LMFDB - Embedded newform 4608.2.a.a.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4608,2,Mod(1,4608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4608, 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("4608.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4608 = 2^{9} \cdot 3^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4608.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:	$36.7950652514$
Analytic rank:	$1$
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:	no (minimal twist has level 1536)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4608.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	0	0
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	0	0
$9$	0	0
$10$	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	0
$16$	0	0
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	5.65685	1.29777	0.648886	−	0.760886i	$-0.275235\pi$
$19$	5.65685	1.29777	0.648886	+	0.760886i	$0.275235\pi$
$20$	0	0
$21$	0	0
$22$	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
$27$	0	0
$28$	0	0
$29$	−0.585786	−0.108778	−0.0543889	−	0.998520i	$-0.517321\pi$
$29$	−0.585786	−0.108778	−0.0543889	+	0.998520i	$0.517321\pi$
$30$	0	0
$31$	−4.58579	−0.823632	−0.411816	−	0.911267i	$-0.635105\pi$
$31$	−4.58579	−0.823632	−0.411816	+	0.911267i	$0.635105\pi$
$32$	0	0
$33$	0	0
$34$	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
$39$	0	0
$40$	0	0
$41$	11.6569	1.82049	0.910247	−	0.414065i	$-0.135891\pi$
$41$	11.6569	1.82049	0.910247	+	0.414065i	$0.135891\pi$
$42$	0	0
$43$	−1.65685	−0.252668	−0.126334	−	0.991988i	$-0.540321\pi$
$43$	−1.65685	−0.252668	−0.126334	+	0.991988i	$0.540321\pi$
$44$	0	0
$45$	0	0
$46$	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
$51$	0	0
$52$	0	0
$53$	−11.8995	−1.63452	−0.817261	−	0.576268i	$-0.804508\pi$
$53$	−11.8995	−1.63452	−0.817261	+	0.576268i	$0.804508\pi$
$54$	0	0
$55$	−1.17157	−0.157975
$56$	0	0
$57$	0	0
$58$	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
$63$	0	0
$64$	0	0
$65$	−1.65685	−0.205507
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	0	0
$70$	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
$75$	0	0
$76$	0	0
$77$	−6.82843	−0.778171
$78$	0	0
$79$	−5.75736	−0.647754	−0.323877	−	0.946099i	$-0.604986\pi$
$79$	−5.75736	−0.647754	−0.323877	+	0.946099i	$0.604986\pi$
$80$	0	0
$81$	0	0
$82$	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	0
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−9.65685	−1.01231
$92$	0	0
$93$	0	0
$94$	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
$99$	0	0
$100$	0	0
$101$	17.5563	1.74692	0.873461	−	0.486894i	$-0.161870\pi$
$101$	17.5563	1.74692	0.873461	+	0.486894i	$0.161870\pi$
$102$	0	0
$103$	−9.07107	−0.893799	−0.446899	−	0.894584i	$-0.647472\pi$
$103$	−9.07107	−0.893799	−0.446899	+	0.894584i	$0.647472\pi$
$104$	0	0
$105$	0	0
$106$	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
$111$	0	0
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	−0.686292	−0.0639970
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.4853	1.14452
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−6.72792	−0.597007	−0.298503	−	0.954409i	$-0.596487\pi$
