LMFDB - Embedded newform 384.3.b.a.319.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,3,Mod(319,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("384.319");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	384.b (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:	$10.4632421514$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			319.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	384.319
Dual form			384.3.b.a.319.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205 q^{3} +4.00000i q^{5} -6.92820i q^{7} +3.00000 q^{9} +O(q^{10})$	$q-1.73205 q^{3} +4.00000i q^{5} -6.92820i q^{7} +3.00000 q^{9} -6.92820 q^{11} -6.92820i q^{15} -18.0000 q^{17} +20.7846 q^{19} +12.0000i q^{21} -41.5692i q^{23} +9.00000 q^{25} -5.19615 q^{27} +4.00000i q^{29} -48.4974i q^{31} +12.0000 q^{33} +27.7128 q^{35} -72.0000i q^{37} +18.0000 q^{41} -62.3538 q^{43} +12.0000i q^{45} -41.5692i q^{47} +1.00000 q^{49} +31.1769 q^{51} +44.0000i q^{53} -27.7128i q^{55} -36.0000 q^{57} -62.3538 q^{59} -72.0000i q^{61} -20.7846i q^{63} +20.7846 q^{67} +72.0000i q^{69} +41.5692i q^{71} -82.0000 q^{73} -15.5885 q^{75} +48.0000i q^{77} +62.3538i q^{79} +9.00000 q^{81} +131.636 q^{83} -72.0000i q^{85} -6.92820i q^{87} +126.000 q^{89} +84.0000i q^{93} +83.1384i q^{95} +110.000 q^{97} -20.7846 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{9}+O(q^{10}) $ 4 * q + 12 * q^9	$ 4 q + 12 q^{9} - 72 q^{17} + 36 q^{25} + 48 q^{33} + 72 q^{41} + 4 q^{49} - 144 q^{57} - 328 q^{73} + 36 q^{81} + 504 q^{89} + 440 q^{97}+O(q^{100}) $ 4 * q + 12 * q^9 - 72 * q^17 + 36 * q^25 + 48 * q^33 + 72 * q^41 + 4 * q^49 - 144 * q^57 - 328 * q^73 + 36 * q^81 + 504 * q^89 + 440 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$-1$	$-1$	$1$

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
$3$	−1.73205	−0.577350
$3$	−1.73205	−0.577350
$4$	0	0
$5$	4.00000i	0.800000i	0.916515	+	0.400000i	$0.130990\pi$
$5$	4.00000i	0.800000i	−0.916515	+	0.400000i	$0.869010\pi$
$6$	0	0
$7$	− 6.92820i	− 0.989743i	−0.868966	−	0.494872i	$-0.835215\pi$
$7$	− 6.92820i	− 0.989743i	0.868966	−	0.494872i	$-0.164785\pi$
$8$	0	0
$9$	3.00000	0.333333
$10$	0	0
$11$	−6.92820	−0.629837	−0.314918	−	0.949119i	$-0.601977\pi$
$11$	−6.92820	−0.629837	−0.314918	+	0.949119i	$0.601977\pi$
$12$	0	0
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	− 6.92820i	− 0.461880i
$16$	0	0
$17$	−18.0000	−1.05882	−0.529412	−	0.848365i	$-0.677587\pi$
$17$	−18.0000	−1.05882	−0.529412	+	0.848365i	$0.677587\pi$
$18$	0	0
$19$	20.7846	1.09393	0.546963	−	0.837157i	$-0.315784\pi$
$19$	20.7846	1.09393	0.546963	+	0.837157i	$0.315784\pi$
$20$	0	0
$21$	12.0000i	0.571429i
$22$	0	0
$23$	− 41.5692i	− 1.80736i	−0.428211	−	0.903679i	$-0.640856\pi$
$23$	− 41.5692i	− 1.80736i	0.428211	−	0.903679i	$-0.359144\pi$
$24$	0	0
$25$	9.00000	0.360000
$26$	0	0
$27$	−5.19615	−0.192450
$28$	0	0
$29$	4.00000i	0.137931i	0.997619	+	0.0689655i	$0.0219698\pi$
$29$	4.00000i	0.137931i	−0.997619	+	0.0689655i	$0.978030\pi$
$30$	0	0
$31$	− 48.4974i	− 1.56443i	−0.623007	−	0.782216i	$-0.714089\pi$
$31$	− 48.4974i	− 1.56443i	0.623007	−	0.782216i	$-0.285911\pi$
$32$	0	0
$33$	12.0000	0.363636
$34$	0	0
$35$	27.7128	0.791795
$36$	0	0
$37$	− 72.0000i	− 1.94595i	−0.230919	−	0.972973i	$-0.574173\pi$
$37$	− 72.0000i	− 1.94595i	0.230919	−	0.972973i	$-0.425827\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	18.0000	0.439024	0.219512	−	0.975610i	$-0.429553\pi$
$41$	18.0000	0.439024	0.219512	+	0.975610i	$0.429553\pi$
$42$	0	0
$43$	−62.3538	−1.45009	−0.725045	−	0.688702i	$-0.758181\pi$
$43$	−62.3538	−1.45009	−0.725045	+	0.688702i	$0.758181\pi$
$44$	0	0
$45$	12.0000i	0.266667i
$46$	0	0
$47$	− 41.5692i	− 0.884451i	−0.896904	−	0.442226i	$-0.854189\pi$
$47$	− 41.5692i	− 0.884451i	0.896904	−	0.442226i	$-0.145811\pi$
$48$	0	0
$49$	1.00000	0.0204082
$50$	0	0
$51$	31.1769	0.611312
$52$	0	0
$53$	44.0000i	0.830189i	0.909778	+	0.415094i	$0.136251\pi$
$53$	44.0000i	0.830189i	−0.909778	+	0.415094i	$0.863749\pi$
$54$	0	0
$55$	− 27.7128i	− 0.503869i
$56$	0	0
$57$	−36.0000	−0.631579
$58$	0	0
$59$	−62.3538	−1.05684	−0.528422	−	0.848982i	$-0.677216\pi$
$59$	−62.3538	−1.05684	−0.528422	+	0.848982i	$0.677216\pi$
$60$	0	0
$61$	− 72.0000i	− 1.18033i	−0.807283	−	0.590164i	$-0.799063\pi$
$61$	− 72.0000i	− 1.18033i	0.807283	−	0.590164i	$-0.200937\pi$
$62$	0	0
$63$	− 20.7846i	− 0.329914i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	20.7846	0.310218	0.155109	−	0.987897i	$-0.450427\pi$
$67$	20.7846	0.310218	0.155109	+	0.987897i	$0.450427\pi$
$68$	0	0
$69$	72.0000i	1.04348i
$70$	0	0
$71$	41.5692i	0.585482i	0.956192	+	0.292741i	$0.0945674\pi$
$71$	41.5692i	0.585482i	−0.956192	+	0.292741i	$0.905433\pi$
$72$	0	0
$73$	−82.0000	−1.12329	−0.561644	−	0.827379i	$-0.689831\pi$
