LMFDB - Embedded newform 6480.2.a.bq.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6480,2,Mod(1,6480)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6480, 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("6480.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6480 = 2^{4} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6480.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:	$51.7430605098$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 3240)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	6480.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.732051 q^{7} +3.73205 q^{11} -5.46410 q^{13} +4.19615 q^{17} -1.00000 q^{19} -0.535898 q^{23} +1.00000 q^{25} -3.73205 q^{29} -4.46410 q^{31} -0.732051 q^{35} -10.7321 q^{37} -4.26795 q^{41} +4.19615 q^{43} -8.73205 q^{47} -6.46410 q^{49} -1.26795 q^{53} +3.73205 q^{55} +9.19615 q^{59} -1.07180 q^{61} -5.46410 q^{65} +7.46410 q^{67} -12.1244 q^{71} -4.73205 q^{73} -2.73205 q^{77} +15.4641 q^{79} +13.1244 q^{83} +4.19615 q^{85} +5.19615 q^{89} +4.00000 q^{91} -1.00000 q^{95} +0.196152 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} - 4 q^{13} - 2 q^{17} - 2 q^{19} - 8 q^{23} + 2 q^{25} - 4 q^{29} - 2 q^{31} + 2 q^{35} - 18 q^{37} - 12 q^{41} - 2 q^{43} - 14 q^{47} - 6 q^{49} - 6 q^{53} + 4 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 - 4 * q^13 - 2 * q^17 - 2 * q^19 - 8 * q^23 + 2 * q^25 - 4 * q^29 - 2 * q^31 + 2 * q^35 - 18 * q^37 - 12 * q^41 - 2 * q^43 - 14 * q^47 - 6 * q^49 - 6 * q^53 + 4 * q^55 + 8 * q^59 - 16 * q^61 - 4 * q^65 + 8 * q^67 - 6 * q^73 - 2 * q^77 + 24 * q^79 + 2 * q^83 - 2 * q^85 + 8 * q^91 - 2 * q^95 - 10 * q^97

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−0.732051	−0.276689	−0.138345	−	0.990384i	$-0.544178\pi$
$7$	−0.732051	−0.276689	−0.138345	+	0.990384i	$0.544178\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.73205	1.12526	0.562628	−	0.826710i	$-0.309790\pi$
$11$	3.73205	1.12526	0.562628	+	0.826710i	$0.309790\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.19615	1.01772	0.508858	−	0.860850i	$-0.330068\pi$
$17$	4.19615	1.01772	0.508858	+	0.860850i	$0.330068\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−0.535898	−0.111743	−0.0558713	−	0.998438i	$-0.517794\pi$
$23$	−0.535898	−0.111743	−0.0558713	+	0.998438i	$0.517794\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−3.73205	−0.693024	−0.346512	−	0.938045i	$-0.612634\pi$
$29$	−3.73205	−0.693024	−0.346512	+	0.938045i	$0.612634\pi$
$30$	0	0
$31$	−4.46410	−0.801776	−0.400888	−	0.916127i	$-0.631298\pi$
$31$	−4.46410	−0.801776	−0.400888	+	0.916127i	$0.631298\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.732051	−0.123739
$36$	0	0
$37$	−10.7321	−1.76434	−0.882169	−	0.470933i	$-0.843917\pi$
$37$	−10.7321	−1.76434	−0.882169	+	0.470933i	$0.843917\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.26795	−0.666542	−0.333271	−	0.942831i	$-0.608152\pi$
$41$	−4.26795	−0.666542	−0.333271	+	0.942831i	$0.608152\pi$
$42$	0	0
$43$	4.19615	0.639907	0.319954	−	0.947433i	$-0.396333\pi$
$43$	4.19615	0.639907	0.319954	+	0.947433i	$0.396333\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.73205	−1.27370	−0.636850	−	0.770988i	$-0.719763\pi$
$47$	−8.73205	−1.27370	−0.636850	+	0.770988i	$0.719763\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−1.26795	−0.174166	−0.0870831	−	0.996201i	$-0.527755\pi$
$53$	−1.26795	−0.174166	−0.0870831	+	0.996201i	$0.527755\pi$
$54$	0	0
$55$	3.73205	0.503230
$56$	0	0
$57$	0	0
$58$	0	0
$59$	9.19615	1.19724	0.598619	−	0.801034i	$-0.295717\pi$
$59$	9.19615	1.19724	0.598619	+	0.801034i	$0.295717\pi$
$60$	0	0
$61$	−1.07180	−0.137230	−0.0686148	−	0.997643i	$-0.521858\pi$
$61$	−1.07180	−0.137230	−0.0686148	+	0.997643i	$0.521858\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.46410	−0.677738
$66$	0	0
$67$	7.46410	0.911885	0.455943	−	0.890009i	$-0.349302\pi$
$67$	7.46410	0.911885	0.455943	+	0.890009i	$0.349302\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.1244	−1.43890	−0.719448	−	0.694546i	$-0.755605\pi$
$71$	−12.1244	−1.43890	−0.719448	+	0.694546i	$0.755605\pi$
$72$	0	0
$73$	−4.73205	−0.553845	−0.276922	−	0.960892i	$-0.589314\pi$
$73$	−4.73205	−0.553845	−0.276922	+	0.960892i	$0.589314\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.73205	−0.311346
$78$	0	0
$79$	15.4641	1.73985	0.869924	−	0.493186i	$-0.164168\pi$
