LMFDB - Embedded newform 48.24.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,24,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	48.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:	$160.897937926$
Analytic rank:	$1$
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} - 313478447x - 3858843765 $ x^3 - x^2 - 313478447*x - 3858843765
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{21}\cdot 3^{5} $
Twist minimal:	no (minimal twist has level 24)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-12.3098$ of defining polynomial
Character	$\chi$	$=$	48.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+177147. q^{3} +7.68230e7 q^{5} -1.22170e9 q^{7} +3.13811e10 q^{9} +O(q^{10})$	$q+177147. q^{3} +7.68230e7 q^{5} -1.22170e9 q^{7} +3.13811e10 q^{9} -1.06854e12 q^{11} -2.56600e12 q^{13} +1.36090e13 q^{15} +2.26622e14 q^{17} +1.03625e14 q^{19} -2.16421e14 q^{21} -2.81634e15 q^{23} -6.01915e15 q^{25} +5.55906e15 q^{27} -2.91373e16 q^{29} +1.27270e17 q^{31} -1.89289e17 q^{33} -9.38547e16 q^{35} +1.71452e17 q^{37} -4.54560e17 q^{39} -3.42183e18 q^{41} -4.70389e18 q^{43} +2.41079e18 q^{45} -1.30939e19 q^{47} -2.58762e19 q^{49} +4.01454e19 q^{51} +9.52257e19 q^{53} -8.20887e19 q^{55} +1.83568e19 q^{57} -3.82902e19 q^{59} -2.25139e20 q^{61} -3.83382e19 q^{63} -1.97128e20 q^{65} +3.91941e20 q^{67} -4.98907e20 q^{69} -2.24642e21 q^{71} +1.20194e21 q^{73} -1.06627e21 q^{75} +1.30544e21 q^{77} -7.35678e21 q^{79} +9.84771e20 q^{81} -1.06765e22 q^{83} +1.74098e22 q^{85} -5.16159e21 q^{87} -1.82064e21 q^{89} +3.13489e21 q^{91} +2.25455e22 q^{93} +7.96076e21 q^{95} +1.10091e23 q^{97} -3.35320e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9}+O(q^{10}) $ 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9	$ 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9} + 66881746140 q^{11} + 5138719820610 q^{13} - 13680206481402 q^{15} - 232964235839898 q^{17} + 39051643356612 q^{19} - 591782205641184 q^{21} + 43\!\cdots\!64 q^{23}+ \cdots + 20\!\cdots\!60 q^{99}+O(q^{100}) $ 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9 + 66881746140 * q^11 + 5138719820610 * q^13 - 13680206481402 * q^15 - 232964235839898 * q^17 + 39051643356612 * q^19 - 591782205641184 * q^21 + 4358324846210664 * q^23 - 5164619678396379 * q^25 + 16677181699666569 * q^27 - 25471683594266694 * q^29 + 125107985305907448 * q^31 + 11847900683462580 * q^33 + 1311200449040041536 * q^35 - 1292646678108250614 * q^37 + 910308800061599670 * q^39 + 144934201638163518 * q^41 - 8133920080752699684 * q^43 - 2423407537560920094 * q^45 - 10029621278906529216 * q^47 + 41823513653599245723 * q^49 - 41268915486330411006 * q^51 - 28251334125947614062 * q^53 - 85062431363722250328 * q^55 + 6917881465693745964 * q^57 - 187335386683361480100 * q^59 - 555296430938057984862 * q^61 - 104832442382718822048 * q^63 - 1392409010623622551476 * q^65 - 1602075967132026564156 * q^67 + 772064171531680495608 * q^69 - 3528346267933953055080 * q^71 + 2100403635635469857646 * q^73 - 914896882168883350713 * q^75 + 8279040903546572168064 * q^77 - 12175210588212413874744 * q^79 + 2954312706550833698643 * q^81 - 30717412717926068652348 * q^83 + 44709116429093587923108 * q^85 - 4512232333673562042018 * q^87 + 58208079477660202949598 * q^89 - 63276323944720307992512 * q^91 + 22162504272985586690856 * q^93 - 79678214445310528084968 * q^95 + 76406389646793178923846 * q^97 + 2098820062373345659260 * 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
$3$	177147.	0.577350
$3$	177147.	0.577350
$4$	0	0
$5$	7.68230e7	0.703617	0.351809	−	0.936072i	$-0.385567\pi$
$5$	7.68230e7	0.703617	0.351809	+	0.936072i	$0.385567\pi$
$6$	0	0
$7$	−1.22170e9	−0.233527	−0.116764	−	0.993160i	$-0.537252\pi$
$7$	−1.22170e9	−0.233527	−0.116764	+	0.993160i	$0.537252\pi$
$8$	0	0
$9$	3.13811e10	0.333333
$10$	0	0
$11$	−1.06854e12	−1.12921	−0.564607	−	0.825360i	$-0.690972\pi$
$11$	−1.06854e12	−1.12921	−0.564607	+	0.825360i	$0.690972\pi$
$12$	0	0
$13$	−2.56600e12	−0.397108	−0.198554	−	0.980090i	$-0.563625\pi$
$13$	−2.56600e12	−0.397108	−0.198554	+	0.980090i	$0.563625\pi$
$14$	0	0
$15$	1.36090e13	0.406233
$16$	0	0
$17$	2.26622e14	1.60376	0.801880	−	0.597485i	$-0.203833\pi$
$17$	2.26622e14	1.60376	0.801880	+	0.597485i	$0.203833\pi$
$18$	0	0
$19$	1.03625e14	0.204078	0.102039	−	0.994780i	$-0.467463\pi$
$19$	1.03625e14	0.204078	0.102039	+	0.994780i	$0.467463\pi$
$20$	0	0
$21$	−2.16421e14	−0.134827
$22$	0	0
$23$	−2.81634e15	−0.616333	−0.308167	−	0.951332i	$-0.599715\pi$
$23$	−2.81634e15	−0.616333	−0.308167	+	0.951332i	$0.599715\pi$
$24$	0	0
$25$	−6.01915e15	−0.504923
$26$	0	0
$27$	5.55906e15	0.192450
$28$	0	0
$29$	−2.91373e16	−0.443479	−0.221739	−	0.975106i	$-0.571173\pi$
$29$	−2.91373e16	−0.443479	−0.221739	+	0.975106i	$0.571173\pi$
$30$	0	0
$31$	1.27270e17	0.899634	0.449817	−	0.893121i	$-0.351489\pi$
$31$	1.27270e17	0.899634	0.449817	+	0.893121i	$0.351489\pi$
$32$	0	0
$33$	−1.89289e17	−0.651952
$34$	0	0
$35$	−9.38547e16	−0.164314
$36$	0	0
$37$	1.71452e17	0.158425	0.0792123	−	0.996858i	$-0.474760\pi$
$37$	1.71452e17	0.158425	0.0792123	+	0.996858i	$0.474760\pi$
$38$	0	0
$39$	−4.54560e17	−0.229270
$40$	0	0
$41$	−3.42183e18	−0.971055	−0.485527	−	0.874221i	$-0.661372\pi$