$73$	−82.0000	−1.12329	−0.561644	+	0.827379i	$0.689831\pi$
$74$	0	0
$75$	−15.5885	−0.207846
$76$	0	0
$77$	48.0000i	0.623377i
$78$	0	0
$79$	62.3538i	0.789289i	0.918834	+	0.394644i	$0.129132\pi$
$79$	62.3538i	0.789289i	−0.918834	+	0.394644i	$0.870868\pi$
$80$	0	0
$81$	9.00000	0.111111
$82$	0	0
$83$	131.636	1.58597	0.792987	−	0.609238i	$-0.208525\pi$
$83$	131.636	1.58597	0.792987	+	0.609238i	$0.208525\pi$
$84$	0	0
$85$	− 72.0000i	− 0.847059i
$86$	0	0
$87$	− 6.92820i	− 0.0796345i
$88$	0	0
$89$	126.000	1.41573	0.707865	−	0.706348i	$-0.249658\pi$
$89$	126.000	1.41573	0.707865	+	0.706348i	$0.249658\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	84.0000i	0.903226i
$94$	0	0
$95$	83.1384i	0.875141i
$96$	0	0
$97$	110.000	1.13402	0.567010	−	0.823711i	$-0.308100\pi$
$97$	110.000	1.13402	0.567010	+	0.823711i	$0.308100\pi$
$98$	0	0
$99$	−20.7846	−0.209946
$100$	0	0
$101$	92.0000i	0.910891i	0.890264	+	0.455446i	$0.150520\pi$
$101$	92.0000i	0.910891i	−0.890264	+	0.455446i	$0.849480\pi$
$102$	0	0
$103$	62.3538i	0.605377i	0.953090	+	0.302688i	$0.0978842\pi$
$103$	62.3538i	0.605377i	−0.953090	+	0.302688i	$0.902116\pi$
$104$	0	0
$105$	−48.0000	−0.457143
$106$	0	0
$107$	−90.0666	−0.841744	−0.420872	−	0.907120i	$-0.638276\pi$
$107$	−90.0666	−0.841744	−0.420872	+	0.907120i	$0.638276\pi$
$108$	0	0
$109$	− 144.000i	− 1.32110i	−0.750782	−	0.660550i	$-0.770323\pi$
$109$	− 144.000i	− 1.32110i	0.750782	−	0.660550i	$-0.229677\pi$
$110$	0	0
$111$	124.708i	1.12349i
$112$	0	0
$113$	−126.000	−1.11504	−0.557522	−	0.830162i	$-0.688248\pi$
$113$	−126.000	−1.11504	−0.557522	+	0.830162i	$0.688248\pi$
$114$	0	0
$115$	166.277	1.44589
$116$	0	0
$117$	0	0
$118$	0	0
$119$	124.708i	1.04796i
$120$	0	0
$121$	−73.0000	−0.603306
$122$	0	0
$123$	−31.1769	−0.253471
$124$	0	0
$125$	136.000i	1.08800i
$126$	0	0
$127$	159.349i	1.25471i	0.778732	+	0.627357i	$0.215863\pi$
$127$	159.349i	1.25471i	−0.778732	+	0.627357i	$0.784137\pi$
$128$	0	0
$129$	108.000	0.837209
$130$	0	0
$131$	−200.918	−1.53372	−0.766862	−	0.641812i	$-0.778183\pi$
$131$	−200.918	−1.53372	−0.766862	+	0.641812i	$0.778183\pi$
$132$	0	0
$133$	− 144.000i	− 1.08271i
$134$	0	0
$135$	− 20.7846i	− 0.153960i
$136$	0	0
$137$	18.0000	0.131387	0.0656934	−	0.997840i	$-0.479074\pi$
$137$	18.0000	0.131387	0.0656934	+	0.997840i	$0.479074\pi$
$138$	0	0
$139$	−62.3538	−0.448589	−0.224294	−	0.974521i	$-0.572008\pi$
$139$	−62.3538	−0.448589	−0.224294	+	0.974521i	$0.572008\pi$
$140$	0	0
$141$	72.0000i	0.510638i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−16.0000	−0.110345
$146$	0	0
$147$	−1.73205	−0.0117827
$148$	0	0
$149$	− 140.000i	− 0.939597i	−0.882774	−	0.469799i	$-0.844327\pi$
$149$	− 140.000i	− 0.939597i	0.882774	−	0.469799i	$-0.155673\pi$
$150$	0	0
$151$	131.636i	0.871761i	0.900005	+	0.435880i	$0.143563\pi$
$151$	131.636i	0.871761i	−0.900005	+	0.435880i	$0.856437\pi$
$152$	0	0
$153$	−54.0000	−0.352941
$154$	0	0
$155$	193.990	1.25155
$156$	0	0
$157$	− 216.000i	− 1.37580i	−0.725807	−	0.687898i	$-0.758534\pi$
$157$	− 216.000i	− 1.37580i	0.725807	−	0.687898i	$-0.241466\pi$
$158$	0	0
$159$	− 76.2102i	− 0.479310i
$160$	0	0
$161$	−288.000	−1.78882
$162$	0	0
$163$	−228.631	−1.40264	−0.701321	−	0.712845i	$-0.747406\pi$
$163$	−228.631	−1.40264	−0.701321	+	0.712845i	$0.747406\pi$
$164$	0	0
$165$	48.0000i	0.290909i
$166$	0	0
$167$	83.1384i	0.497835i	0.968525	+	0.248917i	$0.0800748\pi$
$167$	83.1384i	0.497835i	−0.968525	+	0.248917i	$0.919925\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	62.3538	0.364642
$172$	0	0
$173$	− 52.0000i	− 0.300578i	−0.988642	−	0.150289i	$-0.951980\pi$
$173$	− 52.0000i	− 0.300578i	0.988642	−	0.150289i	$-0.0480204\pi$
$174$	0	0
$175$	− 62.3538i	− 0.356308i
$176$	0	0
$177$	108.000	0.610169
$178$	0	0
$179$	159.349	0.890216	0.445108	−	0.895477i	$-0.353165\pi$
$179$	159.349	0.890216	0.445108	+	0.895477i	$0.353165\pi$
$180$	0	0
$181$	144.000i	0.795580i	0.917476	+	0.397790i	$0.130223\pi$
$181$	144.000i	0.795580i	−0.917476	+	0.397790i	$0.869777\pi$
$182$	0	0
$183$	124.708i	0.681463i
$184$	0	0
$185$	288.000	1.55676
$186$	0	0
$187$	124.708	0.666886
$188$	0	0
$189$	36.0000i	0.190476i
$190$	0	0
$191$	83.1384i	0.435280i	0.976029	+	0.217640i	$0.0698358\pi$
$191$	83.1384i	0.435280i	−0.976029	+	0.217640i	$0.930164\pi$
$192$	0	0
$193$	94.0000	0.487047	0.243523	−	0.969895i	$-0.421697\pi$
$193$	94.0000	0.487047	0.243523	+	0.969895i	$0.421697\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	188.000i	0.954315i	0.878818	+	0.477157i	$0.158333\pi$
$197$	188.000i	0.954315i	−0.878818	+	0.477157i	$0.841667\pi$
$198$	0	0
$199$	− 159.349i	− 0.800747i	−0.916352	−	0.400374i	$-0.868880\pi$
$199$	− 159.349i	− 0.800747i	0.916352	−	0.400374i	$-0.131120\pi$
$200$	0	0
$201$	−36.0000	−0.179104
$202$	0	0
$203$	27.7128	0.136516
$204$	0	0
