LMFDB - Embedded newform 8112.2.a.bs.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8112,2,Mod(1,8112)] 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("8112.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8112, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8112 = 2^{4} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8112.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,0,0,2,0,0,0,0,0,0,0,-12] 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:	yes
Analytic conductor:	$64.7746461197$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 156)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	8112.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{9} +3.46410 q^{11} +3.46410 q^{15} -6.00000 q^{17} +6.92820 q^{19} +7.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +6.92820 q^{31} +3.46410 q^{33} -3.46410 q^{41} +8.00000 q^{43} +3.46410 q^{45} +3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} +12.0000 q^{55} +6.92820 q^{57} -3.46410 q^{59} +10.0000 q^{61} -13.8564 q^{67} -10.3923 q^{71} +6.92820 q^{73} +7.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} +3.46410 q^{83} -20.7846 q^{85} -6.00000 q^{87} +17.3205 q^{89} +6.92820 q^{93} +24.0000 q^{95} +6.92820 q^{97} +3.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{9} - 12 q^{17} + 14 q^{25} + 2 q^{27} - 12 q^{29} + 16 q^{43} - 14 q^{49} - 12 q^{51} + 12 q^{53} + 24 q^{55} + 20 q^{61} + 14 q^{75} + 16 q^{79} + 2 q^{81} - 12 q^{87} + 48 q^{95}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^9 - 12 * q^17 + 14 * q^25 + 2 * q^27 - 12 * q^29 + 16 * q^43 - 14 * q^49 - 12 * q^51 + 12 * q^53 + 24 * q^55 + 20 * q^61 + 14 * q^75 + 16 * q^79 + 2 * q^81 - 12 * q^87 + 48 * q^95

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	3.46410	0.894427
$16$	0	0
$17$	−6.00000	−1.45521	−0.727607	−	0.685994i	$-0.759367\pi$
$17$	−6.00000	−1.45521	−0.727607	+	0.685994i	$0.759367\pi$
$18$	0	0
$19$	6.92820	1.58944	0.794719	−	0.606977i	$-0.207618\pi$
$19$	6.92820	1.58944	0.794719	+	0.606977i	$0.207618\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	6.92820	1.24434	0.622171	−	0.782881i	$-0.286251\pi$
$31$	6.92820	1.24434	0.622171	+	0.782881i	$0.286251\pi$
$32$	0	0
$33$	3.46410	0.603023
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.46410	−0.541002	−0.270501	−	0.962720i	$-0.587189\pi$
$41$	−3.46410	−0.541002	−0.270501	+	0.962720i	$0.587189\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	3.46410	0.516398
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	−6.00000	−0.840168
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	12.0000	1.61808
$56$	0	0
$57$	6.92820	0.917663
$58$	0	0
$59$	−3.46410	−0.450988	−0.225494	−	0.974245i	$-0.572400\pi$
$59$	−3.46410	−0.450988	−0.225494	+	0.974245i	$0.572400\pi$
$60$	0	0
$61$	10.0000	1.28037	0.640184	−	0.768221i	$-0.278858\pi$
$61$	10.0000	1.28037	0.640184	+	0.768221i	$0.278858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−13.8564	−1.69283	−0.846415	−	0.532524i	$-0.821244\pi$
$67$	−13.8564	−1.69283	−0.846415	+	0.532524i	$0.821244\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	0	0
$73$	6.92820	0.810885	0.405442	−	0.914121i	$-0.367117\pi$
$73$	6.92820	0.810885	0.405442	+	0.914121i	$0.367117\pi$
$74$	0	0
$75$	7.00000	0.808290
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.46410	0.380235	0.190117	−	0.981761i	$-0.439113\pi$
$83$	3.46410	0.380235	0.190117	+	0.981761i	$0.439113\pi$
$84$	0	0
$85$	−20.7846	−2.25441
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	17.3205	1.83597	0.917985	−	0.396615i	$-0.129815\pi$
