LMFDB - Embedded newform 8112.2.a.by.1.1

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			1.1
Root			$1.80194$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -2.80194 q^{5} +4.80194 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -2.80194 q^{5} +4.80194 q^{7} +1.00000 q^{9} -1.46681 q^{11} +2.80194 q^{15} -2.44504 q^{17} -2.54288 q^{19} -4.80194 q^{21} +3.51573 q^{23} +2.85086 q^{25} -1.00000 q^{27} +1.85086 q^{29} +7.63102 q^{31} +1.46681 q^{33} -13.4547 q^{35} -4.55496 q^{37} +1.24698 q^{41} -2.38404 q^{43} -2.80194 q^{45} -12.8170 q^{47} +16.0586 q^{49} +2.44504 q^{51} -8.85086 q^{53} +4.10992 q^{55} +2.54288 q^{57} -2.17629 q^{59} -7.82908 q^{61} +4.80194 q^{63} +3.58211 q^{67} -3.51573 q^{69} +8.83877 q^{71} -7.69202 q^{73} -2.85086 q^{75} -7.04354 q^{77} +4.02177 q^{79} +1.00000 q^{81} +0.652793 q^{83} +6.85086 q^{85} -1.85086 q^{87} +6.29590 q^{89} -7.63102 q^{93} +7.12498 q^{95} -10.0315 q^{97} -1.46681 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9} - q^{11} + 4 q^{15} - 7 q^{17} + 11 q^{19} - 10 q^{21} - 2 q^{23} - 5 q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + q^{33} - 18 q^{35} - 14 q^{37} - q^{41} + 3 q^{43} - 4 q^{45} - 9 q^{47} + 17 q^{49} + 7 q^{51} - 13 q^{53} + 13 q^{55} - 11 q^{57} - 14 q^{59} - 13 q^{61} + 10 q^{63} + 5 q^{67} + 2 q^{69} - 6 q^{71} - 18 q^{73} + 5 q^{75} - 15 q^{77} + 9 q^{79} + 3 q^{81} - 16 q^{83} + 7 q^{85} + 8 q^{87} + 5 q^{89} - 8 q^{93} - 3 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 - q^11 + 4 * q^15 - 7 * q^17 + 11 * q^19 - 10 * q^21 - 2 * q^23 - 5 * q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + q^33 - 18 * q^35 - 14 * q^37 - q^41 + 3 * q^43 - 4 * q^45 - 9 * q^47 + 17 * q^49 + 7 * q^51 - 13 * q^53 + 13 * q^55 - 11 * q^57 - 14 * q^59 - 13 * q^61 + 10 * q^63 + 5 * q^67 + 2 * q^69 - 6 * q^71 - 18 * q^73 + 5 * q^75 - 15 * q^77 + 9 * q^79 + 3 * q^81 - 16 * q^83 + 7 * q^85 + 8 * q^87 + 5 * q^89 - 8 * q^93 - 3 * q^95 - 5 * q^97 - 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
$5$	−2.80194	−1.25306	−0.626532	−	0.779395i	$-0.715526\pi$
$5$	−2.80194	−1.25306	−0.626532	+	0.779395i	$0.715526\pi$
$6$	0	0
$7$	4.80194	1.81496	0.907481	−	0.420093i	$-0.138003\pi$
$7$	4.80194	1.81496	0.907481	+	0.420093i	$0.138003\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.46681	−0.442260	−0.221130	−	0.975244i	$-0.570975\pi$
$11$	−1.46681	−0.442260	−0.221130	+	0.975244i	$0.570975\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.80194	0.723457
$16$	0	0
$17$	−2.44504	−0.593010	−0.296505	−	0.955031i	$-0.595821\pi$
$17$	−2.44504	−0.593010	−0.296505	+	0.955031i	$0.595821\pi$
$18$	0	0
$19$	−2.54288	−0.583376	−0.291688	−	0.956514i	$-0.594217\pi$
$19$	−2.54288	−0.583376	−0.291688	+	0.956514i	$0.594217\pi$
$20$	0	0
$21$	−4.80194	−1.04787
$22$	0	0
$23$	3.51573	0.733080	0.366540	−	0.930402i	$-0.380542\pi$
$23$	3.51573	0.733080	0.366540	+	0.930402i	$0.380542\pi$
$24$	0	0
$25$	2.85086	0.570171
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	1.85086	0.343695	0.171848	−	0.985124i	$-0.445026\pi$
$29$	1.85086	0.343695	0.171848	+	0.985124i	$0.445026\pi$
$30$	0	0
$31$	7.63102	1.37057	0.685286	−	0.728274i	$-0.259677\pi$
$31$	7.63102	1.37057	0.685286	+	0.728274i	$0.259677\pi$
$32$	0	0
$33$	1.46681	0.255339
$34$	0	0
$35$	−13.4547	−2.27426
$36$	0	0
$37$	−4.55496	−0.748831	−0.374415	−	0.927261i	$-0.622157\pi$
$37$	−4.55496	−0.748831	−0.374415	+	0.927261i	$0.622157\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.24698	0.194745	0.0973727	−	0.995248i	$-0.468956\pi$
$41$	1.24698	0.194745	0.0973727	+	0.995248i	$0.468956\pi$
$42$	0	0
$43$	−2.38404	−0.363563	−0.181782	−	0.983339i	$-0.558186\pi$
$43$	−2.38404	−0.363563	−0.181782	+	0.983339i	$0.558186\pi$
$44$	0	0
$45$	−2.80194	−0.417688
$46$	0	0
$47$	−12.8170	−1.86955	−0.934776	−	0.355238i	$-0.884400\pi$
$47$	−12.8170	−1.86955	−0.934776	+	0.355238i	$0.884400\pi$
$48$	0	0
$49$	16.0586	2.29409
$50$	0	0
$51$	2.44504	0.342374
$52$	0	0
$53$	−8.85086	−1.21576	−0.607879	−	0.794030i	$-0.707980\pi$
$53$	−8.85086	−1.21576	−0.607879	+	0.794030i	$0.707980\pi$
$54$	0	0
$55$	4.10992	0.554181
$56$	0	0
$57$	2.54288	0.336812
$58$	0	0
$59$	−2.17629	−0.283329	−0.141665	−	0.989915i	$-0.545245\pi$
$59$	−2.17629	−0.283329	−0.141665	+	0.989915i	$0.545245\pi$
$60$	0	0
$61$	−7.82908	−1.00241	−0.501206	−	0.865328i	$-0.667110\pi$
$61$	−7.82908	−1.00241	−0.501206	+	0.865328i	$0.667110\pi$
$62$	0	0
$63$	4.80194	0.604987
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.58211	0.437624	0.218812	−	0.975767i	$-0.429782\pi$
$67$	3.58211	0.437624	0.218812	+	0.975767i	$0.429782\pi$
$68$	0	0
$69$	−3.51573	−0.423244
$70$	0	0
$71$	8.83877	1.04897	0.524485	−	0.851420i	$-0.324258\pi$
$71$	8.83877	1.04897	0.524485	+	0.851420i	$0.324258\pi$
