LMFDB - Embedded newform 48.16.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,16,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, 16, names="a")

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

chi := DirichletCharacter("48.1");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 48 = 2^{4} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
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:	$68.4928824480$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+2187.00 q^{3} -114810. q^{5} +3.03453e6 q^{7} +4.78297e6 q^{9} +O(q^{10})$

$q+2187.00 q^{3} -114810. q^{5} +3.03453e6 q^{7} +4.78297e6 q^{9} +1.03452e8 q^{11} -1.04366e8 q^{13} -2.51089e8 q^{15} +9.97690e8 q^{17} -4.93402e9 q^{19} +6.63651e9 q^{21} -8.32492e9 q^{23} -1.73362e10 q^{25} +1.04604e10 q^{27} +1.04128e11 q^{29} +2.96697e11 q^{31} +2.26249e11 q^{33} -3.48394e11 q^{35} -1.78337e11 q^{37} -2.28248e11 q^{39} -1.79088e12 q^{41} +2.86346e12 q^{43} -5.49133e11 q^{45} -4.33291e12 q^{47} +4.46080e12 q^{49} +2.18195e12 q^{51} +9.73232e12 q^{53} -1.18773e13 q^{55} -1.07907e13 q^{57} +1.35148e13 q^{59} +5.35266e12 q^{61} +1.45141e13 q^{63} +1.19822e13 q^{65} +5.32339e13 q^{67} -1.82066e13 q^{69} +2.02297e13 q^{71} +2.62642e13 q^{73} -3.79144e13 q^{75} +3.13927e14 q^{77} +3.39031e14 q^{79} +2.28768e13 q^{81} -1.31685e14 q^{83} -1.14545e14 q^{85} +2.27728e14 q^{87} -3.93521e13 q^{89} -3.16701e14 q^{91} +6.48876e14 q^{93} +5.66474e14 q^{95} +1.12875e15 q^{97} +4.94806e14 q^{99} +O(q^{100})$

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
$3$	2187.00	0.577350
$3$	2187.00	0.577350
$4$	0	0
$5$	−114810.	−0.657211	−0.328605	−	0.944467i	$-0.606579\pi$
$5$	−114810.	−0.657211	−0.328605	+	0.944467i	$0.606579\pi$
$6$	0	0
$7$	3.03453e6	1.39269	0.696347	−	0.717705i	$-0.254807\pi$
$7$	3.03453e6	1.39269	0.696347	+	0.717705i	$0.254807\pi$
$8$	0	0
$9$	4.78297e6	0.333333
$10$	0	0
$11$	1.03452e8	1.60064	0.800318	−	0.599576i	$-0.204664\pi$
$11$	1.03452e8	1.60064	0.800318	+	0.599576i	$0.204664\pi$
$12$	0	0
$13$	−1.04366e8	−0.461300	−0.230650	−	0.973037i	$-0.574085\pi$
$13$	−1.04366e8	−0.461300	−0.230650	+	0.973037i	$0.574085\pi$
$14$	0	0
$15$	−2.51089e8	−0.379441
$16$	0	0
$17$	9.97690e8	0.589697	0.294848	−	0.955544i	$-0.404731\pi$
$17$	9.97690e8	0.589697	0.294848	+	0.955544i	$0.404731\pi$
$18$	0	0
$19$	−4.93402e9	−1.26633	−0.633167	−	0.774015i	$-0.718246\pi$
$19$	−4.93402e9	−1.26633	−0.633167	+	0.774015i	$0.718246\pi$
$20$	0	0
$21$	6.63651e9	0.804073
$22$	0	0
$23$	−8.32492e9	−0.509825	−0.254913	−	0.966964i	$-0.582047\pi$
$23$	−8.32492e9	−0.509825	−0.254913	+	0.966964i	$0.582047\pi$
$24$	0	0
$25$	−1.73362e10	−0.568074
$26$	0	0
$27$	1.04604e10	0.192450
$28$	0	0
$29$	1.04128e11	1.12094	0.560472	−	0.828174i	$-0.310620\pi$
$29$	1.04128e11	1.12094	0.560472	+	0.828174i	$0.310620\pi$
$30$	0	0
$31$	2.96697e11	1.93687	0.968434	−	0.249270i	$-0.0801908\pi$
$31$	2.96697e11	1.93687	0.968434	+	0.249270i	$0.0801908\pi$
$32$	0	0
$33$	2.26249e11	0.924127
$34$	0	0
$35$	−3.48394e11	−0.915294
$36$	0	0
$37$	−1.78337e11	−0.308837	−0.154419	−	0.988006i	$-0.549350\pi$
$37$	−1.78337e11	−0.308837	−0.154419	+	0.988006i	$0.549350\pi$
$38$	0	0
$39$	−2.28248e11	−0.266332
$40$	0	0
$41$	−1.79088e12	−1.43611	−0.718056	−	0.695986i	$-0.754968\pi$
$41$	−1.79088e12	−1.43611	−0.718056	+	0.695986i	$0.754968\pi$
$42$	0	0
$43$	2.86346e12	1.60649	0.803244	−	0.595650i	$-0.203105\pi$
$43$	2.86346e12	1.60649	0.803244	+	0.595650i	$0.203105\pi$
$44$	0	0
$45$	−5.49133e11	−0.219070
$46$	0	0
$47$	−4.33291e12	−1.24751	−0.623757	−	0.781618i	$-0.714395\pi$
$47$	−4.33291e12	−1.24751	−0.623757	+	0.781618i	$0.714395\pi$
$48$	0	0
$49$	4.46080e12	0.939598
$50$	0	0
$51$	2.18195e12	0.340461
$52$	0	0
$53$	9.73232e12	1.13801	0.569007	−	0.822333i	$-0.307328\pi$
$53$	9.73232e12	1.13801	0.569007	+	0.822333i	$0.307328\pi$
$54$	0	0
$55$	−1.18773e13	−1.05196
$56$	0	0
$57$	−1.07907e13	−0.731119
$58$	0	0
$59$	1.35148e13	0.707002	0.353501	−	0.935434i	$-0.384991\pi$
$59$	1.35148e13	0.707002	0.353501	+	0.935434i	$0.384991\pi$
$60$	0	0
$61$	5.35266e12	0.218070	0.109035	−	0.994038i	$-0.465224\pi$
$61$	5.35266e12	0.218070	0.109035	+	0.994038i	$0.465224\pi$
$62$	0	0
$63$	1.45141e13	0.464231
$64$	0	0
$65$	1.19822e13	0.303171
$66$	0	0
$67$	5.32339e13	1.07307	0.536534	−	0.843879i	$-0.319733\pi$
$67$	5.32339e13	1.07307	0.536534	+	0.843879i	$0.319733\pi$
$68$	0	0
$69$	−1.82066e13	−0.294348
$70$	0	0
$71$	2.02297e13	0.263968	0.131984	−	0.991252i	$-0.457865\pi$
$71$	2.02297e13	0.263968	0.131984	+	0.991252i	$0.457865\pi$
$72$	0	0
$73$	2.62642e13	0.278254	0.139127	−	0.990275i	$-0.455570\pi$
$73$	2.62642e13	0.278254	0.139127	+	0.990275i	$0.455570\pi$
$74$	0	0
