LMFDB - Embedded newform 4608.2.a.b.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:	$0$
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:	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)

$f(q)$	$=$	$q-0.585786 q^{5} -1.41421 q^{7} +O(q^{10})$	$q-0.585786 q^{5} -1.41421 q^{7} +0.828427 q^{11} +4.82843 q^{13} +0.828427 q^{17} +2.82843 q^{19} +6.82843 q^{23} -4.65685 q^{25} -4.58579 q^{29} +7.07107 q^{31} +0.828427 q^{35} +0.343146 q^{37} -6.48528 q^{41} -1.17157 q^{43} -4.48528 q^{47} -5.00000 q^{49} -10.2426 q^{53} -0.485281 q^{55} -9.65685 q^{59} +11.6569 q^{61} -2.82843 q^{65} -5.65685 q^{67} +8.48528 q^{71} +11.3137 q^{73} -1.17157 q^{77} +14.5858 q^{79} -3.17157 q^{83} -0.485281 q^{85} +17.3137 q^{89} -6.82843 q^{91} -1.65685 q^{95} +3.65685 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5}+O(q^{10}) $ 2 * q - 4 * q^5	$ 2 q - 4 q^{5} - 4 q^{11} + 4 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 12 q^{29} - 4 q^{35} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 8 q^{47} - 10 q^{49} - 12 q^{53} + 16 q^{55} - 8 q^{59} + 12 q^{61} - 8 q^{77} + 32 q^{79} - 12 q^{83} + 16 q^{85} + 12 q^{89} - 8 q^{91} + 8 q^{95} - 4 q^{97}+O(q^{100}) $ 2 * q - 4 * q^5 - 4 * q^11 + 4 * q^13 - 4 * q^17 + 8 * q^23 + 2 * q^25 - 12 * q^29 - 4 * q^35 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 8 * q^47 - 10 * q^49 - 12 * q^53 + 16 * q^55 - 8 * q^59 + 12 * q^61 - 8 * q^77 + 32 * q^79 - 12 * q^83 + 16 * q^85 + 12 * q^89 - 8 * q^91 + 8 * 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
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	6.82843	1.42383	0.711913	−	0.702268i	$-0.247829\pi$
$23$	6.82843	1.42383	0.711913	+	0.702268i	$0.247829\pi$
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−4.58579	−0.851559	−0.425780	−	0.904827i	$-0.640000\pi$
$29$	−4.58579	−0.851559	−0.425780	+	0.904827i	$0.640000\pi$
$30$	0	0
$31$	7.07107	1.27000	0.635001	−	0.772512i	$-0.281000\pi$
$31$	7.07107	1.27000	0.635001	+	0.772512i	$0.281000\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.828427	0.140030
$36$	0	0
$37$	0.343146	0.0564128	0.0282064	−	0.999602i	$-0.491020\pi$
$37$	0.343146	0.0564128	0.0282064	+	0.999602i	$0.491020\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.48528	−1.01283	−0.506415	−	0.862290i	$-0.669030\pi$
$41$	−6.48528	−1.01283	−0.506415	+	0.862290i	$0.669030\pi$
$42$	0	0
$43$	−1.17157	−0.178663	−0.0893316	−	0.996002i	$-0.528473\pi$
$43$	−1.17157	−0.178663	−0.0893316	+	0.996002i	$0.528473\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.48528	−0.654246	−0.327123	−	0.944982i	$-0.606079\pi$
$47$	−4.48528	−0.654246	−0.327123	+	0.944982i	$0.606079\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−10.2426	−1.40693	−0.703467	−	0.710727i	$-0.748366\pi$
$53$	−10.2426	−1.40693	−0.703467	+	0.710727i	$0.748366\pi$
$54$	0	0
$55$	−0.485281	−0.0654353
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−9.65685	−1.25722	−0.628608	−	0.777723i	$-0.716375\pi$
$59$	−9.65685	−1.25722	−0.628608	+	0.777723i	$0.716375\pi$
$60$	0	0
$61$	11.6569	1.49251	0.746254	−	0.665662i	$-0.231851\pi$
$61$	11.6569	1.49251	0.746254	+	0.665662i	$0.231851\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.82843	−0.350823
$66$	0	0
$67$	−5.65685	−0.691095	−0.345547	−	0.938401i	$-0.612307\pi$
$67$	−5.65685	−0.691095	−0.345547	+	0.938401i	$0.612307\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.48528	1.00702	0.503509	−	0.863990i	$-0.332042\pi$
$71$	8.48528	1.00702	0.503509	+	0.863990i	$0.332042\pi$
$72$	0	0
$73$	11.3137	1.32417	0.662085	−	0.749429i	$-0.269672\pi$
$73$	11.3137	1.32417	0.662085	+	0.749429i	$0.269672\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.17157	−0.133513
$78$	0	0
$79$	14.5858	1.64103	0.820515	−	0.571626i	$-0.193687\pi$
$79$	14.5858	1.64103	0.820515	+	0.571626i	$0.193687\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−3.17157	−0.348125	−0.174063	−	0.984735i	$-0.555690\pi$
$83$	−3.17157	−0.348125	−0.174063	+	0.984735i	$0.555690\pi$
$84$	0	0
$85$	−0.485281	−0.0526362
$86$	0	0
$87$	0	0
$88$	0	0
$89$	17.3137	1.83525	0.917625	−	0.397448i	$-0.130104\pi$
