LMFDB - Embedded newform 48.16.a.c.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)

$f(q)$

$=$

$q+2187.00 q^{3} -314490. q^{5} -2.02506e6 q^{7} +4.78297e6 q^{9} +O(q^{10})$

$q+2187.00 q^{3} -314490. q^{5} -2.02506e6 q^{7} +4.78297e6 q^{9} -1.10255e8 q^{11} +5.60479e7 q^{13} -6.87790e8 q^{15} -1.93010e9 q^{17} -2.16319e9 q^{19} -4.42880e9 q^{21} -6.22897e9 q^{23} +6.83864e10 q^{25} +1.04604e10 q^{27} +6.47437e10 q^{29} +2.02376e10 q^{31} -2.41128e11 q^{33} +6.36860e11 q^{35} +4.88968e11 q^{37} +1.22577e11 q^{39} -7.72359e11 q^{41} -1.30677e12 q^{43} -1.50420e12 q^{45} -3.35182e12 q^{47} -6.46710e11 q^{49} -4.22114e12 q^{51} +9.38781e12 q^{53} +3.46741e13 q^{55} -4.73089e12 q^{57} -2.89304e13 q^{59} +4.23931e13 q^{61} -9.68578e12 q^{63} -1.76265e13 q^{65} +5.22472e13 q^{67} -1.36228e13 q^{69} +2.71945e13 q^{71} -9.16042e13 q^{73} +1.49561e14 q^{75} +2.23273e14 q^{77} -6.28821e13 q^{79} +2.28768e13 q^{81} +2.23567e14 q^{83} +6.06999e14 q^{85} +1.41595e14 q^{87} +5.54199e14 q^{89} -1.13500e14 q^{91} +4.42597e13 q^{93} +6.80301e14 q^{95} -1.38887e15 q^{97} -5.27346e14 q^{99} +O(q^{100})$

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
$5$	−314490.	−1.80025	−0.900123	−	0.435636i	$-0.856523\pi$
$5$	−314490.	−1.80025	−0.900123	+	0.435636i	$0.856523\pi$
$6$	0	0
$7$	−2.02506e6	−0.929398	−0.464699	−	0.885469i	$-0.653837\pi$
$7$	−2.02506e6	−0.929398	−0.464699	+	0.885469i	$0.653837\pi$
$8$	0	0
$9$	4.78297e6	0.333333
$10$	0	0
$11$	−1.10255e8	−1.70590	−0.852950	−	0.521993i	$-0.825189\pi$
$11$	−1.10255e8	−1.70590	−0.852950	+	0.521993i	$0.825189\pi$
$12$	0	0
$13$	5.60479e7	0.247733	0.123867	−	0.992299i	$-0.460471\pi$
$13$	5.60479e7	0.247733	0.123867	+	0.992299i	$0.460471\pi$
$14$	0	0
$15$	−6.87790e8	−1.03937
$16$	0	0
$17$	−1.93010e9	−1.14081	−0.570406	−	0.821363i	$-0.693214\pi$
$17$	−1.93010e9	−1.14081	−0.570406	+	0.821363i	$0.693214\pi$
$18$	0	0
$19$	−2.16319e9	−0.555191	−0.277595	−	0.960698i	$-0.589537\pi$
$19$	−2.16319e9	−0.555191	−0.277595	+	0.960698i	$0.589537\pi$
$20$	0	0
$21$	−4.42880e9	−0.536588
$22$	0	0
$23$	−6.22897e9	−0.381468	−0.190734	−	0.981642i	$-0.561087\pi$
$23$	−6.22897e9	−0.381468	−0.190734	+	0.981642i	$0.561087\pi$
$24$	0	0
$25$	6.83864e10	2.24088
$26$	0	0
$27$	1.04604e10	0.192450
$28$	0	0
$29$	6.47437e10	0.696968	0.348484	−	0.937315i	$-0.386697\pi$
$29$	6.47437e10	0.696968	0.348484	+	0.937315i	$0.386697\pi$
$30$	0	0
$31$	2.02376e10	0.132113	0.0660567	−	0.997816i	$-0.478958\pi$
$31$	2.02376e10	0.132113	0.0660567	+	0.997816i	$0.478958\pi$
$32$	0	0
$33$	−2.41128e11	−0.984901
$34$	0	0
$35$	6.36860e11	1.67314
$36$	0	0
$37$	4.88968e11	0.846773	0.423387	−	0.905949i	$-0.360841\pi$
$37$	4.88968e11	0.846773	0.423387	+	0.905949i	$0.360841\pi$
$38$	0	0
$39$	1.22577e11	0.143029
$40$	0	0
$41$	−7.72359e11	−0.619356	−0.309678	−	0.950841i	$-0.600221\pi$
$41$	−7.72359e11	−0.619356	−0.309678	+	0.950841i	$0.600221\pi$
$42$	0	0
$43$	−1.30677e12	−0.733136	−0.366568	−	0.930391i	$-0.619467\pi$
$43$	−1.30677e12	−0.733136	−0.366568	+	0.930391i	$0.619467\pi$
$44$	0	0
$45$	−1.50420e12	−0.600082
$46$	0	0
$47$	−3.35182e12	−0.965044	−0.482522	−	0.875884i	$-0.660279\pi$
$47$	−3.35182e12	−0.965044	−0.482522	+	0.875884i	$0.660279\pi$
$48$	0	0
$49$	−6.46710e11	−0.136219
$50$	0	0
$51$	−4.22114e12	−0.658648
$52$	0	0
$53$	9.38781e12	1.09773	0.548865	−	0.835911i	$-0.315060\pi$
$53$	9.38781e12	1.09773	0.548865	+	0.835911i	$0.315060\pi$
$54$	0	0
$55$	3.46741e13	3.07104
$56$	0	0
$57$	−4.73089e12	−0.320540
$58$	0	0
$59$	−2.89304e13	−1.51343	−0.756717	−	0.653742i	$-0.773198\pi$
$59$	−2.89304e13	−1.51343	−0.756717	+	0.653742i	$0.773198\pi$
$60$	0	0
$61$	4.23931e13	1.72711	0.863557	−	0.504251i	$-0.168231\pi$
$61$	4.23931e13	1.72711	0.863557	+	0.504251i	$0.168231\pi$
$62$	0	0
$63$	−9.68578e12	−0.309799
$64$	0	0
$65$	−1.76265e13	−0.445980
$66$	0	0
$67$	5.22472e13	1.05318	0.526590	−	0.850120i	$-0.323470\pi$
$67$	5.22472e13	1.05318	0.526590	+	0.850120i	$0.323470\pi$
$68$	0	0
$69$	−1.36228e13	−0.220240
$70$	0	0
$71$	2.71945e13	0.354849	0.177425	−	0.984134i	$-0.443223\pi$
$71$	2.71945e13	0.354849	0.177425	+	0.984134i	$0.443223\pi$
$72$	0	0
$73$	−9.16042e13	−0.970496	−0.485248	−	0.874376i	$-0.661271\pi$
$73$	−9.16042e13	−0.970496	−0.485248	+	0.874376i	$0.661271\pi$
$74$	0	0
$75$	1.49561e14	1.29378
$76$	0	0
$77$	2.23273e14	1.58546
$78$	0	0
$79$	−6.28821e13	−0.368404	−0.184202	−	0.982888i	$-0.558970\pi$
$79$	−6.28821e13	−0.368404	−0.184202	+	0.982888i	$0.558970\pi$
$80$	0	0
$81$	2.28768e13	0.111111
$82$	0	0
