LMFDB - Embedded newform 1536.2.a.l.1.1

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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.585786 q^{5} +3.41421 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.585786 q^{5} +3.41421 q^{7} +1.00000 q^{9} +2.00000 q^{11} +2.82843 q^{13} +0.585786 q^{15} +3.65685 q^{17} -5.65685 q^{19} +3.41421 q^{21} +1.17157 q^{23} -4.65685 q^{25} +1.00000 q^{27} +0.585786 q^{29} +4.58579 q^{31} +2.00000 q^{33} +2.00000 q^{35} -9.65685 q^{37} +2.82843 q^{39} -11.6569 q^{41} +1.65685 q^{43} +0.585786 q^{45} +12.4853 q^{47} +4.65685 q^{49} +3.65685 q^{51} +11.8995 q^{53} +1.17157 q^{55} -5.65685 q^{57} -4.00000 q^{59} -9.65685 q^{61} +3.41421 q^{63} +1.65685 q^{65} -8.00000 q^{67} +1.17157 q^{69} +9.17157 q^{71} -1.65685 q^{73} -4.65685 q^{75} +6.82843 q^{77} +5.75736 q^{79} +1.00000 q^{81} +9.31371 q^{83} +2.14214 q^{85} +0.585786 q^{87} +2.00000 q^{89} +9.65685 q^{91} +4.58579 q^{93} -3.31371 q^{95} +13.3137 q^{97} +2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{17} + 4 q^{21} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 12 q^{31} + 4 q^{33} + 4 q^{35} - 8 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} + 8 q^{47} - 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 8 q^{61} + 4 q^{63} - 8 q^{65} - 16 q^{67} + 8 q^{69} + 24 q^{71} + 8 q^{73} + 2 q^{75} + 8 q^{77} + 20 q^{79} + 2 q^{81} - 4 q^{83} - 24 q^{85} + 4 q^{87} + 4 q^{89} + 8 q^{91} + 12 q^{93} + 16 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^15 - 4 * q^17 + 4 * q^21 + 8 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 12 * q^31 + 4 * q^33 + 4 * q^35 - 8 * q^37 - 12 * q^41 - 8 * q^43 + 4 * q^45 + 8 * q^47 - 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^55 - 8 * q^59 - 8 * q^61 + 4 * q^63 - 8 * q^65 - 16 * q^67 + 8 * q^69 + 24 * q^71 + 8 * q^73 + 2 * q^75 + 8 * q^77 + 20 * q^79 + 2 * q^81 - 4 * q^83 - 24 * q^85 + 4 * q^87 + 4 * q^89 + 8 * q^91 + 12 * q^93 + 16 * q^95 + 4 * q^97 + 4 * 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
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	3.41421	1.29045	0.645226	−	0.763992i	$-0.276763\pi$
$7$	3.41421	1.29045	0.645226	+	0.763992i	$0.276763\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0	0
$15$	0.585786	0.151249
$16$	0	0
$17$	3.65685	0.886917	0.443459	−	0.896295i	$-0.353751\pi$
$17$	3.65685	0.886917	0.443459	+	0.896295i	$0.353751\pi$
$18$	0	0
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	0	0
$21$	3.41421	0.745042
$22$	0	0
$23$	1.17157	0.244290	0.122145	−	0.992512i	$-0.461023\pi$
$23$	1.17157	0.244290	0.122145	+	0.992512i	$0.461023\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.585786	0.108778	0.0543889	−	0.998520i	$-0.482679\pi$
$29$	0.585786	0.108778	0.0543889	+	0.998520i	$0.482679\pi$
$30$	0	0
$31$	4.58579	0.823632	0.411816	−	0.911267i	$-0.364895\pi$
$31$	4.58579	0.823632	0.411816	+	0.911267i	$0.364895\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−9.65685	−1.58758	−0.793789	−	0.608194i	$-0.791894\pi$
$37$	−9.65685	−1.58758	−0.793789	+	0.608194i	$0.791894\pi$
$38$	0	0
$39$	2.82843	0.452911
$40$	0	0
$41$	−11.6569	−1.82049	−0.910247	−	0.414065i	$-0.864109\pi$
$41$	−11.6569	−1.82049	−0.910247	+	0.414065i	$0.864109\pi$
$42$	0	0
$43$	1.65685	0.252668	0.126334	−	0.991988i	$-0.459679\pi$
$43$	1.65685	0.252668	0.126334	+	0.991988i	$0.459679\pi$
$44$	0	0
$45$	0.585786	0.0873239
$46$	0	0
$47$	12.4853	1.82117	0.910583	−	0.413327i	$-0.135633\pi$
$47$	12.4853	1.82117	0.910583	+	0.413327i	$0.135633\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	3.65685	0.512062
$52$	0	0
$53$	11.8995	1.63452	0.817261	−	0.576268i	$-0.195492\pi$
$53$	11.8995	1.63452	0.817261	+	0.576268i	$0.195492\pi$
$54$	0	0
$55$	1.17157	0.157975
$56$	0	0
$57$	−5.65685	−0.749269
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−9.65685	−1.23643	−0.618217	−	0.786008i	$-0.712145\pi$
$61$	−9.65685	−1.23643	−0.618217	+	0.786008i	$0.712145\pi$
$62$	0	0
$63$	3.41421	0.430150
$64$	0	0
$65$	1.65685	0.205507
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	1.17157	0.141041
$70$	0	0
$71$	9.17157	1.08847	0.544233	−	0.838934i	$-0.316821\pi$
$71$	9.17157	1.08847	0.544233	+	0.838934i	$0.316821\pi$
$72$	0	0
$73$	−1.65685	−0.193920	−0.0969601	−	0.995288i	$-0.530912\pi$
$73$	−1.65685	−0.193920	−0.0969601	+	0.995288i	$0.530912\pi$
$74$	0	0
$75$	−4.65685	−0.537727
$76$	0	0
$77$	6.82843	0.778171
$78$	0	0
$79$	5.75736	0.647754	0.323877	−	0.946099i	$-0.395014\pi$
$79$	5.75736	0.647754	0.323877	+	0.946099i	$0.395014\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.31371	1.02231	0.511156	−	0.859488i	$-0.329217\pi$