$89$	17.3137	1.83525	0.917625	+	0.397448i	$0.130104\pi$
$90$	0	0
$91$	−6.82843	−0.715814
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.65685	−0.169990
$96$	0	0
$97$	3.65685	0.371297	0.185649	−	0.982616i	$-0.440561\pi$
$97$	3.65685	0.371297	0.185649	+	0.982616i	$0.440561\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.7279	−1.46548	−0.732742	−	0.680507i	$-0.761760\pi$
$101$	−14.7279	−1.46548	−0.732742	+	0.680507i	$0.761760\pi$
$102$	0	0
$103$	6.58579	0.648917	0.324458	−	0.945900i	$-0.394818\pi$
$103$	6.58579	0.648917	0.324458	+	0.945900i	$0.394818\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.3137	1.48043	0.740216	−	0.672369i	$-0.234723\pi$
$107$	15.3137	1.48043	0.740216	+	0.672369i	$0.234723\pi$
$108$	0	0
$109$	12.8284	1.22874	0.614370	−	0.789018i	$-0.289410\pi$
$109$	12.8284	1.22874	0.614370	+	0.789018i	$0.289410\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	7.65685	0.720296	0.360148	−	0.932895i	$-0.382726\pi$
$113$	7.65685	0.720296	0.360148	+	0.932895i	$0.382726\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.17157	−0.107398
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	17.8995	1.58832	0.794162	−	0.607707i	$-0.207910\pi$
$127$	17.8995	1.58832	0.794162	+	0.607707i	$0.207910\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	0	0
$133$	−4.00000	−0.346844
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−14.4853	−1.23756	−0.618781	−	0.785564i	$-0.712373\pi$
$137$	−14.4853	−1.23756	−0.618781	+	0.785564i	$0.712373\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	4.00000	0.334497
$144$	0	0
$145$	2.68629	0.223084
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−0.585786	−0.0479895	−0.0239947	−	0.999712i	$-0.507638\pi$
$149$	−0.585786	−0.0479895	−0.0239947	+	0.999712i	$0.507638\pi$
$150$	0	0
$151$	9.41421	0.766118	0.383059	−	0.923724i	$-0.374871\pi$
$151$	9.41421	0.766118	0.383059	+	0.923724i	$0.374871\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.14214	−0.332704
$156$	0	0
$157$	5.31371	0.424080	0.212040	−	0.977261i	$-0.431989\pi$
$157$	5.31371	0.424080	0.212040	+	0.977261i	$0.431989\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−9.65685	−0.761067
$162$	0	0
$163$	4.48528	0.351314	0.175657	−	0.984451i	$-0.443795\pi$
$163$	4.48528	0.351314	0.175657	+	0.984451i	$0.443795\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.8995	0.904702	0.452351	−	0.891840i	$-0.350585\pi$
$173$	11.8995	0.904702	0.452351	+	0.891840i	$0.350585\pi$
$174$	0	0
$175$	6.58579	0.497839
$176$	0	0
$177$	0	0
$178$	0	0
$179$	25.6569	1.91768	0.958842	−	0.283941i	$-0.0916421\pi$
$179$	25.6569	1.91768	0.958842	+	0.283941i	$0.0916421\pi$
$180$	0	0
$181$	−16.1421	−1.19984	−0.599918	−	0.800062i	$-0.704800\pi$
$181$	−16.1421	−1.19984	−0.599918	+	0.800062i	$0.704800\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.201010	−0.0147786
$186$	0	0
$187$	0.686292	0.0501866
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.31371	0.529201	0.264601	−	0.964358i	$-0.414760\pi$
$191$	7.31371	0.529201	0.264601	+	0.964358i	$0.414760\pi$
$192$	0	0
$193$	11.3137	0.814379	0.407189	−	0.913344i	$-0.366509\pi$
$193$	11.3137	0.814379	0.407189	+	0.913344i	$0.366509\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.41421	0.243253	0.121626	−	0.992576i	$-0.461189\pi$
$197$	3.41421	0.243253	0.121626	+	0.992576i	$0.461189\pi$
$198$	0	0
$199$	20.2426	1.43496	0.717481	−	0.696578i	$-0.245295\pi$
$199$	20.2426	1.43496	0.717481	+	0.696578i	$0.245295\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.48528	0.455178
$204$	0	0
$205$	3.79899	0.265333
$206$	0	0
$207$	0	0
$208$	0	0
$209$	2.34315	0.162079
$210$	0	0
$211$	16.0000	1.10149	0.550743	−	0.834675i	$-0.314345\pi$
$211$	16.0000	1.10149	0.550743	+	0.834675i	$0.314345\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.686292	0.0468047
$216$	0	0
$217$	−10.0000	−0.678844
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	−7.55635	−0.506011	−0.253005	−	0.967465i	$-0.581419\pi$
$223$	−7.55635	−0.506011	−0.253005	+	0.967465i	$0.581419\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−16.1421	−1.07139	−0.535696	−	0.844411i	$-0.679951\pi$
$227$	−16.1421	−1.07139	−0.535696	+	0.844411i	$0.679951\pi$