$83$	2.23567e14	0.904321	0.452161	−	0.891937i	$-0.350653\pi$
$83$	2.23567e14	0.904321	0.452161	+	0.891937i	$0.350653\pi$
$84$	0	0
$85$	6.06999e14	2.05374
$86$	0	0
$87$	1.41595e14	0.402394
$88$	0	0
$89$	5.54199e14	1.32813	0.664065	−	0.747675i	$-0.268830\pi$
$89$	5.54199e14	1.32813	0.664065	+	0.747675i	$0.268830\pi$
$90$	0	0
$91$	−1.13500e14	−0.230243
$92$	0	0
$93$	4.42597e13	0.0762756
$94$	0	0
$95$	6.80301e14	0.999480
$96$	0	0
$97$	−1.38887e15	−1.74531	−0.872657	−	0.488333i	$-0.837605\pi$
$97$	−1.38887e15	−1.74531	−0.872657	+	0.488333i	$0.837605\pi$
$98$	0	0
$99$	−5.27346e14	−0.568633
$100$	0	0
$101$	−4.19546e14	−0.389376	−0.194688	−	0.980865i	$-0.562369\pi$
$101$	−4.19546e14	−0.389376	−0.194688	+	0.980865i	$0.562369\pi$
$102$	0	0
$103$	9.82508e14	0.787149	0.393575	−	0.919293i	$-0.371238\pi$
$103$	9.82508e14	0.787149	0.393575	+	0.919293i	$0.371238\pi$
$104$	0	0
$105$	1.39281e15	0.965991
$106$	0	0
$107$	2.46544e14	0.148428	0.0742141	−	0.997242i	$-0.476355\pi$
$107$	2.46544e14	0.148428	0.0742141	+	0.997242i	$0.476355\pi$
$108$	0	0
$109$	8.45123e14	0.442814	0.221407	−	0.975182i	$-0.428935\pi$
$109$	8.45123e14	0.442814	0.221407	+	0.975182i	$0.428935\pi$
$110$	0	0
$111$	1.06937e15	0.488885
$112$	0	0
$113$	3.00170e15	1.20027	0.600135	−	0.799899i	$-0.295113\pi$
$113$	3.00170e15	1.20027	0.600135	+	0.799899i	$0.295113\pi$
$114$	0	0
$115$	1.95895e15	0.686736
$116$	0	0
$117$	2.68075e14	0.0825777
$118$	0	0
$119$	3.90857e15	1.06027
$120$	0	0
$121$	7.97893e15	1.91009
$122$	0	0
$123$	−1.68915e15	−0.357585
$124$	0	0
$125$	−1.19094e16	−2.23390
$126$	0	0
$127$	−1.75446e15	−0.292157	−0.146078	−	0.989273i	$-0.546665\pi$
$127$	−1.75446e15	−0.292157	−0.146078	+	0.989273i	$0.546665\pi$
$128$	0	0
$129$	−2.85790e15	−0.423276
$130$	0	0
$131$	−1.07149e16	−1.41402	−0.707009	−	0.707205i	$-0.749956\pi$
$131$	−1.07149e16	−1.41402	−0.707009	+	0.707205i	$0.749956\pi$
$132$	0	0
$133$	4.38058e15	0.515993
$134$	0	0
$135$	−3.28968e15	−0.346457
$136$	0	0
$137$	−4.07675e15	−0.384512	−0.192256	−	0.981345i	$-0.561580\pi$
$137$	−4.07675e15	−0.384512	−0.192256	+	0.981345i	$0.561580\pi$
$138$	0	0
$139$	−6.18797e15	−0.523524	−0.261762	−	0.965132i	$-0.584304\pi$
$139$	−6.18797e15	−0.523524	−0.261762	+	0.965132i	$0.584304\pi$
$140$	0	0
$141$	−7.33043e15	−0.557168
$142$	0	0
$143$	−6.17956e15	−0.422608
$144$	0	0
$145$	−2.03613e16	−1.25471
$146$	0	0
$147$	−1.41435e15	−0.0786463
$148$	0	0
$149$	4.10483e15	0.206252	0.103126	−	0.994668i	$-0.467116\pi$
$149$	4.10483e15	0.206252	0.103126	+	0.994668i	$0.467116\pi$
$150$	0	0
$151$	4.30971e15	0.195939	0.0979696	−	0.995189i	$-0.468765\pi$
$151$	4.30971e15	0.195939	0.0979696	+	0.995189i	$0.468765\pi$
$152$	0	0
$153$	−9.23163e15	−0.380271
$154$	0	0
$155$	−6.36453e15	−0.237836
$156$	0	0
$157$	−4.10493e16	−1.39334	−0.696672	−	0.717390i	$-0.745337\pi$
$157$	−4.10493e16	−1.39334	−0.696672	+	0.717390i	$0.745337\pi$
$158$	0	0
$159$	2.05311e16	0.633775
$160$	0	0
$161$	1.26140e16	0.354535
$162$	0	0
$163$	3.67161e15	0.0940698	0.0470349	−	0.998893i	$-0.485023\pi$
$163$	3.67161e15	0.0940698	0.0470349	+	0.998893i	$0.485023\pi$
$164$	0	0
$165$	7.58323e16	1.77306
$166$	0	0
$167$	−3.97703e16	−0.849544	−0.424772	−	0.905300i	$-0.639646\pi$
$167$	−3.97703e16	−0.849544	−0.424772	+	0.905300i	$0.639646\pi$
$168$	0	0
$169$	−4.80445e16	−0.938628
$170$	0	0
$171$	−1.03465e16	−0.185064
$172$	0	0
$173$	3.10617e16	0.509189	0.254595	−	0.967048i	$-0.418058\pi$
$173$	3.10617e16	0.509189	0.254595	+	0.967048i	$0.418058\pi$
$174$	0	0
$175$	−1.38486e17	−2.08267
$176$	0	0
$177$	−6.32707e16	−0.873782
$178$	0	0
$179$	−5.31753e16	−0.675013	−0.337507	−	0.941323i	$-0.609584\pi$
$179$	−5.31753e16	−0.675013	−0.337507	+	0.941323i	$0.609584\pi$
$180$	0	0
$181$	7.28364e16	0.850666	0.425333	−	0.905037i	$-0.360157\pi$
$181$	7.28364e16	0.850666	0.425333	+	0.905037i	$0.360157\pi$
$182$	0	0
$183$	9.27137e16	0.997150
$184$	0	0
$185$	−1.53775e17	−1.52440
$186$	0	0
$187$	2.12804e17	1.94611
$188$	0	0
$189$	−2.11828e16	−0.178863
$190$	0	0
$191$	9.39928e16	0.733406	0.366703	−	0.930338i	$-0.380487\pi$
$191$	9.39928e16	0.733406	0.366703	+	0.930338i	$0.380487\pi$
$192$	0	0
$193$	1.88578e17	1.36085	0.680426	−	0.732816i	$-0.261795\pi$
$193$	1.88578e17	1.36085	0.680426	+	0.732816i	$0.261795\pi$
$194$	0	0
$195$	−3.85491e16	−0.257487
$196$	0	0
$197$	−1.86952e17	−1.15673	−0.578366	−	0.815778i	$-0.696309\pi$
$197$	−1.86952e17	−1.15673	−0.578366	+	0.815778i	$0.696309\pi$
$198$	0	0
$199$	−3.33410e16	−0.191241	−0.0956204	−	0.995418i	$-0.530483\pi$
$199$	−3.33410e16	−0.191241	−0.0956204	+	0.995418i	$0.530483\pi$