$83$	9.31371	1.02231	0.511156	+	0.859488i	$0.329217\pi$
$84$	0	0
$85$	2.14214	0.232347
$86$	0	0
$87$	0.585786	0.0628029
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	0	0
$93$	4.58579	0.475524
$94$	0	0
$95$	−3.31371	−0.339979
$96$	0	0
$97$	13.3137	1.35180	0.675901	−	0.736992i	$-0.263755\pi$
$97$	13.3137	1.35180	0.675901	+	0.736992i	$0.263755\pi$
$98$	0	0
$99$	2.00000	0.201008
$100$	0	0
$101$	−17.5563	−1.74692	−0.873461	−	0.486894i	$-0.838130\pi$
$101$	−17.5563	−1.74692	−0.873461	+	0.486894i	$0.838130\pi$
$102$	0	0
$103$	9.07107	0.893799	0.446899	−	0.894584i	$-0.352528\pi$
$103$	9.07107	0.893799	0.446899	+	0.894584i	$0.352528\pi$
$104$	0	0
$105$	2.00000	0.195180
$106$	0	0
$107$	−15.3137	−1.48043	−0.740216	−	0.672369i	$-0.765277\pi$
$107$	−15.3137	−1.48043	−0.740216	+	0.672369i	$0.765277\pi$
$108$	0	0
$109$	−13.1716	−1.26161	−0.630804	−	0.775942i	$-0.717275\pi$
$109$	−13.1716	−1.26161	−0.630804	+	0.775942i	$0.717275\pi$
$110$	0	0
$111$	−9.65685	−0.916588
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	0.686292	0.0639970
$116$	0	0
$117$	2.82843	0.261488
$118$	0	0
$119$	12.4853	1.14452
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	−11.6569	−1.05106
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	6.72792	0.597007	0.298503	−	0.954409i	$-0.403513\pi$
$127$	6.72792	0.597007	0.298503	+	0.954409i	$0.403513\pi$
$128$	0	0
$129$	1.65685	0.145878
$130$	0	0
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	−19.3137	−1.67471
$134$	0	0
$135$	0.585786	0.0504165
$136$	0	0
$137$	4.34315	0.371060	0.185530	−	0.982639i	$-0.440600\pi$
$137$	4.34315	0.371060	0.185530	+	0.982639i	$0.440600\pi$
$138$	0	0
$139$	19.3137	1.63817	0.819084	−	0.573674i	$-0.194482\pi$
$139$	19.3137	1.63817	0.819084	+	0.573674i	$0.194482\pi$
$140$	0	0
$141$	12.4853	1.05145
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	0.343146	0.0284967
$146$	0	0
$147$	4.65685	0.384091
$148$	0	0
$149$	14.2426	1.16680	0.583401	−	0.812184i	$-0.301722\pi$
$149$	14.2426	1.16680	0.583401	+	0.812184i	$0.301722\pi$
$150$	0	0
$151$	4.58579	0.373186	0.186593	−	0.982437i	$-0.440255\pi$
$151$	4.58579	0.373186	0.186593	+	0.982437i	$0.440255\pi$
$152$	0	0
$153$	3.65685	0.295639
$154$	0	0
$155$	2.68629	0.215768
$156$	0	0
$157$	−20.0000	−1.59617	−0.798087	−	0.602542i	$-0.794154\pi$
$157$	−20.0000	−1.59617	−0.798087	+	0.602542i	$0.794154\pi$
$158$	0	0
$159$	11.8995	0.943691
$160$	0	0
$161$	4.00000	0.315244
$162$	0	0
$163$	6.34315	0.496834	0.248417	−	0.968653i	$-0.420090\pi$
$163$	6.34315	0.496834	0.248417	+	0.968653i	$0.420090\pi$
$164$	0	0
$165$	1.17157	0.0912068
$166$	0	0
$167$	−21.6569	−1.67586	−0.837929	−	0.545779i	$-0.816234\pi$
$167$	−21.6569	−1.67586	−0.837929	+	0.545779i	$0.816234\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0	0
$171$	−5.65685	−0.432590
$172$	0	0
$173$	18.7279	1.42386	0.711929	−	0.702252i	$-0.247822\pi$
$173$	18.7279	1.42386	0.711929	+	0.702252i	$0.247822\pi$
$174$	0	0
$175$	−15.8995	−1.20189
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	0	0
$183$	−9.65685	−0.713855
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	7.31371	0.534831
$188$	0	0
$189$	3.41421	0.248347
$190$	0	0
$191$	−16.9706	−1.22795	−0.613973	−	0.789327i	$-0.710430\pi$
$191$	−16.9706	−1.22795	−0.613973	+	0.789327i	$0.710430\pi$
$192$	0	0
$193$	−21.6569	−1.55889	−0.779447	−	0.626468i	$-0.784500\pi$
$193$	−21.6569	−1.55889	−0.779447	+	0.626468i	$0.784500\pi$
$194$	0	0
$195$	1.65685	0.118650
$196$	0	0
$197$	2.92893	0.208678	0.104339	−	0.994542i	$-0.466727\pi$
$197$	2.92893	0.208678	0.104339	+	0.994542i	$0.466727\pi$
$198$	0	0
$199$	−13.5563	−0.960984	−0.480492	−	0.876999i	$-0.659542\pi$
$199$	−13.5563	−0.960984	−0.480492	+	0.876999i	$0.659542\pi$
$200$	0	0
$201$	−8.00000	−0.564276
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−6.82843	−0.476918
$206$	0	0
$207$	1.17157	0.0814299
$208$	0	0
$209$	−11.3137	−0.782586
$210$	0	0
$211$	−19.3137	−1.32961	−0.664805	−	0.747017i	$-0.731485\pi$
$211$	−19.3137	−1.32961	−0.664805	+	0.747017i	$0.731485\pi$
$212$	0	0
$213$	9.17157	0.628426
$214$	0	0
$215$	0.970563	0.0661918
$216$	0	0
$217$	15.6569	1.06286
$218$	0	0
$219$	−1.65685	−0.111960
$220$	0	0
$221$	10.3431	0.695755
$222$	0	0
$223$	7.89949	0.528989	0.264495	−	0.964387i	$-0.414795\pi$
$223$	7.89949	0.528989	0.264495	+	0.964387i	$0.414795\pi$
$224$	0	0
$225$	−4.65685	−0.310457
$226$	0	0
$227$	29.3137	1.94562	0.972810	−	0.231606i	$-0.0743981\pi$
$227$	29.3137	1.94562	0.972810	+	0.231606i	$0.0743981\pi$
$228$	0	0
$229$	−16.4853	−1.08938	−0.544689	−	0.838638i	$-0.683352\pi$
