LMFDB - Embedded newform 8464.2.a.bc.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8464,2,Mod(1,8464)] 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("8464.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8464, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8464 = 2^{4} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8464.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,6,0,4,0,6,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$67.5853802708$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	8464.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{3} +3.00000 q^{5} -0.449490 q^{7} +3.00000 q^{9} -2.44949 q^{11} +5.89898 q^{13} -7.34847 q^{15} +4.89898 q^{17} +3.55051 q^{19} +1.10102 q^{21} +4.00000 q^{25} +7.89898 q^{29} +10.8990 q^{31} +6.00000 q^{33} -1.34847 q^{35} -8.00000 q^{37} -14.4495 q^{39} +7.89898 q^{41} +6.89898 q^{43} +9.00000 q^{45} -1.55051 q^{47} -6.79796 q^{49} -12.0000 q^{51} -8.79796 q^{53} -7.34847 q^{55} -8.69694 q^{57} -5.34847 q^{59} -5.89898 q^{61} -1.34847 q^{63} +17.6969 q^{65} +13.7980 q^{67} +4.44949 q^{71} -1.89898 q^{73} -9.79796 q^{75} +1.10102 q^{77} +12.0000 q^{79} -9.00000 q^{81} +6.89898 q^{83} +14.6969 q^{85} -19.3485 q^{87} -8.79796 q^{89} -2.65153 q^{91} -26.6969 q^{93} +10.6515 q^{95} +1.89898 q^{97} -7.34847 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 4 q^{7} + 6 q^{9} + 2 q^{13} + 12 q^{19} + 12 q^{21} + 8 q^{25} + 6 q^{29} + 12 q^{31} + 12 q^{33} + 12 q^{35} - 16 q^{37} - 24 q^{39} + 6 q^{41} + 4 q^{43} + 18 q^{45} - 8 q^{47} + 6 q^{49}+ \cdots - 6 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 4 * q^7 + 6 * q^9 + 2 * q^13 + 12 * q^19 + 12 * q^21 + 8 * q^25 + 6 * q^29 + 12 * q^31 + 12 * q^33 + 12 * q^35 - 16 * q^37 - 24 * q^39 + 6 * q^41 + 4 * q^43 + 18 * q^45 - 8 * q^47 + 6 * q^49 - 24 * q^51 + 2 * q^53 + 12 * q^57 + 4 * q^59 - 2 * q^61 + 12 * q^63 + 6 * q^65 + 8 * q^67 + 4 * q^71 + 6 * q^73 + 12 * q^77 + 24 * q^79 - 18 * q^81 + 4 * q^83 - 24 * q^87 + 2 * q^89 - 20 * q^91 - 24 * q^93 + 36 * q^95 - 6 * q^97

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
$3$	−2.44949	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$3$	−2.44949	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−0.449490	−0.169891	−0.0849456	−	0.996386i	$-0.527072\pi$
$7$	−0.449490	−0.169891	−0.0849456	+	0.996386i	$0.527072\pi$
$8$	0	0
$9$	3.00000	1.00000
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	0	0
$13$	5.89898	1.63608	0.818041	−	0.575160i	$-0.195060\pi$
$13$	5.89898	1.63608	0.818041	+	0.575160i	$0.195060\pi$
$14$	0	0
$15$	−7.34847	−1.89737
$16$	0	0
$17$	4.89898	1.18818	0.594089	−	0.804400i	$-0.297513\pi$
$17$	4.89898	1.18818	0.594089	+	0.804400i	$0.297513\pi$
$18$	0	0
$19$	3.55051	0.814543	0.407271	−	0.913307i	$-0.366480\pi$
$19$	3.55051	0.814543	0.407271	+	0.913307i	$0.366480\pi$
$20$	0	0
$21$	1.10102	0.240262
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	7.89898	1.46680	0.733402	−	0.679795i	$-0.237931\pi$
$29$	7.89898	1.46680	0.733402	+	0.679795i	$0.237931\pi$
$30$	0	0
$31$	10.8990	1.95751	0.978757	−	0.205023i	$-0.0657268\pi$
$31$	10.8990	1.95751	0.978757	+	0.205023i	$0.0657268\pi$
$32$	0	0
$33$	6.00000	1.04447
$34$	0	0
$35$	−1.34847	−0.227933
$36$	0	0
$37$	−8.00000	−1.31519	−0.657596	−	0.753371i	$-0.728427\pi$
$37$	−8.00000	−1.31519	−0.657596	+	0.753371i	$0.728427\pi$
$38$	0	0
$39$	−14.4495	−2.31377
$40$	0	0
$41$	7.89898	1.23361	0.616807	−	0.787115i	$-0.288426\pi$
$41$	7.89898	1.23361	0.616807	+	0.787115i	$0.288426\pi$
$42$	0	0
$43$	6.89898	1.05208	0.526042	−	0.850458i	$-0.323675\pi$
$43$	6.89898	1.05208	0.526042	+	0.850458i	$0.323675\pi$
$44$	0	0
$45$	9.00000	1.34164
$46$	0	0
$47$	−1.55051	−0.226165	−0.113083	−	0.993586i	$-0.536072\pi$
$47$	−1.55051	−0.226165	−0.113083	+	0.993586i	$0.536072\pi$
$48$	0	0
$49$	−6.79796	−0.971137
$50$	0	0
$51$	−12.0000	−1.68034
$52$	0	0
$53$	−8.79796	−1.20849	−0.604246	−	0.796798i	$-0.706526\pi$
$53$	−8.79796	−1.20849	−0.604246	+	0.796798i	$0.706526\pi$
$54$	0	0
$55$	−7.34847	−0.990867
$56$	0	0
$57$	−8.69694	−1.15194
$58$	0	0
$59$	−5.34847	−0.696311	−0.348156	−	0.937437i	$-0.613192\pi$
$59$	−5.34847	−0.696311	−0.348156	+	0.937437i	$0.613192\pi$
$60$	0	0
$61$	−5.89898	−0.755287	−0.377643	−	0.925951i	$-0.623265\pi$
$61$	−5.89898	−0.755287	−0.377643	+	0.925951i	$0.623265\pi$
$62$	0	0
$63$	−1.34847	−0.169891
$64$	0	0
$65$	17.6969	2.19504
$66$	0	0
$67$	13.7980	1.68569	0.842844	−	0.538157i	$-0.180879\pi$
$67$	13.7980	1.68569	0.842844	+	0.538157i	$0.180879\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4.44949	0.528057	0.264029	−	0.964515i	$-0.414949\pi$