$89$	17.3205	1.83597	0.917985	+	0.396615i	$0.129815\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	6.92820	0.718421
$94$	0	0
$95$	24.0000	2.46235
$96$	0	0
$97$	6.92820	0.703452	0.351726	−	0.936103i	$-0.385595\pi$
$97$	6.92820	0.703452	0.351726	+	0.936103i	$0.385595\pi$
$98$	0	0
$99$	3.46410	0.348155
$100$	0	0
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−6.92820	−0.663602	−0.331801	−	0.943349i	$-0.607656\pi$
$109$	−6.92820	−0.663602	−0.331801	+	0.943349i	$0.607656\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.46410	−0.312348
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	16.0000	1.41977	0.709885	−	0.704317i	$-0.248747\pi$
$127$	16.0000	1.41977	0.709885	+	0.704317i	$0.248747\pi$
$128$	0	0
$129$	8.00000	0.704361
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	3.46410	0.298142
$136$	0	0
$137$	10.3923	0.887875	0.443937	−	0.896058i	$-0.353581\pi$
$137$	10.3923	0.887875	0.443937	+	0.896058i	$0.353581\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	3.46410	0.291730
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−20.7846	−1.72607
$146$	0	0
$147$	−7.00000	−0.577350
$148$	0	0
$149$	−17.3205	−1.41895	−0.709476	−	0.704730i	$-0.751068\pi$
$149$	−17.3205	−1.41895	−0.709476	+	0.704730i	$0.751068\pi$
$150$	0	0
$151$	6.92820	0.563809	0.281905	−	0.959442i	$-0.409034\pi$
$151$	6.92820	0.563809	0.281905	+	0.959442i	$0.409034\pi$
$152$	0	0
$153$	−6.00000	−0.485071
$154$	0	0
$155$	24.0000	1.92773
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.7846	−1.62798	−0.813988	−	0.580881i	$-0.802708\pi$
$163$	−20.7846	−1.62798	−0.813988	+	0.580881i	$0.802708\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	17.3205	1.34030	0.670151	−	0.742225i	$-0.266230\pi$
$167$	17.3205	1.34030	0.670151	+	0.742225i	$0.266230\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	6.92820	0.529813
$172$	0	0
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−3.46410	−0.260378
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	10.0000	0.739221
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−20.7846	−1.51992
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	−13.8564	−0.997406	−0.498703	−	0.866773i	$-0.666190\pi$
$193$	−13.8564	−0.997406	−0.498703	+	0.866773i	$0.666190\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	17.3205	1.23404	0.617018	−	0.786949i	$-0.288341\pi$
$197$	17.3205	1.23404	0.617018	+	0.786949i	$0.288341\pi$
$198$	0	0
$199$	−8.00000	−0.567105	−0.283552	−	0.958957i	$-0.591513\pi$
$199$	−8.00000	−0.567105	−0.283552	+	0.958957i	$0.591513\pi$
$200$	0	0
$201$	−13.8564	−0.977356
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−12.0000	−0.838116
$206$	0	0
$207$	0	0
$208$	0	0
$209$	24.0000	1.66011
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	−10.3923	−0.712069
$214$	0	0
$215$	27.7128	1.89000
$216$	0	0
$217$	0	0
$218$	0	0
$219$	6.92820	0.468165
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.92820	0.463947	0.231973	−	0.972722i	$-0.425482\pi$
$223$	6.92820	0.463947	0.231973	+	0.972722i	$0.425482\pi$
$224$	0	0
$225$	7.00000	0.466667
$226$	0	0
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	0	0
$229$	−6.92820	−0.457829	−0.228914	−	0.973447i	$-0.573518\pi$
$229$	−6.92820	−0.457829	−0.228914	+	0.973447i	$0.573518\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	12.0000	0.782794
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−10.3923	−0.672222	−0.336111	−	0.941822i	$-0.609112\pi$
$239$	−10.3923	−0.672222	−0.336111	+	0.941822i	$0.609112\pi$
$240$	0	0