$72$	0	0
$73$	−7.69202	−0.900283	−0.450142	−	0.892957i	$-0.648626\pi$
$73$	−7.69202	−0.900283	−0.450142	+	0.892957i	$0.648626\pi$
$74$	0	0
$75$	−2.85086	−0.329188
$76$	0	0
$77$	−7.04354	−0.802686
$78$	0	0
$79$	4.02177	0.452485	0.226242	−	0.974071i	$-0.427356\pi$
$79$	4.02177	0.452485	0.226242	+	0.974071i	$0.427356\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.652793	0.0716533	0.0358267	−	0.999358i	$-0.488594\pi$
$83$	0.652793	0.0716533	0.0358267	+	0.999358i	$0.488594\pi$
$84$	0	0
$85$	6.85086	0.743080
$86$	0	0
$87$	−1.85086	−0.198432
$88$	0	0
$89$	6.29590	0.667364	0.333682	−	0.942686i	$-0.391709\pi$
$89$	6.29590	0.667364	0.333682	+	0.942686i	$0.391709\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.63102	−0.791300
$94$	0	0
$95$	7.12498	0.731008
$96$	0	0
$97$	−10.0315	−1.01854	−0.509270	−	0.860607i	$-0.670085\pi$
$97$	−10.0315	−1.01854	−0.509270	+	0.860607i	$0.670085\pi$
$98$	0	0
$99$	−1.46681	−0.147420
$100$	0	0
$101$	13.8877	1.38188	0.690938	−	0.722914i	$-0.257198\pi$
$101$	13.8877	1.38188	0.690938	+	0.722914i	$0.257198\pi$
$102$	0	0
$103$	17.4034	1.71481	0.857405	−	0.514642i	$-0.172075\pi$
$103$	17.4034	1.71481	0.857405	+	0.514642i	$0.172075\pi$
$104$	0	0
$105$	13.4547	1.31305
$106$	0	0
$107$	−10.5526	−1.02015	−0.510077	−	0.860128i	$-0.670383\pi$
$107$	−10.5526	−1.02015	−0.510077	+	0.860128i	$0.670383\pi$
$108$	0	0
$109$	1.07069	0.102553	0.0512766	−	0.998684i	$-0.483671\pi$
$109$	1.07069	0.102553	0.0512766	+	0.998684i	$0.483671\pi$
$110$	0	0
$111$	4.55496	0.432337
$112$	0	0
$113$	−16.5308	−1.55509	−0.777543	−	0.628830i	$-0.783534\pi$
$113$	−16.5308	−1.55509	−0.777543	+	0.628830i	$0.783534\pi$
$114$	0	0
$115$	−9.85086	−0.918597
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.7409	−1.07629
$120$	0	0
$121$	−8.84846	−0.804406
$122$	0	0
$123$	−1.24698	−0.112436
$124$	0	0
$125$	6.02177	0.538604
$126$	0	0
$127$	9.53750	0.846316	0.423158	−	0.906056i	$-0.360921\pi$
$127$	9.53750	0.846316	0.423158	+	0.906056i	$0.360921\pi$
$128$	0	0
$129$	2.38404	0.209903
$130$	0	0
$131$	5.50902	0.481326	0.240663	−	0.970609i	$-0.422635\pi$
$131$	5.50902	0.481326	0.240663	+	0.970609i	$0.422635\pi$
$132$	0	0
$133$	−12.2107	−1.05880
$134$	0	0
$135$	2.80194	0.241152
$136$	0	0
$137$	−16.1836	−1.38266	−0.691329	−	0.722540i	$-0.742974\pi$
$137$	−16.1836	−1.38266	−0.691329	+	0.722540i	$0.742974\pi$
$138$	0	0
$139$	10.5090	0.891364	0.445682	−	0.895191i	$-0.352961\pi$
$139$	10.5090	0.891364	0.445682	+	0.895191i	$0.352961\pi$
$140$	0	0
$141$	12.8170	1.07939
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−5.18598	−0.430672
$146$	0	0
$147$	−16.0586	−1.32449
$148$	0	0
$149$	14.3502	1.17561	0.587807	−	0.809001i	$-0.299992\pi$
$149$	14.3502	1.17561	0.587807	+	0.809001i	$0.299992\pi$
$150$	0	0
$151$	−1.96615	−0.160003	−0.0800014	−	0.996795i	$-0.525492\pi$
$151$	−1.96615	−0.160003	−0.0800014	+	0.996795i	$0.525492\pi$
$152$	0	0
$153$	−2.44504	−0.197670
$154$	0	0
$155$	−21.3817	−1.71742
$156$	0	0
$157$	10.7017	0.854089	0.427045	−	0.904231i	$-0.359555\pi$
$157$	10.7017	0.854089	0.427045	+	0.904231i	$0.359555\pi$
$158$	0	0
$159$	8.85086	0.701918
$160$	0	0
$161$	16.8823	1.33051
$162$	0	0
$163$	−3.89977	−0.305454	−0.152727	−	0.988268i	$-0.548805\pi$
$163$	−3.89977	−0.305454	−0.152727	+	0.988268i	$0.548805\pi$
$164$	0	0
$165$	−4.10992	−0.319957
$166$	0	0
$167$	−21.0194	−1.62653	−0.813264	−	0.581895i	$-0.802312\pi$
$167$	−21.0194	−1.62653	−0.813264	+	0.581895i	$0.802312\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−2.54288	−0.194459
$172$	0	0
$173$	13.2349	1.00623	0.503115	−	0.864219i	$-0.332187\pi$
$173$	13.2349	1.00623	0.503115	+	0.864219i	$0.332187\pi$
$174$	0	0
$175$	13.6896	1.03484
$176$	0	0
$177$	2.17629	0.163580
$178$	0	0
$179$	−8.52781	−0.637399	−0.318699	−	0.947856i	$-0.603246\pi$
$179$	−8.52781	−0.637399	−0.318699	+	0.947856i	$0.603246\pi$
$180$	0	0
$181$	−3.63640	−0.270291	−0.135146	−	0.990826i	$-0.543150\pi$
$181$	−3.63640	−0.270291	−0.135146	+	0.990826i	$0.543150\pi$
$182$	0	0
$183$	7.82908	0.578743
$184$	0	0
$185$	12.7627	0.938333
$186$	0	0
$187$	3.58642	0.262265
$188$	0	0
$189$	−4.80194	−0.349290
$190$	0	0
$191$	−21.3817	−1.54712	−0.773561	−	0.633722i	$-0.781526\pi$
$191$	−21.3817	−1.54712	−0.773561	+	0.633722i	$0.781526\pi$
$192$	0	0
$193$	−8.42758	−0.606631	−0.303315	−	0.952890i	$-0.598094\pi$
$193$	−8.42758	−0.606631	−0.303315	+	0.952890i	$0.598094\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−26.4765	−1.88637	−0.943186	−	0.332264i	$-0.892187\pi$
$197$	−26.4765	−1.88637	−0.943186	+	0.332264i	$0.892187\pi$
$198$	0	0
$199$	14.2524	1.01032	0.505161	−	0.863025i	$-0.331433\pi$
$199$	14.2524	1.01032	0.505161	+	0.863025i	$0.331433\pi$
$200$	0	0
$201$	−3.58211	−0.252662
$202$	0	0
$203$	8.88769	0.623794