$41$	−3.42183e18	−0.971055	−0.485527	+	0.874221i	$0.661372\pi$
$42$	0	0
$43$	−4.70389e18	−0.771915	−0.385958	−	0.922517i	$-0.626129\pi$
$43$	−4.70389e18	−0.771915	−0.385958	+	0.922517i	$0.626129\pi$
$44$	0	0
$45$	2.41079e18	0.234539
$46$	0	0
$47$	−1.30939e19	−0.772578	−0.386289	−	0.922378i	$-0.626243\pi$
$47$	−1.30939e19	−0.772578	−0.386289	+	0.922378i	$0.626243\pi$
$48$	0	0
$49$	−2.58762e19	−0.945465
$50$	0	0
$51$	4.01454e19	0.925931
$52$	0	0
$53$	9.52257e19	1.41118	0.705589	−	0.708621i	$-0.250682\pi$
$53$	9.52257e19	1.41118	0.705589	+	0.708621i	$0.250682\pi$
$54$	0	0
$55$	−8.20887e19	−0.794534
$56$	0	0
$57$	1.83568e19	0.117825
$58$	0	0
$59$	−3.82902e19	−0.165306	−0.0826532	−	0.996578i	$-0.526339\pi$
$59$	−3.82902e19	−0.165306	−0.0826532	+	0.996578i	$0.526339\pi$
$60$	0	0
$61$	−2.25139e20	−0.662456	−0.331228	−	0.943551i	$-0.607463\pi$
$61$	−2.25139e20	−0.662456	−0.331228	+	0.943551i	$0.607463\pi$
$62$	0	0
$63$	−3.83382e19	−0.0778424
$64$	0	0
$65$	−1.97128e20	−0.279412
$66$	0	0
$67$	3.91941e20	0.392068	0.196034	−	0.980597i	$-0.437194\pi$
$67$	3.91941e20	0.392068	0.196034	+	0.980597i	$0.437194\pi$
$68$	0	0
$69$	−4.98907e20	−0.355840
$70$	0	0
$71$	−2.24642e21	−1.15351	−0.576754	−	0.816918i	$-0.695681\pi$
$71$	−2.24642e21	−1.15351	−0.576754	+	0.816918i	$0.695681\pi$
$72$	0	0
$73$	1.20194e21	0.448405	0.224202	−	0.974543i	$-0.428022\pi$
$73$	1.20194e21	0.448405	0.224202	+	0.974543i	$0.428022\pi$
$74$	0	0
$75$	−1.06627e21	−0.291517
$76$	0	0
$77$	1.30544e21	0.263702
$78$	0	0
$79$	−7.35678e21	−1.10656	−0.553282	−	0.832994i	$-0.686625\pi$
$79$	−7.35678e21	−1.10656	−0.553282	+	0.832994i	$0.686625\pi$
$80$	0	0
$81$	9.84771e20	0.111111
$82$	0	0
$83$	−1.06765e22	−0.909978	−0.454989	−	0.890497i	$-0.650357\pi$
$83$	−1.06765e22	−0.909978	−0.454989	+	0.890497i	$0.650357\pi$
$84$	0	0
$85$	1.74098e22	1.12843
$86$	0	0
$87$	−5.16159e21	−0.256043
$88$	0	0
$89$	−1.82064e21	−0.0695405	−0.0347703	−	0.999395i	$-0.511070\pi$
$89$	−1.82064e21	−0.0695405	−0.0347703	+	0.999395i	$0.511070\pi$
$90$	0	0
$91$	3.13489e21	0.0927355
$92$	0	0
$93$	2.25455e22	0.519404
$94$	0	0
$95$	7.96076e21	0.143593
$96$	0	0
$97$	1.10091e23	1.56270	0.781351	−	0.624092i	$-0.214531\pi$
$97$	1.10091e23	1.56270	0.781351	+	0.624092i	$0.214531\pi$
$98$	0	0
$99$	−3.35320e22	−0.376405
$100$	0	0
$101$	1.11453e23	0.994023	0.497011	−	0.867744i	$-0.334431\pi$
$101$	1.11453e23	0.994023	0.497011	+	0.867744i	$0.334431\pi$
$102$	0	0
$103$	−6.90412e22	−0.491451	−0.245726	−	0.969339i	$-0.579026\pi$
$103$	−6.90412e22	−0.491451	−0.245726	+	0.969339i	$0.579026\pi$
$104$	0	0
$105$	−1.66261e22	−0.0948665
$106$	0	0
$107$	−3.26969e23	−1.50173	−0.750867	−	0.660454i	$-0.770364\pi$
$107$	−3.26969e23	−1.50173	−0.750867	+	0.660454i	$0.770364\pi$
$108$	0	0
$109$	7.48321e22	0.277768	0.138884	−	0.990309i	$-0.455648\pi$
$109$	7.48321e22	0.277768	0.138884	+	0.990309i	$0.455648\pi$
$110$	0	0
$111$	3.03722e22	0.0914664
$112$	0	0
$113$	4.28014e23	1.04968	0.524838	−	0.851202i	$-0.324126\pi$
$113$	4.28014e23	1.04968	0.524838	+	0.851202i	$0.324126\pi$
$114$	0	0
$115$	−2.16360e23	−0.433663
$116$	0	0
$117$	−8.05239e22	−0.132369
$118$	0	0
$119$	−2.76864e23	−0.374521
$120$	0	0
$121$	2.46355e23	0.275125
$122$	0	0
$123$	−6.06166e23	−0.560639
$124$	0	0
$125$	−1.37821e24	−1.05889
$126$	0	0
$127$	−9.18263e22	−0.0587793	−0.0293897	−	0.999568i	$-0.509356\pi$
$127$	−9.18263e22	−0.0587793	−0.0293897	+	0.999568i	$0.509356\pi$
$128$	0	0
$129$	−8.33279e23	−0.445665
$130$	0	0
$131$	−6.46108e23	−0.289524	−0.144762	−	0.989466i	$-0.546242\pi$
$131$	−6.46108e23	−0.289524	−0.144762	+	0.989466i	$0.546242\pi$
$132$	0	0
$133$	−1.26598e23	−0.0476578
$134$	0	0
$135$	4.27064e23	0.135411
$136$	0	0
$137$	−2.25789e24	−0.604528	−0.302264	−	0.953224i	$-0.597742\pi$
$137$	−2.25789e24	−0.604528	−0.302264	+	0.953224i	$0.597742\pi$
$138$	0	0
$139$	7.34683e24	1.66506	0.832530	−	0.553981i	$-0.186892\pi$
$139$	7.34683e24	1.66506	0.832530	+	0.553981i	$0.186892\pi$
$140$	0	0
$141$	−2.31954e24	−0.446048
$142$	0	0
$143$	2.74189e24	0.448420
$144$	0	0
$145$	−2.23842e24	−0.312039
$146$	0	0
$147$	−4.58389e24	−0.545865
$148$	0	0
$149$	−1.23725e25	−1.26130	−0.630648	−	0.776069i	$-0.717211\pi$
$149$	−1.23725e25	−1.26130	−0.630648	+	0.776069i	$0.717211\pi$
$150$	0	0
$151$	−1.28595e25	−1.12458	−0.562289	−	0.826941i	$-0.690079\pi$
$151$	−1.28595e25	−1.12458	−0.562289	+	0.826941i	$0.690079\pi$
$152$	0	0
$153$	7.11164e24	0.534587
$154$	0	0
$155$	9.77725e24	0.632998
$156$	0	0
$157$	−2.55809e25	−1.42912	−0.714562	−	0.699572i	$-0.753374\pi$
$157$	−2.55809e25	−1.42912	−0.714562	+	0.699572i	$0.753374\pi$
$158$	0	0
$159$	1.68690e25	0.814744
$160$	0	0
$161$	3.44073e24	0.143930
$162$	0	0
$163$	−9.46789e24	−0.343634	−0.171817	−	0.985129i	$-0.554964\pi$
$163$	−9.46789e24	−0.343634	−0.171817	+	0.985129i	$0.554964\pi$
$164$	0	0
$165$	−1.45418e25	−0.458725
$166$	0	0
$167$	−5.85691e24	−0.160853	−0.0804265	−	0.996761i	$-0.525628\pi$
$167$	−5.85691e24	−0.160853	−0.0804265	+	0.996761i	$0.525628\pi$
$168$	0	0
$169$	−3.51695e25	−0.842305
$170$	0	0
$171$	3.25185e24	0.0680261
$172$	0	0
$173$	−3.06106e25	−0.560198	−0.280099	−	0.959971i	$-0.590367\pi$
$173$	−3.06106e25	−0.560198	−0.280099	+	0.959971i	$0.590367\pi$
$174$	0	0
$175$	7.35360e24	0.117913
$176$	0	0
