LMFDB - Embedded newform 9072.2.a.cg.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9072,2,Mod(1,9072)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9072, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("9072.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9072 = 2^{4} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9072.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:	$72.4402847137$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.45729.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} + 12x - 3 $ x^4 - x^3 - 11x^2 + 12x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 504)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.399463$ of defining polynomial
Character	$\chi$	$=$	9072.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.600537 q^{5} +1.00000 q^{7} +O(q^{10})$	$q-0.600537 q^{5} +1.00000 q^{7} -1.60054 q^{11} +0.330355 q^{13} +1.44990 q^{17} -2.57918 q^{19} +1.84936 q^{23} -4.63936 q^{25} -3.51901 q^{29} +9.62068 q^{31} -0.600537 q^{35} +0.600537 q^{37} -6.62068 q^{41} -3.62962 q^{43} +3.90954 q^{47} +1.00000 q^{49} +9.27166 q^{53} +0.961181 q^{55} -13.8695 q^{59} -5.18865 q^{61} -0.198390 q^{65} +11.8093 q^{67} -4.17972 q^{71} +4.13969 q^{73} -1.60054 q^{77} -8.13076 q^{79} -5.57918 q^{83} -0.870718 q^{85} +3.83069 q^{89} +0.330355 q^{91} +1.54889 q^{95} -1.94956 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q - 3 * q^5 + 4 * q^7	$ 4 q - 3 q^{5} + 4 q^{7} - 7 q^{11} - 3 q^{13} - 3 q^{17} + 4 q^{19} - 2 q^{23} + 5 q^{25} - 9 q^{29} + 3 q^{31} - 3 q^{35} + 3 q^{37} + 9 q^{41} + 8 q^{43} - 3 q^{47} + 4 q^{49} - 6 q^{53} + 28 q^{55} - 10 q^{59} - 20 q^{61} + q^{65} + 11 q^{67} - 3 q^{71} - 24 q^{73} - 7 q^{77} + 21 q^{79} - 8 q^{83} - 9 q^{85} - 6 q^{89} - 3 q^{91} - 36 q^{95} - 16 q^{97}+O(q^{100}) $ 4 * q - 3 * q^5 + 4 * q^7 - 7 * q^11 - 3 * q^13 - 3 * q^17 + 4 * q^19 - 2 * q^23 + 5 * q^25 - 9 * q^29 + 3 * q^31 - 3 * q^35 + 3 * q^37 + 9 * q^41 + 8 * q^43 - 3 * q^47 + 4 * q^49 - 6 * q^53 + 28 * q^55 - 10 * q^59 - 20 * q^61 + q^65 + 11 * q^67 - 3 * q^71 - 24 * q^73 - 7 * q^77 + 21 * q^79 - 8 * q^83 - 9 * q^85 - 6 * q^89 - 3 * q^91 - 36 * q^95 - 16 * 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
$5$	−0.600537	−0.268568	−0.134284	−	0.990943i	$-0.542873\pi$
$5$	−0.600537	−0.268568	−0.134284	+	0.990943i	$0.542873\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.60054	−0.482580	−0.241290	−	0.970453i	$-0.577570\pi$
$11$	−1.60054	−0.482580	−0.241290	+	0.970453i	$0.577570\pi$
$12$	0	0
$13$	0.330355	0.0916240	0.0458120	−	0.998950i	$-0.485412\pi$
$13$	0.330355	0.0916240	0.0458120	+	0.998950i	$0.485412\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.44990	0.351652	0.175826	−	0.984421i	$-0.443740\pi$
$17$	1.44990	0.351652	0.175826	+	0.984421i	$0.443740\pi$
$18$	0	0
$19$	−2.57918	−0.591705	−0.295852	−	0.955234i	$-0.595604\pi$
$19$	−2.57918	−0.591705	−0.295852	+	0.955234i	$0.595604\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.84936	0.385619	0.192809	−	0.981236i	$-0.438240\pi$
$23$	1.84936	0.385619	0.192809	+	0.981236i	$0.438240\pi$
$24$	0	0
$25$	−4.63936	−0.927871
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.51901	−0.653463	−0.326732	−	0.945117i	$-0.605947\pi$
$29$	−3.51901	−0.653463	−0.326732	+	0.945117i	$0.605947\pi$
$30$	0	0
$31$	9.62068	1.72793	0.863963	−	0.503555i	$-0.167975\pi$
$31$	9.62068	1.72793	0.863963	+	0.503555i	$0.167975\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.600537	−0.101509
$36$	0	0
$37$	0.600537	0.0987276	0.0493638	−	0.998781i	$-0.484281\pi$
$37$	0.600537	0.0987276	0.0493638	+	0.998781i	$0.484281\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−6.62068	−1.03398	−0.516989	−	0.855992i	$-0.672947\pi$
$41$	−6.62068	−1.03398	−0.516989	+	0.855992i	$0.672947\pi$
$42$	0	0
$43$	−3.62962	−0.553512	−0.276756	−	0.960940i	$-0.589259\pi$
$43$	−3.62962	−0.553512	−0.276756	+	0.960940i	$0.589259\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3.90954	0.570265	0.285132	−	0.958488i	$-0.407962\pi$
$47$	3.90954	0.570265	0.285132	+	0.958488i	$0.407962\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.27166	1.27356	0.636780	−	0.771046i	$-0.280266\pi$
$53$	9.27166	1.27356	0.636780	+	0.771046i	$0.280266\pi$
$54$	0	0
$55$	0.961181	0.129606
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−13.8695	−1.80566	−0.902828	−	0.430001i	$-0.858513\pi$
$59$	−13.8695	−1.80566	−0.902828	+	0.430001i	$0.858513\pi$
$60$	0	0
$61$	−5.18865	−0.664339	−0.332169	−	0.943220i	$-0.607781\pi$
$61$	−5.18865	−0.664339	−0.332169	+	0.943220i	$0.607781\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.198390	−0.0246073
$66$	0	0
