LMFDB - Embedded newform 768.3.g.a.511.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [768,3,Mod(511,768)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("768.511"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(768, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	768.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-8,0,0,0,-6,0,0,0,0,0,0,0,-36] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$20.9264843029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			511.2
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	768.511
Dual form			768.3.g.a.511.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} -4.00000 q^{5} +6.92820i q^{7} -3.00000 q^{9} -6.92820i q^{11} -6.92820i q^{15} -18.0000 q^{17} -20.7846i q^{19} -12.0000 q^{21} +41.5692i q^{23} -9.00000 q^{25} -5.19615i q^{27} +4.00000 q^{29} -48.4974i q^{31} +12.0000 q^{33} -27.7128i q^{35} +72.0000 q^{37} -18.0000 q^{41} -62.3538i q^{43} +12.0000 q^{45} -41.5692i q^{47} +1.00000 q^{49} -31.1769i q^{51} -44.0000 q^{53} +27.7128i q^{55} +36.0000 q^{57} -62.3538i q^{59} -72.0000 q^{61} -20.7846i q^{63} -20.7846i q^{67} -72.0000 q^{69} -41.5692i q^{71} +82.0000 q^{73} -15.5885i q^{75} +48.0000 q^{77} +62.3538i q^{79} +9.00000 q^{81} -131.636i q^{83} +72.0000 q^{85} +6.92820i q^{87} -126.000 q^{89} +84.0000 q^{93} +83.1384i q^{95} +110.000 q^{97} +20.7846i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 8 q^{5} - 6 q^{9} - 36 q^{17} - 24 q^{21} - 18 q^{25} + 8 q^{29} + 24 q^{33} + 144 q^{37} - 36 q^{41} + 24 q^{45} + 2 q^{49} - 88 q^{53} + 72 q^{57} - 144 q^{61} - 144 q^{69} + 164 q^{73} + 96 q^{77}+ \cdots + 220 q^{97}+O(q^{100}) $ 2 * q - 8 * q^5 - 6 * q^9 - 36 * q^17 - 24 * q^21 - 18 * q^25 + 8 * q^29 + 24 * q^33 + 144 * q^37 - 36 * q^41 + 24 * q^45 + 2 * q^49 - 88 * q^53 + 72 * q^57 - 144 * q^61 - 144 * q^69 + 164 * q^73 + 96 * q^77 + 18 * q^81 + 144 * q^85 - 252 * q^89 + 168 * q^93 + 220 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	1.73205i	0.577350i
$3$	1.73205i	0.577350i
$4$	0	0
$5$	−4.00000	−0.800000	−0.400000	−	0.916515i	$-0.630990\pi$
$5$	−4.00000	−0.800000	−0.400000	+	0.916515i	$0.630990\pi$
$6$	0	0
$7$	6.92820i	0.989743i	0.868966	+	0.494872i	$0.164785\pi$
$7$	6.92820i	0.989743i	−0.868966	+	0.494872i	$0.835215\pi$
$8$	0	0
$9$	−3.00000	−0.333333
$10$	0	0
$11$	− 6.92820i	− 0.629837i	−0.949119	−	0.314918i	$-0.898023\pi$
$11$	− 6.92820i	− 0.629837i	0.949119	−	0.314918i	$-0.101977\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	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
$19$	− 20.7846i	− 1.09393i	−0.837157	−	0.546963i	$-0.815784\pi$
$19$	− 20.7846i	− 1.09393i	0.837157	−	0.546963i	$-0.184216\pi$
$20$	0	0
$21$	−12.0000	−0.571429
$22$	0	0
$23$	41.5692i	1.80736i	0.428211	+	0.903679i	$0.359144\pi$
$23$	41.5692i	1.80736i	−0.428211	+	0.903679i	$0.640856\pi$
$24$	0	0
$25$	−9.00000	−0.360000
$26$	0	0
$27$	− 5.19615i	− 0.192450i
$28$	0	0
$29$	4.00000	0.137931	0.0689655	−	0.997619i	$-0.478030\pi$
$29$	4.00000	0.137931	0.0689655	+	0.997619i	$0.478030\pi$
$30$	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
$35$	− 27.7128i	− 0.791795i
$36$	0	0
$37$	72.0000	1.94595	0.972973	−	0.230919i	$-0.0741732\pi$
$37$	72.0000	1.94595	0.972973	+	0.230919i	$0.0741732\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−18.0000	−0.439024	−0.219512	−	0.975610i	$-0.570447\pi$
$41$	−18.0000	−0.439024	−0.219512	+	0.975610i	$0.570447\pi$
$42$	0	0
$43$	− 62.3538i	− 1.45009i	−0.688702	−	0.725045i	$-0.741819\pi$
$43$	− 62.3538i	− 1.45009i	0.688702	−	0.725045i	$-0.258181\pi$
$44$	0	0
$45$	12.0000	0.266667
$46$	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
$51$	− 31.1769i	− 0.611312i
$52$	0	0
$53$	−44.0000	−0.830189	−0.415094	−	0.909778i	$-0.636251\pi$
$53$	−44.0000	−0.830189	−0.415094	+	0.909778i	$0.636251\pi$
$54$	0	0
$55$	27.7128i	0.503869i
$56$	0	0
$57$	36.0000	0.631579
$58$	0	0
$59$	− 62.3538i	− 1.05684i	−0.848982	−	0.528422i	$-0.822784\pi$
$59$	− 62.3538i	− 1.05684i	0.848982	−	0.528422i	$-0.177216\pi$
$60$	0	0
$61$	−72.0000	−1.18033	−0.590164	−	0.807283i	$-0.700937\pi$