$177$	−6.78300e24	−0.0954397
$178$	0	0
$179$	−1.16379e26	−1.43902	−0.719509	−	0.694484i	$-0.755633\pi$
$179$	−1.16379e26	−1.43902	−0.719509	+	0.694484i	$0.755633\pi$
$180$	0	0
$181$	7.57421e25	0.824202	0.412101	−	0.911138i	$-0.364795\pi$
$181$	7.57421e25	0.824202	0.412101	+	0.911138i	$0.364795\pi$
$182$	0	0
$183$	−3.98826e25	−0.382469
$184$	0	0
$185$	1.31715e25	0.111470
$186$	0	0
$187$	−2.42155e26	−1.81099
$188$	0	0
$189$	−6.79151e24	−0.0449423
$190$	0	0
$191$	−1.86166e26	−1.09148	−0.545740	−	0.837954i	$-0.683752\pi$
$191$	−1.86166e26	−1.09148	−0.545740	+	0.837954i	$0.683752\pi$
$192$	0	0
$193$	−9.22407e25	−0.479748	−0.239874	−	0.970804i	$-0.577106\pi$
$193$	−9.22407e25	−0.479748	−0.239874	+	0.970804i	$0.577106\pi$
$194$	0	0
$195$	−3.49207e25	−0.161319
$196$	0	0
$197$	−1.78913e26	−0.734990	−0.367495	−	0.930026i	$-0.619784\pi$
$197$	−1.78913e26	−0.734990	−0.367495	+	0.930026i	$0.619784\pi$
$198$	0	0
$199$	−4.49344e26	−1.64350	−0.821749	−	0.569850i	$-0.807001\pi$
$199$	−4.49344e26	−1.64350	−0.821749	+	0.569850i	$0.807001\pi$
$200$	0	0
$201$	6.94312e25	0.226360
$202$	0	0
$203$	3.55971e25	0.103564
$204$	0	0
$205$	−2.62875e26	−0.683251
$206$	0	0
$207$	−8.83798e25	−0.205444
$208$	0	0
$209$	−1.10727e26	−0.230448
$210$	0	0
$211$	−8.43056e26	−1.57256	−0.786282	−	0.617868i	$-0.787997\pi$
$211$	−8.43056e26	−1.57256	−0.786282	+	0.617868i	$0.787997\pi$
$212$	0	0
$213$	−3.97947e26	−0.665979
$214$	0	0
$215$	−3.61367e26	−0.543133
$216$	0	0
$217$	−1.55486e26	−0.210089
$218$	0	0
$219$	2.12920e26	0.258887
$220$	0	0
$221$	−5.81513e26	−0.636866
$222$	0	0
$223$	1.39146e27	1.37393	0.686967	−	0.726688i	$-0.258942\pi$
$223$	1.39146e27	1.37393	0.686967	+	0.726688i	$0.258942\pi$
$224$	0	0
$225$	−1.88887e26	−0.168308
$226$	0	0
$227$	1.65194e26	0.132953	0.0664764	−	0.997788i	$-0.478824\pi$
$227$	1.65194e26	0.132953	0.0664764	+	0.997788i	$0.478824\pi$
$228$	0	0
$229$	2.18267e27	1.58811	0.794053	−	0.607848i	$-0.207967\pi$
$229$	2.18267e27	1.58811	0.794053	+	0.607848i	$0.207967\pi$
$230$	0	0
$231$	2.31255e26	0.152248
$232$	0	0
$233$	−1.75593e27	−1.04692	−0.523461	−	0.852050i	$-0.675359\pi$
$233$	−1.75593e27	−1.04692	−0.523461	+	0.852050i	$0.675359\pi$
$234$	0	0
$235$	−1.00591e27	−0.543599
$236$	0	0
$237$	−1.30323e27	−0.638875
$238$	0	0
$239$	1.43851e27	0.640231	0.320115	−	0.947379i	$-0.396278\pi$
$239$	1.43851e27	0.640231	0.320115	+	0.947379i	$0.396278\pi$
$240$	0	0
$241$	−3.61269e27	−1.46095	−0.730474	−	0.682940i	$-0.760701\pi$
$241$	−3.61269e27	−1.46095	−0.730474	+	0.682940i	$0.760701\pi$
$242$	0	0
$243$	1.74449e26	0.0641500
$244$	0	0
$245$	−1.98789e27	−0.665245
$246$	0	0
$247$	−2.65901e26	−0.0810411
$248$	0	0
$249$	−1.89131e27	−0.525376
$250$	0	0
$251$	−3.43927e27	−0.871401	−0.435700	−	0.900092i	$-0.643499\pi$
$251$	−3.43927e27	−0.871401	−0.435700	+	0.900092i	$0.643499\pi$
$252$	0	0
$253$	3.00939e27	0.695972
$254$	0	0
$255$	3.08409e27	0.651501
$256$	0	0
$257$	−1.90737e27	−0.368303	−0.184151	−	0.982898i	$-0.558954\pi$
$257$	−1.90737e27	−0.368303	−0.184151	+	0.982898i	$0.558954\pi$
$258$	0	0
$259$	−2.09463e26	−0.0369964
$260$	0	0
$261$	−9.14361e26	−0.147826
$262$	0	0
$263$	1.11022e28	1.64406	0.822030	−	0.569443i	$-0.192841\pi$
$263$	1.11022e28	1.64406	0.822030	+	0.569443i	$0.192841\pi$
$264$	0	0
$265$	7.31553e27	0.992929
$266$	0	0
$267$	−3.22520e26	−0.0401493
$268$	0	0
$269$	−6.34014e27	−0.724348	−0.362174	−	0.932110i	$-0.617965\pi$
$269$	−6.34014e27	−0.724348	−0.362174	+	0.932110i	$0.617965\pi$
$270$	0	0
$271$	−1.61436e28	−1.69377	−0.846885	−	0.531775i	$-0.821525\pi$
$271$	−1.61436e28	−1.69377	−0.846885	+	0.531775i	$0.821525\pi$
$272$	0	0
$273$	5.55336e26	0.0535408
$274$	0	0
$275$	6.43173e27	0.570166
$276$	0	0
$277$	1.58309e28	1.29118	0.645591	−	0.763683i	$-0.276611\pi$
$277$	1.58309e28	1.29118	0.645591	+	0.763683i	$0.276611\pi$
$278$	0	0
$279$	3.99386e27	0.299878
$280$	0	0
$281$	1.81205e28	1.25328	0.626639	−	0.779310i	$-0.284430\pi$
$281$	1.81205e28	1.25328	0.626639	+	0.779310i	$0.284430\pi$
$282$	0	0
$283$	−1.61476e28	−1.02935	−0.514677	−	0.857384i	$-0.672088\pi$
$283$	−1.61476e28	−1.02935	−0.514677	+	0.857384i	$0.672088\pi$
$284$	0	0
$285$	1.41023e27	0.0829034
$286$	0	0
$287$	4.18045e27	0.226768
$288$	0	0
$289$	3.13899e28	1.57205
$290$	0	0
$291$	1.95023e28	0.902226
$292$	0	0
$293$	3.88969e28	1.66317	0.831587	−	0.555394i	$-0.187433\pi$
$293$	3.88969e28	1.66317	0.831587	+	0.555394i	$0.187433\pi$
$294$	0	0
$295$	−2.94157e27	−0.116312
$296$	0	0
$297$	−5.94010e27	−0.217317
$298$	0	0
$299$	7.22675e27	0.244751
$300$	0	0
$301$	5.74674e27	0.180263
$302$	0	0
$303$	1.97436e28	0.573899
$304$	0	0
$305$	−1.72958e28	−0.466115
$306$	0	0
$307$	−1.59888e28	−0.399690	−0.199845	−	0.979828i	$-0.564044\pi$
$307$	−1.59888e28	−0.399690	−0.199845	+	0.979828i	$0.564044\pi$
$308$	0	0
$309$	−1.22304e28	−0.283740
$310$	0	0
$311$	5.93462e28	1.27835	0.639173	−	0.769063i	$-0.279277\pi$
$311$	5.93462e28	1.27835	0.639173	+	0.769063i	$0.279277\pi$
$312$	0	0
$313$	9.48882e28	1.89868	0.949342	−	0.314246i	$-0.101752\pi$
$313$	9.48882e28	1.89868	0.949342	+	0.314246i	$0.101752\pi$
$314$	0	0
$315$	−2.94526e27	−0.0547712
$316$	0	0
$317$	−5.11392e28	−0.884245	−0.442123	−	0.896955i	$-0.645774\pi$
$317$	−5.11392e28	−0.884245	−0.442123	+	0.896955i	$0.645774\pi$