$67$	11.8093	1.44274	0.721370	−	0.692550i	$-0.243513\pi$
$67$	11.8093	1.44274	0.721370	+	0.692550i	$0.243513\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−4.17972	−0.496041	−0.248021	−	0.968755i	$-0.579780\pi$
$71$	−4.17972	−0.496041	−0.248021	+	0.968755i	$0.579780\pi$
$72$	0	0
$73$	4.13969	0.484514	0.242257	−	0.970212i	$-0.422112\pi$
$73$	4.13969	0.484514	0.242257	+	0.970212i	$0.422112\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.60054	−0.182398
$78$	0	0
$79$	−8.13076	−0.914782	−0.457391	−	0.889266i	$-0.651216\pi$
$79$	−8.13076	−0.914782	−0.457391	+	0.889266i	$0.651216\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.57918	−0.612395	−0.306197	−	0.951968i	$-0.599057\pi$
$83$	−5.57918	−0.612395	−0.306197	+	0.951968i	$0.599057\pi$
$84$	0	0
$85$	−0.870718	−0.0944426
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.83069	0.406053	0.203026	−	0.979173i	$-0.434922\pi$
$89$	3.83069	0.406053	0.203026	+	0.979173i	$0.434922\pi$
$90$	0	0
$91$	0.330355	0.0346306
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.54889	0.158913
$96$	0	0
$97$	−1.94956	−0.197948	−0.0989741	−	0.995090i	$-0.531556\pi$
$97$	−1.94956	−0.197948	−0.0989741	+	0.995090i	$0.531556\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.5904	−1.75031	−0.875155	−	0.483843i	$-0.839241\pi$
$101$	−17.5904	−1.75031	−0.875155	+	0.483843i	$0.839241\pi$
$102$	0	0
$103$	10.6109	1.04553	0.522764	−	0.852478i	$-0.324901\pi$
$103$	10.6109	1.04553	0.522764	+	0.852478i	$0.324901\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.43949	−0.332508	−0.166254	−	0.986083i	$-0.553167\pi$
$107$	−3.43949	−0.332508	−0.166254	+	0.986083i	$0.553167\pi$
$108$	0	0
$109$	−5.55010	−0.531603	−0.265802	−	0.964028i	$-0.585637\pi$
$109$	−5.55010	−0.531603	−0.265802	+	0.964028i	$0.585637\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.76158	−0.824220	−0.412110	−	0.911134i	$-0.635208\pi$
$113$	−8.76158	−0.824220	−0.412110	+	0.911134i	$0.635208\pi$
$114$	0	0
$115$	−1.11061	−0.103565
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.44990	0.132912
$120$	0	0
$121$	−8.43828	−0.767117
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.78879	0.517765
$126$	0	0
$127$	11.1521	0.989590	0.494795	−	0.869010i	$-0.335243\pi$
$127$	11.1521	0.989590	0.494795	+	0.869010i	$0.335243\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.19987	−0.454314	−0.227157	−	0.973858i	$-0.572943\pi$
$131$	−5.19987	−0.454314	−0.227157	+	0.973858i	$0.572943\pi$
$132$	0	0
$133$	−2.57918	−0.223643
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−8.78798	−0.750808	−0.375404	−	0.926861i	$-0.622496\pi$
$137$	−8.78798	−0.750808	−0.375404	+	0.926861i	$0.622496\pi$
$138$	0	0
$139$	9.01867	0.764954	0.382477	−	0.923965i	$-0.375071\pi$
$139$	9.01867	0.764954	0.382477	+	0.923965i	$0.375071\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.528746	−0.0442159
$144$	0	0
$145$	2.11329	0.175499
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.05124	0.0861209	0.0430604	−	0.999072i	$-0.486289\pi$
$149$	1.05124	0.0861209	0.0430604	+	0.999072i	$0.486289\pi$
$150$	0	0
$151$	12.9786	1.05619	0.528094	−	0.849186i	$-0.322907\pi$
$151$	12.9786	1.05619	0.528094	+	0.849186i	$0.322907\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.77757	−0.464066
$156$	0	0
$157$	−19.7340	−1.57494	−0.787471	−	0.616351i	$-0.788610\pi$
$157$	−19.7340	−1.57494	−0.787471	+	0.616351i	$0.788610\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.84936	0.145750
$162$	0	0
$163$	22.9214	1.79534	0.897672	−	0.440664i	$-0.145257\pi$
$163$	22.9214	1.79534	0.897672	+	0.440664i	$0.145257\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3.23016	0.249957	0.124978	−	0.992159i	$-0.460114\pi$
$167$	3.23016	0.249957	0.124978	+	0.992159i	$0.460114\pi$
$168$	0	0
$169$	−12.8909	−0.991605
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−14.1610	−1.07664	−0.538322	−	0.842739i	$-0.680942\pi$
$173$	−14.1610	−1.07664	−0.538322	+	0.842739i	$0.680942\pi$
$174$	0	0
$175$	−4.63936	−0.350702
$176$	0	0
$177$	0	0
$178$	0	0
$179$	7.06790	0.528280	0.264140	−	0.964484i	$-0.414912\pi$
$179$	7.06790	0.528280	0.264140	+	0.964484i	$0.414912\pi$
$180$	0	0
$181$	−19.6207	−1.45839	−0.729197	−	0.684304i	$-0.760106\pi$
$181$	−19.6207	−1.45839	−0.729197	+	0.684304i	$0.760106\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−0.360644	−0.0265151
$186$	0	0
$187$	−2.32062	−0.169700
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.2192	−1.46301	−0.731505	−	0.681836i	$-0.761182\pi$
$191$	−20.2192	−1.46301	−0.731505	+	0.681836i	$0.761182\pi$
$192$	0	0
$193$	17.7895	1.28051	0.640257	−	0.768161i	$-0.278828\pi$