$127$	−6.72792	−0.597007	−0.298503	+	0.954409i	$0.596487\pi$
$128$	0	0
$129$	0	0
$130$	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	0
$136$	0	0
$137$	−4.34315	−0.371060	−0.185530	−	0.982639i	$-0.559400\pi$
$137$	−4.34315	−0.371060	−0.185530	+	0.982639i	$0.559400\pi$
$138$	0	0
$139$	−19.3137	−1.63817	−0.819084	−	0.573674i	$-0.805518\pi$
$139$	−19.3137	−1.63817	−0.819084	+	0.573674i	$0.805518\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.65685	0.473050
$144$	0	0
$145$	0.343146	0.0284967
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.2426	−1.16680	−0.583401	−	0.812184i	$-0.698278\pi$
$149$	−14.2426	−1.16680	−0.583401	+	0.812184i	$0.698278\pi$
$150$	0	0
$151$	−4.58579	−0.373186	−0.186593	−	0.982437i	$-0.559745\pi$
$151$	−4.58579	−0.373186	−0.186593	+	0.982437i	$0.559745\pi$
$152$	0	0
$153$	0	0
$154$	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
$159$	0	0
$160$	0	0
$161$	−4.00000	−0.315244
$162$	0	0
$163$	−6.34315	−0.496834	−0.248417	−	0.968653i	$-0.579910\pi$
$163$	−6.34315	−0.496834	−0.248417	+	0.968653i	$0.579910\pi$
$164$	0	0
$165$	0	0
$166$	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
$171$	0	0
$172$	0	0
$173$	−18.7279	−1.42386	−0.711929	−	0.702252i	$-0.752178\pi$
$173$	−18.7279	−1.42386	−0.711929	+	0.702252i	$0.752178\pi$
$174$	0	0
$175$	15.8995	1.20189
$176$	0	0
$177$	0	0
$178$	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
$183$	0	0
$184$	0	0
$185$	5.65685	0.415900
$186$	0	0
$187$	−7.31371	−0.534831
$188$	0	0
$189$	0	0
$190$	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
$195$	0	0
$196$	0	0
$197$	−2.92893	−0.208678	−0.104339	−	0.994542i	$-0.533273\pi$
$197$	−2.92893	−0.208678	−0.104339	+	0.994542i	$0.533273\pi$
$198$	0	0
$199$	13.5563	0.960984	0.480492	−	0.876999i	$-0.340458\pi$
$199$	13.5563	0.960984	0.480492	+	0.876999i	$0.340458\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	−6.82843	−0.476918
$206$	0	0
$207$	0	0
$208$	0	0
$209$	11.3137	0.782586
$210$	0	0
$211$	19.3137	1.32961	0.664805	−	0.747017i	$-0.268515\pi$
$211$	19.3137	1.32961	0.664805	+	0.747017i	$0.268515\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.970563	0.0661918
$216$	0	0
$217$	15.6569	1.06286
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.3431	−0.695755
$222$	0	0
$223$	−7.89949	−0.528989	−0.264495	−	0.964387i	$-0.585205\pi$
$223$	−7.89949	−0.528989	−0.264495	+	0.964387i	$0.585205\pi$
$224$	0	0
$225$	0	0
$226$	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
$231$	0	0
$232$	0	0
$233$	−9.31371	−0.610161	−0.305081	−	0.952327i	$-0.598683\pi$
$233$	−9.31371	−0.610161	−0.305081	+	0.952327i	$0.598683\pi$
$234$	0	0
$235$	−7.31371	−0.477094
$236$	0	0
$237$	0	0
$238$	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
$243$	0	0
$244$	0	0
$245$	−2.72792	−0.174281
$246$	0	0
$247$	16.0000	1.01806
$248$	0	0
$249$	0	0
$250$	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
$255$	0	0
$256$	0	0
$257$	1.31371	0.0819469	0.0409734	−	0.999160i	$-0.486954\pi$
$257$	1.31371	0.0819469	0.0409734	+	0.999160i	$0.486954\pi$
$258$	0	0
$259$	32.9706	2.04869
$260$	0	0
$261$	0	0
