LMFDB - Embedded newform 8112.2.a.cg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8112,2,Mod(1,8112)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.7746461197$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.35690 q^{5} +2.24698 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.35690 q^{5} +2.24698 q^{7} +1.00000 q^{9} +4.93900 q^{11} -3.35690 q^{15} +0.911854 q^{17} -3.80194 q^{19} +2.24698 q^{21} -2.02715 q^{23} +6.26875 q^{25} +1.00000 q^{27} -3.93900 q^{29} -8.82908 q^{31} +4.93900 q^{33} -7.54288 q^{35} -8.80194 q^{37} -6.93900 q^{41} +2.28621 q^{43} -3.35690 q^{45} +3.80194 q^{47} -1.95108 q^{49} +0.911854 q^{51} +0.542877 q^{53} -16.5797 q^{55} -3.80194 q^{57} -4.71379 q^{59} +3.67994 q^{61} +2.24698 q^{63} -1.52111 q^{67} -2.02715 q^{69} +2.37867 q^{71} -7.41119 q^{73} +6.26875 q^{75} +11.0978 q^{77} +3.74094 q^{79} +1.00000 q^{81} +2.30798 q^{83} -3.06100 q^{85} -3.93900 q^{87} +10.0586 q^{89} -8.82908 q^{93} +12.7627 q^{95} +16.1293 q^{97} +4.93900 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 6 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 6 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 6 q^{5} + 2 q^{7} + 3 q^{9} + 5 q^{11} - 6 q^{15} - q^{17} - 7 q^{19} + 2 q^{21} + 11 q^{25} + 3 q^{27} - 2 q^{29} - 16 q^{31} + 5 q^{33} - 4 q^{35} - 22 q^{37} - 11 q^{41} + 15 q^{43} - 6 q^{45} + 7 q^{47} - 15 q^{49} - q^{51} - 17 q^{53} - 3 q^{55} - 7 q^{57} - 6 q^{59} - 13 q^{61} + 2 q^{63} + 11 q^{67} - 6 q^{73} + 11 q^{75} + 15 q^{77} - 3 q^{79} + 3 q^{81} + 12 q^{83} - 19 q^{85} - 2 q^{87} - q^{89} - 16 q^{93} + 21 q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 6 * q^5 + 2 * q^7 + 3 * q^9 + 5 * q^11 - 6 * q^15 - q^17 - 7 * q^19 + 2 * q^21 + 11 * q^25 + 3 * q^27 - 2 * q^29 - 16 * q^31 + 5 * q^33 - 4 * q^35 - 22 * q^37 - 11 * q^41 + 15 * q^43 - 6 * q^45 + 7 * q^47 - 15 * q^49 - q^51 - 17 * q^53 - 3 * q^55 - 7 * q^57 - 6 * q^59 - 13 * q^61 + 2 * q^63 + 11 * q^67 - 6 * q^73 + 11 * q^75 + 15 * q^77 - 3 * q^79 + 3 * q^81 + 12 * q^83 - 19 * q^85 - 2 * q^87 - q^89 - 16 * q^93 + 21 * q^95 + 5 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−3.35690	−1.50125	−0.750625	−	0.660729i	$-0.770247\pi$
$5$	−3.35690	−1.50125	−0.750625	+	0.660729i	$0.770247\pi$
$6$	0	0
$7$	2.24698	0.849278	0.424639	−	0.905363i	$-0.360401\pi$
$7$	2.24698	0.849278	0.424639	+	0.905363i	$0.360401\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.93900	1.48916	0.744582	−	0.667531i	$-0.232649\pi$
$11$	4.93900	1.48916	0.744582	+	0.667531i	$0.232649\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.35690	−0.866747
$16$	0	0
$17$	0.911854	0.221157	0.110579	−	0.993867i	$-0.464730\pi$
$17$	0.911854	0.221157	0.110579	+	0.993867i	$0.464730\pi$
$18$	0	0
$19$	−3.80194	−0.872224	−0.436112	−	0.899892i	$-0.643645\pi$
$19$	−3.80194	−0.872224	−0.436112	+	0.899892i	$0.643645\pi$
$20$	0	0
$21$	2.24698	0.490331
$22$	0	0
$23$	−2.02715	−0.422689	−0.211345	−	0.977412i	$-0.567784\pi$
$23$	−2.02715	−0.422689	−0.211345	+	0.977412i	$0.567784\pi$
$24$	0	0
$25$	6.26875	1.25375
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.93900	−0.731454	−0.365727	−	0.930722i	$-0.619180\pi$
$29$	−3.93900	−0.731454	−0.365727	+	0.930722i	$0.619180\pi$
$30$	0	0
$31$	−8.82908	−1.58575	−0.792875	−	0.609384i	$-0.791417\pi$
$31$	−8.82908	−1.58575	−0.792875	+	0.609384i	$0.791417\pi$
$32$	0	0
$33$	4.93900	0.859770
$34$	0	0
$35$	−7.54288	−1.27498
$36$	0	0
$37$	−8.80194	−1.44703	−0.723515	−	0.690309i	$-0.757475\pi$
$37$	−8.80194	−1.44703	−0.723515	+	0.690309i	$0.757475\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.93900	−1.08369	−0.541845	−	0.840478i	$-0.682274\pi$
$41$	−6.93900	−1.08369	−0.541845	+	0.840478i	$0.682274\pi$
$42$	0	0
$43$	2.28621	0.348643	0.174322	−	0.984689i	$-0.444227\pi$
$43$	2.28621	0.348643	0.174322	+	0.984689i	$0.444227\pi$
$44$	0	0
$45$	−3.35690	−0.500416
$46$	0	0
$47$	3.80194	0.554570	0.277285	−	0.960788i	$-0.410565\pi$
$47$	3.80194	0.554570	0.277285	+	0.960788i	$0.410565\pi$
$48$	0	0
$49$	−1.95108	−0.278726
$50$	0	0
$51$	0.911854	0.127685
$52$	0	0
$53$	0.542877	0.0745698	0.0372849	−	0.999305i	$-0.488129\pi$
$53$	0.542877	0.0745698	0.0372849	+	0.999305i	$0.488129\pi$
$54$	0	0
$55$	−16.5797	−2.23561
$56$	0	0
$57$	−3.80194	−0.503579
$58$	0	0
$59$	−4.71379	−0.613683	−0.306842	−	0.951761i	$-0.599272\pi$
$59$	−4.71379	−0.613683	−0.306842	+	0.951761i	$0.599272\pi$
$60$	0	0
$61$	3.67994	0.471168	0.235584	−	0.971854i	$-0.424300\pi$
$61$	3.67994	0.471168	0.235584	+	0.971854i	$0.424300\pi$
$62$	0	0
$63$	2.24698	0.283093
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−1.52111	−0.185833	−0.0929164	−	0.995674i	$-0.529619\pi$
$67$	−1.52111	−0.185833	−0.0929164	+	0.995674i	$0.529619\pi$
$68$	0	0
$69$	−2.02715	−0.244040
$70$	0	0