$318$	0	0
$319$	3.11345e28	0.500783
$320$	0	0
$321$	−5.79215e28	−0.867026
$322$	0	0
$323$	2.34836e28	0.327293
$324$	0	0
$325$	1.54452e28	0.200509
$326$	0	0
$327$	1.32563e28	0.160370
$328$	0	0
$329$	1.59968e28	0.180418
$330$	0	0
$331$	−8.30108e28	−0.873198	−0.436599	−	0.899656i	$-0.643817\pi$
$331$	−8.30108e28	−0.873198	−0.436599	+	0.899656i	$0.643817\pi$
$332$	0	0
$333$	5.38034e27	0.0528082
$334$	0	0
$335$	3.01101e28	0.275865
$336$	0	0
$337$	4.03998e28	0.345649	0.172824	−	0.984953i	$-0.444711\pi$
$337$	4.03998e28	0.345649	0.172824	+	0.984953i	$0.444711\pi$
$338$	0	0
$339$	7.58214e28	0.606031
$340$	0	0
$341$	−1.35993e29	−1.01588
$342$	0	0
$343$	6.50494e28	0.454319
$344$	0	0
$345$	−3.83275e28	−0.250375
$346$	0	0
$347$	2.51625e29	1.53803	0.769015	−	0.639231i	$-0.220747\pi$
$347$	2.51625e29	1.53803	0.769015	+	0.639231i	$0.220747\pi$
$348$	0	0
$349$	−1.57577e28	−0.0901570	−0.0450785	−	0.998983i	$-0.514354\pi$
$349$	−1.57577e28	−0.0901570	−0.0450785	+	0.998983i	$0.514354\pi$
$350$	0	0
$351$	−1.42646e28	−0.0764235
$352$	0	0
$353$	−2.11881e29	−1.06337	−0.531684	−	0.846943i	$-0.678441\pi$
$353$	−2.11881e29	−1.06337	−0.531684	+	0.846943i	$0.678441\pi$
$354$	0	0
$355$	−1.72577e29	−0.811628
$356$	0	0
$357$	−4.90456e28	−0.216230
$358$	0	0
$359$	−2.27264e29	−0.939602	−0.469801	−	0.882772i	$-0.655674\pi$
$359$	−2.27264e29	−0.939602	−0.469801	+	0.882772i	$0.655674\pi$
$360$	0	0
$361$	−2.47092e29	−0.958352
$362$	0	0
$363$	4.36410e28	0.158843
$364$	0	0
$365$	9.23367e28	0.315505
$366$	0	0
$367$	−1.61955e29	−0.519678	−0.259839	−	0.965652i	$-0.583669\pi$
$367$	−1.61955e29	−0.519678	−0.259839	+	0.965652i	$0.583669\pi$
$368$	0	0
$369$	−1.07381e29	−0.323685
$370$	0	0
$371$	−1.16337e29	−0.329548
$372$	0	0
$373$	2.81519e29	0.749645	0.374823	−	0.927097i	$-0.377704\pi$
$373$	2.81519e29	0.749645	0.374823	+	0.927097i	$0.377704\pi$
$374$	0	0
$375$	−2.44146e29	−0.611350
$376$	0	0
$377$	7.47665e28	0.176109
$378$	0	0
$379$	1.02993e29	0.228274	0.114137	−	0.993465i	$-0.463590\pi$
$379$	1.02993e29	0.228274	0.114137	+	0.993465i	$0.463590\pi$
$380$	0	0
$381$	−1.62668e28	−0.0339362
$382$	0	0
$383$	−3.99304e29	−0.784364	−0.392182	−	0.919888i	$-0.628280\pi$
$383$	−3.99304e29	−0.784364	−0.392182	+	0.919888i	$0.628280\pi$
$384$	0	0
$385$	1.00288e29	0.185545
$386$	0	0
$387$	−1.47613e29	−0.257305
$388$	0	0
$389$	2.60627e29	0.428153	0.214076	−	0.976817i	$-0.431326\pi$
$389$	2.60627e29	0.428153	0.214076	+	0.976817i	$0.431326\pi$
$390$	0	0
$391$	−6.38245e29	−0.988451
$392$	0	0
$393$	−1.14456e29	−0.167157
$394$	0	0
$395$	−5.65170e29	−0.778597
$396$	0	0
$397$	−4.73663e29	−0.615713	−0.307856	−	0.951433i	$-0.599612\pi$
$397$	−4.73663e29	−0.615713	−0.307856	+	0.951433i	$0.599612\pi$
$398$	0	0
$399$	−2.24265e28	−0.0275152
$400$	0	0
$401$	8.23465e29	0.953860	0.476930	−	0.878941i	$-0.341749\pi$
$401$	8.23465e29	0.953860	0.476930	+	0.878941i	$0.341749\pi$
$402$	0	0
$403$	−3.26575e29	−0.357252
$404$	0	0
$405$	7.56531e28	0.0781797
$406$	0	0
$407$	−1.83204e29	−0.178895
$408$	0	0
$409$	−1.34587e30	−1.24219	−0.621093	−	0.783737i	$-0.713311\pi$
$409$	−1.34587e30	−1.24219	−0.621093	+	0.783737i	$0.713311\pi$
$410$	0	0
$411$	−3.99979e29	−0.349024
$412$	0	0
$413$	4.67792e28	0.0386035
$414$	0	0
$415$	−8.20201e29	−0.640276
$416$	0	0
$417$	1.30147e30	0.961322
$418$	0	0
$419$	1.23126e30	0.860775	0.430387	−	0.902644i	$-0.358377\pi$
$419$	1.23126e30	0.860775	0.430387	+	0.902644i	$0.358377\pi$
$420$	0	0
$421$	−1.15281e30	−0.762984	−0.381492	−	0.924372i	$-0.624590\pi$
$421$	−1.15281e30	−0.762984	−0.381492	+	0.924372i	$0.624590\pi$
$422$	0	0
$423$	−4.10899e29	−0.257526
$424$	0	0
$425$	−1.36407e30	−0.809775
$426$	0	0
$427$	2.75052e29	0.154701
$428$	0	0
$429$	4.85717e29	0.258895
$430$	0	0
$431$	9.33894e28	0.0471855	0.0235928	−	0.999722i	$-0.492489\pi$
$431$	9.33894e28	0.0471855	0.0235928	+	0.999722i	$0.492489\pi$
$432$	0	0
$433$	−3.50663e30	−1.67988	−0.839941	−	0.542678i	$-0.817411\pi$
$433$	−3.50663e30	−1.67988	−0.839941	+	0.542678i	$0.817411\pi$
$434$	0	0
$435$	−3.96529e29	−0.180156
$436$	0	0
$437$	−2.91843e29	−0.125780
$438$	0	0
$439$	4.23630e30	1.73239	0.866194	−	0.499708i	$-0.166559\pi$
$439$	4.23630e30	1.73239	0.866194	+	0.499708i	$0.166559\pi$
$440$	0	0
$441$	−8.12022e29	−0.315155
$442$	0	0
$443$	−1.24130e30	−0.457333	−0.228667	−	0.973505i	$-0.573437\pi$
$443$	−1.24130e30	−0.457333	−0.228667	+	0.973505i	$0.573437\pi$
$444$	0	0
$445$	−1.39867e29	−0.0489299
$446$	0	0
$447$	−2.19176e30	−0.728210
$448$	0	0
$449$	4.59289e30	1.44962	0.724808	−	0.688951i	$-0.241929\pi$
$449$	4.59289e30	1.44962	0.724808	+	0.688951i	$0.241929\pi$
$450$	0	0
$451$	3.65637e30	1.09653
$452$	0	0
$453$	−2.27802e30	−0.649275
$454$	0	0
$455$	2.40832e29	0.0652503
$456$	0	0
$457$	1.16892e30	0.301127	0.150563	−	0.988600i	$-0.451891\pi$
$457$	1.16892e30	0.301127	0.150563	+	0.988600i	$0.451891\pi$
$458$	0	0
$459$	1.25981e30	0.308644
$460$	0	0
$461$	2.42406e30	0.564915	0.282458	−	0.959280i	$-0.408850\pi$
$461$	2.42406e30	0.564915	0.282458	+	0.959280i	$0.408850\pi$
$462$	0	0
$463$	7.55658e30	1.67550	0.837748	−	0.546056i	$-0.183872\pi$
$463$	7.55658e30	1.67550	0.837748	+	0.546056i	$0.183872\pi$
$464$	0	0
$465$	1.73201e30	0.365462
$466$	0	0
$467$	8.44956e30	1.69703	0.848516	−	0.529170i	$-0.177497\pi$
