LMFDB - Embedded newform 3072.2.a.r.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3072,2,Mod(1,3072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3072 = 2^{10} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3072.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:	$24.5300435009$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{24})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 1 $ x^4 - 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 768)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.93185$ of defining polynomial
Character	$\chi$	$=$	3072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.86370 q^{5} -2.44949 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.86370 q^{5} -2.44949 q^{7} +1.00000 q^{9} -1.46410 q^{11} +4.24264 q^{13} +3.86370 q^{15} -3.46410 q^{17} +7.46410 q^{19} -2.44949 q^{21} -2.82843 q^{23} +9.92820 q^{25} +1.00000 q^{27} +8.76268 q^{29} -7.34847 q^{31} -1.46410 q^{33} -9.46410 q^{35} +0.656339 q^{37} +4.24264 q^{39} +4.53590 q^{41} -3.46410 q^{43} +3.86370 q^{45} +2.82843 q^{47} -1.00000 q^{49} -3.46410 q^{51} -4.62158 q^{53} -5.65685 q^{55} +7.46410 q^{57} +13.8564 q^{59} -0.656339 q^{61} -2.44949 q^{63} +16.3923 q^{65} +14.9282 q^{67} -2.82843 q^{69} -6.41473 q^{71} +4.00000 q^{73} +9.92820 q^{75} +3.58630 q^{77} +2.44949 q^{79} +1.00000 q^{81} +5.46410 q^{83} -13.3843 q^{85} +8.76268 q^{87} -4.92820 q^{89} -10.3923 q^{91} -7.34847 q^{93} +28.8391 q^{95} +1.07180 q^{97} -1.46410 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 4 * q^9	$ 4 q + 4 q^{3} + 4 q^{9} + 8 q^{11} + 16 q^{19} + 12 q^{25} + 4 q^{27} + 8 q^{33} - 24 q^{35} + 32 q^{41} - 4 q^{49} + 16 q^{57} + 24 q^{65} + 32 q^{67} + 16 q^{73} + 12 q^{75} + 4 q^{81} + 8 q^{83} + 8 q^{89} + 32 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^11 + 16 * q^19 + 12 * q^25 + 4 * q^27 + 8 * q^33 - 24 * q^35 + 32 * q^41 - 4 * q^49 + 16 * q^57 + 24 * q^65 + 32 * q^67 + 16 * q^73 + 12 * q^75 + 4 * q^81 + 8 * q^83 + 8 * q^89 + 32 * q^97 + 8 * 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
$5$	3.86370	1.72790	0.863950	−	0.503577i	$-0.167983\pi$
$5$	3.86370	1.72790	0.863950	+	0.503577i	$0.167983\pi$
$6$	0	0
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.46410	−0.441443	−0.220722	−	0.975337i	$-0.570841\pi$
$11$	−1.46410	−0.441443	−0.220722	+	0.975337i	$0.570841\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	3.86370	0.997604
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	7.46410	1.71238	0.856191	−	0.516659i	$-0.172825\pi$
$19$	7.46410	1.71238	0.856191	+	0.516659i	$0.172825\pi$
$20$	0	0
$21$	−2.44949	−0.534522
$22$	0	0
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	0	0
$25$	9.92820	1.98564
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.76268	1.62719	0.813595	−	0.581432i	$-0.197508\pi$
$29$	8.76268	1.62719	0.813595	+	0.581432i	$0.197508\pi$
$30$	0	0
$31$	−7.34847	−1.31982	−0.659912	−	0.751343i	$-0.729406\pi$
$31$	−7.34847	−1.31982	−0.659912	+	0.751343i	$0.729406\pi$
$32$	0	0
$33$	−1.46410	−0.254867
$34$	0	0
$35$	−9.46410	−1.59973
$36$	0	0
$37$	0.656339	0.107901	0.0539507	−	0.998544i	$-0.482819\pi$
$37$	0.656339	0.107901	0.0539507	+	0.998544i	$0.482819\pi$
$38$	0	0
$39$	4.24264	0.679366
$40$	0	0
$41$	4.53590	0.708388	0.354194	−	0.935172i	$-0.384755\pi$
$41$	4.53590	0.708388	0.354194	+	0.935172i	$0.384755\pi$
$42$	0	0
$43$	−3.46410	−0.528271	−0.264135	−	0.964486i	$-0.585087\pi$
$43$	−3.46410	−0.528271	−0.264135	+	0.964486i	$0.585087\pi$
$44$	0	0
$45$	3.86370	0.575967
$46$	0	0
$47$	2.82843	0.412568	0.206284	−	0.978492i	$-0.433863\pi$
$47$	2.82843	0.412568	0.206284	+	0.978492i	$0.433863\pi$
$48$	0	0
$49$	−1.00000	−0.142857
$50$	0	0
$51$	−3.46410	−0.485071
$52$	0	0
$53$	−4.62158	−0.634823	−0.317411	−	0.948288i	$-0.602814\pi$
$53$	−4.62158	−0.634823	−0.317411	+	0.948288i	$0.602814\pi$
$54$	0	0
$55$	−5.65685	−0.762770
$56$	0	0
$57$	7.46410	0.988644
$58$	0	0
$59$	13.8564	1.80395	0.901975	−	0.431788i	$-0.142117\pi$
$59$	13.8564	1.80395	0.901975	+	0.431788i	$0.142117\pi$
$60$	0	0
$61$	−0.656339	−0.0840356	−0.0420178	−	0.999117i	$-0.513379\pi$
$61$	−0.656339	−0.0840356	−0.0420178	+	0.999117i	$0.513379\pi$
$62$	0	0
$63$	−2.44949	−0.308607
$64$	0	0
$65$	16.3923	2.03322
$66$	0	0
$67$	14.9282	1.82377	0.911885	−	0.410445i	$-0.134627\pi$
$67$	14.9282	1.82377	0.911885	+	0.410445i	$0.134627\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	−6.41473	−0.761288	−0.380644	−	0.924722i	$-0.624298\pi$
$71$	−6.41473	−0.761288	−0.380644	+	0.924722i	$0.624298\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	9.92820	1.14641
$76$	0	0
$77$	3.58630	0.408697
$78$	0	0
$79$	2.44949	0.275589	0.137795	−	0.990461i	$-0.455999\pi$