$205$	72.0000i	0.351220i
$206$	0	0
$207$	− 124.708i	− 0.602452i
$208$	0	0
$209$	−144.000	−0.688995
$210$	0	0
$211$	−62.3538	−0.295516	−0.147758	−	0.989024i	$-0.547206\pi$
$211$	−62.3538	−0.295516	−0.147758	+	0.989024i	$0.547206\pi$
$212$	0	0
$213$	− 72.0000i	− 0.338028i
$214$	0	0
$215$	− 249.415i	− 1.16007i
$216$	0	0
$217$	−336.000	−1.54839
$218$	0	0
$219$	142.028	0.648530
$220$	0	0
$221$	0	0
$222$	0	0
$223$	− 311.769i	− 1.39807i	−0.715088	−	0.699034i	$-0.753614\pi$
$223$	− 311.769i	− 1.39807i	0.715088	−	0.699034i	$-0.246386\pi$
$224$	0	0
$225$	27.0000	0.120000
$226$	0	0
$227$	−311.769	−1.37343	−0.686716	−	0.726926i	$-0.740948\pi$
$227$	−311.769	−1.37343	−0.686716	+	0.726926i	$0.740948\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	− 83.1384i	− 0.359907i
$232$	0	0
$233$	126.000	0.540773	0.270386	−	0.962752i	$-0.412849\pi$
$233$	126.000	0.540773	0.270386	+	0.962752i	$0.412849\pi$
$234$	0	0
$235$	166.277	0.707561
$236$	0	0
$237$	− 108.000i	− 0.455696i
$238$	0	0
$239$	− 166.277i	− 0.695719i	−0.937547	−	0.347860i	$-0.886909\pi$
$239$	− 166.277i	− 0.695719i	0.937547	−	0.347860i	$-0.113091\pi$
$240$	0	0
$241$	158.000	0.655602	0.327801	−	0.944747i	$-0.393693\pi$
$241$	158.000	0.655602	0.327801	+	0.944747i	$0.393693\pi$
$242$	0	0
$243$	−15.5885	−0.0641500
$244$	0	0
$245$	4.00000i	0.0163265i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−228.000	−0.915663
$250$	0	0
$251$	187.061	0.745265	0.372632	−	0.927979i	$-0.378455\pi$
$251$	187.061	0.745265	0.372632	+	0.927979i	$0.378455\pi$
$252$	0	0
$253$	288.000i	1.13834i
$254$	0	0
$255$	124.708i	0.489050i
$256$	0	0
$257$	−126.000	−0.490272	−0.245136	−	0.969489i	$-0.578833\pi$
$257$	−126.000	−0.490272	−0.245136	+	0.969489i	$0.578833\pi$
$258$	0	0
$259$	−498.831	−1.92599
$260$	0	0
$261$	12.0000i	0.0459770i
$262$	0	0
$263$	− 498.831i	− 1.89669i	−0.317234	−	0.948347i	$-0.602754\pi$
$263$	− 498.831i	− 1.89669i	0.317234	−	0.948347i	$-0.397246\pi$
$264$	0	0
$265$	−176.000	−0.664151
$266$	0	0
$267$	−218.238	−0.817372
$268$	0	0
$269$	52.0000i	0.193309i	0.995318	+	0.0966543i	$0.0308141\pi$
$269$	52.0000i	0.193309i	−0.995318	+	0.0966543i	$0.969186\pi$
$270$	0	0
$271$	− 34.6410i	− 0.127827i	−0.997955	−	0.0639133i	$-0.979642\pi$
$271$	− 34.6410i	− 0.127827i	0.997955	−	0.0639133i	$-0.0203581\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−62.3538	−0.226741
$276$	0	0
$277$	144.000i	0.519856i	0.965628	+	0.259928i	$0.0836988\pi$
$277$	144.000i	0.519856i	−0.965628	+	0.259928i	$0.916301\pi$
$278$	0	0
$279$	− 145.492i	− 0.521478i
$280$	0	0
$281$	414.000	1.47331	0.736655	−	0.676269i	$-0.236404\pi$
$281$	414.000	1.47331	0.736655	+	0.676269i	$0.236404\pi$
$282$	0	0
$283$	353.338	1.24855	0.624273	−	0.781206i	$-0.285395\pi$
$283$	353.338	1.24855	0.624273	+	0.781206i	$0.285395\pi$
$284$	0	0
$285$	− 144.000i	− 0.505263i
$286$	0	0
$287$	− 124.708i	− 0.434521i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	−190.526	−0.654727
$292$	0	0
$293$	− 532.000i	− 1.81570i	−0.419295	−	0.907850i	$-0.637723\pi$
$293$	− 532.000i	− 1.81570i	0.419295	−	0.907850i	$-0.362277\pi$
$294$	0	0
$295$	− 249.415i	− 0.845476i
$296$	0	0
$297$	36.0000	0.121212
$298$	0	0
$299$	0	0
$300$	0	0
$301$	432.000i	1.43522i
$302$	0	0
$303$	− 159.349i	− 0.525903i
$304$	0	0
$305$	288.000	0.944262
$306$	0	0
$307$	519.615	1.69256	0.846279	−	0.532740i	$-0.178838\pi$
$307$	519.615	1.69256	0.846279	+	0.532740i	$0.178838\pi$
$308$	0	0
$309$	− 108.000i	− 0.349515i
$310$	0	0
$311$	83.1384i	0.267326i	0.991027	+	0.133663i	$0.0426740\pi$
$311$	83.1384i	0.267326i	−0.991027	+	0.133663i	$0.957326\pi$
$312$	0	0
$313$	2.00000	0.00638978	0.00319489	−	0.999995i	$-0.498983\pi$
$313$	2.00000	0.00638978	0.00319489	+	0.999995i	$0.498983\pi$
$314$	0	0
$315$	83.1384	0.263932
$316$	0	0
$317$	− 524.000i	− 1.65300i	−0.562939	−	0.826498i	$-0.690329\pi$
$317$	− 524.000i	− 1.65300i	0.562939	−	0.826498i	$-0.309671\pi$
$318$	0	0
$319$	− 27.7128i	− 0.0868740i
$320$	0	0
$321$	156.000	0.485981
$322$	0	0
$323$	−374.123	−1.15828
$324$	0	0
$325$	0	0
$326$	0	0
$327$	249.415i	0.762738i
$328$	0	0
$329$	−288.000	−0.875380
$330$	0	0
$331$	20.7846	0.0627934	0.0313967	−	0.999507i	$-0.490004\pi$
$331$	20.7846	0.0627934	0.0313967	+	0.999507i	$0.490004\pi$
$332$	0	0
$333$	− 216.000i	− 0.648649i
$334$	0	0
$335$	83.1384i	0.248174i
$336$	0	0
$337$	82.0000	0.243323	0.121662	−	0.992572i	$-0.461178\pi$
$337$	82.0000	0.243323	0.121662	+	0.992572i	$0.461178\pi$
$338$	0	0
$339$	218.238	0.643771
$340$	0	0
$341$	336.000i	0.985337i
$342$	0	0
$343$	− 346.410i	− 1.00994i
$344$	0	0
$345$	−288.000	−0.834783
$346$	0	0
$347$	297.913	0.858538	0.429269	−	0.903177i	$-0.358771\pi$
$347$	297.913	0.858538	0.429269	+	0.903177i	$0.358771\pi$
$348$	0	0
$349$	216.000i	0.618911i	0.950914	+	0.309456i	$0.100147\pi$