$467$	8.44956e30	1.69703	0.848516	+	0.529170i	$0.177497\pi$
$468$	0	0
$469$	−4.78835e29	−0.0915584
$470$	0	0
$471$	−4.53158e30	−0.825106
$472$	0	0
$473$	5.02631e30	0.871657
$474$	0	0
$475$	−6.23733e29	−0.103044
$476$	0	0
$477$	2.98828e30	0.470393
$478$	0	0
$479$	3.10371e30	0.465610	0.232805	−	0.972523i	$-0.425210\pi$
$479$	3.10371e30	0.465610	0.232805	+	0.972523i	$0.425210\pi$
$480$	0	0
$481$	−4.39946e29	−0.0629116
$482$	0	0
$483$	6.09515e29	0.0830983
$484$	0	0
$485$	8.45751e30	1.09954
$486$	0	0
$487$	7.12515e30	0.883509	0.441755	−	0.897136i	$-0.354356\pi$
$487$	7.12515e30	0.883509	0.441755	+	0.897136i	$0.354356\pi$
$488$	0	0
$489$	−1.67721e30	−0.198397
$490$	0	0
$491$	7.11130e29	0.0802624	0.0401312	−	0.999194i	$-0.487222\pi$
$491$	7.11130e29	0.0802624	0.0401312	+	0.999194i	$0.487222\pi$
$492$	0	0
$493$	−6.60316e30	−0.711234
$494$	0	0
$495$	−2.57603e30	−0.264845
$496$	0	0
$497$	2.74446e30	0.269375
$498$	0	0
$499$	−1.69860e31	−1.59197	−0.795983	−	0.605319i	$-0.793046\pi$
$499$	−1.69860e31	−1.59197	−0.795983	+	0.605319i	$0.793046\pi$
$500$	0	0
$501$	−1.03753e30	−0.0928685
$502$	0	0
$503$	−3.55048e30	−0.303567	−0.151784	−	0.988414i	$-0.548502\pi$
$503$	−3.55048e30	−0.303567	−0.151784	+	0.988414i	$0.548502\pi$
$504$	0	0
$505$	8.56216e30	0.699411
$506$	0	0
$507$	−6.23018e30	−0.486305
$508$	0	0
$509$	2.03149e31	1.51552	0.757759	−	0.652534i	$-0.226294\pi$
$509$	2.03149e31	1.51552	0.757759	+	0.652534i	$0.226294\pi$
$510$	0	0
$511$	−1.46841e30	−0.104715
$512$	0	0
$513$	5.76056e29	0.0392749
$514$	0	0
$515$	−5.30396e30	−0.345793
$516$	0	0
$517$	1.39914e31	0.872405
$518$	0	0
$519$	−5.42258e30	−0.323431
$520$	0	0
$521$	−5.19251e30	−0.296308	−0.148154	−	0.988964i	$-0.547333\pi$
$521$	−5.19251e30	−0.296308	−0.148154	+	0.988964i	$0.547333\pi$
$522$	0	0
$523$	−1.56446e31	−0.854269	−0.427135	−	0.904188i	$-0.640477\pi$
$523$	−1.56446e31	−0.854269	−0.427135	+	0.904188i	$0.640477\pi$
$524$	0	0
$525$	1.30267e30	0.0680772
$526$	0	0
$527$	2.88421e31	1.44280
$528$	0	0
$529$	−1.29487e31	−0.620133
$530$	0	0
$531$	−1.20159e30	−0.0551021
$532$	0	0
$533$	8.78042e30	0.385614
$534$	0	0
$535$	−2.51187e31	−1.05664
$536$	0	0
$537$	−2.06162e31	−0.830817
$538$	0	0
$539$	2.76498e31	1.06763
$540$	0	0
$541$	−3.72094e31	−1.37684	−0.688421	−	0.725311i	$-0.741696\pi$
$541$	−3.72094e31	−1.37684	−0.688421	+	0.725311i	$0.741696\pi$
$542$	0	0
$543$	1.34175e31	0.475853
$544$	0	0
$545$	5.74883e30	0.195443
$546$	0	0
$547$	−7.68045e30	−0.250341	−0.125171	−	0.992135i	$-0.539948\pi$
$547$	−7.68045e30	−0.250341	−0.125171	+	0.992135i	$0.539948\pi$
$548$	0	0
$549$	−7.06509e30	−0.220819
$550$	0	0
$551$	−3.01935e30	−0.0905044
$552$	0	0
$553$	8.98778e30	0.258413
$554$	0	0
$555$	2.33328e30	0.0643573
$556$	0	0
$557$	8.22620e30	0.217703	0.108852	−	0.994058i	$-0.465283\pi$
$557$	8.22620e30	0.217703	0.108852	+	0.994058i	$0.465283\pi$
$558$	0	0
$559$	1.20702e31	0.306534
$560$	0	0
$561$	−4.28971e31	−1.04557
$562$	0	0
$563$	3.40786e31	0.797326	0.398663	−	0.917098i	$-0.369474\pi$
$563$	3.40786e31	0.797326	0.398663	+	0.917098i	$0.369474\pi$
$564$	0	0
$565$	3.28813e31	0.738571
$566$	0	0
$567$	−1.20309e30	−0.0259475
$568$	0	0
$569$	−5.79293e31	−1.19980	−0.599898	−	0.800077i	$-0.704792\pi$
$569$	−5.79293e31	−1.19980	−0.599898	+	0.800077i	$0.704792\pi$
$570$	0	0
$571$	−4.88305e31	−0.971350	−0.485675	−	0.874139i	$-0.661426\pi$
$571$	−4.88305e31	−0.971350	−0.485675	+	0.874139i	$0.661426\pi$
$572$	0	0
$573$	−3.29787e31	−0.630167
$574$	0	0
$575$	1.69520e31	0.311201
$576$	0	0
$577$	−1.00413e32	−1.77120	−0.885600	−	0.464449i	$-0.846252\pi$
$577$	−1.00413e32	−1.77120	−0.885600	+	0.464449i	$0.846252\pi$
$578$	0	0
$579$	−1.63402e31	−0.276983
$580$	0	0
$581$	1.30435e31	0.212505
$582$	0	0
$583$	−1.01753e32	−1.59352
$584$	0	0
$585$	−6.18609e30	−0.0931373
$586$	0	0
$587$	3.03475e31	0.439323	0.219662	−	0.975576i	$-0.429505\pi$
$587$	3.03475e31	0.439323	0.219662	+	0.975576i	$0.429505\pi$
$588$	0	0
$589$	1.31883e31	0.183596
$590$	0	0
$591$	−3.16940e31	−0.424346
$592$	0	0
$593$	−2.03496e31	−0.262075	−0.131038	−	0.991377i	$-0.541831\pi$
$593$	−2.03496e31	−0.262075	−0.131038	+	0.991377i	$0.541831\pi$
$594$	0	0
$595$	−2.12695e31	−0.263520
$596$	0	0
$597$	−7.96000e31	−0.948873
$598$	0	0
$599$	1.22803e32	1.40864	0.704321	−	0.709882i	$-0.251252\pi$
$599$	1.22803e32	1.40864	0.704321	+	0.709882i	$0.251252\pi$
$600$	0	0
$601$	2.96909e31	0.327769	0.163884	−	0.986480i	$-0.447598\pi$
$601$	2.96909e31	0.327769	0.163884	+	0.986480i	$0.447598\pi$
$602$	0	0
$603$	1.22995e31	0.130689
$604$	0	0
$605$	1.89257e31	0.193582
$606$	0	0
$607$	−7.58594e31	−0.747033	−0.373516	−	0.927624i	$-0.621848\pi$
$607$	−7.58594e31	−0.747033	−0.373516	+	0.927624i	$0.621848\pi$
$608$	0	0
$609$	6.30592e30	0.0597929
$610$	0	0
$611$	3.35989e31	0.306797
$612$	0	0
$613$	8.28233e31	0.728378	0.364189	−	0.931325i	$-0.381346\pi$
$613$	8.28233e31	0.728378	0.364189	+	0.931325i	$0.381346\pi$
$614$	0	0
$615$	−4.65675e31	−0.394475
$616$	0	0
$617$	8.74714e31	0.713817	0.356908	−	0.934139i	$-0.383831\pi$
$617$	8.74714e31	0.713817	0.356908	+	0.934139i	$0.383831\pi$
$618$	0	0
$619$	−2.22016e32	−1.74559	−0.872793	−	0.488091i	$-0.837693\pi$
$619$	−2.22016e32	−1.74559	−0.872793	+	0.488091i	$0.837693\pi$
$620$	0	0
$621$	−1.56562e31	−0.118613