$262$	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
$267$	0	0
$268$	0	0
$269$	6.24264	0.380621	0.190310	−	0.981724i	$-0.439051\pi$
$269$	6.24264	0.380621	0.190310	+	0.981724i	$0.439051\pi$
$270$	0	0
$271$	19.4142	1.17933	0.589665	−	0.807648i	$-0.299260\pi$
$271$	19.4142	1.17933	0.589665	+	0.807648i	$0.299260\pi$
$272$	0	0
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	−15.3137	−0.910305	−0.455153	−	0.890413i	$-0.650415\pi$
$283$	−15.3137	−0.910305	−0.455153	+	0.890413i	$0.650415\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−39.7990	−2.34926
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.8995	−1.62991	−0.814953	−	0.579527i	$-0.803237\pi$
$293$	−27.8995	−1.62991	−0.814953	+	0.579527i	$0.803237\pi$
$294$	0	0
$295$	2.34315	0.136423
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.31371	0.191637
$300$	0	0
$301$	5.65685	0.326056
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5.65685	0.323911
$306$	0	0
$307$	16.0000	0.913168	0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000	0.913168	0.456584	+	0.889680i	$0.349073\pi$
$308$	0	0
$309$	0	0
$310$	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
$315$	0	0
$316$	0	0
$317$	−5.07107	−0.284820	−0.142410	−	0.989808i	$-0.545485\pi$
$317$	−5.07107	−0.284820	−0.142410	+	0.989808i	$0.545485\pi$
$318$	0	0
$319$	−1.17157	−0.0655955
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.6863	−1.15102
$324$	0	0
$325$	−13.1716	−0.730627
$326$	0	0
$327$	0	0
$328$	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
$333$	0	0
$334$	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
$339$	0	0
$340$	0	0
$341$	−9.17157	−0.496669
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	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
$351$	0	0
$352$	0	0
$353$	13.3137	0.708617	0.354309	−	0.935129i	$-0.384716\pi$
$353$	13.3137	0.708617	0.354309	+	0.935129i	$0.384716\pi$
$354$	0	0
$355$	−5.37258	−0.285147
$356$	0	0
$357$	0	0
$358$	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
$363$	0	0
$364$	0	0
$365$	0.970563	0.0508016
$366$	0	0
$367$	22.7279	1.18639	0.593194	−	0.805060i	$-0.297867\pi$
$367$	22.7279	1.18639	0.593194	+	0.805060i	$0.297867\pi$
$368$	0	0
$369$	0	0
$370$	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
$375$	0	0
$376$	0	0
$377$	−1.65685	−0.0853323
$378$	0	0
$379$	18.3431	0.942224	0.471112	−	0.882073i	$-0.343853\pi$
$379$	18.3431	0.942224	0.471112	+	0.882073i	$0.343853\pi$
$380$	0	0
$381$	0	0
$382$	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
$387$	0	0
$388$	0	0
$389$	22.2426	1.12775	0.563873	−	0.825861i	$-0.309311\pi$
$389$	22.2426	1.12775	0.563873	+	0.825861i	$0.309311\pi$
$390$	0	0
$391$	−4.28427	−0.216665
$392$	0	0
$393$	0	0
$394$	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
$399$	0	0
$400$	0	0
$401$	31.6569	1.58087	0.790434	−	0.612547i	$-0.209855\pi$
$401$	31.6569	1.58087	0.790434	+	0.612547i	$0.209855\pi$
$402$	0	0
$403$	−12.9706	−0.646110
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	13.6569	0.672010
$414$	0	0
$415$	−5.45584	−0.267817
$416$	0	0
$417$	0	0
$418$	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
$423$	0	0
$424$	0	0
$425$	17.0294	0.826049
