LMFDB - Embedded newform 1536.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, 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("1536.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.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:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.334904$ of defining polynomial
Character	$\chi$	$=$	1536.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.08402 q^{5} -5.03127 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.08402 q^{5} -5.03127 q^{7} +1.00000 q^{9} -0.828427 q^{11} +2.94725 q^{13} -2.08402 q^{15} +4.82843 q^{17} +2.82843 q^{19} -5.03127 q^{21} +4.16804 q^{23} -0.656854 q^{25} +1.00000 q^{27} -7.97852 q^{29} +5.03127 q^{31} -0.828427 q^{33} +10.4853 q^{35} +7.11529 q^{37} +2.94725 q^{39} +8.82843 q^{41} -12.4853 q^{43} -2.08402 q^{45} +4.16804 q^{47} +18.3137 q^{49} +4.82843 q^{51} +12.1466 q^{53} +1.72646 q^{55} +2.82843 q^{57} -1.65685 q^{59} +7.11529 q^{61} -5.03127 q^{63} -6.14214 q^{65} -2.34315 q^{67} +4.16804 q^{69} +10.0625 q^{71} +4.00000 q^{73} -0.656854 q^{75} +4.16804 q^{77} -5.03127 q^{79} +1.00000 q^{81} +3.17157 q^{83} -10.0625 q^{85} -7.97852 q^{87} +10.0000 q^{89} -14.8284 q^{91} +5.03127 q^{93} -5.89450 q^{95} +0.343146 q^{97} -0.828427 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 8 q^{17} + 20 q^{25} + 4 q^{27} + 8 q^{33} + 8 q^{35} + 24 q^{41} - 16 q^{43} + 28 q^{49} + 8 q^{51} + 16 q^{59} + 32 q^{65} - 32 q^{67} + 16 q^{73} + 20 q^{75} + 4 q^{81} + 24 q^{83} + 40 q^{89} - 48 q^{91} + 24 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^11 + 8 * q^17 + 20 * q^25 + 4 * q^27 + 8 * q^33 + 8 * q^35 + 24 * q^41 - 16 * q^43 + 28 * q^49 + 8 * q^51 + 16 * q^59 + 32 * q^65 - 32 * q^67 + 16 * q^73 + 20 * q^75 + 4 * q^81 + 24 * q^83 + 40 * q^89 - 48 * q^91 + 24 * 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
$5$	−2.08402	−0.932003	−0.466001	−	0.884784i	$-0.654306\pi$
$5$	−2.08402	−0.932003	−0.466001	+	0.884784i	$0.654306\pi$
$6$	0	0
$7$	−5.03127	−1.90164	−0.950821	−	0.309740i	$-0.899758\pi$
$7$	−5.03127	−1.90164	−0.950821	+	0.309740i	$0.899758\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	2.94725	0.817420	0.408710	−	0.912664i	$-0.365979\pi$
$13$	2.94725	0.817420	0.408710	+	0.912664i	$0.365979\pi$
$14$	0	0
$15$	−2.08402	−0.538092
$16$	0	0
$17$	4.82843	1.17107	0.585533	−	0.810649i	$-0.300885\pi$
$17$	4.82843	1.17107	0.585533	+	0.810649i	$0.300885\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	−5.03127	−1.09791
$22$	0	0
$23$	4.16804	0.869097	0.434549	−	0.900648i	$-0.356908\pi$
$23$	4.16804	0.869097	0.434549	+	0.900648i	$0.356908\pi$
$24$	0	0
$25$	−0.656854	−0.131371
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.97852	−1.48157	−0.740787	−	0.671740i	$-0.765547\pi$
$29$	−7.97852	−1.48157	−0.740787	+	0.671740i	$0.765547\pi$
$30$	0	0
$31$	5.03127	0.903643	0.451822	−	0.892108i	$-0.350774\pi$
$31$	5.03127	0.903643	0.451822	+	0.892108i	$0.350774\pi$
$32$	0	0
$33$	−0.828427	−0.144211
$34$	0	0
$35$	10.4853	1.77234
$36$	0	0
$37$	7.11529	1.16975	0.584874	−	0.811124i	$-0.301144\pi$
$37$	7.11529	1.16975	0.584874	+	0.811124i	$0.301144\pi$
$38$	0	0
$39$	2.94725	0.471938
$40$	0	0
$41$	8.82843	1.37877	0.689384	−	0.724396i	$-0.257881\pi$
$41$	8.82843	1.37877	0.689384	+	0.724396i	$0.257881\pi$
$42$	0	0
$43$	−12.4853	−1.90399	−0.951994	−	0.306117i	$-0.900970\pi$
$43$	−12.4853	−1.90399	−0.951994	+	0.306117i	$0.900970\pi$
$44$	0	0
$45$	−2.08402	−0.310668
$46$	0	0
$47$	4.16804	0.607972	0.303986	−	0.952677i	$-0.401682\pi$
$47$	4.16804	0.607972	0.303986	+	0.952677i	$0.401682\pi$
$48$	0	0
$49$	18.3137	2.61624
$50$	0	0
$51$	4.82843	0.676115
$52$	0	0
$53$	12.1466	1.66846	0.834230	−	0.551417i	$-0.185913\pi$
$53$	12.1466	1.66846	0.834230	+	0.551417i	$0.185913\pi$
$54$	0	0
$55$	1.72646	0.232796
$56$	0	0
$57$	2.82843	0.374634
$58$	0	0
$59$	−1.65685	−0.215704	−0.107852	−	0.994167i	$-0.534397\pi$
$59$	−1.65685	−0.215704	−0.107852	+	0.994167i	$0.534397\pi$
$60$	0	0
$61$	7.11529	0.911020	0.455510	−	0.890231i	$-0.349457\pi$
$61$	7.11529	0.911020	0.455510	+	0.890231i	$0.349457\pi$
$62$	0	0
$63$	−5.03127	−0.633881
$64$	0	0
$65$	−6.14214	−0.761838
$66$	0	0
$67$	−2.34315	−0.286261	−0.143130	−	0.989704i	$-0.545717\pi$
$67$	−2.34315	−0.286261	−0.143130	+	0.989704i	$0.545717\pi$
$68$	0	0
$69$	4.16804	0.501773
$70$	0	0
$71$	10.0625	1.19420	0.597102	−	0.802165i	$-0.296319\pi$
$71$	10.0625	1.19420	0.597102	+	0.802165i	$0.296319\pi$
$72$	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
$75$	−0.656854	−0.0758470
$76$	0	0
$77$	4.16804	0.474993
$78$	0	0
$79$	−5.03127	−0.566062	−0.283031	−	0.959111i	$-0.591340\pi$