$61$	−72.0000	−1.18033	−0.590164	+	0.807283i	$0.700937\pi$
$62$	0	0
$63$	− 20.7846i	− 0.329914i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 20.7846i	− 0.310218i	−0.987897	−	0.155109i	$-0.950427\pi$
$67$	− 20.7846i	− 0.310218i	0.987897	−	0.155109i	$-0.0495729\pi$
$68$	0	0
$69$	−72.0000	−1.04348
$70$	0	0
$71$	− 41.5692i	− 0.585482i	−0.956192	−	0.292741i	$-0.905433\pi$
$71$	− 41.5692i	− 0.585482i	0.956192	−	0.292741i	$-0.0945674\pi$
$72$	0	0
$73$	82.0000	1.12329	0.561644	−	0.827379i	$-0.310169\pi$
$73$	82.0000	1.12329	0.561644	+	0.827379i	$0.310169\pi$
$74$	0	0
$75$	− 15.5885i	− 0.207846i
$76$	0	0
$77$	48.0000	0.623377
$78$	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
$83$	− 131.636i	− 1.58597i	−0.609238	−	0.792987i	$-0.708525\pi$
$83$	− 131.636i	− 1.58597i	0.609238	−	0.792987i	$-0.291475\pi$
$84$	0	0
$85$	72.0000	0.847059
$86$	0	0
$87$	6.92820i	0.0796345i
$88$	0	0
$89$	−126.000	−1.41573	−0.707865	−	0.706348i	$-0.750342\pi$
$89$	−126.000	−1.41573	−0.707865	+	0.706348i	$0.750342\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	84.0000	0.903226
$94$	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
$99$	20.7846i	0.209946i
$100$	0	0
$101$	−92.0000	−0.910891	−0.455446	−	0.890264i	$-0.650520\pi$
$101$	−92.0000	−0.910891	−0.455446	+	0.890264i	$0.650520\pi$
$102$	0	0
$103$	− 62.3538i	− 0.605377i	−0.953090	−	0.302688i	$-0.902116\pi$
$103$	− 62.3538i	− 0.605377i	0.953090	−	0.302688i	$-0.0978842\pi$
$104$	0	0
$105$	48.0000	0.457143
$106$	0	0
$107$	− 90.0666i	− 0.841744i	−0.907120	−	0.420872i	$-0.861724\pi$
$107$	− 90.0666i	− 0.841744i	0.907120	−	0.420872i	$-0.138276\pi$
$108$	0	0
$109$	−144.000	−1.32110	−0.660550	−	0.750782i	$-0.729677\pi$
$109$	−144.000	−1.32110	−0.660550	+	0.750782i	$0.729677\pi$
$110$	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
$115$	− 166.277i	− 1.44589i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 124.708i	− 1.04796i
$120$	0	0
$121$	73.0000	0.603306
$122$	0	0
$123$	− 31.1769i	− 0.253471i
$124$	0	0
$125$	136.000	1.08800
$126$	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
$131$	200.918i	1.53372i	0.641812	+	0.766862i	$0.278183\pi$
$131$	200.918i	1.53372i	−0.641812	+	0.766862i	$0.721817\pi$
$132$	0	0
$133$	144.000	1.08271
$134$	0	0
$135$	20.7846i	0.153960i
$136$	0	0
$137$	−18.0000	−0.131387	−0.0656934	−	0.997840i	$-0.520926\pi$
$137$	−18.0000	−0.131387	−0.0656934	+	0.997840i	$0.520926\pi$
$138$	0	0
$139$	− 62.3538i	− 0.448589i	−0.974521	−	0.224294i	$-0.927992\pi$
$139$	− 62.3538i	− 0.448589i	0.974521	−	0.224294i	$-0.0720077\pi$
$140$	0	0
$141$	72.0000	0.510638
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−16.0000	−0.110345
$146$	0	0
$147$	1.73205i	0.0117827i
$148$	0	0
$149$	140.000	0.939597	0.469799	−	0.882774i	$-0.344327\pi$
$149$	140.000	0.939597	0.469799	+	0.882774i	$0.344327\pi$
$150$	0	0
$151$	− 131.636i	− 0.871761i	−0.900005	−	0.435880i	$-0.856437\pi$
$151$	− 131.636i	− 0.871761i	0.900005	−	0.435880i	$-0.143563\pi$
$152$	0	0
$153$	54.0000	0.352941
$154$	0	0
$155$	193.990i	1.25155i
$156$	0	0
$157$	−216.000	−1.37580	−0.687898	−	0.725807i	$-0.741466\pi$
$157$	−216.000	−1.37580	−0.687898	+	0.725807i	$0.741466\pi$
$158$	0	0
$159$	− 76.2102i	− 0.479310i
$160$	0	0
$161$	−288.000	−1.78882
$162$	0	0
$163$	228.631i	1.40264i	0.712845	+	0.701321i	$0.247406\pi$
$163$	228.631i	1.40264i	−0.712845	+	0.701321i	$0.752594\pi$
$164$	0	0
$165$	−48.0000	−0.290909
$166$	0	0
$167$	− 83.1384i	− 0.497835i	−0.968525	−	0.248917i	$-0.919925\pi$
$167$	− 83.1384i	− 0.497835i	0.968525	−	0.248917i	$-0.0800748\pi$
$168$	0	0
$169$	−169.000	−1.00000
$170$	0	0
$171$	62.3538i	0.364642i
$172$	0	0
$173$	−52.0000	−0.300578	−0.150289	−	0.988642i	$-0.548020\pi$
$173$	−52.0000	−0.300578	−0.150289	+	0.988642i	$0.548020\pi$
$174$	0	0
$175$	− 62.3538i	− 0.356308i
$176$	0	0
$177$	108.000	0.610169
$178$	0	0
$179$	− 159.349i	− 0.890216i	−0.895477	−	0.445108i	$-0.853165\pi$
$179$	− 159.349i	− 0.890216i	0.895477	−	0.445108i	$-0.146835\pi$
$180$	0	0
$181$	−144.000	−0.795580	−0.397790	−	0.917476i	$-0.630223\pi$
$181$	−144.000	−0.795580	−0.397790	+	0.917476i	$0.630223\pi$