$79$	15.4641	1.73985	0.869924	+	0.493186i	$0.164168\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	13.1244	1.44059	0.720293	−	0.693670i	$-0.244008\pi$
$83$	13.1244	1.44059	0.720293	+	0.693670i	$0.244008\pi$
$84$	0	0
$85$	4.19615	0.455137
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.19615	0.550791	0.275396	−	0.961331i	$-0.411191\pi$
$89$	5.19615	0.550791	0.275396	+	0.961331i	$0.411191\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	0.196152	0.0199163	0.00995813	−	0.999950i	$-0.496830\pi$
$97$	0.196152	0.0199163	0.00995813	+	0.999950i	$0.496830\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−14.6603	−1.45875	−0.729375	−	0.684114i	$-0.760189\pi$
$101$	−14.6603	−1.45875	−0.729375	+	0.684114i	$0.760189\pi$
$102$	0	0
$103$	−2.39230	−0.235721	−0.117860	−	0.993030i	$-0.537604\pi$
$103$	−2.39230	−0.235721	−0.117860	+	0.993030i	$0.537604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−15.4641	−1.49497	−0.747486	−	0.664278i	$-0.768739\pi$
$107$	−15.4641	−1.49497	−0.747486	+	0.664278i	$0.768739\pi$
$108$	0	0
$109$	5.92820	0.567819	0.283909	−	0.958851i	$-0.408368\pi$
$109$	5.92820	0.567819	0.283909	+	0.958851i	$0.408368\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−8.19615	−0.771029	−0.385515	−	0.922702i	$-0.625976\pi$
$113$	−8.19615	−0.771029	−0.385515	+	0.922702i	$0.625976\pi$
$114$	0	0
$115$	−0.535898	−0.0499728
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−3.07180	−0.281591
$120$	0	0
$121$	2.92820	0.266200
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−11.6603	−1.03468	−0.517340	−	0.855780i	$-0.673078\pi$
$127$	−11.6603	−1.03468	−0.517340	+	0.855780i	$0.673078\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.73205	−0.675552	−0.337776	−	0.941226i	$-0.609675\pi$
$131$	−7.73205	−0.675552	−0.337776	+	0.941226i	$0.609675\pi$
$132$	0	0
$133$	0.732051	0.0634769
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−5.46410	−0.466830	−0.233415	−	0.972377i	$-0.574990\pi$
$137$	−5.46410	−0.466830	−0.233415	+	0.972377i	$0.574990\pi$
$138$	0	0
$139$	9.00000	0.763370	0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000	0.763370	0.381685	+	0.924292i	$0.375344\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−20.3923	−1.70529
$144$	0	0
$145$	−3.73205	−0.309930
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	14.8564	1.20900	0.604499	−	0.796606i	$-0.293373\pi$
$151$	14.8564	1.20900	0.604499	+	0.796606i	$0.293373\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.46410	−0.358565
$156$	0	0
$157$	−18.0526	−1.44075	−0.720376	−	0.693584i	$-0.756031\pi$
$157$	−18.0526	−1.44075	−0.720376	+	0.693584i	$0.756031\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0.392305	0.0309180
$162$	0	0
$163$	−1.12436	−0.0880663	−0.0440332	−	0.999030i	$-0.514021\pi$
$163$	−1.12436	−0.0880663	−0.0440332	+	0.999030i	$0.514021\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.7321	−0.830471	−0.415236	−	0.909714i	$-0.636301\pi$
$167$	−10.7321	−0.830471	−0.415236	+	0.909714i	$0.636301\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	4.53590	0.344858	0.172429	−	0.985022i	$-0.444838\pi$
$173$	4.53590	0.344858	0.172429	+	0.985022i	$0.444838\pi$
$174$	0	0
$175$	−0.732051	−0.0553378
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−21.0526	−1.57354	−0.786771	−	0.617245i	$-0.788249\pi$
$179$	−21.0526	−1.57354	−0.786771	+	0.617245i	$0.788249\pi$
$180$	0	0
$181$	−18.3205	−1.36175	−0.680876	−	0.732398i	$-0.738401\pi$
$181$	−18.3205	−1.36175	−0.680876	+	0.732398i	$0.738401\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.7321	−0.789036
$186$	0	0
$187$	15.6603	1.14519
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.26795	0.598248	0.299124	−	0.954214i	$-0.403306\pi$
$191$	8.26795	0.598248	0.299124	+	0.954214i	$0.403306\pi$
$192$	0	0
$193$	−16.1962	−1.16582	−0.582912	−	0.812535i	$-0.698087\pi$
$193$	−16.1962	−1.16582	−0.582912	+	0.812535i	$0.698087\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.85641	0.417252	0.208626	−	0.977996i	$-0.433101\pi$
$197$	5.85641	0.417252	0.208626	+	0.977996i	$0.433101\pi$
$198$	0	0
$199$	22.0000	1.55954	0.779769	−	0.626067i	$-0.215336\pi$
$199$	22.0000	1.55954	0.779769	+	0.626067i	$0.215336\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.73205	0.191752
$204$	0	0
$205$	−4.26795	−0.298087
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−3.73205	−0.258151
$210$	0	0
$211$	−22.3205	−1.53661	−0.768304	−	0.640086i	$-0.778899\pi$
