LMFDB - Embedded newform 1664.2.b.d.833.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1664,2,Mod(833,1664)] 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("1664.833"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1664 = 2^{7} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1664.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,6,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$13.2871068963$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			833.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1664.833
Dual form			1664.2.b.d.833.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{3} -1.00000i q^{5} +3.00000 q^{7} -6.00000 q^{9} -6.00000i q^{11} -1.00000i q^{13} +3.00000 q^{15} +1.00000 q^{17} +9.00000i q^{21} +6.00000 q^{23} +4.00000 q^{25} -9.00000i q^{27} +10.0000i q^{29} +18.0000 q^{33} -3.00000i q^{35} -9.00000i q^{37} +3.00000 q^{39} +4.00000 q^{41} -9.00000i q^{43} +6.00000i q^{45} +9.00000 q^{47} +2.00000 q^{49} +3.00000i q^{51} +2.00000i q^{53} -6.00000 q^{55} +12.0000i q^{61} -18.0000 q^{63} -1.00000 q^{65} +6.00000i q^{67} +18.0000i q^{69} +9.00000 q^{71} +12.0000i q^{75} -18.0000i q^{77} -6.00000 q^{79} +9.00000 q^{81} -6.00000i q^{83} -1.00000i q^{85} -30.0000 q^{87} +8.00000 q^{89} -3.00000i q^{91} -18.0000 q^{97} +36.0000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{7} - 12 q^{9} + 6 q^{15} + 2 q^{17} + 12 q^{23} + 8 q^{25} + 36 q^{33} + 6 q^{39} + 8 q^{41} + 18 q^{47} + 4 q^{49} - 12 q^{55} - 36 q^{63} - 2 q^{65} + 18 q^{71} - 12 q^{79} + 18 q^{81} - 60 q^{87}+ \cdots - 36 q^{97}+O(q^{100}) $ 2 * q + 6 * q^7 - 12 * q^9 + 6 * q^15 + 2 * q^17 + 12 * q^23 + 8 * q^25 + 36 * q^33 + 6 * q^39 + 8 * q^41 + 18 * q^47 + 4 * q^49 - 12 * q^55 - 36 * q^63 - 2 * q^65 + 18 * q^71 - 12 * q^79 + 18 * q^81 - 60 * q^87 + 16 * q^89 - 36 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1664\mathbb{Z}\right)^\times$.

$n$	$261$	$769$	$1535$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	3.00000i	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$3$	3.00000i	1.73205i	−0.500000	+	0.866025i	$0.666667\pi$
$4$	0	0
$5$	− 1.00000i	− 0.447214i	−0.974679	−	0.223607i	$-0.928217\pi$
$5$	− 1.00000i	− 0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$	0	0
$7$	3.00000	1.13389	0.566947	−	0.823754i	$-0.308125\pi$
$7$	3.00000	1.13389	0.566947	+	0.823754i	$0.308125\pi$
$8$	0	0
$9$	−6.00000	−2.00000
$10$	0	0
$11$	− 6.00000i	− 1.80907i	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	− 6.00000i	− 1.80907i	0.426401	−	0.904534i	$-0.359781\pi$
$12$	0	0
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	0	0
$15$	3.00000	0.774597
$16$	0	0
$17$	1.00000	0.242536	0.121268	−	0.992620i	$-0.461304\pi$
$17$	1.00000	0.242536	0.121268	+	0.992620i	$0.461304\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	0	0
$21$	9.00000i	1.96396i
$22$	0	0
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	− 9.00000i	− 1.73205i
$28$	0	0
$29$	10.0000i	1.85695i	0.371391	+	0.928477i	$0.378881\pi$
$29$	10.0000i	1.85695i	−0.371391	+	0.928477i	$0.621119\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	18.0000	3.13340
$34$	0	0
$35$	− 3.00000i	− 0.507093i
$36$	0	0
$37$	− 9.00000i	− 1.47959i	−0.672832	−	0.739795i	$-0.734922\pi$
$37$	− 9.00000i	− 1.47959i	0.672832	−	0.739795i	$-0.265078\pi$
$38$	0	0
$39$	3.00000	0.480384
$40$	0	0
$41$	4.00000	0.624695	0.312348	−	0.949968i	$-0.398885\pi$
$41$	4.00000	0.624695	0.312348	+	0.949968i	$0.398885\pi$
$42$	0	0
$43$	− 9.00000i	− 1.37249i	−0.727372	−	0.686244i	$-0.759258\pi$
$43$	− 9.00000i	− 1.37249i	0.727372	−	0.686244i	$-0.240742\pi$
$44$	0	0
$45$	6.00000i	0.894427i
$46$	0	0
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	2.00000	0.285714
$50$	0	0
$51$	3.00000i	0.420084i
$52$	0	0
$53$	2.00000i	0.274721i	0.990521	+	0.137361i	$0.0438619\pi$
$53$	2.00000i	0.274721i	−0.990521	+	0.137361i	$0.956138\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0	0	1.00000			$0$
$59$	0	0	−1.00000			$\pi$
$60$	0	0
$61$	12.0000i	1.53644i	0.640184	+	0.768221i	$0.278858\pi$
$61$	12.0000i	1.53644i	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	−18.0000	−2.26779
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	6.00000i	0.733017i	0.930415	+	0.366508i	$0.119447\pi$