$241$	20.7846	1.33885	0.669427	−	0.742878i	$-0.266540\pi$
$241$	20.7846	1.33885	0.669427	+	0.742878i	$0.266540\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−24.2487	−1.54919
$246$	0	0
$247$	0	0
$248$	0	0
$249$	3.46410	0.219529
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−20.7846	−1.30158
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	20.7846	1.27679
$266$	0	0
$267$	17.3205	1.06000
$268$	0	0
$269$	−30.0000	−1.82913	−0.914566	−	0.404436i	$-0.867468\pi$
$269$	−30.0000	−1.82913	−0.914566	+	0.404436i	$0.867468\pi$
$270$	0	0
$271$	−27.7128	−1.68343	−0.841717	−	0.539919i	$-0.818455\pi$
$271$	−27.7128	−1.68343	−0.841717	+	0.539919i	$0.818455\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.2487	1.46225
$276$	0	0
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	0	0
$279$	6.92820	0.414781
$280$	0	0
$281$	−17.3205	−1.03325	−0.516627	−	0.856210i	$-0.672813\pi$
$281$	−17.3205	−1.03325	−0.516627	+	0.856210i	$0.672813\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	24.0000	1.42164
$286$	0	0
$287$	0	0
$288$	0	0
$289$	19.0000	1.11765
$290$	0	0
$291$	6.92820	0.406138
$292$	0	0
$293$	−3.46410	−0.202375	−0.101187	−	0.994867i	$-0.532264\pi$
$293$	−3.46410	−0.202375	−0.101187	+	0.994867i	$0.532264\pi$
$294$	0	0
$295$	−12.0000	−0.698667
$296$	0	0
$297$	3.46410	0.201008
$298$	0	0
$299$	0	0
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−6.00000	−0.344691
$304$	0	0
$305$	34.6410	1.98354
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−24.2487	−1.36194	−0.680972	−	0.732310i	$-0.738442\pi$
$317$	−24.2487	−1.36194	−0.680972	+	0.732310i	$0.738442\pi$
$318$	0	0
$319$	−20.7846	−1.16371
$320$	0	0
$321$	12.0000	0.669775
$322$	0	0
$323$	−41.5692	−2.31297
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−6.92820	−0.383131
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−13.8564	−0.761617	−0.380808	−	0.924654i	$-0.624354\pi$
$331$	−13.8564	−0.761617	−0.380808	+	0.924654i	$0.624354\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−48.0000	−2.62252
$336$	0	0
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	0	0
$339$	−18.0000	−0.977626
$340$	0	0
$341$	24.0000	1.29967
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	0	0		−	1.00000i	$-0.5\pi$
$349$	0	0			1.00000i	$0.5\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	10.3923	0.553127	0.276563	−	0.960996i	$-0.410804\pi$
$353$	10.3923	0.553127	0.276563	+	0.960996i	$0.410804\pi$
$354$	0	0
$355$	−36.0000	−1.91068
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.2487	−1.27980	−0.639899	−	0.768459i	$-0.721024\pi$
$359$	−24.2487	−1.27980	−0.639899	+	0.768459i	$0.721024\pi$
$360$	0	0
$361$	29.0000	1.52632
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	24.0000	1.25622
$366$	0	0
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−3.46410	−0.180334
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−2.00000	−0.103556	−0.0517780	−	0.998659i	$-0.516489\pi$
$373$	−2.00000	−0.103556	−0.0517780	+	0.998659i	$0.516489\pi$
$374$	0	0
$375$	6.92820	0.357771
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−20.7846	−1.06763	−0.533817	−	0.845600i	$-0.679243\pi$
$379$	−20.7846	−1.06763	−0.533817	+	0.845600i	$0.679243\pi$
$380$	0	0
$381$	16.0000	0.819705
$382$	0	0
$383$	24.2487	1.23905	0.619526	−	0.784976i	$-0.287325\pi$
$383$	24.2487	1.23905	0.619526	+	0.784976i	$0.287325\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.00000	0.406663
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	12.0000	0.605320
$394$	0	0
$395$	27.7128	1.39438
$396$	0	0
$397$	−13.8564	−0.695433	−0.347717	−	0.937600i	$-0.613043\pi$