$211$	−22.3205	−1.53661	−0.768304	+	0.640086i	$0.778899\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.19615	0.286175
$216$	0	0
$217$	3.26795	0.221843
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−22.9282	−1.54232
$222$	0	0
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.66025	0.375684	0.187842	−	0.982199i	$-0.439851\pi$
$227$	5.66025	0.375684	0.187842	+	0.982199i	$0.439851\pi$
$228$	0	0
$229$	−9.85641	−0.651330	−0.325665	−	0.945485i	$-0.605588\pi$
$229$	−9.85641	−0.651330	−0.325665	+	0.945485i	$0.605588\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.73205	0.310007	0.155003	−	0.987914i	$-0.450461\pi$
$233$	4.73205	0.310007	0.155003	+	0.987914i	$0.450461\pi$
$234$	0	0
$235$	−8.73205	−0.569616
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.53590	−0.552141	−0.276071	−	0.961137i	$-0.589032\pi$
$239$	−8.53590	−0.552141	−0.276071	+	0.961137i	$0.589032\pi$
$240$	0	0
$241$	−10.4641	−0.674052	−0.337026	−	0.941495i	$-0.609421\pi$
$241$	−10.4641	−0.674052	−0.337026	+	0.941495i	$0.609421\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.46410	−0.412976
$246$	0	0
$247$	5.46410	0.347672
$248$	0	0
$249$	0	0
$250$	0	0
$251$	27.4641	1.73352	0.866759	−	0.498727i	$-0.166199\pi$
$251$	27.4641	1.73352	0.866759	+	0.498727i	$0.166199\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5.32051	−0.331884	−0.165942	−	0.986135i	$-0.553066\pi$
$257$	−5.32051	−0.331884	−0.165942	+	0.986135i	$0.553066\pi$
$258$	0	0
$259$	7.85641	0.488173
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−15.4641	−0.953557	−0.476779	−	0.879023i	$-0.658196\pi$
$263$	−15.4641	−0.953557	−0.476779	+	0.879023i	$0.658196\pi$
$264$	0	0
$265$	−1.26795	−0.0778895
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.73205	0.227547	0.113774	−	0.993507i	$-0.463706\pi$
$269$	3.73205	0.227547	0.113774	+	0.993507i	$0.463706\pi$
$270$	0	0
$271$	27.7128	1.68343	0.841717	−	0.539919i	$-0.181545\pi$
$271$	27.7128	1.68343	0.841717	+	0.539919i	$0.181545\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.73205	0.225051
$276$	0	0
$277$	1.66025	0.0997550	0.0498775	−	0.998755i	$-0.484117\pi$
$277$	1.66025	0.0997550	0.0498775	+	0.998755i	$0.484117\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.39230	0.142713	0.0713565	−	0.997451i	$-0.477267\pi$
$281$	2.39230	0.142713	0.0713565	+	0.997451i	$0.477267\pi$
$282$	0	0
$283$	−4.53590	−0.269631	−0.134816	−	0.990871i	$-0.543044\pi$
$283$	−4.53590	−0.269631	−0.134816	+	0.990871i	$0.543044\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.12436	0.184425
$288$	0	0
$289$	0.607695	0.0357468
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−3.26795	−0.190916	−0.0954578	−	0.995433i	$-0.530432\pi$
$293$	−3.26795	−0.190916	−0.0954578	+	0.995433i	$0.530432\pi$
$294$	0	0
$295$	9.19615	0.535421
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.92820	0.169342
$300$	0	0
$301$	−3.07180	−0.177055
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.07180	−0.0613709
$306$	0	0
$307$	−30.4449	−1.73758	−0.868790	−	0.495181i	$-0.835102\pi$
$307$	−30.4449	−1.73758	−0.868790	+	0.495181i	$0.835102\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	23.0526	1.30719	0.653595	−	0.756844i	$-0.273260\pi$
$311$	23.0526	1.30719	0.653595	+	0.756844i	$0.273260\pi$
$312$	0	0
$313$	28.7846	1.62700	0.813501	−	0.581563i	$-0.197559\pi$
$313$	28.7846	1.62700	0.813501	+	0.581563i	$0.197559\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.5167	−1.43316	−0.716579	−	0.697506i	$-0.754293\pi$
$317$	−25.5167	−1.43316	−0.716579	+	0.697506i	$0.754293\pi$
$318$	0	0
$319$	−13.9282	−0.779830
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.19615	−0.233480
$324$	0	0
$325$	−5.46410	−0.303094
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.39230	0.352419
$330$	0	0
$331$	29.7846	1.63711	0.818555	−	0.574428i	$-0.194775\pi$
$331$	29.7846	1.63711	0.818555	+	0.574428i	$0.194775\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	7.46410	0.407807
$336$	0	0
$337$	−31.3205	−1.70614	−0.853068	−	0.521799i	$-0.825261\pi$
$337$	−31.3205	−1.70614	−0.853068	+	0.521799i	$0.825261\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−16.6603	−0.902203
$342$	0	0
$343$	9.85641	0.532196
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−20.5885	−1.10525	−0.552623	−	0.833431i	$-0.686373\pi$
$347$	−20.5885	−1.10525	−0.552623	+	0.833431i	$0.686373\pi$
$348$	0	0
$349$	23.7846	1.27316	0.636580	−	0.771210i	$-0.280348\pi$