$67$	6.00000i	0.733017i	−0.930415	+	0.366508i	$0.880553\pi$
$68$	0	0
$69$	18.0000i	2.16695i
$70$	0	0
$71$	9.00000	1.06810	0.534052	−	0.845452i	$-0.320669\pi$
$71$	9.00000	1.06810	0.534052	+	0.845452i	$0.320669\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	0	0
$75$	12.0000i	1.38564i
$76$	0	0
$77$	− 18.0000i	− 2.05129i
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	− 6.00000i	− 0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$	− 6.00000i	− 0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$	0	0
$85$	− 1.00000i	− 0.108465i
$86$	0	0
$87$	−30.0000	−3.21634
$88$	0	0
$89$	8.00000	0.847998	0.423999	−	0.905663i	$-0.360626\pi$
$89$	8.00000	0.847998	0.423999	+	0.905663i	$0.360626\pi$
$90$	0	0
$91$	− 3.00000i	− 0.314485i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.0000	−1.82762	−0.913812	−	0.406138i	$-0.866875\pi$
$97$	−18.0000	−1.82762	−0.913812	+	0.406138i	$0.866875\pi$
$98$	0	0
$99$	36.0000i	3.61814i
$100$	0	0
$101$	− 8.00000i	− 0.796030i	−0.917379	−	0.398015i	$-0.869699\pi$
$101$	− 8.00000i	− 0.796030i	0.917379	−	0.398015i	$-0.130301\pi$
$102$	0	0
$103$	−6.00000	−0.591198	−0.295599	−	0.955312i	$-0.595519\pi$
$103$	−6.00000	−0.591198	−0.295599	+	0.955312i	$0.595519\pi$
$104$	0	0
$105$	9.00000	0.878310
$106$	0	0
$107$	12.0000i	1.16008i	0.814587	+	0.580042i	$0.196964\pi$
$107$	12.0000i	1.16008i	−0.814587	+	0.580042i	$0.803036\pi$
$108$	0	0
$109$	− 7.00000i	− 0.670478i	−0.942133	−	0.335239i	$-0.891183\pi$
$109$	− 7.00000i	− 0.670478i	0.942133	−	0.335239i	$-0.108817\pi$
$110$	0	0
$111$	27.0000	2.56273
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	− 6.00000i	− 0.559503i
$116$	0	0
$117$	6.00000i	0.554700i
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−25.0000	−2.27273
$122$	0	0
$123$	12.0000i	1.08200i
$124$	0	0
$125$	− 9.00000i	− 0.804984i
$126$	0	0
$127$	−6.00000	−0.532414	−0.266207	−	0.963916i	$-0.585770\pi$
$127$	−6.00000	−0.532414	−0.266207	+	0.963916i	$0.585770\pi$
$128$	0	0
$129$	27.0000	2.37722
$130$	0	0
$131$	− 15.0000i	− 1.31056i	−0.755388	−	0.655278i	$-0.772551\pi$
$131$	− 15.0000i	− 1.31056i	0.755388	−	0.655278i	$-0.227449\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−9.00000	−0.774597
$136$	0	0
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	− 9.00000i	− 0.763370i	−0.924292	−	0.381685i	$-0.875344\pi$
$139$	− 9.00000i	− 0.763370i	0.924292	−	0.381685i	$-0.124656\pi$
$140$	0	0
$141$	27.0000i	2.27381i
$142$	0	0
$143$	−6.00000	−0.501745
$144$	0	0
$145$	10.0000	0.830455
$146$	0	0
$147$	6.00000i	0.494872i
$148$	0	0
$149$	22.0000i	1.80231i	0.433497	+	0.901155i	$0.357280\pi$
$149$	22.0000i	1.80231i	−0.433497	+	0.901155i	$0.642720\pi$
$150$	0	0
$151$	9.00000	0.732410	0.366205	−	0.930534i	$-0.380657\pi$
$151$	9.00000	0.732410	0.366205	+	0.930534i	$0.380657\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.0000i	0.957704i	0.877896	+	0.478852i	$0.158947\pi$
$157$	12.0000i	0.957704i	−0.877896	+	0.478852i	$0.841053\pi$
$158$	0	0
$159$	−6.00000	−0.475831
$160$	0	0
$161$	18.0000	1.41860
$162$	0	0
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	0	0
$165$	− 18.0000i	− 1.40130i
$166$	0	0
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.0000i	1.67263i	0.548250	+	0.836315i	$0.315294\pi$
$173$	22.0000i	1.67263i	−0.548250	+	0.836315i	$0.684706\pi$
$174$	0	0
$175$	12.0000	0.907115
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.00000i	0.672692i	0.941739	+	0.336346i	$0.109191\pi$
$179$	9.00000i	0.672692i	−0.941739	+	0.336346i	$0.890809\pi$
$180$	0	0
$181$	− 2.00000i	− 0.148659i	−0.997234	−	0.0743294i	$-0.976318\pi$
$181$	− 2.00000i	− 0.148659i	0.997234	−	0.0743294i	$-0.0236816\pi$
$182$	0	0
$183$	−36.0000	−2.66120
$184$	0	0
$185$	−9.00000	−0.661693
$186$	0	0
$187$	− 6.00000i	− 0.438763i
$188$	0	0
$189$	− 27.0000i	− 1.96396i
$190$	0	0
$191$	−6.00000	−0.434145	−0.217072	−	0.976156i	$-0.569651\pi$
$191$	−6.00000	−0.434145	−0.217072	+	0.976156i	$0.569651\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	0	0
$195$	− 3.00000i	− 0.214834i
$196$	0	0
$197$	1.00000i	0.0712470i	0.999365	+	0.0356235i	$0.0113417\pi$
$197$	1.00000i	0.0712470i	−0.999365	+	0.0356235i	$0.988658\pi$
$198$	0	0
$199$	6.00000	0.425329	0.212664	−	0.977125i	$-0.431786\pi$
$199$	6.00000	0.425329	0.212664	+	0.977125i	$0.431786\pi$