$349$	216.000i	0.618911i	−0.950914	+	0.309456i	$0.899853\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−414.000	−1.17280	−0.586402	−	0.810020i	$-0.699456\pi$
$353$	−414.000	−1.17280	−0.586402	+	0.810020i	$0.699456\pi$
$354$	0	0
$355$	−166.277	−0.468386
$356$	0	0
$357$	− 216.000i	− 0.605042i
$358$	0	0
$359$	540.400i	1.50529i	0.658425	+	0.752646i	$0.271223\pi$
$359$	540.400i	1.50529i	−0.658425	+	0.752646i	$0.728777\pi$
$360$	0	0
$361$	71.0000	0.196676
$362$	0	0
$363$	126.440	0.348319
$364$	0	0
$365$	− 328.000i	− 0.898630i
$366$	0	0
$367$	117.779i	0.320925i	0.987042	+	0.160462i	$0.0512986\pi$
$367$	117.779i	0.320925i	−0.987042	+	0.160462i	$0.948701\pi$
$368$	0	0
$369$	54.0000	0.146341
$370$	0	0
$371$	304.841	0.821674
$372$	0	0
$373$	360.000i	0.965147i	0.875855	+	0.482574i	$0.160298\pi$
$373$	360.000i	0.965147i	−0.875855	+	0.482574i	$0.839702\pi$
$374$	0	0
$375$	− 235.559i	− 0.628157i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−394.908	−1.04197	−0.520986	−	0.853565i	$-0.674436\pi$
$379$	−394.908	−1.04197	−0.520986	+	0.853565i	$0.674436\pi$
$380$	0	0
$381$	− 276.000i	− 0.724409i
$382$	0	0
$383$	− 415.692i	− 1.08536i	−0.839940	−	0.542679i	$-0.817410\pi$
$383$	− 415.692i	− 1.08536i	0.839940	−	0.542679i	$-0.182590\pi$
$384$	0	0
$385$	−192.000	−0.498701
$386$	0	0
$387$	−187.061	−0.483363
$388$	0	0
$389$	52.0000i	0.133676i	0.997764	+	0.0668380i	$0.0212911\pi$
$389$	52.0000i	0.133676i	−0.997764	+	0.0668380i	$0.978709\pi$
$390$	0	0
$391$	748.246i	1.91367i
$392$	0	0
$393$	348.000	0.885496
$394$	0	0
$395$	−249.415	−0.631431
$396$	0	0
$397$	216.000i	0.544081i	0.962286	+	0.272040i	$0.0876984\pi$
$397$	216.000i	0.544081i	−0.962286	+	0.272040i	$0.912302\pi$
$398$	0	0
$399$	249.415i	0.625101i
$400$	0	0
$401$	−18.0000	−0.0448878	−0.0224439	−	0.999748i	$-0.507145\pi$
$401$	−18.0000	−0.0448878	−0.0224439	+	0.999748i	$0.507145\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	36.0000i	0.0888889i
$406$	0	0
$407$	498.831i	1.22563i
$408$	0	0
$409$	−14.0000	−0.0342298	−0.0171149	−	0.999854i	$-0.505448\pi$
$409$	−14.0000	−0.0342298	−0.0171149	+	0.999854i	$0.505448\pi$
$410$	0	0
$411$	−31.1769	−0.0758562
$412$	0	0
$413$	432.000i	1.04600i
$414$	0	0
$415$	526.543i	1.26878i
$416$	0	0
$417$	108.000	0.258993
$418$	0	0
$419$	436.477	1.04171	0.520855	−	0.853645i	$-0.325613\pi$
$419$	436.477	1.04171	0.520855	+	0.853645i	$0.325613\pi$
$420$	0	0
$421$	720.000i	1.71021i	0.518452	+	0.855107i	$0.326509\pi$
$421$	720.000i	1.71021i	−0.518452	+	0.855107i	$0.673491\pi$
$422$	0	0
$423$	− 124.708i	− 0.294817i
$424$	0	0
$425$	−162.000	−0.381176
$426$	0	0
$427$	−498.831	−1.16822
$428$	0	0
$429$	0	0
$430$	0	0
$431$	540.400i	1.25383i	0.779088	+	0.626914i	$0.215682\pi$
$431$	540.400i	1.25383i	−0.779088	+	0.626914i	$0.784318\pi$
$432$	0	0
$433$	334.000	0.771363	0.385681	−	0.922632i	$-0.373966\pi$
$433$	334.000	0.771363	0.385681	+	0.922632i	$0.373966\pi$
$434$	0	0
$435$	27.7128	0.0637076
$436$	0	0
$437$	− 864.000i	− 1.97712i
$438$	0	0
$439$	256.344i	0.583926i	0.956430	+	0.291963i	$0.0943084\pi$
$439$	256.344i	0.583926i	−0.956430	+	0.291963i	$0.905692\pi$
$440$	0	0
$441$	3.00000	0.00680272
$442$	0	0
$443$	−561.184	−1.26678	−0.633391	−	0.773832i	$-0.718338\pi$
$443$	−561.184	−1.26678	−0.633391	+	0.773832i	$0.718338\pi$
$444$	0	0
$445$	504.000i	1.13258i
$446$	0	0
$447$	242.487i	0.542477i
$448$	0	0
$449$	−306.000	−0.681514	−0.340757	−	0.940151i	$-0.610683\pi$
$449$	−306.000	−0.681514	−0.340757	+	0.940151i	$0.610683\pi$
$450$	0	0
$451$	−124.708	−0.276514
$452$	0	0
$453$	− 228.000i	− 0.503311i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	466.000	1.01969	0.509847	−	0.860265i	$-0.329702\pi$
$457$	466.000	1.01969	0.509847	+	0.860265i	$0.329702\pi$
$458$	0	0
$459$	93.5307	0.203771
$460$	0	0
$461$	− 196.000i	− 0.425163i	−0.977143	−	0.212581i	$-0.931813\pi$
$461$	− 196.000i	− 0.425163i	0.977143	−	0.212581i	$-0.0681870\pi$
$462$	0	0
$463$	− 200.918i	− 0.433948i	−0.976177	−	0.216974i	$-0.930381\pi$
$463$	− 200.918i	− 0.433948i	0.976177	−	0.216974i	$-0.0696187\pi$
$464$	0	0
$465$	−336.000	−0.722581
$466$	0	0
$467$	214.774	0.459902	0.229951	−	0.973202i	$-0.426143\pi$
$467$	214.774	0.459902	0.229951	+	0.973202i	$0.426143\pi$
$468$	0	0
$469$	− 144.000i	− 0.307036i
$470$	0	0
$471$	374.123i	0.794316i
$472$	0	0
$473$	432.000	0.913319
$474$	0	0
$475$	187.061	0.393814
$476$	0	0
$477$	132.000i	0.276730i
$478$	0	0
$479$	124.708i	0.260350i	0.991491	+	0.130175i	$0.0415539\pi$
$479$	124.708i	0.260350i	−0.991491	+	0.130175i	$0.958446\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	498.831	1.03278
$484$	0	0
$485$	440.000i	0.907216i
$486$	0	0
$487$	187.061i	0.384110i	0.981384	+	0.192055i	$0.0615152\pi$
$487$	187.061i	0.384110i	−0.981384	+	0.192055i	$0.938485\pi$
$488$	0	0
$489$	396.000	0.809816