$397$	−13.8564	−0.695433	−0.347717	+	0.937600i	$0.613043\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−10.3923	−0.518967	−0.259483	−	0.965748i	$-0.583552\pi$
$401$	−10.3923	−0.518967	−0.259483	+	0.965748i	$0.583552\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.46410	0.172133
$406$	0	0
$407$	0	0
$408$	0	0
$409$	20.7846	1.02773	0.513866	−	0.857870i	$-0.328213\pi$
$409$	20.7846	1.02773	0.513866	+	0.857870i	$0.328213\pi$
$410$	0	0
$411$	10.3923	0.512615
$412$	0	0
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−20.7846	−1.01298	−0.506490	−	0.862246i	$-0.669057\pi$
$421$	−20.7846	−1.01298	−0.506490	+	0.862246i	$0.669057\pi$
$422$	0	0
$423$	3.46410	0.168430
$424$	0	0
$425$	−42.0000	−2.03730
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.1769	−1.50174	−0.750870	−	0.660451i	$-0.770365\pi$
$431$	−31.1769	−1.50174	−0.750870	+	0.660451i	$0.770365\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	−20.7846	−0.996546
$436$	0	0
$437$	0	0
$438$	0	0
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	0	0
$443$	12.0000	0.570137	0.285069	−	0.958507i	$-0.407984\pi$
$443$	12.0000	0.570137	0.285069	+	0.958507i	$0.407984\pi$
$444$	0	0
$445$	60.0000	2.84427
$446$	0	0
$447$	−17.3205	−0.819232
$448$	0	0
$449$	3.46410	0.163481	0.0817405	−	0.996654i	$-0.473952\pi$
$449$	3.46410	0.163481	0.0817405	+	0.996654i	$0.473952\pi$
$450$	0	0
$451$	−12.0000	−0.565058
$452$	0	0
$453$	6.92820	0.325515
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6.92820	0.324088	0.162044	−	0.986784i	$-0.448191\pi$
$457$	6.92820	0.324088	0.162044	+	0.986784i	$0.448191\pi$
$458$	0	0
$459$	−6.00000	−0.280056
$460$	0	0
$461$	−10.3923	−0.484018	−0.242009	−	0.970274i	$-0.577806\pi$
$461$	−10.3923	−0.484018	−0.242009	+	0.970274i	$0.577806\pi$
$462$	0	0
$463$	13.8564	0.643962	0.321981	−	0.946746i	$-0.395651\pi$
$463$	13.8564	0.643962	0.321981	+	0.946746i	$0.395651\pi$
$464$	0	0
$465$	24.0000	1.11297
$466$	0	0
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	27.7128	1.27424
$474$	0	0
$475$	48.4974	2.22521
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	−3.46410	−0.158279	−0.0791394	−	0.996864i	$-0.525217\pi$
$479$	−3.46410	−0.158279	−0.0791394	+	0.996864i	$0.525217\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	0	0
$485$	24.0000	1.08978
$486$	0	0
$487$	−6.92820	−0.313947	−0.156973	−	0.987603i	$-0.550174\pi$
$487$	−6.92820	−0.313947	−0.156973	+	0.987603i	$0.550174\pi$
$488$	0	0
$489$	−20.7846	−0.939913
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	0	0
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	17.3205	0.773823
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−20.7846	−0.924903
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.2487	−1.07481	−0.537403	−	0.843326i	$-0.680594\pi$
$509$	−24.2487	−1.07481	−0.537403	+	0.843326i	$0.680594\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	6.92820	0.305888
$514$	0	0
$515$	−13.8564	−0.610586
$516$	0	0
$517$	12.0000	0.527759
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	−8.00000	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$523$	−8.00000	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−41.5692	−1.81078
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−3.46410	−0.150329
$532$	0	0
$533$	0	0
$534$	0	0
$535$	41.5692	1.79719
$536$	0	0
$537$	−12.0000	−0.517838
$538$	0	0
$539$	−24.2487	−1.04447
$540$	0	0
$541$	20.7846	0.893600	0.446800	−	0.894634i	$-0.352564\pi$
$541$	20.7846	0.893600	0.446800	+	0.894634i	$0.352564\pi$
$542$	0	0
$543$	22.0000	0.944110