$71$	4.44949	0.528057	0.264029	+	0.964515i	$0.414949\pi$
$72$	0	0
$73$	−1.89898	−0.222259	−0.111129	−	0.993806i	$-0.535447\pi$
$73$	−1.89898	−0.222259	−0.111129	+	0.993806i	$0.535447\pi$
$74$	0	0
$75$	−9.79796	−1.13137
$76$	0	0
$77$	1.10102	0.125473
$78$	0	0
$79$	12.0000	1.35011	0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000	1.35011	0.675053	+	0.737769i	$0.264121\pi$
$80$	0	0
$81$	−9.00000	−1.00000
$82$	0	0
$83$	6.89898	0.757261	0.378631	−	0.925548i	$-0.376395\pi$
$83$	6.89898	0.757261	0.378631	+	0.925548i	$0.376395\pi$
$84$	0	0
$85$	14.6969	1.59411
$86$	0	0
$87$	−19.3485	−2.07437
$88$	0	0
$89$	−8.79796	−0.932582	−0.466291	−	0.884631i	$-0.654410\pi$
$89$	−8.79796	−0.932582	−0.466291	+	0.884631i	$0.654410\pi$
$90$	0	0
$91$	−2.65153	−0.277956
$92$	0	0
$93$	−26.6969	−2.76834
$94$	0	0
$95$	10.6515	1.09282
$96$	0	0
$97$	1.89898	0.192812	0.0964061	−	0.995342i	$-0.469265\pi$
$97$	1.89898	0.192812	0.0964061	+	0.995342i	$0.469265\pi$
$98$	0	0
$99$	−7.34847	−0.738549
$100$	0	0
$101$	9.00000	0.895533	0.447767	−	0.894150i	$-0.352219\pi$
$101$	9.00000	0.895533	0.447767	+	0.894150i	$0.352219\pi$
$102$	0	0
$103$	−6.89898	−0.679777	−0.339888	−	0.940466i	$-0.610389\pi$
$103$	−6.89898	−0.679777	−0.339888	+	0.940466i	$0.610389\pi$
$104$	0	0
$105$	3.30306	0.322346
$106$	0	0
$107$	−16.4495	−1.59023	−0.795116	−	0.606457i	$-0.792590\pi$
$107$	−16.4495	−1.59023	−0.795116	+	0.606457i	$0.792590\pi$
$108$	0	0
$109$	0.797959	0.0764306	0.0382153	−	0.999270i	$-0.487833\pi$
$109$	0.797959	0.0764306	0.0382153	+	0.999270i	$0.487833\pi$
$110$	0	0
$111$	19.5959	1.85996
$112$	0	0
$113$	5.00000	0.470360	0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000	0.470360	0.235180	+	0.971952i	$0.424432\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	17.6969	1.63608
$118$	0	0
$119$	−2.20204	−0.201861
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	−19.3485	−1.74459
$124$	0	0
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−6.89898	−0.612185	−0.306093	−	0.952002i	$-0.599022\pi$
$127$	−6.89898	−0.612185	−0.306093	+	0.952002i	$0.599022\pi$
$128$	0	0
$129$	−16.8990	−1.48787
$130$	0	0
$131$	18.2474	1.59429	0.797143	−	0.603790i	$-0.206343\pi$
$131$	18.2474	1.59429	0.797143	+	0.603790i	$0.206343\pi$
$132$	0	0
$133$	−1.59592	−0.138384
$134$	0	0
$135$	0	0
$136$	0	0
$137$	11.8990	1.01660	0.508299	−	0.861181i	$-0.330274\pi$
$137$	11.8990	1.01660	0.508299	+	0.861181i	$0.330274\pi$
$138$	0	0
$139$	−12.6969	−1.07694	−0.538470	−	0.842645i	$-0.680998\pi$
$139$	−12.6969	−1.07694	−0.538470	+	0.842645i	$0.680998\pi$
$140$	0	0
$141$	3.79796	0.319846
$142$	0	0
$143$	−14.4495	−1.20833
$144$	0	0
$145$	23.6969	1.96792
$146$	0	0
$147$	16.6515	1.37340
$148$	0	0
$149$	3.89898	0.319417	0.159708	−	0.987164i	$-0.448945\pi$
$149$	3.89898	0.319417	0.159708	+	0.987164i	$0.448945\pi$
$150$	0	0
$151$	−16.2474	−1.32220	−0.661099	−	0.750298i	$-0.729910\pi$
$151$	−16.2474	−1.32220	−0.661099	+	0.750298i	$0.729910\pi$
$152$	0	0
$153$	14.6969	1.18818
$154$	0	0
$155$	32.6969	2.62628
$156$	0	0
$157$	3.00000	0.239426	0.119713	−	0.992809i	$-0.461803\pi$
$157$	3.00000	0.239426	0.119713	+	0.992809i	$0.461803\pi$
$158$	0	0
$159$	21.5505	1.70907
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−7.55051	−0.591402	−0.295701	−	0.955281i	$-0.595553\pi$
$163$	−7.55051	−0.591402	−0.295701	+	0.955281i	$0.595553\pi$
$164$	0	0
$165$	18.0000	1.40130
$166$	0	0
$167$	−5.10102	−0.394729	−0.197364	−	0.980330i	$-0.563238\pi$
$167$	−5.10102	−0.394729	−0.197364	+	0.980330i	$0.563238\pi$
$168$	0	0
$169$	21.7980	1.67677
$170$	0	0
$171$	10.6515	0.814543
$172$	0	0
$173$	1.20204	0.0913895	0.0456947	−	0.998955i	$-0.485450\pi$
$173$	1.20204	0.0913895	0.0456947	+	0.998955i	$0.485450\pi$
$174$	0	0
$175$	−1.79796	−0.135913
$176$	0	0
$177$	13.1010	0.984733
$178$	0	0
$179$	1.10102	0.0822941	0.0411471	−	0.999153i	$-0.486899\pi$
$179$	1.10102	0.0822941	0.0411471	+	0.999153i	$0.486899\pi$
$180$	0	0
$181$	−7.10102	−0.527815	−0.263907	−	0.964548i	$-0.585011\pi$
$181$	−7.10102	−0.527815	−0.263907	+	0.964548i	$0.585011\pi$
$182$	0	0
$183$	14.4495	1.06814
$184$	0	0
$185$	−24.0000	−1.76452
$186$	0	0
$187$	−12.0000	−0.877527
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.89898	−0.499193	−0.249596	−	0.968350i	$-0.580298\pi$
$191$	−6.89898	−0.499193	−0.249596	+	0.968350i	$0.580298\pi$
$192$	0	0
$193$	−8.10102	−0.583124	−0.291562	−	0.956552i	$-0.594175\pi$
$193$	−8.10102	−0.583124	−0.291562	+	0.956552i	$0.594175\pi$
$194$	0	0