$229$	−16.4853	−1.08938	−0.544689	+	0.838638i	$0.683352\pi$
$230$	0	0
$231$	6.82843	0.449278
$232$	0	0
$233$	9.31371	0.610161	0.305081	−	0.952327i	$-0.401317\pi$
$233$	9.31371	0.610161	0.305081	+	0.952327i	$0.401317\pi$
$234$	0	0
$235$	7.31371	0.477094
$236$	0	0
$237$	5.75736	0.373981
$238$	0	0
$239$	16.9706	1.09773	0.548867	−	0.835910i	$-0.315059\pi$
$239$	16.9706	1.09773	0.548867	+	0.835910i	$0.315059\pi$
$240$	0	0
$241$	−17.6569	−1.13738	−0.568689	−	0.822553i	$-0.692549\pi$
$241$	−17.6569	−1.13738	−0.568689	+	0.822553i	$0.692549\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.72792	0.174281
$246$	0	0
$247$	−16.0000	−1.01806
$248$	0	0
$249$	9.31371	0.590232
$250$	0	0
$251$	−22.0000	−1.38863	−0.694314	−	0.719672i	$-0.744292\pi$
$251$	−22.0000	−1.38863	−0.694314	+	0.719672i	$0.744292\pi$
$252$	0	0
$253$	2.34315	0.147312
$254$	0	0
$255$	2.14214	0.134146
$256$	0	0
$257$	−1.31371	−0.0819469	−0.0409734	−	0.999160i	$-0.513046\pi$
$257$	−1.31371	−0.0819469	−0.0409734	+	0.999160i	$0.513046\pi$
$258$	0	0
$259$	−32.9706	−2.04869
$260$	0	0
$261$	0.585786	0.0362593
$262$	0	0
$263$	−13.6569	−0.842118	−0.421059	−	0.907033i	$-0.638341\pi$
$263$	−13.6569	−0.842118	−0.421059	+	0.907033i	$0.638341\pi$
$264$	0	0
$265$	6.97056	0.428198
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	−6.24264	−0.380621	−0.190310	−	0.981724i	$-0.560949\pi$
$269$	−6.24264	−0.380621	−0.190310	+	0.981724i	$0.560949\pi$
$270$	0	0
$271$	−19.4142	−1.17933	−0.589665	−	0.807648i	$-0.700740\pi$
$271$	−19.4142	−1.17933	−0.589665	+	0.807648i	$0.700740\pi$
$272$	0	0
$273$	9.65685	0.584459
$274$	0	0
$275$	−9.31371	−0.561638
$276$	0	0
$277$	−7.51472	−0.451516	−0.225758	−	0.974183i	$-0.572486\pi$
$277$	−7.51472	−0.451516	−0.225758	+	0.974183i	$0.572486\pi$
$278$	0	0
$279$	4.58579	0.274544
$280$	0	0
$281$	−2.00000	−0.119310	−0.0596550	−	0.998219i	$-0.519000\pi$
$281$	−2.00000	−0.119310	−0.0596550	+	0.998219i	$0.519000\pi$
$282$	0	0
$283$	15.3137	0.910305	0.455153	−	0.890413i	$-0.349585\pi$
$283$	15.3137	0.910305	0.455153	+	0.890413i	$0.349585\pi$
$284$	0	0
$285$	−3.31371	−0.196287
$286$	0	0
$287$	−39.7990	−2.34926
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	13.3137	0.780463
$292$	0	0
$293$	27.8995	1.62991	0.814953	−	0.579527i	$-0.196763\pi$
$293$	27.8995	1.62991	0.814953	+	0.579527i	$0.196763\pi$
$294$	0	0
$295$	−2.34315	−0.136423
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	3.31371	0.191637
$300$	0	0
$301$	5.65685	0.326056
$302$	0	0
$303$	−17.5563	−1.00859
$304$	0	0
$305$	−5.65685	−0.323911
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	9.07107	0.516035
$310$	0	0
$311$	11.3137	0.641542	0.320771	−	0.947157i	$-0.396058\pi$
$311$	11.3137	0.641542	0.320771	+	0.947157i	$0.396058\pi$
$312$	0	0
$313$	16.9706	0.959233	0.479616	−	0.877478i	$-0.340776\pi$
$313$	16.9706	0.959233	0.479616	+	0.877478i	$0.340776\pi$
$314$	0	0
$315$	2.00000	0.112687
$316$	0	0
$317$	5.07107	0.284820	0.142410	−	0.989808i	$-0.454515\pi$
$317$	5.07107	0.284820	0.142410	+	0.989808i	$0.454515\pi$
$318$	0	0
$319$	1.17157	0.0655955
$320$	0	0
$321$	−15.3137	−0.854728
$322$	0	0
$323$	−20.6863	−1.15102
$324$	0	0
$325$	−13.1716	−0.730627
$326$	0	0
$327$	−13.1716	−0.728390
$328$	0	0
$329$	42.6274	2.35013
$330$	0	0
$331$	0	0		−	1.00000i	$-0.5\pi$
$331$	0	0			1.00000i	$0.5\pi$
$332$	0	0
$333$	−9.65685	−0.529192
$334$	0	0
$335$	−4.68629	−0.256039
$336$	0	0
$337$	20.9706	1.14234	0.571170	−	0.820832i	$-0.306490\pi$
$337$	20.9706	1.14234	0.571170	+	0.820832i	$0.306490\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	0	0
$341$	9.17157	0.496669
$342$	0	0
$343$	−8.00000	−0.431959
$344$	0	0
$345$	0.686292	0.0369487
$346$	0	0
$347$	25.3137	1.35891	0.679456	−	0.733717i	$-0.262216\pi$
$347$	25.3137	1.35891	0.679456	+	0.733717i	$0.262216\pi$
$348$	0	0
$349$	−0.686292	−0.0367363	−0.0183682	−	0.999831i	$-0.505847\pi$
$349$	−0.686292	−0.0367363	−0.0183682	+	0.999831i	$0.505847\pi$
$350$	0	0
$351$	2.82843	0.150970
$352$	0	0
$353$	−13.3137	−0.708617	−0.354309	−	0.935129i	$-0.615284\pi$
$353$	−13.3137	−0.708617	−0.354309	+	0.935129i	$0.615284\pi$
$354$	0	0
$355$	5.37258	0.285147
$356$	0	0
$357$	12.4853	0.660791
$358$	0	0
$359$	−15.7990	−0.833839	−0.416919	−	0.908943i	$-0.636890\pi$
$359$	−15.7990	−0.833839	−0.416919	+	0.908943i	$0.636890\pi$
$360$	0	0
$361$	13.0000	0.684211
$362$	0	0
$363$	−7.00000	−0.367405
$364$	0	0
$365$	−0.970563	−0.0508016
$366$	0	0
$367$	−22.7279	−1.18639	−0.593194	−	0.805060i	$-0.702133\pi$
$367$	−22.7279	−1.18639	−0.593194	+	0.805060i	$0.702133\pi$
$368$	0	0
$369$	−11.6569	−0.606832