$79$	2.44949	0.275589	0.137795	+	0.990461i	$0.455999\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.46410	0.599763	0.299882	−	0.953976i	$-0.403053\pi$
$83$	5.46410	0.599763	0.299882	+	0.953976i	$0.403053\pi$
$84$	0	0
$85$	−13.3843	−1.45173
$86$	0	0
$87$	8.76268	0.939458
$88$	0	0
$89$	−4.92820	−0.522388	−0.261194	−	0.965286i	$-0.584116\pi$
$89$	−4.92820	−0.522388	−0.261194	+	0.965286i	$0.584116\pi$
$90$	0	0
$91$	−10.3923	−1.08941
$92$	0	0
$93$	−7.34847	−0.762001
$94$	0	0
$95$	28.8391	2.95883
$96$	0	0
$97$	1.07180	0.108824	0.0544122	−	0.998519i	$-0.482671\pi$
$97$	1.07180	0.108824	0.0544122	+	0.998519i	$0.482671\pi$
$98$	0	0
$99$	−1.46410	−0.147148
$100$	0	0
$101$	−14.4195	−1.43480	−0.717399	−	0.696663i	$-0.754667\pi$
$101$	−14.4195	−1.43480	−0.717399	+	0.696663i	$0.754667\pi$
$102$	0	0
$103$	3.20736	0.316031	0.158016	−	0.987437i	$-0.449490\pi$
$103$	3.20736	0.316031	0.158016	+	0.987437i	$0.449490\pi$
$104$	0	0
$105$	−9.46410	−0.923602
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	1.41421	0.135457	0.0677285	−	0.997704i	$-0.478425\pi$
$109$	1.41421	0.135457	0.0677285	+	0.997704i	$0.478425\pi$
$110$	0	0
$111$	0.656339	0.0622969
$112$	0	0
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	0	0
$115$	−10.9282	−1.01906
$116$	0	0
$117$	4.24264	0.392232
$118$	0	0
$119$	8.48528	0.777844
$120$	0	0
$121$	−8.85641	−0.805128
$122$	0	0
$123$	4.53590	0.408988
$124$	0	0
$125$	19.0411	1.70309
$126$	0	0
$127$	−14.5211	−1.28854	−0.644268	−	0.764799i	$-0.722838\pi$
$127$	−14.5211	−1.28854	−0.644268	+	0.764799i	$0.722838\pi$
$128$	0	0
$129$	−3.46410	−0.304997
$130$	0	0
$131$	−14.9282	−1.30428	−0.652142	−	0.758097i	$-0.726129\pi$
$131$	−14.9282	−1.30428	−0.652142	+	0.758097i	$0.726129\pi$
$132$	0	0
$133$	−18.2832	−1.58536
$134$	0	0
$135$	3.86370	0.332535
$136$	0	0
$137$	14.3923	1.22962	0.614809	−	0.788676i	$-0.289233\pi$
$137$	14.3923	1.22962	0.614809	+	0.788676i	$0.289233\pi$
$138$	0	0
$139$	6.92820	0.587643	0.293821	−	0.955860i	$-0.405073\pi$
$139$	6.92820	0.587643	0.293821	+	0.955860i	$0.405073\pi$
$140$	0	0
$141$	2.82843	0.238197
$142$	0	0
$143$	−6.21166	−0.519445
$144$	0	0
$145$	33.8564	2.81162
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−5.37945	−0.440702	−0.220351	−	0.975421i	$-0.570720\pi$
$149$	−5.37945	−0.440702	−0.220351	+	0.975421i	$0.570720\pi$
$150$	0	0
$151$	−6.59059	−0.536335	−0.268167	−	0.963372i	$-0.586418\pi$
$151$	−6.59059	−0.536335	−0.268167	+	0.963372i	$0.586418\pi$
$152$	0	0
$153$	−3.46410	−0.280056
$154$	0	0
$155$	−28.3923	−2.28052
$156$	0	0
$157$	2.17209	0.173352	0.0866758	−	0.996237i	$-0.472376\pi$
$157$	2.17209	0.173352	0.0866758	+	0.996237i	$0.472376\pi$
$158$	0	0
$159$	−4.62158	−0.366515
$160$	0	0
$161$	6.92820	0.546019
$162$	0	0
$163$	4.53590	0.355279	0.177639	−	0.984096i	$-0.443154\pi$
$163$	4.53590	0.355279	0.177639	+	0.984096i	$0.443154\pi$
$164$	0	0
$165$	−5.65685	−0.440386
$166$	0	0
$167$	7.72741	0.597965	0.298982	−	0.954259i	$-0.403353\pi$
$167$	7.72741	0.597965	0.298982	+	0.954259i	$0.403353\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	7.46410	0.570794
$172$	0	0
$173$	−17.2480	−1.31134	−0.655669	−	0.755048i	$-0.727613\pi$
$173$	−17.2480	−1.31134	−0.655669	+	0.755048i	$0.727613\pi$
$174$	0	0
$175$	−24.3190	−1.83835
$176$	0	0
$177$	13.8564	1.04151
$178$	0	0
$179$	−18.9282	−1.41476	−0.707380	−	0.706833i	$-0.750123\pi$
$179$	−18.9282	−1.41476	−0.707380	+	0.706833i	$0.750123\pi$
$180$	0	0
$181$	7.07107	0.525588	0.262794	−	0.964852i	$-0.415356\pi$
$181$	7.07107	0.525588	0.262794	+	0.964852i	$0.415356\pi$
$182$	0	0
$183$	−0.656339	−0.0485180
$184$	0	0
$185$	2.53590	0.186443
$186$	0	0
$187$	5.07180	0.370887
$188$	0	0
$189$	−2.44949	−0.178174
$190$	0	0
$191$	7.72741	0.559136	0.279568	−	0.960126i	$-0.409809\pi$
$191$	7.72741	0.559136	0.279568	+	0.960126i	$0.409809\pi$
$192$	0	0
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	0	0
$195$	16.3923	1.17388
$196$	0	0
$197$	−8.76268	−0.624315	−0.312158	−	0.950030i	$-0.601052\pi$
$197$	−8.76268	−0.624315	−0.312158	+	0.950030i	$0.601052\pi$
$198$	0	0
$199$	−18.6622	−1.32293	−0.661463	−	0.749977i	$-0.730064\pi$
$199$	−18.6622	−1.32293	−0.661463	+	0.749977i	$0.730064\pi$
$200$	0	0
$201$	14.9282	1.05295
$202$	0	0
$203$	−21.4641	−1.50648
$204$	0	0
$205$	17.5254	1.22402
$206$	0	0
$207$	−2.82843	−0.196589
$208$	0	0
$209$	−10.9282	−0.755920
$210$	0	0
$211$	6.92820	0.476957	0.238479	−	0.971148i	$-0.423351\pi$
$211$	6.92820	0.476957	0.238479	+	0.971148i	$0.423351\pi$
$212$	0	0
$213$	−6.41473	−0.439530
$214$	0	0
$215$	−13.3843	−0.912799
$216$	0	0
$217$	18.0000	1.22192
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	−14.6969	−0.988623
$222$	0	0