$195$	−43.3485	−3.10425
$196$	0	0
$197$	19.8990	1.41774	0.708872	−	0.705337i	$-0.249204\pi$
$197$	19.8990	1.41774	0.708872	+	0.705337i	$0.249204\pi$
$198$	0	0
$199$	23.3485	1.65513	0.827565	−	0.561371i	$-0.189726\pi$
$199$	23.3485	1.65513	0.827565	+	0.561371i	$0.189726\pi$
$200$	0	0
$201$	−33.7980	−2.38392
$202$	0	0
$203$	−3.55051	−0.249197
$204$	0	0
$205$	23.6969	1.65507
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−8.69694	−0.601580
$210$	0	0
$211$	−8.24745	−0.567778	−0.283889	−	0.958857i	$-0.591625\pi$
$211$	−8.24745	−0.567778	−0.283889	+	0.958857i	$0.591625\pi$
$212$	0	0
$213$	−10.8990	−0.746786
$214$	0	0
$215$	20.6969	1.41152
$216$	0	0
$217$	−4.89898	−0.332564
$218$	0	0
$219$	4.65153	0.314321
$220$	0	0
$221$	28.8990	1.94396
$222$	0	0
$223$	−5.55051	−0.371690	−0.185845	−	0.982579i	$-0.559502\pi$
$223$	−5.55051	−0.371690	−0.185845	+	0.982579i	$0.559502\pi$
$224$	0	0
$225$	12.0000	0.800000
$226$	0	0
$227$	−5.79796	−0.384824	−0.192412	−	0.981314i	$-0.561631\pi$
$227$	−5.79796	−0.384824	−0.192412	+	0.981314i	$0.561631\pi$
$228$	0	0
$229$	−5.79796	−0.383140	−0.191570	−	0.981479i	$-0.561358\pi$
$229$	−5.79796	−0.383140	−0.191570	+	0.981479i	$0.561358\pi$
$230$	0	0
$231$	−2.69694	−0.177446
$232$	0	0
$233$	−22.5959	−1.48031	−0.740154	−	0.672438i	$-0.765247\pi$
$233$	−22.5959	−1.48031	−0.740154	+	0.672438i	$0.765247\pi$
$234$	0	0
$235$	−4.65153	−0.303432
$236$	0	0
$237$	−29.3939	−1.90934
$238$	0	0
$239$	1.79796	0.116300	0.0581501	−	0.998308i	$-0.481480\pi$
$239$	1.79796	0.116300	0.0581501	+	0.998308i	$0.481480\pi$
$240$	0	0
$241$	−19.0000	−1.22390	−0.611949	−	0.790897i	$-0.709614\pi$
$241$	−19.0000	−1.22390	−0.611949	+	0.790897i	$0.709614\pi$
$242$	0	0
$243$	22.0454	1.41421
$244$	0	0
$245$	−20.3939	−1.30292
$246$	0	0
$247$	20.9444	1.33266
$248$	0	0
$249$	−16.8990	−1.07093
$250$	0	0
$251$	−29.3485	−1.85246	−0.926229	−	0.376960i	$-0.876969\pi$
$251$	−29.3485	−1.85246	−0.926229	+	0.376960i	$0.876969\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−36.0000	−2.25441
$256$	0	0
$257$	21.8990	1.36602	0.683010	−	0.730409i	$-0.260670\pi$
$257$	21.8990	1.36602	0.683010	+	0.730409i	$0.260670\pi$
$258$	0	0
$259$	3.59592	0.223439
$260$	0	0
$261$	23.6969	1.46680
$262$	0	0
$263$	8.44949	0.521018	0.260509	−	0.965471i	$-0.416110\pi$
$263$	8.44949	0.521018	0.260509	+	0.965471i	$0.416110\pi$
$264$	0	0
$265$	−26.3939	−1.62136
$266$	0	0
$267$	21.5505	1.31887
$268$	0	0
$269$	−1.79796	−0.109623	−0.0548117	−	0.998497i	$-0.517456\pi$
$269$	−1.79796	−0.109623	−0.0548117	+	0.998497i	$0.517456\pi$
$270$	0	0
$271$	−18.4495	−1.12073	−0.560363	−	0.828247i	$-0.689338\pi$
$271$	−18.4495	−1.12073	−0.560363	+	0.828247i	$0.689338\pi$
$272$	0	0
$273$	6.49490	0.393089
$274$	0	0
$275$	−9.79796	−0.590839
$276$	0	0
$277$	13.7980	0.829039	0.414520	−	0.910040i	$-0.363950\pi$
$277$	13.7980	0.829039	0.414520	+	0.910040i	$0.363950\pi$
$278$	0	0
$279$	32.6969	1.95751
$280$	0	0
$281$	2.20204	0.131363	0.0656814	−	0.997841i	$-0.479078\pi$
$281$	2.20204	0.131363	0.0656814	+	0.997841i	$0.479078\pi$
$282$	0	0
$283$	16.6969	0.992530	0.496265	−	0.868171i	$-0.334704\pi$
$283$	16.6969	0.992530	0.496265	+	0.868171i	$0.334704\pi$
$284$	0	0
$285$	−26.0908	−1.54549
$286$	0	0
$287$	−3.55051	−0.209580
$288$	0	0
$289$	7.00000	0.411765
$290$	0	0
$291$	−4.65153	−0.272678
$292$	0	0
$293$	−11.2020	−0.654430	−0.327215	−	0.944950i	$-0.606110\pi$
$293$	−11.2020	−0.654430	−0.327215	+	0.944950i	$0.606110\pi$
$294$	0	0
$295$	−16.0454	−0.934200
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−3.10102	−0.178740
$302$	0	0
$303$	−22.0454	−1.26648
$304$	0	0
$305$	−17.6969	−1.01332
$306$	0	0
$307$	1.14643	0.0654301	0.0327151	−	0.999465i	$-0.489585\pi$
$307$	1.14643	0.0654301	0.0327151	+	0.999465i	$0.489585\pi$
$308$	0	0
$309$	16.8990	0.961349
$310$	0	0
$311$	2.24745	0.127441	0.0637206	−	0.997968i	$-0.479703\pi$
$311$	2.24745	0.127441	0.0637206	+	0.997968i	$0.479703\pi$
$312$	0	0
$313$	−5.00000	−0.282617	−0.141308	−	0.989966i	$-0.545131\pi$
$313$	−5.00000	−0.282617	−0.141308	+	0.989966i	$0.545131\pi$
$314$	0	0
$315$	−4.04541	−0.227933
$316$	0	0
$317$	−18.1010	−1.01665	−0.508327	−	0.861164i	$-0.669736\pi$
$317$	−18.1010	−1.01665	−0.508327	+	0.861164i	$0.669736\pi$
$318$	0	0
$319$	−19.3485	−1.08331
$320$	0	0
$321$	40.2929	2.24893
$322$	0	0
$323$	17.3939	0.967821
$324$	0	0
$325$	23.5959	1.30887
$326$	0	0
$327$	−1.95459	−0.108089
$328$	0	0
$329$	0.696938	0.0384235
$330$	0	0
$331$	24.6969	1.35747	0.678733	−	0.734385i	$-0.262529\pi$
$331$	24.6969	1.35747	0.678733	+	0.734385i	$0.262529\pi$