$75$	−3.79144e13	−0.327978
$76$	0	0
$77$	3.13927e14	2.22920
$78$	0	0
$79$	3.39031e14	1.98626	0.993131	−	0.117005i	$-0.0373293\pi$
$79$	3.39031e14	1.98626	0.993131	+	0.117005i	$0.0373293\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	0	0
$83$	−1.31685e14	−0.532660	−0.266330	−	0.963882i	$-0.585811\pi$
$83$	−1.31685e14	−0.532660	−0.266330	+	0.963882i	$0.585811\pi$
$84$	0	0
$85$	−1.14545e14	−0.387555
$86$	0	0
$87$	2.27728e14	0.647177
$88$	0	0
$89$	−3.93521e13	−0.0943069	−0.0471534	−	0.998888i	$-0.515015\pi$
$89$	−3.93521e13	−0.0943069	−0.0471534	+	0.998888i	$0.515015\pi$
$90$	0	0
$91$	−3.16701e14	−0.642450
$92$	0	0
$93$	6.48876e14	1.11825
$94$	0	0
$95$	5.66474e14	0.832249
$96$	0	0
$97$	1.12875e15	1.41844	0.709219	−	0.704989i	$-0.249048\pi$
$97$	1.12875e15	1.41844	0.709219	+	0.704989i	$0.249048\pi$
$98$	0	0
$99$	4.94806e14	0.533545
$100$	0	0
$101$	3.79528e13	0.0352236	0.0176118	−	0.999845i	$-0.494394\pi$
$101$	3.79528e13	0.0352236	0.0176118	+	0.999845i	$0.494394\pi$
$102$	0	0
$103$	2.07297e14	0.166079	0.0830393	−	0.996546i	$-0.473537\pi$
$103$	2.07297e14	0.166079	0.0830393	+	0.996546i	$0.473537\pi$
$104$	0	0
$105$	−7.61938e14	−0.528445
$106$	0	0
$107$	1.99692e15	1.20221	0.601107	−	0.799169i	$-0.294727\pi$
$107$	1.99692e15	1.20221	0.601107	+	0.799169i	$0.294727\pi$
$108$	0	0
$109$	1.35603e13	0.00710510	0.00355255	−	0.999994i	$-0.498869\pi$
$109$	1.35603e13	0.00710510	0.00355255	+	0.999994i	$0.498869\pi$
$110$	0	0
$111$	−3.90024e14	−0.178307
$112$	0	0
$113$	−7.08794e14	−0.283421	−0.141710	−	0.989908i	$-0.545260\pi$
$113$	−7.08794e14	−0.283421	−0.141710	+	0.989908i	$0.545260\pi$
$114$	0	0
$115$	9.55784e14	0.335063
$116$	0	0
$117$	−4.99179e14	−0.153767
$118$	0	0
$119$	3.02752e15	0.821267
$120$	0	0
$121$	6.52501e15	1.56203
$122$	0	0
$123$	−3.91666e15	−0.829139
$124$	0	0
$125$	5.49410e15	1.03056
$126$	0	0
$127$	1.23021e15	0.204858	0.102429	−	0.994740i	$-0.467339\pi$
$127$	1.23021e15	0.204858	0.102429	+	0.994740i	$0.467339\pi$
$128$	0	0
$129$	6.26239e15	0.927507
$130$	0	0
$131$	−7.94836e15	−1.04892	−0.524460	−	0.851435i	$-0.675733\pi$
$131$	−7.94836e15	−1.04892	−0.524460	+	0.851435i	$0.675733\pi$
$132$	0	0
$133$	−1.49724e16	−1.76362
$134$	0	0
$135$	−1.20095e15	−0.126480
$136$	0	0
$137$	−1.78131e16	−1.68010	−0.840050	−	0.542508i	$-0.817475\pi$
$137$	−1.78131e16	−1.68010	−0.840050	+	0.542508i	$0.817475\pi$
$138$	0	0
$139$	−3.89941e15	−0.329904	−0.164952	−	0.986302i	$-0.552747\pi$
$139$	−3.89941e15	−0.329904	−0.164952	+	0.986302i	$0.552747\pi$
$140$	0	0
$141$	−9.47607e15	−0.720253
$142$	0	0
$143$	−1.07968e16	−0.738373
$144$	0	0
$145$	−1.19550e16	−0.736696
$146$	0	0
$147$	9.75577e15	0.542477
$148$	0	0
$149$	−3.48726e15	−0.175222	−0.0876108	−	0.996155i	$-0.527923\pi$
$149$	−3.48726e15	−0.175222	−0.0876108	+	0.996155i	$0.527923\pi$
$150$	0	0
$151$	−6.85712e15	−0.311756	−0.155878	−	0.987776i	$-0.549821\pi$
$151$	−6.85712e15	−0.311756	−0.155878	+	0.987776i	$0.549821\pi$
$152$	0	0
$153$	4.77192e15	0.196566
$154$	0	0
$155$	−3.40637e16	−1.27293
$156$	0	0
$157$	3.69836e16	1.25534	0.627670	−	0.778480i	$-0.284009\pi$
$157$	3.69836e16	1.25534	0.627670	+	0.778480i	$0.284009\pi$
$158$	0	0
$159$	2.12846e16	0.657032
$160$	0	0
$161$	−2.52622e16	−0.710031
$162$	0	0
$163$	−7.42535e15	−0.190244	−0.0951218	−	0.995466i	$-0.530324\pi$
$163$	−7.42535e15	−0.190244	−0.0951218	+	0.995466i	$0.530324\pi$
$164$	0	0
$165$	−2.59756e16	−0.607347
$166$	0	0
$167$	1.47365e16	0.314789	0.157395	−	0.987536i	$-0.449691\pi$
$167$	1.47365e16	0.314789	0.157395	+	0.987536i	$0.449691\pi$
$168$	0	0
$169$	−4.02937e16	−0.787203
$170$	0	0
$171$	−2.35992e16	−0.422112
$172$	0	0
$173$	3.40039e16	0.557421	0.278710	−	0.960375i	$-0.410093\pi$
$173$	3.40039e16	0.557421	0.278710	+	0.960375i	$0.410093\pi$
$174$	0	0
$175$	−5.26073e16	−0.791153
$176$	0	0
$177$	2.95569e16	0.408188
$178$	0	0
$179$	−3.81276e16	−0.483996	−0.241998	−	0.970277i	$-0.577803\pi$
$179$	−3.81276e16	−0.483996	−0.241998	+	0.970277i	$0.577803\pi$
$180$	0	0
$181$	−5.14124e16	−0.600452	−0.300226	−	0.953868i	$-0.597062\pi$
$181$	−5.14124e16	−0.600452	−0.300226	+	0.953868i	$0.597062\pi$
$182$	0	0
$183$	1.17063e16	0.125903
$184$	0	0
$185$	2.04749e16	0.202971
$186$	0	0
$187$	1.03213e17	0.943889
$188$	0	0
$189$	3.17422e16	0.268024
$190$	0	0
$191$	−6.75568e16	−0.527131	−0.263566	−	0.964641i	$-0.584899\pi$
$191$	−6.75568e16	−0.527131	−0.263566	+	0.964641i	$0.584899\pi$
$192$	0	0
$193$	−2.07235e16	−0.149549	−0.0747746	−	0.997200i	$-0.523824\pi$
$193$	−2.07235e16	−0.149549	−0.0747746	+	0.997200i	$0.523824\pi$
$194$	0	0
$195$	2.62052e16	0.175036
$196$	0	0
$197$	1.71244e16	0.105954	0.0529772	−	0.998596i	$-0.483129\pi$