$200$	0	0
$201$	−18.0000	−1.26962
$202$	0	0
$203$	30.0000i	2.10559i
$204$	0	0
$205$	− 4.00000i	− 0.279372i
$206$	0	0
$207$	−36.0000	−2.50217
$208$	0	0
$209$	0	0
$210$	0	0
$211$	− 27.0000i	− 1.85876i	−0.369129	−	0.929378i	$-0.620344\pi$
$211$	− 27.0000i	− 1.85876i	0.369129	−	0.929378i	$-0.379656\pi$
$212$	0	0
$213$	27.0000i	1.85001i
$214$	0	0
$215$	−9.00000	−0.613795
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.00000i	− 0.0672673i
$222$	0	0
$223$	15.0000	1.00447	0.502237	−	0.864730i	$-0.332510\pi$
$223$	15.0000	1.00447	0.502237	+	0.864730i	$0.332510\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	0	0
$227$	− 24.0000i	− 1.59294i	−0.604681	−	0.796468i	$-0.706699\pi$
$227$	− 24.0000i	− 1.59294i	0.604681	−	0.796468i	$-0.293301\pi$
$228$	0	0
$229$	5.00000i	0.330409i	0.986259	+	0.165205i	$0.0528285\pi$
$229$	5.00000i	0.330409i	−0.986259	+	0.165205i	$0.947172\pi$
$230$	0	0
$231$	54.0000	3.55294
$232$	0	0
$233$	−17.0000	−1.11371	−0.556854	−	0.830611i	$-0.687992\pi$
$233$	−17.0000	−1.11371	−0.556854	+	0.830611i	$0.687992\pi$
$234$	0	0
$235$	− 9.00000i	− 0.587095i
$236$	0	0
$237$	− 18.0000i	− 1.16923i
$238$	0	0
$239$	−15.0000	−0.970269	−0.485135	−	0.874439i	$-0.661229\pi$
$239$	−15.0000	−0.970269	−0.485135	+	0.874439i	$0.661229\pi$
$240$	0	0
$241$	−12.0000	−0.772988	−0.386494	−	0.922292i	$-0.626314\pi$
$241$	−12.0000	−0.772988	−0.386494	+	0.922292i	$0.626314\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.00000i	− 0.127775i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	18.0000	1.14070
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	− 36.0000i	− 2.26330i
$254$	0	0
$255$	3.00000	0.187867
$256$	0	0
$257$	17.0000	1.06043	0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000	1.06043	0.530215	+	0.847863i	$0.322111\pi$
$258$	0	0
$259$	− 27.0000i	− 1.67770i
$260$	0	0
$261$	− 60.0000i	− 3.71391i
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	2.00000	0.122859
$266$	0	0
$267$	24.0000i	1.46878i
$268$	0	0
$269$	20.0000i	1.21942i	0.792624	+	0.609711i	$0.208714\pi$
$269$	20.0000i	1.21942i	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	9.00000	0.544705
$274$	0	0
$275$	− 24.0000i	− 1.44725i
$276$	0	0
$277$	− 8.00000i	− 0.480673i	−0.970690	−	0.240337i	$-0.922742\pi$
$277$	− 8.00000i	− 0.480673i	0.970690	−	0.240337i	$-0.0772579\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−16.0000	−0.954480	−0.477240	−	0.878773i	$-0.658363\pi$
$281$	−16.0000	−0.954480	−0.477240	+	0.878773i	$0.658363\pi$
$282$	0	0
$283$	24.0000i	1.42665i	0.700832	+	0.713326i	$0.252812\pi$
$283$	24.0000i	1.42665i	−0.700832	+	0.713326i	$0.747188\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	0	0
$289$	−16.0000	−0.941176
$290$	0	0
$291$	− 54.0000i	− 3.16554i
$292$	0	0
$293$	− 1.00000i	− 0.0584206i	−0.999573	−	0.0292103i	$-0.990701\pi$
$293$	− 1.00000i	− 0.0584206i	0.999573	−	0.0292103i	$-0.00929925\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−54.0000	−3.13340
$298$	0	0
$299$	− 6.00000i	− 0.346989i
$300$	0	0
$301$	− 27.0000i	− 1.55625i
$302$	0	0
$303$	24.0000	1.37876
$304$	0	0
$305$	12.0000	0.687118
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	− 18.0000i	− 1.02398i
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−19.0000	−1.07394	−0.536972	−	0.843600i	$-0.680432\pi$
$313$	−19.0000	−1.07394	−0.536972	+	0.843600i	$0.680432\pi$
$314$	0	0
$315$	18.0000i	1.01419i
$316$	0	0
$317$	− 22.0000i	− 1.23564i	−0.786318	−	0.617822i	$-0.788015\pi$
$317$	− 22.0000i	− 1.23564i	0.786318	−	0.617822i	$-0.211985\pi$
$318$	0	0
$319$	60.0000	3.35936
$320$	0	0
$321$	−36.0000	−2.00932
$322$	0	0
$323$	0	0
$324$	0	0
$325$	− 4.00000i	− 0.221880i
$326$	0	0
$327$	21.0000	1.16130
$328$	0	0
$329$	27.0000	1.48856
$330$	0	0
$331$	− 6.00000i	− 0.329790i	−0.986311	−	0.164895i	$-0.947272\pi$
$331$	− 6.00000i	− 0.329790i	0.986311	−	0.164895i	$-0.0527285\pi$
$332$	0	0
$333$	54.0000i	2.95918i
$334$	0	0
$335$	6.00000	0.327815
$336$	0	0
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	0	0
$339$	30.0000i	1.62938i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−15.0000	−0.809924
$344$	0	0
$345$	18.0000	0.969087
$346$	0	0
$347$	3.00000i	0.161048i	0.996753	+	0.0805242i	$0.0256594\pi$