$490$	0	0
$491$	48.4974	0.0987728	0.0493864	−	0.998780i	$-0.484273\pi$
$491$	48.4974	0.0987728	0.0493864	+	0.998780i	$0.484273\pi$
$492$	0	0
$493$	− 72.0000i	− 0.146045i
$494$	0	0
$495$	− 83.1384i	− 0.167956i
$496$	0	0
$497$	288.000	0.579477
$498$	0	0
$499$	602.754	1.20792	0.603962	−	0.797013i	$-0.293588\pi$
$499$	602.754	1.20792	0.603962	+	0.797013i	$0.293588\pi$
$500$	0	0
$501$	− 144.000i	− 0.287425i
$502$	0	0
$503$	− 290.985i	− 0.578498i	−0.957254	−	0.289249i	$-0.906594\pi$
$503$	− 290.985i	− 0.578498i	0.957254	−	0.289249i	$-0.0934056\pi$
$504$	0	0
$505$	−368.000	−0.728713
$506$	0	0
$507$	−292.717	−0.577350
$508$	0	0
$509$	− 284.000i	− 0.557957i	−0.960297	−	0.278978i	$-0.910004\pi$
$509$	− 284.000i	− 0.557957i	0.960297	−	0.278978i	$-0.0899958\pi$
$510$	0	0
$511$	568.113i	1.11177i
$512$	0	0
$513$	−108.000	−0.210526
$514$	0	0
$515$	−249.415	−0.484302
$516$	0	0
$517$	288.000i	0.557060i
$518$	0	0
$519$	90.0666i	0.173539i
$520$	0	0
$521$	−846.000	−1.62380	−0.811900	−	0.583796i	$-0.801567\pi$
$521$	−846.000	−1.62380	−0.811900	+	0.583796i	$0.801567\pi$
$522$	0	0
$523$	−145.492	−0.278188	−0.139094	−	0.990279i	$-0.544419\pi$
$523$	−145.492	−0.278188	−0.139094	+	0.990279i	$0.544419\pi$
$524$	0	0
$525$	108.000i	0.205714i
$526$	0	0
$527$	872.954i	1.65646i
$528$	0	0
$529$	−1199.00	−2.26654
$530$	0	0
$531$	−187.061	−0.352282
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 360.267i	− 0.673395i
$536$	0	0
$537$	−276.000	−0.513966
$538$	0	0
$539$	−6.92820	−0.0128538
$540$	0	0
$541$	− 432.000i	− 0.798521i	−0.916837	−	0.399261i	$-0.869267\pi$
$541$	− 432.000i	− 0.798521i	0.916837	−	0.399261i	$-0.130733\pi$
$542$	0	0
$543$	− 249.415i	− 0.459328i
$544$	0	0
$545$	576.000	1.05688
$546$	0	0
$547$	1018.45	1.86188	0.930938	−	0.365178i	$-0.118992\pi$
$547$	1018.45	1.86188	0.930938	+	0.365178i	$0.118992\pi$
$548$	0	0
$549$	− 216.000i	− 0.393443i
$550$	0	0
$551$	83.1384i	0.150886i
$552$	0	0
$553$	432.000	0.781193
$554$	0	0
$555$	−498.831	−0.898794
$556$	0	0
$557$	764.000i	1.37163i	0.727774	+	0.685817i	$0.240555\pi$
$557$	764.000i	1.37163i	−0.727774	+	0.685817i	$0.759445\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−216.000	−0.385027
$562$	0	0
$563$	824.456	1.46440	0.732199	−	0.681091i	$-0.238494\pi$
$563$	824.456	1.46440	0.732199	+	0.681091i	$0.238494\pi$
$564$	0	0
$565$	− 504.000i	− 0.892035i
$566$	0	0
$567$	− 62.3538i	− 0.109971i
$568$	0	0
$569$	306.000	0.537786	0.268893	−	0.963170i	$-0.413342\pi$
$569$	306.000	0.537786	0.268893	+	0.963170i	$0.413342\pi$
$570$	0	0
$571$	−145.492	−0.254803	−0.127401	−	0.991851i	$-0.540664\pi$
$571$	−145.492	−0.254803	−0.127401	+	0.991851i	$0.540664\pi$
$572$	0	0
$573$	− 144.000i	− 0.251309i
$574$	0	0
$575$	− 374.123i	− 0.650649i
$576$	0	0
$577$	722.000	1.25130	0.625650	−	0.780104i	$-0.284834\pi$
$577$	722.000	1.25130	0.625650	+	0.780104i	$0.284834\pi$
$578$	0	0
$579$	−162.813	−0.281197
$580$	0	0
$581$	− 912.000i	− 1.56971i
$582$	0	0
$583$	− 304.841i	− 0.522883i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−755.174	−1.28650	−0.643249	−	0.765657i	$-0.722414\pi$
$587$	−755.174	−1.28650	−0.643249	+	0.765657i	$0.722414\pi$
$588$	0	0
$589$	− 1008.00i	− 1.71138i
$590$	0	0
$591$	− 325.626i	− 0.550974i
$592$	0	0
$593$	1026.00	1.73019	0.865093	−	0.501612i	$-0.167259\pi$
$593$	1026.00	1.73019	0.865093	+	0.501612i	$0.167259\pi$
$594$	0	0
$595$	−498.831	−0.838371
$596$	0	0
$597$	276.000i	0.462312i
$598$	0	0
$599$	− 290.985i	− 0.485784i	−0.970053	−	0.242892i	$-0.921904\pi$
$599$	− 290.985i	− 0.485784i	0.970053	−	0.242892i	$-0.0780960\pi$
$600$	0	0
$601$	−146.000	−0.242928	−0.121464	−	0.992596i	$-0.538759\pi$
$601$	−146.000	−0.242928	−0.121464	+	0.992596i	$0.538759\pi$
$602$	0	0
$603$	62.3538	0.103406
$604$	0	0
$605$	− 292.000i	− 0.482645i
$606$	0	0
$607$	− 561.184i	− 0.924521i	−0.886744	−	0.462261i	$-0.847038\pi$
$607$	− 561.184i	− 0.924521i	0.886744	−	0.462261i	$-0.152962\pi$
$608$	0	0
$609$	−48.0000	−0.0788177
$610$	0	0
$611$	0	0
$612$	0	0
$613$	648.000i	1.05710i	0.848903	+	0.528548i	$0.177263\pi$
$613$	648.000i	1.05710i	−0.848903	+	0.528548i	$0.822737\pi$
$614$	0	0
$615$	− 124.708i	− 0.202777i
$616$	0	0
$617$	−1026.00	−1.66288	−0.831442	−	0.555611i	$-0.812484\pi$
$617$	−1026.00	−1.66288	−0.831442	+	0.555611i	$0.812484\pi$
$618$	0	0
$619$	−311.769	−0.503666	−0.251833	−	0.967771i	$-0.581033\pi$
$619$	−311.769	−0.503666	−0.251833	+	0.967771i	$0.581033\pi$
$620$	0	0
$621$	216.000i	0.347826i
$622$	0	0
$623$	− 872.954i	− 1.40121i
$624$	0	0
$625$	−319.000	−0.510400
$626$	0	0
$627$	249.415	0.397792
$628$	0	0
$629$	1296.00i	2.06041i
$630$	0	0
$631$	408.764i	0.647803i	0.946091	+	0.323902i	$0.104995\pi$
$631$	408.764i	0.647803i	−0.946091	+	0.323902i	$0.895005\pi$
$632$	0	0
$633$	108.000	0.170616
$634$	0	0