$200$	0	0
$201$	1.14265e17	0.608053
$202$	0	0
$203$	−1.31110e17	−0.647760
$204$	0	0
$205$	2.42899e17	1.11499
$206$	0	0
$207$	−2.97930e16	−0.127156
$208$	0	0
$209$	2.38502e17	0.947100
$210$	0	0
$211$	4.60034e17	1.70087	0.850436	−	0.526079i	$-0.176338\pi$
$211$	4.60034e17	1.70087	0.850436	+	0.526079i	$0.176338\pi$
$212$	0	0
$213$	5.94744e16	0.204872
$214$	0	0
$215$	4.10965e17	1.31982
$216$	0	0
$217$	−4.09823e16	−0.122786
$218$	0	0
$219$	−2.00338e17	−0.560316
$220$	0	0
$221$	−1.08178e17	−0.282617
$222$	0	0
$223$	−2.73469e17	−0.667762	−0.333881	−	0.942615i	$-0.608358\pi$
$223$	−2.73469e17	−0.667762	−0.333881	+	0.942615i	$0.608358\pi$
$224$	0	0
$225$	3.27090e17	0.746962
$226$	0	0
$227$	−4.06444e17	−0.868573	−0.434287	−	0.900775i	$-0.643000\pi$
$227$	−4.06444e17	−0.868573	−0.434287	+	0.900775i	$0.643000\pi$
$228$	0	0
$229$	1.30379e17	0.260880	0.130440	−	0.991456i	$-0.458361\pi$
$229$	1.30379e17	0.260880	0.130440	+	0.991456i	$0.458361\pi$
$230$	0	0
$231$	4.88297e17	0.915365
$232$	0	0
$233$	−2.73060e17	−0.479832	−0.239916	−	0.970794i	$-0.577120\pi$
$233$	−2.73060e17	−0.479832	−0.239916	+	0.970794i	$0.577120\pi$
$234$	0	0
$235$	1.05411e18	1.73732
$236$	0	0
$237$	−1.37523e17	−0.212698
$238$	0	0
$239$	−1.21033e18	−1.75760	−0.878802	−	0.477186i	$-0.841657\pi$
$239$	−1.21033e18	−1.75760	−0.878802	+	0.477186i	$0.841657\pi$
$240$	0	0
$241$	3.86893e17	0.527793	0.263896	−	0.964551i	$-0.414992\pi$
$241$	3.86893e17	0.527793	0.263896	+	0.964551i	$0.414992\pi$
$242$	0	0
$243$	5.00315e16	0.0641500
$244$	0	0
$245$	2.03384e17	0.245228
$246$	0	0
$247$	−1.21242e17	−0.137539
$248$	0	0
$249$	4.88942e17	0.522110
$250$	0	0
$251$	9.15208e17	0.920379	0.460190	−	0.887821i	$-0.347781\pi$
$251$	9.15208e17	0.920379	0.460190	+	0.887821i	$0.347781\pi$
$252$	0	0
$253$	6.86776e17	0.650745
$254$	0	0
$255$	1.32751e18	1.18573
$256$	0	0
$257$	−2.15995e18	−1.81947	−0.909734	−	0.415191i	$-0.863715\pi$
$257$	−2.15995e18	−1.81947	−0.909734	+	0.415191i	$0.863715\pi$
$258$	0	0
$259$	−9.90187e17	−0.786989
$260$	0	0
$261$	3.09667e17	0.232323
$262$	0	0
$263$	−5.35139e17	−0.379139	−0.189570	−	0.981867i	$-0.560709\pi$
$263$	−5.35139e17	−0.379139	−0.189570	+	0.981867i	$0.560709\pi$
$264$	0	0
$265$	−2.95237e18	−1.97618
$266$	0	0
$267$	1.21203e18	0.766796
$268$	0	0
$269$	−5.06061e17	−0.302734	−0.151367	−	0.988478i	$-0.548368\pi$
$269$	−5.06061e17	−0.302734	−0.151367	+	0.988478i	$0.548368\pi$
$270$	0	0
$271$	−2.92815e18	−1.65700	−0.828501	−	0.559987i	$-0.810806\pi$
$271$	−2.92815e18	−1.65700	−0.828501	+	0.559987i	$0.810806\pi$
$272$	0	0
$273$	−2.48225e17	−0.132931
$274$	0	0
$275$	−7.53994e18	−3.82272
$276$	0	0
$277$	3.77192e17	0.181119	0.0905596	−	0.995891i	$-0.471134\pi$
$277$	3.77192e17	0.181119	0.0905596	+	0.995891i	$0.471134\pi$
$278$	0	0
$279$	9.67959e16	0.0440378
$280$	0	0
$281$	−2.76493e18	−1.19231	−0.596153	−	0.802871i	$-0.703305\pi$
$281$	−2.76493e18	−1.19231	−0.596153	+	0.802871i	$0.703305\pi$
$282$	0	0
$283$	1.13987e18	0.466075	0.233038	−	0.972468i	$-0.425133\pi$
$283$	1.13987e18	0.466075	0.233038	+	0.972468i	$0.425133\pi$
$284$	0	0
$285$	1.48782e18	0.577050
$286$	0	0
$287$	1.56407e18	0.575628
$288$	0	0
$289$	8.62880e17	0.301451
$290$	0	0
$291$	−3.03746e18	−1.00766
$292$	0	0
$293$	5.84207e18	1.84102	0.920512	−	0.390714i	$-0.127772\pi$
$293$	5.84207e18	1.84102	0.920512	+	0.390714i	$0.127772\pi$
$294$	0	0
$295$	9.09831e18	2.72455
$296$	0	0
$297$	−1.15331e18	−0.328300
$298$	0	0
$299$	−3.49121e17	−0.0945022
$300$	0	0
$301$	2.64627e18	0.681375
$302$	0	0
$303$	−9.17548e17	−0.224807
$304$	0	0
$305$	−1.33322e19	−3.10923
$306$	0	0
$307$	3.59204e18	0.797633	0.398816	−	0.917031i	$-0.369421\pi$
$307$	3.59204e18	0.797633	0.398816	+	0.917031i	$0.369421\pi$
$308$	0	0
$309$	2.14875e18	0.454461
$310$	0	0
$311$	−2.18535e18	−0.440371	−0.220185	−	0.975458i	$-0.570666\pi$
$311$	−2.18535e18	−0.440371	−0.220185	+	0.975458i	$0.570666\pi$
$312$	0	0
$313$	5.59874e18	1.07525	0.537623	−	0.843185i	$-0.319322\pi$
$313$	5.59874e18	1.07525	0.537623	+	0.843185i	$0.319322\pi$
$314$	0	0
$315$	3.04608e18	0.557715
$316$	0	0
$317$	−7.36219e18	−1.28547	−0.642736	−	0.766088i	$-0.722201\pi$
$317$	−7.36219e18	−1.28547	−0.642736	+	0.766088i	$0.722201\pi$
$318$	0	0
$319$	−7.13832e18	−1.18896
$320$	0	0
$321$	5.39192e17	0.0856951
$322$	0	0
$323$	4.17518e18	0.633368
$324$	0	0
$325$	3.83291e18	0.555141
$326$	0	0
$327$	1.84828e18	0.255659
$328$	0	0
$329$	6.78763e18	0.896910
$330$	0	0
$331$	−7.52026e18	−0.949562	−0.474781	−	0.880104i	$-0.657473\pi$