$193$	17.7895	1.28051	0.640257	+	0.768161i	$0.278828\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.04150	−0.572933	−0.286467	−	0.958090i	$-0.592481\pi$
$197$	−8.04150	−0.572933	−0.286467	+	0.958090i	$0.592481\pi$
$198$	0	0
$199$	−14.0332	−0.994790	−0.497395	−	0.867524i	$-0.665710\pi$
$199$	−14.0332	−0.994790	−0.497395	+	0.867524i	$0.665710\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−3.51901	−0.246986
$204$	0	0
$205$	3.97596	0.277693
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.12808	0.285545
$210$	0	0
$211$	−14.6524	−1.00872	−0.504358	−	0.863495i	$-0.668271\pi$
$211$	−14.6524	−1.00872	−0.504358	+	0.863495i	$0.668271\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.17972	0.148656
$216$	0	0
$217$	9.62068	0.653095
$218$	0	0
$219$	0	0
$220$	0	0
$221$	0.478982	0.0322198
$222$	0	0
$223$	14.0539	0.941120	0.470560	−	0.882368i	$-0.344052\pi$
$223$	14.0539	0.941120	0.470560	+	0.882368i	$0.344052\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.81202	0.319385	0.159692	−	0.987167i	$-0.448950\pi$
$227$	4.81202	0.319385	0.159692	+	0.987167i	$0.448950\pi$
$228$	0	0
$229$	3.36837	0.222588	0.111294	−	0.993788i	$-0.464500\pi$
$229$	3.36837	0.222588	0.111294	+	0.993788i	$0.464500\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−20.9137	−1.37010	−0.685051	−	0.728495i	$-0.740220\pi$
$233$	−20.9137	−1.37010	−0.685051	+	0.728495i	$0.740220\pi$
$234$	0	0
$235$	−2.34782	−0.153155
$236$	0	0
$237$	0	0
$238$	0	0
$239$	19.2123	1.24274	0.621370	−	0.783517i	$-0.286577\pi$
$239$	19.2123	1.24274	0.621370	+	0.783517i	$0.286577\pi$
$240$	0	0
$241$	−17.7782	−1.14520	−0.572599	−	0.819836i	$-0.694065\pi$
$241$	−17.7782	−1.14520	−0.572599	+	0.819836i	$0.694065\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−0.600537	−0.0383669
$246$	0	0
$247$	−0.852046	−0.0542144
$248$	0	0
$249$	0	0
$250$	0	0
$251$	20.5531	1.29730	0.648649	−	0.761088i	$-0.275335\pi$
$251$	20.5531	1.29730	0.648649	+	0.761088i	$0.275335\pi$
$252$	0	0
$253$	−2.95997	−0.186092
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25.3885	1.58369	0.791846	−	0.610721i	$-0.209120\pi$
$257$	25.3885	1.58369	0.791846	+	0.610721i	$0.209120\pi$
$258$	0	0
$259$	0.600537	0.0373155
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−30.7114	−1.89375	−0.946874	−	0.321606i	$-0.895777\pi$
$263$	−30.7114	−1.89375	−0.946874	+	0.321606i	$0.895777\pi$
$264$	0	0
$265$	−5.56797	−0.342038
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−13.7402	−0.837757	−0.418878	−	0.908042i	$-0.637577\pi$
$269$	−13.7402	−0.837757	−0.418878	+	0.908042i	$0.637577\pi$
$270$	0	0
$271$	−7.64977	−0.464690	−0.232345	−	0.972633i	$-0.574640\pi$
$271$	−7.64977	−0.464690	−0.232345	+	0.972633i	$0.574640\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.42546	0.447772
$276$	0	0
$277$	−8.38347	−0.503714	−0.251857	−	0.967764i	$-0.581041\pi$
$277$	−8.38347	−0.503714	−0.251857	+	0.967764i	$0.581041\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	25.5573	1.52462	0.762310	−	0.647212i	$-0.224065\pi$
$281$	25.5573	1.52462	0.762310	+	0.647212i	$0.224065\pi$
$282$	0	0
$283$	−10.9226	−0.649283	−0.324641	−	0.945837i	$-0.605244\pi$
$283$	−10.9226	−0.649283	−0.324641	+	0.945837i	$0.605244\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−6.62068	−0.390807
$288$	0	0
$289$	−14.8978	−0.876341
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−27.9988	−1.63571	−0.817853	−	0.575427i	$-0.804836\pi$
$293$	−27.9988	−1.63571	−0.817853	+	0.575427i	$0.804836\pi$
$294$	0	0
$295$	8.32915	0.484942
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0.610947	0.0353320
$300$	0	0
$301$	−3.62962	−0.209208
$302$	0	0
$303$	0	0
$304$	0	0
$305$	3.11598	0.178420
$306$	0	0
$307$	33.0259	1.88489	0.942443	−	0.334367i	$-0.108523\pi$
$307$	33.0259	1.88489	0.942443	+	0.334367i	$0.108523\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.0287	−1.47595	−0.737975	−	0.674828i	$-0.764218\pi$
$311$	−26.0287	−1.47595	−0.737975	+	0.674828i	$0.764218\pi$
$312$	0	0
$313$	−10.9324	−0.617934	−0.308967	−	0.951073i	$-0.599983\pi$
$313$	−10.9324	−0.617934	−0.308967	+	0.951073i	$0.599983\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.9719	0.672407	0.336203	−	0.941789i	$-0.390857\pi$
$317$	11.9719	0.672407	0.336203	+	0.941789i	$0.390857\pi$
$318$	0	0
$319$	5.63230	0.315348
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.73956	−0.208074
$324$	0	0
$325$	−1.53264	−0.0850153
$326$	0	0
$327$	0	0
$328$	0	0
$329$	3.90954	0.215540
$330$	0	0
$331$	−23.2352	−1.27712	−0.638561	−	0.769571i	$-0.720470\pi$
$331$	−23.2352	−1.27712	−0.638561	+	0.769571i	$0.720470\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−7.09194	−0.387474
$336$	0	0
$337$	7.79187	0.424450	0.212225	−	0.977221i	$-0.431929\pi$