$635$	−637.395	−1.00377
$636$	0	0
$637$	0	0
$638$	0	0
$639$	124.708i	0.195161i
$640$	0	0
$641$	558.000	0.870515	0.435257	−	0.900306i	$-0.356657\pi$
$641$	558.000	0.870515	0.435257	+	0.900306i	$0.356657\pi$
$642$	0	0
$643$	20.7846	0.0323244	0.0161622	−	0.999869i	$-0.494855\pi$
$643$	20.7846	0.0323244	0.0161622	+	0.999869i	$0.494855\pi$
$644$	0	0
$645$	432.000i	0.669767i
$646$	0	0
$647$	− 706.677i	− 1.09224i	−0.837708	−	0.546118i	$-0.816105\pi$
$647$	− 706.677i	− 1.09224i	0.837708	−	0.546118i	$-0.183895\pi$
$648$	0	0
$649$	432.000	0.665639
$650$	0	0
$651$	581.969	0.893962
$652$	0	0
$653$	− 436.000i	− 0.667688i	−0.942628	−	0.333844i	$-0.891654\pi$
$653$	− 436.000i	− 0.667688i	0.942628	−	0.333844i	$-0.108346\pi$
$654$	0	0
$655$	− 803.672i	− 1.22698i
$656$	0	0
$657$	−246.000	−0.374429
$658$	0	0
$659$	−505.759	−0.767464	−0.383732	−	0.923444i	$-0.625361\pi$
$659$	−505.759	−0.767464	−0.383732	+	0.923444i	$0.625361\pi$
$660$	0	0
$661$	− 648.000i	− 0.980333i	−0.871629	−	0.490166i	$-0.836936\pi$
$661$	− 648.000i	− 0.980333i	0.871629	−	0.490166i	$-0.163064\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	576.000	0.866165
$666$	0	0
$667$	166.277	0.249291
$668$	0	0
$669$	540.000i	0.807175i
$670$	0	0
$671$	498.831i	0.743414i
$672$	0	0
$673$	−722.000	−1.07281	−0.536404	−	0.843961i	$-0.680218\pi$
$673$	−722.000	−1.07281	−0.536404	+	0.843961i	$0.680218\pi$
$674$	0	0
$675$	−46.7654	−0.0692820
$676$	0	0
$677$	4.00000i	0.00590842i	0.999996	+	0.00295421i	$0.000940356\pi$
$677$	4.00000i	0.00590842i	−0.999996	+	0.00295421i	$0.999060\pi$
$678$	0	0
$679$	− 762.102i	− 1.12239i
$680$	0	0
$681$	540.000	0.792952
$682$	0	0
$683$	685.892	1.00423	0.502117	−	0.864800i	$-0.332555\pi$
$683$	685.892	1.00423	0.502117	+	0.864800i	$0.332555\pi$
$684$	0	0
$685$	72.0000i	0.105109i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−394.908	−0.571502	−0.285751	−	0.958304i	$-0.592243\pi$
$691$	−394.908	−0.571502	−0.285751	+	0.958304i	$0.592243\pi$
$692$	0	0
$693$	144.000i	0.207792i
$694$	0	0
$695$	− 249.415i	− 0.358871i
$696$	0	0
$697$	−324.000	−0.464849
$698$	0	0
$699$	−218.238	−0.312215
$700$	0	0
$701$	1252.00i	1.78602i	0.450037	+	0.893010i	$0.351411\pi$
$701$	1252.00i	1.78602i	−0.450037	+	0.893010i	$0.648589\pi$
$702$	0	0
$703$	− 1496.49i	− 2.12872i
$704$	0	0
$705$	−288.000	−0.408511
$706$	0	0
$707$	637.395	0.901548
$708$	0	0
$709$	− 1152.00i	− 1.62482i	−0.583084	−	0.812412i	$-0.698154\pi$
$709$	− 1152.00i	− 1.62482i	0.583084	−	0.812412i	$-0.301846\pi$
$710$	0	0
$711$	187.061i	0.263096i
$712$	0	0
$713$	−2016.00	−2.82749
$714$	0	0
$715$	0	0
$716$	0	0
$717$	288.000i	0.401674i
$718$	0	0
$719$	− 1122.37i	− 1.56101i	−0.625147	−	0.780507i	$-0.714961\pi$
$719$	− 1122.37i	− 1.56101i	0.625147	−	0.780507i	$-0.285039\pi$
$720$	0	0
$721$	432.000	0.599168
$722$	0	0
$723$	−273.664	−0.378512
$724$	0	0
$725$	36.0000i	0.0496552i
$726$	0	0
$727$	491.902i	0.676620i	0.941035	+	0.338310i	$0.109855\pi$
$727$	491.902i	0.676620i	−0.941035	+	0.338310i	$0.890145\pi$
$728$	0	0
$729$	27.0000	0.0370370
$730$	0	0
$731$	1122.37	1.53539
$732$	0	0
$733$	− 288.000i	− 0.392906i	−0.980513	−	0.196453i	$-0.937058\pi$
$733$	− 288.000i	− 0.392906i	0.980513	−	0.196453i	$-0.0629423\pi$
$734$	0	0
$735$	− 6.92820i	− 0.00942613i
$736$	0	0
$737$	−144.000	−0.195387
$738$	0	0
$739$	436.477	0.590632	0.295316	−	0.955400i	$-0.404575\pi$
$739$	436.477	0.590632	0.295316	+	0.955400i	$0.404575\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 665.108i	− 0.895165i	−0.894243	−	0.447582i	$-0.852285\pi$
$743$	− 665.108i	− 0.895165i	0.894243	−	0.447582i	$-0.147715\pi$
$744$	0	0
$745$	560.000	0.751678
$746$	0	0
$747$	394.908	0.528658
$748$	0	0
$749$	624.000i	0.833111i
$750$	0	0
$751$	1198.58i	1.59598i	0.602672	+	0.797989i	$0.294103\pi$
$751$	1198.58i	1.59598i	−0.602672	+	0.797989i	$0.705897\pi$
$752$	0	0
$753$	−324.000	−0.430279
$754$	0	0
$755$	−526.543	−0.697409
$756$	0	0
$757$	− 576.000i	− 0.760898i	−0.924802	−	0.380449i	$-0.875769\pi$
$757$	− 576.000i	− 0.760898i	0.924802	−	0.380449i	$-0.124231\pi$
$758$	0	0
$759$	− 498.831i	− 0.657221i
$760$	0	0
$761$	18.0000	0.0236531	0.0118265	−	0.999930i	$-0.496235\pi$
$761$	18.0000	0.0236531	0.0118265	+	0.999930i	$0.496235\pi$
$762$	0	0
$763$	−997.661	−1.30755
$764$	0	0
$765$	− 216.000i	− 0.282353i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	670.000	0.871261	0.435631	−	0.900125i	$-0.356525\pi$
$769$	670.000	0.871261	0.435631	+	0.900125i	$0.356525\pi$
$770$	0	0
$771$	218.238	0.283059
$772$	0	0
$773$	1100.00i	1.42303i	0.702672	+	0.711514i	$0.251990\pi$
$773$	1100.00i	1.42303i	−0.702672	+	0.711514i	$0.748010\pi$
$774$	0	0
$775$	− 436.477i	− 0.563196i
$776$	0	0
$777$	864.000	1.11197
$778$	0	0
$779$	374.123	0.480261
$780$	0	0
$781$	− 288.000i	− 0.368758i
$782$	0	0
$783$	− 20.7846i	− 0.0265448i