$79$	−5.03127	−0.566062	−0.283031	+	0.959111i	$0.591340\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	3.17157	0.348125	0.174063	−	0.984735i	$-0.444310\pi$
$83$	3.17157	0.348125	0.174063	+	0.984735i	$0.444310\pi$
$84$	0	0
$85$	−10.0625	−1.09144
$86$	0	0
$87$	−7.97852	−0.855388
$88$	0	0
$89$	10.0000	1.06000	0.529999	−	0.847998i	$-0.322192\pi$
$89$	10.0000	1.06000	0.529999	+	0.847998i	$0.322192\pi$
$90$	0	0
$91$	−14.8284	−1.55444
$92$	0	0
$93$	5.03127	0.521719
$94$	0	0
$95$	−5.89450	−0.604763
$96$	0	0
$97$	0.343146	0.0348412	0.0174206	−	0.999848i	$-0.494455\pi$
$97$	0.343146	0.0348412	0.0174206	+	0.999848i	$0.494455\pi$
$98$	0	0
$99$	−0.828427	−0.0832601
$100$	0	0
$101$	−12.1466	−1.20863	−0.604314	−	0.796746i	$-0.706553\pi$
$101$	−12.1466	−1.20863	−0.604314	+	0.796746i	$0.706553\pi$
$102$	0	0
$103$	3.30481	0.325633	0.162816	−	0.986656i	$-0.447942\pi$
$103$	3.30481	0.325633	0.162816	+	0.986656i	$0.447942\pi$
$104$	0	0
$105$	10.4853	1.02326
$106$	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
$109$	−8.84175	−0.846886	−0.423443	−	0.905923i	$-0.639179\pi$
$109$	−8.84175	−0.846886	−0.423443	+	0.905923i	$0.639179\pi$
$110$	0	0
$111$	7.11529	0.675354
$112$	0	0
$113$	11.6569	1.09658	0.548292	−	0.836287i	$-0.315278\pi$
$113$	11.6569	1.09658	0.548292	+	0.836287i	$0.315278\pi$
$114$	0	0
$115$	−8.68629	−0.810001
$116$	0	0
$117$	2.94725	0.272473
$118$	0	0
$119$	−24.2931	−2.22695
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	8.82843	0.796032
$124$	0	0
$125$	11.7890	1.05444
$126$	0	0
$127$	−13.3674	−1.18616	−0.593081	−	0.805143i	$-0.702088\pi$
$127$	−13.3674	−1.18616	−0.593081	+	0.805143i	$0.702088\pi$
$128$	0	0
$129$	−12.4853	−1.09927
$130$	0	0
$131$	15.3137	1.33796	0.668982	−	0.743278i	$-0.266730\pi$
$131$	15.3137	1.33796	0.668982	+	0.743278i	$0.266730\pi$
$132$	0	0
$133$	−14.2306	−1.23395
$134$	0	0
$135$	−2.08402	−0.179364
$136$	0	0
$137$	12.1421	1.03737	0.518686	−	0.854965i	$-0.326421\pi$
$137$	12.1421	1.03737	0.518686	+	0.854965i	$0.326421\pi$
$138$	0	0
$139$	−8.00000	−0.678551	−0.339276	−	0.940687i	$-0.610182\pi$
$139$	−8.00000	−0.678551	−0.339276	+	0.940687i	$0.610182\pi$
$140$	0	0
$141$	4.16804	0.351013
$142$	0	0
$143$	−2.44158	−0.204175
$144$	0	0
$145$	16.6274	1.38083
$146$	0	0
$147$	18.3137	1.51049
$148$	0	0
$149$	−2.08402	−0.170730	−0.0853648	−	0.996350i	$-0.527206\pi$
$149$	−2.08402	−0.170730	−0.0853648	+	0.996350i	$0.527206\pi$
$150$	0	0
$151$	13.3674	1.08782	0.543910	−	0.839143i	$-0.316943\pi$
$151$	13.3674	1.08782	0.543910	+	0.839143i	$0.316943\pi$
$152$	0	0
$153$	4.82843	0.390355
$154$	0	0
$155$	−10.4853	−0.842198
$156$	0	0
$157$	−15.4514	−1.23315	−0.616577	−	0.787294i	$-0.711481\pi$
$157$	−15.4514	−1.23315	−0.616577	+	0.787294i	$0.711481\pi$
$158$	0	0
$159$	12.1466	0.963285
$160$	0	0
$161$	−20.9706	−1.65271
$162$	0	0
$163$	−18.1421	−1.42100	−0.710501	−	0.703696i	$-0.751532\pi$
$163$	−18.1421	−1.42100	−0.710501	+	0.703696i	$0.751532\pi$
$164$	0	0
$165$	1.72646	0.134405
$166$	0	0
$167$	−14.2306	−1.10120	−0.550598	−	0.834771i	$-0.685600\pi$
$167$	−14.2306	−1.10120	−0.550598	+	0.834771i	$0.685600\pi$
$168$	0	0
$169$	−4.31371	−0.331824
$170$	0	0
$171$	2.82843	0.216295
$172$	0	0
$173$	−6.25206	−0.475336	−0.237668	−	0.971346i	$-0.576383\pi$
$173$	−6.25206	−0.475336	−0.237668	+	0.971346i	$0.576383\pi$
$174$	0	0
$175$	3.30481	0.249820
$176$	0	0
$177$	−1.65685	−0.124537
$178$	0	0
$179$	−20.9706	−1.56741	−0.783707	−	0.621131i	$-0.786673\pi$
$179$	−20.9706	−1.56741	−0.783707	+	0.621131i	$0.786673\pi$
$180$	0	0
$181$	8.84175	0.657202	0.328601	−	0.944469i	$-0.393423\pi$
$181$	8.84175	0.657202	0.328601	+	0.944469i	$0.393423\pi$
$182$	0	0
$183$	7.11529	0.525978
$184$	0	0
$185$	−14.8284	−1.09021
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	−5.03127	−0.365971
$190$	0	0
$191$	26.0196	1.88271	0.941356	−	0.337415i	$-0.109553\pi$
$191$	26.0196	1.88271	0.941356	+	0.337415i	$0.109553\pi$
$192$	0	0
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	0	0
$195$	−6.14214	−0.439847
$196$	0	0
$197$	20.4827	1.45933	0.729664	−	0.683806i	$-0.239676\pi$
$197$	20.4827	1.45933	0.729664	+	0.683806i	$0.239676\pi$
$198$	0	0
$199$	−5.03127	−0.356657	−0.178329	−	0.983971i	$-0.557069\pi$
$199$	−5.03127	−0.356657	−0.178329	+	0.983971i	$0.557069\pi$
$200$	0	0
$201$	−2.34315	−0.165273
$202$	0	0
$203$	40.1421	2.81743
$204$	0	0
$205$	−18.3986	−1.28502
$206$	0	0
$207$	4.16804	0.289699
$208$	0	0
$209$	−2.34315	−0.162079
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	0	0
$213$	10.0625	0.689474
$214$	0	0
$215$	26.0196	1.77452
$216$	0	0
$217$	−25.3137	−1.71841
$218$	0	0