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	0	0
$549$	10.0000	0.426790
$550$	0	0
$551$	−41.5692	−1.77091
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.2487	1.02745	0.513725	−	0.857955i	$-0.328265\pi$
$557$	24.2487	1.02745	0.513725	+	0.857955i	$0.328265\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−20.7846	−0.877527
$562$	0	0
$563$	−36.0000	−1.51722	−0.758610	−	0.651546i	$-0.774121\pi$
$563$	−36.0000	−1.51722	−0.758610	+	0.651546i	$0.774121\pi$
$564$	0	0
$565$	−62.3538	−2.62325
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−13.8564	−0.576850	−0.288425	−	0.957503i	$-0.593132\pi$
$577$	−13.8564	−0.576850	−0.288425	+	0.957503i	$0.593132\pi$
$578$	0	0
$579$	−13.8564	−0.575853
$580$	0	0
$581$	0	0
$582$	0	0
$583$	20.7846	0.860811
$584$	0	0
$585$	0	0
$586$	0	0
$587$	31.1769	1.28681	0.643404	−	0.765526i	$-0.277521\pi$
$587$	31.1769	1.28681	0.643404	+	0.765526i	$0.277521\pi$
$588$	0	0
$589$	48.0000	1.97781
$590$	0	0
$591$	17.3205	0.712470
$592$	0	0
$593$	−24.2487	−0.995775	−0.497888	−	0.867242i	$-0.665891\pi$
$593$	−24.2487	−0.995775	−0.497888	+	0.867242i	$0.665891\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−8.00000	−0.327418
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	−13.8564	−0.564276
$604$	0	0
$605$	3.46410	0.140836
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	13.8564	0.559655	0.279827	−	0.960050i	$-0.409723\pi$
$613$	13.8564	0.559655	0.279827	+	0.960050i	$0.409723\pi$
$614$	0	0
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−31.1769	−1.25514	−0.627568	−	0.778562i	$-0.715949\pi$
$617$	−31.1769	−1.25514	−0.627568	+	0.778562i	$0.715949\pi$
$618$	0	0
$619$	−27.7128	−1.11387	−0.556936	−	0.830555i	$-0.688023\pi$
$619$	−27.7128	−1.11387	−0.556936	+	0.830555i	$0.688023\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	24.0000	0.958468
$628$	0	0
$629$	0	0
$630$	0	0
$631$	20.7846	0.827422	0.413711	−	0.910408i	$-0.364232\pi$
$631$	20.7846	0.827422	0.413711	+	0.910408i	$0.364232\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	55.4256	2.19950
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−10.3923	−0.411113
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	41.5692	1.63933	0.819665	−	0.572843i	$-0.194160\pi$
$643$	41.5692	1.63933	0.819665	+	0.572843i	$0.194160\pi$
$644$	0	0
$645$	27.7128	1.09119
$646$	0	0
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−12.0000	−0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	41.5692	1.62424
$656$	0	0
$657$	6.92820	0.270295
$658$	0	0
$659$	−12.0000	−0.467454	−0.233727	−	0.972302i	$-0.575092\pi$
$659$	−12.0000	−0.467454	−0.233727	+	0.972302i	$0.575092\pi$
$660$	0	0
$661$	13.8564	0.538952	0.269476	−	0.963007i	$-0.413150\pi$
$661$	13.8564	0.538952	0.269476	+	0.963007i	$0.413150\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	6.92820	0.267860
$670$	0	0
$671$	34.6410	1.33730
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	0	0
$675$	7.00000	0.269430
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	10.3923	0.398234
$682$	0	0
$683$	3.46410	0.132550	0.0662751	−	0.997801i	$-0.478889\pi$
$683$	3.46410	0.132550	0.0662751	+	0.997801i	$0.478889\pi$
$684$	0	0
$685$	36.0000	1.37549
$686$	0	0
$687$	−6.92820	−0.264327
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−27.7128	−1.05425	−0.527123	−	0.849789i	$-0.676729\pi$
$691$	−27.7128	−1.05425	−0.527123	+	0.849789i	$0.676729\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−27.7128	−1.05121
$696$	0	0
$697$	20.7846	0.787273
$698$	0	0
$699$	−18.0000	−0.680823
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	12.0000	0.451946