$71$	2.37867	0.282296	0.141148	−	0.989989i	$-0.454921\pi$
$71$	2.37867	0.282296	0.141148	+	0.989989i	$0.454921\pi$
$72$	0	0
$73$	−7.41119	−0.867414	−0.433707	−	0.901054i	$-0.642795\pi$
$73$	−7.41119	−0.867414	−0.433707	+	0.901054i	$0.642795\pi$
$74$	0	0
$75$	6.26875	0.723853
$76$	0	0
$77$	11.0978	1.26472
$78$	0	0
$79$	3.74094	0.420888	0.210444	−	0.977606i	$-0.432509\pi$
$79$	3.74094	0.420888	0.210444	+	0.977606i	$0.432509\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	2.30798	0.253334	0.126667	−	0.991945i	$-0.459572\pi$
$83$	2.30798	0.253334	0.126667	+	0.991945i	$0.459572\pi$
$84$	0	0
$85$	−3.06100	−0.332012
$86$	0	0
$87$	−3.93900	−0.422305
$88$	0	0
$89$	10.0586	1.06621	0.533105	−	0.846049i	$-0.321025\pi$
$89$	10.0586	1.06621	0.533105	+	0.846049i	$0.321025\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−8.82908	−0.915533
$94$	0	0
$95$	12.7627	1.30943
$96$	0	0
$97$	16.1293	1.63768	0.818841	−	0.574021i	$-0.194617\pi$
$97$	16.1293	1.63768	0.818841	+	0.574021i	$0.194617\pi$
$98$	0	0
$99$	4.93900	0.496388
$100$	0	0
$101$	−9.94869	−0.989932	−0.494966	−	0.868912i	$-0.664819\pi$
$101$	−9.94869	−0.989932	−0.494966	+	0.868912i	$0.664819\pi$
$102$	0	0
$103$	10.9879	1.08267	0.541336	−	0.840806i	$-0.317919\pi$
$103$	10.9879	1.08267	0.541336	+	0.840806i	$0.317919\pi$
$104$	0	0
$105$	−7.54288	−0.736109
$106$	0	0
$107$	9.87263	0.954423	0.477211	−	0.878789i	$-0.341648\pi$
$107$	9.87263	0.954423	0.477211	+	0.878789i	$0.341648\pi$
$108$	0	0
$109$	−20.2446	−1.93908	−0.969540	−	0.244934i	$-0.921234\pi$
$109$	−20.2446	−1.93908	−0.969540	+	0.244934i	$0.921234\pi$
$110$	0	0
$111$	−8.80194	−0.835443
$112$	0	0
$113$	9.69202	0.911749	0.455874	−	0.890044i	$-0.349327\pi$
$113$	9.69202	0.911749	0.455874	+	0.890044i	$0.349327\pi$
$114$	0	0
$115$	6.80492	0.634562
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.04892	0.187824
$120$	0	0
$121$	13.3937	1.21761
$122$	0	0
$123$	−6.93900	−0.625669
$124$	0	0
$125$	−4.25906	−0.380942
$126$	0	0
$127$	−13.8116	−1.22558	−0.612792	−	0.790244i	$-0.709954\pi$
$127$	−13.8116	−1.22558	−0.612792	+	0.790244i	$0.709954\pi$
$128$	0	0
$129$	2.28621	0.201289
$130$	0	0
$131$	−2.99462	−0.261641	−0.130821	−	0.991406i	$-0.541761\pi$
$131$	−2.99462	−0.261641	−0.130821	+	0.991406i	$0.541761\pi$
$132$	0	0
$133$	−8.54288	−0.740761
$134$	0	0
$135$	−3.35690	−0.288916
$136$	0	0
$137$	−23.0194	−1.96668	−0.983339	−	0.181781i	$-0.941814\pi$
$137$	−23.0194	−1.96668	−0.983339	+	0.181781i	$0.941814\pi$
$138$	0	0
$139$	−0.982542	−0.0833381	−0.0416690	−	0.999131i	$-0.513268\pi$
$139$	−0.982542	−0.0833381	−0.0416690	+	0.999131i	$0.513268\pi$
$140$	0	0
$141$	3.80194	0.320181
$142$	0	0
$143$	0	0
$144$	0	0
$145$	13.2228	1.09810
$146$	0	0
$147$	−1.95108	−0.160923
$148$	0	0
$149$	−10.2591	−0.840455	−0.420228	−	0.907419i	$-0.638050\pi$
$149$	−10.2591	−0.840455	−0.420228	+	0.907419i	$0.638050\pi$
$150$	0	0
$151$	−20.1685	−1.64129	−0.820646	−	0.571438i	$-0.806386\pi$
$151$	−20.1685	−1.64129	−0.820646	+	0.571438i	$0.806386\pi$
$152$	0	0
$153$	0.911854	0.0737190
$154$	0	0
$155$	29.6383	2.38061
$156$	0	0
$157$	10.4383	0.833070	0.416535	−	0.909120i	$-0.363244\pi$
$157$	10.4383	0.833070	0.416535	+	0.909120i	$0.363244\pi$
$158$	0	0
$159$	0.542877	0.0430529
$160$	0	0
$161$	−4.55496	−0.358981
$162$	0	0
$163$	−11.0465	−0.865231	−0.432615	−	0.901579i	$-0.642409\pi$
$163$	−11.0465	−0.865231	−0.432615	+	0.901579i	$0.642409\pi$
$164$	0	0
$165$	−16.5797	−1.29073
$166$	0	0
$167$	8.10992	0.627564	0.313782	−	0.949495i	$-0.398404\pi$
$167$	8.10992	0.627564	0.313782	+	0.949495i	$0.398404\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.80194	−0.290741
$172$	0	0
$173$	−18.0562	−1.37279	−0.686394	−	0.727230i	$-0.740808\pi$
$173$	−18.0562	−1.37279	−0.686394	+	0.727230i	$0.740808\pi$
$174$	0	0
$175$	14.0858	1.06478
$176$	0	0
$177$	−4.71379	−0.354310
$178$	0	0
$179$	19.8702	1.48517	0.742585	−	0.669751i	$-0.233599\pi$
$179$	19.8702	1.48517	0.742585	+	0.669751i	$0.233599\pi$
$180$	0	0
$181$	−10.0828	−0.749446	−0.374723	−	0.927137i	$-0.622262\pi$
$181$	−10.0828	−0.749446	−0.374723	+	0.927137i	$0.622262\pi$
$182$	0	0
$183$	3.67994	0.272029
$184$	0	0
$185$	29.5472	2.17235
$186$	0	0
$187$	4.50365	0.329339
$188$	0	0
$189$	2.24698	0.163444
$190$	0	0
$191$	6.58748	0.476653	0.238327	−	0.971185i	$-0.423401\pi$
$191$	6.58748	0.476653	0.238327	+	0.971185i	$0.423401\pi$
$192$	0	0
$193$	−10.8672	−0.782242	−0.391121	−	0.920339i	$-0.627913\pi$
$193$	−10.8672	−0.782242	−0.391121	+	0.920339i	$0.627913\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.24160	−0.658437	−0.329218	−	0.944254i	$-0.606785\pi$
$197$	−9.24160	−0.658437	−0.329218	+	0.944254i	$0.606785\pi$
$198$	0	0
$199$	1.56465	0.110915	0.0554574	−	0.998461i	$-0.482338\pi$
$199$	1.56465	0.110915	0.0554574	+	0.998461i	$0.482338\pi$
$200$	0	0