$182$	0	0
$183$	− 124.708i	− 0.681463i
$184$	0	0
$185$	−288.000	−1.55676
$186$	0	0
$187$	124.708i	0.666886i
$188$	0	0
$189$	36.0000	0.190476
$190$	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
$197$	−188.000	−0.954315	−0.477157	−	0.878818i	$-0.658333\pi$
$197$	−188.000	−0.954315	−0.477157	+	0.878818i	$0.658333\pi$
$198$	0	0
$199$	159.349i	0.800747i	0.916352	+	0.400374i	$0.131120\pi$
$199$	159.349i	0.800747i	−0.916352	+	0.400374i	$0.868880\pi$
$200$	0	0
$201$	36.0000	0.179104
$202$	0	0
$203$	27.7128i	0.136516i
$204$	0	0
$205$	72.0000	0.351220
$206$	0	0
$207$	− 124.708i	− 0.602452i
$208$	0	0
$209$	−144.000	−0.688995
$210$	0	0
$211$	62.3538i	0.295516i	0.989024	+	0.147758i	$0.0472056\pi$
$211$	62.3538i	0.295516i	−0.989024	+	0.147758i	$0.952794\pi$
$212$	0	0
$213$	72.0000	0.338028
$214$	0	0
$215$	249.415i	1.16007i
$216$	0	0
$217$	336.000	1.54839
$218$	0	0
$219$	142.028i	0.648530i
$220$	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
$227$	311.769i	1.37343i	0.726926	+	0.686716i	$0.240948\pi$
$227$	311.769i	1.37343i	−0.726926	+	0.686716i	$0.759052\pi$
$228$	0	0
$229$	0	0		−	1.00000i	$-0.5\pi$
$229$	0	0			1.00000i	$0.5\pi$
$230$	0	0
$231$	83.1384i	0.359907i
$232$	0	0
$233$	−126.000	−0.540773	−0.270386	−	0.962752i	$-0.587151\pi$
$233$	−126.000	−0.540773	−0.270386	+	0.962752i	$0.587151\pi$
$234$	0	0
$235$	166.277i	0.707561i
$236$	0	0
$237$	−108.000	−0.455696
$238$	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
$243$	15.5885i	0.0641500i
$244$	0	0
$245$	−4.00000	−0.0163265
$246$	0	0
$247$	0	0
$248$	0	0
$249$	228.000	0.915663
$250$	0	0
$251$	187.061i	0.745265i	0.927979	+	0.372632i	$0.121545\pi$
$251$	187.061i	0.745265i	−0.927979	+	0.372632i	$0.878455\pi$
$252$	0	0
$253$	288.000	1.13834
$254$	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
$259$	498.831i	1.92599i
$260$	0	0
$261$	−12.0000	−0.0459770
$262$	0	0
$263$	498.831i	1.89669i	0.317234	+	0.948347i	$0.397246\pi$
$263$	498.831i	1.89669i	−0.317234	+	0.948347i	$0.602754\pi$
$264$	0	0
$265$	176.000	0.664151
$266$	0	0
$267$	− 218.238i	− 0.817372i
$268$	0	0
$269$	52.0000	0.193309	0.0966543	−	0.995318i	$-0.469186\pi$
$269$	52.0000	0.193309	0.0966543	+	0.995318i	$0.469186\pi$
$270$	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
$275$	62.3538i	0.226741i
$276$	0	0
$277$	−144.000	−0.519856	−0.259928	−	0.965628i	$-0.583699\pi$
$277$	−144.000	−0.519856	−0.259928	+	0.965628i	$0.583699\pi$
$278$	0	0
$279$	145.492i	0.521478i
$280$	0	0
$281$	−414.000	−1.47331	−0.736655	−	0.676269i	$-0.763596\pi$
$281$	−414.000	−1.47331	−0.736655	+	0.676269i	$0.763596\pi$
$282$	0	0
$283$	353.338i	1.24855i	0.781206	+	0.624273i	$0.214605\pi$
$283$	353.338i	1.24855i	−0.781206	+	0.624273i	$0.785395\pi$
$284$	0	0
$285$	−144.000	−0.505263
$286$	0	0
$287$	− 124.708i	− 0.434521i
$288$	0	0
$289$	35.0000	0.121107
$290$	0	0
$291$	190.526i	0.654727i
$292$	0	0
$293$	532.000	1.81570	0.907850	−	0.419295i	$-0.137723\pi$
$293$	532.000	1.81570	0.907850	+	0.419295i	$0.137723\pi$
$294$	0	0
$295$	249.415i	0.845476i
$296$	0	0
$297$	−36.0000	−0.121212
$298$	0	0
$299$	0	0
$300$	0	0
$301$	432.000	1.43522
$302$	0	0
$303$	− 159.349i	− 0.525903i
$304$	0	0
$305$	288.000	0.944262
$306$	0	0
$307$	− 519.615i	− 1.69256i	−0.532740	−	0.846279i	$-0.678838\pi$
$307$	− 519.615i	− 1.69256i	0.532740	−	0.846279i	$-0.321162\pi$
$308$	0	0
$309$	108.000	0.349515
$310$	0	0
$311$	− 83.1384i	− 0.267326i	−0.991027	−	0.133663i	$-0.957326\pi$
$311$	− 83.1384i	− 0.267326i	0.991027	−	0.133663i	$-0.0426740\pi$
$312$	0	0
$313$	−2.00000	−0.00638978	−0.00319489	−	0.999995i	$-0.501017\pi$
$313$	−2.00000	−0.00638978	−0.00319489	+	0.999995i	$0.501017\pi$
$314$	0	0
$315$	83.1384i	0.263932i
$316$	0	0
$317$	−524.000	−1.65300	−0.826498	−	0.562939i	$-0.809671\pi$
$317$	−524.000	−1.65300	−0.826498	+	0.562939i	$0.809671\pi$
$318$	0	0
$319$	− 27.7128i	− 0.0868740i
$320$	0	0
$321$	156.000	0.485981
$322$	0	0
$323$	374.123i	1.15828i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 249.415i	− 0.762738i
$328$	0	0
$329$	288.000	0.875380
$330$	0	0
$331$	20.7846i	0.0627934i	0.999507	+	0.0313967i	$0.00999552\pi$
