LMFDB - Embedded newform 48.13.g.b.31.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,13,Mod(31,48)]

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([1, 0, 0]))

N = Newforms(chi, 13, names="a")

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

chi := DirichletCharacter("48.31");

S:= CuspForms(chi, 13);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 13 $
Character orbit:	$[\chi]$	$=$	48.g (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$43.8717032293$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-2803})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 700x^{2} - 701x + 491401 $ x^4 - x^3 - 700x^2 - 701x + 491401
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10}\cdot 3^{11} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			31.4
Root			$23.1751 + 12.8028i$ of defining polynomial
Character	$\chi$	$=$	48.31
Dual form			48.13.g.b.31.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+420.888i q^{3} +25405.3 q^{5} +30429.0i q^{7} -177147. q^{9} +O(q^{10})$	$q+420.888i q^{3} +25405.3 q^{5} +30429.0i q^{7} -177147. q^{9} +3.28283e6i q^{11} +2.58302e6 q^{13} +1.06928e7i q^{15} +1.46112e7 q^{17} -5.30897e7i q^{19} -1.28072e7 q^{21} -6.24661e7i q^{23} +4.01290e8 q^{25} -7.45591e7i q^{27} -5.58145e8 q^{29} +1.58746e9i q^{31} -1.38171e9 q^{33} +7.73060e8i q^{35} +3.20862e9 q^{37} +1.08716e9i q^{39} -4.72945e9 q^{41} -5.17604e9i q^{43} -4.50048e9 q^{45} +1.61120e10i q^{47} +1.29154e10 q^{49} +6.14968e9i q^{51} +1.07668e10 q^{53} +8.34014e10i q^{55} +2.23448e10 q^{57} +4.85149e10i q^{59} -8.18569e10 q^{61} -5.39041e9i q^{63} +6.56225e10 q^{65} +1.06360e10i q^{67} +2.62913e10 q^{69} +7.93092e10i q^{71} -1.63102e11 q^{73} +1.68898e11i q^{75} -9.98934e10 q^{77} +1.50767e11i q^{79} +3.13811e10 q^{81} +3.77720e11i q^{83} +3.71202e11 q^{85} -2.34917e11i q^{87} +3.51850e11 q^{89} +7.85989e10i q^{91} -6.68142e11 q^{93} -1.34876e12i q^{95} +7.62288e11 q^{97} -5.81544e11i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 22392 q^{5} - 708588 q^{9}+O(q^{10}) $ 4 * q + 22392 * q^5 - 708588 * q^9	$ 4 q + 22392 q^{5} - 708588 q^{9} + 8905960 q^{13} + 72547560 q^{17} + 45034704 q^{21} + 718109132 q^{25} - 728411112 q^{29} - 1714782960 q^{33} - 2724573880 q^{37} - 7164120600 q^{41} - 3966675624 q^{45} + 39425266372 q^{49} + 32619271320 q^{53} + 33276867120 q^{57} - 130736241784 q^{61} + 78103345968 q^{65} - 73307680128 q^{69} - 170497449080 q^{73} - 408892017600 q^{77} + 125524238436 q^{81} + 126782064432 q^{85} + 2012222389896 q^{89} - 935876261520 q^{93} + 1491418808200 q^{97}+O(q^{100}) $ 4 * q + 22392 * q^5 - 708588 * q^9 + 8905960 * q^13 + 72547560 * q^17 + 45034704 * q^21 + 718109132 * q^25 - 728411112 * q^29 - 1714782960 * q^33 - 2724573880 * q^37 - 7164120600 * q^41 - 3966675624 * q^45 + 39425266372 * q^49 + 32619271320 * q^53 + 33276867120 * q^57 - 130736241784 * q^61 + 78103345968 * q^65 - 73307680128 * q^69 - 170497449080 * q^73 - 408892017600 * q^77 + 125524238436 * q^81 + 126782064432 * q^85 + 2012222389896 * q^89 - 935876261520 * q^93 + 1491418808200 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/48\mathbb{Z}\right)^\times$.

$n$	$17$	$31$	$37$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	420.888i	0.577350i
$3$	420.888i	0.577350i
$4$	0	0
$5$	25405.3	1.62594	0.812971	−	0.582305i	$-0.197849\pi$
$5$	25405.3	1.62594	0.812971	+	0.582305i	$0.197849\pi$
$6$	0	0
$7$	30429.0i	0.258643i	0.991603	+	0.129321i	$0.0412799\pi$
$7$	30429.0i	0.258643i	−0.991603	+	0.129321i	$0.958720\pi$
$8$	0	0
$9$	−177147.	−0.333333
$10$	0	0
$11$	3.28283e6i	1.85307i	0.376205	+	0.926536i	$0.377229\pi$
$11$	3.28283e6i	1.85307i	−0.376205	+	0.926536i	$0.622771\pi$
$12$	0	0
$13$	2.58302e6	0.535141	0.267570	−	0.963538i	$-0.413779\pi$
$13$	2.58302e6	0.535141	0.267570	+	0.963538i	$0.413779\pi$
$14$	0	0
$15$	1.06928e7i	0.938738i
$16$	0	0
$17$	1.46112e7	0.605330	0.302665	−	0.953097i	$-0.402124\pi$
$17$	1.46112e7	0.605330	0.302665	+	0.953097i	$0.402124\pi$
$18$	0	0
$19$	− 5.30897e7i	− 1.12847i	−0.825616	−	0.564233i	$-0.809172\pi$
$19$	− 5.30897e7i	− 1.12847i	0.825616	−	0.564233i	$-0.190828\pi$
$20$	0	0
$21$	−1.28072e7	−0.149327
$22$	0	0
$23$	− 6.24661e7i	− 0.421966i	−0.977490	−	0.210983i	$-0.932334\pi$
$23$	− 6.24661e7i	− 0.421966i	0.977490	−	0.210983i	$-0.0676665\pi$
$24$	0	0
$25$	4.01290e8	1.64368
$26$	0	0
$27$	− 7.45591e7i	− 0.192450i
$28$	0	0
$29$	−5.58145e8	−0.938337	−0.469169	−	0.883109i	$-0.655446\pi$
$29$	−5.58145e8	−0.938337	−0.469169	+	0.883109i	$0.655446\pi$
$30$	0	0
$31$	1.58746e9i	1.78868i	0.447392	+	0.894338i	$0.352353\pi$
$31$	1.58746e9i	1.78868i	−0.447392	+	0.894338i	$0.647647\pi$
$32$	0	0
$33$	−1.38171e9	−1.06987
$34$	0	0
$35$	7.73060e8i	0.420538i
$36$	0	0
$37$	3.20862e9	1.25057	0.625285	−	0.780397i	$-0.284983\pi$
$37$	3.20862e9	1.25057	0.625285	+	0.780397i	$0.284983\pi$
$38$	0	0
$39$	1.08716e9i	0.308964i
$40$	0	0
$41$	−4.72945e9	−0.995651	−0.497826	−	0.867277i	$-0.665868\pi$
$41$	−4.72945e9	−0.995651	−0.497826	+	0.867277i	$0.665868\pi$
$42$	0	0
$43$	− 5.17604e9i	− 0.818817i	−0.912351	−	0.409408i	$-0.865735\pi$
$43$	− 5.17604e9i	− 0.818817i	0.912351	−	0.409408i	$-0.134265\pi$
$44$	0	0
$45$	−4.50048e9	−0.541980
$46$	0	0
$47$	1.61120e10i	1.49473i	0.664413	+	0.747365i	$0.268681\pi$