$331$	−7.52026e18	−0.949562	−0.474781	+	0.880104i	$0.657473\pi$
$332$	0	0
$333$	2.33872e18	0.282258
$334$	0	0
$335$	−1.64312e19	−1.89598
$336$	0	0
$337$	−4.01228e18	−0.442759	−0.221380	−	0.975188i	$-0.571056\pi$
$337$	−4.01228e18	−0.442759	−0.221380	+	0.975188i	$0.571056\pi$
$338$	0	0
$339$	6.56472e18	0.692976
$340$	0	0
$341$	−2.23130e18	−0.225372
$342$	0	0
$343$	1.09237e19	1.05600
$344$	0	0
$345$	4.28422e18	0.396487
$346$	0	0
$347$	1.92570e19	1.70654	0.853271	−	0.521467i	$-0.174615\pi$
$347$	1.92570e19	1.70654	0.853271	+	0.521467i	$0.174615\pi$
$348$	0	0
$349$	1.07863e19	0.915550	0.457775	−	0.889068i	$-0.348646\pi$
$349$	1.07863e19	0.915550	0.457775	+	0.889068i	$0.348646\pi$
$350$	0	0
$351$	5.86280e17	0.0476762
$352$	0	0
$353$	3.62430e18	0.282432	0.141216	−	0.989979i	$-0.454899\pi$
$353$	3.62430e18	0.282432	0.141216	+	0.989979i	$0.454899\pi$
$354$	0	0
$355$	−8.55241e18	−0.638816
$356$	0	0
$357$	8.54804e18	0.612146
$358$	0	0
$359$	1.75871e19	1.20778	0.603889	−	0.797069i	$-0.293617\pi$
$359$	1.75871e19	1.20778	0.603889	+	0.797069i	$0.293617\pi$
$360$	0	0
$361$	−1.05017e19	−0.691763
$362$	0	0
$363$	1.74499e19	1.10279
$364$	0	0
$365$	2.88086e19	1.74713
$366$	0	0
$367$	2.74539e19	1.59812	0.799059	−	0.601253i	$-0.205332\pi$
$367$	2.74539e19	1.59812	0.799059	+	0.601253i	$0.205332\pi$
$368$	0	0
$369$	−3.69417e18	−0.206452
$370$	0	0
$371$	−1.90108e19	−1.02023
$372$	0	0
$373$	1.44712e19	0.745914	0.372957	−	0.927849i	$-0.378344\pi$
$373$	1.44712e19	0.745914	0.372957	+	0.927849i	$0.378344\pi$
$374$	0	0
$375$	−2.60458e19	−1.28974
$376$	0	0
$377$	3.62875e18	0.172662
$378$	0	0
$379$	−6.83690e18	−0.312655	−0.156328	−	0.987705i	$-0.549966\pi$
$379$	−6.83690e18	−0.312655	−0.156328	+	0.987705i	$0.549966\pi$
$380$	0	0
$381$	−3.83701e18	−0.168677
$382$	0	0
$383$	−1.45278e19	−0.614058	−0.307029	−	0.951700i	$-0.599335\pi$
$383$	−1.45278e19	−0.614058	−0.307029	+	0.951700i	$0.599335\pi$
$384$	0	0
$385$	−7.02170e19	−2.85422
$386$	0	0
$387$	−6.25022e18	−0.244379
$388$	0	0
$389$	−8.73653e18	−0.328638	−0.164319	−	0.986407i	$-0.552543\pi$
$389$	−8.73653e18	−0.328638	−0.164319	+	0.986407i	$0.552543\pi$
$390$	0	0
$391$	1.20226e19	0.435183
$392$	0	0
$393$	−2.34336e19	−0.816383
$394$	0	0
$395$	1.97758e19	0.663217
$396$	0	0
$397$	4.36058e19	1.40804	0.704021	−	0.710179i	$-0.251386\pi$
$397$	4.36058e19	1.40804	0.704021	+	0.710179i	$0.251386\pi$
$398$	0	0
$399$	9.58032e18	0.297909
$400$	0	0
$401$	−1.31498e19	−0.393854	−0.196927	−	0.980418i	$-0.563096\pi$
$401$	−1.31498e19	−0.393854	−0.196927	+	0.980418i	$0.563096\pi$
$402$	0	0
$403$	1.13427e18	0.0327288
$404$	0	0
$405$	−7.19452e18	−0.200027
$406$	0	0
$407$	−5.39111e19	−1.44451
$408$	0	0
$409$	−1.26643e19	−0.327082	−0.163541	−	0.986537i	$-0.552292\pi$
$409$	−1.26643e19	−0.327082	−0.163541	+	0.986537i	$0.552292\pi$
$410$	0	0
$411$	−8.91585e18	−0.221998
$412$	0	0
$413$	5.85856e19	1.40658
$414$	0	0
$415$	−7.03097e19	−1.62800
$416$	0	0
$417$	−1.35331e19	−0.302257
$418$	0	0
$419$	4.10992e19	0.885581	0.442790	−	0.896625i	$-0.353989\pi$
$419$	4.10992e19	0.885581	0.442790	+	0.896625i	$0.353989\pi$
$420$	0	0
$421$	7.41791e19	1.54229	0.771145	−	0.636660i	$-0.219684\pi$
$421$	7.41791e19	1.54229	0.771145	+	0.636660i	$0.219684\pi$
$422$	0	0
$423$	−1.60317e19	−0.321681
$424$	0	0
$425$	−1.31993e20	−2.55643
$426$	0	0
$427$	−8.58484e19	−1.60518
$428$	0	0
$429$	−1.35147e19	−0.243993
$430$	0	0
$431$	7.56855e19	1.31957	0.659786	−	0.751454i	$-0.270647\pi$
$431$	7.56855e19	1.31957	0.659786	+	0.751454i	$0.270647\pi$
$432$	0	0
$433$	−1.11039e19	−0.186990	−0.0934949	−	0.995620i	$-0.529804\pi$
$433$	−1.11039e19	−0.186990	−0.0934949	+	0.995620i	$0.529804\pi$
$434$	0	0
$435$	−4.45301e19	−0.724409
$436$	0	0
$437$	1.34744e19	0.211787
$438$	0	0
$439$	5.15079e19	0.782330	0.391165	−	0.920321i	$-0.372072\pi$
$439$	5.15079e19	0.782330	0.391165	+	0.920321i	$0.372072\pi$
$440$	0	0
$441$	−3.09319e18	−0.0454064
$442$	0	0
$443$	−4.77898e19	−0.678122	−0.339061	−	0.940764i	$-0.610109\pi$
$443$	−4.77898e19	−0.678122	−0.339061	+	0.940764i	$0.610109\pi$
$444$	0	0
$445$	−1.74290e20	−2.39096
$446$	0	0
$447$	8.97726e18	0.119080
$448$	0	0
$449$	5.91210e19	0.758393	0.379196	−	0.925316i	$-0.376201\pi$
$449$	5.91210e19	0.758393	0.379196	+	0.925316i	$0.376201\pi$
$450$	0	0
$451$	8.51565e19	1.05656
$452$	0	0
$453$	9.42534e18	0.113125
$454$	0	0
$455$	3.56946e19	0.414493
$456$	0	0
$457$	3.59497e19	0.403947	0.201973	−	0.979391i	$-0.435265\pi$
$457$	3.59497e19	0.403947	0.201973	+	0.979391i	$0.435265\pi$
$458$	0	0
$459$	−2.01896e19	−0.219549
$460$	0	0