$426$	0	0
$427$	32.9706	1.59556
$428$	0	0
$429$	0	0
$430$	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	0
$436$	0	0
$437$	6.62742	0.317032
$438$	0	0
$439$	15.8995	0.758841	0.379421	−	0.925224i	$-0.376123\pi$
$439$	15.8995	0.758841	0.379421	+	0.925224i	$0.376123\pi$
$440$	0	0
$441$	0	0
$442$	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
$447$	0	0
$448$	0	0
$449$	7.65685	0.361349	0.180675	−	0.983543i	$-0.442172\pi$
$449$	7.65685	0.361349	0.180675	+	0.983543i	$0.442172\pi$
$450$	0	0
$451$	23.3137	1.09780
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	10.9289	0.509011	0.254506	−	0.967071i	$-0.418087\pi$
$461$	10.9289	0.509011	0.254506	+	0.967071i	$0.418087\pi$
$462$	0	0
$463$	−2.44365	−0.113566	−0.0567830	−	0.998387i	$-0.518084\pi$
$463$	−2.44365	−0.113566	−0.0567830	+	0.998387i	$0.518084\pi$
$464$	0	0
$465$	0	0
$466$	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
$471$	0	0
$472$	0	0
$473$	−3.31371	−0.152364
$474$	0	0
$475$	−26.3431	−1.20871
$476$	0	0
$477$	0	0
$478$	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
$483$	0	0
$484$	0	0
$485$	−7.79899	−0.354134
$486$	0	0
$487$	5.55635	0.251782	0.125891	−	0.992044i	$-0.459821\pi$
$487$	5.55635	0.251782	0.125891	+	0.992044i	$0.459821\pi$
$488$	0	0
$489$	0	0
$490$	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
$495$	0	0
$496$	0	0
$497$	−31.3137	−1.40461
$498$	0	0
$499$	−23.3137	−1.04366	−0.521832	−	0.853048i	$-0.674751\pi$
$499$	−23.3137	−1.04366	−0.521832	+	0.853048i	$0.674751\pi$
$500$	0	0
$501$	0	0
$502$	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
$507$	0	0
$508$	0	0
$509$	−23.2132	−1.02891	−0.514454	−	0.857518i	$-0.672005\pi$
$509$	−23.2132	−1.02891	−0.514454	+	0.857518i	$0.672005\pi$
$510$	0	0
$511$	5.65685	0.250244
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.31371	0.234150
$516$	0	0
$517$	24.9706	1.09820
$518$	0	0
$519$	0	0
$520$	0	0
$521$	10.9706	0.480629	0.240315	−	0.970695i	$-0.422749\pi$
$521$	10.9706	0.480629	0.240315	+	0.970695i	$0.422749\pi$
$522$	0	0
$523$	−26.3431	−1.15191	−0.575953	−	0.817483i	$-0.695369\pi$
$523$	−26.3431	−1.15191	−0.575953	+	0.817483i	$0.695369\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	16.7696	0.730493
$528$	0	0
$529$	−21.6274	−0.940322
$530$	0	0
$531$	0	0
$532$	0	0
$533$	32.9706	1.42811
$534$	0	0
$535$	8.97056	0.387831
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	0	0
$545$	7.71573	0.330506
$546$	0	0
$547$	−28.9706	−1.23869	−0.619346	−	0.785118i	$-0.712602\pi$
$547$	−28.9706	−1.23869	−0.619346	+	0.785118i	$0.712602\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−3.31371	−0.141169
$552$	0	0
$553$	19.6569	0.835894
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.0711	0.553839	0.276919	−	0.960893i	$-0.410686\pi$
$557$	13.0711	0.553839	0.276919	+	0.960893i	$0.410686\pi$
$558$	0	0
$559$	−4.68629	−0.198209
$560$	0	0
$561$	0	0
$562$	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
$567$	0	0
$568$	0	0
$569$	−18.9706	−0.795287	−0.397644	−	0.917540i	$-0.630172\pi$
$569$	−18.9706	−0.795287	−0.397644	+	0.917540i	$0.630172\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	−31.7990	−1.31924