$228$	0	0
$229$	−21.7990	−1.44052	−0.720259	−	0.693705i	$-0.755977\pi$
$229$	−21.7990	−1.44052	−0.720259	+	0.693705i	$0.755977\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	2.62742	0.171394
$236$	0	0
$237$	0	0
$238$	0	0
$239$	23.3137	1.50804	0.754019	−	0.656852i	$-0.228113\pi$
$239$	23.3137	1.50804	0.754019	+	0.656852i	$0.228113\pi$
$240$	0	0
$241$	−9.65685	−0.622053	−0.311026	−	0.950401i	$-0.600673\pi$
$241$	−9.65685	−0.622053	−0.311026	+	0.950401i	$0.600673\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	2.92893	0.187123
$246$	0	0
$247$	13.6569	0.868965
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−21.7990	−1.37594	−0.687970	−	0.725739i	$-0.741498\pi$
$251$	−21.7990	−1.37594	−0.687970	+	0.725739i	$0.741498\pi$
$252$	0	0
$253$	5.65685	0.355643
$254$	0	0
$255$	0	0
$256$	0	0
$257$	8.34315	0.520431	0.260216	−	0.965551i	$-0.416206\pi$
$257$	8.34315	0.520431	0.260216	+	0.965551i	$0.416206\pi$
$258$	0	0
$259$	−0.485281	−0.0301539
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−8.97056	−0.553149	−0.276574	−	0.960993i	$-0.589199\pi$
$263$	−8.97056	−0.553149	−0.276574	+	0.960993i	$0.589199\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.24264	0.136736	0.0683681	−	0.997660i	$-0.478221\pi$
$269$	2.24264	0.136736	0.0683681	+	0.997660i	$0.478221\pi$
$270$	0	0
$271$	−12.2426	−0.743687	−0.371844	−	0.928295i	$-0.621274\pi$
$271$	−12.2426	−0.743687	−0.371844	+	0.928295i	$0.621274\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.85786	−0.232638
$276$	0	0
$277$	−20.8284	−1.25146	−0.625729	−	0.780040i	$-0.715199\pi$
$277$	−20.8284	−1.25146	−0.625729	+	0.780040i	$0.715199\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.3431	0.736330	0.368165	−	0.929760i	$-0.379986\pi$
$281$	12.3431	0.736330	0.368165	+	0.929760i	$0.379986\pi$
$282$	0	0
$283$	−20.9706	−1.24657	−0.623285	−	0.781995i	$-0.714202\pi$
$283$	−20.9706	−1.24657	−0.623285	+	0.781995i	$0.714202\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.17157	0.541381
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	0	0
$292$	0	0
$293$	17.0711	0.997302	0.498651	−	0.866803i	$-0.333829\pi$
$293$	17.0711	0.997302	0.498651	+	0.866803i	$0.333829\pi$
$294$	0	0
$295$	5.65685	0.329355
$296$	0	0
$297$	0	0
$298$	0	0
$299$	32.9706	1.90674
$300$	0	0
$301$	1.65685	0.0954995
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−6.82843	−0.390995
$306$	0	0
$307$	−29.6569	−1.69261	−0.846303	−	0.532702i	$-0.821177\pi$
$307$	−29.6569	−1.69261	−0.846303	+	0.532702i	$0.821177\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4.68629	0.265735	0.132868	−	0.991134i	$-0.457581\pi$
$311$	4.68629	0.265735	0.132868	+	0.991134i	$0.457581\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	31.8995	1.79165	0.895827	−	0.444403i	$-0.146584\pi$
$317$	31.8995	1.79165	0.895827	+	0.444403i	$0.146584\pi$
$318$	0	0
$319$	−3.79899	−0.212703
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.34315	0.130376
$324$	0	0
$325$	−22.4853	−1.24726
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.34315	0.349709
$330$	0	0
$331$	13.6569	0.750649	0.375324	−	0.926894i	$-0.377531\pi$
$331$	13.6569	0.750649	0.375324	+	0.926894i	$0.377531\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3.31371	0.181047
$336$	0	0
$337$	3.31371	0.180509	0.0902546	−	0.995919i	$-0.471232\pi$
$337$	3.31371	0.180509	0.0902546	+	0.995919i	$0.471232\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	5.85786	0.317221
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−17.7990	−0.955500	−0.477750	−	0.878496i	$-0.658547\pi$
$347$	−17.7990	−0.955500	−0.477750	+	0.878496i	$0.658547\pi$
$348$	0	0
$349$	1.31371	0.0703212	0.0351606	−	0.999382i	$-0.488806\pi$
$349$	1.31371	0.0703212	0.0351606	+	0.999382i	$0.488806\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	−4.97056	−0.263810
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−24.4853	−1.29228	−0.646142	−	0.763217i	$-0.723619\pi$
$359$	−24.4853	−1.29228	−0.646142	+	0.763217i	$0.723619\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−6.62742	−0.346895
$366$	0	0
$367$	15.0711	0.786703	0.393352	−	0.919388i	$-0.371315\pi$