$201$	−1.52111	−0.107291
$202$	0	0
$203$	−8.85086	−0.621208
$204$	0	0
$205$	23.2935	1.62689
$206$	0	0
$207$	−2.02715	−0.140896
$208$	0	0
$209$	−18.7778	−1.29889
$210$	0	0
$211$	−23.2446	−1.60022	−0.800112	−	0.599851i	$-0.795226\pi$
$211$	−23.2446	−1.60022	−0.800112	+	0.599851i	$0.795226\pi$
$212$	0	0
$213$	2.37867	0.162984
$214$	0	0
$215$	−7.67456	−0.523401
$216$	0	0
$217$	−19.8388	−1.34674
$218$	0	0
$219$	−7.41119	−0.500802
$220$	0	0
$221$	0	0
$222$	0	0
$223$	10.7385	0.719106	0.359553	−	0.933125i	$-0.382929\pi$
$223$	10.7385	0.719106	0.359553	+	0.933125i	$0.382929\pi$
$224$	0	0
$225$	6.26875	0.417917
$226$	0	0
$227$	−6.97584	−0.463003	−0.231501	−	0.972835i	$-0.574364\pi$
$227$	−6.97584	−0.463003	−0.231501	+	0.972835i	$0.574364\pi$
$228$	0	0
$229$	16.8049	1.11050	0.555250	−	0.831683i	$-0.312622\pi$
$229$	16.8049	1.11050	0.555250	+	0.831683i	$0.312622\pi$
$230$	0	0
$231$	11.0978	0.730184
$232$	0	0
$233$	8.69202	0.569433	0.284717	−	0.958612i	$-0.408100\pi$
$233$	8.69202	0.569433	0.284717	+	0.958612i	$0.408100\pi$
$234$	0	0
$235$	−12.7627	−0.832547
$236$	0	0
$237$	3.74094	0.243000
$238$	0	0
$239$	−22.9191	−1.48252	−0.741258	−	0.671220i	$-0.765771\pi$
$239$	−22.9191	−1.48252	−0.741258	+	0.671220i	$0.765771\pi$
$240$	0	0
$241$	−21.9801	−1.41587	−0.707933	−	0.706280i	$-0.750372\pi$
$241$	−21.9801	−1.41587	−0.707933	+	0.706280i	$0.750372\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	6.54958	0.418437
$246$	0	0
$247$	0	0
$248$	0	0
$249$	2.30798	0.146262
$250$	0	0
$251$	−26.6437	−1.68174	−0.840868	−	0.541241i	$-0.817955\pi$
$251$	−26.6437	−1.68174	−0.840868	+	0.541241i	$0.817955\pi$
$252$	0	0
$253$	−10.0121	−0.629454
$254$	0	0
$255$	−3.06100	−0.191687
$256$	0	0
$257$	23.1444	1.44371	0.721853	−	0.692047i	$-0.243291\pi$
$257$	23.1444	1.44371	0.721853	+	0.692047i	$0.243291\pi$
$258$	0	0
$259$	−19.7778	−1.22893
$260$	0	0
$261$	−3.93900	−0.243818
$262$	0	0
$263$	18.5284	1.14251	0.571255	−	0.820773i	$-0.306457\pi$
$263$	18.5284	1.14251	0.571255	+	0.820773i	$0.306457\pi$
$264$	0	0
$265$	−1.82238	−0.111948
$266$	0	0
$267$	10.0586	0.615577
$268$	0	0
$269$	9.75840	0.594980	0.297490	−	0.954725i	$-0.403851\pi$
$269$	9.75840	0.594980	0.297490	+	0.954725i	$0.403851\pi$
$270$	0	0
$271$	−22.8019	−1.38512	−0.692560	−	0.721361i	$-0.743517\pi$
$271$	−22.8019	−1.38512	−0.692560	+	0.721361i	$0.743517\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	30.9614	1.86704
$276$	0	0
$277$	−23.9705	−1.44025	−0.720123	−	0.693847i	$-0.755914\pi$
$277$	−23.9705	−1.44025	−0.720123	+	0.693847i	$0.755914\pi$
$278$	0	0
$279$	−8.82908	−0.528583
$280$	0	0
$281$	4.12498	0.246076	0.123038	−	0.992402i	$-0.460736\pi$
$281$	4.12498	0.246076	0.123038	+	0.992402i	$0.460736\pi$
$282$	0	0
$283$	15.6558	0.930639	0.465320	−	0.885143i	$-0.345939\pi$
$283$	15.6558	0.930639	0.465320	+	0.885143i	$0.345939\pi$
$284$	0	0
$285$	12.7627	0.755998
$286$	0	0
$287$	−15.5918	−0.920354
$288$	0	0
$289$	−16.1685	−0.951090
$290$	0	0
$291$	16.1293	0.945516
$292$	0	0
$293$	−22.5948	−1.32000	−0.660001	−	0.751265i	$-0.729444\pi$
$293$	−22.5948	−1.32000	−0.660001	+	0.751265i	$0.729444\pi$
$294$	0	0
$295$	15.8237	0.921292
$296$	0	0
$297$	4.93900	0.286590
$298$	0	0
$299$	0	0
$300$	0	0
$301$	5.13706	0.296095
$302$	0	0
$303$	−9.94869	−0.571537
$304$	0	0
$305$	−12.3532	−0.707341
$306$	0	0
$307$	−6.55496	−0.374111	−0.187056	−	0.982349i	$-0.559894\pi$
$307$	−6.55496	−0.374111	−0.187056	+	0.982349i	$0.559894\pi$
$308$	0	0
$309$	10.9879	0.625081
$310$	0	0
$311$	12.0392	0.682682	0.341341	−	0.939940i	$-0.389119\pi$
$311$	12.0392	0.682682	0.341341	+	0.939940i	$0.389119\pi$
$312$	0	0
$313$	−33.8950	−1.91586	−0.957929	−	0.287005i	$-0.907340\pi$
$313$	−33.8950	−1.91586	−0.957929	+	0.287005i	$0.907340\pi$
$314$	0	0
$315$	−7.54288	−0.424993
$316$	0	0
$317$	4.49827	0.252648	0.126324	−	0.991989i	$-0.459682\pi$
$317$	4.49827	0.252648	0.126324	+	0.991989i	$0.459682\pi$
$318$	0	0
$319$	−19.4547	−1.08926
$320$	0	0
$321$	9.87263	0.551036
$322$	0	0
$323$	−3.46681	−0.192899
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−20.2446	−1.11953
$328$	0	0
$329$	8.54288	0.470984
$330$	0	0
$331$	−11.2131	−0.616329	−0.308165	−	0.951333i	$-0.599715\pi$
$331$	−11.2131	−0.616329	−0.308165	+	0.951333i	$0.599715\pi$
$332$	0	0
$333$	−8.80194	−0.482343
$334$	0	0
$335$	5.10620	0.278981
$336$	0	0
$337$	7.04892	0.383979	0.191989	−	0.981397i	$-0.438506\pi$
$337$	7.04892	0.383979	0.191989	+	0.981397i	$0.438506\pi$
$338$	0	0
$339$	9.69202	0.526398
$340$	0	0
$341$	−43.6069	−2.36144
$342$	0	0
$343$	−20.1129	−1.08599
$344$	0	0
$345$	6.80492	0.366365
$346$	0	0
$347$	2.70410	0.145164	0.0725819	−	0.997362i	$-0.476876\pi$
$347$	2.70410	0.145164	0.0725819	+	0.997362i	$0.476876\pi$
$348$	0	0
$349$	0.415502	0.0222413	0.0111207	−	0.999938i	$-0.496460\pi$