$370$	0	0
$371$	40.6274	2.10927
$372$	0	0
$373$	−24.2843	−1.25739	−0.628696	−	0.777651i	$-0.716411\pi$
$373$	−24.2843	−1.25739	−0.628696	+	0.777651i	$0.716411\pi$
$374$	0	0
$375$	−5.65685	−0.292119
$376$	0	0
$377$	1.65685	0.0853323
$378$	0	0
$379$	−18.3431	−0.942224	−0.471112	−	0.882073i	$-0.656147\pi$
$379$	−18.3431	−0.942224	−0.471112	+	0.882073i	$0.656147\pi$
$380$	0	0
$381$	6.72792	0.344682
$382$	0	0
$383$	−19.3137	−0.986884	−0.493442	−	0.869779i	$-0.664262\pi$
$383$	−19.3137	−0.986884	−0.493442	+	0.869779i	$0.664262\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	1.65685	0.0842226
$388$	0	0
$389$	−22.2426	−1.12775	−0.563873	−	0.825861i	$-0.690689\pi$
$389$	−22.2426	−1.12775	−0.563873	+	0.825861i	$0.690689\pi$
$390$	0	0
$391$	4.28427	0.216665
$392$	0	0
$393$	−4.00000	−0.201773
$394$	0	0
$395$	3.37258	0.169693
$396$	0	0
$397$	−4.00000	−0.200754	−0.100377	−	0.994949i	$-0.532005\pi$
$397$	−4.00000	−0.200754	−0.100377	+	0.994949i	$0.532005\pi$
$398$	0	0
$399$	−19.3137	−0.966895
$400$	0	0
$401$	−31.6569	−1.58087	−0.790434	−	0.612547i	$-0.790145\pi$
$401$	−31.6569	−1.58087	−0.790434	+	0.612547i	$0.790145\pi$
$402$	0	0
$403$	12.9706	0.646110
$404$	0	0
$405$	0.585786	0.0291080
$406$	0	0
$407$	−19.3137	−0.957345
$408$	0	0
$409$	1.31371	0.0649587	0.0324794	−	0.999472i	$-0.489660\pi$
$409$	1.31371	0.0649587	0.0324794	+	0.999472i	$0.489660\pi$
$410$	0	0
$411$	4.34315	0.214232
$412$	0	0
$413$	−13.6569	−0.672010
$414$	0	0
$415$	5.45584	0.267817
$416$	0	0
$417$	19.3137	0.945796
$418$	0	0
$419$	−1.31371	−0.0641789	−0.0320894	−	0.999485i	$-0.510216\pi$
$419$	−1.31371	−0.0641789	−0.0320894	+	0.999485i	$0.510216\pi$
$420$	0	0
$421$	22.1421	1.07914	0.539571	−	0.841940i	$-0.318587\pi$
$421$	22.1421	1.07914	0.539571	+	0.841940i	$0.318587\pi$
$422$	0	0
$423$	12.4853	0.607055
$424$	0	0
$425$	−17.0294	−0.826049
$426$	0	0
$427$	−32.9706	−1.59556
$428$	0	0
$429$	5.65685	0.273115
$430$	0	0
$431$	35.1127	1.69132	0.845660	−	0.533723i	$-0.179207\pi$
$431$	35.1127	1.69132	0.845660	+	0.533723i	$0.179207\pi$
$432$	0	0
$433$	−16.6274	−0.799063	−0.399531	−	0.916720i	$-0.630827\pi$
$433$	−16.6274	−0.799063	−0.399531	+	0.916720i	$0.630827\pi$
$434$	0	0
$435$	0.343146	0.0164526
$436$	0	0
$437$	−6.62742	−0.317032
$438$	0	0
$439$	−15.8995	−0.758841	−0.379421	−	0.925224i	$-0.623877\pi$
$439$	−15.8995	−0.758841	−0.379421	+	0.925224i	$0.623877\pi$
$440$	0	0
$441$	4.65685	0.221755
$442$	0	0
$443$	−27.9411	−1.32752	−0.663761	−	0.747944i	$-0.731041\pi$
$443$	−27.9411	−1.32752	−0.663761	+	0.747944i	$0.731041\pi$
$444$	0	0
$445$	1.17157	0.0555379
$446$	0	0
$447$	14.2426	0.673654
$448$	0	0
$449$	−7.65685	−0.361349	−0.180675	−	0.983543i	$-0.557828\pi$
$449$	−7.65685	−0.361349	−0.180675	+	0.983543i	$0.557828\pi$
$450$	0	0
$451$	−23.3137	−1.09780
$452$	0	0
$453$	4.58579	0.215459
$454$	0	0
$455$	5.65685	0.265197
$456$	0	0
$457$	−17.3137	−0.809901	−0.404951	−	0.914339i	$-0.632711\pi$
$457$	−17.3137	−0.809901	−0.404951	+	0.914339i	$0.632711\pi$
$458$	0	0
$459$	3.65685	0.170687
$460$	0	0
$461$	−10.9289	−0.509011	−0.254506	−	0.967071i	$-0.581913\pi$
$461$	−10.9289	−0.509011	−0.254506	+	0.967071i	$0.581913\pi$
$462$	0	0
$463$	2.44365	0.113566	0.0567830	−	0.998387i	$-0.481916\pi$
$463$	2.44365	0.113566	0.0567830	+	0.998387i	$0.481916\pi$
$464$	0	0
$465$	2.68629	0.124574
$466$	0	0
$467$	−9.31371	−0.430987	−0.215494	−	0.976505i	$-0.569136\pi$
$467$	−9.31371	−0.430987	−0.215494	+	0.976505i	$0.569136\pi$
$468$	0	0
$469$	−27.3137	−1.26123
$470$	0	0
$471$	−20.0000	−0.921551
$472$	0	0
$473$	3.31371	0.152364
$474$	0	0
$475$	26.3431	1.20871
$476$	0	0
$477$	11.8995	0.544840
$478$	0	0
$479$	−4.48528	−0.204938	−0.102469	−	0.994736i	$-0.532674\pi$
$479$	−4.48528	−0.204938	−0.102469	+	0.994736i	$0.532674\pi$
$480$	0	0
$481$	−27.3137	−1.24540
$482$	0	0
$483$	4.00000	0.182006
$484$	0	0
$485$	7.79899	0.354134
$486$	0	0
$487$	−5.55635	−0.251782	−0.125891	−	0.992044i	$-0.540179\pi$
$487$	−5.55635	−0.251782	−0.125891	+	0.992044i	$0.540179\pi$
$488$	0	0
$489$	6.34315	0.286847
$490$	0	0
$491$	−0.686292	−0.0309719	−0.0154860	−	0.999880i	$-0.504930\pi$
$491$	−0.686292	−0.0309719	−0.0154860	+	0.999880i	$0.504930\pi$
$492$	0	0
$493$	2.14214	0.0964769
$494$	0	0
$495$	1.17157	0.0526583
$496$	0	0
$497$	31.3137	1.40461
$498$	0	0
$499$	23.3137	1.04366	0.521832	−	0.853048i	$-0.325249\pi$
$499$	23.3137	1.04366	0.521832	+	0.853048i	$0.325249\pi$
$500$	0	0
$501$	−21.6569	−0.967557
$502$	0	0
$503$	−4.48528	−0.199989	−0.0999944	−	0.994988i	$-0.531882\pi$
$503$	−4.48528	−0.199989	−0.0999944	+	0.994988i	$0.531882\pi$
$504$	0	0