$461$	7.56833e18	0.0796605	0.0398303	−	0.999206i	$-0.487318\pi$
$461$	7.56833e18	0.0796605	0.0398303	+	0.999206i	$0.487318\pi$
$462$	0	0
$463$	−4.36170e19	−0.444425	−0.222213	−	0.974998i	$-0.571328\pi$
$463$	−4.36170e19	−0.444425	−0.222213	+	0.974998i	$0.571328\pi$
$464$	0	0
$465$	−1.39192e19	−0.137315
$466$	0	0
$467$	5.47599e19	0.523102	0.261551	−	0.965190i	$-0.415766\pi$
$467$	5.47599e19	0.523102	0.261551	+	0.965190i	$0.415766\pi$
$468$	0	0
$469$	−1.05804e20	−0.978823
$470$	0	0
$471$	−8.97748e19	−0.804447
$472$	0	0
$473$	1.44078e20	1.25066
$474$	0	0
$475$	−1.47933e20	−1.24412
$476$	0	0
$477$	4.49016e19	0.365910
$478$	0	0
$479$	2.20355e20	1.74023	0.870113	−	0.492852i	$-0.164046\pi$
$479$	2.20355e20	1.74023	0.870113	+	0.492852i	$0.164046\pi$
$480$	0	0
$481$	2.74056e19	0.209774
$482$	0	0
$483$	2.75869e19	0.204691
$484$	0	0
$485$	4.36786e20	3.14200
$486$	0	0
$487$	−2.46263e20	−1.71764	−0.858819	−	0.512279i	$-0.828802\pi$
$487$	−2.46263e20	−1.71764	−0.858819	+	0.512279i	$0.828802\pi$
$488$	0	0
$489$	8.02982e18	0.0543112
$490$	0	0
$491$	1.45290e20	0.953069	0.476534	−	0.879156i	$-0.341893\pi$
$491$	1.45290e20	0.953069	0.476534	+	0.879156i	$0.341893\pi$
$492$	0	0
$493$	−1.24962e20	−0.795109
$494$	0	0
$495$	1.65845e20	1.02368
$496$	0	0
$497$	−5.50704e19	−0.329796
$498$	0	0
$499$	2.71400e20	1.57709	0.788544	−	0.614978i	$-0.210835\pi$
$499$	2.71400e20	1.57709	0.788544	+	0.614978i	$0.210835\pi$
$500$	0	0
$501$	−8.69777e19	−0.490484
$502$	0	0
$503$	5.02955e19	0.275276	0.137638	−	0.990483i	$-0.456049\pi$
$503$	5.02955e19	0.275276	0.137638	+	0.990483i	$0.456049\pi$
$504$	0	0
$505$	1.31943e20	0.700973
$506$	0	0
$507$	−1.05073e20	−0.541917
$508$	0	0
$509$	−3.41886e20	−1.71198	−0.855989	−	0.516995i	$-0.827051\pi$
$509$	−3.41886e20	−1.71198	−0.855989	+	0.516995i	$0.827051\pi$
$510$	0	0
$511$	1.85504e20	0.901977
$512$	0	0
$513$	−2.26277e19	−0.106847
$514$	0	0
$515$	−3.08989e20	−1.41706
$516$	0	0
$517$	3.69555e20	1.64627
$518$	0	0
$519$	6.79319e19	0.293981
$520$	0	0
$521$	3.04623e20	1.28080	0.640398	−	0.768043i	$-0.278769\pi$
$521$	3.04623e20	1.28080	0.640398	+	0.768043i	$0.278769\pi$
$522$	0	0
$523$	6.22853e19	0.254462	0.127231	−	0.991873i	$-0.459391\pi$
$523$	6.22853e19	0.254462	0.127231	+	0.991873i	$0.459391\pi$
$524$	0	0
$525$	−3.02869e20	−1.20243
$526$	0	0
$527$	−3.90607e19	−0.150716
$528$	0	0
$529$	−2.27835e20	−0.854482
$530$	0	0
$531$	−1.38373e20	−0.504478
$532$	0	0
$533$	−4.32891e19	−0.153435
$534$	0	0
$535$	−7.75357e19	−0.267207
$536$	0	0
$537$	−1.16294e20	−0.389719
$538$	0	0
$539$	7.13030e19	0.232376
$540$	0	0
$541$	5.09061e20	1.61358	0.806790	−	0.590839i	$-0.201203\pi$
$541$	5.09061e20	1.61358	0.806790	+	0.590839i	$0.201203\pi$
$542$	0	0
$543$	1.59293e20	0.491132
$544$	0	0
$545$	−2.65783e20	−0.797174
$546$	0	0
$547$	5.58818e20	1.63067	0.815333	−	0.578992i	$-0.196554\pi$
$547$	5.58818e20	1.63067	0.815333	+	0.578992i	$0.196554\pi$
$548$	0	0
$549$	2.02765e20	0.575705
$550$	0	0
$551$	−1.40053e20	−0.386950
$552$	0	0
$553$	1.27340e20	0.342394
$554$	0	0
$555$	−3.36307e20	−0.880113
$556$	0	0
$557$	1.40176e20	0.357076	0.178538	−	0.983933i	$-0.442863\pi$
$557$	1.40176e20	0.357076	0.178538	+	0.983933i	$0.442863\pi$
$558$	0	0
$559$	−7.32415e19	−0.181622
$560$	0	0
$561$	4.65402e20	1.12359
$562$	0	0
$563$	3.17601e20	0.746567	0.373284	−	0.927717i	$-0.378232\pi$
$563$	3.17601e20	0.746567	0.373284	+	0.927717i	$0.378232\pi$
$564$	0	0
$565$	−9.44005e20	−2.16078
$566$	0	0
$567$	−4.63268e19	−0.103266
$568$	0	0
$569$	1.25453e20	0.272357	0.136178	−	0.990684i	$-0.456518\pi$
$569$	1.25453e20	0.272357	0.136178	+	0.990684i	$0.456518\pi$
$570$	0	0
$571$	4.47156e19	0.0945557	0.0472779	−	0.998882i	$-0.484945\pi$
$571$	4.47156e19	0.0945557	0.0472779	+	0.998882i	$0.484945\pi$
$572$	0	0
$573$	2.05562e20	0.423432
$574$	0	0
$575$	−4.25977e20	−0.854825
$576$	0	0
$577$	−2.39371e20	−0.468008	−0.234004	−	0.972236i	$-0.575183\pi$
$577$	−2.39371e20	−0.468008	−0.234004	+	0.972236i	$0.575183\pi$
$578$	0	0
$579$	4.12420e20	0.785689
$580$	0	0
$581$	−4.52736e20	−0.840474
$582$	0	0
$583$	−1.03505e21	−1.87262
$584$	0	0
$585$	−8.43070e19	−0.148660
$586$	0	0
$587$	−2.93609e20	−0.504642	−0.252321	−	0.967644i	$-0.581194\pi$
$587$	−2.93609e20	−0.504642	−0.252321	+	0.967644i	$0.581194\pi$
$588$	0	0
$589$	−4.37778e19	−0.0733481
$590$	0	0
$591$	−4.08863e20	−0.667839
$592$	0	0
$593$	5.11155e20	0.814033	0.407017	−	0.913421i	$-0.366569\pi$
$593$	5.11155e20	0.814033	0.407017	+	0.913421i	$0.366569\pi$
$594$	0	0
$595$	−1.22921e21	−1.90874
$596$	0	0
$597$	−7.29167e19	−0.110413