$367$	15.0711	0.786703	0.393352	+	0.919388i	$0.371315\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	14.4853	0.752038
$372$	0	0
$373$	−0.343146	−0.0177674	−0.00888371	−	0.999961i	$-0.502828\pi$
$373$	−0.343146	−0.0177674	−0.00888371	+	0.999961i	$0.502828\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−22.1421	−1.14038
$378$	0	0
$379$	−10.8284	−0.556219	−0.278109	−	0.960549i	$-0.589708\pi$
$379$	−10.8284	−0.556219	−0.278109	+	0.960549i	$0.589708\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	33.6569	1.71978	0.859892	−	0.510475i	$-0.170530\pi$
$383$	33.6569	1.71978	0.859892	+	0.510475i	$0.170530\pi$
$384$	0	0
$385$	0.686292	0.0349767
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−13.0711	−0.662729	−0.331365	−	0.943503i	$-0.607509\pi$
$389$	−13.0711	−0.662729	−0.331365	+	0.943503i	$0.607509\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.54416	−0.429903
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.51472	−0.0756414	−0.0378207	−	0.999285i	$-0.512042\pi$
$401$	−1.51472	−0.0756414	−0.0378207	+	0.999285i	$0.512042\pi$
$402$	0	0
$403$	34.1421	1.70074
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0.284271	0.0140908
$408$	0	0
$409$	20.3431	1.00590	0.502952	−	0.864314i	$-0.332247\pi$
$409$	20.3431	1.00590	0.502952	+	0.864314i	$0.332247\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	13.6569	0.672010
$414$	0	0
$415$	1.85786	0.0911990
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−4.82843	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$419$	−4.82843	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$420$	0	0
$421$	5.51472	0.268771	0.134385	−	0.990929i	$-0.457094\pi$
$421$	5.51472	0.268771	0.134385	+	0.990929i	$0.457094\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−3.85786	−0.187134
$426$	0	0
$427$	−16.4853	−0.797779
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−5.85786	−0.282163	−0.141082	−	0.989998i	$-0.545058\pi$
$431$	−5.85786	−0.282163	−0.141082	+	0.989998i	$0.545058\pi$
$432$	0	0
$433$	−6.00000	−0.288342	−0.144171	−	0.989553i	$-0.546051\pi$
$433$	−6.00000	−0.288342	−0.144171	+	0.989553i	$0.546051\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	19.3137	0.923900
$438$	0	0
$439$	21.2132	1.01245	0.506225	−	0.862401i	$-0.331040\pi$
$439$	21.2132	1.01245	0.506225	+	0.862401i	$0.331040\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−30.4853	−1.44840	−0.724200	−	0.689590i	$-0.757791\pi$
$443$	−30.4853	−1.44840	−0.724200	+	0.689590i	$0.757791\pi$
$444$	0	0
$445$	−10.1421	−0.480783
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.7990	1.21753	0.608765	−	0.793351i	$-0.291665\pi$
$449$	25.7990	1.21753	0.608765	+	0.793351i	$0.291665\pi$
$450$	0	0
$451$	−5.37258	−0.252985
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	−22.2843	−1.04241	−0.521207	−	0.853430i	$-0.674518\pi$
$457$	−22.2843	−1.04241	−0.521207	+	0.853430i	$0.674518\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	23.2132	1.08115	0.540573	−	0.841297i	$-0.318207\pi$
$461$	23.2132	1.08115	0.540573	+	0.841297i	$0.318207\pi$
$462$	0	0
$463$	−3.75736	−0.174619	−0.0873096	−	0.996181i	$-0.527827\pi$
$463$	−3.75736	−0.174619	−0.0873096	+	0.996181i	$0.527827\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.8284	−0.963825	−0.481912	−	0.876219i	$-0.660058\pi$
$467$	−20.8284	−0.963825	−0.481912	+	0.876219i	$0.660058\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−0.970563	−0.0446265
$474$	0	0
$475$	−13.1716	−0.604353
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.1716	−0.967354	−0.483677	−	0.875247i	$-0.660699\pi$
$479$	−21.1716	−0.967354	−0.483677	+	0.875247i	$0.660699\pi$
$480$	0	0
$481$	1.65685	0.0755461
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.14214	−0.0972694
$486$	0	0
$487$	−4.72792	−0.214243	−0.107121	−	0.994246i	$-0.534163\pi$
$487$	−4.72792	−0.214243	−0.107121	+	0.994246i	$0.534163\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	40.2843	1.81800	0.909002	−	0.416792i	$-0.136846\pi$
$491$	40.2843	1.81800	0.909002	+	0.416792i	$0.136846\pi$
$492$	0	0
$493$	−3.79899	−0.171098
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−12.0000	−0.538274