$349$	0.415502	0.0222413	0.0111207	+	0.999938i	$0.496460\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	32.1672	1.71209	0.856044	−	0.516904i	$-0.172916\pi$
$353$	32.1672	1.71209	0.856044	+	0.516904i	$0.172916\pi$
$354$	0	0
$355$	−7.98493	−0.423796
$356$	0	0
$357$	2.04892	0.108440
$358$	0	0
$359$	22.3521	1.17970	0.589850	−	0.807513i	$-0.299187\pi$
$359$	22.3521	1.17970	0.589850	+	0.807513i	$0.299187\pi$
$360$	0	0
$361$	−4.54527	−0.239225
$362$	0	0
$363$	13.3937	0.702989
$364$	0	0
$365$	24.8786	1.30221
$366$	0	0
$367$	2.30260	0.120195	0.0600974	−	0.998193i	$-0.480859\pi$
$367$	2.30260	0.120195	0.0600974	+	0.998193i	$0.480859\pi$
$368$	0	0
$369$	−6.93900	−0.361230
$370$	0	0
$371$	1.21983	0.0633305
$372$	0	0
$373$	−19.2760	−0.998076	−0.499038	−	0.866580i	$-0.666313\pi$
$373$	−19.2760	−0.998076	−0.499038	+	0.866580i	$0.666313\pi$
$374$	0	0
$375$	−4.25906	−0.219937
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−7.33944	−0.377002	−0.188501	−	0.982073i	$-0.560363\pi$
$379$	−7.33944	−0.377002	−0.188501	+	0.982073i	$0.560363\pi$
$380$	0	0
$381$	−13.8116	−0.707591
$382$	0	0
$383$	19.0901	0.975457	0.487728	−	0.872995i	$-0.337826\pi$
$383$	19.0901	0.975457	0.487728	+	0.872995i	$0.337826\pi$
$384$	0	0
$385$	−37.2543	−1.89865
$386$	0	0
$387$	2.28621	0.116214
$388$	0	0
$389$	−23.9879	−1.21624	−0.608118	−	0.793847i	$-0.708075\pi$
$389$	−23.9879	−1.21624	−0.608118	+	0.793847i	$0.708075\pi$
$390$	0	0
$391$	−1.84846	−0.0934807
$392$	0	0
$393$	−2.99462	−0.151059
$394$	0	0
$395$	−12.5579	−0.631859
$396$	0	0
$397$	4.41789	0.221728	0.110864	−	0.993836i	$-0.464638\pi$
$397$	4.41789	0.221728	0.110864	+	0.993836i	$0.464638\pi$
$398$	0	0
$399$	−8.54288	−0.427679
$400$	0	0
$401$	20.4088	1.01917	0.509583	−	0.860421i	$-0.329800\pi$
$401$	20.4088	1.01917	0.509583	+	0.860421i	$0.329800\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−3.35690	−0.166805
$406$	0	0
$407$	−43.4728	−2.15487
$408$	0	0
$409$	22.1491	1.09520	0.547602	−	0.836739i	$-0.315541\pi$
$409$	22.1491	1.09520	0.547602	+	0.836739i	$0.315541\pi$
$410$	0	0
$411$	−23.0194	−1.13546
$412$	0	0
$413$	−10.5918	−0.521188
$414$	0	0
$415$	−7.74764	−0.380317
$416$	0	0
$417$	−0.982542	−0.0481153
$418$	0	0
$419$	−14.7560	−0.720878	−0.360439	−	0.932783i	$-0.617373\pi$
$419$	−14.7560	−0.720878	−0.360439	+	0.932783i	$0.617373\pi$
$420$	0	0
$421$	−8.47219	−0.412909	−0.206455	−	0.978456i	$-0.566193\pi$
$421$	−8.47219	−0.412909	−0.206455	+	0.978456i	$0.566193\pi$
$422$	0	0
$423$	3.80194	0.184857
$424$	0	0
$425$	5.71618	0.277276
$426$	0	0
$427$	8.26875	0.400153
$428$	0	0
$429$	0	0
$430$	0	0
$431$	2.39181	0.115210	0.0576048	−	0.998339i	$-0.481654\pi$
$431$	2.39181	0.115210	0.0576048	+	0.998339i	$0.481654\pi$
$432$	0	0
$433$	−10.4286	−0.501169	−0.250584	−	0.968095i	$-0.580623\pi$
$433$	−10.4286	−0.501169	−0.250584	+	0.968095i	$0.580623\pi$
$434$	0	0
$435$	13.2228	0.633986
$436$	0	0
$437$	7.70709	0.368680
$438$	0	0
$439$	32.5502	1.55353	0.776767	−	0.629787i	$-0.216858\pi$
$439$	32.5502	1.55353	0.776767	+	0.629787i	$0.216858\pi$
$440$	0	0
$441$	−1.95108	−0.0929087
$442$	0	0
$443$	−9.58211	−0.455260	−0.227630	−	0.973748i	$-0.573098\pi$
$443$	−9.58211	−0.455260	−0.227630	+	0.973748i	$0.573098\pi$
$444$	0	0
$445$	−33.7657	−1.60065
$446$	0	0
$447$	−10.2591	−0.485237
$448$	0	0
$449$	28.3937	1.33998	0.669992	−	0.742369i	$-0.266298\pi$
$449$	28.3937	1.33998	0.669992	+	0.742369i	$0.266298\pi$
$450$	0	0
$451$	−34.2717	−1.61379
$452$	0	0
$453$	−20.1685	−0.947600
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.99569	−0.233688	−0.116844	−	0.993150i	$-0.537278\pi$
$457$	−4.99569	−0.233688	−0.116844	+	0.993150i	$0.537278\pi$
$458$	0	0
$459$	0.911854	0.0425617
$460$	0	0
$461$	−1.35258	−0.0629961	−0.0314981	−	0.999504i	$-0.510028\pi$
$461$	−1.35258	−0.0629961	−0.0314981	+	0.999504i	$0.510028\pi$
$462$	0	0
$463$	−3.36467	−0.156369	−0.0781846	−	0.996939i	$-0.524912\pi$
$463$	−3.36467	−0.156369	−0.0781846	+	0.996939i	$0.524912\pi$
$464$	0	0
$465$	29.6383	1.37444
$466$	0	0
$467$	−6.91079	−0.319793	−0.159897	−	0.987134i	$-0.551116\pi$
$467$	−6.91079	−0.319793	−0.159897	+	0.987134i	$0.551116\pi$
$468$	0	0
$469$	−3.41789	−0.157824
$470$	0	0
$471$	10.4383	0.480973
$472$	0	0
$473$	11.2916	0.519188
$474$	0	0
$475$	−23.8334	−1.09355
$476$	0	0
$477$	0.542877	0.0248566
$478$	0	0
$479$	3.51573	0.160638	0.0803189	−	0.996769i	$-0.474406\pi$
$479$	3.51573	0.160638	0.0803189	+	0.996769i	$0.474406\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−4.55496	−0.207258
$484$	0	0
$485$	−54.1444	−2.45857
$486$	0	0
$487$	−12.8479	−0.582193	−0.291096	−	0.956694i	$-0.594020\pi$
$487$	−12.8479	−0.582193	−0.291096	+	0.956694i	$0.594020\pi$
$488$	0	0
$489$	−11.0465	−0.499541
$490$	0	0
$491$	−28.6708	−1.29390	−0.646948	−	0.762534i	$-0.723955\pi$