$204$	0	0
$205$	−3.49396	−0.244029
$206$	0	0
$207$	3.51573	0.244360
$208$	0	0
$209$	3.72992	0.258004
$210$	0	0
$211$	−1.85086	−0.127418	−0.0637091	−	0.997969i	$-0.520293\pi$
$211$	−1.85086	−0.127418	−0.0637091	+	0.997969i	$0.520293\pi$
$212$	0	0
$213$	−8.83877	−0.605623
$214$	0	0
$215$	6.67994	0.455568
$216$	0	0
$217$	36.6437	2.48754
$218$	0	0
$219$	7.69202	0.519779
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−18.6504	−1.24892	−0.624462	−	0.781056i	$-0.714682\pi$
$223$	−18.6504	−1.24892	−0.624462	+	0.781056i	$0.714682\pi$
$224$	0	0
$225$	2.85086	0.190057
$226$	0	0
$227$	−9.75733	−0.647617	−0.323808	−	0.946123i	$-0.604963\pi$
$227$	−9.75733	−0.647617	−0.323808	+	0.946123i	$0.604963\pi$
$228$	0	0
$229$	−2.86294	−0.189188	−0.0945941	−	0.995516i	$-0.530155\pi$
$229$	−2.86294	−0.189188	−0.0945941	+	0.995516i	$0.530155\pi$
$230$	0	0
$231$	7.04354	0.463431
$232$	0	0
$233$	−5.78554	−0.379024	−0.189512	−	0.981878i	$-0.560691\pi$
$233$	−5.78554	−0.379024	−0.189512	+	0.981878i	$0.560691\pi$
$234$	0	0
$235$	35.9124	2.34267
$236$	0	0
$237$	−4.02177	−0.261242
$238$	0	0
$239$	−7.09246	−0.458773	−0.229386	−	0.973335i	$-0.573672\pi$
$239$	−7.09246	−0.458773	−0.229386	+	0.973335i	$0.573672\pi$
$240$	0	0
$241$	3.89977	0.251206	0.125603	−	0.992081i	$-0.459913\pi$
$241$	3.89977	0.251206	0.125603	+	0.992081i	$0.459913\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−44.9952	−2.87464
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.652793	−0.0413691
$250$	0	0
$251$	2.44504	0.154330	0.0771648	−	0.997018i	$-0.475413\pi$
$251$	2.44504	0.154330	0.0771648	+	0.997018i	$0.475413\pi$
$252$	0	0
$253$	−5.15691	−0.324212
$254$	0	0
$255$	−6.85086	−0.429017
$256$	0	0
$257$	−14.1304	−0.881428	−0.440714	−	0.897648i	$-0.645275\pi$
$257$	−14.1304	−0.881428	−0.440714	+	0.897648i	$0.645275\pi$
$258$	0	0
$259$	−21.8726	−1.35910
$260$	0	0
$261$	1.85086	0.114565
$262$	0	0
$263$	23.7235	1.46285	0.731426	−	0.681921i	$-0.238855\pi$
$263$	23.7235	1.46285	0.731426	+	0.681921i	$0.238855\pi$
$264$	0	0
$265$	24.7995	1.52342
$266$	0	0
$267$	−6.29590	−0.385303
$268$	0	0
$269$	−5.91617	−0.360715	−0.180357	−	0.983601i	$-0.557725\pi$
$269$	−5.91617	−0.360715	−0.180357	+	0.983601i	$0.557725\pi$
$270$	0	0
$271$	−3.19029	−0.193796	−0.0968982	−	0.995294i	$-0.530892\pi$
$271$	−3.19029	−0.193796	−0.0968982	+	0.995294i	$0.530892\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.18167	−0.252164
$276$	0	0
$277$	21.9366	1.31804	0.659022	−	0.752124i	$-0.270971\pi$
$277$	21.9366	1.31804	0.659022	+	0.752124i	$0.270971\pi$
$278$	0	0
$279$	7.63102	0.456857
$280$	0	0
$281$	−11.9903	−0.715282	−0.357641	−	0.933859i	$-0.616419\pi$
$281$	−11.9903	−0.715282	−0.357641	+	0.933859i	$0.616419\pi$
$282$	0	0
$283$	14.3666	0.854005	0.427002	−	0.904250i	$-0.359570\pi$
$283$	14.3666	0.854005	0.427002	+	0.904250i	$0.359570\pi$
$284$	0	0
$285$	−7.12498	−0.422047
$286$	0	0
$287$	5.98792	0.353456
$288$	0	0
$289$	−11.0218	−0.648339
$290$	0	0
$291$	10.0315	0.588055
$292$	0	0
$293$	−18.7584	−1.09588	−0.547939	−	0.836519i	$-0.684587\pi$
$293$	−18.7584	−1.09588	−0.547939	+	0.836519i	$0.684587\pi$
$294$	0	0
$295$	6.09783	0.355030
$296$	0	0
$297$	1.46681	0.0851131
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−11.4480	−0.659853
$302$	0	0
$303$	−13.8877	−0.797827
$304$	0	0
$305$	21.9366	1.25609
$306$	0	0
$307$	25.6262	1.46257	0.731283	−	0.682074i	$-0.238922\pi$
$307$	25.6262	1.46257	0.731283	+	0.682074i	$0.238922\pi$
$308$	0	0
$309$	−17.4034	−0.990046
$310$	0	0
$311$	−11.3817	−0.645394	−0.322697	−	0.946502i	$-0.604590\pi$
$311$	−11.3817	−0.645394	−0.322697	+	0.946502i	$0.604590\pi$
$312$	0	0
$313$	27.5743	1.55859	0.779297	−	0.626655i	$-0.215576\pi$
$313$	27.5743	1.55859	0.779297	+	0.626655i	$0.215576\pi$
$314$	0	0
$315$	−13.4547	−0.758088
$316$	0	0
$317$	11.2597	0.632405	0.316203	−	0.948692i	$-0.397592\pi$
$317$	11.2597	0.632405	0.316203	+	0.948692i	$0.397592\pi$
$318$	0	0
$319$	−2.71486	−0.152003
$320$	0	0
$321$	10.5526	0.588987
$322$	0	0
$323$	6.21744	0.345948
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1.07069	−0.0592092
$328$	0	0
$329$	−61.5465	−3.39317
$330$	0	0
$331$	−11.9065	−0.654439	−0.327220	−	0.944948i	$-0.606112\pi$
$331$	−11.9065	−0.654439	−0.327220	+	0.944948i	$0.606112\pi$
$332$	0	0
$333$	−4.55496	−0.249610
$334$	0	0
$335$	−10.0368	−0.548371
$336$	0	0
$337$	17.1672	0.935157	0.467578	−	0.883952i	$-0.345127\pi$
$337$	17.1672	0.935157	0.467578	+	0.883952i	$0.345127\pi$
$338$	0	0
$339$	16.5308	0.897830
$340$	0	0
$341$	−11.1933	−0.606150
$342$	0	0
$343$	43.4989	2.34872
$344$	0	0
$345$	9.85086	0.530352
$346$	0	0
$347$	24.2760	1.30321	0.651603	−	0.758560i	$-0.274097\pi$
$347$	24.2760	1.30321	0.651603	+	0.758560i	$0.274097\pi$
$348$	0	0
$349$	4.57242	0.244756	0.122378	−	0.992484i	$-0.460948\pi$