$219$	4.00000	0.270295
$220$	0	0
$221$	14.2306	0.957253
$222$	0	0
$223$	13.3674	0.895145	0.447572	−	0.894248i	$-0.352289\pi$
$223$	13.3674	0.895145	0.447572	+	0.894248i	$0.352289\pi$
$224$	0	0
$225$	−0.656854	−0.0437903
$226$	0	0
$227$	16.1421	1.07139	0.535696	−	0.844411i	$-0.320049\pi$
$227$	16.1421	1.07139	0.535696	+	0.844411i	$0.320049\pi$
$228$	0	0
$229$	5.38883	0.356104	0.178052	−	0.984021i	$-0.443020\pi$
$229$	5.38883	0.356104	0.178052	+	0.984021i	$0.443020\pi$
$230$	0	0
$231$	4.16804	0.274237
$232$	0	0
$233$	1.31371	0.0860639	0.0430320	−	0.999074i	$-0.486298\pi$
$233$	1.31371	0.0860639	0.0430320	+	0.999074i	$0.486298\pi$
$234$	0	0
$235$	−8.68629	−0.566631
$236$	0	0
$237$	−5.03127	−0.326816
$238$	0	0
$239$	−14.2306	−0.920500	−0.460250	−	0.887789i	$-0.652240\pi$
$239$	−14.2306	−0.920500	−0.460250	+	0.887789i	$0.652240\pi$
$240$	0	0
$241$	16.9706	1.09317	0.546585	−	0.837404i	$-0.315928\pi$
$241$	16.9706	1.09317	0.546585	+	0.837404i	$0.315928\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−38.1662	−2.43835
$246$	0	0
$247$	8.33609	0.530412
$248$	0	0
$249$	3.17157	0.200990
$250$	0	0
$251$	15.1716	0.957621	0.478811	−	0.877918i	$-0.341068\pi$
$251$	15.1716	0.957621	0.478811	+	0.877918i	$0.341068\pi$
$252$	0	0
$253$	−3.45292	−0.217083
$254$	0	0
$255$	−10.0625	−0.630141
$256$	0	0
$257$	−8.34315	−0.520431	−0.260216	−	0.965551i	$-0.583794\pi$
$257$	−8.34315	−0.520431	−0.260216	+	0.965551i	$0.583794\pi$
$258$	0	0
$259$	−35.7990	−2.22444
$260$	0	0
$261$	−7.97852	−0.493858
$262$	0	0
$263$	−8.33609	−0.514025	−0.257013	−	0.966408i	$-0.582738\pi$
$263$	−8.33609	−0.514025	−0.257013	+	0.966408i	$0.582738\pi$
$264$	0	0
$265$	−25.3137	−1.55501
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	7.97852	0.486459	0.243230	−	0.969969i	$-0.421793\pi$
$269$	7.97852	0.486459	0.243230	+	0.969969i	$0.421793\pi$
$270$	0	0
$271$	−3.30481	−0.200753	−0.100377	−	0.994950i	$-0.532005\pi$
$271$	−3.30481	−0.200753	−0.100377	+	0.994950i	$0.532005\pi$
$272$	0	0
$273$	−14.8284	−0.897457
$274$	0	0
$275$	0.544156	0.0328138
$276$	0	0
$277$	25.5139	1.53298	0.766492	−	0.642254i	$-0.222001\pi$
$277$	25.5139	1.53298	0.766492	+	0.642254i	$0.222001\pi$
$278$	0	0
$279$	5.03127	0.301214
$280$	0	0
$281$	11.6569	0.695390	0.347695	−	0.937608i	$-0.386965\pi$
$281$	11.6569	0.695390	0.347695	+	0.937608i	$0.386965\pi$
$282$	0	0
$283$	4.97056	0.295469	0.147735	−	0.989027i	$-0.452802\pi$
$283$	4.97056	0.295469	0.147735	+	0.989027i	$0.452802\pi$
$284$	0	0
$285$	−5.89450	−0.349160
$286$	0	0
$287$	−44.4182	−2.62193
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0.343146	0.0201156
$292$	0	0
$293$	12.1466	0.709610	0.354805	−	0.934940i	$-0.384547\pi$
$293$	12.1466	0.709610	0.354805	+	0.934940i	$0.384547\pi$
$294$	0	0
$295$	3.45292	0.201037
$296$	0	0
$297$	−0.828427	−0.0480702
$298$	0	0
$299$	12.2843	0.710418
$300$	0	0
$301$	62.8169	3.62070
$302$	0	0
$303$	−12.1466	−0.697802
$304$	0	0
$305$	−14.8284	−0.849073
$306$	0	0
$307$	−10.3431	−0.590315	−0.295157	−	0.955449i	$-0.595372\pi$
$307$	−10.3431	−0.590315	−0.295157	+	0.955449i	$0.595372\pi$
$308$	0	0
$309$	3.30481	0.188004
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−15.3137	−0.865582	−0.432791	−	0.901494i	$-0.642471\pi$
$313$	−15.3137	−0.865582	−0.432791	+	0.901494i	$0.642471\pi$
$314$	0	0
$315$	10.4853	0.590779
$316$	0	0
$317$	−23.9356	−1.34436	−0.672178	−	0.740390i	$-0.734641\pi$
$317$	−23.9356	−1.34436	−0.672178	+	0.740390i	$0.734641\pi$
$318$	0	0
$319$	6.60963	0.370068
$320$	0	0
$321$	−4.00000	−0.223258
$322$	0	0
$323$	13.6569	0.759888
$324$	0	0
$325$	−1.93591	−0.107385
$326$	0	0
$327$	−8.84175	−0.488950
$328$	0	0
$329$	−20.9706	−1.15614
$330$	0	0
$331$	21.6569	1.19037	0.595184	−	0.803589i	$-0.297079\pi$
$331$	21.6569	1.19037	0.595184	+	0.803589i	$0.297079\pi$
$332$	0	0
$333$	7.11529	0.389916
$334$	0	0
$335$	4.88317	0.266796
$336$	0	0
$337$	−20.0000	−1.08947	−0.544735	−	0.838608i	$-0.683370\pi$
$337$	−20.0000	−1.08947	−0.544735	+	0.838608i	$0.683370\pi$
$338$	0	0
$339$	11.6569	0.633113
$340$	0	0
$341$	−4.16804	−0.225712
$342$	0	0
$343$	−56.9224	−3.07352
$344$	0	0
$345$	−8.68629	−0.467654
$346$	0	0
$347$	−4.82843	−0.259204	−0.129602	−	0.991566i	$-0.541370\pi$
$347$	−4.82843	−0.259204	−0.129602	+	0.991566i	$0.541370\pi$
$348$	0	0
$349$	27.2404	1.45814	0.729072	−	0.684437i	$-0.239952\pi$
$349$	27.2404	1.45814	0.729072	+	0.684437i	$0.239952\pi$
$350$	0	0
$351$	2.94725	0.157313
$352$	0	0
$353$	−21.3137	−1.13441	−0.567207	−	0.823575i	$-0.691976\pi$
$353$	−21.3137	−1.13441	−0.567207	+	0.823575i	$0.691976\pi$
$354$	0	0
$355$	−20.9706	−1.11300
$356$	0	0