$47$	1.61120e10i	1.49473i	−0.664413	+	0.747365i	$0.731319\pi$
$48$	0	0
$49$	1.29154e10	0.933104
$50$	0	0
$51$	6.14968e9i	0.349487i
$52$	0	0
$53$	1.07668e10	0.485772	0.242886	−	0.970055i	$-0.421906\pi$
$53$	1.07668e10	0.485772	0.242886	+	0.970055i	$0.421906\pi$
$54$	0	0
$55$	8.34014e10i	3.01299i
$56$	0	0
$57$	2.23448e10	0.651520
$58$	0	0
$59$	4.85149e10i	1.15017i	0.818092	+	0.575087i	$0.195032\pi$
$59$	4.85149e10i	1.15017i	−0.818092	+	0.575087i	$0.804968\pi$
$60$	0	0
$61$	−8.18569e10	−1.58883	−0.794413	−	0.607378i	$-0.792222\pi$
$61$	−8.18569e10	−1.58883	−0.794413	+	0.607378i	$0.792222\pi$
$62$	0	0
$63$	− 5.39041e9i	− 0.0862142i
$64$	0	0
$65$	6.56225e10	0.870107
$66$	0	0
$67$	1.06360e10i	0.117579i	0.998270	+	0.0587895i	$0.0187241\pi$
$67$	1.06360e10i	0.117579i	−0.998270	+	0.0587895i	$0.981276\pi$
$68$	0	0
$69$	2.62913e10	0.243622
$70$	0	0
$71$	7.93092e10i	0.619118i	0.950880	+	0.309559i	$0.100181\pi$
$71$	7.93092e10i	0.619118i	−0.950880	+	0.309559i	$0.899819\pi$
$72$	0	0
$73$	−1.63102e11	−1.07776	−0.538881	−	0.842382i	$-0.681153\pi$
$73$	−1.63102e11	−1.07776	−0.538881	+	0.842382i	$0.681153\pi$
$74$	0	0
$75$	1.68898e11i	0.948982i
$76$	0	0
$77$	−9.98934e10	−0.479284
$78$	0	0
$79$	1.50767e11i	0.620217i	0.950701	+	0.310109i	$0.100365\pi$
$79$	1.50767e11i	0.620217i	−0.950701	+	0.310109i	$0.899635\pi$
$80$	0	0
$81$	3.13811e10	0.111111
$82$	0	0
$83$	3.77720e11i	1.15532i	0.816278	+	0.577659i	$0.196034\pi$
$83$	3.77720e11i	1.15532i	−0.816278	+	0.577659i	$0.803966\pi$
$84$	0	0
$85$	3.71202e11	0.984230
$86$	0	0
$87$	− 2.34917e11i	− 0.541749i
$88$	0	0
$89$	3.51850e11	0.707974	0.353987	−	0.935250i	$-0.384826\pi$
$89$	3.51850e11	0.707974	0.353987	+	0.935250i	$0.384826\pi$
$90$	0	0
$91$	7.85989e10i	0.138410i
$92$	0	0
$93$	−6.68142e11	−1.03269
$94$	0	0
$95$	− 1.34876e12i	− 1.83482i
$96$	0	0
$97$	7.62288e11	0.915143	0.457571	−	0.889173i	$-0.348719\pi$
$97$	7.62288e11	0.915143	0.457571	+	0.889173i	$0.348719\pi$
$98$	0	0
$99$	− 5.81544e11i	− 0.617691i
$100$	0	0
$101$	−6.15857e11	−0.580165	−0.290083	−	0.957002i	$-0.593683\pi$
$101$	−6.15857e11	−0.580165	−0.290083	+	0.957002i	$0.593683\pi$
$102$	0	0
$103$	− 1.96625e12i	− 1.64670i	−0.567531	−	0.823352i	$-0.692101\pi$
$103$	− 1.96625e12i	− 1.64670i	0.567531	−	0.823352i	$-0.307899\pi$
$104$	0	0
$105$	−3.25372e11	−0.242798
$106$	0	0
$107$	− 8.12975e11i	− 0.541719i	−0.962619	−	0.270860i	$-0.912692\pi$
$107$	− 8.12975e11i	− 0.541719i	0.962619	−	0.270860i	$-0.0873079\pi$
$108$	0	0
$109$	1.01705e11	0.0606433	0.0303217	−	0.999540i	$-0.490347\pi$
$109$	1.01705e11	0.0606433	0.0303217	+	0.999540i	$0.490347\pi$
$110$	0	0
$111$	1.35047e12i	0.722017i
$112$	0	0
$113$	1.43265e12	0.688128	0.344064	−	0.938946i	$-0.388196\pi$
$113$	1.43265e12	0.688128	0.344064	+	0.938946i	$0.388196\pi$
$114$	0	0
$115$	− 1.58697e12i	− 0.686092i
$116$	0	0
$117$	−4.57575e11	−0.178380
$118$	0	0
$119$	4.44604e11i	0.156564i
$120$	0	0
$121$	−7.63856e12	−2.43388
$122$	0	0
$123$	− 1.99057e12i	− 0.574840i
$124$	0	0
$125$	3.99244e12	1.04659
$126$	0	0
$127$	− 1.29868e12i	− 0.309513i	−0.987953	−	0.154757i	$-0.950541\pi$
$127$	− 1.29868e12i	− 0.309513i	0.987953	−	0.154757i	$-0.0494594\pi$
$128$	0	0
$129$	2.17853e12	0.472744
$130$	0	0
$131$	− 8.55265e12i	− 1.69228i	−0.532959	−	0.846141i	$-0.678920\pi$
$131$	− 8.55265e12i	− 1.69228i	0.532959	−	0.846141i	$-0.321080\pi$
$132$	0	0
$133$	1.61547e12	0.291869
$134$	0	0
$135$	− 1.89420e12i	− 0.312913i
$136$	0	0
$137$	1.31315e13	1.98605	0.993026	−	0.117894i	$-0.0376144\pi$
$137$	1.31315e13	1.98605	0.993026	+	0.117894i	$0.0376144\pi$
$138$	0	0
$139$	7.03741e12i	0.975718i	0.872922	+	0.487859i	$0.162222\pi$
$139$	7.03741e12i	0.975718i	−0.872922	+	0.487859i	$0.837778\pi$
$140$	0	0
$141$	−6.78136e12	−0.862983
$142$	0	0
$143$	8.47963e12i	0.991655i
$144$	0	0
$145$	−1.41799e13	−1.52568
$146$	0	0
$147$	5.43592e12i	0.538728i
$148$	0	0
$149$	4.92128e12	0.449739	0.224869	−	0.974389i	$-0.427804\pi$
$149$	4.92128e12	0.449739	0.224869	+	0.974389i	$0.427804\pi$
$150$	0	0
$151$	4.04020e12i	0.340833i	0.985372	+	0.170416i	$0.0545113\pi$
$151$	4.04020e12i	0.340833i	−0.985372	+	0.170416i	$0.945489\pi$
$152$	0	0
$153$	−2.58833e12	−0.201777
$154$	0	0
$155$	4.03299e13i	2.90828i
$156$	0	0
$157$	1.84621e13	1.23277	0.616386	−	0.787444i	$-0.288596\pi$
$157$	1.84621e13	1.23277	0.616386	+	0.787444i	$0.288596\pi$
$158$	0	0
$159$	4.53163e12i	0.280461i
$160$	0	0
$161$	1.90078e12	0.109138
$162$	0	0
$163$	8.20492e12i	0.437470i	0.975784	+	0.218735i	$0.0701931\pi$
$163$	8.20492e12i	0.437470i	−0.975784	+	0.218735i	$0.929807\pi$
$164$	0	0
$165$	−3.51027e13	−1.73955
$166$	0	0
$167$	− 3.50541e13i	− 1.61599i	−0.589187	−	0.807997i	$-0.700552\pi$
$167$	− 3.50541e13i	− 1.61599i	0.589187	−	0.807997i	$-0.299448\pi$
$168$	0	0
$169$	−1.66261e13	−0.713624
$170$	0	0
$171$	9.40468e12i	0.376155i
$172$	0	0
$173$	3.02565e13	1.12861	0.564303	−	0.825568i	$-0.309145\pi$
$173$	3.02565e13	1.12861	0.564303	+	0.825568i	$0.309145\pi$
$174$	0	0
$175$	1.22109e13i	0.425127i
$176$	0	0
$177$	−2.04194e13	−0.664053
$178$	0	0
$179$	1.66584e13i	0.506425i	0.967411	+	0.253212i	$0.0814871\pi$
$179$	1.66584e13i	0.506425i	−0.967411	+	0.253212i	$0.918513\pi$
$180$	0	0
$181$	−2.11600e12	−0.0601789	−0.0300895	−	0.999547i	$-0.509579\pi$
$181$	−2.11600e12	−0.0601789	−0.0300895	+	0.999547i	$0.509579\pi$