$223$	14.5211	0.972403	0.486201	−	0.873847i	$-0.338382\pi$
$223$	14.5211	0.972403	0.486201	+	0.873847i	$0.338382\pi$
$224$	0	0
$225$	9.92820	0.661880
$226$	0	0
$227$	11.3205	0.751369	0.375684	−	0.926748i	$-0.377408\pi$
$227$	11.3205	0.751369	0.375684	+	0.926748i	$0.377408\pi$
$228$	0	0
$229$	−16.8690	−1.11474	−0.557368	−	0.830265i	$-0.688189\pi$
$229$	−16.8690	−1.11474	−0.557368	+	0.830265i	$0.688189\pi$
$230$	0	0
$231$	3.58630	0.235961
$232$	0	0
$233$	12.9282	0.846955	0.423477	−	0.905907i	$-0.360809\pi$
$233$	12.9282	0.846955	0.423477	+	0.905907i	$0.360809\pi$
$234$	0	0
$235$	10.9282	0.712877
$236$	0	0
$237$	2.44949	0.159111
$238$	0	0
$239$	−7.72741	−0.499844	−0.249922	−	0.968266i	$-0.580405\pi$
$239$	−7.72741	−0.499844	−0.249922	+	0.968266i	$0.580405\pi$
$240$	0	0
$241$	−26.7846	−1.72535	−0.862674	−	0.505760i	$-0.831212\pi$
$241$	−26.7846	−1.72535	−0.862674	+	0.505760i	$0.831212\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.86370	−0.246843
$246$	0	0
$247$	31.6675	2.01495
$248$	0	0
$249$	5.46410	0.346273
$250$	0	0
$251$	−6.53590	−0.412542	−0.206271	−	0.978495i	$-0.566133\pi$
$251$	−6.53590	−0.412542	−0.206271	+	0.978495i	$0.566133\pi$
$252$	0	0
$253$	4.14110	0.260349
$254$	0	0
$255$	−13.3843	−0.838155
$256$	0	0
$257$	−22.7846	−1.42126	−0.710632	−	0.703563i	$-0.751591\pi$
$257$	−22.7846	−1.42126	−0.710632	+	0.703563i	$0.751591\pi$
$258$	0	0
$259$	−1.60770	−0.0998973
$260$	0	0
$261$	8.76268	0.542396
$262$	0	0
$263$	−22.6274	−1.39527	−0.697633	−	0.716455i	$-0.745763\pi$
$263$	−22.6274	−1.39527	−0.697633	+	0.716455i	$0.745763\pi$
$264$	0	0
$265$	−17.8564	−1.09691
$266$	0	0
$267$	−4.92820	−0.301601
$268$	0	0
$269$	10.2784	0.626687	0.313344	−	0.949640i	$-0.398551\pi$
$269$	10.2784	0.626687	0.313344	+	0.949640i	$0.398551\pi$
$270$	0	0
$271$	3.20736	0.194834	0.0974168	−	0.995244i	$-0.468942\pi$
$271$	3.20736	0.194834	0.0974168	+	0.995244i	$0.468942\pi$
$272$	0	0
$273$	−10.3923	−0.628971
$274$	0	0
$275$	−14.5359	−0.876548
$276$	0	0
$277$	12.5249	0.752545	0.376273	−	0.926509i	$-0.377206\pi$
$277$	12.5249	0.752545	0.376273	+	0.926509i	$0.377206\pi$
$278$	0	0
$279$	−7.34847	−0.439941
$280$	0	0
$281$	0.928203	0.0553720	0.0276860	−	0.999617i	$-0.491186\pi$
$281$	0.928203	0.0553720	0.0276860	+	0.999617i	$0.491186\pi$
$282$	0	0
$283$	9.85641	0.585903	0.292951	−	0.956127i	$-0.405363\pi$
$283$	9.85641	0.585903	0.292951	+	0.956127i	$0.405363\pi$
$284$	0	0
$285$	28.8391	1.70828
$286$	0	0
$287$	−11.1106	−0.655840
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	1.07180	0.0628298
$292$	0	0
$293$	10.8332	0.632884	0.316442	−	0.948612i	$-0.397512\pi$
$293$	10.8332	0.632884	0.316442	+	0.948612i	$0.397512\pi$
$294$	0	0
$295$	53.5370	3.11705
$296$	0	0
$297$	−1.46410	−0.0849558
$298$	0	0
$299$	−12.0000	−0.693978
$300$	0	0
$301$	8.48528	0.489083
$302$	0	0
$303$	−14.4195	−0.828381
$304$	0	0
$305$	−2.53590	−0.145205
$306$	0	0
$307$	22.9282	1.30858	0.654291	−	0.756243i	$-0.272967\pi$
$307$	22.9282	1.30858	0.654291	+	0.756243i	$0.272967\pi$
$308$	0	0
$309$	3.20736	0.182461
$310$	0	0
$311$	−32.4254	−1.83867	−0.919337	−	0.393471i	$-0.871274\pi$
$311$	−32.4254	−1.83867	−0.919337	+	0.393471i	$0.871274\pi$
$312$	0	0
$313$	3.85641	0.217977	0.108988	−	0.994043i	$-0.465239\pi$
$313$	3.85641	0.217977	0.108988	+	0.994043i	$0.465239\pi$
$314$	0	0
$315$	−9.46410	−0.533242
$316$	0	0
$317$	6.69213	0.375867	0.187934	−	0.982182i	$-0.439821\pi$
$317$	6.69213	0.375867	0.187934	+	0.982182i	$0.439821\pi$
$318$	0	0
$319$	−12.8295	−0.718312
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	−25.8564	−1.43869
$324$	0	0
$325$	42.1218	2.33650
$326$	0	0
$327$	1.41421	0.0782062
$328$	0	0
$329$	−6.92820	−0.381964
$330$	0	0
$331$	1.07180	0.0589113	0.0294556	−	0.999566i	$-0.490623\pi$
$331$	1.07180	0.0589113	0.0294556	+	0.999566i	$0.490623\pi$
$332$	0	0
$333$	0.656339	0.0359671
$334$	0	0
$335$	57.6781	3.15129
$336$	0	0
$337$	9.85641	0.536913	0.268456	−	0.963292i	$-0.413486\pi$
$337$	9.85641	0.536913	0.268456	+	0.963292i	$0.413486\pi$
$338$	0	0
$339$	18.0000	0.977626
$340$	0	0
$341$	10.7589	0.582627
$342$	0	0
$343$	19.5959	1.05808
$344$	0	0
$345$	−10.9282	−0.588355
$346$	0	0
$347$	2.53590	0.136134	0.0680671	−	0.997681i	$-0.478317\pi$
$347$	2.53590	0.136134	0.0680671	+	0.997681i	$0.478317\pi$
$348$	0	0
$349$	27.4249	1.46802	0.734010	−	0.679139i	$-0.237647\pi$
$349$	27.4249	1.46802	0.734010	+	0.679139i	$0.237647\pi$
$350$	0	0
$351$	4.24264	0.226455
$352$	0	0
$353$	−0.928203	−0.0494033	−0.0247016	−	0.999695i	$-0.507864\pi$
$353$	−0.928203	−0.0494033	−0.0247016	+	0.999695i	$0.507864\pi$
$354$	0	0
$355$	−24.7846	−1.31543
$356$	0	0
$357$	8.48528	0.449089
$358$	0	0