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.92820	0.260194	0.130097	−	0.991501i	$-0.458471\pi$
$709$	6.92820	0.260194	0.130097	+	0.991501i	$0.458471\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.3923	−0.388108
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	20.7846	0.772988
$724$	0	0
$725$	−42.0000	−1.55984
$726$	0	0
$727$	16.0000	0.593407	0.296704	−	0.954970i	$-0.404113\pi$
$727$	16.0000	0.593407	0.296704	+	0.954970i	$0.404113\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−48.0000	−1.77534
$732$	0	0
$733$	6.92820	0.255899	0.127950	−	0.991781i	$-0.459160\pi$
$733$	6.92820	0.255899	0.127950	+	0.991781i	$0.459160\pi$
$734$	0	0
$735$	−24.2487	−0.894427
$736$	0	0
$737$	−48.0000	−1.76810
$738$	0	0
$739$	27.7128	1.01943	0.509716	−	0.860343i	$-0.329750\pi$
$739$	27.7128	1.01943	0.509716	+	0.860343i	$0.329750\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−10.3923	−0.381257	−0.190628	−	0.981662i	$-0.561053\pi$
$743$	−10.3923	−0.381257	−0.190628	+	0.981662i	$0.561053\pi$
$744$	0	0
$745$	−60.0000	−2.19823
$746$	0	0
$747$	3.46410	0.126745
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−28.0000	−1.02173	−0.510867	−	0.859660i	$-0.670676\pi$
$751$	−28.0000	−1.02173	−0.510867	+	0.859660i	$0.670676\pi$
$752$	0	0
$753$	12.0000	0.437304
$754$	0	0
$755$	24.0000	0.873449
$756$	0	0
$757$	−38.0000	−1.38113	−0.690567	−	0.723269i	$-0.742639\pi$
$757$	−38.0000	−1.38113	−0.690567	+	0.723269i	$0.742639\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−31.1769	−1.13016	−0.565081	−	0.825035i	$-0.691155\pi$
$761$	−31.1769	−1.13016	−0.565081	+	0.825035i	$0.691155\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−20.7846	−0.751469
$766$	0	0
$767$	0	0
$768$	0	0
$769$	27.7128	0.999350	0.499675	−	0.866213i	$-0.333453\pi$
$769$	27.7128	0.999350	0.499675	+	0.866213i	$0.333453\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	0	0
$773$	51.9615	1.86893	0.934463	−	0.356060i	$-0.115880\pi$
$773$	51.9615	1.86893	0.934463	+	0.356060i	$0.115880\pi$
$774$	0	0
$775$	48.4974	1.74208
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	−6.00000	−0.214423
$784$	0	0
$785$	6.92820	0.247278
$786$	0	0
$787$	−27.7128	−0.987855	−0.493928	−	0.869503i	$-0.664439\pi$
$787$	−27.7128	−0.987855	−0.493928	+	0.869503i	$0.664439\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	20.7846	0.737154
$796$	0	0
$797$	30.0000	1.06265	0.531327	−	0.847167i	$-0.321693\pi$
$797$	30.0000	1.06265	0.531327	+	0.847167i	$0.321693\pi$
$798$	0	0
$799$	−20.7846	−0.735307
$800$	0	0
$801$	17.3205	0.611990
$802$	0	0
$803$	24.0000	0.846942
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−30.0000	−1.05605
$808$	0	0
$809$	−18.0000	−0.632846	−0.316423	−	0.948618i	$-0.602482\pi$
$809$	−18.0000	−0.632846	−0.316423	+	0.948618i	$0.602482\pi$
$810$	0	0
$811$	−6.92820	−0.243282	−0.121641	−	0.992574i	$-0.538816\pi$
$811$	−6.92820	−0.243282	−0.121641	+	0.992574i	$0.538816\pi$
$812$	0	0
$813$	−27.7128	−0.971931
$814$	0	0
$815$	−72.0000	−2.52205
$816$	0	0
$817$	55.4256	1.93910
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−3.46410	−0.120898	−0.0604490	−	0.998171i	$-0.519253\pi$
$821$	−3.46410	−0.120898	−0.0604490	+	0.998171i	$0.519253\pi$
$822$	0	0
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	0	0
$825$	24.2487	0.844232
$826$	0	0
$827$	17.3205	0.602293	0.301147	−	0.953578i	$-0.402631\pi$
$827$	17.3205	0.602293	0.301147	+	0.953578i	$0.402631\pi$
$828$	0	0
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	42.0000	1.45521
$834$	0	0