$197$	1.71244e16	0.105954	0.0529772	+	0.998596i	$0.483129\pi$
$198$	0	0
$199$	1.05743e17	0.606533	0.303267	−	0.952906i	$-0.401923\pi$
$199$	1.05743e17	0.606533	0.303267	+	0.952906i	$0.401923\pi$
$200$	0	0
$201$	1.16423e17	0.619536
$202$	0	0
$203$	3.15980e17	1.56113
$204$	0	0
$205$	2.05611e17	0.943828
$206$	0	0
$207$	−3.98178e16	−0.169942
$208$	0	0
$209$	−5.10432e17	−2.02694
$210$	0	0
$211$	3.13093e17	1.15759	0.578795	−	0.815473i	$-0.303523\pi$
$211$	3.13093e17	1.15759	0.578795	+	0.815473i	$0.303523\pi$
$212$	0	0
$213$	4.42423e16	0.152402
$214$	0	0
$215$	−3.28754e17	−1.05580
$216$	0	0
$217$	9.00334e17	2.69747
$218$	0	0
$219$	5.74397e16	0.160650
$220$	0	0
$221$	−1.04125e17	−0.272027
$222$	0	0
$223$	−3.35017e17	−0.818052	−0.409026	−	0.912523i	$-0.634131\pi$
$223$	−3.35017e17	−0.818052	−0.409026	+	0.912523i	$0.634131\pi$
$224$	0	0
$225$	−8.29187e16	−0.189358
$226$	0	0
$227$	−4.67560e17	−0.999180	−0.499590	−	0.866262i	$-0.666516\pi$
$227$	−4.67560e17	−0.999180	−0.499590	+	0.866262i	$0.666516\pi$
$228$	0	0
$229$	3.61186e17	0.722711	0.361355	−	0.932428i	$-0.382314\pi$
$229$	3.61186e17	0.722711	0.361355	+	0.932428i	$0.382314\pi$
$230$	0	0
$231$	6.86559e17	1.28703
$232$	0	0
$233$	3.61787e17	0.635747	0.317873	−	0.948133i	$-0.397031\pi$
$233$	3.61787e17	0.635747	0.317873	+	0.948133i	$0.397031\pi$
$234$	0	0
$235$	4.97461e17	0.819880
$236$	0	0
$237$	7.41462e17	1.14677
$238$	0	0
$239$	1.06842e18	1.55152	0.775762	−	0.631026i	$-0.217366\pi$
$239$	1.06842e18	1.55152	0.775762	+	0.631026i	$0.217366\pi$
$240$	0	0
$241$	−1.00588e18	−1.37220	−0.686102	−	0.727505i	$-0.740680\pi$
$241$	−1.00588e18	−1.37220	−0.686102	+	0.727505i	$0.740680\pi$
$242$	0	0
$243$	5.00315e16	0.0641500
$244$	0	0
$245$	−5.12144e17	−0.617514
$246$	0	0
$247$	5.14943e17	0.584160
$248$	0	0
$249$	−2.87995e17	−0.307531
$250$	0	0
$251$	−1.74766e18	−1.75754	−0.878768	−	0.477249i	$-0.841634\pi$
$251$	−1.74766e18	−1.75754	−0.878768	+	0.477249i	$0.841634\pi$
$252$	0	0
$253$	−8.61227e17	−0.816044
$254$	0	0
$255$	−2.50509e17	−0.223755
$256$	0	0
$257$	−2.77264e17	−0.233558	−0.116779	−	0.993158i	$-0.537257\pi$
$257$	−2.77264e17	−0.233558	−0.116779	+	0.993158i	$0.537257\pi$
$258$	0	0
$259$	−5.41170e17	−0.430116
$260$	0	0
$261$	4.98042e17	0.373648
$262$	0	0
$263$	−1.73303e18	−1.22783	−0.613914	−	0.789373i	$-0.710406\pi$
$263$	−1.73303e18	−1.22783	−0.613914	+	0.789373i	$0.710406\pi$
$264$	0	0
$265$	−1.11737e18	−0.747914
$266$	0	0
$267$	−8.60631e16	−0.0544481
$268$	0	0
$269$	−5.11271e16	−0.0305850	−0.0152925	−	0.999883i	$-0.504868\pi$
$269$	−5.11271e16	−0.0305850	−0.0152925	+	0.999883i	$0.504868\pi$
$270$	0	0
$271$	−2.08455e17	−0.117962	−0.0589812	−	0.998259i	$-0.518785\pi$
$271$	−2.08455e17	−0.117962	−0.0589812	+	0.998259i	$0.518785\pi$
$272$	0	0
$273$	−6.92625e17	−0.370918
$274$	0	0
$275$	−1.79346e18	−0.909279
$276$	0	0
$277$	3.29723e18	1.58326	0.791628	−	0.611003i	$-0.209234\pi$
$277$	3.29723e18	1.58326	0.791628	+	0.611003i	$0.209234\pi$
$278$	0	0
$279$	1.41909e18	0.645623
$280$	0	0
$281$	−3.98328e18	−1.71768	−0.858841	−	0.512242i	$-0.828815\pi$
$281$	−3.98328e18	−1.71768	−0.858841	+	0.512242i	$0.828815\pi$
$282$	0	0
$283$	2.19051e18	0.895668	0.447834	−	0.894117i	$-0.352196\pi$
$283$	2.19051e18	0.895668	0.447834	+	0.894117i	$0.352196\pi$
$284$	0	0
$285$	1.23888e18	0.480499
$286$	0	0
$287$	−5.43448e18	−2.00006
$288$	0	0
$289$	−1.86704e18	−0.652258
$290$	0	0
$291$	2.46858e18	0.818935
$292$	0	0
$293$	−5.27183e18	−1.66132	−0.830662	−	0.556778i	$-0.812038\pi$
$293$	−5.27183e18	−1.66132	−0.830662	+	0.556778i	$0.812038\pi$
$294$	0	0
$295$	−1.55164e18	−0.464649
$296$	0	0
$297$	1.08214e18	0.308042
$298$	0	0
$299$	8.68837e17	0.235182
$300$	0	0
$301$	8.68925e18	2.23735
$302$	0	0
$303$	8.30029e16	0.0203364
$304$	0	0
$305$	−6.14539e17	−0.143318
$306$	0	0
$307$	−4.77067e18	−1.05935	−0.529677	−	0.848199i	$-0.677687\pi$
$307$	−4.77067e18	−1.05935	−0.529677	+	0.848199i	$0.677687\pi$
$308$	0	0
$309$	4.53359e17	0.0958856
$310$	0	0
$311$	6.46666e18	1.30310	0.651549	−	0.758607i	$-0.274120\pi$
$311$	6.46666e18	1.30310	0.651549	+	0.758607i	$0.274120\pi$
$312$	0	0
$313$	−6.00129e18	−1.15256	−0.576278	−	0.817254i	$-0.695496\pi$
$313$	−6.00129e18	−1.15256	−0.576278	+	0.817254i	$0.695496\pi$
$314$	0	0
$315$	−1.66636e18	−0.305098
$316$	0	0
$317$	−9.53943e17	−0.166563	−0.0832814	−	0.996526i	$-0.526540\pi$
$317$	−9.53943e17	−0.166563	−0.0832814	+	0.996526i	$0.526540\pi$
$318$	0	0
$319$	1.07722e19	1.79422
$320$	0	0
$321$	4.36726e18	0.694098
$322$	0	0
$323$	−4.92262e18	−0.746753
$324$	0	0
$325$	1.80931e18	0.262052
$326$	0	0
$327$	2.96563e16	0.00410213
$328$	0	0
$329$	−1.31483e19	−1.73741
$330$	0	0
$331$	5.29071e18	0.668042	0.334021	−	0.942566i	$-0.391594\pi$