$498$	0	0
$499$	16.2843	0.728984	0.364492	−	0.931207i	$-0.381243\pi$
$499$	16.2843	0.728984	0.364492	+	0.931207i	$0.381243\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−11.7990	−0.526091	−0.263045	−	0.964783i	$-0.584727\pi$
$503$	−11.7990	−0.526091	−0.263045	+	0.964783i	$0.584727\pi$
$504$	0	0
$505$	8.62742	0.383915
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−40.8701	−1.81153	−0.905767	−	0.423777i	$-0.860704\pi$
$509$	−40.8701	−1.81153	−0.905767	+	0.423777i	$0.860704\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.85786	−0.169998
$516$	0	0
$517$	−3.71573	−0.163418
$518$	0	0
$519$	0	0
$520$	0	0
$521$	12.1421	0.531957	0.265978	−	0.963979i	$-0.414305\pi$
$521$	12.1421	0.531957	0.265978	+	0.963979i	$0.414305\pi$
$522$	0	0
$523$	41.4558	1.81274	0.906369	−	0.422488i	$-0.138843\pi$
$523$	41.4558	1.81274	0.906369	+	0.422488i	$0.138843\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.85786	0.255173
$528$	0	0
$529$	23.6274	1.02728
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−31.3137	−1.35635
$534$	0	0
$535$	−8.97056	−0.387831
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4.14214	−0.178414
$540$	0	0
$541$	0.828427	0.0356169	0.0178084	−	0.999841i	$-0.494331\pi$
$541$	0.828427	0.0356169	0.0178084	+	0.999841i	$0.494331\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−7.51472	−0.321895
$546$	0	0
$547$	−18.1421	−0.775702	−0.387851	−	0.921722i	$-0.626782\pi$
$547$	−18.1421	−0.775702	−0.387851	+	0.921722i	$0.626782\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−12.9706	−0.552565
$552$	0	0
$553$	−20.6274	−0.877167
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10.7279	0.454557	0.227278	−	0.973830i	$-0.427017\pi$
$557$	10.7279	0.454557	0.227278	+	0.973830i	$0.427017\pi$
$558$	0	0
$559$	−5.65685	−0.239259
$560$	0	0
$561$	0	0
$562$	0	0
$563$	32.8284	1.38355	0.691777	−	0.722112i	$-0.256828\pi$
$563$	32.8284	1.38355	0.691777	+	0.722112i	$0.256828\pi$
$564$	0	0
$565$	−4.48528	−0.188697
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−37.1127	−1.55585	−0.777923	−	0.628360i	$-0.783726\pi$
$569$	−37.1127	−1.55585	−0.777923	+	0.628360i	$0.783726\pi$
$570$	0	0
$571$	36.0000	1.50655	0.753277	−	0.657704i	$-0.228472\pi$
$571$	36.0000	1.50655	0.753277	+	0.657704i	$0.228472\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−31.7990	−1.32611
$576$	0	0
$577$	8.68629	0.361615	0.180808	−	0.983518i	$-0.442129\pi$
$577$	8.68629	0.361615	0.180808	+	0.983518i	$0.442129\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.48528	0.186081
$582$	0	0
$583$	−8.48528	−0.351424
$584$	0	0
$585$	0	0
$586$	0	0
$587$	12.9706	0.535352	0.267676	−	0.963509i	$-0.413744\pi$
$587$	12.9706	0.535352	0.267676	+	0.963509i	$0.413744\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.6569	1.29999	0.649996	−	0.759938i	$-0.274771\pi$
$593$	31.6569	1.29999	0.649996	+	0.759938i	$0.274771\pi$
$594$	0	0
$595$	0.686292	0.0281352
$596$	0	0
$597$	0	0
$598$	0	0
$599$	3.51472	0.143608	0.0718038	−	0.997419i	$-0.477124\pi$
$599$	3.51472	0.143608	0.0718038	+	0.997419i	$0.477124\pi$
$600$	0	0
$601$	26.6274	1.08615	0.543077	−	0.839683i	$-0.317259\pi$
$601$	26.6274	1.08615	0.543077	+	0.839683i	$0.317259\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	6.04163	0.245627
$606$	0	0
$607$	9.89949	0.401808	0.200904	−	0.979611i	$-0.435612\pi$
$607$	9.89949	0.401808	0.200904	+	0.979611i	$0.435612\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−21.6569	−0.876143
$612$	0	0
$613$	40.6274	1.64093	0.820463	−	0.571700i	$-0.193716\pi$
$613$	40.6274	1.64093	0.820463	+	0.571700i	$0.193716\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.3137	1.18013	0.590063	−	0.807357i	$-0.299103\pi$
$617$	29.3137	1.18013	0.590063	+	0.807357i	$0.299103\pi$
$618$	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
$623$	−24.4853	−0.980982
$624$	0	0
$625$	19.9706	0.798823
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.284271	0.0113346
$630$	0	0
$631$	−44.7279	−1.78059	−0.890295	−	0.455384i	$-0.849502\pi$
$631$	−44.7279	−1.78059	−0.890295	+	0.455384i	$0.849502\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10.4853	−0.416096
$636$	0	0