$598$	0	0
$599$	2.14049e20	0.316091	0.158045	−	0.987432i	$-0.449481\pi$
$599$	2.14049e20	0.316091	0.158045	+	0.987432i	$0.449481\pi$
$600$	0	0
$601$	−4.24612e19	−0.0611553	−0.0305777	−	0.999532i	$-0.509735\pi$
$601$	−4.24612e19	−0.0611553	−0.0305777	+	0.999532i	$0.509735\pi$
$602$	0	0
$603$	2.49897e20	0.351060
$604$	0	0
$605$	−2.50929e21	−3.43864
$606$	0	0
$607$	1.75100e20	0.234084	0.117042	−	0.993127i	$-0.462659\pi$
$607$	1.75100e20	0.234084	0.117042	+	0.993127i	$0.462659\pi$
$608$	0	0
$609$	−2.86737e20	−0.373985
$610$	0	0
$611$	−1.87862e20	−0.239073
$612$	0	0
$613$	−6.98870e20	−0.867846	−0.433923	−	0.900950i	$-0.642871\pi$
$613$	−6.98870e20	−0.867846	−0.433923	+	0.900950i	$0.642871\pi$
$614$	0	0
$615$	5.31221e20	0.643742
$616$	0	0
$617$	−1.22078e20	−0.144378	−0.0721888	−	0.997391i	$-0.522998\pi$
$617$	−1.22078e20	−0.144378	−0.0721888	+	0.997391i	$0.522998\pi$
$618$	0	0
$619$	5.30802e20	0.612707	0.306354	−	0.951918i	$-0.400891\pi$
$619$	5.30802e20	0.612707	0.306354	+	0.951918i	$0.400891\pi$
$620$	0	0
$621$	−6.51573e19	−0.0734135
$622$	0	0
$623$	−1.12228e21	−1.23436
$624$	0	0
$625$	1.65839e21	1.78068
$626$	0	0
$627$	5.21605e20	0.546808
$628$	0	0
$629$	−9.43759e20	−0.966009
$630$	0	0
$631$	1.55994e21	1.55915	0.779573	−	0.626311i	$-0.215436\pi$
$631$	1.55994e21	1.55915	0.779573	+	0.626311i	$0.215436\pi$
$632$	0	0
$633$	1.00609e21	0.981999
$634$	0	0
$635$	5.51760e20	0.525954
$636$	0	0
$637$	−3.62467e19	−0.0337460
$638$	0	0
$639$	1.30071e20	0.118283
$640$	0	0
$641$	1.38155e21	1.22725	0.613624	−	0.789598i	$-0.289711\pi$
$641$	1.38155e21	1.22725	0.613624	+	0.789598i	$0.289711\pi$
$642$	0	0
$643$	−1.66386e21	−1.44389	−0.721947	−	0.691949i	$-0.756752\pi$
$643$	−1.66386e21	−1.44389	−0.721947	+	0.691949i	$0.756752\pi$
$644$	0	0
$645$	8.98780e20	0.762001
$646$	0	0
$647$	4.89973e20	0.405873	0.202937	−	0.979192i	$-0.434951\pi$
$647$	4.89973e20	0.405873	0.202937	+	0.979192i	$0.434951\pi$
$648$	0	0
$649$	3.18972e21	2.58177
$650$	0	0
$651$	−8.96283e19	−0.0708904
$652$	0	0
$653$	1.85767e21	1.43588	0.717942	−	0.696103i	$-0.245084\pi$
$653$	1.85767e21	1.43588	0.717942	+	0.696103i	$0.245084\pi$
$654$	0	0
$655$	3.36974e21	2.54558
$656$	0	0
$657$	−4.38140e20	−0.323499
$658$	0	0
$659$	−1.98175e21	−1.43024	−0.715120	−	0.699002i	$-0.753628\pi$
$659$	−1.98175e21	−1.43024	−0.715120	+	0.699002i	$0.753628\pi$
$660$	0	0
$661$	−1.98920e21	−1.40335	−0.701677	−	0.712495i	$-0.747565\pi$
$661$	−1.98920e21	−1.40335	−0.701677	+	0.712495i	$0.747565\pi$
$662$	0	0
$663$	−2.36586e20	−0.163169
$664$	0	0
$665$	−1.37765e21	−0.928915
$666$	0	0
$667$	−4.03287e20	−0.265871
$668$	0	0
$669$	−5.98077e20	−0.385533
$670$	0	0
$671$	−4.67405e21	−2.94628
$672$	0	0
$673$	−1.59503e21	−0.983232	−0.491616	−	0.870812i	$-0.663594\pi$
$673$	−1.59503e21	−0.983232	−0.491616	+	0.870812i	$0.663594\pi$
$674$	0	0
$675$	7.15346e20	0.431259
$676$	0	0
$677$	−1.18527e21	−0.698878	−0.349439	−	0.936959i	$-0.613628\pi$
$677$	−1.18527e21	−0.698878	−0.349439	+	0.936959i	$0.613628\pi$
$678$	0	0
$679$	2.81254e21	1.62209
$680$	0	0
$681$	−8.88893e20	−0.501471
$682$	0	0
$683$	2.44489e21	1.34929	0.674644	−	0.738144i	$-0.264297\pi$
$683$	2.44489e21	1.34929	0.674644	+	0.738144i	$0.264297\pi$
$684$	0	0
$685$	1.28210e21	0.692216
$686$	0	0
$687$	2.85139e20	0.150619
$688$	0	0
$689$	5.26167e20	0.271944
$690$	0	0
$691$	1.13986e21	0.576454	0.288227	−	0.957562i	$-0.406934\pi$
$691$	1.13986e21	0.576454	0.288227	+	0.957562i	$0.406934\pi$
$692$	0	0
$693$	1.06791e21	0.528486
$694$	0	0
$695$	1.94605e21	0.942472
$696$	0	0
$697$	1.49073e21	0.706568
$698$	0	0
$699$	−5.97183e20	−0.277031
$700$	0	0
$701$	3.11199e19	0.0141303	0.00706517	−	0.999975i	$-0.497751\pi$
$701$	3.11199e19	0.0141303	0.00706517	+	0.999975i	$0.497751\pi$
$702$	0	0
$703$	−1.05773e21	−0.470121
$704$	0	0
$705$	2.30535e21	1.00304
$706$	0	0
$707$	8.49605e20	0.361886
$708$	0	0
$709$	2.13474e21	0.890220	0.445110	−	0.895476i	$-0.353165\pi$
$709$	2.13474e21	0.890220	0.445110	+	0.895476i	$0.353165\pi$
$710$	0	0
$711$	−3.00763e20	−0.122801
$712$	0	0
$713$	−1.26060e20	−0.0503970
$714$	0	0
$715$	1.94341e21	0.760797
$716$	0	0
$717$	−2.64700e21	−1.01475
$718$	0	0
$719$	8.33849e20	0.313055	0.156528	−	0.987674i	$-0.449970\pi$
$719$	8.33849e20	0.313055	0.156528	+	0.987674i	$0.449970\pi$
$720$	0	0
$721$	−1.98963e21	−0.731575
$722$	0	0
$723$	8.46136e20	0.304721
$724$	0	0
$725$	4.42759e21	1.56182
$726$	0	0
$727$	−4.62186e21	−1.59702	−0.798508	−	0.601985i	$-0.794377\pi$
$727$	−4.62186e21	−1.59702	−0.798508	+	0.601985i	$0.794377\pi$
$728$	0	0