$349$	23.7846	1.27316	0.636580	+	0.771210i	$0.280348\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.2679	−1.23843	−0.619214	−	0.785222i	$-0.712549\pi$
$353$	−23.2679	−1.23843	−0.619214	+	0.785222i	$0.712549\pi$
$354$	0	0
$355$	−12.1244	−0.643494
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−18.8038	−0.992429	−0.496215	−	0.868200i	$-0.665277\pi$
$359$	−18.8038	−0.992429	−0.496215	+	0.868200i	$0.665277\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.73205	−0.247687
$366$	0	0
$367$	−37.3205	−1.94811	−0.974057	−	0.226301i	$-0.927337\pi$
$367$	−37.3205	−1.94811	−0.974057	+	0.226301i	$0.927337\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0.928203	0.0481899
$372$	0	0
$373$	23.2679	1.20477	0.602384	−	0.798206i	$-0.294217\pi$
$373$	23.2679	1.20477	0.602384	+	0.798206i	$0.294217\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	20.3923	1.05026
$378$	0	0
$379$	−28.2487	−1.45104	−0.725519	−	0.688202i	$-0.758400\pi$
$379$	−28.2487	−1.45104	−0.725519	+	0.688202i	$0.758400\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−23.3205	−1.19162	−0.595811	−	0.803125i	$-0.703169\pi$
$383$	−23.3205	−1.19162	−0.595811	+	0.803125i	$0.703169\pi$
$384$	0	0
$385$	−2.73205	−0.139238
$386$	0	0
$387$	0	0
$388$	0	0
$389$	12.2487	0.621034	0.310517	−	0.950568i	$-0.399498\pi$
$389$	12.2487	0.621034	0.310517	+	0.950568i	$0.399498\pi$
$390$	0	0
$391$	−2.24871	−0.113722
$392$	0	0
$393$	0	0
$394$	0	0
$395$	15.4641	0.778083
$396$	0	0
$397$	28.2487	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$397$	28.2487	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.14359	−0.206921	−0.103461	−	0.994634i	$-0.532992\pi$
$401$	−4.14359	−0.206921	−0.103461	+	0.994634i	$0.532992\pi$
$402$	0	0
$403$	24.3923	1.21507
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−40.0526	−1.98533
$408$	0	0
$409$	−4.78461	−0.236584	−0.118292	−	0.992979i	$-0.537742\pi$
$409$	−4.78461	−0.236584	−0.118292	+	0.992979i	$0.537742\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−6.73205	−0.331263
$414$	0	0
$415$	13.1244	0.644249
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.3205	−1.13928	−0.569641	−	0.821894i	$-0.692918\pi$
$419$	−23.3205	−1.13928	−0.569641	+	0.821894i	$0.692918\pi$
$420$	0	0
$421$	39.9282	1.94598	0.972991	−	0.230844i	$-0.0741487\pi$
$421$	39.9282	1.94598	0.972991	+	0.230844i	$0.0741487\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.19615	0.203543
$426$	0	0
$427$	0.784610	0.0379699
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−21.7321	−1.04680	−0.523398	−	0.852088i	$-0.675336\pi$
$431$	−21.7321	−1.04680	−0.523398	+	0.852088i	$0.675336\pi$
$432$	0	0
$433$	4.53590	0.217981	0.108991	−	0.994043i	$-0.465238\pi$
$433$	4.53590	0.217981	0.108991	+	0.994043i	$0.465238\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.535898	0.0256355
$438$	0	0
$439$	−20.8564	−0.995422	−0.497711	−	0.867343i	$-0.665826\pi$
$439$	−20.8564	−0.995422	−0.497711	+	0.867343i	$0.665826\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.12436	−0.433511	−0.216756	−	0.976226i	$-0.569547\pi$
$443$	−9.12436	−0.433511	−0.216756	+	0.976226i	$0.569547\pi$
$444$	0	0
$445$	5.19615	0.246321
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−41.4449	−1.95590	−0.977952	−	0.208830i	$-0.933035\pi$
$449$	−41.4449	−1.95590	−0.977952	+	0.208830i	$0.933035\pi$
$450$	0	0
$451$	−15.9282	−0.750030
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4.00000	0.187523
$456$	0	0
$457$	4.58846	0.214639	0.107319	−	0.994225i	$-0.465773\pi$
$457$	4.58846	0.214639	0.107319	+	0.994225i	$0.465773\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.66025	0.123900	0.0619502	−	0.998079i	$-0.480268\pi$
$461$	2.66025	0.123900	0.0619502	+	0.998079i	$0.480268\pi$
$462$	0	0
$463$	22.3923	1.04066	0.520329	−	0.853966i	$-0.325809\pi$
$463$	22.3923	1.04066	0.520329	+	0.853966i	$0.325809\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	11.5167	0.532927	0.266464	−	0.963845i	$-0.414145\pi$
$467$	11.5167	0.532927	0.266464	+	0.963845i	$0.414145\pi$
$468$	0	0
$469$	−5.46410	−0.252309
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.6603	0.720059
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.33975	−0.0612146	−0.0306073	−	0.999531i	$-0.509744\pi$
$479$	−1.33975	−0.0612146	−0.0306073	+	0.999531i	$0.509744\pi$
$480$	0	0
$481$	58.6410	2.67380
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0.196152	0.00890682
$486$	0	0