$182$	0	0
$183$	− 3.44526e13i	− 0.917310i
$184$	0	0
$185$	8.15160e13	2.03335
$186$	0	0
$187$	4.79661e13i	1.12172i
$188$	0	0
$189$	2.26876e12	0.0497758
$190$	0	0
$191$	6.49884e11i	0.0133855i	0.999978	+	0.00669276i	$0.00213039\pi$
$191$	6.49884e11i	0.0133855i	−0.999978	+	0.00669276i	$0.997870\pi$
$192$	0	0
$193$	−6.13766e13	−1.18757	−0.593784	−	0.804624i	$-0.702367\pi$
$193$	−6.13766e13	−1.18757	−0.593784	+	0.804624i	$0.702367\pi$
$194$	0	0
$195$	2.76198e13i	0.502357i
$196$	0	0
$197$	3.94508e13	0.674929	0.337465	−	0.941338i	$-0.390431\pi$
$197$	3.94508e13	0.674929	0.337465	+	0.941338i	$0.390431\pi$
$198$	0	0
$199$	− 9.27979e13i	− 1.49424i	−0.664690	−	0.747119i	$-0.731437\pi$
$199$	− 9.27979e13i	− 1.49424i	0.664690	−	0.747119i	$-0.268563\pi$
$200$	0	0
$201$	−4.47657e12	−0.0678842
$202$	0	0
$203$	− 1.69838e13i	− 0.242694i
$204$	0	0
$205$	−1.20153e14	−1.61887
$206$	0	0
$207$	1.10657e13i	0.140655i
$208$	0	0
$209$	1.74284e14	2.09113
$210$	0	0
$211$	− 1.36693e14i	− 1.54901i	−0.632570	−	0.774503i	$-0.718000\pi$
$211$	− 1.36693e14i	− 1.54901i	0.632570	−	0.774503i	$-0.282000\pi$
$212$	0	0
$213$	−3.33803e13	−0.357448
$214$	0	0
$215$	− 1.31499e14i	− 1.33135i
$216$	0	0
$217$	−4.83048e13	−0.462628
$218$	0	0
$219$	− 6.86478e13i	− 0.622246i
$220$	0	0
$221$	3.77410e13	0.323937
$222$	0	0
$223$	− 1.47042e14i	− 1.19567i	−0.801618	−	0.597837i	$-0.796027\pi$
$223$	− 1.47042e14i	− 1.19567i	0.801618	−	0.597837i	$-0.203973\pi$
$224$	0	0
$225$	−7.10873e13	−0.547895
$226$	0	0
$227$	− 9.06590e13i	− 0.662607i	−0.943524	−	0.331303i	$-0.892512\pi$
$227$	− 9.06590e13i	− 0.662607i	0.943524	−	0.331303i	$-0.107488\pi$
$228$	0	0
$229$	−1.09436e14	−0.758836	−0.379418	−	0.925225i	$-0.623876\pi$
$229$	−1.09436e14	−0.758836	−0.379418	+	0.925225i	$0.623876\pi$
$230$	0	0
$231$	− 4.20440e13i	− 0.276715i
$232$	0	0
$233$	−1.18077e14	−0.737956	−0.368978	−	0.929438i	$-0.620292\pi$
$233$	−1.18077e14	−0.737956	−0.368978	+	0.929438i	$0.620292\pi$
$234$	0	0
$235$	4.09331e14i	2.43034i
$236$	0	0
$237$	−6.34561e13	−0.358083
$238$	0	0
$239$	6.10449e13i	0.327538i	0.986499	+	0.163769i	$0.0523652\pi$
$239$	6.10449e13i	0.327538i	−0.986499	+	0.163769i	$0.947635\pi$
$240$	0	0
$241$	2.96421e14	1.51289	0.756443	−	0.654059i	$-0.226935\pi$
$241$	2.96421e14	1.51289	0.756443	+	0.654059i	$0.226935\pi$
$242$	0	0
$243$	1.32079e13i	0.0641500i
$244$	0	0
$245$	3.28119e14	1.51717
$246$	0	0
$247$	− 1.37132e14i	− 0.603888i
$248$	0	0
$249$	−1.58978e14	−0.667023
$250$	0	0
$251$	2.47270e14i	0.988848i	0.869221	+	0.494424i	$0.164621\pi$
$251$	2.47270e14i	0.988848i	−0.869221	+	0.494424i	$0.835379\pi$
$252$	0	0
$253$	2.05066e14	0.781934
$254$	0	0
$255$	1.56235e14i	0.568246i
$256$	0	0
$257$	−3.52097e14	−1.22198	−0.610990	−	0.791638i	$-0.709228\pi$
$257$	−3.52097e14	−1.22198	−0.610990	+	0.791638i	$0.709228\pi$
$258$	0	0
$259$	9.76352e13i	0.323451i
$260$	0	0
$261$	9.88737e13	0.312779
$262$	0	0
$263$	− 4.27923e14i	− 1.29310i	−0.762872	−	0.646549i	$-0.776212\pi$
$263$	− 4.27923e14i	− 1.29310i	0.762872	−	0.646549i	$-0.223788\pi$
$264$	0	0
$265$	2.73535e14	0.789837
$266$	0	0
$267$	1.48090e14i	0.408749i
$268$	0	0
$269$	−2.89207e14	−0.763300	−0.381650	−	0.924307i	$-0.624644\pi$
$269$	−2.89207e14	−0.763300	−0.381650	+	0.924307i	$0.624644\pi$
$270$	0	0
$271$	− 3.66165e14i	− 0.924403i	−0.886775	−	0.462202i	$-0.847060\pi$
$271$	− 3.66165e14i	− 0.924403i	0.886775	−	0.462202i	$-0.152940\pi$
$272$	0	0
$273$	−3.30814e13	−0.0799112
$274$	0	0
$275$	1.31737e15i	3.04587i
$276$	0	0
$277$	−7.14235e14	−1.58111	−0.790556	−	0.612390i	$-0.790208\pi$
$277$	−7.14235e14	−1.58111	−0.790556	+	0.612390i	$0.790208\pi$
$278$	0	0
$279$	− 2.81213e14i	− 0.596225i
$280$	0	0
$281$	5.50508e14	1.11822	0.559108	−	0.829095i	$-0.311144\pi$
$281$	5.50508e14	1.11822	0.559108	+	0.829095i	$0.311144\pi$
$282$	0	0
$283$	9.13704e14i	1.77864i	0.457290	+	0.889318i	$0.348820\pi$
$283$	9.13704e14i	1.77864i	−0.457290	+	0.889318i	$0.651180\pi$
$284$	0	0
$285$	5.67678e14	1.05933
$286$	0	0
$287$	− 1.43913e14i	− 0.257518i
$288$	0	0
$289$	−3.69136e14	−0.633576
$290$	0	0
$291$	3.20838e14i	0.528358i
$292$	0	0
$293$	−3.80614e14	−0.601560	−0.300780	−	0.953694i	$-0.597247\pi$
$293$	−3.80614e14	−0.601560	−0.300780	+	0.953694i	$0.597247\pi$
$294$	0	0
$295$	1.23254e15i	1.87011i
$296$	0	0
$297$	2.44765e14	0.356624
$298$	0	0
$299$	− 1.61351e14i	− 0.225811i
$300$	0	0
$301$	1.57502e14	0.211781
$302$	0	0
$303$	− 2.59207e14i	− 0.334959i
$304$	0	0
$305$	−2.07960e15	−2.58334
$306$	0	0
$307$	− 7.42024e14i	− 0.886314i	−0.896444	−	0.443157i	$-0.853858\pi$
$307$	− 7.42024e14i	− 0.886314i	0.896444	−	0.443157i	$-0.146142\pi$
$308$	0	0
$309$	8.27572e14	0.950725
$310$	0	0
$311$	− 4.82312e14i	− 0.533047i	−0.963828	−	0.266523i	$-0.914125\pi$
$311$	− 4.82312e14i	− 0.533047i	0.963828	−	0.266523i	$-0.0858750\pi$
$312$	0	0
$313$	7.81453e14	0.831069	0.415535	−	0.909577i	$-0.363595\pi$
$313$	7.81453e14	0.831069	0.415535	+	0.909577i	$0.363595\pi$
$314$	0	0
$315$	− 1.36945e14i	− 0.140179i
$316$	0	0
$317$	1.04341e15	1.02825	0.514126	−	0.857715i	$-0.328116\pi$
$317$	1.04341e15	1.02825	0.514126	+	0.857715i	$0.328116\pi$
$318$	0	0
$319$	− 1.83230e15i	− 1.73881i
$320$	0	0
$321$	3.42172e14	0.312762
$322$	0	0
$323$	− 7.75703e14i	− 0.683094i
$324$	0	0
$325$	1.03654e15	0.879602