$729$	1.09419e20	0.0370370
$730$	0	0
$731$	2.52220e21	0.836370
$732$	0	0
$733$	3.02391e21	0.982403	0.491201	−	0.871046i	$-0.336558\pi$
$733$	3.02391e21	0.982403	0.491201	+	0.871046i	$0.336558\pi$
$734$	0	0
$735$	4.44800e20	0.141583
$736$	0	0
$737$	−5.76052e21	−1.79662
$738$	0	0
$739$	−5.97168e21	−1.82500	−0.912501	−	0.409075i	$-0.865852\pi$
$739$	−5.97168e21	−1.82500	−0.912501	+	0.409075i	$0.865852\pi$
$740$	0	0
$741$	−2.65156e20	−0.0794082
$742$	0	0
$743$	2.73039e20	0.0801324	0.0400662	−	0.999197i	$-0.487243\pi$
$743$	2.73039e20	0.0801324	0.0400662	+	0.999197i	$0.487243\pi$
$744$	0	0
$745$	−1.29093e21	−0.371304
$746$	0	0
$747$	1.06932e21	0.301440
$748$	0	0
$749$	−4.99266e20	−0.137949
$750$	0	0
$751$	−1.84036e20	−0.0498429	−0.0249215	−	0.999689i	$-0.507934\pi$
$751$	−1.84036e20	−0.0498429	−0.0249215	+	0.999689i	$0.507934\pi$
$752$	0	0
$753$	2.00156e21	0.531381
$754$	0	0
$755$	−1.35536e21	−0.352739
$756$	0	0
$757$	−2.94398e21	−0.751131	−0.375565	−	0.926796i	$-0.622551\pi$
$757$	−2.94398e21	−0.751131	−0.375565	+	0.926796i	$0.622551\pi$
$758$	0	0
$759$	1.50198e21	0.375708
$760$	0	0
$761$	−1.39071e20	−0.0341077	−0.0170539	−	0.999855i	$-0.505429\pi$
$761$	−1.39071e20	−0.0341077	−0.0170539	+	0.999855i	$0.505429\pi$
$762$	0	0
$763$	−1.71142e21	−0.411550
$764$	0	0
$765$	2.90326e21	0.684580
$766$	0	0
$767$	−1.62148e21	−0.374928
$768$	0	0
$769$	−7.43550e21	−1.68602	−0.843010	−	0.537898i	$-0.819218\pi$
$769$	−7.43550e21	−1.68602	−0.843010	+	0.537898i	$0.819218\pi$
$770$	0	0
$771$	−4.72380e21	−1.05047
$772$	0	0
$773$	−2.24172e21	−0.488917	−0.244459	−	0.969660i	$-0.578610\pi$
$773$	−2.24172e21	−0.488917	−0.244459	+	0.969660i	$0.578610\pi$
$774$	0	0
$775$	1.38398e21	0.296051
$776$	0	0
$777$	−2.16554e21	−0.454369
$778$	0	0
$779$	1.67076e21	0.343861
$780$	0	0
$781$	−2.99833e21	−0.605337
$782$	0	0
$783$	6.77242e20	0.134131
$784$	0	0
$785$	1.29096e22	2.50836
$786$	0	0
$787$	−3.80561e21	−0.725460	−0.362730	−	0.931894i	$-0.618155\pi$
$787$	−3.80561e21	−0.725460	−0.362730	+	0.931894i	$0.618155\pi$
$788$	0	0
$789$	−1.17035e21	−0.218896
$790$	0	0
$791$	−6.07861e21	−1.11553
$792$	0	0
$793$	2.37604e21	0.427863
$794$	0	0
$795$	−6.45684e21	−1.14095
$796$	0	0
$797$	−7.33682e21	−1.27224	−0.636122	−	0.771589i	$-0.719462\pi$
$797$	−7.33682e21	−1.27224	−0.636122	+	0.771589i	$0.719462\pi$
$798$	0	0
$799$	6.46937e21	1.10093
$800$	0	0
$801$	2.65072e21	0.442710
$802$	0	0
$803$	1.00998e22	1.65557
$804$	0	0
$805$	−3.96698e21	−0.638251
$806$	0	0
$807$	−1.10676e21	−0.174783
$808$	0	0
$809$	−1.11403e22	−1.72696	−0.863481	−	0.504381i	$-0.831721\pi$
$809$	−1.11403e22	−1.72696	−0.863481	+	0.504381i	$0.831721\pi$
$810$	0	0
$811$	9.59033e21	1.45941	0.729704	−	0.683763i	$-0.239658\pi$
$811$	9.59033e21	1.45941	0.729704	+	0.683763i	$0.239658\pi$
$812$	0	0
$813$	−6.40386e21	−0.956671
$814$	0	0
$815$	−1.15469e21	−0.169349
$816$	0	0
$817$	2.82678e21	0.407030
$818$	0	0
$819$	−5.42867e20	−0.0767475
$820$	0	0
$821$	1.24329e22	1.72583	0.862914	−	0.505352i	$-0.168637\pi$
$821$	1.24329e22	1.72583	0.862914	+	0.505352i	$0.168637\pi$
$822$	0	0
$823$	−1.29492e22	−1.76500	−0.882501	−	0.470310i	$-0.844142\pi$
$823$	−1.29492e22	−1.76500	−0.882501	+	0.470310i	$0.844142\pi$
$824$	0	0
$825$	−1.64899e22	−2.20705
$826$	0	0
$827$	−5.99673e21	−0.788175	−0.394088	−	0.919073i	$-0.628939\pi$
$827$	−5.99673e21	−0.788175	−0.394088	+	0.919073i	$0.628939\pi$
$828$	0	0
$829$	1.19803e22	1.54635	0.773174	−	0.634195i	$-0.218668\pi$
$829$	1.19803e22	1.54635	0.773174	+	0.634195i	$0.218668\pi$
$830$	0	0
$831$	8.24920e20	0.104569
$832$	0	0
$833$	1.24822e21	0.155401
$834$	0	0
$835$	1.25074e22	1.52939
$836$	0	0
$837$	2.11693e20	0.0254252
$838$	0	0
$839$	1.44455e21	0.170419	0.0852096	−	0.996363i	$-0.472844\pi$
$839$	1.44455e21	0.170419	0.0852096	+	0.996363i	$0.472844\pi$
$840$	0	0
$841$	−4.43744e21	−0.514236
$842$	0	0
$843$	−6.04691e21	−0.688378
$844$	0	0
$845$	1.51095e22	1.68976
$846$	0	0
$847$	−1.61578e22	−1.77524
$848$	0	0
$849$	2.49289e21	0.269089
$850$	0	0
$851$	−3.04577e21	−0.323017
$852$	0	0
$853$	−5.94981e21	−0.619991	−0.309996	−	0.950738i	$-0.600328\pi$
$853$	−5.94981e21	−0.619991	−0.309996	+	0.950738i	$0.600328\pi$
$854$	0	0
$855$	3.25386e21	0.333160
$856$	0	0
$857$	5.36930e21	0.540209	0.270104	−	0.962831i	$-0.412942\pi$
$857$	5.36930e21	0.540209	0.270104	+	0.962831i	$0.412942\pi$
$858$	0	0
$859$	−1.42295e22	−1.40683	−0.703415	−	0.710779i	$-0.748343\pi$
$859$	−1.42295e22	−1.40683	−0.703415	+	0.710779i	$0.748343\pi$
$860$	0	0
$861$	3.42062e21	0.332339
$862$	0	0