$359$	−18.8380	−0.994234	−0.497117	−	0.867684i	$-0.665608\pi$
$359$	−18.8380	−0.994234	−0.497117	+	0.867684i	$0.665608\pi$
$360$	0	0
$361$	36.7128	1.93225
$362$	0	0
$363$	−8.85641	−0.464841
$364$	0	0
$365$	15.4548	0.808942
$366$	0	0
$367$	16.3886	0.855476	0.427738	−	0.903903i	$-0.359311\pi$
$367$	16.3886	0.855476	0.427738	+	0.903903i	$0.359311\pi$
$368$	0	0
$369$	4.53590	0.236129
$370$	0	0
$371$	11.3205	0.587731
$372$	0	0
$373$	−13.2827	−0.687753	−0.343877	−	0.939015i	$-0.611740\pi$
$373$	−13.2827	−0.687753	−0.343877	+	0.939015i	$0.611740\pi$
$374$	0	0
$375$	19.0411	0.983279
$376$	0	0
$377$	37.1769	1.91471
$378$	0	0
$379$	13.3205	0.684229	0.342114	−	0.939658i	$-0.388857\pi$
$379$	13.3205	0.684229	0.342114	+	0.939658i	$0.388857\pi$
$380$	0	0
$381$	−14.5211	−0.743937
$382$	0	0
$383$	−28.8391	−1.47361	−0.736804	−	0.676106i	$-0.763666\pi$
$383$	−28.8391	−1.47361	−0.736804	+	0.676106i	$0.763666\pi$
$384$	0	0
$385$	13.8564	0.706188
$386$	0	0
$387$	−3.46410	−0.176090
$388$	0	0
$389$	9.52056	0.482711	0.241356	−	0.970437i	$-0.422408\pi$
$389$	9.52056	0.482711	0.241356	+	0.970437i	$0.422408\pi$
$390$	0	0
$391$	9.79796	0.495504
$392$	0	0
$393$	−14.9282	−0.753028
$394$	0	0
$395$	9.46410	0.476191
$396$	0	0
$397$	−23.0807	−1.15839	−0.579193	−	0.815190i	$-0.696632\pi$
$397$	−23.0807	−1.15839	−0.579193	+	0.815190i	$0.696632\pi$
$398$	0	0
$399$	−18.2832	−0.915307
$400$	0	0
$401$	−18.3923	−0.918468	−0.459234	−	0.888315i	$-0.651876\pi$
$401$	−18.3923	−0.918468	−0.459234	+	0.888315i	$0.651876\pi$
$402$	0	0
$403$	−31.1769	−1.55303
$404$	0	0
$405$	3.86370	0.191989
$406$	0	0
$407$	−0.960947	−0.0476324
$408$	0	0
$409$	−9.07180	−0.448571	−0.224286	−	0.974523i	$-0.572005\pi$
$409$	−9.07180	−0.448571	−0.224286	+	0.974523i	$0.572005\pi$
$410$	0	0
$411$	14.3923	0.709920
$412$	0	0
$413$	−33.9411	−1.67013
$414$	0	0
$415$	21.1117	1.03633
$416$	0	0
$417$	6.92820	0.339276
$418$	0	0
$419$	0.392305	0.0191653	0.00958267	−	0.999954i	$-0.496950\pi$
$419$	0.392305	0.0191653	0.00958267	+	0.999954i	$0.496950\pi$
$420$	0	0
$421$	19.6975	0.959995	0.479998	−	0.877270i	$-0.340638\pi$
$421$	19.6975	0.959995	0.479998	+	0.877270i	$0.340638\pi$
$422$	0	0
$423$	2.82843	0.137523
$424$	0	0
$425$	−34.3923	−1.66827
$426$	0	0
$427$	1.60770	0.0778018
$428$	0	0
$429$	−6.21166	−0.299902
$430$	0	0
$431$	−4.34418	−0.209252	−0.104626	−	0.994512i	$-0.533364\pi$
$431$	−4.34418	−0.209252	−0.104626	+	0.994512i	$0.533364\pi$
$432$	0	0
$433$	−29.8564	−1.43481	−0.717404	−	0.696658i	$-0.754670\pi$
$433$	−29.8564	−1.43481	−0.717404	+	0.696658i	$0.754670\pi$
$434$	0	0
$435$	33.8564	1.62329
$436$	0	0
$437$	−21.1117	−1.00991
$438$	0	0
$439$	−5.83272	−0.278381	−0.139190	−	0.990266i	$-0.544450\pi$
$439$	−5.83272	−0.278381	−0.139190	+	0.990266i	$0.544450\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	0	0
$443$	−35.3205	−1.67813	−0.839064	−	0.544033i	$-0.816897\pi$
$443$	−35.3205	−1.67813	−0.839064	+	0.544033i	$0.816897\pi$
$444$	0	0
$445$	−19.0411	−0.902635
$446$	0	0
$447$	−5.37945	−0.254439
$448$	0	0
$449$	−6.39230	−0.301672	−0.150836	−	0.988559i	$-0.548196\pi$
$449$	−6.39230	−0.301672	−0.150836	+	0.988559i	$0.548196\pi$
$450$	0	0
$451$	−6.64102	−0.312713
$452$	0	0
$453$	−6.59059	−0.309653
$454$	0	0
$455$	−40.1528	−1.88239
$456$	0	0
$457$	−20.7846	−0.972263	−0.486132	−	0.873886i	$-0.661592\pi$
$457$	−20.7846	−0.972263	−0.486132	+	0.873886i	$0.661592\pi$
$458$	0	0
$459$	−3.46410	−0.161690
$460$	0	0
$461$	31.1870	1.45252	0.726262	−	0.687418i	$-0.241256\pi$
$461$	31.1870	1.45252	0.726262	+	0.687418i	$0.241256\pi$
$462$	0	0
$463$	29.2180	1.35788	0.678938	−	0.734196i	$-0.262440\pi$
$463$	29.2180	1.35788	0.678938	+	0.734196i	$0.262440\pi$
$464$	0	0
$465$	−28.3923	−1.31666
$466$	0	0
$467$	−30.2487	−1.39974	−0.699872	−	0.714269i	$-0.746760\pi$
$467$	−30.2487	−1.39974	−0.699872	+	0.714269i	$0.746760\pi$
$468$	0	0
$469$	−36.5665	−1.68848
$470$	0	0
$471$	2.17209	0.100085
$472$	0	0
$473$	5.07180	0.233201
$474$	0	0
$475$	74.1051	3.40018
$476$	0	0
$477$	−4.62158	−0.211608
$478$	0	0
$479$	29.0421	1.32697	0.663485	−	0.748190i	$-0.269077\pi$
$479$	29.0421	1.32697	0.663485	+	0.748190i	$0.269077\pi$
$480$	0	0
$481$	2.78461	0.126967
$482$	0	0
$483$	6.92820	0.315244
$484$	0	0
$485$	4.14110	0.188038
$486$	0	0
$487$	−11.4896	−0.520642	−0.260321	−	0.965522i	$-0.583828\pi$
$487$	−11.4896	−0.520642	−0.260321	+	0.965522i	$0.583828\pi$
$488$	0	0
$489$	4.53590	0.205120
$490$	0	0
$491$	5.85641	0.264296	0.132148	−	0.991230i	$-0.457813\pi$
$491$	5.85641	0.264296	0.132148	+	0.991230i	$0.457813\pi$
$492$	0	0
$493$	−30.3548	−1.36711
$494$	0	0
$495$	−5.65685	−0.254257
$496$	0	0
$497$	15.7128	0.704816
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	7.72741	0.345235