$326$	0	0
$327$	4.28064e13i	0.0350124i
$328$	0	0
$329$	−4.90273e14	−0.386601
$330$	0	0
$331$	− 1.92106e15i	− 1.46074i	−0.683052	−	0.730370i	$-0.739348\pi$
$331$	− 1.92106e15i	− 1.46074i	0.683052	−	0.730370i	$-0.260652\pi$
$332$	0	0
$333$	−5.68397e14	−0.416857
$334$	0	0
$335$	2.70211e14i	0.191176i
$336$	0	0
$337$	9.36358e14	0.639238	0.319619	−	0.947546i	$-0.396445\pi$
$337$	9.36358e14	0.639238	0.319619	+	0.947546i	$0.396445\pi$
$338$	0	0
$339$	6.02985e14i	0.397291i
$340$	0	0
$341$	−5.21135e15	−3.31455
$342$	0	0
$343$	8.14179e14i	0.499983i
$344$	0	0
$345$	6.67938e14	0.396115
$346$	0	0
$347$	− 2.27514e15i	− 1.30326i	−0.758537	−	0.651630i	$-0.774086\pi$
$347$	− 2.27514e15i	− 1.30326i	0.758537	−	0.651630i	$-0.225914\pi$
$348$	0	0
$349$	3.45575e15	1.91245	0.956223	−	0.292638i	$-0.0945332\pi$
$349$	3.45575e15	1.91245	0.956223	+	0.292638i	$0.0945332\pi$
$350$	0	0
$351$	− 1.92588e14i	− 0.102988i
$352$	0	0
$353$	1.44427e15	0.746450	0.373225	−	0.927741i	$-0.378252\pi$
$353$	1.44427e15	0.746450	0.373225	+	0.927741i	$0.378252\pi$
$354$	0	0
$355$	2.01488e15i	1.00665i
$356$	0	0
$357$	−1.87129e14	−0.0903923
$358$	0	0
$359$	9.15233e13i	0.0427528i	0.999771	+	0.0213764i	$0.00680484\pi$
$359$	9.15233e13i	0.0427528i	−0.999771	+	0.0213764i	$0.993195\pi$
$360$	0	0
$361$	−6.05199e14	−0.273436
$362$	0	0
$363$	− 3.21498e15i	− 1.40520i
$364$	0	0
$365$	−4.14367e15	−1.75238
$366$	0	0
$367$	− 2.12524e15i	− 0.869785i	−0.900482	−	0.434893i	$-0.856786\pi$
$367$	− 2.12524e15i	− 0.869785i	0.900482	−	0.434893i	$-0.143214\pi$
$368$	0	0
$369$	8.37807e14	0.331884
$370$	0	0
$371$	3.27624e14i	0.125641i
$372$	0	0
$373$	−6.36322e14	−0.236278	−0.118139	−	0.992997i	$-0.537693\pi$
$373$	−6.36322e14	−0.236278	−0.118139	+	0.992997i	$0.537693\pi$
$374$	0	0
$375$	1.68037e15i	0.604251i
$376$	0	0
$377$	−1.44170e15	−0.502142
$378$	0	0
$379$	− 5.84379e14i	− 0.197178i	−0.995128	−	0.0985892i	$-0.968567\pi$
$379$	− 5.84379e14i	− 0.197178i	0.995128	−	0.0985892i	$-0.0314330\pi$
$380$	0	0
$381$	5.46599e14	0.178698
$382$	0	0
$383$	− 1.65996e15i	− 0.525902i	−0.964809	−	0.262951i	$-0.915304\pi$
$383$	− 1.65996e15i	− 0.525902i	0.964809	−	0.262951i	$-0.0846958\pi$
$384$	0	0
$385$	−2.53783e15	−0.779287
$386$	0	0
$387$	9.16920e14i	0.272939i
$388$	0	0
$389$	5.74758e15	1.65878	0.829388	−	0.558673i	$-0.188689\pi$
$389$	5.74758e15	1.65878	0.829388	+	0.558673i	$0.188689\pi$
$390$	0	0
$391$	− 9.12704e14i	− 0.255429i
$392$	0	0
$393$	3.59971e15	0.977040
$394$	0	0
$395$	3.83029e15i	1.00844i
$396$	0	0
$397$	5.62930e14	0.143784	0.0718922	−	0.997412i	$-0.477096\pi$
$397$	5.62930e14	0.143784	0.0718922	+	0.997412i	$0.477096\pi$
$398$	0	0
$399$	6.79932e14i	0.168511i
$400$	0	0
$401$	2.82505e15	0.679454	0.339727	−	0.940524i	$-0.389665\pi$
$401$	2.82505e15	0.679454	0.339727	+	0.940524i	$0.389665\pi$
$402$	0	0
$403$	4.10044e15i	0.957193i
$404$	0	0
$405$	7.97246e14	0.180660
$406$	0	0
$407$	1.05334e16i	2.31740i
$408$	0	0
$409$	−4.75723e15	−1.01628	−0.508141	−	0.861274i	$-0.669667\pi$
$409$	−4.75723e15	−1.01628	−0.508141	+	0.861274i	$0.669667\pi$
$410$	0	0
$411$	5.52689e15i	1.14665i
$412$	0	0
$413$	−1.47626e15	−0.297484
$414$	0	0
$415$	9.59610e15i	1.87848i
$416$	0	0
$417$	−2.96197e15	−0.563331
$418$	0	0
$419$	− 4.49305e15i	− 0.830341i	−0.909744	−	0.415171i	$-0.863722\pi$
$419$	− 4.49305e15i	− 0.830341i	0.909744	−	0.415171i	$-0.136278\pi$
$420$	0	0
$421$	−1.87384e15	−0.336543	−0.168271	−	0.985741i	$-0.553818\pi$
$421$	−1.87384e15	−0.336543	−0.168271	+	0.985741i	$0.553818\pi$
$422$	0	0
$423$	− 2.85420e15i	− 0.498244i
$424$	0	0
$425$	5.86332e15	0.994971
$426$	0	0
$427$	− 2.49083e15i	− 0.410938i
$428$	0	0
$429$	−3.56898e15	−0.572532
$430$	0	0
$431$	1.39845e15i	0.218164i	0.994033	+	0.109082i	$0.0347910\pi$
$431$	1.39845e15i	0.218164i	−0.994033	+	0.109082i	$0.965209\pi$
$432$	0	0
$433$	1.17138e15	0.177734	0.0888672	−	0.996043i	$-0.471675\pi$
$433$	1.17138e15	0.177734	0.0888672	+	0.996043i	$0.471675\pi$
$434$	0	0
$435$	− 5.96814e15i	− 0.880852i
$436$	0	0
$437$	−3.31631e15	−0.476174
$438$	0	0
$439$	6.53887e15i	0.913515i	0.889591	+	0.456758i	$0.150989\pi$
$439$	6.53887e15i	0.913515i	−0.889591	+	0.456758i	$0.849011\pi$
$440$	0	0
$441$	−2.28792e15	−0.311035
$442$	0	0
$443$	− 5.99316e15i	− 0.792928i	−0.918050	−	0.396464i	$-0.870237\pi$
$443$	− 5.99316e15i	− 0.792928i	0.918050	−	0.396464i	$-0.129763\pi$
$444$	0	0
$445$	8.93887e15	1.15112
$446$	0	0
$447$	2.07131e15i	0.259657i
$448$	0	0
$449$	1.43999e15	0.175745	0.0878724	−	0.996132i	$-0.471993\pi$
$449$	1.43999e15	0.175745	0.0878724	+	0.996132i	$0.471993\pi$
$450$	0	0
$451$	− 1.55260e16i	− 1.84501i
$452$	0	0
$453$	−1.70047e15	−0.196780
$454$	0	0
$455$	1.99683e15i	0.225047i
$456$	0	0
$457$	−9.31415e15	−1.02246	−0.511230	−	0.859444i	$-0.670810\pi$
$457$	−9.31415e15	−1.02246	−0.511230	+	0.859444i	$0.670810\pi$
$458$	0	0
$459$	− 1.08940e15i	− 0.116496i
$460$	0	0
$461$	2.81747e15	0.293531	0.146765	−	0.989171i	$-0.453114\pi$
$461$	2.81747e15	0.293531	0.146765	+	0.989171i	$0.453114\pi$
$462$	0	0
$463$	1.38808e16i	1.40905i	0.709678	+	0.704526i	$0.248840\pi$
$463$	1.38808e16i	1.40905i	−0.709678	+	0.704526i	$0.751160\pi$
$464$	0	0
$465$	−1.69744e16	−1.67910
$466$	0	0
$467$	− 3.56873e15i	− 0.344043i	−0.985093	−	0.172022i	$-0.944970\pi$