$487$	−6.53590	−0.296170	−0.148085	−	0.988975i	$-0.547311\pi$
$487$	−6.53590	−0.296170	−0.148085	+	0.988975i	$0.547311\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	18.8038	0.848606	0.424303	−	0.905520i	$-0.360519\pi$
$491$	18.8038	0.848606	0.424303	+	0.905520i	$0.360519\pi$
$492$	0	0
$493$	−15.6603	−0.705302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	8.87564	0.398127
$498$	0	0
$499$	31.9282	1.42930	0.714651	−	0.699481i	$-0.246585\pi$
$499$	31.9282	1.42930	0.714651	+	0.699481i	$0.246585\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	16.3923	0.730897	0.365448	−	0.930832i	$-0.380916\pi$
$503$	16.3923	0.730897	0.365448	+	0.930832i	$0.380916\pi$
$504$	0	0
$505$	−14.6603	−0.652373
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−27.8564	−1.23471	−0.617357	−	0.786683i	$-0.711796\pi$
$509$	−27.8564	−1.23471	−0.617357	+	0.786683i	$0.711796\pi$
$510$	0	0
$511$	3.46410	0.153243
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.39230	−0.105418
$516$	0	0
$517$	−32.5885	−1.43324
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−32.2487	−1.41284	−0.706421	−	0.707792i	$-0.749691\pi$
$521$	−32.2487	−1.41284	−0.706421	+	0.707792i	$0.749691\pi$
$522$	0	0
$523$	−20.3923	−0.891693	−0.445847	−	0.895109i	$-0.647097\pi$
$523$	−20.3923	−0.891693	−0.445847	+	0.895109i	$0.647097\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−18.7321	−0.815981
$528$	0	0
$529$	−22.7128	−0.987514
$530$	0	0
$531$	0	0
$532$	0	0
$533$	23.3205	1.01012
$534$	0	0
$535$	−15.4641	−0.668571
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−24.1244	−1.03911
$540$	0	0
$541$	−10.3205	−0.443713	−0.221857	−	0.975079i	$-0.571212\pi$
$541$	−10.3205	−0.443713	−0.221857	+	0.975079i	$0.571212\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.92820	0.253936
$546$	0	0
$547$	0.784610	0.0335475	0.0167737	−	0.999859i	$-0.494661\pi$
$547$	0.784610	0.0335475	0.0167737	+	0.999859i	$0.494661\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	3.73205	0.158991
$552$	0	0
$553$	−11.3205	−0.481397
$554$	0	0
$555$	0	0
$556$	0	0
$557$	29.4641	1.24843	0.624217	−	0.781251i	$-0.285418\pi$
$557$	29.4641	1.24843	0.624217	+	0.781251i	$0.285418\pi$
$558$	0	0
$559$	−22.9282	−0.969760
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.9090	1.09193	0.545966	−	0.837807i	$-0.316163\pi$
$563$	25.9090	1.09193	0.545966	+	0.837807i	$0.316163\pi$
$564$	0	0
$565$	−8.19615	−0.344815
$566$	0	0
$567$	0	0
$568$	0	0
$569$	45.7321	1.91719	0.958594	−	0.284777i	$-0.0919197\pi$
$569$	45.7321	1.91719	0.958594	+	0.284777i	$0.0919197\pi$
$570$	0	0
$571$	−32.1769	−1.34656	−0.673281	−	0.739387i	$-0.735116\pi$
$571$	−32.1769	−1.34656	−0.673281	+	0.739387i	$0.735116\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−0.535898	−0.0223485
$576$	0	0
$577$	−6.87564	−0.286237	−0.143118	−	0.989706i	$-0.545713\pi$
$577$	−6.87564	−0.286237	−0.143118	+	0.989706i	$0.545713\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.60770	−0.398594
$582$	0	0
$583$	−4.73205	−0.195982
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.4449	−1.33914	−0.669571	−	0.742748i	$-0.733522\pi$
$587$	−32.4449	−1.33914	−0.669571	+	0.742748i	$0.733522\pi$
$588$	0	0
$589$	4.46410	0.183940
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−45.7128	−1.87720	−0.938600	−	0.345007i	$-0.887877\pi$
$593$	−45.7128	−1.87720	−0.938600	+	0.345007i	$0.887877\pi$
$594$	0	0
$595$	−3.07180	−0.125931
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.48334	−0.0606076	−0.0303038	−	0.999541i	$-0.509647\pi$
$599$	−1.48334	−0.0606076	−0.0303038	+	0.999541i	$0.509647\pi$
$600$	0	0
$601$	−4.32051	−0.176237	−0.0881186	−	0.996110i	$-0.528085\pi$
$601$	−4.32051	−0.176237	−0.0881186	+	0.996110i	$0.528085\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.92820	0.119048
$606$	0	0
$607$	−32.1962	−1.30680	−0.653401	−	0.757012i	$-0.726658\pi$
$607$	−32.1962	−1.30680	−0.653401	+	0.757012i	$0.726658\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	47.7128	1.93025
$612$	0	0
$613$	33.4641	1.35160	0.675801	−	0.737084i	$-0.263798\pi$
$613$	33.4641	1.35160	0.675801	+	0.737084i	$0.263798\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.7846	1.31986	0.659929	−	0.751328i	$-0.270586\pi$
$617$	32.7846	1.31986	0.659929	+	0.751328i	$0.270586\pi$
$618$	0	0
$619$	28.9282	1.16272	0.581361	−	0.813646i	$-0.302520\pi$
$619$	28.9282	1.16272	0.581361	+	0.813646i	$0.302520\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−3.80385	−0.152398