$331$	20.7846i	0.0627934i	−0.999507	+	0.0313967i	$0.990004\pi$
$332$	0	0
$333$	−216.000	−0.648649
$334$	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
$339$	− 218.238i	− 0.643771i
$340$	0	0
$341$	−336.000	−0.985337
$342$	0	0
$343$	346.410i	1.00994i
$344$	0	0
$345$	288.000	0.834783
$346$	0	0
$347$	297.913i	0.858538i	0.903177	+	0.429269i	$0.141229\pi$
$347$	297.913i	0.858538i	−0.903177	+	0.429269i	$0.858771\pi$
$348$	0	0
$349$	216.000	0.618911	0.309456	−	0.950914i	$-0.399853\pi$
$349$	216.000	0.618911	0.309456	+	0.950914i	$0.399853\pi$
$350$	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
$355$	166.277i	0.468386i
$356$	0	0
$357$	216.000	0.605042
$358$	0	0
$359$	− 540.400i	− 1.50529i	−0.658425	−	0.752646i	$-0.728777\pi$
$359$	− 540.400i	− 1.50529i	0.658425	−	0.752646i	$-0.271223\pi$
$360$	0	0
$361$	−71.0000	−0.196676
$362$	0	0
$363$	126.440i	0.348319i
$364$	0	0
$365$	−328.000	−0.898630
$366$	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
$371$	− 304.841i	− 0.821674i
$372$	0	0
$373$	−360.000	−0.965147	−0.482574	−	0.875855i	$-0.660298\pi$
$373$	−360.000	−0.965147	−0.482574	+	0.875855i	$0.660298\pi$
$374$	0	0
$375$	235.559i	0.628157i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 394.908i	− 1.04197i	−0.853565	−	0.520986i	$-0.825564\pi$
$379$	− 394.908i	− 1.04197i	0.853565	−	0.520986i	$-0.174436\pi$
$380$	0	0
$381$	−276.000	−0.724409
$382$	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
$387$	187.061i	0.483363i
$388$	0	0
$389$	−52.0000	−0.133676	−0.0668380	−	0.997764i	$-0.521291\pi$
$389$	−52.0000	−0.133676	−0.0668380	+	0.997764i	$0.521291\pi$
$390$	0	0
$391$	− 748.246i	− 1.91367i
$392$	0	0
$393$	−348.000	−0.885496
$394$	0	0
$395$	− 249.415i	− 0.631431i
$396$	0	0
$397$	216.000	0.544081	0.272040	−	0.962286i	$-0.412302\pi$
$397$	216.000	0.544081	0.272040	+	0.962286i	$0.412302\pi$
$398$	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
$405$	−36.0000	−0.0888889
$406$	0	0
$407$	− 498.831i	− 1.22563i
$408$	0	0
$409$	14.0000	0.0342298	0.0171149	−	0.999854i	$-0.494552\pi$
$409$	14.0000	0.0342298	0.0171149	+	0.999854i	$0.494552\pi$
$410$	0	0
$411$	− 31.1769i	− 0.0758562i
$412$	0	0
$413$	432.000	1.04600
$414$	0	0
$415$	526.543i	1.26878i
$416$	0	0
$417$	108.000	0.258993
$418$	0	0
$419$	− 436.477i	− 1.04171i	−0.853645	−	0.520855i	$-0.825613\pi$
$419$	− 436.477i	− 1.04171i	0.853645	−	0.520855i	$-0.174387\pi$
$420$	0	0
$421$	−720.000	−1.71021	−0.855107	−	0.518452i	$-0.826509\pi$
$421$	−720.000	−1.71021	−0.855107	+	0.518452i	$0.826509\pi$
$422$	0	0
$423$	124.708i	0.294817i
$424$	0	0
$425$	162.000	0.381176
$426$	0	0
$427$	− 498.831i	− 1.16822i
$428$	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
$435$	− 27.7128i	− 0.0637076i
$436$	0	0
$437$	864.000	1.97712
$438$	0	0
$439$	− 256.344i	− 0.583926i	−0.956430	−	0.291963i	$-0.905692\pi$
$439$	− 256.344i	− 0.583926i	0.956430	−	0.291963i	$-0.0943084\pi$
$440$	0	0
$441$	−3.00000	−0.00680272
$442$	0	0
$443$	− 561.184i	− 1.26678i	−0.773832	−	0.633391i	$-0.781662\pi$
$443$	− 561.184i	− 1.26678i	0.773832	−	0.633391i	$-0.218338\pi$
$444$	0	0
$445$	504.000	1.13258
$446$	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
$451$	124.708i	0.276514i
$452$	0	0
$453$	228.000	0.503311
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−466.000	−1.01969	−0.509847	−	0.860265i	$-0.670298\pi$
$457$	−466.000	−1.01969	−0.509847	+	0.860265i	$0.670298\pi$
$458$	0	0
$459$	93.5307i	0.203771i
$460$	0	0
$461$	−196.000	−0.425163	−0.212581	−	0.977143i	$-0.568187\pi$
$461$	−196.000	−0.425163	−0.212581	+	0.977143i	$0.568187\pi$
$462$	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
$467$	− 214.774i	− 0.459902i	−0.973202	−	0.229951i	$-0.926143\pi$
$467$	− 214.774i	− 0.459902i	0.973202	−	0.229951i	$-0.0738566\pi$
$468$	0	0
$469$	144.000	0.307036
$470$	0	0
$471$	− 374.123i	− 0.794316i
$472$	0	0
$473$	−432.000	−0.913319
$474$	0	0
$475$	187.061i	0.393814i
$476$	0	0
$477$	132.000	0.276730
$478$	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
$483$	− 498.831i	− 1.03278i
$484$	0	0
$485$	−440.000	−0.907216