$582$	0	0
$583$	−23.7990	−0.985653
$584$	0	0
$585$	0	0
$586$	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
$591$	0	0
$592$	0	0
$593$	2.00000	0.0821302	0.0410651	−	0.999156i	$-0.486925\pi$
$593$	2.00000	0.0821302	0.0410651	+	0.999156i	$0.486925\pi$
$594$	0	0
$595$	−7.31371	−0.299833
$596$	0	0
$597$	0	0
$598$	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
$603$	0	0
$604$	0	0
$605$	4.10051	0.166709
$606$	0	0
$607$	15.8995	0.645341	0.322670	−	0.946511i	$-0.395419\pi$
$607$	15.8995	0.645341	0.322670	+	0.946511i	$0.395419\pi$
$608$	0	0
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	43.9411	1.76900	0.884502	−	0.466537i	$-0.154499\pi$
$617$	43.9411	1.76900	0.884502	+	0.466537i	$0.154499\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	6.82843	0.273575
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	35.3137	1.40805
$630$	0	0
$631$	−33.0711	−1.31654	−0.658269	−	0.752783i	$-0.728711\pi$
$631$	−33.0711	−1.31654	−0.658269	+	0.752783i	$0.728711\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	3.94113	0.156399
$636$	0	0
$637$	13.1716	0.521877
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−42.9706	−1.69724	−0.848618	−	0.529007i	$-0.822565\pi$
$641$	−42.9706	−1.69724	−0.848618	+	0.529007i	$0.822565\pi$
$642$	0	0
$643$	−24.2843	−0.957678	−0.478839	−	0.877903i	$-0.658942\pi$
$643$	−24.2843	−0.957678	−0.478839	+	0.877903i	$0.658942\pi$
$644$	0	0
$645$	0	0
$646$	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
$651$	0	0
$652$	0	0
$653$	−23.4142	−0.916269	−0.458134	−	0.888883i	$-0.651482\pi$
$653$	−23.4142	−0.916269	−0.458134	+	0.888883i	$0.651482\pi$
$654$	0	0
$655$	2.34315	0.0915543
$656$	0	0
$657$	0	0
$658$	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
$663$	0	0
$664$	0	0
$665$	11.3137	0.438727
$666$	0	0
$667$	−0.686292	−0.0265733
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	−7.21320	−0.277226	−0.138613	−	0.990347i	$-0.544264\pi$
$677$	−7.21320	−0.277226	−0.138613	+	0.990347i	$0.544264\pi$
$678$	0	0
$679$	−45.4558	−1.74444
$680$	0	0
$681$	0	0
$682$	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
$687$	0	0
$688$	0	0
$689$	−33.6569	−1.28222
$690$	0	0
$691$	17.6569	0.671698	0.335849	−	0.941916i	$-0.390977\pi$
$691$	17.6569	0.671698	0.335849	+	0.941916i	$0.390977\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	11.3137	0.429153
$696$	0	0
$697$	−42.6274	−1.61463
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−7.41421	−0.280031	−0.140015	−	0.990149i	$-0.544715\pi$
$701$	−7.41421	−0.280031	−0.140015	+	0.990149i	$0.544715\pi$
$702$	0	0
$703$	−54.6274	−2.06031
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	−5.37258	−0.201205
$714$	0	0
$715$	−3.31371	−0.123926
$716$	0	0
$717$	0	0
$718$	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
$723$	0	0
$724$	0	0
$725$	2.72792	0.101312
$726$	0	0
$727$	13.7574	0.510232	0.255116	−	0.966910i	$-0.417886\pi$
$727$	13.7574	0.510232	0.255116	+	0.966910i	$0.417886\pi$
$728$	0	0
$729$	0	0
$730$	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
$735$	0	0
$736$	0	0
$737$	16.0000	0.589368
$738$	0	0
$739$	−34.6274	−1.27379	−0.636895	−	0.770951i	$-0.719782\pi$