$349$	4.57242	0.244756	0.122378	+	0.992484i	$0.460948\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	6.07606	0.323396	0.161698	−	0.986840i	$-0.448303\pi$
$353$	6.07606	0.323396	0.161698	+	0.986840i	$0.448303\pi$
$354$	0	0
$355$	−24.7657	−1.31443
$356$	0	0
$357$	11.7409	0.621396
$358$	0	0
$359$	−14.9661	−0.789883	−0.394942	−	0.918706i	$-0.629235\pi$
$359$	−14.9661	−0.789883	−0.394942	+	0.918706i	$0.629235\pi$
$360$	0	0
$361$	−12.5338	−0.659673
$362$	0	0
$363$	8.84846	0.464424
$364$	0	0
$365$	21.5526	1.12811
$366$	0	0
$367$	−37.0834	−1.93574	−0.967868	−	0.251459i	$-0.919089\pi$
$367$	−37.0834	−1.93574	−0.967868	+	0.251459i	$0.919089\pi$
$368$	0	0
$369$	1.24698	0.0649152
$370$	0	0
$371$	−42.5013	−2.20656
$372$	0	0
$373$	−36.5090	−1.89037	−0.945183	−	0.326542i	$-0.894117\pi$
$373$	−36.5090	−1.89037	−0.945183	+	0.326542i	$0.894117\pi$
$374$	0	0
$375$	−6.02177	−0.310963
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.5851	−1.36558	−0.682792	−	0.730613i	$-0.739235\pi$
$379$	−26.5851	−1.36558	−0.682792	+	0.730613i	$0.739235\pi$
$380$	0	0
$381$	−9.53750	−0.488621
$382$	0	0
$383$	−14.3502	−0.733261	−0.366630	−	0.930367i	$-0.619489\pi$
$383$	−14.3502	−0.733261	−0.366630	+	0.930367i	$0.619489\pi$
$384$	0	0
$385$	19.7356	1.00582
$386$	0	0
$387$	−2.38404	−0.121188
$388$	0	0
$389$	−22.6582	−1.14881	−0.574407	−	0.818570i	$-0.694767\pi$
$389$	−22.6582	−1.14881	−0.574407	+	0.818570i	$0.694767\pi$
$390$	0	0
$391$	−8.59611	−0.434724
$392$	0	0
$393$	−5.50902	−0.277894
$394$	0	0
$395$	−11.2687	−0.566992
$396$	0	0
$397$	−7.90754	−0.396868	−0.198434	−	0.980114i	$-0.563586\pi$
$397$	−7.90754	−0.396868	−0.198434	+	0.980114i	$0.563586\pi$
$398$	0	0
$399$	12.2107	0.611301
$400$	0	0
$401$	−2.93661	−0.146647	−0.0733236	−	0.997308i	$-0.523361\pi$
$401$	−2.93661	−0.146647	−0.0733236	+	0.997308i	$0.523361\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.80194	−0.139229
$406$	0	0
$407$	6.68127	0.331178
$408$	0	0
$409$	−11.7549	−0.581244	−0.290622	−	0.956838i	$-0.593862\pi$
$409$	−11.7549	−0.581244	−0.290622	+	0.956838i	$0.593862\pi$
$410$	0	0
$411$	16.1836	0.798278
$412$	0	0
$413$	−10.4504	−0.514231
$414$	0	0
$415$	−1.82908	−0.0897862
$416$	0	0
$417$	−10.5090	−0.514629
$418$	0	0
$419$	−7.34183	−0.358672	−0.179336	−	0.983788i	$-0.557395\pi$
$419$	−7.34183	−0.358672	−0.179336	+	0.983788i	$0.557395\pi$
$420$	0	0
$421$	−25.6963	−1.25236	−0.626181	−	0.779677i	$-0.715383\pi$
$421$	−25.6963	−1.25236	−0.626181	+	0.779677i	$0.715383\pi$
$422$	0	0
$423$	−12.8170	−0.623184
$424$	0	0
$425$	−6.97046	−0.338117
$426$	0	0
$427$	−37.5948	−1.81934
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.94198	0.430720	0.215360	−	0.976535i	$-0.430907\pi$
$431$	8.94198	0.430720	0.215360	+	0.976535i	$0.430907\pi$
$432$	0	0
$433$	2.91484	0.140078	0.0700391	−	0.997544i	$-0.477688\pi$
$433$	2.91484	0.140078	0.0700391	+	0.997544i	$0.477688\pi$
$434$	0	0
$435$	5.18598	0.248649
$436$	0	0
$437$	−8.94007	−0.427661
$438$	0	0
$439$	−9.05861	−0.432344	−0.216172	−	0.976355i	$-0.569357\pi$
$439$	−9.05861	−0.432344	−0.216172	+	0.976355i	$0.569357\pi$
$440$	0	0
$441$	16.0586	0.764696
$442$	0	0
$443$	−11.2325	−0.533672	−0.266836	−	0.963742i	$-0.585978\pi$
$443$	−11.2325	−0.533672	−0.266836	+	0.963742i	$0.585978\pi$
$444$	0	0
$445$	−17.6407	−0.836250
$446$	0	0
$447$	−14.3502	−0.678741
$448$	0	0
$449$	−28.7579	−1.35717	−0.678585	−	0.734522i	$-0.737407\pi$
$449$	−28.7579	−1.35717	−0.678585	+	0.734522i	$0.737407\pi$
$450$	0	0
$451$	−1.82908	−0.0861282
$452$	0	0
$453$	1.96615	0.0923777
$454$	0	0
$455$	0	0
$456$	0	0
$457$	19.0761	0.892341	0.446170	−	0.894948i	$-0.352788\pi$
$457$	19.0761	0.892341	0.446170	+	0.894948i	$0.352788\pi$
$458$	0	0
$459$	2.44504	0.114125
$460$	0	0
$461$	−31.7332	−1.47796	−0.738981	−	0.673727i	$-0.764692\pi$
$461$	−31.7332	−1.47796	−0.738981	+	0.673727i	$0.764692\pi$
$462$	0	0
$463$	−36.4784	−1.69530	−0.847648	−	0.530559i	$-0.821982\pi$
$463$	−36.4784	−1.69530	−0.847648	+	0.530559i	$0.821982\pi$
$464$	0	0
$465$	21.3817	0.991550
$466$	0	0
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	17.2010	0.794271
$470$	0	0
$471$	−10.7017	−0.493109
$472$	0	0
$473$	3.49694	0.160790
$474$	0	0
$475$	−7.24937	−0.332624
$476$	0	0
$477$	−8.85086	−0.405253
$478$	0	0
$479$	5.61655	0.256627	0.128313	−	0.991734i	$-0.459044\pi$
$479$	5.61655	0.256627	0.128313	+	0.991734i	$0.459044\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−16.8823	−0.768172
$484$	0	0
$485$	28.1075	1.27630
$486$	0	0
$487$	−9.75733	−0.442147	−0.221073	−	0.975257i	$-0.570956\pi$
$487$	−9.75733	−0.442147	−0.221073	+	0.975257i	$0.570956\pi$
$488$	0	0
$489$	3.89977	0.176354
$490$	0	0
$491$	7.38835	0.333432	0.166716	−	0.986005i	$-0.446684\pi$
$491$	7.38835	0.333432	0.166716	+	0.986005i	$0.446684\pi$