$337$	7.79187	0.424450	0.212225	+	0.977221i	$0.431929\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.3983	−0.833862
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.5240	−0.994419	−0.497209	−	0.867631i	$-0.665642\pi$
$347$	−18.5240	−0.994419	−0.497209	+	0.867631i	$0.665642\pi$
$348$	0	0
$349$	−1.91700	−0.102614	−0.0513072	−	0.998683i	$-0.516339\pi$
$349$	−1.91700	−0.102614	−0.0513072	+	0.998683i	$0.516339\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.73956	0.0925872	0.0462936	−	0.998928i	$-0.485259\pi$
$353$	1.73956	0.0925872	0.0462936	+	0.998928i	$0.485259\pi$
$354$	0	0
$355$	2.51007	0.133221
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.07616	−0.162354	−0.0811768	−	0.996700i	$-0.525868\pi$
$359$	−3.07616	−0.162354	−0.0811768	+	0.996700i	$0.525868\pi$
$360$	0	0
$361$	−12.3478	−0.649885
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−2.48604	−0.130125
$366$	0	0
$367$	12.3574	0.645052	0.322526	−	0.946561i	$-0.395468\pi$
$367$	12.3574	0.645052	0.322526	+	0.946561i	$0.395468\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.27166	0.481360
$372$	0	0
$373$	24.7999	1.28409	0.642044	−	0.766668i	$-0.278087\pi$
$373$	24.7999	1.28409	0.642044	+	0.766668i	$0.278087\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1.16252	−0.0598730
$378$	0	0
$379$	11.9650	0.614602	0.307301	−	0.951612i	$-0.400574\pi$
$379$	11.9650	0.614602	0.307301	+	0.951612i	$0.400574\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11.7162	0.598669	0.299335	−	0.954148i	$-0.403235\pi$
$383$	11.7162	0.598669	0.299335	+	0.954148i	$0.403235\pi$
$384$	0	0
$385$	0.961181	0.0489863
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−38.2719	−1.94046	−0.970232	−	0.242178i	$-0.922138\pi$
$389$	−38.2719	−1.94046	−0.970232	+	0.242178i	$0.922138\pi$
$390$	0	0
$391$	2.68139	0.135604
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.88282	0.245681
$396$	0	0
$397$	12.8396	0.644402	0.322201	−	0.946671i	$-0.395577\pi$
$397$	12.8396	0.644402	0.322201	+	0.946671i	$0.395577\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.3140	−1.11431	−0.557155	−	0.830408i	$-0.688107\pi$
$401$	−22.3140	−1.11431	−0.557155	+	0.830408i	$0.688107\pi$
$402$	0	0
$403$	3.17824	0.158320
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.961181	−0.0476440
$408$	0	0
$409$	−7.53210	−0.372438	−0.186219	−	0.982508i	$-0.559623\pi$
$409$	−7.53210	−0.372438	−0.186219	+	0.982508i	$0.559623\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−13.8695	−0.682474
$414$	0	0
$415$	3.35050	0.164470
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−28.2491	−1.38006	−0.690029	−	0.723781i	$-0.742402\pi$
$419$	−28.2491	−1.38006	−0.690029	+	0.723781i	$0.742402\pi$
$420$	0	0
$421$	19.0679	0.929313	0.464656	−	0.885491i	$-0.346178\pi$
$421$	19.0679	0.929313	0.464656	+	0.885491i	$0.346178\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−6.72660	−0.326288
$426$	0	0
$427$	−5.18865	−0.251097
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−22.9786	−1.10684	−0.553421	−	0.832902i	$-0.686678\pi$
$431$	−22.9786	−1.10684	−0.553421	+	0.832902i	$0.686678\pi$
$432$	0	0
$433$	−29.0806	−1.39752	−0.698762	−	0.715354i	$-0.746265\pi$
$433$	−29.0806	−1.39752	−0.698762	+	0.715354i	$0.746265\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.76984	−0.228173
$438$	0	0
$439$	14.7355	0.703289	0.351644	−	0.936134i	$-0.385623\pi$
$439$	14.7355	0.703289	0.351644	+	0.936134i	$0.385623\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.5095	−1.63960	−0.819799	−	0.572652i	$-0.805915\pi$
$443$	−34.5095	−1.63960	−0.819799	+	0.572652i	$0.805915\pi$
$444$	0	0
$445$	−2.30047	−0.109053
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−11.5508	−0.545115	−0.272557	−	0.962140i	$-0.587869\pi$
$449$	−11.5508	−0.545115	−0.272557	+	0.962140i	$0.587869\pi$
$450$	0	0
$451$	10.5966	0.498977
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.198390	−0.00930068
$456$	0	0
$457$	37.3989	1.74945	0.874724	−	0.484621i	$-0.161043\pi$
$457$	37.3989	1.74945	0.874724	+	0.484621i	$0.161043\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	18.2786	0.851318	0.425659	−	0.904884i	$-0.360042\pi$
$461$	18.2786	0.851318	0.425659	+	0.904884i	$0.360042\pi$
$462$	0	0
$463$	−8.49220	−0.394666	−0.197333	−	0.980336i	$-0.563228\pi$
$463$	−8.49220	−0.394666	−0.197333	+	0.980336i	$0.563228\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14.6746	−0.679060	−0.339530	−	0.940595i	$-0.610268\pi$
$467$	−14.6746	−0.679060	−0.339530	+	0.940595i	$0.610268\pi$
$468$	0	0
$469$	11.8093	0.545305
$470$	0	0
$471$	0	0
$472$	0	0
$473$	5.80934	0.267114
$474$	0	0
$475$	11.9657	0.549026
$476$	0	0
$477$	0	0
$478$	0	0