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−45.0333	−1.79560
$630$	0	0
$631$	24.1769	0.962468	0.481234	−	0.876592i	$-0.340189\pi$
$631$	24.1769	0.962468	0.481234	+	0.876592i	$0.340189\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−11.6603	−0.462723
$636$	0	0
$637$	35.3205	1.39945
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−43.7321	−1.72731	−0.863656	−	0.504082i	$-0.831831\pi$
$641$	−43.7321	−1.72731	−0.863656	+	0.504082i	$0.831831\pi$
$642$	0	0
$643$	−18.3397	−0.723249	−0.361624	−	0.932324i	$-0.617778\pi$
$643$	−18.3397	−0.723249	−0.361624	+	0.932324i	$0.617778\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.1051	1.02630	0.513149	−	0.858300i	$-0.328479\pi$
$647$	26.1051	1.02630	0.513149	+	0.858300i	$0.328479\pi$
$648$	0	0
$649$	34.3205	1.34720
$650$	0	0
$651$	0	0
$652$	0	0
$653$	38.2487	1.49679	0.748394	−	0.663255i	$-0.230825\pi$
$653$	38.2487	1.49679	0.748394	+	0.663255i	$0.230825\pi$
$654$	0	0
$655$	−7.73205	−0.302116
$656$	0	0
$657$	0	0
$658$	0	0
$659$	33.4641	1.30358	0.651788	−	0.758401i	$-0.274019\pi$
$659$	33.4641	1.30358	0.651788	+	0.758401i	$0.274019\pi$
$660$	0	0
$661$	47.1051	1.83218	0.916088	−	0.400976i	$-0.131329\pi$
$661$	47.1051	1.83218	0.916088	+	0.400976i	$0.131329\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0.732051	0.0283877
$666$	0	0
$667$	2.00000	0.0774403
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	24.2487	0.934719	0.467360	−	0.884067i	$-0.345205\pi$
$673$	24.2487	0.934719	0.467360	+	0.884067i	$0.345205\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	37.7128	1.44942	0.724711	−	0.689053i	$-0.241973\pi$
$677$	37.7128	1.44942	0.724711	+	0.689053i	$0.241973\pi$
$678$	0	0
$679$	−0.143594	−0.00551061
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−23.3205	−0.892334	−0.446167	−	0.894950i	$-0.647211\pi$
$683$	−23.3205	−0.892334	−0.446167	+	0.894950i	$0.647211\pi$
$684$	0	0
$685$	−5.46410	−0.208773
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.92820	0.263944
$690$	0	0
$691$	−5.21539	−0.198403	−0.0992014	−	0.995067i	$-0.531629\pi$
$691$	−5.21539	−0.198403	−0.0992014	+	0.995067i	$0.531629\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	9.00000	0.341389
$696$	0	0
$697$	−17.9090	−0.678350
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−35.7321	−1.34958	−0.674790	−	0.738009i	$-0.735766\pi$
$701$	−35.7321	−1.34958	−0.674790	+	0.738009i	$0.735766\pi$
$702$	0	0
$703$	10.7321	0.404767
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.7321	0.403620
$708$	0	0
$709$	17.4641	0.655878	0.327939	−	0.944699i	$-0.393646\pi$
$709$	17.4641	0.655878	0.327939	+	0.944699i	$0.393646\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	2.39230	0.0895925
$714$	0	0
$715$	−20.3923	−0.762629
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0.0192379	0.000717452 0	0.000358726	−	1.00000i	$-0.499886\pi$
$719$	0.0192379	0.000717452 0	0.000358726		1.00000i	$0.499886\pi$
$720$	0	0
$721$	1.75129	0.0652214
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.73205	−0.138605
$726$	0	0
$727$	−17.4641	−0.647708	−0.323854	−	0.946107i	$-0.604979\pi$
$727$	−17.4641	−0.647708	−0.323854	+	0.946107i	$0.604979\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	17.6077	0.651244
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.8564	1.02610
$738$	0	0
$739$	−4.85641	−0.178646	−0.0893229	−	0.996003i	$-0.528470\pi$
$739$	−4.85641	−0.178646	−0.0893229	+	0.996003i	$0.528470\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−26.4449	−0.970168	−0.485084	−	0.874468i	$-0.661211\pi$
$743$	−26.4449	−0.970168	−0.485084	+	0.874468i	$0.661211\pi$
$744$	0	0
$745$	8.00000	0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	11.3205	0.413642
$750$	0	0
$751$	−51.7128	−1.88703	−0.943514	−	0.331334i	$-0.892501\pi$
$751$	−51.7128	−1.88703	−0.943514	+	0.331334i	$0.892501\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14.8564	0.540680
$756$	0	0
$757$	−13.4641	−0.489361	−0.244681	−	0.969604i	$-0.578683\pi$
$757$	−13.4641	−0.489361	−0.244681	+	0.969604i	$0.578683\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.9474	0.396844	0.198422	−	0.980117i	$-0.436418\pi$
$761$	10.9474	0.396844	0.198422	+	0.980117i	$0.436418\pi$
$762$	0	0
$763$	−4.33975	−0.157109
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−50.2487	−1.81438
$768$	0	0
$769$	3.53590	0.127508	0.0637539	−	0.997966i	$-0.479693\pi$