$492$	0	0
$493$	−4.52542	−0.203815
$494$	0	0
$495$	4.10992	0.184727
$496$	0	0
$497$	42.4432	1.90384
$498$	0	0
$499$	−43.2814	−1.93754	−0.968771	−	0.247956i	$-0.920241\pi$
$499$	−43.2814	−1.93754	−0.968771	+	0.247956i	$0.920241\pi$
$500$	0	0
$501$	21.0194	0.939077
$502$	0	0
$503$	−10.7670	−0.480078	−0.240039	−	0.970763i	$-0.577160\pi$
$503$	−10.7670	−0.480078	−0.240039	+	0.970763i	$0.577160\pi$
$504$	0	0
$505$	−38.9124	−1.73158
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.5448	1.84144	0.920720	−	0.390223i	$-0.127602\pi$
$509$	41.5448	1.84144	0.920720	+	0.390223i	$0.127602\pi$
$510$	0	0
$511$	−36.9366	−1.63398
$512$	0	0
$513$	2.54288	0.112271
$514$	0	0
$515$	−48.7633	−2.14877
$516$	0	0
$517$	18.8001	0.826829
$518$	0	0
$519$	−13.2349	−0.580948
$520$	0	0
$521$	25.7198	1.12680	0.563402	−	0.826183i	$-0.309492\pi$
$521$	25.7198	1.12680	0.563402	+	0.826183i	$0.309492\pi$
$522$	0	0
$523$	−8.59286	−0.375739	−0.187870	−	0.982194i	$-0.560158\pi$
$523$	−8.59286	−0.375739	−0.187870	+	0.982194i	$0.560158\pi$
$524$	0	0
$525$	−13.6896	−0.597464
$526$	0	0
$527$	−18.6582	−0.812763
$528$	0	0
$529$	−10.6396	−0.462593
$530$	0	0
$531$	−2.17629	−0.0944430
$532$	0	0
$533$	0	0
$534$	0	0
$535$	29.5676	1.27832
$536$	0	0
$537$	8.52781	0.368002
$538$	0	0
$539$	−23.5550	−1.01458
$540$	0	0
$541$	−31.3534	−1.34799	−0.673995	−	0.738736i	$-0.735423\pi$
$541$	−31.3534	−1.34799	−0.673995	+	0.738736i	$0.735423\pi$
$542$	0	0
$543$	3.63640	0.156053
$544$	0	0
$545$	−3.00000	−0.128506
$546$	0	0
$547$	−19.9342	−0.852325	−0.426163	−	0.904647i	$-0.640135\pi$
$547$	−19.9342	−0.852325	−0.426163	+	0.904647i	$0.640135\pi$
$548$	0	0
$549$	−7.82908	−0.334137
$550$	0	0
$551$	−4.70650	−0.200503
$552$	0	0
$553$	19.3123	0.821242
$554$	0	0
$555$	−12.7627	−0.541747
$556$	0	0
$557$	−33.2349	−1.40821	−0.704104	−	0.710097i	$-0.748651\pi$
$557$	−33.2349	−1.40821	−0.704104	+	0.710097i	$0.748651\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.58642	−0.151419
$562$	0	0
$563$	−3.87130	−0.163156	−0.0815779	−	0.996667i	$-0.525996\pi$
$563$	−3.87130	−0.163156	−0.0815779	+	0.996667i	$0.525996\pi$
$564$	0	0
$565$	46.3183	1.94862
$566$	0	0
$567$	4.80194	0.201662
$568$	0	0
$569$	20.1457	0.844551	0.422276	−	0.906468i	$-0.361231\pi$
$569$	20.1457	0.844551	0.422276	+	0.906468i	$0.361231\pi$
$570$	0	0
$571$	32.1269	1.34447	0.672234	−	0.740338i	$-0.265335\pi$
$571$	32.1269	1.34447	0.672234	+	0.740338i	$0.265335\pi$
$572$	0	0
$573$	21.3817	0.893231
$574$	0	0
$575$	10.0228	0.417981
$576$	0	0
$577$	16.7506	0.697338	0.348669	−	0.937246i	$-0.386634\pi$
$577$	16.7506	0.697338	0.348669	+	0.937246i	$0.386634\pi$
$578$	0	0
$579$	8.42758	0.350238
$580$	0	0
$581$	3.13467	0.130048
$582$	0	0
$583$	12.9825	0.537682
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−6.73795	−0.278105	−0.139053	−	0.990285i	$-0.544406\pi$
$587$	−6.73795	−0.278105	−0.139053	+	0.990285i	$0.544406\pi$
$588$	0	0
$589$	−19.4047	−0.799559
$590$	0	0
$591$	26.4765	1.08910
$592$	0	0
$593$	−18.1172	−0.743985	−0.371992	−	0.928236i	$-0.621325\pi$
$593$	−18.1172	−0.743985	−0.371992	+	0.928236i	$0.621325\pi$
$594$	0	0
$595$	32.8974	1.34866
$596$	0	0
$597$	−14.2524	−0.583310
$598$	0	0
$599$	26.7851	1.09441	0.547204	−	0.836999i	$-0.315692\pi$
$599$	26.7851	1.09441	0.547204	+	0.836999i	$0.315692\pi$
$600$	0	0
$601$	−4.70171	−0.191787	−0.0958934	−	0.995392i	$-0.530571\pi$
$601$	−4.70171	−0.191787	−0.0958934	+	0.995392i	$0.530571\pi$
$602$	0	0
$603$	3.58211	0.145875
$604$	0	0
$605$	24.7928	1.00797
$606$	0	0
$607$	−27.6396	−1.12186	−0.560929	−	0.827864i	$-0.689556\pi$
$607$	−27.6396	−1.12186	−0.560929	+	0.827864i	$0.689556\pi$
$608$	0	0
$609$	−8.88769	−0.360147
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−48.1782	−1.94590	−0.972950	−	0.231017i	$-0.925795\pi$
$613$	−48.1782	−1.94590	−0.972950	+	0.231017i	$0.925795\pi$
$614$	0	0
$615$	3.49396	0.140890
$616$	0	0
$617$	−30.3043	−1.22000	−0.610002	−	0.792400i	$-0.708831\pi$
$617$	−30.3043	−1.22000	−0.610002	+	0.792400i	$0.708831\pi$
$618$	0	0
$619$	−10.9041	−0.438272	−0.219136	−	0.975694i	$-0.570324\pi$
$619$	−10.9041	−0.438272	−0.219136	+	0.975694i	$0.570324\pi$
$620$	0	0
$621$	−3.51573	−0.141081
$622$	0	0
$623$	30.2325	1.21124
$624$	0	0
$625$	−31.1269	−1.24508
$626$	0	0
$627$	−3.72992	−0.148959
$628$	0	0
$629$	11.1371	0.444064
$630$	0	0
$631$	−10.4523	−0.416101	−0.208050	−	0.978118i	$-0.566712\pi$
$631$	−10.4523	−0.416101	−0.208050	+	0.978118i	$0.566712\pi$
$632$	0	0
$633$	1.85086	0.0735649
$634$	0	0
$635$	−26.7235	−1.06049
$636$	0	0
$637$	0	0
$638$	0	0
$639$	8.83877	0.349656
$640$	0	0
$641$	−17.5942	−0.694929	−0.347464	−	0.937693i	$-0.612957\pi$
$641$	−17.5942	−0.694929	−0.347464	+	0.937693i	$0.612957\pi$
$642$	0	0
$643$	22.6058	0.891486	0.445743	−	0.895161i	$-0.352940\pi$