$479$	37.4303	1.71023	0.855117	−	0.518435i	$-0.173485\pi$
$479$	37.4303	1.71023	0.855117	+	0.518435i	$0.173485\pi$
$480$	0	0
$481$	0.198390	0.00904582
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.17078	0.0531626
$486$	0	0
$487$	−33.3216	−1.50994	−0.754972	−	0.655757i	$-0.772350\pi$
$487$	−33.3216	−1.50994	−0.754972	+	0.655757i	$0.772350\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−39.8911	−1.80026	−0.900131	−	0.435620i	$-0.856529\pi$
$491$	−39.8911	−1.80026	−0.900131	+	0.435620i	$0.856529\pi$
$492$	0	0
$493$	−5.10221	−0.229792
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.17972	−0.187486
$498$	0	0
$499$	−22.0741	−0.988171	−0.494086	−	0.869413i	$-0.664497\pi$
$499$	−22.0741	−0.988171	−0.494086	+	0.869413i	$0.664497\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−9.30820	−0.415032	−0.207516	−	0.978232i	$-0.566538\pi$
$503$	−9.30820	−0.415032	−0.207516	+	0.978232i	$0.566538\pi$
$504$	0	0
$505$	10.5637	0.470077
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.76931	0.255720	0.127860	−	0.991792i	$-0.459189\pi$
$509$	5.76931	0.255720	0.127860	+	0.991792i	$0.459189\pi$
$510$	0	0
$511$	4.13969	0.183129
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−6.37226	−0.280795
$516$	0	0
$517$	−6.25736	−0.275198
$518$	0	0
$519$	0	0
$520$	0	0
$521$	13.2989	0.582634	0.291317	−	0.956627i	$-0.405907\pi$
$521$	13.2989	0.582634	0.291317	+	0.956627i	$0.405907\pi$
$522$	0	0
$523$	−4.27523	−0.186943	−0.0934713	−	0.995622i	$-0.529796\pi$
$523$	−4.27523	−0.186943	−0.0934713	+	0.995622i	$0.529796\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.9490	0.607629
$528$	0	0
$529$	−19.5799	−0.851298
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.18718	−0.0947372
$534$	0	0
$535$	2.06554	0.0893011
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−1.60054	−0.0689400
$540$	0	0
$541$	−12.2130	−0.525076	−0.262538	−	0.964922i	$-0.584560\pi$
$541$	−12.2130	−0.525076	−0.262538	+	0.964922i	$0.584560\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.33304	0.142772
$546$	0	0
$547$	−38.1820	−1.63254	−0.816272	−	0.577668i	$-0.803963\pi$
$547$	−38.1820	−1.63254	−0.816272	+	0.577668i	$0.803963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	9.07616	0.386658
$552$	0	0
$553$	−8.13076	−0.345755
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−36.8404	−1.56098	−0.780490	−	0.625169i	$-0.785030\pi$
$557$	−36.8404	−1.56098	−0.780490	+	0.625169i	$0.785030\pi$
$558$	0	0
$559$	−1.19906	−0.0507150
$560$	0	0
$561$	0	0
$562$	0	0
$563$	19.1253	0.806036	0.403018	−	0.915192i	$-0.367961\pi$
$563$	19.1253	0.806036	0.403018	+	0.915192i	$0.367961\pi$
$564$	0	0
$565$	5.26165	0.221359
$566$	0	0
$567$	0	0
$568$	0	0
$569$	25.9153	1.08642	0.543212	−	0.839596i	$-0.317208\pi$
$569$	25.9153	1.08642	0.543212	+	0.839596i	$0.317208\pi$
$570$	0	0
$571$	39.1357	1.63778	0.818889	−	0.573951i	$-0.194590\pi$
$571$	39.1357	1.63778	0.818889	+	0.573951i	$0.194590\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.57985	−0.357805
$576$	0	0
$577$	9.72008	0.404652	0.202326	−	0.979318i	$-0.435150\pi$
$577$	9.72008	0.404652	0.202326	+	0.979318i	$0.435150\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5.57918	−0.231463
$582$	0	0
$583$	−14.8396	−0.614595
$584$	0	0
$585$	0	0
$586$	0	0
$587$	13.8218	0.570485	0.285242	−	0.958455i	$-0.407926\pi$
$587$	13.8218	0.570485	0.285242	+	0.958455i	$0.407926\pi$
$588$	0	0
$589$	−24.8135	−1.02242
$590$	0	0
$591$	0	0
$592$	0	0
$593$	22.1361	0.909022	0.454511	−	0.890741i	$-0.349814\pi$
$593$	22.1361	0.909022	0.454511	+	0.890741i	$0.349814\pi$
$594$	0	0
$595$	−0.870718	−0.0356960
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.56118	0.186365	0.0931824	−	0.995649i	$-0.470296\pi$
$599$	4.56118	0.186365	0.0931824	+	0.995649i	$0.470296\pi$
$600$	0	0
$601$	20.4232	0.833081	0.416541	−	0.909117i	$-0.363242\pi$
$601$	20.4232	0.833081	0.416541	+	0.909117i	$0.363242\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	5.06750	0.206023
$606$	0	0
$607$	−14.2241	−0.577339	−0.288670	−	0.957429i	$-0.593213\pi$
$607$	−14.2241	−0.577339	−0.288670	+	0.957429i	$0.593213\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.29154	0.0522499
$612$	0	0
$613$	−31.3892	−1.26780	−0.633899	−	0.773416i	$-0.718546\pi$
$613$	−31.3892	−1.26780	−0.633899	+	0.773416i	$0.718546\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.03533	−0.202715	−0.101357	−	0.994850i	$-0.532319\pi$
$617$	−5.03533	−0.202715	−0.101357	+	0.994850i	$0.532319\pi$
$618$	0	0
$619$	38.3605	1.54184	0.770920	−	0.636933i	$-0.219797\pi$
$619$	38.3605	1.54184	0.770920	+	0.636933i	$0.219797\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.83069	0.153473
$624$	0	0
$625$	19.7204	0.788816