$502$	0	0
$503$	−23.3853	−1.04270	−0.521349	−	0.853343i	$-0.674571\pi$
$503$	−23.3853	−1.04270	−0.521349	+	0.853343i	$0.674571\pi$
$504$	0	0
$505$	−55.7128	−2.47919
$506$	0	0
$507$	5.00000	0.222058
$508$	0	0
$509$	−7.24693	−0.321215	−0.160607	−	0.987018i	$-0.551345\pi$
$509$	−7.24693	−0.321215	−0.160607	+	0.987018i	$0.551345\pi$
$510$	0	0
$511$	−9.79796	−0.433436
$512$	0	0
$513$	7.46410	0.329548
$514$	0	0
$515$	12.3923	0.546070
$516$	0	0
$517$	−4.14110	−0.182126
$518$	0	0
$519$	−17.2480	−0.757102
$520$	0	0
$521$	−21.3205	−0.934068	−0.467034	−	0.884239i	$-0.654678\pi$
$521$	−21.3205	−0.934068	−0.467034	+	0.884239i	$0.654678\pi$
$522$	0	0
$523$	31.4641	1.37583	0.687915	−	0.725792i	$-0.258526\pi$
$523$	31.4641	1.37583	0.687915	+	0.725792i	$0.258526\pi$
$524$	0	0
$525$	−24.3190	−1.06137
$526$	0	0
$527$	25.4558	1.10887
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	13.8564	0.601317
$532$	0	0
$533$	19.2442	0.833558
$534$	0	0
$535$	−15.4548	−0.668170
$536$	0	0
$537$	−18.9282	−0.816812
$538$	0	0
$539$	1.46410	0.0630633
$540$	0	0
$541$	−28.1827	−1.21167	−0.605835	−	0.795590i	$-0.707161\pi$
$541$	−28.1827	−1.21167	−0.605835	+	0.795590i	$0.707161\pi$
$542$	0	0
$543$	7.07107	0.303449
$544$	0	0
$545$	5.46410	0.234056
$546$	0	0
$547$	23.1769	0.990973	0.495487	−	0.868616i	$-0.334990\pi$
$547$	23.1769	0.990973	0.495487	+	0.868616i	$0.334990\pi$
$548$	0	0
$549$	−0.656339	−0.0280119
$550$	0	0
$551$	65.4056	2.78637
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	2.53590	0.107643
$556$	0	0
$557$	−28.0069	−1.18669	−0.593345	−	0.804949i	$-0.702193\pi$
$557$	−28.0069	−1.18669	−0.593345	+	0.804949i	$0.702193\pi$
$558$	0	0
$559$	−14.6969	−0.621614
$560$	0	0
$561$	5.07180	0.214131
$562$	0	0
$563$	1.46410	0.0617045	0.0308523	−	0.999524i	$-0.490178\pi$
$563$	1.46410	0.0617045	0.0308523	+	0.999524i	$0.490178\pi$
$564$	0	0
$565$	69.5467	2.92585
$566$	0	0
$567$	−2.44949	−0.102869
$568$	0	0
$569$	−3.46410	−0.145223	−0.0726113	−	0.997360i	$-0.523133\pi$
$569$	−3.46410	−0.145223	−0.0726113	+	0.997360i	$0.523133\pi$
$570$	0	0
$571$	−31.7128	−1.32714	−0.663570	−	0.748114i	$-0.730959\pi$
$571$	−31.7128	−1.32714	−0.663570	+	0.748114i	$0.730959\pi$
$572$	0	0
$573$	7.72741	0.322817
$574$	0	0
$575$	−28.0812	−1.17107
$576$	0	0
$577$	−21.8564	−0.909894	−0.454947	−	0.890518i	$-0.650342\pi$
$577$	−21.8564	−0.909894	−0.454947	+	0.890518i	$0.650342\pi$
$578$	0	0
$579$	6.00000	0.249351
$580$	0	0
$581$	−13.3843	−0.555273
$582$	0	0
$583$	6.76646	0.280238
$584$	0	0
$585$	16.3923	0.677738
$586$	0	0
$587$	−42.9282	−1.77184	−0.885918	−	0.463841i	$-0.846471\pi$
$587$	−42.9282	−1.77184	−0.885918	+	0.463841i	$0.846471\pi$
$588$	0	0
$589$	−54.8497	−2.26004
$590$	0	0
$591$	−8.76268	−0.360449
$592$	0	0
$593$	37.7128	1.54868	0.774340	−	0.632770i	$-0.218082\pi$
$593$	37.7128	1.54868	0.774340	+	0.632770i	$0.218082\pi$
$594$	0	0
$595$	32.7846	1.34404
$596$	0	0
$597$	−18.6622	−0.763792
$598$	0	0
$599$	−6.96953	−0.284767	−0.142384	−	0.989812i	$-0.545477\pi$
$599$	−6.96953	−0.284767	−0.142384	+	0.989812i	$0.545477\pi$
$600$	0	0
$601$	19.7128	0.804102	0.402051	−	0.915617i	$-0.368297\pi$
$601$	19.7128	0.804102	0.402051	+	0.915617i	$0.368297\pi$
$602$	0	0
$603$	14.9282	0.607923
$604$	0	0
$605$	−34.2185	−1.39118
$606$	0	0
$607$	31.8434	1.29248	0.646241	−	0.763133i	$-0.276340\pi$
$607$	31.8434	1.29248	0.646241	+	0.763133i	$0.276340\pi$
$608$	0	0
$609$	−21.4641	−0.869769
$610$	0	0
$611$	12.0000	0.485468
$612$	0	0
$613$	−6.11012	−0.246785	−0.123393	−	0.992358i	$-0.539377\pi$
$613$	−6.11012	−0.246785	−0.123393	+	0.992358i	$0.539377\pi$
$614$	0	0
$615$	17.5254	0.706691
$616$	0	0
$617$	−39.8564	−1.60456	−0.802279	−	0.596949i	$-0.796379\pi$
$617$	−39.8564	−1.60456	−0.802279	+	0.596949i	$0.796379\pi$
$618$	0	0
$619$	−12.0000	−0.482321	−0.241160	−	0.970485i	$-0.577528\pi$
$619$	−12.0000	−0.482321	−0.241160	+	0.970485i	$0.577528\pi$
$620$	0	0
$621$	−2.82843	−0.113501
$622$	0	0
$623$	12.0716	0.483638
$624$	0	0
$625$	23.9282	0.957128
$626$	0	0
$627$	−10.9282	−0.436430
$628$	0	0
$629$	−2.27362	−0.0906553
$630$	0	0
$631$	23.5612	0.937955	0.468977	−	0.883210i	$-0.344623\pi$
$631$	23.5612	0.937955	0.468977	+	0.883210i	$0.344623\pi$
$632$	0	0
$633$	6.92820	0.275371
$634$	0	0
$635$	−56.1051	−2.22646
$636$	0	0
$637$	−4.24264	−0.168100
$638$	0	0
$639$	−6.41473	−0.253763
$640$	0	0
$641$	−25.6077	−1.01144	−0.505722	−	0.862697i	$-0.668774\pi$
$641$	−25.6077	−1.01144	−0.505722	+	0.862697i	$0.668774\pi$
$642$	0	0
$643$	−28.5359	−1.12535	−0.562673	−	0.826680i	$-0.690227\pi$
$643$	−28.5359	−1.12535	−0.562673	+	0.826680i	$0.690227\pi$
$644$	0	0
$645$	−13.3843	−0.527005
$646$	0	0
$647$	−40.9107	−1.60836	−0.804182	−	0.594383i	$-0.797396\pi$