$467$	− 3.56873e15i	− 0.344043i	0.985093	−	0.172022i	$-0.0550299\pi$
$468$	0	0
$469$	−3.23643e14	−0.0304109
$470$	0	0
$471$	7.77047e15i	0.711741i
$472$	0	0
$473$	1.69921e16	1.51733
$474$	0	0
$475$	− 2.13044e16i	− 1.85484i
$476$	0	0
$477$	−1.90731e15	−0.161924
$478$	0	0
$479$	1.55486e15i	0.128730i	0.997926	+	0.0643649i	$0.0205022\pi$
$479$	1.55486e15i	0.128730i	−0.997926	+	0.0643649i	$0.979498\pi$
$480$	0	0
$481$	8.28794e15	0.669231
$482$	0	0
$483$	8.00018e14i	0.0630111i
$484$	0	0
$485$	1.93662e16	1.48797
$486$	0	0
$487$	− 1.38493e16i	− 1.03814i	−0.854732	−	0.519069i	$-0.826279\pi$
$487$	− 1.38493e16i	− 1.03814i	0.854732	−	0.519069i	$-0.173721\pi$
$488$	0	0
$489$	−3.45335e15	−0.252574
$490$	0	0
$491$	7.19285e15i	0.513348i	0.966498	+	0.256674i	$0.0826266\pi$
$491$	7.19285e15i	0.513348i	−0.966498	+	0.256674i	$0.917373\pi$
$492$	0	0
$493$	−8.15516e15	−0.568003
$494$	0	0
$495$	− 1.47743e16i	− 1.00433i
$496$	0	0
$497$	−2.41330e15	−0.160130
$498$	0	0
$499$	− 2.17699e15i	− 0.141011i	−0.997511	−	0.0705056i	$-0.977539\pi$
$499$	− 2.17699e15i	− 0.141011i	0.997511	−	0.0705056i	$-0.0224613\pi$
$500$	0	0
$501$	1.47538e16	0.932994
$502$	0	0
$503$	− 2.71554e16i	− 1.67667i	−0.545153	−	0.838337i	$-0.683528\pi$
$503$	− 2.71554e16i	− 1.67667i	0.545153	−	0.838337i	$-0.316472\pi$
$504$	0	0
$505$	−1.56461e16	−0.943315
$506$	0	0
$507$	− 6.99772e15i	− 0.412011i
$508$	0	0
$509$	1.15373e16	0.663432	0.331716	−	0.943379i	$-0.392372\pi$
$509$	1.15373e16	0.663432	0.331716	+	0.943379i	$0.392372\pi$
$510$	0	0
$511$	− 4.96304e15i	− 0.278755i
$512$	0	0
$513$	−3.95832e15	−0.217173
$514$	0	0
$515$	− 4.99532e16i	− 2.67744i
$516$	0	0
$517$	−5.28931e16	−2.76984
$518$	0	0
$519$	1.27346e16i	0.651601i
$520$	0	0
$521$	3.19685e16	1.59844	0.799219	−	0.601041i	$-0.205247\pi$
$521$	3.19685e16	1.59844	0.799219	+	0.601041i	$0.205247\pi$
$522$	0	0
$523$	3.45198e16i	1.68678i	0.537303	+	0.843389i	$0.319443\pi$
$523$	3.45198e16i	1.68678i	−0.537303	+	0.843389i	$0.680557\pi$
$524$	0	0
$525$	−5.13941e15	−0.245447
$526$	0	0
$527$	2.31946e16i	1.08274i
$528$	0	0
$529$	1.80126e16	0.821945
$530$	0	0
$531$	− 8.59428e15i	− 0.383391i
$532$	0	0
$533$	−1.22163e16	−0.532814
$534$	0	0
$535$	− 2.06539e16i	− 0.880804i
$536$	0	0
$537$	−7.01132e15	−0.292384
$538$	0	0
$539$	4.23990e16i	1.72911i
$540$	0	0
$541$	−3.99117e15	−0.159190	−0.0795952	−	0.996827i	$-0.525363\pi$
$541$	−3.99117e15	−0.159190	−0.0795952	+	0.996827i	$0.525363\pi$
$542$	0	0
$543$	− 8.90600e14i	− 0.0347443i
$544$	0	0
$545$	2.58385e15	0.0986025
$546$	0	0
$547$	2.53627e16i	0.946828i	0.880840	+	0.473414i	$0.156979\pi$
$547$	2.53627e16i	0.946828i	−0.880840	+	0.473414i	$0.843021\pi$
$548$	0	0
$549$	1.45007e16	0.529609
$550$	0	0
$551$	2.96317e16i	1.05888i
$552$	0	0
$553$	−4.58770e15	−0.160415
$554$	0	0
$555$	3.43092e16i	1.17396i
$556$	0	0
$557$	−1.53368e16	−0.513574	−0.256787	−	0.966468i	$-0.582664\pi$
$557$	−1.53368e16	−0.513574	−0.256787	+	0.966468i	$0.582664\pi$
$558$	0	0
$559$	− 1.33698e16i	− 0.438182i
$560$	0	0
$561$	−2.01884e16	−0.647625
$562$	0	0
$563$	3.46282e16i	1.08737i	0.839288	+	0.543687i	$0.182972\pi$
$563$	3.46282e16i	1.08737i	−0.839288	+	0.543687i	$0.817028\pi$
$564$	0	0
$565$	3.63969e16	1.11886
$566$	0	0
$567$	9.54896e14i	0.0287381i
$568$	0	0
$569$	−2.09807e16	−0.618223	−0.309112	−	0.951026i	$-0.600032\pi$
$569$	−2.09807e16	−0.618223	−0.309112	+	0.951026i	$0.600032\pi$
$570$	0	0
$571$	5.15575e15i	0.148756i	0.997230	+	0.0743782i	$0.0236972\pi$
$571$	5.15575e15i	0.148756i	−0.997230	+	0.0743782i	$0.976303\pi$
$572$	0	0
$573$	−2.73528e14	−0.00772814
$574$	0	0
$575$	− 2.50670e16i	− 0.693579i
$576$	0	0
$577$	4.12378e16	1.11748	0.558741	−	0.829342i	$-0.311285\pi$
$577$	4.12378e16	1.11748	0.558741	+	0.829342i	$0.311285\pi$
$578$	0	0
$579$	− 2.58327e16i	− 0.685643i
$580$	0	0
$581$	−1.14937e16	−0.298814
$582$	0	0
$583$	3.53457e16i	0.900171i
$584$	0	0
$585$	−1.16248e16	−0.290036
$586$	0	0
$587$	− 8.40853e15i	− 0.205538i	−0.994705	−	0.102769i	$-0.967230\pi$
$587$	− 8.40853e15i	− 0.205538i	0.994705	−	0.102769i	$-0.0327702\pi$
$588$	0	0
$589$	8.42776e16	2.01846
$590$	0	0
$591$	1.66044e16i	0.389671i
$592$	0	0
$593$	4.13687e16	0.951357	0.475678	−	0.879619i	$-0.342203\pi$
$593$	4.13687e16	0.951357	0.475678	+	0.879619i	$0.342203\pi$
$594$	0	0
$595$	1.12953e16i	0.254564i
$596$	0	0
$597$	3.90576e16	0.862699
$598$	0	0
$599$	− 6.80753e15i	− 0.147377i	−0.997281	−	0.0736883i	$-0.976523\pi$
$599$	− 6.80753e15i	− 0.147377i	0.997281	−	0.0736883i	$-0.0234770\pi$
$600$	0	0
$601$	−5.31746e16	−1.12839	−0.564193	−	0.825643i	$-0.690813\pi$
$601$	−5.31746e16	−1.12839	−0.564193	+	0.825643i	$0.690813\pi$
$602$	0	0
$603$	− 1.88414e15i	− 0.0391930i
$604$	0	0
$605$	−1.94060e17	−3.95734
$606$	0	0
$607$	3.71049e16i	0.741821i	0.928669	+	0.370910i	$0.120954\pi$
$607$	3.71049e16i	0.741821i	−0.928669	+	0.370910i	$0.879046\pi$
$608$	0	0
$609$	7.14829e15	0.140119
$610$	0	0
$611$	4.16177e16i	0.799891i
$612$	0	0
$613$	1.16050e16	0.218716	0.109358	−	0.994002i	$-0.465120\pi$
$613$	1.16050e16	0.218716	0.109358	+	0.994002i	$0.465120\pi$
$614$	0	0
$615$	− 5.05711e16i	− 0.934655i
$616$	0	0
$617$	1.60639e16	0.291165	0.145582	−	0.989346i	$-0.453494\pi$
$617$	1.60639e16	0.291165	0.145582	+	0.989346i	$0.453494\pi$
$618$	0	0
$619$	2.26811e16i	0.403200i	0.979468	+	0.201600i	$0.0646141\pi$