$643$	22.6058	0.891486	0.445743	+	0.895161i	$0.352940\pi$
$644$	0	0
$645$	−6.67994	−0.263022
$646$	0	0
$647$	24.7918	0.974665	0.487333	−	0.873216i	$-0.337970\pi$
$647$	24.7918	0.974665	0.487333	+	0.873216i	$0.337970\pi$
$648$	0	0
$649$	3.19221	0.125305
$650$	0	0
$651$	−36.6437	−1.43618
$652$	0	0
$653$	−21.8106	−0.853513	−0.426757	−	0.904367i	$-0.640344\pi$
$653$	−21.8106	−0.853513	−0.426757	+	0.904367i	$0.640344\pi$
$654$	0	0
$655$	−15.4359	−0.603132
$656$	0	0
$657$	−7.69202	−0.300094
$658$	0	0
$659$	−16.5526	−0.644796	−0.322398	−	0.946604i	$-0.604489\pi$
$659$	−16.5526	−0.644796	−0.322398	+	0.946604i	$0.604489\pi$
$660$	0	0
$661$	15.9541	0.620541	0.310271	−	0.950648i	$-0.399580\pi$
$661$	15.9541	0.620541	0.310271	+	0.950648i	$0.399580\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	34.2137	1.32675
$666$	0	0
$667$	6.50711	0.251956
$668$	0	0
$669$	18.6504	0.721066
$670$	0	0
$671$	11.4838	0.443327
$672$	0	0
$673$	−7.38835	−0.284800	−0.142400	−	0.989809i	$-0.545482\pi$
$673$	−7.38835	−0.284800	−0.142400	+	0.989809i	$0.545482\pi$
$674$	0	0
$675$	−2.85086	−0.109729
$676$	0	0
$677$	−22.1454	−0.851118	−0.425559	−	0.904931i	$-0.639922\pi$
$677$	−22.1454	−0.851118	−0.425559	+	0.904931i	$0.639922\pi$
$678$	0	0
$679$	−48.1704	−1.84861
$680$	0	0
$681$	9.75733	0.373902
$682$	0	0
$683$	9.10023	0.348211	0.174105	−	0.984727i	$-0.444297\pi$
$683$	9.10023	0.348211	0.174105	+	0.984727i	$0.444297\pi$
$684$	0	0
$685$	45.3454	1.73256
$686$	0	0
$687$	2.86294	0.109228
$688$	0	0
$689$	0	0
$690$	0	0
$691$	13.7711	0.523876	0.261938	−	0.965085i	$-0.415638\pi$
$691$	13.7711	0.523876	0.261938	+	0.965085i	$0.415638\pi$
$692$	0	0
$693$	−7.04354	−0.267562
$694$	0	0
$695$	−29.4456	−1.11694
$696$	0	0
$697$	−3.04892	−0.115486
$698$	0	0
$699$	5.78554	0.218829
$700$	0	0
$701$	46.5090	1.75662	0.878311	−	0.478090i	$-0.158671\pi$
$701$	46.5090	1.75662	0.878311	+	0.478090i	$0.158671\pi$
$702$	0	0
$703$	11.5827	0.436850
$704$	0	0
$705$	−35.9124	−1.35254
$706$	0	0
$707$	66.6878	2.50805
$708$	0	0
$709$	−7.24565	−0.272116	−0.136058	−	0.990701i	$-0.543443\pi$
$709$	−7.24565	−0.272116	−0.136058	+	0.990701i	$0.543443\pi$
$710$	0	0
$711$	4.02177	0.150828
$712$	0	0
$713$	26.8286	1.00474
$714$	0	0
$715$	0	0
$716$	0	0
$717$	7.09246	0.264873
$718$	0	0
$719$	−25.5147	−0.951536	−0.475768	−	0.879571i	$-0.657830\pi$
$719$	−25.5147	−0.951536	−0.475768	+	0.879571i	$0.657830\pi$
$720$	0	0
$721$	83.5701	3.11231
$722$	0	0
$723$	−3.89977	−0.145034
$724$	0	0
$725$	5.27652	0.195965
$726$	0	0
$727$	14.4873	0.537303	0.268651	−	0.963238i	$-0.413422\pi$
$727$	14.4873	0.537303	0.268651	+	0.963238i	$0.413422\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.82908	0.215596
$732$	0	0
$733$	−37.5036	−1.38523	−0.692614	−	0.721308i	$-0.743541\pi$
$733$	−37.5036	−1.38523	−0.692614	+	0.721308i	$0.743541\pi$
$734$	0	0
$735$	44.9952	1.65967
$736$	0	0
$737$	−5.25428	−0.193544
$738$	0	0
$739$	43.1876	1.58868	0.794341	−	0.607472i	$-0.207816\pi$
$739$	43.1876	1.58868	0.794341	+	0.607472i	$0.207816\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	13.4765	0.494405	0.247202	−	0.968964i	$-0.420489\pi$
$743$	13.4765	0.494405	0.247202	+	0.968964i	$0.420489\pi$
$744$	0	0
$745$	−40.2083	−1.47312
$746$	0	0
$747$	0.652793	0.0238844
$748$	0	0
$749$	−50.6728	−1.85154
$750$	0	0
$751$	−35.5894	−1.29868	−0.649338	−	0.760500i	$-0.724954\pi$
$751$	−35.5894	−1.29868	−0.649338	+	0.760500i	$0.724954\pi$
$752$	0	0
$753$	−2.44504	−0.0891023
$754$	0	0
$755$	5.50902	0.200494
$756$	0	0
$757$	−12.2107	−0.443807	−0.221903	−	0.975069i	$-0.571227\pi$
$757$	−12.2107	−0.443807	−0.221903	+	0.975069i	$0.571227\pi$
$758$	0	0
$759$	5.15691	0.187184
$760$	0	0
$761$	−30.9071	−1.12038	−0.560190	−	0.828364i	$-0.689272\pi$
$761$	−30.9071	−1.12038	−0.560190	+	0.828364i	$0.689272\pi$
$762$	0	0
$763$	5.14138	0.186130
$764$	0	0
$765$	6.85086	0.247693
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−11.9892	−0.432343	−0.216172	−	0.976355i	$-0.569357\pi$
$769$	−11.9892	−0.432343	−0.216172	+	0.976355i	$0.569357\pi$
$770$	0	0
$771$	14.1304	0.508892
$772$	0	0
$773$	−43.9661	−1.58135	−0.790676	−	0.612235i	$-0.790271\pi$
$773$	−43.9661	−1.58135	−0.790676	+	0.612235i	$0.790271\pi$
$774$	0	0
$775$	21.7549	0.781460
$776$	0	0
$777$	21.8726	0.784676
$778$	0	0
$779$	−3.17092	−0.113610
$780$	0	0
$781$	−12.9648	−0.463918
$782$	0	0
$783$	−1.85086	−0.0661442
$784$	0	0
$785$	−29.9855	−1.07023
$786$	0	0
$787$	30.3763	1.08280	0.541399	−	0.840766i	$-0.317895\pi$
$787$	30.3763	1.08280	0.541399	+	0.840766i	$0.317895\pi$
$788$	0	0
$789$	−23.7235	−0.844578
$790$	0	0
$791$	−79.3798	−2.82242
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−24.7995	−0.879549
$796$	0	0
$797$	52.5763	1.86235	0.931173	−	0.364577i	$-0.118786\pi$