$491$	−28.6708	−1.29390	−0.646948	+	0.762534i	$0.723955\pi$
$492$	0	0
$493$	−3.59179	−0.161766
$494$	0	0
$495$	−16.5797	−0.745203
$496$	0	0
$497$	5.34481	0.239748
$498$	0	0
$499$	−33.5555	−1.50215	−0.751076	−	0.660215i	$-0.770465\pi$
$499$	−33.5555	−1.50215	−0.751076	+	0.660215i	$0.770465\pi$
$500$	0	0
$501$	8.10992	0.362324
$502$	0	0
$503$	−21.5633	−0.961461	−0.480730	−	0.876868i	$-0.659628\pi$
$503$	−21.5633	−0.961461	−0.480730	+	0.876868i	$0.659628\pi$
$504$	0	0
$505$	33.3967	1.48613
$506$	0	0
$507$	0	0
$508$	0	0
$509$	8.89008	0.394046	0.197023	−	0.980399i	$-0.436873\pi$
$509$	8.89008	0.394046	0.197023	+	0.980399i	$0.436873\pi$
$510$	0	0
$511$	−16.6528	−0.736676
$512$	0	0
$513$	−3.80194	−0.167860
$514$	0	0
$515$	−36.8853	−1.62536
$516$	0	0
$517$	18.7778	0.825846
$518$	0	0
$519$	−18.0562	−0.792580
$520$	0	0
$521$	19.3478	0.847642	0.423821	−	0.905746i	$-0.360688\pi$
$521$	19.3478	0.847642	0.423821	+	0.905746i	$0.360688\pi$
$522$	0	0
$523$	12.5948	0.550731	0.275366	−	0.961340i	$-0.411201\pi$
$523$	12.5948	0.550731	0.275366	+	0.961340i	$0.411201\pi$
$524$	0	0
$525$	14.0858	0.614753
$526$	0	0
$527$	−8.05084	−0.350700
$528$	0	0
$529$	−18.8907	−0.821334
$530$	0	0
$531$	−4.71379	−0.204561
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−33.1414	−1.43283
$536$	0	0
$537$	19.8702	0.857464
$538$	0	0
$539$	−9.63640	−0.415069
$540$	0	0
$541$	−29.0019	−1.24689	−0.623445	−	0.781867i	$-0.714267\pi$
$541$	−29.0019	−1.24689	−0.623445	+	0.781867i	$0.714267\pi$
$542$	0	0
$543$	−10.0828	−0.432693
$544$	0	0
$545$	67.9590	2.91104
$546$	0	0
$547$	−27.7006	−1.18439	−0.592197	−	0.805793i	$-0.701739\pi$
$547$	−27.7006	−1.18439	−0.592197	+	0.805793i	$0.701739\pi$
$548$	0	0
$549$	3.67994	0.157056
$550$	0	0
$551$	14.9758	0.637992
$552$	0	0
$553$	8.40581	0.357452
$554$	0	0
$555$	29.5472	1.25421
$556$	0	0
$557$	−6.80971	−0.288537	−0.144268	−	0.989539i	$-0.546083\pi$
$557$	−6.80971	−0.288537	−0.144268	+	0.989539i	$0.546083\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	4.50365	0.190144
$562$	0	0
$563$	19.2524	0.811390	0.405695	−	0.914008i	$-0.367030\pi$
$563$	19.2524	0.811390	0.405695	+	0.914008i	$0.367030\pi$
$564$	0	0
$565$	−32.5351	−1.36876
$566$	0	0
$567$	2.24698	0.0943643
$568$	0	0
$569$	7.84846	0.329025	0.164512	−	0.986375i	$-0.447395\pi$
$569$	7.84846	0.329025	0.164512	+	0.986375i	$0.447395\pi$
$570$	0	0
$571$	−29.8568	−1.24947	−0.624735	−	0.780837i	$-0.714793\pi$
$571$	−29.8568	−1.24947	−0.624735	+	0.780837i	$0.714793\pi$
$572$	0	0
$573$	6.58748	0.275196
$574$	0	0
$575$	−12.7077	−0.529947
$576$	0	0
$577$	−8.97823	−0.373769	−0.186884	−	0.982382i	$-0.559839\pi$
$577$	−8.97823	−0.373769	−0.186884	+	0.982382i	$0.559839\pi$
$578$	0	0
$579$	−10.8672	−0.451627
$580$	0	0
$581$	5.18598	0.215151
$582$	0	0
$583$	2.68127	0.111047
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.8538	0.984553	0.492277	−	0.870439i	$-0.336165\pi$
$587$	23.8538	0.984553	0.492277	+	0.870439i	$0.336165\pi$
$588$	0	0
$589$	33.5676	1.38313
$590$	0	0
$591$	−9.24160	−0.380149
$592$	0	0
$593$	−11.9866	−0.492230	−0.246115	−	0.969241i	$-0.579154\pi$
$593$	−11.9866	−0.492230	−0.246115	+	0.969241i	$0.579154\pi$
$594$	0	0
$595$	−6.87800	−0.281971
$596$	0	0
$597$	1.56465	0.0640367
$598$	0	0
$599$	−29.1142	−1.18958	−0.594788	−	0.803883i	$-0.702764\pi$
$599$	−29.1142	−1.18958	−0.594788	+	0.803883i	$0.702764\pi$
$600$	0	0
$601$	37.1366	1.51483	0.757417	−	0.652932i	$-0.226461\pi$
$601$	37.1366	1.51483	0.757417	+	0.652932i	$0.226461\pi$
$602$	0	0
$603$	−1.52111	−0.0619442
$604$	0	0
$605$	−44.9614	−1.82794
$606$	0	0
$607$	14.2325	0.577680	0.288840	−	0.957377i	$-0.406731\pi$
$607$	14.2325	0.577680	0.288840	+	0.957377i	$0.406731\pi$
$608$	0	0
$609$	−8.85086	−0.358655
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−30.3139	−1.22437	−0.612184	−	0.790715i	$-0.709709\pi$
$613$	−30.3139	−1.22437	−0.612184	+	0.790715i	$0.709709\pi$
$614$	0	0
$615$	23.2935	0.939285
$616$	0	0
$617$	24.9638	1.00500	0.502501	−	0.864576i	$-0.332413\pi$
$617$	24.9638	1.00500	0.502501	+	0.864576i	$0.332413\pi$
$618$	0	0
$619$	35.6122	1.43138	0.715688	−	0.698420i	$-0.246113\pi$
$619$	35.6122	1.43138	0.715688	+	0.698420i	$0.246113\pi$
$620$	0	0
$621$	−2.02715	−0.0813466
$622$	0	0
$623$	22.6015	0.905509
$624$	0	0
$625$	−17.0465	−0.681861
$626$	0	0
$627$	−18.7778	−0.749912
$628$	0	0
$629$	−8.02608	−0.320021
$630$	0	0
$631$	23.8829	0.950763	0.475382	−	0.879780i	$-0.342310\pi$
$631$	23.8829	0.950763	0.475382	+	0.879780i	$0.342310\pi$
$632$	0	0
$633$	−23.2446	−0.923889
$634$	0	0
$635$	46.3642	1.83991
$636$	0	0
$637$	0	0
$638$	0	0
$639$	2.37867	0.0940986
$640$	0	0
$641$	−24.6577	−0.973920	−0.486960	−	0.873424i	$-0.661894\pi$
$641$	−24.6577	−0.973920	−0.486960	+	0.873424i	$0.661894\pi$
$642$	0	0
$643$	47.8165	1.88570	0.942850	−	0.333218i	$-0.108134\pi$