$619$	2.26811e16i	0.403200i	−0.979468	+	0.201600i	$0.935386\pi$
$620$	0	0
$621$	−4.65742e15	−0.0812074
$622$	0	0
$623$	1.07065e16i	0.183112i
$624$	0	0
$625$	3.45791e15	0.0580141
$626$	0	0
$627$	7.33543e16i	1.20731i
$628$	0	0
$629$	4.68817e16	0.757007
$630$	0	0
$631$	1.57639e16i	0.249739i	0.992173	+	0.124870i	$0.0398512\pi$
$631$	1.57639e16i	0.249739i	−0.992173	+	0.124870i	$0.960149\pi$
$632$	0	0
$633$	5.75327e16	0.894319
$634$	0	0
$635$	− 3.29934e16i	− 0.503251i
$636$	0	0
$637$	3.33607e16	0.499342
$638$	0	0
$639$	− 1.40494e16i	− 0.206373i
$640$	0	0
$641$	−3.67271e16	−0.529467	−0.264733	−	0.964322i	$-0.585284\pi$
$641$	−3.67271e16	−0.529467	−0.264733	+	0.964322i	$0.585284\pi$
$642$	0	0
$643$	− 1.38084e17i	− 1.95378i	−0.213739	−	0.976891i	$-0.568564\pi$
$643$	− 1.38084e17i	− 1.95378i	0.213739	−	0.976891i	$-0.431436\pi$
$644$	0	0
$645$	5.53464e16	0.768654
$646$	0	0
$647$	− 8.12543e16i	− 1.10770i	−0.832618	−	0.553848i	$-0.813159\pi$
$647$	− 8.12543e16i	− 1.10770i	0.832618	−	0.553848i	$-0.186841\pi$
$648$	0	0
$649$	−1.59266e17	−2.13136
$650$	0	0
$651$	− 2.03309e16i	− 0.267098i
$652$	0	0
$653$	−9.85937e16	−1.27166	−0.635828	−	0.771830i	$-0.719341\pi$
$653$	−9.85937e16	−1.27166	−0.635828	+	0.771830i	$0.719341\pi$
$654$	0	0
$655$	− 2.17283e17i	− 2.75155i
$656$	0	0
$657$	2.88931e16	0.359254
$658$	0	0
$659$	3.62039e16i	0.442020i	0.975271	+	0.221010i	$0.0709354\pi$
$659$	3.62039e16i	0.442020i	−0.975271	+	0.221010i	$0.929065\pi$
$660$	0	0
$661$	8.25017e14	0.00989132	0.00494566	−	0.999988i	$-0.498426\pi$
$661$	8.25017e14	0.00989132	0.00494566	+	0.999988i	$0.498426\pi$
$662$	0	0
$663$	1.58848e16i	0.187025i
$664$	0	0
$665$	4.10415e16	0.474562
$666$	0	0
$667$	3.48652e16i	0.395947i
$668$	0	0
$669$	6.18884e16	0.690323
$670$	0	0
$671$	− 2.68723e17i	− 2.94421i
$672$	0	0
$673$	1.18441e16	0.127471	0.0637353	−	0.997967i	$-0.479699\pi$
$673$	1.18441e16	0.127471	0.0637353	+	0.997967i	$0.479699\pi$
$674$	0	0
$675$	− 2.99198e16i	− 0.316327i
$676$	0	0
$677$	−1.18328e17	−1.22901	−0.614504	−	0.788914i	$-0.710644\pi$
$677$	−1.18328e17	−1.22901	−0.614504	+	0.788914i	$0.710644\pi$
$678$	0	0
$679$	2.31957e16i	0.236695i
$680$	0	0
$681$	3.81573e16	0.382556
$682$	0	0
$683$	7.66855e16i	0.755421i	0.925924	+	0.377710i	$0.123289\pi$
$683$	7.66855e16i	0.755421i	−0.925924	+	0.377710i	$0.876711\pi$
$684$	0	0
$685$	3.33610e17	3.22920
$686$	0	0
$687$	− 4.60604e16i	− 0.438114i
$688$	0	0
$689$	2.78110e16	0.259956
$690$	0	0
$691$	1.88583e17i	1.73234i	0.499745	+	0.866172i	$0.333427\pi$
$691$	1.88583e17i	1.73234i	−0.499745	+	0.866172i	$0.666573\pi$
$692$	0	0
$693$	1.76958e16	0.159761
$694$	0	0
$695$	1.78788e17i	1.58646i
$696$	0	0
$697$	−6.91028e16	−0.602697
$698$	0	0
$699$	− 4.96973e16i	− 0.426059i
$700$	0	0
$701$	−8.11290e16	−0.683704	−0.341852	−	0.939754i	$-0.611054\pi$
$701$	−8.11290e16	−0.683704	−0.341852	+	0.939754i	$0.611054\pi$
$702$	0	0
$703$	− 1.70345e17i	− 1.41123i
$704$	0	0
$705$	−1.72283e17	−1.40316
$706$	0	0
$707$	− 1.87399e16i	− 0.150055i
$708$	0	0
$709$	−1.40986e16	−0.110994	−0.0554970	−	0.998459i	$-0.517674\pi$
$709$	−1.40986e16	−0.110994	−0.0554970	+	0.998459i	$0.517674\pi$
$710$	0	0
$711$	− 2.67079e16i	− 0.206739i
$712$	0	0
$713$	9.91623e16	0.754761
$714$	0	0
$715$	2.15428e17i	1.61237i
$716$	0	0
$717$	−2.56931e16	−0.189104
$718$	0	0
$719$	− 4.74847e16i	− 0.343700i	−0.985123	−	0.171850i	$-0.945026\pi$
$719$	− 4.74847e16i	− 0.343700i	0.985123	−	0.171850i	$-0.0549745\pi$
$720$	0	0
$721$	5.98311e16	0.425908
$722$	0	0
$723$	1.24760e17i	0.873466i
$724$	0	0
$725$	−2.23978e17	−1.54233
$726$	0	0
$727$	4.63265e16i	0.313778i	0.987616	+	0.156889i	$0.0501466\pi$
$727$	4.63265e16i	0.313778i	−0.987616	+	0.156889i	$0.949853\pi$
$728$	0	0
$729$	−5.55906e15	−0.0370370
$730$	0	0
$731$	− 7.56280e16i	− 0.495654i
$732$	0	0
$733$	−1.20719e16	−0.0778307	−0.0389154	−	0.999243i	$-0.512390\pi$
$733$	−1.20719e16	−0.0778307	−0.0389154	+	0.999243i	$0.512390\pi$
$734$	0	0
$735$	1.38101e17i	0.875940i
$736$	0	0
$737$	−3.49162e16	−0.217882
$738$	0	0
$739$	− 2.12001e17i	− 1.30158i	−0.759257	−	0.650790i	$-0.774438\pi$
$739$	− 2.12001e17i	− 1.30158i	0.759257	−	0.650790i	$-0.225562\pi$
$740$	0	0
$741$	5.77172e16	0.348655
$742$	0	0
$743$	1.85818e17i	1.10448i	0.833687	+	0.552238i	$0.186226\pi$
$743$	1.85818e17i	1.10448i	−0.833687	+	0.552238i	$0.813774\pi$
$744$	0	0
$745$	1.25027e17	0.731249
$746$	0	0
$747$	− 6.69120e16i	− 0.385106i
$748$	0	0
$749$	2.47380e16	0.140112
$750$	0	0
$751$	− 4.17076e16i	− 0.232475i	−0.993221	−	0.116237i	$-0.962917\pi$
$751$	− 4.17076e16i	− 0.232475i	0.993221	−	0.116237i	$-0.0370833\pi$
$752$	0	0
$753$	−1.04073e17	−0.570911
$754$	0	0
$755$	1.02643e17i	0.554174i
$756$	0	0
$757$	−9.83821e16	−0.522806	−0.261403	−	0.965230i	$-0.584185\pi$
$757$	−9.83821e16	−0.522806	−0.261403	+	0.965230i	$0.584185\pi$
$758$	0	0
$759$	8.63098e16i	0.451450i
$760$	0	0
$761$	3.36969e17	1.73493	0.867466	−	0.497496i	$-0.165747\pi$
$761$	3.36969e17	1.73493	0.867466	+	0.497496i	$0.165747\pi$
$762$	0	0
$763$	3.09478e15i	0.0156849i
$764$	0	0
$765$	−6.57573e16	−0.328077
$766$	0	0
$767$	1.25315e17i	0.615505i
$768$	0	0
$769$	−1.47438e17	−0.712939	−0.356469	−	0.934307i	$-0.616020\pi$
$769$	−1.47438e17	−0.712939	−0.356469	+	0.934307i	$0.616020\pi$
$770$	0	0
$771$	− 1.48194e17i	− 0.705511i
$772$	0	0