$637$	−24.1421	−0.956546
$638$	0	0
$639$	0	0
$640$	0	0
$641$	28.8284	1.13865	0.569327	−	0.822111i	$-0.307204\pi$
$641$	28.8284	1.13865	0.569327	+	0.822111i	$0.307204\pi$
$642$	0	0
$643$	−19.1127	−0.753731	−0.376866	−	0.926268i	$-0.622998\pi$
$643$	−19.1127	−0.753731	−0.376866	+	0.926268i	$0.622998\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	38.4264	1.51070	0.755349	−	0.655323i	$-0.227467\pi$
$647$	38.4264	1.51070	0.755349	+	0.655323i	$0.227467\pi$
$648$	0	0
$649$	−8.00000	−0.314027
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−32.3848	−1.26731	−0.633657	−	0.773614i	$-0.718447\pi$
$653$	−32.3848	−1.26731	−0.633657	+	0.773614i	$0.718447\pi$
$654$	0	0
$655$	4.28427	0.167400
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−14.3431	−0.558730	−0.279365	−	0.960185i	$-0.590124\pi$
$659$	−14.3431	−0.558730	−0.279365	+	0.960185i	$0.590124\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.34315	0.0908633
$666$	0	0
$667$	−31.3137	−1.21247
$668$	0	0
$669$	0	0
$670$	0	0
$671$	9.65685	0.372799
$672$	0	0
$673$	−18.6863	−0.720304	−0.360152	−	0.932894i	$-0.617275\pi$
$673$	−18.6863	−0.720304	−0.360152	+	0.932894i	$0.617275\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	21.0711	0.809827	0.404913	−	0.914355i	$-0.367302\pi$
$677$	21.0711	0.809827	0.404913	+	0.914355i	$0.367302\pi$
$678$	0	0
$679$	−5.17157	−0.198467
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.8579	0.606784	0.303392	−	0.952866i	$-0.401881\pi$
$683$	15.8579	0.606784	0.303392	+	0.952866i	$0.401881\pi$
$684$	0	0
$685$	8.48528	0.324206
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−49.4558	−1.88412
$690$	0	0
$691$	−30.8284	−1.17277	−0.586384	−	0.810033i	$-0.699449\pi$
$691$	−30.8284	−1.17277	−0.586384	+	0.810033i	$0.699449\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.37258	−0.355522
$696$	0	0
$697$	−5.37258	−0.203501
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−30.7279	−1.16058	−0.580289	−	0.814411i	$-0.697060\pi$
$701$	−30.7279	−1.16058	−0.580289	+	0.814411i	$0.697060\pi$
$702$	0	0
$703$	0.970563	0.0366055
$704$	0	0
$705$	0	0
$706$	0	0
$707$	20.8284	0.783334
$708$	0	0
$709$	35.1716	1.32090	0.660448	−	0.750872i	$-0.270366\pi$
$709$	35.1716	1.32090	0.660448	+	0.750872i	$0.270366\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	48.2843	1.80826
$714$	0	0
$715$	−2.34315	−0.0876287
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.48528	0.316448	0.158224	−	0.987403i	$-0.449423\pi$
$719$	8.48528	0.316448	0.158224	+	0.987403i	$0.449423\pi$
$720$	0	0
$721$	−9.31371	−0.346861
$722$	0	0
$723$	0	0
$724$	0	0
$725$	21.3553	0.793117
$726$	0	0
$727$	−40.5269	−1.50306	−0.751530	−	0.659699i	$-0.770684\pi$
$727$	−40.5269	−1.50306	−0.751530	+	0.659699i	$0.770684\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−0.970563	−0.0358976
$732$	0	0
$733$	3.17157	0.117145	0.0585724	−	0.998283i	$-0.481345\pi$
$733$	3.17157	0.117145	0.0585724	+	0.998283i	$0.481345\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.68629	−0.172622
$738$	0	0
$739$	31.3137	1.15189	0.575947	−	0.817487i	$-0.304634\pi$
$739$	31.3137	1.15189	0.575947	+	0.817487i	$0.304634\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	48.2843	1.77138	0.885689	−	0.464279i	$-0.153686\pi$
$743$	48.2843	1.77138	0.885689	+	0.464279i	$0.153686\pi$
$744$	0	0
$745$	0.343146	0.0125719
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−21.6569	−0.791324
$750$	0	0
$751$	−17.4142	−0.635454	−0.317727	−	0.948182i	$-0.602919\pi$
$751$	−17.4142	−0.635454	−0.317727	+	0.948182i	$0.602919\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.51472	−0.200701
$756$	0	0
$757$	26.7696	0.972956	0.486478	−	0.873693i	$-0.338281\pi$
$757$	26.7696	0.972956	0.486478	+	0.873693i	$0.338281\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−28.8284	−1.04503	−0.522515	−	0.852630i	$-0.675006\pi$
$761$	−28.8284	−1.04503	−0.522515	+	0.852630i	$0.675006\pi$
$762$	0	0
$763$	−18.1421	−0.656789
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−46.6274	−1.68362
$768$	0	0
$769$	−27.3137	−0.984958	−0.492479	−	0.870324i	$-0.663909\pi$
