	$0.450566\pi$
$984$	0	0
$985$	3.77192e61	1.62600
$986$	0	0
$987$	5.93939e61	2.45607
$988$	0	0
$989$	−2.47007e60	−0.0979908
$990$	0	0
$991$	1.65448e60	0.0629727	0.0314864	−	0.999504i	$-0.489976\pi$
$991$	1.65448e60	0.0629727	0.0314864	+	0.999504i	$0.489976\pi$
$992$	0	0
$993$	7.98259e61	2.91531
$994$	0	0
$995$	−1.90200e61	−0.666557
$996$	0	0
$997$	3.62868e60	0.122039	0.0610193	−	0.998137i	$-0.480565\pi$
$997$	3.62868e60	0.122039	0.0610193	+	0.998137i	$0.480565\pi$
$998$	0	0
$999$	−1.92170e61	−0.620289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.42.a.b.1.1		2
4.3	odd	2		2.42.a.b.1.2	✓	2
12.11	even	2		18.42.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.42.a.b.1.2	✓	2	4.3	odd	2
16.42.a.b.1.1		2	1.1	even	1	trivial
18.42.a.c.1.2		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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 42 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-1.05814e10 q^{3} -2.54286e14 q^{5} -2.96016e17 q^{7} +7.54941e19 q^{9} +O(q^{10})\)	\(q-1.05814e10 q^{3} -2.54286e14 q^{5} -2.96016e17 q^{7} +7.54941e19 q^{9} +1.09620e21 q^{11} +8.09333e22 q^{13} +2.69071e24 q^{15} -5.33128e24 q^{17} +3.50762e25 q^{19} +3.13228e27 q^{21} -6.68932e27 q^{23} +1.91866e28 q^{25} -4.12899e29 q^{27} -1.63586e30 q^{29} -2.77961e30 q^{31} -1.15993e31 q^{33} +7.52726e31 q^{35} +4.65417e31 q^{37} -8.56392e32 q^{39} -8.83679e32 q^{41} +3.69256e32 q^{43} -1.91971e34 q^{45} +1.89619e34 q^{47} +4.30577e34 q^{49} +5.64127e34 q^{51} -2.01460e35 q^{53} -2.78747e35 q^{55} -3.71157e35 q^{57} -5.07603e34 q^{59} -6.57027e36 q^{61} -2.23474e37 q^{63} -2.05802e37 q^{65} -5.28414e37 q^{67} +7.07827e37 q^{69} +1.04768e38 q^{71} -2.83453e38 q^{73} -2.03022e38 q^{75} -3.24491e38 q^{77} +6.59452e38 q^{79} +1.61558e39 q^{81} -1.30773e39 q^{83} +1.35567e39 q^{85} +1.73098e40 q^{87} +2.16819e39 q^{89} -2.39575e40 q^{91} +2.94123e40 q^{93} -8.91937e39 q^{95} -8.40052e39 q^{97} +8.27563e40 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 41\!\cdots\!86 q^{9}+O(q^{10}) \) 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9	\( 2 q - 8863347528 q^{3} + 97599184325580 q^{5} - 21\!\cdots\!56 q^{7}+ \cdots + 12\!\cdots\!88 q^{99}+O(q^{100}) \) 2 * q - 8863347528 * q^3 + 97599184325580 * q^5 - 217106802373088656 * q^7 + 41972929616872509786 * q^9 - 110594481484939565784 * q^11 + 173293106838448696895068 * q^13 + 3295287287252802285241680 * q^15 - 10287966260741677078429404 * q^17 - 206254882139859790701457000 * q^19 + 3267849131609686416057835584 * q^21 - 14925421378885128304361915568 * q^23 + 97534966241249753807444705150 * q^25 - 533156288863055843864697754320 * q^27 - 1375143011941884388614889894020 * q^29 + 3626496922711160469799126663616 * q^31 - 13672735988327258412044854187424 * q^33 + 103039524095905042910481769779360 * q^35 + 120356679589378915210866994599436 * q^37 - 697708668559917206762862430490352 * q^39 + 1270177139906840106861139346878164 * q^41 + 6045433421160616098770468790853352 * q^43 - 30992658299909156185379775337031460 * q^45 + 25416478966330542868882280891861664 * q^47 + 4716677344275473650585885334355954 * q^49 + 47896619495707425086522465903395056 * q^51 - 301376484284740318156836692232048852 * q^53 - 703399094180619772764269188861842960 * q^55 - 785787658029123941344672503009410400 * q^57 + 3282010250697938085995473868454788040 * q^59 - 4485157745840460017677520841591879236 * q^61 - 24992547810032633572829803615671997008 * q^63 + 11919812053150705903267451825832510120 * q^65 - 55055736076082381526441630307313037256 * q^67 + 56632222305135551516324020091548201152 * q^69 + 48857884738531480102553331099997138416 * q^71 - 485616958845162818165813572349849688812 * q^73 - 68411444542067712502035991381122300600 * q^75 - 419718086944271580508987308621981603648 * q^77 + 905766854341990828970937020702446890080 * q^79 + 2631580786121491108655810678136598411122 * q^81 - 1271755206567258248006583231290595969768 * q^83 - 388512546938141834318136915482004571560 * q^85 + 17757719150026706057484126704955838888080 * q^87 + 2692414558304473820098690391828757436980 * q^89 - 16669534332125295311104493248216990683104 * q^91 + 40418607330508619745600934617627243192576 * q^93 - 93840167647611953736551449995129877215600 * q^95 - 31989468315501178009691057062669320691004 * q^97 + 123209315561064920195916977740234432814088 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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