$773$	−7.63214e16	−0.357741	−0.178871	−	0.983873i	$-0.557244\pi$
$773$	−7.63214e16	−0.357741	−0.178871	+	0.983873i	$0.557244\pi$
$774$	0	0
$775$	6.37031e17i	2.94002i
$776$	0	0
$777$	−4.10935e16	−0.186744
$778$	0	0
$779$	2.51085e17i	1.12356i
$780$	0	0
$781$	−2.60359e17	−1.14727
$782$	0	0
$783$	4.16148e16i	0.180583i
$784$	0	0
$785$	4.69035e17	2.00441
$786$	0	0
$787$	1.41260e15i	0.00594526i	0.999996	+	0.00297263i	$0.000946218\pi$
$787$	1.41260e15i	0.00594526i	−0.999996	+	0.00297263i	$0.999054\pi$
$788$	0	0
$789$	1.80108e17	0.746571
$790$	0	0
$791$	4.35941e16i	0.177979i
$792$	0	0
$793$	−2.11438e17	−0.850246
$794$	0	0
$795$	1.15128e17i	0.456013i
$796$	0	0
$797$	2.67852e17	1.04507	0.522535	−	0.852618i	$-0.324986\pi$
$797$	2.67852e17	1.04507	0.522535	+	0.852618i	$0.324986\pi$
$798$	0	0
$799$	2.35416e17i	0.904805i
$800$	0	0
$801$	−6.23292e16	−0.235991
$802$	0	0
$803$	− 5.35437e17i	− 1.99717i
$804$	0	0
$805$	4.82901e16	0.177453
$806$	0	0
$807$	− 1.21724e17i	− 0.440692i
$808$	0	0
$809$	1.01877e17	0.363400	0.181700	−	0.983354i	$-0.441840\pi$
$809$	1.01877e17	0.363400	0.181700	+	0.983354i	$0.441840\pi$
$810$	0	0
$811$	− 7.85544e16i	− 0.276087i	−0.990426	−	0.138043i	$-0.955919\pi$
$811$	− 7.85544e16i	− 0.276087i	0.990426	−	0.138043i	$-0.0440813\pi$
$812$	0	0
$813$	1.54115e17	0.533704
$814$	0	0
$815$	2.08449e17i	0.711301i
$816$	0	0
$817$	−2.74794e17	−0.924007
$818$	0	0
$819$	− 1.39236e16i	− 0.0461367i
$820$	0	0
$821$	4.02278e17	1.31361	0.656806	−	0.754060i	$-0.271907\pi$
$821$	4.02278e17	1.31361	0.656806	+	0.754060i	$0.271907\pi$
$822$	0	0
$823$	− 3.37251e17i	− 1.08531i	−0.839955	−	0.542655i	$-0.817419\pi$
$823$	− 3.37251e17i	− 1.08531i	0.839955	−	0.542655i	$-0.182581\pi$
$824$	0	0
$825$	−5.54465e17	−1.75853
$826$	0	0
$827$	− 9.50449e16i	− 0.297095i	−0.988905	−	0.148548i	$-0.952540\pi$
$827$	− 9.50449e16i	− 0.297095i	0.988905	−	0.148548i	$-0.0474598\pi$
$828$	0	0
$829$	−3.46796e17	−1.06843	−0.534216	−	0.845348i	$-0.679393\pi$
$829$	−3.46796e17	−1.06843	−0.534216	+	0.845348i	$0.679393\pi$
$830$	0	0
$831$	− 3.00613e17i	− 0.912855i
$832$	0	0
$833$	1.88709e17	0.564835
$834$	0	0
$835$	− 8.90560e17i	− 2.62751i
$836$	0	0
$837$	1.18359e17	0.344231
$838$	0	0
$839$	− 2.21502e17i	− 0.635046i	−0.948251	−	0.317523i	$-0.897149\pi$
$839$	− 2.21502e17i	− 0.635046i	0.948251	−	0.317523i	$-0.102851\pi$
$840$	0	0
$841$	−4.22890e16	−0.119523
$842$	0	0
$843$	2.31702e17i	0.645602i
$844$	0	0
$845$	−4.22391e17	−1.16031
$846$	0	0
$847$	− 2.32434e17i	− 0.629505i
$848$	0	0
$849$	−3.84567e17	−1.02690
$850$	0	0
$851$	− 2.00430e17i	− 0.527698i
$852$	0	0
$853$	7.90888e16	0.205315	0.102658	−	0.994717i	$-0.467265\pi$
$853$	7.90888e16	0.205315	0.102658	+	0.994717i	$0.467265\pi$
$854$	0	0
$855$	2.38929e17i	0.611606i
$856$	0	0
$857$	5.37483e17	1.35669	0.678344	−	0.734745i	$-0.262698\pi$
$857$	5.37483e17	1.35669	0.678344	+	0.734745i	$0.262698\pi$
$858$	0	0
$859$	1.69328e17i	0.421474i	0.977543	+	0.210737i	$0.0675864\pi$
$859$	1.69328e17i	0.421474i	−0.977543	+	0.210737i	$0.932414\pi$
$860$	0	0
$861$	6.05711e16	0.148678
$862$	0	0
$863$	− 1.82015e17i	− 0.440599i	−0.975432	−	0.220299i	$-0.929297\pi$
$863$	− 1.82015e17i	− 0.440599i	0.975432	−	0.220299i	$-0.0707034\pi$
$864$	0	0
$865$	7.68677e17	1.83505
$866$	0	0
$867$	− 1.55365e17i	− 0.365795i
$868$	0	0
$869$	−4.94943e17	−1.14931
$870$	0	0
$871$	2.74730e16i	0.0629213i
$872$	0	0
$873$	−1.35037e17	−0.305048
$874$	0	0
$875$	1.21486e17i	0.270694i
$876$	0	0
$877$	5.53756e16	0.121709	0.0608543	−	0.998147i	$-0.480618\pi$
$877$	5.53756e16	0.121709	0.0608543	+	0.998147i	$0.480618\pi$
$878$	0	0
$879$	− 1.60196e17i	− 0.347311i
$880$	0	0
$881$	−3.93941e17	−0.842510	−0.421255	−	0.906942i	$-0.638410\pi$
$881$	−3.93941e17	−0.842510	−0.421255	+	0.906942i	$0.638410\pi$
$882$	0	0
$883$	8.76120e14i	0.00184841i	1.00000		0.000924207i	$0.000294184\pi$
$883$	8.76120e14i	0.00184841i	−1.00000		0.000924207i	$0.999706\pi$
$884$	0	0
$885$	−5.18761e17	−1.07971
$886$	0	0
$887$	2.89268e17i	0.593961i	0.954883	+	0.296981i	$0.0959797\pi$
$887$	2.89268e17i	0.593961i	−0.954883	+	0.296981i	$0.904020\pi$
$888$	0	0
$889$	3.95176e16	0.0800533
$890$	0	0
$891$	1.03019e17i	0.205897i
$892$	0	0
$893$	8.55382e17	1.68675
$894$	0	0
$895$	4.23212e17i	0.823416i
$896$	0	0
$897$	6.79109e16	0.130372
$898$	0	0
$899$	− 8.86031e17i	− 1.67838i
$900$	0	0
$901$	1.57316e17	0.294052
$902$	0	0
$903$	6.62907e16i	0.122272i
$904$	0	0
$905$	−5.37577e16	−0.0978474
$906$	0	0
$907$	3.50586e17i	0.629725i	0.949137	+	0.314863i	$0.101958\pi$
$907$	3.50586e17i	0.629725i	−0.949137	+	0.314863i	$0.898042\pi$
$908$	0	0
$909$	1.09097e17	0.193388
$910$	0	0
$911$	− 5.90209e17i	− 1.03251i	−0.856434	−	0.516256i	$-0.827325\pi$
$911$	− 5.90209e17i	− 1.03251i	0.856434	−	0.516256i	$-0.172675\pi$
$912$	0	0
$913$	−1.23999e18	−2.14089
$914$	0	0
$915$	− 8.75281e17i	− 1.49149i
$916$	0	0
$917$	2.60249e17	0.437696
$918$	0	0
$919$	− 3.01626e17i	− 0.500697i	−0.968156	−	0.250349i	$-0.919455\pi$
$919$	− 3.01626e17i	− 0.500697i	0.968156	−	0.250349i	$-0.0805453\pi$
$920$	0	0
$921$	3.12309e17	0.511714
$922$	0	0
$923$	2.04857e17i	0.331315i
$924$	0	0
$925$	1.28759e18	2.05554
$926$	0	0
$927$	3.48315e17i	0.548901i
$928$	0	0
$929$	7.23264e17	1.12513	0.562565	−	0.826753i	$-0.309815\pi$
$929$	7.23264e17	1.12513	0.562565	+	0.826753i	$0.309815\pi$