$835$	60.0000	2.07639
$836$	0	0
$837$	6.92820	0.239474
$838$	0	0
$839$	10.3923	0.358782	0.179391	−	0.983778i	$-0.442587\pi$
$839$	10.3923	0.358782	0.179391	+	0.983778i	$0.442587\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	−17.3205	−0.596550
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−20.0000	−0.686398
$850$	0	0
$851$	0	0
$852$	0	0
$853$	−13.8564	−0.474434	−0.237217	−	0.971457i	$-0.576235\pi$
$853$	−13.8564	−0.474434	−0.237217	+	0.971457i	$0.576235\pi$
$854$	0	0
$855$	24.0000	0.820783
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	38.1051	1.29711	0.648557	−	0.761166i	$-0.275373\pi$
$863$	38.1051	1.29711	0.648557	+	0.761166i	$0.275373\pi$
$864$	0	0
$865$	−20.7846	−0.706698
$866$	0	0
$867$	19.0000	0.645274
$868$	0	0
$869$	27.7128	0.940093
$870$	0	0
$871$	0	0
$872$	0	0
$873$	6.92820	0.234484
$874$	0	0
$875$	0	0
$876$	0	0
$877$	41.5692	1.40369	0.701846	−	0.712328i	$-0.252359\pi$
$877$	41.5692	1.40369	0.701846	+	0.712328i	$0.252359\pi$
$878$	0	0
$879$	−3.46410	−0.116841
$880$	0	0
$881$	6.00000	0.202145	0.101073	−	0.994879i	$-0.467773\pi$
$881$	6.00000	0.202145	0.101073	+	0.994879i	$0.467773\pi$
$882$	0	0
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	0	0
$885$	−12.0000	−0.403376
$886$	0	0
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	3.46410	0.116052
$892$	0	0
$893$	24.0000	0.803129
$894$	0	0
$895$	−41.5692	−1.38951
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−41.5692	−1.38641
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	76.2102	2.53331
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	0	0
$909$	−6.00000	−0.199007
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	0	0
$915$	34.6410	1.14520
$916$	0	0
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−31.1769	−1.02288	−0.511441	−	0.859319i	$-0.670888\pi$
$929$	−31.1769	−1.02288	−0.511441	+	0.859319i	$0.670888\pi$
$930$	0	0
$931$	−48.4974	−1.58944
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−72.0000	−2.35465
$936$	0	0
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	10.0000	0.326338
$940$	0	0
$941$	10.3923	0.338779	0.169390	−	0.985549i	$-0.445820\pi$
$941$	10.3923	0.338779	0.169390	+	0.985549i	$0.445820\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−45.0333	−1.46339	−0.731693	−	0.681634i	$-0.761270\pi$
$947$	−45.0333	−1.46339	−0.731693	+	0.681634i	$0.761270\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−24.2487	−0.786318
$952$	0	0
$953$	6.00000	0.194359	0.0971795	−	0.995267i	$-0.469018\pi$
$953$	6.00000	0.194359	0.0971795	+	0.995267i	$0.469018\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−20.7846	−0.671871
$958$	0	0
$959$	0	0
$960$	0	0
$961$	17.0000	0.548387
$962$	0	0
$963$	12.0000	0.386695
$964$	0	0
$965$	−48.0000	−1.54517
$966$	0	0
$967$	34.6410	1.11398	0.556990	−	0.830519i	$-0.311956\pi$
$967$	34.6410	1.11398	0.556990	+	0.830519i	$0.311956\pi$
$968$	0	0
$969$	−41.5692	−1.33540
$970$	0	0
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−24.2487	−0.775785	−0.387893	−	0.921705i	$-0.626797\pi$
$977$	−24.2487	−0.775785	−0.387893	+	0.921705i	$0.626797\pi$
$978$	0	0
$979$	60.0000	1.91761
$980$	0	0
$981$	−6.92820	−0.221201
$982$	0	0
$983$	−31.1769	−0.994389	−0.497195	−	0.867639i	$-0.665636\pi$
$983$	−31.1769	−0.994389	−0.497195	+	0.867639i	$0.665636\pi$
$984$	0	0
$985$	60.0000	1.91176
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	−20.0000	−0.635321	−0.317660	−	0.948205i	$-0.602897\pi$
$991$	−20.0000	−0.635321	−0.317660	+	0.948205i	$0.602897\pi$
$992$	0	0
$993$	−13.8564	−0.439720
$994$	0	0
$995$	−27.7128	−0.878555