$347$	3.00000i	0.161048i	−0.996753	+	0.0805242i	$0.974341\pi$
$348$	0	0
$349$	9.00000i	0.481759i	0.970555	+	0.240879i	$0.0774359\pi$
$349$	9.00000i	0.481759i	−0.970555	+	0.240879i	$0.922564\pi$
$350$	0	0
$351$	−9.00000	−0.480384
$352$	0	0
$353$	22.0000	1.17094	0.585471	−	0.810693i	$-0.300910\pi$
$353$	22.0000	1.17094	0.585471	+	0.810693i	$0.300910\pi$
$354$	0	0
$355$	− 9.00000i	− 0.477670i
$356$	0	0
$357$	9.00000i	0.476331i
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	19.0000	1.00000
$362$	0	0
$363$	− 75.0000i	− 3.93648i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	30.0000	1.56599	0.782994	−	0.622030i	$-0.213692\pi$
$367$	30.0000	1.56599	0.782994	+	0.622030i	$0.213692\pi$
$368$	0	0
$369$	−24.0000	−1.24939
$370$	0	0
$371$	6.00000i	0.311504i
$372$	0	0
$373$	− 24.0000i	− 1.24267i	−0.783544	−	0.621336i	$-0.786590\pi$
$373$	− 24.0000i	− 1.24267i	0.783544	−	0.621336i	$-0.213410\pi$
$374$	0	0
$375$	27.0000	1.39427
$376$	0	0
$377$	10.0000	0.515026
$378$	0	0
$379$	30.0000i	1.54100i	0.637442	+	0.770498i	$0.279993\pi$
$379$	30.0000i	1.54100i	−0.637442	+	0.770498i	$0.720007\pi$
$380$	0	0
$381$	− 18.0000i	− 0.922168i
$382$	0	0
$383$	−21.0000	−1.07305	−0.536525	−	0.843884i	$-0.680263\pi$
$383$	−21.0000	−1.07305	−0.536525	+	0.843884i	$0.680263\pi$
$384$	0	0
$385$	−18.0000	−0.917365
$386$	0	0
$387$	54.0000i	2.74497i
$388$	0	0
$389$	4.00000i	0.202808i	0.994845	+	0.101404i	$0.0323335\pi$
$389$	4.00000i	0.202808i	−0.994845	+	0.101404i	$0.967667\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	0	0
$393$	45.0000	2.26995
$394$	0	0
$395$	6.00000i	0.301893i
$396$	0	0
$397$	6.00000i	0.301131i	0.988600	+	0.150566i	$0.0481095\pi$
$397$	6.00000i	0.301131i	−0.988600	+	0.150566i	$0.951890\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.0000	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$401$	−14.0000	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	− 9.00000i	− 0.447214i
$406$	0	0
$407$	−54.0000	−2.67668
$408$	0	0
$409$	18.0000	0.890043	0.445021	−	0.895520i	$-0.353196\pi$
$409$	18.0000	0.890043	0.445021	+	0.895520i	$0.353196\pi$
$410$	0	0
$411$	− 30.0000i	− 1.47979i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	27.0000	1.32220
$418$	0	0
$419$	3.00000i	0.146560i	0.997311	+	0.0732798i	$0.0233466\pi$
$419$	3.00000i	0.146560i	−0.997311	+	0.0732798i	$0.976653\pi$
$420$	0	0
$421$	− 17.0000i	− 0.828529i	−0.910156	−	0.414265i	$-0.864039\pi$
$421$	− 17.0000i	− 0.828529i	0.910156	−	0.414265i	$-0.135961\pi$
$422$	0	0
$423$	−54.0000	−2.62557
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	36.0000i	1.74216i
$428$	0	0
$429$	− 18.0000i	− 0.869048i
$430$	0	0
$431$	−3.00000	−0.144505	−0.0722525	−	0.997386i	$-0.523019\pi$
$431$	−3.00000	−0.144505	−0.0722525	+	0.997386i	$0.523019\pi$
$432$	0	0
$433$	−25.0000	−1.20142	−0.600712	−	0.799466i	$-0.705116\pi$
$433$	−25.0000	−1.20142	−0.600712	+	0.799466i	$0.705116\pi$
$434$	0	0
$435$	30.0000i	1.43839i
$436$	0	0
$437$	0	0
$438$	0	0
$439$	36.0000	1.71819	0.859093	−	0.511819i	$-0.171028\pi$
$439$	36.0000	1.71819	0.859093	+	0.511819i	$0.171028\pi$
$440$	0	0
$441$	−12.0000	−0.571429
$442$	0	0
$443$	− 15.0000i	− 0.712672i	−0.934358	−	0.356336i	$-0.884026\pi$
$443$	− 15.0000i	− 0.712672i	0.934358	−	0.356336i	$-0.115974\pi$
$444$	0	0
$445$	− 8.00000i	− 0.379236i
$446$	0	0
$447$	−66.0000	−3.12169
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	− 24.0000i	− 1.13012i
$452$	0	0
$453$	27.0000i	1.26857i
$454$	0	0
$455$	−3.00000	−0.140642
$456$	0	0
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	0	0
$459$	− 9.00000i	− 0.420084i
$460$	0	0
$461$	− 19.0000i	− 0.884918i	−0.896789	−	0.442459i	$-0.854106\pi$
$461$	− 19.0000i	− 0.884918i	0.896789	−	0.442459i	$-0.145894\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	12.0000i	0.555294i	0.960683	+	0.277647i	$0.0895545\pi$
$467$	12.0000i	0.555294i	−0.960683	+	0.277647i	$0.910445\pi$
$468$	0	0
$469$	18.0000i	0.831163i
$470$	0	0
$471$	−36.0000	−1.65879
$472$	0	0
$473$	−54.0000	−2.48292
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 12.0000i	− 0.549442i
$478$	0	0
$479$	−15.0000	−0.685367	−0.342684	−	0.939451i	$-0.611336\pi$
$479$	−15.0000	−0.685367	−0.342684	+	0.939451i	$0.611336\pi$
$480$	0	0
$481$	−9.00000	−0.410365
$482$	0	0
$483$	54.0000i	2.45709i