$331$	5.29071e18	0.668042	0.334021	+	0.942566i	$0.391594\pi$
$332$	0	0
$333$	−8.52983e17	−0.102946
$334$	0	0
$335$	−6.11179e18	−0.705232
$336$	0	0
$337$	6.22409e18	0.686834	0.343417	−	0.939183i	$-0.388416\pi$
$337$	6.22409e18	0.686834	0.343417	+	0.939183i	$0.388416\pi$
$338$	0	0
$339$	−1.55013e18	−0.163633
$340$	0	0
$341$	3.06938e19	3.10022
$342$	0	0
$343$	−8.70190e17	−0.0841217
$344$	0	0
$345$	2.09030e18	0.193448
$346$	0	0
$347$	3.22035e18	0.285386	0.142693	−	0.989767i	$-0.454424\pi$
$347$	3.22035e18	0.285386	0.142693	+	0.989767i	$0.454424\pi$
$348$	0	0
$349$	−6.17407e18	−0.524060	−0.262030	−	0.965060i	$-0.584392\pi$
$349$	−6.17407e18	−0.524060	−0.262030	+	0.965060i	$0.584392\pi$
$350$	0	0
$351$	−1.09170e18	−0.0887772
$352$	0	0
$353$	−6.14267e18	−0.478681	−0.239341	−	0.970936i	$-0.576931\pi$
$353$	−6.14267e18	−0.478681	−0.239341	+	0.970936i	$0.576931\pi$
$354$	0	0
$355$	−2.32257e18	−0.173483
$356$	0	0
$357$	6.62118e18	0.474159
$358$	0	0
$359$	5.81103e18	0.399066	0.199533	−	0.979891i	$-0.436057\pi$
$359$	5.81103e18	0.399066	0.199533	+	0.979891i	$0.436057\pi$
$360$	0	0
$361$	9.16338e18	0.603603
$362$	0	0
$363$	1.42702e19	0.901841
$364$	0	0
$365$	−3.01539e18	−0.182872
$366$	0	0
$367$	2.77147e17	0.0161330	0.00806650	−	0.999967i	$-0.497432\pi$
$367$	2.77147e17	0.0161330	0.00806650	+	0.999967i	$0.497432\pi$
$368$	0	0
$369$	−8.56574e18	−0.478704
$370$	0	0
$371$	2.95330e19	1.58490
$372$	0	0
$373$	2.30498e19	1.18809	0.594047	−	0.804430i	$-0.297529\pi$
$373$	2.30498e19	1.18809	0.594047	+	0.804430i	$0.297529\pi$
$374$	0	0
$375$	1.20156e19	0.594991
$376$	0	0
$377$	−1.08674e19	−0.517091
$378$	0	0
$379$	1.34397e19	0.614604	0.307302	−	0.951612i	$-0.400574\pi$
$379$	1.34397e19	0.614604	0.307302	+	0.951612i	$0.400574\pi$
$380$	0	0
$381$	2.69048e18	0.118275
$382$	0	0
$383$	−2.64151e19	−1.11651	−0.558254	−	0.829670i	$-0.688529\pi$
$383$	−2.64151e19	−1.11651	−0.558254	+	0.829670i	$0.688529\pi$
$384$	0	0
$385$	−3.60420e19	−1.46505
$386$	0	0
$387$	1.36958e19	0.535496
$388$	0	0
$389$	3.95275e19	1.48688	0.743442	−	0.668800i	$-0.233192\pi$
$389$	3.95275e19	1.48688	0.743442	+	0.668800i	$0.233192\pi$
$390$	0	0
$391$	−8.30569e18	−0.300642
$392$	0	0
$393$	−1.73831e19	−0.605595
$394$	0	0
$395$	−3.89242e19	−1.30539
$396$	0	0
$397$	2.82890e18	0.0913457	0.0456729	−	0.998956i	$-0.485457\pi$
$397$	2.82890e18	0.0913457	0.0456729	+	0.998956i	$0.485457\pi$
$398$	0	0
$399$	−3.27447e19	−1.01822
$400$	0	0
$401$	−4.63681e19	−1.38879	−0.694395	−	0.719594i	$-0.744328\pi$
$401$	−4.63681e19	−1.38879	−0.694395	+	0.719594i	$0.744328\pi$
$402$	0	0
$403$	−3.09650e19	−0.893477
$404$	0	0
$405$	−2.62648e18	−0.0730234
$406$	0	0
$407$	−1.84493e19	−0.494336
$408$	0	0
$409$	−1.17092e19	−0.302415	−0.151207	−	0.988502i	$-0.548316\pi$
$409$	−1.17092e19	−0.302415	−0.151207	+	0.988502i	$0.548316\pi$
$410$	0	0
$411$	−3.89573e19	−0.970007
$412$	0	0
$413$	4.10112e19	0.984638
$414$	0	0
$415$	1.51187e19	0.350070
$416$	0	0
$417$	−8.52801e18	−0.190470
$418$	0	0
$419$	4.20102e19	0.905210	0.452605	−	0.891711i	$-0.350495\pi$
$419$	4.20102e19	0.905210	0.452605	+	0.891711i	$0.350495\pi$
$420$	0	0
$421$	−1.59718e19	−0.332077	−0.166039	−	0.986119i	$-0.553098\pi$
$421$	−1.59718e19	−0.332077	−0.166039	+	0.986119i	$0.553098\pi$
$422$	0	0
$423$	−2.07242e19	−0.415838
$424$	0	0
$425$	−1.72962e19	−0.334991
$426$	0	0
$427$	1.62428e19	0.303705
$428$	0	0
$429$	−2.36127e19	−0.426300
$430$	0	0
$431$	9.76365e19	1.70229	0.851143	−	0.524934i	$-0.175910\pi$
$431$	9.76365e19	1.70229	0.851143	+	0.524934i	$0.175910\pi$
$432$	0	0
$433$	−6.91455e19	−1.16441	−0.582204	−	0.813043i	$-0.697809\pi$
$433$	−6.91455e19	−1.16441	−0.582204	+	0.813043i	$0.697809\pi$
$434$	0	0
$435$	−2.61455e19	−0.425332
$436$	0	0
$437$	4.10753e19	0.645609
$438$	0	0
$439$	−5.48990e19	−0.833836	−0.416918	−	0.908944i	$-0.636890\pi$
$439$	−5.48990e19	−0.833836	−0.416918	+	0.908944i	$0.636890\pi$
$440$	0	0
$441$	2.13359e19	0.313199
$442$	0	0
$443$	−3.80504e19	−0.539923	−0.269961	−	0.962871i	$-0.587011\pi$
$443$	−3.80504e19	−0.539923	−0.269961	+	0.962871i	$0.587011\pi$
$444$	0	0
$445$	4.51802e18	0.0619795
$446$	0	0
$447$	−7.62664e18	−0.101164
$448$	0	0
$449$	1.11994e18	0.0143663	0.00718317	−	0.999974i	$-0.497714\pi$
$449$	1.11994e18	0.0143663	0.00718317	+	0.999974i	$0.497714\pi$
$450$	0	0
$451$	−1.85270e20	−2.29869
$452$	0	0
$453$	−1.49965e19	−0.179992
$454$	0	0
$455$	3.63604e19	0.422225
$456$	0	0
$457$	−9.70734e19	−1.09076	−0.545379	−	0.838189i	$-0.683614\pi$
$457$	−9.70734e19	−1.09076	−0.545379	+	0.838189i	$0.683614\pi$
$458$	0	0
$459$	1.04362e19	0.113487
$460$	0	0
$461$	−9.78165e19	−1.02957	−0.514784	−	0.857320i	$-0.672128\pi$