$739$	−34.6274	−1.27379	−0.636895	+	0.770951i	$0.719782\pi$
$740$	0	0
$741$	0	0
$742$	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
$747$	0	0
$748$	0	0
$749$	52.2843	1.91043
$750$	0	0
$751$	−30.7279	−1.12128	−0.560639	−	0.828060i	$-0.689444\pi$
$751$	−30.7279	−1.12128	−0.560639	+	0.828060i	$0.689444\pi$
$752$	0	0
$753$	0	0
$754$	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
$759$	0	0
$760$	0	0
$761$	−24.3431	−0.882438	−0.441219	−	0.897399i	$-0.645454\pi$
$761$	−24.3431	−0.882438	−0.441219	+	0.897399i	$0.645454\pi$
$762$	0	0
$763$	44.9706	1.62804
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	−18.7279	−0.673597	−0.336798	−	0.941577i	$-0.609344\pi$
$773$	−18.7279	−0.673597	−0.336798	+	0.941577i	$0.609344\pi$
$774$	0	0
$775$	21.3553	0.767106
$776$	0	0
$777$	0	0
$778$	0	0
$779$	65.9411	2.36259
$780$	0	0
$781$	18.3431	0.656369
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.7157	0.418152
$786$	0	0
$787$	−10.3431	−0.368693	−0.184347	−	0.982861i	$-0.559017\pi$
$787$	−10.3431	−0.368693	−0.184347	+	0.982861i	$0.559017\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20.4853	0.728373
$792$	0	0
$793$	−27.3137	−0.969938
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.5858	−1.15425	−0.577124	−	0.816657i	$-0.695825\pi$
$797$	−32.5858	−1.15425	−0.577124	+	0.816657i	$0.695825\pi$
$798$	0	0
$799$	−45.6569	−1.61522
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−3.31371	−0.116938
$804$	0	0
$805$	2.34315	0.0825850
$806$	0	0
$807$	0	0
$808$	0	0
$809$	38.9706	1.37013	0.685066	−	0.728481i	$-0.259773\pi$
$809$	38.9706	1.37013	0.685066	+	0.728481i	$0.259773\pi$
$810$	0	0
$811$	29.6569	1.04139	0.520697	−	0.853742i	$-0.325672\pi$
$811$	29.6569	1.04139	0.520697	+	0.853742i	$0.325672\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.71573	0.130156
$816$	0	0
$817$	−9.37258	−0.327905
$818$	0	0
$819$	0	0
$820$	0	0
$821$	44.8701	1.56598	0.782988	−	0.622037i	$-0.213695\pi$
$821$	44.8701	1.56598	0.782988	+	0.622037i	$0.213695\pi$
$822$	0	0
$823$	3.21320	0.112005	0.0560026	−	0.998431i	$-0.482164\pi$
$823$	3.21320	0.112005	0.0560026	+	0.998431i	$0.482164\pi$
$824$	0	0
$825$	0	0
$826$	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
$831$	0	0
$832$	0	0
$833$	−17.0294	−0.590035
$834$	0	0
$835$	12.6863	0.439027
$836$	0	0
$837$	0	0
$838$	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
$843$	0	0
$844$	0	0
$845$	2.92893	0.100758
$846$	0	0
$847$	23.8995	0.821196
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	37.5980	1.28432	0.642161	−	0.766570i	$-0.278038\pi$
$857$	37.5980	1.28432	0.642161	+	0.766570i	$0.278038\pi$
$858$	0	0
$859$	−11.0294	−0.376320	−0.188160	−	0.982138i	$-0.560252\pi$
$859$	−11.0294	−0.376320	−0.188160	+	0.982138i	$0.560252\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	0	0
$868$	0	0
$869$	−11.5147	−0.390610
$870$	0	0
$871$	22.6274	0.766701
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	−22.6863	−0.764321	−0.382160	−	0.924096i	$-0.624820\pi$
$881$	−22.6863	−0.764321	−0.382160	+	0.924096i	$0.624820\pi$
$882$	0	0
$883$	−41.6569	−1.40186	−0.700932	−	0.713228i	$-0.747233\pi$