$357$	−24.2931	−1.28573
$358$	0	0
$359$	6.60963	0.348843	0.174421	−	0.984671i	$-0.444195\pi$
$359$	6.60963	0.348843	0.174421	+	0.984671i	$0.444195\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	−10.3137	−0.541329
$364$	0	0
$365$	−8.33609	−0.436331
$366$	0	0
$367$	16.8203	0.878011	0.439006	−	0.898484i	$-0.355331\pi$
$367$	16.8203	0.878011	0.439006	+	0.898484i	$0.355331\pi$
$368$	0	0
$369$	8.82843	0.459590
$370$	0	0
$371$	−61.1127	−3.17281
$372$	0	0
$373$	−18.9043	−0.978828	−0.489414	−	0.872052i	$-0.662789\pi$
$373$	−18.9043	−0.978828	−0.489414	+	0.872052i	$0.662789\pi$
$374$	0	0
$375$	11.7890	0.608782
$376$	0	0
$377$	−23.5147	−1.21107
$378$	0	0
$379$	16.4853	0.846792	0.423396	−	0.905945i	$-0.360838\pi$
$379$	16.4853	0.846792	0.423396	+	0.905945i	$0.360838\pi$
$380$	0	0
$381$	−13.3674	−0.684831
$382$	0	0
$383$	5.89450	0.301195	0.150598	−	0.988595i	$-0.451880\pi$
$383$	5.89450	0.301195	0.150598	+	0.988595i	$0.451880\pi$
$384$	0	0
$385$	−8.68629	−0.442694
$386$	0	0
$387$	−12.4853	−0.634663
$388$	0	0
$389$	2.08402	0.105664	0.0528320	−	0.998603i	$-0.483175\pi$
$389$	2.08402	0.105664	0.0528320	+	0.998603i	$0.483175\pi$
$390$	0	0
$391$	20.1251	1.01777
$392$	0	0
$393$	15.3137	0.772474
$394$	0	0
$395$	10.4853	0.527572
$396$	0	0
$397$	−15.4514	−0.775483	−0.387741	−	0.921768i	$-0.626745\pi$
$397$	−15.4514	−0.775483	−0.387741	+	0.921768i	$0.626745\pi$
$398$	0	0
$399$	−14.2306	−0.712421
$400$	0	0
$401$	−4.14214	−0.206848	−0.103424	−	0.994637i	$-0.532980\pi$
$401$	−4.14214	−0.206848	−0.103424	+	0.994637i	$0.532980\pi$
$402$	0	0
$403$	14.8284	0.738657
$404$	0	0
$405$	−2.08402	−0.103556
$406$	0	0
$407$	−5.89450	−0.292180
$408$	0	0
$409$	7.65685	0.378607	0.189304	−	0.981919i	$-0.439377\pi$
$409$	7.65685	0.378607	0.189304	+	0.981919i	$0.439377\pi$
$410$	0	0
$411$	12.1421	0.598927
$412$	0	0
$413$	8.33609	0.410192
$414$	0	0
$415$	−6.60963	−0.324454
$416$	0	0
$417$	−8.00000	−0.391762
$418$	0	0
$419$	20.8284	1.01754	0.508768	−	0.860904i	$-0.330101\pi$
$419$	20.8284	1.01754	0.508768	+	0.860904i	$0.330101\pi$
$420$	0	0
$421$	5.38883	0.262636	0.131318	−	0.991340i	$-0.458079\pi$
$421$	5.38883	0.262636	0.131318	+	0.991340i	$0.458079\pi$
$422$	0	0
$423$	4.16804	0.202657
$424$	0	0
$425$	−3.17157	−0.153844
$426$	0	0
$427$	−35.7990	−1.73243
$428$	0	0
$429$	−2.44158	−0.117881
$430$	0	0
$431$	15.9570	0.768624	0.384312	−	0.923203i	$-0.374439\pi$
$431$	15.9570	0.768624	0.384312	+	0.923203i	$0.374439\pi$
$432$	0	0
$433$	−20.6274	−0.991290	−0.495645	−	0.868525i	$-0.665068\pi$
$433$	−20.6274	−0.991290	−0.495645	+	0.868525i	$0.665068\pi$
$434$	0	0
$435$	16.6274	0.797224
$436$	0	0
$437$	11.7890	0.563945
$438$	0	0
$439$	−8.48419	−0.404928	−0.202464	−	0.979290i	$-0.564895\pi$
$439$	−8.48419	−0.404928	−0.202464	+	0.979290i	$0.564895\pi$
$440$	0	0
$441$	18.3137	0.872081
$442$	0	0
$443$	−1.51472	−0.0719665	−0.0359832	−	0.999352i	$-0.511456\pi$
$443$	−1.51472	−0.0719665	−0.0359832	+	0.999352i	$0.511456\pi$
$444$	0	0
$445$	−20.8402	−0.987921
$446$	0	0
$447$	−2.08402	−0.0985708
$448$	0	0
$449$	34.4853	1.62746	0.813731	−	0.581242i	$-0.197433\pi$
$449$	34.4853	1.62746	0.813731	+	0.581242i	$0.197433\pi$
$450$	0	0
$451$	−7.31371	−0.344389
$452$	0	0
$453$	13.3674	0.628053
$454$	0	0
$455$	30.9028	1.44874
$456$	0	0
$457$	−12.3431	−0.577388	−0.288694	−	0.957421i	$-0.593221\pi$
$457$	−12.3431	−0.577388	−0.288694	+	0.957421i	$0.593221\pi$
$458$	0	0
$459$	4.82843	0.225372
$460$	0	0
$461$	−29.8301	−1.38933	−0.694663	−	0.719336i	$-0.744446\pi$
$461$	−29.8301	−1.38933	−0.694663	+	0.719336i	$0.744446\pi$
$462$	0	0
$463$	23.4299	1.08888	0.544440	−	0.838800i	$-0.316742\pi$
$463$	23.4299	1.08888	0.544440	+	0.838800i	$0.316742\pi$
$464$	0	0
$465$	−10.4853	−0.486243
$466$	0	0
$467$	−1.79899	−0.0832473	−0.0416237	−	0.999133i	$-0.513253\pi$
$467$	−1.79899	−0.0832473	−0.0416237	+	0.999133i	$0.513253\pi$
$468$	0	0
$469$	11.7890	0.544366
$470$	0	0
$471$	−15.4514	−0.711962
$472$	0	0
$473$	10.3431	0.475578
$474$	0	0
$475$	−1.85786	−0.0852447
$476$	0	0
$477$	12.1466	0.556153
$478$	0	0
$479$	41.9766	1.91796	0.958981	−	0.283471i	$-0.0914858\pi$
$479$	41.9766	1.91796	0.958981	+	0.283471i	$0.0914858\pi$
$480$	0	0
$481$	20.9706	0.956175
$482$	0	0
$483$	−20.9706	−0.954194
$484$	0	0
$485$	−0.715123	−0.0324721
$486$	0	0
$487$	6.75773	0.306222	0.153111	−	0.988209i	$-0.451071\pi$
$487$	6.75773	0.306222	0.153111	+	0.988209i	$0.451071\pi$
$488$	0	0
$489$	−18.1421	−0.820416
$490$	0	0
$491$	−9.65685	−0.435808	−0.217904	−	0.975970i	$-0.569922\pi$
$491$	−9.65685	−0.435808	−0.217904	+	0.975970i	$0.569922\pi$
$492$	0	0
$493$	−38.5237	−1.73502
$494$	0	0
$495$	1.72646	0.0775986