$332$	0	0
$333$	−24.0000	−1.31519
$334$	0	0
$335$	41.3939	2.26159
$336$	0	0
$337$	−29.8990	−1.62870	−0.814351	−	0.580373i	$-0.802907\pi$
$337$	−29.8990	−1.62870	−0.814351	+	0.580373i	$0.802907\pi$
$338$	0	0
$339$	−12.2474	−0.665190
$340$	0	0
$341$	−26.6969	−1.44572
$342$	0	0
$343$	6.20204	0.334879
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.34847	0.394486	0.197243	−	0.980355i	$-0.436801\pi$
$347$	7.34847	0.394486	0.197243	+	0.980355i	$0.436801\pi$
$348$	0	0
$349$	10.6969	0.572594	0.286297	−	0.958141i	$-0.407576\pi$
$349$	10.6969	0.572594	0.286297	+	0.958141i	$0.407576\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.6969	0.835464	0.417732	−	0.908570i	$-0.362825\pi$
$353$	15.6969	0.835464	0.417732	+	0.908570i	$0.362825\pi$
$354$	0	0
$355$	13.3485	0.708463
$356$	0	0
$357$	5.39388	0.285474
$358$	0	0
$359$	8.44949	0.445947	0.222974	−	0.974825i	$-0.428424\pi$
$359$	8.44949	0.445947	0.222974	+	0.974825i	$0.428424\pi$
$360$	0	0
$361$	−6.39388	−0.336520
$362$	0	0
$363$	12.2474	0.642824
$364$	0	0
$365$	−5.69694	−0.298191
$366$	0	0
$367$	−15.3485	−0.801184	−0.400592	−	0.916257i	$-0.631195\pi$
$367$	−15.3485	−0.801184	−0.400592	+	0.916257i	$0.631195\pi$
$368$	0	0
$369$	23.6969	1.23361
$370$	0	0
$371$	3.95459	0.205312
$372$	0	0
$373$	28.4949	1.47541	0.737705	−	0.675123i	$-0.235910\pi$
$373$	28.4949	1.47541	0.737705	+	0.675123i	$0.235910\pi$
$374$	0	0
$375$	7.34847	0.379473
$376$	0	0
$377$	46.5959	2.39981
$378$	0	0
$379$	−28.4495	−1.46135	−0.730676	−	0.682724i	$-0.760795\pi$
$379$	−28.4495	−1.46135	−0.730676	+	0.682724i	$0.760795\pi$
$380$	0	0
$381$	16.8990	0.865761
$382$	0	0
$383$	−13.1010	−0.669431	−0.334715	−	0.942319i	$-0.608640\pi$
$383$	−13.1010	−0.669431	−0.334715	+	0.942319i	$0.608640\pi$
$384$	0	0
$385$	3.30306	0.168340
$386$	0	0
$387$	20.6969	1.05208
$388$	0	0
$389$	30.2929	1.53591	0.767954	−	0.640505i	$-0.221275\pi$
$389$	30.2929	1.53591	0.767954	+	0.640505i	$0.221275\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−44.6969	−2.25466
$394$	0	0
$395$	36.0000	1.81136
$396$	0	0
$397$	−8.79796	−0.441557	−0.220778	−	0.975324i	$-0.570860\pi$
$397$	−8.79796	−0.441557	−0.220778	+	0.975324i	$0.570860\pi$
$398$	0	0
$399$	3.90918	0.195704
$400$	0	0
$401$	−15.4949	−0.773778	−0.386889	−	0.922126i	$-0.626450\pi$
$401$	−15.4949	−0.773778	−0.386889	+	0.922126i	$0.626450\pi$
$402$	0	0
$403$	64.2929	3.20266
$404$	0	0
$405$	−27.0000	−1.34164
$406$	0	0
$407$	19.5959	0.971334
$408$	0	0
$409$	11.1010	0.548910	0.274455	−	0.961600i	$-0.411503\pi$
$409$	11.1010	0.548910	0.274455	+	0.961600i	$0.411503\pi$
$410$	0	0
$411$	−29.1464	−1.43769
$412$	0	0
$413$	2.40408	0.118297
$414$	0	0
$415$	20.6969	1.01597
$416$	0	0
$417$	31.1010	1.52302
$418$	0	0
$419$	−33.3939	−1.63140	−0.815699	−	0.578477i	$-0.803647\pi$
$419$	−33.3939	−1.63140	−0.815699	+	0.578477i	$0.803647\pi$
$420$	0	0
$421$	33.5959	1.63736	0.818682	−	0.574247i	$-0.194705\pi$
$421$	33.5959	1.63736	0.818682	+	0.574247i	$0.194705\pi$
$422$	0	0
$423$	−4.65153	−0.226165
$424$	0	0
$425$	19.5959	0.950542
$426$	0	0
$427$	2.65153	0.128317
$428$	0	0
$429$	35.3939	1.70883
$430$	0	0
$431$	−19.3485	−0.931983	−0.465991	−	0.884789i	$-0.654302\pi$
$431$	−19.3485	−0.931983	−0.465991	+	0.884789i	$0.654302\pi$
$432$	0	0
$433$	−2.10102	−0.100969	−0.0504843	−	0.998725i	$-0.516076\pi$
$433$	−2.10102	−0.100969	−0.0504843	+	0.998725i	$0.516076\pi$
$434$	0	0
$435$	−58.0454	−2.78306
$436$	0	0
$437$	0	0
$438$	0	0
$439$	2.89898	0.138361	0.0691804	−	0.997604i	$-0.477962\pi$
$439$	2.89898	0.138361	0.0691804	+	0.997604i	$0.477962\pi$
$440$	0	0
$441$	−20.3939	−0.971137
$442$	0	0
$443$	22.8990	1.08796	0.543982	−	0.839097i	$-0.316916\pi$
$443$	22.8990	1.08796	0.543982	+	0.839097i	$0.316916\pi$
$444$	0	0
$445$	−26.3939	−1.25119
$446$	0	0
$447$	−9.55051	−0.451724
$448$	0	0
$449$	1.89898	0.0896184	0.0448092	−	0.998996i	$-0.485732\pi$
$449$	1.89898	0.0896184	0.0448092	+	0.998996i	$0.485732\pi$
$450$	0	0
$451$	−19.3485	−0.911084
$452$	0	0
$453$	39.7980	1.86987
$454$	0	0
$455$	−7.95459	−0.372917
$456$	0	0
$457$	−8.10102	−0.378950	−0.189475	−	0.981886i	$-0.560679\pi$
$457$	−8.10102	−0.378950	−0.189475	+	0.981886i	$0.560679\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	42.3939	1.97448	0.987240	−	0.159240i	$-0.0509045\pi$
$461$	42.3939	1.97448	0.987240	+	0.159240i	$0.0509045\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	0	0
$465$	−80.0908	−3.71412
$466$	0	0
$467$	−21.7980	−1.00869	−0.504345	−	0.863502i	$-0.668266\pi$
$467$	−21.7980	−1.00869	−0.504345	+	0.863502i	$0.668266\pi$
$468$	0	0