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.870718	0.0347178
$630$	0	0
$631$	−24.0768	−0.958480	−0.479240	−	0.877684i	$-0.659088\pi$
$631$	−24.0768	−0.958480	−0.479240	+	0.877684i	$0.659088\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6.69725	−0.265772
$636$	0	0
$637$	0.330355	0.0130891
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−41.5600	−1.64152	−0.820760	−	0.571273i	$-0.806450\pi$
$641$	−41.5600	−1.64152	−0.820760	+	0.571273i	$0.806450\pi$
$642$	0	0
$643$	1.84869	0.0729052	0.0364526	−	0.999335i	$-0.488394\pi$
$643$	1.84869	0.0729052	0.0364526	+	0.999335i	$0.488394\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−38.8480	−1.52727	−0.763637	−	0.645646i	$-0.776588\pi$
$647$	−38.8480	−1.52727	−0.763637	+	0.645646i	$0.776588\pi$
$648$	0	0
$649$	22.1987	0.871374
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−40.6450	−1.59056	−0.795281	−	0.606241i	$-0.792677\pi$
$653$	−40.6450	−1.59056	−0.795281	+	0.606241i	$0.792677\pi$
$654$	0	0
$655$	3.12271	0.122014
$656$	0	0
$657$	0	0
$658$	0	0
$659$	38.4280	1.49694	0.748471	−	0.663167i	$-0.230788\pi$
$659$	38.4280	1.49694	0.748471	+	0.663167i	$0.230788\pi$
$660$	0	0
$661$	−11.4568	−0.445619	−0.222809	−	0.974862i	$-0.571523\pi$
$661$	−11.4568	−0.445619	−0.222809	+	0.974862i	$0.571523\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.54889	0.0600635
$666$	0	0
$667$	−6.50793	−0.251988
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.30463	0.320597
$672$	0	0
$673$	46.9866	1.81120	0.905601	−	0.424131i	$-0.139420\pi$
$673$	46.9866	1.81120	0.905601	+	0.424131i	$0.139420\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	16.0788	0.617960	0.308980	−	0.951068i	$-0.400012\pi$
$677$	16.0788	0.617960	0.308980	+	0.951068i	$0.400012\pi$
$678$	0	0
$679$	−1.94956	−0.0748174
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.20464	−0.199150	−0.0995750	−	0.995030i	$-0.531748\pi$
$683$	−5.20464	−0.199150	−0.0995750	+	0.995030i	$0.531748\pi$
$684$	0	0
$685$	5.27750	0.201643
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3.06294	0.116689
$690$	0	0
$691$	−1.56676	−0.0596024	−0.0298012	−	0.999556i	$-0.509487\pi$
$691$	−1.56676	−0.0596024	−0.0298012	+	0.999556i	$0.509487\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.41604	−0.205442
$696$	0	0
$697$	−9.59933	−0.363601
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−5.94479	−0.224532	−0.112266	−	0.993678i	$-0.535811\pi$
$701$	−5.94479	−0.224532	−0.112266	+	0.993678i	$0.535811\pi$
$702$	0	0
$703$	−1.54889	−0.0584176
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−17.5904	−0.661555
$708$	0	0
$709$	−13.0783	−0.491166	−0.245583	−	0.969376i	$-0.578979\pi$
$709$	−13.0783	−0.491166	−0.245583	+	0.969376i	$0.578979\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	17.7921	0.666321
$714$	0	0
$715$	0.317531	0.0118750
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10.5600	−0.393821	−0.196910	−	0.980422i	$-0.563091\pi$
$719$	−10.5600	−0.393821	−0.196910	+	0.980422i	$0.563091\pi$
$720$	0	0
$721$	10.6109	0.395172
$722$	0	0
$723$	0	0
$724$	0	0
$725$	16.3259	0.606330
$726$	0	0
$727$	10.1610	0.376852	0.188426	−	0.982087i	$-0.439661\pi$
$727$	10.1610	0.376852	0.188426	+	0.982087i	$0.439661\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−5.26258	−0.194644
$732$	0	0
$733$	26.4561	0.977177	0.488589	−	0.872514i	$-0.337512\pi$
$733$	26.4561	0.977177	0.488589	+	0.872514i	$0.337512\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−18.9013	−0.696237
$738$	0	0
$739$	−24.3880	−0.897127	−0.448563	−	0.893751i	$-0.648064\pi$
$739$	−24.3880	−0.897127	−0.448563	+	0.893751i	$0.648064\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	16.4392	0.603097	0.301548	−	0.953451i	$-0.402497\pi$
$743$	16.4392	0.603097	0.301548	+	0.953451i	$0.402497\pi$
$744$	0	0
$745$	−0.631308	−0.0231293
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−3.43949	−0.125676
$750$	0	0
$751$	0.130758	0.00477142	0.00238571	−	0.999997i	$-0.499241\pi$
$751$	0.130758	0.00477142	0.00238571	+	0.999997i	$0.499241\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−7.79415	−0.283658
$756$	0	0
$757$	36.1017	1.31214	0.656069	−	0.754701i	$-0.272218\pi$
$757$	36.1017	1.31214	0.656069	+	0.754701i	$0.272218\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.22055	0.334245	0.167122	−	0.985936i	$-0.446553\pi$
$761$	9.22055	0.334245	0.167122	+	0.985936i	$0.446553\pi$
$762$	0	0
$763$	−5.55010	−0.200927
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−4.58186	−0.165442
$768$	0	0
$769$	−45.7320	−1.64914	−0.824568	−	0.565762i	$-0.808582\pi$
$769$	−45.7320	−1.64914	−0.824568	+	0.565762i	$0.808582\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−25.9709	−0.934109	−0.467054	−	0.884229i	$-0.654685\pi$