$496$	0	0
$497$	−50.6274	−2.27095
$498$	0	0
$499$	−41.6569	−1.86482	−0.932408	−	0.361406i	$-0.882297\pi$
$499$	−41.6569	−1.86482	−0.932408	+	0.361406i	$0.882297\pi$
$500$	0	0
$501$	−14.2306	−0.635776
$502$	0	0
$503$	18.3986	0.820354	0.410177	−	0.912006i	$-0.365467\pi$
$503$	18.3986	0.820354	0.410177	+	0.912006i	$0.365467\pi$
$504$	0	0
$505$	25.3137	1.12645
$506$	0	0
$507$	−4.31371	−0.191579
$508$	0	0
$509$	−16.3146	−0.723132	−0.361566	−	0.932346i	$-0.617758\pi$
$509$	−16.3146	−0.723132	−0.361566	+	0.932346i	$0.617758\pi$
$510$	0	0
$511$	−20.1251	−0.890282
$512$	0	0
$513$	2.82843	0.124878
$514$	0	0
$515$	−6.88730	−0.303491
$516$	0	0
$517$	−3.45292	−0.151859
$518$	0	0
$519$	−6.25206	−0.274435
$520$	0	0
$521$	20.8284	0.912510	0.456255	−	0.889849i	$-0.349190\pi$
$521$	20.8284	0.912510	0.456255	+	0.889849i	$0.349190\pi$
$522$	0	0
$523$	18.8284	0.823310	0.411655	−	0.911340i	$-0.364951\pi$
$523$	18.8284	0.823310	0.411655	+	0.911340i	$0.364951\pi$
$524$	0	0
$525$	3.30481	0.144234
$526$	0	0
$527$	24.2931	1.05823
$528$	0	0
$529$	−5.62742	−0.244670
$530$	0	0
$531$	−1.65685	−0.0719014
$532$	0	0
$533$	26.0196	1.12703
$534$	0	0
$535$	8.33609	0.360400
$536$	0	0
$537$	−20.9706	−0.904947
$538$	0	0
$539$	−15.1716	−0.653486
$540$	0	0
$541$	−11.2833	−0.485109	−0.242554	−	0.970138i	$-0.577985\pi$
$541$	−11.2833	−0.485109	−0.242554	+	0.970138i	$0.577985\pi$
$542$	0	0
$543$	8.84175	0.379436
$544$	0	0
$545$	18.4264	0.789301
$546$	0	0
$547$	31.7990	1.35963	0.679813	−	0.733385i	$-0.262061\pi$
$547$	31.7990	1.35963	0.679813	+	0.733385i	$0.262061\pi$
$548$	0	0
$549$	7.11529	0.303673
$550$	0	0
$551$	−22.5667	−0.961373
$552$	0	0
$553$	25.3137	1.07645
$554$	0	0
$555$	−14.8284	−0.629432
$556$	0	0
$557$	14.5882	0.618120	0.309060	−	0.951043i	$-0.399986\pi$
$557$	14.5882	0.618120	0.309060	+	0.951043i	$0.399986\pi$
$558$	0	0
$559$	−36.7973	−1.55636
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	−0.828427	−0.0349140	−0.0174570	−	0.999848i	$-0.505557\pi$
$563$	−0.828427	−0.0349140	−0.0174570	+	0.999848i	$0.505557\pi$
$564$	0	0
$565$	−24.2931	−1.02202
$566$	0	0
$567$	−5.03127	−0.211294
$568$	0	0
$569$	23.4558	0.983320	0.491660	−	0.870787i	$-0.336390\pi$
$569$	23.4558	0.983320	0.491660	+	0.870787i	$0.336390\pi$
$570$	0	0
$571$	23.3137	0.975648	0.487824	−	0.872942i	$-0.337791\pi$
$571$	23.3137	0.975648	0.487824	+	0.872942i	$0.337791\pi$
$572$	0	0
$573$	26.0196	1.08698
$574$	0	0
$575$	−2.73780	−0.114174
$576$	0	0
$577$	0	0		−	1.00000i	$-0.5\pi$
$577$	0	0			1.00000i	$0.5\pi$
$578$	0	0
$579$	−4.00000	−0.166234
$580$	0	0
$581$	−15.9570	−0.662010
$582$	0	0
$583$	−10.0625	−0.416748
$584$	0	0
$585$	−6.14214	−0.253946
$586$	0	0
$587$	−4.97056	−0.205157	−0.102579	−	0.994725i	$-0.532709\pi$
$587$	−4.97056	−0.205157	−0.102579	+	0.994725i	$0.532709\pi$
$588$	0	0
$589$	14.2306	0.586361
$590$	0	0
$591$	20.4827	0.842544
$592$	0	0
$593$	−2.97056	−0.121986	−0.0609932	−	0.998138i	$-0.519427\pi$
$593$	−2.97056	−0.121986	−0.0609932	+	0.998138i	$0.519427\pi$
$594$	0	0
$595$	50.6274	2.07552
$596$	0	0
$597$	−5.03127	−0.205916
$598$	0	0
$599$	−0.715123	−0.0292191	−0.0146096	−	0.999893i	$-0.504651\pi$
$599$	−0.715123	−0.0292191	−0.0146096	+	0.999893i	$0.504651\pi$
$600$	0	0
$601$	−4.68629	−0.191158	−0.0955789	−	0.995422i	$-0.530470\pi$
$601$	−4.68629	−0.191158	−0.0955789	+	0.995422i	$0.530470\pi$
$602$	0	0
$603$	−2.34315	−0.0954203
$604$	0	0
$605$	21.4940	0.873855
$606$	0	0
$607$	−25.1564	−1.02107	−0.510533	−	0.859858i	$-0.670552\pi$
$607$	−25.1564	−1.02107	−0.510533	+	0.859858i	$0.670552\pi$
$608$	0	0
$609$	40.1421	1.62664
$610$	0	0
$611$	12.2843	0.496968
$612$	0	0
$613$	−10.5682	−0.426846	−0.213423	−	0.976960i	$-0.568461\pi$
$613$	−10.5682	−0.426846	−0.213423	+	0.976960i	$0.568461\pi$
$614$	0	0
$615$	−18.3986	−0.741904
$616$	0	0
$617$	−43.9411	−1.76900	−0.884502	−	0.466537i	$-0.845501\pi$
$617$	−43.9411	−1.76900	−0.884502	+	0.466537i	$0.845501\pi$
$618$	0	0
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	4.16804	0.167258
$622$	0	0
$623$	−50.3127	−2.01574
$624$	0	0
$625$	−21.2843	−0.851371
$626$	0	0
$627$	−2.34315	−0.0935762
$628$	0	0
$629$	34.3557	1.36985
$630$	0	0
$631$	35.2189	1.40204	0.701021	−	0.713140i	$-0.252728\pi$
$631$	35.2189	1.40204	0.701021	+	0.713140i	$0.252728\pi$
$632$	0	0
$633$	8.00000	0.317971
$634$	0	0
$635$	27.8579	1.10551
$636$	0	0
$637$	53.9751	2.13857
$638$	0	0
$639$	10.0625	0.398068
$640$	0	0
$641$	32.8284	1.29664	0.648322	−	0.761366i	$-0.275471\pi$
$641$	32.8284	1.29664	0.648322	+	0.761366i	$0.275471\pi$
$642$	0	0
$643$	3.51472	0.138607	0.0693035	−	0.997596i	$-0.477922\pi$