$486$	0	0
$487$	− 187.061i	− 0.384110i	−0.981384	−	0.192055i	$-0.938485\pi$
$487$	− 187.061i	− 0.384110i	0.981384	−	0.192055i	$-0.0615152\pi$
$488$	0	0
$489$	−396.000	−0.809816
$490$	0	0
$491$	48.4974i	0.0987728i	0.998780	+	0.0493864i	$0.0157266\pi$
$491$	48.4974i	0.0987728i	−0.998780	+	0.0493864i	$0.984273\pi$
$492$	0	0
$493$	−72.0000	−0.146045
$494$	0	0
$495$	− 83.1384i	− 0.167956i
$496$	0	0
$497$	288.000	0.579477
$498$	0	0
$499$	− 602.754i	− 1.20792i	−0.797013	−	0.603962i	$-0.793588\pi$
$499$	− 602.754i	− 1.20792i	0.797013	−	0.603962i	$-0.206412\pi$
$500$	0	0
$501$	144.000	0.287425
$502$	0	0
$503$	290.985i	0.578498i	0.957254	+	0.289249i	$0.0934056\pi$
$503$	290.985i	0.578498i	−0.957254	+	0.289249i	$0.906594\pi$
$504$	0	0
$505$	368.000	0.728713
$506$	0	0
$507$	− 292.717i	− 0.577350i
$508$	0	0
$509$	−284.000	−0.557957	−0.278978	−	0.960297i	$-0.589996\pi$
$509$	−284.000	−0.557957	−0.278978	+	0.960297i	$0.589996\pi$
$510$	0	0
$511$	568.113i	1.11177i
$512$	0	0
$513$	−108.000	−0.210526
$514$	0	0
$515$	249.415i	0.484302i
$516$	0	0
$517$	−288.000	−0.557060
$518$	0	0
$519$	− 90.0666i	− 0.173539i
$520$	0	0
$521$	846.000	1.62380	0.811900	−	0.583796i	$-0.198433\pi$
$521$	846.000	1.62380	0.811900	+	0.583796i	$0.198433\pi$
$522$	0	0
$523$	− 145.492i	− 0.278188i	−0.990279	−	0.139094i	$-0.955581\pi$
$523$	− 145.492i	− 0.278188i	0.990279	−	0.139094i	$-0.0444190\pi$
$524$	0	0
$525$	108.000	0.205714
$526$	0	0
$527$	872.954i	1.65646i
$528$	0	0
$529$	−1199.00	−2.26654
$530$	0	0
$531$	187.061i	0.352282i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	360.267i	0.673395i
$536$	0	0
$537$	276.000	0.513966
$538$	0	0
$539$	− 6.92820i	− 0.0128538i
$540$	0	0
$541$	−432.000	−0.798521	−0.399261	−	0.916837i	$-0.630733\pi$
$541$	−432.000	−0.798521	−0.399261	+	0.916837i	$0.630733\pi$
$542$	0	0
$543$	− 249.415i	− 0.459328i
$544$	0	0
$545$	576.000	1.05688
$546$	0	0
$547$	− 1018.45i	− 1.86188i	−0.365178	−	0.930938i	$-0.618992\pi$
$547$	− 1018.45i	− 1.86188i	0.365178	−	0.930938i	$-0.381008\pi$
$548$	0	0
$549$	216.000	0.393443
$550$	0	0
$551$	− 83.1384i	− 0.150886i
$552$	0	0
$553$	−432.000	−0.781193
$554$	0	0
$555$	− 498.831i	− 0.898794i
$556$	0	0
$557$	764.000	1.37163	0.685817	−	0.727774i	$-0.259445\pi$
$557$	764.000	1.37163	0.685817	+	0.727774i	$0.259445\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−216.000	−0.385027
$562$	0	0
$563$	− 824.456i	− 1.46440i	−0.681091	−	0.732199i	$-0.738494\pi$
$563$	− 824.456i	− 1.46440i	0.681091	−	0.732199i	$-0.261506\pi$
$564$	0	0
$565$	504.000	0.892035
$566$	0	0
$567$	62.3538i	0.109971i
$568$	0	0
$569$	−306.000	−0.537786	−0.268893	−	0.963170i	$-0.586658\pi$
$569$	−306.000	−0.537786	−0.268893	+	0.963170i	$0.586658\pi$
$570$	0	0
$571$	− 145.492i	− 0.254803i	−0.991851	−	0.127401i	$-0.959336\pi$
$571$	− 145.492i	− 0.254803i	0.991851	−	0.127401i	$-0.0406636\pi$
$572$	0	0
$573$	−144.000	−0.251309
$574$	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
$579$	162.813i	0.281197i
$580$	0	0
$581$	912.000	1.56971
$582$	0	0
$583$	304.841i	0.522883i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 755.174i	− 1.28650i	−0.765657	−	0.643249i	$-0.777586\pi$
$587$	− 755.174i	− 1.28650i	0.765657	−	0.643249i	$-0.222414\pi$
$588$	0	0
$589$	−1008.00	−1.71138
$590$	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
$595$	498.831i	0.838371i
$596$	0	0
$597$	−276.000	−0.462312
$598$	0	0
$599$	290.985i	0.485784i	0.970053	+	0.242892i	$0.0780960\pi$
$599$	290.985i	0.485784i	−0.970053	+	0.242892i	$0.921904\pi$
$600$	0	0
$601$	146.000	0.242928	0.121464	−	0.992596i	$-0.461241\pi$
$601$	146.000	0.242928	0.121464	+	0.992596i	$0.461241\pi$
$602$	0	0
$603$	62.3538i	0.103406i
$604$	0	0
$605$	−292.000	−0.482645
$606$	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
$613$	−648.000	−1.05710	−0.528548	−	0.848903i	$-0.677263\pi$
$613$	−648.000	−1.05710	−0.528548	+	0.848903i	$0.677263\pi$
$614$	0	0
$615$	124.708i	0.202777i
$616$	0	0
$617$	1026.00	1.66288	0.831442	−	0.555611i	$-0.187516\pi$
$617$	1026.00	1.66288	0.831442	+	0.555611i	$0.187516\pi$
$618$	0	0