$622$	0	0
$623$	2.22427e30	0.0162396
$624$	0	0
$625$	−3.41245e31	−0.240130
$626$	0	0
$627$	−1.96150e31	−0.133049
$628$	0	0
$629$	3.88548e31	0.254075
$630$	0	0
$631$	1.40992e31	0.0888906	0.0444453	−	0.999012i	$-0.485848\pi$
$631$	1.40992e31	0.0888906	0.0444453	+	0.999012i	$0.485848\pi$
$632$	0	0
$633$	−1.49345e32	−0.907920
$634$	0	0
$635$	−7.05438e30	−0.0413581
$636$	0	0
$637$	6.63984e31	0.375452
$638$	0	0
$639$	−7.04952e31	−0.384503
$640$	0	0
$641$	3.15907e32	1.66223	0.831116	−	0.556100i	$-0.187703\pi$
$641$	3.15907e32	1.66223	0.831116	+	0.556100i	$0.187703\pi$
$642$	0	0
$643$	1.98687e32	1.00865	0.504327	−	0.863513i	$-0.331741\pi$
$643$	1.98687e32	1.00865	0.504327	+	0.863513i	$0.331741\pi$
$644$	0	0
$645$	−6.40150e31	−0.313578
$646$	0	0
$647$	−1.47618e32	−0.697814	−0.348907	−	0.937157i	$-0.613447\pi$
$647$	−1.47618e32	−0.697814	−0.348907	+	0.937157i	$0.613447\pi$
$648$	0	0
$649$	4.09148e31	0.186666
$650$	0	0
$651$	−2.75438e31	−0.121295
$652$	0	0
$653$	2.61131e32	1.11009	0.555044	−	0.831821i	$-0.312701\pi$
$653$	2.61131e32	1.11009	0.555044	+	0.831821i	$0.312701\pi$
$654$	0	0
$655$	−4.96359e31	−0.203714
$656$	0	0
$657$	3.77182e31	0.149468
$658$	0	0
$659$	3.91233e32	1.49711	0.748555	−	0.663073i	$-0.230748\pi$
$659$	3.91233e32	1.49711	0.748555	+	0.663073i	$0.230748\pi$
$660$	0	0
$661$	2.94764e32	1.08932	0.544662	−	0.838656i	$-0.316658\pi$
$661$	2.94764e32	1.08932	0.544662	+	0.838656i	$0.316658\pi$
$662$	0	0
$663$	−1.03013e32	−0.367695
$664$	0	0
$665$	−9.72566e30	−0.0335328
$666$	0	0
$667$	8.20608e31	0.273331
$668$	0	0
$669$	2.46494e32	0.793242
$670$	0	0
$671$	2.40570e32	0.748054
$672$	0	0
$673$	−5.79662e32	−1.74181	−0.870907	−	0.491448i	$-0.836468\pi$
$673$	−5.79662e32	−1.74181	−0.870907	+	0.491448i	$0.836468\pi$
$674$	0	0
$675$	−3.34608e31	−0.0971725
$676$	0	0
$677$	4.87131e32	1.36734	0.683669	−	0.729793i	$-0.260383\pi$
$677$	4.87131e32	1.36734	0.683669	+	0.729793i	$0.260383\pi$
$678$	0	0
$679$	−1.34498e32	−0.364933
$680$	0	0
$681$	2.92637e31	0.0767604
$682$	0	0
$683$	3.47690e32	0.881769	0.440884	−	0.897564i	$-0.354665\pi$
$683$	3.47690e32	0.881769	0.440884	+	0.897564i	$0.354665\pi$
$684$	0	0
$685$	−1.73458e32	−0.425356
$686$	0	0
$687$	3.86653e32	0.916893
$688$	0	0
$689$	−2.44350e32	−0.560390
$690$	0	0
$691$	7.86137e32	1.74382	0.871909	−	0.489669i	$-0.162882\pi$
$691$	7.86137e32	1.74382	0.871909	+	0.489669i	$0.162882\pi$
$692$	0	0
$693$	4.09661e31	0.0879007
$694$	0	0
$695$	5.64406e32	1.17156
$696$	0	0
$697$	−7.75461e32	−1.55734
$698$	0	0
$699$	−3.11058e32	−0.604440
$700$	0	0
$701$	8.41904e32	1.58309	0.791545	−	0.611111i	$-0.209277\pi$
$701$	8.41904e32	1.58309	0.791545	+	0.611111i	$0.209277\pi$
$702$	0	0
$703$	1.77666e31	0.0323310
$704$	0	0
$705$	−1.78194e32	−0.313847
$706$	0	0
$707$	−1.36162e32	−0.232131
$708$	0	0
$709$	−4.07757e32	−0.672929	−0.336465	−	0.941696i	$-0.609231\pi$
$709$	−4.07757e32	−0.672929	−0.336465	+	0.941696i	$0.609231\pi$
$710$	0	0
$711$	−2.30864e32	−0.368855
$712$	0	0
$713$	−3.58436e32	−0.554474
$714$	0	0
$715$	2.10640e32	0.315516
$716$	0	0
$717$	2.54827e32	0.369637
$718$	0	0
$719$	1.02178e33	1.43540	0.717701	−	0.696352i	$-0.245195\pi$
$719$	1.02178e33	1.43540	0.717701	+	0.696352i	$0.245195\pi$
$720$	0	0
$721$	8.43477e31	0.114767
$722$	0	0
$723$	−6.39977e32	−0.843479
$724$	0	0
$725$	1.75382e32	0.223923
$726$	0	0
$727$	5.66215e32	0.700383	0.350191	−	0.936678i	$-0.386117\pi$
$727$	5.66215e32	0.700383	0.350191	+	0.936678i	$0.386117\pi$
$728$	0	0
$729$	3.09032e31	0.0370370
$730$	0	0
$731$	−1.06600e33	−1.23797
$732$	0	0
$733$	5.44573e32	0.612859	0.306430	−	0.951893i	$-0.400866\pi$
$733$	5.44573e32	0.612859	0.306430	+	0.951893i	$0.400866\pi$
$734$	0	0
$735$	−3.52148e32	−0.384080
$736$	0	0
$737$	−4.18806e32	−0.442728
$738$	0	0
$739$	−1.30807e31	−0.0134036	−0.00670180	−	0.999978i	$-0.502133\pi$
$739$	−1.30807e31	−0.0134036	−0.00670180	+	0.999978i	$0.502133\pi$
$740$	0	0
$741$	−4.71036e31	−0.0467891
$742$	0	0
$743$	−2.01540e33	−1.94084	−0.970422	−	0.241414i	$-0.922389\pi$
$743$	−2.01540e33	−1.94084	−0.970422	+	0.241414i	$0.922389\pi$
$744$	0	0
$745$	−9.50497e32	−0.887469
$746$	0	0
$747$	−3.35040e32	−0.303326
$748$	0	0
$749$	3.99458e32	0.350695
$750$	0	0
$751$	1.65714e32	0.141091	0.0705456	−	0.997509i	$-0.477526\pi$
$751$	1.65714e32	0.141091	0.0705456	+	0.997509i	$0.477526\pi$
$752$	0	0
$753$	−6.09256e32	−0.503104
$754$	0	0
$755$	−9.87907e32	−0.791272
$756$	0	0
$757$	−3.14060e32	−0.244012	−0.122006	−	0.992529i	$-0.538933\pi$
$757$	−3.14060e32	−0.244012	−0.122006	+	0.992529i	$0.538933\pi$
$758$	0	0
$759$	5.33104e32	0.401820
$760$	0	0
$761$	−5.10481e32	−0.373299	−0.186649	−	0.982427i	$-0.559763\pi$
$761$	−5.10481e32	−0.373299	−0.186649	+	0.982427i	$0.559763\pi$
$762$	0	0
$763$	−9.14223e31	−0.0648665
$764$	0	0
$765$	5.46338e32	0.376144
$766$	0	0
$767$	9.82528e31	0.0656445
$768$	0	0
$769$	−1.46960e33	−0.952899	−0.476449	−	0.879202i	$-0.658076\pi$
$769$	−1.46960e33	−0.952899	−0.476449	+	0.879202i	$0.658076\pi$
$770$	0	0
$771$	−3.37886e32	−0.212640
$772$	0	0
$773$	3.12910e33	1.91142	0.955709	−	0.294313i	$-0.0950910\pi$
$773$	3.12910e33	1.91142	0.955709	+	0.294313i	$0.0950910\pi$
$774$	0	0
$775$	−7.66056e32	−0.454246
$776$	0	0
$777$	−3.71057e31	−0.0213599
$778$	0	0