$484$	0	0
$485$	18.0000i	0.817338i
$486$	0	0
$487$	−24.0000	−1.08754	−0.543772	−	0.839233i	$-0.683004\pi$
$487$	−24.0000	−1.08754	−0.543772	+	0.839233i	$0.683004\pi$
$488$	0	0
$489$	−18.0000	−0.813988
$490$	0	0
$491$	15.0000i	0.676941i	0.940977	+	0.338470i	$0.109909\pi$
$491$	15.0000i	0.676941i	−0.940977	+	0.338470i	$0.890091\pi$
$492$	0	0
$493$	10.0000i	0.450377i
$494$	0	0
$495$	36.0000	1.61808
$496$	0	0
$497$	27.0000	1.21112
$498$	0	0
$499$	24.0000i	1.07439i	0.843459	+	0.537194i	$0.180516\pi$
$499$	24.0000i	1.07439i	−0.843459	+	0.537194i	$0.819484\pi$
$500$	0	0
$501$	72.0000i	3.21672i
$502$	0	0
$503$	24.0000	1.07011	0.535054	−	0.844818i	$-0.320291\pi$
$503$	24.0000	1.07011	0.535054	+	0.844818i	$0.320291\pi$
$504$	0	0
$505$	−8.00000	−0.355995
$506$	0	0
$507$	− 3.00000i	− 0.133235i
$508$	0	0
$509$	− 14.0000i	− 0.620539i	−0.950649	−	0.310270i	$-0.899581\pi$
$509$	− 14.0000i	− 0.620539i	0.950649	−	0.310270i	$-0.100419\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.00000i	0.264392i
$516$	0	0
$517$	− 54.0000i	− 2.37492i
$518$	0	0
$519$	−66.0000	−2.89708
$520$	0	0
$521$	29.0000	1.27051	0.635257	−	0.772301i	$-0.280894\pi$
$521$	29.0000	1.27051	0.635257	+	0.772301i	$0.280894\pi$
$522$	0	0
$523$	− 12.0000i	− 0.524723i	−0.964970	−	0.262362i	$-0.915499\pi$
$523$	− 12.0000i	− 0.524723i	0.964970	−	0.262362i	$-0.0845013\pi$
$524$	0	0
$525$	36.0000i	1.57117i
$526$	0	0
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	12.0000	0.518805
$536$	0	0
$537$	−27.0000	−1.16514
$538$	0	0
$539$	− 12.0000i	− 0.516877i
$540$	0	0
$541$	− 25.0000i	− 1.07483i	−0.843317	−	0.537417i	$-0.819400\pi$
$541$	− 25.0000i	− 1.07483i	0.843317	−	0.537417i	$-0.180600\pi$
$542$	0	0
$543$	6.00000	0.257485
$544$	0	0
$545$	−7.00000	−0.299847
$546$	0	0
$547$	3.00000i	0.128271i	0.997941	+	0.0641354i	$0.0204289\pi$
$547$	3.00000i	0.128271i	−0.997941	+	0.0641354i	$0.979571\pi$
$548$	0	0
$549$	− 72.0000i	− 3.07289i
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−18.0000	−0.765438
$554$	0	0
$555$	− 27.0000i	− 1.14609i
$556$	0	0
$557$	23.0000i	0.974541i	0.873251	+	0.487271i	$0.162007\pi$
$557$	23.0000i	0.974541i	−0.873251	+	0.487271i	$0.837993\pi$
$558$	0	0
$559$	−9.00000	−0.380659
$560$	0	0
$561$	18.0000	0.759961
$562$	0	0
$563$	− 3.00000i	− 0.126435i	−0.998000	−	0.0632175i	$-0.979864\pi$
$563$	− 3.00000i	− 0.126435i	0.998000	−	0.0632175i	$-0.0201362\pi$
$564$	0	0
$565$	− 10.0000i	− 0.420703i
$566$	0	0
$567$	27.0000	1.13389
$568$	0	0
$569$	17.0000	0.712677	0.356339	−	0.934357i	$-0.384025\pi$
$569$	17.0000	0.712677	0.356339	+	0.934357i	$0.384025\pi$
$570$	0	0
$571$	21.0000i	0.878823i	0.898286	+	0.439411i	$0.144813\pi$
$571$	21.0000i	0.878823i	−0.898286	+	0.439411i	$0.855187\pi$
$572$	0	0
$573$	− 18.0000i	− 0.751961i
$574$	0	0
$575$	24.0000	1.00087
$576$	0	0
$577$	−16.0000	−0.666089	−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000	−0.666089	−0.333044	+	0.942911i	$0.608076\pi$
$578$	0	0
$579$	42.0000i	1.74546i
$580$	0	0
$581$	− 18.0000i	− 0.746766i
$582$	0	0
$583$	12.0000	0.496989
$584$	0	0
$585$	6.00000	0.248069
$586$	0	0
$587$	− 30.0000i	− 1.23823i	−0.785299	−	0.619116i	$-0.787491\pi$
$587$	− 30.0000i	− 1.23823i	0.785299	−	0.619116i	$-0.212509\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−3.00000	−0.123404
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	− 3.00000i	− 0.122988i
$596$	0	0
$597$	18.0000i	0.736691i
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	− 36.0000i	− 1.46603i
$604$	0	0
$605$	25.0000i	1.01639i
$606$	0	0
$607$	−24.0000	−0.974130	−0.487065	−	0.873366i	$-0.661933\pi$
$607$	−24.0000	−0.974130	−0.487065	+	0.873366i	$0.661933\pi$
$608$	0	0
$609$	−90.0000	−3.64698
$610$	0	0
$611$	− 9.00000i	− 0.364101i
$612$	0	0
$613$	− 6.00000i	− 0.242338i	−0.992632	−	0.121169i	$-0.961336\pi$
$613$	− 6.00000i	− 0.242338i	0.992632	−	0.121169i	$-0.0386643\pi$
$614$	0	0
$615$	12.0000	0.483887
$616$	0	0
$617$	−28.0000	−1.12724	−0.563619	−	0.826035i	$-0.690591\pi$
$617$	−28.0000	−1.12724	−0.563619	+	0.826035i	$0.690591\pi$
$618$	0	0
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	− 54.0000i	− 2.16695i
$622$	0	0
$623$	24.0000	0.961540
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 9.00000i	− 0.358854i
$630$	0	0