$505$	−10.2843	−0.457644
$506$	0	0
$507$	−5.00000	−0.222058
$508$	0	0
$509$	23.2132	1.02891	0.514454	−	0.857518i	$-0.327995\pi$
$509$	23.2132	1.02891	0.514454	+	0.857518i	$0.327995\pi$
$510$	0	0
$511$	−5.65685	−0.250244
$512$	0	0
$513$	−5.65685	−0.249756
$514$	0	0
$515$	5.31371	0.234150
$516$	0	0
$517$	24.9706	1.09820
$518$	0	0
$519$	18.7279	0.822065
$520$	0	0
$521$	−10.9706	−0.480629	−0.240315	−	0.970695i	$-0.577251\pi$
$521$	−10.9706	−0.480629	−0.240315	+	0.970695i	$0.577251\pi$
$522$	0	0
$523$	26.3431	1.15191	0.575953	−	0.817483i	$-0.304631\pi$
$523$	26.3431	1.15191	0.575953	+	0.817483i	$0.304631\pi$
$524$	0	0
$525$	−15.8995	−0.693911
$526$	0	0
$527$	16.7696	0.730493
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−32.9706	−1.42811
$534$	0	0
$535$	−8.97056	−0.387831
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	9.31371	0.401170
$540$	0	0
$541$	−5.17157	−0.222343	−0.111172	−	0.993801i	$-0.535460\pi$
$541$	−5.17157	−0.222343	−0.111172	+	0.993801i	$0.535460\pi$
$542$	0	0
$543$	8.48528	0.364138
$544$	0	0
$545$	−7.71573	−0.330506
$546$	0	0
$547$	28.9706	1.23869	0.619346	−	0.785118i	$-0.287398\pi$
$547$	28.9706	1.23869	0.619346	+	0.785118i	$0.287398\pi$
$548$	0	0
$549$	−9.65685	−0.412144
$550$	0	0
$551$	−3.31371	−0.141169
$552$	0	0
$553$	19.6569	0.835894
$554$	0	0
$555$	−5.65685	−0.240120
$556$	0	0
$557$	−13.0711	−0.553839	−0.276919	−	0.960893i	$-0.589314\pi$
$557$	−13.0711	−0.553839	−0.276919	+	0.960893i	$0.589314\pi$
$558$	0	0
$559$	4.68629	0.198209
$560$	0	0
$561$	7.31371	0.308785
$562$	0	0
$563$	−28.6274	−1.20650	−0.603251	−	0.797551i	$-0.706128\pi$
$563$	−28.6274	−1.20650	−0.603251	+	0.797551i	$0.706128\pi$
$564$	0	0
$565$	3.51472	0.147865
$566$	0	0
$567$	3.41421	0.143383
$568$	0	0
$569$	18.9706	0.795287	0.397644	−	0.917540i	$-0.369828\pi$
$569$	18.9706	0.795287	0.397644	+	0.917540i	$0.369828\pi$
$570$	0	0
$571$	−20.0000	−0.836974	−0.418487	−	0.908223i	$-0.637439\pi$
$571$	−20.0000	−0.836974	−0.418487	+	0.908223i	$0.637439\pi$
$572$	0	0
$573$	−16.9706	−0.708955
$574$	0	0
$575$	−5.45584	−0.227524
$576$	0	0
$577$	28.2843	1.17749	0.588745	−	0.808319i	$-0.299622\pi$
$577$	28.2843	1.17749	0.588745	+	0.808319i	$0.299622\pi$
$578$	0	0
$579$	−21.6569	−0.900028
$580$	0	0
$581$	31.7990	1.31924
$582$	0	0
$583$	23.7990	0.985653
$584$	0	0
$585$	1.65685	0.0685025
$586$	0	0
$587$	−8.68629	−0.358522	−0.179261	−	0.983802i	$-0.557371\pi$
$587$	−8.68629	−0.358522	−0.179261	+	0.983802i	$0.557371\pi$
$588$	0	0
$589$	−25.9411	−1.06889
$590$	0	0
$591$	2.92893	0.120480
$592$	0	0
$593$	−2.00000	−0.0821302	−0.0410651	−	0.999156i	$-0.513075\pi$
$593$	−2.00000	−0.0821302	−0.0410651	+	0.999156i	$0.513075\pi$
$594$	0	0
$595$	7.31371	0.299833
$596$	0	0
$597$	−13.5563	−0.554824
$598$	0	0
$599$	28.4853	1.16388	0.581939	−	0.813233i	$-0.302294\pi$
$599$	28.4853	1.16388	0.581939	+	0.813233i	$0.302294\pi$
$600$	0	0
$601$	5.65685	0.230748	0.115374	−	0.993322i	$-0.463193\pi$
$601$	5.65685	0.230748	0.115374	+	0.993322i	$0.463193\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	−4.10051	−0.166709
$606$	0	0
$607$	−15.8995	−0.645341	−0.322670	−	0.946511i	$-0.604581\pi$
$607$	−15.8995	−0.645341	−0.322670	+	0.946511i	$0.604581\pi$
$608$	0	0
$609$	2.00000	0.0810441
$610$	0	0
$611$	35.3137	1.42864
$612$	0	0
$613$	34.6274	1.39859	0.699294	−	0.714834i	$-0.253498\pi$
$613$	34.6274	1.39859	0.699294	+	0.714834i	$0.253498\pi$
$614$	0	0
$615$	−6.82843	−0.275349
$616$	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
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	1.17157	0.0470136
$622$	0	0
$623$	6.82843	0.273575
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	−11.3137	−0.451826
$628$	0	0
$629$	−35.3137	−1.40805
$630$	0	0
$631$	33.0711	1.31654	0.658269	−	0.752783i	$-0.271289\pi$
$631$	33.0711	1.31654	0.658269	+	0.752783i	$0.271289\pi$
$632$	0	0
$633$	−19.3137	−0.767651
$634$	0	0
$635$	3.94113	0.156399
$636$	0	0
$637$	13.1716	0.521877
$638$	0	0
$639$	9.17157	0.362822
$640$	0	0
$641$	42.9706	1.69724	0.848618	−	0.529007i	$-0.177435\pi$
$641$	42.9706	1.69724	0.848618	+	0.529007i	$0.177435\pi$
$642$	0	0
$643$	24.2843	0.957678	0.478839	−	0.877903i	$-0.341058\pi$
$643$	24.2843	0.957678	0.478839	+	0.877903i	$0.341058\pi$
$644$	0	0
$645$	0.970563	0.0382159
$646$	0	0
$647$	−17.1716	−0.675084	−0.337542	−	0.941310i	$-0.609596\pi$
$647$	−17.1716	−0.675084	−0.337542	+	0.941310i	$0.609596\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	15.6569	0.613640
$652$	0	0
$653$	23.4142	0.916269	0.458134	−	0.888883i	$-0.348518\pi$
$653$	23.4142	0.916269	0.458134	+	0.888883i	$0.348518\pi$