$930$	0	0
$931$	− 6.85672e17i	− 1.05298i
$932$	0	0
$933$	2.02999e17	0.307755
$934$	0	0
$935$	1.21859e18i	1.82385i
$936$	0	0
$937$	1.33864e17	0.197800	0.0989000	−	0.995097i	$-0.468468\pi$
$937$	1.33864e17	0.197800	0.0989000	+	0.995097i	$0.468468\pi$
$938$	0	0
$939$	3.28905e17i	0.479818i
$940$	0	0
$941$	−1.08680e18	−1.56535	−0.782675	−	0.622431i	$-0.786145\pi$
$941$	−1.08680e18	−1.56535	−0.782675	+	0.622431i	$0.786145\pi$
$942$	0	0
$943$	2.95430e17i	0.420131i
$944$	0	0
$945$	5.76387e16	0.0809325
$946$	0	0
$947$	6.82098e17i	0.945686i	0.881147	+	0.472843i	$0.156772\pi$
$947$	6.82098e17i	0.945686i	−0.881147	+	0.472843i	$0.843228\pi$
$948$	0	0
$949$	−4.21297e17	−0.576754
$950$	0	0
$951$	4.39159e17i	0.593662i
$952$	0	0
$953$	−6.09235e17	−0.813256	−0.406628	−	0.913594i	$-0.633295\pi$
$953$	−6.09235e17	−0.813256	−0.406628	+	0.913594i	$0.633295\pi$
$954$	0	0
$955$	1.65105e16i	0.0217641i
$956$	0	0
$957$	7.71192e17	1.00390
$958$	0	0
$959$	3.99579e17i	0.513678i
$960$	0	0
$961$	−1.73236e18	−2.19936
$962$	0	0
$963$	1.44016e17i	0.180573i
$964$	0	0
$965$	−1.55929e18	−1.93092
$966$	0	0
$967$	1.12645e18i	1.37770i	0.724905	+	0.688848i	$0.241883\pi$
$967$	1.12645e18i	1.37770i	−0.724905	+	0.688848i	$0.758117\pi$
$968$	0	0
$969$	3.26484e17	0.394384
$970$	0	0
$971$	6.84108e17i	0.816224i	0.912932	+	0.408112i	$0.133813\pi$
$971$	6.84108e17i	0.816224i	−0.912932	+	0.408112i	$0.866187\pi$
$972$	0	0
$973$	−2.14142e17	−0.252362
$974$	0	0
$975$	4.36268e17i	0.507839i
$976$	0	0
$977$	1.47795e18	1.69939	0.849693	−	0.527278i	$-0.176787\pi$
$977$	1.47795e18	1.69939	0.849693	+	0.527278i	$0.176787\pi$
$978$	0	0
$979$	1.15506e18i	1.31193i
$980$	0	0
$981$	−1.80167e16	−0.0202144
$982$	0	0
$983$	1.41377e18i	1.56696i	0.621416	+	0.783481i	$0.286558\pi$
$983$	1.41377e18i	1.56696i	−0.621416	+	0.783481i	$0.713442\pi$
$984$	0	0
$985$	1.00226e18	1.09739
$986$	0	0
$987$	− 2.06350e17i	− 0.223204i
$988$	0	0
$989$	−3.23327e17	−0.345513
$990$	0	0
$991$	− 4.02772e17i	− 0.425223i	−0.977137	−	0.212612i	$-0.931803\pi$
$991$	− 4.02772e17i	− 0.425223i	0.977137	−	0.212612i	$-0.0681969\pi$
$992$	0	0
$993$	8.08552e17	0.843359
$994$	0	0
$995$	− 2.35756e18i	− 2.42954i
$996$	0	0
$997$	4.89766e17	0.498675	0.249338	−	0.968417i	$-0.419787\pi$
$997$	4.89766e17	0.498675	0.249338	+	0.968417i	$0.419787\pi$
$998$	0	0
$999$	− 2.39232e17i	− 0.240672i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.13.g.b.31.4	yes	4
3.2	odd	2		144.13.g.f.127.2		4
4.3	odd	2	inner	48.13.g.b.31.2	✓	4
8.3	odd	2		192.13.g.b.127.3		4
8.5	even	2		192.13.g.b.127.1		4
12.11	even	2		144.13.g.f.127.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.13.g.b.31.2	✓	4	4.3	odd	2	inner
48.13.g.b.31.4	yes	4	1.1	even	1	trivial
144.13.g.f.127.1		4	12.11	even	2
144.13.g.f.127.2		4	3.2	odd	2
192.13.g.b.127.1		4	8.5	even	2
192.13.g.b.127.3		4	8.3	odd	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:	\( N \)	\(=\)	\( 48 = 2^{4} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	48.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+420.888i q^{3} +25405.3 q^{5} +30429.0i q^{7} -177147. q^{9} +O(q^{10})\)	\(q+420.888i q^{3} +25405.3 q^{5} +30429.0i q^{7} -177147. q^{9} +3.28283e6i q^{11} +2.58302e6 q^{13} +1.06928e7i q^{15} +1.46112e7 q^{17} -5.30897e7i q^{19} -1.28072e7 q^{21} -6.24661e7i q^{23} +4.01290e8 q^{25} -7.45591e7i q^{27} -5.58145e8 q^{29} +1.58746e9i q^{31} -1.38171e9 q^{33} +7.73060e8i q^{35} +3.20862e9 q^{37} +1.08716e9i q^{39} -4.72945e9 q^{41} -5.17604e9i q^{43} -4.50048e9 q^{45} +1.61120e10i q^{47} +1.29154e10 q^{49} +6.14968e9i q^{51} +1.07668e10 q^{53} +8.34014e10i q^{55} +2.23448e10 q^{57} +4.85149e10i q^{59} -8.18569e10 q^{61} -5.39041e9i q^{63} +6.56225e10 q^{65} +1.06360e10i q^{67} +2.62913e10 q^{69} +7.93092e10i q^{71} -1.63102e11 q^{73} +1.68898e11i q^{75} -9.98934e10 q^{77} +1.50767e11i q^{79} +3.13811e10 q^{81} +3.77720e11i q^{83} +3.71202e11 q^{85} -2.34917e11i q^{87} +3.51850e11 q^{89} +7.85989e10i q^{91} -6.68142e11 q^{93} -1.34876e12i q^{95} +7.62288e11 q^{97} -5.81544e11i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 22392 q^{5} - 708588 q^{9}+O(q^{10}) \) 4 * q + 22392 * q^5 - 708588 * q^9	\( 4 q + 22392 q^{5} - 708588 q^{9} + 8905960 q^{13} + 72547560 q^{17} + 45034704 q^{21} + 718109132 q^{25} - 728411112 q^{29} - 1714782960 q^{33} - 2724573880 q^{37} - 7164120600 q^{41} - 3966675624 q^{45} + 39425266372 q^{49} + 32619271320 q^{53} + 33276867120 q^{57} - 130736241784 q^{61} + 78103345968 q^{65} - 73307680128 q^{69} - 170497449080 q^{73} - 408892017600 q^{77} + 125524238436 q^{81} + 126782064432 q^{85} + 2012222389896 q^{89} - 935876261520 q^{93} + 1491418808200 q^{97}+O(q^{100}) \) 4 * q + 22392 * q^5 - 708588 * q^9 + 8905960 * q^13 + 72547560 * q^17 + 45034704 * q^21 + 718109132 * q^25 - 728411112 * q^29 - 1714782960 * q^33 - 2724573880 * q^37 - 7164120600 * q^41 - 3966675624 * q^45 + 39425266372 * q^49 + 32619271320 * q^53 + 33276867120 * q^57 - 130736241784 * q^61 + 78103345968 * q^65 - 73307680128 * q^69 - 170497449080 * q^73 - 408892017600 * q^77 + 125524238436 * q^81 + 126782064432 * q^85 + 2012222389896 * q^89 - 935876261520 * q^93 + 1491418808200 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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