$619$	− 311.769i	− 0.503666i	−0.967771	−	0.251833i	$-0.918967\pi$
$619$	− 311.769i	− 0.503666i	0.967771	−	0.251833i	$-0.0810333\pi$
$620$	0	0
$621$	216.000	0.347826
$622$	0	0
$623$	− 872.954i	− 1.40121i
$624$	0	0
$625$	−319.000	−0.510400
$626$	0	0
$627$	− 249.415i	− 0.397792i
$628$	0	0
$629$	−1296.00	−2.06041
$630$	0	0
$631$	− 408.764i	− 0.647803i	−0.946091	−	0.323902i	$-0.895005\pi$
$631$	− 408.764i	− 0.647803i	0.946091	−	0.323902i	$-0.104995\pi$
$632$	0	0
$633$	−108.000	−0.170616
$634$	0	0
$635$	− 637.395i	− 1.00377i
$636$	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
$643$	− 20.7846i	− 0.0323244i	−0.999869	−	0.0161622i	$-0.994855\pi$
$643$	− 20.7846i	− 0.0323244i	0.999869	−	0.0161622i	$-0.00514482\pi$
$644$	0	0
$645$	−432.000	−0.669767
$646$	0	0
$647$	706.677i	1.09224i	0.837708	+	0.546118i	$0.183895\pi$
$647$	706.677i	1.09224i	−0.837708	+	0.546118i	$0.816105\pi$
$648$	0	0
$649$	−432.000	−0.665639
$650$	0	0
$651$	581.969i	0.893962i
$652$	0	0
$653$	−436.000	−0.667688	−0.333844	−	0.942628i	$-0.608346\pi$
$653$	−436.000	−0.667688	−0.333844	+	0.942628i	$0.608346\pi$
$654$	0	0
$655$	− 803.672i	− 1.22698i
$656$	0	0
$657$	−246.000	−0.374429
$658$	0	0
$659$	505.759i	0.767464i	0.923444	+	0.383732i	$0.125361\pi$
$659$	505.759i	0.767464i	−0.923444	+	0.383732i	$0.874639\pi$
$660$	0	0
$661$	648.000	0.980333	0.490166	−	0.871629i	$-0.336936\pi$
$661$	648.000	0.980333	0.490166	+	0.871629i	$0.336936\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−576.000	−0.866165
$666$	0	0
$667$	166.277i	0.249291i
$668$	0	0
$669$	540.000	0.807175
$670$	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
$675$	46.7654i	0.0692820i
$676$	0	0
$677$	−4.00000	−0.00590842	−0.00295421	−	0.999996i	$-0.500940\pi$
$677$	−4.00000	−0.00590842	−0.00295421	+	0.999996i	$0.500940\pi$
$678$	0	0
$679$	762.102i	1.12239i
$680$	0	0
$681$	−540.000	−0.792952
$682$	0	0
$683$	685.892i	1.00423i	0.864800	+	0.502117i	$0.167445\pi$
$683$	685.892i	1.00423i	−0.864800	+	0.502117i	$0.832555\pi$
$684$	0	0
$685$	72.0000	0.105109
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	394.908i	0.571502i	0.958304	+	0.285751i	$0.0922430\pi$
$691$	394.908i	0.571502i	−0.958304	+	0.285751i	$0.907757\pi$
$692$	0	0
$693$	−144.000	−0.207792
$694$	0	0
$695$	249.415i	0.358871i
$696$	0	0
$697$	324.000	0.464849
$698$	0	0
$699$	− 218.238i	− 0.312215i
$700$	0	0
$701$	1252.00	1.78602	0.893010	−	0.450037i	$-0.148589\pi$
$701$	1252.00	1.78602	0.893010	+	0.450037i	$0.148589\pi$
$702$	0	0
$703$	− 1496.49i	− 2.12872i
$704$	0	0
$705$	−288.000	−0.408511
$706$	0	0
$707$	− 637.395i	− 0.901548i
$708$	0	0
$709$	1152.00	1.62482	0.812412	−	0.583084i	$-0.198154\pi$
$709$	1152.00	1.62482	0.812412	+	0.583084i	$0.198154\pi$
$710$	0	0
$711$	− 187.061i	− 0.263096i
$712$	0	0
$713$	2016.00	2.82749
$714$	0	0
$715$	0	0
$716$	0	0
$717$	288.000	0.401674
$718$	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
$723$	273.664i	0.378512i
$724$	0	0
$725$	−36.0000	−0.0496552
$726$	0	0
$727$	− 491.902i	− 0.676620i	−0.941035	−	0.338310i	$-0.890145\pi$
$727$	− 491.902i	− 0.676620i	0.941035	−	0.338310i	$-0.109855\pi$
$728$	0	0
$729$	−27.0000	−0.0370370
$730$	0	0
$731$	1122.37i	1.53539i
$732$	0	0
$733$	−288.000	−0.392906	−0.196453	−	0.980513i	$-0.562942\pi$
$733$	−288.000	−0.392906	−0.196453	+	0.980513i	$0.562942\pi$
$734$	0	0
$735$	− 6.92820i	− 0.00942613i
$736$	0	0
$737$	−144.000	−0.195387
$738$	0	0
$739$	− 436.477i	− 0.590632i	−0.955400	−	0.295316i	$-0.904575\pi$
$739$	− 436.477i	− 0.590632i	0.955400	−	0.295316i	$-0.0954249\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	665.108i	0.895165i	0.894243	+	0.447582i	$0.147715\pi$
$743$	665.108i	0.895165i	−0.894243	+	0.447582i	$0.852285\pi$
$744$	0	0
$745$	−560.000	−0.751678
$746$	0	0
$747$	394.908i	0.528658i
$748$	0	0
$749$	624.000	0.833111
$750$	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
$755$	526.543i	0.697409i
$756$	0	0
$757$	576.000	0.760898	0.380449	−	0.924802i	$-0.375769\pi$
$757$	576.000	0.760898	0.380449	+	0.924802i	$0.375769\pi$
$758$	0	0
$759$	498.831i	0.657221i
$760$	0	0
$761$	−18.0000	−0.0236531	−0.0118265	−	0.999930i	$-0.503765\pi$