$647$	−40.9107	−1.60836	−0.804182	+	0.594383i	$0.797396\pi$
$648$	0	0
$649$	−20.2872	−0.796342
$650$	0	0
$651$	18.0000	0.705476
$652$	0	0
$653$	−24.9754	−0.977362	−0.488681	−	0.872463i	$-0.662522\pi$
$653$	−24.9754	−0.977362	−0.488681	+	0.872463i	$0.662522\pi$
$654$	0	0
$655$	−57.6781	−2.25367
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	6.51626	0.253453	0.126727	−	0.991938i	$-0.459553\pi$
$661$	6.51626	0.253453	0.126727	+	0.991938i	$0.459553\pi$
$662$	0	0
$663$	−14.6969	−0.570782
$664$	0	0
$665$	−70.6410	−2.73934
$666$	0	0
$667$	−24.7846	−0.959664
$668$	0	0
$669$	14.5211	0.561417
$670$	0	0
$671$	0.960947	0.0370969
$672$	0	0
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	0	0
$675$	9.92820	0.382137
$676$	0	0
$677$	3.30890	0.127171	0.0635857	−	0.997976i	$-0.479746\pi$
$677$	3.30890	0.127171	0.0635857	+	0.997976i	$0.479746\pi$
$678$	0	0
$679$	−2.62536	−0.100752
$680$	0	0
$681$	11.3205	0.433803
$682$	0	0
$683$	25.1769	0.963368	0.481684	−	0.876345i	$-0.340025\pi$
$683$	25.1769	0.963368	0.481684	+	0.876345i	$0.340025\pi$
$684$	0	0
$685$	55.6076	2.12466
$686$	0	0
$687$	−16.8690	−0.643594
$688$	0	0
$689$	−19.6077	−0.746994
$690$	0	0
$691$	26.3923	1.00401	0.502005	−	0.864865i	$-0.332596\pi$
$691$	26.3923	1.00401	0.502005	+	0.864865i	$0.332596\pi$
$692$	0	0
$693$	3.58630	0.136232
$694$	0	0
$695$	26.7685	1.01539
$696$	0	0
$697$	−15.7128	−0.595165
$698$	0	0
$699$	12.9282	0.488990
$700$	0	0
$701$	−14.9743	−0.565573	−0.282787	−	0.959183i	$-0.591259\pi$
$701$	−14.9743	−0.565573	−0.282787	+	0.959183i	$0.591259\pi$
$702$	0	0
$703$	4.89898	0.184769
$704$	0	0
$705$	10.9282	0.411580
$706$	0	0
$707$	35.3205	1.32836
$708$	0	0
$709$	−49.0913	−1.84366	−0.921832	−	0.387590i	$-0.873308\pi$
$709$	−49.0913	−1.84366	−0.921832	+	0.387590i	$0.873308\pi$
$710$	0	0
$711$	2.44949	0.0918630
$712$	0	0
$713$	20.7846	0.778390
$714$	0	0
$715$	−24.0000	−0.897549
$716$	0	0
$717$	−7.72741	−0.288585
$718$	0	0
$719$	2.27362	0.0847919	0.0423959	−	0.999101i	$-0.486501\pi$
$719$	2.27362	0.0847919	0.0423959	+	0.999101i	$0.486501\pi$
$720$	0	0
$721$	−7.85641	−0.292588
$722$	0	0
$723$	−26.7846	−0.996130
$724$	0	0
$725$	86.9977	3.23101
$726$	0	0
$727$	8.10634	0.300648	0.150324	−	0.988637i	$-0.451968\pi$
$727$	8.10634	0.300648	0.150324	+	0.988637i	$0.451968\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	12.0000	0.443836
$732$	0	0
$733$	7.07107	0.261176	0.130588	−	0.991437i	$-0.458314\pi$
$733$	7.07107	0.261176	0.130588	+	0.991437i	$0.458314\pi$
$734$	0	0
$735$	−3.86370	−0.142515
$736$	0	0
$737$	−21.8564	−0.805091
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	31.6675	1.16333
$742$	0	0
$743$	−27.3233	−1.00240	−0.501198	−	0.865333i	$-0.667107\pi$
$743$	−27.3233	−1.00240	−0.501198	+	0.865333i	$0.667107\pi$
$744$	0	0
$745$	−20.7846	−0.761489
$746$	0	0
$747$	5.46410	0.199921
$748$	0	0
$749$	9.79796	0.358010
$750$	0	0
$751$	−46.1886	−1.68545	−0.842723	−	0.538348i	$-0.819049\pi$
$751$	−46.1886	−1.68545	−0.842723	+	0.538348i	$0.819049\pi$
$752$	0	0
$753$	−6.53590	−0.238181
$754$	0	0
$755$	−25.4641	−0.926734
$756$	0	0
$757$	−33.6365	−1.22254	−0.611270	−	0.791422i	$-0.709341\pi$
$757$	−33.6365	−1.22254	−0.611270	+	0.791422i	$0.709341\pi$
$758$	0	0
$759$	4.14110	0.150313
$760$	0	0
$761$	−52.2487	−1.89401	−0.947007	−	0.321212i	$-0.895910\pi$
$761$	−52.2487	−1.89401	−0.947007	+	0.321212i	$0.895910\pi$
$762$	0	0
$763$	−3.46410	−0.125409
$764$	0	0
$765$	−13.3843	−0.483909
$766$	0	0
$767$	58.7878	2.12270
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	−22.7846	−0.820568
$772$	0	0
$773$	48.9155	1.75937	0.879684	−	0.475560i	$-0.157754\pi$
$773$	48.9155	1.75937	0.879684	+	0.475560i	$0.157754\pi$
$774$	0	0
$775$	−72.9571	−2.62070
$776$	0	0
$777$	−1.60770	−0.0576757
$778$	0	0
$779$	33.8564	1.21303
$780$	0	0
$781$	9.39182	0.336066
$782$	0	0
$783$	8.76268	0.313153
$784$	0	0
$785$	8.39230	0.299534
$786$	0	0
$787$	−23.4641	−0.836405	−0.418202	−	0.908354i	$-0.637340\pi$
$787$	−23.4641	−0.836405	−0.418202	+	0.908354i	$0.637340\pi$
$788$	0	0
$789$	−22.6274	−0.805557
$790$	0	0
$791$	−44.0908	−1.56769
$792$	0	0
$793$	−2.78461	−0.0988844
$794$	0	0
$795$	−17.8564	−0.633301
$796$	0	0
$797$	21.9439	0.777292	0.388646	−	0.921387i	$-0.372943\pi$
$797$	21.9439	0.777292	0.388646	+	0.921387i	$0.372943\pi$
$798$	0	0
$799$	−9.79796	−0.346627
$800$	0	0
$801$	−4.92820	−0.174129
$802$	0	0
$803$	−5.85641	−0.206668
$804$	0	0
$805$	26.7685	0.943466
$806$	0	0
$807$	10.2784	0.361818
$808$	0	0
$809$	−6.67949	−0.234838	−0.117419	−	0.993082i	$-0.537462\pi$
$809$	−6.67949	−0.234838	−0.117419	+	0.993082i	$0.537462\pi$
$810$	0	0
$811$	54.3923	1.90997	0.954986	−	0.296651i	$-0.0958697\pi$