$631$	21.0000	0.835997	0.417998	−	0.908448i	$-0.362732\pi$
$631$	21.0000	0.835997	0.417998	+	0.908448i	$0.362732\pi$
$632$	0	0
$633$	81.0000	3.21946
$634$	0	0
$635$	6.00000i	0.238103i
$636$	0	0
$637$	− 2.00000i	− 0.0792429i
$638$	0	0
$639$	−54.0000	−2.13621
$640$	0	0
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	0	0
$643$	6.00000i	0.236617i	0.992977	+	0.118308i	$0.0377472\pi$
$643$	6.00000i	0.236617i	−0.992977	+	0.118308i	$0.962253\pi$
$644$	0	0
$645$	− 27.0000i	− 1.06312i
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	28.0000i	1.09572i	0.836569	+	0.547862i	$0.184558\pi$
$653$	28.0000i	1.09572i	−0.836569	+	0.547862i	$0.815442\pi$
$654$	0	0
$655$	−15.0000	−0.586098
$656$	0	0
$657$	0	0
$658$	0	0
$659$	36.0000i	1.40236i	0.712984	+	0.701180i	$0.247343\pi$
$659$	36.0000i	1.40236i	−0.712984	+	0.701180i	$0.752657\pi$
$660$	0	0
$661$	42.0000i	1.63361i	0.576913	+	0.816805i	$0.304257\pi$
$661$	42.0000i	1.63361i	−0.576913	+	0.816805i	$0.695743\pi$
$662$	0	0
$663$	3.00000	0.116510
$664$	0	0
$665$	0	0
$666$	0	0
$667$	60.0000i	2.32321i
$668$	0	0
$669$	45.0000i	1.73980i
$670$	0	0
$671$	72.0000	2.77953
$672$	0	0
$673$	1.00000	0.0385472	0.0192736	−	0.999814i	$-0.493865\pi$
$673$	1.00000	0.0385472	0.0192736	+	0.999814i	$0.493865\pi$
$674$	0	0
$675$	− 36.0000i	− 1.38564i
$676$	0	0
$677$	38.0000i	1.46046i	0.683202	+	0.730229i	$0.260587\pi$
$677$	38.0000i	1.46046i	−0.683202	+	0.730229i	$0.739413\pi$
$678$	0	0
$679$	−54.0000	−2.07233
$680$	0	0
$681$	72.0000	2.75905
$682$	0	0
$683$	30.0000i	1.14792i	0.818884	+	0.573959i	$0.194593\pi$
$683$	30.0000i	1.14792i	−0.818884	+	0.573959i	$0.805407\pi$
$684$	0	0
$685$	10.0000i	0.382080i
$686$	0	0
$687$	−15.0000	−0.572286
$688$	0	0
$689$	2.00000	0.0761939
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	108.000i	4.10258i
$694$	0	0
$695$	−9.00000	−0.341389
$696$	0	0
$697$	4.00000	0.151511
$698$	0	0
$699$	− 51.0000i	− 1.92900i
$700$	0	0
$701$	34.0000i	1.28416i	0.766637	+	0.642081i	$0.221929\pi$
$701$	34.0000i	1.28416i	−0.766637	+	0.642081i	$0.778071\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	27.0000	1.01688
$706$	0	0
$707$	− 24.0000i	− 0.902613i
$708$	0	0
$709$	− 10.0000i	− 0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$	− 10.0000i	− 0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	0	0
$714$	0	0
$715$	6.00000i	0.224387i
$716$	0	0
$717$	− 45.0000i	− 1.68056i
$718$	0	0
$719$	−30.0000	−1.11881	−0.559406	−	0.828894i	$-0.688971\pi$
$719$	−30.0000	−1.11881	−0.559406	+	0.828894i	$0.688971\pi$
$720$	0	0
$721$	−18.0000	−0.670355
$722$	0	0
$723$	− 36.0000i	− 1.33885i
$724$	0	0
$725$	40.0000i	1.48556i
$726$	0	0
$727$	30.0000	1.11264	0.556319	−	0.830969i	$-0.312213\pi$
$727$	30.0000	1.11264	0.556319	+	0.830969i	$0.312213\pi$
$728$	0	0
$729$	27.0000	1.00000
$730$	0	0
$731$	− 9.00000i	− 0.332877i
$732$	0	0
$733$	31.0000i	1.14501i	0.819901	+	0.572506i	$0.194029\pi$
$733$	31.0000i	1.14501i	−0.819901	+	0.572506i	$0.805971\pi$
$734$	0	0
$735$	6.00000	0.221313
$736$	0	0
$737$	36.0000	1.32608
$738$	0	0
$739$	12.0000i	0.441427i	0.975339	+	0.220714i	$0.0708386\pi$
$739$	12.0000i	0.441427i	−0.975339	+	0.220714i	$0.929161\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.0000	0.550297	0.275148	−	0.961402i	$-0.411273\pi$
$743$	15.0000	0.550297	0.275148	+	0.961402i	$0.411273\pi$
$744$	0	0
$745$	22.0000	0.806018
$746$	0	0
$747$	36.0000i	1.31717i
$748$	0	0
$749$	36.0000i	1.31541i
$750$	0	0
$751$	−36.0000	−1.31366	−0.656829	−	0.754039i	$-0.728103\pi$
$751$	−36.0000	−1.31366	−0.656829	+	0.754039i	$0.728103\pi$
$752$	0	0
$753$	36.0000	1.31191
$754$	0	0
$755$	− 9.00000i	− 0.327544i
$756$	0	0
$757$	− 34.0000i	− 1.23575i	−0.786276	−	0.617876i	$-0.787994\pi$
$757$	− 34.0000i	− 1.23575i	0.786276	−	0.617876i	$-0.212006\pi$
$758$	0	0
$759$	108.000	3.92015
$760$	0	0
$761$	−20.0000	−0.724999	−0.362500	−	0.931984i	$-0.618077\pi$
$761$	−20.0000	−0.724999	−0.362500	+	0.931984i	$0.618077\pi$
$762$	0	0
$763$	− 21.0000i	− 0.760251i
$764$	0	0
$765$	6.00000i	0.216930i
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−32.0000	−1.15395	−0.576975	−	0.816762i	$-0.695767\pi$
$769$	−32.0000	−1.15395	−0.576975	+	0.816762i	$0.695767\pi$
$770$	0	0
$771$	51.0000i	1.83672i
$772$	0	0
$773$	− 1.00000i	− 0.0359675i	−0.999838	−	0.0179838i	$-0.994275\pi$