$643$	3.51472	0.138607	0.0693035	+	0.997596i	$0.477922\pi$
$644$	0	0
$645$	26.0196	1.02452
$646$	0	0
$647$	−44.4182	−1.74626	−0.873130	−	0.487487i	$-0.837914\pi$
$647$	−44.4182	−1.74626	−0.873130	+	0.487487i	$0.837914\pi$
$648$	0	0
$649$	1.37258	0.0538786
$650$	0	0
$651$	−25.3137	−0.992122
$652$	0	0
$653$	30.5452	1.19533	0.597663	−	0.801747i	$-0.296096\pi$
$653$	30.5452	1.19533	0.597663	+	0.801747i	$0.296096\pi$
$654$	0	0
$655$	−31.9141	−1.24699
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	−9.65685	−0.376178	−0.188089	−	0.982152i	$-0.560229\pi$
$659$	−9.65685	−0.376178	−0.188089	+	0.982152i	$0.560229\pi$
$660$	0	0
$661$	−27.2404	−1.05953	−0.529764	−	0.848145i	$-0.677720\pi$
$661$	−27.2404	−1.05953	−0.529764	+	0.848145i	$0.677720\pi$
$662$	0	0
$663$	14.2306	0.552670
$664$	0	0
$665$	29.6569	1.15004
$666$	0	0
$667$	−33.2548	−1.28763
$668$	0	0
$669$	13.3674	0.516812
$670$	0	0
$671$	−5.89450	−0.227555
$672$	0	0
$673$	37.3137	1.43834	0.719169	−	0.694835i	$-0.244523\pi$
$673$	37.3137	1.43834	0.719169	+	0.694835i	$0.244523\pi$
$674$	0	0
$675$	−0.656854	−0.0252823
$676$	0	0
$677$	−22.2091	−0.853566	−0.426783	−	0.904354i	$-0.640353\pi$
$677$	−22.2091	−0.853566	−0.426783	+	0.904354i	$0.640353\pi$
$678$	0	0
$679$	−1.72646	−0.0662555
$680$	0	0
$681$	16.1421	0.618568
$682$	0	0
$683$	−22.4853	−0.860375	−0.430188	−	0.902739i	$-0.641553\pi$
$683$	−22.4853	−0.860375	−0.430188	+	0.902739i	$0.641553\pi$
$684$	0	0
$685$	−25.3045	−0.966834
$686$	0	0
$687$	5.38883	0.205597
$688$	0	0
$689$	35.7990	1.36383
$690$	0	0
$691$	33.1716	1.26191	0.630953	−	0.775821i	$-0.282664\pi$
$691$	33.1716	1.26191	0.630953	+	0.775821i	$0.282664\pi$
$692$	0	0
$693$	4.16804	0.158331
$694$	0	0
$695$	16.6722	0.632412
$696$	0	0
$697$	42.6274	1.61463
$698$	0	0
$699$	1.31371	0.0496890
$700$	0	0
$701$	4.52560	0.170930	0.0854649	−	0.996341i	$-0.472762\pi$
$701$	4.52560	0.170930	0.0854649	+	0.996341i	$0.472762\pi$
$702$	0	0
$703$	20.1251	0.759032
$704$	0	0
$705$	−8.68629	−0.327145
$706$	0	0
$707$	61.1127	2.29838
$708$	0	0
$709$	−19.6194	−0.736823	−0.368411	−	0.929663i	$-0.620098\pi$
$709$	−19.6194	−0.736823	−0.368411	+	0.929663i	$0.620098\pi$
$710$	0	0
$711$	−5.03127	−0.188687
$712$	0	0
$713$	20.9706	0.785354
$714$	0	0
$715$	5.08831	0.190292
$716$	0	0
$717$	−14.2306	−0.531451
$718$	0	0
$719$	−6.60963	−0.246497	−0.123249	−	0.992376i	$-0.539331\pi$
$719$	−6.60963	−0.246497	−0.123249	+	0.992376i	$0.539331\pi$
$720$	0	0
$721$	−16.6274	−0.619237
$722$	0	0
$723$	16.9706	0.631142
$724$	0	0
$725$	5.24073	0.194636
$726$	0	0
$727$	36.9454	1.37023	0.685114	−	0.728436i	$-0.259752\pi$
$727$	36.9454	1.37023	0.685114	+	0.728436i	$0.259752\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−60.2843	−2.22969
$732$	0	0
$733$	−43.1974	−1.59553	−0.797767	−	0.602966i	$-0.793985\pi$
$733$	−43.1974	−1.59553	−0.797767	+	0.602966i	$0.793985\pi$
$734$	0	0
$735$	−38.1662	−1.40778
$736$	0	0
$737$	1.94113	0.0715023
$738$	0	0
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	8.33609	0.306234
$742$	0	0
$743$	17.6835	0.648745	0.324373	−	0.945929i	$-0.394847\pi$
$743$	17.6835	0.648745	0.324373	+	0.945929i	$0.394847\pi$
$744$	0	0
$745$	4.34315	0.159121
$746$	0	0
$747$	3.17157	0.116042
$748$	0	0
$749$	20.1251	0.735355
$750$	0	0
$751$	−5.03127	−0.183594	−0.0917969	−	0.995778i	$-0.529261\pi$
$751$	−5.03127	−0.183594	−0.0917969	+	0.995778i	$0.529261\pi$
$752$	0	0
$753$	15.1716	0.552883
$754$	0	0
$755$	−27.8579	−1.01385
$756$	0	0
$757$	17.1778	0.624339	0.312170	−	0.950026i	$-0.398944\pi$
$757$	17.1778	0.624339	0.312170	+	0.950026i	$0.398944\pi$
$758$	0	0
$759$	−3.45292	−0.125333
$760$	0	0
$761$	−42.7696	−1.55040	−0.775198	−	0.631719i	$-0.782350\pi$
$761$	−42.7696	−1.55040	−0.775198	+	0.631719i	$0.782350\pi$
$762$	0	0
$763$	44.4853	1.61048
$764$	0	0
$765$	−10.0625	−0.363812
$766$	0	0
$767$	−4.88317	−0.176321
$768$	0	0
$769$	12.0000	0.432731	0.216366	−	0.976312i	$-0.430580\pi$
$769$	12.0000	0.432731	0.216366	+	0.976312i	$0.430580\pi$
$770$	0	0
$771$	−8.34315	−0.300471
$772$	0	0
$773$	−32.2717	−1.16073	−0.580365	−	0.814356i	$-0.697090\pi$
$773$	−32.2717	−1.16073	−0.580365	+	0.814356i	$0.697090\pi$
$774$	0	0
$775$	−3.30481	−0.118712
$776$	0	0
$777$	−35.7990	−1.28428
$778$	0	0
$779$	24.9706	0.894663
$780$	0	0
$781$	−8.33609	−0.298289
$782$	0	0
$783$	−7.97852	−0.285129
$784$	0	0
$785$	32.2010	1.14930
$786$	0	0
$787$	52.7696	1.88103	0.940516	−	0.339750i	$-0.110343\pi$
$787$	52.7696	1.88103	0.940516	+	0.339750i	$0.110343\pi$
$788$	0	0
$789$	−8.33609	−0.296773
$790$	0	0
$791$	−58.6488	−2.08531
$792$	0	0
$793$	20.9706	0.744687
$794$	0	0
$795$	−25.3137	−0.897785
$796$	0	0