$883$	−41.6569	−1.40186	−0.700932	+	0.713228i	$0.747233\pi$
$884$	0	0
$885$	0	0
$886$	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
$891$	0	0
$892$	0	0
$893$	70.6274	2.36346
$894$	0	0
$895$	2.34315	0.0783227
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2.68629	0.0895928
$900$	0	0
$901$	43.5147	1.44969
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−4.97056	−0.165227
$906$	0	0
$907$	27.5980	0.916376	0.458188	−	0.888855i	$-0.348499\pi$
$907$	27.5980	0.916376	0.458188	+	0.888855i	$0.348499\pi$
$908$	0	0
$909$	0	0
$910$	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
$915$	0	0
$916$	0	0
$917$	13.6569	0.450989
$918$	0	0
$919$	−53.3553	−1.76003	−0.880015	−	0.474946i	$-0.842468\pi$
$919$	−53.3553	−1.76003	−0.880015	+	0.474946i	$0.842468\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	25.9411	0.853863
$924$	0	0
$925$	44.9706	1.47862
$926$	0	0
$927$	0	0
$928$	0	0
$929$	49.5980	1.62726	0.813628	−	0.581385i	$-0.197489\pi$
$929$	49.5980	1.62726	0.813628	+	0.581385i	$0.197489\pi$
$930$	0	0
$931$	26.3431	0.863362
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	−16.3848	−0.534128	−0.267064	−	0.963679i	$-0.586054\pi$
$941$	−16.3848	−0.534128	−0.267064	+	0.963679i	$0.586054\pi$
$942$	0	0
$943$	13.6569	0.444728
$944$	0	0
$945$	0	0
$946$	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
$951$	0	0
$952$	0	0
$953$	2.97056	0.0962260	0.0481130	−	0.998842i	$-0.484679\pi$
$953$	2.97056	0.0962260	0.0481130	+	0.998842i	$0.484679\pi$
$954$	0	0
$955$	9.94113	0.321687
$956$	0	0
$957$	0	0
$958$	0	0
$959$	14.8284	0.478835
$960$	0	0
$961$	−9.97056	−0.321631
$962$	0	0
$963$	0	0
$964$	0	0
$965$	12.6863	0.408386
$966$	0	0
$967$	−10.2426	−0.329381	−0.164691	−	0.986345i	$-0.552663\pi$
$967$	−10.2426	−0.329381	−0.164691	+	0.986345i	$0.552663\pi$
$968$	0	0
$969$	0	0
$970$	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
$975$	0	0
$976$	0	0
$977$	−26.2843	−0.840908	−0.420454	−	0.907314i	$-0.638129\pi$
$977$	−26.2843	−0.840908	−0.420454	+	0.907314i	$0.638129\pi$
$978$	0	0
$979$	−4.00000	−0.127841
$980$	0	0
$981$	0	0
$982$	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
$987$	0	0
$988$	0	0
$989$	−1.94113	−0.0617242
$990$	0	0
$991$	7.89949	0.250936	0.125468	−	0.992098i	$-0.459957\pi$
$991$	7.89949	0.250936	0.125468	+	0.992098i	$0.459957\pi$
$992$	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
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.a.b.1.2	✓	2	24.11	even	2
1536.2.a.e.1.1	yes	2	3.2	odd	2
1536.2.a.g.1.2	yes	2	24.5	odd	2
1536.2.a.l.1.1	yes	2	12.11	even	2
1536.2.d.a.769.1		4	48.11	even	4
1536.2.d.a.769.4		4	48.35	even	4
1536.2.d.f.769.2		4	48.29	odd	4
1536.2.d.f.769.3		4	48.5	odd	4
4608.2.a.a.1.2		2	1.1	even	1	trivial
4608.2.a.e.1.2		2	4.3	odd	2
4608.2.a.n.1.1		2	8.5	even	2
4608.2.a.r.1.1		2	8.3	odd	2
4608.2.d.c.2305.2		4	16.11	odd	4
4608.2.d.c.2305.3		4	16.3	odd	4
4608.2.d.o.2305.2		4	16.5	even	4
4608.2.d.o.2305.3		4	16.13	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:	\( N \)	\(=\)	\( 4608 = 2^{9} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4608.a (trivial)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more