$654$	0	0
$655$	−2.34315	−0.0915543
$656$	0	0
$657$	−1.65685	−0.0646400
$658$	0	0
$659$	−16.6863	−0.650006	−0.325003	−	0.945713i	$-0.605365\pi$
$659$	−16.6863	−0.650006	−0.325003	+	0.945713i	$0.605365\pi$
$660$	0	0
$661$	16.6863	0.649022	0.324511	−	0.945882i	$-0.394800\pi$
$661$	16.6863	0.649022	0.324511	+	0.945882i	$0.394800\pi$
$662$	0	0
$663$	10.3431	0.401694
$664$	0	0
$665$	−11.3137	−0.438727
$666$	0	0
$667$	0.686292	0.0265733
$668$	0	0
$669$	7.89949	0.305412
$670$	0	0
$671$	−19.3137	−0.745597
$672$	0	0
$673$	36.6274	1.41188	0.705942	−	0.708270i	$-0.250524\pi$
$673$	36.6274	1.41188	0.705942	+	0.708270i	$0.250524\pi$
$674$	0	0
$675$	−4.65685	−0.179242
$676$	0	0
$677$	7.21320	0.277226	0.138613	−	0.990347i	$-0.455736\pi$
$677$	7.21320	0.277226	0.138613	+	0.990347i	$0.455736\pi$
$678$	0	0
$679$	45.4558	1.74444
$680$	0	0
$681$	29.3137	1.12330
$682$	0	0
$683$	−25.3137	−0.968602	−0.484301	−	0.874901i	$-0.660926\pi$
$683$	−25.3137	−0.968602	−0.484301	+	0.874901i	$0.660926\pi$
$684$	0	0
$685$	2.54416	0.0972072
$686$	0	0
$687$	−16.4853	−0.628953
$688$	0	0
$689$	33.6569	1.28222
$690$	0	0
$691$	−17.6569	−0.671698	−0.335849	−	0.941916i	$-0.609023\pi$
$691$	−17.6569	−0.671698	−0.335849	+	0.941916i	$0.609023\pi$
$692$	0	0
$693$	6.82843	0.259390
$694$	0	0
$695$	11.3137	0.429153
$696$	0	0
$697$	−42.6274	−1.61463
$698$	0	0
$699$	9.31371	0.352277
$700$	0	0
$701$	7.41421	0.280031	0.140015	−	0.990149i	$-0.455285\pi$
$701$	7.41421	0.280031	0.140015	+	0.990149i	$0.455285\pi$
$702$	0	0
$703$	54.6274	2.06031
$704$	0	0
$705$	7.31371	0.275450
$706$	0	0
$707$	−59.9411	−2.25432
$708$	0	0
$709$	31.1127	1.16846	0.584231	−	0.811587i	$-0.301396\pi$
$709$	31.1127	1.16846	0.584231	+	0.811587i	$0.301396\pi$
$710$	0	0
$711$	5.75736	0.215918
$712$	0	0
$713$	5.37258	0.201205
$714$	0	0
$715$	3.31371	0.123926
$716$	0	0
$717$	16.9706	0.633777
$718$	0	0
$719$	25.1716	0.938741	0.469371	−	0.883001i	$-0.344481\pi$
$719$	25.1716	0.938741	0.469371	+	0.883001i	$0.344481\pi$
$720$	0	0
$721$	30.9706	1.15340
$722$	0	0
$723$	−17.6569	−0.656665
$724$	0	0
$725$	−2.72792	−0.101312
$726$	0	0
$727$	−13.7574	−0.510232	−0.255116	−	0.966910i	$-0.582114\pi$
$727$	−13.7574	−0.510232	−0.255116	+	0.966910i	$0.582114\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.05887	0.224096
$732$	0	0
$733$	40.4853	1.49536	0.747679	−	0.664060i	$-0.231168\pi$
$733$	40.4853	1.49536	0.747679	+	0.664060i	$0.231168\pi$
$734$	0	0
$735$	2.72792	0.100621
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	34.6274	1.27379	0.636895	−	0.770951i	$-0.280218\pi$
$739$	34.6274	1.27379	0.636895	+	0.770951i	$0.280218\pi$
$740$	0	0
$741$	−16.0000	−0.587775
$742$	0	0
$743$	40.0000	1.46746	0.733729	−	0.679442i	$-0.237778\pi$
$743$	40.0000	1.46746	0.733729	+	0.679442i	$0.237778\pi$
$744$	0	0
$745$	8.34315	0.305669
$746$	0	0
$747$	9.31371	0.340771
$748$	0	0
$749$	−52.2843	−1.91043
$750$	0	0
$751$	30.7279	1.12128	0.560639	−	0.828060i	$-0.310556\pi$
$751$	30.7279	1.12128	0.560639	+	0.828060i	$0.310556\pi$
$752$	0	0
$753$	−22.0000	−0.801725
$754$	0	0
$755$	2.68629	0.0977642
$756$	0	0
$757$	15.5147	0.563892	0.281946	−	0.959430i	$-0.409020\pi$
$757$	15.5147	0.563892	0.281946	+	0.959430i	$0.409020\pi$
$758$	0	0
$759$	2.34315	0.0850508
$760$	0	0
$761$	24.3431	0.882438	0.441219	−	0.897399i	$-0.354546\pi$
$761$	24.3431	0.882438	0.441219	+	0.897399i	$0.354546\pi$
$762$	0	0
$763$	−44.9706	−1.62804
$764$	0	0
$765$	2.14214	0.0774491
$766$	0	0
$767$	−11.3137	−0.408514
$768$	0	0
$769$	−13.6569	−0.492479	−0.246239	−	0.969209i	$-0.579195\pi$
$769$	−13.6569	−0.492479	−0.246239	+	0.969209i	$0.579195\pi$
$770$	0	0
$771$	−1.31371	−0.0473121
$772$	0	0
$773$	18.7279	0.673597	0.336798	−	0.941577i	$-0.390656\pi$
$773$	18.7279	0.673597	0.336798	+	0.941577i	$0.390656\pi$
$774$	0	0
$775$	−21.3553	−0.767106
$776$	0	0
$777$	−32.9706	−1.18281
$778$	0	0
$779$	65.9411	2.36259
$780$	0	0
$781$	18.3431	0.656369
$782$	0	0
$783$	0.585786	0.0209343
$784$	0	0
$785$	−11.7157	−0.418152
$786$	0	0
$787$	10.3431	0.368693	0.184347	−	0.982861i	$-0.440983\pi$
$787$	10.3431	0.368693	0.184347	+	0.982861i	$0.440983\pi$
$788$	0	0
$789$	−13.6569	−0.486197
$790$	0	0
$791$	20.4853	0.728373
$792$	0	0
$793$	−27.3137	−0.969938
$794$	0	0
$795$	6.97056	0.247220
$796$	0	0
$797$	32.5858	1.15425	0.577124	−	0.816657i	$-0.304175\pi$
$797$	32.5858	1.15425	0.577124	+	0.816657i	$0.304175\pi$
$798$	0	0
$799$	45.6569	1.61522
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	−3.31371	−0.116938
$804$	0	0
$805$	2.34315	0.0825850
$806$	0	0
$807$	−6.24264	−0.219751
$808$	0	0