$797$	52.5763	1.86235	0.931173	+	0.364577i	$0.118786\pi$
$798$	0	0
$799$	31.3381	1.10866
$800$	0	0
$801$	6.29590	0.222455
$802$	0	0
$803$	11.2828	0.398160
$804$	0	0
$805$	−47.3032	−1.66722
$806$	0	0
$807$	5.91617	0.208259
$808$	0	0
$809$	49.4215	1.73757	0.868783	−	0.495193i	$-0.164903\pi$
$809$	49.4215	1.73757	0.868783	+	0.495193i	$0.164903\pi$
$810$	0	0
$811$	−1.36526	−0.0479406	−0.0239703	−	0.999713i	$-0.507631\pi$
$811$	−1.36526	−0.0479406	−0.0239703	+	0.999713i	$0.507631\pi$
$812$	0	0
$813$	3.19029	0.111888
$814$	0	0
$815$	10.9269	0.382753
$816$	0	0
$817$	6.06233	0.212094
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−0.665939	−0.0232414	−0.0116207	−	0.999932i	$-0.503699\pi$
$821$	−0.665939	−0.0232414	−0.0116207	+	0.999932i	$0.503699\pi$
$822$	0	0
$823$	10.0592	0.350642	0.175321	−	0.984511i	$-0.443904\pi$
$823$	10.0592	0.350642	0.175321	+	0.984511i	$0.443904\pi$
$824$	0	0
$825$	4.18167	0.145587
$826$	0	0
$827$	−37.3038	−1.29718	−0.648590	−	0.761138i	$-0.724641\pi$
$827$	−37.3038	−1.29718	−0.648590	+	0.761138i	$0.724641\pi$
$828$	0	0
$829$	−42.6209	−1.48028	−0.740142	−	0.672451i	$-0.765242\pi$
$829$	−42.6209	−1.48028	−0.740142	+	0.672451i	$0.765242\pi$
$830$	0	0
$831$	−21.9366	−0.760973
$832$	0	0
$833$	−39.2640	−1.36042
$834$	0	0
$835$	58.8950	2.03815
$836$	0	0
$837$	−7.63102	−0.263767
$838$	0	0
$839$	4.14005	0.142930	0.0714652	−	0.997443i	$-0.477233\pi$
$839$	4.14005	0.142930	0.0714652	+	0.997443i	$0.477233\pi$
$840$	0	0
$841$	−25.5743	−0.881874
$842$	0	0
$843$	11.9903	0.412968
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−42.4898	−1.45997
$848$	0	0
$849$	−14.3666	−0.493060
$850$	0	0
$851$	−16.0140	−0.548953
$852$	0	0
$853$	−17.3502	−0.594059	−0.297030	−	0.954868i	$-0.595996\pi$
$853$	−17.3502	−0.594059	−0.297030	+	0.954868i	$0.595996\pi$
$854$	0	0
$855$	7.12498	0.243669
$856$	0	0
$857$	41.0180	1.40115	0.700575	−	0.713579i	$-0.252927\pi$
$857$	41.0180	1.40115	0.700575	+	0.713579i	$0.252927\pi$
$858$	0	0
$859$	−6.59286	−0.224945	−0.112473	−	0.993655i	$-0.535877\pi$
$859$	−6.59286	−0.224945	−0.112473	+	0.993655i	$0.535877\pi$
$860$	0	0
$861$	−5.98792	−0.204068
$862$	0	0
$863$	−16.6455	−0.566619	−0.283310	−	0.959028i	$-0.591432\pi$
$863$	−16.6455	−0.566619	−0.283310	+	0.959028i	$0.591432\pi$
$864$	0	0
$865$	−37.0834	−1.26087
$866$	0	0
$867$	11.0218	0.374319
$868$	0	0
$869$	−5.89918	−0.200116
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−10.0315	−0.339513
$874$	0	0
$875$	28.9162	0.977545
$876$	0	0
$877$	−54.4965	−1.84022	−0.920108	−	0.391666i	$-0.871899\pi$
$877$	−54.4965	−1.84022	−0.920108	+	0.391666i	$0.871899\pi$
$878$	0	0
$879$	18.7584	0.632705
$880$	0	0
$881$	−9.00670	−0.303444	−0.151722	−	0.988423i	$-0.548482\pi$
$881$	−9.00670	−0.303444	−0.151722	+	0.988423i	$0.548482\pi$
$882$	0	0
$883$	−18.8907	−0.635722	−0.317861	−	0.948137i	$-0.602965\pi$
$883$	−18.8907	−0.635722	−0.317861	+	0.948137i	$0.602965\pi$
$884$	0	0
$885$	−6.09783	−0.204976
$886$	0	0
$887$	46.9124	1.57517	0.787583	−	0.616209i	$-0.211332\pi$
$887$	46.9124	1.57517	0.787583	+	0.616209i	$0.211332\pi$
$888$	0	0
$889$	45.7985	1.53603
$890$	0	0
$891$	−1.46681	−0.0491401
$892$	0	0
$893$	32.5921	1.09065
$894$	0	0
$895$	23.8944	0.798702
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.1239	0.471059
$900$	0	0
$901$	21.6407	0.720957
$902$	0	0
$903$	11.4480	0.380966
$904$	0	0
$905$	10.1890	0.338693
$906$	0	0
$907$	35.3013	1.17216	0.586080	−	0.810253i	$-0.300671\pi$
$907$	35.3013	1.17216	0.586080	+	0.810253i	$0.300671\pi$
$908$	0	0
$909$	13.8877	0.460626
$910$	0	0
$911$	9.80731	0.324931	0.162465	−	0.986714i	$-0.448055\pi$
$911$	9.80731	0.324931	0.162465	+	0.986714i	$0.448055\pi$
$912$	0	0
$913$	−0.957524	−0.0316894
$914$	0	0
$915$	−21.9366	−0.725202
$916$	0	0
$917$	26.4540	0.873588
$918$	0	0
$919$	−18.4655	−0.609120	−0.304560	−	0.952493i	$-0.598509\pi$
$919$	−18.4655	−0.609120	−0.304560	+	0.952493i	$0.598509\pi$
$920$	0	0
$921$	−25.6262	−0.844413
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−12.9855	−0.426961
$926$	0	0
$927$	17.4034	0.571603
$928$	0	0
$929$	25.6267	0.840785	0.420393	−	0.907342i	$-0.361892\pi$
$929$	25.6267	0.840785	0.420393	+	0.907342i	$0.361892\pi$
$930$	0	0
$931$	−40.8351	−1.33831
$932$	0	0
$933$	11.3817	0.372618
$934$	0	0
$935$	−10.0489	−0.328635
$936$	0	0
$937$	7.54932	0.246625	0.123313	−	0.992368i	$-0.460648\pi$
$937$	7.54932	0.246625	0.123313	+	0.992368i	$0.460648\pi$
$938$	0	0
$939$	−27.5743	−0.899854
$940$	0	0
$941$	12.6418	0.412110	0.206055	−	0.978540i	$-0.433937\pi$
$941$	12.6418	0.412110	0.206055	+	0.978540i	$0.433937\pi$
$942$	0	0
$943$	4.38404	0.142764
$944$	0	0
$945$	13.4547	0.437682
$946$	0	0
$947$	27.9801	0.909233	0.454616	−	0.890687i	$-0.349776\pi$