$773$	−25.9709	−0.934109	−0.467054	+	0.884229i	$0.654685\pi$
$774$	0	0
$775$	−44.6338	−1.60329
$776$	0	0
$777$	0	0
$778$	0	0
$779$	17.0759	0.611809
$780$	0	0
$781$	6.68979	0.239380
$782$	0	0
$783$	0	0
$784$	0	0
$785$	11.8510	0.422979
$786$	0	0
$787$	−50.7641	−1.80955	−0.904773	−	0.425894i	$-0.859960\pi$
$787$	−50.7641	−1.80955	−0.904773	+	0.425894i	$0.859960\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−8.76158	−0.311526
$792$	0	0
$793$	−1.71410	−0.0608694
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−26.5953	−0.942054	−0.471027	−	0.882119i	$-0.656117\pi$
$797$	−26.5953	−0.942054	−0.471027	+	0.882119i	$0.656117\pi$
$798$	0	0
$799$	5.66844	0.200535
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−6.62573	−0.233817
$804$	0	0
$805$	−1.11061	−0.0391439
$806$	0	0
$807$	0	0
$808$	0	0
$809$	21.1540	0.743735	0.371867	−	0.928286i	$-0.378718\pi$
$809$	21.1540	0.743735	0.371867	+	0.928286i	$0.378718\pi$
$810$	0	0
$811$	15.4615	0.542927	0.271464	−	0.962449i	$-0.412492\pi$
$811$	15.4615	0.542927	0.271464	+	0.962449i	$0.412492\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−13.7652	−0.482172
$816$	0	0
$817$	9.36145	0.327516
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−26.5686	−0.927252	−0.463626	−	0.886031i	$-0.653452\pi$
$821$	−26.5686	−0.927252	−0.463626	+	0.886031i	$0.653452\pi$
$822$	0	0
$823$	25.0938	0.874714	0.437357	−	0.899288i	$-0.355915\pi$
$823$	25.0938	0.874714	0.437357	+	0.899288i	$0.355915\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.8156	0.445642	0.222821	−	0.974859i	$-0.428474\pi$
$827$	12.8156	0.445642	0.222821	+	0.974859i	$0.428474\pi$
$828$	0	0
$829$	37.9832	1.31921	0.659605	−	0.751613i	$-0.270724\pi$
$829$	37.9832	1.31921	0.659605	+	0.751613i	$0.270724\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.44990	0.0502361
$834$	0	0
$835$	−1.93983	−0.0671305
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−5.91275	−0.204131	−0.102065	−	0.994778i	$-0.532545\pi$
$839$	−5.91275	−0.204131	−0.102065	+	0.994778i	$0.532545\pi$
$840$	0	0
$841$	−16.6166	−0.572986
$842$	0	0
$843$	0	0
$844$	0	0
$845$	7.74144	0.266313
$846$	0	0
$847$	−8.43828	−0.289943
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.11061	0.0380712
$852$	0	0
$853$	−19.4522	−0.666030	−0.333015	−	0.942922i	$-0.608066\pi$
$853$	−19.4522	−0.666030	−0.333015	+	0.942922i	$0.608066\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.85004	0.233993	0.116996	−	0.993132i	$-0.462673\pi$
$857$	6.85004	0.233993	0.116996	+	0.993132i	$0.462673\pi$
$858$	0	0
$859$	−6.09483	−0.207953	−0.103977	−	0.994580i	$-0.533157\pi$
$859$	−6.09483	−0.207953	−0.103977	+	0.994580i	$0.533157\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−33.6568	−1.14569	−0.572846	−	0.819663i	$-0.694161\pi$
$863$	−33.6568	−1.14569	−0.572846	+	0.819663i	$0.694161\pi$
$864$	0	0
$865$	8.50423	0.289152
$866$	0	0
$867$	0	0
$868$	0	0
$869$	13.0136	0.441455
$870$	0	0
$871$	3.90128	0.132190
$872$	0	0
$873$	0	0
$874$	0	0
$875$	5.78879	0.195697
$876$	0	0
$877$	16.3781	0.553049	0.276525	−	0.961007i	$-0.410817\pi$
$877$	16.3781	0.553049	0.276525	+	0.961007i	$0.410817\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−22.0864	−0.744108	−0.372054	−	0.928211i	$-0.621346\pi$
$881$	−22.0864	−0.744108	−0.372054	+	0.928211i	$0.621346\pi$
$882$	0	0
$883$	−13.1872	−0.443784	−0.221892	−	0.975071i	$-0.571223\pi$
$883$	−13.1872	−0.443784	−0.221892	+	0.975071i	$0.571223\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	29.4347	0.988321	0.494160	−	0.869371i	$-0.335476\pi$
$887$	29.4347	0.988321	0.494160	+	0.869371i	$0.335476\pi$
$888$	0	0
$889$	11.1521	0.374030
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.0834	−0.337428
$894$	0	0
$895$	−4.24453	−0.141879
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−33.8553	−1.12914
$900$	0	0
$901$	13.4430	0.447850
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.7829	0.391678
$906$	0	0
$907$	−18.0783	−0.600280	−0.300140	−	0.953895i	$-0.597033\pi$
$907$	−18.0783	−0.600280	−0.300140	+	0.953895i	$0.597033\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	8.48099	0.280988	0.140494	−	0.990082i	$-0.455131\pi$
$911$	8.48099	0.280988	0.140494	+	0.990082i	$0.455131\pi$
$912$	0	0
$913$	8.92968	0.295529
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−5.19987	−0.171715
$918$	0	0
$919$	−22.0092	−0.726017	−0.363008	−	0.931786i	$-0.618250\pi$
$919$	−22.0092	−0.726017	−0.363008	+	0.931786i	$0.618250\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1.38079	−0.0454493
$924$	0	0
$925$	−2.78610	−0.0916065
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−42.8215	−1.40493	−0.702464	−	0.711719i	$-0.747917\pi$