$461$	−9.78165e19	−1.02957	−0.514784	+	0.857320i	$0.672128\pi$
$462$	0	0
$463$	2.17159e19	0.221269	0.110634	−	0.993861i	$-0.464712\pi$
$463$	2.17159e19	0.221269	0.110634	+	0.993861i	$0.464712\pi$
$464$	0	0
$465$	−7.44974e19	−0.734927
$466$	0	0
$467$	1.98443e20	1.89565	0.947826	−	0.318788i	$-0.103276\pi$
$467$	1.98443e20	1.89565	0.947826	+	0.318788i	$0.103276\pi$
$468$	0	0
$469$	1.61540e20	1.49446
$470$	0	0
$471$	8.08830e19	0.724771
$472$	0	0
$473$	2.96230e20	2.57140
$474$	0	0
$475$	8.55373e19	0.719372
$476$	0	0
$477$	4.65494e19	0.379338
$478$	0	0
$479$	3.78661e19	0.299043	0.149522	−	0.988758i	$-0.452227\pi$
$479$	3.78661e19	0.299043	0.149522	+	0.988758i	$0.452227\pi$
$480$	0	0
$481$	1.86123e19	0.142467
$482$	0	0
$483$	−5.52484e19	−0.409936
$484$	0	0
$485$	−1.29592e20	−0.932212
$486$	0	0
$487$	−3.89265e19	−0.271505	−0.135752	−	0.990743i	$-0.543345\pi$
$487$	−3.89265e19	−0.271505	−0.135752	+	0.990743i	$0.543345\pi$
$488$	0	0
$489$	−1.62392e19	−0.109837
$490$	0	0
$491$	−1.40516e20	−0.921753	−0.460876	−	0.887464i	$-0.652465\pi$
$491$	−1.40516e20	−0.921753	−0.460876	+	0.887464i	$0.652465\pi$
$492$	0	0
$493$	1.03888e20	0.661016
$494$	0	0
$495$	−5.68087e19	−0.350652
$496$	0	0
$497$	6.13875e19	0.367627
$498$	0	0
$499$	1.53703e20	0.893157	0.446578	−	0.894745i	$-0.352642\pi$
$499$	1.53703e20	0.893157	0.446578	+	0.894745i	$0.352642\pi$
$500$	0	0
$501$	3.22287e19	0.181744
$502$	0	0
$503$	−5.62265e18	−0.0307738	−0.0153869	−	0.999882i	$-0.504898\pi$
$503$	−5.62265e18	−0.0307738	−0.0153869	+	0.999882i	$0.504898\pi$
$504$	0	0
$505$	−4.35737e18	−0.0231493
$506$	0	0
$507$	−8.81222e19	−0.454492
$508$	0	0
$509$	3.29307e20	1.64899	0.824495	−	0.565869i	$-0.191459\pi$
$509$	3.29307e20	1.64899	0.824495	+	0.565869i	$0.191459\pi$
$510$	0	0
$511$	7.96993e19	0.387523
$512$	0	0
$513$	−5.16115e19	−0.243706
$514$	0	0
$515$	−2.37998e19	−0.109149
$516$	0	0
$517$	−4.48247e20	−1.99682
$518$	0	0
$519$	7.43666e19	0.321827
$520$	0	0
$521$	1.71775e20	0.722231	0.361116	−	0.932521i	$-0.382396\pi$
$521$	1.71775e20	0.722231	0.361116	+	0.932521i	$0.382396\pi$
$522$	0	0
$523$	1.59414e20	0.651275	0.325637	−	0.945495i	$-0.394421\pi$
$523$	1.59414e20	0.651275	0.325637	+	0.945495i	$0.394421\pi$
$524$	0	0
$525$	−1.15052e20	−0.456773
$526$	0	0
$527$	2.96011e20	1.14216
$528$	0	0
$529$	−1.97331e20	−0.740078
$530$	0	0
$531$	6.46410e19	0.235667
$532$	0	0
$533$	1.86907e20	0.662478
$534$	0	0
$535$	−2.29266e20	−0.790108
$536$	0	0
$537$	−8.33851e19	−0.279435
$538$	0	0
$539$	4.61477e20	1.50395
$540$	0	0
$541$	4.74121e19	0.150283	0.0751415	−	0.997173i	$-0.476059\pi$
$541$	4.74121e19	0.150283	0.0751415	+	0.997173i	$0.476059\pi$
$542$	0	0
$543$	−1.12439e20	−0.346671
$544$	0	0
$545$	−1.55686e18	−0.00466955
$546$	0	0
$547$	2.80986e20	0.819934	0.409967	−	0.912100i	$-0.365540\pi$
$547$	2.80986e20	0.819934	0.409967	+	0.912100i	$0.365540\pi$
$548$	0	0
$549$	2.56016e19	0.0726900
$550$	0	0
$551$	−5.13770e20	−1.41949
$552$	0	0
$553$	1.02880e21	2.76626
$554$	0	0
$555$	4.47787e19	0.117185
$556$	0	0
$557$	−3.49642e20	−0.890654	−0.445327	−	0.895368i	$-0.646913\pi$
$557$	−3.49642e20	−0.890654	−0.445327	+	0.895368i	$0.646913\pi$
$558$	0	0
$559$	−2.98847e20	−0.741073
$560$	0	0
$561$	2.25726e20	0.544955
$562$	0	0
$563$	−2.68215e20	−0.630479	−0.315240	−	0.949012i	$-0.602085\pi$
$563$	−2.68215e20	−0.630479	−0.315240	+	0.949012i	$0.602085\pi$
$564$	0	0
$565$	8.13766e19	0.186267
$566$	0	0
$567$	6.94203e19	0.154744
$568$	0	0
$569$	−7.95691e20	−1.72744	−0.863719	−	0.503974i	$-0.831871\pi$
$569$	−7.95691e20	−1.72744	−0.863719	+	0.503974i	$0.831871\pi$
$570$	0	0
$571$	−8.84470e20	−1.87030	−0.935152	−	0.354246i	$-0.884738\pi$
$571$	−8.84470e20	−1.87030	−0.935152	+	0.354246i	$0.884738\pi$
$572$	0	0
$573$	−1.47747e20	−0.304339
$574$	0	0
$575$	1.44323e20	0.289618
$576$	0	0
$577$	3.15722e20	0.617286	0.308643	−	0.951178i	$-0.400125\pi$
$577$	3.15722e20	0.617286	0.308643	+	0.951178i	$0.400125\pi$
$578$	0	0
$579$	−4.53224e19	−0.0863423
$580$	0	0
$581$	−3.99601e20	−0.741832
$582$	0	0
$583$	1.00682e21	1.82154
$584$	0	0
$585$	5.73107e19	0.101057
$586$	0	0
$587$	−7.33397e20	−1.26053	−0.630265	−	0.776380i	$-0.717054\pi$
$587$	−7.33397e20	−1.26053	−0.630265	+	0.776380i	$0.717054\pi$
$588$	0	0
$589$	−1.46391e21	−2.45272
$590$	0	0
$591$	3.74511e19	0.0611728
$592$	0	0
$593$	−3.25571e20	−0.518485	−0.259242	−	0.965812i	$-0.583473\pi$
$593$	−3.25571e20	−0.518485	−0.259242	+	0.965812i	$0.583473\pi$
$594$	0	0
$595$	−3.47589e20	−0.539746
$596$	0	0
$597$	2.31261e20	0.350182
$598$	0	0
$599$	−6.61173e20	−0.976368	−0.488184	−	0.872741i	$-0.662341\pi$
$599$	−6.61173e20	−0.976368	−0.488184	+	0.872741i	$0.662341\pi$
$600$	0	0
$601$	−2.49467e20	−0.359297	−0.179649	−	0.983731i	$-0.557496\pi$