$643$	47.8165	1.88570	0.942850	+	0.333218i	$0.108134\pi$
$644$	0	0
$645$	−7.67456	−0.302186
$646$	0	0
$647$	5.38942	0.211880	0.105940	−	0.994373i	$-0.466215\pi$
$647$	5.38942	0.211880	0.105940	+	0.994373i	$0.466215\pi$
$648$	0	0
$649$	−23.2814	−0.913876
$650$	0	0
$651$	−19.8388	−0.777543
$652$	0	0
$653$	−5.03790	−0.197148	−0.0985741	−	0.995130i	$-0.531428\pi$
$653$	−5.03790	−0.197148	−0.0985741	+	0.995130i	$0.531428\pi$
$654$	0	0
$655$	10.0526	0.392789
$656$	0	0
$657$	−7.41119	−0.289138
$658$	0	0
$659$	43.9812	1.71326	0.856632	−	0.515927i	$-0.172553\pi$
$659$	43.9812	1.71326	0.856632	+	0.515927i	$0.172553\pi$
$660$	0	0
$661$	38.2194	1.48656	0.743280	−	0.668980i	$-0.233269\pi$
$661$	38.2194	1.48656	0.743280	+	0.668980i	$0.233269\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	28.6775	1.11207
$666$	0	0
$667$	7.98493	0.309178
$668$	0	0
$669$	10.7385	0.415176
$670$	0	0
$671$	18.1752	0.701647
$672$	0	0
$673$	−6.66487	−0.256912	−0.128456	−	0.991715i	$-0.541002\pi$
$673$	−6.66487	−0.256912	−0.128456	+	0.991715i	$0.541002\pi$
$674$	0	0
$675$	6.26875	0.241284
$676$	0	0
$677$	4.80194	0.184553	0.0922767	−	0.995733i	$-0.470586\pi$
$677$	4.80194	0.184553	0.0922767	+	0.995733i	$0.470586\pi$
$678$	0	0
$679$	36.2422	1.39085
$680$	0	0
$681$	−6.97584	−0.267315
$682$	0	0
$683$	11.2591	0.430816	0.215408	−	0.976524i	$-0.430892\pi$
$683$	11.2591	0.430816	0.215408	+	0.976524i	$0.430892\pi$
$684$	0	0
$685$	77.2737	2.95247
$686$	0	0
$687$	16.8049	0.641148
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−24.7144	−0.940179	−0.470090	−	0.882619i	$-0.655778\pi$
$691$	−24.7144	−0.940179	−0.470090	+	0.882619i	$0.655778\pi$
$692$	0	0
$693$	11.0978	0.421572
$694$	0	0
$695$	3.29829	0.125111
$696$	0	0
$697$	−6.32736	−0.239666
$698$	0	0
$699$	8.69202	0.328762
$700$	0	0
$701$	25.8920	0.977927	0.488964	−	0.872304i	$-0.337375\pi$
$701$	25.8920	0.977927	0.488964	+	0.872304i	$0.337375\pi$
$702$	0	0
$703$	33.4644	1.26213
$704$	0	0
$705$	−12.7627	−0.480671
$706$	0	0
$707$	−22.3545	−0.840728
$708$	0	0
$709$	−0.0851621	−0.00319833	−0.00159916	−	0.999999i	$-0.500509\pi$
$709$	−0.0851621	−0.00319833	−0.00159916	+	0.999999i	$0.500509\pi$
$710$	0	0
$711$	3.74094	0.140296
$712$	0	0
$713$	17.8979	0.670280
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−22.9191	−0.855931
$718$	0	0
$719$	−27.0508	−1.00883	−0.504413	−	0.863463i	$-0.668291\pi$
$719$	−27.0508	−1.00883	−0.504413	+	0.863463i	$0.668291\pi$
$720$	0	0
$721$	24.6896	0.919490
$722$	0	0
$723$	−21.9801	−0.817451
$724$	0	0
$725$	−24.6926	−0.917061
$726$	0	0
$727$	47.1584	1.74901	0.874503	−	0.485019i	$-0.161187\pi$
$727$	47.1584	1.74901	0.874503	+	0.485019i	$0.161187\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.08469	0.0771050
$732$	0	0
$733$	9.68186	0.357608	0.178804	−	0.983885i	$-0.442777\pi$
$733$	9.68186	0.357608	0.178804	+	0.983885i	$0.442777\pi$
$734$	0	0
$735$	6.54958	0.241585
$736$	0	0
$737$	−7.51275	−0.276736
$738$	0	0
$739$	36.3139	1.33583	0.667915	−	0.744238i	$-0.267187\pi$
$739$	36.3139	1.33583	0.667915	+	0.744238i	$0.267187\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−41.7536	−1.53179	−0.765896	−	0.642965i	$-0.777704\pi$
$743$	−41.7536	−1.53179	−0.765896	+	0.642965i	$0.777704\pi$
$744$	0	0
$745$	34.4386	1.26173
$746$	0	0
$747$	2.30798	0.0844445
$748$	0	0
$749$	22.1836	0.810571
$750$	0	0
$751$	22.1927	0.809823	0.404911	−	0.914356i	$-0.367302\pi$
$751$	22.1927	0.809823	0.404911	+	0.914356i	$0.367302\pi$
$752$	0	0
$753$	−26.6437	−0.970950
$754$	0	0
$755$	67.7036	2.46399
$756$	0	0
$757$	9.23729	0.335735	0.167868	−	0.985810i	$-0.446312\pi$
$757$	9.23729	0.335735	0.167868	+	0.985810i	$0.446312\pi$
$758$	0	0
$759$	−10.0121	−0.363416
$760$	0	0
$761$	7.50173	0.271937	0.135969	−	0.990713i	$-0.456585\pi$
$761$	7.50173	0.271937	0.135969	+	0.990713i	$0.456585\pi$
$762$	0	0
$763$	−45.4892	−1.64682
$764$	0	0
$765$	−3.06100	−0.110671
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−5.14005	−0.185355	−0.0926774	−	0.995696i	$-0.529543\pi$
$769$	−5.14005	−0.185355	−0.0926774	+	0.995696i	$0.529543\pi$
$770$	0	0
$771$	23.1444	0.833524
$772$	0	0
$773$	8.50173	0.305786	0.152893	−	0.988243i	$-0.451141\pi$
$773$	8.50173	0.305786	0.152893	+	0.988243i	$0.451141\pi$
$774$	0	0
$775$	−55.3473	−1.98813
$776$	0	0
$777$	−19.7778	−0.709524
$778$	0	0
$779$	26.3817	0.945221
$780$	0	0
$781$	11.7482	0.420385
$782$	0	0
$783$	−3.93900	−0.140768
$784$	0	0
$785$	−35.0404	−1.25065
$786$	0	0
$787$	−7.78554	−0.277525	−0.138762	−	0.990326i	$-0.544312\pi$
$787$	−7.78554	−0.277525	−0.138762	+	0.990326i	$0.544312\pi$
$788$	0	0
$789$	18.5284	0.659629
$790$	0	0
$791$	21.7778	0.774329
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−1.82238	−0.0646332
$796$	0	0
$797$	21.3840	0.757462	0.378731	−	0.925507i	$-0.376361\pi$