$761$	−18.0000	−0.0236531	−0.0118265	+	0.999930i	$0.503765\pi$
$762$	0	0
$763$	− 997.661i	− 1.30755i
$764$	0	0
$765$	−216.000	−0.282353
$766$	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
$771$	− 218.238i	− 0.283059i
$772$	0	0
$773$	−1100.00	−1.42303	−0.711514	−	0.702672i	$-0.751990\pi$
$773$	−1100.00	−1.42303	−0.711514	+	0.702672i	$0.751990\pi$
$774$	0	0
$775$	436.477i	0.563196i
$776$	0	0
$777$	−864.000	−1.11197
$778$	0	0
$779$	374.123i	0.480261i
$780$	0	0
$781$	−288.000	−0.368758
$782$	0	0
$783$	− 20.7846i	− 0.0265448i
$784$	0	0
$785$	864.000	1.10064
$786$	0	0
$787$	893.738i	1.13563i	0.823157	+	0.567813i	$0.192210\pi$
$787$	893.738i	1.13563i	−0.823157	+	0.567813i	$0.807790\pi$
$788$	0	0
$789$	−864.000	−1.09506
$790$	0	0
$791$	− 872.954i	− 1.10361i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	304.841i	0.383448i
$796$	0	0
$797$	−724.000	−0.908407	−0.454203	−	0.890898i	$-0.650076\pi$
$797$	−724.000	−0.908407	−0.454203	+	0.890898i	$0.650076\pi$
$798$	0	0
$799$	748.246i	0.936478i
$800$	0	0
$801$	378.000	0.471910
$802$	0	0
$803$	− 568.113i	− 0.707488i
$804$	0	0
$805$	1152.00	1.43106
$806$	0	0
$807$	90.0666i	0.111607i
$808$	0	0
$809$	558.000	0.689740	0.344870	−	0.938650i	$-0.387923\pi$
$809$	558.000	0.689740	0.344870	+	0.938650i	$0.387923\pi$
$810$	0	0
$811$	1351.00i	1.66584i	0.553390	+	0.832922i	$0.313334\pi$
$811$	1351.00i	1.66584i	−0.553390	+	0.832922i	$0.686666\pi$
$812$	0	0
$813$	60.0000	0.0738007
$814$	0	0
$815$	− 914.523i	− 1.12211i
$816$	0	0
$817$	−1296.00	−1.58629
$818$	0	0
$819$	0	0
$820$	0	0
$821$	292.000	0.355664	0.177832	−	0.984061i	$-0.443092\pi$
$821$	292.000	0.355664	0.177832	+	0.984061i	$0.443092\pi$
$822$	0	0
$823$	1447.99i	1.75941i	0.475520	+	0.879705i	$0.342260\pi$
$823$	1447.99i	1.75941i	−0.475520	+	0.879705i	$0.657740\pi$
$824$	0	0
$825$	−108.000	−0.130909
$826$	0	0
$827$	− 810.600i	− 0.980169i	−0.871675	−	0.490085i	$-0.836966\pi$
$827$	− 810.600i	− 0.980169i	0.871675	−	0.490085i	$-0.163034\pi$
$828$	0	0
$829$	−720.000	−0.868516	−0.434258	−	0.900788i	$-0.642989\pi$
$829$	−720.000	−0.868516	−0.434258	+	0.900788i	$0.642989\pi$
$830$	0	0
$831$	− 249.415i	− 0.300139i
$832$	0	0
$833$	−18.0000	−0.0216086
$834$	0	0
$835$	332.554i	0.398268i
$836$	0	0
$837$	−252.000	−0.301075
$838$	0	0
$839$	− 956.092i	− 1.13956i	−0.821797	−	0.569781i	$-0.807028\pi$
$839$	− 956.092i	− 1.13956i	0.821797	−	0.569781i	$-0.192972\pi$
$840$	0	0
$841$	−825.000	−0.980975
$842$	0	0
$843$	− 717.069i	− 0.850616i
$844$	0	0
$845$	676.000	0.800000
$846$	0	0
$847$	505.759i	0.597118i
$848$	0	0
$849$	−612.000	−0.720848
$850$	0	0
$851$	2992.98i	3.51702i
$852$	0	0
$853$	72.0000	0.0844080	0.0422040	−	0.999109i	$-0.486562\pi$
$853$	72.0000	0.0844080	0.0422040	+	0.999109i	$0.486562\pi$
$854$	0	0
$855$	− 249.415i	− 0.291714i
$856$	0	0
$857$	−306.000	−0.357060	−0.178530	−	0.983935i	$-0.557134\pi$
$857$	−306.000	−0.357060	−0.178530	+	0.983935i	$0.557134\pi$
$858$	0	0
$859$	1101.58i	1.28240i	0.767372	+	0.641202i	$0.221564\pi$
$859$	1101.58i	1.28240i	−0.767372	+	0.641202i	$0.778436\pi$
$860$	0	0
$861$	216.000	0.250871
$862$	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
$867$	60.6218i	0.0699213i
$868$	0	0
$869$	432.000	0.497123
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−330.000	−0.378007
$874$	0	0
$875$	942.236i	1.07684i
$876$	0	0
$877$	−360.000	−0.410490	−0.205245	−	0.978711i	$-0.565799\pi$
$877$	−360.000	−0.410490	−0.205245	+	0.978711i	$0.565799\pi$
$878$	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
$883$	311.769i	0.353079i	0.984294	+	0.176540i	$0.0564904\pi$
$883$	311.769i	0.353079i	−0.984294	+	0.176540i	$0.943510\pi$
$884$	0	0
$885$	−432.000	−0.488136
$886$	0	0
$887$	− 498.831i	− 0.562380i	−0.959652	−	0.281190i	$-0.909271\pi$
$887$	− 498.831i	− 0.562380i	0.959652	−	0.281190i	$-0.0907290\pi$
$888$	0	0
$889$	−1104.00	−1.24184
$890$	0	0
$891$	− 62.3538i	− 0.0699819i
$892$	0	0
$893$	−864.000	−0.967525
$894$	0	0
$895$	637.395i	0.712173i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 193.990i	− 0.215784i
$900$	0	0
$901$	792.000	0.879023
$902$	0	0