$469$	−6.20204	−0.286384
$470$	0	0
$471$	−7.34847	−0.338600
$472$	0	0
$473$	−16.8990	−0.777016
$474$	0	0
$475$	14.2020	0.651634
$476$	0	0
$477$	−26.3939	−1.20849
$478$	0	0
$479$	−21.1010	−0.964130	−0.482065	−	0.876135i	$-0.660113\pi$
$479$	−21.1010	−0.964130	−0.482065	+	0.876135i	$0.660113\pi$
$480$	0	0
$481$	−47.1918	−2.15176
$482$	0	0
$483$	0	0
$484$	0	0
$485$	5.69694	0.258685
$486$	0	0
$487$	22.2474	1.00813	0.504064	−	0.863666i	$-0.331838\pi$
$487$	22.2474	1.00813	0.504064	+	0.863666i	$0.331838\pi$
$488$	0	0
$489$	18.4949	0.836368
$490$	0	0
$491$	5.10102	0.230206	0.115103	−	0.993354i	$-0.463280\pi$
$491$	5.10102	0.230206	0.115103	+	0.993354i	$0.463280\pi$
$492$	0	0
$493$	38.6969	1.74282
$494$	0	0
$495$	−22.0454	−0.990867
$496$	0	0
$497$	−2.00000	−0.0897123
$498$	0	0
$499$	35.5959	1.59349	0.796746	−	0.604314i	$-0.206553\pi$
$499$	35.5959	1.59349	0.796746	+	0.604314i	$0.206553\pi$
$500$	0	0
$501$	12.4949	0.558231
$502$	0	0
$503$	13.1464	0.586170	0.293085	−	0.956086i	$-0.405318\pi$
$503$	13.1464	0.586170	0.293085	+	0.956086i	$0.405318\pi$
$504$	0	0
$505$	27.0000	1.20148
$506$	0	0
$507$	−53.3939	−2.37131
$508$	0	0
$509$	22.5959	1.00155	0.500773	−	0.865579i	$-0.333049\pi$
$509$	22.5959	1.00155	0.500773	+	0.865579i	$0.333049\pi$
$510$	0	0
$511$	0.853572	0.0377598
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−20.6969	−0.912016
$516$	0	0
$517$	3.79796	0.167034
$518$	0	0
$519$	−2.94439	−0.129244
$520$	0	0
$521$	27.1010	1.18732	0.593659	−	0.804717i	$-0.297683\pi$
$521$	27.1010	1.18732	0.593659	+	0.804717i	$0.297683\pi$
$522$	0	0
$523$	−31.5959	−1.38159	−0.690797	−	0.723049i	$-0.742740\pi$
$523$	−31.5959	−1.38159	−0.690797	+	0.723049i	$0.742740\pi$
$524$	0	0
$525$	4.40408	0.192210
$526$	0	0
$527$	53.3939	2.32587
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−16.0454	−0.696311
$532$	0	0
$533$	46.5959	2.01829
$534$	0	0
$535$	−49.3485	−2.13352
$536$	0	0
$537$	−2.69694	−0.116381
$538$	0	0
$539$	16.6515	0.717232
$540$	0	0
$541$	40.3939	1.73667	0.868334	−	0.495980i	$-0.165191\pi$
$541$	40.3939	1.73667	0.868334	+	0.495980i	$0.165191\pi$
$542$	0	0
$543$	17.3939	0.746443
$544$	0	0
$545$	2.39388	0.102542
$546$	0	0
$547$	7.30306	0.312256	0.156128	−	0.987737i	$-0.450099\pi$
$547$	7.30306	0.312256	0.156128	+	0.987737i	$0.450099\pi$
$548$	0	0
$549$	−17.6969	−0.755287
$550$	0	0
$551$	28.0454	1.19477
$552$	0	0
$553$	−5.39388	−0.229371
$554$	0	0
$555$	58.7878	2.49540
$556$	0	0
$557$	25.6969	1.08881	0.544407	−	0.838821i	$-0.316755\pi$
$557$	25.6969	1.08881	0.544407	+	0.838821i	$0.316755\pi$
$558$	0	0
$559$	40.6969	1.72130
$560$	0	0
$561$	29.3939	1.24101
$562$	0	0
$563$	42.0908	1.77392	0.886958	−	0.461850i	$-0.152814\pi$
$563$	42.0908	1.77392	0.886958	+	0.461850i	$0.152814\pi$
$564$	0	0
$565$	15.0000	0.631055
$566$	0	0
$567$	4.04541	0.169891
$568$	0	0
$569$	−21.4949	−0.901113	−0.450556	−	0.892748i	$-0.648774\pi$
$569$	−21.4949	−0.901113	−0.450556	+	0.892748i	$0.648774\pi$
$570$	0	0
$571$	37.5505	1.57144	0.785720	−	0.618582i	$-0.212293\pi$
$571$	37.5505	1.57144	0.785720	+	0.618582i	$0.212293\pi$
$572$	0	0
$573$	16.8990	0.705965
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−25.5959	−1.06557	−0.532786	−	0.846250i	$-0.678855\pi$
$577$	−25.5959	−1.06557	−0.532786	+	0.846250i	$0.678855\pi$
$578$	0	0
$579$	19.8434	0.824662
$580$	0	0
$581$	−3.10102	−0.128652
$582$	0	0
$583$	21.5505	0.892531
$584$	0	0
$585$	53.0908	2.19504
$586$	0	0
$587$	7.59592	0.313517	0.156759	−	0.987637i	$-0.449896\pi$
$587$	7.59592	0.313517	0.156759	+	0.987637i	$0.449896\pi$
$588$	0	0
$589$	38.6969	1.59448
$590$	0	0
$591$	−48.7423	−2.00499
$592$	0	0
$593$	−36.8990	−1.51526	−0.757630	−	0.652685i	$-0.773643\pi$
$593$	−36.8990	−1.51526	−0.757630	+	0.652685i	$0.773643\pi$
$594$	0	0
$595$	−6.60612	−0.270825
$596$	0	0
$597$	−57.1918	−2.34071
$598$	0	0
$599$	13.3485	0.545404	0.272702	−	0.962099i	$-0.412083\pi$
$599$	13.3485	0.545404	0.272702	+	0.962099i	$0.412083\pi$
$600$	0	0
$601$	4.79796	0.195713	0.0978564	−	0.995201i	$-0.468801\pi$
$601$	4.79796	0.195713	0.0978564	+	0.995201i	$0.468801\pi$
$602$	0	0
$603$	41.3939	1.68569
$604$	0	0
$605$	−15.0000	−0.609837
$606$	0	0
$607$	16.4495	0.667664	0.333832	−	0.942633i	$-0.391658\pi$
$607$	16.4495	0.667664	0.333832	+	0.942633i	$0.391658\pi$
$608$	0	0
$609$	8.69694	0.352418
$610$	0	0
$611$	−9.14643	−0.370025
$612$	0	0
$613$	−17.8990	−0.722933	−0.361466	−	0.932385i	$-0.617724\pi$
$613$	−17.8990	−0.722933	−0.361466	+	0.932385i	$0.617724\pi$
$614$	0	0
$615$	−58.0454	−2.34062
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	0	0