$784$	0	0
$785$	864.000	1.10064
$786$	0	0
$787$	−893.738	−1.13563	−0.567813	−	0.823157i	$-0.692210\pi$
$787$	−893.738	−1.13563	−0.567813	+	0.823157i	$0.692210\pi$
$788$	0	0
$789$	864.000i	1.09506i
$790$	0	0
$791$	872.954i	1.10361i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	304.841	0.383448
$796$	0	0
$797$	− 724.000i	− 0.908407i	−0.890898	−	0.454203i	$-0.849924\pi$
$797$	− 724.000i	− 0.908407i	0.890898	−	0.454203i	$-0.150076\pi$
$798$	0	0
$799$	748.246i	0.936478i
$800$	0	0
$801$	378.000	0.471910
$802$	0	0
$803$	568.113	0.707488
$804$	0	0
$805$	− 1152.00i	− 1.43106i
$806$	0	0
$807$	− 90.0666i	− 0.111607i
$808$	0	0
$809$	−558.000	−0.689740	−0.344870	−	0.938650i	$-0.612077\pi$
$809$	−558.000	−0.689740	−0.344870	+	0.938650i	$0.612077\pi$
$810$	0	0
$811$	1351.00	1.66584	0.832922	−	0.553390i	$-0.186666\pi$
$811$	1351.00	1.66584	0.832922	+	0.553390i	$0.186666\pi$
$812$	0	0
$813$	60.0000i	0.0738007i
$814$	0	0
$815$	− 914.523i	− 1.12211i
$816$	0	0
$817$	−1296.00	−1.58629
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 292.000i	− 0.355664i	−0.984061	−	0.177832i	$-0.943092\pi$
$821$	− 292.000i	− 0.355664i	0.984061	−	0.177832i	$-0.0569083\pi$
$822$	0	0
$823$	− 1447.99i	− 1.75941i	−0.475520	−	0.879705i	$-0.657740\pi$
$823$	− 1447.99i	− 1.75941i	0.475520	−	0.879705i	$-0.342260\pi$
$824$	0	0
$825$	108.000	0.130909
$826$	0	0
$827$	−810.600	−0.980169	−0.490085	−	0.871675i	$-0.663034\pi$
$827$	−810.600	−0.980169	−0.490085	+	0.871675i	$0.663034\pi$
$828$	0	0
$829$	− 720.000i	− 0.868516i	−0.900788	−	0.434258i	$-0.857011\pi$
$829$	− 720.000i	− 0.868516i	0.900788	−	0.434258i	$-0.142989\pi$
$830$	0	0
$831$	− 249.415i	− 0.300139i
$832$	0	0
$833$	−18.0000	−0.0216086
$834$	0	0
$835$	−332.554	−0.398268
$836$	0	0
$837$	252.000i	0.301075i
$838$	0	0
$839$	956.092i	1.13956i	0.821797	+	0.569781i	$0.192972\pi$
$839$	956.092i	1.13956i	−0.821797	+	0.569781i	$0.807028\pi$
$840$	0	0
$841$	825.000	0.980975
$842$	0	0
$843$	−717.069	−0.850616
$844$	0	0
$845$	676.000i	0.800000i
$846$	0	0
$847$	505.759i	0.597118i
$848$	0	0
$849$	−612.000	−0.720848
$850$	0	0
$851$	−2992.98	−3.51702
$852$	0	0
$853$	− 72.0000i	− 0.0844080i	−0.999109	−	0.0422040i	$-0.986562\pi$
$853$	− 72.0000i	− 0.0844080i	0.999109	−	0.0422040i	$-0.0134379\pi$
$854$	0	0
$855$	249.415i	0.291714i
$856$	0	0
$857$	306.000	0.357060	0.178530	−	0.983935i	$-0.442866\pi$
$857$	306.000	0.357060	0.178530	+	0.983935i	$0.442866\pi$
$858$	0	0
$859$	1101.58	1.28240	0.641202	−	0.767372i	$-0.278436\pi$
$859$	1101.58	1.28240	0.641202	+	0.767372i	$0.278436\pi$
$860$	0	0
$861$	216.000i	0.250871i
$862$	0	0
$863$	1080.80i	1.25238i	0.779672	+	0.626188i	$0.215386\pi$
$863$	1080.80i	1.25238i	−0.779672	+	0.626188i	$0.784614\pi$
$864$	0	0
$865$	208.000	0.240462
$866$	0	0
$867$	−60.6218	−0.0699213
$868$	0	0
$869$	− 432.000i	− 0.497123i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	330.000	0.378007
$874$	0	0
$875$	942.236	1.07684
$876$	0	0
$877$	− 360.000i	− 0.410490i	−0.978711	−	0.205245i	$-0.934201\pi$
$877$	− 360.000i	− 0.410490i	0.978711	−	0.205245i	$-0.0657992\pi$
$878$	0	0
$879$	921.451i	1.04829i
$880$	0	0
$881$	−414.000	−0.469921	−0.234960	−	0.972005i	$-0.575496\pi$
$881$	−414.000	−0.469921	−0.234960	+	0.972005i	$0.575496\pi$
$882$	0	0
$883$	−311.769	−0.353079	−0.176540	−	0.984294i	$-0.556490\pi$
$883$	−311.769	−0.353079	−0.176540	+	0.984294i	$0.556490\pi$
$884$	0	0
$885$	432.000i	0.488136i
$886$	0	0
$887$	498.831i	0.562380i	0.959652	+	0.281190i	$0.0907290\pi$
$887$	498.831i	0.562380i	−0.959652	+	0.281190i	$0.909271\pi$
$888$	0	0
$889$	1104.00	1.24184
$890$	0	0
$891$	−62.3538	−0.0699819
$892$	0	0
$893$	− 864.000i	− 0.967525i
$894$	0	0
$895$	637.395i	0.712173i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	193.990	0.215784
$900$	0	0
$901$	− 792.000i	− 0.879023i
$902$	0	0
$903$	− 748.246i	− 0.828622i
$904$	0	0
$905$	−576.000	−0.636464
$906$	0	0
$907$	769.031	0.847884	0.423942	−	0.905689i	$-0.360646\pi$
$907$	769.031	0.847884	0.423942	+	0.905689i	$0.360646\pi$
$908$	0	0
$909$	276.000i	0.303630i
$910$	0	0
$911$	1163.94i	1.27765i	0.769353	+	0.638824i	$0.220579\pi$
$911$	1163.94i	1.27765i	−0.769353	+	0.638824i	$0.779421\pi$
$912$	0	0
$913$	−912.000	−0.998905
$914$	0	0
$915$	−498.831	−0.545170
$916$	0	0
$917$	1392.00i	1.51799i
$918$	0	0
$919$	1711.27i	1.86210i	0.364897	+	0.931048i	$0.381104\pi$
$919$	1711.27i	1.86210i	−0.364897	+	0.931048i	$0.618896\pi$
$920$	0	0
$921$	−900.000	−0.977199
$922$	0	0
$923$	0	0
$924$	0	0
$925$	− 648.000i	− 0.700541i
$926$	0	0
$927$	187.061i	0.201792i
$928$	0	0
$929$	270.000	0.290635	0.145318	−	0.989385i	$-0.453580\pi$
$929$	270.000	0.290635	0.145318	+	0.989385i	$0.453580\pi$
$930$	0	0
$931$	20.7846	0.0223250
$932$	0	0
$933$	− 144.000i	− 0.154341i
$934$	0	0
$935$	498.831i	0.533509i