$769$	3.53590	0.127508	0.0637539	+	0.997966i	$0.479693\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	21.8038	0.784230	0.392115	−	0.919916i	$-0.371744\pi$
$773$	21.8038	0.784230	0.392115	+	0.919916i	$0.371744\pi$
$774$	0	0
$775$	−4.46410	−0.160355
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.26795	0.152915
$780$	0	0
$781$	−45.2487	−1.61913
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−18.0526	−0.644323
$786$	0	0
$787$	31.3731	1.11833	0.559165	−	0.829057i	$-0.311122\pi$
$787$	31.3731	1.11833	0.559165	+	0.829057i	$0.311122\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6.00000	0.213335
$792$	0	0
$793$	5.85641	0.207967
$794$	0	0
$795$	0	0
$796$	0	0
$797$	14.5359	0.514888	0.257444	−	0.966293i	$-0.417120\pi$
$797$	14.5359	0.514888	0.257444	+	0.966293i	$0.417120\pi$
$798$	0	0
$799$	−36.6410	−1.29627
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17.6603	−0.623217
$804$	0	0
$805$	0.392305	0.0138269
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−1.98076	−0.0696399	−0.0348199	−	0.999394i	$-0.511086\pi$
$809$	−1.98076	−0.0696399	−0.0348199	+	0.999394i	$0.511086\pi$
$810$	0	0
$811$	−13.7846	−0.484043	−0.242022	−	0.970271i	$-0.577810\pi$
$811$	−13.7846	−0.484043	−0.242022	+	0.970271i	$0.577810\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−1.12436	−0.0393845
$816$	0	0
$817$	−4.19615	−0.146805
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−23.5885	−0.823243	−0.411621	−	0.911355i	$-0.635037\pi$
$821$	−23.5885	−0.823243	−0.411621	+	0.911355i	$0.635037\pi$
$822$	0	0
$823$	12.1436	0.423299	0.211650	−	0.977346i	$-0.432116\pi$
$823$	12.1436	0.423299	0.211650	+	0.977346i	$0.432116\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	44.1051	1.53369	0.766843	−	0.641835i	$-0.221827\pi$
$827$	44.1051	1.53369	0.766843	+	0.641835i	$0.221827\pi$
$828$	0	0
$829$	45.7846	1.59017	0.795083	−	0.606501i	$-0.207427\pi$
$829$	45.7846	1.59017	0.795083	+	0.606501i	$0.207427\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−27.1244	−0.939803
$834$	0	0
$835$	−10.7321	−0.371398
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−31.1962	−1.07701	−0.538505	−	0.842622i	$-0.681011\pi$
$839$	−31.1962	−1.07701	−0.538505	+	0.842622i	$0.681011\pi$
$840$	0	0
$841$	−15.0718	−0.519717
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.8564	0.579878
$846$	0	0
$847$	−2.14359	−0.0736547
$848$	0	0
$849$	0	0
$850$	0	0
$851$	5.75129	0.197152
$852$	0	0
$853$	1.41154	0.0483303	0.0241652	−	0.999708i	$-0.492307\pi$
$853$	1.41154	0.0483303	0.0241652	+	0.999708i	$0.492307\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.9282	0.646575	0.323288	−	0.946301i	$-0.395212\pi$
$857$	18.9282	0.646575	0.323288	+	0.946301i	$0.395212\pi$
$858$	0	0
$859$	46.9615	1.60231	0.801153	−	0.598459i	$-0.204220\pi$
$859$	46.9615	1.60231	0.801153	+	0.598459i	$0.204220\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	19.5167	0.664355	0.332177	−	0.943217i	$-0.392217\pi$
$863$	19.5167	0.664355	0.332177	+	0.943217i	$0.392217\pi$
$864$	0	0
$865$	4.53590	0.154225
$866$	0	0
$867$	0	0
$868$	0	0
$869$	57.7128	1.95777
$870$	0	0
$871$	−40.7846	−1.38193
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.732051	−0.0247478
$876$	0	0
$877$	−50.2487	−1.69678	−0.848389	−	0.529373i	$-0.822427\pi$
$877$	−50.2487	−1.69678	−0.848389	+	0.529373i	$0.822427\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	20.6603	0.696062	0.348031	−	0.937483i	$-0.386850\pi$
$881$	20.6603	0.696062	0.348031	+	0.937483i	$0.386850\pi$
$882$	0	0
$883$	−29.1244	−0.980113	−0.490056	−	0.871691i	$-0.663024\pi$
$883$	−29.1244	−0.980113	−0.490056	+	0.871691i	$0.663024\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−42.1962	−1.41681	−0.708404	−	0.705807i	$-0.750585\pi$
$887$	−42.1962	−1.41681	−0.708404	+	0.705807i	$0.750585\pi$
$888$	0	0
$889$	8.53590	0.286285
$890$	0	0
$891$	0	0
$892$	0	0
$893$	8.73205	0.292207
$894$	0	0
$895$	−21.0526	−0.703709
$896$	0	0
$897$	0	0
$898$	0	0
$899$	16.6603	0.555651
$900$	0	0
$901$	−5.32051	−0.177252
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.3205	−0.608994
$906$	0	0
$907$	−31.8564	−1.05777	−0.528887	−	0.848692i	$-0.677391\pi$
$907$	−31.8564	−1.05777	−0.528887	+	0.848692i	$0.677391\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.66025	−0.220664	−0.110332	−	0.993895i	$-0.535191\pi$
$911$	−6.66025	−0.220664	−0.110332	+	0.993895i	$0.535191\pi$
$912$	0	0
$913$	48.9808	1.62103