$797$	21.3840	0.757462	0.378731	+	0.925507i	$0.376361\pi$
$798$	0	0
$799$	3.46681	0.122647
$800$	0	0
$801$	10.0586	0.355403
$802$	0	0
$803$	−36.6039	−1.29172
$804$	0	0
$805$	15.2905	0.538920
$806$	0	0
$807$	9.75840	0.343512
$808$	0	0
$809$	2.28621	0.0803788	0.0401894	−	0.999192i	$-0.487204\pi$
$809$	2.28621	0.0803788	0.0401894	+	0.999192i	$0.487204\pi$
$810$	0	0
$811$	17.8079	0.625320	0.312660	−	0.949865i	$-0.398780\pi$
$811$	17.8079	0.625320	0.312660	+	0.949865i	$0.398780\pi$
$812$	0	0
$813$	−22.8019	−0.799699
$814$	0	0
$815$	37.0820	1.29893
$816$	0	0
$817$	−8.69202	−0.304095
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.9054	−0.485302	−0.242651	−	0.970114i	$-0.578017\pi$
$821$	−13.9054	−0.485302	−0.242651	+	0.970114i	$0.578017\pi$
$822$	0	0
$823$	7.24831	0.252660	0.126330	−	0.991988i	$-0.459680\pi$
$823$	7.24831	0.252660	0.126330	+	0.991988i	$0.459680\pi$
$824$	0	0
$825$	30.9614	1.07794
$826$	0	0
$827$	54.1191	1.88191	0.940953	−	0.338536i	$-0.109932\pi$
$827$	54.1191	1.88191	0.940953	+	0.338536i	$0.109932\pi$
$828$	0	0
$829$	−5.79178	−0.201157	−0.100578	−	0.994929i	$-0.532069\pi$
$829$	−5.79178	−0.201157	−0.100578	+	0.994929i	$0.532069\pi$
$830$	0	0
$831$	−23.9705	−0.831526
$832$	0	0
$833$	−1.77910	−0.0616422
$834$	0	0
$835$	−27.2241	−0.942130
$836$	0	0
$837$	−8.82908	−0.305178
$838$	0	0
$839$	−5.29350	−0.182752	−0.0913760	−	0.995816i	$-0.529127\pi$
$839$	−5.29350	−0.182752	−0.0913760	+	0.995816i	$0.529127\pi$
$840$	0	0
$841$	−13.4843	−0.464975
$842$	0	0
$843$	4.12498	0.142072
$844$	0	0
$845$	0	0
$846$	0	0
$847$	30.0954	1.03409
$848$	0	0
$849$	15.6558	0.537305
$850$	0	0
$851$	17.8428	0.611644
$852$	0	0
$853$	13.5961	0.465522	0.232761	−	0.972534i	$-0.425224\pi$
$853$	13.5961	0.465522	0.232761	+	0.972534i	$0.425224\pi$
$854$	0	0
$855$	12.7627	0.436475
$856$	0	0
$857$	−23.8323	−0.814097	−0.407048	−	0.913407i	$-0.633442\pi$
$857$	−23.8323	−0.814097	−0.407048	+	0.913407i	$0.633442\pi$
$858$	0	0
$859$	26.9861	0.920754	0.460377	−	0.887723i	$-0.347714\pi$
$859$	26.9861	0.920754	0.460377	+	0.887723i	$0.347714\pi$
$860$	0	0
$861$	−15.5918	−0.531367
$862$	0	0
$863$	−27.0291	−0.920080	−0.460040	−	0.887898i	$-0.652165\pi$
$863$	−27.0291	−0.920080	−0.460040	+	0.887898i	$0.652165\pi$
$864$	0	0
$865$	60.6128	2.06090
$866$	0	0
$867$	−16.1685	−0.549112
$868$	0	0
$869$	18.4765	0.626772
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.1293	0.545894
$874$	0	0
$875$	−9.57002	−0.323526
$876$	0	0
$877$	−40.8780	−1.38035	−0.690176	−	0.723642i	$-0.742467\pi$
$877$	−40.8780	−1.38035	−0.690176	+	0.723642i	$0.742467\pi$
$878$	0	0
$879$	−22.5948	−0.762103
$880$	0	0
$881$	−22.4964	−0.757921	−0.378961	−	0.925413i	$-0.623718\pi$
$881$	−22.4964	−0.757921	−0.378961	+	0.925413i	$0.623718\pi$
$882$	0	0
$883$	4.16315	0.140101	0.0700505	−	0.997543i	$-0.477684\pi$
$883$	4.16315	0.140101	0.0700505	+	0.997543i	$0.477684\pi$
$884$	0	0
$885$	15.8237	0.531908
$886$	0	0
$887$	26.3002	0.883075	0.441537	−	0.897243i	$-0.354433\pi$
$887$	26.3002	0.883075	0.441537	+	0.897243i	$0.354433\pi$
$888$	0	0
$889$	−31.0344	−1.04086
$890$	0	0
$891$	4.93900	0.165463
$892$	0	0
$893$	−14.4547	−0.483709
$894$	0	0
$895$	−66.7023	−2.22961
$896$	0	0
$897$	0	0
$898$	0	0
$899$	34.7778	1.15990
$900$	0	0
$901$	0.495024	0.0164916
$902$	0	0
$903$	5.13706	0.170951
$904$	0	0
$905$	33.8468	1.12511
$906$	0	0
$907$	57.9114	1.92292	0.961458	−	0.274952i	$-0.0886620\pi$
$907$	57.9114	1.92292	0.961458	+	0.274952i	$0.0886620\pi$
$908$	0	0
$909$	−9.94869	−0.329977
$910$	0	0
$911$	0.286799	0.00950208	0.00475104	−	0.999989i	$-0.498488\pi$
$911$	0.286799	0.00950208	0.00475104	+	0.999989i	$0.498488\pi$
$912$	0	0
$913$	11.3991	0.377255
$914$	0	0
$915$	−12.3532	−0.408383
$916$	0	0
$917$	−6.72886	−0.222206
$918$	0	0
$919$	31.1239	1.02668	0.513342	−	0.858184i	$-0.328407\pi$
$919$	31.1239	1.02668	0.513342	+	0.858184i	$0.328407\pi$
$920$	0	0
$921$	−6.55496	−0.215993
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−55.1771	−1.81421
$926$	0	0
$927$	10.9879	0.360891
$928$	0	0
$929$	−7.62671	−0.250224	−0.125112	−	0.992143i	$-0.539929\pi$
$929$	−7.62671	−0.250224	−0.125112	+	0.992143i	$0.539929\pi$
$930$	0	0
$931$	7.41789	0.243112
$932$	0	0
$933$	12.0392	0.394147
$934$	0	0
$935$	−15.1183	−0.494421
$936$	0	0
$937$	−5.67324	−0.185337	−0.0926683	−	0.995697i	$-0.529540\pi$
$937$	−5.67324	−0.185337	−0.0926683	+	0.995697i	$0.529540\pi$
$938$	0	0
$939$	−33.8950	−1.10612
$940$	0	0
$941$	−41.5394	−1.35415	−0.677073	−	0.735916i	$-0.736752\pi$
$941$	−41.5394	−1.35415	−0.677073	+	0.735916i	$0.736752\pi$
$942$	0	0
$943$	14.0664	0.458064
$944$	0	0
$945$	−7.54288	−0.245370
$946$	0	0
$947$	−47.3110	−1.53740	−0.768700	−	0.639610i	$-0.779096\pi$
$947$	−47.3110	−1.53740	−0.768700	+	0.639610i	$0.779096\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	4.49827	0.145866