$619$	25.7980	1.03691	0.518454	−	0.855106i	$-0.326508\pi$
$619$	25.7980	1.03691	0.518454	+	0.855106i	$0.326508\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.95459	0.158437
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	21.3031	0.850762
$628$	0	0
$629$	−39.1918	−1.56268
$630$	0	0
$631$	−4.69694	−0.186982	−0.0934911	−	0.995620i	$-0.529803\pi$
$631$	−4.69694	−0.186982	−0.0934911	+	0.995620i	$0.529803\pi$
$632$	0	0
$633$	20.2020	0.802959
$634$	0	0
$635$	−20.6969	−0.821333
$636$	0	0
$637$	−40.1010	−1.58886
$638$	0	0
$639$	13.3485	0.528057
$640$	0	0
$641$	15.8990	0.627972	0.313986	−	0.949428i	$-0.398336\pi$
$641$	15.8990	0.627972	0.313986	+	0.949428i	$0.398336\pi$
$642$	0	0
$643$	−15.1464	−0.597317	−0.298658	−	0.954360i	$-0.596539\pi$
$643$	−15.1464	−0.597317	−0.298658	+	0.954360i	$0.596539\pi$
$644$	0	0
$645$	−50.6969	−1.99619
$646$	0	0
$647$	4.65153	0.182871	0.0914353	−	0.995811i	$-0.470855\pi$
$647$	4.65153	0.182871	0.0914353	+	0.995811i	$0.470855\pi$
$648$	0	0
$649$	13.1010	0.514260
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	29.6969	1.16213	0.581066	−	0.813857i	$-0.302636\pi$
$653$	29.6969	1.16213	0.581066	+	0.813857i	$0.302636\pi$
$654$	0	0
$655$	54.7423	2.13896
$656$	0	0
$657$	−5.69694	−0.222259
$658$	0	0
$659$	25.7980	1.00495	0.502473	−	0.864593i	$-0.332424\pi$
$659$	25.7980	1.00495	0.502473	+	0.864593i	$0.332424\pi$
$660$	0	0
$661$	29.6969	1.15508	0.577539	−	0.816363i	$-0.304013\pi$
$661$	29.6969	1.15508	0.577539	+	0.816363i	$0.304013\pi$
$662$	0	0
$663$	−70.7878	−2.74917
$664$	0	0
$665$	−4.78775	−0.185661
$666$	0	0
$667$	0	0
$668$	0	0
$669$	13.5959	0.525649
$670$	0	0
$671$	14.4495	0.557816
$672$	0	0
$673$	−17.7980	−0.686061	−0.343030	−	0.939324i	$-0.611453\pi$
$673$	−17.7980	−0.686061	−0.343030	+	0.939324i	$0.611453\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−11.8990	−0.457315	−0.228657	−	0.973507i	$-0.573434\pi$
$677$	−11.8990	−0.457315	−0.228657	+	0.973507i	$0.573434\pi$
$678$	0	0
$679$	−0.853572	−0.0327571
$680$	0	0
$681$	14.2020	0.544223
$682$	0	0
$683$	−25.3939	−0.971670	−0.485835	−	0.874051i	$-0.661484\pi$
$683$	−25.3939	−0.971670	−0.485835	+	0.874051i	$0.661484\pi$
$684$	0	0
$685$	35.6969	1.36391
$686$	0	0
$687$	14.2020	0.541842
$688$	0	0
$689$	−51.8990	−1.97719
$690$	0	0
$691$	24.2929	0.924144	0.462072	−	0.886842i	$-0.347106\pi$
$691$	24.2929	0.924144	0.462072	+	0.886842i	$0.347106\pi$
$692$	0	0
$693$	3.30306	0.125473
$694$	0	0
$695$	−38.0908	−1.44487
$696$	0	0
$697$	38.6969	1.46575
$698$	0	0
$699$	55.3485	2.09347
$700$	0	0
$701$	−21.7980	−0.823298	−0.411649	−	0.911343i	$-0.635047\pi$
$701$	−21.7980	−0.823298	−0.411649	+	0.911343i	$0.635047\pi$
$702$	0	0
$703$	−28.4041	−1.07128
$704$	0	0
$705$	11.3939	0.429118
$706$	0	0
$707$	−4.04541	−0.152143
$708$	0	0
$709$	5.00000	0.187779	0.0938895	−	0.995583i	$-0.470070\pi$
$709$	5.00000	0.187779	0.0938895	+	0.995583i	$0.470070\pi$
$710$	0	0
$711$	36.0000	1.35011
$712$	0	0
$713$	0	0
$714$	0	0
$715$	−43.3485	−1.62114
$716$	0	0
$717$	−4.40408	−0.164473
$718$	0	0
$719$	−34.4949	−1.28644	−0.643221	−	0.765680i	$-0.722402\pi$
$719$	−34.4949	−1.28644	−0.643221	+	0.765680i	$0.722402\pi$
$720$	0	0
$721$	3.10102	0.115488
$722$	0	0
$723$	46.5403	1.73085
$724$	0	0
$725$	31.5959	1.17344
$726$	0	0
$727$	−20.4495	−0.758430	−0.379215	−	0.925309i	$-0.623806\pi$
$727$	−20.4495	−0.758430	−0.379215	+	0.925309i	$0.623806\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	33.7980	1.25006
$732$	0	0
$733$	−14.7980	−0.546575	−0.273288	−	0.961932i	$-0.588111\pi$
$733$	−14.7980	−0.546575	−0.273288	+	0.961932i	$0.588111\pi$
$734$	0	0
$735$	49.9546	1.84260
$736$	0	0
$737$	−33.7980	−1.24496
$738$	0	0
$739$	−48.0454	−1.76738	−0.883689	−	0.468074i	$-0.844948\pi$
$739$	−48.0454	−1.76738	−0.883689	+	0.468074i	$0.844948\pi$
$740$	0	0
$741$	−51.3031	−1.88467
$742$	0	0
$743$	14.4495	0.530100	0.265050	−	0.964235i	$-0.414611\pi$
$743$	14.4495	0.530100	0.265050	+	0.964235i	$0.414611\pi$
$744$	0	0
$745$	11.6969	0.428543
$746$	0	0
$747$	20.6969	0.757261
$748$	0	0
$749$	7.39388	0.270166
$750$	0	0
$751$	−28.7423	−1.04882	−0.524412	−	0.851465i	$-0.675715\pi$
$751$	−28.7423	−1.04882	−0.524412	+	0.851465i	$0.675715\pi$
$752$	0	0
$753$	71.8888	2.61977
$754$	0	0
$755$	−48.7423	−1.77392
$756$	0	0
$757$	1.89898	0.0690196	0.0345098	−	0.999404i	$-0.489013\pi$
$757$	1.89898	0.0690196	0.0345098	+	0.999404i	$0.489013\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−12.3939	−0.449278	−0.224639	−	0.974442i	$-0.572120\pi$
$761$	−12.3939	−0.449278	−0.224639	+	0.974442i	$0.572120\pi$