$769$	−27.3137	−0.984958	−0.492479	+	0.870324i	$0.663909\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−29.7574	−1.07030	−0.535149	−	0.844758i	$-0.679745\pi$
$773$	−29.7574	−1.07030	−0.535149	+	0.844758i	$0.679745\pi$
$774$	0	0
$775$	−32.9289	−1.18284
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−18.3431	−0.657211
$780$	0	0
$781$	7.02944	0.251533
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.11270	−0.111097
$786$	0	0
$787$	18.8284	0.671161	0.335580	−	0.942012i	$-0.391068\pi$
$787$	18.8284	0.671161	0.335580	+	0.942012i	$0.391068\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−10.8284	−0.385015
$792$	0	0
$793$	56.2843	1.99871
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.5269	−0.939631	−0.469816	−	0.882765i	$-0.655680\pi$
$797$	−26.5269	−0.939631	−0.469816	+	0.882765i	$0.655680\pi$
$798$	0	0
$799$	−3.71573	−0.131453
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.37258	0.330751
$804$	0	0
$805$	5.65685	0.199378
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−21.5147	−0.756417	−0.378209	−	0.925720i	$-0.623460\pi$
$809$	−21.5147	−0.756417	−0.378209	+	0.925720i	$0.623460\pi$
$810$	0	0
$811$	−16.4853	−0.578877	−0.289438	−	0.957197i	$-0.593468\pi$
$811$	−16.4853	−0.578877	−0.289438	+	0.957197i	$0.593468\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−2.62742	−0.0920344
$816$	0	0
$817$	−3.31371	−0.115932
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−35.0122	−1.22193	−0.610967	−	0.791656i	$-0.709219\pi$
$821$	−35.0122	−1.22193	−0.610967	+	0.791656i	$0.709219\pi$
$822$	0	0
$823$	−31.0711	−1.08307	−0.541535	−	0.840678i	$-0.682157\pi$
$823$	−31.0711	−1.08307	−0.541535	+	0.840678i	$0.682157\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−34.6274	−1.20411	−0.602057	−	0.798453i	$-0.705652\pi$
$827$	−34.6274	−1.20411	−0.602057	+	0.798453i	$0.705652\pi$
$828$	0	0
$829$	25.5147	0.886163	0.443081	−	0.896481i	$-0.353885\pi$
$829$	25.5147	0.886163	0.443081	+	0.896481i	$0.353885\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.14214	−0.143516
$834$	0	0
$835$	−7.02944	−0.243264
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−55.1127	−1.90270	−0.951351	−	0.308110i	$-0.900304\pi$
$839$	−55.1127	−1.90270	−0.951351	+	0.308110i	$0.900304\pi$
$840$	0	0
$841$	−7.97056	−0.274847
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−6.04163	−0.207838
$846$	0	0
$847$	14.5858	0.501174
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.34315	0.0803220
$852$	0	0
$853$	21.3137	0.729767	0.364884	−	0.931053i	$-0.381109\pi$
$853$	21.3137	0.729767	0.364884	+	0.931053i	$0.381109\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	10.4853	0.358170	0.179085	−	0.983834i	$-0.442686\pi$
$857$	10.4853	0.358170	0.179085	+	0.983834i	$0.442686\pi$
$858$	0	0
$859$	14.8284	0.505939	0.252970	−	0.967474i	$-0.418593\pi$
$859$	14.8284	0.505939	0.252970	+	0.967474i	$0.418593\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	4.00000	0.136162	0.0680808	−	0.997680i	$-0.478312\pi$
$863$	4.00000	0.136162	0.0680808	+	0.997680i	$0.478312\pi$
$864$	0	0
$865$	−6.97056	−0.237006
$866$	0	0
$867$	0	0
$868$	0	0
$869$	12.0833	0.409897
$870$	0	0
$871$	−27.3137	−0.925490
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−8.00000	−0.270449
$876$	0	0
$877$	−35.9411	−1.21365	−0.606823	−	0.794837i	$-0.707556\pi$
$877$	−35.9411	−1.21365	−0.606823	+	0.794837i	$0.707556\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−15.6569	−0.527493	−0.263746	−	0.964592i	$-0.584958\pi$
$881$	−15.6569	−0.527493	−0.263746	+	0.964592i	$0.584958\pi$
$882$	0	0
$883$	−15.7990	−0.531678	−0.265839	−	0.964017i	$-0.585649\pi$
$883$	−15.7990	−0.531678	−0.265839	+	0.964017i	$0.585649\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	52.2843	1.75553	0.877767	−	0.479088i	$-0.159032\pi$
$887$	52.2843	1.75553	0.877767	+	0.479088i	$0.159032\pi$
$888$	0	0
$889$	−25.3137	−0.848995
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−12.6863	−0.424531
$894$	0	0
$895$	−15.0294	−0.502379
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−32.4264	−1.08148
$900$	0	0
$901$	−8.48528	−0.282686
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.45584	0.314323
$906$	0	0