$797$	−30.5452	−1.08197	−0.540983	−	0.841033i	$-0.681948\pi$
$797$	−30.5452	−1.08197	−0.540983	+	0.841033i	$0.681948\pi$
$798$	0	0
$799$	20.1251	0.711975
$800$	0	0
$801$	10.0000	0.353333
$802$	0	0
$803$	−3.31371	−0.116938
$804$	0	0
$805$	43.7031	1.54033
$806$	0	0
$807$	7.97852	0.280857
$808$	0	0
$809$	−3.45584	−0.121501	−0.0607505	−	0.998153i	$-0.519349\pi$
$809$	−3.45584	−0.121501	−0.0607505	+	0.998153i	$0.519349\pi$
$810$	0	0
$811$	49.4558	1.73663	0.868315	−	0.496014i	$-0.165203\pi$
$811$	49.4558	1.73663	0.868315	+	0.496014i	$0.165203\pi$
$812$	0	0
$813$	−3.30481	−0.115905
$814$	0	0
$815$	37.8086	1.32438
$816$	0	0
$817$	−35.3137	−1.23547
$818$	0	0
$819$	−14.8284	−0.518147
$820$	0	0
$821$	36.4397	1.27175	0.635877	−	0.771790i	$-0.280638\pi$
$821$	36.4397	1.27175	0.635877	+	0.771790i	$0.280638\pi$
$822$	0	0
$823$	−13.3674	−0.465957	−0.232978	−	0.972482i	$-0.574847\pi$
$823$	−13.3674	−0.465957	−0.232978	+	0.972482i	$0.574847\pi$
$824$	0	0
$825$	0.544156	0.0189451
$826$	0	0
$827$	−53.9411	−1.87572	−0.937858	−	0.347018i	$-0.887194\pi$
$827$	−53.9411	−1.87572	−0.937858	+	0.347018i	$0.887194\pi$
$828$	0	0
$829$	2.94725	0.102362	0.0511811	−	0.998689i	$-0.483701\pi$
$829$	2.94725	0.102362	0.0511811	+	0.998689i	$0.483701\pi$
$830$	0	0
$831$	25.5139	0.885068
$832$	0	0
$833$	88.4264	3.06379
$834$	0	0
$835$	29.6569	1.02632
$836$	0	0
$837$	5.03127	0.173906
$838$	0	0
$839$	41.9766	1.44919	0.724597	−	0.689172i	$-0.242026\pi$
$839$	41.9766	1.44919	0.724597	+	0.689172i	$0.242026\pi$
$840$	0	0
$841$	34.6569	1.19506
$842$	0	0
$843$	11.6569	0.401483
$844$	0	0
$845$	8.98986	0.309261
$846$	0	0
$847$	51.8911	1.78300
$848$	0	0
$849$	4.97056	0.170589
$850$	0	0
$851$	29.6569	1.01662
$852$	0	0
$853$	−15.4514	−0.529045	−0.264523	−	0.964379i	$-0.585214\pi$
$853$	−15.4514	−0.529045	−0.264523	+	0.964379i	$0.585214\pi$
$854$	0	0
$855$	−5.89450	−0.201588
$856$	0	0
$857$	30.4853	1.04136	0.520679	−	0.853753i	$-0.325679\pi$
$857$	30.4853	1.04136	0.520679	+	0.853753i	$0.325679\pi$
$858$	0	0
$859$	21.4558	0.732064	0.366032	−	0.930602i	$-0.380716\pi$
$859$	21.4558	0.732064	0.366032	+	0.930602i	$0.380716\pi$
$860$	0	0
$861$	−44.4182	−1.51377
$862$	0	0
$863$	14.2306	0.484415	0.242207	−	0.970224i	$-0.422129\pi$
$863$	14.2306	0.484415	0.242207	+	0.970224i	$0.422129\pi$
$864$	0	0
$865$	13.0294	0.443014
$866$	0	0
$867$	6.31371	0.214425
$868$	0	0
$869$	4.16804	0.141391
$870$	0	0
$871$	−6.90584	−0.233995
$872$	0	0
$873$	0.343146	0.0116137
$874$	0	0
$875$	−59.3137	−2.00517
$876$	0	0
$877$	22.3572	0.754950	0.377475	−	0.926020i	$-0.376792\pi$
$877$	22.3572	0.754950	0.377475	+	0.926020i	$0.376792\pi$
$878$	0	0
$879$	12.1466	0.409694
$880$	0	0
$881$	−38.9706	−1.31295	−0.656476	−	0.754347i	$-0.727954\pi$
$881$	−38.9706	−1.31295	−0.656476	+	0.754347i	$0.727954\pi$
$882$	0	0
$883$	24.7696	0.833562	0.416781	−	0.909007i	$-0.363158\pi$
$883$	24.7696	0.833562	0.416781	+	0.909007i	$0.363158\pi$
$884$	0	0
$885$	3.45292	0.116069
$886$	0	0
$887$	−8.33609	−0.279898	−0.139949	−	0.990159i	$-0.544694\pi$
$887$	−8.33609	−0.279898	−0.139949	+	0.990159i	$0.544694\pi$
$888$	0	0
$889$	67.2548	2.25565
$890$	0	0
$891$	−0.828427	−0.0277534
$892$	0	0
$893$	11.7890	0.394504
$894$	0	0
$895$	43.7031	1.46083
$896$	0	0
$897$	12.2843	0.410160
$898$	0	0
$899$	−40.1421	−1.33882
$900$	0	0
$901$	58.6488	1.95388
$902$	0	0
$903$	62.8169	2.09041
$904$	0	0
$905$	−18.4264	−0.612514
$906$	0	0
$907$	−23.7990	−0.790232	−0.395116	−	0.918631i	$-0.629296\pi$
$907$	−23.7990	−0.790232	−0.395116	+	0.918631i	$0.629296\pi$
$908$	0	0
$909$	−12.1466	−0.402876
$910$	0	0
$911$	−34.3557	−1.13825	−0.569127	−	0.822249i	$-0.692719\pi$
$911$	−34.3557	−1.13825	−0.569127	+	0.822249i	$0.692719\pi$
$912$	0	0
$913$	−2.62742	−0.0869548
$914$	0	0
$915$	−14.8284	−0.490213
$916$	0	0
$917$	−77.0474	−2.54433
$918$	0	0
$919$	15.0938	0.497899	0.248950	−	0.968516i	$-0.419915\pi$
$919$	15.0938	0.497899	0.248950	+	0.968516i	$0.419915\pi$
$920$	0	0
$921$	−10.3431	−0.340818
$922$	0	0
$923$	29.6569	0.976167
$924$	0	0
$925$	−4.67371	−0.153671
$926$	0	0
$927$	3.30481	0.108544
$928$	0	0
$929$	−17.7990	−0.583966	−0.291983	−	0.956424i	$-0.594315\pi$
$929$	−17.7990	−0.583966	−0.291983	+	0.956424i	$0.594315\pi$
$930$	0	0
$931$	51.7990	1.69764
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.33609	0.272619
$936$	0	0
$937$	4.62742	0.151171	0.0755856	−	0.997139i	$-0.475917\pi$
$937$	4.62742	0.151171	0.0755856	+	0.997139i	$0.475917\pi$
$938$	0	0
$939$	−15.3137	−0.499744
$940$	0	0
$941$	−51.3854	−1.67512	−0.837558	−	0.546348i	$-0.816018\pi$
$941$	−51.3854	−1.67512	−0.837558	+	0.546348i	$0.816018\pi$
$942$	0	0
$943$	36.7973	1.19828