$811$	54.3923	1.90997	0.954986	+	0.296651i	$0.0958697\pi$
$812$	0	0
$813$	3.20736	0.112487
$814$	0	0
$815$	17.5254	0.613887
$816$	0	0
$817$	−25.8564	−0.904601
$818$	0	0
$819$	−10.3923	−0.363137
$820$	0	0
$821$	−25.1784	−0.878734	−0.439367	−	0.898308i	$-0.644797\pi$
$821$	−25.1784	−0.878734	−0.439367	+	0.898308i	$0.644797\pi$
$822$	0	0
$823$	−9.97382	−0.347666	−0.173833	−	0.984775i	$-0.555615\pi$
$823$	−9.97382	−0.347666	−0.173833	+	0.984775i	$0.555615\pi$
$824$	0	0
$825$	−14.5359	−0.506075
$826$	0	0
$827$	44.7846	1.55731	0.778657	−	0.627450i	$-0.215901\pi$
$827$	44.7846	1.55731	0.778657	+	0.627450i	$0.215901\pi$
$828$	0	0
$829$	−1.61729	−0.0561706	−0.0280853	−	0.999606i	$-0.508941\pi$
$829$	−1.61729	−0.0561706	−0.0280853	+	0.999606i	$0.508941\pi$
$830$	0	0
$831$	12.5249	0.434482
$832$	0	0
$833$	3.46410	0.120024
$834$	0	0
$835$	29.8564	1.03322
$836$	0	0
$837$	−7.34847	−0.254000
$838$	0	0
$839$	26.0106	0.897987	0.448994	−	0.893535i	$-0.351783\pi$
$839$	26.0106	0.897987	0.448994	+	0.893535i	$0.351783\pi$
$840$	0	0
$841$	47.7846	1.64775
$842$	0	0
$843$	0.928203	0.0319690
$844$	0	0
$845$	19.3185	0.664577
$846$	0	0
$847$	21.6937	0.745404
$848$	0	0
$849$	9.85641	0.338271
$850$	0	0
$851$	−1.85641	−0.0636368
$852$	0	0
$853$	47.0208	1.60996	0.804980	−	0.593301i	$-0.202176\pi$
$853$	47.0208	1.60996	0.804980	+	0.593301i	$0.202176\pi$
$854$	0	0
$855$	28.8391	0.986276
$856$	0	0
$857$	43.1769	1.47490	0.737448	−	0.675404i	$-0.236031\pi$
$857$	43.1769	1.47490	0.737448	+	0.675404i	$0.236031\pi$
$858$	0	0
$859$	−32.2487	−1.10031	−0.550156	−	0.835062i	$-0.685432\pi$
$859$	−32.2487	−1.10031	−0.550156	+	0.835062i	$0.685432\pi$
$860$	0	0
$861$	−11.1106	−0.378649
$862$	0	0
$863$	37.1213	1.26362	0.631812	−	0.775122i	$-0.282312\pi$
$863$	37.1213	1.26362	0.631812	+	0.775122i	$0.282312\pi$
$864$	0	0
$865$	−66.6410	−2.26586
$866$	0	0
$867$	−5.00000	−0.169809
$868$	0	0
$869$	−3.58630	−0.121657
$870$	0	0
$871$	63.3350	2.14602
$872$	0	0
$873$	1.07180	0.0362748
$874$	0	0
$875$	−46.6410	−1.57675
$876$	0	0
$877$	−35.7071	−1.20574	−0.602871	−	0.797839i	$-0.705977\pi$
$877$	−35.7071	−1.20574	−0.602871	+	0.797839i	$0.705977\pi$
$878$	0	0
$879$	10.8332	0.365396
$880$	0	0
$881$	−36.6410	−1.23447	−0.617234	−	0.786780i	$-0.711747\pi$
$881$	−36.6410	−1.23447	−0.617234	+	0.786780i	$0.711747\pi$
$882$	0	0
$883$	−5.60770	−0.188714	−0.0943570	−	0.995538i	$-0.530080\pi$
$883$	−5.60770	−0.188714	−0.0943570	+	0.995538i	$0.530080\pi$
$884$	0	0
$885$	53.5370	1.79963
$886$	0	0
$887$	36.5665	1.22778	0.613891	−	0.789391i	$-0.289603\pi$
$887$	36.5665	1.22778	0.613891	+	0.789391i	$0.289603\pi$
$888$	0	0
$889$	35.5692	1.19295
$890$	0	0
$891$	−1.46410	−0.0490492
$892$	0	0
$893$	21.1117	0.706475
$894$	0	0
$895$	−73.1330	−2.44457
$896$	0	0
$897$	−12.0000	−0.400668
$898$	0	0
$899$	−64.3923	−2.14760
$900$	0	0
$901$	16.0096	0.533358
$902$	0	0
$903$	8.48528	0.282372
$904$	0	0
$905$	27.3205	0.908164
$906$	0	0
$907$	−42.3923	−1.40761	−0.703807	−	0.710392i	$-0.748518\pi$
$907$	−42.3923	−1.40761	−0.703807	+	0.710392i	$0.748518\pi$
$908$	0	0
$909$	−14.4195	−0.478266
$910$	0	0
$911$	−22.0726	−0.731298	−0.365649	−	0.930753i	$-0.619153\pi$
$911$	−22.0726	−0.731298	−0.365649	+	0.930753i	$0.619153\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	−2.53590	−0.0838342
$916$	0	0
$917$	36.5665	1.20753
$918$	0	0
$919$	−0.582009	−0.0191987	−0.00959936	−	0.999954i	$-0.503056\pi$
$919$	−0.582009	−0.0191987	−0.00959936	+	0.999954i	$0.503056\pi$
$920$	0	0
$921$	22.9282	0.755510
$922$	0	0
$923$	−27.2154	−0.895805
$924$	0	0
$925$	6.51626	0.214253
$926$	0	0
$927$	3.20736	0.105344
$928$	0	0
$929$	−45.3205	−1.48692	−0.743459	−	0.668782i	$-0.766816\pi$
$929$	−45.3205	−1.48692	−0.743459	+	0.668782i	$0.766816\pi$
$930$	0	0
$931$	−7.46410	−0.244626
$932$	0	0
$933$	−32.4254	−1.06156
$934$	0	0
$935$	19.5959	0.640855
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	0	0
$939$	3.85641	0.125849
$940$	0	0
$941$	−36.2891	−1.18299	−0.591495	−	0.806309i	$-0.701462\pi$
$941$	−36.2891	−1.18299	−0.591495	+	0.806309i	$0.701462\pi$
$942$	0	0
$943$	−12.8295	−0.417785
$944$	0	0
$945$	−9.46410	−0.307867
$946$	0	0
$947$	29.0718	0.944706	0.472353	−	0.881409i	$-0.343405\pi$
$947$	29.0718	0.944706	0.472353	+	0.881409i	$0.343405\pi$
$948$	0	0
$949$	16.9706	0.550888
$950$	0	0
$951$	6.69213	0.217007
$952$	0	0
$953$	46.3923	1.50279	0.751397	−	0.659850i	$-0.229380\pi$
$953$	46.3923	1.50279	0.751397	+	0.659850i	$0.229380\pi$
$954$	0	0
$955$	29.8564	0.966131
$956$	0	0
$957$	−12.8295	−0.414717
$958$	0	0
$959$	−35.2538	−1.13840
$960$	0	0
$961$	23.0000	0.741935
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	23.1822	0.746262
$966$	0	0