$601$	−2.49467e20	−0.359297	−0.179649	+	0.983731i	$0.557496\pi$
$602$	0	0
$603$	2.54616e20	0.357689
$604$	0	0
$605$	−7.49136e20	−1.02659
$606$	0	0
$607$	−1.08917e21	−1.45606	−0.728031	−	0.685544i	$-0.759564\pi$
$607$	−1.08917e21	−1.45606	−0.728031	+	0.685544i	$0.759564\pi$
$608$	0	0
$609$	6.91048e20	0.901320
$610$	0	0
$611$	4.52208e20	0.575478
$612$	0	0
$613$	1.99450e20	0.247674	0.123837	−	0.992303i	$-0.460480\pi$
$613$	1.99450e20	0.247674	0.123837	+	0.992303i	$0.460480\pi$
$614$	0	0
$615$	4.49672e20	0.544919
$616$	0	0
$617$	4.21127e20	0.498052	0.249026	−	0.968497i	$-0.419890\pi$
$617$	4.21127e20	0.498052	0.249026	+	0.968497i	$0.419890\pi$
$618$	0	0
$619$	1.27272e21	1.46910	0.734552	−	0.678552i	$-0.237392\pi$
$619$	1.27272e21	1.46910	0.734552	+	0.678552i	$0.237392\pi$
$620$	0	0
$621$	−8.70816e19	−0.0981159
$622$	0	0
$623$	−1.19415e20	−0.131341
$624$	0	0
$625$	−1.01717e20	−0.109218
$626$	0	0
$627$	−1.11632e21	−1.17025
$628$	0	0
$629$	−1.77925e20	−0.182120
$630$	0	0
$631$	3.42884e20	0.342711	0.171355	−	0.985209i	$-0.445185\pi$
$631$	3.42884e20	0.342711	0.171355	+	0.985209i	$0.445185\pi$
$632$	0	0
$633$	6.84735e20	0.668335
$634$	0	0
$635$	−1.41241e20	−0.134635
$636$	0	0
$637$	−4.65555e20	−0.433436
$638$	0	0
$639$	9.67578e19	0.0879893
$640$	0	0
$641$	1.00075e21	0.888980	0.444490	−	0.895784i	$-0.353385\pi$
$641$	1.00075e21	0.888980	0.444490	+	0.895784i	$0.353385\pi$
$642$	0	0
$643$	1.35729e20	0.117785	0.0588927	−	0.998264i	$-0.481243\pi$
$643$	1.35729e20	0.117785	0.0588927	+	0.998264i	$0.481243\pi$
$644$	0	0
$645$	−7.18985e20	−0.609567
$646$	0	0
$647$	−2.15462e21	−1.78479	−0.892397	−	0.451251i	$-0.850978\pi$
$647$	−2.15462e21	−1.78479	−0.892397	+	0.451251i	$0.850978\pi$
$648$	0	0
$649$	1.39813e21	1.13165
$650$	0	0
$651$	1.96903e21	1.55738
$652$	0	0
$653$	−2.06101e21	−1.59306	−0.796529	−	0.604601i	$-0.793333\pi$
$653$	−2.06101e21	−1.59306	−0.796529	+	0.604601i	$0.793333\pi$
$654$	0	0
$655$	9.12551e20	0.689362
$656$	0	0
$657$	1.25621e20	0.0927515
$658$	0	0
$659$	1.61309e21	1.16418	0.582089	−	0.813125i	$-0.302236\pi$
$659$	1.61309e21	1.16418	0.582089	+	0.813125i	$0.302236\pi$
$660$	0	0
$661$	1.60888e21	1.13504	0.567521	−	0.823359i	$-0.307903\pi$
$661$	1.60888e21	1.13504	0.567521	+	0.823359i	$0.307903\pi$
$662$	0	0
$663$	−2.27721e20	−0.157055
$664$	0	0
$665$	1.71898e21	1.15907
$666$	0	0
$667$	−8.66859e20	−0.571485
$668$	0	0
$669$	−7.32683e20	−0.472303
$670$	0	0
$671$	5.53742e20	0.349051
$672$	0	0
$673$	1.67294e21	1.03126	0.515628	−	0.856812i	$-0.327559\pi$
$673$	1.67294e21	1.03126	0.515628	+	0.856812i	$0.327559\pi$
$674$	0	0
$675$	−1.81343e20	−0.109326
$676$	0	0
$677$	7.27000e20	0.428666	0.214333	−	0.976761i	$-0.431242\pi$
$677$	7.27000e20	0.428666	0.214333	+	0.976761i	$0.431242\pi$
$678$	0	0
$679$	3.42523e21	1.97545
$680$	0	0
$681$	−1.02255e21	−0.576877
$682$	0	0
$683$	−2.85909e21	−1.57787	−0.788936	−	0.614475i	$-0.789368\pi$
$683$	−2.85909e21	−1.57787	−0.788936	+	0.614475i	$0.789368\pi$
$684$	0	0
$685$	2.04512e21	1.10418
$686$	0	0
$687$	7.89913e20	0.417257
$688$	0	0
$689$	−1.01572e21	−0.524965
$690$	0	0
$691$	−1.71274e21	−0.866178	−0.433089	−	0.901351i	$-0.642576\pi$
$691$	−1.71274e21	−0.866178	−0.433089	+	0.901351i	$0.642576\pi$
$692$	0	0
$693$	1.50150e21	0.743065
$694$	0	0
$695$	4.47691e20	0.216817
$696$	0	0
$697$	−1.78675e21	−0.846870
$698$	0	0
$699$	7.91229e20	0.367049
$700$	0	0
$701$	−8.95519e20	−0.406621	−0.203311	−	0.979114i	$-0.565170\pi$
$701$	−8.95519e20	−0.406621	−0.203311	+	0.979114i	$0.565170\pi$
$702$	0	0
$703$	8.79920e20	0.391091
$704$	0	0
$705$	1.08795e21	0.473358
$706$	0	0
$707$	1.15169e20	0.0490557
$708$	0	0
$709$	−3.99642e21	−1.66657	−0.833286	−	0.552842i	$-0.813543\pi$
$709$	−3.99642e21	−1.66657	−0.833286	+	0.552842i	$0.813543\pi$
$710$	0	0
$711$	1.62158e21	0.662088
$712$	0	0
$713$	−2.46998e21	−0.987464
$714$	0	0
$715$	1.23958e21	0.485267
$716$	0	0
$717$	2.33664e21	0.895772
$718$	0	0
$719$	2.66669e20	0.100117	0.0500583	−	0.998746i	$-0.484059\pi$
$719$	2.66669e20	0.100117	0.0500583	+	0.998746i	$0.484059\pi$
$720$	0	0
$721$	6.29049e20	0.231297
$722$	0	0
$723$	−2.19986e21	−0.792242
$724$	0	0
$725$	−1.80519e21	−0.636779
$726$	0	0
$727$	1.69447e21	0.585499	0.292750	−	0.956189i	$-0.405430\pi$
$727$	1.69447e21	0.585499	0.292750	+	0.956189i	$0.405430\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	2.85684e21	0.947341
$732$	0	0
$733$	3.16239e21	1.02739	0.513695	−	0.857973i	$-0.328276\pi$
$733$	3.16239e21	1.02739	0.513695	+	0.857973i	$0.328276\pi$
$734$	0	0
$735$	−1.12006e21	−0.356522
$736$	0	0
$737$	5.50714e21	1.71759
$738$	0	0
$739$	−2.01381e21	−0.615440	−0.307720	−	0.951477i	$-0.599566\pi$