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.bs.1.2		2
4.3	odd	2		2028.2.a.g.1.2		2
12.11	even	2		6084.2.a.v.1.1		2
13.5	odd	4		624.2.c.f.337.1		2
13.8	odd	4		624.2.c.f.337.2		2
13.12	even	2	inner	8112.2.a.bs.1.1		2
39.5	even	4		1872.2.c.c.1585.2		2
39.8	even	4		1872.2.c.c.1585.1		2
52.3	odd	6		2028.2.i.i.529.2		4
52.7	even	12		2028.2.q.c.361.1		2
52.11	even	12		2028.2.q.b.1837.1		2
52.15	even	12		2028.2.q.c.1837.1		2
52.19	even	12		2028.2.q.b.361.1		2
52.23	odd	6		2028.2.i.i.529.1		4
52.31	even	4		156.2.b.a.25.1	✓	2
52.35	odd	6		2028.2.i.i.2005.2		4
52.43	odd	6		2028.2.i.i.2005.1		4
52.47	even	4		156.2.b.a.25.2	yes	2
52.51	odd	2		2028.2.a.g.1.1		2
104.5	odd	4		2496.2.c.e.961.2		2
104.21	odd	4		2496.2.c.e.961.1		2
104.83	even	4		2496.2.c.l.961.2		2
104.99	even	4		2496.2.c.l.961.1		2
156.47	odd	4		468.2.b.a.181.1		2
156.83	odd	4		468.2.b.a.181.2		2
156.155	even	2		6084.2.a.v.1.2		2
260.47	odd	4		3900.2.j.h.649.4		4
260.83	odd	4		3900.2.j.h.649.1		4
260.99	even	4		3900.2.c.c.3301.2		2
260.187	odd	4		3900.2.j.h.649.3		4
260.203	odd	4		3900.2.j.h.649.2		4
260.239	even	4		3900.2.c.c.3301.1		2
364.83	odd	4		7644.2.e.g.4705.2		2
364.307	odd	4		7644.2.e.g.4705.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
156.2.b.a.25.1	✓	2	52.31	even	4
156.2.b.a.25.2	yes	2	52.47	even	4
468.2.b.a.181.1		2	156.47	odd	4
468.2.b.a.181.2		2	156.83	odd	4
624.2.c.f.337.1		2	13.5	odd	4
624.2.c.f.337.2		2	13.8	odd	4
1872.2.c.c.1585.1		2	39.8	even	4
1872.2.c.c.1585.2		2	39.5	even	4
2028.2.a.g.1.1		2	52.51	odd	2
2028.2.a.g.1.2		2	4.3	odd	2
2028.2.i.i.529.1		4	52.23	odd	6
2028.2.i.i.529.2		4	52.3	odd	6
2028.2.i.i.2005.1		4	52.43	odd	6
2028.2.i.i.2005.2		4	52.35	odd	6
2028.2.q.b.361.1		2	52.19	even	12
2028.2.q.b.1837.1		2	52.11	even	12
2028.2.q.c.361.1		2	52.7	even	12
2028.2.q.c.1837.1		2	52.15	even	12
2496.2.c.e.961.1		2	104.21	odd	4
2496.2.c.e.961.2		2	104.5	odd	4
2496.2.c.l.961.1		2	104.99	even	4
2496.2.c.l.961.2		2	104.83	even	4
3900.2.c.c.3301.1		2	260.239	even	4
3900.2.c.c.3301.2		2	260.99	even	4
3900.2.j.h.649.1		4	260.83	odd	4
3900.2.j.h.649.2		4	260.203	odd	4
3900.2.j.h.649.3		4	260.187	odd	4
3900.2.j.h.649.4		4	260.47	odd	4
6084.2.a.v.1.1		2	12.11	even	2
6084.2.a.v.1.2		2	156.155	even	2
7644.2.e.g.4705.1		2	364.307	odd	4
7644.2.e.g.4705.2		2	364.83	odd	4
8112.2.a.bs.1.1		2	13.12	even	2	inner
8112.2.a.bs.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8112.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{9} +3.46410 q^{11} +3.46410 q^{15} -6.00000 q^{17} +6.92820 q^{19} +7.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +6.92820 q^{31} +3.46410 q^{33} -3.46410 q^{41} +8.00000 q^{43} +3.46410 q^{45} +3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} +12.0000 q^{55} +6.92820 q^{57} -3.46410 q^{59} +10.0000 q^{61} -13.8564 q^{67} -10.3923 q^{71} +6.92820 q^{73} +7.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} +3.46410 q^{83} -20.7846 q^{85} -6.00000 q^{87} +17.3205 q^{89} +6.92820 q^{93} +24.0000 q^{95} +6.92820 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{9} - 12 q^{17} + 14 q^{25} + 2 q^{27} - 12 q^{29} + 16 q^{43} - 14 q^{49} - 12 q^{51} + 12 q^{53} + 24 q^{55} + 20 q^{61} + 14 q^{75} + 16 q^{79} + 2 q^{81} - 12 q^{87} + 48 q^{95}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^9 - 12 * q^17 + 14 * q^25 + 2 * q^27 - 12 * q^29 + 16 * q^43 - 14 * q^49 - 12 * q^51 + 12 * q^53 + 24 * q^55 + 20 * q^61 + 14 * q^75 + 16 * q^79 + 2 * q^81 - 12 * q^87 + 48 * q^95

Introduction

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