$809$	−38.9706	−1.37013	−0.685066	−	0.728481i	$-0.740227\pi$
$809$	−38.9706	−1.37013	−0.685066	+	0.728481i	$0.740227\pi$
$810$	0	0
$811$	−29.6569	−1.04139	−0.520697	−	0.853742i	$-0.674328\pi$
$811$	−29.6569	−1.04139	−0.520697	+	0.853742i	$0.674328\pi$
$812$	0	0
$813$	−19.4142	−0.680886
$814$	0	0
$815$	3.71573	0.130156
$816$	0	0
$817$	−9.37258	−0.327905
$818$	0	0
$819$	9.65685	0.337438
$820$	0	0
$821$	−44.8701	−1.56598	−0.782988	−	0.622037i	$-0.786305\pi$
$821$	−44.8701	−1.56598	−0.782988	+	0.622037i	$0.786305\pi$
$822$	0	0
$823$	−3.21320	−0.112005	−0.0560026	−	0.998431i	$-0.517836\pi$
$823$	−3.21320	−0.112005	−0.0560026	+	0.998431i	$0.517836\pi$
$824$	0	0
$825$	−9.31371	−0.324262
$826$	0	0
$827$	37.9411	1.31934	0.659671	−	0.751554i	$-0.270696\pi$
$827$	37.9411	1.31934	0.659671	+	0.751554i	$0.270696\pi$
$828$	0	0
$829$	20.7696	0.721356	0.360678	−	0.932690i	$-0.382545\pi$
$829$	20.7696	0.721356	0.360678	+	0.932690i	$0.382545\pi$
$830$	0	0
$831$	−7.51472	−0.260683
$832$	0	0
$833$	17.0294	0.590035
$834$	0	0
$835$	−12.6863	−0.439027
$836$	0	0
$837$	4.58579	0.158508
$838$	0	0
$839$	11.5147	0.397532	0.198766	−	0.980047i	$-0.436307\pi$
$839$	11.5147	0.397532	0.198766	+	0.980047i	$0.436307\pi$
$840$	0	0
$841$	−28.6569	−0.988167
$842$	0	0
$843$	−2.00000	−0.0688837
$844$	0	0
$845$	−2.92893	−0.100758
$846$	0	0
$847$	−23.8995	−0.821196
$848$	0	0
$849$	15.3137	0.525565
$850$	0	0
$851$	−11.3137	−0.387829
$852$	0	0
$853$	−10.6274	−0.363876	−0.181938	−	0.983310i	$-0.558237\pi$
$853$	−10.6274	−0.363876	−0.181938	+	0.983310i	$0.558237\pi$
$854$	0	0
$855$	−3.31371	−0.113326
$856$	0	0
$857$	−37.5980	−1.28432	−0.642161	−	0.766570i	$-0.721962\pi$
$857$	−37.5980	−1.28432	−0.642161	+	0.766570i	$0.721962\pi$
$858$	0	0
$859$	11.0294	0.376320	0.188160	−	0.982138i	$-0.439748\pi$
$859$	11.0294	0.376320	0.188160	+	0.982138i	$0.439748\pi$
$860$	0	0
$861$	−39.7990	−1.35635
$862$	0	0
$863$	48.9706	1.66698	0.833489	−	0.552537i	$-0.186340\pi$
$863$	48.9706	1.66698	0.833489	+	0.552537i	$0.186340\pi$
$864$	0	0
$865$	10.9706	0.373010
$866$	0	0
$867$	−3.62742	−0.123194
$868$	0	0
$869$	11.5147	0.390610
$870$	0	0
$871$	−22.6274	−0.766701
$872$	0	0
$873$	13.3137	0.450601
$874$	0	0
$875$	−19.3137	−0.652923
$876$	0	0
$877$	−0.686292	−0.0231744	−0.0115872	−	0.999933i	$-0.503688\pi$
$877$	−0.686292	−0.0231744	−0.0115872	+	0.999933i	$0.503688\pi$
$878$	0	0
$879$	27.8995	0.941027
$880$	0	0
$881$	22.6863	0.764321	0.382160	−	0.924096i	$-0.375180\pi$
$881$	22.6863	0.764321	0.382160	+	0.924096i	$0.375180\pi$
$882$	0	0
$883$	41.6569	1.40186	0.700932	−	0.713228i	$-0.252767\pi$
$883$	41.6569	1.40186	0.700932	+	0.713228i	$0.252767\pi$
$884$	0	0
$885$	−2.34315	−0.0787640
$886$	0	0
$887$	−18.3431	−0.615903	−0.307951	−	0.951402i	$-0.599643\pi$
$887$	−18.3431	−0.615903	−0.307951	+	0.951402i	$0.599643\pi$
$888$	0	0
$889$	22.9706	0.770408
$890$	0	0
$891$	2.00000	0.0670025
$892$	0	0
$893$	−70.6274	−2.36346
$894$	0	0
$895$	−2.34315	−0.0783227
$896$	0	0
$897$	3.31371	0.110642
$898$	0	0
$899$	2.68629	0.0895928
$900$	0	0
$901$	43.5147	1.44969
$902$	0	0
$903$	5.65685	0.188248
$904$	0	0
$905$	4.97056	0.165227
$906$	0	0
$907$	−27.5980	−0.916376	−0.458188	−	0.888855i	$-0.651501\pi$
$907$	−27.5980	−0.916376	−0.458188	+	0.888855i	$0.651501\pi$
$908$	0	0
$909$	−17.5563	−0.582307
$910$	0	0
$911$	51.3137	1.70010	0.850050	−	0.526703i	$-0.176572\pi$
$911$	51.3137	1.70010	0.850050	+	0.526703i	$0.176572\pi$
$912$	0	0
$913$	18.6274	0.616478
$914$	0	0
$915$	−5.65685	−0.187010
$916$	0	0
$917$	−13.6569	−0.450989
$918$	0	0
$919$	53.3553	1.76003	0.880015	−	0.474946i	$-0.157532\pi$
$919$	53.3553	1.76003	0.880015	+	0.474946i	$0.157532\pi$
$920$	0	0
$921$	−16.0000	−0.527218
$922$	0	0
$923$	25.9411	0.853863
$924$	0	0
$925$	44.9706	1.47862
$926$	0	0
$927$	9.07107	0.297933
$928$	0	0
$929$	−49.5980	−1.62726	−0.813628	−	0.581385i	$-0.802511\pi$
$929$	−49.5980	−1.62726	−0.813628	+	0.581385i	$0.802511\pi$
$930$	0	0
$931$	−26.3431	−0.863362
$932$	0	0
$933$	11.3137	0.370394
$934$	0	0
$935$	4.28427	0.140111
$936$	0	0
$937$	−8.62742	−0.281845	−0.140923	−	0.990021i	$-0.545007\pi$
$937$	−8.62742	−0.281845	−0.140923	+	0.990021i	$0.545007\pi$
$938$	0	0
$939$	16.9706	0.553813
$940$	0	0
$941$	16.3848	0.534128	0.267064	−	0.963679i	$-0.413946\pi$
$941$	16.3848	0.534128	0.267064	+	0.963679i	$0.413946\pi$
$942$	0	0
$943$	−13.6569	−0.444728
$944$	0	0
$945$	2.00000	0.0650600
$946$	0	0
$947$	10.6274	0.345345	0.172672	−	0.984979i	$-0.444760\pi$
$947$	10.6274	0.345345	0.172672	+	0.984979i	$0.444760\pi$
$948$	0	0
$949$	−4.68629	−0.152123
$950$	0	0
$951$	5.07107	0.164441
$952$	0	0