$903$	748.246i	0.828622i
$904$	0	0
$905$	576.000	0.636464
$906$	0	0
$907$	769.031i	0.847884i	0.905689	+	0.423942i	$0.139354\pi$
$907$	769.031i	0.847884i	−0.905689	+	0.423942i	$0.860646\pi$
$908$	0	0
$909$	276.000	0.303630
$910$	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
$915$	498.831i	0.545170i
$916$	0	0
$917$	−1392.00	−1.51799
$918$	0	0
$919$	− 1711.27i	− 1.86210i	−0.364897	−	0.931048i	$-0.618896\pi$
$919$	− 1711.27i	− 1.86210i	0.364897	−	0.931048i	$-0.381104\pi$
$920$	0	0
$921$	900.000	0.977199
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−648.000	−0.700541
$926$	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
$931$	− 20.7846i	− 0.0223250i
$932$	0	0
$933$	144.000	0.154341
$934$	0	0
$935$	− 498.831i	− 0.533509i
$936$	0	0
$937$	674.000	0.719317	0.359658	−	0.933084i	$-0.382893\pi$
$937$	674.000	0.719317	0.359658	+	0.933084i	$0.382893\pi$
$938$	0	0
$939$	− 3.46410i	− 0.00368914i
$940$	0	0
$941$	−1444.00	−1.53454	−0.767269	−	0.641326i	$-0.778385\pi$
$941$	−1444.00	−1.53454	−0.767269	+	0.641326i	$0.778385\pi$
$942$	0	0
$943$	− 748.246i	− 0.793474i
$944$	0	0
$945$	−144.000	−0.152381
$946$	0	0
$947$	− 741.318i	− 0.782806i	−0.920219	−	0.391403i	$-0.871990\pi$
$947$	− 741.318i	− 0.782806i	0.920219	−	0.391403i	$-0.128010\pi$
$948$	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	$-0.503006\pi$
$953$	−18.0000	−0.0188877	−0.00944386	+	0.999955i	$0.503006\pi$
$954$	0	0
$955$	− 332.554i	− 0.348224i
$956$	0	0
$957$	48.0000	0.0501567
$958$	0	0
$959$	− 124.708i	− 0.130039i
$960$	0	0
$961$	−1391.00	−1.44745
$962$	0	0
$963$	270.200i	0.280581i
$964$	0	0
$965$	−376.000	−0.389637
$966$	0	0
$967$	− 1697.41i	− 1.75534i	−0.479269	−	0.877668i	$-0.659098\pi$
$967$	− 1697.41i	− 1.75534i	0.479269	−	0.877668i	$-0.340902\pi$
$968$	0	0
$969$	−648.000	−0.668731
$970$	0	0
$971$	907.595i	0.934701i	0.884072	+	0.467350i	$0.154791\pi$
$971$	907.595i	0.934701i	−0.884072	+	0.467350i	$0.845209\pi$
$972$	0	0
$973$	432.000	0.443988
$974$	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
$979$	872.954i	0.891679i
$980$	0	0
$981$	432.000	0.440367
$982$	0	0
$983$	1496.49i	1.52237i	0.648534	+	0.761186i	$0.275383\pi$
$983$	1496.49i	1.52237i	−0.648534	+	0.761186i	$0.724617\pi$
$984$	0	0
$985$	752.000	0.763452
$986$	0	0
$987$	498.831i	0.505401i
$988$	0	0
$989$	2592.00	2.62083
$990$	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
$995$	− 637.395i	− 0.640598i
$996$	0	0
$997$	−648.000	−0.649950	−0.324975	−	0.945723i	$-0.605356\pi$
$997$	−648.000	−0.649950	−0.324975	+	0.945723i	$0.605356\pi$
$998$	0	0
$999$	− 374.123i	− 0.374497i

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	768.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} -4.00000 q^{5} +6.92820i q^{7} -3.00000 q^{9} -6.92820i q^{11} -6.92820i q^{15} -18.0000 q^{17} -20.7846i q^{19} -12.0000 q^{21} +41.5692i q^{23} -9.00000 q^{25} -5.19615i q^{27} +4.00000 q^{29} -48.4974i q^{31} +12.0000 q^{33} -27.7128i q^{35} +72.0000 q^{37} -18.0000 q^{41} -62.3538i q^{43} +12.0000 q^{45} -41.5692i q^{47} +1.00000 q^{49} -31.1769i q^{51} -44.0000 q^{53} +27.7128i q^{55} +36.0000 q^{57} -62.3538i q^{59} -72.0000 q^{61} -20.7846i q^{63} -20.7846i q^{67} -72.0000 q^{69} -41.5692i q^{71} +82.0000 q^{73} -15.5885i q^{75} +48.0000 q^{77} +62.3538i q^{79} +9.00000 q^{81} -131.636i q^{83} +72.0000 q^{85} +6.92820i q^{87} -126.000 q^{89} +84.0000 q^{93} +83.1384i q^{95} +110.000 q^{97} +20.7846i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 8 q^{5} - 6 q^{9} - 36 q^{17} - 24 q^{21} - 18 q^{25} + 8 q^{29} + 24 q^{33} + 144 q^{37} - 36 q^{41} + 24 q^{45} + 2 q^{49} - 88 q^{53} + 72 q^{57} - 144 q^{61} - 144 q^{69} + 164 q^{73} + 96 q^{77}+ \cdots + 220 q^{97}+O(q^{100}) \) 2 * q - 8 * q^5 - 6 * q^9 - 36 * q^17 - 24 * q^21 - 18 * q^25 + 8 * q^29 + 24 * q^33 + 144 * q^37 - 36 * q^41 + 24 * q^45 + 2 * q^49 - 88 * q^53 + 72 * q^57 - 144 * q^61 - 144 * q^69 + 164 * q^73 + 96 * q^77 + 18 * q^81 + 144 * q^85 - 252 * q^89 + 168 * q^93 + 220 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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