$952$	0	0
$953$	−34.3435	−1.11249	−0.556247	−	0.831017i	$-0.687759\pi$
$953$	−34.3435	−1.11249	−0.556247	+	0.831017i	$0.687759\pi$
$954$	0	0
$955$	−22.1135	−0.715576
$956$	0	0
$957$	−19.4547	−0.628882
$958$	0	0
$959$	−51.7241	−1.67026
$960$	0	0
$961$	46.9527	1.51460
$962$	0	0
$963$	9.87263	0.318141
$964$	0	0
$965$	36.4802	1.17434
$966$	0	0
$967$	48.5096	1.55996	0.779982	−	0.625802i	$-0.215228\pi$
$967$	48.5096	1.55996	0.779982	+	0.625802i	$0.215228\pi$
$968$	0	0
$969$	−3.46681	−0.111370
$970$	0	0
$971$	−41.4650	−1.33068	−0.665338	−	0.746542i	$-0.731712\pi$
$971$	−41.4650	−1.33068	−0.665338	+	0.746542i	$0.731712\pi$
$972$	0	0
$973$	−2.20775	−0.0707772
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.09677	0.0670816	0.0335408	−	0.999437i	$-0.489322\pi$
$977$	2.09677	0.0670816	0.0335408	+	0.999437i	$0.489322\pi$
$978$	0	0
$979$	49.6795	1.58776
$980$	0	0
$981$	−20.2446	−0.646360
$982$	0	0
$983$	25.2336	0.804826	0.402413	−	0.915458i	$-0.368172\pi$
$983$	25.2336	0.804826	0.402413	+	0.915458i	$0.368172\pi$
$984$	0	0
$985$	31.0231	0.988478
$986$	0	0
$987$	8.54288	0.271923
$988$	0	0
$989$	−4.63448	−0.147368
$990$	0	0
$991$	−12.0489	−0.382746	−0.191373	−	0.981517i	$-0.561294\pi$
$991$	−12.0489	−0.382746	−0.191373	+	0.981517i	$0.561294\pi$
$992$	0	0
$993$	−11.2131	−0.355838
$994$	0	0
$995$	−5.25236	−0.166511
$996$	0	0
$997$	−1.43403	−0.0454160	−0.0227080	−	0.999742i	$-0.507229\pi$
$997$	−1.43403	−0.0454160	−0.0227080	+	0.999742i	$0.507229\pi$
$998$	0	0
$999$	−8.80194	−0.278481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.cg.1.2		3
4.3	odd	2		507.2.a.i.1.3	✓	3
12.11	even	2		1521.2.a.s.1.1		3
13.12	even	2		8112.2.a.cp.1.2		3
52.3	odd	6		507.2.e.l.22.1		6
52.7	even	12		507.2.j.i.361.4		12
52.11	even	12		507.2.j.i.316.3		12
52.15	even	12		507.2.j.i.316.4		12
52.19	even	12		507.2.j.i.361.3		12
52.23	odd	6		507.2.e.i.22.3		6
52.31	even	4		507.2.b.f.337.3		6
52.35	odd	6		507.2.e.l.484.1		6
52.43	odd	6		507.2.e.i.484.3		6
52.47	even	4		507.2.b.f.337.4		6
52.51	odd	2		507.2.a.l.1.1	yes	3
156.47	odd	4		1521.2.b.k.1351.3		6
156.83	odd	4		1521.2.b.k.1351.4		6
156.155	even	2		1521.2.a.n.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.i.1.3	✓	3	4.3	odd	2
507.2.a.l.1.1	yes	3	52.51	odd	2
507.2.b.f.337.3		6	52.31	even	4
507.2.b.f.337.4		6	52.47	even	4
507.2.e.i.22.3		6	52.23	odd	6
507.2.e.i.484.3		6	52.43	odd	6
507.2.e.l.22.1		6	52.3	odd	6
507.2.e.l.484.1		6	52.35	odd	6
507.2.j.i.316.3		12	52.11	even	12
507.2.j.i.316.4		12	52.15	even	12
507.2.j.i.361.3		12	52.19	even	12
507.2.j.i.361.4		12	52.7	even	12
1521.2.a.n.1.3		3	156.155	even	2
1521.2.a.s.1.1		3	12.11	even	2
1521.2.b.k.1351.3		6	156.47	odd	4
1521.2.b.k.1351.4		6	156.83	odd	4
8112.2.a.cg.1.2		3	1.1	even	1	trivial
8112.2.a.cp.1.2		3	13.12	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.35690 q^{5} +2.24698 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.35690 q^{5} +2.24698 q^{7} +1.00000 q^{9} +4.93900 q^{11} -3.35690 q^{15} +0.911854 q^{17} -3.80194 q^{19} +2.24698 q^{21} -2.02715 q^{23} +6.26875 q^{25} +1.00000 q^{27} -3.93900 q^{29} -8.82908 q^{31} +4.93900 q^{33} -7.54288 q^{35} -8.80194 q^{37} -6.93900 q^{41} +2.28621 q^{43} -3.35690 q^{45} +3.80194 q^{47} -1.95108 q^{49} +0.911854 q^{51} +0.542877 q^{53} -16.5797 q^{55} -3.80194 q^{57} -4.71379 q^{59} +3.67994 q^{61} +2.24698 q^{63} -1.52111 q^{67} -2.02715 q^{69} +2.37867 q^{71} -7.41119 q^{73} +6.26875 q^{75} +11.0978 q^{77} +3.74094 q^{79} +1.00000 q^{81} +2.30798 q^{83} -3.06100 q^{85} -3.93900 q^{87} +10.0586 q^{89} -8.82908 q^{93} +12.7627 q^{95} +16.1293 q^{97} +4.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 6 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 6 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 6 q^{5} + 2 q^{7} + 3 q^{9} + 5 q^{11} - 6 q^{15} - q^{17} - 7 q^{19} + 2 q^{21} + 11 q^{25} + 3 q^{27} - 2 q^{29} - 16 q^{31} + 5 q^{33} - 4 q^{35} - 22 q^{37} - 11 q^{41} + 15 q^{43} - 6 q^{45} + 7 q^{47} - 15 q^{49} - q^{51} - 17 q^{53} - 3 q^{55} - 7 q^{57} - 6 q^{59} - 13 q^{61} + 2 q^{63} + 11 q^{67} - 6 q^{73} + 11 q^{75} + 15 q^{77} - 3 q^{79} + 3 q^{81} + 12 q^{83} - 19 q^{85} - 2 q^{87} - q^{89} - 16 q^{93} + 21 q^{95} + 5 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 6 * q^5 + 2 * q^7 + 3 * q^9 + 5 * q^11 - 6 * q^15 - q^17 - 7 * q^19 + 2 * q^21 + 11 * q^25 + 3 * q^27 - 2 * q^29 - 16 * q^31 + 5 * q^33 - 4 * q^35 - 22 * q^37 - 11 * q^41 + 15 * q^43 - 6 * q^45 + 7 * q^47 - 15 * q^49 - q^51 - 17 * q^53 - 3 * q^55 - 7 * q^57 - 6 * q^59 - 13 * q^61 + 2 * q^63 + 11 * q^67 - 6 * q^73 + 11 * q^75 + 15 * q^77 - 3 * q^79 + 3 * q^81 + 12 * q^83 - 19 * q^85 - 2 * q^87 - q^89 - 16 * q^93 + 21 * q^95 + 5 * q^97 + 5 * q^99

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