$947$	27.9801	0.909233	0.454616	+	0.890687i	$0.349776\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−11.2597	−0.365119
$952$	0	0
$953$	4.00239	0.129650	0.0648251	−	0.997897i	$-0.479351\pi$
$953$	4.00239	0.129650	0.0648251	+	0.997897i	$0.479351\pi$
$954$	0	0
$955$	59.9101	1.93864
$956$	0	0
$957$	2.71486	0.0877589
$958$	0	0
$959$	−77.7126	−2.50947
$960$	0	0
$961$	27.2325	0.878468
$962$	0	0
$963$	−10.5526	−0.340052
$964$	0	0
$965$	23.6136	0.760148
$966$	0	0
$967$	12.2239	0.393094	0.196547	−	0.980494i	$-0.437027\pi$
$967$	12.2239	0.393094	0.196547	+	0.980494i	$0.437027\pi$
$968$	0	0
$969$	−6.21744	−0.199733
$970$	0	0
$971$	23.6401	0.758648	0.379324	−	0.925264i	$-0.376157\pi$
$971$	23.6401	0.758648	0.379324	+	0.925264i	$0.376157\pi$
$972$	0	0
$973$	50.4637	1.61779
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−18.7313	−0.599266	−0.299633	−	0.954055i	$-0.596864\pi$
$977$	−18.7313	−0.599266	−0.299633	+	0.954055i	$0.596864\pi$
$978$	0	0
$979$	−9.23490	−0.295149
$980$	0	0
$981$	1.07069	0.0341844
$982$	0	0
$983$	12.0954	0.385785	0.192892	−	0.981220i	$-0.438213\pi$
$983$	12.0954	0.385785	0.192892	+	0.981220i	$0.438213\pi$
$984$	0	0
$985$	74.1855	2.36375
$986$	0	0
$987$	61.5465	1.95905
$988$	0	0
$989$	−8.38165	−0.266521
$990$	0	0
$991$	28.5526	0.907002	0.453501	−	0.891256i	$-0.350175\pi$
$991$	28.5526	0.907002	0.453501	+	0.891256i	$0.350175\pi$
$992$	0	0
$993$	11.9065	0.377841
$994$	0	0
$995$	−39.9342	−1.26600
$996$	0	0
$997$	−23.9347	−0.758019	−0.379010	−	0.925393i	$-0.623735\pi$
$997$	−23.9347	−0.758019	−0.379010	+	0.925393i	$0.623735\pi$
$998$	0	0
$999$	4.55496	0.144112

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8112.2.a.by.1.1		3
4.3	odd	2		507.2.a.j.1.3	✓	3
12.11	even	2		1521.2.a.q.1.1		3
13.12	even	2		8112.2.a.cf.1.3		3
52.3	odd	6		507.2.e.k.22.1		6
52.7	even	12		507.2.j.h.361.5		12
52.11	even	12		507.2.j.h.316.2		12
52.15	even	12		507.2.j.h.316.5		12
52.19	even	12		507.2.j.h.361.2		12
52.23	odd	6		507.2.e.j.22.3		6
52.31	even	4		507.2.b.g.337.2		6
52.35	odd	6		507.2.e.k.484.1		6
52.43	odd	6		507.2.e.j.484.3		6
52.47	even	4		507.2.b.g.337.5		6
52.51	odd	2		507.2.a.k.1.1	yes	3
156.47	odd	4		1521.2.b.m.1351.2		6
156.83	odd	4		1521.2.b.m.1351.5		6
156.155	even	2		1521.2.a.p.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.3	✓	3	4.3	odd	2
507.2.a.k.1.1	yes	3	52.51	odd	2
507.2.b.g.337.2		6	52.31	even	4
507.2.b.g.337.5		6	52.47	even	4
507.2.e.j.22.3		6	52.23	odd	6
507.2.e.j.484.3		6	52.43	odd	6
507.2.e.k.22.1		6	52.3	odd	6
507.2.e.k.484.1		6	52.35	odd	6
507.2.j.h.316.2		12	52.11	even	12
507.2.j.h.316.5		12	52.15	even	12
507.2.j.h.361.2		12	52.19	even	12
507.2.j.h.361.5		12	52.7	even	12
1521.2.a.p.1.3		3	156.155	even	2
1521.2.a.q.1.1		3	12.11	even	2
1521.2.b.m.1351.2		6	156.47	odd	4
1521.2.b.m.1351.5		6	156.83	odd	4
8112.2.a.by.1.1		3	1.1	even	1	trivial
8112.2.a.cf.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -2.80194 q^{5} +4.80194 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -2.80194 q^{5} +4.80194 q^{7} +1.00000 q^{9} -1.46681 q^{11} +2.80194 q^{15} -2.44504 q^{17} -2.54288 q^{19} -4.80194 q^{21} +3.51573 q^{23} +2.85086 q^{25} -1.00000 q^{27} +1.85086 q^{29} +7.63102 q^{31} +1.46681 q^{33} -13.4547 q^{35} -4.55496 q^{37} +1.24698 q^{41} -2.38404 q^{43} -2.80194 q^{45} -12.8170 q^{47} +16.0586 q^{49} +2.44504 q^{51} -8.85086 q^{53} +4.10992 q^{55} +2.54288 q^{57} -2.17629 q^{59} -7.82908 q^{61} +4.80194 q^{63} +3.58211 q^{67} -3.51573 q^{69} +8.83877 q^{71} -7.69202 q^{73} -2.85086 q^{75} -7.04354 q^{77} +4.02177 q^{79} +1.00000 q^{81} +0.652793 q^{83} +6.85086 q^{85} -1.85086 q^{87} +6.29590 q^{89} -7.63102 q^{93} +7.12498 q^{95} -10.0315 q^{97} -1.46681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 3 q^{9} - q^{11} + 4 q^{15} - 7 q^{17} + 11 q^{19} - 10 q^{21} - 2 q^{23} - 5 q^{25} - 3 q^{27} - 8 q^{29} + 8 q^{31} + q^{33} - 18 q^{35} - 14 q^{37} - q^{41} + 3 q^{43} - 4 q^{45} - 9 q^{47} + 17 q^{49} + 7 q^{51} - 13 q^{53} + 13 q^{55} - 11 q^{57} - 14 q^{59} - 13 q^{61} + 10 q^{63} + 5 q^{67} + 2 q^{69} - 6 q^{71} - 18 q^{73} + 5 q^{75} - 15 q^{77} + 9 q^{79} + 3 q^{81} - 16 q^{83} + 7 q^{85} + 8 q^{87} + 5 q^{89} - 8 q^{93} - 3 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 4 * q^5 + 10 * q^7 + 3 * q^9 - q^11 + 4 * q^15 - 7 * q^17 + 11 * q^19 - 10 * q^21 - 2 * q^23 - 5 * q^25 - 3 * q^27 - 8 * q^29 + 8 * q^31 + q^33 - 18 * q^35 - 14 * q^37 - q^41 + 3 * q^43 - 4 * q^45 - 9 * q^47 + 17 * q^49 + 7 * q^51 - 13 * q^53 + 13 * q^55 - 11 * q^57 - 14 * q^59 - 13 * q^61 + 10 * q^63 + 5 * q^67 + 2 * q^69 - 6 * q^71 - 18 * q^73 + 5 * q^75 - 15 * q^77 + 9 * q^79 + 3 * q^81 - 16 * q^83 + 7 * q^85 + 8 * q^87 + 5 * q^89 - 8 * q^93 - 3 * q^95 - 5 * q^97 - q^99

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