$779$	−3.54586e32	−0.198171
$780$	0	0
$781$	2.40040e33	1.30256
$782$	0	0
$783$	−1.61976e32	−0.0853476
$784$	0	0
$785$	−1.96520e33	−1.00556
$786$	0	0
$787$	2.60994e32	0.129695	0.0648473	−	0.997895i	$-0.479344\pi$
$787$	2.60994e32	0.129695	0.0648473	+	0.997895i	$0.479344\pi$
$788$	0	0
$789$	1.96672e33	0.949199
$790$	0	0
$791$	−5.22905e32	−0.245128
$792$	0	0
$793$	5.77706e32	0.263066
$794$	0	0
$795$	1.29592e33	0.573268
$796$	0	0
$797$	−2.78741e33	−1.19793	−0.598964	−	0.800776i	$-0.704421\pi$
$797$	−2.78741e33	−1.19793	−0.598964	+	0.800776i	$0.704421\pi$
$798$	0	0
$799$	−2.96735e33	−1.23903
$800$	0	0
$801$	−5.71335e31	−0.0231802
$802$	0	0
$803$	−1.28433e33	−0.506345
$804$	0	0
$805$	2.64327e32	0.101272
$806$	0	0
$807$	−1.12314e33	−0.418203
$808$	0	0
$809$	−2.12201e33	−0.767962	−0.383981	−	0.923341i	$-0.625447\pi$
$809$	−2.12201e33	−0.767962	−0.383981	+	0.923341i	$0.625447\pi$
$810$	0	0
$811$	−3.83715e33	−1.34980	−0.674900	−	0.737909i	$-0.735813\pi$
$811$	−3.83715e33	−1.34980	−0.674900	+	0.737909i	$0.735813\pi$
$812$	0	0
$813$	−2.85980e33	−0.977899
$814$	0	0
$815$	−7.27352e32	−0.241787
$816$	0	0
$817$	−4.87439e32	−0.157531
$818$	0	0
$819$	9.83761e31	0.0309118
$820$	0	0
$821$	−3.04214e32	−0.0929463	−0.0464732	−	0.998920i	$-0.514798\pi$
$821$	−3.04214e32	−0.0929463	−0.0464732	+	0.998920i	$0.514798\pi$
$822$	0	0
$823$	5.54831e33	1.64840	0.824200	−	0.566299i	$-0.191625\pi$
$823$	5.54831e33	1.64840	0.824200	+	0.566299i	$0.191625\pi$
$824$	0	0
$825$	1.13936e33	0.329186
$826$	0	0
$827$	1.88657e33	0.530102	0.265051	−	0.964234i	$-0.414611\pi$
$827$	1.88657e33	0.530102	0.265051	+	0.964234i	$0.414611\pi$
$828$	0	0
$829$	1.16951e33	0.319615	0.159808	−	0.987148i	$-0.448913\pi$
$829$	1.16951e33	0.319615	0.159808	+	0.987148i	$0.448913\pi$
$830$	0	0
$831$	2.80439e33	0.745464
$832$	0	0
$833$	−5.86411e33	−1.51630
$834$	0	0
$835$	−4.49945e32	−0.113179
$836$	0	0
$837$	7.07501e32	0.173135
$838$	0	0
$839$	−4.39094e32	−0.104543	−0.0522715	−	0.998633i	$-0.516646\pi$
$839$	−4.39094e32	−0.104543	−0.0522715	+	0.998633i	$0.516646\pi$
$840$	0	0
$841$	−3.46774e33	−0.803326
$842$	0	0
$843$	3.20999e33	0.723580
$844$	0	0
$845$	−2.70183e33	−0.592660
$846$	0	0
$847$	−3.00972e32	−0.0642490
$848$	0	0
$849$	−2.86051e33	−0.594298
$850$	0	0
$851$	−4.82867e32	−0.0976423
$852$	0	0
$853$	5.99154e33	1.17930	0.589650	−	0.807659i	$-0.299266\pi$
$853$	5.99154e33	1.17930	0.589650	+	0.807659i	$0.299266\pi$
$854$	0	0
$855$	2.49817e32	0.0478643
$856$	0	0
$857$	−4.62186e33	−0.862061	−0.431030	−	0.902337i	$-0.641850\pi$
$857$	−4.62186e33	−0.862061	−0.431030	+	0.902337i	$0.641850\pi$
$858$	0	0
$859$	2.53686e32	0.0460654	0.0230327	−	0.999735i	$-0.492668\pi$
$859$	2.53686e32	0.0460654	0.0230327	+	0.999735i	$0.492668\pi$
$860$	0	0
$861$	7.40554e32	0.130924
$862$	0	0
$863$	1.06204e33	0.182818	0.0914089	−	0.995813i	$-0.470863\pi$
$863$	1.06204e33	0.182818	0.0914089	+	0.995813i	$0.470863\pi$
$864$	0	0
$865$	−2.35160e33	−0.394165
$866$	0	0
$867$	5.56064e33	0.907621
$868$	0	0
$869$	7.86104e33	1.24955
$870$	0	0
$871$	−1.00572e33	−0.155693
$872$	0	0
$873$	3.45477e33	0.520900
$874$	0	0
$875$	1.68376e33	0.247279
$876$	0	0
$877$	1.34383e34	1.92242	0.961209	−	0.275821i	$-0.0889497\pi$
$877$	1.34383e34	1.92242	0.961209	+	0.275821i	$0.0889497\pi$
$878$	0	0
$879$	6.89048e33	0.960234
$880$	0	0
$881$	−5.30079e33	−0.719644	−0.359822	−	0.933021i	$-0.617163\pi$
$881$	−5.30079e33	−0.719644	−0.359822	+	0.933021i	$0.617163\pi$
$882$	0	0
$883$	6.44487e33	0.852444	0.426222	−	0.904618i	$-0.359844\pi$
$883$	6.44487e33	0.852444	0.426222	+	0.904618i	$0.359844\pi$
$884$	0	0
$885$	−5.21090e32	−0.0671530
$886$	0	0
$887$	7.46152e33	0.936927	0.468463	−	0.883483i	$-0.344808\pi$
$887$	7.46152e33	0.936927	0.468463	+	0.883483i	$0.344808\pi$
$888$	0	0
$889$	1.12184e32	0.0137266
$890$	0	0
$891$	−1.05227e33	−0.125468
$892$	0	0
$893$	−1.35685e33	−0.157666
$894$	0	0
$895$	−8.94061e33	−1.01252
$896$	0	0
$897$	1.28020e33	0.141307
$898$	0	0
$899$	−3.70830e33	−0.398969
$900$	0	0
$901$	2.15802e34	2.26319
$902$	0	0
$903$	1.01802e33	0.104075
$904$	0	0
$905$	5.81873e33	0.579922
$906$	0	0
$907$	6.80142e33	0.660870	0.330435	−	0.943829i	$-0.392805\pi$
$907$	6.80142e33	0.660870	0.330435	+	0.943829i	$0.392805\pi$
$908$	0	0
$909$	3.49752e33	0.331341
$910$	0	0
$911$	−1.66289e34	−1.53603	−0.768017	−	0.640429i	$-0.778757\pi$
$911$	−1.66289e34	−1.53603	−0.768017	+	0.640429i	$0.778757\pi$
$912$	0	0
$913$	1.14083e34	1.02756
$914$	0	0
$915$	−3.06390e33	−0.269112
$916$	0	0
$917$	7.89350e32	0.0676117
$918$	0	0
$919$	4.33927e33	0.362483	0.181242	−	0.983439i	$-0.441988\pi$
$919$	4.33927e33	0.362483	0.181242	+	0.983439i	$0.441988\pi$
$920$	0	0
$921$	−2.83236e33	−0.230761
$922$	0	0
$923$	5.76433e33	0.458068
$924$	0	0
$925$	−1.03199e33	−0.0799922
$926$	0	0
$927$	−2.16659e33	−0.163817
$928$	0	0
$929$	−1.59855e34	−1.17909	−0.589544	−	0.807736i	$-0.700693\pi$
$929$	−1.59855e34	−1.17909	−0.589544	+	0.807736i	$0.700693\pi$
$930$	0	0
$931$	−2.68141e33	−0.192949
$932$	0	0
$933$	1.05130e34	0.738053
$934$	0	0
$935$	−1.86031e34	−1.27424
$936$	0	0
$937$	3.33418e33	0.222835	0.111418	−	0.993774i	$-0.464461\pi$
$937$	3.33418e33	0.222835	0.111418	+	0.993774i	$0.464461\pi$
$938$	0	0
$939$	1.68092e34	1.09621
$940$	0	0