$773$	− 1.00000i	− 0.0359675i	0.999838	−	0.0179838i	$-0.00572471\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	81.0000	2.90586
$778$	0	0
$779$	0	0
$780$	0	0
$781$	− 54.0000i	− 1.93227i
$782$	0	0
$783$	90.0000	3.21634
$784$	0	0
$785$	12.0000	0.428298
$786$	0	0
$787$	− 24.0000i	− 0.855508i	−0.903895	−	0.427754i	$-0.859305\pi$
$787$	− 24.0000i	− 0.855508i	0.903895	−	0.427754i	$-0.140695\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	30.0000	1.06668
$792$	0	0
$793$	12.0000	0.426132
$794$	0	0
$795$	6.00000i	0.212798i
$796$	0	0
$797$	− 20.0000i	− 0.708436i	−0.935163	−	0.354218i	$-0.884747\pi$
$797$	− 20.0000i	− 0.708436i	0.935163	−	0.354218i	$-0.115253\pi$
$798$	0	0
$799$	9.00000	0.318397
$800$	0	0
$801$	−48.0000	−1.69600
$802$	0	0
$803$	0	0
$804$	0	0
$805$	− 18.0000i	− 0.634417i
$806$	0	0
$807$	−60.0000	−2.11210
$808$	0	0
$809$	−11.0000	−0.386739	−0.193370	−	0.981126i	$-0.561942\pi$
$809$	−11.0000	−0.386739	−0.193370	+	0.981126i	$0.561942\pi$
$810$	0	0
$811$	12.0000i	0.421377i	0.977553	+	0.210688i	$0.0675706\pi$
$811$	12.0000i	0.421377i	−0.977553	+	0.210688i	$0.932429\pi$
$812$	0	0
$813$	− 45.0000i	− 1.57822i
$814$	0	0
$815$	6.00000	0.210171
$816$	0	0
$817$	0	0
$818$	0	0
$819$	18.0000i	0.628971i
$820$	0	0
$821$	19.0000i	0.663105i	0.943437	+	0.331552i	$0.107572\pi$
$821$	19.0000i	0.663105i	−0.943437	+	0.331552i	$0.892428\pi$
$822$	0	0
$823$	0	0		−	1.00000i	$-0.5\pi$
$823$	0	0			1.00000i	$0.5\pi$
$824$	0	0
$825$	72.0000	2.50672
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	2.00000i	0.0694629i	0.999397	+	0.0347314i	$0.0110576\pi$
$829$	2.00000i	0.0694629i	−0.999397	+	0.0347314i	$0.988942\pi$
$830$	0	0
$831$	24.0000	0.832551
$832$	0	0
$833$	2.00000	0.0692959
$834$	0	0
$835$	− 24.0000i	− 0.830554i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−48.0000	−1.65714	−0.828572	−	0.559883i	$-0.810846\pi$
$839$	−48.0000	−1.65714	−0.828572	+	0.559883i	$0.810846\pi$
$840$	0	0
$841$	−71.0000	−2.44828
$842$	0	0
$843$	− 48.0000i	− 1.65321i
$844$	0	0
$845$	1.00000i	0.0344010i
$846$	0	0
$847$	−75.0000	−2.57703
$848$	0	0
$849$	−72.0000	−2.47103
$850$	0	0
$851$	− 54.0000i	− 1.85110i
$852$	0	0
$853$	− 15.0000i	− 0.513590i	−0.966466	−	0.256795i	$-0.917333\pi$
$853$	− 15.0000i	− 0.513590i	0.966466	−	0.256795i	$-0.0826666\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.0000	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$857$	34.0000	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$858$	0	0
$859$	− 36.0000i	− 1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$	− 36.0000i	− 1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$	0	0
$861$	36.0000i	1.22688i
$862$	0	0
$863$	21.0000	0.714848	0.357424	−	0.933942i	$-0.383655\pi$
$863$	21.0000	0.714848	0.357424	+	0.933942i	$0.383655\pi$
$864$	0	0
$865$	22.0000	0.748022
$866$	0	0
$867$	− 48.0000i	− 1.63017i
$868$	0	0
$869$	36.0000i	1.22122i
$870$	0	0
$871$	6.00000	0.203302
$872$	0	0
$873$	108.000	3.65525
$874$	0	0
$875$	− 27.0000i	− 0.912767i
$876$	0	0
$877$	39.0000i	1.31694i	0.752609	+	0.658468i	$0.228795\pi$
$877$	39.0000i	1.31694i	−0.752609	+	0.658468i	$0.771205\pi$
$878$	0	0
$879$	3.00000	0.101187
$880$	0	0
$881$	−19.0000	−0.640126	−0.320063	−	0.947396i	$-0.603704\pi$
$881$	−19.0000	−0.640126	−0.320063	+	0.947396i	$0.603704\pi$
$882$	0	0
$883$	27.0000i	0.908622i	0.890843	+	0.454311i	$0.150115\pi$
$883$	27.0000i	0.908622i	−0.890843	+	0.454311i	$0.849885\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	− 54.0000i	− 1.80907i
$892$	0	0
$893$	0	0
$894$	0	0
$895$	9.00000	0.300837
$896$	0	0
$897$	18.0000	0.601003
$898$	0	0
$899$	0	0
$900$	0	0
$901$	2.00000i	0.0666297i
$902$	0	0
$903$	81.0000	2.69551
$904$	0	0
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	27.0000i	0.896520i	0.893903	+	0.448260i	$0.147956\pi$
$907$	27.0000i	0.896520i	−0.893903	+	0.448260i	$0.852044\pi$
$908$	0	0
$909$	48.0000i	1.59206i
$910$	0	0
$911$	−6.00000	−0.198789	−0.0993944	−	0.995048i	$-0.531691\pi$
$911$	−6.00000	−0.198789	−0.0993944	+	0.995048i	$0.531691\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	36.0000i	1.19012i
$916$	0	0
$917$	− 45.0000i	− 1.48603i
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 9.00000i	− 0.296239i
$924$	0	0
$925$	− 36.0000i	− 1.18367i