$762$	0	0
$763$	−0.358674	−0.0129849
$764$	0	0
$765$	44.0908	1.59411
$766$	0	0
$767$	−31.5505	−1.13922
$768$	0	0
$769$	−16.4949	−0.594821	−0.297411	−	0.954750i	$-0.596123\pi$
$769$	−16.4949	−0.594821	−0.297411	+	0.954750i	$0.596123\pi$
$770$	0	0
$771$	−53.6413	−1.93185
$772$	0	0
$773$	−10.0000	−0.359675	−0.179838	−	0.983696i	$-0.557557\pi$
$773$	−10.0000	−0.359675	−0.179838	+	0.983696i	$0.557557\pi$
$774$	0	0
$775$	43.5959	1.56601
$776$	0	0
$777$	−8.80816	−0.315991
$778$	0	0
$779$	28.0454	1.00483
$780$	0	0
$781$	−10.8990	−0.389996
$782$	0	0
$783$	0	0
$784$	0	0
$785$	9.00000	0.321224
$786$	0	0
$787$	12.6515	0.450978	0.225489	−	0.974246i	$-0.427602\pi$
$787$	12.6515	0.450978	0.225489	+	0.974246i	$0.427602\pi$
$788$	0	0
$789$	−20.6969	−0.736831
$790$	0	0
$791$	−2.24745	−0.0799101
$792$	0	0
$793$	−34.7980	−1.23571
$794$	0	0
$795$	64.6515	2.29295
$796$	0	0
$797$	7.59592	0.269061	0.134531	−	0.990909i	$-0.457047\pi$
$797$	7.59592	0.269061	0.134531	+	0.990909i	$0.457047\pi$
$798$	0	0
$799$	−7.59592	−0.268724
$800$	0	0
$801$	−26.3939	−0.932582
$802$	0	0
$803$	4.65153	0.164149
$804$	0	0
$805$	0	0
$806$	0	0
$807$	4.40408	0.155031
$808$	0	0
$809$	−5.59592	−0.196742	−0.0983710	−	0.995150i	$-0.531363\pi$
$809$	−5.59592	−0.196742	−0.0983710	+	0.995150i	$0.531363\pi$
$810$	0	0
$811$	−18.9444	−0.665227	−0.332614	−	0.943063i	$-0.607931\pi$
$811$	−18.9444	−0.665227	−0.332614	+	0.943063i	$0.607931\pi$
$812$	0	0
$813$	45.1918	1.58495
$814$	0	0
$815$	−22.6515	−0.793449
$816$	0	0
$817$	24.4949	0.856968
$818$	0	0
$819$	−7.95459	−0.277956
$820$	0	0
$821$	−3.49490	−0.121973	−0.0609864	−	0.998139i	$-0.519425\pi$
$821$	−3.49490	−0.121973	−0.0609864	+	0.998139i	$0.519425\pi$
$822$	0	0
$823$	46.0454	1.60504	0.802521	−	0.596624i	$-0.203492\pi$
$823$	46.0454	1.60504	0.802521	+	0.596624i	$0.203492\pi$
$824$	0	0
$825$	24.0000	0.835573
$826$	0	0
$827$	−0.247449	−0.00860463	−0.00430232	−	0.999991i	$-0.501369\pi$
$827$	−0.247449	−0.00860463	−0.00430232	+	0.999991i	$0.501369\pi$
$828$	0	0
$829$	33.0000	1.14614	0.573069	−	0.819507i	$-0.305753\pi$
$829$	33.0000	1.14614	0.573069	+	0.819507i	$0.305753\pi$
$830$	0	0
$831$	−33.7980	−1.17244
$832$	0	0
$833$	−33.3031	−1.15388
$834$	0	0
$835$	−15.3031	−0.529584
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−20.9444	−0.723081	−0.361540	−	0.932356i	$-0.617749\pi$
$839$	−20.9444	−0.723081	−0.361540	+	0.932356i	$0.617749\pi$
$840$	0	0
$841$	33.3939	1.15151
$842$	0	0
$843$	−5.39388	−0.185775
$844$	0	0
$845$	65.3939	2.24962
$846$	0	0
$847$	2.24745	0.0772233
$848$	0	0
$849$	−40.8990	−1.40365
$850$	0	0
$851$	0	0
$852$	0	0
$853$	32.8990	1.12644	0.563220	−	0.826307i	$-0.309562\pi$
$853$	32.8990	1.12644	0.563220	+	0.826307i	$0.309562\pi$
$854$	0	0
$855$	31.9546	1.09282
$856$	0	0
$857$	−6.39388	−0.218411	−0.109205	−	0.994019i	$-0.534831\pi$
$857$	−6.39388	−0.218411	−0.109205	+	0.994019i	$0.534831\pi$
$858$	0	0
$859$	−11.3485	−0.387205	−0.193602	−	0.981080i	$-0.562017\pi$
$859$	−11.3485	−0.387205	−0.193602	+	0.981080i	$0.562017\pi$
$860$	0	0
$861$	8.69694	0.296391
$862$	0	0
$863$	−3.75255	−0.127738	−0.0638692	−	0.997958i	$-0.520344\pi$
$863$	−3.75255	−0.127738	−0.0638692	+	0.997958i	$0.520344\pi$
$864$	0	0
$865$	3.60612	0.122612
$866$	0	0
$867$	−17.1464	−0.582323
$868$	0	0
$869$	−29.3939	−0.997119
$870$	0	0
$871$	81.3939	2.75793
$872$	0	0
$873$	5.69694	0.192812
$874$	0	0
$875$	1.34847	0.0455866
$876$	0	0
$877$	16.8990	0.570638	0.285319	−	0.958433i	$-0.407900\pi$
$877$	16.8990	0.570638	0.285319	+	0.958433i	$0.407900\pi$
$878$	0	0
$879$	27.4393	0.925504
$880$	0	0
$881$	47.0908	1.58653	0.793265	−	0.608877i	$-0.208380\pi$
$881$	47.0908	1.58653	0.793265	+	0.608877i	$0.208380\pi$
$882$	0	0
$883$	16.0454	0.539971	0.269985	−	0.962864i	$-0.412981\pi$
$883$	16.0454	0.539971	0.269985	+	0.962864i	$0.412981\pi$
$884$	0	0
$885$	39.3031	1.32116
$886$	0	0
$887$	14.2020	0.476858	0.238429	−	0.971160i	$-0.423368\pi$
$887$	14.2020	0.476858	0.238429	+	0.971160i	$0.423368\pi$
$888$	0	0
$889$	3.10102	0.104005
$890$	0	0
$891$	22.0454	0.738549
$892$	0	0
$893$	−5.50510	−0.184221
$894$	0	0
$895$	3.30306	0.110409
$896$	0	0
$897$	0	0
$898$	0	0
$899$	86.0908	2.87129
$900$	0	0
$901$	−43.1010	−1.43590
$902$	0	0
$903$	7.59592	0.252776
$904$	0	0
$905$	−21.3031	−0.708138
$906$	0	0
$907$	−10.9444	−0.363402	−0.181701	−	0.983354i	$-0.558160\pi$
$907$	−10.9444	−0.363402	−0.181701	+	0.983354i	$0.558160\pi$
$908$	0	0
$909$	27.0000	0.895533
$910$	0	0
$911$	−2.89898	−0.0960475	−0.0480237	−	0.998846i	$-0.515292\pi$