$953$	−2.97056	−0.0962260	−0.0481130	−	0.998842i	$-0.515321\pi$
$953$	−2.97056	−0.0962260	−0.0481130	+	0.998842i	$0.515321\pi$
$954$	0	0
$955$	−9.94113	−0.321687
$956$	0	0
$957$	1.17157	0.0378716
$958$	0	0
$959$	14.8284	0.478835
$960$	0	0
$961$	−9.97056	−0.321631
$962$	0	0
$963$	−15.3137	−0.493477
$964$	0	0
$965$	−12.6863	−0.408386
$966$	0	0
$967$	10.2426	0.329381	0.164691	−	0.986345i	$-0.447337\pi$
$967$	10.2426	0.329381	0.164691	+	0.986345i	$0.447337\pi$
$968$	0	0
$969$	−20.6863	−0.664539
$970$	0	0
$971$	6.68629	0.214573	0.107287	−	0.994228i	$-0.465784\pi$
$971$	6.68629	0.214573	0.107287	+	0.994228i	$0.465784\pi$
$972$	0	0
$973$	65.9411	2.11398
$974$	0	0
$975$	−13.1716	−0.421828
$976$	0	0
$977$	26.2843	0.840908	0.420454	−	0.907314i	$-0.361871\pi$
$977$	26.2843	0.840908	0.420454	+	0.907314i	$0.361871\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	0	0
$981$	−13.1716	−0.420536
$982$	0	0
$983$	52.2843	1.66761	0.833805	−	0.552060i	$-0.186158\pi$
$983$	52.2843	1.66761	0.833805	+	0.552060i	$0.186158\pi$
$984$	0	0
$985$	1.71573	0.0546677
$986$	0	0
$987$	42.6274	1.35685
$988$	0	0
$989$	1.94113	0.0617242
$990$	0	0
$991$	−7.89949	−0.250936	−0.125468	−	0.992098i	$-0.540043\pi$
$991$	−7.89949	−0.250936	−0.125468	+	0.992098i	$0.540043\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7.94113	−0.251751
$996$	0	0
$997$	−20.9706	−0.664144	−0.332072	−	0.943254i	$-0.607748\pi$
$997$	−20.9706	−0.664144	−0.332072	+	0.943254i	$0.607748\pi$
$998$	0	0
$999$	−9.65685	−0.305529

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.a.l.1.1	yes	2
3.2	odd	2		4608.2.a.e.1.2		2
4.3	odd	2		1536.2.a.e.1.1	yes	2
8.3	odd	2		1536.2.a.g.1.2	yes	2
8.5	even	2		1536.2.a.b.1.2	✓	2
12.11	even	2		4608.2.a.a.1.2		2
16.3	odd	4		1536.2.d.f.769.2		4
16.5	even	4		1536.2.d.a.769.1		4
16.11	odd	4		1536.2.d.f.769.3		4
16.13	even	4		1536.2.d.a.769.4		4
24.5	odd	2		4608.2.a.r.1.1		2
24.11	even	2		4608.2.a.n.1.1		2
48.5	odd	4		4608.2.d.c.2305.2		4
48.11	even	4		4608.2.d.o.2305.2		4
48.29	odd	4		4608.2.d.c.2305.3		4
48.35	even	4		4608.2.d.o.2305.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.b.1.2	✓	2	8.5	even	2
1536.2.a.e.1.1	yes	2	4.3	odd	2
1536.2.a.g.1.2	yes	2	8.3	odd	2
1536.2.a.l.1.1	yes	2	1.1	even	1	trivial
1536.2.d.a.769.1		4	16.5	even	4
1536.2.d.a.769.4		4	16.13	even	4
1536.2.d.f.769.2		4	16.3	odd	4
1536.2.d.f.769.3		4	16.11	odd	4
4608.2.a.a.1.2		2	12.11	even	2
4608.2.a.e.1.2		2	3.2	odd	2
4608.2.a.n.1.1		2	24.11	even	2
4608.2.a.r.1.1		2	24.5	odd	2
4608.2.d.c.2305.2		4	48.5	odd	4
4608.2.d.c.2305.3		4	48.29	odd	4
4608.2.d.o.2305.2		4	48.11	even	4
4608.2.d.o.2305.3		4	48.35	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.585786 q^{5} +3.41421 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.585786 q^{5} +3.41421 q^{7} +1.00000 q^{9} +2.00000 q^{11} +2.82843 q^{13} +0.585786 q^{15} +3.65685 q^{17} -5.65685 q^{19} +3.41421 q^{21} +1.17157 q^{23} -4.65685 q^{25} +1.00000 q^{27} +0.585786 q^{29} +4.58579 q^{31} +2.00000 q^{33} +2.00000 q^{35} -9.65685 q^{37} +2.82843 q^{39} -11.6569 q^{41} +1.65685 q^{43} +0.585786 q^{45} +12.4853 q^{47} +4.65685 q^{49} +3.65685 q^{51} +11.8995 q^{53} +1.17157 q^{55} -5.65685 q^{57} -4.00000 q^{59} -9.65685 q^{61} +3.41421 q^{63} +1.65685 q^{65} -8.00000 q^{67} +1.17157 q^{69} +9.17157 q^{71} -1.65685 q^{73} -4.65685 q^{75} +6.82843 q^{77} +5.75736 q^{79} +1.00000 q^{81} +9.31371 q^{83} +2.14214 q^{85} +0.585786 q^{87} +2.00000 q^{89} +9.65685 q^{91} +4.58579 q^{93} -3.31371 q^{95} +13.3137 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} + 4 q^{15} - 4 q^{17} + 4 q^{21} + 8 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{29} + 12 q^{31} + 4 q^{33} + 4 q^{35} - 8 q^{37} - 12 q^{41} - 8 q^{43} + 4 q^{45} + 8 q^{47} - 2 q^{49} - 4 q^{51} + 4 q^{53} + 8 q^{55} - 8 q^{59} - 8 q^{61} + 4 q^{63} - 8 q^{65} - 16 q^{67} + 8 q^{69} + 24 q^{71} + 8 q^{73} + 2 q^{75} + 8 q^{77} + 20 q^{79} + 2 q^{81} - 4 q^{83} - 24 q^{85} + 4 q^{87} + 4 q^{89} + 8 q^{91} + 12 q^{93} + 16 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^9 + 4 * q^11 + 4 * q^15 - 4 * q^17 + 4 * q^21 + 8 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^29 + 12 * q^31 + 4 * q^33 + 4 * q^35 - 8 * q^37 - 12 * q^41 - 8 * q^43 + 4 * q^45 + 8 * q^47 - 2 * q^49 - 4 * q^51 + 4 * q^53 + 8 * q^55 - 8 * q^59 - 8 * q^61 + 4 * q^63 - 8 * q^65 - 16 * q^67 + 8 * q^69 + 24 * q^71 + 8 * q^73 + 2 * q^75 + 8 * q^77 + 20 * q^79 + 2 * q^81 - 4 * q^83 - 24 * q^85 + 4 * q^87 + 4 * q^89 + 8 * q^91 + 12 * q^93 + 16 * q^95 + 4 * q^97 + 4 * q^99

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