$926$	0	0
$927$	36.0000	1.18240
$928$	0	0
$929$	16.0000	0.524943	0.262471	−	0.964940i	$-0.415462\pi$
$929$	16.0000	0.524943	0.262471	+	0.964940i	$0.415462\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.00000	−0.196221
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	− 57.0000i	− 1.86012i
$940$	0	0
$941$	11.0000i	0.358590i	0.983795	+	0.179295i	$0.0573816\pi$
$941$	11.0000i	0.358590i	−0.983795	+	0.179295i	$0.942618\pi$
$942$	0	0
$943$	24.0000	0.781548
$944$	0	0
$945$	−27.0000	−0.878310
$946$	0	0
$947$	− 18.0000i	− 0.584921i	−0.956278	−	0.292461i	$-0.905526\pi$
$947$	− 18.0000i	− 0.584921i	0.956278	−	0.292461i	$-0.0944741\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	66.0000	2.14020
$952$	0	0
$953$	−55.0000	−1.78162	−0.890812	−	0.454371i	$-0.849864\pi$
$953$	−55.0000	−1.78162	−0.890812	+	0.454371i	$0.849864\pi$
$954$	0	0
$955$	6.00000i	0.194155i
$956$	0	0
$957$	180.000i	5.81857i
$958$	0	0
$959$	−30.0000	−0.968751
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	− 72.0000i	− 2.32017i
$964$	0	0
$965$	− 14.0000i	− 0.450676i
$966$	0	0
$967$	−15.0000	−0.482367	−0.241184	−	0.970479i	$-0.577536\pi$
$967$	−15.0000	−0.482367	−0.241184	+	0.970479i	$0.577536\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	9.00000i	0.288824i	0.989518	+	0.144412i	$0.0461290\pi$
$971$	9.00000i	0.288824i	−0.989518	+	0.144412i	$0.953871\pi$
$972$	0	0
$973$	− 27.0000i	− 0.865580i
$974$	0	0
$975$	12.0000	0.384308
$976$	0	0
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	0	0
$979$	− 48.0000i	− 1.53409i
$980$	0	0
$981$	42.0000i	1.34096i
$982$	0	0
$983$	−21.0000	−0.669796	−0.334898	−	0.942254i	$-0.608702\pi$
$983$	−21.0000	−0.669796	−0.334898	+	0.942254i	$0.608702\pi$
$984$	0	0
$985$	1.00000	0.0318626
$986$	0	0
$987$	81.0000i	2.57826i
$988$	0	0
$989$	− 54.0000i	− 1.71710i
$990$	0	0
$991$	48.0000	1.52477	0.762385	−	0.647124i	$-0.224028\pi$
$991$	48.0000	1.52477	0.762385	+	0.647124i	$0.224028\pi$
$992$	0	0
$993$	18.0000	0.571213
$994$	0	0
$995$	− 6.00000i	− 0.190213i
$996$	0	0
$997$	18.0000i	0.570066i	0.958518	+	0.285033i	$0.0920045\pi$
$997$	18.0000i	0.570066i	−0.958518	+	0.285033i	$0.907995\pi$
$998$	0	0
$999$	−81.0000	−2.56273

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1664.2.b.d.833.2	yes	2
4.3	odd	2		1664.2.b.a.833.1	✓	2
8.3	odd	2		1664.2.b.a.833.2	yes	2
8.5	even	2	inner	1664.2.b.d.833.1	yes	2
16.3	odd	4		3328.2.a.k.1.1		1
16.5	even	4		3328.2.a.l.1.1		1
16.11	odd	4		3328.2.a.b.1.1		1
16.13	even	4		3328.2.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1664.2.b.a.833.1	✓	2	4.3	odd	2
1664.2.b.a.833.2	yes	2	8.3	odd	2
1664.2.b.d.833.1	yes	2	8.5	even	2	inner
1664.2.b.d.833.2	yes	2	1.1	even	1	trivial
3328.2.a.a.1.1		1	16.13	even	4
3328.2.a.b.1.1		1	16.11	odd	4
3328.2.a.k.1.1		1	16.3	odd	4
3328.2.a.l.1.1		1	16.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1664 = 2^{7} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1664.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} -1.00000i q^{5} +3.00000 q^{7} -6.00000 q^{9} -6.00000i q^{11} -1.00000i q^{13} +3.00000 q^{15} +1.00000 q^{17} +9.00000i q^{21} +6.00000 q^{23} +4.00000 q^{25} -9.00000i q^{27} +10.0000i q^{29} +18.0000 q^{33} -3.00000i q^{35} -9.00000i q^{37} +3.00000 q^{39} +4.00000 q^{41} -9.00000i q^{43} +6.00000i q^{45} +9.00000 q^{47} +2.00000 q^{49} +3.00000i q^{51} +2.00000i q^{53} -6.00000 q^{55} +12.0000i q^{61} -18.0000 q^{63} -1.00000 q^{65} +6.00000i q^{67} +18.0000i q^{69} +9.00000 q^{71} +12.0000i q^{75} -18.0000i q^{77} -6.00000 q^{79} +9.00000 q^{81} -6.00000i q^{83} -1.00000i q^{85} -30.0000 q^{87} +8.00000 q^{89} -3.00000i q^{91} -18.0000 q^{97} +36.0000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{7} - 12 q^{9} + 6 q^{15} + 2 q^{17} + 12 q^{23} + 8 q^{25} + 36 q^{33} + 6 q^{39} + 8 q^{41} + 18 q^{47} + 4 q^{49} - 12 q^{55} - 36 q^{63} - 2 q^{65} + 18 q^{71} - 12 q^{79} + 18 q^{81} - 60 q^{87}+ \cdots - 36 q^{97}+O(q^{100}) \) 2 * q + 6 * q^7 - 12 * q^9 + 6 * q^15 + 2 * q^17 + 12 * q^23 + 8 * q^25 + 36 * q^33 + 6 * q^39 + 8 * q^41 + 18 * q^47 + 4 * q^49 - 12 * q^55 - 36 * q^63 - 2 * q^65 + 18 * q^71 - 12 * q^79 + 18 * q^81 - 60 * q^87 + 16 * q^89 - 36 * q^97

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