$929$	−42.8215	−1.40493	−0.702464	+	0.711719i	$0.747917\pi$
$930$	0	0
$931$	−2.57918	−0.0845293
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.39362	0.0455761
$936$	0	0
$937$	8.65749	0.282828	0.141414	−	0.989951i	$-0.454835\pi$
$937$	8.65749	0.282828	0.141414	+	0.989951i	$0.454835\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−51.8947	−1.69172	−0.845859	−	0.533406i	$-0.820912\pi$
$941$	−51.8947	−1.69172	−0.845859	+	0.533406i	$0.820912\pi$
$942$	0	0
$943$	−12.2441	−0.398721
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23.0788	−0.749962	−0.374981	−	0.927033i	$-0.622351\pi$
$947$	−23.0788	−0.749962	−0.374981	+	0.927033i	$0.622351\pi$
$948$	0	0
$949$	1.36757	0.0443932
$950$	0	0
$951$	0	0
$952$	0	0
$953$	22.9656	0.743927	0.371964	−	0.928247i	$-0.378685\pi$
$953$	22.9656	0.743927	0.371964	+	0.928247i	$0.378685\pi$
$954$	0	0
$955$	12.1424	0.392918
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−8.78798	−0.283779
$960$	0	0
$961$	61.5576	1.98573
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.6832	−0.343905
$966$	0	0
$967$	−58.9393	−1.89536	−0.947680	−	0.319222i	$-0.896578\pi$
$967$	−58.9393	−1.89536	−0.947680	+	0.319222i	$0.896578\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	25.8074	0.828198	0.414099	−	0.910232i	$-0.364097\pi$
$971$	25.8074	0.828198	0.414099	+	0.910232i	$0.364097\pi$
$972$	0	0
$973$	9.01867	0.289125
$974$	0	0
$975$	0	0
$976$	0	0
$977$	44.9549	1.43823	0.719117	−	0.694889i	$-0.244547\pi$
$977$	44.9549	1.43823	0.719117	+	0.694889i	$0.244547\pi$
$978$	0	0
$979$	−6.13116	−0.195953
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−8.67152	−0.276579	−0.138289	−	0.990392i	$-0.544160\pi$
$983$	−8.67152	−0.276579	−0.138289	+	0.990392i	$0.544160\pi$
$984$	0	0
$985$	4.82922	0.153872
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−6.71248	−0.213445
$990$	0	0
$991$	3.39241	0.107763	0.0538817	−	0.998547i	$-0.482841\pi$
$991$	3.39241	0.107763	0.0538817	+	0.998547i	$0.482841\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8.42747	0.267169
$996$	0	0
$997$	28.0403	0.888045	0.444023	−	0.896016i	$-0.353551\pi$
$997$	28.0403	0.888045	0.444023	+	0.896016i	$0.353551\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9072.2.a.cg.1.2		4
3.2	odd	2		9072.2.a.cj.1.3		4
4.3	odd	2		4536.2.a.y.1.2		4
9.2	odd	6		1008.2.r.l.337.1		8
9.4	even	3		3024.2.r.m.2017.3		8
9.5	odd	6		1008.2.r.l.673.1		8
9.7	even	3		3024.2.r.m.1009.3		8
12.11	even	2		4536.2.a.z.1.3		4
36.7	odd	6		1512.2.r.e.1009.3		8
36.11	even	6		504.2.r.e.337.4	yes	8
36.23	even	6		504.2.r.e.169.4	✓	8
36.31	odd	6		1512.2.r.e.505.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.2.r.e.169.4	✓	8	36.23	even	6
504.2.r.e.337.4	yes	8	36.11	even	6
1008.2.r.l.337.1		8	9.2	odd	6
1008.2.r.l.673.1		8	9.5	odd	6
1512.2.r.e.505.3		8	36.31	odd	6
1512.2.r.e.1009.3		8	36.7	odd	6
3024.2.r.m.1009.3		8	9.7	even	3
3024.2.r.m.2017.3		8	9.4	even	3
4536.2.a.y.1.2		4	4.3	odd	2
4536.2.a.z.1.3		4	12.11	even	2
9072.2.a.cg.1.2		4	1.1	even	1	trivial
9072.2.a.cj.1.3		4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9072 = 2^{4} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9072.a (trivial)

\(f(q)\)	\(=\)	\(q-0.600537 q^{5} +1.00000 q^{7} +O(q^{10})\)	\(q-0.600537 q^{5} +1.00000 q^{7} -1.60054 q^{11} +0.330355 q^{13} +1.44990 q^{17} -2.57918 q^{19} +1.84936 q^{23} -4.63936 q^{25} -3.51901 q^{29} +9.62068 q^{31} -0.600537 q^{35} +0.600537 q^{37} -6.62068 q^{41} -3.62962 q^{43} +3.90954 q^{47} +1.00000 q^{49} +9.27166 q^{53} +0.961181 q^{55} -13.8695 q^{59} -5.18865 q^{61} -0.198390 q^{65} +11.8093 q^{67} -4.17972 q^{71} +4.13969 q^{73} -1.60054 q^{77} -8.13076 q^{79} -5.57918 q^{83} -0.870718 q^{85} +3.83069 q^{89} +0.330355 q^{91} +1.54889 q^{95} -1.94956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q - 3 * q^5 + 4 * q^7	\( 4 q - 3 q^{5} + 4 q^{7} - 7 q^{11} - 3 q^{13} - 3 q^{17} + 4 q^{19} - 2 q^{23} + 5 q^{25} - 9 q^{29} + 3 q^{31} - 3 q^{35} + 3 q^{37} + 9 q^{41} + 8 q^{43} - 3 q^{47} + 4 q^{49} - 6 q^{53} + 28 q^{55} - 10 q^{59} - 20 q^{61} + q^{65} + 11 q^{67} - 3 q^{71} - 24 q^{73} - 7 q^{77} + 21 q^{79} - 8 q^{83} - 9 q^{85} - 6 q^{89} - 3 q^{91} - 36 q^{95} - 16 q^{97}+O(q^{100}) \) 4 * q - 3 * q^5 + 4 * q^7 - 7 * q^11 - 3 * q^13 - 3 * q^17 + 4 * q^19 - 2 * q^23 + 5 * q^25 - 9 * q^29 + 3 * q^31 - 3 * q^35 + 3 * q^37 + 9 * q^41 + 8 * q^43 - 3 * q^47 + 4 * q^49 - 6 * q^53 + 28 * q^55 - 10 * q^59 - 20 * q^61 + q^65 + 11 * q^67 - 3 * q^71 - 24 * q^73 - 7 * q^77 + 21 * q^79 - 8 * q^83 - 9 * q^85 - 6 * q^89 - 3 * q^91 - 36 * q^95 - 16 * q^97

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