$944$	0	0
$945$	10.4853	0.341086
$946$	0	0
$947$	4.97056	0.161522	0.0807608	−	0.996734i	$-0.474265\pi$
$947$	4.97056	0.161522	0.0807608	+	0.996734i	$0.474265\pi$
$948$	0	0
$949$	11.7890	0.382687
$950$	0	0
$951$	−23.9356	−0.776164
$952$	0	0
$953$	−9.79899	−0.317420	−0.158710	−	0.987325i	$-0.550734\pi$
$953$	−9.79899	−0.317420	−0.158710	+	0.987325i	$0.550734\pi$
$954$	0	0
$955$	−54.2254	−1.75469
$956$	0	0
$957$	6.60963	0.213659
$958$	0	0
$959$	−61.0904	−1.97271
$960$	0	0
$961$	−5.68629	−0.183429
$962$	0	0
$963$	−4.00000	−0.128898
$964$	0	0
$965$	8.33609	0.268348
$966$	0	0
$967$	−18.5467	−0.596423	−0.298211	−	0.954500i	$-0.596390\pi$
$967$	−18.5467	−0.596423	−0.298211	+	0.954500i	$0.596390\pi$
$968$	0	0
$969$	13.6569	0.438721
$970$	0	0
$971$	35.4558	1.13783	0.568916	−	0.822396i	$-0.307363\pi$
$971$	35.4558	1.13783	0.568916	+	0.822396i	$0.307363\pi$
$972$	0	0
$973$	40.2502	1.29036
$974$	0	0
$975$	−1.93591	−0.0619989
$976$	0	0
$977$	30.7696	0.984405	0.492203	−	0.870481i	$-0.336192\pi$
$977$	30.7696	0.984405	0.492203	+	0.870481i	$0.336192\pi$
$978$	0	0
$979$	−8.28427	−0.264766
$980$	0	0
$981$	−8.84175	−0.282295
$982$	0	0
$983$	−3.45292	−0.110131	−0.0550655	−	0.998483i	$-0.517537\pi$
$983$	−3.45292	−0.110131	−0.0550655	+	0.998483i	$0.517537\pi$
$984$	0	0
$985$	−42.6863	−1.36010
$986$	0	0
$987$	−20.9706	−0.667500
$988$	0	0
$989$	−52.0392	−1.65475
$990$	0	0
$991$	−28.6093	−0.908804	−0.454402	−	0.890797i	$-0.650147\pi$
$991$	−28.6093	−0.908804	−0.454402	+	0.890797i	$0.650147\pi$
$992$	0	0
$993$	21.6569	0.687259
$994$	0	0
$995$	10.4853	0.332406
$996$	0	0
$997$	−9.55688	−0.302669	−0.151335	−	0.988483i	$-0.548357\pi$
$997$	−9.55688	−0.302669	−0.151335	+	0.988483i	$0.548357\pi$
$998$	0	0
$999$	7.11529	0.225118

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.a.n.1.2	yes	4
3.2	odd	2		4608.2.a.t.1.3		4
4.3	odd	2		1536.2.a.m.1.2	✓	4
8.3	odd	2	inner	1536.2.a.n.1.3	yes	4
8.5	even	2		1536.2.a.m.1.3	yes	4
12.11	even	2		4608.2.a.ba.1.3		4
16.3	odd	4		1536.2.d.g.769.3		8
16.5	even	4		1536.2.d.g.769.2		8
16.11	odd	4		1536.2.d.g.769.6		8
16.13	even	4		1536.2.d.g.769.7		8
24.5	odd	2		4608.2.a.ba.1.2		4
24.11	even	2		4608.2.a.t.1.2		4
48.5	odd	4		4608.2.d.p.2305.6		8
48.11	even	4		4608.2.d.p.2305.5		8
48.29	odd	4		4608.2.d.p.2305.4		8
48.35	even	4		4608.2.d.p.2305.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.m.1.2	✓	4	4.3	odd	2
1536.2.a.m.1.3	yes	4	8.5	even	2
1536.2.a.n.1.2	yes	4	1.1	even	1	trivial
1536.2.a.n.1.3	yes	4	8.3	odd	2	inner
1536.2.d.g.769.2		8	16.5	even	4
1536.2.d.g.769.3		8	16.3	odd	4
1536.2.d.g.769.6		8	16.11	odd	4
1536.2.d.g.769.7		8	16.13	even	4
4608.2.a.t.1.2		4	24.11	even	2
4608.2.a.t.1.3		4	3.2	odd	2
4608.2.a.ba.1.2		4	24.5	odd	2
4608.2.a.ba.1.3		4	12.11	even	2
4608.2.d.p.2305.3		8	48.35	even	4
4608.2.d.p.2305.4		8	48.29	odd	4
4608.2.d.p.2305.5		8	48.11	even	4
4608.2.d.p.2305.6		8	48.5	odd	4

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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.08402 q^{5} -5.03127 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.08402 q^{5} -5.03127 q^{7} +1.00000 q^{9} -0.828427 q^{11} +2.94725 q^{13} -2.08402 q^{15} +4.82843 q^{17} +2.82843 q^{19} -5.03127 q^{21} +4.16804 q^{23} -0.656854 q^{25} +1.00000 q^{27} -7.97852 q^{29} +5.03127 q^{31} -0.828427 q^{33} +10.4853 q^{35} +7.11529 q^{37} +2.94725 q^{39} +8.82843 q^{41} -12.4853 q^{43} -2.08402 q^{45} +4.16804 q^{47} +18.3137 q^{49} +4.82843 q^{51} +12.1466 q^{53} +1.72646 q^{55} +2.82843 q^{57} -1.65685 q^{59} +7.11529 q^{61} -5.03127 q^{63} -6.14214 q^{65} -2.34315 q^{67} +4.16804 q^{69} +10.0625 q^{71} +4.00000 q^{73} -0.656854 q^{75} +4.16804 q^{77} -5.03127 q^{79} +1.00000 q^{81} +3.17157 q^{83} -10.0625 q^{85} -7.97852 q^{87} +10.0000 q^{89} -14.8284 q^{91} +5.03127 q^{93} -5.89450 q^{95} +0.343146 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 4 * q^9	\( 4 q + 4 q^{3} + 4 q^{9} + 8 q^{11} + 8 q^{17} + 20 q^{25} + 4 q^{27} + 8 q^{33} + 8 q^{35} + 24 q^{41} - 16 q^{43} + 28 q^{49} + 8 q^{51} + 16 q^{59} + 32 q^{65} - 32 q^{67} + 16 q^{73} + 20 q^{75} + 4 q^{81} + 24 q^{83} + 40 q^{89} - 48 q^{91} + 24 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 4 * q^9 + 8 * q^11 + 8 * q^17 + 20 * q^25 + 4 * q^27 + 8 * q^33 + 8 * q^35 + 24 * q^41 - 16 * q^43 + 28 * q^49 + 8 * q^51 + 16 * q^59 + 32 * q^65 - 32 * q^67 + 16 * q^73 + 20 * q^75 + 4 * q^81 + 24 * q^83 + 40 * q^89 - 48 * q^91 + 24 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

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Database

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