$941$	7.67430e33	0.488381	0.244190	−	0.969727i	$-0.421478\pi$
$941$	7.67430e33	0.488381	0.244190	+	0.969727i	$0.421478\pi$
$942$	0	0
$943$	9.63704e33	0.598493
$944$	0	0
$945$	−5.21744e32	−0.0316222
$946$	0	0
$947$	−9.39902e33	−0.555978	−0.277989	−	0.960584i	$-0.589668\pi$
$947$	−9.39902e33	−0.555978	−0.277989	+	0.960584i	$0.589668\pi$
$948$	0	0
$949$	−3.08418e33	−0.178065
$950$	0	0
$951$	−9.05916e33	−0.510519
$952$	0	0
$953$	−2.17432e34	−1.19606	−0.598032	−	0.801472i	$-0.704050\pi$
$953$	−2.17432e34	−1.19606	−0.598032	+	0.801472i	$0.704050\pi$
$954$	0	0
$955$	−1.43018e34	−0.767985
$956$	0	0
$957$	5.51539e33	0.289127
$958$	0	0
$959$	2.75847e33	0.141174
$960$	0	0
$961$	−3.81570e33	−0.190658
$962$	0	0
$963$	−1.02606e34	−0.500578
$964$	0	0
$965$	−7.08621e33	−0.337559
$966$	0	0
$967$	3.22790e34	1.50146	0.750732	−	0.660607i	$-0.229701\pi$
$967$	3.22790e34	1.50146	0.750732	+	0.660607i	$0.229701\pi$
$968$	0	0
$969$	4.16005e33	0.188962
$970$	0	0
$971$	1.08922e34	0.483162	0.241581	−	0.970381i	$-0.422334\pi$
$971$	1.08922e34	0.483162	0.241581	+	0.970381i	$0.422334\pi$
$972$	0	0
$973$	−8.97562e33	−0.388836
$974$	0	0
$975$	2.73606e33	0.115764
$976$	0	0
$977$	3.09917e33	0.128073	0.0640365	−	0.997948i	$-0.479603\pi$
$977$	3.09917e33	0.128073	0.0640365	+	0.997948i	$0.479603\pi$
$978$	0	0
$979$	1.94543e33	0.0785262
$980$	0	0
$981$	2.34831e33	0.0925895
$982$	0	0
$983$	−3.58644e34	−1.38133	−0.690665	−	0.723175i	$-0.742682\pi$
$983$	−3.58644e34	−1.38133	−0.690665	+	0.723175i	$0.742682\pi$
$984$	0	0
$985$	−1.37447e34	−0.517151
$986$	0	0
$987$	2.83378e33	0.104164
$988$	0	0
$989$	1.32478e34	0.475757
$990$	0	0
$991$	1.99976e34	0.701667	0.350833	−	0.936438i	$-0.385898\pi$
$991$	1.99976e34	0.701667	0.350833	+	0.936438i	$0.385898\pi$
$992$	0	0
$993$	−1.47051e34	−0.504141
$994$	0	0
$995$	−3.45200e34	−1.15639
$996$	0	0
$997$	−3.69076e34	−1.20815	−0.604076	−	0.796927i	$-0.706458\pi$
$997$	−3.69076e34	−1.20815	−0.604076	+	0.796927i	$0.706458\pi$
$998$	0	0
$999$	9.53111e32	0.0304888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.24.a.l.1.3		3
4.3	odd	2		24.24.a.b.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.24.a.b.1.3	✓	3	4.3	odd	2
48.24.a.l.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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

\(f(q)\)	\(=\)	\(q+177147. q^{3} +7.68230e7 q^{5} -1.22170e9 q^{7} +3.13811e10 q^{9} +O(q^{10})\)	\(q+177147. q^{3} +7.68230e7 q^{5} -1.22170e9 q^{7} +3.13811e10 q^{9} -1.06854e12 q^{11} -2.56600e12 q^{13} +1.36090e13 q^{15} +2.26622e14 q^{17} +1.03625e14 q^{19} -2.16421e14 q^{21} -2.81634e15 q^{23} -6.01915e15 q^{25} +5.55906e15 q^{27} -2.91373e16 q^{29} +1.27270e17 q^{31} -1.89289e17 q^{33} -9.38547e16 q^{35} +1.71452e17 q^{37} -4.54560e17 q^{39} -3.42183e18 q^{41} -4.70389e18 q^{43} +2.41079e18 q^{45} -1.30939e19 q^{47} -2.58762e19 q^{49} +4.01454e19 q^{51} +9.52257e19 q^{53} -8.20887e19 q^{55} +1.83568e19 q^{57} -3.82902e19 q^{59} -2.25139e20 q^{61} -3.83382e19 q^{63} -1.97128e20 q^{65} +3.91941e20 q^{67} -4.98907e20 q^{69} -2.24642e21 q^{71} +1.20194e21 q^{73} -1.06627e21 q^{75} +1.30544e21 q^{77} -7.35678e21 q^{79} +9.84771e20 q^{81} -1.06765e22 q^{83} +1.74098e22 q^{85} -5.16159e21 q^{87} -1.82064e21 q^{89} +3.13489e21 q^{91} +2.25455e22 q^{93} +7.96076e21 q^{95} +1.10091e23 q^{97} -3.35320e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9}+O(q^{10}) \) 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9	\( 3 q + 531441 q^{3} - 77225166 q^{5} - 3340627872 q^{7} + 94143178827 q^{9} + 66881746140 q^{11} + 5138719820610 q^{13} - 13680206481402 q^{15} - 232964235839898 q^{17} + 39051643356612 q^{19} - 591782205641184 q^{21} + 43\!\cdots\!64 q^{23}+ \cdots + 20\!\cdots\!60 q^{99}+O(q^{100}) \) 3 * q + 531441 * q^3 - 77225166 * q^5 - 3340627872 * q^7 + 94143178827 * q^9 + 66881746140 * q^11 + 5138719820610 * q^13 - 13680206481402 * q^15 - 232964235839898 * q^17 + 39051643356612 * q^19 - 591782205641184 * q^21 + 4358324846210664 * q^23 - 5164619678396379 * q^25 + 16677181699666569 * q^27 - 25471683594266694 * q^29 + 125107985305907448 * q^31 + 11847900683462580 * q^33 + 1311200449040041536 * q^35 - 1292646678108250614 * q^37 + 910308800061599670 * q^39 + 144934201638163518 * q^41 - 8133920080752699684 * q^43 - 2423407537560920094 * q^45 - 10029621278906529216 * q^47 + 41823513653599245723 * q^49 - 41268915486330411006 * q^51 - 28251334125947614062 * q^53 - 85062431363722250328 * q^55 + 6917881465693745964 * q^57 - 187335386683361480100 * q^59 - 555296430938057984862 * q^61 - 104832442382718822048 * q^63 - 1392409010623622551476 * q^65 - 1602075967132026564156 * q^67 + 772064171531680495608 * q^69 - 3528346267933953055080 * q^71 + 2100403635635469857646 * q^73 - 914896882168883350713 * q^75 + 8279040903546572168064 * q^77 - 12175210588212413874744 * q^79 + 2954312706550833698643 * q^81 - 30717412717926068652348 * q^83 + 44709116429093587923108 * q^85 - 4512232333673562042018 * q^87 + 58208079477660202949598 * q^89 - 63276323944720307992512 * q^91 + 22162504272985586690856 * q^93 - 79678214445310528084968 * q^95 + 76406389646793178923846 * q^97 + 2098820062373345659260 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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