$739$	−2.01381e21	−0.615440	−0.307720	+	0.951477i	$0.599566\pi$
$740$	0	0
$741$	1.12618e21	0.337265
$742$	0	0
$743$	−2.11411e21	−0.620458	−0.310229	−	0.950662i	$-0.600406\pi$
$743$	−2.11411e21	−0.620458	−0.310229	+	0.950662i	$0.600406\pi$
$744$	0	0
$745$	4.00372e20	0.115157
$746$	0	0
$747$	−6.29844e20	−0.177553
$748$	0	0
$749$	6.05970e21	1.67432
$750$	0	0
$751$	−4.95087e21	−1.34086	−0.670428	−	0.741974i	$-0.733890\pi$
$751$	−4.95087e21	−1.34086	−0.670428	+	0.741974i	$0.733890\pi$
$752$	0	0
$753$	−3.82213e21	−1.01471
$754$	0	0
$755$	7.87266e20	0.204889
$756$	0	0
$757$	−4.85879e21	−1.23968	−0.619839	−	0.784729i	$-0.712802\pi$
$757$	−4.85879e21	−1.23968	−0.619839	+	0.784729i	$0.712802\pi$
$758$	0	0
$759$	−1.88350e21	−0.471143
$760$	0	0
$761$	−3.69092e21	−0.905211	−0.452605	−	0.891711i	$-0.649505\pi$
$761$	−3.69092e21	−0.905211	−0.452605	+	0.891711i	$0.649505\pi$
$762$	0	0
$763$	4.11491e19	0.00989523
$764$	0	0
$765$	−5.47864e20	−0.129185
$766$	0	0
$767$	−1.41049e21	−0.326140
$768$	0	0
$769$	−2.33815e21	−0.530181	−0.265091	−	0.964224i	$-0.585402\pi$
$769$	−2.33815e21	−0.530181	−0.265091	+	0.964224i	$0.585402\pi$
$770$	0	0
$771$	−6.06376e20	−0.134845
$772$	0	0
$773$	−5.20682e21	−1.13560	−0.567801	−	0.823166i	$-0.692206\pi$
$773$	−5.20682e21	−1.13560	−0.567801	+	0.823166i	$0.692206\pi$
$774$	0	0
$775$	−5.14361e21	−1.10028
$776$	0	0
$777$	−1.18354e21	−0.248328
$778$	0	0
$779$	8.83624e21	1.81860
$780$	0	0
$781$	2.09279e21	0.422516
$782$	0	0
$783$	1.08922e21	0.215726
$784$	0	0
$785$	−4.24608e21	−0.825023
$786$	0	0
$787$	1.92196e21	0.366381	0.183191	−	0.983077i	$-0.441357\pi$
$787$	1.92196e21	0.366381	0.183191	+	0.983077i	$0.441357\pi$
$788$	0	0
$789$	−3.79013e21	−0.708887
$790$	0	0
$791$	−2.15085e21	−0.394718
$792$	0	0
$793$	−5.58635e20	−0.100596
$794$	0	0
$795$	−2.44368e21	−0.431809
$796$	0	0
$797$	−6.23880e20	−0.108184	−0.0540921	−	0.998536i	$-0.517226\pi$
$797$	−6.23880e20	−0.108184	−0.0540921	+	0.998536i	$0.517226\pi$
$798$	0	0
$799$	−4.32290e21	−0.735655
$800$	0	0
$801$	−1.88220e20	−0.0314356
$802$	0	0
$803$	2.71707e21	0.445384
$804$	0	0
$805$	2.90035e21	0.466640
$806$	0	0
$807$	−1.11815e20	−0.0176583
$808$	0	0
$809$	5.72858e21	0.888042	0.444021	−	0.896016i	$-0.353552\pi$
$809$	5.72858e21	0.888042	0.444021	+	0.896016i	$0.353552\pi$
$810$	0	0
$811$	−4.80014e21	−0.730462	−0.365231	−	0.930917i	$-0.619010\pi$
$811$	−4.80014e21	−0.730462	−0.365231	+	0.930917i	$0.619010\pi$
$812$	0	0
$813$	−4.55892e20	−0.0681056
$814$	0	0
$815$	8.52505e20	0.125030
$816$	0	0
$817$	−1.41284e22	−2.03435
$818$	0	0
$819$	−1.51477e21	−0.214150
$820$	0	0
$821$	8.89775e21	1.23511	0.617556	−	0.786527i	$-0.288123\pi$
$821$	8.89775e21	1.23511	0.617556	+	0.786527i	$0.288123\pi$
$822$	0	0
$823$	1.39946e21	0.190749	0.0953744	−	0.995441i	$-0.469595\pi$
$823$	1.39946e21	0.190749	0.0953744	+	0.995441i	$0.469595\pi$
$824$	0	0
$825$	−3.92231e21	−0.524973
$826$	0	0
$827$	−6.61343e21	−0.869232	−0.434616	−	0.900616i	$-0.643116\pi$
$827$	−6.61343e21	−0.869232	−0.434616	+	0.900616i	$0.643116\pi$
$828$	0	0
$829$	4.14735e21	0.535318	0.267659	−	0.963514i	$-0.413750\pi$
$829$	4.14735e21	0.535318	0.267659	+	0.963514i	$0.413750\pi$
$830$	0	0
$831$	7.21105e21	0.914093
$832$	0	0
$833$	4.45049e21	0.554078
$834$	0	0
$835$	−1.69190e21	−0.206883
$836$	0	0
$837$	3.10355e21	0.372750
$838$	0	0
$839$	−9.57811e21	−1.12997	−0.564983	−	0.825103i	$-0.691117\pi$
$839$	−9.57811e21	−1.12997	−0.564983	+	0.825103i	$0.691117\pi$
$840$	0	0
$841$	2.21350e21	0.256513
$842$	0	0
$843$	−8.71142e21	−0.991705
$844$	0	0
$845$	4.62612e21	0.517358
$846$	0	0
$847$	1.98003e22	2.17544
$848$	0	0
$849$	4.79065e21	0.517114
$850$	0	0
$851$	1.48465e21	0.157453
$852$	0	0
$853$	−8.15361e21	−0.849634	−0.424817	−	0.905279i	$-0.639662\pi$
$853$	−8.15361e21	−0.849634	−0.424817	+	0.905279i	$0.639662\pi$
$854$	0	0
$855$	2.70943e21	0.277416
$856$	0	0
$857$	−1.53552e22	−1.54490	−0.772449	−	0.635076i	$-0.780969\pi$
$857$	−1.53552e22	−1.54490	−0.772449	+	0.635076i	$0.780969\pi$
$858$	0	0
$859$	4.87617e21	0.482092	0.241046	−	0.970514i	$-0.422510\pi$
$859$	4.87617e21	0.482092	0.241046	+	0.970514i	$0.422510\pi$
$860$	0	0
$861$	−1.18852e22	−1.15474
$862$	0	0
$863$	4.76667e21	0.455128	0.227564	−	0.973763i	$-0.426924\pi$
$863$	4.76667e21	0.455128	0.227564	+	0.973763i	$0.426924\pi$
$864$	0	0
$865$	−3.90399e21	−0.366343
$866$	0	0
$867$	−4.08321e21	−0.376581
$868$	0	0
$869$	3.50734e22	3.17928
$870$	0	0
$871$	−5.55580e21	−0.495006
$872$	0	0
$873$	5.39878e21	0.472812
$874$	0	0
$875$	1.66720e22	1.43525
$876$	0	0
$877$	1.57944e22	1.33661	0.668306	−	0.743887i	$-0.267020\pi$
$877$	1.57944e22	1.33661	0.668306	+	0.743887i	$0.267020\pi$
$878$	0	0
$879$	−1.15295e22	−0.959165