$911$	−2.89898	−0.0960475	−0.0480237	+	0.998846i	$0.515292\pi$
$912$	0	0
$913$	−16.8990	−0.559275
$914$	0	0
$915$	43.3485	1.43306
$916$	0	0
$917$	−8.20204	−0.270855
$918$	0	0
$919$	−17.3485	−0.572273	−0.286137	−	0.958189i	$-0.592371\pi$
$919$	−17.3485	−0.572273	−0.286137	+	0.958189i	$0.592371\pi$
$920$	0	0
$921$	−2.80816	−0.0925322
$922$	0	0
$923$	26.2474	0.863945
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	−20.6969	−0.679777
$928$	0	0
$929$	−32.4949	−1.06612	−0.533062	−	0.846076i	$-0.678959\pi$
$929$	−32.4949	−1.06612	−0.533062	+	0.846076i	$0.678959\pi$
$930$	0	0
$931$	−24.1362	−0.791033
$932$	0	0
$933$	−5.50510	−0.180229
$934$	0	0
$935$	−36.0000	−1.17733
$936$	0	0
$937$	−36.8990	−1.20544	−0.602719	−	0.797954i	$-0.705916\pi$
$937$	−36.8990	−1.20544	−0.602719	+	0.797954i	$0.705916\pi$
$938$	0	0
$939$	12.2474	0.399680
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−55.1918	−1.79349	−0.896747	−	0.442544i	$-0.854076\pi$
$947$	−55.1918	−1.79349	−0.896747	+	0.442544i	$0.854076\pi$
$948$	0	0
$949$	−11.2020	−0.363634
$950$	0	0
$951$	44.3383	1.43777
$952$	0	0
$953$	−5.69694	−0.184542	−0.0922710	−	0.995734i	$-0.529413\pi$
$953$	−5.69694	−0.184542	−0.0922710	+	0.995734i	$0.529413\pi$
$954$	0	0
$955$	−20.6969	−0.669737
$956$	0	0
$957$	47.3939	1.53203
$958$	0	0
$959$	−5.34847	−0.172711
$960$	0	0
$961$	87.7878	2.83186
$962$	0	0
$963$	−49.3485	−1.59023
$964$	0	0
$965$	−24.3031	−0.782343
$966$	0	0
$967$	−26.6515	−0.857055	−0.428528	−	0.903529i	$-0.640968\pi$
$967$	−26.6515	−0.857055	−0.428528	+	0.903529i	$0.640968\pi$
$968$	0	0
$969$	−42.6061	−1.36871
$970$	0	0
$971$	56.7423	1.82095	0.910474	−	0.413566i	$-0.135717\pi$
$971$	56.7423	1.82095	0.910474	+	0.413566i	$0.135717\pi$
$972$	0	0
$973$	5.70714	0.182963
$974$	0	0
$975$	−57.7980	−1.85102
$976$	0	0
$977$	17.6969	0.566175	0.283088	−	0.959094i	$-0.408641\pi$
$977$	17.6969	0.566175	0.283088	+	0.959094i	$0.408641\pi$
$978$	0	0
$979$	21.5505	0.688757
$980$	0	0
$981$	2.39388	0.0764306
$982$	0	0
$983$	−26.8990	−0.857944	−0.428972	−	0.903318i	$-0.641124\pi$
$983$	−26.8990	−0.857944	−0.428972	+	0.903318i	$0.641124\pi$
$984$	0	0
$985$	59.6969	1.90210
$986$	0	0
$987$	−1.70714	−0.0543390
$988$	0	0
$989$	0	0
$990$	0	0
$991$	48.9898	1.55621	0.778106	−	0.628133i	$-0.216181\pi$
$991$	48.9898	1.55621	0.778106	+	0.628133i	$0.216181\pi$
$992$	0	0
$993$	−60.4949	−1.91975
$994$	0	0
$995$	70.0454	2.22059
$996$	0	0
$997$	−34.3939	−1.08927	−0.544633	−	0.838675i	$-0.683331\pi$
$997$	−34.3939	−1.08927	−0.544633	+	0.838675i	$0.683331\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8464.2.a.bc.1.1		2
4.3	odd	2		4232.2.a.q.1.2	yes	2
23.22	odd	2		8464.2.a.y.1.1		2
92.91	even	2		4232.2.a.p.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4232.2.a.p.1.2	✓	2	92.91	even	2
4232.2.a.q.1.2	yes	2	4.3	odd	2
8464.2.a.y.1.1		2	23.22	odd	2
8464.2.a.bc.1.1		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:	\( N \)	\(=\)	\( 8464 = 2^{4} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8464.a (trivial)

\(f(q)\)	\(=\)	\(q-2.44949 q^{3} +3.00000 q^{5} -0.449490 q^{7} +3.00000 q^{9} -2.44949 q^{11} +5.89898 q^{13} -7.34847 q^{15} +4.89898 q^{17} +3.55051 q^{19} +1.10102 q^{21} +4.00000 q^{25} +7.89898 q^{29} +10.8990 q^{31} +6.00000 q^{33} -1.34847 q^{35} -8.00000 q^{37} -14.4495 q^{39} +7.89898 q^{41} +6.89898 q^{43} +9.00000 q^{45} -1.55051 q^{47} -6.79796 q^{49} -12.0000 q^{51} -8.79796 q^{53} -7.34847 q^{55} -8.69694 q^{57} -5.34847 q^{59} -5.89898 q^{61} -1.34847 q^{63} +17.6969 q^{65} +13.7980 q^{67} +4.44949 q^{71} -1.89898 q^{73} -9.79796 q^{75} +1.10102 q^{77} +12.0000 q^{79} -9.00000 q^{81} +6.89898 q^{83} +14.6969 q^{85} -19.3485 q^{87} -8.79796 q^{89} -2.65153 q^{91} -26.6969 q^{93} +10.6515 q^{95} +1.89898 q^{97} -7.34847 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 4 q^{7} + 6 q^{9} + 2 q^{13} + 12 q^{19} + 12 q^{21} + 8 q^{25} + 6 q^{29} + 12 q^{31} + 12 q^{33} + 12 q^{35} - 16 q^{37} - 24 q^{39} + 6 q^{41} + 4 q^{43} + 18 q^{45} - 8 q^{47} + 6 q^{49}+ \cdots - 6 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 4 * q^7 + 6 * q^9 + 2 * q^13 + 12 * q^19 + 12 * q^21 + 8 * q^25 + 6 * q^29 + 12 * q^31 + 12 * q^33 + 12 * q^35 - 16 * q^37 - 24 * q^39 + 6 * q^41 + 4 * q^43 + 18 * q^45 - 8 * q^47 + 6 * q^49 - 24 * q^51 + 2 * q^53 + 12 * q^57 + 4 * q^59 - 2 * q^61 + 12 * q^63 + 6 * q^65 + 8 * q^67 + 4 * q^71 + 6 * q^73 + 12 * q^77 + 24 * q^79 - 18 * q^81 + 4 * q^83 - 24 * q^87 + 2 * q^89 - 20 * q^91 - 24 * q^93 + 36 * q^95 - 6 * q^97

Introduction

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