$863$	1.26362e22	1.20652	0.603261	−	0.797543i	$-0.293868\pi$
$863$	1.26362e22	1.20652	0.603261	+	0.797543i	$0.293868\pi$
$864$	0	0
$865$	−9.76859e21	−0.916666
$866$	0	0
$867$	1.88712e21	0.174043
$868$	0	0
$869$	6.93307e21	0.628459
$870$	0	0
$871$	2.92835e21	0.260907
$872$	0	0
$873$	−6.64292e21	−0.581772
$874$	0	0
$875$	2.41171e22	2.07618
$876$	0	0
$877$	1.94622e22	1.64700	0.823502	−	0.567313i	$-0.192017\pi$
$877$	1.94622e22	1.64700	0.823502	+	0.567313i	$0.192017\pi$
$878$	0	0
$879$	1.27766e22	1.06292
$880$	0	0
$881$	−4.01382e20	−0.0328275	−0.0164138	−	0.999865i	$-0.505225\pi$
$881$	−4.01382e20	−0.0328275	−0.0164138	+	0.999865i	$0.505225\pi$
$882$	0	0
$883$	1.87591e22	1.50837	0.754183	−	0.656664i	$-0.228033\pi$
$883$	1.87591e22	1.50837	0.754183	+	0.656664i	$0.228033\pi$
$884$	0	0
$885$	1.98980e22	1.57302
$886$	0	0
$887$	3.48285e21	0.270712	0.135356	−	0.990797i	$-0.456782\pi$
$887$	3.48285e21	0.270712	0.135356	+	0.990797i	$0.456782\pi$
$888$	0	0
$889$	3.55288e21	0.271530
$890$	0	0
$891$	−2.52228e21	−0.189544
$892$	0	0
$893$	7.25062e21	0.535783
$894$	0	0
$895$	1.67231e22	1.21519
$896$	0	0
$897$	−7.63527e20	−0.0545608
$898$	0	0
$899$	1.31026e21	0.0920787
$900$	0	0
$901$	−1.81195e22	−1.25230
$902$	0	0
$903$	5.78740e21	0.393392
$904$	0	0
$905$	−2.29063e22	−1.53141
$906$	0	0
$907$	−2.79349e22	−1.83693	−0.918464	−	0.395505i	$-0.870570\pi$
$907$	−2.79349e22	−1.83693	−0.918464	+	0.395505i	$0.870570\pi$
$908$	0	0
$909$	−2.00668e21	−0.129792
$910$	0	0
$911$	1.46026e22	0.929059	0.464530	−	0.885558i	$-0.346223\pi$
$911$	1.46026e22	0.929059	0.464530	+	0.885558i	$0.346223\pi$
$912$	0	0
$913$	−2.46494e22	−1.54268
$914$	0	0
$915$	−2.91575e22	−1.79512
$916$	0	0
$917$	2.16983e22	1.31419
$918$	0	0
$919$	1.53275e22	0.913281	0.456640	−	0.889651i	$-0.349053\pi$
$919$	1.53275e22	0.913281	0.456640	+	0.889651i	$0.349053\pi$
$920$	0	0
$921$	7.85579e21	0.460514
$922$	0	0
$923$	1.52420e21	0.0879079
$924$	0	0
$925$	3.34387e22	1.89752
$926$	0	0
$927$	4.69931e21	0.262383
$928$	0	0
$929$	1.72599e22	0.948243	0.474121	−	0.880459i	$-0.342766\pi$
$929$	1.72599e22	0.948243	0.474121	+	0.880459i	$0.342766\pi$
$930$	0	0
$931$	1.39895e21	0.0756277
$932$	0	0
$933$	−4.77936e21	−0.254248
$934$	0	0
$935$	−6.69247e22	−3.50348
$936$	0	0
$937$	2.57190e21	0.132497	0.0662486	−	0.997803i	$-0.478897\pi$
$937$	2.57190e21	0.132497	0.0662486	+	0.997803i	$0.478897\pi$
$938$	0	0
$939$	1.22444e22	0.620793
$940$	0	0
$941$	−1.86347e22	−0.929823	−0.464911	−	0.885357i	$-0.653914\pi$
$941$	−1.86347e22	−0.929823	−0.464911	+	0.885357i	$0.653914\pi$
$942$	0	0
$943$	4.81101e21	0.236264
$944$	0	0
$945$	6.66178e21	0.321997
$946$	0	0
$947$	2.87836e22	1.36937	0.684684	−	0.728840i	$-0.259940\pi$
$947$	2.87836e22	1.36937	0.684684	+	0.728840i	$0.259940\pi$
$948$	0	0
$949$	−5.13422e21	−0.240424
$950$	0	0
$951$	−1.61011e22	−0.742167
$952$	0	0
$953$	−5.82222e21	−0.264175	−0.132088	−	0.991238i	$-0.542168\pi$
$953$	−5.82222e21	−0.264175	−0.132088	+	0.991238i	$0.542168\pi$
$954$	0	0
$955$	−2.95598e22	−1.32031
$956$	0	0
$957$	−1.56115e22	−0.686444
$958$	0	0
$959$	8.25565e21	0.357365
$960$	0	0
$961$	−2.30557e22	−0.982546
$962$	0	0
$963$	1.17921e21	0.0494761
$964$	0	0
$965$	−5.93059e22	−2.44987
$966$	0	0
$967$	1.36325e22	0.554470	0.277235	−	0.960802i	$-0.410582\pi$
$967$	1.36325e22	0.554470	0.277235	+	0.960802i	$0.410582\pi$
$968$	0	0
$969$	9.13112e21	0.365675
$970$	0	0
$971$	2.33092e22	0.919144	0.459572	−	0.888141i	$-0.348003\pi$
$971$	2.33092e22	0.919144	0.459572	+	0.888141i	$0.348003\pi$
$972$	0	0
$973$	1.25310e22	0.486562
$974$	0	0
$975$	8.38258e21	0.320511
$976$	0	0
$977$	−3.08656e21	−0.116216	−0.0581080	−	0.998310i	$-0.518507\pi$
$977$	−3.08656e21	−0.116216	−0.0581080	+	0.998310i	$0.518507\pi$
$978$	0	0
$979$	−6.11032e22	−2.26566
$980$	0	0
$981$	4.04220e21	0.147605
$982$	0	0
$983$	−4.05804e22	−1.45937	−0.729684	−	0.683784i	$-0.760333\pi$
$983$	−4.05804e22	−1.45937	−0.729684	+	0.683784i	$0.760333\pi$
$984$	0	0
$985$	5.87944e22	2.08240
$986$	0	0
$987$	1.48445e22	0.517831
$988$	0	0
$989$	8.13981e21	0.279668
$990$	0	0
$991$	2.37255e22	0.802903	0.401451	−	0.915880i	$-0.368506\pi$
$991$	2.37255e22	0.802903	0.401451	+	0.915880i	$0.368506\pi$
$992$	0	0
$993$	−1.64468e22	−0.548230
$994$	0	0
$995$	1.04854e22	0.344280
$996$	0	0
$997$	2.95460e21	0.0955620	0.0477810	−	0.998858i	$-0.484785\pi$
$997$	2.95460e21	0.0955620	0.0477810	+	0.998858i	$0.484785\pi$
$998$	0	0
$999$	5.11477e21	0.162962

Twists

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

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