$880$	0	0
$881$	−1.12789e22	−0.922461	−0.461230	−	0.887280i	$-0.652592\pi$
$881$	−1.12789e22	−0.922461	−0.461230	+	0.887280i	$0.652592\pi$
$882$	0	0
$883$	−4.48307e21	−0.360471	−0.180236	−	0.983623i	$-0.557686\pi$
$883$	−4.48307e21	−0.360471	−0.180236	+	0.983623i	$0.557686\pi$
$884$	0	0
$885$	−3.39343e21	−0.268265
$886$	0	0
$887$	8.56246e21	0.665536	0.332768	−	0.943009i	$-0.392017\pi$
$887$	8.56246e21	0.665536	0.332768	+	0.943009i	$0.392017\pi$
$888$	0	0
$889$	3.73312e21	0.285304
$890$	0	0
$891$	2.36664e21	0.177848
$892$	0	0
$893$	2.13786e22	1.57977
$894$	0	0
$895$	4.37743e21	0.318087
$896$	0	0
$897$	1.90015e21	0.135783
$898$	0	0
$899$	3.08945e22	2.17112
$900$	0	0
$901$	9.70983e21	0.671082
$902$	0	0
$903$	1.90034e22	1.29173
$904$	0	0
$905$	5.90266e21	0.394624
$906$	0	0
$907$	−7.34638e21	−0.483079	−0.241540	−	0.970391i	$-0.577652\pi$
$907$	−7.34638e21	−0.483079	−0.241540	+	0.970391i	$0.577652\pi$
$908$	0	0
$909$	1.81527e20	0.0117412
$910$	0	0
$911$	1.18999e22	0.757106	0.378553	−	0.925580i	$-0.376422\pi$
$911$	1.18999e22	0.757106	0.378553	+	0.925580i	$0.376422\pi$
$912$	0	0
$913$	−1.36230e22	−0.852594
$914$	0	0
$915$	−1.34400e21	−0.0827447
$916$	0	0
$917$	−2.41195e22	−1.46083
$918$	0	0
$919$	−1.92629e22	−1.14777	−0.573886	−	0.818935i	$-0.694565\pi$
$919$	−1.92629e22	−1.14777	−0.573886	+	0.818935i	$0.694565\pi$
$920$	0	0
$921$	−1.04334e22	−0.611618
$922$	0	0
$923$	−2.11129e21	−0.121768
$924$	0	0
$925$	3.09170e21	0.175442
$926$	0	0
$927$	9.91495e20	0.0553596
$928$	0	0
$929$	8.78109e21	0.482426	0.241213	−	0.970472i	$-0.422455\pi$
$929$	8.78109e21	0.482426	0.241213	+	0.970472i	$0.422455\pi$
$930$	0	0
$931$	−2.20096e22	−1.18985
$932$	0	0
$933$	1.41426e22	0.752343
$934$	0	0
$935$	−1.18499e22	−0.620334
$936$	0	0
$937$	−2.35424e22	−1.21284	−0.606421	−	0.795144i	$-0.707395\pi$
$937$	−2.35424e22	−1.21284	−0.606421	+	0.795144i	$0.707395\pi$
$938$	0	0
$939$	−1.31248e22	−0.665429
$940$	0	0
$941$	−2.31660e22	−1.15592	−0.577961	−	0.816064i	$-0.696152\pi$
$941$	−2.31660e22	−1.15592	−0.577961	+	0.816064i	$0.696152\pi$
$942$	0	0
$943$	1.49090e22	0.732166
$944$	0	0
$945$	−3.64433e21	−0.176148
$946$	0	0
$947$	−2.02578e22	−0.963755	−0.481877	−	0.876239i	$-0.660045\pi$
$947$	−2.02578e22	−0.963755	−0.481877	+	0.876239i	$0.660045\pi$
$948$	0	0
$949$	−2.74108e21	−0.128359
$950$	0	0
$951$	−2.08627e21	−0.0961651
$952$	0	0
$953$	−1.66351e22	−0.754794	−0.377397	−	0.926052i	$-0.623181\pi$
$953$	−1.66351e22	−0.754794	−0.377397	+	0.926052i	$0.623181\pi$
$954$	0	0
$955$	7.75620e21	0.346436
$956$	0	0
$957$	2.35589e22	1.03589
$958$	0	0
$959$	−5.40544e22	−2.33987
$960$	0	0
$961$	6.45637e22	2.75146
$962$	0	0
$963$	9.55119e21	0.400738
$964$	0	0
$965$	2.37927e21	0.0982853
$966$	0	0
$967$	2.10794e22	0.857355	0.428677	−	0.903458i	$-0.358980\pi$
$967$	2.10794e22	0.857355	0.428677	+	0.903458i	$0.358980\pi$
$968$	0	0
$969$	−1.07658e22	−0.431138
$970$	0	0
$971$	3.09261e22	1.21950	0.609749	−	0.792594i	$-0.291270\pi$
$971$	3.09261e22	1.21950	0.609749	+	0.792594i	$0.291270\pi$
$972$	0	0
$973$	−1.18329e22	−0.459456
$974$	0	0
$975$	3.95696e21	0.151296
$976$	0	0
$977$	7.85725e21	0.295843	0.147922	−	0.988999i	$-0.452742\pi$
$977$	7.85725e21	0.295843	0.147922	+	0.988999i	$0.452742\pi$
$978$	0	0
$979$	−4.07105e21	−0.150951
$980$	0	0
$981$	6.48584e19	0.00236837
$982$	0	0
$983$	−1.27981e22	−0.460251	−0.230125	−	0.973161i	$-0.573914\pi$
$983$	−1.27981e22	−0.460251	−0.230125	+	0.973161i	$0.573914\pi$
$984$	0	0
$985$	−1.96605e21	−0.0696343
$986$	0	0
$987$	−2.87554e22	−1.00309
$988$	0	0
$989$	−2.38381e22	−0.819028
$990$	0	0
$991$	5.39385e22	1.82535	0.912675	−	0.408686i	$-0.134013\pi$
$991$	5.39385e22	1.82535	0.912675	+	0.408686i	$0.134013\pi$
$992$	0	0
$993$	1.15708e22	0.385694
$994$	0	0
$995$	−1.21404e22	−0.398620
$996$	0	0
$997$	−1.99678e22	−0.645828	−0.322914	−	0.946428i	$-0.604663\pi$
$997$	−1.99678e22	−0.645828	−0.322914	+	0.946428i	$0.604663\pi$
$998$	0	0
$999$	−1.86547e21	−0.0594358

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.16.a.d.1.1		1
3.2	odd	2		144.16.a.j.1.1		1
4.3	odd	2		6.16.a.b.1.1	✓	1
12.11	even	2		18.16.a.b.1.1		1
20.3	even	4		150.16.c.a.49.1		2
20.7	even	4		150.16.c.a.49.2		2
20.19	odd	2		150.16.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.16.a.b.1.1	✓	1	4.3	odd	2
18.16.a.b.1.1		1	12.11	even	2
48.16.a.d.1.1		1	1.1	even	1	trivial
144.16.a.j.1.1		1	3.2	odd	2
150.16.a.f.1.1		1	20.19	odd	2
150.16.c.a.49.1		2	20.3	even	4
150.16.c.a.49.2		2	20.7	even	4

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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	48.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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