$907$	30.8284	1.02364	0.511821	−	0.859092i	$-0.328971\pi$
$907$	30.8284	1.02364	0.511821	+	0.859092i	$0.328971\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−59.5980	−1.97457	−0.987285	−	0.158963i	$-0.949185\pi$
$911$	−59.5980	−1.97457	−0.987285	+	0.158963i	$0.949185\pi$
$912$	0	0
$913$	−2.62742	−0.0869548
$914$	0	0
$915$	0	0
$916$	0	0
$917$	10.3431	0.341561
$918$	0	0
$919$	15.5563	0.513157	0.256578	−	0.966523i	$-0.417405\pi$
$919$	15.5563	0.513157	0.256578	+	0.966523i	$0.417405\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	40.9706	1.34856
$924$	0	0
$925$	−1.59798	−0.0525413
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−19.8579	−0.651515	−0.325758	−	0.945453i	$-0.605619\pi$
$929$	−19.8579	−0.651515	−0.325758	+	0.945453i	$0.605619\pi$
$930$	0	0
$931$	−14.1421	−0.463490
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−0.402020	−0.0131475
$936$	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
$941$	−30.0416	−0.979329	−0.489665	−	0.871911i	$-0.662881\pi$
$941$	−30.0416	−0.979329	−0.489665	+	0.871911i	$0.662881\pi$
$942$	0	0
$943$	−44.2843	−1.44209
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.3431	1.24598	0.622992	−	0.782228i	$-0.285917\pi$
$947$	38.3431	1.24598	0.622992	+	0.782228i	$0.285917\pi$
$948$	0	0
$949$	54.6274	1.77328
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−34.4853	−1.11709	−0.558544	−	0.829475i	$-0.688640\pi$
$953$	−34.4853	−1.11709	−0.558544	+	0.829475i	$0.688640\pi$
$954$	0	0
$955$	−4.28427	−0.138636
$956$	0	0
$957$	0	0
$958$	0	0
$959$	20.4853	0.661504
$960$	0	0
$961$	19.0000	0.612903
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.62742	−0.213344
$966$	0	0
$967$	−48.5269	−1.56052	−0.780260	−	0.625455i	$-0.784913\pi$
$967$	−48.5269	−1.56052	−0.780260	+	0.625455i	$0.784913\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	19.1716	0.615245	0.307623	−	0.951508i	$-0.400467\pi$
$971$	19.1716	0.615245	0.307623	+	0.951508i	$0.400467\pi$
$972$	0	0
$973$	−22.6274	−0.725402
$974$	0	0
$975$	0	0
$976$	0	0
$977$	40.8284	1.30622	0.653109	−	0.757264i	$-0.273465\pi$
$977$	40.8284	1.30622	0.653109	+	0.757264i	$0.273465\pi$
$978$	0	0
$979$	14.3431	0.458409
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−26.3431	−0.840216	−0.420108	−	0.907474i	$-0.638008\pi$
$983$	−26.3431	−0.840216	−0.420108	+	0.907474i	$0.638008\pi$
$984$	0	0
$985$	−2.00000	−0.0637253
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.00000	−0.254385
$990$	0	0
$991$	−22.1005	−0.702046	−0.351023	−	0.936367i	$-0.614166\pi$
$991$	−22.1005	−0.702046	−0.351023	+	0.936367i	$0.614166\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−11.8579	−0.375920
$996$	0	0
$997$	6.97056	0.220760	0.110380	−	0.993889i	$-0.464793\pi$
$997$	6.97056	0.220760	0.110380	+	0.993889i	$0.464793\pi$
$998$	0	0
$999$	0	0

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.585786 q^{5} -1.41421 q^{7} +O(q^{10})\)	\(q-0.585786 q^{5} -1.41421 q^{7} +0.828427 q^{11} +4.82843 q^{13} +0.828427 q^{17} +2.82843 q^{19} +6.82843 q^{23} -4.65685 q^{25} -4.58579 q^{29} +7.07107 q^{31} +0.828427 q^{35} +0.343146 q^{37} -6.48528 q^{41} -1.17157 q^{43} -4.48528 q^{47} -5.00000 q^{49} -10.2426 q^{53} -0.485281 q^{55} -9.65685 q^{59} +11.6569 q^{61} -2.82843 q^{65} -5.65685 q^{67} +8.48528 q^{71} +11.3137 q^{73} -1.17157 q^{77} +14.5858 q^{79} -3.17157 q^{83} -0.485281 q^{85} +17.3137 q^{89} -6.82843 q^{91} -1.65685 q^{95} +3.65685 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5}+O(q^{10}) \) 2 * q - 4 * q^5	\( 2 q - 4 q^{5} - 4 q^{11} + 4 q^{13} - 4 q^{17} + 8 q^{23} + 2 q^{25} - 12 q^{29} - 4 q^{35} + 12 q^{37} + 4 q^{41} - 8 q^{43} + 8 q^{47} - 10 q^{49} - 12 q^{53} + 16 q^{55} - 8 q^{59} + 12 q^{61} - 8 q^{77} + 32 q^{79} - 12 q^{83} + 16 q^{85} + 12 q^{89} - 8 q^{91} + 8 q^{95} - 4 q^{97}+O(q^{100}) \) 2 * q - 4 * q^5 - 4 * q^11 + 4 * q^13 - 4 * q^17 + 8 * q^23 + 2 * q^25 - 12 * q^29 - 4 * q^35 + 12 * q^37 + 4 * q^41 - 8 * q^43 + 8 * q^47 - 10 * q^49 - 12 * q^53 + 16 * q^55 - 8 * q^59 + 12 * q^61 - 8 * q^77 + 32 * q^79 - 12 * q^83 + 16 * q^85 + 12 * q^89 - 8 * q^91 + 8 * q^95 - 4 * q^97

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