$967$	10.3800	0.333797	0.166899	−	0.985974i	$-0.446625\pi$
$967$	10.3800	0.333797	0.166899	+	0.985974i	$0.446625\pi$
$968$	0	0
$969$	−25.8564	−0.830627
$970$	0	0
$971$	22.2487	0.713995	0.356998	−	0.934105i	$-0.383800\pi$
$971$	22.2487	0.713995	0.356998	+	0.934105i	$0.383800\pi$
$972$	0	0
$973$	−16.9706	−0.544051
$974$	0	0
$975$	42.1218	1.34898
$976$	0	0
$977$	−35.1769	−1.12541	−0.562705	−	0.826658i	$-0.690239\pi$
$977$	−35.1769	−1.12541	−0.562705	+	0.826658i	$0.690239\pi$
$978$	0	0
$979$	7.21539	0.230605
$980$	0	0
$981$	1.41421	0.0451524
$982$	0	0
$983$	23.7370	0.757093	0.378547	−	0.925582i	$-0.376424\pi$
$983$	23.7370	0.757093	0.378547	+	0.925582i	$0.376424\pi$
$984$	0	0
$985$	−33.8564	−1.07875
$986$	0	0
$987$	−6.92820	−0.220527
$988$	0	0
$989$	9.79796	0.311557
$990$	0	0
$991$	−12.2474	−0.389053	−0.194527	−	0.980897i	$-0.562317\pi$
$991$	−12.2474	−0.389053	−0.194527	+	0.980897i	$0.562317\pi$
$992$	0	0
$993$	1.07180	0.0340124
$994$	0	0
$995$	−72.1051	−2.28589
$996$	0	0
$997$	17.6269	0.558250	0.279125	−	0.960255i	$-0.409956\pi$
$997$	17.6269	0.558250	0.279125	+	0.960255i	$0.409956\pi$
$998$	0	0
$999$	0.656339	0.0207656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3072.2.a.r.1.4		4
3.2	odd	2		9216.2.a.bc.1.1		4
4.3	odd	2		3072.2.a.l.1.4		4
8.3	odd	2	inner	3072.2.a.r.1.1		4
8.5	even	2		3072.2.a.l.1.1		4
12.11	even	2		9216.2.a.bi.1.1		4
16.3	odd	4		3072.2.d.h.1537.1		8
16.5	even	4		3072.2.d.h.1537.4		8
16.11	odd	4		3072.2.d.h.1537.8		8
16.13	even	4		3072.2.d.h.1537.5		8
24.5	odd	2		9216.2.a.bi.1.4		4
24.11	even	2		9216.2.a.bc.1.4		4
32.3	odd	8		768.2.j.e.577.1	yes	8
32.5	even	8		768.2.j.f.193.2	yes	8
32.11	odd	8		768.2.j.e.193.1	✓	8
32.13	even	8		768.2.j.f.577.2	yes	8
32.19	odd	8		768.2.j.f.577.4	yes	8
32.21	even	8		768.2.j.e.193.3	yes	8
32.27	odd	8		768.2.j.f.193.4	yes	8
32.29	even	8		768.2.j.e.577.3	yes	8
96.5	odd	8		2304.2.k.e.1729.1		8
96.11	even	8		2304.2.k.l.1729.4		8
96.29	odd	8		2304.2.k.l.577.4		8
96.35	even	8		2304.2.k.l.577.3		8
96.53	odd	8		2304.2.k.l.1729.3		8
96.59	even	8		2304.2.k.e.1729.2		8
96.77	odd	8		2304.2.k.e.577.2		8
96.83	even	8		2304.2.k.e.577.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
768.2.j.e.193.1	✓	8	32.11	odd	8
768.2.j.e.193.3	yes	8	32.21	even	8
768.2.j.e.577.1	yes	8	32.3	odd	8
768.2.j.e.577.3	yes	8	32.29	even	8
768.2.j.f.193.2	yes	8	32.5	even	8
768.2.j.f.193.4	yes	8	32.27	odd	8
768.2.j.f.577.2	yes	8	32.13	even	8
768.2.j.f.577.4	yes	8	32.19	odd	8
2304.2.k.e.577.1		8	96.83	even	8
2304.2.k.e.577.2		8	96.77	odd	8
2304.2.k.e.1729.1		8	96.5	odd	8
2304.2.k.e.1729.2		8	96.59	even	8
2304.2.k.l.577.3		8	96.35	even	8
2304.2.k.l.577.4		8	96.29	odd	8
2304.2.k.l.1729.3		8	96.53	odd	8
2304.2.k.l.1729.4		8	96.11	even	8
3072.2.a.l.1.1		4	8.5	even	2
3072.2.a.l.1.4		4	4.3	odd	2
3072.2.a.r.1.1		4	8.3	odd	2	inner
3072.2.a.r.1.4		4	1.1	even	1	trivial
3072.2.d.h.1537.1		8	16.3	odd	4
3072.2.d.h.1537.4		8	16.5	even	4
3072.2.d.h.1537.5		8	16.13	even	4
3072.2.d.h.1537.8		8	16.11	odd	4
9216.2.a.bc.1.1		4	3.2	odd	2
9216.2.a.bc.1.4		4	24.11	even	2
9216.2.a.bi.1.1		4	12.11	even	2
9216.2.a.bi.1.4		4	24.5	odd	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:	\( N \)	\(=\)	\( 3072 = 2^{10} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3072.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.86370 q^{5} -2.44949 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.86370 q^{5} -2.44949 q^{7} +1.00000 q^{9} -1.46410 q^{11} +4.24264 q^{13} +3.86370 q^{15} -3.46410 q^{17} +7.46410 q^{19} -2.44949 q^{21} -2.82843 q^{23} +9.92820 q^{25} +1.00000 q^{27} +8.76268 q^{29} -7.34847 q^{31} -1.46410 q^{33} -9.46410 q^{35} +0.656339 q^{37} +4.24264 q^{39} +4.53590 q^{41} -3.46410 q^{43} +3.86370 q^{45} +2.82843 q^{47} -1.00000 q^{49} -3.46410 q^{51} -4.62158 q^{53} -5.65685 q^{55} +7.46410 q^{57} +13.8564 q^{59} -0.656339 q^{61} -2.44949 q^{63} +16.3923 q^{65} +14.9282 q^{67} -2.82843 q^{69} -6.41473 q^{71} +4.00000 q^{73} +9.92820 q^{75} +3.58630 q^{77} +2.44949 q^{79} +1.00000 q^{81} +5.46410 q^{83} -13.3843 q^{85} +8.76268 q^{87} -4.92820 q^{89} -10.3923 q^{91} -7.34847 q^{93} +28.8391 q^{95} +1.07180 q^{97} -1.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 8 q^{11} + 16 q^{19} + 12 q^{25} + 4 q^{27} + 8 q^{33} - 24 q^{35} + 32 q^{41} - 4 q^{49} + 16 q^{57} + 24 q^{65} + 32 q^{67} + 16 q^{73} + 12 q^{75} + 4 q^{81} + 8 q^{83} + 8 q^{89} + 32 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^11 + 16 * q^19 + 12 * q^25 + 4 * q^27 + 8 * q^33 - 24 * q^35 + 32 * q^41 - 4 * q^49 + 16 * q^57 + 24 * q^65 + 32 * q^67 + 16 * q^73 + 12 * q^75 + 4 * q^81 + 8 * q^83 + 8 * q^89 + 32 * q^97 + 8 * q^99

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