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.66025	0.186918
$918$	0	0
$919$	16.7128	0.551305	0.275652	−	0.961257i	$-0.411106\pi$
$919$	16.7128	0.551305	0.275652	+	0.961257i	$0.411106\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	66.2487	2.18060
$924$	0	0
$925$	−10.7321	−0.352868
$926$	0	0
$927$	0	0
$928$	0	0
$929$	29.9808	0.983637	0.491818	−	0.870698i	$-0.336332\pi$
$929$	29.9808	0.983637	0.491818	+	0.870698i	$0.336332\pi$
$930$	0	0
$931$	6.46410	0.211852
$932$	0	0
$933$	0	0
$934$	0	0
$935$	15.6603	0.512145
$936$	0	0
$937$	32.6410	1.06634	0.533168	−	0.846010i	$-0.321001\pi$
$937$	32.6410	1.06634	0.533168	+	0.846010i	$0.321001\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	7.46410	0.243323	0.121661	−	0.992572i	$-0.461178\pi$
$941$	7.46410	0.243323	0.121661	+	0.992572i	$0.461178\pi$
$942$	0	0
$943$	2.28719	0.0744811
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.0718	0.619750	0.309875	−	0.950777i	$-0.399713\pi$
$947$	19.0718	0.619750	0.309875	+	0.950777i	$0.399713\pi$
$948$	0	0
$949$	25.8564	0.839334
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−39.0333	−1.26441	−0.632207	−	0.774800i	$-0.717851\pi$
$953$	−39.0333	−1.26441	−0.632207	+	0.774800i	$0.717851\pi$
$954$	0	0
$955$	8.26795	0.267545
$956$	0	0
$957$	0	0
$958$	0	0
$959$	4.00000	0.129167
$960$	0	0
$961$	−11.0718	−0.357155
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−16.1962	−0.521373
$966$	0	0
$967$	−1.01924	−0.0327765	−0.0163882	−	0.999866i	$-0.505217\pi$
$967$	−1.01924	−0.0327765	−0.0163882	+	0.999866i	$0.505217\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	39.1962	1.25786	0.628932	−	0.777460i	$-0.283492\pi$
$971$	39.1962	1.25786	0.628932	+	0.777460i	$0.283492\pi$
$972$	0	0
$973$	−6.58846	−0.211216
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.3923	−0.652408	−0.326204	−	0.945299i	$-0.605770\pi$
$977$	−20.3923	−0.652408	−0.326204	+	0.945299i	$0.605770\pi$
$978$	0	0
$979$	19.3923	0.619781
$980$	0	0
$981$	0	0
$982$	0	0
$983$	5.66025	0.180534	0.0902670	−	0.995918i	$-0.471228\pi$
$983$	5.66025	0.180534	0.0902670	+	0.995918i	$0.471228\pi$
$984$	0	0
$985$	5.85641	0.186601
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.24871	−0.0715049
$990$	0	0
$991$	33.1051	1.05162	0.525809	−	0.850602i	$-0.323763\pi$
$991$	33.1051	1.05162	0.525809	+	0.850602i	$0.323763\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	22.0000	0.697447
$996$	0	0
$997$	−43.2679	−1.37031	−0.685155	−	0.728397i	$-0.740265\pi$
$997$	−43.2679	−1.37031	−0.685155	+	0.728397i	$0.740265\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.bq.1.1		2
3.2	odd	2		6480.2.a.bg.1.1		2
4.3	odd	2		3240.2.a.l.1.2	yes	2
12.11	even	2		3240.2.a.g.1.2	✓	2
36.7	odd	6		3240.2.q.bb.1081.1		4
36.11	even	6		3240.2.q.bf.1081.1		4
36.23	even	6		3240.2.q.bf.2161.1		4
36.31	odd	6		3240.2.q.bb.2161.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3240.2.a.g.1.2	✓	2	12.11	even	2
3240.2.a.l.1.2	yes	2	4.3	odd	2
3240.2.q.bb.1081.1		4	36.7	odd	6
3240.2.q.bb.2161.1		4	36.31	odd	6
3240.2.q.bf.1081.1		4	36.11	even	6
3240.2.q.bf.2161.1		4	36.23	even	6
6480.2.a.bg.1.1		2	3.2	odd	2
6480.2.a.bq.1.1		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6480.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.732051 q^{7} +3.73205 q^{11} -5.46410 q^{13} +4.19615 q^{17} -1.00000 q^{19} -0.535898 q^{23} +1.00000 q^{25} -3.73205 q^{29} -4.46410 q^{31} -0.732051 q^{35} -10.7321 q^{37} -4.26795 q^{41} +4.19615 q^{43} -8.73205 q^{47} -6.46410 q^{49} -1.26795 q^{53} +3.73205 q^{55} +9.19615 q^{59} -1.07180 q^{61} -5.46410 q^{65} +7.46410 q^{67} -12.1244 q^{71} -4.73205 q^{73} -2.73205 q^{77} +15.4641 q^{79} +13.1244 q^{83} +4.19615 q^{85} +5.19615 q^{89} +4.00000 q^{91} -1.00000 q^{95} +0.196152 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 2 q^{7} + 4 q^{11} - 4 q^{13} - 2 q^{17} - 2 q^{19} - 8 q^{23} + 2 q^{25} - 4 q^{29} - 2 q^{31} + 2 q^{35} - 18 q^{37} - 12 q^{41} - 2 q^{43} - 14 q^{47} - 6 q^{49} - 6 q^{53} + 4 q^{55}+ \cdots - 10 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 2 * q^7 + 4 * q^11 - 4 * q^13 - 2 * q^17 - 2 * q^19 - 8 * q^23 + 2 * q^25 - 4 * q^29 - 2 * q^31 + 2 * q^35 - 18 * q^37 - 12 * q^41 - 2 * q^43 - 14 * q^47 - 6 * q^49 - 6 * q^53 + 4 * q^55 + 8 * q^59 - 16 * q^61 - 4 * q^65 + 8 * q^67 - 6 * q^73 - 2 * q^77 + 24 * q^79 + 2 * q^83 - 2 * q^85 + 8 * q^91 - 2 * q^95 - 10 * q^97

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