$936$	0	0
$937$	−674.000	−0.719317	−0.359658	−	0.933084i	$-0.617107\pi$
$937$	−674.000	−0.719317	−0.359658	+	0.933084i	$0.617107\pi$
$938$	0	0
$939$	−3.46410	−0.00368914
$940$	0	0
$941$	− 1444.00i	− 1.53454i	−0.641326	−	0.767269i	$-0.721615\pi$
$941$	− 1444.00i	− 1.53454i	0.641326	−	0.767269i	$-0.278385\pi$
$942$	0	0
$943$	− 748.246i	− 0.793474i
$944$	0	0
$945$	−144.000	−0.152381
$946$	0	0
$947$	741.318	0.782806	0.391403	−	0.920219i	$-0.371990\pi$
$947$	741.318	0.782806	0.391403	+	0.920219i	$0.371990\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	907.595i	0.954358i
$952$	0	0
$953$	18.0000	0.0188877	0.00944386	−	0.999955i	$-0.496994\pi$
$953$	18.0000	0.0188877	0.00944386	+	0.999955i	$0.496994\pi$
$954$	0	0
$955$	−332.554	−0.348224
$956$	0	0
$957$	48.0000i	0.0501567i
$958$	0	0
$959$	− 124.708i	− 0.130039i
$960$	0	0
$961$	−1391.00	−1.44745
$962$	0	0
$963$	−270.200	−0.280581
$964$	0	0
$965$	376.000i	0.389637i
$966$	0	0
$967$	1697.41i	1.75534i	0.479269	+	0.877668i	$0.340902\pi$
$967$	1697.41i	1.75534i	−0.479269	+	0.877668i	$0.659098\pi$
$968$	0	0
$969$	648.000	0.668731
$970$	0	0
$971$	907.595	0.934701	0.467350	−	0.884072i	$-0.345209\pi$
$971$	907.595	0.934701	0.467350	+	0.884072i	$0.345209\pi$
$972$	0	0
$973$	432.000i	0.443988i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1710.00	1.75026	0.875128	−	0.483892i	$-0.160777\pi$
$977$	1710.00	1.75026	0.875128	+	0.483892i	$0.160777\pi$
$978$	0	0
$979$	−872.954	−0.891679
$980$	0	0
$981$	− 432.000i	− 0.440367i
$982$	0	0
$983$	− 1496.49i	− 1.52237i	−0.648534	−	0.761186i	$-0.724617\pi$
$983$	− 1496.49i	− 1.52237i	0.648534	−	0.761186i	$-0.275383\pi$
$984$	0	0
$985$	−752.000	−0.763452
$986$	0	0
$987$	498.831	0.505401
$988$	0	0
$989$	2592.00i	2.62083i
$990$	0	0
$991$	90.0666i	0.0908846i	0.998967	+	0.0454423i	$0.0144697\pi$
$991$	90.0666i	0.0908846i	−0.998967	+	0.0454423i	$0.985530\pi$
$992$	0	0
$993$	−36.0000	−0.0362538
$994$	0	0
$995$	637.395	0.640598
$996$	0	0
$997$	648.000i	0.649950i	0.945723	+	0.324975i	$0.105356\pi$
$997$	648.000i	0.649950i	−0.945723	+	0.324975i	$0.894644\pi$
$998$	0	0
$999$	374.123i	0.374497i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.3.b.a.319.2	yes	4
3.2	odd	2		1152.3.b.h.703.1		4
4.3	odd	2	inner	384.3.b.a.319.4	yes	4
8.3	odd	2	inner	384.3.b.a.319.1	✓	4
8.5	even	2	inner	384.3.b.a.319.3	yes	4
12.11	even	2		1152.3.b.h.703.2		4
16.3	odd	4		768.3.g.b.511.2		2
16.5	even	4		768.3.g.a.511.2		2
16.11	odd	4		768.3.g.a.511.1		2
16.13	even	4		768.3.g.b.511.1		2
24.5	odd	2		1152.3.b.h.703.3		4
24.11	even	2		1152.3.b.h.703.4		4
48.5	odd	4		2304.3.g.n.1279.2		2
48.11	even	4		2304.3.g.n.1279.1		2
48.29	odd	4		2304.3.g.g.1279.2		2
48.35	even	4		2304.3.g.g.1279.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.3.b.a.319.1	✓	4	8.3	odd	2	inner
384.3.b.a.319.2	yes	4	1.1	even	1	trivial
384.3.b.a.319.3	yes	4	8.5	even	2	inner
384.3.b.a.319.4	yes	4	4.3	odd	2	inner
768.3.g.a.511.1		2	16.11	odd	4
768.3.g.a.511.2		2	16.5	even	4
768.3.g.b.511.1		2	16.13	even	4
768.3.g.b.511.2		2	16.3	odd	4
1152.3.b.h.703.1		4	3.2	odd	2
1152.3.b.h.703.2		4	12.11	even	2
1152.3.b.h.703.3		4	24.5	odd	2
1152.3.b.h.703.4		4	24.11	even	2
2304.3.g.g.1279.1		2	48.35	even	4
2304.3.g.g.1279.2		2	48.29	odd	4
2304.3.g.n.1279.1		2	48.11	even	4
2304.3.g.n.1279.2		2	48.5	odd	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:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	384.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.73205 q^{3} +4.00000i q^{5} -6.92820i q^{7} +3.00000 q^{9} +O(q^{10})\)	\(q-1.73205 q^{3} +4.00000i q^{5} -6.92820i q^{7} +3.00000 q^{9} -6.92820 q^{11} -6.92820i q^{15} -18.0000 q^{17} +20.7846 q^{19} +12.0000i q^{21} -41.5692i q^{23} +9.00000 q^{25} -5.19615 q^{27} +4.00000i q^{29} -48.4974i q^{31} +12.0000 q^{33} +27.7128 q^{35} -72.0000i q^{37} +18.0000 q^{41} -62.3538 q^{43} +12.0000i q^{45} -41.5692i q^{47} +1.00000 q^{49} +31.1769 q^{51} +44.0000i q^{53} -27.7128i q^{55} -36.0000 q^{57} -62.3538 q^{59} -72.0000i q^{61} -20.7846i q^{63} +20.7846 q^{67} +72.0000i q^{69} +41.5692i q^{71} -82.0000 q^{73} -15.5885 q^{75} +48.0000i q^{77} +62.3538i q^{79} +9.00000 q^{81} +131.636 q^{83} -72.0000i q^{85} -6.92820i q^{87} +126.000 q^{89} +84.0000i q^{93} +83.1384i q^{95} +110.000 q^{97} -20.7846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{9}+O(q^{10}) \) 4 * q + 12 * q^9	\( 4 q + 12 q^{9} - 72 q^{17} + 36 q^{25} + 48 q^{33} + 72 q^{41} + 4 q^{49} - 144 q^{57} - 328 q^{73} + 36 q^{81} + 504 q^{89} + 440 q^{97}+O(q^{100}) \) 4 * q + 12 * q^9 - 72 * q^17 + 36 * q^25 + 48 * q^33 + 72 * q^41 + 4 * q^49 - 144 * q^57 - 328 * q^73 + 36 * q^81 + 504 * q^89 + 440 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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